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Dynamically selecting critical KV caches during decoding helps maintain performance. Existing methods use random linear hashing to identify important tokens, but this approach is inefficient due to the orthogonal distribution of queries and keys within two narrow cones in LLMs. We introduce Spotlight Attention, a novel method that employs non-linear hashing functions to optimize the embedding distribution of queries and keys, enhancing coding efficiency and robustness. We also developed a lightweight, stable training framework using a Bradley-Terry ranking-based loss, enabling optimization of the non-linear hashing module on GPUs with 16GB memory in 8 hours. Experimental results show that Spotlight Attention drastically improves retrieval precision while shortening the length of the hash code at least $5 \times$ compared to traditional linear hashing. Finally, we exploit the computational advantages of bitwise operations by implementing specialized CUDA kernels, achieving hashing retrieval for 512K tokens in under $100\mu s$ on a single A100 GPU, with end-to-end throughput up to $3 \times$ higher than vanilla decoding. All the training and evaluation stuff can be found at https://github.com/wenhaoli-xmu/spotlight. + +# 1 Introduction + +Large Language Models (LLMs) are propelling groundbreaking advancements in various natural language tasks, significantly enhancing applications such as content creation and chat assistance. Generally, the inference process of LLMs can be divided into (1) the pre-filling phase calculates the key-value (KV) cache for input tokens in the prompt prior to autoregressive generation, and (2) the decoding phase auto-regressively generates tokens, producing one token per forward pass based on the KV cache. Among them, the decoding phase serves as the primary inference bottleneck due to the frequent exchanges between on-board and on-chip memory for model parameters and KV cache, which limits GPU scalability [3] more so than the pre-filling phase that processes input prompts in parallel. For example, deploying LLaMA2-7B [21] on an A100 GPU for a single request achieves nearly $100\%$ GPU utilization during the pre-filling phase but drops to below $10\%$ on average during decoding, which largely restrains the inference efficiency of LLMs. + +![](images/57f21152d5589dae641648eb4a2e2cf6c0006054fc6ef58e4df94e64e0a00ced.jpg) +Figure 1: Overview. (Left) Architecture. Comparison of our Spotlight Attention versus normal attention. Spotlight Attention adds an additional hash code-based retrieval mechanism for each layer. (Middle) Performance. Spotlight attention achieves the most accurate retrieval and generates the closest response compared to the original model on QA datasets. (Right) Visualization. For arbitrarily complex attention patterns, our method estimates the top-k sequences well, with an average correctness rate of more than half for different models. + +![](images/92ef0b82091450855243625b1c0735ef499f7acdecf82d914a66f2aabaecf1d9.jpg) + +![](images/11975bb8161c0f4ae759e31b828ba4c8de70b0ff6bc10a0c901f02dc94c489b9.jpg) + +![](images/845afa23c51edb343977b00c06792610930bf1c005032d64a158919737e5a4ce.jpg) + +To alleviate this inference bottleneck, extensive research has focused on heuristically eliminating the KV cache burden based on attention scores [25, 23, 17]. While effective for short sequences, such irreversible removal of KV cache can significantly degrade performance on long-sequence tasks, especially in Needle-in-a-Haystack scenarios [14]. To explain, tokens initially considered unimportant and removed might later attain higher attention scores during the prolonged decoding phase, which is crucial for output quality [20, 15]. To overcome this limitation, recent works have turned to retaining all KV cache tokens and dynamically selecting important tokens for computation during decoding [20, 10], which is the focus of this paper. + +Despite the convincing performance of such on-the-fly KV cache selection, how to effectively pick up those important tokens remains challenging. As a pioneering effort, Quest [20] selects tokens by matching queries and keys in a block-wise manner. While effective, such coarse-grained selection hardly guarantees precise localization of important tokens. MagicPIG [10] advances Quest by implementing token-level cache retrieval, specifically utilizing Local Sensitive Hashing (LSH) to encode queries and keys into hash codes and pinpointing the best matches as the selected tokens. However, as depicted in Figure 2a, the efficacy of such linear hashing heavily depends on the hash code length, e.g., a hash code length of 1,024 bits per query/key is necessitated to achieve promising token retrieval. Considering the already substantial size of the KV cache, storing these lengthy hash codes markedly impairs deployment efficiency. Moreover, employing a linear projection with a large output dimension for key processing incurs significant computational overhead. + +Delving deeper, prior work [10] has discovered that the queries and keys within LLMs typically form nearly orthogonal cones within the embedding space, as depicted in Figure 2b. Given the truth that linear hashing function partitions the embedding space using random hyperplanes, such orthogonal distribution of queries and keys barely lead to satisfying encoding quality, which can even result in a collapse of the hashing outcomes, i.e., identical codes for all queries and keys, as shown in Figure 2c. Therefore, extremely long hash codes are necessary to mine meaningful information and accurately match essential tokens. MagicPIG attempted to mitigate this issue by normalizing the keys before retrieval. However, this approach remains suboptimal as it overlooks the query distribution and introduces bias to the retrieval process. + +To address the aforementioned limitation, we propose Spotlight Attention, a novel method that replaces random hyperplanes with curved surfaces for space partitioning via a non-linear MLP hashing function. As depicted in Figure 2d, this non-linearity can better fit skewed distributions, thereby improving code quality. In particular, we utilize the Bradley-Terry ranking objective [7] to optimize the non-linear MLP layer, wherein the learning target involves minimizing the difference between the estimated top- $k$ indices and the ground truth top- $k$ indices obtained via the vanilla attention scores. This learning process is exceptionally efficient, with the LLM backbone remaining frozen and requiring only a minimal amount of calibration data. As a result, the optimized non-linear + +![](images/8f98aa767946566aa3b866b3be9632d168e821c962db1cc1be755964a0acc0fc.jpg) +(a) + +![](images/3d14b55900d1dd99b68deb2b734d0e52f53f3c910a3c06ccffec32157c72f784.jpg) +(b) + +![](images/d0861207ffe8cf707c21e48ca43bd96e8c48dfdc359c6728a9bc9371dc538513.jpg) +(c) +Figure 2: Motivation. (a) The empirical evaluation shows that upgrading the hashing function from linear to MLP can bring a huge improvement, (b) this is because query and key are usually distributed in two small cones in the space [10]. (c) In this situation, it is difficult for the space to be uniformly partitioned by linear boundaries, (d) which can be well solved by using an MLP hashing function. + +![](images/c10abafbe15da49cedd376d14f99e3c309f5f41261556ae0b29c845ce6a350c4.jpg) +(d) + +hashing function can match the performance of linear hashing while using $5 \times$ shorter hash codes, achieving higher efficiency than MagicPIG. We further implemented CUDA kernels for the hash code processing, including bit-packing and bitwise NXOR GEMM operators, achieving significant latency reductions in practice. For example, our method achieves up to $3 \times$ increase in Qwen2.5-7B [24] inference throughput for both 32K and 128K sequences, with only $\sim 2\%$ performance degradation on the LLaMA3 [13] series and no loss on Qwen2.5 [24] series. + +Our contributions are threefold: + +- We propose Spotlight Attention for accelerating LLM inference, which employs non-linear hashing function to encode and match queries and key values within LLMs, thereby efficiently selecting critical KV cache for model inference. +- We develop a lightweight and robust training framework based on the Bradley-Terry ranking objective, which effectively optimizes the non-linear hashing function using only a small amount of calibration data. +- Extensive experiments demonstrate that Spotlight Attention can drastically reduce LLM inference latency while maintaining the strongest performance retention in comparison with state-of-the-art methods. + +# 2 Related Work + +This section covers the spectrum of studies on LLM KV cache pruning that are closely related to our work, which we heuristically categorize into static pruning, dynamic pruning with permanent eviction, and dynamic pruning without permanent eviction. + +Static KV cache pruning. These methods compress the KV cache once after the pre-filling phase, using the compressed cache for subsequent decoding. For example, FastGen [12] introduces a pattern-aware approach by identifying five fundamental attention structures and applying targeted selection strategies. SnapKV [16] further simplifies FastGen by focusing solely on retrieving tokens based on their importance scores, showing that only a subset of prompt tokens carry critical information for response generation and retain their significance during the whole decoding phase. However, without pruning during decoding, these methods are primarily suited for scenarios with long prompts and relatively short responses. + +Dynamic pruning with permanent eviction. This category of methods performs dynamic KV cache pruning during the decoding phase, permanently removing pruned KV cache tokens from memory. For example, H2O [25] leverages cumulative attention scores to retain high-impact tokens. NACL [9] identifies a fundamental limitation in H2O, namely their dependence on potentially biased local attention statistics. To overcome this issue, they develop an alternative approach, implementing a diversified random eviction strategy. Keyformer [2] highlights that token removal distorts the underlying softmax probability distribution. Considering the pivotal role of softmax distributions in token significance evaluation, they incorporate regularization techniques to mitigate these distributional perturbations. Unlike static KV cache selection, these methods enable dynamic pruning during decoding, making them better suited for tasks requiring extensive generation. However, they assume that critical information is concentrated in a small subset of KV cache tokens, a condition that does not always hold. As MagicPIG [10] points out, token importance can vary significantly + +![](images/d43e452690274fd95bde671e221be67d7d4b844fd40e145a386d738f7aac7704.jpg) + +![](images/8ead7c28b1d58a0a7c51d3b9460dec1ab451a3e0c194e0272d99235cf4bce12b.jpg) + +![](images/efedd65140a471382119eb7906db8deb8e87dea9f47a91e5d2161186e363cba9.jpg) +Figure 3: Optimization. (Left) Reconstruction Loss. This loss minimizes the MSE between the estimated and ground-truth attention scores. It has two main drawbacks. First, it is highly sensitive to score magnitudes and prone to outliers. Second, it wastes most of the hashing function's capacity on preserving order within the top- $k$ and non-top- $k$ sets. (Right) Our Ranking Loss. Our loss adopts the Bradley-Terry ranking objective, which is robust to score magnitude and outliers, and provides supervision focused solely on distinguishing between top- $k$ and non-top- $k$ sets. + +![](images/24ece74e633c7b31bbc5e20fb04e824424285586cb42ba91ea388793ea98aac6.jpg) + +across tasks, leading to premature eviction of tokens before they are needed. For example, H2O may fail to answer questions like $a$ is $b$ , $c$ is $d$ , $a$ is? due to forgetting earlier facts. + +Dynamic pruning without permanent eviction. The limited applicability of permanent token eviction methods has led to a shift toward non-permanent eviction approaches. These methods assume the importance of KV cache tokens varies with each query, requiring importance estimation at every decoding step. Instead of permanently evicting unimportant tokens, they exclude them from attention calculations for that step only. While this improves accuracy, it demands frequent importance estimation, unlike permanent eviction methods that prune tokens in batches after many steps. Research has therefore focused on optimizing the efficiency and accuracy of these estimations. Quest [20] groups KV cache tokens into blocks, estimating block importance via the dot product between queries and block representations derived from the minimum and maximum key values. Although efficient, this approach suffers from internal fragmentation, as entire blocks are processed even if only a few tokens are important. MagicPIG [10] eliminates this issue by mapping queries and keys to hash codes for token-level retrieval via Hamming distance, avoiding fragmentation but reducing efficiency. Building on MagicPIG, our method significantly shortens hash code, drastically reducing computation while preserving accuracy. + +# 3 Methodology + +# 3.1 Preliminary + +Attention computing. We first present the basic preliminaries for attention computation and KV cache pruning during the decoding phase of LLMs. We define the query, key, and value inputs to the attention module as $Q, K, V \in \mathbb{R}^{1 \times d}$ , where $d$ is the embedding dimension. We use $\oplus$ to denote concatenation, and $K_{\mathrm{cache}}, V_{\mathrm{cache}} \in \mathbb{R}^{n \times d}$ represent the key-value cache generated during the pre-filling phase and previous decoding steps. With these definitions, the standard attention is calculated as follows: + +$$ +\mathcal {A} = \operatorname {s o f t m a x} \left(\frac {f (Q , K _ {\text {c a c h e}}) \oplus Q K ^ {\top}}{\sqrt {d}}\right) \left(V _ {\text {c a c h e}} \oplus V\right), \tag {1} +$$ + +where $f(X, X') = XX'^\top$ is inner-product. + +KV cache pruning. As the decoding sequence length increases, the size of the KV cache $\{K,V\}_{\mathrm{cache}}$ can grow exceedingly large, creating an LLM inference bottleneck. KV cache pruning that selectively preserves only essential portions of the cache for computation serves as an efficient way to alleviate + +Table 1: KV retrieval accuracy. Measured by IoU↑ / PPL↓ changes before and after training. (1) Training is essential for efficient MLP hashing. (2) Limited improvement from training linear hashing highlights the necessity of MLP hashing. See Appendix B.1 for per-head IoU scores. + +
MethodOriginalOracle Top-2%LSH Top-2%MLP Hashing Top-2% (ours)
BeforeAfterBeforeAfter
LLaMA2-7B- / 5.581.00 / 5.690.17 / 5.860.20 / 5.840.05 / 20.310.41 / 5.72
LLaMA2-7B-Chat- / 7.101.00 / 6.870.17 / 7.340.19 / 7.450.05 / 21.340.42 / 6.98
LLaMA3-8B- / 6.451.00 / 6.630.15 / 7.120.18 / 7.070.07 / 148.20.34 / 6.69
Qwen2.5-7B- / 7.171.00 / 7.280.13 / 8.810.16 / 8.730.09 / 22.070.35 / 7.31
+ +this problem. Given a desired cache budget $K$ , it first identifies the indices $\mathcal{I}$ of top- $K$ important tokens and then extracts a subset of the KV cache $\{K, V\}_{\mathrm{subset}}$ for attention computation as + +$$ +\{K, V \} _ {\text {s u b s e t}} = \operatorname {g a t h e r} (\{K, V \} _ {\text {c a c h e}}, \mathcal {I}). \tag {2} +$$ + +As previously discussed, existing methods for pruning the KV cache either permanently eliminate cache entries not in the set $\mathcal{I}$ [25, 16] or retain all caches but dynamically determine $\mathcal{I}$ during the decoding process [20, 10]. In this paper, we focus on the latter due to its superior performance preservation. + +# 3.2 Revisiting Token-level Cache Retrieval + +Token-level cache retrieval refers to dynamically selecting cached entries at the granularity of individual tokens [10]. + +Oracle top-k retrieval. We conducted a preliminary experiment to assess the upper-bound performance, using full-precision attention scores $f(Q, K_{\mathrm{cache}})$ to select key tokens. Surprisingly, by pruning layers beyond the first two, we discarded up to $98\%$ of the KV cache with only a 0.1 perplexity increase on PG19, revealing significant untapped potential. This suggests that top- $k$ retrieval enables near-lossless compression, contrasting sharply with prior findings [10]. + +LSH top-k retrieval. Although oracle attention scores accurately identify key tokens, computing $f$ is impractical for real-world applications. To address this, MagicPIG approximates $f$ with $\tilde{f}$ using Locality-Sensitive Hashing (LSH) to efficiently retrieve critical KV entries. Specifically, a linear hash function $\mathcal{H}$ computes: + +$$ +\tilde {f} (X, X ^ {\prime}) = \mathcal {H} (X) \otimes \mathcal {H} \left(X ^ {\prime}\right), \tag {3} +$$ + +where $\otimes$ denotes a matrix multiplication-like operation, substituting floating-point multiplication with NXOR. The indices $\mathcal{I}$ of the top- $k$ largest values in $\tilde{f}(Q, K_{\mathrm{cache}})$ are used to retrieve the most relevant KV entries. + +LSH groups similar vectors into the same bucket with high probability, making it ideal for dense vector spaces. A common variant employs random hyperplanes, where distinct hyperplanes create linear decision boundaries, assigning data on either side to bit-0 or bit-1. The bits sequence from all hyperplanes forms the hash code. In practice, these steps can be simplified to a single matrix multiplication followed by a sign operation. For a vector $x \in \mathbb{R}^d$ , we apply a random projection matrix $R \in \mathbb{R}^{d \times d_{\mathcal{H}}}$ to obtain a hash code by taking the sign of the resulting product: + +$$ +\mathcal {H} (x) = \operatorname {s i g n} (x R). \tag {4} +$$ + +MLP hashing top-k retrieval. As shown in Figure 2, queries and keys typically lie within two cone-shaped regions in high-dimensional space [10]. This distribution causes uneven partitioning in LSH, reducing encoding efficiency. To address this, we propose MLP hashing, a learned non-linear hashing network tailored to query and key distributions. This approach enhances hash code information density, enabling effective partitioning of skewed data through non-linear decision boundaries. + +Specifically, we replace the projection matrix $R$ in Eq. (4) with a two-layer MLP: + +$$ +\operatorname {M L P} (x) = W _ {2} \left(\operatorname {S i L U} \left(W _ {1} x + b _ {1}\right)\right), \tag {5} +$$ + +where $W_{1}, b_{1}$ , and $W_{2}$ are learnable parameters. Hash codes are then computed as: + +$$ +\mathcal {H} (x) = \operatorname {s i g n} (\mathrm {M L P} (x)). \tag {6} +$$ + +Table 2: Perplexity comparison with Quest. Perplexity evaluation on PG19 (#1), ProofPile (#2), and CodeParrot (#3) datasets. All models truncate inputs to their maximum supported token length. Spotlight Attention achieves performance comparable to Quest with a $10 \times$ smaller token budget. + +
MethodConfigurationFrozen LayersLLaMA2-7BLLaMA2-7B-ChatLLaMA3-8BQwen2.5-7B
#1#2#3#1#2#3#1#2#3#1#2#3
Vanilla Oracle Top-K0% PrunedN/A6.8794.2773.6799.2125.9434.7868.6043.5175.21911.1123.8334.951
98% Pruned[0,1]6.9414.3173.7299.2245.8914.7958.8813.5905.35310.4353.5874.733
Quest75% Pruned[0,1]7.1164.4043.8549.2825.9304.8799.9124.0245.89310.4854.2815.482
87.5% Pruned7.7354.7544.0549.8206.2265.08412.4344.9276.64613.5965.4716.213
93.7% Pruned9.7465.7754.57112.0587.4405.68617.3206.7498.69319.8527.2747.823
96.9% Pruned15.4948.5785.99618.71911.0837.34527.51011.63114.20531.58313.20812.526
Spotlight (ours)80% Pruned[0,1]6.8874.2783.6829.1075.8604.7678.6123.5195.2289.7663.4654.618
90% Pruned6.9084.2853.6899.0585.7964.7548.6513.5295.2399.7833.4674.621
95% Pruned6.9594.3043.7039.0675.7484.7528.7343.5525.2859.8253.4754.627
98% Pruned7.1064.3643.7689.2625.7704.8068.9773.6215.4349.9303.4974.645
+ +# 3.3 Optimization. + +Optimizing the MLP hashing function $\mathcal{H}$ to capture the query and key distribution is more critical than the hashing network design and forms the core contribution of this work. We next introduce two intuitive training objectives and explain their limitations in this context. + +$\times$ Language modeling loss. A natural approach to optimize $\mathcal{H}$ is to minimize the language modeling loss directly through a few hundred post-training steps. However, this method has significant limitations. First, it requires full forward and backward passes through the entire LLM, which is computationally costly. More critically, differentiating the top- $k$ operator in Eq. (2) requires a complex "soft top- $k$ " approximation, which is challenging to implement. + +$\times$ Reconstruction loss. An alternative approach uses MSE to align $\tilde{f}$ with $f$ , enabling layer-wise optimization and reducing computational cost. However, this method has a key limitation: the objective is to select which KV cache entries to retain, not to rank their relative importance. Training with this loss misallocates capacity by prioritizing the ranking of excluded entries, deviating from the core goal, and causing significant performance degradation. See Figure 3 (Left) for its limitations. + +$\checkmark$ Our ranking loss. To address this issue, we adopt a Bradley-Terry ranking objective, inspired by RankNet [8]. As shown in Figure 3 (Right), during training, we identify top- $k$ and non-top- $k$ index sets based on attention scores. These indices split the estimated scores derived from Hamming distances of query and key hash codes into sets $B$ and $C$ . An optimizer then updates the hashing function parameters to ensure each score in $B$ exceeds every score in $C$ : + +$$ +\mathcal {L} _ {\text {r a n k}} = - \frac {1}{k (n - k)} \sum_ {i, j} \log (\text {s i g m o i d} (\beta (B _ {i} - C _ {j}) - \alpha)), \tag {7} +$$ + +where $\beta$ and $\alpha$ are positive constants used to amplify the separation between $B$ and $C$ , facilitating convergence. The core of this loss design lies in filtering out supervising signals related to internal ranking within $B$ and $C$ , effectively addressing the issue of capacity misallocation. We provide the pseudo-code for our ranking loss in Appendix A.3 to aid readers familiar with code. + +Make hashing function differentiable. After computing the loss, errors can be backpropagated to the MLP hashing function. However, the sign function's non-differentiability blocks gradient flow. To resolve this, we substitute the sign function with a soft sign function during training: + +$$ +\operatorname {s o f t s i g n} (x) = \frac {\gamma x}{1 + \gamma | x |}, \tag {8} +$$ + +where $\gamma \in \mathbb{R}$ is a hyperparameter controlling the extent of smoothing. This soft sign function is used only during training. In inference, the non-differentiable sign function is reinstated. + +![](images/e3fd3018038432aa3f410e136f26fd1697f6cc2de8310b4f6c85c90b8b315db5.jpg) +Figure 4: NIAH results. Using LLaMA3-8B [13] as the base model, we compared the retrieval accuracy of MagicPIG with our method. Our approach, which relies solely on hash code-based retrieval without local windows or sink tokens, achieves comparable response accuracy. + +![](images/cbf9e895e008c5f562add8602c85d506c3024ad8e87dea00444e3d89dd22673c.jpg) + +![](images/86830c00683bb63fb1305926fcce689e6e67786bb6a7252c9f9be8cba38b3273.jpg) + +Table 3: Perplexity versus MagicPIG. Comparison of perplexity on PG19 (#1), ProofPile (#2), and CodeParrot (#3). Due to the time-consuming evaluation process of MagicPIG, we sampled only 10 data points from each dataset for testing. + +
MethodConfigurationLLaMA3-8B
FrozenLocal (64)Sink (4)Retrieve (2%)#1#2#3
Vanilla Oracle Top-K[0,1]✓9.562.832.26
9.892.892.28
MagicPIG[0,16]✓✓✓12.653.542.91
✓✓16.946.274.57
✓✓50.968.526.67
✓✓42.128.6510.43
✓NaNNaNNaN
Spotlight (ours)[0,1]✓✓✓9.872.892.29
✓✓13.895.503.98
✓9.992.912.30
+ +Table 4: QA response fidelity. On Long-Bench, output fidelity (measured by Rouge-L between compressed and vanilla model outputs) shows our method achieves performance closest to the vanilla model. + +
MethodConfigurationSimilarity
LocalSinkRetrieveFrozen
VanillaXXXX1.00
Oracle Top-K1630.66
LSH Top-KXX163[0,1]0.37
Quest10240.56
Quest2560.34
MagicPIG644Dynamic[0,16]0.44
Spotlight (ours)XX163[0,1]0.58
+ +![](images/d9fe460af03c8568f07052abf85c4a8b820f74e831035165c10a006ca41de926.jpg) +Figure 5: Downstream QA tasks. (Left) Relative score of each method compared to the vanilla baseline; each point denotes a subtask. (Right) Absolute score comparison. + +# 4 Experimentation + +Our evaluation spans multiple dimensions. (1) We first assess the errors introduced by retrieval and sparsification, measured by retrieval accuracy and perplexity score. (2) We then evaluate long-context key information retrieval using the Needle-in-a-Haystack [14] benchmark. (3) Next, we evaluated downstream QA tasks on LongBench [6], comparing response similarity between compressed and original models using Rouge-L. The key insight from this Rouge-L comparison is straightforward: higher-quality compression produces more concise outputs. (4) We also measure end-to-end throughput gains and the execution efficiency of our CUDA ops. + +To evaluate generalization, we tested the performance of the Qwen2.5 series [24] across various model sizes and LLaMA3-8B [13] across diverse training corpora. Additionally, we conducted ablation studies to assess the impact of loss functions and attention estimation methods, with details provided in Appendix B.5 and B.6. + +Experimental setups. We employ LLaMA3-8B [13], LLaMA2-7B, LLaMA2-7B-Chat [21], and Qwen2.5 models (1.5B, 7B, 14B) [24] as base models. The baseline methods compared include (1) oracle top- $k$ retrieval, (2) linear LSH top- $k$ , (3) Quest, and (4) MagicPIG. Their implementation details are provided in Appendix A.2. + +Our MLP hashing function employs 128-dimensional input, intermediate, and output layers, with a distinct MLP for each head in every layer, producing a 128-bit hash code—much shorter than MagicPIG's minimum of 720 bits. Only the hashing functions are trainable. Training data consists of 8,192 samples, evenly drawn from the Book and Arxiv datasets [22]. + +![](images/91f08f1a5e851fd535102bc624936f9865209f612857bd1fe73d89127cb10859.jpg) + +![](images/726201a08517fcdd2a6cf0dde5899ac572a7bb14015234a58fcf7d0a4a7f7243.jpg) + +![](images/b0a7e5c69088ffa36b1295e3ea2e7d3ba338ae0c5ce27e898e9742b7e877a8d0.jpg) + +![](images/654db1a9985f74393d8c4f30b1fc198f1385272c60b6fa409f7b960870fbf791.jpg) + +![](images/012052ca94e5ffcf8ab19a32f44e65baac2ffd778cc593319f0d86d00a0d0141.jpg) +Figure 6: Efficiency. (Upper Left) End-to-end throughput comparison with fixed context length across varying batch sizes. (Bottom Left) End-to-end throughput comparison with fixed batch size across different context lengths. (Upper Right) Hash code size comparison between MagicPIG and our method, alongside the execution latency of the two most computationally intensive operations in our method. (Bottom Right) Complexity comparison of different computational steps. + +![](images/5a3ec547a054f3f981b06a171940d6feb63fd5e8a46fa364605072e032b164e7.jpg) + +
Symbol Def.Inference Sub-StepComplexity
h: head count① Bitwise encodingO(hd)
d: embed dim② Similarity searchO(n hd)
n: sequence length③ Top-k gatheringO(n h log k)
④ Sparse attentionO(khd)
+ +To improve efficiency, hidden states for all layers are precomputed and stored, enabling independent layer-wise training without joint fine-tuning. Training uses $\gamma = 64$ , a learning rate of $1 \times 10^{-3}$ , $\beta = 1$ , and $\alpha = 3$ , for one epoch. Additional details are provided in Appendix A.1. The pruning rate remains fixed at $98\%$ during training, irrespective of evaluation settings. + +# 4.1 Main Results + +KV retrieval accuracy. This experiment compares linear LSH and MLP hashing for KV retrieval. We used the first sample from PG19 as test data and assessed KV retrieval accuracy with the average IoU score across all heads and layers. IoU is computed as the intersection of the top- $k$ indices retrieved by the algorithm and the oracle top- $k$ indices, divided by their union. + +For LSH, we initialized the projection matrix via QR decomposition (see Appendix A.8). For MLP hashing, parameters were initialized with random Gaussian distributions. Results in Table 1 show that LSH performs best without training but shows minimal improvement post-training. Conversely, MLP hashing markedly improves retrieval accuracy after training, achieving the highest performance. + +Language modeling. We evaluated three language modeling benchmarks—PG19 [19], ProofPile [5], and CodeParrot [18]—with 100, 79, and 100 samples, respectively, using perplexity to detect minor errors from sparsification. Additional experimental details are in Appendix A.5. + +Results in Table 2 show our method outperforms Quest with a tenfold reduction in token budget for most models. Comparisons with MagicPIG (Table 3) indicate that MagicPIG depends heavily on local windows and sink tokens, failing without them, while our method autonomously identifies these elements, demonstrating significantly higher retrieval accuracy. + +MagicPIG excels at retrieving facts in scenarios like Needle-in-a-Haystack [14], its contribution should not be dismissed based solely on its limitations in this language modeling test. + +Needle-in-a-Haystack. We use the offline evaluation version of NIAH [1], which differs from ChatGPT scoring by using the Rouge score to measure output accuracy. We utilize LLaMA3-8B [13] as the base model, evaluating context lengths ranging from 256 to 8192, with a step size of 256 for testing. The Needle dataset is highly prompt-sensitive, we provide the needle, retrieval question, and haystack context in Appendix A.4. Additionally, to reduce output variability, all evaluated methods use greedy search. As shown in Figure 4, Spotlight Attention achieves performance on par with the original model. + +Downstream QA tasks. We evaluated the performance of various compression methods on Long-Bench [6] subtasks. Quest employs a token budget of 1,024, while MagicPIG uses 4 sink tokens, a 64-token local window, and a 1,500-bit hash code per key, following their default configurations. Our method uses a token budget of 163. All experimental results reported in the main text use LLaMA3-8B as the base model, scores of other models are provided in Appendix B.4, and more experimental setup details are in Appendix A.6. Figure 5 (Right) compares absolute scores across all + +subtasks. More importantly, Figure 5 (Left) shows relative scores, revealing that our method's scores closely align with those of the vanilla model. + +We also assessed output fidelity using Rouge-L to measure similarity between the vanilla model's outputs and those of these methods. Results are presented in Table 4. Our method's outputs are the most similar to the vanilla model, with the longest consecutive subsequence match approaching $60\%$ . In contrast, Quest requires retrieving six times more tokens to achieve comparable similarity. These findings hold for nearly all subtasks, with detailed scores of each subtask provided in Appendix B.2. + +# 4.2 Efficiency + +All efficiency experiments were performed on Qwen2.5-7B [24] using eight A100 GPUs. For enhanced flexibility, experiments utilized the HuggingFace Transformers framework, optimized with pipeline parallelization and KV cache pre-allocation to boost throughput. + +End-to-end throughput evaluation. To evaluate model throughput at extended context lengths (e.g., 2M tokens), we expanded positional encoding, disregarding output quality. We selected the first sample from the PG19 test set [19], repeating it to reach a 2M-token context. GPU execution time was measured using CUDA events. We generated eight consecutive tokens, computed throughput by dividing by generation time, and averaged three runs per data point. Results in Figure 6 (Left) show that our approach consistently delivers throughput gains, especially at larger input scales. + +Kernel evaluation. We implemented CUDA kernels for both bit-packing and NXOR GEMM. Bit-packing compresses 32 Torch boolean values into a single unsigned 32-bit integer, as detailed in Appendix A.7. For NXOR GEMM, we utilized the standard library's popcount to count bit-1s in each NXOR result. For top- $k$ gathering and sparse attention, we employed Torch and FlashAttention implementations, respectively. As shown in Figure 6 (Upper Right), compared to dense vectors, our hashing-based similarity search significantly reduces storage and computation, enabling bit-packing and similarity search within $100\mu s$ for context lengths up to 512K. + +Table 5: Ablation on model size. Various Qwen2.5 model sizes, augmented with Spotlight Attention, demonstrated better perplexity across diverse language modeling tasks. + +
ModelMethodPG19MathCode
Qwen2.5-1.5BVanilla13.8284.1815.081
Spotlight Top-2% (ours)13.5104.1435.064
Qwen2.5-7BVanilla11.1123.8334.951
Spotlight Top-2% (ours)9.9303.4974.645
Qwen2.5-14BVanilla8.4163.2304.472
Spotlight Top-2% (ours)8.2613.1964.440
+ +Table 6: Ablation on training tasks. Alongside the standard ArXiv+Books training data, we trained models on the C4 and GitHub Code datasets, achieving comparable perplexity (PPL) results. + +
LLaMA3-8BTraining CorpusPG19MathCode
Oracle Top-2%-8.8813.5905.353
SpotlightArxiv + Book (default)8.9773.6215.434
C48.9583.6315.417
Top-2% (ours)Github Code8.8913.6115.393
+ +# 4.3 Ablations + +In the main text, we present ablation studies on model size and training tasks only. Additional ablation experiments, including loss functions and attention estimation methods, are detailed in Appendix B.5 and B.6, respectively. + +All ablation experiments use language modeling perplexity as the evaluation metric. We assessed performance on PG19 [19], Proof-Pile (Math) [5], and CodeParrot (Code) [18], with sample sizes of 100, 79, and 100, respectively. + +Model size. We compare Qwen2.5 [24] models of 1.5B, 7B, and 14B parameters, all trained with the standard recipe. As shown in Table 5, Spotlight Attention achieves consistently strong performance across different model sizes. + +Training tasks. To validate our method's applicability across diverse tasks, we trained models on the GitHub Code [18] and C4 [18] datasets, in addition to the ArXiv and Books [22] datasets used previously. As presented in Table 6, training with GitHub Code or C4 unexpectedly outperformed the ArXiv+Books combination, demonstrating the robust adaptability of our training framework. However, due to the already-completion of the main results, we did not re-evaluate all benchmarks with these checkpoints despite their superior performance. + +# 5 Conclusion and Limitation + +We introduce Spotlight Attention, an advancement over Quest and MagicPIG, incorporating a nonlinear hashing function and an optimized framework. This approach addresses the underfitting of MagicPIG's linear hashing while significantly reducing hash code length. Spotlight Attention performs well on downstream tasks but has limitations. The IoU remains around $40\%$ despite non-linear hashing, indicating potential for improvement. + +# 6 Acknowledgments + +This work was supported by the National Science Fund for Distinguished Young Scholars (No.62025603), the National Natural Science Foundation of China (No. U21B2037, No. U22B2051, No. U23A20383, No. 62176222, No. 62176223, No. 62176226, No. 62072386, No. 62072387, No. 62072389, No. 62002305 and No. 62272401), the Natural Science Foundation of Fujian Province of China (No. 2021J06003, No.2022J06001), the National Natural Science Foundation of China No.624B2119, and the China Postdoctoral Science Foundation (No.BX20250384). + +# References + +[1] Github repository: 66ring/llmtest_needleinahaystack-local, 2023. +[2] M. Adnan, A. Arunkumar, G. Jain, P. Nair, I. Soloveychik, and P. Kamath. Keyformer: Kv cache reduction through key tokens selection for efficient generative inference. $MLSys$ , 2024. +[3] A. Agrawal, A. Panwar, J. Mohan, N. Kwatra, B. S. Gulavani, and R. Ramjee. Sarathi: Efficient llm inference by piggybacking decodes with chunked prefills. arXiv, 2023. +[4] A. Andoni, P. Indyk, T. Laarhoven, I. Razenshteyn, and L. Schmidt. Practical and optimal lsh for angular distance. In NeurIPS, 2015. +[5] Z. Azerbayev, E. Ayers, and B. Piotrowski. Github repository: hoskison-center/proof-pile, 2022. +[6] Y. Bai, X. Lv, J. Zhang, H. Lyu, J. Tang, Z. Huang, Z. Du, X. Liu, A. Zeng, L. Hou, et al. LongBench: A bilingual, multitask benchmark for long context understanding. In ACL, 2024. +[7] R. A. Bradley and M. E. Terry. Rank analysis of incomplete block designs: I. the method of paired comparisons. Biometrika, 1952. +[8] Z. Cao, T. Qin, T.-Y. Liu, M.-F. Tsai, and H. Li. Learning to rank: from pairwise approach to listwise approach. In ICML, 2007. +[9] Y. Chen, G. Wang, J. Shang, S. Cui, Z. Zhang, T. Liu, S. Wang, Y. Sun, et al. NACL: A general and effective KV cache eviction framework for LLM at inference time. In NACL, 2024. +[10] Z. Chen, R. Sadhukhan, Z. Ye, Y. Zhou, J. Zhang, N. Nolte, Y. Tian, M. Douze, L. Bottou, Z. Jia, and B. Chen. MagicPIG: LSH sampling for efficient LLM generation. In NeurIPS, 2024. +[11] L. Gao, J. Tow, B. Abbasi, S. Biderman, S. Black, A. DiPofi, C. Foster, et al. Github repository: Eleutherai/lm-evaluation-harness, 2024. +[12] S. Ge, Y. Zhang, L. Liu, M. Zhang, J. Han, and J. Gao. Model tells you what to discard: Adaptive KV cache compression for LLMs. In ICLR, 2024. +[13] A. Grattafori, A. Dubey, A. Jauhri, A. Pandey, A. Kadian, A. Al-Dahle, A. Letman, et al. The llama 3 herd of models. arXiv, 2024. +[14] G. Kamradt. Github repository: gkamradt/llmtest_needleinahaystack, 2023. +[15] H. Li, Y. Li, A. Tian, T. Tang, Z. Xu, X. Chen, N. Hu, W. Dong, Q. Li, and L. Chen. A survey on large language model acceleration based on kv cache management. arXiv, 2025. +[16] Y. Li, Y. Huang, B. Yang, B. Venkitesh, A. Locatelli, H. Ye, T. Cai, P. Lewis, and D. Chen. SnapKV: LLM knows what you are looking for before generation. In NeurlPS, 2024. +[17] M. Oren, M. Hassid, N. Yarden, Y. Adi, and R. Schwartz. Transformers are multi-state rnns. arXiv, 2024. +[18] Z. Peitian. Huggingface dataset: namespace-pt/long-llm-data, 2024. +[19] J. W. Rae, A. Potapenko, S. M. Jayakumar, C. Hillier, and T. P. Lillicrap. Compressive transformers for long-range sequence modelling. In ICLR, 2020. +[20] J. Tang, Y. Zhao, K. Zhu, G. Xiao, B. Kasikci, and S. Han. QUEST: Query-Aware Sparsity for Efficient Long-Context LLM Inference. In ICML, 2024. +[21] H. Touvron, L. Martin, K. Stone, T. Scialom, et al. Llama 2: Open foundation and fine-tuned chat models. arXiv, 2023. +[22] M. Weber, D. Y. Fu, Q. G. Anthony, Y. Oren, S. Adams, A. Alexandrov, X. Lyu, et al. Redpajama: an open dataset for training large language models. In NeurIPS, 2024. +[23] G. Xiao, Y. Tian, B. Chen, S. Han, and M. Lewis. Efficient streaming language models with attention sinks. In ICLR, 2024. + +[24] A. Yang, B. Yang, B. Zhang, B. Hui, B. Zheng, B. Yu, C. Li, D. Liu, F. Huang, et al. Qwen2.5 technical report. arXiv, 2025. +[25] Z. Zhang, Y. Sheng, T. Zhou, T. Chen, L. Zheng, R. Cai, Z. Song, Y. Tian, C. Re, C. Barrett, Z. Wang, and B. Chen. H2o: Heavy-hitter oracle for efficient generative inference of large language models. In NeurIPS, 2023. + +# NeurIPS Paper Checklist + +The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit. + +Please read the checklist guidelines carefully for information on how to answer these questions. 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Licenses for existing assets + +Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? + +# Answer: [Yes] + +Justification: All assets used are well-known, properly referenced, and their information is readily accessible. + +# Guidelines: + +- The answer NA means that the paper does not use existing assets. +- The authors should cite the original paper that produced the code package or dataset. +- The authors should state which version of the asset is used and, if possible, include a URL. +- The name of the license (e.g., CC-BY 4.0) should be included for each asset. +- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. +- If assets are released, the license, copyright information, and terms of use in the package should be provided. 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Crowdsourcing and research with human subjects + +Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? + +Answer: [NA] + +Justification: Our paper does not involve crowdsourcing nor research with human subjects. + +Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. +- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. +- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. + +# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects + +Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? + +Answer: [NA] + +Justification: Our paper does not involve crowdsourcing nor research with human subjects. + +Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. +- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. +- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. +- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: Our study focuses on open-source LLMs, but LLMs are not involved in the research methodology. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Implementation Details + +# A.1 Training Setup + +Table 7 summarizes the training configuration. For further details, including low-level specifics and weight files, refer to our open-source code. + +Table 7: Detailed training configuration. + +
GeneralLearning RateGradient
PrecisionNum ItersBatch SizeMax LRMin LRWarm Up ItersWarm Up MethodAnnealingAccumulationClipping
bf168,19210.001081linearcosine11.0
OptimizerData
Optimizerβ1β2Weight DecayCorpusArxiv SamplesBook SamplesLLaMA2 TruncLLaMA3/Qwen TruncTrunc Side
adamw0.90.980.1arxiv, book4,0964,0964.0968,192right
+ +# A.2 Baseline Methods + +Oracle top-k. As introduced in Section 3.2, this baseline employs original attention to identify top- $k$ indices, serving as a theoretical upper bound on performance. We provide pseudo-code below to illustrate a concrete implementation of the oracle top- $k$ algorithm. + +LSH top-k. This method uses training-free angular LSH [4] with the same hash code length as Spotlight Attention, providing a reference for MagicPIG's linear hashing under our framework. Initialization details for angular LSH are in Appendix A.8. + +Quest. [20] We adopt Quest's official implementation with default hyperparameters, modifying only the token budget. Beyond the default 1024-token budget, we test ultra-low budgets to align with Spotlight Attention. For instance, for LLaMA2-7B and LLaMA3-8B, we use budgets of 128 and 256 tokens, compared to Spotlight Attention's 81 and 163 tokens. + +MagicPIG. [10] We adopt MagicPIG's official implementation with default hyperparameters ( $K = 15$ , $L = 100$ ), retaining 64 local tokens and 4 initial tokens. For all experiments except the efficiency test, we use the official Python evaluation code to conduct the experiments. + +# A.3 Ranking Loss + +We provide the ranking loss calculation in the following pseudo-code. To support longer sequence lengths during training, we employ three optimization techniques: (1) Random query selection, where only queries specified by query_index are optimized, rather than all queries. (2) Random top- $k$ selection, where max_top is randomly sampled from the top- $k$ set for optimization. (3) Random non-top- $k$ selection, where max_oth is randomly sampled from the non-top- $k$ set for optimization. These techniques enhance training efficiency in long-context scenarios. + +```txt +1 def ranking_loss( +2 draft_attn, +3 true_attn, +4 query_index, +5 max_top, +6 max_oth, +7 maskout, +8 beta: float $= 1.0$ +9 alpha: float $= 0.0$ .. +10 +11 loss $=$ torch.tensor(0,dtype $\equiv$ torch.float32) +12 criterion $=$ torch(nn.BCEWithLogitsLoss() +13 +14 # PREPARE & APPLY MASK +15 num_kv $=$ true_attn.shape[-1] +16 mask $=$ torch.triu(torch.ones((num_kv, num_kv), dtype $\equiv$ torch(bool device $\equiv$ true_attn_device), diagonal $\coloneqq 1$ [None, None, :,:] +17 if query_index is not None: +18 mask $=$ mask[., query_index, :] +19 true_attn $=$ torchmasked_fill(true_attn, mask, value $\equiv$ torch+finfo(true_attn.dtype).min) +20 indices $=$ torch.argsort(true_attn, dim=-1, descending=True) +21 +22 +23 top_cnt $=$ int(indices.shape[-1] \* (1 - maskout)) +24 top Indices $=$ indices[.,top_cnt] +25 oth Indices $=$ indices[.,top_cnt:] +26 +``` + +```python +27 if max_top is not None: top_rnd Indices = torch.randomperm(top_cnt, dtype=torch.int64, device=indices_device) [max_top] topIndices = topIndices[., top_rnd Indices] +30 if max_oth is not None: oth_rnd indices = torch.randomperm(indices.shape[-1] - top_cnt, dtype=torch.int64, device=indices_device)[ max_oth] oth Indices = oth Indices[., oth_rnd indices] +33 +34 top_mask = torch.gather mask.expand_as(true_attn), dim=-1, index=topIndices)[., :, None] +35 oth_mask = torch.gather mask.expand_as(true_attn), dim=-1, index=oth Indices)[., None, :] +36 +37 top_draft_attn = torch.gather(draft_attn, dim=-1, index=topIndices)[., :, None] +38 oth_draft_attn = torch.gather(draft_attn, dim=-1, index=oth Indices)[., None, :] +39 +40 residual = top_draft_attn - oth_draft_attn +41 residual_mask = (top_mask | oth_mask).expand_as(residual). flatten(-3) +42 +43 logits = residual Flatten(-3) [residual_mask(bool()) ] +44 labels = torch. ones_like(logits) +45 loss += criterion(logits * beta - alpha, labels).cpu() +46 +47 diff = torch.count_nonzero(logits < 0) / logits.numel() +48 +49 return diff, loss +``` + +# A.4 Needle-in-a-Haystack + +NIAH is a prompt-sensitive test focusing on relative performance changes before and after applying Spotlight Attention, rather than absolute performance. The haystack, needle, and question used in our evaluation are depicted in Figure 7. The prompt, selected from a set proven effective in practice, is shown in Figure 8. + +# Haystack, Needle and Question + +Haystack + +Paul Graham Essays + +Needle + +The best thing to do in San Francisco is eat a sandwich and sit in Dolores Park on a sunny day. + +Question + +What is the best thing to do in San Francisco? + +Figure 7: The haystack, needle, and retrieval question for the NIAH. + +# Prompt + +LLaMA2-7B / LLaMA3-8B + +You are a helpful AI bot that answers questions for a user. Keep your response short and direct. Don't give information outside the document or repeat your findings. The document definitely contains the answer, and I'm $100\%$ sure. So try your best to find it. + +LLaMA2-7B-Chat + +[INST] You are a helpful AI bot that answers questions for a user. Keep your response short and direct. Don't give information outside the document or repeat your findings. The document definitely contains the answer, and I'm $100\%$ sure. So try your best to find it. [/INST] + +Figure 8: Prompts used in NIAH test. + +# A.5 Language Modeling Perplexity + +For Quest, we used the first 128 tokens for pre-filling with full attention. For MagicPIG, we pre-filled the first 1024 tokens with full attention to compute the key's mean value. Our Spotlight Attention method employed sparse KV retrieval throughout without pre-filling. For LLaMA2 models, we evaluated the first 4K tokens per sample; for LLaMA3 and Qwen2.5 models, we used the first 8K tokens. We compared Quest and MagicPIG separately, as Quest and our method use a fixed token budget, while MagicPIG dynamically selects tokens and uses local windows and sink tokens. + +In comparison with Quest, we calculated the token budget by multiplying the token sequence length by the pruning rate, with a minimum budget of 20. + +# A.6 Downstream QA Tasks + +For contexts exceeding 8K tokens, we truncated 8K tokens from right to left. We evaluated all LongBench [6] subdatasets using the official test scripts. The LLaMA2 chat model employs the official chat template, whereas other models do not. + +# A.7 Bit-Packing CUDA Kernel + +Bit-packing (Figure 9) is crucial because PyTorch lacks a native bit type, and boolean values are stored as full bytes. Without compaction, storage usage would increase substantially. The bit-packing program groups 32 boolean values and iteratively packs them into a single uint_32t, as detailed in the pseudocode accompanying Figure 9. + +![](images/c27a722caf3a757668e6ea33ef1bde00500a46a4d8303946fb6c41723cf9ec8b.jpg) +① Shift logical left +Figure 9: Bit-packing. + +```python +1 def bit Packing(tensor): +2 # WE PRESENT THE LOGIC HERE, +3 # WITH THE ACTUAL IMPLEMENTATION WRITTEN IN CUDA. +4 n, d = tensor.shape +5 assert d % 32 == 0 +6 +7 tensor = tensor.chunk(32, dim=-1) +8 output = torch.zeros((n, d // 32), dtype=uint32) +9 for x in tensor: +10 output <= 1 +11 output |= x & 0x01 +12 +13 return output +``` + +# A.8 LSH Weight Initialization + +For the linear hashing function, we employ angular LSH with a rotation matrix as the initial value, outperforming standard random initialization. To generate a $d$ -dimensional rotation matrix, we use QR decomposition. We first create a random matrix $R \in \mathbb{R}^{d \times d}$ , with each element independently sampled from a standard normal distribution: + +$$ +R \in \mathbb {R} ^ {n \times n}, R _ {i, j} \sim \mathcal {N} (0, 1). \tag {9} +$$ + +We then perform QR decomposition on $R$ , yielding an orthogonal matrix $\mathbf{Q}$ and an upper triangular matrix $\mathbf{R}$ : + +$$ +R = \mathbf {Q} \mathbf {R}, \quad \mathbf {Q} ^ {\top} \mathbf {Q} = \mathbf {I}. \tag {10} +$$ + +The matrix $\mathbf{Q}$ is not necessarily in the special orthogonal group $\mathbb{S}0(d)$ , as its determinant can be either $+1$ or $-1$ . To ensure $\mathbf{Q} \in \mathbb{S}0(d)$ , we flip the sign of the first column of $\mathbf{Q}$ if $\operatorname*{det}(\mathbf{Q}) < 0$ . + +# B More Experimental Results + +# B.1 Per-Head Retrieval Accuracy + +The IoU reported in the experiments section is averaged across all heads and layers. Given the varied behavior of LLM heads, individual IoU scores differ significantly. Figure 10 presents detailed per-head IoU comparisons for various models, using Spotlight Attention and linear hashing functions, both before and after training. + +# B.2 Detailed QA Response Fidelity Scores + +In the main text, we report the average Rouge-L score across all subdatasets as the similarity metric. However, scores vary significantly across individual subdatasets. Figure 11 provides detailed similarity scores for each method compared to the original LLaMA3-8B model. To avoid confusion, we assign a Rouge-L score of 1 to identical outputs, except in special cases like outputs containing only $\backslash n$ characters, where tokenization issues may prevent a perfect score. + +![](images/0b5528b9c075fa501da1e29cfeda04315e71c19b56f8008147cf8163a6c52114.jpg) +Figure 10: Per-head retrieval accuracy. Measured by Intersection over Union (IoU) of top- $k$ sets predicted by oracle and Spotlight. (1) The results of linear hashing reveal that most heads maintain low IoU both pre- and post-training, suggesting their latent distributions are challenging to approximate with linear functions. (2) Random parameter initialization results in low before-training IoU for Spotlight Attention, which improves significantly after training. + +# B.3 Few-Shot Learning + +To evaluate Spotlight Attention with short context lengths, we restrict the KV cache to 20 tokens and assess performance on 5-shot learning datasets from LM-Eval-Harness [11], including GLUE, SuperGLUE, OpenBookQA, HellaSwag, PiQA, Winogrande, ARC-E, ARC-C, MathQA, and MMLU. + +As shown in Table 8, Spotlight Attention delivers strong performance. Comparisons with Quest and MagicPIG are omitted, as their large local windows or budgets consume most of the prompt length on most datasets, rendering such comparisons less meaningful. Instead, we emphasize the relative performance between Spotlight Attention and the original model. + +Table 8: A 5-shot learning comparison between original model and Spotlight Attention under a fixed budget of 20 tokens was conducted across: GLUE (1), SuperGLUE (2), OpenBookQA (3), HellaSwag (4), PiQA (5), Winogrande (6), ARC-E (7), ARC-C (8), MathQA (9), and MMLU (10). + +
#1#2#3#4#5#6#7#8#9#10Win Rate
LLaMA2-7B0.4890.6460.3420.6040.7760.7330.7930.4780.2620.46856%
+Spotlight0.4940.6320.3360.5810.7860.7480.7930.4690.2840.46644%
LLaMA2-7B-Chat0.6250.7170.3460.5980.7630.7290.8110.4740.2950.48289%
+Spotlight0.5820.6640.350.5780.7590.7030.8090.4670.2750.48211%
LLaMA3-8B0.6180.7270.3780.6310.8040.7720.850.5340.4180.66338%
+Spotlight0.6200.7360.3780.6240.8040.7730.8510.5330.4110.66462%
+ +# B.4 Downstream QA Tasks + +The main text primarily reports LLaMA3-8B scores on LongBench. Here, we additionally present results for LLaMA2-7B and LLaMA2-7B-Chat, alongside comparisons of our method and Quest under varying token budgets, as shown in Table 9. + +# B.5 Ablation on Loss Function + +In the previous section, we analyzed the limitations of reconstruction loss. Here, we compare its empirical performance with our proposed ranking loss. As shown in Table 10, our ranking loss significantly outperforms reconstruction loss, which provides minimal to no benefit. + +![](images/6df06db751d8e328758fe9480bbe790c6ae2b858f1c539e8f921993e0313e39e.jpg) +Figure 11: Detailed similarity between (i) the outputs of different models (including LLaMA3-8B itself) and (ii) those of LLaMA3-8B. + +Table 9: Absolute scores on LongBench. Performance comparison of different methods on Long-Bench's long-text downstream tasks: NarrativeQA (1-1), Qasper (1-2), MultiFieldQA-en (1-3), MultiFieldQA-zh (1-4), HotpotQA (2-1), 2WikiMultihopQA (2-2), MuSiQue (2-3), DuReader (2-4), GovReport (3-1), QMSum (3-2), MultiNews (3-3), VCSUM (3-4), TREC (4-1), TriviaQA (4-2), SAMSum (4-3), LSHT (4-4), PassageCount (5-1), PassageRetrieval-en (5-2), PassageRetrieval-zh (5-3), LCC (6-1), and RepoBench-P (6-2). (1) With $98\%$ tokens pruned and no local window or global sink tokens, Spotlight Attention outperforms Quest and MagicPIG. (2) Spotlight Attention achieves performance on par with the original model, even in tasks like summarization and few-shot learning. (3) On subsets of Chinese (1-4, 2-4, 3-4), other LLaMA2-7B-Chat-based models generated answers in English, while Quest produced responses in Chinese, achieving overwhelmingly high scores. + +
MethodConfigurationSingle-Doc.Multi-Doc.SummarizationFev-ShotSyntheticCode
#1-1#2-1#3-1#4Avg.#2-1#2-2#2-3#2-4Avg.#3-1#2-3#3-3#4Avg.#4-1#2-4#3-4Avg.#5-1#5-2#3-3Avg.#6-1#6-2Avg.
LLaMA2-7B8.737.1815.4213.8411.296.558.272.9111.387.2715.0619.806.039.3012.5468.0030.6230.8318.2536.921.266.978.005.4163.6656.6360.14
+Quest1024 Token Budget8.789.7817.9514.8612.848.918.512.9512.538.2218.3820.849.638.4014.3166.0067.0630.3517.5042.221.696.477.385.1863.8858.8261.35
+MagiPIG9.174.6913.695.768.326.005.162.078.055.326.7317.783.273.347.7845.0031.5314.538.0024.760.834.061.502.1347.8844.1646.02
+Spotlight90% Pruned (≤ 409)10.1613.7614.5611.818.998.992.5312.3310.9916.2316.3219.118.948.3466.5068.0030.9019.0039.900.991.001.001.001.001.001.00
+Spotlight98% Pruned (≤ 81)10.768.0916.6312.9512.106.467.982.9811.257.0916.3219.3110.248.5113.5967.5038.0810.0038.562.078.276.795.7164.1056.7660.43
LLaMA2-7B-Chat18.7124.8331.638.3620.8831.7628.2212.662.4818.7827.1820.3626.170.2418.4864.5077.8540.7615.7049.561.642.753.422.6054.5748.7451.65
+Quest1024 Token Budget19.1413.7614.5611.818.998.992.5312.3310.9916.2316.3219.118.948.3466.5068.0030.9019.003.9549.562.086.477.385.1863.88
+Quest128 Token Budget14.1112.7817.599.4213.4728.3821.038.998.5316.7311.5518.4420.918.3914.8238.5066.4431.6910.5036.780.723.507.303.7043.2139.5541.38
+MagiPIG18.8423.7928.809.4320.2131.9728.2311.963.1718.8326.3220.3125.490.1418.6565.5085.5940.5216.2551.961.523.504.243.0857.9452.1755.05
+Spotlight90% Pruned (≤ 409)18.8423.7928.809.4320.2131.9728.2312.662.4818.7826.3220.3125.490.1418.6565.5085.5940.5216.2551.961.523.504.243.0857.9452.1755.05
+ +# B.6 Ablation on Attention Estimation Methods + +We evaluated two approaches: hashing with Hamming distance (our default choice) and down-projection with inner product. At a 16x compression rate, results in Table 11 show that hashing outperforms down-projection, demonstrating greater efficiency when the dimensionality is low. + +Table 10: Ablation on training loss. Compared to attention reconstruction loss, our proposed ranking loss yields significantly improved training outcomes. + +
LossPG19MathCode
Attn. Recon. Loss21.3418.85611.573
Ranking Loss (ours)8.9773.6215.434
+ +Table 11: Ablation on estimation method. We compared two attention estimation methods: hashing with Hamming distance (our default choice) and downprojection with inner product. Results indicate that hashing with Hamming distance performs better. + +
Estimation MethodTrainingPG19MathCode
Down Proj. (16×) + Inner Prod.✓14.7435.4257.619
Hashing + Hamming Dist. (ours)8.9773.6215.434
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Despite the effectiveness of existing methods in evaluating image authenticity and locating forgeries, these approaches often lack human interpretability and do not fully address the growing complexity of synthetic data. To tackle these challenges, we introduce FakeVLM, a specialized large multimodal model designed for both general synthetic image and DeepFake detection tasks. FakeVLM not only excels in distinguishing real from fake images but also provides clear, natural language explanations for image artifacts, enhancing interpretability. Additionally, we present FakeClue, a comprehensive dataset containing over 100,000 images across seven categories, annotated with fine-grained artifact clues in natural language. FakeVLM demonstrates performance comparable to expert models while eliminating the need for additional classifiers, making it a robust solution for synthetic data detection. Extensive evaluations across multiple datasets confirm the superiority of FakeVLM in both authenticity classification and artifact explanation tasks, setting a new benchmark for synthetic image detection. The code, model weights, and dataset can be found here: https://github.com/opendatalab/FakeVLM. + +# 1 Introduction + +As AI-generated content technologies advance, synthetic images are increasingly integrated into our daily lives [1, 2, 3, 4]. Technologies like Diffusion [5, 1, 6] and Flux can generate highly realistic images. However, synthetic image data also poses significant risks, including potential misuse and social disruption [7, 8]. For instance, synthetic fake news [9, 10, 11] and forged facial scams [12, 13] make it increasingly challenging to discern the truth and establish trust in information. In response to these threats, the field of synthetic data detection has garnered widespread attention in recent years. However, traditional synthetic detection methods typically focus on simple authenticity judgments [14, 15, 16, 17, 18, 19], ultimately providing only a forgery probability or classification result. This approach still has significant limitations in terms of human interpretability. As a result, users find it challenging to understand the reasons behind the system's decisions, affecting the decision-making process's transparency and trustworthiness. + +The rapid development of large multimodal models has accelerated progress in synthetic data detection [20, 21, 22, 23, 24, 25]. These models, particularly, can provide explanations for authenticity judgments in natural language, thus laying the groundwork for enhancing the interpretability of synthetic + +detection. For instance, Jia et al. [26] explored ChatGPT's ability to assess synthetic data, highlighting the potential of large models in this domain. LOKI [27] and Fakebench [28] further delved into the capabilities of large models in offering explanations for image detail artifacts. However, these studies primarily focus on pre-trained large models, utilizing strategies such as different prompt wordings to enhance model performance on this task rather than developing a specialized multimodal model for this specific domain. Moreover, existing general large models still show a significant performance gap compared to expert models or human users in detection tasks. + +Researchers have further explored and designed multimodal models specifically for synthetic data detection tasks [25, 24, 27, 29, 30]. For instance, works like DD-VQA [23] and FFAA [24] focus primarily on the performance of large models in Deepfake detection tasks, especially in the context of artifact explanation. However, their performance on more general types of synthetic images still requires further investigation. Other works, such as Fakeshield [20] and ForgeryGPT [22], effectively examine the ability of large models to localize forgery artifacts and explain manipulated synthetic data. However, artifacts in forged synthetic images often concentrate in transitional areas, exhibiting more noticeable edge artifacts. In contrast, direct synthetic images are more likely to show structural, distortion, or physical artifacts, highlighting significant differences between the two. Additionally, current large models still lag behind expert models in pure authenticity classification. For example, studies such as X2-DFD [25] and FFAA [24] attempt to integrate traditional expert-based synthetic detection methods to improve classification accuracy without fully unlocking the potential of large models in synthetic detection tasks. + +To address the challenges outlined above, we introduce FakeVLM, a large multimodal model specifically designed for fake image detection and artifact explanation. FakeVLM focuses on artifacts generated from synthetic image models rather than forgery artifacts. Moreover, it is not limited to facing Deepfake tasks but extends to more general synthetic data detection. Notably, FakeVLM achieves performance comparable to expert models based on binary classification without requiring additional classifiers or expert models. Additionally, we introduce FakeClue, a dataset containing over 100,000 real and synthetic images, along with corresponding artifact cues for the synthetic images. FakeClue includes images from seven different categories (e.g., animal, human, object, scenery, satellite, document, face manipulation), and leverages category-specific knowledge to annotate image artifacts in natural language using multiple LMMs. Our main contributions are as follows: + +- We propose FakeVLM, a multimodal large model designed for both general synthetic and deepfake image detection tasks. It excels at distinguishing real from fake images while also providing excellent interpretability for artifact details in synthetic images. +- We introduce the FakeClue dataset, which includes a rich variety of image categories and fine-grained artifact annotations in natural language. +- Our method has been extensively evaluated on multiple datasets, achieving outstanding performance in both synthetic detection and abnormal artifact explanation tasks. + +# 2 Related Work + +# 2.1 Synthetic Image Detection + +Traditional synthetic detection tasks are typically approached as binary classification tasks [31, 26, 32, 33, 34, 35], treating them as either true or false, based on data-driven methods, with various detectors such as CNNs [36] and transformers [37]. Some research [38, 39, 40] has explored methods like learning anomalies in the frequency domain, reconstruction learning, and adversarial learning. The focus of traditional synthetic detection tasks has mainly been on addressing the generalization issue [15, 16], where the training and testing domains differ, in order to counter the rapidly evolving synthetic models. However, these methods only provide authenticity predictions without offering detailed explanations behind the predictions. + +Some works have gone beyond the fundamental true/false classification problem, focusing on interpretable synthetic detection. For example, gradient-based methods are used to visualize highlighted regions of predictions [41, 42, 43]. Alternatively, model designs like DFGNN [44] enhance interpretability by applying interpretable GNNs to deepfake detection tasks. Additionally, research has explored the detection and localization of forgeries by constructing image artifacts or modifying labels [17, 18, 19]. While these methods enhance model interpretability, using natural language to describe the identified reasons remains underexplored. + +![](images/c9f11916672711a7e46d89215a87b7b2c12a2fa4913f1015449e0a77dbc00cd6.jpg) +Figure 1: Construction pipeline of FakeClue dataset, including data collection from open source and self-synthesized datasets, pre-processing with categorization, label prompt design based on category knowledge(Face M: Face manipulation), and multiple LMMs annotation with result aggregation. + +# 2.2 Synthetic Image Detection via LMMs + +Recently, the development of large multimodal models (LMMs) has been rapid [45, 46, 47, 48, 49, 50]. Models such as the closed-source GPT-4o [51] and open-source models like InternVL [52] and Qwen2-VL [53] have demonstrated outstanding performance across various tasks, showcasing impressive capabilities. In the domain of synthetic data detection, several works like [26], Fakebench [28], and LOKI [27] have investigated the potential of LMMs, where these large models not only deliver accurate synthetic detection judgments but also offer natural language explanations for their true/false predictions, enhancing the interpretability of synthetic detection. However, these studies mainly focus on evaluating pretrained large models rather than training expert multimodal models. Furthermore, existing general models still lag behind expert models or humans in detection tasks. + +An increasing number of studies have further explored the use of LMMs for synthetic detection, such as DD-VQA [23] and FFAA [24] have explored the performance of large models in the deepfake domain. However, their performance on general synthetic images remains less explored. Fakeshield [20], SIDA [21] and ForgeryGPT [22] investigate the ability of LMMs to detect/explain artifacts in manipulated synthetic data. However, tampering artifacts (mainly transitional), while direct image synthesis artifacts tend to involve structural distortions or other types of image warping, which differ significantly. Furthermore, current LMMs underperform expert models in simple real/false classification tasks [25, 26]. For example, studies like X2-DFD [25] and FFAA [24] have combined traditional expert synthetic detection methods with large models to improve classification accuracy. + +# 3 Dataset + +# 3.1 Overview + +We introduce FakeClue, a multimodal synthetic data detection benchmark for general and DeepFake detection. It includes two tasks: synthetic detection and artifact explanation, requiring models to determine image authenticity and explain its artifacts. FakeClue covers 7 image categories with over 100k samples. Utilizing a multi-LMM labeling strategy with category priors, FakeClue provides image-caption pairs for the image and its natural language artifact explanation. It emphasizes direct synthesis over tampered image artifacts. Training/test sets are randomly split; the test set contains 5,000 diverse image samples. Detailed dataset information is provided in the supplementary materials. + +# 3.2 Construction of FakeClue + +Data Collection: As shown in the data collection phase of Figure 1, FakeClue has two data sources, including open synthetic datasets and our newly synthesized data for specialized types. For open synthetic datasets, we extracted approximately 80K data from GenImage [54], FF++ [55] and Chameleon [56], maintaining a 1:1 ratio of fake to real data. For specialized types of data, such as remote sensing and document images, we generated the data ourselves. We collected remote sensing images from public datasets [57, 58, 59, 60, 61, 62], using GAN and Diffusion-based methods [63, 64], covering urban, suburban, and natural scenes. For document images, we adopted a layout-first, content-rendering approach, generating synthetic images of newspapers, papers, and magazines with real-world data sourced from the M6Doc dataset [65]. + +Data Pre-Processing: At this stage, we first categorize the collected raw image data based on authenticity labels. Since the data from GenImage and Chameleon lack category information, we use a classification model [66] to divide the data into four categories: animal, object, human, and scene. The $\mathrm{FF} + +$ dataset corresponds to the face manipulation category, while the newly synthesized satellite and document data also have clear category divisions. The distribution and proportions of these categories are shown in Figure 1. The labels obtained during the data preprocessing stage will serve as the foundation for the subsequent Label Prompt Design. + +Label Prompt Design based on category knowledge: To overcome the hallucinations and limited synthetic detection capabilities of large models, we inject external knowledge to aid in artifact detection. Pre-processed authenticity labels serve as prior prompt knowledge. For real images, we analyze the plausibility of the image as a photographic result, while for fake images, we focus on detecting artifacts throughout the image. And classification labels are used as category-specific knowledge. This knowledge comprises predefined focal points and common artifact types pertinent to each category, guiding the model's attention to key areas, such as the artifact detection cues for facial images, as shown in Fig. 1. Accordingly, we design 14 distinct types of prompts to address various authenticity and category labels. In addition, for synthetic images that may have high quality and no visible artifacts (e.g., images from the Chameleon dataset), we use special prompts to prevent the model from forcing interpretation (see Appendix H for detailed prompts for different categories). + +Multiple LMMs Annotation: To mitigate the bias or hallucination effects of a single multimodal model, we adopt a strategy of annotating and then aggregating results from multiple high-performance open-source large models. For a given image $I_{i}$ from the set $\{I_i\}_{i = 1}^N$ , we first use three large multimodal models—Qwen2-VL, InternVL, and Deepseek—to generate a set of candidate artifact captions $C_i = \{A_i^1,A_i^2,A_i^3\}$ . Each caption $A_{i}^{k}\in C_{i}$ potentially highlights different aspects of image artifacts. + +To synthesize a unified annotation $A_{i}$ from candidate captions $C_i = \{A_i^1, A_i^2, A_i^3\}$ , we employ Qwen2-VL for aggregation, denoted as $\mathcal{M}_{\mathrm{agg}}$ . This model is prompted with specific instructions $P_{\mathrm{instr}}$ (detailed in the Appendix H) to identify consistent artifacts across $C_i$ and remove redundant information. Thus, the final aggregated annotation $A_{i}$ for image $I_{i}$ is obtained as: + +$$ +A _ {i} = \mathcal {M} _ {\text {a g g}} \left(\mathrm {C} _ {i}, P _ {\text {i n s t r}}\right) \tag {1} +$$ + +Where $A_{i}^{k}$ is the $k$ -th candidate caption for $I_{i}$ , and $\mathcal{M}_{\mathrm{agg}}$ aggregates $C_{i}$ using prompt $P_{\mathrm{instr}}$ (enforcing consistency/non-redundancy) into the final annotation $A_{i}$ . + +The model extracts common points from multiple model responses, filters out irrelevant observations that appear in only one model (unless they are critical, like glaring artifacts) and organizes them hierarchically by categories (e.g., texture, geometry, lighting), before outputting in a fixed format. + +Table 1: Comparison with existing synthetic detection datasets (DF: DeepFake, Gen: General, Syn: Synthesis, Tam: Tampering). + +
DatasetFieldArtifactAnnotatorCategoryNumber
DD-VQA [23]DFSynHuman5k
FF-VQA [24]DFSynGPT95K
LOKI [27]GenSynHuman✓3k
Fakebench [28]GenSynHuman✓6k
MMTD-Set [20]GenTamGPT100k
FakeClueDF+GenSynMulti-LMMs✓100k
+ +# 3.3 Comparisons with Existing Datasets + +Table 1 presents a comparison of the synthetic detection performance of FakeClue and existing evaluated LMMs datasets. FakeClue offers broader domain coverage and, unlike DD-VQA, is not confined to DeepFake detection, featuring well-defined categories including specialized types like satellite images and documents. In contrast, LOKI and Fakebench are limited by the number of annotations and can only serve as evaluation sets. Compared to the recent MMTD-Set dataset, FakeClue focuses more on directly synthesized image artifacts rather than tampered artifacts. + +# 4 Method + +In this section, we first analyze the challenges of large multimodal models in synthetic image detection (Section 4.1). Then, we provide a detailed description of the FakeVLM architecture (Section 4.2) and training strategy (Section 4.3). + +![](images/172bc93522aa4d3d7617915db3e2c1cb830583730d2eb2967717bf35ee72239e.jpg) +Figure 2: Overview of FakeVLM, our proposed framework for detecting synthetic images and explaining their artifacts. Built upon LLaVA, FakeVLM integrates multiple captioning models to assess key visual aspects. + +# 4.1 Re-thinking LMMs' Challenges in Synthetic Image Detection + +Recent studies have shown that directly using LMMs for synthetic image detection still face significant challenges [27, 28, 26]. Although large models possess strong text explanation capabilities, when tasked with determining whether an image was AI-generated or identifying forged images from a set, pretrained LMMs often fail to achieve satisfactory performance. This phenomenon highlights the difficulty of relying on LMMs for authenticity judgment, which is closely related to the fact that these models are not inherently designed for synthetic data detection tasks. Nevertheless, through extensive pretraining tasks, multimodal large models have developed strong visual feature extraction abilities and alignment with text. This raises the question: do the internal representations of these large models potentially encode information that can distinguish real images from synthetic ones? + +Inspired by Zhang et al. [67], we explored a simple yet effective method on FakeClue: extracting visual features from the last layer of a pre-trained LMM and training a lightweight linear classifier to determine image authenticity. If the representations learned by the model indeed contain discriminative information related to authenticity, even a simple classifier can perform initial detection using these features. As shown in Figure 3, the results confirm that the LMM has potential in distinguishing authenticity. However, in contrast to the conclusions of Zhang et al. [67], framing the task as "Does the image look real/fake?" by using fixed answers like "Real" or "Fake" not only limits the model's ability to provide textual explanations but also results in + +![](images/96a2b55b63f64b6ecae1d8b52327ca480dd00b10105518dd51dad2e194feb48d.jpg) + +suboptimal performance. This is likely due to large models' challenges in aligning complex visual content with such binary answers. Building on the dataset we constructed, we found that framing the task as visual question answering, where the model is required to provide not just "Real"/"Fake" answers but also explain the image "artifacts", leads to better alignment between the response text and the image content in Figure 3 (For detailed results, please refer to Appendix B). This approach not only improves the performance of artifact explanation but also significantly enhances the overall performance of synthetic image detection. + +# 4.2 Model Architecture + +Our approach follows the architecture of LLaVA-v1.5, as illustrated in the framework diagram (see Figure 2 in the top-right corner), which consists of three core components: i) a Global Image Encoder, ii) an MLP Projector, and iii) a Large Language Model (LLM). We detail each component as follows: + +Global Image Encoder: We employ the pretrained vision backbone of CLIP-ViT(L-14) [68] as our global image encoder. The encoder processes input images with a resolution of $336 \times 336$ to preserve synthetic artifact details, resulting in 576 patches per image. + +$$ +V = \operatorname {C L I P} - \operatorname {V i T} (I) \in \mathbb {R} ^ {N \times d _ {v}} \tag {2} +$$ + +where $N = \frac{HW}{P^2}$ denotes the number of patches ( $P = 14$ ), and $d_v = 1024$ the feature dimension. + +Multi-modal Projector: A two-layer MLP adaptor bridges visual and textual modalities: + +$$ +H = \operatorname {G e L U} \left(V W _ {1} + b _ {1}\right) +$$ + +$$ +Z = H W _ {2} + b _ {2} \tag {3} +$$ + +where $W_{1} \in \mathbb{R}^{1024 \times 4096}$ , $W_{2} \in \mathbb{R}^{4096 \times 4096}$ are learnable parameters. The projected features $Z \in \mathbb{R}^{N \times 4096}$ combine with text embeddings of the task prompt $P$ through concatenation. + +Large Language Model: We utilize Vicuna-v1.5-7B, a 7B language model, as our base LLM. Vicuna-v1.5 is renowned for its strong instruction-following capabilities and robust performance across diverse tasks. To further enhance its reasoning abilities on synthetic data, we perform full-parameter fine-tuning, optimizing the following objective: + +$$ +\mathcal {L} (\theta) = - \sum_ {t = 1} ^ {T _ {i}} \log p _ {\theta} \left(a _ {i, t} \mid a _ {i, < t}, [ Z; E (P) ]\right) \tag {4} +$$ + +where $E(\cdot)$ denotes text embeddings. We update all parameters $\theta$ of the LLM during training, enabling full adaptation to synthetic data reasoning while preserving original instruction-following capabilities via full-parameter optimization + +By integrating these components, our pipeline leverages the strengths of multimodal models and fine-tunes LLaVA to achieve robust performance in detecting and explaining synthetic data. + +# 4.3 Training strategy + +As described in the construction process of FakeClue, we leverage category knowledge and utilize multiple LMMs to annotate image artifacts in natural language. As shown on the left side of Fig. 2, we obtain high-quality artifact descriptions as target outputs for the model. Then, we use the QA pairs obtained after the multi-model annotation and summarization steps as our training data. Each data sample consists of: (1) an image $I$ ; (2) a standardized prompt $P$ : "Does the image look real/fake?"; (3) the aggregated answer $A$ from the multi-model annotations. + +Our model, initialized from LLaVA-1.5 7B weights [45], underwent full-parameter fine-tuning on our constructed QA dataset. The training is conducted for two epochs on eight NVIDIA A100 GPUs with a batch size of 32 per GPU using a 2e-5 learning rate with $3\%$ linear warmup and cosine decay. This full fine-tuning adapted the model to synthetic data detection/explanation nuances while preserving its general instruction-following capabilities. + +# 5 Experiment + +In this section, we introduce three additional datasets used in the experiments, alongside FakeClue, and describe our experimental setup. We then present FakeVLM's performance on general synthetic and DeepFake detection tasks, as well as its ability to explain image artifacts. Finally, we conduct ablation studies and further exploratory experiments to assess the model's performance. + +# 5.1 Other Benchmarks + +LOKI [27] is a recently proposed benchmark for evaluating multimodal large models in general synthetic detection tasks. Beyond just distinguishing real from fake, LOKI also includes human manually annotated fine-grained image artifacts, enabling a thorough exploration of the model's ability to explain image artifacts. This inclusion allows us to verify, to some extent, whether our model's perception of artifacts aligns with human cognition. + +$\mathbf{FF} + +$ [55] is a widely used benchmark dataset for facial forgery detection, containing face images and videos generated by different types of forgery techniques. The dataset includes forged data created using four common forgery methods: DeepFakes, Face2Face, FaceSwap, and NeuralTextures. We used the commonly employed C23 versions. + +DD-VQA [23] is a new face-domain artifact explanation dataset leveraging human common-sense perception for authenticity assessment. It features artifacts like blurred hairlines and unnatural skin shadows. Built on $\mathrm{FF}++$ data, DD-VQA employs manual artifact annotations in a VQA format, requiring models to answer common-sense questions about artifacts. + +# 5.2 Experimental setup + +Task Settings. LOKI serves as the evaluation set, while training is conducted on the FakeClue dataset, with testing performed on LOKI. For $\mathrm{FF}++$ and DD-VQA, we use their default training-test splits for evaluation. Evaluation metrics cover two tasks: detection and artifact explanation. Classification accuracy is represented by Acc, Auc, and F1 scores, while artifact explanation accuracy is measured using CSS and ROUGE_L. + +Compared Baselines. For tasks requiring both synthetic detection and artifact explanation (e.g., FakeClue, DD-VQA, LOKI), we compared various general-purpose LMMs, including closed-source models like GPT-4 and open-source ones such as Qwen2-VL, LLaVA, InternVL2, and Deepseek-VL2. We also included the Common-DF method from the DD-VQA dataset. For pure synthetic data detection, we further compared with recent SOTA expert methods. + +# 5.3 Universal synthetic detection + +Table 2 compares FakeVLM with leading general-purpose LMMs and expert models, demonstrating its superior performance in both synthetic detection and artifact explanation. Specifically, compared to the current powerful open-source model Qwen2-VL-72B and leading expert model NPR or AIDE, which is also trained on FakeClue, FakeVLM achieves an average improvement of $7.7\%$ in Acc and $3.6\%$ in F1 on both FakeClue and LOKI. Additionally, LOKI includes an extra evaluation metric for human performance, with an Acc of $80.1\%$ , whereas FakeVLM achieves an Acc of $84.3\%$ , surpassing + +Table 2: The experimental results on the FakeClue and LOKI datasets include both Detection and Artifact Explanation performance. * denotes methods trained on FakeClue. + +
MethodFakeClue (Ours)LOKI Evaluations (ICLR 2025)
Acc ↑F1 ↑ROUGE_L ↑CSS ↑Acc ↑F1 ↑ROUGE_L ↑CSS ↑
Deepseek-VL2-small [69]40.454.217.150.425.338.716.439.1
Deepseek-VL2 [69]47.554.117.250.543.139.216.938.8
InternVL2-8B [52]50.649.018.058.152.634.017.947.2
InternVL2-40B [52]50.746.317.655.250.737.618.447.3
Qwen2-VL-7B [53]45.759.226.656.557.135.018.238.4
Qwen2-VL-72B [53]57.856.517.554.455.440.917.343.2
GPT-4o (2024-08-06) [51]47.442.013.440.763.457.214.735.4
CNNSpot [70]43.19.8--43.111.4--
FreqNet [71]48.739.3--58.950.6--
Fatformer [72]54.545.1--58.848.4--
UnivFD [15]63.146.8--49.035.8--
AIDE* [56]85.994.5--65.680.2--
NPR* [73]90.291.6--77.480.0--
FakeVLM98.698.158.087.784.383.720.158.2
+ +![](images/d1b2a717715dd22d945bac53dfbc29689e6463f8c4453e98df9fea63f33795ac.jpg) + +This is a fake image. The dog's ears are disproportionately large and misaligned; the fur texture is inconsistent, with some areas appearing overly smooth or blurry; the background is too uniform and lacks realistic details, such as shadows or texture variations... + +FakeVLM + +This is a real image. It is of a French Bulldog, and the lighting, texture of the fur, and the dog's posture look natural. The dog's expression also seems authentic, further contributing to its realism... + +ChatGPT + +![](images/602725990e77ec888c01c18cf83486c5bc55129c22f9261ecf23f910a6995b91.jpg) + +This is a fake image. The labels and letters on the devices are blurry, and the details appear distorted. The reflections on the metal surfaces seem too perfect, which is unusual for real-world objects... + +FakeVLM + +This is a real image. The image appears to be of two vintage-style, silver-colored devices, possibly record players or similar objects. The wood grain of the surface and the reflections on the devices further contribute to its authenticity... ChatGPT + +![](images/3e54e89042a9c7390d13e0016fa2b5dc3fd9f90295b1000e97c0885df24f8b0f.jpg) +Figure 4: Synthetic image detection cases, covering animals, people, objects, documents, and remote sensing (red denotes incorrect, green denotes correct detection). FakeVLM outperforms GPT in precision, comprehensiveness, and relevance, demonstrating superior detection and interpretation. + +This is a fake image. The outlines of buildings and vehicles are blurry and distorted. The boundaries between vegetation, land, and buildings are blurred, with texture mixing and unnatural transitions, such as unnatural vegetation... FakeVLM + +FakeVLM + +This is a real image. The image appears to be a real aerial view image taken from a satellite. The details, such as the road markings, buildings, and greenery, seem natural. The presence of pedestrians and vehicles adds to the realism... ChatGPT + +![](images/1c3ef8495ed900b6155fb629afa85954e3b9c38f9316134761fa4cd86a2f7037.jpg) + +This is a fake image. Although it does not show obvious artifacts or direct signs of composition, there are underlying characteristic inconsistencies in its features that suggest it is artificially created. + +FakeVLM + +# + +![](images/cc2273f741cd008c020ab05c9821ededc1bb613a18ab5f031f73f00723174384.jpg) + +This is a real image. It appears to be a genuine photograph of a young woman in an outdoor setting, possibly taken on a sunny day. The lighting, natural movement of her hair, and the background all contribute to the authenticity of the image.... + +ChatGPT + +# + +This image is fake. The handwritten content suddenly appears amidst document. The text contains nonsensical phrases that fail to meet readability standards; the layout is misaligned, deviating from a professionally formatted document... + +FakeVLM + +明 + +This is a real image. It is a composite of multiple elements, including an advertisement for the Xiaomi Mi 10 phone, some text in Chinese, and parts of a letter or email message... + +ChatGPT + +3 + +![](images/c85bf4344d6b25d2014b52c6b8e76080612bdd9b14bf6d72ec93c1de02785b8e.jpg) + +This is a fake image. The smoke from the volcano is too uniform and lacks natural irregularities; the lava flow is too smooth and lacks the rough, uneven texture of real lava... + +FakeVLM + +# + +This is a fake image. The most noticeable oddity is the cloud of smoke, which looks very stylized and somewhat too clean and fluffy compared to natural volcanic eruptions. The way the smoke puffs out doesn't match the natural irregularity... + +ChatGPT + +# + +human performance. This is likely attributed to FakeVLM's ability to capture deep, image-level features that are imperceptible to the human eye for accurate authenticity judgment. Moreover, FakeVLM's strong ROUGE_L and CSS scores on the human-annotated LOKI dataset highlight alignment with human artifact perception, a noteworthy achievement given its training on entirely LMM-generated data, which is largely thanks to our meticulously designed data annotation pipeline. + +Figure 4 presents FakeVLM's qualitative evaluation. FakeVLM identifies synthesis issues (e.g., visual artifacts, texture distortions, structural anomalies) with detailed natural language explanations. Unlike probability-threshold methods, FakeVLM's intuitive, interpretable descriptions, enhance detection transparency, enabling confident and reliable synthetic content assessment. + +We also present the generalization experiment results of FakeVLM on DMImage [74] in Table 3, following Huang et al. [21]. The experimental results show that the performance gap between FakeVLM and other expert models is not significant, with FakeVLM even outperforming some of them. FakeVLM does not rely on additional classifiers or expert models, yet it achieves performance comparable to or even exceeding that of expert classifiers while retaining its language capability for artifact explanation. This demonstrates the potential of large models in synthetic detection. + +Table 3: Comparison with other detection methods on the DMImage [74] dataset, using the original weights for each method. + +
MethodRealFakeOverall
AccF1AccF1AccF1
CNNSpot [70]87.888.428.444.240.643.3
Gram-Net [75]62.854.178.888.167.479.4
Fusing [8]87.786.115.527.240.436.5
LNP [76]63.167.456.972.558.268.3
UnivFD [15]89.488.344.961.253.960.7
AntifakePrompt [77]91.392.589.391.290.691.2
SIDA [21]92.993.190.791.091.892.4
FakeVLM98.299.189.794.694.094.3
+ +# 5.4 DeepFake detection + +We evaluate FakeVLM's performance on DeepFake detection, with results on DD-VQA presented in Table 4. FakeVLM surpasses both general-purpose multimodal models and the specialized vision-language model, Common-DF, with improvements of $5.7\%$ in Acc, $3\%$ in F1, and $9.5\%$ in ROUGE_L. + +Table 4: The experimental results were evaluated on the DD-VQA datasets. Common-DF-T and Common-DF-I represent text or image contrastive losses, respectively, while Common-DF-TI denotes both text and image contrastive losses. + +
MethodDD-VQA (ECCV 2024)
Acc ↑F1 ↑ROUGE_L ↑CSS ↑
InternVL2-8B [52]56.953.114.351.1
InternVL2-40B [52]52.557.722.254.5
Qwen2-VL-7B [53]45.758.926.656.5
Qwen2-VL-72B [53]59.557.920.556.6
GPT-4o [51]53.231.713.042.7
Common-DF-T [78]83.787.657.7-
Common-DF-I [78]84.988.458.8-
Common-DF-TI [78]87.590.160.9-
FakeVLM93.293.170.486.6
+ +Table 5 shows FakeVLM's evaluation on the FF++ DeepFake detection dataset, including subcategories like DeepFakes, Face2Face, FaceSwap, and NeuralTextures. The experimental results demonstrate that FakeVLM continues to exhibit strong performance in these tasks, comparable to or even surpassing leading deepfake expert models. It not only leads in overall detection but also exhibits balanced performance across categories, avoiding overfitting. + +Table 5: Performance evaluation of FakeVLM on the FF++ DeepFake detection dataset. The results highlight FakeVLM's robust detection performance, on par with or outperforming top expert models. + +
MethodFF++ (ICCV 2019) - AUC(%)
FF-DFFF-F2FFF-FSFF-NTAverage
FWA [79]92.190.088.481.287.7
Face X-ray [80]97.998.798.792.995.9
SRM [81]97.397.097.493.095.8
CDFA [82]99.986.993.380.790.2
FakeVLM97.296.096.895.096.3
+ +# 5.5 Ablation Study and More Exploration + +Impact of explanatory text. To validate the effectiveness of our explanatory VQA text paradigm, we conduct ablation experiments comparing two variants under the same training settings: (1) LLaVA + Linear Head: Full parameter fine-tuning with a linear classification head trained for binary prediction; (2) LLaVA + Explanatory Text: Training LLaVA to generate explanatory answers instead of fixed labels. Both variants trained on FakeClue, tested on LOKI; all parameters trainable. As shown in Table 6, the explanatory text paradigm demonstrates advantages in out-of-distribution (OOD) generalization on the LOKI benchmark. + +Table 6: Ablation study comparing linear classification and explanatory text paradigms. + +
MethodLOKI Evaluations (ICLR 2025)
Acc ↑F1 ↑ROUGE_L ↑CSS ↑
LLaVA + Linear Head81.677.8--
LLaVA + Explanatory Text84.380.160.958.2
+ +Performance on Real Images. In practical applications, authentic images dominate, necessitating models that can avoid misidentifying artifacts and incorrectly filtering genuine images. Fig 5 illustrates FakeVLM's real image performance, showing it integrates features like object structure, lighting, color, and fine textures for authenticity assessment. This holistic approach enhances the model's reliability and robustness in distinguishing between synthetic and real images. Additional experiments, including robustness studies on image perturbations, are provided in Appendix C. + +![](images/9c712091586e922bbe22f90b65eb905655aec3e9522bcd93b335d2525927518c.jpg) + +This is a real image. The monkey's physical features are proportioned correctly, with symmetrical ears, paws, and body. The fur is clear and well-defined, exhibiting a realistic texture that matches natural monkey fur. The color of the monkey's fur appears natural and consistent with real-world monkey species. The background and environment, such as the trees and foliage, are also realistic, with no obvious distortions or unnatural elements. + +![](images/3c5f07fe6f586fc16f5609708cca5c1fc36db7084eedfd8776cd87616ad04b39.jpg) +Figure 5: Performance of FakeVLM on real images. + +This is a real image. The colors and textures, from the architecture to the sky. The buildings, lamp posts, and other objects are placed logically. The objects in the scene are grounded in realistic physics, with no floating or unsupported elements. The scene includes natural details and imperfections, such as the texture of the stone and the slight variations in the structure's surface. + +![](images/9d5270ec6cedc055235e7d80f1d6a1b54503f3c6c148127d8f482ef6c09413b1.jpg) + +FakeVLM + +![](images/367bbc0502e054ff729f06afb1b33ed459d87934270e1e5b412307968237c257.jpg) + +This is a real image. The radiator has a realistic texture and material, with a metallic finish that matches the natural texture of a radiator. The blinds have a natural, woven texture that looks realistic. The lighting and shadows are applied in a consistent, believable manner, with realistic shading based on the light source and surroundings. + +![](images/d04155b9851339f1d84109c226c5fd197a407a444a8ea804936db0f92a960f85.jpg) + +FakeVLM + +# 6 Conclusion + +The rapid growth of AI-generated images has posed challenges to the authenticity of information, driving the demand for reliable and transparent detection methods. As image synthesis detection techniques have advanced alongside large multimodal models (LMMs), approaches have shifted from non-LMMs to methods based on LMMs. Our proposed FakeVLM is a large model that integrates both synthetic image detection and artifact explanation. Through an effective training strategy, FakeVLM leverages the potential of large models for synthetic detection without relying on expert classifiers. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: LLMs are just used for editing. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Appendices + +# A Dataset visualization examples + +Figure 6 shows some example images from the FakeClue dataset, which includes seven major categories. Our dataset also contains a rich variety of synthetic images with different qualities and resolutions, some of which exhibit noticeable artifacts, while others are difficult to detect with the naked eye. The typical artifact characteristics vary across different categories of images, which is why we use different prompts tailored to each category in order to enhance the model's ability to capture these artifacts. + +![](images/b51f0b54231b3fe236f13a9cd57d6746d20fe9e51d0343d5e1b81582bc4537fb.jpg) +Figure 6: Real and fake image examples from the FakeClue dataset, categorized by Document, Object, Animal, Human, Scene, Satellite, and Face Manipulation. + +# B Different training strategies + +We explore different training strategies on FakeClue and evaluate them on both the FakeClue test set and the additional LOKI test set. As shown in Figure 7, a simple linear layer can partially activate the large model's capability in synthetic detection. Compared to the direct Real/Fake QA approach, the VQA format, which includes artifact explanations alongside authenticity judgments, achieves the best performance. This improvement is likely due to better alignment between visual image content and textual explanations. + +![](images/96404680f0bd04653c81866976bc18f1074d453ca0149dc324eeebc75ee0b3cb.jpg) +Figure 7: Performance of different training strategies. + +# C Robustness Study + +We further evaluated the robustness of FakeVLM to common image distortions, such as JPEG compression, resizing, and Gaussian noise. Table 7 presents the performance of our model on FakeClue under eight distortion scenarios: JPEG compression (quality levels 70 and 80), resizing (scaling factors of 0.5 and 0.75), Gaussian noise (variances of 5 and 10), as well as additional transformations such as flipping, rotating, sharpening, and adjusting contrast or blur levels. Although the model was not explicitly trained on distorted data, FakeVLM demonstrated resilience to these low-level distortions, highlighting its robustness and practical applicability. + +Table 7: Performance of FakeVLM under different perturbations on FakeClue. + +
MethodDetectionExplanation
Acc ↑F1 ↑ROUGE_L ↑CSS ↑
JPEG 7091.288.755.785.3
JPEG 8091.088.556.385.9
Resize 0.596.494.957.087.0
Resize 0.7598.197.457.787.5
Gaussian 1092.189.856.186.4
Gaussian 594.592.556.586.6
Flip horizontal98.498.354.083.3
Rotate1590.989.644.076.6
Sharpen1.597.497.254.383.6
Contrast0.796.496.153.082.8
Contrast1.398.298.054.083.3
Blur389.587.850.381.2
Origin image98.698.158.087.7
+ +# D Category generalization test + +In the LOKI evaluation set, different category divisions allow us to assess the model's performance across various categories. Table 8 presents the performance of FakeVLM and GPT-4o on different image categories. The results indicate that GPT-4o exhibits a noticeable category bias, whereas FakeVLM, trained on diverse data, achieves a more balanced performance across all categories. Additionally, its relatively lower performance on human portraits may be attributed to the rapid advancements in generative models within this domain. + +# E Performance Summary + +We visualize the performance of FakeVLM and several leading LMMs across three datasets—DDVQA [23], FakeClue, and LOKI [27]—using a radar chart, as shown in Figure 8. The results indicate that FakeVLM demonstrates a clear advantage over existing general-purpose multimodal models in both synthetic detection and artifact explanation tasks. + +# F Limitations and Future Works + +Despite the strong performance of FakeVLM and the scalability of the FakeClue dataset, certain limitations warrant attention. First, as FakeClue's annotations are primarily derived from multi-LMM aggregation, potential biases intrinsic to the annotating models and insufficient capture of fine-grained artifacts may persist. Future iterations should incorporate heterogeneous annotation strategies, including human expert validation, to enhance dataset robustness. Second, FakeVLM exhibits diminished sensitivity to high-fidelity synthetic images with imperceptible artifacts. Detecting such subtle inconsistencies demands more advanced methodologies capable of analyzing latent statistical irregularities beyond conventional artifact cues. + +Table 8: Judgment questions results of different models on the LOKI Image modality. * denotes the closed-source models. + +
OverallSceneAnimalPersonObjectMedicineDocSatellite
Expert (AIDE)63.1-89.962.596.553.449.739.3
MiniCPM-V-2.644.852.034.453.131.553.851.538.3
Phi-3.5-Vision52.550.841.771.534.157.354.360.5
LLaVA-OneVision-7B49.859.241.958.137.352.353.050.1
InternLM-XComposer2.546.452.740.056.732.556.149.838.2
mPLUG-Owl3-7B45.952.137.352.931.455.353.838.1
Qwen2-VL-7B47.854.738.957.930.356.059.636.9
LongVA-7B46.257.637.452.534.154.449.839.7
Mantis-8B54.654.952.254.853.553.151.963.3
Idefics2-8B45.051.835.352.329.252.353.940.6
InternVL2-8B49.758.839.454.437.853.960.244.2
Llama-3-LongVILA-8B49.849.850.550.647.250.049.950.0
VILA1.5-13B49.352.038.654.231.050.156.662.4
InternVL2-26B44.351.635.450.828.251.354.437.6
VILA1.5-40B48.853.739.350.033.452.559.950.6
InternVL2-40B49.655.737.359.234.855.564.840.8
Qwen2-VL-72B53.255.943.466.938.055.973.738.2
LLaVA-OneVision-72b46.354.731.653.127.852.167.936.6
Claude-3.5-Sonnet*53.651.651.655.251.451.959.150.9
Gemini-1.5-Pro*43.553.735.751.530.350.047.238.1
GPT-4o*63.470.169.784.470.354.360.145.0
FakeVLM84.398.989.773.894.169.385.097.2
+ +![](images/3cf3239cb20780503d43dc849e5ec824dfd28409e43a8b6edc6b611593a0d108.jpg) +Figure 8: The performances of the 7 leading LMMs on DD-VQA [23], FakeClue and LOKI [27]. + +# G Broader Impacts + +This study investigates the use of Large Multimodal Models (LMMs) for synthetic image detection and artifact interpretation, offering both significant societal benefits and potential ethical challenges. On the positive side, tools such as FakeVLM enhance the identification of AI-generated content and provide interpretable artifact explanations, thereby promoting media authenticity, digital trust, and public awareness. These capabilities are critical in countering misinformation, forged visual + +content, and deceptive practices, with added value in user education through transparency of detection rationale. + +However, the approach also raises important concerns. As generative techniques evolve in parallel with detection methods, a dynamic adversarial escalation may ensue, potentially resulting in more evasive synthetic content. Moreover, false positives risk unjust censorship or reputational harm, while false negatives permit malicious media to persist. Overreliance on automated systems may also weaken critical user discernment. Additionally, the computational intensity of LMMs could restrict their adoption to resource-rich institutions, exacerbating inequality in digital verification capabilities. + +Thus, while the proposed framework represents a step toward securing online visual ecosystems, responsible deployment necessitates further research, ethical foresight, and inclusive public education to mitigate associated risks. + +# H Label Prompt + +At the end of the supplementary materials, we provide specific examples of the Label Prompts mentioned in the main text. These examples cover different categories, including Animal, Scene, Object, Human, Satellite, and Document, with variations for both real and fake labels. Additionally, we detail the specific prompt designed for the aggregation of outputs from the three initial multimodal models. + +# Model output merging + +You are an expert in AI-generated content analysis. Merge the following three model responses into one unified answer. All responses agree on the image's authenticity (real/fake). Prioritize explanations mentioned by at least two models and omit points unique to a single model. + +Follow these steps: + +- Extract Common Ground: Identify overlapping details (e.g., shadows, outlines, object alignment) across all three responses. +- Filter Minority Claims: Discard observations mentioned by only one model unless they are critical (e.g., glaring artifacts). +- Structure Hierarchically: Group explanations by category (e.g., lighting, geometry, textures) for clarity. +- Maintain Original Format: Begin with "This is a [real/fake] image." followed by a concise, semicolon-separated list of consolidated evidence. +- Avoid Redundancy: Rephrase overlapping points to eliminate repetition while preserving technical accuracy. +- Ensure Logical Consistency: If any response contains nonsensical, contradictory, or infinite loop reasoning, disregard that portion of the answer. + +Output format: + +For a real image: + +"This is a real image. [Explanation summarizing common realistic features, such as clear outlines, organized objects, realistic shadows, and aligned roads.] + +For a fake image: + +"This is a fake image. [Explanation summarizing common unrealistic features, such as blurry outlines, disorganized objects, unrealistic shadows, and misaligned roads.] + +# animal + +# Real + +Describes realistic areas of real animal imagery. + +Note: + +- Animal structure and physical feature accuracy: Check if the animal's features like ears, paws, wings, etc., are correctly proportioned and symmetrical. Ensure the animal's body parts are connected in a natural and realistic manner. +- Surface texture and color consistency: Examine if the animal's texture is realistic, with clear, well-defined fur, feathers, or skin. The texture should match natural details, and the colors should be accurate and natural, without unnatural blending or over-saturation. +- Object or environmental realism: Check the background and surroundings for natural elements, ensuring the animal is integrated with its environment in a way that reflects the real world. The connections between objects, like water, plants, or terrain, should be believable. +- Symmetry and consistency: Look for symmetry in the animal's body parts, with natural connections between joints, paws, or wings. The overall shape and structure should be consistent without any noticeable discrepancies. +- Overall color and image processing accuracy: Ensure the overall color balance in the image is natural and consistent with real-world animal features, avoiding any unrealistic color shifts or over-saturation. + +# Output format: + +Please output your answer in non-segmented text format, not Markdown format, as follows: "This is a real image. [Explanation of the realistic aspects found in the image, such as accurate animal structure, natural texture, realistic environment, etc.]" + +# Fake + +Describes unrealistic areas of synthetic animal imagery. + +Note: + +- Animal structure and physical feature anomalies: Check if the animal's features like ears, paws, wings, etc., are mismatched or distorted. Ensure the proportions of animal parts (like paws, wings, and tails) are accurate and symmetrical. Look for unnatural connections between body parts. +- Surface texture and color distortion: Examine if the animal's texture is inconsistent, such as blurry patches or unnatural blending. Ensure the fur, feathers, or skin texture matches realistic details. Look for overly saturated or unnatural colors in the animal's appearance. +- Object or environmental anomalies: Check the background for abnormal noise, holes, or disjointed textures. Ensure objects in the environment (like water, plants, or terrain) are realistically connected to the animal. Look for unnatural reflections or highlights, especially in aquatic settings. +- Symmetry and consistency issues: Look for asymmetry or noticeable discrepancies in the animal's parts. Ensure the connections between body parts (like joints, paws, or wings) are clearly defined and consistent. +- Overall color and image processing distortion: Ensure the overall color balance in the image looks natural and not over-saturated. Watch for color shifts that don't align with real-world animal features. + +# Output format: + +Please output your answer in non-segmented text format, not Markdown format, as follows: "This is a fake image. [Explanation of the issues found in the image, such as animal structure anomalies, texture inconsistencies, unnatural environment, etc.]". + +# scene + +# Real + +Describes realistic areas of real animal imagery. + +Note: + +- Physical logic accuracy: Check if all objects in the scene are grounded in realistic physics, with no floating or unsupported elements. Ensure that all objects are placed in logical positions and follow natural laws, such as trees with visible roots or furniture resting on solid surfaces. +- Structural and environmental consistency: Look for harmony between the objects and their environment. Ensure that the placement of structures like houses or roads makes sense within the context of the surrounding landscape. For instance, a house should have a stable foundation, and roads should blend naturally with the terrain. +- Consistent lighting and shadows: Observe if the lighting and shadows align logically with one another, with clear and natural light sources. Ensure that shadows cast by objects are consistent in direction and intensity, and that lighting across different parts of the image (such as the foreground and background) is coherent. +- Natural color and texture fidelity: Check if the colors are realistic and not overly saturated. Ensure that materials like grass, water, or rocks have natural, believable textures. The terrain and other elements should reflect real-world characteristics without unnatural distortions. +- Clear edges and smooth transitions: Ensure that all transitions between objects and environments are realistic and clear, without unnatural blending. For example, a shoreline should transition naturally into water, and terrain features should merge in a way that makes sense within the landscape. +- Natural details and imperfections: Look for small, realistic imperfections or variations in the scene, such as asymmetry in plant growth, irregularities in rock formations, or natural inconsistencies in structure shapes. Ensure that the details align with how things appear in the real world, avoiding perfect symmetry or overly manufactured features. + +# Output format: + +Please output your answer in non-segmented text format, not Markdown format, as follows: "This is a real image. [Explanation of the realistic aspects found in the image, such as physical accuracy, consistent lighting, natural textures, etc.]" + +# Fake + +Describes unrealistic areas of synthetic scene imagery. + +Note: + +- Physical logic anomalies: Check if objects appear to defy natural laws, such as floating chairs or trees with exposed roots. Ensure elements of the scene are grounded in realistic physics, with no floating or unsupported objects. Look for any objects in impossible positions or states that would not naturally occur. +- Structural and environmental contradictions: Look for mismatches between the objects and their environments. For instance, check if a house is placed in an odd location, like on a rocky surface with no foundation, or if a smooth road appears in an otherwise rugged landscape. Ensure that the placement of objects makes sense within the context of the environment. +- Inconsistent lighting and shadows: Observe if lighting and shadows do not match, such as light sources coming from inconsistent directions, strange reflections, or broken shadows. Check if areas of the image, particularly the upper and lower portions, have lighting that doesn't align with one another. +- Color and texture distortion: Check if the colors are overly saturated or unrealistic. For instance, unnatural shades of green for grass or unrealistic wave textures. Pay attention to materials that appear distorted or do not resemble real-world textures, such as block-like rocks or unnatural terrain features. +- Edge and transition anomalies: Look for unnatural or blurred boundaries between objects and environments. Check for smooth or illogical transitions, such as a shoreline blending too smoothly into water or an unnatural merge between terrain features. Make sure all transitions between objects and the environment are clear and realistic. +- Detail violations of reality: Examine whether details are too perfect or unnatural. For instance, overly symmetrical flower arrangements, distorted structures like leaning towers, or plants that grow in unrealistic patterns. Ensure that all details in the scene align with natural or realistic conditions. + +# Output format: + +Please output your answer in non-segmented text format, not Markdown format, as follows: + +"This is a fake image. [Explanation of the issues found in the image, such as physical anomalies, structural contractions, color distortions, etc.]" + +# object + +# Real + +Describes realistic areas of real object imagery. + +Note: + +- Material and texture authenticity: Check if the surface texture of objects reflects real-world materials accurately, such as leaves that look like real leaves or a keyboard with clearly defined key textures. Ensure the textures appear clear and natural, corresponding to the material's properties. +- Consistent shapes and structures: Ensure that objects have natural and realistic shapes, with no distortion or unnatural features. For instance, a keyboard should have a well-defined spacebar, and a boat should not be stacked on top of another in a way that defies real-world physics. +- Accurate color and lighting: Observe whether the colors of objects are natural and not overly saturated, and ensure that the lighting and shadows are applied in a consistent, believable manner. Objects should have realistic shading based on light sources and their surroundings. +- Natural color and texture fidelity: Check if the colors are realistic and not overly saturated. Ensure that materials like grass, water, or rocks have natural, believable textures. The terrain and other elements should reflect real-world characteristics without unnatural distortions. +- Sharp and clear details: Ensure that all details in the image, such as text on a device or the edges of objects, are sharp, well-defined, and easy to interpret. There should be no blurriness or loss of clarity. +- Smooth transitions and connections: Check if the object's transition into the background or other objects is seamless and natural, without awkward or forced connections. For example, the object should appear logically integrated into its environment, with smooth connections to surrounding elements. + +# Output format: + +Please output your answer in non-segmented text format, not Markdown format, as follows: "This is a real image. [Explanation of the realistic aspects found in the image, such as texture authenticity, consistent shapes, accurate lighting, etc.]" + +# Fake + +Describes unrealistic areas of synthetic object imagery. + +Note: + +- Material and texture anomalies: Look for objects where the surface texture does not match reality, such as a leaf that looks more like clay or a keyboard with distorted key textures. Ensure the textures are clear and realistic, reflecting the true material properties. +- Irregular shapes and structures: Check if objects have unnatural or distorted shapes, like a keyboard with a misshapen spacebar or a boat stacked on another boat. Ensure that the structure and geometry of the object align with what is physically plausible. +- **Color and lighting distortions:** Watch for overly saturated colors that make objects appear unrealistic, such as an overly bright or unnatural color palette. Ensure that lighting and shadows are applied in a consistent and believable way. +- Blurry and unclear details: Look for details that are fuzzy or difficult to interpret, such as unreadable text on an electronic device or unclear edges on objects. Ensure that the object's features are sharp and well-defined. +- Unnatural transitions and connections: Check if the object's transition to the background or other objects seems awkward or unnatural, such as an illogical connection between the ocean and a distant mountain. Ensure all object transitions and connections are smooth and realistic. + +# Output format: + +Please output your answer in non-segmented text format, not Markdown format, as follows: "This is a fake image. [Explanation of the issues found in the image, such as texture anomalies, irregular shapes, lighting inconsistencies, etc.]" + +# human + +# Real + +Describes realistic areas of real human portrait imagery. + +Note: + +- Texture clarity and definition: Check if the skin, hair, and other body parts have clear, well-defined textures, with no unnatural blending or merging. Ensure the edges between areas, like fingers and hair, are sharp and distinct, reflecting real-world detail. +- Natural proportions and symmetry: Ensure that all body parts are proportioned correctly and symmetrically, following realistic anatomical rules. For example, fingers should not be too long, eyes should be aligned, and the mouth should have natural symmetry. +- Accurate color and saturation: Watch for natural skin tones and realistic clothing colors, avoiding overly saturated or unnatural hues. Ensure the transitions between different areas of the image, such as between skin and clothing, are smooth and true to life. +- Sharp details and no artifacts: Ensure that all details, such as teeth, pupils, and small facial features, are crisp and clear. There should be no noticeable artifacts, such as blurry features or missing details, like a poorly rendered neck. +- High image quality: Check if the image maintains a high level of clarity, with no visible noise, distortion, or blurring in the foreground or background. The image should be uniformly sharp, with consistent quality throughout. +- Logical placement and accurate perspectives: Ensure that all elements within the image, including clothing, hair, and body parts, are correctly positioned according to realistic perspectives and lighting. There should be no illogical placement, such as mismatched highlights or shadows on clothing. + +# Output format: + +Please output your answer in non-segmented text format, not Markdown format, as follows: "This is a real image. [Explanation of the realistic aspects found in the image, such as clear textures, accurate proportions, natural color, etc.]" + +# Fake + +Describes unrealistic areas of synthetic human portrait imagery. + +Note: + +- Texture blending and blurring: Look for blurred edges or unnatural fusion between different parts of the image, such as fingers merging or hair blending into the skin. Ensure that details like skin, hair, and fingers are clearly defined and separated, without textures merging unnaturally. +- Structural distortion and asymmetry: Check for unnatural proportions or distorted body parts, such as a finger that is too long, an asymmetrical mouth, or eyes that are misaligned. Ensure all body parts are correctly positioned and proportioned, following realistic anatomical rules. Ensure that the structure and geometry of the object align with what is physically plausible. +- Color and saturation anomalies: Watch for overly saturated colors or unrealistic transitions, such as unnatural skin tones or overly vivid clothing details. Ensure the overall color scheme and texture transitions appear natural and true to life. Ensure that lighting and shadows are applied in a consistent and believable way. +- Detail errors and artifacts: Look for unnatural details, such as odd textures on the teeth or irregularities in the pupils, and check for missing or incomplete details like poorly rendered necks or blurry features. Ensure all details are sharp and well-defined, with no noticeable artifacts or unrealistic elements. +- Overall quality issues: Check if the image quality is compromised, such as noticeable noise, image distortion, or a blurry foreground/background. Ensure the image maintains a high level of clarity, with a consistent level of sharpness across the image. +- Object placement and logical errors Look for any errors in the placement or logic of the image, such as clothing with mismatched highlights and shadows or an incorrect position of elements like skirts or hair. Ensure that all elements within the image align logically with realistic perspectives and lighting. + +# Output format: + +Please output your answer in non-segmented text format, not Markdown format, as follows: "This is a fake image. [Explanation of the issues found in the image, such as texture blending, structural errors, color anomalies, etc.]" + +# satellite + +# Real + +Describes realistic areas of real satellite imagery. + +Note: + +- Clear outlines of buildings and vehicles: Check if the outlines of buildings and vehicles are sharp and well-defined, without blurring or distortion. +- Organized arrangement of objects: Check if objects on the ground are arranged in a consistent and orderly fashion, with a logical spatial layout. +- Natural boundary transitions: Observe if the boundaries between vegetation, land, and buildings are clear, with smooth and realistic transitions. +- Realistic shadows and lighting: Check if shadows are positioned logically, following physical laws, and are consistent with the light sources in the image. +- Natural textures and curves: Look for textures and curves that are typical in real satellite images, with no unusual patterns or distortions. +- Proper alignment of roads: Check if the roads are aligned and show natural curves, without any distortion, misalignment, or unrealistic interruptions. + +# Output format: + +Please output your answer in non-segmented text format, not Markdown format, as follows: "This is a real image. [Explanation of the realistic aspects found in the image, such as clear outlines, organized objects, realistic shadows, aligned roads, etc.]". + +# Fake + +Describes unrealistic areas of synthetic satellite imagery. + +Note: + +- Blurry outlines of buildings and vehicles: Check if the outlines of buildings and vehicles are clear, especially if their edges are blurred or distorted. +- Disorganized arrangement of objects: Check if objects on the ground are arranged irregularly, appearing chaotic and lacking logical spatial layout. +- Unnatural boundary transitions: Observe if the boundaries between vegetation, land, and buildings are blurred, with any texture mixing or unnatural transitions. +- Unrealistic shadows and lighting: Check if shadows appear in illogical positions, especially whether shadows of trees or buildings follow physical laws. +- Abnormal textures and curves: CLook for any unnatural curves, stripes, or textures that do not match those found in real satellite images. +- Distortion or misalignment of roads: Check if the roads in the image are distorted or misaligned, particularly if road lines are clear, the direction appears natural, or if there are unreasonable curves or interruptions. + +# Output format: + +Please output your answer in non-segmented text format, not Markdown format, as follows: "This is a fake image. [Explanation of the issues found in the image, such as blurry outlines, disorganized objects, unrealistic shadows, misaligned roads, etc.]". + +# doc + +# Real + +Describes realistic features of real document images. + +Note: + +- Coherent and legible text: Check if the text in the document is semantically coherent, logically structured, and meets standard readability, avoiding nonsensical or illogical phrases. +- Well-aligned text layout: Check if the text is arranged in a neat and organized manner, mimicking the alignment and structure of genuine documents with proper margins, spacing, and indentation. +- Natural fonts and typography: Observe if the fonts appear consistent and realistic, without any noticeable distortions, deformations, or irregularities. +- Consistent formatting and structure: Look for logical paragraph breaks, appropriate headers, and overall formatting that reflects a professionally produced document. +- Attention to typographic details: Verify the clarity of characters, letter spacing, and overall text sharpness, ensuring the document appears authentic and naturally printed. + +# Output format: + +Please output your answer in non-segmented text format, not Markdown format, as follows: "This is a real image. [Explanation of the realistic aspects found in the document, such as coherent text, well-aligned layout, natural fonts, and consistent formatting.]" + +# Fake + +Describes unrealistic features of fake document images. + +Note: + +- Incoherent and illegible text: Check if the text in the document lacks semantic coherence, contains nonsensical or illogical phrases, and fails to meet standard readability, making it appear artificial. +- Misaligned and disorganized text layout: Check if the text arrangement appears cluttered, lacks proper margins, spacing, or indentation, and does not follow the structured alignment of genuine documents. +- Unnatural fonts and typography: Observe if the fonts exhibit inconsistencies, distortions, deformations, or irregularities, making the document look artificial or computer-generated. +- Inconsistent formatting and structure: Look for unnatural paragraph breaks, misplaced headers, or formatting anomalies that disrupt the logical flow of a professionally produced document. +- Lack of typographic detail: Verify if the characters appear unclear, have irregular letter spacing, or exhibit a lack of sharpness, reducing the overall authenticity of the document. + +# Output format: + +Please output your answer in non-segmented text format, not Markdown format, as follows: "This is a fake image. [Explanation of the unrealistic aspects found in the document, such as incoherent text, misaligned layout, unnatural fonts, and inconsistent formatting]." + +# face manipulation + +# Real + + You are verifying a KNOWN AUTHENTIC photo. Confirm: + +- Nostril Geometry: Natural asymmetric shape with pore-level detail. +- Skin Texture: Gradual tone transitions with microscopic skin imperfections. +- Eye Reflection: Physically accurate light interactions in cornea and conjunctiva. +- Lip Texture: Visible lip striations with natural moisture gradient. +- Shadow Integrity: Consistent ambient occlusion in nasal folds and facial contours. +- Biological Signatures: Micro-movements in facial muscles and natural blink patterns. + +# Output format: + +Please output your answer in non-segmented text format, not Markdown format, as follows: "This is a real image. [Authenticity evidence]" + +# Examples: + +Please give me your analysis of the given picture. + +# ASSISTANT: + +# Fake + + You are analyzing a KNOWN SYNTHETIC image. Check these artifacts: + +- Nostril Area: Identify blurriness, pixelation, or AI-generated irregularities. +- Skin Texture: Detect artificial color transitions, wax-like smoothing, or patchy texture. +- Eye Region: Find unnatural reflections, mismatched pupil details, or AI-generated artifacts. +- Lip Contours: Check for bleeding colors, mismatched texture, or edge inconsistencies. +- Facial Contours: Identify over-smoothed jawlines, unnatural shadow transitions, or warping. +- Global Consistency: Detect mismatched lighting direction, floating hairs, or texture repetition. + +# Output format: + +Please output your answer in non-segmented text format, not Markdown format, as follows: "This is a fake image. 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Despite progress, existing methods are often vulnerable to spurious correlations that mislead models and compromise robustness. To address this, we propose SPROD, a novel prototype-based OOD detection approach that explicitly addresses the challenge posed by unknown spurious correlations. Our post-hoc method refines class prototypes to mitigate bias from spurious features without additional data or hyperparameter tuning, and is broadly applicable across diverse backbones and OOD detection settings. We conduct a comprehensive spurious correlation OOD detection benchmarking, comparing our method against existing approaches and demonstrating its superior performance across challenging OOD datasets, such as CelebA, Waterbirds, UrbanCars, Spurious Imagenet, and the newly introduced Animals MetaCoCo. On average, SPROD improves AUROC by $4.8\%$ and FPR@95 by $9.4\%$ over the second best. + +# 1 Introduction + +Machine learning systems in real-world applications often encounter out-of-distribution (OOD) inputs, which are samples from distributions different from the training data. These inputs require cautious handling to prevent overconfident mispredictions during inference [1]. This makes OOD detection crucial, as it aims to identify whether an input belongs to the known distribution or not. Yet, deep neural networks, widely used in vision tasks [2-5], tend to make high-confidence predictions even on OOD inputs, demonstrating their inability to recognize data outside the training distribution as OOD [6, 7]. The reliability of OOD detection is especially critical in applications like healthcare and autonomous driving, where overconfident predictions on unfamiliar data could have serious consequences [8, 9]. + +Recent research on OOD detection aims to ensure the reliable deployment of DNNs [7, 10-13]. Despite many effective methods, their robustness can be undermined by spurious correlations in the training data [14]. Studies indicate that models often rely on features that are statistically informative but not causally representative of the object itself [15-17]. These misleading cues can act as shortcuts, allowing models to achieve high accuracy without learning the core, causally relevant features [18]. While spurious correlations have been well-explored in classification tasks [17, 19, 20], their impact on OOD detection remains underexplored. Recently, [14] underscores the impact of spurious correlations on OOD detection and introduces a formalization that categorizes OOD samples into two types: spurious OODs (SP-OODs), which contain spurious attributes but lack core features, + +![](images/20ea62686522855e6e5859f53cccc65206b79f6fadc2cf9746ece40028ee3f8a.jpg) + +![](images/cdccc0cdfa2ff88cb891f6711d7f395b8d229c5b1719a167350e187dfd6e35dd.jpg) + +![](images/d8f3a55c97ffb9c897c664a2f1735531ec745de504f4e732f2a6708836027282.jpg) + +![](images/967f9ea6f368b544e2fc9ccffbb479b061980a595fe817d04bb39f5ca5ff605c.jpg) + +![](images/1b22d71314fe6efd2424c39b51cd34ae88256596c1453735e9dd92de153d030c.jpg) + +![](images/35c010f10489072c4905000c08d8e5cc348ac4963f8bff6db51297d2b217e717.jpg) + +![](images/85fb98b4853c71d9ed425dbdd799cc2e6c0581cff67d2610198dd01a12e5795b.jpg) + +![](images/6cc005088fd6d50cb67bd727e1827096edb8bd638fda15cd4f5fbe5eaab1bd70.jpg) + +![](images/0fb841a148b4ebd76da476a23e504abbbe6269eea39d453b71550ad45947efd3.jpg) + +![](images/bbc241d383776ae1b66d08069a8ddb67f4ba13178d9978c218f9658bfc2b47c3.jpg) + +![](images/58528dd5ad655b0794160a1f71c4a17cf46ce0694cd9ea9d4683e0aefca4e682.jpg) +Figure 1: The challenge of spurious correlations in OOD detection. ID classes (dog, fox, wolf) appear in correlated backgrounds (grass, autumn, snow), with majority groups relying on context shortcuts (blue frames). SP-OOD samples share the same contextual backgrounds, making detection more difficult. NSP-OOD samples differ in context and lack both spurious and core features. + +![](images/6bab5700cef38e9fddd1d8cbb57157bd2e71749e9fe8a313eaf7e70ca91d77c2.jpg) + +![](images/14ef5e0a597f66c036aaa5c0bd9581990bae8435346d43c8f10331bd144c93b8.jpg) + +![](images/956e5f457006e34e8f752de956f23f39d6764b14e119ee4ef4f0162291de0578.jpg) + +![](images/77e17aceca74c2cee178aea52dda56eff67637a329722434cb05776df78acce8.jpg) + +and non-spurious OODs (NSP-OODs), which lack both attributes and align with the traditional OOD setting. Figure 1 shows an example of this problem setting. + +Recent efforts to mitigate spurious correlations in OOD detection can be grouped into several broad categories. Outlier Exposure (OE) techniques reduce reliance on spurious correlations by incorporating synthetic OOD samples in their training [21-24]. Other methods focus on modifying training objectives to explicitly discourage models from depending on spurious features [25, 26]. However, these methods usually require retraining and generating additional OOD data. In contrast, several post-hoc approaches have been proposed that bypass the limitations of training-heavy approaches and offer fast and light alternatives [27-32]. Despite their promise, these methods are often tested on limited synthetic datasets or specific backbones, and some rely on multiple modalities. + +To address these limitations, we propose Spurious-Aware Prototype Refinement for Reliable Out-of-Distribution Detection (SPROD) for robust OOD detection, especially in the presence of unknown spurious correlations. It follows a three-stage process: (1) initial prototype construction, (2) classification-aware prototype calculation, and (3) group prototype refinement. SPROD can be easily applied to any pretrained feature extractor without fine-tuning on target datasets, offering a straightforward, hyperparameter-free approach that is both efficient and adaptable across diverse OOD detection tasks. Moreover, our work offers a comprehensive evaluation of OOD detection in the presence of spurious correlations, benchmarking existing methods across multiple challenging datasets, including Waterbirds [33], CelebA [34], UrbanCars [35], Spurious ImageNet [36], and the newly introduced Animals MetaCoCo. SPROD achieves state-of-the-art performance and consistently exhibits robust behavior across a wide range of benchmarks and experimental conditions. The main contributions of this work are as follows: + +- We propose SPROD, a post-hoc OOD detection method that directly addresses unknown spurious correlations by design and outperforms the state-of-the-art. We provide theoretical insight into how the proposed method mitigates spurious bias. +- SPROD is a fast, simple, and general approach applicable to diverse pretrained feature extractors and OOD detection settings. +- SPROD does not assume access to group annotations to achieve robustness to spurious correlations and does not require either OOD or ID validation data for hyperparameter tuning. Moreover, it maintains strong performance even in low-training data regimes. +- This work conducts and introduces comprehensive benchmarking across multiple SP-OOD datasets, including the newly introduced Animals MetaCoCo, a realistic, multiclass dataset with diverse spurious attributes. +- Finally, our study sheds new light on key factors influencing SP-OOD detection, such as the impact of backbone fine-tuning and the choice of scoring mechanisms. + +# 2 Related Work + +OOD detection methods can be categorized into training-time and post-hoc approaches [37]. Training-time methods leverage auxiliary OOD samples (Outlier Exposure) [12, 38, 39] or apply regularization [40-43] to enhance OOD detection. Post-hoc methods, in contrast, derive OOD scores from base classifiers without modifying training [37]. Overall, post-hoc methods offer simplicity and competitive performance [37], making them practical under limited data or training resources. Among post-hoc methods, several approaches apply transformations to model logits to derive OOD scores. MSP [7] uses the maximum softmax probability, the energy-based method [13] computes the log-sum-exp of logits, MLS [44] uses the maximum logit and introduces KL Matching (KLM) based on KL divergence, and GEN [45] employs generalized entropy of softmax outputs. + +Another class of post-hoc methods detects OOD samples via feature-space distances. MDS [11] fits class-conditional Gaussians to pre-logit features and computes Mahalanobis distances, refined by RMDS [46] with an unconditional Gaussian on ID data. KNN [47] uses distances to nearest ID samples. SHE [48] scores samples by their distance to stored ID feature templates. NNGuide [49] leverages nearest-neighbor guidance to adjust test features toward the ID manifold. Relation [50] constructs a graph over training embeddings and detects outliers via relational anomalies. NECO [51] scores samples by their feature alignment with class weight vectors, leveraging neural collapse geometry. SCALE [52] separates ID and OOD samples by scaling penultimate-layer activations. FDBD [53] measures features' regularized mean distance to the classifier's decision boundaries. NCI [54] scores samples by their distance to class weight vectors, filtered by feature norms. + +Prototype-based methods shape class representations for distance-based OOD scoring. Classical approaches like MDS [11] and its variants [46] are closely related, as they model each class by a centroid in feature space, optionally using class-conditional covariances to compute Mahalanobis distances. Recent works extend this via explicit training objectives. CIDER [55] learns hyperspherical embeddings by jointly enforcing intra-class compactness and inter-class dispersion, thereby improving ID and OOD separability. PALM [56] represents each class as a mixture of learnable prototypes and optimizes a maximum-likelihood and contrastive objective, updating prototypes and backbone features jointly during training. PROWL [57] also leverages prototype representations, but for pixel-level OOD detection in segmentation. While these methods share a prototypical framework, SPROD differs as a post-hoc method operating on pretrained backbones and is explicitly designed to mitigate the negative effects of unknown spurious correlations. Appendix H further analyzes a variant, SPROD-KMeans, which connects to mixture-of-prototypes ideas while remaining fully post-hoc. + +A few methods combine information from both feature and logit spaces. ReAct [58] thresholds activations before applying energy-based scoring. ViM [59] adds a virtual logit from the residual norm between input features and the ID subspace and applies softmax over extended logits. ASH [60] prunes high-magnitude activations and rescales remaining features before logit computation, improving energy-based OOD separability. Some methods also exploit gradient space for OOD scoring [61, 62]. GradNorm [61] computes the KL divergence to a uniform distribution and uses the gradient norm (w.r.t. the penultimate layer) as the score. + +Spurious correlations in training data degrade OOD detection performance, as shown in [14]. Evaluations of popular methods [7, 10, 11, 13, 63] reveal that as spurious correlations increase, detection performance drops, and SP-OOD samples become especially challenging to detect. Feature-based methods like MDS [11] outperform others, especially for NSP samples. Recent work addresses spurious correlations in OOD detection through various strategies. OE methods synthesize OOD samples in ways that reduce reliance on background cues, encouraging models to focus on core semantic features [21-24]. Training-time regularization mitigates spurious cues by reweighting samples or augmenting non-semantic features [25, 26]. Post-hoc methods improve inference by modifying inputs to isolate semantics or reduce background influence [27, 28]. + +Recent advances in vision-language models [64] have led to a category of zero-shot OOD detection methods [29-32] that use textual inputs, such as class names or attribute descriptions, to define ID data and identify OOD samples. While some report results on Waterbirds SP-OOD, this is not their main focus. Moreover, lacking training data with spurious correlations, their zero-shot setting does not fully capture the SP-OOD challenge. In contrast to these approaches, which rely on explicit text for OOD scoring, our method refines visual prototypes using only ID features and class labels from the training set. A detailed review of studies regarding SP-OOD can be found in Appendix B. + +![](images/c424bdf62d79206713073c36430290877787751d671143963e7431682b9bd7f4.jpg) +(a) + +![](images/781be58e9c6c8b207ee6de44832a16ba64dbc5009e5b3381822771dbaab4eafb.jpg) +(b) +Figure 2: (a) A far-OOD sample may receive a high softmax score, similar to a near-boundary ID sample. (b) Distances to class prototypes offer a more consistent separation of OOD samples. (c) In the SP-OOD setting, the problem is even more severe: A biased decision boundary causes the OOD sample to receive high softmax confidence, while a minority ID sample receives lower confidence. + +![](images/35f8917ba7217159bf46937fa6cb95fefeec581f69f56110ac3977d66ff7f234.jpg) +(c) + +# 3 Preliminaries + +# 3.1 Problem Setup + +This paper addresses Out-of-Distribution (OOD) detection under spurious correlations in a supervised classification setting. Let $\mathcal{X}$ be the input space and $\mathcal{Y} = \{1,\dots ,C\}$ the label set. The in-distribution (ID) training data $\mathcal{D}_{\mathrm{in}} = \{(x_i,y_i)\}_{i = 1}^N$ comprises samples from a joint distribution $P_{\mathcal{X},\mathcal{Y}}$ . A neural network $f_{\theta}$ maps each input $x_{i}$ to a feature embedding $h_i = f_\theta (x_i)$ . This network is typically pretrained or fine-tuned on the training data. OOD detection aims to identify test samples from distributions not seen during training, including those from unseen classes. + +Spurious correlations in OOD detection were first formalized by [14]. According to this framework, each input can be decomposed into: (i) core features, which are causally related to the label, and (ii) spurious features, which are correlated with the label but not causally relevant. Imbalances in the training data often result in dominant core-spurious combinations (majority groups), while rarer combinations (minority groups) remain underrepresented. This bias encourages models to rely disproportionately on spurious cues. In controllable settings, the proportion of majority group samples within a class is captured by the correlation rate. As illustrated in Figure 1, this setup gives rise to two types of OOD examples: Spurious OOD (SP-OOD), which share spurious features with ID data but differ in core features (e.g., a cat on grass, where grass is spuriously associated with ID classes dog, wolf, fox); and Non-spurious OOD (NSP-OOD), which differ in both core and spurious features (e.g., a cat on a laptop). + +# 3.2 Score Calculation + +A key element in OOD detection is the scoring function $S(x)$ , which assigns a scalar value reflecting how likely an input belongs to the in-distribution (ID). This score is typically derived from a model's learned representations or its predictive outputs. OOD detection is performed based on this score function. An effective OOD method offers distinct and well-separated distributions of scores for ID and OOD samples. + +Prior studies show that feature-based scores, derived from intermediate representations [10, 13, 47], typically outperform those based on output probabilities [7], especially in the presence of spurious correlations [14]. Our experiments further support this trend in certain settings, suggesting that output-based methods (especially those relying on softmax probabilities) may be less reliable when model confidence is influenced by spurious features. To better understand this difference, consider two probabilistic perspectives in classification: discriminative models directly estimate $p(y|x)$ and focus on separating classes, while generative models estimate $p(x|y)$ and capture how data $x$ is distributed for each class $y$ . Most distance-based OOD detection methods can be viewed as approximating a generative approach in feature space. + +While directly modeling $p(y|x)$ is generally effective for classifying ID samples in standard settings, discriminative approaches have been reported to be more sensitive under distribution shifts, such as in continual learning [65] or in the presence of spurious correlations [66]. Furthermore, their utility for + +![](images/b66acedd7787e0831c625ca631ea9352e8a4f8bc2269a0b6bc0ed1c5f05be291.jpg) +Figure 3: Overview of the three main stages of SPROD. In the first stage, class prototypes are computed, though they may be biased due to spurious correlations. In the second stage, group prototypes are constructed for the misclassified and correctly classified samples of each class. Finally, in the third stage, class samples are reassigned to their nearest group prototypes, and based on these assignments, refined minority and majority prototypes are recalculated. + +![](images/d34972ee2cdf6b309422da76306aaf7f64d5d3b19c8722dc113b7b51a3819faa.jpg) + +![](images/ce912d973bf0f274d6815f553451df908791cc72741e8db66bc68c2745671000.jpg) + +![](images/5613a5110ca98f8b8c8b7aa479ce64e19d01c53a9bea47a88ab4bd73502584e3.jpg) + +OOD detection can be limited, particularly when methods are optimized solely on ID data without exposure to OOD examples. The softmax function forces a normalized probability distribution over the known classes, which can degrade OOD detection performance. For instance, an OOD sample that lies far from all class distributions may receive a high softmax probability if it is only slightly distant from a decision boundary. Conversely, an ID sample situated near the boundary between multiple classes could receive a low maximum softmax probability, potentially leading to its misclassification as OOD. In contrast, distance-based approaches, which rely on the class-conditional distribution $p(x|y)$ , are by nature more robust in these scenarios. OOD samples that share few characteristics with any known class typically exhibit low likelihood under all class-conditional distributions $p(x|y)$ , and can be reliably identified, regardless of their proximity to decision boundaries. + +Figure 2 highlights the limitations of softmax-based OOD scoring in a controlled toy dataset. The challenge becomes more pronounced in the presence of spurious correlations, as shown in Figure 2c. In this scenario, a discriminative model, potentially biased by spurious features, may assign high confidence to an SP-OOD sample that shares these spurious cues with an ID class, while simultaneously assigning low confidence to a minority ID sample that lacks them. Motivated by these observations, we design a generative distance-based approach that is more robust to unknown spurious correlations. + +# 4 SPROD + +In this section, we introduce Spurious-Aware Prototype Refinement for Reliable Out-of-Distribution Detection (SPROD). SPROD adapts the prototypical framework [67] for robust OOD detection by constructing class prototypes designed to be resilient to spurious correlations. The core method involves a three-stage process, which is shown in Figure 3 and detailed in the following subsections. + +# 4.1 Stage 1: Initial Prototype Construction + +Given a pretrained feature extractor $f_{\theta}$ , we first obtain feature embeddings $h_i = f_{\theta}(x_i)$ for each training sample $x_i$ . To ensure uniformity in feature representation, these embeddings are normalized to have unit norm $z_i = h_i / \| h_i \|_2$ . For each ID class $c \in \mathcal{V}$ , an initial prototype $\mathbf{p}_c$ is computed as the mean of these normalized embeddings: $\mathbf{p}_c = 1 / N_c \sum_{i : y_i = c} z_i$ , where $N_c$ is the number of samples in class $c$ . Each $\mathbf{p}_c$ serves as an initial estimate of the class centroid in the normalized feature space. A query sample $x_q$ (with normalized embedding $z_q$ ) is typically classified to the class $c$ whose prototype $\mathbf{p}_c$ is the closest (e.g., using Euclidean distance $d(z_q, \mathbf{p}_c)$ ). We use $d(z_q, \mathbf{p}_c)$ as the scoring function for OOD detection. While our empirical results demonstrate the effectiveness of the naive prototypical method in both SP-OOD and NSP-OOD settings, this approach remains vulnerable to biases from spurious correlations in the training data. As a result, the prototypes become skewed toward majority groups within each class (see part I of Figure 3). This bias leads to a scenario where SP-OOD samples (represented by purple squares in Figure 3) may be erroneously classified as ID due to their proximity to these biased prototypes. These limitations motivate the subsequent refinement stages. + +# 4.2 Stage 2: Classification-Aware Prototype Calculation + +To mitigate biases from spurious correlations present in the Stage 1 prototypes, we begin by analyzing how these initial prototypes classify the training data itself. Inspired by [20], our debiasing process starts with classifying the training samples based on the initial prototypes and partitioning the samples based on their prediction outcomes. For each class $c$ , this identifies a set of correctly classified samples, $S_{c}^{\mathrm{corr}}$ , and (multiple) sets of misclassified samples, $\{S_{c\rightarrow m}^{\mathrm{misc}}\}_{m\neq c}$ , where $m$ is the incorrectly predicted class. The core assumption is that samples in $S_{c\rightarrow m}^{\mathrm{misc}}$ belong to subgroups of class $c$ that share spurious features with class $m$ , leading to their misclassification. + +Formally, for each class $c$ , we compute the prototype $\mathbf{p}_c^{\mathrm{corr}}$ by averaging over embeddings of correctly classified samples $S_c^{\mathrm{corr}}$ . For the misclassified samples of class $c$ , we compute the set of misclassified group prototypes $\{\mathbf{p}_{c\rightarrow m}^{\mathrm{misc}}\}_{m\neq c}$ by averaging over samples in $\{S_{c\rightarrow m}^{\mathrm{misc}}\}_{m\neq c}$ individually. The number of misclassified group prototypes for class $c$ , denoted by $C_c^{\mathrm{misc}}$ , corresponds to the number of other classes that training samples from class $c$ have been misclassified as during evaluation. + +This procedure expands the number of prototypes per class from 1 up to $C$ (total number of classes), helping incorporate diverse subgroup characteristics within each class. However, this approach has a potential limitation. Specifically, samples within the minority group may still contribute to the prototype for the correctly classified (majority) group if they were initially classified correctly, resulting in slightly biased prototypes (see part II of Figure 3). Hence, we further refine the group prototypes in the third stage. + +# 4.3 Stage 3: Group Prototype Refinement + +In the third stage, we refine the group prototypes computed in Stage 2 to further reduce the remaining bias within them. Inspired by the reassignment step in K-means clustering, we first reassign samples within each class $\{z_i\mid y_i = c\}$ to either majority $S_{c}^{\mathrm{maj}}$ or minority groups $\{S_{c\to m}^{\mathrm{min}}\}_{m\neq c}$ based on their proximity to the corresponding prototypes ( $\mathbf{p}_c^{\mathrm{corr}}$ for majority members and $\{\mathbf{p}_{c\rightarrow m}^{\mathrm{misc}}\}_{m\neq c}$ for minority members). Following this reassignment, refined prototypes are computed as the mean of the updated group members: + +$$ +\mathbf {p} _ {c} ^ {\text {m a j}} = \frac {1}{| \mathcal {S} _ {c} ^ {\text {m a j}} |} \sum_ {z _ {i} \in \mathcal {S} _ {c} ^ {\text {m a j}}} z _ {i}, \quad \mathbf {p} _ {c \rightarrow m} ^ {\min } = \frac {1}{| \mathcal {S} _ {c \rightarrow m} ^ {\min } |} \sum_ {z _ {i} \in \mathcal {S} _ {c \rightarrow m} ^ {\min }} z _ {i} \quad \forall m \neq c \tag {1} +$$ + +The refined prototypes $\mathbf{p}_c^{\mathrm{maj}}$ and $\mathbf{p}_{c\rightarrow m}^{\mathrm{min}}$ , further reduce bias through the proposed refitting process. During classification and OOD detection, the query embedding $z_{q}$ is compared to all group-specific prototypes, and the final prediction is based on the nearest prototype, regardless of group type. This multiple-prototype approach reduces the likelihood of OOD samples being erroneously classified due to shared spurious attributes with any single prototype (see part III of Figure 3). For each sample, the OOD score is simply calculated based on the distance to the nearest group prototype. + +Algorithm 1 Spurious-Aware Prototype Refinement +1: Input: Training samples $\{(x_i, y_i)\}_{i=1}^N$ , feature extractor $f_\theta$ +2: Output: Refined class prototypes +3: Get feature embedding $h_i = f_\theta(x_i)$ and $z_i = \frac{h_i}{||h_i||_2} \forall x_i$ +4: for each class $c = 1, \ldots, C$ : +5: $\mathbf{p}_c = \frac{1}{N_c} \sum_{i : y_i = c} z_i$ +6: Classify all training samples using initial prototypes +7: for each class $c = 1, \ldots, C$ : +8: Separate samples into correctly classified $S_c^{\text{corr}}$ and misclassified $\{S_{c \rightarrow m}^{\text{misc}}\}_{m \neq c}$ +9: Compute $\mathbf{p}_c^{\text{corr}}$ based on $S_c^{\text{corr}}$ and $\{\mathbf{p}_{c \rightarrow m}^{\text{misc}}\}_{m \neq c}$ based on $\{S_{c \rightarrow m}^{\text{misc}}\}_{m \neq c}$ +10: for each class $c = 1, \ldots, C$ : +11: Construct majority $S_c^{\text{maj}}$ and minority $\{S_{c \rightarrow m}^{\text{min}}\}_{m \neq c}$ groups based on proximity to $\mathbf{p}_c^{\text{corr}}$ or $\{\mathbf{p}_{c \rightarrow m}^{\text{misc}}\}_{m \neq c}$ +12: Compute $\mathbf{p}_c^{\text{maj}}$ based on $S_c^{\text{maj}}$ and $\{\mathbf{p}_{c \rightarrow m}^{\text{min}}\}_{m \neq c}$ based on $\{S_{c \rightarrow m}^{\text{min}}\}_{m \neq c}$ + +The overall procedure of our prototype refinement strategy is outlined in Algorithm 1 and a theoretical justification of how the proposed procedure mitigates spurious bias is provided in Appendix A. + +# 5 Experiments + +# 5.1 Experimental Setup + +We evaluate our method against a comprehensive suite of 19 post-hoc OOD detection approaches: MSP [7], MDS [11], RMDS [46], Energy [13], GradNorm [61], ReAct [58], MaxLogit [44], MLS & KLM [44], VIM [59], KNN [47], SHE [48], ASH [60], NECO [51], NNGuide [49], Relation [50], SCALE [52], fDBD [53], and NCI [54]. We assess performance primarily using the Area Under the Receiver Operating Characteristic curve (AUROC), a threshold-independent metric, and the False Positive Rate at $95\%$ True Positive Rate (FPR@95). Additional metrics, including the Area Under the Precision-Recall curve (AUPR), are provided in Appendix G. We repeat all experiments five times with different random seeds and report the mean and standard deviation. + +In this experiments section, we focus on the more challenging SP-OOD scenarios, particularly those with the highest degree of spurious correlation in each dataset, as our method achieves near-perfect performance on far NSP-OOD samples. Detailed results for the NSP-OOD setting are provided in Appendix E. Further analyses of SPROD's stages, along with broader evaluations across various transformer-based and convolutional backbones, are included in Appendix F and Appendix G. For consistency, the results in this section primarily use the widely adopted ResNet-50 [68] backbone, with ResNet-18 additionally used in one analysis. Beyond SP-OOD benchmarks, we also evaluate our approach under conventional (standard) OOD settings to further demonstrate its general applicability. + +# 5.2 Datasets + +Table 1: Overview of datasets used for SP-OOD evaluation. "# Groups" denotes the number of distinct subpopulations based on class and spurious attribute combinations. "NA" indicates cases where such grouping is not explicitly defined. + +
DatasetType# Classes# Spurious Attr.# GroupsSP-OOD
Waterbirds (WB) [33]Synthetic2 (Bird Type)2 Backgrounds4Places [69] background
CelebA (CA) [34]Real-world2 (Hair Color)2 Genders4Bald male (no hair)
UrbanCars (UC) [35]Synthetic2 (Car Type)2 Backgrounds × 2 Objects8Background / Background + Object
Animals MetaCoCo (AMC) [ours]Real-world24 (Animal Type)8 BackgroundsNALeave-2-out (class-based)
Sp-ImageNet100 (SpI) [36]Real-world100 (ImageNet classes)NA (spurious visual features)NASpurious ImageNet [36]
+ +For evaluating SP-OOD detection, we utilize five diverse datasets, whose properties are summarized in Table 1. Additional details and visual examples for all datasets, including the NSP datasets and their evaluation setup, are provided in Appendix D. The datasets include Waterbirds (WB) [33] and CelebA (CA) [34]. While widely adopted, these datasets present limitations in terms of scale and realism (CelebA, in particular, is noted for its label noise [70]), making them insufficient for comprehensive evaluation. To address this, our evaluation incorporates three additional datasets designed to test robustness under diverse conditions, including multi-class scenarios, multiple spurious attributes, and real-world complexity. + +UrbanCars (UC) [35] is a binary classification dataset (urban vs. country cars) with two spurious attributes: background and a co-occurring object, both correlated with the class, making it a challenging multi-spurious benchmark. Next, we introduce Animals MetaCoCo (AMC), a realistic SP-OOD benchmark created by subsampling and curating animal categories from MetaCoCo [71]. It contains 26 distinct classes, each with 8 different background types serving as imbalance shortcut attributes. We use a class-level leave-2-out setting, where two classes are held out as SP-OOD and the rest are treated as ID. The significant similarity in background distributions between the ID and OOD splits makes Animals MetaCoCo a particularly challenging multi-class benchmark for SP-OOD detection (Figure 1). Last, we also use the Spurious ImageNet dataset introduced in [36] as SP-OOD data. This dataset consists of real-world images that contain only spurious features, such as bird feeders or graffiti, without the actual class objects, yet are consistently misclassified as specific ImageNet classes. These OOD samples are constructed for a subset of 100 ImageNet classes identified to rely on harmful spurious correlations. We refer to this subset of classes as Sp-ImageNet100 (SpI), a name we use for clarity, and treat it as the ID data. + +For conventional OOD evaluations, we follow the same settings as in OpenOOD [37]. Specifically, we use CIFAR-10 [72], CIFAR-100 [72], and ImageNet-1k [73], along with their respective near-OOD and far-OOD datasets, to ensure consistency with widely adopted benchmarks in the field. + +Table 2: Comparative performance of post-hoc OOD detection methods on SP-OOD benchmarks using a ResNet-50 backbone. Left: AUROC scores (higher is better); Right: FPR@95 scores (lower is better). Feature-based methods are indicated in blue, output-based methods in red, gradient-based in black, and hybrid methods in green. For each experiment, the top-performing method is shown in bold, and the second-best is underlined. + +
AUROC↑
MethodWBCAUCAMCSpIAvg.
MSP[7]62.3±0.646.0±1.438.5±0.379.7±0.483.1±0.361.9
Energy[13]62.0±2.645.4±3.438.4±2.179.9±0.680.6±0.461.3
MLS[44]62.2±2.345.3±3.238.4±1.480.2±0.681.9±0.361.6
KLM[44]51.2±0.741.7±2.557.0±0.274.2±0.679.6±0.860.7
GEN[45]62.3±0.646.0±1.438.5±0.380.2±0.080.8±0.461.6
GNorm[61]79.5±0.438.0±1.346.6±0.474.2±0.585.2±0.264.7
ReAct[58]72.9±3.645.6±5.341.3±3.180.1±0.683.6±0.764.7
VIM[59]79.6±2.550.4±3.160.7±1.778.6±0.677.4±0.969.3
ASH[60]78.5±3.247.3±2.839.6±1.778.0±0.286.6±0.766.0
MDS[11]90.2±0.157.8±0.591.8±0.162.9±0.858.4±0.172.2
RMDS[46]59.4±0.133.6±1.447.4±0.281.9±0.468.8±0.158.2
KNN[47]98.6±0.054.5±0.591.1±0.179.7±0.677.4±0.080.3
SHE[48]88.3±0.242.7±0.673.2±0.154.8±0.783.0±0.168.4
NECO[51]53.5±1.639.5±3.235.1±1.580.2±0.167.2±0.355.1
NNGuide[49]70.6±2.949.8±4.243.6±2.179.4±0.085.1±0.865.7
Relation[50]80.7±0.260.4±2.596.0±0.574.5±0.381.8±0.778.7
SCALE[52]89.0±2.944.9±3.254.4±2.178.4±0.486.270.6
tDBD[53]71.1±0.551.3±1.347.4±0.279.9±0.084.2±0.366.8
NCI[54]84.0±0.146.4±2.454.8±0.878.3±0.184.9±0.269.7
SPROD98.8±0.061.6±0.997.4±0.082.4±0.585.3±0.085.1
+ +
FPR@95↓
MethodWBCAUCAMCSpIAvg.
MSP[7]87.9±0.898.7±0.597.3±0.383.8±0.774.1±1.288.4
Energy[13]89.2±3.298.6±0.795.5±3.184.8±0.876.3±0.988.9
MLS[44]88.1±2.098.8±0.696.7±2.084.4±0.874.6±0.988.5
KLM[44]89.1±0.798.7±0.597.1±0.380.5±0.876.1±1.788.3
GEN[45]87.9±0.898.7±0.597.3±0.384.8±0.176.3±0.789.0
GNorm[61]84.2±0.798.8±0.497.1±0.184.2±0.654.7±0.683.8
ReAct[58]86.9±7.096.3±2.495.5±3.283.9±0.857.5±1.684.0
VIM[59]61.4±3.596.2±0.469.0±1.586.6±0.779.5±0.578.5
ASH[60]85.2±7.096.9±1.496.1±1.587.9±0.452.9±3.183.8
MDS[11]49.2±0.296.0±0.539.0±0.393.0±0.390.5±0.173.5
RMDS[46]91.7±0.299.6±0.195.3±0.183.4±0.988.1±0.191.6
KNN[47]4.8±0.194.4±1.042.5±0.279.9±1.170.4±0.258.4
SHE[48]33.2±0.596.4±0.576.5±0.293.9±0.352.6±0.870.5
NECO[51]90.5±2.298.8±0.696.7±1.978.2±0.089.9±0.890.8
NNGuide[49]77.7±6.097.6±1.291.6±3.686.3±0.152.2±1.481.1
Relation[50]73.8±0.595.4±0.224.2±1.784.6±0.278.0±1.171.2
SCALE[52]61.0±22.598.7±0.694.6±3.187.7±0.353.0±1.579.0
fDBD[53]85.5±0.898.6±0.596.1±0.485.4±0.270.8±1.087.3
NCI[54]41.1±0.199.4±0.392.2±0.685.8±0.363.8±0.976.5
SPROD4.7±0.193.7±0.919.0±0.469.5±1.258.0±0.149.0
+ +![](images/d310af7933a28335d77d9b2130ebde8ae89be23059dbee9b70054b574a5ac7e9.jpg) +(a) ResNet-50 backbone. + +![](images/988bddd61c811c98796a9077a850a196587314e806b05ed6b27a742e2903dc3d.jpg) + +![](images/1e7de07f4febc0a2ae3523e3cdfb89c5da98234d66a09279474b765e7776ea1a.jpg) +(b) ResNet-18 backbone. + +![](images/5796c04be03f2379c23e2fabe8af96d0c5cfb0c9322c49a0ed6fd8f48dfc9a0d.jpg) +Figure 4: Effect of backbone fine-tuning and spurious correlation on SP-OOD detection using the Waterbirds dataset. Left: ResNet-50; right: ResNet-18. Each pair shows results under $50\%$ (left) and $90\%$ (right) spurious correlation in ID data. Fine-tuned models are marked with a hatch texture. + +# 5.3 Results + +The primary results, summarized in Table 2, demonstrate the efficacy of SPROD across various datasets using a ResNet-50 backbone, evaluated with both AUROC and FPR@95 metrics. Additional results on other backbone architectures are provided in Appendix G. SPROD consistently achieves superior performance compared to the 19 post-hoc baselines in all conducted experiments. In contrast, competing methods exhibit more variable performance, excelling in only a subset of the experimental settings. Generally, feature-based OOD detection methods tend to outperform output-based methods; however, this advantage appears to diminish on larger-scale datasets with multiple classes. Overall, the average performance across all datasets reveals a notable margin by which SPROD surpasses the other evaluated baselines: Specifically, SPROD reaches an average of $85.1\%$ AUROC, $4.8\%$ higher (absolute) than the second best, KNN. For FPR@95, SPROD reaches $49.1\%$ error rate on average, $9.3\%$ better than the second best, KNN. All other compared approaches are more than $20\%$ worse. + +![](images/3a1e4e4909d94a6769eae2d8e8e2bfedf7a3b19bcfc57d011a8f159829c8b1da.jpg) +(a) + +![](images/10d45ee77193d1191d6e0dd9966089cae7ad0414d58c822e8627c8598fdde7ed.jpg) +Figure 5: Comparison of generative and discriminative scoring for OOD detection using SPROD. (a) Histograms of ID and OOD sample scores using the distance-based generative approach and the softmax-based discriminative approach, both computed with SPROD on the Waterbirds dataset. (b) Performance comparison between the generative (distance-based) and discriminative (softmax-based) scoring variants of SPROD across the five SP-OOD benchmark datasets. + +![](images/93df1a94dcc2f74fb50adccee086a84fd01b46bc6c6aee678e80f1718888bac5.jpg) +(b) + +Table 3: Comparison with the best-performing methods reported in the OpenOOD [37] on conventional (standard) OOD datasets (AUROC†). The highest scores are highlighted in bold. + +
MethodCIFAR-10 NearCIFAR-10 FarCIFAR-100 NearCIFAR-100 FarImageNet NearImageNet FarAvg (%)
RMDS [46]89.8092.2080.1582.9276.9986.3884.74
MLS [44]87.5291.1081.0579.6776.4689.5784.23
VIM [59]88.6893.4874.9881.7072.0892.6883.27
KNN [47]90.6492.9680.1882.4071.1090.1884.58
ASH [60]75.1278.4978.2080.5878.1795.7481.05
SPROD89.0491.7881.8079.9375.0695.2985.15
+ +Next, we investigate the impact of backbone fine-tuning and correlation rate on OOD detection performance, using the Waterbirds dataset with ResNet-50 (Figure 4a) and ResNet-18 (Figure 4b) backbones. Fine-tuning the backbone on this dataset generally degrades OOD detection performance, a finding that contrasts with some common assumptions. This negative effect appears to be more pronounced for feature-based methods. As expected, increasing the spurious correlation rate of ID data from $50\%$ to $90\%$ leads to a general decline in performance across methods; however, this degradation is more noticeable for output-based techniques. Furthermore, the results suggest that employing a lighter, less expressive backbone (ResNet-18 compared to ResNet-50) does not lead to a substantial performance drop in OOD detection. Across these variations, SPROD demonstrates consistent robustness and maintains its performance as the spurious correlation rate increases. + +To investigate the impact of the scoring mechanism, as discussed in Section 3.2, we conduct an ablation study using SPROD as the base method. This controlled experiment compares the effectiveness of deriving OOD scores in a discriminative manner $p(y|z)$ versus a generative manner $p(z|y)$ . Both baselines utilize the same samples, feature embeddings, and refined prototypes from SPROD. For the discriminative score, we apply a softmax function to the negative distances between a sample's embedding $z$ and the class prototypes. The default SPROD approach, which uses the negative of the minimum distance to class prototypes, serves as the generative baseline (aligning with a log-likelihood under an exponential family distribution assumption). + +Figure 5a presents histograms of the scores generated by both approaches, showing that the generative scoring method yields more distinctly separated distributions for ID and OOD samples. Furthermore, Figure 5b compares the OOD detection performance of these two scoring variants across our five SP-OOD benchmark settings. The results indicate that applying the softmax function for discriminative scoring substantially degrades performance, particularly in WaterBirds, CelebA and UrbanCars datasets. Conversely, on Animals MetaCoCo and Spurious ImageNet, the performance degradation from using softmax is less pronounced. This observation aligns with the trends in Table 2, where traditional output-based methods tend to perform relatively better in these datasets. + +To further assess the generality of SPROD beyond spurious correlation settings, we evaluate it on conventional OOD benchmarks using the standardized OpenOOD protocol [37]. We compare SPROD with the top-performing post-hoc methods reported in the OpenOOD study. Table 3 shows that SPROD attains performance comparable to or exceeding these methods and achieves the highest average score across all settings, confirming that SPROD performs reliably on standard OOD benchmarks and is not limited to spurious correlation scenarios. + +While the simplicity of its prototypical framework makes SPROD a computationally efficient post-hoc method, its sample efficiency in SP-OOD settings also deserves investigation. To this end, we evaluate SPROD alongside the second top-performing method from Table 2 under low-shot conditions. In this setup, we reduce the number of ID training samples while carefully preserving the original level of spurious correlation present in the experimental setting. The results, presented in Figure 6, demonstrate that both SPROD and KNN maintain strong performance even in this data-scarce regime, highlighting the sample efficiency of these post-hoc approaches. Interestingly, for the CelebA dataset, both methods exhibit improved OOD detection performance in the low-shot setting compared to when trained on the full dataset. + +![](images/6bf9c5c057a64db873e8f2b01742f40520b70cd1485b1e055050394501660227.jpg) +Figure 6: Performance of SPROD and KNN in low-shot SP-OOD settings across three datasets. Dashed lines indicate performance with the full training set, while solid lines show performance using varying numbers of samples per minority group. + +![](images/4a0bd897e17d6bb0cc50ef463402f3ce5965cfdcea65c4445252f4f0590fe10c.jpg) + +![](images/11422f3cc040eb48e72a1a672c3f9ad7843a01d1cbc9fad3cbefd0843392309c.jpg) + +Zero-shot OOD detection methods have reported results on Waterbirds using CLIP-B/16, but since they operate in a zero-shot setting, spurious correlations in the training set are not meaningful for them. Still, we compare: MCM [29] and CMA [32] achieve 98.36 and 98.62 AUROC using text inputs, while our text-free, vision-only method outperforms both with 99.01. COVER [28] also evaluates this setting but over a different operating range, achieving lower AUROC scores (90.52 vs. 90.31 for MCM), highlighting our method's superior performance. Additional comparisons with existing SP-OOD methods (specifically CLIP-based approaches) are provided in Appendix C. + +# 6 Discussion + +This paper introduces SPROD, a prototype-based method enhancing out-of-distribution (OOD) detection robustness against unknown spurious correlations. SPROD refines class prototypes by identifying and then adjusting for potential subgroups influenced by spurious features, thereby aligning representations with core, invariant class characteristics. A key strength is SPROD's efficiency and adaptability: as a post-hoc method, it integrates with various pre-trained feature extraction models without requiring retraining, additional OOD data, or hyperparameter tuning. + +Experimental results consistently demonstrate the superiority of SPROD across ten convolutional and transformer-based backbones and five diverse SP-OOD benchmarks, including the newly introduced Animals MetaCoCo dataset. Our evaluations also reveal that fine-tuning the feature backbone on ID data can degrade SP-OOD detection performance. Furthermore, investigations into scoring mechanisms highlighted the advantages of distance-based approaches over softmax-based scoring for SP-OOD detection, particularly affirming the design choices in SPROD. + +SPROD offers a scalable solution for improving robustness in OOD detection. While demonstrating significant advancements, SPROD's current formulation relies on class labels from the ID training data to construct class-conditional prototypes. This reliance aligns with the typical assumptions of the spurious correlation setting, which presumes access to data samples with spuriously correlated labels. SPROD is not always the single best-performing method on every metric or in every setting (particularly in the conventional setting), but it remains competitive with the best-performing approaches. 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URL https://arxiv.org/abs/2110.11334. + +# A Theoretical Analysis of SPROD + +SPROD employs a two-step prototype refinement strategy to approximate group-specific representations within each class. The first step creates new prototypes for misclassified training samples, motivated by the observation that minority group instances (those that deviate from dominant spurious patterns) are more prone to misclassification. However, some correctly classified minority samples may still be incorrectly assigned to biased prototypes. + +To address this, the second step reassigns each training sample to its nearest prototype and recalculates prototypes based on these updated assignments. This reassignment helps produce cleaner subgroup representations by reducing the influence of spurious-majority samples on minority feature structure. + +By applying these two steps, SPROD aims to obtain nearly pure group-specific prototypes. In the remainder of this section, we provide a theoretical formulation to explain why such subgroup-specific prototypes improve robustness to spurious OOD (SP-OOD) samples. + +# A.0.1 Feature Decomposition + +Without loss of generality, we assume a binary classification problem with classes $c \in \{0,1\}$ . Each input sample $x_{i}$ is mapped to a normalized feature embedding $z_{i} \in \mathbb{R}^{d}$ . We consider a general decomposition of $z_{i}$ into three semantically distinct components: spurious, core, and irrelevant features. We model $z_{i}$ as a linear combination of three functionally distinct components that span the embedding space: + +$$ +z _ {i} = \sum_ {j = 1} ^ {n _ {u _ {c} ^ {\mathrm {s p}}}} \alpha_ {i, j} ^ {\mathrm {s p}} \vec {\mathbf {u}} _ {c, j} ^ {\mathrm {s p}} + \sum_ {j = 1} ^ {n _ {u} ^ {\mathrm {c o r e}}} \beta_ {i, j} ^ {\mathrm {c o r e}} \vec {\mathbf {u}} _ {c, j} ^ {\mathrm {c o r e}} + \sum_ {j = 1} ^ {n _ {u} ^ {\mathrm {i r r}}} \gamma_ {i, j} ^ {\mathrm {i r r}} \vec {\mathbf {u}} _ {j} ^ {\mathrm {i r r}}, +$$ + +where the sets $\{\vec{\mathbf{u}}_{c,j}^{\mathrm{sp}}\}$ , $\{\vec{\mathbf{u}}_{c,j}^{\mathrm{core}}\}$ , and $\{\vec{\mathbf{u}}_j^{\mathrm{irr}}\}$ form orthonormal bases for the spurious, core, and irrelevant subspaces, respectively. Each set of coefficients $\{\alpha_{i,j}^{\mathrm{sp}}\}$ , $\{\beta_{i,j}^{\mathrm{core}}\}$ , and $\{\gamma_{i,j}^{\mathrm{irr}}\}$ is specific to instance $i$ . These subspaces span mutually exclusive semantic roles: + +- Spurious subspace: captures features correlated with the label during training but not causally predictive. +- Core subspace: captures features that are causally predictive of the label. +- Irrelevant subspace: captures features unrelated to the task. + +By definition, the basis vectors across subspaces are orthogonal: if an irrelevant basis vector were correlated with a core basis, it would be predictive and thus belong in the core subspace. + +This decomposition can be expressed compactly in matrix form: + +$$ +z _ {i} = U _ {c} ^ {\mathrm {s p}} \vec {\alpha} _ {i} ^ {\mathrm {s p}} + U _ {c} ^ {\mathrm {c o r e}} \vec {\beta} _ {i} ^ {\mathrm {c o r e}} + U ^ {\mathrm {i r r}} \vec {\gamma} _ {i} ^ {\mathrm {i r r}}, +$$ + +where each $U^{(\cdot)}\in \mathbb{R}^{d\times n_u^{(\cdot)}}$ is a matrix of orthonormal basis vectors for the corresponding subspace, and each coefficient vector (e.g., $\vec{\alpha}_i^{\mathrm{sp}}\in \mathbb{R}^{n_{u_c^{\mathrm{sp}}}}$ ) represents the coordinates of $z_{i}$ in that subspace. + +# A.0.2 Distributional Assumptions + +We assume the coefficient vectors for each instance $z_{i}$ are drawn from class-conditional distributions: + +$$ +\vec {\boldsymbol {\alpha}} _ {i} ^ {\mathrm {s p}} \sim P _ {c} ^ {\mathrm {s p}}, \quad \vec {\boldsymbol {\beta}} _ {i} ^ {\mathrm {c o r e}} \sim P _ {c} ^ {\mathrm {c o r e}}, +$$ + +where $P_{c}^{\mathrm{sp}}$ and $P_{c}^{\mathrm{core}}$ are distributions over the spurious and core subspace coefficients for class $c$ . + +The irrelevant component is shared across classes: + +$$ +\vec {\gamma} _ {i} ^ {\mathrm {i r r}} \sim P ^ {\mathrm {i r r}}. +$$ + +# A.0.3 Group Definitions + +Within each class $c$ , we define majority and minority groups based on the alignment of spurious features. + +Majority group samples $z_{i}\in S_{c}^{\mathrm{maj}}$ satisfy: + +$$ +\vec {\boldsymbol {\alpha}} _ {i} ^ {\mathrm {s p}} \sim P _ {c} ^ {\mathrm {s p}}, \quad \vec {\boldsymbol {\beta}} _ {i} ^ {\mathrm {c o r e}} \sim P _ {c} ^ {\mathrm {c o r e}}. +$$ + +Minority group samples $z_{i}\in S_{c}^{\mathrm{min}}$ satisfy: + +$$ +\vec {\alpha} _ {i} ^ {\mathrm {s p}} \sim P _ {1 - c} ^ {\mathrm {s p}}, \quad \vec {\beta} _ {i} ^ {\mathrm {c o r e}} \sim P _ {c} ^ {\mathrm {c o r e}}. +$$ + +Let $C_c$ denote the number of samples in class $c$ , and $|S_c^{\mathrm{maj}}|$ the number of majority samples. Then: + +$$ +r _ {c} = \frac {\left| \mathcal {S} _ {c} ^ {\text {m a j}} \right|}{C _ {c}} \in [ 0, 1 ] +$$ + +quantifies the class-conditional spurious correlation strength. + +# A.0.4 OOD Sample Definition + +OOD samples may contain components beyond the core, spurious, and irrelevant subspaces. To model this, we extend the feature decomposition to include an additional subspace $U^{\mathrm{ext}}$ , capturing external factors not observed during training. We define an OOD sample as: + +$$ +z _ {\mathrm {O O D}} = U _ {c} ^ {\mathrm {s p}} \vec {\alpha} _ {\mathrm {O O D}} ^ {\mathrm {s p}} + U _ {c} ^ {\mathrm {i r r}} \vec {\gamma} _ {\mathrm {O O D}} ^ {\mathrm {i r r}} + U ^ {\mathrm {e x t}} \vec {\delta} _ {\mathrm {O O D}} ^ {\mathrm {e x t}}, +$$ + +where $\vec{\alpha}_{\mathrm{OOD}}^{\mathrm{sp}} \sim P_c^{\mathrm{sp}}$ , $\vec{\gamma}_{\mathrm{OOD}}^{\mathrm{irr}} \sim P_{\mathrm{irr}}^{\mathrm{irr}}$ , and $\vec{\delta}_{\mathrm{OOD}}^{\mathrm{ext}}$ is unconstrained. Hard OOD samples may also include core components drawn from a shifted distribution $\vec{\beta}_{\mathrm{OOD}}^{\mathrm{core}} \sim Q_{c}^{\mathrm{core}} \neq P_{c}^{\mathrm{core}}$ . Although it is possible to analyze the general form with core and external components, for simplicity we focus on near-OOD samples with no core or external components, i.e., + +$$ +z _ {\mathrm {O O D}} = U _ {c} ^ {\mathrm {s p}} \vec {\alpha} _ {\mathrm {O O D}} ^ {\mathrm {s p}} + U _ {c} ^ {\mathrm {i r r}} \vec {\gamma} _ {\mathrm {O O D}} ^ {\mathrm {i r r}}, \quad \text {w i t h} \vec {\beta} _ {\mathrm {O O D}} ^ {\mathrm {c o r e}} = \vec {0}, \vec {\delta} _ {\mathrm {O O D}} ^ {\mathrm {e x t}} = \vec {0}. +$$ + +We define spurious-OOD groups based on the source of the spurious component: + +$$ +\mathcal {S} _ {c} ^ {\mathrm {O O D}} = \left\{z _ {\mathrm {O O D}} \left| \vec {\alpha} _ {\mathrm {O O D}} ^ {\mathrm {s p}} \sim P _ {c} ^ {\mathrm {s p}}, \vec {\gamma} _ {\mathrm {O O D}} ^ {\mathrm {i r r}} \sim P ^ {\mathrm {i r r}} \right. \right\}, \quad c \in \{0, 1 \}. +$$ + +These near-OOD samples resemble class $c$ only through spurious features and match the in-distribution (ID) irrelevant component distribution. + +# A.0.5 Prototype Calculation + +We define the expected coefficient vectors for each subspace: + +$$ +\vec {\boldsymbol {\mu}} _ {c} ^ {\text {c o r e}} = \mathbb {E} _ {\vec {\boldsymbol {\beta}} \sim P _ {c} ^ {\text {c o r e}}} [ \vec {\boldsymbol {\beta}} ], \quad \vec {\boldsymbol {\mu}} _ {c} ^ {\text {s p}} = \mathbb {E} _ {\vec {\boldsymbol {\alpha}} \sim P _ {c} ^ {\text {s p}}} [ \vec {\boldsymbol {\alpha}} ], \quad \vec {\boldsymbol {\mu}} ^ {\text {i r r}} = \mathbb {E} _ {\vec {\boldsymbol {\gamma}} \sim P ^ {\text {i r r}}} [ \vec {\boldsymbol {\gamma}} ]. +$$ + +Using these, we define the majority and minority subgroup prototypes for class $c \in \{0,1\}$ as: + +$$ +\vec {\mathbf {p}} _ {c} ^ {\text {m a j}} = U _ {c} ^ {\text {s p}} \vec {\boldsymbol {\mu}} _ {c} ^ {\text {s p}} + U _ {c} ^ {\text {c o r e}} \vec {\boldsymbol {\mu}} _ {c} ^ {\text {c o r e}} + U ^ {\text {i r r}} \vec {\boldsymbol {\mu}} ^ {\text {i r r}}, +$$ + +$$ +\vec {\mathbf {p}} _ {c} ^ {\min } = U _ {c} ^ {\mathrm {s p}} \vec {\boldsymbol {\mu}} _ {1 - c} ^ {\mathrm {s p}} + U _ {c} ^ {\mathrm {c o r e}} \vec {\boldsymbol {\mu}} _ {c} ^ {\mathrm {c o r e}} + U ^ {\mathrm {i r r}} \vec {\boldsymbol {\mu}} ^ {\mathrm {i r r}}. +$$ + +The overall class prototype is a convex combination of subgroup prototypes: + +$$ +\vec {\mathbf {p}} _ {c} = r _ {c} \vec {\mathbf {p}} _ {c} ^ {\operatorname {m a j}} + (1 - r _ {c}) \vec {\mathbf {p}} _ {c} ^ {\min }, +$$ + +where $r_c \in [0,1]$ denotes the proportion of majority group samples in class $c$ . + +# A.0.6 Bias in Prototype Distances Under Strong Spurious Correlation + +In standard prototype-based OOD detection, the class prototype $\vec{\mathbf{p}}_c$ is a convex combination of majority and minority subgroup prototypes. In spurious-dominated regimes where $r_c\approx 1$ , we have: + +$$ +\vec {\mathbf {p}} _ {c} \approx \vec {\mathbf {p}} _ {c} ^ {\text {m a j}} = U _ {c} ^ {\text {s p}} \vec {\boldsymbol {\mu}} _ {c} ^ {\text {s p}} + U _ {c} ^ {\text {c o r e}} \vec {\boldsymbol {\mu}} _ {c} ^ {\text {c o r e}} + U ^ {\text {i r r}} \vec {\boldsymbol {\mu}} ^ {\text {i r r}}. +$$ + +Case 1: Spurious-OD Sample. Let $z_{\mathrm{OOD}} \in S_c^{\mathrm{OOD}}$ be an OOD sample with spurious alignment to class $c$ , but no core features: + +$$ +z _ {\mathrm {O O D}} = U _ {c} ^ {\mathrm {s p}} \vec {\alpha} _ {\mathrm {O O D}} + U ^ {\mathrm {i r r}} \vec {\gamma} _ {\mathrm {O O D}}, \quad \text {w i t h} \vec {\alpha} _ {\mathrm {O O D}} \sim P _ {c} ^ {\mathrm {s p}}, \vec {\gamma} _ {\mathrm {O O D}} \sim P ^ {\mathrm {i r r}}. +$$ + +The expected squared distance to the biased class prototype becomes: + +$$ +\mathbb {E} \left[ \| z _ {\mathrm {O O D}} - \vec {\mathbf {p}} _ {c} \| ^ {2} \right] = \underbrace {\| \vec {\boldsymbol {\mu}} _ {c} ^ {\text {c o r e}} \| ^ {2}} _ {\text {c o r e b i a s}} + \underbrace {\operatorname {T r} (\Sigma_ {c} ^ {\text {s p}})} _ {\text {s p u r i o u s v a r i a n c e}} + \underbrace {\operatorname {T r} (\Sigma^ {\text {i r r}})} _ {\text {i r r e l e v a n t v a r i a n c e}}. +$$ + +Case 2: Minority In-Distribution Sample. Now consider an ID sample from the minority group: + +$$ +z _ {\min } = U _ {c} ^ {\mathrm {s p}} \vec {\alpha} _ {\min } + U _ {c} ^ {\mathrm {c o r e}} \vec {\beta} _ {\min } + U ^ {\mathrm {i r r}} \vec {\gamma} _ {\min }, +$$ + +with: + +$$ +\vec {\alpha} _ {\min } \sim P _ {1 - c} ^ {\mathrm {s p}}, \quad \vec {\beta} _ {\min } \sim P _ {c} ^ {\mathrm {c o r e}}, \quad \vec {\gamma} _ {\min } \sim P ^ {\mathrm {i r r}}. +$$ + +Its expected squared distance to the majority prototype is: + +$$ +\mathbb {E} \left[ \| z _ {\min } - \vec {\mathbf {p}} _ {c} ^ {\operatorname {m a j}} \| ^ {2} \right] = \underbrace {\| \vec {\boldsymbol {\mu}} _ {1 - c} ^ {\operatorname {s p}} - \vec {\boldsymbol {\mu}} _ {c} ^ {\operatorname {s p}} \| ^ {2}} _ {\text {s p u r i o u s b i a s}} + \underbrace {\operatorname {T r} (\Sigma_ {c} ^ {\operatorname {c o r e}})} _ {\text {c o r e v a r i a n c e}} + \underbrace {\operatorname {T r} (\Sigma^ {\operatorname {i r r}})} _ {\text {i r r e l e v a n t v a r i a n c e}}. +$$ + +Key Insight: Although $z_{\mathrm{min}}$ is an ID sample, its distance to the biased prototype includes a spurious bias term, while the OOD sample differs only in the core direction. Depending on the relative magnitudes of the core and spurious bias terms, this highlights the potential for erroneous OOD detection when prototype estimates are biased toward spurious features. + +For example, in the Waterbird dataset, the background (water or land) represents the spurious feature dimension characterized by distinct mean vectors $\vec{\mu}_c^{\mathrm{sp}}$ and $\vec{\mu}_{1 - c}^{\mathrm{sp}}$ . The difference $\| \vec{\mu}_c^{\mathrm{sp}} - \vec{\mu}_{1 - c}^{\mathrm{sp}}\|$ may exceed the core bias in OOD distances, causing minority samples with conflicting backgrounds to lie farther from prototypes than some OOD samples. + +# A.0.7 Why SPROD Mitigates Distance Bias + +If prototypes can be accurately estimated to match each group's distribution, then each ID sample $z_{i} \in S_{c}^{g}$ can be compared to its corresponding prototype $\hat{\mathbf{p}}(z_{i}) = \vec{\mathbf{p}}_{c}^{g}$ , where $g \in \{\mathrm{maj}, \mathrm{min}\}$ . + +Because the prototype shares the same spurious basis alignment, the spurious bias is eliminated. The expected squared distance becomes: + +$$ +\mathbb {E} \left[ \| z _ {i} - \hat {\mathbf {p}} (z _ {i}) \| ^ {2} \right] = \mathrm {T r} (\Sigma_ {c} ^ {\mathrm {c o r e}}) + \mathrm {T r} (\Sigma^ {\mathrm {i r r}}). +$$ + +In contrast, the standard prototype-based approach introduces a systematic bias for minority samples: + +$$ +\mathbb {E} \left[ \| z _ {\min } - \vec {\mathbf {p}} _ {c} ^ {\operatorname {m a j}} \| ^ {2} \right] = \| \vec {\boldsymbol {\mu}} _ {1 - c} ^ {\mathrm {s p}} - \vec {\boldsymbol {\mu}} _ {c} ^ {\mathrm {s p}} \| ^ {2} + \operatorname {T r} \left(\Sigma_ {c} ^ {\text {c o r e}}\right) + \operatorname {T r} \left(\Sigma^ {\text {i r r}}\right). +$$ + +For an SP-OOD sample, the prototype it is compared against (either group) contains a core component it lacks. Hence the expected distance becomes: + +$$ +\mathbb {E} \left[ \| z _ {\mathrm {O O D}} - \hat {\mathbf {p}} \| ^ {2} \right] = \| \vec {\boldsymbol {\mu}} _ {c} ^ {\mathrm {c o r e}} \| ^ {2} + \operatorname {T r} (\Sigma_ {c} ^ {\mathrm {c o r e}}) + \operatorname {T r} (\Sigma^ {\mathrm {i r r}}). +$$ + +This is strictly greater than the refined ID distance, which includes neither the spurious bias nor the core bias term: + +$$ +\mathbb {E} \left[ \| z _ {\mathrm {O O D}} - \hat {\mathbf {p}} \| ^ {2} \right] > \mathbb {E} \left[ \| z _ {i} - \hat {\mathbf {p}} (z _ {i}) \| ^ {2} \right]. +$$ + +Conclusion: SPROD removes the spurious bias term for ID samples and reduces overlap with OOD samples by accounting for their missing core features. This systematic (in expectation) bias elimination leads to tighter ID clusters and more reliable OOD detection. + +# B Spurious Correlation in OOD Detection: Literature Review + +This work [14] was the first to introduce the problem of spurious correlations in OOD detection, showing that detection performance degrades as spurious correlation increases, especially for SP-OOD samples. It also encouraged the community to report results of OOD methods under this challenging setting. Recently, several works have addressed the challenge of spurious correlations in out-of-distribution (OOD) detection, employing different strategies that can be broadly categorized into outlier exposure based, training-time regularization, and post-hoc methods. + +Outlier exposure based methods include RONF [21], which improves synthetic outlier generation and model fine-tuning using only ID data. It introduces Boundary Feature Mixup to create more realistic virtual outliers by interpolating near decision boundaries, and Optimal Parameter Learning to suppress spurious feature learning during training. At inference, it uses a custom Energy with Energy Discrepancy score to better separate ID from OOD samples without relying on external OOD datasets. Similarly, KIRBY [22] generates hard negative samples by removing class-discriminative regions identified via Class Activation Maps (CAM) and inpainting these regions with background-like content, creating semantically degraded but visually plausible images as surrogate OOD examples. A lightweight rejection network trained on features from both clean and modified images enables strong OOD detection without requiring real OOD data or backbone retraining. Although not directly targeting spurious correlations, ImOOD [23] evaluates robustness under spurious settings, focusing on long-tailed datasets where class imbalance biases OOD detection toward frequent (head) classes. It learns a bias correction term to shift OOD scores per input, improving separation especially for rare (tail) classes. + +Training-time regularization methods seek to reduce reliance on spurious cues during model training. BackMix [24] regularizes models by mixing foreground objects with different backgrounds, breaking spurious correlations between objects and backgrounds. Using CAM to estimate foreground regions, it replaces background patches with those from other images while preserving labels, thus improving robustness primarily against background spuriousity. RW [25] introduces a nuisance-aware training framework that reweights the training loss to reduce correlations between class labels and nuisance attributes. It further applies a penalty based on the Hilbert-Schmidt Independence Criterion (HSIC) to explicitly remove nuisance information from learned features, enhancing semantic OOD detection under shared nuisance conditions. NsED [26] decomposes images into semantic (phase) and style (amplitude) components via Discrete Fourier Transform, generating augmented samples by mixing amplitude spectra across images. Training with a robust loss that minimizes worst-case classification error over these augmented samples results in features less sensitive to style variations. + +Post-hoc methods operate after model training to improve OOD detection. Projection Regret (PR) [27] uses partial diffusion and denoising to project inputs onto the ID manifold, measuring semantic novelty by comparing the original image with its projection. A second projection step recursively removes background bias, isolating true semantic differences and enabling object-level OOD detection focused on meaningful changes rather than background similarity. CoVer [28] enhances detection by evaluating model confidence across multiple corrupted versions of the same input (e.g., fog, blur, contrast shifts). Instead of relying on a single prediction, it averages confidence scores over these variants, improving robustness. + +Finally, several zero-shot OOD detection approaches, while not explicitly targeting spurious correlations, report results under limited spurious settings. Maximum Concept Matching (MCM) [29] leverages CLIP's vision-language embeddings for zero-shot detection without fine-tuning or extra data. Each ID class is represented by a text prompt encoded as a concept prototype, and test images are scored based on cosine similarity with these prototypes. A softmax scaling sharpens distinctions between ID and OOD samples. Another zero-shot multimodal method [30] improves performance by prompting a large language model to generate rich descriptors for each ID class, filtered by consistency on retrieval tasks to reduce hallucinations. It combines CLIP-based similarity scores between test images, filtered descriptors, and detected object labels to compute a robust matching score. NegLabel [31] defines negative labels semantically distant from ID classes and computes softmax-based similarity scores that favor ID labels while penalizing negatives, optionally averaging over label groups to reduce noise. Lastly, CMA [32] embeds both ID class labels and neutral, class-agnostic Agents in a shared semantic space, leveraging a triangular similarity relationship to reduce confidence on OOD images while maintaining high confidence on ID data. + +Together, these approaches represent diverse strategies to tackle the spurious correlation problem in OOD detection, from data augmentation and loss regularization to post-hoc corrections and zero-shot semantic reasoning. + +While existing approaches have made significant strides in mitigating spurious correlations in OOD detection, each comes with certain trade-offs and constraints. Outlier exposure and training-time regularization methods often involve non-trivial training overhead, including access to auxiliary data or retraining backbone models with certain objectives, which limits their practicality in deployment settings. Moreover, some of these methods, such as BackMix [24], primarily address specific types of spurious features (e.g., background), which may not generalize to more complex real-world scenarios involving multiple or less structured spurious cues. Post-hoc methods offer a more efficient alternative by avoiding retraining and operating directly on pre-trained models. However, many still rely on careful tuning of multiple hyperparameters or complex transformation pipelines (e.g., CoVer's corruption sets [28] or PR's iterative projections [27]), making them sensitive to design choices and dataset specifics. Furthermore, several zero-shot approaches utilize vision-language models like CLIP, benefiting from rich semantic priors but introducing dependencies on text prompts and external descriptors, which can be limiting when only visual features are available or when text-image alignment is imperfect. + +# C Comparison with SP-OOD Methods + +Most existing OOD detection methods are not directly comparable to SPROD, as they often rely on different assumptions, such as access to auxiliary OOD data, or necessitate retraining the feature backbone with specific objectives. These design choices contrast with SPROD's post-hoc nature and its objectives of simplicity, efficiency, and broad applicability without model retraining. Nonetheless, to provide a broader context, we include results from a selection of methods that, while not perfectly aligned, have been evaluated under similar SP-OOD conditions. For fairness and consistency, we report the results as published in their original papers, rather than re-implementing them, thereby representing each method according to its best-reported performance. As shown in Table 4, under the most comparable conditions presented, SPROD achieves highly competitive performance, notably without requiring model retraining (when using a ResNet-18 backbone, as applicable to some comparisons) or access to an auxiliary text modality (in the case of methods leveraging CLIP). + +To further investigate the sample efficiency of SPROD, we conduct a low-shot experiment using feature embeddings from a frozen CLIP ViT-B/16 vision encoder, evaluated on the Waterbirds dataset. Two experimental settings for spurious correlation are considered: a $50\%$ correlation rate (where majority and minority group sample sizes are equal) and a $90\%$ correlation rate (where majority group samples are nine times more numerous than minority group samples). Within each setting, we vary the number of available samples per minority group and evaluate performance using both Euclidean and cosine distance metrics for score calculation. The results are presented in Figure 7. Initially, with very few samples per minority group, the $90\%$ correlation setting exhibits slightly higher performance, which may be due to the larger initial population of majority group samples aiding prototype stability. However, as the number of samples per minority group increases (e.g., to four samples), the $50\%$ correlation setting surpasses the $90\%$ setting in performance. Notably, the performance in both low-shot variants quickly becomes competitive with the performance achieved + +Table 4: A comparative analysis of AUROC and FPR@95 performance metrics for different methods and models evaluated on the Waterbirds dataset. To ensure a fair comparison, we report our results using both the pretrained CLIP and ResNet-18 models, aligning with the settings of the compared methods. + +
MethodBackboneAUROC ↑FPR@95 ↓Notes
Backmix [24] + EnergyWideResNet40-480.681.7
ImOOD [23]83.2757.69
RW [25] + MDResNet-18<90-Exact AUROC not clear from plots
SPROD98.287.27
CoVer [28]90.5233.17Also reports MCM: 90.31 / 25.66
MCM [29]98.365.87
Dai et al. [30]CLIP-B/1698.624.56
Neglabel [31]94.679.5Also reports MCM: 93.30 / 14.45
CMA [32]99.013.22
SPROD99.012.94
+ +![](images/9c0ca7187348d98d1cc7adea88c4123a46937af143eac0166242db0454c59993.jpg) +(a) AUROC + +![](images/7cbc9c7be2fc8c9e01c5cd20ed3617c75aee22cd08b48896703823fcad5bc5de.jpg) +(b) FPR@95 +Figure 7: Low-shot SP-OOD detection performance of SPROD on the Waterbirds dataset using features from a frozen CLIP ViT-B/16 vision encoder. Performance (AUROC and FPR@95) is shown as a function of the number of samples per minority group for two spurious correlation rates (50% and 90%) and two distance metrics (Euclidean and Cosine). Dashed lines indicate performance with the full training set. + +![](images/972ca0c0f7cee5b448052ee75a7f879b9b25a19c3a0cd657f3b69a64c06f1404.jpg) +(a) AUROC + +![](images/272103c5eb1d3dbe94390a5f1f6a96ce4016ff74aacc9700d9c702fc4abd01b0.jpg) +(b) FPR@95 +Figure 8: Impact of softmax normalization on low-shot SP-OOD detection performance of SPROD on the Waterbirds dataset using features from a frozen CLIP ViT-B/16 vision encoder. OOD scores are derived by applying softmax to the negative distances. Performance (AUROC and FPR@95) is shown as a function of the number of samples per minority group for two spurious correlation rates (50% and 90%) and two distance metrics (Euclidean and Cosine). Dashed lines indicate performance with the full training set using softmax-normalized scores. + +using the full dataset, underscoring the sample efficiency of the prototypical approach, even with CLIP features. + +We extend this low-shot analysis by examining the impact of applying a softmax normalization to the distance-based OOD scores. Figure 8 illustrates the results of this modification. A considerable + +performance degradation is observed when softmax normalization is applied to the distance scores, further highlighting the potential sensitivity of softmax-based scoring mechanisms. In this softmax-normalized setting, Euclidean distance appears to yield relatively better performance compared to cosine distance, an observation that differs from the direct distance-based scoring results shown in Figure 7, where both distance metrics performed equally. + +# D Dataset Details and Examples + +Datasets available for studying spurious correlations are generally limited. In the context of OOD detection, this limitation becomes even more pronounced, as the task requires datasets with SP-OOD samples. As a result, the datasets employed must be thoughtfully considered to ensure meaningful evaluation. In this study, our goal is to address a broader range of spurious features present in data, beyond just background features. A distinctive aspect of our experimental design is the inclusion of three additional settings to explore: + +1. Multi-spurious feature setting: Scenarios where multiple spurious features (e.g., both background and co-occurring objects) are simultaneously present, increasing the complexity of the detection task. To the best of our knowledge, this is the first time such a setting has been explored in the context of OOD detection. +2. Multi-class setting: Scenarios with more than two classes, where inter-class relationships and spurious correlations introduce additional challenges. +3. Realistic dataset setting: Beyond existing datasets in the literature, where spurious correlations are often predefined and controlled, we also focus on scenarios leveraging realistic datasets such as AnimalsMetaCoCo and Sp-ImageNet100 [36]. These datasets include diverse spurious correlations and more closely mimic real-world data distributions. + +These aspects are relatively underexplored in the context of SP-OOD detection. By incorporating these settings, we aim to demonstrate the effectiveness of our proposed approach. + +To evaluate the proposed approach, we used the following datasets: + +- Waterbirds [17]: This synthetic dataset is generated by combining the CUB [74] (bird classes) and the Places [69] (background scenes) datasets for a binary classification task, with labels $y \in \{ \text{waterbird}, \text{landbird} \}$ . Spurious correlations are introduced between the background $e \in \{ \text{water}, \text{land} \}$ and the label. The dataset consists of four groups, as depicted in Figure 9, with the minority and majority groups highlighted by red and green borders, respectively. Two different correlation values, $r \in \{ 0.5, 0.9 \}$ , are employed, where $r$ denotes the probability that the environment $e$ aligns with the label $y$ . Specifically, we have: + +$$ +r = P \left(e = _ {\text {w a t e r}} \mid y = _ {\text {w a t e r b i r d}}\right) = P \left(e = _ {\text {l a n d}} \mid y = _ {\text {l a n d b i r d}}\right). +$$ + +The distribution of samples within each group and class is provided in Table 5. For SP-OOD, we select samples from the Places dataset [69], following the previous works [14, 25]. + +CelebA [34]: This dataset is used for a binary classification task with labels $y \in \{\text{blond hair, non-blond hair}\}$ , where spurious correlations with gender $\in \{\text{male, female}\}$ are present. It is a real-world dataset, making it suitable for evaluating models in realistic settings. In the dataset, most females have blond hair, and most males have non-blond hair, forming the majority groups, as shown in Figure 10 with colored borders. The minority groups are the opposite combinations. The distribution of groups across classes with varying correlation levels is presented in Table 6. In this setting, SP-OOD samples are those with no hair, but still exhibiting the spurious gender-related features. For our SP-OOD samples, we used bald males, who lack core features but retain spurious (gender) cues. + +- UrbanCars [35]: This dataset, introduced in [35], is synthetically generated by combining the Stanford Cars dataset [75] (which includes both urban and country cars) with co-occurring objects from either urban or country environments, sourced from the LVIS dataset [76]. The co-occurring objects are positioned to the right of the cars, and both the car and the object are placed onto background images from the Places dataset [69], representing either urban or country scenes. The dataset is considered particularly challenging due to the presence of two spurious features. By varying the combinations of cars, co-occurring + +objects, and backgrounds, the dataset is divided into eight distinct groups, as illustrated in Figure 11. Notably, the "all-country" and "all-urban" groups dominate the dataset, while the other groups are underrepresented, as highlighted by the border colors. The exact number of samples in each group of our generated dataset, which exhibits a $95\%$ correlation, is detailed in Table 7, with two groups being especially underrepresented. For the SP-OOD analysis, we sample combinations of backgrounds and co-occurring objects, called the BG & CoObj setting, as well as backgrounds alone, referred to as the BG setting. In the results section, we reported the more challenging scenario(BG & CoObj) as it presented greater difficulty, as expected. + +- **AnimalsMetaCoCo:** To create a multi-class, multi-valued spurious attribute setting that reflects realistic and challenging scenarios, we selected a subset of 26 animal classes from the MetaCoCo dataset [71]. The samples were first cleaned through relabeling, removal of duplicates, and deletion of irrelevant images. Subsequently, we defined subattributes to create a total of 8 major concepts for modeling attribute imbalance and spurious features, as illustrated in the first Figure of the paper. We introduce this new dataset as **AnimalsMeta-CoCo**, a refined subset tailored for multi-class, multi-valued spurious feature scenarios. Details of the classes, attributes, and group sample distributions are provided in Table 8. This dataset represents a more realistic scenario, where each class is associated with at least one spurious feature similar to those present in ID data. While the strength of spurious correlations may not be as pronounced as in the other three datasets, AnimalsMetaCoCo introduces several unique and challenging aspects. Specifically, its spurious attributes can take on multiple distinct values, each exhibiting varying degrees of imbalance across classes. Moreover, the class distributions themselves are inherently imbalanced. To construct SP-OOD scenarios, we adopt a leave-2-out strategy: in each round, two classes are treated as SP-OOD, characterized by semantic shifts while retaining at least one spurious attribute shared with ID data, and the remaining classes serve as ID. This setup significantly increases the difficulty of the detection task due to overlapping spurious patterns across environments. + +- Sp-ImageNet100: We evaluate models on the Spurious ImageNet (SpI) dataset introduced in [36]. This dataset contains real-world images (from OpenImages [77] and Flickr) that include only spurious features, such as bird feeders or graffiti, without the actual class object. These images are consistently misclassified as specific ImageNet classes, revealing harmful spurious correlations. + +SpI focuses on 100 ImageNet classes (we call Sp-ImageNet100) where such correlations are prevalent. The authors distinguish between two types of harmful spurious features: + +- Spurious Class Extension: A spurious feature alone causes a class prediction (e.g., bird feeder $\rightarrow$ hummingbird). +- Spurious Shared Feature: A feature shared across classes biases prediction toward one due to imbalance (e.g., water jet $\rightarrow$ fireboat over fountain). + +We note that most spurious features in SpI are of the class extension type. This is less aligned with our goal of identifying underrepresented groups based on shared spurious features and misclassification signals [20], and without group supervision. + +Nevertheless, we include SpI as a challenging, naturally occurring OOD benchmark for evaluating model robustness to spurious correlations. + +For NSP-OOD, we utilized the SVHN [78], LSUN [79], and iSUN [80] datasets, which are commonly employed in prior works [14, 25]. These datasets are used consistently as NSP-OOD for all the mentioned ID datasets. + +# E Performance on Non-Spurious OOD Datasets + +We evaluate the performance of post-hoc methods in the non-spurious OOD (NSP-OOD) detection setting. For this evaluation, the OOD samples are drawn from a collection of standard benchmark datasets: SVHN [78], LSUN (resized)[79], and iSUN[80]. The ID context for each evaluation varies across the five datasets used in our primary spurious correlation experiments: Waterbirds (WB), CelebA (CA), UrbanCars (UC), Animals MetaCoCo (AMC), and Spurious ImageNet (SpI). Specifically, for each of these five ID contexts, the pretrained ResNet-50 backbone is employed, and + +![](images/6078e5bfd21ef78bc8bef5791643ab73b45a65fd8b1b2ed11a893f7a5f55266f.jpg) +y: waterbird e: water + +![](images/da7480fd0700f7a9d70ae6e668ddc95e98dcfe45e8512836a017a45a33126d47.jpg) +y: waterbird e: land + +![](images/11bb14ceaa5822091afcbb69c3f718bfa369db9d9c9ebf9545994648fbc33e92.jpg) +Spurious OOD-land + +![](images/9fde9d8fba10257f3ba92a48fde35857d9737e4eecac8c831529b8e2ae33732b.jpg) +y:landbird e:water + +![](images/e9af5dba9791fc3bcaa8b4bf1185b41853c7972907108edd9da18e19221bee0e.jpg) +y:landbird e: land + +![](images/ecfb722313bc542c257a906bf8465b416928c14e57cfa3be70c8a5942d0bb564.jpg) +Spurious OOD-water +Figure 9: Representative examples from the Waterbirds [17] dataset. The dataset consists of four groups: (Waterbird, Water), (Waterbird, Land), (Landbird, Water), and (Landbird, Land). Minority groups, indicated with red borders, are underrepresented, while majority groups, indicated with green borders, are more prevalent. Spurious OOD samples include only background features (land or water) without core bird-related features. + +![](images/2e869c375c479e5c4afac4a4e367a704a6b1dda2935a093fc523e622d519a2e4.jpg) +y: blond hair e: female + +![](images/2390a7954146e401dfd6905de17cdd49a4415b6882425dc69821299e8342bb8c.jpg) +y: blond hair e: male + +![](images/fca68c1e29fed60728827d78f0bb64991e5a89363ef7b5412cab087808e090ae.jpg) +Spurious OOD-female + +![](images/dbdb7ee9a4eeb1291ed27b9ae708bc493a5937bd22572f4299c0701781515592.jpg) +y: non-blond hair e: female +Figure 10: Representative examples from the CelebA [34] dataset, which is divided into four groups: (Blond hair, Female), (Blond hair, Male), (Non-blond, Female), and (Non-blond, Male). Minority groups, marked with red borders, are underrepresented, while majority groups, highlighted with green borders, are more prevalent. Spurious OOD samples contain only spurious features (gender) without core hair color attributes. + +![](images/c52b6fce3d921ea05f1bcf266d480d4a5fd95887af0aeb7f5d940689eded75d5.jpg) +y: non-blond hair e: male + +![](images/32c14b9e36084e91fe1a9691663c07d3b98105afa28414978eb0a32c6daa1bd4.jpg) +Spurious OOD-male + +Table 5: Group-wise distribution of the Waterbirds [17] training set across land and water attributes at varying correlation levels. The distribution reflects the degree of alignment between bird classes (landbird or waterbird) and their respective backgrounds, with higher correlation level indicating a stronger dependence between the bird label and the background. + +
CorrelationSP-FeatureLandbirdWaterbirdTotal (Row)
50%Land54418532397
Water54518532398
Total (Col)108937064795
90%Land9973691366
Water11133183429
Total (Col)110836874795
+ +Table 6: Distribution of the CelebA [34] training set across male and female attributes at varying correlation levels. The correlation levels indicate the strength of the spurious relationship between gender and the presence of blond hair, with higher correlations reflecting a stronger association between these attributes in the dataset. + +
CorrelationSP-FeatureBlondNon-blondTotal (Row)
50%Male138713872774
Female138713872774
Total (Col)277427745548
90%Male29624682764
Female24743102784
Total (Col)277027785548
+ +Table 7: Distribution of UrbanCars [35] training samples with $95\%$ correlation across groups, categorized by background and co-occurring object features within each class. This dataset features six minority groups out of eight possible combinations, highlighting its challenging nature due to the underrepresentation of most groups. + +
SP-FeaturesCountry CarUrban CarTotal (Row)
Country BG, Country CoObj3606103616
Country BG, Urban CoObj190190380
Urban BG, Country CoObj190189379
Urban BG, Urban CoObj1036053615
Total (Col)399639947990
+ +the post-hoc OOD detection methods are subsequently set up using the features derived from this backbone. + +The performance metrics reported in Tables 9 and 10 are the average performance when distinguishing each ID dataset from the NSP-OOD samples, averaged across SVHN, LSUN, and iSUN. This setup assesses the general OOD detection capability of the post-hoc methods when the primary challenge is not spurious correlations shared between ID and OOD, but the ID datasets still contain inherent biases. As observed in the tables, distance-based methods such as SPROD and KNN consistently achieve near-perfect scores across all metrics (AUROC, FPR@95, AUPR-In, and AUPR-Out) and ID contexts. Other methods like VIM and MDS also show strong performance. + +# F Ablation Study on SPROD Stages + +To understand the contribution of each stage in our proposed SPROD method, we conduct an ablation study. We evaluate the performance of: + +Table 8: Group-wise distribution of 26 selected animal classes from the MetaCoCo [71] dataset called Animals MetaCoCo across various environments. + +
Class \Attributeat homeautumndimgrassin cageon snowrockwaterTotal
bear01097621909898362962
cat284159150439951021213861736
cow571371196930122382211387
crab00437300153102371
dog1062211745611112081454401966
dolphin0011300015425553
elephant10898247434059383751359
fox019562367661601361451131
frog02323530003223421429
giraffe047915039700782271331
goose0105135378540832631018
horse573661286720129594571868
kangaroo0841902140556197701
lion05605853736792751711716
lizard0130423033503022421054
monkey018384592761004632551753
ostrich01641252351597673144976
owl014114713136927887712
rabbit01473163710513491731218
rat110001235241066392
seal05731158192662405471318
sheep0626308650992072372064
squirrel02123241801321881181100
tiger021214435661762363231462
tortoise012914028400157234944
wolf019297198901511881881104
Total722493824219893100022793845652731625
+ +![](images/69361d62f2b416b61a017f5c7f0a815adb210b35dfcafe26337b99c9aa436e58.jpg) +y: country car +e1: country bg +e2: country ob + +![](images/3ab59de27a623fe4cdd1a09938ff546c07d0f098461cfa7d7ce9a6ab1b5f9b8a.jpg) + +![](images/0eda314f704744c0fd5e45e8f35d7141e1f8ea3dc6c34bf84f015fee241ff1cb.jpg) + +![](images/0c6254d04908d2d5fe6928bb750f92c8df256780102b002b8877086bcd1aa0da.jpg) + +![](images/f8f145b8eeb8942f06b202f70cc38a39e4137c852f61e190f0de6247306cc701.jpg) +Spurious OOD - BG urban bg + +![](images/f103d09c9589fdc9285a87db186e1f7f450d0f5bda6b3bf88f9c96354613aa1b.jpg) +BG & CoObj urban $\mathrm{bg + }$ urban obj + +![](images/e30797841c5e5d71ea6585bb46b50ef09c066995ee1b79377b47d4e52b9494f5.jpg) +y: urban car +e1: country bg +e2: country obj +Figure 11: Representative examples from the UrbanCars [35] dataset, which has 8 groups, each pairing country and urban car classes with two spurious features: background and co-occurring object, both of which can be urban or country related. The "all-country" and "all-urban" groups (green borders) dominate the dataset, while the remaining groups are significantly underrepresented (red borders). Spurious OOD samples, containing only spurious features (no cars), consist of combinations of background and co-occurring objects (BG & CoObj) as well as background-only (BG) samples. + +![](images/9365f0185c6b73627092569489a74dc0f854f25c405543c5baf0aa89544243f0.jpg) +y: country car +e1: country bg +e2: urban obj +y: urban car +e1: country bg +e2: urban obj + +![](images/b3b80dadd15d73ba350aa4ed129b81162e1e53a85dd34abf40f1f1145b664b2f.jpg) +y: country car +e1: urban bg +e2: country obj +y: urban car +e1: urban bg +e2: country obj + +![](images/f2b54297eaa72564d7033991e873b994cd308e0e2e8b17dfa906e4068b330543.jpg) +y: country car +e1: urban bg +e2: urban obj +y: urban car +e1: urban bg +e2: urban obj + +![](images/8670c3a48b3e22fcbe0f86764e0ed1a5f7db5a8b21eb38c7c9ab29b4a04abde9.jpg) +Spurious OOD - BG country bg + +![](images/be026d6ceb3aa8c4167b4daaad64794d75bf0a3483a03c93f437a36f707441be.jpg) +BG & CoObj country bg + country obj + +Table 9: Average NSP-OOD detection performance (AUROC and FPR@95) using a ResNet-50 backbone. Cell values are performance metrics averaged over SVHN, LSUN, and iSUN as OOD. + +
AUROC↑
MethodWBCAUCAMCSpIAvg.
MSP86.2±1.077.5±0.952.6±4.092.6±0.294.9±0.380.8
Energy85.0±6.475.4±6.353.2±8.096.8±0.197.7±0.381.6
MLS85.7±4.276.4±4.153.0±6.396.2±0.196.3±0.381.5
KLM57.7±2.052.5±2.147.5±1.690.1±0.396.1±0.568.8
GNorm90.7±0.670.8±0.964.1±2.197.3±0.199.0±0.084.4
ReAct86.8±5.266.1±7.550.0±8.796.3±0.199.0±0.379.6
VIM100.0±0.0100.0±0.099.6±0.499.9±0.099.6±0.099.8
MDS100.0±0.0100.0±0.0100.0±0.099.4±0.097.5±0.099.4
RMDS58.6±0.468.6±2.948.9±0.693.0±0.297.1±0.173.2
KNN100.0±0.0100.0±0.0100.0±0.0100.0±0.099.8±0.0100.0
SHE96.4±0.298.6±0.299.1±0.194.3±0.298.5±0.197.4
SPROD100.0±0.0100.0±0.0100.0±0.0100.0±0.099.9±0.0100.0
+ +
MethodWBCAUCAMCSpIAvg.
MSP76.5±1.685.2±2.795.6±0.446.1±1.534.8±3.167.6
Energy74.5±24.884.4±14.590.8±9.719.9±1.113.4±2.256.6
MLS76.3±15.086.9±9.693.8±5.026.3±1.324.5±2.561.6
KLM80.5±1.484.9±2.695.5±0.557.8±1.425.5±6.268.8
GNorm66.4±2.785.2±2.193.2±0.613.8±0.64.8±0.352.7
ReAct73.5±23.488.0±10.791.6±6.723.3±0.84.3±2.056.1
VIM0.0±0.00.0±0.01.9±1.60.0±0.00.0±0.00.4
MDS0.0±0.00.0±0.00.0±0.00.7±0.011.9±0.42.5
RMDS90.5±0.389.2±1.795.4±0.451.3±1.617.7±0.468.8
KNN0.0±0.00.0±0.00.0±0.00.0±0.00.6±0.00.1
SHE15.9±0.67.0±0.84.9±0.429.8±1.07.7±0.813.1
SPROD0.0±0.00.0±0.00.0±0.00.0±0.00.0±0.00.0
+ +Table 10: Average NSP-OOD detection performance (AUPR-In and AUPR-Out) using a ResNet-50 backbone. Cell values are performance metrics averaged over SVHN, LSUN, and iSUN as OOD. AUPR-IN↑ AUPR-OUT↑ + +
MethodWBCAUCAMCSpIAvg.
MSP95.8±0.367.7±1.443.4±3.996.0±0.195.6±0.279.7
Energy95.5±1.866.2±5.044.2±6.298.3±0.197.9±0.280.4
MLS95.7±1.366.5±4.043.9±5.698.0±0.196.8±0.280.2
KLM77.1±1.127.8±1.034.4±1.393.9±0.296.6±0.466.0
GNorm97.2±0.249.7±1.159.6±2.098.4±0.199.0±0.080.8
ReAct96.1±1.553.9±4.640.6±5.998.0±0.099.0±0.377.5
VIM100.0±0.0100.0±0.099.2±0.7100.0±0.099.7±0.099.8
MDS100.0±0.0100.0±0.0100.0±0.099.7±0.098.0±0.099.5
RMDS78.0±0.246.1±3.935.8±0.596.4±0.197.4±0.170.7
KNN100.0±0.0100.0±0.0100.0±0.0100.0±0.099.8±0.0100.0
SHE98.7±0.196.0±0.498.6±0.189.6±0.598.6±0.196.4
SPROD100.0±0.0100.0±0.0100.0±0.0100.0±0.099.9±0.0100.0
+ +
MethodWBCAUCAMCSpIAvg.
MSP55.8±1.985.9±0.765.0±2.285.4±0.693.9±0.577.2
Energy56.4±18.083.6±6.866.3±7.992.6±0.497.5±0.379.3
MLS56.3±12.184.7±4.465.7±5.591.3±0.495.7±0.478.7
KLM38.4±1.776.9±0.962.7±0.679.6±0.694.9±0.670.5
GNorm65.1±1.783.8±0.771.1±1.594.2±0.298.9±0.082.6
ReAct57.9±15.978.6±7.465.0±8.091.8±0.299.0±0.378.5
VIM100.0±0.0100.0±0.099.8±0.299.9±0.099.2±0.099.8
MDS100.0±0.0100.0±0.0100.0±0.098.8±0.096.5±0.099.1
RMDS31.3±0.381.8±1.864.0±0.683.5±0.596.3±0.171.4
KNN100.0±0.0100.0±0.0100.0±0.0100.0±0.099.8±0.0100.0
SHE91.6±0.499.4±0.199.5±0.097.0±0.198.3±0.197.2
SPROD100.0±0.0100.0±0.0100.0±0.099.9±0.0100.0±0.0100.0
+ +- Stage 2: Classification-Aware Prototype Calculation +- Stage 3: Group Prototype Refinement + +For this analysis, we select the Waterbirds dataset, as its synthetic nature allows for precise control over spurious correlations and clearly exemplifies the SP-OOD challenge by design. We evaluate on two versions of Waterbirds: one with a $50\%$ spurious correlation rate (where spurious features are less effective) and another with a $90\%$ correlation rate (representing a strong spurious bias). Experiments are conducted using features from pretrained ResNet-50 and ResNet-18 backbones, without fine-tuning on Waterbirds, to isolate the effect of the prototype refinement stages. + +The results of this ablation study are presented in Figure 12 and Figure 13. As seen, the performance of the simple initial prototypical approach (Stage 1) performs competitively, especially on the $50\%$ correlation setting. This suggests that a basic prototypical method, which computes distances to class means in feature space, is a competitive baseline for OOD detection that has been somewhat overlooked in existing literature. When the spurious correlation rate is increased to $90\%$ , we observe a general reduction in OOD detection performance across all three variants. This is expected, as stronger spurious correlations make it more difficult to distinguish true class features from misleading cues. However, Stage 3 (the full SPROD method) is significantly more robust to the increase of spurious correlation. + +# G Backbone Experiments + +To assess the generality and robustness of SPROD across different neural network architectures, we evaluate its performance using a wide range of feature backbones. This analysis complements the main paper's results, which primarily rely on ResNet-50 [68], and demonstrates that our method remains effective across both convolutional and transformer-based representations. + +We include a comprehensive selection of backbones commonly used in the OOD detection literature. This includes all major variants of the ResNet [68] family, ResNet-18 (R18), ResNet-34 (R34), ResNet-50 (R50), and ResNet-101 (R101), owing to their widespread adoption and varying representational capacities. Alongside these, we evaluate several modern transformer-based architectures with diverse embedding sizes and training paradigms, such as the self-supervised DINOv2-S (DINOv2) [81], the standard supervised ViT-S (ViT) [82], and more hierarchical or data-efficient + +![](images/20c308d0ddb0c88eef28644876827d0fcd7ee67f8ba825543919c9d574ddb9b7.jpg) +(a) AUROC + +![](images/741507109df9bf27b0212a18434051e20f31919d448476bb221f4e7ff54ee32f.jpg) +(b) FPR@95 +Figure 12: Ablation study of SPROD stages on the Waterbirds dataset using ResNet-50 features. Results are shown for two spurious correlation settings: $50\%$ and $90\%$ . + +![](images/651e340332306c9b30048033c927cff711572ea3d10f71e5c93f42360b89785d.jpg) +(a) AUROC +Figure 13: Ablation study of SPROD stages on the Waterbirds dataset using ResNet-18 features. Results are shown for two spurious correlation settings: $50\%$ and $90\%$ . + +![](images/78585348fd084e7fb867f26be76cfa01a1c26baaa4cd8843f15c8e4eb2b6be34.jpg) +(b) FPR@95 + +designs like Swin-B (Swin) [83], DeiT-B (DeiT) [82], ConvNeXt-B (CvNXt) [84], and BiT-R50x1 (BiT) [85]. + +We analyze the performance of each method using four complementary metrics (AUROC, FPR@95, AUPR-IN, and AUPR-OUT) to evaluate different aspects of OOD detection performance. Tables 11, 12, 13, and 14 summarize the performance of our post-hoc method across various backbone architectures, averaged over the five SP-OOD datasets studied in this work. Detailed results for each individual backbone across the different datasets are presented in the subsequent tables. + +Overall, we observe that SPROD consistently achieves strong performance across all backbone architectures, often with a notable margin. KNN emerges as the second-best method, suggesting that simpler approaches can be counterintuitively effective; however, its performance is highly sensitive to the choice of hyperparameters. MDS is also among the top-performing methods; it is a metric-based approach similar to SPROD but employs a more complex model by estimating class-specific covariance matrices. This added complexity, while potentially beneficial, may increase the risk of overfitting, especially in OOD settings. In addition, output-based methods show notable drops in performance on several benchmarks, highlighting their susceptibility to distributional shifts. Notably, SPROD is the only method that maintains stable performance across all evaluated settings, without experiencing major degradation under any backbone or dataset configuration. Among the evaluated backbones, ConvNeXt-B and Swin-B stand out as frozen feature extractors with superior performance for post-hoc OOD detection. + +Table 11: AUROC performance of all evaluated methods across various backbone architectures. Results are averaged over five SP-OOD datasets. + +
MethodR18R34R50R101DINOv2ViTSwinDeiTCvNXtBiTAvg.
MSP61.161.961.963.370.265.470.869.772.065.066.1
Energy61.561.661.362.770.565.870.470.071.564.766.0
MLS61.761.961.663.170.565.870.770.171.364.866.2
KLM56.658.860.761.570.264.568.662.469.360.663.3
GNorm64.364.264.766.469.966.456.762.167.854.263.7
ReAct62.462.164.764.170.067.372.169.870.760.566.4
VIM66.168.069.372.178.273.577.672.980.077.873.5
MDS65.670.572.274.283.182.980.071.384.884.276.9
RMDS54.857.260.258.363.361.169.665.569.566.562.6
KNN77.778.680.379.979.780.384.378.986.181.680.7
SHE69.472.468.473.583.079.884.981.884.673.577.1
SPROD83.083.285.185.987.285.190.184.689.887.186.1
+ +Table 12: FPR@95 performance of all evaluated methods across various backbone architectures. Results are averaged over five SP-OOD datasets. + +
MethodR18R34R50R101DINOv2ViTSwinDeiTCvNXtBiTAvg.
MSP90.888.988.486.581.186.471.780.467.884.282.6
Energy88.586.988.987.577.984.973.778.668.282.881.8
MLS89.687.488.586.878.384.971.578.267.882.781.6
KLM91.390.388.387.480.685.772.680.566.883.582.7
GNorm86.184.083.882.980.285.379.282.370.787.782.2
ReAct85.684.484.083.578.386.872.678.369.486.180.9
VIM83.081.978.574.356.775.068.382.647.461.871.0
MDS82.080.873.567.751.461.970.782.650.448.466.9
RMDS93.292.291.689.481.083.872.985.267.782.283.9
KNN64.762.458.457.351.866.753.471.946.951.058.5
SHE68.265.770.567.054.263.146.466.950.574.062.6
SPROD53.154.649.046.643.051.135.355.135.242.946.6
+ +# H Analysis of Mixture of Prototypes + +To evaluate the role of prototype augmentation and refinement, here we analyze two new variants of SPROD, comparing them against our standard method (referred to as SPROD-Default in this section). The first is SPROD-KMeans, a clustering baseline where embeddings of training samples within each class are clustered using K-Means. The number of centroids per class is set to match the number of group prototypes that SPROD-Default would derive. This tests whether the benefits of SPROD-Default stem solely from using multiple prototypes per class or from its proposed classification-driven refinement strategy. Recent works, such as Prototypical Learning with a Mixture of Prototypes (PALM) [56], demonstrate the importance of multiple prototypes to capture intra-class diversity. SPROD-Kmeans serves as a simpler post-hoc method based on the K-Means algorithm in this context. + +We also introduce SPROD-Converged, which iteratively refines the group prototypes derived from SPROD-Default's Stage 2 by repeatedly applying the Stage 3 reassignment and recalculation steps until centroid convergence, similar to the K-Means algorithm. The centroid initialization mechanism distinguishes SPROD-Converged from SPROD-Kmeans, where K-Means clustering is typically initialized using a method like K-Means++ directly on the raw class samples. + +As shown in Tables 35 and 36, SPROD-Default generally outperforms the two other variants when the amount of spurious correlation in the ID data is high (e.g., Waterbirds, CelebA, and UrbanCars); on other datasets, their performances are competitive. Proto-KMeans is slightly less effective in separating SP-OOD samples, likely due to its reliance on purely geometric clustering, whereas SPROD with classification-aware refinement slightly enhances the robustness when the spurious correlation is high. + +Table 13: AUPR-IN performance of all evaluated methods across various backbone architectures. Results are averaged over five SP-OOD datasets. + +
MethodR18R34R50R101DINOv2ViTSwinDeiTCvNXtBiTAvg.
MSP48.449.349.850.959.354.461.159.064.151.354.8
Energy47.948.348.750.159.954.859.358.763.150.454.1
MLS48.448.649.350.659.254.860.159.063.050.654.4
KLM41.043.845.245.256.149.352.448.452.845.548.0
GNorm50.350.251.051.358.555.538.842.951.540.549.0
ReAct49.950.752.753.159.656.163.059.362.647.855.5
VIM51.653.955.459.766.762.168.963.172.564.461.8
MDS52.260.261.664.171.874.572.060.877.075.367.0
RMDS42.245.345.145.550.249.559.053.759.248.849.9
KNN63.666.068.067.764.670.874.669.276.569.869.1
SHE52.857.252.556.675.171.575.073.580.161.265.5
SPROD72.774.076.577.178.977.184.176.584.378.378.0
+ +Table 14: AUPR-OUT performance of all evaluated methods across various backbone architectures. Results are averaged over five SP-OOD datasets. + +
MethodR18R34R50R101DINOv2ViTSwinDeiTCvNXtBiTAvg.
MSP72.273.073.074.379.975.381.378.981.976.376.6
Energy73.173.673.074.080.476.181.580.081.976.677.0
MLS72.973.573.174.280.476.081.679.881.876.677.0
KLM70.871.573.073.879.374.579.675.480.974.075.3
GNorm74.374.875.076.179.576.076.176.679.271.875.9
ReAct73.874.175.275.480.276.782.180.081.473.577.2
VIM77.879.080.181.886.682.584.880.289.286.182.8
MDS77.879.781.382.688.787.983.978.390.388.583.9
RMDS68.970.471.872.476.474.079.875.780.774.974.5
KNN85.685.987.487.789.486.990.785.891.189.588.0
SHE80.182.679.182.688.386.490.784.687.682.484.4
SPROD88.888.790.190.792.990.393.990.793.892.791.3
+ +# I Reproducibility and Resources + +The data (cleaned AnimalsMetaCoCo dataset) and code for our approach are available at the following GitHub repository: https://github.com/ReihanehZohrabi/SPROD. In our benchmarking, we utilized components of the OpenOOD v1.5 [86, 37, 87] framework to obtain results for previously proposed OOD detection methods. + +Our experiments were designed to be post-hoc and computationally efficient. All experiments were conducted on a single GeForce RTX 3090 Ti GPU, demonstrating the method's practicality in terms of resource requirements. + +Table 15: AUROC and FPR@95 performance of all methods using ResNet-18 as the feature backbone. AUROC↑ FPR@95↓ + +
MethodWBCAUCAMCSpIAvg.
MSP68.1±0.747.6±2.940.2±0.572.4±0.677.3±0.261.1
Energy69.4±4.348.5±4.243.2±5.071.0±0.875.6±0.361.5
MLS69.2±3.648.1±4.042.7±3.871.6±0.876.8±0.361.7
KLM43.9±0.541.1±1.457.7±0.468.2±0.572.3±0.456.6
GNorm81.9±0.342.0±2.448.3±0.467.5±0.682.0±0.264.3
ReAct75.9±4.140.8±3.744.0±5.571.3±0.879.8±0.462.4
VIM77.2±3.052.7±3.863.9±3.871.8±0.864.8±0.266.1
MDS74.4±0.158.9±0.589.5±0.160.1±0.844.9±0.165.6
RMDS46.3±0.137.1±0.744.7±0.274.8±0.571.0±0.154.8
KNN93.9±0.058.8±0.492.1±0.873.3±0.670.3±0.177.7
SHE91.0±0.147.4±0.274.2±0.152.3±0.982.3±0.269.4
SPROD98.3±0.063.6±1.097.3±0.075.5±0.580.1±0.083.0
+ +
MethodWBCAUCAMCSpIAvg.
MSP89.7±0.698.0±0.595.5±0.689.2±0.481.7±0.890.8
Energy85.7±8.096.9±1.091.3±5.190.8±0.477.9±1.388.5
MLS86.7±6.297.8±0.693.0±3.690.3±0.480.3±0.689.6
KLM91.2±0.598.0±0.595.5±0.685.8±0.486.0±1.691.3
GNorm85.9±0.898.1±0.495.1±0.591.4±0.460.1±1.086.1
ReAct82.0±9.198.0±0.892.1±4.590.9±0.564.8±1.585.6
VIM69.0±5.695.1±1.071.4±5.789.8±0.589.8±0.583.0
MDS69.1±0.194.5±0.558.5±0.592.8±0.394.9±0.082.0
RMDS95.0±0.099.0±0.397.2±0.085.8±0.789.2±0.293.2
KNN25.1±0.193.7±1.141.7±2.385.7±0.777.2±0.264.7
SHE27.7±0.393.9±0.868.4±0.494.3±0.456.6±0.568.2
SPROD7.3±0.191.8±1.118.8±0.479.5±0.667.9±0.253.1
+ +Table 16: AUROC and FPR@95 performance of all methods using ResNet-34 as the feature backbone. AUROC† FPR@95↓ + +
MethodWBCAUCAMCSpIAvg.
MSP66.0±0.741.9±1.542.7±0.677.5±0.481.6±0.261.9
Energy64.9±6.843.2±4.842.5±4.478.2±0.679.4±0.261.6
MLS65.0±5.742.6±4.142.5±3.478.4±0.680.8±0.261.9
KLM42.2±0.845.0±1.356.2±0.572.2±0.578.2±0.858.8
GNorm77.9±0.435.4±0.950.0±0.573.7±0.484.1±0.164.2
ReAct72.3±5.934.4±5.943.5±5.477.9±0.682.4±0.462.1
VIM80.8±4.350.3±3.262.4±3.375.5±0.770.9±0.268.0
MDS85.7±0.060.8±0.889.8±0.161.4±0.854.8±0.170.5
RMDS52.4±0.133.3±1.244.1±0.279.3±0.677.0±0.157.2
KNN97.8±0.056.4±0.788.8±0.275.9±0.773.9±0.178.6
SHE87.9±0.240.8±0.381.2±0.268.5±0.283.4±0.272.4
SPROD98.2±0.061.4±1.095.3±0.178.1±0.683.0±0.183.2
+ +
MethodWBCAUCAMCSpIAvg.
MSP91.2±0.498.4±0.495.5±0.785.9±0.573.7±2.588.9
Energy87.1±0.297.3±1.393.1±4.885.3±0.771.6±2.086.9
MLS88.6±7.098.0±0.894.1±3.784.7±0.771.8±1.387.4
KLM92.9±0.598.4±0.595.5±0.884.1±0.680.6±2.790.3
GNorm89.2±0.598.5±0.595.3±0.585.0±0.552.1±0.984.0
ReAct86.2±9.896.3±2.394.5±4.085.5±0.759.6±1.784.4
VIM64.6±8.595.4±0.972.2±4.789.7±0.587.6±0.181.9
MDS61.9±0.195.1±0.459.6±0.393.8±0.393.8±0.080.8
RMDS94.2±0.199.1±0.196.4±0.183.5±0.988.0±0.192.2
KNN9.3±0.194.9±0.850.2±0.383.9±0.973.6±0.162.4
SHE32.7±0.396.6±0.260.5±0.785.1±0.453.4±0.865.7
SPROD7.5±0.194.5±0.731.6±0.578.4±0.960.8±0.354.6
+ +Table 17: AUROC and FPR@95 performance of all methods using ResNet-50 as the feature backbone. AUROC† FPR@95↓ + +
MethodWBCAUCAMCSpIAvg.
MSP62.3±0.646.0±1.438.5±0.379.7±0.483.1±0.361.9
Energy62.0±2.645.4±3.438.4±2.179.9±0.680.6±0.461.3
MLS62.2±2.345.3±3.238.4±1.480.2±0.681.9±0.361.6
KLM51.2±0.741.7±2.557.0±0.274.2±0.679.6±0.860.7
GNorm79.5±0.438.0±1.346.6±0.474.2±0.585.2±0.264.7
ReAct72.9±3.645.6±5.341.3±3.180.1±0.683.6±0.764.7
VIM79.6±2.550.4±3.160.7±1.778.6±0.677.4±0.969.3
MDS90.2±0.157.8±0.591.8±0.162.9±0.858.4±0.172.2
RMDS59.4±0.133.6±1.447.4±0.281.9±0.468.8±0.158.2
KNN98.6±0.054.5±0.591.1±0.179.7±0.677.4±0.080.3
SHE88.3±0.242.7±0.673.2±0.154.8±0.783.0±0.168.4
SPROD98.8±0.061.6±0.997.4±0.082.4±0.585.3±0.085.1
+ +
MethodWBCAUCAMCSpIAvg.
MSP87.9±0.898.7±0.597.3±0.383.8±0.774.1±1.288.4
Energy89.2±3.298.6±0.795.5±3.184.8±0.876.3±0.988.9
MLS88.1±2.098.8±0.696.7±2.084.4±0.874.6±0.988.5
KLM89.1±0.798.7±0.597.1±0.380.5±0.876.1±1.788.3
GNorm84.2±0.798.8±0.497.1±0.184.2±0.654.7±0.683.8
ReAct86.9±7.096.5±2.495.5±3.283.9±0.857.5±1.684.0
VIM61.4±3.596.2±0.469.0±1.586.6±0.779.5±0.578.5
MDS49.2±0.296.0±0.539.0±0.393.0±0.390.5±0.173.5
RMDS91.7±0.299.6±0.195.3±0.183.4±0.988.1±0.191.6
KNN4.8±0.194.4±1.042.5±0.279.9±1.170.4±0.258.4
SHE33.2±0.596.4±0.576.5±0.293.9±0.352.6±0.870.5
SPROD4.7±0.193.7±0.919.0±0.469.5±1.258.0±0.149.0
+ +Table 18: AUROC and FPR@95 performance of all methods using ResNet-101 as the feature backbone. + +
AUROC↑
MethodWBCAUCAMCSpIAvg.
MSP65.5±0.644.9±3.141.3±1.281.2±0.583.8±0.263.3
Energy63.9±2.945.5±3.941.1±2.581.8±0.781.3±0.462.7
MLS64.3±2.345.3±3.741.2±2.182.0±0.782.7±0.263.1
KLM49.7±0.745.0±2.556.4±1.276.1±0.680.1±0.361.5
GNorm76.5±0.545.4±1.850.0±0.975.1±0.685.0±0.266.4
ReAct72.0±3.739.0±7.943.5±3.881.3±0.984.9±0.764.1
VIM88.5±1.748.3±3.763.8±1.479.4±0.780.5±0.672.1
MDS96.1±0.053.4±0.493.2±0.063.7±0.964.7±0.174.2
RMDS60.4±0.131.1±0.443.2±0.183.8±0.573.2±0.158.3
KNN98.9±0.049.6±5.389.5±0.181.5±0.680.2±0.179.9
SHE83.5±0.464.6±1.080.3±0.257.9±1.181.0±0.273.5
SPROD99.0±0.062.8±1.297.9±0.083.1±0.586.8±0.185.9
+ +
FPR@95↓
MethodWBCAUCAMCSpIAvg.
MSP87.7±0.798.9±0.395.9±0.581.0±0.969.0±1.186.5
Energy90.4±5.698.2±0.995.2±2.279.2±1.274.4±1.087.5
MLS89.4±3.298.8±0.495.7±1.679.2±1.170.9±0.786.8
KLM89.0±0.798.9±0.395.8±0.578.6±0.974.5±1.187.4
GNorm85.7±0.998.8±0.495.7±0.680.2±0.853.9±1.382.9
ReAct86.9±8.797.2±1.794.6±2.979.9±1.159.0±2.283.5
VIM42.2±4.297.2±0.668.3±2.687.0±0.876.9±0.974.3
MDS23.6±0.295.8±0.640.4±0.392.9±0.385.8±0.167.7
RMDS89.3±0.299.5±0.196.0±0.179.4±1.282.7±0.289.4
KNN3.1±0.194.9±1.343.2±0.276.8±1.368.3±0.157.3
SHE41.5±0.689.0±1.159.4±0.790.1±0.755.0±0.467.0
SPROD3.3±0.194.8±1.111.4±0.270.6±1.252.9±0.246.6
+ +Table 19: AUROC and FPR@95 performance of all methods using DINOv2-S as the feature backbone. AUROC↑ FPR@95↓ + +
MethodWBCAUCAMCSpIAvg.
MSP80.1±2.034.0±3.364.3±0.587.6±0.484.9±0.470.2
Energy77.4±3.036.5±9.263.2±9.988.8±0.486.6±0.870.5
MLS78.0±2.635.9±8.963.5±9.188.8±0.486.5±0.670.5
KLM64.7±1.454.1±3.665.2±6.382.8±0.784.4±0.370.2
GNorm82.7±1.532.8±2.463.5±0.583.4±0.786.9±0.669.9
ReAct76.5±2.835.2±8.262.9±9.688.7±0.486.7±0.770.0
VIM97.8±0.439.0±8.283.0±7.286.5±0.784.9±0.378.2
MDS99.1±0.055.0±0.794.1±0.083.8±0.883.7±0.183.1
RMDS65.2±0.126.4±0.950.3±0.289.8±0.685.0±0.163.3
KNN99.3±0.043.4±0.583.9±0.491.2±0.380.5±0.879.7
SHE99.1±0.054.6±0.992.5±0.184.3±1.184.7±0.283.0
SPROD99.6±0.063.4±1.096.1±0.091.2±0.385.5±0.187.2
+ +
MethodWBCAUCAMCSpIAvg.
MSP80.1±2.699.0±0.291.8±1.671.6±1.263.0±2.881.1
Energy79.0±7.597.9±1.587.5±7.969.3±1.756.0±3.777.9
MLS78.2±7.298.4±1.088.0±7.669.3±1.657.8±3.178.3
KLM82.7±2.799.0±0.293.2±3.263.7±1.264.4±2.280.6
GNorm79.6±2.799.0±0.291.9±1.673.1±1.457.3±2.380.2
ReAct79.8±7.197.8±1.587.7±7.568.9±1.757.2±3.478.3
VIM6.0±1.797.4±1.156.6±13.167.1±2.256.6±1.156.7
MDS2.5±0.196.4±0.430.4±0.269.2±2.358.6±0.251.4
RMDS85.9±0.199.8±0.194.8±0.159.7±2.264.6±0.281.0
KNN1.6±0.196.9±0.444.3±1.448.8±2.167.6±2.551.8
SHE2.9±0.198.2±0.235.4±0.372.7±1.261.9±0.854.2
SPROD1.3±0.094.7±0.724.4±0.240.0±1.354.4±0.243.0
+ +Table 20: AUROC and FPR@95 performance of all methods using ViT-S as the feature backbone. AUROC↑ FPR@95↓ + +
MethodWBCAUCAMCSpIAvg.
MSP71.5±1.939.2±1.247.8±0.483.2±0.285.2±0.365.4
Energy69.0±9.840.3±10.548.9±5.085.4±0.485.6±0.765.8
MLS69.3±9.139.8±9.948.7±4.685.3±0.485.7±0.565.8
KLM62.6±2.550.6±4.747.2±5.677.9±0.584.4±0.664.5
GNorm76.0±1.438.4±1.047.9±0.683.0±0.486.7±0.366.4
ReAct70.3±7.849.1±9.649.0±5.284.9±0.483.4±1.067.3
VIM84.5±7.944.0±9.767.8±6.085.9±0.485.5±0.573.5
MDS95.1±0.059.6±1.094.3±0.081.6±0.583.9±0.082.9
RMDS60.5±0.129.2±1.042.1±0.187.2±0.386.4±0.161.1
KNN94.0±0.053.2±4.191.2±0.083.3±0.579.6±0.180.3
SHE94.5±0.251.5±0.691.2±0.278.3±2.083.4±0.279.8
SPROD97.0±0.060.9±1.196.3±0.085.0±0.386.1±0.185.1
+ +
MethodWBCAUCAMCSpIAvg.
MSP88.7±1.299.0±0.296.0±0.681.3±0.667.1±1.686.4
Energy87.9±8.498.3±1.593.6±3.778.2±1.266.5±2.184.9
MLS88.3±7.398.6±1.093.9±3.078.6±1.165.0±1.484.9
KLM89.7±2.199.1±0.296.2±2.578.3±0.765.4±2.185.7
GNorm88.4±1.299.0±0.295.9±0.580.6±0.862.5±0.785.3
ReAct86.4±11.297.0±2.994.1±3.079.6±1.176.7±2.386.8
VIM63.6±15.197.5±1.478.1±6.772.1±1.263.8±1.375.0
MDS30.9±0.195.6±0.342.3±0.275.6±1.265.1±0.261.9
RMDS91.2±0.199.8±0.196.8±0.171.0±1.760.1±0.183.8
KNN39.7±0.195.8±0.550.1±0.175.4±1.372.7±0.266.7
SHE28.6±1.898.2±0.242.2±1.680.2±1.666.3±1.463.1
SPROD9.1±0.295.0±0.925.0±0.166.6±1.160.0±0.251.1
+ +Table 21: AUROC and FPR@95 performance of all methods using Swin-B as the feature backbone. AUROC↑ FPR@95↓ + +
MethodWBCAUCAMCSpIAvg.
MSP82.8±0.930.9±2.057.2±0.492.7±0.290.6±0.470.8
Energy82.4±1.531.0±2.957.5±2.793.6±0.387.6±0.670.4
MLS82.6±1.330.9±2.857.5±2.292.9±0.389.6±0.470.7
KLM57.3±1.752.8±3.550.5±0.592.5±0.390.1±0.368.6
GNorm63.4±1.127.6±1.252.4±0.484.0±0.956.0±0.856.7
ReAct83.6±1.734.3±2.958.0±2.693.6±0.390.8±0.472.1
VIM90.6±0.339.8±3.276.9±1.488.1±0.392.6±0.177.6
MDS85.9±0.160.8±1.086.7±0.274.7±0.892.1±0.080.0
RMDS82.6±0.129.9±1.349.5±0.193.2±0.292.8±0.069.6
KNN90.4±0.055.4±0.891.8±0.292.5±0.391.3±0.084.3
SHE97.3±0.049.6±0.996.4±0.192.9±0.488.2±0.184.9
SPROD99.0±0.066.6±1.298.7±0.092.7±0.293.7±0.090.1
+ +
MethodWBCAUCAMCSpIAvg.
MSP76.1±2.799.7±0.193.0±0.651.8±2.038.1±2.071.7
Energy76.3±6.399.4±0.391.9±3.247.3±2.353.5±1.873.7
MLS75.7±4.799.6±0.292.9±2.648.5±2.241.0±1.471.5
KLM79.3±2.199.7±0.192.9±0.841.3±1.749.8±5.372.6
GNorm75.9±2.499.7±0.193.1±0.646.5±2.280.9±2.279.2
ReAct75.9±6.499.0±0.491.8±2.648.0±2.248.3±1.472.6
VIM58.2±2.097.7±0.665.0±1.982.7±1.237.8±0.668.3
MDS67.5±0.294.3±1.057.3±0.592.9±0.641.5±0.170.7
RMDS67.4±0.199.9±0.095.4±0.264.3±2.337.7±0.272.9
KNN43.5±0.193.9±0.542.2±0.447.9±2.039.7±0.253.4
SHE16.4±0.498.6±0.224.1±0.850.4±3.742.4±0.546.4
SPROD2.8±0.192.3±0.98.4±0.243.6±1.229.6±0.235.3
+ +Table 22: AUROC and FPR@95 performance of all methods using DeiT-B as the feature backbone. AUROC↑ FPR@95↓ + +
MethodWBCAUCAMCSpIAvg.
MSP75.6±0.943.4±2.253.3±0.188.5±0.487.8±0.469.7
Energy77.2±3.144.3±4.052.4±3.190.0±0.686.2±1.170.0
MLS76.9±2.544.0±3.752.6±2.589.6±0.687.4±0.470.1
KLM47.0±0.643.2±2.049.6±0.385.8±0.486.6±0.462.4
GNorm67.3±0.843.2±1.653.3±0.276.5±1.270.3±2.562.1
ReAct77.1±3.041.9±4.052.7±3.390.2±0.687.3±1.069.8
VIM82.7±1.746.2±3.365.6±2.482.5±0.487.4±0.272.9
MDS74.9±0.150.8±0.678.2±0.165.6±0.987.2±0.171.3
RMDS69.2±0.131.7±0.350.0±0.387.3±0.389.1±0.165.5
KNN90.0±0.646.4±4.387.6±0.083.9±0.686.5±0.178.9
SHE93.9±0.054.1±0.794.0±0.080.1±1.386.9±0.181.8
SPROD94.1±0.054.8±0.695.8±0.088.9±0.389.5±0.184.6
+ +
MethodWBCAUCAMCSpIAvg.
MSP86.7±0.698.9±0.294.9±0.658.9±1.762.5±2.380.4
Energy80.3±8.698.2±1.094.3±3.351.5±2.368.6±4.878.6
MLS82.0±5.998.5±0.694.7±2.652.7±2.263.2±2.978.2
KLM87.3±0.898.8±0.294.9±0.461.2±1.360.2±4.980.5
GNorm84.4±0.798.9±0.294.6±0.452.5±2.281.0±4.682.3
ReAct80.6±7.998.4±0.993.5±3.752.7±2.266.2±4.878.3
VIM79.3±2.496.7±0.783.0±4.089.1±0.864.8±0.982.6
MDS81.3±0.195.6±0.676.0±0.294.3±0.565.8±0.282.6
RMDS86.9±0.199.5±0.195.1±0.186.6±0.858.0±0.285.2
KNN58.6±2.095.3±0.757.6±0.276.0±1.771.9±0.171.9
SHE44.0±0.598.3±0.338.1±0.892.1±1.262.1±1.366.9
SPROD42.3±0.295.3±0.630.6±0.253.4±1.154.1±0.155.1
+ +Table 23: AUROC and FPR@95 performance of all methods using ConvNeXt-B as the feature backbone. + +
AUROC↑
MethodWBCAUCAMCSpIAvg.
MSP88.1±0.730.4±2.557.5±0.491.1±0.293.1±0.272.0
Energy86.1±4.029.1±4.755.7±5.793.6±0.292.9±0.371.5
MLS86.6±3.828.9±4.556.0±5.091.7±0.293.1±0.371.3
KLM65.4±1.848.2±2.850.0±0.492.6±0.290.4±0.469.3
GNorm71.6±1.247.0±1.846.1±0.585.0±0.789.4±0.467.8
ReAct85.4±4.226.1±3.755.7±5.593.2±0.292.9±0.270.7
VIM97.6±0.532.4±4.382.6±3.294.7±0.292.7±0.180.0
MDS97.7±0.052.3±1.095.7±0.188.4±0.489.7±0.084.8
RMDS85.4±0.126.9±0.848.9±0.294.1±0.292.4±0.069.5
KNN96.7±0.056.2±0.793.4±0.191.0±0.393.1±0.086.1
SHE95.1±0.162.0±1.199.1±0.175.4±0.891.5±0.184.6
SPROD99.1±0.064.0±0.999.7±0.092.0±0.594.3±0.089.8
+ +
FPR@95↓
MethodWBCAUCAMCSpIAvg.
MSP59.0±3.499.6±0.193.7±1.155.0±1.531.5±1.167.8
Energy65.1±10.299.4±0.691.1±7.353.3±1.932.3±1.868.2
MLS62.4±9.299.6±0.492.1±5.852.9±1.832.0±2.567.8
KLM59.7±3.799.6±0.193.6±0.941.6±1.339.7±5.466.8
GNorm61.9±2.899.6±0.194.4±0.860.5±1.737.0±2.470.7
ReAct67.2±8.499.5±0.491.5±6.756.3±1.832.4±2.369.4
VIM9.4±4.398.6±0.651.5±6.737.5±2.039.8±2.847.4
MDS10.0±0.196.5±1.028.3±0.463.8±2.053.2±0.250.4
RMDS50.6±0.199.9±0.194.9±0.254.6±2.338.5±0.167.7
KNN15.7±0.194.6±1.323.8±0.466.6±1.733.8±0.146.9
SHE22.0±0.697.0±0.53.6±0.591.5±0.638.4±0.850.5
SPROD2.6±0.193.8±1.70.5±0.051.4±1.827.5±0.135.2
+ +Table 24: AUROC and FPR@95 performance of all methods using BiT-R50x1 as the feature backbone. + +
AUROC↑
MethodWBCAUCAMCSpIAvg.
MSP69.5±1.741.8±5.446.8±1.084.3±0.282.5±0.365.0
Energy66.8±2.343.1±9.444.2±7.087.6±0.381.6±0.664.7
MLS67.0±2.142.9±9.344.4±6.387.4±0.382.2±0.564.8
KLM48.4±1.041.9±6.051.1±1.178.6±0.483.0±0.460.6
GNorm49.7±1.535.7±2.860.6±1.671.3±0.653.9±1.554.2
ReAct57.0±2.336.0±8.046.0±7.581.9±0.581.4±1.060.5
VIM93.8±1.648.0±10.273.3±7.587.5±0.386.6±0.277.8
MDS99.1±0.061.9±0.798.5±0.078.2±0.883.3±0.184.2
RMDS70.0±0.137.2±2.155.4±0.385.4±0.384.5±0.166.5
KNN98.5±0.049.5±4.095.4±0.085.2±0.579.3±0.181.6
SHE80.0±0.847.3±0.597.1±0.179.6±0.863.0±0.373.5
SPROD98.5±0.067.1±1.098.6±0.087.1±0.484.3±0.187.1
+ +
FPR@95↓
MethodWBCAUCAMCSpIAvg.
MSP84.7±1.498.5±0.695.2±0.676.0±0.666.7±1.984.2
Energy89.2±2.897.8±1.594.9±5.464.2±1.268.1±2.782.8
MLS88.2±2.898.1±1.395.1±4.865.4±1.166.6±2.482.7
KLM85.2±1.298.4±0.695.1±0.574.8±0.763.8±2.883.5
GNorm85.1±1.298.6±0.695.0±0.770.1±0.989.6±1.687.7
ReAct94.3±1.497.4±1.995.2±4.774.2±1.169.4±3.486.1
VIM27.9±5.297.2±1.361.0±12.072.0±1.451.0±0.561.8
MDS2.5±0.195.7±0.55.9±0.183.4±1.054.6±0.248.4
RMDS83.3±0.198.5±0.490.4±0.182.1±1.256.8±0.282.2
KNN3.7±0.195.5±0.422.4±0.264.3±1.768.9±0.151.0
SHE83.5±0.998.0±0.415.5±0.884.6±1.488.3±0.574.0
SPROD4.9±0.195.6±0.55.9±0.151.2±1.156.8±0.342.9
+ +Table 25: AUPR-IN and AUPR-OUT performance of all methods using ResNet-18 as the feature backbone. + +
AUPR-IN↑
MethodWBCAUCAMCSpIAvg.
MSP62.2±0.923.1±2.428.9±0.280.8±0.547.2±0.648.4
Energy62.7±4.323.6±3.730.2±1.879.8±0.643.3±0.547.9
MLS62.6±4.023.5±3.630.0±1.580.2±0.645.9±0.548.4
KLM34.4±0.316.8±0.440.9±0.570.9±0.641.8±1.041.0
GNorm76.0±0.416.7±0.835.1±0.273.7±0.450.0±0.550.3
ReAct71.6±3.918.1±1.630.7±2.280.4±0.648.7±0.849.9
VIM71.5±3.528.3±4.142.1±3.080.6±0.535.5±0.551.6
MDS61.6±0.129.9±0.586.8±0.267.0±0.915.9±0.052.2
RMDS36.6±0.115.3±0.233.1±0.282.5±0.443.7±0.142.2
KNN88.1±0.227.4±0.588.7±1.480.3±0.533.4±0.163.6
SHE81.6±0.217.5±0.157.1±0.260.1±0.747.9±0.352.8
SPROD97.1±0.140.4±1.196.6±0.181.0±0.548.4±0.172.7
+ +
AUPR-OUT↑
MethodWBCAUCAMCSpIAvg.
MSP72.7±0.477.3±1.359.7±0.559.1±0.992.4±0.272.2
Energy74.5±5.478.3±2.263.0±5.257.3±1.092.4±0.173.1
MLS74.3±4.277.7±1.862.2±4.058.0±1.092.4±0.172.9
KLM60.6±0.575.0±0.868.6±0.459.4±0.890.4±0.370.8
GNorm82.5±0.375.8±1.362.9±0.555.3±1.095.2±0.174.3
ReAct79.2±5.075.0±1.963.1±5.557.3±1.194.3±0.173.8
VIM83.3±2.580.9±1.679.4±2.858.3±1.087.3±0.177.8
MDS82.8±0.183.4±0.292.5±0.149.9±0.780.4±0.077.8
RMDS59.4±0.172.8±0.460.9±0.162.3±0.989.0±0.068.9
KNN96.2±0.084.1±0.395.0±0.561.8±0.990.7±0.085.6
SHE95.0±0.180.6±0.484.8±0.145.0±1.095.3±0.080.1
SPROD98.9±0.085.6±0.698.1±0.066.9±0.794.4±0.088.8
+ +Table 26: AUPR-IN and AUPR-OUT performance of all methods using ResNet-34 as the feature backbone. + +
AUPR-IN↑
MethodWBCAUCAMCSpIAvg.
MSP60.1±0.918.6±1.130.7±0.284.0±0.352.9±0.449.3
Energy57.8±5.819.0±2.930.4±1.785.4±0.448.8±0.648.3
MLS57.8±5.318.8±2.830.4±1.484.8±0.451.3±0.548.6
KLM33.9±0.418.3±0.440.1±0.374.3±0.652.2±2.943.8
GNorm69.5±0.314.5±0.336.8±0.378.5±0.351.7±0.250.2
ReAct68.6±4.814.3±1.232.0±2.585.4±0.453.3±1.050.7
VIM74.8±4.924.7±4.042.5±2.983.9±0.543.7±0.553.9
MDS82.3±0.133.9±0.587.8±0.171.2±0.825.9±0.160.2
RMDS41.1±0.114.4±0.331.7±0.186.2±0.353.2±0.245.3
KNN96.5±0.127.0±0.483.6±0.383.2±0.539.6±0.266.0
SHE75.9±0.315.3±0.169.0±0.474.2±0.551.6±0.457.2
SPROD97.0±0.140.4±0.694.5±0.184.6±0.453.4±0.274.0
+ +
AUPR-OUT↑
MethodWBCAUCAMCSpIAvg.
MSP70.9±0.674.8±0.860.8±0.664.1±0.994.3±0.173.0
Energy71.4±7.576.2±2.561.7±5.364.9±1.193.8±0.173.6
MLS71.3±6.075.5±1.861.4±4.065.1±1.094.1±0.173.5
KLM58.5±0.676.3±0.667.3±0.762.7±0.992.6±0.471.5
GNorm79.0±0.573.0±0.663.6±0.562.6±1.095.8±0.074.8
ReAct75.5±6.774.1±3.961.3±5.464.4±1.195.2±0.274.1
VIM85.7±3.380.2±1.178.7±2.360.8±1.089.6±0.179.0
MDS88.8±0.084.0±0.592.2±0.149.7±0.883.6±0.079.7
RMDS63.0±0.171.0±0.560.4±0.266.5±1.091.3±0.070.4
KNN98.5±0.082.7±0.492.7±0.163.5±0.992.3±0.085.9
SHE93.3±0.177.0±0.388.5±0.158.6±0.595.7±0.182.6
SPROD98.9±0.084.2±0.796.3±0.068.5±0.995.4±0.088.7
AUPR-IN↑
MethodWBCAUCAMCSpIAvg.
MSP52.4±1.123.9±0.928.3±0.186.4±0.358.2±0.649.8
Energy52.1±3.423.4±2.628.2±0.786.7±0.452.9±0.948.7
MLS52.1±3.323.4±2.528.2±0.586.9±0.455.8±0.549.3
KLM39.5±0.517.1±0.741.0±0.776.0±0.752.3±1.245.2
GNorm70.2±0.515.8±0.234.5±0.378.4±0.556.3±0.351.0
ReAct67.8±3.220.2±2.329.7±1.187.0±0.458.7±1.552.7
VIM72.7±3.627.3±3.039.2±1.286.5±0.451.5±1.255.4
MDS88.0±0.130.2±0.588.4±0.172.6±0.728.9±0.161.6
RMDS45.3±0.114.4±0.534.4±0.188.5±0.343.1±0.145.1
KNN97.8±0.124.4±0.585.6±0.286.2±0.446.0±0.268.0
SHE77.0±0.416.2±0.358.2±0.261.6±0.749.4±0.352.5
SPROD98.1±0.140.6±1.296.7±0.188.9±0.458.3±0.276.5
+ +Table 27: AUPR-IN and AUPR-OUT performance of all methods using ResNet-50 as the feature backbone. + +
AUPR-OUT↑
MethodWBCAUCAMCSpIAvg.
MSP70.9±0.475.8±1.057.9±0.266.5±0.993.9±0.173.0
Energy70.4±2.475.8±1.958.5±3.066.5±1.093.7±0.273.0
MLS70.8±1.875.6±1.758.1±2.066.9±1.094.0±0.173.1
KLM64.7±0.574.6±1.566.8±0.265.7±0.993.3±0.373.0
GNorm81.5±0.573.7±0.960.7±0.363.1±0.996.1±0.175.0
ReAct76.2±4.677.7±3.560.0±3.666.5±1.095.5±0.275.2
VIM85.9±1.679.5±1.378.7±1.064.2±1.092.4±0.380.1
MDS92.5±0.082.8±0.394.6±0.051.2±0.985.6±0.081.3
RMDS68.0±0.170.7±0.662.8±0.168.3±0.989.0±0.071.8
KNN99.1±0.082.3±0.494.7±0.067.6±1.093.4±0.087.4
SHE93.5±0.177.4±0.382.6±0.146.4±0.895.6±0.079.1
SPROD99.2±0.084.3±0.698.1±0.073.0±0.996.0±0.090.1
+ +Table 28: AUPR-IN and AUPR-OUT performance of all methods using ResNet-101 as the feature backbone. + +
AUPR-IN↑
MethodWBCAUCAMCSpIAvg.
MSP55.8±0.923.6±2.929.6±0.688.0±0.357.4±0.350.9
Energy53.9±3.223.9±3.429.6±1.188.0±0.455.0±0.750.1
MLS54.1±2.923.8±3.329.7±1.088.1±0.457.5±0.450.6
KLM38.1±0.418.7±1.240.2±0.678.2±0.650.6±0.845.2
GNorm64.2±0.519.0±0.637.5±0.778.8±0.457.1±0.251.3
ReAct67.2±3.016.7±2.631.4±1.887.6±0.662.8±1.953.1
VIM83.7±2.227.1±3.843.4±1.887.2±0.556.9±1.159.7
MDS95.2±0.124.4±0.591.5±0.173.9±0.835.6±0.164.1
RMDS44.8±0.013.8±0.231.6±0.189.8±0.347.4±0.245.5
KNN97.9±0.120.2±5.482.3±0.488.0±0.449.9±0.267.7
SHE68.5±0.634.9±1.166.8±0.365.7±0.747.1±0.356.6
SPROD98.1±0.144.2±1.297.3±0.087.8±0.458.0±0.277.1
+ +
AUPR-OUT↑
MethodWBCAUCAMCSpIAvg.
MSP72.7±0.675.3±1.459.8±1.168.8±1.094.9±0.174.3
Energy70.7±3.976.0±2.060.0±2.969.6±1.293.9±0.274.0
MLS71.4±2.575.6±1.759.8±2.269.9±1.294.5±0.174.2
KLM64.2±0.775.8±1.167.3±1.267.9±1.093.6±0.273.8
GNorm79.8±0.676.1±1.163.0±1.065.6±1.196.0±0.176.1
ReAct75.5±5.175.0±4.261.4±3.969.2±1.295.7±0.375.4
VIM92.3±1.278.4±1.680.0±1.064.8±1.193.5±0.281.8
MDS97.0±0.081.5±0.495.2±0.051.3±0.988.2±0.182.6
RMDS69.9±0.169.8±0.260.2±0.171.2±1.090.8±0.072.4
KNN99.4±0.080.9±1.893.7±0.170.0±1.194.3±0.087.7
SHE90.8±0.287.1±0.588.2±0.151.6±1.295.1±0.182.6
SPROD99.4±0.084.8±0.798.5±0.074.5±0.996.5±0.090.7
+ +Table 29: AUPR-IN and AUPR-OUT performance of all methods using DINOv2-S as the feature backbone. + +
AUPR-IN↑
MethodWBCAUCAMCSpIAvg.
MSP75.8±2.414.8±1.054.0±0.692.1±0.360.0±1.159.3
Energy71.7±3.315.9±2.853.1±9.393.4±0.265.6±1.559.9
MLS72.2±3.015.7±2.753.3±8.893.4±0.261.4±1.559.2
KLM64.7±1.623.5±1.947.2±6.284.4±0.860.8±1.656.1
GNorm79.4±1.713.8±0.547.1±0.486.3±0.766.0±1.358.5
ReAct70.7±2.915.2±2.252.9±9.293.4±0.265.6±1.359.6
VIM95.8±0.916.8±3.374.4±9.491.9±0.454.7±1.066.7
MDS98.8±0.029.6±0.791.9±0.189.9±0.548.7±0.271.8
RMDS51.2±0.113.0±0.335.5±0.293.9±0.357.5±0.350.2
KNN98.9±0.116.0±0.268.5±1.294.2±0.245.6±1.064.6
SHE98.6±0.144.7±1.388.9±0.285.3±0.458.1±0.475.1
SPROD99.4±0.045.0±1.195.3±0.093.9±0.261.1±0.278.9
+ +
AUPR-OUT↑
MethodWBCAUCAMCSpIAvg.
MSP82.3±1.671.9±1.372.8±0.377.1±0.895.5±0.279.9
Energy81.1±3.273.7±4.373.3±8.478.0±1.196.1±0.480.4
MLS81.7±2.873.1±4.073.3±7.878.2±1.095.9±0.380.4
KLM75.7±3.379.2±1.869.2±5.377.4±0.995.1±0.279.3
GNorm83.8±1.471.5±1.172.7±0.373.6±1.196.1±0.279.5
ReAct80.5±3.073.3±4.073.1±8.178.0±1.196.0±0.480.2
VIM98.5±0.375.1±3.688.9±4.874.7±1.295.9±0.186.6
MDS99.3±0.081.2±0.395.3±0.072.3±1.395.6±0.088.7
RMDS73.6±0.167.8±0.365.0±0.180.4±1.195.4±0.076.4
KNN99.5±0.077.8±0.390.9±0.284.5±0.894.3±0.389.4
SHE99.4±0.079.0±0.495.3±0.172.3±1.495.5±0.188.3
SPROD99.7±0.084.6±0.497.1±0.087.3±0.596.0±0.092.9
+ +Table 30: AUPR-IN and AUPR-OUT performance of all methods using ViT-S as the feature backbone. + +
AUPR-IN↑
MethodWBCAUCAMCSpIAvg.
MSP67.1±2.216.8±0.735.0±0.389.4±0.263.9±1.254.4
Energy63.2±10.119.0±0.735.5±2.890.6±0.365.6±1.754.8
MLS63.3±9.818.8±6.535.4±2.690.5±0.366.1±1.554.8
KLM53.6±2.921.7±2.131.2±3.679.3±0.760.5±2.149.3
GNorm73.1±1.416.4±0.534.6±0.587.8±0.465.6±0.655.5
ReAct64.0±8.124.4±0.035.5±2.990.8±0.265.8±2.056.1
VIM81.0±8.821.8±8.751.7±5.891.3±0.364.6±1.362.1
MDS94.4±0.138.1±1.293.8±0.187.1±0.458.9±0.274.5
RMDS46.6±0.113.6±0.330.9±0.192.0±0.264.5±0.249.5
KNN92.8±0.130.6±6.989.3±0.188.3±0.452.8±0.270.8
SHE92.1±0.338.9±1.087.8±0.380.4±0.758.5±0.271.5
SPROD95.1±0.142.0±1.495.5±0.188.4±0.364.4±0.377.1
+ +
AUPR-OUT↑
MethodWBCAUCAMCSpIAvg.
MSP75.0±1.573.4±0.662.4±0.570.5±0.695.2±0.275.3
Energy73.4±8.874.5±4.864.3±4.572.9±0.995.4±0.376.1
MLS73.6±8.174.0±4.364.0±4.172.8±0.895.5±0.276.0
KLM71.1±6.977.8±2.459.2±4.669.3±0.795.3±0.274.5
GNorm77.5±1.373.2±0.662.5±0.570.8±0.895.9±0.176.0
ReAct74.5±7.778.5±4.664.1±4.672.0±0.994.3±0.476.7
VIM87.1±7.176.4±4.279.2±4.474.4±0.895.6±0.282.5
MDS95.7±0.083.1±0.595.2±0.070.3±0.895.3±0.087.9
RMDS69.3±0.168.8±0.459.1±0.176.7±0.795.9±0.074.0
KNN94.4±0.080.8±1.293.4±0.072.1±0.893.6±0.086.9
SHE95.3±0.277.7±0.393.8±0.270.1±0.995.0±0.186.4
SPROD97.8±0.083.3±0.597.2±0.077.0±0.696.0±0.090.3
+ +Table 31: AUPR-IN and AUPR-OUT performance of all methods using Swin-B as the feature backbone. + +
AUPR-IN↑
MethodWBCAUCAMCSpIAvg.
MSP79.8±1.014.0±0.647.1±0.494.8±0.270.0±1.861.1
Energy79.0±1.814.0±0.747.5±2.595.9±0.260.3±1.859.3
MLS79.2±1.614.0±0.747.5±2.395.0±0.264.9±1.660.1
KLM40.2±1.024.9±2.135.9±0.594.0±0.367.2±1.052.4
GNorm43.1±0.812.7±0.235.1±0.283.8±1.019.2±0.338.8
ReAct80.9±1.715.2±0.848.5±2.396.1±0.274.5±1.463.0
VIM89.5±0.418.3±1.665.9±2.193.6±0.277.0±0.468.9
MDS83.7±0.134.4±0.879.9±0.385.3±0.576.6±0.272.0
RMDS72.7±0.113.7±0.335.4±0.196.4±0.176.6±0.259.0
KNN86.8±0.129.8±0.888.2±0.295.2±0.273.0±0.374.6
SHE96.6±0.127.2±0.995.4±0.295.9±0.259.7±0.275.0
SPROD98.3±0.147.8±1.498.3±0.095.2±0.280.9±0.284.1
+ +
AUPR-OUT↑
MethodWBCAUCAMCSpIAvg.
MSP84.5±0.969.7±0.869.0±0.486.2±0.597.3±0.181.3
Energy84.4±1.970.0±1.369.5±2.987.4±0.696.1±0.281.5
MLS84.6±1.469.8±1.269.3±2.287.1±0.697.0±0.181.6
KLM72.6±1.377.4±1.865.6±0.585.2±0.497.0±0.279.6
GNorm76.3±0.969.2±0.667.3±0.481.3±0.986.6±0.576.1
ReAct85.2±2.171.4±1.469.8±2.787.0±0.697.0±0.282.1
VIM91.7±0.274.8±1.686.0±0.973.5±0.897.9±0.084.8
MDS88.1±0.184.2±0.791.1±0.158.3±1.197.8±0.083.9
RMDS86.5±0.069.0±0.563.9±0.181.8±0.598.0±0.079.8
KNN93.3±0.082.5±0.594.8±0.185.5±0.697.4±0.090.7
SHE98.0±0.077.2±0.597.5±0.183.8±0.896.8±0.090.7
SPROD99.3±0.086.3±0.699.2±0.086.6±0.498.1±0.093.9
+ +Table 32: AUPR-IN and AUPR-OUT performance of all methods using DeiT-B as the feature backbone. + +
AUPR-IN↑
MethodWBCAUCAMCSpIAvg.
MSP72.9±1.221.4±1.440.5±0.292.4±0.367.9±1.159.0
Energy73.7±3.422.2±3.539.8±2.492.4±0.565.6±1.858.7
MLS73.6±3.222.2±3.439.9±2.292.4±0.567.1±1.159.0
KLM35.3±0.318.6±0.635.7±0.487.5±0.565.1±1.748.4
GNorm51.7±1.117.1±0.537.1±0.177.9±1.130.9±2.442.9
ReAct73.6±3.420.3±2.940.3±2.693.1±0.469.4±1.759.3
VIM81.7±1.923.5±3.452.1±2.790.1±0.368.0±0.763.1
MDS69.6±0.219.4±0.168.7±0.379.1±0.667.0±0.360.8
RMDS56.3±0.113.9±0.236.2±0.392.8±0.269.3±0.353.7
KNN88.6±0.818.5±2.682.9±0.289.5±0.466.7±0.369.2
SHE93.8±0.035.3±0.692.4±0.180.6±1.065.4±0.473.5
SPROD93.6±0.132.1±0.795.2±0.092.1±0.269.7±0.376.5
+ +
AUPR-OUT↑
MethodWBCAUCAMCSpIAvg.
MSP77.9±0.774.7±1.066.1±0.380.8±0.794.9±0.278.9
Energy80.1±3.675.6±2.265.7±3.383.5±0.994.9±0.580.0
MLS79.9±2.875.2±1.865.8±2.682.9±0.895.1±0.279.8
KLM64.3±0.574.7±1.264.0±0.278.3±0.695.8±0.375.4
GNorm75.5±0.675.6±0.866.6±0.375.1±1.290.3±1.276.6
ReAct80.1±3.474.7±2.266.1±3.683.5±0.995.4±0.580.0
VIM84.2±1.577.9±1.577.3±2.066.1±0.995.7±0.180.2
MDS79.8±0.180.8±0.484.4±0.150.8±1.195.7±0.078.3
RMDS75.5±0.170.1±0.164.1±0.272.3±0.696.6±0.075.7
KNN91.4±0.579.3±1.491.3±0.072.0±1.095.1±0.085.8
SHE94.3±0.079.3±0.496.0±0.057.6±1.195.7±0.084.6
SPROD94.8±0.081.9±0.597.0±0.083.3±0.696.7±0.090.7
+ +Table 33: AUPR-IN and AUPR-OUT performance of all methods using ConvNeXt-B as the feature backbone. + +
AUPR-IN↑
MethodWBCAUCAMCSpIAvg.
MSP85.4±1.014.1±1.148.5±0.694.5±0.178.2±0.864.1
Energy82.7±4.813.5±1.146.0±4.196.1±0.177.5±1.363.1
MLS83.1±4.713.4±1.046.1±3.895.1±0.177.5±1.363.0
KLM44.3±1.223.9±1.335.6±0.594.1±0.366.2±0.852.8
GNorm51.0±1.421.0±1.131.1±0.287.8±0.666.4±1.451.5
ReAct82.1±5.412.8±0.746.1±4.196.1±0.175.8±1.362.6
VIM96.6±0.614.4±1.673.0±4.696.8±0.181.5±0.572.5
MDS97.0±0.125.0±0.894.9±0.192.6±0.375.3±0.277.0
RMDS73.2±0.113.0±0.334.4±0.296.8±0.178.6±0.259.2
KNN94.3±0.128.3±0.385.7±0.294.9±0.279.3±0.276.5
SHE92.8±0.149.7±1.098.7±0.183.5±0.775.6±0.180.1
SPROD98.4±0.145.0±0.999.6±0.095.3±0.683.2±0.284.3
+ +
AUPR-OUT↑
MethodWBCAUCAMCSpIAvg.
MSP89.6±0.669.5±0.968.6±0.584.0±0.597.7±0.181.9
Energy88.4±3.769.5±2.267.9±0.185.7±0.697.9±0.181.9
MLS88.9±3.269.3±1.968.0±5.485.0±0.698.0±0.181.8
KLM80.0±1.175.0±1.565.4±0.387.6±0.496.7±0.280.9
GNorm82.1±0.675.0±0.864.0±0.578.1±0.996.9±0.179.2
ReAct87.7±3.568.3±1.867.9±5.985.1±0.697.9±0.181.4
VIM98.1±0.471.5±2.089.8±2.088.9±0.597.6±0.189.2
MDS98.1±0.080.5±0.796.7±0.079.2±0.796.8±0.090.3
RMDS89.6±0.067.9±0.363.9±0.284.4±0.597.7±0.080.7
KNN97.8±0.082.7±0.696.6±0.180.5±0.798.1±0.091.1
SHE96.8±0.182.8±0.799.5±0.061.1±0.997.7±0.187.6
SPROD99.4±0.085.3±0.799.8±0.086.0±0.798.4±0.093.8
+ +Table 34: AUPR-IN and AUPR-OUT performance of all methods using BiT-R50x1 as the feature backbone. + +
AUPR-IN↑
MethodWBCAUCAMCSpIAvg.
MSP59.3±3.018.8±3.933.4±0.889.8±0.255.4±0.651.3
Energy56.4±2.920.5±7.532.0±3.791.3±0.352.0±0.850.4
MLS56.5±2.920.5±7.532.0±3.591.3±0.352.7±0.750.6
KLM35.7±0.518.9±2.036.8±0.679.7±0.656.5±1.045.5
GNorm38.2±0.814.8±0.855.0±1.674.1±0.720.5±0.740.5
ReAct47.6±2.914.9±2.133.7±4.286.9±0.455.9±0.947.8
VIM90.7±2.325.7±9.955.3±9.892.3±0.258.1±0.464.4
MDS98.8±0.141.0±1.198.3±0.085.5±0.652.7±0.375.3
RMDS47.2±0.113.5±0.635.4±0.291.2±0.256.6±0.348.8
KNN97.4±0.120.9±8.193.1±0.189.4±0.448.2±0.269.8
SHE75.5±0.824.4±1.095.8±0.179.6±0.530.7±0.261.2
SPROD97.3±0.149.8±1.198.3±0.089.9±0.356.1±0.378.3
+ +
AUPR-OUT↑
MethodWBCAUCAMCSpIAvg.
MSP76.0±1.174.7±2.362.6±0.873.5±0.694.8±0.276.3
Energy72.8±2.575.7±4.160.8±6.479.5±0.894.3±0.376.6
MLS73.3±2.375.5±4.060.9±5.778.7±0.794.6±0.376.6
KLM65.3±0.574.2±3.164.5±0.971.0±0.794.9±0.274.0
GNorm65.6±1.172.9±1.768.1±1.067.6±0.884.6±0.871.8
ReAct64.7±2.074.2±4.661.7±6.872.6±0.994.2±0.573.5
VIM95.8±1.078.0±4.384.8±4.675.4±0.896.5±0.186.1
MDS99.4±0.083.9±0.498.8±0.065.1±1.095.5±0.088.5
RMDS73.4±0.169.1±1.164.7±0.271.4±0.795.8±0.074.9
KNN99.0±0.080.3±0.897.2±0.076.9±0.893.9±0.089.5
SHE81.9±0.676.7±0.398.2±0.067.6±0.987.6±0.282.4
SPROD99.1±0.085.5±0.599.0±0.083.8±0.595.9±0.092.7
+ +Table 35: Comparison of SPROD variants on SP-OOD datasets using AUROC and FPR@95 metrics. All methods utilize a pretrained ResNet-50 backbone. Values are averaged over five runs. + +
AUROC↑
MethodWBCAUCAMCSpIAvg.
SPROD-Default98.8±0.061.6±0.997.4±0.082.1±0.085.3±0.085.0
SPROD-Converged98.5±0.059.1±0.997.1±0.183.0±0.585.3±0.084.6
SPROD-KMeans98.3±0.157.3±2.296.8±0.182.8±0.585.9±0.084.2
+ +
MethodWBCAUCAMCSpIAvg.
SPROD-Default4.7±0.193.7±0.919.0±0.470.1±0.058.0±0.149.1
SPROD-Converged6.4±0.193.9±1.019.1±0.471.0±1.258.1±0.149.7
SPROD-KMeans6.0±0.593.7±1.219.3±0.471.3±1.156.4±0.149.3
+ +Table 36: Comparison of SPROD variants on SP-OOD datasets using AUPR-IN and AUPR-OUT metrics. All methods utilize a pretrained ResNet-50 backbone. Values are averaged over five runs. + +
MethodWBCAUCAMCSpIAvg.
SPROD-Default98.1±0.140.6±1.296.7±0.188.9±0.058.3±0.076.5
SPROD-Converged97.5±0.136.5±1.296.0±0.188.2±0.455.8±0.274.8
SPROD-KMeans97.6±0.132.1±4.296.0±0.188.4±0.458.6±0.374.5
+ +
MethodWBCAUCAMCSpIAvg.
SPROD-Default99.2±0.084.3±0.698.1±0.073.0±0.896.0±0.090.1
SPROD-Converged99.1±0.083.6±0.698.0±0.073.2±0.896.0±0.090.0
SPROD-KMeans99.1±0.083.4±0.698.0±0.073.4±0.896.3±0.090.0
+ +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract and introduction clearly state the paper's main contributions, which include: (1) the proposal of SPROD, a novel post-hoc prototype-based OOD detection method designed to be robust against unknown spurious correlations; (2) the method's ability to operate without additional OOD data, hyperparameter tuning, or group annotations; (3) its broad applicability and superior performance, demonstrated through comprehensive experiments on diverse SP-OOD datasets, including the newly introduced AnimalsMetaCoCo dataset; and (4) insights into factors like backbone fine-tuning and scoring mechanisms in the context of SP-OOD. These claims are consistently addressed and supported by the methodology (Section 4), experimental results (Section 5.3, Table 2, Figures 4-6), and dataset descriptions (Section 5.2). + +# Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: Section 6, discusses limitations. + +# Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best + +judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. + +# 3. Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +Answer: [NA] + +Justification: The paper proposes an algorithmic method (SPROD) and provides extensive empirical validation. It does not present new theoretical results in the form of theorems or formal proofs. The motivation in Section 3.2 discusses probabilistic concepts but does not derive new theoretical claims. Section 6 suggests that theoretical justifications are a direction for future work. + +Guidelines: + +- The answer NA means that the paper does not include theoretical results. +- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced. +- All assumptions should be clearly stated or referenced in the statement of any theorems. +- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. +- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. +- Theorems and Lemmas that the proof relies upon should be properly referenced. + +# 4. Experimental result reproducibility + +Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? + +Answer: [Yes] + +Justification: The paper provides substantial information to facilitate the reproduction of its main experimental results. 1. The proposed SPROD algorithm is described in detail in Section 4 and 1, with its three stages (Initial Prototype Construction, Classification-Aware Prototype Calculation, and Group Prototype Refinement) clearly outlined. Algorithm 1 provides pseudocode for SPROD. 2. 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Typically, they appear as likelihoods, where their advantageous computational, geometric and statistical properties are exploited for fast estimation algorithms, representational properties and statistical guarantees. Here, we investigate the use of squared families as prior beliefs in Bayesian inference. We find that they can form helpful conjugate families, often allowing for closed-form and tractable Bayesian inference and marginal likelihoods. We apply such conjugate families to Bayesian regression in feature space using end-to-end learnable neural network features. Such a setting allows for a rich multi-modal alternative to Gaussian processes with neural network features, often called deep kernel learning. We demonstrate our method on few shot learning, outperforming existing neural methods based on Gaussian processes and normalising flows. $^1$ + +# 1 Introduction + +Squared families are useful likelihoods Families $\{q(\cdot \mid f)\}_{f\in \mathcal{H}}$ of probability densities of the form $q(\cdot \mid f)\propto \| f(\cdot)\| _2^2$ , where $f\in \mathcal{H}$ and $\mathcal{H}$ is some suitably nice space of vector-valued functions, have recently emerged as promising likelihoods in several different communities in machine learning. For example, functions of the form $f(\cdot) = \Theta \psi (\cdot)$ for some $\Theta \in \mathbb{R}^{m\times n}$ and $\psi (\cdot) = (k(\cdot ,a_1),\dots ,k(\cdot ,a_N))^{\top}$ , where $(a_{i})_{i = 1}^{N}$ are data points and $k$ is the kernel of a reproducing kernel Hilbert space (RKHS), are obtained as minimisers of regularised empirical risk problems [Marteau-Ferey et al., 2020]. When appropriately normalised and applied to divergence minimisation, they can be used to model probability densities [Rudi and Ciliberto, 2021]. When $m = 1$ , such density models can also be applied to modelling intensities (of Poisson point processes) [Flaxman et al., 2017], and Bayesian variants which use Gaussian processes (GPs) instead of RKHS models are also readily available [Walder and Bishop, 2017]. Separately to kernel methods, probabilistic circuits [Choi et al., 2020] can be chosen for $f$ , which can be squared [Sladek, 2023, Loconte et al., 2023a,b, Lo Conte and Vergari, 2025], or squared and summed [Lo Conte et al., 2024], to obtain tractable and expressive models for probability density functions. Finally, neural networks can be used for $f$ , and when they are two-layered and fully-connected, they often admin tractable and closed-form normalising constants, marginal distributions and conditional distributions [Tsuchida et al., 2023]. + +![](images/6872975aebc40af52e4e8793e063bf09840ddcc5a7c47dcb01ee62a7d74c426b.jpg) +Figure 1: (Left) Empirical Bayesian estimation of the rate parameter $\lambda$ of a Poisson likelihood in a hierarchical model. Here the mode index $j = 1,2,3$ is observed, but not the mode rate $\lambda_{j}$ . A classical empirical Bayesian approach might be to place a shared conjugate prior over $\lambda_{j}$ (in this case, the Gamma density, see Appendix A), and maximise the marginal likelihood with respect to the prior hyperparameters. Classical conjugate family priors are only capable of expressing a unimodal prior belief, unless they are mixture models. (Right) At inference time, we are presented $N$ datapoints from a single mode — Mode 1. GSF priors (solid curves) are multimodal, and are able to adapt their trimodal prior to the observations quickly, whereas the unimodal Gamma prior (dashed curves) have to slowly adapt their density to pick out the correct mode. GSFs significantly generalise mixture models of simpler base conjugate priors. + +![](images/8655f9d2d0c5c3fac1dca5a3d0d0ae50e2fe3b1058df24b4a0d596d5314758a1.jpg) + +Are squared families useful priors? In this work, we examine the use of such families as priors, and more specifically, families of conjugate priors. To the best of our knowledge, we are the first to consider such families as priors. Note that this question is distinct from a Bayesian treatment of squared family likelihoods [Walder and Bishop, 2017, for example], where a squared family is used for a likelihood and some prior belief over $f$ (for example, $f \sim \mathcal{GP}$ ) is employed. Rather, this question is about using some other (not necessarily squared family likelihood) and a squared family prior. We first construct a hierarchy of families, called generalised squared families (GSF). We then show when a prior belongs to the GSF, the marginal likelihoods, posterior parameter and posterior predictive distributions all admit closed-form expressions. Furthermore, the posterior distributions remain within the GSF. This allows for rich and multimodal expression of prior beliefs within a conjugate family (see Figure 1, for example). We apply our conjugate families to the problem of Bayesian regression in feature space, which offers a model called GSF processes (GSFP) with rich multi-modal uncertainty and capabilities well-suited for few-task learning. + +# 2 Background + +Conjugate priors Let $\mathbb{P}$ be a family of probability measures each supported on a subset of $\mathbb{Q}$ . We call $\mathbb{P}$ a conjugate class for a likelihood $p(\boldsymbol{u} \mid \boldsymbol{\omega})$ if for all $\pi \in \mathbb{P}$ , the posterior probability measure $\pi'$ defined by $\pi'(d\omega \mid \boldsymbol{u}) = p(\boldsymbol{u} \mid \boldsymbol{\omega})\pi(d\omega) / \int_{\Omega} p(\boldsymbol{u} \mid \boldsymbol{\omega}')\pi(d\omega')$ is also an element of $\mathbb{P}$ . An immediate but unhelpful conjugate class is the class $\mathbb{Q}$ of all probability measures, since for any likelihood a prior in $\mathbb{Q}$ results in a posterior in $\mathbb{Q}$ [Robert, 2007, page 98]. Often when authors speak of conjugate families, they implicitly and loosely also require that such conjugate families have closed-form or tractable normalising constants or parameters. In order to facilitate such closed-form updates, conjugate families are therefore required to be small and analytically tractable. Exponential families of priors matched to exponential family likelihoods are arguably the most well-known examples of conjugate priors (see Robert [2007, §3.3.3] and Appendix A). We explicitly differentiate between conjugate families and conjugate families with closed-form normalising constants. We will consider two distinct types of conjugate families: one for classes of squared families, and one for their corresponding base measures. Both conjugacies are with respect to the same likelihood function. + +Gaussian processes Gaussian processes (GPs) are flexible nonparametric models for probabilistic inference over functions. Typically one specifies that some function of interest a priori follows a GP, takes some observations from the likelihood given the value of the GP, and then computes the posterior conditioned on the observations under the likelihood. If the likelihood is Gaussian, then the posterior process is also a GP because the Gaussian distribution is a conjugate prior for the Gaussian likelihood. There are multiple ways to mathematically construct GPs, two of which are the function space view and the weight space view [Rasmussen and Williams, 2006, § 2.2 and § 2.1]. + +![](images/95fd5e98cd8425f49d5b5ed4ef157a09e8fbe733faa46c8298eb5005c2eabf5b.jpg) +Figure 2: Given base measure $\mu$ and feature $\psi$ , a parameter $M \in \mathbb{M}$ indexes a prior $Q(\cdot \mid M, \mu, \psi)$ from a generalised squared family (GSF) $\mathbb{Q}_{\mathbb{M}, \mu, \psi}$ (see Definition 1). Given some observed data $U$ and likelihood $p(U \mid \cdot)$ , the same parameter $M \in \mathbb{M}$ indexes the posterior $Q(\cdot \mid M, \nu, \psi)$ , where $\nu = p(U \mid \cdot) \mu(\cdot)$ (see Proposition 3). The posterior and prior both belong to the conjugate class $\mathbb{Q}_{M, \wedge, \psi}$ , a GSF (see Definition 1). In certain cases of linear regression in feature space, where the prior is placed over coefficients and the likelihood is Gaussian with expectation described by the regression function, the same parameter $M \in \mathbb{M}$ indexes the posterior predictive distribution $Q(\cdot \mid M, \nu'', \psi')$ , for some appropriate $\nu''$ and $\psi'$ (see Corollary 6). The posterior predictive, posterior parameter, and prior distributions all belong to the conjugate class $\mathbb{Q}_{M, \wedge, \psi}$ , also a GSF. Closed-form expressions for all densities in the GSF $\mathbb{Q}_{\mathbb{M}, \wedge, \psi}$ as well as the marginal likelihood (see Proposition 2) allow for joint learning of $M$ , $\mu$ and $\psi$ of the prior distribution via type II maximum likelihood, or empirical Bayes (see § 4), and for closed-form posterior updates. + +Function-space view A (vector valued) Gaussian process $\{\mathbf{f}(\boldsymbol{x})\}_{\boldsymbol{x}\in \mathbb{X}}$ , or simply $\mathbf{f}$ , is a collection of (possibly infinitely many) random $d_{2}$ -dimensional vectors such that every finite subset $\{\mathbf{f}(\boldsymbol{x}_i)\}_{i=1}^N$ follows an $Nd_{2}$ -variate Gaussian distribution. A GP is characterised by its vector-valued mean function $m: \mathbb{X} \to \mathbb{R}^{d_2}$ and positive semidefinite PSD matrix-valued covariance (or kernel) function $\pmb{K}: \mathbb{X} \times \mathbb{X} \to \mathbb{S}_{+}^{d_2}$ , where $\mathbb{S}_{+}^{d_2}$ is the set of $d_{2} \times d_{2}$ PSD matrices. We write $\mathbf{f} \sim \mathcal{GP}(\boldsymbol{m}, \boldsymbol{K})$ to mean that $\mathbf{f}$ is a GP with mean and covariance functions $\boldsymbol{m}$ and $\pmb{K}$ respectively, meaning that for any finite set of indices $\boldsymbol{X} = \{\mathbf{x}_i\}_{i=1}^N$ , $\operatorname{vec}\left(\mathbf{f}(\boldsymbol{X})\right) \in \mathbb{R}^{Nd_2}$ has mean matrix $\operatorname{vec}\left(\boldsymbol{m}(\boldsymbol{X})\right) \in \mathbb{R}^{Nd_2}$ and block partitioned covariance matrix with $ij$ th block $\pmb{K}(\boldsymbol{x}_i, \boldsymbol{x}_j) \in \mathbb{S}_{+}^{d_2}$ . + +Weight-space view Finite-feature GPs can be constructed by specifying a feature mapping $\gamma$ , defining a function $\mathbf{f}(\boldsymbol{x}) = \boldsymbol{\gamma}(\boldsymbol{x})\boldsymbol{\Omega}^{\top}$ and choosing a Gaussian distribution over the parameters $\Omega$ . Let $\Omega^{\top} \in \mathbb{R}^{d_1 \times d_2}$ follow a matrix Gaussian distribution such that $\mathrm{vec}(\Omega^{\top})$ has mean zero and covariance $\boldsymbol{B} \otimes \boldsymbol{C}$ with $\boldsymbol{B} \in \mathbb{S}_{+}^{d_2}$ and $\boldsymbol{C} \in \mathbb{S}_{+}^{d_1}$ , with the notation $\otimes$ denoting the Kronecker product. Let $\gamma: \mathbb{X} \to \mathbb{R}^{d_1}$ be a finite feature mapping and define $\Gamma = \gamma(\boldsymbol{X})$ . Since Gaussian random vectors are closed under addition, one may verify that $\mathrm{vec}\left(\mathbf{f}(\boldsymbol{X})\right) = \mathrm{vec}\left(\Gamma\boldsymbol{\Omega}^{\top}\right)$ follows a $Nd_2$ -variate Gaussian distribution. The covariance matrix is given by + +$$ +\mathbb {E} \left[ \operatorname {v e c} \left(\boldsymbol {\Gamma} \boldsymbol {\Omega} ^ {\top}\right) \operatorname {v e c} \left(\boldsymbol {\Gamma} \boldsymbol {\Omega} ^ {\top}\right) ^ {\top} \right] = \boldsymbol {B} \otimes \left(\boldsymbol {\Gamma} \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top}\right). +$$ + +We note that this is a special-case way of handling vector-valued GPs, corresponding with the so-called coregionalisation model [Alvarez et al., 2012]. Our later discussions likely extend to more involved vector-valued setups, with more involved notations. + +Conditionals and marginals of GPs One attractive property of GPs is that marginal, conditional and evaluations of linear operators of GPs are also GPs. This in particular implies that if one uses a GP function prior or Gaussian parameter prior and one makes observations from a Gaussian likelihood, the posterior parameter distribution (if applicable) and the posterior predictive distribution are also Gaussian and available in closed-form. Furthermore, the marginal likelihood is available in closed-form and is Gaussian. Access to the marginal likelihood allows for empirical Bayes, or type II maximum likelihood for estimation of hyperparameters of the kernel and mean functions. When combined with deep learning, this approach is known as deep kernel learning [Wilson et al., 2016]. This allows for deep neural network features to be used end-to-end in probabilistic regression models. Unfortunately, while expressive deep learning features are used, which leads to improved point estimates, the uncertainty itself is limited to a unimodal Gaussian distribution. + +Generalised squared families Let $\psi : \mathbb{Q} \to \mathbb{R}^n$ be some feature mapping, and let $\mu$ be a measure supported on $\mathbb{Q}$ . We construct densities $q(\omega \mid \Theta, \mu, \psi)$ (with respect to measure $\mu$ ) which are proportional to the squared Euclidean norm of $\Theta \psi(\omega)$ . That is, + +$$ +q (\boldsymbol {\omega} \mid \boldsymbol {\Theta}, \mu , \boldsymbol {\psi}) = \frac {\left\| \boldsymbol {\Theta} \boldsymbol {\psi} (\boldsymbol {\omega}) \right\| _ {2} ^ {2}}{z (\boldsymbol {\Theta})}, \qquad z (\boldsymbol {\Theta}) = \int_ {\Omega} \left\| \boldsymbol {\Theta} \boldsymbol {\psi} (\boldsymbol {\omega}) \right\| _ {2} ^ {2} d \mu (\boldsymbol {\omega}). +$$ + +The normalising constant $z(\Theta)$ satisfies a convenient parameter-integral factorisation + +$$ +z (\boldsymbol {\Theta}) = \operatorname {T r} \left(\boldsymbol {\Theta} ^ {\top} \boldsymbol {\Theta} \left(\overbrace {\int_ {\Omega} \psi (\omega) \psi (\omega) ^ {\top} \mu (d \omega)} ^ {K _ {\mu , \psi} \triangleq}\right)\right), +$$ + +which follows from the cyclic property of the trace and linearity of the integral. The squared family kernel $K_{\mu, \psi}$ is a PSD matrix, as is $\Theta^{\top} \Theta$ , and the normalising constant may be viewed as an inner product of PSD matrices. The squared family kernel is available in closed-form for many combinations of $\Omega$ , $\mu$ and $\psi$ , with examples from the neural network Gaussian process [Neal, 1995] and neural tangent kernel literature [Han et al., 2022, table 1], random feature literature [Rahimi and Recht, 2007], and classical exponential families [Nielsen and Garcia, 2024] (see Appendix B). The parameter factor $\Theta^{\top} \Theta$ involves no integration, and the integral factor $K_{\mu, \psi}$ does not depend on any parameters. This factorisation extends beyond the normalising constant to other integrals (such as those involved in the marginal likelihood, posterior parameter, and posterior predictive distributions) and is very helpful, allowing factorisation into $\Theta^{\top} \Theta$ and a parameter-independent integral. + +The numerator $\left\| \Theta \psi (\omega)\right\| _2^2 = \operatorname {Tr}\left(\Theta^\top \Theta \psi (\omega)\psi (\omega)^\top\right)$ may also be understood as an inner product of a PSD parameter matrix and a rank 1 feature matrix. Hence we may also parameterise $q(\omega \mid \Theta ,\mu ,\psi)$ and its corresponding probability measure $Q(d\omega \mid M,\mu ,\psi)$ in terms of $M = \Theta^{\top}\Theta$ + +$$ +Q (d \boldsymbol {\omega} \mid M, \mu , \psi) = \frac {\operatorname {T r} \left(M \psi (\boldsymbol {\omega}) \psi (\boldsymbol {\omega}) ^ {\top}\right)}{z (M)} \mu (d \boldsymbol {\omega}), \quad z (M) = \operatorname {T r} \left(M K _ {\mu , \psi}\right). \tag {1} +$$ + +Distributions of the form (1) generalise mixture models, which are obtained when $M$ is diagonal. + +# 3 Conjugacy of Generalised Squared Families + +In order to better describe convenient properties of Bayesian updates, we introduce various families of probability densities. We later describe precisely how in some sense these families are closed under Bayesian updating, and how this closure is computationally attractive. + +Definition 1. Let $\mathbb{M}$ be a set of $n\times n$ PSD matrices, $\wedge$ be a set of nonnegative measures, and $\Psi$ be a set of feature mappings of the form $\psi :\mathbb{Q}\to \mathbb{R}^n$ . A generalised squared family (GSF) is a set of probability measures of the form (1), + +$$ +\mathbb {Q} _ {\mathbb {M}, \mathbb {A}, \Psi} = \{Q (d \boldsymbol {\omega} \mid \boldsymbol {M}, \mu , \psi) \} _ {\boldsymbol {M} \in \mathbb {M}, \mu \in \mathbb {A}, \psi \in \Psi}, +$$ + +indexed by $M \in \mathbb{M}, \mu \in \mathbb{A}, \psi \in \mathbb{P}$ . We allow any of the indexing sets $\mathbb{M}, \mathbb{A}, \mathbb{P}$ to be a singleton, and with an abuse of notation, write the singleton's element in place of the set. For example, + +$$ +\mathbb {Q} _ {M, \mathbb {A}, \psi} = \left\{Q (d \boldsymbol {\omega} \mid M, \mu , \boldsymbol {\psi}) \right\} _ {\mu \in \mathbb {A}} \quad a n d \quad \mathbb {Q} _ {M, \mathbb {A}, \Psi} = \left\{Q (d \boldsymbol {\omega} \mid M, \mu , \boldsymbol {\psi}) \right\} _ {\mu \in \mathbb {A}, \psi \in \Psi}. +$$ + +We note that by Von Neumann's trace inequality, $\mathbb{M}$ may be taken to be the set of all non-zero PSD matrices as long as the squared family kernel $K_{\mu,\psi}$ is PD, since $\operatorname{Tr}\left(MK_{\mu,\psi}\right) = z(M) > 0$ . The inclusions $\mathbb{Q}_{\mathbb{M},\mu,\psi} \subseteq \mathbb{Q}_{\mathbb{M},\mathbb{A},\psi} \subseteq \mathbb{Q}_{\mathbb{M},\mathbb{A},\psi}$ , together with a summary of some of our later results, are visualised in Figure 2. Previous work [Tsuchida et al., 2025] called families of the form $\mathbb{Q}_{\mathbb{M},\mu,\psi}$ , where $\mathbb{M}$ is the set of rank 1 PSD matrices, squared families. Throughout this paper, we use $Q$ for probability measures belonging to GSFs, and $P$ and $p$ for arbitrary probability measures and densities (which may also belong to GSFs, where stated). + +![](images/007df93f850bcaeae107734baf70430668375b7682c0798755fba04c18e9dafc.jpg) +Figure 3: (Left) In § 3.1, we consider an abstract model where random variables following GSFs $\omega$ are latent under an observation model $\mathbf{u} \sim p(\mathbf{u} \mid \boldsymbol{\omega})$ . (Middle) In § 3.2 specialise on a setting where we expand the observed node $\mathbf{u}$ so that observations take the form of $(\mathbf{x}, \mathbf{y})$ pairs, and assume an observation model of the form $\mathbf{y} \sim p(\mathbf{y} \mid \Omega \gamma(\mathbf{x}))$ . Here $\boldsymbol{\omega} = \mathrm{vec}(\Omega^{\top})$ is a parameter and $\gamma$ is some deterministic feature mapping (for example, from a deep neural network). Finally in § 3.3, we focus on the setting where the likelihood $p(\mathbf{y} \mid \Omega \gamma(\mathbf{x}))$ and base measure $\mu$ are both Gaussian, allowing us to generalise finite-feature Gaussian process regression. (Right) Our results extend to the case of a hierarchical model with a shared prior over $J$ datasets. We apply this to few-shot learning in § 4. + +# 3.1 Marginal likelihood and posterior distributions + +We first consider an abstract setting where there are two variables of interest — a variable which we observe and a latent variable which we do not observe but would like to infer. The latent variable $\omega$ a priori is distributed $\omega \sim Q(d\omega \mid M, \mu, \psi)$ according to an element of a GSF, and is mapped to the observed variable $\mathbf{U} = \{\mathbf{u}_i\}_{i=1}^N$ implicitly through some arbitrary likelihood function $p(\mathbf{u} \mid \omega)$ . This model is depicted in the left part of Figure 3. + +The main trick in performing posterior updates in GSFs is that the base measure and the likelihood combine to give a new base measure. We derive the marginal likelihood in the main text to expose the simplicity of this idea. Here we make use of the parameter-integral factorisation. The proofs of later results follow a similar manipulation (see Appendices C, D). + +Proposition 2. Consider the left model of Figure 3. Define a measure $\nu(d\omega) = p(\mathbf{U} \mid \omega)\mu(d\omega)$ as the product of the likelihood and base measure. The marginal likelihood is the ratio of normalising constants, $p(\mathbf{U}) = \operatorname{Tr}\left(M\mathbf{K}_{\nu,\psi}\right) / \operatorname{Tr}\left(M\mathbf{K}_{\mu,\psi}\right)$ (where $\mathbf{K}_{\nu,\psi}$ depends on $\mathbf{U}$ through $\nu$ ). + +Proof. The marginal likelihood $p(\mathbf{U})$ is given by + +$$ +\int_ {\Omega} p (\boldsymbol {U} \mid \boldsymbol {\omega}) Q (d \boldsymbol {\omega} \mid \boldsymbol {M}, \mu , \boldsymbol {\psi}) = \frac {\mathrm {T r} \left(M \int_ {\Omega} \boldsymbol {\psi} (\boldsymbol {\omega}) \boldsymbol {\psi} (\boldsymbol {\omega}) ^ {\top} \overbrace {p (\boldsymbol {U} \mid \boldsymbol {\omega}) \mu (d \boldsymbol {\omega})} ^ {\nu (d \boldsymbol {\omega})}\right)}{\mathrm {T r} \left(\boldsymbol {M K} _ {\mu , \boldsymbol {\psi}}\right)} = \frac {\mathrm {T r} \left(\boldsymbol {M K} _ {\nu , \boldsymbol {\psi}}\right)}{\mathrm {T r} \left(\boldsymbol {M K} _ {\mu , \boldsymbol {\psi}}\right)}. +$$ + +![](images/f14baaefc1821cfcc5df7bd4e63ac090635b1771a6ad08a659519a1da3daa5dd.jpg) + +The marginal likelihood does not belong to a GSF. The posterior distribution does belong to a GSF, even for finite $N$ . We may form a GSF which includes both the prior and the posterior, with an updated base measure and an unchanged parameter. + +Proposition 3. Consider the model in Figure 3 (Left). Define a measure $\nu(d\omega) = p(U \mid \omega)\mu(d\omega)$ . The posterior $P(d\omega \mid U) = Q(d\omega \mid M, \nu, \psi)$ belongs to a GSF with base measure $\nu$ , feature mapping $\psi$ and parameter $M$ . + +Proposition 3 implies that the GSF $\mathbb{Q}_{M,\mathbb{A},\psi}$ is a conjugate class for likelihood $p(\boldsymbol{u} \mid \boldsymbol{\omega})$ , where $\mathbb{A}$ is the set of all measures that can be expressed in the form $\left(\prod_{i=1}^{h} p(\boldsymbol{u}_i \mid \boldsymbol{\omega})\right) \mu(d\boldsymbol{\omega})$ for some integer $h \geq 0$ and $\boldsymbol{u}_i$ belonging to the support of the likelihood. We note that the posterior predictive is $p(\boldsymbol{U}_* \mid \boldsymbol{U})$ is also available, since it is essentially of the same form as the marginal likelihood, replacing the GSF prior with a GSF posterior and the training likelihood with the test likelihood. + +Proposition 4. Consider the model in Figure 3 (Left). Define a measure $\nu(d\omega) = p(\mathbf{U} \mid \omega)\mu(d\omega)$ and $\nu'(d\omega) = p(\mathbf{U}_* \mid \omega)\nu(d\omega)$ , where, abusing notation, the train likelihood $p(\mathbf{U} \mid \omega)$ and test likelihood $p(\mathbf{U}_* \mid \omega)$ are not necessarily the same. The posterior predictive is the ratio of normalising constants, $p(\mathbf{U}_* \mid \mathbf{U}) = \operatorname{Tr}\left(M\mathbf{K}_{\nu',\psi}\right) / \operatorname{Tr}\left(M\mathbf{K}_{\nu,\psi}\right)$ . + +The following observation illustrates that it is easy to obtain closed-form expressions for the marginal likelihood, posterior parameter and posterior predictive distributions, whenever the base measure is itself conjugate to the likelihood (for exponential family examples, see Appendix A). + +Remark 5. If $\mathbb{A}$ is a conjugate class for $p(\boldsymbol{u} \mid \boldsymbol{\omega})$ , then $\nu \in \mathbb{A}$ (up to a normalising constant), so if the squared family kernel $\pmb{K}_{\mu, \psi}$ is available in closed-form for all $\mu \in \mathbb{A}$ , then so is $\pmb{K}_{\nu, \psi}$ . + +Note that we here refer to two different conjugate updates: One as if the prior were base measure $\mu$ belonging to $\mathbb{A}$ and another for the true prior belonging to a GSF $\mathbb{Q}_{M,\mathbb{A},\psi}$ . As described in § 2 and Appendix B, closed-form expressions for the squared family kernel are available for many of combinations of $\mathbb{O}$ , $\mu$ and $\psi$ [Neal, 1995, Williams, 1996, Rahimi and Recht, 2007, Cho and Saul, 2009, Nielsen and Garcia, 2024, Han et al., 2022, Tsuchida et al., 2023, for example]. + +# 3.2 Regression models + +We now consider a more specific instance of the previously described model, as shown in the middle of Figure 3. We consider Bayesian learning of vector-valued functions of the form + +$$ +\mathbf {f} (\boldsymbol {x}) = \boldsymbol {\Omega} \boldsymbol {\gamma} (\boldsymbol {x}), \quad \boldsymbol {\omega} \sim Q (d \boldsymbol {\omega} \mid M, \mu , \psi), \boldsymbol {\omega} = \operatorname {v e c} \left(\boldsymbol {\Omega} ^ {\top}\right) \in \mathbb {R} ^ {d}, \quad \text {w h e r e} \quad d = d _ {1} d _ {2}, \tag {2} +$$ + +where $\boldsymbol{\Omega}^{\top} \in \mathbb{R}^{d_1 \times d_2}$ are readout parameters and $\gamma: \mathbb{X} \to \mathbb{R}^{d_1}$ is a feature mapping (from e.g. a deep neural network). We assume our observations are independent samples $(\boldsymbol{X}, \boldsymbol{Y}) = \{(x_i, y_i)\}_{i=1}^N$ from $p(\boldsymbol{y} \mid \boldsymbol{\Omega} \boldsymbol{\gamma}(\boldsymbol{x}))$ and some distribution over $\mathbf{x}$ . Let $\Gamma = \gamma(\boldsymbol{X})$ . The marginal likelihood, + +$$ +p (\boldsymbol {Y} \mid \boldsymbol {\Gamma}) = \int_ {\Omega} p (\boldsymbol {Y} \mid \boldsymbol {\Gamma} \boldsymbol {\Omega} ^ {\top}) Q (d \boldsymbol {\omega} \mid \boldsymbol {M}, \mu , \boldsymbol {\psi}), +$$ + +serves two purposes. First, as a normalisation constant for the posterior, and second, as an objective function for point estimation of $M$ , $\mu$ , and $\psi$ . The posterior parameter distribution is given by + +$$ +P (d \boldsymbol {\Omega} \mid \boldsymbol {Y}, \boldsymbol {\Gamma}) = \frac {p (\boldsymbol {Y} \mid \boldsymbol {\Gamma} \boldsymbol {\Omega} ^ {\top}) Q (d \boldsymbol {\omega} \mid \boldsymbol {M} , \mu , \boldsymbol {\psi})}{p (\boldsymbol {Y} \mid \boldsymbol {\Gamma})}, +$$ + +which in turn induces a posterior over noisy evaluations $\mathbf{Y}_{*}$ of $\mathbf{f}$ at new test points (features) $X_{*}(\Gamma_{*})$ . This posterior is called the (noisy) posterior predictive distribution, + +$$ +p \big (\boldsymbol {Y} _ {*} \mid \boldsymbol {Y}, \boldsymbol {\Gamma}, \boldsymbol {\Gamma} _ {*} \big) = \int_ {\Omega} p (\boldsymbol {Y} _ {*} \mid \boldsymbol {\Gamma} _ {*} \boldsymbol {\Omega} ^ {\top}) P (d \boldsymbol {\Omega} \mid \boldsymbol {Y}, \boldsymbol {\Gamma}), +$$ + +and is used to form predictions on test data $\mathbf{X}_{*}$ . Note that the test likelihood $p(\mathbf{Y}_{*} \mid \boldsymbol{\Gamma}_{*}\boldsymbol{\Omega}^{\top})$ may be different to the train likelihood $p(\mathbf{Y} \mid \boldsymbol{\Gamma}\boldsymbol{\Omega}^{\top})$ , e.g. with heteroskedastic noise. The noiseless posterior predictive $p(\mathbf{f}(\mathbf{X}_{*}) \mid \mathbf{Y},\boldsymbol{\Gamma},\boldsymbol{\Gamma}_{*})$ is obtained whenever the test likelihood has zero variance. + +Marginal likelihood, posterior parameter, and posterior predictive distributions The results of Propositions 2, 3 and 4 apply directly to the marginal likelihood, posterior parameter and posterior predictive distributions. That is, defining $\nu(d\omega) = p(\boldsymbol{Y} \mid \Gamma \Omega^{\top}) \mu(d\omega)$ and $\nu'(d\omega) = p(\boldsymbol{Y}_{*} \mid \Gamma_{*}\Omega^{\top}) \nu(d\omega)$ , the marginal likelihood is $p(\boldsymbol{Y} \mid \Gamma, M) = \operatorname{Tr}(MK_{\nu,\psi}) / \operatorname{Tr}(MK_{\mu,\psi})$ , the posterior parameter distribution $P(d\Omega \mid \boldsymbol{Y}, \Gamma) = Q(d\omega \mid M, \nu, \psi)$ belongs to a GSF and the posterior predictive distribution is $p(\boldsymbol{Y}_{*} \mid \Gamma_{*}, \boldsymbol{Y}, \Gamma) = \operatorname{Tr}(MK_{\nu',\psi}) / \operatorname{Tr}(MK_{\nu,\psi})$ . Note also that Remark 5 still applies; if $\mu, \nu$ and $\nu'$ remain inside the same conjugate class and $K_{.,\psi}$ is available within that family, then closed-form normalising constants are available. We utilise this observation in the following subsection, focusing on conjugacy of the Gaussian family with Gaussian likelihood. + +# 3.3 Generalised squared family processes + +We now specialise our choice of base measure and likelihood. Take the Radon-Nikodym derivative of the base measure $\mu$ with respect to Lebesgue measure to be a matrix Gaussian, + +$$ +\frac {d \mu}{d \lambda} (\boldsymbol {\Omega}) \triangleq p (\boldsymbol {\Omega}) = \mathcal {N} (\boldsymbol {\omega} \mid \operatorname {v e c} (\boldsymbol {M}), \boldsymbol {B} \otimes \boldsymbol {C}). \tag {3} +$$ + +Take the train likelihood to be multivariate Gaussian with homoskedastic noise with variance $v^2$ , + +$$ +p (\boldsymbol {Y} \mid \boldsymbol {\Gamma} \boldsymbol {\Omega} ^ {\top}) = \mathcal {N} \left(\operatorname {v e c} (\boldsymbol {Y}) \mid \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma}\right) \boldsymbol {\omega}, v ^ {2} \boldsymbol {I} _ {N d _ {2}}\right) \tag {4} +$$ + +Take the test likelihood to be multivariate Gaussian with homoskedastic noise with variance $v'^2$ , + +$$ +p \left(\boldsymbol {Y} _ {*} \mid \boldsymbol {\Gamma} _ {*} \boldsymbol {\Omega} ^ {\top}\right) = \mathcal {N} \left(\operatorname {v e c} \left(\boldsymbol {Y} _ {*}\right) \mid \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {\omega}, v ^ {\prime 2} \boldsymbol {I} _ {N _ {*} d _ {2}}\right). \tag {5} +$$ + +With a GSF prior, we refer to such a model as a GSF process (GSFP). + +Completing the square and parameter-integral factorisation By Proposition 4 and Gaussian conjugacy, in order to compute the posterior predictive distribution, we need only reverse the factorisation $\nu'(d\omega) = p(Y_* \mid \Gamma_* \Omega^\top)p(Y \mid \Gamma \Omega^\top)\mu(d\omega)$ as $\nu'(d\omega) = \mu(d\omega \mid Y, Y_*)p(Y_ \mid Y)p(Y)$ , which is Gaussian, and same for $\nu(d\omega) = p(Y \mid \Gamma \Omega^\top)\mu(d\omega)$ . The mean and covariance parameters of these Gaussian measures can be found using the same techniques as in vector-valued Bayesian linear regression. We show only the noise free posterior predictive distribution here, with more details in Appendix D. + +Corollary 6. Consider the special model (3), (4) and (5) applied to Figure 3 (Right). Define $\nu'(d\omega) = p(\mathbf{Y}_* \mid \Gamma_* \Omega^\top)p(\mathbf{Y} \mid \Gamma \Omega^\top)\mu(\Omega)$ and + +$$ +\boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime} = \operatorname {v e c} (\boldsymbol {M}) + (\boldsymbol {B} \otimes \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top}) (\boldsymbol {B} \otimes \boldsymbol {\Gamma} \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top} + v ^ {2} \boldsymbol {I} _ {N d _ {2}}) ^ {- 1} \left(\operatorname {v e c} (\boldsymbol {Y}) - \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma}\right) \operatorname {v e c} (\boldsymbol {M})\right) +$$ + +$$ +\boldsymbol {C} ^ {\prime} = \boldsymbol {B} \otimes \boldsymbol {C} - (\boldsymbol {B} \otimes \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top}) (\boldsymbol {B} \otimes \boldsymbol {\Gamma} \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top} + v ^ {2} \boldsymbol {I} _ {N d _ {2}}) ^ {- 1} (\boldsymbol {B} \otimes \boldsymbol {\Gamma} \boldsymbol {C}). +$$ + +Suppose that $C'$ and $\Gamma_*$ are full-rank and $N_* \geq d_1$ . Then the noise free predictive distribution $P\big(d\pmb{f}(\pmb{X}_*) \mid \pmb{Y}, \Gamma, \Gamma_*\big) = Q\big(d\pmb{f}(\pmb{X}_*) \mid \pmb{M}, \nu'', \psi'\big)$ belongs to a GSF with modified feature mapping and modified base measure, + +$$ +\psi^ {\prime} \left(\boldsymbol {f} \left(\boldsymbol {X} _ {*}\right)\right) \triangleq \psi \left(\left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) ^ {\dagger} \boldsymbol {f} \left(\boldsymbol {X} _ {*}\right)\right) = \psi \left(\operatorname {v e c} \left(\boldsymbol {\Gamma} _ {*} ^ {\dagger} \boldsymbol {f} \left(\boldsymbol {X} _ {*}\right)\right)\right) +$$ + +$$ +\nu^ {\prime \prime} (d \boldsymbol {f} (\boldsymbol {X} _ {*})) \triangleq \mathcal {N} \left(\operatorname {v e c} (\boldsymbol {f} (\boldsymbol {X} _ {*})) \mid (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}) \boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime}, (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}) \boldsymbol {C} ^ {\prime} (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top})\right) d \boldsymbol {f} (\boldsymbol {X} _ {*}). \tag {6} +$$ + +Corollary 6 implies that under the Gaussian likelihood and base measure setting, the GSF $\mathbb{Q}_{M,\mathbb{A},\Psi}$ is a conjugate class, where $\Psi$ is the class of transformations obtained by composition of $\psi$ with linear transformations and $\Lambda$ involves deformation via Gaussian conjugate updates, as per (6). + +# 3.4 Related work + +To the best of our knowledge, we are the first to consider squared families as priors and posteriors. The nature of the results concerning likelihoods is fundamentally different to that of priors and posteriors, because for example, one does not attempt to estimate parameters in forming a posterior, one merely attempts to form a posterior belief. Nevertheless, existing work on special cases of squared family likelihoods is relevant in that expressive power, flexibility and tractability of densities is intuitively relevant to both likelihoods and posteriors, and uncertainty quantification more generally. + +Kernel methods Building off the general results of modelling non-negative functions using squared elements of RKHS [Marteau-Ferey et al., 2020], Rudi and Ciliberto [2021] model probability distributions, with tractable marginalisation and closure under multiplication. They fit likelihood models within this class by minimising a regularised L2 distance between the target likelihood and the model. While these models are nonparametric in construction, estimates resulting from minimising the objective satisfy a representer theorem type instantiation in finite dimensions. The authors show that the model admits both favourable representational capability (in being able to represent a $\beta$ -times differentiable target density with error less than with number of parameters scaling like $\mathcal{O}(\epsilon^{-r / \beta}(\log (1 / \epsilon))^{r / 2})$ , where $r$ is the dimension of the domain of the distribution), as well as statistical estimation properties (as likelihoods, the proposed estimate converges at a rate of $N^{-\frac{\beta}{2\beta + r}}(\log N)^{r / 2}$ , where $N$ is the number of datapoints). + +Probabilistic circuits Probabilistic circuits [Choi et al., 2020] provide a computational framework for tractable probabilistic modelling. They are constructed as graphs of connected units, and by constraining the graph, allow for tractable marginalisation. However, they need to impose some structure on the units or the connections in order to ensure nonnegativity. Squared probabilistic circuits [Sladek, 2023, Loconte et al., 2023a,b, Lo Conte and Vergari, 2025] bypass this constraint by squaring the output of the circuit. Sums of squared circuits can also be instantiated [Lo Conte et al., 2024]. The focus on such works is typically in showing tractable (polynomial or indeed quadratic time/memory in the number of units) computation of normalising constants or marginalisation, as well as the inclusions of the function spaces imposed by different classes of probabilistic circuits. + +Neural networks and finite feature models Previous work [Tsuchida et al., 2025] has modelled likelihoods as proportional to the squared Euclidean norm of a function of the form $\Theta \psi(x)$ , where $\Theta$ is a matrix and $\psi$ is a vector-valued function. Provided the features $\psi$ are rich enough, such models admit a universal approximation property, typically approximating the squared L2 distance between a square root of a target density and the model at a rate of $\mathcal{O}(1/n)$ , where $n$ is a parameter count. As likelihoods, they learn target densities with a KL divergence to the target density decreasing at a rate of $\mathcal{O}(1/\sqrt{N})$ , where $N$ is the number of samples. Such families of models have tractable normalising constants, Fisher information and statistical divergences. Furthermore, in some special cases, such families have not only tractable but indeed closed-form normalising constants, Fisher information and statistical divergences. An example studied previously was squared neural families [Tsuchida et al., 2023], where is a single hidden layer neural network. + +Applications Poisson point processes (PPPs) are helpful variations of density modelling, where one models an intensity instead of a density. Whereas in density modelling, realisations are i.i.d. draws from a target density, in intensity modelling, realisations are conditionally i.i.d. draws from a target density given the number of realisations. The number of realisations follows a Poisson distribution with an expected number of points equal to the integral of the intensity over the domain. Flaxman et al. [2017] used (frequentist) squared elements of RKHSs to model intensity functions, whereas Walder and Bishop [2017] used a (Bayesian) squared Gaussian process prior to model intensity functions. We note that placing a prior over a function which when squared gives a intensity/density (likelihood) is completely disjoint to the problem we are solving here, which is to use a squared function as a prior density. Probabilistic circuits have been applied in converting knowledge graph embeddings into generative models [Lo Conte et al., 2023a]. Squared neural family likelihoods have been applied to label distribution learning [Zhang et al., 2025], which is a kind of variation on classical regression where instead of the target label being a single class, the target variable is a discrete distribution (i.e. an element of the simplex). Hence, here squared neural family models model a distribution over distributions with a closed-form normalising constant, expectation, variance, and covariance conditioned on input samples. Probabilistic modelling is leveraged to provide confidence intervals, conformal predictions, active learning, and model ensembling. + +# 4 Experiments on few-shot learning + +Few shot learning setting The hyperparameters of a nonparametric probabilistic model (such as a GP or GSFP) can be adapted to a single training dataset $(\mathbf{X},\mathbf{Y})$ with $N$ examples, by maximising the marginal likelihood $p(\mathbf{Y}\mid \mathbf{X})$ under the model with respect to the model hyperparameters, also referred to as type II maximum marginal likelihood [Rasmussen and Williams, 2006, § 5.4]. It may also happen that a meta dataset, consisting of $J$ datasets $\{(X_j,Y_j)\}_{j = 1}^J$ , is assumed to arise as $J$ samples from a model sharing the same hyperparameters, in which case the marginal likelihood $p\big(\{\mathbf{Y}_j\}_{j = 1}^J\mid \{\mathbf{X}_j\}_{j = 1}^J\big)$ still serves as a natural objective for tuning model hyperparameters [Rasmussen and Williams, 2006, p. 115]. In this case, each of the $(X_{j},Y_{j})$ consists of $N_{j}$ examples, where $N_{j}$ may be constant $N$ across all $J$ datasets, but not necessarily. At inference time, when a new support dataset $(X_1^*,Y_1^*)$ and a new set of query inputs $X_{2}^{*}$ are given, one may form the posterior predictive $p(Y_2^*\mid X_2^*,X_1^*,Y_1^*)$ using the hyperparameters obtained by maximising the marginal likelihood of the tasks. This classical setting arises from point estimation of the hyperparameters under the assumption of shared point hyperparameters (see right side of Figure 3), and has traditionally been referred to under the umbrella of multi-task learning [Caruana, 1997, Minka and Picard, 1997]. + +This setting has recently received renewed attention under the name of few-shot learning, and GP models have been empirically shown to outperform other more recent models on modern benchmarks, + +![](images/5ea9f4e1cdaf1737d39b369bff9132a23f1d2fff7bfde3000b155cadf514e59e.jpg) + +![](images/0a5834b88dd28a2c58dd900149b6403caaeae93bf968ca53066b286cc14d87e3.jpg) + +![](images/9db2eefa262e37248001f4a85e4a17053a2499672354ce2c9cff058894b319e8.jpg) + +![](images/c6a8b788f44c7f66c1db918e201afb8bb8252af52ef619b6d81472c24ddaa810.jpg) +Figure 4: When can a multimodal prior be helpful? (Top) GP using deep learning features (deep kernel learning [Wilson et al., 2016]). (Bottom) GSFP using deep learning features. (Left) Prior predictive, trained using points (white) drawn from a multimodal distribution over functions (with mean given by red and blue). The GFSP is bimodal, whereas the GP is always unimodal. (Middle) Posteriors for both models given some ambiguous points, which may belong to either red or blue modes. The GP incorrectly places all of its belief over one of the modes, whereas the GFSP reserves belief for both modes. (Right) Given additional data from $x \leq 0$ , both identify the correct mode. + +![](images/3fff4cd7e484df53cd8d6535d01eb7d736ebac66ec74ea99c64a3d0d903b394d.jpg) + +![](images/9c04e507d7f3a7157d468d6f6e0247a4affbef9188caa9a41efc1696be94ce34.jpg) + +both in the regression and classification setting [Patacchiola et al., 2020] (Deep kernel transfer, or DKT). In the regression setting, further enhancements to GP models, which use continuous normalising flows to transform GP marginal predictive posterior distributions into more complicated distributions, have further been used to improve upon GPs [Sendera et al., 2021] (Non-Gaussian Gaussian processes, or NGGP). Such enhancements address GP prior limitations; namely lack of an ability to probabilistically model skewed, non-Gaussian-tailed or multimodal beliefs. Our GSFP also offer a means to address such limitations (see Figure 4), and are therefore an excellent candidate for few-shot learning. We consider the same scalar-valued $(d_{2} = 1)$ regression benchmarks as Sendera et al. [2021], who in turn add to the regression benchmarks of Patacchiola et al. [2020]. + +Implementation details We learn a prior predictive distribution, which is further broken into a prior parameter distribution belonging to a GSF $\mathbb{Q}_{\mathbb{M},\Lambda ,\Psi}$ and a (deterministic) feature extractor $\gamma$ , by attempting to maximise the marginal likelihood (see Proposition 2) with respect to some tuneable decision variables. The decision variables are an unconstrained $\Theta$ , with $\Theta^{\top}\Theta \in \mathbb{M}$ and a feature mapping $\psi$ (which together define the prior over $\Omega$ in (2)), as well as the tuneable parameters of the neural network $\gamma$ . We further parameterise the feature mapping $\psi$ as the hidden layer of a learnable neural network, $\psi (\omega) = \sigma \bigl (W\omega +b\bigr)$ , with decision variables $W$ and $b$ . Updates within the set $\wedge$ happen automatically as part of the Bayesian inference pipeline and are not part of the decision variables in the marginal likelihood (see Proposition 2). We reimplemented competitor models DKT [Patacchiola et al., 2020] and NGGP [Sendera et al., 2021], as we were not able to get the originally supplied code to run, and through extensive hyperparameter tuning were able to obtain results better than what they originally reported in all but two datasets (NDX100 and EEG). Full hyperparameter details and run times are given in Appendix F. Results are shown in Table 4. We observe that GFSP obtains the lowest NLL on 11 out of 12 settings, although sometimes is well within one standard deviation of the best competing method. While our method is designed for situations in which a multimodal belief is helpful, e.g. few-shot learning, we can also apply it to vanilla regression. See Appendix G. + +# 5 Discussion, limitations and conclusion + +Limitations The main computational hurdle in working with GSFs is computing the (exact) normalising constant via the squared family kernel, which has a time complexity of $\mathcal{O}(n^2 d)$ , where $n$ is the size of the hidden layer and $d$ is the dimension of the random variable. In practice, the training time of GSFP is slightly longer than that of DKT, but substantially less than that of NGGP (see Table F.9). The inference time is much lower than NGGP, but slightly more than DKT. As with competitors DKT and NGGP, which are also based on (potentially finite-feature) GP methods, our + +Table 1: Benchmark results showing test NLLs for regression tasks across 6 meta datasets. For each meta dataset, we train 5 meta models with different random seed via maximum marginal likelihood over the $J$ datasets $\left\{ {\left( {\mathbf{X}}_{j},{\mathbf{Y}}_{j}\right) }\right\} {}_{j = 1}^{J}$ . At testing time,we condition (train) on a support (few shot) dataset $\left( {{\mathbf{X}}_{1}^{ * },{\mathbf{Y}}_{1}^{ * }}\right)$ and evaluate the test NLL on the query dataset $\left( {{\mathbf{X}}_{2}^{ * },{\mathbf{Y}}_{2}^{ * }}\right)$ . We evaluate the test NLL on these few shot datasets for 500 random shuffles of the test meta dataset. In total, $5 \times {500} = 2,{500}$ models are trained. Each entry shows the mean $\pm$ standard deviation over 2,500 model evaluations. Each testing scenario includes in-range and out-of-range, where the support and query set is either similar to or different to the $J$ training datasets. Best results in bold,second best underlined. + +
MethodsSines[Patacchiola et al., 2020]Mixed-Noise Sines [Sendera et al., 2021]QMUL [Gong et al., 1996]
NameKernelNLLIn↓NLLOut↓NLLIn↓NLLOut↓NLLIn↓NLLOut↓
DKTRBF-0.71 ± 0.06-0.62 ± 0.080.76 ± 0.061.07 ± 0.13-0.73 ± 0.19-0.69 ± 0.21
Spectral-0.82 ± 0.05-0.79 ± 0.060.38 ± 0.171.78 ± 0.49-0.67 ± 0.24-0.64 ± 0.20
NN Linear-0.76 ± 0.090.30 ± 1.070.47 ± 0.252.31 ± 0.72-0.66 ± 0.490.87 ± 1.18
NGGPRBF-0.74 ± 0.06-0.56 ± 0.140.28 ± 0.132.45 ± 0.87-0.40 ± 0.690.15 ± 0.70
Spectral-0.84 ± 0.05-0.73 ± 0.070.38 ± 0.171.78 ± 0.49-0.99 ± 0.31-0.77 ± 0.31
NN Linear-0.80 ± 0.060.27 ± 1.260.24 ± 0.892.44 ± 1.320.04 ± 1.341.41 ± 1.27
GSFPNN Linear-0.85 ± 0.06-0.83 ± 0.070.46 ± 0.100.98 ± 0.21-1.09 ± 0.12-1.15 ± 0.14
MethodsNDX100 [Qin et al., 2017]EEG [Fernandez-Fraga et al., 2019]Power [Hebrail and Berard, 2006]
NameKernelNLLIn↓NLLOut↓NLLIn↓NLLOut↓NLLIn↓NLLOut↓
DKTRBF-1.16 ± 0.03-1.17 ± 0.03-1.10 ± 0.54-1.16 ± 1.09-0.47 ± 0.62-0.34 ± 0.70
Spectral-0.46 ± 0.43-0.63 ± 0.30-1.32 ± 0.310.84 ± 0.67-0.47 ± 0.62-0.35 ± 0.70
NN Linear-2.16 ± 1.61-2.41 ± 0.96-1.10 ± 0.52-1.02 ± 1.06-0.42 ± 0.68-0.33 ± 0.73
NGGPRBF1.16 ± 0.011.15 ± 0.01-0.96 ± 0.470.05 ± 0.321.15 ± 0.021.15 ± 0.02
Spectral-1.32 ± 0.310.84 ± 0.67-1.07 ± 0.30-0.19 ± 0.371.35 ± 0.011.35 ± 0.01
NN Linear-2.07 ± 1.26-2.32 ± 0.62-1.05 ± 0.59-1.18 ± 0.89-0.27 ± 0.40-0.27 ± 0.40
GSFPNN Linear-2.49 ± 1.28-2.75 ± 0.53-1.49 ± 0.44-1.38 ± 0.81-0.71 ± 0.39-0.65 ± 0.45
+ +method has $\mathcal{O}(N^2)$ and $\mathcal{O}(N^3)$ memory and time complexity for inference (or $\mathcal{O}(nN)$ and $\mathcal{O}(n^3)$ for finite-feature models), where $N$ is the number of training points. + +Computational complexity of Bayesian inference In the general case, Bayesian inference is computationally difficult, unless some heavy restriction is placed on the densities (e.g. log concave). This is especially difficult when the dimension is large, due to the curse of dimensionality and the exponential growth of volume. In practice, if one desires an exact computation, one needs to restrict the class of probability distributions. When we restrict the class to GSFs, we are able to compute exact posteriors. The complexity of this calculation is governed by the inner product of the parameter matrix $M$ and the kernel matrix $K_{\mu,\psi}$ (as in (1)). Assuming the kernel matrix is known, this inner product has complexity $\mathcal{O}(n^2)$ , where $n$ is the number of parameters in the GSF. Crucially, dependence of the complexity on $d$ only appears through the calculation of the kernel matrix $K_{\mu,\psi}$ . In practice, the kernel matrices $K_{\mu,\psi}$ can be computed exactly in linear time in $d$ . This leads to a combined complexity of $\mathcal{O}(dn^2)$ , i.e. exact inference linear in dimension $d$ , which is a huge improvement over inexact inference exponential in dimension $d$ . This improvement comes with the restriction of the class of densities to GSFs, however this restriction is not too severe because GSFs are rich density approximators. More precisely navigating the trade-off between universal approximation (making $n$ large) and computational efficiency (making $n$ small) is an important direction for future work. + +Conclusion GSF likelihoods appear in disguise in a wide range of applications, including Poisson point processes [Flaxman et al., 2017, Walder and Bishop, 2017, Tsuchida et al., 2024] generative models [Lo Conte et al., 2023a], and most fundamentally, density estimation [Rudi and Ciliberto, 2021]. Here we considered the orthogonal setting of GSF priors. Using the parameter-integral factorisation, we found that GSFs form conjugate priors and that for many likelihoods of interest, admit closed-form Bayesian updates. Such a closed-form update allows one to generalise Gaussian parameter prior and Gaussian likelihood regression in feature space models to GSF priors, allowing for multimodal and rich expressions of uncertainty. We empirically demonstrated that GSFPs perform better than or as well as other models on a range of benchmark few-shot regression problems. In future, it should be possible to show asymptotic normality of maximum marginal likelihood estimates, as well as provide generalisation bounds for the problem of density estimation under a hierarchical model, by using the fact (see Footnote 2) that the marginal likelihood asymptotically belongs to a GSF and using known results for GSF likelihoods. + +# Acknowledgements + +DS was partially supported by the Responsible AI Research Centre (RAIR). + +# References + +Maurizio A Alvarez, Lorenzo Rosasco, Neil D Lawrence, et al. Kernels for vector-valued functions: A review. Foundations and Trends® in Machine Learning, 4(3):195-266, 2012. +Christopher M Bishop. Pattern recognition and machine learning, volume 4. Springer, 2006. +Lawrence D Brown. 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Let $\pmb{t} : \mathbb{X} \to \mathbb{R}^{a_2}$ with $\mathbb{X} \subseteq \mathbb{R}^{a_1}$ , and let $H$ be a nonnegative measure. Let $A$ be such that + +$$ +P (d \boldsymbol {x} \mid \boldsymbol {\eta}) = \exp (\boldsymbol {\eta} ^ {\top} \boldsymbol {t} (\boldsymbol {x}) - A (\boldsymbol {\eta})) H (d \boldsymbol {x}) +$$ + +integrates to 1. The set $\{P(dx\mid \eta)\}_{\eta \in \{\eta^{\prime}|0 < \int \exp (\eta^{\prime \top}t(x))H(dx) < \infty \}}$ is called an exponential family. A conjugate family for an exponential family likelihood has densities of the form + +$$ +p (\boldsymbol {\eta} \mid \boldsymbol {\chi}, r) = \frac {\exp \left(\boldsymbol {\chi} ^ {\top} \boldsymbol {\eta} - r A (\boldsymbol {\eta})\right)}{\int \exp \left(\boldsymbol {\chi} ^ {\top} \boldsymbol {\eta} - r A (\boldsymbol {\eta})\right) d \boldsymbol {\eta}} \propto \exp \left(- r A (\boldsymbol {\eta})\right) \exp (\boldsymbol {\chi} ^ {\top} \boldsymbol {\eta}) +$$ + +That is, a conjugate family is $\{p(\boldsymbol{\eta} \mid \boldsymbol{\chi}, r)\}_{\boldsymbol{\chi} \in \mathbb{R}^{a_2}, r > 0}$ . Note that the right expression is in the form of an exponential family with canonical parameters $\widetilde{\boldsymbol{\eta}} = (\boldsymbol{\chi}, r)$ and sufficient statistic $\widetilde{\boldsymbol{t}}(\boldsymbol{\eta}) = (\boldsymbol{\eta}, -A(\boldsymbol{\eta}))$ . The conjugate prior has the exponential family form + +$$ +p (\boldsymbol {\eta} \mid \widetilde {\boldsymbol {\eta}}) = \exp \left(\widetilde {\boldsymbol {\eta}} ^ {\top} \widetilde {t} (\boldsymbol {\eta}) - \widetilde {A} (\widetilde {\boldsymbol {\eta}})\right), \quad \widetilde {A} (\widetilde {\boldsymbol {\eta}}) = \log \int \exp \left(\widetilde {\boldsymbol {\eta}} ^ {\top} \widetilde {t} (\boldsymbol {\eta})\right) d \boldsymbol {\eta}. +$$ + +Given iid observations $X \in \mathbb{R}^{N \times a_1}$ from the likelihood, the posterior is obtained by updating the parameters of the prior with the sufficient statistics, + +$$ +p (\boldsymbol {\eta} \mid \boldsymbol {X}, \boldsymbol {\chi}, r) = p \left(\boldsymbol {\eta} \mid \boldsymbol {\chi} + \sum_ {i = 1} ^ {N} \boldsymbol {t} \left(\boldsymbol {x} _ {i}\right), r + N\right). +$$ + +# A.2 Examples + +Poisson-Gamma Take $\mathbb{X} = \{0,1,\ldots\}$ , $h(x) = \frac{1}{x}$ and $t(x) = x$ . This results in a Poisson likelihood, with rate $\lambda = \exp \eta$ and $A(\eta) = \exp(\eta)$ . The conjugate family is a set of distributions of the form + +$$ +p (\eta \mid \chi , r) \propto \exp (A (- r \eta)) \exp (\chi \eta) = \exp (- r \exp (\eta)) \exp (\chi \eta). +$$ + +Taking $\lambda = \exp \eta$ , and accounting for the Jacobian of the transformation, we identify the family of densities of the form + +$$ +p (\lambda \mid \chi , r) \propto \lambda^ {\chi - 1} \exp (- r \lambda), +$$ + +which are Gamma densities with rate $\lambda$ and shape $\chi$ . These have a known (divisive) normalising constant given by $\Gamma(\chi)/r^{\chi}$ , where here $\Gamma$ denotes the Gamma function. The posterior density is then + +$$ +p (\lambda \mid \boldsymbol {X}, \chi , r) \propto \lambda^ {\chi + \sum_ {i = 1} ^ {N} \boldsymbol {t} (\boldsymbol {x} _ {i}) - 1} \exp (- (r + N) \lambda), +$$ + +with known normalising constant. The corresponding prior distribution over $\eta = \log \lambda$ , as an exponential family, is the exponential-gamma distribution, with + +$$ +p (\eta \mid \boldsymbol {X}, \chi , r) = \frac {r ^ {\chi}}{\Gamma (\chi)} \exp (\chi \eta) \exp (- r \exp (\eta)). +$$ + +An example of Poisson-Gamma updates is shown in Figure 5. + +Gaussian-Gaussian Take $\mathbb{X} = \mathbb{R}$ , $h(x) = \exp \left(-x^{2} / (2s^{2})\right)\left(2\pi s^{2}\right)^{-1 / 2}$ , and $t(x) = x / s$ for some known standard deviation $s > 0$ . This results in a Gaussian likelihood, with known variance $s^2$ , mean $s\eta$ and $A(\eta) = \eta^2 /2$ . The conjugate family is a set of distributions of the form + +$$ +p (\eta \mid \chi , r) \propto \exp (- r \eta^ {2} / 2) \exp (\chi \eta), +$$ + +which are Gaussian densities with variance $1 / r$ and mean $\chi /r$ . These have a known (divisive) normalising constant, and + +$$ +p (\eta \mid \chi , r) = \frac {1}{\sqrt {2 \pi r ^ {- 1}}} \exp \left(- \frac {1}{2 r ^ {- 1}} (\eta - \chi / r) ^ {2}\right). +$$ + +![](images/f8a736f38fa06125d190a03b68c6ac4aacfba89687143397c4ce3d2244f68a13.jpg) +Figure 5: Bayesian inference in a well-specified Poisson model, using a Gamma conjugate family for rate $\lambda$ (or, in canonical parameterisation, an exponential-Gamma conjugate family for $\eta = \log \lambda$ ). Shown are the posterior distributions for the parameter given $N = 0, 20 \dots, 100$ samples from the likelihood. + +![](images/dd8460909b41259109aa4a14e0c2f0dbffd12e4112bfe58d14b68b329f7744cf.jpg) + +![](images/f623a4228575037ec1bbe663ffb42fc17d3810861cfc139fc3d064839a33f615.jpg) +Figure 6: Bayesian inference in a well-specified Gaussian model (with known variance), using a Gaussian conjugate family for mean $m$ (in canonical parameterisation, the conjugate family is also Gaussian). Shown are the posterior distributions for the parameter given $N = 0,20,\ldots ,100$ samples from the likelihood. + +![](images/593895801dfdc6841ca82321ccd66f616a62c91826a83c78c802781682c2d0ad.jpg) + +Under the change of variables $m = s\eta$ for mean $m$ + +$$ +p (m \mid \chi , r) = \mathcal {N} (\chi s / r, s ^ {2} / r) \quad \text {a n d} +$$ + +$$ +p (m \mid \boldsymbol {X}, \chi , r) = \mathcal {N} \left(\left(\chi + \sum_ {i = 1} ^ {N} x _ {i}\right) s / (r + N), s ^ {2} / (r + N)\right). +$$ + +An example of Gaussian-Gaussian updates is shown in Figure 6. + +# B Examples of squared family kernels + +# B.1 Exponential family base measures and exponential features + +Take the base measure $\mu$ to belong to any exponential family with sufficient statistic $\widetilde{t}$ . Take the feature $\psi$ to be $\psi(\omega) = \exp\left(W\widetilde{t}(\omega)/2\right)$ . Then the $ij$ th entry of the squared family kernel $K_{\mu,\psi}$ is + +$$ +\left(\boldsymbol {K} _ {\mu , \psi}\right) _ {i j} = \int_ {\Omega} \exp \left(\left(\boldsymbol {w} _ {i} + \boldsymbol {w} _ {j}\right) ^ {\top} \widetilde {\boldsymbol {t}} (\boldsymbol {\omega}) / 2\right) \mu (d \boldsymbol {\omega}), +$$ + +where $\boldsymbol{w}_i$ is the $i$ th row of $\boldsymbol{W}$ . This is simply the moment generating function of the sufficient statistic $\widetilde{\boldsymbol{t}}$ under the exponential family, which is available in closed-form in terms of the log normalising constant $\widetilde{A}(\widetilde{\eta})$ of the exponential family, + +$$ +\left(\boldsymbol {K} _ {\mu , \psi}\right) _ {i j} = \exp \left(\widetilde {A} \left(\widetilde {\boldsymbol {\eta}} + (\boldsymbol {w} _ {i} + \boldsymbol {w} _ {j}) / 2\right) - \widetilde {A} (\widetilde {\boldsymbol {\eta}})\right), +$$ + +where $\eta$ is the canonical parameter of the exponential family. Recall Remark 5, that if $\mu$ itself belongs to a conjugate prior family for the likelihood $p(\boldsymbol{U} \mid \cdot)$ , then $\nu(\cdot) = \mu(\cdot)p(\boldsymbol{U} \mid \cdot)$ remains within the conjugate family, up to a multiplication by the marginal likelihood, and therefore the squared family kernel $K_{\nu,\psi}$ is available in closed-form for all posterior updates. By Appendix A, $\mu$ does indeed belong to a conjugate family. Log normalising constants are available for many exponential families which are known to be conjugate priors for exponential family likelihoods [Diaconis and Ylvisaker, 1979] (see e.g. Appendix A and closed-form normalising constants in Nielsen and Garcia [2024]). + +# B.2 Normalised base measures and random Fourier features + +Bochner's theorem says that every stationary kernel is the Fourier transform of a nonnegative probability measure, and vice versa [Rahimi and Recht, 2007, for example]. Hence for real-valued stationary squared family kernels and probability base measures $\mu$ , + +$$ +\begin{array}{l} \left(\boldsymbol {K} _ {\mu , \psi}\right) _ {i j} = \int_ {\Omega} \exp \left(\mathrm {i} \left(\boldsymbol {w} _ {i} - \boldsymbol {w} _ {j}\right) ^ {\top} \boldsymbol {\omega}\right) \mu (d \boldsymbol {\omega}) \\ = \int_ {\Omega} \psi_ {i} (\omega) \psi_ {j} (\omega) \mu (d \omega), \\ \end{array} +$$ + +where $\psi_{i}(\omega) = \cos (\pmb{w}_{i}^{\top}\pmb {\omega}) + \mathrm{i}\sin (\pmb{w}_{i}^{\top}\pmb {\omega})$ and i is the imaginary unit. This allows the use of base measures with cosine and sine features, provided the base measure has a closed-form Fourier transform. For example, the Gaussian, Laplace and Cauchy base measures all have closed-form kernels (corresponding with Gaussian, Cauchy and Laplacian kernels respectively). Real-valued cosine and sine transforms are also available for many densities, including the Gaussian density. + +# B.3 Gaussian family base measures and neural network features + +Integrals of the form of the squared family kernel arise as so-called neural network Gaussian process kernels, when the features are hidden layers of neural networks and the base measure is Gaussian. That is, the features take the form $\psi (\omega) = \sigma (W\omega +b)$ for some weight and bias parameters $W$ and $b$ , and some activation function $\sigma$ , and + +$$ +\left(\boldsymbol {K} _ {\mu , \psi}\right) _ {i j} = \mathbb {E} _ {\boldsymbol {\omega} \sim \mathcal {N}} \left[ \sigma \left(\boldsymbol {w} _ {i} ^ {\top} \boldsymbol {\omega} + b _ {i}\right) \sigma \left(\boldsymbol {w} _ {j} ^ {\top} \boldsymbol {\omega} + b _ {j}\right) \right]. +$$ + +Example activation functions with known closed form include the error function [Williams, 1996], the ReLU (and other rectified monomial) function [Cho and Saul, 2009] and sine and cosines [Pearce et al., 2020]. Further examples are given in Table 1 of Tsuchida et al. [2023] and Table 1 of Han et al. [2022], including monomials, GeLU, snake and approximations to well-behaved functions. + +# B.4 Table of some closed-form squared families + +
Domain ΩBase measure μFeature ψ(ω)Reference/Notes
RdN(0, Cov)erf(Wω)Williams [1996]
RdN(0, Cov)ReLU(Wω)Cho and Saul [2009]
RdN(0, Cov)GELU(Wω)Tsuchida et al. [2021]
RdN(mean, Cov)Snake(Wω + b)Tsuchida et al. [2023]
RdN(mean, Cov)exp(Wω + b)Nielsen and Garcia [2024]
{0,1,2,...}(ω!)-1νexp(Wω + b)Nielsen and Garcia [2024]
Sd-1Uniformexp(Wω + b)Brown [1986]
+ +Table 2: All of these settings admit closed-form squared family kernels $K_{\mu, \psi}$ . In each case, the closed-form kernel is computed in linear time in the dimension $d$ . Here erf denotes the error function, $\mathbb{S}^{d-1}$ denotes the unit hypersphere, Snake denotes the snake activation function [Ziyin et al., 2020]. Even further examples are given in Han et al. [2022, table 1] and Tsuchida et al. [2023, table 1]. + +# C Proofs + +Proposition 3. Consider the model in Figure 3 (Left). Define a measure $\nu(d\omega) = p(\mathbf{U} \mid \omega)\mu(d\omega)$ . The posterior $P(d\omega \mid \mathbf{U}) = Q(d\omega \mid \mathbf{M}, \nu, \psi)$ belongs to a GSF with base measure $\nu$ , feature mapping $\psi$ and parameter $\mathbf{M}$ . + +Proof. The posterior is the product of the likelihood and prior divided by the marginal likelihood, + +$$ +\begin{array}{l} \frac {1}{\operatorname {T r} \left(M K _ {\mu , \psi}\right) p (U)} \langle M \psi (\omega), \psi (\omega) \rangle p (U \mid \omega) \mu (d \omega) \\ \propto \langle M \psi (\omega), \psi (\omega) \rangle p (U \mid \omega) \mu (d \omega) \\ \propto Q (d \boldsymbol {\omega} \mid M, \nu , \psi). \\ \end{array} +$$ + +![](images/80c713474aa7d17a32009c84f9b58aa87c3e098ef9f29f8f9cc0973c4c6130da.jpg) + +Proposition 4. Consider the model in Figure 3 (Left). Define a measure $\nu(d\omega) = p(\mathbf{U} \mid \omega)\mu(d\omega)$ and $\nu'(d\omega) = p(\mathbf{U}_* \mid \omega)\nu(d\omega)$ , where, abusing notation, the train likelihood $p(\mathbf{U} \mid \omega)$ and test likelihood $p(\mathbf{U}_* \mid \omega)$ are not necessarily the same. The posterior predictive is the ratio of normalising constants, $p(\mathbf{U}_* \mid \mathbf{U}) = \operatorname{Tr}\left(M\mathbf{K}_{\nu',\psi}\right) / \operatorname{Tr}\left(M\mathbf{K}_{\nu,\psi}\right)$ . + +Proof. This follows by applying Proposition 2 twice, or directly. Directly, the posterior predictive is obtained by marginalising out the predictive distribution given the parameters with respect to the parameter posterior. That is, + +$$ +\begin{array}{l} p \left(\boldsymbol {U} _ {*} \mid \boldsymbol {U}\right) = \int_ {\Omega} p \left(\boldsymbol {U} _ {*} \mid \boldsymbol {\omega}\right) P (d \boldsymbol {\omega} \mid \boldsymbol {u}) \\ = \int_ {\Omega} p (\boldsymbol {U} _ {*} \mid \boldsymbol {\omega}) Q (d \boldsymbol {\omega} \mid \boldsymbol {M}, \nu \psi) \\ = \frac {1}{\operatorname {T r} \left(\boldsymbol {M} \boldsymbol {K} _ {\nu , \psi}\right)} \int_ {\Omega} \operatorname {T r} \left(\boldsymbol {M} \boldsymbol {\psi} (\omega) \boldsymbol {\psi} (\omega) ^ {\top}\right) p \left(\boldsymbol {U} _ {*} \mid \omega\right) \nu (d \omega) \\ = \frac {\operatorname {T r} \left(M K _ {\nu^ {\prime} , \psi}\right)}{\operatorname {T r} \left(M K _ {\nu , \psi}\right)}. \\ \end{array} +$$ + +![](images/db5499be1eee708fc51ba5e5e4d221e96fa11f5331d903273418fc37ed871e60.jpg) + +# D Squared probability process calculations + +The derivation of the following two corollaries may appear long and symbol-laden but is essentially just mixing the parameter-integral factorisation (see § 2) with the standard "completing the square" technique in Gaussian linear regression [Bishop, 2006, §2.3.1] [Rasmussen and Williams, 2006, §2.1.1]. + +Corollary 7. Consider the special model (3), (4) and (5) applied to Figure 3 (Right). Define $\nu'(d\omega) = p(Y_* \mid \Gamma_*\Omega^\top)p(Y \mid \Gamma\Omega^\top)\mu(\Omega)$ . The Radon-Nikodym derivatives of $\nu$ and $\nu'$ are the same as in the finite-feature Gaussian process regression setting. That is, + +$$ +\frac {d \nu}{d \lambda} (d \omega) = \mathcal {N} \big (\boldsymbol {\omega} \mid \boldsymbol {m} _ {Y} ^ {\prime}, \boldsymbol {C} ^ {\prime} \big) \mathcal {N} _ {Y}, \quad w h e r e +$$ + +$$ +\boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime} = \operatorname {v e c} (\boldsymbol {M}) + (\boldsymbol {B} \otimes \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top}) (\boldsymbol {B} \otimes \boldsymbol {\Gamma} \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top} + v ^ {2} \boldsymbol {I} _ {N d _ {2}}) ^ {- 1} \big (\operatorname {v e c} (\boldsymbol {Y}) - (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma}) \operatorname {v e c} (\boldsymbol {M}) \big) +$$ + +$$ +\boldsymbol {C} ^ {\prime} = \boldsymbol {B} \otimes \boldsymbol {C} - (\boldsymbol {B} \otimes \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top}) (\boldsymbol {B} \otimes \boldsymbol {\Gamma} \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top} + v ^ {2} \boldsymbol {I} _ {N d _ {2}}) ^ {- 1} (\boldsymbol {B} \otimes \boldsymbol {\Gamma} \boldsymbol {C}), +$$ + +and + +$$ +\begin{array}{l} \frac {d \nu^ {\prime}}{d \lambda} (d \boldsymbol {\omega}) = \mathcal {N} \left(\operatorname {v e c} \left(\boldsymbol {Y} _ {*}\right) \mid \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime}, v ^ {\prime 2} \boldsymbol {I} _ {N _ {*} d _ {2}} + \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {C} ^ {\prime} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}\right)\right) \\ \mathcal {N} \left(\omega \mid m _ {\boldsymbol {Y}, \boldsymbol {Y} _ {*}} ^ {\prime \prime}, C ^ {\prime \prime}\right) \mathcal {N} _ {\boldsymbol {Y}}, \quad w h e r e \\ \end{array} +$$ + +$$ +\begin{array}{l} \boldsymbol {m} _ {\boldsymbol {Y}, \boldsymbol {Y} _ {*}} ^ {\prime \prime} = \boldsymbol {C} ^ {\prime} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}\right) \left(\left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {C} ^ {\prime} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}\right) + v ^ {\prime 2} \boldsymbol {I} _ {N _ {*} d _ {2}}\right) ^ {- 1} \left(\operatorname {v e c} \left(\boldsymbol {Y} _ {*}\right) - \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime}\right) \\ + m _ {Y} ^ {\prime} \\ \end{array} +$$ + +$$ +\boldsymbol {C} ^ {\prime \prime} = \boldsymbol {C} ^ {\prime} - \boldsymbol {C} ^ {\prime} (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}) \left(\left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {C} ^ {\prime} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}\right) + v ^ {\prime 2} \boldsymbol {I} _ {N _ {*} d _ {2}}\right) ^ {- 1} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {C} ^ {\prime}. +$$ + +Here $\mathcal{N}_{\pmb{Y}}$ hides an inconsequential closed-form factor which depends only on $\pmb{Y}$ . Hence the noisy posterior predictive distribution is available as $p\big(\pmb{Y}_* \mid \pmb{Y}, \pmb{\Gamma}, \pmb{\Gamma}_*\big) = \mathrm{Tr}\left(M\pmb{K}_{\nu',\psi}\right) / \mathrm{Tr}\left(M\pmb{K}_{\nu,\psi}\right)$ , whenever $\pmb{K}_{\cdot,\psi}$ can be computed in closed-form for Gaussian base measures. + +Noise-free posterior predictive In general, the noisy posterior predictive distribution $p\big(Y_{*} \mid Y, \Gamma, \Gamma_{*}\big)$ is not a GSF. However, if the posterior predictive distribution is evaluated on a large enough test set of size $N_{*} \geq d_{1}$ , and the resulting feature matrix $\Gamma_{*}$ is full-rank, then the noiseless posterior predictive distribution is itself a GSF. + +Corollary 8. Suppose additionally that $C'$ and $\Gamma_*$ are full-rank and $N_* \geq d_1$ . Then + +$$ +\begin{array}{l} \lim _ {v ^ {\prime 2} \rightarrow 0 ^ {+}} \boldsymbol {m} _ {\boldsymbol {Y}, \boldsymbol {Y} _ {*}} ^ {\prime \prime} = (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}) ^ {\dagger} \operatorname {v e c} (\boldsymbol {Y} _ {*}) = (\boldsymbol {I} _ {d _ {2}} \otimes (\boldsymbol {\Gamma} _ {*} ^ {\top} \boldsymbol {\Gamma} _ {*}) ^ {- 1} \boldsymbol {\Gamma} _ {*} ^ {\top}) \operatorname {v e c} (\boldsymbol {Y} _ {*}) \\ \lim_{v^{\prime 2}\to 0^{+}}C^{\prime \prime} = 0, \\ \end{array} +$$ + +and the noise free predictive distribution $P\big(d\pmb {f}(\pmb{X}_{*})\mid \pmb {Y},\pmb {\Gamma},\pmb{\Gamma}_{*}\big) = Q\big(d\pmb {f}(\pmb{X}_{*})\mid \pmb {M},\nu^{\prime \prime},\psi^{\prime}\big)$ belongs to a GSF with modified feature mapping and modified base measure, + +$$ +\psi^ {\prime} \left(\boldsymbol {f} \left(\boldsymbol {X} _ {*}\right)\right) \triangleq \psi \left(\left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) ^ {\dagger} \boldsymbol {f} \left(\boldsymbol {X} _ {*}\right)\right) = \psi \left(\operatorname {v e c} \left(\boldsymbol {\Gamma} _ {*} ^ {\dagger} \boldsymbol {f} \left(\boldsymbol {X} _ {*}\right)\right)\right) +$$ + +$$ +\nu^ {\prime \prime} (d \boldsymbol {f} (\boldsymbol {X} _ {*})) \triangleq \mathcal {N} \left(\operatorname {v e c} (\boldsymbol {f} (\boldsymbol {X} _ {*})) \mid (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}) \boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime}, (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}) \boldsymbol {C} ^ {\prime} (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top})\right) d \boldsymbol {f} (\boldsymbol {X} _ {*}). \tag {7} +$$ + +Proof. Take the train likelihood to be multivariate Gaussian with homoskedastic noise with variance $v^2$ , + +$$ +\begin{array}{l} p (\boldsymbol {Y} \mid \boldsymbol {\Gamma} \boldsymbol {\Omega} ^ {\top}) = \frac {1}{(2 \pi v ^ {2}) ^ {d _ {1} d _ {2} / 2}} \exp \left(- \frac {1}{2 v ^ {2}} \operatorname {T r} \left(\left(\boldsymbol {Y} - \boldsymbol {\Gamma} \boldsymbol {\Omega} ^ {\top}\right) ^ {\top} \left(\boldsymbol {Y} - \boldsymbol {\Gamma} \boldsymbol {\Omega} ^ {\top}\right)\right)\right) \\ = \frac {1}{(2 \pi v ^ {2}) ^ {d _ {1} d _ {2} / 2}} \exp \left(- \frac {1}{2 v ^ {2}} \left(\operatorname {v e c} (\boldsymbol {Y}) - (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma}) \boldsymbol {\omega}\right) ^ {\top} \left(\operatorname {v e c} (\boldsymbol {Y}) - (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma}) \boldsymbol {\omega}\right)\right) \\ = \mathcal {N} (\operatorname {v e c} (\boldsymbol {Y}) \mid (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma}) \boldsymbol {\omega}, v ^ {2} \boldsymbol {I} _ {N d _ {2}}) \\ \end{array} +$$ + +Take the Radon-Nikodym derivative of the base measure $\mu$ with respect to Lebesgue measure $\lambda$ to be the density of a matrix Gaussian distribution, + +$$ +\begin{array}{l} \frac {d \mu}{d \lambda} (\boldsymbol {\Omega}) \triangleq p (\boldsymbol {\Omega}) = \frac {\exp \left(- \frac {1}{2} \operatorname {T r} \left(\boldsymbol {B} ^ {\dagger} (\boldsymbol {\Omega} ^ {\top} - \boldsymbol {M}) ^ {\top} \boldsymbol {C} ^ {\dagger} (\boldsymbol {\Omega} ^ {\top} - \boldsymbol {M})\right)\right)}{(2 \pi) ^ {d _ {1} d _ {2} / 2} | \boldsymbol {B} | ^ {d _ {1} / 2} | \boldsymbol {C} | ^ {d _ {2} / 2}} \\ = \mathcal {N} (\boldsymbol {\omega} \mid \operatorname {v e c} (M), \boldsymbol {B} \otimes \boldsymbol {C}) \\ \end{array} +$$ + +We convert the product $p(\mathbf{Y} \mid \Gamma \boldsymbol{\Omega}^{\top}) p(\boldsymbol{\Omega})$ into a product of two Gaussian probability density functions. In order to do so, we write $p(\mathbf{Y} \mid \Gamma \boldsymbol{\Omega}^{\top}) p(\boldsymbol{\Omega}) = p(\mathbf{Y}) p(\boldsymbol{\Omega} \mid \mathbf{Y}) = p(\mathbf{Y}, \boldsymbol{\Omega})$ and note that $\mathbf{Y}$ and $\boldsymbol{\Omega}$ are jointly Gaussian. Since $\mathbf{Y} = \Gamma \boldsymbol{\Omega}^{\top} + \mathbf{E}$ , the parameters of the marginal distribution of $\mathbf{Y}$ can be simply read off as + +$$ +\mathbb {E} \left[ \operatorname {v e c} (\boldsymbol {\Upsilon}) \right] = \operatorname {v e c} (\boldsymbol {\Gamma} \boldsymbol {M}), +$$ + +$$ +\begin{array}{l} \mathbb {E} \left[ \operatorname {v e c} (\boldsymbol {\Upsilon}) \operatorname {v e c} (\boldsymbol {\Upsilon}) ^ {\top} \right] = v ^ {2} \boldsymbol {I} _ {N d _ {2}} + \mathbb {E} \left[ \operatorname {v e c} \left(\boldsymbol {\Gamma} \boldsymbol {\Omega} ^ {\top}\right) \operatorname {v e c} \left(\boldsymbol {\Gamma} \boldsymbol {\Omega} ^ {\top}\right) ^ {\top} \right] \\ = v ^ {2} \boldsymbol {I} _ {N d _ {2}} + \mathbb {E} \left[ \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma}\right) \operatorname {v e c} \left(\boldsymbol {\Omega} ^ {\top}\right) \operatorname {v e c} \left(\boldsymbol {\Omega} ^ {\top}\right) ^ {\top} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma}\right) ^ {\top} \right] \\ = v ^ {2} \boldsymbol {I} _ {N d _ {2}} + (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma}) (\boldsymbol {B} \otimes \boldsymbol {C}) (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} ^ {\top}) \\ = v ^ {2} \boldsymbol {I} _ {N d _ {2}} + \boldsymbol {B} \otimes (\boldsymbol {\Gamma C} \boldsymbol {\Gamma} ^ {\top}), \\ \end{array} +$$ + +so that + +$$ +p (\boldsymbol {Y}) = \mathcal {N} \big (\operatorname {v e c} (\boldsymbol {Y}) | \underbrace {\operatorname {v e c} (\boldsymbol {\Gamma} \boldsymbol {M})} _ {(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma}) \operatorname {v e c} (\boldsymbol {M})}, v ^ {2} \boldsymbol {I} _ {N d _ {2}} + \boldsymbol {B} \otimes (\boldsymbol {\Gamma} \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top}) \big). +$$ + +For $p(\Omega \mid Y)$ , we compute the unnormalised posterior Gaussian probability density function by completing the square of the quadratic, following a typical calculation. This yields + +$$ +p (\boldsymbol {\Omega} \mid \boldsymbol {Y}) = \mathcal {N} \left(\omega \mid m _ {\boldsymbol {Y}} ^ {\prime}, C _ {\boldsymbol {Y}} ^ {\prime}\right), +$$ + +where + +$$ +\boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime} = \operatorname {v e c} (\boldsymbol {M}) + (\boldsymbol {B} \otimes \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top}) (\boldsymbol {B} \otimes \boldsymbol {\Gamma} \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top} + v ^ {2} \boldsymbol {I} _ {N d _ {2}}) ^ {- 1} \big (\operatorname {v e c} (\boldsymbol {Y}) - (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma}) \operatorname {v e c} (\boldsymbol {M}) \big) +$$ + +$$ +\boldsymbol {C} _ {\boldsymbol {Y}} ^ {\prime} = \boldsymbol {B} \otimes \boldsymbol {C} - (\boldsymbol {B} \otimes \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top}) (\boldsymbol {B} \otimes \boldsymbol {\Gamma} \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top} + v ^ {2} \boldsymbol {I} _ {N d _ {2}}) ^ {- 1} (\boldsymbol {B} \otimes \boldsymbol {\Gamma} \boldsymbol {C}). +$$ + +Therefore, + +$$ +p (\boldsymbol {Y} \mid \Gamma \boldsymbol {\Omega} ^ {\top}) \frac {d \mu}{d \lambda} (\boldsymbol {\Omega}) = \mathcal {N} \left(\operatorname {v e c} (\boldsymbol {Y}) \mid \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma}\right) \operatorname {v e c} (\boldsymbol {M}), v ^ {2} \boldsymbol {I} _ {N d _ {2}} + \boldsymbol {B} \otimes \left(\boldsymbol {\Gamma} \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top}\right)\right) \mathcal {N} \left(\omega \mid \boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime}, \boldsymbol {C} _ {\boldsymbol {Y}} ^ {\prime}\right), \tag {8} +$$ + +Applying the same result again for $p(\mathbf{Y}_{*} \mid \boldsymbol{\Gamma}_{*}\boldsymbol{\Omega}^{\top}) = \mathcal{N}\big(\operatorname{vec}(\mathbf{Y}_{*}) \mid (\mathbf{I}_{d_{2}} \otimes \boldsymbol{\Gamma}_{*})\boldsymbol{\omega}, v'^{2}\mathbf{I}_{N_{*}d_{2}}\big)$ , we may form the probability density function corresponding with the base measure $\nu'$ , + +$$ +\begin{array}{l} p (\boldsymbol {Y} _ {*} \mid \boldsymbol {\Gamma} _ {*} \boldsymbol {\Omega} ^ {\top}) p (\boldsymbol {Y} \mid \boldsymbol {\Gamma} \boldsymbol {\Omega} ^ {\top}) \frac {d \mu}{d \lambda} (\boldsymbol {\Omega}) \\ = \mathcal {N} \left(\operatorname {v e c} (\boldsymbol {Y}) \mid \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma}\right) \operatorname {v e c} (\boldsymbol {M}), v ^ {2} \boldsymbol {I} _ {N d _ {2}} + \boldsymbol {B} \otimes \left(\boldsymbol {\Gamma} \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top}\right)\right) \\ \mathcal {N} \left(\operatorname {v e c} \left(Y _ {*}\right) \mid \left(I _ {d _ {2}} \otimes \Gamma_ {*}\right) m _ {\boldsymbol {Y}} ^ {\prime}, v ^ {\prime 2} I _ {N _ {*} d _ {2}} + \left(I _ {d _ {2}} \otimes \Gamma_ {*}\right) C _ {\boldsymbol {Y}} ^ {\prime} \left(I _ {d _ {2}} \otimes \Gamma_ {*} ^ {\top}\right)\right) \mathcal {N} \left(\omega \mid m _ {\boldsymbol {Y}, \boldsymbol {Y} _ {*}} ^ {\prime \prime}, C _ {\boldsymbol {Y}, \boldsymbol {Y} _ {*}} ^ {\prime \prime}\right). \tag {9} \\ \end{array} +$$ + +where + +$$ +\begin{array}{l} \boldsymbol {m} _ {\boldsymbol {Y}, \boldsymbol {Y} _ {*}} ^ {\prime \prime} = \boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime} + \boldsymbol {C} _ {\boldsymbol {Y}} ^ {\prime} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}\right) \left(\left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {C} _ {\boldsymbol {Y}} ^ {\prime} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}\right) + v ^ {\prime 2} \boldsymbol {I} _ {N _ {*} d _ {2}}\right) ^ {- 1} \left(\operatorname {v e c} \left(\boldsymbol {Y} _ {*}\right) - \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime}\right) \\ C _ {\boldsymbol {Y}, \boldsymbol {Y} _ {*}} ^ {\prime \prime} = C _ {\boldsymbol {Y}} ^ {\prime} - C _ {\boldsymbol {Y}} ^ {\prime} (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}) \big ((\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}) C _ {\boldsymbol {Y}} ^ {\prime} (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}) + v ^ {\prime 2} \boldsymbol {I} _ {N _ {*} d _ {2}} \big) ^ {- 1} (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}) C _ {\boldsymbol {Y}} ^ {\prime}. \\ \end{array} +$$ + +Limiting base measure $\nu'$ Let $C_{\mathbf{Y}}' = \mathbf{L}\mathbf{L}^\top$ be a decomposition in terms of $\pmb{L}$ such that $\pmb{L} \in \mathbb{R}^{d_1 d_2 \times d_1 d_2}$ and suppose $C_{\mathbf{Y}}'$ is invertible. We may use the limit definition of the pseudo-inverse to find + +$$ +\begin{array}{l} \lim _ {v ^ {\prime 2} \rightarrow 0 ^ {+}} C _ {\boldsymbol {Y}} ^ {\prime} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}\right)\left(\left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) C _ {\boldsymbol {Y}} ^ {\prime} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}\right) + v ^ {\prime 2} \boldsymbol {I} _ {N _ {*} d _ {2}}\right) ^ {- 1} \\ = \lim _ {v ^ {\prime 2} \rightarrow 0 ^ {+}} \boldsymbol {L L} ^ {\top} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}\right)\left(\left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {L L} ^ {\top} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}\right) + v ^ {\prime 2} \boldsymbol {I} _ {N _ {*} d _ {2}}\right) ^ {- 1} \\ = L \left(\left(I _ {d _ {2}} \otimes \Gamma_ {*}\right) L\right) ^ {\dagger} \\ \end{array} +$$ + +The matrix $(\mathbf{I}_{d_2} \otimes \mathbf{\Gamma}_*) \in \mathbb{R}^{N_* d_2 \times d_1 d_2}$ has linearly independent columns, assuming $N_*$ is large enough. $\mathbf{L}$ has linearly independent rows (since $\mathbf{C}_{\mathbf{Y}}'$ is invertible). Therefore + +$$ +\begin{array}{l} L \bigl (\left(I _ {d _ {2}} \otimes \Gamma_ {*}\right) L \bigr) ^ {\dagger} \\ = \boldsymbol {L} \boldsymbol {L} ^ {\dagger} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) ^ {\dagger} \\ = \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) ^ {\dagger} \\ \end{array} +$$ + +We may obtain the base measure $\lim_{v^{\prime 2}\to 0^{+}}\nu^{\prime}(d\omega)$ which has parameters + +$$ +\boldsymbol {l} _ {\boldsymbol {Y}, \boldsymbol {Y} ^ {*}} \triangleq \lim _ {v ^ {\prime 2} \rightarrow 0 ^ {+}} \boldsymbol {m} _ {\boldsymbol {Y}, \boldsymbol {Y} _ {*}} ^ {\prime \prime} = \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) ^ {\dagger} \operatorname {v e c} \left(\boldsymbol {Y} _ {*}\right) +$$ + +$$ +\lim _ {v ^ {\prime 2} \rightarrow 0 ^ {+}} C _ {\mathbf {Y}, \mathbf {Y} _ {*}} ^ {\prime \prime} = \mathbf {0}. +$$ + +We therefore have that + +$$ +\begin{array}{l} \lim _ {v ^ {\prime 2} \rightarrow 0 ^ {+}} v ^ {\prime} (d \omega) = \mathcal {N} \big (\operatorname {v e c} (\boldsymbol {Y}) \mid \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma}\right) \operatorname {v e c} (\boldsymbol {M}), v ^ {2} \boldsymbol {I} _ {N d _ {2}} + \boldsymbol {B} \otimes \left(\boldsymbol {\Gamma} \boldsymbol {C} \boldsymbol {\Gamma} ^ {\top}\right) \big) \\ \mathcal {N} \big (\operatorname {v e c} \left(\boldsymbol {Y} _ {*}\right) \mid \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime}, \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {C} _ {\boldsymbol {Y}} ^ {\prime} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}\right) \big) \delta \left(\omega - l _ {\boldsymbol {Y}, \boldsymbol {Y} ^ {*}}\right). \\ \end{array} +$$ + +Integration against this measure can be understood as an evaluation functional, and since $l_{Y,Y^{*}}$ is linear in $\mathbf{Y}_{*}$ , the evaluation is also GSF. More concretely, + +$$ +\begin{array}{l} z (\boldsymbol {\Theta}) \propto \int_ {\mathbb {Q}} \| \boldsymbol {\Theta} \psi (\boldsymbol {\omega}) \| _ {2} ^ {2} \delta (\boldsymbol {\omega} - \boldsymbol {l} _ {\boldsymbol {Y}, \boldsymbol {Y} ^ {*}}) d \boldsymbol {\omega} \\ \mathcal {N} \left(\boldsymbol {f} \left(\boldsymbol {X} _ {*}\right) \mid \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime}, \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {C} _ {\boldsymbol {Y}} ^ {\prime} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}\right)\right) \\ = \| \Theta \psi (l _ {\boldsymbol {Y}, \boldsymbol {Y} ^ {*}}) \| _ {2} ^ {2} \mathcal {N} \left(\operatorname {v e c} (\boldsymbol {f} (\boldsymbol {X} _ {*})) \mid (\boldsymbol {I} _ {d _ {2}} \otimes \Gamma_ {*}) \boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime}, (\boldsymbol {I} _ {d _ {2}} \otimes \Gamma_ {*}) \boldsymbol {C} _ {\boldsymbol {Y}} ^ {\prime} (\boldsymbol {I} _ {d _ {2}} \otimes \Gamma_ {*} ^ {\top})\right) \\ = \left\| \boldsymbol {\Theta} \psi \big (\left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} \right) ^ {\dagger} \operatorname {v e c} \left(\boldsymbol {f} \left(\boldsymbol {X} _ {*}\right)\right) \big) \right\| _ {2} ^ {2} \\ \mathcal {N} \big (\operatorname {v e c} (\boldsymbol {f} (\boldsymbol {X} _ {*})) \mid (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}) \boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime}, (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}) \boldsymbol {C} _ {\boldsymbol {Y}} ^ {\prime} (\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}) \big). \\ \end{array} +$$ + +Hence the posterior predictive bleongs to a GSF with base measure $\nu''(d\pmb{f}(\pmb{X}_*)) =$ + +$$ +\mathcal {N} \Big (\operatorname {v e c} \left(\boldsymbol {f} \left(\boldsymbol {X} _ {*}\right)\right) \mid \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {m} _ {\boldsymbol {Y}} ^ {\prime}, \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*}\right) \boldsymbol {C} _ {\boldsymbol {Y}} ^ {\prime} \left(\boldsymbol {I} _ {d _ {2}} \otimes \boldsymbol {\Gamma} _ {*} ^ {\top}\right) \Big) d \boldsymbol {f} \left(\boldsymbol {X} _ {*}\right). +$$ + +# E Algorithms + +In response to reviewer feedback, we include a description of the algorithms used in our experiments. During the experiments, we run Algorithm 3 on the meta dataset then run Algorithm 2 on the support/query dataset. + +Algorithm 1 Marginal likelihood (i.e., Proposition 2) +Input: Trainable final layer parameter $\Theta$ , trainable hidden network $\psi : \mathbb{R}^d \to \mathbb{R}^n$ , trainable base measure $\mu$ , frozen likelihood $p(\boldsymbol{U} \mid \boldsymbol{\omega})$ +Output: Marginal likelihood $p(\boldsymbol{U}) \in [0, \infty)$ +1: Set $M = \Theta^\top \Theta$ . +2: Compute $K_{\mu, \psi}$ +3: Set $\nu(d\omega) = p(\boldsymbol{U} \mid \boldsymbol{\omega})\mu(d\omega)$ , then compute $K_{\nu, \psi}$ +4: Set $z(M) = \operatorname{Tr}(MK_{\mu, \psi})$ +5: return $p(\boldsymbol{U}) = \operatorname{Tr}(MK_{\mu, \psi}) / z(\boldsymbol{M})$ +▷ as per Proposition 2 + +Algorithm 2 GSFP posterior predictive update (i.e., Corollary 6) +Input: Frozen Gaussian likelihood $p(\mathbf{Y} \mid \boldsymbol{\gamma}(\mathbf{X})\omega)$ on support set. Query $\mathbf{X}^*$ . Prior element of a GSF parameterised by: final layer parameter $\Theta \in \mathbb{R}^{m \times n}$ , hidden GSF network $\psi : \mathbb{R}^d \to \mathbb{R}^n$ , Gaussian base measure $\mu$ , deep feature extractor $\gamma : \mathbb{X} \to \mathbb{R}^d$ . +Output: Posterior element of a GSF, parameterised by: final layer parameter, hidden network, and base measure. +1: Set $M = \Theta^\top \Theta$ . +2: Set $p(z \mid x^*)$ to be the Gaussian posterior predictive. ▷ as per (6) +3: Set $\psi'$ ▷ as per Corollary 6 +4: Set $z(M) = \operatorname{Tr}(MK_{\nu'', \psi'})$ ▷ as per (1) +5: return Posterior element of a GSF parameterised by $M, \nu''$ and $\psi'$ . That is, $P(df \mid M, \nu'', \psi') = \operatorname{Tr}(M\psi'(f)\psi'(f)^\top) / z(M)\nu''(df)$ ▷ as per (1) and Proposition 4 + +(Here $\pmb{f}$ is shorthand for $\pmb{f}(\pmb{X}_{*})$ ) + +Algorithm 3 GSFP Few-shot learning pretraining (i.e. paragraph 1 of Section 4) +Input: Meta-datasets $(X_{j},Y_{j})$ for $j = 1,\dots ,J$ . Batch size $B\in \{1,\ldots ,J\}$ . Trainable final layer parameter $\Theta \in \mathbb{R}^{m\times n}$ . Trainable hidden network $\psi :\mathbb{R}^d\to \mathbb{R}^n$ . Trainable Gaussian base measure $\mu$ . Deep feature extractor $\gamma :\mathbb{X}\rightarrow \mathbb{R}^d$ . +Output: Trained GSFP prior distribution +1: for each batch $b = 1,\ldots ,\lfloor J / b\rfloor +1$ do +2: Set $U_{i} = (Y_{i},\gamma (X_{i}))$ , for each index $i$ in the batch. Compute Gaussian likelihood $p(Y_i|$ $\gamma X_{i}\omega_{i})$ as per (4), for each index $i$ in the batch +3: Compute gradients of the marginal likelihood (Algorithm 1) on input $(\Theta ,\psi ,p(Y_{i}|$ $\gamma (X_i)\omega))$ for each $i$ in the batch, with respect to parameters $\Theta$ and parameters of $\psi ,\mu$ and $\gamma$ using automatic differentiation. +4: Update $\Theta$ , and parameters of $\psi ,\mu$ and $\gamma$ , using the computed gradients (e.g. using SGD update rule). +5: Optionally store gradient statistics (e.g. for Adam or other optimisers). +6: end for +7: return Trained GSFP prior distribution described by $\Theta ,\psi ,\mu$ and $\gamma$ . + +# F Experiment details - few-shot regression + +Here we describe the six datasets (two of which come with licenses, the rest without), the implementation details, and the run times for each method. + +# F.1 Sines + +Sines dataset [Finn et al., 2017] is comprised of input $\times$ from the interval $[-5, 5]$ with corresponding label $y = A \cdot \sin(x + p) + \epsilon$ , where amplitude $A \sim U[0.1, 5.0]$ and phase $p \sim U[0, \pi]$ are uniformly distributed on their respective intervals, and $\epsilon \sim \mathcal{N}(0, 0.1)$ is a Gaussian noise with 0 mean and a standard deviation of 0.1. + +In training, 10 input-label pairs are sampled with five each for support and query sets. Evaluation is performed on 500 inference iterations on 200 data points, with a 5/195 split ratio for support and query sets. Testing inputs are sampled from $[-5, 5]$ for in-distribution evaluation and $[-5, 10]$ for out-of-distribution evaluation. + +# F.2 Mixed-Noise Sines + +Mixed-Noise Sines [Sendera et al., 2021] is a variant of sines experiment by utilising input-dependent noise: $y = A \cdot \sin (x + p) + |x - p| \cdot \epsilon$ where $|\cdot|$ is an absolute value function. + +# F.3 NDX100 + +NASDAQ100 Small Dataset [Qin et al., 2017] contains stock prices of 81 major corporations and index values of NASDAQ 100 from July 26 to December 22 in 2016. The data are recorded at a frequency of one point per minute, resulting in 390 data points collected in a standard trade day, with exceptions of 210 and 180 points on November 25 and December 22 separately. + +Training sets and in-distribution evaluation sets are obtained by partitioning NASDAQ 100 index with a 70/30 split. Out-of-distribution evaluation is performed on the entire data series of Yahoo stock prices. During both training and evaluation, 10 input-label pairs are sampled with a fixed frequency of 1 point per 10 minutes. Both in-distribution and out-of-distribution evaluations report average performances over 500 sampled data sequences. + +# F.4 EEG + +EEG Steady-State Visual Evoked Potential Signals Dataset [Fernandez-Fraga et al., 2019] contains the electroencephalogram (EEG) data from 29 subjects performing different visual tests. The time-series data signals of various lengths (owing to different test durations) are recorded at a frequency of $128\mathrm{Hz}$ . This dataset is available under a Creative Commons Attribution 4.0 International (CC-BY-4.0) licence. + +Electrode AF4 data obtained from the first Five Box Visual Test 1 on Subject 1 from Group A (file name A001SB1_1.csv) is used for training and in-distribution evaluation following a partition ratio of 70/30. Out-of-distribution evaluation is performed on the entire electrode AF4 data obtained from the first Five Box Visual Test 1 on Subject 3 from Group A (file name A003SB1_1.csv). During both training and evaluation, 10 input-label pairs are sampled with a frequency of $12.8\mathrm{Hz}$ . Both in-distribution and out-of-distribution evaluations report average performances over 500 sampled data sequences. + +# F.5 QMUL + +Queen Mary University of London Multiview Face Dataset (QMUL) [Gong et al., 1996] contains normalised facial images of 48 people. There are 133 facial images associated with each person, representing a slice of a view sphere where yaw and pitch span $\alpha \in \{10^{\circ} \times i : i \in \{-9, -8, \dots, 8, 9\}\}$ and $\beta \in \{10^{\circ} \times j : j \in \{-3, -2, \dots, 2, 3\}\}$ respectively. + +We use the subset comprising of only 37 people, whose facial images are grayscale, for training and evaluation following a split ratio of $32/5$ . Randomly sampled head trajectory (a sequence of yaw-pitch pairs) is obtained as: $\{(\alpha, 10 \cdot \lfloor \mathsf{A} \cdot \sin(\alpha / 10 + \mathsf{p}) + 3 \rfloor - 30)\}$ , where amplitude $\mathsf{A} \sim U[0.1, 5.0]$ + +and phase $\mathsf{p} \sim U[0, \pi]$ are uniformly distributed. Images and tilts along the trajectory are used as input-label pairs for training/evaluation. Note in out-of-distribution experiments, yaw is limited to a subset $\{10^{\circ} \times i : i \in \{-9, -8, \dots, -1, 0\}\}$ during training, but has the entire set in the evaluation. Both in-distribution and out-of-distribution evaluations report average performances over 500 sampled data sequences. + +# F.6 Power + +Power [Hebrail and Berard, 2006] dataset contains 2,075,259 electricity consumption readings in a house in Sceaux (France), spanning 47 months between December 2006 and November 2010. The time-series measurements of electricity consumption (watt per hour) are recorded at a frequency of one reading per minute on individual metres. This dataset is licenced under a Creative Commons Attribution 4.0 International (CC-BY-4.0) licence. + +Readings from Sub_metering_3 are used for training and evaluation of the models. Following the setting in Sendera et al. [2021], $70\%$ of the data, starting from the beginning of the time series, is used for both training and in-distribution evaluation, while the remaining $30\%$ is used for out-of-distribution evaluation. During both training and evaluation, 10 input-label pairs are sampled with a frequency of 1 reading per 10 minutes. Both in-distribution and out-of-distribution evaluations report average performances over 500 sampled data sequences. + +# F.7 Implementation details + +GSFP is implemented using Pytorch and GPytorch frameworks. Features $\gamma(\boldsymbol{x})$ of inputs $\boldsymbol{x}$ are obtained using a Multilayer Perceptron (MLP) in Sines, Mixed-Noise-Sines, NDX100 and EEG experiments, and a Convolutional Neural Network (CNN) for image data in QMUL experiment. The MLP has two fully-connected layers, with the first layer mapping $\mathbb{X} \to \mathbb{R}^d$ followed by a ReLU activation function [Hahnloser et al., 2000], and the second one mapping $\mathbb{R}^d \to \mathbb{R}^d$ . The CNN has three convolutional layers with fixed kernel size of 3, stride of 2 and dilation of 2. Input channel number of the first convolutional layer is 3, and upsized to 36 in the two subsequent convolutional layers. Output channel number is 36 across all convolutional layers. ReLU activation function is appended to each of the first two convolutional layers. Output of the last convolutional layer is flattened into a $d$ -dimensional vector. A scaled cosine function $\cos(\cdot)/\sqrt{d}$ is appended as the final activation. + +The squared neural network consists of a hidden layer $(\psi : \mathbb{Q} \to \mathbb{R}^n$ , where $\Omega = \mathbb{R}^d$ ) and readout parameters $(\Theta \in \mathbb{R}^{m \times n})$ . The hidden layer is a fully connected layer $\psi(\omega) = \sigma(W\omega + b)$ with a weight matrix $W \in \mathbb{R}^{n \times d}$ initialised from a standard normal distribution $\mathcal{N}(0, I)$ , and a bias vector $b \in \mathbb{R}^n$ initialised to be all ones. For the activation function $\sigma$ , we use the Snake activation function [Ziyin et al., 2020], + +$$ +\operatorname {S n a k e} _ {a} (z) = z + \frac {1}{a} \sin^ {2} (a z) = z - \frac {1}{2 a} \cos (2 a z) + \frac {1}{2 a z}. +$$ + +The dimensions of data, latent feature and output used across the experiments are summarised in Table 3. + +Table 3: Summary of dimensions used across the experiments. Recall that $d$ is the dimension of the feature $\gamma$ in the regression model $f(\pmb{x}) = \pmb{\omega}^{\top}\pmb{\gamma}(\pmb{x})$ (see (2)), and $n$ and $m$ are the dimensions of the readout parameter $\Theta \in \mathbb{R}^{m\times n}$ in the GSF model (see § 2). + +
DimensionsDatasets
SinesMixed-Noise SinesNDX100EEGPowerQMUL
d40405552,916
n1010222500
m1010111500
+ +Both DKT [Patacchiola et al., 2020] and NGGP [Sendera et al., 2021] are integrated into our code to facilitate their evaluations under the same experimental settings as those of GSFP. Their implementations follow the details specified in their original works, with only exceptions of increasing the dimension of the range of the feature extractor or feature mapping function $(\gamma)$ from 1 to 5 in the NDX100, EEG, and Power datasets. + +# F.8 Training details + +All models, including DKT [Patacchiola et al., 2020], NGGP [Sendera et al., 2021] and GFSP, are trained with Adam optimiser [Kingma, 2014] using a fixed learning rate of 0.001 and default beta coefficients of $\beta_{1} = 0.9$ and $\beta_{2} = 0.999$ across all experiments. All models are trained on $J = 10,000$ training datasets that are randomly sampled from the meta dataset as detailed in § F.1-F.6. The evaluation results of each meta model is averaged over 500 test datasets, with each entry representing the mean $\pm$ the standard deviation. Optimal performance of each meta model is obtained with grid search at a frequency of 500 training datasets. Both training and evaluation are conducted with a single RTX 2080TI GPU and an Intel(R) Xeon(R) Gold 6242 CPU @ 2.80GHz. + +# F.9 Run times + +Table 4: Benchmark results showing training time (mins) and inference speed (tests / s) on 1 meta dataset across regression tasks. The results are evaluated on 5 meta models, and each entry shows the mean ± standard deviation over these 5 meta models. + +
MethodsTraining Time (mins) ↓
NameKernelSinesMixed-Noise SinesNDX100EEGQMULPower
DKTRBF1.04 ± 0.021.02 ± 0.051.05 ± 0.020.98 ± 0.013.06 ± 0.111.01 ± 0.05
Spectral1.07 ± 0.031.11 ± 0.011.16 ± 0.011.15 ± 0.013.34 ± 0.011.08 ± 0.01
NN Linear1.02 ± 0.010.98 ± 0.030.97 ± 0.010.95 ± 0.013.12 ± 0.030.98 ± 0.01
NGGPRBF41.59 ± 0.6043.57 ± 0.5853.41 ± 2.5955.97 ± 5.5457.68 ± 0.8875.30 ± 2.89
Spectral37.40 ± 1.4135.19 ± 0.6653.57 ± 0.8764.33 ± 7.2257.13 ± 1.4170.52 ± 3.48
NN Linear44.01 ± 0.3942.53 ± 1.6545.62 ± 2.8263.75 ± 5.2165.93 ± 0.7361.92 ± 3.57
GSFPNN Linear4.09 ± 0.324.14 ± 0.254.39 ± 0.043.46 ± 0.109.67 ± 0.023.45 ± 0.07
MethodsInference Speed (tests / s) ↑
NameKernelSinesMixed-Noise SinesNDX100EEGQMULPower
DKTRBF163.73 ± 1.17167.84 ± 1.98170.96 ± 1.28174.98 ± 1.8859.92 ± 4.26169.38 ± 0.44
Spectral166.22 ± 0.52163.67 ± 0.64172.98 ± 0.75175.61 ± 1.4460.06 ± 1.46174.97 ± 3.25
NN Linear184.35 ± 1.46185.00 ± 1.31195.49 ± 0.66196.61 ± 0.5962.25 ± 1.97195.17 ± 1.12
NGGPRBF5.49 ± 0.075.30 ± 0.054.33 ± 0.874.11 ± 0.463.84 ± 0.243.85 ± 0.28
Spectral7.09 ± 0.356.80 ± 0.854.40 ± 0.584.33 ± 0.133.87 ± 0.093.78 ± 0.16
NN Linear4.96 ± 0.275.18 ± 0.175.86 ± 0.945.07 ± 0.293.06 ± 0.163.97 ± 0.32
GSFPNN Linear99.56 ± 0.4999.08 ± 0.97150.09 ± 1.86149.61 ± 1.317.14 ± 0.03149.41 ± 1.80
+ +# G Experimental details - vanilla regression + +Here we describe the nine datasets selected by [Hernández-Lobato and Adams, 2015] for regression experiments, as well as the implementation details, experimental performances in terms of NLL, and run times for each method. + +# G.1 Boston Housing + +The Boston Housing dataset [Harrison Jr and Rubinfeld, 1978] consists of 506 instances with 14 variables describing the socioeconomic characteristics of neighbourhoods in Boston, Massachusetts. We use 13 features as input variables and the remaining variable, MEDV (median value of owner-occupied homes in $1000's), serves as the regression target. This dataset is licensed under an Apache 2.0 open source license. + +# G.2 Concrete Compressive Strength + +The Concrete Compressive Strength dataset [Yeh, 1998] consists of 1,030 instances with 8 variables describing the compositional structure, the curation duration, and the compressive strength of concrete samples. We use 7 features as input variables and the remaining variable, concrete compressive strength (MPa), serves as the regression target. This dataset is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) license. + +# G.3 Energy Efficiency + +The Energy Efficiency dataset [Tsanas and Xifara, 2012] consists of 768 samples with 8 features describing the design details of the buildings. We use these 8 features as input variables and the variable, cooling load, serves as the regression target. This dataset is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) license. + +# G.4 Kin8nm + +The Kin8nm dataset is part of the Kin family of datasets [Corke, 2002] that consists of 8,192 instances with 8 features simulating the forward kinetics of a robotic arm. We use these 8 features as input variables and the remaining variable (y) serves as the regression target. + +# G.5 Condition Based Maintenance of Naval Propulsion Plants + +The Condition Based Maintenance of Naval Propulsion Plant dataset [Coraddu et al., 2014] consists of 11,934 samples generated from a numerical simulation of a naval vessel with a gas turbine propulsion plant. We use the 16 steady-state sensor and operational measurements as input variables, and the compressor degradation serves as the regression target. This dataset is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) license. + +# G.6 Combined Cycle Power Plant + +The Combined Cycle Power Plant dataset [Tüfekci, 2014] consists of 9,568 data points collected from a fully operational power plant over a period of 2006-2011. We use the hourly average readings from the four sensors as input variables and the remaining variable, net hourly electrical energy output (EP), serves as the regression target. This dataset is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) license. + +# G.7 Physicochemical Properties of Protein Tertiary Structure + +The Physicochemical Properties of Protein Tertiary Structure dataset [Rana, 2013] consists of 45,730 samples of decoy structures. Each sample has nine physicochemical descriptor features, which we use as input variables. The remaining variable, root-mean-square deviation (RMSD), is the regression target. This dataset is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) license. We use the first 15,000 samples for our regression experiment. + +# G.8 Wine Quality + +The Wine Quality dataset [Cortez et al., 2009] has two sub-datasets regarding red wine and white wine from the Portuguese "Vinho Verde" region. Following [Hernández-Lobato and Adams, 2015], we choose the red wine sub-dataset for our regression experiment. The red wine sub-dataset has 1,599 samples, each of which is described with 11 physicochemical features. The quality variable, taking ordinal integers ranging from 0 to 10, serves as the regression target. This dataset is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) license. + +# G.9 Yacht Hydrodynamics + +The Yacht Hydrodynamics dataset [Gerritsma et al., 1981] consists of 308 samples with 6 features derived from hull geometry and hydrodynamic context. The output variable, resistance, serves as the regression target. This dataset is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) license. + +# G.10 Implementation details + +Most of the implementation details for GSFP, DKT, and NGGP in the regression experiments follow those in Appendix F.7, with differences as follows. (a) The feature extractor is the two-layer MLP; (b) We fix $d = 5$ , $n = 2$ , and $m = 1$ . Both (a) and (b) are applied across the nine regression experiments. We originally considered the same few-shot learning benchmark (all datasets) as in Sendera et al. [2021]. During the rebuttal period we applied DKT, NGGP and GSFP to vanilla regression datasets Boston, Concrete, Energy, Kin8nm, Naval, Power Plant, Protein, Wine and Yacht. In applying our multimodal, non-Gaussian model to vanilla regression, we repeat the message of Sendera et al. [2021]: "the main goal of [NNGP [Sendera et al., 2021]] was to show improvement of NGGP over standard GPs in the case of a few-shot regression task...Intuition is that NGGP may be superior to standard GPs in a simple regression setting for datasets with non-Gaussian characteristics, but do not expect any improvement otherwise". We stress that we did not perform hyperparameter tuning during the rebuttal period, and the results which follow are merely to demonstrate the performance of the models previously tuned for few-shot regression in vanilla regression settings. As such, we caution against general takeaways. + +# G.11 Training details + +All models, including DKT [Patacchiola et al., 2020], NGGP [Sendera et al., 2021] and GFSP, are trained with Adam optimiser [Kingma, 2014] using a fixed learning rate of 0.001 and default beta coefficients of $\beta_{1} = 0.9$ and $\beta_{2} = 0.999$ across all experiments. The train/test split ratio of 80/20 is applied across the datasets as described in § G.1- G.9. Five models are trained for each method on each dataset using random seeds ranging from 1 to 5. All models are trained on the train set for 10,000 iterations. The evaluation result of each method is averaged over 5 models, with each entry representing the mean $\pm$ standard deviation. Optimal performance of each meta model is obtained with grid search at a frequency of 100 training iterations. Both training and evaluation are conducted with a single RTX 2080TI GPU and an Intel(R) Xeon(R) Gold 6242 CPU @ 2.80GHz. + +# G.12 Results + +As shown in Table 5, we observe that GSFP obtains the lower NLL on 7 out of 9 datasets, and the second lowest NLL on the Protein dataset. For Kin8nm, Naval, Power Plant and Protein datasets, DKT and NGGP with spectral kernels cannot be trained due to the large number of trainable parameters associated with the spectral kernel, which cannot fit in our available GPU memory. + +# G.13 Run times + +Table 5: Benchmark results showing test NLLs for regression tasks across 9 datasets described in § G.1-G.9. For each dataset, we train 5 models with different random seeds (1-5) via maximum marginal likelihood over the train set. At testing time, we condition on the train set and evaluate the NLL on the test set for each model. Each entry shows the mean ± standard deviation over 5 model evaluations. Best results in bold, second best underlined. + +
DatasetsMethods
DKT [Patacchiola et al., 2020]NGGP [Sendera et al., 2021]GSFP
RBFSpectralNN LinearRBFSpectralNN LinearNN Linear
Boston0.335 ± 0.1980.491 ± 0.1790.478 ± 0.1290.271 ± 0.2290.439 ± 0.2280.288 ± 0.084-0.345 ± 0.272
Concrete0.300 ± 0.1230.302 ± 0.1350.442 ± 0.0820.647 ± 0.2550.447 ± 0.2470.411 ± 0.407-0.072 ± 0.175
Energy-0.302 ± 0.173-0.277 ± 0.094-0.474 ± 0.1180.481 ± 0.1350.597 ± 0.0270.364 ± 0.107-1.342 ± 0.393
Kin8nm0.153 ± 0.011-0.388 ± 0.0330.310 ± 0.150-0.655 ± 0.1130.021 ± 0.080
Naval-1.070 ± 0.011-0.994 ± 0.1560.405 ± 0.474-0.830 ± 0.135-2.019 ± 0.219
Power Plant-0.220 ± 0.193--0.028 ± 0.022-0.114 ± 0.137--0.156 ± 0.136-0.552 ± 0.137
Protein1.024 ± 0.045-1.144 ± 0.0240.418 ± 0.048-0.844 ± 0.0970.818 ± 0.152
Wine1.162 ± 0.0341.192 ± 0.0501.170 ± 0.031-1.381 ± 0.264-1.444 ± 0.276-1.666 ± 0.6640.755 ± 0.138
Yacht-1.850 ± 0.251-1.278 ± 1.219-1.496 ± 0.256-2.350 ± 0.5340.542 ± 0.1590.166 ± 0.591-2.873 ± 0.238
+ +Table 6: Benchmark results showing training time (hours) and inference speed (tests / s) for each model across regression tasks. The results are evaluated on 5 models, and each entry shows the mean ± standard deviation over these 5 models. + +
DatasetsTraining Time (hours) ↓
Methods
DKT [Patacchiola et al., 2020]NGGP [Sendera et al., 2021]GSFP
RBFSpectralNN LinearRBFSpectralNN LinearNN Linear
Boston0.034 ± 0.0010.037 ± 0.0010.032 ± 0.0025.042 ± 0.6194.024 ± 0.1984.828 ± 0.3530.100 ± 0.003
Concrete0.091 ± 0.0070.112 ± 0.0050.115 ± 0.01110.660 ± 1.20910.592 ± 1.4009.229 ± 1.2580.176 ± 0.003
Energy0.048 ± 0.0000.053 ± 0.0010.045 ± 0.0017.585 ± 0.7057.741 ± 0.6456.866 ± 0.4500.121 ± 0.004
Kin8nm1.923 ± 0.016-0.727 ± 0.03562.107 ± 7.461-64.435 ± 5.7471.238 ± 0.002
Naval3.149 ± 0.075-1.367 ± 0.052127.623 ± 11.349-108.597 ± 5.0252.025 ± 0.008
Power Plant1.664 ± 0.271-0.839 ± 0.06176.962 ± 4.722-68.406 ± 5.1271.519 ± 0.003
Protein5.415 ± 0.559-2.303 ± 0.092144.444 ± 10.479-111.314 ± 12.8362.868 ± 0.020
Wine0.099 ± 0.0080.175 ± 0.0170.101 ± 0.00816.680 ± 2.62118.150 ± 3.62318.791 ± 4.2150.206 ± 0.001
Yacht0.033 ± 0.0000.035 ± 0.0000.032 ± 0.0004.677 ± 0.1084.432 ± 0.2064.867 ± 0.2110.109 ± 0.001
+ +
Inference Time (seconds) ↓
DatasetsMethods
DKT [Patacchiola et al., 2020]NGGP [Sendera et al., 2021]GSFP
RBFSpectralNN LinearRBFSpectralNN LinearNN Linear
Boston0.30 ± 0.000.30 ± 0.000.27 ± 0.007.09 ± 0.537.36 ± 0.076.85 ± 1.220.52 ± 0.01
Concrete1.10 ± 0.001.43 ± 0.051.55 ± 0.1714.11 ± 2.3312.95 ± 1.5515.74 ± 1.491.04 ± 0.00
Energy0.70 ± 0.000.79 ± 0.040.55 ± 0.008.20 ± 0.828.38 ± 0.938.64 ± 0.031.02 ± 0.02
Kin8nm30.96 ± 0.19-20.54 ± 1.34118.48 ± 11.49-113.30 ± 12.798.53 ± 0.01
Naval67.41 ± 2.61-42.08 ± 3.60178.25 ± 3.59-155.69 ± 2.9715.03 ± 0.19
Power Plant42.51 ± 0.56-27.32 ± 2.16144.27 ± 4.71-134.66 ± 10.579.67 ± 0.02
Protein118.11 ± 0.02-70.83 ± 2.72297.26 ± 14.70-211.55 ± 4.8915.10 ± 0.03
Wine1.70 ± 0.002.98 ± 0.101.62 ± 0.0017.08 ± 0.7225.11 ± 1.9521.76 ± 5.182.10 ± 0.01
Yacht0.17 ± 0.000.17 ± 0.000.15 ± 0.005.59 ± 1.326.45 ± 0.745.24 ± 1.660.41 ± 0.01
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However, existing estimation theory typically requires stringent smoothness conditions on the underlying Brenier potentials and assumes bounded distribution supports, limiting practical applicability. In this paper, we introduce a unified theoretical framework for semi-dual OT map estimation that relaxes both of these restrictions. Building on sieved convex conjugate, our framework has two key contributions: (i) a new map stability bounds that holds without any second-order regularity assumptions on the true Brenier potentials, and (ii) an oracle inequality that cleanly decomposes the estimation error into statistical error, sieved bias, and approximation error. Specifically, our approximation error is measured in the $L^{\infty}$ norm rather than Sobolev norm in the existing results, aligning more naturally with classical approximation theory. Leveraging these tools, we provide statistical error of semi-dual estimators with mild and verifiable conditions on the true OT map. Moreover, we establish the first theoretical guarantee for deep neural network OT map estimator between general distributions, with Tanh network function class as an example. + +# 1 Introduction + +Optimal Transport (OT) provides a powerful framework for transforming one probability measure into another. Concretely, given two distributions $\mu$ and $\nu$ on $\mathbb{R}^d$ , the OT problem seeks a map $T: \mathbb{R}^d \to \mathbb{R}^d$ that transports $\mu$ to $\nu$ while minimizing a cost function. Beyond its elegant mathematical foundation, OT has found broad applications in Ai4Science [39, 7, 29], computer vision [3, 23, 43, 12], and nonparametric statistical inference [10, 26, 24, 18], powering advances in scientific discovery and machine/deep learning. Despite its broad impact, rigorous theoretical guarantees for OT estimators under mild and realistic regularity conditions remain limited. + +This paper aims to contribute to this direction by focusing on the dual-type OT map estimators, formally introduced in Section 2, which are based on the characterization of the optimal transport map as the gradient of a convex function (the so-called Brenier potential). The seminal work of [27] established minimax optimality for a wavelet-truncated dual-type OT estimator, but under restrictive conditions: $\mu$ is supported on $[0,1]^d$ with a bounded density, $\nu$ is compactly supported, and the true + +OT map is both $\alpha$ -Hölder smooth and strongly convex. These assumptions exclude many practical OT applications. + +In parallel, Gunsilius [25] introduced an alternative analysis using the Poincaré inequality, a fundamental tool in PDEs and functional analysis. While still assuming compact and convex supports, this approach removed the need for Hölder smoothness or strong convexity of the Brenier potential. Building on this, [21] extended the theory to sub-exponential $\mu$ , under $(\alpha, a)$ -convexity and $(\beta, a)$ -smoothness conditions on the Brenier potential. More recently, [20] relaxed the Poincaré inequality assumption to accommodate heavy-tailed $\mu$ and $\nu$ , and eliminated the $(\alpha, a)$ -convexity condition via a new sieved OT estimator, characterizing how tail thickness of $\mu$ and $\nu$ influences convergence. + +We also note recent advances in plug-in type OT estimators, which directly compute the OT map between estimated distributions [10, 17, 34, 37, 5]. In parallel with these theoretical advances, significant efforts have focused on improving the computational efficiency of OT, including entropic regularization [15, 1, 22, 11, 19] and sum-of-squares formulations [35, 41]. Meanwhile, neural-network-based OT estimators have advanced rapidly, powering large-scale applications such as image generation and translation [33, 30, 4, 31, 12]. + +Despite these advances, a few core regularity assumptions on the true Brenier potential $\varphi_0$ and distributions still limit the applicability of dual-type estimators. To clarify these limitations, we decompose the overall estimation error into two components: statistical error and approximation error. The statistical error measures the discrepancy between the estimator and the best possible function in $\mathcal{F}$ , reflecting the randomness from finite samples. The approximation error captures the gap between the true Brenier potential and its best approximation within $\mathcal{F}$ , since we do not assume that the true Brenier potential $\varphi_0$ lies in $\mathcal{F}$ . Existing analyses of each error component often rely on restrictive smoothness or convexity assumptions, which we discuss in more below. + +First, bounding the statistical error in existing works requires a $(\beta, b)$ -smoothness condition on $\varphi_0$ , i.e., a polynomial bound on its Hessian. While this condition is verifiable under strong log-concavity of $\mu$ and $\nu$ [8, 14, 13, 28], its validity beyond this class remains unclear. Moreover, Proposition 3.12 of [20] shows that this smoothness requirement constrains $\nu$ to have tails no much heavier than those of $\mu$ , excluding many practical scenarios, such as multivariate quantile [9, 10, 24], where we transport from $\mathbb{U}([0,1]^d)$ to distributions with unbounded support, or transporting from a Gaussian (with exponential tails) to a Student's $t$ -distribution (with polynomial tails). + +Second, controlling the approximation error typically relies on $(\alpha, a)$ -convexity of $\varphi_0$ , i.e. a generalization of strong convexity, which similarly imposes constraints on the relative tail behavior of $\mu$ and $\nu$ [20]. For instance, OT map from an unbounded $\mu$ to a bounded $\nu$ fails this condition. This is especially problematic in generative modeling, where the OT map is learned from Gaussian noise to image data of bounded values [40, 30, 38, 32]. Moreover, existing approximation error bounds assume that both $\mu$ and $\nu$ have bounded support, [27, 21]. + +These limitations raise a fundamental question in broadening the scope of optimal transport: + +Can one bound the statistical and approximation errors of dual-type OT estimators for general distributions without such restrictive assumptions? + +In this work, we provide a positive answer to the question posed above. Our two core innovations are a novel map stability inequality and a refined oracle inequality. Building on these tools, we are the first to establish statistical and approximation error bounds for dual-type OT estimators under minimal regularity assumptions. Table 1 provides a side-by-side comparison of the regularity conditions required by our results versus those in the existing literature. + +Table 1: Regularity assumptions on the true Brenier potential $\varphi_0$ and $\mu$ required to establish statistical and approximation error bounds ( $\mathcal{E}_{stat}$ and $\mathcal{E}_{app}$ ) in prior work and in our results. + +
Assumption on φ0Distributional Assumption
(β,b)-smooth (for Estat)(α,a)-convex (for App)Tail thickness of μPoincaré inequality on μ
[27]b=0a=0CompactImplicitly Used
[25]Not RequiredNot RequiredCompactRequired
[21]RequiredRequiredSub-exponentialRequired
[20]RequiredRequiredGeneralPoincaré-type inequality
OursNot RequiredNot RequiredGeneralNot Required
+ +Specifically, we summarize our main contributions as follows: + +1. A new map stability inequality without $(\beta, b)$ -smoothness. Stability inequalities are key to linking OT estimation error to the semi-dual objective. Existing results rely on $(\beta, b)$ -smoothness of $\varphi_0$ . In Proposition 3.3, we derive a new inequality that removes this requirement, assuming only a well-behaved OT map (Assumption 3.2), which holds unconditionally when $\nu$ is bounded and under mild conditions for unbounded distributions. +2. A novel oracle inequality without $(\alpha, a)$ -convexity. Proposition 3.4 provides an oracle inequality that avoids the $(\alpha, a)$ -convexity assumption on $\varphi_0$ and decomposes the estimation error into statistical error, sieved bias, and approximation error. Unlike prior works [21, 20], which involve hard-to-analyze gradient terms like $\inf_{\varphi \in \mathcal{F}}\| \nabla \varphi -\nabla \varphi_0\|_{L^2 (\mu)}$ , our bound relies on a more tractable supremum error over a compact region $B$ : $\inf_{\varphi \in \mathcal{F}}\| \varphi -\varphi_0\|_{L^{\infty}(B)}$ , making it more suitable for analysis using classical approximation theory. +3. First non-asymptotic bounds under minimal smoothness. With our new map stability inequality, Theorem 3.6 establishes the first non-asymptotic statistical error bound for dual-type OT estimators requiring only a first-order condition on $\varphi_0$ , without second-order smoothness assumptions. +4. First theoretical guarantee for neural OT estimators. With our new tools, we derive the first convergence guarantee for neural OT estimators using smooth networks. For tanh neural networks (TNNs), Theorem 3.7 shows that if $\varphi_0\in C^\alpha (\mathbb{R}^d)$ for $\alpha \geq 2$ , then the neural OT estimator $\nabla \hat{\varphi}_n$ between unbounded distributions satisfies: + +$$ +\mathbb {E} \| \nabla \hat {\varphi} _ {n} - \nabla \varphi_ {0} \| _ {L ^ {2} (\mu)} ^ {2} \lesssim_ {\log n} n ^ {- \frac {\alpha}{d + 2 \alpha}}. \tag {1} +$$ + +Notably, this rate matches that of [25], but under significantly weaker assumptions: we do not require compact support for $\mu$ and $\nu$ , nor a Poincaré inequality on $\mu$ . + +Notations and Terminologies Throughout this paper, we use the following notations. For $\alpha \in \mathbb{R}$ , $\lfloor \alpha \rfloor = \{x \in \mathbb{Z} : x < \alpha\}$ . For $v \in \mathbb{R}^d$ , we write $\| v \|_2$ for its $\ell_2$ norm and $\| v \|_\infty$ for its $\ell_\infty$ norm. For a matrix $A \in \mathbb{R}^{d_1 \times d_2}$ , let $\| A \|_\infty := \max_{i,j} |A_{i,j}|$ . If $A$ is symmetric, $\| A \|_{op}$ denotes its operator norm, $\lambda_{\min}(A)$ denotes its smallest eigenvalue. For a function $f: \mathbb{R}^d \to \mathbb{R}^m$ and a measure $\mu$ , we write $\mu f, \mathbb{E}_\mu f(X)$ , or $\int f \, \mathrm{d}\mu$ for its integral, and $\| f \|_{L^2(\mu)} := (\int \| f \|_2^2 \, \mathrm{d}\mu)^{1/2}$ for the $L^2$ norm. For $f: \mathbb{R}^d \to \mathbb{R}$ and a set $B \subset \mathbb{R}^d$ , we denote $\| f \|_{L^\infty(B)} := \sup_{x \in B} |f(x)|$ . + +We write $a \lesssim b$ to mean $a \leq C b$ for some constant $C > 0$ independent of $a$ and $b$ , and $a \lesssim_{\log n} b$ to omit polylogarithmic factors in $n$ . We write $a \asymp b$ if $a \lesssim b$ and $b \lesssim a$ . We use $a \vee b$ and $a \wedge b$ to denote the maximum and minimum of $a$ and $b$ , respectively. For sequences $a_n$ and $b_n$ , we write $b_n = \mathcal{O}(a_n)$ if there exists a constant $c > 0$ such that $|b_n| \leq c a_n$ , and write $b_n = \mathcal{O}_{\log}(a_n)$ if there exists constants $c_1, c_2 > 0$ such that $|b_n| \leq c_1 a_n \log(n)^{c_2}$ . We use $\mathbb{I}$ to denote indicator function. + +For $x \in \mathbb{R}^d$ , write $\langle x \rangle \coloneqq 1 + \| x\|_2$ . When $x \in \mathbb{R}$ , we set $\langle x \rangle \coloneqq 1 + |x|$ . We define $\log_{+}(x) \coloneqq \max \{\log x, 1\}$ . A function $\varphi \in C^2$ is called $(\beta, \mathbf{b})$ -smooth if $\|\nabla^2\varphi(x)\|_{op} \leq \beta \langle x \rangle^b$ for all $x$ , and $(\alpha, \mathbf{a})$ -convex if $\lambda_{\min}(\nabla^2\varphi(x)) \geq \alpha \langle x \rangle^a$ . + +A probability measure $\mu$ is $(\lambda, \mathbf{K})$ -sub-Weibull if for $X \sim \mu$ , $\mathbb{P}(\|X\|_2 \geq t) \lesssim \exp(-(t / K)^\lambda)$ for all $t > 0$ . We refer to such a distribution as $\lambda$ -sub-Weibull, or simply sub-Weibull. + +# 2 Background on Optimal Transport + +Let $\mu$ and $\nu$ be two probability measures on $\mathbb{R}^d$ . A measurable map $T: \mathbb{R}^d \to \mathbb{R}^d$ is called a push-forward from $\mu$ to $\nu$ , if for any measurable set $A \subset \mathbb{R}^d$ , $\mu(T^{-1}(A)) = \nu(A)$ . We denote this by $\nu = T_{\#}\mu$ . In Monge's problem, the optimal transport map is defined as the push-forward that minimizes the transport cost from $\mu$ to $\nu$ : + +$$ +\min _ {T: \mathbb {R} ^ {d} \rightarrow \mathbb {R} ^ {d}} \int_ {\mathbb {R} ^ {d}} \| x - T (x) \| _ {2} ^ {2} \mu (\mathrm {d} x), \quad \text {s . t .} \quad T _ {\#} \mu = \nu , \tag {2} +$$ + +However, Monge's problem may not admit a solution, for example, when $\mu$ is discrete and $\nu$ is continuous. A natural relaxation is the Kantorovich formulation, which considers couplings $\pi \in \Pi (\mu ,\nu)$ with marginals $\mu$ and $\nu$ : + +$$ +\min _ {\pi} \iint \| x - y \| _ {2} ^ {2} \pi (\mathrm {d} x, \mathrm {d} y), \quad \text {s . t .} \quad \pi \in \Pi (\mu , \nu). \tag {3} +$$ + +Kantorovich formulation is always feasible [42]. As a linear program, its semi-dual form reads: + +$$ +\min _ {\varphi \in L ^ {1} (\mu)} \left\{\int \varphi d \mu + \int \varphi^ {*} d \nu \right\} = \min _ {\varphi \in L ^ {1} (\mu)} \mu \varphi + \nu \varphi^ {*}. \tag {4} +$$ + +where $\varphi^{*}$ is the convex conjugate (a.k.a. Legendre-Fenchel conjugate) of $\varphi$ : + +$$ +\varphi^ {*} (y) = \sup _ {x \in \mathbb {R} ^ {d}} \left\{\langle x, y \rangle - \varphi (x) \right\}, \quad \text {f o r a l l} y \in \mathbb {R} ^ {d}. \tag {5} +$$ + +The equivalence between Monge's problem (2), Kantorovich's formulation (3), and its semi-dual form (4) was established in the seminal work of Brenier [6]: + +Theorem 2.1 (Brenier's Theorem). Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^d$ , both with finite second moments, and assume that $\mu$ does not assign mass to small sets. Then there exists a convex function $\varphi_0$ that uniquely (up to constants) solves (4). Moreover, $\nabla \varphi_0$ solves Monge's problem (2), and $(Id \times \nabla \varphi_0)_{\#} \mu$ solves the Kantorovich formulation (3). + +Brenier's Theorem reveals that the OT map is the gradient of a convex function $\varphi_0$ , called the Brenier potential. Furthermore, since $\varphi_0$ is the minimizer of (4), we can estimate it using empirical measures $\mu_n$ and $\nu_m$ of $\mu$ and $\nu$ , respectively. For a chosen function class $\mathcal{F}$ , we define the Brenier potential estimator and the associated dual-type OT estimator as: + +$$ +\hat {\varphi} _ {n, m} = \underset {\varphi \in \mathcal {F}} {\arg \min } \mu_ {n} \varphi + \nu_ {m} \varphi^ {*}, \quad \text {w i t h O T m a p e s t i m a t o r} \quad \nabla \hat {\varphi} _ {n, m}. \tag {6} +$$ + +However, the supremum over the entire space $\mathbb{R}^d$ in the convex conjugate $\varphi^{*}$ makes the optimization procedure unstable and sensitive. It also introduces theoretical challenges when analyzing the convergence of dual-type OT estimator $\hat{\varphi}_{n,m}$ . To address this issue, [20] proposed the concept of a sieved convex conjugate, which restricts the supremum to a bounded region. Specifically, for some sieve radius $\tilde{R} > 0$ , define: + +$$ +\varphi_ {\tilde {R}} ^ {*} (y) := \sup _ {x: \| x \| _ {2} \leq \tilde {R}} \langle x, y \rangle - \varphi (x), \quad \text {f o r a l l} y \in \mathbb {R} ^ {d}. \tag {7} +$$ + +Note that $\varphi^{*}(\nabla \varphi (x)) = \langle x,\nabla \varphi (x)\rangle -\varphi (x)$ . By restricting $x$ to $B(0,\tilde{R})$ , we equivalently restrict the domain of $\varphi_{\tilde{R}}^{*}(y)$ to the compact set $\nabla \varphi (B(0,\tilde{R}))$ . This compactness facilitates us to analyze the estimation error of sieved OT estimator. + +Given a sieve radius $\tilde{R}$ , the corresponding sieved estimator is defined as + +$$ +\tilde {\varphi} _ {n, m} := \underset {\varphi \in \mathcal {F}} {\operatorname {a r g m i n}} \mu_ {n} \varphi + \nu_ {m} \varphi_ {\tilde {R}} ^ {*}, \quad \text {w i t h s i e v e d e s t i m a t o r} \quad \nabla \tilde {\varphi} _ {n, m}. \tag {8} +$$ + +Note that when $\tilde{R} = \infty$ , we recover the original dual-type estimator in Equation (6). In the following, we may abbreviate it to $\tilde{\varphi}_n$ when $n = m$ . + +# 3 Main Results + +In Sections 3.1 and 3.2, we develop two core analytical tools: a new stability inequality for the sieved estimator (Proposition 3.3), and a refined oracle inequality that cleanly decomposes the estimation error (Proposition 3.4). Building on these results, Section 3.3 establishes non-asymptotic statistical error rates for general function classes (Theorem 3.6). Finally, in Section 3.4, we show how the overall estimation error of neural OT estimators between general distributions can be controlled, using tanh-activated neural networks a concrete example (Theorem 3.7). + +# 3.1 A New Map Stability Inequality + +From a statistical learning perspective, map stability inequalities are essential for linking the semidual objective to estimation error. For the original dual-type estimator, denote its objective is $S_{\mu ,\nu}(\varphi)\coloneqq \mu \varphi +\nu \varphi^{*}$ . Existing results [27, 21, 20] typically show that for all $\varphi \in \mathcal{F}$ + +$$ +\left\| \nabla \varphi - \nabla \varphi_ {0} \right\| _ {L ^ {2} (\mu)} ^ {2} \lesssim \mathcal {S} _ {\mu , \nu} (\varphi) - \mathcal {S} _ {\mu , \nu} (\varphi_ {0}). \tag {9} +$$ + +where $\nabla \varphi_0$ is the true OT map to be estimated. + +However, these results apply only to the original estimator (6), not the sieved version in (8), which often performs better in practice. Moreover, while neither side of the inequality involves second-order derivatives, prior analyses still require $(\beta, b)$ -smoothness of $\varphi_0$ , limiting their generality. + +To accommodate sieved estimator and relax their assumptions, we consider the following conditions: + +Assumption 3.1 (Envelope for Function Class). Let $\mathcal{F} \subset C^2(\mathbb{R}^d)$ be a function class. We assume there exists a non-decreasing and pointwise finite function $U_2: [0, \infty) \to [1, \infty)$ such that: + +$$ +\sup _ {\varphi \in \mathcal {F}} \sup _ {\| x \| \leq R} \| \nabla^ {2} \varphi (x) \| _ {o p} \leq U _ {2} (R), \quad \sup _ {\varphi \in \mathcal {F}} \| \nabla \varphi (0) \| _ {2} \leq U _ {2} (R). +$$ + +Assumption 3.1 depends only on $\mathcal{F}$ and is independent of $\varphi_0, \mu$ and $\nu$ . It is satisfied by many common function classes, for example, + +1. Quadratic function class: When $\mu$ and $\nu$ are both Gaussian (or from the same elliptical family), the Brenier potential is quadratic and lies in the following quadratic family: + +$$ +\mathcal {F} _ {q u a d} = \left\{x \mapsto x ^ {\top} B x + \langle b, x \rangle : B \in \mathbb {S} _ {+} ^ {d}, \| B \| _ {o p} \leq r _ {1}, \| b \| _ {2} \leq r _ {2} \right\}. +$$ + +2. Smooth Neural Networks: Neural network function classes with all parameters bounded and activation functions that are $C^2$ -smooth, such as sigmoid, tanh, or softmax. +3. Reproducing Kernel Hilbert Spaces (RKHS): Let $K: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$ be a positive-definite kernel on its domain $\mathcal{X} \times \mathcal{X}$ , and let $\mathcal{H}_K$ be the corresponding RKHS with norm $\| \cdot \|_{\mathcal{H}_K}$ . We assume $\mathcal{F}$ is $C^4$ and take $\mathcal{F}$ to be the unit ball of $\mathcal{H}_K$ , i.e., $\mathcal{F} = \{\varphi \in \mathcal{H}_K : \| \varphi \|_{\mathcal{H}_K} \leq 1\}$ . + +Assumption 3.1 and 3.2 together play a role similar to Assumption A.1 in [21], but are strictly weaker. As a result, the function classes discussed in Section 4 of [21], including parametric family, wavelet expansions, RKHS, and Barron Spaces, apply directly to our setting. We refer readers to Section 4 of [21] for a detailed discussion. + +Assumption 3.2 (Envelope for OT map). There is a function $u \in L^4(\mu)$ , such that for all $\varphi \in \mathcal{F}$ + +$$ +\| \nabla \varphi (x) - \nabla \varphi_ {0} (x) \| _ {2} \leq u (x), \quad f o r a l l x. +$$ + +Define $U_{1}(R) \coloneqq \sup_{\| x \|_{2} \leq R} u(x)$ . Additionally, let $\Phi \in L^{2}(\mu)$ be an envelope function for $\mathcal{F} \cup \{\varphi_0\}$ , i.e. + +$$ +\left| \varphi (x) \right| \leq \Phi (x), \quad f o r a l l x \in \mathbb {R} ^ {d} a n d a l l \varphi \in \mathcal {F} \cup \varphi_ {0}. +$$ + +A sufficient condition for Assumption 3.2 is that both Proposition A.2 and Assumption 3.1 hold. Specifically, Proposition A.2 ensures $\| \nabla \varphi_0(x)\| _2$ to be bounded by some function $L_{1}(x)$ . Meanwhile, Assumption 3.1 provides an envelope for $\| \nabla \varphi (x)\| _2$ , denoted as $L_{2}(x)$ . Then Assumption 3.2 follows via the triangle inequality $\| \nabla \varphi (x) - \nabla \varphi_0(x)\| _2\leq L_1(x) + L_2(x)$ . While Assumption 3.1 solely relies on $\mathcal{F}$ , typical distributions for Proposition A.2 to hold include: $\mu$ can be normal or Student-t, and $\nu$ can be any measure with certain tail decay. We refer to Appendix A for further details and an explicit example verifying Assumption 3.2. + +We are now ready to present our new stability inequality for the sieved estimator: + +Proposition 3.3 (Map Stability Inequality). Suppose Assumptions 3.1 and 3.2 hold. For any $\varepsilon >0$ define $R_{\varepsilon}$ and sieve radius $\tilde{R}_{\varepsilon}$ such that + +$$ +\mu \left(\| X \| _ {2} > R _ {\varepsilon}\right) \leq \varepsilon , \quad \tilde {R} _ {\varepsilon} \geq \sup _ {\| x \| _ {2} \leq R _ {\varepsilon}} \left\{\| x \| _ {2} + \frac {u (x)}{U _ {2} \left(\| x \| _ {2} + u (x)\right)} \right\} \tag {10} +$$ + +For any $\varphi \in \mathcal{F}$ , define the truncated excess risk: + +$$ +r _ {\varepsilon} (\varphi) := \int_ {B (0, R _ {\varepsilon})} \varphi (x) + \varphi_ {R _ {\varepsilon}} ^ {*} (\nabla \varphi_ {0} (x)) \mu (d x) - \int_ {B (0, R _ {\varepsilon})} \varphi_ {0} (x) + \varphi_ {0} ^ {*} (\nabla \varphi_ {0} (x)) \mu (d x). \tag {11} +$$ + +Then the following inequality holds: for any $\varphi \in \mathcal{F}$ + +$$ +\left\| \nabla \varphi - \nabla \varphi_ {0} \right\| _ {L ^ {2} (\mu)} ^ {2} \leq 2 U _ {2} \left(R _ {\varepsilon} + U _ {1} \left(R _ {\varepsilon}\right)\right) \cdot r _ {\varepsilon} (\varphi) + \| u \| _ {L ^ {4} (\mu)} ^ {2} \cdot \varepsilon^ {\frac {1}{2}}, \tag {12} +$$ + +We first clarify the roles of $R_{\varepsilon}$ and $\tilde{R}_{\varepsilon}$ , which enable a more tractable approximation error in the next section. Specifically, the hyper-balls $B(0,R_{\varepsilon})$ and $\nabla \varphi_0(B(0,R_\varepsilon))$ serve as bounded pseudosupports for $\mu$ and $\nu$ , respectively, over which the behavior of $\varphi$ and its convex conjugate can be effectively controlled. Meanwhile, the sieve radius $\tilde{R}_{\varepsilon}$ is introduced to handle any mismatch between $\nabla \varphi \bigl (B(0,R_{\varepsilon})\bigr)$ and the pseudo-support of $\nu$ , by better regulating the range of the sieved conjugate. In practice, if $\nabla \varphi$ closely approximates $\nabla \varphi_0$ in $L^{\infty}$ norm (so that $u\approx 0$ ), the mismatch is negligible and one can take $\tilde{R}_{\varepsilon}\approx R_{\varepsilon}$ . In that case, it suffices to compute the convex conjugate over the pseudo-support of $\mu$ alone. + +For simplicity, a sufficient choice for $\tilde{R}_{\varepsilon}$ is $\tilde{R}_{\varepsilon} = R_{\varepsilon} + U_1(R_{\varepsilon})$ . In particular, if $\mu$ is supported on $B(0,R)$ , one may take $R_{\varepsilon} = R$ . + +Equation (12) decomposes the estimation error into two components. The first arises from the truncated excess risk over the high-probability region $B(0, R_{\varepsilon})$ , which can be well-controlled (see Proposition 3.4). Its coefficient $U_{2}(R_{\varepsilon} + U_{1}(R_{\varepsilon}))$ originates from a second-order Taylor expansion of $\varphi$ , capturing the "uniform curvature" of $\mathcal{F}$ on the ball $B(0, R_{\varepsilon} + U_{1}(R_{\varepsilon}))$ . The second component accounts for the residual risk outside this region. As a result, $\varepsilon$ must be carefully selected to balance the trade-off: a smaller $\varepsilon$ reduces the residual risk but increases the truncated excess risk. + +We defer the proof of Proposition 3.3 to Appendix B. + +Comparison with existing map stability results. The most closely related results are Proposition 10 of [27], Proposition 1 of [21], and Lemma 3.15 of [20], all of which require $(\beta, b)$ -smoothness assumptions on the true Brenier potential $\varphi_0$ and the candidate class $\mathcal{F}$ . In contrast, our Proposition 3.3 substantially relaxes these requirements by assuming only an upper bound on the true OT map $\nabla \varphi_0$ via function $u(x)$ . + +Another key distinction is the introduction of the sieved convex conjugate $\varphi_{\tilde{R}_{\xi}}^{*}$ , essential for providing a more accessible approximation error in the next section. This sieving remains necessary even when $\mu$ has bounded support, unless additional regularity conditions are imposed on $\mu$ and $\nu$ . + +Compared to map stability results for plug-in type estimators, our bound also holds under milder assumptions. For instance, Theorem 2.1 of [17] requires $\nabla \varphi_0$ to be Lipschitz, while Theorem 3 of [5] assumes that the Brenier potential is strongly convex. + +# 3.2 A New Oracle Inequality + +Now, we present our second contribution of a new oracle inequality that helps cleanly decompose the estimation error into statistical error, sieved bias, and approximation error. + +Proposition 3.4 (Oracle Inequality). Under Assumptions 3.1 and 3.2, and let $R_{\varepsilon}, \tilde{R}_{\varepsilon}$ be as defined in Equation (10). For the sieved OT estimator $\tilde{\varphi}_{n,m}$ from Equation (8), its truncated excess risk $\tilde{r}_{n,m,\varepsilon} := r_{\varepsilon}(\tilde{\varphi}_{n,m})$ , defined in Equation (11), admits the decomposition: + +$$ +\tilde {r} _ {n, m, \varepsilon} := r _ {\varepsilon} \left(\tilde {\varphi} _ {n, m}\right) \leq \mathcal {E} _ {\text {s t a t}} + \mathcal {E} _ {\text {s i e v e}} + \mathcal {E} _ {\text {a p p}}, \tag {13} +$$ + +where + +$$ +\mathcal {E} _ {\text {s t a t}} := \sup _ {f \in \bar {\mathcal {F}}} \int_ {B (0, R _ {\varepsilon})} f d (\mu_ {n} - \mu) + \sup _ {g \in \bar {\mathcal {G}}} \int_ {B (0, R _ {\varepsilon})} g d (\nu_ {m} - \nu), \tag {14a} +$$ + +$$ +\mathcal {E} _ {\text {s i e v e}} := 2 \mathbb {E} _ {X \sim \mu_ {n}} [ \Phi (X) \cdot \mathbb {I} (\| X \| _ {2} > R _ {\varepsilon}) ] + 2 \mathbb {E} _ {Y \sim \nu_ {m}} [ G (Y) \cdot \mathbb {I} (\| (\nabla \varphi_ {0}) ^ {- 1} (Y) \| _ {2} > R _ {\varepsilon}) ], \tag {14b} +$$ + +$$ +\mathcal {E} _ {\text {a p p}} := 2 \inf _ {\varphi \in \mathcal {F}} \| \varphi - \varphi_ {0} \| _ {L ^ {\infty} (B (0, \tilde {R} _ {\varepsilon}))}. \tag {14c} +$$ + +Here, $\bar{\mathcal{F}} := \{\varphi_1 - \varphi_2 : \varphi_1, \varphi_2 \in \mathcal{F}\}$ , $\bar{\mathcal{G}} := \{\varphi_{1,\tilde{R}_{\varepsilon}}^* - \varphi_{2,\tilde{R}_{\varepsilon}}^* : \varphi_1, \varphi_2 \in \mathcal{F}\}$ and $G(y) := \tilde{R}_{\varepsilon} \| y \|_2 + \sup_{x \in B(0,\tilde{R}_{\varepsilon})} \Phi(x)$ . + +Thus, with a careful choice of $\varepsilon$ and $R_{\varepsilon}$ (which can be selected in a distribution-free manner, as discussed in Section 3.3), Propositions 3.3 and 3.4 together provide a complete characterization of how the sample sizes, function class $\mathcal{F}$ , and the sieved convex conjugate collectively influence the estimation error of the sieved dual-type estimators. + +Specifically, this result decomposes the truncated excess risk into three components. The first term, $\mathcal{E}_{stat}$ , reflects the statistical error from replacing the underlying distributions $\mu$ and $\nu$ with their + +estimates $\mu_{n}$ and $\nu_{m}$ . Importantly, these estimates need not be empirical measures, allowing for smoothed or bootstrap-based alternatives. The second term, $\mathcal{E}_{\text{sieve}}$ , accounts for the bias introduced by replacing the original convex conjugate (Equation (7)) with its sieved version. This bias vanishes when $R_{\varepsilon} = \infty$ , where the sieved and original convex conjugates coincide. + +The third term, $\mathcal{E}_{app}$ , captures the approximation error of the function class $\mathcal{F}$ in representing the true potential $\varphi_0$ over the compact domain $B(0,\tilde{R}_{\varepsilon})$ . As usual, enlarging $\mathcal{F}$ reduces $\mathcal{E}_{app}$ at the cost of increasing $\mathcal{E}_{stat}$ . Notably, we express the approximation error in the $L^{\infty}$ norm on a compact set, aligning naturally with classical approximation theory. In contrast, prior works [27, 21, 20] typically rely on gradient-based errors like $\inf_{\varphi \in \mathcal{F}}\| \nabla \varphi -\nabla \varphi_0\|_{L^2 (\mu)}$ , which remain challenging to analyze for rich function classes such as deep neural networks. + +Proof ideas of Propositions 3.4 The sieve technique allows us to treat $B(0, R_{\varepsilon})$ and $\nabla \varphi_0(B(0, \tilde{R}_{\varepsilon}))$ as bounded pseudo-supports of $\mu$ and $\nu$ , or equivalently, as bounded pseudo-domains for $\varphi_0$ and $\varphi_0^*$ . This reduction localizes the approximation problem to compact sets, where classical approximation theory applies. Another key benefit of this sieving approach is that it eliminates the need for the $(\alpha, a)$ -convexity assumption required to control the approximation error in prior works. + +The sieve bias is then controlled via tail probability bounds, while the statistical error is handled using standard tools from empirical process theory. The full proof is deferred to Appendix C. + +# 3.3 Non-Asymptotic Bounds of Statistical Error and Sieved Bias + +In this section, we derive bounds for the statistical error $\mathcal{E}_{stat}$ and the sieved bias $\mathcal{E}_{sieve}$ in Proposition 3.4, under minimal assumptions on the source distribution $\mu$ . Notably, our results do not require smoothness of the true OT map $\nabla \varphi_0$ , improving upon existing analyses. + +We begin with the following assumption on the covering entropy of the function class $\mathcal{F}$ : + +Assumption 3.5 (Covering Entropy of $\mathcal{F}$ ). For some $\eta \geq 0$ , $\gamma \in [0,2)$ , $\gamma' \geq 1$ and $D_{\mathcal{F}} > 1$ , the covering entropy of $\mathcal{F}$ satisfies: for any $h \geq 0$ , + +$$ +\log \mathcal {N} (h, \mathcal {F}, L ^ {\infty} ([ - R, R ] ^ {d}) \leq D _ {\mathcal {F}} \cdot h ^ {- \gamma} \cdot \log_ {+} (1 / h) ^ {\gamma^ {\prime}} \cdot R ^ {\eta}. \tag {15} +$$ + +We focus on the Donsker regime ( $\gamma \in [0,2)$ ) for theoretical simplicity, which already captures a wide range of function classes, such as parametric families, wavelets, reproducing kernel Hilbert space (RKHS), and both shallow and deep neural networks. Here, $D_{\mathcal{F}}$ represents the "effective dimension" of $\mathcal{F}$ , and $n / D_{\mathcal{F}}$ can be viewed as the effective sample size from the statistical learning theory perspective. Because most covering entropy results are stated for the $L^{\infty}([0,1]^d)$ norm, we introduce an additional $R^{\eta}$ factor to translate them to our unbounded setting in $L^{\infty}([-R,R]^{d})$ . Specifically, the exponent $\eta$ measures how the envelope of $\mathcal{F}$ grows (e.g. $\eta = 2$ if the envelope scales like $x \mapsto x^2$ ). + +Theorem 3.6 (Statistical Error with Empirical Measures). Under assumptions of Proposition 3.4, and Assumption 3.5, the sieved estimator $\tilde{\varphi}_{n,m}$ from Equation (8), computed with empirical measures $\mu_{n}$ and $\nu_{m}$ , satisfies: + +$$ +\mathbb {E} \left[ \mathcal {E} _ {\text {s t a t}} \right] \lesssim \sqrt {\frac {D _ {\mathcal {F}}}{n}} \left(R _ {\varepsilon} ^ {\frac {\eta}{2}} + \sup _ {x \in B (0, R _ {\varepsilon})} \Phi (x) ^ {\frac {1}{2}}\right) \log (n) ^ {\frac {\gamma^ {\prime}}{2}} + \sqrt {\frac {D _ {\mathcal {F}}}{m}} \left(\tilde {R} _ {\varepsilon} ^ {\frac {\eta}{2}} + M ^ {\frac {1}{2}}\right) \log (m) ^ {\frac {\gamma^ {\prime}}{2}}, \tag {16} +$$ + +and + +$$ +\mathbb {E} \left[ \mathcal {E} _ {\text {s i e v e}} \right] \leq 2 \left(\| \Phi \| _ {L ^ {2} (\mu)} + \| G \| _ {L ^ {2} (\nu)}\right) \cdot \varepsilon^ {\frac {1}{2}}, \tag {17} +$$ + +where $M \coloneqq \tilde{R}_{\varepsilon} \sup_{x \in B(0, R_{\varepsilon})} \| \nabla \varphi_0(x) \|_2 + \sup_{x \in B(0, \tilde{R}_{\varepsilon})} \Phi(x)$ , and the suppressed constant depends only on $\gamma, \gamma'$ . + +We defer the proof to Appendix D, where we also establish corresponding high-probability guarantees. + +In Theorem 3.6, we establish a standard convergence rate for estimation error, $\mathbb{E}[\mathcal{E}_{\text{stat}}] = \mathcal{O}_{\log}(\sqrt{D_{\mathcal{F}} / n} + \sqrt{D_{\mathcal{F}} / m})$ , which is common in empirical process theory. Meanwhile, the multiplicative prefactors in $\mathcal{E}_{\text{stat}}$ , $R_{\varepsilon}^{\frac{n}{2}} + \sup_{x \in B(0, R_{\varepsilon})} \Phi(x)^{\frac{1}{2}}$ and $\tilde{R}_{\varepsilon}^{\frac{n}{2}} + M^{\frac{1}{2}}$ , naturally capture the "diameter" of $\mathcal{F}$ on the pseudo-supports of $\mu$ and $\nu$ . + +Moreover, it tells that the sieve bias $\mathcal{E}$ is governed by the tail probability $\varepsilon$ defined in Equation (10). + +Practical choice of $R_{\varepsilon}$ and $\tilde{R}_{\varepsilon}$ . As shown in [20], $R_{\varepsilon}$ can be selected in a distribution-free manner. Here, we extend this idea to provide a practical, joint selection strategy for both $R_{\varepsilon}$ and $\tilde{R}_{\varepsilon}$ : + +$$ +R _ {\varepsilon} = \max _ {1 \leq i \leq n} \| X _ {i} \| _ {2}, \quad \tilde {R} _ {\varepsilon} = R _ {\varepsilon} + C \max _ {1 \leq j \leq m} \| Y _ {j} \| _ {2}, \tag {18} +$$ + +where $C \geq 0$ is a tuning parameter designed to account for the potential mismatch between the estimated and true OT maps, specifically the discrepancy $\| \nabla \tilde{\varphi}_{n,m} - \nabla \varphi_0\|_{L^\infty (B(0,R_\varepsilon))}$ . + +The choice of $R_{\varepsilon}$ can also be justified by our stability and oracle bounds (Propositions 3.3 and 3.4). In particular, if $n = m$ , $\mu$ is sub-Weibull, and both $\Phi(\cdot)$ , $\|\nabla \varphi_0(\cdot)\|_2$ grow at most polynomially, then the estimation error of the sieved estimator reads: + +$$ +\mathbb {E} \| \nabla \tilde {\varphi} _ {n} - \nabla \varphi_ {0} \| _ {L ^ {2} (\mu)} ^ {2} = \mathcal {O} _ {\log} \left(\left(n / D _ {\mathcal {F}}\right) ^ {- \frac {1}{2}} + \mathcal {E} _ {a p p} + \varepsilon^ {\frac {1}{2}}\right). \tag {19} +$$ + +By exchangability of $X_{i}$ 's, choosing $R_{\varepsilon} \coloneqq \max_{1 \leq i \leq n} \|X_{i}\|_{2}$ yields $\varepsilon \asymp n^{-1}$ , so the sieved bias $\varepsilon^{\frac{1}{2}}$ is dominated by the statistical error term $(n / D_{\mathcal{F}})^{-\frac{1}{2}}$ . + +While a complete theoretical justification for our choice of $\tilde{R}_{\varepsilon}$ remains an open question, numerical results in Section 4 provide strong empirical support: our sieved estimator consistently outperforms the original dual-type estimator of [27, 25, 21] across a range of settings. + +# 3.4 Estimation Error of Neural OT Estimator + +In this section, we derive the first non-asymptotic error bounds for dual-type OT map estimators parameterized by deep neural networks between general distributions. + +ReLU networks, despite their popularity, lack the smoothness needed to recover the gradient of the estimated Brenier potential. Moreover, our analysis (Assumption 3.1) requires $\mathcal{F}$ to be $C^2$ . Consequently, smooth activations are both practically and theoretically preferable. + +While input convex neural networks (ICNNs) [2] exploit convexity of Brenier potential and have been applied at scale [33, 4, 31], their approximation theory remains underdeveloped. This gap limits rigorous control of approximation error in existing analyses of dual-type estimators [27, 25, 21, 20]. By contrast, our approach removes the convexity assumption on $\mathcal{F}$ , allowing us to employ standard fully connected networks. + +Accordingly, we adopt the tanh-activated fully-connected neural network (TNN) as our function class $\mathcal{F}$ , due to its $C^\infty$ smoothness and non-asymptotic universal approximation guarantees [16]: + +$$ +f (x) = \mathcal {L} _ {L} \circ \sigma \circ \mathcal {L} _ {L - 1} \circ \dots \circ \sigma \circ \mathcal {L} _ {1} (x), \tag {20} +$$ + +where $\sigma(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$ is applied entrywise, and $\mathcal{L}_{\ell}(x) = A_{\ell}x + b_{\ell}$ for $\ell \in [L]$ , where $A_{l} \in \mathbb{R}^{p_{l+1} \times p_{l}}$ , $b_{l} \in \mathbb{R}^{p_{l}}$ , with $p_1 = d$ , $p_{L+1} = 1$ . + +We define the TNN class $\mathcal{F}$ with bounded parameters as: + +$$ +\mathcal {F} (L, W, \kappa) := \left\{f \text {o f f o r m (2 0)}: \| A _ {\ell} \| _ {\infty} \vee \| b _ {\ell} \| _ {\infty} \leq \kappa , \quad p _ {l} \leq W \text {f o r a l l} \ell \in [ L ] \right\}, \tag {21} +$$ + +where $\kappa > 0$ is a truncation threshold used for technical convenience. + +Theorem 3.7 (OT Estimation via Tanh Neural Network). Let $n = m$ , and suppose $\mu$ is $(\lambda, K)$ -sub-Weibull, and the true Brenier potential $\varphi_0 \in C^\alpha(\mathbb{R}^d)$ with $\alpha \geq 2$ is $(\beta, b)$ -smooth. By setting $R_{\varepsilon}, \tilde{R}_{\varepsilon}$ according to Equation (10), there exists a deep TNN function class $\mathcal{F} := \mathcal{F}(L, W, \kappa)$ where $L, W, \kappa$ depend on $n, d, \alpha, \beta, b$ and are of order $3 \leq L = \mathcal{O}(1)$ , $W = \mathcal{O}(n^{\frac{d}{d + 2\alpha}})$ , $\kappa = \mathcal{O}(n^{\frac{d(d + (|\alpha| + 2)^2 + 4) + 2}{2(d + 2\alpha)}})$ , such that the sieved estimator $\nabla \tilde{\varphi}_n$ with sieve radius + +$$ +\tilde {R} _ {n} = C _ {K, d, \alpha} \cdot (\log n) ^ {\frac {1}{\lambda}}, \quad \text {f o r s o m e c o n s t a n t} \quad C _ {K, d, \alpha} \geq 4 K \frac {2 \alpha}{d + 2 \alpha} + 2 \tag {22} +$$ + +satisfies + +$$ +\mathbb {E} \| \nabla \tilde {\varphi} _ {n} - \nabla \varphi_ {0} \| _ {L ^ {2} (\mu)} ^ {2} \lesssim_ {\log n} n ^ {- \frac {\alpha}{d + 2 \alpha}}. \tag {23} +$$ + +Remark 3.8 (On mild regularity conditions). The sub-Weibull and $(\beta, b)$ -smoothness assumptions are used in our work primarily for analytical convenience, not because they are fundamentally + +required for the methodology. Notably, both conditions are widely used in the OT literature, see [21, 20, 27, 25, 34] for $(\beta, b)$ -smoothness and [21, 17, 20] for sub-Weibull condition. + +The $(\beta, b)$ -smoothness of $\varphi_0$ is needed solely to control approximation error through Lemma F.4, ensuring the error grows polynomially in $\tilde{R}_{\varepsilon}$ . Meanwhile, the sub-Weibull $\mu$ guarantees that $R_{\varepsilon}, \tilde{R}_{\varepsilon}$ scale polynomially in $\log(1/\varepsilon)$ . Together, these two results ensure that the relevant Sobolev norm is bounded by a polynomial in $\log(1/\varepsilon)$ , and the consequent approximation error has the order of $\varepsilon$ . + +If $(\beta, b)$ -smoothness were lifted, one could still obtain approximation bounds, but they may degrade to $O(\varepsilon^a)$ for some $a < 1$ . This complicates the statistical error, approximation error, and sieved estimation bias-variance trade-off by introducing additional convergence exponent. Likewise, relaxing source distribution to be polynomially tails causes $R_{\varepsilon}, \tilde{R}_{\varepsilon}$ to scale polynomially in $1 / \varepsilon$ . While the proofs follow similar steps, the resulting convergence rate becomes much more complex (See Equation 3.10 in [20] for an illustration of the added technical complications). + +Remark 3.9 (On the choice of activation). While ReLU networks are widely used in practice, our framework currently focuses on the tanh activation due to theoretical considerations. Specifically, our estimator is defined via the gradient of a potential function. Although PyTorch can handle ReLU networks computationally, the ReLU activation is not globally differentiable in mathematics, making this definition is ill-posed. Moreover, even under a weak derivative framework, ReLU networks lack second-order weak derivatives, whereas the existing optimal transport theory fundamentally relies on second-order regularity for statistical analysis. + +The rate in Theorem 3.7 matches that of [25], but under substantially milder assumptions. Although slower than the minimax rate in [27], this is expected: we do not assume $(\alpha, a)$ -convexity of the true Brenier potential $\varphi_0$ , compact support of $\mu$ and $\nu$ , or a Poincaré inequality on $\mu$ . Moreover, the estimators in [27, 25, 21] are either computationally intensive or NP-hard. In contrast, our deep TNN estimator is both implementable and scalable in practice. + +We focus on the TNN architecture due to the current lack of approximation theory for general smooth neural networks. The proof of Theorem 3.7 is provided in Appendix E. With minor modifications, the result also holds with high probability. + +# 4 Numerical Examples + +In this section, we present numerical simulations to evaluate the performance of our sieved-TNN estimator and compare it against the original dual-type estimators from [27, 25, 21]. We consider four synthetic OT problems that pose challenges for existing theoretical frameworks: + +- Non- $(\beta, b)$ -smooth maps: $\mathcal{N}(0,1) \to t_6$ , Uniform(0,1) $\to \mathcal{N}(0,1)$ . +- Non- $(\alpha, a)$ -convex maps: $t_6 \to \mathcal{N}(0,1)$ , $\mathcal{N}(0,1) \to \mathrm{Uniform}(0,1)$ . + +We evaluate performance in dimensions $d = 5, 10, 20$ , using i.i.d. samples $X_{i} \sim \mu$ and $Y_{j} \sim \nu$ with sample sizes $n = m \in \{64, 128, 256, 512, 1024, 2048\}$ . Experiments were conducted on a server with Intel Xeon Gold 6342 processors, requiring approximately 4,000 CPU core hours in total. + +Implementation setup We implement our estimator in PyTorch [36], following algorithms from [20] with objective function in Equation (8). The candidate class $\mathcal{F}$ consists of TNNs with two hidden layers of width 10 (for $d = 5$ ), 20 (for $d = 10$ ), and 30 (for $d = 20$ ). Training proceeds in two phases: a warm-up phase of $\max \{\lfloor 10000 / n\rfloor ,50\}$ epochs at learning rate $5\times 10^{-3}$ , followed by 300 epochs at $10^{-3}$ . Mini-batch size is 64 for both $\mu$ and $\nu$ , and the sieved convex conjugate subproblem is solved using 300 inner iterations (see Algorithm 2 in [20]). In each trial, $10\%$ of the samples are held out for validation. We defer implementation details to Appendix G. + +Sieve radius As discussed in Equation (18), we choose the sieve radius $\tilde{R}$ as + +$$ +\tilde {R} = \max _ {1 \leq i \leq n} \| X _ {i} \| _ {2} + C \cdot \max _ {1 \leq j \leq n} \| Y _ {j} \| _ {2}, \quad C \in \{0, 1, 2, 3, \infty \}. +$$ + +Here, $C = 0$ corresponds to the setting in [20], while $C = \infty$ recovers the original dual-type estimator studied in [27, 25, 21], enabling direct comparison. + +Evaluation We assess the estimation error using unexplained variance proportion (UVP) [30]: $L^2$ -UVP( $\nabla \hat{\varphi}_n$ ) := $\| \nabla \hat{\varphi}_n - \nabla \varphi_0 \|_{L^2(\mu)}^2 / \mathrm{Var}_{\nu} (\| Y \|_2)$ . Lower values indicate better performance. For each experiment, we approximate the $L^2$ -UVP using an independent set of $10^6$ samples. + +![](images/7b2d2a59f3f927f6dfa14d4129b1cbe6cea74a06f25146cecd5bc2c9f25ecaa3.jpg) +(a) $\mathcal{N}\to t_6$ + +![](images/f8c5e68b007204af2a05dbee3d4b87f648fbb6355fe7721bb7914848469d4fb1.jpg) +(b) Unif $\rightarrow \mathcal{N}$ + +![](images/1e96345efadd786682132e89800a2547b2721a6c99af8ce07d2f40f0605d318d.jpg) +(c) $t_6\to \mathcal{N}$ + +![](images/09a7c28cd42cb9781726afa173261328e7101e38d0d86cb8bfadba362bf02e71.jpg) +(d) $\mathcal{N}\to \mathrm{Unif}$ +Figure 1: $L^2$ -UVP when $d = 5$ . Each curve shows the mean $L^2$ -UVP over 50 random trials with one standard deviation. + +Due to space constraints, we report only the $L^2$ -UVP error versus sample size $n$ for $d = 5$ in Figure 1 here. The full set of numerical results is available in Appendix G. + +Across all settings, our sieved TNN estimator consistently converges and outperforms the original dual-type estimator $(C = \infty)$ , particularly in the small-sample regime. As shown in Figure 4a, sieve-based estimators yield lower $L^2$ -UVP errors for small $n$ , while the classical estimator only catches up as $n$ grows large. This aligns with our theory: larger sieve radii may degrade convergence, and are only needed when the mismatching $\| \nabla \tilde{\varphi}_{n,m} - \nabla \varphi_0\|_{L^\infty}$ , is large. + +These results also offer practical guidance: a modest sieve constant (e.g., $C = 0$ or 1) suffices, and performance of our sieved estimator remains stable across a range of $C$ . + +# 5 Conclusion + +In this paper, we develop new map stability and oracle inequalities for sieved dual-type OT estimators, without restrictive smoothness and convexity assumptions on the true Brenier potential $\varphi_0$ in the literature. Under mild regularity conditions, we establish the first non-asymptotic error bounds for neural OT estimators, using tanh networks as a concrete example. Numerical experiments further confirm the strong empirical performance of our sieved approach. Altogether, our unified framework advances the theoretical foundations of optimal transport and paves the way for future developments in both methodology and applications. + +Acknowledgment: This research was partially supported the U.S. National Science Foundation under the grants DMS 2514400 and DMS 2210775, and by the U.S. National Institutes of Health under the grants R01GM163244 and 1R01GM152812. + +# References + +[1] Jason Altschuler, Jonathan Niles-Weed, and Philippe Rigollet. Near-linear time approximation algorithms for optimal transport via sinkhorn iteration. Advances in neural information processing systems, 30, 2017. +[2] Brandon Amos, Lei Xu, and J Zico Kolter. Input convex neural networks. In International Conference on Machine Learning, pages 146-155. PMLR, 2017. +[3] Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial networks. In International conference on machine learning, pages 214-223. 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Learning generative models with sinkhorn divergences. In International Conference on Artificial Intelligence and Statistics, pages 1608-1617. PMLR, 2018. +[24] Promit Ghosal and Bodhisattva Sen. Multivariate ranks and quantiles using optimal transport: Consistency, rates and nonparametric testing. The Annals of Statistics, 50(2):1012-1037, 2022. +[25] Florian F Gunsilius. On the convergence rate of potentials of brenier maps. *Econometric Theory*, 38(2):381-417, 2022. +[26] Marc Hallin, Eustasio del Barrio, Juan Cuesta-Albertos, and Carlos Matrán. Distribution and quantile functions, ranks and signs in dimension $d$ : A measure transportation approach. The Annals of Statistics, 49(2):1139 - 1165, 2021. +[27] Jan-Christian Hütter and Philippe Rigollet. Minimax estimation of smooth optimal transport maps. 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In International Conference on Machine Learning, pages 28128–28150. PMLR, 2023. + +[38] Litu Rout, Alexander Korotin, and Evgeny Burnaev. Generative modeling with optimal transport maps. 2022. +[39] Geoffrey Schiebinger, Jian Shu, Marcin Tabaka, Brian Cleary, Vidya Subramanian, Aryeh Solomon, Joshua Gould, Siyan Liu, Stacie Lin, Peter Berube, et al. Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming. Cell, 176(4):928–943, 2019. +[40] Ilya O. Tolstikhin, Olivier Bousquet, Sylvain Gelly, and Bernhard Scholkopf. Wasserstein auto-encoders. In International Conference on Learning Representations, 2018. +[41] Adrien Vacher, Boris Muzellec, Alessandro Rudi, Francis Bach, and Francois-Xavier Vialard. A dimension-free computational upper-bound for smooth optimal transport estimation. In Conference on Learning Theory, pages 4143-4173. PMLR, 2021. +[42] Cédric Villani. Optimal Transport: Old and New, volume 338. Springer, 2009. +[43] Fangneng Zhan, Yingchen Yu, Kaiwen Cui, Gongjie Zhang, Shijian Lu, Jianxiong Pan, Chang-gong Zhang, Feiying Ma, Xuansong Xie, and Chunyan Miao. Unbalanced feature transport for exemplar-based image translation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 15028-15038, 2021. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: We provide exact Theorems/Propositions numbers for our main claims in the abstract and introduction, which correspond precisely with our four subsections in Section 3. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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We show that under the similar common assumptions — Polyak-Lojasiewicz condition, smoothness, and Lipschitz continuous for losses — rates of $O\left(\log^2(n)/n^2\right)$ are at most achievable. To our knowledge, our analysis also provides the tightest high-probability bounds for gradient-based generalization gaps in nonconvex settings. + +# 1 Introduction + +Algorithmic stability is a fundamental concept in learning theory [3], which can be traced back to the foundational works of Vapnik and Chervonenkis [46] and has a deep connection with learnability [37, 40, 39]. Roughly speaking, an algorithm is stable if a substitution of single example in the training dataset leads to a minor change of the output model. With such algorithmic stability, the generalization ability of model is guaranteed. Owing to the relationship, the algorithmic stability is recognized as a powerful tool to explore the generalization. + +In practice, researchers built the generalization bounds expressed in expectation or in-high probability. While providing in-expectation bounds is relatively straightforward, their high probability counterparts are more crucial for practical optimization algorithms [12, 4, 20]. Because we often train models single time in practice, and high probability bounds offer more informative insights. Therefore, we focus on improving high probability risk bounds through an exploration of algorithmic stability. + +Exploring the high-probability generalization gap bounded via algorithmic stability was firstly chased to Bousquet and Elisseeff [3], and was further developed by Feldman and Vondrak [11, 12]. Within these literature, the optimal high-probability bound of generalization error consists of the "sampling error" $O(\log (n) / \sqrt{n})$ plus the algorithmic stability parameter for bounded loss function. Owing to the existence of sampling error, deriving the bound within the framework in [4] can not be improved, since the lower bound of sampling error is proven to be $O(1 / \sqrt{n})$ . To resolve this, Klochkov and Zhivotovsky [20] propose to alternatively explore the excess risk, which directly reveals the performance of learned model, and can be decomposed into optimization and generalization errors. Without considering the sampling error, Klochkov and Zhivotovsky [20] improve the high-probability bound to excess risk of $O(\log (n) / n)$ plus algorithmic stability parameter, under a Bernstein type condition [49]. Moreover, under strongly convexity condition, the algorithmic stability parameter of projected gradient descent (PGD) is $O(\log (n) / n)$ [20]. Following this, the algorithmic stability- + +based excess risk bounds were further explored by Yuan and Li [52, 53], Yi et al. [51], while their results are all weaker than $O(\log (n) / n)$ . Thus, a natural question is: + +Can algorithmic stability-based technique provide high probability excess risk bounds better than $O(1/n)$ ? + +The main results of this paper answer this question positively. Roughly speaking, under proper regularity conditions, we establish the first high probability excess risk bounds that are dimension-free with the rate $O\left(\log^2 (n) / n^2\right)$ for models obtained by empirical risk minimization (ERM), PGD and stochastic gradient descent (SGD). To do so, our core technique is generalizing the standard concept of algorithmic stability of loss function to its gradient, i.e., the "uniform stability in gradients" in Definition 2 [24]. With this, we can connect it with a sharper generalization bound, compared with the classical ones [4]. + +Next, we outline our idea to the sharper bound to excess risk. Notably, under some regularity conditions e.g. strongly convexity or Polyak-Lojasiewicz (PL) condition [20], the loss function is upper bounded by the square of its gradient norm. Then, by invoking the algorithmic stability of gradient, we prove a generalization bound w.r.t. gradient of loss function. With this, an upper bound to the gradient of excess risk is obtained, which naturally leads to the bound of excess risk, under strongly convexity of PL conditions. During the proof, our generalization bound w.r.t. gradient consists of algorithmic stability coefficient, gradient of loss function on trained parameters, and terms in $O(1/n)$ . Intuitively, the gradient related term on trained model tends to close zero. Thus, combining the derived generalization bound on gradient and analysis on optimization error induces our sharper bound $O\left(\log^2(n)/n^2\right)$ on excess risk. + +Finally, we summarize our contributions as follows, and compare our results with the previous ones in Table 1. + +- Under proper regularity conditions, we sharpen the algorithmic stability-based generalization bound i.e., from $\tilde{O}(1/n)$ [20] to $\tilde{O}\left(1/n^{2}\right)$ via the stability of gradients. To the best of our knowledge, this is optimal dimension-free results based on algorithmic stability up till now. +- With our framework, we derive the first dimension-free high probability excess risk bounds of $O\left(\log^2 (n) / n^2\right)$ for ERM, PGD, and SGD, addressing an open problem posed in Xu and Zeevi [50]. For SGD, we have also shown that under the same assumptions, an equally tight bound can be achieved with fewer iterations. + +Notations. In this paper, we consider a set of training data independent and identically distributed (i.i.d.) observations $S = \{z_1, \ldots, z_n\}$ sampled from a probability distribution $\rho$ defined on $\mathcal{Z}$ , $S' = \{z_1', \ldots, z_n'\}$ be its independent copy. For any $i \in [n]$ , define $S^{(i)} = \{z_i, \ldots, z_{i-1}, z_i', z_{i+1}, \ldots, z_n\}$ by replacing the $i$ -th sample in $S$ with another i.i.d. sample $z_i'$ . Based on the training set $S$ , our goal is to build a model with parameter $\mathbf{w} \in \mathcal{W} \subset \mathbb{R}^d$ . We denote the performance of model with parameters $\mathbf{w}$ evaluated on $z$ by loss function $f(\mathbf{w}; z)$ , where $f: \mathcal{W} \times \mathcal{Z} \mapsto \mathbb{R}_+$ . Then the population risk and the empirical risk of $\mathbf{w} \in \mathcal{W}$ , are respectively denoted as + +$$ +F (\mathbf {w}) := \mathbb {E} _ {z \sim \rho} [ f (\mathbf {w}; z) ], \quad F _ {S} (\mathbf {w}) := \frac {1}{n} \sum_ {i = 1} ^ {n} f (\mathbf {w}; z _ {i}). +$$ + +We respectively denote $\mathbf{w}^{*}\in \arg \min_{\mathbf{w}\in \mathcal{W}}F(\mathbf{w})$ , and $\mathbf{w}_S^*\in \arg \min_{\mathbf{w}\in \mathcal{W}}F_S(\mathbf{w})$ , and learned model parameters $A(S)\in \mathcal{W}$ be the output of a (possibly randomized) algorithm $A$ on the training set $S$ . In general, we will care the generalization error of model $A(S)$ defined as the gap between population risk and empirical risk i.e., $F(A(S)) - F_S(A(S))$ . Besides, the excess risk $F(A(S)) - F(\mathbf{w}^{*})$ decides the performance of model is more important in practice. Finally, the notation $\tilde{O} (\cdot)$ hides logarithmic factors and we denote $L_{p}$ -norm of real value $Y$ as $\| Y\| _p\coloneqq (\mathbb{E}[|Y|^p])^{\frac{1}{p}}$ . Similarly, let $\| \cdot \|$ denote the norm in a Hilbert space $\mathcal{H}$ . For a random variable $X$ taking values in a Hilbert space, its $L_{p}$ -norm is defined by $\| \| \mathbf{X}\| \| _p\coloneqq (\mathbb{E}[\| \mathbf{X}\| ^p])^{\frac{1}{p}}$ . + +# 2 Related Work + +Algorithmic stability is a classical approach in generalization analysis, which can be traced back to the foundational works of Vapnik and Chervonenkis [46]. It gave the generalization bound by + +Table 1: Summary of high probability excess risk bounds. All conclusions herein assume Lipschitz continuity, and all SGD algorithms presuppose bounded variance of the gradients; therefore, these two assumptions are omitted in the table. Abbreviations: uniform convergence $\rightarrow$ UC, algorithmic stability $\rightarrow$ AS, strongly convex $\rightarrow$ SC, low noise [55] $\rightarrow$ LN, Polyak-Lojasiewicz condition $\rightarrow$ PL. + +
ReferenceAlgorithmMethodAssumptionsIterationsSample SizeBounds
[55]ERMUCSmooth, SC, LN-Ω(γ2d/μ2)O(1/n2)
[50]ERMUCSmooth, PL, LN-Ω(γ2d/μ2)O(1/n2)
PGDUCSmooth, PL, LNT ≈ log(n)Ω(γ2d/μ2)O(1/n2)
[29]SGDUCSmooth, PL, LNT ≈ n2Ω(γ2d/μ2)O(1/n2)
ASSmooth, SCT ≈ n2-O(1/n)
[20]ERMASSC--O(1/n)
PGDASSmooth, SCT ≈ log(n)-O(1/n)
This workERMASSmooth, PL, LN-Ω(γ2/μ2)O(1/n2)
PGDASSmooth, PL, LNT ≈ log(n)Ω(γ2/μ2)O(1/n2)
PGDASSmooth, PL, LNT ≈ log(n)Ω(γ2/μ2)O(1/n2)
SGDASSmooth, PL, LNT ≈ n2Ω(γ2/μ2)O(1/n2)
SGDASSmooth, PLT ≈ nΩ(γ2/μ2)O(1/n)
+ +analyzing the sensitivity of a particular learning algorithm when changing one data point in the dataset. Modern method of stability analysis was established by Bousquet and Elisseeff [3], where they presented the concept of uniform stability. + +Since then, a lot of works based on uniform stability have emerged. Some of the existing results built their bound in expectation [17, 51, 5, 8], while it is weaker than the high probability bound as we discussed. To resolve this, Bousquet and Elisseeff [3], Elisseeff et al. [9], Feldman and Vondrak [11, 12], Bousquet et al. [4], Klochkov and Zhivotovsky [20], Yuan and Li [52, 53], Fan and Lei [10] considered high probability bounds. Besides that, some other generalized algorithmic stability measures are further developed to study the generalization. For instances, uniform argument stability [31, 1], uniform stability in gradients [24, 10], on average stability [40, 21], hypothesis stability [3, 5], hypothesis set stability [13], pointwise uniform stability [10], PAC-Bayesian stability [28], locally elastic stability [8], and collective stability [32]. However, the optimal generalization bound among these results is $O(\log (n) / n)$ , which is weaker than our $O\left(\log^2 (n) / n^2\right)$ in this paper. + +Notably, the $O\left(\log^2 (n) / n^2\right)$ generalization bound under PL condition or strongly convexity has also been developed by uniform convergence technique as in Zhang et al. [55], Xu and Zeevi [50]. However, in contrast to ours, their results are restricted to the number of samples $n = \Omega (d)$ . + +# 3 Stability and Generalization + +In this section, we first introduce the stability of gradient, and connect it to the generalization bound w.r.t. gradient. Following this, under proper regularity condition, i.e., PL condition, we can extrapolate the generalization of gradient to the loss function. With the desired generalization bound, by combining it with the optimization error, we can further bound the excess risk. + +Firstly, let us introduce some basic definitions here. Before presenting our main results (Theorem 1 and Theorem 2) i.e., the generalization bound w.r.t. gradient, we emphasis that they do not require the smoothness assumption and PL condition. This indicates their potential applications within the nonconvex problems as well. + +Definition 1. Let $f:\mathcal{W}\mapsto \mathbb{R}$ . Let $M,\gamma ,\mu >0$ + +- We say $f$ is $M$ -Lipschitz if + +$$ +\left| f (\mathbf {w}) - f \left(\mathbf {w} ^ {\prime}\right) \right| \leq M \| \mathbf {w} - \mathbf {w} ^ {\prime} \| _ {2}, \quad \forall \mathbf {w}, \mathbf {w} ^ {\prime} \in \mathcal {W}. +$$ + +- We say $f$ is $\gamma$ -smooth if + +$$ +\| \nabla f (\mathbf {w}) - \nabla f \left(\mathbf {w} ^ {\prime}\right) \| _ {2} \leq \gamma \| \mathbf {w} - \mathbf {w} ^ {\prime} \| _ {2}, \quad \forall \mathbf {w}, \mathbf {w} ^ {\prime} \in \mathcal {W}. +$$ + +- Let $f^{*} = \min_{\mathbf{w} \in \mathcal{W}} f(\mathbf{w})$ . We say $f$ satisfies the Polyak-Lojasiewicz (PL) condition with parameter $\mu > 0$ on $\mathcal{W}$ if + +$$ +f (\mathbf {w}) - f ^ {*} \leq \frac {1}{2 \mu} \| \nabla f (\mathbf {w}) \| _ {2} ^ {2}, \quad \forall \mathbf {w} \in \mathcal {W}. +$$ + +# 3.1 Sharper Generalization Bounds in Gradients + +Let us start with the notation of algorithmic stability w.r.t. gradient. + +Definition 2 (Uniform Stability in Gradients). Let $A$ be a randomized algorithm. We say $A$ is $\beta$ -uniformly-stable in gradients if for all neighboring datasets $S, S^{(i)}$ , we have + +$$ +\sup _ {z} \mathbb {E} _ {A} \left[ \left\| \nabla f (A (S); z) - \nabla f (A (S ^ {(i)}); z) \right\| _ {2} ^ {2} \right] \leq \beta^ {2}. \tag {1} +$$ + +Remark 1. The notation of gradient-based stability was firstly introduced by Lei [24], Fan and Lei [10] to describe the generalization performance for nonconvex problems. Because in this regime, the exploration usually focuses on the first-order critical condition i.e., $\| \nabla F(A(S))\| _2$ . Then, with the stability on gradient, the "generalization of gradient" can be expressed $\| \nabla F(A(S)) - \nabla F_S(A(S))\| _2$ . Thus, combining it with a triangle inequality and standard results on gradient of empirical risk $\| \nabla F_S(A(S))\| _2$ characterizes the desired $\| \nabla F(A(S))\| _2$ . Further more, under a small $\| \nabla F(A(S))\| _2$ , the PL condition (see Definition 1) leads to the desired bound on excess risk. This explains why the gradient-based stability is the core idea of this paper. + +With the uniform stability in gradients, we prove that it implies the generalization of gradient in the following theorem. + +Theorem 1 (Generalization via Stability in Gradients [10]). Assume for any $z$ , $f(\cdot, z)$ is $M$ -Lipschitz. If $A$ is $\beta$ -uniformly-stable in gradients, then for any $\delta \in (0, 1)$ , the following inequality holds with probability at least $1 - \delta$ + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} \right] \leq 2 \beta + \frac {4 M \left(1 + e \sqrt {2 \log (e / \delta)}\right)}{\sqrt {n}} \\ + 8 \times 2 ^ {\frac {1}{4}} (\sqrt {2} + 1) \sqrt {e} \beta \left\lceil \log_ {2} (n) \right\rceil \log \left(e / \delta\right). \\ \end{array} +$$ + +Remark 2. Notably, our Theorem 1 is similar to Theorem 3 in [10], while ours has a constant-level improvement owing to a sharper concentration inequality i.e., Lemma 7 in Appendix. However, the generalization bounds in both of the aforementioned theorems have dependence $O(M / \sqrt{n})$ , which leads to a $O(M^2 / n)$ generalization bound of loss function, and it is weaker than our desired result. Thus, we derive the following sharper generalization bound of gradients under same assumptions. + +Theorem 2 (Sharper Generalization via Stability in Gradients). Assume for any $z$ , $f(\cdot, z)$ is $M$ -Lipschitz. If $A$ is $\beta$ -uniformly-stable in gradients, then for any $\delta \in (0,1)$ , the following inequality holds with probability at least $1 - \delta$ + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} \right] \\ \leq \sqrt {\frac {4 \mathbb {E} _ {Z , A} \left[ \| \nabla f (A (S) ; Z) \| _ {2} ^ {2} \right] \log (6 / \delta)}{n}} + \sqrt {\frac {\left(\frac {1}{2} \beta^ {2} + 3 2 n \beta^ {2} \log (3 / \delta)\right) \log (6 / \delta)}{n}} \\ + \frac {M \log (6 / \delta)}{n} + 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} \beta \lceil \log_ {2} (n) \rceil \log (3 e / \delta) + 3 2 \sqrt {e} \beta \lceil \log_ {2} (n) \rceil \sqrt {\log (3 e / \delta)}. \\ \end{array} +$$ + +Remark 3. We begin by comparing the generalization bound of gradient in the proposed Theorem 2 with Theorem 1. The critical difference is that constants in $1 / \sqrt{n}$ is improved from $O\left(M\sqrt{\log(e / \delta)}\right)$ to $O\left(\sqrt{\mathbb{E}_Z\left[\|\nabla f(A(S);Z)\|_2^2\right]\log(1 / \delta)} + \beta\log(1 / \delta)\right)$ , since $M$ is the maximal gradient norm. Notably, the proved dependence $\mathbb{E}_Z\left[\|\nabla f(A(S);Z)\|_2^2\right]$ is interpreted as the + +squared gradient norm of the loss functions on test data, under the optimized model parameters $A(S)$ . Thus, it is supposed to be small, compared with constant $M$ , since the optimization algorithms often provide parameters approaching the optimal solution. + +Moreover, our bound is not restricted to any specific algorithm and without dependence on the number of optimization iterations. Clearly, this is an improvement to the results constructed under a specific algorithm, e.g., the $O(T / n)$ generalization bound under SGD in Bassily et al. [1], Lei [24], which is only applied to SGD, and becomes vacuous when iteration steps $T$ exceeds $n$ . + +Next, we give the proof sketch to our Theorem 2, which is motivated by the analysis in Klochkov and Zhivotovsky [20]. Similar to the standard result in Bousquet and Elisseeff [3], which says the generalization of loss function is implied by its stability and the variance of stability. Similar result is generalized to the gradient. During our proof, such algorithmic stability of gradient is characterized by the gap $\mathbf{q}_i(S) = \mathbf{h}_i(S) - \mathbb{E}_{S\setminus \{z_i\},A}[\mathbf{h}_i(S)]$ with $\mathbf{h}_i(S) = \mathbb{E}_{z_i'}\left[\mathbb{E}_Z\left[\nabla f(A(S^{(i)}),Z)\right] - \nabla f(A(S^{(i)}),z_i)\right]$ and its variance. The stability is $\beta$ appears in Theorem 2. While instead of controlling its variance with maximal gradient norm, we can link it to the coefficient $\mathbb{E}_Z\left[\| \nabla f(A(S);Z)\| _2^2\right]$ , which implies our result. We refer readers for more details to the proof in Appendix. + +To further characterize the $\mathbb{E}_Z\left[\| \nabla f(A(S);Z)\| _2^2\right]$ , we need the strong growth condition (SGC) imposed in Solodov [42], Vaswani et al. [48], Lei [24], which is satisfied many important loss function, e.g., squared-hinge loss with finite support [48]. + +Definition 3 (Strong Growth Condition). The SGC holds, if + +$$ +\mathbb {E} _ {Z} \left[ \| \nabla f (\mathbf {w}; Z) \| _ {2} ^ {2} \right] \leq \lambda \| \nabla F (\mathbf {w}) \| _ {2} ^ {2}. +$$ + +Proposition 1 (SGC case). Let assumptions in Theorem 2 hold and suppose SGC holds. Then for any $\delta >0$ , $\eta >0$ , with probability at least $1 - \delta$ , we have + +$$ +\mathbb {E} _ {A} \left[ \| \nabla F (A (S)) \| _ {2} \right] \lesssim (1 + \eta) \mathbb {E} _ {A} \left[ \| \nabla F _ {S} (A (S)) \| _ {2} \right] + \frac {1 + \eta}{\eta} \left(\frac {\lambda M \log (1 / \delta)}{n} + \beta \log (n) \log (1 / \delta)\right). +$$ + +Proposition 1 build a connection between the population gradient error and the empirical ones under Lipschitz, nonconvex, nonsmooth and SGC conditions. The conclusion is directly obtained from the triangle inequality, Young's inequality, and SGC. Since the model parameters $A(S)$ is optimized, the $\| F_S(A(S))\|$ is supposed to be small. Thus, the Proposition 1 is a valuable upper bound on that of population error, for algorithm with stability w.r.t. gradient (small $\beta$ ). + +Remark 4. Finally, we make a discussion to the priority of our Theorem 2, solely in that of gradient generalization. In a word, we address an open problem posed by [50], namely achieving a bound to generalization error of gradient independent of the dimension $d$ . Concretely, Xu and Zeevi [50] prove a generalization bound via uniform convergence technique [47] is + +$$ +\begin{array}{l} \left\| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \right\| _ {2} \\ \lesssim \sqrt {\frac {\mathbb {E} _ {Z} \left[ \| \nabla f (\mathbf {w} ^ {*} ; Z) \| _ {2} ^ {2} \right] \log (1 / \delta)}{n}} + \frac {\log (1 / \delta)}{n} + \max \left\{\| A (S) - \mathbf {w} ^ {*} \| _ {2}, \frac {1}{n} \right\} \left(\sqrt {\frac {d}{n}} + \frac {d}{n}\right), \tag {2} \\ \end{array} +$$ + +which is the optimal result when we only consider the order of $n$ . The uniform convergence results are related to the dimension $d$ , which are unacceptable in high-dimensional learning problems. Note that (2) requires an additional smoothness-type assumption. As a comparison, when $f$ is $\gamma$ -smooth and $A$ is a deterministic algorithm, our result in Theorem 2 becomes + +$$ +\begin{array}{l} \left\| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \right\| _ {2} \\ \lesssim \beta \log n \log (1 / \delta) + \frac {\log (1 / \delta)}{n} + \sqrt {\frac {\mathbb {E} _ {Z} \left[ \| \nabla f (\mathbf {w} ^ {*} ; Z) \| _ {2} ^ {2} \right] \log (1 / \delta)}{n}} + \| A (S) - \mathbf {w} ^ {*} \| \sqrt {\frac {\log (1 / \delta)}{n}}. \\ \end{array} +$$ + +Above inequality also holds in nonconvex problems and implies that when the uniformly stable in gradients parameter $\beta$ is smaller than $1 / \sqrt{n}$ , our bound is tighter than (2) and is dimension independent. + +# 3.2 Sharper Excess Risk Bounds + +In this subsection, we proceed to the desired upper bound of excess risk. The result is obtained by applying PL condition to Theorem 2 as in the following Theorem. + +Theorem 3. Let assumptions in Theorem 2 hold. Suppose the function $f$ is $\gamma$ -smooth and the population risk $F$ satisfies the PL condition with parameter $\mu$ . $\mathbf{w}^*$ is the projection of $A(S)$ onto the solution set $\arg \min_{\mathbf{w} \in \mathcal{W}} F(\mathbf{w})$ . Then for any $\delta \in (0,1)$ , when $n \geq \frac{32\gamma^2 \log(6/\delta)}{\mu^2}$ , with probability at least $1 - \delta$ , we have + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ F (A (S)) \right] - F (\mathbf {w} ^ {*}) \\ \lesssim \frac {\mathbb {E} _ {A} \left[ | | \nabla F _ {S} (A (S)) | | _ {2} ^ {2} \right]}{\mu} + \frac {\gamma F (\mathbf {w} ^ {*}) \log (1 / \delta)}{\mu n} + \frac {M ^ {2} \log^ {2} (1 / \delta)}{\mu n ^ {2}} + \frac {\beta^ {2} \log^ {2} n \log^ {2} (1 / \delta)}{\mu}. \\ \end{array} +$$ + +Remark 5. Notably, in Theorem 3, the PL condition is imposed to the population risk. This can be hold for many cases within classical learning theory. For instance, the classical linear regression setup: $f(w;(x,y)) = ||y - \langle w,x\rangle ||^2$ , where $y = \langle w^{*},x\rangle +v$ , with $v\sim N(0,1)$ and $x\sim N(0,\sigma^2 I_{d\times d})$ . In this case, we have $\mu = 2$ . + +Theorem 3 implies that excess risk can be bound by the optimization gradient error $\| \nabla F_S(A(S))\| _2$ and uniform stability in gradients $\beta$ . Notably, the theorem is applied to any algorithm with uniform stability in gradient. Latter, we will analyze the stability for specific algorithm in the next Section. Note that the assumption $F(\mathbf{w}^{*}) < O(1 / n)$ is common and can be found in Srebro et al. [43], Lei and Ying [26], Liu et al. [30], Zhang et al. [55], Zhang and Zhou [54] when analyzes sharper bounds. In our bound, when $F(\mathbf{w}^{*}) < O(1 / n)$ , and the stability $\beta = O(1 / n)$ , we can obtain the desired $O(1 / n^2)$ upper bound for excess risk when the empirical risk is sufficiently minimized by algorithm $A$ . + +To further compare our result with the optimal results based on algorithmic stability, i.e., [20], they only impose the assumption of bounded loss function, which is relative weaker than ours (smoothness and PL condition). However, our result is supposed to be sharper than their $\tilde{O} (\beta +1 / n)$ , since ours has the potential to be $\tilde{O} (1 / n^2)$ but their results are at most $\tilde{O} (1 / n)$ even if the algorithm is stable enough. More than that, the imposed smoothness and PL conditions in our theorem are also used in Klochkov and Zhivotovsky [20] to further analysis the algorithmic stability for PGD, whereas their excess risk bound becomes $\tilde{O} (1 / n)$ . However, our sharper bound indicates that they do not fully leverage these assumptions. The improvement is mainly from the novel stability in gradient during our analysis. This is also why our work can fully utilize these assumptions. + +In addition, our results provide a more granular analysis dependent on optimal parameters. On one hand, when the algorithm's stability $\beta = O(1 / \sqrt{n})$ the upper bound, according to [20], can at most reach the order of $\tilde{O}(1 / \sqrt{n})$ due to the algorithm's stability constraints. In contrast, our result shows that even under the assumption that $(F(\mathbf{w}^{*}) = O(1)$ , treating $F(\mathbf{w}^{*})$ as a constant, we can achieve an order of $\tilde{O}(1 / n)$ under the same algorithmic stability. On the other hand, their result is insensitive to the stability parameter being smaller than $O(1 / n)$ and their best rates can only up to $\tilde{O}(1 / n)$ . Our results can be up to $\tilde{O}(1 / n^2)$ under some specific assumptions. We will discuss it in Section 4. + +While Theorem 3 considers the algorithm's uniform stability in gradients rather than the classic uniform stability, calculating the uniform stability in gradients is not more challenging than the classic uniform stability. We discuss uniform stability in gradients for common algorithms such as ERM, PGD, and SGD in Section 4. Our results can be easily extended to other stable algorithms. Due to the smoothness property linking uniform stability in gradients with uniform argument stability, many works [1, 12, 17] that explore uniform argument stability can also utilize our method. + +Finally, we notice that there are currently several studies based on Klochkov and Zhivotovskiy [20] addressing different settings, such as differentially private models [18] and pairwise learning [27]. These studies also utilize the smoothness assumption in their optimization analysis. However, since they rely on the method established in Klochkov and Zhivotovskiy [20] for the generalization aspect of their research, they can only achieve an $\tilde{O}(1/n)$ bound at most. In contrast, they can easily achieve better results without making additional assumptions using our method. + +# 4 Application + +In this section, we take ERM, PGD, and SGD as examples to provide a detailed discussion of how our algorithm can be applied to common methods, resulting in tighter upper bounds. + +# 4.1 Empirical Risk Minimizer + +Empirical risk minimizer is one of the classical approaches for solving stochastic optimization (also referred to as sample average approximation (SAA)) in machine learning community. The following lemma shows the uniform stability in gradient for ERM under the PL condition and $\gamma$ -smoothness assumptions. + +Lemma 1 (Stability of ERM). Suppose the objective function $f$ is $M$ -Lipschitz and $\gamma$ -smooth, the empirical risk $F_{S}$ and $F_{S^{(i)}}$ satisfy the PL condition with parameter $\mu$ . Let $\hat{\mathbf{w}}^*(S^{(i)})$ be the ERM of $F_{S^{(i)}}(\mathbf{w})$ that denotes the empirical risk on the samples $S^{(i)} = \{z_1, \dots, z_i', \dots, z_n\}$ and $\hat{\mathbf{w}}^*(S)$ be the ERM of $F_{S}(\mathbf{w})$ on the samples $S = \{z_1, \dots, z_i, \dots, z_n\}$ . For any $S^{(i)}$ and $S$ , there holds the following uniform stability bound of ERM: + +$$ +\forall z \in \mathcal {Z}, \quad \left\| \nabla f (\hat {\mathbf {w}} ^ {*} (S ^ {(i)}); z) - \nabla f (\hat {\mathbf {w}} ^ {*} (S); z) \right\| _ {2} ^ {2} \leq \frac {1 6 M ^ {2} \gamma^ {2}}{n ^ {2} \mu^ {2}}. +$$ + +Then, we present the application of our main sharper Theorem 2. In the PL condition and smooth case, we provide a up to $\tilde{O}\left(1 / n^2\right)$ high probability excess risk guarantee valid for any algorithms depending on the optimal population error $F(\mathbf{w}^{*})$ . + +Theorem 4. Let assumptions in Theorem 3 and Lemma 1 hold. Suppose the function $f$ is nonnegative. Then for any $\delta \in (0,1)$ , when $n \geq \frac{32\gamma^2\log(6 / \delta)}{\mu^2}$ , with probability at least $1 - \delta$ , we have + +$$ +F (\hat {\mathbf {w}} ^ {*} (S)) - F (\mathbf {w} ^ {*}) \lesssim \frac {\gamma F (\mathbf {w} ^ {*}) \log (1 / \delta)}{\mu n} + \frac {M ^ {2} \gamma^ {2} \log^ {2} n \log^ {2} (1 / \delta)}{\mu^ {3} n ^ {2}}. +$$ + +Remark 6. Theorem 4 shows that when the objective function $f$ is $M$ -Lipschitz, $\gamma$ -smooth and nonnegative, at the same time, both the empirical risk $F_{S}$ and the population risk $F$ satisfies PL condition with parameter $\mu$ , then the high probability risk bounds can be up to $\tilde{O}\left(1 / n^{2}\right)$ for ERM. The most related work to ours is Zhang et al. [55]. They also obtain the $\tilde{O}\left(1 / n^{2}\right)$ -type bounds for ERM by uniform convergence of gradients approach under the same assumptions. However, they need the sample number $n = \Omega (\gamma^2 d / \mu^2)$ , which is related to the dimension $d$ . Our risk bounds are dimension independent and only require the sample number $n = \Omega (\gamma^2 /\mu^2)$ . Comparing with Klochkov and Zhivotovsky [20], we add two assumptions, smoothness and $F(\mathbf{w}^{*}) < O(1 / n)$ , the later of which is a common assumption towards sharper risk bounds [43, 26, 30, 55, 54] called low-noise, but our bounds are also tighter, from $\tilde{O}(1 / n)$ to $\tilde{O}\left(1 / n^{2}\right)$ . + +Our results are asymptotically optimal, which aligns with existing theories. According to the classical asymptotic theory, under some local regularity assumptions, when $n \to \infty$ , it is shown in the asymptotic statistics monographs [45] that + +$$ +\sqrt {n} \left(\hat {\mathbf {w}} ^ {*} (S) - \mathbf {w} ^ {*}\right) \xrightarrow {\rho} \mathcal {N} \left(0, \boldsymbol {H} ^ {- 1} \boldsymbol {Q} \boldsymbol {H} ^ {- 1}\right), \tag {3} +$$ + +where $\hat{\mathbf{w}}^{*}(S)$ denotes the ERM algorithm, $H = \nabla^2 F(\mathbf{w}^*)$ , $Q$ is the covariance matrix of the loss gradient at $\mathbf{w}^*$ (also called Fisher's information matrix): $Q = \mathbb{E}[\nabla f(\mathbf{w}^*;z)\nabla f(\mathbf{w}^*;z)^T]$ ( $A^T$ denotes the transpose of a matrix $A$ ), and $\xrightarrow{\rho}$ means convergence in distribution. The second-order Taylor expansion of the population risk around $\mathbf{w}^*$ then allows to derive the same asymptotic law for the scaled excess risk $2n(F(\hat{\mathbf{w}}^{*}(S)) - F(\mathbf{w}^{*}))$ . Under suitable conditions, this asymptotic rate is usually theoretically optimal [44]. For example, when $f(\mathbf{w};z)$ is a negative log-likelihood, this asymptotic rate matches the Hajek-LeCam asymptotic minimax lower bound [16, 23]. We then analyze the result in Theorem 4. In the proof of Theorem 3, before we use the self-bounded smoothness property $\| \nabla f(\mathbf{w}^*;z)\|^2 \leq 4\gamma f(\mathbf{w}^*;z)$ , we get the following result for Theorem 4 + +$$ +F (\hat {\mathbf {w}} ^ {*} (S)) - F (\mathbf {w} ^ {*}) \lesssim \frac {\mathbb {E} [ \| \nabla f (\mathbf {w} ^ {*} ; z) \| ^ {2} ] \log (1 / \delta)}{\mu n} + \frac {M ^ {2} \gamma^ {2} \log^ {2} (n) \log^ {2} (1 / \delta)}{\mu^ {3} n ^ {2}}. +$$ + +Our result is the finite sample version of the asymptotic rate (3), which characterizes the critical sample size sufficient to enter this "asymptotic regime". This is because the excess risk error $F(\hat{\mathbf{w}}^{*}(S)) - F(\mathbf{w}^{*})$ can be approximated by the quadratic form $(\hat{\mathbf{w}}^{*}(S) - \mathbf{w}^{*})^{T}\mathbf{H}(\hat{\mathbf{w}}^{*}(S) - \mathbf{w}^{*})$ . $1 / \mu$ is a natural proxy for the inverse Hessian $H^{-1}$ , and $\mathbb{E}[\| \nabla f(\mathbf{w}^*;z)\|^2 ]$ is a natural proxy for Fisher's information matrix $Q$ . Furthermore, when discussing sample complexity, Xu and Zeevi [50] constructed a simple linear model to demonstrate the constant-level optimality of the sample complexity lower bound $\Omega (d\beta^{2} / \mu^{2})$ under such conditions. Our theorem further reveals, through the use of stability methods, that this complexity lower bound can be independent of the dimensionality $d$ . + +# 4.2 Projected Gradient Descent + +Note that when the objective function $f$ is smooth and the empirical risk $F_{S}$ satisfies the PL condition, the optimization error can be ignored. However, Klochkov and Zhivotovsky [20] does not use smoothness assumption for their generalization bounds, which only derive high probability excess risk bound of order $\tilde{O}(1/n)$ after $T = O(\log(n))$ steps. In this subsection, we provide sharper risk bound under the same iteration steps, which is because our generalization analysis also fully utilized the smooth assumptions. Here we introduce the procedure of the PGD algorithm. + +Let $\mathbf{w}_1 \in \mathbb{R}^d$ be an initial point and $\{\eta_t\}_t$ be a sequence of positive step sizes. PGD updates parameters by + +$$ +\mathbf {w} _ {t + 1} = \Pi_ {\mathcal {W}} \left(\mathbf {w} _ {t} - \eta_ {t} \nabla F _ {S} \left(\mathbf {w} _ {t}\right)\right), +$$ + +where $\nabla F_S(\mathbf{w}_t)$ denotes a subgradient of $F_{S}$ w.r.t. $\mathbf{w}_t$ and $\Pi_{\mathcal{W}}$ is the projection operator onto $\mathcal{W}$ . Lemma 2 (Stability of Projected Gradient Descent [12, 17]). Suppose the objective function $f$ is $M$ -Lipschitz and $\gamma$ -smooth, the empirical risk $F_{S}$ and $F_{S^{(i)}}$ satisfy the PL condition with parameter $\mu$ . Let $\mathbf{w}_t^\prime$ be the output of $F_{S^{(i)}}(\mathbf{w})$ on $t$ -th iteration on the samples $S^{(i)} = \{z_1,\dots,z_i',\dots,z_n\}$ in running PGD, and $\mathbf{w}_t$ be the output of $F_{S}(\mathbf{w})$ on $t$ -th iteration on the samples $S = \{z_1,\dots,z_i,\dots,z_n\}$ in running PGD. Let the constant step size $\eta_t = \frac{1}{\gamma}$ . For any $S^{(i)}$ and $S$ , there holds the following uniform stability bound of PGD: + +$$ +\forall z \in \mathcal {Z}, \quad \left\| \nabla f (\mathbf {w} _ {t} ^ {i}; z) - \nabla f (\mathbf {w} _ {t}; z) \right\| _ {2} ^ {2} \leq \frac {4 M ^ {2} \gamma^ {2}}{n ^ {2} \mu^ {2}}. +$$ + +The derivations of Feldman and Vondrak [12] in Section 4.1.2 (See also [17] in Section 3.4) imply that if the objective function $f$ is $\gamma$ -smooth and $M$ -Lipschitz, and the empirical risk $F_{S}$ and $F_{S^{(i)}}$ satisfy the PL condition with parameter $\mu$ , then PGD with the constant step size $\eta = \frac{1}{\gamma}$ is $\left(\frac{2M}{n\mu}\right)$ -uniformly argument stable for any number of steps, which means that PGD is $\left(\frac{2M\gamma}{n\mu}\right)$ -uniformly-stable in gradients regardless of iteration steps. + +Theorem 5. Let assumptions in Theorem 3 and Lemma 1 hold. Suppose the function $f$ is nonnegative and $\mathbf{w}^*$ belongs to the projected set $\mathcal{W}$ . Let $\{\mathbf{w}_t\}_t$ be the sequence produced by PGD with $\eta_t = 1 / \gamma$ . Then for any $\delta \in (0,1)$ , when $n \geq \frac{32\gamma^2\log(6 / \delta)}{\mu^2}$ , with probability at least $1 - \delta$ , we have + +$$ +F \left(\mathbf {w} _ {T + 1}\right) - F \left(\mathbf {w} ^ {*}\right) \lesssim \left(1 - \frac {\mu}{\gamma}\right) ^ {2 T} + \frac {\gamma F \left(\mathbf {w} ^ {*}\right) \log (1 / \delta)}{\mu n} + \frac {M ^ {2} \gamma^ {2} \log^ {2} n \log^ {2} (1 / \delta)}{\mu^ {3} n ^ {2}}. +$$ + +Furthermore, assume $F(\mathbf{w}^{*}) < O\left(\frac{1}{n}\right)$ and let $T \asymp \log (n)$ , we have + +$$ +F (\mathbf {w} _ {T + 1}) - F (\mathbf {w} ^ {*}) \lesssim \frac {M ^ {2} \gamma^ {2} \log^ {2} n \log^ {2} (1 / \delta)}{\mu^ {3} n ^ {2}}. +$$ + +Remark 7. Theorem 5 shows that under the same assumptions as Klochkov and Zhivotovsky [20], our bound is $O\left(\frac{F(\mathbf{w}^{*})\log(1 / \delta)}{n} +\frac{\log^{2}(n)\log^{2}(1 / \delta)}{n^{2}}\right)$ . Comparing with their bound $O\left(\frac{\log(n)\log(1 / \delta)}{n}\right)$ , we are sharper because $F(\mathbf{w}^{*})$ is the minimal population risk, which is a common assumption towards sharper risk bounds [43, 26, 30, 55, 54]. We use this assumption to demonstrate that under low-noise conditions, our bounds can achieve the tightest possible rate of $\tilde{O}(1 / n^2)$ . This is because we also fully utilized the properties of smooth in the process of analyzing generalization errors. Considering that smooth is a very common assumption in optimization communities, our work can better utilize existing optimization work and obtain better excess risk bounds under the same assumptions. + +# 4.3 Stochastic Gradient Descent + +Stochastic gradient descent optimization algorithm has been widely used in machine learning due to its simplicity in implementation, low memory requirement and low computational complexity per iteration, as well as good practical behavior. We provide the excess risk bounds for SGD using our method in this subsection. Here we introduce the procedure of the standard SGD algorithm. + +Let $\mathbf{w}_1 \in \mathbb{R}^d$ be an initial point and $\{\eta_t\}_t$ be a sequence of positive step sizes. SGD updates parameters by + +$$ +\mathbf {w} _ {t + 1} = \Pi_ {\mathcal {W}} \left(\mathbf {w} _ {t} - \eta_ {t} \nabla f \left(\mathbf {w} _ {t}; z _ {i _ {t}}\right)\right), +$$ + +where $\nabla f(\mathbf{w}_t;z_{i_t})$ denotes a subgradient of $f$ w.r.t. $\mathbf{w}_t$ and $i_t$ is independently drawn from the uniform distribution over $[n]\coloneqq \{1,2,\ldots ,n\}$ . + +Lemma 3 (Stability of SGD). Suppose the objective function $f$ is $M$ -Lipschitz and $\gamma$ -smooth, the empirical risk $F_{S}$ and $F_{S^{(i)}}$ satisfy the PL condition with parameter $\mu$ . Let $\mathbf{w}_t^i$ be the output of $F_{S^{(i)}}(\mathbf{w})$ on $t$ -th iteration on the samples $S^{(i)} = \{z_1, \dots, z_i', \dots, z_n\}$ in running SGD with $\eta_t = \frac{2t + 1}{2\mu(t + 1)^2}$ and $\mathbf{w}_t$ be the output of $F_{S}(\mathbf{w})$ on $t$ -th iteration on the samples $S = \{z_1, \dots, z_i, \dots, z_n\}$ in running SGD with $\eta_t = \frac{2t + 1}{2\mu(t + 1)^2}$ . For any $S^{(i)}$ and $S$ and any $z \in \mathcal{Z}$ , there holds the following uniform stability bound of SGD: + +$$ +\mathbb {E} _ {A} \left[ \left\| \nabla f (\mathbf {w} _ {t}; z) - \nabla f (\mathbf {w} _ {t} ^ {i}; z) \right\| _ {2} ^ {2} \right] \leq \frac {6 M ^ {2} \gamma^ {3}}{t \mu^ {3}} + \frac {4 8 M ^ {2} \gamma^ {2}}{n ^ {2} \mu^ {2}}. +$$ + +Next, we introduce a necessary assumption in stochastic optimization theory. + +Assumption 1. Assume the existence of $\sigma >0$ satisfying + +$$ +\mathbb {E} _ {i _ {t}} \left[ \left\| \nabla f \left(\mathbf {w} _ {t}; z _ {i _ {t}}\right) - \nabla F _ {S} \left(\mathbf {w} _ {t}\right) \right\| _ {2} ^ {2} \right] \leq \sigma^ {2}, \quad \forall t \in \mathbb {N}, \tag {4} +$$ + +where $\mathbb{E}_{i_t}$ denotes the expectation w.r.t. $i_t$ . + +Remark 8. Assumption 1 is a standard assumption from the stochastic optimization theory [34, 14, 15, 21, 56, 2, 25], which essentially bounds the variance of the stochastic gradients for dataset $S$ . + +Theorem 6. Let Assumptions in Theorem 2 and Lemma 3 hold. Suppose Assumption 1 holds and the function $f$ is nonnegative. Let $\{\mathbf{w}_t\}_t$ be the sequence produced by SGD with $\eta_t = \frac{2t + 1}{2\mu(t + 1)^2}$ . Then for any $\delta > 0$ , when $n \geq \frac{32\gamma^2\log(6 / \delta)}{\mu^2}$ , with probability at least $1 - \delta$ , we have + +$$ +\mathbb {E} _ {A} \left[ F \left(\mathbf {w} _ {T + 1}\right) \right] - F \left(\mathbf {w} ^ {*}\right) \lesssim \frac {\gamma F \left(\mathbf {w} ^ {*}\right) \log (1 / \delta)}{\mu n} + \left(\frac {\gamma}{T \mu} + \frac {1}{n ^ {2}}\right) \frac {M ^ {2} \gamma^ {2} \log^ {2} n \log^ {2} (1 / \delta)}{\mu^ {3}}. +$$ + +Furthermore, assume $T \asymp n^2$ and $F(\mathbf{w}^{*}) < O\left(\frac{1}{n}\right)$ , we have + +$$ +F \left(\mathbf {w} _ {T + 1}\right) - F \left(\mathbf {w} ^ {*}\right) \lesssim \frac {M ^ {2} \gamma^ {3} \log^ {2} n \log^ {2} (1 / \delta)}{\mu^ {4} n ^ {2}}. +$$ + +Remark 9. Theorem 6 implies that high probability risk bounds for SGD optimization algorithm can be up to $\tilde{O}(1/n^2)$ and the rate is dimension-free in high-dimensional learning problems. We compare Theorem 6 with most related work. For algorithmic stability, high probability risk bounds in Fan and Lei [10] is up to $\tilde{O}(1/n)$ when choosing optimal iterate number $T$ for SGD optimization algorithm. To the best of knowledge, we are faster than all the existing bounds. The best high probability risk bounds $\tilde{O}(1/n^2)$ are given by [29] via uniform convergence, which require the same iterate number $T \asymp n^2$ and the sample number $n = \Omega(\gamma^2 d/\mu^2)$ depending on dimension $d$ . + +Although a considerable amount of theoretical (eg: [27]) and practical (eg: [35]) evidence indicates that multi-pass SGD can enhance the model's generalization performance. Someone may only utilize one-pass SGD in practice due to the large volume of data, which means that $T \asymp n$ . In fact, when $T \asymp n$ and $F(\mathbf{w}^{*}) = O(1)$ , our bound is $\tilde{O}(1/n)$ , which is also the sharpest hight probability bound under $T \asymp n$ iterations. For comparison, there are two results in stability analysis that are similar to ours. One is the $\tilde{O}(1/n)$ result when $T \asymp n$ [26], but it pertains to the expected version and also needs $F(\mathbf{w}^{*}) = 0$ . The high-probability version is significantly more challenging. Currently, the best result under the high-probability version is also $\tilde{O}(1/n)$ [29], but [29] requires $T \asymp n^2$ iterations. + +# 5 Conclusion + +In this paper, we derive sharper generalization bounds in gradients in nonconvex problems, which can further be used to obtain sharper high probability excess risk bounds for stable optimization algorithms. In application, we study three common algorithms: ERM, PGD, SGD. To the best of our knowledge, we provide the sharpest high probability dimension independent $\tilde{O}(1/n^2)$ -type for these algorithms. + +# Acknowledgments and Disclosure of Funding + +This research was supported by National Key Research and Development Program of China (NO. 2024YFE0203200), National Natural Science Foundation of China (No.62476277), CCF-ALIMAMA TECH Kangaroo Fund(No.CCF-ALIMAMA OF 2024008), and Huawei-Renmin University joint program on Information Retrieval. We also acknowledge the support provided by the fund for building worldclass universities (disciplines) of Renmin University of China and by the funds from Beijing Key Laboratory of Big Data Management and Analysis Methods, Gaoling School of Artificial Intelligence, Renmin University of China, from Engineering Research Center of NextGeneration Intelligent Search and Recommendation, Ministry of Education, from Intelligent Social Governance Interdisciplinary Platform, Major Innovation & Planning Interdisciplinary Platform for the "DoubleFirst Class" Initiative, Renmin University of China, from Public Policy and Decision-making Research Lab of Renmin University of China, and from Public Computing Cloud, Renmin University of China. + +# References + +[1] R. Bassily, V. Feldman, C. Guzmán, and K. Talwar. Stability of stochastic gradient descent on nonsmooth convex losses. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [NA] + +Justification: The core method development in this paper does not involve LLMs as any important, original, or non-standard components. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Additional definitions and lemmata + +Lemma 4 (Equivalence of tails and moments for random vectors [1]). Let $X$ be a random variable with + +$$ +\left\| X \right\| _ {p} \leq \sqrt {p} a + p b +$$ + +for some $a, b \geq 0$ and for any $p \geq 2$ . Then for any $\delta \in (0,1)$ we have, with probability at least $1 - \delta$ , + +$$ +| X | \leq e \left(a \sqrt {\log \left(\frac {e}{\delta}\right)} + b \log \frac {e}{\delta}\right). +$$ + +Lemma 5 (Vector Bernstein's inequality [36, 41]). Let $\{X_i\}_{i=1}^n$ be a sequence of i.i.d. random variables taking values in a real separable Hilbert space. Assume that $\mathbb{E}[X_i] = \mu$ , $\mathbb{E}[\|X_i - \mu\|^2] = \sigma^2$ , and $\|X_i\| \leq M$ , $\forall 1 \leq i \leq n$ , then for all $\delta \in (0,1)$ , with probability at least $1 - \delta$ we have + +$$ +\left\| \frac {1}{n} \sum_ {i = 1} ^ {n} X _ {i} - \mu \right\| \leq \sqrt {\frac {2 \sigma^ {2} \log (\frac {2}{\delta})}{n}} + \frac {M \log \frac {2}{\delta}}{n}. +$$ + +Definition 4 (Weakly self-Bounded Function). Assume that $a, b > 0$ . A function $f: \mathcal{Z}^n \mapsto [0, +\infty)$ is said to be $(a, b)$ -weakly self-bounded if there exist functions $f_i: \mathcal{Z}^{n-1} \mapsto [0, +\infty)$ that satisfies for all $Z^n \in \mathcal{Z}^n$ , + +$$ +\sum_ {i = 1} ^ {n} \left(f _ {i} \left(Z ^ {n}\right) - f \left(Z ^ {n}\right)\right) ^ {2} \leq a f \left(Z ^ {n}\right) + b. +$$ + +Lemma 6 ([20]). Suppose that $z_{1}, \ldots, z_{n}$ are independent random variables and the function $f: \mathcal{Z}^{n} \mapsto [0, +\infty)$ is $(a, b)$ -weakly self-bounded and the corresponding function $f_{i}$ satisfy $f_{i}(Z^{n}) \geq f(Z^{n})$ for $\forall i \in [n]$ and any $Z^{n} \in \mathcal{Z}^{n}$ . Then, for any $t > 0$ , + +$$ +P r (\mathbb {E} f (z _ {1}, \dots , z _ {n}) \geq f (z _ {1}, \dots , z _ {n}) + t) \leq \exp \left(- \frac {t ^ {2}}{2 a \mathbb {E} f (z _ {1} , \dots , z _ {n}) + 2 b}\right). +$$ + +Definition 5. A Rademacher random variable is a Bernoulli variable that takes values $\pm 1$ with probability $\frac{1}{2}$ each. + +# B Summary of Our High Probability Excess Risk Bounds. + +Our high probability excess risk bounds can be summarized in Table 1. + +# C Proofs of Section 3 + +# C.1 An Improved Moment Bound for Sums of Vector-valued Functions + +In this section, we present our sharper moment bound for sums of vector-valued functions of $n$ independent variables, which is constant-level improvement comparing with [10]. With this inequality, we further prove the connection between algorithmic stability w.r.t. gradient and generalization. Notably, our bound relies on the "bounded difference" property, which is similar to uniform stability in gradient. Thus the bound can be further applied in deriving generalization bound on gradient, under stability condition. + +Lemma 7. Let $\mathbf{Z} = (Z_1, \ldots, Z_n)$ be a vector of independent random variables each taking values in $\mathcal{Z}$ . $A$ is a independent random variable taking values in $\mathcal{A}$ and let $\mathbf{g}_1, \ldots, \mathbf{g}_n$ be some functions: $\mathbf{g}_i : \mathcal{Z}^n \times \mathcal{A} \mapsto \mathcal{H}$ such that the following holds for any $i \in [n]$ : + +$\| \mathbb{E}[\mathbf{g}_i(\mathbf{Z})|Z_i]\| \leq G$ a.s. +$\mathbb{E}\left[\mathbf{g}_i(\mathbf{Z})|Z_{[n]\setminus \{i\}}\right] = 0$ a.s., +- $\mathbf{g}_i$ satisfies the bounded difference property with $\beta$ , namely, for any $i = 1, \ldots, n$ , the following inequality holds + +$$ +\begin{array}{l} \sup _ {z _ {1}, \dots , z _ {n}, z _ {j} ^ {\prime}} \mathbb {E} _ {A} \left[ \| \mathbf {g} _ {i} \left(z _ {1}, \dots , z _ {j - 1}, z _ {j}, z _ {j + 1}, \dots , z _ {n}\right) \right. \tag {5} \\ \left. - \mathbf {g} _ {i} \left(z _ {1}, \dots , z _ {j - 1}, z _ {j} ^ {\prime}, z _ {j + 1}, \dots , z _ {n}\right) \| \right] \leq \beta . \\ \end{array} +$$ + +Then, for any $p \geq 2$ , we have + +$$ +\begin{array}{l} \left\| \left\| \sum_ {i = 1} ^ {n} \mathbf {g} _ {i} \right\| \right\| _ {p} \leq 4 \times 2 ^ {\frac {1}{2 p}} \left(\sqrt {\frac {p}{e}}\right) (\sqrt {2 p} + 1) n \beta \lceil \log_ {2} n \rceil \\ + 2 (\sqrt {2 p} + 1) \sqrt {n} G. \\ \end{array} +$$ + +Remark 10. We compare with existing results. The proof is motivated by [4, 20]. [52, 53] have also explored several related problems based on this approach. However, all of them focus specifically on upper bounds for sums of real-valued functions. The result most closely related to Lemma 7 is provided by [10]. Under the same assumptions, [10] established the following inequality2 + +$$ +\left\| \left\| \sum_ {i = 1} ^ {n} \mathbf {g} _ {i} \right\| \right\| _ {p} \leq 4 (\sqrt {2} + 1) n p \beta \lceil \log_ {2} n \rceil + 2 (\sqrt {2} + 1) \sqrt {n p} G. \tag {6} +$$ + +It is easy to verify that our result is tighter than result provided by [10] for both the first and second term. Comparing Lemma 7 with (6), the larger $p$ is, the tighter our result is relative to (6). In the worst case, when $p = 2$ , the constant of our second term is 0.879 times tighter than (6), and the constant of our first term is 0.634 times tighter than (6). This is because we derive the optimal Marcinkiewicz-Zygmund's inequality for random variables taking values in a Hilbert space in the proof. + +Although Lemma 7 seems to the constant-level improvement, considering that this theorem has other broad applications, we report this result as well. On the other hand, the proof was challenging, as it involved establishing the best constant in Marcinkiewicz-Zygmund's inequality for random variables taking values in a Hilbert space, which has its foundations in Khintchine-Kahane's inequality. To prove the best constant, we utilized Stirling's formula for the Gamma function to construct appropriate functions, establishing both upper and lower bounds. Then using Mean Value Theorem, this approach ultimately led to the convergence of the constant as $p$ approaches infinity. + +The proof of Lemma 7 is motivated by [4], which need the Marcinkiewicz-Zygmund's inequality for random variables taking values in a Hilbert space and the McDiarmid's inequality for vector-valued functions. + +Firstly, we derive the optimal constants in the Marcinkiewicz-Zygmund's inequality for random variables taking values in a Hilbert space. + +Lemma 8 (Marcinkiewicz-Zygmund's Inequality for Random Variables Taking Values in a Hilbert Space). Let $\mathbf{X}_1, \ldots, \mathbf{X}_n$ be random variables taking values in a Hilbert space with $\mathbb{E}[\mathbf{X}_i] = 0$ for all $i \in [n]$ . Then for $p \geq 2$ we have + +$$ +\left\| \left\| \sum_ {i = 1} ^ {n} \mathbf {X} _ {i} \right\| \right\| _ {p} \leq 2 \cdot 2 ^ {\frac {1}{2 p}} \sqrt {\frac {n p}{e}} \left(\frac {1}{n} \sum_ {i = 1} ^ {n} \left\| \left\| \mathbf {X} _ {i} \right\| \right\| _ {p} ^ {p}\right) ^ {\frac {1}{p}}. +$$ + +Remark 11. Comparing with Marcinkiewicz-Zygmund's inequality given by [10], we provide best constants. Next, we give the proof of Lemma 8. + +The Marcinkiewicz-Zygmund's inequality can be proved by using its connection to Khintchine-Kahane's inequality. Thus, we introduce the best constants in Khintchine-Kahane's inequality for random variables taking values from a Hilbert space here. + +Lemma 9 (Best constants in Khintchine-Kahane's inequality in Hilbert space [22, 33]). For all $p \in [2,\infty)$ and for all choices of Hilbert space $\mathcal{H}$ , finite sets of vectors $\mathbf{X}_i, \ldots, \mathbf{X}_n \in \mathcal{X} \in \mathcal{H}$ , and independent Rademacher variables $r_1, \ldots, r_n$ , + +$$ +\left[ \mathbb {E} \left\| \sum_ {i = 1} ^ {n} r _ {i} \mathbf {X} _ {i} \right\| ^ {p} \right] ^ {\frac {1}{p}} \leq C _ {p} \cdot \left[ \sum_ {i = 1} ^ {n} \| \mathbf {X} _ {i} \| ^ {2} \right] ^ {\frac {1}{2}}, +$$ + +where $C_p = 2^{\frac{1}{2}}\left\{\frac{\Gamma\left(\frac{p + 1}{2}\right)}{\sqrt{\pi}}\right\}^{\frac{1}{p}}$ + +2They assume $n = 2^k$ $k\in \mathbb{N}$ . Here we give the version of their result with general $n$ + +Proof of Lemma 8. The symmetrization argument goes as follows: Let $(r_1, \ldots, r_n)$ be i.i.d. with $\mathbb{P}(r_i = 1) = \mathbb{P}(r_i = -1) = 1/2$ and besides such that $r_1, \ldots, r_n$ and $(\mathbf{X}_1, \ldots, \mathbf{X}_n)$ are independent. Then by independence and symmetry, according to Lemma 1.2.6 of [7], conditioning on $(\mathbf{X}_1, \ldots, \mathbf{X}_n)$ yields + +$$ +\mathbb {E} \left[ \left\| \sum_ {i = 1} ^ {n} \mathbf {X} _ {i} \right\| ^ {p} \right] = 2 ^ {p} \mathbb {E} \left[ \left\| \sum_ {i = 1} ^ {n} r _ {i} \mathbf {X} _ {i} \right\| ^ {p} \right] \leq 2 ^ {p} \mathbb {E} \left[ \mathbb {E} \left[ \left\| \sum_ {i = 1} ^ {n} r _ {i} \mathbf {X} _ {i} \right\| ^ {p} \mid \mathbf {X} _ {1}, \dots , \mathbf {X} _ {n} \right] \right]. \tag {7} +$$ + +As for the conditional expectation in (7), notice that by independence + +$$ +\mathbb {E} \left[ \left\| \sum_ {i = 1} ^ {n} r _ {i} \mathbf {X} _ {i} \right\| ^ {p} \mid \mathbf {X} _ {1} = \mathbf {x} _ {1}, \dots , \mathbf {X} _ {n} = \mathbf {x} _ {n} \right] = \mathbb {E} \left[ \left\| \sum_ {i = 1} ^ {n} r _ {i} \mathbf {x} _ {i} \right\| ^ {p} \right] \tag {8} +$$ + +According to Lemma 9, for $v_{n}$ -almost every $\mathbf{x}_1,\ldots ,\mathbf{x}_n\in \mathcal{X}^n$ , where $v_{n}\coloneqq \mathbb{P}\circ (\mathbf{X}_{1},\dots ,\mathbf{X}_{n})^{-1}$ denotes the distribution of $(\mathbf{X}_1,\dots ,\mathbf{X}_n)$ , we have + +$$ +\left[ \mathbb {E} \left\| \sum_ {i = 1} ^ {n} r _ {i} \mathbf {x} _ {i} \right\| ^ {p} \right] \leq C \cdot \left[ \sum_ {i = 1} ^ {n} \| \mathbf {x} _ {i} \| ^ {2} \right] ^ {\frac {p}{2}}, \tag {9} +$$ + +where $C = 2^{\frac{p}{2}} \frac{\Gamma\left(\frac{p + 1}{2}\right)}{\sqrt{\pi}}$ and $C$ is optimal. This means that for any constant $C'$ such that + +$$ +\left[ \mathbb {E} \left\| \sum_ {i = 1} ^ {n} r _ {i} \mathbf {x} _ {i} \right\| ^ {p} \right] \leq C ^ {\prime} \cdot \left[ \sum_ {i = 1} ^ {n} \| \mathbf {x} _ {i} \| ^ {2} \right] ^ {\frac {p}{2}}, \tag {10} +$$ + +for all $n\in \mathbb{N}$ and for each collection of vectors $\mathbf{x}_1,\ldots ,\mathbf{x}_n$ , it follows that $C^\prime \geq C$ + +From (8) and (9), we can infer that + +$$ +\mathbb {E} \left[ \left\| \sum_ {i = 1} ^ {n} r _ {i} \mathbf {X} _ {i} \right\| ^ {p} \middle | \mathbf {X} _ {1} = \mathbf {x} _ {1}, \dots , \mathbf {X} _ {n} = \mathbf {x} _ {n} \right] \leq C \cdot \left[ \sum_ {i = 1} ^ {n} \| \mathbf {X} _ {i} \| ^ {2} \right] ^ {\frac {p}{2}}. +$$ + +Taking expectations in the above inequalities and (7) yield that + +$$ +\mathbb {E} \left[ \left\| \sum_ {i = 1} ^ {n} \mathbf {X} _ {i} \right\| ^ {p} \right] \leq C \cdot \mathbb {E} \left[ \sum_ {i = 1} ^ {n} \| \mathbf {X} _ {i} \| ^ {2} \right] ^ {\frac {p}{2}}. \tag {11} +$$ + +To see optimality let the above statement hold for some constants $C'$ in place of $C$ . Then if we choose $\mathbf{X}_i \coloneqq \mathbf{x}_i r_i, 1 \leq i \leq n$ with arbitrary reals vectors $\mathbf{x}_1, \ldots, \mathbf{x}_n$ , it follows that + +$$ +\mathbb {E} \left[ \left\| \sum_ {i = 1} ^ {n} r _ {i} \mathbf {x} _ {i} \right\| ^ {p} \right] \leq C ^ {\prime} \cdot \mathbb {E} \left[ \sum_ {i = 1} ^ {n} \| \mathbf {x} _ {i} \| ^ {2} \right] ^ {\frac {p}{2}}, +$$ + +whence we can conclude from (10) that $C' \geq C$ . Thus we obtain that $C' = C$ . + +Notice that by Holder's inequality + +$$ +\left[ \sum_ {i = 1} ^ {n} \| \mathbf {X} _ {i} \| ^ {2} \right] ^ {\frac {p}{2}} \leq n ^ {p / 2 - 1} \sum_ {i = 1} ^ {n} \| \mathbf {X} _ {i} \| ^ {p}. \tag {12} +$$ + +Plugging (12) into (11), we have + +$$ +\mathbb {E} \left[ \left\| \sum_ {i = 1} ^ {n} \mathbf {X} _ {i} \right\| ^ {p} \right] \leq C \cdot 2 ^ {p} n ^ {p / 2 - 1} \cdot \mathbb {E} \left[ \sum_ {i = 1} ^ {n} \| \mathbf {X} _ {i} \| ^ {p} \right], +$$ + +where $C = 2^{\frac{p}{2}}\frac{\Gamma\left(\frac{p + 1}{2}\right)}{\sqrt{\pi}}$ is a constant. + +Next, we use the following form of Stirling's formula for the Gamma-function, which follows from (6.1.5), (6.1.15) and (6.1.38) in [6] to bound the constant $C$ . For every $x > 0$ , there exists a $\mu(x) \in (0, 1/(12x))$ such that + +$$ +\Gamma (x) = \sqrt {2 \pi} x ^ {x - 1 / 2} e ^ {- x} e ^ {\mu (x)}. +$$ + +Thus + +$$ +C = 2 ^ {\frac {p}{2}} \frac {\Gamma \left(\frac {p + 1}{2}\right)}{\sqrt {\pi}} = g (p) \sqrt {2} e ^ {- p / 2} p ^ {p / 2}, +$$ + +with $g(p) = \left(1 + \frac{1}{p}\right)^{p / 2}e^{v(p) - 1 / 2}$ , where $0 < v(p) < 1 / (6(p + 1))$ . By Taylor's formula we have that + +$$ +\log (1 + x) = \sum_ {m = 1} ^ {\infty} \frac {1}{m} (- 1) ^ {m - 1} x ^ {m}, \quad \forall x \in (- 1, 1 ], +$$ + +and that for every $k\in \mathbb{N}_0$ + +$$ +\sum_ {m = 1} ^ {2 k} \frac {1}{m} (- 1) ^ {m - 1} x ^ {m} \leq \log (1 + x) \leq \sum_ {m = 1} ^ {2 k + 1} \frac {1}{m} (- 1) ^ {m - 1} x ^ {m}, \forall x \geq 0. +$$ + +Therefor we obtain with $k = 1$ that + +$$ +\log g (p) = \frac {p}{2} \log (1 + \frac {1}{p}) + v (p) - \frac {1}{2} \leq - \frac {1}{4 p} + \frac {1}{6 p ^ {2}} + \frac {1}{6 (p + 1)} \leq - \frac {1}{1 8 p}, +$$ + +where the last equality follows from elementary calculus. Similarly, + +$$ +\log g (p) = \frac {p}{2} \log \left(1 + \frac {1}{p}\right) + v (p) - \frac {1}{2} \geq - \frac {1}{4 p} + v (p) \geq - \frac {1}{4 p}, +$$ + +Thus, we have + +$$ +e ^ {- \frac {1}{4 p}} \sqrt {2} e ^ {- p / 2} p ^ {p / 2} < C < e ^ {- \frac {1}{1 8 p}} \sqrt {2} e ^ {- p / 2} p ^ {p / 2}, +$$ + +which implies that $C$ is strictly smaller than $\sqrt{2} e^{-p / 2}p^{p / 2}$ for all $p\geq 2$ + +Since $C = \frac{1}{g(p)}\sqrt{2} e^{-p / 2}p^{p / 2}$ and $g(p)\geq e^{-\frac{1}{4p}}$ , we can obtain that the relative error between $C$ and $\sqrt{2} e^{-p / 2}p^{p / 2}$ is equal to + +$$ +\frac {1}{g (p)} - 1 \leq e ^ {- \frac {1}{4 p}} - 1 \leq \frac {1}{4 p} e ^ {\frac {1}{4 p}} +$$ + +using Mean Value Theorem. This implies that the corresponding relative errors between $C$ and $\sqrt{2} e^{-p/2} p^{p/2}$ converge to zero as $p$ tends to infinity. + +The proof is complete. + +![](images/69e2ce8f77cda847b676b94f54501c1719a483bea926f65941cfa35643ec7427.jpg) + +Then we introduce the McDiarmid's inequality for vector-valued functions. We firstly consider real-valued functions, which follows from the standard tail-bound of McDiarmid's inequality and Proposition 2.5.2 in [49]. + +Lemma 10 (McDiarmid's Inequality for real-valued functions). Let $Z_{i}, \ldots, Z_{n}$ be independent random variables, and $f: \mathcal{Z}^{n} \mapsto \mathbb{R}$ such that the following inequality holds for any $z_{i}, \ldots, z_{i-1}, z_{i+1}, \ldots, z_{n}$ + +$$ +\sup _ {z _ {i}, z _ {i} ^ {\prime}} | f (z _ {1}, \ldots , z _ {i - 1}, z _ {i}, z _ {i + 1}, \ldots , z _ {n}) - f (z _ {1}, \ldots , z _ {i - 1}, z _ {i} ^ {\prime}, z _ {i + 1}, \ldots , z _ {n}) | \leq \beta , +$$ + +Then for any $p > 1$ we have + +$$ +\left\| f \left(Z _ {1}, \dots , Z _ {n}\right) - \mathbb {E} f \left(Z _ {1}, \dots , Z _ {n}\right) \right\| _ {p} \leq \sqrt {2 p n} \beta . +$$ + +To derive the McDiarmid's inequality for vector-valued functions, we need the expected distance between $\mathbf{f}(Z_1,\ldots ,Z_n)$ and its expectation. + +Lemma 11 ([38]). Let $Z_{i}, \ldots, Z_{n}$ be independent random variables, and $\mathbf{f}: \mathcal{Z}^{n} \mapsto \mathcal{H}$ is a function into a Hilbert space $\mathcal{H}$ such that the following inequality holds for any $z_{i}, \ldots, z_{i-1}, z_{i+1}, \ldots, z_{n}$ + +$$ +\sup _ {z _ {i}, z _ {i} ^ {\prime}} \| \mathbf {f} (z _ {1}, \ldots , z _ {i - 1}, z _ {i}, z _ {i + 1}, \ldots , z _ {n}) - \mathbf {f} (z _ {1}, \ldots , z _ {i - 1}, z _ {i} ^ {\prime}, z _ {i + 1}, \ldots , z _ {n}) \| \leq \beta , +$$ + +Then we have + +$$ +\mathbb {E} \left[ \left\| \mathbf {f} \left(Z _ {1}, \dots , Z _ {n}\right) - \mathbb {E} \mathbf {f} \left(Z _ {1}, \dots , Z _ {n}\right) \right\| \right] \leq \sqrt {n} \beta . +$$ + +Now, we can easily derive the $p$ -norm McDiarmid's inequality for vector-valued functions which refines from [10] with better constants. + +Lemma 12 (McDiarmid's inequality for vector-valued functions). Let $Z_{i}, \ldots, Z_{n}$ be independent random variables, and $\mathbf{f}: \mathcal{Z}^n \times \mathcal{A} \mapsto \mathcal{H}$ is a function into a Hilbert space $\mathcal{H}$ such that the following inequality holds for any $z_{i}, \ldots, z_{i-1}, z_{i+1}, \ldots, z_{n}$ + +$$ +\sup _ {z _ {i}, z _ {i} ^ {\prime}} \mathbb {E} _ {A} \| \mathbf {f} \left(z _ {1}, \dots , z _ {i - 1}, z _ {i}, z _ {i + 1}, \dots , z _ {n}\right) - \mathbf {f} \left(z _ {1}, \dots , z _ {i - 1}, z _ {i} ^ {\prime}, z _ {i + 1}, \dots , z _ {n}\right) \| \leq \beta , \tag {13} +$$ + +Then for any $p > 1$ we have + +$$ +\left\| \left\| \mathbb {E} _ {A} \mathbf {f} \left(Z _ {1}, \dots , Z _ {n}\right) - \mathbb {E} \mathbf {f} \left(Z _ {1}, \dots , Z _ {n}\right) \right\| \right\| _ {p} \leq (\sqrt {2 p} + 1) \sqrt {n} \beta . +$$ + +Proof of Lemma 12. Define a real-valued function $h: \mathcal{Z}^n \mapsto \mathbb{R}$ as + +$$ +h \left(z _ {1}, \dots , z _ {n}\right) = \| \mathbb {E} _ {A} \mathbf {f} \left(z _ {1}, \dots , z _ {n}\right) - \mathbb {E} [ \mathbf {f} \left(Z _ {1}, \dots , Z _ {n}\right) ] \|. +$$ + +We notice that this function satisfies the increment condition. For any $i$ and $z_{1},\ldots ,z_{i - 1},z_{i + 1},\ldots ,z_{n}$ we have + +$$ +\begin{array}{l} \sup _ {z _ {i}, z _ {i} ^ {\prime}} | h (z _ {1}, \dots , z _ {i - 1}, z _ {i}, z _ {i + 1}, \dots , z _ {n}) - h (z _ {1}, \dots , z _ {i - 1}, z _ {i} ^ {\prime}, z _ {i + 1}, \dots , z _ {n}) | \\ = \sup _ {z _ {i}, z _ {i} ^ {\prime}} | | \mathbb {E} _ {A} \mathbf {f} (z _ {1}, \dots , z _ {n}) - \mathbb {E} [ \mathbf {f} (Z _ {1}, \dots , Z _ {n}) ] | | - | | \mathbb {E} _ {A} \mathbf {f} (z _ {1}, \dots , z _ {i - 1}, z _ {i} ^ {\prime}, z _ {i + 1}, \dots , z _ {n}) - \mathbb {E} [ \mathbf {f} (Z _ {1}, \dots , Z _ {n}) ] | | | \\ \leq \sup _ {z _ {i}, z _ {i} ^ {\prime}} \mathbb {E} _ {A} | | \mathbf {f} (z _ {1}, \ldots , z _ {n}) - \mathbf {f} (z _ {1}, \ldots , z _ {i - 1}, z _ {i} ^ {\prime}, z _ {i + 1}, \ldots , z _ {n}) | | \leq \beta . \\ \end{array} +$$ + +Therefore, we can apply Lemma 10 to the real-valued function $h$ and derive the following inequality + +$$ +\left\| h \left(Z _ {1}, \dots , Z _ {n}\right) - \mathbb {E} \left[ h \left(Z _ {1}, \dots , Z _ {n}\right) \right] \right\| _ {p} \leq \sqrt {2 p n} \beta . +$$ + +According to Lemma 11, we know the following inequality $\mathbb{E}[h(Z_1,\dots,Z_n)]\leq \sqrt{n}\beta$ . Combining the above two inequalities together and we can derive the following inequality + +$$ +\begin{array}{l} \| \left\| \mathbb {E} _ {A} \mathbf {f} (Z _ {1}, \dots , Z _ {n}) - \mathbb {E} \mathbf {f} (Z _ {1}, \dots , Z _ {n}) \right\| \| _ {p} \\ \leq \| h (Z _ {1}, \dots , Z _ {n}) - \mathbb {E} [ h (Z _ {1}, \dots , Z _ {n}) ] \| _ {p} + \| \mathbb {E} [ h (Z _ {1}, \dots , Z _ {n}) ] \| _ {p} \\ \leq (\sqrt {2 p} + 1) \sqrt {n} \beta . \\ \end{array} +$$ + +The proof is complete. + +Proof of Lemma 7. For $\mathbf{g}(Z_1,\ldots ,Z_n)$ and $D\subset [n]$ , we write $\| \|\mathbf{g}\|\|_{p}(Z_{D}) = (\mathbb{E}[\| f\|^{p}Z_{D}])^{\frac{1}{p}}$ . Without loss of generality, we suppose that $n = 2^k$ . Otherwise, we can add extra functions equal to zero, increasing the number of therms by at most two times. + +Consider a sequence of partitions $\mathcal{P}_0, \ldots, \mathcal{P}_k$ with $\mathcal{P}_0 = \{\{i\} : i \in [n]\}$ , $\mathcal{P}_k$ with $\mathcal{P}_n = \{[n]\}$ , and to get $\mathcal{P}_l$ from $\mathcal{P}_{l+1}$ we split each subset in $\mathcal{P}_{l+1}$ into two equal parts. We have + +$$ +\mathcal {P} _ {0} = \{\{1 \}, \ldots , \{2 ^ {k} \} \}, \quad \mathcal {P} _ {1} = \{\{1, 2 \}, \{3, 4 \}, \ldots , \{2 ^ {k} - 1, 2 ^ {k} \} \}, \quad \mathcal {P} _ {k} = \{\{1, \ldots , 2 ^ {k} \} \}. +$$ + +We have $|\mathcal{P}_l| = 2^{k - l}$ and $|P| = 2^l$ for each $P \in \mathcal{P}_l$ . For each $i \in [n]$ and $l = 0, \dots, k$ , denote by $P^l(i) \in \mathcal{P}_l$ the only set from $\mathcal{P}_l$ that contains $i$ . In particular, $P^0(i) = \{i\}$ and $P^K(i) = [n]$ . + +For each $i\in [n]$ and every $l = 0,\dots ,k$ consider the random variables + +$$ +\mathbf {g} _ {i} ^ {l} = \mathbf {g} _ {i} ^ {l} \left(Z _ {i}, Z _ {[ n ] \backslash P ^ {l} (i)}\right) = \mathbb {E} \left[ \mathbf {g} _ {i} \mid Z _ {i}, Z _ {[ n ] \backslash P ^ {l} (i)} \right], +$$ + +i.e. conditioned on $z_{i}$ and all the variables that are not in the same set as $Z_{i}$ in the partition $\mathcal{P}_l$ . In particular, $\mathbf{g}_i^0 = \mathbf{g}_i$ and $\mathbf{g}_i^k = \mathbb{E}[\mathbf{g}_i|Z_i]$ . We can write a telescopic sum for each $i \in [n]$ , + +$$ +\mathbf {g} _ {i} - \mathbb {E} [ \mathbf {g} _ {i} | Z _ {i} ] = \sum_ {l = 1} ^ {k - 1} \mathbf {g} _ {i} ^ {l} - \mathbf {g} _ {i} ^ {l + 1}. +$$ + +Then, by the triangle inequality + +$$ +\left\| \left\| \sum_ {i = 1} ^ {n} \mathbf {g} _ {i} \right\| \right\| _ {p} \leq \left\| \left\| \sum_ {i = 1} ^ {n} \mathbb {E} [ \mathbf {g} _ {i} | Z _ {i} ] \right\| \right\| _ {p} + \sum_ {l = 0} ^ {k - 1} \left\| \left\| \sum_ {i = 1} ^ {n} \mathbf {g} _ {i} ^ {l} - \mathbf {g} _ {i} ^ {l + 1} \right\| \right\| _ {p}. \tag {14} +$$ + +To bound the first term, since $\| \mathbb{E}[\mathbf{g}_i|Z_i]\| \leq G$ , we can check that the vector-valued function $\mathbf{f}(Z_1,\ldots,Z_n) = \sum_{i=1}^n \mathbb{E}[\mathbf{g}_i|Z_i]$ satisfies (13) with $\beta = 2G$ , and $\mathbb{E}[\mathbb{E}[\mathbf{g}_i|Z_i]] = 0$ , applying Lemma 12 with $\beta = 2G$ , we have + +$$ +\left\| \left\| \sum_ {i = 1} ^ {n} \mathbb {E} [ \mathbf {g} _ {i} | Z _ {i} ] \right\| \right\| _ {p} \leq 2 (\sqrt {2 p} + 1) \sqrt {n} G. \tag {15} +$$ + +Then we start to bound the second term of the right hand side of (14). Observe that + +$$ +\mathbf {g} _ {i} ^ {l + 1} \left(Z _ {i}, Z _ {[ n ] \backslash P ^ {l + 1} (i)}\right) = \mathbb {E} \left[ \mathbf {g} _ {i} ^ {l} \left(Z _ {i}, Z _ {[ n ] \backslash P ^ {l} (i)}\right) \mid Z _ {i}, Z _ {[ n ] \backslash P ^ {l + 1} (i)} \right], +$$ + +where the expectation is taken with respect to the variables $Z_{j}, j \in P^{l + 1}(i) \backslash P^{l}(i)$ . Changing any $Z_{j}$ would change $\mathbf{g}_i^l$ by $\beta$ . Therefore, we apply Lemma 12 with $\mathbf{f} = \mathbf{g}_i^l$ where there are $2^l$ random variables and obtain a uniform bound + +$$ +\left\| \left\| \mathbf {g} _ {i} ^ {l} - \mathbf {g} _ {i} ^ {l + 1} \right\| \right\| _ {p} \left(Z _ {i}, Z _ {[ n ] \backslash P ^ {l + 1} (i)}\right) \leq (\sqrt {2 p} + 1) \sqrt {2 ^ {l}} \beta , \quad \forall p \geq 2, +$$ + +Taking integration over $(Z_{i}, Z_{[n] \setminus P^{l+1}(i)})$ , we have $\left\| \left\| \mathbf{g}_{i}^{l} - \mathbf{g}_{i}^{l+1} \right\| \right\|_{p} \leq (\sqrt{2p} + 1) \sqrt{2^{l}} \beta$ as well. + +Next, we turn to the sum $\sum_{i\in P^l}\mathbf{g}_i^l -\mathbf{g}_i^{l + 1}$ for any $P^l\in \mathcal{P}_l$ . Since $\mathbf{g}_i^l -\mathbf{g}_i^{l + 1}$ for $i\in P^l$ depends only on $Z_{i},Z_{[n]\backslash P^{l}}$ , the terms are independent and zero mean conditioned on $Z_{[n]\backslash P^l}$ . Applying Lemma 8, we have for any $p\geq 2$ + +$$ +\left\| \left\| \sum_ {i \in P ^ {l}} \mathbf {g} _ {i} ^ {l} - \mathbf {g} _ {i} ^ {l + 1} \right\| \right\| _ {p} ^ {p} (Z _ {[ n ] \backslash P ^ {l}}) \leq \left(2 \cdot 2 ^ {\frac {1}{2 p}} \sqrt {\frac {2 ^ {l} p}{e}}\right) ^ {p} \frac {1}{2 ^ {l}} \sum_ {i \in P ^ {l}} \left\| \left\| \mathbf {g} _ {i} ^ {l} - \mathbf {g} _ {i} ^ {l + 1} \right\| \right\| _ {p} ^ {p} (Z _ {[ n ] \backslash P ^ {l}}). +$$ + +Integrating with respect to $(Z_{[n]\setminus P^l)}$ and using $\left\| \left\| \mathbf{g}_i^l -\mathbf{g}_i^{l + 1}\right\| \right\| _p\leq (\sqrt{2p} +1)\sqrt{2^l}\beta$ , we have + +$$ +\begin{array}{l} \left\| \left\| \sum_ {i \in P ^ {l}} \mathbf {g} _ {i} ^ {l} - \mathbf {g} _ {i} ^ {l + 1} \right\| \right\| _ {p} \leq \left(2 \cdot 2 ^ {\frac {1}{2 p}} \sqrt {\frac {2 ^ {l} p}{e}}\right) \frac {1}{2 ^ {l}} \times 2 ^ {l} (\sqrt {2 p} + 1) \sqrt {2 ^ {l}} \beta \\ = 2 ^ {1 + \frac {1}{2 p}} \left(\sqrt {\frac {p}{e}}\right) (\sqrt {2 p} + 1) 2 ^ {l} \beta . \\ \end{array} +$$ + +Then using triangle inequality over all sets $P^l \in \mathcal{P}_l$ , we have + +$$ +\begin{array}{l} \left\| \left\| \sum_ {i \in [ n ]} \mathbf {g} _ {i} ^ {l} - \mathbf {g} _ {i} ^ {l + 1} \right\| \right\| _ {p} \leq \sum_ {P ^ {l} \in \mathcal {P} _ {l}} \left\| \left\| \sum_ {i \in P ^ {l}} \mathbf {g} _ {i} ^ {l} - \mathbf {g} _ {i} ^ {l + 1} \right\| \right\| _ {p} \\ \leq 2 ^ {k - l} \times 2 ^ {1 + \frac {1}{2 p}} \left(\sqrt {\frac {p}{e}}\right) (\sqrt {2 p} + 1) 2 ^ {l} \beta \\ \leq 2 ^ {1 + \frac {1}{2 p}} \left(\sqrt {\frac {p}{e}}\right) (\sqrt {2 p} + 1) 2 ^ {k} \beta . \\ \end{array} +$$ + +Recall that $2^k \leq n$ due to the possible extension of the sample. Then we have + +$$ +\sum_ {l = 0} ^ {k - 1} \left\| \left\| \sum_ {i = 1} ^ {n} \mathbf {g} _ {i} ^ {l} - \mathbf {g} _ {i} ^ {i + 1} \right\| \right\| _ {p} \leq 2 ^ {2 + \frac {1}{2 p}} \left(\sqrt {\frac {p}{e}}\right) (\sqrt {2 p} + 1) n \beta \lceil \log_ {2} n \rceil . +$$ + +We can plug the above bound together with (15) into (14), to derive the following inequality + +$$ +\left\| \left\| \sum_ {i = 1} ^ {n} \mathbf {g} _ {i} \right\| \right\| _ {p} \leq 2 (\sqrt {2 p} + 1) \sqrt {n} G + 2 ^ {2 + \frac {1}{2 p}} \left(\sqrt {\frac {p}{e}}\right) (\sqrt {2 p} + 1) n \beta \lceil \log_ {2} n \rceil . +$$ + +The proof is completed. + +![](images/4465e219d4c75602aabc814dabea6cbb8d93805d06f9ee64136b62abd15f9a64.jpg) + +# C.2 Proofs of Subsection 3.1 + +Proof of Theorem 1. Let $S = \{z_1, \ldots, z_n\}$ be a set of independent random variables each taking values in $\mathcal{Z}$ and $S' = \{z_1', \ldots, z_n'\}$ be its independent copy. For any $i \in [n]$ , define $S^{(i)} = \{z_i, \ldots, z_{i-1}, z_i', z_{i+1}, \ldots, z_n\}$ be a dataset replacing the $i$ -th sample in $S$ with another i.i.d. sample $z_i'$ . Then we can firstly write the following decomposition + +$$ +\begin{array}{l} n \nabla F (A (S)) - n \nabla F _ {S} (A (S)) \\ = \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z} \left[ \nabla f (A (S); Z) ] - \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \nabla f (A (S ^ {(i)}), Z) \right] \right] \\ + \sum_ {i = 1} ^ {n} \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \mathbb {E} _ {Z} \left[ \nabla f (A (S ^ {(i)}), Z) \right] - \nabla f (A (S ^ {(i)}), z _ {i}) \right] \\ + \sum_ {i = 1} ^ {n} \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \nabla f (A (S ^ {(i)}), z _ {i}) \right] - \sum_ {i = 1} ^ {n} \nabla f (A (S), z _ {i}). \\ \end{array} +$$ + +We denote that $\mathbf{g}_i(S) = \mathbb{E}_{z_i'}\left[\mathbb{E}_Z\left[\nabla f(A(S^{(i)}),Z)\right] - \nabla f(A(S^{(i)}),z_i)\right]$ , thus we have + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| n \nabla F (A (S)) - n \nabla F _ {S} (A (S)) \| _ {2} \right] \\ = \mathbb {E} _ {A} \left\| \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z} \left[ \nabla f (A (S); Z) \right] - \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \nabla f (A (S ^ {(i)}), Z) \right] \right] \\ + \sum_ {i = 1} ^ {n} \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \mathbb {E} _ {Z} \left[ \nabla f (A (S ^ {(i)}), Z) \right] - \nabla f (A (S ^ {(i)}), z _ {i}) \right] \tag {16} \\ + \sum_ {i = 1} ^ {n} \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \nabla f (A (S ^ {(i)}), z _ {i}) \right] - \sum_ {i = 1} ^ {n} \nabla f (A (S), z _ {i}) \Bigg \| _ {2} \\ \leq 2 n \beta + \mathbb {E} _ {A} \left[ \left\| \sum_ {i = 1} ^ {n} \mathbf {g} _ {i} (S) \right\| _ {2} \right], \\ \end{array} +$$ + +where the inequality holds from the definition of uniform stability in gradients. + +According to our assumptions, we get $\| \mathbf{g}_i(S)\| _2\leq 2M$ and + +$$ +\begin{array}{l} \mathbb {E} _ {z _ {i}} \left[ \mathbf {g} _ {i} (S) \right] = \mathbb {E} _ {z _ {i}} \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \mathbb {E} _ {Z} \left[ \nabla f (A (S ^ {(i)}); Z) \right] - \nabla f (A (S ^ {(i)}); z _ {i}) \right] \\ = \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \mathbb {E} _ {Z} \left[ \nabla f (A (S ^ {(i)}); Z) \right] - \mathbb {E} _ {z _ {i}} \left[ \nabla f (A (S ^ {(i)}); z _ {i}) \right] \right] = 0, \\ \end{array} +$$ + +where this equality holds from the fact that $z_{i}$ and $Z$ follow from the same distribution. For any $i \in [n]$ , any $j \neq i$ and any $z_{j}^{\prime \prime}$ , we have + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \left\| \mathbf {g} _ {i} \left(z _ {1}, \dots , z _ {j - 1}, z _ {j}, z _ {j + 1}, \dots , z _ {n}\right) - \mathbf {g} _ {i} \left(z _ {1}, \dots , z _ {j - 1}, z _ {j} ^ {\prime \prime}, z _ {j + 1}, \dots , z _ {n}\right) \right\| _ {2} \right] \\ \leq \mathbb {E} _ {A} \left[ \left\| \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \mathbb {E} _ {Z} \left[ \nabla f (A (S ^ {(i)}); Z) \right] - \nabla f (A (S ^ {(i)}); z _ {i}) \right] - \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \mathbb {E} _ {Z} \left[ \nabla f (A (S _ {j} ^ {(i)}); Z) \right] - \nabla f (A (S _ {j} ^ {(i)}; z _ {i}) \right] \right\| _ {2} \right] \\ \leq \mathbb {E} _ {A} \left[ \left\| \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \mathbb {E} _ {Z} \left[ \nabla f (A (S ^ {(i)}); Z) - \nabla f (A (S _ {j} ^ {(i)}); Z) \right] \right] \right\| _ {2} \right] \\ + \mathbb {E} _ {A} \left[ \left\| \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \mathbb {E} _ {Z} \left[ \nabla f (A (S ^ {(i)}); Z) \right] - \nabla f (A (S _ {j} ^ {(i)}); z _ {i}) \right] \right\| _ {2} \right] \\ \end{array} +$$ + +$$ +\leq 2 \beta , +$$ + +where $S^{(i)} = \{z_i, \dots, z_{i-1}, z_i', z_{i+1}, \dots, z_n\}$ . Since the noise introduced by the algorithm's inherent randomness $A$ is typically assumed to be independent of the dataset $S$ , we have verified that three conditions in Lemma 7 are satisfied for $\mathbf{g}_i(S)$ . We have the following result for any $p > 2$ + +$$ +\left\| \left\| \sum_ {i = 1} ^ {n} \mathbf {g} _ {i} (S) \right\| \right\| _ {p} \leq 4 (\sqrt {2 p} + 1) \sqrt {n} M + 8 \times 2 ^ {\frac {1}{4}} \left(\sqrt {\frac {p}{e}}\right) (\sqrt {2 p} + 1) n \beta \lceil \log_ {2} n \rceil . +$$ + +According to Lemma 4 for any $\delta \in (0,1)$ , with probability at least $1 - \delta$ , we have + +$$ +\begin{array}{l} \left\| \sum_ {i = 1} ^ {n} \mathbf {g} _ {i} (S) \right\| _ {2} \\ \leq 4 \sqrt {n} M + 8 \times 2 ^ {\frac {3}{4}} \sqrt {e} n \beta \left\lceil \log_ {2} n \right\rceil \log \left(e / \delta\right) + \left(4 e \sqrt {2 n} M + 8 \times 2 ^ {\frac {1}{4}} \sqrt {e} n \beta \left\lceil \log_ {2} n \right\rceil\right) \sqrt {\log e / \delta}. \\ \end{array} +$$ + +We can finally combine the above inequality and (16) to derive that, for any $\delta \in (0,1)$ , with probability at least $1 - \delta$ , we have + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} \right] \\ \leq 2 \beta + \frac {4 M \left(1 + e \sqrt {2 \log (e / \delta)}\right)}{\sqrt {n}} + 8 \times 2 ^ {\frac {1}{4}} (\sqrt {2} + 1) \sqrt {e} \beta \lceil \log_ {2} n \rceil \log (e / \delta). \\ \end{array} +$$ + +The proof is completed. + +![](images/67e72af85d6ac8a7aacc417f2dfb4b82ecfadec440a67c16ffd67b64e0d16684.jpg) + +Proof of Theorem 2. We can firstly write the following decomposition + +$$ +\begin{array}{l} n \nabla F (A (S)) - n \nabla F _ {S} (A (S)) \\ = \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z} \left[ \nabla f (A (S); Z) - \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \nabla f (A (S ^ {(i)}), Z) \right] \right] \\ + \sum_ {i = 1} ^ {n} \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \mathbb {E} _ {Z} \left[ \nabla f (A (S ^ {(i)}), Z) \right] - \nabla f (A (S ^ {(i)}), z _ {i}) \right] \\ + \sum_ {i = 1} ^ {n} \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \nabla f (A (S ^ {(i)}), z _ {i}) \right] - \sum_ {i = 1} ^ {n} \nabla f (A (S), z _ {i}). \\ \end{array} +$$ + +We denote that $\mathbf{h}_i(S) = \mathbb{E}_{z_i'}\left[\mathbb{E}_Z\left[\nabla f(A(S^{(i)}),Z)\right] - \nabla f(A(S^{(i)}),z_i)\right]$ , we have + +$$ +\begin{array}{l} n \nabla F (A (S)) - n \nabla F _ {S} (A (S)) - \sum_ {i = 1} ^ {n} \mathbf {h} _ {i} (S) \\ = \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z} \left[ \nabla f (A (S); Z) - \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \nabla f (A (S ^ {(i)}), Z) \right] \right] \\ + \sum_ {i = 1} ^ {n} \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \nabla f (A (S ^ {(i)}), z _ {i}) \right] - \sum_ {i = 1} ^ {n} \nabla f (A (S), z _ {i}), \\ \end{array} +$$ + +which implies that + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left\| n \nabla F (A (S)) - n \nabla F _ {S} (A (S)) - \sum_ {i = 1} ^ {n} \mathbf {h} _ {i} (S) \right\| _ {2} \\ = \mathbb {E} _ {A} \left\| \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z} \left[ \nabla f (A (S); Z) - \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \nabla f (A (S ^ {(i)}), Z) \right] \right] \right. \tag {17} \\ + \sum_ {i = 1} ^ {n} \mathbb {E} _ {z _ {i} ^ {\prime}} \left[ \nabla f (A (S ^ {(i)}), z _ {i}) \right] - \sum_ {i = 1} ^ {n} \nabla f (A (S), z _ {i}) \| _ {2} \\ \leq 2 n \beta , \\ \end{array} +$$ + +where the inequality holds from the definition of uniform stability in gradients. + +Then, for any $i = 1, \ldots, n$ , we define $\mathbf{q}_i(S) = \mathbf{h}_i(S) - \mathbb{E}_{S \setminus \{z_i\}, A}[\mathbf{h}_i(S)]$ . It is easy to verify that $\mathbb{E}_{S \setminus \{z_i\}}[\mathbf{q}_i(S)] = \mathbf{0}$ and $\mathbb{E}_{z_i}[\mathbf{q}_i(S)] = \mathbb{E}_{z_i}[\mathbf{h}_i(S)] - \mathbb{E}_{z_i} \mathbb{E}_{S \setminus \{z_i\}, A}[\mathbf{h}_i(S)] = \mathbf{0} - \mathbf{0} = \mathbf{0}$ . Also, for any $j \in [n]$ with $j \neq i$ and $z_j'' \in \mathcal{Z}$ , we have the following inequality + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \mathbf {q} _ {i} (S) - \mathbf {q} _ {i} \left(z _ {1}, \dots , z _ {j - 1}, z _ {j} ^ {\prime \prime}, z _ {j + 1}, \dots , z _ {n}\right) \| _ {2} \right] \\ \leq \mathbb {E} _ {A} \left[ \| \mathbf {h} _ {i} (S) - \mathbf {h} _ {i} \left(z _ {1}, \dots , z _ {j - 1}, z _ {j} ^ {\prime \prime}, z _ {j + 1}, \dots , z _ {n}\right) \| _ {2} \right] \\ + \mathbb {E} _ {A} \left[ \| \mathbb {E} _ {S \setminus \{z _ {i} \}} [ \mathbf {h} _ {i} (S) ] - \mathbb {E} _ {S \setminus \{z _ {i} \}} [ \mathbf {h} _ {i} (z _ {1}, \dots , z _ {j - 1}, z _ {j} ^ {\prime \prime}, z _ {j + 1}, \dots , z _ {n}) ] \| _ {2} \right]. \\ \end{array} +$$ + +For the first term $\mathbb{E}[\| \mathbf{h}_i(S) - \mathbf{h}_i(z_1,\dots ,z_{j - 1},z_j^{\prime \prime},z_{j + 1},\dots ,z_n)\| _2]$ , it can be bounded by $2\beta$ according to the definition of uniform stability. Similar result holds for the second term $\mathbb{E}[\| \mathbb{E}_{S\setminus \{z_i\}}[\mathbf{h}_i(S)]-$ $\mathbb{E}_{S\setminus \{z_i\}}[\mathbf{h}_i(z_1,\dots ,z_{j - 1},z_j^{\prime \prime},z_{j + 1},\dots ,z_n)]\| _2]$ according to the uniform stability. By a combination of the above analysis, we get $\mathbb{E}_A[\| \mathbf{q}_i(S) - \mathbf{q}_i(z_1,\dots ,z_{j - 1},z_j^{\prime \prime},z_{j + 1},\dots ,z_n)\| _2]\leq 4\beta$ + +Thus, we have verified that three conditions in Lemma 7 are satisfied for $\mathbf{q}_i(S)$ . We have the following result for any $p \geq 2$ + +$$ +\left\| \left\| \sum_ {i = 1} ^ {n} \mathbf {q} _ {i} (S) \right\| \right\| _ {p} \leq 2 ^ {4 + \frac {1}{4}} \left(\sqrt {\frac {p}{e}}\right) (\sqrt {2 p} + 1) n \beta \lceil \log_ {2} n \rceil . +$$ + +According to Lemma 4 for any $\delta \in (0,1)$ , with probability at least $1 - \delta /3$ , we have + +$$ +\left\| \sum_ {i = 1} ^ {n} \mathbf {q} _ {i} (S) \right\| _ {2} \leq 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} \beta \lceil \log_ {2} n \rceil \log (3 e / \delta) + 1 6 \times 2 ^ {\frac {1}{4}} \sqrt {e} \beta \lceil \log_ {2} n \rceil \sqrt {\log 3 e / \delta}. \tag {18} +$$ + +Furthermore, we can derive that + +$$ +\begin{array}{l} n \nabla F (A (S)) - n \nabla F _ {S} (A (S)) - \sum_ {i = 1} ^ {n} \mathbf {h} _ {i} (S) + \sum_ {i = 1} ^ {n} \mathbf {q} _ {i} (S) \\ = n \nabla F (A (S)) - n \nabla F _ {S} (A (S)) - \sum_ {i = 1} ^ {n} \mathbb {E} _ {S \backslash \{z _ {i} \}, A} [ \mathbf {h} _ {i} (S) ] \\ = n \nabla F (A (S)) - n \nabla F _ {S} (A (S)) - n \mathbb {E} _ {S ^ {\prime}, A} [ \nabla F (A (S ^ {\prime})) ] + n \mathbb {E} _ {S, A} [ \nabla F (A (S)) ]. \\ \end{array} +$$ + +Due to the i.i.d. property between $S$ and $S'$ , we know that $\mathbb{E}_{S'}[\nabla F(A(S'))] = \mathbb{E}_S[\nabla F(A(S))]$ . Thus, combined above equality, (17) and (18), we have + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| n \nabla F (A (S)) - n \nabla F _ {S} (A (S)) - n \mathbb {E} _ {S, A} [ \nabla F (A (S)) ] + n \mathbb {E} _ {S ^ {\prime}, A} [ \nabla F _ {S} (A (S ^ {\prime})) ] \| _ {2} \right] \\ \leq \mathbb {E} _ {A} \left[ \left\| n \nabla F (A (S)) - n \nabla F _ {S} (A (S)) - \sum_ {i = 1} ^ {n} \mathbf {h} _ {i} (S) \right\| _ {2} \right] \\ + \mathbb {E} _ {A} \left[ \left\| \sum_ {i = 1} ^ {n} \mathbf {h} _ {i} (S) - n \mathbb {E} _ {S, A} [ \nabla F (A (S)) ] + n \mathbb {E} _ {S ^ {\prime}, A} F _ {S} [ A (S ^ {\prime}) ] \right\| _ {2} \right] \tag {19} \\ = \mathbb {E} _ {A} \left[ \left\| n \nabla F (A (S)) - n \nabla F _ {S} (A (S)) - \sum_ {i = 1} ^ {n} \mathbf {h} _ {i} (S) \right\| _ {2} \right] + \mathbb {E} _ {A} \left[ \left\| \sum_ {i = 1} ^ {n} \mathbf {q} _ {i} (S) \right\| _ {2} \right] \\ \leq 2 n \beta + 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} n \beta \lceil \log_ {2} n \rceil \log (3 e / \delta) + 1 6 \times 2 ^ {\frac {1}{4}} \sqrt {e} n \beta \lceil \log_ {2} n \rceil \sqrt {\log 3 e / \delta} \\ \leq 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} n \beta \lceil \log_ {2} n \rceil \log (3 e / \delta) + 3 2 \sqrt {e} n \beta \lceil \log_ {2} n \rceil \sqrt {\log 3 e / \delta}. \\ \end{array} +$$ + +This implies that for any $\delta \in (0,1)$ , with probability at least $1 - \delta / 3$ , we have + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} \right] \\ \leq \left\| \mathbb {E} _ {S ^ {\prime}, A} \left[ \nabla F _ {S} (A (S ^ {\prime})) \right] - \mathbb {E} _ {S, A} \left[ \nabla F (A (S)) \right] \right\| _ {2} \tag {20} \\ + 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} \beta [ \log_ {2} n ] \log (3 e / \delta) + 3 2 \sqrt {e} \beta [ \log_ {2} n ] \sqrt {\log 3 e / \delta}. \\ \end{array} +$$ + +Next, we need to bound the term $\| \mathbb{E}_{S',A}[\nabla F_S(A(S'))] - \mathbb{E}_{S,A}[\nabla F(A(S))]\|_2$ . There holds that $\| \mathbb{E}_{S,A}\mathbb{E}_{S'}[\nabla F_S(A(S'))]\|_2 = \| \mathbb{E}_{S,A}[\nabla F(A(S))]\|_2$ . Then, by the Bernstein inequality in Lemma 5, we obtain the following inequality with probability at least $1 - \delta /3$ + +$$ +\left\| \mathbb {E} _ {S ^ {\prime}, A} \left[ \nabla F _ {S} \left(A \left(S ^ {\prime}\right)\right) \right] - \mathbb {E} _ {S, A} \left[ \nabla F (A (S)) \right] \right\| _ {2} \leq \sqrt {\frac {2 \mathbb {E} _ {z _ {i}} \left[ \| \mathbb {E} _ {S ^ {\prime} , A} \nabla f (A \left(S ^ {\prime}\right) ; z _ {i}) \| _ {2} ^ {2} \right] \log (6 / \delta)}{n}} + \frac {M \log (6 / \delta)}{n}. \tag {21} +$$ + +Then using Jensen's inequality, we have + +$$ +\begin{array}{l} \mathbb {E} _ {z _ {i}} \left[ \left\| \mathbb {E} _ {S ^ {\prime}, A} \nabla f (A (S ^ {\prime}); z _ {i}) \right\| _ {2} ^ {2} \right] \leq \mathbb {E} _ {z _ {i}} \mathbb {E} _ {S ^ {\prime}, A} \| \nabla f (A (S ^ {\prime}); z _ {i}) \| _ {2} ^ {2} \tag {22} \\ = \mathbb {E} _ {Z} \mathbb {E} _ {S ^ {\prime}, A} \| \nabla f (A (S ^ {\prime}); Z) \| _ {2} ^ {2} = \mathbb {E} _ {Z} \mathbb {E} _ {S, A} \| \nabla f (A (S); Z) \| _ {2} ^ {2}. \\ \end{array} +$$ + +Combining (20), (21) with (22), we finally obtain that with probability at least $1 - 2\delta /3$ + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} \right] \\ \leq \sqrt {\frac {2 \mathbb {E} _ {Z} \mathbb {E} _ {S , A} \| \nabla f (A (S) ; Z) \| _ {2} ^ {2} \log (6 / \delta)}{n}} + \frac {M \log (6 / \delta)}{n} \tag {23} \\ + 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} \beta [ \log_ {2} n ] \log (3 e / \delta) + 3 2 \sqrt {e} \beta [ \log_ {2} n ] \sqrt {\log 3 e / \delta}. \\ \end{array} +$$ + +Next, since $S = \{z_{i},\ldots ,z_{n}\}$ , we define $p = p(z_{1},\dots ,z_{n}) = \mathbb{E}_{Z,A}[\| \nabla f(A(S);Z)\|_{2}^{2}]$ and $p_i = p_i(z_1,\ldots ,z_n) = \sup_{z_i\in \mathcal{Z}}p(z_i,\ldots ,z_n)$ . So there holds $p_i\geq p$ for any $i = 1,\dots ,n$ and any + +$\{z_1,\ldots ,z_n\} \in \mathcal{Z}^n$ . Also, there holds that + +$$ +\begin{array}{l} \sum_ {i = 1} ^ {n} (p _ {i} - p) ^ {2} \\ = \sum_ {i = 1} ^ {n} \left(\sup _ {z _ {i} \in \mathcal {Z}} \mathbb {E} _ {Z, A} [ \| \nabla f (A (S); Z) \| _ {2} ^ {2} ] - \mathbb {E} _ {Z, A} [ \| \nabla f (A (S); Z) \| _ {2} ^ {2} ]\right) ^ {2} \\ \leq \sum_ {i = 1} ^ {n} \left(\mathbb {E} _ {Z, A} \left[ \sup _ {z _ {i} \in \mathcal {Z}} \| \nabla f (A (S); Z) \| _ {2} ^ {2} - \| \nabla f (A (S); Z) \| _ {2} ^ {2} \right]\right) ^ {2} \\ = \sum_ {i = 1} ^ {n} \left(\mathbb {E} _ {Z, A} \left[ \left(\sup _ {z _ {i} \in \mathcal {Z}} \| \nabla f (A (S); Z) \| _ {2} - \| \nabla f (A (S); Z) \| _ {2}\right) \left(\sup _ {z _ {i} \in \mathcal {Z}} \| \nabla f (A (S); Z) \| _ {2} + \| \nabla f (A (S); Z) \| _ {2}\right) \right]\right) ^ {2} \\ \leq \sum_ {i = 1} ^ {n} \beta^ {2} \left(\mathbb {E} _ {Z, A} \left[ \| \nabla f (A (S); Z) \| _ {2} + \sup _ {z _ {i} \in \mathcal {Z}} \| \nabla f (A (S); Z) \| _ {2} \right]\right) ^ {2} \\ \leq n \beta^ {2} \left(2 \mathbb {E} _ {Z, A} [ \| \nabla f (A (S); Z) \| _ {2} + \beta ]\right) ^ {2} \\ \leq 8 n \beta^ {2} p + 2 n \beta^ {4}, \tag {24} \\ \end{array} +$$ + +where the first inequality follows from the Jensen's inequality. The second and third inequalities follow from the definition of uniform stability in gradients. The last inequality holds from that $(a + b)^2 \leq 2a^2 + 2b^2$ . + +From (24), we know that $p$ is $(8n\beta^2, 2n\beta^4)$ weakly self-bounded. Thus, by Lemma 6, we obtain that with probability at least $1 - \delta / 3$ , + +$$ +\begin{array}{l} \mathbb {E} _ {Z} \mathbb {E} _ {S, A} [ \| \nabla f (A (S); Z) \| _ {2} ^ {2} ] - \mathbb {E} _ {Z, A} [ \| \nabla f (A (S); Z) \| _ {2} ^ {2} ] \\ \leq \sqrt {\left(1 6 n \beta^ {2} \mathbb {E} _ {S} \mathbb {E} _ {Z} \left[ \| \nabla f (A (S) ; Z) \| _ {2} ^ {2} \right] + 4 n \beta^ {4}\right) \log \left(3 / \delta\right)} \\ = \sqrt {\left(\mathbb {E} _ {S} \mathbb {E} _ {Z , A} [ \| \nabla f (A (S) ; Z) \| _ {2} ^ {2} ] + \frac {1}{4} \beta^ {2}\right) 1 6 n \beta^ {2} \log (3 / \delta)} \\ \leq \frac {1}{2} (\mathbb {E} _ {S} \mathbb {E} _ {Z, A} [ \| \nabla f (A (S); Z) \| _ {2} ^ {2} ] + \frac {1}{4} \beta^ {2}) + 8 n \beta^ {2} \log (3 / \delta), \\ \end{array} +$$ + +where the last inequality follows from that $\sqrt{ab} \leq \frac{a + b}{2}$ for all $a, b > 0$ . Thus, we have + +$$ +\mathbb {E} _ {Z} \mathbb {E} _ {S, A} [ \| \nabla f (A (S); Z) \| _ {2} ^ {2} ] \leq 2 \mathbb {E} _ {Z, A} [ \| \nabla f (A (S); Z) \| _ {2} ^ {2} ] + \frac {1}{4} \beta^ {2} + 1 6 n \beta^ {2} \log (3 / \delta). \tag {25} +$$ + +Substituting (25) into (23), we finally obtain that with probability at least $1 - \delta$ + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} \right] \\ \leq \sqrt {\frac {2 \left(2 \mathbb {E} _ {Z , A} [ \| \nabla f (A (S) ; Z) \| _ {2} ^ {2} ] + \frac {1}{4} \beta^ {2} + 1 6 n \beta^ {2} \log (3 / \delta)\right) \log (6 / \delta)}{n}} + \frac {M \log (6 / \delta)}{n} \tag {26} \\ + 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} \beta \lceil \log_ {2} n \rceil \log (3 e / \delta) + 3 2 \sqrt {e} \beta \lceil \log_ {2} n \rceil \sqrt {\log 3 e / \delta}. \\ \end{array} +$$ + +According to inequality $\sqrt{a + b} = \sqrt{a} +\sqrt{b}$ for any $a,b > 0$ , with probability at least $1 - \delta$ , we have + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} \right] \\ \leq \sqrt {\frac {4 \mathbb {E} _ {Z , A} [ \| \nabla f (A (S) ; Z) \| _ {2} ^ {2} ] \log (6 / \delta)}{n}} + \sqrt {\frac {\left(\frac {1}{2} \beta^ {2} + 3 2 n \beta^ {2} \log (3 / \delta)\right) \log (6 / \delta)}{n}} + \frac {M \log (6 / \delta)}{n} \\ + 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} \beta [ \log_ {2} n ] \log (3 e / \delta) + 3 2 \sqrt {e} \beta [ \log_ {2} n ] \sqrt {\log 3 e / \delta}. \\ \end{array} +$$ + +The proof is complete. + +Proof of Proposition 1. According to the proof in Theorem 2, we have the following inequality with probability at least $1 - \delta$ + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} \right] \\ \leq \sqrt {\frac {2 \left(2 \mathbb {E} _ {Z , A} [ \| \nabla f (A (S) ; Z) \| _ {2} ^ {2} ] + \frac {1}{4} \beta^ {2} + 1 6 n \beta^ {2} \log (3 / \delta)\right) \log (6 / \delta)}{n}} \tag {27} \\ + \frac {M \log (6 / \delta)}{n} + 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} \beta \lceil \log_ {2} n \rceil \log (3 e / \delta) + 3 2 \sqrt {e} \beta \lceil \log_ {2} n \rceil \sqrt {\log 3 e / \delta}. \\ \end{array} +$$ + +Since SGC implies that $\mathbb{E}_Z[\| \nabla f(\mathbf{w};Z)\| _2^2 ]\leq \lambda \| \nabla F(\mathbf{w})\| _2^2$ , according to inequalities $\sqrt{ab}\leq \eta a + \frac{1}{\eta} b$ and $\sqrt{a + b}\leq \sqrt{a} +\sqrt{b}$ for any $a,b,\eta >0$ , we have the following inequality with probability at least $1 - \delta$ + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} \right] \\ \leq \sqrt {\frac {2 \left(2 \rho \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) \| _ {2} ^ {2} \right] + \frac {1}{4} \beta^ {2} + 1 6 n \beta^ {2} \log (3 / \delta)\right) \log (6 / \delta)}{n}} \\ + \frac {M \log (6 / \delta)}{n} + 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} \beta \lceil \log_ {2} n \rceil \log (3 e / \delta) + 3 2 \sqrt {e} \beta \lceil \log_ {2} n \rceil \sqrt {\log 3 e / \delta} \\ \end{array} +$$ + +$$ +\begin{array}{l} \leq \sqrt {\frac {\left(\frac {1}{2} \beta^ {2} + 3 2 n \beta^ {2} \log (3 / \delta)\right) \log (6 / \delta)}{n}} + \frac {\eta}{1 + \eta} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) \| _ {2} \right] + \frac {1 + \eta}{\eta} \frac {4 \lambda M \log (6 / \delta)}{n} \\ + \frac {M \log (6 / \delta)}{n} + 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} \beta \lceil \log_ {2} n \rceil \log (3 e / \delta) + 3 2 \sqrt {e} \beta \lceil \log_ {2} n \rceil \sqrt {\log 3 e / \delta}. \\ \end{array} +$$ + +which implies that + +$$ +\mathbb {E} _ {A} \left[ \| \nabla F (A (S)) \| _ {2} \right] \leq (1 + \eta) \mathbb {E} _ {A} \left[ \| \nabla F _ {S} (A (S)) \| _ {2} \right] + C \frac {1 + \eta}{\eta} \left(\frac {\lambda M}{n} \log (6 / \delta) + \beta \log n \log (1 / \delta)\right). +$$ + +The proof is complete. + +![](images/fcccf42ec263ccba9c0efabbf440ee2947be3c8aad91259eded7e8c5c4c8fe71.jpg) + +Proof of Remark 4. According to the proof in Theorem 2, we have the following inequality that with probability at least $1 - \delta$ + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} \right] \\ \leq \sqrt {\frac {4 \mathbb {E} _ {Z , A} [ \| \nabla f (A (S) ; Z) \| _ {2} ^ {2} ] \log (6 / \delta)}{n}} + \sqrt {\frac {\left(\frac {1}{2} \beta^ {2} + 3 2 n \beta^ {2} \log (3 / \delta)\right) \log (6 / \delta)}{n}} + \frac {M \log (6 / \delta)}{n} \\ + 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} \beta \lceil \log_ {2} n \rceil \log (3 e / \delta) + 3 2 \sqrt {e} \beta \lceil \log_ {2} n \rceil \sqrt {\log 3 e / \delta}. \tag {28} \\ \end{array} +$$ + +Since $f(\mathbf{w})$ is $\gamma$ -smooth, we have + +$$ +\begin{array}{l} \mathbb {E} _ {Z, A} \left[ \| \nabla f (A (S); Z) \| _ {2} ^ {2} \right] \\ \leq \mathbb {E} _ {Z, A} [ \| \nabla f (A (S); Z) - \nabla f (\mathbf {w} ^ {*}; Z) \| _ {2} ^ {2} + \| \nabla f (\mathbf {w} ^ {*}; Z) \| _ {2} ^ {2} ] \tag {29} \\ \leq \gamma^ {2} \mathbb {E} _ {A} \left[ \| A (S) - \mathbf {w} ^ {*} \| _ {2} ^ {2} \right] + \mathbb {E} _ {Z} [ \| \nabla f (\mathbf {w} ^ {*}; Z) \| _ {2} ^ {2} ] \\ \end{array} +$$ + +Plugging (29) into (28), we have + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} \right] \\ \leq \sqrt {\frac {4 (\gamma^ {2} \mathbb {E} _ {A} [ \| A (S) - \mathbf {w} ^ {*} \| _ {2} ^ {2} ] + \mathbb {E} _ {Z} [ \| \nabla f (\mathbf {w} ^ {*} ; Z) \| _ {2} ^ {2} ]) \log (6 / \delta)}{n}} + \sqrt {\frac {\left(\frac {1}{2} \beta^ {2} + 3 2 n \beta^ {2} \log (3 / \delta)\right) \log (6 / \delta)}{n}} \\ + \frac {M \log (6 / \delta)}{n} + 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} \beta \lceil \log_ {2} n \rceil \log (3 e / \delta) + 3 2 \sqrt {e} \beta \lceil \log_ {2} n \rceil \sqrt {\log 3 e / \delta} \\ \leq 2 \gamma \sqrt {\frac {\mathbb {E} _ {A} \left[ \| A (S) - \mathbf {w} ^ {*} \| _ {2} ^ {2} \right] \log \left(6 / \delta\right)}{n}} + \sqrt {\frac {4 \mathbb {E} _ {Z} \left[ \| \nabla f (\mathbf {w} ^ {*} ; Z) \| _ {2} ^ {2} \right] \log \left(6 / \delta\right)}{n}} \\ + \sqrt {\frac {\left(\frac {1}{2} \beta^ {2} + 3 2 n \beta^ {2} \log (3 / \delta)\right) \log (6 / \delta)}{n}} + \frac {M \log (6 / \delta)}{n} \\ + 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} \beta \lceil \log_ {2} n \rceil \log (3 e / \delta) + 3 2 \sqrt {e} \beta \lceil \log_ {2} n \rceil \sqrt {\log 3 e / \delta}, \tag {30} \\ \end{array} +$$ + +where the second inequality holds because $\sqrt{a + b} \leq \sqrt{a} + \sqrt{b}$ for any $a, b > 0$ , which means that + +$$ +\mathbb {E} _ {A} [ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} ] +$$ + +$$ +\lesssim \beta \log n \log (1 / \delta) + \frac {\log (1 / \delta)}{n} + \sqrt {\frac {\mathbb {E} _ {Z} [ \nabla \| f (\mathbf {w} ^ {*} ; Z) \| _ {2} ^ {2} ] \log (1 / \delta)}{n}} + \sqrt {\frac {\mathbb {E} _ {A} [ \| A (S) - \mathbf {w} ^ {*} \| _ {2} ^ {2} ] \log (1 / \delta)}{n}}. +$$ + +The proof is complete. + +![](images/c999719d39b33cef13481b0cfb75c14a772a70568365b84afd7fccd59d5e0027.jpg) + +# C.3 Proofs of Subsection 3.2 + +Lemma 13. Assume for any $z$ , $f(\cdot, z)$ is $M$ -Lipschitz. If $A$ is $\beta$ -uniformly-stable in gradients, then for any $\delta \in (0, 1)$ , the following inequality holds with probability at least $1 - \delta$ + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} ^ {2} \right] \\ \leq \sqrt {\frac {4 \mathbb {E} _ {Z} \left[ \| \nabla f (A (S) ; Z) \| _ {2} ^ {2} \right] \log (6 / \delta)}{n}} + \sqrt {\frac {\left(\frac {1}{2} \beta^ {2} + 3 2 n \beta^ {2} \log (3 / \delta)\right) \log (6 / \delta)}{n}} \\ + \frac {M \log (6 / \delta)}{n} + 1 6 \times 2 ^ {\frac {3}{4}} \sqrt {e} \beta \left\lceil \log_ {2} (n) \right\rceil \log (3 e / \delta) + 3 2 \sqrt {e} \beta \left\lceil \log_ {2} (n) \right\rceil \sqrt {\log (3 e / \delta)}. \\ \end{array} +$$ + +Proof of Lemma 13. According to proof of Theorem 2, similar to (19), we have the following inequality. + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| n \nabla F (A (S)) - n \nabla F _ {S} (A (S)) - n \mathbb {E} _ {S, A} [ \nabla F (A (S)) ] + n \mathbb {E} _ {S ^ {\prime}, A} [ \nabla F _ {S} (A (S ^ {\prime})) ] \| _ {2} ^ {2} \right] \\ \leq 2 \mathbb {E} _ {A} \left[ \left\| n \nabla F (A (S)) - n \nabla F _ {S} (A (S)) - \sum_ {i = 1} ^ {n} \mathbf {h} _ {i} (S) \right\| _ {2} ^ {2} \right] \\ + 2 \mathbb {E} _ {A} \left[ \left\| \sum_ {i = 1} ^ {n} \mathbf {h} _ {i} (S) - n \mathbb {E} _ {S, A} [ \nabla F (A (S)) ] + n \mathbb {E} _ {S ^ {\prime}, A} F _ {S} [ A (S ^ {\prime}) ] \right\| _ {2} ^ {2} \right] \\ = 2 \mathbb {E} _ {A} \left[ \left\| n \nabla F (A (S)) - n \nabla F _ {S} (A (S)) - \sum_ {i = 1} ^ {n} \mathbf {h} _ {i} (S) \right\| _ {2} ^ {2} \right] + 2 \mathbb {E} _ {A} \left[ \left\| \sum_ {i = 1} ^ {n} \mathbf {q} _ {i} (S) \right\| _ {2} ^ {2} \right] \\ \leq 8 n ^ {2} \beta^ {2} + 1 0 2 4 \sqrt {2} e n ^ {2} \beta^ {2} \left(\lceil \log_ {2} n \rceil\right) ^ {2} \log^ {2} (3 e / \delta) + 5 1 2 \sqrt {2} e n ^ {2} \beta^ {2} \left(\lceil \log_ {2} n \rceil\right) ^ {2} \log (3 e / \delta) \\ \leq 8 n ^ {2} \beta^ {2} + 2 0 4 8 \sqrt {2} e n ^ {2} \beta^ {2} (\lceil \log_ {2} n \rceil) ^ {2} \log^ {2} (3 e / \delta), \\ \end{array} +$$ + +where the second inequality follows from the definition of uniform stability in gradients and Cauchy-Bunyakovsky-Schwarz inequality. + +This implies that for any $\delta \in (0,1)$ , with probability at least $1 - \delta / 3$ , we have + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} ^ {2} \right] \\ \leq \left\| \mathbb {E} _ {S ^ {\prime}, A} \left[ \nabla F _ {S} \left(A \left(S ^ {\prime}\right)\right)\right] - \mathbb {E} _ {S, A} \left[ \nabla F (A (S)) \right]\right\| _ {2} ^ {2} + 8 \beta^ {2} + 2 0 4 8 \sqrt {2} e \beta^ {2} \left(\left\lceil \log_ {2} n \right\rceil\right) ^ {2} \log^ {2} \left(3 e / \delta\right). \tag {31} \\ \end{array} +$$ + +According to (21) (22) and (25), using Cauchy-Bunyakovsky-Schwarz inequality, for any $\delta \in (0,1)$ with probability at least $1 - 2\delta /3$ , we have + +$$ +\begin{array}{l} \left\| \mathbb {E} _ {S ^ {\prime}, A} \left[ \nabla F _ {S} \left(A \left(S ^ {\prime}\right)\right) \right] - \mathbb {E} _ {S, A} \left[ \nabla F (A (S)) \right] \right\| _ {2} ^ {2} \\ \leq \frac {\left(8 \mathbb {E} _ {Z , A} [ \| \nabla f (A (S) ; Z) \| _ {2} ^ {2} ] + \beta^ {2} + 6 4 n \beta^ {2} \log (3 / \delta)\right) \log (6 / \delta)}{n} + \frac {2 M ^ {2} \log (6 / \delta)}{n ^ {2}}. \\ \end{array} +$$ + +Combining above inequality with (31), with probability at least $1 - \delta$ , we have + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} ^ {2} \right] \\ \leq \frac {\left(8 \mathbb {E} _ {Z , A} [ \| \nabla f (A (S) ; Z) \| _ {2} ^ {2} ] + \beta^ {2} + 6 4 n \beta^ {2} \log (3 / \delta)\right) \log (6 / \delta)}{n} + \frac {2 M ^ {2} \log (6 / \delta)}{n ^ {2}} \\ + 8 \beta^ {2} + 2 0 4 8 \sqrt {2} e \beta^ {2} \left(\left\lceil \log_ {2} n \right\rceil\right) ^ {2} \log^ {2} (3 e / \delta). \\ \end{array} +$$ + +The proof is complete. + +![](images/7f68b7e06234d847ab2532130c8fe150d5adeceff0696bf8aa257d44a2684a39.jpg) + +Proof of Theorem 3. Since $f(\mathbf{w})$ is $\gamma$ -smooth, we have + +$$ +\begin{array}{l} \mathbb {E} _ {Z, A} \left[ \| \nabla f (A (S); Z) \| _ {2} ^ {2} \right] \\ \leq \mathbb {E} _ {Z, A} [ \| \nabla f (A (S); Z) - \nabla f (\mathbf {w} ^ {*}; Z) \| _ {2} ^ {2} + \| \nabla f (\mathbf {w} ^ {*}; Z) \| _ {2} ^ {2} ] \tag {32} \\ \leq \gamma^ {2} \mathbb {E} _ {A} \left[ \| A (S) - \mathbf {w} ^ {*} \| _ {2} ^ {2} \right] + \mathbb {E} _ {Z} [ \| \nabla f (\mathbf {w} ^ {*}; Z) \| _ {2} ^ {2} ] \\ \end{array} +$$ + +Combining above inequality with Lemma 13, with probability least $1 - \delta$ , we have + +$$ +\mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} ^ {2} \right] +$$ + +$$ +\begin{array}{l} \leq \frac {8 \gamma^ {2} \mathbb {E} _ {A} \left[ \| A (S) - \mathbf {w} ^ {*} \| _ {2} ^ {2} \right] \log (6 / \delta)}{n} + \frac {8 \mathbb {E} _ {Z} \left[ \| \nabla f (\mathbf {w} ^ {*} ; Z) \| _ {2} ^ {2} \right] \log (6 / \delta)}{n} \\ + 6 5 \beta^ {2} \log (3 / \delta) \log (6 / \delta) + \frac {2 M ^ {2} \log (6 / \delta)}{n ^ {2}} + 8 \beta^ {2} + 2 0 4 8 \sqrt {2} e \beta^ {2} (\lceil \log_ {2} n \rceil) ^ {2} \log^ {2} (3 e / \delta). \tag {33} \\ \end{array} +$$ + +When $F(\mathbf{w})$ satisfies the PL condition and $\mathbf{w}^*$ is the projection of $A(S)$ onto the solution set $\arg \min_{\mathbf{w} \in \mathcal{W}} F(\mathbf{w})$ , there holds the following error bound property (refer to Theorem 2 in [19]) + +$$ +\| \nabla F (A (S)) \| _ {2} \geq \mu \| A (S) - \mathbf {w} ^ {*} \| _ {2}. +$$ + +Thus, we have + +$$ +\mu^ {2} \mathbb {E} _ {A} \left[ \| A (S) - \mathbf {w} ^ {*} \| _ {2} ^ {2} \right] \leq \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) \| _ {2} ^ {2} \right] +$$ + +$$ +\leq 2 \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} ^ {2} \right] + 2 \mathbb {E} _ {A} \left[ \| \nabla F _ {S} (A (S)) \| _ {2} ^ {2} \right] +$$ + +$$ +\begin{array}{l} \leq 2 \mathbb {E} _ {A} [ \| \nabla F _ {S} (A (S)) \| _ {2} ^ {2} ] + \frac {1 6 \gamma^ {2} \mathbb {E} _ {A} \left[ \| A (S) - \mathbf {w} ^ {*} \| _ {2} ^ {2} \right] \log (6 / \delta)}{n} + \frac {1 6 \mathbb {E} _ {Z} \left[ \| \nabla f (\mathbf {w} ^ {*} ; Z) \| _ {2} ^ {2} \right] \log (6 / \delta)}{n} \\ + 1 3 0 \beta^ {2} \log (3 / \delta) \log (6 / \delta) + \frac {4 M ^ {2} \log (6 / \delta)}{n ^ {2}} + 1 6 \beta^ {2} + 4 0 9 6 \sqrt {2} e \beta^ {2} (\lceil \log_ {2} n \rceil) ^ {2} \log^ {2} (3 e / \delta). \\ \end{array} +$$ + +When $n \geq \frac{32\gamma^2\log(6 / \delta)}{\mu^2}$ , we have $\frac{16\gamma^2\log(6 / \delta)}{n} \leq \frac{\mu^2}{2}$ , then we can derive that + +$$ +\begin{array}{l} \mu^ {2} \mathbb {E} _ {A} \left[ \| A (S) - \mathbf {w} ^ {*} \| _ {2} ^ {2} \right] \\ \leq 2 \mathbb {E} _ {A} \left[ \| \nabla F _ {S} (A (S)) \| _ {2} ^ {2} \right] + \frac {\mu^ {2}}{2} \mathbb {E} _ {A} \left[ \| A (S) - \mathbf {w} ^ {*} \| _ {2} ^ {2} \right] + \frac {1 6 \mathbb {E} _ {Z} \left[ \| \nabla f (\mathbf {w} ^ {*} ; Z) \| _ {2} ^ {2} \right] \log (6 / \delta)}{n} \\ + 1 3 0 \beta^ {2} \log (3 / \delta) \log (6 / \delta) + \frac {4 M ^ {2} \log (6 / \delta)}{n ^ {2}} + 1 6 \beta^ {2} + 4 0 9 6 \sqrt {2} e \beta^ {2} (\lceil \log_ {2} n \rceil) ^ {2} \log^ {2} (3 e / \delta). \\ \end{array} +$$ + +This implies that + +$$ +\begin{array}{l} \mathbb {E} _ {A} [ \| A (S) - \mathbf {w} ^ {*} \| _ {2} ^ {2} ] \\ \leq \frac {2}{\mu^ {2}} \left(2 \mathbb {E} _ {A} [ \| \nabla F _ {S} (A (S)) \| _ {2} ^ {2} ] + \frac {1 6 \mathbb {E} _ {Z} \left[ \| \nabla f (\mathbf {w} ^ {*} ; Z) \| _ {2} ^ {2} \right] \log (6 / \delta)}{n} \right. \\ + 1 3 0 \beta^ {2} \log (3 / \delta) \log (6 / \delta) + \frac {4 M ^ {2} \log (6 / \delta)}{n ^ {2}} + 1 6 \beta^ {2} + 4 0 9 6 \sqrt {2} e \beta^ {2} (\lceil \log_ {2} n \rceil) ^ {2} \log^ {2} (3 e / \delta)). \tag {34} \\ \end{array} +$$ + +Then, substituting (34) into (33), when $n \geq \frac{32\gamma^2 \log(6 / \delta)}{\mu^2}$ , with probability at least $1 - \delta$ + +$$ +\mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} ^ {2} \right] +$$ + +$$ +\begin{array}{l} \leq \mathbb {E} _ {A} \left[ \| \nabla F _ {S} (A (S)) \| _ {2} ^ {2} \right] + \frac {1 6 \mathbb {E} _ {Z} \left[ \| \nabla f (\mathbf {w} ^ {*} ; Z) \| _ {2} ^ {2} \right] \log (6 / \delta)}{n} + 1 3 0 \beta^ {2} \log (3 / \delta) \log (6 / \delta) \tag {35} \\ + \frac {4 M ^ {2} \log (6 / \delta)}{n ^ {2}} + 1 6 \beta^ {2} + 4 0 9 6 \sqrt {2} e \beta^ {2} (\lceil \log_ {2} n \rceil) ^ {2} \log^ {2} (3 e / \delta). \\ \end{array} +$$ + +Since $F$ satisfies the PL condition with $\mu$ , we have + +$$ +\mathbb {E} _ {A} [ F (A (S)) ] - F (\mathbf {w} ^ {*}) \leq \frac {\mathbb {E} _ {A} \left[ \| \nabla F (A (S)) \| _ {2} ^ {2} \right]}{2 \mu}, \quad \forall \mathbf {w} \in \mathcal {W}. \tag {36} +$$ + +So to bound $\mathbb{E}_A[F(A(S))] - F(\mathbf{w}^*)$ , we need to bound the term $\mathbb{E}_A\left[\| \nabla F(A(S))\|_2^2\right]$ . And there holds + +$$ +\mathbb {E} _ {A} \left[ \| \nabla F (A (S)) \| _ {2} ^ {2} \right] = 2 \mathbb {E} _ {A} \left[ \| \nabla F (A (S)) - \nabla F _ {S} (A (S)) \| _ {2} ^ {2} \right] + 2 \mathbb {E} _ {A} \left[ \| \nabla F _ {S} (A (S)) \| _ {2} ^ {2} \right]. \tag {37} +$$ + +On the other hand, when $f$ is nonenegative and $\gamma$ -smooth, from Lemma 4.1 of [43], we have + +$$ +\left\| \nabla f (\mathbf {w} ^ {*}; z) \right\| _ {2} ^ {2} \leq 4 \gamma f (\mathbf {w} ^ {*}; z), +$$ + +which implies that + +$$ +\mathbb {E} _ {Z} \left[ \| \nabla f \left(\mathbf {w} ^ {*}; Z\right) \| _ {2} ^ {2} \right] \leq 4 \gamma \mathbb {E} _ {Z} f \left(\mathbf {w} ^ {*}; Z\right) = 4 \gamma F \left(\mathbf {w} ^ {*}\right). \tag {38} +$$ + +Combing (35),(36), (37) and (38), using Cauchy-Bunyakovsky-Schwarz inequality, we can derive that + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ F (A (S)) \right] - F (\mathbf {w} ^ {*}) \\ \lesssim \frac {\mathbb {E} _ {A} \left[ \| \nabla F _ {S} (A (S)) \| _ {2} ^ {2} \right]}{\mu} + \frac {\gamma F (\mathbf {w} ^ {*}) \log (1 / \delta)}{\mu n} + \frac {M ^ {2} \log^ {2} (1 / \delta)}{\mu n ^ {2}} + \frac {\beta^ {2} \log^ {2} n \log^ {2} (1 / \delta)}{\mu}. \\ \end{array} +$$ + +The proof is complete. + +# D Proofs of ERM + +Proof of Lemma 1. Since $F_{S^{(i)}}(\mathbf{w}) = \frac{1}{n}\left(f(\mathbf{w};z_i') + \sum_{j\neq i}f(\mathbf{w},z_j)\right)$ , we have + +$$ +\begin{array}{l} F _ {S} \left(\hat {\mathbf {w}} ^ {*} (S ^ {(i)}\right)) - F _ {S} \left(\hat {\mathbf {w}} ^ {*} (S)\right) \\ = \frac {f (\hat {\mathbf {w}} ^ {*} (S ^ {(i)} ; z _ {i}) - f (\hat {\mathbf {w}} ^ {*} (S) ; z _ {i})}{n} + \frac {\sum_ {j \neq i} (f (\hat {\mathbf {w}} ^ {*} (S ^ {(i)} ; z _ {j}) - f (\hat {\mathbf {w}} ^ {*} (S) ; z _ {j}))}{n} \\ = \frac {f (\hat {\mathbf {w}} ^ {*} (S ^ {(i)}); z _ {i}) - f (\hat {\mathbf {w}} ^ {*} (S) ; z _ {i})}{n} + \frac {f (\hat {\mathbf {w}} ^ {*} (S) ; z _ {i} ^ {\prime}) - f (\hat {\mathbf {w}} ^ {*} (S ^ {(i)} ; z _ {i} ^ {\prime})}{n} \\ + \left(F _ {S ^ {(i)}} \left(\hat {\mathbf {w}} ^ {*} (S ^ {(i)})\right) - F _ {S ^ {(i)}} \left(\hat {\mathbf {w}} ^ {*} (S)\right)\right) \\ \leq \frac {f (\hat {\mathbf {w}} ^ {*} (S ^ {(i)}); z _ {i}) - f (\hat {\mathbf {w}} ^ {*} (S) ; z _ {i})}{n} + \frac {f (\hat {\mathbf {w}} ^ {*} (S) ; z _ {i} ^ {\prime}) - f (\hat {\mathbf {w}} ^ {*} (S ^ {(i)} ; z _ {i} ^ {\prime})}{n} \\ \leq \frac {2 M}{n} \left\| \hat {\mathbf {w}} ^ {*} (S ^ {(i)}) - \hat {\mathbf {w}} ^ {*} (S) \right\| _ {2}, \\ \end{array} +$$ + +where the first inequality follows from the fact that $\hat{\mathbf{w}}^{*}(S^{(i)})$ is the ERM of $F_{S^{(i)}}$ and the second inequality follows from the Lipschitz property. Furthermore, for $\hat{\mathbf{w}}^{*}(S^{(i)})$ , the PL condition and smoothness property of $F_{S}$ imply that its closest optima point of $F_{S}$ is $\hat{\mathbf{w}}^{*}(S)$ (the global minimizer of $F_{S}$ is unique [50]). Then, $F_{S}$ satisfies the quadratic growth property [19], which means that + +$$ +F _ {S} \left(\hat {\mathbf {w}} ^ {*} \left(S ^ {(i)}\right)\right) - F _ {S} \left(\hat {\mathbf {w}} ^ {*} (S)\right) \geq \frac {\mu}{2} \left\| \hat {\mathbf {w}} ^ {*} \left(S ^ {(i)}\right) - \hat {\mathbf {w}} ^ {*} (S) \right\| _ {2} ^ {2}. +$$ + +Then we get + +$$ +\frac {\mu}{2} \| \hat {\mathbf {w}} ^ {*} (S ^ {(i)}) - \hat {\mathbf {w}} ^ {*} (S) \| _ {2} ^ {2} \leq F _ {S} (\hat {\mathbf {w}} ^ {*} (S ^ {(i)})) - F _ {S} (\hat {\mathbf {w}} ^ {*} (S)) \leq \frac {2 M}{n} \| \hat {\mathbf {w}} ^ {*} (S ^ {(i)}) - \hat {\mathbf {w}} ^ {*} (S) \| _ {2}, +$$ + +which implies that $\| \hat{\mathbf{w}}^* (S^{(i)}) - \hat{\mathbf{w}}^* (S)\| _2\leq \frac{4M}{n\mu}$ . Combined with the smoothness property of $f$ we obtain that for any $S^{(i)}$ and $S$ + +$$ +\forall z \in \mathcal {Z}, \left\| \nabla f (\hat {\mathbf {w}} ^ {*} (S ^ {(i)}); z) - \nabla f (\hat {\mathbf {w}} ^ {*} (S); z) \right\| _ {2} ^ {2} \leq \frac {1 6 M ^ {2} \gamma^ {2}}{n ^ {2} \mu^ {2}}. +$$ + +The proof is complete. + +![](images/99bfce39c71327e0427bc00731d20d55db6fbe12c64b09ebbefa05eb3aee37b7.jpg) + +Proof of Theorem 4. From Lemma 1, we have $\| \nabla f(\hat{\mathbf{w}}^* (S);z) - \nabla f(\hat{\mathbf{w}}^* (S';z))\| _2\leq \frac{4M\gamma}{n\mu}$ . Since $\nabla F_{S}(\hat{\mathbf{w}}^{*}) = 0$ , we have $\| \nabla F_S(\hat{\mathbf{w}}^*)\| _2 = 0$ . According to Theorem 3, since ERM is a deterministic algorithm, we can derive that + +$$ +F (\hat {\mathbf {w}} ^ {*} (S)) - F (\mathbf {w} ^ {*}) = \mathbb {E} [ F (\hat {\mathbf {w}} ^ {*} (S)) ] - F (\mathbf {w} ^ {*}) \lesssim \frac {\gamma F (\mathbf {w} ^ {*}) \log (1 / \delta)}{\mu n} + \frac {M ^ {2} \gamma^ {2} \log^ {2} n \log^ {2} (1 / \delta)}{\mu^ {3} n ^ {2}}. +$$ + +![](images/b353ada393c0cb68cdbbb4d8f93597fe017a6d68efe127710807dc77b0b72a2d.jpg) + +# E Proofs of PGD + +Proof of Theorem 5. According to smoothness assumption and $\eta = 1 / \gamma$ , we can derive that + +$$ +\begin{array}{l} F _ {S} \left(\mathbf {w} _ {t + 1}\right) - F _ {S} \left(\mathbf {w} _ {t}\right) \\ \leq \left\langle \mathbf {w} _ {t + 1} - \mathbf {w} _ {t}, \nabla F _ {S} (\mathbf {w} _ {t}) \right\rangle + \frac {\gamma}{2} \| \mathbf {w} _ {t + 1} - \mathbf {w} _ {t} \| _ {2} ^ {2} \\ = - \eta_ {t} \| \nabla F _ {S} (\mathbf {w} _ {t}) \| _ {2} ^ {2} + \frac {\gamma}{2} \eta_ {t} ^ {2} \| \nabla F _ {S} (\mathbf {w} _ {t}) \| _ {2} ^ {2} \\ = \left(\frac {\gamma}{2} \eta_ {t} ^ {2} - \eta_ {t}\right) \| \nabla F _ {S} (\mathbf {w} _ {t}) \| _ {2} ^ {2} \\ \leq - \frac {1}{2} \eta_ {t} \| \nabla F _ {S} (\mathbf {w} _ {t}) \| _ {2} ^ {2}. \\ \end{array} +$$ + +According to above inequality and the assumptions that $F_{S}$ satisfies the PL condition with parameter $\mu$ , we can prove that + +$$ +F _ {S} \left(\mathbf {w} _ {t + 1}\right) - F _ {S} \left(\mathbf {w} _ {t}\right) \leq - \frac {1}{2} \eta_ {t} \| \nabla F _ {S} \left(\mathbf {w} _ {t}\right) \| _ {2} ^ {2} \leq - \mu \eta_ {t} \left(F _ {S} \left(\mathbf {w} _ {t}\right) - F _ {S} \left(\hat {\mathbf {w}} ^ {*}\right)\right), +$$ + +which implies that + +$$ +F _ {S} \left(\mathbf {w} _ {t + 1}\right) - F _ {S} \left(\hat {\mathbf {w}} ^ {*}\right) \leq (1 - \mu \eta_ {t}) \left(F _ {S} \left(\mathbf {w} _ {t}\right) - F _ {S} \left(\hat {\mathbf {w}} ^ {*}\right)\right). +$$ + +According to the property for $\gamma$ -smooth for $F_{S}$ and the PL condition property with parameter $\mu$ for $F_{S}$ , we have + +$$ +\frac {1}{2 \gamma} \| \nabla F _ {S} (\mathbf {w}) \| _ {2} ^ {2} \leq F _ {S} (\mathbf {w}) - F _ {S} (\hat {\mathbf {w}} ^ {*}) \leq \frac {1}{2 \mu} \| \nabla F _ {S} (\mathbf {w}) \| _ {2} ^ {2}, +$$ + +which means that $\frac{\mu}{\gamma} \leq 1$ . + +Then If $\eta_t = 1 / \gamma, 0 \leq 1 - \mu \eta_t < 1$ , taking over $T$ iterations, we get + +$$ +F _ {S} \left(\mathbf {w} _ {t + 1}\right) - F _ {S} \left(\hat {\mathbf {w}} ^ {*}\right) \leq \left(1 - \mu \eta_ {t}\right) ^ {T} \left(F _ {S} \left(\mathbf {w} _ {t}\right) - F _ {S} \left(\hat {\mathbf {w}} ^ {*}\right)\right). \tag {39} +$$ + +Combined (39), the smoothness of $F_{S}$ and the nonnegative property of $f$ , it can be derive that + +$$ +\left\| \nabla F _ {S} \left(\mathbf {w} _ {T + 1}\right)\right) \| _ {2} ^ {2} = O \left(\left(1 - \frac {\mu}{\gamma}\right) ^ {T}\right). +$$ + +On the other hand, from Lemma 2, we have $\| \nabla f(\mathbf{w}_{T + 1}(S);z) - \nabla f(\mathbf{w}_{T + 1}(S');z)\| _2^2\leq \frac{4M^2\gamma^2}{n^2\mu^2}$ . Since $\| \nabla F_S(\mathbf{w}_{T + 1})\| _2 = O\left((1 - \frac{\mu}{\gamma})^T\right)$ and PGD is a deterministic algorithm, according to Theorem 3, we can derive that + +$$ +\begin{array}{l} F \left(\mathbf {w} _ {T + 1}\right) - F \left(\mathbf {w} ^ {*}\right) = \mathbb {E} \left[ F \left(\mathbf {w} _ {T + 1}\right) \right] - F \left(\mathbf {w} ^ {*}\right) \\ \lesssim \left(1 - \frac {\mu}{\gamma}\right) ^ {2 T} + \frac {\gamma F \left(\mathbf {w} ^ {*}\right) \log \left(1 / \delta\right)}{\mu n} + \frac {M ^ {2} \gamma^ {2} \log^ {2} n \log^ {2} \left(1 / \delta\right)}{\mu^ {3} n ^ {2}}. \\ \end{array} +$$ + +Let $T \asymp \log n$ , we have + +$$ +F \left(\mathbf {w} _ {T + 1}\right) - F \left(\mathbf {w} ^ {*}\right) \lesssim \frac {\gamma F \left(\mathbf {w} ^ {*}\right) \log (1 / \delta)}{\mu n} + \frac {M ^ {2} \gamma^ {2} \log^ {2} n \log^ {2} (1 / \delta)}{\mu^ {3} n ^ {2}}. +$$ + +The proof is complete. + +![](images/bbbccdab55432b8900a6370dedfd3ce68152da583272f8d5c355887ee5cc7565.jpg) + +# F Proofs of SGD + +We first introduce the necessary lemma for the optimization error bound. + +Lemma 14 ([19]). Let $\{\mathbf{w}_t\}_t$ be the sequence produced by SGD with $\eta_t = \frac{2t + 1}{2\mu(t + 1)^2}$ . Suppose Assumption 1 hold. Assume for all $z$ , the function $\mathbf{w} \mapsto f(\mathbf{w}; z)$ is $M$ -Lipschitz and $\gamma$ -smooth and assume $F_S$ satisfies PL condition with parameter $\mu$ . There holds that + +$$ +\mathbb {E} _ {A} \left[ F _ {S} (\mathbf {w} _ {T + 1}) \right] - F _ {S} (\hat {\mathbf {w}} ^ {*}) \leq \frac {M ^ {2} \gamma}{2 T \mu^ {2}}. +$$ + +Proof of Lemma 3. We have known that $F_{S^{(i)}}(\mathbf{w}) = \frac{1}{n}\left(f(\mathbf{w};z_i') + \sum_{j\neq i}f(\mathbf{w};z_j)\right)$ . We denote $\hat{\mathbf{w}}^* (S^{(i)})$ be the ERM of $F_{S^{(i)}}(\mathbf{w})$ and $\hat{\mathbf{w}}_S^*$ be the ERM of $F_{S}(\mathbf{w})$ . From Lemma 1, we know that + +$$ +\forall z \in \mathcal {Z}, \quad \left\| \nabla f (\hat {\mathbf {w}} ^ {*} (S ^ {(i)}); z) - \nabla f (\hat {\mathbf {w}} ^ {*} (S); z) \right\| _ {2} ^ {2} \leq \frac {1 6 M ^ {2} \gamma^ {2}}{n ^ {2} \mu^ {2}}. +$$ + +Also, for $\mathbf{w}_t$ , the PL condition property of $F_{S}$ implies that its closest optima point of $F_{S}$ is $\hat{\mathbf{w}}^{*}(S)$ (the global minimizer of $F_{S}$ is unique [50]). Then, there holds that + +$$ +\frac {\mu}{2} \mathbb {E} _ {A} \left[ \| \mathbf {w} _ {t} - \hat {\mathbf {w}} ^ {*} (S) \| _ {2} ^ {2} \right] \leq \mathbb {E} _ {A} \left[ F _ {S} (\mathbf {w} _ {t}) \right] - F _ {S} (\hat {\mathbf {w}} ^ {*} (S)) \leq \frac {M ^ {2} \gamma}{2 t \mu^ {2}}. +$$ + +Thus we have $\mathbb{E}_A[\| \mathbf{w}_t - \hat{\mathbf{w}}^* (S)\| _2^2 ]\leq \frac{M^2\gamma}{t\mu^3}$ . A similar relation holds between $\hat{\mathbf{w}}^{*}(S^{(i)})$ and $\mathbf{w}_t^i$ . Combined with the Lipschitz property of $f$ we obtain that for $\forall z\in \mathcal{Z}$ , there holds that + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ \left\| \nabla f (\mathbf {w} _ {t}; z) - \nabla f (\mathbf {w} _ {t} ^ {i}, z) \right\| _ {2} ^ {2} \right] \\ \leq 3 \mathbb {E} _ {A} \left[ \| \nabla f (\mathbf {w} _ {t}; z) - \nabla f (\hat {\mathbf {w}} ^ {*} (S); z) \| _ {2} ^ {2} \right] + 3 \left\| \nabla f (\hat {\mathbf {w}} ^ {*} (S); z) - \nabla f (\hat {\mathbf {w}} ^ {*} (S ^ {(i)}); z) \right\| _ {2} ^ {2} \\ + 3 \mathbb {E} _ {A} \left[ \left\| \nabla f (\hat {\mathbf {w}} ^ {*} (S ^ {(i)}); z) - \nabla f (\mathbf {w} _ {t} ^ {i}, z) \right\| _ {2} ^ {2} \right] \\ \leq 3 \gamma^ {2} \mathbb {E} _ {A} \left[ \| \mathbf {w} _ {t} - \hat {\mathbf {w}} ^ {*} (S) \| _ {2} ^ {2} \right] + \frac {4 8 M ^ {2} \gamma^ {2}}{n ^ {2} \mu^ {2}} + 3 \gamma^ {2} \mathbb {E} _ {A} \left[ \| \hat {\mathbf {w}} ^ {*} (S ^ {(i)}) - \mathbf {w} _ {t} ^ {i} \| _ {2} ^ {2} \right] \\ \leq \frac {6 M ^ {2} \gamma^ {3}}{t \mu^ {3}} + \frac {4 8 M ^ {2} \gamma^ {2}}{n ^ {2} \mu^ {2}}. \\ \end{array} +$$ + +The proof is complete. + +![](images/13bb97a5a2f1823eb7ec67c71791b6fcf4a8bdf6e5d891cb44cf17bb91e92469.jpg) + +Proof of Theorem 6. From Lemma 3, we have + +$$ +\mathbb {E} _ {A} \left[ \left\| \nabla f \left(\mathbf {w} _ {T + 1}; z\right) - \nabla f \left(\mathbf {w} _ {T + 1} ^ {i}; z\right) \right\| _ {2} ^ {2} \right] \leq \frac {6 M ^ {2} \gamma^ {3}}{T \mu^ {3}} + \frac {4 8 M ^ {2} \gamma^ {2}}{n ^ {2} \mu^ {2}}. \tag {40} +$$ + +On the other hand, according to the smoothness property of $F_{S}$ and Lemma 14, we have + +$$ +\mathbb {E} _ {A} \left[ \| \nabla F _ {S} (\mathbf {w} _ {T + 1}) \| _ {2} ^ {2} \right] \leq \frac {M ^ {2} \gamma^ {2}}{T \mu^ {2}}. \tag {41} +$$ + +Using Theorem 3, with probability at least $1 - \delta$ , we have + +$$ +\begin{array}{l} \mathbb {E} _ {A} \left[ F \left(\mathbf {w} _ {T + 1}\right) \right] - F \left(\mathbf {w} ^ {*}\right) \\ \lesssim \frac {M ^ {2} \gamma^ {2}}{T \mu^ {3}} + \frac {\gamma F (\mathbf {w} ^ {*}) \log (1 / \delta)}{\mu n} + \frac {M ^ {2} \log^ {2} (1 / \delta)}{\mu n ^ {2}} + \left(\frac {\gamma}{T \mu} + \frac {1}{n ^ {2}}\right) \frac {M ^ {2} \gamma^ {2} \log^ {2} n \log^ {2} (1 / \delta)}{\mu^ {3}} \\ \lesssim \frac {\gamma F (\mathbf {w} ^ {*}) \log (1 / \delta)}{\mu n} + \left(\frac {\gamma}{T \mu} + \frac {1}{n ^ {2}}\right) \frac {M ^ {2} \gamma^ {2} \log^ {2} n \log^ {2} (1 / \delta)}{\mu^ {3}}. \\ \end{array} +$$ + +Furthermore, choosing $T \asymp n^2$ , we finally obtain that when $n \geq \frac{32\gamma^2 \log(6 / \delta)}{\mu^2}$ , with probability at least $1 - 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For fully observable systems, the state-of-the-art approaches leverage an unstable/stable subspace decomposition to achieve sample complexity that depends only on the number of unstable modes, independent of the dimension of the system state. However, it remains open whether such sample complexity can be achieved for partially observable systems because such systems do not admit a uniquely identifiable unstable subspace. In this paper, we propose LTS-P, a novel technique that leverages compressed singular value decomposition (SVD) on the "lifted" Hankel matrix to estimate the unstable subsystem up to an unknown transformation. Then, we design a stabilizing controller that integrates a robust stabilizing controller for the unstable mode and a small-gain-type assumption on the stable subspace. We show that LTS-P achieves state-of-the-art, dimension-free sample complexity that scales only with the number of unstable modes. This substantially reduces data requirements for stabilizing high-dimensional systems, particularly those dominated by stable dynamics. + +# 1 Introduction + +Learning-based control of unknown dynamical systems is of critical importance for many autonomous control systems [3, 30, 6, 24, 12]. Despite recent advances, many existing methods make strong assumptions such as open-loop stability, availability of an initial stabilizing controller, and fully observable systems [15, 41]. However, these assumptions may not hold in practice. Motivated by this gap, this paper studies the problem of stabilizing an unknown, partially-observable, unstable system without access to an initial stabilizing controller. In particular, we consider the following linear time-invariant (LTI) system: + +$$ +x _ {t + 1} = A x _ {t} + B u _ {t} \tag {1} +$$ + +$$ +y _ {t} = C x _ {t} + D u _ {t} + v _ {t}, +$$ + +where $x_{t} \in \mathbb{R}^{n}$ and $u_{t} \in \mathbb{R}^{d_{u}}$ , $y_{t} \in \mathbb{R}^{d_{y}}$ are the state, control input, and observed output at time step $t \in \{0, \dots, T - 1\}$ , respectively. The system also has additive observation noise $v_{t} \sim \mathcal{N}(0, \sigma_v^2 I)$ . While there are works studying system identification for partially observable LTI systems [15, 43, 41, 34, 51], they do not address the problem of stabilization, and many assume the system is open-loop stable [15, 41, 34]. Other adaptive control approaches can address the learn-to-stabilize problem for fully observable systems based on Lyapunov methods [35, 37], but there are few systematic approaches to construct a Lyapunov function in a way that optimizes sample complexities or transient performance during learning. + +In the special case of fully-observable LTI system $(C = I)$ , Chen and Hazan [9] reveals that the transient performance during the learn-to-stabilize process suffers from exponential blow-up, i.e. the + +system state can blow up exponentially in the state dimension. This phenomenon arises because stabilization requires accurate identification of the full system dynamics, which in turn necessitates at least $n$ samples along a single trajectory. During this identification phase, the system can remain unstable and grow exponentially. To relieve this problem, Hu et al. [19] proposed a framework that separates the unstable component and focuses on stabilizing these subsystems. This reduces the sample complexity to only grow with the number of unstable eigenvalues, rather than the full state dimension $n$ . This result was later extended to noisy setting in Zhang et al. [50]. To date, this dependence on the number of unstable eigenvalues remains the best sample complexity for the learn-to-stabilize problem in the fully observable case. + +In contrast, we address a more challenging setting of partially observable systems, and answer the following research question: Is it possible to stabilize a partially observable LTI system by only identifying an unstable component (to be properly defined) with a sample complexity independent of the overall state dimension? This question introduces two key technical challenges. First, partially observable systems typically require a dynamic controller, which renders the stable feedback controllers in the existing approach inapplicable [19]. Second, the construction in "unstable subspace" in Hu et al. [19] is not uniquely defined in the partially observable case. This is because the state and the $A, B, C$ matrices are not uniquely identifiable from the learner's perspective, as any similarity transformation of the matrices can yield the same input-output behavior. + +Contribution. In this paper, we answer the above question in the affirmative. Firstly, we propose a novel definition of "unstable component" to be a low-rank version of the transfer function of the original system that only retains the unstable eigenvalues. Based on this unstable component, we propose LTS-P, which leverages compressed singular value decomposition (SVD) on a "lifted" Hankel matrix to estimate the unstable component. Then, we design a robust stabilizing controller for the unstable component, and show that it stabilizes the full system under a small-gain-type assumption on the $H_{\infty}$ norm of the stable component. Importantly, our theoretical analysis shows that the sample complexity of the proposed approach only scales with the dimension of unstable component, i.e. the number of unstable eigenvalues, as opposed to the dimension of the full state space. We also conduct simulations to validate the effectiveness of LTS-P, showcasing its ability to efficiently stabilize partially observable LTI systems with reduced samples (Figures 1 and 2). + +Moreover, the technical innovations underlying our approach are of independent interest. We show that by using compressed singular value decomposition (SVD) on a properly defined lifted Hankel matrix, we can estimate the unstable component of the system. While this is conceptually related to classical model reduction techniques [13, Chapter 4.6], our setting departs significantly: unlike prior work for stable dynamics and bounded Hankel matrices [13], we address systems with unstable modes, where the associated Hankel matrices may grow unboundedly over time. This distinction renders standard identification and reduction techniques inapplicable [15, 41, 34]. Interestingly, the $H_{\infty}$ norm condition on the stable component, derived from the small gain theorem, is a necessary and sufficient condition for stabilization. This characterization reveals the exact subspace of the system that must be estimated for stabilization to be possible. As a result, our analysis not only supports the optimality of LTS-P in terms of sample complexity but also informs the fundamental limit of stabilizability. + +Related Work. Our work is closely related to learn-to-stabilize on multiple trajectories and learn-to-control with known stabilizing controllers. In addition, we will also briefly cover existing literature in learn-to-stabilize on a single trajectory, adaptive control, and system identification. + +Learn-to-stabilize on multiple trajectories. There are also works that do not assume open-loop stability and learn the full system dynamics before designing a stabilizing controller, typically incurring a sample complexity of $\widetilde{\Theta} (\mathrm{poly}(n))$ [12, 45, 51]. Recently, a model-free approach via the policy gradient method offers a novel perspective with the same sample complexity [36]. While these techniques are developed for fully observable systems, the proposed algorithm, LTS-P, tackles a significantly more challenging setting of partially observable, unstable systems. As detailed above, this setting introduces fundamental technical challenges that require novel algorithmic and analytical tools beyond those used in fully observable systems. Moreover, compared with those works, the sample complexity of the proposed algorithm does not depend on $n$ and only scales with the number of unstable modes. + +Learn to control with known stabilizing controller. Extensive research has been conducted on stabilizing LTI systems under stochastic noise [5, 22, 25, 30]. One branch of research uses the + +model-free approach to learn the optimal controller [16, 21, 30, 46, 49]. Those algorithms typically require a known stabilizing controller as an initialization policy for the learning process. Another line of research utilizes model-based approaches, which require an initial stabilizing controller to learn the system dynamics before designing the controllers [10, 32, 38, 52]. On the other hand, we focus on learn-to-stabilize. Our method can be used as the initial policy in these methods to remove their requirement for initial stabilizing controllers. + +Learn-to-stabilize on a single trajectory. Learning to stabilize a linear system in an infinite time horizon is a classic problem in control [7, 26, 27]. Early works achieved the regret bounds of $2^{O(n)}O(\sqrt{T})$ , which rely on assumptions of observability and strictly stable transition matrices [1, 20]. + +Subsequent work improved the regret to $2^{\tilde{O}(n)} + \tilde{O}(\mathrm{poly}(n)\sqrt{T})$ [9, 28]. Recently, Hu et al. [19] proposed an algorithm that requires $\tilde{O}(k)$ samples, where $k$ is the number of unstable modes. While these techniques are developed for fully observable systems using a single trajectory, we consider a different problem of learning to stabilize a partially observable system using multiple trajectories. + +Adaptive control. Adaptive control is a well-established methodology for controlling systems with uncertain or time-varying parameters [8, 35, 37]. Existing literature has established techniques for stabilizing unknown systems with asymptotic stability [9, 14, 29, 44, 45]. Other works use past trajectory to estimate the system dynamics and then design the controller [4, 11, 31]. While these works often assume stability, fully observable systems and are based on Lyapunov methods, we focus on partially observable, unstable systems and build our algorithm based on compressed SVD on the lifted Hankel matrix. + +System identification. Existing literature has developed a variety of techniques to estimate system parameters [33, 40, 43, 42, 47]. Hankel matrix are also used in techniques such as Eigensystem Realization Algorithm (ERA), subspace identification, etc [23, 39]. Our work utilizes a similar approach to partially determine the system parameters before constructing the stabilizing controller. While these works focus on identifying the system dynamics, we close the loop and establish state-of-the-art and optimal sample complexity guarantees for the stabilization problem. + +# 2 Problem Statement + +Notations. We use $\|\cdot\|$ to denote the $L^2$ -norm for vectors and the spectral norm for matrices. We use $M^*$ to represent the conjugate transpose of $M$ . We use $\sigma_{\min}(\cdot)$ and $\sigma_{\max}(\cdot)$ to denote the smallest and largest singular value of a matrix, and $\kappa(\cdot)$ to denote the condition number of a matrix. We use the standard big $O(\cdot), \Omega(\cdot), \Theta(\cdot)$ notation to highlight dependence on a certain parameter, hiding all other parameters. We use $f \lesssim g$ , $f \gtrsim g$ , $f \asymp g$ to mean $f = O(g)$ , $f = \Omega(g)$ , $f = \Theta(g)$ respectively while only hiding numeric constants. We provide an indexing of notations at Appendix I. + +For simplicity, we primarily deal with the system where $D = 0$ . For the case where $D \neq 0$ , we can easily estimate $D$ in the process and subtract $Du_{t}$ to obtain a new observation measure not involving control input. We briefly introduce the method for estimating $D$ and how to apply the proposed algorithm in the case when $D \neq 0$ in Appendix F. + +Learn-to-stabilize. As the unknown system as defined in (1) can be unstable, the goal of the learn-to-stabilize problem is to return a controller that stabilizes the system using data collected from interacting with the system on $M$ rollouts. More specifically, in each rollout, the learner can determine $u_{t}$ and observe $y_{t}$ for a rollout of length $T$ starting from $x_0$ , which we assume $x_0 = 0$ for simplicity of proof. + +Goal. The sample complexity of stabilization is the number of samples, $MT$ , needed for the learner to return a stabilizing controller. Standard system identification and certainty equivalence controller design need at least $\Theta (n)$ samples for stabilization, as $\Theta (n)$ is the number of samples needed to learn the full dynamical system. In this paper, our goal is to study whether it is possible to stabilize the system with sample complexity independent from $n$ . + +# 2.1 Background on $H_{\infty}$ control + +In this section, we briefly introduce the background of $H$ -infinity control. First, we define the open loop transfer function of system (1) from $u_{t}$ to $y_{t}$ to be + +$$ +F ^ {\text {f u l l}} (z) = C (z I - A) ^ {- 1} B, \tag {2} +$$ + +which reflects the cumulative output of the system in the infinite time horizon. Next, we introduce the $\mathcal{H}_{\infty}$ space on transfer functions in the $z$ -domain. + +Definition 2.1 ( $\mathcal{H}_{\infty}$ -space). Let $\mathcal{H}_{\infty}$ denote the Banach space of matrix-valued functions that are analytic and bounded outside of the unit sphere. Let $\mathcal{R}\mathcal{H}_{\infty}$ denote the real and rational subspace of $\mathcal{H}_{\infty}$ . The $\mathcal{H}_{\infty}$ -norm is defined as + +$$ +\| f \| _ {\mathcal {H} _ {\infty}} := \sup _ {z \in \mathbb {C}, | z | \geq 1} \sigma_ {\max } \left(f (z)\right) = \sup _ {z \in \mathbb {C}, | z | = 1} \sigma_ {\max } \left(f (z)\right), +$$ + +where the second equality is a simple application of the Maximum modulus principle. We also denote $\mathbb{C}_{\geq 1} = \{z\in \mathbb{C}:|z|\geq 1\}$ be the complement of the unit disk in the complex domain. For any transfer function $G$ , we say it is internally stable if $G\in \mathcal{R}\mathcal{H}_{\infty}$ . + +The $H_{\infty}$ norm of a transfer function is crucial in robust control, as it represents the amount of modeling error the system can tolerate without losing stability, due to the small gain theorem [53]. Abundant research has been done in $H_{\infty}$ control design to minimize the $H_{\infty}$ norm of transfer functions [53]. In this work, $H_{\infty}$ control plays an important role as we treat the stable component (to be defined later) of the system as a modeling error and show that the control we design can stabilize despite the modeling error. + +# 3 Algorithm Idea + +In this paper, we assume the matrix $A$ does not have marginally stable eigenvalues, and the eigenvalues are ranked in decreasing order of magnitude, i.e. $|\lambda_1| \geq |\lambda_2| \geq \dots \geq |\lambda_k| > 1 > |\lambda_{k+1}| \geq \dots \geq |\lambda_n|$ . The high-level idea of the paper is to first decompose the system dynamics into the unstable and stable components (Section 3.1), estimate the unstable component via a low-rank approximation of the lifted Hankel matrix (Section 3.2), and design a robust stabilizing for the unstable component that stabilizes the whole system (Section 3.3). + +# 3.1 Decomposition of Dynamics + +Given the eigenvalues, we have the following decomposition for the system dynamics matrix: + +$$ +A = \underbrace {\left[ Q _ {1} Q _ {2} \right]} _ {:= Q} \left[ \begin{array}{c c} N _ {1} & 0 \\ 0 & N _ {2} \end{array} \right] \underbrace {\left[ \begin{array}{l} R _ {1} \\ R _ {2} \end{array} \right]} _ {:= R}, \tag {3} +$$ + +where $R = Q^{-1}$ , and the columns of $Q_{1} \in \mathbb{R}^{n \times k}$ are an orthonormal basis for the invariant subspace of the unstable eigenvalues $\lambda_{1}, \ldots, \lambda_{k}$ , with $N_{1}$ inheriting eigenvalues $\lambda_{1}, \ldots, \lambda_{k}$ from $A$ . Similarly, columns of $Q_{2} \in \mathbb{R}^{n \times (n - k)}$ form an orthonormal basis for the invariant subspace of the unstable eigenvalues $\lambda_{k + 1}, \ldots, \lambda_{n}$ and $N_{2}$ inherit all the stable eigenvalues $\lambda_{k + 1}, \ldots, \lambda_{n}$ . + +Given the decomposition of the matrix $A$ , our key idea is to only estimate the unstable component of the dynamics, which we define below. Consider the transfer function of the original system: + +$$ +\begin{array}{l} F ^ {\mathrm {f u l l}} (z) = C (z I - A) ^ {- 1} B = C \left(z I - [ Q _ {1} Q _ {2} ] \left[ \begin{array}{c c} N _ {1} & 0 \\ 0 & N _ {2} \end{array} \right] \left[ \begin{array}{c} R _ {1} \\ R _ {2} \end{array} \right]\right) ^ {- 1} B \\ = C \left(\left[ Q _ {1} Q _ {2} \right] \left[ \begin{array}{c c} z I - N _ {1} & 0 \\ 0 & z I - N _ {2} \end{array} \right] \left[ \begin{array}{c} R _ {1} \\ R _ {2} \end{array} \right]\right) ^ {- 1} B \\ = \underbrace {C Q _ {1} \left(z I - N _ {1}\right) ^ {- 1} R _ {1} B} _ {:= F (z)} + \underbrace {C Q _ {2} \left(z I - N _ {2}\right) ^ {- 1} R _ {2} B} _ {:= \Delta (z)} = F (z) + \Delta (z). \tag {4} \\ \end{array} +$$ + +Therefore, the original system is an additive decomposition into the unstable transfer function $F(z)$ , which we refer to as the unstable component, and the stable transfer function $\Delta(z)$ , which we refer to as the stable component. + +# 3.2 Approximate low-rank factorization of the lifted Hankel matrix + +In this section, we define a "lifted" Hankel matrix, show it admits a rank $k$ approximation, based on which the unstable component can be estimated. + +If each rollout has length $T$ , we can decompose $T \coloneqq m + p + q + 2$ and estimate the following "lifted" Hankel matrix where the $(i,j)$ -th block is $[H]_{ij} = CA^{m + i + j - 2}B$ where $i = 1,\dots ,p, j = 1,\dots ,q$ . In other words, + +$$ +H = \left[ \begin{array}{c c c c} C A ^ {m} B & C A ^ {m + 1} B & \dots & C A ^ {m + q - 1} B \\ C A ^ {m + 1} B & C A ^ {m + 2} B & \dots & C A ^ {m + q} B \\ \dots & \dots & \dots & \dots \\ C A ^ {m + p - 1} B & C A ^ {m + p} B & \dots & C A ^ {m + p + q - 2} B \end{array} \right], \tag {5} +$$ + +for some $m, p, q$ that we will select later. We call this Hankel matrix "lifted" as it starts with $CA^m B$ . This "lifting" is essential to our approach, as raising $A$ to the power of $m$ can separate the stable and unstable components and facilitate better estimation of the unstable component, which will become clear later on. + +Define + +$$ +\mathcal {O} = \left[ \begin{array}{c} C A ^ {m / 2} \\ C A ^ {m / 2 + 1} \\ \dots \\ C A ^ {m / 2 + p - 1} \end{array} \right], \quad \mathcal {C} = \left[ A ^ {m / 2} B, A ^ {m / 2 + 1} B, \dots , A ^ {m / 2 + q - 1} B \right]. \tag {6} +$$ + +Then we have the factorization $H = \mathcal{OC}$ , indicating that $H$ is of rank at most $n$ . + +Rank $k$ approximation. We now show that $H$ has a rank $k$ approximation corresponding to the unstable component. Given the decomposition of $A$ in (3), we can write each block of the lifted Hankel matrix as + +$$ +C A ^ {\ell} B = C [ Q _ {1} Q _ {2} ] \left[ \begin{array}{c c} N _ {1} ^ {\ell} & 0 \\ 0 & N _ {2} ^ {\ell} \end{array} \right] \left[ \begin{array}{c} R _ {1} \\ R _ {2} \end{array} \right] B = [ C Q _ {1} \quad C Q _ {2} ] \left[ \begin{array}{c c} N _ {1} ^ {\ell} & 0 \\ 0 & N _ {2} ^ {\ell} \end{array} \right] \left[ \begin{array}{c} R _ {1} B \\ R _ {2} B \end{array} \right]. +$$ + +If $\ell$ is reasonably large, using the fact that $N_1^\ell \gg N_2^\ell \approx 0$ , we can have $CA^{\ell}B \approx CQ_{1}N_{1}^{\ell}R_{1}B$ . Therefore, we know that when $m$ is reasonably large, we have $H$ can be approximately factorized as $H \approx \tilde{\mathcal{O}}\tilde{\mathcal{C}}$ where: + +$$ +\tilde {\mathcal {O}} = \left[ \begin{array}{c} C Q _ {1} N _ {1} ^ {m / 2} \\ C Q _ {1} N _ {1} ^ {m / 2 + 1} \\ \dots \\ C Q _ {1} N _ {1} ^ {m / 2 + p - 1} \end{array} \right], \quad \tilde {\mathcal {C}} = \left[ N _ {1} ^ {m / 2} R _ {1} B, \dots , N _ {1} ^ {m / 2 + q - 1} R _ {1} B \right]. \tag {7} +$$ + +As $\tilde{\mathcal{O}}$ has $k$ columns, $\tilde{\mathcal{O}}\tilde{\mathcal{C}}$ has (at most) rank- $k$ . We also use the notation + +$$ +\tilde {H} = \tilde {\mathcal {O}} \tilde {\mathcal {C}}, \tag {8} +$$ + +to denote this rank- $k$ approximation of Hankel. As from $H$ to $\tilde{H}$ , the only thing that are omitted are of order $O(N_2^m)$ , it is reasonable to expect that $\| H - \tilde{H} \| \leq O(\lambda_{k + 1}^m)$ , i.e. this rank $k$ approximation has exponentially decaying error in $m$ , as shown in Lemma C.3. + +Estimating unstable component of dynamics. In the actual algorithm (to be introduced in Section 4), $\tilde{H}$ is to be estimated and therefore not known perfectly. However, to illustrate the methodology of the proposed method, for this subsection, we consider $\tilde{H}$ to be known perfectly and show that the unstable component $F(z)$ can be recovered perfectly. + +Suppose we have the following factorization of $\tilde{H}$ for some $\bar{\mathcal{O}}\in \mathbb{R}^{d_y\times k},\bar{\mathcal{C}}\in \mathbb{R}^{k\times d_u}$ , (which has infinite possible solutions), $\tilde{H} = \bar{\mathcal{O}}\bar{\mathcal{C}}$ . We show in Lemma D.1 there exists an invertible $S$ , such that $\bar{\mathcal{O}} = \tilde{\mathcal{O}} S$ , $\bar{\mathcal{C}} = S^{-1}\tilde{\mathcal{C}}$ . Therefore, from the construction of $\tilde{\mathcal{O}}$ and $\tilde{\mathcal{C}}$ in (7), we see that $\bar{\mathcal{O}}_1 = CQ_1N_1^{m / 2}S$ , where $\bar{\mathcal{O}}_1$ represent the first block submatrix of $\bar{\mathcal{O}}$ . Solving for $\bar{N}_1$ for the equation $\bar{\mathcal{O}}_{2:p} = \bar{\mathcal{O}}_{1:p - 1}\bar{N}_1$ , we can get $\bar{N}_1 = S^{-1}N_1S$ . After which we can get $\bar{C} = \bar{\mathcal{O}}_1\bar{N}_1^{-m / 2} = CQ_1S$ , and $\bar{B} = \bar{N}_1^{-m / 2}\bar{C}_1 = S^{-1}R_1B$ . In summary, we get the following realization of the system: + +$$ +z _ {t + 1} = S ^ {- 1} N _ {1} S z _ {t} + S ^ {- 1} R _ {1} B u _ {t}, \tag {9a} +$$ + +$$ +y _ {t} = C Q _ {1} S z _ {t}. \tag {9b} +$$ + +whose transfer function is exactly $F(z)$ , the unstable component of the original system. + +Algorithm 1 LTS-P: learning the Hankel Matrix +1: for $i = 1:M$ do +2: Generate and apply $T$ Gaussian control inputs $[u_0^{(i)},\dots,u_{T - 1}^{(i)}]\stackrel {\mathrm{i.i.d}}{\sim}\mathcal{N}(0,\sigma_u^2)$ . Collect $y^{(i)} = [y_0^{(i)},\ldots ,y_{T - 1}^{(i)}]$ +3: end for +4: Compute $\hat{\Phi}$ with (17). +5: Recover $\hat{H}$ with (18). +6: Compute $\hat{H}$ from $\hat{H}$ with (19). +7: Compute $\hat{\mathcal{O}}$ and $\hat{C}$ with (20). +8: Recover $\hat{N}_1,\hat{N}_{1,O}^{\frac{m}{2}},\hat{N}_{1,C}^{\frac{m}{2}}$ with (21), (22), (23), respectively. +9: Compute $\hat{C},\hat{B}$ with (24). +10: Design $H_{\infty}$ controller with $(\hat{N}_1,\hat{C},\hat{B})$ + +# 3.3 Robust stabilizing controller design + +After the unstable component $F(z)$ is estimated, we can treat the unobserved part of the system as a disturbance $\Delta(z)$ and then synthesize a robust stabilizing controller. Suppose we design a controller $u(z) = K(z)y(z)$ for $F(z)$ , and denote its sensitivity function as: + +$$ +F _ {K} (z) = \bigl (I - F (z) K (z) \bigr) ^ {- 1}. +$$ + +Now let's look at what if we applied the same $K(z)$ to the full system $F^{\mathrm{full}}(z)$ in (4). In this case, the closed loop system is stable if and only if the transfer function $F_{K}^{\mathrm{full}}(z) = (I - F^{\mathrm{full}}(z)K(z))^{-1}$ is analytic for all $|z| \geq 1$ , see e.g. Chapter 5 of [53]. Note that, + +$$ +\begin{array}{l} F _ {K} ^ {\mathrm {f u l l}} (z) = \left(I - F ^ {\mathrm {f u l l}} (z) K (z)\right) ^ {- 1} = \left(I - F (z) K (z) - \Delta (z) K (z)\right) ^ {- 1} \\ = \left(F _ {K} (z) ^ {- 1} - \Delta (z) K (z)\right) ^ {- 1} = F _ {K} (z) \left(I - \Delta (z) K (z) F _ {K} (z)\right) ^ {- 1}. \tag {10} \\ \end{array} +$$ + +The Small Gain Theorem shows a necessary and sufficient condition for the system to have a stabilizing controller: + +Lemma 3.1 (Theorem 8.1 of [53]). Given $\gamma >0$ , the closed loop transfer function defined in (10) is internally stable for any $\| \Delta (z)\|_{H_\infty}\leq \gamma$ if and only if $\| K(z)F_K(z)\|_{H_\infty} < \frac{1}{\gamma}$ . + +Since $\Delta(z)$ is stable and therefore has bounded $H_{\infty}$ norm, it suffices to find a controller such that + +$$ +\left\| K (z) F _ {K} (z) \right\| _ {H _ {\infty}} < \frac {1}{\left\| \Delta (z) \right\| _ {H _ {\infty}}}, \tag {11} +$$ + +in order to stabilize the original full system. + +# 4 Algorithm + +Based on the ideas developed in Section 3, we now design an algorithm that learns to stabilize from data. The pseudocode is provided in Algorithm 1. + +Step 1: Approximate Low-Rank Lifted Hankel Estimation. Section 3 shows that if $\tilde{H}$ defined in (8) is known, then we can design a controller satisfying (11) to stabilize the system. In this section, we discuss a method to estimate $\tilde{H}$ with singular value decomposition (SVD) of the lifted Hankel matrix. + +Data collection and notation. Consider we sample $M$ trajectories, each with length $T$ . To simplify notation, for each trajectory $i \in \{1, \dots, M\}$ , we organize input and output data as + +$$ +y ^ {(i)} = \left[ \begin{array}{l l l l} y _ {0} ^ {(i)} & y _ {1} ^ {(i)} & \dots & y _ {T - 1} ^ {(i)} \end{array} \right], \quad u ^ {(i)} = \left[ \begin{array}{l l l l} u _ {0} ^ {(i)} & u _ {1} ^ {(i)} & \dots & u _ {T - 1} ^ {(i)} \end{array} \right], \tag {12} +$$ + +where each $u_{j}^{(i)} \sim \mathcal{N}(0, \sigma_{u}^{2})$ are independently selected. We also define the a new matrix for the observation noise as + +$$ +v ^ {(i)} = \left[ \begin{array}{l l l l} v _ {0} ^ {(i)} & v _ {1} ^ {(i)} & \dots & v _ {T - 1} ^ {(i)} \end{array} \right]. \tag {13} +$$ + +Substituting the above into (1), we obtain + +$$ +y _ {t} ^ {(i)} = C x _ {t} ^ {(i)} + v _ {t} ^ {(i)} = \sum_ {j = 0} ^ {T} C A ^ {j} \left(B u _ {t - j - 1} ^ {(i)} + w _ {t - j - 1} ^ {(i)}\right) + v _ {t} ^ {(i)}. \tag {14} +$$ + +We further define an upper-triangular Toeplitz matrix and a matrix of observed system dynamics: + +$$ +U ^ {(i)} = \left[ \begin{array}{c c c c c} u _ {0} ^ {(i)} & u _ {1} ^ {(i)} & u _ {2} ^ {(i)} & \dots & u _ {T - 1} ^ {(i)} \\ 0 & u _ {0} ^ {(i)} & u _ {1} ^ {(i)} & \dots & u _ {T - 2} ^ {(i)} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & u _ {0} ^ {(i)} \end{array} \right], \quad \Phi = \left[ \begin{array}{l l l l l} 0 & C B & C A B & \dots & C A ^ {T - 2} B \end{array} \right]. \tag {15} +$$ + +Note that the lifted Hankel matrix $H$ in (5) can be recovered from $\Phi$ above as they contain the same block submatrices. The measurement data (14) in each rollout can be written as + +$$ +y ^ {(i)} = \Phi U ^ {(i)} + v ^ {(i)}. \tag {16} +$$ + +Therefore, we can estimate $\Phi$ by the following ordinary least square problem: + +$$ +\hat {\Phi} := \arg \min _ {X \in \mathbb {R} ^ {d _ {y} \times (T * d _ {u})}} \sum_ {i = 1} ^ {M} \left\| y ^ {(i)} - X U ^ {(i)} \right\| _ {F} ^ {2}. \tag {17} +$$ + +We then estimate the lifted Hankel matrix as follows: + +$$ +\hat {H} := \left[ \begin{array}{c c c c} \hat {\Phi} _ {2 + m} & \hat {\Phi} _ {3 + m} & \dots & \hat {\Phi} _ {2 + p + m} \\ \hat {\Phi} _ {3 + m} & \hat {\Phi} _ {4 + m} & \dots & \hat {\Phi} _ {3 + p + m} \\ \vdots & \vdots & \vdots & \vdots \\ \hat {\Phi} _ {2 + q + m} & \hat {\Phi} _ {3 + q + m} & \dots & \hat {\Phi} _ {2 + p + q + m} \end{array} \right]. \tag {18} +$$ + +Let $\hat{H} = \hat{U}_H\hat{\Sigma}_H\hat{V}_H^*$ denote the singular value decomposition of $\hat{H}$ , and we define the $k$ -th order estimation of $\hat{H}$ : + +$$ +\hat {\tilde {H}} := \hat {U} \hat {\Sigma} \hat {V} ^ {*}, \tag {19} +$$ + +where $\hat{U} (\hat{V})$ is the matrix of the top $k$ left (right) singular vector in $\hat{U}_H(\hat{V}_H)$ , and $\hat{\Sigma}$ is the matrix of the top $k$ singular values in $\hat{\Sigma}_H$ . + +Step 2: Esitmating unstable transfer function $\mathbf{F}$ . With the SVD $\hat{\tilde{H}} = \hat{U} \hat{\Sigma} \hat{V}^*$ , we further do the following factorization: + +$$ +\hat {\mathcal {O}} = \hat {U} \hat {\Sigma} ^ {1 / 2}, \quad \hat {\mathcal {C}} = \hat {\Sigma} ^ {1 / 2} \hat {V} ^ {*}. \tag {20} +$$ + +With the above, we can estimate $\bar{N}_1,\bar{C},\bar{B}$ similar to the procedure introduced in Section 3.2: + +$$ +\hat {N} _ {1} := \left(\hat {\mathcal {O}} _ {1: p - 1} ^ {*} \hat {\mathcal {O}} _ {1: p - 1}\right) ^ {- 1} \hat {\mathcal {O}} _ {1: p - 1} ^ {*} \hat {\mathcal {O}} _ {2: p}. \tag {21} +$$ + +To reduce the error of estimating $CQ_{1}$ and $R_{1}B$ and avoid compounding error caused by raising $\hat{N}_{1}$ to the $m / 2$ 'th power, we also directly estimate $\bar{N}_1^{\frac{m}{2}}$ from both $\hat{\mathcal{O}}$ and $\hat{\mathcal{C}}$ . + +$$ +\widehat {N _ {1 , O} ^ {\frac {m}{2}}} := \left(\hat {\mathcal {O}} _ {1: p - \frac {m}{2}} ^ {*} \hat {\mathcal {O}} _ {1: p - \frac {m}{2}}\right) ^ {- 1} \hat {\mathcal {O}} _ {1: p - \frac {m}{2}} ^ {*} \hat {\mathcal {O}} _ {\frac {m}{2} + 1: p}, \tag {22} +$$ + +$$ +\widehat {N _ {1 , C} ^ {\frac {m}{2}}} := \hat {\mathcal {C}} _ {\frac {m}{2} + 1: p} \hat {\mathcal {C}} _ {1: p - \frac {m}{2}} ^ {*} \left(\hat {\mathcal {C}} _ {1: p - \frac {m}{2}} \hat {\mathcal {C}} _ {1: p - \frac {m}{2}} ^ {*}\right) ^ {- 1}. \tag {23} +$$ + +In practice, the estimation obtained from (22) and (23) are very similar, and we can use either one to estimate $\bar{N}_1^{\frac{m}{2}}$ . Lastly, we estimate $\bar{C}$ and $\bar{B}$ as follows: + +$$ +\hat {C} := \hat {\mathcal {O}} _ {1} \widehat {N _ {1 , O} ^ {- \frac {m}{2}}}, \quad \hat {B} := \widehat {N _ {1 , C} ^ {- \frac {m}{2}}} \hat {C} _ {1}, \tag {24} +$$ + +where for simplicity, we use $\widehat{N_{1,O}^{-\frac{m}{2}}}$ , $\widehat{N_{1,C}^{-\frac{m}{2}}}$ to denote the inverse of $\widehat{N_{1,O}^{\frac{m}{2}}}$ , $\widehat{N_{1,C}^{\frac{m}{2}}}$ . We will provide accuracy bounds for those estimations in Appendix B. With the above estimations, we are ready to design a controller with the following transfer function, + +$$ +\hat {F} (z) = \hat {C} \left(z I - \hat {N} _ {1}\right) ^ {- 1} \hat {B}. \tag {25} +$$ + +Step 3: Designing robust controller. After estimating the system dynamics and obtain the estimated transfer function in (25) as discussed in Section 3.3, we can design a stabilizing controller by existing $H_{\infty}$ control methods to minimize $\| K(z)F_K(z)\|_{H_\infty}$ . The details on designing the robust controller can be found in robust control documentations, e.g. Chapter 7 of Dullerud and Paganini [13]. + +What if $k$ is not known a priori? The proposed method requires knowledge of $k$ , the number of unstable eigenvalues. If $k$ is not known, we show in Lemma B.1 and Lemma C.3 that the first $k$ singular values of $\hat{H}$ and $H$ increase exponentially with $m$ , and the remaining singular values decrease exponentially with $m$ . Therefore, with a reasonably sized $m$ and if $p, q$ are chosen to be larger than $k$ , there will be a large spectral gap among the singular values of $\hat{H}$ . The learner can use the location of the spectral gap to determine the value of $k$ . + +# 5 Main Results + +In this section, we provide the sample complexity needed for the proposed algorithm to return a stabilizing controller. We first introduce a standard assumption on controllability and observability. + +Assumption 5.1. The LTI system $(A,B)$ is controllable, and $(A,C)$ is observable. + +We also need an assumption on the existence of a controller that meets the small gain theorem's criterion (11). + +Assumption 5.2. There exists a controller $K(z)$ that stabilizes plant $F(z)$ , such that for some fixed $\epsilon_{*} > 0$ , $\| K(z)F_{K}(z)\|_{H_{\infty}} < \frac{1}{\|\Delta(z)\|_{H_{\infty}} + 3\epsilon_{*}}$ . + +To state our main result, we introduce some system theoretic quantity. + +Controllability/observability Gramian for unstable subsystem. For the unstable subsystem $N_{1}, R_{1}B, CQ_{1}$ , its $\alpha$ -observability Gramian is $\mathcal{G}_{\mathrm{ob}} = [(CQ_{1})^{*}(CQ_{1}N_{1})^{*}\dots (CQ_{1}N_{1}^{\alpha - 1})^{*}]^{*}$ , and its $\alpha$ -controllability Gramian is $\mathcal{G}_{\mathrm{con}} = [R_{1}B, N_{1}R_{1}B, \dots, N_{1}^{\alpha - 1}R_{1}B]$ . Per Lemma H.1 and Assumption 5.1, we know the $N_{1}, R_{1}B, CQ_{1}$ subsystem is both controllable and observable, and hence we can select $\alpha$ to be the smallest integer such that both $\mathcal{G}_{\mathrm{ob}}, \mathcal{G}_{\mathrm{con}}$ are rank- $k$ . Note that we always have $\alpha \leq k$ . Our main theorem will use the following system-theoretic parameters: $\alpha, \kappa(\mathcal{G}_{\mathrm{ob}}), \kappa(\mathcal{G}_{\mathrm{con}})$ (the condition numbers of $\mathcal{G}_{\mathrm{ob}}, \mathcal{G}_{\mathrm{con}}$ respectively), and $\sigma_{\min}(\mathcal{G}_{\mathrm{ob}}), \sigma_{\min}(\mathcal{G}_{\mathrm{con}})$ (the smallest singular values of $\mathcal{G}_{\mathrm{ob}}, \mathcal{G}_{\mathrm{con}}$ , respectively). + +Umbrella upper bound $L$ . We use a constant $L$ to upper bound the norms $\| A\| ,\| B\| ,\| C\| ,\| N_1\| ,\| R_1B\| ,\| CQ_1\|$ . We will use the Jordan form decomposition for $N_{1} = P_{1}\Lambda P_{1}^{-1}$ , and let $L$ upper bound $\| P_1\| \| P_1^{-1}\| = \kappa (P_1)$ . We also use $L$ in $\sup_{|z| = 1}\| (zI - N_1)^{-1}\| \leq \frac{L}{|\lambda_k| - 1}$ . Lastly, we use $L$ to upper bound the constant in Gelfand's formula (Lemma H.2) for $N_{2}$ and $N_{1}^{-1}$ . Specifically, we will use $\| N_2^t\| \leq L\left(\frac{|\lambda_{k + 1}| + 1}{2}\right)^t,\| (N_1^{-1})^t\| \leq L\left(\frac{\frac{1}{|\lambda_k|} + 1}{2}\right)^t,\forall t\geq 0$ . + +With the above preparations, we are ready to state the main theorem. + +Theorem 5.3. Suppose Assumption 5.1, Assumption 5.2 holds and $N_{1}$ is diagonalizable. In the regime where $|\lambda_k| - 1$ , $1 - |\lambda_{k + 1}|$ , $\epsilon_{*}$ are small, we set (recall throughout the paper $\asymp$ only hides numerical constants) + +$$ +m \asymp \max \left(\alpha , \frac {\log L}{1 - | \lambda_ {k + 1} |}, \frac {1}{| \lambda_ {k} | - 1} \log \frac {\kappa (\mathcal {G} _ {\mathrm {o b}}) \kappa (\mathcal {G} _ {\mathrm {c o n}}) L}{\min \left(\sigma_ {\min } (\mathcal {G} _ {\mathrm {o b}}) , \sigma_ {\min } (\mathcal {G} _ {\mathrm {c o n}})\right) (| \lambda_ {k} | - 1) \epsilon_ {*}}\right), +$$ + +and we set $p = q = m$ (hence each trajectory's length is $T = 3m + 2$ ) and the number of trajectories + +$$ +M \asymp \left(\frac {\sigma_ {v}}{\sigma_ {u}}\right) ^ {2} \left(d _ {u} + d _ {y}\right) \log \frac {m}{\delta} + d _ {u} m. +$$ + +for some $\delta \in (0,1)$ . Then, with probability at least $1 - \delta$ , $\hat{K}(z)$ obtained by Algorithm 1 stabilizes the original dynamical system (2). + +The total number of samples needed for the algorithm is $MT = (3m + 2)M$ . In the bound in Theorem 5.3, the only constant that explicitly grows with $k$ is the controllability/observability index $\alpha = O(k)$ which appears in the bound for $m$ . Therefore, the sample complexity $MT = O(m^2) = O(\alpha^2) = O(k^2)$ grows quadratically with $k$ and independent from system state dimension $n$ . In the regime $k \ll n$ , this significantly reduces the sample complexity of stabilization compared to methods that estimate the full system dynamics. To the best of our knowledge, this is the first result that achieves stabilization of an unknown partially observable LTI system with sample complexity independent from the system state dimension $n$ . To validate the claim in Theorem 5.3, we use the LTS-P to stabilize a randomly generated partially observable system and compare the result with two state-of-the-art benchmarks [43, 51] in Appendix A. In Appendix B, we provide a high-level overview of the proof of Theorem 5.3, with detailed proof steps in Appendices C to E. + +Dependence on system theoretic parameters. As the system becomes less observable and controllable, $\kappa (\mathcal{G}_{\mathrm{con}}),\kappa (\mathcal{G}_{\mathrm{ob}})$ increases and $\sigma_{\mathrm{min}}(\mathcal{G}_{\mathrm{con}}),\sigma_{\mathrm{min}}(\mathcal{G}_{\mathrm{ob}})$ decreases, increasing $m$ and the sample complexity $MT$ grows in the order of $(\log \frac{\kappa(\mathcal{G}_{\mathrm{ob}})\kappa(\mathcal{G}_{\mathrm{con}})}{\min(\sigma_{\mathrm{min}}(\mathcal{G}_{\mathrm{ob}}),\sigma_{\mathrm{min}}(\mathcal{G}_{\mathrm{con}}))})^2$ . Moreover, when $|\lambda_{k + 1}|,|\lambda_k|$ are closer to 1, the sample complexity also increases in the order of $(\max (\frac{1}{1 - |\lambda_{k + 1}|},\frac{1}{|\lambda_k| - 1}\log \frac{1}{|\lambda_k| - 1}))^2$ . This makes sense as the unstable and stable components of the system would become closer to marginal stability and harder to distinguish apart. Lastly, if the $\epsilon_*$ in Assumption 5.2 becomes smaller, we have a smaller margin for stabilization, which also increases the sample complexity in the order of $(\log \frac{1}{\epsilon})^2$ . + +Non-diagonalization case. Theorem 5.3 assumes $N_{1}$ is diagonalizable, which is only used in a single place in the proof in Lemma D.1. More specifically, we need to upper bound $J^{t}(J^{*})^{-t}$ for some Jordan block $J$ and integer $t$ , and in the diagonalizable case an upper bound is 1 as the Jordan block is size 1. In the case that $N_{1}$ is not diagonalizable, a similar bound can still be proven with dependence on the size of the largest Jordan block of $N_{1}$ , denoted as $\gamma$ . Eventually, this will lead to an additional multiplicative factor $\gamma$ in the sample complexity on $m$ (cf. (124)), whereas the requirements for $p, q, M$ is the same. The derivations can be found in Appendix G. + +Necessity of Assumption 5.2. Assumption 5.2 is needed if one only estimates the unstable component of the system, treating the stable component as unknown. This is because the small gain theorem is an if-and-only-if condition. If Assumption 5.2 is not true, then no matter what controller $K(z)$ one designs, there must exist an adversarially chosen stable component $\Delta(z)$ (not known to the learner) causing the system to be unstable, even if $F(z)$ is perfectly learned. For a specific construction of such a stability-breaking $\Delta(z)$ , see the proof of the small gain theorem, e.g. Theorem 8.1 of Zhou and Doyle [53]. Although Hu et al. [19], Zhang et al. [50] do not explicitly impose this assumption, they impose similar assumptions on the eigenvalues, e.g. the $|\lambda_1| |\lambda_{k+1}| < 1$ in Theorem 4.2 of Zhang et al. [50], and we believe the small gain theorem and Assumption 5.2 is the intrinsic reason underlying those eigenvalue assumptions in Zhang et al. [50]. We believe this requirement of Assumption 5.2 is the fundamental limit of the unstable + stable decomposition approach. One way to break this fundamental limit is that instead of doing unstable + stable decomposition, we learn a larger component corresponding to eigenvalues $\lambda_1, \ldots, \lambda_k$ for some $\tilde{k} > k$ , which effectively means we learn the unstable component and part of the stable component of the system. Doing so, Assumption 5.2 will be easier to satisfy as when $\tilde{k}$ approaches $n$ , $\| \Delta(z) \|_{H_\infty}$ will approach 0. + +Relation to hardness of control results. Recently, there has been related work on the hardness of the learn-to-stabilize problem [9, 48]. Our setting does not conflict with these hardness results for the following reasons. Firstly, our result focuses on the $k \ll n$ regime. In the regime $k = n$ , our result does not improve over other approaches. Secondly, our Assumption 5.2 is related to the co-stabilizability concept in Zeng et al. [48], as Assumption 5.2 effectively means the existence of a controller that stabilizes the system for all possible $\Delta(z)$ . In some sense, our results complements these results showing when is learn-to-stabilize easy under certain assumptions. + +# 6 Conclusion + +In this paper, we examined the necessary and sufficient conditions for the stabilization of a partially observable LTI system and proposed a novel SVD-based robust controller design framework. Future work includes studying the transfer function from the process noise $w_{t}$ to $y_{t}$ , and improving the complexity bound in Theorem 5.3. + +# 7 Acknowledgement + +This work is supported in part by NSF 2154171, NSF CAREER 2339112, CMU CyLab seed funding, in part by a grant from the Commonwealth of Pennsylvania, Department of Community and Economic Development, in part by National Science Foundation under Grant No. 2442948, and in part by the Department of the Navy, Office of Naval Research, under award number N00014-23-1-2252. The views expressed are those of the authors and do not reflect the official policy or position of the US Navy, Department of Defense or the US Government. + +# References + +[1] Yasin Abbasi-Yadkori and Csaba Szepesvári. Regret bounds for the adaptive control of linear quadratic systems. In Sham M. Kakade and Ulrike von Luxburg, editors, Proceedings of the 24th Annual Conference on Learning Theory, volume 19 of Proceedings of Machine Learning Research, pages 1-26, Budapest, Hungary, 09-11 Jun 2011. PMLR. +[2] L. V. Ahlfors. Complex Analysis. McGraw-Hill Book Company, 2 edition, 1966. +[3] Randal W. Beard, George N. Saridis, and John T. Wen. Galerkin approximations of the generalized hamilton-jacobi-bellman equation. Automatica, 33(12):2159-2177, 1997. ISSN 0005-1098. doi: https://doi.org/10.1016/S0005-1098(97)00128-3. +[4] Julian Berberich, Johannes Kohler, Matthias Muller, and Frank Allgower. Data-driven model predictive control with stability and robustness guarantees. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: This paper does not involve LLM usage. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Simulations + +![](images/9c80a5eb746daee14ba2aa75ef6fb25f619ad443fb6441221596c16f0e94a70f.jpg) +(a) $\sigma = 0.4$ + +![](images/7861c323c263510822c4d2866777346f009fabce275b5df97e1cb9340f47f08d.jpg) +(b) $\sigma = 0.6$ + +![](images/4f6cf19ff2475fe3e4fa252ed2d20f60841672f3e734fa39fe8aa0a0d2c97cd4.jpg) +(c) $\sigma = 0.8$ + +![](images/689bc171aafbc347a59015662d324d08b9ece9b12b17bb990de95d9a0a219df5.jpg) +Figure 1: The above figure shows the length of rollouts needed to identify and stabilize an unstable system with the unstable dimension $k = 5$ . The solid line shows the average length of rollouts the learner takes to stabilize the system. The shaded area shows the standard deviation of the length of rollouts. The proposed method requires the shortest rollouts and has the smallest standard deviation. +(a) $\sigma = 0.4$ +Figure 2: The above figure shows the number of rollouts needed to identify and stabilize an unstable system with the unstable dimension $k = 5$ . The solid line shows the average number of rollouts the learner takes to stabilize the system. The shaded area shows the standard deviation of the number of rollouts. The proposed method requires the least number rollouts and has small standard deviation. + +![](images/915458b61845b220847f23aedb2d92b3a0c44335918db6b466cba572a629db79.jpg) +(b) $\sigma = 0.6$ + +![](images/b4c96c84685521afc6196bbbf6218eb9529e15f0e364046321f6bb47659b72d2.jpg) +(c) $\sigma = 0.8$ + +In this section, we compare our algorithm to the two existing benchmarks: nuclear norm regularization in Sun et al. [43] and the Ho-Kalman algorithm in Zheng and Li [51]. We use a more complicated system than (1): + +$$ +x _ {t + 1} = A x _ {t} + B u _ {t} + w _ {t} \tag {26} +$$ + +$$ +y _ {t} = C x _ {t} + D u _ {t} + v _ {t}, +$$ + +where there are both process noise $w_{t}$ and observation noise $v_{t}$ . We fix the unstable part of the LTI system to dimension 5, i.e. $k = 5$ , and observe the effect of increasing dimension $n$ on the system. For each dimension $n$ , we randomly generate a matrix with $k$ unstable eigenvalues $\lambda_{i} \sim \mathrm{unif}(1.1,2)$ + +and $n - k$ stable eigenvalues $\lambda_{j} \sim \mathrm{unif}(0, 0.5)$ . The basis for these eigenvalues is also initialized randomly. In both parts of the simulations, we fix $m = 4$ and $p = q = \left\lfloor \frac{T - 4}{2} \right\rfloor$ for the proposed algorithm, LTS-P. + +In the first part of the simulations, we fix the number of rollouts to be 4 times the dimension of the system and let the system run with an increasing length of rollouts until a stabilizing controller is generated. The system is simulated in three different settings, each with noise $w_{t}, v_{t} \sim \mathcal{N}(0, \sigma)$ , with $\sigma = 0.4, 0.6, 0.8$ . For each of the three algorithms, the simulation is repeated for 30 trials, and the result is shown in Figure 1. The proposed method requires the shortest rollouts. The algorithm in Sun et al. [43] roughly have the same length of rollouts across different dimensions. This is to be expected, as Sun et al. [43] only uses the last data point of each rollout, so a longer trajectory does not generate more data. However, unlike the proposed algorithm, Sun et al. [43] still needs to estimate the entire system, which is why it still needs a longer rollout. The length of the rollouts of Zheng and Li [51] grows linearly with the dimension of the system. We also see fluctuations of the length of rollouts required for stabilization in Figure 1 because the matrix generated for each $n$ has a randomly generated eigenbasis, which influences the amount of data needed for stabilization, as shown in Theorem 5.3. Overall, this simulation shows that the length of rollouts of the proposed algorithm remains $O(\mathrm{poly}(k))$ regardless of the dimension of the system $n$ . + +In the second part of the simulation, we fix the length of each trajectory to 70 and examine the number of trajectories needed before the system stabilizes. We do not increase the length of trajectory in this part of the simulation with increasing $n$ , since Sun et al. [43] only uses the last data point, and does not show significant performance improvement with longer trajectories, as shown in the first part of the simulation, so it would be an unfair comparison if the trajectory is overly long. Similar to the first part, the system is simulated in three different settings and the result is shown in Figure 2. Overall, the proposed method requires the least number of trajectories. When the $n \gtrsim k$ , the proposed algorithm requires a similar number of data with Zheng and Li [51], as the proposed algorithm is the same to that in Zheng and Li [51] when $k = n$ . When $n \gg k$ , the proposed algorithm outperforms both benchmarks. + +![](images/1a53fd1c7540182f7640e4fc61c4a932650a7fc73bfb04178e98a0d08323a002.jpg) +Figure 3: The above figure shows the probability of stabilization for a randomly generated matrix with $n = 15$ and $k = 5$ . + +In the last simulation, we fix a randomly generated matrix with dimension $n = 15$ and an unstable subspace of dimension $k = 5$ . We further set the variance of the process and observation noise to $\sigma = 0.4$ . As shown in Figure 3, the stabilization probability is most enhanced by adding more trajectories, but longer trajectories also demonstrate a significant effect for stabilization, as proven in Theorem 5.3. + +# B Proof Outline + +In this section, we will give a high-level overview of the key proof ideas for the main theorem. The full proof details can be found in Appendix C (for Step 1), Appendix D (for Step 2), and Appendix E (for Step 3). + +Proof Structure. The proof is largely divided into three steps following the idea developed in Section 3. In the first step, we examine how accurately the learner can estimate the low-rank lifted Hankel matrix $\tilde{H}$ . In the second step, we examine how accurately the learner estimates the transfer function of the unstable component from $\tilde{H}$ . In the last step, we provide a stability guarantee. + +Step 1. We show that $\hat{\tilde{H}}$ obtained in (19) is an accurate estimate of $\tilde{H}$ . + +Lemma B.1. With probability at least $1 - \delta$ , the estimation $\hat{\tilde{H}}$ obtained in (19) satisfies + +$$ +\left\| \Delta_ {H} \right\| := \left\| \hat {\tilde {H}} - \tilde {H} \right\| < \epsilon := 2 \hat {\epsilon} + 2 \tilde {\epsilon}, +$$ + +where $\hat{\epsilon}$ is the estimation error for $\| H - \hat{H}\|$ that decays in the order of $O\left(\frac{1}{\sqrt{M}}\right)$ ; $\tilde{\epsilon}$ is the error $\left\| H - \tilde{H}\right\|$ that decays in the order of $O\left((\frac{|\lambda_{k + 1}| + 1}{2})^m\right)$ . See Theorem C.2 and Lemma C.3 for the exact constants. + +We later (in Step 2, 3) will show that for a sufficiently small $\epsilon$ , the robust controller produced by the algorithm will stabilize the system in (1). We will invoke Lemma B.1 at the end of the proof in Step 3 to pick $m, p, q, M$ values that achieve the required $\epsilon$ (in Appendix E). + +Step 2. In this step, we will show that $N_{1}, R_{1}B, CQ_{1}$ can be estimated up to an error of $O(\epsilon)$ and up to a similarity transformation, given that $\left\| \hat{\tilde{H}} - \tilde{H} \right\| < \epsilon$ . + +Lemma B.2. The estimated system dynamics $\tilde{N}_1$ satisfies the following bound for some invertible matrix $\hat{S}$ : + +$$ +\| \hat {S} ^ {- 1} \hat {N} _ {1} \hat {S} - N _ {1} \| \leq \frac {4 L c _ {2} \epsilon}{c _ {1} \tilde {o} _ {\mathrm {m i n}} (m)} := \epsilon_ {N}. +$$ + +where $c_2, c_1$ are constants depending on system theoretic parameters, and $\tilde{o}_{\mathrm{min}}(m)$ is a quantity that grows with $m$ . See Lemma D.1 for the definition of these quantities. + +Lemma B.3. Up to transformation $\hat{S}$ (same as the one in Lemma B.2), we can bound the difference between $\hat{C}$ and the unstable component of the system $CQ_{1}$ as follows: + +$$ +\left\| C Q _ {1} - \hat {C} \hat {S} \right\| \leq 2 \left(\frac {4 c _ {2} L}{c _ {1} \tilde {o} _ {\min } (m)} + \frac {1}{\tilde {c} _ {\min } (m)}\right) \epsilon := \epsilon_ {C}. +$$ + +where $\tilde{c}_{\min}(m)$ is a quantity that grows with $m$ defined in Lemma D.1. + +Lemma B.4. Up to transformation $\hat{S}^{-1}$ ( $\hat{S}$ is the same as the one in Lemma B.2), we can bound the difference between $\hat{B}$ and the unstable component of the system $R_{1}B$ as follows: + +$$ +\left\| R _ {1} B - \hat {S} ^ {- 1} \hat {B} \right\| < 8 \left(\frac {c _ {2} L}{c _ {1} \tilde {c} _ {\min } (m)} + \frac {1}{\tilde {o} _ {\min } (m)}\right) \epsilon := \epsilon_ {B}. +$$ + +Step 3. We show that, under sufficiently accurate estimation, the controller $\hat{K}(z)$ returned by the algorithm stabilizes plant $F(z)$ and hence $\| \hat{K}(z) F_{\hat{K}}(z) \|_{H_{\infty}}$ is well defined. This is done via the following helper proposition. + +Proposition B.5. Recall the definition of the estimated transfer function in (25). Suppose the estimation errors $\epsilon_N, \epsilon_C, \epsilon_B$ are small enough such that $\sup_{|z|=1} \left\| F(z) - \hat{F}(z) \right\| < \epsilon_*$ . Then, the following two statements hold: (a) If $K(z)$ stabilizes $F(z)$ and $\| K(z)F_K(z)\|_{H_\infty} < \frac{1}{\epsilon_*}$ , then $K(z)$ stabilizes plant $\hat{F}(z)$ as well; (b) similarly, if $\hat{K}(z)$ stabilizes $\hat{F}(z)$ with $\| \hat{K}(z)\hat{F}_{\hat{K}}(z)\|_{H_\infty} < \frac{1}{\epsilon_*}$ , then $\hat{K}(z)$ stabilizes plant $F(z)$ as well. + +Then, we upper bound $\| \hat{K} (z)F_{\hat{K}}(z)\|_{H_{\infty}}$ and use small gain theorem (Lemma 3.1) to show that the stable mode $\Delta (z)$ does not affect stability either, and therefore, the controller $\hat{K} (z)$ stabilizes the full system, $F^{\mathrm{full}}(z)$ in (4). Leveraging the error bounds in Step 2 and Step 1, we will also provide the final sample complexity bound. The detailed proof can be found in Appendix E. + +# C Proof of Step 1: Bounding Hankel Matrix Error + +To further simplify notation, define + +$$ +Y = \left[ \begin{array}{l l l} y ^ {(1)} & \dots & y ^ {(M)} \end{array} \right], \tag {27} +$$ + +$$ +U = \left[ \begin{array}{l l l} U ^ {(1)} & \dots & U ^ {(M)} \end{array} \right], \tag {28} +$$ + +$$ +v = \left[ \begin{array}{l l l} v ^ {(1)} & \dots & v ^ {(M)} \end{array} \right], \tag {29} +$$ + +so (17) can be simplified to + +$$ +\hat {\Phi} = \arg \min _ {X \in \mathbb {R} ^ {d _ {y} \times (T * d _ {u})}} \| Y - X U \| _ {F} ^ {2}, \tag {30} +$$ + +respectively. An analytic solution to (30) is $\hat{\Phi} \coloneqq YU^{\dagger}$ , where $U^{\dagger} \coloneqq U^{*}(UU^{*})^{-1}$ is the right pseudo-inverse of $U$ . In this paper, we will use $\hat{\tilde{\Phi}}$ to recover $\tilde{H}$ , and we want to bound the error between $\tilde{H}$ and $\hat{\tilde{H}}$ . + +$$ +\begin{array}{l} \hat {\Phi} - \Phi = Y U ^ {*} (U U ^ {*}) ^ {- 1} - \Phi \\ = (Y - \Phi U) U ^ {*} \left(U U ^ {*}\right) ^ {- 1} \tag {31} \\ = v U ^ {*} \left(U U ^ {*}\right) ^ {- 1}, \\ \end{array} +$$ + +where the third equality used (16). We can bound the estimation error (31) by + +$$ +\left\| \hat {\Phi} - \Phi \right\| \leq \left\| \left(U U ^ {*}\right) ^ {- 1} \right\| \| v U ^ {*} \|. \tag {32} +$$ + +We then prove a lemma that links the difference between $\Phi$ and Hankel matrix $H$ . + +Lemma C.1. Given two Hankel matrices $H^{\clubsuit}$ , $H^{\diamond}$ constructed from $\Phi^{\bullet}$ , $\Phi^{\diamond}$ with (18), respectively, then + +$$ +\left\| H ^ {\clubsuit} - H ^ {\diamond} \right\| \leq \sqrt {\min \{p , q \}} \left\| \Phi_ {2 + m: T} ^ {\clubsuit} - \Phi_ {2 + m: T} ^ {\diamond} \right\|. +$$ + +Proof. From the construction in (18), we see that each column of $H^{\clubsuit} - H^{\diamond}$ is a submatrix of $\Phi_{2 + m:T}^{\clubsuit} - \Phi_{2 + m:T}^{\diamond}$ , then + +$$ +\left\| H ^ {\clubsuit} - H ^ {\diamond} \right\| \leq \sqrt {q} \left\| \Phi_ {2 + m: T} ^ {\clubsuit} - \Phi_ {2 + m: T} ^ {\diamond} \right\|. \tag {33} +$$ + +Similarly, each row of $H^{\clubsuit} - H^{\diamond}$ is a submatrix of $\Phi_{2 + m:T}^{\clubsuit} - \Phi_{2 + m:T}^{\diamond}$ . Therefore, + +$$ +\left\| H ^ {\clubsuit} - H ^ {\diamondsuit} \right\| \leq \sqrt {p} \left\| \Phi_ {2 + m: T} ^ {\clubsuit} - \Phi_ {2 + m: T} ^ {\diamondsuit} \right\|. \tag {34} +$$ + +Combining (33) and (34) gives the desired result. + +We then bound the gap between $H^{(k)}$ and $\tilde{H}$ by adapting Theorem 3.1 of Zheng and Li [51]. For completeness, we put the theorem and proof below: + +Theorem C.2. For any $\delta > 0$ , the following inequality holds when $M > 8d_u T + 4(d_u + d_y + 4) \log(3T / \delta)$ with probability at least $1 - \delta$ : + +$$ +\left\| H - \hat {H} \right\| \leq \frac {8 \sigma_ {v} \sqrt {T (T + 1) \left(d _ {u} + d _ {y}\right) \min \{p , q \} \log (2 7 T / \delta)}}{\sigma_ {u} \sqrt {M}} := \hat {\epsilon} +$$ + +Proof. By Lemma C.1, we have + +$$ +\left\| H - \hat {H} \right\| \leq \sqrt {\min \{p , q \}} \left\| \Phi_ {2 + m: T} - \hat {\Phi} _ {2 + m: T} \right\|. +$$ + +Bounding $\| (UU^{*})\|^{-1}$ and $\| vU^{*}\|$ with Proposition 3.1 and Proposition 3.2 of [51] gives the result in the theorem statement. + +Lemma C.3. + +$$ +\left\| H - \tilde {H} \right\| \leq \sqrt {q} p \leq L ^ {3} (\frac {| \lambda_ {k + 1} | + 1}{2}) ^ {m} := \tilde {\epsilon} +$$ + +Proof. We first bound $H$ and $\tilde{H}$ entry-wise, + +$$ +\begin{array}{l} H _ {i, j} - \tilde {H} _ {i, j} = C A ^ {m + i + j - 2} B - C Q _ {1} N _ {1} ^ {m + i + j - 2} R _ {1} B \\ = C \left(Q _ {1} N _ {1} ^ {m + i + j - 2} R _ {1} + Q _ {2} N _ {2} ^ {m + i + j - 2} R _ {2}\right) B - C Q _ {1} N _ {1} ^ {m + i + j - 2} R _ {1} B \\ = C Q _ {2} N _ {2} ^ {m + i + j - 2} R _ {2} B. \\ \end{array} +$$ + +Applying Gelfand's formula (Lemma H.2), we obtain + +$$ +\left\| H _ {i, j} - \tilde {H} _ {i, j} \right\| \leq \| B \| \| C \| \zeta_ {\epsilon_ {2}} \left(N _ {2}\right) \left(\left| \lambda_ {k + 1} \right| + \epsilon_ {4}\right) ^ {m + i + j - 2} \leq L ^ {3} \left(\frac {\left| \lambda_ {k + 1} \right| + 1}{2}\right) ^ {m + i + j - 2}. \tag {35} +$$ + +Moreover, we have the following well-known inequality (see, for example, [17]), for $\Lambda \in \mathbb{R}^{n_1\times n_2}$ + +$$ +\frac {1}{\sqrt {n _ {1}}} \| \Lambda \| _ {1} \leq \| \Lambda \| _ {2} \leq \sqrt {n _ {2}} \| \Lambda \| _ {1}. \tag {36} +$$ + +Applying (36) to (35) gives us the desired inequality. + +We are now ready to prove Lemma B.1. + +Proof of Lemma B.1. We first provide a bound on $\hat{\tilde{H}} -\hat{H}$ . As $\hat{\tilde{H}}$ is the rank- $k$ approximation of $\hat{H}$ , we have $\left\| \hat{\tilde{H}} -\hat{H}\right\| \leq \left\| \tilde{H} -\hat{H}\right\|$ , then + +$$ +\left\| \hat {\tilde {H}} - \hat {H} \right\| \leq \left\| \tilde {H} - H \right\| + \left\| H - \hat {H} \right\| = \tilde {\epsilon} + \hat {\epsilon}. \tag {37} +$$ + +where we use Theorem C.2 and Lemma C.3. Therefore, we obtain the following bound + +$$ +\left\| \hat {\tilde {H}} - \tilde {H} \right\| \leq \left\| \hat {\tilde {H}} - \hat {H} \right\| + \left\| \hat {H} - H \right\| + \left\| H - \tilde {H} \right\| \leq 2 \hat {\epsilon} + 2 \tilde {\epsilon} := \epsilon . \tag {38} +$$ + +![](images/0b9e1db9dd93ebc423a2a31fc5820071fecabbc30d29486037e4c0abf959dfaa.jpg) + +# D Proof of Step 2: Bounding Dynamics Error + +In the previous section, we proved Lemma B.1 and provided a bound for $\| \tilde{H} -\tilde{H}\|$ . In this section, we use this bound to derive the error bound for our system estimation. We start with the following lemma that bounds the difference between $\tilde{\mathcal{O}}$ and $\hat{\mathcal{O}}$ and the difference between $\tilde{\mathcal{C}}$ and $\hat{\mathcal{C}}$ up to a transformation. + +Lemma D.1. Given $N_{1}$ is diagonalizable, and recall $N_{1} = P_{1}\Lambda P_{1}^{-1}$ denotes the eigenvalue decomposition of $N_{1}$ , where $\Lambda = \mathrm{diag}(\lambda_1,\ldots ,\lambda_k)$ is the diagonal matrix of eigenvalues. Consider $\hat{\tilde{H}} = \hat{\mathcal{O}}\hat{\mathcal{C}}$ and $\tilde{H} = \tilde{\mathcal{O}}\tilde{C}$ , where we recall $\hat{\mathcal{O}},\hat{\mathcal{C}}$ are obtained by doing $SVD\hat{\tilde{H}} = \hat{U}\hat{\Sigma}\hat{V}^{*}$ and setting $\hat{\mathcal{O}} = \hat{U}\hat{\Sigma}^{1 / 2},\hat{\mathcal{C}} = \hat{\Sigma}^{1 / 2}\hat{V}^{*}$ (cf. (20)). Suppose $\| \Delta_H\| = \left\| \tilde{H} -\hat{\tilde{H}}\right\| \leq \epsilon$ , then there exists an invertible $k$ -by- $k$ matrix $\hat{S}$ such that + +$$ +(a) \left\| \tilde {\mathcal {O}} - \hat {\mathcal {O}} \hat {S} \right\| \leq \frac {1}{\sigma_ {\min } (\tilde {\mathcal {C}})} \epsilon a n d \left\| \tilde {\mathcal {C}} - \hat {S} ^ {- 1} \hat {\mathcal {C}} \right\| \leq \frac {4}{\sigma_ {\min } (\tilde {\mathcal {O}})} \epsilon , +$$ + +(b) there exists $c_{1}, c_{2} > 0$ such that $c_{1}I \preceq \hat{S} \preceq c_{2}I$ , where given $\underline{\sigma}_{s}, \overline{\sigma}_{s}$ in (49), + +$$ +c _ {1} ^ {2} = \frac {\underline {{\sigma}} _ {s}}{2 \sqrt {2} \sigma_ {\max} (P _ {1}) ^ {2}}, \qquad c _ {2} ^ {2} = 2 \sqrt {2} \frac {\overline {{\sigma}} _ {s}}{\sigma_ {\min} (P _ {1}) ^ {2}}. +$$ + +(c) we can bound $\sigma_{\mathrm{min}}(\tilde{\mathcal{O}})$ and $\sigma_{\mathrm{min}}(\tilde{\mathcal{C}})$ as follows: + +$$ +\sigma_ {\mathrm {m i n}} (\tilde {\mathcal {O}}) \geq \sigma_ {\mathrm {m i n}} (\tilde {\mathcal {O}} _ {1: \alpha}) \geq \sigma_ {\mathrm {m i n}} (\mathcal {G} _ {\mathrm {o b}}) \sigma_ {\mathrm {m i n}} (N _ {1} ^ {m / 2}) := \tilde {o} _ {\mathrm {m i n}} (m), +$$ + +$$ +\sigma_ {\mathrm {m i n}} (\tilde {\mathcal {C}}) \geq \sigma_ {\mathrm {m i n}} (\tilde {\mathcal {C}} _ {1: \alpha}) \geq \sigma_ {\mathrm {m i n}} (\mathcal {G} _ {\mathrm {c o n}}) \sigma_ {\mathrm {m i n}} (N _ {1} ^ {m / 2}) := \tilde {c} _ {\mathrm {m i n}} (m). +$$ + +Proof. Proof of part(a). Recall the definition of $\tilde{H}$ + +$$ +\begin{array}{l} \tilde {H} = [ C Q _ {1} N _ {1} ^ {m + i + j} R _ {1} B ] _ {i \in \{0, \dots , p - 1 \}, j \in \{0, \dots , q - 1 \}} \\ = \underbrace {\left[ \begin{array}{c} C Q _ {1} N _ {1} ^ {\frac {m}{2}} \\ \cdots \\ C Q _ {1} N _ {1} ^ {\frac {m}{2} + p - 1} \end{array} \right]} _ {\tilde {\mathcal {O}}} \underbrace {\left[ \begin{array}{c c c} N _ {1} ^ {\frac {m}{2}} R _ {1} B & \ldots & N _ {1} ^ {\frac {m}{2} + q - 1} R _ {1} B \end{array} \right]} _ {\tilde {\mathcal {C}}}. \\ \end{array} +$$ + +We can represent $\tilde{\mathcal{O}}$ and $\tilde{\mathcal{C}}$ as follows: + +$$ +\tilde {\mathcal {O}} = \underbrace {\left[ \begin{array}{c c c c} C Q _ {1} P _ {1} & & & \\ & C Q _ {1} P _ {1} & & \\ & & \dots & \\ & & & C Q _ {1} P _ {1} \end{array} \right] \left[ \begin{array}{c} \Lambda^ {\frac {m}{2}} \\ \dots \\ \Lambda^ {\frac {m}{2} + p - 1} \end{array} \right]} _ {:= G _ {1}} P _ {1} ^ {- 1}. \tag {39} +$$ + +$$ +\tilde {\mathcal {C}} = P _ {1} \underbrace {\left[ \begin{array}{l l l} \Lambda^ {\frac {m}{2}} & \dots & \Lambda^ {\frac {m}{2} + q - 1} \end{array} \right] \left[ \begin{array}{l l l} P _ {1} ^ {- 1} R _ {1} B & & \\ & \dots & \\ & & P _ {1} ^ {- 1} R _ {1} B \end{array} \right]} _ {:= G _ {2} ^ {*}}. \tag {40} +$$ + +Next, note that + +$$ +\hat {\tilde {H}} = \hat {U} \hat {\Sigma} \hat {V} ^ {*} = \hat {U} \hat {\Sigma} ^ {\frac {1}{2}} \hat {\Sigma} ^ {\frac {1}{2}} \hat {V} ^ {*} = \hat {\mathcal {O C}} = \tilde {\mathcal {O C}} + \Delta_ {H} = G _ {1} G _ {2} ^ {*} + \Delta_ {H}. \tag {41} +$$ + +Therefore, we have + +$$ +\begin{array}{l} G _ {1} G _ {2} ^ {*} G _ {2} = \underbrace {\hat {U} \hat {\Sigma} ^ {\frac {1}{2}}} _ {\hat {\mathcal {O}}} \hat {\Sigma} ^ {\frac {1}{2}} \hat {V} ^ {*} G _ {2} - \Delta_ {H} G _ {2} \\ \Rightarrow G _ {1} = \hat {\mathcal {O}} \hat {\Sigma} ^ {\frac {1}{2}} \hat {V} ^ {*} G _ {2} \left(G _ {2} ^ {*} G _ {2}\right) ^ {- 1} - \Delta_ {H} G _ {2} \left(G _ {2} ^ {*} G _ {2}\right) ^ {- 1} \tag {42} \\ \Rightarrow \tilde {\mathcal {O}} = G _ {1} P _ {1} ^ {- 1} = \hat {\mathcal {O}} \underbrace {\hat {\Sigma} ^ {\frac {1}{2}} \hat {V} ^ {*} G _ {2} (G _ {2} ^ {*} G _ {2}) ^ {- 1} P _ {1} ^ {- 1}} _ {:= \hat {S}} - \Delta_ {H} G _ {2} (G _ {2} ^ {*} G _ {2}) ^ {- 1} P _ {1} ^ {- 1}. \\ \end{array} +$$ + +where we have defined the matrix $\hat{S}$ above. We can also expand $\hat{S}$ as follows: + +$$ +\begin{array}{l} \hat {S} = \hat {\Sigma} ^ {\frac {1}{2}} \hat {V} ^ {*} G _ {2} P _ {1} ^ {*} (P _ {1} ^ {*}) ^ {- 1} (G _ {2} ^ {*} G _ {2}) ^ {- 1} P _ {1} ^ {- 1} \\ = \hat {\Sigma} ^ {\frac {1}{2}} \hat {V} ^ {*} G _ {2} P _ {1} ^ {*} \left(P _ {1} G _ {2} ^ {*} G _ {2} P _ {1} ^ {*}\right) ^ {- 1} \\ = \hat {\mathcal {C}} \tilde {\mathcal {C}} ^ {*} \left(\tilde {\mathcal {C}} \tilde {\mathcal {C}} ^ {*}\right) ^ {- 1}. \tag {43} \\ \end{array} +$$ + +Further, the last term in the last equation in (42) can be written as, + +$$ +\begin{array}{l} \Delta_ {H} G _ {2} \left(G _ {2} ^ {*} G _ {2}\right) ^ {- 1} P _ {1} ^ {- 1} = \Delta_ {H} G _ {2} P _ {1} ^ {*} \left(P _ {1} ^ {*}\right) ^ {- 1} \left(G _ {2} ^ {*} G _ {2}\right) ^ {- 1} P _ {1} ^ {- 1} \\ = \Delta_ {H} \tilde {\mathcal {C}} ^ {*} \left(\tilde {\mathcal {C}} \tilde {\mathcal {C}} ^ {*}\right) ^ {- 1}. \tag {44} \\ \end{array} +$$ + +Putting (42), (44) together, we have + +$$ +\left\| \tilde {\mathcal {O}} - \hat {\mathcal {O}} \hat {S} \right\| \leq \frac {1}{\sigma_ {\min } (\tilde {\mathcal {C}})} \epsilon , \tag {45} +$$ + +which proves one side of part (a). To finish part(a), we are left to bound $\left\| \tilde{\mathcal{C}} - \hat{S}^{-1}\hat{\mathcal{C}} \right\|$ . With the expression of $\hat{S}$ in (43), we obtain + +$$ +\left\| \hat {\mathcal {C}} - \hat {S} ^ {- 1} \tilde {\mathcal {C}} \right\| = \left\| \hat {\mathcal {C}} - \tilde {\mathcal {C}} \tilde {\mathcal {C}} ^ {*} (\hat {\mathcal {C}} \tilde {\mathcal {C}} ^ {*}) ^ {- 1} \tilde {\mathcal {C}} \right\|. +$$ + +Let $E \coloneqq \hat{\mathcal{O}}\hat{S} - \tilde{\mathcal{O}}$ so $\tilde{\mathcal{O}} = \hat{\mathcal{O}}\hat{S} - E$ . By (42) and (44), we obtain + +$$ +E = \Delta_ {H} \tilde {\mathcal {C}} ^ {*} (\tilde {\mathcal {C}} \tilde {\mathcal {C}} ^ {*}) ^ {- 1}, \tag {46} +$$ + +and + +$$ +\left\| E \tilde {\mathcal {C}} \right\| = \left\| \Delta_ {H} \tilde {\mathcal {C}} ^ {*} (\tilde {\mathcal {C}} \tilde {\mathcal {C}} ^ {*}) ^ {- 1} \tilde {\mathcal {C}} \right\| \leq \epsilon . +$$ + +Therefore, we can obtain the following inequalities: + +$$ +\begin{array}{l} \epsilon \geq \left\| \tilde {\mathcal {O}} \tilde {\mathcal {C}} - \hat {\mathcal {O}} \hat {\mathcal {C}} \right\| = \left\| (\hat {\mathcal {O}} \hat {S} - E) \tilde {\mathcal {C}} - \hat {\mathcal {O}} \hat {\mathcal {C}} \right\| = \left\| \hat {\mathcal {O}} (\hat {S} \tilde {\mathcal {C}} - \hat {\mathcal {C}}) - E \tilde {\mathcal {C}} \right\| \geq \left\| \hat {\mathcal {O}} (\hat {S} \tilde {\mathcal {C}} - \hat {\mathcal {C}}) \right\| - \left\| E \tilde {\mathcal {C}} \right\| \\ \Rightarrow \left\| \hat {\mathcal {O}} \hat {S} (\tilde {\mathcal {C}} - \hat {S} ^ {- 1} \hat {\mathcal {C}}) \right\| \leq 2 \epsilon \\ \Rightarrow \left\| \tilde {\mathcal {C}} - \hat {S} ^ {- 1} \hat {\mathcal {C}} \right\| \leq \frac {2}{\sigma_ {\min} (\hat {\mathcal {O}} \hat {S})} \epsilon \leq \frac {2}{\sigma_ {\min} (\tilde {\mathcal {O}}) - \left\| \tilde {\mathcal {O}} - \hat {\mathcal {O}} \hat {S} \right\|} \epsilon . \\ \end{array} +$$ + +Substituting (45), we get + +$$ +\left\| \tilde {\mathcal {C}} - \hat {S} ^ {- 1} \hat {\mathcal {C}} \right\| \leq \frac {2 \epsilon}{\sigma_ {\min} (\tilde {\mathcal {O}}) - \frac {1}{\sigma_ {\min} (\tilde {\mathcal {C}})} \epsilon} \leq \frac {4}{\sigma_ {\min} (\tilde {\mathcal {O}})} \epsilon . +$$ + +where the last inequality requires + +$$ +\epsilon < \frac {1}{2} \sigma_ {\min } (\tilde {\mathcal {O}}) \sigma_ {\min } (\tilde {\mathcal {C}}). \tag {47} +$$ + +which we will verify at the end of the proof of Theorem 5.3 that our selection of algorithm parameters guarantees is true. + +Proof of part(b). We first examine the relationship between $G_{1}$ and $G_{2}$ . Recall $\alpha$ is the maximum of the controllability and observability index. Therefore, the matrix + +$$ +d _ {u} * \alpha \left\{\left[ \begin{array}{c} \overbrace {B ^ {*} R _ {1} ^ {*} (P _ {1} ^ {- 1}) ^ {*}} ^ {k} \\ B ^ {*} R _ {1} ^ {*} (P _ {1} ^ {- 1}) ^ {*} \Lambda^ {*} \\ \vdots \\ B ^ {*} R _ {1} ^ {*} (P _ {1} ^ {- 1}) ^ {*} (\Lambda^ {\alpha - 1}) ^ {*} \end{array} \right], \right. +$$ + +which is exactly $\mathcal{G}_{\mathrm{con}}^{*}(P_{1}^{*})^{-1}$ , is full column rank. For any full column rank matrix $\Gamma$ , let's $\Gamma^{\dagger} = (\Gamma^{*}\Gamma)^{-1}\Gamma^{*}$ denote its Moore-Penrose inverse that satisfies $\Gamma^{\dagger}\Gamma = I$ . With this, for nonnegative integer $j$ we define the following matrix, which is essentially $[G_1]_{j\alpha:(j + 1)\alpha -1}$ "divided by" $[G_2]_{j\alpha :(j + 1)\alpha -1}$ in the Moore-Penrose sense: + +$$ +S _ {C B} ^ {(j)} = \left[ \begin{array}{c} C Q _ {1} P _ {1} \Lambda^ {\frac {m}{2} + j \alpha} \\ C Q _ {1} P _ {1} \Lambda^ {\frac {m}{2} + j \alpha + 1} \\ \vdots \\ C Q _ {1} P _ {1} \Lambda^ {\frac {m}{2} + (j + 1) \alpha - 1} \end{array} \right] \left[ \begin{array}{c} B ^ {*} R _ {1} ^ {*} (P _ {1} ^ {- 1}) ^ {*} (\Lambda^ {\frac {m}{2} + j \alpha}) ^ {*} \\ B ^ {*} R _ {1} ^ {*} (P _ {1} ^ {- 1}) ^ {*} (\Lambda^ {\frac {m}{2} + j \alpha + 1}) ^ {*} \\ \vdots \\ B ^ {*} R _ {1} ^ {*} (P _ {1} ^ {- 1}) ^ {*} (\Lambda^ {\frac {m}{2} + (j + 1) \alpha - 1}) ^ {*} \end{array} \right] ^ {\dagger} +$$ + +$$ += \underbrace {\left[ \begin{array}{c} C Q _ {1} P _ {1} \\ C Q _ {1} P _ {1} \Lambda \\ \vdots \\ C Q _ {1} P _ {1} \Lambda^ {\alpha - 1} \end{array} \right]} _ {= \mathcal {G} _ {\mathrm {o b}} P _ {1}} \Lambda^ {\frac {m}{2} + j \alpha} (\Lambda^ {- \frac {m}{2} - j \alpha}) ^ {*} \underbrace {\left[ \begin{array}{c} B ^ {*} R _ {1} ^ {*} (P _ {1} ^ {- 1}) ^ {*} \\ B ^ {*} R _ {1} ^ {*} (P _ {1} ^ {- 1}) ^ {*} \Lambda^ {*} \\ \vdots \\ B ^ {*} R _ {1} ^ {*} (P _ {1} ^ {- 1}) ^ {*} (\Lambda^ {\alpha - 1}) ^ {*} \end{array} \right] ^ {\dagger}} _ {= (\mathcal {G} _ {\mathrm {c o n}} ^ {*} (P _ {1} ^ {*}) ^ {- 1}) ^ {- 1}}. \tag {48} +$$ + +As in the main Theorem 5.3 we assumed the $N_{1}$ is diagonalizable, $\Lambda$ is a diagonal matrix with possibly complex entries. As such, $\Lambda^{\frac{m}{2} +j\alpha}(\Lambda^{-\frac{m}{2} -j\alpha})^{*}$ is a diagonal matrix with all entries having modulus 1. Therefore, we have + +$$ +\sigma_ {\min } \left(S _ {C B} ^ {(j)}\right) \geq \frac {\sigma_ {\min } \left(\mathcal {G} _ {\mathrm {o b}}\right)}{\sigma_ {\max } \left(\mathcal {G} _ {\mathrm {c o n}}\right)} \sigma_ {\min } \left(P _ {1}\right) ^ {2} := \underline {{\sigma}} _ {s}, \tag {49} +$$ + +$$ +\sigma_ {\max } (S _ {C B} ^ {(j)}) \leq \frac {\sigma_ {\max } (\mathcal {G} _ {\mathrm {o b}})}{\sigma_ {\min } (\mathcal {G} _ {\mathrm {c o n}})} \sigma_ {\max } (P _ {1}) ^ {2} := \overline {{\sigma}} _ {s}, +$$ + +for all $j$ . Recall the condition in the main theorem 5.3 requires $p = q$ . Without loss of generality, we can assume $\alpha$ divides $p$ and $q$ ,4 then $G_{1}$ and $G_{2}$ can be related in the following way: + +$$ +G _ {1} = \underbrace {\left[ \begin{array}{c c c c} S _ {C B} ^ {(0)} & & & \\ & S _ {C B} ^ {(1)} & & \\ & & \dots & \\ & & & S _ {C B} ^ {\left(\frac {p}{\alpha}\right)} \end{array} \right]} _ {:= \mathcal {S} _ {C B}} G _ {2}. \tag {50} +$$ + +With the above relation, we are now ready to bound the norm of $\hat{S}$ . We calculate + +$$ +\hat {S} ^ {*} \hat {S} = \left(P _ {1} ^ {- 1}\right) ^ {*} \left(G _ {2} ^ {*} G _ {2}\right) ^ {- 1} G _ {2} ^ {*} \hat {V} \hat {\Sigma} \hat {V} ^ {*} G _ {2} \left(G _ {2} ^ {*} G _ {2}\right) ^ {- 1} P _ {1} ^ {- 1}. \tag {51} +$$ + +If we examine the middle term $\hat{V}\hat{\Sigma}\hat{V}^{*}$ , we can expand it as + +$$ +\begin{array}{l} \hat {V} \hat {\Sigma} \hat {V} ^ {*} = (\hat {V} \hat {\Sigma} ^ {2} \hat {V} ^ {*}) ^ {\frac {1}{2}} \\ = (\hat {V} \hat {\Sigma} \hat {U} ^ {*} \hat {U} \hat {\Sigma} \hat {V} ^ {*}) ^ {\frac {1}{2}} \\ = \left(\hat {\tilde {H}} ^ {*} \hat {\tilde {H}}\right) ^ {\frac {1}{2}} \\ = \left(\left(\tilde {H} + \Delta_ {H}\right) ^ {*} \left(\tilde {H} + \Delta_ {H}\right)\right) ^ {\frac {1}{2}}. \tag {52} \\ \end{array} +$$ + +Using Lemma H.3 on (52), we obtain + +$$ +\begin{array}{l} \left(\frac {1}{2} \tilde {H} ^ {*} \tilde {H} - \Delta_ {H} ^ {*} \Delta_ {H}\right) ^ {\frac {1}{2}} \preceq \hat {V} \hat {\Sigma} \hat {V} ^ {*} \preceq \left(2 \tilde {H} ^ {*} \tilde {H} + 2 \Delta_ {H} ^ {*} \Delta_ {H}\right) ^ {\frac {1}{2}} \\ \Rightarrow \left(\frac {1}{2} \left(G _ {1} G _ {2} ^ {*}\right) ^ {*} G _ {1} G _ {2} ^ {*} - \Delta_ {H} ^ {*} \Delta_ {H}\right) ^ {\frac {1}{2}} \preceq \hat {V} \hat {\Sigma} \hat {V} ^ {*} \preceq \left(2 \left(G _ {1} G _ {2} ^ {*}\right) ^ {*} G _ {1} G _ {2} ^ {*} + 2 \Delta_ {H} ^ {*} \Delta_ {H}\right) ^ {\frac {1}{2}} \quad \text {s u b s i t u t e (4 1)} \\ \Rightarrow \left(\frac {1}{2} G _ {2} G _ {1} ^ {*} G _ {1} G _ {2} ^ {*} - \Delta_ {H} ^ {*} \Delta_ {H}\right) ^ {\frac {1}{2}} \preceq \hat {V} \hat {\Sigma} \hat {V} ^ {*} \preceq \left(2 G _ {2} G _ {1} ^ {*} G _ {1} G _ {2} ^ {*} + 2 \Delta_ {H} ^ {*} \Delta_ {H}\right) ^ {\frac {1}{2}} \\ \Rightarrow \left(\frac {1}{2} G _ {2} G _ {2} ^ {*} \mathcal {S} _ {C B} ^ {*} \mathcal {S} _ {C B} G _ {2} G _ {2} ^ {*} - \Delta_ {H} ^ {*} \Delta_ {H}\right) ^ {\frac {1}{2}} \preceq \hat {V} \hat {\Sigma} \hat {V} ^ {*} \preceq \left(2 G _ {2} G _ {2} ^ {*} \mathcal {S} _ {C B} ^ {*} \mathcal {S} _ {C B} G _ {2} G _ {2} ^ {*} + 2 \Delta_ {H} ^ {*} \Delta_ {H}\right) ^ {\frac {1}{2}} \quad \text {s u b s t i t u t e (5 0)}. \\ \end{array} +$$ + +Therefore, we can bound $\hat{V}\hat{\Sigma}\hat{V}^{*}$ : + +$$ +\frac {1}{\sqrt {2}} \underline {{\sigma}} _ {s} G _ {2} G _ {2} ^ {*} - \epsilon I \preceq \hat {V} \hat {\Sigma} \hat {V} ^ {*} \preceq \sqrt {2} \bar {\sigma} _ {s} G _ {2} G _ {2} ^ {*} + \sqrt {2} \epsilon I \tag {53} +$$ + +where we used $\underline{\sigma}_s^2 I \preceq S_{CB}^* S_{CB} \preceq \overline{\sigma}_s^2 I$ and $\sigma_{\mathrm{min}}(S_{CB}) \geq \underline{\sigma}_s$ , as $S_{CB}$ is the block-diagonal matrix of $S_{CB}^{(j)}$ (cf. (49), (50)). Therefore, plugging (53) into (51), we are ready to bound $\hat{S}^*\hat{S}$ has follows: + +$$ +\begin{array}{l} \hat {S} ^ {*} \hat {S} \preceq \sqrt {2} \left(P _ {1} ^ {- 1}\right) ^ {*} \left(G _ {2} ^ {*} G _ {2}\right) ^ {- 1} G _ {2} ^ {*} \left(\bar {\sigma} _ {s} G _ {2} G _ {2} ^ {*} + \epsilon I\right) G _ {2} \left(G _ {2} ^ {*} G _ {2}\right) ^ {- 1} P _ {1} ^ {- 1} \\ = \sqrt {2} \bar {\sigma} _ {s} \left(P _ {1} ^ {- 1}\right) ^ {*} P _ {1} ^ {- 1} + \sqrt {2} \epsilon \left(\tilde {\mathcal {C}} \tilde {\mathcal {C}} ^ {*}\right) ^ {- 1} \tag {54} \\ \preceq \sqrt {2} \left(\frac {\overline {{\sigma}} _ {s}}{\sigma_ {\min } (P _ {1}) ^ {2}} + \frac {2 \epsilon}{\sigma_ {\min } (\tilde {\mathcal {C}}) ^ {2}}\right) I. \\ \end{array} +$$ + +Similarly, + +$$ +\hat {S} ^ {*} \hat {S} \succeq \left(\frac {\underline {{\sigma}} _ {s}}{\sqrt {2} \sigma_ {\max } (P _ {1}) ^ {2}} - \frac {\epsilon}{\sigma_ {\min } (\tilde {\mathcal {C}}) ^ {2}}\right) I. \tag {55} +$$ + +If we further pick $\epsilon$ such that + +$$ +\epsilon \leq \min \left(\frac {\underline {{\sigma}} _ {s} \sigma_ {\operatorname* {m i n}} (\tilde {\mathcal {C}}) ^ {2}}{2 \sqrt {2} \sigma_ {\operatorname* {m a x}} (P _ {1}) ^ {2}}, \frac {\overline {{\sigma}} _ {s} \sigma_ {\operatorname* {m i n}} (\tilde {\mathcal {C}}) ^ {2}}{2 \sqrt {2} \sigma_ {\operatorname* {m i n}} (P _ {1}) ^ {2}}\right) = c _ {1} ^ {2} \sigma_ {\operatorname* {m i n}} (\tilde {\mathcal {C}}) ^ {2}, \tag {56} +$$ + +(54) and (55) can be simplified as + +$$ +\underbrace {\frac {\sigma_ {s}}{2 \sqrt {2} \sigma_ {\max } (P _ {1}) ^ {2}}} _ {c _ {1} ^ {2}} I \preceq \hat {S} ^ {*} \hat {S} \preceq \underbrace {2 \sqrt {2} \frac {\overline {{\sigma}} _ {s}}{\sigma_ {\min } (P _ {1}) ^ {2}}} _ {c _ {2} ^ {2}} I. \tag {57} +$$ + +Proof of part (c). Observe that $\tilde{\mathcal{O}}_{1:\alpha} = \mathcal{G}_{\mathrm{ob}}N_1^{m / 2}$ . Therefore, + +$$ +\sigma_ {\mathrm {m i n}} (\tilde {\mathcal {O}}) \geq \sigma_ {\mathrm {m i n}} (\tilde {\mathcal {O}} _ {1: \alpha}) \geq \sigma_ {\mathrm {m i n}} (\mathcal {G} _ {\mathrm {o b}}) \sigma_ {\mathrm {m i n}} (N _ {1} ^ {m / 2}) := \tilde {o} _ {\mathrm {m i n}} (m). +$$ + +Similarly, we have + +$$ +\sigma_ {\min } (\tilde {\mathcal {C}}) \geq \sigma_ {\min } (\tilde {\mathcal {C}} _ {1: \alpha}) \geq \sigma_ {\min } (\mathcal {G} _ {\mathrm {c o n}}) \sigma_ {\min } \left(N _ {1} ^ {m / 2}\right) := \tilde {c} _ {\min } (m). +$$ + +We are now ready to prove Lemma B.2 + +Proof of Lemma B.2. Let $E = \tilde{\mathcal{O}} - \hat{\mathcal{O}}\hat{S}$ , by Lemma D.1, $\|E\| \leq \frac{\epsilon}{\sigma_{\min}(\tilde{\mathcal{C}})}$ . Moreover, $\tilde{\mathcal{O}}_{1:p-1} = \hat{\mathcal{O}}_{1:p-1}\hat{S} + E_{1:p-1}$ and $\tilde{\mathcal{O}}_{2:p} = \hat{\mathcal{O}}_{2:p}\hat{S} + E_{2:p}$ . Given that $\tilde{\mathcal{O}}_{2:p} = \tilde{\mathcal{O}}_{1:p-1}N_1$ , we have $N_1 = (\tilde{\mathcal{O}}_{1:p-1}^* \tilde{\mathcal{O}}_{1:p-1})^{-1} \tilde{\mathcal{O}}_{1:p-1}^* \tilde{\mathcal{O}}_{2:p}$ , where we have used by the condition in the main theorem, + +$$ +p - 1 \geq \alpha \tag {58} +$$ + +so that $\tilde{\mathcal{O}}_{1:p - 1}$ is full column rank. Recall $N_{1}$ is estimated by + +$$ +\hat {N} _ {1} := \left(\hat {\mathcal {O}} _ {1: p - 1} ^ {*} \hat {\mathcal {O}} _ {1: p - 1}\right) ^ {- 1} \hat {\mathcal {O}} _ {1, p - 1} ^ {*} \hat {\mathcal {O}} _ {2: p}. \tag {59} +$$ + +Given that $\tilde{\mathcal{O}}_{2:p} = \tilde{\mathcal{O}}_{1:p - 1}N_1$ , we have + +$$ +\hat {\mathcal {O}} _ {2: p} \hat {S} + E _ {2: p} = \left(\hat {\mathcal {O}} _ {1: p - 1} \hat {S} + E _ {1: p - 1}\right) N _ {1} \tag {60} +$$ + +$$ +\Rightarrow \hat {\mathcal {O}} _ {2: p} + E _ {2: p} \hat {S} ^ {- 1} = \hat {\mathcal {O}} _ {1: p - 1} \hat {S} N _ {1} \hat {S} ^ {- 1} + E _ {1: p - 1} N _ {1} \hat {S} ^ {- 1} \tag {61} +$$ + +$$ +\Rightarrow \hat {\mathcal {O}} _ {2: p} = \hat {\mathcal {O}} _ {1: p - 1} \hat {S} N _ {1} \hat {S} ^ {- 1} + \tilde {E}, \tag {62} +$$ + +where $\tilde{E} \coloneqq E_{1:p-1}N_1\hat{S}^{-1} - E_{2:p}\hat{S}^{-1}$ . Substituting (62) into (59), we obtain + +$$ +\begin{array}{l} \hat {N} _ {1} = \left(\hat {\mathcal {O}} _ {1: p - 1} ^ {*} \hat {\mathcal {O}} _ {1: p - 1}\right) ^ {- 1} \hat {\mathcal {O}} _ {1: p - 1} ^ {*} \left(\hat {\mathcal {O}} _ {1: p - 1} \hat {S} N _ {1} \hat {S} ^ {- 1} + \tilde {E}\right) \\ = \hat {S} N _ {1} \hat {S} ^ {- 1} + \left(\hat {\mathcal {O}} _ {1: p - 1} ^ {*} \hat {\mathcal {O}} _ {1: p - 1}\right) ^ {- 1} \hat {\mathcal {O}} _ {1, p - 1} ^ {*} \tilde {E}. \\ \end{array} +$$ + +Therefore, + +$$ +\begin{array}{l} \left\| N _ {1} - \hat {S} ^ {- 1} \hat {N} _ {1} \hat {S} \right\| \leq \left\| \hat {S} ^ {- 1} \left(\hat {\mathcal {O}} _ {1: p - 1} ^ {*} \hat {\mathcal {O}} _ {1: p - 1}\right) ^ {- 1} \hat {\mathcal {O}} _ {1, p - 1} ^ {*} \tilde {E} \hat {S} \right\| \\ = \left\| \hat {S} ^ {- 1} \left(\hat {\mathcal {O}} _ {1: p - 1} ^ {*} \hat {\mathcal {O}} _ {1: p - 1}\right) ^ {- 1} \hat {\mathcal {O}} _ {1, p - 1} ^ {*} \left(E _ {1: p - 1} N _ {1} - E _ {2: p}\right) \right\| \\ \leq \frac {\epsilon}{c _ {1} \sigma_ {\min } (\hat {\mathcal {O}} _ {1 , p - 1})} (1 + L) \\ \leq \frac {2 L \epsilon}{c _ {1} \sigma_ {\min } (\hat {S} ^ {- 1}) (\tilde {o} _ {\min } (m) - \| \tilde {\mathcal {O}} - \hat {\mathcal {O}} \hat {S} \|)} \\ \leq \frac {4 L c _ {2} \epsilon}{c _ {1} \tilde {o} _ {\min } (m)}. \\ \end{array} +$$ + +where the last inequality requires (47). + +![](images/8ce753c38612d61e2d3c53c26cc439051fc9b2ffd8d45202e9619311cf688a29.jpg) + +Following similar arguments from the above proof for the estimation of $\widehat{N_{1,O}^{\frac{m}{2}}}$ in (22), we have the following collarary. + +Corollary D.2. We have + +$$ +\left\| I - \hat {S} ^ {- 1} \widehat {N _ {1 , O} ^ {\frac {m}{2}}} \hat {S} N _ {1} ^ {- \frac {m}{2}} \right\| \leq \frac {4 c _ {2} \epsilon}{c _ {1} \tilde {\sigma} _ {\operatorname* {m i n}} (m)}. +$$ + +Proof. Let $E = \tilde{\mathcal{O}} - \hat{\mathcal{O}}\hat{S}$ , then $\|E\| \leq \frac{\epsilon}{\sigma_{\min}(\tilde{\mathcal{C}})}$ . Moreover, $\tilde{\mathcal{O}}_{1:p-\frac{m}{2}} = \hat{\mathcal{O}}_{1:p-\frac{m}{2}}\hat{S} + E_{1:p-\frac{m}{2}}$ and $\tilde{\mathcal{O}}_{\frac{m}{2}+1:p} = \hat{\mathcal{O}}_{\frac{m}{2}+1:p}\hat{S} + E_{\frac{m}{2}+1:p}$ . Given that $\tilde{\mathcal{O}}_{\frac{m}{2}+1:p} = \tilde{\mathcal{O}}_{1:p-\frac{m}{2}}N_1^{\frac{m}{2}}$ , we have $N_1^{\frac{m}{2}} = (\tilde{\mathcal{O}}_{1:p-\frac{m}{2}}\tilde{\mathcal{O}}_{1:p-\frac{m}{2}})^{-1}\tilde{\mathcal{O}}_{1:p-\frac{m}{2}}^* \tilde{\mathcal{O}}_{\frac{m}{2}+1:p}$ , where we have used by the condition in the main theorem, + +$$ +p \geq \alpha + \frac {m}{2} \tag {63} +$$ + +so that $\tilde{\mathcal{O}}_{1:p - \frac{m}{2}}$ is full column rank. Recall in (22), we estimate $N_1^{\frac{m}{2}}$ by + +$$ +\widehat {N _ {1 , O} ^ {\frac {m}{2}}} := \left(\hat {\mathcal {O}} _ {1: p - \frac {m}{2}} ^ {*} \hat {\mathcal {O}} _ {1: p - \frac {m}{2}}\right) ^ {- 1} \hat {\mathcal {O}} _ {1: p - \frac {m}{2}} ^ {*} \hat {\mathcal {O}} _ {\frac {m}{2} + 1: p}. \tag {64} +$$ + +Given that $\tilde{\mathcal{O}}_{\frac{m}{2} +1:p} = \tilde{\mathcal{O}}_{1:p - \frac{m}{2}}N_1^{\frac{m}{2}}$ , we have + +$$ +\begin{array}{l} \hat {\mathcal {O}} _ {\frac {m}{2} + 1: p} \hat {S} + E _ {\frac {m}{2} + 1: p} = \left(\hat {\mathcal {O}} _ {1: p - \frac {m}{2}} \hat {S} + E _ {1: p - \frac {m}{2}}\right) N _ {1} ^ {\frac {m}{2}} (65) \\ \Rightarrow \hat {\mathcal {O}} _ {\frac {m}{2} + 1: p} + E _ {\frac {m}{2} + 1: p} \hat {S} ^ {- 1} = \hat {\mathcal {O}} _ {1: p - \frac {m}{2}} \hat {S} N _ {1} ^ {\frac {m}{2}} \hat {S} ^ {- 1} + E _ {1: p - \frac {m}{2}} N _ {1} ^ {\frac {m}{2}} \hat {S} ^ {- 1} (66) \\ \Rightarrow \hat {\mathcal {O}} _ {\frac {m}{2} + 1: p} = \hat {\mathcal {O}} _ {1: p - \frac {m}{2}} \hat {S} N _ {1} ^ {\frac {m}{2}} \hat {S} ^ {- 1} + \tilde {E}, (67) \\ \end{array} +$$ + +where $\tilde{E} := E_{1:p - \frac{m}{2}}N_1^{\frac{m}{2}}\hat{S}^{-1} - E_{\frac{m}{2} + 1:p}\hat{S}^{-1}$ . Substituting (67) into (64), we obtain + +$$ +\begin{array}{l} \widehat {N _ {1 , O} ^ {\frac {m}{2}}} = \left(\hat {\mathcal {O}} _ {1: p - \frac {m}{2}} ^ {*} \hat {\mathcal {O}} _ {1: p - \frac {m}{2}}\right) ^ {- 1} \hat {\mathcal {O}} _ {1: p - \frac {m}{2}} ^ {*} \left(\hat {\mathcal {O}} _ {1: p - \frac {m}{2}} \hat {S} N _ {1} ^ {\frac {m}{2}} \hat {S} ^ {- 1} + \tilde {E}\right) (68) \\ = \hat {S} N _ {1} ^ {\frac {m}{2}} \hat {S} ^ {- 1} + \left(\hat {\mathcal {O}} _ {1: p - \frac {m}{2}} ^ {*} \hat {\mathcal {O}} _ {1: p - \frac {m}{2}}\right) ^ {- 1} \hat {\mathcal {O}} _ {1: p - \frac {m}{2}} ^ {*} \tilde {E}. (69) \\ \end{array} +$$ + +Therefore, + +$$ +\begin{array}{l} \left\| I - \hat {S} ^ {- 1} \widehat {N _ {1 , O} ^ {\frac {m}{2}}} \hat {S} N _ {1} ^ {- \frac {m}{2}} \right\| = \left\| \hat {S} ^ {- 1} (\hat {\mathcal {O}} _ {1: p - \frac {m}{2}} ^ {*} \hat {\mathcal {O}} _ {1: p - \frac {m}{2}}) ^ {- 1} \hat {\mathcal {O}} _ {1: p - \frac {m}{2}} ^ {*} \tilde {E} \hat {S} N _ {1} ^ {- \frac {m}{2}} \right\| \\ = \left\| \hat {S} ^ {- 1} \left(\hat {\mathcal {O}} _ {1: p - \frac {m}{2}} ^ {*} \hat {\mathcal {O}} _ {1: p - \frac {m}{2}}\right) ^ {- 1} \hat {\mathcal {O}} _ {1: p - \frac {m}{2}} ^ {*} \left(E _ {1: p - \frac {m}{2}} N _ {1} ^ {\frac {m}{2}} \hat {S} ^ {- 1} - E _ {\frac {m}{2} + 1: p} \hat {S} ^ {- 1}\right) \hat {S} N _ {1} ^ {- \frac {m}{2}} \right\| \\ \leq \frac {\epsilon}{c _ {1} \sigma_ {\operatorname* {m i n}} (\hat {\mathcal {O}} _ {1 : p - \frac {m}{2}})} \left(1 + L \left(\frac {\frac {1}{| \lambda_ {k} |} + 1}{2}\right) ^ {\frac {m}{2}}\right) \\ \leq \frac {2 \epsilon}{c _ {1} \sigma_ {\min } \left(\hat {\mathcal {O}} _ {1 : p - \frac {m}{2}}\right)} \tag {70} \\ \end{array} +$$ + +$$ +\begin{array}{l} \leq \frac {2 \epsilon}{c _ {1} \sigma_ {\min } (\hat {S} ^ {- 1}) (\tilde {o} _ {\min } (m) - \| \tilde {\mathcal {O}} - \hat {\mathcal {O}} \hat {S} \|)} \\ \leq \frac {2 \epsilon}{c _ {1} \sigma_ {\min} (\hat {S} ^ {- 1}) (\tilde {o} _ {\min} (m) - \frac {1}{\sigma_ {\min} (\tilde {\mathcal {C}})} \epsilon)} \\ \leq \frac {4 c _ {2} \epsilon}{c _ {1} \tilde {\sigma} _ {\min } (m)}, \tag {71} \\ \end{array} +$$ + +where (70) requires + +$$ +m > \frac {- \log (L)}{\log \left( \right.\frac {1}{\left| \lambda_ {K} \right|} + 1}{2}} \tag {72} +$$ + +which we will verify at the end of the proof of Theorem 5.3, and (71) requires (47). + +![](images/9f63cc61407940504e0b5c564e3d055339c1d8936ff5e0f970341e9134350a06.jpg) + +By using an identical argument, we can also prove a similar corollary for $\widehat{N_{1,C}^{\frac{m}{2}}}$ . The proof is omitted. Corollary D.3. Assuming $p, q > \frac{m}{2} + 1$ , we have + +$$ +\left\| I - N _ {1} ^ {- \frac {m}{2}} \hat {S} ^ {- 1} \widehat {N _ {1 , C} ^ {\frac {m}{2}}} \hat {S} \right\| \leq \frac {4 c _ {2} \epsilon}{c _ {1} \tilde {c} _ {\operatorname* {m i n}} (m)}. +$$ + +With the above two corallaries, we are now ready to prove Lemma B.3. + +Proof of Lemma B.3. By the definition of $\tilde{\mathcal{O}}$ in (7), we have + +$$ +\tilde {\mathcal {O}} _ {1} = C Q _ {1} N _ {1} ^ {\frac {m}{2}}. \tag {73} +$$ + +Furthermore, we recall by the way $\hat{C}$ is estimated in (24), we obtain + +$$ +\hat {\mathcal {O}} _ {1} = \widehat {C N _ {1 , O} ^ {\frac {m}{2}}}. \tag {74} +$$ + +Let $E_{O_1} \coloneqq \tilde{\mathcal{O}}_1 - \hat{\mathcal{O}}_1\hat{S}$ . Substituting it in (74) leads to + +$$ +\tilde {\mathcal {O}} _ {1} = \hat {\mathcal {O}} _ {1} \hat {S} + E _ {O _ {1}} = \widehat {C N _ {1 , O} ^ {\frac {m}{2}}} \hat {S} + E _ {O _ {1}}. +$$ + +Substituting in (73), we have + +$$ +\begin{array}{l} C Q _ {1} N _ {1} ^ {\frac {m}{2}} = \widehat {C N _ {1 , O} ^ {\frac {m}{2}}} \hat {S} + E _ {O _ {1}} \\ \Rightarrow C Q _ {1} = \hat {C} \hat {S} \hat {S} ^ {- 1} \widehat {N _ {1 , O} ^ {\frac {m}{2}}} \hat {S} N _ {1} ^ {- \frac {m}{2}} + E _ {O _ {1}} N _ {1} ^ {- \frac {m}{2}}. \tag {75} \\ \end{array} +$$ + +By Corollary D.2, we obtain + +$$ +\begin{array}{l} \left\| C Q _ {1} - \hat {C} \hat {S} \right\| \leq \left\| \hat {C} \hat {S} \right\| \left\| \hat {S} ^ {- 1} \widehat {N _ {1 , O} ^ {\frac {m}{2}}} \hat {S} N _ {1} ^ {- m / 2} - I \right\| + \| E _ {O _ {1}} \| \left\| N _ {1} ^ {- m / 2} \right\| \\ \leq \left\| \hat {C} \hat {S} \right\| \frac {4 c _ {2} \epsilon}{c _ {1} \tilde {\sigma} _ {\mathrm {m i n}} (m)} + \| E _ {O _ {1}} \| \left\| N _ {1} ^ {- m / 2} \right\| \\ \leq \left(\left\| \hat {C} \hat {S} \right\| \frac {4 c _ {2}}{c _ {1} \tilde {\sigma} _ {\min } (m)} + \frac {1}{\tilde {c} _ {\min } (m)}\right) \epsilon (76) \\ \leq \left(\frac {4 c _ {2} \left(\left\| C Q _ {1} - \hat {C} \hat {S} \right\| + \| C Q _ {1} \|\right)}{c _ {1} \tilde {o} _ {\min } (m)} + \frac {1}{\tilde {c} _ {\min } (m)}\right) \epsilon (77) \\ \leq 2 \left(\frac {4 c _ {2} L}{c _ {1} \tilde {o} _ {\min } (m)} + \frac {1}{\tilde {c} _ {\min } (m)}\right) \epsilon . (78) \\ \end{array} +$$ + +where (76) is by Lemma D.1 and requires (72), and (78) requires + +$$ +\epsilon < \frac {c _ {1} \tilde {o} _ {\min } (m)}{8 c _ {2}}. \tag {79} +$$ + +![](images/8e2df47f184cb9108841f41d51c8a34e18dc2d963e5f157d366746de65409385.jpg) + +The proof of Lemma B.4 undergoes a very similar procedure. + +Proof of Lemma B.4. By the definition of $\tilde{\mathcal{C}}$ in (7), we have + +$$ +\tilde {\mathcal {C}} _ {1} = N _ {1} ^ {\frac {m}{2}} R _ {1} B. \tag {80} +$$ + +Further, by (24), we have + +$$ +\widehat {\mathcal {C}} _ {1} = \widehat {N _ {1 , C} ^ {\frac {m}{2}}} \hat {B}. \tag {81} +$$ + +Let $E_{C_1} \coloneqq \tilde{\mathcal{C}}_1 - \hat{S}^{-1}\hat{\mathcal{C}}_1$ . Substitute in (81), we get + +$$ +\tilde {\mathcal {C}} _ {1} = \hat {S} ^ {- 1} \hat {\mathcal {C}} _ {1} + E _ {C _ {1}} = \hat {S} ^ {- 1} \widehat {N _ {1 , C} ^ {\frac {m}{2}}} \hat {B} + E _ {C _ {1}}. +$$ + +Substitute in (80), we obtain + +$$ +N _ {1} ^ {\frac {m}{2}} R _ {1} B = \hat {S} ^ {- 1} \widehat {N _ {1 , C} ^ {\frac {m}{2}}} \hat {B} + E _ {C _ {1}} +$$ + +$$ +R _ {1} B = N _ {1} ^ {- \frac {m}{2}} \hat {S} ^ {- 1} \widehat {N _ {1 , C} ^ {\frac {m}{2}}} \hat {S} \hat {S} ^ {- 1} \hat {B} + N _ {1} ^ {- \frac {m}{2}} E _ {C _ {1}}. \tag {82} +$$ + +By Corollary D.3, we obtain + +$$ +\begin{array}{l} \left\| R _ {1} B - \hat {S} ^ {- 1} \hat {B} \right\| \leq \left\| N _ {1} ^ {- \frac {m}{2}} \hat {S} ^ {- 1} \widehat {N _ {1 , C} ^ {\frac {m}{2}}} \hat {S} - I \right\| \left\| \hat {S} ^ {- 1} \hat {B} \right\| + \left\| N _ {1} ^ {- \frac {m}{2}} \right\| \| E _ {C _ {1}} \| \\ \leq 4 \left(\frac {c _ {2} \left\| \hat {S} ^ {- 1} \hat {B} \right\|}{c _ {1} \tilde {c} _ {\min } (m)} + \frac {1}{\tilde {o} _ {\min } (m)}\right) \epsilon (83) \\ \leq 4 \left(\frac {c _ {2} \left(\left\| R _ {1} B - \hat {S} ^ {- 1} \hat {B} \right\| + \| R _ {1} B \|\right)}{c _ {1} \tilde {c} _ {\min } (m)} + \frac {1}{\tilde {o} _ {\min } (m)}\right) \epsilon (84) \\ \leq 8 \left(\frac {c _ {2} L}{c _ {1} \tilde {c} _ {\min } (m)} + \frac {1}{\tilde {o} _ {\min } (m)}\right) \epsilon , (85) \\ \end{array} +$$ + +where (83) is by Lemma D.1 and requires (72), and (85) requires + +$$ +\epsilon < \frac {c _ {1} \tilde {c} _ {\min } (m)}{8 c _ {2}}. \tag {86} +$$ + +# E Proof of Step 3 + +Before we analyze the analytic condition of the transfer functions, we need to analyze the values of $F(z)$ and $\hat{F}(z)$ on the unit circle through the following lemma: + +Lemma E.1. We have + +$$ +\sup _ {| z | = 1} \left\| F (z) - \hat {F} (z) \right\| \leq \frac {1}{| \lambda_ {k} | - 1} \Big (8 L ^ {2} \epsilon_ {C} + \frac {4 L ^ {4} \epsilon_ {N}}{| \lambda_ {k} | - 1} + L ^ {2} \epsilon_ {B} \Big). +$$ + +Proof. We have that + +$$ +\begin{array}{l} \sup _ {| z | = 1} \left\| \hat {F} (z) - F (z) \right\| = \sup _ {| z | = 1} \left\| \hat {C} \hat {S} (z I - \hat {S} ^ {- 1} \hat {N} _ {1} \hat {S}) ^ {- 1} \hat {S} ^ {- 1} \hat {B} - C Q _ {1} (z I - N _ {1}) ^ {- 1} R _ {1} B \right\| \\ \leq \sup _ {| z | = 1} \left( \right.\left\| \hat {C} \hat {S} (z I - \hat {S} ^ {- 1} \hat {N} _ {1} \hat {S}) ^ {- 1} \hat {S} ^ {- 1} \hat {B} - C Q _ {1} (z I - \hat {S} ^ {- 1} \hat {N} _ {1} \hat {S}) ^ {- 1} \hat {S} ^ {- 1} \hat {B} \right\| \\ \left. + \left\| C Q _ {1} \left(z I - \hat {S} ^ {- 1} \hat {N} _ {1} \hat {S}\right) ^ {- 1} \hat {S} ^ {- 1} \hat {B} - C Q _ {1} \left(z I - N _ {1}\right) ^ {- 1} \hat {S} ^ {- 1} \hat {B} \right\| + \left\| C Q _ {1} \left(z I - N _ {1}\right) ^ {- 1} \hat {S} ^ {- 1} \hat {B} - C Q _ {1} \left(z I - N _ {1}\right) ^ {- 1} R _ {1} B \right\|\right) \\ \leq \sup _ {| z | = 1} \left(\left\| \hat {C} \hat {S} - C Q _ {1} \right\| \left\| (z I - \hat {S} ^ {- 1} \hat {N} _ {1} \hat {S}) ^ {- 1} \hat {S} ^ {- 1} \hat {B} \right\| + \| C Q _ {1} \| \left\| (z I - \hat {S} ^ {- 1} \hat {N} _ {1} \hat {S}) ^ {- 1} - (z I - N _ {1}) ^ {- 1} \right\| \left\| \hat {S} ^ {- 1} \hat {B} \right\|\right. \\ \end{array} +$$ + +$$ +\left. + \left\| C Q _ {1} \left(z I - N _ {1}\right) ^ {- 1} \right\| \left\| \hat {S} ^ {- 1} \hat {B} - R _ {1} B \right\|\right). \tag {87} +$$ + +Moreover, + +$$ +\left\| \left(z I - \hat {S} ^ {- 1} \hat {N} _ {1} \hat {S}\right) ^ {- 1} - \left(z I - N _ {1}\right) ^ {- 1} \right\| \leq \left\| \left(z I - \hat {S} ^ {- 1} \hat {N} _ {1} \hat {S}\right) ^ {- 1} \right\| \left\| \hat {S} ^ {- 1} \hat {N} _ {1} \hat {S} - N _ {1} \right\| \| (z I - N _ {1}) ^ {- 1} \| \tag {88} +$$ + +Substituting (88) into (87), we obtain + +$$ +\begin{array}{l} \left\| \hat {F} (z) - F (z) \right\| _ {H _ {\infty}} \leq \sup _ {| z | = 1} \left(\epsilon_ {C} \left\| (z I - \hat {S} ^ {- 1} \hat {N} _ {1} \hat {S}) ^ {- 1} \hat {S} ^ {- 1} \hat {B} \right\| + \epsilon_ {N} \| C Q _ {1} \| \left\| (z I - \hat {S} ^ {- 1} \hat {N} _ {1} \hat {S}) ^ {- 1} \right\| \left\| (z I - N _ {1}) ^ {- 1} \right\| \left\| \hat {S} ^ {- 1} \hat {B} \right\|\right. \\ \left. + \epsilon_ {B} \left\| C Q _ {1} \left(z I - N _ {1}\right) ^ {- 1} \right\|\right) \\ \leq \sup _ {| z | = 1} \left(4 \left\| \hat {S} ^ {- 1} \hat {B} \right\| \left\| (z I - N _ {1}) ^ {- 1} \right\| \epsilon_ {C} + 2 \| C Q _ {1} \| \left\| \hat {S} ^ {- 1} \hat {B} \right\| \left\| (z I - N _ {1}) ^ {- 1} \right\| ^ {2} \epsilon_ {N} + \| C Q _ {1} \| \left\| (z I - N _ {1}) ^ {- 1} \right\| \right. (89) \\ \leq \sup _ {| z | = 1} \left(8 L \left\| (z I - N _ {1}) ^ {- 1} \right\| \epsilon_ {C} + 4 \| C Q _ {1} \| L \left\| (z I - N _ {1}) ^ {- 1} \right\| ^ {2} \epsilon_ {N} + \| C Q _ {1} \| \left\| (z I - N _ {1}) ^ {- 1} \right\| \epsilon_ {B}\right) (90) \\ \end{array} +$$ + +where (89) requires $\left\| (zI - N_1 + N_1 - \hat{S}^{-1}\hat{N}_1\hat{S})^{-1}\right\|\leq \frac{1}{\sigma_{\min}(zI - N_1 + N_1 - \hat{S}^{-1}\hat{N}_1\hat{S})}\leq$ $\frac{1}{\sigma_{\min}(zI - N_1) - \epsilon_N}\leq \frac{2}{\sigma_{\min}(zI - N_1)} = 2\left\| (zI - N_1)^{-1}\right\|$ , which in turn requires, by Lemma B.2, + +$$ +\epsilon_ {N} \leq \frac {1}{2} \inf _ {| z | = 1} \sigma_ {\min } (z I - N _ {1}) \quad \Leftarrow \quad \epsilon \leq \frac {c _ {1} \tilde {o} _ {\min } (m)}{8 L c _ {2}} \inf _ {| z | = 1} \sigma_ {\min } (z I - N _ {1}) \Leftarrow \epsilon \leq \frac {c _ {1} \tilde {o} _ {\min } (m)}{8 L ^ {2} c _ {2}} (| \lambda_ {k} | - 1), \tag {91} +$$ + +where the last step uses $\inf_{|z| = 1}\sigma_{\min}(zI - N_1) = \frac{1}{\sup_{|z| = 1}\|(zI - N_1)^{-1}\|} \geq \frac{|\lambda_k| - 1}{L}$ . Eq. (90) uses $\left\| \hat{S}^{-1}\hat{B}\right\| \leq \| R_1B\| + \epsilon_B \leq 2L$ , which requires + +$$ +\epsilon_ {B} \leq L \quad \Leftarrow \quad \epsilon \leq \min \left(\frac {L}{1 6} \frac {c _ {1} \tilde {c} _ {\min } (m)}{c _ {2} L}, \frac {L}{1 6} \tilde {o} _ {\min } (m)\right). \tag {92} +$$ + +We will verify (91) and (92) at the end of the proof of Theorem 5.3 to make sure it is met by our choice of algorithm parameters. Lastly, substituting $\max (\| R_1B\| ,\| CQ_1\|)\leq L$ and using $\sup_{|z| = 1}\| (zI - N_1)^{-1}\| \leq \frac{L}{|\lambda_k| - 1}$ give the desired result. + +We are now ready to prove Proposition B.5, + +Proof of Proposition B.5. We only prove part (a). The proof of part (b) is similar and is hence omitted. Suppose $K(z)$ is a controller that satisfies the condition in part(a). Our goal is to show this $K(z)$ stabilizes $\hat{F}(z)$ , or in other words, $\hat{F}_K(z)$ is analytic on $\mathbb{C}_{\geq 1}$ . Let $\Delta_F(z) \coloneqq F(z) - \hat{F}(z)$ , we have + +$$ +\begin{array}{l} I - \hat {F} (z) K (z) = I - F (z) K (z) + \Delta_ {F} (z) K (z) \\ = (I + \Delta_ {F} (z) K (z) (I - F (z) K (z)) ^ {- 1}) (I - F (z) K (z)). \\ \end{array} +$$ + +Note that to show that $(I - \hat{F}(z)K(z))^{-1}$ is analytic on $\mathbb{C}_{\geq 1}$ , we only need to show $(I - \hat{F}(z)K(z))$ has the same number of zeros on $\mathbb{C}_{\geq 1}$ as that of $I - F(z)K(z)$ , because we know $(I - F(z)K(z))$ is non-singular (i.e. has no zeros) on $\mathbb{C}_{\geq 1}$ . Given (91), we also know $I - F(z)K(z)$ and $I - \hat{F}(z)K(z)$ has the same number of poles on $\mathbb{C}_{\geq 1}$ . By Rouche's Theorem (Theorem H.4), $I - \hat{F}(z)K(z)$ has the same number of zeros on $\mathbb{C}_{\geq 1}$ as $I - F(z)K(z)$ if + +$$ +\begin{array}{l} \left\| \left(I - \hat {F} (z) K (z)\right) - \left(I - F (z) K (z)\right) \right\| < \| \left(I - F (z) K (z)\right) \|, \forall | z | = 1 \\ \Leftarrow \sup _ {| z | = 1} \| \Delta_ {F} (z) K (z) (I - F (z) K (z)) ^ {- 1} \| < 1 \\ \Leftrightarrow \sup _ {| z | = 1} \| \Delta_ {F} (z) \| \| K (z) (I - F (z) K (z)) ^ {- 1} \| \leq \epsilon_ {*} \sup _ {| z | = 1} \| K (z) F _ {K} (z) \| < 1, \\ \end{array} +$$ + +where the last condition is satisfied by the condition in part (a) of this proposition. Therefore, $I - \hat{F}(z)K(z)$ has no zeros on $\mathbb{C}_{\geq 1}$ and the proof is concluded. + +□ + +We are now ready to prove Theorem 5.3. + +Proof of Theorem 5.3. Suppose for now the number of samples is large enough such that the condition in Proposition B.5 holds: $\sup_{|z| = 1}\left\| F(z) - \hat{F} (z)\right\| < \epsilon_{*}$ . At the end of the proof we will provide a sample complexity bound for which this holds. + +By Assumption 5.2, there exists $K(z)$ that stabilizes $F(z)$ with + +$$ +\| K (z) F _ {K} (z) \| _ {H _ {\infty}} \leq \frac {1}{\| \Delta (z) \| _ {H _ {\infty}} + 3 \epsilon_ {*}}. +$$ + +Therefore, by Proposition B.5, $K(z)$ stabilizes plant $\hat{F} (z)$ as well. Therefore, $\| K(z)\hat{F}_K(z)\|_{H_\infty}$ is well defined and can be evaluated by only taking sup over $|z| = 1$ : + +$$ +\begin{array}{l} \left\| K (z) \hat {F} _ {K} (z) \right\| _ {H _ {\infty}} = \left\| K (z) \left(I - \hat {F} (z) K (z)\right) ^ {- 1} \right\| _ {H _ {\infty}} \\ = \sup _ {| z | = 1} \left\| K (z) \left(I - \hat {F} (z) K (z)\right) ^ {- 1} \right\| _ {2} \\ = \sup _ {| z | = 1} \left\| K (z) (I - F (z) K (z) + F (z) K (z) - \hat {F} (z) K (z)) ^ {- 1} \right\| _ {2} \\ = \sup _ {| z | = 1} \left\| K (z) (I - F (z) K (z) + \Delta_ {F} (z) K (z)) ^ {- 1} \right\| _ {2} \\ = \sup _ {| z | = 1} \left\| K (z) \underbrace {(I - F (z) K (z)) ^ {- 1}} _ {F _ {K} (z)} (I + \Delta_ {F} (z) K (z) \underbrace {(I - F (z) K (z)) ^ {- 1}} _ {F _ {K} (z)}) ^ {- 1} \right\| _ {2} \\ = \sup _ {| z | = 1} \| K (z) F _ {K} (z) (I + \Delta_ {F} (z) K (z) F _ {K} (z)) ^ {- 1} \| _ {2} \\ \leq \sup _ {| z | = 1} \| K (z) F _ {K} (z) \| \sum_ {i = 0} ^ {\infty} \| \Delta_ {F} (z) \| ^ {i} \| K (z) F _ {K} (z) \| ^ {i} \\ = \sup _ {| z | = 1} \frac {\| K (z) F _ {K} (z) \|}{1 - \| \Delta_ {F} (z) \| \| K (z) F _ {K} (z) \|} \\ = \frac {\| K (z) F _ {K} (z) \| _ {H _ {\infty}}}{1 - \epsilon_ {*} \| K (z) F _ {K} (z) \| _ {H _ {\infty}}} \\ \leq \frac {1}{\left\| \Delta (z) \right\| _ {H _ {\infty}} + 2 \epsilon_ {*}}, \tag {93} \\ \end{array} +$$ + +where we substituted in Assumption 5.2 in the last inequality. In Algorithm 1, we find a controller $\hat{K}(z)$ that stabilizes $\hat{F}(z)$ and minimizes $\| \hat{K}(z) \hat{F}_{\hat{K}}(z) \|_{H_{\infty}}$ . As $K(z)$ stabilizes $\hat{F}(z)$ and satisfies the upper bound in (93), the $\hat{K}(z)$ returned by the algorithm must stabilize $\hat{F}(z)$ and is such that $\| \hat{K}(z) \hat{F}_{\hat{K}}(z) \|_{H_{\infty}}$ is upper bounded by the RHS of (93). By Proposition B.5, we know $\hat{K}(z)$ stabilizes $F(z)$ as well. Therefore, $\| \hat{K}(z) F_{\hat{K}}(z) \|_{H_{\infty}}$ is well defined and can be evaluated on taking the sup over $|z| = 1$ : + +$$ +\begin{array}{l} \| \hat {K} (z) F _ {\hat {K}} (z) \| _ {H _ {\infty}} = \left\| \hat {K} (z) (I - F (z) \hat {K} (z)) ^ {- 1} \right\| _ {H _ {\infty}} \\ = \sup _ {| z | = 1} \left\| \hat {K} (z) (I - F (z) \hat {K} (z)) ^ {- 1} \right\| _ {2} \\ = \sup _ {| z | = 1} \left\| \hat {K} (z) (I - \hat {F} (z) \hat {K} (z) + \hat {F} (z) \hat {K} (z) - F (z) \hat {K} (z)) ^ {- 1} \right\| _ {2} \\ = \sup _ {| z | = 1} \left\| \hat {K} (z) (I - \hat {F} (z) \hat {K} (z) - \Delta_ {F} (z) \hat {K} (z)) ^ {- 1} \right\| _ {2} \\ \end{array} +$$ + +$$ +\begin{array}{l} = \sup _ {| z | = 1} \left\| \hat {K} (z) \underbrace {(I - \hat {F} (z) \hat {K} (z)) ^ {- 1}} _ {\hat {F} _ {\hat {K}} (z)} (I - \Delta_ {F} (z) \hat {K} (z) \underbrace {(I - \hat {F} (z) \hat {K} (z)) ^ {- 1}} _ {\hat {F} _ {\hat {K}} (z)}) ^ {- 1} \right\| _ {2} \\ = \sup _ {| z | = 1} \left\| \hat {K} (z) \hat {F} _ {\hat {K}} (z) \left(I - \Delta_ {F} (z) \hat {K} (z) \hat {F} _ {\hat {K}} (z)\right) ^ {- 1} \right\| _ {2} \\ \leq \left\| \hat {K} (z) \hat {F} _ {\hat {K}} (z) \right\| _ {H _ {\infty}} \sup _ {| z | = 1} \sum_ {i = 0} ^ {\infty} \| \Delta_ {F} (z) \| ^ {i} \left\| \hat {K} (z) \hat {F} _ {\hat {K}} (z) \right\| ^ {i} \\ = \frac {\left\| \hat {K} (z) \hat {F} _ {\hat {K}} (z) \right\| _ {H _ {\infty}}}{1 - \epsilon_ {*} \left\| \hat {K} (z) \hat {F} _ {\hat {K}} (z) \right\| _ {H _ {\infty}}}. \tag {94} \\ \end{array} +$$ + +As $\hat{K}(z)$ is such that $\| \hat{K}(z) \hat{F}_{\hat{K}}(z) \|_{H_{\infty}}$ satisfies the upper bound by the RHS in (93), we have + +$$ +\frac {\left\| \hat {K} (z) \hat {F} _ {\hat {K}} (z) \right\| _ {H _ {\infty}}}{1 - \epsilon_ {*} \left\| \hat {K} (z) \right\| \hat {F} _ {K} (z)} _ {H _ {\infty}} \leq \frac {\frac {1}{\| \Delta (z) \| _ {H _ {\infty}} + 2 \epsilon_ {*}}}{1 - \epsilon_ {*} \frac {1}{\| \Delta (z) \| _ {H _ {\infty}} + 2 \epsilon_ {*}}} \leq \frac {1}{\| \Delta (z) \| _ {H _ {\infty}} + \epsilon_ {*}} < \frac {1}{\| \Delta (z) \| _ {H _ {\infty}}}. \tag {95} +$$ + +Therefore, by small gain theorem (Lemma 3.1), we know the controller $\hat{K} (z)$ stabilizes the original full system. + +For the remaining of the proof, we analyze the number of samples required to satisfy the condition in Proposition B.5, i.e. + +$$ +\begin{array}{l} \sup _ {| z | = 1} \left\| F (z) - \hat {F} (z) \right\| < \epsilon_ {*} \\ \Leftarrow \frac {1}{| \lambda_ {k} | - 1} \left(8 L ^ {2} \epsilon_ {C} + \frac {4 L ^ {4} \epsilon_ {N}}{| \lambda_ {k} | - 1} + L ^ {2} \epsilon_ {B}\right) < \epsilon_ {*} \\ \Leftarrow \left\{ \begin{array}{l l} \epsilon_ {N} & < \frac {\left(| \lambda_ {k} | - 1\right) ^ {2} \epsilon_ {*}}{1 2 L ^ {4}} \\ \epsilon_ {B} & < \frac {\left(| \lambda_ {k} | - 1\right) \epsilon_ {*}}{3 L ^ {2}} \\ \epsilon_ {C} & < \frac {\left(| \lambda_ {k} | - 1\right) \epsilon_ {*}}{2 4 L ^ {2}} \end{array} \right. \\ \end{array} +$$ + +where the middle step is by Lemma E.1. By Lemma B.2, Lemma B.4 and Lemma B.3, we obtain the following sufficient condition, + +$$ +\begin{array}{l} \frac {4 L c _ {2}}{c _ {1} \tilde {o} _ {\mathrm {m i n}} (m)} \epsilon \leq \frac {\left(| \lambda_ {k} | - 1\right) ^ {2} \epsilon_ {*}}{1 2 L ^ {4}}, \\ 8 \left(\frac {c _ {2} L}{c _ {1} \tilde {c} _ {\mathrm {m i n}} (m)} + \frac {1}{\tilde {o} _ {\mathrm {m i n}} (m)}\right) \epsilon < \frac {(| \lambda_ {k} | - 1) \epsilon_ {*}}{3 L ^ {2}}, \\ 2 \left(\frac {4 c _ {2} L}{c _ {1} \tilde {\sigma} _ {\operatorname* {m i n}} (m)} + \frac {1}{\tilde {c} _ {\operatorname* {m i n}} (m)}\right) \epsilon < \frac {(| \lambda_ {k} | - 1) \epsilon_ {*}}{2 4 L ^ {2}}. \\ \end{array} +$$ + +By simple computation, we obtain the following sufficient condition: + +$$ +\begin{array}{l} \epsilon < \min \left\{\frac {c _ {1} \tilde {o} _ {\mathrm {m i n}} (m) (| \lambda_ {k} | - 1) ^ {2} \epsilon_ {*}}{4 8 c _ {2} L ^ {5}}, \frac {c _ {1} \tilde {c} _ {\mathrm {m i n}} (m) (| \lambda_ {k} | - 1) \epsilon_ {*}}{4 8 c _ {2} L ^ {3}}, \frac {\tilde {o} _ {\mathrm {m i n}} (m) (| \lambda_ {k} | - 1) \epsilon_ {*}}{4 8 L ^ {2}}, \frac {c _ {1} \tilde {o} _ {\mathrm {m i n}} (m) (| \lambda_ {k} | - 1) \epsilon_ {*}}{3 8 4 c _ {2} L ^ {3}}, \right. \\ \left. \frac {\tilde {c} _ {\operatorname* {m i n}} (m) \left(\left| \lambda_ {k} \right| - 1\right) \epsilon_ {*}}{9 6 L ^ {3}} \right\}, \\ \end{array} +$$ + +which can be simplified into the following sufficient condition: + +$$ +\epsilon < \frac {c _ {1}}{3 8 4 c _ {2} L ^ {5}} \min \left(\tilde {o} _ {\min } (m), \tilde {c} _ {\min } (m)\right) \min \left(\left| \lambda_ {k} \right| - 1\right) ^ {2}, \left| \lambda_ {k} \right| - 1) \epsilon_ {*} \tag {96} +$$ + +Moreover, we also need to satisfy (47),(56),(79),(86),(91),(92), which requires + +$$ +\begin{array}{l} \epsilon < \left\{ \begin{array}{l} \frac {1}{4} \tilde {o} _ {\min } (m) \tilde {c} _ {\min } (m), c _ {1} ^ {2} \tilde {c} _ {\min } (m) ^ {2}, \frac {c _ {1} \tilde {o} _ {\min } (m)}{8 c _ {2}}, \frac {c _ {1} \tilde {c} _ {\min } (m)}{8 c _ {2}}, \\ \tilde {c} _ {\min } (m) \end{array} \right. \tag {97} \\ \left. \frac {c _ {1} \tilde {o} _ {\operatorname* {m i n}} (m)}{8 L ^ {2} c _ {2}} (| \lambda_ {k} | - 1), \frac {L}{1 6} \frac {c _ {1} \tilde {e} _ {\operatorname* {m i n}} (m)}{c _ {2} L}, \frac {L}{1 6} \tilde {o} _ {\operatorname* {m i n}} (m) \right\}. \\ \end{array} +$$ + +A sufficient condition that merges both (96) and (97) is: + +$$ +\epsilon < \frac {c _ {1}}{3 8 4 c _ {2} L ^ {5}} \min \left(\tilde {o} _ {\min } (m), \tilde {c} _ {\min } (m)\right) \min \left(\left(\left| \lambda_ {k} \right| - 1\right) ^ {2}, \left| \lambda_ {k} \right| - 1\right) \min \left(\epsilon_ {*}, 1\right). \tag {98} +$$ + +Plugging in the definition of $c_{1}, c_{2}, \tilde{o}_{\min}(m), \tilde{c}_{\min}(m)$ , we have $\frac{c_{1}}{c_{2}} = \sqrt{\frac{\sigma_{\min}(P_{1})^{2} \sigma_{s}}{8 \sigma_{\max}(P_{1})^{2} \tilde{\sigma}_{s}}} = \sqrt{\frac{\sigma_{\min}(P_{1})^{4} \sigma_{\min}(\mathcal{G}_{\mathrm{ob}}) \sigma_{\min}(\mathcal{G}_{\mathrm{con}})}{8 \sigma_{\max}(P_{1})^{4} \sigma_{\max}(\mathcal{G}_{\mathrm{ob}}) \sigma_{\max}(\mathcal{G}_{\mathrm{con}})}} = \frac{1}{\kappa(P_{1})^{2} \sqrt{8 \kappa(\mathcal{G}_{\mathrm{ob}}) \kappa(\mathcal{G}_{\mathrm{con}})}}$ , where we recall $\kappa(\cdot)$ means the condition number of a matrix. Further, as $\|(N_{1}^{-1})^{m/2}\| \leq L(\frac{1}{|\lambda_{k}|} + 1/2)^{m/2}$ , we have $\sigma_{\min}(N_{1}^{m/2}) \geq \frac{1}{L} \left( \frac{2}{|\lambda_{k}| + 1} \right)^{m/2}$ . Therefore, the condition (98) can be replaced with the following sufficient condition + +$$ +\epsilon \lesssim \frac {1}{\sqrt {\kappa (\mathcal {G} _ {\mathrm {o b}}) \kappa (\mathcal {G} _ {\mathrm {c o n}})} L ^ {8}} \min \left(\sigma_ {\min } \left(\mathcal {G} _ {\mathrm {o b}}\right), \sigma_ {\min } \left(\mathcal {G} _ {\mathrm {c o n}}\right)\right) \left(\frac {2 \left| \lambda_ {k} \right|}{\left| \lambda_ {k} \right| + 1}\right) ^ {m / 2} \min \left(\left| \lambda_ {k} \right| - 1\right) ^ {2}, \left| \lambda_ {k} \right| - 1) \min \left(\epsilon_ {*}, 1\right). \tag {99} +$$ + +where the $\lesssim$ above only hides numerical constants. + +To meet (58), (63), we set $p = q = \max(\alpha + \frac{m}{2}, m)$ . We also set $m$ large enough so that the right hand side of (99) is lower bounded by $\sqrt{m^3}$ . Using the simple fact that for $a > 1$ , $a^{m/2} = \sqrt{a}^{m/2} \sqrt{a}^{m/2} \geq \sqrt{a}^{m/2} m^{1.5} e^{1.5} \left( \frac{1}{6} \log a \right)^{1.5}$ , and replacing $a$ with $\frac{2|\lambda_k|}{|\lambda_k| + 1}$ , a sufficient condition for such an $m$ is + +$$ +m \gtrsim \frac {1}{\log \frac {2 | \lambda_ {k} |}{| \lambda_ {k} | + 1}} \log \frac {\sqrt {\kappa (\mathcal {G} _ {\mathrm {o b}}) \kappa (\mathcal {G} _ {\mathrm {c o n}})} L ^ {8}}{\min \left(\sigma_ {\min } (\mathcal {G} _ {\mathrm {o b}}), \sigma_ {\min } (\mathcal {G} _ {\mathrm {c o n}})\right) \left(\log \frac {2 | \lambda_ {k} |}{| \lambda_ {k} | + 1})\right) ^ {3 / 2} \min \left(\left(| \lambda_ {k} | - 1\right) ^ {2}, \left| \lambda_ {k} \right| - 1\right) \min \left(\epsilon_ {*}, 1\right)}, +$$ + +Recall that $\epsilon = 2\hat{\epsilon} + 2\tilde{\epsilon}$ . With the above condition on $m$ , it now suffices to require $\max(\hat{\epsilon}, \tilde{\epsilon}) \lesssim \sqrt{m^3}$ . Plugging in the definition of $\hat{\epsilon}, \tilde{\epsilon}$ in Theorem C.2, Lemma C.3, we need + +$$ +M > 8 d _ {u} T + 4 \left(d _ {u} + d _ {y} + 4\right) \log (3 T / \delta) \tag {101} +$$ + +$$ +\frac {8 \sigma_ {v} \sqrt {T (T + 1) \left(d _ {u} + d _ {y}\right) \min \{p , q \} \log (2 7 T / \delta)}}{\sigma_ {u} \sqrt {M}} \lesssim \sqrt {m ^ {3}} \tag {102} +$$ + +$$ +L ^ {3} \left(\frac {\left| \lambda_ {k + 1} \right| + 1}{2}\right) ^ {m} \lesssim \sqrt {m ^ {3}}. \tag {103} +$$ + +To satisfy (102), recall $T = m + p + q + 2$ . Let us also require + +$$ +m \geq 2 \alpha \tag {104} +$$ + +so that $p = q = \max \left( \alpha + \frac{m}{2}, m \right) = m$ , so we have $T(T + 1) \min (p, q) \lesssim m^3$ . Therefore, it suffices to require $M \gtrsim \left( \frac{\sigma_v}{\sigma_u} \right)^2 (d_u + d_y) \log \frac{m}{\delta}$ to satisfy (102). + +To satisfy (103), we need + +$$ +m \gtrsim \frac {\log \frac {1}{L}}{\log \frac {| \lambda_ {k + 1} | + 1}{2}}. \tag {105} +$$ + +To summarize, collecting the requirements on $m$ (100), (104), (105) and also (72), we have the final complexity on $m$ : + +$$ +\begin{array}{l} m \gtrsim \max (\alpha , \frac {\log \frac {1}{L}}{\log \frac {| \lambda_ {k + 1} | + 1}{2}}, \frac {\log \left(\frac {1}{L}\right)}{\log \left(\frac {\frac {1}{| \lambda_ {k} |} + 1}{2}\right)}, \tag {106} \\ \frac {1}{\log \frac {2 | \lambda_ {k} |}{| \lambda_ {k} | + 1}} \log \frac {\sqrt {\kappa (\mathcal {G} _ {\mathrm {o b}}) \kappa (\mathcal {G} _ {\mathrm {c o n}})} L ^ {8}}{\min \left(\sigma_ {\min } (\mathcal {G} _ {\mathrm {o b}}) , \sigma_ {\min } (\mathcal {G} _ {\mathrm {c o n}})\right) \left(\log \frac {2 | \lambda_ {k} |}{| \lambda_ {k} | + 1})\right) ^ {3 / 2} \min \left(\left(| \lambda_ {k} | - 1\right) ^ {2}, \left| \lambda_ {k} \right| - 1\right) \min \left(\epsilon_ {*}, 1\right)}. \\ \end{array} +$$ + +The final requirement on $p, q$ is, + +$$ +p = q = m. \tag {108} +$$ + +The final complexity on the number of trajectories is + +$$ +M \gtrsim \left(\frac {\sigma_ {v}}{\sigma_ {u}}\right) ^ {2} \left(d _ {u} + d _ {y}\right) \log \frac {m}{\delta} + d _ {u} m \tag {109} +$$ + +Lastly, in the most interesting regime that $|\lambda_k| - 1, 1 - |\lambda_{k + 1}|$ , $\epsilon_*$ is small, we have $\log \frac{2|\lambda_k|}{|\lambda_k| + 1} \asymp |\lambda_k| - 1$ , $\min((|\lambda_k| - 1)^2, |\lambda_k| - 1) = (|\lambda_k| - 1)^2$ , $\min(\epsilon_*|1) = \epsilon_*$ . In this case, (106) can be simplified: + +$$ +m \gtrsim \max \left(\alpha , \frac {\log L}{1 - | \lambda_ {k + 1} |}, \frac {1}{| \lambda_ {k} | - 1} \log \frac {\kappa (\mathcal {G} _ {\mathrm {o b}}) \kappa (\mathcal {G} _ {\mathrm {c o n}}) L}{\min \left(\sigma_ {\min } (\mathcal {G} _ {\mathrm {o b}}) , \sigma_ {\min } (\mathcal {G} _ {\mathrm {c o n}})\right) \left(| \lambda_ {k} | - 1\right) \epsilon_ {*}}\right). \tag {110} +$$ + +![](images/cab97e29b717acc33347a36f96675b0c12f2dbbdc0dd0260081e5ea0219f4537.jpg) + +# F System identification when $D \neq 0$ + +In the case when $D \neq 0$ in (1), the transfer function becomes + +$$ +F (z) = D + C Q _ {1} \left(z I - N _ {1}\right) ^ {- 1} R _ {1} B, \tag {111} +$$ + +and the estimated transfer function becomes $\hat{F} (z) = \hat{D} +\hat{C}\hat{S} (zI - \hat{S}^{-1}\hat{N}_1\hat{S})^{-1}\hat{S}^{-1}\hat{B}$ , so the gap between estimated transfer function and the ground-truth transfer function can be bounded as follows + +$$ +\left\| F (z) - \hat {F} (z) \right\| \leq \left\| D - \hat {D} \right\| + \left\| C Q _ {1} \left(z I - N _ {1}\right) ^ {- 1} R _ {1} B - \hat {C} \hat {S} \left(z I - \hat {S} ^ {- 1} \hat {N} _ {1} \hat {S}\right) ^ {- 1} \hat {S} ^ {- 1} \hat {B} \right\|. \tag {112} +$$ + +The second term in (112) can be bounded by Lemma E.1. Therefore, in this section, we focus on providing a bound for $\left\| D - \hat{D} \right\|$ . The recursive relationship of the process and measurement data will change from (14) to the following: + +$$ +\begin{array}{l} y _ {t} ^ {(i)} = C x _ {t} ^ {(i)} + D u _ {t} ^ {(i)} + v _ {t} ^ {(i)} \\ = \sum_ {j = 0} ^ {T} C A ^ {j} \left(B u _ {t - j - 1} ^ {(i)} + w _ {t - j - 1} ^ {(i)}\right) + D u _ {t} ^ {(i)} + v _ {t} ^ {(i)}. \tag {113} \\ \end{array} +$$ + +Therefore, we can estimate each block of the Hankel matrix as follows: + +$$ +\begin{array}{l} \Phi_ {D} = \left[ \begin{array}{c c c c c} D & C B & C A B & \dots & C A ^ {T - 2} B \end{array} \right] \tag {114} \\ = \left[ \begin{array}{c c c c} D & 0 & \ldots & 0 \end{array} \right] + \Phi , \\ \end{array} +$$ + +from which we can easily obtain the $D$ matrix and use $\Phi$ to design the controller, as in the rest of the main text. Fortunately, from (113), we see that $\Phi_D$ can also be estimated via (17), i.e. + +$$ +\hat {\Phi} _ {D} := \arg \min _ {X \in \mathbb {R} ^ {d _ {y} \times (T * d _ {u})}} \sum_ {i = 1} ^ {M} \left\| y ^ {(i)} - X U ^ {(i)} \right\| _ {F} ^ {2}. \tag {115} +$$ + +In particular, we see that even in the case where $D \neq 0$ , the estimation error of $D$ does not affect the estimation of $\Phi$ or $\hat{\tilde{\Phi}}$ . Therefore, the error bound of $N_{1}, CQ_{1}, R_{1}B$ in Lemma B.2, Lemma B.3, Lemma B.4 still holds. The error of estimating $D$ can be bounded as follows: + +Lemma F.1 (Corollary 3.1 of [51]). Let $\hat{D}$ denote the first block submatrix in (115), then + +$$ +\left\| \hat {D} - D \right\| \leq \frac {L}{\sqrt {M}}, +$$ + +where $L$ is a constant depending on $A, B, C, D$ , the dimension constants $n, m, d_u, d_y$ , and the variance of control and system noise $\sigma_u, \sigma_v, \sigma_w$ . + +Therefore, we have proved a bound in the place of Lemma E.1. The rest of the proof will follow the same line as Appendix E. + +# G Bounding $\hat{S}$ when $N_{1}$ is not diagonalizable + +In proving Theorem 5.3, we used the diagonalizability assumption in Lemma D.1. More specifically, it was used in (48) to upper and lower bound $\Lambda^{\frac{m}{2} +j\alpha}(\Lambda^{-\frac{m}{2} -j\alpha})^{*}$ for $j = 0,1,\ldots ,\frac{p}{\alpha}$ , which is reflected in the value of $\bar{\sigma}_s,\underline{\sigma}_s$ . In the diagonalizable case, $\Lambda^{\frac{m}{2} +j\alpha}(\Lambda^{-\frac{m}{2} -j\alpha})^{*}$ is a diagonal matrix with all entries having modulus 1 regardless of the value of $j$ . Now in the non-diagonalizable case, we bound $\Lambda^{\frac{m}{2} +j\alpha}(\Lambda^{-\frac{m}{2} -j\alpha})^{*}$ , provide new values for $\bar{\sigma}_s,\underline{\sigma}_s$ , and analyze how it affects the final sample complexity. + +Consider + +$$ +\Lambda = \left[ \begin{array}{c c c c c} J _ {1} & 0 & 0 & \dots & 0 \\ 0 & J _ {2} & 0 & \dots & 0 \\ 0 & 0 & J _ {3} & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & \dots & \dots & 0 & J _ {k, J} \end{array} \right] \tag {116} +$$ + +where $k_{J}$ is the number of Jordan blocks, and each $J_{i}$ is a Jordan block that is either a scalar or a square matrix with eigenvalues on the diagonal, 1's on superdiagonal, and zeros everywhere else. + +Without loss of generality, assume $J_{1}$ is the largest Jordan block with eigenvalue $\lambda$ satisfying $|\lambda| > 1$ and size $\gamma$ , then $J_{1} = \lambda I + \Gamma$ , where $\Gamma$ is the nilpotent super-diagonal matrix of 1's such that $\Gamma^i = 0$ for all $i \geq \gamma$ . Therefore, we have + +$$ +J _ {1} ^ {\frac {m}{2} + j \alpha} = (\lambda I + \Gamma) ^ {\frac {m}{2} + j \alpha} = \lambda^ {\frac {m}{2} + j \alpha} \sum_ {i = 0} ^ {\gamma} \binom {\frac {m}{2} + j \alpha} {i} \frac {\Gamma^ {i}}{\lambda^ {i}}, +$$ + +and + +$$ +(J_{1}^{-\frac{m}{2} -j\alpha})^{*} = \bar{\lambda}^{-\frac{m}{2} -j\alpha}(I + \frac{1}{\bar{\lambda}}\Gamma^{*})^{-\frac{m}{2} -j\alpha} = \bar{\lambda}^{-\frac{m}{2} -j\alpha}\left(\sum_{i = 0}^{\gamma}\binom {\frac{m}{2} +j\alpha}{i}\frac{(\Gamma^{*})^{i}}{\bar{\lambda}^{i}}\right)^{-1}, +$$ + +where $\bar{\lambda}$ is the conjugate of $\lambda$ . Because $\lambda^{\frac{m}{2} + j\alpha} \bar{\lambda}^{-\frac{m}{2} - j\alpha}$ has modulus 1, we have + +$$ +\sigma_ {\max } \left(J _ {1} ^ {\frac {m}{2} + j \alpha} \left(J _ {1} ^ {- \frac {m}{2} - j \alpha}\right) ^ {*}\right) \leq \frac {\sigma_ {\max } \left(\sum_ {i = 0} ^ {\gamma} \binom {\frac {m}{2} + j \alpha} {i} \frac {\Gamma^ {i}}{\lambda^ {i}}\right)}{\sigma_ {\min } \left(\sum_ {i = 0} ^ {\gamma} \binom {\frac {m}{2} + j \alpha} {i} \frac {\Gamma^ {i}}{\lambda^ {i}}\right)}. \tag {117} +$$ + +We can then calculate, + +$$ +\begin{array}{l} \sigma_ {\max } \left(\sum_ {i = 0} ^ {\gamma} \binom {\frac {m}{2} + j \alpha} {i} \frac {\Gamma^ {i}}{\lambda^ {i}}\right) \leq \sum_ {i = 0} ^ {\gamma} \binom {\frac {m}{2} + j \alpha} {i} \\ \leq \gamma \left( \begin{array}{c} \frac {m}{2} + j \alpha \\ \gamma \end{array} \right) \\ \leq \gamma (\frac {m}{2} + j \alpha) ^ {\gamma} \\ \leq \gamma (2 m) ^ {\gamma} \\ \end{array} +$$ + +where the second inequality requires $m \geq 4\gamma$ , and the last inequality uses $j\alpha \leq p = m$ . To calculate the smallest singular value, note that $\sum_{i=0}^{\gamma}\binom{\frac{m}{2}+j\alpha}{i}\frac{\Gamma^{i}}{\lambda^{i}}$ is an upper triangular matrix with diagonal entries being 1, therefore its smallest singular value is lower bounded by 1. Therefore, we have + +$$ +\sigma_ {\max } \left(J _ {1} ^ {\frac {m}{2} + j \alpha} \left(J _ {1} ^ {- \frac {m}{2} - j \alpha}\right) ^ {*}\right) \leq \gamma (2 m) ^ {\gamma}, \tag {118} +$$ + +$$ +\sigma_ {\max } \left(J _ {1} ^ {\frac {m}{2} + j \alpha} \left(J _ {1} ^ {- \frac {m}{2} - j \alpha}\right) ^ {*}\right) \geq \frac {1}{\gamma (2 m) ^ {\gamma}}. \tag {119} +$$ + +As such, the constants $\bar{\sigma}_s,\underline{\sigma}_s$ in (49) need to be modified to + +$$ +\bar {\sigma} _ {s} ^ {\prime} = \bar {\sigma} _ {s} \gamma (2 m) ^ {\gamma}, \tag {120} +$$ + +$$ +\underline {{\sigma}} _ {s} ^ {\prime} = \underline {{\sigma}} _ {s} \frac {1}{\gamma (2 m) ^ {\gamma}}. \tag {121} +$$ + +This will affect the sample complexity calculation step in (99), which will become + +$$ +\epsilon \lesssim \frac {1}{\sqrt {\kappa (\mathcal {G} _ {\mathrm {o b}}) \kappa (\mathcal {G} _ {\mathrm {c o n}})} L ^ {8} \gamma (2 m) ^ {\gamma}} \min \left(\sigma_ {\min } \left(\mathcal {G} _ {\mathrm {o b}}\right), \sigma_ {\min } \left(\mathcal {G} _ {\mathrm {c o n}}\right)\right) \left(\frac {2 | \lambda_ {k} |}{| \lambda_ {k} | + 1}\right) ^ {m / 2} \min \left(\left(| \lambda_ {k} | - 1\right) ^ {2}, \left| \lambda_ {k} \right| - 1\right) \min \left(\epsilon_ {*}, 1\right), \tag {122} +$$ + +where the only difference is the additional factor in the denominator highlighted in blue. As this factor is only polynomial in $m$ , we can merge it with the exponential factor $\left(\frac{2|\lambda_k|}{|\lambda_k| + 1}\right)^{m / 2}$ such that + +$$ +\frac {\left(\frac {2 | \lambda_ {k} |}{| \lambda_ {k} | + 1}\right) ^ {m / 2}}{\gamma (2 m) ^ {\gamma}} \geq (\sqrt {\frac {2 | \lambda_ {k} |}{| \lambda_ {k} | + 1}}) ^ {m / 2} \frac {1}{\gamma} (\frac {e \log \frac {2 | \lambda_ {k} |}{| \lambda_ {k} | + 1}}{8 \gamma}) ^ {\gamma}. +$$ + +Therefore, (100) will be changed into: + +$$ +m \gtrsim \frac {1}{\log \frac {2 | \lambda_ {k} |}{| \lambda_ {k} | + 1}} \log \frac {\sqrt {\kappa (\mathcal {G} _ {\mathrm {o b}}) \kappa (\mathcal {G} _ {\mathrm {c o n}})} L ^ {8}}{\min \left(\sigma_ {\min } \left(\mathcal {G} _ {\mathrm {o b}}\right), \sigma_ {\min } \left(\mathcal {G} _ {\mathrm {c o n}}\right)\right) \frac {1}{\gamma} \left(\frac {e \log \frac {2 | \lambda_ {k} |}{| \lambda_ {k} | + 1}}{8 \gamma}\right) ^ {\gamma} \left(\log \frac {2 | \lambda_ {k} |}{| \lambda_ {k} | + 1}\right)) ^ {3 / 2} \min \left(\left(| \lambda_ {k} | - 1\right) ^ {2}, \left| \lambda_ {k} \right| - 1\right) \min \left(\epsilon_ {*}, 1\right)} \tag {123} +$$ + +and lastly, the final simplified complexity on $m$ will be changed into + +$$ +m \gtrsim \max (\alpha , \frac {\log L}{1 - | \lambda_ {k + 1} |}, \frac {\gamma}{| \lambda_ {k} | - 1} \log \frac {\kappa \left(\mathcal {G} _ {\mathrm {o b}}\right) \kappa \left(\mathcal {G} _ {\mathrm {c o n}}\right) L \gamma}{\min \left(\sigma_ {\min } \left(\mathcal {G} _ {\mathrm {o b}}\right) , \sigma_ {\min } \left(\mathcal {G} _ {\mathrm {c o n}}\right)\right) \left(| \lambda_ {k} | - 1\right) \epsilon_ {*}}). \tag {124} +$$ + +which only adds a multiplicative factor in $\gamma$ and the additional $\gamma$ in the log. Given such a change in the bound for $m$ , the bound for other algorithm parameters $p, q, M$ is the same (except for the changes caused by their dependence on $m$ ). + +# H Additional Helper Lemmas + +Lemma H.1. Given Assumption 5.1 is satisfied, then $N_{1}, CQ_{1}$ is observable, $N_{1}, R_{1}B$ is controllable. + +Proof. Let $w$ denote any unit eigenvector of $N_{1}$ with eigenvalue $\lambda$ , then + +$$ +N _ {1} w = \lambda w \quad \Rightarrow \quad A Q _ {1} w = Q _ {1} N _ {1} w = \lambda (Q _ {1} w). +$$ + +Therefore, $Q_{1}w$ is an eigenvector of $A$ . As $A, C$ is observable, by PBH test, this leads to + +$$ +\| C Q _ {1} w \| > 0. +$$ + +By PBH Test, this directly leads to $(N_{1},CQ_{1})$ is observable. The controllability part is similar. + +![](images/992f436b2ee52018fa026b9995eae56fb2c06e9f81ea98a745cd8e877db1aa43.jpg) + +Lemma H.2 (Gelfand's formula). For any square matrix $X$ , we have + +$$ +\rho (X) = \lim _ {t \rightarrow \infty} \left\| X ^ {t} \right\| ^ {1 / t}. +$$ + +In other words, for any $\epsilon > 0$ , there exists a constant $\zeta_{\epsilon}(X)$ such that + +$$ +\sigma_ {\max } (X ^ {t}) = \| X \| \leq \zeta_ {\epsilon} (X) (\rho (X) + \epsilon) ^ {t}. +$$ + +Further, if $X$ is invertible, let $\lambda_{\min}(X)$ denote the eigenvalue of $X$ with minimum modulus, then + +$$ +\sigma_ {\min } (X ^ {t}) \geq \frac {1}{\zeta_ {\epsilon} (X ^ {- 1})} \left(\frac {| \lambda_ {\min } (X) |}{1 + \epsilon | \lambda_ {\min } (X) |}\right) ^ {t}. +$$ + +The proof can be found in existing literatures (e.g. [18]). + +Lemma H.3. For two matrices $H, E$ , $\frac{1}{2} H^{*}H - E^{*}E \preceq (H + E)^{*}(H + E) \preceq 2H^{*}H + 2E^{*}E$ . + +Proof. For the upper bound, notice that + +$$ +2 H ^ {*} H + 2 E ^ {*} E - (H + E) ^ {*} (H + E) = H ^ {*} H + E ^ {*} E - H ^ {*} E - E ^ {*} H = (H - E) ^ {*} (H - E) ^ {*} \succeq 0 +$$ + +Therefore, we have + +$$ +(H + E) ^ {*} (H + E) \preceq 2 H ^ {*} H + 2 E ^ {*} E. \tag {125} +$$ + +For the lower bound, we have + +$$ +H ^ {*} H = (H + E - E) ^ {*} (H + E - E) ^ {*} \preceq 2 (H + E) ^ {*} (H + E) + 2 E ^ {*} E +$$ + +where in the last inequality, we used (125). Rearrange the terms, we get the lower bound. + +$$ +\frac {1}{2} H ^ {*} H - E ^ {*} E \preceq (H + E) ^ {*} (H + E). +$$ + +![](images/2956602f3549ce214a802ef2539aec0a6c5ceab4bae5a3c4582ff04cf3a0b2c5.jpg) + +Theorem H.4 (Rouche's theorem). Let $D \subset \mathbb{C}$ be a simply connected domain, $f$ and $g$ two meromorphic functions on $D$ with a finite set of zeros and poles $F$ . Let $\gamma$ be a positively oriented simple closed curve which avoids $F$ and bounds a compact set $K$ . If $|f - g| < |g|$ along $\gamma$ , then + +$$ +N _ {f} - P _ {f} = N _ {g} - P _ {g} +$$ + +where $N_{f}$ (resp. $P_{f}$ ) denotes the number of zeros (resp. poles) of $f$ within $K$ , counted with multiplicity (similarly for $g$ ). + +For proof, see e.g. [2]. + +# I Indexing + +For the convenience of readers, we provide a table summarizing all constants appearing in the bounds. + +Table 1: Lists of parameters and constants appearing in the bound. + +
ConstantAppearanceExplanation
TSection 1Length of each roll out.
MSection 3Number of roll outs.
mSection 3number of open-loop steps for converging to unstable state space.
p,qSection 3dimension of Hankel matrix to be estimated.
+ +Table 2: System theoretic parameters. + +
ConstantAppearanceExplanation
H(5)Ground truth Hankel matrix
O, C(6)factorization of H.
O, C(7)rank-k approximation of O, C, respectively.
H(8)rank-k approximation of H.
H(18)numerical approximation of H.
H(19)rank-k approximation of H.
O, C(20)factorization of H.
GobSection 5Observability Gramian
GconSection 5Controllability Gramian
δmin(m)Lemma D.1a positive real number such that σmin(δ) ≥ δmin(m).
cmin(m)Lemma D.1a positive real number such that σmin(C) ≥ cmin(m).
αAppendix CControllability index.
C,C,CB(48)modified controllability and observability matrices.
ζε(·)Lemma H.2Gelfand constant for the norm of matrix exponents
+ +Table 3: Shorthand notations (introduced in proofs). + +
ConstantAppearanceExplanation
σsLemma D.1σs := σmin(Gob) / σmax(Gcon) σmin(P1)2
σsLemma D.1σs := σmax(Gob) / σmin(Gcon) σmax(P1)2
εNLemma B.2εN := 4Lc2ε / c1δmin(m)
εCLemma B.3εC := 2(4c2L / c1δmin(m) + 1 / cmin(m))ε
εBLemma B.4εB := 8(c2L / c1δmin(m) + 1 / δmin(m))ε
ε*Assumption 5.2
εLemma B.1ε := 2ˆ + 2ˆ
ˆTheorem C.2ˆ := 8σv√T(T+1)(du+dy) min{p,q} log(27T/δ) / σu√M
ˆLemma C.3ˆ := L3(λk+1+1/2)m
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However, existing models and benchmarks fail to capture the complexity and requirements of professional video generation. Towards that goal, we introduce Stable Cinemetrics, a structured evaluation framework that formalizes filmmaking controls into four disentangled, hierarchical taxonomies: Setup, Event, Lighting, and Camera. Together, these taxonomies define 76 fine-grained control nodes grounded in industry practices. Using these taxonomies, we construct a benchmark of prompts aligned with professional use cases and develop an automated pipeline for prompt categorization and question generation, enabling independent evaluation of each control dimension. We conduct a large-scale human study spanning $10+$ models and 20K videos, annotated by a pool of $80+$ film professionals. Our analysis, both coarse and fine-grained reveal that even the strongest current models exhibit significant gaps, particularly in Events and Camera-related controls. To enable scalable evaluation, we train an automatic evaluator, a vision-language model aligned with expert annotations that outperforms existing zero-shot baselines. SCINE is the first approach to situate professional video generation within the landscape of video generative models, introducing taxonomies centered around cinematic controls and supporting them with structured evaluation pipelines and detailed analyses to guide future research. + +# 1 Introduction + +The field of video generative models has made significant progress in recent years [37], drawing substantial interest from both academia and industry. This can be evidenced by the growing number of benchmarks [25, 32, 2, 35], datasets [41, 58], and both open- [3, 56, 14, 65] and closed- [6, 54, 36] source models that have collectively driven the field forward. The foundational nature of these models makes them useful for several downstream tasks, including video editing [26], 3D generation [55] and robotics [69]. This widespread adoption of video generative models underpins the growing assertion that they represent a revolution for professional video generation. + +Generative vision offers tremendous potential for media creation, but a fundamental question remains: how can we shift generative video from casual, exploratory synthesis to a medium that supports professional-grade, controllable cinematic outputs? The important distinction between casual and professional generative video lies in the critical gap of cinematic control [7]: while today's models can generate videos of "an astronaut riding a horse", professional creation necessitates granular control over cinematic elements such as the framing of the shot, position of the key light, and even whether the astronaut smiles before or after the horse gallops away - a truly professional video generation system must put every one of those cinematic choices back in the creator's hands. The need for this exact control over every cinematic element, from the timing of a smile to the quality of + +![](images/c150f61fa92cff11ff299cd776026bb0f503cc6e3b93858fc57dce1745a6009a.jpg) + +![](images/cc1e518b277b02107a59a9d8131fbfa9ba81311e33fddb5f659f8f3e973b445d.jpg) +Figure 1: Stable Cinemetrics introduces structured taxonomies grounded in the controls required for professional video generation. These taxonomies form the foundation of our prompt based benchmark that mirrors real-world shot creation, progressing from scriptwriting to on-screen visuals. Every control element in a prompt is automatically categorized back to the taxonomy, enabling the generation of isolated evaluation questions for independent investigation into each element. This supports large scale human evaluation enabling both coarse and fine-grained insights into the capabilities of current models for professional video generation. To drive scalable annotations, we develop our own VLMs that outperform existing models in alignment with human judgements. + +![](images/174957b007f44fa8ac1309b540864182ed46ea70a4fa1f9ffc021fe722c66f28.jpg) + +![](images/1d83d3123f5b07da33a32228ff7861019348907f3d264bda432c6f57fe4e4e94.jpg) + +light, is the very reason filmmakers shoot multiple takes, selecting only the frames where everything comes together to tell the story most effectively [10]. + +In this work, we investigate the intersection of video generative models and the nuanced control mechanisms associated with professional video production. Despite rapid advancements in video generative modeling, the field still lacks both a clear definition of essential cinematic controls and standardized evaluation protocols for benchmarking progress at this crucial intersection. To bridge this gap, we present SCINE (Stable Cinemetrics) - an evaluation suite specifically designed to characterize this intersection, enabling us to directly address the question: "Are Current Video Generative Models Ready for Professional Use?" + +The core focus of SCINE is to develop a taxonomy, at the intersection of generative vision and professional video generation, built on the principles of control and specificity [4, 28, 43]. These principles are important in film-making, as every decision carries cinematic meaning; a low angle conveys power while low-key lighting evokes drama. SCINE captures these principles by organizing control knobs into four cinematic pillars - Setup, Events, Lighting, and Camera - enabling the evaluation of video generative models along the same axes that a professional would. This disentanglement also allows evaluation around personalization: which aligns with the collaborative nature of real productions, in comparison to the monolithic nature of current video generation models. Our taxonomy is hierarchical, branching from coarse cinematic concepts to leaf-level controls that naturally map onto computer vision concepts such as object semantics and scene geometry [49]. + +Leveraging our taxonomy, we generate two prompt types: story-driven and visual exposition, to mirror professional workflows. Story-driven prompts act as mini-screenplays [17], specifying characters, dialogue, actions, and emotions. We enrich these with visual exposition cues, by sampling control nodes from our taxonomy, emulating the transition from script to shot in filmmaking [46]. Sampling control nodes allows automated (a) prompt categorization: mapping each control element to the taxonomy, and (b) generation of targeted evaluation questions for each element, allowing disentagged evaluation of each cinematic control. The structured nature of our taxonomy supports scalable human annotation: we evaluate $10+$ models across 20K generated videos with feedback from 80+ professionals. This enables analysis across control dimensions where we observe substantial variance in performance across taxonomy pillars, even for top-performing models such as WAN-14B and Minimax. Our taxonomy facilitates both coarse insights, showing that models struggle most with + +Events and Camera and fine-grained comparisons, such as better performance on shot size over camera framing, and on natural over artificial lighting. To support automatic evaluation of fine-grained cinematic controls in generated videos, we train a vision-language model (VLM) that aligns with the large-scale human annotations. Our model outperforms existing baselines, achieving an overall accuracy of $72.36\%$ with human annotators. An overview of SCINE, outlining its contributions is shown in Figure 1. + +# 2 Related Work + +Video Generative Models and Evaluations. Video generative models can broadly be classified into two categories based on their input conditioning: image-to-video (I2V) and text-to-video (T2V). While I2V approaches such as Stable Video Diffusion (SVD) [3] have been widely adopted by the community, the focus of our work is to evaluate T2V models for professional use. We choose text as an input modality, since it is an effective and free-form way of describing the controls defined in our taxonomy. The Sora Preview [6] served as a catalyst for a wave of T2V model releases across the closed [54, 11] and open-source communities. State of the art open-source models include Wan [56], HunyuanVideo [14] and StepVideo [12]. Several T2V evaluation benchmarks have also emerged, with VBenchmark [25, 67] gaining broad adoption. VBenchmark evaluates T2V models by developing text prompts that evaluate generated videos across dimensions such as temporal flickering, aesthetic quality and, motion smoothness, while employing automatic metrics. Additional T2V benchmarks include - VideoPhy [2], which evaluates physical plausibility by measuring adherence of generated videos to real-world physics, and T2V-CompBench [48], which studies compositional consistency [23] in video generation. Prior work lacks the shot-level structure and cinematic detail needed for professional control; motivating our taxonomy and benchmark to capture the nuanced elements of industry-standard filmmaking. Existing benchmarks lack cinematic depth; for example, a prompt such as "A man is walking" from VBenchmark-2 [67] omits key details like character appearance or camera movement; all essential to setup a cinematic shot. Existing benchmarks are static, relying on fixed prompt sets that limit extensibility. In contrast, SCINE's taxonomy-guided prompt generation enables future-proof evaluation, allowing prompt complexity to scale with model capabilities. + +Structured Video Generation and Shot-Level Control. MovieNet [24] provides large-scale movie annotations including scene boundaries, cinematic styles, and character metadata.Storyboard-driven approaches like VDS [44] and VAST [66] generate structured video via intermediate pose/layout representations. Multi-shot generation has been addressed through systems like VideoGen-of-Thought [68] and MovieAgent [62], which use hierarchical reasoning to plan and synthesize sequences. However, these efforts overlook the fine-grained structure of a single shot which is core unit of cinematic composition. They often emphasize isolated factors, without modeling the full set of interdependent creative controls. In contrast, SCINE introduces a unified taxonomy that captures the complete spectrum of shot-level cinematic elements. + +# 3 Stable Cinematics + +The subsequent sections detail our proposed taxonomy (Section 3.1), and its underlying design principles. Next, we develop our benchmark comprising of prompts designed for professional use (Section 3.2). Section 3.3 describes how prompt categorization and question generation are performed, enabling large-scale evaluations of video generative models for professional use. + +# 3.1 Taxonomy Design + +Our taxonomies are developed in iteration with industry professionals, including organizations established by the Big Five Studios [47], independent cinematographers and screenwriters, and an Academy Award winning Visual Effects + +![](images/8567b23dd4dc14f4e53df1fc65227b53038be629f5f4286368f0350a37dc98f8.jpg) +Figure 3: The Events taxonomy captures the narrative dimension of a shot which includes actions, emotions, and their fine grained portrayal as they evolve over time within a shot. + +![](images/0d9671ee135a246bd8ee52469e8a44062f50ddd705acbaf4c129199cddc9de01.jpg) +(a) The Setup taxonomy outlines the visual components within the frame, including subjects, props, and environmental context. + +![](images/3826a1753bdc19c7b339328453138e3b75733700aca6b7155b29a54383e5be0c.jpg) +(b) The Camera taxonomy defines all controls related to camera configuration during a shot setup. + +![](images/c76eeddeebf3bb23a0cc0c8e63659ac1e40bef5fb5e4a1d315131f43080f3f04.jpg) +(c) The Lighting taxonomy specifies the illumination of shot, through light sources, their properties, and their interaction with the scene. +Figure 2: Setup, Camera and Lighting Taxonomies that structure the visual elements of a shot. + +Artist. The central guiding question in our taxonomy development was: "What controls do professionals require when setting up a shot?" + +A shot is the atomic unit of filmmaking; an uninterrupted sequence without cuts in which cinematic meaning emerges from the coordination of multiple cinematic choices. The average shot length (ASL) in feature films is 5–10 seconds [8], which closely aligns with the temporal limits of current video generation models. This duration is not a limitation; it is a compact canvas where rich and enough narrative and visual complexity can unfold. A single shot entails numerous controls to convey intent, emotion, and story. This motivates our decision to design the taxonomy at the shot level, where fine-grained control is paramount. Control is the key distinction between casual and professional video creation. In casual settings, users often accept model outputs with minimal intervention, delegating the key creative decisions to the model. In contrast, professional use demands precise, deliberate control at every stage of the generative process. In fact, pixel-level control is common in film-making [29, 61], underscoring the importance of fine-grained adjustments in achieving the desired visual effect. Professionals such as cinematographers and directors are primarily responsible for defining a shot's creative intent. While filmmaking is collaborative, these roles offer a practical abstraction for modeling control. A key insight is that, despite overlap, they are sufficiently disentangled to support distinct control dimensions: screenwriters rarely specify lighting or camera movement, and production designers are typically not associated with emotional tone or narrative pacing. These factors motivate the development of our 4 control pillars, each contributing to the composition of a shot. Our taxonomies are structured as hierarchical trees, where leaf nodes correspond to the most granular control parameters, each associated with a set of values. We describe each pillar below: + +Setup. Setup (Figure 2a) encompasses all visible elements within the frame. We organize it into three top-level groups: (1) Scene aggregates environmental controls, including Texture, Geometry, and Set Design. Texture covers aspects such as contrast and color palette, which govern the surface feel of the shot. Geometry captures dominant shapes and the spatial arrangement of elements within the scene. Set Design comprises Props and their attributes; the Backdrop, which establishes the macro context of the set; and Environment, which defines micro-level elements contributing to the "feel" of the shot. (2) Subjects refer to the focal characters within a shot, defined by attributes such as costumes and accessories. (3) Text Generation refers to on-screen typography such as titles or lettering, designed to appear as integrated graphical elements. Each node in Setup has cinematic meaning: a dawn (Time of Day) setup combined with mist (Elements) can suggest danger, while a clean (Organization), symmetrical hallway (Balance) conveys order. + +Table 1: Structured prompt upsampling with SCINE taxonomies. We show how control nodes from our taxonomies enable the generation of visually expressive (SCINE Visuals) prompts from narrative scripts (SCINE Scripts). The table also demonstrates how a single script can yield multiple visual interpretations, enabled by our taxonomy guided prompt generation pipeline. This aligns with filmmaking principles, where a script can be visually realized in diverse ways depending on the creative choices made by the filmmakers. + +
Baseline script: A man serves dinner to his family.
Taxonomy BranchBaseline choice → narrative impactAlternative choice → narrative impact
Depth of Field (Camera)Shallow → isolates food/serving hand, ro- mantic warmthDeep → every family member equally sharp, ensemble clarity
Camera Movement (Camera)Static tripod + gentle dolly-in → calm focus on gesture; subtle emphasisHandheld tracking → urgency, energetic fam- ily chaos
Lighting Source (Lighting)Warm tungsten practicals → cozy, inviting domestic glowCool morning daylight through windows → brisk freshness and emotional distance
Backdrop / Time of Day (Setup)Evening interior → nostalgic comfort, winding-down moodBright morning interior → optimism and up- beat tempo
Props (Setup)Earth-tone wooden utensils → homelySilver cutlery → formal, upscale
Upsampled prompt: A man serves dinner to his family with shallow depth of field on a static tripod with a gentle dolly-in, under warm tungsten interior lighting in the evening, in a cozy earth-tone kitchen with wooden utensils.
+ +Lighting. "Lighting is the key to turning amateur footage into professional stories and presentation" - Jay Holben [21]. Motivated by this principle, we define the following groups for Lighting (Figure 2b): (1) Source, the origin of illumination within the shot; (2) Color Temperature, which controls the warmth of the light; (3) Lighting Conditions, preset configurations describing scene-wide illumination; (4) Effects, visual outcomes resulting from light interacting with the scene; (5) Position, the spatial relation of the light source to the subject; and (6) Advanced Controls such as flickering modulation and the use of color gels to adjust lighting hue. Each control knob corresponds to distinct cinematic expressions: a shot with only a back light (Position) evokes mystery, while hard shadows are often used to amplify tension. + +Camera. The camera taxonomy (Figure 2c) encompasses all camera-related control dimensions involved in a shot. We organize these into 4 high-level groups: (1) Intrinsics: optical and exposure parameters governing the light captured by the camera; (2) Extrinsics: position and orientation of the camera relative to the subject; (3) Trajectory: motion of the camera and the supporting gear that enables it; and (4) Creative Intent: compositional choices that shape the narrative or emotional tone of a shot. Prior works have primarily focused on camera motion control [19, 22]; however, we show that a much broader range of camera parameters can be independently manipulated while setting up a shot. Each parameter has tangible cinematic impact; for example, a shallow depth of field can isolate the subject from the background to direct emotional focus while, an insert framing spotlights narrative details with precision. + +Events. Events (Figure 3) encodes the narrative substance of a shot, namely the depicted actions, emotions, and dialogues - which are further decomposed into dependent nodes for fine-grained control. These dependent nodes represent attributes that cannot exist independently of their parent categories; for instance, these nodes can specify the type of interaction or the delivery mode of a dialogue. Emotions may appear explicitly (visible tears) or implicitly (a clenched jaw), while actions can be stand-alone or interactive. The Portrayed As category captures aspects such as: Temporal, which refers to the unfolding pattern of the event (e.g., laughter erupting simultaneously vs. sequentially), and Contextual, which indicates whether the event occurs in the foreground or background. Advanced Controls refines pacing and the story structure of the shot, such as a turning point or climax. While recent works [59, 16] evaluate T2V models on sequential event generation, we show that, from a professional standpoint, Events encompass a much broader and richer evaluative space. + +The taxonomies define a total of 76 leaf-level controls that can be independently adjusted when crafting a shot. We structure our taxonomies as hierarchical trees to enable disentanglement and multi-level abstraction of cinematic controls. Attributes within each branch are highly correlated, while branches remain independent, ensuring, for example, adjusting Depth of Field does not affect Camera Movement. The tree structure naturally supports multi-level abstraction, aligning with how filmmakers conceptualize scenes, starting from high-level intent and refining toward specific implementation. For example, a directive such as "set a tense alley at night" can be decomposed + +into an EXT Setting, a cool Color Temperature and a Deep Depth of Field. This structure also allows easier scalability; adding a new detail like floating embed sparks, fits cleanly under (Environment $\rightarrow$ Elements) without disrupting the rest of the taxonomy. An alternate structure such as a flat, linear list would not support such extensibility. + +Developing the taxonomy is a non-trivial task. This required multiple iterations with experts since professionals interpret and prioritize controls differently. Furthermore, shot creation is a multi-stage process, starting from script-writing to setup design to camera blocking, making unification under a single structured framework challenging. While some taxonomies [40] focus only on filmmaking aspects, they lack structure and are not aligned with generative modeling. Our goal instead was to impose structure, and develop a taxonomy that bridges professional filmmaking and generative video models. Details of each control node and their corresponding values are provided in Appendix A.1. + +# 3.2 Designing Prompts for Professional Use + +The taxonomies form the foundation for constructing prompts tailored to professional use. Our core approach in creating prompts involves sampling values from the control nodes and creating prompts that reflect realistic cinematic intent. Mirroring the filmmaking process, we first generate narrative scripts and then inject visual elements into these scripts: + +Scripts. These prompts, referred to as SCINE-Scripts, contain the narrative content of individual shots. We collaborate with a professional screenwriter to create seed prompts that meet strict constraints: a single shot, under 10 seconds, with no reliance on off-screen elements. These seed prompts, along with sampled nodes from the Events taxonomy are provided as input to a LLM, for prompt generation. We use the Events taxonomy for these prompts because it directly encodes narrative beats i.e., what happens in a shot. It covers nodes such as physical + +![](images/07db11181bd9dc525cc13e6c9ce973860d0677be0c616a5c93d2d043cb241dc9.jpg) +Figure 4: t-SNE visualization showing substantial overlap between ground truth screenplays and prompts in SCINE Scripts, in comparison to existing prompt based benchmarks such as VBench-2.0 + +dynamics (actions) and verbal interactions (dialogues), which are crucial to the story conveyed in a shot. To ensure prompt diversity, we vary parameters across multiple LLM invocations, sampling emotions from Plutchik's model [42], alternating actions, dialogue structures, genres, and subject composition. We use LLMs, as prior work [39, 50] have shown their effectiveness in screenplay generation. t-SNE visualization (Figure 4) of SCINE-Scripts embeddings [57] shows substantial overlap with ground-truth screenplays [45], whereas prompts from VBench-2.0 [67] and MovieGenBench [11] exhibit minimal overlap. + +Visual Exposition. We refer to these set of prompts as SCINE-Visuals, which enrich SCINE-Scripts with visual elements from the Camera, Lighting, and Setup taxonomies. In contrast to Events, these taxonomies offer fine-grained control over the visual style and composition of a shot. For each base prompt in SCINE-Scripts, we sample values from one or more control nodes and inject them to expand the prompt with structured visual specifications. SCINE-Visuals highlight a key advantage of our taxonomy: structured prompt upsampling. Unlike existing prompt upsampling techniques that delegate all creative decisions to the LLM, our method constrains generation within the taxonomy, enabling more controlled and interpretable prompt expansion. Table 1 provides a breakdown of how SCINE-Scripts is subsequently upsampled via the taxonomy to generate SCINE-Visuals. + +# 3.3 Category and Question Generation + +Next, we extract categories and generate questions for each prompt. Categories link prompts back to the taxonomy, allowing fine-grained evaluations across different levels of abstraction; a single prompt can map to multiple categories across different taxonomies. For each category, we generate targeted questions that are shown to human annotators during video evaluations. These questions are specific + +![](images/1c614b0d8998cd1dbe015f412b57a3ccfd2eea85cfb3f648897fb5b0c64fb9a6.jpg) +(a) Overall results on SCINE Scripts, across genres. Minimax and Wan 14B emerge as the strongest models. The strongest genre is Biography while all models consistently struggle at Comedy. +Figure 5: Overall results on SCINE Scripts and Visuals. + +![](images/4254986d86743e1966338713c40391a2ad4569252bd10cbb7f619c550a297853.jpg) +(b) Overall results on SCINE Visuals across four pillars of professional control. Events emerges as the most challenging category across models, while Setup yields the strongest performance. + +in nature and target only a single control node, enabling its isolated evaluation. Unlike high-level prompt adherence questions, which lack fine-grained attribution, our framework supports per-control annotations. A minimal example is shown below : + +Prompt: A tight close-up focuses on a fireplace, its embers flickering brightly. + +- Category: Lighting $\rightarrow$ Advanced Controls $\rightarrow$ Motion | Question : Does the scene exhibit dynamic flickering effects in its lighting that align with the description? +- Category: Camera $\rightarrow$ Creative Intent $\rightarrow$ Shot Size | Question: Does the video include a tight close-up shot that captures the detailed framing? + +Additional details are presented in Appendix A.3. + +# 4 Are current Video Generative Models Ready for Professional Use? + +We now evaluate state of the art text-to-video (T2V) models against the professional standards defined in our taxonomy pillars. Our analysis reveals both strengths and persistent challenges of current models, offering an overview of how current models align with professional quality expectations. + +# 4.1 Experimental Setup + +**Prompts.** The SCINE benchmark comprises two prompt categories, Scripts and Visuals, each aligned with distinct professional roles (Table 2). The Visuals prompts are created by systematically upsampling Scripts using our taxonomies leading to a total of 2,089 prompts. We categorize prompts by difficulty: we create basic prompts by limiting the number of sampled control nodes, while in advanced prompts, we do not impose any restriction. + +Models. We evaluate 13 state-of-the-art T2V models, both open-source (WAN 1B/14B [56], Hunyuan-Video [14], Step Video [12], CogVideoX 5B [65], LTX-Video [18], Pyramid Flow [27], Easy Animate 5.1 [63], Mochi [52]), VChitect-2.0 [15] and closed source (Minimax [38], Luma Ray 2 [30], Pika 2.2 [31]). Our goal is to assess each model's suitability for role-specific professional tasks, like evaluating narrative fidelity from a screenwriter's perspective. Unless otherwise noted, we use default sampling parameters and maintain a consistent seed per prompt, across models for fair comparison. + +Human Annotation Setup. To ensure high-quality evaluation, we work with a pool of 84 expert annotators with an average of 6.5 years of experience in film production, across roles such as cinematographers, film editors, screenwriters, visual communication designers, and directors. Annotators were shown a prompt along with two generated video samples. For each prompt, they were presented with taxonomy derived evaluation categories and + +Table 2: The SCINE benchmark includes prompts tailored to professional roles, where each prompt is paired with multiple, fine grained evaluation questions. + +
Target RoleTarget Taxonomies# of PromptsAvg. Questions per Prompt
ScreenwritersSCINE-Scripts
Events11332.57 ± 0.98
CinematographersSCINE-Visuals
Camera, Lighting3555.42 ± 4.47
Production DesignersSetup2984.00 ± 3.35
DirectorsAll30310.48 ± 5.07
+ +corresponding questions. Each video was rated independently on a 1–5 scale, where 1 indicated complete misalignment with the category and 5 indicated a perfect match. Although the evaluation was non-comparative, our UX ablation studies showed that displaying two videos side by side improved annotator calibration, especially when selecting middle range scores. To promote consistency and reduce subjectivity, we developed a comprehensive annotation guide covering each control node in the taxonomy. We collect 3 votes for every video-question pair across 13,457 unique questions, collecting a total of 248,536 pairwise annotations. We observe an intra-class correlation coefficient (ICC) [51] of $80.4\%$ for the 1-5 ratings at the model-pair level, and $95.5\%$ when the models are considered individually. Further, we conduct Wilcoxon signed-rank tests [60] across 45 model pairs where we observe statistically significant preferences in 37 of them, which highlights that the annotators agree on model preferences. + +# 4.2 Results and Analysis + +SCINE Scripts We first evaluate models on narrative event generation, i.e. the story dimension of a shot. Figure 5a compares model performance across different genres, focusing on the accuracy and coherence of generated events. Minimax and WAN-14B emerge as the overall top performers, while LTX-Video consistently underperforms. We observe that models generally perform better on the Biography genre, whereas Comedy proves challenging. In Figure 6, we zoom into sub-categories within Events $\rightarrow$ Types. Minimax leads in nearly all categories, showing the largest margins in Dialogues and Change in Environment, + +but falls short in Advanced Controls, where WAN-14B outperforms. Further, models are better at stand-alone actions compared to interactive actions and portray implicit emotions better than explicit ones. Within Event Types, Actions exhibit lowest variance across models. Performance on Causal and Sequential events is highly correlated $(\rho = 0.94)$ , as is performance on Concurrent and Overlapping events $(\rho = 0.86)$ across models. Despite variation across models, all show limitations in multiple Event aspects, highlighting opportunities for improvement. + +SCINE Visuals Figure 5b demonstrates overall results on SCINE Visuals. Consistent with the pair-wise rankings in Figure 1, WAN-14B and Minimax emerge as top performers across + +![](images/1896d22089604faf2751f04f5d9c4831d3438e36ad1ae1af35856f14a91f0a27.jpg) +Figure 6: Fine-grained evaluation on Events. Models handle environmental changes well but struggle with dialogues and shot pacing. Standalone actions outperform interactive, and implicit emotions are easier than explicit. + +all pillars. We find that current models struggle most with Events and Camera, while elements of Setup and Lighting are comparatively easier to capture. Only the top three models- WAN-14B, WAN-1B, and Minimax - reliably depict Events, with a substantial performance gap from the rest. While Camera scores are low across the board, the narrow spread suggests that all models face similar limitations. Lighting shows the most consistent performance, with most models achieving relatively high scores whereas Setup yields the highest absolute scores for the top-performing models. + +Cinematographer. We evaluate this role by creating prompts that inject control nodes from the Camera and Lighting taxonomy. Within Camera, Extrinsics and Trajectory have the lowest average performance and the narrowest intermodel spread. For Lighting, the primary bottleneck is Lighting Position. We further present results, split by prompt difficulty in Figure 7. Across all models, performance degrades on advanced prompts, indicating that under conditions resembling professional workflows where cinematographers have a large amount of control, current models struggle. The biggest perfor + +![](images/1243cdaab8e75a4810493d416caa91b62f80ddc856bbf87ae743b2386862c4d0.jpg) +Figure 7: Split results on basic vs. advanced prompts for Camera and Lighting. All models show performance drops on advanced prompts, with the largest decline in Lighting Source. + +mance drops occur in Lighting Source, Color Temperature, and Creative Intent. Lighting Position and + +![](images/0b2d2f7bc4f28a2710178c77855d38d6c1a464d920e29e9cfbd466ad91b1fdaf.jpg) +(a) Fine-grained results on Subjects. Models perform well on hair and accessories, but struggle with personality and makeup. + +![](images/30bab113b5ef6c676ba06f70b3d9ee96e40abeda473f23760385bc4546a00c55.jpg) +(b) Fine-grained results on Set Design. Models perform better at Backdrop in comparison to Environment, and struggle most with styling the shot appropriately. + +![](images/1a03115774053fb4633c12d6252e9bc58de3c79bbfd751216c26232685e9179c.jpg) +Figure 8: Fine-grained results on Setup across Subjects and Set Design. +Figure 9: Director results: Joint specification of all controls, mirroring real-world shot creation leads to a performance drop on all models, compared to evaluation in a standalone manner. + +Advanced Controls show the smallest performance drops, but remain the weakest categories overall, highlighting persistent challenges regardless of prompt complexity. Hunyuan and WAN-1B exhibit consistent performance across complexity levels. + +Production Designer. Models perform strongest on the Setup pillar; within Setup, models achieve comparable performance on Subject and Scene generation, but show a drop in Text Generation. In Subjects (Figure 8a), model performances largely vary, with the highest scores for Hair and Accessories, and the weakest in Personality and Make-up. In Set Design (Figure 8b), performance trends follow the order: Backdrop >Props >Environment. Within Props, models perform well in Material but struggle at generating intricate Patterns. For Environment, current limitations lie in adhering to a coherent Style and Backgrounds, whereas better performance is seen in organizing Space within a frame. + +Director. Prompts targeting this role differ from prior categories as they evaluate models across all taxonomies simultaneously. Model performance declines on average when all controls are defined jointly (Figure 9). The largest performance drop is observed in Camera, followed by Setup and Lighting. Wan 14B is the only model to show improved performance on Director prompts for Lighting and Setup, compared to its Standalone results. + +Our evaluation identifies a three-tier hierarchy among current T2V models: Minimax and WAN-14B at the top, followed by Luma Ray 2, Hunyuan, and WAN-1B, with the remaining models forming the third tier. While overall performance varies, most models struggle with the fine-grained elements critical to professional video generation. For example, atomic events are handled reasonably well, but models falter on concurrent and causal events, which demand deeper temporal reasoning. Similarly, high-level cues like lighting conditions are better captured than nuanced aspects like precise light positioning. In summary, even top performing models exhibit substantial room for improvement across all dimensions of our taxonomy. No model achieves consistently strong performance across all aspects of shot composition, underscoring the challenge of aligning generative video models with professional standards. Additional results and analysis are presented in Appendix A.2. + +# 5 Scalable Evaluation of Professional Videos + +In the previous section, we evaluated video generative models using expert annotations across 76 control nodes defined by our taxonomy. While human evaluation remains the gold standard, it is costly and difficult to scale, and defining reliable automatic metrics for each control node is non-trivial. Recent advances in vision-language models (VLMs) [13, 33, 1] offer a scalable alternative, showing strong performance in video understanding tasks. In this section, we leverage these models to perform automatic evaluation of professional video generation. + +Zero-shot VLM Evaluations. The rise of multimodal VLMs has enabled progress on vision-language tasks, including video understanding, making them natural candidates for evaluating professional video generation. We use expert annotations as ground truth and measure VLM alignment by prompting models with a video, its associated prompt, and a specific question tied to a taxonomy node, asking for a 1–5 rating similar to our user study. We explicitly instruct the VLM to ignore factors unrelated to the specified category when evaluating the video. We determine VLM preferences by independently scoring each video and selecting the higher-scoring one. This design mitigates hallucination and order-sensitivity issues commonly observed when prompting with both videos simultaneously. + +We use Qwen2.5-VL-Instruct models due to their strong video understanding capabilities. To study the effect of model scale, we evaluate 3 sizes: 7, 32 and 72B. Figure 10 shows that increasing model size does not yield significant improvements in alignment with human judgment. Our results reveal overall poor agreement, consistent with prior findings [20, 64] that highlight the need for fine-tuning on in-distribution, human-labeled data. + +Aligning Human and VLM ratings. We adopt Qwen-2.5-VL-7B [53] as the base model for fine-tuning; our training and validation dataset consist of 44,062 and 12,763 samples, respectively. We aggregate annotator scores (3/video pair) into binary preferences, excluding ties. Videos are pre + +![](images/82043a69562b1aa51b53cbd1aad1cd8fe5ac2c043d8a3809198582167736655c.jpg) +Figure 10: Our trained VLM shows consistent alignment with human annotations across video generation models, outperforming baselines, most notably on WAN-14B. + +processed at 2 fps at their native resolution. Each sample consists of a prompt, two videos, and a binary label. The trained model acts as a classifier; we modify the model architecture to output a scalar score using a linear projection over the final layer's last token. The model takes as input a single video, its prompt, and the evaluation question as input. For training, we use the Bradley-Terry objective [5] due to its sample efficiency over regression [34]. The model is trained for 1 epoch with a batch size of 8 and learning rate 6e-5. Similar to zero-shot evaluation, we compute pairwise preference accuracy against the average of the annotators as the target metric. Our fine-tuned model achieves an overall accuracy of $72.36\%$ , outperforming all zero-shot VLM baselines. This represents an absolute improvement of $\sim 20\%$ , over the baseline 7B model. Our model (Figure 10) shows consistent performance across videos generated by different models, highlighting its ability to generalize across different video qualities. Additional VLM results are presented in Appendix A.7. + +# 6 Conclusion and Future Work + +Stable Cinemetrics probes at the intersection of professional video generation and generative video models, grounding prompts, evaluations, and analysis in our structured taxonomies. Our findings reveal where current state-of-the-art models perform well, and where substantial improvements are needed. Our prompt suite offers a strong testbed for future video generative models and can be easily extended as models improve, owing to the flexibility of our taxonomy. We envision several extensions of our work; while our current focus is on evaluation, the taxonomy can also support analyzing video datasets for cinematic diversity or serve as a structure for video captioning. While today's text-to-video models are not yet usable in a fully zero-shot capacity, our findings identify the main challenging pillars for professional filmmaking, illuminating the need for potential solutions like fine-tuning and customization that can bring these models closer to real production use. We hope SCINE encourages deeper exploration at the intersection of filmmaking and video generative models, fostering closer collaboration between artists and models. + +# 7 Acknowledgements + +We thank Robert Legato, Hanno Basse and Heather Ferreira for their valuable input on our work. We are also grateful to the team at MovieLabs for their feedback on our taxonomies. A special thanks to Cedric Wagrez for his invaluable assistance with the human annotations! + +# 8 Limitations + +Although our taxonomy was developed in consultation with domain experts, it is limited by the scope of our collaborator network. Filmmaking terminology and interpretive nuance vary across regions and cultures, greater expert diversity would enable broader incorporation of global cinematic controls into the taxonomy. Some taxonomy nodes (e.g., Color Temperature, ISO) were abstracted for evaluation, as we found it difficult for annotators to consistently perceive fine-grained values (such as 2000K or ISO 800). Prompt generation is based on LLMs, whose proprietary nature and potential biases can influence the language and structure of the prompts. Our zero-shot VLM evaluations were bounded by compute and data resources, limiting the scale and scope of the experiments. + +# References + +[1] Shuai Bai, Keqin Chen, Xuejing Liu, Jialin Wang, Wenbin Ge, Sibo Song, Kai Dang, Peng Wang, Shijie Wang, Jun Tang, Humen Zhong, Yuanzhi Zhu, Mingkun Yang, Zhaohai Li, Jianqiang Wan, Pengfei Wang, Wei Ding, Zheren Fu, Yiheng Xu, Jiabo Ye, Xi Zhang, Tianbao Xie, Zesen Cheng, Hang Zhang, Zhibo Yang, Haiyang Xu, and Junyang Lin. 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In Ruslan Salakhutdinov, Zico Kolter, Katherine Heller, Adrian Weller, Nuria Oliver, Jonathan Scarlett, and Felix Berkenkamp, editors, Proceedings of the 41st International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, pages 61885-61896. PMLR, 21-27 Jul 2024. URL https://proceedings.mlr.press/v235/zhou24f.html. + +# NeurIPS Paper Checklist + +The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit. + +Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist: + +- You should answer [Yes], [No], or [NA]. +- [NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available. +- Please provide a short (1–2 sentence) justification right after your answer (even for NA). + +The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper. + +The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes]" is generally preferable to "[No]", it is perfectly acceptable to answer "[No]" provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No]" or "[NA]" is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found. + +IMPORTANT, please: + +- Delete this instruction block, but keep the section heading "NeurIPS Paper Checklist", +- Keep the checklist subsection headings, questions/answers and guidelines below. +- Do not modify the questions and only use the provided macros for your answers. + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract and introduction accurately capture the paper's contribution and scope by clearly stating the claims made and outlining primary contributions. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: We discuss the limitations of our current work. + +# Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. + +# 3. Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +# Answer: [NA] + +Justification: The paper does not present theoretical results, assumptions, or proofs. Instead, it proposes a taxonomy for professional movie elements and provides statistical analysis on how current state-of-the-art text-to-video (T2V) models perform across different pillars of the taxonomy. + +# Guidelines: + +- The answer NA means that the paper does not include theoretical results. +- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced. +- All assumptions should be clearly stated or referenced in the statement of any theorems. +- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. +- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. +- Theorems and Lemmas that the proof relies upon should be properly referenced. + +# 4. Experimental result reproducibility + +Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? + +# Answer: [Yes] + +Justification: The paper provides all necessary information to reproduce the main experimental results. Details for creating the taxonomy are available in Section 3. The setup for VLM evaluation is described in Section 5. + +# Guidelines: + +- The answer NA means that the paper does not include experiments. +- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. +- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. +- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. +- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example +(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm. +(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully. +(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset). +(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. + +# 5. Open access to data and code + +Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? + +Answer: [Yes] + +Justification: We will release our proposed taxonomy upon acceptance. + +Guidelines: + +- The answer NA means that paper does not include experiments requiring code. +- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details. +- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark). +- The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details. +- The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc. + +- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why. +- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable). +- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. + +# 6. Experimental setting/details + +Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results? + +Answer: [Yes] + +Justification: The paper provides all necessary information to reproduce the main experimental results. Details for creating the taxonomy are available in Section 3. The setup for VLM evaluation is described in Section 5. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. +- The full details can be provided either with the code, in appendix, or as supplemental material. + +# 7. Experiment statistical significance + +Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? + +Answer: [Yes] + +Justification: We report statistically significant annotator agreements within the annotators. Additional details are present in the Appendix. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. +- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). +- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) +- The assumptions made should be given (e.g., Normally distributed errors). +- It should be clear whether the error bar is the standard deviation or the standard error of the mean. +- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a $96\%$ CI, if the hypothesis of Normality of errors is not verified. +- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). +- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. + +# 8. Experiments compute resources + +Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? + +# Answer: [Yes] + +Justification: Yes, we report these numbers in the Appendix. + +# Guidelines: + +- The answer NA means that the paper does not include experiments. +- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. +- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. +- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). + +# 9. Code of ethics + +Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines? + +# Answer: [Yes] + +Justification: The paper complies with NeurIPS Code of Ethics. + +# Guidelines: + +- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics. +- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. +- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). + +# 10. Broader impacts + +Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? + +# Answer: [Yes] + +Justification: Yes, we report the impacts of our current work in the Supplementary. + +# Guidelines: + +- The answer NA means that there is no societal impact of the work performed. +- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. +- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. +- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. +- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. +- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). + +# 11. Safeguards + +Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? + +Answer: [NA] + +Justification: We do not anticipate any high risk misuse of our current work. + +Guidelines: + +- The answer NA means that the paper poses no such risks. +- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. +- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. +- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort. + +# 12. Licenses for existing assets + +Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? + +Answer: [Yes] + +Justification: We cite and credit papers that introduce the pre-trained models we train on, as well as the LLM used for generating prompts, and the metrics we use to compare across models. + +Guidelines: + +- The answer NA means that the paper does not use existing assets. +- The authors should cite the original paper that produced the code package or dataset. +- The authors should state which version of the asset is used and, if possible, include a URL. +- The name of the license (e.g., CC-BY 4.0) should be included for each asset. +- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. +- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. +- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. +- If this information is not available online, the authors are encouraged to reach out to the asset's creators. + +# 13. New assets + +Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? + +Answer: [Yes] + +Justification: We will release our taxonomy which is well documented. + +Guidelines: + +- The answer NA means that the paper does not release new assets. +- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. +- The paper should discuss whether and how consent was obtained from people whose asset is used. + +- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file. + +# 14. Crowdsourcing and research with human subjects + +Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? + +Answer: [Yes] + +Justification: We discuss the full text of instructions given to participants and include screenshots in the Supplementary. + +Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. +- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. +- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. + +# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects + +Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? + +Answer: [No] + +Justification: The user study involves participants score/rank videos generated with a predetermined list of prompts and questions in the professional pillars. As such, we believe the study does not pose any risk or harm to participants. + +Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. +- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. +- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. +- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [Yes] + +Justification: We provide information on using LLMs in helping us creating prompt set and questions in Section 3.2 and 3.3. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Appendix + +Taxonomy Details 23 + +Additional Results and Analysis 24 + +Details on Prompt Generation 30 + +Distribution of Taxonomy Categories in SCINE Prompts 33 + +Annotation Details 33 + +Statistical Tests 37 + +Additional VLM Results 40 + +Additional Results on Recent Models 41 + +Broader Impact. 41 + +# A.1 Taxonomy Details + +We provide additional details on control nodes and their associated values in the taxonomies. Some nodes accept open-ended values, for example, a range of stand-alone actions. To simplify evaluation, we abstract certain values that can be fine-grained in future works. For instance, we group Aperture into wide/medium/narrow, though exact f-stop values can also be studied in the future. Similarly, color palette is treated as a discrete value in our current work, but can be decomposed into hue, brightness, and saturation. Table 3 - 6 details the control nodes and their values of the Camera, Lighting, Setup and Events Taxonomies, respectively. + +Table 3: Camera Taxonomy Control Nodes and Values + +
NameDescriptionPotential Values
Lens SizeDefines the focal length and field of view of the camera lens.Standard, Fisheye, Wide, Medium, Long Lens, Telephoto
Depth of FieldControls the range of focus in the image, affecting subject isolation.Deep, Shallow, Soft, Rack, Split Diopter, Tilt Shift
ApertureThe camera lens opening that controls the amount of light propagated through the camera.Wide, Medium, Narrow
Shutter SpeedThe duration for which the camera sensor is exposed to light.Slow, Medium, Fast
ISOSensitivity of the camera sensor to light.Low, Medium, High
AngleDefines the camera's viewpoint in relation to the subject.Low, High, Aerial, Overhead, Dutch, Eye-level, Shoulder, Hip, Knee, Ground, Continuous Values
StaticA fixed camera position without any movement.-
2DCamera movements restricted to horizontal or vertical axes.Pan left, Pan right, Tilt up, Tilt down, Zoom in, Zoom out
3DCamera movements that incorporate spatial depth and multi-axis motion.Push In, Pull Out, Dolly Zoom, Camera Roll, Tracking, Trucking, Arc, Crane
GearSpecifies the support systems and stabilization equipment used to facilitate camera movement.Handheld, Tripod, Pedestal, Cranes, Over-head Rigs, Dolly, Stabilizer, Snorricam, Vehicle Mount, Drones, Motion Control, Steadicam
Shot SizeDetermines how much of the subject and surroundings are visible in the frame.Establishing, Master, Wide, Full, Medium-Full, Medium, Medium-Close-up, Close-up, Extreme Close-up
FramingPlacements and composition of subjects within the frame.Single, Two Shot, Crowd, OTS, PoV, Insert
+ +Table 4: Lighting Taxonomy Control Nodes and Values + +
NameDescriptionPotential Values
Natural LightNatural sources of light, such as sunlight, moonlight, or firelight.Sunlight, Moonlight, Firelight
Artificial/Practices LightMan-made light sources that illuminate the scene.LED, HMI, Tungsten, Fluorescent, HID
Color TemperatureDefines the hue of the light, typically measured in Kelvin, affecting the scene's mood.Warm, Cool, Cold
Lighting ConditionsDescribes various lighting scenarios or ambient conditions present in a scene.Candlelight, Golden Hour, White Fluorescent, Clear Daylight, Overcast
Soft ShadowsSubtle and diffused shadows resulting from indirect or scattered light.Diffused Light, High Key Lighting, Reflectors
Hard ShadowsSharp, well-defined shadows generated by a direct light source.Direct Light, Low Key Lighting
ReflectionThe effect of light bouncing off surfaces to create a reflective appearance.-
Lighting PositionSpecifies the placement or direction of the light source relative to the subject.Back Light, Fill Light, Top Light, Side Light, Key Light
MotionDynamic changes or movement in the lighting effect.Flickering, Pulsing
Color GelsColored filters applied to lights to modify or enhance the color of the illumination.-
+ +![](images/35bcd1989d66cec8c18d51b7fb09761b48bc13ac49511d54f4a4b9ff40aca1cc.jpg) +Figure 11: Model performance on Events across temporal portrayal of Actions. Atomic actions are handled well, whereas models struggle with causal and overlapping Events. + +# A.2 Additional Results and Analysis + +# A.2.1 Events + +Figure 11 shows Events performance across 13 models and 6 Temporal portrayal of Actions. Models handle atomic and concurrent actions well, but struggle with causal, overlapping, and cyclic events. Figure 12 shows Events performance across 13 models and 12 genres. Biography and Adventure are strongest whereas Comedy and Horror are the weakest. Minimax leads in 6/12 genres, Luma Ray 2 tops Action and Drama, and WAN-14B is the most consistent, with the lowest standard deviation. + +Figure 13 shows performance of 10 open-source models across 19 emotion classes. Models perform best on remorse and ecstasy, but fare poorly on aggressiveness and rage. As shown in Figure 14, dialogue performance is weaker in comparison to Emotions and Actions. Models particularly struggle with multi-turn dialogues or when non-verbal reactions follow. Since T2V models do not generate audio, we evaluate whether the correct character delivers the line and/or with appropriate visual expression. Most models fail to localize the speaker, often attributing a single dialogue to multiple characters. + +![](images/1a23758e4e112de2d319a027f95a797600ab7bb014345985df725e95ce7c73f8.jpg) +Figure 12: Model performance on Events across genres. Across 13 models and 12 genres, portrayal of Events in Biography and Adventure are the strongest, while Comedy and Horror are the weakest. + +![](images/a96e2cce6284f691e53fc99ba4f999661eb0dea071274b565e52874710450c01.jpg) +Figure 13: Model performance on Emotions. Among 10 models and 19 emotions, Remorse is best portrayed, while Aggressiveness is the weakest. + +![](images/4629ed88c65c21cd3c9d283d1c0f23f8514edbf5f5eaa390b2ebf0cc2a8d47b4.jpg) +Figure 14: Model performance on Dialogues. Compared to Actions and Emotions, models struggle at Dialogues. Within Dialogues, performance drop is seen during multi-turn conversations. + +![](images/ad46fcd007e1441a53c99dd0dc57af3c21065c46e2c4d2feab2ee965d8ad835c.jpg) +Figure 15: Model performances across Lighting Source. Strobes and Sunlight emerge strongest, whereas HMI and Fluorescent are points of weaknesses. + +![](images/03f6ae705f917a709c81648f547e891e868715961790edbb38f3d1f5c4eb26c9.jpg) +Figure 16: Model performances across Camera Angles. The Dutch angle poses a common challenge to all current video generative models + +Table 5: Setup Taxonomy Control Nodes and Values + +
NameDescriptionPotential Values
ContrastDetermines the difference between light and dark areas to enhance visual impact.Low, High
BlurIntroduces softness to parts of the image to guide focus or create mood.Gaussian, Radial, Motion
NoiseAdds random variations in brightness or color, mimicking film grain or digital sensor noise.Gaussian, Salt and Pepper, Poisson
Film GrainEmulates the granular texture of traditional film photography for a classic look.-
Color PaletteDefines the overall range and harmony of colors in the scene, influencing its mood.Open Set
LinesDirectional elements that guide the viewer's gaze within the shot.Horizontal, Vertical, Diagonal
Regular ShapesStructured, geometric forms such as squares, circles, and triangles that add order to the design.Square, Circle, Triangle
Natural ShapesUnstructured shapes that naturally emerge in the scene, without any geometric constraints.Water-like, Cloud-like
Frame BalanceRefers to the distribution of visual weight across the composition, ensuring a harmonious layout.Rule of Thirds, Symmetry, Right Heavy, Left Heavy
Positional AccuracyThe absolute position of an object or a subject in a scene.Open Set
Relative PositioningThe relative positioning of an object in relationship to other objects in the scene.Open Set
DepthControls the perception of distance between elements, enhancing the three-dimensional feel of the scene.Deep, Flat, Limited, Ambiguous
SettingDefines if the scene is happening indoors or outdoors.INT/EXT
Time of DayThe time of day the scene is set in.DAY, NIGHT, MORNING, EVENING, DAWN, DUSK, LATE NIGHT, MIDDAY, SUNRISE, SUNSET, AFTERNOON
LocationThe specific place or setting of the scene.Open Set
Negative SpaceDefines if there is a lot of empty space.-
PositiveDefines how the space is occupied in the environment.Clean, Cluttered
MoodThe emotional atmosphere or feeling created by the environment.Open Set
ScaleThe relative size or extent of the environment.Open Set
StyleThe artistic or visual style of the backdrop.Open Set
BackgroundThe part of the scene that is behind the main subject and does not need to be exactly described.Open Set
ElementsThe natural or artificial components of the backdrop.Rain, Snow, Fog, Wind, Thunder, Smoke, Dust, Ash, Fire
Prop DescriptionA general description of the prop.Open Set
Prop ClassThe category or type of the prop.Open Set
Prop MaterialThe substance(s) the prop is made of.Wood, Glass, Gold, Paper, Plastic
Prop PatternThe design on the prop.Grid, Checker, Stripes, Zigzag, Dots, Bricks, Metal, Hexagons
Prop UtilityThe purpose or function of the prop, whether it just exists in the scene or will it be used by the subject.Decorative, Functional
Subject ClassThe category or type of the subjects.Open Set
Subject AccessoriesItems worn or carried by the subjects that enhance their appearance or functionality.Open Set
Subject CostumeThe clothing worn by the subjects, especially for a performance or to create a specific character.Open Set
Subject HairThe style and appearance of the subjects' hair.Open Set
Subject MakeupCosmetics applied to the subjects' face or body to enhance or alter their appearance.Open Set
Subject PoseThe position or stance of the subjects, especially for a photograph or portrait.Open Set
Subject SilhouetteThe outline or shape of the subjects against a light background.Open Set
Subject ProportionsThe relative size and scale of the subjects' body parts or features.Open Set
Text GenerationThe process of creating written content to be displayed on the video.Open Set
+ +# A.2.2 Camera and Lighting + +On Lighting Source (Figure 15), Sunlight, Strobes, and Firelight are handled more reliably, while HMI, Fluorescent, and Tungsten lighting show lower performance. As shown in Figure 16, Aerial and Knee level camera angles are depicted better, while the Dutch and Shoulder-level angles show lower performance. On Shot Sizes (Figure 17), Medium-Wide and Master shots have stronger performance in comparison to Full and Extreme Close-Up shots. + +Table 6: Events Taxonomy Control Nodes and Values + +
NameDescriptionPotential Values
Standalone ActionsIf the action is stand-aloneOpen Set
Interactive ActionsIf the action involves subject-subject or object-subject interactionOpen Set
Temporal (Actions)How actions unfold across time.Atomic, Concurrent, Sequential, Causal, Overlapping, Cyclic, Reverse
Foreground (Actions)Describes if the action is taking place in the foregroundLocal, Global, Focal
Background (Actions)Describes if the is taking place in the background-
UncertaintyThe probabilistic nature of the action outcomeProbabilistic, Deterministic, Mixed
Implicit EmotionsEmotions that are suggested or implied rather than directly stated.Open Set
Explicit EmotionsEmotions that are clearly and directly shown or stated within the scene.Open Set
Temporal (Emotions)How emotions evolve across time.Atomic, Concurrent, Sequential, Overlapping, Causal
Foreground (Emotions)Describes if the emotion is taking place in the foregroundLocal, Global, Focal
Background (Emotions)Describes if the emotion is taking place in the background-
Type of Dialogue DeliveryHow the dialogue is deliveredDash, Ellipsis, Monologue
Foreground (Emotions)Describes if the dialogue is being spoken in the foregroundLocal, Global, Focal
Background (Emotions)Describes if the dialogue is being spoken the background-
Change in EnvironmentChange of environment or occurrences within a shotOpen Set
Story StructureKey narrative elements that shape the scene's progression.Turning Point, Climax, Foreshadowing, Conflict
PaceHow fast the events are happening in a shotSlow, Fast
RegularityHow regularly the events are happening in a shotRegular, Irregular
+ +![](images/b90c79c51a9c46032a659458806c3a5d7a6bf716cfc175c7585afa1e0c3a744b.jpg) +Figure 17: Model performances across Camera Shot Size. Models perform well on Master and Establishing shots and struggle at medium-wide and extreme-close-up shots. + +# A.2.3 Setup + +For the Setup taxonomy, we also analyze performance at the value level. In Balance (Figure 18), models handle rule of thirds framing more effectively but struggle with symmetrical compositions. For Time of Day (Figure 19), among 11 categories, Sunrise and Morning are portrayed well, while Afternoon remains challenging. + +![](images/5011f4370918a2e83493e99a9fbdbd2be9df8280c2b756132bede95e707e1190.jpg) +Figure 18: Between different frame compositions, models are better at Rule of Thirds but struggle at maintaining Symmetry. + +![](images/926feb395bddebe0b0544d34f16b07617f2122ec33613e0f34736d5ef1578459.jpg) +Figure 19: Across Time of Day setups, Sunrise shots are handled better, while Afternoon remains more challenging for models. + +![](images/2b8512ac227eb3e65e5a2d24a5291ea66afa9c60b7ed3adebc0f6cdf80c8a413.jpg) +Figure 20: 1. Prompt Generation Pipeline SCINE Scripts are created by passing seed prompts and sampled Events taxonomy nodes to an LLM, forming the narrative component of our benchmark. SCINE Visuals are then generated through structured upsampling, where nodes from the Camera, Lighting, and Setup taxonomies are sampled and injected into each SCINE Script to create prompts that capture visual exposition. 2. Automatic Categorization and Question Generation Given a SCINE prompt and taxonomy, we categorize each taxonomy element present in the prompt and generate a corresponding question to enable isolated evaluation of each control node. + +Table 7: A working example of a prompt with its corresponding categories and questions. Each question targets a single control node from the taxonomy, enabling human annotators to perform fine-grained, independent evaluations per node. + +
PromptFinal CategoryQuestion
In a stark white laboratory illuminated by cool LEDs casting clinical precision, a scientist carefully drops a single blue chemical into a beaker, the camera framing an intimate close-up as soft depth of field blurs the sterile environment behind. A back light carves a subtle halo around the glassware moments before the liquid erupts into bright green, intensified by a strategic neon-tinted color gel that makes the reaction glow like bottled lightning.Camera → Creative Intent → Shot SizeDoes the generated video clearly exhibit a well-executed close-up shot that captures the subject with the intended intimacy and detail?
Camera → Intrinsicity → Depth of FieldDoes the video effectively showcase a soft depth of field that isolates the subject while smoothly blurring the background?
Lighting → Sources → Artificial/Practicals LightIs the effect of artificial LED source clearly visible and does it emulate the clinical, cool lighting effect as described in the scene?
Lighting → Color TemperatureDoes the video convey a cool color temperature in its lighting setup that reinforces the clinical precision suggested in the prompt?
Lighting → Lighting PositionIs a back lighting effect evident in the video, such that it effectively carves a halo or outline around the subject as described?
Lighting → Advanced Controls → Color GelsDoes the video incorporate a neon-tinted color gel effect that intensifies the lighting during the chemical reaction as detailed in the prompt?
+ +# A.3 Details on Prompt Generation + +The overall evaluation pipeline, depicting prompt generation, categorization and question generation, is presented in Figure 20. + +Table 7 presents an example of a prompt and the corresponding categories and generated questions. + +Below, we show an example of the instruction given to an LLM to upsample a SCINE-Script into SCINE-Visuals by incorporating control nodes from the Camera and Lighting taxonomies. + +# SCINE Scripts - Cinematographer + +# System Prompt : + +You are a world-class cinematographer known for your visionary storytelling, mastery of light, and camera. You have decades of experience working on award-winning films across genres, collaborating with top directors and production teams. Your insights blend technical expertise with artistic sensibility. When describing scenes or advising on visual storytelling, you use cinematic terminology with clarity and inspiration. Think like Roger Deakins, Emmanuel Lubezki, and Greig Fraser—your visual choices always elevate the emotional tone and narrative arc of a project. + +User Prompt : + +# GOAL + +You will be given a prompt and 2 taxonomies that define camera and lighting controls commonly used by cinematic professionals. Your objective is to enrich the given prompt by sampling relevant nodes from both the taxonomies. As a cinematographer, your role is to "shoot" this scene using the best possible cinematic expression, utilizing the camera and lighting control options provided in the taxonomy. + +PROMPT: {prompt} + +# MOST IMPORTANT INFORMATION + +1. Only Use Nodes from the Provided Taxonomies: - You must never introduce nodes that are not present in the given taxonomies. - While the values within each node can be flexible-allowing for creativity and imagination grounded in your professional experience. For example, the node "Color Gel" is defined, but has no values. It is up to you to define these values. - The structure must strictly adhere to the nodes defined in the taxonomy. Think expansively within the bounds of each node, but never go beyond them. +2. Preserve the Original Prompt Content : - Do NOT remove or add any of the original content from the input prompt. - Your only task is to enrich the prompt by layering in camera and lighting related information. The core semantics and narrative of the prompt must remain entirely intact. +3. Do NOT include the path through which you sample the nodes in the prompt. That is, do NOT add the paths from the taxonomy using $->$ . + +GUIDELINES 1. Input Prompt - The input prompt describes a single continuous event, intended to occur within one uninterrupted shot. Therefore, do not include any cuts or multiple camera setups. Assume this is a one-shot sequence. + +2. Each node in the taxonomies contains: - Description: A definition of what the node represents. +- Example: An example of how the node may appear in a prompt. +- Values: A non-exhaustive list of possible values for the node. Some notation: a. OPEN SET - Indicates the node supports a wide range of possible values. +b. [] - Indicates the node may have multiple values, which are not predefined and should be selected based on your reasoning and cinematic knowledge. +3. Enriched Prompt - Your enriched version will serve as input to a text-to-video model. It must be fluent, natural, and interpretable by the model, while incorporating cinematic elements effectively. + +CAMERA TAXONOMY The Camera Taxonomy defines elements related to the camera's intrinsics, extrinsics, and its cinematic use: {camera_taxonomy} + +LIGHTING TAXONOMY The Lighting taxonomy broadly defines all elements of lighting, including source, position of lighting, along with its effects such as shadows and reflections, along with color temperature, lighting motion such as flickering etc :{lighting_taxonomy} + +When incorporating lighting into your enriched prompt, remember that a cinematographer can shape the look and feel of a shot by selectively illuminating different depth planes of the scene. Lighting can be applied to the foreground, mid-ground, background, and the subject itself—either individually or in combination. Your choices should support the emotional tone, visual focus, and narrative intent of the shot. + +Below, we show an example of the instruction given to an LLM to categorize and generate evaluation questions for an input prompt using the Camera taxonomy. + +# Camera Categorization and Question Generation + +# GOAL + +You are an expert prompt evaluator. Your task is to analyze a video generation prompt and categorize it based on a predefined taxonomy. + +# PROMPT: {prompt} + +Available Categories (with Examples) + +The category presented to you is that of Camera. The Camera taxonomy broadly defines everything related to the camera - the intrinsics, the extrinsics and the cinematic use of camera. {camera_taxonomy} + +# Notes about the Taxonomy + +Each node in the taxonomy contains : + +1. Description : Definition of what that node represents. +2. Example : An example of the presence of a node in the form of a prompt. +3. Values : A non-exhaustive list of values of these nodes. Values are a list of values that this node can have. Some nomenclature : +a. OPEN SET indicates that this node contains a large number of values. +b. [] indicates that this node may have multiple values, but are not defined explicitly and it is up to your reasoning and knowledge. + +# Examples of Categorization + +1. Static Medium-Close-Up of David's face showing quiet devastation. Quick Push In as tears well up in his eyes. Shot with a medium ISO to capture the dim apartment lighting. +Static - Camera -> Trajectory -> Camera Movement -> Static Push In - Camera -> Trajectory -> Camera Movement -> 3D Medium-Close up - Camera -> Creative Intent -> Shot Size Medium ISO - Camera -> Intrinsicis -> Exposure -> ISO +2. Wide shot of a bustling city street at night. The neon lights of the shops and restaurants cast a colorful glow on the wet pavement. People walk by, their faces illuminated by the bright signs. The camera pans up to reveal the towering skyscrapers that loom overhead, their windows reflecting the city lights. + +Wide shot - Camera -> Creative Intent -> Shot Size Pans Up - Camera -> Trajectory -> Camera Movement -> 2D + +# TASK + +Analyze the given prompt and return the following structured output in a valid JSON format: + +Words: Extract important keywords or key phrases from the prompt using the following guidance: + +- Identify named entities related to a camera in professional use as you would in NER (Named Entity Recognition). +- Extract noun phrases or descriptive terms that relate to a camera. +- Prefer multi-word expressions where meaningful related to a camera. +- Avoid generic or uninformative words like “a”, “video”, “the”, etc. + +Categories: For each word or phrase, assign the most appropriate category from the taxonomy. A dictionary of relevant categories from the taxonomy. + +- For each relevant category, assign a score between '0' and '1' representing how strongly the prompt matches the category. +- Provide a reason for each score, referring to the words or phrases extracted and how they relate to the category. +- Generate a question that helps a human evaluator determine whether this category is visually present in the generated video. Use your reasoning to guide the question. The + +Table 8: Lexical Diversity of SCINE Scripts. Compared to existing prompt-based benchmarks, SCINE-Scripts demonstrate higher lexical diversity across multiple metrics. + +
BenchmarkTTR ↑Distinct Bi-Grams ↑Jaccard Distance ↑
VBench [25]0.14890.46050.9384
MovieGenBench [11]0.16600.53110.9285
EvalCrafter [35]0.22700.60380.9413
T2V-CompBench [48]0.14350.47810.9350
SCINE Scripts0.17600.61770.9445
+ +evaluator will use this question to rate the video on a scale from 1 (not at all) to 5 (strongly represented). + +- The generated question should evaluate quality, consistency and presence of the node in the video. + +# Important Guidelines: + +- The camera information should be explicitly mentioned in the prompt. Do NOT imply, assume or derive anything. Only consider a word or a phrase a match, if it is explicitly mentioned in the prompt. +- Each prompt can have multiple nodes of the Camera taxonomy. You should capture all of the nodes in the prompt and map it back to the taxonomy. +- You must always traverse from the root node, which is Camera in this case. That is, the 'category' should always start as (Camera -> ..) +- You will never create a node that is not in the taxonomy. These nodes can have multiple values, as previously explained and you are expected to be imaginative about the values. But the nodes, should always come from the given taxonomy. +- Since the taxonomy is of Camera, we do not care about objects, subjects, lighting, events, actions or emotions. Your sole focus should be about camera terms that are present in the prompt in accordance with the taxonomy. You will NOT ask any question related to objects, subjects, lighting, events, actions or emotions. + +Table 8 compares SCINE-Scripts with existing prompt-based video generation benchmarks. We compute token level metrics: Type-Token Ratio (TTR), Distinct Bi-Grams, and average pairwise Jaccard Distance, and find that SCINE-Scripts exhibits strong lexical diversity. + +# A.4 Distribution of Taxonomy Categories in SCINE Prompts + +Figures 21, 22, and 23 show the distribution of activated nodes in SCINE Visuals, aggregated at the node level, across the roles of Cinematographer, Production Designer, and Director, respectively. As shown, our prompts cover a broad distribution of nodes across all the taxonomies. + +# A.5 Annotation Details + +Figure 24 shows the annotation interface used by human annotators during evaluation. We also present the distribution of annotators' years of experience in film production in Figure 25. While annotations for cinematic controls can be subjective, especially given the large number of control nodes, we try our best to mitigate this by providing clear rating guidelines to annotators for each control node. Table 9 presents a minimal example of the rating guidelines shared with the annotators. + +![](images/aea4d67eb00deca51eb778a6ed380b04ac0b830d67cc349d9c6bfeefc1bb689c.jpg) +(a) Level 1 Activations + +![](images/8b941241b1dbf1a8fbaae43ded7f4e4a613be11d7373ee13dcac8028eb5af5c8.jpg) +(b) Level 2 Activations + +![](images/cd555cd41eb692b3a8da456900d2f8d06c054264f5630962de61122d65bd58b0.jpg) +(c) Level 3 Activations + +![](images/c02ad73a30216fda6399fe7086bb5e4f71d1195d3398cf55b65a43fd8f1cb506.jpg) +(d) Level 4 Activations +Figure 21: Node activations in Camera and Lighting taxonomies for the Cinematographer role in SCINE Visuals. + +![](images/1afcbfd3823988bebfdb482b3bb62bb2a865ea2aa692199475fedf64a67cdc94.jpg) +(a) Level 2 Activations + +![](images/3733d3a9cd837d396cc53569a051259467a9a2210027e2e5bbb9190a14e14d44.jpg) +(b) Level 3 Activations + +![](images/b34cabdfc5c35fdf444fe623ddcbc22436a99a9ea79f24e80c0e87609bbc0c4a.jpg) +(c) Level 4 Activations +Figure 22: Node activations in Setup taxonomy for the Production Designer role in SCINE Visuals. + +![](images/4cb457afad8aaacd5f57b09cff53d49c1f7bc846ce7ff24d67087284a70f456f.jpg) +(d) Level 5 Activations + +![](images/3e3712af3537f26a2097208e4e9ce9adfcd6eee1c23ec1d19ace3ef75ee14328.jpg) +(a) Level 1 Activations + +![](images/b2880449dce5651444729551a6cd13efdc13d53c233fbf2ae4fdf98851fe9041.jpg) +(b) Level 2 Activations + +![](images/9ba9b8ec0b2adbffb5506b023df625b9f8f6279c01de43db0880e296daa734ff.jpg) +(c) Level 3 Activations + +![](images/e71b2ba2c9e7d75144502773a97f509582741c80e29a07e813550c7d4650c087.jpg) +(d) Level 4 Activations +Figure 23: Node activations in All taxonomies for the Director role in SCINE Visuals. + +![](images/68ae18bb8b5393dce5dd7da58d216a5513911d50e56e36bfb45a99c61dc69bfd.jpg) + +![](images/7b51af9400b7fd42c7878370f4cd9ef601a4adb5f47180a56ca96d67989b9d5b.jpg) +Figure 24: User Interface used by annotators to perform evaluations. + +![](images/d166a439ff1565eaba06efe28f24ab2c43b8dc8943a7fc404e0b915b05450baf.jpg) +Figure 25: Distribution of the years of film production experience amongst human annotators in our evaluation setup. + +Table 9: Examples of rating guidelines provided to human annotators for different control nodes, across all taxonomies. + +
DimensionScoreWhat to Look For
Low Angle1Camera is at or above eye level, not low angle at all.
2Slight upward tilt, but still feels neutral.
3Below subject, mild upward view, light impact.
4Clear low angle, subject looks larger or imposing.
5Strong low angle, subject dominates, towering presence.
Overlapping Actions1No action or one action is present.
2Actions are isolated or unrelated.
3Timing is off, they start or end awkwardly.
4Some overlap, but hard to follow.
5Fluid overlap, actions feel natural and dynamic together.
Back Light Position1Light clearly comes from front or side, no rim light or background separation.
2Some edge lighting, but not consistent or strong, subject may still blend into background.
3Back light is partially visible, outline is hinted but not clear on full subject.
4Back light is clearly present, rim light separates subject from background.
5Strong back light effect, glowing edges around hair or shoulders. Subject clearly pops against the background. Perfect match.
Symmetrical Frame Balance1Composition is clearly asymmetrical.
2Some repeating elements, but no visual mirror.
3Partial symmetry or mirrored clutter that's not clean.
4Almost perfect symmetry, small inconsistencies exist.
5Clear and precise symmetry, mirrored subjects, reflections, or centered framing. Strong and intentional.
+ +# A.6 Statistical tests + +Pairwise t-tests show that the vast majority of model comparisons in our human evaluation, across all taxonomies are statistically significant at the $5\%$ level $(p < 0.05)$ . Figure (26 - 28) presents the t-test results of Events, Lighting and Camera, and Setup, respectively. + +![](images/f0c5abe2d5cc456832dcefedb0942677109cbc00c670374d8405fd6b169341d2.jpg) +(a) Pairwise t-tests across models on emotions + +![](images/8084f0db7afaf6e1bd0220e540e693e25bda6b35f717483a65e8f8121bb0ed75.jpg) +(b) Pairwise t-tests across models on dialogues + +![](images/b0767223ab3a5e0eb42c4b4cc1ca0b0c071eee2496e72050380cd9ee48e0f3e1.jpg) +(c) Pairwise t-tests across models on actions +Figure 26: Statistical comparison matrices for Events: Emotions, Dialogue, and Actions. + +![](images/ebe5f3b056a2e8b165496b1b8088ac537f6331e6d6729ef93108af37a9faba30.jpg) +(a) Pairwise t-tests across models on Color Temperature + +![](images/9dddd9934e302c8693027b1ac154d92370daa70aa7ccb3e6669e7d26f1238352.jpg) +(b) Pairwise t-tests across models on Lighting Effects + +![](images/79d7d837d3a501af4cbf68474d0ed5dafeb46ea8eab2244d8eaf33a0f8e0b5e9.jpg) +(c) Pairwise t-tests across models on Extrinsics + +![](images/1dce95be22b24e278a92cbd3f832ddb43f59c002d6000a3beb3836afc661381b.jpg) +(d) Pairwise t-tests across models on Intrinsics + +![](images/0fb987fb9843f00ea2b1734942e07e4e35d7d819300e0e1399ffb0830972c609.jpg) +(e) Pairwise t-tests across models on Trajectory +Figure 27: Statistical comparison matrices for Camera and Lighting + +![](images/6295d0a5ac7d5cb4f1b1b544779126098d47fa75ab04745d2ed7e88062bd2376.jpg) +(a) Pairwise t-tests across models on Subject Class + +![](images/da5367aee94c069aea7879c2c5dae49db9d4ec92ed9b97b829629806bd270310.jpg) +(b) Pairwise t-tests across models on Subjet Costume + +![](images/669a7fa44eef29b2e25c0ef8b875b644d5d22a2c1da2ed128c6d7d8f03a5cc8f.jpg) +(c) Pairwise t-tests across models on Elements + +![](images/224f4acf172c0a3424aa478128737739219fd34b173c27d908471fe66dea0ccf.jpg) +(d) Pairwise t-tests across models on Time of Day + +![](images/b6a8d6060822720d231b07fb9f2c9855c797fe2276d766a583f3fa51f053da24.jpg) +(e) Pairwise t-tests across models on Location +Figure 28: Statistical comparison matrices for Setup + +Table 10: Top 10 nodes where the VLM shows the strongest performance. Score indicates the accuracy of the VLM's choice with the most common preference among human annotators. + +
NodeScore
Setup→Scene→Set Design→Environment→Mood0.88
Events→Adv.Controls→Rhythm→Pace0.82
Setup→Scene→Set Design→Environment→Style0.82
Setup→Scene→Set Design→Props→Utility0.80
Events→Types→Emotions→Exp Types→Explicit0.79
Setup→Scene→Geometry→Frame→Shapes→Regular0.78
Lighting→Lighting Effects→Shadows→Soft0.76
Setup→Scene→Geometry→Space→Spatial Loc.→Rel(Pos.0.75
Events→Types→Actions→Int TYPES→Standalone0.75
Events→Types→Emotions→Exp Types→Explicit0.75
+ +Table 11: Bottom 10 nodes where the VLM shows the weakest performance. Score indicates the accuracy of the VLM's choice with the most common preference among human annotators. + +
NodeScore
Setup→Subjects→Makeup0.33
Setup→Scene→Set Design→Environment→Background0.33
Events→Types→Actions→Portrayed as→Contextual→Background0.46
Setup→Subjects→Accessories0.53
Lighting→Lighting Effects→Reflection0.55
Setup→Scene→Texture→Color Palette0.55
Camera→Intrinsics→Exposure→Shutter Speed0.57
Lighting→Adv.Controls→Color Gels0.57
Setup→Scene→Texture→Blur0.57
Lighting→Color Temperature0.58
+ +# A.7 Additional VLM results + +Node-specific results of VLM evaluator Table 10 and 11 list the set of nodes on which our VLM evaluator has the strongest and weakest performance, respectively. + +Comparison with Closed Source Models We extend our validation to closed-source, flagship SOTA models. Specifically, we evaluate two recent models from the Gemini family with distinct purposes: Gemini-2.0-Flash, optimized for fast inference, and Gemini-2.5-Pro-Preview-05-06, optimized for complex reasoning. We use the same human-aligned preference accuracy metric as with open-source models. Due to the lack of public details on model sizes, we cannot draw conclusions about scaling effects. However, Gemini-2.5-Pro consistently outperforms open-source models, including QwenVL-2.5-72B, across all categories. Notably, as shown in Figure 29, our 7B model outperforms Gemini-Flash across all categories and performs competitively with Gemini-2.5-Pro. This highlights the strength and scalability of our approach for professional video evaluation. + +Reliability in VLMs A reliable VLM-as-a-Judge should produce consistent scores when given the same video, prompt, and focus aspect. In this analysis, we evaluate the raw scores generated by VLMs rather than preference rankings, and measure their stability under Best-of-5 sampling. Since VLMs are probabilistic, we evaluate reliability via the standard deviation of scores across runs. We use temperature $= 0$ to sample to make ensure that the highest probability is selected at each sampling step. We exclude our model from this analysis, as its architecture includes a dedicated value head, unlike zero-shot VLMs that produce rewards as text. Our results show that Qwen-2.5VL-3B exhibits a high variance, making it unreliable under repeated sampling. In contrast, the flagship models and the strongest open-source model, QwenVL-2.5-72B, demonstrate high reliability, with consistently low variance (Table 12). + +![](images/0e6cb60c0f802d96f8acb2fc2042b925c0bb3f15579380fb5a149e0f999b451b.jpg) +Zero-Shot VLMs vs Our Model's Human Preference Accuracy +Figure 29: Preference Accuracy of open and closed-sourced VLMs in rating videos generated for Professional Use + +Table 12: Measuring VLM Reliability across best-of-5 sampling + +
ModelStandard-Deviation ↓Krippendorff-alpha ↑
Qwen2.5-VL-3B2.340.36
Qwen2.5-VL-7B0.370.84
Qwen2.5-VL-32B0.470.65
Qwen2.5-VL-72B0.230.95
Gemini2.5-Flash0.200.90
Gemini2.5-Pro0.140.95
+ +Table 13: Average scores across taxonomy categories on recently released video generative models. + +
ModelCameraEventsLightingSetupOverall Avg.
Veo 3 Fast3.6863.3823.9744.0783.780
Wan 2.2 14B3.5843.0604.0094.1143.692
Wan 2.1 14B3.2392.8213.8273.9753.466
Wan 2.2 5B3.1942.7053.8383.9133.412
Wan 2.1 1B3.2402.5793.6983.7423.315
+ +# A.8 Additional Results on Recent Models + +We also perform small-scale evaluations on recently released models, specifically Veo 3 (Fast) [9] and the Wan 2.2 family of models. Results are presented in Table 13. + +# A.9 Broader Impact + +We hope SCINE encourages the generative AI and computer vision communities to engage more deeply with the elements required to produce a professional cinematic shot. While our taxonomy is currently used for evaluation, it also offers a structured foundation for broader tasks such as captioning, creating control aware video datasets, and guiding model training toward explicit cinematic intent. We also envision that our fine-grained control nodes can drive new directions in open computer vision problems, such as estimating camera intrinsics from video, inferring lighting position, and understanding frame compositionality. 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Coreset selection aims to accelerate training by identifying small, representative subsets of data that approximate the performance of the full dataset. Among various approaches, gradient-based methods stand out due to their strong theoretical underpinnings and practical benefits, particularly under limited data budgets. However, these methods face challenges such as naive stochastic gradient descent (SGD) acting as a surprisingly strong baseline and the breakdown of representativeness due to loss curvature mismatches over time. + +In this work, we propose a novel framework that addresses these limitations. First, we establish a connection between posterior sampling and loss landscapes, enabling robust coreset selection even in high-data-corruption scenarios. Second, we introduce a smoothed loss function based on posterior sampling onto the model weights, enhancing stability and generalization while maintaining computational efficiency. We also present a novel convergence analysis for our sampling-based coreset selection method. Finally, through extensive experiments, we demonstrate how our approach achieves faster training and enhanced generalization across diverse datasets than the current state of the art. (Code are available in: https://github.com/changwk1001/stable-coreset.git) + +# 1 Introduction + +Modern deep learning thrives on massive datasets, but training on all available data can be prohibitively slow. Coreset selection aims to address this issue by selecting a small, representative subset of training data that can be used in lieu of the full dataset, thus accelerating training while preserving model performance. Recent gradient-based coreset methods (Mirzasoleiman et al., 2020; Killamsetty et al., 2021a; Pooladzandi et al., 2022; Yang et al., 2023) have shown promise by approximating the full-data gradient using a subset. In practice, however, we observe a critical failure mode in these approaches: the loss landscape induced by the selected subset is often misaligned with that of the full dataset. This misalignment becomes especially pronounced under realistic conditions like label noise, adversarial corruptions, or very small subset budgets, leading to poor generalization despite the subset's seeming adequacy in matching gradients at selection time. + +To illustrate this misalignment problem, consider a model selecting a coreset on a noisy dataset. Gradient-based selection methods tend to prioritize examples with large gradient magnitudes. If some training labels are corrupted (or the model is under-trained, the coreset size is small), such methods can inadvertently overweight outliers or mislabeled examples simply because they induce large immediate loss gradients. The resulting subset produces a distorted loss surface: the model trained on + +![](images/ba55637e0a5c7e068aa6de4e90299590f30467ab71589e5216a3e4cff4b7f87f.jpg) +(a) + +![](images/e35c6b0dd7aab2761b6203ce0be2531d541955fd510c2bb8b417698b1901b919.jpg) +(b) +Figure 1: (Left) The smoothed surface: The loss landscape generated by coresets selected by (a) Craig (Mirzasoleiman et al., 2020) The loss landscape generated by the coreset of (b) our method. Our method smoothens the induced landscape. The graph is generated with CIFAR10 data and ResNet20 using $1\%$ data budget. (Right) Better match in loss landscape: The black line in both the figures represent the loss landscape of the full training set for MNIST with LeNet using $1\%$ data budget for coreset selection. (d) Our method matches the underlying landscape better than (c) Craig. The mean square error between the two landscapes for Craig and our method are 0.0703 and 0.0662 respectively. (see details in Appendix B.4 and Figure 6.) + +![](images/adf765beb90c4686a4cc87afc1d761dd10324280c6921dd7c6970bca827e6929.jpg) +(c) + +![](images/1afbd88e668bab1997ab1b8fc350311d3a813f0a83c687f65de79ba84983cb8d.jpg) +(d) + +this subset will descend along directions that differ from those it would follow if it saw all data (See Section A.8). In other words, the subset's gradient and curvature information (its geometry) deviate from the full data's. In fact, even without noisy labels, minor aberrations in gradient approximations (e.g. due to small coreset budget) can distort the induced loss landscape (see Figure 1 (Left)). We term this phenomenon loss landscape misalignment. Over time this discrepancy can magnify – the model's update on the subset can lead it into a region of parameter space that is suboptimal for the true objective. Empirically, we observe that with $20\% -50\%$ label noise on benchmarks like CIFAR-10, CIFAR-100, and TinyImageNet, the performance of many gradient-matching based methods degrades catastrophically – in some cases barely above chance – revealing their brittleness in the face of corrupted data. + +Prior attempts to address this instability have looked to second-order information. By incorporating Hessian estimates, one can in principle better align a coreset's loss landscape with the full data's landscape. Indeed, recent methods have tried to match not only gradients but also curvatures (e.g. using approximate Hessians) to select more robust subsets (Yang et al., 2023). Unfortunately, these second-order approaches come with serious drawbacks. Computing or approximating Hessians in deep networks is extremely costly and often numerically unstable. The required computations can nullify the very speed-ups that coreset selection aims to achieve (as noted by Okanovic et al. (2023); Mahmood et al. (2025)). Further, we empirically also observe these methods are still susceptible to noise in the labels and can even diverge due to sensitivity to noisy curvature estimates. + +We propose a new perspective: posterior smoothing for coreset selection. Rather than deterministically matching gradients or explicitly calculating Hessians, we sample model weight perturbations from a Gaussian posterior centered at the current parameters, and use these perturbed weights to estimate the landscape and evaluate the coreset selection criteria. Intuitively, this strategy smoothens out the loss landscape by marginalizing over a local neighborhood in weight space, effectively simulating a Bayesian posterior over models. By averaging gradient information across multiple weight samples, our method obtains a Monte Carlo smoothing of the coreset objective, and yields a more stable and geometry-aligned selection criterion. This yields a smoothed loss function that closely preserves the underlying structure of the true loss landscape while damping spurious oscillations due to noise or small sample size. + +The proposed posterior-sampling strategy is both scalable and theoretically grounded. It requires no explicit Hessian computations; the added overhead is merely a handful of forward passes for sampling, which is much faster compared to full gradient evaluation. We provide a rigorous analysis showing that posterior smoothing provably improves alignment between the induced and full-data loss landscapes. In particular, under standard smoothness assumptions, we prove that our posterior sampling leads to tighter approximation of the true gradient and Hessian of the entire dataset (see Theorem 3.2). This result formalizes the notion of "geometry alignment": the subset sees nearly the same curvature as the full data (see Figure 1 (Right) for a real-world illustration). Finally, we derive convergence guarantees for training on smoothed coresets: under standard assumptions (e.g. smoothness and bounded curvature), we prove that optimizing on the coreset will converge to a + +solution that is $\epsilon$ -close to the full data optimum. As a side effect, we improve the best known previous rate under the same set of assumptions by $O(1 / \sqrt{M})$ for multiplicative noise, where $M$ is the number of samples used in Monte Carlo smoothing. + +Empirical results on a wide range of datasets strongly support our claims. Our posterior-based coreset selection consistently outperforms state-of-the-art methods in both accuracy and convergence speed, while using a fraction of the data. The gains are most pronounced in high-noise settings – for instance, with $50\%$ corrupted labels on SNLI, our approach outperforms the next best method by $7\%$ absolute accuracy. Interestingly, for this setting the next best method is Random Sampling — a notoriously strong baseline in data selection. We find that our method handily beats the baselines across almost all settings across multiple datasets (SNLI, TinyImageNet, ImageNet-1k, CIFAR-100, CIFAR-10, MNIST), across different architectures (LeNet on MNIST to ResNet-50 on TinyImageNet/Imagenet-1k and RoBERTa on SNLI), across different subset sizes and different noise corruption levels. Further, our memory footprint is smaller, especially compared to the next most frequently best method (Crest (Yang et al., 2023)) which requires expensive Hessian approximations and sometimes has larger memory footprint (see Figure 2 (Right)) with longer run time than our method and sometimes even full data training. To further solidify the supremacy of our method, we also achieve $20\% - 200\%$ speedup for time-to-highest-accuracy compared to Crest across several different datasets. Our ablation studies reveal practical insights as well. For example, we find that posterior sampling through normalization layers (rather than into all model weights or final-layer weights) provides an optimal trade-off between perturbation and stability, further boosting performance (See Table 2). + +To summarize, we make the following technical contributions in this paper: + +- Robust coreset selection via posterior sampling. We introduce a novel posterior smoothing framework for loss landscape alignment under subset selection. We prove that sampling weights from a local Gaussian posterior yields more faithful gradient and curvature estimates for the subset, resulting in provable improvements in both Hessian alignment and Newton-step similarity between the coreset and full dataset (see Theorem 3.2). This perspective opens up a new family of coreset methods that rigorously account for model uncertainty during selection. +- Extended convergence theory. We provide a comprehensive convergence analysis for sampling-based coreset SGD, strengthening and improving prior theoretical results.(see Theorem 3.3) Our analysis accounts for the dual sources of randomness - subset selection and weight sampling - and characterizes how different noise structures (e.g. spherical vs. Hessian-informed Gaussian) influence convergence rates. +- Exhaustive empirical validation. Through extensive experiments on vision and NLP benchmarks, we demonstrate that our approach consistently outperforms existing coreset methods across both clean and corrupted datasets across different noise levels and model architectures. Notably, under severe label noise (20–50% of labels corrupted on CIFAR-10, CIFAR-100, TinyImageNet, ImageNet-1k, SNLI), our method remains remarkably robust while prior gradient-based approaches fail. We also highlight key implementation findings – for example, the importance of imposing posteriors in solely the normalization layers – that further enhance coreset effectiveness. Together, our results establish a new state of the art with minimal overhead and memory footprint in both accuracy and stability for coreset selection in deep learning. + +For a detailed review of related work, please refer to Appendix A.1. + +# 2 Background + +The classic training objective is to optimize the Empirical Risk Minimization (ERM) on the training dataset. + +$$ +w ^ {*} = \underset {w} {\operatorname {a r g m i n}} l (w) = \underset {w} {\operatorname {a r g m i n}} \sum_ {i = 1} ^ {n} l _ {i} (w). \tag {1} +$$ + +For tractability and better generalization, in practice, we use SGD (stochastic gradient descent) for optimization of (1) can lead to large memory burden and can be inefficient due to the calculation time. + +Algorithm 1 Ensemble Coreset $(r,E,T,B,w_i,\eta,P)$ +1: Parameters: Subsample size $R$ , Ensemble size $M$ , Max epochs $T$ , Number of Batch $B$ , Initial model parameters $w_0$ , Learning rate $\eta$ , Number of batch selection $P$ , Gaussian Prior $\delta$ , minibatch size $m$ +2: for $t = 1$ to $T$ do +3: for $p = 1$ to $P$ do +4: Select random subset $V_p \subseteq S$ , $|V_p| = R$ +5: Select $S_p \in \arg \min_{S_p \subseteq V_p} \sum_{i \in V_p} \min_{j \in S_p} E_\delta ||\nabla l_j(w_t + \delta) - \nabla l_i(w_t + \delta)||$ , $|S_p| \leq m$ +6: end for +7: $S_t = \bigcup_{p \in [P]} \{S_p\}$ +8: for $b = 1$ to $B$ do +9: Sample batch $S_b \subseteq S_t$ , $|S_b| = m$ +10: end for +11: end for + +The iterative update of the model parameter can be written as: + +$$ +w _ {t + 1} = w _ {t} - \eta \nabla l _ {s ^ {\prime}} (w _ {t}). \tag {2} +$$ + +where $s$ is the random subset sampled from the whole training dataset and the gradient $\nabla l_{s'}(w_t)$ is evaluated on the subset. Several studies have investigated the convergence rate of stochastic gradient descent (SGD) under different settings. In particular, Ghadimi & Lan (2013) established a convergence rate of $\mathcal{O}\left(\frac{1}{\sqrt{t}}\right)$ , while Xiao et al. (2015) extended this analysis to the mini-batch setting, obtaining $\mathcal{O}\left(\frac{1}{\sqrt{Rt}}\right)$ , where $R$ represents the mini-batch size. + +Building up on the SGD, the gradient-based coreset selections (Mirzasoleiman et al., 2020; Killamsetty et al., 2021a; Pooladzandi et al., 2022) aim to find a subset of data whose gradient aligns with the gradient direction of whole training dataset. The formulation is as follows. + +$$ +S ^ {*} = \underset {S ^ {\prime} \subseteq S, \gamma_ {j} \geq 0} {\operatorname {a r g m i n}} | S ^ {\prime} | \text {s . t} \max _ {w _ {t} \in W} \left| \left| \sum_ {i \in S} \nabla l _ {i} \left(w _ {t}\right) - \sum_ {j \in S ^ {\prime}} \gamma_ {j} \nabla l _ {j} \left(w _ {t}\right) \right| \right| \leq \epsilon . \tag {3} +$$ + +Here $\gamma_{i}$ is the weight for the specific sample $i$ . The goal is to jointly optimize for both the subset $S^{\prime}$ out of the full dataset $S$ and the weights $\gamma_{i}$ while ensuring an error of at most $\epsilon$ . Practically, the inner optimization over $W$ is often omitted by setting $W = \{w_{t}\}$ , the singleton set of the current iterate in the SGD. As pointed out by Mirzasoleiman et al. (2020), we can transform the problem (3) into a submodular cover problem (see Appendix A.6 for details): + +$$ +S ^ {*} = \underset {S ^ {\prime} \subseteq S} {\operatorname {a r g m i n}} | S ^ {\prime} | \text {s . t} \sum_ {j \in S ^ {\prime}} \min _ {i \in S} \left| \left| \nabla l _ {i} \left(w _ {t}\right) - \nabla l _ {j} \left(w _ {t}\right) \right| \right| \leq \epsilon . \tag {4} +$$ + +# 3 Posterior-Stable Coreset Selection: New Paradigm for Landscape-Aligned Subsets + +The stability in optimization and its importance for generalization is widely studied in the Nguyen et al. (2022); Chen et al. (2018); Lei (2023); Harrison et al. (2022); Attia & Koren (2022). For example, Bisla et al. (2022); Duchi et al. (2012) suggest making the model more stable by sampling model weights with Gaussian posterior. This leads to smoothening of the loss surface and the induced stability can help with generalization. Liu et al. (2022) adapted a similar idea to sharpness-aware minimization algorithm to help stabilize the training process and gain performance advantage. + +We build on these previous works and seek a related but different goal. Subset selection can lead to highly unstable and non-smooth loss surfaces (Shin et al., 2023) (see Figure 1). How do we choose a coreset that improves stability of the selection process so that the induced loss landscape matches with the underlying ERM loss landscape? Towards this end, we first propose a new definition for the stability of the selected coresets: + +![](images/41da18cda051bdcfc27c1c3d8490fbd69b62abaeffafd6b98a7cc4844ee00770.jpg) +Figure 2: (Left) Gradient match: The average gradient estimation error for our method and Craig selection method using LeNet on MNIST. The error is calculated with $\left|\frac{1}{|S|}\sum_{i\in S}\nabla l_i(w_t) - \frac{1}{|S'|}\sum_{j\in S'}\gamma_j\nabla l_j(w_t)\right|$ , where $S$ is the training set and $S'$ is the subset selected. Our method generally produces smaller gradient errors and better gradient estimation compared to the Craig method. (Right) Memory buffer: The memory consumption for TinyImagenet. Our method shows less memory during the training process. Crest (Yang et al. (2023)) requires to maintain the information of Hessian during training and the intermediate calculation such as checking the threshold calculation of Hessian norm also require large memory buffer. For the plot, we use running average to average over the time step and show the mean value. + +![](images/558842c4620921ef8a31898e119a3faf45a0077e094fa75c4652ca487b0a79bf.jpg) + +Definition 3.1. $(\sigma, \epsilon, \bar{w})$ -Stability. A subset $S'$ is called $(\sigma, \epsilon, \bar{w})$ -stable if there exists $\sigma, \epsilon$ such that + +$$ +E _ {w \sim N (\bar {w}, \sigma I)} \| \nabla l _ {S ^ {\prime}} (w) - \nabla l (w) \| _ {2} ^ {2} \leq \epsilon , \tag {5} +$$ + +where $\nabla l_{S'}(w)$ is the weighted gradient over the subset $S'$ and $\nabla l(w)$ is the full gradient over the entire dataset. This definition captures the essence of stability by quantifying how much the gradient varies when considering perturbations around the model parameter $\bar{w}$ . Similar stability frameworks have been explored in the literature (Bisla et al., 2022; Liu et al., 2022; Duchi et al., 2012) highlighting the crucial role of stability in ensuring that machine learning models for generalization and robustness. A key side-effect of the stability is the smoothening of the function (See Theorem 1 in Bisla et al. (2022)). + +Algorithm Our goal is to select the $(\sigma, \epsilon, \bar{w})$ -stable coresets instead of solving the classic coreset problem (3) by replacing the maximization over the domain $W$ with the stability constraint. Similar to the transformed coreset problem (4), we write our coreset selection optimization problem: + +$$ +S ^ {*} = \underset {S ^ {\prime} \subseteq S} {\operatorname {a r g m i n}} | S ^ {\prime} | \text {s . t} \sum_ {i \in S} \underset {j \in S ^ {\prime}} {\operatorname * {m i n}} E _ {\delta} \| \nabla l _ {j} (\bar {w} + \delta) - \nabla l _ {i} (\bar {w} + \delta) \| \leq \epsilon , \tag {6} +$$ + +where $\delta \sim N(0, \sigma I)$ as $\sigma$ is a hyperparameter or a design variable that controls the perturbation of the weights $\bar{w}$ . By optimizing this selection process, we aim to ensure that the coreset $S'$ retains the essential characteristics of the original dataset while adhering to the stability constraints characterized. + +Our algorithm is presented in Algorithm 1. (For time complexity analysis, please refer to appendix C) In each epoch $t$ , create a pseudo dataset by subsampling stable gradient-matched data points, $P$ times. Instead of solving the constrained selection to match full gradient up to $\epsilon$ , we select $m$ data points each time as is standard in other implementations of similar algorithms. This creates a shadow dataset $S_{t}$ of size $P * m$ . We then use the shadow dataset $S_{t}$ to compute the gradients and optimize the model parameters using batched stochastic gradient descent on $S_{t}$ . This iterative approach allows for refining the model with each epoch, progressively improving its convergence properties and stability. (For more detailed discussion, see Appendix A.7 and Appendix C) + +For deep neural networks (DNNs), adding stability to the entire $w_{t}$ can be prohibitively expensive. Previous works have tried to mitigate this by focusing on the parameters in the last layer of the neural network (Yang et al., 2023; Killamsetty et al., 2021a; Pooladzandi et al., 2022). However, more recent research (Mahmood et al., 2025) has shown that such strategies can be detrimental because of implicit regularization properties of the SGD. Motivated by the recent studies (Mueller et al., 2023; Frankle et al., 2021; Xu et al., 2019) on importance of normalization layers for stability and improved + +![](images/cc2c612b5e273de03b2975f86bb16e587edd209de4d52f65e42900ce90de67f9.jpg) +(a) CIFAR-10 + +![](images/8cc5ee6d2291fc73b90e730d4c4b635d63748a3645e8217b34bcb3ca1ee69371.jpg) +(b) CIFAR-100 +Figure 3: Time-to-accuracy: We plot the time taken to achieve certain validation accuracy against the state-of-the-art Crest Yang et al. (2023) which uses $2^{\text{nd}}$ order information for coreset selection and show that our proposed method is more efficient and faster in achieving the same validation accuracies. + +![](images/fc163142adab671e0ec7105e2f03c5c986d4c4b471b6c50f722c386b30f86aea.jpg) +(c) TinyImagenet + +![](images/8844108fb1028d3a797e769d9ba45f52e6e7c00c83c4b7d134c6960cc43f9910.jpg) +(d) Imagenet-1k + +predictive performance in DNNs, we focus on our attention for sampling in the batch normalization layers. + +We now discuss theoretical properties of stable coresets. Our algorithm can be viewed as coreset selection on the smoothed loss function that leads to stable and better performance on we shall see in the sequel. Furthermore, we establish the relationship between the posterior sampling and the loss landscape. Previous works have proposed that in order to successfully select key samples in training dataset, the loss landscape over selected samples must match to the loss landscape of the whole training dataset. To do so, these works relied on calculating or approximating the parameter Hessian matrix which can be prohibitively expensive and can even offset the speedup obtained by using coresets while also requiring additional memory. Instead, our sampling based strategy is a cheaper and faster way to match the underlying loss landscape without direct calculation Hessians. To understand the relationship between the Gaussian perturbation and loss landscape matching. We propose a our theory in the following. (The proof is in appendix A.2) + +Theorem 3.2. Suppose a subset $S' \subset S$ is $(\sigma, \epsilon, w)$ -stable and let the Hessian difference be $H_{S',w} - H_{S,w} =: \mathcal{E}$ , then, + +(1) The Hessian difference matrix $\mathcal{E}$ satisfies: + +$$ +\| \mathcal {E} \| \leq \mathcal {O} \left(\epsilon^ {\frac {1}{2}}\right) a n d \operatorname {t r} \left(\mathcal {E} ^ {2}\right) \leq \mathcal {O} \left(\frac {\epsilon}{\sigma}\right). +$$ + +(2) The difference between newton step of two subset is bounded. + +$$ +\left\| H _ {S ^ {\prime}, w} ^ {- 1} \nabla l _ {S ^ {\prime}} (w) - H _ {S, w} ^ {- 1} \nabla l _ {S} (w) \right\| \leq \mathcal {O} \left(\epsilon^ {\frac {1}{2}}\right). +$$ + +Discussion: Theorem 3.2 formalizes the notion that posterior sampling provides an effective means to ensure the alignment of the loss landscape of the chosen coreset with that of the underlying data. Specifically, if the gradients of a selected subset match those of the full dataset under Gaussian posterior sampling, the discrepancy between their Hessians remains small. This aligns with the findings of Shin et al. (2023), which emphasize the necessity of Hessian similarity across subsets. Furthermore, our approach implicitly ensures that inverse-Hessian weighted gradients are also aligned, satisfying the selection criteria established by Pooladzandi et al. (2022). By leveraging posterior sampling, we circumvent the computational challenges associated with direct Hessian comparisons while preserving theoretical rigor. For even greater control over this alignment, more sophisticated posterior distributions can be designed to precisely fine-tune the loss landscape matching between the coreset and the full training dataset (see appendix A.5). + +Convergence Analysis Lastly, we also provide convergence analysis for the coresets Mini-batch Stochastic GD along the lines of SGD analysis of Yang et al. (2023) except that we generalize it for the stability setting and improve on their convergence rate. + +Theorem 3.3. Say $w \sim N(w_t, \sigma_2 I)$ at epoch $t$ . Assume the $l(\cdot)$ is $\beta$ -smooth, and the expectation in (6) is calculated by taking $M$ samples. We consider noise in the gradients resulting from the random sampling of batches and coreset selection as $\xi_1$ and $\xi_2$ respectively: + +$$ +\nabla l (w) = \nabla l _ {S} (w) + \xi_ {1} + \xi_ {2} +$$ + +(1) (Absolute noise) If the noise in coreset selection is of the form: + +$$ +E [ | | \xi_ {2} | | ] \leq \epsilon , +$$ + +![](images/99c46000cba761af1b61db41387d4392bacd5de44649e285f931d21eeae3f111.jpg) +Figure 4: Strong against all baseline: Performance at different corrupt ratio with respect to MNIST, EMINST, CIFAR-10 and CIFAR-100. Our method (purple line) consistently outperform others method at different corrupt ratio, and the performance drop is less when raising the corrupt ratio. Note that the Random sampling method still suffers sharp drop in the high corrupt ratio region in CIFAR-100 dataset. + +![](images/97e9e837b4868667218ad19395b417616f163ece01a5a83e1a4cc6338443cee6.jpg) + +![](images/f7169a942c29e067ef2b83340a55e619b1a64cb5fea37bec48b66da140eb00c0.jpg) + +![](images/8aede46d66d05f551a13faa760642a11321e628babdd203b9895ab3872eb3c37.jpg) + +then by setting the learning rate to be $\eta = \min \left\{\frac{1}{\sqrt{T}},\frac{1}{\beta}\right\}$ and $\sigma_2^2 d = \frac{1}{M\sqrt{T}}$ , We can have convergence rate $\frac{1}{\sqrt{T}}$ + +$$ +\left. E _ {t} \left\| \nabla l \left(w _ {t}\right) \right\| ^ {2} \leq \mathcal {O} \left(\frac {1}{\sqrt {T}} \left(l \left(w _ {0}\right) - l \left(w ^ {*}\right) + \frac {\beta^ {2}}{2 M} + \frac {\beta \epsilon^ {2}}{M} + \frac {\beta \sigma_ {1} ^ {2}}{M R}\right)\right) \right. \tag {7} +$$ + +2. (Multiplicative noise) If the noise in estimation of gradients is of the form below + +$$ +E [ \| \xi_ {2} \| ] \leq \epsilon \| \nabla l (w) \| \tag {8} +$$ + +If we set $\sigma_2^2 d = \frac{1}{M\sqrt{T}}$ and $\eta = \min \left\{\frac{1}{\sqrt{T}},\frac{1}{\beta}\right\}$ , we can have convergence rate $\frac{1}{\sqrt{MRT}}$ . + +$$ +\frac {1}{T} \sum_ {t = 0} ^ {T - 1} | | \nabla l \left(w _ {t}\right) | | ^ {2} = \mathcal {O} \left(\frac {1}{\sqrt {M R T}} \left(2 \left(l \left(w _ {0}\right) - l \left(w ^ {*}\right)\right) + \beta^ {2} + 2 \beta \sigma_ {1} ^ {2}\right)\right) \tag {9} +$$ + +Discussion Theorem 3.3 studies the tradeoff between noise levels and corresponding convergence rates. In case 1 of relatively larger absolute noise, we improve certain terms over previously known standard SGD rates by a factor of $M$ , though we have an additional term due to the noise injected into the model weights. These trade-off between different forms of noise actually give us some room for engineering the overall training process. For example, we could avoid the additional term by switching to the naive SGD when close to end of the training. Another observation is that the trade-off will be more favorable towards our method when the random sampling noise $\sigma_{1}$ becomes large which can happen in the corrupt learning situation (data corruption) making our method more stable in such a situation. In the second case of smaller (proportional to the gradient magnitude) noise, we improve the convergence rate from $\mathcal{O}\left(\frac{1}{\sqrt{RT}}\right)$ (Yang et al., 2023) to $\mathcal{O}\left(\frac{1}{\sqrt{MR T}}\right)$ . + +Selective Sampling and Stability Batch Normalization (BN) and related normalization techniques are central to the training and generalization of deep learning models, influencing both the loss landscape and optimization behavior (Santurkar et al., 2019; Sun et al., 2020; Wang et al., 2021). Recent studies also emphasize the role of noise in normalization layers (Kosson et al., 2023; Liang et al., 2019), underscoring their importance in stabilizing learning. + +Our method involves sampling around model weights to capture curvature information without explicitly computing the Hessian. However, full-model sampling is computationally expensive, and larger variance (e.g., $\sigma_2^2 d$ ) leads to smoother loss surfaces but slower convergence (see Theorem 3.3). To balance efficiency and stability, we restrict sampling to the batch normalization layers. This choice is supported by recent findings that highlight the unique role of BN layers in controlling sharpness and enabling efficient optimization (Mueller et al., 2023; Frankle et al., 2021; Xu et al., 2019). While this design is empirically motivated, developing a full theoretical understanding of BN's stabilizing effect remains an open question. + +# 4 Experiments: When Does Posterior Sampling Help Most + +To evaluate our method. We test our method on ResNet models (ResNet20, ResNet18 and ResNet 50) and transformer based models (ViT, RoBERTa and ELECTRA-small-discriminator) with vision datasets (MNIST, EMNIST, CIFAR10, CIFAR100, TinyImagenet,Imagenet) and language datasets + +(SNLI, REC-50). It achieves state-of-the-art accuracy with improved time and memory efficiency, and demonstrates greater robustness to label noise than baselines. All results are averaged over three random seeds for reproducibility. To reduce compute overhead, we apply Gaussian sampling only to batch normalization layers. Each experiment uses 4 sampled models, with $\sigma$ selected via cross-validation from 0.1, 0.01, 0.001. Sampling is performed dynamically during forward passes, eliminating the need for separate model copies and adding at most one extra layer's memory overhead. This makes our method scalable to larger networks. (See Appendix B for full setup details and Appendix A.9 for more experiment setting such as training budget and architectures.) + +Gradient Matching: Importance of loss landscape. As noted by (Yang et al., 2023; Shin et al., 2023), naive optimization of the coreset function (3) to match the gradients without accounting for the loss landscape as done by Craig (Mirzasoleiman et al., 2020) may not actually yield the best quality subset. We verify this, and show that even on a simple dataset like MNIST, coresets outputted by Craig as vastly inferior to the ones we generate in matching the gradient of the full dataset. The results are presented in Figure 1. + +
DatasetCorruptionOur MethodRandomCrest
SNLI0.00.9132±0.00130.9046±0.00200.9098±0.0022
0.10.8664±0.00120.8324±0.00540.8254±0.0028
0.30.7841±0.00210.7529±0.00310.7587±0.0042
0.50.6062±0.00160.5316±0.00240.5104±0.0055
TinyImageNet0.00.5732±0.00110.5520±0.00940.5609±0.0040
0.10.5526±0.00410.5176±0.00570.5150±0.0641
0.30.4832±0.00430.4193±0.00630.4760±0.0061
0.50.3644±0.00580.2857±0.01370.3567±0.0069
ImageNet-1k0.00.7091±0.00040.7074±0.00040.7136±0.0015
0.10.6977±0.00160.6905±0.00160.6946±0.0035
0.30.6837±0.00070.6514±0.00010.6606±0.0024
0.50.6388±0.00080.5939±0.00170.6051±0.0009
+ +Table 1: Large scale experiment: Test accuracy under varying corruption levels on SNLI, TinyImageNet, and ImageNet-1k. Our method consistently outperforms Random and Crest. We did not compare to Craig and Glister on TinyImagent as we run into Out-Of-Memory errors while using code base mentioned in Appendix B. Note that despite Crest marginally outperform our method in Imagenet-1k at zero corruption, the time taken is double of our method (Crest: 42 hours, Ours: 20 hours) along with triple memory consumption. + +The role of sampling in batch normalization. Our decision to conduct sampling on BN layers is motivated by both computational efficiency and their fundamental role in shaping model behavior. Empirical studies show that sampling on various layers improves performance, but BN layers consistently yield the best results across different models and datasets. Given this trade-off between computational cost and effectiveness, we focus on BN layer sampling to demonstrate the robustness of our method. The table 2 (Left) is performance comparison between sampling at different layers of ResNet20 for CIFAR-10 dataset. + +Test performance: Robustness and Accuracy. We perform an extensive study across different datasets and label corruption ratios to show the advantage of our method. We summarize the numerical results in Figure 4. For raw numbers, see Table 4 in the appendix. We can observe that our method outperforms established benchmarks across several different settings, especially when the label corruption ratio is high. Most methods suffer a drastic performance drop when the corrupt ratio is increased due to the stability. Naive SGD has been shown to be a strong baseline Okanovic et al. (2023) with little overhead, and our experiments confirm its effectiveness over several traditional baselines. However, its performance also deteriorates significantly with increasing corruption. We attribute this to increased stability of our method compared to other methods. In fact, we also observed that some methods can fail the training as the instability arising out of coreset selections for noisy data can cause the loss to go infinitely large (see caption in Table 4). To further demonstrate the advantage of our method, we conduct an experiment on even higher corrupt ratio (0.6 - 0.9) in table 2 (Right) and find that our method offers strong performance compared to others. Finally, many methods in coreset selection could not scale to large datasets such as SNLI, TinyImagenet andImagenet-1k (see Table 1). + +Training dynamics: Memory Footprint and Time-to-Accuracy. As shown in Figure 2, it requires significantly less memory than Crest which performs $2nd$ best accuracy-wise after our method in + +
Layer0.00.10.30.5
All0.8654 ± 0.00940.8479 ± 0.00810.8079 ± 0.00840.7288 ± 0.0140
BN0.8757 ± 0.00290.8544 ± 0.00340.8120 ± 0.00580.7318 ± 0.0143
FC0.8705 ± 0.00190.8525 ± 0.00330.8099 ± 0.00430.7232 ± 0.0171
+ +
DatasetCorrupt RatioOur MethodRandomCrest
CIFAR-100.60.6704 ± 0.00630.6422 ± 0.01850.3374 ± 0.0912
0.70.5511 ± 0.02210.5221 ± 0.03410.2853 ± 0.0328
0.80.3701 ± 0.00530.3011 ± 0.00840.1520 ± 0.0276
0.90.0953 ± 0.01830.0950 ± 0.02860.0938 ± 0.0108
CIFAR-1000.60.3911 ± 0.00630.2528 ± 0.01310.3306 ± 0.0054
0.70.2810 ± 0.00480.1504 ± 0.00360.2442 ± 0.0028
0.80.1680 ± 0.01070.0863 ± 0.02110.1613 ± 0.0084
0.90.0761 ± 0.00290.0367 ± 0.00870.0707 ± 0.0061
+ +predictive performance on non-corrupted data,. Crest consumes up to three times more memory than full training due to tracking additional statistics(their algorithm maintains Hessian and accumulated gradient steps for adaptive coreset updates). While their strategy reduces re-selection frequency for coresets, the computational overhead often negates its theoretical potential benefits (Figure 3) and lead to slower training. Our method consistently outperforms Crest across various settings, from small-scale models (MNIST + LeNet) to large-scale architectures (TinyImagenet / Imagenet / SNLI + ResNet50 / RoBERTa). + +Design of more effective posteriors The choice of posterior can influence the tradeoff between fidelity to the loss landscape and stability. We test with different posteriors - (a) the spherical Gaussian prior with a fixed hyper-parameter $\sigma$ , Gaussian with the Hessian-inverse as the covariance matrix, and an Ensemble of models with 4 different random seed with coresets selected as per Algorithm 1. For Ensemble, we average the gradients of the final layer and use it for coreset calculation. As shown in Table 3, the Spherical-Gaussian posterior achieves the highest final accuracy across various scenarios. We analyze potential failure modes associated with different posteriors, with Figure 5 (in Appendix) illustrating key failure reasons. For CIFAR-10 we observe that Hessian-Gaussian, Ensemble, and Gaussian posteriors require 5962.6s, 3454.2s (parallel), and 2073.5s, respectively, compared to 3553.5s for full training. Notably, only the Spherical-Gaussian posterior offers a speed-up. While training multiple models with different random seeds can be parallelized, the increased memory overhead outweighs the benefits, making it a less favorable trade-off. + +Table 2: (Left) Sampling at different layers: CIFAR-10 test accuracy under varying corruption ratios. BN: BatchNorm; FC: Fully Connected; All: All parameters perturbed. (Right) Robust at even higher corruption: Performance comparison on CIFAR-10 and CIFAR-100 under corruption ratios from 0.6 to 0.9. Our method consistently outperforms Random selection and Crest. + +
DatasetCorrupt RatioHessian-GaussianEnsembleSpherical-Gaussian
CIFAR-100.00.8721 ± 0.00350.8738 ± 0.00100.8757 ± 0.0029
0.10.8523 ± 0.00220.8506 ± 0.00100.8544 ± 0.0034
0.30.8041 ± 0.02400.8036 ± 0.00100.8120 ± 0.0058
0.50.5870 ± 0.01560.7216 ± 0.00300.7318 ± 0.0143
CIFAR-1000.00.6841 ± 0.00450.6968 ± 0.00700.6986 ± 0.0025
0.10.6438 ± 0.00430.6618 ± 0.00300.6644 ± 0.0034
0.30.5731 ± 0.00320.6066 ± 0.00200.6085 ± 0.0044
0.50.4340 ± 0.00510.4965 ± 0.00600.5014 ± 0.0068
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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Appendix + +# A.1 Related work + +Coreset methods To facilitate the training of deep learning models, various methods have been proposed to select informative or representative samples based on the uncertainty of models toward these samples such as Sachdeva et al. (2021) and Coleman et al. (2020). Another line of work selects samples based on their loss difference or degree of error. Methods such as Forgetting Events Toneva et al. (2019), GraNd Paul et al. (2023) have employed this strategy to prioritize samples. These methods aim to reduce the variance of the gradient, thereby improving efficiency of reducing the loss. Another line of work use Bayesian perspective to perform coreset selection Zhang et al. (2021); Guha et al. (2021) which aims to select subset that is equally representative across assigned posteriors. + +Additionally, some methods emphasize the centrality of features or embeddings, forming subsets that best represent clusters of samples. Examples include Herding Chen et al. (2012), K-Center Greedy Ding et al. (2019), and Prototypes Sorscher et al. (2023). Another category of methods bases their selection strategy on observations from a validation set Killamsetty et al. (2021b), leveraging additional information to identify the samples most beneficial for training. + +Despite lacking rigorous theoretical proof for the advantages of selection strategies, these approaches have demonstrated empirical improvements in speed or performance. Gradient-based methods Mirzasoleiman et al. (2020); Killamsetty et al. (2021a); Pooladzandi et al. (2022) on the other hand, select samples that best approximate the gradient of the entire dataset (training or validation sets). These methods offer theoretically sound support and provide guarantees for the training process. Additionally, several works Zhang et al. (2021); Pooladzandi et al. (2022); Shin et al. (2023) in this direction propose to further match the subset with the loss landscape using information in Hessian matrix and show better convergent result and generalization performance. Our method builds on this direction and addresses the shortcomings of previous work. + +Smoothness and Stability The optimization of deep learning models has been active area of studied to understand the reasons behind their superior performance in practice. For instance, the Random Weight Perturbation (RWP) algorithm has been shown to smooth the objective function (Bisla et al., 2022; Li et al., 2024) and improve generalization error (Zhou et al., 2019; Jin et al., 2019; Wang & Mao, 2022) despite its simplicity. By perturbing model weights, several studies have demonstrated an increased ability to escape minima with poor generalization and enhance the stability of the optimization process. Our work draws connection to this approach. Instead of explicitly optimizing using a smoothed objective, we select samples that capture similar features, thereby achieving comparable effects. This novel perspective enables us to leverage the benefits of smoothness without directly modifying the optimization objective. + +Bayesian Methods in Deep Learning Various parameter distributions have been proposed in deep learning to address tasks such as uncertainty estimation, out-of-distribution detection, and classification. Markov chain Monte Carlo (MCMC) methods Chen et al. (2014); Welling & Teh (2011) leverage gradient information for inference, while Laplace approximation-based approaches MacKay (1992); Kirkpatrick et al. (2017); Ritter et al. (2018a) employ Gaussian distributions with the Fisher information matrix or Hessian as the covariance matrix. Other methods explore different strategies: Maddox et al. (2019) average models across different time steps, and Fort et al. (2020) utilize models trained independently with different random seeds. In our work, we evaluate models derived from various posteriors to analyze the trade-offs associated with different sampling strategies. + +# A.2 Second order proof + +Theorem A.1. Suppose a subset $S' \subset S$ is $\{\sigma, \epsilon, \bar{w}\}$ -stable and let the Hessian difference be $H_{S',\bar{w}} - H_{S,\bar{w}} \eqqcolon \mathcal{E}$ , and model has $d$ parameters then, + +(1) The gradient at difference of subset at $w$ is upper bounded + +$$ +\left\| \nabla l _ {S} (\bar {w}) - \nabla l _ {S ^ {\prime}} (\bar {w}) \right\| \leq \frac {1}{2} \left(c _ {1} \sigma^ {2} d + \sqrt {c _ {1} ^ {2} \sigma^ {4} d ^ {2} + 4 \epsilon}\right) = \mathcal {O} \left(\epsilon^ {\frac {1}{2}}\right) \tag {10} +$$ + +(1) The Hessian difference matrix $\mathcal{E}$ satisfies: + +$$ +\left\| \mathcal {E} \right\| \leq c _ {1} \sigma d + \frac {1}{\sigma} \sqrt {c _ {1} ^ {2} \sigma^ {4} d ^ {2} + \sigma \left(\epsilon - c _ {2} ^ {2} \sigma^ {2} d\right)} a n d \operatorname {t r} \left(\mathcal {E} ^ {2}\right) \leq \frac {\epsilon - c _ {2} ^ {2} \sigma^ {2} d}{\sigma \left(1 - 2 c _ {1} \sigma d\right)} \tag {11} +$$ + +(2) The difference between newton step of two subset is bounded. $(\lambda_{\mathrm{max}},\lambda_{\mathrm{min}})$ are largest and smallest eigenvalue in $H_{S,w^*}$ + +$$ +\begin{array}{l} \| H _ {S ^ {\prime}, \bar {w}} ^ {- 1} \nabla l _ {S ^ {\prime}} (\bar {w}) - H _ {S, \bar {w}} ^ {- 1} \nabla l _ {S} (\bar {w}) \| \leq \frac {1}{\lambda_ {\min}} \frac {1}{2} (c _ {1} \sigma^ {2} d + \sqrt {c _ {1} ^ {2} \sigma^ {4} d ^ {2} + 4 \epsilon}) + c \frac {\lambda_ {\max} ^ {\frac {1}{2}}}{\lambda_ {\min} ^ {2}} (c _ {1} \sigma d + \\ \frac {1}{\sigma} \sqrt {c _ {1} ^ {2} \sigma^ {4} d ^ {2} + \sigma (\epsilon - c _ {2} ^ {2} \sigma^ {2} d)} + \frac {\lambda_ {\max} ^ {\frac {1}{2}}}{\lambda_ {\min} ^ {2}} \frac {1}{2} (c _ {1} \sigma^ {2} d + \sqrt {c _ {1} ^ {2} \sigma^ {4} d ^ {2} + 4 \epsilon}) (c _ {1} \sigma d \\ + \frac {1}{\sigma} \sqrt {c _ {1} ^ {2} \sigma^ {4} d ^ {2} + \sigma \left(\epsilon - c _ {2} ^ {2} \sigma^ {2} d\right))} + \mathcal {O} (| | \mathcal {E} | | ^ {2}) \\ \end{array} +$$ + +Proof. Let $f(\bar{w}) = \nabla l_{S'}(\bar{w}) - \nabla l_S(\bar{w})$ and $\nabla f(\bar{w}) = \nabla^2 l_{S'}(\bar{w}) - \nabla^2 l_S(\bar{w})$ (difference in terms of Hessian) + +Assumption 1 Suppose we have the $\nabla f(\bar{w})$ being $2c_{1}$ -Lipschitz Hessian i.e., + +$$ +\left| \left| \nabla f (w) - \nabla f (\bar {w}) \right| \right| \leq 2 c _ {1} \| w - \bar {w} \| \tag {12} +$$ + +Assumption 2 Suppose we have the $\nabla f(\bar{w})$ being bounded below and above + +$$ +2 c _ {2} I \preceq \nabla f (w), \quad \forall w. \tag {13} +$$ + +Assumption 3 (Symmetric and non-singular) $\nabla f(w)$ is symmetric and non-singular. + +With assumption 1 and assumption 2, we can bound the $f(\bar{w})$ by the following through Hefferon (2017): + +$$ +c _ {2} \| w - \bar {w} \| \leq \| f (w) - f (\bar {w}) - \nabla f (\bar {w}) (w - \bar {w}) \| \leq c _ {1} \| w - \bar {w} \| \quad \forall w, \bar {w} \tag {14} +$$ + +The assumption is aiming to capture the degree of change of function using polynomial terms. For more discuss about the assumption of the theory, please refer to appendix D + +we start with the following + +$$ +\begin{array}{l} \epsilon \geq \int f (w) ^ {2} p (w) d w \\ = \int (f (\bar {w}) + f (w) - f (\bar {w})) ^ {2} p (w) d w \tag {15} \\ = f (\bar {w}) ^ {2} + 2 f (\bar {w}) \int (f (w) - f (\bar {w})) p (w) d w + \int (f (w) - f (\bar {w})) ^ {2} p (w) d w \\ \geq | f (\bar {w}) | ^ {2} - c _ {1} \sigma^ {2} d | f (\bar {w}) | + \int (f (w) - f (\bar {w})) ^ {2} p (w) d w \\ \end{array} +$$ + +For last inequality, we use the following: + +$$ +\begin{array}{l} f (\bar {w}) \int (f (w) - f (\bar {w})) p (w) d w = f (\bar {w}) \int (f (w) - f (\bar {w}) - \nabla f (\bar {w}) (w - \bar {w}) + \nabla f (\bar {w}) (w - \bar {w})) p (w) d w \\ = f (\bar {w}) \int ((f (w) - f (\bar {w}) - \nabla f (\bar {w}) (w - \bar {w})) p (w) d w + f (\bar {w}) \int \nabla f (\bar {w}) (w - \bar {w})) p (w) d w \\ = f (\bar {w}) \int ((f (w) - f (\bar {w}) - \nabla f (\bar {w}) (w - \bar {w})) p (w) d w \\ \leq | | f (\bar {w}) | | \int | | (f (w) - f (\bar {w}) - \nabla f (\bar {w}) (w - \bar {w}) | | p (w) d w \\ \leq | | f (\bar {w}) | | \int c _ {1} | | w - \bar {w} | | p (w) d w \\ = c _ {1} | | f (\bar {w}) | | E _ {p} [ | | w - \bar {w} | | ] \\ \leq c _ {1} | | f (\bar {w}) | | \sqrt {E _ {p} | | w - \bar {w} | | ^ {2}} \\ = c _ {1} \left\| f (\bar {w}) \right\| \sigma \sqrt {d} \tag {16} \\ \end{array} +$$ + +We first focus on the first two terms + +$$ +\epsilon \geq \| f (\bar {w}) \| ^ {2} - c _ {1} \sigma^ {2} d \| f (\bar {w}) \| \tag {17} +$$ + +By solving the equation, we can obtain upper bound on the $|f(\bar{w})|$ as following: + +$$ +\left\| f (\bar {w}) \right\| \leq \frac {1}{2} \left(c _ {1} \sigma^ {2} d + \sqrt {c _ {1} ^ {2} \sigma^ {4} d ^ {2} + 4 \epsilon}\right) = \mathcal {O} \left(\epsilon^ {\frac {1}{2}}\right) \tag {18} +$$ + +as a check, if we use linear approximation (i.e., $c_{1} = 0$ ) we will have + +$$ +\left\| f (\bar {w}) \right\| \leq \sqrt {\epsilon} \tag {19} +$$ + +Now, obtain upper bound on the graident difference at $\bar{w}$ , we continue on the cross terms + +$$ +\begin{array}{l} \epsilon \geq \int (f (w) - f (\bar {w})) ^ {2} p (w) d w \\ = \int (f (w) - f (\bar {w}) - \nabla f (\bar {w}) (w - \bar {w}) + \nabla f (\bar {w}) (w - \bar {w})) ^ {2} p (w) d w \\ = \int (f (w) - f (\bar {w}) - \nabla f (\bar {w}) (w - \bar {w})) ^ {2} p (w) d w \\ + 2 \int (f (w) - f (\bar {w}) - \nabla f (\bar {w}) (w - \bar {w})) \nabla f (\bar {w}) (w - \bar {w}) p (w) d w + \int (w - \bar {w}) \nabla f (\bar {w}) ^ {2} (w - \bar {w}) p (w) d w \tag {20} \\ \end{array} +$$ + +we continue by noticing that first term is lower bounded in our assumption. + +$$ +\begin{array}{l} \epsilon \geq c _ {2} ^ {2} \sigma^ {2} d \\ + 2 \int (f (w) - f (\bar {w}) - \nabla f (\bar {w}) (w - \bar {w})) \nabla f (\bar {w}) (w - \bar {w}) p (w) d w + \int (w - \bar {w}) \nabla f (\bar {w}) ^ {2} (w - \bar {w}) p (w) d w \tag {21} \\ \end{array} +$$ + +The first term is obtained through following: + +$$ +\begin{array}{l} \int | | f (w) - f (\bar {w}) - \nabla f (\bar {w}) (w - \bar {w}) | | ^ {2} p (w) d w \geq \int c _ {2} | | w - \bar {w} | | ^ {2} p (w) d w \\ = c _ {2} E _ {p} \left[ \left\| w - \bar {w} \right\| ^ {2} \right] \tag {22} \\ = c _ {2} \sigma^ {2} d \\ \end{array} +$$ + +Now address the cross term, + +$$ +\begin{array}{l} 2 \int (f (w) - f (\bar {w}) - \nabla f (\bar {w}) (w - \bar {w})) \nabla f (\bar {w}) (w - \bar {w}) p (w) d w \geq - 2 c _ {1} \int \| w - \bar {w} \| ^ {2} \mathcal {E} (w - \bar {w}) | p (w) d w \\ \geq - 2 c _ {1} \max _ {i} \lambda_ {\varepsilon , i} \int (w - \bar {w}) ^ {2} p (w) d w \\ \geq - 2 c _ {1} \sigma^ {2} d \max _ {i} \lambda_ {\mathcal {E}, i} \tag {23} \\ \end{array} +$$ + +Now, we address the last term + +$$ +\int (w - \bar {w}) \mathcal {E} ^ {2} (w - \bar {w}) 2 \pi^ {- \frac {d}{2}} \det (\sigma I) ^ {- \frac {1}{2}} \exp \left(- \frac {1}{2} (w - \bar {w}) (\sigma I) ^ {- 1} (w - \bar {w})\right) d w \tag {24} +$$ + +By change of variable $u = \nabla f(\bar{w})(w - \bar{w})$ , we can obtain the following by assuming that $\mathcal{E}$ is not degenerate: + +$$ +\begin{array}{l} \int \left| \det \left(\mathcal {E} ^ {- 1}\right) \right| u ^ {T} u 2 \pi^ {- \frac {d}{2}} \det (\sigma I) ^ {- \frac {1}{2}} \exp \left(- \frac {1}{2} u ^ {T} \mathcal {E} _ {1} ^ {- 1} (\sigma I) ^ {- 1} \mathcal {E} ^ {- 1} u\right) d u \\ = \left| \det \left(\mathcal {E} ^ {- 1}\right) \right| \frac {\det \left(\mathcal {E} (\sigma I) \mathcal {E}\right) ^ {\frac {1}{2}}}{\det \left((\sigma I)\right) ^ {\frac {1}{2}}} \operatorname {t r} \left(\mathcal {E} (\sigma I) \mathcal {E}\right) \tag {25} \\ = \operatorname {t r} \left(\sigma I \mathcal {E} ^ {2}\right) \\ \geq \sigma \operatorname {t r} \left(\mathcal {E} ^ {2}\right) \\ \geq \sigma \max _ {i} \lambda_ {\mathcal {E}, i} ^ {2} \\ \end{array} +$$ + +We assume the $H^{-1}$ and $\nabla f(\bar{w})$ are symmetry matrix. Here, we utilize two identities for the first equality and second inequality: + +$$ +\operatorname {t r} (A B C) = \operatorname {t r} (B C A) \tag {26} +$$ + +The second identity follows the following derivation by first noting that symmetry matrix can be decompose into the spectral form $A = \sum_{i} \lambda_{A,i} v_i v_i^T$ + +$$ +\begin{array}{l} \operatorname {t r} (A B) = \sum_ {i} \lambda_ {B, i} \left(v _ {i} ^ {T} A v _ {i}\right) \\ \geq \min _ {i} \lambda_ {B, i} \sum_ {i} \left(v _ {i} ^ {T} A v _ {i}\right) \tag {27} \\ = \min _ {i} \lambda_ {B, i} \operatorname {t r} (A) \\ \end{array} +$$ + +Now, we put everything together, + +$$ +\epsilon \geq c _ {2} ^ {2} \sigma^ {2} d - 2 c _ {1} \sigma^ {2} d \max _ {i} \lambda_ {\mathcal {E}, i} + \sigma \max _ {i} \lambda_ {\mathcal {E}, i} ^ {2} \tag {28} +$$ + +we can solve for the largest difference in eigenvalue + +$$ +c _ {1} \sigma d + \frac {1}{\sigma} \sqrt {c _ {1} ^ {2} \sigma^ {4} d ^ {2} + \sigma \left(\epsilon - c _ {2} ^ {2} \sigma^ {2} d\right)} \geq \max _ {i} \lambda_ {\mathcal {E}, i} \tag {29} +$$ + +Therefore, we can have that + +$$ +\left\| H _ {S ^ {\prime}, \bar {w}} - H _ {S, \bar {w}} \right\| \leq \mathcal {O} \left(\epsilon^ {\frac {1}{2}}\right) \tag {30} +$$ + +We can also obtain upper bound on the difference in Hessian in terms of trace + +$$ +\begin{array}{l} \epsilon - c _ {2} ^ {2} \sigma^ {2} d \geq - 2 c _ {1} \sigma^ {2} d \max _ {i} \lambda_ {\varepsilon_ {1, i}} + \sigma \max _ {i} \lambda_ {\varepsilon_ {1, i}} ^ {2} \\ \geq - 2 c _ {1} \sigma^ {2} d \operatorname {t r} \left(\mathcal {E} ^ {2}\right) + \sigma \operatorname {t r} \left(\mathcal {E} ^ {2}\right) \tag {31} \\ \geq \sigma (1 - 2 c _ {1} \sigma d) \operatorname {t r} \left(\mathcal {E} ^ {2}\right) \\ \end{array} +$$ + +and therefore + +$$ +\frac {\epsilon - c _ {2} ^ {2} \sigma^ {2} d}{\sigma \left(1 - 2 c _ {1} \sigma d\right)} \geq \operatorname {t r} \left(\mathcal {E} ^ {2}\right) \tag {32} +$$ + +Now, to prove the newton step is also similar, we write the overall into the following equation + +$$ +\left\| H _ {S ^ {\prime}, \bar {w}} ^ {- 1} \nabla_ {s} ^ {\prime} l (\bar {w}) - H _ {S, \bar {w}} ^ {- 1} \nabla l _ {S} (\bar {w}) \right\| = \left\| \left(H _ {S, \bar {w}} + \mathcal {E}\right) ^ {- 1} \left(\nabla l _ {S} (\bar {w}) + \mathcal {E} _ {2}\right) - H _ {S, \bar {w}} ^ {- 1} \nabla l _ {S} (\bar {w}) \right\| \tag {33} +$$ + +To obtain the upper bound of the Hessian inverse, we use inverse perturbation theory + +$$ +\left(H _ {S, \bar {w}} + \mathcal {E}\right) ^ {- 1} = H _ {S, \bar {w}} ^ {- 1} - H _ {S, \bar {w}} ^ {- 1} \mathcal {E} H _ {S, \bar {w}} ^ {- 1} + \mathcal {O} \left(\left| \left| \mathcal {E} \right| \right| ^ {2}\right) \tag {34} +$$ + +Now, we obtain upper bound on the magnitude of eigenvalue of the difference on the Hessian for the subset.(i.e., $\max_i |\lambda_{\mathcal{E},i}| \leq c_1 \sigma d + \frac{1}{\sigma} \sqrt{c_1^2 \sigma^4 d^2 + \sigma (\epsilon - c_2^2 \sigma^2 d)}$ and the gradient difference $\| \nabla l_{s'}(w) - \nabla l_s(w) \| \leq \frac{1}{2} (c_1 \sigma^2 d + \sqrt{c_1^2 \sigma^4 d^2 + 4\epsilon})$ . We can follow to upper bound the desired quantity $\| H_{S',\bar{w}}^{-1} \nabla l_{S'}(w) - H_{S,\bar{w}}^{-1} \nabla l_s(w) \|_2^2$ . At this point, we assume that the gradient has bounded magnitude $\| \nabla l_S(w) \| \leq c, \forall S, w$ + +Put in the above, and assume that the gradient has bounded magnitude $\| \nabla l_S(w)\| \leq c\forall S,w$ + +$$ +\leq \frac {1}{\lambda_ {\min }} \frac {1}{2} (c _ {1} \sigma^ {2} d + \sqrt {c _ {1} ^ {2} \sigma^ {4} d ^ {2} + 4 \epsilon}) + c \frac {\lambda_ {\max } ^ {\frac {1}{2}}}{\lambda_ {\min } ^ {2}} (c _ {1} \sigma d + \frac {1}{\sigma} \sqrt {c _ {1} ^ {2} \sigma^ {4} d ^ {2} + \sigma (\epsilon - c _ {2} ^ {2} \sigma^ {2} d)}) + \frac {\lambda_ {\max } ^ {\frac {1}{2}}}{\lambda_ {\min } ^ {2}} \frac {1}{2} (c _ {1} \sigma^ {2} d + \sqrt {c _ {1} ^ {2} \sigma^ {4} d ^ {2} + 4 \epsilon}) (c _ {1} \sigma d + \frac {1}{\sigma} \sqrt {c _ {1} ^ {2} \sigma^ {4} d ^ {2} + 4 \epsilon}). +$$ + +$$ +\begin{array}{l} \left\| H _ {S ^ {\prime}, \bar {w}} ^ {- 1} \nabla_ {s} ^ {\prime} l (\bar {w}) - H _ {S, \bar {w}} ^ {- 1} \nabla l _ {s} (\bar {w}) \right\| \\ = \| \left(H _ {S, \bar {w}} + \mathcal {E}\right) ^ {- 1} \left(\nabla l _ {S} (\bar {w}) + \mathcal {E} _ {2}\right) - H _ {S, \bar {w}} ^ {- 1} \nabla l _ {s} (\bar {w}) \| \\ \leq \left\| H _ {S, \bar {w}} ^ {- 1} \mathcal {E} _ {2} - H _ {S, \bar {w}} ^ {- 1} \mathcal {E} H _ {S, \bar {w}} ^ {- 1} \nabla l _ {S} (\bar {w}) + H _ {S, \bar {w}} ^ {- 1} \mathcal {E} H _ {S, \bar {w}} ^ {- 1} \mathcal {E} _ {2}\right) + \mathcal {O} (| | \mathcal {E} | | ^ {2}) \| \\ \leq \| H _ {S, \bar {w}} ^ {- 1} \mathcal {E} _ {2} \| + \| H _ {S, \bar {w}} ^ {- 1} \mathcal {E} H _ {S, \bar {w}} ^ {- 1} \nabla l _ {S} (\bar {w}) \| + \| H _ {S, \bar {w}} ^ {- 1} \mathcal {E} H _ {S, \bar {w}} ^ {- 1} \mathcal {E} _ {2}) \| + \mathcal {O} (| | \mathcal {E} | | ^ {2}) \\ + \frac {1}{\sigma} \sqrt {c _ {1} ^ {2} \sigma^ {4} d ^ {2} + \sigma (\epsilon - c _ {2} ^ {2} \sigma^ {2} d))} + \mathcal {O} (| | \mathcal {E} | | ^ {2}) \\ \leq \mathcal {O} \left(\epsilon^ {\frac {1}{2}}\right) \tag {35} \\ \end{array} +$$ + +At this point, we can conclude that + +$$ +\left\| H _ {S ^ {\prime}, \bar {w}} ^ {- 1} \nabla_ {s} ^ {\prime} l (\bar {w}) - H _ {S, \bar {w}} ^ {- 1} \nabla l _ {s} (\bar {w}) \right\| _ {2} ^ {2} \leq \mathcal {O} (\epsilon) \tag {36} +$$ + +# A.3 Proof for Theorem 3.3 + +Next, the assumptions required for the proof are listed below: + +Assumption 1. Bounded variance and unbiased estimator of the random sampling and coreset selection subroutine: + +$$ +E [ \| \nabla l _ {S} (w) - \nabla l (w) \| ^ {2} ] \leq \sigma_ {1} ^ {2} \tag {37} +$$ + +$$ +E [ \nabla l _ {S} (w) ] = \nabla l (w) \tag {38} +$$ + +Assumption 2. $\alpha$ -lipschitz continuity: + +$$ +l (w) - l (v) \leq \alpha \| w - v \| _ {2} \tag {39} +$$ + +Assumption 3. $\beta$ -smoothness: + +$$ +\left| \left| \nabla l (w) - \nabla l (v) \right| \right| \leq \beta \| w - v \| _ {2} \tag {40} +$$ + +Assumption 4. $\epsilon_{i}\in \Re^{d}$ sampled i.i.d from diagonal gaussian: + +$$ +\epsilon_ {i} \sim N (0, \sigma_ {2} I), \forall i = 1 \dots M \tag {41} +$$ + +There exist two randomnesses in our formulation. One is the subset obtained through random sampling from the whole dataset. The other randomness results from the noise inject to model weight. + +$$ +\begin{array}{l} l \left(w _ {t + 1}\right) \leq l \left(w _ {t}\right) + \nabla l \left(w _ {t}\right) ^ {T} \left(w _ {t + 1} - w _ {t}\right) + \frac {\beta}{2} \left\| w _ {t + 1} - w _ {t} \right\| ^ {2} \\ \leq l \left(w _ {t}\right) - \eta_ {t} \nabla l \left(w _ {t}\right) ^ {T} \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {s} l \left(w _ {t} + \epsilon_ {i}\right) + \frac {\beta \eta_ {t} ^ {2}}{2} | | \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l \left(w _ {t} + \epsilon_ {i}\right) | | ^ {2} \tag {42} \\ \end{array} +$$ + +We rewrite the last term as follows: + +$$ +\frac {\beta \eta_ {t} ^ {2}}{2} | | \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l (w _ {t} + \epsilon_ {i}) | | ^ {2} = \frac {\beta \eta_ {t} ^ {2}}{2} (| | \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l (w _ {t} + \epsilon_ {i}) - \nabla l (w _ {t}) | | ^ {2} + \frac {2}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l (w _ {t} + \epsilon_ {i}) \nabla l (w _ {t}) - | | \nabla l (w _ {t}) | | ^ {2}) \tag {43} +$$ + +Input the above into the original formulation: + +$$ +\begin{array}{l} l (w _ {t + 1}) \leq l (w _ {t}) - \eta_ {t} \nabla l (w _ {t}) ^ {T} \sum_ {i = 1} ^ {M} \nabla_ {S} l (w _ {t} + \epsilon_ {i}) + \frac {\beta \eta_ {t} ^ {2}}{2} | | \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l (w _ {t} + \epsilon_ {i}) | | ^ {2} \\ \leq l (w _ {t}) - \eta_ {t} (1 - \eta_ {t} \beta) \nabla l (w _ {t}) ^ {T} \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l (w _ {t} + \epsilon_ {i}) + \frac {\beta \eta_ {t} ^ {2}}{2} (| | \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l (w _ {t} + \epsilon_ {i}) - \nabla l (w _ {t}) | | ^ {2} - | | \nabla l (w _ {t}) | | ^ {2}) \\ \leq l (w _ {t}) - \eta_ {t} (1 - \eta_ {t} \beta) \nabla l (w _ {t}) ^ {T} \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l (w _ {t} + \epsilon_ {i}) - \frac {\beta \eta_ {t} ^ {2}}{2} | | \nabla l (w _ {t}) | | ^ {2} + \\ \end{array} +$$ + +$$ +\left. \beta \eta_ {t} ^ {2} \right| | \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l \left(w _ {t} + \epsilon_ {i}\right) - \nabla l \left(w _ {t} + \epsilon_ {i}\right) | | ^ {2} + \beta \eta_ {t} ^ {2} | | \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla l \left(w _ {t} + \epsilon_ {i}\right) - \nabla l \left(w _ {t}\right) | | ^ {2} \tag {44} +$$ + +The last two terms are achieved through the identity $||a - b|| \leq 2(||a - c|| + ||b - c||)$ . We now want to resolve the second term in the formulation. + +$$ +\begin{array}{l} \nabla l (w _ {t}) ^ {T} \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l (w _ {t} + \epsilon_ {i}) = \nabla l (w _ {t}) ^ {T} (\frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l (w _ {t} + \epsilon_ {i}) - \nabla l (w _ {t}) + \nabla l (w _ {t})) \\ = \left\| \nabla l (w _ {t}) \right\| ^ {2} + \nabla l (w _ {t}) ^ {T} \left(\frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l (w _ {t} + \epsilon_ {i}) - \nabla l (w _ {t})\right) \\ = | | \nabla l (w _ {t}) | | ^ {2} + \\ \end{array} +$$ + +$$ +\nabla l \left(w _ {t}\right) ^ {T} \left(\frac {1}{M} \sum_ {i = 1} ^ {M} \left(\nabla_ {S} l \left(w _ {t} + \epsilon_ {i}\right) - \nabla l \left(w _ {t} + \epsilon_ {i}\right)\right) + \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla l \left(w _ {t} + \epsilon_ {i}\right) - \nabla l \left(w _ {t}\right)\right) \tag {45} +$$ + +The term can be simplified by taking the expectation over the randomness in a stochastic subset. + +$$ +\begin{array}{l} E _ {S} \left[ \nabla l \left(w _ {t}\right) ^ {T} \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l \left(w _ {t} + \epsilon_ {i}\right) \right] = \left\| \nabla l \left(w _ {t}\right) \right\| ^ {2} + \nabla l \left(w _ {t}\right) ^ {T} \left(\frac {1}{M} \sum_ {i = 1} ^ {M} \nabla l \left(w _ {t} + \epsilon_ {i}\right) - \nabla l \left(w _ {t}\right)\right) \\ \leq \left| | \nabla l \left(w _ {t}\right) \right| ^ {2} + \frac {1}{2} \left| | \nabla l \left(w _ {t}\right) \right| ^ {2} + \frac {1}{2} \left(\left| \right| \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla l \left(w _ {t} + \epsilon_ {i}\right) - \nabla l \left(w _ {t}\right) \right. \tag {46} \\ \end{array} +$$ + +The last term is achieved through the Cauchy-Schwarz inequality. Therefore, + +$$ +- E _ {s} \left[ \nabla l \left(w _ {t}\right) ^ {T} \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l \left(w _ {t} + \epsilon_ {i}\right) \right] \leq - \frac {1}{2} \left\| \nabla l \left(w _ {t}\right) \right\| ^ {2} + \frac {1}{2} \left(\left\| \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla l \left(w _ {t} + \epsilon_ {i}\right) - \nabla l \left(w _ {t}\right) \right\| ^ {2}\right) \tag {47} +$$ + +We further simplified the formulation by taking the expectation over the randomness in noise injected to the model weight + +$$ +- E _ {s, \epsilon_ {i}, i = 1 \dots d} \left[ \nabla l \left(w _ {t}\right) ^ {T} \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l \left(w _ {t} + \epsilon_ {i}\right) \right] \leq - \frac {1}{2} \left\| \nabla l \left(w _ {t}\right) \right\| ^ {2} + \frac {1}{2} \beta^ {2} \sigma_ {2} ^ {2} d \tag {48} +$$ + +Next, we first solve the fifth term in the original formulation taking the expectation with respect to $\epsilon_{i}$ for each $i$ . + +$$ +\begin{array}{l} \beta \eta_ {t} ^ {2} E [ | | \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla l (w _ {t} + \epsilon_ {i}) - \nabla l (w _ {t}) | | ^ {2} ] \leq \beta \eta_ {t} ^ {2} E [ | | \frac {1}{M} \sum_ {i = 1} ^ {M} \beta \epsilon_ {i} | | ^ {2} ] \\ \leq \beta^ {3} \eta_ {t} ^ {2} \frac {1}{M} \sum_ {i = 1} ^ {M} E [ \| \epsilon_ {i} \| ^ {2} ] \tag {49} \\ \leq \beta^ {3} \eta_ {t} ^ {2} \sigma_ {2} ^ {2} d \\ \end{array} +$$ + +Finally, we deal with the term. + +$$ +\beta \eta_ {t} ^ {2} | | \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l \left(w _ {t} + \epsilon_ {i}\right) - \nabla l \left(w _ {t} + \epsilon_ {i}\right) | | ^ {2} \tag {50} +$$ + +Here, we assume two different errors that can arise in practice. The first is the random sampling batch created by sampling from the entire dataset. The second error originates from the coreset approximation. We formulate as follows: + +$$ +\nabla l (w) = \nabla_ {S} l (w) + \xi_ {1} + \xi_ {2} \tag {51} +$$ + +The $\xi_{1}$ is the result of the stochasticity of the random batch generation. The $\xi_{2}$ is the error that originates from the selection of the coreset. We consider two different forms of error in $\xi_{2}$ . One is the absolute error and the other is that the error is proportional to the gradient. We formulate as follows: + +$$ +E [ | | \xi_ {2} | | ] \leq \epsilon \quad \text {o r} \quad E [ | | \xi_ {2} | | ] \leq \epsilon | | \nabla l (w) | |, \quad \epsilon > 0 \tag {52} +$$ + +Note: we assume here that these two errors $\xi_{1},\xi_{2}$ are independent to each other. + +Situation 1 We first consider the case where the error is absolute. + +$$ +\begin{array}{l} \beta \eta_ {t} ^ {2} E _ {S, \epsilon_ {i}, i = 1 \dots M} E | | \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l (w _ {t} + \epsilon_ {i}) - \nabla l (w _ {t} + \epsilon_ {i}) | | ^ {2} \leq \beta \eta_ {t} ^ {2} \frac {1}{M ^ {2}} \sum_ {i = 1} ^ {M} E | | \nabla_ {S} l (w _ {t} + \epsilon_ {i}) - \nabla l (w _ {t} + \epsilon_ {i}) | | ^ {2} \\ = \beta \eta_ {t} ^ {2} \frac {1}{M ^ {2}} \sum_ {i = 1} ^ {M} E [ | | \xi_ {1, i} | | ^ {2} + 2 \xi_ {1, i} \xi_ {2, i} + | | \xi_ {2, i} | | ^ {2} ] \\ = \beta \eta_ {t} ^ {2} \frac {1}{M} (| | \xi_ {1} | | ^ {2} + 2 \xi_ {1} \xi_ {2} + | | \xi_ {2} | | ^ {2}) \\ \leq \beta \eta_ {t} ^ {2} \frac {1}{M} \left(\frac {\sigma_ {1} ^ {2}}{R} + \epsilon^ {2}\right) \tag {53} \\ \end{array} +$$ + +The R is the batch size for each batch of random sampling. The cross terms are eliminated due to the assumption 5. + +Integrate those terms into the original formulation. + +$$ +l \left(w _ {t + 1}\right) \leq l \left(w _ {t}\right) - \eta_ {t} \left\| \nabla l \left(w _ {t}\right) \right\| ^ {2} + \frac {\eta_ {t} \left(1 - \eta_ {t} \beta\right)}{2} \beta^ {2} \sigma_ {2} ^ {2} d + \frac {\beta \eta_ {t} ^ {2} \sigma_ {1} ^ {2}}{M R} + \frac {\beta \eta_ {t} ^ {2} \epsilon^ {2}}{M} + \beta^ {3} \eta_ {t} ^ {2} \sigma_ {2} ^ {2} d \tag {54} +$$ + +Here, we pick $\eta_t = \eta$ (fixed step size) and $\eta \leq \frac{1}{\beta}$ . Rearrange and sum over time step, and we will have as follows: + +$$ +\eta \sum_ {t = 0} ^ {T - 1} \left| \left| \nabla l \left(w _ {t}\right) \right| \right| ^ {2} \leq l \left(w _ {0}\right) - l \left(w ^ {*}\right) + \frac {\beta^ {2} \sigma_ {2} ^ {2} d}{2} \sum_ {t = 0} ^ {T - 1} \eta (1 + \eta \beta) + \frac {\beta \epsilon^ {2}}{M} \sum_ {t = 0} ^ {T - 1} \eta^ {2} + \frac {\beta \sigma_ {1} ^ {2}}{M R} \sum_ {t = 0} ^ {T - 1} \eta^ {2} \tag {55} +$$ + +Here, we divide on both sides by $T\eta$ + +$$ +\begin{array}{l} \frac {1}{T} \sum_ {t = 0} ^ {T - 1} \left\| \nabla l \left(w _ {t}\right) \right\| ^ {2} \leq \frac {1}{T \eta} \left(l \left(w _ {0}\right) - l \left(w ^ {*}\right)\right) + \frac {\beta^ {2} \sigma_ {2} ^ {2} d}{2 T} \sum_ {t = 0} ^ {T - 1} (1 + \eta \beta) + \frac {\beta \epsilon^ {2}}{M} \eta + \frac {\beta \sigma_ {1} ^ {2}}{M R} \eta \tag {56} \\ = \frac {1}{T \eta} (l (w _ {0}) - l (w ^ {*})) + \frac {\beta^ {2} \sigma_ {2} ^ {2} d}{2} + \frac {\beta^ {3} \sigma_ {2} ^ {2} d}{2} \eta + \frac {\beta \epsilon^ {2}}{M} \eta + \frac {\beta \sigma_ {1} ^ {2}}{M R} \eta \\ \end{array} +$$ + +As we have control for both $\eta$ and $\sigma_2^2 d$ , we pick $\sigma_2^2 d = \frac{1}{M\sqrt{T}}$ and $\eta = \min \left\{\frac{1}{\sqrt{T}}, \frac{1}{\beta}\right\}$ and, therefore, + +$$ +\frac {1}{T} \sum_ {t = 0} ^ {T - 1} | | \nabla l \left(w _ {t}\right) \| ^ {2} \leq \frac {1}{T} \left(l \left(w _ {0}\right) - l \left(w ^ {*}\right)\right) \max \left\{\sqrt {T}, \beta \right\} + \frac {1}{\sqrt {T}} \left(\frac {\beta^ {2}}{2 M} + \frac {\beta \epsilon^ {2}}{M} + \frac {\beta \sigma_ {1} ^ {2}}{M R}\right) + \frac {1}{T} \frac {\beta^ {3}}{2 M} \tag {57} +$$ + +If we stop at any specific time step with probability $\frac{1}{T}$ , and we observe that the average gradient exists convergent rate $\frac{1}{\sqrt{T}}$ for $T$ large enough which is: + +$$ +\begin{array}{l} E _ {t} \| \nabla l \left(w _ {t}\right) \| ^ {2} \leq \frac {1}{\sqrt {T}} \left(l \left(w _ {0}\right) - l \left(w ^ {*}\right)\right) + \frac {1}{\sqrt {T}} \left(\frac {\beta^ {2}}{2 M} + \frac {\beta \epsilon^ {2}}{M} + \frac {\beta \sigma_ {1} ^ {2}}{M R}\right) + \frac {1}{T} \frac {\beta^ {3}}{2 M} \tag {58} \\ = \mathcal {O} (\frac {1}{\sqrt {T}} (l (w _ {0}) - l (w ^ {*}) + \frac {\beta^ {2}}{2 M} + \frac {\beta \epsilon^ {2}}{M} + \frac {\beta \sigma_ {1} ^ {2}}{M R})) \\ \end{array} +$$ + +Situation 2 We now consider the case where the error is proportional to the magnitude of the gradient. i.e., + +$$ +E [ \| \xi_ {2} \| ] \leq \epsilon \| \nabla l (w) \|, \quad \epsilon > 0 \tag {59} +$$ + +We analyze the term as follows: + +$$ +\begin{array}{l} \beta \eta_ {t} ^ {2} E [ \| \frac {1}{M} \sum_ {i = 1} ^ {M} \nabla_ {S} l \left(w _ {t} + \epsilon_ {i}\right) - \nabla l \left(w _ {t} + \epsilon_ {i}\right) \| ^ {2} ] \leq \beta \eta_ {t} ^ {2} \frac {1}{M} E [ \| \xi_ {1} \| ^ {2} + 2 \xi_ {1} \xi_ {2} + \| \xi_ {2} \| ^ {2} ] \tag {60} \\ \leq \beta \eta_ {t} ^ {2} \frac {1}{M} \left(\frac {\sigma_ {1} ^ {2}}{R} + \epsilon^ {2} | | \nabla l (w _ {t}) | | ^ {2}\right) \\ \end{array} +$$ + +Integrate the term to previous result. + +$$ +l \left(w _ {t + 1}\right) \leq l \left(w _ {t}\right) - \left(\eta_ {t} - \frac {\beta \eta_ {t} ^ {2} \epsilon^ {2}}{M}\right) | | \nabla l \left(w _ {t}\right) | | ^ {2} + \frac {\eta_ {t} (1 - \eta_ {t} \beta)}{2} \beta^ {2} \sigma_ {2} ^ {2} d + \frac {\beta \eta_ {t} ^ {2} \sigma_ {1} ^ {2}}{M R} + \beta^ {3} \eta_ {t} ^ {2} \sigma_ {2} ^ {2} d \tag {61} +$$ + +Set $\eta_t = \eta$ . We rearrange and perform same operation and we get: + +$$ +\begin{array}{l} \sum_ {t = 0} ^ {T - 1} \left(\eta - \frac {\beta \eta^ {2} \epsilon^ {2}}{M}\right) \| \nabla l \left(w _ {t}\right) \| ^ {2} \leq l \left(w _ {0}\right) - l \left(w ^ {*}\right) + \frac {\beta^ {2} \sigma_ {2} ^ {2} d}{2} \sum_ {t = 0} ^ {T - 1} \eta (1 + \eta \beta) + \frac {\beta \sigma_ {1} ^ {2}}{M R} \sum_ {t = 0} ^ {T - 1} \eta^ {2} \tag {62} \\ = l \left(w _ {0}\right) - l \left(w ^ {*}\right) + \frac {\beta^ {2} \sigma_ {2} ^ {2} d}{2} T \eta (1 + \eta \beta) + \frac {\beta \sigma_ {1} ^ {2}}{M R} T \eta^ {2} \\ \end{array} +$$ + +Divide by $T\left( {\eta - \frac{{\beta }^{{2\epsilon }^{2}}}{M}}\right)$ on both sides and choose step size such that $\left( {1 - \frac{{\beta }^{2}\eta {\epsilon }^{2}}{M}}\right) \geq \frac{1}{2}$ + +$$ +\frac {1}{T} \sum_ {t = 0} ^ {T - 1} | | \nabla l (w _ {t}) | | ^ {2} \leq \frac {2}{T} \left(l \left(w _ {0}\right) - l \left(w ^ {*}\right)\right) + \beta^ {2} \sigma_ {2} ^ {2} d (1 + \eta \beta) + \frac {2 \beta \sigma_ {1} ^ {2}}{M R} \eta \tag {63} +$$ + +We pick $\sigma_2^2 d = \frac{1}{\sqrt{MR}}$ and $\eta = \frac{\sqrt{MR}}{\sqrt{T}}$ and we will have + +$$ +\begin{array}{l} \frac {1}{T} \sum_ {t = 0} ^ {T - 1} | | \nabla l \left(w _ {t}\right) | | ^ {2} \leq \frac {2}{T} \left(l \left(w _ {0}\right) - l \left(w ^ {*}\right)\right) + \frac {\beta^ {2}}{\sqrt {M R T}} \left(1 + \beta \frac {\sqrt {M R}}{\sqrt {T}}\right) + \frac {2 \beta \sigma_ {1} ^ {2}}{\sqrt {M R T}} \tag {64} \\ = \mathcal {O} (\frac {1}{\sqrt {M R T}} (2 (l (w _ {0}) - l (w ^ {*})) + \beta^ {2} + 2 \beta \sigma_ {1} ^ {2})) \\ \end{array} +$$ + +Similar to the first situation, we will have convergence rate with $\frac{1}{\sqrt{MRT}}$ which is $\frac{1}{\sqrt{M}}$ faster than the naive SGD. + +# A.4 Data summary + +
DatasetCorrupt RatioRandomCrestGlisterCraigOurs
MNIST00.9921±0.00080.9892±0.00050.9997±0.0001(*)0.0972±0.00.9914±0.0003
0.10.9854±0.00090.8384±0.08120.7676±0.02710.0961±0.00090.9876±0.0009
0.30.9772±0.00080.3374±0.33850.1815±0.06640.5196±0.23460.9809±0.0025
0.50.962±0.00190.3277±0.20620.0725±0.01090.4264±0.01310.9715±0.0021
EMNIST00.8713±0.00260.7955±0.01960.8246±0.0049(*)0.0219±0.00.8715±0.0015
0.10.8649±0.00160.7788±0.01320.6762±0.00940.0218±0.00070.8679±0.0012
0.30.8557±0.00070.7621±0.00440.4748±0.02360.301±0.25530.8396±0.0016
0.50.8409±0.00240.7146±0.00510.2627±0.01720.321±0.01110.8100±0.0029
CIFAR-1000.8660±0.00150.8724±0.0040.8207±0.00970.7618±0.0080.8757±0.0029
0.10.8481±0.0010.8440±0.0030.6350±0.02610.7490±0.00660.8544±0.0034
0.30.8063±0.0100.7843±0.0040.3953±0.05480.7050±0.01120.8120±0.0058
0.50.7257±0.0150.6652±0.0130.2175±0.02920.6320±0.01860.7318±0.0143
CIFAR-10000.6659±0.00410.6728±0.0030.6004±0.00520.5665±0.00530.6986±0.0025
0.10.6268±0.0050.6391±0.0040.4538±0.00760.4629±0.01020.6644±0.0034
0.30.5119±0.0140.5712±0.0060.2846±0.00740.2968±0.01620.6085±0.0044
0.50.3293±0.0140.4305±0.0260.1578±0.02180.1603±0.00480.5014±0.0068
+ +Table 4: Performance comparison of different methods across datasets and corruption ratios. Results are reported as mean ± standard deviation. Each experiments are average over 5 different random seed. The best performance for each setting is highlighted in bold. The * mark the situation in which has diverging behavior during the optimization. The situation usually occurs in Craig method for high corruption scenario. + +# A.5 Discussion about different posterior + +Hessian inverse covariance is the one containing exact information about loss landscape of the models at certain training points and it is also the posterior used in deriving the theory for sampling. However, calculation of Hessian can introduce large memory footprint as it will require us to keep track of the Hessian during training as listed in works Yang et al. (2023). Not only it can cause large memory footprint, it is also hard to calculate and require steps of approximation. The selection strategy involving calculation of Hessian will inevitably be slowed down as it requires at least one forward, backward propagation and the intermediate calculation. Another issue about Hessian inverse is that despite it contains the curvatures information, it can easily fail to reflect on the true loss landscape from sampling viewpoint(see 5). For convex optimization view point, Hessian indeed captures the global information about the loss landscape, but for non-convex region, it can only express the local curvature information and sampling according to the local information can easily lead to sampling of high loss region. + +Direct training with different random seeds. The posterior consist of models trained with different random seeds is studied in Fort et al. (2020) and shown be simple and strong base line for posterior for uncertainty measurement. The prediction of models trained with different random seeds provides strong diversity compared to the sampling methods which explore only the local region. In practice, it is easy to implement and applied to various to different leaning scenarios. The shortcoming of the method is that it requires much larger memory than other method as it stores and trains multiple models at the same time. Additionally, it did not necessarily violate the above observation as the models involved in the posteriors are trained simultaneously and have low loss guarantee. + +Diagonal Gaussian Diagonal Gaussian posteriors is well studied and shown to offer generalization guarantee. It provides easy control for the region to explore. Despite the fact that it did not contain local curvatures information, it can still fit in the observation from our theory as long as we select proper range for the variance. Compared to the two previously mentioned method, it offers better speed and memory consumption as we only need to sample independent variables for the construction of the whole distribution. In addition to the advantages mentioned, there is subtle connection between this posterior and optimization and we will illustrate in the following. + +# A.6 Details about Greedy selection + +Greedy selection method has been studied in many different prior works Khanna et al. (2017); Elenberg et al. (2017) to set up basis for its correctness and its applicability for different functions. As pointed out in Mirzasoleiman et al. (2020), one can transform the coreset problem on gradient + +
Speed upCIFAR10CIFAR100TinyImagenet
Crest0.461223961.2203575341.328621908
Our1.713768991.2273727981.448220165
+ +Table 5: Our method obtain better speed up compare to the benchmark method and the results also generalized to different datasets and architectures. The speed up is calculated as (Full training time / Method training time). The results are average over 5 different random seeds. + +![](images/9dd93dac38f3a8030ef782f6b773151ecaf683edaeb2452c2715e18d23f5545c.jpg) +Figure 5: (Left) We calculate the Hessian for sampling at red dot and sampling (yellow dot) using the Hessian inverse as a covariance matrix. The motivation for performing sampling of this kind is that one expect model to be sample lying on the low loss region as it put larger probability density in the small eigenvalue direction. However, when the Hessian loss its ability to represent the true loss curvature (non-convex setting), the Hessian posterior can sample models in regions of high loss, even when the Hessian is computed correctly. (Right) For ensemble method, we find its performance is competitive to Gaussian posterior. However, we find that there exist performance divergence in the different models trained with different random seed and the coreset selected show different performance gain for different models. The coreset selected through this posterior may not be able offer best performance gain in the ensemble. + +![](images/ed43a2430c91c719074b8729fdc312a28ac1e6d8986eafe2865a95665b497a23.jpg) + +into the following submodular cover problem with constant $C$ : + +$$ +S ^ {*} = \underset {S ^ {\prime} \subseteq S} {\operatorname {a r g m i n}} | S ^ {\prime} | \text {s . t} \sum_ {j \in S ^ {\prime}} \min _ {i \in S} \left| \left| \nabla l _ {i} \left(w _ {t}\right) - \nabla l _ {j} \left(w _ {t}\right) \right| \right| \leq \epsilon \tag {65} +$$ + +The $\gamma$ in origin problem will be calculated as the number of times a specific sample $j \in S'$ is used to achieve minimum distance in the argument right hand side in this transformed problem. Greedy algorithm is used to calculate the sample being selected in which achieve time complexity $\mathcal{O}(nk)$ in existing work Mirzasoleiman et al. (2020); Killamsetty et al. (2021a); Pooladzandi et al. (2022), where $n$ is the size of the training set and $k$ is the number of samples being selected. + +Despite linear complexity in terms of the sample selected, the calculation of the difference norm of the right hand side is still expansive due to the high dimensional properties of deep learning models. Several works Killamsetty et al. (2021b,a); Mirzasoleiman et al. (2020); Pooladzandi et al. (2022) demonstrate experimentally and theoretically that one can use the gradient with respect to the last layer for the calculation of gradient difference as it captures the norm of difference properly and greatly speed up the process for practical application, though a recent work has argued against it. Lastly, we complete the formulation with the formulation with fix sample size selected $k$ as following: + +$$ +S ^ {*} = \underset {S ^ {\prime} \subseteq S} {\operatorname {a r g m a x}} C - \sum_ {j \in S ^ {\prime}} \underset {i \in S} {\operatorname {a r g m i n}} \left\| \nabla l _ {i} \left(w _ {t}\right) ^ {L} - \nabla l _ {j} \left(w _ {t}\right) ^ {L} \right\| \tag {66} +$$ + +$$ +\begin{array}{c c} \text {s . t} & | S ^ {\prime} | \leq k \end{array} +$$ + +# A.7 Details about the algorithm design + +In this section, we will briefly discuss the design of the our algorithm. First of all, instead of selecting the entire training set like Mirzasoleiman et al. (2020), we adapt from Yang et al. (2023) to select + +![](images/14c9ba36a70e9b86c6431a7c1109b38db939de1c070258a1defe8267a019dacf.jpg) +Figure 6: The loss landscape generated by coreset of Craig method and the coreset of our method. The graph is generated with CIFAR10 data and ResNet20 using $1\%$ data budget. + +![](images/e7a9346105da65065f2886a8e16376f337077bd29cd4b5fd3485ea339f71692e.jpg) + +from mini-batch and union the mini-batch to obtain coreset with specified size. This help to reduce the selection time as mentioned in the Yang et al. (2023). For the selection, we calculate the expected version as 6 to ensure the properties obtained in the theory remain true. We did not perform threshold check listed in the Yang et al. (2023), we instead perform update on each epoch. + +# A.8 Toy model: Trajectory difference + +![](images/56a6cd5fcbf1932f623c2bd8205aa6215daca1950f697aea6aa11395914785a8.jpg) +Figure 7: The trajectory difference between gradient descent, Crest and gradient descent. To mimic the gradient mismatch resulting from the label noise, we inject high frequency function into the Crest and our method. To mimic the smoothed version of the loss landscape, we reduce the magnitude of the gradient in our method. We can find that the injected noise can drastically change the trajectory of different methods while the smoothed version can help better recover the ground truth trajectory. The fluctuation from noise becomes more significant around minima as the gradient around minima becomes smaller under this simulated experiment. + +# A.9 More learning scenarios. (Different training budget, corruption level, and architecture.) + +In this section, we perform experiments on even more learning condition such as different corruption level, training budget, and architecture. For architecture, We extend our result to Vision transformer to identify whether or not attention based models can work properly with our method and the results show that our method already outperform others under various structures. For other learning setting, we also find our method consistently outperform other benchmark which further justify the robustness of our method. + +
BudgetCorrupt RatioOur MethodRandomCrest
0.20.00.8969 ± 0.00260.8968 ± 0.00260.8083 ± 0.0125
0.10.8795 ± 0.00260.8788 ± 0.00320.8331 ± 0.0454
0.30.8495 ± 0.00530.8483 ± 0.00270.7632 ± 0.0396
0.50.8022 ± 0.00200.8011 ± 0.00320.5308 ± 0.0103
0.010.00.6683 ± 0.00440.6573±0.02110.4370 ± 0.0184
0.10.6216 ± 0.02290.6147 ± 0.02720.3631 ± 0.0496
0.30.5667 ± 0.00370.5431 ± 0.02720.3022 ± 0.0234
0.50.4801 ± 0.01500.4412 ± 0.01720.1988 ± 0.0231
+ +Table 6: CIFAR-10. Performance comparison of different methods across varying corruption ratios and budget levels. The best-performing method for each setting is highlighted in bold. + +
BudgetCorrupt RatioOur MethodRandomCrest
0.20.00.7337 ± 0.00130.7291 ± 0.00100.7172 ± 0.0029
0.10.6712 ± 0.00220.6566 ± 0.00450.6357 ± 0.0033
0.30.5521 ± 0.00210.5246 ± 0.00610.5081 ± 0.0064
0.50.4535 ± 0.00140.3884 ± 0.00590.3500 ± 0.0094
0.010.00.2660 ± 0.00790.2521 ± 0.00390.1672 ± 0.0126
0.10.2396 ± 0.00290.2231 ± 0.00330.1469 ± 0.0039
0.30.1773 ± 0.00500.1566 ± 0.00250.1191 ± 0.0038
0.50.1235 ± 0.00260.1134 ± 0.00510.0823 ± 0.0021
+ +Table 7: CIFAR-100. Performance comparison of different methods with varying corruption ratios and budget levels. The best-performing method for each setting is highlighted in **bold**. + +
Corrupt Ratio0.00.10.30.5
Our Method0.8294 ± 0.00110.8077 ± 0.00080.7734 ± 0.00090.7064 ± 0.0026
Crest0.8274 ± 0.00380.8002 ± 0.00420.7467 ± 0.01370.6653 ± 0.0112
Random0.8200 ± 0.00340.7980 ± 0.00780.7479 ± 0.00400.6806 ± 0.0030
+ +Table 8: Performance comparison of different selection methods using pretrained ViT-Base on CIFAR-100. Our method consistently outperforms both Crest and random sampling. We train with a learning rate of 0.0003, weight decay of 0.1, and a warm-up scheduling for 20 epochs. The full training process consists of 100 epochs to fit within the rebuttal time constraints. + +
Method0.00.10.30.5
Ours0.8773±0.00460.8893±0.01720.8787±0.00610.8767±0.0042
Crest0.8707±0.01630.8727±0.01100.8593±0.01030.8727±0.0253
Random0.7520±0.00400.7447±0.01420.7373±0.01860.7400±0.0243
+ +Table 9: additional experiment on the TREC-50 language dataset using ELECTRA-small-discriminator, a compact transformer model with approximately 14 million parameters. Alongside SNLI (used in the main paper), this dataset represents a language inference-style task. We finetune the pretrained model for 50 epochs under a 10 percent data budget, using AdamW with a learning rate of 1e-4, weight decay of 0.01, and standard default settings. We also apply learning rate warm-up for 1 epoch to stabilize training. + +
Corruption RatioRBFNN(Tukan et al., 2023)CraigCRESTOurs
0.00.6449±0.00380.5665±0.00530.6728±0.00300.6986±0.0025
0.10.5248±0.01330.4629±0.01020.6391±0.00400.6644±0.0034
0.30.3422±0.01400.2968±0.01620.5712±0.00600.6085±0.0044
0.50.2868±0.01900.1603±0.00480.4305±0.02600.5014±0.0068
+ +Table 10: CIFAR-100 test accuracy (mean ± std) under different corruption ratios. We compare our method with sensitivity based method (RBFNN(Tukan et al., 2023)) on CIFAR100 dataset under different corruption level. + +
Corruption RatioRBFNN(Tukan et al., 2023)CraigCRESTOurs
0.00.8255±0.00400.7618±0.00800.8724±0.00400.8757±0.0029
0.10.7924±0.01130.7490±0.00660.8440±0.00300.8544±0.0034
0.30.7280±0.01180.7050±0.01120.7843±0.00400.8120±0.0058
0.50.6216±0.02410.6320±0.01860.6652±0.01300.7318±0.0143
+ +Table 11: CIFAR-10 test accuracy (mean ± std) under different corruption ratios. We compare our method with sensitivity based method (RBFNN(Tukan et al., 2023)) on CIFAR10 dataset under different corruption level. + +# B Experiment details + +# B.1 Code base + +We develop our method base on code provided in Crest (Yang et al., 2023). For Craig and Glister, we use code based from CORD (https://github.com/decile-team/cords) with training hyperparameter changed to our setting. + +# B.2 Datasets and architectures + +In our work, we conduct experiments on various image datasets. MNIST (Deng, 2012), EMNIST (Cohen et al., 2017), CIFAR10, CIFAR100 (Krizhevsky, 2009), and TinyImagenet (Russakovsky et al., 2015), SNLI (Bowman et al., 2015) and Imagenet-1k dataset. For MNIST and EMNIST datasets, we use Lenet. For CIFAR10, CIFAR100, TinyImagenet and Imagenet-1k, we use respectively ResNet20, ResNet18 and ResNet50. For SNLI, we use pretrain RoBERTa (Liu et al., 2019) model. To create data corruption, we pick specified portion of training samples and flip the corresponding label to other classes to ensure the corrupt ratio is rigorous. For all experiments except for TinyImagenet, we run on single A10 GPU. For TinyImagenet, we use single NVIDIA A100 GPU. + +# B.3 Training hyperparameter + +For all experiments, we fix the peak learning rate at 0.1 and total training epoch to 200. The batch size is set to 128. For the first 20 epochs, we use linear warm up until learning rate reach 0.1 and decrease the learning rate by factor 0.1 at 120 epoch and 170 epoch. These hyperparameters were consistent with those in Yang et al. (2023) and were chosen to ensure fair comparisons across methods and datasets. + +# B.4 Experiment details about loss landscape and its matching + +We generate the loss landscape plot using technics in Li et al. (2018). We purturb the model weights using + +$$ +f (\alpha , \beta) = l \left(w ^ {*} + \alpha \delta + \beta \eta\right) \tag {67} +$$ + +In which the $\delta$ and $\eta$ are two randomly initialized vectors with magnitude scaled to models parameters. The plots are generated using 20 by 20 grid and for each grid we calculate the loss on the whole training dataset, Craig subset, and our subset with the same parameter. In the plot, we incorporate our method to the Craig method and use Gaussian noise 0.01. We select all at once as Craig does to verify that the method will bring smoothness to the loss surface and we observe that there exist more sharp corner for the loss surface created by Craig and the loss surface generated by our method is smoother than Craig method. The loss for Craig and our method are scaled loss using the $\gamma$ constant obtain through the greedy selection subroutine. Both plot are generated using $1\%$ data budget for each selection methods. The 3D plot is in Figure 6. + +# B.5 Evaluation + +We evaluated different methods on various datasets under different corruption ratios, with a fixed data budget of $10\%$ . We recorded the final test accuracy and measured time as the process wall time (i.e., from the start to the end of the process). To ensure reproducibility, all experiments were conducted using 5 different random seeds, and results were averaged across these runs. + +# C Time complexity analysis + +For Craig, there exist the need for forward propagation for each sample, and the corresponding time complexity is $\mathcal{O}(dn)$ . (d:model parameter, n:size of the dataset) The method is to select $q$ fraction $(0 < q < 1)$ of the whole dataset with size all at once. The greedy selection strategy for selecting $qn$ samples out n samples is $\mathcal{O}(qn^2)$ . Therefore, the complete time complexity is $\mathcal{O}(qn^2 + dn)$ for each epoch. + +For Crest, instead of selecting from whole dataset, they select from subset with size M, and the corresponding time complexity for forward propagation is $\mathcal{O}(dR)$ . (d:model parameter, n:size of the dataset) The method is to select $m$ point from the subset. The greedy selection strategy for selecting $Rm$ samples out n samples is $\mathcal{O}(Rm)$ . Therefore, the time complexity is $\mathcal{O}(P(Rm + dR))$ where the $P$ is the number of subset that is manually chosen. The overall time complexity for Crest method need to multiple by the times they update the coreset within one epoch for fair comparison, but the average number of update for coreset cannot be calculated as they utilize adaptive strategy which depend on the curvature of the loss landscape. + +For our method, we also select from subset with size $\mathbf{R}$ but we need multiple forward propagation (M times) and the corresponding time complexity is $\mathcal{O}(MdR)$ . We then select $m$ points from the subset and the corresponding time complexity is $\mathcal{O}(mR)$ . We need to perform multiple time to have $q$ portion of the whole dataset. The overall time complexity is $\mathcal{O}(q\frac{n}{R} (MdR + mR))$ for one epoch. + +For CIFAR-10 dataset, for each epoch, the running time for forward pass to obtain the gradient is 0.136 second for each batch, and the running time for the greedy selection is 0.000612 second for each batch. Hence, a much larger time is spent on the forward pass instead of the greedy selection part of the algorithm. The current state of the art method CREST requires expensive tracking of the Hessian which results in significantly longer training time and memory footprint. As a result, our method remains competitive in terms of the total training time while offering improved selection quality. + +# D Assumptions in the theory + +# Theorem 4.2 relies on assumptions of third-order smoothness and Hessian symmetry of the loss function + +The assumptions of third-order smoothness and Hessian symmetry are commonly used in deep learning theory to facilitate theoretical analysis. One key observation in deep learning is that the loss landscape often exhibits a degree of continuity, meaning that small changes in the parameter space generally lead to gradual changes in the loss function. This aligns with empirical findings on neural network optimization, where sharp transitions in loss are rare under typical training conditions. + +The symmetry of the Hessian follows naturally from the continuity and differentiability of the loss function. There are several important results derived using these assumptions. [A, B, C, D, E, F, G](Martens, 2010; Kiros, 2013; Ghorbani et al., 2019; Kunin et al., 2021; Barshan et al., 2020; Yao et al., 2020; Bottou et al., 2018) While non-linear activation functions introduce complexities, prior works suggest that, in practice, the loss function remains smooth enough for such assumptions to be reasonable.[C, H] (Ghorbani et al., 2019; Liu et al., 2023). Additionally, there are lines of research trying to approximate the Hessian using Fisher Information matrix (FIM) such as (Pascanu & Bengio, 2014; Liao et al., 2018; Sen et al., 2024). This implicitly assumes that the Hessian is symmetric as FIM is symmetric according to its definition. Also, there are works(Kirkpatrick et al., 2017; Ritter et al., 2018b) using Hessain as a precision matrix in probabilistic models, which implicitly assume symmetry in its structure and receive success in capturing or improving the behavior of deep learning. + +Similarly, the third-order smoothness assumption extends this notion by ensuring that second-order derivatives do not change abruptly, which aligns with empirical observations about the optimization dynamics of deep networks. These smoothness and regularity conditions are standard in optimization theory((Jin et al., 2017a; Allen-Zhu & Li, 2018; Carmon et al., 2017; Criscitiello & Boumal, 2021; Jin et al., 2017b; Bottou et al., 2018)) and are widely used to analyze generalization and convergence properties of deep learning models. + +Thus, while these assumptions may not hold universally in all settings (and we are not aware of any assumptions that hold universally for all models), they are reasonable approximations that enable theoretical insights into the learning dynamics of deep neural networks. We hope the reviewer agrees that our results are novel and useful within the context of current understanding of deep neural networks. + +# E Adacore comparison + +AdaCore is related to our work in terms of its motivation to capture loss curvature. However, we were unable to include it in our experiments due to the lack of an accessible or functional implementation. We explored multiple sources, including the official AdaCore repository (https://github.com/opooladz/AdaCore.git), which has always been empty ever since it was created, as well as public coreset libraries such as CORD (https://github.com/decile-team/cords) and DeepCore (https://github.com/PatrickZH/DeepCore.git). We did not find a working implementation of AdaCore in any of these repositories. + +We also attempted to reimplement the method based on the paper, but were unable to reproduce the reported performance. The method requires several manually tuned hyperparameters and includes steps involving Hessian computation, which are both computationally intensive and memory demanding. This introduces a significant runtime and scalability barrier, particularly problematic for the large-scale or noisy settings we focus on, and undermines the motivation for using coresets to speed up training. + +Additionally, we note that AdaCore has not been included in recent coreset benchmarks, such as those by (Yang et al., 2023; Okanovic et al., 2023) which includes the authors of Adacore themselves, where efficiency and scalability are prioritized. 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Several recent works have proposed mechanisms to address this, but they are often complex and fail to highlight the causes underlying this difficulty. In this work, we conduct a series of empirical analyses which suggest that the combination of non-stationarity with gradient pathologies, due to suboptimal architectural choices, underlie the challenges of scale. We propose a series of direct interventions that stabilize gradient flow, enabling robust performance across a range of network depths and widths. Our interventions are simple to implement and compatible with well-established algorithms, and result in an effective mechanism that enables strong performance even at large scales. We validate our findings on a variety of agents and suites of environments. We make our code publicly available. + +"We must be able to look at the world and see it as a dynamic process, not a static picture." — David Bohm + +# 1 Introduction + +Recent advances in deep reinforcement learning (deep RL) have demonstrated the ability of deep neural networks to solve complex decision-making tasks from robotics to game play and resource optimization [Mnih et al., 2015, Vinyals et al., 2019, Bellemare et al., 2020, Fawzi et al., 2022]. Motivated by successes in supervised and generative learning, recent works have explored scaling architectures in deep RL, showing gains in representation quality and generalization across tasks [Farebrother et al., 2023, Taiga et al., 2023]. However, scaling neural networks in deep RL remains fundamentally challenging [Ceron et al., 2024b,a]. A central cause of this instability lies in the unique optimization challenges of RL. Unlike supervised learning, where data distributions are fixed, deep RL involves policy-dependent data that constantly change during training [Lyle et al., 2022]. Each update of the policy $\pi_{\theta}$ alters future states and rewards, making the training objective inherently non-stationary. Value-based methods exacerbate these issues via bootstrapping, recursively using predicted values as targets. + +Estimation errors compound over time [Fujimoto et al., 2018], especially under sparse or delayed rewards [Zheng et al., 2018], leading to unstable updates, policy collapse, or value divergence [Van Hasselt et al., 2016, Lyle et al., 2023, 2024]. These dynamics are tightly coupled with architectural vulnerabilities. Deep networks face well known pathologies such as vanishing/exploding gradients [Pascanu et al., 2013], ill-conditioned Jacobians [Pennington et al., 2017], and activation saturation [Glorot and Bengio, 2010]. In deep RL, these are magnified by the "deadly triad" [Sutton + +and Barto, 2018, Van Hasselt et al., 2018], off-policy corrections, and changing targets. As networks scale, the risk of signal distortion and misalignment increases, resulting in underutilized capacity and brittle learning [Obando Ceron et al., 2023, Ceron et al., 2024a]. + +One overlooked source of these failures lies in how gradients propagate through the network. Specifically, the gradient decomposition, the layer-wise structure of backpropagation as a chain of Jacobians and weights determine how information flows during learning [Lee et al., 2020]. While gradient signal preservation has been studied in supervised learning [Schoenholz et al., 2017, Jacot et al., 2018], its role in deep RL, where both inputs and targets shift continually, remains poorly understood. + +In this work, we investigate how gradient decomposition interacts with non-stationarity and network scaling in deep RL. We demonstrate that in non-stationary settings like RL – where targets are bootstrapped, policies evolve continually, and data distributions shift – gradient signals progressively degrade across depth. This motivates the need for methods that explicitly preserve the structure of gradient information across layers. We explore this through a series of controlled experiments and ablations across multiple algorithms and environments, demonstrating that actively encouraging gradient propagation significantly improves stability and performance, even with large networks. Our work offers a promising approach for scaling deep RL architectures, yielding substantial performance gains across a variety of agents and training regimes. + +# 2 Preliminaries + +Deep Reinforcement Learning A deep reinforcement learning agent interacts with an environment through sequences of actions $(a\in \mathcal{A})$ , which produce corresponding sequences of observations $(s\in S)$ and rewards $(r\in \mathbb{R})$ , resulting in trajectories of the form $\tau \coloneqq \{s_0,a_0,r_0,s_1,a_1,r_2,\ldots \}$ . The agent's behavior is often represented by a neural network with parameters $\theta$ , composed of convolutional layers $\{\phi_1,\phi_2,\dots ,\phi_{L_c}\}$ and dense (fully connected) layers $\{\psi_1,\psi_2,\dots ,\psi_{L_d}\}$ , where $\psi_{L_d}$ has an output dimensionality of $|\mathcal{A}|$ . At every timestep $t$ , an observation $s_t\in S$ is fed through the network to obtain an estimate of the long-term value of each action: $Q_{\theta}(s_t,\cdot) = \psi_{L_d}(\psi_{L_d - 1}(\dots (\phi_{L_c}(\dots (\phi_1(s_t))\dots))\dots))$ . The agent's policy $\pi_{\theta}(\cdot |s_t)$ specifies the probability of selecting each action, for instance by taking the softmax over the estimated values as in Eq. 1. The training objective is typically defined as the maximization of expected cumulative reward as in Eq. 2, + +$$ +\pi_ {\theta} \left(a _ {t} \mid s _ {t}\right) = \frac {e ^ {Q _ {\theta} \left(s _ {t} , a _ {t}\right)}}{\sum_ {a \in \mathcal {A}} e ^ {Q _ {\theta} \left(s _ {t} , a\right)}} \tag {2} +$$ + +where $\gamma \in [0,1)$ is a discount factor and $\tau$ denotes a trajectory generated by following policy $\pi_{\theta}$ . Optimization proceeds by minimizing a surrogate loss $\mathcal{L}(\theta)$ , which may be derived from temporal-difference (TD) errors, policy gradients, or actor-critic estimators [Sutton and Barto, 2018]. In TD-based methods, the TD error at timestep $t$ is defined as: + +$$ +\delta_ {t} = r _ {t} + \gamma V _ {\theta} (s _ {t + 1}) - V _ {\theta} (s _ {t}), +$$ + +where $V_{\theta}(s) = \mathbb{E}_{a\sim \pi_{\theta}(a|s)}Q_{\theta}(s,a)$ . The recurrent nature of $\delta_t$ introduces dependencies on both current estimates and future rewards, making $\mathcal{L}(\theta)$ inherently non-stationary. As the policy $\pi_{\theta}$ evolves, the data distribution used for training shifts, further complicating optimization. Training is performed by collecting trajectories, computing gradients $\nabla \mathcal{L}(\theta)$ , and updating parameters via $\theta \gets \theta - \eta \nabla \mathcal{L}(\theta)$ , where $\eta > 0$ is the learning rate. Following conventions from supervised learning, deep RL algorithms often use adaptive variants of stochastic gradient descent, such as Adam [Kingma and Ba, 2014] or RMSprop [Hinton, 2012], which adjust learning rates based on running estimates of gradient statistics. The gradients with respect to each layer are denoted by; + +$$ +\nabla \phi_ {i} = \frac {\partial \mathcal {L}}{\partial \phi_ {i}}, \quad \nabla \psi_ {j} = \frac {\partial \mathcal {L}}{\partial \psi_ {j}}, +$$ + +where $\phi_{i}$ and $\psi_{j}$ represent the parameters (i.e., weight matrices or bias vectors) of layer $i$ and $j$ respectively. The structure and magnitude of these gradients $(\nabla \phi_{i}$ and $\nabla \psi_{j})$ are influenced by the loss function, data distribution collected from the environment, and the architecture itself. These per-layer gradients determine how effectively different parts of the network adapt during training. + +While training large models in supervised learning settings present challenges, advances in initialization, normalization, and scaling strategies have enabled relatively stable optimization [Ioffe and + +Szegedy, 2015, Ba et al., 2016, Glorot and Bengio, 2010]. Scaling up model size has been a central driver of progress across domains, improving generalization, enhancing representation learning, and boosting downstream performance [Kaplan et al., 2020]. + +Deep RL differs substantially from supervised learning. First, the data distribution is non-stationary, continually shifting as $\pi_{\theta}$ updates. Second, learning signals are often sparse, delayed, or noisy, which introduces variance in the estimated gradients [Han et al., 2022, Fujimoto et al., 2018]. These factors destabilize optimization and lead to loss surfaces with sharp curvature and complex local structure [Ilyas et al., 2020, Achiam et al., 2019]. Moreover, increasing model capacity often degrades performance unless regularization or architectural interventions are applied [Gogianu et al., 2021, Bjorck et al., 2021, Schwarzer et al., 2023, Wang et al., 2025]. + +These challenges are further compounded by both architectural and environmental factors. Network depth, width, initialization, and nonlinearity affect how gradients are propagated across layers. Meanwhile, reward sparsity, exploration difficulty, and transition stochasticity impose additional structure on the optimization landscape. The resulting geometry reflects the joint dynamics of policy, environment, and architecture, making deep RL optimization uniquely complex. + +Gradient Propagation Training deep networks poses fundamental challenges for effective gradient propagation [Glorot and Bengio, 2010]. As network depth increases, gradients may either vanish or explode as they are backpropagated through multiple layers, impeding the optimization of early layers and destabilizing learning dynamics [Ba et al., 2016]. These issues arise from repeated applications of the chain rule. For a network with intermediate hidden representations $\{h_0, h_1, \ldots, h_L\}$ , where $h_k \in \mathbb{R}^{d_k}$ , the gradient of the loss $\mathcal{L}$ with respect to a hidden layer $h_\ell$ is: + +$$ +\frac {\partial \mathcal {L}}{\partial h _ {\ell}} = \left(\prod_ {k = \ell + 1} ^ {L} \frac {\partial h _ {k}}{\partial h _ {k - 1}}\right) \frac {\partial \mathcal {L}}{\partial h _ {L}}, +$$ + +where each $\frac{\partial h_k}{\partial h_{k-1}} \in \mathbb{R}^{d_k \times d_{k-1}}$ is the Jacobian. If the singular values of these Jacobians are not properly controlled, their repeated multiplication can cause the norm of the gradient to shrink or grow exponentially with $\mathcal{L}$ . This severely impairs convergence, as earlier layers receive little to no useful gradient signal or become numerically unstable [Ioffe and Szegedy, 2015, He et al., 2016]. + +In addition to depth, the width of the network also influences gradient propagation. Consider a fully connected layer with weight matrix $W \in \mathbb{R}^{m \times n}$ and input vector $h \in \mathbb{R}^n$ . The output is $Wh \in \mathbb{R}^m$ , and under the assumption that $W$ and $h$ have i.i.d. zero-mean entries with finite variance $\sigma_W$ and $\sigma_h$ , respectively, the variance of the output is given by $\mathrm{Var}[Wh] = n\sigma_W\sigma_h$ . Thus, scaling the width $n$ without adjusting $\sigma_W$ and $\sigma_h$ leads to instability in forward and backward signal propagation affecting gradient norms and optimization trajectories. + +Beyond depth and width, the choice of nonlinearity also plays a central role in determining how gradients propagate. In a typical feedforward network, hidden activations evolve as $h_k = \zeta(W_k h_{k-1})$ , where $\zeta(\cdot)$ is a nonlinear activation function (e.g., ReLU, tanh, sigmoid), and $W_k$ is the weight matrix at layer $k$ . During backpropagation, the gradient with respect to a hidden layer includes the product of the Jacobian of the linear transformation and the derivative of the nonlinearity: + +$$ +\frac {\partial \mathcal {L}}{\partial h _ {k - 1}} = W _ {k} ^ {\top} \left(\zeta^ {\prime} \left(W _ {k} h _ {k - 1}\right) \odot \frac {\partial \mathcal {L}}{\partial h _ {k}}\right), +$$ + +where $\zeta'(\cdot)$ denotes the elementwise derivative of the activation function, and $\odot$ represents elementwise multiplication. For ReLU, $\zeta'(x) = 1_{x > 0}$ , so the gradient is entirely blocked wherever the neuron is inactive. This leads to the well-known dying ReLU problem, where a significant portion of the network ceases to update and becomes untrainable [Lu et al., 2019, Shin and Karniadakis, 2020]. + +# 3 Diagnosis: Gradients Under Non-stationarity and Scale + +A fundamental premise of modern deep learning is that scaling model capacity yields consistent gains in performance [Kaplan et al., 2020, Chowdhery et al., 2023]. This has held true in large-scale supervised learning, where training data distributions are stationary and i.i.d., and gradient descent operates under relatively stable conditions. However, in non-stationary settings, such as RL, gradient-based optimization faces severe challenges that scaling alone may exacerbate [Ceron et al., 2024a,b]. + +In this section, we diagnose how gradient pathologies emerge and intensify across different settings, with a focus on architectural scaling in width and depth (network scales used specified in Table 1). + +# 3.1 Gradient Pathologies + +We train neural networks of varying depths and widths and analyze their training dynamics. + +Supervised Learning (Stationary and Non-Stationary) We use the CIFAR-10 image classification benchmark [Krizhevsky et al., 2009], where the input-output mapping remains fixed over time. Models consist of standard 6-layer convolutional neural networks (CNN) followed by a multi-layer perceptron (MLP). We vary the depth and width of the MLP to explore how model scale influences learning behavior. To introduce non-stationarity, we periodically shuffle the training labels during training, following the setup by Sokar et al. [2023]. This creates a loss landscape that changes over time, echoing the challenges of deep RL. Fig. 1 illustrates the contrast in training behavior and gradient flow between stationary and non-stationary supervised learning. Under non-stationarity, deep networks fail to recover accuracy, which aligns with a marked degradation in gradient magnitudes. + +![](images/ed52625b0a901e332f227cf08edb9bfb979ada5c1b612a22996730bc4ee99fa4.jpg) +Figure 1: Training dynamics under stationary and non-stationary supervised learning. (Left) In the stationary setting, both shallow and deep models fit the data effectively across widths. Under non-stationarity only shallow networks partially recover during training, while deeper ones collapse. (Right) This collapse correlates with degraded gradient flow. In stationary settings, gradient norms remains stable across all network scales (shaded boxes) while in non-stationary settings (solid-colored boxes), gradient magnitudes diminish with depth and width, suggesting poor adaptability. + +![](images/15e052169d9684eefd4a63f8eb13f412aba311847621c0ef06f867ae818f6e1d.jpg) + +![](images/54c9365b5c113a2f72730c6c568ce5a72a87c99f198013c865d61544775f1312.jpg) + +![](images/63d212ccb55d8114ea74271fbbd949d0a36e562651f0edddb1a7af9c8265de58.jpg) + +Reinforcement Learning As discussed in Sec. 2, RL introduces fundamentally different sources of non-stationarity due to the policy-dependent data distribution and moving target estimates. To study gradient dynamics, we use PQN [Gallici et al., 2025], a recent value-based algorithm that achieves strong performance without relying on a target network or replay buffer. PQN ensures stability and convergence using Layer Normalization [Ba et al., 2016] and supports GPU-based training through vectorized environments for online parallel data collection. In Sec. C.1 we extend our investigation to DQN [Mnih et al., 2015] and Rainbow [Hessel et al., 2018], demonstrating the generality of our observations. As shown in Fig. 2, deeper networks trained with PQN exhibit a collapse in both episode returns and gradient norms², highlighting the fragility of deep models under non-stationarity. + +# 3.2 Training Degradation + +In Fig. 3 we evaluate diagnostic metrics capturing expressivity and training dynamics, revealing that deeper networks exhibit pronounced training pathologies and degraded performance. We first measure the fraction of dormant neurons, defined as units with near-zero activations over a batch of trajectories [Sokar et al., 2023], and find that dormant neurons grow with depth, signaling underutilized capacity. Next, we assess representational expressivity using $SRank$ , the effective rank of penultimate-layer activations [Kumar et al., 2020], observing that deeper networks tend to collapse state representations into lower-dimensional, and less expressive (as evidenced by declining returns) subspaces. To study loss curvature, we compute the Hessian trace of the temporal-difference loss. This metric serves as a proxy for sharpness or smoothness in optimization [Ghorbani et al., 2019], similarly to tracking the largest eigenvalue. Fig. 3 shows that only shallow networks exhibit high Hessian trace values, + +![](images/67ec732451a618c7e8991cd5b423c5b8634a875b0dd0e84d004de1b0b479c122.jpg) +Figure 2: Mean episode returns and gradient norms across increasing MLP depths and widths on two ALE games using PQN. (Left) Only shallow networks achieve high episode returns; performance collapses for deeper networks. (Right) The collapse correlates with vanishing gradient norms, suggesting that deeper models fail to adapt to non-stationarity in deep RL. + +![](images/c29a3d0cdef4d4c8d29f0868062eb6a4100d522fa0e77223a094b45e53dd32f4.jpg) + +![](images/a32a202d067ca10b7aa9aafb418519559edf8cda7cccf1236772c98487513662.jpg) +Figure 3: Training pathologies emerge as MLP depth increases. Deeper networks exhibit a higher fraction of inactive neurons, reduced representation rank (SRank), vanishing Hessian trace (loss curvature), and degraded learning performance (mean Q-values and episode returns). These trends indicate that scaling depth limits expressivity and plasticity, impairing policy quality. + +suggesting access to sharper regions of the loss surface with pronounced directions of improvement. In contrast, deeper architectures consistently show near-zero trace, indicating poorly conditioned geometry that hinders effective gradient-based updates. These findings suggest a breakdown in representation, plasticity, and optimization as networks scale, ultimately impeding learning. + +# Key observations on gradients under non-stationarity and scale: + +- Non-stationarity amplifies gradient degradation in deeper and wider networks. +- In deep RL, deeper models suffer from vanishing gradients, reduced activations, and loss of representational expressivity. +- The flat loss curvature intensifies with depth, correlating with poor learning. + +# 4 Stabilizing Gradients + +Having identified the pathologies that emerge in non-stationary regimes, particularly under large-scale architectures, we investigate strategies to mitigate these instabilities. We focus on two complementary interventions: skip connections [He et al., 2016] and optimizers [Martens and Grosse, 2015], as these directly improve gradient flow. We continue to use PQN as our base RL algorithm and evaluate on the Atari-10 suite [Aitchison et al., 2023]. In Sec. 5, we demonstrate that the effectiveness of our proposed gradient interventions generalize beyond this specific algorithm and environment suite. + +# 4.1 Intervention 1: Multi-Skip Residuals for Gradient Stability + +Gradient instability in deep networks is often aggravated by increasing depth, non-linear activations, and misaligned curvature across layers. While standard residual connections offer some relief by introducing shortcut paths for gradient flow [He et al., 2016], they typically span only one or two layers, which can be insufficient in the presence of severe gradient disruption due to non-stationarity. + +![](images/9697a57277bc3a02c496abb27e966894dbfb4a1773abb57fa799778f9483962c.jpg) +Figure 4: (Left) MLP architectures and (right) scaling strategies studied. + +![](images/0f696f920f6a1be5d98a7fd364502271b5b6a0b364bdfc0d533000675d9b7b62.jpg) + +We introduce multi-skip residual connections, in which the flattened convolutional features are broadcast directly to all subsequent MLP layers. This design ensures that gradients can propagate from any depth back to the shared encoder without obstruction. + +We compare our network architecture against the standard fully connected baseline across varying depths. As shown in Fig. 5 (left), performance collapses with increased depth in the baseline, while the multi-skip architecture maintains stable learning and continues to improve across widths. This improvement is accompanied by consistently higher gradient magnitudes. Complete results across all network depths and widths are presented in Sec. C.4. + +![](images/e1f0377de9db7f715a84a290465dfe68a5cd86d4306b90fdc5abc4b355582918.jpg) +Figure 5: Gradient-stabilizing interventions improve scalability in deep RL. (Left) Standard fully connected networks trained with PQN collapse at greater depths due to vanishing gradients. In contrast, multi-skip architectures maintain gradient flow and scale effectively. (Right) The default RAdam optimizer leads to instability in deep networks, while switching to the Kron optimizer preserves gradient signal and enables stable learning without architectural changes. + +![](images/f67cc2e0342038cbfce5063408b546f2f9c00a6e1cca6c15a71081043f6b9515.jpg) + +![](images/a86756a843e8c6c75e062f9cf71c466fc1e2857c17a29335554df0f6e98b4144.jpg) + +![](images/cb02d6362c359a8f5cf8036cae8b61acb7f5c54ad927dbc7aa8ccad0a64b5887.jpg) + +# 4.2 Intervention 2: Second-Order Optimizers for Non-Stationarity + +First-order optimizers such as SGD and Adam rely on local gradient estimates and fixed heuristics (e.g., momentum, adaptive step sizes) [Kingma and Ba, 2014], which are agnostic to curvature and often brittle under shifting data distributions. In contrast, second-order methods adjust parameter updates using curvature information, enabling more informed and stable adaptation. + +Let $\mathcal{L}(\theta)$ denote the loss function, and $g = \nabla \mathcal{L}(\theta)$ its gradient. A second-order update takes the form $\theta_{t + 1} = \theta_t - \eta H^{-1}g$ , where $H$ is the curvature matrix, typically the Hessian or the Fisher Information Matrix (FIM) [Martens, 2020]. Directly inverting $H$ is computationally infeasible in deep neural networks so Kronecker-factored approximations, such as K-FAC [Martens and Grosse, 2015], address this challenge by approximating $H$ using low-rank Kronecker products. + +Kronecker-factored optimizer (Kron for short) approximates the FIM and applies structured preconditioning that captures inter-parameter dependencies, unlike Adam's diagonal scaling. This yields directionally aware preconditioning that better aligns with the curvature of the loss surface [Martens, 2020]. In non-stationary settings, such as deep RL, where both the data distribution and curvature + +evolve over time, curvature-aware updates can help preserve gradient signal by maintaining stable update magnitudes and directions. As shown in Fig. 5 (right), replacing RAdam with Kron prevents performance collapse at greater depths, even in standard MLP architectures. Complete results across all network depths and widths are presented in Sec. C.6. + +# 4.3 Combining Gradient Interventions + +We combine both gradient interventions to PQN and evaluate it on the full ALE suite (57 games), across 3 seeds and 200M frames. Fig. 6 shows that our augmented agent outperforms the baseline in $90\%$ of the environments, achieving a median relative improvement of $83.27\%$ . Notably, the baseline PQN is itself competitive with strong agents such as Rainbow [Gallici et al., 2025], highlighting the effectiveness of our interventions. Detailed per-environment learning curves can be found in Sec. E.1. + +In Fig. 7 we validate the effectiveness of the combined gradient interventions in the non-stationary SL setting we used as motivation in Sec. 3. The results verify that these interventions enable high accuracy and sustained adaptability across depths and widths, even under dynamic label reshuffling. + +![](images/8558039e975781bef2dd829b85fa7bfa6e523bb3c329cc22029d2546c6d3bfd6.jpg) +Figure 6: Gradient-stabilized PQN achieves superior scalability. (Left) On Atari-10, the combined interventions lead to high HNS even at greater depths, outperforming either intervention alone (see Fig. 5) and increased gradient gradient flow. (Right) On the full ALE suite, our agent outperforms the baseline in $90\%$ of the games with a median performance improvement of $83.27\%$ . + +![](images/13ef38cdd2c46abc92a89576d18e638f06c341bb6b6fd37774726a6b601ea50f.jpg) +Figure 7: Gradient interventions enable rapid recovery in non-stationary SL. (Left) Models with combined gradient interventions rapidly recover accuracy after label reshuffling, demonstrating robust adaptation in non-stationary settings. (Right) This is supported by stable gradient flow across depth. Dashed curves and shaded boxes indicate MLP baselines. + +# 4.4 Alternative Gradient-Stabilization Methods + +To ensure that our findings were not specific to a narrow choice of interventions, we conducted a broader exploration of alternative strategies for improving gradient stability in deep RL. We tested a variety of approaches inspired by prior work on optimization and representation stability in both supervised and RL settings (see Sec. C.7 for more details). + +As summarized in Tab. 3, none of these methods consistently improved stability or performance compared to our proposed combination of multi-skip residuals and Kronecker-factored optimization. In many cases, the alternatives yielded either negligible gains or degraded performance as network depth increased, reinforcing that architectural and curvature-aware interventions are key to preserving gradient flow at scale. + +# 5 Beyond the ALE and PQN + +To evaluate the generality of our findings, we extend our analyses. Specifically, we: (i) apply our proposed methods to PPO [Schulman et al., 2017] on the full ALE and on continuous control tasks in Isaac Gym [Makoviychuk et al., 2021]; (ii) assess the impact of richer convolutional encoders by replacing the standard CNN backbone used in the ALE with the Impala CNN architecture [Espeholt et al., 2018]; (iii) augment Simba [Lee et al., 2025] with our proposed techniques and evaluate performance on the DeepMind Control Suite (DMC) [Tassa et al., 2018]; and (iv) investigate whether our interventions can stabilize and scale a wide range of Q-learning algorithms in challenging offline RL benchmarks [Park et al., 2025]. + +![](images/6e3e9d8b8af9bcd1c56238017533f007c9636fa2402892b3c03084593db0c9fd.jpg) +Figure 8: PPO with gradient interventions. Left: On the full ALE suite, applying the combined gradient interventions to PPO yields a median performance improvement of $31.40\%$ and outperforms the baseline in $83.64\%$ of the games. Right: In the Cartpole and Anymal tasks from IsaacGym, only the augmented PPO maintains stable performance across depths and widths. + +![](images/0756d9f0572dce552c9c67c328431334019c6ad5b24a164c2f4a18670a8f10a7.jpg) + +PPO with Gradient Interventions. Fig. 8 (left) shows that augmenting PPO with the same strategies as in PQN (Layer Normalization by default on PQN, multi-skip residual connections, and Kronecker-factored optimization) significantly boots performance. On the ALE benchmark, the augmented PPO outperforms the baseline in $83.64\%$ of the environments, achieving a median relative improvement of $31.40\%$ . In Isaac Gym's continuous control tasks, including Cartpole and Anymal (Fig. 8, right), the baseline PPO collapses as model size increases, while the augmented variant remains stable and achieves superior performance at all depths and widths. + +Gradient Interventions in Scaled Encoder Variants The Impala CNN is a scalable convolutional architecture that has demonstrated strong performance gains in agents such as Impala [Espeholt et al., 2018] and Rainbow [Hessel et al., 2018]. We investigate whether, given its capacity to extract richer representations from visual input, combining Impala CNN with our gradient flow interventions enables effective scaling of the MLP component. As shown in Fig. 9, PPO and PQN benefit significantly from replacing the standard CNN with the Impala CNN. For PQN, the Impala encoder enables successful scaling of the MLP, in contrast to the performance collapse seen without our interventions. These results suggest that the expressivity of richer visual encoders is more effectively leveraged by deeper networks when gradient flow is preserved. + +![](images/0cd0318ba3435caa14e6c73670702b5ec1e78a294d5af919963dc18e6d21c8a7.jpg) +Figure 9: Scaling performance with standard vs. Impala CNN encoders on PQN (left) and PPO (right). Each agent is evaluated using both the Atari CNN (left sub-panels) and the Impala CNN (right sub-panels) as the encoder. Gradient interventions enable successful scaling in both cases. + +![](images/ebcf84004857399f9ed571a6af67c467ce025f98e3907fd2a929833e1a2d6af8.jpg) + +Simba with Kron Optimizer. Simba [Lee et al., 2025] is a scalable actor-critic framework that integrates normalization, residual connections, and LayerNorm. We augment Simba by replacing its + +![](images/126ca88c5373814f8621340c0562563584413ef92ba05b8dd3a4c1c50c157214.jpg) +Figure 10: Performance comparison between AdamW (dashed lines) and Kron (solid lines) optimizers using the SimBa architecture with SAC and DDPG, averaged over 5 random seeds. As model size increases, AdamW leads to consistent performance degradation, while Kron enables stable and improved learning with larger networks. + +default AdamW optimizer with Kron while keeping all other hyperparameters fixed. We evaluate SAC [Haarnoja et al., 2018] and DDPG [Lillicrap et al., 2015] on challenging DMC tasks, using the Simba architectures of varying depth and width. Despite its design for scalability, default Simba collapses across all tasks as networks grow as shown in Fig. 10 (additional results in Sec. E.2). In contrast, the Kron-augmented version successfully scales in both depth and width, achieving consistent and stable performance gains. These findings underscore the generality of our approach as effectively enabling parameter scaling in deep RL agents. + +Gradient Interventions for Scalable Offline Q-Learning. Park et al. [2025] highlight significant challenges in scaling Q-learning algorithms for complex offline RL tasks, demonstrating that many standard offline RL baselines fail to learn effective policies, even on large, high-quality datasets. Their key finding was that performance improvements were primarily driven by techniques that shorten the effective credit assignment horizon, such as n-step returns and hierarchical methods. This led to their proposal of two new high-performing algorithms, SHARSA and DSHARSA, which are designed to operate with shorter effective horizons. This finding motivates a parallel investigation: can our proposed gradient interventions, which are designed to stabilize and accelerate deep network training, also address the scaling limitations of offline Q-learning? To test this, we augment the full suite of baselines and novel algorithms from Park et al. [2025] with our proposed gradient interventions. The results, presented in Fig. 11, show that our methods provide a complementary path to scalability. Applying our interventions generally improves the performance of the baselines across all tasks. + +The performance gains are particularly pronounced in the most sparse-reward task, humanoidmaze-giantNavigate, where our gradient interventions enable multiple methods to achieve near-optimal performance, whereas their baseline counterparts largely fail. Furthermore, this stability extends to generalization across task difficulty. When moving from puzzle-4x5-play to the harder puzzle-4x6-play task, many baselines exhibit a sharp performance degradation. In contrast, the performance of several algorithms with our interventions remains consistent and high, demonstrating improved robustness. Finally, we note that while the primary focus of this paper is to address gradient pathologies arising from scaling and non-stationarity, these results highlight that our interventions are also highly beneficial in offline deep RL, where inputs are stationary. + +# 6 Related Work + +A central challenge in scaling deep RL lies in the inefficient use of model capacity. Increasing parameter counts often fails to yield proportional gains due to under-utilization. Sokar et al. [2023] and Liu et al. [2025b] show that online RL induces a growing fraction of inactive neurons, a phenomenon also observed in offline settings. Ceron et al. [2024a] report that up to $95\%$ of parameters can be pruned post-training with negligible performance drop, underscoring substantial redundancy. These findings have motivated techniques such as weight resetting [Schwarzer et al., 2023], tokenized computation [Sokar et al., 2025], and sparse architectures [Ceron et al., 2024b, Willi et al., 2024, Liu + +![](images/4b15fd7355516098e12dadce72dcadcf7a10960b6a01c789079660ee9da5f522.jpg) +Figure 11: Performance of offline Q-learning algorithms with and without our gradient interventions. We compare the original algorithms from Park et al. [2025] against our augmented versions. The results, averaged over 3 seeds demonstrate a general improvement in scalability. + +et al., 2025a, Ma et al., 2025], along with auxiliary objectives to promote capacity utilization [Farebrother et al., 2023]. While scaling model size offers greater expressivity, its benefits depend on appropriate training strategies [Ota et al., 2021]. Architectural interventions such as SimBa [Lee et al., 2025] improve robustness by regularizing signal propagation through components such as observation normalization, residual feedforward blocks, and layer normalization. Complementarily, BRO [Nauman et al., 2024] shows that scaling the critic network yields substantial gains in sample and compute efficiency, provided it is paired with strong regularization and optimistic exploration strategies. + +Gradient flow, however, remains a central bottleneck. We complement prior efforts by explicitly targeting vanishing gradients as a mechanism for improving scalability. Our approach builds on the role of LayerNorm in stabilizing training and enhancing plasticity [Lyle et al., 2024], and leverages its theoretical effect on gradient preservation as formalized in PQN [Gallici et al., 2025]. Optimization-level interventions such as second-order methods [Martens and Grosse, 2015, Muppidi et al., 2024] and adaptive optimizers [Ellis et al., 2024, Bengio et al., 2021, Wu et al., 2017] also address instability under non-stationarity. Our approach integrates architectural and optimizer-level interventions to enable stable gradient flow and unlock parameter scaling in deep RL agents. + +# 7 Discussion + +Our analyses in Sec. 3 suggest that the difficulty in scaling networks in deep RL stems from the interaction between inherent non-stationarity and gradient pathologies that worsen with network size. In Sec. 4, we introduced targeted interventions to address these challenges, and in Sec. 4.3, we demonstrated their effectiveness. We validated the generality of our approach across agents and environment suites, consistently observing similar trends. These findings reaffirm the critical role of network design and optimization dynamics in training scalable RL agents. While our proposed solutions may not be optimal, they establish a strong baseline and provide a foundation for future work on gradient stabilization in deep RL. More broadly, our findings suggest that scaling limitations in deep RL are not solely attributable to algorithmic instability or insufficient exploration, but also stem from gradient pathologies amplified by architectural and optimization choices. Addressing these issues directly, without altering the learning algorithm, yields substantial gains in scalability and performance. This suggests that ensuring stable gradient flow is a necessary precondition for effective parameter scaling in deep RL. + +Limitations. Our study is constrained by computational resources, which limited our ability to explore architectures beyond a certain size. While our interventions show consistent improvements across agents and environments, further scaling remains an open question. While using second order optimizers introduced additional computational overhead (see Tab. 12), this cost is mitigated by leveraging vectorized environments and efficient deep RL algorithms, narrowing the gap relative to standard methods. These limitations highlight promising directions for future work, including the development of more computationally efficient gradient stabilization strategies and scalable optimization techniques. + +Acknowledgment The authors would like to thank João Guilherme Madeira Araujo, Evan Walters, Olya Mastikhina, Dhruv Sreenivas, Ali Saheb Pasand, Ayoub Echchahed and Gandharv Patil for valuable discussions during the preparation of this work. João Araujo deserves a special mention for providing us valuable feed-back on an early draft of the paper. We want to acknowledge funding support from Natural Sciences and Engineering Research Council (NSERC) of Canada, Google Research, Fonds de recherche du Québec (FRQNT) and The Canadian Institute for Advanced Research (CIFAR) and compute support from Digital Research Alliance of Canada, Mila IDT, and NVidia. We would also like to thank the Python community Van Rossum and Drake Jr [1995], Oliphant [2007] for developing tools that enabled this work, including NumPy Harris et al. [2020], matplotlib Hunter [2007], Jupyter Kluyver et al. [2016], and Pandas McKinney [2013]. + +Broader Impact This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here. + +# References + +Zaheer Abbas, Rosie Zhao, Joseph Modayil, Adam White, and Marlos C Machado. Loss of plasticity in continual deep reinforcement learning. 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Adbelief optimizer: Adapting step sizes by the belief in observed gradients. Advances in neural information processing systems, 33:18795-18806, 2020. + +# A Environment Details + +Throughout the paper, we evaluate the deep reinforcement learning agents' performance on the Atari-10 suite [Aitchison et al., 2023], a curated subset of games from the Arcade Learning Environment (ALE) [Bellemare et al., 2013]. Atari-10 consists of 10 games selected to capture the maximum variance in algorithm performance, achieving over $90\%$ correlation with results on the full ALE benchmark. This makes it a computationally efficient yet representative testbed for deep reinforcement learning. We follow the experimental protocol of Obando Ceron et al. [2023], Ceron et al. [2024b], Agarwal et al. [2021], running each experiment with three random seeds and reporting the aggregate human-normalized score across games. + +The games in Atari-10 are: + +- Amidar, Battle Zone, Bowling, Double Dunk, Frostbite, Kung Fu Master, Name This Game, Phoenix, Q*Bert and River Raid. + +Additionally, to further support the generality of our findings, we evaluate the proposed combined gradient interventions on the full ALE benchmark. We also assess their effectiveness on continuous control tasks from the IsaacGym simulator [Makoviychuk et al., 2021] and the DeepMind Control Suite (DMC) [Tassa et al., 2018], extending our analysis to robotics-based environments. We conduct experiments on the 4 challenging tasks of DMC: + +- Humanoid Walk, Humanoid Run, Dog Trot and Dog Run. + +# B Network Sizes + +Throughout the paper, we experiment with models of varying depths and widths. Unless stated otherwise (e.g. in Sec. 5, where we evaluate the Impala CNN), the convolutional feature extractors are kept fixed. Consequently, our experiments focus primarily on scaling strategies and architectural variations in the MLP components of the networks. + +To enable meaningful comparisons across different learning regimes, the MLP architectures are kept consistent across supervised learning (SL), non-stationary SL, and reinforcement learning (RL) experiments. This consistency ensures that observed differences in gradient behavior arise from the learning setting itself, rather than confounding factors due to domain-specific architectures. + +Table 1 provides detailed information on the number of parameters for each depth-width configuration, categorized as small, medium, or large, as used throughout the paper. + +Table 1: Number of parameters (in millions) for different MLP architectures. + +
Depth / WidthSmallMediumLarge
Small2.3911.9027.70
Medium3.4521.3553.93
Large4.5030.7980.15
+ +# C Additional Experiments + +# C.1 Scaling with DQN and Rainbow + +To further support our hypothesis on the emergence of gradient pathologies in deep reinforcement learning, we investigate whether similar issues arise in algorithms beyond PQN and PPO, as discussed in the main paper. Specifically, we study the effects of architectural scaling on two widely used value-based algorithms: DQN [Mnih et al., 2015] and Rainbow [Hessel et al., 2018]. + +DQN is a foundational deep RL algorithm that learns action-value functions using temporal difference updates and experience replay, serving as a standard baseline for value-based methods. Rainbow extends DQN by integrating several enhancements, such as double Q-learning, prioritized experience + +replay, dueling networks, multi-step learning, distributional value functions, and noisy exploration, to achieve improved sample efficiency and stability. + +In Fig. 12, we report the performance of DQN and Rainbow as we scale the depth and width of their networks. As with PQN and PPO, we observe consistent degradation in performance at larger scales. In Fig. 13, we present the corresponding gradient behavior, which reveals the same vanishing and destabilization phenomena discussed in this work. These findings reinforce the generality of the identified gradient pathologies across both policy-based and value-based deep RL algorithms. + +![](images/2854d9fccb9815ac97d0de191d399dc9d6487a0ed31821d7858c1bd0a727ffdf.jpg) +(a) DQN HNS + +![](images/d17608742d8cf29ba4ae4295cc4367dea4eaf5e72d876cb414e49baa10522688.jpg) +(b) Rainbow HNS + +![](images/7af619cfda69ff6c64ee6cfbd2aa224cff29128d4169765c70d1e25c346b7abf.jpg) +Figure 12: Median human normalized scores for DQN (left) and Rainbow (right) as a function of total network parameters. Lines of different colors denote varying network depths, while marker shapes indicate different widths. For both agents, performance consistently declines as network size increases, highlighting the adverse effects of scaling. +Figure 13: Gradient magnitudes during training for DQN (top) and Rainbow (bottom). As network depth increases, gradient flow systematically diminishes, ultimately collapsing to near-zero values. This consistent decay mirrors the performance degradation observed at larger scales. + +# C.2 Combining Gradient Interventions in Non-stationary Supervised Learning + +Building on our findings in Sec. 4.3, we extend our analysis by applying the proposed combined gradient interventions to the same image classification models used in Sec. 3. Specifically, we train the models in the non-stationary supervised learning setup, where the CIFAR-10 labels are iteratively shuffled, following the experimental design from Sokar et al. [2023]. As demonstrated in Sec. 3, while models in standard supervised learning settings are able to scale effectively and maintain high performance, introducing non-stationarity leads to failure in adaptation for baselines that use fully connected layers and the Adam optimizer. This issue is exacerbated as model scale increases. + +Our results, presented in Fig. 7, show that combining the multi-skip architecture for the MLP component with the Kronecker-factored optimizer and Layer Normalization enables near-perfect + +continuous adaptation. The models quickly adapt to the changing optimization problem following label reshuffling, with gradient magnitudes remaining stable throughout the process. + +# C.3 Architecture and Optimizer Ablations + +In this work, we introduce the multi-skip architecture, an extension of the standard residual MLP design, and propose the use of the Kronecker-factored optimizer for online deep RL. While these techniques form the basis of our primary interventions, our broader goal is not to prescribe a fixed set of methods, but rather to motivate a general class of architectural and optimization interventions that promote healthy gradient flow in deep networks. To this end, we expand the scope of our evaluation by incorporating a wider range of baselines. Specifically, we compare various optimizer choices, including Adam and AdaBelief [Zhuang et al., 2020], alongside MLP architectures such as the standard residuapl MLP [He et al., 2016] and DenseNet [Huang et al., 2017]. These architectures have been previously explored in the context of scaling networks in online deep RL [Lee et al., 2025, Ota et al., 2024], providing a relevant basis for comparison. + +We also evaluated state-of-the-art optimizers that have demonstrated success in training large-scale models such as transformers in supervised learning. Specifically, we tested Shampoo [Gupta et al., 2018], a second-order optimizer that maintains and preconditions gradients using full-matrix statistics per layer, and Apollo [Ma, 2020], an adaptive optimizer that leverages curvature information without explicitly computing or storing second-order matrices. + +Despite extensive hyperparameter tuning for both methods, we were unable to achieve strong performance in the online deep RL setting. This suggests that further investigation is needed to understand the key properties required for these optimizers to be effective in this regime. Asadi et al. [2023], Ceron and Castro [2021] demonstrate that optimizer behavior plays a critical role in the training dynamics of online deep RL methods, with Asadi et al. [2023] showing that stale optimizer states can hinder learning, and Ceron and Castro [2021] revealing that optimizer sensitivity interacts with the choice of loss function, particularly when comparing Huber and MSE losses. + +We present the results for PPO and PQN across all tested optimizers in Fig. 14. + +![](images/6b83cc53ddce6832b6c86554534a01d13fc78dd91685968034ebd58952ac7621.jpg) +Figure 14: Median human normalized scores on Atari-10 for PPO (top row) and PQN (bottom row), comparing a range of optimizers including RAdam, AdaBelief, Shampoo, Apollo, and Kron (shown in the main curves). While adaptive optimizers like AdaBelief show some robustness, only Kron consistently enables stable and performant training as models scale. Each curve represents the mean performance across three random seeds per algorithm, with shaded areas indicating $95\%$ bootstrap confidence intervals. + +# C.4 Results with the Multi-Skip Architecture. + +We present the full learning curves comparing the proposed multi-skip architecture to the baseline fully connected architecture across all depths and widths studied in the paper. We follow the experimental protocol of Obando Ceron et al. [2023], Ceron et al. [2024b], Agarwal et al. [2021], running each experiment with three random seeds. + +![](images/7c1e5808d29d653c9efc48581de315e112959f967b359276dd81dbda2b981c91.jpg) +Figure 15: Median human-normalized scores with PQN on the Atari-10 benchmark, comparing the baseline agent and the proposed multi-skip architecture across varying depths and widths. The multi-skip architecture not only improves performance at shallow depths, but also enables PQN to remain trainable across all scales considered, whereas the baseline MLP rapidly collapses as depth and width increase. Each curve represents the mean performance across three random seeds per algorithm, with shaded areas indicating $95\%$ bootstrap confidence intervals. + +# C.5 Ablation on the number of skip connections. + +To isolate the effect of skip length on performance, we fix the main network of our proposed MultiSkip architecture (in large size, which includes 5 residual blocks) and vary how many of these blocks receive skip connections from the encoder. When $\mathrm{Skip} = k$ , we apply the encoder features as skip connections to the first $k$ residual blocks immediately following the encoder, while the remaining $(5 - k)$ blocks operate without direct encoder input. The table below reports human-normalized scores on the Atari-10 benchmark Agarwal et al. [2021]. + +Table 2: Human-normalized scores on the Atari-10 benchmark, varying the number of residual blocks $(k)$ that receive skip connections from the encoder. Performance generally improves as more connections are added. + +
EnvironmentSkip=1Skip=2Skip=3Skip=4Skip=5
Amidar-v50.200.170.190.200.36
BattleZone-v50.010.670.620.600.69
Bowling-v50.070.040.040.080.23
DoubleDunk-v5-2.09-2.00-1.55-1.36-1.32
Frostbite-v50.670.700.790.920.88
KungFuMaster-v50.930.950.931.120.98
NameThisGame-v50.790.650.720.851.24
Phoenix-v50.690.680.680.660.66
Qbert-v50.841.011.060.911.09
Riverraid-v50.420.440.650.690.99
Aggregate (mean)0.250.330.410.470.58
+ +Performance steadily improves as more skip connections are added, peaking when all 5 blocks are connected. This supports our original design decision to broadcast features to all MultiSkip blocks. + +# C.6 Results with the Kron Optimizer. + +We present the full learning curves comparing the Kron optimizer to the baseline RAdam optimizer originally used in PQN [Gallici et al., 2025], across all depths and widths studied in the paper. We follow the experimental protocol of Obando Ceron et al. [2023], Ceron et al. [2024b], Agarwal et al. [2021], running each experiment with three random seeds. + +![](images/48e63c23400ddca43b7e939c6bbfdba96f957ea437f037487674c168f8ad969f.jpg) +Figure 16: Median human-normalized scores with PQN on the Atari-10 benchmark, comparing the Kron optimizer to the baseline RAdam optimizer across varying depths and widths. Similar to the multi-skip architecture, Kron not only improves performance at shallow depths, but also enables PQN to remain trainable across all scales considered. In contrast, performance with RAdam rapidly collapses as depth and width increase. Each curve represents the mean performance across the random seeds per algorithm, with shaded areas indicating $95\%$ bootstrap confidence intervals. + +# C.7 Justification for Selected Interventions. + +The choice of Kronecker-factored optimization and the MultiSkip architecture was the result of a systematic exploration of candidate interventions aimed at improving gradient flow, plasticity, and stability in deep RL. We evaluated a wide range of alternative methods from prior work known to mitigate optimization pathologies [Moalla et al., 2024, Juliani and Ash, 2024]. + +Our evaluated methods, summarized in Tab. 3, included: + +- Second-order and adaptive optimizers: Apollo [Ma, 2020], Shampoo [Gupta et al., 2018], AdaBelief [Zhuang et al., 2020]. +- Regularization: L2 norm penalties [Kumar et al., 2023], weight clipping, and weight decay [Elsayed et al., 2024]. +- Activation functions: GELU [Hendrycks, 2016] and CReLU [Abbas et al., 2023]. +- Learning rate schedules: Cosine annealing and cyclic schedulers. +- Learning rate scaling: Multiplying and dividing the default learning rate (2.5e-4) by 10 to compensate for increased network scale. + +As shown in the table, none of these interventions consistently improved performance compared to our proposed combination. This motivated our decision to focus on the combination of Kronecker-factored optimization and the multi-skip architecture. + +Table 3: Comparison of mean human-normalized scores on Atari-10 for alternative interventions. Our proposed method (Ours) is compared against ablations using only a single alternative intervention. The results (3 seeds per experiment) show that no single alternative consistently matches the performance of our combined approach. + +
EnvironmentOursCosine LRCyclic LRGELUCReLUL2 NormWeight ClipWeight DecayLR=2.08e-5LR=3.00e-3
Amidar-v50.355897-0.0016630.0022760.0151430.0165140.0292930.0122830.0130130.0188480.078660
BattleZone-v50.6945660.031297-0.0002870.015505-0.004594-0.0045940.0356040.0614460.0198120.575407
Bowling-v50.2318310.050145-0.0087210.028706-0.009811-0.0007270.0305230.051235-0.0864830.067587
DoubleDunk-v5-1.318182-2.318182-2.318182-2.363636-2.318182-2.363636-2.454545-2.454545-2.454545-2.090909
Frostbite-v50.881087-0.0019210.0277080.008890-0.0109380.0047550.0753720.0217360.0080340.110036
KungFuMaster-v50.975251-0.0110550.003181-0.000823-0.011500-0.011278-0.011500-0.011055-0.0110551.362077
NameThisGame-v51.242066-0.030191-0.132767-0.261400-0.198430-0.068147-0.197995-0.148227-0.1344170.747251
Phoenix-v50.6551410.072687-0.087237-0.090014-0.090168-0.068722-0.090940-0.093949-0.1023580.411968
Qbert-v51.0948940.0138140.010052-0.0028330.0083590.0061020.0162590.0001770.0013050.112282
Riverraid-v50.993568-0.0001270.3451000.304636-0.056846-0.0584940.293514-0.0444560.0577650.237365
+ +# D Hyper-parameters + +Below, we provide details of the hyperparameters used throughout the paper for each algorithm. In general, they match those proposed in the corresponding original papers. + +Table 4: PQN Hyperparameters + +
HyperparameterValue / Description
Learning rate2.5e-4
Anneal lrFalse (no learning rate annealing)
Num envs128 (parallel environments)
Num steps32 (steps per rollout per environment)
Gamma0.99 (discount factor)
Num minibatches32
Update epochs2 (policy update epochs)
Max grad norm10.0 (gradient clipping)
Start e1.0 (initial exploration rate)
End e0.005 (final exploration rate)
Exploration fraction0.10 (exploration annealing fraction)
Q lambda0.65 (Q(λ) parameter)
Use lnTrue (use layer normalization)
Activation fnrelu (activation function)
+ +Table 5: PPO Hyperparameters + +
HyperparameterValue / Description
Learning rate2.5e-4
Num envs8
Num steps128 (steps per rollout per environment)
Anneal lrTrue (learning rate annealing enabled)
Gamma0.99 (discount factor)
Gae lambda0.95 (GAE parameter)
Num minibatches4
Update epochs4
Norm advTrue (normalize advantages)
Clip coef0.1 (PPO clipping coefficient)
Clip vlossTrue (clip value loss)
Ent coef0.01 (entropy regularization coefficient)
Vf coef0.5 (value function loss coefficient)
Max grad norm0.5 (gradient clipping threshold)
Use lnFalse (no layer normalization)
Activation fnrelu (activation function)
Shared cnnTrue (shared CNN between policy and value networks)
+ +Table 6: PPO Hyperparameters for IsaacGym + +
HyperparameterValue / Description
Total timesteps30,000,000
Learning rate0.0026
Num envs4096 (parallel environments)
Num steps16 (steps per rollout)
Anneal lrFalse (disable learning rate annealing)
Gamma0.99 (discount factor)
Gae lambda0.95 (GAE lambda)
Num minibatches2
Update epochs4 (update epochs per PPO iteration)
Norm advTrue (normalize advantages)
Clip coef0.2 (policy clipping coefficient)
Clip vlossFalse (disable value function clipping)
Ent coef0.0 (entropy coefficient)
Vf coef2.0 (value function loss coefficient)
Max grad norm1.0 (max gradient norm)
Use lnFalse (no layer normalization)
Activation fnrelu (activation function)
+ +Table 7: DQN Hyperparameters + +
HyperparameterValue / Description
Learning rate1e-4
Num envs1
Buffer size1,000,000 (replay memory size)
Gamma0.99 (discount factor)
Tau1.0 (target network update rate)
Target network frequency1000 (timesteps per target update)
Batch size32
Start e1.0 (initial exploration epsilon)
End e0.01 (final exploration epsilon)
Exploration fraction0.10 (fraction of total timesteps for decay)
Learning starts80,000 (timesteps before training starts)
Train frequency4 (training frequency)
Use InFalse (no layer normalization)
Activation fnrelu (activation function)
+ +Table 8: Rainbow Hyperparameters + +
HyperparameterValue / Description
Learning rate6.25e-5
Num envs1
Buffer size1,000,000 (replay memory size)
Gamma0.99 (discount factor)
Tau1.0 (target network update rate)
Target network frequency8000 (timesteps per target update)
Batch size32
Start e1.0 (initial exploration epsilon)
End e0.01 (final exploration epsilon)
Exploration fraction0.10 (fraction of total timesteps for decay)
Learning starts80,000 (timesteps before training starts)
Train frequency4 (training frequency)
N step3 (n-step Q-learning horizon)
Prioritized replay alpha0.5
Prioritized replay beta0.4
Prioritized replay eps1e-6
N atoms51 (number of atoms in distributional RL)
V min-10 (value distribution lower bound)
V max10 (value distribution upper bound)
Use lnFalse (no layer normalization)
Activation fnrelu (activation function)
+ +Table 9: Image Classification Hyperparameters (CIFAR-10) + +
HyperparameterValue
Batch size256
Epochs100
Learning rate0.00025
+ +Table 10: SAC Hyperparameters + +
HyperparameterValue / Description
Critic block typeSimBa
Critic num blocks{2, 4, 6, 8}
Critic hidden dim{512, 1024, 1536, 2048}
Target critic momentum (τ)5e-3
Actor block typeSimBa
Actor num blocks{1, 2, 3, 4}
Actor hidden dim{128, 256, 384, 512}
Initial temperature (α0)1e-2
Temperature learning rate1e-4
Target entropy (H*)|A|/2
Batch size256
Optimizer{AdamW, Kron}
AdamW's learning rate1e-4
Kron's learning rate5e-5
Optimizer momentum (β1, β2)(0.9, 0.999)
Weight decay (λ)1e-2
Discount (γ)Heuristic
Replay ratio2
Clipped Double QFalse
+ +Table 11: DDPG Hyperparameters + +
HyperparameterValue / Description
Critic block typeSimBa
Critic num blocks{2, 4, 6, 8}
Critic hidden dim{512, 1024, 1536, 2048}
Critic learning rate1e-4
Target critic momentum (τ)5e-3
Actor block typeSimBa
Actor num blocks{1, 2, 3, 4}
Actor hidden dim{128, 256, 384, 512}
Actor learning rate1e-4
Exploration noiseN(0, 0.12)
Batch size256
Optimizer{AdamW, Kron}
AdamW's learning rate1e-4
Kron's learning rate5e-5
Optimizer momentum (β1, β2)(0.9, 0.999)
Weight decay (λ)1e-2
Discount (γ)Heuristic
Replay ratio2
Clipped Double QFalse
+ +# E Compute Details + +All experiments were conducted on a single-GPU setup using an NVIDIA RTX 8000, 12 CPU workers, and 50GB of RAM. + +Table 12: Training times across model scales for two optimizers K-FAC shows increased cost as depth and width grow. + +
DepthWidthOptimizerTime
RAdam
SmallSmallAdam51m
SmallMediumAdam53m
SmallLargeAdam57m
MediumSmallAdam1h 4m
MediumMediumAdam1h 10m
MediumLargeAdam1h 11m
LargeSmallAdam1h 18m
LargeMediumAdam1h 18m
LargeLargeAdam1h 27m
Kron
SmallSmallKron1h 59m
SmallMediumKron2h 27m
SmallLargeKron3h 38m
MediumSmallKron2h 44m
MediumMediumKron3h 32m
MediumLargeKron5h 59m
LargeSmallKron3h 27m
LargeMediumKron4h 36m
LargeLargeKron7h 42m
+ +# E.1 Results on the Full ALE + +In this section, we provide the full training curves corresponding to the aggregated results shown in Sec. 4.3, where we evaluate the performance of the PQN and PPO agents on the full set of environments from the ALE after applying our two proposed gradient interventions. The per-environment learning curves are presented in Fig. 17 for PQN and Fig. 18 for PPO. We follow the experimental protocol of Obando Ceron et al. [2023], Ceron et al. [2024b], Agarwal et al. [2021], running each experiment with three random seeds. + +![](images/8d303452e6525ea1fec690985d19f68c887a6918174296585a953eaad2a86e12.jpg) +Figure 17: Mean human-normalized score on the full ALE suite, comparing the baseline PQN agent (light curves) with the augmented agent using our combined gradient interventions (dark curves). + +![](images/f10eac81498a41ba85db19cbc299053cdec1b83c0a8c90bb5d44fae3eea3e723.jpg) +Figure 18: Mean human-normalized score on the full ALE suite, comparing the baseline PPO agent (light curves) with the augmented agent using our combined gradient interventions (dark curves). + +# E.2 Simba on DMC + +In this section, we present the full results accompanying the experiments combining Simba [Lee et al., 2025] with our proposed gradient interventions, as introduced in Sec. 4.3. For these experiments, we retain Simba's original architectural choices but replace the AdamW optimizer with Kron. + +We compare Simba using both SAC and DDPG as the underlying RL algorithms. While SAC generally outperforms DDPG, we consistently observe that scaling depth and width, either independently or jointly, leads to a degradation in performance with Simba. However, this degradation is mitigated, and in many cases reversed, when using the Kron optimizer, resulting in improved performance as model capacity increases. + +The following figures illustrate these findings: + +![](images/d6fb2780640e54607c238cfc185eb8d6c0b45a5f108110b2ff8029bf6a657f55.jpg) +Figure 19: SAC scaling depth + +![](images/0c934d9a4ebde5f8dc73b707d85858366868a38c794971a59aae77d570c0925d.jpg) +Figure 20: SAC scaling width + +![](images/a0d00f35b97a050a121ac1c84ca0f9096c8e0cff466e9c214e3e082705f3b01b.jpg) +Figure 21: SAC scaling both depth and width + +![](images/9c7d225c33f3b8c5ffbfbcbc264b7ef2aa57ed345e8450e148bfdefd849377fc.jpg) +Figure 22: DDPG scaling depth + +![](images/68e888c294dd4151ca09426070a4fd53a1049920b669c0c8ced8f6485b8d827d.jpg) +Figure 23: DDPG scaling width + +![](images/87b264f8f353154937a2743ddea452e49cd977986e2eb909e79302d79cf48475.jpg) +Figure 24: DDPG scaling both depth and width + +# NeurIPS Paper Checklist + +# 1. 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diff --git a/stableguardtowardsunifiedcopyrightprotectionandtamperlocalizationinlatentdiffusionmodels/full.md b/stableguardtowardsunifiedcopyrightprotectionandtamperlocalizationinlatentdiffusionmodels/full.md new file mode 100644 index 0000000000000000000000000000000000000000..9b3e4d91be5e172c13d90c5bddb7830ce7300a44 --- /dev/null +++ b/stableguardtowardsunifiedcopyrightprotectionandtamperlocalizationinlatentdiffusionmodels/full.md @@ -0,0 +1,790 @@ +# StableGuard: Towards Unified Copyright Protection and Tamper Localization in Latent Diffusion Models + +Haoxin Yang $^{1}$ Bangzhen Liu $^{1}$ Xuemiao Xu $^{1\dagger}$ Cheng Xu $^{2\dagger}$ Yuyang Yu $^{1}$ Zikai Huang $^{1}$ Yi Wang $^{3}$ Shengfeng He $^{2}$ + +$^{1}$ South China University of Technology $^{2}$ Singapore Management University + +$^{3}$ Dongguan University of Technology + +harxis@outlook.com xuemx@scut.edu.cn + +{liubz.scut, cschengxu, yyyy0611,zikaihuang0428}@gmail.com + +yiwang@dgut.edu.cn shengfenghe@smu.edu.sg + +https://github.com/Harxis/StableGuard + +# Abstract + +The advancement of diffusion models has enhanced the realism of AI-generated content but also raised concerns about misuse, necessitating robust copyright protection and tampering localization. Although recent methods have made progress toward unified solutions, their reliance on post hoc processing introduces considerable application inconvenience and compromises forensic reliability. We propose StableGuard, a novel framework that seamlessly integrates a binary watermark into the diffusion generation process, ensuring copyright protection and tampering localization in Latent Diffusion Models through an end-to-end design. We develop a Multiplexing Watermark VAE (MPW-VAE) by equipping a pretrained Variational Autoencoder (VAE) with a lightweight latent residual-based adapter, enabling the generation of paired watermarked and watermark-free images. These pairs, fused via random masks, create a diverse dataset for training a tampering-agnostic forensic network. To further enhance forensic synergy, we introduce a Mixture-of-Experts Guided Forensic Network (MoE-GFN) that dynamically integrates holistic watermark patterns, local tampering traces, and frequency-domain cues for precise watermark verification and tampered region detection. The MPW-VAE and MoEGFN are jointly optimized in a self-supervised, end-to-end manner, fostering a reciprocal training between watermark embedding and forensic accuracy. Extensive experiments demonstrate that StableGuard consistently outperforms state-of-the-art methods in image fidelity, watermark verification, and tampering localization. + +# 1 Introduction + +The unprecedented capabilities of latent diffusion models (LDMs) have revolutionized image synthesis, redefining artistic creation and visual content production [1-6]. However, as these models become more accessible, concerns over unauthorized use and malicious content tampering have intensified, necessitating solutions to safeguard intellectual property and ensure content authenticity. + +Traditional copyright protection relies on watermarking techniques [7-9], which embed identifiable information into images through post-hoc processing. While effective in some cases, these methods introduce computational overhead and image degradation due to their detachment from the generative process. Recent approaches to diffusion-native watermarking integrate embedding mechanisms directly within the generation pipeline [10-14], but they fail to support advanced forensic needs + +such as quantifying tampering severity or precisely localizing manipulations [15-20]. Although recent works have attempted to unify copyright protection and tampering localization in a single framework [21-23], they remain inherently post-hoc, with generation and forensics optimized independently. This decoupled optimization prevents mutual enhancement between the two tasks, thereby limiting both the overall forensic robustness and practical applicability. The absence of a seamless framework that integrates these functionalities in an LDM-native manner remains a major gap, hindering real-world deployment in security-sensitive applications. + +To bridge the gap between copyright protection and content forensics in LDMs, we propose StableGuard, a unified framework that integrates holistic watermark directly into the generative process. Our core insight is twofold: (i) holistically distributed watermarks exhibit inherent robustness against localized manipulations due to spatial redundancy, and (ii) the subtle perturbations introduced by these watermarks can serve as reliable cues for tampering localization through missing feature detection. These complementary properties make holistic watermarking a compelling foundation for jointly addressing ownership verification and fine-grained forensic analysis in a unified LDM-native manner. StableGuard achieves this integration through two key components: the Multiplexing Watermark Variational Autoencoder (MPW-VAE) for embedding watermarks during generation, and the Mixture-of-Experts Guided Forensic Network (MoE-GFN) for tamper-agnostic forensic analysis. MPW-VAE extends the standard LDM architecture by incorporating a trainable watermark adapter within the VAE decoder and employs a multiplexing strategy to generate both visually indistinguishable watermarked and watermark-free images, facilitating self-supervised training + +without manual annotations. MoE-GFN performs watermark extraction and tampering localization via a Mixture-of-Forensic-Experts (MoFE) block comprising three specialized experts: a Watermark Extraction Expert, a Tampering Localization Expert, and a Boundary Enhancement Expert. These experts are dynamically fused by a Dynamic Soft Router, allowing for precise and holistic forensic predictions. The MPW-VAE and MoE-GFN are trained jointly in an end-to-end manner, enabling mutual reinforcement between watermark embedding and forensic performance. Extensive experiments across multiple benchmarks demonstrate that StableGuard outperforms existing methods in both watermark retrieval and tampering localization, validating its effectiveness for real-world security-sensitive applications. + +In summary, our main contributions are as follows: + +- We introduce StableGuard, a unified proactive forensics framework for LDMs, seamlessly integrating copyright protection and tampering localization into LDM-based image generation. +- We propose a simple yet effective self-supervised forensic learning framework that embeds an imperceptible, holistic bit watermark via a multiplexing watermark VAE. This enables precise copyright protection and tampering localization without requiring labeled data. +- We develop a tampering-agnostic mixture-of-experts forensic network that synergistically integrates holistic, subtle, and boundary features to extract forensic cues, ensuring robust watermark retrieval and precise tampering localization across various attack scenarios. +- Extensive experiments demonstrate that StableGuard outperforms state-of-the-art methods in both forensic accuracy and robustness while preserving the visual fidelity of generated images. + +![](images/3158208a89c7bbaa3cda7e23d42f86f6acd2d1df172750d689e8800446142220.jpg) +Figure 1: (a) Unlike other post-hoc methods that add external watermarking, introducing overhead and detaching from the generative process. (b) StableGuard embeds a holistic binary watermark directly into the diffusion pipeline and adaptively extracts intrinsic perturbation patterns via collaborative forensic experts, enabling unified, precise, and seamless protection within LDMs. + +# 2 Related Work + +# 2.1 Image Watermark + +Image watermarking has long served as a foundational technique for intellectual property protection [24-28]. Traditional approaches typically rely on encoder and decoder architectures [7, 8], flow-based models [21, 29], or per-image encoding strategies [7, 30]. These methods are often implemented as separate steganographic modules applied after image generation, introducing additional processing overhead and potential degradation in visual quality. In parallel, diffusion models, particularly latent diffusion models (LDMs), have recently achieved remarkable progress across diverse domains [1, 2, 31-35]. Motivated by this success, recent studies have explored embedding watermarks directly into the generative process [10-12, 14, 36]. For example, Tree-Ring [36] modifies the initial noise vector, CROSS [14] employs image-to-image steganography, and methods such as Stable Signature [10], WOUAF [11], and WaDiff [12] fine-tune LDMs to embed watermarks in latent space. While these LDM-native approaches improve integration and visual fidelity, they generally neglect tampering localization—a key aspect for assessing manipulation severity and enabling forensic content reuse. Our work introduces a unified framework that jointly addresses copyright protection and tampering localization in LDMs, leveraging mutual optimization between watermark and forensic analysis to deliver robust, high-fidelity image forensics. + +# 2.2 Tampering Localization + +Tampering localization aims to detect and localize manipulated regions, crucial for verifying image authenticity and quantifying alterations [15-21, 37, 38]. Existing methods fall into two broad categories: passive detection and proactive defense. Passive methods rely solely on visual artifacts. For instance, MVSSNet [15] captures boundary inconsistencies, IML-ViT [16] extracts multi-scale features via a transformer backbone, and PSCC-Net [17], ObjectFormer [18], and HDF-Net [19] enhance spatial awareness using complementary cues. However, these approaches often require paired supervision and struggle to generalize across manipulation types. Proactive defenses embed auxiliary signals into the image to aid future tamper detection. Traditional fragile watermark offers block-level localization but lacks adaptability [39-43]. Recent methods like MaLP [44], EditGuard [21], OmniGuard [22] and WAM [23] use learned templates or invertible flows to embed watermarks for both copyright and localization. Other approaches [45, 46] explore semi-supervised tampering localization, offering post-hoc forensic analysis without prior watermarking. Despite their advantages, these post-hoc designs require external embedding networks, introduce application complexity, and are prone to quality degradation, especially for LDM-generated content. Moreover, methods like EditGuard and OmniGuard rely on reference images [21, 22], making them sensitive to auxiliary input quality. In contrast, our proposed StableGuard integrates watermark directly into the LDM process, eliminating the need for post-processing. This diffusion-native approach unifies copyright protection and tampering localization within a single, end-to-end framework where both tasks reinforce one another. + +# 3 Proposed Method + +# 3.1 Preliminaries of Latent Diffusion Models + +Latent Diffusion Models (LDMs) are a class of generative models that synthesize images by iterative denoising a latent representation [1-5]. Unlike conventional diffusion models [47, 48] that operate in pixel space, LDMs first encode images into a lower-dimensional latent space, enabling more efficient training and inference while preserving image fidelity. The generative process begins by sampling a latent vector $z_{T} \sim \mathcal{N}(0, I)$ , which is progressively denoised through the recurrence: + +$$ +z _ {t - 1} = \frac {1}{\sqrt {\alpha_ {t}}} \left(z _ {t} - \frac {1 - \alpha_ {t}}{\sqrt {1 - \bar {\alpha} _ {t}}} \epsilon_ {\theta} \left(z _ {t}, t\right)\right), \tag {1} +$$ + +where $z_{t}$ is the latent at timestep $t$ , $\alpha_{t}$ is the noise schedule, and $\bar{\alpha}_{t} = \prod_{s=1}^{t} \alpha_{s}$ . The model $\epsilon_{\theta}$ predicts the added noise at each step. This iterative refinement continues until the clean latent $z_{0}$ is recovered. Finally, $z_{0}$ is decoded into image space via a VAE decoder: + +$$ +X = \operatorname {V A E} \operatorname {D e c o d e r} \left(z _ {0}\right). \tag {2} +$$ + +![](images/7f1904b3fa6afa5120b3794b7e9a5318a37b11a64efbe7a12e5322191e42414f.jpg) +Figure 2: Overview of the StableGuard Framework. StableGuard integrates an MPW-VAE and an MoE-GFN in a self-supervised, end-to-end pipeline. The MPW-VAE generates visually consistent watermarked and non-watermarked image pairs from the same latent code by introducing a lightweight, residual-based watermark adapter into the LDM decoder. These image pairs, augmented via random masking, form a diverse training set for MoE-GFN training. MoE-GFN adaptively fuses features from multiple forensic experts to extract robust, unified signals from holistic watermark perturbations. Joint optimization of both components enables reciprocal enhancement of watermark embedding and forensic analysis, supporting accurate copyright verification and tampering localization. + +# 3.2 Overview + +To enable imperceptible watermark embedding and robust forensic analysis within LDMs, we propose StableGuard, a unified framework for copyright protection and tampering localization that integrates seamlessly into the LDM pipeline. As illustrated in Fig. 2, StableGuard consists of two core components: the Multiplexing Watermark VAE (MPW-VAE) and the Mixture-of-Experts Guided Forensic Network (MoE-GFN). MPW-VAE introduces a hierarchy of lightweight, latent residual-based watermark adapters into the LDM's VAE decoder, enabling holistic binary watermark embedding while preserving perceptual fidelity. Crucially, watermark adapters can be toggled on or off, facilitating the generation of paired watermarked and clean samples from shared latent, supporting fully self-supervised training of the forensic module. For downstream analysis, MoE-GFN leverages the subtle perturbations introduced by the watermark to perform both extraction and tampering localization. Central to this network is the Mixture-of-Forensic-Experts (MoFE) Block, comprising three expert branches specialized in capturing complementary forensic cues. Their outputs are adaptively fused through a dynamic soft routing mechanism, allowing the network to selectively emphasize relevant features and achieve robust, accurate forensic predictions. + +# 3.3 Multiplexing Watermark VAE + +Most existing passive tampering localization methods depend on manually annotated masks for supervised learning, which is not only labor-intensive but also exhibits poor generalization across diverse manipulation scenarios [15-19]. To address these challenges, we propose the MPW-VAE, which incorporates a toggleable lightweight watermark injection branch into a pre-trained LDM VAE decoder. This multiplexing design enables the generation of visually indistinguishable watermarked and watermark-free image pairs from the same latent input, providing a scalable foundation for self-supervised learning in forensic tasks. + +To preserve visual fidelity, watermark embedding is achieved through latent residual-based adapters inserted after each block of the VAE decoder. Each adapter encodes the watermark via two fully connected layers, reshapes the embedding to match the feature dimensions, and concatenates it with the decoder features. The fused features are then processed through two convolutional blocks, followed by a residual connection to the input. This design ensures coherent and unobtrusive watermark integration (see Fig. 2(b)). Leveraging this residual mechanism, MPW-VAE produces watermarked images with imperceptible yet traceable watermark signals, while maintaining high visual similarity to the watermark-free outputs generated by the original decoder. + +To simulate diverse and realistic tampering scenarios, we synthesize training samples by randomly fusing watermarked images $Y$ with their watermark-free counterparts, either the VAE-reconstructed image $\hat{X}$ or the original image $X$ , using a random binary mask $M$ : + +$$ +\left\{ \begin{array}{l l} \hat {Y} = (1 - M) \odot Y + M \odot X, & p \leq 0. 5 \\ \hat {Y} = (1 - M) \odot Y + M \odot \hat {X}, & p > 0. 5 \end{array} \right. \tag {3} +$$ + +where $\hat{Y}$ denotes the synthesized training image, and $p$ controls whether the spliced region is drawn from the VAE reconstruction or the original image. Specifically, we use $X$ (a real image) simulates human-made manipulations produced by editing tools (e.g., Photoshop), while using $\hat{X}$ (the vanilla LDM VAE reconstruction) emulates AI-generated forgeries increasingly common with generative models. To diversify tampering patterns, we adopt a hybrid masking strategy: with $50\%$ probability, we apply random binary masks; otherwise, we use semantic masks drawn from a pool of SAM-generated segmentations [49]. These SAM-based masks are non-image-paired, ensuring shape diversity without extra supervision. This augmentation strategy enables the network to learn to distinguish watermarked from non-watermarked regions in a self-supervised manner, thereby supporting robust joint learning of watermark extraction and tampering localization. + +# 3.4 Mixture-of-Experts Guided Forensic Network + +The embedding of holistic watermarks into generated images enables not only robust watermark verification but also tampering localization by identifying disruptions in the underlying perturbation patterns. To effectively harness the synergy between these two tasks, we propose the Mixture-of-Experts Guided Forensic Network (MoE-GFN). Inspired by the mixture-of-experts paradigm [50], MoE-GFN integrates task-specialized modules to enhance forensic performance through collaborative feature modeling. As illustrated in Fig. 2(a), MoE-GFN adopts a UNet-based architecture[51], where the decoder is augmented with a Mixture-of-Forensic-Experts (MoFE) Block (Fig. 2(c)). This block comprises three dedicated experts—Watermark Extraction, Tampering Localization, and Boundary Enhancement—each designed to capture complementary forensic cues. Their outputs are fused via a Dynamic Soft Router, which adaptively integrates task-relevant features based on the input. The complete pipeline includes a Stem module[52] for dimensionality reduction, a UNet backbone for multi-scale feature extraction, and a Forensic Decoder for final prediction. + +Watermark Extraction Expert. To recover globally embedded watermarks even under local modifications, we employ a transformer-based expert that captures long-range dependencies. By modeling global correlations, this module provides robust watermark reconstruction and indirectly facilitates tampering localization by highlighting regions where watermark patterns are missing. Given input features $x \in \mathbb{R}^{b,c,h,w}$ , we reshape them to $x_{\mathrm{flat}} \in \mathbb{R}^{b,hw,c}$ and process them through a linear projection (Proj), followed by Multi-Head Self-Attention (MHSA) and a Feed-Forward Network (FFN) [53]. The final output is computed with residual addition: + +$$ +x _ {\mathrm {w m}} = \operatorname {F F N} (\mathrm {M H S A} (\operatorname {P r o j} (x _ {\mathrm {f l a t}}))) + x _ {\mathrm {f l a t}}. \tag {4} +$$ + +Tampering Localization Expert. In contrast to watermark extraction, tampering localization demands sensitivity to subtle, local inconsistencies. To this end, we propose a sub-patch transformer that partitions feature maps into smaller patches and applies localized attention to enhance the detection of fine-grained manipulation artifacts. Formally, the input $x$ is partitioned into $n \times n$ patches and reshaped to $x_{\mathrm{sub}} \in \mathbb{R}^{b \cdot \frac{hw}{n^2}, n^2, c}$ . It is then processed as: + +$$ +x _ {\text {t a m p}} = \operatorname {F F N} (\mathrm {M H S A} (\operatorname {P r o j} (x _ {\text {s u b}}))) + x _ {\text {s u b}}. \tag {5} +$$ + +Boundary Enhancement Expert. To further refine tampering boundaries, we incorporate frequency-domain cues through a Boundary Enhancement Expert, which highlights high-frequency components + +indicative of manipulation edges. This expert applies a transformer in the Fourier domain to capture boundary-specific anomalies that may be missed in the spatial domain. Given input $x$ , we compute its Fourier transform, reshape the frequency representation, and process it through transformer layers before applying the inverse transform: + +$$ +x _ {\mathrm {F F T}} = \operatorname {R e s h a p e} (\operatorname {F F T} (x), (b, h w, c)), +$$ + +$$ +\begin{array}{l} x _ {\text {F F T}} = \operatorname {I n t e m p l e} (\mathrm {F F T} (x), (x, \dots , x)), \\ x _ {\text {b o u n d}} = \operatorname {i F F T} (\operatorname {F F N} (\operatorname {M H S A} (\operatorname {P r o j} (x _ {\mathrm {F F T}}))) + x _ {\mathrm {F F T}}). \end{array} \tag {6} +$$ + +where FFT and iFFT denote the Fourier transform and inverse Fourier transform, respectively. + +Dynamic Soft Router. To adaptively fuse the outputs of the three expert branches, we introduce a Dynamic Soft Router, which predicts expert-specific fusion weights conditioned on the input. This mechanism enables MoE-GFN to dynamically modulate its focus based on the type and spatial characteristics of the tampering. Given a shared input feature map $x$ , the router generates soft weights $R = \operatorname{Softmax}(f(x))$ , where $f(\cdot)$ is a lightweight multi-layer perceptron. The unified forensic representation is computed as a weighted sum: + +$$ +x _ {\text {u n i f i e d}} = \sum_ {n = 1} ^ {3} R _ {n} \odot \operatorname {E x p e r t} _ {n} (x). \tag {7} +$$ + +By enabling input-adaptive feature integration, the Dynamic Soft Router enhances the model's ability to generalize across diverse manipulation patterns, yielding more resilient forensic predictions. + +# 3.5 Loss Function + +StableGuard is trained in a fully self-supervised, end-to-end manner, enabling mutual enhancement between watermark embedding and its watermark extraction and tampering localization tasks. During MPW-VAE training, we freeze the parameters of the pre-trained LDM VAE decoder and update only the watermark adapter. This strategy ensures high perceptual fidelity between the original and watermarked images while effectively encoding watermark information. The total training objective comprises three components: a similarity loss to maintain image quality, a watermark loss to ensure accurate extraction, and a tampering loss to localize manipulated regions precisely. + +Similarity Loss. To preserve visual consistency between the watermarked image $Y$ and the reconstructed watermark-free image $\hat{X}$ , we combine L1 distance with perceptual similarity [54]: + +$$ +\mathcal {L} _ {\text {s i m}} = \left| \left| \hat {X} - Y \right| \right| _ {1} + \operatorname {P S} (\hat {X}, Y). \tag {8} +$$ + +Watermark Loss. We supervise watermark extraction using a binary cross-entropy loss: + +$$ +\mathcal {L} _ {\mathrm {w m}} = - \frac {1}{L} \sum_ {l} \left(W _ {l} \cdot \log \left(\hat {W} _ {l}\right) + (1 - W _ {l}) \cdot \log \left(1 - \hat {W} _ {l}\right)\right), \tag {9} +$$ + +where $W$ and $\dot{W}$ are the ground truth and predicted watermark bits, and $L$ is the total bit length. + +Tampering Loss. To achieve precise localization of tampered regions, we combine weighted binary cross-entropy (WBCE) with Dice loss [55]: + +$$ +\mathcal {L} _ {\text {t a m p e r}} = \lambda_ {0} \mathcal {L} _ {\mathrm {w b c e}} + (1 - \lambda_ {0}) \mathcal {L} _ {\mathrm {d i c e}}, \tag {10} +$$ + +$$ +\mathcal {L} _ {\mathrm {w b c e}} = - \frac {1}{N} \sum_ {i, j} \left(\lambda_ {1} \cdot M _ {i, j} \cdot \log \left(\sigma \left(\hat {M} _ {i, j}\right)\right) + \lambda_ {2} \cdot \left(1 - M _ {i, j}\right) \cdot \log \left(1 - \sigma \left(\hat {M} _ {i, j}\right)\right)\right), \tag {11} +$$ + +$$ +\mathcal {L} _ {\mathrm {d i c e}} = 1 - \frac {2 \sum_ {i , j} M _ {i , j} \cdot \sigma \left(\hat {M} _ {i , j}\right)}{\sum_ {i , j} \left(M _ {i , j}\right) ^ {2} + \sum_ {i , j} \left(\sigma \left(\hat {M} _ {i , j}\right)\right) ^ {2}}, \tag {12} +$$ + +where $M$ and $\hat{M}$ denote the ground truth and predicted tampering masks, $\sigma$ is the sigmoid function, $N$ is the total number of pixels, $i,j$ are the pixel indices, and $\lambda_1,\lambda_2$ are weighting factors for foreground and background. + +Total Loss. The final training objective for the StableGuard framework is defined as: + +$$ +\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\text {s i m}} + \mathcal {L} _ {\text {w m}} + \mathcal {L} _ {\text {t a m p e r}}. \tag {13} +$$ + +![](images/6cd68579a458b16f6dd9685dd6937315dedf0fff868104e3f715aaadb94a12d2.jpg) +Figure 3: Comparison between the original Stable Diffusion VAE and a variant using our MPW-VAE alongside other watermarking methods [21-23]. To better visualization, pixel-wise differences are amplified $100 \times$ . StableGuard embeds holistic watermarks that remain visually imperceptible while preserving image quality. + +# 4 Experiments + +# 4.1 Settings + +Datasets. We train StableGuard on the COCO training set [56] in a fully self-supervised manner, without requiring any manual tampering annotations. For evaluation, watermark extraction is tested on both the COCO test set and a custom text-to-image (T2I) dataset, measuring performance on real and LDM-generated content. To assess tampering localization, we focus on robustness against AI-generated manipulations using a large-scale AIGC benchmark of 35,000 images—25,000 from COCO and 10,000 from T2I dataset. Semantic masks are generated with SAM [49], followed by region edits using Stable Diffusion [1], SDXL [2], Kandinsky [57], ControlNet [4], and LaMa [58]. Further details are provided in the Appendices. + +Implementation Details. StableGuard is implemented in PyTorch, built upon Stable Diffusion $2.1^{1}$ . During training, we freeze the base model and optimize only the Watermark Adapter and MoE-GFN using the Adam optimizer [59] with a learning rate of $1 \times 10^{-4}$ . All experiments are run for 10 epochs on two NVIDIA RTX 4090D GPUs with a batch size of 16. The tampering localization expert operates on sub-patches of size $n = 8$ . Loss weights are set as $\lambda_0 = 0.2$ (Eq. (10)), $\lambda_1 = 2$ , and $\lambda_2 = 0.5$ (Eq. (11)). + +# 4.2 Comparison on Watermarking + +The effectiveness of our proposed StableGuard for watermark embedding and extraction is validated by experiments on two datasets: the COCO dataset and the T2I dataset. To intuitively demonstrate StableGuard's watermark injection capability, we showcase two watermark embedding examples from the T2I dataset and compare with three post-hoc watermarking methods [21-23] in Fig. 3. The corresponding prompts for generating images are "A dragon perched atop a mountain peak" and "A colorful hot air balloon floating over a mountain range", respectively. As observed, the images before and after watermark embedding exhibit high visual consistency, with pixel-wise differences at a negligible level. The difference maps reveal globally distributed pixels across the entire image, demonstrating that StableGuard indeed effectively embeds holistic imperceptible watermarks while maintaining high-fidelity image generation results. + +We also quantitatively compare the performance of StableGuard with eight state-of-the-art watermarking techniques under different watermark bit lengths (B.L.). The compared methods include five post-hoc watermarking approaches: HiDDeN [7], SepMark [8], EditGuard [21], OmniGuard [22] and WAM [23], as well as three diffusion-based methods: Stable Signature [10], WOUAF [11], and WaDiff [12]. As shown in Tab. 1, our approach maintains competitive with existing solutions on effectively preserving the perceptual quality of the original content after watermark embedding, where the embedded images exhibit close visual similarity with their non-watermarked counterparts. + +Table 1: Quantitative comparison on COCO [56] and the T2I dataset. The best and second-best results are bolded and underlined, respectively. + +
MethodB.L.COCO Dataset [56]T2I-Dataset
PSNR↑SSIM↑LPIPS↓FID↓Bit Acc↑PSNR↑SSIM↑LPIPS↓FID↓Bit Acc↑
HiDDeN [7]3231.950.8790.10420.098.8032.030.8810.10119.998.79
SepMark [8]3235.380.9290.09020.398.8535.420.9310.08920.198.86
WOUAF [11]3231.200.9060.11521.699.1331.140.9080.11321.799.14
Stable Signature [10]4830.120.8900.12020.999.1230.060.8920.11820.899.10
WaDiff [12]4835.550.9600.08519.898.1435.490.9620.08319.698.13
EditGuard [21]6432.750.9370.10520.099.7732.870.9390.10319.999.78
OmniGuard [22]10037.540.9500.07220.198.1137.320.9440.07020.098.09
WAM [23]3238.200.9510.06719.998.1737.800.9460.07319.698.33
Ours3240.500.9700.06219.599.9740.530.9720.06019.499.98
4840.420.9690.06319.799.9640.470.9710.06119.699.98
6440.400.9680.06519.899.9540.440.9700.06319.799.96
12840.100.9660.07019.999.8740.110.9680.06919.899.88
25640.050.9640.07320.199.8340.080.9650.06220.099.84
+ +![](images/1497af0e284606d320e3e4afbd6125b0bb037fb9f7e4c02447d5366c65895804.jpg) +Figure 4: The visualization of tampering localization results on the AIGC tampering dataset. (a) is the original image, (b) is the watermarked image by StableGuard, (c) is the tampered image, (d)-(m) are the tampering localization results of MVSS-Net [15], IML-ViT [16], PSCC-Net [17], ObjectFormer [18], HDF-Net [19], EditGuard [21], OmniGuard [22], WAM [23], our StableGuard, and the ground truth, respectively. Please zoom in for best view. + +Meanwhile, the accuracy of the watermark extracted from these images consistently outperforms existing techniques. + +Unlike conventional post-hoc approaches, our method is seamlessly integrated into the LDM generation pipeline, exploiting an end-to-end model to directly embed watermarks into generated images. Furthermore, compared to those diffusion-native watermarking methods, our tailored MPW-VAE employs a latent residual-based watermark adapter after the VAE decoder block, which is jointly trained with the forensics tasks, thus achieving adaptive holistic watermark injection while minimally altering the original image content. + +# 4.3 Comparison on Tampering Localization + +The quantitative results of various methods on the AIGC tampering dataset are presented in Tab. 2. As shown, existing passive localization techniques exhibit notably inferior performance when compared to StableGuard, highlighting their inability to effectively detect the latest AIGC tampering methods. In contrast, StableGuard demonstrates robust detection capabilities across a range of AIGC tampering techniques, validating the effectiveness of our approach. When compared to EditGuard, OmniGuard and WAM, StableGuard outperforms them on most metrics. This improvement is a direct result of StableGuard's design, which integrates the unique characteristics of both watermarking and tampering detection. Unlike other methods, which identify tampered regions by detecting anomalies + +Table 2: Localization precision comparison on the AIGC tampering dataset. + +
MethodSD Inpainting [1]SD XL [2]Kandinsky [57]ControlNet [4]LAMA [58]
F1↑AUC↑IoU↑F1↑AUC↑IoU↑F1↑AUC↑IoU↑F1↑AUC↑IoU↑F1↑AUC↑IoU↑
MVSS-Net [15]0.8620.9340.7910.8480.9290.7750.8480.9280.7750.8560.9300.7820.8600.9340.789
IML-ViT [16]0.9070.9230.8790.9040.9210.8760.9060.9230.8770.8980.8830.8400.8980.8940.880
PSCC-Net [17]0.8980.9760.8290.8990.9770.8300.8940.9750.8250.8990.9770.8300.8980.9760.829
ObjectFormer [18]0.4760.7220.3980.4790.7190.3980.4720.7180.3980.4670.7240.3900.5030.7380.425
HDF-Net [19]0.5560.7620.4680.7630.4700.5600.5440.7590.4570.5510.7640.4630.5650.7670.476
EditGuard [21]0.9370.9770.9110.9380.9760.9130.9350.9660.9130.9390.9690.9070.9390.9770.917
OmniGuard [22]0.8530.9640.8100.8670.9730.8240.8680.9660.8300.8580.9650.8150.8640.9690.823
WAM [23]0.9240.9770.8680.9180.9760.8620.9210.9760.8650.9170.9770.8600.9220.9670.864
Ours0.9800.9930.9620.9810.9910.9610.9800.9920.9600.9810.9930.9630.9790.9930.961
+ +Table 3: Robustness comparison on the AIGC tampering dataset under different image degradation conditions (Bit Acc / F1 Score). + +
MethodCleanGaussian NoiseJPEG CompressionPoisson Noise
σ=1σ=3σ=5Q=70Q=80Q=90
Stable Signature [10]99.10 / -98.51 / -98.23 / -97.90 / -96.12 / -96.73 / -97.19 / -97.47 / -
WOUAF [11]99.14 / -98.37 / -98.35 / -98.41 / -95.49 / -96.78 / -96.99 / -97.67 / -
MVSS-Net [15]- / 0.862- / 0.849- / 0.878- / 0.824- / 0.627- / 0.702- / 0.773- / 0.844
HDF-Net [19]- / 0.595- / 0.588- / 0.574- / 0.538- / 0.409- / 0.495- / 0.516- / 0.568
EditGuard [21]99.78 / 0.93898.91 / 0.89798.13 / 0.88198.11 / 0.86696.77 / 0.57797.12 / 0.77697.13 / 0.80099.75 / 0.887
OmniGuard [22]98.11 / 0.86397.25 / 0.82196.78 / 0.79996.54 / 0.78094.32 / 0.51295.67 / 0.63496.12 / 0.68997.45 / 0.835
WAM [23]98.17 / 0.92097.56 / 0.88197.12 / 0.86096.89 / 0.84595.78 / 0.58996.45 / 0.70296.78 / 0.75598.12 / 0.903
Ours99.98 / 0.97299.90 / 0.94099.80 / 0.93999.69 / 0.92899.73 / 0.90899.88 / 0.92799.97 / 0.95199.65 / 0.947
+ +in image-based positional watermarks, StableGuard adopts a more precise method for tampering localization. + +We also compare the tampering localization maps generated by different methods, as shown in Fig. 4. As illustrated, StableGuard consistently achieves superior tampering detection accuracy compared to other methods. This is largely attributed to the self-supervised tampering detection framework we developed, which incorporates holistic watermarks through our MPW-VAE. This framework allows the forensic model to effectively differentiate between watermarked and non-watermarked regions. Moreover, the MoE-GFN we designed plays a pivotal role in extracting and fusing diverse forensic features, further enhancing the accuracy of tampering localization. + +# 4.4 Robustness Analysis + +To assess robustness against image degradations, we evaluate StableGuard on the AIGC tampering dataset under Gaussian noise, JPEG compression, and Poisson noise, following the protocol of prior works [19, 21, 22]. As shown in Tab. 3, StableGuard consistently outperforms competing methods, maintaining strong watermark extraction and tampering localization under all conditions. This robustness stems from our holistic watermark embedding strategy and the MoE-guided forensic network, which distributes watermark signals throughout the image and adaptively integrates diverse forensic cues. These results highlight StableGuard's strong potential for practical deployment. + +# 4.5 Ablation Study + +We conduct a comprehensive ablation study to evaluate the contributions of our core components: MPW-VAE, the MoE-GFN architecture, and the joint optimization strategy. + +Effectiveness of MPW-VAE. To assess MPW-VAE's impact on tampering localization, we replace it with WOUAF [11], a state-of-the-art post-hoc watermarking method. Using WOUAF, we generate watermarked images (via Eq. (3)) and train MoE-GFN for tampering localization. As shown in Tab. 4 (w/o MPW-VAE), performance degrades substantially. This gap stems from two key advantages of MPW-VAE: 1) its diffusion-native, residual-based watermarking enables high-fidelity paired image generation, which significantly benefits forensic training; and 2) joint optimization with the forensic network creates a synergistic effect, enhancing both watermark embedding and tampering detection. + +Effectiveness of MoE-GFN Components. We evaluate various architectural variants of MoEGFN. Removing the entire MoFE block (w/o MoFE) and reverting to a plain UNet leads to subpar performance due to the lack of expert-driven feature extraction. Excluding individual experts—Watermark Extraction (w/o WEE), Tampering Localization (w/o TLE), or Boundary Enhancement (w/o BEE)—each results in notable performance drops, confirming their complementary roles + +Table 4: Quantitative ablation study on the AIGC tampering dataset, where the $\dagger$ represents the structure used in this paper. + +
MethodF1↑AUC↑IoU↑Bit Acc↑Param↓Flops↓
w/o MPW-VAE0.8110.7960.77499.1352.02M78.51G
w/o MoFE0.9310.9200.90595.1238.11M45.72G
w/o WEE0.9690.9580.94598.6948.51M70.21G
w/o TLE0.9520.9400.93098.9048.51M70.21G
w/o BEE0.9620.9500.94099.1145.23M62.71G
w/o DSR0.9660.9550.94898.9751.26M75.74G
w/o JOS0.9210.9190.90899.1452.02M78.51G
Enc0.9740.9700.96099.79125.51M96.15G
Enc & Dec0.9820.9880.97699.96139.41M104.93G
Dec†0.9800.9920.96199.9852.02M78.51G
+ +in capturing global watermark signals, local manipulations, and boundary refinements, respectively. Replacing the Dynamic Soft Router(w/o DSR) with simple feature summation further degrades performance, highlighting its importance in adaptively fusing expert outputs based on input-specific forensic cues. While the removal of single experts (rows 3-5 in Tab. 4) leads to only modest declines in F1, AUC, IoU, and Bit Accuracy, this stems from the model operating near a performance ceiling, where marginal contributions appear limited in isolation. Crucially, the sharp performance drop when removing the entire MoFE block (row 2) highlights that these experts are complementary rather than redundant: their integration is essential for robustness and accuracy. + +We also explore the placement of the MoFE block within the UNet. Placing it in the encoder (Enc) improves localization but increases computational cost and underperforms compared to the decoder placement. Using MoFE in both encoder and decoder (Enc & Dec) yields marginal gains but at the expense of significantly higher complexity. Thus, placing the MoFE block in the decoder (Dec) strikes the best balance between performance and efficiency. + +Effectiveness of Joint Optimization Strategy. To isolate the benefit of joint self-supervised optimization, we compare StableGuard with a variant where watermark embedding/extraction and tampering localization are trained separately. Results in Tab. 4 (w/o JOS) show that decoupled training leads to weaker localization. In contrast, our unified optimization promotes stronger interaction between the embedding and forensic tasks, yielding consistent watermarks that enhance localization precision. + +# 5 Conclusion + +In this paper, we introduce StableGuard, a unified framework for copyright protection and tampering localization in the context of LDMs. By injecting a global watermark during the diffusion process and leveraging its intrinsic perturbation patterns, StableGuard achieves reliable and robust watermark extraction and tampering localization simultaneously. At the core of our method is the Multiplexing Watermark VAE (MPW-VAE), which generates visually consistent paired watermarked and watermark-free images to facilitate self-supervised training of the Mixture-of-Experts Guided Forensic Network (MoE-GFN). This network is composed of specialized experts that handle holistic watermark analysis, subtle artifact recognition, and boundary transition detection, with the Mixture-of-Forensic-Experts (MoFE) block ensuring precise and efficient forensic analysis. Extensive experimental evaluations validate StableGuard's superior performance, demonstrating its effectiveness and adaptability across a wide range of image scenarios, and highlighting its potential to advance both watermarking and tamper localization in diffusion models. + +Acknowledgement. This research is supported by the "Leading Talent" under Guangdong Special Support Program (2024TX08X048), China National Key R&D Program (2024YFB4709200), KeyArea Research and Development Program of Guangzhou City (No.2023B01J0022), Guangdong Provincial Natural Science Foundation for Outstanding Youth Team Project (No. 2024B1515040010), NSFC Key Project (No. U23A20391), Guangdong Basic and Applied Basic Research Foundation (No. 2023B1515120058), Guangdong Natural Science Funds for Distinguished Young Scholars (Grant 2023B1515020097), the GuangDong Basic and Applied Basic Research Foundation (2025A1515010124), the National Research Foundation Singapore under the AI Singapore Programme (AISG Award No: AISG4-TC-2025-018-SGKR), and the Lee Kong Chian Fellowships. + +# References + +[1] Rombach, R., A. Blattmann, D. Lorenz, et al. High-resolution image synthesis with latent diffusion models. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Appendix + +# A Problem Definition + +Our goal is to imprint an imperceptible holistic watermark $W$ during the generation process of a latent diffusion model for an image $X$ , and comprehensively leverage the intrinsic patterns of this watermark to achieve reliable and robust copyright protection and tampering localization within a unified framework. Let the LDMs follow a generative process parameterized by $\mathcal{G}_{\phi}$ . The watermark embedding task can then be formulated as: + +$$ +X = \mathcal {G} _ {\phi} (z, W), \tag {14} +$$ + +where $\mathcal{G}_{\phi}$ represents the LDM-based image generation function, and $z$ is the latent input. The embedded watermark $W$ remains imperceptible yet robust against various tampering manipulations $T$ , such as copy-paste or splicing. + +To detect tampering and recover the watermark, we employ a forensic network $\mathcal{F}_{\psi}$ , which takes the potentially manipulated image $T(X)$ as input and simultaneously produces the extracted watermark $\hat{W}$ and a tampering localization map $\hat{M}$ : + +$$ +\hat {W}, \hat {M} = \mathcal {F} _ {\psi} (T (X)). \tag {15} +$$ + +Here, $\hat{W}$ is expected to be reliably recovered with its integrity preserved, while $\hat{M}$ precisely identifies the manipulated regions, ensuring a robust and effective framework for both watermark extraction and tampering detection. + +# B Comparison of Existing Methods + +In this section, we provide a concise overview of the key comparison methods, summarized in Tab. 5. As illustrated, this work focuses on two primary tasks: tampering localization and copyright protection for content generated by diffusion models. The compared copyright protection approaches encompass both traditional post-hoc image watermarking techniques and more recent diffusion-native watermarking methods that integrate directly with the generative process. Tampering detection methods are categorized into passive localization and proactive protection strategies. Our proposed approach offers a unified, diffusion-native solution for both tampering detection and copyright protection, achieving state-of-the-art performance across both tasks. + +Table 5: Comparative analysis of existing methods for copyright protection and tampering localization. + +
MethodReferenceCopyright ProtectionTampering LocalizationJoint OptimizationWatermarking Type
HiDDeN [7]ECCV'18✓Post-hoc
SepMark [8]MM'23✓Post-hoc
Stable Signature [10]ICCV'23✓Diffusion-native
WOUAF [11]CVPR'24✓Diffusion-native
WaDiff [12]ECCV'24✓Diffusion-native
MVSS-Net [15]TPAMI'22✓-
IML-ViT [16]arXiv'23✓-
PSCC-Net [17]TCSVT'22✓-
ObjectFormer [18]CVPR'22✓-
HDF-Net [19]TPAMI'24✓-
EditGuard [21]CVPR'24✓✓Post-hoc
OmniGuard [22]CVPR'25✓✓Post-hoc
WAM [23]ICLR'25✓✓✓Post-hoc
StableGuard (Ours)NeurIPS'25✓✓✓Diffusion-native
+ +# C In-depth Analysis of Mixture-of-Experts Guided Forensic Network + +In this paper, we introduce a Mixture-of-Experts Guided Forensic Network (MoE-GFN), which trains specialized expert modules to address distinct forensic sub-tasks, thereby enhancing the overall forensic performance. At the core of MoE-GFN is the Mixture-of-Forensic-Experts (MoFE) block, + +comprising three domain-specific experts and a dynamic soft router. As illustrated in Fig. 5, we visualize the feature activations of each expert alongside the weight distributions produced by the dynamic router. The watermark extraction expert exhibits strong activation in untampered regions rich in watermark signals $(x_{\mathrm{wm}})$ , effectively recovering the embedded global watermark and demonstrating robustness to local tampering. The tampering localization expert focuses on corrupted regions $(x_{\mathrm{tamp}})$ , identifying missing watermark areas and enabling precise localization of tampering. The boundary enhancement expert targets the edges of tampered regions $(x_{\mathrm{bound}})$ , extracting fine-grained features to refine boundary detection. Unlike static fusion strategies, the dynamic router assigns adaptive weights to each expert's features based on input characteristics $(R_{\mathrm{wm}}, R_{\mathrm{tamp}}, R_{\mathrm{bound}})$ , facilitating a more effective integration of forensic cues and significantly improving both watermark extraction and tampering localization accuracy $(x_{\mathrm{unified}})$ . In the Ablation Study section in the main text, we also analyzed in detail the role of each expert and dynamic fusion strategy of MoFE and the final impact on the quantization results. + +![](images/4e560a77c2f64d5fdf0f49bf3ae89c0bd25421def264fcd618bc36f15b126dfe.jpg) + +![](images/8549aa614d12983bdf2d0311d35a6f0a81597c7634f8d4d6d98cc89eaec70429.jpg) + +![](images/89e6ac38ec6e16143d1488939ed0a69df2c0826b2885920e1e26e50a9693fb31.jpg) + +![](images/f591d7568267be38b8f51dacd9c616f2e70771f5f587a515fe06dd341d75524a.jpg) + +![](images/931878faf4597e67170e5c29a6b2a2a48a500b93a1d97f23d4f4233e7d817092.jpg) + +![](images/e6c956d1445fe8c1767c18888fc1a5197751ccdaf01ede587d13d7849692793d.jpg) +(a) Original +(f) Tamper +Figure 5: The visualizations of the distinct features extracted by each expert and their corresponding soft weights. The Watermark Extraction Expert captures global watermark patterns, the Tampering Localization Expert focuses on fine-grained artifacts, and the Boundary Enhancement Expert sharpens tampering boundaries. These complementary features are adaptively fused via dynamic weighting, enabling comprehensive and precise forensic analysis. + +![](images/0c027a753ee53d2f2f28cdadad58d4c6d6bfa101bb0d2d4e0481d65d5e5bdcba.jpg) +(b) $x_{\mathrm{wm}}$ +(g) $R_{\mathrm{wm}}$ + +![](images/d502d428db275c987938cf72e799aec061d9421a6fc58a6916522987cff31d1b.jpg) +(c) $x_{\mathrm{tamp}}$ +(h) $R_{\mathrm{tamp}}$ + +![](images/f6e727a14a1ee16675c813fff4ffcea701ad828d99569026a858d5fb7cca049e.jpg) +(d) $x_{\mathrm{bound}}$ +(i) $R_{\mathrm{bound}}$ + +![](images/2911e353e3ce943ffef4f2f10cedb0aa6c8195fa263f7d989c6d5691adead029.jpg) +(e) $x_{\mathrm{unified}}$ +(j) GT + +# D Training Process of StableGuard + +We present the detailed training procedure of StableGuard in Algorithm 1. The entire process is fully self-supervised and trained in an end-to-end manner, without requiring any manually annotated data. Both the embedding of diffusion-native watermarks and the training of the joint model for watermark and tampering forensics are accomplished within a unified framework. + +Algorithm 1 Training Process of StableGuard. +Output: MPW-VAE decoder $\mathcal{D}_{\phi}$ , MoE-GFN $\mathcal{F}_{\psi}$ . +```txt +1: for $i = 1$ to steps do +``` + +Input: Input image $X$ , iteration steps steps, LDM VAE encoder $\mathcal{E}_{\mu}$ , LDM VAE decoder $\mathcal{D}_{\nu}$ . +```txt +2: Inverse latent $z = \mathcal{E}_{\mu}(X)$ +``` + +```txt +3: Decode watermark-free image $\tilde{X} = \mathcal{D}_{\nu}(z)$ +``` + +```txt +4: Decode watermarked image $Y = \mathcal{D}_{\phi}(z)$ +``` + +```txt +5: Generate random tamper mask $M$ +``` + +```txt +6: Generate tamper image $\hat{Y}$ by Eq. (3). +``` + +```txt +7: Predict watermark and tamper area $\hat{W}$ , $\hat{M} = \mathcal{F}_{\psi}(\hat{Y})$ +``` + +```txt +8: Comptue loss by Eq. (13). +``` + +10: end for +```txt +9: Update $\mathcal{D}_{\phi}$ and $\mathcal{F}_{\psi}$ . +``` + +# E More Implementation Details and Experiments + +# E.1 More Implementation Details + +In our experiments, we utilized two datasets: one consisting of real images from COCO [56], and the other comprising generated images from a Text-to-Image dataset. For the real image dataset, we first reconstructed each image using our MPW-VAE and then embedded a holistic watermark into the reconstructed image. For the generated dataset, we began by obtaining the denoised latent representation through stable diffusion, which was then decoded into the image space using the MPW-VAE decoder to produce the watermarked image. + +Text-to-Image (T2I) Dataset. The T2I dataset is designed to evaluate the watermarking effectiveness of StableGuard in the LDM framework. To construct this dataset, we first generate the latent embedding of each image using the original pre-trained Stable Diffusion [1] denoising UNet, condition on its corresponding textual prompt. We then decode this embedding through our MPW-VAE, producing a set of watermarked images. Specifically, we leverage $\mathrm{ChatGPT}^2$ to generate 1,000 textual prompts, with each prompt yielding 10 images, resulting in a dataset of 10,000 images. + +AIGC Tampering Dataset. The AIGC tampering dataset is collected to benchmark the performance of various tampering localization techniques against manipulations introduced by state-of-the-art image editing methods [1, 2, 4, 57, 58]. This large-scale dataset consists of 35,000 images, comprising 25,000 from COCO and 10,000 from our T2I dataset. We utilize the Segment Anything Model (SAM) [49] to generate semantic masks by segmenting images, after which we apply Stable Diffusion Inpainting [1], Stable Diffusion XL [2], Kandinsky Inpainting [57], ControlNet [4], and LAMA [58] to edit the segmented regions. During the tampering process, we applied modifications to selected regions and composited the altered content onto the watermarked image using a corresponding binary mask following by [21, 22]. + +Metrics. We leverage several widely-used metrics to collaboratively evaluate the pixel-level fidelity and perceptual quality of the generated images, including Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index Measure (SSIM) [60], Learned Perceptual Image Patch Similarity (LPIPS) [54], and Fréchet Inception Distance (FID) [61]. To quantify the performance of watermark extraction, we adopt the Bit Accuracy metric to calculate the proportion of correctly extracted watermark bits over the total embedded bits. For tampering detection, we used the F1 Score, the Area Under Curve (AUC), and IoU metrics to assess the accuracy and completeness of the detected area. The F1 Score is the harmonic mean of detection precision and recall, and AUC represents the area under the ROC Curve. IoU is calculated by the intersection over the union between the detected tampered areas and the ground-truth regions. + +Forensic Decoder. The forensic decoder consists of two parallel decoding heads: 1) The mask prediction head is responsible for localizing tampered regions and is implemented as a lightweight two-layer convolutional network. 2) The watermark prediction head is used for recovering the embedded watermark and comprises a two-layer convolutional network followed by a fully connected layer. This modular design allows the network to jointly optimize for both localization and watermark reconstruction. + +# E.2 Tampering Localization Results on Traditional Tampering Dataset + +Our proposed StableGuard framework can be naturally extended to conventional natural images through reconstruction using a variational autoencoder, demonstrating strong generalization across a wide range of image domains. To assess its effectiveness beyond synthetic settings, we conducted comprehensive evaluations on five widely used tampering datasets: CASIA [62], NIST [63], Columbia [64], and Coverage [65]. The corresponding quantitative results and visual comparisons are reported in Tab. 6 and Fig. 6. + +Although existing passive tampering localization methods are trained on meticulously constructed paired datasets, they consistently fall short when compared to our StableGuard and its predecessor EditGuard, OmniGuard and WAM. This performance gap reveals the inherent limitations of passive, post-hoc approaches, which often lack robustness when faced with novel manipulations or diverse + +![](images/ffabd16185e36405135f0ceed3fc759d3d3aaaedbd15dbdcb7f69edadd5464b0.jpg) +Figure 6: The visualization comparison of tampering localization results on the traditional tampering dataset. (a) is the tampered image, (b)-(m) are the tampering localization results of MVSS-Net [15], PSCC-Net [17], ObjectFormer [18], TruFor [66], NCL-IML [67], IML-ViT [16], HDF-Net [19], EditGuard [21], OmniGuard [22], WAM [23], our StableGuard, and the ground truth, respectively. + +Table 6: Localization precision comparison with other competitive methods on five common datasets [62-65, 68]. + +
MethodCAISA1.0 [62]NIST16 [63]Columbia [64]Coverage [65]IMD20 [68]
F1↑AUC↑IoU↑F1↑AUC↑IoU↑F1↑AUC↑IoU↑F1↑AUC↑IoU↑F1↑AUC↑IoU↑
Mantra-Net [69]0.2150.6610.1200.4280.8280.2720.3510.6930.2130.2620.7120.1590.2710.7310.157
SPAN [70]0.2520.6760.1440.2050.6000.1140.4650.7410.3030.2580.6590.1480.2170.7010.134
CAT-Net [71]0.2580.7170.1840.6950.9480.5730.4180.4600.2640.2920.7390.1940.1840.4520.102
MVSS-Net [15]0.2630.6380.2070.8330.9820.7560.4240.7640.2760.2290.8160.1520.3180.7030.234
PSCC-Net [17]0.2120.6110.1360.4790.8460.3150.5320.7580.3630.2970.7230.1740.2290.7070.145
ObjectFormer [18]0.2080.7390.1350.7200.9580.6420.5290.7650.3740.2770.7560.1650.2510.8020.166
TruFor [66]0.4450.8440.2860.3600.8060.2200.5660.8690.3950.1190.8370.0630.2170.8100.122
NCL-IML [67]0.1690.7660.1230.8850.9810.8340.4200.7120.2530.2620.8240.1720.3390.8270.245
HDF-Net [19]0.3950.7470.3030.8950.9890.8400.6000.8200.5130.3450.8110.2330.3550.8140.257
EditGuard [21]0.9140.9510.8650.9210.9570.8810.9530.9540.9430.9450.9640.9280.9060.9260.851
OmniGuard [22]0.8780.9420.8120.9100.9530.8570.9410.9500.9180.9200.9580.8950.8700.9120.801
WAM [23]0.8720.9300.8030.9080.9500.8540.9380.9470.9120.9180.9550.8910.8650.9080.792
Ours0.9330.9700.9000.9560.9830.9260.9950.9970.9900.9860.9940.9730.9490.9750.919
+ +Table 7: Statistics of runtime latency and VRAM on the AIGC tampering dataset. + +
MetricVanilla VAEMPW-VAEMoE-GFN
Latency12.69 ms14.99 ms91.88 ms
VRAM1681.93 MB2033.97 MB428.32 MB
+ +image distributions. In contrast, our findings underscore the advantages of proactive forensic methods, particularly those grounded in watermarking strategies that are embedded during the generative process. These methods exhibit greater resilience and accuracy in both tampering detection and localization. + +In addition, StableGuard demonstrates superior performance over other methods across a variety of practical scenarios. It achieves this improvement while eliminating the need for complex and often restrictive post-processing procedures. This enhancement is largely attributed to the Mixture of Experts Guided Forensic Network, which is designed to extract and integrate diverse forensic cues. By employing specialized expert modules and adaptive feature routing, the network enables StableGuard to jointly perform watermark recovery and tampering localization in a coherent and dependable manner. + +Table 8: Comparison of watermark extraction accuracy (↑) under different tampering rates on the AIGC tampering dataset. + +
Method10%30%50%70%90%
EditGuard [21]99.7899.6097.6690.9569.13
OmniGuard [22]98.1198.0296.8491.3383.90
WAM [23]98.1797.1194.8993.7488.53
Ours99.9899.9899.9699.2789.58
+ +Table 9: Comparison of false positive rate (FPR ↓) on the AIGC tampering dataset. + +
TypeEditGuard [21]OmniGuard [22]WAM [23]Ours
Partially tampered0.04220.01910.00450.0023
Fully untampered0.00280.00240.00190.0016
+ +# E.3 Time Complexity Analysis + +We conducted additional experiments to evaluate the inference latency and memory footprint of our framework. The detailed results are summarized in Tab. 7. Specifically, we measured the runtime and memory usage of our MPW-VAE in comparison to the baseline Stable Diffusion VAE (at a resolution of $512 \times 512$ ), as well as the runtime of the MoE-GFN forensic network under identical settings. The results show that MPW-VAE introduces only negligible computational overhead relative to the original diffusion VAE. Moreover, the MoE-GFN model achieves competitive runtime efficiency, with both inference latency and memory consumption well within the bounds required for practical deployment. + +# E.4 Forensic Accuracy under Varying Tampering Ratios + +We further analyze watermark extraction accuracy under different levels of tampering, ranging from $10\%$ to $90\%$ . The results in Tab. 8 demonstrate that our method consistently outperforms existing approaches across all tampering ratios. This robustness stems from three main factors: 1) The MPW-VAE effectively encodes holistic watermark features at multiple scales during decoding, enhancing the resilience of watermark embedding. 2) The global self-attention mechanism in our watermark extraction expert captures long-range dependencies and contextual cues that remain intact even when large regions are tampered. 3) The use of randomly generated masks with varying coverage during training, including large-area tampering, encourages the model to generalize across diverse manipulation ratios and ensures reliable watermark extraction even under severe distortions. + +# E.5 True Positive Rate (TPR) under Tampering Scenarios + +Our framework is designed as a proactive watermarking system [21-23], where the watermark plays a central role in simultaneously enabling copyright protection and tampering localization. Unlike passive detection methods, we assume that images are watermarked at creation time; thus, clean images without watermarks fall outside our system's operational scope and can be conservatively regarded as "fully tampered." To assess practical reliability, we measure false positives in two scenarios: (1) untampered regions within partially tampered watermarked images, and (2) fully untampered, watermarked images. As shown in Tab. 9, our method achieves the lowest false positive rate (measured as the ratio of false positive pixels to the total number of untampered pixels) in both cases, demonstrating its robustness and reliability. + +# E.6 Performance across Different Tampering Types + +We also conducted a per-type analysis of tampering detection accuracy, as reported in Tab. 10. Our approach consistently delivers strong performance across diverse manipulation types, underscoring its robustness and generalizability. + +Table 10: Tampering performance of our method under different tampering types on the AIGC tampering dataset. + +
TypeSD-Inpainting [1]SD-XL [2]Kandinsky [57]ControlNet [4]LAMA [58]
F1↑AUC↑IoU↑F1↑AUC↑IoU↑F1↑AUC↑IoU↑F1↑AUC↑IoU↑F1↑AUC↑IoU↑
Splicing0.9750.9910.9630.9840.9940.9650.9850.9930.9640.9840.9940.9630.9820.9950.965
Copy-and-paste0.9780.9900.9580.9760.9860.9590.9800.9890.9610.9780.9870.9570.9760.9910.960
Removal0.9800.9940.9650.9820.9890.9660.9780.9900.9620.9810.9930.9640.9840.9880.968
Inpainting0.9870.9970.9620.9820.9950.9540.9770.9960.9530.9810.9980.9680.9740.9980.951
+ +Table 11: Performance under various degradations on AIGC tampering dataset. + +
TypeEditGuard [21]OmniGuard [22]WAM [23]Ours
Bit Acc.↑F1↑Bit Acc.↑F1↑Bit Acc.↑F1↑Bit Acc.↑F1↑
Clean99.780.93898.110.86398.170.92099.980.972
JPEG (Q=30)62.300.23063.340.31180.960.53298.870.866
WebP (Q=50)52.880.23356.160.30784.870.53898.950.703
HEIC (Q=50)66.530.33559.410.32387.460.40298.360.748
Resale (0.25)41.770.25654.950.34395.980.82197.550.858
Brightness (0.8-1.2)91.350.50489.550.74194.710.81698.940.860
Contrast (0.8-1.2)90.840.78891.450.83695.000.82497.930.836
Saturation (0.8-1.2)92.930.81793.440.85695.690.81497.980.911
+ +# E.7 Robustness under Real-World Image Degradations + +To evaluate real-world applicability, we conducted additional experiments with diverse image degradation operations, including: (1) aggressive compression, (2) severe resolution downsampling, and (3) color transformations. The parameter ranges for each transformation are summarized in Tab. 11. As shown in the results, our framework maintains high watermark extraction accuracy and reasonable tampering localization performance. This observation aligns with our design intuition: watermark extraction leverages global cues that are relatively robust to distortions, whereas tampering localization relies on local consistency, making it more sensitive to compression and resampling artifacts. Importantly, the decline in localization accuracy remains moderate and within acceptable limits, highlighting the resilience of our method in real-world scenarios. + +# E.8 More Visual Results + +Fig. 7 and Fig. 8 present additional visual results of tampering localization on the AIGC tampering dataset of COCO [56] and T2I Dataset, respectively. Please zoom in for the best view. + +# F Disscussion and Limitations + +StableGuard is specifically designed for image latent diffusion models. However, since watermarking is applied solely through the variational autoencoder decoder within the latent diffusion framework, our approach can be naturally extended to video latent diffusion models. Given the rapid progress in this area, adapting our method to the video domain represents an important direction for future research. + +While our approach maintains strong performance under certain levels of image degradation, it is not entirely immune to the decline in forensic accuracy caused by such distortions. Addressing the sensitivity of forensic performance to image degradation remains a current limitation of our method and a key focus for future investigation. + +![](images/001be48e341fd5efa5f5723aca9649d617667514fbf80b8e3cfc2cf2c714e1e1.jpg) +Figure 7: The visualization of tampering localization results on the AIGC tampering dataset of COCO part. (a) is the original image, (b) is the watermarked image by StableGuard, (c) is the tampered image, (d)-(m) are the tampering localization results of MVSS-Net [15], IML-ViT [16], PSCC-Net [17], ObjectFormer [18], HDF-Net [19], EditGuard [21], OmniGuard [22], WAM [23], our StableGuard, and the ground truth, respectively. + +![](images/ce284d9734b3f2d65f3ad0149d8a7a93da7475bf62081e2c4d9511c3cf311f24.jpg) +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +(i) +(i) +(k) +(1) +(m) +Figure 8: The visualization of tampering localization results on the AIGC tampering dataset of T2I dataset part. (a) is the original image, (b) is the watermarked image by StableGuard, (c) is the tampered image, (d)-(m) are the tampering localization results of MVSS-Net [15], IML-ViT [16], PSCC-Net [17], ObjectFormer [18], HDF-Net [19], EditGuard [21], OmniGuard [22], WAM [23], our StableGuard, and the ground truth, respectively. \ No newline at end of file diff --git a/stableguardtowardsunifiedcopyrightprotectionandtamperlocalizationinlatentdiffusionmodels/images.zip b/stableguardtowardsunifiedcopyrightprotectionandtamperlocalizationinlatentdiffusionmodels/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..3cb5f7760599908ce57b07c67c5a5a6c60e77b6e --- /dev/null +++ b/stableguardtowardsunifiedcopyrightprotectionandtamperlocalizationinlatentdiffusionmodels/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fbb03d9b395feb70a6900cc1ebf1b9a3da5b7c97b6723c2c03d6cb53e3b14e2f +size 1372791 diff --git a/stableguardtowardsunifiedcopyrightprotectionandtamperlocalizationinlatentdiffusionmodels/layout.json b/stableguardtowardsunifiedcopyrightprotectionandtamperlocalizationinlatentdiffusionmodels/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..3de256b12085185a5cb1c9e261485632ee73d70e --- /dev/null +++ b/stableguardtowardsunifiedcopyrightprotectionandtamperlocalizationinlatentdiffusionmodels/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:86ee155c559c64fd77d429cde22629e231d6acaba3b1ca229abcbfa25c493c24 +size 841624 diff --git a/stablematchingwithtiesapproximationratiosandlearning/0f22fdda-0bf1-4b58-ba48-7917eea00447_content_list.json b/stablematchingwithtiesapproximationratiosandlearning/0f22fdda-0bf1-4b58-ba48-7917eea00447_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..3f96c8f04a2add2f1c464fc9638590c3e3954f52 --- /dev/null +++ b/stablematchingwithtiesapproximationratiosandlearning/0f22fdda-0bf1-4b58-ba48-7917eea00447_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5aa4508f7b7cf52767a9599797b315515a84cf28638f08a85903ec6084b00218 +size 263644 diff --git a/stablematchingwithtiesapproximationratiosandlearning/0f22fdda-0bf1-4b58-ba48-7917eea00447_model.json b/stablematchingwithtiesapproximationratiosandlearning/0f22fdda-0bf1-4b58-ba48-7917eea00447_model.json new file mode 100644 index 0000000000000000000000000000000000000000..f11b842a23ad1450180e5e703ba4e64a1915babb --- /dev/null +++ b/stablematchingwithtiesapproximationratiosandlearning/0f22fdda-0bf1-4b58-ba48-7917eea00447_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2c3b43e20b6a1222f63555ccab8232d6b870f4a99559eda43410a63c0b10591d +size 324775 diff --git a/stablematchingwithtiesapproximationratiosandlearning/0f22fdda-0bf1-4b58-ba48-7917eea00447_origin.pdf b/stablematchingwithtiesapproximationratiosandlearning/0f22fdda-0bf1-4b58-ba48-7917eea00447_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..89858d18521550ff7bbf86deb3f64d45aa4c287c --- /dev/null +++ b/stablematchingwithtiesapproximationratiosandlearning/0f22fdda-0bf1-4b58-ba48-7917eea00447_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e5062e3741fbaa948a9ee844f636cc3465871b3716042e7574e89af6dcf32efc +size 656789 diff --git a/stablematchingwithtiesapproximationratiosandlearning/full.md b/stablematchingwithtiesapproximationratiosandlearning/full.md new file mode 100644 index 0000000000000000000000000000000000000000..48bd0768f9fe0b4709ffe96c147786078c86e4c7 --- /dev/null +++ b/stablematchingwithtiesapproximationratiosandlearning/full.md @@ -0,0 +1,1363 @@ +# Stable Matching with Ties: Approximation Ratios and Learning + +Shiyun Lin* + +Center for Statistical Science +School of Mathematical Sciences, Peking University +shiyunlin@stu.pku.edu.cn + +Nadav Merlis + +Technion - Israel Institute of Technology nmerlis@technion.ac.il + +Simon Mauras + +INRIA, FairPlay Joint Team simon.mauras@inria.fr + +Vianney Perchet +CREST, ENSAE, IP Paris +Criteo AI Lab, FairPlay Joint Team +Vianney.perchet@normalesup.org + +# Abstract + +We study matching markets with ties, where workers on one side of the market may have tied preferences over jobs, determined by their matching utilities. Unlike classical two-sided markets with strict preferences, no single stable matching exists that is utility-maximizing for all workers. To address this challenge, we introduce the Optimal Stable Share (OSS)-ratio, which measures the ratio of a worker's maximum achievable utility in any stable matching to their utility in a given matching. We prove that distributions over only stable matchings can incur linear utility losses, i.e., an $\Omega(N)$ OSS-ratio, where $N$ is the number of workers. To overcome this, we design an algorithm that efficiently computes a distribution over (possibly non-stable) matchings, achieving an asymptotically tight $\mathcal{O}(\log N)$ OSS-ratio. When exact utilities are unknown, our second algorithm guarantees workers a logarithmic approximation of their optimal utility under bounded instability. Finally, we extend our offline approximation results to a bandit learning setting where utilities are only observed for matched pairs. In this setting, we consider worker-optimal stable regret, design an adaptive algorithm that smoothly interpolates between markets with strict preferences and those with statistical ties, and establish a lower bound revealing the fundamental trade-off between strict and tied preference regimes. + +# 1 Introduction + +Two-sided matching markets are prevalent in various contexts, such as matching students to schools [2, 3], doctors to hospitals [50], or workers to jobs [5]. In this paper, we model the market as a company that assigns jobs to workers. Each participant has a preference ordering over the other side of the market. For example, jobs rank workers by ability, while workers rank jobs by personal preference. Stability ensures a fair equilibrium where workers receive sufficiently desirable jobs while respecting the preferences and priorities of all parties. When preferences are strict, the deferred acceptance algorithm [23] efficiently computes a worker-optimal stable matching – no worker can get a better job without violating stability. + +In online marketplaces, for example, the online crowd-sourcing platform Amazon Mechanical Turk, workers are usually uncertain of their preferences over jobs at the beginning, since they do not have hands-on experience. However, there are numerous similar tasks to be delegated + +on the platform and, fortunately, the uncertain preferences can thus be learnt during the iterative matchings. Recent research has explored this scenario within the framework of multi-player multi-armed bandits [42, 43, 8, 37]. Under the strict preferences assumption, these works combine bandit learning algorithms with the deferred acceptance procedure to guide the market toward the worker-optimal stable matching. + +However, in real-life scenarios, workers could be indifferent between some jobs due to inherent uncertainty or coarse evaluations. For instance, conference management systems like the Toronto Paper Matching System (TPMS): while the system generates continuous scores to evaluate the suitability of each reviewer for a paper, which theoretically avoids ties, the bidding process introduces unavoidable indifference through discrete categorical ratings (e.g., "Eager", "Willing", "In a pinch", "Not willing"), creating natural ties in preferences. The challenge becomes even more pronounced in learning-based matching markets, where statistically indistinguishable utility estimates produce effective ties between options. This presents a fundamental limitation for bandit learning approaches, as standard algorithms typically fail to provide meaningful regret guarantees when facing such indifference structures in the preference landscape. In particular, when utility differences become small (statistically indistinguishable), existing regret bounds break down completely, and handling this regime was previously considered impossible [42]. + +With indifferent preferences, a stable matching can be obtained by arbitrarily breaking ties and applying the deferred-acceptance algorithm. However, the resulting matching is no longer worker-optimal, as different tie-breaking rules may lead to different stable matchings preferred by different workers – potentially creating dramatic utility disparities across outcomes. This challenge is particularly acute in bandit learning settings, where statistically indistinguishable utilities for one worker may lead to arbitrarily large regret for others due to the cascading effects of tie-breaking decisions. In fair resource allocation, fractional matching is a standard technique for balancing competing interests when a single integral matching is infeasible [33, 27, 9]. The Birkhoff-von Neumann (BvN) theorem [10, 58] establishes that such a fractional matching is equivalent to a probability distribution over integral matchings. + +These observations motivate our core research question: For markets with tied preferences, can we approximate a stable solution by considering distributions over matchings, while guaranteeing all workers a fair, minimum level of satisfaction? + +To answer this question, we define a worker's optimal-stable-share (OSS) as her maximum achievable utility across all stable matchings. We then introduce the OSS-ratio as a fairness metric, which measures the fraction of the OSS that each worker is guaranteed to receive under any allocation. + +We begin by analyzing the offline setting with known preferences, establishing tight OSS-ratio bounds across different matching classes. These results naturally extend to settings with preference uncertainty. Building on these offline results, we further formulate the problem within a multi-player multi-armed bandit framework for online learning scenarios, and show how our approximation guarantees provide the crucial foundation for achieving sublinear regret in matching markets with indifference. + +# 1.1 Main Contributions + +Offline Approximation Oracle and Matched Upper and Lower Bounds. We first demonstrate that restricting to stable matchings yields only a trivial (and tight) lower bound on the OSS-ratio (Theorem 1), motivating our study for broader matching classes. We then establish a logarithmic lower bound for general matchings (Theorem 2) and construct an approximation oracle (Algorithm 1) achieving this bound while maintaining internal stability (Theorem 3). + +Robustness to Approximated Preferences. We prove our positive results are robust to utility uncertainty: when exact utilities are unknown but lie within a given uncertainty set, we maintain the same guarantees with only an additive error bounded by the maximum uncertainty (Theorem 6). This holds especially for rectangular uncertainty sets, which model utility matrices estimated from data. + +Bandit Learning in Matching Markets with Indifference. Building on our offline approximation results, we introduce $\alpha$ -approximation stable regret $Reg_{i}^{\alpha}(T)$ , using an $\alpha$ -fraction of the optimal-stable-share as a tractable benchmark for markets with (statistical) ties. Our adaptive algorithm ETCO (Algorithm 3) seamlessly handles both strict and tied preferences. Theorem 7 establishes its regret bounds, which match the lower bound [52] in markets with large preference gaps. Theorem 8 further reveals a fundamental trade-off: no algorithm can simultaneously achieve optimal regret in both large-gap (standard regret) and small/no-gap (approximation regret) regimes. + +# 1.2 Techniques Involved and Developed + +The upper bound on the approximation ratio is the first key technical contribution of our paper. We establish this result via three main steps: 1) Introducing a novel component – the duplication index – into the algorithm design; 2) Constructing a directed forest where edges encode conflicts between workers competing for the same job copies across different matchings; 3) Leveraging the tree structure and stability constraints to derive the upper bound inductively. + +In the bandit learning setting, the primary technical challenge and key contribution lie in the lower bound proof. To establish this result, we carefully construct two instances with 4 workers and 4 jobs, where the utility matrices differ in only one critical entry that determines whether meaningful ties exist. This construction reveals how ties in one worker's preferences propagate to affect other workers' regret. Furthermore, we employ an information-theoretic argument to demonstrate that the algorithm must sample this critical entry sufficiently often to avoid incurring linear regret. To our knowledge, we are the first to provably show a tradeoff between standard regret and approximation regret in bandit settings. + +# 1.3 Related Work + +Stable Matching with Ties. A natural extension of Gale and Shapley's work [23] considers settings with tied or incomplete preferences. Irving [29] introduced three stability notions - weak, strong, and super-stability - with weak stability being the most studies [25, 26, 35, 46], as it always guarantees existence, unlike strong or super-stability. However, weakly stable matchings may vary in size, and finding a maximum one is NP-hard [30], while verifying weak stability is NP-complete [46]. Unlike prior work focused on maximizing matching size, we instead study fair job allocations, ensuring each worker receives a utility within a guaranteed fraction of their optimal stable matching, and we characterize the approximation ratio of such allocations. + +Fairness in Two-sided Matching. Recent work has increasingly addressed fairness in two-sided markets. In fair division, Freeman et al. [21] introduces double envy-freeness up to one match (DEF1) and double maximin share guarantee (DMMS) for many-to-many matching, while Igarashi et al. [28] studies many-to-one matching, enforcing EF1 for one side while preserving stability. In machine learning, Karni et al. [32] incorporates preference-informed individual fairness (PIIF) [34], requiring allocations to satisfy individual fairness [18] while respecting preferences. Our work diverges by focusing on one-to-one markets, where standard notions like EF1 and MMS are inapplicable. We propose a novel share-based fairness concept (OSS-ratio) to measure workers' gains relative to their optimal-stable-share. Our algorithm returns a random matching that is ex-ante stable (no justified envy) and ex-post internally stable, achieving a best-of both-worlds guarantee. + +Bandit Learning in Matching Markets. Das and Kamenica [16] first formalized bandit problems in matching markets, with subsequent work [42, 43, 8, 52, 37] exploring this model. In this setting, players (with unknown utilities) and arms (with known preferences) form a two-sided market. Player-optimal stable regret [42] measures the utility difference between a player's outcome and their optimal stable match. Yet, existing results are limited to markets with strict preferences, as stable regret becomes linear and ill-defined when ties exist. Kong et al. [38] recently studied indifference cases, but their player-pessimistic regret benchmark cannot recover optimal stable matches in tie-free settings. Our work bridges this gap by: (1) establishing a tight logarithmic OSS-ratio for offline matching with ties, (2) introducing approximation regret as a tractable objective for tied markets, and (3) developing an adaptive algorithm that achieves optimal regret bounds in both tied and tie-free settings. + +# 2 Preliminaries + +We model the matching market as a company that assigns jobs to workers. There are $N$ workers, $\mathcal{W} = \{w_1, w_2, \dots, w_N\}$ and $K$ jobs, $\mathcal{A} = \{a_1, a_2, \dots, a_K\}$ . The company assigns jobs to workers such that each job is assigned to at most one worker and each worker performs at most one job. The assignment is therefore a matching $\mu$ . We shall use $\mu(w)$ to represent the allocated job to worker $w$ , and $\mu(a)$ to denote the worker with job $a$ . If a worker $w$ or a job $a$ remains unmatched, we will use the notation $\mu(w) = \bot$ or $\mu(a) = \bot$ . + +For every job, the company has a strict rating over the workers based on their expertise and ability on this job. Specifically, if $w \succ_{a} w'$ , worker $w$ performs job $a$ strictly better than $w'$ . On the other hand, + +workers also have preferences over the jobs, and it is possible that a worker is indifferent among several jobs. The preferences of workers on jobs are represented through a utility matrix $\mathbf{U}$ , where $\mathbf{U}(w,a) \in [0,1]$ denotes the preference of worker $w$ on job $a$ . If $\mathbf{U}(w,a) > \mathbf{U}(w,a')$ , worker $w$ prefers job $a$ over $a'$ , and $\mathbf{U}(w,a) = \mathbf{U}(w,a')$ implies that $w$ is indifferent between jobs $a$ and $a'$ . For simplicity, we will assume that a worker $w$ will refuse to be matched with job $a$ if it has utility $\mathbf{U}(w,a) = 0$ ; stated otherwise, either $\mathbf{U}(w,\bot)$ is positive but infinitely small or $\mathbf{U}(w,\bot) = 0$ and ties are broken in favor of $\bot$ . As a consequence, a problem instance $(\mathbf{U},P_{a})$ is defined by a utility matrix $\mathbf{U}$ and a preference profile $P_{a}$ representing the preferences of jobs over workers. + +Stability is a key concept in two-sided matching markets, which ensures there is no justified envy in the market, i.e., the only jobs a worker prefers over her own job are the ones that she is less suitable to face than the currently assigned worker. When preferences include ties, multiple stability notions arise, and we focus on weak stability [29]. A matching $\mu$ is weakly stable if no worker-job pair exists where both strictly prefer each other over their allocated partners: + +Definition 1 (Weak Stability). A matching $\mu$ is weakly stable if there is no blocking pair $(w, a)$ such that $w \succ_{a} \mu(a)$ and $U(w, a) > U(w, \mu(w))$ . + +If a matching is weakly stable, there exists a tie-breaking mechanism such that this matching is stable in the resulting instance with strict preferences. Conversely, any stable matching that is generated using a tie-breaking mechanism is also weakly stable in the original instance. Without causing ambiguity, we will refer to weak stable as stable for brevity. Furthermore, internally stable matching [44] refers to a matching where there are no blocking pairs when only considering the matched workers and jobs. + +Definition 2 (Internal Stability). A matching $\mu$ is internally stable if there is no internally blocking pair $(w, a)$ such that 1) both $w$ and $a$ are matched in $\mu$ , and 2) $w \succ_{a} \mu(a)$ and $U(w, a) > U(w, \mu(w))$ . + +Given a problem instance, we define the following classes of matchings: $\mathcal{M} \coloneqq \{\mu : \mu \text{ is a matching}\}$ , $S \coloneqq \{\mu : \mu \text{ is a stable matching}\}$ , and $\mathcal{I} \coloneqq \{\mu : \mu \text{ is an internally stable matching}\}$ . + +In a matching market with ties, stable matchings are not unique, given different tie-breaking mechanisms. A job $a$ is a valid stable match of worker $w$ if there exists a stable matching that matches $w$ with $a$ . We say $a$ is the optimal stable match of worker $w$ if it is the most preferred valid stable match, i.e., there exists a matching $\mu^{*} \in S$ such that $\mu^{*}(w) = a$ and $U(w, \mu^{*}(w)) = \max_{\mu \in S} U(w, \mu(w))$ . We call $U(w, \mu^{*}(w))$ the optimal stable share (OSS) for worker $w$ , denoted as $U^{*}(w)$ . + +The canonical results in two-sided matching markets are the Gale-Shapley theorem and algorithm (GS) [23], which guarantee both the existence of stable matchings and an efficient $\mathcal{O}(n^2)$ computation. The GS algorithm operates through an iterative proposal process. First, workers sequentially propose to their most preferred available jobs. Each job tentatively accepts its most preferred proposal and rejects others. After that, rejected workers continue proposing to their next preferences. The process terminates when no rejections occur, yielding a stable matching. In markets with strict preferences, GS produces a matching that is optimal for all proposers. However, when preferences contain ties, this optimality no longer holds uniformly. + +Example 1 (Stable matching with indifference). Let $\mathcal{W} = \{w_1, w_2, w_3\}$ be workers and $\mathcal{A} = \{a_1, a_2\}$ be jobs with $w_1 \succ w_2 \succ w_3$ for all jobs. The utility matrix that encodes the preference of workers over jobs is given by: + +$$ +\boldsymbol {U} = \left[ \begin{array}{c c} 1 & 1 \\ 1 & 0 \\ 0 & 1 \end{array} \right] +$$ + +There are 2 stable matchings in this instance: $\mu_1 = \{(w_1, a_1), (w_3, a_2)\}$ , $\mu_2 = \{(w_1, a_2), (w_2, a_1)\}$ . There are 4 extra non-empty internally stable matchings, where exactly one worker is assigned a job of utility 1, and unmatched workers/jobs cannot be involved in blocking pairs. All workers have an OSS of 1. More precisely, $w_1$ receives utility $U^*(w_1) = U(w_1, a_1) = U(w_1, a_2) = 1$ in both stable matchings, $w_2$ receives utility $U^*(w_2) = U(w_2, a_1) = 1$ in $\mu_2$ , and $w_3$ receives utility $U^*(w_3) = U(w_3, a_2) = 1$ in $\mu_1$ . + +Example 1 demonstrates that different workers may achieve their optimal outcomes in different stable matchings. However, it is impossible to simultaneously guarantee all workers their OSS with a single matching (even non-stable). Based on this impossibility result, a natural question arises as to whether an allocation exists such that every worker is at least satisfied at a certain level. Formally, given a problem instance and a class of matchings $\mathcal{C}$ , we are interested in the following optimal stable share-ratio (OSS-ratio): + +$$ +R _ {\mathcal {C}} := \min _ {D \in \Delta (\mathcal {C})} \max _ {w \in \mathcal {W}} \frac {\mathbf {U} ^ {*} (w)}{\mathbf {U} _ {D} (w)}, \tag {1} +$$ + +where $\Delta (\mathcal{C})$ is the set of distributions over $\mathcal{C}$ and $U_{D}(w)$ is worker $w$ 's expected utility given a distribution $D$ , i.e., $U_{D}(w) = \mathbb{E}_{\mu \sim D}\left[U(w,\mu (w))\right]$ . When we are constrained to the set of matchings, stable matchings and internally stable matchings, $R_{\mathcal{M}}$ , $R_{\mathcal{S}}$ and $R_{\mathcal{I}}$ are defined accordingly. + +The OSS-ratio adopts a worst-case perspective by taking the maximum over workers, ensuring every worker receives a fair share of their optimal stable utility. Formally, if $\max_U R_{\mathcal{M}} \leq \alpha$ , then every worker $w_i$ is guaranteed at least $\frac{1}{\alpha} U^*(w_i)$ in expectation, regardless of the market's preference structure. The minimum over distributions reflects a central planner's optimization: the distribution represents a rotating schedule (e.g., matchings in the support correspond to daily assignments), and restricted support encodes practical constraints. For instance, limiting support to internally stable matchings ensures no justified envy arises between co-present workers in any schedule realization. + +# 3 Approximation Ratios for Stable Matching with Ties + +In this section, we aim to characterize the scale of the OSS-ratio $R_{\mathcal{C}}$ from the worker's perspective, which allows for ties, while additional findings related to the job side are provided in Appendix J. As a first observation, $S \subset \mathcal{I} \subset \mathcal{M}$ implies $R_{\mathcal{M}} \leq R_{\mathcal{I}} \leq R_{\mathcal{S}}$ , and $R_{\mathcal{S}} \leq N$ , since uniformly selecting a worker and their favored stable matching achieves this bound. + +# 3.1 Lower Bound + +We first prove that the trivial upper bound on $R_{\mathcal{S}}$ is asymptotically tight. + +Theorem 1. There exists an instance, such that for any distribution over stable matchings, one worker only receives a $2 / N$ fraction of their optimal stable share, i.e., $R_{S} \geq \frac{N}{2} = \Omega(N)$ . + +To prove Theorem 1, we construct an instance with $N / 2$ highly-skilled workers and $N / 2$ regular workers, such that every stable matching can satisfy at most one regular worker at a time, proving that $R_{S} \geq N / 2$ . The formal proof is deferred to Appendix B. + +However, a closer look at our instance reveals that all regular workers can be satisfied in a single (non-stable) matching (See Remark 4 in Appendix B). Thus, we turn our attention to distribution over (possibly non-stable) matchings, and the ratio $R_{\mathcal{M}}$ . Theorem 2 shows that if we extend the support of $D$ to include all matchings, i.e., $D \in \Delta(\mathcal{M})$ , the ratio $R_{\mathcal{M}}$ is still lower bounded by $\log N$ . + +Theorem 2. There exists an instance s.t. for any distribution over (possibly non-stable) matchings, one worker only receives a $1 / \Omega (\log N)$ fraction of their optimal stable share, i.e., $R_{\mathcal{M}} = \Omega (\log N)$ . + +To prove Theorem 2, we recursively construct instances with global ranking of jobs over workers, and each worker could be assigned to a job they like, but such that the number of workers grows logarithmically faster than the number of valuable jobs, proving that each worker can only receive a logarithmic fraction of their optimal stable share. The full proof could be found in Appendix B. + +# 3.2 Upper Bound + +We show that the logarithmic ratio obtained in Theorem 2 is asymptotically tight, even if we consider distributions over internally stable matchings. + +Theorem 3. For any problem instance, there exists a distribution $D$ over internally stable matchings s.t. all workers only receive a $1 / \mathcal{O}(\log N)$ fraction of their optimal stable share, i.e., $R_{\mathcal{I}} = \mathcal{O}(\log N)$ . + +We prove Theorem 3 by constructing an offline approximation oracle (Algorithm 1), which generates a uniform distribution over $m$ internally stable matchings $\tilde{\mu}_1, \ldots, \tilde{\mu}_m$ . Each worker $w$ is matched in exactly one matching $\tilde{\mu}_i$ , the key technical insight is that setting $m > \log_2 N + 1$ ensures $U_D(w) = U(w, \tilde{\mu}_i(w)) / m \geq U^*(w) / m$ . To prove this, we construct a directed forest where nodes represent workers who prefer a stable matching over the algorithm's output, and edges capture conflicts where workers compete for the same job copies under different matchings. By exploiting the tree structure and stability constraint, the proof shows that if any worker were worse off, the graph would imply an exponential growth in the number of workers. For more details, please refer to Appendix C. + +Remark 1. The distribution computed by Algorithm 1 is not only "ex-post" internally stable, but also "ex-ante" (externally) stable, in the sense that no worker has justified envy towards any other worker's (randomized) allocation. + +Remark 2. In Algorithm 1, each worker is assigned a job with a probability of $1 / m$ . Under such an allocation, some matchings in the support only assign a subset of jobs. In practice, if some job $a$ is not allocated in a matching $\tilde{\mu}_j$ , but is allocated to worker $w$ in $\tilde{\mu}_i$ , we can give $a$ to $w$ in $\tilde{\mu}_j$ without breaking internal stability of $\tilde{\mu}_j$ . This post-processing is a Pareto improvement of our solution. + +# Algorithm 1 Internally Stable Matchings for Matching Market with Indifference + +Input: $N$ workers, $K$ jobs, Utility matrix $\mathbf{U}$ that encodes the preference of workers over jobs, strict preference list $P_{a}$ of jobs over workers, a positive number $m$ . + +1: For each job $a \in \mathcal{A}$ , duplicate it $m$ times and denote the $i$ -th copy as $a^{(i)}$ . +2: Each replica $a^{(i)}$ shares the same preference $P_{a}$ as the original job $a$ . +3: For each worker $w$ , define an ordering $P_w$ , by sorting jobs $a_k^{(i)}$ by decreasing utility $U(w, a)$ , breaking ties in favour of lower duplication index $i$ , then in favour of lower index $k$ . That is, + +$$ +a _ {k} ^ {(i)} \succ_ {P _ {w}} a _ {\ell} ^ {(j)} \quad \Leftrightarrow \quad \left\{ \begin{array}{l l} U (w, a _ {k}) > U (w, a _ {\ell}) & \text {o r} \\ U (w, a _ {k}) = U (w, a _ {\ell}) \text {a n d} i < j & \text {o r} \\ U (w, a _ {k}) = U (w, a _ {\ell}) \text {a n d} i = j \text {a n d} k < \ell \end{array} \right. +$$ + +4: Run Gale-Shapley algorithm on $P_w$ and $P_a$ to compute a worker-optimal stable matching $\tilde{\mu}$ . + +5: For each $i \in [m]$ , build a matching $\tilde{\mu}_i$ , which matches each job $a$ with $\tilde{\mu}_i(a) \coloneqq \tilde{\mu}(a^{(i)})$ . + +Output: The distribution $D$ which selects each matching $\tilde{\mu}_i$ with probability $1 / m$ . + +Finally, we show that Algorithm 1 cannot be manipulated by a worker who mis-reports her preferences to obtain a distribution that gives them a higher utility, whereas the proof is deferred to Appendix C.3. + +Theorem 4. Algorithm 1 is dominant strategy incentive compatible: for every utility matrices $\mathbf{U}$ and $\mathbf{U}'$ that differ only on the row of worker $w$ , let $D$ and $D'$ be the distributions computed by Algorithm 1, then $\mathbf{U}_D(w) \geq \mathbf{U}_{D'}(w)$ . + +# 4 Robustness and $\epsilon$ -Stability + +In Section 3, we present an asymptotically tight algorithm for approximating the optimal stable share in markets with ties under stability. However, exact stability often proves too rigid for real-world applications where preferences may fluctuate slightly. We therefore introduce $\epsilon$ -stability, which tolerates blocking pairs with utility gains below a threshold $\epsilon$ . This relaxation yields robust matching resilient to preference perturbations while maintaining theoretical guarantees. + +Definition 3 ( $\epsilon$ -Stability). Given $\epsilon \geq 0$ , a matching $\mu$ is $\epsilon$ -stable if there is no $\epsilon$ -blocking pair $(w, a)$ such that $w \succ_{a} \mu(a)$ and $U(w, a) > U(w, \mu(w)) + \epsilon$ . + +The notion of $\epsilon$ -stability is a relaxation of weak stability, where setting $\epsilon = 0$ makes it equivalent to weak stability (Definition 1). In general, $\epsilon$ -stable matching is not unique, and there is not a single $\epsilon$ -stable matching that simultaneously maximizes the utilities for all workers. Therefore, similar to matching markets with ties, we define $S_{\epsilon} := \{\mu : \mu \text{ is an } \epsilon$ -stable matching\}, and we call $a$ a valid $\epsilon$ -stable match of worker $w$ if there exists an $\epsilon$ -stable matching matches $w$ with $a$ , and it is the optimal $\epsilon$ -stable match of worker $w$ if it is the most preferred valid $\epsilon$ -stable match, i.e., there exists a matching $\mu_{\epsilon}^{*} \in S_{\epsilon}$ such that $\mu_{\epsilon}^{*}(w) = a$ and $U(w, \mu_{\epsilon}^{*}(w)) = \max_{\mu \in S_{\epsilon}} U(w, \mu(w))$ . And we say $U(w, \mu_{\epsilon}^{*}(w))$ is the optimal $\epsilon$ -stable share for worker $w$ , denoted as $U_{\epsilon}^{*}(w)$ . + +Algorithm 2 (see Appendix D) generalizes Algorithm 1 with a different workers' preference profiles generation. It outputs a randomized matching that achieves an expected utility within a log $N$ factor of the optimal $\epsilon$ -stable share, plus an $\epsilon$ -additive error. + +Theorem 5. Given any utility matrix $\mathbf{U}$ , parameter $m = \lfloor \log_2 N + 2 \rfloor$ , and the instability tolerance $\epsilon \geq 0$ , Algorithm 2 computes a distribution $D \in \Delta(\mathcal{I})$ , such that $U_D(w) \geq \frac{U_\epsilon^*(w)}{m} - \epsilon, \forall w \in \mathcal{W}$ . + +The proof of Theorem 5 is deferred to Appendix D.2. Interestingly, the distribution $D$ randomizes over internally stable matchings, which do not depend on $\epsilon$ . + +In labor markets, worker preferences are typically estimated with uncertainty via i.i.d. observations, we construct utility uncertainty sets using concentration inequalities. Theorem 6 shows that for any utility matrix in such a set $\mathcal{U}$ , Algorithm 2 produces a random matching guaranteeing each + +worker a logarithmic approximation to their optimal share within $\mathcal{U}$ , where the proof is deferred to Appendix D.4. + +Theorem 6. Given an uncertainty set $\mathcal{U}$ , the optimal stable share within $\mathcal{U}$ is + +$$ +\mathcal {U} ^ {*} (w) := \sup _ {\boldsymbol {U} \in \mathcal {U}} \max _ {\mu \in \mathcal {S} ^ {U}} \boldsymbol {U} (w, \mu (w)), \quad \forall w \in \mathcal {W}. \tag {2} +$$ + +We define the center $\hat{U}$ of the set $\mathcal{U}$ as $\hat{U}(w, a) = \frac{\inf_{U \in \mathcal{U}} U(w, a) + \sup_{U \in \mathcal{U}} U(w, a)}{2}$ , and the uncertainty parameter as $\epsilon = 2 \cdot \sup_{U_1, U_2 \in \mathcal{U}} \|U_1 - U_2\|_{\max}$ . Algorithm 2 with input $\hat{U}$ , $m = \lfloor \log_2 N + 2 \rfloor$ , and $\epsilon$ outputs a distribution $D \in \Delta(\mathcal{I})$ such that $U_D(w) \geq \frac{\mathcal{U}^*(w)}{m} - \epsilon, \forall w \in \mathcal{W}$ . + +Example 2 illustrates an application of Theorem 6 to batch learning problems. + +Example 2 (Batch learning). Suppose that we have a dataset of size $T$ , where each data point $\mathbf{U}$ is a noisy observation of the ground-truth utility matrix $\tilde{\mathbf{U}}$ , i.e., each $\mathbf{U}(i,j)$ is sampled from a 1-sub-Gaussian distribution with mean $\tilde{\mathbf{U}}(i,j)$ . Given a parameter $\delta$ , set $\epsilon = 2\sqrt{\ln(\frac{1}{\delta}) / T}$ , and define the uncertainty set for each entry $(w,a)$ as $\mathcal{U}_{w,a} = \left\{\mathbf{U}(w,a):|\mathbf{U}(w,a) - \hat{\mathbf{U}}(w,a)|\leq \epsilon /2\right\}$ , and $\mathcal{U} = \bigotimes_{(w,a)\in \mathcal{W}\times \mathcal{A}}\mathcal{U}_{w,a}$ , where $\hat{\mathbf{U}}$ is the empirical mean utility matrix computed from the dataset. The OSS within the uncertainty set $\mathcal{U}^* (w)$ could be defined as in Eq.(2). By Lemma 2, we know that with probability $1 - \delta$ , the ground-truth utility matrix $\tilde{\mathbf{U}}\in \mathcal{U}$ , and hence $\tilde{\mathbf{U}}^{*}(w)\leq \mathcal{U}^{*}(w)$ . Therefore, by running Algorithm 2 with the empirical mean utility matrix as input, and set $\epsilon = 2\sqrt{\ln(\frac{1}{\delta}) / T}$ , $m = \lfloor \log_2N + 2\rfloor$ , we have w.p. $1 - \delta$ that the corresponding output distribution $D$ over matchings satisfies $U_{D}(w)\geq \frac{\tilde{\mathbf{U}}^{*}(w)}{[\log_{2}N + 2]} -2\sqrt{\ln(\frac{1}{\delta}) / T}$ for all $w\in \mathcal{W}$ . + +# 5 Bandit Learning in Matching Markets + +Example 2 demonstrates the application of our offline oracle to learning problems. We now transition to an online learning setting, framing the matching market as a multi-player bandit problem to show how the offline results naturally connect learning scenarios both with and without statistical ties. + +In online marketplaces, companies can evaluate workers through interviews, but typically lack prior knowledge of worker preferences over jobs. Still, by leveraging repeated matching opportunities, these preferences can be learned through ex-post evaluations. Recent work models this as a multi-armed bandit (MAB) problem [42, 43, 8, 37], where workers ("players") and jobs ("arms") interact over $T$ rounds. Each round, the company outputs a matching $\mu_t$ assigning jobs to workers and observes 1-subgaussian rewards $X_i(t)$ for matched pairs $(w_i, \mu_t(w_i))$ with mean $U(w_i, \mu_t(w_i)) \in [0,1]$ . Following bandit matching literature, we assume $N \leq K$ (more jobs than workers) to ensure matching feasibility. If $N > K$ , we can extend the problem by adding zero-utility jobs or randomly assigning unmatched workers. + +The company seeks to learn the worker-optimal stable matching $\mu^{*}(w_{i})$ through interactions. Specifically, it aims to minimize the worker-optimal stable regret for each $w_{i} \in \mathcal{W}$ , defined as the cumulative reward difference between being matched with $\mu_{i}^{*}$ and that $w_{i}$ receives over $T$ rounds: + +$$ +\operatorname {R e g} _ {i} (T) = T \cdot \mathbf {U} ^ {*} \left(w _ {i}\right) - \mathbb {E} \left[ \sum_ {t = 1} ^ {T} X _ {i} (t) \right]. \tag {3} +$$ + +The expectation is taken over the randomness of the received reward and the allocation strategy. + +Prior work on minimizing worker-optimal stable regret focuses exclusively on tie-free markets [42, 8, 37], rendering their results inapplicable when preferences contain ties. Crucially, existing regret bounds scale as $1 / \Delta^2$ , where $\Delta$ is the minimum utility gap across all workers $w$ and jobs $a$ , i.e., $\Delta = \min_w\min_{a,a'}|U(w,a) - U(w,a')|^2$ . As shown in Example 2 in [42], this dependence is fundamental - achieving sublinear regret requires $\Delta = \omega (1 / \sqrt{T})$ . + +When the benchmark is unachievable (computationally or statistically), prior work adopts $\alpha$ -approximation regret to ensure sublinear regret relative to an $\alpha$ -fraction of the benchmark [31, 54, 14]. + +In our setting, since ties prevent all workers from simultaneously achieving their optimal stable share, we assume access to an offline oracle that, given utility matrix $\mathbf{U}$ , outputs a randomized matching guaranteeing each worker at least an $1 / \alpha$ of $\mathbf{U}^{*}(w)$ in expectation, with additional error $\epsilon$ . Formally, + +Definition 4 (( $\alpha$ , $\epsilon$ -Approximation Oracle). An $(\alpha, \epsilon)$ -approximation oracle takes a rectangular uncertainty set $\mathcal{U}$ with width $\epsilon$ as input and returns a (randomized) matching $\tilde{\mu}$ satisfying: $\mathbb{E}[U_{\tilde{\mu}}(w)] \geq \alpha^{\mathcal{U}}(w) \cdot \mathcal{U}^{*}(w) - \epsilon$ for every worker $w$ , where $\alpha^{\mathcal{U}} \in (0,1]^N$ is a worker-specific approximation ratio vector (often simplified to $\alpha$ ). If $\alpha^{\mathcal{U}}(w) = \alpha$ is uniform across workers and independent of $\mathcal{U}$ , we call it an $(\alpha, \epsilon)$ -approximation oracle, + +For example, Algorithm 2 guarantees that for any input utility matrix $U$ , $\alpha^U(w) \geq 1 / \lfloor \log_2 N + 2 \rfloor$ . With ties, our regret metric should not compare against the OSS each time, but against an $\alpha$ -fraction of the optimal stable share, since the offline oracle can only guarantee this fraction in expectation: + +$$ +\operatorname {R e g} _ {i} ^ {\alpha} (T) = \alpha T \cdot \mathbf {U} ^ {*} \left(w _ {i}\right) - \mathbb {E} \left[ \sum_ {t = 1} ^ {T} X _ {i} (t) \right], \tag {4} +$$ + +where $\alpha \in (0,1]$ is the approximation ratio given by the offline oracle. When we want to emphasize that the observations $X(t)$ come from a distribution $\pmb{\nu}$ , we write $Reg_{i}(T;\pmb{\nu})$ and $Reg_{i}^{\alpha}(T;\pmb{\nu})$ . + +For markets without ties, [37] achieves a stable regret of $\mathcal{O}(K\ln T / \Delta^2)$ , matching the $\Omega (N\ln T / \Delta^2)$ lower bound [52] in $T$ and $\Delta$ . We seek a best-of-both-worlds guarantee, i.e., an algorithm that attains $Reg_{i}(T) = \mathcal{O}(\ln T / \Delta^{2})$ when $\Delta = \omega (1 / \sqrt{T})$ , and $Reg_{i}^{\alpha}(T) = o(T)$ when $\Delta = \mathcal{O}(1 / \sqrt{T})$ . + +# 5.1 Algorithm: Explore-then-Choose-Oracle + +We present our algorithm, Explore-then-Choose-Oracle (ETCO, Algorithm 3 in Appendix E), and summarize it here. The algorithm consists of two phases. In each round of the exploration phase, the company allocates a job to every worker in a round-robin way to estimate their utilities accurately. In the second phase, the company checks for plausible ties in utilities. If none exists, it computes a matching using GS algorithm; otherwise, it uses the approximation oracle. In subsequent rounds, jobs are allocated based on the chosen oracle's output. + +In the exploration phase, the company allocates jobs to workers in a round-robin way, according to the index of the workers. In this way, every $K$ rounds, each worker is matched to every job exactly once. The maximal number of exploration rounds is bounded by a parameter $T_{0}$ . After each allocation, based on the observation, we update the estimated utility $\hat{U}(i,\mu_t(i)) = \frac{\hat{U}(i,\mu_t(i))\cdot T_{i,\mu_t(i)} + X_{i,\mu_t(i)}(t)}{T_{i,\mu_t(i)} + 1}$ , and the observation count of worker $w_{i}$ and job $\mu_t(i)$ as $T_{i,\mu_t(i)} = T_{i,\mu_t(i)} + 1$ . The company also builds a confidence set for each utility estimate, ensuring the true expected utility is included with high probability. Particularly, the confidence interval (CI) for worker $w_{i}$ 's preference utility over job $a_{j}$ is $[LCB_{i,j},UCB_{i,j}]$ , with the upper and lower confidence bounds defined as + +$$ +U C B _ {i, j} = \hat {\boldsymbol {U}} (i, j) + \sqrt {\frac {6 \ln T}{\operatorname* {m a x} \left\{T _ {i , j} , 1 \right\}}}, \quad L C B _ {i, j} = \hat {\boldsymbol {U}} (i, j) - \sqrt {\frac {6 \ln T}{\operatorname* {m a x} \left\{T _ {i , j} , 1 \right\}}}. \tag {5} +$$ + +When confidence sets for jobs $a_{j}$ and $a_{j'}$ are disjoint ( $LCB_{i,j} > UCB_{i,j'}$ or vice versa), we can determine worker $w_{i}$ 's strict preference between them. If all top- $N$ job CIs for $w_{i}$ become disjoint, we recover the true preference with high probability. If this occurs for all workers before the exploration phase $T_{0}$ ends, we switch to the Gale-Shapley oracle for exploitation, as no top- $N$ ties exist w.h.p. Otherwise, remaining CI overlaps indicate potential ties, triggering our approximation oracle instead. + +# 5.2 Theoretical Analysis + +Before stating the regret guarantee for ETCO algorithm, we first give a formal definition of the minimum preference gap, which measures the hardness of the learning problem. + +Definition 5 (Minimum Preference Gap). For each worker $w_i$ and job $a_j \neq a_{j'}$ , let $\Delta_{i,j,j'} = |U(i,j) - U(i,j')|$ be the preference gap for $w_i$ between $a_j$ and $a_{j'}$ . Let $r_i$ be the preference ranking of worker $w_i$ and $r_{i,k}$ be the $k$ -th preferred job in $w_i$ 's ranking for $k \in [K]$ . Define $\Delta_{\min} = \min_{i \in [N]; k \in [N]} \Delta_{i,r_{i,k},r_{i,k+1}}$ as the minimum preference gap among all workers and their first $(N+1)$ -ranked jobs. + +Next, we present upper bounds for the worker-optimal stable regret for each worker when using ETCO. + +Theorem 7 (Upper Bound). Following the ETCO algorithm with exploration phase of length $T_0$ and an $\left(\alpha, 2\sqrt{\frac{6K\ln T}{T_0}}\right)$ -approximation oracle, for $w_i \in \mathcal{W}$ , we have that + +$$ +\operatorname {R e g} _ {i} (T) = \mathcal {O} \left(\frac {K \ln T}{\Delta_ {\min } ^ {2}}\right) \quad i f \quad \Delta_ {\min } > \sqrt {\frac {9 6 K \ln T}{T _ {0}}} = \Omega \left(\sqrt {\frac {K \ln T}{T _ {0}}}\right), \tag {6} +$$ + +$$ +R e g _ {i} ^ {\alpha} (T) \leq 2 \alpha T _ {0} + \mathcal {O} \left(T \sqrt {\frac {K \ln T}{T _ {0}}}\right) \quad i f \quad \Delta_ {\min } \leq \sqrt {\frac {9 6 K \ln T}{T _ {0}}} = \mathcal {O} \left(\sqrt {\frac {K \ln T}{T _ {0}}}\right). \tag {7} +$$ + +See Appendix G for the proof. Our bound exhibits two regimes: (1) large- $\Delta$ regime: when $\Delta_{\mathrm{min}}$ is large, the exploration phase learns the top- $(N + 1)$ job preferences w.h.p. before $T_0$ , enabling exact worker-optimal stability via Gale-Shapley in exploitation. This reduces to ETGS [37] under centralization; (2) small- $\Delta$ / tied regime: for small $\Delta_{\mathrm{min}}$ or exact ties, worker-optimal stability is unattainable; instead, implementing an approximation oracle guarantees an $\alpha$ -approximation regret sublinear in $T$ . + +When $\Delta_{\mathrm{min}}$ is sufficiently large, our upper bound matches the $\Omega (N\ln T / \Delta_{\mathrm{min}}^2)$ lower bound [52] for serial dictatorship markets (where all jobs share identical preferences). This tightness, however, comes at a fundamental trade-off: Theorem 8 shows that extending sublinear regret guarantees to wider ranges of $\Delta_{\mathrm{min}}$ unavoidably worsens approximation regret in small- or no-gap regimes. + +Prior to presenting our trade-off lower bound, we formally define two key concepts. The Pareto-optimal stable matching set $S_{opt}^{U}$ , comprising stable matchings where no worker's utility can be strictly improved without harming another worker, and the relevant preference utility gap $\Delta_{rel}$ , representing the maximum utility perturbation that preserves $S_{opt}^{U}$ . + +Definition 6 (Pareto-optimal Stable Matching Set). Given a utility matrix $\mathbf{U}$ , the worker-optimal Pareto-optimal stable matching set $S_{opt}^{\mathbf{U}}$ is the set of all matchings $\mu$ such that: 1) $\mu$ is stable; 2) If there exists a stable matching $\mu'$ and a worker $w$ such that $\mathbf{U}(w, \mu'(w)) > \mathbf{U}(w, \mu(w))$ , then for some $w' \neq w$ , it holds that $\mathbf{U}(w', \mu'(w')) < \mathbf{U}(w', \mu(w'))$ . + +Definition 7 (Relevant Utility Gap). Given a utility matrix $\mathbf{U}$ , the relevant preference gap $\Delta_{rel}$ is + +$$ +\Delta_ {r e l} := \inf \left\{\varepsilon : \exists i \in [ N ], j \in [ K ], \tilde {\boldsymbol {U}} (i, j) \in [ \boldsymbol {U} (i, j) - \varepsilon , \boldsymbol {U} (i, j) + \varepsilon ] s. t. \mathcal {S} _ {o p t} ^ {\boldsymbol {U}} \neq \mathcal {S} _ {o p t} ^ {\tilde {\boldsymbol {U}}} \right\}. \tag {8} +$$ + +By definition, $\Delta_{\mathrm{rel}} \geq 0$ . When $S_{opt}^{U}$ contains multiple matchings, $\Delta_{\mathrm{rel}} = 0$ since any perturbation acts as a tie-breaker, eliminating at least one matching from $S_{opt}^{U}$ (by the uniqueness of worker-optimal stable matching in tie-free markets). Furthermore, $\Delta_{\mathrm{rel}} \geq \Delta_{\mathrm{min}}$ because perturbations smaller than $\Delta_{\mathrm{min}}$ cannot alter any worker's top- $N$ preferences or the worker-optimal matching3. + +Theorem 8 (Trade-off between Regret and Approximation Regret). Let $\delta \in (0, \frac{1}{2})$ and fix $N = K = 4$ . Consider the class of instances with a large relevant utility gap, denoted as $\mathcal{E}_{\ell}(T)$ , i.e., for any instance $\nu \in \mathcal{E}_{\ell}(T)$ , we have $\Delta_{rel}^{\nu} \geq c T^{-1/2 + \delta}$ for some absolute $c > 0$ . Assume that an algorithm $\pi$ guarantees sublinear regret for all workers, for all $\nu \in \mathcal{E}_{\ell}(T)$ . Then there exists an instance such that this algorithm suffers $\Omega(T^{1 - 2\delta})$ approximation regret for some worker when $\Delta_{rel} = 0$ w.r.t the best approximation ratio $\alpha^{*}$ for this instance, i.e., + +$$ +\begin{array}{l} I f \forall w _ {i} \in \mathcal {W}, \quad \lim _ {T \to \infty} \sup _ {\nu \in \mathcal {E} _ {\ell} (T)} \frac {\sup _ {\nu \in \mathcal {E} _ {\ell} (T)} R e g _ {i} (T ; \nu)}{T} = 0, \\ \Longrightarrow \exists w _ {i} \in \mathcal {W}, \nu^ {\prime} \quad s. t. \quad \Delta_ {r e l} ^ {\nu^ {\prime}} = 0, a n d R e g _ {i} ^ {\alpha^ {*} (w _ {i})} (T; \nu^ {\prime}) = \Omega \left(T ^ {1 - 2 \delta}\right), \tag {9} \\ \end{array} +$$ + +where $\alpha^{*}(w_{i}) = \max \left\{\alpha (w_{i}):\alpha (w)\geq 1 / R_{\mathcal{M}}^{U},\forall w\in \mathcal{W}\right\}$ , for any $w_{i}\in \mathcal{W}$ , and $R_{\mathcal{M}}^U$ is the OSS-ratio on matchings with a given utility matrix $\pmb{U}$ . + +The proof appears in Appendix H. We construct two serial dictatorship instances with 4 workers and 4 jobs each, whose utility matrices differ in only one entry (for the highest-priority worker), yielding $\Delta_{\mathrm{rel}} = 0$ . The first instance evaluates $\alpha^{*}$ -approximation regret, while the second analyzes standard stable regret. Crucially, this single entry difference completely alters the benchmark utilities for the other three workers. Thus, one of the two cases happens: (1) under-sampling: without + +enough samples of the differing entry, at least one worker incurs linear approximation regret; (2) over-sampling: After $T_0$ samples of the differing entry, at least one of the remaining workers suffers $\Omega(T_0)$ approximation regret. + +Theorem 8 establishes an inherent trade-off between large-gap and small / no-gap regimes: as $\delta \rightarrow 0$ , sublinear regret in the former necessitates linear approximation regret in the latter. Consequently, the exploration length $T_{0}$ of ETCO algorithm critically determines regime-specific performance. We provide two $T_{0}$ choices and their corresponding regret bounds. + +Corollary 1. Following ETCO algorithm with $T_0 = T^{2/3}$ ( $K \ln T$ ) for $w_i \in \mathcal{W}$ , we have + +$$ +R e g _ {i} (T) = \mathcal {O} \left(\frac {K \ln T}{\Delta_ {\min} ^ {2}}\right) i f \Delta_ {\min } = \tilde {\Omega} \left(T ^ {- \frac {1}{3}}\right); R e g _ {i} ^ {\alpha} (T) = \mathcal {O} \left((K \ln T) ^ {\frac {1}{3}} T ^ {\frac {2}{3}}\right) i f \Delta_ {\min } = \tilde {\mathcal {O}} \left(T ^ {- \frac {1}{3}}\right). +$$ + +Choosing $T_0 = T^{2/3} (K \ln T)^{1/3}$ yields the optimal approximation regret upper bound in the small gap regime when implementing explore-then-commit type algorithms. However, this choice is not satisfiable when $\Delta_{\min} \in [\tilde{\Omega} (T^{-1/2}), \tilde{\mathcal{O}} (T^{-1/3})]$ , since setting $T_0$ as such cannot guarantee detection of instances when $\Delta_{\min}$ falls in this intermediate regime. For these cases, we must resort to the approximation oracle during exploitation. Crucially, since the oracle's solution differs by a constant factor from the Gale-Shapley optimal, each exploitation round incurs constant regret when measured against Eq.(3), resulting in an overall linear regret. + +Corollary 2. Following ETCO algorithm with $T_0 = \frac{T}{2\ln T}$ , for $w_i \in \mathcal{W}$ , we have + +$$ +R e g _ {i} (T) = \mathcal {O} \left(\frac {K \ln T}{\Delta_ {\min} ^ {2}}\right) i f \Delta_ {\min } = \tilde {\Omega} \left(T ^ {- \frac {1}{2}}\right); R e g _ {i} ^ {\alpha - \frac {1}{\ln T}} (T) = \mathcal {O} \left(\sqrt {K T} \ln T\right) i f \Delta_ {\min } = \tilde {\mathcal {O}} \left(T ^ {- \frac {1}{2}}\right). +$$ + +Choosing $T_0 = T / (2\ln T)$ yields the optimal regret that matches the lower bound for any $\Delta_{\mathrm{min}} = \tilde{\Omega}\left(T^{-1/2}\right)$ . However, for $\Delta_{\mathrm{min}} = \tilde{\mathcal{O}}\left(T^{-1/2}\right)$ , we can only guarantee sublinear approximation regret with an approximation ratio of $\alpha - 1 / \ln T$ even when using an offline $\alpha$ -approximation oracle. + +Remark 3. The approximation regret lower bound in Theorem 8 is both non-trivial and potentially of independent interest for bandit theory. While combinatorial bandits typically use approximation regret to circumvent computational limits (with statistical lower bounds focusing on 1-regret [15, 39, 48]), our result reveals a fundamental distinction: in matching markets, this approximation factor persists even given unlimited computational resources. + +# 6 Conclusion + +In this paper, we study stable matching with one-sided indifference, modeled as a company assigning workers to jobs. Using a utility matrix to encode workers' potentially tied preferences over jobs, we define the optimal stable share (OSS) for each worker as the maximum utility achievable in any stable matching. To address fairness, we introduce the OSS-ratio, quantifying the fraction of the OSS a worker obtains under random matchings. We first analyze distributions over stable matchings, showing that a linear approximation to the OSS is trivial and asymptotically tight. For general matchings, we prove that no better than logarithmic approximation is possible. To achieve this bound, we propose a polynomial-time algorithm computing a distribution over internally stable matchings, which is asymptotically optimal in OSS ratio and dominant strategy incentive-compatible. Next, we extend our framework to settings where the utility matrix is uncertain but lies within a given uncertainty set. By incorporating $\epsilon$ -stable matchings and relating them to perturbations of the utility matrix, we derive a logarithmic approximation with an additive $\epsilon$ error, matching the deterministic case. Finally, we explore online learning, where existing stable regret frameworks fail to handle tied preferences. Leveraging the OSS-ratio, we define $\alpha$ -approximation stable regret and provide an algorithm whose upper bound matches the lower bound in the no-tied case. We further derive approximation regret bounds for small or no utility gaps and establish a fundamental trade-off between regret types, highlighting the need for careful exploration stopping time decisions. + +Our work establishes the first instance-independent worker-optimal stable regret bound in bandit learning for matching markets, achieved through centralized job allocation. However, real-world marketplaces typically operate in decentralized settings where workers cannot directly coordinate. While the Gale-Shapley algorithm naturally decentralizes, extending our approximation guarantees to decentralized bandit learning remains an open challenge, which is an important direction for future research. Furthermore, exploring the application of our proposed algorithms to real-world datasets would be a valuable next step, as it would help address the practical challenge of stable matching when ties exist in preference rankings. + +# Acknowledgements + +This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101034255. Shiyun Lin acknowledges the financial support from the China Scholarship Council (Grant No.202306010152). Vianney Perchet's research was supported in part by the French National Research Agency (ANR) in the framework of the PEPR IA FOUNDRY project (ANR-23-PEIA-0003) and through the grant DOOM ANR-23-CE23-0002. It was also funded by the European Union (ERC, Ocean, 101071601). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Further Related Work + +Fractional Matchings Aside from integral matchings, fractional matchings have also attracted research interests due to their practical implications. For instance, in our running example, consider a time-sharing scenario [51], where each worker could spend five days a week at work. An integral matching requires every worker to work full-time on a single job, while fractional matchings allow them to switch among different jobs, making it natural in such situations. By the well-known Birkhoff-von-Neumann (BvN) theorem [10, 58], a fractional matching could be written as a convex combination of several integral matchings. + +In the context of stable matching, fractional matching has also been studied. Considering purely ordinal preferences, several notions of stability have been proposed, such as strong stability [51], ex-post stability [51], and fractional stability [57]. In these works, the stable matching problem is formulated as a linear program, Teo and Sethuraman [55] showed that any fractional solution in the stable matching polytope is a convex combination of integral stable matchings. On the other hand, concerning purely cardinal preferences, Anshelevich et al. [4] proposes the notions of stability and $\varepsilon$ -stability, while Caragiannis et al. [12] shows that the set of stable fractional matchings that satisfies the notion can be non-convex. + +In this paper, we consider a two-sided market where the worker side has cardinal preferences while the job side has ordinal preferences. We do not concern the notions of fractional stable matchings, instead, we focus on finding a distribution over integral matchings such that it is fair in the sense that every worker could receive a certain fraction of its optimal stable share in expectation. + +Fair Division Fair division is the problem of dividing a set of items among several people in a fair manner. Steinhaus [53] pioneers this line of research and defines a share-based notion, i.e., proportionality, where each player gets a $1/N$ fraction of all items. Foley [20] and Varian [56] define envy-freeness, where no player prefers the bundle allocated to another player, and this notion is later generalized by Weller [59]. In two-sided matching markets, a stable matching eliminates justified envy [1]. Regarding the problem of sharing indivisible goods, share-based guarantees such as MMS [11] and envy-based guarantees such as EF1 [41, 11] or EFX [13] are proposed. Recent works have studied best-of-both-world fairness [6, 7, 19], providing random allocation with fairness guarantees both in expectation and for every realization. + +Approximation Regret in Bandit Learning In combinatorial bandit problems, approximation regret is often considered instead of the standard regret [31, 54, 22, 14, 47, 49]. The reason mainly lies in the complex reward structure and the computational intractability of the problem, i.e., rewards are often dependent on the combination of actions, leading to an exponentially large action space, which makes it computationally prohibitive to find the exact solution. + +Besides computational intractability, there is another more fundamental reason for using the approximation regret framework in this paper. The stable regret is defined for each worker, which means the company aims to solve a multi-objective optimization problem while it couldn't satisfy everyone simultaneously. Consequently, the approximation regret serves as a compromise between fairness and efficiency. + +# B Lower bounds on OSS-ratio + +# B.1 Lower Bound for Distributions over Weakly Stable Matchings + +The proof idea of Theorem 1 could be illustrated through Figure 1. + +Proof of Theorem 1. Assume that $N$ is even, let $\mathcal{W} = \{w_1, w_2, \dots, w_N\}$ and $\mathcal{A} = \{a_1, a_2, \dots, a_K\}$ with $K = \frac{N}{2} + 1$ and $w_1 \succ w_2 \succ \dots \succ w_N$ for all the jobs. The utility matrix that encodes the preference of workers over jobs is as follows: + +![](images/42372cabc4232ffb169d5a203546bff18e156b93302f43a143bc614acd09d6c6.jpg) +Figure 1: Lower bound on $R_{S}$ . All jobs have the same ordering over workers, from top to bottom. Any stable matching can be obtained by letting the first worker pick a job, then the second, etc. Hence, each stable matching contains at most one blue edge. + +$$ +\boldsymbol {U} = \left( \begin{array}{c c c c c} 1 & 0 & \dots & 0 & 1 \\ 0 & 1 & \dots & 0 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & 1 \\ 1 & 0 & \dots & 0 & 0 \\ 0 & 1 & \dots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & 0 \end{array} \right) \left\{ \begin{array}{l} \frac {N}{2} \\ \frac {N}{2} \end{array} \right.. +$$ + +In any stable matching, every worker $w_{i}$ in $\{w_{1}, w_{2}, \dots, w_{N/2}\}$ must be assigned to $a_{i}$ or $a_{\frac{N}{2} + 1}$ , leading to a utility of 1 for them. Without loss of generality, let $\mu_{i}$ be the matching such that $\mu(w_{i}) = a_{\frac{N}{2} + 1}$ . Then in $\mu_{i}$ , only worker $w_{\frac{N}{2} + i}$ would receive a utility of 1, by matching it to job $a_{i}$ , while all workers in $\{w_{N/2 + 1}, \dots, w_{N/2 + i - 1}, w_{N/2 + i + 1}, \dots, w_{N}\}$ would be unmatched and receive a utility of 0. Indeed, for any $j \neq i$ , the unique optimal match of worker $w_{N/2 + j}$ is already taken by worker $w_{j}$ . + +For every stable matching, at most one of the workers in $\{w_{N / 2 + 1}, w_{N / 2 + 2}, \dots, w_N\}$ could be assigned to their optimal match. Since there are $N / 2$ such workers, then for any distribution $D$ , there must be at least one of the workers for which the probability to be optimally matched is smaller than $2 / N$ , for this worker, it holds that $\frac{\mathbf{U}^{*}(w)}{\mathbf{U}_{D}(w)} \geq \frac{N}{2}$ , which implies $R_{\mathcal{S}} \geq N / 2$ . + +![](images/1e88454ba35a2809cc1e74ced3565c9e1b3b2c9f227c8abb7d1405f57c19eae6.jpg) + +Remark 4. Theorem 1 shows that if we only consider random allocations of stable matchings, then in the worst case, workers could only expect $\mathcal{O}(1/N)$ profit share compared to their benchmark. On the other hand, define + +$$ +\mu_ {1} = \left\{\left(w _ {i}, a _ {i}\right): i \in \{1, 2, \dots , N / 2 \} \right\}, +$$ + +$$ +\mu_ {2} = \left\{\left(w _ {i + N / 2}, a _ {i}\right): i \in \{1, 2, \dots , N / 2 \} \right\}. +$$ + +Here, $\mu_{1}$ is a stable matching while $\mu_{2}$ is non-stable. We construct a distribution $D$ as follows: + +$$ +\mathbb {P} (D = \mu_ {1}) = \frac {1}{2}, \quad \mathbb {P} (D = \mu_ {2}) = \frac {1}{2}, +$$ + +then all workers have $\frac{U^{*}(w)}{U_{D}(w)} = 2$ . This result implies that if we consider possibly non-stable matchings for the support of $D$ , there is space for improvement on the OSS-ratio. + +# B.2 Lower Bound for Distributions over Matchings + +The proof idea of Theorem 2 could be illustrated through Figure 2. + +![](images/bfc7485c4b72bc6590aada30e0903f4f5db6facceffd1488f9b232320283560d.jpg) +Figure 2: Lower bound on $R_{\mathcal{M}}$ . In each example, left nodes represent workers while right nodes represent jobs. If there is an edge connecting a left node $w$ and a right node $a$ , we have $U(w, a) = 1$ , and $U(w, a) = 0$ otherwise. All the right nodes without edges connecting to them are hidden from the graph. + +Proof of Theorem 2. Consider the following sequence of problem instances, as depicted in Figure 2. In each bipartite graph, left nodes represent the set of workers while the right nodes represent the set of jobs. Jobs share a global preference over the workers, with the topmost node being the most preferred and the preference decreasing from top to bottom. From the worker side, the utility matrix is binary. Specifically, for a worker-job pair $(w,a)$ , $U(w,a) = 1$ if $(w,a)$ is connected, while $U(w,a) = 0$ otherwise. For example, $I_{1}$ is the graph representation of the problem instance of Example 1. + +Instance $I_{n}$ is constructed recursively. Given $I_{n-1}$ , we first duplicate this instance, denoted the replications as upper class and lower class, respectively. Take $K_{n-1}$ as the number of right nodes and $N_{n-1}$ as the number of left nodes in $I_{n-1}$ , and denote these nodes as $\left\{a_{1}^{u}, a_{2}^{u}, \dots, a_{K_{n-1}}^{u}\right\}$ , $\left\{w_{1}^{u}, w_{2}^{u}, \dots, w_{N_{n-1}}^{u}\right\}$ and $\left\{a_{1}^{\ell}, a_{2}^{\ell}, \dots, a_{K_{n-1}}^{\ell}\right\}$ , $\left\{w_{1}^{\ell}, w_{2}^{\ell}, \dots, w_{N_{n-1}}^{\ell}\right\}$ for the upper and lower classes, respectively. Then, we introduce $K_{n-1}$ prioritized workers in $I_{n}$ , who are uniformly more preferred by the jobs than the workers in the upper and lower classes. In particular, denote the set of prioritized workers as $\left\{w_{1}, w_{2}, \dots, w_{K_{n-1}}\right\}$ , we have + +$$ +w _ {1} \succ w _ {2} \succ \dots \succ w _ {K _ {n - 1}} \succ w _ {1} ^ {u} \succ w _ {2} ^ {u} \succ \dots \succ w _ {N _ {n - 1}} ^ {u} \succ w _ {1} ^ {\ell} \succ w _ {2} ^ {\ell} \succ \dots \succ w _ {N _ {n - 1}} ^ {\ell}. +$$ + +And for each $w_{i}$ , we have $\pmb{U}(w_{i},a_{i}^{u}) = \pmb{U}(w_{i},a_{i}^{\ell}) = 1$ , and $\pmb{U}(w_{i},a) = 0$ otherwise. We first prove by induction that the optimal-stable-share is $\pmb{U}^{*}(w) = 1$ for any worker $w$ . For $I_0$ , the unique matching is stable. Suppose that for any worker $w$ in $I_{n - 1}$ , there exists at least one stable matching $\mu$ such that $\pmb{U}(w,\mu (w)) = 1$ . Then in $I_{n}$ , all the prioritized workers could be matched in any stable matching. Furthermore, as long as they simultaneously choose to be matched to the jobs in the same class, the other class is free, and hence by induction assumption, every worker in that class gets a chance to be matched in at least one stable matching. Therefore, by breaking ties for the upper (lower) class, all workers in the lower (upper) class can be matched. + +On the other hand, given that $\mathbf{U}^{*}(w) = 1$ for any $w$ , the ratio $R_{\mathcal{M}}$ is equal to $\min_D \max_w \frac{1}{\mathbf{U}_D(w)}$ for these instances. Then, proving a lower bound on this quantity is equivalent to establishing an upper bound on $\max_D \min_w \mathbf{U}_D(w)$ . In particular, we have $\max_D \min_w \mathbf{U}_D(w) \leq \max_D \frac{\sum_w \mathbf{U}_D(w)}{N}$ , where $N$ is the number of left nodes. Now notice that $\sum_w \mathbf{U}_D(w)$ is the expected size of the matching under distribution $D$ , since the utility is 1 when a worker is matched and 0 otherwise. We have $\sum_w \mathbf{U}_D(w) \leq K$ from the fact that the size of any matching is bounded by $K$ . Combining the above derivation, we have + +$$ +R _ {\mathcal {M}} \geq \frac {N}{K}. +$$ + +Finally, in instance $I_{n}$ , by the recursive construction, we have + +$$ +K _ {n} = 2 \cdot K _ {n - 1}, +$$ + +$$ +N _ {n} = 2 \cdot N _ {n - 1} + K _ {n - 1}. +$$ + +Solving the recursive equation with the initial condition $K_{0} = N_{0} = 1$ , we know that there are $2^{n}$ right nodes and $N = (n + 2)2^{n - 1}$ left nodes in $I_{n}$ , which implies that $R_{\mathcal{M}} \geq n / 2 + 1$ . We rewrite $N = (n + 2)2^{n - 1}$ to obtain $2^{n} = 2N / (n + 2)$ and we deduce that + +$$ +N / n \leq 2 ^ {n} \leq 2 N. +$$ + +Taking a logarithm in the inequalities, we have + +$$ +\log_ {2} N - \log_ {2} n \leq n \leq 1 + \log_ {2} N. +$$ + +And thus + +$$ +n \geq \log_ {2} N - \log_ {2} n \geq \log_ {2} N - \log_ {2} (1 + \log_ {2} N). +$$ + +Therefore, $n = \Omega (\log N)$ and hence $R_{\mathcal{M}}$ is $\Omega (\log N)$ . + +![](images/b3ce0aa93a3759a5e21c6fad3ea3eb91b7c9a849a2c19bed199b8014e734c726.jpg) + +# C Upper Bound on OSS-ratio + +# C.1 Procedure Illustration of Algorithm 1 + +We use Example 3 to illustrate the procedure stated in Algorithm 1. + +Example 3. Let $\mathcal{W} = \{w_1, w_2, w_3\}$ , $\mathcal{A} = \{a_1, a_2, a_3\}$ . We consider the following preference list $P_a$ of jobs over workers, and utility matrix $\mathbf{U}$ that encodes the preference of workers over jobs: + +$$ +a _ {1}: w _ {2} \succ w _ {1} \succ w _ {3}, +$$ + +$$ +a _ {2} \quad : w _ {1} \succ w _ {3} \succ w _ {2}, +$$ + +$$ +a _ {3} \quad : w _ {1} \succ w _ {2} \succ w _ {3}. +$$ + +$$ +\boldsymbol {U} = \left[ \begin{array}{c c c} 1 & 1 & 0 \\ 0. 5 & 0. 1 & 0. 1 \\ 0 & 0. 8 & 0 \end{array} \right]. +$$ + +If $m = 2$ , the preference profile $P_w$ generated from the algorithm is + +$$ +w _ {1}: a _ {1} ^ {(1)} \succ a _ {2} ^ {(1)} \succ a _ {1} ^ {(2)} \succ a _ {2} ^ {(2)} \succ a _ {3} ^ {(1)} \succ a _ {3} ^ {(2)}, +$$ + +$$ +w _ {2}: a _ {1} ^ {(1)} \succ a _ {1} ^ {(2)} \succ a _ {2} ^ {(1)} \succ a _ {3} ^ {(1)} \succ a _ {2} ^ {(2)} \succ a _ {3} ^ {(2)}, +$$ + +$$ +w _ {3}: a _ {2} ^ {(1)} \succ a _ {2} ^ {(2)} \succ a _ {1} ^ {(1)} \succ a _ {3} ^ {(1)} \succ a _ {1} ^ {(2)} \succ a _ {3} ^ {(2)}. +$$ + +Running Gale-Shapley algorithm on $P_w$ and $P_a$ , the worker-optimal stable matching would be $\tilde{\mu} = \{(w_1, a_2^{(1)}), (w_2, a_1^{(1)}), (w_3, a_2^{(2)})\}$ , and we can recover two internally stable matchings from $\tilde{\mu}$ , i.e., $\tilde{\mu}_1 = \{(w_1, a_2), (w_2, a_1)\}$ and $\tilde{\mu}_2 = \{(w_3, a_2)\}$ . + +# C.2 Proof of Theorem 3 + +In Algorithm 1, each worker $w$ is matched in exactly one matching $\tilde{\mu}_i$ , where we call $i$ the index of $w$ , denoted $\mathrm{index}(w)$ . In other words, the index of a worker is the index of the job she receives, that is, $\mathrm{index}(w) = i$ if worker $w$ receives $a_j^{(i)}$ for some $j$ . + +Definition 8. Given a problem instance $(\mathbf{U}, P_{a})$ , run Algorithm 1 with duplication number $m$ to generate the output distribution $D$ . Then, for any stable matching $\mu$ with respect to $(\mathbf{U}, P_{a})$ , we define a graph $G_{\mu} = (V_{\mu}, E_{\mu})$ where + +$$ +V _ {\mu} := \{w \in \mathcal {W}: U (w, \mu (w)) \geq m \cdot U _ {D} (w) \}, +$$ + +$$ +E _ {\mu} := \left\{\left(w, w ^ {\prime}\right) \in V _ {\mu} ^ {2}: \mu (w) = \tilde {\mu} _ {j} \left(w ^ {\prime}\right) \text {w h e r e} j = \operatorname {i n d e x} \left(w ^ {\prime}\right) < \operatorname {i n d e x} (w) \right\}. +$$ + +Informally, $G_{\mu}$ is the graph of workers who (weakly) prefer $\mu$ to their match in distribution $D$ , where an edge $(w, w')$ means that $w'$ received a job that $w$ would have liked. Next, we show properties on the graph $G_{\mu}$ , which we illustrate in Figure 3. + +Proposition 1. For any stable matching $\mu$ , the following holds + +- $G_{\mu}$ is a directed forest (there is no cycle and each vertex has at most one incoming edge), +- For every worker $w \in V_{\mu}$ with $i = \mathrm{index}(w)$ , and for every $1 \leq j < i$ , there is a worker $w' \in V_{\mu}$ with $j = \mathrm{index}(w')$ such that $(w, w') \in E_{\mu}$ . + +![](images/b01dfeb37dfe5697daeaa26bfbe32c4acb3d17d48bf72799b52056547e658b83.jpg) +Figure 3: The graph $G_{\mu} = (V_{\mu}, E_{\mu})$ is a directed forest. The matching $\tilde{\mu}$ , computed in Algorithm 1 matches each worker to a single (copy of) job in $\tilde{\mu}$ . In the stable matching $\mu$ , each worker $w$ is connected to all copies of $\mu(w)$ which have lower index. + +Proof. The graph $G_{\mu}$ is a directed forest by construction. Indeed, it has no cycle because edges connect workers to lower index workers. And every node has at most one incoming edge because $\tilde{\mu}$ matches each worker to at most one job, and $\mu$ matches each job to at most one worker. + +Fix a worker $w \in V_{\mu}$ with $i = \mathrm{index}(w)$ , let $1 \leq j < i$ , and let $a = \mu(w)$ . By definition of $V_{\mu}$ , worker $w$ weakly prefers $\mu$ to $D$ , that is $U(w, a) \geq m \cdot U_D(w) = U(w, \tilde{\mu}_i(w))$ . By definition of $w$ 's preference list $P_w$ in Algorithm 1, the lexicographic ordering gives that + +$$ +a ^ {(j)} \succ_ {P _ {w}} \tilde {\mu} (w). +$$ + +Because $\tilde{\mu}$ is a stable matching, it should not be blocked by the pair $(w, a^{(j)})$ . Thus, there exists a worker $w' \in \mathcal{W}$ such that $\tilde{\mu}(w') = a^{(j)}$ and + +$$ +w ^ {\prime} \succ_ {a} w. +$$ + +Finally, because $\mu$ is a stable matching, it should not be blocked by the pair $(w', a)$ , thus + +$$ +\boldsymbol {U} \left(w ^ {\prime}, \mu \left(w ^ {\prime}\right)\right) \geq \boldsymbol {U} \left(w ^ {\prime}, a\right) = m \cdot \boldsymbol {U} _ {D} \left(w ^ {\prime}\right), +$$ + +proving that $w' \in V_{\mu}$ . Hence, there is an edge $(w, w') \in E_{\mu}$ , which concludes the proof. + +![](images/1643bbaa4c860980288bec87286eb45b720a8a1efa7593a1697e7f161c13d0d9.jpg) + +Proposition 2. In the graph $G_{\mu}$ , each node of index $i \geq 1$ can reach $2^{i - 1}$ nodes (including itself). + +Proof. We show that the property holds by induction on $i$ . The property trivially holds for $i = 1$ . Let $i > 1$ such that there is a worker $w \in V_{\mu}$ with $\mathrm{index}(w) = i$ . Using Proposition 1, there is an edge $(w, w_j) \in E_{\mu}$ with $\mathrm{index}(w_j) = j$ for every $1 \leq j < i$ . Because the graph is a directed forest, the set of nodes reachable from each $w_j$ are disjoint. Thus, the number of nodes reachable from $w$ (including itself) is $1 + \sum_{j=1}^{i-1} 2^{j-1} = 2^{i-1}$ . + +Finally, we conclude with the proof that Algorithm 1 computes a distribution over internally stable matching which guarantees each worker a logarithmic fraction of their optimal stable share. + +Proof of Theorem 3. Algorithm 1 first computes a stable matching $\tilde{\mu}$ for the instance with duplicated jobs, then build $m$ matchings $\tilde{\mu}_1,\ldots ,\tilde{\mu}_m$ . If there were a pair $(w,a)$ with index $(w) = i$ which blocks matching $\tilde{\mu}_i$ , that is $U(w,a) > U(w,\tilde{\mu}_i(w))$ and $w\succ_{a}\tilde{\mu}_{i}(a)$ , then $(w,a^{(i)})$ would block $\tilde{\mu}$ , which is a contradiction. Thus, each matching $\tilde{\mu}_i$ is internally stable. + +Now, let us assume, that there is a stable matching $\mu$ in which a worker $w$ with $\mathrm{index}(w) = i$ receives $a = \mu(w)$ having utility $U(w, a) > U(w, \tilde{\mu}_i(w))$ . In the matching $\tilde{\mu}$ , job $a^{(m)}$ must be matched to some worker $w'$ such that $w' \succ_a w$ , otherwise $(w, a)$ would block $\tilde{\mu}$ . Moreover, we must have $\mathbf{U}(w', \mu(w')) \geq m \cdot U_D(w')$ otherwise $(w', a)$ would be blocking $\mu$ . Thus, there is a node $w' \in V_\mu$ of index $m$ , which proves that there exists at least $2^{m-1}$ nodes, and thus that $N \geq 2^{m-1}$ . By contrapositive, if we set $m > 1 + \log_2 N$ , then we have $m \cdot U_D(w) \geq U^*(w)$ for every worker $w$ , which concludes the proof. + +# C.3 Dominant Strategy Incentive Compatibility of Algorithm 1 + +Proof of Theorem 4. We will use the fact that when workers have strict preferences, Gale and Shapley's worker-proposing deferred acceptance procedure is dominant strategy incentive compatible, i.e., it is always optimal for workers to report their true preferences [17]. + +First, notice that for each worker $w$ , the ranking $P_w$ used in Algorithm 1 aligns with her utility, ensuring that all copies of a higher-utility job are ranked above copies of lower-utility ones. + +To see that it is optimal for a worker $w$ to report her true vector of utility, we will give her more strategic power, and we will let her choose her ranking $P_w'$ over all the duplicated jobs. By the incentive compatibility property of the deferred acceptance procedure with strict preferences, she cannot obtain any job ranked above $\tilde{\mu}(w)$ in $P_w$ . And because $P_w$ is consistent with $w$ 's utility, it is optimal to report $P_w' = P_w$ . + +# D $\epsilon$ -Oracle for Approximated Worker Optimal Stable Matching + +# D.1 $\epsilon$ -Oracle + +# Algorithm 2 $\epsilon$ -Oracle for Approximated Worker Optimal Stable Matching + +Input: $N$ workers, $K$ jobs, Utility matrix $\mathbf{U}$ that encodes the preference of workers over jobs, strict preference profile $P_{a}$ of jobs, an integer $m\geq 1$ , and the instability tolerance $\epsilon \geq 0$ . + +1: For each job $a \in \mathcal{A}$ , duplicate it $m$ times and denote the $i$ -th copy as $a^{(i)}$ . +2: Each replica $a^{(i)}$ shares the same preference $P_{a}$ as the original job $a$ . +3: For every worker $w$ and job $a^{(i)}$ , define the utility + +$$ +\boldsymbol {U} (w, a ^ {(i)}) := \boldsymbol {U} (w, a) - (i - 1) \epsilon +$$ + +and use it to generate the workers' preference profile $P_{w}$ (breaking ties in favor of lower indices). + +4: Run Gale-Shapley algorithm on $P_w$ and $P_a$ to compute a worker-optimal stable matching $\tilde{\mu}$ . +5: For each $i \in [m]$ , build a matching $\tilde{\mu}_i$ , which matches each job $a$ with $\tilde{\mu}_i(a) \coloneqq \tilde{\mu}(a^{(i)})$ . + +Output: The distribution $D$ which selects each matching $\tilde{\mu}_i$ with probability $1 / m$ . + +# D.2 Proof of Theorem 5 + +Similarly to the proof of Theorem 3, we run Algorithm 2 and define the index of a worker as the index of the job she receives in $\tilde{\mu}$ , that is, $\mathrm{index}(w) = i$ if worker $w$ receives $a_{j}^{(i)}$ for some $j$ . + +Definition 9. Given a problem instance $(U, P_{a})$ , run Algorithm 2 with duplication number $m$ and instability tolerance $\epsilon$ to generate the output distribution $D$ . Then, for any $\epsilon$ -stable matching $\mu$ with respect to $(U, P_{a})$ , we define a graph $G_{\mu} = (V_{\mu}, E_{\mu})$ where + +$$ +V _ {\mu} := \{w \in \mathcal {W} \mid \boldsymbol {U} (w, \mu (w)) \geq m \cdot \boldsymbol {U} _ {D} (w) - \epsilon \}, +$$ + +$$ +E _ {\mu} := \left\{\left(w, w ^ {\prime}\right) \in V _ {\mu} ^ {2} \mid \mu (w) = \tilde {\mu} _ {j} \left(w ^ {\prime}\right) \text {w h e r e} j = \operatorname {i n d e x} \left(w ^ {\prime}\right) < \operatorname {i n d e x} (w) \right\}. +$$ + +Once again, $G_{\mu}$ is the graph of workers who prefer $\mu$ to their match in distribution $D$ , where an edge $(w, w')$ means that $w'$ received a job that $w$ would have liked. Next, we show properties on the graph $G_{\mu}$ . + +Proposition 3. For any stable matching $\mu$ , we have that + +- $G_{\mu}$ is a directed forest (there is no cycle and each vertex has at most one incoming edge), +- For every worker $w \in V_{\mu}$ with $i = \mathrm{index}(w)$ , and for every $1 \leq j < i$ , there is a worker $w' \in V_{\mu}$ with $j = \mathrm{index}(w')$ such that $(w, w') \in E_{\mu}$ . + +Proof. The proof is almost identical to that of Proposition 1. Fix a worker $w \in V_{\mu}$ with $i = \mathrm{index}(w)$ , let $1 \leq j < i$ , and let $a = \mu(w)$ . By definition of $V_{\mu}$ , worker $w$ prefers $\mu$ to $D$ , that is $U(w, a) \geq$ + +$m \cdot U_D(w) - \epsilon = U(w, \tilde{\mu}_i(w)) - \epsilon$ . By definition of $w$ 's preference list $P_w$ in Algorithm 2, the lexicographic ordering gives that + +$$ +a ^ {(j)} \succ_ {P _ {w}} \tilde {\mu} (w). +$$ + +Because $\tilde{\mu}$ is a $\epsilon$ -stable matching, it should not be blocked by the pair $(w, a^{(j)})$ . Thus, there exists a worker $w' \in \mathcal{W}$ such that $\tilde{\mu}(w') = a^{(j)}$ and + +$$ +w ^ {\prime} \succ_ {a} w. +$$ + +Finally, because $\mu$ is a stable matching, it should not be blocked by the pair $(w', a)$ , thus + +$$ +\boldsymbol {U} (w ^ {\prime}, \mu (w)) \geq \boldsymbol {U} (w ^ {\prime}, a) - \epsilon = m \cdot \boldsymbol {U} _ {D} (w ^ {\prime}) - \epsilon , +$$ + +proving that $w' \in V_{\mu}$ . Hence, there is an edge $(w, w') \in E_{\mu}$ , which concludes the proof. + +![](images/f0682c948862ad0407fe1768152a372f21aacb1d2a50038583600ca0c92ba92e.jpg) + +We will once again use Proposition 2 to give a lower on the number of nodes in the graph $G_{\mu}$ . Finally, we conclude with the proof that Algorithm 2 computes a distribution over internally $\epsilon$ -stable matching which guarantees each worker a logarithmic fraction of their optimal stable share. + +Proof of Theorem 5. Algorithm 2 first computes a stable matching $\tilde{\mu}$ for the instance with duplicated jobs, then build $m$ matchings $\tilde{\mu}_1,\ldots ,\tilde{\mu}_m$ . If there were a pair $(w,a)$ with $\mathrm{index}(w) = i$ which $\epsilon$ -blocks matching $\tilde{\mu}_i$ , that is $\boldsymbol {U}(\boldsymbol {w},a) > \boldsymbol {U}(\boldsymbol {w},\tilde{\mu}_i(\boldsymbol {w})) + \epsilon$ and $w\succ_{a}\tilde{\mu}_{i}(a)$ , then $(w,a^{(i)})$ would block $\tilde{\mu}$ , which is a contradiction. Thus, each matching $\tilde{\mu}_i$ is internally stable. + +Now, let us assume, that there is a stable matching $\mu$ in which a worker $w$ with index $(w) = i$ receives $a = \mu(w)$ having utility $U(w, a) > U(w, \tilde{\mu}_i(w)) + m\epsilon$ . In the matching $\tilde{\mu}$ , job $a^{(m)}$ must be matched to some worker $w'$ such that $w' \succ_a w$ , otherwise $(w, a)$ would block $\tilde{\mu}$ . Moreover, we must have $\mathbf{U}(w', \mu(w')) \geq m \cdot \mathbf{U}_D(w') - \epsilon$ otherwise $(w', a)$ would be blocking $\mu$ . Using Proposition 2, there is a node $w' \in V_\mu$ of index $m$ , which proves that there exists at least $2^{m-1}$ nodes, and thus that $N \geq 2^{m-1}$ . By contrapositive, if we set $m > 1 + \log_2 N$ , then we have $U_D(w) \geq U^*(w) / m - \epsilon$ for every worker $w$ , which concludes the proof. + +# D.3 Robustness of $\epsilon$ -stable matching + +Lemma 1. Fix the preferences of jobs over workers. Given two utility matrices $\mathbf{U}_1$ and $\mathbf{U}_2$ such that $\left\| \mathbf{U}_1 - \mathbf{U}_2 \right\|_{\max} < \frac{\epsilon}{2}$ , then any stable matching $\mu$ for $\mathbf{U}_1$ , is also $\epsilon$ -stable with respect to $\mathbf{U}_2$ . + +Proof. If a matching $\mu$ is stable with respect to $U_{1}$ , then for any $(w, a)$ pair such that $w \succ_{a} \mu(a)$ , we must have + +$$ +\boldsymbol {U} _ {1} (w, a) \leq \boldsymbol {U} _ {1} (w, \mu (w)), \tag {10} +$$ + +from the definition of stable matching (Definition 1). + +Since $\| \pmb{U}_1 - \pmb{U}_2 \|_{\max} \leq \frac{\epsilon}{2}$ , we have that for any $(w, a)$ pair, $|U_1(w, a) - U_2(w, a)| \leq \frac{\epsilon}{2}$ . Therefore, + +$$ +\boldsymbol {U} _ {2} (w, a) \leq \boldsymbol {U} _ {1} (w, a) + \frac {\epsilon}{2} \leq \boldsymbol {U} _ {1} (w, \mu (w)) + \frac {\epsilon}{2} \leq \boldsymbol {U} _ {2} (w, \mu (w)) + \epsilon , \tag {11} +$$ + +where the first and the last inequality come from $\| U_1 - U_2\|_{\max} \leq \frac{\epsilon}{2}$ , while the second inequality holds according to Eq.(10). Therefore, combining $w\succ_{a}\mu (a)$ and Eq.(11), we can conclude that matching $\mu$ is $\epsilon$ -stable with respect to $U_{2}$ . + +# D.4 Proof of Theorem 6 + +For convenience, we denote $\mathcal{S}^U$ (resp. $\mathcal{S}_{\epsilon}^{U}$ ) the stable matchings ( $\epsilon$ -stable matchings) with respect to $U$ . + +Proof. By running Algorithm 2 with $\pmb{U} = \hat{\pmb{U}}, \epsilon = \epsilon, m = \lfloor \log_2 N + 2 \rfloor$ , from Theorem 5, we get + +$$ +\boldsymbol {U} _ {D} (w) \geq \frac {\hat {\boldsymbol {U}} _ {\epsilon} ^ {*} (w)}{m} - \epsilon , \quad \forall w \in \mathcal {W}. \tag {12} +$$ + +By construction, any utility matrix $U \in \mathcal{U}$ satisfies $\| U - \hat{U} \|_{\max} \leq \frac{\epsilon}{2}$ . From Lemma 1, we know that for any matching $\mu \in S^U$ , $\forall U \in \mathcal{U}$ , we have $\mu \in S_{\epsilon}^{\hat{U}}$ , that is $\bigcup_{U \in \mathcal{U}} S^U \subseteq S_{\epsilon}^{\hat{U}}$ . Therefore, for the optimal stable share, we have + +$$ +\mathcal {U} ^ {*} (w) \leq \hat {\mathcal {U}} _ {\epsilon} ^ {*} (w), \quad \forall w \in \mathcal {W}. \tag {13} +$$ + +Combining Eq.(12) and (13), the conclusion holds. + +# E Explore-then-Choose-Oracle Algorithm + +Algorithm 3 is the full version of the Explore-then-Choose-Oracle algorithm. + +Algorithm 3 Explore-then-Choose-Oracle (Full version) +Input: $N$ workers, $K$ jobs, horizon $T$ , exploration length $T_0 < T$ , preference profile $P_{a}$ for all jobs $a \in \mathcal{K}$ , approximation stable-matching oracle $\mathbb{O}$ . +1: Initialize: $\hat{U}(i,j) = 0, T_{i,j} = 0, \forall i \in [N], j \in [K]$ . +2: Initialize: $F_i \gets \text{False}$ . Whether the CIs of the first ( $N + 1$ )-ranked jobs are disjoint. +3: Set $t = 1, T_0 \gets K[T_0 / K]$ , $t_m = 0$ . To have full rounds of round-robin. +4: while $t \leq T_0$ and $\exists i \in [N]$ s.t. $F_i == \text{False}$ do Phase 1, round-robin exploration. +5: Match $\mu_t(i) \gets a_{((t + i - 1) \mod K) + 1}, \forall i \in [N]$ . +6: Observe $X_{i,\mu_t(i)}(t)$ and update $\hat{U}(i,\mu_t(i)), T_{i,\mu_t(i)}$ as follows: + $\hat{U}(i,\mu_t(i)) = \frac{\hat{U}(i,\mu_t(i)) \cdot T_{i,\mu_t(i)} + X_{i,\mu_t(i)}(t)}{T_{i,\mu_t(i)} + 1}, T_{i,\mu_t(i)} = T_{i,\mu_t(i)} + 1$ . +7: $t \gets t + 1$ +8: if $t \mod K == 0$ then Completed a full round of round-robin +9: $t_m \gets t_m + 1$ . +10: Compute $UCB_{i,j}$ and $LCB_{i,j}$ for all $i \in [N], j \in [K]$ . See Equation (5) +11: for $i = 1, 2, \dots, N$ do +12: $\hat{U}_{\text{sort}}(i, \cdot) \gets \text{Sort}(\hat{U}(i, \cdot), \text{decreasing})$ +13: $\Delta_{i,\min} \gets \min\left\{\hat{U}_{\text{sort}}(i,j) - \hat{U}_{\text{sort}}(i,j + 1), j \in [N]\right\}$ +14: if $\Delta_{i,\min} > 2\sqrt{\frac{6\ln T}{t_m}}$ then +15: $F_i \gets \text{True}$ . +16: end if +17: end for +18: if $F_i == \text{True}, \forall i \in [N]$ , then No ties - standard oracle +19: Compute preference list $P_w$ for all $w \in \mathcal{W}$ according to $\hat{U}$ +20: $\hat{\mu}^* \gets \text{worker-optimal stable matching w.r.t } P_w$ and $P_a$ (using GS algorithm) +21: else if $t = T_0$ then Potential ties - approximation oracle +22: $\hat{\mu}^* \gets \mathbb{O}(\hat{U})$ for $\bar{U}$ s.t. $\bar{U}(i,j) = UCB_{i,j}$ for all $i \in [N], j \in [K]$ +23: end if +24: end if +25: end while +26: while $t \leq T$ do Phase 2, exploitation with the chosen oracle. +27: Match $\mu_t(i) \gets \hat{\mu}^*(i), \forall i$ . +28: $t \gets t + 1$ +29: end while + +# F Technical Lemmas + +Lemma 2 (Corollary 5.5 in Lattimore and Szepesváři [40]). Assume that $X_{1}, X_{2}, \dots, X_{n}$ are independent, $\sigma$ -subgaussian random variables centered around $\mu$ . Then for any $\varepsilon > 0$ , + +$$ +\mathbb {P} \left(\frac {1}{n} \sum_ {i = 1} ^ {n} X _ {i} \geq \mu + \varepsilon\right) \leq \exp \left(- \frac {n \varepsilon^ {2}}{2 \sigma^ {2}}\right), \quad \mathbb {P} \left(\frac {1}{n} \sum_ {i = 1} ^ {n} X _ {i} \leq \mu - \varepsilon\right) \leq \exp \left(- \frac {n \varepsilon^ {2}}{2 \sigma^ {2}}\right). +$$ + +Lemma 3 (Divergence Decomposition, Lemma 15.1 in Lattimore and Szepesvári [40]). For two bandit instances $\nu = \{\nu_{ij} : i \in [N], j \in [K]\}$ , and $\nu' = \{\nu_{ij}' : i \in [N], j \in [K]\}$ , fix some policy $\pi$ and let $\mathbb{P}_{\nu,\pi}$ and $\mathbb{P}_{\nu',\pi}$ be the probability measures induced by the $T$ -round interconnection of $\pi$ and $\nu$ (respectively, $\pi$ and $\nu'$ ), the following divergence decomposition holds, + +$$ +D \left(\mathbb {P} _ {\nu , \pi}, \mathbb {P} _ {\nu^ {\prime}, \pi}\right) = \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {K} \mathbb {E} _ {\nu , \pi} N _ {i j} (T) \cdot D \left(\nu_ {i j}, \nu_ {i j} ^ {\prime}\right). \tag {14} +$$ + +Lemma 4 (Data-processing Inequality, Lemma 1 in Garivier et al. [24]). Consider a measurable space $(\Omega, \mathcal{F})$ equipped with two distributions $\mathbb{P}_1$ and $\mathbb{P}_2$ , and any $\mathcal{F}$ -measurable random variable $Z: \Omega \to [0,1]$ . We denote respectively by $\mathbb{E}_1$ and $\mathbb{E}_2$ the expectations under $\mathbb{P}_1$ and $\mathbb{P}_2$ . Then, + +$$ +K L \left(\mathbb {P} _ {1}, \mathbb {P} _ {2}\right) \geq k l \left(\mathbb {E} _ {1} [ Z ], \mathbb {E} _ {2} [ Z ]\right), +$$ + +where $kl$ denotes the KL divergence for Bernoulli distributions, i.e., $\forall p,q\in [0,1]^2,kl(p,q) = p\ln \frac{p}{q} + (1 - p)\ln \frac{1 - p}{1 - q}$ . + +# G Proof of Theorem 7 + +For convenience, let $\hat{U}^{(t)}(i,j), T_{i,j}^{(t)}, UCB_{i,j}^{(t)}, LCB_{i,j}^{(t)}$ be the value of $\hat{U}(i,j), T_{i,j}, UCB_{i,j}, LCB_{i,j}$ at the end of round $t$ . Define $\mathcal{F} = \left\{\exists t \in [T], i \in [N], j \in [K] : |\hat{U}^{(t)}(i,j) - U(i,j)| > \sqrt{\frac{6\ln T}{T_{i,j}^{(t)}}}\right\}$ as the bad event that some preference is not estimated well during the horizon. + +Lemma 5. + +$$ +\mathbb {P} (\mathcal {F}) \leq 2 N K / T. +$$ + +Proof. + +$$ +\begin{array}{l} \mathbb {P} (\mathcal {F}) = \mathbb {P} \left(\exists 1 \leq t \leq T, i \in [ N ], j \in [ K ]: | \hat {\boldsymbol {U}} ^ {(t)} (i, j) - \boldsymbol {U} (i, j) | > \sqrt {\frac {6 \ln T}{T _ {i , j} ^ {(t)}}}\right) \\ \leq \sum_ {t = 1} ^ {T} \sum_ {i \in [ N ]} \sum_ {j \in [ K ]} \mathbb {P} \left(| \hat {\boldsymbol {U}} ^ {(t)} (i, j) - \boldsymbol {U} (i, j) | > \sqrt {\frac {6 \ln T}{T _ {i , j} ^ {(t)}}}\right) \\ \leq \sum_ {t = 1} ^ {T} \sum_ {i \in [ N ]} \sum_ {j \in [ K ]} \sum_ {s = 1} ^ {t} \mathbb {P} \left(T _ {i, j} ^ {(t)} = s, | \hat {\boldsymbol {U}} ^ {(t)} (i, j) - \boldsymbol {U} (i, j) | > \sqrt {\frac {6 \ln T}{s}}\right) \\ \leq \sum_ {t = 1} ^ {T} \sum_ {i \in [ N ]} \sum_ {j \in [ K ]} t \cdot 2 \exp (- 3 \ln T) \\ \leq 2 N K / T, \\ \end{array} +$$ + +where the second last inequality results from Lemma 2. + +Lemma 6. Conditional on $\neg \mathcal{F}$ , $UCB_{i,j}^{(t)} < LCB_{i,j'}^{(t)}$ implies $\pmb{U}(i,j) < \pmb{U}(i,j')$ . + +Proof. According to the definition of LCB and UCB, we have that conditional on $\neg \mathcal{F}$ + +$$ +L C B _ {i, j} ^ {(t)} = \hat {\boldsymbol {U}} _ {i, j} ^ {(t)} - \sqrt {\frac {6 \ln T}{T _ {i , j} ^ {(t)}}} \leq \boldsymbol {U} (i, j) \leq \hat {\boldsymbol {U}} ^ {(t)} (i, j) + \sqrt {\frac {6 \ln T}{T _ {i , j} ^ {(t)}}} = U C B _ {i, j} ^ {(t)}. +$$ + +Therefore, if $UCB_{i,j}^{(t)} < LCB_{i,j'}^{(t)}$ , we have that + +$$ +\boldsymbol {U} (i, j) \leq U C B _ {i, j} ^ {(t)} \leq L C B _ {i, j ^ {\prime}} ^ {(t)} \leq \boldsymbol {U} (i, j ^ {\prime}). +$$ + +Lemma 7. In round $t$ , let $T_{i}^{(t)} = \min_{j\in [K]}T_{i,j}^{(t)}$ . Conditional on $\neg \mathcal{F}$ , if $T_{i}^{(t)} > 96\ln T / \Delta_{\mathrm{min}}^{2}$ , we have $LCB_{i,\rho_{i,k}}^{(t)} > UCB_{i,\rho_{i,k + 1}}^{(t)}$ for any $k\in [N]$ , and $LCB_{i,\rho_{i,N}}^{(t)} > UCB_{i,\rho_{i,k}}^{(t)}$ for any $N + 1\leq k\leq K$ . + +Proof. We prove it by contradiction, suppose that there exists $k \in [N]$ such that $LCB_{i,\rho_{i,k}}^{(t)} \leq UCB_{i,\rho_{i,k+1}}^{(t)}$ or there exists $N + 1 \leq k \leq K$ such that $LCB_{i,\rho_{i,N}}^{(t)} \leq UCB_{i,\rho_{i,k}}^{(t)}$ . Without loss of generality, denote $j$ as the arm on the LHS and $j'$ as the arm on the RHS. + +Conditional on $\neg \mathcal{F}$ and by the definition of $LCB$ and $UCB$ , we have that + +$$ +\boldsymbol {U} (i, j) - 2 \sqrt {\frac {6 \ln T}{T _ {i} ^ {(t)}}} \leq L C B _ {i, j} ^ {(t)} \leq U C B _ {i, j ^ {\prime}} ^ {(t)} \leq \boldsymbol {U} (i, j ^ {\prime}) + 2 \sqrt {\frac {6 \ln T}{T _ {i} ^ {(t)}}}. +$$ + +Therefore, $\Delta_{i,j,j'} = U(i,j) - U_{i,j'} \leq 4\sqrt{\frac{6\ln T}{T_i^{(t)}}}$ , which implies that $T_i^{(t)} \leq \frac{96\ln T}{\Delta_{i,j,j'}^2} \leq \frac{96\ln T}{\Delta_{\min}^2}$ , which is a contradiction. + +Lemma 8. Conditional on $\neg \mathcal{F}$ , if $\Delta_{\mathrm{min}} > \sqrt{\frac{96K\ln T}{T_0}}$ , Algorithm 3 would enter the exploitation phase and choose the Gale-Shapley oracle at some $t \leq T_0$ . + +Proof. If $\Delta_{\mathrm{min}} > \sqrt{\frac{96K\ln T}{T_0}}$ , we have $T_0 > \frac{96K\ln T}{\Delta_{\mathrm{min}}^2}$ . Since for every worker, Algorithm 3 allocates jobs in a round-robin fashion, we have that $T_i^{(t)} > \frac{96\ln T}{\Delta_{\mathrm{min}}^2}$ . + +By Lemma 7, we know that for any worker $w_{i}$ , $LCB_{i,\rho_{i,k}}^{(t)} > UCB_{i,\rho_{i,k+1}}^{(t)}$ for any $k \in [N]$ , and $LCB_{i,\rho_{i,N}}^{(t)} > UCB_{i,\rho_{i,k}}^{(t)}$ for any $N + 1 \leq k \leq K$ , i.e., the preference utility for the first $N$ -ranked jobs for every worker has been estimated well enough with the confidence intervals disjoint. The flag $F_{i}$ would be set as True as in Line 15 in Algorithm 3 and we would enter Phase 2 at some time $t \leq T_{0}$ . + +Lemma 9. Given a utility matrix $\mathbf{U}_{N\times K}$ without ties, the worker-optimal stable matching job of each worker must be its first $N$ -ranked. + +Proof. We implement the Gale-Shapley algorithm with the workers as the proposing side. Once a job is proposed, it has a temporary worker. By contradiction, once $N$ jobs have been proposed, we have $N$ workers occupied. Therefore, each worker would be allocated with a job and the Gale-Shapley algorithm would stop. Since in the deferred-acceptance procedure, workers propose to jobs one by one according to their preference list, then the worker-optimal stable matching job of each worker must be its first $N$ -ranked. + +Proof of Theorem 7. We consider the two cases separately. + +Case 1. $\Delta_{\mathrm{min}} > \sqrt{\frac{96K\ln T}{T_0}}$ + +Let $\Delta_{i,\max} = \max_{j\in [K]}\left[U^{*}(w_{i}) - U(i,j)\right]$ be the maximum worker-optimal stable regret that may be suffered by $w_{i}$ in all rounds, we have $\Delta_{i,\max}\leq 1$ . The worker-optimal stable regret for each + +worker $w_{i}$ by following Algorithm 3 satisfies + +$$ +\begin{array}{l} R e g _ {i} (T) = \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \left(\boldsymbol {U} ^ {*} \left(w _ {i}\right) - X _ {i} (t)\right) \right] \\ \leq \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \mathbb {1} \left\{\mu_ {t} (i) \neq \mu^ {*} (i) \right\} \cdot \Delta_ {i, \max } \right] (15) \\ \leq \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \mathbb {1} \left\{\mu_ {t} (i) \neq \mu^ {*} (i) \right\} | \neg \mathcal {F} \right] \cdot \Delta_ {i, \max } + \mathbb {P} (\mathcal {F}) \cdot T \cdot \Delta_ {i, \max } \\ \leq \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \mathbb {1} \left\{\mu_ {t} (i) \neq \mu^ {*} (i) \right\} | \neg \mathcal {F} \right] \cdot \Delta_ {i, m a x} + 2 N K \Delta_ {i, \max } (16) \\ \leq \left\lceil \frac {9 6 K \ln T}{\Delta_ {\min } ^ {2}} \right\rceil \cdot \Delta_ {i, m a x} + 2 N K \Delta_ {i, \max } (17) \\ = O \left(\frac {K \ln T}{\Delta_ {\mathrm {m i n}} ^ {2}}\right), \\ \end{array} +$$ + +where Eq.(15) comes from the fact that in a matching market without ties, there is a unique worker-optimal stable matching and hence a unique optimal stable match $\mu^{*}(i)$ for worker $i$ , Eq.(16) holds based on Lemma 5, Eq.(17) holds according to Lemma 8 and 9 and the fact that Gale-Shapley algorithm could always output the worker-optimal stable matching with respect to the given utility matrix by treating worker as the proposing side. + +Case 2. $\Delta_{\mathrm{min}} \leq \sqrt{\frac{96K\ln T}{T_0}}$ . + +The objective function is the approximation regret $Reg_{i}^{\alpha}(T)$ . Denote $\mathcal{F}_d^{(t)}$ as the event that $LCB_{i,\rho_{i,k}}^{(t)} > UCB_{i,\rho_{i,k+1}}^{(t)}$ for all $k \in [N]$ , and $LCB_{i,\rho_{i,N}}^{(t)} > UCB_{i,\rho_{i,k}}^{(t)}$ for all $N + 1 \leq k \leq K$ . We have + +$$ +\begin{array}{l} R e g _ {i} ^ {\alpha} (T) = \mathbb {E} \left[ \alpha T \cdot \boldsymbol {U} ^ {*} (w _ {i}) - \sum_ {t = 1} ^ {T} X _ {i} (t) \mid \mathcal {F} \right] \cdot \mathbb {P} (\mathcal {F}) \\ + \mathbb {E} \left[ \alpha T \cdot \boldsymbol {U} ^ {*} (w _ {i}) - \sum_ {t = 1} ^ {T} X _ {i} (t) | \neg \mathcal {F} \right] \cdot \mathbb {P} (\neg \mathcal {F}) \\ \leq \alpha T \cdot \mathbb {P} (\mathcal {F}) + \mathbb {E} \left[ \alpha T \cdot \boldsymbol {U} ^ {*} \left(w _ {i}\right) - \sum_ {t = 1} ^ {T} X _ {i} (t) | \neg \mathcal {F} \right] (18) \\ \leq 2 \alpha N K + \mathbb {E} \left[ \alpha T \cdot \boldsymbol {U} ^ {*} \left(w _ {i}\right) - \sum_ {t = 1} ^ {T} X _ {i} (t) \mid \neg \mathcal {F} \right] (19) \\ \leq 2 \alpha N K + \mathbb {E} \left[ \left(\alpha T \cdot \boldsymbol {U} ^ {*} (w _ {i}) - \sum_ {t = 1} ^ {T} X _ {i} (t)\right) \mathbb {1} \left\{\mathcal {F} _ {d} ^ {(T _ {0})} \right\} | \neg \mathcal {F} \right] \\ + \mathbb {E} \left[ \left(\alpha T \cdot \boldsymbol {U} ^ {*} (w _ {i}) - \sum_ {t = 1} ^ {T} X _ {i} (t)\right) \mathbb {1} \left\{\neg \mathcal {F} _ {d} ^ {(T _ {0})} \right\} | \neg \mathcal {F} \right] \\ \leq 2 \alpha N K + \alpha T _ {0} + \mathbb {E} \left[ \left(\alpha T \cdot \boldsymbol {U} ^ {*} \left(w _ {i}\right) - \sum_ {t = 1} ^ {T} X _ {i} (t)\right) \mathbb {1} \left\{\neg \mathcal {F} _ {d} ^ {\left(T _ {0}\right)} \right\} | \neg \mathcal {F} \right], (20) \\ \end{array} +$$ + +where Eq.(18) comes from the fact that $U^{*}(w_{i}) \leq 1$ and $X_{i}(t) \geq 0$ . Eq.(19) holds according to Lemma 5. Eq.(20) comes from the fact that when the good event $\neg \mathcal{F}$ that all utilities are well estimated and the top $(N + 1)$ -ranked CIs are disjoint before $T_{0}$ , the Gale-Shapley algorithm would give us the OSS in the exploitation phase, and since $\alpha \in (0,1]$ , the approximation regret would be no larger than $\alpha T_{0} + (\alpha - 1) \cdot (T - T_{0}) \leq \alpha T_{0}$ . + +Moreover, conditional on $\neg \mathcal{F}$ , we have the ground-truth utility matrix $\pmb{U}$ lies in the uncertainty set constructed by the empirical mean utility matrix $\hat{\pmb{U}}^{(T_0)}$ and the $UCB^{(T_0)}$ and $LCB^{(T_0)}$ , i.e., for any $(i,j)\in [N]\times [K],|\hat{\pmb{U}}^{(T_0)}(i,j) - \pmb {U}(i,j)|\leq \sqrt{\frac{6K\ln T}{T_0}}$ . If we implement an $(\alpha ,\epsilon)$ -oracle, with $\epsilon$ being $2\sqrt{\frac{6K\ln T}{T_0}}$ , follow a similar proof as that for Theorem 6, in each round $t$ in the exploitation phase, we have that + +$$ +\alpha \mathbf {U} ^ {*} (w _ {i}) - \mathbb {E} X _ {i} (t) \leq 2 \sqrt {\frac {6 K \ln T}{T _ {0}}}. \tag {21} +$$ + +Since there are in total $T - T_{0}$ rounds of exploitation, we have that + +$$ +\mathbb {E} \left[ \left(\alpha T \cdot \boldsymbol {U} ^ {*} \left(w _ {i}\right) - \sum_ {t = 1} ^ {T} X _ {i} (t)\right) \mathbb {1} \left\{\neg \mathcal {F} _ {d} ^ {\left(T _ {0}\right)} \right\} | \neg \mathcal {F} \right] \leq \alpha T _ {0} + 2 \sqrt {\frac {6 K \ln T}{T _ {0}}} (T - T _ {0}). \tag {22} +$$ + +Therefore, combining Eq.(20) and (22), we have + +$$ +R e g _ {i} ^ {\alpha} (T) \leq 2 \alpha N K + 2 \alpha T _ {0} + 2 \sqrt {\frac {6 K \ln T}{T _ {0}}} (T - T _ {0}). +$$ + +![](images/3be223a6264376805e5171e5f6f5849aa5329dc5ec6abb4122866e3b19d654f2.jpg) + +# H Proof of Theorem 8 + +Proof. Let $\mathcal{W} = \{w_1, w_2, w_3, w_4\}$ and $\mathcal{A} = \{a_1, a_2, a_3, a_4\}$ and $w_1 \succ w_2 \succ w_3 \succ w_4$ for all the jobs. Throughout the proof, we assume that all observations are Gaussian of unit variance, that is, when matching $w_i$ to $a_j$ at round $t$ , we observe $X_i(t) \sim \mathcal{N}(U(i,j), 1)$ . Consider two instances $\pmb{\nu}$ and $\pmb{\nu}'$ with the following mean utility matrices $\pmb{U}$ and $\pmb{U}'$ , respectively. + +$$ +\boldsymbol {U} = \left[ \begin{array}{c c c c} \frac {1}{2} & \frac {1}{2} & 0 & 0 \\ \frac {1}{2} & 0 & \frac {1}{2} & 0 \\ \frac {1}{2} & 0 & 0 & \frac {1}{4} \\ 0 & 0 & \frac {1}{2} & 0 \end{array} \right], \quad \boldsymbol {U} ^ {\prime} = \left[ \begin{array}{c c c c} \frac {1}{2} + \gamma & \frac {1}{2} & 0 & 0 \\ \frac {1}{2} & 0 & \frac {1}{2} & 0 \\ \frac {1}{2} & 0 & 0 & \frac {1}{4} \\ 0 & 0 & \frac {1}{2} & 0 \end{array} \right], +$$ + +where $\gamma < \frac{1}{4}$ + +Lemma 10 (Properties of Instances $\pmb{\nu}$ and $\pmb{\nu}^{\prime}$ ). Based on the utility matrices $\mathbf{U}$ and $\mathbf{U}^{\prime}$ , we have the following properties of $\pmb{\nu}$ and $\pmb{\nu}^{\prime}$ : + +1. Under $\pmb{\nu}$ , the optimal stable shares are $U^{*}(w) = \frac{1}{2}, \forall w \in \mathcal{W}$ ; Under $\pmb{\nu}'$ , the optimal stable shares are $(U')^{*}(w_{1}) = \frac{1}{2} + \gamma, (U')^{*}(w_{2}) = \frac{1}{2}, (U')^{*}(w_{3}) = \frac{1}{4}$ , and $(U')^{*}(w_{4}) = 0$ . +2. The relevant utility gaps for the two instances are $\Delta_{rel}^{\nu} = 0$ , and $\Delta_{rel}^{\nu'} = \gamma$ . +3. Given an offline oracle that could compute the best approximation ratio, the benchmark utilities for the four workers (after multiplying the approximation ratio) are $(\frac{1}{2}, \frac{3}{8}, \frac{3}{8}, \frac{3}{8})$ under $\pmb{\nu}$ and $(\frac{1}{2} + \gamma, \frac{1}{2}, \frac{1}{4}, 0)$ under $\pmb{\nu}'$ . + +We provide the proof of Lemma 10 in Appendix I. + +For a worker $w_{i}, i \in [N]$ , job $a_{j}, j \in [K]$ and time slot $t \in [T]$ , denote $N_{ij}(t) \in \mathbb{N} \cup \{0\}$ as the number of times worker $w_{i}$ is matched to job $a_{j}$ , up to and including time $t$ , and denote the past information as $I_{t} := (\mu_{1}, X(1), \mu_{2}, X(2), \dots, \mu_{t-1}, X(t-1))$ , where $X(t) = (X_{1}(t), X_{2}(t), \dots, X_{N}(t))$ is the realized reward vector for all $N$ workers in round $t$ . Finally, let $\mathbb{P}_{\nu, \pi}$ be the joint probability measure over the history and $\mathbb{E}_{\nu, \pi}$ be the expectation induced by instance $\nu$ and policy $\pi$ , and $\mathbb{P}_{\nu', \pi}$ , $\mathbb{E}_{\nu', \pi}$ be defined similarly. By divergence decomposition theorem [40, restated in Lemma 3], we have that + +$$ +D _ {\mathrm {K L}} \left(\mathbb {P} _ {\boldsymbol {\nu}, \pi}, \mathbb {P} _ {\boldsymbol {\nu} ^ {\prime}, \pi}\right) = \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {K} \mathbb {E} _ {\boldsymbol {\nu}, \pi} N _ {i j} (T) \cdot D _ {\mathrm {K L}} \left(\boldsymbol {\nu} _ {i j}, \boldsymbol {\nu} _ {i j} ^ {\prime}\right), +$$ + +where $\pmb{\nu}_{ij}$ is the distribution of utilities obtained when worker $w_{i}$ is matched to job $a_{j}$ in the environment $\pmb{\nu}$ . + +Since the only change in utility distribution happens in $(w_{1},a_{1})$ pair, we have that + +$$ +D _ {\mathrm {K L}} \left(\mathbb {P} _ {\boldsymbol {\nu}, \pi}, \mathbb {P} _ {\boldsymbol {\nu} ^ {\prime}, \pi}\right) = D _ {\mathrm {K L}} \left(\boldsymbol {\nu} _ {1 1}, \boldsymbol {\nu} _ {1 1} ^ {\prime}\right) \mathbb {E} _ {\boldsymbol {\nu}, \pi} [ N _ {1 1} (T) ] = \mathbb {E} _ {\boldsymbol {\nu}, \pi} [ N _ {1 1} (T) ] \cdot \frac {\gamma^ {2}}{2}, \tag {23} +$$ + +where the second equality comes from the fact that for two Gaussian distributions with means $\frac{1}{2}$ and $\frac{1}{2} + \gamma$ and variance 1, the KL divergence is $\frac{\gamma^2}{2}$ . + +By data-processing inequality [40, restated in Lemma 4], we know that for all $\sigma (I_T)$ -measurable random variable $Z\in [0,1]$ , we have that + +$$ +D _ {\mathrm {K L}} \left(\mathbb {P} _ {\nu , \pi}, \mathbb {P} _ {\nu^ {\prime}, \pi}\right) \geq \operatorname {k l} \left(\mathbb {E} _ {\nu , \pi} (Z), \mathbb {E} _ {\nu^ {\prime}, \pi} (Z)\right). \tag {24} +$$ + +where $\mathrm{kl}$ denotes the KL divergence between two Bernoulli distributions, i.e., $\forall p,q\in [0,1]^2,\mathrm{kl}(p,q) = p\ln \frac{p}{q} + (1 - p)\ln \frac{1 - p}{1 - q}$ . + +Let $Z = \frac{N_{21}(T) + N_{23}(T)}{T}$ , then $Z \in [0, 1]$ , by Pinsker's inequality, we have that + +$$ +\operatorname {k l} \left(\mathbb {E} _ {\boldsymbol {\nu}, \pi} (Z), \mathbb {E} _ {\boldsymbol {\nu} ^ {\prime}, \pi} (Z)\right) \geq 2 \cdot \left[ \mathbb {E} _ {\boldsymbol {\nu}, \pi} (Z) - \mathbb {E} _ {\boldsymbol {\nu} ^ {\prime}, \pi} (Z) \right] ^ {2}. \tag {25} +$$ + +Combining Eq.(23), (24), (25), we have that + +$$ +\mathbb {E} _ {\nu , \pi} [ N _ {1 1} (T) ] \cdot \frac {\gamma^ {2}}{2} \geq 2 \cdot \left[ \mathbb {E} _ {\nu , \pi} (Z) - \mathbb {E} _ {\nu^ {\prime}, \pi} (Z) \right] ^ {2}. \tag {26} +$$ + +We now divide into two cases, depending on the asymptotic number of matches $\mathbb{E}_{\pmb{\nu},\pi}[N_{11}(T)]$ + +Case I: $\lim \inf_{T\to \infty}\frac{\mathbb{E}_{\pmb{\nu},\pi}[N_{11}(T)]}{T^{1 - 2\delta}} = 0$ . We assume that both $Reg_{i}(T;\pmb{\nu}^{\prime})$ and $Reg_{i}^{\pmb{\alpha}^{*}(w_{i})}(T;\pmb{\nu})$ are sublinear for all workers and show that we have a contradiction. + +Since $\gamma = cT^{-\frac{1}{2} +\delta}$ , by Eq. (26), we have + +$$ +\lim _ {T \rightarrow \infty} \frac {\left| \mathbb {E} _ {\boldsymbol {\nu} , \pi} [ N _ {2 1} (T) + N _ {2 3} (T) ] - \mathbb {E} _ {\boldsymbol {\nu} ^ {\prime} , \pi} [ N _ {2 1} (T) + N _ {2 3} (T) ] \right|}{T} = 0. \tag {27} +$$ + +In $\pmb{\nu}^{\prime}$ , if the ground-truth utility matrix $\pmb{U}^{\prime}$ is known and we would like to achieve the benchmark utility for $w_{2}$ , we need $N_{21}(T) + N_{23}(T) = T$ , since worker $w_{2}$ could only get positive utilities from jobs $a_{1}$ and $a_{3}$ and her benchmark utility is $\frac{1}{2}$ . In particular, the regret for this worker is + +$$ +R e g _ {i} (T; \boldsymbol {\nu} ^ {\prime}) = \frac {1}{2} (T - \mathbb {E} _ {\boldsymbol {\nu} ^ {\prime}, \pi} [ N _ {2 1} (T) + N _ {2 3} (T) ]), +$$ + +and to guarantee sublinear regret for $w_{2}$ for any large enough $T$ , we must have + +$$ +\liminf_{T\to \infty}\frac{\mathbb{E}_{\boldsymbol{\nu}^{\prime},\pi}[N_{21}(T) + N_{23}(T)]}{T} = 1. +$$ + +Therefore, to satisfy Eq.(27), we must also have + +$$ +\lim _ {T \rightarrow \infty} \inf _ {\pi} \frac {\mathbb {E} _ {\nu , \pi} [ N _ {2 1} (T) + N _ {2 3} (T) ]}{T} = 1. \tag {28} +$$ + +On the other hand, since $w_{4}$ only gets positive utilities in $U(4,3)$ , to achieve the benchmark utility in $\pmb{\nu}$ , we need $N_{43}(T) = \frac{3}{4} T$ . Therefore, to guarantee sublinear approximation regret for $w_{4}$ , we must have + +$$ +\liminf_{T\to \infty}\frac{\mathbb{E}_{\boldsymbol{\nu},\pi}[N_{43}(T)]}{T}\geq \frac{3}{4}, +$$ + +which implies $\lim \sup_{T\to \infty}\frac{\mathbb{E}_{\nu,\pi}[N_{23}(T)]}{T}\leq \frac{1}{4}$ , since the total number of times that jobs $a_3$ being allocated cannot be more than the horizon $T$ . Therefore, $\lim \inf_{T\to \infty}\frac{\mathbb{E}_{\nu,\pi}[N_{21}(T)]}{T}\geq \frac{3}{4}$ according to Eq.(28). Using again the fact that a job can be allocated no more than $T$ times, we get + +$$ +\lim _ {T \rightarrow \infty} \sup _ {\pi} \frac {\mathbb {E} _ {\boldsymbol {\nu} , \pi} [ N _ {3 1} (T) ]}{T} \leq \frac {1}{4}. \tag {29} +$$ + +Finally, we write the regret of $w_{3}$ as + +$$ +\begin{array}{l} R e g _ {3} ^ {\alpha} (T) = \frac {3 T}{8} - \frac {1}{2} \mathbb {E} _ {\boldsymbol {\nu}, \pi} [ N _ {3 1} (T) ] - \frac {1}{4} \mathbb {E} _ {\boldsymbol {\nu}, \pi} [ N _ {3 4} (T) ] \\ = \frac {3 T}{8} - \frac {1}{4} \mathbb {E} _ {\boldsymbol {\nu}, \pi} [ N _ {3 1} (T) ] - \frac {1}{4} (\mathbb {E} _ {\boldsymbol {\nu}, \pi} [ N _ {3 4} (T) ] + \mathbb {E} _ {\boldsymbol {\nu}, \pi} [ N _ {3 1} (T) ]) \\ \geq \frac {T}{8} - \frac {1}{4} \mathbb {E} _ {\boldsymbol {\nu}, \pi} [ N _ {3 1} (T) ], \\ \end{array} +$$ + +where the inequality is since $w_{3}$ is matched at most $T$ times, namely, $\mathbb{E}_{\nu,\pi}[N_{34}(T)] + \mathbb{E}_{\nu,\pi}[N_{31}(T)] \leq T$ . Combining with Eq. (29), we have $\lim \inf_{T \to \infty} \frac{\text{Reg}_3^\alpha(T;\nu,\pi)}{T} \geq \frac{1}{8} - \frac{1}{4} \cdot \frac{1}{4} = \frac{1}{16}$ , which implies worker $w_{3}$ suffers linear approximation regret in $\pmb{\nu}$ . + +Thus, to summarize, assuming that all workers in both problem exhibit sublinear regret for this case leads to a contradiction. + +Case II: $\lim \inf_{T\to \infty}\frac{\mathbb{E}\nu,\pi[N_{11}(T)]}{T^{1 - 2\delta}} >0$ Our goal is to prove a lower bound on the regret in $\pmb{\nu}$ for some worker. + +For a fixed $T$ , denote for brevity $N = \mathbb{E}_{\nu, \pi}[N_{11}(T)]$ , and assume with contradiction that all workers $w_2, w_3, w_4$ suffer a regret smaller than $N/32$ . Denote the cumulative allocation given by the algorithm by $D \in [0, T]^{4 \times 4}$ , namely $D(i, j) = \sum_{t=1}^{T} \mathbb{1}\{(w_i, a_j) \in \mu_t\}$ . In particular, we know that $D(1,1) = N$ and that for all $i \in \{2, 3, 4\}$ , it holds that + +$$ +\forall i \in \{2, 3, 4 \}, \quad \sum_ {j = 1} ^ {4} \boldsymbol {U} (i, j) D (i, j) \geq \frac {3 T}{8} - \frac {N}{3 2} \tag {30} +$$ + +We now state a set of assumptions on the matching that the policy outputs at each round. Each of these assumptions never decreases the worker utility. Thus, and since we want to prove a contradiction in the utility lower bound of Eq. (30), they could be assumed without loss of generality. + +1. When $w_{1}$ is not matched to $a_{1}$ , it is always matched to $a_{2}$ ( $D(1, 1) + D(1, 2) = T$ ) - so that it suffers zero regret. +2. $a_3$ is always matched to either $w_2$ or $w_4$ ( $D(2, 3) + D(4, 3) = T$ ). +3. $a_1$ is always assigned to one of the first three workers $(D(1, 1) + D(2, 1) + D(3, 1) = T)$ . +4. If $a_1$ is not assigned to $w_3$ , then $a_4$ is assigned to $w_3$ ( $D(3, 1) + D(3, 4) = T$ ). + +Notice that changing each individual allocation to follow this condition can only require unmatched a worker from a job that yields her no utility and matching all conditions is feasible (e.g., by $\mu = \{(w_{1},a_{1}),(w_{3},a_{4}),(w_{4},a_{3})\}$ ). + +We now modify the matching allocation $D$ to allocation $\bar{D}$ while maintaining the above properties as follows: + +We initialize $\bar{D} = D$ +- If $D(4,3) \leq \frac{3T}{4} + \frac{N}{16}$ , we set $\bar{D}(4,3) = \frac{3T}{4} + \frac{N}{16}$ and $\bar{D}(2,3) = \frac{T}{4} - \frac{N}{16}$ ; otherwise, we leave $\bar{D}(4,3) = D(4,3)$ . By Eq. (30), we know that $D(4,3) \geq \frac{3T}{4} - \frac{N}{16}$ , and combined with Assumption 2, this change can only decrease the allocation to worker $w_{2}$ by + +$$ +D (2, 3) - \bar {D} (2, 3) = \bar {D} (4, 3) - D (4, 3) \leq \frac {N}{8} +$$ + +This decreases the utility of worker $w_{2}$ by $\frac{1}{2}\big(D(2,3) - \bar{D}(2,3)\big) \leq \frac{N}{16}$ , and after this modification, the cumulative utility of worker $w_{4}$ is $\frac{3T}{8} + \frac{N}{32}$ . + +- By Assumption 1, we know that $D(1,1) = N$ and $D(1,2) = T - N$ . We also know by assumption 4 that in all rounds where $a_1$ was assigned to $w_1$ , $a_4$ was assigned to $w_3$ , and therefore, $D(3,4) \geq N$ . In $\bar{D}$ , we move all the $N$ assignments of $(w_1, a_1)$ to $(w_1, a_2)$ , so that $\bar{D}(1,1) = 0$ and $\bar{D}(1,2) = T$ ; in particular, $w_1$ still gets its OSS. We split the allocation of $a_1$ evenly between $w_2$ and $w_3$ by letting: + +1. $\bar{D}(2,1) = \min \left\{T - \bar{D}(2,3), D(2,1) + N/2\right\}$ , thus making sure that the utility of $w_2$ is at least either $T/2 \geq 3T/8 + N/16$ or + +$$ +\frac {1}{2} \big (\bar {D} (2, 1) + \bar {D} (2, 3) \big) \geq \frac {1}{2} \left(D (2, 1) + \frac {N}{2} + D (2, 3) - \frac {N}{8}\right) \geq \frac {3 T}{8} - \frac {N}{3 2} + \frac {3 N}{1 6} = \frac {3 T}{8} + \frac {5 N}{3 2}, +$$ + +where in the last inequality we again used the assumption on the regret of $w_{2}$ in Eq.(30). + +2. We move the matches from $(w_{1},a_{1})$ that were not allocated to $D(2,1)$ as follows: + +$$ +\bar {D} (3, 1) = D (3, 1) + N - \left(\bar {D} (2, 1) - D (2, 1)\right) \geq D (3, 1) + N / 2, \quad \text {a n d}, +$$ + +$$ +\bar {D} (3, 4) = D (3, 4) - \left(N - \left(\bar {D} (2, 1) - D (2, 1)\right)\right) \geq D (3, 4) - N / 2. +$$ + +Both allocations are valid since $D(3,4) \geq N$ due to Assumption 4. In particular, this shift from $a_4$ to $a_1$ increases the utility of $w_3$ by at least $\left(\frac{1}{2} - \frac{1}{4}\right)\frac{N}{2} = \frac{N}{8}$ , ensuring it a total utility of at least $3T / 8 + 3N / 32$ . + +Notice that all changes either kept $\bar{D}$ a doubly-stochastic matrix or decreased the sum of a row - we can w.l.o.g increase another element of $\bar{D}$ or use partial matchings. + +Importantly, at the end of this process, the utility of $w_{1}$ under the matching distribution $\bar{D}$ remained $T / 2$ , while the utility of all other workers increased by at least $N / 16$ - contradicting the fact that no matching distribution can collect more than $3T / 8$ to all workers $w_{2}, w_{3}, w_{4}$ . Thus, Eq. (30) cannot hold and at least one worker must suffer a regret of at least $N / 32$ . Finally, since $\lim \inf_{T\to \infty}\frac{\mathbb{E}_{\nu,\pi}[N_{11}(T)]}{T^{1 - 2\delta}} > 0$ , we get the same for the regret of one of the workers, concluding the proof. + +# I Proof of Lemma 10 + +Proof. We proof the three properties as follows. + +# 1. Optimal Stable Share + +Under $\pmb{\nu}$ , consider the following stable matchings: $\mu_{1} = \{(w_{1},a_{2}),(w_{2},a_{1}),(w_{3},a_{4}),(w_{4},a_{3})\}$ and $\mu_{2} = \{(w_{1},a_{2}),(w_{2},a_{3}),(w_{3},a_{1}),(w_{4},a_{4})\}$ . The optimal stable share is $U^{*}(w) = \frac{1}{2}, \forall w \in \mathcal{W}$ with $w_{1}$ and $w_{2}$ receives it in both matchings, $w_{3}$ receives it in matching $\mu_{2}$ and $w_{4}$ receives it in matching $\mu_{1}$ . Under $\pmb{\nu}'$ , the optimal stable shares are $(u')^{*}(w_{1}) = \frac{1}{2} + \gamma$ , $(u')^{*}(w_{2}) = \frac{1}{2}$ , $(u')^{*}(w_{3}) = \frac{1}{4}$ , and $(u')^{*}(w_{4}) = 0$ , achieved through the stable matching $\mu' = \{(w_{1},a_{1}),(w_{2},a_{3}),(w_{3},a_{4}),(w_{4},a_{2})\}$ . + +# 2. Relevant Utility Gap + +Besides, the relevant utility gap for $\pmb{\nu}$ is $\Delta_{\mathrm{rel}}^{\nu} = 0$ since both $\mu_{1}$ and $\mu_{2}$ belongs to the Pareto-optimal stable matching set $S_{opt}^{U}$ . On the other hand, since the jobs have global preference rankings over the workers, i.e., serial dictatorship, $\mu'$ is the unique stable matching with respect to $\pmb{\nu}'$ , i.e., $S_{opt}^{U'} = \{\mu'\}$ . A perturbation of $-\gamma$ in $U(1,1)$ or $\gamma$ in $U(1,2)$ brings ties for worker $w_{1}$ , and hence would change $S_{opt}^{U'}$ to $S_{opt}^{U}$ . On the other hand, a perturbation of $\frac{1}{4}$ in $U(3,2)$ or $-\frac{1}{4}$ in $U(3,4)$ would make $S_{opt}^{U'}$ include both $\mu'$ and $\mu_2' = \{(w_1,a_1),(w_2,a_3),(w_3,a_2),(w_4,a_4)\}$ . All other entries need a perturbation of scale larger than $\frac{1}{4}$ to change the Pareto-optimal stable matching set. Since $\gamma < \frac{1}{4}$ , we have that $\Delta_{\mathrm{rel}}^{\nu'} = \gamma$ . + +# 3. Benchmark Utility + +Let $\gamma = cT^{-1/2 + \delta}$ for some $\delta \in (0, \frac{1}{2})$ . Then, under $\pmb{\nu}$ , we aim to minimize the approximation regret $Reg_{i}^{\alpha}(T)$ for every worker $w_{i}$ , while under $\pmb{\nu}'$ , the objective is to minimize regret $Reg_{i}(T)$ for each worker $w_{i}$ . Given an offline oracle that could compute the best possible approximation ratio, the benchmark utilities for the four workers (after multiplying the approximation ratio) are $\left(\frac{1}{2}, \frac{3}{8}, \frac{3}{8}, \frac{3}{8}\right)$ under $\pmb{\nu}$ and $\left(\frac{1}{2} + \gamma, \frac{1}{2}, \frac{1}{4}, 0\right)$ under $\pmb{\nu}'$ . For $\pmb{\nu}'$ , by serial dictatorship, the allocation scheme is equivalent to letting the workers choose their favorite jobs one by one. Doing so leads to the matching $\mu = \{(w_{1}, a_{2}), (w_{2}, a_{3}), (w_{3}, a_{1}), (w_{4}, a_{4})\}$ , which is unique since for all workers, when they choose their jobs, only a single unallocated job maximizes their utility. Hence, the benchmark utilities are immediately determined by the utility under this matching. + +For $\pmb{\nu}$ , since all workers but $w_{1}$ have no utility from job $a_{2}$ , and $\pmb{U}^{*}(w_{1}) = \pmb{U}(1,2)$ , it is always optimal to assign $a_{2}$ to $w_{1}$ deterministically and the benchmark utility for $w_{1}$ is $1/2$ . For the other players, if we match $w_{2}$ to $a_{1}$ , then for $w_{3}$ and $w_{4}$ , there are two possible matchings, i.e., $\{(w_{3},a_{4}),(w_{4},a_{3})\}$ and $\{(w_{3},a_{3}),(w_{4},a_{4})\}$ , but the first one is always better since it gives both players higher utilities. Similarly, if we match $w_{2}$ to $a_{3}$ , it is always better to select the matching $\{(w_{3},a_{1}),(w_{4},a_{4})\}$ rather than $\{(w_{3},a_{4}),(w_{4},a_{3})\}$ , and if $w_{2}$ is matched to $a_{4}$ , then we choosing $\{(w_{3},a_{1}),(w_{4},a_{3})\}$ yields higher utilities than choosing $\{(w_{3},a_{3}),(w_{4},a_{1})\}$ . Since all three three players have an OSS of $1/2$ , to maximize the OSS ratio, we need to compute a distribution $D$ over the three matchings $\mu_{a} = \{(w_{2},a_{1}),(w_{3},a_{4}),(w_{4},a_{3})\}$ , $\mu_{b} = \{(w_{2},a_{3}),(w_{3},a_{1}),(w_{4},a_{4})\}$ and $\mu_{c} = \{(w_{2},a_{4}),(w_{3},a_{1}),(w_{4},a_{3})\}$ , such that $\min \{u_D(w_2),u_D(w_3),u_D(w_4)\}$ is maximized. Noticing that each of the three matchings yields the OSS to two players and a lower utility for the third one, we can conclude that the optimal balance would be $u_D(w_2) = u_D(w_3) = u_D(w_4)$ - otherwise, we could increase the OSS-ratio by moving utility from the highest rewarded player to the lowest rewarded one. This condition is satisfied iff + +$$ +\mathbb {P} (\mu_ {a}) = \frac {1}{2}, \quad \mathbb {P} (\mu_ {b}) = \frac {1}{4}, \quad \mathbb {P} (\mu_ {c}) = \frac {1}{4}, +$$ + +and the benchmark utility for any of these three players is $\frac{3}{8}$ . + +![](images/1be958854fec87a014e2e4e351498c00576996c51cf704129c847fbbd1ff824f.jpg) + +# J Discussion Regarding Stable Matching with One-sided Ties and Job Utilities + +In this paper, we consider a matching market where one side has possibly tied cardinal preferences and the other side has strict ordinal preferences. It directly generalizes to the setting that both sides have cardinal preferences but only one side admits ties, by recovering an ordinal preference list from the utility, and we can define the OSS-ratio for the jobs in a similar fashion, denoted as $R_{\mathcal{M}}^a$ ; in the following, we rename the OSS-ratio for the workers as $R_{\mathcal{M}}^w$ for distinguishment. We claim that the setting with one-sided ties is not only practical in reality, but also important for our theoretical results, if we want to consider the OSS-ratio for both sides of the market. + +Stable Matching Without Ties The distributive lattice structure is a striking feature for a matching market without ties [36], which reveals that all workers could be optimally matched simultaneously, by simply running the deferred-acceptance algorithm with worker proposing, denoted as $\mu_w$ . Conversely, job-proposing deferred-acceptance algorithm, denoted as $\mu_a$ , gives every job its corresponding optimal stable match. Therefore, construct the distribution $D$ as follows: + +$$ +\mathbb {P} (D = \mu_ {w}) = \frac {1}{2}, \quad \mathbb {P} (D = \mu_ {a}) = \frac {1}{2}. +$$ + +Since $\mu_w$ and $\mu_a$ are both stable matchings, with distribution $D$ , we know that $R_{\mathcal{M}}^w \leq R_S^w \leq \frac{1}{2}$ , and the same result holds for $R_{\mathcal{M}}^a$ . This implies that both $R_{\mathcal{M}}^w$ and $R_{\mathcal{M}}^a$ are $\Theta(1)$ , and $\Omega(1)$ is a trivial lower bound for the two ratios. + +Stable Matching With One-sided Ties The lattice structure is absent when ties exist in the preference profiles [45]. When only one side of the market admits ties, we have proved that $R_{\mathcal{M}}^w = \Theta (\log N)$ . On the other hand, for $R_{\mathcal{M}}^a$ , we have the following result. + +Theorem 9. For a matching market where the workers have ties while the jobs have strict preferences, there exists an instance such that $R_{\mathcal{M}}^a = \Omega (N)$ . + +Proof. Consider the following utility matrix for a market with $N$ workers and $N$ jobs. The matrix encodes the preferences of the workers and the jobs simultaneously. + +$$ +\boldsymbol {U} = \left[ \begin{array}{c c c c} 1 & 1 & \dots & 1 \\ \varepsilon_ {1} & \varepsilon_ {1} & \dots & \varepsilon_ {1} \\ \varepsilon_ {2} & \varepsilon_ {2} & \dots & \varepsilon_ {2} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 0 \end{array} \right], +$$ + +where $\varepsilon_1 > \varepsilon_2 > \dots > 0$ . $u_{i,j} \geq u_{i,j'}$ implies $a_j \succsim u_i$ , $a_{j'}$ and $u_{i,j} \geq u_{i',j}$ implies $w_i \succsim a_j$ , $w_{i'}$ . In this example, every worker is indifferent among all the jobs, while every job has the preference profile that $w_1 \succ w_2 \succ \dots \succ w_N$ . The optimal stable share for each job is 1, and it is achieved by an appropriate tie-breaking of $w_1$ . However, in any matching, exactly one job would receive a utility of 1. When $\varepsilon_1, \varepsilon_2, \dots$ approach 0, we have that $\max_a \frac{U^*(a)}{U_D(a)} \geq N$ , with the equality achieved when we consider the distribution $D$ that assigns probability $1/N$ on matching $\mu_j$ , where $\mu_j$ refers to a matching in which job $a_j$ gets a utility of 1. Therefore, $\lim_{\varepsilon \to 0} R_{\mathcal{M}}^a = \Omega(N)$ . + +Stable Matching with Two-sided Ties When both sides of the market have ties, by symmetry, the OSS-ratio for the worker side and the job side would be of the same order. + +Theorem 10. For a matching market where both sides admit ties, there exists an instance, such that $R_{\mathcal{M}}^{w} = R_{\mathcal{M}}^{a} = \Omega(N)$ . + +Proof. Consider the following $N \times N$ utility matrix which simultaneously encodes the preferences of the workers and the jobs. + +$$ +\boldsymbol {U} = \left[ \begin{array}{c c c c} 1 & 1 & \dots & 1 \\ 1 & 0 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 0 & \dots & 0 \end{array} \right], +$$ + +where the entries of this utility matrix share the same correspondence to the preference profile as those in the example in Theorem 9. In this example, worker $w_{1}$ is indifferent among all the jobs, while all the other workers share the same preference over the jobs, that is, $a_{1} \succ a_{2} \sim a_{3} \sim \dots \sim a_{N}$ . The preference of jobs over workers is symmetrically derived. Every matching that involves all the workers and jobs is a stable matching, which gives a utility of 1 to worker $w_{1}$ and job $a_{1}$ , while for the remaining workers and jobs, at most one worker and one job would receive a utility of 1, and all the other workers get 0. For every worker and every job, there exists a tie-breaking mechanism such that it gets a utility of 1. Therefore, the optimal stable share is $U^{*}(w) = U^{*}(a) = 1$ for any $w$ and $a$ . And any distribution $D$ over matchings gives $\frac{U^{*}(w_{1})}{U_{D}(w_{1})} = \frac{U^{*}(a_{1})}{U_{D}(a_{1})} = 1$ , $\max_{w \in \{w_{2}, w_{3}, \dots, w_{N}\}} \frac{U^{*}(w)}{U_{D}(w)} \geq N - 1$ , and $\max_{a \in \{a_{2}, a_{3}, \dots, a_{N}\}} \frac{U^{*}(a)}{U_{D}(a)} \geq N - 1$ , with the equality achieved when we adopt the random allocation that assigns probability $1/(N - 1)$ on matching $\mu_{i}$ , in which workers $w_{1}$ and $w_{i}$ , jobs $a_{1}$ and $a_{i}$ receive a utility of 1, while all the other workers and jobs get 0. + +# K Discussion Regarding Two-sided Bandit Learning in Matching Markets + +In this paper, we consider bandit learning for one side of the market, where the preferences of jobs over workers are assumed to be known. A possible future direction would be to consider two-sided bandit learning for matching markets. + +For a matching market without ties, a fundamental result indicates that the worker-optimal stable matching is necessarily job-pessimial, and no stable matching simultaneously maximizes utility for both sides. Therefore, a reasonable benchmark would be to consider the optimal stable matching for workers and the pessimal stable matching for jobs, leading to the following regret definition: + +$$ +R e g _ {i} (T) = T \cdot \bar {\boldsymbol {U}} (w _ {i}) - \mathbb {E} \left[ \sum_ {t = 1} ^ {T} X _ {i} (t) \right], \quad \forall w _ {i} \in \mathcal {W}, +$$ + +$$ +R e g _ {j} (T) = T \cdot \underline {{U}} (a _ {j}) - \mathbb {E} \left[ \sum_ {t = 1} ^ {T} X _ {j} (t) \right], \quad \forall a _ {j} \in \mathcal {A}, +$$ + +where $\bar{U}(w_i)$ represents the utility from the worker-optimal stable matching and $\underline{U}(a_j)$ denotes the utility from the job-pessimial stable matching. + +Previous work [60] studied regret minimization for both sides of the market in a setting without ties, adopting the regret definitions above. Zhang and Fang [60] establish an $\mathcal{O}\left(K\log T / \Delta^2\right)$ regret bound for every agent, measured against their respective benchmark (worker-optimal and + +job-pessimial). Our algorithm directly generalizes to the same setting and matches the theoretical guarantees of Zhang and Fang [60] when there are no ties. This is because the initial exploration phase will find the strict preference list for both sides simultaneously w.h.p. and thus, after committing, the resulting Gale-Shapley output will be the worker-optimal / job-pessimial stable matching. + +In contrast, introducing ties to the market significantly complicates the analysis. The set of stable matchings expands substantially, and no single matching simultaneously satisfies the benchmarks defined for both sides. To extend our results to markets with ties, we propose using the Optimal Stable Share (OSS) as the benchmark for workers as defined in our paper. It is thus natural to introduce the equivalent Pessimal Stable Share (PSS) – the minimum utility a job can receive across all stable matchings. 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Existing work either requires interpolation or focuses only on univariate inputs. This paper presents new and somewhat surprising theoretical results for multivariate inputs. On two natural settings (1) generalization gap for flat solutions, and (2) mean-squared error (MSE) in nonparametric function estimation by stable minima, we prove upper and lower bounds, which establish that while flatness does imply generalization, the resulting rates of convergence necessarily deteriorate exponentially as the input dimension grows. This gives an exponential separation between the flat solutions compared to low-norm solutions (i.e., weight decay), which are known not to suffer from the curse of dimensionality. In particular, our minimax lower bound construction, based on a novel packing argument with boundary-localized ReLU neurons, reveals how flat solutions can exploit a kind of "neural shattering" where neurons rarely activate, but with high weight magnitudes. This leads to poor performance in high dimensions. We corroborate these theoretical findings with extensive numerical simulations. To the best of our knowledge, our analysis provides the first systematic explanation for why flat minima may fail to generalize in high dimensions. + +# 1 Introduction + +Modern deep learning is inherently overparameterized. In this regime, there are typically infinitely many global (i.e., zero-loss or interpolating) minima to the training objective, yet gradient-descent (GD) training seems to successfully avoid "bad" minima, finding those that generalize well. Understanding this phenomenon boils down to understanding the implicit biases of training algorithms [Zhang et al., 2021]. A large body of work has focused on understanding this phenomenon in the interpolation regime [Du et al., 2018, Liu et al., 2022], and the related concept of "benign overfitting" [Belkin et al., 2019, Bartlett et al., 2020, Frei et al., 2022]. + +While these directions have been fruitful, there is increasing evidence that rectified linear unit (ReLU) neural networks do not benignly overfit [Mallinar et al., 2022, Haas et al., 2023], particularly in the case of learning problems with noisy data [Joshi et al., 2023, Qiao et al., 2024]. Furthermore, for noisy labels, it takes many iterations of GD to actually interpolate the labels [Zhang et al., 2021]. This discounts theories based on interpolation to explain the generalization performance of practical neural networks, which would have entered the so-called edge-of-stability regime [Cohen et al., 2020], or stopped long before interpolating the training data. + +![](images/23d5d5662b52d9c75a97be8cdb9ff5539d697d68e25f549ed870e603d340b154.jpg) +Figure 1: The "neural shattering" phenomenon: From empirical observations to its geometric origin and theoretical consequences. Left panel: Training with a large learning rate and gradient descent empirically results in "neural shattering": Neurons develop large weights despite activating on very few inputs, leading to a high MSE of $\approx 1.105$ (red points). In contrast, explicit $\ell^2$ -regularization prevents this, achieving a much lower MSE of $\approx 0.055$ (orange points). Middle panel: The number of distinct directions, or "caps", on a high-dimensional sphere grows exponentially. Consequently, the data sites are spread thinly across these caps. This makes it trivial for a ReLU neuron to find a direction that isolates only a few data points. This sparse activation pattern allows neurons to use large weight magnitudes for this local fitting without impacting the global loss curvature, thus "tricking" the flatness criterion. Right panel: Visualization of "hard-to-learn" function from our minimax lower bound construction, built from localized ReLU neurons described in the middle panel. + +![](images/118644a37fecf2352ca63f5c2bcd8691185f91fbea20df52813e5de0e64adf9d.jpg) + +![](images/807d2218868ff5198e0258a14faf0cbbc3c7c9377fffd88c3d5a40f80bb539ee.jpg) + +To that end, it has been observed that an important factor that affects/characterizes the implicit bias of GD training is the notion of dynamical stability [Wu et al., 2018]. Intuitively, the (dynamical) stability of a particular minimum refers to the ability of the training algorithm to "stably converge" to that minimum. The stability of a minimum is intimately related to the flatness of the loss landscape about the minima [Mulayoff et al., 2021]. A number of recent works have focused on understanding linear stability, i.e., the stability of an algorithm's linearized dynamics about a minimum, in order to characterize the implicit biases of training algorithms [Wu et al., 2018, Nar and Sastry, 2018, Mulayoff et al., 2021, Ma and Ying, 2021, Nacson et al., 2023]. Minima that exhibit linear stability are often referred to as stable minima. In particular, Mulayoff et al. [2021] and Nacson et al. [2023] focus on the interpolation regime of two-layer overparameterized ReLU neural networks in the univariate input and multivariate input settings, respectively. Roughly speaking, the main takeaway from their work is that stability / flatness in parameter space implies a bounded-variation-type of smoothness in function space. + +Moving beyond the interpolation regime, Qiao et al. [2024] extend the framework of Mulayoff et al. [2021] and provide generalization and risk bounds for stable minima in the non-interpolation regime for univariate inputs. They show that for univariate nonparametric regression, the functions realized by stable minima cannot overfit in the sense that the generalization gap vanishes as the number of training examples grows. Furthermore, they show that the learned functions achieve near-optimal estimation error rates for functions of second-order bounded variation on an interval strictly inside the data support. While this work is a good start, it begs the questions of (i) what happens in the multivariate / high-dimensional case and (ii) what happens off of this interval (i.e., how does the network extrapolate). Indeed, these are key to understanding the implicit bias of GD trained neural networks, especially since learning high dimensions seems to always amount to extrapolation [Balestriero et al., 2021]. These two questions motivate the present paper in which we provide a precise answer to the following fundamental question. + +How well do stable minima of two-layer overparameterized ReLU neural networks perform in the high-dimensional and non-interpolation regime? + +We provide several new theoretical results for stable minima in this scenario, which are corroborated by numerical simulations. Some of our findings are surprising given the current state of understanding of stable minima. Notably, we show that, while flatness does imply generalization, the resulting sample complexity grows exponentially with the input dimension. This gives an exponential separation between flat solutions and low-norm solutions (weight decay) which are known not to suffer from the curse of dimensionality [Bach, 2017, Parhi and Nowak, 2023b]. + +# 1.1 Contributions + +In this paper, we provide new theoretical results for stable minima of two-layer ReLU neural networks, particularly in the high-dimensional and non-interpolation regime. Our primary contributions lie in the rigorous analysis of the generalization and statistical properties of stable minima and the resulting insights into their high-dimensional behavior. In particular, our contributions include the following. + +1. We establish that the functions realized by stable minima are regular in the sense of a weighted variation norm (Theorem 3.2 and Corollary 3.3). This norm defines a data-dependent function class that captures the inductive bias of stable minima.1 Furthermore, this regularity admits an analytic description as a form of weighted total variation in the domain of the Radon transform. These results synthesize and extend previous work [cf., Nacson et al., 2023, Qiao et al., 2024] by removing interpolation assumptions and generalizing them to multivariate inputs. +2. We analyze the generalization properties of stable minima in both a statistical learning setting and a nonparametric regression setting defined using the smoothness class above. + +- We establish that stable minima provably cannot overfit in the sense that their generalization gap (i.e., a uniform convergence bound) tends to 0 as the number of training examples $n \to \infty$ at a rate $n^{-\frac{1}{2d + 4}}$ up to logarithmic factors (Theorem 3.5). +- For high-dimensional $(d > 1)$ nonparametric regression, we show that stable minima (up to logarithmic factors) achieve an estimation error rate, in mean-squared error (MSE), upper bounded by $n^{-\frac{1}{2d + 4}}$ (Theorem 3.6). +- We prove a minimax lower bound of rate $n^{-\frac{2}{d + 1}}$ up to a constant (Theorem 3.7) on both the MSE and the generalization gap, which certifies that stable minima are not immune to the curse of dimensionality. This gives an exponential separation between flat solutions and low-norm solutions (weight decay) [Bach, 2017, Parhi and Nowak, 2023b]. +- By specializing the MSE upper bound to the univariate case ( $d = 1$ ), we show that stable minima (up to logarithmic factors) achieve an upper bound of $n^{-\frac{1}{6}}$ . Furthermore, by a construction specific to the univariate case, we have a sharper lower bound of $n^{-\frac{1}{2}}$ when $d = 1$ . These results should be contrasted to those of Qiao et al. [2024], who derive matching upper and lower bounds of $n^{-\frac{4}{5}}$ on an interval strictly inside the data support. Note that our results hold over the full domain, therefore capturing how the networks extrapolate. Thus, our results provide a more realistic characterization of the statistical properties of stable minima in the univariate case than in prior work. + +3. In Section 4, we corroborate our theoretical results with extensive numerical simulations. As a by-product, we uncover and characterize a phenomenon we refer to as "neural shattering" that is inherent to stable minima in high dimensions. This refers to the observation that each neuron in a flat solution has very few activated data points, which means that the activation boundaries of the ReLU neurons in the solutions shatter the data set into small pieces. This leads to poor performance in high dimensions. We also highlight that this observation exactly matches the construction of "hard-to-learn" functions for our minimax lower bound. Thus, our empirical validation combined with our theoretical analysis offers fresh insights into how high-dimensionality impacts neural network optimization and generalization. Indeed, our results reveal a subtle mechanism that leads to poor performance specifically in high dimensions. + +These results are based on two novel technical innovations in the analysis of minima stability in comparison to prior works, which we summarize below. + +Statistical bounds on the full input domain. The data-dependent nature of the stable minima function class implies that there are regions of the input domain where neuron activations are sparse for stable minima. This is because the functions in this class have local smoothness that can become arbitrarily irregular near the boundary of the data support. This makes it challenging to study the statistical performance of stable minima in the irregular regions. This was bypassed in the univariate case by Qiao et al. [2024] by restricting their attention to an interval strictly inside the data support, completely ignoring these hard-to-handle regions. Our analysis overcomes this via a novel technique + +that balances the error strictly inside the data support with the error close to the boundary. This allows us to establish meaningful statistical bounds on the full input domain. + +ReLU-specific minimax lower bound construction. We develop a novel minimax lower bound construction (see proof of Theorem 3.7) using functions built from sums of ReLU neurons. These neurons are strategically chosen to have activation regions near the boundary of the input domain. This exploits the "on/off" nature of ReLUs and high-dimensional geometry to create "hard-to-learn" functions. The data-dependent weighting allows these sparsely active, high-magnitude neurons to exist within the stable minima function class. This construction is fundamentally different from classical nonparametric techniques and is tightly linked to our experimental findings on neural shattering (see Figure 1). + +# 1.2 Related Work + +Stable minima and function spaces. Many works have investigated characterizations of the implicit bias of GD training from the perspective of dynamical stability [Wu et al., 2018, Nar and Sastry, 2018, Mulayoff et al., 2021, Ma and Ying, 2021, Nacson et al., 2023, Wang et al., 2022, Qiao et al., 2024]. In particular, Mulayoff et al. [2021] characterized the function-space implicit bias of minima stability for two-layer overparameterized univariate ReLU networks in the interpolation regime. This was extended to the multivariate case by Nacson et al. [2023] and, in the univariate case, this was extended to the non-interpolation regime by Qiao et al. [2024] with the addition of generalization guarantees. In this paper, we extend these works to the high-dimensional and non-interpolation regime and characterize the generalization and statistical properties of stable minima. + +Nonparametric function estimation with neural networks. It is well known that neural networks are minimax optimal estimators for a wide variety of functions [Suzuki, 2018, Schmidt-Hieber, 2020, Kohler and Langer, 2021, Parhi and Nowak, 2023b, Zhang and Wang, 2023, Yang and Zhou, 2024, Qiao et al., 2024]. Outside of the univariate work of Qiao et al. [2024], all prior works construct their estimators via empirical risk minimization problems. Thus, they do not incorporate the training dynamics that arise when training neural networks in practice. Thus, the results of this paper provide more practically relevant results on nonparametric function estimation, providing estimation error rates achieved by local minima that GD training can stably converge to. + +Loss curvature and generalization. A long-standing theory to explain why overparameterized neural networks generalize well is that the flat minima found by GD training generalize well [Hochreiter and Schmidhuber, 1997, Keskar et al., 2017]. Although there is increasing theoretical evidence for this phenomenon in various settings [Ding et al., 2024, Qiao et al., 2024], there is also evidence that sharp minima can also generalize [Dinh et al., 2017]. Thus, this paper adds complementary results to this list where we establish that, while flatness does imply generalization for two-layer ReLU networks, the resulting sample complexity grows exponentially with the input dimension. + +# 2 Preliminaries, Notation, and Problem Setup + +We investigate learning with two-layer ReLU neural networks. Our focus is on understanding the generalization and statistical performance of solutions obtained through GD training, particularly those that are stable. + +Neural networks. We consider two-layer ReLU neural networks with $K$ neurons. Such a network implements a function $f_{\theta}:\mathbb{R}^{d}\to \mathbb{R}$ of the form + +$$ +f _ {\boldsymbol {\theta}} (\boldsymbol {x}) = \sum_ {k = 1} ^ {K} v _ {k} \phi \left(\boldsymbol {w} _ {k} ^ {\mathsf {T}} \boldsymbol {x} - b _ {k}\right) + \beta , \tag {1} +$$ + +where $\pmb{\theta} = \{K\} \cup \{v_k, \pmb{w}_k, b_k\}_{k=1}^K \cup \{\beta\}$ denotes the collection of all neural network parameters, including the width $K \in \mathbb{N}$ . Here, $v_k \in \mathbb{R}$ denotes the output-layer weights, $\pmb{w}_k \in \mathbb{R}^d$ denotes the input-layer weights, $b_k \in \mathbb{R}$ denotes the input-layer biases, and $\beta \in \mathbb{R}$ denotes the output-layer bias. + +Data fitting and loss function. We consider the problem of fitting the data $\mathcal{D} = \{(x_i, y_i)\}_{i=1}^n$ , where $x_i \in \mathbb{R}^d$ and $y_i \in \mathbb{R}$ . We consider the empirical risk minimization problem with squared-error loss $\mathcal{L}(\boldsymbol{\theta}) = \frac{1}{2n} \sum_{i=1}^{n} (y_i - f_\boldsymbol{\theta}(\boldsymbol{x}_i))^2$ . + +Gradient descent and minima stability. We aim to minimize $\mathcal{L}(\cdot)$ via GD training, i.e., we consider the iteration $\theta_{t + 1} = \theta_t - \eta \nabla_\theta \mathcal{L}(\theta_t)$ , for $t = 0,1,2,\ldots$ , where $\eta >0$ is the step size / learning rate, $\nabla_{\theta}$ denotes the gradient operator with respect to $\theta$ , $\nabla_{\theta}^{2}$ denotes the Hessian operator with respect to $\theta$ , and the iteration is initialized with some initial condition $\theta_0$ . The analysis of these dynamics in generality is intractable in most cases. Thus, following the work of Wu et al. [2018], many works [e.g., Nar and Sastry, 2018, Mulayoff et al., 2021, Ma and Ying, 2021, Wang et al., 2022, Nacson et al., 2023, Qiao et al., 2024] have considered the behavior of this iteration using linearized dynamics about a minimum. Following Mulayoff et al. [2021], we consider the Taylor series expansion of the loss function about a minimum $\theta^{\star}$ . That is, + +$$ +\mathcal {L} (\boldsymbol {\theta}) \approx \mathcal {L} \left(\boldsymbol {\theta} ^ {\star}\right) + \left(\boldsymbol {\theta} - \boldsymbol {\theta} ^ {\star}\right) ^ {\top} \nabla_ {\boldsymbol {\theta}} \mathcal {L} \left(\boldsymbol {\theta} ^ {\star}\right) + \frac {1}{2} \left(\boldsymbol {\theta} - \boldsymbol {\theta} ^ {\star}\right) ^ {\top} \nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L} \left(\boldsymbol {\theta} ^ {\star}\right) \left(\boldsymbol {\theta} - \boldsymbol {\theta} ^ {\star}\right). \tag {2} +$$ + +As the GD iteration approaches a minimum $\theta^{\star}$ , it is well approximated by the linearized dynamics + +$$ +\boldsymbol {\theta} _ {t + 1} = \boldsymbol {\theta} _ {t} - \eta \left[ \nabla_ {\boldsymbol {\theta}} \mathcal {L} \left(\boldsymbol {\theta} ^ {\star}\right) + \nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L} \left(\boldsymbol {\theta} ^ {\star}\right) \left(\boldsymbol {\theta} _ {t} - \boldsymbol {\theta} ^ {\star}\right) \right], \quad t = 0, 1, 2, \dots . \tag {3} +$$ + +A minimum is said to be linearly stable if the GD iterates are "trapped" once they enter a neighborhood of the minimum. See Wu et al. [2018], Ma and Ying [2021], or Chemnitz and Engel [2025] for various rigorous definitions of linear stability that have appeared in the literature. It turns out that stability is tightly connected to the flatness of the minimum. Indeed, many equivalences have been proven, e.g., Mulayoff et al. [2021, Lemma 1], Qiao et al. [2024, Lemma 2.2], or Chemnitz and Engel [2025, Section 2.3]. We have the following proposition from Chemnitz and Engel [2025, p. 7]. + +Proposition 2.1. Suppose that $\eta < 2$ . A minimum $\theta^{\star}$ is linearly stable if and only if + +$$ +\lambda_ {\max } \left(\nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L} \left(\boldsymbol {\theta} ^ {\star}\right)\right) \leq \frac {2}{\eta}. \tag {4} +$$ + +Thus, we see that the stability of a minimum is equivalent to the flatness of the minimum under the assumption that the step size $\eta$ satisfies $\eta < 2$ . Thus, we make this assumption in the remainder of this paper. Given a data set $\mathcal{D}$ , we refer to the class of neural network parameters + +$$ +\Theta_ {\text {f l a t}} (\eta ; \mathcal {D}) := \left\{\boldsymbol {\theta}: \lambda_ {\max } \left(\nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L} (\boldsymbol {\theta})\right) \leq \frac {2}{\eta} \right\}, \tag {5} +$$ + +as the collection of flat / stable minima or flat / stable solutions. This parameter class is further motivated by empirical observations that GD often operates in the edge-of-stability regime, where $\lambda_{\max}(\nabla_{\pmb{\theta}}^{2}\mathcal{L}(\pmb{\theta}_{t}))$ hovers around $2 / \eta$ [Cohen et al., 2020, Damian et al., 2024]. + +# 3 Main Results + +In this section, we characterize the implicit bias of stable solutions. It turns out that every function $f_{\theta}$ , with $\pmb{\theta} \in \Theta_{\mathrm{flat}}(\eta; \mathcal{D})$ , is regular in the sense of a weighted variation norm. In particular, the weight function is a data-dependent quantity. This weight function reveals that neural networks can learn features that are intrinsic within the structure of the training data. To that end, given a data set $\mathcal{D} = \{(x_i, y_i)\}_{i=1}^n \subset \mathbb{R}^d \times \mathbb{R}$ , we consider a weight function $g: \mathbb{S}^{d-1} \times \mathbb{R} \to \mathbb{R}$ , where $\mathbb{S}^{d-1} := \{ \pmb{u} \in \mathbb{R}^d : \| \pmb{u} \| = 1 \}$ denotes the unit sphere. This weight is defined by $g(\pmb{u}, t) := \min \{ \tilde{g}(\pmb{u}, t), \tilde{g}(-\pmb{u}, -t) \}$ , where + +$$ +\tilde {g} (\boldsymbol {u}, t) := \mathbb {P} \left(\boldsymbol {X} ^ {\top} \boldsymbol {u} > t\right) ^ {2} \cdot \mathbb {E} \left[ \boldsymbol {X} ^ {\top} \boldsymbol {u} - t \mid \boldsymbol {X} ^ {\top} \boldsymbol {u} > t \right] \cdot \sqrt {1 + \left\| \mathbb {E} \left[ \boldsymbol {X} \mid \boldsymbol {X} ^ {\top} \boldsymbol {u} > t \right] \right\| ^ {2}}. \tag {6} +$$ + +Here, $X$ is a random vector drawn uniformly at random from the training examples $\{\pmb{x}_i\}_{i=1}^n$ . Note that the distribution $\mathbb{P}_X$ from which $\{\pmb{x}_i\}_{i=1}^n$ are drawn i.i.d. controls the regularity of $g$ . + +With this weight function in hand, we define a (semi)norm on functions of the form + +$$ +f _ {\nu , \boldsymbol {c}, c _ {0}} (\boldsymbol {x}) = \int_ {\mathbb {S} ^ {d - 1} \times [ - R, R ]} \phi \left(\boldsymbol {u} ^ {\mathsf {T}} \boldsymbol {x} - t\right) \mathrm {d} \nu (\boldsymbol {u}, t) + \boldsymbol {c} ^ {\mathsf {T}} \boldsymbol {x} + c _ {0}, \quad \boldsymbol {x} \in \mathbb {R} ^ {d}, \tag {7} +$$ + +where $R > 0$ , $c \in \mathbb{R}^d$ , and $c_0 \in \mathbb{R}$ . Functions of this form are "infinite-width" neural networks. We define the weighted variation (semi)norm as + +$$ +| f | _ {\mathrm {V} _ {g}} := \inf _ {\substack {\nu \in \mathcal {M} (\mathbb {S} ^ {d - 1} \times [ - R, R ]) \\ \boldsymbol {c} \in \mathbb {R} ^ {d}, c _ {0} \in \mathbb {R}}} \| g \cdot \nu \| _ {\mathcal {M}} \quad \text {s.t.} \quad f = f _ {\nu , \boldsymbol {c}, c _ {0}}, \tag{8} +$$ + +where, if there does not exist a representation of $f$ in the form of (7), then the seminorm4 is understood to take the value $+\infty$ . Here, $\mathcal{M}(\mathbb{S}^{d-1} \times [-R,R])$ denotes the Banach space of (Radon) measures and, for $\mu \in \mathcal{M}(\mathbb{S}^{d-1} \times [-R,R])$ , $\| \mu \|_{\mathcal{M}} \coloneqq \int_{\mathbb{S}^{d-1} \times [-R,R]} \mathrm{d}|\mu|(\boldsymbol{u},t)$ is the measure-theoretic total-variation norm. + +With this seminorm, we define the Banach space of functions $\mathrm{V}_g(\mathbb{B}_R^d)$ on the ball $\mathbb{B}_R^d \coloneqq \{\pmb{x} \in \mathbb{R}^d : \| \pmb{x} \|_2 \leq R\}$ as the set of all functions $f$ such that $|f|_{\mathrm{V}_g}$ is finite. When $g \equiv 1, | \cdot |_{\mathrm{V}_g}$ and $\mathrm{V}_g(\mathbb{B}_R^d)$ coincide with the variation (semi)norm and variation norm space of Bach [2017]. + +Example 3.1. Since we are interested in functions defined on $\mathbb{B}_R^d$ , for a finite-width neural network $f_{\theta}(\pmb{x}) = \sum_{k=1}^{K} v_k \phi(\pmb{w}_k^\top \pmb{x} - b_k) + \beta$ , we observe that it has the equivalent implementation as $f_{\theta}(\pmb{x}) = \sum_{j=1}^{J} a_j \phi(\pmb{u}_j^\top \pmb{x} - t_j) + \pmb{c}^\top \pmb{x} + c_0$ , where $a_j \in \mathbb{R}$ , $\pmb{u}_j \in \mathbb{S}^{d-1}$ , $t_j \in \mathbb{R}$ , $\pmb{c} \in \mathbb{R}^d$ , and $c_0 \in \mathbb{R}$ . Indeed, this is due to the fact that the ReLU is homogeneous, which allows us to absorb the magnitude of the input weights into the output weights (i.e., each $a_j = |v_{k_j}||\pmb{w}_{k_j}||_2$ for some $k_j \in \{1, \dots, K\}$ ). Furthermore, any ReLUs in the original parameterization whose activation threshold is outside $\mathbb{B}_R^d$ can be implemented by an affine function on $\mathbb{B}_R^d$ , which gives rise to the $c^\top \pmb{x} + c_0$ term in the implementation. If this new implementation is in "reduced form", i.e., the collection $\{(u_j, t_j)\}_{j=1}^J$ are distinct, then we have that $|f_{\theta}|_{\mathrm{V}_g} = \sum_{j=1}^{J} |a_j| g(u_j, t_j)$ . + +This example reveals that this seminorm is a weighted path norm of a neural network and, in fact, coincides with the path norm when $g \equiv 1$ [Neyshabur et al., 2015]. It also turns out that the data-dependent regularity induced by this seminorm is tightly linked to the flatness of a neural network minimum. We summarize this fact in the next theorem. + +Theorem 3.2. Suppose that $f_{\theta}$ is a two-layer neural network such that the loss $\mathcal{L}(\cdot)$ is twice differentiable at $\pmb{\theta}$ . Then, $|f_{\theta}|_{\mathrm{V}_g} \leq \frac{\lambda_{\max}(\nabla_\theta^2\mathcal{L}(\pmb{\theta}))}{2} - \frac{1}{2} + (R + 1)\sqrt{2\mathcal{L}(\pmb{\theta})}$ . + +The proof of this theorem appears in Appendix C. This theorem reveals that flatness implies regularity in the sense of the variation space $\mathrm{V}_g(\mathbb{B}_R^d)$ . In particular, we also have an immediate corollary for stable minima thanks to Proposition 2.1. + +Corollary 3.3. For any $\theta \in \Theta_{\mathrm{flat}}(\eta; \mathcal{D})$ , $|f_{\theta}|_{\mathrm{V}_g} \leq \frac{1}{\eta} - \frac{1}{2} + (R + 1)\sqrt{2\mathcal{L}(\theta)}$ . + +The main takeaway messages from Theorem 3.2 and Corollary 3.3 are that flat / stable solutions are smooth in the sense of $\mathrm{V}_g(\mathbb{B}_R^d)$ . In particular, we see that the Banach space $\mathrm{V}_g(\mathbb{B}_R^d)$ is the natural function space to study stable minima. This framework provides the mathematical foundation and sets the stage to investigate the generalization and statistical performance of stable minima. + +We also note that, from Corollary 3.3 and Example 3.1, for stable solutions $f_{\theta}$ , as the step size $\eta$ grows, the function $f_{\theta}$ becomes smoother, eventually approaching an affine function as $\eta \to \infty$ . This can be viewed as an example of the simplicity bias phenomenon of GD training [Arpit et al., 2017, Kalimeris et al., 2019, Valle-Perez et al., 2019]. + +Finally, we note that the function-space regularity induced by $\mathrm{V}_g(\mathbb{B}_R^d)$ has an equivalent analytic description via a weighted norm in the domain of the Radon transform of the function. This analytic description is based on the $\mathcal{R}$ -(semi)norm/second-order Radon-domain total variation inductive bias of infinite-width two-layer neural networks [Ongie et al., 2020, Parhi and Nowak, 2021, Bartolucci et al., 2023]. Before stating our theorem, we first recall the definition of the Radon transform. The Radon transform of a function $f\in L^{1}(\mathbb{R}^{d})$ is given by + +$$ +\mathcal {R} \{f \} (\boldsymbol {u}, t) = \int_ {\boldsymbol {u} ^ {\top} \boldsymbol {x} = t} f (\boldsymbol {x}) \mathrm {d} \boldsymbol {x}, \quad (\boldsymbol {u}, t) \in \mathbb {S} ^ {d - 1} \times \mathbb {R}, \tag {9} +$$ + +where the integration is against the $(d - 1)$ -dimensional Lebesgue measure on the hyperplane $\pmb{u}^{\top}\pmb{x} = t$ . Thus, we see that the Radon transform integrates functions along hyperplanes. + +Theorem 3.4. For every $f \in \mathrm{V}_g(\mathbb{B}_R^d)$ , consider the canonical extension $f_{\mathrm{ext}}: \mathbb{R}^d \to \mathbb{R}$ via its integral representation (7). It holds that $|f|_{\mathrm{V}_g} = \| g \cdot \mathcal{R}(-\Delta)^{\frac{d + 1}{2}} f_{\mathrm{ext}} \|_{\mathcal{M}}$ , where fractional powers of the Laplacian are understood via the Fourier transform. + +The proof of this theorem appears in Appendix D. We remark that the operators that appear in the theorem must be understood in the distributional sense (i.e., by duality). We refer the reader to Parhi and Unser [2024] for rigorous details about the distributional extension of the Radon transform. We also remark that a version of this theorem also appeared in Nacson et al. [2023, Theorem 1], but we note that their problem setting was the implicit bias of minima stability in the interpolation regime. + +# 3.1 Stable Solutions Generalize But Suffer the Curse of Dimensionality + +In the remainder of this paper, we focus on the scenario where inputs $\{\pmb{x}_i\}_{i=1}^n$ are drawn i.i.d. uniformly from the unit ball $\mathbb{B}_1^d$ (i.e., $R = 1$ ). Under this assumption, the population version of the weight function, which we denote as $g_P$ , has a well-defined asymptotic behavior. As detailed in Appendix E, for $|t| \geq 1$ , $g_P(\pmb{u}, t) = 0$ , and as $|t| \to 1^{-}$ , $g_P(\pmb{u}, t) \asymp (1 - |t|)^{d+2}$ . While the actual weight function $g$ in our analysis remains the empirical one derived from the data, this population behavior serves as a crucial analytical guide. Our proofs will show that the empirical $g$ concentrates around this population version (Appendix E.2). For clarity in expressing our main results and their dependence on dimensionality, we will characterize the function space of stable minima with respect to a canonical weight function $g(\pmb{u}, t) \coloneqq (1 - |t|)^{d+2}$ , which captures this essential asymptotic property. + +With this in hand, we can now characterize the generalization gap of stable minima, which is defined to be the absolute difference between the training loss and the population risk. We are able to characterize the generalization gap under mild conditions on the joint distribution of the training examples and the labels. + +Theorem 3.5. Let $\mathcal{P}$ denote the joint distribution of $(\pmb{x},y)$ . Assume that $\mathcal{P}$ is supported on $\mathbb{B}_1^d\times [-D,D]$ for some $D > 0$ and that the marginal distribution of $\pmb{x}$ is $\mathrm{Uniform}(\mathbb{B}_1^d)$ . Fix a data set $\mathcal{D} = \{(\pmb{x}_i,y_i)\}_{i = 1}^n$ , where each $(\pmb{x}_i,y_i)$ is drawn i.i.d. from $\mathcal{P}$ . Then, with probability $\geq 1 - \delta$ we have that for the plug-in risk estimator $\hat{R} (f)\coloneqq \frac{1}{n}\sum_{i = 1}^{n}(f(\pmb{x}_i) - y_i)^2$ + +$$ +\begin{array}{l} \sup _ {\boldsymbol {\theta} \in \Theta_ {\text {f l a t}} (\eta ; \mathcal {D})} \operatorname {G e n e r a l i z a t i o n G a p} (f _ {\boldsymbol {\theta}}; \hat {R}) := \left| \underset {(\boldsymbol {x}, y) \sim \mathcal {P}} {\mathbb {E}} \left[ (f _ {\boldsymbol {\theta}} (\boldsymbol {x}) - y) ^ {2} \right] - \hat {R} (f _ {\boldsymbol {\theta}}) \right| \\ \lesssim_ {d} \left(\frac {1}{\eta} - \frac {1}{2} + 4 M\right) ^ {\frac {d}{d ^ {2} + 4 d + 3}} M ^ {2} n ^ {- \frac {1}{2 d + 4}}, \tag {10} \\ \end{array} +$$ + +where $M := \max \{D, \| f_{\theta} \|_{L^{\infty}(\mathbb{B}_1^d)}, 1\}$ and $\lesssim_d$ hides constants (which could depend on $d$ ) and logarithmic factors in $n$ and $(1/\delta)$ . Furthermore, for any $L \geq D$ , it holds that + +$$ +\inf _ {\tilde {R}} \sup _ {\mathcal {P}} \underset {\mathcal {D} \sim \mathcal {P} ^ {\otimes n}} {\mathbb {E}} \left[ \sup _ {\substack {\boldsymbol {\theta} \in \Theta_ {\text {flat}} (\eta ; \mathcal {D}) \\ \| f _ {\boldsymbol {\theta}} \| _ {L ^ {\infty} \left(\mathbb {B} _ {1} ^ {d}\right)} \leq L}} \text {G e n e r a l i z a t i o n G a p} (f _ {\boldsymbol {\theta}}; \tilde {R}) \right] \gtrsim_ {d} L ^ {2} n ^ {- \frac {2}{d + 1}}, \tag{11} +$$ + +where the inf is over all risk estimators, $\gtrsim_d$ hides constants (which could depend on $d$ ), and the sup is over all distributions that satisfy the above hypotheses. + +The proof of this theorem appears in Appendix F. While this theorem does show that as $n \to \infty$ , the generalization gap vanishes, it reveals that the sample complexity grows exponentially with the input dimension (as seen by the $n^{-\frac{1}{2d + 4}}$ term in the upper bound and the $n^{-\frac{2}{d + 1}}$ term in the lower bound). This suggests that the curse-of-dimensionality is intrinsic to the stable minima set $\Theta_{\mathrm{flat}}(\eta; \mathcal{D})$ — not an artifact of our mathematical analysis nor the naive plug-in empirical risk estimator being suboptimal. On the other hand, for low-norm solutions (in the sense that they minimize the weight-decay objective), it can be shown that the generalization gap decays at a rate of $\widetilde{O}(n^{-\frac{1}{4}})$ , where $\widetilde{O}(\cdot)$ hides logarithmic factors [cf., Bach, 2017, Parhi and Nowak, 2023b]. This uncovers an exponential gap between flat and low-norm solutions, and, in particular, that stable solutions suffer the curse of dimensionality. When $d = 1$ , this result also provides a strict generalization of Qiao et al. [2024, Theorem 4.3], as they measure the error strictly inside the input domain, rather than on the full input domain. Thus, our result also characterizes how stable solutions extrapolate. + +# 3.2 Nonparametric Function Estimation With Stable Minima + +We now turn to the problem of nonparametric function estimation. As we have seen that $\mathrm{V}_g(\mathbb{B}_1^d)$ is a natural model class for stable minima, this raises two fundamental questions: (i) How well do stable minima estimate functions in $\mathrm{V}_g(\mathbb{B}_1^d)$ from noisy data? and (ii) What is the best performance any estimation method could hope to achieve for functions in $\mathrm{V}_g(\mathbb{B}_1^d)$ . In this section we provide answers to both these questions by deriving a mean-squared error upper bound for stable minima and a minimax lower bound for this function class. + +Theorem 3.6. Fix a step size $\eta >0$ and noise level $\sigma >0$ . Given a ground truth function $f_{0}\in \mathrm{V}_{g}(\mathbb{B}_{1}^{d})$ such that $\| f_0\|_{L^{\infty}}\leq B$ and $|f_0|_{\mathrm{V}_g}\leq \widetilde{O}\left(\frac{1}{\eta} -\frac{1}{2} +2\sigma\right)$ , suppose that we are given a data set $y_{i} = f_{0}(\pmb{x}_{i}) + \varepsilon_{i}$ , where $\pmb{x}_i$ are i.i.d. $\operatorname {Uniform}(\mathbb{B}_1^d)$ and $\varepsilon_{i}$ are i.i.d. $\mathcal{N}(0,\sigma^2)$ . Then, with probability $\geq 1 - \delta$ , we have that + +$$ +\frac {1}{n} \sum_ {i = 1} ^ {n} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - f _ {0} \left(\boldsymbol {x} _ {i}\right)\right) ^ {2} \lesssim_ {d} \left(\frac {1}{\eta} - \frac {1}{2} + 2 \sigma\right) ^ {\frac {d}{(2 d ^ {2} + 6 d + 3) (d + 2)}} B ^ {2} \left(\frac {\sigma^ {2}}{n}\right) ^ {\frac {1}{2 d + 4}}, \tag {12} +$$ + +for any $\pmb{\theta} \in \Theta_{\mathrm{flat}}(\eta; \mathcal{D})$ that is optimized, i.e., $(f_{\pmb{\theta}}(\pmb{x}_i) - y_i)^2 \leq (f_0(\pmb{x}_i) - y_i)^2$ , for $i = 1, \dots, n$ . + +The proof of this theorem appears in Appendix G. This theorem shows that optimized stable minima incur an estimation error rate that decays as $\widetilde{O}(n^{-\frac{1}{2d + 4}})$ , which suffers the curse of dimensionality. The optimized assumption is mild as it only asks that the error for each data point is smaller than the label noise $\sigma^2$ , which is easy to achieve in practice with GD training, especially in the overparameterized regime. The next theorem shows that the curse of dimensionality is actually necessary for this function class. + +Theorem 3.7. Consider the same data-generating process as in Theorem 3.6. We have the following minimax lower bounds. + +$$ +\inf _ {\hat {f}} \sup _ {\substack {f \in \mathrm {V} _ {g} (\mathbb {B} _ {1} ^ {d}) \\ \| f \| _ {L ^ {\infty}} \leq B, | f | _ {\mathrm {V} _ {g}} \leq C}} \mathbb {E} \| \hat {f} - f \| _ {L ^ {2}} ^ {2} \gtrsim_ {d} \left\{ \begin{array}{l l} \min (B, C) ^ {2} \left(\frac {\sigma^ {2}}{n}\right) ^ {\frac {2}{d + 1}}, & d > 1, \\ \min (B, C) ^ {2} \left(\frac {\sigma^ {2}}{n}\right) ^ {\frac {1}{2}}, & d = 1. \end{array} \right. \tag{13} +$$ + +where $\gtrsim_d$ hides constants (that could depend on $d$ ). + +The proof of this theorem appears in Appendix H. Our proof relies on two high-dimensional constructions. The first construction is to pack the unit sphere $\mathbb{S}^{d - 1}$ with $M = \exp (\Omega (d))$ pairwise-disjoint spherical caps, each specified by a unit vector $\pmb{u}_i$ as its center. Then, for every center $\pmb{u}_i$ the ReLU neuron $\varphi_{i}(\pmb {x}) = c\phi (\pmb{u}_{i}^{\top}\pmb {x} - t)$ is active only on its outward-facing cap, and attains its peak value $\min \{B,C\}$ by choosing a suitable $t$ . The second construction is to observe that since the weight function $g(\pmb {u},t)$ decreases quickly as $|t|\to 1$ (see Appendix E), the regularity constraint $|\cdot |_{\mathrm{V}_g}\leq C$ allows us to combine an exponential number of such atoms to construct a family of "hard-to-learn" functions. Traditional lower-bound constructions satisfy regularity by shrinking bump amplitudes + +(vertical changes), whereas our approach fundamentally differs by shifting and resizing bump supports (horizontal changes). Our experiments reveal that stable minima actually favor these kinds of hard-to-learn functions and we refer to this as the neural shattering phenomenon. + +# 4 Experiments + +In this section, we empirically validate our claims that (i) stable minima are not immune to the curse of dimensionality and (ii) the "neural shattering" phenomenon occurs. All synthetic data points are generated by uniformly sampling $\pmb{x}$ from $\mathbb{B}_1^d$ and $y_{i} = f_{0}(\pmb{x}_{i}) + \mathcal{N}(0,\sigma^{2})$ , where the ground-truth function $f_{0}(\pmb{x}) = \pmb{w}^{\top}\pmb{x}$ for some fixed vector $\pmb{w}$ with $\| \pmb{w}\| = 1$ . All the models are two-layer ReLU neural networks with width four times the training data size. The networks are randomly initialized by the standard Kaiming initialization [He et al., 2015]. We also use gradient clipping with threshold 50 to avoid divergence for large learning rates. + +**Curse of dimensionality.** In this experiment, we train neural networks with GD and vary the data set sizes in $\{32, 64, 128, 256, 512\}$ and dimensions in $\{1, 2, 3, 4, 5\}$ , with noise level $\sigma = 1$ . For each data set size, dimension, and training parameters ( $\eta = 0.2$ without weight decay and $\eta = 0.01$ with weight-decay 0.1), we conduct 5 experiments and take the median. The log-log curves are displayed in Figure 2. + +![](images/37f8addad53390d5f0454c421cd1fe6755a1fe9b0af08226296e00eccbbdf39f.jpg) +Figure 2: Empirical validation of the curse of dimensionality. Left panel: The slope of log MSE versus $\log n$ for training with vanilla gradient descent rapidly decreases with dimension, falling to about 0.1 at $d = 5$ . Right panel: Training with $\ell^2$ (weight decay) results in slopes above 0.5 in the log-log scale. + +![](images/e67113d3edc4d12a545cfd15800f939024d80b73c82ef084c3905e286463c8e1.jpg) + +Neural shattering. As briefly illustrated in the right panel of Figure 1, Figure 3 presents more detailed experiments. We train a two-layer ReLU network of width 2048 on 512 noisy samples $(\sigma = 1)$ of a 10-dimensional linear target. Under a large step size $\eta = 0.9$ (no weight decay), gradient descent enters a flat / stable minimum $(\lambda_{\max}(\nabla_{\theta}^{2}\mathcal{L}(\theta))$ oscillates around $2 / \eta \approx 2.2$ , signaling edge-of-stability dynamics). This drastically reduces each neuron's data-activation rate to $\leq 10\%$ , rather than reducing their weight norms. The network overfits (train MSE $\approx 1.105$ , matching the noise level). In contrast, with $\eta = 0.01$ plus $\ell^2$ -weight-decay $\lambda = 0.1$ , all neurons remain active and weight norms stay tightly bounded, so the model avoids overfitting (train MSE $\approx 0.055$ ). + +# 5 Discussion and Conclusion + +This paper presents a nuanced conclusion on the link between minima stability and generalization: Stable solutions do generalize, but when data is distributed uniformly on a ball, this generalization ability is severely weakened by the Curse of Dimensionality (CoD). Our analysis pinpoints the mechanism behind this failure. The implicit regularization from GD is not uniform across the input + +![](images/aada48975a26fcbdb1179b1f93bafd2b2044cd843f80d633b87c0284dbded779.jpg) + +![](images/584407775f217145fcce8c4a32519c8688c7af7e18eaa4b39bb73d8162268794.jpg) + +![](images/91ee21c290cf96538f5b3e2b7e42a079b5010a4c6684bcc493bda878a63e686f.jpg) + +![](images/5f971a044592310c342bc16b0bf8a0ee9f88d80508d83d152568c1403954eee2.jpg) +Figure 3: The top-left plot illustrates the neural shattering phenomenon: after large-step training each ReLU neuron (orange) is active on only a tiny fraction of the data (small horizontal support) yet its weight norm remains large, exactly as in our sphere-packing lower-bound construction where each outward-facing ReLU atom fires on very few inputs but retains full peak amplitude. + +![](images/450cb0a2fa6d977c5a8f675dc6fee5ec2a8353dcac81620c0473adeb8f8390e4.jpg) + +![](images/27bbd932bb523d9719e8d6a11dc57634698c50aa3096016a4c85a5dfa62b9b83.jpg) + +domain. While it imposes strong regularity in the strict interior of the data support, this guarantee collapses at the boundary. This localized failure of regularization is precisely what enables "neural shattering", a phenomenon where neurons satisfy the stability condition not by shrinking their weights, but by minimizing their activation frequency. This causes the CoD: The intrinsic geometry of a high-dimensional ball provides an exponential increase in available directions for shattering to occur, while the boundary regularization simultaneously weakens exponentially as the input dimension $d$ grows. This mechanism, confirmed by both our lower bounds and experiments, explains why stable solutions exhibit poor generalization in high dimensions. + +Several simplifications limit the scope of these results. The theory treats only two-layer ReLU networks and relies on the idealized assumption that samples are drawn uniformly from the unit ball. 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In International Conference on Learning Representations, 2023. + +# Supplementary Materials + +# A Additional Experiments 15 + +A.1 Empirical Evidence: High Dimensionality Yields Neural Shattering 15 +A.2 Neural Shattering and Learning Rate (dim=5) 16 +A.3 Neural Shattering in the Underparametrized Regime 17 +A.4 Neural Shattering for GELU Networks 18 +A.5 Empirical Analysis of the Curse of Dimensionality (I) 19 +A.6 Empirical Analysis of the Curse of Dimensionality (II) 20 +A.7 Empirical Analysis of the Curse of Dimensionality (III) 22 + +# B Overview of the Proofs 24 + +B.1 Proof Overview of Theorem 3.2. 24 +B.2 Proof Overview of Theorem 3.5 25 +B.3 Proof Overview of Theorem 3.6 25 +B.4 Proof Overview of Theorem 3.7 26 +B.5 Discussion of the Proofs 27 + +# C Proof of Theorem 3.2: Stable Minima Regularity 28 + +# D Proof of Theorem 3.4: Radon-Domain Characterization of Stable Minima 30 + +# E Characterization of the Weight Function for the Uniform Distribution 30 + +E.1 The Computation of the Population Weight Function 30 +E.2 Empirical Process for the Weight Function 34 + +# F Proof of Theorem 3.5: Generalization Gap of Stable Minima 36 + +F.1 Definition of the Variation Space of ReLU Neural Networks 36 +F.2 Metric Entropy and Variation Spaces 36 +F.3 Generalization Gap of Unweighted Variation Function Class 37 +F.4 Concentration Property on the Ball: Uniform Distribution 38 +F.5 Upper Bound of Generalization Gap of Stable Minima 38 + +# G Proof of Theorem 3.6: Estimation Error Rate for Stable Minima 42 + +G.1 Computation of Local Gaussian Complexity 42 +G.2 Proof of the Estimation Error Upper Bound 43 + +# H Proof of Theorem 3.7: Minimax Lower Bound 44 + +H.1 The Multivariate Case 44 +H.2 Why Classical Bump-Type Constructions Are Ineffective 48 +H.3 The Univariate Case 49 + +# I Lower Bound on Generalization Gap 51 + +I.1 The Lower Bound Construction Can be Realized by Stable Minima 51 + +# J Technical Lemmas 56 + +J.1 Information-Theoretic tools 56 +J.2 Poissonization and Le Cam's Inequality 57 + +# A Additional Experiments + +# A.1 Empirical Evidence: High Dimensionality Yields Neural Shattering + +![](images/07b2b99229cd14c228db60267e2114146406d988d29a0243d463f4facc3aa7b7.jpg) + +![](images/d816afcd8a7f32c78a24ab4b8ed88287502bfab02fbf6e77435cbb7a8d87cc1c.jpg) + +![](images/7be8a938bb06f14c3d31b6b70b6505f19cb48f802f137a9c94ca7e16309e4146.jpg) + +![](images/7c3063b097bce046675bfcc6f2792c3fa28c6b784f30ef9b3728e373af11de7b.jpg) + +![](images/319e00811325caffb203ed6031abe29d007b09c47c4609387a845791e860c20c.jpg) +Figure 4: Comparison across input dimension $d$ for a two-layer ReLU network of width 1024 trained on 512 samples for 20000 epochs with learning rate $\eta = 0.5$ . At $d = 1$ , all neurons extrapolate (0% active), while as $d$ increases the fraction of neurons surviving training rises dramatically (up to 65% at $d = 6$ ). Simultaneously, the training loss monotonically decreases whereas the training MSE increases with $d$ , demonstrating that neural shattering under large learning rates may be the key driver of the curse of dimensionality in stable minima. + +![](images/1e876de4f1e30cf29c774ea2b2a92e423724ed44eadc468cb307816d4ae1657c.jpg) + +# A.2 Neural Shattering and Learning Rate (dim=5) + +![](images/cd8ef561f83a2e6d5f73d3453c4c4db774a7ec0f64b7aae36c15a7a8d77cf7e8.jpg) + +![](images/ebe3719f7d5762ce9fa944b9a08873e7f2a30b84f2a6fecf0daf82825e80ad82.jpg) + +![](images/3719d8dc24292f269e85a6b328f2bd2693175f59ad7b9a7714d5353dfa2e28f3.jpg) + +![](images/8e21d8330f379a91a36edb47155839a8ddec71fc099bb9cc5b8e3c11f2686633.jpg) + +![](images/5a6fe636dc6567016e3be7c11444399c23e969e7d97ef55e8a9934aaf90d9d37.jpg) + +![](images/c99a96fd8b88c8ec5d0efff1a3e72211164df657cd1ec49456248a432652b60b.jpg) + +![](images/9e09bd746c7934a248dc376d1bacc25fa7426c4b5e03595ace946f9f4589693f.jpg) +Figure 5: Effect of increasing learning rate $\eta$ on shattering ( $\eta \times \mathrm{epochs} = 10000$ ): as $\eta$ grows, the stability/flatness constraint forces an ever larger fraction of neurons to activate only on a small subset of the data (neural shattering). To further decrease the training loss, gradient descent correspondingly increases the weight norms of the remaining active neurons. + +![](images/cb0c860fbd570dfebad40711cab8bb5ca8f7c564db267703b5a327b3a64a1f6b.jpg) + +# A.3 Neural Shattering in the Underparametrized Regime + +![](images/0c97cdb1a5de150f4bf511f844fc0b5704dbf617fdc7d3059ff03146409a0249.jpg) +(a) $W = 32$ weight decay $= 0$ + +![](images/d2a6c5effb26f0a8180362b78406cdb6d3d53b669c277281fde8cc95b2b3196e.jpg) +(b) $W = 48$ ,weightdecay $= 0$ + +![](images/6a0b34dce86bca103882e86e1057147fdc531ba32ccd225552e4b540f6692ca2.jpg) +(c) $W = 64$ weight decay $= 0$ + +![](images/3beb338806ac31edcd15b9b55f55be9fccc520c3dc9472550164b4188cbdd3be.jpg) +(d) $W = 32$ ,weightdecay $= 0.05$ + +![](images/c169d79263876a59a94adcb001606b7879688870f428522a4fd11b42326ff0ac.jpg) +(e) $W = 48$ weight decay $= 0.05$ + +![](images/1eddd4dcd14118e20307a6ccef8b488c4d6b0eced6c1b32b1198f767af5277e5.jpg) +(f) $W = 64$ weight decay $= 0.05$ + +![](images/73780560f1c1c6848329eae822db68ebf4c1d4f4482e110c46718ae422ff269d.jpg) +Figure 6: Persistence of neural shattering across width in the underparametrized regime. Each panel plots per-neuron activation rate versus weight norm after training a two-layer ReLU network on $n = 1024$ samples with learning rate $\eta = 0.5$ . Columns correspond to widths $W \in \{32, 48, 64\}$ ; the top row uses no weight decay, and the bottom row uses mild decay ( $\lambda = 0.05$ ). The parameter count is $Wd + W + 1$ , so $W = 64$ is mildly overparameterized while $W = 32, 48$ are strictly underparameterized. Across all widths, we observe clear neural shattering: neurons with smaller activation fractions carry larger weight norms. This monotone trend is especially visible in the stable-minima panels ( $\lambda = 0$ ), exactly as predicted by our theory. The weight-decay panels serve only as a high-activation baseline to calibrate what "few activations" means, underscoring how exceptionally low the activation rates are at stable minima. +(a) $d = 32$ +Figure 7: Each panel shows the relation between neuron activation rate and weight norm after training a two-layer ReLU network of width $W = 16$ on $n = 1024$ samples drawn from a linear target with learning rate $\eta = 0.5$ for $d \in \{64, 48, 32\}$ . This observation indicates that neural shattering is a generic feature of stable minima, robust even when in small network. + +![](images/33165e9ab65d685e0baad906e9791cb178e68a7f20190b8905a6ae11c872733b.jpg) +(b) $d = 48$ + +![](images/6c7315848af885a1ac45eec6915e68ea0a26abdf43889f8df0a5b22f0479fee6.jpg) +(c) $d = 64$ + +# A.4 Neural Shattering for GELU Networks + +This set of stable minima serves as a prism through which we can understand the emergent behaviors of the learning process. The data-dependent weight function $g(\boldsymbol{u}, t)$ , which is central to our analysis, arises directly from the structure of the loss Hessian and provides a static characterization of the implicit bias of gradient dynamics. A smaller value of $g(\boldsymbol{u}, t)$ for a neuron $(\boldsymbol{u}, t)$ implies that the stability condition imposes a weaker regularization, allowing for larger weight magnitudes for that neuron. + +This static view is intimately connected to the underlying learning dynamics. In high-dimensional spaces, a neuron's activation boundary can easily drift to a region where it activates on only a small fraction of the data points. For such a neuron, the gradients it receives are sparse and localized. If the few data points it activates on are already well-fitted, the local gradient signal can vanish, causing the neuron's parameters to become effectively "stuck" or stable. The small value of $g(\boldsymbol{u}, t)$ in these boundary regions creates "space" within the class of stable functions for these trapped, high-magnitude, yet sparsely-activating neurons to exist, a possibility our lower bound construction then formalizes and exploits. + +The ReLU activation function is analytically convenient for this analysis because its hard-sparsity property: a strictly zero gradient for non-activating inputs. This leads to a sparse loss Hessian, allowing for a clean derivation of the weight function $g(\boldsymbol{u}, t)$ . However, the underlying "stuck neuron" dynamic is not necessarily unique to ReLU. Activations like GELU provide a non-zero gradient for negative inputs, but this signal is weak and decays quickly away from the activation boundary. It is therefore plausible that this weak gradient is insufficient to pull a "stuck" neuron back from the data boundary once it has drifted there and its activation rate has diminished. This suggests that the fundamental mechanism enabling "neural shattering" may persist. This hypothesis motivates an empirical investigation into whether the same phenomena of neural shattering also manifests in networks trained with GELU activations. + +![](images/697927f7a435e6e404908adaad187a1505684b518278e41cea44a5d57251be34.jpg) + +![](images/437ea62a79beb21587252182ad6984a80428bd476c801fd71f61a9ff4cd5582f.jpg) + +![](images/3a7ba92f6305770f09578f32b33ef38e4c82bf5192d554e34c23eba4e5fc4a22.jpg) +Figure 8: Comparison across input dimension $d$ for a two-layer GELU network of width 1024 trained on 512 samples for 20000 epochs with learning rate $\eta = 0.5$ . The neural shattering behavior observed for ReLU networks in Figure 4 also appears here with GeLU activations. In particular, we can see the trend more clearly: neurons with lower activation rates tend to develop larger weight norms, highlighting that the neural shattering mechanism extends beyond piecewise-linear activations. + +![](images/ff06dfc55fe4ea88ff77ab97d77148a114dd21b13bee819404715709a80c01e0.jpg) + +# A.5 Empirical Analysis of the Curse of Dimensionality (I) + +We conduct the following experiments in the setting where the ground-truth function is linear with Gaussian noise $\sigma^2 = 1$ . The width of neural network is 4 times of the number of samples. + +![](images/4360c90f912a113dbd666267b8ba909f2b981cd8ec8ecbf58578c381c6530ffb.jpg) + +![](images/26c94c955b330979c7ec8e497a6773b2652133841d95bae1e530e9b0259c61c6.jpg) + +![](images/a1881edf7cb626f0f872310ab9551d29e56fb4bdfbc99aa0014f2dbd1bc5baa9.jpg) + +![](images/cc08c4bba05bc2a21b40e2178a9a2206de7540ad6ea37fcc690605c187aa1aab.jpg) + +![](images/5df8489260a87cc89bc767b44e470b03d5022398d88f53b3ee8af22464f3c797.jpg) +Figure 9: Log-log plots of the mean squared error (MSE) versus sample size $n$ (Part I). Each curve is regressed by the median result over five random initializations (lighter markers), while the shallow markers denote the other runs. As the input dimension increases, the slope of the fitted regression line becomes progressively shallower, indicating slower error decay. + +![](images/3ffaa75abc91d9fe95bac027d64674ad1f6d63cc33edaea5a93693fd6a87738f.jpg) + +![](images/01894a50da17eb49d7617963ea7eb8ba65c7890402c07108e107ebf1aa6c7e88.jpg) + +![](images/8e1078a953c5b063a5a1956ad92f3a777cad1ecb4bd9c8fed41bab730db46609.jpg) + +![](images/f69740ab9f8f67935b237b1ed4df52f22315023f98c6571b0334e220c5bb7dd8.jpg) +Figure 10: Log-log plots of the mean squared error (MSE) versus sample size $n$ , illustrating the curse of dimensionality in stable minima (Part II). We can see in dimension 5, the slope is almost flat and even the large-step size cannot save the results (even worse than small step-size). + +![](images/0681124c152c322fd0486a2ca20ab9cd67d6d8e4833200cbb33d41976e45827d.jpg) + +# A.6 Empirical Analysis of the Curse of Dimensionality (II) + +We conduct the following experiments in the setting where the ground-truth function is linear with Gaussian noise $\sigma^2 = 0.25$ . The width of neural network is 2 times of the number of samples. + +![](images/b528341072af7d80c95bb3c1c9b5cb9e12ab774609ea3a05532700a3525f5d4d.jpg) +Figure 11: Log-log plots of the mean squared error (MSE) versus sample size $n$ (Part III). Compared to the previous experiments, this setup reduces the noise level to $\sigma = 0.5$ , applies weight decay $\lambda = 0.01$ , and constrains the model width to $2n$ . + +![](images/4ae8c5cb8813251a5acd69793ab7d1c0f8a3b5e3855bc362fc22bc5dd86ea5b5.jpg) + +![](images/8a173e71e866617e4fc7045aa74d4acc6ba76f9961e44c84bd4c0004bd6e0f7e.jpg) + +![](images/ed962025b9ff3a31877059005c4e78354b2c63374e0bd0ea964e2ee1a094ba50.jpg) + +![](images/87941d0896c345b097f0a9dce667950fd8ee1b27a6f06d5da4044b2e0284b347.jpg) + +![](images/c0d4925066bc96f21755e460c691e7cb7d95609f73a0fad024a2c90084953b0c.jpg) + +![](images/a52d5191e4bf4ac30fe81b398ec68db164cef19e945e69521c7c269cb921bd14.jpg) +Figure 12: Log-log plots of the mean squared error (MSE) versus sample size $n$ (Part IV). The log-log MSE vs. $n$ curves still exhibit progressively flattening slopes as the input dimension grows, demonstrating the enduring curse of dimensionality in stable minima. + +![](images/ff1f7598f045c094f8ac24f8667d1ffa9cc9efd180de066c234878e79b1bdb40.jpg) + +# A.7 Empirical Analysis of the Curse of Dimensionality (III) + +We conduct the following experiments in the setting where the ground-truth function is Hölder(1/2) $f(\pmb{x}) = \frac{1}{d}\sum_{i=1}^{d}|\pmb{u}_j^\top\pmb{x}|^{1/2} + 1$ with Gaussian noise $\sigma^2 = 0.25$ , where $\pmb{u}_j$ is uniformly sampled from $\mathbb{S}^{d-1}$ . The width of neural network is 2 times of the number of samples. + +![](images/318411f149565fd42ee62f33452d2e4a6a2fce04d728aa63099a8b92f0a345a8.jpg) +Figure 13: Log-log plots of the mean squared error (MSE) versus sample size $n$ (Part V). We can see the generalization slopes of stable minima degrades as dimension increase from 1 to 4. + +![](images/b3709065fe72baa5dec79da668cb85210dccf4c4ac07c54aec88863e1b4e6629.jpg) + +![](images/02c19e1fadc0f98c01355ceb2704b664e80660e760b30f0f269df2be6962c578.jpg) + +![](images/ead7fb66b8ef0abefaa5482d66f7bdacef0ef11869495bc78be9614a959d3765.jpg) + +![](images/01d71fc4418f06ed16fe9b147d0e5e9cbf6564dbf35e7091075aa3e4fd7e7b7b.jpg) +Figure 14: Log-log plots of the mean squared error (MSE) versus sample size $n$ (Part VI). The panels on the left are the log-log plots for stable minima trained in $\eta \in \{0.05, 0.1, 0.2, 0.4\}$ , while the panels on the left are the log-log plots for low-norm solutions trained in weight decay $\lambda = 0.01$ . + +![](images/cfe869586404d261bb3483021303ec16508b19149078791c2ff3aa96243fe722.jpg) + +![](images/6d341feeb435e9463d3a129a14737dfa2b525325d6250be23eb3157bb94c0bf8.jpg) + +![](images/97f2effd26f8934006c716ca125fce2056782da6ef995a9d3faa5b6f90678b79.jpg) + +![](images/4ecef57c44516a4d02c628d6f641440f4f88261355d367827f5e27c12f933185.jpg) +Figure 15: Log-log plots of the mean squared error (MSE) versus sample size $n$ (Part VII). The panels on the left are the log-log plots for stable minima trained in different learning rate $\eta \in \{0.05, 0.1, 0.2, 0.4\}$ , while The panels on the left are the log-log plots for low-norm solutions trained in weight decay $\lambda = 0.01$ . + +![](images/9d83638c847d60dfd2468a52651368522a503d431e3ac4e69bfea2df22251e05.jpg) + +# B Overview of the Proofs + +In this section, we provide an overview of the proofs of the claims in the paper. The full proofs are deferred to later appendices. We introduce the following notations we use in our proofs and their overviews. + +- Let $\varphi(\varepsilon)$ and $\psi(\varepsilon)$ be two functions in variable of $\varepsilon$ . For constants $a, b \in \mathbb{R}$ (independent of $\varepsilon$ ), the notation + +$$ +\varphi (\varepsilon) \stackrel {a} {\underset {b} {\sim}} \psi (\varepsilon) +$$ + +means that $\varphi(\varepsilon) \leq a\psi(\varepsilon)$ and $b\psi(\varepsilon) \leq \varphi(\varepsilon)$ . We may directly use the notation $\asymp$ if the constants are hidden (we may use the simplified version when the constants are justified). + +- $f(x) = O(g(x))$ means there exist constants $c > 0$ and $x_0 > 0$ such that + +$$ +0 \leq f (x) \leq c g (x), \quad \forall x \geq x _ {0}. +$$ + +Intuitively, for sufficiently large $x$ , $f(x)$ grows at most as fast as $g(x)$ , up to a constant factor. We may also use $f(x) \lesssim g(x)$ . + +- $f(x) = \Omega(g(x))$ means there exist constants $c' > 0$ and $x_1 > 0$ such that + +$$ +0 \leq c ^ {\prime} g (x) \leq f (x), \quad \forall x \geq x _ {1}. +$$ + +Intuitively, for sufficiently large $x$ , $f(x)$ grows at least as fast as $g(x)$ , up to a constant factor. + +- $f(x) = \Theta(g(x))$ means there exist constants $c_1, c_2 > 0$ and $x_2 > 0$ such that + +$$ +0 \leq c _ {1} g (x) \leq f (x) \leq c _ {2} g (x), \quad \forall x \geq x _ {2}. +$$ + +Equivalently, + +$$ +f (x) = \Theta (g (x)) \Longleftrightarrow [ f (x) = O (g (x)) ] \text {a n d} [ f (x) = \Omega (g (x)) ]. +$$ + +Intuitively, for sufficiently large $x$ , $f(x)$ grows at the same rate as $g(x)$ , up to constant factors. + +# B.1 Proof Overview of Theorem 3.2. + +We consider the neural network of the form: + +$$ +f _ {\boldsymbol {\theta}} (\boldsymbol {x}) = \sum_ {k = 1} ^ {K} v _ {k} \phi \left(\boldsymbol {w} _ {k} ^ {\mathsf {T}} \boldsymbol {x} - b _ {k}\right) + \beta . \tag {14} +$$ + +The Hessian matrix of the loss function, obtained through direct computation, is expressed as: + +$$ +\nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L} (\boldsymbol {\theta}) = \frac {1}{n} \sum_ {i = 1} ^ {n} \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) ^ {\top} + \frac {1}{n} \sum_ {i = 1} ^ {n} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - y _ {i}\right) \nabla_ {\boldsymbol {\theta}} ^ {2} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right). \tag {15} +$$ + +Consider $\pmb{v}$ to be the unit eigenvector (i.e., $\| \pmb{v} \|_2 = 1$ ) corresponding to the largest eigenvalue of the matrix $\frac{1}{n} \sum_{i=1}^{n} (\nabla_\theta f_\theta(\pmb{x}_i)) (\nabla_\theta f_\theta(\pmb{x}_i))^{\top}$ . Consequently, the maximum eigenvalue of the Hessian of the loss can be lower-bounded as follows: + +$$ +\begin{array}{l} \lambda_ {\max } \left(\nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L} (\boldsymbol {\theta})\right) \geq \boldsymbol {v} ^ {\top} \nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L} (\boldsymbol {\theta}) \boldsymbol {v} \\ = \underbrace {\lambda_ {\max } \left(\frac {1}{n} \sum_ {i = 1} ^ {n} \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) ^ {\top}\right)} _ {\text {(T e r m A)}} \tag {16} \\ + \underbrace {\frac {1}{n} \sum_ {i = 1} ^ {n} (f _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {i}) - y _ {i}) \boldsymbol {v} ^ {\mathsf {T}} \nabla_ {\boldsymbol {\theta}} ^ {2} f _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {i}) \boldsymbol {v}} _ {\mathrm {(T e r m B)}}. \\ \end{array} +$$ + +Regarding (Term A), its maximum eigenvalue at a given $\theta$ can be related to the $\mathrm{V}_g$ seminorm of the associated function $f = f_{\theta}$ . Letting $\Omega = \mathbb{B}^d(\mathbf{0}, R)$ , Nacson et al. [2023, Appendix F.2] demonstrate that: + +$$ +(\text {T e r m A}) = \lambda_ {\max } \left(\frac {1}{n} \sum_ {i = 1} ^ {n} \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) ^ {\top}\right) \geq 1 + 2 \sum_ {k = 1} ^ {K} \left| v _ {k} \right| \| \boldsymbol {w} _ {k} \| _ {2} \tilde {g} \left(\bar {\boldsymbol {w}} _ {k}, \bar {b} _ {k}\right), \tag {17} +$$ + +where $\bar{\pmb{w}}_k = \pmb {w}_k / \| \pmb {w}_k\| _2\in \mathbb{S}^{d - 1}$ $\bar{b}_k = b_k / \| \pmb {w}_k\| _2$ and + +$$ +\tilde {g} (\bar {\boldsymbol {w}}, \bar {b}) = \mathbb {P} \left(\boldsymbol {X} ^ {\top} \bar {\boldsymbol {w}} > \bar {b}\right) ^ {2} \cdot \mathbb {E} \left[ \boldsymbol {X} ^ {\top} \bar {\boldsymbol {w}} - \bar {b} \mid \boldsymbol {X} ^ {\top} \bar {\boldsymbol {w}} > \bar {b} \right] \cdot \sqrt {1 + \left\| \mathbb {E} \left[ \boldsymbol {X} \mid \boldsymbol {X} ^ {\top} \bar {\boldsymbol {w}} > \bar {b} \right] \right\| ^ {2}}. \tag {18} +$$ + +For (Term B), an upper bound can be established using the training loss $\mathcal{L}(\pmb{\theta})$ via the Cauchy-Schwarz inequality. This also employs a notable uniform upper bound for $\pmb{v}^{\top}\nabla_{\pmb{\theta}}^{2}f_{\pmb{\theta}}(\pmb{x}_{n})\pmb{v}$ , as detailed in Lemma C.1: + +$$ +| (\text {T e r m B}) | \leq \sqrt {\frac {1}{n} \sum_ {i = 1} ^ {n} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - y _ {i}\right) ^ {2}} \cdot \sqrt {\frac {1}{n} \sum_ {i = 1} ^ {n} \left(\boldsymbol {v} ^ {\top} \nabla_ {\boldsymbol {\theta}} ^ {2} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) \boldsymbol {v}\right) ^ {2}} \leq 2 (R + 1) \sqrt {2 \mathcal {L} (\boldsymbol {\theta})}. \tag {19} +$$ + +# B.2 Proof Overview of Theorem 3.5 + +This proof establishes an upper bound on the generalization gap for stable minima in $\Theta_{\mathrm{flat}}(\eta; \mathcal{D})$ . The strategy leverages the structural properties of these solutions, which are captured by a data-dependent weighted variation norm. + +First, we recall from Corollary 3.3 that any stable solution $f_{\theta}$ has a bounded norm, $|f_{\theta}|_{\mathrm{V}_g} \leq A$ . The weight function $g$ is determined by the training data. This data-dependent nature is central to our analysis. To bound the generalization gap, we must translate the constraint on the weighted norm $|f|_{\mathrm{V}_g}$ into a bound on the standard, unweighted norm $|f|_{\mathrm{V}}$ . This is possible only in regions where the weight function $g$ is bounded away from zero. This naturally suggests a decomposition of the input domain into two parts: a "well-behaved" region where $g$ has a positive lower bound, and the remaining region where $g$ may be arbitrarily close to zero. + +To facilitate a tractable analysis, we introduce the deterministic population weight function $g_{P}$ as a reference. We then bridge the two using empirical process theory. As established in Appendix E.2 (Theorem E.5), the uniform deviation between $g$ and its population counterpart is bounded by a statistical error term $\epsilon_{n} = \tilde{O}(\sqrt{d / n})$ with high probability. This allows us to leverage the well-behaved properties of $g_{P}$ to characterize the behavior of the empirical function $g$ . + +For inputs from $\mathrm{Uniform}(\mathbb{B}_1^d)$ , this population function behaves like $(1 - |t|)^{d + 2}$ (see Appendix E), where $|t|$ is the distance of a neuron's activation boundary from the origin. This behavior motivates a specific geometric decomposition: an inner core $\mathbb{B}_{1 - \varepsilon}^d$ (where $|t| < 1 - \varepsilon$ ) and an outer annulus $\mathbb{A}_{\varepsilon}^d$ . + +For the inner core $\mathbb{B}_{1 - \varepsilon}^{d}$ , the key step is to translate the bound on $|f|_{\mathrm{V}_g}$ to a bound on the standard (unweighted) norm $|f|_{\mathrm{V}}$ . To do this, we need a lower bound on the empirical weight $g$ within the core. Using $g_{P}$ as our analytical proxy, we establish that $g_{\min} \geq g_{P,\min} - \varepsilon_n \approx \varepsilon^{d + 2} - \varepsilon_n$ . This step requires a validity condition: $\varepsilon$ must be large enough such that the geometric term $\varepsilon^{d + 2}$ dominates the statistical error $\varepsilon_n$ . With the unweighted norm now bounded, we utilize metric entropy arguments (e.g., Proposition F.4 and results from Parhi and Nowak [2023b]) to bound the generalization error in the core that scales with $O\left(\varepsilon^{-\frac{d(d + 2)}{2d + 3}}n^{-\frac{d + 3}{4d + 6}}\right)$ . In the annulus $\mathbb{A}_{\varepsilon}^{d}$ , the contribution is small, scaling with $O(\varepsilon)$ . + +# B.3 Proof Overview of Theorem 3.6 + +The proof for Theorem 3.6 establishes an upper bound on the mean squared error (MSE) for estimating a true function $f_{0}$ using a stable minimum $f_{\theta}$ . The overall strategy shares similarities with the proof of the generalization gap, particularly in its treatment of the data-dependent function class. + +The argument begins by leveraging the property that a stable minimum $\theta \in \Theta_{\mathrm{flat}}(\eta; \mathcal{D})$ corresponds to a neural network $f_{\theta}$ with a bounded weighted variation norm $|f_{\theta}|_{\mathrm{V}_g}$ , where $g$ is the empirical + +weight function. The theorem also assumes the ground truth $f_{0}$ lies in a similar space. A key condition is that $f_{\theta}$ is "optimized" such that its empirical loss is no worse than that of $f_{0}$ . This is crucial as it allows us to bound the empirical MSE primarily by an empirical process term involving the noise terms $\varepsilon_{i}$ . + +To bound this empirical process, the proof again decomposes the input domain $\mathbb{B}_1^d$ into a strict interior ball $\mathbb{B}_{1 - \varepsilon}^d$ and an annulus $\mathbb{A}_{\varepsilon}^{d}$ . In the outer shell, the contribution to the MSE is controlled by the function's $L^{\infty}$ bound. For the strict interior $\mathbb{B}_{1 - \varepsilon}^d$ , we analyze the difference function $f_{\Delta} = f_{\theta} - f_0$ . Consistent with our generalization analysis, we use the results from Appendix E.2 to ensure the empirical weight function $g$ can be reliably analyzed via its population counterpart. This allows us to bound the unweighted variation norm of $f_{\Delta}$ over the core, which is then used to bound the empirical process via local Gaussian complexities (as detailed in Appendix G). + +The bounds from the annulus and the core are then summed. The resulting expression for the total MSE is minimized by choosing an optimal $\varepsilon$ . This balancing yields the final estimation error rate presented in Theorem 3.6, connecting the stability-induced regularity and the "optimized" nature of the solution to its statistical performance. + +# B.4 Proof Overview of Theorem 3.7 + +The proof establishes the minimax lower bound by constructing a packing set of functions within the specified function class $\mathrm{V}_g(\mathbb{B}_1^d)$ and then applying Fano's Lemma. The construction differs for multivariate $(d > 1)$ and univariate $(d = 1)$ cases. + +Multivariate Case $(d > 1)$ The core idea is to use highly localized ReLU atoms that have a small $\mathrm{V}_g$ norm due to the weighting $g(\pmb {u},t)$ vanishing near the boundary $(|t|\to 1)$ , yet can be combined to form a sufficiently rich and separated set of functions. + +![](images/feb51553a83bf68999ca071f2a36b48f9a98c35ae0dee60371353cc63ce7292f.jpg) +Figure 16: The ReLU atoms only activate on the localized spherical cap and with $L^{\infty}(\mathbb{B}_1^d)$ -norm equal to 1. As dimension increases, more data points will concentrate on the boundary region and the choice of directions increase exponentially. + +![](images/027539f3a30a4a4aa4876db1930360c00728a033922dfb3fd166a246e6362377.jpg) + +1. Atom Construction: We utilize ReLU atoms $\Phi_{\pmb{u},\varepsilon^2}(\pmb{x}) = \varepsilon^{-2}\phi(\pmb{u}^\top \pmb{x} - (1 - \varepsilon^2))$ as defined in Construction H.4 (see Eq. (105) for the unnormalized version). These atoms are $L^{\infty}$ -normalized, have an $L^2(\mathbb{B}_1^d)$ -norm $\| \Phi_{\pmb{u},\varepsilon^2} \|_{L^2} \asymp \varepsilon^{\frac{d+1}{2}}$ (Lemma H.1), and a weighted variation norm $|\Phi_{\pmb{u},\varepsilon^2}|_{\mathrm{V}_g} = \varepsilon^{2d+2}$ (Lemma H.2, Eq. (116)). The small $\mathrm{V}_g$ norm is crucial. +2. Packing Set: Using a packing of $K \asymp \varepsilon^{-(d - 1)}$ disjoint spherical caps on $\mathbb{S}^{d - 1}$ (Lemma H.3), we construct a family of functions $f_{\xi}(\pmb{x}) = \sum_{i=1}^{K} \xi_i \Phi_{\pmb{u}_i, \varepsilon^2}(\pmb{x})$ for $\xi \in \{-1,1\}^K$ . By Varshamov-Gilbert lemma (Lemma J.2), we can find a subset $\Xi \subset \{-1,1\}^K$ such that $\log |\Xi| \asymp K \asymp \varepsilon^{-(d - 1)}$ and for any distinct $f_{\xi}, f_{\xi'} \in \{f_{\zeta}\}_{\zeta \in \Xi}$ , their $L^2$ -distance is $\| f_{\xi} - f_{\xi'} \|_{L^2} \gtrsim \varepsilon$ . The total variation norm $|f_{\xi}|_{\mathrm{V}_g} \leq K \varepsilon^{2d + 2} \asymp \varepsilon^{d + 3}$ , which is significantly smaller than 1 when $\varepsilon < 1$ . + +3. Leveraging Fano's Lemma: (Proposition H.5) The KL divergence between distributions induced by $f_{\xi}$ and $f_{\xi'}$ is $\mathrm{KL}(P_{\xi} \| P_{\xi'}) \asymp n\varepsilon^2 / \sigma^2$ . To apply Fano's Lemma (see Lemma J.1), we need to satisfy the condition (141) that $n\varepsilon^2 / \sigma^2 \lesssim \log |\Xi| \asymp \varepsilon^{-(d-1)}$ , which implies $\varepsilon \asymp (\sigma^2 / n)^{\frac{1}{d+1}}$ and the minimax risk is then given by $\mathbb{E} \| \hat{f} - f \|_{L^2}^2 \gtrsim \varepsilon^2 \asymp (\sigma^2 / n)^{\frac{2}{d+1}}$ . + +Univariate Case $(d = 1)$ The high-dimensional spherical cap packing is not applicable. Instead, we use scaled bump functions and exploit the simplified $1\mathrm{D}\mathrm{V}_g$ norm. + +1. Function Class: For $d = 1$ , if we assume $f$ is smooth, then $|f|_{\mathrm{V}_g} = \| f'' \cdot g \|_{\mathcal{M}} = \int_{-1}^{1} |f''(x)| (1 - |x|)^3 \, \mathrm{d}x$ (from Theorem 3.4 and leading to the class in Eq. (121)). +2. Atom Construction: We construct functions $\Phi_k(x)$ as smooth bump functions, each supported on a distinct interval of width $\varepsilon^2$ near the boundary (e.g., $x \in [1 - \varepsilon + (k - 1)\varepsilon^2, 1 - \varepsilon + k\varepsilon^2]$ ). These are scaled such that $\|\Phi_k\|_{L^2} \asymp \varepsilon$ . Due to the $(1 - |x|)^3 \lesssim \varepsilon^3$ factor in the $\mathrm{V}_g$ norm and $\int_{-1}^{1} |\Phi_k''(x)| \, \mathrm{d}x \asymp 1 / \varepsilon^2$ , the weighted variation is $|\Phi_k|_{\mathrm{V}_g} \asymp \varepsilon^3 \cdot (1 / \varepsilon^2) = \varepsilon$ . +3. Packing Set: A family $f_{\xi}(x) = \sum_{k=1}^{K} \xi_k \Phi_k(x)$ is formed with $K \asymp 1 / \varepsilon$ terms. Using Varshamov-Gilbert (Lemma J.2), we find a subset $\Xi$ with $\log |\Xi| \asymp K \asymp 1 / \varepsilon$ such that for distinct $f_{\xi}, f_{\xi'}$ , the $L^2$ -distance is $\| f_{\xi} - f_{\xi'} \|_{L^2} \gtrsim \sqrt{K} \varepsilon \asymp \sqrt{1 / \varepsilon} \cdot \varepsilon = \sqrt{\varepsilon}$ . +4. Leveraging Fano's Lemma: The KL divergence is $\mathrm{KL}(P_{\xi}\| P_{\xi^{\prime}})\asymp n(\sqrt{\varepsilon})^{2} / \sigma^{2} = n\varepsilon /\sigma^{2}$ . Fano's condition (141) $n\varepsilon /\sigma^2\lesssim \log |\Xi |\asymp 1 / \varepsilon$ implies $\varepsilon \asymp (\sigma^2 /n)^{1 / 2}$ . The minimax risk is then $\mathbb{E}\| \hat{f} -f\|_{L^2}^2\gtrsim (\sqrt{\varepsilon})^2 = \varepsilon \asymp (\sigma^2 /n)^{1 / 2}$ . + +# B.5 Discussion of the Proofs + +A notable feature in the proofs for the generalization gap upper bound (Theorem 3.5) and the MSE upper bound (Theorem 3.6) is the strategy of decomposing the domain $\mathbb{B}_1^d$ into an inner core $\mathbb{B}_{1 - \varepsilon}^d$ and an annulus $\mathbb{A}_{\varepsilon}^{d}$ . This decomposition, involving a trade-off by treating the boundary region differently, is not merely a technical convenience but is fundamentally motivated by the characteristics of the function class $\mathrm{V}_g(\mathbb{B}_1^d)$ and the nature of "hard-to-learn" functions within it. + +The necessity for this approach is starkly illustrated by our minimax lower bound construction in Theorem 3.7 (see Appendix H for construction details) and Proposition I.1. The hard-to-learn functions used to establish this lower bound are specifically constructed using ReLU neurons that activate only near the boundary of the unit ball (i.e., for $\pmb{x}$ such that $\pmb{u}^{\mathrm{T}}\pmb{x} \approx 1$ ). The crucial insight here is the behavior of the weight function $g(\pmb{u}, t) \asymp (1 - |t|)^{d + 2}$ (see Appendix E). For these boundary-activating neurons, $|t|$ is close to 1, making $g(\pmb{u}, t)$ exceptionally small. This allows for functions that are potentially complex or have large unweighted magnitudes near the boundary (the annulus) to still possess a small weighted variation norm $|f|_{\mathrm{V}_g}$ , thus qualifying them as members of the function class under consideration. Our lower bound construction focuses almost exclusively on these boundary phenomena, as they represent the primary source of difficulty for estimation within this specific weighted variation space. + +The upper bound proofs implicitly acknowledge this. By isolating the annulus $\mathbb{A}_{\varepsilon}^{d}$ , the analysis effectively concedes that this region might harbor complex behavior. The error contribution from this annulus is typically bounded by simpler means, often proportional to its small volume (controlled by $\varepsilon$ ) and the $L^{\infty}$ norm of the functions. The more sophisticated analysis, involving metric entropy or Gaussian complexity arguments (which depend on an unweighted variation norm that becomes large as $|f|_{\mathrm{V}_g} / \varepsilon^{d + 2}$ when restricted to the strict interior $\mathbb{B}_{1 - \varepsilon}^{d}$ ), is applied to the "better-behaved" interior region. The parameter $\varepsilon$ is then chosen optimally to balance the error from the boundary (which increases with $\varepsilon$ ) against the error from the interior (where the complexity term effectively increases as $\varepsilon$ shrinks). + +This methodological alignment between our upper and lower bounds underscores a self-consistency in our analysis. Both components of the argument effectively exploit the geometric properties stemming from the uniform data distribution on a sphere and the specific decay characteristics of the data-dependent weight function $g$ near the boundary. The strategy of "sacrificing the boundary" in the upper bounds is thus a direct and necessary consequence of where the challenging functions identified by the lower bound constructions. + +# C Proof of Theorem 3.2: Stable Minima Regularity + +In this section, we prove the regularity constraint of stable minima. We begin by upper bounding the operator norm of the Hessian matrix. In other words, we upper bound $|v^{\mathsf{T}}\nabla_{\theta}^{2}f_{\theta}(x)v|$ under the constraint that $\| v\| _2 = 1$ . + +Lemma C.1. Assume $f_{\pmb{\theta}}(\pmb{x}) = \sum_{k=1}^{K} v_k \phi(\pmb{w}_k^\top \pmb{x} + b_k) + \beta$ is a two-layer ReLU network with input $\pmb{x} \in \mathbb{R}^d$ such that $\| \pmb{x} \|_2 \leq R$ . Let $\theta$ represent all parameters $\{\pmb{w}_k, b_k, v_k, \beta\}_{k=1}^K$ . Assume $f_{\pmb{\theta}}(\pmb{x})$ is twice differentiable with respect to $\theta$ at $\pmb{x}$ . Then for any vector $\pmb{v}$ corresponding to a perturbation in $\theta$ such that $\| \pmb{v} \|_2 = 1$ , it holds that: + +$$ +\left| \boldsymbol {v} ^ {\top} \nabla_ {\boldsymbol {\theta}} ^ {2} f _ {\boldsymbol {\theta}} (\boldsymbol {x}) \boldsymbol {v} \right| \leq 2 (R + 1). \tag {20} +$$ + +Proof. Let the parameters be $\theta = (\pmb{w}_1^{\top},\dots,\pmb{w}_K^{\top},b_1,\dots,b_K,v_1,\dots,v_K,\beta)^{\top}$ . The total number of parameters is $N = K\times d + K + K + 1 = K(d + 2) + 1$ . Let the corresponding perturbation vector be $\pmb{v}\in \mathbb{R}^{N}$ , structured as: $\pmb{v} = (\pmb{\alpha}_{1}^{\top},\dots,\pmb{\alpha}_{K}^{\top},\delta_{1},\dots,\delta_{K},\gamma_{1},\dots,\gamma_{K},\iota)^{\top}$ , where $\pmb{\alpha}_{k}\in \mathbb{R}^{d}$ corresponds to $\pmb{w}_k$ , $\delta_k\in \mathbb{R}$ corresponds to $b_{k}$ , $\gamma_{k}\in \mathbb{R}$ corresponds to $v_{k}$ , and $\iota \in \mathbb{R}$ corresponds to $\beta$ . The normalization constraint is + +$$ +\left\| \boldsymbol {v} \right\| _ {2} ^ {2} = \sum_ {k = 1} ^ {K} \left\| \boldsymbol {\alpha} _ {k} \right\| _ {2} ^ {2} + \sum_ {k = 1} ^ {K} \delta_ {k} ^ {2} + \sum_ {k = 1} ^ {K} \gamma_ {k} ^ {2} + \iota^ {2} = 1 \tag {21} +$$ + +We need to compute the Hessian matrix $\nabla_{\pmb{\theta}}^{2}f_{\pmb{\theta}}(\pmb{x})$ . Let $z_{k} = \pmb{w}_{k}^{\top}\pmb{x} + b_{k}$ and $1_{k} = 1(z_{k} > 0)$ . Since we assume twice differentiability, $z_{k} \neq 0$ for all $k$ , the Hessian $\nabla_{\pmb{\theta}}^{2}f_{\pmb{\theta}}(\pmb{x})$ is block diagonal, with $K$ blocks corresponding to each neuron. The $k$ -th block, $\nabla_{(\theta_k)}^{2}f_{\pmb{\theta}}(\pmb{x})$ , involves derivatives with respect to $\theta_{k} = (\pmb{w}_{k}^{\top}, b_{k}, v_{k})^{\top}$ . The relevant non-zero second partial derivatives defining this block are: + +$\frac{\partial^2f_{\pmb{\theta}}}{\partial\pmb{w}_k\partial v_k} = \frac{\partial}{\partial v_k} (\nabla_{\pmb{w}_k}f_{\pmb{\theta}}) = \frac{\partial}{\partial v_k} (v_k\phi '(z_k)\pmb {x}) = \phi '(z_k)\pmb {x} = 1_k\pmb{x}$ +- $\frac{\partial^2 f_{\pmb{\theta}}}{\partial b_k \partial v_k} = \frac{\partial}{\partial v_k} \left( \frac{\partial f_{\pmb{\theta}}}{\partial b_k} \right) = \frac{\partial}{\partial v_k} \left( v_k \phi'(z_k) \right) = \phi'(z_k) = 1_k$ + +All other second derivatives within the block are zero, as are derivatives between different blocks or involving $\beta$ . The $k$ -th block of the Hessian is thus: + +$$ +\nabla_ {\left(\theta_ {k}\right)} ^ {2} f _ {\boldsymbol {\theta}} (\boldsymbol {x}) = \left( \begin{array}{c c c} \frac {\partial^ {2} f _ {\boldsymbol {\theta}}}{\left(\partial \boldsymbol {w} _ {k}\right) ^ {2}} & \frac {\partial^ {2} f _ {\boldsymbol {\theta}}}{\partial \boldsymbol {w} _ {k} \partial b _ {k}} & \frac {\partial^ {2} f _ {\boldsymbol {\theta}}}{\partial \boldsymbol {w} _ {k} \partial v _ {k}} \\ \frac {\partial^ {2} f _ {\boldsymbol {\theta}}}{\partial b _ {k} \partial \boldsymbol {w} _ {k}} & \frac {\partial^ {2} f _ {\boldsymbol {\theta}}}{\left(\partial b _ {k}\right) ^ {2}} & \frac {\partial^ {2} f _ {\boldsymbol {\theta}}}{\partial b _ {k} \partial v _ {k}} \\ \frac {\partial^ {2} f _ {\boldsymbol {\theta}}}{\partial v _ {k} \partial \boldsymbol {w} _ {k}} & \frac {\partial^ {2} f _ {\boldsymbol {\theta}}}{\partial v _ {k} \partial b _ {k}} & \frac {\partial^ {2} f _ {\boldsymbol {\theta}}}{\left(\partial v _ {k}\right) ^ {2}} \end{array} \right) = \left( \begin{array}{c c c} \mathbf {0} _ {d \times d} & \mathbf {0} _ {d} & 1 _ {k} \boldsymbol {x} \\ \mathbf {0} _ {d} ^ {\top} & 0 & 1 _ {k} \\ 1 _ {k} \boldsymbol {x} ^ {\top} & 1 _ {k} & 0 \end{array} \right) \tag {22} +$$ + +where $\mathbf{0}_{d\times d}$ is the $d\times d$ zero matrix and $\mathbf{0}_d$ is the $d$ -dimensional zero vector. + +The quadratic form $\pmb{v}^{\top}\nabla_{\pmb{\theta}}^{2}f_{\pmb{\theta}}(\pmb{x})\pmb{v}$ becomes: + +$$ +\begin{array}{l} \boldsymbol {v} ^ {\mathsf {T}} \nabla_ {\boldsymbol {\theta}} ^ {2} f _ {\boldsymbol {\theta}} (\boldsymbol {x}) \boldsymbol {v} = \sum_ {k = 1} ^ {K} \left( \begin{array}{c c c} \boldsymbol {\alpha} _ {k} ^ {\mathsf {T}} & \delta_ {k} & \gamma_ {k} \end{array} \right) \left( \begin{array}{c c c} \boldsymbol {0} _ {d \times d} & \boldsymbol {0} _ {d} & 1 _ {k} \boldsymbol {x} \\ \boldsymbol {0} _ {d} ^ {\mathsf {T}} & 0 & 1 _ {k} \\ 1 _ {k} \boldsymbol {x} ^ {\mathsf {T}} & 1 _ {k} & 0 \end{array} \right) \left( \begin{array}{c} \boldsymbol {\alpha} _ {k} \\ \delta_ {k} \\ \gamma_ {k} \end{array} \right) \\ = \sum_ {k = 1} ^ {K} \left(\boldsymbol {\alpha} _ {k} ^ {\mathsf {T}} \left(1 _ {k} \boldsymbol {x}\right) \gamma_ {k} + \delta_ {k} \left(1 _ {k}\right) \gamma_ {k} + \gamma_ {k} \left(1 _ {k} \boldsymbol {x} ^ {\mathsf {T}}\right) \boldsymbol {\alpha} _ {k} + \gamma_ {k} \left(1 _ {k}\right) \delta_ {k}\right) \tag {23} \\ = \sum_ {k = 1} ^ {K} 2 \cdot 1 _ {k} \cdot \gamma_ {k} \left(\boldsymbol {\alpha} _ {k} ^ {\mathsf {T}} \boldsymbol {x} + \delta_ {k}\right) \\ \end{array} +$$ + +Now, we bound the absolute value: + +$$ +\begin{array}{l} | \boldsymbol {v} ^ {\mathsf {T}} \nabla_ {\boldsymbol {\theta}} ^ {2} f _ {\boldsymbol {\theta}} (\boldsymbol {x}) \boldsymbol {v} | = \left| \sum_ {k = 1} ^ {K} 2 \cdot 1 _ {k} \cdot \gamma_ {k} \left(\boldsymbol {\alpha} _ {k} ^ {\mathsf {T}} \boldsymbol {x} + \delta_ {k}\right) \right| \leq \sum_ {k = 1} ^ {K} 2 \cdot 1 _ {k} \cdot | \gamma_ {k} | | \boldsymbol {\alpha} _ {k} ^ {\mathsf {T}} \boldsymbol {x} + \delta_ {k} | \\ \leq \sum_ {k = 1} ^ {K} 2 | \gamma_ {k} | \left(| \boldsymbol {\alpha} _ {k} ^ {\mathsf {T}} \boldsymbol {x} | + | \delta_ {k} |\right) \quad \leq \sum_ {k = 1} ^ {K} 2 | \gamma_ {k} | \left(\| \boldsymbol {\alpha} _ {k} \| _ {2} \| \boldsymbol {x} \| _ {2} + | \delta_ {k} |\right) \quad (\text {C a u c h y - S c h w a r z}) \\ = 2 R \sum_ {k = 1} ^ {K} | \gamma_ {k} | \| \boldsymbol {\alpha} _ {k} \| _ {2} + 2 \sum_ {k = 1} ^ {K} | \gamma_ {k} | | \delta_ {k} | \\ \leq 2 R \sqrt {\sum_ {k = 1} ^ {K} \gamma_ {k} ^ {2}} \sqrt {\sum_ {k = 1} ^ {K} \| \boldsymbol {\alpha} _ {k} \| _ {2} ^ {2}} + 2 \sqrt {\sum_ {k = 1} ^ {K} \gamma_ {k} ^ {2}} \sqrt {\sum_ {k = 1} ^ {K} \delta_ {k} ^ {2}} \quad (\text {C a u c h y - S c h w a r z o n s u m s}) \\ \leq 2 R \sqrt {\sum \gamma_ {k} ^ {2}} \cdot \sqrt {1} + 2 \sqrt {\sum \gamma_ {k} ^ {2}} \cdot \sqrt {1} \\ = 2 (R + 1) \sqrt {\sum \gamma_ {k} ^ {2}} \leq 2 (R + 1). \\ \end{array} +$$ + +![](images/d1ee3ec487d4634f20c49435c5260288354bab3184a9c9054a51e990d7ab86b0.jpg) + +Now we are ready to prove Theorem 3.2. + +Proof of Theorem 3.2. Without loss of generality, we consider neural networks of the following form: + +$$ +f _ {\boldsymbol {\theta}} (\boldsymbol {x}) = \sum_ {k = 1} ^ {K} v _ {k} \phi \left(\boldsymbol {w} _ {k} ^ {\mathsf {T}} \boldsymbol {x} - b _ {k}\right) + \beta . \tag {24} +$$ + +The Hessian matrix of the loss function, obtained through direct computation, is expressed as: + +$$ +\nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L} (\boldsymbol {\theta}) = \frac {1}{n} \sum_ {i = 1} ^ {n} \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) ^ {\top} + \frac {1}{n} \sum_ {i = 1} ^ {n} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - y _ {i}\right) \nabla_ {\boldsymbol {\theta}} ^ {2} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right). \tag {25} +$$ + +Let $\pmb{v}$ be the unit eigenvector (i.e., $\| \pmb{v} \|_2 = 1$ ) corresponding to the largest eigenvalue of the matrix $\frac{1}{n} \sum_{i=1}^{n} (\nabla_\theta f_\theta(\pmb{x}_i)) (\nabla_\theta f_\theta(\pmb{x}_i))^{\top}$ , the maximum eigenvalue of the Hessian matrix of the loss can be lower-bounded as follows: + +$$ +\begin{array}{l} \lambda_ {\max } \left(\nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L} (\boldsymbol {\theta})\right) \geq \boldsymbol {v} ^ {\top} \nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L} (\boldsymbol {\theta}) \boldsymbol {v} \\ = \underbrace {\lambda_ {\max } \left(\frac {1}{n} \sum_ {i = 1} ^ {n} \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) ^ {\mathsf {T}}\right)} _ {\text {(T e r m A)}} \tag {26} \\ + \underbrace {\frac {1}{n} \sum_ {i = 1} ^ {n} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - y _ {i}\right) \boldsymbol {v} ^ {\mathsf {T}} \nabla_ {\boldsymbol {\theta}} ^ {2} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) \boldsymbol {v}} _ {\text {(T e r m B)}}. \\ \end{array} +$$ + +Regarding (Term A), its maximum eigenvalue at a given $\pmb{\theta}$ can be related to the $\mathrm{V}_g$ norm of the associated function $f = f_{\pmb{\theta}}$ . Considering the domain $\mathbb{B}_R^d$ , Nacson et al. [2023, Appendix F.2] demonstrate that: + +$$ +\left(\text {T e r m A}\right) = \lambda_ {\max } \left(\frac {1}{n} \sum_ {i = 1} ^ {n} \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) ^ {\mathsf {T}}\right) \geq 1 + 2 \sum_ {k = 1} ^ {K} \left| v _ {k} \right| \| \boldsymbol {w} _ {k} \| _ {2} \tilde {g} \left(\bar {\boldsymbol {w}} _ {k}, \bar {b} _ {k}\right), \tag {27} +$$ + +where $\bar{\pmb{w}}_k = \pmb {w}_k / \| \pmb {w}_k\| _2\in \mathbb{S}^{d - 1}$ $\bar{b}_k = b_k / \| \pmb {w}_k\| _2$ and + +$$ +\tilde {g} (\bar {\boldsymbol {w}}, \bar {b}) = \mathbb {P} \left(\boldsymbol {X} ^ {\top} \bar {\boldsymbol {w}} > \bar {b}\right) ^ {2} \cdot \mathbb {E} \left[ \boldsymbol {X} ^ {\top} \bar {\boldsymbol {w}} - \bar {b} \mid \boldsymbol {X} ^ {\top} \bar {\boldsymbol {w}} > \bar {b} \right] \cdot \sqrt {1 + \| \mathbb {E} [ \boldsymbol {X} \mid \boldsymbol {X} ^ {\top} \bar {\boldsymbol {w}} > \bar {b} ] \| ^ {2}}. \tag {28} +$$ + +For (Term B), an upper bound can be established using the training loss $\mathcal{L}(\pmb{\theta})$ via the Cauchy-Schwarz inequality. This also employs a notable uniform upper bound for $|\pmb{v}^{\top}\nabla_{\pmb{\theta}}^{2}f_{\pmb{\theta}}(\pmb{x}_{n})\pmb{v}|$ , as detailed in Lemma C.1: + +$$ +| (\text {T e r m B}) | \leq \sqrt {\frac {1}{n} \sum_ {i = 1} ^ {n} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - y _ {i}\right) ^ {2}} \cdot \sqrt {\frac {1}{n} \sum_ {i = 1} ^ {n} \left(\boldsymbol {v} ^ {\top} \nabla_ {\boldsymbol {\theta}} ^ {2} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) \boldsymbol {v}\right) ^ {2}} \leq 2 (R + 1) \sqrt {2 \mathcal {L} (\boldsymbol {\theta})}. \tag {29} +$$ + +Finally, the proof of Theorem 3.2 is complete by plugging (27) and (29) into (26). + +![](images/537fe0a0cba781a0f24b66c9c716e433be056560221e48732da50289e6473393.jpg) + +# D Proof of Theorem 3.4: Radon-Domain Characterization of Stable Minima + +In this part, we prove Theorem 3.4 by extending the unweighted case to the weighted one. + +Proof of Theorem 3.4. In the unweighted scenario, i.e., $g \equiv 1$ , it was established by Parhi and Nowak [2023b, Lemma 2] that if $f \in \mathrm{V}(\mathbb{B}_R^d) \coloneqq \mathrm{V}_1(\mathbb{B}_R^d)$ with integral representation + +$$ +f (\boldsymbol {x}) = \int_ {\mathbb {S} ^ {d - 1} \times [ - R, R ]} \phi \left(\boldsymbol {u} ^ {\mathsf {T}} \boldsymbol {x} - t\right) \mathrm {d} \nu (\boldsymbol {u}, t) + \boldsymbol {c} ^ {\mathsf {T}} \boldsymbol {x} + c _ {0}, \quad \boldsymbol {x} \in \mathbb {B} _ {R} ^ {d}, \tag {30} +$$ + +where $\nu, c,$ and $c_{0}$ solve (8) (with $g\equiv 1$ ) that + +$$ +\nu = \mathcal {R} (- \Delta) ^ {\frac {d + 1}{2}} f _ {\text {e x t}}, \tag {31} +$$ + +where we recall that $f_{\mathrm{ext}}$ is the canonical extension of $f$ from $\mathbb{B}_R^d$ to $\mathbb{R}^d$ via the formula (30) and $\nu \in \mathcal{M}(\mathbb{S}^{d - 1} \times \mathbb{R})$ with $\operatorname{supp} \nu = \mathbb{S}^{d - 1} \times [-R, R]$ (i.e., we can identify $\nu$ with a measure in $\mathcal{M}(\mathbb{S}^{d - 1} \times [-R, R])$ ). Since the weighted variation seminorm $|\cdot|_{\mathrm{V}_g}$ is simply (cf., (8)) + +$$ +| f | _ {\mathrm {V} _ {g}} = \inf _ {\substack {\nu \in \mathcal {M} (\mathbb {S} ^ {d - 1} \times [ - R, R ]) \\ \boldsymbol {c} \in \mathbb {R} ^ {d}, c _ {0} \in \mathbb {R}}} \| g \cdot \nu \| _ {\mathcal {M}} \quad \text {s.t.} \quad f = f _ {\nu , \boldsymbol {c}, c _ {0}}, \tag{32} +$$ + +we readily see that $|f|_{\mathrm{V}_g} = \| g \cdot \mathcal{R}(-\Delta)^{\frac{d + 1}{2}} f_{\mathrm{ext}}\|_{\mathcal{M}}$ . + +![](images/555476b5da25216f7dcdfc3cea541c9af33612502929f5fe1d46213a1779eda6.jpg) + +Remark D.1. The unweighted variation seminorm exactly corresponds to the second-order Radon-domain total variation of the function [Ongie et al., 2020, Parhi and Nowak, 2021, 2022, 2023b,a]. Thus, the weighted variation seminorm is a weighted variant of the second-order Radon-domain total variation. + +# E Characterization of the Weight Function for the Uniform Distribution + +Recall that, given a data set $\mathcal{D} = \{(x_i,y_i)\}_{i = 1}^n\subset \mathbb{R}^d\times \mathbb{R}$ , we consider a weight function $g:\mathbb{S}^{d - 1}\times \mathbb{R}\to \mathbb{R}$ , where $\mathbb{S}^{d - 1}\coloneqq \{\pmb {u}\in \mathbb{R}^d:\| \pmb {u}\| = 1\}$ denotes the unit sphere. This weight is defined by $g(\pmb {u},t)\coloneqq \min \{\tilde{g} (\pmb {u},t),\tilde{g} (-\pmb {u}, - t)\}$ , where + +$$ +\tilde {g} (\boldsymbol {u}, t) := \mathbb {P} \left(\boldsymbol {X} ^ {\top} \boldsymbol {u} > t\right) ^ {2} \cdot \mathbb {E} \left[ \boldsymbol {X} ^ {\top} \boldsymbol {u} - t \mid \boldsymbol {X} ^ {\top} \boldsymbol {u} > t \right] \cdot \sqrt {1 + \left\| \mathbb {E} \left[ \boldsymbol {X} \mid \boldsymbol {X} ^ {\top} \boldsymbol {u} > t \right] \right\| ^ {2}}. \tag {33} +$$ + +Here, $\mathbf{X}$ is a random vector drawn uniformly at random from the training examples $\{\pmb{x}_i\}_{i=1}^n$ . Note that the distribution $\mathbb{P}_X$ for which the $\{\pmb{x}_i\}_{i=1}^n$ are drawn i.i.d. from controls the regularity of $g$ . + +In this section, We first analyze the properties of the population version $g_{P}$ by assuming that the random vector $\mathbf{X}$ is uniformly sampled from the $d$ -dimensional unit ball $\mathbb{B}_1^d = \{\pmb{x} \in \mathbb{R}^d : \| \pmb{x} \|_2 \leq 1\}$ . Then we analyze the gap between the empirical $g$ and the population $g_{P}$ using the empirical process. + +# E.1 The Computation of the Population Weight Function + +We focus on the marginal distribution of a single coordinate and related conditional expectations. Let $X_{1}$ be the first coordinate of $X$ . Due to symmetry, all coordinates have the same marginal distribution. + +The following proposition calculates the marginal probability density function of the first coordinate (and also other coordinates) of the random vector $X$ . + +Proposition E.1 (Marginal PDF of a Coordinate). Let $\mathbf{X}$ follow the uniform distribution in $\mathbb{B}_1^d$ . The probability density function (PDF) of its first coordinate $X_1$ is given by: + +$$ +f _ {X _ {1}} (t) = c _ {1} (d) \left(1 - t ^ {2}\right) ^ {\alpha}, \quad t \in [ - 1, 1 ] \tag {34} +$$ + +where $\alpha = \frac{d - 1}{2}$ and the normalization constant is + +$$ +c _ {1} (d) = \frac {\Gamma \left(\frac {d}{2} + 1\right)}{\sqrt {\pi} \Gamma \left(\frac {d + 1}{2}\right)}. \tag {35} +$$ + +Proof. The volume of the unit ball is $V_{d} = \frac{\pi^{d / 2}}{\Gamma(d / 2 + 1)}$ . The uniform density is $f_{\mathbf{X}}(\pmb {x}) = 1 / V_{d}$ for $\pmb {x}\in \mathbb{B}_1^d$ . The marginal PDF is found by integrating out the other coordinates: + +$$ +f _ {X _ {1}} (t) = \int_ {\{\pmb {x} ^ {\prime} \in \mathbb {R} ^ {d - 1}: \| \pmb {x} ^ {\prime} \| ^ {2} \leq 1 - t ^ {2} \}} \frac {1}{V _ {d}} \mathrm {d} \pmb {x} ^ {\prime} = \frac {\operatorname {V o l} _ {d - 1} (\sqrt {1 - t ^ {2}})}{V _ {d}} +$$ + +where $\mathrm{Vol}_{d - 1}(R)$ is the volume of a $(d - 1)$ -ball of radius $R$ . Using $V_{d - 1} = \frac{\pi^{(d - 1) / 2}}{\Gamma((d - 1) / 2 + 1)} = \frac{\pi^{(d - 1) / 2}}{\Gamma((d + 1) / 2)}$ , we get + +$$ +f _ {X _ {1}} (t) = \frac {V _ {d - 1} (1 - t ^ {2}) ^ {(d - 1) / 2}}{V _ {d}} = \frac {\pi^ {(d - 1) / 2}}{\Gamma ((d + 1) / 2)} \frac {\Gamma (d / 2 + 1)}{\pi^ {d / 2}} (1 - t ^ {2}) ^ {(d - 1) / 2} +$$ + +which simplifies to the stated result. For $d = 2$ , $\alpha = 1/2$ , $c_{1}(2) = \frac{\Gamma(2)}{\sqrt{\pi} \Gamma(3/2)} = \frac{1}{\sqrt{\pi} (\sqrt{\pi}/2)} = \frac{2}{\pi}$ . For $d = 3$ , $\alpha = 1$ , $c_{1}(3) = \frac{\Gamma(5/2)}{\sqrt{\pi} \Gamma(2)} = \frac{3 \sqrt{\pi}/4}{\sqrt{\pi}} = \frac{3}{4}$ . + +Given the marginal probability density function, the tail probability follows from direct calculation. + +Proposition E.2 (Tail Probability). Let $\mathbf{X}$ be a random vector uniformly distributed in the $d$ -dimensional unit ball $\mathbb{B}_1^d = \{\pmb{x} \in \mathbb{R}^d : \| \pmb{x} \|_2 \leq 1\}$ . Let $X_1$ be its first coordinate whose tail probability is defined as $Q(x) = \mathbb{P}(X_1 > x)$ for $x \in [-1, 1]$ . Then there exists a fixed $x_0 \in [0, 1)$ (specifically, we choose $x_0 = 3/4$ , which implies $(1 - x) \in (0, 1/4]$ ) such that for all $x \in [x_0, 1)$ : + +$$ +Q (x) \underset {c _ {2} (d)} {\overset {c _ {3} (d)} {\sim}} (1 - x) ^ {\frac {d + 1}{2}}. +$$ + +Or equivalently, + +$$ +c _ {2} (d) (1 - x) ^ {\frac {d + 1}{2}} \leq Q (x) \leq c _ {3} (d) (1 - x) ^ {\frac {d + 1}{2}}, +$$ + +where the constants $c_{2}(d)$ and $c_{3}(d)$ are given by: + +$$ +c _ {2} (d) = \frac {c _ {1} (d)}{d + 1} \left(\frac {7}{4}\right) ^ {\frac {d + 1}{2}} +$$ + +$$ +c _ {3} (d) = \frac {c _ {1} (d)}{d + 1} 2 ^ {\frac {d + 2}{2}} +$$ + +and $c_{1}(d) = \frac{\Gamma\left(\frac{d}{2} + 1\right)}{\sqrt{\pi}\Gamma\left(\frac{d + 1}{2}\right)}$ is the normalization constant from the marginal PDF of $X_{1}$ . + +Proof. The tail probability $Q(x)$ is given by the integral of the marginal PDF $f_{X_1}(t) = c_1(d)(1 - t^2)^\alpha$ for $t \in [-1, 1]$ , where $\alpha = \frac{d - 1}{2}$ . + +$$ +Q (x) = \int_ {x} ^ {1} c _ {1} (d) (1 - t ^ {2}) ^ {\alpha} \mathrm {d} t +$$ + +Let $s = t^2$ , so $\mathrm{d}t = \mathrm{d}s / (2\sqrt{s})$ . The limits of integration for $s$ become $x^2$ to 1. + +$$ +Q (x) = c _ {1} (d) \int_ {x ^ {2}} ^ {1} (1 - s) ^ {\alpha} \frac {d s}{2 \sqrt {s}} +$$ + +Now, let $u = 1 - s$ . Then $s = 1 - u$ and $\mathrm{d}s = -\mathrm{d}u$ . Let $\delta_s = 1 - x^2$ . The limits for $u$ become $\delta_s$ to 0. + +$$ +Q (x) = \frac {c _ {1} (d)}{2} \int_ {0} ^ {\delta_ {s}} (1 - u) ^ {- 1 / 2} u ^ {\alpha} \mathrm {d} u +$$ + +For $u \in [0, \delta_s]$ , we have $1 \leq (1 - u)^{-1/2} \leq (1 - \delta_s)^{-1/2}$ , because $(1 - u)$ is decreasing and non-negative. The integral $\int_0^{\delta_s} u^\alpha \, \mathrm{d}u = \frac{\delta_s^{\alpha + 1}}{\alpha + 1}$ . Substituting these bounds for the term $(1 - u)^{-1/2}$ : + +$$ +\frac {c _ {1} (d)}{2 (\alpha + 1)} \delta_ {s} ^ {\alpha + 1} \leq Q (x) \leq \frac {c _ {1} (d)}{2 (\alpha + 1)} \left(1 - \delta_ {s}\right) ^ {- 1 / 2} \delta_ {s} ^ {\alpha + 1} +$$ + +We use $\alpha = \frac{d - 1}{2}$ , so $2(\alpha + 1) = 2\left(\frac{d - 1}{2} + 1\right) = 2\left(\frac{d + 1}{2}\right) = d + 1$ . The exponent $\alpha + 1 = \frac{d + 1}{2}$ . Thus, + +$$ +\frac {c _ {1} (d)}{d + 1} \delta_ {s} ^ {\frac {d + 1}{2}} \leq Q (x) \leq \frac {c _ {1} (d)}{d + 1} (1 - \delta_ {s}) ^ {- 1 / 2} \delta_ {s} ^ {\frac {d + 1}{2}} +$$ + +We choose $x_0 = 3/4$ , so we consider $x \in [3/4, 1)$ , which means $(1 - x) \in (0, 1/4]$ . For $x \in [3/4, 1)$ we have $1 + x \in [7/4, 2)$ . The term $\delta_s = 1 - x^2 = (1 - x)(1 + x)$ . Given the range for $1 + x$ , for $(1 - x) \in (0, 1/4]$ : + +$$ +\frac {7}{4} (1 - x) \leq \delta_ {s} < 2 (1 - x) +$$ + +- Lower Bound for $Q(x)$ : Using $\delta_s \geq \frac{7}{4}(1 - x)$ from the above range: + +$$ +Q (x) \geq \frac {c _ {1} (d)}{d + 1} \left(\frac {7}{4} (1 - x)\right) ^ {\frac {d + 1}{2}} = \frac {c _ {1} (d)}{d + 1} \left(\frac {7}{4}\right) ^ {\frac {d + 1}{2}} (1 - x) ^ {\frac {d + 1}{2}} +$$ + +This establishes the lower bound with $c_2(d) = \frac{c_1(d)}{d + 1} \left( \frac{7}{4} \right)^{\frac{d + 1}{2}}$ . + +- Upper Bound for $Q(x)$ : For the term $\delta_s^{\frac{d + 1}{2}}$ , we use $\delta_s < 2(1 - x)$ , so $\delta_s^{\frac{d + 1}{2}} < (2(1 - x))^{\frac{d + 1}{2}}$ . For the term $(1 - \delta_s)^{-1/2}$ : Since $(1 - x) \in (0, 1/4]$ , $\delta_s < 2(1 - x) \leq 2(1/4) = 1/2$ . So, $1 - \delta_s > 1 - 1/2 = 1/2$ . This implies $(1 - \delta_s)^{-1/2} < (1/2)^{-1/2} = \sqrt{2}$ . Combining these for the upper bound of $Q(x)$ : + +$$ +Q (x) \leq \frac {c _ {1} (d)}{d + 1} 2 ^ {1 / 2} \cdot 2 ^ {\frac {d + 1}{2}} (1 - x) ^ {\frac {d + 1}{2}} = \frac {c _ {1} (d)}{d + 1} 2 ^ {\frac {d + 2}{2}} (1 - x) ^ {\frac {d + 1}{2}} +$$ + +This establishes the upper bound with $c_{3}(d) = \frac{c_{1}(d)}{d + 1} 2^{\frac{d + 2}{2}}$ . + +Thus, for $x\in [3 / 4,1)$ (i.e., $1 - x\in (0,1 / 4]$ ): + +$$ +c _ {2} (d) (1 - x) ^ {\frac {d + 1}{2}} \leq Q (x) \leq c _ {3} (d) (1 - x) ^ {\frac {d + 1}{2}} +$$ + +This corresponds to $Q(x) \asymp_{c_2(d)}^{c_3(d)} (1 - x)^{\frac{d + 1}{2}}$ . The constants $c_2(d)$ and $c_3(d)$ depend only on the dimension $d$ (via $c_1(d)$ ) and the exponents derived from $d$ ) and are valid for the specified range of $x$ . + +Based on the tail probability, we calculate the expectation conditional on the tail events. + +Proposition E.3 (Conditional Expectation). For $x \in [3/4, 1)$ , the conditional expectation $\mathbb{E}[X_1 | X_1 > x]$ is bounded by + +$$ +1 - c _ {5} (d) (1 - x) \leq \mathbb {E} \left[ X _ {1} \mid X _ {1} > x \right] \leq 1 - c _ {4} (d) (1 - x), \tag {36} +$$ + +where the constants $c_{4}(d)$ and $c_{5}(d)$ are given by: + +$$ +c _ {4} (d) = \frac {2 (d + 1)}{d + 3} \frac {(7 / 4) ^ {(d - 1) / 2}}{2 ^ {(d + 2) / 2}}, +$$ + +$$ +c _ {5} (d) = \frac {2 (d + 1)}{d + 3} \frac {2 ^ {(d - 1) / 2}}{(7 / 4) ^ {(d + 1) / 2}}. +$$ + +Proof. We consider $1 - \mathbb{E}[X_1\mid X_1 > x] = \mathbb{E}[1 - X_1\mid X_1 > x]$ + +$$ +\begin{array}{l} \mathbb {E} [ 1 - X _ {1} \mid X _ {1} > x ] = \frac {1}{Q (x)} \int_ {x} ^ {1} (1 - t) f _ {X _ {1}} (t) \mathrm {d} t \\ = \frac {c _ {1} (d)}{Q (x)} \int_ {x} ^ {1} (1 - t) (1 - t ^ {2}) ^ {\alpha} d t \\ = \frac {c _ {1} (d)}{Q (x)} \int_ {x} ^ {1} (1 - t) ^ {\alpha + 1} (1 + t) ^ {\alpha} \mathrm {d} t \\ \end{array} +$$ + +Let $I_1(x) = \int_x^1 (1 - t)^{\alpha + 1}(1 + t)^\alpha dt$ . We consider $x \in [3/4, 1)$ . For $t \in [x, 1]$ , we have $1 + t \in [1 + x, 2]$ . Since $x \geq 3/4, 1 + x \geq 7/4$ . Thus, $(7/4)^\alpha \leq (1 + t)^\alpha \leq 2^\alpha$ for $t \in [x, 1]$ (assuming $\alpha \geq 0$ , which holds for $d \geq 1$ ). The integral $\int_x^1 (1 - t)^{\alpha + 1} dt = \left[-\frac{(1 - t)^{\alpha + 2}}{\alpha + 2}\right]_x^1 = \frac{(1 - x)^{\alpha + 2}}{\alpha + 2}$ . So, $I_1(x)$ is bounded by: + +$$ +(7 / 4) ^ {\alpha} \frac {(1 - x) ^ {\alpha + 2}}{\alpha + 2} \leq I _ {1} (x) \leq 2 ^ {\alpha} \frac {(1 - x) ^ {\alpha + 2}}{\alpha + 2} +$$ + +Let $N_{1}(x) = c_{1}(d)I_{1}(x)$ . Then, using $\alpha + 2 = (d + 3)/2$ : + +$$ +\frac {2 c _ {1} (d) (7 / 4) ^ {(d - 1) / 2}}{d + 3} (1 - x) ^ {\frac {d + 3}{2}} \leq N _ {1} (x) \leq \frac {2 c _ {1} (d) 2 ^ {(d - 1) / 2}}{d + 3} (1 - x) ^ {\frac {d + 3}{2}} +$$ + +From Proposition E.2, for $x \in [3/4, 1)$ , $Q(x)$ is bounded by: + +$$ +c _ {2} (d) (1 - x) ^ {\frac {d + 1}{2}} \leq Q (x) \leq c _ {3} (d) (1 - x) ^ {\frac {d + 1}{2}} +$$ + +where $c_{2}(d) = \frac{c_{1}(d)}{d + 1}\left(\frac{7}{4}\right)^{\frac{d + 1}{2}}$ and $c_{3}(d) = \frac{c_{1}(d)}{d + 1} 2^{\frac{d + 2}{2}}$ . Therefore, $\mathbb{E}[1 - X_1 \mid X_1 > x] = \frac{N_1(x)}{Q(x)}$ is bounded by: + +- Lower bound: + +$$ +\begin{array}{l} \frac {\frac {2 c _ {1} (d) (7 / 4) ^ {(d - 1) / 2}}{d + 3} (1 - x) ^ {\frac {d + 3}{2}}}{c _ {3} (d) (1 - x) ^ {\frac {d + 1}{2}}} = \frac {2 c _ {1} (d) (7 / 4) ^ {(d - 1) / 2} / (d + 3)}{\frac {c _ {1} (d)}{d + 1} 2 ^ {\frac {d + 2}{2}}} (1 - x) \\ = \frac {2 (d + 1)}{d + 3} \frac {(7 / 4) ^ {(d - 1) / 2}}{2 ^ {(d + 2) / 2}} (1 - x) = c _ {4} (d) (1 - x) \\ \end{array} +$$ + +- Upper bound: + +$$ +\begin{array}{l} \frac {\frac {2 c _ {1} (d) 2 ^ {(d - 1) / 2}}{d + 3} (1 - x) ^ {\frac {d + 3}{2}}}{c _ {2} (d) (1 - x) ^ {\frac {d + 1}{2}}} = \frac {2 c _ {1} (d) 2 ^ {(d - 1) / 2} / (d + 3)}{\frac {c _ {1} (d)}{d + 1} \left(\frac {7}{4}\right) ^ {\frac {d + 1}{2}}} (1 - x) \\ = \frac {2 (d + 1)}{d + 3} \frac {2 ^ {(d - 1) / 2}}{(7 / 4) ^ {(d + 1) / 2}} (1 - x) = c _ {5} (d) (1 - x) \\ \end{array} +$$ + +So, for $x\in [3 / 4,1)$ + +$$ +c _ {4} (d) (1 - x) \leq \mathbb {E} [ 1 - X _ {1} \mid X _ {1} > x ] \leq c _ {5} (d) (1 - x) +$$ + +This implies: + +$$ +1 - c _ {5} (d) (1 - x) \leq \mathbb {E} \left[ X _ {1} \mid X _ {1} > x \right] \leq 1 - c _ {4} (d) (1 - x) +$$ + +This completes the proof. + +Finally, we combine the results and characterize the asymptotic behavior of the weight function $g$ . Proposition E.4 (Asymptotic Behavior of $g^{+}(x)$ ). Let the function $g^{+}(x)$ be defined as: for $x \in (-1, 1)$ , + +$$ +g ^ {+} (x) = \mathbb {P} \left(X _ {1} > x\right) ^ {2} \cdot \mathbb {E} \left[ X _ {1} - x \mid X _ {1} > x \right] \cdot \sqrt {1 + \left(\mathbb {E} \left[ X _ {1} \mid X _ {1} > x \right]\right) ^ {2}}. \tag {37} +$$ + +Then for $x\in [3 / 4,1)$ , we have: + +$$ +c _ {L} ^ {(g)} (d) (1 - x) ^ {d + 2} \leq g ^ {+} (x) \leq c _ {U} ^ {(g)} (d) (1 - x) ^ {d + 2}, \tag {38} +$$ + +where $c_{L}^{(g)}(d)$ and $c_{U}^{(g)}(d)$ are positive constants depending on dimension $d$ , defined in the proof (39). + +Proof. Let $Q(x) = \mathbb{P}(X_1 > x)$ and $E(x) = \mathbb{E}[X_1 \mid X_1 > x]$ . The function is $g^{+}(x) = Q(x)^{2} \cdot (E(x) - x) \cdot \sqrt{1 + E(x)^{2}}$ . Now, we establish precise bounds for $x \in [3/4, 1)$ . Let $(1 - x)$ be the variable. + +1. Bounds for $Q(x)^2$ : From Proposition E.2, $c_2(d)(1 - x)^{\frac{d + 1}{2}} \leq Q(x) \leq c_3(d)(1 - x)^{\frac{d + 1}{2}}$ . So, $A_L(d)(1 - x)^{d + 1} \leq Q(x)^2 \leq A_U(d)(1 - x)^{d + 1}$ , where + +$$ +A _ {L} (d) = \left(c _ {2} (d)\right) ^ {2} = \left(\frac {c _ {1} (d)}{d + 1} \left(\frac {7}{4}\right) ^ {\frac {d + 1}{2}}\right) ^ {2}, +$$ + +$$ +A _ {U} (d) = \left(c _ {3} (d)\right) ^ {2} = \left(\frac {c _ {1} (d)}{d + 1} 2 ^ {\frac {d + 2}{2}}\right) ^ {2}. +$$ + +2. Bounds for $E(x) - x = \mathbb{E}[X_1 - x \mid X_1 > x]$ : From Proposition E.3, we have $(1 - x) - c_5(d)(1 - x) \leq E(x) - x \leq (1 - x) - c_4(d)(1 - x)$ . So, $B_L(d)(1 - x) \leq E(x) - x \leq B_U(d)(1 - x)$ , where + +$$ +B _ {L} (d) = 1 - c _ {5} (d) = 1 - \frac {2 (d + 1)}{d + 3} \frac {2 ^ {(d - 1) / 2}}{(7 / 4) ^ {(d + 1) / 2}}, +$$ + +$$ +B _ {U} (d) = 1 - c _ {4} (d) = 1 - \frac {2 (d + 1)}{d + 3} \frac {(7 / 4) ^ {(d - 1) / 2}}{2 ^ {(d + 2) / 2}}. +$$ + +Since $E(x) - x = \mathbb{E}[X_1 - x \mid X_1 > x]$ must be positive (as $X_1 > x$ ), we take $B_L(d) = \max(0, 1 - c_5(d))$ . + +3. Bounds for $\sqrt{1 + E(x)^2}$ : We know $1 - c_{5}(d)(1 - x) \leq E(x) \leq 1 - c_{4}(d)(1 - x)$ . For $x \in [3/4, 1)$ , $(1 - x) \in (0, 1/4]$ and the upper bound of $E(x)$ is given by + +$$ +E (x) \leq 1 - c _ {4} (d) (1 - x) < 1. +$$ + +The lower bound is also given in this way + +$$ +E (x) = \mathbb {E} \left[ X _ {1} \mid X _ {1} > x \right] \geq x \geq 3 / 4. +$$ + +Therefore, we deduce that + +$$ +C _ {L} (d) \leq \sqrt {1 + E (x) ^ {2}} \leq C _ {U} (d), +$$ + +where + +$$ +C _ {L} (d) = \frac {5}{4}, C _ {U} (d) = \sqrt {2}. +$$ + +Combining these bounds, for $x\in [3 / 4,1)$ + +$$ +g ^ {+} (x) \geq A _ {L} (d) B _ {L} (d) C _ {L} (d) (1 - x) ^ {d + 1} (1 - x) = c _ {L} ^ {(g)} (d) (1 - x) ^ {d + 2}, +$$ + +$$ +g ^ {+} (x) \leq A _ {U} (d) B _ {U} (d) C _ {U} (d) (1 - x) ^ {d + 1} (1 - x) = c _ {U} ^ {(g)} (d) (1 - x) ^ {d + 2}. +$$ + +The constants are: + +$$ +c _ {L} ^ {(g)} (d) = \left(c _ {2} (d)\right) ^ {2} \cdot \left(1 - c _ {5} (d)\right) \cdot 5 / 4, \tag {39} +$$ + +$$ +c _ {U} ^ {(g)} (d) = \left(c _ {3} (d)\right) ^ {2} \cdot \left(1 - c _ {4} (d)\right) \cdot \sqrt {2}. +$$ + +![](images/71b82209527d535582268efb134c50073b333d0e9cdcfdb922f5c5e56d8eb3c6.jpg) + +# E.2 Empirical Process for the Weight Function + +In this section, we discuss the empirical process of $g_{P}$ . Now we may relax the assumption by just assuming that $X$ is random variable with $\mathrm{supp}(X) \subseteq \mathbb{B}_1^d$ . Fix dimension $d \in \mathbb{N}$ , sample size $n \in \mathbb{N}$ , and let $X_1, \ldots, X_n$ be i.i.d. copies of $X$ . We use the notation $\hat{g}_n$ to denote empirical weight function $g$ as we defined previously. + +For $\pmb{u} \in \mathbb{S}^{d-1}$ and $t \in [-1, 1]$ , define + +$$ +p (\boldsymbol {u}, t) := \mathbb {P} \left(\boldsymbol {X} ^ {\top} \boldsymbol {u} > t\right), \qquad s (\boldsymbol {u}, t) := \mathbb {E} \left[ \left(\boldsymbol {X} ^ {\top} \boldsymbol {u} - t\right) _ {+} \right], +$$ + +and recall the population weight $g_{P}(\boldsymbol{u}, t)$ . By the bounds $0 \leq (\boldsymbol{X}^{\top} \boldsymbol{u} - t)_{+} \leq 2$ and $\| \mathbb{E}[X \mid X^{\top} \boldsymbol{u} > t] \| \leq 1$ (valid on $\mathbb{B}_1^d$ ), we have the pointwise comparison + +$$ +g _ {P} (\boldsymbol {u}, t) \asymp p (\boldsymbol {u}, t) s (\boldsymbol {u}, t) \quad \text {w i t h a b s o l u t e c o n s t a n t s}, \tag {40} +$$ + +i.e., there exist universal $c, C \in (0, \infty)$ such that $c p s \leq g_{P} \leq C p s$ for all $(\boldsymbol{u}, t) \in \mathbb{S}^{d-1} \times [-1, 1]$ . Consider the empirical plug-ins + +$$ +\hat {p} _ {n} (\boldsymbol {u}, t) := \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \left\{\boldsymbol {X} _ {i} ^ {\top} \boldsymbol {u} > t \right\}, \quad \hat {s} _ {n} (\boldsymbol {u}, t) := \frac {1}{n} \sum_ {i = 1} ^ {n} \left(\boldsymbol {X} _ {i} ^ {\top} \boldsymbol {u} - t\right) _ {+}, \quad \hat {g} _ {n} (\boldsymbol {u}, t) := \hat {p} _ {n} (\boldsymbol {u}, t) \hat {s} _ {n} (\boldsymbol {u}, t). +$$ + +Note that $\hat{g}_n$ involves no division by $\hat{p}_n$ , hence avoids any small-mass instability. We now give a self-contained proof of a sharp, distribution-free uniform deviation bound. + +Theorem E.5 (Distribution-free uniform deviation for $\hat{g}_n$ ). There exists a universal constant $C > 0$ such that, for every $\delta \in (0,1)$ , + +$$ +\mathbb {P} \left(\sup _ {\boldsymbol {u} \in \mathbb {S} ^ {d - 1}, t \in [ - 1, 1 ]} \left| \hat {g} _ {n} (\boldsymbol {u}, t) - g _ {P} (\boldsymbol {u}, t) \right| > C \sqrt {\frac {d + \log (2 / \delta)}{n}}\right) \leq \delta . +$$ + +Proof. Using (40), it is enough (up to absolute constants) to control $\left|\hat{p}_n(\boldsymbol{u},t)\hat{s}_n(\boldsymbol{u},t) - p(\boldsymbol{u},t)s(\boldsymbol{u},t)\right|$ uniformly over $(\boldsymbol{u},t) \in \mathbb{S}^{d-1} \times [-1,1]$ . Observe that $0 \leq s, \hat{s}_n \leq 2$ and $0 \leq p, \hat{p}_n \leq 1$ , so + +$$ +\left| \hat {p} _ {n} \hat {s} _ {n} - p s \right| \leq | \hat {p} _ {n} - p | s + | \hat {s} _ {n} - s | \hat {p} _ {n} \leq 2 | \hat {p} _ {n} - p | + | \hat {s} _ {n} - s |. \tag {41} +$$ + +We thus seek uniform bounds for the two empirical processes appearing on the right-hand side. The argument proceeds in two steps: + +- Halfspaces. The class $\{x \mapsto \mathbb{1} \{x^{\top} \boldsymbol{u} > t\} : \boldsymbol{u} \in \mathbb{S}^{d - 1}, t \in \mathbb{R}\}$ has VC-dimension $d + 1$ . Hence, by the VC uniform convergence inequality for $\{0, 1\}$ -valued classes (e.g., [Vapnik, 1998]), there exists a universal constant $C_1 > 0$ such that, for all $\delta \in (0, 1)$ , + +$$ +\mathbb {P} \left(\sup _ {\boldsymbol {u} \in \mathbb {S} ^ {d - 1}, t \in [ - 1, 1 ]} \left| \hat {p} _ {n} (\boldsymbol {u}, t) - p (\boldsymbol {u}, t) \right| > C _ {1} \sqrt {\frac {d + \log (1 / \delta)}{n}}\right) \leq \delta . \tag {42} +$$ + +- ReLU class. Let $\mathcal{F} \coloneqq \{f_{\boldsymbol{u},t}(\boldsymbol{x}) = (\boldsymbol{u}^{\top}\boldsymbol{x} - t)_{+} : \boldsymbol{u} \in \mathbb{S}^{d - 1}, t \in [-1,1]\}$ . Since $\|\boldsymbol{x}\| \leq 1$ and $t \in [-1,1]$ , we have $f \in [0,2]$ . Consider the subgraph family + +$$ +\mathsf {s u b G} (\mathcal {F}) = \left\{\left(\boldsymbol {x}, y\right) \in \mathbb {R} ^ {d} \times \mathbb {R}: y \leq \left(\boldsymbol {u} ^ {\top} \boldsymbol {x} - t\right) _ {+} \right\}. +$$ + +For $y \leq 0$ membership is automatic; for $y > 0$ it is equivalent to the affine halfspace condition $\mathbf{u}^{\top}\mathbf{x} - t - y \geq 0$ in $\mathbb{R}^{d + 1}$ . Thus $\mathrm{VCdim}(\mathrm{subG}(\mathcal{F})) \leq d + 2$ , whence $\mathrm{Pdim}(\mathcal{F}) \leq d + 2$ . Standard pseudo-dimension bounds (see [Haussler, 1992, Thm. 3, 6, 7]) give a universal $C_2 > 0$ with + +$$ +\mathbb {P} \left(\sup _ {\boldsymbol {u} \in \mathbb {S} ^ {d - 1}, t \in [ - 1, 1 ]} \left| \hat {s} _ {n} (\boldsymbol {u}, t) - s (\boldsymbol {u}, t) \right| > C _ {2} \sqrt {\frac {d + \log (1 / \delta)}{n}}\right) \leq \delta . \tag {43} +$$ + +Combining Step (I) and Step (II) with a union bound and the previous inequality yields + +$$ +\mathbb {P} \left(\sup _ {\boldsymbol {u}, t} \left| \hat {p} _ {n} \hat {s} _ {n} - p s \right| > C ^ {\prime} \sqrt {\frac {d + \log (2 / \delta)}{n}}\right) \leq \delta \tag {44} +$$ + +for an absolute constant $C' > 0$ . Finally, the equivalence (40) transfers this bound to $\sup_{\boldsymbol{u},t} \left| \hat{g}_n - g_P \right|$ , up to a universal multiplicative factor and the same failure probability. + +# F Proof of Theorem 3.5: Generalization Gap of Stable Minima + +Let $\mathcal{P}$ denote the joint distribution of $(x,y)$ . Assume that $\mathcal{P}$ is supported on $\mathbb{B}_1^d\times [-D,D]$ for some $D > 0$ . Let $f$ be a function. The population risk or expected risk of $f$ is defined to be + +$$ +R (f) = \underset {(\boldsymbol {x}, y) \sim \mathcal {P}} {\mathbb {E}} \left[ (f (\boldsymbol {x}) - y) ^ {2} \right] \tag {45} +$$ + +Let $\mathcal{D} = \{(x_i, y_i)\}_{i=1}^n$ be a data set where each $(x_i, y_i)$ is drawn i.i.d. from $\mathcal{P}$ . Then the empirical risk is defined to be + +$$ +\hat {R} (f) = \frac {1}{n} \sum_ {i = 1} ^ {n} \left(f \left(\boldsymbol {x} _ {i}\right) - y _ {i}\right) ^ {2} \tag {46} +$$ + +The generalization gap is defined to be + +$$ +\operatorname {G e n e r a l i z a t i o n} \operatorname {G a p} (f; \hat {R}) := | R (f) - \hat {R} (f) |. \tag {47} +$$ + +The generalization gap measures the difference between the train loss and the expected testing error. The smaller the generalization gap, the less likely the model overfits. + +# F.1 Definition of the Variation Space of ReLU Neural Networks + +Recall the notion in Section 3, the weighted variation (semi)norm is defined to be + +$$ +| f | _ {\mathrm {V} _ {g}} := \inf _ {\substack {\nu \in \mathcal {M} (\mathbb {S} ^ {d - 1} \times [ - R, R ]) \\ c \in \mathbb {R} ^ {d}, c _ {0} \in \mathbb {R}}} \| g \cdot \nu \| _ {\mathcal {M}} \quad \text {s.t.} \quad f = f _ {\nu , c, c _ {0}}, \tag{48} +$$ + +and now we define the unweighted variation norm or simply variation norm to be + +$$ +| f | _ {\mathrm {V}} := \inf _ {\substack {\nu \in \mathcal {M} (\mathbb {S} ^ {d - 1} \times [ - R, R ]) \\ c \in \mathbb {R} ^ {d}, c _ {0} \in \mathbb {R}}} \| \nu \| _ {\mathcal {M}} \quad \text {s.t.} \quad f = f _ {\nu , c, c _ {0}}. \tag{49} +$$ + +This definition is identical to the one in [Parhi and Nowak, 2023b, Section V.B]. The following example for unweighted variation norm is similar to Example 3.1. + +Example F.1. Since we are interested in functions defined on $\mathbb{B}_R^d$ , for a finite-width neural network $f_{\theta}(\pmb{x}) = \sum_{k=1}^{K} v_k \phi(\pmb{w}_k^\top \pmb{x} - b_k) + \beta$ , we observe that it has the equivalent implementation as $f_{\theta}(\pmb{x}) = \sum_{j=1}^{J} a_j \phi(\pmb{u}_j^\top \pmb{x} - t_j) + \pmb{c}^\top \pmb{x} + c_0$ , where $a_j \in \mathbb{R}$ , $\pmb{u}_j \in \mathbb{S}^{d-1}$ , $t_j \in \mathbb{R}$ , $\pmb{c} \in \mathbb{R}^d$ , and $c_0 \in \mathbb{R}$ . Indeed, this is due to the fact that the ReLU is homogeneous, which allows us to absorb the magnitude of the input weights into the output weights (i.e., each $a_j = |v_{k_j} \| \pmb{w}_{k_j} \|_2$ for some $k_j \in \{1, \dots, K\}$ ). Furthermore, any ReLUs in the original parameterization whose activation threshold8 is outside $\mathbb{B}_R^d$ can be implemented by an affine function on $\mathbb{B}_R^d$ , which gives rise to the $c^\top \pmb{x} + c_0$ term in the implementation. If this new implementation is in "reduced form", i.e., the collection $\{(u_j, t_j)\}_{j=1}^J$ are distinct, then we have that $|f_{\theta}|_{\mathrm{V}} = \sum_{j=1}^{J} |a_j|$ . + +The bounded variation function class is defined w.r.t. the unweighted variation norm. + +Definition F.2. For the compact region $\Omega = \mathbb{B}_R^d$ , we define the bounded variation function class as + +$$ +\mathrm {V} _ {C} (\Omega) := \left\{f: \Omega \rightarrow \mathbb {R} \mid f = \int_ {\mathbb {S} ^ {d - 1} \times [ - R, R ]} \phi \left(\boldsymbol {u} ^ {\mathsf {T}} \boldsymbol {x} - t\right) \mathrm {d} \nu (\boldsymbol {u}, t) + \boldsymbol {c} ^ {\mathsf {T}} \boldsymbol {x} + b, | f | _ {\mathrm {V}} \leq C \right\}. \tag {50} +$$ + +# F.2 Metric Entropy and Variation Spaces + +Metric entropy quantifies the compactness of a set $A$ in a metric space $(X, \rho_X)$ . Below we introduce the definition of covering numbers and metric entropy. + +Definition F.3 (Covering Number and Entropy). Let $A$ be a compact subset of a metric space $(X, \rho_X)$ . For $t > 0$ , the covering number $N(A, t, \rho_X)$ is the minimum number of closed balls of radius $t$ needed to cover $A$ : + +$$ +N (t, A, \rho_ {X}) := \min \left\{N \in \mathbb {N}: \exists x _ {1}, \dots , x _ {N} \in X \text {s . t .} A \subset \bigcup_ {i = 1} ^ {N} \mathbb {B} \left(x _ {i}, t\right) \right\}, \tag {51} +$$ + +where $\mathbb{B}(x_i,t) = \{y\in X:\rho_X(y,x_i)\leq t\}$ . The metric entropy of $A$ at scale $t$ is defined as: + +$$ +H _ {t} (A) _ {X} := \log N (t, A, \rho_ {X}). \tag {52} +$$ + +The metric entropy of the bounded variation function class has been studied in previous works. More specifically, we will directly use the one below in future analysis. + +Proposition F.4 (Parhi and Nowak 2023b, Appendix D). The metric entropy of $\mathrm{V}_C(\mathbb{B}_R^d)$ (see Definition F.2) with respect to the $L^{\infty}(\mathbb{B}_R^d)$ -distance $\| \cdot \|_{\infty}$ satisfies + +$$ +\log N \left(t, \mathrm {V} _ {C} \left(\mathbb {B} _ {R} ^ {d}\right), \| \cdot \| _ {\infty}\right) \lesssim_ {d} \left(\frac {C}{t}\right) ^ {\frac {2 d}{d + 3}}. \tag {53} +$$ + +where $\lesssim_d$ hides constants (which could depend on $d$ ) and logarithmic factors. + +# F.3 Generalization Gap of Unweighted Variation Function Class + +As a middle step towards bounding the generalization gap of the weighted variation function class, we first bound the generalization gap of the unweighted variation function class according to a metric entropy analysis. + +Lemma F.5. Let $\mathcal{F}_{M,C} = \{f\in \mathrm{V}_C(\mathbb{B}_R^d)\mid \| f\|_{\infty}\leq M\}$ with $M\geq D$ where $D$ refers to Theorem 3.5. Then with probability at least $1 - \delta$ + +$$ +\sup _ {f \in \mathcal {F} _ {M, C}} \left| R (f) - \widehat {R} _ {n} (f) \right| \lesssim_ {d} C ^ {\frac {d}{2 d + 3}} M ^ {\frac {3 (d + 2)}{2 d + 3}} n ^ {- \frac {d + 3}{4 d + 6}}. \tag {54} +$$ + +Proof. According to Proposition F.4, one just needs $N(t)$ balls to cover $\mathcal{F}$ in $\| \cdot \|_{\infty}$ with radius $t > 0$ such that where + +$$ +\log N (t) \lesssim_ {d} \left(\frac {C}{t}\right) ^ {\frac {2 d}{d + 3}}. +$$ + +Then for any $f,g\in \mathcal{F}_{M,C}$ and any $(x,y)$ + +$$ +\left| \left(f (\boldsymbol {x}) - y\right) ^ {2} - \left(g (\boldsymbol {x}) - y\right) ^ {2} \right| = | f (\boldsymbol {x}) - g (\boldsymbol {x}) | | f (\boldsymbol {x}) + g (\boldsymbol {x}) - 2 y | \leq 4 M \| f - g \| _ {\infty}. +$$ + +Hence replacing $f$ by a centre $f_{i}$ within $t$ changes both the empirical and true risks by at most $4Mt$ . + +For any fixed centre $\bar{f}$ in the covering, Hoeffding's inequality implies that with probability at least $>1 - \delta$ , we have + +$$ +\left| R (\bar {f}) - \hat {R} (\bar {f}) \right| \leq 4 M ^ {2} \sqrt {\frac {\log (2 / \delta)}{n}} \tag {55} +$$ + +because each squared error lies in $[0,4M^2]$ . Then we take all the centers with union bound to deduce that with probability at least $1 - \delta /2$ , for any center $\bar{f}$ in the set of covering index, we have + +$$ +\begin{array}{l} \left| R (\bar {f}) - \hat {R} (\bar {f}) \right| \leq 4 M ^ {2} \sqrt {\frac {\log (4 N (t) / \delta)}{n}} \tag {56} \\ \stackrel {< } {_ \approx_ {d}} M ^ {2} \cdot \left(\frac {C}{t}\right) ^ {\frac {d}{d + 3}} n ^ {- \frac {1}{2}}. \\ \end{array} +$$ + +According to the definition of covering sets, for any $f \in \mathcal{F}_{M,C}$ , we have that $\| f - \bar{f} \|_{\infty} \leq t$ for some center $\bar{f}$ . Then we have + +$$ +\begin{array}{l} \left| R (f) - \hat {R} (f) \right| \\ \leq | R (\bar {f}) - \hat {R} (\bar {f}) | + O (M t) \tag {57} \\ \leq M ^ {2} \cdot \left(\frac {C}{t}\right) ^ {\frac {d}{d + 3}} n ^ {- \frac {1}{2}} + O (M t). \\ \end{array} +$$ + +After tuning $t$ to be the optimal choice, we deduce that (54). + +# F.4 Concentration Property on the Ball: Uniform Distribution + +In the following analysis, we will handle the interior and boundary of the unit ball separately. In this part, we define the annulus of a ball rigorously and provide a high-probability bound on the number of samples falling in the annulus. + +Definition F.6. Let $\mathbb{B}_1^d$ be the unit ball. The $\varepsilon$ -annulus is a subset of $\mathbb{B}_1^d$ defined as + +$$ +\mathbb {A} _ {\varepsilon} ^ {d} := \left\{\boldsymbol {x} \in \mathbb {B} _ {1} ^ {d} \mid \| \boldsymbol {x} \| _ {2} \geq 1 - \varepsilon \right\} +$$ + +and the closure of its complement is called $\varepsilon$ -strict interior and denoted by $\mathbb{I}_{\varepsilon}^{d}$ . + +Lemma F.7 (High-Probability Upper Bound on Annulus). Let $d \in \mathbb{N}$ and $\varepsilon \in (0,1)$ . Let + +$$ +\boldsymbol {x} _ {1}, \dots , \boldsymbol {x} _ {n} \sim \operatorname {U n i f o r m} \left(\mathbb {B} _ {1} ^ {d}\right). +$$ + +Define $n_A \coloneqq |\{i \mid \boldsymbol{x}_i \in \mathbb{A}_{\varepsilon}^d\}|$ and $p = \mathbb{P}\big(\boldsymbol{X} \in \mathbb{A}_{\varepsilon}^{d}\big) = 1 - (1 - \varepsilon)^{d} = \Theta(\varepsilon)$ . Then for any $\delta \in (0,1)$ , with probability at least $1 - \delta$ , + +$$ +\frac {n _ {A}}{n} \leq p + \sqrt {\frac {3 p \log (1 / \delta)}{n}}. \tag {58} +$$ + +Proof. For each $i = 1, \ldots, n$ , consider a Bernoulli random variable + +$$ +U = \mathbb {1} \left\{\boldsymbol {X} \in \mathbb {A} _ {\varepsilon} ^ {d} \right\}, +$$ + +so that $\mathbb{E}[U] = p$ and regards $U_{i}$ as a sample. Then we may take $n_A = \sum_{i=1}^n U_i$ . By the multiplicative Chernoff bound for the upper tail of a sum of independent Bernoulli variables, + +$$ +\mathbb {P} \left(n _ {A} > (1 + \gamma) n p\right) \leq \exp \left(- \frac {\gamma^ {2}}{3} n p\right), \quad \forall \gamma > 0. +$$ + +Set the right-hand side equal to $\delta$ and solve for $\gamma$ : + +$$ +\exp \left(- \frac {\gamma^ {2}}{3} n p\right) = \delta \quad \Longrightarrow \quad - \frac {\gamma^ {2}}{3} n p = \ln \delta \quad \Longrightarrow \quad \gamma = \sqrt {\frac {3 \ln (1 / \delta)}{n p}}. +$$ + +If $\gamma > 1$ , note that trivially $n_A / n \leq 1 \leq p + \sqrt{\frac{3p \ln(1 / \delta)}{n}}$ , so the claimed bound holds in all cases. Otherwise, plugging this choice of $\gamma$ into the Chernoff bound gives + +$$ +\mathbb {P} \left(n _ {A} \leq n p (1 + \gamma)\right) \geq 1 - \delta , +$$ + +i.e. with probability at least $1 - \delta$ + +$$ +n _ {A} \leq n p + \sqrt {3 n p \ln (1 / \delta)}, +$$ + +and dividing by $n$ yields the stated inequality. + +![](images/b5856625dc314f57fd1ba0765ddf8f2ab586467ddce10c53d4d8b14f53378cd0.jpg) + +# F.5 Upper Bound of Generalization Gap of Stable Minima + +Let $f = f_{\theta}$ be a stable solution of the loss function $\mathcal{L}(\theta)$ , trained by gradient descent with learning rate $\eta$ . Then we have + +$$ +\begin{array}{l} \frac {2}{\eta} \geq \lambda_ {\max } \left(\nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L} (\boldsymbol {\theta})\right) \geq \boldsymbol {v} ^ {\top} \nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L} (\boldsymbol {\theta}) \boldsymbol {v} \\ = \underbrace {\lambda_ {\max } \left(\frac {1}{n} \sum_ {i = 1} ^ {n} \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) ^ {\top}\right)} _ {\text {(T e r m A)}} \tag {59} \\ + \underbrace {\frac {1}{n} \sum_ {i = 1} ^ {n} (f _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {i}) - y _ {i}) \boldsymbol {v} ^ {\mathsf {T}} \nabla_ {\boldsymbol {\theta}} ^ {2} f _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {i}) \boldsymbol {v}} _ {\text {(T e r m B)}}. \\ \end{array} +$$ + +For (Term A), we have + +$$ +\lambda_ {\max } \left(\frac {1}{n} \sum_ {i = 1} ^ {n} \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) \left(\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right)\right) ^ {\mathsf {T}}\right) \geq 1 + 2 \left| f _ {\boldsymbol {\theta}} \right| _ {\mathrm {V} _ {g}}. \tag {60} +$$ + +For (Term B), we have + +$$ +| (\text {T e r m B}) | \leq \sqrt {\frac {1}{n} \sum_ {i = 1} ^ {n} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - y _ {i}\right) ^ {2}} \cdot \sqrt {\frac {1}{n} \sum_ {i = 1} ^ {n} \left(\boldsymbol {v} ^ {\top} \nabla_ {\boldsymbol {\theta}} ^ {2} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) \boldsymbol {v}\right) ^ {2}} \leq 4 \sqrt {2 \mathcal {L} (\boldsymbol {\theta})}. \tag {61} +$$ + +Let $M = \max \{\| f\|_{\infty},D,1\}$ . Then we have + +$$ +\sqrt {2 \mathcal {L} (\boldsymbol {\theta})} = \sqrt {\frac {1}{n} \sum_ {i = 1} ^ {n} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - y _ {i}\right) ^ {2}} \leq 2 M. +$$ + +Combining these inequalities together, we may deduce that + +$$ +\left| f _ {\theta} \right| _ {\mathrm {V} _ {g}} \leq \frac {1}{\eta} - \frac {1}{2} + 4 M. \tag {62} +$$ + +With all the preparations, we are ready to prove the generalization gap upper bound for stable minima. + +Theorem F.8. (First part of Theorem 3.5) Let $\mathcal{P}$ denote the joint distribution of $(\pmb{x},y)$ . Assume that $\mathcal{P}$ is supported on $\mathbb{B}_1^d\times [-D,D]$ for some $D > 0$ and that the marginal distribution of $\pmb{x}$ is $\mathrm{Uniform}(\mathbb{B}_1^d)$ . Fix a data set $\mathcal{D} = \{(x_i,y_i)\}_{i = 1}^n$ , where each $(x_{i},y_{i})$ is drawn i.i.d. from $\mathcal{P}$ , and $\mathcal{D}$ yields the empirical weight function $g$ defined in (6). Then, with probability at least $1 - \delta$ , we have that for the plug-in risk estimator $\hat{R} (f)\coloneqq \frac{1}{n}\sum_{i = 1}^{n}(f(x_i) - y_i)^2$ , + +Suppose that $\hat{R}$ is a generalization of $R$ . Then $|R(f)| \leqslant d$ , and $A^{\frac{d}{d^2 + 4d + 3}} B^2 n^{-\frac{1}{2d + 4}}$ . + +$$ +\begin{array}{c} f \in \mathrm {V} _ {g} (\mathbb {B} _ {1} ^ {d}) \\ f | _ {\mathrm {V} _ {g}} \leq A, \| f \| _ {L ^ {\infty}} \leq B \end{array} +$$ + +where $B$ is assumed $>1$ and $\lesssim_d$ hides constants (which could depend on $d$ ) and logarithmic factors in $n$ and $(1/\delta)$ . In particular, Theorem 3.2 and (62) imply that that + +$$ +f _ {\boldsymbol {\theta}} \in \left\{f \in \mathrm {V} _ {g} \left(\mathbb {B} _ {1} ^ {d}\right) \mid | f | _ {\mathrm {V} _ {g}} \leq \frac {1}{\eta} - \frac {1}{2} + 4 M, \| f \| _ {L ^ {\infty} \left(\mathbb {B} _ {1} ^ {d}\right)} \leq M \right\} \tag {63} +$$ + +for every + +$$ +\boldsymbol {\theta} \in \left\{\boldsymbol {\theta} \in \Theta_ {\text {f l a t}} (\eta ; \mathcal {D}) \mid \| f \| _ {L ^ {\infty} \left(\mathbb {B} _ {1} ^ {d}\right)} \leq M \right\}. \tag {64} +$$ + +Therefore, we may conclude that + +$\sup_{\pmb {\theta}\in \Theta_{\mathrm{flat}}(\eta ;\mathcal{D})}$ GeneralizationGap $(f_{\pmb{\theta}};\hat{R}):= |R(f_{\pmb{\theta}})-\hat{R}(f_{\pmb{\theta}})|$ + +$$ +\lesssim_ {d} \left(\frac {1}{\eta} - \frac {1}{2} + 4 M\right) ^ {\frac {d}{d ^ {2} + 4 d + 3}} M ^ {2} n ^ {- \frac {1}{2 d + 4}}, \tag {65} +$$ + +where $M\coloneqq \max \left\{D,\| f_{\pmb{\theta}}\|_{L^{\infty}(\mathbb{B}_1^d)},1\right\}$ + +Proof. For any fixed $\varepsilon < 1/4$ , we may decompose $\mathbb{B}_1^d$ into $\varepsilon$ -annulus and $\varepsilon$ -strict interior + +$$ +\mathbb {B} _ {1} ^ {d} = \mathbb {A} _ {\varepsilon} ^ {d} \cup \mathbb {I} _ {\varepsilon} ^ {d}. +$$ + +According to the law of total expectation, the population risk is decomposed into + +$$ +\underset {(\boldsymbol {x}, y) \sim \mathcal {P}} {\mathbb {E}} \left[ (f (\boldsymbol {x}) - y) ^ {2} \right] = \mathbb {P} (\boldsymbol {x} \in \mathbb {A} _ {\varepsilon} ^ {d}) \cdot \mathbb {E} _ {\mathbb {A}} \left[ (f (\boldsymbol {x}) - y) ^ {2} \right] + \mathbb {P} (\boldsymbol {x} \in \mathbb {I} _ {\varepsilon} ^ {d}) \cdot \mathbb {E} _ {\mathbb {I}} \left[ (f (\boldsymbol {x}) - y) ^ {2} \right], \tag {66} +$$ + +where $\mathbb{E}_{\mathbb{A}}$ means that $\{\pmb{x},y\}$ is a new sample from the data distribution conditioned on $\pmb{x} \in \mathbb{A}_{\varepsilon}^{d}$ and $\mathbb{E}_{\mathbb{I}}$ means that $(\pmb{x},y)$ is a new sample from the data distribution conditioned on $\pmb{x} \in \mathbb{I}_{\varepsilon}^{d}$ . + +Similarly, we also have this decomposition for empirical risk + +$$ +\begin{array}{l} \frac {1}{n} \sum_ {i = 1} ^ {n} (f (\boldsymbol {x} _ {i}) - y _ {i}) ^ {2} = \frac {1}{n} \left(\sum_ {i \in I} (f (\boldsymbol {x} _ {i}) - y _ {i}) ^ {2} + \sum_ {j \in A} (f (\boldsymbol {x} _ {j}) - y _ {j}) ^ {2}\right) \tag {67} \\ = \frac {n _ {I}}{n} \frac {1}{n _ {I}} \sum_ {i \in I} (f (\pmb {x} _ {i}) - y _ {i}) ^ {2} + \frac {n _ {A}}{n} \frac {1}{n _ {A}} \sum_ {j \in A} (f (\pmb {x} _ {j}) - y _ {j}) ^ {2}, \\ \end{array} +$$ + +where $I$ is the set of data points with $\pmb{x}_i\in \mathbb{I}_\varepsilon^d$ and $A$ is the set of data points with $\pmb {x}_i\in \mathbb{A}_\varepsilon^d$ . Then the generalization gap can be decomposed into + +$$ +\begin{array}{l} \left| R (f) - \hat {R} (f) \right| \leq \mathbb {P} (\boldsymbol {x} \in \mathbb {A} _ {\varepsilon} ^ {d}) \cdot \mathbb {E} _ {\mathbb {A}} \left[ \left(f _ {\boldsymbol {\theta}} (\boldsymbol {x}) - y\right) ^ {2} \right] + \frac {n _ {A}}{n} \frac {1}{n _ {A}} \sum_ {j \in A} \left(f \left(\boldsymbol {x} _ {j}\right) - y _ {j}\right) ^ {2} + (68) \\ + \left| \mathbb {P} \left(\boldsymbol {x} \in \mathbb {I} _ {\varepsilon} ^ {d}\right) - \frac {n _ {I}}{n} \right| \frac {1}{n _ {I}} \sum_ {i \in I} \left(f \left(\boldsymbol {x} _ {i}\right) - y _ {i}\right) ^ {2} (69) \\ + \mathbb {P} (\boldsymbol {x} \in \mathbb {I} _ {\varepsilon} ^ {d}) \cdot \left| \mathbb {E} _ {\mathbb {I}} \left[ (f (\boldsymbol {x}) - y) ^ {2} \right] - \frac {1}{n _ {I}} \sum_ {i \in I} (f (\boldsymbol {x} _ {i}) - y _ {i}) ^ {2} \right|. (70) \\ \end{array} +$$ + +Using the property that the marginal distribution of $\pmb{x}$ is $\mathrm{Uniform}(\mathbb{B}_1^d)$ and its concentration property (see Lemma F.7), with probability at least $1 - \delta /2$ : + +$$ +(6 8) \lesssim_ {d} O \left(B ^ {2} \varepsilon\right), \tag {71} +$$ + +where $\lesssim_d$ hides the constants that could depend on $d$ and logarithmic factors of $1 / \delta$ . + +For the term (69), with probability $1 - \delta /3$ + +$$ +\left\{ \begin{array}{l l} \left| \mathbb {P} \left(\boldsymbol {x} \in \mathbb {I} _ {\varepsilon} ^ {d}\right) - \frac {n _ {I}}{n} \right| & \lesssim \sqrt {\frac {\varepsilon \log (3 / \delta)}{n}}, \\ \frac {1}{n _ {I}} \sum_ {i \in I} \left(f \left(\boldsymbol {x} _ {i}\right) - y _ {i}\right) ^ {2} & \leq 4 B ^ {2} \end{array} \right. \tag {72} +$$ + +so we may also conclude that + +$$ +(6 9) \lesssim M ^ {2} \sqrt {\frac {\varepsilon \log (3 / \delta)}{n}} \tag {73} +$$ + +For the part of the interior (70), the scalar $\mathbb{P}(\pmb{x}\in \mathbb{I}_{\varepsilon}^{d})$ is less than 1 with high-probability. Therefore, we just need to deal with the term + +$$ +\mathbb {E} _ {\mathbb {I}} \left[ (f (\boldsymbol {x}) - y) ^ {2} \right] - \frac {1}{n _ {I}} \sum_ {i \in I} \left(f \left(\boldsymbol {x} _ {i}\right) - y _ {i}\right) ^ {2}. \tag {74} +$$ + +Since both the distribution and sample points only support in $\mathbb{I}_{\varepsilon}^{d}$ , we may consider $f$ by its restrictions in $\mathbb{I}_{\varepsilon}^{d}$ , which are denoted by $f^{\varepsilon}$ . Furthermore, according to the definition, we have + +$$ +\begin{array}{l} f (\boldsymbol {x}) = \int_ {\mathbb {S} ^ {d - 1} \times [ - 1, 1 ]} \phi \left(\boldsymbol {u} ^ {\mathsf {T}} \boldsymbol {x} - t\right) \mathrm {d} \nu (\boldsymbol {u}, t) + \boldsymbol {c} ^ {\mathsf {T}} \boldsymbol {x} + b \\ = \int_ {\mathbb {S} ^ {d - 1} \times [ - 1 + \varepsilon , 1 - \varepsilon ]} \phi (\boldsymbol {u} ^ {\mathsf {T}} \boldsymbol {x} - t) \mathrm {d} \nu (\boldsymbol {u}, t) + \underbrace {\int_ {\mathbb {S} ^ {d - 1} \times [ - 1 , - 1 + \varepsilon) \cup (1 - \varepsilon , 1 ]} \phi (\boldsymbol {u} ^ {\mathsf {T}} \boldsymbol {x} - t) \mathrm {d} \nu (\boldsymbol {u} , t)} _ {\text {A n n u l u s R e L U}} \\ + \boldsymbol {c} ^ {\top} \boldsymbol {x} + b \tag {75} \\ \end{array} +$$ + +where the Annulus ReLU term is totally linear in the strictly interior i.e. there exists $c', b'$ such that + +$$ +\boldsymbol {c} ^ {\prime \mathrm {T}} \boldsymbol {x} + b ^ {\prime} = \int_ {\mathbb {S} ^ {d - 1} \times [ - 1, - 1 + \varepsilon) \cup (1 - \varepsilon , 1 ]} \phi (\boldsymbol {u} ^ {\mathrm {T}} \boldsymbol {x} - t) \mathrm {d} \nu (\boldsymbol {u}, t), \quad \forall \boldsymbol {x} \in \mathbb {I} _ {\varepsilon} ^ {d}. \tag {76} +$$ + +Therefore, we may write + +$$ +f (\boldsymbol {x}) = f ^ {\varepsilon} (\boldsymbol {x}) = \int_ {\mathbb {S} ^ {d - 1} \times [ - 1 + \varepsilon , 1 - \varepsilon ]} \phi \left(\boldsymbol {u} ^ {\mathsf {T}} \boldsymbol {x} - t\right) \mathrm {d} \nu (\boldsymbol {u}, t) + \left(\boldsymbol {c} + \boldsymbol {c} ^ {\prime}\right) ^ {\mathsf {T}} \boldsymbol {x} + \boldsymbol {b} + \boldsymbol {b} ^ {\prime}, \quad \boldsymbol {x} \in \mathbb {I} _ {\varepsilon} ^ {d}. \tag {77} +$$ + +The core of the argument is to rigorously bound the interior generalization gap. Recall that a stable minima $\pmb{\theta} \in \Theta_{\mathrm{flat}}(\eta; \mathcal{D})$ satisfies $|f|_{\mathrm{V}_g} \leq A$ with respect to the empirical weight function $g$ . To analyze the complexity of its restriction $f^{\varepsilon}$ on the core $\mathbb{I}_{\varepsilon}^{d}$ , we need a lower bound on $g_{\min}^{\varepsilon} \coloneqq \inf_{|t| \leq 1 - \varepsilon} g(\boldsymbol{u}, t)$ . This quantity is a random variable. + +From empirical process we discussed in Section E.2, especially Theorem E.5, we know that with probability at least $1 - \delta /3$ + +$$ +\sup _ {\boldsymbol {u}, t} | g (\boldsymbol {u}, t) - g _ {P} (\boldsymbol {u}, t) | \lesssim_ {d} \sqrt {\frac {d + \log (6 / \delta)}{n}} =: \epsilon_ {n}. \tag {78} +$$ + +This implies a lower bound on the empirical minimum weight in the core with probability at least $1 - \delta / 3$ , + +$$ +g _ {\min } ^ {\varepsilon} = \inf _ {| t | \leq 1 - \varepsilon} g (\boldsymbol {u}, t) \geq \underbrace {\inf _ {| t | \leq 1 - \varepsilon} g _ {P} (\boldsymbol {u} , t)} _ {g _ {P, \min } ^ {\varepsilon}} - \epsilon_ {n} = g _ {P, \min } ^ {\varepsilon} - \epsilon_ {n}. \tag {79} +$$ + +Here, $g_{\tilde{P},\min}^{\varepsilon}\asymp \varepsilon^{d + 2}$ is the minimum of the population weight function in the core. + +For the bound $|f^{\varepsilon}|_{\mathrm{V}} \leq A / g_{\min}^{\varepsilon} \leq A / (g_{P,\min} - \epsilon_n)$ to be meaningful with high probability, we must operate in a regime where $g_{\min} \geq \epsilon_n$ . We enforce a stricter validity condition for our proof + +$$ +g _ {\min } \geq 2 \epsilon_ {n} \quad \Longrightarrow \quad \varepsilon^ {d + 2} \underset {\approx_ {d}} {\geq} n ^ {- \frac {1}{2}}. \tag {80} +$$ + +Therefore, we may choose + +$$ +\varepsilon \asymp \left(A ^ {\frac {d}{d ^ {2} + 4 d + 3}} \cdot \sqrt {\frac {d + \log (6 / \delta)}{n}}\right) ^ {\frac {1}{d + 2}} \tag {81} +$$ + +Under this condition, we have $g_{\mathrm{min}}^{\varepsilon} \geq g_{P,\mathrm{min}}^{\varepsilon} - \epsilon_n \geq g_{P,\mathrm{min}}^{\varepsilon} / 2 \asymp \varepsilon^{d + 2}$ . Thus, for any stable solution $f$ , its restriction $f^{\varepsilon}$ has a controlled unweighted variation norm with high probability: + +$$ +\left| f ^ {\varepsilon} \right| _ {\mathrm {V} \left(\mathbb {B} _ {1 - \varepsilon} ^ {d}\right)} \leq \frac {A}{g _ {\min } ^ {\varepsilon}} \leq \frac {A}{g _ {P , \min } ^ {\varepsilon} / 2} \asymp \frac {A}{\varepsilon^ {d + 2}} =: C _ {\varepsilon}. +$$ + +We can now apply the generalization bound from Lemma F.5 to the class $\mathrm{V}_{C_{\varepsilon}}(\mathbb{B}_{1 - \varepsilon}^{d})$ by plugging in (81), with probability $1 - \delta /3$ + +$$ +\text {I n t e r i o r G a p} (7 0) \lesssim_ {\approx d} \left(C _ {\varepsilon}\right) ^ {\frac {d}{2 d + 3}} B ^ {\frac {3 (d + 2)}{2 d + 3}} n ^ {- \frac {d + 3}{4 d + 6}} \tag {82} +$$ + +$$ += \left(A ^ {1 - \frac {d}{d ^ {2} + 4 d + 3}} \sqrt {\frac {n}{d + \log (6 / \delta)}}\right) ^ {\frac {d}{2 d + 3}} B ^ {\frac {3 (d + 2)}{2 d + 3}} n ^ {- \frac {d + 3}{4 d + 6}} \tag {83} +$$ + +$$ +\stackrel {< } {\approx} _ {d} A ^ {\frac {d}{d ^ {2} + 4 d + 3}} B ^ {\frac {3 (d + 2)}{2 d + 3}} n ^ {- \frac {3}{4 d + 6}} \tag {84} +$$ + +where $\lesssim_d$ hides the constants that could depend on $d$ and logarithmic factors of $1 / \delta$ . + +Now we combine the upper bounds (71), (73) and (84) to deduce an upper bound of the generalization gap. With probability $1 - \delta$ , we have + +$$ +\left| R (f) - \hat {R} (f) \right| \lesssim_ {d} A ^ {\frac {d}{d ^ {2} + 4 d + 3}} B ^ {2} n ^ {- \frac {1}{2 d + 4}} + A ^ {\frac {d}{d ^ {2} + 4 d + 3}} B ^ {\frac {3 (d + 2)}{2 d + 3}} n ^ {- \frac {3}{4 d + 6}}. \tag {85} +$$ + +Since $n^{-\frac{1}{2d + 4}} > n^{-\frac{3}{4d + 6}}$ and $B^2 > B^{\frac{3(d + 2)}{2d + 3}}$ with the assumption $M \geq 1$ , we conclude that + +$$ +\left| R (f) - \hat {R} (f) \right| \lesssim_ {d} \left(\frac {1}{\eta} - \frac {1}{2} + 4 B\right) ^ {\frac {d}{d ^ {2} + 4 d + 3}} M ^ {2} n ^ {- \frac {1}{2 d + 4}}, \tag {86} +$$ + +which finishes the proof. + +![](images/49cfda790dd028f0bd6bd68ce569ad518e8881ba10d3e8b77d4adaa247741555.jpg) + +Remark F.9. For the generalization gap lower bound (second part of Theorem 3.5), we defer the proof to Appendix I as it relies on a construction that is used to prove Theorem 3.7 from Appendix H. + +# G Proof of Theorem 3.6: Estimation Error Rate for Stable Minima + +# G.1 Computation of Local Gaussian Complexity + +It is known from Wainwright 2019 that a tight analysis of MSE results from local gaussian complexity. We begin with the following proposition that connects the local gaussian complexity to the critical radius. + +Proposition G.1 (Wainwright 2019, Chapter 13). Let $\mathcal{F}$ be a convex model class that contains the constant function 1. Fix design points $x_{1},\ldots ,x_{n}$ in the region of interest and denote the empirical norm + +$$ +\| f \| _ {n} ^ {2} := \frac {1}{n} \sum_ {i = 1} ^ {n} f \left(\boldsymbol {x} _ {i}\right) ^ {2}. +$$ + +For any radius $r > 0$ write + +$$ +\mathcal {F} (r) := \left\{f \in \mathcal {F}: \| f \| _ {n} \leq r \right\}, \quad \widehat {\mathcal {G}} _ {n} (r, \mathcal {F}) := \sup _ {f \in \mathcal {F} (r)} \frac {1}{n} \sum_ {i = 1} ^ {n} \varepsilon_ {i} f (\boldsymbol {x} _ {i}), +$$ + +where $\varepsilon_1,\ldots ,\varepsilon_n\stackrel {i.i.d.}{\sim}\mathcal{N}(0,\sigma^2)$ and $\mathcal{G}_n(r,\mathcal{F}):= \mathbb{E}\widehat{\mathcal{G}}_n(r,\mathcal{F})$ + +If $\delta$ satisfies the integral inequality + +$$ +\frac {1 6}{\sqrt {n}} \int_ {0} ^ {r} \sqrt {\log N (t , \partial \mathcal {F} , \| \cdot \| _ {n})} d t \leq \frac {r}{4}, \tag {87} +$$ + +where $\partial \mathcal{F} \coloneqq \left\{f_1 - f_2 : f_1, f_2 \in \mathcal{F}\right\}$ , then the local empirical Gaussian complexity obeys + +$$ +\frac {\mathcal {G} _ {n} (r , \mathcal {F})}{r} \leq \frac {r}{2 \sigma}. \tag {88} +$$ + +Moreover, with probability at least $1 - \delta$ one has + +$$ +\widehat {\mathcal {G}} _ {n} (r, \mathcal {F}) \leq \frac {r ^ {2}}{2 \sigma} + r \frac {\sqrt {\log (1 / \delta)}}{\sqrt {n}} \quad (\delta > 0). \tag {89} +$$ + +As a result, we can derive an upper bound for the local empirical Gaussian complexity of the variation function class through a careful analysis of the critical radius. + +Lemma G.2. Let $\mathcal{F}_{B,C}(\mathbb{B}_R^d) = \{f\in \mathrm{V}_C(\mathbb{B}_R^d)\mid \| f\|_{L^\infty (\mathbb{B}_R^d)}\leq B\}$ . Then with probability at least $1 - \delta$ , we have + +$$ +\frac {1}{n} \sum_ {i = 1} ^ {n} \varepsilon_ {i} \left(f _ {1} \left(\boldsymbol {x} _ {i}\right) - f _ {2} \left(\boldsymbol {x} _ {i}\right)\right) \lesssim_ {d} C ^ {\frac {2 d}{2 d + 3}} n ^ {- \frac {d + 3}{2 d + 3}} + C ^ {\frac {d}{2 d + 3}} n ^ {- \frac {3 d + 6}{4 d + 6}} \sqrt {\log (1 / \delta)}, \tag {90} +$$ + +for any two $f_{1},f_{2}\in \mathcal{F}_{B,C}$ + +Proof. As $\partial \mathcal{F}_{B,C} = 2\mathcal{F}_{B,C}\subset \mathcal{F}_{2B,2C}$ , bounding the entropy of $\mathcal{F}_{2B,2C}$ suffices. Using $\| f\| _n\leq$ $\| f\|_{L^{\infty}(\mathbb{B}_1^d)}$ and referring to Proposition F.4, we have, up to logarithmic factors, + +$$ +\log N \big (t, \mathcal {F} _ {2 B, 2 C}, \| \cdot \| _ {n} \big) \lesssim_ {d} \left(\frac {C}{t}\right) ^ {\frac {2 d}{d + 3}}. +$$ + +Plugging this entropy bound into the left side of (87) and integrating, + +$$ +\frac {1 6}{\sqrt {n}} \int_ {0} ^ {r} \left(\frac {C}{t}\right) ^ {\frac {d}{d + 3}} \mathrm {d} t \lesssim_ {d} \frac {C ^ {\frac {d}{d + 3}}}{\sqrt {n}} \int_ {0} ^ {r} t ^ {- \frac {d}{d + 3}} \mathrm {d} t = \frac {C ^ {\frac {d}{d + 3}} r ^ {\frac {3}{d + 3}}}{\sqrt {n}}. +$$ + +Hence inequality (87) is met provided + +$$ +\frac {C ^ {\frac {d}{d + 3}} r ^ {\frac {3}{d + 3}}}{\sqrt {n}} \lesssim_ {d} \frac {r}{4}, \quad \Longleftrightarrow \quad r ^ {\frac {d}{d + 3}} \gtrsim_ {d} C ^ {\frac {d}{d + 3}} n ^ {- 1 / 2}. +$$ + +Solving for $r^2$ (and keeping only dominant terms) yields + +$$ +r _ {n} ^ {2} \asymp_ {d} C ^ {\frac {2 d}{2 d + 3}} n ^ {- \frac {d + 3}{2 d + 3}}. +$$ + +With this choice of $r_n$ , Proposition G.1 guarantees + +$$ +\mathcal {G} _ {n} \big (\mathcal {F} _ {B, C} (r _ {n}) \big) \lesssim_ {d} r _ {n}, +$$ + +and the high-probability version (89) holds verbatim. + +![](images/935ea6c56b750939cdc09181da6fb51514aef42c42728a1a549a623c7baafd73.jpg) + +# G.2 Proof of the Estimation Error Upper Bound + +Given the local gaussian complexity upper bound, together with the assumption of solutions being "optimized", we can prove the following MSE upper bound. + +Theorem G.3 (Restate Theorem 3.6). Fix a step size $\eta >0$ and noise level $\sigma >0$ . Given a ground truth function $f_0\in \mathrm{V}_g(\mathbb{B}_1^d)$ such that $\| f_0\|_{L^\infty}\leq B$ and $|f_0|_{\mathrm{V}_g}\leq \widetilde{O}\left(\frac{1}{\eta} -\frac{1}{2} +2\sigma\right)$ , suppose that we are given a data set $y_{i} = f_{0}(\pmb{x}_{i}) + \varepsilon_{i}$ , where $\pmb{x}_i$ are i.i.d. Uniform $(\mathbb{B}_1^d)$ and $\varepsilon_{i}$ are i.i.d. $\mathcal{N}(0,\sigma^2)$ . Then, with probability at least $1 - \delta$ , we have that + +$$ +\frac {1}{n} \sum_ {i = 1} ^ {n} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - f _ {0} \left(\boldsymbol {x} _ {i}\right)\right) ^ {2} \lesssim_ {d} \left(\frac {1}{\eta} - \frac {1}{2} + 2 \sigma\right) ^ {\frac {d}{(2 d ^ {2} + 6 d + 3) (d + 2)}} B ^ {2} \left(\frac {\sigma^ {2}}{n}\right) ^ {\frac {1}{2 d + 4}}, \tag {91} +$$ + +for any $\pmb{\theta} \in \Theta_{\mathrm{flat}}(\eta; \mathcal{D})$ that is optimized, i.e., $(f_{\pmb{\theta}}(\pmb{x}_i) - y_i)^2 \leq (f_0(\pmb{x}_i) - y_i)^2$ , for $i = 1, \dots, n$ . Here, $\lesssim_d$ hides constants (that could depend on $d$ ) and logarithmic factors in $n$ and $(1/\delta)$ . + +Proof of Theorem 3.6. The empirical Mean Squared Error (MSE) we want to bound is $\mathrm{MSE}(f_{\theta}, f_0) = \frac{1}{n} \sum_{i=1}^{n} (f_{\theta}(\pmb{x}_i) - f_0(\pmb{x}_i))^2$ . + +First, we establish bounds on the regularity of $f_{\pmb{\theta}}(\pmb{x}) - f_0(\pmb{x})$ . The condition that $f_{\pmb{\theta}}$ is "optimized" means $(f_{\pmb{\theta}}(\pmb{x}_i) - y_i)^2 \leq (f_0(\pmb{x}_i) - y_i)^2$ for all $i$ . Summing over $i$ and dividing by $n$ , we have $\frac{1}{n} \sum_{i=1}^{n} (f_{\pmb{\theta}}(\pmb{x}_i) - y_i)^2 \leq \frac{1}{n} \sum_{i=1}^{n} (f_0(\pmb{x}_i) - y_i)^2 = \frac{1}{n} \sum_{i=1}^{n} \varepsilon_i^2$ . Since $\varepsilon_i \sim \mathcal{N}(0, \sigma^2)$ , $\mathbb{E}[\frac{1}{n} \sum_{i=1}^{n} \varepsilon_i^2] = \sigma^2$ . Standard concentration inequalities (e.g., for sums of $\chi^2(1)$ scaled variables) show that $\frac{1}{n} \sum_{i=1}^{n} \varepsilon_i^2 \lesssim \sigma^2$ with high probability (hiding logarithmic factors in $1/\delta$ , which are absorbed into $\lesssim_d$ ). Thus, $2\mathcal{L}(\pmb{\theta}) \lesssim \sigma^2$ . For $\pmb{\theta} \in \Theta_{\mathrm{flat}}(\eta; \mathcal{D})$ , by Corollary 3.3 (with $R = 1$ for $\mathbb{B}_1^d$ , so $R + 1 = 2$ ), we have + +$$ +\left| f _ {\boldsymbol {\theta}} \right| _ {\mathrm {V} _ {g}} \leq \frac {1}{\eta} - \frac {1}{2} + 2 \sqrt {2 \mathcal {L} (\boldsymbol {\theta})} \leq \frac {1}{\eta} - \frac {1}{2} + 2 \sigma . \tag {92} +$$ + +Let $C \coloneqq \frac{1}{\eta} - \frac{1}{2} + 2\sigma$ . The theorem assumes $|f_0|_{\mathrm{V}_g} \leq C$ . Thus, we have $|f_0|_{\mathrm{V}_g} \lesssim C$ . The difference $f_\theta(\boldsymbol{x}) - f_0(\boldsymbol{x})$ then satisfies + +$$ +\left| f _ {\boldsymbol {\theta}} - f _ {0} \right| _ {\mathrm {V} _ {g}} \leq \left| f _ {\boldsymbol {\theta}} \right| _ {\mathrm {V} _ {g}} + \left| f _ {0} \right| _ {\mathrm {V} _ {g}} \leq 2 C. \tag {93} +$$ + +Also, $\| f_{\pmb{\theta}} - f_0\|_{L^{\infty}(\mathbb{B}_1^d)}\leq \| f_{\pmb{\theta}}\|_{L^{\infty}(\mathbb{B}_1^d)} + \| f_0\|_{L^{\infty}(\mathbb{B}_1^d)}\leq B + B = 2B.$ + +The optimized condition $(f_{\pmb{\theta}}(\pmb{x}_i) - y_i)^2 \leq (f_0(\pmb{x}_i) - y_i)^2$ implies $((f_{\pmb{\theta}}(\pmb{x}_i) - f_0(\pmb{x}_i)) - \varepsilon_i)^2 \leq \varepsilon_i^2$ . Expanding this gives $(f_{\pmb{\theta}}(\pmb{x}_i) - f_0(\pmb{x}_i))^2 - 2(f_{\pmb{\theta}}(\pmb{x}_i) - f_0(\pmb{x}_i))\varepsilon_i + \varepsilon_i^2 \leq \varepsilon_i^2$ , which simplifies to + +$$ +\left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - f _ {0} \left(\boldsymbol {x} _ {i}\right)\right) ^ {2} \leq 2 \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - f _ {0} \left(\boldsymbol {x} _ {i}\right)\right) \varepsilon_ {i}. \tag {94} +$$ + +This inequality is crucial and holds for each data point. + +We decompose the MSE based on the location of data points. Let $\mathbb{A}_{\varepsilon}^{d} := \{\pmb{x} \in \mathbb{B}_{1}^{d} : \| \pmb{x} \|_{2} \geq 1 - \varepsilon\}$ be the annulus and $\mathbb{B}_{1 - \varepsilon}^{d}$ be the inner core. Let $S_{A} := \{i : \pmb{x}_{i} \in \mathbb{A}_{\varepsilon}^{d}\}$ and $S_{I} := \{i : \pmb{x}_{i} \in \mathbb{B}_{1 - \varepsilon}^{d}\}$ . The total empirical MSE is + +$$ +\begin{array}{l} \operatorname {M S E} \left(f _ {\boldsymbol {\theta}}, f _ {0}\right) = \frac {1}{n} \sum_ {i \in S _ {A}} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - f _ {0} \left(\boldsymbol {x} _ {i}\right)\right) ^ {2} + \frac {1}{n} \sum_ {i \in S _ {I}} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - f _ {0} \left(\boldsymbol {x} _ {i}\right)\right) ^ {2} \\ \leq \frac {n _ {A}}{n} \left(\frac {1}{n _ {A}} \sum_ {i \in S _ {A}} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - f _ {0} \left(\boldsymbol {x} _ {i}\right)\right) ^ {2}\right) + \frac {n _ {I}}{n} \left(\frac {1}{n _ {I}} \sum_ {i \in S _ {I}} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - f _ {0} \left(\boldsymbol {x} _ {i}\right)\right) ^ {2}\right) \tag {95} \\ \leq \frac {n _ {A}}{n} \underbrace {\left(\frac {1}{n _ {A}} \sum_ {i \in S _ {A}} (f _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {i}) - f _ {0} (\boldsymbol {x} _ {i})) ^ {2}\right)} _ {\mathrm {M S E} _ {\mathcal {S}}} + \underbrace {\frac {1}{n _ {I}} \sum_ {i \in S _ {I}} (f _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {i}) - f _ {0} (\boldsymbol {x} _ {i})) ^ {2}} _ {\mathrm {M S E} _ {\mathcal {I}}} \\ \end{array} +$$ + +The contribution from the shell, $\mathrm{MSE}_{\mathcal{S}}$ , is bounded using the $L^{\infty}$ norm of $f_{\theta} - f_0$ and the concentration of points in the shell. Let $n_A \coloneqq |S_A|$ . By Lemma F.7, $n_A / n \lesssim \varepsilon$ with high probability. + +$$ +\mathrm {M S E} _ {\mathcal {S}} \leq \frac {n _ {A}}{n} \| f _ {\boldsymbol {\theta}} - f _ {0} \| _ {L ^ {\infty}} ^ {2} \leq \frac {n _ {A}}{n} (2 B) ^ {2} \lesssim_ {d} B ^ {2} \varepsilon . \tag {96} +$$ + +For the inner core's contribution, $\mathrm{MSE}_{\mathcal{I}}$ , we use Equation (94): + +$$ +\mathrm {M S E} _ {\mathcal {I}} = \frac {1}{n} \sum_ {i \in S _ {I}} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - f _ {0} \left(\boldsymbol {x} _ {i}\right)\right) ^ {2} \leq \frac {2}{n} \sum_ {i \in S _ {I}} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - f _ {0} \left(\boldsymbol {x} _ {i}\right)\right) \varepsilon_ {i}. \tag {97} +$$ + +Let $n_I \coloneqq |S_I|$ . The empirical process term is $2\frac{n_I}{n}\left(\frac{1}{n_I}\sum_{i\in S_I}(f_\theta (\pmb{x}_i) - f_0(\pmb{x}_i))\varepsilon_i\right)$ . The function $f_{\theta} - f_0$ restricted to $\mathbb{B}_{1 - \varepsilon}^d$ has an unweighted variation norm. As shown in Appendix E, for $\pmb{x} \in \mathrm{Uniform}(\mathbb{B}_1^d)$ , the population weight function $g_P(\pmb {u},t)\asymp (1 - |t|)^{d + 2}$ . For activation hyperplanes relevant to $\mathbb{B}_{1 - \varepsilon}^d$ (i.e., $|t|\leq 1 - \varepsilon$ ), $g_{P}(\pmb {u},t)\gtrsim \varepsilon^{d + 2}$ . Thus, the unweighted variation of $f_{\theta} - f_0$ on $\mathbb{B}_{1 - \varepsilon}^d$ is + +$$ +\left| f _ {\boldsymbol {\theta}} - f _ {0} \right| _ {\mathrm {V} \left(\mathbb {B} _ {1 - \varepsilon} ^ {d}\right)} \lesssim \left| f _ {\boldsymbol {\theta}} - f _ {0} \right| _ {\mathrm {V} _ {g _ {P}}} / \varepsilon^ {d + 2} \lesssim C / \varepsilon^ {d + 2}. \tag {98} +$$ + +We apply Lemma G.2 to bound $\frac{1}{n_I} \sum_{i \in S_I} (f_\theta(\pmb{x}_i) - f_0(\pmb{x}_i)) \varepsilon_i$ . The function $h(\pmb{x}) = f_\theta(\pmb{x}) - f_0(\pmb{x})$ has unweighted variation $\lesssim C / \varepsilon^{d + 2}$ and $L^\infty$ norm $\leq 2B$ . Therefore, we have that + +$$ +\mathrm {M S E} _ {\mathcal {I}} \lesssim \left(\frac {C}{\varepsilon^ {d + 2}}\right) ^ {\frac {2 d}{2 d + 3}} \left(\frac {\sigma^ {2}}{n}\right) ^ {\frac {d + 3}{2 d + 3}}. \tag {99} +$$ + +Combining the bounds for $\mathrm{MSE}_S$ and $\mathrm{MSE}_{\mathcal{I}}$ : + +$$ +\operatorname {M S E} \left(f _ {\boldsymbol {\theta}}, f _ {0}\right) \lesssim B ^ {2} \varepsilon + C ^ {\frac {2 d}{2 d + 3}} \varepsilon^ {- (d + 2) \frac {2 d}{2 d + 3}} \left(\frac {\sigma^ {2}}{n}\right) ^ {\frac {d + 3}{2 d + 3}}. \tag {100} +$$ + +Similarly to the proof of Theorem 3.5 in Appendix F.5, we require that + +$$ +\frac {1}{\inf _ {| t | \leq 1 - \varepsilon} g _ {P} (\boldsymbol {u} , t)} \asymp \varepsilon^ {d + 2} \gtrsim_ {d} \sqrt {\frac {1}{n}}. \tag {101} +$$ + +to filling the gap between the empirical weighted function $g$ and the population $g_{P}$ with high probability, because with high probability, + +$$ +\sup _ {\boldsymbol {u}, t} | g (\boldsymbol {u}, t) - g _ {P} (\boldsymbol {u}, t) | \lesssim_ {d} \sqrt {\frac {1}{n}} \tag {102} +$$ + +where $\lesssim_d$ hides constants (which could depend on $d$ ) and logarithmic factors, as stated by Theorem E.5 in Section E.2. Therefore, we may choose + +$$ +\varepsilon \asymp \left(C ^ {\frac {2 d}{2 d ^ {2} + 6 d + 3}} \cdot \frac {\sigma^ {2}}{n}\right) ^ {\frac {1}{2 d + 4}}, \tag {103} +$$ + +and plug it into (100) to have + +$$ +\operatorname {M S E} \left(f _ {\theta}, f _ {0}\right) \lesssim_ {d} \left(\frac {1}{\eta} - \frac {1}{2} + 2 \sigma\right) ^ {\frac {d}{(2 d ^ {2} + 6 d + 3) (d + 2)}} \left(B ^ {2} \left(\frac {\sigma^ {2}}{n}\right) ^ {\frac {1}{2 d + 4}} + \left(\frac {\sigma^ {2}}{n}\right) ^ {\frac {3}{2 d + 3}}\right). \tag {104} +$$ + +Since $\frac{1}{2d + 4} < \frac{3}{2d + 3}$ , we conclude that + +$$ +\mathrm {M S E} (f _ {\boldsymbol {\theta}}, f _ {0}) \lesssim_ {d} \left(\frac {1}{\eta} - \frac {1}{2} + 2 \sigma\right) ^ {\frac {d}{(2 d ^ {2} + 6 d + 3) (d + 2)}} B ^ {2} \left(\frac {\sigma^ {2}}{n}\right) ^ {\frac {1}{2 d + 4}}, +$$ + +which completes the proof. + +![](images/0abe631a8ce1a4188c293824123ebb2c97234afcdd414184e154adbadae9eacb.jpg) + +# H Proof of Theorem 3.7: Minimax Lower Bound + +# H.1 The Multivariate Case + +In this section, we assume that $d > 1$ and all the norms and semi-norms are restricted to the unit ball $\mathbb{B}_1^d$ . Let $\pmb{u} \in \mathbb{S}^{d-1}$ be a unit vector. Let $\varepsilon \in \mathbb{R}_+$ be a constant with $\varepsilon \leq 1/2$ . Consider the ReLU atom: + +$$ +\varphi_ {\boldsymbol {u}, \varepsilon^ {2}} (\boldsymbol {x}) = \phi \left(\boldsymbol {u} ^ {\mathsf {T}} \boldsymbol {x} - \left(1 - \varepsilon^ {2}\right)\right). \tag {105} +$$ + +Lemma H.1. The $L^2$ -norm of $\varphi_{\pmb{u},\varepsilon^2}$ over $\mathbb{B}_1^d$ is given by + +$$ +\left\| \varphi_ {\boldsymbol {u}, \varepsilon^ {2}} \right\| _ {L ^ {2} \left(\mathbb {B} _ {1} ^ {d}\right)} \underset {c _ {7} (d)} {\overset {c _ {8} (d)} {\widetilde {\longrightarrow}}} \varepsilon^ {\frac {d + 5}{2}}, \tag {106} +$$ + +where $c_7(d)$ and $c_8(d)$ are constants depending on $d$ (the concrete definition is (113)). Recall that $\asymp_{c_7(d)}^{c_8(d)}$ means + +$$ +c _ {7} (d) \varepsilon^ {\frac {d + 5}{2}} \leq \left\| \varphi_ {\boldsymbol {u}, \varepsilon^ {2}} \right\| _ {L ^ {2} \left(\mathbb {B} _ {1} ^ {d}\right)} \leq c _ {8} (d) \varepsilon^ {\frac {d + 5}{2}}. +$$ + +Proof. The squared $L^2$ norm of $\varphi_{\pmb{u},\varepsilon^2}$ over the unit ball $\mathbb{B}_1^d$ is defined as: + +$$ +\left\| \varphi_ {\boldsymbol {u}, \varepsilon^ {2}} \right\| _ {L ^ {2} \left(\mathbb {B} _ {1} ^ {d}\right)} ^ {2} = \int_ {\mathbb {B} _ {1} ^ {d}} \left| \varphi_ {\varepsilon^ {2}} (\boldsymbol {x}) \right| ^ {2} d \boldsymbol {x} +$$ + +Substituting the definition of $\varphi_{\varepsilon^2}(\boldsymbol{w},\boldsymbol{x})$ and using the property of the ReLU function that $\phi (z) = z$ for $z > 0$ and $\phi (z) = 0$ for $z\leq 0$ , we get: + +$$ +\begin{array}{l} \left\| \varphi_ {\boldsymbol {u}, \varepsilon^ {2}} \right\| _ {L ^ {2} \left(\mathbb {B} _ {1} ^ {d}\right)} ^ {2} = \int_ {\mathbb {B} _ {1} ^ {d}} [ \phi (\boldsymbol {u} ^ {\mathsf {T}} \boldsymbol {x} - (1 - \varepsilon^ {2})) ] ^ {2} d \boldsymbol {x} \tag {107} \\ = \int_ {\left\{\boldsymbol {x} \in \mathbb {B} _ {1} ^ {d}: \boldsymbol {u} ^ {\top} \boldsymbol {x} > 1 - \varepsilon^ {2} \right\}} \left(\boldsymbol {u} ^ {\top} \boldsymbol {x} - (1 - \varepsilon^ {2})\right) ^ {2} d \boldsymbol {x} \\ \end{array} +$$ + +To simplify the integral, we perform a rotation of the coordinate system such that $\pmb{w}$ aligns with the $d$ -th standard basis vector $e_d = (0, \dots, 0, 1)$ . In these new coordinates, $\pmb{u}^{\top} \pmb{x} = X_d$ . The unit ball remains the unit ball under rotation. The integral becomes: + +$$ +I = \| \varphi_ {\boldsymbol {u}, \varepsilon^ {2}} \| _ {L ^ {2} \left(\mathbb {B} _ {1} ^ {d}\right)} ^ {2} = \int_ {\left\{X \in \mathbb {B} _ {1} ^ {d}: X _ {d} > 1 - \varepsilon^ {2} \right\}} \left(X _ {d} - (1 - \varepsilon^ {2})\right) ^ {2} d X +$$ + +We can write the volume element $\mathrm{d}X$ as $\mathrm{d}X^{\prime}\mathrm{d}X_{d}$ , where $X^{\prime}\in \mathbb{R}^{d - 1}$ represents the first $d - 1$ coordinates. The condition $X\in \mathbb{B}_1^d$ translates to $\| X^{\prime}\| _2^2 +X_d^2\leq 1$ . The integral can be written as an iterated integral: + +$$ +I = \int_ {1 - \varepsilon^ {2}} ^ {1} \left(\int_ {\| X ^ {\prime} \| _ {2} ^ {2} \leq 1 - X _ {d} ^ {2}} (X _ {d} - (1 - \varepsilon^ {2})) ^ {2} \mathrm {d} X ^ {\prime}\right) \mathrm {d} X _ {d} +$$ + +The inner integral is over a $(d - 1)$ -dimensional ball in $\mathbb{R}^{d - 1}$ with radius $R = \sqrt{1 - X_d^2}$ . The integrand $(X_d - (1 - \varepsilon^2))^2$ is constant with respect to $X'$ . Therefore, the inner integral evaluates to: + +$$ +\left(X _ {d} - \left(1 - \varepsilon^ {2}\right)\right) ^ {2} \cdot \operatorname {V o l} _ {d - 1} (R) +$$ + +where $\mathrm{Vol}_{d - 1}(R)$ is the volume of the $(d - 1)$ -dimensional ball of radius $R$ . This volume is given by $V_{d - 1}R^{d - 1}$ , with $V_{d - 1} = \frac{\pi^{(d - 1) / 2}}{\Gamma(\frac{d + 1}{2})}$ . So, the inner integral is $(X_d - (1 - \varepsilon^2))^2 V_{d - 1}(1 - X_d^{2})^{(d - 1) / 2}$ , and the outer integral becomes: + +$$ +I = V _ {d - 1} \int_ {1 - \varepsilon^ {2}} ^ {1} \left(X _ {d} - \left(1 - \varepsilon^ {2}\right)\right) ^ {2} \left(1 - X _ {d} ^ {2}\right) ^ {\frac {d - 1}{2}} d X _ {d} \tag {108} +$$ + +Let $X_{d} = 1 - \delta$ performing a change of variable. Then $\mathrm{d}X_{d} = -\mathrm{d}\delta$ . The integration limits change + +$$ +\begin{array}{l} I = V _ {d - 1} \int_ {\varepsilon^ {2}} ^ {0} ((1 - \delta) - (1 - \varepsilon^ {2})) ^ {2} (1 - (1 - \delta) ^ {2}) ^ {\frac {d - 1}{2}} (- \mathrm {d} \delta) \\ = V _ {d - 1} \int_ {0} ^ {\varepsilon^ {2}} \left(\varepsilon^ {2} - \delta\right) ^ {2} \left(1 - \left(1 - 2 \delta + \delta^ {2}\right)\right) ^ {\frac {d - 1}{2}} d \delta \tag {109} \\ = V _ {d - 1} \int_ {0} ^ {\varepsilon^ {2}} (\varepsilon^ {2} - \delta) ^ {2} (2 \delta - \delta^ {2}) ^ {\frac {d - 1}{2}} d \delta \\ \end{array} +$$ + +Since we assumed $\varepsilon^2 < \frac{1}{4}$ , for the integration range $[0, \varepsilon^2]$ , we may write $2\delta - \delta^2 = (2 - \delta)\delta \asymp_{7/4}^{2} 2\delta$ . + +$$ +\left(\frac {7}{4}\right) ^ {\frac {d - 1}{2}} \delta^ {\frac {d - 1}{2}} \leq \left(2 \delta - \delta^ {2}\right) ^ {\frac {d - 1}{2}} \leq 2 ^ {\frac {d - 1}{2}} \delta^ {\frac {d - 1}{2}} +$$ + +The integral is approximated by: + +$$ +V _ {d - 1} \left(\frac {7}{4}\right) ^ {\frac {d - 1}{2}} \int_ {0} ^ {\varepsilon^ {2}} (\varepsilon^ {2} - \delta) ^ {2} \delta^ {\frac {d - 1}{2}} d \delta \leq I \leq V _ {d - 1} 2 ^ {\frac {d - 1}{2}} \int_ {0} ^ {\varepsilon^ {2}} (\varepsilon^ {2} - \delta) ^ {2} \delta^ {\frac {d - 1}{2}} d \delta \tag {110} +$$ + +Consider another change of variable: $\delta = \varepsilon^2 s$ . Then $\mathrm{d}\delta = \varepsilon^2\mathrm{d}s$ . The limits change + +$$ +\begin{array}{l} \int_ {0} ^ {\varepsilon^ {2}} \left(\varepsilon^ {2} - \delta\right) ^ {2} \delta^ {\frac {d - 1}{2}} d \delta = \int_ {0} ^ {1} \left(\varepsilon^ {2} - \varepsilon^ {2} s\right) ^ {2} \left(\varepsilon^ {2} s\right) ^ {\frac {d - 1}{2}} \left(\varepsilon^ {2} d s\right) \\ = \int_ {0} ^ {1} (\varepsilon^ {2}) ^ {2} (1 - s) ^ {2} (\varepsilon^ {2}) ^ {(d - 1) / 2} s ^ {\frac {d - 1}{2}} \varepsilon^ {2} d s \\ = \left(\varepsilon^ {2}\right) ^ {2 + \frac {d - 1}{2} + 1} \int_ {0} ^ {1} (1 - s) ^ {2} s ^ {\frac {d - 1}{2}} d s \tag {111} \\ = \varepsilon^ {d + 5} \int_ {0} ^ {1} (1 - s) ^ {2} s ^ {\frac {d - 1}{2}} d s \\ = \underbrace {\left(\int_ {0} ^ {1} (1 - s) ^ {2} s ^ {\frac {d - 1}{2}} \mathrm {d} s\right)} _ {\text {c o n s t a n t}} \varepsilon^ {d + 5} \\ \end{array} +$$ + +The $L^2$ norm is the square root of $I$ is given by + +$$ +c _ {7} (d) \varepsilon^ {\frac {d + 5}{2}} \leq \left\| \varphi_ {\boldsymbol {u}, \varepsilon^ {2}} \right\| _ {L ^ {2} \left(\mathbb {B} _ {1} ^ {d}\right)} = \sqrt {I} \leq c _ {8} (d) \varepsilon^ {\frac {d + 5}{2}} \tag {112} +$$ + +where $c_{7}(d)$ and $c_{8}(d)$ are constants defined by + +$$ +c _ {7} (d) = \sqrt {\frac {V _ {d - 1} \left(\frac {7}{4}\right) ^ {\frac {d - 1}{2}} \left(\int_ {0} ^ {1} (1 - s) ^ {2} s ^ {\frac {d - 1}{2}} \mathrm {d} s\right)}{} \tag {113} +$$ + +$$ +c _ {8} (d) = \sqrt {V _ {d - 1} 2 ^ {\frac {d - 1}{2}} \left(\int_ {0} ^ {1} (1 - s) ^ {2} s ^ {\frac {d - 1}{2}} \mathrm {d} s\right)} +$$ + +This completes the proof. + +Lemma H.2. Let $\varphi_{\pmb{u},\varepsilon^2}$ be a ReLU atom defined in (105). Then + +$$ +\left| \varphi_ {\boldsymbol {u}, \varepsilon^ {2}} \right| _ {\mathrm {V} _ {g}} = \varepsilon^ {2 d + 4}. \tag {114} +$$ + +Proof. We decode the definition (see Example 3.1) and compute directly the weighted function $g(\pmb{u}, 1 - \varepsilon^2) = (\varepsilon^2)^{d + 2} = \varepsilon^{2d + 4}$ . + +Let $\mathbb{S}^{d - 1}$ be the unit sphere in $\mathbb{R}^d$ . For $0 < \varepsilon < 1$ and $\boldsymbol{w} \in \mathbb{S}^{d - 1}$ , define the spherical cap $C(\boldsymbol{u}, \varepsilon^2)$ as + +$$ +C (\boldsymbol {u}, \varepsilon^ {2}) = \left\{\boldsymbol {x} \in \mathbb {S} ^ {d - 1}: \boldsymbol {u} ^ {\top} \boldsymbol {x} \geq 1 - \varepsilon^ {2} \right\}. \tag {115} +$$ + +Lemma H.3. Let $N_{max}(\varepsilon, d)$ denote the maximum number of points $\mathbf{u}_1, \ldots, \mathbf{u}_N \in \mathbb{S}^{d-1}$ such that the caps $C(\mathbf{u}_i, \varepsilon^2)$ are mutually disjoint. Then, as $\varepsilon \to 0$ , + +$$ +N _ {m a x} (\varepsilon , d) \asymp \varepsilon^ {- (d - 1)} +$$ + +where the implicit constants depend only on the dimension $d$ . + +Proof. The spherical cap $C(\pmb{u}, \varepsilon^2)$ has an angular radius $\vartheta = \arccos(1 - \varepsilon^2)$ , satisfying $\vartheta = \Theta(\varepsilon)$ for small $\varepsilon$ . The condition that caps $C(\pmb{u}_i, \varepsilon^2)$ and $C(\pmb{u}_j, \varepsilon^2)$ are disjoint requires the angular separation $\phi_{ij}$ between their centers $\pmb{w}_i$ and $\pmb{w}_j$ to be at least $2\vartheta$ . Thus, $N_{max}(\varepsilon, d)$ is the maximum size $M(\mathbb{S}^{d-1}, 2\vartheta)$ of a $2\vartheta$ -separated set (packing number) on $\mathbb{S}^{d-1}$ . + +The upper bound $N_{max}(\varepsilon, d) = O(\varepsilon^{-(d-1)})$ follows from a surface area argument: $N$ disjoint caps $C(\pmb{u}_i, \varepsilon^2)$ , each with surface area $\Theta(\vartheta^{d-1}) = \Theta(\varepsilon^{d-1})$ , must fit within the total surface area of $\mathbb{S}^{d-1}$ . + +For the lower bound, we relate the packing number $M(\mathbb{S}^{d - 1},\alpha)$ to the covering number $N(\mathbb{S}^{d - 1},\alpha)$ , the minimum number of caps of angular radius $\alpha$ needed to cover $\mathbb{S}^{d - 1}$ . It is a standard result that these quantities are closely related, for instance, $M(\mathbb{S}^{d - 1},\alpha)\geq N(\mathbb{S}^{d - 1},\alpha)$ can be shown via a greedy packing argument [Vershynin, 2018, see discussions in Chapter 4]. Furthermore, the asymptotic behavior of the covering number is known to be $N(\mathbb{S}^{d - 1},\alpha)\asymp \alpha^{-(d - 1)}$ for small $\alpha$ [Vershynin, 2018, Corollary 4.2.14]. Setting the minimum separation $\alpha = 2\vartheta = \Theta (\varepsilon)$ , we obtain the lower bound: + +$$ +N _ {m a x} (\varepsilon , d) = M \left(\mathbb {S} ^ {d - 1}, 2 \vartheta\right) \geq N \left(\mathbb {S} ^ {d - 1}, 2 \vartheta\right) \asymp (2 \vartheta) ^ {- (d - 1)} = \Omega \left(\varepsilon^ {- (d - 1)}\right), +$$ + +where the implicit constants depend only on the dimension $d$ . Combining the upper and lower bounds, we conclude that $N_{max}(\varepsilon, d) \asymp \varepsilon^{-(d-1)}$ . + +**Construction H.4.** We construct a suitable packing set in $\mathcal{F} = \{f \in \mathrm{V}_g(\mathbb{B}_1^d) : \| f \|_{L^\infty} \leq 1, |f|_{\mathrm{V}_g} \leq 1\}$ based on a weighted ReLU atoms. Let $\varphi_{u,\varepsilon^2}$ be the ReLU atom defined in (105), and according to Lemma H.1 and Lemma H.2: + +$$ +\Phi_ {\boldsymbol {u}, \varepsilon^ {2}} := \varepsilon^ {- 2} \varphi_ {\boldsymbol {u}, \varepsilon^ {2}} \Rightarrow \left\{ \begin{array}{l} \| \Phi_ {\boldsymbol {u}, \varepsilon^ {2}} \| _ {L ^ {\infty} \left(\mathbb {B} _ {1} ^ {d}\right)} = 1, \\ \| \Phi_ {\boldsymbol {u}, \varepsilon^ {2}} \| _ {L ^ {2} \left(\mathbb {B} _ {1} ^ {d}\right)} \asymp \varepsilon^ {\frac {d + 1}{2}}, \\ | \Phi_ {\boldsymbol {u}, \varepsilon^ {2}} | _ {\mathrm {V} _ {g}} = \varepsilon^ {2 d + 2}. \end{array} \right. \tag {116} +$$ + +According to Lemma H.1, there exists $N = c_{N}(d)\varepsilon^{-d + 1}$ spherical caps $\pmb{u}_{1},\dots ,\pmb{u}_{N}$ such that the caps $C(\pmb {u}_i,\varepsilon^2)$ are mutually disjoint, for some constant $c_{N}(d)\leq 1$ that may depend on the dimension $d$ . For convenience, we simply denote $\Phi_i = \Phi_{\pmb {u}_i,\varepsilon^2}$ . Therefore, we have $|N\Phi_i|_{\mathrm{V}_g} = c_N(d)\varepsilon^{d + 3} < 1$ referring to (116). For each $\xi = (\xi_1,\ldots ,\xi_N)\in \{-1,1\} ^N$ , define + +$$ +f _ {\xi} (\boldsymbol {x}) = \sum_ {i = 1} ^ {N} \xi_ {i} \Phi_ {i} (\boldsymbol {x}). +$$ + +According to the conventions, each $f_{\xi}$ belongs to $\mathcal{F}$ . Since the supports of the ridge functions are disjoint, for any $\xi, \xi' \in \{-1, 1\}^N$ we have + +$$ +\| f _ {\xi} - f _ {\xi^ {\prime}} \| _ {L ^ {2} (\mathbb {B} _ {1} ^ {d})} ^ {2} = \sum_ {i \in S} \| 2 \Phi_ {i} \| _ {L ^ {2} (\mathbb {B} _ {1} ^ {d})} ^ {2} \asymp \varepsilon^ {\frac {d + 1}{2}} d _ {H} (\xi , \xi^ {\prime}), +$$ + +where $d_H(\xi, \xi')$ denotes the Hamming distance between $\xi$ and $\xi'$ , and $S$ is the set of indices where $\xi_i \neq \xi_i'$ . By the Varshamov-Gilbert lemma, there exists a subset $\Xi \subset \{-1, 1\}^N$ with + +$$ +\log | \Xi | = K \asymp \varepsilon^ {- d + 1}. +$$ + +for some constant, and such that for any distinct $\xi ,\xi^{\prime}\in \Xi$ , the Hamming distance $d_H$ + +$$ +d _ {H} (\xi , \xi^ {\prime}) \gtrsim K. +$$ + +Thus, for any distinct $\xi ,\xi^{\prime}\in \Xi$ we obtain + +$$ +\left\| f _ {\xi} - f _ {\xi^ {\prime}} \right\| _ {L ^ {2} (\mathbb {B} _ {1} ^ {d})} \gtrsim \varepsilon^ {\frac {d + 1}{2}} \sqrt {K} \asymp \varepsilon^ {\frac {d + 1}{2}} \varepsilon^ {\frac {- d + 1}{2}} = \varepsilon . +$$ + +Proposition H.5 (Minimax Lower Bound via Fano's Lemma). Consider the problem of estimating a function $f \in \mathcal{F} = \{f \in \mathrm{V}_g(\mathbb{B}_1^d) : \| f \|_{L^\infty} \leq 1, |f|_{\mathrm{V}_g} \leq 1\}$ with + +$$ +y _ {i} = f \left(\boldsymbol {x} _ {i}\right) + \varepsilon_ {i}, i = 1, \dots , n +$$ + +where $\{\varepsilon_i\}_{i=1}^n$ are i.i.d. $\mathcal{N}(0, \sigma^2)$ random variables and $\{\pmb{x}_i\}_{i=1}^n \subset \mathbb{B}_1^d$ are i.i.d. uniform random variables on $\mathbb{B}_1^d$ . The lower bound of the minimax non-parametric risk is given by + +$$ +\inf_{\hat{f}}\sup_{f\in \mathcal{F}}\mathbb{E}\left\| \hat{f} -f\right\|_{L^{2}(\mathbb{B}_{1}^{d})}^{2}\gtrsim \left(\frac{\sigma^{2}}{n}\right)^{\frac{2}{d + 1}}. +$$ + +Proof. We use the standard Fano's lemma argument. By our Construction (105), we have a packing set $\{f_{\xi} : \xi \in \Xi\}$ in $\mathcal{F}$ with the following properties: + +1. The $L^2$ -distance between any two distinct functions is at least $\delta$ , where $\delta \asymp \varepsilon$ . +2. The size of the packing set satisfies $\log |\Xi| \gtrsim K \asymp \varepsilon^{-(d-1)}$ . + +For Gaussian noise with variance $\sigma^2$ , the Kullback-Leibler divergence between the distributions induced by two functions $f_{\xi}$ and $f_{\xi'}$ is + +$$ +\mathrm {K L} (P _ {\xi} \| P _ {\xi^ {\prime}}) = \frac {n}{2 \sigma^ {2}} \| f _ {\xi} - f _ {\xi^ {\prime}} \| _ {L ^ {2} (\mathbb {B} _ {1} ^ {d})} ^ {2} = \frac {n \delta^ {2}}{2 \sigma^ {2}}. +$$ + +In order to use Fano's lemma J.1 effectively, we need to satisfy the requirement (141), where in this context is + +$$ +\frac {n \delta^ {2}}{2 \sigma^ {2}} \lesssim \log | \Theta | +$$ + +for some small constant $\alpha > 0$ , then the minimax risk is bounded from below by a constant multiple of $\delta^2$ . + +Substituting $\delta \asymp \varepsilon$ and $\log |\Xi| \gtrsim \varepsilon^{-(d-1)}$ , the condition becomes + +$$ +\frac {n \varepsilon^ {2}}{\sigma^ {2}} \lesssim \varepsilon^ {- (d - 1)}, +$$ + +or equivalently, + +$$ +n \lesssim \frac {\sigma^ {2}}{\varepsilon^ {d + 1}}. +$$ + +Solving for $\varepsilon$ , we have + +$$ +\varepsilon^ {d + 1} \asymp \frac {\sigma^ {2}}{n} \quad \Longrightarrow \quad \varepsilon \asymp \left(\frac {\sigma^ {2}}{n}\right) ^ {\frac {1}{d + 1}}. \tag {117} +$$ + +Then, the separation becomes + +$$ +\delta \asymp \varepsilon \asymp \left(\frac {\sigma^ {2}}{n}\right) ^ {\frac {1}{d + 1}}. +$$ + +Therefore, Fano's lemma J.1 (particularly (140)) yields + +$$ +\inf_{\hat{f}}\sup_{f\in \mathcal{F}}\mathbb{E}\left\| \hat{f} -f\right\|_{L^{2}(\mathbb{B}_{1}^{d})}^{2}\gtrsim \delta^{2}\asymp \left(\frac{\sigma^{2}}{n}\right)^{\frac{2}{d + 1}}, +$$ + +which is the desired result. + +Corollary H.6. Let $\{f\in \mathrm{V}_g(\mathbb{B}_1^d):\| f\|_{L^\infty}\leq B,|f|_{\mathrm{V}_g}\leq C\}$ . Then + +$$ +\inf_{\hat{f}}\sup_{f\in \mathcal{F}}\mathbb{E}\| \hat{f} -f\|_{L^{2}}^{2}\gtrsim \min (B,C)^{2}\left(\frac{\sigma^{2}}{n}\right)^{\frac{2}{d + 1}}. +$$ + +Proof. We just need to replace $f_{\xi}$ in Construction 105 by $\min(B, C)f_{\xi}$ and adapt it to the proof of Proposition H.5. + +# H.2 Why Classical Bump-Type Constructions Are Ineffective + +The minimax lower bound construction in this paper crucially hinges on exploiting the properties of the data-dependent weighted variation norm, denoted as $|\cdot|_{\mathrm{V}_g}$ , where the weight function is $g(\boldsymbol{u}, t)$ . A key characteristic of $g(\boldsymbol{u}, t)$ (when data is, for instance, uniform on the unit ball $\mathbb{B}_1^d$ ) is that $g(\boldsymbol{u}, t) \asymp (1 - |t|)^{d + 2}$ . This implies that $g(\boldsymbol{u}, t)$ becomes very small as $|t| \to 1$ , i.e., for activations near the boundary of the domain. This property allows for the construction of functions with significant magnitudes near the boundary using a relatively small variation norm. Therefore, any effective construction for the lower bound must create functions that are highly localized near this boundary. + +![](images/8103e5f115ed6ee95154788c9cd34c2bac82477e0a1d00a0b383ccce99b8b5ab.jpg) +Figure 17: Isotropic Locality is Costly: An isotropic bump function, by definition, must be localized (decay rapidly) in all directions around its center. Suppose we place such a bump centered at a point $\pmb{x}_0$ near the boundary in direction $\pmb{u}_0$ (i.e., $\pmb{u}_0^\top \pmb{x}_0 \approx 1 - \varepsilon^2$ , here $\pmb{u}_0 = (1,0)$ ). To achieve localization in directions orthogonal to $\pmb{u}_0$ , one would need to combine ReLU activations whose ridges are oriented appropriately. More critically, to achieve localization in the direction parallel to $\pmb{u}_0$ (i.e., to ensure the bump decays as we move radially inward from $\pmb{x}_0$ ), we would need ReLU activations whose ridges $\{\pmb{x} : \pmb{u}_0^\top \pmb{x} = t\}$ have $t < 1 - \varepsilon^2$ and are potentially much closer to the origin (i.e., $t$ is significantly smaller than 1). + +For these ReLU activations whose ridges are not very close to the boundary (i.e., $t$ is not close to 1), the weight function $g(\pmb{u}_0, t)$ will not be small. Consequently, constructing a sharply localized bump isotropically would require a substantial sum of weighted coefficients in the $\mathrm{V}_g$ norm to cancel out the function in regions away from its intended support while maintaining a significant peak. This large variation norm would make such functions "too regular" or "too expensive" to serve as effective elements in a packing set for Fano's Lemma, especially when aiming to show a rate degradation due to dimensionality. + +In essence, isotropic bump functions do not efficiently leverage the anisotropic nature of the ReLU activation and the specific properties of the $g(\pmb{u}, t)$ weighting. The construction used in this paper, which employs ReLU atoms active only on thin spherical caps near the boundary (an anisotropic construction), is far more effective. It allows for localization and significant function magnitude primarily by choosing the activation threshold $t$ to be very close to 1 (making $g(\pmb{u}, t)$ small), rather than by intricate cancellations of many neurons with large weighted coefficients. This is why such anisotropic, boundary-localized constructions are essential for revealing the curse of dimensionality in this setting. + +# H.3 The Univariate Case + +The minimax lower bound construction detailed above, which leverages a packing argument with boundary-localized ReLU neurons exploiting the multiplicity of available directions on $\mathbb{S}^{d-1}$ , is particularly effective in establishing the curse of dimensionality for $d > 1$ . However, the geometric foundation of this approach, specifically the ability to pack an exponential number of disjoint spherical caps, does not directly translate to the univariate case ( $d = 1$ ) where the notion of distinct directional activation regions fundamentally changes. Consequently, the lower bound for $d = 1$ necessitates a separate construction or modification of the argument. Fortunately, in the one-dimensional setting, the distinction between isotropic and anisotropic function characteristics, which posed challenges for classical approaches in higher dimensions under the specific data-dependent weighted norm, becomes moot. This simplification allows us to directly employ classical bump function constructions, suitably adapted to the function class, to establish the minimax rates in one dimension. + +According to Theorem 3.4, we have + +$$ +\left| f \right| _ {\mathrm {V} _ {g}} = \left\| g \cdot \mathcal {R} (- \Delta) ^ {\frac {d + 1}{2}} f \right\| _ {\mathcal {M}} \tag {118} +$$ + +When $d = 1$ and $f$ is smooth, (118) is simplified to be + +$$ +| f | _ {\mathrm {V} _ {g}} = \| f ^ {\prime \prime} \cdot g \| _ {\mathcal {M}} = \int_ {- 1} ^ {1} | f ^ {\prime \prime} (x) | g (x) \mathrm {d} x = \int_ {- 1} ^ {1} | f ^ {\prime \prime} (x) | (1 - | x |) ^ {3} \mathrm {d} x \tag {119} +$$ + +and so is the unweighted variation seminorm + +$$ +| f | _ {\mathrm {V}} = \| f ^ {\prime \prime} \| _ {\mathcal {M}} = \int_ {- 1} ^ {1} | f ^ {\prime \prime} (x) | \mathrm {d} x =: \mathrm {T V} ^ {2} (f) \tag {120} +$$ + +which is also known as the second-order total variation seminorm. Therefore, the function class of stable minima in univariate case is characterized into + +$$ +\mathcal {F} _ {B, C} := \left\{f: [ - 1, 1 ] \rightarrow \mathbb {R} \mid \| f \| _ {L ^ {\infty}} \leq B, \| f ^ {\prime \prime} \cdot (1 - | \cdot |) ^ {3} \| _ {\mathcal {M}} \leq C \right\}. \tag {121} +$$ + +Using this characterization, it is more convenient to smooth bump functions to construct a minimax risk lower bound for stable minima class. + +Construction H.7. Consider a smooth compact support function: + +$$ +\Phi (x) = \left\{ \begin{array}{l l} c \exp \left(- \frac {1}{1 - x ^ {2}}\right) & | x | < 1 \\ 0 & \text {o t h e r w i s e} \end{array} . \right. \tag {122} +$$ + +By adjusting the constant $c$ , we may assume + +$$ +\mathrm {T V} ^ {2} (\Phi) := \int_ {- 1} ^ {1} | \Phi^ {\prime \prime} (t) | \mathrm {d} t = 1 \tag {123} +$$ + +and let $D$ be a constant such that $\| \Phi (x)\|_{L^2} = \sqrt{2} D$ . We can construct a translated and scaled version: + +$$ +\Phi_ {a, b} (x) = \Phi \left(\frac {2 x - (a + b)}{b - a}\right) \quad \text {f o r} a < b. \tag {124} +$$ + +and in particular, $\Phi_{a,b}$ has the following properties by directly computations: + +$$ +\left\{ \begin{array}{l l} \operatorname {s u p p} \left(\Phi_ {a, b}\right) & = (a, b), \\ \operatorname {T V} ^ {2} \left(\Phi_ {a, b}\right) & = \frac {1}{b - a}, \\ \| \Phi_ {a, b} \| _ {L ^ {2} ([ - 1, 1 ])} & = \sqrt {b - a} D. \end{array} \right. \tag {125} +$$ + +Proposition H.8. Consider the problem of estimating a function $f \in \mathcal{F}_{1,1}$ with + +$$ +y _ {i} = f (x _ {i}) + \varepsilon_ {i}, i = 1, \dots , n +$$ + +where $\{\varepsilon_i\}_{i=1}^n$ are i.i.d. $\mathcal{N}(0, \sigma^2)$ random variables and $\{x_i\}_{i=1}^N \subset [-1,1]$ are i.i.d. uniform random variables on $[-1,1]$ . The lower bound of the minimax non-parametric risk is given by + +$$ +\inf_{\hat{f}}\sup_{f\in \mathcal{F}_{1,1}}\mathbb{E}\left\| f - \hat{f}\right\|_{L^{2}([-1,1])}^{2}\gtrsim \left(\frac{\sigma^{2}}{n}\right)^{\frac{1}{2}} +$$ + +Proof. For any $\varepsilon > 0$ , we may construct a family. Let $a_{k} = 1 - \varepsilon + k\varepsilon^{2}$ , $k = 0, \dots, \left\lfloor \frac{1}{\varepsilon} \right\rfloor$ . We denote $K = \left\lfloor \frac{1}{\varepsilon} \right\rfloor$ . For each $k = 1, \dots, K$ , we define $\Phi_{k} := \Phi_{a_{k-1}, a_{k}}$ . Since $a_{k} - a_{k-1} = \varepsilon^{2}$ , we have the following properties + +$\| \Phi_k\|_{L^2} = D\cdot \varepsilon$ +- $\mathrm{TV}^2 (\Phi_k)\asymp \frac{1}{\varepsilon^2}\Rightarrow \int_{-1}^1 |f''(t)|g(t)\mathrm{d}x\lesssim \varepsilon$ because $g(t) < \varepsilon^3$ ， $\forall t\in [a_{k - 1},a_k]$ + +Let $\{\Phi_1,\dots ,\Phi_K\}$ $K\succ \lfloor \frac{1}{\varepsilon}\rfloor$ be such a family of function classes, and any $K$ -terms combination $\{\Phi_1,\ldots ,\Phi_K\}$ is in $\mathcal{F}_{1,1}$ . Then we let + +$$ +\phi : \{\pm 1 \} ^ {K} \rightarrow \mathcal {F} _ {1, 1}, \quad \xi = \left(a _ {k}\right) _ {k = 1} ^ {K} \mapsto \sum_ {k = 1} ^ {K} a _ {k} \Phi_ {k} =: f _ {\xi}. \tag {126} +$$ + +For any two indexes $\xi_1, \xi_2$ in $\{\pm 1\}^K$ , we have that + +$$ +\left\| f _ {\xi_ {1}} - f _ {\xi_ {2}} \right\| _ {L ^ {2}} = \varepsilon \sqrt {d _ {H} \left(\xi_ {1} , \xi_ {2}\right)}. \tag {127} +$$ + +where $d_H$ is the Hamming distance. Then, using Varshamov-Gilbert's lemma (Lemma J.2), the pruned cube of $\{f_1,\dots,f_M\}$ has a size $M \geq 2^{K/8}$ , and each has the property that if $i \neq j$ , + +$$ +\left\| f _ {i} - f _ {j} \right\| _ {L ^ {2} ([ - 1, 1 ])} \geq D \cdot \sqrt {\frac {K}{8}} \cdot \varepsilon \asymp \sqrt {\varepsilon}, +$$ + +and thus for any $i \neq j$ + +$$ +\mathrm {K L} (P _ {f _ {i}} \| P _ {f _ {j}}) = \frac {n \varepsilon}{2 \sigma^ {2}} +$$ + +On the other hand, to satisfy the Fano inequality (141): + +$$ +\frac {n \varepsilon}{2 \sigma^ {2}} = \mathrm {K L} \left(P _ {f _ {i}} \| P _ {f _ {j}}\right) \lesssim \log M \asymp \frac {1}{\varepsilon} +$$ + +we let + +$$ +\varepsilon \asymp \left(\frac {\sigma^ {2}}{n}\right) ^ {\frac {1}{2}}. +$$ + +and thus Fano's lemma (Lemma J.1, particularly (140)) implies that + +$$ +\inf_{\hat{f}}\sup_{f\in \mathcal{F}_{1,1}}\mathbb{E}\| f - \hat{f}\|_{L^{2}([-1,1])}^{2}\gtrsim \left(\frac{\sigma^{2}}{n}\right)^{\frac{1}{2}}. +$$ + +![](images/8be7b3d49d4f36c7385d6333f69726ca4589b745c3dda1c3e7aca8f79ac2835d.jpg) + +Note that by rescale the functions in the lower bound construction, we can deduce a more general result. + +Corollary H.9. For general case $\mathcal{F}_{B,C}$ , we can scale the construction functions by $\min(B, C)$ to deduce the result: + +$$ +\inf _ {\hat {f}} \sup _ {f \in \mathcal {F} _ {B, C}} \mathbb {E} \| f - \hat {f} \| _ {L ^ {2} ([ - 1, 1 ])} ^ {2} \gtrsim \min (B, C) ^ {2} \left(\frac {\sigma^ {2}}{n}\right) ^ {\frac {1}{2}} \tag {128} +$$ + +# I Lower Bound on Generalization Gap + +In this section, we derive a lower bound for the generalization gap. + +# I.1 The Lower Bound Construction Can be Realized by Stable Minima + +Recall the notations in Construction H.4, for $\varepsilon \in (0,1)$ and a unit vector $u\in \mathbb{S}^{d - 1}$ , define the (ball) cap + +$$ +C (\boldsymbol {u}, \varepsilon) := \{x \in \mathbb {B} _ {1} ^ {d}: \boldsymbol {u} ^ {\mathsf {T}} \boldsymbol {x} \geq 1 - \varepsilon \}. +$$ + +Fix a dimension $d \geq 2$ and a fixed cap $C = C(\pmb{u}, \varepsilon^2)$ , the mass under $\mathrm{Uniform}(\mathbb{B}_1^d)$ satisfies the two-sided bound + +$$ +\underbrace {\frac {2}{d + 1} \frac {v _ {d - 1}}{v _ {d}}} _ {L _ {d} ^ {c a p}} \varepsilon^ {d + 1} \leq \mathbb {P} _ {\boldsymbol {X} \sim \operatorname {U n i f o r m} \left(\mathbb {B} _ {1} ^ {d}\right)} (\boldsymbol {X} \in C) \leq \underbrace {\frac {2 ^ {\frac {d + 1}{2}}}{d + 1} \frac {v _ {d - 1}}{v _ {d}}} _ {U _ {d} ^ {c a p}} \varepsilon^ {d + 1}. \tag {129} +$$ + +where writing $v_{k}\coloneqq \mathrm{Vol}(Bb_{1}^{k})$ + +Indeed, writing $h = \varepsilon^2$ and parameterizing $x = tu + \sqrt{1 - t^2} z$ with $t \in [1 - h, 1]$ and $z \in \mathbb{S}^{d-2}$ , we have + +$$ +\operatorname {V o l} (C) = v _ {d - 1} \int_ {1 - h} ^ {1} \left(1 - t ^ {2}\right) ^ {\frac {d - 1}{2}} d t = v _ {d - 1} \int_ {0} ^ {h} \left(s (2 - s)\right) ^ {\frac {d - 1}{2}} d s, +$$ + +where $s = 1 - t$ . Using $1 \leq 2 - s \leq 2$ on $[0, h]$ yields + +$$ +v _ {d - 1} \int_ {0} ^ {h} s ^ {\frac {d - 1}{2}} d s \leq \operatorname {V o l} (C) \leq v _ {d - 1} 2 ^ {\frac {d - 1}{2}} \int_ {0} ^ {h} s ^ {\frac {d - 1}{2}} d s = \frac {2}{d + 1} v _ {d - 1} h ^ {\frac {d + 1}{2}} \leq \frac {2 ^ {\frac {d + 1}{2}}}{d + 1} v _ {d - 1} h ^ {\frac {d + 1}{2}}. +$$ + +Dividing by $v_{d} = \mathrm{Vol}(\mathbb{B}_{1}^{d})$ and recalling $h = \varepsilon^2$ gives the stated probability bounds. + +Proposition I.1 (Many caps does not have a sample point w.h.p. via Poissonization). Fix $d \geq 2$ and $\varepsilon = \kappa n^{-1/(d+1)}$ with a constant $\kappa \in (0,1]$ . Let $\{C(\pmb{u}_j, \varepsilon^2)\}_{j=1}^m$ be any family of pairwise-disjoint caps as in (129), and draw $\pmb{X}_1, \ldots, \pmb{X}_n \stackrel{\text{i.i.d.}}{\sim}$ uniform $(\mathbb{B}_1^d)$ . For each $j$ , write $Z_j := \# \{i \leq n : \pmb{X}_i \in C(\pmb{u}_j, \varepsilon^2)\}$ and $p_j := \mathbb{P}(\pmb{X} \in C(\pmb{u}_j, \varepsilon^2))$ . Then there exist absolute constants $c_*, C_* > 0$ (depending only on $d$ and $\kappa$ ) such that, for any $\delta \in (0,1)$ , + +$$ +\mathbb {P} \Big (\# \{1 \leq j \leq m: Z _ {j} = 0 \} \geq (q _ {*} - \delta) m \Big) \geq 1 - \exp (- c _ {*} \delta^ {2} m) - C _ {*} m \varepsilon^ {2 (d + 1)}, +$$ + +where $q_{*} = e^{-\lambda_{+}}$ and $\lambda_{+} = U_{d}^{cap}\kappa^{d + 1}$ is a constant independent of $n$ . In particular, with probability at least $1 - \exp (-cm)$ (for some $c > 0$ ), there exists a subset $\Gamma \subset \{1,\dots ,m\}$ with $|\Gamma |\geq c_0m$ such that $Z_{j}\leq 1$ for every $j\in \Gamma$ , where $c_{0}\in (0,q_{*})$ depends only on $d,\kappa$ . + +Proof. Introduce a Poisson variable $N \sim \mathrm{Poi}(n)$ independent of the data. Conditionally on $N$ , draw $\mathbf{X}_1, \ldots, \mathbf{X}_N \stackrel{\mathrm{i.i.d.}}{\sim} \mathrm{Uniform}(\mathbb{B}_1^d)$ . Let $\widetilde{Z}_j := \# \{i \leq N : \mathbf{X}_i \in C(\mathbf{u}_j, \varepsilon^2)\}$ . By standard Poisson thinning, $\widetilde{Z}_1, \ldots, \widetilde{Z}_m$ are independent with $\widetilde{Z}_j \sim \mathrm{Poi}(\lambda_j)$ and + +$$ +\lambda_ {j} = \mathbb {E} [ \widetilde {Z} _ {j} ] = n p _ {j}, \qquad n L _ {d} ^ {c a p} \varepsilon^ {d + 1} \leq \lambda_ {j} \leq n U _ {d} ^ {c a p} \varepsilon^ {d + 1}. +$$ + +At the critical scaling $\varepsilon = \kappa n^{-1 / (d + 1)}$ , we thus have constants + +$$ +\underbrace {L _ {d} ^ {c a p} \kappa^ {d + 1}} _ {\lambda_ {-}} \leq \lambda_ {j} \leq \underbrace {U _ {d} ^ {c a p} \kappa^ {d + 1}} _ {\lambda_ {+}}. +$$ + +For each $j$ , set $A_{j} \coloneqq \mathbb{1}\{\widetilde{Z}_{j} = 0\}$ . Then $A_{1}, \ldots, A_{m}$ are independent Bernoulli random variables with + +$$ +\mathbb {E} [ A _ {j} ] = \mathbb {P} (\operatorname {P o i} (\lambda_ {j}) = 0) = e ^ {- \lambda_ {j}} \geq \underbrace {e ^ {- \lambda_ {+}}} _ {q _ {*}}. +$$ + +Therefore, by Hoeffding's inequality for independent bounded variables, + +$$ +\mathbb {P} \left(\frac {1}{m} \sum_ {j = 1} ^ {m} A _ {j} < q _ {*} - \delta\right) \leq \exp (- 2 \delta^ {2} m) \quad \text {f o r a l l} \delta \in (0, 1). +$$ + +Equivalently, + +$$ +\mathbb {P} \Big (\# \{j: \widetilde {Z} _ {j} = 0 \} \geq (q _ {*} - \delta) m \Big) \geq 1 - \exp (- 2 \delta^ {2} m). +$$ + +To proceed de-Poissonization, we compare $(Z_{1},\ldots ,Z_{m})$ under the fixed- $n$ model (a multinomial random variable) to the corresponding joint Poisson variable $(\widetilde{Z}_1,\dots ,\widetilde{Z}_m)$ . By Le Cam's inequality for Poisson approximation, the total variation distance between the joint law of the Bernoulli multivariables $(Z_{1},\ldots ,Z_{m})$ and that of independent $\mathrm{Poi}(\lambda_j)$ variables is bounded by + +$$ +\sup _ {E} \left| \mathbb {P} _ {\text {m u l t i}} \left(\left(Z _ {1}, \dots , Z _ {m}\right) \in E\right) - \mathbb {P} _ {\text {P o i}} \left(\left(\widetilde {Z} _ {1}, \dots , \widetilde {Z} _ {m}\right) \in E\right) \right| \leq \sum_ {j = 1} ^ {m} p _ {j} ^ {2} \leq m \left(U _ {d} ^ {\text {c a p}} \varepsilon^ {d + 1}\right) ^ {2}. \tag {130} +$$ + +Applying this to the event $E = \{\# \{j: Z_j = 0\} \geq (q_* - \delta)m\}$ and combining with (130) yields + +$$ +\mathbb {P} \Big (\# \{j: Z _ {j} = 0 \} \geq (q _ {*} - \delta) m \Big) \geq 1 - \exp (- 2 \delta^ {2} m) - (U _ {d} ^ {c a p}) ^ {2} m \varepsilon^ {2 (d + 1)}. +$$ + +Setting $c_* := 2$ and $C_* := (U_d^{cap})^2$ gives the stated bound. In particular, choosing any fixed $\delta \in (0, q_*)$ and defining $c_0 = q_* - \delta \in (0, q_*)$ proves that with probability at least $1 - \exp(-cm) - C_* m \varepsilon^{2(d+1)}$ there exists $\Gamma \subset \{1, \ldots, m\}$ , $|\Gamma| \geq c_0 m$ , such that $Z_j \leq 1$ for all $j \in \Gamma$ . + +Proposition I.1 ensures that, at the critical scaling $\varepsilon \asymp n^{-1 / (d + 1)}$ , there exists (with overwhelmingly high probability) a large subcollection of caps, each containing no sample. + +We now show that if a neural network does not have any activated datapoint, the operator norm of its Hessian is constantly 1. + +Proposition I.2. Let $f_{\pmb{\theta}}(\pmb{x}) = \sum_{k=1}^{K} v_k \phi(\pmb{w}_k^\top \pmb{x} - b_k) + \beta$ be network defined in (1). Let $\mathcal{D} = \{(x_i, y_i)\}_{i=1}^n$ be a data set such that each neuron of $f_{\pmb{\theta}}$ contains no activated datapoint, i.e. for each $k$ , $\sum_{i=1}^n \mathbb{1}\{w_k^\top x_i - b_k\} = 0$ , and $f_{\pmb{\theta}}$ interpolates $\mathcal{D}$ in the sense that $f_{\pmb{\theta}}(x_i) = y_i = 0$ for each $i$ . Then $\lambda_{\max}(\nabla_\theta^2 \mathcal{L}) = 1$ . + +Proof. By direct computation, the Hessian $\nabla_{\pmb{\theta}}^2\mathcal{L}$ is given by + +$$ +\nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L} = \frac {1}{n} \sum_ {i = 1} ^ {n} \nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) \nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) ^ {\top} + \frac {1}{n} \sum_ {i = 1} ^ {n} \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) - y _ {i}\right) \nabla_ {\boldsymbol {\theta}} ^ {2} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right). \tag {131} +$$ + +Since the model interpolates $f_{\pmb{\theta}}(\pmb{x}_i) = y_i$ for all $i$ , we have + +$$ +\nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L} = \frac {1}{n} \sum_ {i = 1} ^ {n} \nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) \nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {i}\right) ^ {\top}. \tag {132} +$$ + +Consider the tangent features matrix that is defined by + +$$ +\boldsymbol {\Phi} = \left[ \nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {1}\right), \nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {2}\right), \dots , \nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {n}\right) \right]. \tag {133} +$$ + +Then we have $\nabla_{\pmb{\theta}}^2\mathcal{L} = \Phi \Phi^\top / n$ , and the operator norm is computed by + +$$ +\lambda_ {\max } \left(\nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L}\right) = \max _ {\boldsymbol {\gamma} \in \mathbb {S} ^ {(d + 2) K}} \frac {1}{n} \| \boldsymbol {\Phi} ^ {\top} \boldsymbol {\gamma} \| ^ {2} = \max _ {\boldsymbol {u} \in \mathbb {S} ^ {n - 1}} \frac {1}{n} \| \boldsymbol {\Phi} \boldsymbol {u} \| ^ {2} \tag {134} +$$ + +Furthermore, we have + +$$ +\nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} (\boldsymbol {x}) = \left( \begin{array}{l} \nabla_ {\boldsymbol {W}} (f _ {\boldsymbol {\theta}}) \\ \nabla_ {\boldsymbol {b}} (f _ {\boldsymbol {\theta}}) \\ \nabla_ {\boldsymbol {v}} (f _ {\boldsymbol {\theta}}) \\ \nabla_ {\boldsymbol {\beta}} (f _ {\boldsymbol {\theta}}) \end{array} \right) \tag {135} +$$ + +For the parameters $[w_k, b_k, v_k]$ associated to the neuron of index $k$ , + +$$ +\frac {\partial f _ {\boldsymbol {\theta}} (\boldsymbol {x})}{\partial v _ {k}} = \mathbb {1} \left\{\boldsymbol {w} _ {k} ^ {\top} \boldsymbol {x} > b _ {k} \right\} \left(\boldsymbol {w} _ {k} ^ {\top} \boldsymbol {x} - b _ {k}\right), \quad \frac {\partial f _ {\boldsymbol {\theta}} (\boldsymbol {x})}{\partial \boldsymbol {w} _ {k}} = \mathbb {1} \left\{\boldsymbol {w} _ {k} ^ {\top} \boldsymbol {x} > b _ {k} \right\} v _ {k} \boldsymbol {x}, +$$ + +$$ +\frac {\partial f _ {\boldsymbol {\theta}} (\boldsymbol {x})}{\partial b _ {k}} = \mathbb {1} \left\{\boldsymbol {w} _ {k} ^ {\intercal} \boldsymbol {x} > b _ {k} \right\} v _ {k}, \quad \frac {\partial f _ {\boldsymbol {\theta}} (\boldsymbol {x})}{\partial \beta} = 1. +$$ + +Since there is no data point activating, we have that + +$$ +\nabla_ {\left(\boldsymbol {w} _ {k}, b _ {k}, v _ {k}, \beta\right)} f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {k}\right) = \binom {\mathbf {0}} {1}, \tag {136} +$$ + +After subsisition by (136), (134) is of the form + +$$ +\boldsymbol {\Phi} = \left( \begin{array}{c c c c} \mathbf {0} & \mathbf {0} & \dots & \mathbf {0} \\ \vdots & \vdots & \dots & \vdots \\ \mathbf {0} & \mathbf {0} & \dots & \mathbf {0} \\ 1 & 1 & \dots & 1 \end{array} \right). \tag {137} +$$ + +Let $\pmb{u} = (u_{1},\dots ,u_{n})\in \mathbb{S}^{n - 1}$ and plug (137) in (134) to have + +$$ +\lambda_ {\max } (\nabla_ {\boldsymbol {\theta}} ^ {2} \mathcal {L}) = \max _ {\boldsymbol {u} \in \mathbb {S} ^ {n - 1}} \frac {1}{n} \| \boldsymbol {\Phi} \boldsymbol {u} \| ^ {2} = \max _ {\boldsymbol {u} \in \mathbb {S} ^ {n - 1}} \frac {\left(\sum_ {i = 1} ^ {n} u _ {i}\right) ^ {2}}{n} = 1. +$$ + +□ + +We now establish that such a specially constructed interpolation solution is indeed stable. As shown in the proof, for an interpolation solution where none of the hidden neurons are active on the training data, the Hessian of the loss has an operator norm of exactly 1, i.e., $\lambda_{\mathrm{max}}(\nabla^2\mathcal{L}(\pmb {\theta})) = 1$ . The primary contribution to this norm comes from the gradient of the output layer bias. According to the stability condition defined in Proposition 2.1 ( $\lambda_{\mathrm{max}}\leq 2 / \eta$ ), this solution is guaranteed to be in $\Theta_{\mathrm{flat}}(\eta ;\mathcal{D})$ so long as the step size satisfies $\eta \leq 2$ . Since we assume that $\eta < 2$ in this paper (cf. Proposition 2.1 and the discussion below it), we have that this interpolating solution is indeed stable. + +For brevity, we write $\mathcal{F}_{\mathrm{flat}}(\eta ;\mathcal{D})\coloneqq \{f_{\theta}\mid \theta \in \Theta_{\mathrm{flat}}(\eta ;\mathcal{D})\}$ in the sequel. + +Corollary 1.3 (Stronger Version of Minimax Lower Bound). Consider the problem of estimating a function $f \in \mathcal{F} = \{f \mid f \in \mathcal{F}_{\mathrm{flat}}(\eta; \mathcal{D}), \| f \|_{L^{\infty}(\mathbb{B}_1^d)} \leq L\}$ with + +$$ +y _ {i} = f \left(\boldsymbol {x} _ {i}\right) +$$ + +where $\{\pmb{x}_i\}_{i=1}^n \subset \mathbb{B}_1^d$ are i.i.d. uniform random variables on $\mathbb{B}_1^d$ . The lower bound of the minimax nonparametric risk is given by + +$$ +\inf_{\hat{f}}\sup_{f\in \mathcal{F}}\underset {\mathcal{D}\sim \mathcal{P}^{\otimes n}}{\mathbb{E}}\| \hat{f} (\mathcal{D}) - f\|_{L^{2}(\mathbb{B}_{1}^{d})}^{2}\gtrsim_{d}L^{2}\left(\frac{1}{n}\right)^{\frac{2}{d + 1}}. +$$ + +where $\hat{f}$ refers to a estimator and $\hat{f} (\mathcal{D})$ means the estimation based on the data set $\mathcal{D}$ . + +Proof. The core of the proof is to construct two functions, $f_{1}$ and $f_{2}$ , which belong to the function class $\mathcal{F}$ but are far apart in $L^2$ norm. We will show that for a typical random data set $\{\pmb{x}_i\}_{i=1}^n$ , any estimator $\hat{f}(\mathcal{D})$ cannot distinguish between them, as they produce identical observations on $\mathcal{D}$ . This implies a lower bound on the minimax risk. + +We set the critical scaling for our construction to be: + +$$ +\varepsilon = n ^ {- 1 / (d + 1)}. \tag {138} +$$ + +Following the geometric packing argument from Lemma H.3, we can find a set of $N \asymp \varepsilon^{-(d-1)}$ pairwise-disjoint spherical caps $\{C_j\}_{j=1}^N$ , where each cap $C_j = C(\pmb{u}_j, \varepsilon^2)$ is defined by a unique direction $\pmb{u}_j \in \mathbb{S}^{d-1}$ . + +Let $\{\pmb{x}_i\}_{i=1}^n$ be the randomly drawn data set's inputs. Let $\mathcal{E}$ be the event that there exists a subset of indices $\Gamma \subset \{1, \dots, N\}$ such that: + +(i) $|\Gamma| \geq c_0 N$ for some constant $c_0 > 0$ . +(ii) For every $j\in \Gamma$ , the cap $C_j$ is empty, i.e., $C_j\cap \{\pmb {x}_i\}_{i = 1}^n = \emptyset$ + +According to Proposition I.1, this event $\mathcal{E}$ occurs with high probability, i.e., $\mathbb{P}(\mathcal{E})\geq 1 - \exp (-c_1N)$ for some constant $c_{1} > 0$ . From now on, we condition our entire analysis on this high-probability event $\mathcal{E}$ occurring. + +Conditioned on the event $\mathcal{E}$ , we now define two functions. Let $j \in \Gamma$ be an index corresponding to one of the empty caps, $C_j$ . + +1. Let the first function be the zero function: + +$$ +f _ {1} (\boldsymbol {x}) = 0. +$$ + +Clearly, $f_{1}\in \mathcal{F}_{\mathrm{flat}}(\eta ;\mathcal{D})$ and $\| f_1\|_{L^\infty} = 0\leq 1$ + +2. Let the second function be a combination appropriately scaled ReLU atoms supported on the empty cap $C_j$ : + +$$ +f _ {2} (\boldsymbol {x}) = L \sum_ {j \in \Gamma} \varepsilon^ {- 2} \phi \left(\boldsymbol {u} _ {j} ^ {\mathsf {T}} \boldsymbol {x} - \left(1 - \varepsilon^ {2}\right)\right). +$$ + +On the event $\mathcal{E}$ , both functions produce the exact same observations. For any $\pmb{x}_i$ : + +$$ +y _ {i, 1} = f _ {1} \left(\boldsymbol {x} _ {i}\right) = 0 \quad \text {a n d} \quad y _ {i, 2} = f _ {2} \left(\boldsymbol {x} _ {i}\right) = 0. +$$ + +Therefore, the data set generated by both functions is identical: $\mathcal{D} = \{(\pmb{x}_1,0),\dots ,(\pmb{x}_n,0)\}$ + +Since the data set $\mathcal{D}$ is fixed by the condition $\mathcal{E}$ and the cap $C_j$ is empty for $j\in \Gamma$ , the neuron implementing $f_{\Gamma}$ is never active on any data point $\pmb{x}_i\in \{\pmb{x}_i\}_{i = 1}^n$ . This implies $f_{2}(\pmb{x}_{i}) = 0$ for all $\pmb{x}_i$ . The corresponding labels are $y_{i} = 0$ . Therefore, $f_{2}$ perfectly interpolates the data $(x_{i},0)$ . According to Proposition I.2, any such network that interpolates the data and has no active neurons on the data set has $\lambda_{\mathrm{max}}(\nabla^{2}\mathcal{L}(\pmb{\theta}_{2})) = 1$ , where $\pmb{\theta}_{2}$ is the parameter vector that implements $f_{2}$ . Since $\eta < 2$ , this solution is stable. Thus, $f_{2}\in \mathcal{F}_{\mathrm{flat}}(\eta ;\mathcal{D})$ . Moreover, $\| f_2\|_{L^\infty} = L$ , since the spherical caps $\{C_j\}$ are disjoint, at any point $\pmb{x}$ at most one of the scaled ReLU atoms is non-zero. In summary, both $f_{1}$ and $f_{2}$ are valid functions in the class $\mathcal{F}$ . + +An estimator $\hat{f}$ takes as input the data set $\mathcal{D}$ , which is identical for both potential ground-truth functions $f_{1}$ and $f_{2}$ , and produces an estimate function, which we denote by $\hat{f}(\mathcal{D})$ . The performance of this estimator is measured by its population risk. The estimator's objective is to minimize this risk under a worst-case choice of the ground truth from the set $\{f_{1}, f_{2}\}$ . + +For a given estimate $\hat{f} (\mathcal{D})$ , the worst-case risk over this set is + +$$ +\operatorname {R i s k} (\hat {f} (\mathcal {D})) = \max \left\{\left\| \hat {f} (\mathcal {D}) - f _ {1} \right\| _ {L ^ {2} \left(\mathbb {B} _ {1} ^ {d}\right)} ^ {2}, \left\| \hat {f} (\mathcal {D}) - f _ {2} \right\| _ {L ^ {2} \left(\mathbb {B} _ {1} ^ {d}\right)} ^ {2} \right\}, +$$ + +The minimax risk for this problem is the minimal possible worst-case risk achievable by any estimator. It is lower-bounded by considering the optimal decision rule conditioned on the event $\mathcal{E}$ : + +$$ +\inf_{\hat{f}}\sup_{f\in \{f_{1},f_{2}\}}\mathbb{E}_{\mathcal{D}}\left[\left\| \hat{f} (\mathcal{D}) - f\right\|_{L^{2}(\mathbb{B}_{1}^{d})}^{2}\right]\geq \inf_{\hat{f}}\operatorname {Risk}(\hat{f} (\mathcal{D}))\cdot \mathbb{P}(\mathcal{E}). +$$ + +The function $\hat{f}(\mathcal{D})^*$ that minimizes $\max \left\{\left\| \hat{f}(\mathcal{D}) - f_1 \right\|_{L^2(\mathbb{B}_1^d)}^2, \left\| \hat{f}(\mathcal{D}) - f_2 \right\|_{L^2(\mathbb{B}_1^d)}^2 \right\}$ is the average of $f_1$ and $f_2$ in the Hilbert space $L^2(\mathbb{B}_1^d)$ . This optimal estimate is $\hat{f}(\mathcal{D})^* = (f_1 + f_2)/2$ . The minimal possible worst-case risk is thus achieved at this midpoint: + +$$ +\inf _ {\hat {f} (\mathcal {D}) \in \mathcal {F}} \operatorname {R i s k} (\hat {f} (\mathcal {D})) = \left\| \hat {f} (\mathcal {D}) ^ {*} - f _ {1} \right\| _ {L ^ {2} (\mathbb {B} _ {1} ^ {d})} ^ {2} = \left\| \frac {f _ {1} + f _ {2}}{2} - f _ {1} \right\| _ {L ^ {2} (\mathbb {B} _ {1} ^ {d})} ^ {2} = \left\| \frac {f _ {2} - f _ {1}}{2} \right\| _ {L ^ {2} (\mathbb {B} _ {1} ^ {d})} ^ {2}. +$$ + +According to the computation in Construction H.4, we may conclude that + +$$ +\| f _ {2} - f _ {1} \| _ {L ^ {2} \left(\mathbb {B} _ {1} ^ {d}\right)} ^ {2} = \left\| L \sum_ {j \in \Gamma} \varepsilon^ {- 2} \phi \left(\boldsymbol {u} _ {j} ^ {\intercal} \boldsymbol {x} - (1 - \varepsilon^ {2})\right) \right\| _ {L ^ {2} \left(\mathbb {B} _ {1} ^ {d}\right)} ^ {2} \asymp L ^ {2} \varepsilon^ {2} \asymp L ^ {2} \left(\frac {1}{n}\right) ^ {\frac {2}{d + 1}} +$$ + +This completes the proof. + +Theorem I.4. Let $\mathcal{P}$ denote any joint distribution of $(\pmb{x},y)$ where the marginal distribution of $\pmb{x}$ is $\mathrm{Uniform}(\mathbb{B}_1^d)$ and $y$ satisfies the $\mathbb{P}_{\mathcal{P}}[-D\leq y\leq D] = 1$ . + +Let $\mathcal{D} = \{(\pmb{x}_j, y_j)\}_{j=1}^n$ be a data set of $n$ i.i.d. samples from $\mathcal{P}$ , and that $\tilde{R}$ is any risk estimator that takes any $f$ and $\mathcal{D}$ as input, then outputs a scalar that aims at estimating the risk $R_{\mathcal{P}}(f) := \mathbb{E}_{(\pmb{x}, y) \sim \mathcal{P}}[(f(\pmb{x}) - y)^2]$ . Moreover, let $\mathcal{F}$ be the function class we defined in Corollary I.3. + +Then + +$$ +\inf _ {\tilde {R}} \sup _ {\mathcal {P}} \mathbb {E} \left[ \sup _ {\substack {f \in \mathcal {F} _ {\text {flat}} (\eta ; \mathcal {D}) \\ \| f \| _ {L ^ {\infty} \left(\mathbb {B} _ {1} ^ {d}\right)} \leq L}} \left| R _ {\mathcal {P}} (f) - \tilde {R} (f; \mathcal {D}) \right| \right] \gtrsim_ {d} L ^ {2} n ^ {- \frac {2}{d + 1}}. \tag{139} +$$ + +where we assume that $L \geq D$ . + +Proof. Let the $\mathbb{E}[\cdot ]$ be the short-hand for the expectation over the random training data set $\mathcal{D}$ + +$$ +\begin{array}{l} \inf _ {\tilde {R}} \sup _ {\mathcal {P}} \mathbb {E} \left[ \sup _ {f \in \mathcal {F} _ {\text {f l a t}} (\eta ; \mathcal {D})} \left| R _ {\mathcal {P}} (f) - \tilde {R} (f; \mathcal {D}) \right| \right] \\ \geq \inf_{\tilde{R}}\sup_{\substack{\mathcal{P} = \operatorname {Unif}(\mathbb{B}_{1}^{d})\times f_{0}\\ f_{0}\in \mathcal{F}_{\text{flat}}(\eta ;\mathcal{D})}}\mathbb{E}\left[\sup_{f\in \mathcal{F}_{\text{flat}}(\eta ;\mathcal{D})}\Big|R_{\mathcal{P}}(f) - \tilde{R} (f;\mathcal{D})\Big|\right] \\ \geq \inf_{\tilde{R}}\sup_{\substack{\mathcal{P} = \operatorname {Unif}(\mathbb{B}_{1}^{d})\times f_{0}\\ f_{0}\in \mathcal{F}_{\text{flat}}(\eta ;\mathcal{D})}}\frac{1}{2}\mathbb{E}\left[R_{\mathcal{P}}(\hat{f}_{\text{ERM}}(\tilde{R} (\cdot ,\mathcal{D})) - R_{\mathcal{P}}(f_{0})\right] \\ \geq \inf_{\hat{f}}\sup_{\substack{\mathcal{P} = \operatorname {Unif}(\mathbb{B}_{1}^{d})\times f_{0}\\ f_{0}\in \mathcal{F}_{\text{flat}}(\eta ;\mathcal{D})}}\frac{1}{2}\mathbb{E}\left[R_{\mathcal{P}}(\hat{f} (\mathcal{D})) - R_{\mathcal{P}}(f_{0})\right] \\ = \frac{1}{2}\inf_{\hat{f}}\sup_{\substack{\mathcal{P} = \operatorname {Unif}(\mathbb{B}_{1}^{d})\times f_{0}\\ f_{0}\in \mathcal{F}_{\text{flat}}(\eta ;\mathcal{D})}}\mathbb{E}\left[||\hat{f} (\mathcal{D}) - f_{0}||_{L^{2}(\mathbb{B}_{1}^{d})}^{2}\right] \\ \end{array} +$$ + +(Corollary I.3) $\longrightarrow \gtrsim_{d} L^{2} n^{-\frac{2}{d + 1}}$ . + +The first inequality restricts $\mathcal{P}$ further to deterministic labels with labeling functions in $\mathcal{F}$ . Check that any function in $\mathcal{F}$ is bounded between $[-M, M]$ . The second inequality uses the fact that $f_0 \in \mathcal{F}_{\mathrm{flat}}(\eta; \mathcal{D})$ , and the following decomposition + +$$ +\begin{array}{l} R _ {\mathcal {P}} \left(\hat {f} _ {\mathrm {E R M}} (\tilde {R})\right) - R _ {\mathcal {P}} \left(f _ {0}\right) \\ = R _ {\mathcal {P}} \left(\hat {f} _ {\mathrm {E R M}} (\tilde {R})\right) - \tilde {R} \left(\hat {f} _ {\mathrm {E R M}}; \mathcal {D}\right) + \tilde {R} \left(\hat {f} _ {\mathrm {E R M}}; \mathcal {D}\right) - \tilde {R} \left(f _ {0}; \mathcal {D}\right) + \tilde {R} \left(f _ {0}; \mathcal {D}\right) - R _ {\mathcal {P}} \left(f _ {0}\right) \\ \leq \left| R _ {\mathcal {P}} \left(\hat {f} _ {\mathrm {E R M}} (\tilde {R})\right) - \tilde {R} \left(\hat {f} _ {\mathrm {E R M}}; \mathcal {D}\right) \right| + \left| \tilde {R} \left(f _ {0}; \mathcal {D}\right) - R _ {\mathcal {P}} \left(f _ {0}\right) \right| \\ \leq 2 \sup _ {f \in \mathcal {F} _ {\text {f l a t}} (\eta ; \mathcal {D})} \left| R _ {\mathcal {P}} (f) - \tilde {R} (f; \mathcal {D}) \right|, \\ \end{array} +$$ + +where we used $\tilde{R} (\hat{f}_{\mathrm{ERM}};\mathcal{D}) - \tilde{R} (f_0;\mathcal{D})\leq 0$ from the definition of $\hat{f}_{\mathrm{ERM}}$ + +$$ +\hat{f}_{\text{ERM}}(\tilde{R} (\cdot ,\mathcal{D})):= \operatorname *{argmin}_{f\in \mathcal{F}_{\text{flat}}(\eta ;\mathcal{D})}\tilde{R} (f;\mathcal{D}). +$$ + +The third inequality enlarges the set of ERM estimators to any function of the data $\hat{f}$ that output. The subsequent identity uses the fact that $R_{\mathcal{P}}(f_0) = 0$ + +This completes the proof for the lower bound on generalization gap stated in Theorem 3.5. + +# J Technical Lemmas + +# J.1 Information-Theoretic tools + +Fano's Lemma provides a powerful method for establishing such minimax lower bounds by relating the estimation problem to a hypothesis testing problem. It leverages information-theoretic concepts, particularly the Kullback-Leibler (KL) divergence. + +Lemma J.1 (Fano's Lemma (Statistical Estimation Context)). Consider a finite set of functions (or parameters) $\{f_1, f_2, \ldots, f_M\} \subset \mathcal{F}$ , with $N \geq 2$ . Let $P_{f_j}$ denote the probability distribution of the observed data $\mathcal{D}$ when the true underlying function is $f_j$ . Suppose that for any estimator $\hat{f}$ , the loss function $L(f_j, \hat{f})$ satisfies $L(f_j, \hat{f}) \geq s^2 / 2 > 0$ if $\hat{f}$ is not close to $f_j$ (e.g., if we make a wrong decision in a multi-hypothesis test where closeness is defined by a metric $d(f_j, f_k) \geq s$ ). More specifically, for function estimation with squared $L^2$ -norm loss, if we have a packing set $\{f_1, \ldots, f_M\} \subset \mathcal{F}$ such that $\|f_j - f_k\|_{L^2}^2 \geq s^2$ for all $j \neq k$ , then the minimax risk is bounded as: + +$$ +\inf _ {\hat {f}} \sup _ {f \in \mathcal {F}} \mathbb {E} \| \hat {f} - f \| _ {L ^ {2}} ^ {2} \geq \frac {s ^ {2}}{4} \left(1 - \frac {\max _ {j \neq k} \mathrm {K L} \left(P _ {f _ {j}} \| P _ {f _ {k}}\right) + \log 2}{\log M}\right), \tag {140} +$$ + +provided the term in the parenthesis is positive. $\mathrm{KL}(P_{f_j}\| P_{f_k})$ denotes the Kullback-Leibler divergence between the distributions $P_{f_j}$ and $P_{f_k}$ . For this bound to be non-trivial (e.g., $\gtrsim s^2$ ), we typically require that the number of well-separated functions $M$ is large enough such that + +$$ +\log \frac {M}{2} > \max _ {j \neq k} \mathrm {K L} \left(P _ {f _ {j}} \| P _ {f _ {k}}\right). \tag {141} +$$ + +One can refer to Wasserman [2020, Theorem 12, Corollary 13] or Tsybakov [2009, Chapter 2] for more details. + +Our application of Fano's Lemma (for proving Proposition H.5) involves: + +1. Constructing a suitable finite subset of functions $\{f_1, \ldots, f_M\}$ within the class $\mathcal{F}$ such that they are well-separated in the metric defined by the loss function (e.g., pairwise $L^2$ -distance $s$ ). This is often achieved using techniques like the Varshamov-Gilbert lemma (Lemma J.2) for constructing packings. +2. Bounding the KL divergence (or another information measure like $\chi^2$ -divergence) between the probability distributions generated by pairs of these functions. For $n$ i.i.d. observations with additive Gaussian noise $\tilde{\mathcal{N}}(0, \sigma^2)$ , and if using the empirical $L^2$ norm $\| \cdot \|_{L^2(\mathbb{P}_n)}$ based on fixed data points $\pmb{x}_i$ , this divergence is often related to $\frac{1}{2\sigma^2} \sum_{i=1}^{n} (f_j(\pmb{x}_i) - f_k(\pmb{x}_i))^2$ . More generally, for population norms, it's often $\frac{n \| f_j - f_k \|_{L^2}^2}{2\sigma^2}$ . +3. Choosing $M$ and $s$ (or the parameters defining the packing) to maximize the lower bound, typically by ensuring that the KL divergence term does not dominate $\log M$ . + +Lemma J.2 (Varshamov-Gilbert Lemma). Let + +$$ +\Xi = \left\{\xi = \left(\xi_ {1}, \dots , \xi_ {N}\right): \xi_ {j} \in \{0, 1 \} \right\}. +$$ + +Suppose $N \geq 8$ . Then there exist + +$$ +\xi^ {0}, \xi^ {1}, \dots , \xi^ {M} \in \Xi +$$ + +such that + +1. $\xi_0 = (0,\dots ,0)$ +2. $M\geq 2^{N / 8}$ +3. for all $0 \leq j < k \leq M$ , the Hamming distance satisfies + +$$ +d _ {H} \left(\xi^ {j}, \xi^ {k}\right) \geq \frac {N}{8}. +$$ + +We call $\{\xi^0,\xi^1,\ldots ,\xi^M\}$ a pruned hypercube. + +One can refer to Tsybakov [2009, Lemma 2.9] and Wasserman [2020, Lemma 15] for more details. + +# J.2 Poissonization and Le Cam's Inequality + +For a random variable $S$ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with values in a measurable space $(E, \mathcal{E})$ , the law is $\mathcal{L}(S) \coloneqq \mathbb{P} \circ S^{-1}$ . For two laws $\mu, \nu$ on the same space, define the total variation distance + +$$ +d _ {\mathrm {T V}} (\mu , \nu) := \sup _ {A \in \mathcal {E}} | \mu (A) - \nu (A) |. +$$ + +When $E$ is countable, $d_{\mathrm{TV}}(\mu ,\nu) = \frac{1}{2}\sum_{x\in E}|\mu (\{x\}) - \nu (\{x\})|$ . + +Lemma J.3 (Poissonization [Barbour et al., 1992, Ch. 1]). Let $N \sim \mathrm{Poi}(\lambda)$ and, conditional on $N$ , let $W_{1},\ldots ,W_{N}$ be i.i.d. taking values in $\{0,1,\dots,m\}$ with $\mathbb{P}\{W = j\} = p_j$ for $j = 0,1,\dots,m$ , where $p_0 \coloneqq 1 - \sum_{j = 1}^{m}p_{j} \geq 0$ . Define $\widetilde{Z}_j \coloneqq \# \{1 \leq i \leq N : W_i = j\}$ for $j = 1,\dots,m$ . Then $\widetilde{Z}_{1},\ldots ,\widetilde{Z}_{m}$ are independent and $\widetilde{Z}_j \sim \mathrm{Poi}(\lambda p_j)$ . + +Remark J.4. This standard Poissonization trick replaces the fixed sample size $n$ by a Poisson random size $N \sim \mathrm{Poi}(n)$ , making the cell counts independent. See Barbour, Holst and Janson [Barbour et al., 1992, Ch. 1] for a general treatment and applications in occupancy problems. + +Lemma J.5 (Le Cam's inequality for Poisson approximation [Cam, 1960, Arratia et al., 1989, Barbour et al., 1992]). Let $(Z_{1},\ldots,Z_{m}) \sim \mathrm{Mult}(n;p_{1},\ldots,p_{m},p_{0})$ with $p_0 = 1 - \sum_{j=1}^{m} p_j$ . Let $Y_{1},\ldots,Y_{m}$ be independent with $Y_{j} \sim \mathrm{Poi}(np_{j})$ . Then there exists a universal constant $C > 0$ such that + +$$ +d _ {\mathrm {T V}} \big (\mathcal {L} (Z _ {1}, \dots , Z _ {m}), \mathcal {L} (Y _ {1}) \otimes \dots \otimes \mathcal {L} (Y _ {m}) \big) \leq C \sum_ {j = 1} ^ {m} p _ {j} ^ {2}. +$$ + +Remark J.6. Lemma J.5 is a classical result of Le Cam [Cam, 1960], with modern proofs and refinements given by [Arratia et al., 1989] and by [Barbour et al., 1992, Sec. 1.3]. It provides a quantitative control of the total variation distance between the multinomial occupancy vector and the independent Poisson approximation. We use this bound to justify the de-Poissonization step in the proof of Proposition I.1. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The main claims in the abstract and introduction accurately reflect the paper's contributions and scope. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [NA] + +Justification: The core method development of this paper comes from human brains. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. \ No newline at end of file diff --git a/stableminimaofreluneuralnetworkssufferfromthecurseofdimensionalitytheneuralshatteringphenomenon/images.zip b/stableminimaofreluneuralnetworkssufferfromthecurseofdimensionalitytheneuralshatteringphenomenon/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..437ab112046c83652a45fa6d589ea8907c690bbc --- /dev/null +++ b/stableminimaofreluneuralnetworkssufferfromthecurseofdimensionalitytheneuralshatteringphenomenon/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:35ff6bfb00d02323ab47559ba1d64e207c490845f8c13b692725471e3754f928 +size 3473168 diff --git a/stableminimaofreluneuralnetworkssufferfromthecurseofdimensionalitytheneuralshatteringphenomenon/layout.json b/stableminimaofreluneuralnetworkssufferfromthecurseofdimensionalitytheneuralshatteringphenomenon/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..e782303830417c4b594e1e5b2be76d3f16f383d5 --- /dev/null +++ b/stableminimaofreluneuralnetworkssufferfromthecurseofdimensionalitytheneuralshatteringphenomenon/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:119839862081a11141477f2bd2e475fdbc8fa164e6327692c7df4cff25033d59 +size 3082863 diff --git a/stablepartdiffusion4dmultiviewrgbandkinematicpartsvideogeneration/515fbc30-9f51-4587-abc4-03e2e759328b_content_list.json b/stablepartdiffusion4dmultiviewrgbandkinematicpartsvideogeneration/515fbc30-9f51-4587-abc4-03e2e759328b_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..5e95c79c0f48c48117c88661b9e62b7eb6164fd8 --- /dev/null +++ b/stablepartdiffusion4dmultiviewrgbandkinematicpartsvideogeneration/515fbc30-9f51-4587-abc4-03e2e759328b_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:343df7158b0a268f14cdabb40b5a3aca99a96563a552b52a5011d6a30164f9c5 +size 137603 diff --git a/stablepartdiffusion4dmultiviewrgbandkinematicpartsvideogeneration/515fbc30-9f51-4587-abc4-03e2e759328b_model.json b/stablepartdiffusion4dmultiviewrgbandkinematicpartsvideogeneration/515fbc30-9f51-4587-abc4-03e2e759328b_model.json new file mode 100644 index 0000000000000000000000000000000000000000..8bb002cc4c8a9dbfa9b7b360a1757fdea1cd1e87 --- /dev/null +++ b/stablepartdiffusion4dmultiviewrgbandkinematicpartsvideogeneration/515fbc30-9f51-4587-abc4-03e2e759328b_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a30e8e2076c314407ce2340f74db774d8cdb3e2f8f699e18062c0d10e6042a29 +size 184319 diff --git a/stablepartdiffusion4dmultiviewrgbandkinematicpartsvideogeneration/515fbc30-9f51-4587-abc4-03e2e759328b_origin.pdf b/stablepartdiffusion4dmultiviewrgbandkinematicpartsvideogeneration/515fbc30-9f51-4587-abc4-03e2e759328b_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5c4123e6874f467e2a2c40edf7f003245d1aab1c --- /dev/null +++ b/stablepartdiffusion4dmultiviewrgbandkinematicpartsvideogeneration/515fbc30-9f51-4587-abc4-03e2e759328b_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:737d66c4803908da67851d1072bce2310cd8b8e8eedec7fe8d95520e7ec9f047 +size 9969474 diff --git a/stablepartdiffusion4dmultiviewrgbandkinematicpartsvideogeneration/full.md b/stablepartdiffusion4dmultiviewrgbandkinematicpartsvideogeneration/full.md new file mode 100644 index 0000000000000000000000000000000000000000..5c6daeb53ae8b58ae6efaa4616dff8285b14c100 --- /dev/null +++ b/stablepartdiffusion4dmultiviewrgbandkinematicpartsvideogeneration/full.md @@ -0,0 +1,625 @@ +# Stable Part Diffusion 4D: Multi-View RGB and Kinematic Parts Video Generation + +Hao Zhang $^{1,2*}$ Chun-Han Yao $^{1}$ Simon Donne $^{1}$ Narendra Ahuja $^{2}$ Varun Jampani $^{1}$ $^{1}$ Stability AI University of Illinois Urbana-Champaign + +![](images/e209a2878559ffaae5a7a95c8dd16396acc032299f4bae470eae797aaf12a42b.jpg) +Figure 1: Left: Stable Part Diffusion 4D (SP4D) takes a monocular input video and generates novel-view RGB videos (bottom-left) as well as consistent part segmentation videos across all views. Right: SPD also supports single image input and synthesizes multi-view RGB images and corresponding part decompositions. These results can be lifted to 3D to produce riggable meshes with part-aware geometry and articulated structure. + +# Abstract + +We present Stable Part Diffusion 4D (SP4D), a framework for generating paired RGB and kinematic part videos from monocular inputs. Unlike conventional part segmentation methods that rely on appearance-based semantic cues, SP4D learns to produce kinematic parts — structural components aligned with object articulation and consistent across views and time. SP4D adopts a dual-branch diffusion model that jointly synthesizes RGB frames and corresponding part segmentation maps. To simplify architecture and flexibly enable different part counts, we introduce a spatial color encoding scheme that maps part masks to continuous RGB-like images. This encoding allows the segmentation branch to share the latent VAE from the RGB branch, while enabling part segmentation to be recovered via straightforward post-processing. A Bidirectional Diffusion Fusion (BiDiFuse) module enhances cross-branch consistency, supported by a contrastive part consistency loss to promote spatial and temporal alignment of part predictions. We demonstrate that the generated 2D part maps can be lifted to 3D to derive skeletal structures and harmonic skinning weights with few manual adjustments. To train and evaluate SP4D, we construct KinematicParts20K, a curated dataset of over 20K rigged objects selected and processed from Objaverse XL (Deitke et al., 2023), each paired with multi-view RGB and part video sequences. Experiments show that SP4D generalizes strongly to diverse scenarios, including real-world videos, novel generated objects, and rare articulated poses, producing kinematic-aware outputs suitable for downstream animation and motion-related tasks. + +![](images/8561a910416e6af0b710f5df016ad4aa75e068e81f5702cb339078825a562165.jpg) +RGB + +![](images/df31052a569d73ed841e13e6384ec874d48a5c9500c0c5c51c381804878d6033.jpg) +SAM2 + +![](images/5d28822bd6ad4557a8dbe92bd736953f7529d55c6f271b4cd9df969a81c1e575.jpg) +Ours +Figure 2: Limitations of traditional 2D and 3D part decomposition methods. Left: Appearance-based 2D segmentation methods like SAM2 fail to produce kinematic parts. Middle: SOTA 3D rigging methods (Song et al., 2025) lack the capability to infer kinematic part structures from appearance and generalize poorly to diverse shapes. Right: Existing 3D part segmentation models (Tang et al., 2024a; Yang et al., 2024) focus on semantic regions and are not suited for kinematic decomposition. + +![](images/2984e73a96031b94b0359777c67ed4afdafdc6b28fe36b07c8199630228def1d.jpg) +SOTA 3D Rigging Method + +![](images/b3344c357bf0e8051bf16a7546d35378c18d72a877474c6db9173a349df66fae.jpg) + +![](images/d79c2f4664ffa120894531bc967de66b591bc772e482374764f8e4023dbc8379.jpg) + +![](images/f19bf5f87eafe37bdf3c25ae718e7a6a06cbdb23bac92f20867c25a338e953e4.jpg) + +![](images/165901bd808126bb2bb7b086c05096bc1338d9d0b4f4766ec9d42b81e4341585.jpg) +SOTA 3D Part Segmentation + +![](images/d13d90b9c47376235f6601327c3870c00375f57d3c6829ae02a470b010234fc2.jpg) + +![](images/59e92703d7ad8f781fc4f5488190957e61f492cb68db21f52b0bfaa8730d31a4.jpg) + +# 1 Introduction + +Generating kinematic-aware and structure-consistent videos from monocular inputs is a fundamental challenge in computer vision and graphics, with wide applications in animation, AR/VR, robotics, and simulation. A key aspect of this is understanding how an object moves, articulates, and preserves spatial part relationships over time. While conventional video generation methods focus on realistic RGB synthesis, they often overlook internal articulation and fail to model a consistent structure. + +Recent 4D generation approaches (Liang et al., 2024a; Zhang et al., 2024; Li et al., 2024a; Ren et al., 2024; Xie et al., 2025; Yang et al., 2025c; Zhao et al., 2025a; Zhu et al., 2025; Yao et al., 2025) have made notable progress in reconstructing dynamic 3D sequences from monocular video, but primarily concentrate on surface-level geometry. These methods do not provide meaningful structural part decomposition and are not optimized for articulated modeling. Auto-rigging methods are a traditional option for extracting kinematic parts. However, learning-based rigging methods (Xu et al., 2020; Liu et al., 2025; Song et al., 2025; Zhang et al., 2025; Deng et al., 2025) operate on static 3D meshes and rely on explicit supervision such as skeletal annotations or pre-rigged models. However, these methods are fundamentally constrained by the limited scale and diversity of high-quality 3D rigging datasets, making it difficult to leverage large-scale 2D visual data and powerful pretrained image/video models. As a result, they struggle to generalize to novel object categories and rare articulated poses as shown in fig. 2. + +Meanwhile, part segmentation methods (Amir et al., 2021; Tang et al., 2024a; Yang et al., 2024, 2025b) often rely on semantic labels or appearance cues, leading to predictions that are temporally unstable or inconsistent across viewpoints. Most of these methods focus on semantic segmentation (e.g., head, tail, leg), which does not necessarily reflect the physical articulation or structural function of an object as shown in fig. 2. In contrast, kinematic part segmentation identifies physically meaningful regions that move together over time—providing essential structure for downstream animation, motion retargeting, or deformation modeling as shown in fig. 1. + +In this work, we introduce Stable Part Diffusion 4D (SP4D), a novel framework for jointly generating RGB and kinematic part videos from monocular inputs. SP4D builds on multi-view video diffusion (Yao et al., 2025) and adopts a dual-branch architecture: one UNet generates multi-view RGB frames, while the other produces spatially and temporally consistent part segmentation maps. Unlike conventional approaches that rely on predefined semantic categories or part counts, we encode part masks as continuous RGB-like images using a spatial color encoding scheme. This allows the part branch to share the same VAE encoder and decoder with the RGB branch, and enables discrete part maps to be recovered via simple clustering in post-processing. + +To ensure coherence between appearance and structure, we introduce a novel Bidirectional Diffusion Fusion (BiDiFuse) module, inspired by Vainer et al. (2024), which facilitates information exchange between the RGB and part branches during the denoising process. This cross-branch communication encourages mutual guidance and alignment between modalities. Crucially, because of the parts' spatial color encoding, the diffusion model lacks explicit supervision to enforce consistent part appearance across different views and time steps. The resulting temporal inconsistency leads to severe degradation in structural coherence. To address this, we introduce a contrastive part consistency loss, which aligns latent part features corresponding to the same physical regions across views and time. This loss plays a central role in enabling the model to learn stable, kinematically meaningful part representations that remain consistent throughout the generated video. + +Although our framework does not explicitly output 3D models, it enables a lightweight rigging pipeline by lifting the 2D part maps to 3D. From the recovered part regions, we estimate harmonic skinning weights without requiring explicit skeleton annotations—allowing the generated videos to support animation-aware applications with minimal manual intervention. + +To support training and evaluation, we curate KinematicParts20K, a dataset of over 20K rigged objects selected and processed from Objaverse XL (Deitke et al., 2023), annotated with skinning weights. We adopt a two-stage training strategy: the model is first trained on ObjaverseDy (Xie et al., 2025) with RGB supervision only and the BiDiFuse module bypassed; it is then fine-tuned on KinematicParts20K with supervision on both branches. This strategy leverages the generalization strength of pre-trained RGB diffusion models while gradually introducing structure-aware learning. + +# Our main contributions are as follows: + +- We propose Stable Part Diffusion 4D (SP4D), the first framework to generate multi-view, temporally consistent kinematic part decompositions jointly with RGB videos from monocular inputs. +- We introduce a compact architecture with spatial color encoding for encoder-decoder sharing, efficient joint modeling with our novel Bidirectional Diffusion Fusion (BiDiFuse) and a contrastive part consistency loss to explicitly enforce cross-view and temporal alignment of part features. +- We establish a simple yet effective 2D-to-Kinematic Mesh pipeline by lifting part maps to 3D and estimating harmonic skinning weights, enabling skeleton-free animation-ready outputs. +- We curate KinematicParts20K, a large-scale dataset of over 20K rigged objects with paired RGB and part video annotations to support training and evaluation. +- Our method demonstrates strong generalization across real-world and synthetic scenarios, and offers a promising direction for leveraging 2D data and pretrained priors to solve long-standing challenges in 3D rigging, with clear benefits for downstream animation and motion-related tasks. + +# 2 Related Work. + +3D and 4D Generation. We focus on diffusion-based 3D and 4D generation, which typically yield higher-quality assets than feed-forward techniques (Hong et al., 2024; Jiang et al., 2024; Wang et al., 2024a; Zou et al., 2024; Wei et al., 2024; Tochilkin et al., 2024; Ren et al., 2024; Chen et al., 2024b; Zuo et al., 2024) and are not as class-bound as a GAN or VAE. We identify three main approaches: SDS-based and photogrammetry-based methods leverage recent advantages in image and video diffusion models, compared to directly performing diffusion in 3D. The seminal Dreamfusion (Poole et al., 2023) used Score Distillation Sampling to refine a random initialization using the diffusion model; it is training-free but takes a very high inference cost. Follow-up works have improved both the quality and the inference speed significantly (Yi et al., 2024; Tang et al., 2024b; Shi et al., 2024b; Wang et al., 2024b; Li et al., 2024c; Weng et al., 2023; Pan et al., 2024; Chen et al., 2024a; Sun et al., 2024; Sargent et al., 2024; Liang et al., 2024b; Zhou et al., 2024; Guo et al., 2023), but these methods are still considered impractically slow for many contexts. An alternative way of leveraging existing image and video models is to synthesize multi-view imagery and subsequently perform photogrammetry to extract 3D structure (Liu et al., 2023b, 2024b; Long et al., 2024; Voleti et al., 2024; Ye et al., 2024; Karnewar et al., 2023; Li et al., 2024b; Shi et al., 2024a, 2023; Wang and Shi, 2023; Liu et al., 2023a, 2024a)). Finally, more recent methods perform the modeling directly in 3D (Zhao et al., 2025b; Xiang et al., 2024), often relying on powerful VAEs to compress the data dimensionality and make the problem tractable. + +Image-based approaches tend to be much more data-efficient, as they can leverage the strong priors of the underlying diffusion models, at the price of costlier inference or training. However, both families of approaches omit a key aspect for practical usability: rigging and skinning of the objects, to turn them into animation-compatible assets. We propose to generate kinematics-aware part segmentation for the resulting objects, which feeds more cleanly into downstream pipelines. + +Rigging and animation. Preparing raw 3D objects for animation involves two steps: rigging (building a piecewise-rigid skeleton of bones) and skinning (defining how each part of the object deforms in function of the movement of these bones). We focus on the former, as a bad rig precludes proper skinning. Although we do not provide an end-to-end rigging and skinning pipeline, the kinematic + +parts we segment are a proxy to both: they form local clusters that directly correlate to the most influential bone; we posit that bones and even the skeletal tree structure can be extracted from these. + +Learning-based rigging methods (Xu et al., 2020; Liu et al., 2025; Song et al., 2025; Zhang et al., 2025; Deng et al., 2025) have shown promise in predicting skeleton structure and inferring skinning weights, but are typically trained on very limited datasets, restricting their generalization to unseen object categories and poses. Moreover, most of them operate on static geometry and fail to capture and/or leverage the dynamic articulation present in real-world videos. Our approach addresses these limitations by leveraging the rich video diffusion priors and learning movement-aware part decomposition from videos, thereby enabling broader generalization and dynamic rigging capabilities from very little training data. + +Part Decomposition. Parts are useful intermediate representations for recognition, generation, and animation. Earlier 3D segmentation methods (Qi et al., 2017; Li et al., 2018; Qian et al., 2022) rely on static geometry and annotated datasets, which limits their generalization to unseen or dynamic objects. More recent works use 2D semantic features for co-part segmentation (Hung et al., 2019; Amir et al., 2021; Tang et al., 2024a; Yang et al., 2024), but these tend to be view-inconsistent and temporally unstable. Moreover, semantic parts are not always meaningful for animation, where rigid or articulable components are preferred. As shown in fig. 2, the semantic and kinematic parts differ both visually and functionally. Our goal is to identify physically coherent regions that move consistently over time. To our knowledge, no prior work explicitly tackles kinematic part segmentation—likely due to the lack of suitable training data. We address this by leveraging the pretrained SV4D (Yao et al., 2025) video diffusion model, and extending it with a parallel part segmentation branch trained in the multi-view video space, following Vainer et al. (2024). + +# 3 Method + +# 3.1 Preliminaries: SV4D 2.0 Network Architecture + +Our method builds on the SV4D 2.0 framework (Yao et al., 2025), a state-of-the-art multi-view video diffusion model designed for 4D content generation. SV4D 2.0 synthesizes multi-frame, multi-view videos using a spatio-temporally consistent latent diffusion architecture. It takes either a monocular video or a single still image as input. At its core, SV4D 2.0 represents video as a latent tensor (in 4D, as indexed by spatial, temporal, and view dimensions) and applies denoising through a UNet-based architecture composed of spatial, temporal, and view-level attention blocks. The architecture initializes frame attention modules from Stable Video Diffusion (Blattmann et al., 2023) and spatial-view components from SV3D (Voleti et al., 2024), benefiting from strong spatio-temporal priors. The resulting model supports long-range, self-consistent video synthesis and handles both large deformations and occlusions robustly. It also introduces learnable $\alpha$ -blending strategies to combine temporal and spatial-view features during fusion, enabling smooth integration of multiple priors while preserving the pre-trained knowledge. The model is conditioned on both camera and frame embeddings, allowing flexible synthesis under diverse trajectories and temporal contexts. During training, SV4D 2.0 applies random view masking, which reduces reliance on explicit multi-view supervision and allows inference without external view-conditioning models. To improve performance across sparse or nonuniform camera layouts, the model replaces traditional view attention with 3D attention layers that jointly reason over spatial and view axes. + +# 3.2 Stable Part Diffusion 4D + +Problem Setting. Stable Part Diffusion 4D (SP4D) aims to generate multi-view, temporally coherent kinematic part segmentation videos alongside consistent RGB videos, conditioned on a monocular RGB video input. Formally, conditioned on input frames $J = \{J_{f}\}_{f = 1}^{F}$ , the model aims to produce + +$$ +M = \{M _ {v, f} \} _ {v = 1, f = 1} ^ {V, F}, \quad P = \{P _ {v, f} \} _ {v = 1, f = 1} ^ {V, F}, +$$ + +where $M_{v,f}$ and $P_{v,f}$ represent the generated RGB image and its corresponding part segmentation at view $v$ and frame $f$ , respectively. + +The goal is to produce photo-realistic video sequences $M$ that are consistent across views and time, while also generating part segmentations $P$ that reflect view-invariant kinematic structure. Unlike + +![](images/3f54054b4755e5a68dd93800a1b33fbc23ae46fa9dbcd17908422fa0b9cbc75f.jpg) +Figure 3: Stable Part Diffusion 4D model architecture. Our model builds upon SV4D 2.0 and extends it with a parallel part segmentation branch and a BiDiFuse module that enables bidirectional feature exchange between RGB and part branches. The network jointly generates multi-view videos for appearance and kinematics-aware part segmentation. Key components include: (1) spatial color encoding for part masks, enabling shared VAE encoder/decoder; (2) BiDiFuse for cross-branch consistency; and (3) a contrastive loss for spatial-temporal part alignment. We use a two-stage training strategy: first, training the RGB branch on ObjaverseDy, then fine-tuning the full model with BiDiFuse on KinematicParts20K with supervision on both branches. + +traditional semantic segmentation, these part representations are not predefined by category, but instead capture locally rigid, articulated motion-consistent regions. + +Network Architecture. Our network architecture extends SV4D 2.0 with a dual-branch UNet structure to jointly generate RGB and kinematic part segmentation videos. We adopt the full SV4D 2.0 backbone for RGB generation, including its spatial, temporal, and 3D attention mechanisms, and replicate this backbone to form a second branch for part segmentation, as seen in fig. 3. + +Each branch processes half of the latent channels and shares positional embeddings (e.g., camera intrinsics and temporal indices) as input. Inspired by Vainer et al. (2024), the two branches operate independently but are connected through dedicated Bidirectional Diffusion Fusion (BiDiFuse) layers inserted at every block of the network. Given intermediate features $h^{\mathrm{RGB}}$ and $h^{\mathrm{Part}}$ at any resolution, we compute updated representations using a fusion module: + +$$ +h _ {\text {f u s e d}} ^ {\mathrm {R G B}} = h ^ {\mathrm {R G B}} + \mathcal {F} \left(\left[ h ^ {\mathrm {R G B}}, h ^ {\text {P a r t}} \right]\right), \quad h _ {\text {f u s e d}} ^ {\text {P a r t}} = h ^ {\text {P a r t}} + \mathcal {F} \left(\left[ h ^ {\mathrm {R G B}}, h ^ {\text {P a r t}} \right]\right) \tag {1} +$$ + +where $\mathcal{F}$ is a lightweight fusion function composed of two $1\times 1$ convolutions with ReLU activations. This module encourages bidirectional feature sharing while maintaining branch-specific learning. + +The forward pass proceeds as follows: the input latent is split along the channel dimension and passed through two identical UNet backbones. After each encoder block, the intermediate features are fused via BiDiFuse. The same process applies to the middle block and each decoder stage, with skip connections preserved within corresponding branches. The final outputs from both branches are separately passed through a shared VAE decoder to produce RGB and part predictions independently. + +Spatial Color Encoding. To enable decoder sharing between RGB and part branches, we represent part segmentation maps as continuous RGB-like images using a spatial color encoding scheme. To assign temporally consistent colors, we first normalize the 3D coordinates of each point on the mesh or reconstructed surface to a unit cube. Then we compute the coordinates of the 3D center of each part in the first frame and use its normalized $(x,y,z)$ coordinate as the color code for all frames and views. This ensures that the same part is assigned the same color across all frames and views, maintaining identity consistency over time. Unlike schemes that randomly assign colors to parts per iteration, our deterministic encoding significantly reduces computational overhead, as random coloring would require regenerating encoded part images at every step. Our approach enables the + +diffusion model to treat part segmentation as an image generation task, facilitating compatibility with the RGB branch and enabling unified training within a shared latent space. + +Back-Mapping from RGB Image to Part Mask. To recover discrete part masks from the generated spatial encoding, we avoid clustering the generated colors, which can be noisy. Instead, we apply SAM (Segment Anything Model) in auto-generation mode to produce per-view segmentation masks — we found this remarkably effective at providing clean candidate segments. For each segment, we then compute the mode of the underlying RGB pixel values and assign this color to the entire mask. This procedure robustly eliminates pixel-level noise and ensures clean, discrete part representations. Then we apply clustering (McInnes et al., 2017) on all images to produce part masks. We do not use SAM2 (the video tracking version of SAM) as it only supports parts that are visible in the first frame, and thus fails to capture parts that only first appear later in the video. + +Contrastive Part Consistency loss. The spatial color encoding represents parts as RGB-like images, enabling a shared encoder and decoder between both branches. However, this model lacks an explicit supervision to ensure that the same kinematic part maintains a consistent appearance across different viewpoints and time steps. Without regularization, the model may produce temporally or spatially inconsistent segmentations. To address this, we extract part-specific features by aggregating pixel-level features within each predicted part region, and project them into a shared embedding space. For each training batch, we collect a set of part features $\{f_i\}_{i=1}^N$ , where each $f_i$ corresponds to one part instance (across view and frame). Features with the same part identity but from different frames or views are considered positive pairs, while features from different parts serve as negatives. We adopt an InfoNCE-style contrastive loss defined over all part pairs (Oord et al., 2018): + +$$ +\mathcal {L} _ {\text {c o n t r a s t}} = - \mathbb {E} _ {i \in \mathcal {P}, j \in \mathcal {P} _ {i} ^ {+}} \left[ \log \frac {\exp \left(\sin \left(f _ {i} , f _ {j}\right) / \tau\right)}{\sum_ {k \in \mathcal {P} \backslash \{i \}} \exp \left(\sin \left(f _ {i} , f _ {k}\right) / \tau\right)} \right] \tag {2} +$$ + +where $\mathcal{P}$ is the set of all valid part features, $\mathcal{P}_i^+$ is the set of positive indices for part $i$ , $\mathrm{sim}(\cdot ,\cdot)$ denotes cosine similarity, and $\tau$ is a temperature hyperparameter ( $\tau = 0.07$ by default). This loss encourages the same part to be consistently encoded across views and frames, while remaining distinct from different parts. + +# 3.3 KinematicParts20K Dataset + +We curate the KinematicParts20K dataset from the ObjaverseDy++ (Yao et al., 2025) dataset to support the training and evaluation of our part-aware generation framework. We first filter objects that include rigging annotations with well-structured skeletal hierarchies and bone transformations. + +Bone Merging to Control Granularity. For objects with excessively many bones, we automatically reduce skeletal complexity. For each pair of connected bones, we compute (1) the average relative 3D displacement between the two bones across all frames, and (2) the cosine similarity between their 2D part segmentation masks based on Dino features. If both the motion difference and feature dissimilarity fall below predefined thresholds, we merge the two bones. We set an upper bound of 100 bones per object; if an object cannot be reduced to within this limit, it is discarded from the dataset. + +Multiview Rendering and Part Label Generation. For each selected object, we render 24 frames from 24 camera views uniformly distributed along a horizontal circle. We also render per-bone 2D skinning weight maps. To compute the part segmentation masks, we use a per-pixel argmax over all bone-specific weight maps within each view. The resulting part maps provide high-quality multiview kinematic part segmentation labels aligned with the rigging annotations, enabling supervised training of SP4D using part labels that reflect true kinematic decomposition. After all the filtering steps, we are left with almost 20,000 training objects. + +# 3.4 2D-to-Kinematic Mesh Generation + +Lifting from 2D. We propose a simple yet effective pipeline that converts a single image into a fully riggable 3D asset with geometry, part decomposition, and skinning weights - only missing skeletal connectivity. We first apply our Stable Part Diffusion 4D (SP4D) model to generate multi-view sequences of RGB frames and corresponding part segmentation (cleaned by SAM) from a single input image. For recovering geometry, we use Hunyuan 3D 2.0 (Zhao et al., 2025b), a state-of-the-art images-to-3D framework, to turn the multi-view RGB images generated by SP4D into textured + +Table 1: Quantitative comparison of kinematic parts on KinematicParts20K val set for multi-view (static object) and multi-frame (static camera). The Hungarian algorithm aligns predictions to the ground-truth, ignoring parts missing in the first image. SAM2* uses ground-truth point prompts per part. + +
MethodMulti-viewMulti-frame
mIoUARIF1mAccmIoUARIF1mAcc
SAM20.150.050.310.210.160.050.320.22
SAM2*0.220.080.370.260.340.160.450.34
DeepViT0.170.060.330.230.180.060.340.24
Ours w/o PCP Loss0.380.150.460.490.440.220.520.56
Ours w/o BiDiFuse0.570.510.600.620.610.580.640.68
Ours (Full)0.680.600.700.740.700.630.720.77
+ +Table 2: User study on kinematic part segmentation. Participants rated three methods on part clarity, view consistency, and rigging suitability. The study was conducted on 20 randomly selected samples from the validation set. + +
MethodOursSAM2DeepViT
Clarity4.422.132.01
Consistency4.092.001.86
Rigging4.261.751.69
Average4.261.961.85
+ +3D geometry. Once we obtain the 3D mesh, we reuse this geometry to associate texture information from both the RGB and part segmentation views separately, following Hunyuan 3D 2.0. Through HDBSCAN (McInnes et al., 2017), we assign each vertex its discrete ID for the part segmentation. + +Harmonic Skinning Weight Computation. Given the 3D part labels, we compute continuous skinning weights using harmonic field estimation. We extract the boundary $\partial \Omega_p$ of part $p$ by identifying mesh edges that connect two vertices belonging to different parts; the binary indicator function $b_{p}(x)$ indicates whether vertex $x$ belongs to part $p$ . We then solve: + +$$ +\Delta w _ {p} (x) = 0 \quad \text {f o r a l l i n t e r i o r v e r t i c e s ,} \quad \text {s u b j e c t t o} \quad w _ {p} (x) = b _ {p} (x) \text {o n} \partial \Omega_ {p} +$$ + +where $\Delta$ is the mesh Laplacian operator and $w_{p}(x)$ denotes the smooth harmonic field corresponding to part $p$ . The harmonic solution to this Laplace equation propagates part influence across the surface, yielding soft per-vertex part assignments which we interpret as skinning weights. + +# 4 Experiments + +We demonstrate that SP4D performs robustly and generalizes across a wide variety of articulated objects with diverse shapes and motions, including both synthetic models and real-world videos in fig. 4. We conduct comprehensive experiments to evaluate the effectiveness of our method, including comparisons with state-of-the-art approaches for part segmentation, as well as ablation studies on key design choices. We report both quantitative metrics (mIoU, ARI, F1 Score, mAcc) in table 1 and a user study in table 2 to assess quality from a rigging perspective. Details on implementation, datasets, training regime, evaluation protocols, metric definitions and more experiments on 3D segmentation and rigging can be found in the Appendix. + +# 4.1 Part Decomposition Comparison + +2D Part Decomposition. We compare SP4D with two representative 2D part segmentation baselines. The first is SAM2 (Ravi et al., 2024), a tracking-based method that generates part masks in the first image and propagates them to the others. We also include a stronger variant, SAM2*, where point prompts from the ground-truth part centroids are used to initialize tracking. The second baseline is DeepViT (Amir et al., 2021), an unsupervised segmentation method that leverages features from a self-supervised DINO-ViT model (Caron et al., 2021). We apply K-Means clustering on intermediate feature maps to obtain part-level masks across views. + +As shown in table 1, SP4D significantly outperforms all baselines in both multi-view and multi-frame settings. SAM2 performs poorly due to its dependency on appearance and semantic cues, which rarely align with kinematic part boundaries. SAM2*, despite being guided by ground-truth points, suffers from the same fundamental limitation. While DeepViT captures coarse semantic structures, it lacks any awareness of object articulation or motion consistency. In contrast, SP4D generates parts directly via kinematic-aware diffusion, leveraging geometry and view consistency to produce temporally stable, kinematic-aware decompositions. This advantage is further supported by fig. 5, which presents qualitative comparisons across multiple views, clearly showing SP4D's superior structural alignment. Additional user preference results in table 2 also confirm the perceptual quality of SP4D's outputs. + +![](images/f5792ef755e19ddc57a738bd05bfda63f49a220282809e973b148afb7defa7af.jpg) +Figure 4: Multi-view kinematic part video results on synthetic and real-world videos. We show qualitative results of our SP4D model on both the validation set of KinematicParts20K and real-world DAVIS videos. Each group presents two time frames across two novel views. The input video frame is noted with purple boxes. SP4D produces temporally and spatially consistent part decompositions across diverse object categories and motions. + +![](images/f221fe9fd4de4f0046c92d7f010a3050386fe491b33e01314424bbcd0c4d03de.jpg) +Figure 5: Visual comparison of part segmentation. We show results across three views for various articulated objects. The rows contain input RGB image (top), our SP4D-generated part segmentation (middle), and the SAM2 baseline (bottom). Compared to SAM2, SP4D produces more structured part decompositions that align with object articulation and are consistent across views. + +3D Part Decomposition. Recent state-of-the-art 3D part segmentation methods, such as SAMesh (Tang et al., 2024a) and SamPart3D (Yang et al., 2024), rely on 2D segmentation cues + +to supervise 3D decomposition in different ways. SAMesh fuses 2D segmentations (from SAM) of multiple rendered views using visibility-weighted voting. In contrast, SamPart3D distills dense visual features from DINOv2 into a 3D point-based backbone, and leverages SAM masks through a scale-conditioned MLP to achieve granularity-controllable part grouping via clustering. + +Despite their differences, both methods fundamentally depend on the quality of 2D segmentations. When appearance-based segmenters like SAM or DINOv2 fail to produce meaningful part boundaries—particularly for kinematic or textureless regions—the resulting 3D decomposition is unreliable and misaligned with object articulation. As illustrated in fig. 2, these approaches often struggle to produce structurally coherent part segmentation under such challenging conditions. Additional comparisons are provided in the supplementary material. + +# 4.2 Ablation Study + +We conduct ablation studies to evaluate the contribution of two core components in our framework: the BiDiFuse cross-branch fusion module and the part consistency loss. Results are reported under both multi-view and multi-frame settings (see table 1). Removing the part consistency loss leads to noticeable performance degradation, especially in terms of ARI. Without this loss, the model loses explicit guidance to maintain spatial and temporal coherence of part assignments across views or frames, resulting in fragmented or inconsistent segmentations. This highlights the importance of encouraging feature-level alignment among corresponding parts throughout the video sequence. + +Disabling the BiDiFuse module also causes substantial drops across all metrics. Since BiDiFuse facilitates bidirectional interaction between the RGB and part branches. Without it, the network lacks effective cross-modal information exchange, leading to suboptimal alignment between both branches, particularly in view-consistency and part boundary sharpness. Crucially, the segmentation branch can no longer effectively leverage the prior of the RGB model. These results confirm that both components are essential for achieving robust, consistent, and rigging-friendly part decompositions. + +# 5 Broader Societal Impact + +SP4D has the potential to substantially reduce the manual work required for rigging in animation and 3D asset production, benefiting creators in film, gaming, education, AR/VR, and robotics. Particularly for those with limited access to professional modeling pipelines, our method broadens accessibility to animation-ready assets and opens new opportunities in educational content creation, interactive media, and rapid prototyping. SP4D's ability to generalize across real-world footage and synthetic objects further supports its potential in democratizing digital content creation. + +However, the ability to synthesize from minimal visual input introduces risks. These include the creation of synthetic humans or avatars for deceptive purposes. While our method does not focus on facial reenactment or human identity synthesis, downstream misuse remains a concern. We recommend clear disclosure, attribution mechanisms for automatically generated 3D content, and ethical oversight in such applications. We emphasize that all training uses CC-licensed assets, carefully filtered to respect creator rights. + +# 6 Conclusion + +We propose SP4D to jointly generate multi-view parts video with aligned RGB frames from a monocular input video. Uniquely, we predict kinematic rather than semantic parts, based on segments in articulated motion skeletons. This closes a significant gap in the 3D generation pipeline, drastically reducing the manual annotation required to prepare the generated objects for animation. + +By leveraging a synchronized two-branch architecture, we maximally leverage the prior of the pretrained RGB model; this results in a robust and generalizable approach despite the data scarcity for training. Both a quantitative comparison with representative baselines and a user study show the clear benefit of our approach over existing semantic-oriented part segmentation for the same task. + +# References + +Shir Amir, Yossi Gandelsman, Shai Bacon, and Tali Dekel. 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Triplane meets gaussian splatting: Fast and generalizable single-view 3D reconstruction with transformers. In CVPR, 2024. +Qi Zuo, Xiaodong Gu, Lingteng Qiu, Yuan Dong, Weihao Yuan, Rui Peng, Siyu Zhu, Liefeng Bo, Zilong Dong, Qixing Huang, et al. Videomv: Consistent multi-view generation based on large video generative model. 2024. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes]. + +Justification: Any significant claim made in our manuscript is at least outlined in the abstract and motivated in the introduction. 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Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: LLMs were not involved in the core method development of our research. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Stable Part Diffusion 4D: Multi-View RGB and Kinematic Parts Video Generation Supplementary Material + +In the appendix, we provide the following supplementary materials: (1) Implementation Details, (2) our newly introduced dataset, KinematicParts20K, (3) Additional Qualitative Results, and (4) Evaluation Details. + +# A Implementation Details + +Our model is implemented by directly extending the SV4D 2.0 framework (Yao et al., 2025). We retain the original U-Net architecture, latent VAE encoding, and diffusion setup, and introduce two key modifications: (1) an architecturally identical second branch that generates part segmentation outputs jointly with the existing RGB branch, and (2) Bidirectional Diffusion Fusion (BiDiFuse) modules inserted between each corresponding layer pair to enable cross-branch feature sharing. In the first stage, the RGB branch is trained following SV4D 2.0. The training setup — including optimizer, noise schedule, loss functions, and sampling strategy — follows SV4D 2.0 exactly. We adopt the EDM (Karras et al., 2022) training framework with an L2 loss and precompute VAE latents and CLIP features for all training images to accelerate convergence. The obtained network parameters are used to initialize both the RGB and part generation branches. + +We train the full SP4D model with BiDiFuse and our proposed contrastive part consistency loss on the KinematicParts20K dataset (as discussed below) for 40K iterations. Training is performed on 32 NVIDIA H100 GPUs with an effective batch size of 32, using 12 views and 4 frames per object sampled from the rendered dataset. + +# B KinematicParts20K Dataset + +Our dataset is constructed by further filtering the SV4D 2.0 dataset, which is based on CC-licensed dynamic 3D assets from Objaverse and ObjaverseXL. We select only objects that contain rigging annotations, including bone hierarchies and skinning weights. To mitigate overly fine-grained or noisy bone structures, we apply a bone merging procedure based on two criteria: (1) the relative transformation between connected bones across all frames, and (2) the similarity of their projected part appearance in 2D using DINO features. Bone pairs with low motion discrepancy and high appearance similarity are merged. Objects with more than 100 bones after merging are discarded. + +All objects are scaled to unit bounding boxes and rendered at $576 \times 576$ resolution using Blender's Cycles renderer under a curated set of HDRI environment maps. We adopt orbit rendering with 24 azimuthal views and 24 video frames per object. In addition to RGB, we simultaneously render per-bone skinning weight maps. For each view and frame, we generate pixel-wise part segmentation labels by taking the argmax over the bone-specific skinning maps, resulting in multi-view, multi-frame kinematic part masks for supervision. + +# C More Qualitative Results + +We show fixed-view cross-frame part tracking, fixed-frame cross-view part tracking, 3D decomposition, rigging, and animation results for synthetic data, real-world data, and zero-shot generated data. Please refer to the summary video in the supplementary material. + +Evaluation on 3D Segmentation. It is also important to position SP4D in the broader context of 3D kinematic segmentation rather than only comparing against 2D segmentation baselines. State-of-the-art 3D segmentation methods, such as Segment Anything Mesh Tang et al. (2024a) and SAMPart3D Yang et al. (2024), are built upon 2D semantic segmentation backbones (e.g., SAM, DINOv2) that are primarily texture- or appearance-driven, and thus not explicitly designed for kinematic reasoning. This limitation is evident in our visual comparisons (Figure.2), where appearance-based cues alone fail to recover accurate part structures for novel or textureless objects. + +To quantitatively assess this gap, we conduct a comprehensive evaluation on the KinematicParts20K test set using these SOTA 3D segmentation baselines Tang et al. (2024a); Yang et al. (2024). We report mean Intersection-over-Union (mIoU), Adjusted Rand Index (ARI), F1 score, mean Accuracy (mAcc), and User Study ratings (following the evaluation criteria in Supplementary Section D.2). As shown in Table 3, SP4D substantially outperforms the baselines across all metrics, highlighting its capability to capture kinematic structure rather than relying solely on appearance cues. + +Table 3: Comparison of SP4D with SOTA 3D segmentation methods on KinematicParts20K-test. SP4D achieves significantly higher scores across all metrics, indicating stronger kinematic reasoning capabilities. + +
MethodmIoUARIF1mAccUser Study
Segment Any Mesh0.150.060.290.201.98
SAMPart3D0.130.050.270.181.75
Ours (Full)0.640.580.670.724.13
+ +Evaluation beyond segmentation accuracy. To further assess the usefulness of our kinematic representation beyond segmentation accuracy, we conduct additional experiments on rigging precision and animation plausibility. (i) Rigging precision. We evaluate the predicted skinning weights on the KinematicParts20K-test split, which contains ground-truth rigging annotations. We compare SP4D against two state-of-the-art auto-rigging methods Song et al. (2025); Zhang et al. (2025), reporting precision scores in Table 4. SP4D achieves the highest precision (72.7), outperforming Magic Articulate (63.7) and UniRig (64.3), demonstrating the accuracy of our learned kinematic decomposition when ground-truth supervision is available. (ii) Animation plausibility for generated objects. For generated meshes (e.g., dinosaurs, robotic arms) without ground-truth rigging, we conduct a user study to evaluate animation plausibility. Participants were shown animations produced by SP4D and the SOTA baselines Song et al. (2025); Zhang et al. (2025), and asked to rate the plausibility on a 1–5 Likert scale. SP4D achieves a significantly higher score (4.1) than Magic Articulate (2.7) and UniRig (2.3), confirming better generalization to unseen object categories and poses. + +Notably, as shown in Figure 2 (middle), Magic Articulate, despite being trained on large-scale rigged meshes from Articulation-XL, performs well on seen categories but struggles with unusual generated shapes. In contrast, SP4D leverages strong priors from a 2D diffusion model and learns kinematic decomposition robustly, enabling accurate rigging for both real-world and synthetic objects. This highlights a key motivation for our approach: learning kinematic structure from 2D multi-view supervision yields superior generalization to novel inputs. + +Table 4: Comparison of SP4D with SOTA Auto-rigging Methods. + +
MethodKinematicPart20K-testGenerated Objects
PrecisionUser StudyUser Study
Magic Articulate63.73.82.7
UniRig64.33.92.3
Ours (Full)72.74.34.1
+ +# D Evaluation Details + +# D.1 Quantitative Metrics. + +To evaluate the quality of kinematic part decomposition across multi-view and multi-frame settings, we report four standard metrics. Since the predicted part masks are label-free, we apply the Hungarian algorithm to align predicted and ground-truth parts based on respective IoU, for those metrics which require correspondences. The following metrics are computed: + +- mIoU - Mean intersection-over-union across matched part masks. +- ARI - Adjusted Rand Index, which captures clustering similarity independent of label permutation. + +![](images/079dd7932e529fee9664be463ecaf5e90d24e2233ff4a932b13fa9462c4b4055.jpg) +Figure 6: User study interface for evaluating multi-view kinematic part segmentation. Participants are presented with video results generated by different methods and asked to rank them based on part consistency, structural correctness, and motion coherence. The study compares SP4D with baseline methods to assess perceptual quality and kinematic alignment. + +![](images/1b748d3034857e7d48f6e8070dc5ef80c833fa968c180283d36b025ed485940a.jpg) + +![](images/0fc6db22ee0346cb51971fa838055e6bbd9ba169a8a565be2d2291f186fbf18f.jpg) + +- F1 Score - The harmonic mean of precision and recall, reflecting pixel-level agreement. +- mAcc - Mean class-wise accuracy, indicating the average recall per ground-truth part. + +# D.2 User Study on Multi-view Kinematic Part Decomposition for Rigging + +To evaluate the practical utility of different multi-view kinematic part decomposition methods for rigging tasks, we conducted a user study focusing on the perceived quality of part segmentation from a rigging perspective. These decompositions aim to separate articulated object parts (e.g., limbs, joints, head) to facilitate automatic or semi-automatic skeleton binding and animation control. + +Study Setup. We randomly selected 20 sets of decomposition results, each containing visualizations from different methods applied to the same object. For each set, we generated animated GIFs showing the part decomposition from multiple viewpoints, allowing participants to better understand spatial consistency and articulation structure. All visualizations were presented anonymously to avoid bias. The study was conducted via a Google Form and received responses from 20 participants. + +Evaluation Criteria. Participants were instructed to rate each method based on the following three criteria: + +- Clarity of part boundaries - Are the decomposed part regions cleanly separated with well-defined borders? +- Consistency across views - Do the decomposed parts remain stable and coherent when viewed from different angles? +- Suitability for rigging/animation - Are the decomposed parts appropriate for assigning joints and performing realistic articulated motion? + +Each criterion was rated on a scale from 1 (Poor) to 5 (Excellent). + +Goal. The objective of this study is to assess the effectiveness of part decomposition methods in real-world rigging scenarios, providing insight into their strengths and limitations for downstream animation applications. + +# E Additional Discussion + +Dual-branch architecture design. We investigate both single-branch and dual-branch architectures for jointly predicting multi-view RGB sequences and kinematic part sequences. In the single-branch variant, the two modalities are concatenated into a shared latent representation and split prior to decoding. This configuration exhibits slower convergence and lower performance than the dual-branch counterpart under the same training schedule. We attribute this to the fundamentally different nature of the tasks: RGB synthesis focuses on high-frequency appearance modeling, whereas kinematic part segmentation emphasizes structural reasoning and temporal-spatial consistency. A single-branch network that forces both tasks to share all intermediate features is prone to cross-task interference, particularly degrading consistency in multi-frame outputs. + +Our BiDiFuse dual-branch UNet addresses this issue by maintaining task-specific feature streams while enabling bidirectional cross-modal feature exchange. This architecture preserves modality-specific learning, reduces interference, and improves overall performance. We include a detailed discussion of this design choice and relate it to recent work on unified dense prediction in video diffusion models Yang et al. (2025a). + +Avoiding category-specific pose or rigging priors. While 2D/3D pose and rigging priors encode rich kinematic information, SP4D is deliberately designed without reliance on human- or animal-specific templates to ensure category-agnostic applicability. Template-based approaches often generalize poorly to objects with unconventional topology, such as loose clothing, handheld tools (e.g., shields, skis), or non-biological categories (e.g., crabs, robotic arms, mechanical assemblies). In our experiments, these priors frequently failed to produce meaningful part structures for such diverse object types. + +By contrast, SP4D learns kinematic decomposition directly from 2D multi-view supervision, without assuming a fixed skeleton topology, enabling robust generalization to both natural and synthetic domains. Nevertheless, integrating lightweight 2D/3D structural priors into diffusion-based generation remains an interesting direction for future research. + +Relation to optimization-based methods. We also relate SP4D to prior part-aware rendering approaches Zhang et al. (b,a); Yang et al. (2022); Noguchi et al. (2022); Yang et al. (2021). These methods are typically optimization-based pipelines for per-instance 3D reconstruction from multiview videos, often requiring ground-truth camera poses, complete multi-view coverage of the same object, and extensive per-instance optimization (e.g., $48+$ GPU hours on an A100). Their kinematic reasoning is constrained by the motion observed in the input video; for example, if a limb remains static throughout, the model may fail to segment it. Such methods are not designed for category-level generalization and cannot perform feedforward inference from monocular inputs such as a single image or single-view video. + +In contrast, SP4D is a feedforward, category-agnostic generative model capable of producing consistent RGB renderings and kinematic part decompositions within seconds, given only a single image or video. While both SP4D and part-aware rendering approaches can output kinematic segmentations, they address fundamentally different problem settings and exhibit markedly different capabilities. + +Limitations and future work. Our method inherits the camera parameterization design from SV4D 2.0 Yao et al. (2025), which models only azimuth and elevation with a simple lens model. This limits our ability to handle videos with strong perspective distortion or complex camera trajectories. Moreover, SP4D is primarily trained under the assumption that each scene contains a single object. In scenarios where multiple objects appear simultaneously, the model may struggle to handle all of them at once. Extending SP4D to support full 6-DoF camera motion and multi-object scenarios remains a promising direction for future research. 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Roth, Dominik K. Klein, Maximilian Kannapinn, Jan Peters, Oliver Weeger + +Technical University of Darmstadt, Darmstadt, Germany {roth,klein,kannapinn,weeger}@cps.tu-darmstadt.de, peters@ias.tu-darmstadt.de + +# Abstract + +In recent years, nonlinear dynamic system identification using artificial neural networks has garnered attention due to its broad potential applications across science and engineering. However, purely data-driven approaches often struggle with extrapolation and may yield physically implausible forecasts. Furthermore, the learned dynamics can exhibit instabilities, making it difficult to apply such models safely and robustly. This article introduces stable port-Hamiltonian neural networks, a machine learning architecture that incorporates physical biases of energy conservation and dissipation while ensuring global Lyapunov stability of the learned dynamics. Through illustrative and real-world examples, we demonstrate that these strong inductive biases facilitate robust learning of stable dynamics from sparse data, while avoiding instability and surpassing purely data-driven approaches in accuracy and physically meaningful generalization. Furthermore, the model's applicability and potential for data-driven surrogate modeling are showcased on multiphysics simulation data. + +# 1 Introduction + +Recent years have seen a strongly growing interest in using machine learning (ML) to identify the dynamics of physical systems directly from observations [5, 28, 31, 32, 40, 42, 53]. This approach has diverse applications, ranging from modeling phenomena with partially or entirely unknown underlying physics to creating surrogate and reduced-order models for rapid simulations and digital twins [23, 53, 54]. In the latter application, data-driven methods can address the drawbacks of traditional numerical methods, which often require tremendous computational resources and expertise [1, 53]. However, in turn, classical ML algorithms typically require large amounts of training data and lack robustness, as they often suffer from poor extrapolation and are prone to making physically implausible forecasts [1, 33, 53, 54]. To overcome these limitations, a general consensus is to avoid discarding centuries' worth of established knowledge in physics, which is an implicit consequence of relying solely on data-driven methods. Instead, there is a growing emphasis on incorporating physical priors and inductive biases into ML approaches [33, 54]. This physics-guided ML promises improved accuracy, robustness, and interpretability while reducing data requirements [33, 53]. + +Several physics-guided architectures have been proposed for learning dynamical systems. These approaches often parameterize an ordinary differential equation (ODE) to identify continuous-time dynamics. To ensure adherence to the laws of thermodynamics, many architectures build upon frameworks such as Hamiltonian mechanics [9, 16, 20], Lagrangian mechanics [10, 34], Poisson systems [19], GENERIC [18, 56], port-Hamiltonian systems (PHS) [11, 13, 37, 38, 57], or the generalized Onsager principle [55]. The latter three allow for the modeling of dissipative dynamics, while the former are generally limited to energy-conserving systems. Additionally, PHS explicitly incorporate external excitations, making them particularly suited for modeling controlled systems. + +The mentioned physical modeling frameworks have implications for the stability of the learned dynamics. For example, in Hamiltonian neural networks, stable equilibria correspond to extrema in + +the Hamiltonian energy function. However, these effects are often not discussed in the referenced works. Nevertheless, stability is an important and fundamental property of any dynamic system. Guarantees on the stability of the dynamics may improve the model's robustness, help trust their predictions, and can themselves be viewed as physically motivated biases [14]. + +Given the importance of the stability property, multiple other works have proposed methods to enhance the stability of learned dynamics, often using neural ordinary differential equations (NODEs) [7] as a basis. Massaroli et al. [36] propose a provably stable NODE variant that guarantees the existence of locally asymptotically stable equilibria. Multiple studies have combined Lyapunov's theory with neural networks (NNs) to guarantee the stability of discrete-time dynamics [14, 29] or to estimate the region of attraction for given dynamical systems [3, 44]. Kolter and Manek [27] describe another Lyapunov-stable architecture based on NODEs. They concurrently learn unconstrained dynamics with a NODE and a convex Lyapunov function. Global stability around an equilibrium point is then ensured by projecting the NODE-dynamics onto the space of stable dynamics as given by the Lyapunov function. Takeishi and Kawahara [50] extend this approach from a stable equilibrium to stable invariant sets. Similar projection-based techniques have also been proposed to learn input-output stable dynamics [26] and dissipative systems [39]. However, these projection methods can introduce discontinuities, hindering training strategies that rely on integrating trajectories rather than matching state-derivative pairs [46]. Other approaches employ physical modeling frameworks to formulate the learning problem directly in the subspace of stable dynamics. Rettberg et al. [43] propose identifying a linear latent PHS, which makes enforcing global stability relatively straightforward. Yu et al. [55] propose a model based on the generalized Onsager principle that can guarantee bounded solutions and the existence of locally asymptotically stable equilibria. While their evolution equation is similar to PHS, their approach does not consider arbitrary, time-dependent external input signals. + +This work exploits the connections between stability and physical frameworks and proposes stable port-Hamiltonian neural network (sPHNN) for learning nonlinear and globally stable dynamical systems. The method is projection-free and can incorporate input signals. We show that the approach is capable of learning equilibrium points, though our numerical evaluations focus on practical applications where the equilibrium is known a priori. In such cases, this prior knowledge can be seamlessly integrated into the proposed architecture, improving prediction accuracy while simultaneously reducing data requirements. + +# 2 Background + +# 2.1 Stability of dynamical systems + +We consider autonomous dynamical systems as first-order ODEs, where the state $\boldsymbol{x}(t) \in \mathbb{R}^n$ evolves according to a vector-valued function $\boldsymbol{f}: \mathbb{R}^n \to \mathbb{R}^n$ : + +$$ +\dot {\boldsymbol {x}} (t) = \boldsymbol {f} (\boldsymbol {x} (t)). \tag {1} +$$ + +Suppose $\pmb{f}$ is sufficiently smooth to ensure a unique solution $\pmb{x}(t)$ for any initial condition $\pmb{x}(t_0)$ . A solution $\hat{\pmb{x}}(t)$ is stable in the sense of Lyapunov if, for any small perturbation in the initial condition, the perturbed solution $\bar{\pmb{x}}(t)$ remains close for all times, that is, if for any $\epsilon > 0$ , there exists a $\delta > 0$ such that $\| \hat{\pmb{x}}(t_0) - \bar{\pmb{x}}(t_0) \| < \delta$ implies $\| \hat{\pmb{x}}(t) - \bar{\pmb{x}}(t) \| < \epsilon$ for all $t > t_0$ [35, 52]. A stronger version of stability, namely asymptotic stability, is obtained if additionally the perturbed solution $\bar{\pmb{x}}(t)$ converges to $\hat{\pmb{x}}(t)$ for $t \to \infty$ for initial conditions $\| \hat{\pmb{x}}(t_0) - \bar{\pmb{x}}(t_0) \| < \Delta$ with some $\Delta > 0$ [52]. + +Lyapunov stability is a local property, in the sense that for sufficiently large perturbations, the perturbed solution may stray arbitrarily far even from asymptotically stable solutions [49]. Quantifying the permissible extent of perturbations on the initial condition such that the perturbed solution still converges to the unperturbed solution leads to the concept of a basin of attraction [30]. Asymptotically stable equilibria, for which the basin of attraction extends to the entire phase space, are called globally asymptotically stable [30]. A system with a globally stable equilibrium cannot have any other equilibria, as their existence would imply that there are solutions (namely, the other equilibria) that do not converge to the globally stable equilibrium. + +In the following, we consider the case of an equilibrium solution $\pmb{x}(t) = \mathbf{0}$ , i.e., $\pmb{f}(\mathbf{0}) = \mathbf{0}$ . Placing the equilibrium at the origin is done for ease of notation and is without loss of generality, as arbitrary equilibria can be translated to the origin with a time-independent transformation. The stability of the + +equilibrium can be shown through the use of Lyapunov's stability theory by finding a suitable scalar Lyapunov function $V$ [24, 52]. Let $V: D \to \mathbb{R}$ be continuously differentiable and positive definite, that is, $V(\mathbf{0}) = 0$ and $V(\mathbf{x}) > 0$ in $D \setminus \{\mathbf{0}\}$ , where $D \subseteq \mathbb{R}^n$ is a neighborhood of the origin. If the value of $V$ is nonincreasing along trajectories of the system in $D$ , then the equilibrium is stable in the sense of Lyapunov. Mathematically, this is the case if $\dot{V}(\mathbf{x})$ is negative semi-definite: + +$$ +\dot {V} (\boldsymbol {x}) = \frac {\partial V}{\partial \boldsymbol {x}} ^ {\textsf {T}} \dot {\boldsymbol {x}} = \frac {\partial V}{\partial \boldsymbol {x}} ^ {\textsf {T}} \boldsymbol {f} (\boldsymbol {x}) \leq 0 \quad \forall \boldsymbol {x} \in D \setminus \{\mathbf {0} \}. \tag {2} +$$ + +In the case of a negative definiteness $(\dot{V}(\pmb{x}) < 0$ in $D \setminus \{\mathbf{0}\}$ ), asymptotic stability can be concluded. To obtain global asymptotic stability, these conditions must be satisfied globally, that is, for all $\pmb{x} \in D = \mathbb{R}^n$ , and the Lyapunov function must be radially unbounded, i.e., $V(\pmb{x}) \to \infty$ for $\| \pmb{x} \| \to \infty$ [24]. The latter condition avoids the existence of solutions that do not converge to the equilibrium by diverging to infinity without violating the decrease condition in Equation (2). + +# 2.2 Port-Hamiltonian systems + +PHS theory provides a framework for modeling physical systems that generalizes the underlying mathematical structure of Hamiltonian dynamics. It incorporates the modeling of energy-dissipating elements, which are typically absent in classical Hamiltonian systems [51]. Additionally, PHS theory has roots in control theory, emphasizing the interactions of dynamic systems with their environment. In this work, we consider PHS of the form + +$$ +\dot {\boldsymbol {x}} = \left[ \boldsymbol {J} (\boldsymbol {x}) - \boldsymbol {R} (\boldsymbol {x}) \right] \frac {\partial \mathcal {H}}{\partial \boldsymbol {x}} (\boldsymbol {x}) + \boldsymbol {G} (\boldsymbol {x}) \boldsymbol {u} (t). \tag {3} +$$ + +Here, $\pmb{x}(t) \in \mathbb{R}^n$ describes the system's state, $\pmb{u}(t) \in \mathbb{R}^m$ its input, and $\mathcal{H}: \mathbb{R}^n \to \mathbb{R}$ denotes the Hamiltonian, which typically represents the total energy of the system [4, 51]. The skew-symmetric structure matrix $\pmb{J}: \mathbb{R}^n \to \mathbb{R}^{n \times n}$ , $\pmb{J}(\pmb{x}) = -\pmb{J}^\intercal(\pmb{x})$ describes the conservative energy flux within the system. The symmetric positive semi-definite matrix $\pmb{R}: \mathbb{R}^n \to \mathbb{R}^{n \times n}$ , $\pmb{R}(\pmb{x}) = \pmb{R}^\intercal(\pmb{x}) \succeq 0$ is called dissipation matrix and describes the energy losses. Finally, the input matrix $\pmb{G}: \mathbb{R}^n \to \mathbb{R}^{n \times m}$ describes how energy enters and exits the system via the inputs. PHS extend the energy conservation of Hamiltonian systems to the energy dissipation inequality: + +$$ +\dot {\mathcal {H}} = \underbrace {\frac {\partial \mathcal {H} ^ {\intercal} \boldsymbol {J}}{\partial \boldsymbol {x}} \frac {\partial \mathcal {H}}{\partial \boldsymbol {x}}} _ {= 0} - \underbrace {\frac {\partial \mathcal {H} ^ {\intercal} \boldsymbol {R}}{\partial \boldsymbol {x}} \frac {\partial \mathcal {H}}{\partial \boldsymbol {x}}} _ {\geq 0} + \underbrace {\frac {\partial \mathcal {H} ^ {\intercal} \boldsymbol {G} \boldsymbol {u} (t)}{\partial \boldsymbol {x}} \leq s (\boldsymbol {x} , \boldsymbol {u})}. \tag {4} +$$ + +In the language of system theory, the system is dissipative with respect to the supply rate $s(\pmb{x}, \pmb{u})$ . The formal definition of dissipativity requires $\mathcal{H} \geq 0$ , which will be ensured later. Since the supply rate $s(\pmb{x}, \pmb{u})$ is linear in the input $\pmb{u}$ , it vanishes for zero-input $s(\pmb{x}, \pmb{0}) = 0$ . The inequality thus ensures that the system cannot gain energy in the absence of an external excitation $(\pmb{u} = \pmb{0})$ . Besides this physical motivation, the dissipativity of PHS plays a crucial role in stability analysis. For the unforced system with $s = 0$ , the Hamiltonian inherently fulfills the decrease condition $\dot{\mathcal{H}} \leq 0$ . Thus, by using $\mathcal{H}$ as a Lyapunov function, it can be shown that any strict minimum in the Hamiltonian implies a stable equilibrium of the unforced dynamics [51]. Furthermore, the equilibrium is asymptotically stable if the system continuously dissipates energy $(R > 0)$ for states near the minimum. + +The use of the port-Hamiltonian formulation as a basis for an ML model has multiple favorable consequences. When learning system dynamics from data, the structure of Equation (3) provides both a physical bias and the opportunity for interpreting the resulting trained model, while enabling the modeling of a wide range of relevant systems. Equation (4) highlights the distinctive roles played by the three matrix-valued functions. The structure matrix $J(\boldsymbol{x})$ results in conservative dynamics, the dissipation matrix $R(\boldsymbol{x})$ contributes only to the dissipative dynamics, and the input matrix $G(\boldsymbol{x})$ captures the energy flow over the system boundaries due to the inputs. This knowledge improves the interpretability of the trained model components. Furthermore, it allows new ways of incorporating prior knowledge into the system. For example, the learned system can be constrained to be conservative by fixing $R = 0$ . Furthermore, the connection of PHS to the decrease condition from Lyapunov's theory provides the opportunity to introduce further desirable stability constraints. + +To illustrate the above concepts with a simple example, consider the dynamics of a three-dimensional spinning rigid body described by Euler's rotation equations. Expressing the angular velocity vector + +![](images/4143972a4d819688d9351e8da1f3a6dc26ef0af78a37ea9f26a5916222f5c989.jpg) +Figure 1. Spinning rigid body: Interquartile mean (lines) and range (shaded regions) of the energy $E$ computed from the predicted states. Left: Models without stability bias; Right: Models with stability bias. + +$\omega(t) \in \mathbb{R}^3$ in the principal-axis coordinate frame, the evolution equation is given by + +$$ +\boldsymbol {I} \dot {\boldsymbol {\omega}} + \boldsymbol {\omega} \times (\boldsymbol {I} \boldsymbol {\omega}) = - \mu \boldsymbol {\omega} \quad \forall t > 0. \tag {5} +$$ + +Here, $I = \mathrm{diag}(I_1, I_2, I_3)$ denotes the inertia matrix and $\mu$ is a damping coefficient. By introducing + +$$ +\mathcal {H} (\omega) = \frac {1}{2} I _ {1} \omega_ {1} ^ {2} + \frac {1}{2} I _ {2} \omega_ {2} ^ {2} + \frac {1}{2} I _ {3} \omega_ {3} ^ {2}, \tag {6} +$$ + +which describes the kinetic energy $E$ , the system can be rewritten in the form of Equation (3), with state-dependent structure matrix $J$ , constant dissipation matrix $R$ , and vanishing input matrix $G$ : + +$$ +\left[ \begin{array}{l} \dot {\omega} _ {1} \\ \dot {\omega} _ {2} \\ \dot {\omega} _ {3} \end{array} \right] = \left(\underbrace {\left[ \begin{array}{c c c} 0 & - \frac {I _ {3}}{I _ {1} I _ {2}} \omega_ {3} & \frac {I _ {2}}{I _ {3} I _ {1}} \omega_ {2} \\ \frac {I _ {3}}{I _ {1} I _ {2}} \omega_ {3} & 0 & - \frac {I _ {1}}{I _ {2} I _ {3}} \omega_ {1} \\ - \frac {I _ {2}}{I _ {3} I _ {1}} \omega_ {2} & \frac {I _ {1}}{I _ {2} I _ {3}} \omega_ {1} & 0 \end{array} \right]} _ {J (\boldsymbol {\omega})} - \underbrace {\mu \left[ \begin{array}{c c c} \frac {1}{I _ {1} ^ {2}} & 0 & 0 \\ 0 & \frac {1}{I _ {2} ^ {2}} & 0 \\ 0 & 0 & \frac {1}{I _ {3} ^ {2}} \end{array} \right]}\right) \frac {\partial \mathcal {H}}{\partial \boldsymbol {\omega}}. \tag {7} +$$ + +In anticipation of the numerical experiment in Section 4.1, Figure 1 shows the predicted energy $E$ from various models trained on trajectories from the spinning body system. Those incorporating both, physical biases from the PHS-formulation and stability (bPHNN, sPHNN, and sPHNN-LM) perform best, highlighting the importance of these constraints for learning system dynamics. + +# 3 Stable port-Hamiltonian neural networks + +Our goal is to learn the dynamics of physical systems in the form of Equation (3) with desirable stability properties from observations. To this end, we parameterize the components $\mathcal{H}, J, R$ and $G$ of PHS with NNs. + +# 3.1 Stability of port-Hamiltonian dynamics + +To guarantee the existence of stable equilibria of port-Hamiltonian systems, it is sufficient to enforce the existence of strict local minima in the Hamiltonian. However, these stability implications are only local, meaning the size and shape of the corresponding basin of attraction are not constrained by Equation (3). Small perturbations might be enough to stray far from the stable equilibria. Furthermore, the general PHS can represent dynamics such as unbounded trajectories, which must be considered unstable from a practical standpoint. The following describes the necessary requirements for achieving global stability. Essentially, the Hamiltonian $\mathcal{H}$ is required to be a suitable Lyapunov function, which includes positive definiteness and radial unboundedness. These properties are achieved by constraining the Hamiltonian to be convex and ensuring the existence of a strict minimum. This approach is similar to the construction of a Lyapunov function by Kolter and Manek [27], but differs in the way the minimum is enforced. + +Theorem 3.1. Consider the PHS Equation (3) in the unforced case $\pmb{u}(t) = \mathbf{0}$ : + +$$ +\dot {\boldsymbol {x}} = \left[ \boldsymbol {J} (\boldsymbol {x}) - \boldsymbol {R} (\boldsymbol {x}) \right] \frac {\partial \mathcal {H}}{\partial \boldsymbol {x}} (\boldsymbol {x}), \quad \text {w i t h} \quad \boldsymbol {J} = - \boldsymbol {J} ^ {\intercal}, \boldsymbol {R} = \boldsymbol {R} ^ {\intercal} \succeq 0. \tag {8} +$$ + +Suppose the Hamiltonian $\mathcal{H}(\pmb{x})$ is convex, twice continuously differentiable, and fulfills: + +$$ +\mathcal {H} (\mathbf {0}) = 0, \quad \left. \frac {\partial \mathcal {H}}{\partial \boldsymbol {x}} \right| _ {\boldsymbol {x} = \boldsymbol {0}} = \boldsymbol {0}, \quad \left. \frac {\partial^ {2} \mathcal {H}}{\partial \boldsymbol {x} \partial \boldsymbol {x}} \right| _ {\boldsymbol {x} = \boldsymbol {0}} \succ 0. \tag {9} +$$ + +Then, $\mathcal{H}$ is a suitable Lyapunov function for showing stability of the equilibrium at $\pmb{x}(t) = \mathbf{0}$ , and all solutions are bounded. Furthermore, the equilibrium is globally asymptotically stable if $\pmb{R}(\pmb{x}) \succ 0$ . + +A proof of Theorem 3.1 is provided in Appendix A. However, its implications also follow a clear physical intuition: If a dynamical system continuously loses energy via internal dissipation $(\pmb{R} \succ 0)$ , it will eventually settle into an energy minimum corresponding to an asymptotically stable equilibrium. The convexity requirement, together with Equation (9), guarantees the existence of a unique strict and global minimum at the origin. Additionally, it ensures that the energy for states at infinity is unbounded, i.e., $\mathcal{H}(\pmb{x}) \to \infty$ as $\| \pmb{x} \| \to \infty$ . Thus, no finite state has more energy than a state at infinity, and all trajectories must be bounded. As a result, all solutions eventually converge to the global energy minimum. While convexity is sufficient to ensure global asymptotic stability, it is not a necessary condition. To enhance model expressiveness when needed, techniques such as input warping [27] can be employed, but were not required in the applications considered in this work. + +For conservative systems with $R = 0$ , energy is not dissipated, and attractive equilibria cannot exist. Therefore, a conservative system cannot possess asymptotically stable equilibria. Nevertheless, the minimum in the energy still represents a stable equilibrium, and all trajectories stay bounded. The mixed case, where $R \succeq 0$ , only allows the same conclusions as the conservative case. However, unless the dissipation matrix vanishes in a large region in phase space, it is unlikely that a system with $R \succeq 0$ is not globally asymptotically stable. + +# 3.2 Neural network model architecture + +Having established the theoretical requirements for stability of PHS, we now outline how these conditions can be met with a dedicated NN model. We call the resulting architecture sPHNN. Its computation graph is illustrated in Figure 2a. Figure 2b shows the decomposition of a sPHNN with randomly initialized NNs into conservative and dissipative dynamics, highlighting the model's physical bias and inherent interpretability. + +Hamiltonian To fulfill the constraints of Theorem 3.1, the Hamiltonian $\mathcal{H}(\pmb{x})$ is represented by a fully input convex neural network (FICNN) [2]. FICNNs can approximate any convex function [8]. They are defined by the recurrence relation + +$$ +\boldsymbol {z} _ {1} = \sigma_ {0} \left(\boldsymbol {W} _ {0} \boldsymbol {x} + \boldsymbol {b} _ {0}\right), +$$ + +$$ +\boldsymbol {z} _ {i + 1} = \sigma_ {i} \left(\boldsymbol {U} _ {i} \boldsymbol {z} _ {i} + \boldsymbol {W} _ {i} \boldsymbol {x} + \boldsymbol {b} _ {i}\right), i = 1, 2, \dots , k - 1, \tag {10} +$$ + +$$ +f (\boldsymbol {x}) = z _ {k}, +$$ + +with the pass-through weight matrices $\mathbf{W}_i$ that map the input $\mathbf{x}$ directly to the activation of the $k - 1$ hidden layers, bias vectors $\mathbf{b}_i$ , and layer outputs $\mathbf{z}_i$ . If the weight matrices $\mathbf{U}_i$ have non-negative + +![](images/9bca18be12e2773470768c0b48b7dc27fb77581bcf5a74784345ded00bd4ca2f.jpg) +(a) Computation graph. + +![](images/829f9a6b06c3462b259ff3ba505364b63e8f6e740850fc549815c40d0896360e.jpg) +(b) Composition of the total dynamics. +Figure 2. sPHNN model architecture: (a) Computation graph of sPHNN. The FICNN parameterizing $f$ is normalized to obtain $\mathcal{H}$ and the outputs of the FFNNs for $J, L$ , and $G$ are reshaped to be skew-symmetric, lower triangular, and rectangular matrices, respectively. (b) Dynamics of a randomly initialized sPHNN for $n = 2$ , showing the built-in interpretability through the separation into conservative and dissipative dynamics. + +components, the first activation function $\sigma_0$ is convex, and all subsequent $\sigma_{i}$ are convex and nondecreasing, then the output $f(x)$ is convex in the input $x$ . Since the Hamiltonian should be twice continuously differentiable, the same must hold for the activation functions. One suitable choice is the softplus activation. The conditions for stability in Equation (9) are then fulfilled by normalizing the FICNN $f$ : + +$$ +\mathcal {H} (\boldsymbol {x}) = f (\boldsymbol {x}) \underbrace {- f \left(\boldsymbol {x} ^ {*}\right) - \frac {\partial f}{\partial \boldsymbol {x}} \Big | _ {\boldsymbol {x} ^ {*}} ^ {\intercal} \left(\boldsymbol {x} - \boldsymbol {x} ^ {*}\right)} _ {f _ {\text {n o r m}} \left(\boldsymbol {x}, \boldsymbol {x} ^ {*}\right)} + \underbrace {\epsilon \left\| \boldsymbol {x} - \boldsymbol {x} ^ {*} \right\| ^ {2}} _ {f _ {\text {r e g}} \left(\boldsymbol {x}, \boldsymbol {x} ^ {*}\right)}. \tag {11} +$$ + +Consequently, the Hamiltonian $\mathcal{H}$ and its gradient vanish at $x^{*}$ due to the normalization term $f_{\mathrm{norm}}$ . Moreover, the regularization term $f_{\mathrm{reg}}$ with an arbitrarily small $\epsilon > 0$ ensures positive definiteness of the Hamiltonian's Hessian. However, in practice, $f_{\mathrm{reg}}$ can be omitted since the FICNN provides enough bias towards local strict convexity. After training, the stability guarantee can then be recovered by verifying the Hamiltonian's positive definiteness at $x^{*}$ . Choosing $x^{*} = 0$ fulfills all requirements of Theorem 3.1. However, the presented approach is more general. As the assumption that the equilibrium is located at the origin is only made for ease of notation, the normalization approach in Equation (11) can position the stable equilibrium anywhere in phase space and is not restricted to the origin. If the true equilibrium position is known beforehand, fixing $x^{*}$ introduces an additional inductive bias. While this information can often be obtained from prior physical knowledge, it may not always be available. In such cases, the equilibrium can be inferred directly from data by treating $x^{*}$ as a trainable parameter during optimization. + +Structure, dissipation and input matrices The matrix-valued functions $J(x)$ , $R(x)$ and $G(x)$ are parameterized by feedforward neural networks (FFNNs), where the outputs are suitably reshaped into matrices. As $J$ is constrained to be skew-symmetric, the respective FFNN's output vector is mapped to the space of skew-symmetric matrices via a linear transformation. Similarly, the positive definiteness (or semi-definiteness) of $R$ is ensured via the Cholesky factorization $R = LL^{\top}$ , where the reshaped FFNN output $L$ is a lower triangular matrix with positive (or non-negative) elements along the main diagonal. For some applications, $J$ , $R$ and $G$ are independent of $x$ , and thus, learning constant matrices is sufficient. In particular, this is the case when the conservative dynamics of the system under consideration are Hamiltonian in the given coordinates. Furthermore, by fixing $J$ to the symplectic matrix (and choosing $R = G = 0$ ), the sPHNN can be reduced to a Hamiltonian NN. + +# 3.3 Training stable port-Hamiltonian neural networks + +There are two fundamentally different approaches for training time-continuous dynamic models such as NODEs and sPHNNs: derivative and trajectory fitting. The former directly compares the predicted state derivatives $\dot{\pmb{x}}$ from Equation (3) to the true ones. While this approach is efficient, it requires the true time derivatives to be available as training data, which is typically not the case for real-world measurement data. Furthermore, derivative fitting is not applicable when augmented states are used [12], i.e., when $\pmb{x}$ is padded with additional dimensions compared to the observable states to enrich the representable dynamics. Further, a model proficient in predicting $\dot{\pmb{x}}$ does not necessarily produce accurate predictions for trajectories $\pmb{x}(t)$ , as small derivative errors can accumulate exponentially when integrating an ODE [17]. On the other hand, trajectory fitting directly compares integrated model trajectories $\pmb{x}(t)$ with ground-truth trajectories, ensuring long-term accuracy. Here, the optimization process necessitates either the propagation of the loss gradients through the ODE solver or the use of the adjoint sensitivity method [7], which computes the gradients by solving a modified ODE. In both cases, trajectory fitting incurs significant computational costs due to gradient propagation and model integration, making trajectory fitting much slower than derivative fitting. + +A distinct advantage of sPHNNs is the favorable influence the stability constraint exerts on the training process when using trajectory fitting. NODEs and similar architectures can suffer from exploding gradients, where small parameter changes lead to large loss gradients. This can slow the training process or prevent it from converging. A potential cause is the crossing between the boundaries of local basins of attraction during training, leading to significant differences in long-term behavior [41]. Since sPHNNs have only one global basin of attraction, this source of training difficulties is eliminated. In our numerical experiments, we observe good convergence of the training error of sPHNNs where NODEs struggle without the use of regularization terms or gradient clipping. + +# 4 Experiments + +We evaluate the proposed model on multiple example problems, including measurement data from a real physical system and multiphysics simulation data. The performance of sPHNNs is compared to that of NODEs, PHNNs, and bounded PHNNs (bPHNNs). The latter two are PHS-based NN models that do not enforce or relax the convexity constraint of sPHNNs, and thus do not guarantee global stability. In PHNNs, the Hamiltonian is modeled by an unconstrained FFNN, whereas in bPHNNs, it is constructed to be positive and radially unbounded, which ensures bounded trajectories. The latter architecture is intentionally designed to resemble OnsagerNet [55], but with the key difference that it supports arbitrary time-dependent inputs, making it suitable for our experimental setting. For examples that purely rely on derivative fitting, we further include stable NODEs (sNODEs) [27]. Using trajectory fitting, we could not achieve convergence of sNODEs, excluding them from most experiments. This is likely due to difficulties caused by discontinuous dynamics stemming from the projection-based stability enforcement, as also mentioned by [46]. Finally, we include a variant of sPHNNs with learnable minima (sPHNN-LM). In contrast to sPHNNs, which have $x^{*}$ fixed to an a-priori known equilibrium position, sPHNN-LMs have to infer the equilibrium position from the training data and start out with a randomly initialized $x^{*}$ . All models are trained by minimizing the mean squared error using the ADAM optimizer [25]. Model hyperparameters and computational costs are listed in Appendix B. Predictions are generated by rollout with a numerical integrator using the Runge-Kutta scheme Tsim5, with adaptive step size and a known input function $u(t)$ . Code and model weights are available on GitHub1. + +# 4.1 Spinning rigid body + +We generate data of a damped spinning rigid body by numerically integrating Euler's rotation equations, which were already introduced in Section 2.2, Equation (5). Here, accurately capturing the dynamics of the angular velocity vector $\pmb{x} = \pmb{\omega}$ with a PHS requires a state-dependent structure matrix $\pmb{J}(\pmb{x})$ . We train ten instances per model type using derivative fitting with $(\omega, \dot{\omega})$ pairs sampled from the trajectories. Figure 1 shows the rotational energy $E(\omega)$ computed from model predictions $\omega$ via Equation (6). The PHNN, NODE, and sNODE models show deviations from ground truth, with the former two not converging to zero. Furthermore, the NODEs show unphysical behavior with at times increasing energy and exhibit high variance between instances. In contrast, sPHNNs and bPHNNs match the true energy well, highlighting the importance of providing both stability and physical biases. The sPHNN-LM variant recovered the actual equilibrium position accurately, with an interquartile mean distance between the true and inferred minimum across all instances of 0.010. As a result, its predictive performance is nearly identical to that of the sPHNN with fixed $\pmb{x}^*$ . + +# 4.2 Cascaded tanks + +Next, the sPHNN model is evaluated using the cascaded tanks dataset [47, 48]. It contains measurements of a physical fluid level control system consisting of a pump and two tanks, see Appendix D for details. The training data consists of a single trajectory with 1024 pairs of pump voltages $u$ and water level measurements $y$ of the lower tank. We model the system using a two-dimensional state vector $\boldsymbol{x} = [x_{1}, x_{2}]^{\top}$ , where $y = x_{2}$ . Assuming the augmented state $x_{1}$ describes the water level in the upper tank, it is straightforward to determine the equilibrium of the autonomous system. With zero input, the tanks will eventually drain; therefore, the stable equilibrium is given by $\boldsymbol{x}^{*} = \mathbf{0}$ . We thus fix the minimum of the sPHNN's Hamiltonian to the origin. To investigate the effects of a potential lack of this knowledge, we also train the sPHNN-LM variant. Its minimum location $\boldsymbol{x}^{*}$ is randomly initialized and optimized during training. Apart from this, both sPHNN versions are identical and use a constant fixed symplectic matrix as the structure matrix, as well as constant but learnable dissipation and input matrices. The same structure applies to PHNN and bPHNN. + +We train 20 instances per model type and show a statistical evaluation of the root mean squared errors (RMSEs) in Figure 3a. Overall, sPHNN, sPHNN-LM, and bPHNN achieve comparable accuracy, while NODE and PHNN perform worse on average across both trajectories. In addition to their better performance, the models with stability bias exhibit lower variance across model instances, indicating more consistent results. To evaluate the models' extrapolation capabilities, we extend the evaluation data with $400\mathrm{s}$ of zero-input. The resulting predictions are depicted in + +![](images/1754d9c9fa870b5d1b7f4c6bc3e64e92ea3f47cb4e395c59254e5a09c6679ac2.jpg) +(a) Training and test RMSE. + +![](images/b7ae9b3022514d905e028795c8d903259557dfc3f74d5113b957a48fffc6324b.jpg) +(b) Relaxation. +Figure 3. Cascaded tanks: (a) RMSE of the models on training and test trajectory. (b): Predictions for the extended test trajectory. At $t = 4096 \, \text{s}$ , the pump is turned off. Lines correspond to the interquartile mean and shaded areas represent the interquartile range of the predictions from the 20 model instances. + +Figure 3b. Assuming perfectly calibrated sensors, the true system response is an eventual zero output as both tanks drain. By construction, this behavior is guaranteed for sPHNNs. The 20 sPHNN-LM instances reliably identified the systems equilibrium to lie between $-0.196\mathrm{V}$ to $0.432\mathrm{V}$ , which aligns with physical intuition. In contrast the bPHNN, NODE and PHNN models exhibit poor physical extrapolation, predicting equilibria far from zero, diverging water levels, or oscillatory behavior. In an additional experiment presented in Appendix D, the sPHNN models are also shown to have successfully learned the hard saturation effect that occurs when a water tank overflows — a task at which both NODE and PHNN fail. + +# 4.3 Thermal food processing surrogate + +With the applicability of sPHNNs to real-world data established, this section focuses on one of the promising applications of data-driven dynamic system identification: the efficient construction of surrogate models. Here, data from a conjugate heat transfer simulation of a convection oven coupled with a soft matter model for meats from Kannapinn et al. [21, 23] is employed. The data consists of trajectories representing the temperature histories $T_{A}$ and $T_{B}$ at two probe points within the meat. These result from a predefined excitation signal $T_{\mathrm{oven}}$ , which controls the oven temperature. The surrogate's task is to predict the probe temperatures given the oven temperature as input. + +We fit the models to initially $n_D = 2$ trajectories consisting of 280 samples each (see Appendix E). Per model, we vary the dimensionality of the state $\pmb{x} \in \mathbb{R}^n$ with $n = 2 + n_A$ by adding 0 to 3 augmented dimensions $n_A$ . The minimum $\pmb{x}^*$ of the sPHNNs' Hamiltonian is set to the point in phase space corresponding to the ambient temperature, as motivated by physical intuition about the stable thermal equilibrium. The initial values of the augmented states are fixed to zero. In total, 20 instances are trained per model type and augmented dimension. + +The RMSEs evaluated on 15 test trajectories (see Appendix E) are depicted in Figure 4a. The sPHNN predictions tend to improve with more augmented dimensions, achieving errors comparable to those obtained by Kannapinn et al. [21] using commercial software. This appears intuitive, as additional hidden state variables may be required to accurately describe the evolution of $T_{A}$ and $T_{B}$ that results from the multiphysics, conjugate heat transfer process, see also [12]. However, for bPHNNs and sPHNN-LMs no clear trend emerges. The NODEs and PHNNs even tend to deteriorate in performance with more augmented dimensions, likely due to instabilities in the learned dynamics. This becomes apparent in Figure 4c (top row), which depicts the models' predictions for an extended test case. While the sPHNNs consistently converge to temperatures close to thermal equilibrium, the predictions of NODEs and PHNNs tend to become unstable when one or more augmented dimensions are used. Although all models improve with more training data (Figure 4b), these instabilities persist when extrapolating in time (Figure 4c, bottom row $t > 1395\mathrm{s}$ ). In contrast, the stability constraint of the sPHNNs and sPHNN-LMs allows safe use of the additional flexibility provided by a higher-dimensional state space, avoiding the pitfalls observed in the unconstrained models. Given sufficient training data, the sPHNN-LM variant performs on par with the sPHNN, whereas in the scarce-data regime its errors increase and are comparable to those of the bPHNNs. Although the bPHNNs also remain stable, they are consistently outperformed by the sPHNNs for $n_{A} = 3$ , regardless of dataset size, and especially when only a single training trajectory is available $n_{D} = 1$ , see Figure 4b and Figure 4c (bottom left). + +![](images/f51cc905aee998f3472121cae20494182433a6f96622a520e621ff832eff2299.jpg) +(a) Test RMSEs for $n_D = 2$ training trajectories. + +![](images/0d41f0bd9c3931ad908f914dbc4d7d17f219c574096f8237a5cbc748122480dd.jpg) + +![](images/df1efd118f2824da120afcd12ea7aeed0156c2b3e2f101a7c11c530d4dd67e0d.jpg) +(b) Test RMSEs for $n_A = 3$ augmented dimensions. +(c) Model predictions for different combinations of augmented dimensions and training data. +Figure 4. Thermal food processing surrogate: (a) and (b) RMSEs evaluated on 15 test trajectories for various numbers of augmented dimensions $n_A$ and training trajectories $n_D$ . (c) Interquartile mean and range of the $T_A$ predictions for a custom test case. Top row: Varying number of augmented dimensions with fixed number $n_D = 2$ of training trajectories. Bottom row: Varying number of training trajectories with fixed number $n_A = 3$ of augmented dimensions. The time $t = 1395 \, \text{s}$ marks the length of the training trajectories. + +Table 1. Median RMSE of 20 instances per models using $n_A = 3$ augmented dimensions, trained on $n_D = 2$ noisy trajectories and evaluated on noise-free training and test sets. + +
ModelTrainingTest
5% noise15% noise25% noise5% noise15% noise25% noise
sPHNN0.3320.8541.3541.1702.5953.177
sPHNN-LM0.6381.0961.6222.4202.8883.677
bPHNN0.5001.0492.2071.5722.8285.231
PHNN2.8286.7505.77928.23325.63124.224
NODE0.9931.9742.4233.5773.4824.362
+ +To assess the models' robustness to noise, we conducted an additional experiment using $n_A = 3$ augmented dimensions and $n_D = 2$ training trajectories, introducing zero-mean Gaussian noise for both the input $T_{\mathrm{oven}}$ and the outputs $T_A$ and $T_B$ . The noise amplitudes were varied between $5\%$ and $25\%$ of the original signals' standard deviations. Table 1 reports the average RMSE over 20 model instances per model type, evaluated on noise-free test data. To isolate the effect of noise and eliminate generalization error, we also evaluated the RMSE on the noise-free training data. All models demonstrate a degree of robustness to noise, since the integration step involved in trajectory fitting acts as a low-pass filter, effectively smoothing the learned dynamics. As a result, high-frequency Gaussian noise is attenuated, which helps prevent overfitting and improves generalization. Nevertheless, the sPHNN consistently outperforms the baselines on both the training and test sets. + +# 4.4 Additive manufacturing surrogate + +Finally, we evaluate the applicability of sPHNNs to higher-dimensional state spaces and build a reduced order model (ROM) as a surrogate of a 3D partial differential equation problem. To this end, we generate field data by numerically solving the heat conduction equation for a moving heat + +![](images/3bdcc37611413d59a0f565de00c241cbbbbbf022760b3ddfa8056c618dabd146.jpg) +Figure 5. Heat source field and temperature field predictions on the cuboid domain for a test case with $v = 12.5 \, \mathrm{mm} \, \mathrm{s}^{-1}$ , $Q = 400 \, \mathrm{W}$ . The instances selected for this evaluation resulted in the median test error for the corresponding model type. Colors are clipped to remain in the legend's range. + +source on a cuboid domain with convective and radiative thermal boundary conditions. The finite element simulation is designed to model the evolution of the temperature field of a metal additive manufacturing process, see [22] for details. In total, 25 trajectories are obtained by varying the heat source speed $v$ from $10\mathrm{mm}\mathrm{s}^{-1}$ to $20\mathrm{mm}\mathrm{s}^{-1}$ and power $Q$ from $300\mathrm{W}$ to $500\mathrm{W}$ . All trajectories span $20\mathrm{s}$ , with a non-zero heat source lasting for the initial $2.5\mathrm{s}$ to $6\mathrm{s}$ , depending on $v$ . However, directly using the discretized field data would lead to very high-dimensional models and render the training and inference inefficient. We thus apply a proper orthogonal decomposition (POD) to map the temperature and heat source fields onto 40-dimensional latent spaces, respectively (see Appendix F for details). The models are trained on the latent representations of the two trajectories with $(v,Q) = (10\mathrm{mm}\mathrm{s}^{-1},300\mathrm{W})$ and $(20\mathrm{mm}\mathrm{s}^{-1},500\mathrm{W})$ using derivative fitting and subsequent fine-tuning with trajectory fitting. + +Figure 5 depicts the heat source field (top row) and true and predicted temperature fields from the different model types. While the sPHNN's total test RMSE peaks below $30\mathrm{K}$ (see Appendix F, Figure F.2) and decreases as the system approaches thermal equilibrium, the errors for NODE and PHNN increase rapidly, driven by instabilities in the learned dynamics. These instabilities are likely caused by the sparsity of the training data, which leaves gaps in the phase space, allowing unstable dynamics to emerge. The stability constraint of sPHNN and sPHNN-LM mitigates these issues, ensuring robust performance even with scarce training data, while outperforming the bPHNN thanks to their stronger inductive bias. + +# 5 Conclusion + +This work proposes stable port-Hamiltonian neural networks (sPHNNs) for accurate, robust, and reliable identification of stable nonlinear dynamics. They leverage the port-Hamiltonian framework, which provides a basis for model interpretation and ensures adherence to the laws of thermodynamics in terms of energy conservation or dissipation balances with respect to a learned Hamiltonian energy function. Additionally, the approach ensures global asymptotic stability of the identified dynamics, without requiring projection, by constraining the Hamiltonian to be a convex, positive definite Lyapunov function. While this limits sPHNNs to modeling systems that appear globally stable within the region of interest, it provides a strong inductive bias that enables robust learning and reasonable generalization from sparse data. We demonstrate the approach's viability using data from synthetic examples, real-world measurements, and complex multiphysics simulation models. The resulting models, built with small neural network architectures, exhibit low variance across instances, accurate prediction, and robust generalization. Furthermore, the proposed method can successfully exploit augmented dimensions where unconstrained alternative models struggle with instabilities or data scarcity. While the approach can identify the equilibrium point from data, we find that providing the equilibrium as a prior can further enhance prediction accuracy and reduce data requirements. Future work will extend the approach to multistable systems by allowing Hamiltonians with multiple minima, while retaining favorable stability properties. + +# Acknowledgments and Disclosure of Funding + +The authors acknowledge the financial support provided by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, research grant number 492770117 and the CRC/TRR 361 CREATOR), the Hessian Center for Artificial Intelligence (Hessian.AI, project EnDyRo), and the Graduate School Computational Engineering at TU Darmstadt. Furthermore, we would like to thank An Thai Le, Daniel Palenicek, and Joe Watson from TU Darmstadt (IAS) for their constructive feedback on the manuscript. + +# References + +[1] A. Abbasi, P. N. Kambali, P. Shahidi, and C. Nataraj. "Physics-Informed Machine Learning for Modeling Multidimensional Dynamics". In: Nonlinear Dynamics (2024). DOI: 10.1007/s11071-024-10163-3. +[2] B. Amos, L. Xu, and J. Z. Kolter. "Input Convex Neural Networks". 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DOI: 10.48550/arXiv.2002.08860. + +# A Proof of stability + +This section provides a proof of Theorem 3.1, which describes the essential requirements for achieving stable dynamics with port-Hamiltonian systems. Due to the relevance of convexity to the following discussion, the following Lemma first provides some widely known equivalent formulations of convexity for scalar-valued multivariate functions. + +Lemma A.1. Consider a twice continuously differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ , $x \mapsto f(x)$ . If $f$ is convex, the following statements hold: + +1. $\lambda f(\pmb{x}_1) + (1 - \lambda)f(\pmb{x}_2)\geq f(\lambda \pmb{x}_1 + (1 - \lambda)\pmb{x}_2)\quad \forall \pmb{x}_1,\pmb{x}_2\in \mathbb{R}^n,\lambda \in [0,1],$ +2. $\left.\frac{\partial f}{\partial\pmb{x}}\right|_{\pmb{x}_1}^{\mathsf{T}}(\pmb{x}_2 - \pmb{x}_1)\leq f(\pmb{x}_2) - f(\pmb{x}_1)\quad \forall \pmb{x}_1,\pmb{x}_2\in \mathbb{R}^n,$ +3. $\frac{\partial^2 f}{\partial \boldsymbol{x} \partial \boldsymbol{x}}(\boldsymbol{x}) \succeq 0 \quad \forall \boldsymbol{x} \in \mathbb{R}^n$ . + +For a proof, the reader is referred to [6]. Note that for strict convexity, the inequalities in statements 1 and 2 must be strict for all $x_{1} \neq x_{2}$ , and the Hessian in statement 3 must be positive definite. With the notion of convexity established, attention can now be directed to the proof of Theorem 3.1. For convenience, the statement of the theorem is repeated verbatim below: + +Theorem A.2. Consider the PHS Equation (3) in the unforced case $\pmb{u}(t) = \mathbf{0}$ : + +$$ +\dot {\boldsymbol {x}} = \left[ \boldsymbol {J} (\boldsymbol {x}) - \boldsymbol {R} (\boldsymbol {x}) \right] \frac {\partial \mathcal {H}}{\partial \boldsymbol {x}} (\boldsymbol {x}), \quad \text {w i t h} \quad \boldsymbol {J} = - \boldsymbol {J} ^ {\intercal}, \boldsymbol {R} = \boldsymbol {R} ^ {\intercal} \succeq 0. \tag {A.1} +$$ + +Suppose the Hamiltonian $\mathcal{H}(\pmb{x})$ is convex, twice continuously differentiable, and fulfills: + +$$ +\mathcal {H} (\mathbf {0}) = 0, \quad \left. \frac {\partial \mathcal {H}}{\partial \boldsymbol {x}} \right| _ {\boldsymbol {x} = \boldsymbol {0}} = \boldsymbol {0}, \quad \left. \frac {\partial^ {2} \mathcal {H}}{\partial \boldsymbol {x} \partial \boldsymbol {x}} \right| _ {\boldsymbol {x} = \boldsymbol {0}} \succ 0. \tag {A.2} +$$ + +Then, the system in Equation (A.1) has a stable equilibrium at $\pmb{x}(t) = \mathbf{0}$ , and all solutions are bounded. Furthermore, the equilibrium is globally asymptotically stable if $\pmb{R}(\pmb{x}) > 0$ . + +Proof. Since the gradient of the Hamiltonian vanishes and its Hessian is positive definite at $\mathbf{x} = \mathbf{0}$ , it has a strict local minimum at the origin. This implies the existence of an $R > 0$ such that + +$$ +\mathcal {H} (\boldsymbol {x}) > \mathcal {H} (\boldsymbol {0}) = 0 \quad \forall \boldsymbol {x} \in \{\| \boldsymbol {x} \| \leq R, \boldsymbol {x} \neq \boldsymbol {0} \}. \tag {A.3} +$$ + +Consider the function $g(\pmb{x}) = \frac{a}{r} \| \pmb{x} \|$ for some $r \in (0, R]$ with $a = \inf_{\| \pmb{x} \| = r} \mathcal{H}(\pmb{x}) > 0$ . We claim that $g$ is a lower bound for $\mathcal{H}$ , i.e. $g(\pmb{x}) \leq \mathcal{H}(\pmb{x})$ for all $\| \pmb{x} \| > r$ . + +Suppose there existed an $\tilde{\pmb{x}}\in \mathbb{R}^n$ that violated the claim, that is $\| \tilde{\pmb{x}}\| >r$ and $g(\tilde{\pmb{x}}) > \mathcal{H}(\tilde{\pmb{x}})$ . Due to the convexity of $\mathcal{H}$ , it holds + +$$ +\lambda \mathcal {H} (\tilde {\boldsymbol {x}}) + (1 - \lambda) \mathcal {H} (\mathbf {0}) \geq \mathcal {H} (\lambda \tilde {\boldsymbol {x}} + (1 - \lambda) \cdot \mathbf {0}) \quad \forall \lambda \in [ 0, 1 ] \tag {A.4} +$$ + +and thus $\lambda g(\tilde{\pmb{x}}) > \lambda \mathcal{H}(\tilde{\pmb{x}})\geq \mathcal{H}(\lambda \tilde{\pmb{x}})$ . Choosing $\lambda = \frac{r}{\|\tilde{\pmb{x}}\|}$ and applying the definition of $g$ we obtain + +$$ +\frac {r}{\| \tilde {\boldsymbol {x}} \|} g (\tilde {\boldsymbol {x}}) = a > \mathcal {H} \left(\frac {r}{\| \tilde {\boldsymbol {x}} \|} \tilde {\boldsymbol {x}}\right) \geq a. \tag {A.5} +$$ + +This is a contradiction, and therefore $g$ is a lower bound. + +With the bound established, showing that $\mathcal{H}$ is radially unbounded is straightforward. For any path $\pmb{x}(t)$ with $\lim_{t\to \infty}\| \pmb{x}(t)\| = \infty$ , once $\| \pmb{x}(t)\| > r$ we have + +$$ +\lim _ {t \rightarrow \infty} \mathcal {H} (\boldsymbol {x} (t)) \geq \lim _ {t \rightarrow \infty} g (\boldsymbol {x} (t)) = \infty . \tag {A.6} +$$ + +We would like to remark that the same conclusion could alternatively be reached by showing that $\mathcal{H}$ has at least one bounded sublevel set due to its strict minimum and then concluding radial unboundedness (i.e., coercivity) using Corollary 8.7.1 in [45] together with the standard equivalence for proper closed convex functions. However, the presented approach using a lower bound can directly be extended to show global positive definiteness of $\mathcal{H}$ . For this, note that for all $\| \pmb{x} \| > r$ the lower bound implies $0 < g(\pmb{x}) \leq \mathcal{H}(\pmb{x})$ . Together with Equation (A.3), this implies that $\mathcal{H}$ is indeed positive definite for all $\pmb{x} \in \mathbb{R}^n$ . + +From the port-Hamiltonian structure of Equation (A.1) we immediately obtain the decrease condition $\dot{\mathcal{H}}\leq 0$ (see Equation (4)). Using the $\mathcal{H}$ as a Lyapunov function, we obtain all requirements to conclude local stability of the equilibrium. Furthermore, the decrease condition implies that a solution starting at $x^{*}$ cannot leave the sublevel set $\Omega_{\mathcal{H}} = \{\pmb {x}\in \mathbb{R}^n |\mathcal{H}(\pmb {x})\leq \mathcal{H}(\pmb{x}^*)\}$ . Since $\Omega_{\mathcal{H}}\subseteq \Omega_g = \{\pmb {x}\in \mathbb{R}^n |g(\pmb {x})\leq \mathcal{H}(\pmb{x}^*)\}$ and $\Omega_{g}$ is clearly bounded, all solutions of Equation (A.1) must be bounded as well. + +Finally, to show that global asymptotic stability follows from $R \succ 0$ , it remains to be demonstrated that $\mathcal{H} < 0$ for all $x \neq 0$ . This is implied by Equation (4) if $\frac{\partial \mathcal{H}}{\partial x} \neq 0$ for all $x \neq 0$ . Proving that the gradient of $\mathcal{H}$ vanishes only at the origin is done by contradiction. Assume that there exists an $\tilde{x} \neq 0$ such that $\left.\frac{\partial \mathcal{H}}{\partial x}\right|_{\tilde{x}} = 0$ . Furthermore, let $x^{*} = 0$ . Since $\mathcal{H}$ is convex it follows (see Lemma A.1) that + +$$ +\mathcal {H} \left(\boldsymbol {x} ^ {*}\right) - \mathcal {H} (\tilde {\boldsymbol {x}}) \geq \left. \frac {\partial \mathcal {H}}{\partial \boldsymbol {x}} \right| _ {\tilde {\boldsymbol {x}}} ^ {\intercal} \left(\boldsymbol {x} ^ {*} - \tilde {\boldsymbol {x}}\right) = 0, \tag {A.7} +$$ + +$$ +\mathcal {H} (\tilde {\boldsymbol {x}}) - \mathcal {H} \left(\boldsymbol {x} ^ {*}\right) \geq \left. \frac {\partial \mathcal {H}}{\partial \boldsymbol {x}} \right| _ {\boldsymbol {x} ^ {*}} ^ {\top} \left(\tilde {\boldsymbol {x}} - \boldsymbol {x} ^ {*}\right) = 0. \tag {A.8} +$$ + +This implies $\mathcal{H}(\boldsymbol{x}^*) = \mathcal{H}(\tilde{\boldsymbol{x}}) = 0$ . However, since $\mathcal{H}$ is globally positive definite and thus only vanishes at the origin, we have $\boldsymbol{x}^* = \tilde{\boldsymbol{x}}$ . This is a contradiction as $\tilde{\boldsymbol{x}} \neq \mathbf{0}$ , which concludes the proof. + +# B Experiment hyperparameters + +In each experiment presented within this manuscript, the different models were trained for the same number of steps using identical learning rates. The mean squared error loss function and the ADAM [25] optimizer were used throughout. The model and training hyperparameters, along with the number of model instances trained for statistical evaluation, are provided in Table 2. Throughout the experiments presented in Sections 4.1 to 4.3, all NNs consisted of two hidden layers, each containing 16 neurons. The network width was increased to 32 neurons for the additive manufacturing surrogate in Section 4.4. All network weights were initialized using the Glorot uniform [15] scheme. + +The PHS-based models (sPHNN, sPHNN-LM, PHNN, bPHNN) share the same structure, differing only in how the Hamiltonian $\mathcal{H}$ is parametrized. Specifically, the sPHNNs use an FICNN with a normalization term, as described in Section 3. The only difference for the sPHNN-LM variant is that the minimum $x^{*}$ of the Hamiltonian is learned as an additional parameter. The PHNNs, by contrast, use an unconstrained FFNN without any normalization. Finally, the bPHNNs ensures positivity and radially unboundedness of the Hamiltonian by adopting the method described by Yu et al. [55] for the OnsagerNets Energy function. Here, the output of an unconstrained network is squared and combined with a quadratic regularization term. However, unlike OnsagerNet, we used standard FFNNs instead of ResNets. + +The models were trained on a Windows machine with Intel i7-13700K (13th Generation) and 32 GB of memory. Table 3 provides representative computational costs for training and prediction. Here, Derivative evaluation refers to the average time taken by each model to compute the state derivative $\dot{\pmb{x}}$ + +Table 2. Hyperparameters per experiment. TF and DF refer to trajectory and derivative fitting, respectively. The number of instances trained per model type is listed in the "# inst." column, and $n$ and $m$ denote the number of dimensions of state and input. The final three columns list the parametrization of the matrices for sPHNN, sPHNN-LM, bPHNN, and PHNN as either state-dependent matrices (s.) or constant matrices (c.). + +
Experiment# training stepslearning rate# inst.nmmatrices
JRG
Randomly initialized model--12-s.s.-
Spinning rigid body50000 DF\( 1 \times 10^{-3} \)103-s.c.-
Cascaded tanks50000 TF\( 5 \times 10^{-4} \)2021c.c.c.
Food processing surrogate30000 TF\( 1 \times 10^{-4} \)202 to 51c.c.c.
Additive manufacturing surrogate20000 DF and 10000 TF\( 1 \times 10^{-3} \) \( 1 \times 10^{-5} \)204040c.c.c.
+ +Table 3. Representative computational costs for training and prediction. The top two rows report the training times for derivative and trajectory fitting, respectively. The bottom two rows show the time for a single derivative evaluation and a full trajectory integration using data and models from Section 4.3. + +
sPHNNsPHNN-LMsNODEbPHNNPHNNNODE
Derivative fit Section 4.1 (min)0.5770.6440.6720.6350.5390.346
Trajectory fit Section 4.3 (min)37.66339.160-38.32919.11117.951
Derivative evaluation (ms)0.0070.007-0.0070.0070.005
Model integration (ms)0.3650.393-0.4220.2370.221
+ +for a given state $\pmb{x}$ and input $\pmb{u}$ , which needs to be evaluated for time integration. In contrast, Model integration denotes the time required to generate a single trajectory prediction via integration. While the latter better reflects practical use cases, the reported integration time also depends on the integration scheme and chosen step size. Since we use an adaptive step size controller, the step size in turn depends on the learned dynamics. + +# C Spinning rigid body system + +The governing equations of the three-dimensional spinning rigid body used in the experiment in Section 4.1 are described in Section 2.2. We generated $(\omega, \bar{\omega})$ pairs for training by evaluating Equation (5) for varying $\omega$ . To simulate a measurement process, we selected $\omega$ values from $50~\mathrm{s}$ long numerically integrated trajectories, using 10 randomly chosen initial conditions $\omega(0) \in [0,1]^3$ with parameters $\mathbf{I} = \mathrm{diag}(1,2,3)$ and $\mu = 0.01$ . A representative trajectory of $\omega$ is shown in Figure C.1. + +![](images/7c7475306cb0ac7303520be072fde3ea0bb3ed407394b7c9f1f7a0d2e9ce0ec8.jpg) +Figure C.1. Representative, extended training trajectory for spinning rigid body, including interquartile mean and range of model predictions for $\omega_{3}$ . Left: Models without stability bias; Right: Models with stability bias. + +![](images/fff04a417b2c547f7657d6a2f88df0b03ba3466bda8483283646c5147dfe225e.jpg) + +# D Cascaded tanks data + +The cascaded tanks dataset [47, 48] used in Section 4.2 consists of measurements of a fluid level control system. Liquid flows from an upper tank through an outlet into a lower tank, exits via a similar outlet into a reservoir, and is pumped back to the upper tank, see Figure D.1a. The system's input $u$ is the voltage applied to the pump, with higher values resulting in a faster volume flow. When the pump speed is large enough, an overflow can occur in the upper tank. A capacitive sensor measures the fluid level in the lower tank, and the resulting voltage represents the system's output $y$ . The dataset consists of two trajectories comprising 1024 pairs $(u, y)$ each and Figure D.1b depicts the training trajectory. The initial water level in the upper tank is unknown, but it is the same for both trajectories. + +Besides the experiments presented in Section 4.2, we further investigated the models' capabilities to learn the hard saturation that the cascaded tanks data system exhibits at an output value of $10\mathrm{V}$ . This occurs when the tanks overflow in response to a large input voltage over an extended period of time (see Figure D.1b) [48]. We explore the models' behavior in this regime by constructing a pump voltage signal that linearly increases from $1\mathrm{V}$ initially to $8\mathrm{V}$ at $2800\mathrm{s}$ . The resulting predictions + +![](images/77df4dd7123cb8ff1debd8a4fa12d7e09c1afb001085dca8b949f9b36441ebc2.jpg) + +![](images/b834394313f41408bbcdfed1761bda257591ce6765c917fd64a7affd28f9326f.jpg) +(b) Cascaded tanks training data (overflow highlighted). + +![](images/6acebe16ec72018ada83b5b82a17eab4e17f9def11a3bd7614ec4dc4b872a71d.jpg) +(a) Experimental setup +(c) Saturation. +Figure D.1. Cascaded tanks: (a) System setup, (b) Training trajectory. (c) Predictions for a linearly increasing pump voltage. Due to an overflow, the true system exhibits a hard saturation at $x_{2} = 10\mathrm{V}$ . Lines correspond to the interquartile mean, whereas shaded areas represent the interquartile range of the predictions from the 20 model instances. + +are shown in Figure D.1c. For pump speeds up to $\sim 7\mathrm{V}$ ( $\sim 2400\mathrm{s}$ ), the sPHNN, sPHNN-LM, and bPHNN instances capture the saturation effect, while the NODE and PHNN instances fail to do so. + +# E Thermal food processing surrogate data + +This section provides additional details about the data used in Section 4.3, which is a subset of the data generated and used by Kannapinn et al. [21] in the creation of ROMs. The dataset describes the thermal processing of chicken breasts in a convection oven and was generated via the simulation of a soft-matter model for meats implemented in COMSOL Multiphysics. Each trajectory in the dataset describes the temperature history at two points inside the full order model. These result from a predefined excitation signal that controls the oven temperature. The probe points are situated in the center and close to the surface of the food item, with the corresponding temperatures labeled as $T_{A}$ and $T_{B}$ , respectively. To facilitate cross-referencing, the alphanumeric identifiers used by Kannapinn et al. [21] for the individual trajectories are retained. The training data consists of trajectories 745 and 795. They use an amplitude-modulated pseudo-random binary sequence (APRBS) and a multi-sine as the excitation signal $T_{\mathrm{oven}}$ . Figure E.1 (left and middle) illustrates $T_{\mathrm{oven}}$ along with the core and surface temperatures for both trajectories. This selection provides a fair evaluation of the models across the relevant temperature spectrum. For studying the impact of the number of training trajectories on the models, we additionally included 13 trajectories with identifiers 835, 851, 853, 861, 868, 873, 874, 888, 899, 902, 905, 906 and 917. In the experiment, the test group AP15 [21] is utilized as test data. It comprises 15 trajectories with APRBS excitations, specifically selected to ensure an even distribution of amplitudes and median values. Figure E.1 (right) shows an exemplary test trajectory, including the predictions for $T_{A}$ from all models using 3 augmented dimensions. All trajectories span $1395\mathrm{s}$ and consist of 280 samples. The data is normalized for training to ensure the numerical values are of magnitude one. For this, an affine transformation is applied to $T_{\mathrm{oven}}$ , $T_{A}$ , and $T_{B}$ individually. It shifts the ambient temperature to a value of zero and rescales the signal to have unit variance. The scaling and shifting factors are calculated using only the training data to ensure that the test data does not influence the training procedure. + +![](images/eeef8ca1e5c9881159d03ff98102ab6a1b7fc04502b72a3bdf9708a7ff1bb668.jpg) +Figure E.1. Left and middle: Training trajectories. Right: Exemplary test trajectory with interquartile mean and range of the model predictions for $T_{A}$ . + +![](images/de079aaf7f09c5fb933227dc82d52161fbd983e3fd4f3d0fe351d6dc365a45d8.jpg) + +![](images/f050b1288b6762ab9b029fb36385bebe81b0247572caa7b3440ba4abecac06a1.jpg) + +# F Thermal field data + +# F.1 Dimensionality reduction + +The following describes the POD method for dimensionality reduction of the field data applied in Section 4.4. The process is first described for general field data given as a matrix $\mathbf{A} = [a_0, \dots, a_M]^\mathrm{T}$ of $M$ snapshots $\mathbf{a}_i \in \mathbb{R}^N$ , where $M$ is the number of snapshots and $N$ is the number of finite element nodes. To identify the dominant modes of the data, we perform a singular value decomposition (SVD) of $\mathbf{A}$ : + +$$ +\boldsymbol {A} = \boldsymbol {U} \boldsymbol {\Sigma} \boldsymbol {V} ^ {\intercal} \in \mathbb {R} ^ {M \times N}, \tag {F.1} +$$ + +where $\pmb{U} \in \mathbb{R}^{M \times M}$ and $\pmb{V} \in \mathbb{R}^{N \times N}$ are unitary matrices and $\pmb{\Sigma} \in \mathbb{R}^{M \times N}$ is a rectangular diagonal matrix containing the singular values $\sigma_{1} \geq \dots \geq \sigma_{M}$ on its diagonal. Dimensionality reduction is achieved by truncating the SVD $\pmb{A} \approx \tilde{\pmb{U}}\tilde{\pmb{\Sigma}}\tilde{\pmb{V}}^{\top}$ , where $\tilde{\pmb{U}} \in \mathbb{R}^{M \times n}$ and $\tilde{\pmb{V}} \in \mathbb{R}^{N \times n}$ contain the first $n$ columns of $\pmb{U}$ and $\pmb{V}$ , and $\tilde{\Sigma} = \mathrm{diag}(\sigma_{1},\ldots,\sigma_{n})$ is the truncated matrix of the dominant singular values. The dominant modes in the data are then given by $\pmb{M} = c\tilde{\pmb{V}}$ . Here, the scaling factor $c$ is the standard deviation of the elements in $\tilde{\pmb{U}}\tilde{\pmb{\Sigma}}$ . This scaling ensures that the latent trajectories will have unit variance. To map a latent state $\pmb{x} \in \mathbb{R}^{n}$ to the full-order field $\pmb{X} \in \mathbb{R}^{N}$ we compute: + +$$ +\boldsymbol {X} = \boldsymbol {M} \boldsymbol {x}. \tag {F.2} +$$ + +Conversely, to map from the full-order field to the latent space, we solve the least-squares regression problem: + +$$ +\boldsymbol {x} = \underset {\boldsymbol {x} ^ {\prime}} {\arg \min } \| \boldsymbol {X} - \boldsymbol {M} \boldsymbol {x} ^ {\prime} \| = \frac {1}{c ^ {2}} \boldsymbol {M} ^ {\intercal} \boldsymbol {X}. \tag {F.3} +$$ + +In Section 4.4, we consider two fields: The thermal field describing the state of the system and the excitation given by the heat source field. For each, a separate SVD is performed to compute $M_T$ and $M_q$ using the respective snapshots from the two training trajectories. We thus obtain two pairs of encoders $\psi$ and decoders $\phi$ : + +$$ +\psi_ {T} (\boldsymbol {T}) = \underset {\boldsymbol {x}} {\arg \min } \| \boldsymbol {T} - \boldsymbol {M} _ {T} \boldsymbol {x} \|, \quad \phi_ {T} (\boldsymbol {x}) = \boldsymbol {M} _ {T} \boldsymbol {x}, \tag {F.4} +$$ + +$$ +\psi_ {q} (\boldsymbol {q}) = \underset {\boldsymbol {u}} {\arg \min } \| \boldsymbol {q} - \boldsymbol {M} _ {q} \boldsymbol {u} \|, \quad \phi_ {q} (\boldsymbol {u}) = \boldsymbol {M} _ {q} \boldsymbol {u}. \tag {F.5} +$$ + +To fully leverage the benefits of sPHNNs, it is essential to ensure that the absence of a heat source is encoded as the zero-vector in the corresponding latent space. Since the mapping from the latent space Equation (F.2) is linear, and $q = 0$ corresponds to the absence of a heat source in the full-order space, this requirement is automatically fulfilled. However, for applications where a nonzero state $\tilde{\pmb{x}}$ describes the absence of an external energy source or for general nonlinear latent mappings (such as autoencoders), the condition can be met by applying a constant translation in the latent space, which shifts the encoded state $\tilde{\pmb{x}} = \psi (\tilde{\pmb{x}})$ to the origin. In Section 4.4, we use this method to ensure that the thermal equilibrium corresponds to the origin of the temperature latent space. While this is not strictly necessary, knowledge of the equilibrium location in the latent space enables its integration + +into the sPHNN's architecture by fixing the stable equilibrium position. Consequently, we arrive at the representation of the latent state + +$$ +\boldsymbol {x} = \psi_ {T} (\boldsymbol {T}) - \psi_ {T} \left(\boldsymbol {T} _ {\mathrm {e q}}\right), \quad \boldsymbol {T} \approx \phi_ {T} \left(\boldsymbol {x} + \psi_ {T} \left(\boldsymbol {T} _ {\mathrm {e q}}\right)\right). \tag {F.6} +$$ + +where $T_{\mathrm{eq}}$ is the full-order state vector describing thermal equilibrium. The complete model architecture is represented in Figure F.1. There, the function $f$ is used as a placeholder for an arbitrary evolution equation, to generalize to all model architectures used in Section 4.4. + +![](images/bd564de9d8e73b0d56747de5d7cf4da00f192708efac9e92140d1428d93b1edd.jpg) +Figure F.1. Model setup used in Section 4.4. The initial temperature field $T(t_0)$ and input field $q(t)$ are mapped via encoders $\phi_T, \phi_q$ into the latent space, where the latent dynamics $f$ are integrated. Here, $f$ is used as a placeholder for a specific model, e.g., NODE or sPHNN. Finally, the predicted latent trajectory $x(t)$ is mapped to the full-order field via the decoder $\phi_T$ . + +Finally, we note that alternative dimensionality reduction techniques could be applied. In preliminary experiments, we explored the use of autoencoders for this purpose. Although autoencoders offer advantages over the linear SVD regarding achievable compression rates, they exhibited poor extrapolation performance when trained on sparse data. This issue could potentially be mitigated by training the autoencoders with additional synthetic data, which would not necessitate costly simulations; however, further research is needed to explore this approach. + +# F.2 Additional results + +We briefly present additional results from Section 4.4 to demonstrate that the differences in predictions across the various model architectures remain consistent across multiple instances. Figure F.2 depicts the RMSEs of each model from Section 4.4 computed over all 25 trajectories and the spatial domain. The behavior of the models with stability bias can be distinguished from the NODE and PHNN, which suffer from instabilities that result in rapidly increasing errors. While both sPHNN variants and bPHNNs exhibit decreasing errors as the system returns to thermal equilibrium, sPHNNs achieve a peak RMSE nearly an order of magnitude lower, due to their strong inductive bias from enforcing global stability. The sPHNN-LM performs almost identically to the sPHNN initially, while showing slightly larger errors in the convergence to equilibrium. This is due to slight errors in the learned equilibrium position, which can be addressed by increasing the amount of training data, as demonstrated in Section 4.3. + +![](images/b4f571e868c4e11a27575bd453696c3123f555deb302a1202de29a6ae8e39767.jpg) +Figure F.2. Interquartile mean and range of the RMSEs evaluated with all 25 trajectories of the thermal field data. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: Our claim, that the enforcement of global stability and physics-based conditions results in a strong inductive bias and increased performance in applicable cases, is highlighted in the abstract and introduction, and demonstrated by our empirical results. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [NA] + +Justification: This work does not involve large language models in any part of the core research methodology. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. \ No newline at end of file diff --git a/stableporthamiltonianneuralnetworks/images.zip b/stableporthamiltonianneuralnetworks/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..205194a47e3e96de02d1a5c4c5f5e1c77478a792 --- /dev/null +++ b/stableporthamiltonianneuralnetworks/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6ec31e9162c7bfee7d899d844860e304a6f58f5ff6422b47c19e1354f1b62aaa +size 740809 diff --git a/stableporthamiltonianneuralnetworks/layout.json b/stableporthamiltonianneuralnetworks/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..64f67a92ff6760daa850c59642800c156a25e287 --- /dev/null +++ b/stableporthamiltonianneuralnetworks/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:73c11c0001f2b1e811ba5201aae48ba7fd1673690ca78a1aecaf5ccda04a9e02 +size 962775 diff --git a/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/a251582d-ed41-42cc-af7f-2f2d6708606e_content_list.json b/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/a251582d-ed41-42cc-af7f-2f2d6708606e_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..64fcef0750a100e88e7a1226c531c06272498c00 --- /dev/null +++ b/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/a251582d-ed41-42cc-af7f-2f2d6708606e_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b2b265b961413bd93c6b9e7bab67b4918eadee8cee36f93f71558dbfc6a9477d +size 223406 diff --git a/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/a251582d-ed41-42cc-af7f-2f2d6708606e_model.json b/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/a251582d-ed41-42cc-af7f-2f2d6708606e_model.json new file mode 100644 index 0000000000000000000000000000000000000000..5ffac11537c3b3e92d0ecf39fd6529f29040004b --- /dev/null +++ b/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/a251582d-ed41-42cc-af7f-2f2d6708606e_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:27083e7f72d0851ca3b89033527fbe7036ef146a557de83ea0cf1c470cd1d08d +size 274206 diff --git a/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/a251582d-ed41-42cc-af7f-2f2d6708606e_origin.pdf b/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/a251582d-ed41-42cc-af7f-2f2d6708606e_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..095aeb77ca4cd62eb85b89a8ee81a087ee78895f --- /dev/null +++ b/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/a251582d-ed41-42cc-af7f-2f2d6708606e_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2f4bc89d3b5daec2cef9affb39639fe0dec0c46d6ee639b9ccaad527795418d8 +size 843345 diff --git a/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/full.md b/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/full.md new file mode 100644 index 0000000000000000000000000000000000000000..a82a35af89cc94b915c3423609f36d351a42a088 --- /dev/null +++ b/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/full.md @@ -0,0 +1,1224 @@ +# Stab-SGD: Noise-Adaptivity in Smooth Optimization with Stability Ratios + +David A. R. Robin + +INRIA - ENS Paris PSL Research University + +Killian Bakong +INRIA - ENS Paris +PSL Research University + +Kevin Scaman +INRIA - ENS Paris +PSL Research University + +# Abstract + +In the context of smooth stochastic optimization with first order methods, we introduce the stability ratio of gradient estimates, as a measure of local relative noise level, from zero for pure noise to one for negligible noise. We show that a schedule-free variant (Stab-SGD) of stochastic gradient descent obtained by just shrinking the learning rate by the stability ratio achieves real adaptivity to noise levels (i.e. without tuning hyperparameters to the gradient's variance), with all key properties of a good schedule-free algorithm: neither plateau nor explosion at initialization, and no saturation of the loss. We believe this theoretical development reveals the importance of estimating the local stability ratio in the construction of well-behaved (last-iterate) schedule-free algorithms, particularly when hyperparameter-tuning budgets are a small fraction of the total budget, since noise-adaptivity and cheaper horizon-free tuning are most crucial in this regime. + +We consider the standard Machine Learning setup, where the task of learning a function $f: \mathbb{R}^q \to \mathbb{R}^k$ from samples $(X \in \mathbb{R}^q, Y \in \mathcal{Y}) \sim \mathcal{D}$ is decomposed into a parameterization $F: \mathbb{R}^d \times \mathbb{R}^q \to \mathbb{R}^k$ and a loss function $\ell: \mathcal{Y} \times \mathbb{R}^k \to \mathbb{R}$ , with the aim to minimize $\mathbb{E}[\ell(Y, f(x)) | X = x]$ . A parameter $\theta \in \mathbb{R}^d$ yields a predicted function $f_\theta = F(\theta, -): \mathbb{R}^q \to \mathbb{R}^k$ , whose quality is evaluated according to $\mathcal{L}(\theta) = \mathbb{E}_{X,Y}[\ell(Y, F(\theta, X))]$ defining a loss function $\mathcal{L}: \mathbb{R}^d \to \mathbb{R}$ to be minimized. + +Typical scenarios include least-squares regression $\ell(u, v) = \|u - v\|_2^2$ for $\mathcal{Y} = \mathbb{R}^k$ , with functional optimum $x \mapsto \mathbb{E}[Y | X = x]$ ; and classification with cross-entropy $\ell(y, u) = -u_y + \log \sum_i \exp(u_i)$ for $\mathcal{Y} = [k]$ . The success of deep learning has taken this long past the historically well-studied linear case of $d = q \times k$ , with impressive empirical performance lacking a strong theoretical support. + +Using small batches of data to estimate gradients is one of the keys used to scale up such settings, leading to stochastic iterative algorithms. This randomness induces failures of constant-step gradient descents, which saturate and fail to minimize the loss past a threshold (e.g. Wilson and Martinez [2001]). This leads to the use of schedulers to shrink the learning rate over time. Setting it too low slows down optimization, and too high recovers saturated losses, thus even more hyperparameters are added to define schedulers of varying decay rates such as $\eta_t = \eta_0 \cdot t^{-\alpha}$ for $\alpha \in [0,1]$ . + +Related works. The elimination of such hyperparameters, by a theory-backed choice of algorithm, has naturally been an active study of research. Such tentatives include the early "Adagrad" [Duchi et al., 2011] and "Adadelta" [Zeiler, 2012] adaptive algorithms, but also "AC-SA" [Lan, 2012, Sec 3.1] and its more recent variants such as "Schedule-free SGD" [Defazio et al., 2024]. One branch of this effort chose to model the loss $\mathcal{L}$ as Lipschitz, i.e. having bounded gradients, see for instance the "COCOB" [Orabona and Tommasi, 2017, Thm 1] and "D-Adapt" algorithms [Defazio and Mishchenko, 2023, Thm 3] with known Lipschitz constant. Despite the immediate incompatibility with the least-squares objective, this modeling choice is supported by the Lipschitz-continuity of the ReLU non-linearity $x\mapsto \max (0,x)$ which is not differentiable (and thus not smooth) at the origin. + +The Lipschitz-model, typically used with convexity of $\mathcal{L}$ in addition, does not produce guarantees for the last iterate, but for the average $\frac{1}{T}\sum_{t}x_{t}$ or ergodic average $\sum_{t}\eta_{t}x_{t} / \sum_{t}\eta_{t}$ of iterates [Garrigos and Gower, 2023, Thm 9.6 - 9.12]. On the contrary, there is growing evidence that such aggregation is not mandatory1 (e.g. the same reference Orabona and Tommasi [2017, Algorithm 2] from the Lipschitz-model branch does not use averaging on neural network experiments), and possibly detrimental in non-convex cases [Zhou et al., 2020, Figure 4]. Other requirements such as bounded domain are also questionable. A second branch of this research effort thus focuses on a smooth model of the loss $\mathcal{L}$ , i.e. Lipschitz-continuous gradients, which yield good last-iterate predictions (see Garrigos and Gower [2023, Thm 4.3] for the deterministic case and Bach and Moulines [2011, Thm 4] for the stochastic case with power schedule). By continuous-differentiability, these losses have gradients converging to zero near the global minimum which naturally leads to smaller steps, contrary to Lipschitz losses. This lack of averaging is also supported, outside the convex case using Jensen's inequality, by the lack of guarantees on the loss of the average iterate, even if the averaged loss is controlled. + +Although this smooth model does not immediately fit the ReLU-based networks, experiments with smooth non-linearities often match performance of ReLU networks [Clevert et al., 2016, Elfwing et al., 2018, Sitzmann et al., 2020]. Moreover, any continuously differentiable function is smooth on compact domains, which supports the idea that this model will also be a good description of training dynamics naturally constrained to a compact set, e.g. by a regularization. + +Contributions. We introduce in Sec. 1 the stability ratio, as a measure of gradient stochasticity, and as a shrinkage of SGD learning rates to obtain an algorithm adaptive to noise levels, formalized in Sec. 2. We show that this ratio is computable from samples and give an estimator. We prove in Sec. 3 how this adaptively achieves the optimal last-iterate rates of SGD at various noise levels, without the need to tune the learning rate to the (unknown) noise level or training horizon. We validate these statements with experiments in convex and deep learning scenarios in Sec. 4. + +# 1 Stability Ratio: ensuring (strict) expected loss decrease + +In gradient descents with large amounts of noise, a common practice is to shrink the step-size, backed by the standard intuition that lower learning rates are required to converge to low loss values. To quantify how much lower, we define a measure of "relative" or "normalized" noise level (between zero and one), inspired by classical smooth stochastic analysis, and show shrinking by this quantity achieved the desired adaptive result. For a random variable $X \in \mathbb{R}^d$ (not identically zero) with $0 < \mathbb{E}\left[\|X\|_2^2\right] < +\infty$ , we denote as "Stability Ratio" the quantity $\operatorname{Stab}(X) \in [0,1]$ defined by + +$$ +\operatorname {S t a b} (X) = \frac {\| \mathbb {E} [ X ] \| _ {2} ^ {2}}{\mathbb {E} [ \| X \| _ {2} ^ {2} ]} +$$ + +Note, for $\mu = \mathbb{E}[X] \neq 0$ , that $\mathbb{V}[X] = \sigma^2$ implies $\mathrm{Stab}(X) = 1 / (1 + \sigma^2 / \| \mu \|_2^2)$ , thus smaller variance leads to a stability ratio closer to 1. On the other hand, near-zero mean and non-negligible variance give stability ratios approaching zero: these are the estimates causing instabilities in the loss. The lower the stability ratio of the gradient, the lower the step-size must be taken to avoid instability. + +For an SGD sequence $(\theta_t \in \mathbb{R}^d)$ , using unbiased stochastic gradient estimates $G_{t+1} \approx \nabla \mathcal{L}(\theta_t)$ to compute $\theta_{t+1} = \theta_t - \eta_t G_{t+1}$ , the loss variation for a $\beta$ -smooth function is at most + +$$ +\mathcal {L} (\theta_ {t + 1}) - \mathcal {L} (\theta_ {t}) \leq - \eta_ {t} \cdot (\nabla \mathcal {L} (\theta_ {t}) \cdot G _ {t + 1}) + \frac {\beta}{2} \eta_ {t} ^ {2} \| G _ {t + 1} \| _ {2} ^ {2} +$$ + +When $G_{t + 1} = \nabla \mathcal{L}(\theta_t)$ , this is minimized at $\eta_t = 1 / \beta$ , as in classical smooth deterministic analysis. In the stochastic case, taking the expectation and minimizing immediately gives $\eta_t = \mathrm{Stab}(G_{t + 1}) / \beta$ . Moreover, $\eta_t \leq \mathrm{Stab}(G_{t + 1}) / \beta$ ensures that $\mathbb{E}_{G_{t + 1}}[\mathcal{L}(\theta_{t + 1})] - \mathcal{L}(\theta_t) \leq -\eta_t\| \nabla \mathcal{L}(\theta_t)\| _2^2 /2$ , and thus a decrease similar to that of gradient flow. Convergence is slowed down by a factor Stab $(G_{t + 1})$ , that is equal to 1 in the deterministic regime, and small in the high variance regime (where $\| \nabla \mathcal{L}(\theta_t)\| _2^2\approx 0$ and $\mathbb{E}\left[\| G_{t + 1}\| _2^2\right]\gg 1$ ). In what follows, we refer to SGD with such adaptive step-sizes as Stab-SGD, and discuss how to estimate this stability ratio in practice in Sec. 2.2. + +# 1.1 Adaptivity to noise level of stability-adjusted learning rates + +Two typical regimes of SGD are depicted in Figure 1.1, with quadratic problems and injected additive gaussian noise $\varepsilon \sim \mathcal{N}(0,\sigma_0^2 I)$ for gradient estimates (varying $\sigma_0$ ), for a total variance of $\sigma^2 = \sigma_0^2 d$ . + +$$ +\mathcal {L}: x \in \mathbb {R} ^ {d} \mapsto \frac {1}{2} \sum_ {i < d} \frac {1}{1 + i} x _ {i} ^ {2}, \quad x ^ {0} = (1) _ {i \in [ d ]} \in \mathbb {R} ^ {d}, \quad d = 2 5 0 \tag {QSC} +$$ + +$$ +\mathcal {L}: x \in \mathbb {R} ^ {d} \mapsto \frac {1}{2} \sum_ {i < d} 2 ^ {- i} x _ {i} ^ {2}, \quad x ^ {0} = \left(2 ^ {- i}\right) _ {i \in [ d ]} \in \mathbb {R} ^ {d}, \quad d = 2 5 \tag {QWC} +$$ + +Both losses are smooth with parameter $\beta = 1$ . Problem QSC has $\mathcal{L}(x^0) \approx 3.05$ , and is $\mu_0$ -strongly convex with $\mu_0 = 1/250 = 4 \cdot 10^{-3}$ . On the other hand, Problem QWC has $\| x^0 - x^\star \|_2^2 \approx 1.33$ , $\mathcal{L}(x^0) \approx 0.5714$ , and is $\mu_1$ -strongly convex with $\mu_1 = 2^{-25} \approx 3 \cdot 10^{-8}$ , which is too small to play a quantitative role in experiments, hence it is likely better described by weakly-convex smooth theory. + +![](images/479f73e4eea39e7bc71cb56cb785428a9364ada32f9de33928e0d7d7d6bd6d40.jpg) +(a) Problem QSC, low noise $(\sigma^2 = 10^{-16} \cdot d)$ + +![](images/55ed499057b0bc9c58a9f4a7c46e5cfe7a2c8bee860416f48db11f35963285e3.jpg) +Figure 1: Mean excess loss of SGD and variants. The hyperparameter of SGD (both constant and scheduled) is tuned by grid search at $10^{3}$ iterations (vertical dashed line). Bounds of Sec. 3 are presented with dotted lines. Details of all experimental protocols deferred to Sec. 4. + +![](images/04b5711fbb1cd9a2c989a2fb285d679224f2e833886c898d2082d709af87e5bf.jpg) +(b) Problem QWC, high noise $(\sigma^2 = d)$ + +![](images/83f0387df0df599e58b954f0499d11923c2d549df67d06263151364242adcafa.jpg) + +In (near-)deterministic settings, large step sizes are necessary, and decreasing too much gives slow asymptotic convergence (see Fig. 1a). Fast-decreasing schedulers emulating these large (near-constant) learning rates need huge initial learning rates, causing initial explosions which are prohibitive in deep learning. On the contrary, in more noisy settings (see Fig. 1b), shrinking the learning rate sufficiently is necessary, and constant learning rates trying to lower the saturation threshold will use much lower learning rates causing large initial plateaus. In both cases, the trajectory of Stab-SGD seems a more reasonable balance to strive for: no explosion, no initial plateau, no saturation. + +![](images/6c7496ce74e478e874b235ba09a569a1d9086a49c80cbf06940a59cc0b862dca.jpg) +(a) Constant $\eta_{t} = \eta_{0}$ + +![](images/56244c8221a048aa17657f701a64a7c3b90a52ae37eda22c0228670c048081bd.jpg) +(b) Schedule $\eta_t = \eta_0 \cdot t^{-1/2}$ + +![](images/85952d95bb3d11f696f8fa3a21f054a24c191c36b514b49f53e2145050754fd8.jpg) +(c) Schedule $\eta_t = \eta_0 \cdot t^{-1}$ + +![](images/4af7465ea51d227e3eb781d8a831e64c494d0242afe04131bffb12e1d5bb9fbf.jpg) +Figure 2: Mean excess loss of various SGD schedulers on Problem QWC, $\sigma^2 = d$ . Horizon-dependent hyperparameters are still needed, with high sensitivity to perturbations of the noise-dependent $\eta_0$ . + +The use of schedulers does not eliminate the need to tune the learning rate (see Fig. 2b) and selection of learning rate decrease speed is not trivial (compare with Fig. 2c). Typical prescriptions are tuned to the target horizon $T \in \mathbb{N}$ , e.g. with the constant but horizon-dependent rate $\eta_t = C\sigma^{-1}T^{-1 / 2}$ . + +# 2 Stab-SGD: Stochastic Gradient Descent with stability-adapted step-sizes + +We build our formal statements in the rigorous formalism of stochastic processes, motivated by the crucial part that the step-sizes $\eta_t$ must depend on the local stability ratio of gradient estimates, which itself is a function of the iterates, therefore the step-sizes are random and must be handled carefully. + +We take $(\Omega, \mathcal{A}, \mathcal{P})$ to be a probability space, with a filtration $(\mathcal{F}_n)_{n \in \mathbb{N}}$ of $\mathcal{A}$ . A sequence of random variables $(X_n)_{n \in \mathbb{N}}$ is said to be "adapted" to $\mathcal{F}$ if $X_i$ is $\mathcal{F}_i$ -measurable for all $i \in \mathbb{N}$ . + +Intuition. The standard informal interpretation is that $\mathcal{F}$ models the passage of time, and $X$ is adapted to $\mathcal{F}$ if $X_{i}$ is "known" at time $i\in \mathbb{N}$ . In our case, if the sequence of iterates $(\theta_n)_n$ is adapted to $\mathcal{F}$ , then any deterministic function $Y_{t} = \phi (\theta_{t},\dots ,\theta_{0})$ of previous iterates is adapted to $\mathcal{F}$ as well. + +Definition 1 (Stochastic Gradient Descent, with unbiased gradients and stochastic step sizes). + +A stochastic gradient descent of $\mathcal{L}:\mathbb{R}^d\to \mathbb{R}$ is an $\widetilde{\mathcal{F}}$ -adapted sequence of random variables $(\theta_{n}\in \mathbb{R}^{d})_{n\in \mathbb{N}}$ together with two $\mathcal{F}$ -adapted sequences $(G_{n}\in \mathbb{R}^{d})_{n\in \mathbb{N}}$ and $(\eta_{n}\in \mathbb{R}_{+})_{n\in \mathbb{N}}$ , such that for all $t\in \mathbb{N}$ , it holds $\theta_{t + 1} = \theta_t - \eta_t\cdot G_{t + 1}$ and $\mathbb{E}\left[G_{t + 1}\mid \mathcal{F}_t\right] = \nabla \mathcal{L}(\theta_t)$ + +Definition 2 (Conditional Stability Ratio). The Stability Ratio of a random variable $X \in \mathbb{R}^d$ conditionally on $\mathcal{F}_t$ is defined for any $t \in \mathbb{N}$ as: $\mathrm{Stab}(X \mid \mathcal{F}_t) = \left\| \mathbb{E}[X \mid \mathcal{F}_t] \right\|_2^2 / \mathbb{E}\left[\| X \|_2^2 \mid \mathcal{F}_t\right]$ . + +# 2.1 Stab-SGD: A noise-adaptive algorithm with stability oracles + +The Stab-SGD iterates $(\theta_t \in \mathbb{R}^d)_{t \in \mathbb{N}}$ of loss $\mathcal{L}: \mathbb{R}^d \to \mathbb{R}$ are defined as + +$$ +\theta_ {t + 1} = \theta_ {t} - \eta_ {t} \cdot G _ {t + 1} \quad \eta_ {t} = \frac {1}{\beta} \operatorname {S t a b} \left(G _ {t + 1} \mid \mathcal {F} _ {t}\right) +$$ + +for any adapted sequence $(G_{t})_{t}$ satisfying $\mathbb{E}[G_{t + 1}|\mathcal{F}_t] = \nabla \mathcal{L}(\theta_t)$ and $\mathbb{V}[G_{t + 1}|\mathcal{F}_t] < +\infty$ + +Note that Stab-SGD has a single hyperparameter $\beta \in \mathbb{R}_+$ , which must be set below the smoothness constant of $\mathcal{L}$ . There is no noise-hyperparameter and no horizon-hyperparameter, contrary to SGD bounds typically $^4$ using step-size $\eta_t \propto \sigma^{-1} / \sqrt{T}$ to give bounds at horizon $T \in \mathbb{N}$ under variance $\sigma^2$ . Stab-SGD is a noise-adaptive algorithm (conditionally on access to stability ratios), in the sense that it depends on the realized noise level only through the stability ratio, which can be adaptively estimated at every step. This single algorithm adaptively achieves all the convergence rates of Table 1. + +Table 1: Convergence rate of Stab-SGD under affine variance $\mathbb{V}\left\lbrack {{G}_{t + 1} \mid {\mathcal{F}}_{t}}\right\rbrack \leq \alpha \parallel \nabla \mathcal{L}\left( {\theta }_{t}\right) {\begin{Vmatrix}2 \end{Vmatrix}}^{2} + {\sigma }^{2}$ . + +
NoiseE[ L(θT+1)] - L*E[ 1/T ∑t<T ||∇L(θt)||2/2] Non-convex β-smooth
Convex β-smoothμ-strongly convex β-smooth
σ2=0O(T-1)O(exp(-1/1+αμβT))O(T-1)
σ2>0O(T-1/3)O(T-1)O(T-1/2)
+ +Rates in Table 1 are presented in expectation for the last iterate. In particular, the $\mathcal{O}(T^{-1/3})$ rate in the weakly-convex smooth setting matches Bach and Moulines [2011, Theorem 4] (conjectured to be the optimal horizon-free last-iterate rate for SGD with schedule $\eta_t = \eta_0 t^\kappa$ and achieved for $\kappa = -2/3$ , see reference for details). The weakly-convex case additionally assumes that there exists $\theta^\star \in \mathbb{R}^d$ such that $\mathcal{L}(\theta^\star) = \mathcal{L}^\star = \inf \mathcal{L}$ , see Theorem 1 for the complete statement. + +# 2.2 Estimations of Stability Ratio from samples + +A natural estimator for $\operatorname{Stab}(X)$ consists in replacing expectations with averages over $n$ iid samples. Unfortunately, this estimator is strongly biased towards 1 when the number of samples is small. We thus propose another estimator using Jackknife resampling for the numerator [Quenouille, 1956]. + +3Without loss of generality, we can assume that no $G_{t + 1}$ is identically zero by skipping such iterations. +4See Garrigos and Gower [2023, Thm 5.5] after canceling gradients with respect to step-size. +A slightly altered $\eta_t = \min(1/2\beta, \eta_0 / \sqrt{t})$ was shown to break this conjecture in Liu and Zhou [2023], reaching improved rate $\mathcal{O}(\log(T) / \sqrt{T})$ . But it does not reach the $\sigma = 0$ or $\mu > 0$ fast rates without modifying $\eta_t$ . Thus the question of getting improved rate for the bottom-left case while retaining adaptivity is left open. + +Definition 3. The Jackknife estimator of Stab $(X)$ from iid samples $(X_{i}\in \mathbb{R}^{d})_{i\in [n]}$ is + +$$ +R _ {n} = \frac {1}{n - 1} \frac {\sum_ {i} \sum_ {j \neq i} \langle X _ {i} , X _ {j} \rangle}{\sum_ {i} \| X _ {i} \| _ {2} ^ {2}} +$$ + +This can be computed by constructing the sequences $(M_i\in \mathbb{R}^d)_{i\in [n + 1]}$ and $(Z_{i}\in \mathbb{R}^{d})_{i\in [n + 1]}$ from $M_0 = 0\in \mathbb{R}^d$ and $Z_{0} = 0\in \mathbb{R}$ , as $M_{i + 1} = M_{i} + (X_{i} - M_{i}) / (i + 1)$ to compute the mean, and $Z_{i + 1} = Z_{i} + (\| X_{i}\|_{2}^{2} - Z_{i}) / (i + 1)$ for the second moment, then $R_{n} = \frac{n}{n - 1} (\| M_n\| _2^2 -Z_n) / Z_n$ . This gives a numerically stable algorithm with $\mathcal{O}(1)$ space complexity to estimate the stability ratio. + +Lemma 1 (Relative error of stability estimation). Let $(X_{i}\in \mathbb{R}^{d})_{i\in [n]}$ be iid random variables. Define $J_{n} = \frac{1}{n(n - 1)}\sum_{i\neq j}X_{i}\cdot X_{j}\in \mathbb{R}$ and $Z_{n} = \frac{1}{n}\sum_{i}\| X_{i}\|_{2}^{2}\in \mathbb{R}_{+}$ , then $R_{n} = \operatorname {clip}_{[0,1]}(J_n / Z_n)\in [0,1]$ . + +Let $\mu = \mathbb{E}\left[X\right]$ , and $\sigma^2 = \mathbb{E}\left[\| X - \mu \| _2^2\right]$ , and $\kappa = \mathbb{E}\left[\| X\| _2^4\right] / \mathbb{E}\left[\| X\| _2^2\right]^2$ . If $R_{\star} = \mathrm{Stab}(X)\neq 0$ + +$$ +\mathbb {E} \left[ \left(\frac {R _ {n} - R _ {\star}}{R _ {\star}}\right) ^ {2} \right] \leq R _ {\star} ^ {- 1} \frac {4 4 + 4 \kappa}{n - 1} + R _ {\star} ^ {- 2} \exp \left(- \frac {n}{8 \kappa}\right) +$$ + +In particular (when clipping to $[0,1])$ , $R_{n}\to R^{\star} = \mathrm{Stab}(X)$ with high probability, so this estimator is consistent. This lemma is a direct consequence of Lemma A.9. Note that for isotropic multivariate normal random variables $X\in \mathbb{R}^d$ , such as $X\sim \mathcal{N}(0,\sigma^2 I)$ , it holds $\kappa \leq 1 + 3 / d$ (for any $\sigma$ ). Thus the number of samples needed to estimate a stability ratio $R_{\star} > 0$ is often of order $n\propto R_{\star}^{-1}$ . The kurtosis $\kappa$ is used to quantify the number of samples needed to estimate the variance. + +# 3 Convergence analysis + +The tactic used for all following proofs closely tracks the continuous-time analogue by integration along gradient flows (i.e. $\left[\mathrm{d}\Phi (\mathcal{L}_t)\cdot \partial_t\mathcal{L}\leq -1\Rightarrow \Phi (\mathcal{L}_t)\leq \Phi (\mathcal{L}_0) - t\right]$ for any desingularizer $\Phi :\mathbb{R}_{+}^{*}\to \mathbb{R}$ , such as $\Phi = \log$ ). This is done by leveraging the "sufficient decrease" inequality $^8$ $\mathbb{E}\left[\mathcal{L}(\theta_{t + 1})\mid \mathcal{F}_t\right] - \mathcal{L}(\theta_t)\leq -\frac{1}{2}\eta_t\| \nabla \mathcal{L}(\theta_t)\| _2^2$ (obtained by construction of Stab-SGD) together with the variance control assumption $\mathbb{V}\left[G_{t + 1}\mid \mathcal{F}_t\right]\leq \alpha \| \nabla \mathcal{L}(\theta_t)\| _2^2 +\sigma^2$ , to obtain an "average sufficient decrease" inequality $\mathbb{E}\left[\mathcal{L}(\theta_{t + 1}) - \mathbb{E}\left[\mathcal{L}(\theta_t)\right]\leq -\frac{1}{2\beta}\psi \left(\mathbb{E}\left[||\nabla \mathcal{L}(\theta_t)\| _2^2\right]\right)$ , for a well-chosen convex and increasing function $\psi$ , namely $\psi :u\mapsto u^{2} / (\sigma^{2} + (1 + \alpha)u)$ for this affine variance control. + +This result can be composed with any bound of the form $\mathbb{E}\left[\|\nabla\mathcal{L}(\theta_t)\|_2^2\right] \geq \varphi(\mathbb{E}\left[\mathcal{L}(\theta_t)\right] - \mathcal{L}^\star)$ , to bound the optimization gap $\Delta_t = \mathbb{E}\left[\mathcal{L}(\theta_t)\right] - \mathcal{L}^\star$ as $\Delta_t \leq \Phi^{-1}(\Phi(\Delta_0) + t/(2\beta))$ , where $\Phi$ is obtained by integration of $\mathrm{d}\Phi(u) = 1 / (\psi \circ \varphi)(u)$ . Different assumptions, leading to various choices of $\varphi$ , yield different convergence speeds, as integrated into the function $\Phi$ . In particular, local Kurdyka-Lojasiewicz inequalities $\|\nabla\mathcal{L}(\theta)\|_2^2 \geq \varphi(\mathcal{L}(\theta) - \mathcal{L}^\star)$ for convex functions $\varphi$ immediately satisfy the previous condition in expectation (such as $\varphi(x) = 2\mu x$ for $\mu$ -strong convexity). + +# 3.1 Convergence statements with stability oracles + +Assumption 1 (Stab-SGD with stability oracle and affinely-bounded variance). + +This set of assumptions is satisfied if there are constants $\beta \in \mathbb{R}_+^*$ , $\alpha \in \mathbb{R}_+$ and $\sigma \in \mathbb{R}_+$ such that: + +- $\mathcal{L}:\mathbb{R}^d\to \mathbb{R}$ is differentiable and uniformly $\beta$ -smooth +- $(\theta_t \in \mathbb{R}^d, G_t \in \mathbb{R}^d, \eta_t \in \mathbb{R}_+^*)_{t \in \mathbb{N}}$ is an SGD of $\mathcal{L}$ (Definition 1) +- $\forall t\in \mathbb{N}$ , $\mathbb{V}[G_{t + 1}|\mathcal{F}_t]\leq \alpha \| \nabla \mathcal{L}(\theta_t)\| _2^2 +\sigma^2$ (affinely bounded variance) +- $\forall t\in \mathbb{N}$ , $\eta_t = \mathrm{Stab}(G_{t + 1}|\mathcal{F}_t) / \beta$ (strong stability condition) + +In such a case, the sequence of random variables $\theta :\mathbb{N}\to \mathbb{R}^d$ are called Stab-SGD iterates. + +The following theorems are derived from Corollary A.1, Corollary A.2, and Proposition A.3. + +Theorem 1 (Weakly convex smooth rate). If $\mathcal{L}:\mathbb{R}^d\to \mathbb{R}$ is convex, uniformly $\beta$ -smooth, and if there exists $\theta^{\star}\in \Theta$ such that $\mathcal{L}^{\star} = \mathcal{L}(\theta^{\star})$ , then for any Stab-SGD iterates $\theta :\mathbb{N}\rightarrow \mathbb{R}^d$ satisfying Assumption 1, and if + +$$ +T \geq \frac {2}{3} \frac {\beta D _ {0} ^ {4} \sigma^ {2}}{\varepsilon^ {3}} + (1 + \alpha) \frac {\beta D _ {0} ^ {2}}{\varepsilon} +$$ + +then $\mathbb{E}\left[\mathcal{L}(\theta_{T + 1})\right] \leq \mathcal{L}^{\star} + \varepsilon$ , where $D_0^2 = \mathbb{E}\left[\| \theta_0 - \theta^\star \|_2^2\right]$ measures initial distance to optimum. + +Theorem 2 (Strongly convex smooth rate). If $\mathcal{L}:\mathbb{R}^d\to \mathbb{R}$ is uniformly $\beta$ -smooth and $\mu$ -strongly convex, then any Stab-SGD iterates $\theta :\mathbb{N}\rightarrow \mathbb{R}^d$ satisfying Assumption 1, and if + +$$ +T \geq \frac {\sigma^ {2} \beta}{2 \mu^ {2} \varepsilon} + (1 + \alpha) \frac {\beta}{\mu} \log \left(\frac {\Delta_ {0}}{\varepsilon}\right) +$$ + +then $\mathbb{E}\left[\mathcal{L}(\theta_{T + 1})\right] \leq \mathcal{L}^{\star} + \varepsilon$ , where $\Delta_0 = \mathbb{E}\left[\mathcal{L}(\theta_0)\right] - \mathcal{L}^{\star}$ measures the initial optimization gap. + +Theorem 3 (Non-convex rate). If $\mathcal{L}:\mathbb{R}^d\to \mathbb{R}$ is uniformly $\beta$ -smooth, for any Stab-SGD iterates $\theta :\mathbb{N}\rightarrow \mathbb{R}^d$ satisfying Assumption 1, + +$$ +\forall T \in \mathbb {N}, \quad \mathbb {E} \left[ \frac {1}{T} \sum_ {t < T} \| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2} \right] \leq (1 + \alpha) \frac {2 \beta \Delta_ {0}}{T} + \sqrt {\frac {2 \beta \Delta_ {0} \sigma^ {2}}{T}} +$$ + +where $\Delta_0 = \mathbb{E}\left[\mathcal{L}(\theta_0)\right] - \mathcal{L}^\star$ measures the optimization gap at initialization. + +# 3.2 Inline Stability Estimation + +To incorporate the estimation of the gradient's stability ratio in the algorithm at little overhead cost, we propose Algorithm 1, with access to noisy gradients but without a stability oracle. + +This algorithm uses three parameters to control estimation overhead: + +- a sample overhead $\zeta \in \mathbb{R}_{+}^{*}$ (of order 10 to 100) +- a time step $\kappa \in \mathbb{R}_{+}^{*}$ +- a time exponent $\gamma \in [0,1]$ + +The most conservative configuration $(\gamma = 0, \kappa = 1)$ estimates the stability ratio at every step. However, if the stability ratio is expected to be relatively continuous, then a looser configuration $(\gamma = 1)$ will perform only logarithmically many estimations with respect to the horizon, which is a minimal overhead. + +In looser configurations, incorrect ratios could yield temporary saturations (overestimation), or temporary slowdowns (underestimation). + +Input: $x_0 \in \mathbb{R}^d$ , $\eta_0 \in \mathbb{R}_+^*$ , $\zeta \geq 1$ , $\kappa \in \mathbb{R}_+^*$ , $\gamma \in [0,1]$ ( $S, T) \gets (S_0 = 1 \in [0,1], T_0 = 1 \in \mathbb{R}_+^*)$ + +for $k\in \mathbb{N}$ do + +if $(k = 0)$ or $(k\geq T)$ then + +$$ +n \leftarrow \lceil \zeta / S \rceil \in \mathbb {N} +$$ + +$$ +(M _ {0}, Z _ {0}) \leftarrow (0 \in \mathbb {R} ^ {d}, 0 \in \mathbb {R}) +$$ + +for $i\in [n]$ do + +$$ +v _ {i} \leftarrow G _ {k, i} \in \mathbb {R} ^ {d} [ \text {e s t i m a t e o f} \nabla \mathcal {L} (x _ {k}) ] +$$ + +$$ +M _ {i + 1} \leftarrow M _ {i} + \left(v _ {i} - M _ {i}\right) / (i + 1) +$$ + +$$ +Z _ {i + 1} \leftarrow Z _ {i} + \left(\left\| v _ {i} \right\| _ {2} ^ {2} - Z _ {i}\right) / (i + 1) +$$ + +end + +$$ +S \leftarrow \frac {n}{n - 1} \frac {\| M _ {i} \| _ {2} ^ {2} - Z _ {i}}{Z _ {i}} +$$ + +$$ +T \leftarrow T + \kappa \cdot T ^ {\gamma} +$$ + +end + +$$ +m _ {k} \leftarrow G _ {k} \in \mathbb {R} ^ {d} [ \text {f r e s h e s t i m a t e o f} \nabla \mathcal {L} (x _ {k}) ] +$$ + +$$ +x _ {k + 1} \leftarrow x _ {k} - \left(\eta_ {0} \cdot S\right) \cdot m _ {k} +$$ + +end + +# Algorithm 1: Inline Stab-SGD + +While we can't guarantee the quality of the looser configurations without additional assumptions such as continuity of noise variance, we observe empirically that loose options such as $(\gamma = 1,\kappa = 0.5)$ still display all key properties of Stab-SGD: no initial plateau or explosion, and no saturation. + +Note on overhead cost. For a total of $T$ gradients queried at stability ratios above $s_{\star} > 0$ , at most a fraction $c / (1 + c) \in ]0,1[$ of queries are dedicated to stability estimation, where $c \in \mathbb{R}_+$ can be controlled by tuning $\kappa$ (e.g. set to $c \leq 1$ ). If $\gamma = 1$ , then $c \leq \zeta s_{\star}^{-1}\kappa^{-1}\log (T) / T$ is vanishing with $T$ . If $\gamma = 0$ , then $c \leq \zeta s_{\star}^{-1}\kappa^{-1}$ . For the exponent $\alpha$ , the movement's characteristic timescale is estimated using $\mathbb{E}\left[\| G_{t + k + 1}\| |\mathcal{F}_t\right] \leq (\|\nabla \mathcal{L}(\theta_t)\|^2 + \sigma^2)^{1/2}$ (unrigorously) without expectations for a quick approximation, and smoothness as $\| \nabla \mathcal{L}(x)\|_2^2 \leq 2\beta (\mathcal{L}(x) - \mathcal{L}^\star)$ with $\mathcal{L}^\star = 0$ for simplicity, + +$$ +\left\| \theta_ {t + \Delta t} - \theta_ {t} \right\| \leq \sum_ {k \leq \Delta t} \eta_ {t} \| G _ {t + k + 1} \| \lesssim \sum_ {k \leq \Delta t} \frac {1}{\beta} \frac {\left\| \nabla \mathcal {L} \left(\theta_ {t + k}\right) \right\| ^ {2}}{\left(\left\| \nabla \mathcal {L} \left(\theta_ {t + k}\right) \right\| ^ {2} + \sigma^ {2}\right) ^ {1 / 2}} \leq \frac {2}{\sigma} \sum_ {k \leq \Delta t} \mathcal {L} (\theta_ {t + k}) +$$ + +If $\mathcal{L}(\theta_t) \leq C_0 t^{-1/3}$ , this bound is at most $C_1 \Delta t \cdot t^{-1/3}$ , so the unit-scale movements' characteristic time is at most $\Delta t \approx C_1^{-1} t^{1/3}$ . This quick calculation suggests that even in the worst case, $\gamma = 1/3$ should remain a safe option. Similarly, a rate $\mathcal{L}(\theta_t) \leq C_0 t^{-1}$ could use $\gamma = 1$ safely, but we conjecture that such loose settings will be useable far outside this regime. Characterisation of precise noise-continuity hypotheses under which such choices are provably safe is left for future work. + +# 4 Experiments + +Methods. We perform experiments in two stages, first training for $T_0 \in \mathbb{N}$ (tuning horizon) iterations on a grid of hyperparameters ( $\log \eta_0$ from -7 to +5 by increments of 0.5, a total of $k = 25$ values). We then select the best hyperparameter (at $T_0$ ) and train with this value for $T \in \mathbb{N}$ iterations. The fraction of the total budget spent on hyperparameter tuning is thus $kT_0 / (kT_0 + T)$ , and the tuning overhead (excess cost of tuning relative to training) is $kT_0 / T$ . These quantities are rarely reported on large-scale experiments failing to take hyperparameter-tuning costs into account, but there is a common intuition that popular algorithms require a massive fraction of budget allocated to tuning. + +# 4.1 Comparisons with concurrent schedule-free algorithms + +Cheap regime: low noise, strong convexity. Fig. 3 presents loss as a function of tuning horizon. + +Vertical gaps within curves indicate the final gap in loss if less budget is spent on tuning. + +The sensitivity of SGD is visible on the right (the noisedominated regime). The long-horizon optimal learning rate cannot be selected well on short tuning horizons (which do not enter the noise regime), a property that is likely shared by deep learning settings. + +![](images/b8fcfefec67f5b37bf787fc3a64a32aa297fa78d5939dfdfa336772fc451dc8b.jpg) +Figure 3: Mis-tuning cost on Problem QSC, $\sigma^2 = 10^{-16} \cdot d$ . + +![](images/6117804cef82588187254ca795cc8b7c33bbecd556e560902a9b400f8b19325e.jpg) +Figure 4: Evolution of the loss on Problem QSC, $\sigma^2 = 10^{-16} \cdot d$ . Tuning horizon as dashed line. + +Figure 4 depicts the evolution of the loss over time for a tuning horizon at $T_0 = 10^3$ . Algorithms designed for the noisy regime alone (such as COCOB and D-Adapt, which use iterate-averaging) fail to take advantage of strong convexity, leaving them 8 orders of magnitude behind at $10^4$ iterations. + +Expensive regime: smooth with high noise. Figure 5 presents mean loss as a function of tuning horizon for Problem QwC. Each training run at $10^{7}$ iterations takes about one hour on our CPUs. + +Slope indicates sensitivity of the hyperparameter to the tuning horizon. Algorithms with large slopes are only usable if essentially all budget is spent tuning the sensitive parameter. + +At high noise with this training horizon $(10^{7})$ , SGD only outperforms Stab-SGD if at least $71\%$ of the total budget is spent on hyperparameter tuning, i.e. if an extra $+250\%$ of the training budget is spent tuning at $T_{0} = 10^{6}$ horizon. + +![](images/80a34ec300b588598815ee8944217a407b812c5b4d8cdd554f946130bbbed61e.jpg) +Figure 5: Mis-tuning cost on Problem QWC, $\sigma^2 = d$ + +![](images/d74df86445483ef11bdba25a9d0807c58c27090eb8d55db1c525081676dac3f1.jpg) + +Algorithms previously well-performing (such as "Schedule-Free SGD") are not as good in this regime, sometimes even indistinguishable from equivalently-tuned SGD. On the contrary, algorithms designed for this setting (e.g. COCOB) perform much better. This leaderboard reversal induces a difficulty to choose the best algorithm with unknown noise level. Stab-SGD gives consistent performance in both settings. The price of this adaptivity is apparent in both cases, but not necessarily prohibitive. + +![](images/822c57db2b7d505ff904b880f6c5f41d86ea5f2a43e504819b29d021d937d199.jpg) +Figure 6: Evolution of the loss on Problem QWC, $\sigma^2 = d$ . Tuning horizon as dashed line. + +![](images/b6b85a7986591370b2442751c354a72cfd48bf26a977484a5bd7c387107a108a.jpg) + +Although its asymptotic performance is slightly suboptimal compared to other methods, and does not achieve the minimax optimal rate of averaging methods, the complete absence of noise-dependent tuning of the hyperparameter, and reasonable properties (no plateau, no explosion, no saturation) of the Stab-SGD trajectory make it an interesting research direction for schedule-free settings aiming for those properties, particularly when the hyperparameter-tuning cost is taken into account. + +The proof of last-iterate expected loss matching these observations also highlights the importance of the stability ratio of gradients in the development of smooth optimization with last-iterate guarantees, possibly better suited to the study of non-convex models such as neural networks. + +We conjecture that it will be possible to construct accelerated noise-adaptive algorithms which will be competitive not only on low tuning budgets, but also on high tuning budgets (right end of Figure 5) where Stab-SGD and its stability-oracle-free variant Algorithm 1 are found to be lacking, possibly due to a suboptimal asymptotic rate. Nonetheless, works on accelerated stochastic algorithms typically use hyperparameters with convoluted dependence on noise parameters, see for instance Jain et al. [2018, Thm 1] with impressive speed but four noise-dependent hyperparameters for the case of quadratic problems alone. Therefore, we suspect that an accelerated noise-adaptive horizon-free extension of Stab-SGD could be a vastly more complicated algorithm than the ones presented here. + +# 4.2 ResNet training experiments on CIFAR-10 + +Methods. We perform experiments on the CIFAR-10 image classification dataset [Krizhevsky, 2009] with the ResNet-56 architecture $^{14}$ [He et al., 2015a, Sec 4.2]. We compare with the aforementioned original ResNet publication, which uses a learning rate $10^{-1}$ for 32k iterations, then $10^{-2}$ for the next 16k and $10^{-3}$ for the last 16k, totaling 64k iterations (thresholds depicted by dashed vertical lines). We use batches of size 128 sampled without replacement for each epoch (391 batches / epoch). We restrict the hyperparameter search for $\log_{10}(\eta_0)$ to a grid from $-3$ to $+1$ by steps of 0.5, informed by choices in the original reference. We use an $\ell_2^2$ weight decay with $\lambda = 10^{-4}$ for all runs. + +We run Algorithm 1 with $\eta_0 = 10^{+1}$ , with the configuration $\kappa = 10^{-1}$ , $\gamma = 1$ and $\zeta = 100$ . To evaluate the overhead cost of stability-estimation, we provide both curves: oracle where the number of iterations is the number of weight-updating steps (effective iterations); and raw where iterations correspond to the total number of gradients queried, including gradients used for stability estimation. + +![](images/d4467db9ece04e0329d7b6c93289468aa66c0a736bc1ced764395104e63821cc.jpg) +Figure 7: ResNet-56 on CIFAR-10. Evolution of accuracy and loss, presented as medians and quartiles for error bars, for 20 seeds of Algorithm 1. Average runtime of 4h to 5h per seed on GPU. + +![](images/422e102f160de0ffa9311eaf92d30d71fbfa34a044ce9edf2e76a504e4576eb6.jpg) + +![](images/2126cad2a958fd9c5a7d82d18a4b4bbe446eb07771302e299c51608dc7ec5796.jpg) + +The results presented in Figure 7 show performance comparable between the oracle variant and SGD with tuned schedule. Without the need to tune a scheduler, this algorithm has correctly used a first large step-size then much lower, allowing it to break past the mid-training plateau incurred by SGD (visible at $32\mathrm{k}$ iterations). Nonetheless, the variance across seeds is significantly increased, and taking into account the cost of stability-ratio estimation (with the raw variant) we can estimate that it needs on the order of twice as many iterations for similar performance in this experiment. For context, the choice of scheduler must have been guided by experiments, say $k \in \mathbb{N}^*$ runs15, thus the total cost comparison with noise-dependent scheduler tuning is between $k \times T$ for the scheduled SGD, and $2T$ for Algorithm 1 (raw), which is in favor of the adaptive algorithm presented here as soon as $k > 2$ . + +Although not competitive on such problems at this stage of development, Alg. 1 remains a promising research direction, since it maintains in this non-convex setting the desired properties: no initial explosion or plateau, and no saturation requiring large learning rate shrinkage. It reaches lower loss than SGD with $\eta = 10^{-1}$ (before first threshold) without tuning a threshold (at $32\mathrm{k}$ ) or shrinking factor ( $\times 0.1$ ). + +Fig. 8 shows evolution of the Stability Ratio along trajectories. Shrinkage behavior is consistent with the original: small variations up to $10^{4}$ then decreasing by several orders of magnitude. The original tuned schedule shrinked learning rates at $32\mathrm{k}$ and $48\mathrm{k}$ . More details in Appendix C.2. + +![](images/b9082d3c846febdd9c801a34f6fab4ea4705b14babb454ff6486a1b25c7a7df9.jpg) +Figure 8: Stability along trajectory + +![](images/f725734692208f8381c500b24c570db4ff862e8e9ef66f115480f01e2882fb0b.jpg) +Figure 9: ResNet-56, loss and accuracy as a function of learning rate for SGD. + +![](images/d2ad1530caf36657c4cd08372b437334ce84b200689fcf623bf4e104733b4629.jpg) + +This is consistent with convex experiments, indicating that Stab-SGD enables selection of a larger base learning rate, which is automatically adapted to the noise level. Indeed, with the initial Stability Ratio around $10^{-1}$ , the effective learning rate of the first $10^{3}$ iterations is around $10^{0}$ , which is not far from the optimum observed for SGD over that period (see Fig. 9). The performance of the oracle variant (i.e. ignoring stability-estimation costs) showcases the competitive behavior that could be reachable for future works achieving cheaper stability estimations. + +Conclusion. We introduced the Stability Ratio, a natural measure of local relative noise of stochastic gradient estimates, yielding a schedule-free variant of SGD achieving real adaptivity to the noise level. We presented new theoretical tools to analyze this stochastic-step algorithm in convex, strongly convex and non-convex settings, with strong last-iterate guarantees in expectation, obtained by a stochastic version of Kurdyka-Lojasiewicz integration. We validated the adaptivity of this proposed algorithm with convex experiments showing that it outperforms algorithms not achieving the fast rate on strongly convex problems (such as COCOB or D-Adapt, developed for less regular settings), and that it remains in the competitive range without the need for a noise-dependent tuning of hyperparameters. We measured performance on ResNet networks for CIFAR-10 which further strengthened that when taking hyperparameter-tuning budgets into account, this last-iterate noise-adaptive algorithm retains reasonable performance on non-convex deep learning problems. This shows that future algorithms leveraging this idea together with improved estimates of the stability ratio along a training trajectory will likely be able to outperform extensively-tuned learning rate schedulers in deep learning scenarios. + +# Acknowledgements + +This work was supported by the French government managed by the Agence Nationale de la Recherche (ANR) through France 2030 program with the reference ANR-23-PEIA-005 (REDEEM project). It was also funded in part by the Groupe La Poste, sponsor of the Inria Foundation, in the framework of the FedMalin Inria Challenge. + +# References + +Francis Bach and Eric Moulines. Non-asymptotic analysis of stochastic approximation algorithms for machine learning. In J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, and K.Q. Weinberger, editors, Advances in Neural Information Processing Systems, volume 24. Curran Associates, Inc., 2011. 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Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: Schedule-free and noise-adaptive results of Sec. 3 are present in the abstract, along with matching experiments, as claimed. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: See end of Section 4.1 + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. + +# 3. Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +Answer: [Yes] + +Justification: See Section 3, where Theorem 1, Theorem 2 and Theorem 3 use assumptions. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [NA] + +Justification: No LLMs were used in the making of this paper. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Appendix + +In all the appendix, $\mathcal{L}:\mathbb{R}^d\to \mathbb{R}^d$ is a $\beta$ -smooth function for some $\beta \in \mathbb{R}_+^*$ , and $(\theta_t\in \mathbb{R}^d)_{t\in \mathbb{N}}$ is a stochastic gradient descent of $\mathcal{L}$ (Definition 1) with gradient estimates $(G_{t + 1}\in \mathbb{R}^{d})_{t\in \mathbb{N}}$ and step-sizes $(\eta_t\in \mathbb{R}_+^*)_{t\in \mathbb{N}}$ satisfying $\theta_{t + 1} = \theta_t - \eta_t\cdot G_{t + 1}$ and $\mathbb{E}[G_{t + 1}|\mathcal{F}_t] = \nabla \mathcal{L}(\theta_t)$ , where $\mathcal{F}$ is the time filtration. + +When appropriate, the variable $T \in \mathbb{N}$ denotes a horizon, $\mathcal{L}^{\star} = \inf \mathcal{L}$ is the infimum of the loss, and $\theta^{\star} \in \mathbb{R}^{d}$ is a global optimum $\mathcal{L}(\theta^{\star}) = \mathcal{L}^{\star}$ when it is assumed to exist. + +# A.1 Rates with stability oracle + +Lemma A.1 (Base reduction). + +If for all $t \leq T$ , it holds $\eta_t \leq \beta^{-1} \operatorname{Stab}(G_{t+1} \mid \mathcal{F}_t)$ (weak stability condition), then it holds + +$$ +\forall t \leq T, \quad \mathbb {E} [ \mathcal {L} (\theta_ {t + 1}) | \mathcal {F} _ {t} ] \leq \mathcal {L} (\theta_ {t}) - \frac {\eta_ {t}}{2} \| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2} +$$ + +Proof. By $\beta$ -smoothness of $\mathcal{L}$ , then simplifying conditional expectations, + +$$ +\begin{array}{l} \mathcal {L} \left(\theta_ {t + 1}\right) \leq \mathcal {L} \left(\theta_ {t}\right) - \eta_ {t} \nabla \mathcal {L} \left(\theta_ {t}\right) \cdot G _ {t + 1} + \frac {1}{2} \beta \eta_ {t} ^ {2} \| G _ {t + 1} \| _ {2} ^ {2} \\ \mathbb {E} \left[ \mathcal {L} \left(\theta_ {t + 1}\right) \mid \mathcal {F} _ {t} \right] \leq \mathbb {E} \left[ \mathcal {L} \left(\theta_ {t}\right) - \eta_ {t} \nabla \mathcal {L} \left(\theta_ {t}\right) \cdot G _ {t + 1} + \frac {1}{2} \beta \eta_ {t} ^ {2} \| G _ {t + 1} \| _ {2} ^ {2} \Bigg | \mathcal {F} _ {2} \right] \\ \leq \mathcal {L} \left(\theta_ {t}\right) - \eta_ {t} \nabla \mathcal {L} \left(\theta_ {t}\right) \cdot \mathbb {E} \left[ G _ {t + 1} \mid \mathcal {F} _ {t} \right] + \frac {1}{2} \beta \eta_ {t} ^ {2} \mathbb {E} \left[ \| G _ {t + 1} \| _ {2} ^ {2} \mid \mathcal {F} _ {t} \right] \\ \leq \mathcal {L} (\theta_ {t}) - \eta_ {t} \| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2} + \frac {1}{2} \beta \eta_ {t} ^ {2} \frac {\| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2}}{\operatorname {S t a b} \left(G _ {t + 1} \mid \mathcal {F} _ {t}\right)} \\ \leq \mathcal {L} (\theta_ {t}) - \eta_ {t} \| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2} + \frac {1}{2} \eta_ {t} \| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2} \\ \leq \mathcal {L} (\theta_ {t}) - \frac {\eta_ {t}}{2} \| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2} \\ \end{array} +$$ + +![](images/3f1485c359c29c9ba4ee5997aac4140942d931ecf6dd804b37b1ecb2a09f6239.jpg) + +Lemma A.2. For all $(\sigma, \alpha) \in \mathbb{R}_+^2$ , the function $\psi: \mathbb{R}_+ \to \mathbb{R}_+$ is strictly increasing and convex. + +$$ +\psi : u \mapsto \frac {u ^ {2}}{\sigma^ {2} + (1 + \alpha) u} +$$ + +Proof. By continuity at zero and twice-differentiability of $\psi$ on $\mathbb{R}_+^*$ , it suffices to check, for every $u\in \mathbb{R}_+^*$ , that $\mathrm{d}\psi (u) > 0$ (strict increase) and $\mathrm{d}^2\psi (u)\geq 0$ (convexity). Write $c = 1 + \alpha$ and compute + +$$ +\mathrm {d} \psi (u) = \frac {2 u (\sigma^ {2} + c u) - c u ^ {2}}{(\sigma^ {2} + c u) ^ {2}} = \frac {2 u \sigma^ {2} + c u ^ {2}}{(\sigma^ {2} + c u) ^ {2}} > 0 +$$ + +Then the second derivative of $\psi$ is observed to be non-negative, which concludes the proof. + +$$ +\begin{array}{l} \mathrm {d} ^ {2} \psi (u) = \frac {(2 \sigma^ {2} + 2 c u) (\sigma^ {2} + c u) ^ {2} - (2 u \sigma^ {2} + c u ^ {2}) \cdot 2 c (\sigma^ {2} + c u)}{(\sigma^ {2} + c u) ^ {4}} \\ = \frac {(2 \sigma^ {2} + 2 c u) (\sigma^ {2} + c u) - 2 c (2 u \sigma^ {2} + c u ^ {2})}{(\sigma^ {2} + c u) ^ {3}} \\ = \frac {2 \sigma^ {4} + 4 \sigma^ {2} c u + 2 c ^ {2} u ^ {2} - (4 \sigma^ {2} c u + 2 c ^ {2} u ^ {2})}{(\sigma^ {2} + c u) ^ {3}} = \frac {2 \sigma^ {4}}{(\sigma^ {2} + c u) ^ {3}} \geq 0 \\ \end{array} +$$ + +![](images/c1b0cfa9b628fbebc1ca3a301667cf71fad252ef7a4cba58b83e91322a43f130.jpg) + +Lemma A.3. For all $(\sigma, \alpha) \in \mathbb{R}_+^2$ , the function $\psi : u \in \mathbb{R}_+ \mapsto u^2 \cdot (\sigma^2 + (1 + \alpha)u)^{-1}$ admits an inverse $\psi^{-1} : \mathbb{R}_+ \to \mathbb{R}_+$ , and for all $x \in \mathbb{R}_+$ , it holds $\psi^{-1}(x) \leq (1 + \alpha)x + \sigma \sqrt{x}$ . + +Proof. By Lemma A.2, $\psi$ is strictly increasing and has $\psi(u) \to 0$ when $u \to 0$ , and $\psi(u) \to +\infty$ when $u \to \infty$ , therefore $\psi$ is bijective and admits an inverse, which is also strictly increasing. + +Moreover, defining $z = (1 + \alpha)x + \sigma \sqrt{x}$ , observe that + +$$ +\begin{array}{l} \psi (z) = \frac {z ^ {2}}{\sigma^ {2} + (1 + \alpha) z} = \frac {(1 + \alpha) x ^ {2} + 2 (1 + \alpha) x \sqrt {x} + \sigma^ {2} x}{\sigma^ {2} + (1 + \alpha) ^ {2} x + (1 + \alpha) \sigma \sqrt {x}} \\ = x + \frac {(1 + \alpha) x \sqrt {x}}{\sigma^ {2} + (1 + \alpha) ^ {2} x + (1 + \alpha) \sigma \sqrt {x}} \geq x \\ \end{array} +$$ + +Therefore $x \leq \psi(z)$ , which implies $\psi^{-1}(x) \leq z$ and concludes the proof. + +![](images/4778c1a2eb588cd794d142218f6b0074bdbdc77b86a2212759230d08023e3573.jpg) + +# Lemma A.4 (Key reduction). + +If for all $t \leq T$ , it holds $\mathbb{V}[G_{t+1} | \mathcal{F}_t] \leq \alpha \| \nabla \mathcal{L}(\theta_t) \|_2^2 + \sigma^2$ (affinely bounded variance), and $\eta_t = \beta^{-1} \operatorname{Stab}(G_{t+1} | \mathcal{F}_t)$ (strong stability condition), then it holds + +$$ +\forall t \leq T, \quad \mathbb {E} [ \mathcal {L} (\theta_ {t + 1}) ] - \mathbb {E} [ \mathcal {L} (\theta_ {t}) ] \leq - \frac {1}{2 \beta} \psi (\mathbb {E} [ \| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2} ]) +$$ + +where $\psi : u \mapsto u^2 / (\sigma^2 + (1 + \alpha)u)$ is a convex and increasing function. + +Proof. Starting from Lemma A.1, and using the affinely bounded variance assumption to obtain the inequality $\mathbb{E}\left[\| G_{t + 1}\| _2^2\big|\mathcal{F}_t\right]\leq \| \mathbb{E}\left[G_{t + 1}\mid \mathcal{F}_t\right]\| _2^2 +\mathbb{V}\left[G_{t + 1}\mid \mathcal{F}_t\right]\leq (1 + \alpha)\| \nabla \mathcal{L}(\theta_t)\| _2^2 +\sigma^2$ substituted in the stability ratio, we obtain + +$$ +\begin{array}{l} \mathbb {E} \left[ \mathcal {L} \left(\theta_ {t + 1}\right) \mid \mathcal {F} _ {t} \right] - \mathcal {L} \left(\theta_ {t}\right) \leq - \frac {\eta_ {t}}{2} \| \nabla \mathcal {L} \left(\theta_ {t}\right) \| _ {2} ^ {2} \\ \leq - \frac {1}{2 \beta} \frac {\| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2}}{\sigma^ {2} + (1 + \alpha) \| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2}} \| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2} \\ \leq - \frac {1}{2 \beta} \psi \left(\| \nabla \mathcal {L} \left(\theta_ {t}\right) \| _ {2} ^ {2}\right) \\ \end{array} +$$ + +Therefore, taking expectations, and using convexity of $\psi$ (Lemma A.2) as $\mathbb{E}[\psi(U)] \geq \psi(\mathbb{E}[U])$ , + +$$ +\mathbb {E} \left[ \mathcal {L} \left(\theta_ {t + 1}\right) \right] - \mathbb {E} \left[ \mathcal {L} \left(\theta_ {t}\right) \right] \leq - \frac {1}{2 \beta} \mathbb {E} \left[ \psi \left(\| \nabla \mathcal {L} \left(\theta_ {t}\right) \| _ {2} ^ {2}\right) \right] \leq - \frac {1}{2 \beta} \psi \left(\mathbb {E} \left[ \| \nabla \mathcal {L} \left(\theta_ {t}\right) \| _ {2} ^ {2} \right]\right) +$$ + +![](images/e1866a1b3ee4170272d7d096121da1479371813a663c8febb47066cebe933338.jpg) + +# Lemma A.5 (KL stochastic integration). + +If for all $t \leq T$ , it holds $\mathbb{V}[G_{t+1} | \mathcal{F}_t] \leq \alpha \| \nabla \mathcal{L}(\theta_t) \|_2^2 + \sigma^2$ (affinely bounded variance), and $\eta_t = \beta^{-1} \operatorname{Stab}(G_{t+1} | \mathcal{F}_t)$ (strong stability condition), and if it holds for some increasing function $\varphi: \mathbb{R}_+ \to \mathbb{R}_+$ that $\mathbb{E}\left[\| \nabla \mathcal{L}(\theta_t) \|_2^2\right] \geq \varphi(\mathbb{E}\left[\mathcal{L}(\theta_t)\right] - \mathcal{L}^\star)$ , then it holds + +$$ +\forall t \leq T, \quad \mathbb {E} \left[ \mathcal {L} \left(\theta_ {t + 1}\right) \right] - \mathcal {L} ^ {\star} \leq \Phi^ {- 1} \left(\Phi \left(\mathbb {E} \left[ \mathcal {L} \left(\theta_ {0}\right) \right] - \mathcal {L} ^ {\star}\right) + \frac {t}{2 \beta}\right) +$$ + +where $\Phi :\mathbb{R}_+^*\to \mathbb{R}$ is the $^{16}$ function defined as $\mathrm{d}\Phi (u) = -\left(\sigma^2 +(1 + \alpha)\varphi (u)\right)\cdot \varphi (u)^{-2}.$ + +In particular, if $T \geq 2\beta$ ( $\Phi(\varepsilon) - \Phi(\Delta_0)$ ) for $\Delta_0 = \mathbb{E}[\mathcal{L}(\theta_0)] - \mathcal{L}^\star$ , then $\mathbb{E}[\mathcal{L}(\theta_{T+1})] \leq \mathcal{L}^\star + \varepsilon$ . + +Proof. Starting from Lemma A.4, and using the last assumption since $\psi$ is increasing, + +$$ +\mathbb {E} \left[ \mathcal {L} \left(\theta_ {t + 1}\right) \right] - \mathbb {E} \left[ \mathcal {L} \left(\theta_ {t}\right) \right] \leq - \frac {1}{2 \beta} \psi \left(\mathbb {E} \left[ \| \nabla \mathcal {L} \left(\theta_ {t}\right) \| _ {2} ^ {2} \right]\right) \leq - \frac {1}{2 \beta} (\psi \circ \varphi) (\mathbb {E} \left[ \mathcal {L} \left(\theta_ {t}\right) \right] - \mathcal {L} ^ {\star}) +$$ + +Note that by definition $\mathrm{d}\Phi (u) = -1 / (\psi \circ \varphi)(u)$ . We will use this to simplify the above equation, but also to observe that $\mathrm{d}\Phi$ is increasing since $(\psi \circ \varphi)$ is increasing as a composition of increasing + +functions. Therefore, $\Phi$ is a convex function (since it has increasing derivative) which can be used as $\Phi(y) - \Phi(x) \geq \mathrm{d}\Phi(x) \cdot (y - x)$ to further simplify + +$$ +\mathrm {d} \Phi (\mathbb {E} [ \mathcal {L} (\theta_ {t}) ] - \mathcal {L} ^ {\star}) \cdot (\mathbb {E} [ \mathcal {L} (\theta_ {t + 1}) ] - \mathbb {E} [ \mathcal {L} (\theta_ {t}) ]) \geq \frac {1}{2 \beta} +$$ + +$$ +\Phi \left(\mathbb {E} \left[ \mathcal {L} \left(\theta_ {t + 1}\right) \right] - \mathcal {L} ^ {\star}\right) - \Phi \left(\mathbb {E} \left[ \mathcal {L} \left(\theta_ {t}\right) \right] - \mathcal {L} ^ {\star}\right) \geq \frac {1}{2 \beta} +$$ + +Observing that $\Phi$ is decreasing (since it has negative derivate), this implies + +$$ +\mathbb {E} \left[ \mathcal {L} \left(\theta_ {t + 1}\right) \right] - \mathcal {L} ^ {\star} \leq \Phi^ {- 1} \left(\Phi \left(\mathbb {E} \left[ \mathcal {L} \left(\theta_ {0}\right) \right] - \mathcal {L} ^ {\star}\right) + \frac {t}{2 \beta}\right) +$$ + +Defining $\Delta_0 = \mathbb{E}\left[\mathcal{L}(\theta_0)\right] - \mathcal{L}^\star$ and injecting $T\geq 2\beta (\Phi (\varepsilon) - \Phi (\Delta_0))$ in the previous equation yields the final claim, by decrease of $\Phi$ + +$$ +\mathbb {E} \left[ \mathcal {L} \left(\theta_ {T + 1}\right) \right] - \mathcal {L} ^ {\star} \leq \Phi^ {- 1} \left(\Phi \left(\Delta_ {0}\right) + \frac {T}{2 \beta}\right) \leq \varepsilon +$$ + +![](images/d40125c8ae517138cb5e96aa1887ea6f4e8be5bf2acc4fbd973630babd8f87fc.jpg) + +# Lemma A.6 (Squared distance to optimum is a submartingale). + +If $\mathcal{L}:\mathbb{R}^d\to \mathbb{R}$ is convex, and there exists $\theta^{\star}\in \mathbb{R}^{d}$ such that $\mathcal{L}(\theta^{\star}) = \mathcal{L}^{\star}$ , and if for all $t\leq T$ , it holds $\eta_t\leq \beta^{-1}\operatorname {Stab}(G_{t + 1}\mid \mathcal{F}_t)$ (weak stability condition), then it holds + +$$ +\forall t \leq T, \quad \mathbb {E} \left[ \| \theta_ {t} - \theta^ {\star} \| _ {2} ^ {2} \right] \leq \mathbb {E} \left[ \| \theta_ {0} - \theta^ {\star} \| _ {2} ^ {2} \right] +$$ + +Proof. Define the random variable $D_{t} \in \mathbb{R}$ as $D_{t}^{2} = \| \theta_{t} - \theta^{\star}\|_{2}^{2}$ . Observe that expanding the square, + +$$ +D _ {t + 1} ^ {2} - D _ {t} ^ {2} = - 2 \eta_ {t} G _ {t + 1} \cdot (\theta_ {t} - \theta^ {\star}) + \eta_ {t} ^ {2} \| G _ {t + 1} \| _ {2} ^ {2} +$$ + +Thus taking conditional expectations and using the weak stability condition, + +$$ +\begin{array}{l} \mathbb {E} \left[ D _ {t + 1} ^ {2} \mid \mathcal {F} _ {t} \right] - D _ {t} ^ {2} = - 2 \eta_ {t} \nabla \mathcal {L} (\theta_ {t}) \cdot (\theta_ {t} - \theta^ {\star}) + \eta_ {t} ^ {2} \| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2} / \operatorname {S t a b} (G _ {t + 1} \mid \mathcal {F} _ {t}) \\ \leq - 2 \eta_ {t} \nabla \mathcal {L} \left(\theta_ {t}\right) \cdot \left(\theta_ {t} - \theta^ {\star}\right) + \frac {\eta_ {t}}{\beta} \| \nabla \mathcal {L} \left(\theta_ {t}\right) \| _ {2} ^ {2} \\ \end{array} +$$ + +By convexity of $\mathcal{L}$ , the first term can be bounded with $\mathcal{L}^{\star} - \mathcal{L}(\theta_t) \geq -\nabla \mathcal{L}(\theta_t) \cdot (\theta_t - \theta^{\star})$ , and the second term can be bounded by $\beta$ -smoothness of $\mathcal{L}$ as $\|\nabla \mathcal{L}(\theta_t)\|_2^2 \leq 2\beta (\mathcal{L}(\theta_t) - \mathcal{L}^{\star})$ , thus + +$$ +\mathbb {E} \left[ D _ {t + 1} ^ {2} \mid \mathcal {F} _ {t} \right] - D _ {t} ^ {2} \leq - 2 \eta_ {t} (\mathcal {L} ^ {\star} - \mathcal {L} (\theta_ {t})) + 2 \eta_ {t} (\mathcal {L} (\theta_ {t}) - \mathcal {L} ^ {\star}) \leq 0 +$$ + +Hence $\mathbb{E}\left[D_{t + 1}^2\mid \mathcal{F}_t\right]\leq D_t^2$ and by induction $\mathbb{E}\left[D_{t + 1}^2\right]\leq \mathbb{E}\left[D_0^2\right]$ , which concludes the proof. + +# Corollary A.1 (Convex smooth rate). + +If $\mathcal{L}:\mathbb{R}^d\to \mathbb{R}$ is convex and there exists $\theta^{\star}\in \mathbb{R}^{d}$ such that $\mathcal{L}(\theta^{\star}) = \mathcal{L}^{\star}$ , and if for all $t\leq T$ , it holds $\eta_t = \beta^{-1}\mathrm{Stab}(G_{t + 1}\mid \mathcal{F}_t)$ (strong stability condition), and $\mathbb{V}[G_{t + 1}|\mathcal{F}_t]\leq \alpha \| \nabla \mathcal{L}(\theta_t)\| _2^2 +\sigma^2$ (affinely bounded variance), then it holds + +$$ +\forall t \leq T, \quad \mathbb {E} \left[ \mathcal {L} \left(\theta_ {t + 1}\right) \right] - \mathcal {L} ^ {\star} \leq \Phi^ {- 1} \left(\Phi \left(\mathbb {E} \left[ \mathcal {L} \left(\theta_ {0}\right) \right] - \mathcal {L} ^ {\star}\right) + \frac {t}{2 \beta}\right) +$$ + +where $\Phi : u \mapsto \frac{C^2\sigma^2}{3u^3} + (1 + \alpha)\frac{C}{2u}$ for $C = \mathbb{E}\left[\|\theta_0 - \theta^\star\|_2^2\right] \in \mathbb{R}_+$ . + +Therefore, $T \geq \frac{2}{3} \beta C^2 \sigma^2 (\varepsilon^{-3} - \Delta_0^{-3}) + (1 + \alpha) \beta C (\varepsilon^{-1} - \Delta_0^{-1})$ implies $\mathbb{E}\left[\mathcal{L}(\theta_{T+1})\right] \leq \mathcal{L}^\star + \varepsilon$ , which is a rate of $\mathcal{O}(T^{-1/3})$ if with additive noise $\sigma^2 > 0$ , and $\mathcal{O}(T^{-1})$ in the case $\sigma^2 = 0$ . + +Proof. Define $C = \mathbb{E}\left[\| \theta_0 - \theta^\star \| _2^2\right]$ and $\varphi :u\mapsto u^2 /C$ . In order to use Lemma A.5, let us show that $\mathbb{E}\left[\| \nabla \mathcal{L}(\theta_t)\| _2^2\right]\geq \varphi (\mathbb{E}\left[\mathcal{L}(\theta_t) - \mathcal{L}^\star \right])$ . By convexity of $\mathcal{L}$ and then by Cauchy-Schwarz inequality. + +$$ +\begin{array}{l} \mathcal {L} \left(\theta_ {t}\right) - \mathcal {L} ^ {\star} \leq \nabla \mathcal {L} \left(\theta_ {t}\right) \cdot \left(\theta_ {t} - \theta^ {\star}\right) \\ \mathbb {E} \left[ \mathcal {L} (\boldsymbol {\theta} _ {t}) - \mathcal {L} ^ {\star} \right] ^ {2} \leq \mathbb {E} \left[ \| \nabla \mathcal {L} (\boldsymbol {\theta} _ {t}) \| _ {2} ^ {2} \right] \cdot \mathbb {E} \left[ \| \boldsymbol {\theta} _ {t} - \boldsymbol {\theta} ^ {\star} \| _ {2} ^ {2} \right] \\ \end{array} +$$ + +Using additionally Lemma A.6 to get $\mathbb{E}\left[\| \theta_t - \theta^\star \|_2^2\right] \leq \mathbb{E}\left[\| \theta_0 - \theta^\star \|_2^2\right]$ , this concludes the first part of the proof, that $\mathbb{E}\left[\| \nabla \mathcal{L}(\theta_t) \|_2^2\right] \geq \varphi(\mathbb{E}\left[\mathcal{L}(\theta_t) - \mathcal{L}^\star\right])$ . + +For the second part of the proof, apply Lemma A.5, with desingularizer $\Phi$ obtained by integration + +$$ +\mathrm {d} \Phi (u) = - \frac {\sigma^ {2} + (1 + \alpha) \varphi (u)}{\varphi (u) ^ {2}} = - \frac {\sigma^ {2} + (1 + \alpha) C ^ {- 1} u ^ {2}}{C ^ {- 2} u ^ {4}} +$$ + +$$ +\Phi (u) = \frac {C ^ {2} \sigma^ {2}}{3 u ^ {3}} + (1 + \alpha) \frac {C}{2 u} +$$ + +Bound inversion: the condition to obtain $\mathbb{E}\left[\mathcal{L}(\theta_{T + 1})\right]\leq \mathcal{L}^{\star} + \varepsilon$ with $T$ as a function of $\varepsilon$ , i.e. + +$$ +T _ {\varepsilon} \geq \frac {2}{3} \beta C ^ {2} \sigma^ {2} \left(\varepsilon^ {- 3} - \Delta_ {0} ^ {- 3}\right) + (1 + \alpha) \beta C \left(\varepsilon^ {- 1} - \Delta_ {0} ^ {- 1}\right) +$$ + +can be rewritten with $\varepsilon$ as a function of $T$ , as in the original statement of Corollary A.1, in the form + +$$ +\varepsilon_ {T} \leq \Phi^ {- 1} \left(\Phi (\Delta_ {0}) + \frac {T}{2 \beta}\right) +$$ + +with $a = C^2\sigma^2 /3$ and $b = (1 + \alpha)C / 2$ defining $\Phi (u) = au^{-3} + bu^{-1}$ . This expression can be simplified at $y = \Phi (\Delta_0) + \frac{T}{2\beta}$ with the intermediate variables $p = -\frac{b^2}{3y^2}$ and $q = \frac{2b^3}{27y^3} +\frac{a}{y}$ using + +$$ +\Phi^ {- 1} (y) = \sqrt [ 3 ]{\frac {q}{2} + \sqrt {\left(\frac {q}{2}\right) ^ {2} + \left(\frac {p}{3}\right) ^ {3}}} + \sqrt [ 3 ]{\frac {q}{2} - \sqrt {\left(\frac {q}{2}\right) ^ {2} + \left(\frac {p}{3}\right) ^ {3}}} + \frac {b}{3 y} \quad (\mathrm {C V X - I N V}) +$$ + +This expression of $\varepsilon_T = \Phi^{-1}(y)$ is not any easier to use, hence our statement in the other $T_{\varepsilon}$ form. + +# Corollary A.2 (Strongly-convex smooth rate). + +If $\mathcal{L}$ is $\mu$ -strongly convex, and if for all $t \leq T$ , it holds $\mathbb{V}[G_{t+1} \mid \mathcal{F}_t] \leq \alpha \| \nabla \mathcal{L}(\theta_t) \|_2^2 + \sigma^2$ (affinely bounded variance), and $\eta_t = \beta^{-1} \operatorname{Stab}(G_{t+1} \mid \mathcal{F}_t)$ (strong stability condition), then it holds + +$$ +\forall t \leq T, \quad \mathbb {E} \left[ \mathcal {L} \left(\theta_ {t + 1}\right) \right] - \mathcal {L} ^ {\star} \leq \Phi^ {- 1} \left(\Phi \left(\mathbb {E} \left[ \mathcal {L} \left(\theta_ {0}\right) \right] - \mathcal {L} ^ {\star}\right) + \frac {t}{2 \beta}\right) +$$ + +where $\Phi :u\mapsto \frac{\sigma^2}{4\mu^2}\frac{1}{u} -\frac{1 + \alpha}{2\mu}\log (u).$ + +For $T \geq \frac{\sigma^2\beta}{2\mu^2} (\varepsilon^{-1} - \Delta_0^{-1}) + (1 + \alpha)\frac{\beta}{\mu}\log (\Delta_0 / \varepsilon)$ , where $\Delta_0 = \mathbb{E}\left[\mathcal{L}(\theta_0)\right] - \mathcal{L}^\star$ , this implies that $\mathbb{E}\left[\mathcal{L}(\theta_{T + 1})\right] - \mathcal{L}^\star \leq \varepsilon$ . This is a rate of $\mathcal{O}(T^{-1})$ with additive noise $\sigma^2 > 0$ and a linear rate $\mathcal{O}(\exp (-\kappa T / (1 + \alpha)))$ for $\kappa = \mu /\beta$ in the noiseless / multiplicative-noise case $\sigma^2 = 0$ . + +Proof. The proof is a straightforward application of Lemma A.5 with $\varphi : u \mapsto 2\mu u$ , which satisfies $\mathbb{E}\left[\|\nabla \mathcal{L}(\theta_t)\|_2^2\right] \geq \varphi(\mathbb{E}\left[\mathcal{L}(\theta_t) - \mathcal{L}^\star\right])$ , because by strong convexity of $\mathcal{L}$ , it holds for all $x \in \mathbb{R}^d$ that $\|\nabla \mathcal{L}(x)\|_2^2 \geq 2\mu(\mathcal{L}(x) - \mathcal{L}^\star)$ . It remains to compute the desingularizer $\Phi$ by integration + +$$ +\mathrm {d} \Phi (u) = - \frac {\sigma^ {2} + (1 + \alpha) \varphi (u)}{\varphi (u) ^ {2}} = - \frac {\sigma^ {2} + 2 (1 + \alpha) \mu u}{4 \mu^ {2} u ^ {2}} +$$ + +$$ +\Phi (u) = \frac {\sigma^ {2}}{4 \mu^ {2}} \frac {1}{u} - \frac {1 + \alpha}{2 \mu} \log (u) +$$ + +# Proposition A.3 (Non-convex rate). + +If for all $t \leq T$ , it holds $\mathbb{V}[G_{t+1} | \mathcal{F}_t] \leq \alpha \| \nabla \mathcal{L}(\theta_t) \|_2^2 + \sigma^2$ (affinely bounded variance), and $\eta_t = \beta^{-1} \operatorname{Stab}(G_{t+1} | \mathcal{F}_t)$ (strong stability condition), then writing $\Delta_0 = \mathbb{E}[\mathcal{L}(\theta_0)] - \mathcal{L}^\star$ , it holds + +$$ +\mathbb {E} \left[ \frac {1}{T} \sum_ {t < T} \| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2} \right] \leq (1 + \alpha) \frac {2 \beta \Delta_ {0}}{T} + \sqrt {\frac {2 \beta \Delta_ {0} \sigma^ {2}}{T}} +$$ + +Proof. Starting from Lemma A.4 (valid by strong stability condition and affinely bounded variance), + +$$ +\mathbb {E} \left[ \mathcal {L} \left(\theta_ {t + 1}\right) \mid \mathcal {F} _ {t} \right] - \mathcal {L} \left(\theta_ {t}\right) \leq - \frac {1}{2 \beta} \psi \left(\| \nabla \mathcal {L} \left(\theta_ {t}\right) \| _ {2} ^ {2}\right) +$$ + +where $\psi : u \mapsto u^2 \cdot (\sigma^2 + (1 + \alpha) u)^{-1}$ is a convex increasing function. Thus, taking total expectations and summing over iterates $t \in [T]$ to telescope $(a)$ , and then using convexity of $\psi$ to bound $(b)$ , + +$$ +\psi \left(\mathbb {E} \left[ \frac {1}{T} \sum_ {t < T} \| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2} \right]\right) \leq_ {(b)} \mathbb {E} \left[ \frac {1}{T} \sum_ {t < T} \psi \left(\| \nabla \mathcal {L} (\theta_ {t}) \| _ {2} ^ {2}\right) \right] \leq_ {(a)} \frac {2 \beta \Delta_ {0}}{T} +$$ + +where $\Delta_0 = \mathbb{E}\left[\mathcal{L}(\theta_0)\right] - \mathcal{L}^*$ is the expected initial optimization error. It remains to use the bound $\psi^{-1}(x) \leq (1 + \alpha)x + \sigma \sqrt{x}$ (Lemma A.3), to obtain + +$$ +\mathbb {E} \left[ \frac {1}{T} \sum_ {t < T} \| \nabla \mathcal {L} (\theta_ {t}) \| ^ {2} \right] \leq \frac {2 (1 + \alpha) \beta \Delta_ {0}}{T} + \sqrt {\frac {2 \beta \Delta_ {0} \sigma^ {2}}{T}}. +$$ + +![](images/078d410f1e954b45f49914d904b90e9d5d4474ae01ecd867cb39b1ad8683a742.jpg) + +We thus recover the classical deterministic and stochastic regimes in, respectively, $O(1 / T)$ and $O(1 / \sqrt{T})$ depending on whether the additive variance term $\sigma^2$ is positive or equal to 0. + +The same analysis would hold in a more general setting in which $\operatorname{Stab}(G_{t+1} \mid \mathcal{F}_t) \geq \varphi(\|\nabla \mathcal{L}(\theta_t)\|^2)$ and $x \mapsto x \cdot \varphi(x)$ is a positive, increasing and convex function. + +# A.2 Estimation of stability ratio + +Lemma A.7. Let $B > 0$ and $(X_{i})_{i\in [n]}$ be i.i.d. real random variables such that, for all $i\in [n]$ , it holds $\mathbb{E}[X_i] = 0$ and $X_{i}\leq B$ almost surely. Then, for any $t > 0$ , we have + +$$ +\mathbb {P} \left(\frac {1}{n} \sum_ {i \in [ n ]} X _ {i} \geq t\right) \leq \exp \left(- \frac {n t ^ {2}}{2 B ^ {2} f (\mathbb {V} [ X _ {1} ] / B ^ {2})}\right), \tag {1} +$$ + +where $f(x) = (1 + x)^2 / 4$ if $x < 1$ , and $f(x) = x$ otherwise. (In particular, $\forall x, f(x) \leq 1 + x$ ) + +Proof. Use Fan et al. [2015, Corollary 2.7] with $U_{i - 1} = B$ , note that $B^{2}f(\mathbb{V}[X] / B^{2}) = C_{i - 1}^{2}$ exactly matches the definition in the reference's notation, thus following the reference and simplifying constants $C_i$ , we get for $v^2 = n\sum_{i = 1}^n C_{i - 1}^2 = nB^2 f(\mathbb{V}[X_i] / B^2)$ , that it holds + +$$ +\mathbb {P} \left(\sum_ {i \in [ n ]} X _ {i} \geq x\right) \leq \exp \left(- \frac {x ^ {2}}{2 v ^ {2}}\right) +$$ + +The result follows using $x = nt$ . + +![](images/effb924aebb5440c848fc184cbca48ff9b7744a54b3d33b691f4fa9c3ea10b08.jpg) + +Lemma A.8. Let $(D_{i})_{i\in [n]}$ be non-negative i.i.d. random variables with $\mathbb{E}\left[D_i^2\right] < +\infty$ and $\mathbb{E}\left[D_i\right] = D\in \mathbb{R}_+^*$ . Then, for $\kappa = \mathbb{E}\left[D_i^2\right] / \mathbb{E}\left[D_i\right]^2\in [1,\infty [$ , it holds + +$$ +\mathbb {P} \left(\frac {1}{n} \sum_ {i \in [ n ]} D _ {i} \leq D / 2\right) \leq \exp \left(- \frac {n}{8 \kappa}\right) +$$ + +Proof. Let $X_{i} = D - D_{i}$ . Observe that $\mathbb{E}[X_i] = 0$ , and $X_{i} \leq D$ almost surely. Additionally, by expanding the square, $\mathbb{V}[X_i] = \mathbb{E}\left[D_i^2\right] - D^2$ . + +Apply Lemma A.7 with $B = D$ and $t = D / 2$ and use $f(x)\leq 1 + x$ to simplify the denominator with $D^{2}f(\mathbb{V}[D_{i}] / D^{2})\leq D^{2} + \mathbb{V}[D_{i}] = \mathbb{E}\left[D_{i}^{2}\right]$ . Therefore, + +$$ +\mathbb {P} \left(\frac {1}{n} \sum_ {i \in [ n ]} D _ {i} \leq D / 2\right) \leq \exp \left(- \frac {n D ^ {2} / 4}{2 \mathbb {E} \left[ D _ {i} ^ {2} \right]}\right) = \exp \left(- \frac {n}{8 \mathbb {E} \left[ D _ {i} ^ {2} \right] / D ^ {2}}\right) +$$ + +![](images/5539a989c50c34416f612171aced7036c0007df343bdf791c71d0e4121be83c5.jpg) + +Lemma A.9 (Relative error of stability estimation). Let $(X_{i}\in \mathbb{R}^{d})_{i\in [n]}$ be iid random variables. Define $J = \frac{1}{n(n - 1)}\sum_{i\neq j}X_i\cdot X_j\in \mathbb{R},Z = \frac{1}{n}\sum_{i}\| X_{i}\|_{2}^{2}\in \mathbb{R}_{+},$ then $S = \operatorname {clip}_{[0,1]}(J / Z)\in [0,1]$ + +Write $\mu = \mathbb{E}\left[X\right]\in \mathbb{R}^d$ and $\sigma^2 = \mathbb{E}\left[\| X - \mu \| _2^2\right]$ , and $\kappa = \mathbb{E}\left[\| X\| _2^4\right] / \sigma^4$ . If $R = \| \mu \| _2^2 /\sigma^2\neq 0,$ and if $n\geq 1 + a / R$ for a constant $a\geq 1$ , then + +$$ +\mathbb {E} \left[ \left| \frac {S - R}{R} \right| ^ {2} \right] \leq \frac {4 8 + 4 (\kappa - 1)}{a} + \frac {1}{R ^ {2}} \exp \left(- \frac {n}{8 \kappa}\right) +$$ + +At $\kappa = 3$ (for a centered gaussian) and neglecting the fast-decreasing second term, this is a relative squared error of $56 / a$ , i.e. a relative error of order $7.48 / \sqrt{a}$ , which is below 1 as low as $a = 100$ . + +Proof. Let $N = \| \mu \|_2^2$ and $D = \mathbb{E}\left[\| X\|_2^2\right]$ be the numerator and denominator in $R = N / D$ . Note that $\mathbb{E}[J] = N$ and $\mathbb{E}[Z] = D$ . Proceed then by case disjunction: if on one hand $Z \leq D / 2$ , then $|S - R| \leq 1$ (both are in [0, 1]), while on the other hand if $Z \geq D / 2$ , then + +$$ +\begin{array}{l} | S - R | \leq \left| \frac {J}{Z} - R \right| = \left| \frac {J - N}{Z} + N \left(\frac {1}{Z} - \frac {1}{D}\right) \right| \leq \frac {| J - N |}{Z} + \frac {N}{D} \frac {| Z - D |}{Z} \\ \leq \frac {| J - N |}{D / 2} + \frac {N}{D} \frac {| Z - D |}{(D / 2)} = 2 R \frac {| J - N |}{N} + 2 R \frac {| Z - D |}{D} \\ \end{array} +$$ + +Therefore joining both cases after taking squares, + +$$ +\left| \frac {S - R}{R} \right| ^ {2} \leq 4 \left| \frac {J - N}{N} \right| ^ {2} + 4 \left| \frac {Z - D}{D} \right| ^ {2} + \frac {1}{R ^ {2}} \mathbb {1} \{Z \leq D / 2 \} +$$ + +Hence, after taking expectations and applying Lemma A.12 (numerator sample control) and Lemma A.11 (denominator variance), it holds for $n \geq 1 + a / R$ that + +$$ +\mathbb {E} \left[ \left| \frac {S - R}{R} \right| ^ {2} \right] \leq 4 \left(\frac {4}{a ^ {2}} + \frac {8}{a}\right) + 4 \frac {\kappa - 1}{n} + \frac {1}{R ^ {2}} \mathbb {P} (Z \leq D / 2) +$$ + +Additionally, by Lemma A.8, $\mathbb{P}(Z\leq D / 2)\leq \exp \left(-\frac{n}{8s}\right)$ where $s = \mathbb{E}\left[\| X_i\| ^4\right] / D^2 = \kappa$ . Thus, + +$$ +\mathbb {E} \left[ \left| \frac {S - R}{R} \right| ^ {2} \right] \leq 4 \left(\frac {4}{a ^ {2}} + \frac {8}{a}\right) + 4 \frac {R (\kappa - 1)}{a} + \frac {1}{R ^ {2}} \exp \left(- \frac {n}{8 \kappa}\right) +$$ + +The result follows by using $a \geq 1$ and $R \leq 1$ . + +Lemma A.10 (Uncentered kurtosis of isotropic normal distribution). Let $X \in \mathbb{R}^d$ be a random variable with $X \sim \mathcal{N}(0, \sigma^2 I)$ . It holds $\mathbb{E}\left[\| X\|_2^2\right] = d\sigma^2$ and $\mathbb{E}\left[\| X\|_2^4\right] / \mathbb{E}\left[\| X\|_2^2\right]^2 = \frac{d - 1}{d} + \frac{3}{d}$ + +Proof. By expanding the sum, + +$$ +\begin{array}{l} \mathbb {E} \left[ | | X | | _ {2} ^ {2} \right] = \mathbb {E} \left[ \sum_ {i} X _ {i} ^ {2} \right] = \sum_ {i} \mathbb {E} \left[ X _ {i} ^ {2} \right] = d \sigma^ {2} \\ \mathbb {E} \left[ \| X \| _ {2} ^ {4} \right] = \mathbb {E} \left[ \left(\sum_ {i} X _ {i} ^ {2}\right) ^ {2} \right] = \sum_ {i, j} \mathbb {E} \left[ X _ {i} ^ {2} X _ {j} ^ {2} \right] \\ = \sum_ {i} \mathbb {E} \left[ X _ {i} ^ {4} \right] + \sum_ {i \neq j} \mathbb {E} \left[ X _ {i} ^ {2} \right] \mathbb {E} \left[ X _ {j} ^ {2} \right] = d \cdot 3 \cdot \sigma^ {4} + d (d - 1) \sigma^ {4} \\ \end{array} +$$ + +The result follows by taking the quotient of both. + +Lemma A.11 (Kurtosis bound for the denominator). + +Let $(X_{i}\in \mathbb{R}^{d})_{i\in [n]}$ be iid random variables with $\mathbb{E}\left[\| X\| _2^2\right] = Q$ and $Z = \frac{1}{n}\sum_{i\in [n]}\| X_i\|^2$ . Then + +$$ +\mathbb {E} \left[ | Z - Q | ^ {2} \right] = \frac {1}{n} \left(\mathbb {E} \left[ \| X \| _ {2} ^ {4} \right] - \mathbb {E} \left[ \| X \| _ {2} ^ {2} \right] ^ {2}\right) +$$ + +and thus for $\kappa = \frac{\mathbb{E}[\|X\|_2^4]}{\mathbb{E}[\|X\|_2^2]^2}$ (uncentered kurtosis of $X$ ), it holds $\mathbb{P}(|Z - Q| > \tau Q) \leq \frac{\kappa - 1}{n\tau^2}$ . + +Proof of the expectation is just expansion of the square and linearity of expectation. The second proposition is Chebyshev's inequality. + +Lemma A.12 (Numerator sample control). + +Let $(X_{i}\in \mathbb{R}^{d})_{i\in [n]}$ be iid random variables with $\mathbb{E}[X] = \mu$ and $J = \frac{1}{n(n - 1)}\sum_{i\neq j}X_i\cdot X_j$ . If $\mu \neq 0$ and if $n\geq 1 + c\cdot \mathbb{E}\left[\| X - \mu \| _2^2\right] / \| \mu \| _2^2$ then it holds + +$$ +\frac {\mathbb {E} \left[ | J - \| \mu \| _ {2} ^ {2} | ^ {2} \right]}{\| \mu \| _ {2} ^ {4}} \leq \frac {4}{c ^ {2}} + \frac {8}{c} +$$ + +This is an immediate corollary of the following lemma. + +Lemma A.13 (Variance bound for the Jackknife numerator). + +Let $(X_{i}\in \mathbb{R}^{d})_{i\in [n]}$ be iid random variables with $\mathbb{E}[X] = \mu \in \mathbb{R}^d$ , and $J = \frac{1}{n(n - 1)}\sum_{i\neq j}X_i\cdot X_j$ Then it holds $\mathbb{E}[J] = \| \mu \| _2^2$ , and + +$$ +\mathbb {E} \left[ \left| J - \| \mu \| _ {2} ^ {2} \right| ^ {2} \right] \leq 4 \frac {\mathbb {E} \left[ \| X - \mu \| _ {2} ^ {2} \right] ^ {2}}{n (n - 1)} + \frac {8}{n} \mathbb {E} \left[ \| X - \mu \| _ {2} ^ {2} \right] \cdot \| \mu \| _ {2} ^ {2} +$$ + +Proof. + +$$ +\begin{array}{l} J - \mathbb {E} [ J ] = \frac {1}{n (n - 1)} \sum_ {i \neq j} \left(X _ {i} \cdot X _ {j} - \mu^ {2}\right) \\ = \frac {1}{n (n - 1)} \sum_ {i \neq j} \left(\left(X _ {i} - \mu\right) \cdot \left(X _ {j} - \mu\right) + \left(X _ {i} + X _ {j}\right) \mu - 2 \mu^ {2}\right) \\ = \frac {1}{n (n - 1)} \left(\left(\sum_ {i \neq j} \left(X _ {i} - \mu\right) \cdot \left(X _ {j} - \mu\right)\right) + \left(2 (n - 1) \sum_ {i} X _ {i} \cdot \mu\right) - 2 n (n - 1) \mu^ {2}\right) \\ = \frac {1}{n (n - 1)} \left(\sum_ {i \neq j} \left(X _ {i} - \mu\right) \cdot \left(X _ {j} - \mu\right)\right) + 2 \left(\frac {1}{n} \sum_ {i} X _ {i} - \mu\right) \cdot \mu \\ := A + B \\ \end{array} +$$ + +As a sanity check, observe that $\mathbb{E}[J - \mathbb{E}[J]] = 0$ because $\mathbb{E}[A] = 0$ and $\mathbb{E}[B] = 0$ . + +We will use the (crude) bound $\mathbb{E}\left[|J - \mu^2|^2\right] \leq 2\mathbb{E}\left[A^2\right] + 2\mathbb{E}\left[B^2\right]$ . Let us compute each. + +$$ +\mathbb {E} \left[ B ^ {2} \right] = 4 \mathbb {E} \left[ \left(\left(\frac {1}{n} \sum_ {i} X _ {i} - \mu\right) \cdot \mu\right) ^ {2} \right] \leq 4 \mathbb {E} \left[ \left\| \frac {1}{n} \sum_ {i} X _ {i} - \mu \right\| _ {2} ^ {2} \right] \cdot \mu^ {2} \leq 4 \frac {\sigma^ {2}}{n} \mu^ {2} +$$ + +On the other hand, by Lemma A.14, $\mathbb{E}\left[A^2\right] \leq 2\mathbb{E}\left[\|X - \mu\|_2^2\right]^2 / (n(n-1))$ . Thus the conclusion, + +$$ +\mathbb {E} \left[ \left| J - \| \mu \| _ {2} ^ {2} \right| ^ {2} \right] \leq 2 \mathbb {E} \left[ A ^ {2} \right] + 2 \mathbb {E} \left[ B ^ {2} \right] \leq 4 \frac {\mathbb {E} \left[ \| X - \mu \| _ {2} ^ {2} \right] ^ {2}}{n (n - 1)} + \frac {8}{n} \mathbb {E} \left[ \| X - \mu \| _ {2} ^ {2} \right] \cdot \| \mu \| _ {2} ^ {2} +$$ + +□ + +![](images/0a3299ddf568001cfee557cf281d5497bddd818c109100fda0b01217f2d89588.jpg) +Figure 10: Empirical measurements of $\mathbb{E}\left[|J_n - \mathbb{E}[J_n]|^2\right]$ as a function of $n$ (mean and 5-sigma confidence interval for the mean, $10^{3}$ samples) vs Lemma A.13 upper-bound, for isotropic gaussians in dimension $d = 10$ with noise $\sigma_0 = 10^2$ per coordinate, thus $\sigma^2 = d\sigma_0^2 = 10^5$ , and $\|\mu\|_2^2 = d$ . + +Lemma A.14 (Variance of the squared-mean U-statistic). + +Let $(C_i\in \mathbb{R}^d)_{i\in [n]}$ be iid random variables with $\mathbb{E}[C_i] = 0$ and $\mathbb{E}\left[\| C_i\| _2^2\right] = \sigma^2\in \mathbb{R}_+$ . Define $A = \frac{1}{n(n - 1)}\sum_{i\neq j}C_i\cdot C_j$ . Then $\mathbb{E}\left[A^2\right]\leq 2\sigma^4 /(n(n - 1))$ . + +This is the usual analysis of variance of a U-statistic by intersection disjunction, see for instance the lecture notes Jordan [2007] for Berkeley's Stat 210B, or the more conventional reference Asymptotic Statistics [Vaart, 1998]. An empirical verification and tightness evaluation is performed in Figure 11. + +Proof. Starting from the definition of $A$ + +$$ +A ^ {2} = \frac {1}{n ^ {2} (n - 1) ^ {2}} \sum_ {i \neq j} \sum_ {k \neq l} \left(C _ {i} \cdot C _ {j}\right) \left(C _ {k} \cdot C _ {l}\right) +$$ + +Proceed by case disjunction: + +- if $\{i,j\} \cap \{k,l\} = \emptyset$ , then $\mathbb{E}\left[(C_i \cdot C_j)(C_k \cdot C_l)\right] = \mathbb{E}\left[C_i \cdot C_j\right]\mathbb{E}\left[C_k \cdot C_l\right] = 0$ . +- if $\# (\{i,j\} \cap \{k,l\}) = 2$ , then $\mathbb{E}\left[(C_i\cdot C_j)(C_k\cdot C_l)\right] = \mathbb{E}\left[(C_i\cdot C_j)^2\right]$ , and by Cauchy-Schwarz inequality, it holds $\mathbb{E}\left[(C_i\cdot C_j)^2\right] \leq \mathbb{E}\left[C_i^2 C_j^2\right] \leq \mathbb{E}\left[C_i^2\right]\mathbb{E}\left[C_j^2\right] \leq \sigma^4$ . +- if $\# (\{i,j\} \cap \{k,l\}) = 1$ , then without loss of generality $i = k$ and $j \neq l$ . Therefore + +$$ +\mathbb {E} \left[ \left(C _ {i} \cdot C _ {j}\right) \left(C _ {k} \cdot C _ {l}\right) \right] = \mathbb {E} \left[ \left(\left(C _ {i} \cdot C _ {l}\right) C _ {i}\right) \cdot C _ {j} \right] = \mathbb {E} \left[ \left(C _ {i} \cdot C _ {l}\right) C _ {i} \right] \cdot \mathbb {E} \left[ C _ {j} \right] = 0 +$$ + +It remains to take expectations and count the number of size-2 intersections. + +$$ +\mathbb {E} \left[ A ^ {2} \right] \leq \frac {1}{n ^ {2} (n - 1) ^ {2}} \sum_ {i \neq j} 2 \sigma^ {4} \leq \frac {2 \sigma^ {4}}{n (n - 1)} +$$ + +![](images/76e6084a14894aae8cedeaeaca4daa6128bb6afa249b7fecdab38f3b800d234a.jpg) +Figure 11: Empirical measurements of $\mathbb{E}\left[A_n^2\right]$ as a function of $n$ (mean and 5-sigma confidence interval for the mean, 500 samples) versus upper-bound used in Lemma A.14, for isotropic gaussians in dimension $d = 10$ with noise $\sigma_0 = 100$ per coordinate, thus $\mathbb{E}\left[\|C_i\|_2^2\right] = \sigma^2 = d\sigma_0^2 = 10^5$ . + +# B Influence of learning rate parameters $(\eta$ -scan) on Problem QWC + +We present results of all algorithms on Problem QwC at various noise levels, for all learning rate parameters tried in our experimental protocol. Flatter lines indicate less sensibility to the hyperparameter, aligned minima indicate ability to tune on short horizons. The smooth standard limit step $\beta^{-1} = 1$ is displayed as a vertical dotted line, for all algorithms with smooth claims. + +# B.1 SGD with constant, scheduled, or stability-adjusted learning rates + +![](images/4a5165469bb5ffd1156e298a98923374079c3bf42961a6f3222419f984a9ffdd.jpg) +(a) SGD: $\eta_t = \eta_0$ + +![](images/fa769551701bc8ad1dea0f06e99747bc8542ed484b5f85fa9112263291dc40e2.jpg) +(b) SGD: $\eta_t = \eta_0 \cdot t^{-1/2}$ + +![](images/959e44bb794e26812820d1310fd51586f0041243ec29fb29782055f729eefcf9.jpg) +(c) Stab-SGD + +![](images/378db5c61fbb36517cda43373259fd78746a568566ed365924bfde569604379c.jpg) +Figure 12: Problem QwC with additive gaussian noise of variance $\sigma^2 = d$ . Excess loss versus base learning rate $\eta_0 \in \mathbb{R}_+^*$ and training time $T \in \mathbb{N}$ . + +![](images/38b97fe9f8115f0a35c631c1c6c847f60655db9a08a2c562a2101d1874fa599e.jpg) +(a) SGD: $\eta_t = \eta_0$ + +![](images/92e22014c50049902432dc79b055b81176108e9909d15c92b1c863ca53769301.jpg) +(b) SGD $\eta_t = \eta_0 \cdot t^{-1/2}$ + +![](images/fd8f895c744155ef721d19d42a7765ac0d220d6e5083b028f69a5050146f345a.jpg) +Figure 13: Problem QWC with additive gaussian noise of variance $\sigma^2 = 10^{-4} \cdot d$ . Excess loss versus base learning rate $\eta_0 \in \mathbb{R}_+^*$ and training time $T \in \mathbb{N}$ . + +![](images/087f57f0f2ca67dd12df158e75ea1534ce24949948fffa850728e649e9ee302f.jpg) +(c) Stab-SGD + +![](images/40682a6368c5c0e38661b782b75e7af2cf3c776383177e571e3186d86d78df53.jpg) +(a) SGD: $\eta_t = \eta_0$ + +![](images/01e53ae4092c44ef3ecd876cacb307d1e448079588681f712523502f689f269a.jpg) +(b) SGD: $\eta_t = \eta_0 \cdot t^{-1/2}$ + +![](images/a97147a3153e019dd5f7c2328573c805996fae2736823530be335fcf4bfefa37.jpg) +Figure 14: Problem QWC with additive gaussian noise of variance $\sigma^2 = 10^{-16} \cdot d$ . Excess loss versus base learning rate $\eta_0 \in \mathbb{R}_+^*$ and training time $T \in \mathbb{N}$ . + +![](images/b89d45d318b977a4a82fa09e6b02175c926ad051665111427c5a383731a6eb4c.jpg) +(c) Stab-SGD + +Fig. 12, Fig. 13 and Fig. 14 show evolution of the mean excess loss as a function of the base learning rate $\eta_0$ (before applying any scheduler) and the total training time $T \in \mathbb{N}$ (a.k.a. "horizon"). The dependence of the optimal base learning rate on the horizon $T$ is visible for both SGD with constant learning rate and with $t^{-1/2}$ schedule. Additionally, these optimal base learning rates are seen to shift between the two figures, when the noise levels vary. In particular, this means that the learning rate of mini-batch SGD must be re-tuned if the batch size (i.e. noise level) is altered. + +Consistently with other experiments, the optimal learning rates for long horizons $(T \geq 10^{7})$ are associated with a long plateau at initialization. This implies that models tuned for long horizons are essentially unusable at mid-training (no better than initialization), thus it is meaningless to consider an "optimal trajectory", or a horizon-independent "optimal learning rate"; on the contrary, the horizon plays a central role in evaluating the quality of the model. This effect is much less pronounced with Stab-SGD, with little to no movement around the prescribed rate $\eta_0 = \beta = 10^0$ across noise levels. + +# B.2 D-Adapt + +We repeat the experiment at multiple noise levels with the D-adapt algorithm, Defazio and Mishchenko [2023, Algorithm 2]. We run the experiment with the hyperparameters $D = 2$ and $D = 200$ separately, and sweep over all "learning rates" $G^{-1}$ for each case. + +![](images/921cc18697cff801bab2c2b20f8c72a98d924cafa9f068039e4826eab5d240e3.jpg) +(a) $\sigma^2 = d$ +Figure 15: Performance of D-adapt algorithm, for $D = 2$ , on Problem QwC at various noise levels. + +![](images/1580266ca2d1253bdbe4a384e89f15e81c0e4b5f16876255d967dfea0d8d6f22.jpg) +(b) $\sigma^2 = 10^{-4} \cdot d$ + +![](images/ee967b655d2022f9cf14bc958f30b533411d7947a7eda61514aa12c1bac5b1a7.jpg) +(c) $\sigma^2 = 10^{-16} \cdot d$ + +![](images/8e16a8f13204b8bac259b1823039b98edfa922c9b9fd7c6965ec0195ccbd2956.jpg) + +![](images/1d0dffdc4847a64723dd8c937356310f317352e751e28e6a79a214c8f8d11828.jpg) +(a) $\sigma^2 = d$ + +![](images/83667afc24e443b965f8099f14500de757a78871516104f1e4e6519c865c330f.jpg) +(b) $\sigma^2 = 10^{-4} \cdot d$ + +![](images/c74b29ac3dde8b6d102b3b548aced50bb486db8e27e7e470bd757e3c8853b425.jpg) +(c) $\sigma^2 = 10^{-16} \cdot d$ + +![](images/0d6be9511c95b4177a669329a95134a9abe05348afdee790aaf05499f2e41d64.jpg) +Figure 16: Performance of D-adapt algorithm, for $D = 200$ , on Problem QWC at various noise levels. + +# B.3 "Schedule-free SGD" + +We repeat the experiments with the "Schedule-free SGD" algorithm from "The Road Less Scheduled", as it is described in the main text: Defazio et al. [2024, Sec. 2, Eq 3-5], i.e. with hyperparameters $\beta = 0.9$ and $x$ -step schedule $c_{t} = 1 / (t + 1)$ as prescribed in Sec. 2 §2. + +![](images/fa413a819d39ba70c8880abcd1a2d438eca84659160f9a542c1cd2216f1e78a4.jpg) +(a) $\sigma^2 = d$ +Figure 17: Performance of the "Schedule-free SGD" algorithm, on Problem QWC at various noise levels. We observe saturation at all noise levels, this is inconsistent with the idea that this algorithm can be used instead of a scheduler for SGD. + +![](images/2aacd1c4602d03aa0db731409a444edbd4b8bea60d66c3c11f6fb28496fe7a06.jpg) +(b) $\sigma^2 = 10^{-4} \cdot d$ + +![](images/f1c999b09b0af5823e5a5c929d3d7783ec9cb72461be3636d86be8ed5dd45c6b.jpg) +(c) $\sigma^2 = 10^{-16} \cdot d$ + +![](images/b13554afbb1821afe89cc392ab5f0a9775d9f8474c5d280620f246cb0a342c40.jpg) + +To contrast this with the theoretical predictions in the reference, note that Defazio et al. [2024, Thm 1] only gives convergence (in the Lipschitz model) with the horizon-dependent hyperparameter $\gamma = DT^{-1/2}$ . The smooth result Defazio et al. [2024, Appendix Corollary 2] uses a time-varying parameter $\beta_{t}$ , such as $\beta_{t} = 1/(5(t+1))$ (obtained by injecting the bounds on $w_{t}$ and $\alpha_{t}$ of Corollary 2 into their definition in Thm 5), to guarantee speed $\mathcal{O}(D^{2}\beta/T^{2} + D\sigma/\sqrt{T})$ , and uses an "optimistic online learning algorithm" for $z$ - the one given in appendix Sec D.1 uses a vanishing learning rate. + +# B.4 COCOB - Coin-betting approach + +We perform the same experiments with the Continuous Coin-Betting algorithm (COCOB) Orbona and Tommasi [2017, Algorithm 1], designed for the setting of convex online learning with Lipschitz losses and almost surely bounded gradients. Although this experiment uses smooth losses with gaussian noise (unbounded with finite variance), the performance of both this algorithm and its "Backprop" version more adapted to the non-convex setting remain competitive. + +![](images/87de2b81dad92dcabdf3fb4b80cb91d3d6d03128de66c329bdfc9a347df8edaa.jpg) +(a) $\sigma^2 = d$ + +![](images/e39e92ecc860d82b8c34d7a55a93d5a739bb080a684b9a79ab80156c6c9eccb0.jpg) +(b) $\sigma^2 = 10^{-4} \cdot d$ + +![](images/b53a197de9b7947d0e16d400ce776b628952a323d9b1380f68705c23a3dcb34c.jpg) +(c) $\sigma^2 = 10^{-16} \cdot d$ + +![](images/b0cca3c79983804856ebea574db5f816f7843617901770462911722165fdebc9.jpg) +Figure 18: Performance of the "COCOB" algorithm, on Problem QWC at various noise levels. The algorithm uses a hyperparameter $(L_{i})_{i}\in \mathbb{R}_{+}^{d}$ , which we set identically for all directions for this experiment, this being the only reasonable choice without a canonical basis. + +As observed in Fig. 18, the alignment of the optimal hyperparameter across training horizons is excellent, despite the mismatch in settings (Lipschitz objective in the theory, versus quadratic loss in the experiment, which is uniformly 1-smooth but not Lipschitz on the entire domain). The value of the limit learning rate however is perhaps not so intuitive, since it is no longer directly linked to $\beta^{-1}$ . + +Fig. 19 shows the results of COCOB-Backprop Orabona and Tommasi [2017, Algorithm 2]. + +![](images/f19b591dd8ebf343e45b004ac6039d3c3fa68ab39bffe937d8a7479dfa215c4f.jpg) +(a) $\sigma^2 = d$ +Figure 19: Performance of the "COCOB-Backprop" algorithm, on Problem QWC at various noise levels. The vertical line depicts the default value $(\alpha = 10^{+2})$ suggested to make this algorithm completely "parameter-free" (in the sense that it has no parameters to tune). + +![](images/eb6c197eeeefddbcae106686e6dbb7f691b6515cf1b9126f384b3aa09a925955.jpg) +(b) $\sigma^2 = 10^{-4} \cdot d$ + +![](images/9c182edae103d9e924cbbdaf6bf4f5a0f3182166ee63206717b30a880d2b7e40.jpg) +(c) $\sigma^2 = 10^{-16} \cdot d$ + +![](images/6fa825b613306b72a601e7b8e48c46d22b2e565bd0aa27bffc144e2401f0d150.jpg) + +The sensitivity to the hyperparameter is essentially non-existent except near the initialization. The performance does not quite match that of SGD. For instance at $\sigma^2 = 10^{-16} \cdot d$ , SGD (both constant-step and $t^{-1/2}$ -scheduled) reach $10^{-17}$ after $10^7$ iterations (cf. Figure 14), while COCOB-Backprop reaches only $10^{-15}$ . The observation of such a gap on a single problem does not allow general conclusions on the behavior of the algorithm (usually evaluated only in worst-case performance) but remains marginally informative. The gap in performance was most apparent on Problem QSC. + +# C ResNet Training Experiments + +Methods (additional details). Consistently with the original experimental protocol He et al. [2015a, Section 3.4], we use the initialization taken from He et al. [2015b], also known as "Kaiming" initialization. This explains in particular the large initial loss, due to large values in the last layer at initialization under such scheme. Since the number of samples is not perfectly divisible by the batch size, our last batch in each epoch is smaller, we do not use a multiplicative correction for this altered size. We present in the following pictures results over 20 random seeds. Since one in those twenty essentially failed to train (loss nearly stalled at initial value), we present median and quartiles for error bars instead of means, which are less sensitive to large but rare values. + +# C.1 Loss and accuracy across multiple runs (full scale) + +![](images/7e9d2dbd312cde769e70f2a5141fc07181289d4d80865a84838ea95b746a5d19.jpg) +Figure 20: median (and quartiles as error bars) of the training loss as a function of iterations. + +![](images/3d6dde304e4fedaeccf449dbf5ad057b492b63b6fd88bdbd12469595098b721f.jpg) + +![](images/3e876638bda14e0a6cd30e187def920bc9f289b80d468907d801d69881e11044.jpg) +Figure 21: Median (and quartiles as error bars) of the test accuracy as a function of iterations. + +# C.2 Stability ratio along trajectory, and kurtosis estimations + +Fig. 22 shows the Stability Ratio and estimated kurtosis of gradients along the trajectory. Except for one run with very high kurtosis $(>40)$ , all observed values are below 10 for most of the trajectory, leading to an error of $44 + 4\kappa \leq 84$ (Lemma 1) which is below our choice of $\zeta = 100$ . + +![](images/bc0c1198695df8a3fc4276872e55ed446e989c20d6ecb5efad3c160aaa78c58d.jpg) +Figure 22: Stability ratio and kurtosis along trajectory (10 random seeds). + +![](images/0c632a064e5f6e7877334b214127526b512ed7ee539e71862b0f33850b68a46e.jpg) \ No newline at end of file diff --git a/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/images.zip b/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..5ce5bc855cd4a0fd881dea324de604d3344d0c10 --- /dev/null +++ b/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fc173aaed95e37f8e1510a09db92ddf3598d57bd374fb707e1d863f62a26ec4a +size 1954033 diff --git a/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/layout.json b/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..1bb5e1332ef9e55d6d750c6e1350f0a68ac72a27 --- /dev/null +++ b/stabsgdnoiseadaptivityinsmoothoptimizationwithstabilityratios/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e4b8f843479d526a7c8463d01485d4f8be0ece38efe41766c08d91375c53269c +size 1504261 diff --git a/stackelberglearningwithoutcomebasedpayment/3541914b-95a0-49da-aaea-f9fdb12ab8b0_content_list.json b/stackelberglearningwithoutcomebasedpayment/3541914b-95a0-49da-aaea-f9fdb12ab8b0_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..7be6e6e014c7af6775c8c3a0cf541ed61028d7a7 --- /dev/null +++ b/stackelberglearningwithoutcomebasedpayment/3541914b-95a0-49da-aaea-f9fdb12ab8b0_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3e0540e0f8db2bd8eb1f71c940889eca5fa5d22402bb0ccd60959da31b62487c +size 119708 diff --git a/stackelberglearningwithoutcomebasedpayment/3541914b-95a0-49da-aaea-f9fdb12ab8b0_model.json b/stackelberglearningwithoutcomebasedpayment/3541914b-95a0-49da-aaea-f9fdb12ab8b0_model.json new file mode 100644 index 0000000000000000000000000000000000000000..f9f96f2fb3857609e98a2084c52a0a0265303d6e --- /dev/null +++ b/stackelberglearningwithoutcomebasedpayment/3541914b-95a0-49da-aaea-f9fdb12ab8b0_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1330b2626135fef5a6a1423f58cfaab4278e1862de19abdc3b495b8e90cdb19c +size 158443 diff --git a/stackelberglearningwithoutcomebasedpayment/3541914b-95a0-49da-aaea-f9fdb12ab8b0_origin.pdf b/stackelberglearningwithoutcomebasedpayment/3541914b-95a0-49da-aaea-f9fdb12ab8b0_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ec988eea4fe220c68dbb087661c7e97a34d1114f --- /dev/null +++ b/stackelberglearningwithoutcomebasedpayment/3541914b-95a0-49da-aaea-f9fdb12ab8b0_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2e3a86d2a4f4422c6da9930342c1fc1e65503caaf2b06a34fb2a55fc4ab7c123 +size 496481 diff --git a/stackelberglearningwithoutcomebasedpayment/full.md b/stackelberglearningwithoutcomebasedpayment/full.md new file mode 100644 index 0000000000000000000000000000000000000000..4a577003530040ce205c67667ae654254f6c034e --- /dev/null +++ b/stackelberglearningwithoutcomebasedpayment/full.md @@ -0,0 +1,637 @@ +# Stackelberg Learning with Outcome-based Payment + +Tom Yan + +Carnegie Mellon University + +tyyan@cmu.edu + +Chicheng Zhang + +University of Arizona + +chichengz@cs.arizona.edu + +# Abstract + +With businesses starting to deploy agents to act on their behalf, an emerging challenge that businesses have to contend with is how to incentivize other agents with differing interests to work alongside its own agent. In present day commerce, payment is a common way that different parties use to economically align their interests. In this paper, we study how one could analogously learn such payment schemes for aligning agents in the decentralized multi-agent setting. We model this problem as a Stackelberg Markov game, in which the leader can commit to a policy and also designate a set of outcome-based payments. We are interested in answering the question: when do efficient learning algorithms exist? To this end, we characterize the computational and statistical complexity of planning and learning in general-sum and cooperative games. In general-sum games, we find that planning is computationally intractable. In cooperative games, we show that learning can be statistically hard without payment and efficient with payment, showing that payment is necessary for learning even with aligned rewards. Altogether, our work aims to consolidate our theoretical understanding of outcome-based payment algorithms that can economically align decentralized agents. + +# 1 Introduction + +Increasingly, we are seeing businesses deploying agents to carry out tasks on their behalf. In the coming agentic era, we will inevitably have multiple, decentralized agents interacting together. An emerging challenge that businesses may have to face is how to incentivize other agents to work alongside its agent. This challenge requires addressing a central difficulty in decentralized multi-agent systems, which is that of differing interests. + +In present day commerce, payment is a standard way that two parties use to resolve this challenge and more closely align their business interests. This inspires us to study the overarching question in this paper: how can we analogously implement such payment schemes in the multi-agent setting and enable economic alignment? That is, if I am a business looking to use payment to incentivize another business (and/or its agent) to work with my agent, how can I learn a good policy for my agent along with a payment scheme to go with it? + +On a technical level, this setting may be viewed as a Stackelberg Markov game. In this paper, we study the two-player Stackelberg game, where one player (leader) commits to a policy taking into account the best response to the policy by the other player (follower). We focus on Stackelberg Markov games in particular as agents will be interacting over multiple turns and potentially long horizons. Finally, to model the payment aspect, the leader is able to also increase the reward of the follower in the Markov game, which may be viewed as a form of reward shaping in line with the existing formulation in the literature [Ben-Porat et al., 2024; Bollini et al., 2024; Ivanov et al., 2024; Scheid et al., 2024; Wu et al., 2024]. + +In this work, we aim to consolidate the theoretical foundations of Stackelberg learning with payment, as complexity results have yet to be established for two-player Stackelberg Markov games. We focus + +
Planning, LearningWithout PaymentWith Payment
DAGTreeDAG
Cooperative✓, × (Theorem 5.15.2)✓, ✓✓, ✓ (Theorem 6.46.5)
General Sum×, × (Proposition 4.1)✓, ✓ (Proposition 4.3)×, × (Theorem 4.2)
+ +Table 1: Planning & learning settings where computationally and statistically efficient algorithms exist. + +on a fundamental question: is there an efficient algorithm that can provably compute or learn the optimal policy and payment? Indeed, this is an important question to address as businesses in the future would want payment schemes with provable guarantees, so as to ensure that their expenditure is optimal. + +Contributions: We analyze the planning and learning setting through both the computational and statistical lens. Please see Table $\square$ for an overview of our results. + +1. We begin by considering planning in general-sum games. Is there an efficient algorithm that can return the optimal policy and payment? We prove that such a computationally efficient algorithm cannot exist unless NP=P, and identify the structural property of the MDP that results in this hardness. To complement the negative results, we develop an efficient algorithm, applicable when this property is removed. +2. Next, we turn to Cooperative games, which is a broad subclass of Markov games useful for modeling e.g. the interaction between AI service-providers and their users. Moreover, planning is computationally efficient in this setting, making it plausible that efficient learning algorithms may be attainable. As the rewards are already aligned, we begin by considering learning in the Stackelberg game without payment. Surprisingly, however, we find that an efficient algorithm cannot exist, this time in the statistical sense. We identify structural properties of the MDP that result in statistical hardness, and develop an efficient algorithm for when such properties are removed to complement our negative results. +3. Finally, we study learning in Cooperative games with payment. Can payment be used to alleviate the statistical hardness of learning? We answer this in the affirmative by showing that we can adapt existing no-regret RL algorithms to enable sample-efficient learning. In closing, we also use this setting to contrast the two different payment settings we study. We derive matching upper and lower regret bounds for when the leader has to make payments upfront versus on-the-fly, allowing us to quantitatively assess the benefits of being able to make payments on-the-fly. + +# 2 Formulation + +# 2.1 Stackelberg Markov Game + +We consider the standard two-player, episodic finite-horizon Markov game $M$ parameterized by $\langle S,A,B,H,P,\rho ,r^L,r^F\rangle$ with state space $S$ , initial state distribution $s_0\sim \rho$ , transitions $P$ and episode length $H$ . The leader has action set $A$ and reward $r^L\in [-1,1]$ , the follower action set $B$ and reward $r^{F}\in [-1,1]$ . In the case that the game is cooperative, $r^L = r^F$ . + +In the problem of online learning for Stackelberg Markov games, the learner plays the role of the leader, where apriori the reward functions $r^L$ , $r^F$ and the transitions are unknown to the leader. At each episode $k \in [T]$ , the leader commits first to a policy $\pi_k$ . The follower best responds to $\pi_k$ with $\mu(\pi_k) \in \operatorname{argmax}_{\mu} V^{\pi_k, \mu}(s_0; r^F)$ . One may view best response as the equilibrium behavior of the follower to the leader policy. + +After the episode, the leader and the follower observe the resultant trajectory $\tau_{k} = \left\{(s_{i},a_{i},b_{i},r^{L}(s_{i},a_{i},b_{i}),r^{F}(s_{i},a_{i},b_{i})\right\}_{i = 1}^{H}$ realized by the chosen policies in $M$ : $a_{i}\sim \pi_{k}(s_{i}),b_{i}\sim \mu (\pi_{k})(s_{i}),s_{i + 1}\sim P_{i}(\cdot |s_{i},a_{i},b_{i})$ + +This trajectory is the outcome of the policies' interaction, which in turn determines the outcome-based payment the follower receives. + +Leader Payment: Following existing formulations in prior literature, the leader can increase $r^F$ by creating outcome-based payment $b_i^k (s_i,a_i,b_i)$ , if state-actions $s_i,a_i,b_i$ are realized during the episode, $s_i,a_i,b_i\in \tau_k$ . This results in a modified Markov game where the leader is able to additionally assign payment, with the payment function having signature $b_{i}^{k}:S\times A\times B\to \mathbb{R}^{+}$ . + +We note that the outcome-based payment need not correspond to direct monetary transfer. For example, we may be interested in modeling the setting where the leader is an AI-service-provider and the follower is a customer user. The leader spends money to improve its agent, and this improved agent adds additional value (e.g. more saved time) for the user during its use. But during this interaction, there is no direct transfer of money from the company to the user. + +Thus, to model indirect payments in addition to direct ones, we introduce a final piece of notation, multiplier $\kappa \in \mathbb{R}^{+}$ . $\kappa \cdot b_{i}^{k}(s_{i},a_{i},b_{i})$ corresponds to the proportional cost to the leader in creating payment (reward) $b_{i}^{k}(s_{i},a_{i},b_{i})$ for the follower. We believe proportionality is a natural assumption to make, and verily $\kappa = 1$ corresponds to direct payment. + +# 2.2 Payment Settings + +To complete the formulation, we touch on the two types of payment settings considered in this paper. + +Trajectory Payment: The first is the existing payment setting commonly studied in prior literature, which we term trajectory payment. Here, a payment is made by the leader for every state-action on the realized trajectory. This form of payment is considered in principal-agent contracting literature, where the trajectory informing how much the leader will be paying ex-post [Dutting et al., 2021]. + +Moreover, this form of payment corresponds to the trendy outcome-based pricing model, which is experiencing rapid adoption by several notable SaaS companies due to the rising usage of AI agents [Stripe, 2025, Intercom, 2025, Zendesk, 2025]. Indeed, this marks a fundamental paradigm shift in software pricing in industry, moving from seat-based subscriptions (traditional SaaS) and usage-based models (cloud infrastructure) to now outcome-based pricing in the agent era [Boston Consulting Group, 2025, Sequoia Capital, 2025]. This also makes it imperative then to bolster our theoretical understanding of outcome based pricing, which we study in this paper. + +Upfront Payment: In this paper, we will also consider a setting that we term upfront payment. As the name suggests, the leader pays for every state-action in the MDP, regardless of the realized trajectory. Note that the follower is still paid based on the realized trajectory. This is more realistic in settings where the leader pays indirectly to the benefit of the follower, and is bound by temporal constraints such that the payment cannot be made on-the-fly. + +For a motivating example, consider the AI-service provider setting discussed earlier. The company invests before deployment to improve the agent's functionality, which means that the user (follower) gains added value (reward) on the trajectory realized during the agent's use. However, the key temporal constraint is that the company cannot improve its agent on-the-fly, as the users are using it. Thus, this makes upfront payment a more realistic model of the leader's expenditure. The leader had to invest upfront to improve the agent's capabilities in all states, even though this includes off-trajectory states that are not visited during the interaction with the user. For instance, suppose the agent is a computer-using-agent [Anthropic, 2024]. The user may use it to handle emails, and the agent would act in states of the computer corresponding to the inbox. However, even though the company had also invested to improve the agent's capabilities in coding, the user may not invoke the agent to do so (perhaps due to excessive risk). And so, the agent would not have acted in other states of the computer corresponding to the codebase. + +More generally, there is sizable body of economics contracting literature studying settings where only ex-ante (upfront) payment is possible. Some reasons for this include non-enforceable contracts, where the principal can renegel upon observing the outcome [Hart and Moore 1988]. Another cause for this may be non-verifiable outcomes; that is, when outcomes cannot be verified, ex-post contracts become unenforceable as there is no way to condition legally binding payments [Aghion and Holden 2011]. Finally, one other reason may simply be that the agent is risk-averse, thus preferring upfront payment in face of stochastic outcomes [Laffont and Martimort 2002]. + +Leader Optimization: Putting it all together, we can now write down the resulting Stackelberg game under the two payment settings. + +Definition 2.1. In Stackelberg Markov games with trajectory payment, the leader optimizes: + +$$ +\max _ {\pi , b \geq 0} V ^ {\pi , \mu (\pi)} \left(s _ {0}; r ^ {L} - \kappa \cdot b\right) \tag {1} +$$ + +$$ +s. t. \quad \mu (\pi) \in \operatorname {a r g m a x} _ {\mu^ {\prime}} V ^ {\pi , \mu^ {\prime}} \left(s _ {0}; r ^ {F} + b\right) +$$ + +In Stackelberg Markov games with upfront payment, the leader optimizes: + +$$ +\max _ {\pi , b \geq 0} \left(V ^ {\pi , \mu (\pi)} \left(s _ {0}; r ^ {L}\right) - \kappa \cdot \sum_ {s, a, b \in S \times A \times B} b (s, a, b)\right) \tag {2} +$$ + +$$ +s. t. \quad \mu (\pi) \in \operatorname {a r g m a x} _ {\mu^ {\prime}} V ^ {\pi , \mu^ {\prime}} \left(s _ {0}; r ^ {F} + b\right) +$$ + +Before moving on, we highlight the generality of the class of games we are studying. The class of Stackelberg Markov games with payment generalizes Stackelberg Markov games. Indeed, constraining the leader to zero payment (i.e. $(\pi, b) = (\pi, 0)$ ) corresponds to the leader's policy space in Stackelberg Markov Games. Analogously, for Cooperative Stackelberg Markov games with payment studied in the later sections, this class of games generalizes Cooperative Stackelberg Markov games. + +# 3 Related Works + +As we focus on Stackelberg Markov games with payment, our paper is most related to two lines of work. The first is the line of work studying the complexity of Stackelberg policy computation in Markov games. And the second is algorithms for computing optimal payment schemes in MDPs. We cover both lines of work below, and include a discussion on additional related works in Appendix E. + +Stackelberg Optimal Policies in Markov Games without Payment: Due to the wide applicability of the Stackelberg Markov games, there has been a long line of work seeking to understand how to compute optimal leader policies with provable guarantees. + +For planning, Conitzer and Sandholm [2006], Letchford et al. [2012], Letchford and Conitzer [2010] study the computational tractability of optimal Stackelberg policy computation in Markov games and subclasses thereof. For stochastic MDPs, they establish that computing the optimal Stackelberg policy is NP-Hard. + +For learning, Zhao et al. [2023] studies the statistical complexity in cooperative bandit games. Bai et al. [2021] studies the statistical complexity in bandit-RL games, a particular subclass of Markov games. Our work differs from this line of work in focusing on Markov games, which are more general than bandit-RL games and have longer horizon than bandit settings. Moreover, the leader is allowed to use payments to shape the follower's rewards. As we will see, this turns out to be crucial for improved exploration during learning in certain settings. + +Learning the Optimal Payment Scheme in MDPs: Recently, there has been burgeoning interest in computing optimal payment schemes for contracting agents to act in MDP environments, wherein the leader may increase the follower's rewards as a form of reward shaping to incentivize the follower to play policies desirable to the leader. + +The single-agent MDP setting, where only the follower acts in the MDP and the leader incentivizes, is formulated by Ben-Porat et al. [2024], Chen et al. [2022]. This is followed by a series of interesting work by Bollini et al. [2024], Ivanov et al. [2024], Wu et al. [2024], studying learning under a variety of different payment functions taking as input the state, the state-action or the state next-state. Our work adds to this line of work by focusing on two-player Markov games, which generalize the single-player setting. Furthermore, while previous works mostly focus on trajectory payment, we also consider upfront payment, applicable in settings where the leader cannot pay on the fly due to temporal constraints. We derive tight regret guarantees to contrast the two differing payment settings. + +The paper closest in formulation to that of ours is that by Scheid et al. [2024], who considers the same state-action based payment function in the bandit setting. Our work differs in focusing on Markov games, with a longer horizon than that in bandit settings. This in turn introduces difficulty in terms of exploration, and requires a more nuanced optimal payment computation beyond the binary search approach used in [Scheid et al., 2024]. + +Finally, as payment may be viewed as strategic reward shaping, our analysis is also related to existing RL literature that seeks to theoretically quantify the benefits of reward shaping [Ng et al., 1999]. Gupta et al. [2022] quantifies how statistical sample complexity is improved by reward shaping in the single-agent setting. By contrast, in our work, we study improved sample complexity in two-player cooperative Stackelberg Markov games. + +# 4 Planning in General-sum Games + +In this section, we ask: is there an efficient algorithm that can compute the optimal policy and payment in general-sum games? We investigate the computational complexity of such an algorithm, starting with the planning setting, where the Markov game dynamics and reward functions are known. + +Our main finding is that there is no such computationally efficient algorithm unless NP=P. Outcome-based payment does not alleviate the computational intractability of computing the optimal Stackelberg policy, even in planning [Conitzer and Sandholm 2006]. We identify that when the MDP has DAG structure, this leads to computational intractability. Later in the section, we complement this negative result with a positive result for when the MDP has tree structure. All proofs in this section may be found in Appendix A. + +# 4.1 Hardness Results + +We first derive a result showing that it is NP-Hard to compute the optimal leader policy even in deterministic MDPs, without payment. Note that in [Conitzer and Sandholm, 2006], computational intractability is demonstrated in stochastic MDPs. + +Proposition 4.1. Under Markov games that are deterministic DAGs, it is NP-Hard to compute the optimal policy: + +$$ +\max _ {\pi} V ^ {\pi , \mu (\pi)} (s _ {0}; r ^ {L}) +$$ + +$$ +s. t. \quad \mu (\pi) \in \operatorname {a r g m a x} _ {\mu^ {\prime}} V ^ {\pi , \mu^ {\prime}} \left(s _ {0}; r ^ {F}\right) +$$ + +Helpfully, deterministic MDPs allow us to provide guarantees for both two payment settings. As we show in the proof, the optimal payment scheme pays zero in off-policy states, which can be readily characterized in deterministic MDPs. This result is intuitive as paying in off-policy states only incentivizes the follower to deviate off-policy, which is undesirable and increases leader total payment. With this result, we can derive that the optimal payment scheme is the same under trajectory and upfront payment. Thus, we use same construction, which provides a reduction to the PARTITION problem, to prove computational intractability under both payment settings. + +Theorem 4.2. Under Markov games that are deterministic DAGs, it is NP-Hard to compute the optimal policy and optimal trajectory payment: + +$$ +\max _ {\pi , b \geq 0} V ^ {\pi , \mu (\pi)} \left(s _ {0}; r ^ {L} - \kappa \cdot b\right) +$$ + +$$ +s. t. \quad \mu (\pi) \in \operatorname {a r g m a x} _ {\mu^ {\prime}} V ^ {\pi , \mu^ {\prime}} \left(s _ {0}; r ^ {F} + b\right) +$$ + +and it is also NP-Hard to compute the optimal policy and optimal upfront payment: + +$$ +\max _ {\pi , b \geq 0} \quad \left(V ^ {\pi , \mu (\pi)} (s _ {0}; r ^ {L}) - \kappa \cdot \sum_ {s, a, b \in S \times A \times B} b (s, a, b)\right) +$$ + +$$ +s. t. \quad \mu (\pi) \in \operatorname {a r g m a x} _ {\mu^ {\prime}} V ^ {\pi , \mu^ {\prime}} \left(s _ {0}; r ^ {F} + b\right) +$$ + +In closing, we note that the optimal objective value of the subset of Markov games used to reduce to the PARTITION problem is an integral multiple of $1/2$ . Due to this, we have that computational intractability in planning implies computational intractability in learning. In more detail, let $M^*$ be the optimal objective value, which is an integer multiple of $1/2$ . Suppose by contradiction that we had an algorithm with sublinear regret $T^\alpha (\alpha < 1)$ . We can then set $T$ large enough such that $T^\alpha / T < 1/2$ . This allows us to infer $M^*$ exactly by rounding to the nearest $1/2$ , giving us a computationally efficient algorithm for answering the decision version of the PARTITION problem, which is a contradiction. + +# Algorithm 1 Planning Algorithm for MDP with Deterministic Tree Structure + +Require: Pre-computed policy $\pi^{-} \in \mathrm{argmin}_{\pi} V^{\pi, \mu(\pi)}(s_0; r^F)$ (efficiently computed via Nash-VI) + +for all root to leaf paths $\tau = s_1, a_1, b_1, s_2, a_2, b_2, \ldots, s_H, a_H, b_H$ do + +Define $\pi(s_i) = a_i$ for $s_i, a_i \in \tau$ . In every other state $s_i' \notin \tau$ , let $\pi(s_i') = \pi^{-}(s_i')$ . + +Compute $\mu (\pi)$ and compute follower Q-values, $Q^{\pi ,\mu (\pi)}(\cdot ,\cdot ,\cdot)$ + +Solve for the minimal payment scheme using LP: + +$$ +\begin{array}{l} b ^ {\tau} (\pi) = \operatorname {a r g m i n} _ {b} \quad \sum_ {s _ {i}, a _ {i}, b _ {i} \in \tau} b (s _ {i}, a _ {i}, b _ {i}) \\ \text {s . t .} \quad \sum_ {i \geq h, s _ {i}, a _ {i}, b _ {i} \in \tau} r ^ {F} \left(s _ {i}, a _ {i}, b _ {i}\right) + b \left(s _ {i}, a _ {i}, b _ {i}\right) \geq \max _ {b _ {h} ^ {\prime} \neq b _ {h}} Q ^ {\pi , \mu (\pi)} \left(s _ {h}, a _ {h}, b _ {h} ^ {\prime}; r ^ {F}\right) \tag {3} \\ \end{array} +$$ + +end for + +Output the leader policy $\pi$ and payment scheme of the path $\tau$ with maximal return $\sum_{s_i, a_i, b_i \in \tau} r^L(s_i, a_i, b_i) - \kappa \cdot b^\tau(\pi)$ . + +# 4.2 Positive Results + +To complement our negative results, we show that positive results are attainable in MDPs without DAG structure. That is, in general-sum games where the MDP has tree structure, there is a polynomial-time algorithm for learning the optimal leader policy and payment. We describe our planning algorithm, Algorithm $\boxed{1}$ that forms the crux of our approach to learning in this setting, and is applicable under both trajectory and upfront payment. + +Proposition 4.3. Under Markov games that are deterministic trees, there exists a polynomial-time planning algorithm that computes the optimal policy and payment. + +Remark 4.4. To complete the result, we note in Appendix A that there is a simple exploration strategy using payment for general-sum, deterministic trees, as exploration needs to only recover rewards. This strategy allows us to reduce learning to planning, and then apply Algorithm 7 + +Before moving on, we note that in this general-sum game, the leader behaves in a zero-sum like manner in off-policy states in Algorithm [1]. This incentivizes the follower to take the desired policy and allows the leader to minimize the total payment needed to incentivize such policy. + +Finally, due to the intractability of computing a global Stackelberg optimum, it is natural to consider computing a local Stackelberg optimum instead, so that the policy and payment scheme does attain some guarantees. Building on existing results on first order methods in Stackelberg games [Shen et al., 2024], we derive a first order approach to this end. Note that while our paper is concerned with global Stackelberg optimality guarantees, we use this to illustrate that a more relaxed solution concept can be computed, if desired. + +# 5 Learning in Cooperative Games without Payment + +The computational intractability in the general-sum case prompts us to investigate whether efficient algorithms are attainable in significant subclasses of Markov games. Cooperative games are a broad subclass of Markov games useful for modeling e.g. the aforementioned AI-service based setting. Indeed, since the goal of the assistant agent is to aid the user, their rewards are aligned. And so, such settings correspond to a two-player cooperative game, making it an important subclass of Markov games to understand. + +Moreover, on a technical level, it seems that there is hope for efficient algorithms as planning is efficient in cooperative games (e.g. via Nash-VI as in [Bai and Jin 2020]). And so, in this section, we study the question: is there an efficient learning algorithm in cooperative games? We delve into this by first considering cooperative games without payment, which has yet to be addressed in the prior literature. Since the rewards are already aligned, we might expect that there are efficient learning algorithms. To our surprise, however, we find that learning in Cooperative Markov games + +can be prohibitively hard, this time in the statistical sense. All proofs in this section may be found in Appendix B. + +Structural properties of MDP: We identify the specific MDP properties under which exploration can be statistically intractable, along with complementary positive results. In a nutshell, we find that if the MDP has deterministic tree structure, then efficient algorithms are possible. However, allowing for stochastic or DAG transitions leads to statistical hardness. + +Theorem 5.1. There exists a turn-based Stochastic Tree Markov game such that: any (possibly randomized) algorithm that returns the optimal leader policy with probability at least $1/2$ requires at least $\Omega(2^{|S|})$ number of episodes. + +Theorem 5.2. There exists a turn-based Deterministic DAG Markov game such that: any (possibly randomized) algorithm that returns the optimal leader policy with probability at least $1/2$ requires at least $\Omega(2^{|H|})$ number of episodes. + +Proposition 5.3. Under Markov games that are deterministic trees, then there exists a polynomial-time algorithm that can learn a near-optimal leader policy. + +We remark that the statistical intractability results are based on a "needle-in-the-haystack" construction, where only a specific combination of leader actions is optimal. Structural properties of the MDP like stochastic or DAG transitions allow us to embed this construction in the MDP. Combined with the follower best responding instead of coordinating exploration with the leader, we can show that an exponential number of samples is needed by the leader to find the right combination, even if the rewards are already aligned. + +Relaxing Follower Best Response behavior: As the statistical hardness is due to both the structural property of the MDP and the best response nature of the follower, a natural question one may ask is: can relaxing the latter alleviate statistical hardness and allow for efficient learning across all MDPs? + +The natural way to relax best response is to consider best response under $\lambda$ -entropy-regularization, which generalizes follower best response (corresponding to when $\lambda = \infty$ ). This behavior model is often used to model human behavior in human-AI interaction and behavioral economics literature [Ziebart et al., 2010, Reddy et al., 2018, McKelvey and Palfrey, 1995]. However, we again find that learning with this follower behavior does not allow for more sample efficient exploration: + +Theorem 5.4. There exists a turn-based Deterministic DAG Markov game such that: any (possibly randomized) algorithm that outputs the optimal policy given $\lambda$ -Entropy-regularized best response with probability at least $1/2$ requires at least $\Omega(\exp(\lambda^2 H/8))$ episodes if $\lambda \leq 1$ and $\Omega(\exp(H/8))$ episodes if $\lambda > 1$ . + +In closing, we offer a conceptual interpretation of the technical results in this section, using the example of the assistant agent and the user. Our results suggest that the service provider company can have difficulty exploring, due to the user's best response. Indeed, users are simply looking to use the agent wherever it is at its best, and will not use the agent for the sake of its improvement. In particular, this means that users are not willing to use the agent in states that it currently does not currently excel in. Even though, these are precisely the states that the agent needs to obtain more training samples in. And so, this suggests that if the company wants to efficiently explore to learn an even better agent, incentivized exploration is needed. + +# 6 Learning in Cooperative Games with Payment + +In sum, we know from the previous section that in Stackelberg games, coordinated exploration is necessary for efficient learning. And so, in this section, we study how payment can be used to align the follower and enable efficient leader exploration. Our overall finding is that payment can lead to efficient exploration, and alleviate the statistical hardness in cooperative games without payment. All proofs in this section may be found in Appendix C. + +# 6.1 Regret Guarantees in Cooperative Games + +We study regret guarantees under the standard reinforcement learning setup with unknown transitions and unknown rewards, which can be stochastic. + +Learning protocol: At each episode $k \in [T]$ , the leader commits first to a policy $\pi^k$ and a payment function $b^k$ . The follower best responds to $\pi^k$ with $\mu(\pi^k) \in \operatorname{argmax}_{\mu} V^{\pi_k, \mu}(s_0; r^F + b^k)$ . After the episode, the leader and the follower observe the resultant trajectory $\tau_k = \{(s_i, a_i, b_i, r^L(s_i, a_i, b_i))\}_{i=1}^H$ realized by the chosen policies in $M$ (recall that $r^L = r^F$ ). The goal of the learner is to minimize its Stackelberg regret, defined as follows: + +Definition 6.1. In Stackelberg games with trajectory payment, the Stackelberg regret is defined as: + +$$ +\mathcal {R} (T) = \sum_ {k = 1} ^ {T} V ^ {\pi^ {*}, \mu (\pi^ {*}; r ^ {F} + b ^ {*})} (s _ {0}; r ^ {L} - \kappa \cdot b ^ {*}) - V ^ {\pi^ {k}, \mu (\pi^ {k}; r ^ {F} + b ^ {k})} (s _ {0}; r ^ {L} - \kappa \cdot b ^ {k}) +$$ + +The regret under upfront payment regret may be defined analogously. + +Towards analyzing Stackelberg regret, we characterize the optimal policy and trajectory payment when $r^L = r^F$ ; we can analogously show the same result under upfront payment. + +Lemma 6.2. For any $\pi^{*}, b^{*}$ such that: + +$$ +\begin{array}{l} \pi^ {*}, b ^ {*} = \operatorname {a r g m a x} _ {\pi , b} \quad V ^ {\pi , \mu (\pi ; r ^ {F} + b)} \left(s _ {0}; r ^ {L} - \kappa \cdot b\right) \tag {4} \\ s. t. \quad \mu (\pi ; r ^ {F} + b) \in \operatorname {a r g m a x} _ {\mu^ {\prime}} V ^ {\pi , \mu^ {\prime}} (s _ {0}; r ^ {F} + b) \\ \end{array} +$$ + +If $r^L = r^F$ , then we must have $\pi^{*}, \cdot = \operatorname{argmax}_{\pi, \mu} V^{\pi, \mu}(s_0; r^L)$ and $b^{*} = 0$ . + +With this, we have that the optimal payment scheme in any cooperative game must be zero, as one would intuitively expect with already aligned rewards. This allows us to decompose Stackelberg regret into regret due to sub-optimality in policy and regret due to payment used during exploration, which will be responsible for the differing rates between trajectory and upfront payment. + +Moreover, we note an interesting contrast due to this result. As we just saw, learning can be prohibitively hard in the absence of payment. Hence, we have that payment is not necessary in planning, but is crucial for learning (efficiently). + +The crux of our positive results is that we can apply the canonical optimism under uncertainty principle to achieving sublinear Stackelberg regret. This follows from the observation that payment enables optimism in learning, which the leader can operationalize by setting payments according to its bonuses. This incentivizes the follower to also explore optimistically. A key lemma for bounding the policy regret portion of Stackelberg regret goes as follows. + +Lemma 6.3. Suppose we can construct an optimistic MDP $M_{k}$ of the true MDP $M$ . Let the optimal leader policy under $M_{k}$ be $\pi_{k}$ , then: + +$$ +\sum_ {k = 1} ^ {T} V _ {M} ^ {\pi^ {*}, \mu_ {M} (\pi^ {*})} (s _ {0}; r ^ {L}) - V _ {M} ^ {\pi^ {k}, \mu_ {M} (\pi^ {k})} (s _ {0}; r ^ {L}) \leq \sum_ {k = 1} ^ {T} V _ {M _ {k}} ^ {\pi^ {k}, \mu_ {M _ {k}} (\pi^ {k})} (s _ {0}; r ^ {L}) - V _ {M} ^ {\pi^ {k}, \mu_ {M _ {k}} (\pi^ {k})} (s _ {0}; r ^ {L}) +$$ + +Note that because the leader knows $M_{k}$ , they know the policy $\mu_{M_k}(\pi^k)$ that they would like to incentivize the follower to play. Using this, we show that one can also bound the regret due to the cumulative payment, to obtain the following regret guarantees. + +Theorem 6.4. UCB-VI-FP (Algorithm 2) incurs $O(T^{1/2})$ regret under trajectory payment. This is tight as there exists a subset of Markov games, where any learning algorithm must incur $\Omega(T^{1/2})$ regret. + +Theorem 6.5. There exists an algorithm, leveraging UCB-VI-FP as subroutine, that incurs $O(T^{2/3})$ regret under upfront payment. + +# 6.2 Contrasting Trajectory Payment with Upfront Payment + +Finally, as positive results are attainable in Cooperative Markov games, we can analyze the difference in regret rates under the two different payment settings. What is the benefit afforded by settings where the leader can pay on-the-fly? Towards answering this question, we analyze the simple setup of unknown, deterministic rewards. Helpfully, this learning task already a sizable contrast in terms of regret between the two settings. We provide tight bounds on regret guarantees under both payment settings to contrast the two payment settings. + +Algorithm 2 UCB-VI with Follower Payment (UCB-VI-FP) +Initialize $Q_{h}(s,a,b) = H$ for all $h\in [H],s,a,b\in S_h\times A\times B.$ +for $k = 1,\dots,T$ do construct $M_{k}$ +for $h = H,\ldots ,1;s,a,b\in S_h\times A\times B$ do $\triangleright$ standard bonus for stochastic rewards +Compute estimated transitions from data in buffer: $\hat{P}_h(s'|s,a,b) = \frac{N_h^k(s,a,b,s')}{N_h^k(s,a,b)}$ +Compute optimistic rewards of $M_{k}$ from reward samples in buffer: $\hat{r}_h^k (s,a,b) =$ $\bar{r}_h^k (s,a,b) + c\sqrt{\frac{H^2}{N_h^k(s,a,b)}}$ $Q_{h}(s,a,b) = \min (H,\hat{r}_{h}^{k}(s,a,b) + \hat{P}_{h}^{k}V_{h + 1}(s,a,b))$ $V_{h}(s) = \max_{a,b}Q_{h}(s,a,b)$ +end for +Leader commits to Stackelberg policy $\pi^k:\pi^k (s_h) = \mathrm{argmax}_a\max_bQ_h(s_h,a,b).$ +Set outcome-based payment scheme: $\beta_h^k (s_h,a_h,b_h) = 2\cdot c\sqrt{\frac{H^2|S|}{N_h^k(s,a,b)}}.$ +for $h = 1,\dots,H$ do Leader plays $a_h^k\sim \pi^k (s_h^k)$ , follower plays $b_{h}^{k}$ via $\mu (\pi^k)$ Transition to $s_{h + 1}^{k}\sim P(\cdot |s_{h}^{k},a_{h}^{k},b_{h}^{k})$ and save data $(s_h^k,a_h^k,b_h^k,s_{h + 1}^k)$ in buffer end for +end for + +Proposition 6.6. UCB-VI-FP with indicator bonus incurs constant $O(|S||A||B|)$ regret under trajectory payment, where we designate reward under indicator bonus to be $\hat{r}(s,a,b) = 1 \{ if (s,a,b) is unvisited \}$ and $r(s,a,b)$ o.w. + +As the regret bound is constant in $T$ , we have that the bound must be tight. Next, we derive regret rates under upfront payment, whose regret lower bound requires a significantly nuanced probabilistic argument using Yao's lemma. + +Proposition 6.7. There exists an algorithm, leveraging UCB-VI-FP with indicator bonus as subroutine, that incurs $O(T^{1/2})$ regret under upfront payment. + +Proposition 6.8. There exists a subset of Markov Game instances such that any learning algorithm has to incur $\Omega (T^{1 / 2})$ regret under upfront payment. + +The construction of the negative result reveals the key difference in two payment schemes. In a nutshell, upfront payment is affected by difficult-to-reach states ( $\epsilon$ -significant states [Jin et al., 2020]). On the other hand, trajectory payment is unaffected as the payment is made only if the follower does reach such a state. That is, the leader's payment for actions in that statement is weighted by the visitation probability. + +And so, the key difficulty in exploration under upfront payment is that when payment is needed to incentivize the follower to reach insignificant states, a lot of the payment can be wasted even if the follower is aligned, due to the low visitation probability. This is directly responsible for the sizable change in the regret guarantee, going from $O(1)$ to $\Omega (T^{1 / 2})$ . Overall, this suggests that if the leader cannot pay on-the-fly, the payment scheme should factor in the reachability of states. + +# 7 Discussion + +In this work, we study learning in Stackelberg Markov games with payment. To consolidate the theoretical foundations of this setting, we chart the computational and statistical complexity of both planning and learning. + +Future Work: Due to the intractability of general-sum settings, we believe that there is much more work to be done in analyzing more specific subclasses of Markov games. Which other subclasses of Markov games are such that efficient algorithms are attainable? + +Limitations: One key assumption underlying our paper is that the follower's action can be observed by the leader. We believe that this can be realistic for modeling certain digital settings (such as computers), wherein the agent's actions can be readily tracked (computer-using-agent's actions can be logged and monitored) [Anthropic, 2024; Sumers et al., 2025]. With that said, handling the case + +for when the follower's action is not observable is very important, especially in physical environment where monitoring is not possible. And we believe that results from the setting we study can serve as a stepping stone towards results in partial information settings with unobserved actions. + +Another key underlying assumption is that the leader can readily observe the follower's reward, either directly or through the follower's report. It is conceivable that in cases the leader cannot observe the reward directly, the follower may not report their reward truthfully. In such settings, we note two observations. Let $(\pi^{*}(r), b^{*}(r))$ denote an optimal policy under reported follower reward $r$ . Let $r^{F}$ denote the true reward and $r^{\prime F}$ the reported reward. + +First, if we are in the cooperative setting, we observe that there is no incentive for the follower to misreport. Because the leader payment is zero, truthful reporting yields the highest return: $V^{\pi^{*}(r^{F}),\mu (\pi^{*}(r^{F}))}(s_{0};r^{F}) \geq V^{\pi^{*}(r^{\prime F}),\mu (\pi^{*}(r^{\prime F}))}(s_{0};r^{F}).$ + +Second, in the general-sum bandit setting with direct payment considered by [Scheid et al. 2024], the payment can now be nonzero but the follower's gain from misreporting is bounded. + +Proposition 7.1. Suppose the follower can misreport $r^F$ up to $\Delta$ , $\| r'^F - r^F \|_1 \leq \Delta$ . In the bandit setting, the follower's return can change by at most: + +$$ +\left| V ^ {\pi^ {*} \left(r ^ {F}\right), \mu \left(\pi^ {*} \left(r ^ {F}\right)\right)} \left(s _ {0}; r ^ {F} + b ^ {*} \left(r ^ {F}\right)\right) - V ^ {\pi^ {*} \left(r ^ {\prime F}\right), \mu \left(\pi^ {*} \left(r ^ {\prime F}\right)\right)} \left(s _ {0}; r ^ {F} + b ^ {*} \left(r ^ {\prime F}\right)\right) \right| \leq 2 \Delta +$$ + +and the leader's return can change by at most: + +$$ +\left| V ^ {\pi^ {*} (r ^ {F}), \mu (\pi^ {*} (r ^ {F}))} (s _ {0}; r ^ {L} - b ^ {*} (r ^ {F})) - V ^ {\pi^ {*} (r ^ {\prime F}), \mu (\pi^ {*} (r ^ {\prime F}))} (s _ {0}; r ^ {L} - b ^ {*} (r ^ {\prime F})) \right| \leq 2 \Delta +$$ + +However, an open question is whether such a bound carries over to the Markov game case. How much could the follower gain from misreporting $r^F$ up to $\Delta$ ? Are there algorithms that can induce truthfulness, while still attaining some optimality guarantees? We believe there is a fruitful line of work to be done to handle cases where the leader cannot directly observe and/or verify the follower rewards. + +# 8 Acknowledgement + +TY is grateful for the support of the NSF GRFP. CZ is supported by NSF Award IIS-2440266. + +# References + +Omer Ben-Porat, Yishay Mansour, Michal Moshkovitz, and Boaz Taitler. Principal-agent reward shaping in mdps. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 9502-9510, 2024. +Matteo Bollini, Francesco Bacchiocchi, Matteo Castiglioni, Alberto Marchesi, and Nicola Gatti. Contracting with a reinforcement learning agent by playing trick or treat. arXiv preprint arXiv:2410.13520, 2024. +Dima Ivanov, Paul Dütting, Inbal Talgam-Cohen, Tonghan Wang, and David C Parkes. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. \ No newline at end of file diff --git a/stackelberglearningwithoutcomebasedpayment/images.zip b/stackelberglearningwithoutcomebasedpayment/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..87b073c9d3c4b07208ab9a9f145854e4d2046b74 --- /dev/null +++ b/stackelberglearningwithoutcomebasedpayment/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cfb3fde61d59589dddbf9964cab7a60c6791300f423cedf07679401074ec7552 +size 140787 diff --git a/stackelberglearningwithoutcomebasedpayment/layout.json b/stackelberglearningwithoutcomebasedpayment/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..272034f01b431e3b248f3719f68fe94353aed7ed --- /dev/null +++ b/stackelberglearningwithoutcomebasedpayment/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b9f5c9e4bcb817ef39843ad70ef2b927a09bb4aec21636ec76d6bb81b0abc688 +size 646803 diff --git a/stackelbergselfannotationarobustapproachtodataefficientllmalignment/84c2cbd6-978c-4a8c-ad88-8527b2ad072b_content_list.json b/stackelbergselfannotationarobustapproachtodataefficientllmalignment/84c2cbd6-978c-4a8c-ad88-8527b2ad072b_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..8e8f21545eabe728a20c47d39e9306d5379c3911 --- /dev/null +++ b/stackelbergselfannotationarobustapproachtodataefficientllmalignment/84c2cbd6-978c-4a8c-ad88-8527b2ad072b_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d96219b1066a7ee484d9f972a1f8f77c48c7ec0c0259060ead79b32351910456 +size 276836 diff --git a/stackelbergselfannotationarobustapproachtodataefficientllmalignment/84c2cbd6-978c-4a8c-ad88-8527b2ad072b_model.json b/stackelbergselfannotationarobustapproachtodataefficientllmalignment/84c2cbd6-978c-4a8c-ad88-8527b2ad072b_model.json new file mode 100644 index 0000000000000000000000000000000000000000..26cf1b4f44e67512d03b41a44e566a09c85e6ade --- /dev/null +++ b/stackelbergselfannotationarobustapproachtodataefficientllmalignment/84c2cbd6-978c-4a8c-ad88-8527b2ad072b_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f28c791d3568f14d21f963e91aa901d4f2579e9ae1ee279dffee06b45a98e411 +size 343803 diff --git a/stackelbergselfannotationarobustapproachtodataefficientllmalignment/84c2cbd6-978c-4a8c-ad88-8527b2ad072b_origin.pdf b/stackelbergselfannotationarobustapproachtodataefficientllmalignment/84c2cbd6-978c-4a8c-ad88-8527b2ad072b_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5dbc0fc5af02dc86beba03d3985e76c49cae37f0 --- /dev/null +++ b/stackelbergselfannotationarobustapproachtodataefficientllmalignment/84c2cbd6-978c-4a8c-ad88-8527b2ad072b_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3adfc7c69c2fde12fcbe68d1141721ee671cc737194d676725042402cfc8d78b +size 585539 diff --git a/stackelbergselfannotationarobustapproachtodataefficientllmalignment/full.md b/stackelbergselfannotationarobustapproachtodataefficientllmalignment/full.md new file mode 100644 index 0000000000000000000000000000000000000000..e2b5fafec73b40a25bf4747135f1afca64c7f7df --- /dev/null +++ b/stackelbergselfannotationarobustapproachtodataefficientllmalignment/full.md @@ -0,0 +1,1513 @@ +# Stackelberg Self-Annotation: A Robust Approach to Data-Efficient LLM Alignment + +Xu Chu* 1,2,3, Zhixin Zhang $^{1,3}$ , Tianyu Jia $^{1,3}$ , Yujie Jin $^{1,3}$ + +1Key Laboratory of High Confidence Software Technologies, Ministry of Education + +$^{2}$ Center on Frontiers of Computing Studies, Peking University + +$^{3}$ School of Computer Science, Peking University + +https://github.com/EunTilofy/SSAPO + +# Abstract + +Aligning large language models (LLMs) with human preferences typically demands vast amounts of meticulously curated data, which is both expensive and prone to labeling noise. We propose Stackelberg Game Preference Optimization (SGPO), a robust alignment framework that models alignment as a two-player Stackelberg game between a policy (leader) and a worst-case preference distribution (follower). The proposed SGPO guarantees $\mathcal{O}(\epsilon)$ -bounded regret within an $\epsilon$ -Wasserstein ball, offering formal robustness to (self-)annotation noise. We instantiate SGPO with Stackelberg Self-Annotated Preference Optimization (SSAPO), which uses minimal human-labeled "seed" preferences and iteratively self-annotates new prompts. In each iteration, SSAPO applies a distributionally robust reweighting of synthetic annotations, ensuring that noisy or biased self-labels do not derail training. Remarkably, using only 2K seed preferences—about 1/30 of standard human labels—SSAPO achieves strong win rates against GPT-4 across multiple benchmarks within three iterations. These results highlight that a principled Stackelberg formulation yields data-efficient alignment for LLMs, significantly reducing reliance on costly human annotations. + +# 1 Introduction + +Large language models (LLMs) have demonstrated remarkable capabilities across a broad range of tasks, but aligning their outputs with human preferences remains a core challenge for safety and usability [1, 2, 3]. Traditional alignment paradigms, such as Reinforcement Learning from Human Feedback (RLHF) [3, 4] or Direct Preference Optimization (DPO) [5], typically rely on large amounts of meticulously curated preference data. Such data collection is costly, time-consuming, and inevitably prone to labeling noise or bias, which can in turn degrade model performance once integrated at scale [6]. Consequently, an important question arises: How can we achieve robust alignment of LLMs without relying on vast, error-prone human-labeled datasets? + +One promising direction is to reduce the need for human-annotated samples by having the model itself generate preference labels on newly sampled prompts—so-called "self-annotation" [7, 8, 9]. However, most self-annotation approaches overlook the fact that synthetic labels may be systematically biased or noisy. If these errors go unchecked, they can compound over iterative rounds of training, ultimately harming rather than helping alignment [10, 11]. + +In this work, we address this problem by framing preference alignment as a two-player Stackelberg game between a policy (leader) and a worst-case preference distribution (follower). Our formulation, which we call Stackelberg Game Preference Optimization (SGPO), explicitly guards against plausible shifts or adversarial corruption in the preference data by operating within an $\epsilon$ -Wasserstein ball [12] around the empirical distribution. We prove that the resulting policy achieves $\mathcal{O}(\epsilon)$ -bounded + +regret, a theoretical guarantee of robustness to noise or distribution mismatch [13]. This stands in contrast to standard DPO, whose regret can grow linearly with such shifts [5]. We delay a more thorough related work section in the Appendix A. + +To instantiate SGPO with a concrete algorithm, we then present Stackelberg Self-Annotated Preference Optimization (SSAPO). Starting from a small set of human-labeled "seed" preferences, SSAPO self-annotates new prompts by generating candidate responses and ranking them internally. Crucially, it couples this self-annotation with a distributionally robust reweighting [13] that prevents noisy synthetic labels from overwhelming the training updates. Remarkably, we find that using only 2K seed preference pairs (around 1/30 of the usual scale), SSAPO outperforms or matches methods that rely on significantly more human labels. On multiple alignment benchmarks—including AlpacaEval [14] and MT-Bench [15]—SSAPO rapidly achieves competitive or superior performance within just three rounds of iterative self-annotation. + +We summarize our contributions as follows. 1. Stackelberg formulation of preference alignment: We recast alignment as a two-player game and prove the existence of a robust equilibrium with $\mathcal{O}(\epsilon)$ -bounded regret under $\epsilon$ -Wasserstein preference shifts. 2. Robust self-annotation algorithm (SSAPO): We instantiate our framework by combining minimal seed labels with iterative synthetic annotations. Our distributionally robust reweighting attenuates the impact of potential labeling noise. 3. Data efficiency and empirical results: Experiments show that SSAPO maintains high-level performance despite using only a fraction of typical human annotations, achieving strong results against GPT-4 in head-to-head comparisons. + +# 2 Theoretical Foundation: SGPO Framework + +This section formalizes Stackelberg Game Preference Optimization (SGPO) and establishes its guarantees. We begin with DPO preliminaries (Section 2.1), then cast SGPO as a two-player Stackelberg game over gap distributions (Section 2.2). We prove existence of a Stackelberg equilibrium and local convergence of a practical alternating scheme (Section 2.3), and finally quantify regret and contrast SGPO with DPO (Section 2.4). All proofs are deferred to Appendix D. + +# 2.1 Preliminaries: Preference Datasets and DPO + +Preference-ranked dataset. We use $D = \{(x^{i},y_{w}^{i},y_{\ell}^{i})\}_{i = 1}^{N}$ , where $x^{i}$ is a prompt and $(y_w^i,y_\ell^i)$ are the winner/loser responses (from human or partially self-annotated feedback). + +RLHF and KL regularization. Classical RLHF [4] optimizes a policy $\pi_{\theta}$ under a KL penalty to $\pi_{\mathrm{ref}}$ + +$$ +\max _ {\theta \in \Theta} \mathbb {E} _ {x \sim \mathcal {D}, y \sim \pi_ {\theta} (\cdot | x)} \left[ R (x, y) - \beta D _ {\mathrm {K L}} \left(\pi_ {\theta} (\cdot | x) \| \pi_ {\text {r e f}} (\cdot | x)\right) \right], \tag {1} +$$ + +where $\beta > 0$ controls regularization and $\mathcal{D}$ is the prompt distribution. + +Direct Preference Optimization (DPO). Under the Bradley-Terry model $p(y_w \succ y_\ell \mid x) = \sigma(R(x, y_w) - R(x, y_\ell))$ with $\sigma(z) = 1/(1 + e^{-z})$ , first-order optimality of a related KL-regularized objective yields + +$$ +R (x, y) = \beta \log \frac {\pi_ {\theta} (y \mid x)}{\pi_ {\text {r e f}} (y \mid x)} + \beta \log Z (x), \tag {2} +$$ + +with partition function $Z$ . Plugging this into the BT likelihood gives + +$$ +\mathcal {L} _ {\mathrm {D P O}} (\theta) = \mathbb {E} _ {(x, y _ {w}, y _ {\ell}) \sim D} \left[ \log \sigma \left(R (x, y _ {w}) - R (x, y _ {\ell})\right) \right]. \tag {3} +$$ + +DPO is simple but provides no explicit protection against shifts away from the empirical preference distribution. This motivates a robust formulation. + +# 2.2 SGPO: A Two-Player Stackelberg Game + +SGPO imposes robustness over preference gaps. For a policy $\pi$ , define the gap map + +$$ +\Delta R _ {\pi} (x, y _ {w}, y _ {\ell}) := R _ {\pi} (x, y _ {w}) - R _ {\pi} (x, y _ {\ell}) \in \mathbb {R}. +$$ + +Let $\hat{P} = \frac{1}{N}\sum_{i=1}^{N}\delta_{(x^i,y_w^i,y_\ell^i)}$ and let $(\Delta R_{\pi})_{\#}\hat{P}$ be its push-forward, i.e., + +$$ +\left(\Delta R _ {\pi}\right) _ {\#} \hat {P} = \frac {1}{N} \sum_ {i = 1} ^ {N} \delta_ {\Delta R _ {\pi} \left(x ^ {i}, y _ {w} ^ {i}, y _ {\ell} ^ {i}\right)} \in \mathcal {P} (\mathbb {R}). +$$ + +We measure uncertainty via the 1-Wasserstein ball $\mathcal{U}_{\epsilon}(\nu) \coloneqq \{\alpha \in \mathcal{P}(\mathbb{R}) : W_1(\alpha, \nu) \leq \epsilon\}$ centered at a gap distribution $\nu$ . Hereafter, for simplicity, we write $\pi$ instead of $\pi_{\theta}$ , the dependence on $\theta$ is implicitly assumed throughout the paper. + +Leader objective in gap space. Let $f(\xi) = \log \sigma (\xi)$ , a concave and 1-Lipschitz function on $\mathbb{R}$ . The SGPO leader solves + +$$ +\max _ {\pi \in \Pi} \min _ {\alpha \in \mathcal {U} _ {\epsilon} \left(\left(\Delta R _ {\pi}\right) _ {\#} \hat {P}\right)} \mathbb {E} _ {\xi \sim \alpha} [ f (\xi) ]. \tag {4} +$$ + +This objective optimizes the worst-case preference likelihood over perturbations in gap space rather than in token space. See Section 3 for implementation details. + +Follower best response in gap space. For any center $\nu \in \mathcal{P}(\mathbb{R})$ and radius $\epsilon > 0$ , define the follower best-response set + +$$ +\mathcal {A} _ {\epsilon} (\nu) := \arg \min _ {\alpha \in \mathcal {P} (\mathbb {R})} \left\{\mathbb {E} _ {\xi \sim \alpha} [ f (\xi) ]: W _ {1} (\alpha , \nu) \leq \epsilon \right\}. \tag {5} +$$ + +When $\nu = (\Delta R_{\pi})_{\#}\hat{P}$ , any $\alpha^{\star} \in \mathcal{A}_{\epsilon}(\nu)$ is a follower best response against $\pi$ . + +This formalization induces the Stackelberg equilibrium: + +Definition 2.1 (Stackelberg equilibrium). A pair $(\pi^{*},\alpha^{*})$ is a Stackelberg equilibrium if + +$$ +\pi^ {*} \in \underset {\pi \in \Pi} {\operatorname {a r g m a x}} \min _ {\alpha \in \mathcal {U} _ {\epsilon} ((\Delta R _ {\pi}) _ {\#} \hat {P})} \mathbb {E} _ {\xi \sim \alpha} [ f (\xi) ], \quad \alpha^ {*} \in \mathcal {A} _ {\epsilon} \bigl ((\Delta R _ {\pi^ {*}}) _ {\#} \hat {P} \bigr). \tag {6} +$$ + +This definition links the leader's robust optimization (4) with the follower's DRO problem (5). We next establish existence and analyze a practical alternating scheme. + +# 2.3 Existence and Convergence of a Stackelberg Equilibrium + +We first state mild conditions ensuring existence, then analyze an alternating best-response with a proximal leader step. + +Assumptions. (i) $\Pi$ is compact. (ii) For each $(x,y)$ , $\pi \mapsto R_{\pi}(x,y)$ is continuous; hence $\pi \mapsto (\Delta R_{\pi})_{\#}\hat{P}$ is continuous in the weak topology. (iii) $f(\xi) = \log \sigma (\xi)$ is continuous and 1-Lipschitz on $\mathbb{R}$ . + +Theorem 2.2 (Existence of a Stackelberg equilibrium). Under the assumptions above, problem (4) admits at least one solution $(\pi^{*},\alpha^{*})$ + +Proof sketch. For fixed $\pi$ , the inner problem has a minimizer by compactness of $\mathcal{U}_{\epsilon}((\Delta R_{\pi})_{\#}\hat{P})$ and continuity in $\alpha$ . Berge's maximum theorem [16, 17] yields upper semicontinuity of $V(\pi) = \min_{\alpha \in \mathcal{U}_{\epsilon}((\Delta R_{\pi})_{\#}\hat{P})} \mathbb{E}[f]$ , and compactness of $\Pi$ gives a maximizer $\pi^{*}$ and a follower best response $\alpha^{*}$ . See Appendix D. + +Alternating best responses with a proximal leader step. Let $\hat{\alpha} (\pi)\coloneqq (\Delta R_{\pi})_{\#}\hat{P}$ . Given $\pi_t$ choose a follower best response + +$$ +\alpha_ {t + 1} \in \mathcal {A} _ {\epsilon} (\hat {\alpha} (\pi_ {t})) \in \arg \min _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha} (\pi_ {t}))} \mathbb {E} _ {\xi \sim \alpha} [ f (\xi) ]. \tag {7} +$$ + +Update the leader via a proximal step: + +$$ +\pi_ {t + 1} \in \operatorname {a r g m a x} _ {\pi \in \Pi} \left\{\min _ {\alpha \in \mathcal {U} _ {\epsilon} (\dot {\alpha} (\pi_ {t}))} \mathbb {E} _ {\xi \sim \alpha} [ f (\xi) ] - \frac {\lambda}{2} \| \pi - \pi_ {t} \| ^ {2} \right\}. \tag {8} +$$ + +This scheme makes the leader step stable while allowing the follower to track the changing center $\hat{\alpha} (\pi_t)$ . The regularization term $\| \pi -\pi_t\|^2$ in the proximal step can be induced in practice with small learning rate and weight decay regularization. + +Theorem 2.3 (Well-posedness and local linear convergence). Suppose the proximal leader objective is $\mu$ -strongly concave in a neighborhood of $\pi^{*}$ , uniformly over $\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))$ , and $\pi \mapsto \hat{\alpha}(\pi)$ is locally Lipschitz in $W_{1}$ . Then the update map $(\pi_{t}, \alpha_{t}) \mapsto (\pi_{t+1}, \alpha_{t+1})$ is a contraction near $(\pi^{*}, \alpha^{*})$ and thus converges locally linearly. + +In practice, one may not directly implement (7)-(8), but the Theorem 2.3 shows that any procedure that approximates these alternating best-response updates can converge to the robust equilibrium. This provides a theoretical grounding for the SSAPO algorithm (to be introduced in the section 3), which combines standard gradient-based optimization with distributionally robust optimization. + +# 2.4 Regret Analysis and Comparison with DPO + +We quantify worst-case performance under gap-space shifts and compare with DPO. Define the performance functional + +$$ +\mathcal {P} (\pi , \alpha) = \mathbb {E} _ {\xi \sim \alpha} [ \log \sigma (\xi) ]. +$$ + +Let $\pi^{\star}$ solve (4). We prove that $\pi^{*}$ maintains high performance on all distributions $\alpha$ within $\epsilon$ -Wasserstein distance of $\hat{\alpha}$ . In particular, the drop from $\hat{P}$ to any $P$ is at most $\mathcal{O}(\epsilon)$ . + +Theorem 2.4 (Worst-case performance in gap space). For every $\alpha \in \mathcal{U}_{\epsilon}\bigl (\hat{\alpha} (\pi^{\star})\bigr)$ + +$$ +\mathcal {P} \left(\pi^ {\star}, \alpha\right) \geq \mathcal {P} \left(\pi^ {\star}, \hat {\alpha} \left(\pi^ {\star}\right)\right) - \epsilon . +$$ + +Gap-space regret. We define the Gap-space regret of a policy $\pi$ on a distribution $\alpha$ as $\operatorname{Regret}(\pi, \alpha) := \max_{\tilde{\pi} \in \Pi} \mathcal{P}(\tilde{\pi}, \alpha) - \mathcal{P}(\pi, \alpha)$ . + +Theorem 2.5 (SGPO regret bound). For $\pi^{\star}$ solving (4), + +$$ +\sup \quad \operatorname {R e g r e t} (\pi^ {\star}, \alpha) \leq 2 \epsilon . +$$ + +$$ +\alpha \in \mathcal {U} _ {\epsilon} \left(\hat {\alpha} (\pi^ {*})\right) +$$ + +Thus, SGPO is robust: under any shift of at most $\epsilon$ , its regret is bounded by a constant factor of $\epsilon$ . + +# 2.4.1 Comparison: DPO's Linear Regret + +Let $\pi_{\mathrm{DPO}} \in \operatorname{argmax}_{\pi} \mathcal{P}\big(\pi, \hat{\alpha}(\pi)\big)$ be a DPO solution on the empirical center. For any target $\alpha^{\star}$ with $\delta := W_1\big(\alpha^{\star}, \hat{\alpha}\big(\pi_{\mathrm{DPO}}\big)\big)$ , we have: + +Theorem 2.6 (DPO regret lower bound). Regret $\left(\pi_{\mathrm{DPO}}, \alpha^{\star}\right) \geq \delta - 2\epsilon$ . In particular, if $\delta \gg \epsilon$ , DPO's regret grows linearly in $\delta$ . + +Corollary 2.7 (SGPO advantage over DPO). If $W_{1}\big(\hat{\alpha} (\pi),\alpha^{\star}\big) = \delta >2\epsilon$ , then + +$$ +\frac {\operatorname {R e g r e t} (\pi_ {\mathrm {D P O}} , \alpha^ {\star})}{\operatorname {R e g r e t} (\pi^ {\star} , \alpha^ {\star})} \geq \frac {\delta - 2 \epsilon}{2 \epsilon}. +$$ + +Thus, SGPO's robust policy can outperform DPO by a factor of $\frac{\delta}{2\epsilon} - 1$ under sufficiently large distribution shift $\delta$ . SGPO builds in-sample performance and out-of-sample robustness into a single objective by optimizing against $\mathcal{U}_{\epsilon}$ in gap space. The $\mathcal{O}(\epsilon)$ worst-case degradation contrasts with DPO's linear sensitivity to distribution mismatch $\delta$ , mirroring our empirical results in Section 4. + +# 3 Practical Instantiation: SSAPO Algorithm + +We now present a practical and computationally tractable realization of the Stackelberg scheme from the theory section, called Stackelberg Self-Annotated Preference Optimization (SSAPO). SSAPO implements the iterative leader-follower updates of Theorem 2.3 and (7)-(8) for preference alignment. + +Notation. Let $\sigma(u) = (1 + e^{-u})^{-1}$ and define the margin random variable + +$$ +\xi := \Delta R _ {\theta} (x, y _ {w}, y _ {\ell}) \quad \text {f o r w i n n e r - l o s e r p a i r s} (y _ {w}, y _ {\ell}), \quad \hat {\alpha} (\pi_ {t}) = \frac {1}{N} \sum_ {i = 1} ^ {N} \delta_ {\hat {\xi} _ {i}}, \hat {\xi} _ {i} := \Delta R _ {\theta_ {t}} (x ^ {i}, y _ {w} ^ {i}, y _ {\ell} ^ {i}). +$$ + +The follower (adversary) chooses a distribution $\alpha$ over $\xi$ inside the $W_{1}$ Wasserstein ball $\mathcal{U}_{\epsilon}(\hat{\alpha} (\pi_t))$ centered at $\hat{\alpha} (\pi_t)$ , while the leader (policy) updates $\theta$ . + +Implementation challenges addressed. 1. Minimal human labels via self-annotation. We bootstrap from a small seed of human-labeled preferences and enlarge the dataset by letting the current policy rank its own responses on unlabeled prompts. 2. Loss re-representation for tractable DRO. The follower minimizes a concave inner objective $\mathbb{E}_{\alpha}[\log \sigma (\xi)]$ . Writing $\ell (\xi)\coloneqq -\log \sigma (\xi)$ , which is convex and 1-Lipschitz, turns the inner problem into $-\sup_{\alpha}\mathbb{E}_{\alpha}[\ell (\xi)]$ and allows a convex PWL surrogate that yields a finite convex program for the follower. 3. Scalability via uniform grouping. For large datasets, we solve the follower subproblem on groups and average the resulting worst-case distributions, trading a small approximation for substantial speed-ups. + +# 3.1 Follower objective: loss re-representation and a closed form + +Since $\log \sigma$ is concave and 1-Lipschitz, define the convex 1-Lipschitz loss + +$$ +\ell (\xi) := - \log \sigma (\xi) = \log (1 + e ^ {- \xi}), \quad \min _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha} (\pi_ {t}))} \mathbb {E} _ {\alpha} [ \log \sigma (\xi) ] = - \sup _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha} (\pi_ {t}))} \mathbb {E} _ {\alpha} [ \ell (\xi) ]. +$$ + +When $\Xi = \mathbb{R}$ (no support restrictions) and the ground metric is the absolute value, the worst-case expectation of any $L$ -Lipschitz function equals the empirical mean plus $L\epsilon$ . Specializing this convex reduction result of Mohajerin Esfahani and Kuhn [13, Thm. 6.3] to $\ell$ (whose Lipschitz constant is 1) gives: + +Lemma 3.1 (Closed-form follower in the unconstrained one-dimensional case; 13, Thm. 6.3). If $\Xi = \mathbb{R}$ and $\mathcal{U}_{\epsilon}$ is a $W_{1}$ ball (absolute ground metric), then + +$$ +\sup _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha} (\pi_ {t}))} \mathbb {E} _ {\alpha} [ \ell (\xi) ] = \frac {1}{N} \sum_ {i = 1} ^ {N} \ell (\hat {\xi} _ {i}) + \epsilon , \quad \Longleftrightarrow \quad \min _ {\alpha} \mathbb {E} _ {\alpha} [ \log \sigma (\xi) ] = \frac {1}{N} \sum_ {i = 1} ^ {N} \log \sigma (\hat {\xi} _ {i}) - \epsilon . +$$ + +Although Lemma 3.1 provides an exact closed form in the unconstrained 1-D setting, in this paper we propose to solve a finite convex program that also returns a discrete worst-case distribution. + +# 3.2 Follower via a max-of-affine surrogate and a finite convex program + +We approximate $\ell$ by a convex piecewise-linear under-approximation + +$$ +\tilde {\ell} (\xi) = \max _ {1 \leq k \leq K} \left\{a _ {k} \xi + b _ {k} \right\} \leq \ell (\xi), \quad \text {w i t h} \quad a _ {k} = \ell^ {\prime} (\xi^ {(k)}) = - \sigma (- \xi^ {(k)}), b _ {k} = \ell (\xi^ {(k)}) - a _ {k} \xi^ {(k)}. +$$ + +Knots $\{\xi^{(k)}\}_{k = 1}^{K}$ are chosen on a window $[a_t, b_t]$ in margin space (empirical quantiles or $[\min_i \hat{\xi}_i - \tau, \max_i \hat{\xi}_i + \tau]$ with small $\tau > 0$ ). Endpoint tangents extend $\tilde{\ell}$ outside $[a_t, b_t]$ , preserving $\tilde{\ell} \leq \ell$ globally. Because $\tilde{\ell} \leq \ell$ , replacing $\ell$ by $\tilde{\ell}$ in (3.1) yields an upper bound on the original inner minimum, which tightens with $K$ : + +Proposition 3.2 (Monotone tightening in $K$ ). Let $\hat{\alpha}$ be a probability measure on $\mathbb{R}$ with finite first moment and let $\mathcal{U}_{\epsilon}(\hat{\alpha})$ be the $W_{1}$ ball of radius $\epsilon \geq 0$ . Let $\ell(\xi) = -\log \sigma(\xi)$ and let $(\tilde{\ell}_K)_{K \geq 1}$ be convex piecewise-linear underestimators of $\ell$ such that $\tilde{\ell}_K \leq \tilde{\ell}_{K+1} \leq \ell$ pointwise and $\tilde{\ell}_K \uparrow \ell$ . Define + +$$ +v ^ {\star} := \inf _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha})} \mathbb {E} _ {\alpha} [ \log \sigma (\xi) ] = - \sup _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha})} \mathbb {E} _ {\alpha} [ \ell (\xi) ], v _ {K} := - \sup _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha})} \mathbb {E} _ {\alpha} [ \tilde {\ell} _ {K} (\xi) ]. +$$ + +Then, for all $K \geq 1$ : (i) $v_{K} \geq v^{\star}$ (valid upper bound); (ii) $v_{K+1} \leq v_{K}$ (monotone in $K$ ); (iii) $v_{K} \downarrow v^{\star}$ as $K \to \infty$ . + +A finite convex program for the worst case (after 13, Thm. 4.4). For losses representable as a pointwise maximum of finitely many affine functions, + +$$ +\sup _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha} (\pi_ {t}))} \mathbb {E} _ {\alpha} \left[ \max _ {k \leq K} \left\{a _ {k} \xi + b _ {k} \right\} \right] +$$ + +admits a finite convex program whose solution is a discrete extremal distribution. Specializing Mohajerin Esfahani and Kuhn [13, Thm. 4.4] to our one-dimensional $\xi$ and absolute ground metric yields: + +Theorem 3.3 (Finite convex program for max-of-affine (PWL convex) losses; specialization of Mohajerin Esfahani and Kuhn [13], Thm. 4.4). Let $\tilde{\ell} (\xi) = \max_{k\leq K}\{a_k\xi +b_k\}$ and $\Xi \subseteq \mathbb{R}$ . Introduce, for each sample $i$ and piece $k$ , a mixing weight $s_{ik}\geq 0$ and a displacement $q_{ik}\in \mathbb{R}$ . Then + +$$ +\sup _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha} (\pi_ {t}))} \mathbb {E} _ {\alpha} [ \tilde {\ell} (\xi) ] = \max _ {\left\{s _ {i k}, q _ {i k} \right\}} \frac {1}{N} \sum_ {i = 1} ^ {N} \sum_ {k = 1} ^ {K} \left[ s _ {i k} \left(a _ {k} \hat {\xi} _ {i} + b _ {k}\right) - a _ {k} q _ {i k} \right] \tag {9} +$$ + +![](images/156b527f946d4ca64f10ead365e5de4a0c8c2b14fae014cd1f020db87095dd01.jpg) +Figure 1: SSAPO workflow. We maintain a large prompt pool and a small set of seed-labeled preferences. The policy self-annotates new prompts by generating and ranking responses, thus expanding the preference database. A follower then identifies a worst-case distribution for these preferences, and the leader (policy) is updated accordingly. This process repeats for iterations. + +subject to the Wasserstein and feasibility constraints + +$$ +\frac {1}{N} \sum_ {i = 1} ^ {N} \sum_ {k = 1} ^ {K} | q _ {i k} | \leq \epsilon , \quad \sum_ {k = 1} ^ {K} s _ {i k} = 1, s _ {i k} \geq 0 (\forall i), \quad a _ {t} s _ {i k} \leq s _ {i k} \hat {\xi} _ {i} - q _ {i k} \leq b _ {t} s _ {i k} (\forall i, k). +$$ + +An extremal discrete measure $\alpha_{t}^{*} = \frac{1}{N}\sum_{i = 1}^{N}\sum_{k = 1}^{K}s_{ik}^{*}\delta_{\xi_{i} - q_{ik}^{*} / s_{ik}^{*}}$ attains the supremum. + +The change of variables $z_{ik} = \hat{\xi}_i - q_{ik} / s_{ik}$ (when $s_{ik} > 0$ ) reveals that $s_{ik}$ splits the unit mass at $\hat{\xi}_i$ across pieces, while $q_{ik}$ transports that mass in margin space. The interval constraints are linear "perspective" constraints enforcing $\xi \in [a_t, b_t]$ . + +# 3.3 SSAPO workflow + +Figure 1 summarizes the SSAPO workflow. Starting with a small seed of human-labeled preferences plus a large unlabeled pool, we proceed in the following loop, at iteration $t$ : + +1. Self-annotation. Sample prompts from the unlabeled pool, generate multiple responses, and let the current policy $\pi_{\theta_t}$ rank them to create new preference pairs $(y_w, y_\ell)$ . +2. Empirical center. Compute $\hat{\xi}_i = \Delta R_{\theta_t}(x^i,y_w^i,y_\ell^i)$ and set $\hat{\alpha} (\pi_t) = \frac{1}{N}\sum_i\delta_{\hat{\xi}_i}$ . +3. Follower (DRO). Build $\tilde{\ell} (\xi) = \max_{k\leq K}(a_k\xi +b_k)$ using tangents at knots $\{\xi^{(k)}\}$ ; choose $[a_t,b_t]$ as above. Solve the convex program in Theorem 3.3 to obtain the worst-case $\alpha_{t}^{*}$ . +4. Leader update. Update $\theta$ by minimizing + +$$ +\theta_ {t + 1} \in \operatorname * {a r g m i n} _ {\theta} \mathbb {E} _ {(x, y _ {w}, y _ {\ell}) \sim \mathcal {D}} \mathbb {E} _ {\xi \sim \alpha_ {t} ^ {*}} \left[ \ell \big (\Delta R _ {\theta} (x, y _ {w}, y _ {\ell}) \big) \right], +$$ + +implemented as standard minibatch SGD on the preference pairs while incorporating the per-sample mixture weights induced by $(s^{*},q^{*})$ . + +Repeating for $T$ total iterations yields the final aligned policy $\pi_{\theta_T}$ . A more explicit version of SSAPO is provided in Algorithm 1 (Appendix E), along with its computational complexity analysis. + +How $\alpha_{t}^{*}$ enters the leader step. The optimizer $(s^{*},q^{*})$ associates to each training pair $i$ a set of active affine pieces (weights $\{s_{ik}^{*}\}_{k}$ ) and a transport direction (through $q_{ik}^{*} / s_{ik}^{*}$ ). In practice we (i) weight the per-pair loss contributions by $\{s_{ik}^{*}\}_{k}$ and (ii) optionally add a proximal penalty nudging the current margin toward $\hat{\xi}_i - q_{ik}^* /s_{ik}^*$ for stability. + +# 3.4 Scalability and Complexity + +Grouping for large $N$ . When $N$ is large (e.g. $10^{5}$ or more preferences), solving the convex program in Step (Worst-Case Distribution) can be expensive. A popular heuristic partitions $\{\hat{\xi}_1,\dots ,\hat{\xi}_N\}$ into $M$ groups (each of size $G = N / M$ ), and solves the finite program (9) separately within each group. + +The resulting distributions $\alpha_{m}^{*}$ are then averaged (or merged proportionally): + +$$ +P _ {\mathrm {f i n a l}} ^ {*} = \frac {1}{M} \sum_ {m = 1} ^ {M} \alpha_ {m} ^ {*}. +$$ + +While not an exact solution to the global $N$ -sample problem, this still confers substantial robustness while reducing complexity from $N$ to $G \ll N$ in each subproblem. In summary, this grouping approach greatly reduces memory/compute cost, and is parallelizable. Section F in the appendix remarks the approximation effects of SSAPO algorithm design on SGPO guarantees + +Complexity. The program in Theorem 3.3 introduces $O(NK)$ variables and $O(NK)$ constraints and becomes a pure LP after linearizing $|q_{ik}|$ via standard slack variables. With warm starts, per-iteration time scales nearly linearly in $N$ and linearly in $K$ . A more detailed algorithmic complexity analysis is in Appendix E. + +Why a finite convex program is necessary in practice. We argue the special case of unconstrained one-dimensional case the follower (that collapses to an exact closed form from Lemma 3.1) is too restrictive for real pipelines: we (i) clip or restrict the support of margins for stability, (ii) require extremal discrete adversaries whose mixture weights can be recycled to form stochastic gradients for the leader, and (iii) need a controllable approximation whose accuracy improves monotonically. Accordingly, SSAPO replaces the inner concave expectation $\min_{\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha} (\theta_{t}))}\mathbb{E}_{\alpha}[\log \sigma (\xi)]$ by a tractable finite convex program obtained from a convex piecewise-linear surrogate $\tilde{\ell}_K(\xi) = \max_{k\leq K}\{a_k\xi +b_k\}$ of $\ell (\xi) = -\log \sigma (\xi)$ . This yields an upper bound $-\sup_{\alpha}\mathbb{E}_{\alpha}[\tilde{\ell}_K(\xi)]$ on the inner minimum that tightens monotonically in $K$ (Proposition 3.2) and becomes exact as $K\to \infty$ . Moreover, specializing Mohajerin Esfahani and Kuhn [13, Thm. 4.4] to our one-dimensional absolute ground metric produces a finite convex program whose optimizer is a discrete extremal distribution supported on at most $NK$ atoms: exactly the structure we need to implement sample reweighting and efficient leader updates. In short, the finite convex program is both theoretically correct (via 13, Thm. 6.3 and Thm. 4.4) and operationally necessary for SSAPO's stability and efficiency. + +# 4 Experiments + +In this section, we present an extensive empirical evaluation of our proposed Stackelberg Self-Annotated Preference Optimization (SSAPO) algorithm. + +# 4.1 Experiment Setup + +We introduce the basic experiment setup in this subsection (Cf. Appendix G for more details). The settings are mostly consistent to the recent literature Kim et al. [9]. Datasets. We used the UltraFeedback dataset [18], containing 60K samples. A seed of 2K human-labeled preferences (3.3% of total 60K data) was used for initial training. The rest (58K samples) were split into three subsets (8K, 20K, and 30K) for self-annotation in iterative stages. + +Models. We use the supervised fine-tuned Mistral-7B-0.1 [19] as the initial model $\pi_{\mathrm{init}}$ and LLaMA-3-8B² for compatibility checks. All models are fine-tuned on UltraChat [20]. + +**Evaluations.** We use AlpacaEval 2.0 [14] for instruction-following tasks and MT-Bench [15] to evaluate multi-turn performance across tasks like math, coding, and writing. Both benchmarks assess the alignment with human preferences and the model's functional proficiency. We stress that AlpacaEval 2.0 is especially useful for measuring how well the model aligns with general user preferences (and controlling for length bias), whereas MT-Bench tests the model's functional capabilities across a broader range of tasks. + +Implementation. We initialize training with DPO on 2K seed samples, followed by 3 iterative stages of self-annotation. In each stage, new preferences are generated via a policy that ranks response pairs. A distributionally robust optimization (DRO) is performed using sequential least squares programming (SLSQP) to adjust the model based on adversarial shifts within a Wasserstein ball. The group size $G$ for parallel computation is set to 100 unless otherwise specified. + +Baselines. We consider the following baselines for comparison: (1) DPO, which performs DPO training only on the seed data. (2) Iter DPO [11], which iteratively generates preference data using an external reward model (PairRM) [21] or LLM-as-judge [22]. (3) SPA [9], which iteratively generates preference data using implicit reward model. + +${}^{2}$ meta-LLaMA/Meta-LLaMA-3-8B-Instruct + +Table 1: Main results. Evaluation results on AlpacaEval 2.0 and MT-Bench with different variants of Mistral-7B-v0.1 and LLaMA3-8B. All models use the same 2K preference data with gold label as seed data. The best and second-best results are highlighted in bold and underlined, respectively. Most of the baseline results are from [9]. + +
ModelsAlpacaEval 2.0MT-Bench
Len-control. Win Rate (%)Win Rate vs. GPT-4 (%)Avg. Score (0-10)
Mistral-7B-DPO9.037.686.81
Mistral-7B-Iter DPO (PairRM)11.879.466.98
Mistral-7B-Iter DPO (LLM-as-judge)9.289.186.67
LLaMA3-8B-DPO20.6118.04-
Mistral-7B-SPA15.3921.136.94
LLaMA3-8B-SPA21.8524.957.86
Mistral-7B-SSAPO (Ours)24.4435.826.68
LLaMA3-8B-SSAPO (Ours)33.3340.128.03
+ +Table 2: Comparison with different variants of Mistral. Evaluation results on AlpacaEval 2.0 and MT-Bench with different variants of Mistral-7B-v0.1. The best scores are highlighted with bold. The baseline results are from [9] and [23]. + +
ModelsGold Label (%)AlpacaEval 2.0MT-Bench
Len-control. Win Rate (%)Win Rate vs. GPT-4 (%)Avg. Score (0-10)
Mistral-7B-v0.1-0.170.503.25
Zephyr-7B-β10011.7510.036.87
Mistral-7B-SFT-7.584.726.34
Mistral-7B-DPO3.39.037.686.81
Mistral-Large (123B)-21.432.7-
Mistral-7B-SSAPO (Ours)3.324.4435.826.68
+ +# 4.2 Main Results + +Table 1 summarizes our primary comparison on AlpacaEval 2.0 and MT-Bench. All models in this comparison use only 2K preference pairs of the UltraFeedback dataset as seed data (3.3% out of 60K), with the remainder self-annotated. Our SSAPO method consistently outperforms DPO and other iterative baselines (e.g., Iter-DPO, SPA) in both the length-controlled (LC) and raw win-rate metrics on AlpacaEval 2.0. For Mistral-7B, SSAPO achieves $24.44\%$ LC win rate and $35.82\%$ raw win rate, compared to only $9.03\%$ and $7.68\%$ with standard DPO. On the larger LLaMA-3-8B model, SSAPO reaches a $33.33\%$ LC win rate and $40.12\%$ raw win rate, surpassing its SPA counterpart by a wide margin. MT-Bench scores corroborate these improvements, indicating that SSAPO yields robust, high-quality responses across diverse tasks. + +To further illustrate SSAPO's data-efficiency and robustness, Table 2 compares various Mistral models, including Mistral-7B-SFT, Mistral-Large (the number of parameters is 123B), and a fully-finetuned Zephyr-7B- $\beta$ variant with $100\%$ labeled data. Remarkably, Mistral-7B-SSAPO outperforms or closely approaches these stronger references in AlpacaEval 2.0, despite using only 2K preference pairs ( $3.3\%$ out of the 60K human-labeled training set). This demonstrates that a principled Stackelberg method can substantially mitigate the reliance on massive human annotations. It also aligns with our theoretical findings (Section 2) that SGPO-based approaches, when instantiated as SSAPO, achieve bounded regret under moderate preference shift. + +# 4.3 Ablation and Sensitivity Analysis + +Table 3: Effect of Wasserstein Radius $\epsilon$ on Performance. Evaluation results on Mistral-7B, showing the impact of varying the Wasserstein radius on the Len-control. Win Rate and Win Rate vs. GPT-4. + +
ε00.010.030.050.1
Len-control. Win Rate (%)19.7624.4422.4223.2019.78
Win Rate vs. GPT-4 (%)26.5835.8232.3032.9225.84
+ +We conduct a series of ablation studies to understand the factors influencing the efficacy and robustness of our Stackelberg Self-Annotated Preference Optimization (SSAPO). Specifically, we vary the + +Table 4: Impact of Tangent Size $(K)$ and Impact of Group Size $(G)$ on Model Performance. Evaluation results on Mistral-7B. + +
Impact of Tangent Size (K)Effect of Group Size (G)
K567G100200300
Len-control. Win Rate (%)22.8923.2019.05CPU Runtime (min)45206630
Win Rate vs. GPT-4 (%)29.1932.9225.84Len-control. Win Rate (%)13.7014.8116.95
Win Rate vs. GPT-4 (%)10.0011.7414.91
+ +![](images/d3265d34b623fd5feca3fd379872dc701d8bae65d56d396abae306b1a53fce6a.jpg) +Figure 2: Improvement during iterations Evaluation results on AlpcaEval 2.0 of initial DPO stage and each iteration, the results of the SFT model are from [9]. + +Wasserstein radius $\epsilon$ , the number of tangents $K$ , and the group size $G$ . We conduct the experiments on the Mistral-7B model for budget consideration. These experiments confirm our method's flexibility and validate the practical design choices guided by our theoretical framework. + +Wasserstein Radius $\epsilon$ . Table 3 demonstrates how performance changes with different Wasserstein radius. When $\epsilon = 0$ , our approach reduces to self-annotated DPO without robust reweighting, yielding weaker results (19.76% LC win rate). As $\epsilon$ increases slightly (e.g., 0.01–0.05), both win-rates improve substantially, with the best outcomes at $\epsilon = 0.01$ . However, overly large $\epsilon$ (e.g., 0.1) can make the adversarial shift too pessimistic, degrading performance. These findings align with our theoretical analysis in Section 2, where moderate $\epsilon$ provides a robust yet not overly conservative solution, thus striking the optimal balance between data fidelity and adversarial resilience. + +Number of Tangents $K$ . Since our piecewise-linear approximation of $-\log \sigma(\cdot)$ uses $K$ linear segments (cf. Section 3), we examine how varying $K$ affects alignment (Table 4 left). At $K = 5$ , the model attains a $22.89\%$ LC win-rate, while increasing to $K = 6$ yields a marginally better $23.20\%$ . Interestingly, moving to $K = 7$ leads to performance drops (19.05%). We hypothesize that while a larger $K$ refines the convex under-approximation, it may also overcomplicate optimization or amplify minor errors in the approximation. Thus, $K = 6$ serves as a sweet spot in our setting, balancing expressiveness and computational stability. + +Group Size $G$ . Our distributionally robust optimization solver randomly partitions data into groups of size $G$ for parallel subproblem solutions. Table 4 (right half) illustrates the trade-off between computational cost and performance. A small group size ( $G = 100$ ) has faster runtime ( $45\mathrm{min}$ ) but yields a $13.70\%$ LC win-rate, whereas a larger $G = 300$ reaches $16.95\%$ yet takes over 10 times longer ( $630\mathrm{min}$ ). This confirms that while bigger groups permit more fine-grained reweighting and hence improved alignment, the overhead grows significantly. In practice, we choose $G = 100$ or $G = 200$ for an acceptable performance-efficiency balance. + +Iterative Performance Gains. Figure 2 provides a direct illustration of iterative improvement over three rounds of SSAPO. Starting from a baseline DPO model, each round not only adds new self-annotated preferences but also reweights them adversarially within an $\epsilon$ -ball. We observe a consistent upward trend in alignment metrics during the first two rounds, validating our claim that robust self-annotation can compensate for scarce human labels while preserving alignment quality. + +Taken together, these ablations highlight the flexibility and effectiveness of SSAPO: Moderate $\epsilon$ balances robustness and data fidelity, confirming our theoretical finding that worst-case reweighting within a bounded radius can significantly enhance alignment without over-penalizing feasible distributions. Piecewise-linear approximations with small $K$ are sufficient to capture the shape of $-\log (\sigma (\cdot))$ , maintaining computational tractability. Group size $G$ offers a controllable trade-off between runtime and fine-grained adversarial reweighting, making the approach scalable to different budget constraints. Iterative self-annotation with minimal seed data substantially boosts alignment, demonstrating that only 2K human-labeled preferences can suffice to achieve high performance. Overall, these experiments affirm our primary contributions: a data-efficient and theoretically grounded approach to preference alignment. + +# Practical Hyperparameter Guidelines + +Wasserstein radius $\epsilon$ . Scale $\epsilon$ with expected self-annotation noise. Capable models: $\epsilon \in [0.005, 0.02]$ ; smaller models: $\epsilon \in [0.01, 0.05]$ . A robust default is $\epsilon = 0.01$ when no validation is available. + +Piecewise approximation tangents $K$ . We recommend $K = 6$ as a stable default. $K = 7$ may hurt due to solver instability rather than approximation error. + +Grouping size $G$ . For parallel DRO, $G \in [100, 1000]$ balances robustness and throughput; we find $G \approx 100 - 300$ a sweet spot in practice. + +# 4.3.1 Robustness to Seed Label Noise (25% flips) + +To assess robustness promised by our $\mathrm{O}(\epsilon)$ -regret guarantee, we flip the preferred/unpreferred labels on $25\%$ of the 2K seed pairs and re-run SSAPO end-to-end. Table 5 shows that Mistral-SSAPO suffers only a $\sim 7 - 13\%$ degradation, while LLaMA-SSAPO improves under noise, indicating DRO regularization and stronger self-annotation can counteract moderate seed noise. + +Table 5: Effect of $25\%$ seed label corruption. Entries show (AlpacaEval 2.0 LC win-rate / Win-rate vs GPT-4), higher is better. + +
ModelNo noise25% noise
Mistral-SSAPO26.90% / 31.93%19.70% / 18.51%
LLaMA-SSAPO33.33% / 40.12%43.74% / 46.70%
+ +Discussion. For Mistral, bounded degradation empirically aligns with our $\mathrm{O}(\epsilon)$ -regret theory. For LLaMA-3-8B, noise acts as implicit regularization: the worst-case distribution explores a wider $\epsilon$ -ball region and mitigates overfitting to small seeds. + +# 5 Conclusion, Limitations and Future Work + +Aiming at a data-efficient alignment method, we have introduced SGPO alignment framework with $\mathcal{O}(\epsilon)$ -bounded regret under moderate noise or distribution shifts. Our practical instantiation, SSAPO, uses self-annotation and distributionally robust reweighting to achieve strong performance with far fewer human labels. The scalability limitation of SSAPO comes from the number of preferences $N$ , we use a simple uniform group trick to balance between robustness and complexity. For further improvement, one may resort to primal-dual or specialized cutting-plane methods [13], or use approximate relaxations with entropic regularization [24]. Our guarantees target training-time robustness to preference noise and mild distribution shifts (e.g., reweightings within a $W_{1}$ $\epsilon$ -ball). This is distinct from inference-time robustness to adversarial prompts or jailbreak attacks. While our DRO step improves alignment under noisy supervision, it does not replace dedicated safety mechanisms for adversarial inputs. We only consider the human-labeled preference restricted scenario, however, SSAPO can also be integrated with prompt-generation procedure such as EVA [25], which could be crucial to scaling large language model based intelligence, considering that high-quality human data is running out in the next few years [26]. + +# Acknowledgments + +We thank the reviewers for their useful feedback that improved this work. 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Advances in Neural Information Processing Systems, 36:74764-74786, 2023. + +# Organization of the Appendix. + +- Section A recap LLM alignment and data-efficient methods, as well as the Game-theoretic alignment methods. And discuss the connection and distinction between SGPO/SSAPO with them +- Section B revisits the core definitions and properties of the 1-Wasserstein metric, including a statement of the Kantorovich-Rubinstein duality. +- Section C restates and discusses the regularity conditions needed for our theoretical guarantees, such as compactness and Lipschitz continuity. +- Section D provides Proofs for the existence and convergence of the Stackelberg equilibrium, as well as the regret bounds for SGPO and comparisons with DPO. +- Section E presents the SSAPO algorithm in pseudocode and includes an analysis of its computational complexity. +- Section F remarks the approximation effects of SSAPO algorithm design on SGPO guarantees +- Section G gives supplementary information on experimental setups, hyperparameter choices, grouping strategies for DRO, and other implementation details. +- Section H illustrates additional qualitative comparisons of model outputs, highlighting the differences between DPO, SPA, and SSAPO in practice. +- Section I discusses the potential broader impact of this work. + +# A More Detailed Related Work + +LLM Alignment and Data-Efficient Methods Aligning large language models (LLMs) with human preferences is central to modern deployments [1, 2, 3]. While Reinforcement Learning with Human Feedback (RLHF) [4] trains a reward model and then maximizes it under KL constraints, it typically requires massive human-annotated data. Recent alternatives focus on directly fine-tuning LLMs from pairwise preference data without an explicit reward model. Notably, Direct Preference Optimization (DPO) [5] derives a closed-form surrogate objective that recovers RLHF's solution but avoids a separate reward modeling stage. Subsequent works simplify or extend this pipeline; for instance, Ethayarajh et al. [27] remove the need for pairwise labels by adopting a human utility model, while there are also works [28, 29, 30] introduce novel optimization objectives to handle different preference formats. Despite progress, these approaches still rely on large-scale preference annotations, making label-efficiency a key challenge. To reduce the reliance on expensive human labels, several methods have explored letting the LLM or an auxiliary model generate and rank unlabeled responses, thereby creating synthetic preference data [21, 8, 11, 9]. However, many of these approaches assume accessibility to a reliable well-aligned "judge", which could be prohibitive costly in realistic scenarios. To address the cost bottleneck, Kim et al. [9] propose a Spread Preference Annotation (SPA) framework that starts from a small seed of human-annotated preferences and iteratively expands the dataset by self-annotation. Our work is closely related to SPA: we replicate its experimental setup by using the same small-scale seed preferences and iterating between new response generation and preference learning. However, our Stackelberg perspective considers the inaccuracy of self-annotation, and explicitly defends against worst-case preference shifts. Empirically, we show that this game-theoretic distributional approach yields stronger label efficiency. + +Game-Theoretic Alignment Methods An emerging body of work has begun to frame preference alignment of LLMs through the lens of games. A conceptual similar work [31] propose Stackelberg Alignment RLHF. However, their nested gradient-based heuristic does not guarantee to converge to the equilibrium. While we prove our updates for the leader and follower converge to an equilibrium. Meanwhile, Ye et al. [25] present a framework that casts prompt-creator and solver asymmetric players in an evolving game, the differences between our work is we focus on evolving the distribution of the responses , while they focus on evolting the distribution of the prompts. SPIN [32] use self-play to iteratively refine a policy without additional human data, however they assume accessible to adequate supervised fine-tuning (SFT) data. Other works adopt Nash or minimax formulations: Melnyk et al. [33] study alignment via an optimal-transport objective to capture distributional preferences, Zhang et al. [34] and Rosset et al. [35] formulate alignment as a two-player game aiming for a Nash policy, and Munos et al. [36] proposes "Nash learning from human feedback" by treating the policy and a competing policy as iterative players. Likewise, Swamy et al. [37], Wu et al. [38] introduce self-play preference + +optimization methods in which two policies repeatedly compete under a constant-sum setting. They demonstrate promising performance on synthetic and text-based benchmarks, but typically set both players as policy vs. policy. By contrast, our SGPO framework focuses on policy vs. distribution: the leader policy maximizes preference likelihood, while the follower adversarially reweights or shifts the empirical preference distribution. This setup offers a distinct distributional robust-control view, leading to tight theoretical guarantees (e.g., $\mathcal{O}(\epsilon)$ -bounded regret) and a practical algorithm (SSAPO) that is readily integrated with self-annotation. Hence, our method complements the "policy vs. policy" family by delivering strong resistance to noisy or distribution-mismatched preferences at small annotation cost. + +# B Preliminaries on the Wasserstein Metric Space + +Wasserstein (or Earth Mover's) distances are widely used in robust optimization and optimal transport to measure how far two probability distributions are from one another [12]. Below, we give a concise overview of the 1-Wasserstein metric on a subset $\Xi \subseteq \mathbb{R}^m$ . We also recap the Kantorovich-Rubinstein duality (Lemma B.2), which is central to several of our regret and robustness proofs in the main text. + +# B.1 Definition of the 1-Wasserstein Metric + +Let $\mathcal{M}(\Xi)$ be the space of all probability distributions supported on $\Xi$ such that + +$$ +\mathbb {E} _ {\xi \sim F} [ \| \xi \| ] = \int_ {\Xi} \| \xi \| d F (\xi) < \infty . +$$ + +In our setting, $\| \cdot \|$ can be any norm on $\mathbb{R}^m$ , typically the Euclidean norm (although other choices are possible). + +Definition B.1 (1-Wasserstein Metric). For two probability distributions $F_{1}, F_{2} \in \mathcal{M}(\Xi)$ , the 1-Wasserstein distance (often just called "the Wasserstein distance") is defined as + +$$ +W _ {1} \left(F _ {1}, F _ {2}\right) := \inf _ {\pi \in \Pi \left(F _ {1}, F _ {2}\right)} \left\{\int_ {\Xi \times \Xi} \left\| \xi_ {1} - \xi_ {2} \right\| \mathrm {d} \pi \left(\xi_ {1}, \xi_ {2}\right) \right\}, \tag {10} +$$ + +where $\Pi(F_1, F_2)$ is the set of all joint distributions on $\Xi \times \Xi$ whose marginals are $F_1$ and $F_2$ , respectively. Intuitively, $\pi$ specifies how "mass" is transported from points in the support of $F_1$ to points in the support of $F_2$ , and $\|\xi_1 - \xi_2\|$ is the cost of moving a unit of mass from $\xi_1$ to $\xi_2$ . + +Domain used in this paper. All Wasserstein balls in our analysis live in $\Xi = \mathbb{R}$ over scalar preference gaps $\xi = \Delta R_{\pi}(y_w, y_\ell)$ induced by a fixed prompt $x$ ; we do not transport $x$ or raw sentences. Robustness is with respect to annotation noise through the induced gap distribution. + +Equivalently, one can interpret the Wasserstein distance as the minimal cost of transforming the distribution $F_{1}$ into $F_{2}$ when the cost of moving a unit mass from $\xi_{1}$ to $\xi_{2}$ is $\| \xi_1 - \xi_2\|$ . This framework underpins many distributionally robust methods, including the SGPO formulation in our paper. + +# B.2 Kantorovich-Rubinstein Duality + +A crucial result for the 1-Wasserstein distance is the Kantorovich-Rubinstein duality (Theorem 5.9 in Villani et al. [12]), which states that the infimum over transport plans (as in Definition B.1) is equivalent to a supremum over 1-Lipschitz test functions. We use this lemma extensively to derive Lipschitz-based bounds in the main proofs (e.g., Theorems 2.5-2.6). + +Lemma B.2 (Kantorovich-Rubinstein Duality). Let $F_{1}, F_{2} \in \mathcal{M}(\Xi)$ with finite first moments. Then the 1-Wasserstein distance (10) admits the following dual representation: + +$$ +W _ {1} \left(F _ {1}, F _ {2}\right) = \sup _ {\| f \| _ {\mathrm {L i p}} \leq 1} \left(\mathbb {E} _ {\xi \sim F _ {1}} [ f (\xi) ] - \mathbb {E} _ {\xi \sim F _ {2}} [ f (\xi) ]\right), \tag {11} +$$ + +where the supremum is taken over all 1-Lipschitz functions $f:\Xi \to \mathbb{R}$ , i.e. functions satisfying + +$$ +\left| f (\xi) - f \left(\xi^ {\prime}\right) \right| \leq \| \xi - \xi^ {\prime} \| \quad \forall \xi , \xi^ {\prime} \in \Xi . +$$ + +Lemma B.2 underpins many of our theoretical arguments, particularly in bounding the impact of perturbations measured in the $W_{1}$ ball $\mathcal{U}_{\epsilon}(\hat{\alpha} (\pi))$ in gap space, via $\left|\mathbb{E}_{\alpha}[f] - \mathbb{E}_{\beta}[f]\right|\leq W_1(\alpha ,\beta)$ for 1-Lipschitz $f$ . As shown in Section D of our paper, it simplifies comparing $\mathbb{E}_P[f]$ and $\mathbb{E}_{\hat{P}}[f]$ when $f$ is Lipschitz in model parameters or responses. + +# C Regularity Conditions for Stackelberg Game Preference Optimization + +Setup. Let $\Pi$ denote a (parameterized) class of policies $\pi$ . Given preference triples $(x^i, y_w^i, y_\ell^i)$ for $i = 1, \ldots, N$ , define the empirical measure on triples $\hat{P} = \frac{1}{N} \sum_{i=1}^{N} \delta_{(x^i, y_w^i, y_\ell^i)}$ and the gap map $\Delta R_{\pi}(x, y_w, y_\ell) := R_{\pi}(x, y_w) - R_{\pi}(x, y_\ell)$ . The corresponding empirical gap distribution is $\hat{\alpha}(\pi) := (\Delta R_{\pi})_{\#} \hat{P} = \frac{1}{N} \sum_{i=1}^{N} \delta_{\Delta R_{\pi}(x^i, y_w^i, y_\ell^i)} \in \mathcal{P}(\mathbb{R})$ . For $\epsilon > 0$ , we denote the 1-Wasserstein ball in gap space by $\mathcal{U}_{\epsilon}(\hat{\alpha}(\pi)) := \{\alpha \in \mathcal{P}(\mathbb{R}) : W_1(\alpha, \hat{\alpha}(\pi)) \leq \epsilon\}$ , where the ground metric is the absolute value on $\mathbb{R}$ . The leader payoff is $\mathcal{P}(\pi, \alpha) := \mathbb{E}_{\xi \sim \alpha}[\log \sigma(\xi)]$ and the follower chooses $\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))$ . + +Standing assumptions. We use the following minimal conditions. + +(A1) Compactness. $\Pi$ is compact (with respect to the topology induced by the model parameterization). +(A2) Continuity of rewards. For each $(x,y)$ , the map $\pi \mapsto R_{\pi}(x,y)$ is continuous. Consequently, for each $i$ , the map $\pi \mapsto \Delta R_{\pi}(x^i,y_w^i,y_\ell^i)$ is continuous. +(A3) Continuity of the push-forward center. The map $\pi \mapsto \hat{\alpha}(\pi) = (\Delta R_{\pi})_{\#} \hat{P}$ is continuous in the topology induced by $W_{1}$ ; in particular, $W_{1}(\hat{\alpha}(\pi_{1}), \hat{\alpha}(\pi_{2})) \to 0$ whenever $\pi_{1} \to \pi_{2}$ . (This actually follows from (A2). See Lemma C.1.) +(A4) Gap-link function. $f(\xi) \coloneqq \log \sigma (\xi)$ is continuous, concave, and 1-Lipschitz on $\mathbb{R}$ . +(A5) (Optional, for local convergence.) There exists $\lambda > 0$ such that the proximal leader objective $G(\pi; \pi_t, \alpha) \coloneqq \mathbb{E}_{\xi \sim \alpha}[f(\Delta R_{\pi}(x, y_w, y_\ell))] - \frac{\lambda}{2} \| \theta(\pi) - \theta(\pi_t) \|^2$ is $\mu$ -strongly concave in $\theta(\pi)$ on a neighborhood of a solution, uniformly over $\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi_t))$ . +(A6) (Optional, for bounded margins or clipping.) Either $|\Delta R_{\pi}(x,y_w,y_\ell)| \leq B$ for all $(x,y_w,y_\ell)$ and $\pi \in \Pi$ , or margins are deterministically clipped to a window $[a,b]$ . All PWL error bounds are computed on this interval. The $R_{\mathrm{max}}$ in the paper can be set to $B / 2$ . + +Lemma C.1 (Continuity of the center in $W_{1}$ ). Under (A2), for any $\pi_1, \pi_2 \in \Pi$ , + +$$ +W _ {1} \big (\hat {\alpha} (\pi_ {1}), \hat {\alpha} (\pi_ {2}) \big) \leq \frac {1}{N} \sum_ {i = 1} ^ {N} \big | \Delta R _ {\pi_ {1}} (x ^ {i}, y _ {w} ^ {i}, y _ {\ell} ^ {i}) - \Delta R _ {\pi_ {2}} (x ^ {i}, y _ {w} ^ {i}, y _ {\ell} ^ {i}) \big |. +$$ + +In particular, $\pi \mapsto \hat{\alpha} (\pi)$ is continuous in $W_{1}$ . + +Proof. Couple the Dirac masses in $\hat{\alpha} (\pi_1)$ and $\hat{\alpha} (\pi_2)$ index-wise. The claim follows because the 1-Wasserstein distance on $\mathbb{R}$ is bounded above by the average transport cost under any coupling. + +Lemma C.2 (log $\sigma$ is 1-Lipschitz and concave). For all $\xi \in \mathbb{R}$ , $\frac{\mathrm{d}}{\mathrm{d}\xi} \log \sigma(\xi) = \sigma(-\xi) \in (0,1)$ and $\frac{\mathrm{d}^2}{\mathrm{d}\xi^2} \log \sigma(\xi) = -\sigma(\xi) \sigma(-\xi) \leq 0$ . Hence $\log \sigma$ is 1-Lipschitz and concave. + +Proof. Direct differentiation; $|\sigma(-\xi)| \leq 1$ gives the Lipschitz constant and the second derivative is nonpositive. + +Lemma C.3 (Compact follower feasible set). For each fixed $\pi$ , the set $\mathcal{U}_{\epsilon}(\hat{\alpha}(\pi)) \subset \mathcal{P}(\mathbb{R})$ is tight, closed in $W_{1}$ , and thus compact. + +Proof. On the Polish space $(\mathbb{R},|\cdot|)$ , closed and $W_{1}$ -bounded sets of probability measures are relatively compact; tightness follows from Markov's inequality under bounded first moments, which hold for all $\alpha$ with $W_{1}(\alpha,\hat{\alpha}(\pi)) \leq \epsilon$ . Closure is standard for $W_{1}$ -balls. + +Remarks. (i) No Lipschitz condition on $R_{\pi}$ in the output space $y$ is needed because robustness is posed in gap space $\mathbb{R}$ . (ii) Assumption (A5) matches practice (small step sizes/weight decay) and is only required for the local rate. (iii) Although neural network parameters $\theta \in \mathbb{R}^d$ are technically unbounded, many theoretical analyses restrict $\theta$ to a large but bounded ball (via a norm constraint) or rely on a coercive objective to prevent unbounded parameter growth. Hence, requiring $\Pi$ to be compact is common in theoretical treatments. In practice, gradient-based optimization does not typically push $\| \theta \|$ to infinity. + +# D Theoretical Results + +# D.1 Preliminaries and basic lemmas + +Notation and spaces. All random variables in this section take values in $(\mathbb{R},|\cdot |)$ equipped with the Borel $\sigma$ -algebra. For a policy $\pi$ and i.i.d. samples $\{(x^i,y_w^i,y_\ell^i)\}_{i = 1}^N\sim \hat{P}$ , define the empirical gap distribution + +$$ +\hat {\alpha} (\pi) := \left(\Delta R _ {\pi}\right) _ {\#} \hat {P} = \frac {1}{N} \sum_ {i = 1} ^ {N} \delta_ {\hat {\xi} _ {i}}, \quad \text {w h e r e} \quad \hat {\xi} _ {i} := \Delta R _ {\pi} \left(x ^ {i}, y _ {w} ^ {i}, y _ {\ell} ^ {i}\right) \in \mathbb {R}. +$$ + +For $\epsilon >0$ , the (1-) Wasserstein ball around $\hat{\alpha} (\pi)$ is + +$$ +\mathcal {U} _ {\epsilon} \big (\hat {\alpha} (\pi) \big) := \Big \{\alpha \in \mathcal {P} (\mathbb {R}): W _ {1} \big (\alpha , \hat {\alpha} (\pi) \big) \leq \epsilon \Big \}. +$$ + +We write $u(\xi) \coloneqq \log \sigma(\xi)$ (concave, 1-Lipschitz) and $\ell(\xi) \coloneqq -u(\xi) = -\log \sigma(\xi)$ (convex, 1-Lipschitz). When a piecewise-linear (PWL) surrogate is used, we set + +$$ +\widetilde {\ell} (\xi) := \max _ {1 \leq k \leq K} \ell_ {k} (\xi), \qquad \ell_ {k} (\xi) := a _ {k} \xi + b _ {k}, +$$ + +chosen as global supporting tangents so that $\widetilde{\ell} (\xi)\leq \ell (\xi)$ for all $\xi$ + +Lemma D.1 (log $\sigma$ is concave and 1-Lipschitz). For every $\xi \in \mathbb{R}$ , $\frac{\mathrm{d}}{\mathrm{d}\xi} \log \sigma(\xi) = \sigma(-\xi) \in (0,1)$ and $\frac{\mathrm{d}^2}{\mathrm{d}\xi^2} \log \sigma(\xi) = -\sigma(\xi) \sigma(-\xi) \leq 0$ . Hence $u(\xi) = \log \sigma(\xi)$ is concave and 1-Lipschitz. + +Proof. Recall $\sigma (\xi) = \frac{1}{1 + e^{-\xi}}$ . Then + +$$ +\frac {\mathrm {d}}{\mathrm {d} \xi} \log \sigma (\xi) = \frac {\sigma^ {\prime} (\xi)}{\sigma (\xi)} = \frac {\sigma (\xi) (1 - \sigma (\xi))}{\sigma (\xi)} = 1 - \sigma (\xi) = \sigma (- \xi) \in (0, 1). +$$ + +Hence $\left| \frac{\mathrm{d}}{\mathrm{d}\xi} \log \sigma(\xi) \right| \leq 1$ for all $\xi$ , so $\log \sigma$ is 1-Lipschitz: $|\log \sigma(\xi) - \log \sigma(\xi')| \leq |\xi - \xi'|$ by the mean-value theorem. Further, + +$$ +\frac {\mathrm {d} ^ {2}}{\mathrm {d} \xi^ {2}} \mathrm {l o g} \sigma (\xi) = \frac {\mathrm {d}}{\mathrm {d} \xi} \sigma (- \xi) = - \sigma (- \xi) \big (1 - \sigma (- \xi) \big) = - \sigma (\xi) \sigma (- \xi) \leq 0, +$$ + +so $\log \sigma$ is concave. + +![](images/90872d88fff1018c204e4c9e5126db324ff6d41e6c271d9be6447cb4321a164b.jpg) + +Kantorovich-Rubinstein (KR) duality for $W_{1}$ . For any 1-Lipschitz $h:\mathbb{R}\to \mathbb{R}$ and $\alpha ,\beta \in \mathcal{P}(\mathbb{R})$ + +$$ +\left| \mathbb {E} _ {\alpha} [ h ] - \mathbb {E} _ {\beta} [ h ] \right| \leq W _ {1} (\alpha , \beta). +$$ + +We use this both as a continuity tool and as a tight transport sensitivity bound. + +Lemma D.2 (Continuity of the empirical-center map). If $\pi \mapsto \Delta R_{\pi}(x^{i},y_{w}^{i},y_{\ell}^{i})$ is continuous for each $i\in [N]$ , then for any $\pi_1,\pi_2$ + +$$ +W _ {1} \big (\hat {\alpha} (\pi_ {1}), \hat {\alpha} (\pi_ {2}) \big) \leq \frac {1}{N} \sum_ {i = 1} ^ {N} \left| \Delta R _ {\pi_ {1}} ^ {i} - \Delta R _ {\pi_ {2}} ^ {i} \right|, +$$ + +hence $\pi \mapsto \hat{\alpha} (\pi)$ is continuous in the $W_{1}$ metric. + +Proof. Write $\hat{\alpha} (\pi_j) = \frac{1}{N}\sum_{i = 1}^N\delta_{\hat{\xi}_i^{(j)}}$ with $\hat{\xi}_i^{(j)}\coloneqq \Delta R_{\pi_j}(x^i,y_w^i,y_\ell^i)$ for $j\in \{1,2\}$ . Define the coupling $\gamma = \frac{1}{N}\sum_{i = 1}^{N}\delta_{(\hat{\xi}_i^{(1)},\hat{\xi}_i^{(2)})}$ . By definition of $W_{1}$ (optimal transport with cost $|\cdot |$ on $\mathbb{R}$ ), + +$$ +W _ {1} \big (\hat {\alpha} (\pi_ {1}), \hat {\alpha} (\pi_ {2}) \big) \leq \int | x - y | \mathrm {d} \gamma (x, y) = \frac {1}{N} \sum_ {i = 1} ^ {N} \left| \hat {\xi} _ {i} ^ {(1)} - \hat {\xi} _ {i} ^ {(2)} \right| = \frac {1}{N} \sum_ {i = 1} ^ {N} \left| \Delta R _ {\pi_ {1}} ^ {i} - \Delta R _ {\pi_ {2}} ^ {i} \right|. +$$ + +If for each $i$ the map $\pi \mapsto \Delta R_{\pi}(x^{i},y_{w}^{i},y_{\ell}^{i})$ is continuous (Assumption (A2)), then $W_{1}\bigl (\hat{\alpha} (\pi_{n}),\hat{\alpha} (\pi)\bigr)\to 0$ whenever $\pi_n\rightarrow \pi$ , i.e., $\pi \mapsto \hat{\alpha} (\pi)$ is continuous in the $W_{1}$ metric. + +Lemma D.3 (Compactness of Wasserstein balls about empirical centers). For fixed $\pi$ , the feasible follower set $\mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))$ is nonempty, tight, closed in $W_{1}$ , hence compact. + +Proof. Fix $\pi$ . Let $\mathcal{B} \coloneqq \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi)) = \{\alpha \in \mathcal{P}(\mathbb{R}) : W_1(\alpha, \hat{\alpha}(\pi)) \leq \epsilon\}$ . + +(i) Nonemptiness. Trivially $\hat{\alpha} (\pi)\in \mathcal{B}$ +(ii) Uniform first-moment bound. On $\mathbb{R}$ with ground metric $|\cdot|$ , we have $W_{1}(\alpha, \delta_{0}) = \int |x| \mathrm{d}\alpha(x)$ . By the triangle inequality, + +$$ +\int | x | \mathrm {d} \alpha (x) = W _ {1} (\alpha , \delta_ {0}) \leq W _ {1} (\alpha , \hat {\alpha} (\pi)) + W _ {1} (\hat {\alpha} (\pi), \delta_ {0}) \leq \epsilon + \int | x | \mathrm {d} \hat {\alpha} (\pi) (x), +$$ + +so the family $\mathcal{B}$ has uniformly bounded first moments. + +(iii) Tightness. For any $R > 0$ + +$$ +\alpha \left(| x | > R\right) \leq \frac {1}{R} \int | x | \mathrm {d} \alpha (x) \leq \frac {\epsilon + \int | x | \mathrm {d} \hat {\alpha} (\pi)}{R} \quad \forall \alpha \in \mathcal {B}, +$$ + +by Markov's inequality. Hence $\mathcal{B}$ is tight. + +(iv) Closedness in $W_{1}$ . If $\alpha_{n} \in \mathcal{B}$ with $W_{1}(\alpha_{n},\alpha) \to 0$ , then $W_{1}(\alpha, \hat{\alpha}(\pi)) \leq \operatorname*{liminf}_{n}\left[W_{1}(\alpha,\alpha_{n}) + W_{1}(\alpha_{n},\hat{\alpha}(\pi))\right] \leq \epsilon$ , so $\alpha \in \mathcal{B}$ and $\mathcal{B}$ is closed. +(v) Compactness. On the Polish space $\mathbb{R}$ , Prokhorov's theorem gives that tight families are relatively compact in the weak topology; the uniform first-moment bound tightens this to relative compactness in $W_{1}$ (since $W_{1}$ convergence is equivalent to weak convergence plus convergence of first moments on $\mathbb{R}$ ). Combining relative compactness with closedness in $W_{1}$ yields compactness of $\mathcal{B}$ in $(\mathcal{P}_1(\mathbb{R}), W_1)$ . + +# D.2 Existence of a Stackelberg solution + +Theorem D.4 (Existence). Assume: (A1) $\Pi$ compact; (A2) each $\pi \mapsto \Delta R_{\pi}(x^{i},y_{w}^{i},y_{\ell}^{i})$ is continuous; (A3) hence $\pi \mapsto \hat{\alpha} (\pi)$ is $W_{1}$ -continuous (Lemma D.2); (A4) $u(\cdot)$ is 1-Lipschitz and concave (Lemma D.1). Then + +$$ +\max _ {\pi \in \Pi} \min _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha} (\pi))} \mathbb {E} _ {\alpha} [ u (\xi) ] +$$ + +admits a solution $(\pi^{\star},\alpha^{\star})$ + +Proof. We verify the conditions of Berge's maximum theorem step by step. + +Step 1 (Follower minimizer exists for each fixed $\pi$ ). For fixed $\pi$ , Lemma D.3 shows the feasible set $\mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))$ is nonempty and compact in $W_{1}$ . By Lemma D.1, $u$ is 1-Lipschitz, hence $\alpha \mapsto \mathbb{E}_{\alpha}[u]$ is continuous under $W_{1}$ (KR inequality). Therefore, the follower problem admits a minimizer $\alpha^{\star}(\pi) \in \operatorname*{argmin}_{\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))} \mathbb{E}_{\alpha}[u]$ . + +Step 2 (Continuity of the feasible-set correspondence in $\pi$ ). By Lemma D.2, $\pi \mapsto \hat{\alpha}(\pi)$ is continuous in $W_{1}$ . The set-valued map $\pi \mapsto \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))$ thus varies continuously in the Hausdorff metric induced by $W_{1}$ (closed balls move continuously with their centers in a metric space), in particular it is upper hemicontinuous and compact valued. + +Step 3 (Upper semicontinuity of the value map). Define $V(\pi) \coloneqq \min_{\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))} \mathbb{E}_{\alpha}[u]$ . By Berge's maximum theorem (compact-valued, upper hemicontinuous correspondence; continuous objective), $V$ is upper semicontinuous on $\Pi$ . + +Step 4 (Maximizer exists). Under (A1), $\Pi$ is compact. Since $V$ is upper semicontinuous on a compact set, it attains its maximum at some $\pi^{\star} \in \Pi$ . By Step 1, there is a realizing follower $\alpha^{\star} \in \operatorname{argmin}_{\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi^{\star}))} \mathbb{E}_{\alpha}[u]$ . + +Therefore the problem admits a solution $(\pi^{\star},\alpha^{\star})$ + +# D.3 Local linear convergence of alternating updates + +Consider the iterates with a (Euclidean) proximal leader step: + +$$ +\alpha_ {t + 1} \in \arg \min _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha} (\pi_ {t}))} \mathbb {E} _ {\alpha} [ u (\xi) ], \qquad \pi_ {t + 1} \in \operatorname * {a r g m a x} _ {\pi \in \Pi} \Big \{\mathbb {E} _ {\alpha_ {t + 1}} [ u (\xi) ] - \frac {\lambda}{2} \| \theta (\pi) - \theta (\pi_ {t}) \| ^ {2} \Big \}. +$$ + +Theorem D.5 (Well-posedness and local linear convergence). Assume (i) the leader's proximal objective is $\mu$ -strongly concave in $\theta(\pi)$ uniformly over $\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi_t))$ on a neighborhood of a solution, and (ii) the map $\pi \mapsto \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))$ is Lipschitz in the $W_1$ -Hausdorff distance near $\pi^{\star}$ . Then the update map $(\pi_t, \alpha_t) \mapsto (\pi_{t+1}, \alpha_{t+1})$ is a contraction in a neighborhood of $(\pi^{\star}, \alpha^{\star})$ , and the iterates converge linearly to $(\pi^{\star}, \alpha^{\star})$ . + +Proof. Let the follower best-response be any measurable selection $\alpha^{\sharp}(\pi) \in \mathrm{argmin}_{\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))} \mathbb{E}_{\alpha}[u]$ , whose existence follows from the measurable maximum theorem since the correspondence is compact valued and upper hemicontinuous. + +Define the proximal leader map at iterate $\pi_t$ : + +$$ +\mathcal {T} (\alpha ; \pi_ {t}) \in \operatorname * {a r g m a x} _ {\pi \in \Pi} G (\pi ; \pi_ {t}, \alpha), \qquad G (\pi ; \pi_ {t}, \alpha) := \mathbb {E} _ {\xi \sim \alpha} [ u (\Delta R _ {\pi}) ] - \frac {\lambda}{2} \| \theta (\pi) - \theta (\pi_ {t}) \| ^ {2}. +$$ + +Step 1 (Follower map is Lipschitz in the center, hence in $\pi$ locally). Fix $\pi_1, \pi_2$ , and let $\alpha_j = \alpha^\sharp(\pi_j)$ . Because $\mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))$ is a closed ball in $(\mathcal{P}_1(\mathbb{R}), W_1)$ , for any $\beta$ we can project it to the closest point in the ball (metric projection is 1-Lipschitz). In particular, for $\beta = \alpha_1$ we have + +$$ +\left. \operatorname {d i s t} _ {W _ {1}} \left(\alpha_ {1}, \mathcal {U} _ {\epsilon} (\hat {\alpha} (\pi_ {2}))\right) \leq W _ {1} \left(\hat {\alpha} (\pi_ {1}), \hat {\alpha} (\pi_ {2})\right). \right. +$$ + +Let $\tilde{\alpha}_{2}\in \mathcal{U}_{\epsilon}(\hat{\alpha} (\pi_{2}))$ be a nearest point to $\alpha_{1}$ . Using 1-Lipschitzness of $u$ and the optimality of $\alpha_{2}$ at $\pi_2$ + +$$ +\mathbb {E} _ {\alpha_ {2}} [ u ] \leq \mathbb {E} _ {\tilde {\alpha} _ {2}} [ u ] \leq \mathbb {E} _ {\alpha_ {1}} [ u ] + W _ {1} (\tilde {\alpha} _ {2}, \alpha_ {1}) \leq \mathbb {E} _ {\alpha_ {1}} [ u ] + W _ {1} (\hat {\alpha} (\pi_ {1}), \hat {\alpha} (\pi_ {2})). +$$ + +Symmetrizing the roles of 1 and 2 gives by triangle inequality + +$$ +W _ {1} \left(\alpha_ {1}, \alpha_ {2}\right) \leq 2 W _ {1} \left(\hat {\alpha} \left(\pi_ {1}\right), \hat {\alpha} \left(\pi_ {2}\right)\right). +$$ + +Thus the follower map is Lipschitz in the empirical center with constant $L_{f}\leq 2$ . By Lemma D.2, + +$$ +W _ {1} \left(\hat {\alpha} (\pi_ {1}), \hat {\alpha} (\pi_ {2})\right) \leq \frac {1}{N} \sum_ {i = 1} ^ {N} \left| \Delta R _ {\pi_ {1}} ^ {i} - \Delta R _ {\pi_ {2}} ^ {i} \right|. +$$ + +Assume (locally around the target) the maps $\theta \mapsto \Delta R_{\pi_\theta}^i$ are $L_{R}$ -Lipschitz for $i = 1,\dots,N$ . Then locally + +$$ +W _ {1} \left(\alpha^ {\sharp} \left(\pi_ {1}\right), \alpha^ {\sharp} \left(\pi_ {2}\right)\right) \leq L _ {f} W _ {1} \left(\hat {\alpha} \left(\pi_ {1}\right), \hat {\alpha} \left(\pi_ {2}\right)\right) \leq L _ {f} L _ {R} \| \theta \left(\pi_ {1}\right) - \theta \left(\pi_ {2}\right) \|. +$$ + +Step 2 (Leader prox map is Lipschitz in $\alpha$ under local strong concavity). By (A5), for each fixed $\pi_t$ the map $\pi \mapsto G(\pi; \pi_t, \alpha)$ is $\mu$ -strongly concave in $\theta(\pi)$ on a neighborhood of the solution, uniformly over $\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi_t))$ . Let $\pi_j^+ := \mathcal{T}(\alpha_j; \pi_t)$ for $j \in \{1, 2\}$ . Since $u$ is 1-Lipschitz and $\Delta R_{\pi}$ is continuous in $\pi$ , there exists $L_{\ell}$ (local) such that + +$$ +\left| \left| \nabla_ {\theta} \mathbb {E} _ {\alpha_ {1}} [ u (\Delta R _ {\pi}) ] - \nabla_ {\theta} \mathbb {E} _ {\alpha_ {2}} [ u (\Delta R _ {\pi}) ] \right| \right| \leq L _ {\ell} W _ {1} (\alpha_ {1}, \alpha_ {2}) \quad \text {f o r} \pi \text {n e a r t h e s o l u t i o n}. +$$ + +By standard stability of maximizers under strong concavity (e.g., by the implicit function theorem or strong monotonicity of the gradient mapping), we obtain the Lipschitz dependence + +$$ +\left\| \theta \left(\pi_ {1} ^ {+}\right) - \theta \left(\pi_ {2} ^ {+}\right) \right\| \leq \frac {L _ {\ell}}{\mu} W _ {1} \left(\alpha_ {1}, \alpha_ {2}\right). +$$ + +Step 3 (Contraction of the composition and linear rate). Set $\alpha_{t} \coloneqq \alpha^{\sharp}(\pi_{t})$ and $\pi_{t+1} \coloneqq \mathcal{T}(\alpha_{t};\pi_{t})$ . Let $(\pi^{\star},\alpha^{\star})$ be a Stackelberg solution; then $\alpha^{\star} = \alpha^{\sharp}(\pi^{\star})$ and $\pi^{\star} = \mathcal{T}(\alpha^{\star};\pi^{\star})$ . Combining Steps 1 and 2, + +$$ +\left\| \theta \left(\pi_ {t + 1}\right) - \theta \left(\pi^ {\star}\right) \right\| \leq \frac {L _ {\ell}}{\mu} W _ {1} \left(\alpha_ {t}, \alpha^ {\star}\right) \leq \frac {L _ {\ell}}{\mu} L _ {f} L _ {R} \left\| \theta \left(\pi_ {t}\right) - \theta \left(\pi^ {\star}\right) \right\|. +$$ + +Choose the proximal weight $\lambda$ (hence the local strong-concavity modulus $\mu$ ) so that $\rho \coloneqq \frac{L_{\ell}L_{f}L_{R}}{\mu} < 1$ . Then Banach's fixed-point theorem yields linear convergence: + +$$ +\left\| \theta \left(\pi_ {t}\right) - \theta \left(\pi^ {\star}\right) \right\| \leq \rho^ {t - t _ {0}} \left\| \theta \left(\pi_ {t _ {0}}\right) - \theta \left(\pi^ {\star}\right) \right\| \quad \text {f o r a l l} t \geq t _ {0} \text {i n t h e n e i g h b o r h o o d}. +$$ + +This also implies well-posedness (local single-valuedness) of the composite best-response map in that neighborhood. + +# D.4 Worst-case performance drop and SGPO regret bound + +Theorem D.6 (Worst-case performance drop). If $\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha} (\pi))$ , then $\mathbb{E}_{\alpha}[u(\xi)]\geq \mathbb{E}_{\hat{\alpha} (\pi)}[u(\xi)] - \epsilon$ + +Proof. By Lemma D.1, $u$ is 1-Lipschitz. For any $\alpha$ with $W_{1}(\alpha ,\hat{\alpha} (\pi))\leq \epsilon$ , the Kantorovich-Rubinstein inequality yields + +$$ +\mathbb {E} _ {\alpha} [ u ] - \mathbb {E} _ {\hat {\alpha} (\pi)} [ u ] \geq - W _ {1} (\alpha , \hat {\alpha} (\pi)) \geq - \epsilon . +$$ + +Rearranging gives the claim. + +![](images/4684f3a4ad3fbe9887648c2d12df6d8ef62ad28b79723ae8af2f41b2c94201d5.jpg) + +Theorem D.7 (SGPO regret bound). Let $\pi^{\star}$ solve $\max_{\pi} \min_{\alpha \in \mathcal{U}_{\epsilon}(\dot{\alpha}(\pi))} \mathbb{E}_{\alpha}[u]$ . Then + +$$ +\sup_{\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha} (\pi^{\star}))}\operatorname {Regret}(\pi^{\star},\alpha)\leq 2\epsilon . +$$ + +Proof. Let $\pi^{\star} \in \operatorname{argmax}_{\pi} \min_{\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))} \mathbb{E}_{\alpha}[u]$ and fix any $\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi^{\star}))$ . Let $\pi_{\alpha}^{\star} \in \operatorname{argmax}_{\pi} \mathbb{E}_{\alpha}[u]$ be the $\alpha$ -optimal policy. By Theorem D.6, + +$$ +\mathbb {E} _ {\alpha} [ u ] _ {\pi_ {\hat {\alpha}} ^ {*}} \leq \mathbb {E} _ {\hat {\alpha} (\pi^ {*})} [ u ] _ {\pi_ {\hat {\alpha}} ^ {*}} + \epsilon , \quad \mathbb {E} _ {\alpha} [ u ] _ {\pi^ {*}} \geq \mathbb {E} _ {\hat {\alpha} (\pi^ {*})} [ u ] _ {\pi^ {*}} - \epsilon . +$$ + +Subtracting gives + +$$ +\operatorname {R e g r e t} (\pi^ {\star}, \alpha) = \mathbb {E} _ {\alpha} [ u ] _ {\pi_ {\alpha} ^ {\star}} - \mathbb {E} _ {\alpha} [ u ] _ {\pi^ {\star}} \leq \underbrace {\left(\mathbb {E} _ {\hat {\alpha} (\pi^ {\star})} [ u ] _ {\pi_ {\alpha} ^ {\star}} - \mathbb {E} _ {\hat {\alpha} (\pi^ {\star})} [ u ] _ {\pi^ {\star}}\right)} _ {\leq 0 \text {b y d e f . o f} \pi^ {\star}} + 2 \epsilon \leq 2 \epsilon . +$$ + +![](images/8a52dda094f935879c8e7050e465a908f06d9734488999e0101eaf6b551449ff.jpg) + +# D.5 DPO regret lower bound under a stability assumption + +Assumption D.8 (Center stability at the robust follower). Let $\alpha^{\star} \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi^{\star}))$ be a follower minimizer for $\pi^{\star}$ and $\pi_{\alpha^{\star}}^{*} \in \operatorname{argmax}_{\pi} \mathbb{E}_{\alpha^{\star}}[u]$ . Assume there exists $\kappa \leq \epsilon$ such that $W_{1}(\hat{\alpha}(\pi_{\alpha^{\star}}^{*}), \hat{\alpha}(\pi_{\mathrm{DPO}})) \leq \kappa$ . + +Theorem D.9 (DPO regret lower bound (stability version)). Let $\delta \coloneqq W_{1}\bigl (\alpha^{\star},\hat{\alpha} (\pi_{\mathrm{DPO}})\bigr)$ . Under Assumption D.8, + +$$ +\operatorname {R e g r e t} \left(\pi_ {\mathrm {D P O}}, \alpha^ {\star}\right) \geq \delta - (\kappa + \epsilon) \geq \delta - 2 \epsilon . +$$ + +Proof. Let $\delta \coloneqq W_{1}(\alpha^{\star},\hat{\alpha} (\pi_{\mathrm{DPO}}))$ and recall Assumption D.8. First, by KR and 1-Lipschitzness of $u$ + +$$ +\mathbb {E} _ {\alpha^ {\star}} [ u ] _ {\pi_ {\alpha^ {\star}} ^ {*}} \geq \mathbb {E} _ {\hat {\alpha} (\pi_ {\mathrm {D P O}})} [ u ] _ {\pi_ {\alpha^ {\star}} ^ {*}} - \delta , \quad \mathbb {E} _ {\alpha^ {\star}} [ u ] _ {\pi_ {\mathrm {D P O}}} \leq \mathbb {E} _ {\hat {\alpha} (\pi_ {\mathrm {D P O}})} [ u ] _ {\pi_ {\mathrm {D P O}}} + \delta . +$$ + +Subtracting, + +$$ +\operatorname {R e g r e t} \left(\pi_ {\mathrm {D P O}}, \alpha^ {\star}\right) \geq \left[ \mathbb {E} _ {\hat {\alpha} \left(\pi_ {\mathrm {D P O}}\right)} [ u ] \pi_ {\alpha^ {\star}} ^ {*} - \mathbb {E} _ {\hat {\alpha} \left(\pi_ {\mathrm {D P O}}\right)} [ u ] \pi_ {\mathrm {D P O}} \right] - 2 \delta . +$$ + +By center stability and optimality of $\pi_{\alpha^{\star}}^{*}$ at $\alpha^{\star}$ , + +$$ +\mathbb {E} _ {\hat {\alpha} \left(\pi_ {\mathrm {D P O}}\right)} [ u ] _ {\pi_ {\alpha^ {*}}} \geq \mathbb {E} _ {\hat {\alpha} \left(\pi_ {\alpha^ {*}} ^ {*}\right)} [ u ] _ {\pi_ {\alpha^ {*}}} - \kappa \geq \mathbb {E} _ {\alpha^ {*}} [ u ] _ {\pi_ {\alpha^ {*}}} - \kappa . +$$ + +Finally, since $W_{1}(\hat{\alpha} (\pi^{\star}),\alpha^{\star})\leq \epsilon$ and $\pi^{\star}$ is optimal at $\hat{\alpha} (\pi^{\star})$ + +$$ +\mathbb {E} _ {\alpha^ {\star}} [ u ] _ {\pi_ {\alpha^ {\star}} ^ {\star}} \geq \mathbb {E} _ {\hat {\alpha} (\pi^ {\star})} [ u ] _ {\pi^ {\star}} - \epsilon . +$$ + +Combining the displays and cancelling the center-optimal term gives $\mathrm{Regret}(\pi_{\mathrm{DPO}},\alpha^{\star})\geq$ $\delta -(\kappa +\epsilon)\geq \delta -2\epsilon$ (using $\kappa \leq \epsilon$ + +Remark. Without Assumption D.8, the lower bound can vanish if $\pi_{\alpha^{\star}}^{*}$ recenters too far from $\hat{\alpha} (\pi_{\mathrm{DPO}})$ ; the stability phrasing makes explicit the (mild) continuity needed for a linear-in- $\delta$ lower bound. + +# D.6 Approximation effects (piecewise, grouping, inner tolerance) + +Let $m(\pi) \coloneqq \min_{\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))} \mathbb{E}_{\alpha}[u]$ denote the true follower value and let $\widetilde{m}_{K,\mathrm{grp},\eta}(\pi)$ denote the value computed with: (i) PWL under-approximation $\widetilde{\ell}_K$ with $K$ pieces, (ii) a restricted feasible set $\widetilde{\mathcal{U}}_K(\hat{\alpha}(\pi)) \subseteq \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))$ (e.g., via grouping), and (iii) inner tolerance $\eta$ . + +Proposition D.10 (Monotone tightening in the number of pieces). Fix a reference distribution $\hat{\alpha}$ on $\mathbb{R}$ with finite first moment and radius $\epsilon \geq 0$ , and let $\mathcal{U}_{\epsilon}(\hat{\alpha})$ denote the associated 1-Wasserstein ambiguity set (absolute ground metric). Let $\ell : \mathbb{R} \to [0, \infty)$ be convex and define $\{\tilde{\ell}_K\}_{K \geq 1}$ as convex piecewise-linear underestimators of $\ell$ of the form + +$$ +\tilde {\ell} _ {K} (\xi) = \max _ {1 \leq k \leq K} \left\{a _ {k} \xi + b _ {k} \right\}, \quad \text {w i t h} \quad \tilde {\ell} _ {K} (\xi) \leq \tilde {\ell} _ {K + 1} (\xi) \leq \ell (\xi) \forall \xi , +$$ + +such that $\tilde{\ell}_K(\xi)\uparrow \ell (\xi)$ pointwise as $K\to \infty$ . Define the exact and surrogate inner values + +$$ +v ^ {\star} := \inf _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha})} \mathbb {E} _ {\alpha} [ \log \sigma (\xi) ] = - \sup _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha})} \mathbb {E} _ {\alpha} [ \ell (\xi) ], \qquad v _ {K} := - \sup _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha})} \mathbb {E} _ {\alpha} [ \tilde {\ell} _ {K} (\xi) ]. +$$ + +Assume $\sup_{\alpha \in \mathcal{U}_{\varepsilon}(\hat{\alpha})}\mathbb{E}_{\alpha}[\ell (\xi)] < \infty$ . Then: + +1. (Validity) $v_{K} \geq v^{\star}$ for all $K \geq 1$ . +2. (Monotonicity) $v_{K + 1} \leq v_K$ for all $K \geq 1$ . +3. (Limit) $v_{K} \downarrow v^{\star}$ as $K \to \infty$ . + +Moreover, if for some $K$ the supremum in the definition of $v_{K}$ is attained by $\alpha_{K} \in \mathcal{U}_{\epsilon}(\hat{\alpha})$ and $\tilde{\ell}_{K}(\xi) = \ell(\xi)$ holds $\alpha_{K}$ -almost surely, then $v_{K} = v^{\star}$ . + +Proof. By definition of $\ell = -\log \sigma$ we have $\inf_{\alpha}\mathbb{E}_{\alpha}[\log \sigma] = -\sup_{\alpha}\mathbb{E}_{\alpha}[\ell]$ , hence $v^{\star} = -\sup_{\alpha}\mathbb{E}_{\alpha}[\ell]$ . Since $\tilde{\ell}_K \leq \ell$ , it follows that $\sup_{\alpha}\mathbb{E}_{\alpha}[\tilde{\ell}_K] \leq \sup_{\alpha}\mathbb{E}_{\alpha}[\ell]$ , which implies $v_K \geq v^{\star}$ , proving (a). Because $\tilde{\ell}_{K+1} \geq \tilde{\ell}_K$ pointwise, also $\sup_{\alpha}\mathbb{E}_{\alpha}[\tilde{\ell}_{K+1}] \geq \sup_{\alpha}\mathbb{E}_{\alpha}[\tilde{\ell}_K]$ , hence $v_{K+1} \leq v_K$ , proving (b). For (c), by monotone convergence, for each fixed $\alpha$ we have $\mathbb{E}_{\alpha}[\tilde{\ell}_K] \uparrow \mathbb{E}_{\alpha}[\ell]$ ; therefore $\sup_{\alpha}\mathbb{E}_{\alpha}[\tilde{\ell}_K] \uparrow \sup_{\alpha}\mathbb{E}_{\alpha}[\ell]$ and taking negatives yields $v_K \downarrow v^{\star}$ . The final claim is immediate from the definitions. + +Remark D.11. In our setting $\ell (\xi) = -\log \sigma (\xi)$ is nonnegative and 1-Lipschitz, so $\sup_{\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha})}\mathbb{E}_{\alpha}[\ell (\xi)] < \infty$ whenever $\mathcal{U}_{\epsilon}(\hat{\alpha})$ is a $W_{1}$ -ball around a measure with finite first moment. + +Proposition D.12 (Error from PWL under-approximation). Let $\Delta_{\mathrm{pl}}(K)\coloneqq \sup_{\xi}\bigl (\ell (\xi) - \widetilde{\ell}_K(\xi)\bigr)\geq 0.$ Then + +$$ +0 \leq \widetilde {m} _ {K} (\pi) - m (\pi) \leq \Delta_ {\mathrm {p l}} (K). +$$ + +Proof. By construction $\tilde{\ell}_K\leq \ell$ pointwise. Since $u = -\ell$ , for any feasible $\alpha$ + +$$ +\mathbb {E} _ {\alpha} [ u ] = - \mathbb {E} _ {\alpha} [ \ell ] \leq - \mathbb {E} _ {\alpha} [ \widetilde {\ell} _ {K} ] \leq - \sup _ {\alpha^ {\prime} \in \mathcal {U} _ {\varepsilon}} \mathbb {E} _ {\alpha^ {\prime}} [ \widetilde {\ell} _ {K} ] = \widetilde {m} _ {K} (\pi), +$$ + +so $\widetilde{m}_K(\pi) \geq m(\pi)$ . Moreover, for any $\alpha$ , $\mathbb{E}_{\alpha}[\ell] - \mathbb{E}_{\alpha}[\widetilde{\ell}_K] \leq \sup_{\xi} \bigl( \ell(\xi) - \widetilde{\ell}_K(\xi) \bigr) =: \Delta_{\mathrm{pl}}(K)$ . Taking the supremum over $\alpha$ and flipping the sign gives + +$$ +0 \leq \tilde {m} _ {K} (\pi) - m (\pi) \leq \Delta_ {\mathrm {p l}} (K). +$$ + +Proposition D.13 (Error from grouping/restiction). Let $d_H$ denote the directed Hausdorff distance (under $W_1$ ) from $\mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))$ to $\widetilde{\mathcal{U}}_K(\hat{\alpha}(\pi))$ . Then + +$$ +0 \leq \tilde {m} _ {K} ^ {\mathrm {g r p}} (\pi) - m (\pi) \leq d _ {H}. +$$ + +Proof. Let $\alpha^{\star} \in \operatorname{argmin}_{\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\pi))} \mathbb{E}_{\alpha}[u]$ and let $\tilde{\alpha}$ be any element of the restricted set $\widetilde{\mathcal{U}}_K(\hat{\alpha}(\pi))$ satisfying $W_{1}(\alpha^{\star}, \tilde{\alpha}) \leq d_{H}$ by definition of the directed Hausdorff distance. Using 1-Lipschitzness of $u$ , + +$$ +\min _ {\tilde {\alpha} \in \widetilde {\mathcal {U}} _ {\kappa} (\hat {\alpha} (\pi))} \mathbb {E} _ {\tilde {\alpha}} [ u ] \leq \mathbb {E} _ {\tilde {\alpha}} [ u ] \leq \mathbb {E} _ {\alpha^ {\star}} [ u ] + W _ {1} (\tilde {\alpha}, \alpha^ {\star}) \leq m (\pi) + d _ {H}. +$$ + +Thus $0 \leq \widetilde{m}_K^{\mathrm{grp}}(\pi) - m(\pi) \leq d_H$ . + +![](images/3bae3d27b868617e044cac7491941c0c107af4ee1b4119a2f828bb77534445a6.jpg) + +# Algorithm 1 Stackelberg Self-Annotated Preference Optimization (SSAPO) + +Require: Seed labeled set $\mathcal{D}_{\mathrm{seed}}$ ; unlabeled data $\mathcal{D}_{\mathrm{unlabeled}}$ ; Wasserstein radius $\epsilon$ ; number of linear pieces $K$ ; max iterations $T$ . + +1: Initialize policy $\theta_0$ , set $\mathcal{D} \gets \mathcal{D}_{\mathrm{seed}}$ . +2: for $t = 0$ to $T - 1$ do +3: (Self-Annotation): From $\mathcal{D}_{\mathrm{unlabeled}}$ , sample prompts, generate & rank responses under $\pi_{\theta_t}$ , add new preference pairs $(y_w, y_\ell)$ to $\mathcal{D}$ . +4: (Form $\hat{\alpha}(\pi_t)$ ): For each $(y_w^i, y_\ell^i) \in \mathcal{D}$ , define $\hat{\xi}_i = R_{\theta_t}(y_w^i) - R_{\theta_t}(y_\ell^i)$ , and let $\hat{\alpha}(\pi_t) = \frac{1}{N} \sum_{i=1}^{N} \delta_{\hat{\xi}_i}$ . +5: (Convex Pieces): Choose $K$ linear functions $\ell_k(\cdot)$ such that $\widetilde{\ell} (\xi) = \max_{1\leq k\leq K}\ell_k(\xi)\leq -\log \sigma (\xi)$ . Choose $K$ knots $\{\xi^{(k)}\}$ in $\xi$ -space over a bounded interval $[a_t,b_t]$ (e.g., $a_{t} = \min_{i}\hat{\xi}_{i} - \tau$ , $b_{t} = \max_{i}\hat{\xi}_{i} + \tau$ with $\tau >0$ , or empirical $(\alpha ,1 - \alpha)$ quantiles). Define $\ell_{k}$ as tangents (or chords) from below to $-\log \sigma (\xi)$ at those knots. +6: (Worst-Case Distribution): Solve the DRO finite convex program + +$$ +\alpha_ {t} ^ {*} \in \arg \max _ {\alpha \in \mathcal {U} _ {\epsilon} (\tilde {\alpha} (\pi_ {t}))} \mathbb {E} _ {\alpha} \left[ \widetilde {\ell} (\xi) \right]. +$$ + +By Theorem 3.3, $\alpha_{t}^{*}$ is discrete with atoms $\left\{\hat{\xi}_i - \frac{q_{ik}^*}{s_{ik}^*}\right\}$ and weights $s_{ik}^{*} / N$ . + +7: (Policy Update): Let $w_{i}^{(t)} \coloneqq \sum_{k=1}^{K} s_{ik}^{*(t)}$ . Update $\theta_{t+1}$ by minimizing the weighted logistic loss + +$$ +\frac {1}{N} \sum_ {i = 1} ^ {N} w _ {i} ^ {(t)} \left[ - \log \sigma \left(\Delta R _ {\theta} \left(x ^ {i}, y _ {w} ^ {i}, y _ {\ell} ^ {i}\right)\right) \right] +$$ + +(optional with KL or weight decay), via standard gradient methods. + +8: end for +9: return $\theta_T$ (final policy). + +Proposition D.14 (Cumulative approximation bound). For all $\pi$ , + +$$ +0 \leq \tilde {m} _ {K, \mathrm {g r p}, \eta} (\pi) - m (\pi) \leq \Delta_ {\mathrm {p l}} (K) + d _ {H} + \eta . +$$ + +Proof. Combine Proposition D.12 (PWL gap $\leq \Delta_{\mathrm{pl}}(K)$ ), Proposition D.13 (restriction gap $\leq d_H$ ), and note that an inner solver with tolerance $\eta$ perturbs the value by at most $\eta$ . Errors add up, giving $0 \leq \widetilde{m}_{K,\mathrm{grp},\eta}(\pi) - m(\pi) \leq \Delta_{\mathrm{pl}}(K) + d_H + \eta$ . + +Theorem D.15 (Effect on regret guarantees under approximations). Let $\tilde{\pi}$ be produced by SSAPO with $K$ PWL pieces, $G$ groups, inner tolerance $\eta$ . Then, compared to Theorem D.7, + +$$ +\sup _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha} (\tilde {\pi}))} \operatorname {R e g r e t} (\tilde {\pi}, \alpha) \leq 2 \epsilon + 2 \left[ \Delta_ {\mathrm {p l}} (K) + d _ {H} + \eta \right]. +$$ + +Proof. Let $\pi^{\star}$ be the SGPO optimizer and $\tilde{\pi}$ the SSAPO solution under approximations (PWL with $K$ pieces, grouping, and inner accuracy $\eta$ ). For any $\alpha \in \mathcal{U}_{\epsilon}(\hat{\alpha}(\tilde{\pi}))$ , + +$$ +\operatorname {R e g r e t} (\tilde {\pi}, \alpha) = \mathbb {E} _ {\alpha} [ u ] _ {\pi_ {\alpha} ^ {\star}} - \mathbb {E} _ {\alpha} [ u ] _ {\tilde {\pi}} \leq \left(\mathbb {E} _ {\hat {\alpha} (\tilde {\pi})} [ u ] _ {\pi_ {\alpha} ^ {\star}} - \mathbb {E} _ {\hat {\alpha} (\tilde {\pi})} [ u ] _ {\tilde {\pi}}\right) + 2 \epsilon +$$ + +by Theorem D.6. Replacing the center objective $m(\cdot)$ by its approximate counterpart $\widetilde{m}_{K,\mathrm{grp},\eta}(\cdot)$ incurs at most $\Delta_{\mathrm{pl}}(K) + d_H + \eta$ at $\tilde{\pi}$ and the same at the comparator, hence the extra $2\big[\Delta_{\mathrm{pl}}(K) + d_H + \eta \big]$ . + +# E SSAPO algorithm and Analysis on Computational Complexity + +# E.1 The SSAPO algorithm + +# E.2 Computational Complexity of SSAPO + +In this subsection, we analyze the computational costs incurred by each step of the Stackelberg Self-Annotated Preference Optimization (SSAPO) algorithm (Algorithm 1). We denote: + +- $N$ : the total number of preference pairs in the dataset $\mathcal{D}$ at a given iteration, +- $K$ : the number of linear pieces used in the convex piecewise approximation of $-\log \sigma(\xi)$ , +- $T$ : the total number of outer iterations for SSAPO. + +We assume each iteration refers to Steps 1-5 of Algorithm 1. + +Step 1 (Self-Annotation) The cost of self-annotation depends on the number of prompts and the policy's inference procedure. Let $M_{t}$ denote the number of new prompts labeled at iteration $t$ . Generating and ranking responses under $\pi_{\theta_t}$ typically dominates this step. If: + +- $G_{t}$ is the number of candidate responses generated per prompt, +- $C_{\mathrm{inference}}$ is the average cost of a single forward pass (token generation) under $\pi_{\theta_t}$ , + +then the time complexity for Step 1 is approximately + +$$ +\mathcal {O} \left(M _ {t} \cdot G _ {t} \cdot C _ {\text {i n f e r e n c e}}\right), +$$ + +plus any overhead for storing new winner-loser pairs in $\mathcal{D}$ . Since the number of newly added preferences grows over iterations, $N$ itself typically increases from iteration to iteration. + +Step 2 (Forming $\hat{\alpha} (\pi_t)$ ) Once $\mathcal{D}$ is updated, we compute $\hat{\xi}_i = R_{\theta_t}(y_w^i) - R_{\theta_t}(y_\ell^i)$ for each pair. The cost here depends on: + +- $N$ , the current size of $\mathcal{D}$ +- $C_{\text{reward}}$ , the average cost to compute $R_{\theta_t}(y) = \beta \log \frac{\pi_{\theta_t}(y|x)}{\pi_{\text{ref}}(y|x)}$ for a given response $y$ . + +Because each preference pair requires evaluating $R_{\theta_t}$ on $(y_w^i, y_\ell^i)$ , this step has complexity + +$$ +\mathcal {O} \left(N \cdot C _ {\text {r e w a r d}}\right). +$$ + +In practical implementations, $R_{\theta_t}(y)$ often just reads off the log-probabilities from $\pi_{\theta_t}$ and $\pi_{\mathrm{ref}}$ at the final tokens, making $C_{\mathrm{reward}}$ similar to a single forward-pass cost per response. + +Step 3 (Convex Piecewise Approximation) We construct $K$ linear functions $\ell_k(\xi)$ such that $\widetilde{\ell} (\xi) = \max_{1\leq k\leq K}\ell_k(\xi)\leq -\log \sigma (\xi)$ . In principle, one can precompute these $K$ pieces over a small interval (e.g., [0,1]) once and reuse them in every iteration. Hence, the complexity for updating or verifying the piecewise function at iteration $t$ is typically: $\mathcal{O}(K)$ , assuming $\{\xi^{(k)}\}_{k = 1}^{K}$ are fixed or can be quickly adapted based on the range of $\{\hat{\xi}_i\}$ . This step is therefore relatively cheap compared to distributionally robust optimization. + +Step 4 (Worst-Case Distribution) Step 4 solves the distributionally robust optimization (DRO) finite convex program + +$$ +\alpha_ {t} ^ {*} = \arg \max _ {\alpha \in \mathcal {U} _ {\epsilon} (\hat {\alpha} (\pi_ {t}))} \mathbb {E} _ {\alpha} [ \widetilde {\ell} (\xi) ]. +$$ + +The naive formulation (per [13]) becomes high-dimensional if $N$ is large, because each sample point $\hat{\xi}_i$ and each piecewise component $\ell_k$ introduces auxiliary variables (such as $s_{ik}, q_{ik}$ ). Concretely, the number of decision variables can scale like $\mathcal{O}(N \cdot K)$ , and the resulting linear or convex program might require $\mathcal{O}((NK)^\gamma)$ time in the worst case for some exponent $\gamma > 1$ (depending on the chosen solver and constraints). + +However, several factors can reduce this cost: + +- Approximate Solvers. In practice, specialized cutting-plane or primal-dual methods solve these DRO problems more efficiently than the worst-case theoretical bound. + +- Grouping Heuristics. If one partitions the $N$ samples into smaller groups (each of size $G < N$ ), the complexity per group is $\mathcal{O}((GK)^{\gamma})$ . Then one aggregates $M = \frac{N}{G}$ group-level solutions. This lowers the complexity significantly if $G \ll N$ . + +Hence, the worst-case step here is often $\mathcal{O}(N\cdot K)$ to $\mathcal{O}((NK)^{\gamma})$ , but can be much more tractable in practice with grouping or approximate methods. Regardless, Step 4 typically dominates the iteration complexity for large $N$ . + +Step 5 (Policy Update) Finally, we minimize + +$$ +\mathbb {E} _ {\xi \sim \alpha_ {t} ^ {*}} \left[ - \log \sigma (\xi) \right], \qquad \alpha_ {t} ^ {*} \in \arg \max _ {\alpha \in \mathcal {U} _ {\epsilon} (\dot {\alpha} (\pi_ {t}))} \mathbb {E} _ {\xi \sim \alpha} \left[ - \log \sigma (\xi) \right] +$$ + +(by the $\ell = -\log \sigma$ reparameterization). In practice we compute gradients via the chain rule $\xi = \Delta R_{\theta}(x,y_w,y_\ell)$ and reweight per-pair contributions to match $\alpha_t^*$ (see Alg. 1). + +Assuming each of the $N$ preference pairs in $\alpha_{t}^{*}$ can be sampled over multiple epochs. In many implementations, $N$ can be large, so the training complexity depends heavily on how many gradient epochs or passes one uses at iteration $t$ . + +Overall Complexity per Iteration Putting the above pieces together, let us summarize the dominating terms: + +1. Self-Annotation (Step 1): $\mathcal{O}(M_t\cdot G_t\cdot C_{\mathrm{inference}})$ +2. Forming $\hat{\alpha} (\pi_t)$ (Step 2): $\mathcal{O}(N\cdot C_{\mathrm{reward}})$ +3. Convex Piecewise Approx. (Step 3): $\mathcal{O}(K)$ +4. Worst-Case Distribution (Step 4): $\mathcal{O}((NK)^\gamma)$ in the naive case, often reduced by grouping, +5. Policy Update (Step 5): $\mathcal{O}(N\cdot C_{\mathrm{reward}}\cdot$ (number of epochs)). + +If we denote the cost of solving the DRO subproblem by $C_{\mathrm{DRO}}(N, K)$ (which could itself be significantly reduced by grouping into subproblems of size $G$ ), then each iteration of SSAPO costs approximately: + +$$ +\mathcal {O} \Big (M _ {t} \cdot G _ {t} \cdot C _ {\text {i n f e r e n c e}} + N \cdot C _ {\text {r e w a r d}} + C _ {\text {D R O}} (N, K) +... \Big). +$$ + +In most scenarios, either the distributionally robust optimization (Step 4) or the gradient-based policy update (Step 5) will be the main bottleneck, depending on solver implementation and whether grouping is employed. + +Total Complexity over $T$ Iterations Over $T$ total iterations, we multiply the above per-iteration cost by $T$ . Additionally, note that $N$ can increase each iteration if new self-annotated preferences are continuously appended to $\mathcal{D}$ . Denoting $N_{t}$ as the dataset size at iteration $t$ , the total complexity from Steps 2-5 is roughly $\sum_{t=0}^{T-1}\left[\mathcal{O}(N_{t} \cdot C_{\text {reward }}) + C_{\text {DRO}}(N_{t}, K)\right]$ , plus the self-annotation cost from Step 1. If $N$ grows in a controlled manner (for example, linearly in $t$ ), the cumulative cost can be bounded accordingly. + +# Practical Guidelines. + +- Grouping for DRO. To handle large $N$ , we recommend partitioning the data into multiple groups $G \ll N$ . The overall complexity then becomes $\mathcal{O}\big(M \cdot C_{\mathrm{DRO}}(G, K)\big)$ , where $M = N / G$ , which can be significantly faster in practice. +- Caching Log-Probabilities. The reward $R_{\theta_t}(y)$ can be computed from log-probabilities of $\pi_{\theta_t}$ and $\pi_{\mathrm{ref}}$ . Caching or reusing these values may reduce $C_{\mathrm{reward}}$ . +- Adjusting $K$ . Increasing $K$ refines the concave approximation but grows the size of the DRO problem. Hence, $K$ is a hyperparameter balancing approximation quality and computational overhead. + +Overall, the time complexity of SSAPO grows with $N$ , $K$ , and the iteration count $T$ . By employing grouping and efficient solvers, We can typically achieve robustness benefits without incurring excessive computational cost. + +# F Approximation Effects of SSAPO Algorithm Design on SGPO Guarantees + +Setup. Section 2 establishes guarantees for the Stackelberg game + +$$ +\max _ {\pi} \min _ {\alpha \in \mathcal {U} _ {\epsilon} ((\Delta R _ {\pi}) _ {\#} \dot {P})} \mathbb {E} _ {\xi \sim \alpha} [ \log \sigma (\xi) ], \tag {12} +$$ + +where $\xi = \Delta R_{\pi}(x,y_w,y_\ell)$ is the reward gap and the Wasserstein ball is taken over the push-forward of the empirical pair distribution by $\Delta R_{\pi}$ . This choice matches the semantics of preference robustness and was the condition attached to acceptance. We abbreviate $\phi (\xi)\triangleq \log \sigma (\xi)$ , note that $\phi$ is 1-Lipschitz and bounded on $[-2R_{\mathrm{max}},2R_{\mathrm{max}}]$ under the standing bounded-reward assumption from Section 2. + +Throughout this section we write $u(\xi) \coloneqq \log \sigma (\xi)$ and identify the PWL loss-approximation gap $\Delta_{\mathrm{pl}}(K)\coloneqq \sup_{\xi \in [a,b]}\left(\ell (\xi) - \tilde{\ell}_K(\xi)\right)$ with the symbol $\delta_K$ used in the main text. Likewise we upper bound the grouping error $\delta_{\mathrm{grp}}(\epsilon)$ by the directed Hausdorff distance $d_H$ between $\mathcal{U}_{\epsilon}$ and its group-restricted surrogate. + +Goal of this section. SSAPO instantiates (12) with three pragmatic approximations: (i) a $K$ -tangent convex under-approximation of the $-\log \sigma$ loss used to form a tractable DRO subproblem, (ii) a group-restricted Wasserstein ball that disallows cross-group transport for scalability, and (iii) inexact solves (tolerance $\eta$ ) of the follower and leader subproblems. We quantify how each approximation perturbs the clean guarantees of Section 2 and provide principles for choosing $(K,G,\eta)$ . + +# F.1 A bias decomposition for SSAPO + +Let $\mathcal{V}(\pi) \triangleq \min_{\alpha \in \mathcal{U}_{\epsilon}} \mathbb{E}_{\alpha}[\phi(\xi)]$ denote the ideal follower value for a fixed policy $\pi$ , and let $\widetilde{\mathcal{V}}(\pi)$ be the value obtained by SSAPO with all approximations enabled. Then + +$$ +\underbrace {\widetilde {\mathcal {V}} (\pi) - \mathcal {V} (\pi)} _ {\text {o p t i m i s m i n d u c e d b y a p p r o x i m a t i o n s}} = \underbrace {\left(\mathcal {V} _ {K} (\pi) - \mathcal {V} (\pi)\right)} _ {\text {p i e c e w i s e l o s s}} + \underbrace {\left(\mathcal {V} _ {K , G} (\pi) - \mathcal {V} _ {K} (\pi)\right)} _ {\text {g r o u p r e s t r i c t i o n}} + \underbrace {\left(\widetilde {\mathcal {V}} (\pi) - \mathcal {V} _ {K , G} (\pi)\right)} _ {\text {s o l v e t o l e r a n c e}}, \tag {13} +$$ + +where $\mathcal{V}_K$ is the value when $\phi$ is replaced by its $K$ -tangent surrogate and $\mathcal{V}_{K,G}$ additionally restricts the follower to a group-wise uncertainty set (defined below). Each term in (13) is nonnegative (the approximations weaken the adversary) and admits a simple Lipschitz control. + +# F.2 Effect of the $K$ -tangent surrogate + +Let $\widetilde{\ell} (\cdot)$ be the convex piecewise-linear under-approximation of $-\log \sigma (\cdot)$ built from K tangents, constructed on $[-B,B]$ with $B = 2R_{\mathrm{max}}$ . Define the uniform approximation error + +$$ +\delta_ {K} \triangleq \sup _ {\xi \in [ - B, B ]} \left| (- \log \sigma (\xi)) - \widetilde {\ell} (\xi) \right|. +$$ + +Because $\phi = \log \sigma = -(-\log \sigma)$ , replacing $\phi$ by $-\tilde{\ell}$ in the follower objective can only increase its minimum: + +$$ +0 \leq \mathcal {V} _ {K} (\pi) - \mathcal {V} (\pi) \leq \delta_ {K}, \tag {14} +$$ + +Thus, the $K$ -tangent surrogate yields a one-sided, additive slack $\delta_K$ in the inner value and therefore at most $\delta_K$ optimism in the leader's objective. In practice, we found $K = 6$ strikes a stable accuracy/conditioning trade-off, whereas $K = 7$ can degrade numerics without reducing $\delta_K$ appreciably (solver instability rather than approximation error). + +Design takeaway. Choose $K$ so that $\delta_K$ is below the statistical noise floor of the preference estimator on $[-2R_{\max}, 2R_{\max}]$ . Empirically, $K = 6$ is a robust default; increasing $K$ past this point can complicate the convex program and harm solver stability. + +# F.3 Effect of group-restricted Wasserstein uncertainty + +Partition the support of the empirical gap distribution $\hat{\alpha} (\pi)$ into $G$ disjoint bins $\{S_g\}_{g = 1}^G$ (uniform in $\xi$ for SSAPO). The group-restricted follower can transport mass only within each $S_{g}$ , + +$$ +\mathcal {U} _ {\epsilon} ^ {\mathrm {g r p}} \big (\hat {\alpha} (\pi) \big) \triangleq \Big \{\alpha = \sum_ {g} \alpha_ {g} \Big | \alpha_ {g} \in \mathcal {U} _ {\epsilon_ {g}} \big (\hat {\alpha} (\pi) \upharpoonright S _ {g} \big), \sum_ {g} \epsilon_ {g} \leq \epsilon \Big \} \subseteq \mathcal {U} _ {\epsilon} \big (\hat {\alpha} (\pi) \big). +$$ + +Let $\Pi (\epsilon)$ be the set of optimal global followers and define the projection gap of the restriction + +$$ +\delta_ {\mathrm {g r p}} (\epsilon) \triangleq \sup _ {\alpha^ {\star} \in \Pi (\epsilon)} \inf _ {\tilde {\alpha} \in \mathcal {U} _ {\epsilon} ^ {\mathrm {g r p}}} W _ {1} \left(\alpha^ {\star}, \tilde {\alpha}\right). +$$ + +By Kantorovich-Rubinstein duality and the 1-Lipschitzness of $\phi$ + +$$ +0 \leq \mathcal {V} _ {K, G} (\pi) - \mathcal {V} _ {K} (\pi) \leq \delta_ {\mathrm {g r p}} (\epsilon). \tag {15} +$$ + +On the real line, uniform (equal-mass) binning gives a simple control $\delta_{\mathrm{grp}}(\epsilon)\leq \bar{w}_G$ , the average within-bin width in $\xi$ ; hence the restriction error decays as $O(1 / G)$ as bins refine. This formalizes the empirical guideline that $G$ between $10^{2}$ and $10^{3}$ preserves robustness while enabling embarrassingly parallel solves. + +Design takeaway. Use $G \in [100,1000]$ (sweet spot 100-300): it keeps $\bar{w}_G$ small, retains near-global robustness, and maximizes parallel throughput. Disallowing cross-group transport weakens the adversary only by at most a bin-width in $W_1$ not by $\epsilon$ itself so the $O(\epsilon)$ regret from Section 2 is intact up to an $O(1 / G)$ term. + +# F.4 Effect of inexact solves + +Suppose each follower problem is solved to absolute tolerance $\eta$ and the leader update attains an $\eta$ -accurate step (e.g., via a proximal DPO update). Then for any $\pi$ , + +$$ +0 \leq \widetilde {\mathcal {V}} (\pi) - \mathcal {V} _ {K, G} (\pi) \leq \eta . \tag {16} +$$ + +and the cumulative leader suboptimality over $T$ rounds contributes at most $O(\eta)$ to the final value, consistent with the linear-convergence picture reported in Section 2. In SSAPO, the follower is solved offline and in parallel with modest wall clock, so $\eta$ can be driven small at negligible training-loop cost. + +# F.5 Putting the pieces together + +Combining (14), (15), and (16) in (13), the approximation-induced optimism in the follower value obeys + +$$ +0 \leq \widetilde {\mathcal {V}} (\pi) - \mathcal {V} (\pi) \leq \underbrace {\delta_ {K}} _ {K - \text {t a n g e n t}} + \underbrace {\delta_ {\text {g r p}} (\epsilon)} _ {\text {g r o u p i n g}} + \underbrace {\eta} _ {\text {t o l e r a n c e}}. \tag {17} +$$ + +Therefore, the leader who maximizes $\widetilde{\mathcal{V}} (\pi)$ enjoys the same $O(\epsilon)$ robustness as in Section 2, up to an additive $O(\delta_K + \delta_{\mathrm{grp}} + \eta)$ slack. Because $\delta_K$ and $\eta$ are user-controlled and $\delta_{\mathrm{grp}}$ shrinks with $G$ , the theory carries over with explicit, tunable error bars. + +# Practical summary. + +- Where the ball lives. All results hinge on placing $\mathcal{U}_{\epsilon}$ on the gap push-forward $(\Delta R_{\pi})_{\#} \hat{P}$ (not on $(x, y_{w}, y_{\ell})$ ). This keeps the geometry 1-D and the Lipschitz constants sharp. +- $K$ tangents. $K = 6$ gives a stable frontier; $K = 7$ may hurt due to conditioning rather than approximation quality. Tune $K$ to make $\delta_K$ sub-dominant to data noise. +- $G$ groups. Choose $G \in [100, 1000]$ (sweet spot 100–300) to make the group-restriction gap $\delta_{\mathrm{grp}}(\epsilon) \lesssim \bar{w}_G$ negligible while exploiting parallelism. +- Tolerance. Solve the offline follower to a tight $\eta$ (cutting-plane typically converges in 10–20 iterations), so training-time overhead is small and approximation slack is dominated by statistical error. +- Scope. These approximations target training-time robustness to noisy preferences; they are orthogonal to inference-time adversarial prompts and do not weaken that disclaimer. + +# G More Details of Experimental Setups + +# G.1 Detailed Experimental Setups + +We introduce more detailed experimental setups in Section 4 as follows. + +Datasets. For preference learning, we employed the UltraFeedback dataset $[18]^3$ , aligning with prior research [35, 9]. Specifically, we extracted a seed dataset comprising 2K samples (3.3% of the total 60K training samples), which included prompts, responses, and ground-truth preference labels. These ground-truth preference labels are referred to as gold labels in Table 1. The remaining training samples were then partitioned into three subsets of 8K, 20K, and 30K samples, retaining only the prompts. These subsets were utilized as the prompt sets for the 1st, 2nd, and 3rd iteration stages, respectively. + +Models. Following previous work [9], we primarily conducted our experiments using the supervised fine-tuned Mistral-7B-0.1 model [19] as the initial model $\pi_{\mathrm{init}}$ . Specifically, we used the open-sourced model4 that follows the recipe of Zephyr [39] and is fine-tuned on the instructions of UltraChat [20]. In Table 1, we also used LLaMA-3-8B5 to validate the compatibility of our method across different models. We used the generally fine-tuned models as there are no models that have been fine-tuned on the UltraChat dataset. + +**Evaluations.** Following standard practices for aligning LLMs, we employed two primary evaluation benchmarks to assess model performance. First, we used AlpacaEval 2.0 [14, 23], a benchmark designed to approximate human preferences in instruction-following tasks. This evaluation involves 805 diverse instructions sourced from multiple datasets, where responses from the model under test are compared against those generated by GPT-4 [40] to determine win rates. To address potential biases related to response length—a known factor influencing LLM preferences [15, 41], we report both the original win rate and a length-controlled (LC) win rate. The LC win rate is calculated using a regression model trained to neutralize the impact of response length, thereby focusing on the quality of the generated content [23]. + +Second, we employed MT-Bench [15] to evaluate the model's capabilities across a broader range of tasks. MT-Bench assesses a chatbot's performance in areas such as math, coding, role-playing, and writing through multi-turn interactions. Responses are scored by GPT-4, providing a comprehensive measure of the model's proficiency in key LLM functionalities. Together, these benchmarks offer a robust evaluation of how well the model aligns with human preferences and its effectiveness in real-world applications. + +Implementation Details. In the initial alignment phase, we train the model using Direct Preference Optimization (DPO) on a seed dataset of 2K samples to obtain the base model $\pi_0$ . Following this, we conduct 3 iterative stages of data expansion. In the $i$ -th iteration $(i = 1,2,3)$ , we generate preference data by independently sampling two responses for each prompt using a temperature of 0.7 and labeling them as chosen or rejected through $R(x,y)$ , resulting in a preference dataset $\{\xi_i\}_{i=1}^N$ ( $N$ is the size of the $i$ -th prompt set). Following SPA [9], we restricted the maximum token length for self-generated responses to 300 tokens. This limit corresponds to approximately 900 characters. To model the worst-case distribution program, we define a set of linear functions $\ell_k(x) = -\frac{K}{k} (x - \frac{k}{K}) - \log \left(\frac{k}{K}\right)$ for $k = 1,\dots,K$ (the family of tangents of the loss function at the $K$ -equipartition of $[0,1]$ ). We solve the associated optimization program using the Sequential Least Squares Programming (SLSQP) method. The group size $G$ is set to 100 unless otherwise specified for parallel computation of the convex program. Finally, we update the policy model by minimizing the reweighted loss to get $\pi_i$ ensuring improved alignment with the desired preferences. + +Hyper-parameters for Different LLMs. For Mistral-7B-0.1, We set learning rate $= 5 \times 10^{-7}$ and DPO hyper-parameter $\beta = 0.1$ throughout the entire preference learning process. We conduct 3 epochs for the initial DPO training and 3 iterations for SSAPO game play (leader-follower updates). + +For LLaMA-3-8B, We set learning rate $= 1 \times 10^{-6}$ and DPO hyper-parameter $\beta = 0.05$ throughout the entire preference learning process. We conduct 1 epoch for the initial DPO training and 2 iteration for SSAPO game play (leader-follower updates). + +# G.2 Construction of Seed Data + +Seed data (e.g. the initial labeled training data) has an impact on the performance of self-training algorithms. In our experiments, we explore two different methods of selecting seed data from the original training set, namely UltraFeedback dataset [18]. Ground truth labels (e.g. the preferred and dispreferred response) of the selected seed data are retained, while the labels of the remaining data are discarded. As a result, our proposed method use only a small proportion of ground truth labels. + +Random sampling. We select seed data from the entire training dataset randomly. + +Farthest point sampling. Intuitively, the distribution of the seed data should "cover" as much part of the entire training dataset as possible, such that subsequent self-training iterations won't suffer from distribution shift. With this motivation, we first leverage the SFT model to obtain an embedding vector for each prompt in the dataset, and then use farthest point sampling algorithm to select a subset of the embeddings (and corresponding preference pairs) as seed data. Specifically, we feed each prompt into the SFT model, and extract the hidden state of the last token of the last layer as the embedding. We use cosine similarity as the distance metric for embedding vectors. + +Stability across runs. Using farthest-point sampling (FPS) for seed selection reduced performance variance by approximately $40\%$ compared to random seeds in our experiments, by covering the preference space more uniformly. + +In our implementation, we find that Mistral-7B model is not sensitive to either seed data sampling method. And LLaMA-3-8B model enjoy a slightly better performance with respect to the farther point sampling. + +# G.3 Infrastructure + +For all experiments, we utilized 4 A800 GPUs. Under this computational resource, generating responses for $10\mathrm{K}$ prompts takes approximately 15 minutes, and preference judging for generated responses takes about 30 minutes. Solving worst-case distribution with Eq. 9 takes about 40 minutes. Policy update takes about 1 hour per $10\mathrm{K}$ training data. + +# H Qualitative Examples + +In this section, we present qualitative examples of the responses from the models trained under different methods: DPO (only train on initial seed data using DPO), SPA[9], and SSAPO (our proposed method). The results are presented below. These examples show that the model trained under SSAPO provides more detailed and precise responses to various questions. + +# Prompt: + +The given text must be classified as offensive or not offensive. + +You acted really crazy last night. + +# DPO response: + +I do not have any information about the specific context or situation in which the statement you acted really crazy last night was made. Therefore, I am unable to determine whether it is classified as offensive or not. Please provide more context or details about the situation so that I can provide a more accurate response. + +# SPA response: + +Based on the given text, it can be classified as "offensive" because it uses a colloquial and informal expression ("acted really crazy") to describe someone's behavior, which may convey a negative or derogatory connotation towards the person's actions. In formal or professional contexts, using such language may be considered inappropriate or unprofessional, and it could potentially offend the person involved or others who may be present. Using precise and objective language to describe behavior or actions is generally recommended in such situations to avoid any potential offense or misunderstanding. + +Therefore, using a more neutral or appropriate expression, such as "displayed unconventional behavior last night," may be a better option in this case. + +# SSAPO response: + +Based on the given text "You acted really crazy last night," it is difficult to determine whether it is classified as offensive or not offensive. The text does not contain any explicit or direct insults, profanities, or derogatory language directed towards the person addressed. The phrase "acted really crazy" could be interpreted as a subjective assessment of the person's behavior or actions during the previous night, and may not necessarily be intended as an insult or criticism. + +However, the tone or context in which the text is used could influence its perceived offensiveness. If the phrase "acted really crazy" is used sarcastically or with a condescending tone, it may be interpreted as an indirect insult or criticism aimed at the person's behavior or character. Alternatively, if the phrase is used casually or with a neutral tone, it may be interpreted as a straightforward observation or comment on the person's actions without any malicious intent. + +In summary, the given text "You acted really crazy last night" may or may not be classified as offensive, depending on the tone, context, and intended meaning behind the phrase. Without additional information or context, it cannot be definitively labeled as either offensive or not offensive. + +# Prompt: + +Verify the correctness of the given statement. + +"For all integers $j$ and $k$ , if $j$ and $k$ are odd, then $jk$ is odd." + +# DPO response: + +The statement "For all integers $j$ and $k$ , if $j$ and $k$ are odd, then $jk$ is odd" is correct. + +Proof: + +Let $j$ and $k$ be any two odd integers. + +Since $j$ and $k$ are both odd, there exist positive integers $a$ and $b$ such that: + +$$ +j = 2 a + 1 \text {a n d} k = 2 b + 1 +$$ + +where a and b are integers. + +Multiplying $j$ and $k$ , we get: + +$$ +\begin{array}{l} \mathrm {j k} = (2 \mathrm {a} + 1) (2 \mathrm {b} + 1) \\ = 2 (2 a b + a + b) + 1 \\ = 2 (2 a ^ {*} 2 b + a + b) + 1 \\ = 2 (2 a ^ {*} 2 b + 1) + (2 ^ {*} a + 2 ^ {*} b + 1) \\ \end{array} +$$ + +Since 2a and 2b are even integers, the product $2(2\mathrm{a}^{*}2\mathrm{b} + 1)$ is an odd integer. Also, $(2^{*}\mathrm{a} + 2^{*}\mathrm{b} + 1)$ is an odd integer since it is of the form $2^{*}\mathrm{c} + 1$ where $\mathbf{c}$ is the sum of two even integers (2a and 2b). Therefore, $\mathrm{jk}$ is an odd integer. + +Hence, for all integers $j$ and $k$ , if $j$ and $k$ are odd, then $jk$ is odd. + +Q.E.D. + +# SPA response: + +The statement "For all integers $j$ and $k$ , if $j$ and $k$ are odd, then $jk$ is odd" is correct. + +To prove this statement, we can use mathematical induction on the sum of the absolute values of $j$ and $k$ , denoted by $|j| + |k|$ . + +First, consider the base case where $|\mathrm{j}| + |\mathrm{k}| = 1$ . In this case, either $\mathrm{j} = 1$ and $\mathrm{k} = 0$ or $\mathrm{j} = 0$ and $\mathrm{k} = 1$ . In both cases, $\mathrm{jk} = 1$ , which is odd. + +Next, assume the statement holds for all pairs of integers $j$ and $k$ such that $|j| + |k| = n$ , where $n$ is a positive integer. Let $j$ and $k$ be any two integers such that $|j| + |k| = n + 2$ . Then, either $|j| = n + 1$ and $|k| = 1$ or vice versa. Without loss of generality, let $|j| = n + 1$ and $|k| = 1$ . Then, $j = n + 1$ and $k = 1$ or $j = -(n + 1)$ and $k = -1$ . In both cases, $jk = (n + 1)^* 1 = n + 1$ , which is odd by the induction hypothesis. + +By mathematical induction, the statement holds for all integers $j$ and $k$ if $j$ and $k$ are odd, and $jk$ is odd. Thus, the statement is correct. + +# SSAPO response: + +The given statement "For all integers $j$ and $k$ , if $j$ and $k$ are odd, then $jk$ is odd" is correct. + +To prove this statement, we need to show that for all integers $j$ and $k$ , if both $j$ and $k$ are odd, then their product $jk$ is also odd. + +Let $j$ and $k$ be two odd integers, i.e., $j = 2n + 1$ and $k = 2m + 1$ , where $n$ and $m$ are any integers. + +Then, their product $\mathrm{jk}$ can be written as: + +$$ +j k = (2 n + 1) * (2 m + 1) = 4 n m + 2 n + 2 m + 1 = 2 (2 n m + n + m) + 1 +$$ + +Since $n, m$ are integers, the product $2nm + n + m$ is also an integer. Therefore, the above expression $jk$ is equal to the sum of an even number $(2(2nm + n + m))$ and an odd number (1). + +As the sum of two odd numbers is always even, the sum $2(2\mathrm{nm} + \mathrm{n} + \mathrm{m}) + 1$ is an odd number. + +Hence, the product $\mathrm{jk} = (2\mathrm{n} + 1)^{*}(2\mathrm{m} + 1) = 2(2\mathrm{nm} + \mathrm{n} + \mathrm{m}) + 1$ is an odd integer, which proves the given statement. + +Therefore, the given statement "For all integers $j$ and $k$ , if $j$ and $k$ are odd, then $jk$ is odd" is correct. + +# Prompt: + +Find the answer that best describes the underlined SAT word. Select the correct option and explain the meaning of the underlined word. + +Despite the _cacophony, the student tried to study. + +A. Loud sounds +B. Difficult subject +C. Late hour +D. Low lighting + +# DPO response: + +The correct option is A. Loud sounds. The word cacophony means "a loud and confusing mixture of sounds." In this sentence, it describes the noise level in the environment where the student is trying to study. Therefore, the underlined word cacophony refers to the loud sounds that are making it difficult for the student to concentrate. + +# SPA response: + +A) Loud sounds (cacophony) + +Explanation: + +The word "cacophony" refers to a loud and unpleasant combination of sounds, typically characterized by a discordant or confusing mixture of noise. In the context of the sentence, "despite the _cacophony" sug- + +gests that the student is trying to study despite the overwhelming and distracting noise around them. The underlined word "cacophony" modifies the phrase "loud sounds" and provides a descriptive adjective. + +Option B) Difficult subject, C) Late hour, and D) Low lighting are not relevant to the given context as they do not refer to loud or unpleasant sounds. Option A) "loud sounds" is a synonym for "cacophony" and accurately describes the situation mentioned in the sentence. + +Therefore, the correct option for the underlined word is A) Loud sounds (cacophony). + +# SSAPO response: + +Option A: "Despite the cacophony, the student tried to study" + +The underlined word "cacophony" means a loud or chaotic combination of sounds. It describes a situation where there is an overwhelming or unpleasant mixture of noise or sounds. + +Cacophony is derived from the Greek words "kakos" meaning "bad" or "ugly" and "phone" meaning "sound". It is commonly used to describe a variety of loud or unpleasant sounds, such as traffic noise, barking dogs, screaming children, or discordant music. In the given sentence, the student is trying to study despite the distracting and disruptive noise or sounds, which creates a cacophony. The word "despite" implies that the student is facing an obstacle or hindrance, which is the excessive noise or chaos, but they still persist in their efforts to concentrate and study. + +Therefore, the meaning of the underlined word "cacophony" is a loud or chaotic combination of sounds, which creates an unpleasant or distracting environment for the student's studying. Option A is the correct answer. Option B refers to a difficult subject, Option C refers to a late hour, and Option D refers to low lighting, none of which are relevant to the given context. + +# I Broader Impact + +Our work aims to improve the data efficiency and robustness of language model alignment by formulating preference optimization as a Stackelberg game and introducing a self-annotation mechanism. By reducing reliance on large-scale human-labeled data, our framework could democratize alignment research and make it more accessible to smaller organizations, labs, and communities (those lack substantial annotation budgets). Moreover, robust optimization against noisy or adversarial preference distributions may help mitigate unintentional bias if the seed data deviate from the true user preference distribution. + +# NeurIPS Paper Checklist + +# 1. 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To maximize throughput, it is common to use short rollouts per policy update, increasing the update-to-data (UTD) ratio. However, we find that, in this setting, standard synchronous resets introduce harmful nonstationarity, skewing the learning signal and destabilizing training. We introduce staggered resets, a simple yet effective technique where environments are initialized and reset at varied points within the task horizon. This yields training batches with greater temporal diversity, reducing the nonstationarity induced by synchronized rollouts. We characterize dimensions along which RL environments can benefit significantly from staggered resets through illustrative toy environments. We then apply this technique to challenging high-dimensional robotics environments, achieving significantly higher sample efficiency, faster wall-clock convergence, and stronger final performance. Finally, this technique scales better with more parallel environments compared to naive synchronized rollouts. + +# 1 Introduction + +Reinforcement Learning (RL) has emerged as a powerful paradigm for tackling complex sequential decision-making problems, particularly in continuous control domains like robotics [8, 11, 10]. However, RL often depends on vast quantities of interaction data, a requirement that can be prohibitively expensive or slow to acquire in real-world settings. Massively parallel simulation environments, especially those accelerated on GPUs [14, 3, 20, 16], have enabled data-generation throughput on orders of magnitude greater than traditional CPU-based setups. The increase in data throughput has enabled far faster training of robotics models with successful sim2real deployments of locomotion [21, 15], state-based manipulation [5, 13], and vision-based manipulation [26, 32, 24]. + +Despite this paradigm shift in data generation capabilities, the core algorithms, particularly on-policy methods like Proximal Policy Optimization (PPO) [22], have often been adapted with only superficial changes—typically larger batch sizes and shorter per-environment rollouts $(K)$ to increase the update-to-data (UTD) ratio [21]. This strategy, while seemingly maximizing hardware utilization, overlooks a critical interaction between the data collection process and the learning algorithm's stability when the task horizon $(H)$ significantly exceeds the rollout length $(K \ll H)$ . + +We argue that stable and efficient learning in this massively parallel regime requires more than just algorithmic re-tuning; it necessitates a modification to the environment interaction protocol itself. We introduce staggered resets, a simple yet highly effective technique that breaks this harmful + +synchronicity. By initializing parallel environments at diverse effective time steps distributed across the task horizon $H$ , staggered resets ensure that each training batch contains a rich, temporally heterogeneous mix of experiences. This provides the learner with a more stationary and representative view of the overall task dynamics within every gradient update. To summarize our contributions: + +- We precisely identify, evidence, and formulate the problem of cyclical batch nonstationarity stemming from synchronous full-episode resets combined with short rollouts ( $K \ll H$ ) in massively parallel on-policy RL, explaining its detrimental impact on learning dynamics. +- We propose staggered resets, an elegant and easily implementable mechanism independent of the RL algorithm itself to ensure temporal diversity within training batches by desynchronizing the effective starting points of parallel environments across the task horizon. +- Through illustrative toy environments, we characterize the conditions under which this nonstationarity is most severe and staggered resets offer maximal benefit. +- We provide compelling empirical evidence on challenging, high-dimensional robotics tasks, demonstrating that staggered resets significantly improve sample efficiency, wall-clock convergence speed, final policy performance, and scalability with increasing parallelism compared to standard synchronous reset protocols. + +# 2 Related Work + +Massively Parallel RL The inherent sample inefficiency of many reinforcement learning algorithms has significant research into scaling and parallelization to improve training time and performance. Early approaches, such as IMPALA and others [17, 2, 18, 6], utilized multiple CPU workers to achieve parallelism and scalability. These typically relied on benchmarks [31, 4, 28] built on top of CPU-based simulators like MuJoCo [27], PyBullet [1], PhysX etc. More recently, GPU-accelerated simulators [14, 3] and other JAX-based GPU-accelerated environments [20, 9] have enabled a much greater degree of parallelism, resulting in considerable speedups for training complex policies. Recent work has sought to replace the popular choice of PPO as an RL algorithm by improving scalability [12, 25]. In contrast our work is algorithm-agnostic and addresses the challenge of handling non-stationary data in synchronous highly-parallel regimes for on-policy algorithms like PPO. + +Nonstationarity in RL Nonstationarity in the data distribution is a recognized challenge within reinforcement learning. Such nonstationarity can stem from various sources, including changes in environment dynamics, the reward function, or, as pertinent to our work, the data collection process itself. Previous research has explored related issues such as catastrophic forgetting in continual learning scenarios [7] and representation collapse arising from biased data. The "primacy bias," wherein early experiences exert a disproportionate influence on RL training, has also been documented [19, 23]. Our work specifically highlights and aims to mitigate the cyclical nonstationarity induced by the interplay of synchronous resets and short rollouts in massively parallel RL settings. + +Short Rollouts in Parallel RL The use of short rollouts ( $K \ll H$ ) in parallel RL is frequently motivated by the desire to increase update frequency (achieving a high update-to-data ratio) and maximize wall-clock training speed [30, 12]. While this strategy can be effective, underlying issues related to data distribution bias are often overlooked. Some implementations incorporate partial resets—resetting environments upon task success, failure, or termination—which can introduce some eventual desynchronization. However, this process can be slow and may prove insufficient to counteract the initial bias, particularly when $K$ is very small. Furthermore, while some simulation environment implementations include versions of the reset staggering technique we propose [21], these are typically not detailed in accompanying publications. To our knowledge, our work is the first to thoroughly investigate and analyze this method, as well as to characterize the environmental conditions under which it proves most effective. + +# 3 Methodology + +# 3.1 Preliminaries: PPO in Massively Parallel Environments + +We consider the standard RL setting where an agent interacts with an environment modeled as a Markov Decision Process (MDP), defined by $(S, A, P, r, \rho_0, \gamma, H)$ , where $S$ is the state space, $A$ is the action space, $P(s'|s, a)$ is the transition probability function, $r(s, a)$ is the reward function, $\rho_0$ is the initial state distribution, $\gamma \in [0, 1)$ is the discount factor, and $H$ is the maximum episode horizon. The goal is to learn a policy $\pi_\theta(a|s)$ , parameterized by $\theta$ , that maximizes the expected discounted cumulative return: + +$$ +J (\pi_ {\theta}) = \mathbb {E} _ {s _ {0} \sim \rho_ {0}, a _ {t} \sim \pi_ {\theta} (\cdot | s _ {t}), s _ {t + 1} \sim P (\cdot | s _ {t}, a _ {t})} \left[ \sum_ {t = 0} ^ {H - 1} \gamma^ {t} r (s _ {t}, a _ {t}) \right]. +$$ + +PPO [22] is an actor-critic method that optimizes this objective using policy gradients. It alternates between collecting trajectories using the current policy $\pi_{\theta_{\mathrm{old}}}$ and updating the policy parameters $\theta$ as well as a value function. + +In the massively parallel setting, $N$ independent copies of the environment are simulated synchronously. During the data collection phase, each of the $N$ environments executes the current policy $\pi_{\theta_{\mathrm{old}}}$ for $K$ steps (the rollout length). This generates a batch of $N\times K$ transitions $\{(s_t,a_t,r_t,s_{t + 1},d_t)\}_{i = 1..N,t = 0..K - 1}$ , where $d_{t}$ indicates if state $s_{t + 1}$ is terminal. This batch is then used to compute policy and value function gradients and perform updates for several epochs. + +# 3.2 The Problem: Cyclical Nonstationarity with Synchronous Resets + +Consider the common scenario in massively parallel RL where the PPO rollout length $K$ is chosen to be much smaller than the task horizon $H$ ( $K \ll H$ ). This strategy aims to increase the update-to-data (UTD) ratio and overall data throughput. In standard synchronous implementations, where all $N$ parallel environments are reset only after completing their full $H$ -step episode duration: + +1. Initial Synchronized Start: All $N$ environments are initially reset, commencing their episodes at effective time $t = 0$ from an initial state distribution $\rho_0$ . +2. First Rollout Batch: For the first PPO update, each environment executes the current policy for $K$ steps. The resulting batch of $N \times K$ transitions exclusively contains data from the time window $[0, K - 1]$ of the episode. +3. Subsequent Rollout Batches: For the second PPO update, environments continue, and the batch is now formed from transitions within the window $[K, 2K - 1]$ . This pattern continues: for the $j$ -th PPO update (assuming no environment has yet completed $H$ steps), the batch consists of transitions exclusively from the window $[(j - 1)K, jK - 1]$ . Environments that terminate prematurely within a $K$ -step window (e.g., due to task success/failure before $H$ steps) are typically reset to $t = 0$ and continue, slightly desynchronizing from the main $H$ -step cycle but not fundamentally altering the batch-wise temporal homogeneity. +4. Synchronized Full Reset: After approximately $m = \lceil H / K \rceil$ rollouts, all (or most) environments will have reached or exceeded $H$ elapsed steps since their last full reset. At this point, they are all synchronously reset back to an effective time $t = 0$ . +5. Cycle Repetition: Consequently, the PPO update following this full synchronous reset (e.g., the $(m + 1)$ -th update) will again process a batch exclusively composed of transitions from the $[0, K - 1]$ window, mirroring the temporal origin of the very first batch. + +The crucial issue stemming from this process is a cyclical nonstationarity of the training batches. While each environment eventually explores states across its entire $H$ -step horizon, each individual batch used for a PPO gradient update is temporally homogeneous. It contains data only from a narrow $K$ -step slice of the overall task. The learner is thus fed a data stream where the underlying state distribution within the batch shifts dramatically and predictably from one update to the next, cycling through different segments of the episode. For instance, one batch might contain only early-episode states, the next only mid-episode states, and another only late-episode states, before abruptly reverting to early-episode states after the synchronous full reset of all environments. This cyclical bias prevents the learner from accurately estimating values and advantages, likely due to catastrophic forgetting + +phenomena (empirically verified in Appendix E). This leads to poor performance, instability, and a failure to learn long-horizon behaviors, effectively negating the benefits of parallelization and high UTD ratios. Empirics we collect in Sections 4 and 5 justify this claim. + +# 3.3 Staggered Resets + +![](images/1c8ed91588e3f65d6b9d95492f325f05cd5ee4647054b85dfa475f23af8a94d1.jpg) +Figure 1: Data collection in massively parallel RL. Rows are environments, columns are time within task horizon $H$ . Colors (red, orange, etc.) mark distinct task stages. (a) Synchronous Resets (Naive): All environments start at $t = 0$ (red stage). Each rollout batch (e.g., Rollout 1: all red; Rollout 2: all orange) is temporally homogeneous. Batch content cycles through stages every $H / K$ rollouts, causing cyclical nonstationarity for the learner. (b) Staggered Resets: Environments start and hence at varied points in the task. Each rollout batch contains a mix of task stages (red, orange, green, blue, purple). This within-batch temporal diversity is maintained across rollouts, yielding a more stationary and representative data distribution. + +To counteract this detrimental cyclical nonstationarity, we introduce staggered resets. The core principle is to ensure that the $N$ parallel environments are not all synchronized to the same portion of the task at the beginning of each data collection phase. Instead, they are deliberately initialized to cover diverse effective time steps within the overall task. The mechanism is straightforward: + +Initial Staggering Strategy: Before the main training loop commences, each of the $N$ parallel environments $i$ is independently advanced (e.g., with random actions or the initial untrained policy) for a specific number of offset steps, $t_i^{\mathrm{offset}}$ . This offset $t_i^{\mathrm{offset}}$ can be sampled uniformly from $[0, H - 1]$ . + +However, to manage the frequency of reset and advance operations (which can be costly, especially in GPU environments if not batched), we discretize the staggering process by dividing the $N$ parallel environments into $N_B$ groups. Each environment $i$ is then assigned an offset $t_i^{\mathrm{offset}}$ sampled from a set of discrete intervals, e.g., $\{0\cdot S,1\cdot S,\ldots ,(N_B - 1)\cdot S\}$ , where $S$ is the stagger step size, chosen such that $N_B\cdot S\approx H$ . A standard choice is $S = K$ (the PPO rollout length), leading to offsets like $\{0\cdot K,1\cdot K,\dots ,\lfloor (H - K) / K\rfloor \cdot K\}$ . This pre-initialization phase positions environments at distinct effective starting points within the task horizon $H$ . + +The choice of $N_B$ (number of distinct offset groups) balances temporal diversity against the complexity of managing reset schedules; a heuristic of $N_B \approx H / K$ often works well, ensuring coverage of the horizon within each PPO update cycle while grouping initial advance operations. + +Synchronous Rollouts with Staggered Starts: Once initially staggered, the standard synchronous PPO data collection proceeds. All $N$ environments execute the current policy $\pi_{\theta_{\mathrm{old}}}$ for $K$ steps. However, because each environment $i$ effectively began its current episode segment at a different offset $t_i^{\mathrm{offset}}$ relative to the global task timeline, the aggregated batch of $N\times K$ transitions now contains experiences from a much wider and more representative range of the task horizon. For instance, if environments are staggered across multiples of $K$ , the collected batch will naturally include data segments corresponding to time windows like $[0,K - 1],[K,2K - 1],\ldots ,[H - K,H - 1]$ , rather than being overwhelmingly biased towards a single segment. This concept is illustrated in Figure 1(b). + +# Handling Resets During Training: + +- End-of-Horizon Resets (End of $H$ ): After an environment accumulates $H$ effective steps, it is reset to $s_0 \sim \rho_0$ . These operations typically occur for entire groups of environments simultaneously, enabling efficient batching. +- Partial Resets (Early Termination): If an environment $j$ terminates early (e.g., success/failure before $H$ lifetime steps), it is flagged. To optimize wall-clock time by maximizing batched operations, it waits for the next scheduled "reset gate" (when a group undergoes a reset by reaching $H$ elapsed lifetime steps). At this gate, environment $j$ is reset to $s_0 \sim \rho_0$ alongside others, starting its new episode from effective time $t = 0$ . This strategy minimizes the number of batched environment reset calls to align with $N_B$ , thereby reducing wall-time costs. In practice, this approach effectively maintains the benefits of staggered data collection and its associated temporal diversity without significant degradation. + +Policy Updates: The aggregated batch $\mathcal{B}$ now contains the temporally diverse transitions due to staggered starts/reset management. $\mathcal{B}$ is then used for policy and value function updates in PPO. + +By ensuring temporal diversity within each batch, staggered resets provide the learner with a data distribution that better approximates the true state visitation distribution $\rho_{\pi}^{(0:H - 1)}$ encountered over complete episodes. This stabilization is crucial for allowing the use of short rollouts $K$ (and thus high update-to-data ratios) without succumbing to the cyclical nonstationarity bias that plagues naive synchronous reset schemes. The improved data quality promotes more stable learning, better value estimates for states across the entire task, and ultimately, enhanced performance on long-horizon tasks. Our empirical results in Section 5 and Section 4 corroborate these benefits. + +# 4 Illustrative Experiments on Toy Environments + +To dissect the impact of staggered resets and pinpoint conditions where they offer maximal benefit, we conducted experiments in configurable 1-dimensional toy environments. These allow controlled variation of factors that often interact in real-world tasks. Understanding these interactions in simplified settings provides insights into the efficacy of staggered resets in more complex, high-dimensional scenarios that often exhibit similar characteristics. + +# 4.1 Toy Environment Design + +Our primary toy environment is a 1D chain of $B$ discrete levels. An episode spans $H$ steps, with each level covering $L = H / B$ steps. The agent's state is its current level index $b_{t} = \lfloor t / L\rfloor$ . In each level $b$ , the agent selects an action $a_{t}$ from a discrete set $A$ . A fixed target action $a_{b}^{*}\in A$ is assigned to each level. Correct actions $(a_{t} = a_{b_{t}}^{*})$ yield a reward of $+0.5$ , incorrect actions $-0.5$ . We manipulate three key environment dynamics to explore when staggered resets are most impactful: + +Rollout Length Ratio $(K / H)$ : We vary the environment task horizon $H$ , exploring scenarios from $K \approx H$ (short effective skill horizon, less temporal bias) to $K \ll H$ (long effective skill horizon where cyclical nonstationarity with naive resets is hypothesized to be severe). + +Reset Dynamics: Upon episode termination, resets can range from deterministic ( $\lambda_R = 0$ , always to level $b_0 = 0$ ) to stochastic ( $\lambda_R > 0$ , where $b_0 \sim \mathrm{Poisson}(\lambda_R)$ centered around $\lambda_R$ ). This tests if inherent start-state randomness mitigates synchronous reset issues versus needing deliberate staggering. The term "Reset Homogeneity" in our plots (Figure 2b) refers to $2 - \lambda_R$ , where higher values mean more deterministic (homogeneous) resets to $b_0 = 0$ . + +Sequential Skill Gating: Progression from level $b$ to $b + 1$ depends on a mastery threshold $k_{\mathrm{mastery}}$ (number of correct actions $a_b^*$ ) and an unconditional progression probability $p_{\mathrm{prog}}$ . The agent advances if mastery is met or a random check $(p_{\mathrm{prog}})$ succeeds. Varying $p_{\mathrm{prog}} \in [0,1]$ (where $p_{\mathrm{prog}} = 0$ implies hard gating requiring $k_{\mathrm{mastery}} > 0$ ) creates tasks from simple linear progression to those requiring sequential skill acquisition. + +# 4.2 Results on Toy Environments + +Experiments on these toy environments (Figure 2) characterize how different environmental factors influence the efficacy of staggered resets compared to naive synchronous rollouts, particularly highlighting the impact of data non-stationarity. + +![](images/c7bb7ad54fbd330c4c9067ee0b3a6bfa39c179968d92bedd9944783e19e921ca.jpg) +(a) + +![](images/d3fbc5a6bf91d475059bf7c43e55aa6a8806e95e5531d1cddcdd83b1b33b4cda.jpg) +(b) +Figure 2: PPO with Non-Staggered (blue) vs. Staggered (red) resets on toy environments (mean $\pm 1$ std dev). Staggered resets show robust performance as (a) horizon $H$ increases, (b) reset homogeneity $(2 - \lambda_{R})$ increases, and (c) progression probability $(p_{\mathrm{prog}})$ varies, unlike non-staggered PPO which degrades especially with longer horizons, more deterministic resets, and easier skill gates $(p_{\mathrm{prog}} > 0)$ . + +![](images/faabafcc2bd0687344ab18a1fcf83cd9fb85050571c97d59e938234330a7a28f.jpg) +(c) + +Horizon Length (Figure 2a): With fixed short rollouts $(K)$ , each batch of $N \times K$ transitions is temporally homogeneous, covering only a $K$ -step slice of the $H$ -step task. As $H$ increases, the number of distinct temporal slices in the data collection cycle $(H / K)$ also grows. This means the periodicity of revisiting any specific task segment (e.g., the initial states) becomes longer. The learner struggles to consolidate information and may forget what it learned about earlier segments by the time the data cycle returns to them, especially with many intermediate, distinct batch types (see Figure 12 for a visualization of this forgetting process). Staggered resets, by providing temporally diverse data within each batch, maintain high performance irrespective of $H$ , effectively facilitating learning across the entire task horizon even when $K \ll H$ . + +Reset Stochasticity/Homogeneity (Figure 2b): This experiment, conducted with minimal skill gating $(p_{\mathrm{prog}} = 1.0)$ , varies the determinism of reset locations. As resets become more concentrated at the episode's start (higher "Reset Homogeneity," i.e., $\lambda_R \approx 0$ ), the performance of non-staggered PPO deteriorates. While some inherent randomness in reset locations ( $\lambda_R > 0$ ) can offer minor desynchronization and marginal performance improvements for the naive method, deliberate staggering consistently yields substantially better results. This indicates that relying on incidental environmental randomness is insufficient to overcome the core non-stationarity induced by synchronous rollouts. + +Skill Gating Dynamics (Figure 2c): This experiment tests how $(p_{\mathrm{prog}})$ (ease of progressing without mastery) affects data collection strategies. Staggered PPO performs robustly across all $(p_{\mathrm{prog}})$ levels, its comprehensive sampling ensuring sufficient data for learning all skills. Non-staggered PPO's behavior is more complex: it performs best with stringent skill gates $((p_{\mathrm{prog}} = 0))$ , where the environment imposes a curriculum forcing mastery of early skills to reach later states. As $(p_{\mathrm{prog}})$ increases (gates soften), non-staggered PPO's performance degrades. With relaxed gates, random advancement occurs, but the established data collection bias towards initial states prevents learning a coherent policy for the full task, as sporadic later-state exposure without mastery is insufficient. Consequently, the advantage of staggered resets widens as $(p_{\mathrm{prog}})$ increases. This highlights that the cyclical non-stationarity of naive rollouts is a key limiter, especially when environmental structures (like hard gates) that aid exploration are weak. + +The environmental factors varied in our toy experiments—horizon length, reset stochasticity, and skill gating—mirror complexities encountered in high-dimensional robotics tasks, such as those in ManiSkill3. For instance, long manipulation sequences often mean the task horizon $H$ is much larger than practical PPO rollout lengths $K$ , exacerbating the long periodicity issue seen in Figure 2a. "Skill gates" in toy tasks are analogous to critical bottleneck sub-tasks in robotics, like achieving a stable grasp in StackCube-v1 before attempting to lift and place. If an agent can bypass such a bottleneck (akin to high $p_{\mathrm{prog}}$ ) but its experience is fed in temporally homogeneous batches, learning a robust overall strategy becomes difficult. Similarly, the degree of stochasticity in initial object poses or slight variations in robot starting conditions in ManiSkill3 tasks relates to the reset stochasticity (Figure 2b). Highly deterministic setups in robotics can make the cyclical batch problem more pronounced. + +# 5 Experiments on High-Dimensional Robotics Tasks + +# 5.1 Experimental Setup + +Our evaluation suite includes several challenging robotics tasks from ManiSkill3 [26], a GPU-accelerated robotics framework based on SAPIEN [29]. We test on StackCube-v1, a manipulation task requiring an agent to stack one cube onto another; PushT, where a T-shaped block must be pushed to a target pose; TwoRobotPushCube, where two robots work together to move a cube to a goal; Unitree Transport Box, a humanoid task where a box must be transported to a table; and Anymal Reach C, where an Anymal C robot must move to a specific goal location. We also test on MS-HumanoidWalk, a humanoid walking control task. These environments involve high-dimensional continuous state and action spaces, providing a suitable testbed for evaluating the effectiveness of staggered resets. + +# 5.2 State Visitation Dynamics in High-Dimensional Robotics + +![](images/41cc4adea89db5c05128c0ef9032993542b7c83f15aab9569cae7d0441ef8632.jpg) +Figure 3: State visitation KDEs in StackCube-v1 over five rollouts. (a) Long Rollout ( $K = 100$ ): stable, broad coverage. (b) Naive Short Rollout ( $K = 25$ ): cyclical non-stationarity, narrow/erratic coverage. (c) Staggered Short Rollout ( $K = 25$ ): stable, diverse coverage, emulating (a) despite short trajectories. + +A central premise of this work is that the common practice of using short rollouts ( $K \ll H$ ) with naive synchronous resets in massively parallel RL leads to a problematic data generation process. This process is characterized by a temporally unstable and skewed state visitation distribution, which we hypothesize significantly impedes learning. Our proposed staggered resets intervention aims to rectify this by enabling short rollouts to yield data distributions more similar to those from longer, more informative trajectories. + +To empirically assess this hypothesis, we visualize and analyze state visitation patterns on the challenging StackCube-v1 robotics manipulation task. Figure 3 presents Kernel Density Estimates (KDEs) of visited states in a given rollout buffer, projected onto their first two principal components derived from the aggregate data of all rollouts across an entire training run. Each row corresponds to a different data collection strategy, and columns depict the evolution of the state distribution over five consecutive PPO data collection rollouts. + +The Long Rollout PPO baseline ( $K = 100$ ), shown in Figure 3a, serves as an empirical ideal, exhibiting broad and stable state coverage across all five rollouts. This pattern signifies a rich and temporally consistent data stream, achieved here at the expense of a lower UTD ratio. + +Figure 3b, representing the Naive Short Rollout PPO strategy ( $K = 25$ ), illustrates the nonstationarity in consecutive rollout buffers induced by synchronous resets. Initial rollouts show a state distribution highly concentrated near the environment's reset distribution. As training progresses, the distribu + +tion expands, but its shape and locus shift dramatically between rollouts. After the environments synchronously reset (following Rollout 4), the state distribution at Rollout 5 closely mirrors that of Rollout 1. This cyclical nonstationarity means the learner receives a constantly changing and biased view of the transition space, hindering its ability to form accurate value estimates and a robust policy. + +Figure 3c depicts the state visitation dynamics for our Staggered Short Rollout PPO strategy (also $K = 25$ ). The distributions achieved by staggered resets qualitatively mirror the desirable breadth and consistency of the long rollout baseline (a). This empirical evidence strongly supports our central claim: staggered resets effectively counteract the cyclical non-stationarity induced by synchronous resets in short-rollout regimes. + +# 5.3 Performance on High-Dimensional Robotics Tasks + +To validate the effectiveness and generality of staggered resets, we conduct experiments on two distinct sets of high-dimensional robotics benchmarks, using two different state-of-the-art on-policy algorithms: Proximal Policy Optimization (PPO) and Split and Aggregate Policy Gradients (SAPG). + +![](images/bbd0fbf0f27e268bb3e3f6865d9eb574b2eabf11a6c339262bfa45e441faab23.jpg) + +![](images/7ca78d3c53f34822df1a5a6b18ae656bb9b44eccb1900ee7af5c99490f4481d7.jpg) + +![](images/19a3ce2aad17ca5d71603ca25adfd390143cd1647883feedf78c78f04c84e635.jpg) + +![](images/dcc5b17ee5dca9cca1bc3a11d5b9ea4dce92b839bbdb4fdd8b6c62d3f511da0c.jpg) + +![](images/4de641d4deb5b9f31d7d5577580475532e25a806add9c913d26929cb34c9fce1.jpg) + +![](images/05eb97d8de7c2c143bb07d1d49d4c007cb53c5a8bd7fde4f4f71dd9e0e4c4575.jpg) + +![](images/a3129e57f7bf5944a0d43ff3b0d9ab236d182464589aa3964530d4717eefa614.jpg) +Figure 4: Staggered Resets (Ours, red) consistently outperform Synchronous Resets (blue) across different on-policy algorithms and task suites. Plots show the average evaluation metric (success rate or reward) vs. environment steps. Shaded areas show the standard deviation over 10 seeds. (Top & Middle Rows) On PPO with diverse ManiSkill3 tasks (StackCube, PushT, etc.), staggered resets consistently improve learning speed, final performance, and stability. Performance is comparable only on the locomotion task MS-HumanoidWalk, where natural desynchronization reduces the severity of the problem. (Bottom Row) To demonstrate algorithm-agnosticism, we evaluate on SAPG with challenging AllegroKuka manipulation tasks. Staggered resets again yield substantial gains in sample efficiency and final reward, confirming the generality of our approach. + +![](images/d5d0a4a0a1634e0f364ec700625191cc8d5e957dbc870c7884c6969c5f4c60bd.jpg) + +# 5.3.1 Performance Improvements with PPO on ManiSkill3 Tasks + +As shown in the top and middle rows of Figure 4, staggered resets achieve substantially faster convergence, higher final success rates, and greater stability on a wide range of PPO-based manipulation tasks. For instance, in PushT and StackCube-v1, staggering leads to significantly higher and + +more stable final performance. Similarly, in AnymalC-Reach and TwoRobotPickCube, learning is markedly quicker and reaches a better asymptotic success rate. Crucially, with an increased evaluation of 10 seeds, the variance across runs is also substantially lower with staggered resets, confirming that our method improves overall training stability. + +Interestingly, on the locomotion task MS-HumanoidWalk, both methods perform comparably. As discussed in our toy experiments, locomotion environments often feature shorter effective skill horizons and highly stochastic reset behaviors (e.g., the agent falling at unpredictable times). These factors induce a degree of natural desynchronization, making the cyclical batch nonstationarity less severe and thereby reducing the marginal benefit of explicit staggering. This aligns with prior work [21] that employed a form of staggering in locomotion contexts, where our analysis suggests the benefits are less pronounced compared to more complex, longer-horizon manipulation tasks. + +# 5.3.2 Validating Algorithm-Agnosticism with SAPG + +A central claim of our work is that the benefits of staggered resets are algorithm-agnostic, stemming from the data distribution rather than the specifics of the learning update. To substantiate this, we conduct additional experiments with a more recent on-policy algorithm, SAPG [25], on the challenging A1legroKuka dexterous manipulation tasks. + +The results, shown in the bottom row of Figure 4, demonstrate that staggered resets significantly improve the performance of SAPG. Both AllegroKukaThrow and AllegroKukaRegrasping show dramatic improvements in sample efficiency and final asymptotic reward. Notably, these environments feature frequent early resets (e.g., on task success or object drop) and significant domain randomization, which can naturally induce some desynchronization and reduce the cyclicity problem. Despite this, staggered resets still yield a pronounced performance improvement, underscoring the robustness and general applicability of our method for improving data quality in massively parallel on-policy RL. + +# 5.4 Scaling with Parallel Environments and Overcoming Performance Saturation + +A key challenge in massively parallel RL is effectively utilizing increased parallelism. SAPG [25], shows that beyond a threshold, additional environments may not reduce wall-clock time and may even hurt performance due to higher gradient variance or communication overhead. We address this issue and show that staggered resets improve convergence speed across a wide range of $N$ , yielding a higher marginal utility of increased environment parallelism. + +Figure 5 shows the wall-clock time required to reach a fixed convergence threshold (70% success rate) as a function of the number of parallel environments $N$ . Results are shown for two representative tasks: StackCube-v1 (Figure 5a) and Unitree G1 Transport Box (Figure 5b). + +For Naive PPO (blue curves), increasing parallel environments $(N)$ initially speeds up wall-clock convergence. However, this benefit saturates or even reverses at larger $N$ (e.g., beyond $N \approx 1024$ for StackCube-v1, Figure 5a; degrading sharply at $N = 6144$ for Unittree G1 Transport Box, Figure 5b). Hence, while data throughput rises, learning efficiency diminishes due to temporally homogeneous from many synchronously reset environments, offsetting parallelism gains. Conversely, Staggered PPO (red curves) demonstrates superior scaling, with wall-clock convergence time continuing to decrease as $N$ increases across all tested values, even beyond 6000 environments. + +![](images/9c670a12a16338f2d28d5ae444e36ef0c8465cf34c31f1a96bfcb72b398ccb0e.jpg) +(a) + +![](images/60bc1dd5fd7c78f384dd74185f55d68bb7b8d5db65a726694ede89b4976a5b5a.jpg) +(b) +Figure 5: Wall-clock time to convergence versus number of parallel environments $(N)$ for (a) StackCube-v1 and (b) Unitree G1 Transport Box + +# 6 Discussion + +Our investigation reveals a significant challenge in modern massively parallel on-policy RL. The cyclical nonstationarity introduced by synchronous environment resets when coupled with short + +rollouts $(K\ll H)$ inadvertently biases the learning signal by repeatedly oversampling states from the initial segments of episodes. This bias can detrimentally affect learning stability, convergence speed, and ultimate policy quality, leading to issues like value function divergence and catastrophic forgetting, as we quantitatively demonstrate in Appendix E. + +Staggered resets directly address this nonstationarity without any changes to the core learning algorithm. By deliberately initializing parallel environments at varied effective time steps within the task horizon, we ensure that each training batch encompasses a temporally diverse set of experiences. This creates a more stationary and representative data distribution for the learner. The state visitation KDEs (Figure 3) offer compelling visual evidence: staggered resets with short rollouts yield broad and stable state coverage, closely emulating the desirable properties of much longer rollouts, whereas naive short rollouts suffer from erratic, cycling state distributions. + +Crucially, our work also provides empirical justification for focusing on the short-rollout ( $K \ll H$ ) regime. As shown in our appendix experiments (Table 1), this regime is not merely a heuristic but is often optimal for wall-clock efficiency. On challenging tasks like StackCube-v1, short rollouts ( $K = 8 - 16$ ) converge $2 - 3 \times$ faster than long rollouts while achieving the same or better final performance, reinforcing that addressing issues within this specific setting is of high practical importance. + +The illustrative toy experiments (Section 4) further characterize the conditions where staggered resets are most impactful. Specifically, tasks with longer horizons relative to the rollout length, more deterministic (homogeneous) reset states, and weaker intrinsic task curricula (e.g., where agents are not strictly forced to master early skills to progress) show pronounced benefits from the explicit temporal diversification that staggering provides. + +A key practical advantage of staggered resets is their ability to enhance the scalability of on-policy RL in massively parallel settings (Section 5.4, Figure 5). While naive PPO often encounters diminishing returns or even performance degradation as the number of parallel environments grows—likely due to increasingly redundant data—staggered resets facilitate continued improvements in wall-clock convergence time. This indicates a more effective utilization of parallel compute resources, as the increased data volume is also more diverse and informative. + +In conclusion, staggered resets provide a robust, easily implementable, and computationally inexpensive method to significantly enhance the performance and scalability of on-policy RL. Its benefits are algorithm-agnostic, improving performance for both PPO and more recent methods like SAPG (Section 5.3.2). By directly addressing the issue of cyclical data nonstationarity, this technique allows for more stable value estimation, faster convergence, and better final policies, paving the way for more effective learning in complex, long-horizon tasks. + +# 7 Acknowledgements + +Stone Tao is supported in part by the NSF Graduate Research Fellowship Program grant under grant No. DGE2038238. + +# References + +[1] Erwin Coumans and Yunfei Bai. 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In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2020. +[30] Jie Xu, Viktor Makoviychuk, Yashraj Narang, Fabio Ramos, Wojciech Matusik, Animesh Garg, and Miles Macklin. Accelerated policy learning with parallel differentiable simulation. In International Conference on Learning Representations, 2021. +[31] Tianhe Yu, Deirdre Quillen, Zhanpeng He, Ryan Julian, Karol Hausman, Chelsea Finn, and Sergey Levine. Meta-world: A benchmark and evaluation for multi-task and meta reinforcement learning. In Conference on Robot Learning (CoRL), 2019. URL https://arxiv.org/abs/1910.10897. + +[32] Kevin Zakka, Baruch Tabanpour, Qiayuan Liao, Mustafa Haiderbhai, Samuel Holt, Jing Yuan Luo, Arthur Allshire, Erik Frey, Koushil Sreenath, Lueder A. Kahrs, Carlo Sferrazza, Yuval Tassa, and Pieter Abbeel. Mujoco playground: An open-source framework forgpu-accelerated robot learning and sim-to-real transfer., 2025. URL https://github.com/google-deepmind/mujoco Playground. + +# A Motivating Experiments + +# A.1 Empirical Motivation for the $K \ll H$ Regime + +To motivate the massively parallel short-rollout regime, we sweep $K \in \{1, 2, 4, 8, 16, 32, 64, 100\}$ on StackCube-v1 ( $H = 100$ ). Each configuration is trained for 100M steps across three seeds with identical hyperparameters. Short rollouts ( $K = 8 - 16$ ) reach the same or better final reward as long rollouts while converging $2 - 3 \times$ faster in wall-clock time, confirming the practical relevance of $K \ll H$ in massively parallel on-policy RL. + +Table 1: Performance sweep over rollout length $K$ on StackCube-v1. Best results $(K = 8 - 16)$ in bold. + +
(a) Final reward after 100M environment steps.
Metric1248163264100
Staggered Naive0.38±0.060.47±0.010.70±0.000.74±0.000.72±0.030.75±0.020.71±0.030.70±0.02
0.27±0.170.36±0.120.47±0.020.62±0.010.72±0.020.79±0.030.71±0.030.70±0.02
(b) Environment steps (millions) to reach >75% success rate.
Metric1248163264100
Staggered NaiveDNCDNCDNC16.2±0.217.2±0.123.2±1.450.1±5.649.6±9.9
DNCDNCDNC35.8±4.518.5±2.124.1±1.950.5±6.249.6±9.9
(c) Wall-clock time (minutes) to reach >75% success rate.
Metric1248163264100
Staggered NaiveDNCDNCDNC16.6±3.215.3±1.022.5±2.340.0±8.742.6±8.7
DNCDNCDNC26.2±2.131.5±13.630.0±11.941.1±4.543.2±9.9
+ +Observation. The fastest convergence is achieved with short rollouts ( $K = 8 - 16$ ) on an NVIDIA RTX-4090 GPU, matching Rudin et al. (2021) and Singla et al. (2024). Training with $K = 1 - 4$ fails to converge (DNC). Final reward saturates beyond $K = 8$ , reinforcing that modern massively parallel on-policy systems naturally operate in the $K \ll H$ regime. + +# B Implementation Details + +# B.1 Staggered Reset Implementation Details + +The staggered reset mechanism aims to distribute the effective starting timesteps of the $N$ parallel environments across the task horizon $H$ . This was achieved by dividing environments into $N_B = \lceil H / K\rceil$ groups, with each group $j$ starting its first "effective" episode step after an initial offset of $j\cdot K$ simulation steps (typically performed with random actions or the initial policy). This ensures that each PPO batch contains data from various segments of the task horizon. + +# B.2 Details on Toy Environments + +The toy environments described in Section 4 were designed to isolate and study the effects of data nonstationarity under different environmental conditions. See Table 2 for the hyperparameters chosen for PPO in the toy environment. We describe more concretely the ablation and environment implementation details below. + +# B.2.1 Environment Dynamics + +The environment is a 1-dimensional chain of $B$ discrete levels or "blocks". An episode lasts for a maximum of $H$ time steps. Each level $b \in \{0, \dots, B - 1\}$ covers $L = H / B$ steps. The agent's state $s_t$ is its current level index $b_t = \lfloor \text{Elapsed\_steps}_t / L \rfloor$ . At each time step $t$ , the agent, being in + +level $b_{t}$ , chooses an action $a_{t}$ from a discrete set of $A_{c}$ categories (e.g., $A_{c} = 20$ ). Each level $b$ has a pre-assigned target action $a_{b}^{*} \in \{0, \dots, A_{c} - 1\}$ . The reward function is: + +$$ +r (s _ {t}, a _ {t}) = \left\{ \begin{array}{l l} + 0. 5 & \text {i f} a _ {t} = a _ {b _ {t}} ^ {*} \\ - 0. 5 & \text {i f} a _ {t} \neq a _ {b _ {t}} ^ {*} \end{array} \right. +$$ + +The episode terminates if elapsed_steps $t \geq H$ . + +# B.2.2 Further Details on Ablations on Toy Environments + +The following parameters were varied to create the different experimental conditions shown in Figure 2: + +- Horizon Length ( $H$ vs. $K$ ): (Figure 2a) The task horizon $H$ (max_steps in code) was varied across values [50, 100, 200, 300, 400, 500]. The PPO rollout length $K$ (num_steps in PPO loop, i.e., buffer size per environment before update) was kept fixed at $K = 5$ . The block length $L$ was also fixed at 5. For this experiment, skill gating was moderate $(p_{\mathrm{prog}} = 0.5, k_{\mathrm{mastery}} = 3)$ and reset was deterministic $(\lambda_R = 0)$ . +- Reset Stochasticity/Homogeneity $(\lambda_R)$ : (Figure 2b) Upon episode termination, the reset mechanism was varied. The parameter $\lambda_R$ (reset_stochasticity_lambda in code) controls the mean of a Poisson distribution from which the starting block $b_0$ is sampled, i.e., $b_0 \sim \mathrm{Poisson}(\lambda_R)$ , clamped to $[0, B - 1]$ . $\lambda_R = 0$ corresponds to a deterministic reset to $b_0 = 0$ . The "Reset Homogeneity" axis in the plot is $2.0 - \lambda_R$ for visualization purposes (higher values = more deterministic starts at $b_0 = 0$ ). $\lambda_R$ was varied in $[0.0, 0.1, \dots, 1.0]$ . For this experiment, $H = 50$ , $L = 5$ , $K = 5$ , $p_{\mathrm{prog}} = 1.0$ (easy progression), $k_{\mathrm{mastery}} = 3$ . +- Skill Gating Dynamics ( $p_{\mathrm{prog}}$ ): (Figure 2c) Progression from the current block $b$ to $b + 1$ (when enough steps within block $b$ have nominally passed to enter $b + 1$ ) occurs if either: + +1. The agent has achieved "mastery" in block $b$ , defined as making at least $k_{\mathrm{mastery}}$ correct actions $a_b^*$ within block $b$ during the current episode. ( $k_{\mathrm{mastery}} = 3$ was used). +2. A random chance $p_{\mathrm{prog}}$ for unconditional progression succeeds. + +The probability $p_{\mathrm{prog}}$ (progression\_prob in code) was varied in [0.0, 0.1, ..., 1.0]. $p_{\mathrm{prog}} = 0$ means hard gating requiring mastery. For this experiment, $H = 200$ , $L = 5$ , $K = 5$ , $\lambda_R = 0$ . + +# B.3 Implementation Details and Hyperparameters for ManiSkill Experiments + +This section details the Proximal Policy Optimization (PPO) configuration for experiments on ManiSkill robotics tasks (Section 5), utilizing ManiSkill3 [26] for GPU-accelerated simulation. The exact PPO implementation is based on the one provided by ManiSkill3 baselines, which is based on LeanRL and CleanRL. Table 3 provides a comprehensive list of hyperparameters. + +Table 2: PPO and Environment Hyperparameters for Toy Environment Experiments. Shaded rows categorize parameters. Values listed are defaults; specific sweeps varied $H$ , $p_{\mathrm{prog}}$ , or $\lambda_R$ as detailed in text and figures. + +
HyperparameterValue
PPO Algorithm Core Settings
Learning Rate3 × 10-4
Discount Factor (γ)0.99
GAE Lambda (λ)0.95
PPO Rollout Length (K)5 steps per environment
Number of Parallel Environments (N)512
Total Training Updates150
Update Epochs4
Number of minibatches4 (Minibatch size: (512 × 5)/4 = 640)
PPO Clipping Coefficient (ε)0.2
Value Function Loss Coefficient0.5
Entropy Bonus Coefficient0.01
Max Gradient Norm0.5
Network Architecture (Actor & Critic MLP)
InputCurrent block index (integer state)
Embedding LayerInput block index to 64-dim embedding
Hidden Layers4
Units per Hidden Layer256
Activation FunctionReLU
Policy OutputCategorical distribution over actions
Value OutputScalar state value
Optimization
OptimizerAdam
Toy Environment Base Parameters (Defaults for Sweeps)
Episode Horizon (H)Varied (e.g., 50, 100, 200, 375 for specific experiments)
Block Length (L)5 steps
Number of Action Categories (Ac)20
Reward for Correct Action+0.5
Reward for Incorrect Action-0.5
Success DefinitionAgent is in the final block at episode end
Skill Gating: Progression Prob. (pprog)Varied (0.0 to 1.0)
Skill Gating: Mastery Threshold (kmastery)3 correct actions
Reset Stochasticity (λR)Varied (Poisson mean for start block, 0.0 for deterministic)
Staggered Resets Mechanism (When Enabled)
Number of Stagger Blocks (NB)[H/K] = [Episode Horizon/5]
Stagger Step Size (S)K = 5
+ +Table 3: PPO Hyperparameters for ManiSkill Experiments. Common settings are listed first, followed by per-environment variations where applicable. Shaded rows categorize parameters. + +
HyperparameterValue / Per-Environment Specification
PPO Algorithm Core Settings
Learning Rate3 × 10-4
Update Epochs4
Number of minibatches32
PPO Clipping Coefficient (ε)0.2
Value Function Loss Coefficient0.5
Entropy Bonus Coefficient0.005
Max Gradient Norm0.5
Advantage NormalizationTrue (Per minibatch)
Target KL for Early Stopping0.1
Network Architecture (Actor & Critic MLP)
Hidden Layers3
Units per Hidden Layer[256, 256, 256]
Activation FunctionTanh
Weight InitializationOrthogonal
Policy OutputGaussian mean, learnable state-independent log std. dev.
Optimization
OptimizerAdam
Adam Epsilon1 × 10-5
Learning Rate AnnealingFalse
Environment Interaction & Data Collection (Common)
Total Training Timesteps2 × 108
Partial Resets (Training)True
Evaluation Environments128
Evaluation Partial ResetsFalse
Observation NormalizationVia environment wrappers / running mean & std
Reward ScalingEnvironment-dependent (aim for std approx. 1)
Staggered Resets Mechanism (When Enabled)
Staggering ModeUniform distribution of start times
Number of Stagger Blocks (NB)[H/K] (Task Horizon / Rollout Length)
Stagger Step Size (S)K (Rollout Length)
Per-Environment Specific Hyperparameters
ParameterValues for: StackCube / PushT / AnymalC / HumanoidWalk / TwoRobotCube / UnitreeBox
Rollout Length (K)8/8/16/64/16/32
Task Horizon (H)100/100/200/1000/100/500
Discount Factor (γ)0.8/0.99/0.99/0.97/0.8/0.8
GAE Lambda (λ)0.9/0.9/0.95/0.9/0.9/0.9
Num. Parallel Env. (N)4096/4096/512/4096/2048/1024
+ +# B.4 Implementation Details and Hyperparameters for SAPG Experiments + +This section details the configuration for the experiments on the AllegroKuka dexterous manipulation tasks (Section 5.3), which use the Split and Aggregate Policy Gradients (SAPG) algorithm. The experiments were conducted using the IsaacGym simulator. To ensure a fair comparison and isolate the effect of staggered resets, we adopt the official hyperparameters reported in the original SAPG paper [25]. Table 4 provides a comprehensive list of these hyperparameters. Our method introduces only the staggered reset mechanism on top of this baseline. + +Table 4: SAPG Hyperparameters for AllegroKuka Experiments, based on [25]. Shaded rows categorize parameters. + +
HyperparameterValue
SAPG Algorithm Core Settings
Learning Rate1 × 10-4
Update Epochs2
Mini-batch SizeNum. Parallel Env. × 4
Clipping Coefficient (ε)0.1
Value Function Loss Coefficient4.0
Entropy Bonus Coefficient0 (default, see [25] for tuning)
Max Gradient Norm1.0
KL Threshold for LR Update0.016
Bounds Loss Coefficient0.0001
Network Architecture (Recurrent Actor & Critic)
Observation PreprocessorMLP with hidden layers [768, 512, 256]
Policy/Value Core1-layer LSTM with 768 hidden units
Activation FunctionELU
Policy OutputGaussian mean, learnable state-independent std. dev.
Optimization
OptimizerAdam
Environment Interaction & Data Collection
Number of Parallel Env. (N)24576
Rollout Length (K)16
Discount Factor (γ)0.99
GAE Lambda (λ)0.95
LSTM Sequence Length16
Staggered Resets Mechanism (When Enabled)
Staggering ModeUniform distribution of start times
Number of Stagger Blocks (NB)[H/K] (Task Horizon / Rollout Length)
Stagger Step Size (S)K (Rollout Length)
+ +# C Additional Results on Toy Environments + +To further illustrate the impact of synchronous versus staggered resets on the data distribution and learning progress, we visualize the training dynamics in one of the toy environments (specifically, $H = 200$ , $L = 5$ , $K = 5$ , $p_{\mathrm{prog}} = 0.5$ , $\lambda_R = 0$ ). Figure 6 shows the training average accuracy and the mean distribution of environments across different blocks (states) over the course of training updates. For these experiments, we define accuracy as the rolling percentage of correct one-hot action guesses from the PPO agent. + +![](images/060eea7a8bfdffa4e81dcab0368da16cb5f340f757975fdb110798614904d065.jpg) +(a) + +![](images/2c277ba65bb537e5741f0b1a494b4923932b95a5f3fa7dec4855b25148549e96.jpg) + +![](images/54036740c0437dc19199931c157930ec25d517a58d43af83360926a3909ea708.jpg) +(c) + +![](images/3493bb4b6907eb860c6f8868f396ca416186c4c5c74f5ece390d4adaa4fc5d35.jpg) +(b) +(d) +Figure 6: Comparison of training dynamics in a toy environment with (a, c) naive synchronous resets versus (b, d) staggered resets. (a) & (b): Training average accuracy over 150 PPO updates. With naive resets (a), accuracy is unstable and struggles to converge. With staggered resets (b), accuracy rises quickly and stabilizes at a high level. (c) & (d): Heatmaps showing the mean number of environments occupying each block (y-axis) at each PPO update index (x-axis). (c) With naive synchronous resets, environments progress through blocks in tight, synchronized waves. After approximately 40 updates (when $H / K = 200 / 5 = 40$ rollouts complete an episode), all environments abruptly reset to block 0, leading to a cyclical pattern where training batches are temporally homogeneous (all early-episode, then all mid-episode, etc.). (d) With staggered resets, the distribution of environments across blocks is far more uniform at any given update index. This indicates that each training batch contains a mix of experiences from different stages of the episode, leading to a more stationary data distribution for the learner. + +Subplots (c) and (d) in Figure 6 are heatmaps where the x-axis represents the PPO training update index, the y-axis represents the block index (state) within the toy environment's episode, and the color intensity indicates the mean number of parallel environments present in that block at that specific training update. + +Naive Synchronous Resets (Figure 6c) The heatmap for naive synchronous resets clearly shows distinct diagonal bands. Each band signifies that the cohort of parallel environments is synchronously progressing through the episode's blocks. A crucial observation is the abrupt termination of these bands followed by an immediate restart from block 0 (the bottom of the y-axis). This occurs approximately every 40 updates, corresponding to the episode horizon ( $H = 200$ ) divided by the + +rollout length ( $K = 5$ ). This pattern visually confirms the cyclical nonstationarity discussed in Section 3.2: at any given update, the training batch is predominantly composed of states from a narrow segment of the episode, and this segment predictably cycles. For instance, for updates 1-5, data is from blocks near 0-4; for updates 35-40, data is from blocks near 35-39; then at update 41, data abruptly shifts back to blocks 0-4. + +Staggered Resets (Figure 6d) In contrast, the heatmap for staggered resets shows a much more diffuse and uniform pattern. While environments still progress through blocks (indicated by the general upward-right trend), there is no global, abrupt reset of all environments. At any given PPO update index, environments are distributed across a wide range of blocks. This means that each training batch collected under staggered resets contains a temporally diverse mix of experiences—some from early parts of episodes, some from middle, and some from later parts. This significantly reduces the cyclical nonstationarity of the data fed to the PPO algorithm. + +Impact on Learning (Figure 6a and 6b) The consequences of these different state visitation dynamics are evident in the training accuracy plots. With naive synchronous resets (Figure 6a), the learning curve for average accuracy is highly erratic, exhibiting periodic dips and slow overall improvement, struggling to consistently achieve high accuracy. This instability likely results from the PPO learner trying to adapt to the rapidly shifting data distributions. Conversely, with staggered resets (Figure 6b), the average accuracy rises smoothly and rapidly, quickly converging to a high and stable level. This demonstrates that providing the learner with more temporally diverse and stationary batches facilitates more effective and stable learning. + +# C.1 State Visitation Distribution Analysis + +We provide further detail on the state visitation dynamics in our toy environments by visualizing the mean environment block distribution over training updates (similar to Figure 6c and 6d) while systematically varying key environmental parameters. For all visualizations in this section, the PPO rollout length $K = 5$ , number of parallel environments $N = 512$ , total training updates shown are 150, environment block length $L = 5$ , number of action categories $A_{c} = 20$ , and mastery threshold $k_{\mathrm{mastery}} = 3$ , unless specified otherwise. Base PPO hyperparameters are detailed in Appendix B.2. Each figure's top row shows results for naive synchronous resets, and the bottom row shows results for staggered resets. + +# C.2 Impact of Progression Probability $(p_{\mathrm{prog}})$ + +Figure 7 illustrates how the probability of unconditional progression, $p_{\mathrm{prog}}$ , affects state visitation. The experiment uses a fixed horizon $H = 200$ and deterministic resets $(\lambda_R = 0)$ . + +# Non-Staggered Resets (Top Row): + +- $p_{\mathrm{prog}} = 0.0$ (Left): With hard skill gating, environments get stuck at early blocks if they fail to achieve mastery. The heatmap shows most environments concentrated at very low block indices, with only very few managing to progress. The cyclical reset pattern is still evident for those that do run the full course or get reset. +- $p_{\mathrm{prog}} = 0.5$ (Center): Partial random progression allows more environments to reach later blocks, but the density remains higher at earlier stages due to the gating. The cyclical nature of resets is clear. +- $p_{\mathrm{prog}} = 1.0$ (Right): Environments progress freely through blocks irrespective of mastery. This results in the most pronounced cyclical bands, as all environments synchronously march through the episode blocks and reset together. + +Staggered Resets (Bottom Row): Across all values of $p_{\mathrm{prog}}$ , staggered resets maintain a significantly more uniform distribution of environments across different blocks at any given training update. While a lower $p_{\mathrm{prog}}$ means individual environments might take longer or struggle more to traverse all blocks within their own episode lifetime, the staggering ensures that the batch fed to the learner still contains diverse experiences. The overall "texture" of the heatmap becomes denser as $p_{\mathrm{prog}}$ increases, indicating that more environments are successfully exploring the full range of blocks over time, but the crucial within-batch temporal diversity is preserved by the staggered mechanism itself. + +![](images/db22475fbf4015afee5634d4f1cdb2b451ac8a09a18dc725dce4c6961fe0208a.jpg) + +![](images/f8a592b424e2e0fcf6fd1e60cbac64224ca00d6842b5693900a55d0c1a6f85f1.jpg) + +![](images/a99e0713c88de7e664c45c8f46ec0959f98d208f00afb1a6e54b5e7d59c06096.jpg) + +![](images/c8db229c610469cc397ed0d107d2ba6e04a4559778c10597547c666bb72eb4fa.jpg) +Figure 7: Mean environment block distribution over training for varying progression probabilities $(p_{\mathrm{prog}})$ . Top Row (Non-Staggered): From left to right, $p_{\mathrm{prog}} = 0.0, 0.5, 1.0$ . Bottom Row (Staggered): From left to right, $p_{\mathrm{prog}} = 0.0, 0.5, 1.0$ . With non-staggered resets, low $p_{\mathrm{prog}}$ (hard skill gating) leads to environments bunching at early blocks, with sparse exploration of later stages. As $p_{\mathrm{prog}}$ increases, environments progress more freely, but the strong cyclical reset pattern remains. With staggered resets, coverage is more uniform across updates regardless of $p_{\mathrm{prog}}$ , though higher $p_{\mathrm{prog}}$ allows environments to explore the full range of blocks more rapidly within their individual (staggered) episode timelines. + +![](images/9c2f5ac4366b8de6049af89ce0f76965afea9de4c53183850278a7a607fdddf7.jpg) + +![](images/e23b1f2baa4de01ed5c785e666f7fe269ab1d0d5a2f52ed44db2ea5bd36d6836.jpg) + +# C.3 Impact of Episode Horizon Length $(H)$ + +Figure 8 shows the effect of varying the episode horizon length $H$ . For these plots, progression probability $p_{\mathrm{prog}} = 0.5$ and reset stochasticity $\lambda_R = 0$ are fixed. The number of blocks $B = H / L$ changes with $H$ . + +# Non-Staggered Resets (Top Row): + +- $H = 10$ (Left): With a very short horizon, the full episode cycle $H / K = 10 / 5 = 2$ updates. The cyclical pattern appears as very rapid, almost vertical bands. While cyclical, all parts of this very short episode are revisited extremely frequently. The y-axis shows only 2 blocks (10/5). +- $H = 150$ (Center): The cycle period is $150 / 5 = 30$ updates. Clear diagonal bands show synchronous progression and reset. The y-axis spans 30 blocks. +- $H = 375$ (Right): The cycle period becomes $375 / 5 = 75$ updates. The bands are elongated, indicating a longer time between revisiting the same episode phase. The y-axis spans 75 blocks. This long periodicity is hypothesized to be particularly detrimental. + +Staggered Resets (Bottom Row): Staggered resets consistently provide diverse state visitations within each batch, regardless of the horizon length $H$ . The heatmaps show that environments are distributed across the available blocks (which scale with $H$ ) at each update. This ensures the learner receives a more stationary data stream, which is particularly beneficial for longer horizons where the non-staggered approach suffers from infrequent revisitation of early-episode states. + +# C.4 Impact of Reset Stochasticity $(\lambda_R)$ + +Figure 9 visualizes the influence of reset stochasticity, $\lambda_R$ , which controls the mean of a Poisson distribution for sampling the starting block upon reset. Here, $H = 200$ and $p_{\mathrm{prog}} = 0.5$ . + +![](images/31b2c880692fa4ed7d95cc00dc21e6bebb13a34a389c668d70eff40107774e7b.jpg) + +![](images/403a5700a4a87afd5f0c93eb3a53ec39fce937faf636c4ee74eb3382042d8d7c.jpg) + +![](images/0aeabc7e54a92de1ec02d8beef180a2916fafe7d9cc40ac60d602d74d8aec816.jpg) + +![](images/54850dfbcfd4cca6965321724d4170411f308b5954798b45ae59899377a1fb73.jpg) +Figure 8: Mean environment block distribution over training for varying episode horizon lengths $(H)$ . Top Row (Non-Staggered): From left to right, $H = 10, 150, 375$ . (Note: y-axis Block Index scales with $H$ ). Bottom Row (Staggered): From left to right, $H = 10, 150, 375$ . For non-staggered resets, shorter horizons $(H = 10)$ lead to very rapid cycles $(H / K = 10 / 5 = 2$ updates per cycle). As $H$ increases, the period of these cycles becomes longer, potentially exacerbating learning instability. Staggered resets maintain uniform coverage irrespective of $H$ . + +![](images/afe2dc70c8173fa783f5b13f0f1f708e9273d793a06461d264cc0dbbe805d073.jpg) + +![](images/38f3dfae7eb9617f6145d216ac1da969402bc11a1df7677f6107672e8ce55b3e.jpg) + +![](images/a8f3daf063eeb98179195d8a2d7c64ecd9ff203961156f1a9883f0fd71ba77e7.jpg) + +![](images/1dad7cf7d4f9bade19c72a41f4a334f62678998f3ff315bd3ae733e002d749fc.jpg) + +![](images/04335345199acab5d7f004a574dc6711dbff6b93451352291fb154930637f9e4.jpg) + +![](images/5268feb35585309651382a27b67584897ea23380d25b8beef95009cca9359532.jpg) +Figure 9: Mean environment block distribution over training for varying reset stochasticity $(\lambda_R)$ . Top Row (Non-Staggered): From left to right, $\lambda_R = 0.0, 1.0, 2.0$ . Bottom Row (Staggered): From left to right, $\lambda_R = 0.0, 1.0, 2.0$ . For non-staggered resets, increasing $\lambda_R$ slightly "fuzzes" the start of each cycle after a mass reset, but the overall cyclical progression remains. Staggered resets maintain uniform coverage; $\lambda_R$ primarily influences the initial state distribution within each environment's individual staggered timeline. + +![](images/ef023e620c5f5c5f54bd2f55a437117930b21d89c3ac81fb42132cf5434b6947.jpg) + +![](images/b0e06a4db4b821716e6de4b932a2ddbdcfa0b87f7cac59216f0d88c6ae4a0187.jpg) + +# Non-Staggered Resets (Top Row): + +- $\lambda_{R} = 0.0$ (Left): Deterministic reset to block 0. This results in the sharpest cyclical bands, as all environments restart precisely from the same initial state after completing an episode. + +- $\lambda_{R} = 1.0$ (Center) and $\lambda_{R} = 2.0$ (Right): Stochastic resets mean environments restart from a distribution of initial blocks centered around block 0 (due to Poisson sampling with mean $\lambda_{R}$ , then clamped). This causes the beginning of each major cycle (after most environments have run for $H$ steps) to appear slightly "fuzzier" or more spread out near block 0. However, once this initial phase passes, the environments that didn't terminate early tend to re-synchronize in their progression, and the cyclical bands through the bulk of the episode remain prominent. The inherent reset stochasticity offers only a minor and temporary desynchronization. + +Staggered Resets (Bottom Row): With staggered resets, the distribution of environments across blocks remains largely uniform over updates, irrespective of $\lambda_{R}$ . The primary mechanism for achieving temporal diversity in batches is the explicit staggering of episode start times. The reset stochasticity parameter $\lambda_{R}$ further diversifies the exact starting block for an environment when its individual (staggered) episode concludes and it resets, but it does not fundamentally change the broad, uniform coverage ensured by the staggering mechanism itself. Staggered resets are effective even with deterministic environment resets ( $\lambda_{R} = 0.0$ ). + +These visualizations across different parameter sweeps consistently highlight the ability of staggered resets to create more temporally diverse and stationary training batches compared to naive synchronous resets, providing a more stable learning signal for the on-policy RL agent. + +# D Wall-Time Results and Additional Analysis + +# D.1 Wall-Time Analysis for Staggering Granularity $(N_B)$ + +A critical aspect of staggered resets is balancing the desired temporal diversity of training batches with the computational overhead associated with environment reset operations. While an ideal scenario might involve resetting each environment at a unique timestep (effectively $N_B \approx N$ , or even finer if $N_B \approx H / \mathrm{sim\_dt}$ ), frequent, unbatched 'env.reset()' calls can be costly, especially in GPU-accelerated simulations. The parameter $N_B$ , representing the number of distinct stagger groups or "blocks," controls this trade-off. A smaller $N_B$ means fewer, larger groups of environments are reset/advanced synchronously, reducing reset call frequency but potentially coarsening the approximation of a truly staggered (temporally diverse) data distribution. + +We empirically investigated this trade-off on the StackCube-v1 task ( $H = 100$ , with PPO rollout $K = 8$ , thus $H / K \approx 12.5$ ) by measuring the wall-clock time to reach a $70\%$ success rate while varying $N_B$ . Figure 10 illustrates the results. + +The findings, shown in Figure 10, indicate: + +- Few Blocks ( $N_B \approx 1$ ): When $N_B = 1$ , all environments are effectively synchronized, resembling the naive reset scheme. This results in the slowest wall-clock convergence due to the detrimental effects of cyclical batch nonstationarity. +- Moderate Blocks ( $N_B \approx H / K$ ): As $N_B$ increases, wall-clock time to convergence rapidly decreases. The optimal performance is typically observed when $N_B$ is in the vicinity of $H / K$ . For StackCube-v1 with $K = 8$ , this optimal is around $N_B \approx 10 - 13$ . This granularity provides sufficient temporal diversity in training batches to stabilize learning and accelerate convergence. +- Many Blocks ( $N_B \gg H / K$ ): Further increasing $N_B$ beyond $H / K$ leads to marginal improvements or even a slight degradation in wall-clock convergence time. While providing finer-grained staggering, the overhead of managing and executing resets for many small, distinct groups may start to outweigh the benefits from any additional (and likely minimal) gains in data diversity. + +This analysis demonstrates that a judicious choice of $N_B$ , typically around $H / K$ , allows us to effectively approximate the benefits of a continuously staggered reset distribution (where each environment could theoretically start at any unique step within $H$ ) while maintaining wall-clock efficiency. This approach strikes a practical balance, achieving most of the sample efficiency gains from temporal diversity without incurring the potentially significant computational costs of excessively + +![](images/683738bb14f73cf774c7ebac8871b44f3b9c5469affeba3de9bc1e51b410b8ca.jpg) +Figure 10: Wall-clock time to convergence (70% success on StackCube-v1) as a function of the number of stagger blocks $(N_B)$ . Performance, measured by faster convergence (lower wall-clock time), improves significantly as $N_B$ increases from 1 (naive synchronous resets) towards $N_B \approx H / K$ (here, $H = 100$ , $K = 8$ , so $H / K \approx 12.5$ ). Beyond this point, further increasing $N_B$ yields diminishing returns or even a slight increase in wall-clock time, likely due to the overhead of managing more numerous, smaller reset groups outweighing marginal gains in temporal diversity. This demonstrates that a moderate number of stagger blocks (e.g., $N_B \approx H / K$ ) effectively balances temporal diversity benefits with wall-time efficiency, approximating a continuously staggered reset distribution without incurring prohibitive reset costs. + +frequent or unbatched reset operations. The default setting in our experiments for staggering (see Appendix B.3) is chosen based on this principle, typically defaulting to $\lfloor H / K\rfloor$ + +# D.2 Wall-Clock Training Time for ManiSkill Environments + +Beyond improvements in sample efficiency (i.e., performance per environment step), staggered resets also demonstrate significant advantages in terms of wall-clock training time. Figure 11 presents a comparison of evaluation success rates against wall-clock time for several challenging ManiSkill tasks. + +![](images/b8e2fb16f96549346df8d95106af860bc5cbca76037292eba81f017902f5fa15.jpg) +Staggered Resets (Ours) + +![](images/586105b7f89d18a6389133ece27c5d5a3091c3384946cd06661578f913750d54.jpg) +Synchronous Resets + +![](images/4010413192072bdb1992af9b0a3887f05f7819c7ea7436f8372261d406574b7a.jpg) +Figure 11: Comparison of evaluation success rates versus wall-clock training time for Staggered Resets (Ours, red) and Synchronous Resets (blue) on three ManiSkill tasks: AnymalC-Reach, UnitreeG1TransportBox, and StackCube. The x-axis represents wall-clock time (units may vary per plot, e.g., minutes or hours, but relative comparison is key). Staggered resets consistently achieve higher success rates faster, or reach comparable success rates in significantly less wall-clock time than synchronous resets. This highlights that the benefits of improved data quality and learning stability from staggered resets translate directly into more efficient use of compute time. + +As illustrated in Figure 11, policies trained with staggered resets (red curves) consistently achieve target success rates in substantially less wall-clock time compared to those trained with naive + +synchronous resets (blue curves). For instance, in AnymalC-Reach, staggered resets reach over $80\%$ success much earlier than synchronous resets begin to show significant learning. Similarly, for UnitreeG1TransportBox and StackCube, the learning curves for staggered resets are considerably steeper when plotted against wall time, indicating faster convergence to high-performing policies. + +This empirical evidence supports the conclusion that the improved sample efficiency and learning stability afforded by staggered resets directly translate into reduced overall training time, making the technique not only more data-efficient but also more computationally efficient in practice for achieving desired performance levels on complex robotics tasks. The overhead of managing staggered resets is outweighed by the gains from more effective learning per unit of time. + +# E Quantitative Analysis of Cyclical Nonstationarity Effects + +In the main text (Sections 1, 3.2, and 4.2), we claim that the cyclical nonstationarity induced by synchronous resets can destabilize value function learning, induce policy oscillations, and cause catastrophic forgetting. This appendix provides direct quantitative evidence for these claims using the toy environment described in Section 4.1. We compare two settings: one using our proposed staggered resets and one using naive synchronous resets, with a moderate skill gate (progression\_prob=0.1) to highlight the differences. + +# E.0.1 Metric Computation Details + +To substantiate our claims, we introduce three quantitative metrics computed during the toy environment training runs. The following descriptions detail their implementation. + +Value Function Estimation Error. To measure the stability of the critic network, we track the value function loss at each PPO update. This is computed as the Mean Squared Error (MSE) between the critic's value predictions for the states in the rollout buffer, $V(s_{t})$ , and the empirical Monte-Carlo returns calculated via Generalized Advantage Estimation (GAE). Large spikes in this value indicate that the critic is struggling to predict returns, often due to a sudden shift in the underlying data distribution. + +Policy Oscillations. To quantify the magnitude of policy changes between updates, we measure the approximate KL divergence between the policy before an update $(\pi_{\mathrm{old}})$ and the policy after the update $(\pi_{\mathrm{new}})$ . This is calculated as the mean squared difference between the log-probabilities of the actions taken during the rollout: $0.5 \cdot \mathbb{E}[(\log \pi_{\mathrm{new}}(a|s) - \log \pi_{\mathrm{old}}(a|s))^2]$ . Consistently high or spiky KL values suggest the agent is making large, unstable updates rather than smooth, incremental improvements. + +Catastrophic Forgetting. To directly measure the agent's ability to retain knowledge of early-episode skills while training on late-episode data, we construct a "forgetting matrix." The process is as follows: + +1. At every training update $t$ , we record the agent's average accuracy on each discrete block $b$ of the task, forming an accuracy matrix $A(t,b)$ . +2. For each block, we compute the best-so-far accuracy achieved up to the current update: $A_{\mathrm{best}}(t,b) = \max_{t' \leq t} A(t',b)$ . +3. We define forgetting at update $t$ for block $b$ as the difference between the peak performance and the current performance: Forgetting $(t, b) = A_{\mathrm{best}}(t, b) - A(t, b)$ . + +A high value in this matrix (visualized as a bright color in the plots) signifies that the agent previously had high accuracy on a specific block but has since "forgotten" how to solve it, resulting in a performance drop. A dark matrix indicates stable knowledge retention. + +# E.0.2 Analysis of Results + +Figures 12 and 13 provide compelling evidence for the detrimental effects of cyclical nonstationarity. The analysis below directly compares the corresponding plots from each figure. + +![](images/f9c3baa7cfcfabdf06c507c2c312cecb03e704d17b603d3eeb2f42a08582cffe.jpg) + +![](images/88267c6b54bda2134b9daf2bdd1be067a010fcece01c94cef10ce8af5d2b5964.jpg) +Figure 12: Training is stable with Staggered Resets. The left panel shows value function error (top) and policy KL divergence (bottom). Both metrics remain low and stable, with the value error (MSE) never exceeding 2.5. The right panel shows the forgetting matrix, which is almost completely dark, indicating that the agent successfully retains knowledge of all task stages throughout training. + +![](images/1cd88a070de163b29f8a1678ef784ea32001ff06c03939c949a22991d590cacf.jpg) + +![](images/91bf7e4bb76dd53609e5a7c2dc648df334f7a16bde5ac5c5e43c004b611f5223.jpg) + +![](images/e97cda0c20ffef09c1b198e7aa77030688ed3028d009faca9fc9529b950dee24.jpg) +Figure 13: Training is unstable with Synchronous (Naive) Resets. The left panel shows catastrophic spikes in value function error (top, MSE > 80) that coincide with mass reset events (updates 40, 80, 120), as well as more erratic policy updates (bottom). The forgetting matrix on the right shows bright, cyclical patterns, indicating severe forgetting of early-stage skills as the training data shifts to later stages. + +![](images/c6c1eec22a3561a981ba13e930bfb3e3e5ee1b615622e297a70ebb683c4d2845.jpg) + +Value Function Instability. The value function error, visualized in the top-left plot of each figure, demonstrates the most dramatic effect. With synchronous resets (Figure 13), the value function experiences catastrophic prediction error spikes. The Mean Squared Error (MSE) surges to over 80 at regular intervals. These spikes align perfectly with the mass environment reset cycle (every 40 updates, as $H / K = 200 / 5 = 40$ ), which occurs when the data distribution abruptly shifts from late-episode states back to initial states. This provides clear evidence that the cyclical data stream makes it impossible for the critic to maintain a consistent estimate of state values. In stark contrast, the agent trained with staggered resets (Figure 12) maintains a stable and low value estimation error, never exceeding an MSE of 2.5. + +Policy Oscillations. The policy oscillation metric, measured by the KL divergence between consecutive policy updates (visualized in the bottom-left plot of each figure), also shows a clear difference. While both methods exhibit some variance, the policy under synchronous resets (Figure 13) becomes significantly more erratic after the first mass reset (update 40), exhibiting larger and more frequent spikes in KL divergence. This indicates that the learning algorithm is making large, corrective updates in response to the sudden distributional shift, rather than the smooth, incremental learning facilitated by the stationary data from staggered resets. + +Catastrophic Forgetting. The forgetting matrices, shown in the right panel of each figure, visualize the agent's inability to retain knowledge. For the agent with synchronous resets (Figure 13), the matrix reveals a stark sawtooth pattern of forgetting. As the training data progresses to later blocks in an episode cycle, the agent's accuracy on early blocks (e.g., blocks 0-10) is almost completely lost (bright yellow/orange). When the mass reset occurs, the agent must then relearn these initial skills. Conversely, the forgetting matrix for the staggered resets agent (Figure 12) is almost entirely black, signifying near-perfect knowledge retention across all task stages throughout training. Quantitatively, accuracy with naive resets drops by an average of 0.21 relative to its peak, while with staggered resets, the average drop is only 0.015—a 14-fold improvement in knowledge retention. This directly validates our central hypothesis: the cyclical data stream forces the agent into a destructive learn-forget cycle, which staggered resets completely mitigates. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract and introduction clearly state the problem of cyclical batch nonstationarity, the proposed solution of staggered resets, and the claimed benefits (improved sample efficiency, wall-clock convergence, final performance, scalability), which are then substantiated in the experimental sections (Sections 4 and 5). + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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+version https://git-lfs.github.com/spec/v1 +oid sha256:af39bfebb78c26a875e95a5b77b74ddbb0e330b9abf481793caff5ee2e73bfa0 +size 7017869 diff --git a/startrailconcentricringsequenceparallelismforefficientnearinfinitecontexttransformermodeltraining/full.md b/startrailconcentricringsequenceparallelismforefficientnearinfinitecontexttransformermodeltraining/full.md new file mode 100644 index 0000000000000000000000000000000000000000..09cbcfcb7b61437bdf75eca177367f3abe481470 --- /dev/null +++ b/startrailconcentricringsequenceparallelismforefficientnearinfinitecontexttransformermodeltraining/full.md @@ -0,0 +1,661 @@ +# StarTrail: Concentric Ring Sequence Parallelism for Efficient Near-Infinite-Context Transformer Model Training + +# Ziming Liu* + +National University of Singapore liuziming@comp.nus.edu.sg + +# Shenggan Cheng + +National University of Singapore shenggan@comp.nus.edu.sg + +# Kai Wang + +National University of Singapore kai.wang@comp.nus.edu.sg + +# James Demmel + +University of California, Berkeley demmel@berkeley.edu + +# Shaoyu Wang* + +University of Southern California wangshao@usc.edu + +# Zhongkai Zhao + +National University of Singapore zhongkai.zhao@u.nus.edu + +# Xuanlei Zhao + +National University of Singapore xuanlei@comp.nus.edu.sg + +# Yang You + +National University of Singapore youy@comp.nus.edu.sg + +# Abstract + +Training Transformer models on long sequences in a distributed setting poses significant challenges in terms of efficiency and scalability. Current methods are either constrained by the number of attention heads or excessive communication overheads. To address this problem, we propose StarTrail, a multi-dimensional concentric distributed training system for long sequences, fostering an efficient communication paradigm and providing additional tuning flexibility for communication arrangements. Specifically, StarTrail introduces an extra parallel dimension and divides the peer-to-peer communication into sub-rings to substantially reduce communication volume and avoid bandwidth bottlenecks. Through comprehensive experiments across diverse hardware environments and on both Natural Language Processing (NLP) and Computer Vision (CV) tasks, we demonstrate that our approach significantly surpasses state-of-the-art methods that support Long sequence lengths, achieving performance improvements of up to $77.12\%$ on GPT-style models and up to $114.33\%$ on DiT (Diffusion Transformer) models without affecting the computation results. + +# 1 Introduction + +Over the past decade, Transformer[38] models have made remarkable strides in diverse fields, including computer vision (CV) and natural language processing (NLP). As the technology has evolved, the ability to efficiently process long sequences with Transformer has emerged as a pivotal challenge. For instance, in text summarization, the ability to handle extensive sequences is vital, as the content to be summarized can range from lengthy chapters to entire books [17, 3]. Similarly, chat-based applications, such as ChatGPT [1], require the capacity to process extensive dialogue + +histories to ensure conversational consistency. There are also applications in other fields like video generation[5, 30] and protein structure prediction[15, 7]. + +The long context in the above scenarios has introduced several challenges for model training and inference: 1) Efficiency and Adaptability. The challenge of efficiency predominantly lies in handling long sequences that require quadratic computations during attention, and in addressing the large amount of communication during distributed processing. 2) Memory. Besides the major obstacle of storing the model weight and optimizer states, the activation has also exceeded the capacity of a single GPU and risen as a new memory challenge due to the extreme sequence length. 3) Scalability. Current Transformer models usually require thousands of GPUs for pre-training, even with datasets of regular lengths. For longer sequences, ensuring an acceptable scaling speedup rate with both the sequence length and the number of GPUs increasing is even more critical to reducing time and economic costs. + +Traditional parallelisms such as Data Parallelism[12, 37, 22, 41], Tensor Parallelism[37, 39, 40], and Pipeline Parallelism[13, 10, 23, 25] distribute the model, input batch, and the optimizer states, but can not directly address the large memory requirement of extremely long sequences as the sequence length dimension remains unchanged. To break through this obstacle, Sequence Parallelism has been introduced, splitting the input on the sequence length dimension. Mainstream Sequence Parallelism schemes can generally be classified into two categories, and are usually combined [11] to complement each other's drawbacks. Methods like DeepSpeed Ulysses[14], which are based on all-to-all communication, offer efficiency but require the splitting of attention heads. Consequently, these methods are limited in scalability and can not be scaled to more devices than the number of attention heads. On the other hand, peer-to-peer communication methods[24, 20], such as Ring Attention[24], do allow for near-infinite context lengths; however, they require the transmission of complete keys and values across all GPUs, leading to significantly high communication loads. In sum + +![](images/fc4b65e0f31dcc74f46965551be7a17e0d3edfabcbc57cf48b9fdace9570ed2a.jpg) + +![](images/8d3c0b8d120a0e974ad39f88fcff7e025fe9a586889fd7b0f2e11f582db40c43.jpg) +Figure 1: StarTrail-2 and StarTrail-4 theoretically save around $50\%$ and $75\%$ of total P2P communication volume for various sequence length and achieves up to 2x speedup in end-to-end training + +mary, there remains a deficiency in communication-efficient methods that are capable of supporting near-infinite context lengths. In this paper, we focus on solving the communication inefficiency of ring-style sequence parallelism. + +To solve these challenges, we introduce StarTrail, a novel near-infinite-context Transformer training system with concentric multi-ring sequence parallelism that incorporates an additional parallel dimension into the existing ring-style communication. Specifically, instead of including all GPUs in a single parallel group as done in Ring Attention [24], StarTrail groups the GPUs into teams and divides the peer-to-peer communication within these teams. This approach fosters an efficient communication paradigm and provides extra tuning flexibility for communication arrangements. With very little additional memory cost, StarTrail parallelism significantly reduces the peer-to-peer communication volume, as shown in Figure 1. Compared to previous works, StarTrail is not limited in supported sequence length by attention heads like DeepSpeed Ulysses[14] and Megatron Sequence Parallelism[18], and also shows better communication efficiency and scalability than Ring Attention[24]. We perform experiments on mainstream Transformer models, including GPT-style[33] and DiT-style[31], conducting performance and scaling tests across various computing clusters. Experiment results indicate that our StarTrail system outperforms Ring Attention by up to $77.12\%$ on the GPT model and up to $114.33\%$ on the DiT model, showcasing its efficiency and scalability. + +# 2 Background and Related Works + +# 2.1 Long Sequence Training and Sequence Parallelism + +The key mechanism behind these Transformer-based models is attention[38], which captures the text feature by calculating the attention score between every two single tokens. However, the sequence length can reach hundreds of thousands, when dealing with multi-round chatting, or high-resolution long video generation. It then becomes necessary to distribute the sequence across multiple GPUs. This distribution helps to reduce both the memory and computation demands on any single device. This strategy is also known as sequence parallelism. Presently, Sequence parallelism can be divided into two main categories: attention-head-sharding-based and peer-to-peer-communication-based. The former involves distributing the attention heads of multi-head attention across multiple GPUs, whereas the latter resembles a distributed version of FlashAttention, relying on peer-to-peer communication to transfer keys, values, and intermediate statistics. + +# 2.1.1 Ring-peer-to-peer-communication-based + +The primary method in peer-to-peer-communication-based strategies is Ring Attention[24], which is also the main baseline of this work. Introduced in 2023, Ring Attention[24] innovatively partitions the sequence dimension and utilizes a ring-style peer-to-peer (P2P) communication pattern to transfer Keys and Values across all GPUs. Each GPU receives the key and value matrices from the preceding rank, updates the local attention score, and then forwards them to the next rank, as is shown in Figure 2. This method employs an online-softmax and updates attention scores incrementally, allowing the computation of attention scores without retaining the full sequence length. Thus, it potentially supports infinite context, provided sufficient + +![](images/0d744cbaa5a3c5f4b89f35b117556fc715316ffd11b9c9c8730d79655f09828d.jpg) +Figure 2: An example of Ring Attention Computation on 16 GPUs in two nodes. The Communication is largely limited by the inter-node bottleneck. + +computing resources are available. However, the requirement for the same number of rounds of P2P communication as the number of GPUs renders this approach less efficient in environments with high-latency communication. + +Figure 3: An overview of the StarTrail Training System +![](images/d1b2cb98e40d1aca1f5ed9aa8048b267d8b1749d056d7a88a7bcdc97b04fa1f8.jpg) +(a) StarTrail-2 reduces $60\%$ P2P communication volume on 16 GPUs compared with ring attention + +![](images/ac3d2f9b4cdae7e74d4e3c06104efb72704a093479e871e6632b828033de08c9.jpg) +(b) StarTrail divides one ring into four concentric sub-rings, with every two connected with collective communication. + +# 2.1.2 Attention-Head-Sharding-Based + +There are two representative methods, DeepSpeed-Ulysses and Megatron Sequence Parallelism in this category. DeepSpeed-Ulysses[14] transitions from sequence parallelism to a method akin to tensor parallelism with two all-to-all communication. It divides the query, key, and value matrices across the attention heads, thereby preserving the original attention computation structure. Megatron Sequence Parallelism[18] focuses on minimizing memory usage and reducing the necessity for + +activation recomputation rather than efficiency. The two methods both rely on the number of attention heads to split the sequence, thus limited in scalability, especially when employing techniques like grouped-query attention (GQA) [2] or multi-query attention (MQA) [36]. As these two sequence parallelism methods are orthogonal to the ring-based method, they are usually combined with ring attention to enable longer sequences. + +In this paper, we focus on optimizing ring-style sequence parallelism, noting that integrating it with DeepSpeed-Ulysses Parallelism does not interfere with the underlying ring process. + +# 3 StarTrail Training System + +![](images/b7c4464769f810c8e2e4fc05a661a533aedd93dd21477d6056e44ba4314f71ca.jpg) +Figure 4: An example of StarTrail Attention on 64 GPUs. Each team member forms a sub-ring with members with the same local rank from other teams in the same ring communication group, reducing the communication volume of each team member by $75\%$ . + +# 3.1 Motivation + +Through observation, we identify two main drawbacks of Ring Attention. First, the communication overhead is exceedingly high because every GPU in the system must send and receive keys and values for nearly the entire sequence length before completing the attention computation. Second, variations in bandwidth between and within computing nodes can cause communication bottlenecks. As illustrated in Figure 2, the bandwidths between GPUs 3 and 4 and between GPUs 11 and 12 are lower than those between other GPUs in the ring. Despite this, the system is forced to operate in a complete circle, which can result in unnecessary idle times for other GPUs. To solve these drawbacks, we develop the StarTrail training system, which we will detail in the following section. + +# 3.2 StarTrail Attention + +As discussed in the previous section, a major limitation of Ring Attention is the extensive amount of peer-to-peer (P2P) communication required, which becomes problematic in environments with weak connections between computing nodes. We enhance the ring sequence parallelism by introducing an additional dimension. The fundamental idea of StarTrail is akin to a divide-and-conquer strategy. During attention, each token must compute its attention score with every other token in the sequence. While Ring Attention passes keys and values along a ring of $P$ GPUs over $P - 1$ iterations, our ap + +Figure 5: Meanings of the symbols that are used in this paper + +
PThe number of GPUs
CThe parallel size of StarTrail (team size)
HThe hidden dimension size of the Transformer blocks
NThe total number of tokens within the whole sequence
BThe training batch size
WThe communication bandwidth between GPUs
LThe communication latency between GPUs
+ +proach introduces the concept of a team. In this setting, each team member interacts only with a designated portion of the overall sequence, and the results are later aggregated using collective + +communication. Thus, StarTrail can be divided into three phases: preprocessing, ring-phase, and postprocessing. + +For preprocessing, we duplicate the queries within a team using an all-gather operation. This ensures that when a GPU receives new keys and values, it can compute the attention scores for the entire team's queries. Similarly, gathering the keys and values allows us to reduce the number of P2P communication iterations by transmitting longer sequences per iteration. Following the preprocessing, we enter the ring-style communication phase. With the number of GPUs in one team being $C$ , $CN / P$ tokens are exchanged in each iteration, and each GPU is responsible for computing $N / C$ tokens. This leads to a number of iterations of $\frac{N / C}{CN / P} = \frac{P}{C^2}$ , within a smaller ring, which we refer to as a subring. For convenience, we group $\frac{P}{C^2}$ adjacent teams into a team group for subring communication, where GPUs sharing the same local team rank form the subring. An initial P2P communication step is executed to ensure that each team group has access to the complete set of keys and values for the sequence (details are provided in the Appendix). After completing the subring iterations, each GPU holds $1 / C$ of the overall computation result for its team. With the help of online softmax, we then apply a simple reduce-scatter to combine these results while eliminating the duplicate tokens, which we refer to as the postprocessing. Throughout the attention process, asynchronous communication is employed alongside the early launch of communication kernels to maximize the overlap of computation and communication tasks. Now we will delve into more details in the StarTrail training process. + +# 3.2.1 Configurations of StarTrail Parallelism + +In the StarTrail system, GPUs are grouped into Teams to coordinate computation and communication tasks more efficiently. StarTrail introduces an additional parameter, $C$ , which determines the replication factor of the input and, consequently, the number of GPUs within each team. The range of $C$ is from 1 to $\sqrt{P}$ . When $C$ equals one, the algorithm falls back to Ring Attention. When $C$ equals $\sqrt{P}$ , the algorithm becomes a completely collective-communication-based one with no rings. When $1 < C < \sqrt{P}$ , it becomes a structure with multiple rings looping concurrently. + +Algorithm 1 StarTrail Attention Block (Forward) +Require: Input sequence x, Linear Function query, key, and value, attention parallelism size c, global rank r, global size gs, team process group pg, init send/recv target rsend and rrecv 1: compute the gathered qteam, kteam, vteam = AllGather_QKVmatmul(query, key, value, x, pg) 2: launch the asynchronous send and receive request reqsend and reqrecv, sending kteam, vteam to rsend and receiving knext, vnext from rrecv 3: get the ring P2P target rnext and rlast with get_P2P_ranks(r, gs, c) 4: initialize attention score O, extra statistics lse to zero. // lse stands for log-sum-exp 5: for $1 \leq i \leq$ world_size/c² do 6: wait for reqsend and reqrecv 7: kcurrent = knext, vcurrent = vnext 8: launch reqsend to send kcurrent and vcurrent to rnext, launch reqrecv to receive knext and vnext from rlast 9: calculate lse, O = forward_iteration (lse, O, qteam, kcurrent, vcurrent) 10: end for 11: compute Ofinal = ReduceScatter Combine (lse, O, pg) 12: return Ofinal + +Forward Propagation. In Figure 4, we have an example of one team of four GPUs out of all the 64 GPUs performing StarTrail-style attention. Each training iteration begins with the dataloader splitting the entire input sequence of length $N$ into $N / P$ sub-sequences, which are then loaded onto each GPU. As previously mentioned, the next step involves computing the queries, keys, and values. These are computed separately via matrix multiplication, followed immediately by the launch of the all-gather kernel, which gathers the above QKVs within the team, allowing for the overlap of up to two-thirds of the communication with computation. + +Once this phase is complete, each GPU within the team possesses the same Q, K, and V, each of a length of $\frac{CN}{P}$ . To distribute the communication and computation tasks among the team members, + +we divide the original workload based on four specific ranks assigned to each GPU. These ranks determine each GPU's partners and position within the P2P ring, as is shown in Figure 6. + +Following the setup, the Keys and Values are dispatched to their designated locations within the cluster to establish the initial sub-ring, setting the stage for the multi-ring iteration phase of StarTrail attention. Given that each sub-sequence is $\frac{CN}{P}$ long and each GPU is tasked with computing the attention score for $\frac{1}{C}$ of the whole sequence, it results in $\frac{N/C}{CN/P} - 1 = P/C^2 - 1$ rounds of communication. This implies that there are $P/C^2$ GPUs in one ring. + +![](images/0ea7291a5dc4417644e69e2c7cac816cab411991373547dc1c9a1291b5f61439.jpg) +Figure 6: An example of ring initialization process of 8GPUs and 4 sub-rings in StarTrail. + +The iteration process involves storing the log-sum-exp (lse) and intermediate output O, which are updated step by step. Queries are retained locally, while Keys + +and Values circulate through the ring via P2P communication. After completing the iterations, each team member accumulates the attention scores for the entire team's sub-sequence of Queries with $1 / C$ of the Keys and Values from the full sequence. + +A simple reduce-scatter operation is then employed to amalgamate the intermediate results and distribute them among the team members. Each GPU ultimately contains the final attention score for its portion of the sequence over the entire sequence. + +Backward Propagation. The major distinction between backward and forward propagation is the inability to calculate queries independently during the backward phase. Unlike forward propagation, the backward phase requires the complete set of keys and values to calculate the gradient for queries, and vice versa. To manage this, we have structured the gradient calculation into two loops: the key & value outer loop and the query inner loop. In the outer loop, gradients for keys and values are tracked and maintained fixed on the corresponding GPUs within the sub-rings; these gradients do not transfer between GPUs. The inner loop, however, handles the gradients for queries, which start initialized as zero and are circulated along the sub-rings together with the Queries themselves. During each iteration, the approach mirrors the backward computation method used in FlashAttention[8], where the updated gradient of the current query shard is passed to the next GPU in the ring, while the gradients for keys and Values are retained for subsequent query shards. + +# 3.2.2 Theoretical Analysis + +During the analysis, we will employ a case study using the StarTrail system with an attention parallel size of $C = 4$ on a llama-30B model, which consists of 64 layers. For this model, referred to as model M, the batch size $= B$ is set to 1, the sequence length $= N$ to 65536, the hidden dimension $= H$ to 6656, and the number of GPUs $= P$ to 64. Additionally, the computation will utilize bfloat16 precision. + +Communication Analysis. Let's analyze the communication overhead within one forward Transformer block on a single GPU. For Ring Attention, the communication is primarily due to the ring P2P loop. As the total number of iterations done is $P - 1$ , the total communication overhead can be calculated as: + +$$ +(P - 1) \left(\frac {2 B N H}{P W} + L\right) = \frac {2 B N H (P - 1)}{W P} + (P - 1) L \tag {1} +$$ + +and this overhead can be partially overlapped with the attention computation. + +For StarTrail, the communication overhead comes from both collective and P2P. The collective overhead for all-gather and reduce-scatter is: + +$$ +\frac {4 B N H (C - 1)}{P W} \tag {2} +$$ + +while the P2P communication can be similarly computed as: + +$$ +\left(\frac {P}{C ^ {2}} - 1\right) \left(\frac {2 C B N H}{P W} + L\right) = \frac {\left(P - C ^ {2}\right) 2 B N H}{C P W} + \left(\frac {P}{C ^ {2}} - 1\right) L \tag {3} +$$ + +The advantages of StarTrail over Ring Attention during the ring-P2P phase are evident in three main aspects: 1) Reduced Communication and Latency: Ring Attention requires C times more communication than StarTrail, significantly increasing the bandwidth requirement across the entire cluster. For the llama 30B model M, the total communication volume of ring P2P communication and collective communication volume for Ring Attention and StarTrail can be computed as 1.625 GB and 0.152 GB(collective) + 0.406GB (P2P) = 0.558GB. Furthermore, while Ring Attention necessitates $P - 1$ iterations per attention block, StarTrail only requires $\frac{P}{C^2} - 1$ , reducing the latency overhead by around $C^2$ . 2) Localized Communication: In scenarios like those depicted in Figure 3, StarTrail's ring P2P communication can be confined within the same computing node, where bandwidth is typically much higher than between computing nodes. Conversely, Ring Attention demands inter-node communication during every iteration, which can be less efficient. 3) Enhanced Overlap of Communication and Computation: During each iteration, the communication volume of StarTrail is $C$ times higher than that of Ring Attention, while the computational volume during attention is approximately $C^2$ times greater. This higher computation-to-communication ratio makes it easier for StarTrail to overlap P2P communication with computation, enhancing overall efficiency. + +Memory Analysis. In this section, we estimate the theoretical peak memory requirements necessary to store the model weights, activations, and optimizer states. Our implementation utilizes the Adam Optimizer [16], bfloat16 precision, and Zero-2 optimization [34]. We name the memory cost for the model and optimizer as $M_{m + o}$ . As for the activation, we refer to the size of one single activation of a sub-sequence on one GPU as + +$$ +A = \frac {B \times N \times H}{P} \tag {4} +$$ + +As we use the checkpointing scheme from [20], a model of $Y$ layers needs to save $Y + 1$ activations as checkpoints. Now we calculate the approximate peak memory after Q, K, and V are already calculated and before the attention computation at the last layer of the whole model. For Ring Attention and StarTrail, the peak memories are: + +$$ +P M _ {\text {R i n g}} = M _ {m + o} + (Y + 4) A \tag {5} +$$ + +$$ +P M _ {S t a r} = M _ {m + o} + (Y + 3 C + 1) A \tag {6} +$$ + +, where C is the StarTrail attention dimension. And for the example model M, the peak memory would be $M_{m + o} + 68A$ and $M_{m + o} + 77A$ , and the extra memory cost compared with Ring Attention is a lot less than $13.2\%$ , while the P2P communication volume is reduced by about $75\%$ . In a word, the extra memory cost is acceptable as a tradeoff for the communication reduction. + +# 4 Evaluation + +Table 1: Cluster and Model Configurations. All GPUs are connected with NVLink with computing nodes. + +
GPUdev. × nodeMem. (GB)inter-node bandwidthModel#Heads#LayersDim.
H1008 × 8808*400Gbps InfiniBandGPT 3B12164096
A10016 × 240100Gbps EthernetGPT 7B32324096
A1008 × 440100Gbps EthernetDiT 1B24241536
A1004 × 840100Gbps Ethernet
+ +The computational resources we use in the experiments include a local Nvidia H100 cluster with eight nodes and three Nvidia A100 clusters, as listed in table 1. We utilize two model types of total three settings, as listed in table 1. For the DiT(Diffusion Transformer) model, we use similar configurations as those in Stable Diffusion 3[9]. We utilize the backbone Diffusion Transformer only, without other components like the text and image encoders. During training, both models use bfloat16 precision and a batch size of 1 to accommodate longer input sequences. + +In the evaluation section, we aim to answer three major questions: 1) How much improvement in throughput can StarTrail bring? Additionally, how adaptable is StarTrail to clusters with both good and poor inter-node connections? 2) Is the additional memory cost incurred by StarTrail acceptable considering the throughput improvement it offers? 3) How does StarTrail perform in scenarios of weak and strong scaling? Specifically, does it outperform Ring Attention when scaled to handle longer inputs? + +![](images/8ce5ab82f26fd8f81d4d70b58515484183a6249a3c6c2ad08acf0b9b4c7d94dc.jpg) +Figure 7: Throughput evaluation of Ring Attention and StarTrail on 32 GPUs from three different clusters. We place the performance of StarTrail with both $C = 2$ and $C = 4$ in the figure. The configurations are marked in the titles of the sub-figures. For instance, A100_40GB_8(1B, 512K) represents that the experiment is on machines with 8 Nvidia A100 40GB GPUs in each node, the model used has one billion parameters, and the sequence length is 512k. + +# 4.1 Throughput and Adaptability + +Our first experiment aims to assess the performance of StarTrail and Ring Attention across different clusters with varying environments, testing the adaptability of both methods. There are several factors influencing the efficiency of ring-style attention computation: + +Theoretical Computation-Communication Volume Ratio: Primarily determined by the sequence length used during training. Attention computation exhibits a computational complexity of $O(N^2 \cdot H)$ , whereas P2P communication complexity is $O(N \cdot H)$ . Thus, the model configuration does not impact this ratio; only the sequence length does. A larger $N$ increases the computation-communication ratio, facilitating easier overlap of communication with computation. + +Compute Capability and Connectivity of GPUs: The computing overhead, given a specific volume, affects the computation-communication overhead ratio. Higher compute capabilities make overlapping more challenging. We utilize two sets of GPUs in this evaluation: Nvidia A100 40GB and Nvidia H100 80GB, with the latter offering significantly higher theoretical tflops on bf16 computations. Connectivity is considered in two parts: intra-node and inter-node. Our clusters are equipped with NVLink, ensuring robust intra-node communication. For inter-node communication, our H100 nodes leverage InfiniBand with eight adapters per node for superior inter-node bandwidth, whereas the Google Cloud servers use Ethernet. The diversity in node configurations (8-GPU and 16-GPU nodes) allows us to assess adaptability across different topologies. This evaluation not only highlights the inherent differences between the schemes but also tests their flexibility in various hardware settings. + +The results of our evaluation are illustrated in Figure 7. We measure throughput in thousands of tokens per second. To better demonstrate how to select the optimal configuration of StarTrail under each condition, we included two configurations, Star-2 and Star-4, in the figure. We omit configurations with lower performance for clarity. As indicated in the figure, in all six settings, at least one configuration of StarTrail achieves higher throughput than Ring Attention, with $2.114\mathrm{x}$ , $1.414\mathrm{x}$ , $1.629\mathrm{x}$ , $1.425\mathrm{x}$ , $1.360\mathrm{x}$ , $1.199\mathrm{x}$ , $1.771\mathrm{x}$ , and $1.346\mathrm{x}$ the throughput of Ring Attention. This advantage is primarily due to the additional parallel dimension that StarTrail introduces. Unlike Ring Attention, which requires inter-node P2P communication in each iteration, StarTrail's P2P communication is mostly confined intra-node, except for initial data transfers. This experiment clearly demonstrates StarTrail's superior performance across various environments. Another observation is that the optimal configuration for StarTrail may vary depending on the environment, reflecting differences in the computation-communication ratio and the trade-offs between collective and P2P communication. + +![](images/e07d54611fe84bcb0c6091c3a3f67ad2a24594d7c19e8fe0dffb78bcafb91e39.jpg) +Figure 8: The normalized relative memory cost of different configurations of StarTrail compared with Ring Attention on different clusters. + +# 4.2 Memory Consumption + +The memory results, displayed in Figure 8, reveal that for the configurations yielding the highest throughput, StarTrail consumes between $7.9\%$ and $30.79\%$ more GPU memory than RingAttention, while achieving $1.199\mathrm{x}$ to $2.114\mathrm{x}$ throughput. Moreover, in scenarios involving larger models, the relative increase in memory consumption due to QKV duplication diminishes. This reduction is explained by equation 6. This phenomenon is further evidenced by the fact that the extra memory ratio for experiments with the 7B model is significantly smaller than that for the 3B and 1B model. Considering the substantial throughput gains provided by StarTrail, this tradeoff between memory usage and efficiency is deemed acceptable. + +![](images/c7bc17fb85f6ee481a66061338a8d23016a34c5de950d2951fd67a6b0d102658.jpg) +(a) DiT Strong Scaling on Nvidia A100 40GB GPUs. All configurations include inter-node communication. + +![](images/21ac4943c1edded316388308944acc811e840d4314ef92711e4032b76348d127.jpg) +(b) GPT Strong Scaling on Nvidia H100 80GB GPUs. +Figure 9: Strong scaling experiments with fixed sequence length of $128\mathrm{K}$ + +# 4.3 Strong and Weak Scaling + +In the scaling tests we carry out experiments for both strong and weak scaling. For strong scaling, we fix the sequence length to 128K while increasing the number of GPUs from 8 to 64 for the GPT model and from 8 to 32 for the DiT model. For weak scaling, we scale the sequence length from 128k to 512k for the DiT model and from 64k to 512k for the GPT model proportionally increasing the number of GPUs from 8 to 32. As is depicted in Figure 9 and 10, StarTrail shows obvious advantage over Ring Attention as we increase the number of GPUs. The results for strong scaling can also be explained by the computation-communication ratio. When scaled to more GPUs, the local sequence length on each GPU becomes smaller, and as explained the previous sections, makes it harder to overlap the P2P communication with attention computation. The overall scaling performance is limited by the nature of ring-style communication, but we still consider our improvement over Ring Attention meaningful due to the necessity of using Ring-style Parallelisms during training. + +In summary, StarTrail shows better scalability in both strong and weak scaling experiments, making it a better choice for large-scale Transformer model training. + +# 5 Conclusion + +StarTrail represents an advanced near-infinite-context Transformer model training system, featuring a communication-optimized concentric ring sequence parallelism scheme. Through experiments, we + +![](images/e89c48cd1b9653a66d4178df42f8c1b0c69dfabf3e94463aaa420bfc9cda561a.jpg) +(a) DiT Weak scaling on Nvidia A100 40GB GPUs of sequence length from $128\mathrm{K}$ to $512\mathrm{K}$ . All configurations include inter-node communication. +Figure 10: Weak scaling Experiments + +![](images/b007de3a24a82ec71f668cbddc38f0f2b2c35e553086d8cc6e9c6d7df746d433.jpg) +(b) GPT Weak scaling on Nvidia H100 80GB GPUs of sequence length from 64K to 512K. + +demonstrate that our system not only achieves high efficiency across various training environments but also excels under both strong and weak scaling conditions for both CV and NLP models. Current limitations of StarTrail include that although orthogonal, we can still further improve the co-design of StarTrail and hybrid parallelism in future works. In an era increasingly demanding longer contexts for both NLP and CV, StarTrail is poised to make significant contributions to the industry and inspire innovative research in academia. + +# 6 acknowledgement + +Yang You's research group is being sponsored by NUS startup grant (Presidential Young Professorship), Singapore MOE Tier-1 grant, ByteDance grant, NUS ARTIC grant, Apple grant, Alibaba grant, Google Research and Google grant for TPU usage. + +# References + +[1] Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, et al. Gpt-4 technical report, 2023. +[2] Joshua Ainslie, James Lee-Thorp, Michiel de Jong, Yury Zemlyanskiy, Federico Lebrón, and Sumit Sanghai. Gqa: Training generalized multi-query transformer models from multi-head checkpoints, 2023. +[3] Iz Beltagy, Matthew E. Peters, and Arman Cohan. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: Our key method is modifying the parallelism technique and is not related with LLM itself. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Technical Appendices and Supplementary Material + +# A.1 Sequence Parallelism Dataloger + +The causal masks present a challenge due to the unbalanced computational load across GPUs: subsequences at the beginning of a sequence require significantly more computation than those at the end. To address this imbalance and achieve load equilibrium among GPUs, we modify the ZigZag scheme introduced by [42], illustrated in Figure 11. The figure illustrates the simplest case of zigzag load-balancing. Notably, the effectiveness of this strategy improves as the number of GPUs increases. This improvement correlates with the expanding difference in computation volume between the first and the last token, which escalates as the sequence length extends. This approach ensures that the total workload on each GPU is balanced, eliminating the need for additional communication mechanisms like those employed in DistFlashAttention[20]. + +![](images/9424ca13dedbdf50b7953c944ce84db4dbcb6f006846d603775bca014910aab6.jpg) +Figure 11: A comparison between naive and zigzag dataloader for 8 GPUs with attention parallel dimension of 2. The corresponding initialization can be found in Figure 6 with the same configuration. The improvement of efficiency from load-balancing increases with the number of GPUs. + +# A.2 Details in the training process + +Apart from the attention process described in Algorithm 1, we would like to provide a few other details to be more comprehensive. First, before the concentric ring attention, we need to initialize the the Keys and Values and determine each GPU's position within the rings. + +Initially, for the setup stage, it is essential to establish the sub-rings by rearranging the activation positions. Specifically, during forward propagation, the queries do not require rearrangement; however, the keys and values must be transmitted to their corresponding positions in the ring prior to commencing the loop. As illustrated in Figure 6 and algorithm, this initialization ensures that each team member holds a different shard of keys and values. Moreover, it guarantees that no two teams within the same ring possess identical keys and values. Initially, for the setup stage, it is essential to establish the sub-rings by rearranging the activation positions. Specifically, during forward propagation, the queries do not require rearrangement; however, the keys and values must be transmitted to their corresponding positions in the ring prior to commencing the loop. As illustrated in Figure 6 and algorithm, this initialization ensures that each team member holds a different shard of keys and values. Moreover, it guarantees that no two teams within the same ring possess identical keys + +# Algorithm 2 get_init_send() + +Require: inter-team rank $\mathbf{r}_t$ , intra-team rank $\mathbf{r}_a$ , inter-team dimension $\mathbf{d}_t$ , intra-team dimension $\mathbf{d}_a$ + +1: team group size $= \frac{\mathbf{d}_t}{\mathbf{d}_a}$ +2: target team group rank = $\mathbf{r}_a$ +3: target team = target team group rank * team group size + $\mathbf{r}_t / / \mathbf{d}_a$ +4: target device intra-team rank = $\mathbf{r}_t\% \mathbf{d}_a$ +5: target global rank = target team * $\mathbf{d}_a +$ target device intra-team rank +6: return target global rank + +After the initialization of activations, we can set up the rings by providing the GPUs their last and next GPU within their rings, as is described in Algorithm 3. + +# A.3 StarTrail Runtime + +StarTrail is written in PyTorch[29] and uses the PyTorch torch.autograd.function and NCCL[28] backend for forward and backward implementation. StarTrail also employs multiple techniques during runtime to improve its overall training efficiency. + +Ingetrate Flash Attention. The StarTrail attention mechanism involves multiple iterations that loop over Keys and Values, with each iteration still using traditional self-attention with corresponding Query, Key, and Value (QKV). This approach enables StarTrail to incorporate flash attention effectively, extending its capability by preserving intermediate states across iterations. Additionally, + +# Algorithm 3 get_P2P_config() + +Require: inter-team rank $\mathbf{r}_t$ , intra-team rank $\mathbf{r}_a$ , inter-team dimension $\mathbf{d}_t$ , intra-team dimension $\mathbf{d}_a$ + +1: team group size $= \mathbf{d}_t / \mathbf{d}_a$ +2: self team group rank = $\frac{r_t}{\text{team group size}}$ +3: next team in group $= (r_{t} + 1)\%$ team group size + team group size × self team group rank +4: last team in group $= (r_{t} - 1)\%$ team group size + team group size × self team group rank +5: next device global rank $= r_{a} +$ next team in group $\times d_{a}$ +6: last device global rank $= r_{a} +$ last team in group $\times d_{a}$ +7: return next device global rank, last device global rank + +Table 2: Supported Sequence Length of Ring Attention and StarTrail on one Nvidia A100 80GB GPU. + +
Supported Seq Len on one 80GB A100 GPU (K Tokens)
Model SizeLengthRing AttentionStarTrail
3B128✓✓
256✓✓
512✓✓
7B128✓✓
256✓✓
512XX
13B128✓✓
256XX
512XX
+ +StarTrail enhances the efficiency of the forward process with the help of torch JIT to fuse kernels aside from flash attention. + +Overlap communication with computing. In StarTrail attention, P2P communication and self-attention computing are interleaved across iterations, each incurring considerable time. To mitigate this, StarTrail employs a double buffering technique to asynchronously execute communication and computing kernels, effectively overlapping these processes and enhancing GPU utilization. + +Save recomputation with checkpoints. StarTrail adopts the checkpointing strategy introduced by DistFlashAttn[20], placing checkpoints at the end of the self-attention phase rather than the FFN of each transformer layer. This checkpoint placement effectively obviates the need to recompute the self-attention forward process during the backward pass, avoiding redundant attention computation. + +# B Additional Experiment + +To comprehensively evaluate the memory consumption of StarTrail and Ring Attention, we compared the maximum supported sequence lengths of StarTrail with those reported in the Ring Attention paper [24]. As shown in Table 2, although StarTrail requires slightly more memory, it still supports sequence lengths commonly used in training tasks. + +# B.1 Discussion + +# B.2 Larger Batch Sizes and Models + +We utilized small batches and model sizes in our experiments because these choices do not affect the underlying improvements in communication efficiency and computation-to-communication ratio that StarTrail provides. For larger batch sizes, both communication and computation scale proportionally, leaving the overlapping ability unchanged. Similarly, while larger models involve more layers or + +larger hidden sizes, the key attention computations and corresponding ratios remain unaffected. Hence, our conclusions naturally extend to scenarios with larger batches and models. + +# B.2.1 FlashAttention3 and Hopper GPUs + +In addition to the original FlashAttention [8] used in our experiments, FlashAttention3 [35] has been introduced, specifically designed for Hopper and newer Nvidia GPUs. For FP16 precision, which is utilized in this paper, FlashAttention3 achieves a 1.5-2.0x speedup on Hopper GPUs. As discussed in Section 3, reducing attention computation overhead results in more P2P communication not being overlapped, further emphasizing the need to reduce communication volume. With the increasing adoption of Hopper GPUs, the significance of the StarTrail system will also grow. + +# B.2.2 StarTrail and Other Parallelisms + +Model Parallelism As is well known, tensor parallelism shards activations by attention heads during attention computation, making it easily combinable with StarTrail with minimal effort. However, when combined with tensor parallelism, the need for attention heads can limit the scalability of head-based sequence parallel methods like DeepSpeed-Ulysses. Pipeline parallelism, on the other hand, divides the model across layers without altering the computation patterns within Transformer blocks, making StarTrail orthogonal to it. + +Other Sequence Parallelism StarTrail is orthogonal with other attention-head-sharding-based sequence parallelism approaches, such as DeepSpeed-Ulysses [14]. While DeepSpeed-Ulysses distributes attention heads across different devices, StarTrail can independently partition activations along the sequence length dimension. In future work, we can explore combining StarTrail with DeepSpeed-Ulysses to expand the communication scheduling space, harnessing the scalability of StarTrail alongside the efficiency of DeepSpeed-Ulysses. + +In summary, StarTrail can be seamlessly integrated with other parallel training techniques, enabling the creation of a hybrid distributed training system. + +# B.3 Other Related Works + +Attention Optimization. Traditional full attention mechanisms necessitate $O(n^{2})$ memory for storing the outputs of $QK^T$ , leading to significant computational and memory demands. To address these challenges within the GPU, several approaches have been devised to reduce both memory and computational requirements. Memory-efficient attention[32] introduces a straightforward algorithm that requires only $O(1)$ memory relative to the sequence length, with an extension for self-attention that needs only $O(\log n)$ memory. FlashAttention further minimizes I/O overhead and enhances overall efficiency. Additionally, optimization methods specifically tailored for inference, such as PagedAttention[19], are also being developed to improve the efficiency of attention computations. In this work, we utilize FlashAttention within each iteration to reduce the computation overhead. + +Long-Sequence Training Techniques. Sequence Parallelism[21] was initially introduced to enhance the efficiency of parallel long-sequence training. Ring Attention[24] improved communication efficiency through memory-efficient methods[32], supporting near-infinite sequence lengths. DeepSpeed-Ulysses[14] employs attention head splitting to achieve high efficiency, though it is constrained by the number of heads. Megatron Sequence Parallelism focuses on reducing memory costs during Tensor Parallelism, while DistFlashAttention[20] features a load-balance scheme and a novel gradient checkpoint method. Our work builds on these innovations, introducing a system that supports large-scale training with an efficient communication scheme. + +Techniques for Distributed Model Training. Distributed model training encompasses two primary areas: 1) Memory Management: Various techniques aim to conserve GPU memory during distributed training, such as mixed precision training[26] and the ZeRO series[34]. In this work, we implement ZeRO-2 to manage optimizer states and gradients efficiently. 2) Hybrid Parallelism: Frameworks like Megatron[27] and Colossal AI[4] integrate multiple forms of parallelism. There are various existing Parallelism techniques like Pipeline Parallelism[13, 10, 23, 25] and Tensor Parallelism[37], which can be combined with StarTrail Parallelism to facilitate large-scale training. 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However, their performance is fundamentally limited by the quality and diversity of training data. This often restricts their generalization to tasks outside their training distribution or longer planning horizons. To overcome this challenge, we propose State-Covering Trajectory Stitching (SCoTS), a novel reward-free trajectory augmentation method that incrementally stitches together short trajectory segments, systematically generating diverse and extended trajectories. SCoTS first learns a temporal distance-preserving latent representation that captures the underlying temporal structure of the environment, then iteratively stitches trajectory segments guided by directional exploration and novelty to effectively cover and expand this latent space. We demonstrate that SCoTS significantly improves the performance and generalization capabilities of diffusion planners on offline goal-conditioned benchmarks requiring stitching and long-horizon reasoning. Furthermore, augmented trajectories generated by SCoTS significantly improve the performance of widely used offline goal-conditioned RL algorithms across diverse environments. Our code is available at https://github.com/leekwoon/scots/ + +# 1 Introduction + +In many real-world applications, agents must plan over hundreds of steps, often receiving sparse or delayed feedback until they reach a distant goal. Perfect knowledge of the environment allows powerful planners like MPC (Tassa et al., 2012) and MCTS (Silver et al., 2016, 2017) to excel. However, most real-world tasks instead require learning environment dynamics from data. Model-based reinforcement learning (MBRL) (Sutton, 2018) constructs such world models, offering sample-efficient learning and improved generalization (Ha & Schmidhuber, 2018; Hafner et al., 2019; Kaiser et al., 2020). However, autoregressive predictions from learned models accumulate small errors into a cascade of inaccuracies. This compounding error can cause planners to exploit model inaccuracies and generate trajectories that are suboptimal or even physically infeasible, especially in long-horizon tasks (Talvitie, 2014; Asadi et al., 2018; Janner et al., 2019; Voelcker et al., 2022; Chen et al., 2024a). + +To address these limitations, diffusion planners (Janner et al., 2022; Ajay et al., 2023; Liang et al., 2023; Chen et al., 2024c) have recently emerged as a promising alternative for trajectory generation in sequential decision-making. Instead of rolling out one step at a time, diffusion planners treat each trajectory as a single high-dimensional sample, learning a denoising process that transforms noise drawn from a simple prior into trajectories that match the target distribution (Ho et al., 2020; Song et al., 2021). By operating on entire trajectories simultaneously, these methods inherently prevent the compounding of prediction errors that undermine autoregressive dynamics models. Moreover, the generative nature of diffusion models allows for flexible conditioning and guidance mechanisms, enabling the synthesis of plans with properties like reaching specific goals or maximizing expected returns (Dhariwal & Nichol, 2021). + +Despite these advantages, the effectiveness of diffusion planners remains fundamentally limited by the quality, diversity, and coverage of the offline training data. First, their effective planning horizon is inherently coupled to the maximum trajectory length observed during training, making it challenging to generate coherent plans that significantly exceed this length. Second, their generalization capability is often confined to the specific types of trajectories and transitions represented in the training data. For instance, if the dataset predominantly features certain movement patterns, the planner may struggle to synthesize solutions for novel tasks requiring different compositions of behaviors (as illustrated in Figure 1). While exhaustively collecting data for all conceivable scenarios could mitigate this, such an approach is prohibitively expensive. Trajectory stitching (Ziebart et al., 2008) offers a promising alternative by composing novel, longer sequences from existing short segments. However, existing methods rely heavily on extrinsic rewards for segment selection, and maintaining the dynamic consistency and feasibility of stitched trajectories remains challenging. + +![](images/1341e49d7e9fa9019720e0e1e8587e73946988c593bbcddba16d4b2ad37f3f66.jpg) +(a) Offline Data + +![](images/019601f2934b3770a2a8676e21576f6837d89212db91ac010e157cdbcd9e34c7.jpg) +Figure 1: Improved generalization with SCoTS. (a) Examples from the training dataset, illustrating limited coverage. (b) Plans generated by Hierarchical Diffuser (HD) (Chen et al., 2024c), which fail to generalize well to these out-of-distribution tasks due to insufficient coverage of the training data. (c) Plans generated by HD trained on SCoTS-augmented data, demonstrating significantly improved trajectory stitching capability and generalization to unseen tasks. Each color corresponds to one of 10 plans generated by the planner. + +![](images/1a17139264edd6b14861eb27d21b5df4e79e347a3e31f505cd20842025092258.jpg) +(b) HD + +![](images/c907f25e290a55361c58570b46ae8751eb16ca19d3dab91fe2481e27d935d379.jpg) + +![](images/1f6ac729b3533756c19639f241df89064082445351ba9b5905160bcb21291a07.jpg) +(c) Ours + +![](images/359162443ca0067dfed1092205f2c0961ce0f19d7467fa3ad4d3befe27aad505.jpg) + +In this paper, we propose State-Covering Trajectory Stitching (SCoTS), a reward-free trajectory augmentation framework that systematically extends trajectories to cover diverse, unexplored regions of the state space. Specifically, SCoTS employs a three-stage approach: First, we learn a temporal distance-preserving latent representation by training a model to encode states based on learned optimal temporal distances, facilitating efficient identification of viable trajectory segments. Second, we introduce a novel iterative stitching strategy that balances directed exploration with state-space coverage. In this process, trajectory segments are selected based on their progress along a learned direction in the latent space and their novelty relative to previously explored regions within the rollout. Finally, we refine the resulting stitched trajectories using a diffusion-based refinement procedure. Consequently, the resulting trajectories exhibit broader state-space coverage while preserving dynamic feasibility. + +To summarize, our contribution in this paper is the introduction of SCoTS, a reward-free trajectory augmentation approach designed to generate diverse, long-horizon trajectories that enhance diffusion planners. Extensive experiments across diverse and challenging benchmark tasks show that SCoTS significantly enhances the stitching capabilities and long-horizon generalization of diffusion planners. Furthermore, augmented trajectories generated by SCoTS notably boost the performance of widely used offline goal-conditioned reinforcement learning (GCRL) algorithms in across multiple trajectory stitching benchmarks. + +# 2 Planning with Diffusion Models + +Diffusion-based planners (Janner et al., 2022; Liang et al., 2023; Chen et al., 2024c) provide a promising framework for long-horizon decision-making by modeling entire trajectories as joint distributions. A trajectory $\pmb{\tau}$ is typically represented as a sequence of states $s_t$ and actions $\pmb{a}_t$ over a planning horizon $T$ : + +$$ +\boldsymbol {\tau} = \left[ \begin{array}{l l l l} \boldsymbol {s} _ {1} & \boldsymbol {s} _ {2} & \dots & \boldsymbol {s} _ {T} \\ \boldsymbol {a} _ {1} & \boldsymbol {a} _ {2} & \dots & \boldsymbol {a} _ {T} \end{array} \right], \tag {1} +$$ + +where $s_t$ and $a_t$ denote the state and action at time step $t$ , respectively. Diffusion planners utilize diffusion probabilistic models (Sohl-Dickstein et al., 2015; Ho et al., 2020) to learn a trajectory distribution $p_{\theta}(\pmb{\tau}^{0})$ over noise-free trajectories $\pmb{\tau}^{0}$ . This involves a predefined forward noisng process + +![](images/bf848de9a5fe1d412cb07491ca51827c6f88dd4ca27b12b4738846167ea28457.jpg) +(a) Search + +![](images/8c613eb43d3349f72183b7b5ad48eba67aae9b5a677130d5da69f47b69607c8c.jpg) +(b) Select + +![](images/1fcffa3815cdccac30e01ec17719e666c8fd0f5075bf50a921c9753fc76a252b.jpg) +Figure 2: Overview of the SCoTS stitching process. (a) Temporal Distance-Preserving Search: Given the currently composed trajectory (red), we identify candidate segments (gray) by searching in a latent space learned to preserve temporal distances. Candidates are selected based on proximity to the endpoint of the current trajectory in latent space. (b) Exploratory Segment Selection: Among the retrieved candidate segments, we select the segment (blue) that best balances directional progress toward a randomly sampled latent direction and novelty relative to previously visited states in latent space. (c) Diffusion-based Statching Refinement: To ensure smooth transitions, a diffusion model refines the stitching point between segments, generating dynamically consistent trajectories. + +![](images/d9026bf1547077a8590a6090799aa35a75263c8f22f47f3c4bbd33cec4043af0.jpg) +(c)Stitch + +![](images/2af0da1b17fea7e083408c19ebb0a79db2c10f94ded165e4b854f7293a1df496.jpg) + +![](images/f5ed075526ebbb7875c561d99080eb692f7c9bf8011e53101490a8922b2bbafe.jpg) + +and a learned reverse denoising process. The forward process incrementally adds Gaussian noise to the trajectories through $M$ discrete diffusion timesteps with a variance schedule $\{\beta_i\}_{i=1}^M$ : + +$$ +q \left(\boldsymbol {\tau} ^ {i} \mid \boldsymbol {\tau} ^ {i - 1}\right) := \mathcal {N} \left(\boldsymbol {\tau} ^ {i}; \sqrt {1 - \beta_ {i}} \boldsymbol {\tau} ^ {i - 1}, \beta_ {i} \mathbf {I}\right). \tag {2} +$$ + +A key property is the direct sampling of intermediate trajectories: + +$$ +q \left(\boldsymbol {\tau} ^ {i} \mid \boldsymbol {\tau} ^ {0}\right) = \mathcal {N} \left(\boldsymbol {\tau} ^ {i}; \sqrt {\alpha_ {i}} \boldsymbol {\tau} ^ {0}, (1 - \alpha_ {i}) \mathbf {I}\right), \tag {3} +$$ + +where $\alpha_{i} := \prod_{s=1}^{i}(1 - \beta_{s})$ . The schedule ensures that $\pmb{\tau}^{M}$ approximates a standard Gaussian distribution $\mathcal{N}(\mathbf{0}, \mathbf{I})$ . The reverse process learns to invert this noising process and define following generative process with a standard Gaussian prior $p(\pmb{\tau}^{M})$ : + +$$ +p _ {\theta} \left(\boldsymbol {\tau} ^ {0}\right) = \int p \left(\boldsymbol {\tau} ^ {M}\right) \prod_ {i = 1} ^ {M} p _ {\theta} \left(\boldsymbol {\tau} ^ {i - 1} \mid \boldsymbol {\tau} ^ {i}\right) \mathrm {d} \boldsymbol {\tau} ^ {1: M} \tag {4} +$$ + +with a learnable Gaussian transition: $p_{\theta}(\tau^{i - 1}|\pmb{\tau}^i) = \mathcal{N}(\pmb{\tau}^{i - 1}|\pmb{\mu}_{\theta}(\pmb{\tau}^i,i),\pmb{\Sigma}^i)$ . + +Given an offline dataset $\mathcal{D}$ , diffusion models in practice simplify training by parameterizing a noise-prediction network $\epsilon_{\theta}$ , trained to predict the noise $\epsilon$ added during the forward process (Ho et al., 2020): + +$$ +\mathcal {L} (\theta) := \mathbb {E} _ {i, \epsilon , \tau^ {0}} [ \| \epsilon - \epsilon_ {\theta} (\tau^ {i}, i) \| ^ {2} ], \tag {5} +$$ + +where $i \in \{0,1,\dots,M\}$ is the diffusion timestep, $\epsilon \sim \mathcal{N}(\mathbf{0},\mathbf{I})$ is target noise that was used to corrupt clean trajectory $\pmb{\tau}^{0}$ into $\pmb{\tau}^{i} = \sqrt{\alpha_{i}}\pmb{\tau}^{0} + \sqrt{1 - \alpha_{i}}\pmb{\epsilon}$ . + +Remark. Previous works generally assume the offline dataset $\mathcal{D}$ sufficiently covers diverse trajectories with substantial length. Consequently, these studies have primarily focused on improving network architectures, action generation methods, and planning strategies. In contrast, we explicitly aim to generate an augmented dataset $\mathcal{D}_{\mathrm{aug}}$ that extends trajectory coverage, enabling diffusion planners to generalize effectively beyond their training distribution. + +# 3 State-Covering Trajectory Stitching + +We introduce State-Covering Trajectory Stitching (SCoTS), a novel reward-free trajectory augmentation framework designed to synthesize an augmented dataset $\mathcal{D}_{\mathrm{aug}}$ from an offline dataset $\mathcal{D}$ . The core idea of SCoTS is to iteratively construct long and diverse trajectories by repeatedly stitching short segments guided by latent directional exploration, resulting in significantly improved generalization and extended planning horizons for diffusion planners. SCoTS consists of three stages: (1) learning a temporal distance-preserving embedding for efficient segment retrieval (Section 3.1); (2) iterative trajectory stitching driven by latent directional exploration and novelty-based selection (Section 3.2); and (3) diffusion-based refinement to ensure dynamically consistent transitions (Section 3.3). The overall procedure of SCoTS, including segment search, exploratory selection, and diffusion-based refinement, is illustrated in Figure 2. The detailed algorithm is summarized in Algorithm 1. + +Algorithm 1 Overview of the SCoTS Framework +1: Input: Offline dataset $\mathcal{D}$ , Temporal distance-preserving embedding $\phi$ , Diffusion stretcher $p_{\theta}^{\text{stitcher}}$ +2: Initialize: Augmented dataset $\mathcal{D}_{\text{aug}} = \emptyset$ +3: for $n = 1, \ldots, N_{\text{traj}}$ do +4: // Sample initial segment from offline data +5: $\tau_{\text{comp}} \sim \mathcal{D}$ +6: // Sample a random latent exploration direction +7: $z \sim \mathcal{N}(0, \mathbf{I})$ ; $z \gets z / \|z\|$ +8: for $t = 1, \ldots, N_{\text{stitch}}$ do +9: // Retrieve nearest segments using temporal embedding +10: $\{\tau_j\}_{j=1}^k \gets \text{TopKNeighbors}(\phi(\text{end}(\tau_{\text{comp}})), \phi(\mathcal{D}), k)$ +11: // Compute directional progress and novelty scores +12: Compute scores $S_j = P_j + \beta N_j$ (Eq. (11)) +13: // Select best candidate segment +14: $\tau_{\text{best}} \gets \arg \max_j S_j$ +15: // Diffusion-based stitching refinement +16: $\tau' \sim p_{\theta}^{\text{stitcher}}(\cdot | s_1 = \text{end}(\tau_{\text{comp}}), s_H = \text{end}(\tau_{\text{best}}))$ +17: // Concatenate refined segment to trajectory +18: $\tau_{\text{comp}} \gets [\tau_{\text{comp}}, \tau']$ +19: end for +20: $\mathcal{D}_{\text{aug}} \gets \mathcal{D}_{\text{aug}} \cup \{\tau_{\text{comp}}\}$ +21: end for +22: Train diffusion planner on $\mathcal{D}_{\text{aug}}$ + +# 3.1 Temporal Distance-Preserving Embedding + +Identifying trajectory segments that are suitable for stitching requires accurately measuring their temporal closeness. However, simply using raw state-space distances can yield temporally incoherent results due to potential dynamic inconsistencies arising from ignoring state reachability. To address this, we employ a temporal distance-preserving embedding $\phi : S \to \mathcal{Z}$ , which maps raw states to a latent space $\mathcal{Z}$ designed such that the Euclidean distance $\| \phi(s) - \phi(g) \|_2$ approximates the optimal temporal distance $d^*(s, g)$ , defined as the minimum number of environment steps required to transition from state $s$ to state $g$ . Formally, we parameterize a goal-conditioned value function $V(s, g)$ following (Park et al., 2024a): + +$$ +V (\boldsymbol {s}, \boldsymbol {g}) := - | | \phi (\boldsymbol {s}) - \phi (\boldsymbol {g}) | | _ {2}, \tag {6} +$$ + +which is trained on the offline dataset $\mathcal{D}$ using a temporal difference objective inspired by implicit Q-learning (Kostrikov et al., 2022): + +$$ +\mathcal {L} _ {\phi} := \mathbb {E} _ {\left(\boldsymbol {s}, \boldsymbol {a}, \boldsymbol {s} ^ {\prime}, \boldsymbol {g}\right) \sim \mathcal {D}} \left[ \ell_ {\xi} ^ {2} (- \mathbb {1} (\boldsymbol {s} \neq \boldsymbol {g}) - \gamma | | \bar {\phi} (\boldsymbol {s} ^ {\prime}) - \bar {\phi} (\boldsymbol {g}) | | _ {2} + | | \phi (\boldsymbol {s}) - \phi (\boldsymbol {g}) | | _ {2}) \right], \tag {7} +$$ + +where $\bar{\phi}$ is a target network (Mnih, 2013), $\gamma$ is a discount factor, and $\ell_{\xi}^{2}$ denotes the expectile loss (Kostrikov et al., 2022; Newey & Powell, 1987). + +Remark. We note that the learned latent space is not a perfect metric representation of the MDP (Park et al., 2024a). However, the reliability of SCoTS is grounded in its design, which does not require a globally accurate temporal distance metric. Instead, as we detail next, our framework leverages the latent distance for the more tractable local problem of retrieving promising and reachable candidate segments at each step of the iterative stitching process. This local approach makes the overall framework robust to the inherent imperfections of the embedding. + +# 3.2 Directional and Exploratory Trajectory Stitching + +Given the learned temporal distance-preserving embedding $\phi$ , we iteratively construct extended trajectories via stitching. We start each new trajectory by randomly sampling an initial segment $\pmb{\tau}_{\mathrm{init}}$ from the offline dataset $\mathcal{D}$ . To encourage diverse state coverage, we randomly sample a fixed latent exploration direction $z$ as a unit vector, i.e., $z \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ , $z \gets z / \|z\|$ , for each trajectory rollout. + +At each stitching iteration, let $\tau_{\mathrm{comp}}$ denote the currently composed trajectory. We define $\operatorname{end}(\tau)$ as a function returning the final state of trajectory $\tau$ . We then identify a set of candidate segments $\{\tau_j\}_{j=1}^k$ whose initial states are nearest neighbors to $\operatorname{end}(\tau_{\mathrm{comp}})$ in the latent space: + +$$ +\left\{\tau_ {j} \right\} _ {j = 1} ^ {k} = \operatorname {T o p K N e i g h b o r s} \left(\phi \left(\operatorname {e n d} \left(\tau_ {\text {c o m p}}\right)\right), \phi (\mathcal {D}), k\right), \tag {8} +$$ + +where $\phi (\mathcal{D})$ is a concise representation for the set of latent embeddings of the initial states of all trajectories within the dataset $\mathcal{D}$ , and the distance metric is $\| \phi (\mathrm{end}(\pmb{\tau}_{\mathrm{comp}})) - \phi (\pmb{s}_{1,j})\| _2$ . + +To select the best candidate for stitching, we evaluate each candidate segment $\tau_{j} = (s_{1,j},\dots,s_{H,j})$ based on a composite score balancing directional progress and novelty. The progress score quantifies the alignment in the latent space between the segment direction and the exploration direction $z$ : + +$$ +P _ {j} = \left\langle \phi \left(\operatorname {e n d} \left(\boldsymbol {\tau} _ {j}\right)\right) - \phi \left(\boldsymbol {s} _ {1, j}\right), \boldsymbol {z} \right\rangle . \tag {9} +$$ + +The novelty score promotes exploration and coverage of novel latent states by estimating the entropy of the endpoint of each candidate segment $\tau_{j}$ relative to previously visited latent states. Here, $\nu_{\mathrm{rollout}}$ denotes the collection of latent representations of every state along previously stitched segments. Leveraging a non-parametric particle-based estimator (Liu & Abbeel, 2021) on our temporal distance-preserving embeddings, we compute the novelty score as: + +$$ +N _ {j} = \frac {1}{k _ {\text {d e n s i t y}}} \sum_ {\substack {\phi_ {v} \in k - N N (\phi (\operatorname {e n d} (\boldsymbol {\tau} _ {j})), \mathcal {V} _ {\text {r o l l o u t}}, k _ {\text {d e n s i t y}})}} \left\| \phi (\operatorname {e n d} (\boldsymbol {\tau} _ {j})) - \phi_ {v} \right\| _ {2}. \tag{10} +$$ + +A higher $N_{j}$ indicates greater novelty, signaling that the candidate segment expands coverage by moving towards less-explored regions of the latent space. We combine these two metrics to form the overall selection criterion: + +$$ +S _ {j} = P _ {j} + \beta N _ {j}, \tag {11} +$$ + +where $\beta$ balances progress and novelty. We then stitch the candidate $\pmb{\tau}_{\mathrm{best}}$ with the highest score to $\pmb{\tau}_{\mathrm{comp}}$ . + +# 3.3 Diffusion-based Stitching Refinement + +Although the exploratory selection step identifies segments with desirable progress and novelty, the stitching points, i.e., the connecting states between consecutive trajectory segments, may still exhibit minor dynamic inconsistencies or sub-optimal transitions. To mitigate these issues, we introduce a diffusion-based refinement step. Specifically, we train a diffusion model, termed the *stitcher* $p_{\theta}^{\mathrm{stitcher}}$ , which generates intermediate states conditioned on the boundary states of adjacent segments. Given a selected segment $\tau_{\mathrm{best}}$ , the stitcher produces a refined trajectory $\tau'$ by sampling from: + +$$ +\boldsymbol {\tau} ^ {\prime} \sim p _ {\theta} ^ {\text {s t i t c h e r}} (\cdot \mid s _ {1} = \operatorname {e n d} \left(\boldsymbol {\tau} _ {\text {c o m p}}\right), s _ {H} = \operatorname {e n d} \left(\boldsymbol {\tau} _ {\text {b e s t}}\right)), \tag {12} +$$ + +where $\mathrm{end}(\tau_{\mathrm{comp}})$ denotes the end state of the current composite trajectory $\tau_{\mathrm{comp}}$ , and $\mathrm{end}(\tau_{\mathrm{best}})$ denotes the end state of the newly selected segment $\tau_{\mathrm{best}}$ . This diffusion-based refinement effectively smoothes out transitions, ensuring dynamic coherence and feasibility of the stitched trajectories. + +By iteratively repeating segment search, exploratory selection, and this refinement process, we construct a diverse set of augmented trajectories. To generate corresponding action sequences for these trajectories, we train an inverse dynamics model $\boldsymbol{a}_t = f_{\psi}(\boldsymbol{s}_t, \boldsymbol{s}_{t+1})$ on the offline dataset $\mathcal{D}$ , which infers the actions that transition between consecutive states. The resulting state-action trajectories are aggregated into the augmented dataset $\mathcal{D}_{\mathrm{aug}}$ . This systematic and iterative augmentation approach generates an augmented dataset that broadly covers the state space. Crucially, diffusion planners trained on this augmented data exhibit significantly enhanced trajectory stitching capabilities and improved long-horizon generalization, particularly for tasks requiring extensive trajectory stitching and long-horizon reasoning (Section 4.3). + +# 4 Experiments + +In this section, we empirically validate the effectiveness of our proposed SCoTS framework. Specifically, we aim to investigate (1) whether SCoTS can generate diverse trajectories that extend significantly beyond the planning horizons present in the original offline dataset, (2) whether training + +diffusion planners on these augmented trajectories enhances their capability to produce feasible long-horizon plans in unseen scenarios, and (3) whether the augmented dataset generated by SCoTS provides significant performance improvements for existing offline goal-conditioned reinforcement learning (GCRL) algorithms. Additional results can be found in Appendix C. + +# 4.1 Datasets and Environments + +We evaluate SCoTS on OGBench benchmark (Park et al., 2025), spanning diverse difficulties, environment sizes, agent state dimensions, and training data qualities. Specifically, the benchmark includes three locomotion environments: PointMaze (controlling a 2D point mass) and AntMaze (controlling an 8-DoF quadrupedal Ant). We consider two distinct dataset types, each designed to evaluate specific challenges. The Stitch dataset comprises short, goal-reaching trajectories limited to four cell units, thus requiring the agent to stitch multiple segments (up to 8) for successful inference. In contrast, the Explore dataset assesses learning navigation behaviors from extensive yet low-quality exploratory trajectories, collected by frequently resampling random directions and injecting significant action noise. For each environment, we report the success rate averaged over all evaluation episodes, where an episode is considered successful if the agent reaches sufficiently close to the goal state within a predefined distance threshold. See Appendix A for dataset details. + +# 4.2 Diversity and State Coverage Analysis + +To investigate whether SCoTS effectively promotes diverse state-space coverage through trajectory stitching, we evaluate its performance in the PointMaze-Giant-Stitch environment. As illustrated in Figure 3, we visualize the incremental stitching process for different values of the novelty weighting parameter $\beta \in \{0,2,20\}$ . We observe that when $\beta = 0$ , trajectory stitching predominantly follows latent directional guidance, resulting in trajectories with limited coverage but clear directional distinctions. With a moderate setting $\beta = 2$ , trajectories exhibit a balanced trade-off, achieving substantial state-space coverage with notable diversity. Conversely, at a higher novelty weight $\beta = 20$ , trajectories broadly cover the state space but lose their distinctiveness, leading to overlapping paths across different latent exploration directions. Based on these results, we use $\beta = 2$ across all environments in our experiments. + +![](images/eb0a3cd59f25d965d090dec86c69f5b5eda79f0d979b699973d71372b07e9e5d.jpg) +Figure 3: Effect of novelty score on Trajectory Stitching. Trajectory stitching examples in the PointMaze-Giant-Stitch environment. The original dataset (Stitch) consists of short segments limited to at most four maze cells. Different colors represent trajectories generated from distinct latent exploration directions $z$ . + +# 4.3 Diffusion Planning with SCoTS-Augmented Data + +We next demonstrate how SCoTS-generated trajectories enhance the ability of diffusion planners to generate feasible, long-horizon plans beyond their training distribution. We compare our approach with offline goal-conditioned reinforcement learning (GCRL) methods including goal-conditioned implicit Q-learning (GCIQL) (Kostrikov et al., 2022), Quasimetric RL (QRL) (Wang et al., 2023), Contrastive RL (CRL) (Eysenbach et al., 2022), and Hierarchical implicit Q-learning (HIQL) (Park et al., 2023). We also include diffusion-based generative planning baselines explicitly designed for long-horizon generalization, such as Generative Skill Chaining (GSC) (Mishra et al., 2023) and Compositional Diffuser (CD) (Luo et al., 2025). + +For our experiments, we adopt a hierarchical diffusion planner (HD) (Chen et al., 2024c) that generates plans through a two-level planning process. Specifically, the high-level diffusion model first generates sparse, temporally coarse waypoints, after which a low-level diffusion model fills in the intermediate states between these waypoints, producing a temporally dense trajectory. Initially constrained by limited and short-horizon training data, we augment the original dataset with SCoTS-generated trajectories. After dataset augmentation, we train diffusion planner and employ a value-based low-level controller for action execution, following recent approaches (Yoon et al., 2025a; Lu et al., + +Table 1: Quantitative results on locomotion tasks in OGBench. Results are averaged over 5 random seeds, each with 50 episodes per task. Standard deviations are reported after the $\pm$ sign. + +
EnvTypeSizeGCIQLQRLCRLHIQLGSCCDHDSCoTS
PointMazeStitchMedium21 ±980 ±120 ±174 ±6100±0100±024±3100±0
Large31 ±284 ±150 ±013 ±6100±0100±017±2100±0
Giant0 ±050 ±80 ±00 ±029±368±30±0100±0
AntMazeStitchMedium29 ±659 ±753 ±694 ±197±296±271±197±1
Large7 ±218 ±211 ±267 ±566±286±236±293±1
Giant0 ±00 ±00 ±02 ±220±165±30±087±2
ExploreMedium13 ±21 ±13 ±237 ±1090±281±242±399±1
Large0 ±00 ±00 ±04 ±521±327±113±298±1
Average12.636.58.436.465.377.925.496.8
+ +![](images/5cbb0160f55e8defcc472ed53092ea171697e03f24a4916417029221f9b8cdc3.jpg) +denoising steps +Figure 4: SCoTS enables long-horizon planning. We visualize trajectories generated by a diffusion planner trained on SCoTS-augmented data, evaluated on two challenging AntMaze datasets: Explore (top) and Stitch (bottom). The original Stitch dataset contains trajectories limited to four maze cells per segment, necessitating extensive stitching, whereas the Explore dataset comprises low-quality trajectories with large action noise. Despite these constraints, SCoTS augmentation allows the planner to synthesize trajectories that substantially surpass the horizon and quality of the original data, connecting specified start $\odot$ and goal $\star$ . + +2025). The plans generated by the diffusion planner serve as sequences of subgoals for the low-level controller. At each step, the low-level controller executes actions toward a subgoal selected from the generated plan; after a fixed horizon or once the subgoal is reached, it dynamically updates the subgoal by selecting the next state at a specified horizon further along in the plan generated by the diffusion planner. For each dataset, we upsample the original data to 5M samples. Additional implementation details, including hyperparameters and specifics of the low-level controller, are provided in Appendix B. + +As shown in Table 1, integrating SCoTS consistently enhances the performance of the hierarchical diffusion planner across all tasks, achieving near-optimal success rates. Notably, the advantage of SCoTS becomes especially pronounced as the complexity and scale of the mazes increase, with the gap between SCoTS and other baselines maximized in the largest (Giant) environments. Furthermore, in the challenging Explore dataset of the AntMaze environment consisting of noisy and short-range exploratory trajectories, augmentation via SCoTS significantly improves the planner ability to generate coherent, long-range, goal-directed plans, clearly highlighting the effectiveness of SCoTS. + +# 4.4 Offline GCRL with SCoTS-Augmented Data + +Although SCoTS is primarily designed for diffusion planners, we additionally evaluate whether trajectories augmented by SCoTS can enhance the performance of existing offline goal-conditioned RL (GCRL) algorithms. Specifically, we retrain widely used offline GCRL algorithms, including GCIQL (Kostrikov et al., 2022), CRL (Eysenbach et al., 2022), and HIQL (Park et al., 2023), on the SCoTS-augmented dataset. All hyperparameters remain identical to their original implementations. Additionally, we compare our approach with SynthER (Lu et al., 2023), which employs an + +Table 2: Performance enhancement of offline GCRL algorithms with SCoTS-augmented dataset. Results are averaged over 5 seeds, each with 50 episodes per task. Standard deviations are indicated by $\pm$ sign. + +
EnvTypeSizeGCIQLCRLHIQL
OriginalSynthERSCoTSOriginalSynthERSCoTSOriginalSynthERSCoTS
PointMazeStitchMedium21 ±930 ±379 ±10 ±10 ±046 ±274 ±677 ±482 ±4
Large31 ±235 ±426 ±20 ±00 ±039 ±213 ±616 ±367 ±1
Giant0 ±00 ±00 ±00 ±00 ±018 ±20 ±00 ±027 ±2
AntMazeStitchMedium29 ±631 ±335 ±253 ±648 ±365 ±394 ±191 ±294 ±1
Large7 ±23 ±47 ±111 ±212 ±219 ±167 ±565 ±391 ±2
Giant0 ±00 ±00 ±00 ±00 ±02 ±12 ±20 ±055 ±5
ExploreMedium13 ±212 ±318 ±33 ±23 ±115 ±337 ±1045 ±894 ±1
Large0 ±00 ±00 ±00 ±02 ±119 ±14 ±512 ±377 ±2
Average12.613.920.78.48.127.936.438.373.4
+ +unconditional diffusion model for transition-level data augmentation. Results summarized in Table 2 clearly demonstrate that SCoTS-generated trajectories consistently outperform SynthER and methods trained solely on the original offline datasets, significantly boosting performance across all tested algorithms. This indicates that augmenting data at the trajectory-level with SCoTS, which explicitly considers long-term dynamics and diversity, provides more effective supervision for learning robust trajectory stitching and long-horizon planning capabilities. + +# 4.5 Ablation Studies + +![](images/8e3276bab3af04d4dad4b30e103b2b3098f182d89b563dcc91589080c8f3b7c9.jpg) +Figure 5: Ablation study on low-level controller horizon. Success rates in the AntMaze-Giant-Stitch environment comparing SCoTS against Compositional Diffuser (CD) (Luo et al., 2025), across various low-level controller horizon lengths. + +![](images/176b0d602cd0848d9b175a77b401f265ec016f6b91c1ab9ec52deeae3f68d13b.jpg) +Figure 6: Dynamic MSE comparison at stitching points. Histograms showing the distributions of Dynamic MSE at trajectory stitching points in the AntMaze-Giant-Stitch environment, comparing results with and without the diffusion-based stitching refinement step. + +Ablation study on low-level controller horizon. We investigate how the performance of our approach (SCoTS) is influenced by varying the horizon length of the low-level controller in the AntMaze-Giant-Stitch environment. As shown in Figure 5, SCoTS achieves consistently strong performance across different horizon lengths $H \in \{5,10,15,20,25\}$ , outperforming the Compositional Diffuser (CD) (Luo et al., 2025). These results demonstrate that the diffusion planner trained with SCoTS generates highly feasible subgoals, maintaining robustness and effectiveness regardless of the chosen low-level execution horizon. + +Effectiveness of diffusion-based stitching refinement. To further illustrate the effectiveness of the diffusion-based stitching refinement step in our SCoTS framework, we quantitatively evaluate its impact on dynamic consistency at stitching points. Specifically, we compute the Dynamic Mean Squared Error (Dynamic MSE) (Lu et al., 2023), defined as: + +$$ +\text {D y n a m i c M S E} = \| f ^ {*} (\boldsymbol {s}, \boldsymbol {a}) - \boldsymbol {s} ^ {\prime} \| _ {2} ^ {2}, +$$ + +which measures how closely the generated transitions adhere to the true environment dynamics $f^{*}$ . Figure 6 compares the distribution of Dynamic MSE at stitching points before and after applying refinement on a logarithmic scale. Results clearly show that diffusion-based refinement substantially reduces dynamic inconsistencies, highlighting its critical role in generating dynamically feasible and coherent trajectories. + +Ablation study on replanning. We employ replanning during a rollout, enabling the agent to recover from failures, such as when the diffusion planner generates unreachable subgoals for the low-level controller. In practice, we set a replanning interval (e.g., every 200 steps); further implementation details are provided in Appendix B. In Table 3, we present an ablation study comparing performance with and without replanning on the PointMaze and AntMaze Stitch datasets from OGBench. SCoTS consistently outperforms Compositional Diffuser (CD) (Luo et al., 2025), the best-performing baseline, even without re + +Table 3: Impact of replanning. Success rates on OGBench PointMaze and AntMaze Stitch datasets, comparing SCoTS and CD (Luo et al., 2025). $\checkmark$ indicates with replanning; $\pmb{x}$ indicates without replanning. + +
EnvSizeCDSCoTS
X✓X✓
PointMazeMedium100 ±0100 ±0100 ±0100 ±0
Large100 ±0100 ±0100 ±0100 ±0
Giant53 ±668 ±389 ±2100 ±0
AntMazeMedium92 ±296 ±297 ±197 ±1
Large76 ±286 ±292 ±293 ±1
Giant27 ±465 ±384 ±287 ±2
Average74.785.993.796.2
+ +planning. Additionally, the performance with and without replanning is similar, highlighting the reliability and efficacy of the SCoTS-augmented diffusion planner. + +# 5 Related Work + +Planning with Diffusion Models. Diffusion probabilistic models (Sohl-Dickstein et al., 2015; Ho et al., 2020) have emerged as powerful tools for reinforcement learning, especially in offline settings. These models iteratively denoise sampled data from noise, effectively learning gradients of the data distribution (Song & Ermon, 2019) and demonstrating strong capabilities in modeling complex trajectories. Early work such as Diffuser (Janner et al., 2022) employed unconditional diffusion models guided by learned value estimators (Dhariwal & Nichol, 2021). Subsequent methods like Decision Diffuser (Ajay et al., 2023) and AdaptDiffuser (Liang et al., 2023) introduced classifier-free guidance and progressive fine-tuning. Recent advancements further leveraged hierarchical structures (Chen et al., 2024c; Li et al., 2023), multi-task conditioning (Ni et al., 2023; He et al., 2023; Dong et al., 2024), and multi-agent setups (Zhu et al., 2023). Additionally, diffusion planners have explored integration with tree search methods (Yoon et al., 2025a,b), refined trajectory sampling techniques (Lee et al., 2023b; Feng et al., 2024; Lee & Choi, 2025), and investigated critical design choices to improve robustness (Lu et al., 2025). Despite these advances, diffusion planners still fundamentally depend on the quality and diversity of the offline training datasets, limiting their ability to generate coherent and feasible long-horizon plans beyond their training distribution. Recent approaches such as Generative Skill Chaining (GSC) (Mishra et al., 2023) and Compositional Diffuser (Luo et al., 2025) address this by composing short segments at test time into long-horizon trajectories. Our work presents an orthogonal solution by directly augmenting the offline dataset itself, significantly enhancing the capability of diffusion planners to generalize to diverse and substantially longer trajectories. + +Data Augmentation for RL. Data augmentation is a recognized strategy for improving sample efficiency and generalization in reinforcement learning (RL). In pixel-based RL, techniques like random image transformations (e.g., cropping, translation) have proven effective in works such as CURL (Laskin et al., 2020b), RAD (Laskin et al., 2020a), and DrQ (Yarats et al., 2021). For state-based observations, methods like S4RL (Sinha et al., 2022) and AWM (Ball et al., 2021) often introduce perturbations to states or learned dynamics models to enhance robustness. Recent advances in generative models have enabled trajectory-level augmentation methods, either at the transition level (Lu et al., 2023; Wang et al., 2024) or the full trajectory level (He et al., 2023; Jackson et al., 2024; Lee et al., 2024). For instance, MTDiff-S (He et al., 2023) generates synthetic trajectories for multi-task scenarios, while Policy-Guided Diffusion (PGD) (Jackson et al., 2024) and GTA (Lee et al., 2024) employ generative models to produce high-reward trajectories guided by policies or returns. + +DiffStitch (Li et al., 2024) further systematically connects trajectories based on extrinsic rewards, yet these methods typically require explicit reward signals and are limited to generating short-horizon trajectories. In contrast, our proposed SCoTS method operates in a reward-free manner, systematically synthesizing long-horizon, diverse, and dynamically consistent trajectories to significantly enhance offline datasets, thereby facilitating the generation of feasible plans in downstream tasks requiring extended horizon reasoning. + +Temporal Distance in RL. Temporal distance has been widely adopted as a structural inductive bias in various reinforcement learning (RL) paradigms, including imitation learning (Sermanet et al., 2018), unsupervised skill discovery (Hartikainen et al., 2019; Park et al., 2024b,a), goal-conditioned RL (Durugkar et al., 2021; Eysenbach et al., 2022; Wang et al., 2023; Bae et al., 2024), and curriculum learning (Zhang et al., 2020; Kim et al., 2023). Recent methods such as METRA (Park et al., 2024b), QRL (Wang et al., 2023), HILP (Park et al., 2024a), and TLDR (Bae et al., 2024) particularly focus on learning temporal distance-preserving representations to facilitate diverse skill discovery or efficient goal-reaching behaviors. Distinct from prior methods, our SCoTS framework explicitly leverages temporal distance-preserving representations to identify temporally viable trajectory segments for stitching. This allows systematic synthesis of extended, diverse, and dynamically consistent trajectories, significantly augmenting offline datasets and improving long-horizon generalization for diffusion-based planners. + +# 6 Conclusion + +In this work, we introduced State-Covering Trajectory Stitching (SCoTS), a novel reward-free trajectory augmentation approach designed to enhance the performance and generalization capabilities of diffusion planners. By leveraging temporal distance-preserving embeddings, SCoTS iteratively stitches together short trajectory segments, systematically extending the diversity and horizon of offline data. Empirical results across challenging benchmarks demonstrated that SCoTS-generated trajectories significantly improve the ability of diffusion planners to perform long-horizon planning and generalize to novel tasks. Furthermore, we showed that our augmented dataset notably enhances the performance of widely used offline goal-conditioned reinforcement learning algorithms, highlighting the broad utility of our approach. + +Limitations. While SCoTS achieves strong empirical performance, it exhibits certain limitations. First, generating augmented trajectories through iterative stitching and diffusion-based refinement introduces significant computational overhead, especially due to the additional training of the diffusion-based stitched model and the trajectory augmentation process. Second, our temporal distance-preserving embeddings do not capture the asymmetric temporal distances between states, potentially limiting their effectiveness in highly asymmetric or disconnected Markov Decision Processes (MDPs), such as object manipulation tasks involving irreversible actions or environments containing isolated regions with sparse connectivity. + +Impact Statement. This paper advances the field of diffusion-based planning by introducing a novel trajectory augmentation method, enhancing long-horizon reasoning capabilities in reinforcement learning. While we do not identify direct negative societal impacts stemming from this research, practitioners are encouraged to carefully assess real-world implications, particularly regarding safety and reliability, prior to deploying this method in practical scenarios. + +# Acknowledgements + +This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2019-0-00075, Artificial Intelligence Graduate School Program (KAIST); No. 2022-0-00984, Development of Artificial Intelligence Technology for Personalized Plug-and-Play Explanation and Verification of Explanation; No. RS-2024-00457882, AI Research Hub Project; No.RS-2022-II220184, Development and Study of AI Technologies to Inexpensively Conform to Evolving Policy on Ethics), and by the InnoCORE program of the Ministry of Science and ICT (N10250156). + +# References + +Ajay, A., Du, Y., Gupta, A., Tenenbaum, J. B., Jaakkola, T. S., and Agrawal, P. 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If you obtained IRB approval, you should clearly state this in the paper. + +We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. +- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: This research does not use large language models as part of its methodology. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Details of Datasets + +We evaluate our method on the OGBench benchmark (Park et al., 2025) $^{1}$ . Since our primary goal is to assess trajectory stitching capability and long-horizon reasoning, we specifically utilize the Stitch and Explore datasets. As shown in Figure 7, the Stitch dataset is explicitly designed to challenge trajectory stitching ability, comprising short, goal-reaching trajectories limited to a maxi- + +mum length of four cell units. Consequently, agents must effectively stitch together multiple short segments (up to eight) to successfully complete long-horizon tasks. In contrast, the Explore dataset is designed to test navigation skills learned from extensive yet low-quality trajectories. These trajectories are generated by commanding a low-level policy with random movement directions re-sampled every ten steps, along with significant action noise. Each demonstration trajectory typically spans only two to three blocks, resulting in noisy and clustered paths that pose additional challenges for evaluating the ability to learn effective policies from highly suboptimal data. + +Table 4: Dataset specifications. + +
EnvTypeSize# Transitions# EpisodesData Episode Length
PointMazeStitchMedium1M5,000200
Large1M5,000200
Giant1M5,000200
AntMazeStitchMedium1M5,000200
Large1M5,000200
Giant1M5,000200
ExploreMedium5M10,000500
Large5M10,000500
+ +![](images/18ad8990a26643076c7f51c931996b77f2d28e8852fb006d67b3140851cb51f2.jpg) +(a) PointMaze-Medium-Stitch + +![](images/ea9e92dacc1f011735b5d523a66f6307c986b3972ad08226c7fbc470e8fa9e80.jpg) + +![](images/b86dec5e7b0b0e72b75db9d7b681f308bf56979aa2d72819cec2d55e12c89f92.jpg) +(d) AntMaze-Medium-Stitch +(g) AntMaze-Medium-Explore +Figure 7: Visualization of trajectories from OGBench datasets. Each sub-figure illustrates example trajectories from different combinations of environments and datasets used in our experiments. + +![](images/eaa013f728c18fa92d1e7898b2c6043bccb46189cfc0d16d2fe658f7ca213017.jpg) +(b) PointMaze-Large-Stitch + +![](images/33b2d02d7404e38a1e9de709d2550424ec9e1fd6b7252d6ed770b81f119bb46c.jpg) + +![](images/3195cd296eb5f5b5b2dd8dc583fa0b88d64e64079296d758239de510daaf81a3.jpg) +(e) AntMaze-Large-Stitch +(h) AntMaze-Large-Explore + +![](images/5affe4bac6e84c425f0e3a6a0f7db19ecd9b25000214405f395adde899fb2d0e.jpg) + +![](images/9cd14690ee40d577259030880bf01b842011b500c12bda9548c2b22372d41434.jpg) +(c) PointMaze-Giant-Stitch +(f) AntMaze-Giant-Stitch + +# B Implementation Details + +Network architecture. We utilize DiT1D (Peebles & Xie, 2023) as the neural network backbone for both the diffusion planner and the stretcher, due to its large receptive field and effectiveness in modeling trajectory-level dependencies. Following prior studies (Dong et al., 2023; Lu et al., 2025), we employ a DiT1D architecture with a hidden dimension of 256, a head dimension of 32, and a total of 8 DiT blocks consistently across all environments. + +Table 5: Hyperparameters for SCoTS. + +
ComponentHyperparameterValueTuning Choices
SCoTS: Temporal Distance-Preserving Embedding (φ)
Learning Rate3 × 10-4-
Latent Dimension32-
Batch Size1024-
Training Steps1,000,000-
Network BackboneMLP-
MLP Dimensions(512, 512, 512)-
Expectile (ξ for ℓξ2)0.95-
SCoTS: Inverse Dynamics Model (for actions in Daug)
Network BackboneMLP-
MLP Dimensions(256, 256, 256)-
Training Steps200,000-
SCoTS: Stitching Process Parameters
Top-k Candidates (Search)10-
kdensity (Novelty Score)30-
Novelty Weight (β)2.0-
Augmented Dataset Size~5M transitions-
Nstitch (Stitches per Traj.)Task-dependent (e.g., 40)-
Ntraj (Generated Traj.)Task-dependent (e.g., 5,000)-
SCoTS: Diffusion-based Stitcher (pθstitcher)
Network BackboneDiT1D-
Learning Rate2 × 10-4-
Weight Decay1 × 10-5-
Batch Size64-
Training Steps1,000,000-
SolverDDIM-
Sampling Steps (DDIM)20-
Horizon (Hstitcher)26-
Hierarchical Diffusion Planner (HD)
Network BackboneDiT1D-
Learning Rate2 × 10-4-
Weight Decay1 × 10-5-
Batch Size64-
Training Steps1,000,000-
SolverDDIM-
Sampling Steps (DDIM)101 (Stitch), 401 (Explore)-
Plan Horizon (on original data)501 (M/L), 1001 (G/Explore)-
Plan Horizon (on Daug)26-
Execution Parameters
Low-level Controller HorizonTuned{5,10,15,20,25}
Replanning IntervalTuned{50,100,200}
+ +Details of the low-level controller. A key challenge in diffusion-based planning is balancing global trajectory coherence with effective low-level control in high-dimensional state-action spaces (Chen et al., 2024a,b; Yoon et al., 2025a). Previous approaches, such as PlanDQ (Chen et al., 2024b) and MCTD (Yoon et al., 2025a), address this issue by integrating high-level diffusion planners with separately trained low-level controllers. Similarly, we adopt a hierarchical strategy, where the diffusion planner generates plans based primarily on compact, lower-dimensional state representations (e.g., positions of the agent itself), delegating the fine-grained, low-level action execution to a dedicated low-level controller. In our experiments, we specifically employ GCIQL (Kostrikov et al., 2022) as the learned low-level policy in the PointMaze environments and CRL (Eysenbach et al., 2022) in the AntMaze environments. A detailed visualization of generated subgoals and their corresponding execution rollouts can be seen in Figure 9. Furthermore, an ablation study examining the impact of the horizon length of the low-level controller is presented in Figure 5. + +Implementation details for SCoTS and diffusion planning. In the temporal distance-preserving search stage of SCoTS, we retrieve the top $k = 10$ candidate segments based on their proximity in the learned latent embedding space during each stitching step. For computing the novelty score, we utilize a density estimator parameter $k_{\mathrm{density}} = 30$ and set the novelty weighting factor $\beta = 2.0$ consistently across all tested environments. The horizon length for the diffusion-based stretcher is uniformly set to $H_{\mathrm{stitcher}} = 26$ . + +To generate the augmented dataset $\mathcal{D}$ aug, we perform the stitching procedure $N$ stitch iterations per trajectory, creating a total of $N_{\mathrm{traj}}$ trajectories, thus ensuring the augmented dataset comprises approximately 5 million transitions. Specifically, in the AntMaze-Large-Stitch environment, we set $N_{\mathrm{stitch}} = 40$ and $N_{\mathrm{traj}} = 5000$ . + +For configuring the Hierarchical Diffusion (HD) planner (Chen et al., 2024c), parameters are adapted according to the properties of the training data. When training on the original Stitch and Explore datasets, which contain inherently shorter trajectories (as detailed in Table 4, column "Data Episode Length"), we set the high-level planning horizon to 101 steps for Stitch and 401 steps for Explore, both with temporal jumps of 26 steps between waypoints. However, when utilizing SCoTS-augmented datasets that feature longer and more diverse trajectories, we extend this planning horizon to 501 steps for Medium and Large environments, and to 1001 steps for Giant environments, maintaining the temporal jump of 26 steps. Similarly, for SCoTS-augmented Explore datasets, we also use a planning horizon of 1001 steps with 26-step jumps. + +We apply jumpy denoising with DDIM sampling (Song et al., 2020) using 20 denoising steps across all environments. Additionally, we tune the replanning interval from the set $\{50,100,200\}$ steps and tune the horizon for the low-level controller from $\{5,10,15,20,25\}$ . A full list of the hyperparameters is reported in Table 5. + +Practical implementation of temporal distance-preserving search. Our SCoTS framework relies on a learned latent space $\mathcal{Z}$ where the $L_{2}$ distance, $\| \phi (s) - \phi (g)\| _2$ , approximates the optimal temporal distance $d^{*}(s,g)$ between states (as detailed in Section 3.1). A critical step in SCoTS is the efficient identification of suitable candidate trajectory segments from a large offline dataset $\mathcal{D}$ . This requires a fast nearest neighbor search mechanism within the learned latent space $\mathcal{Z}$ . To achieve this, we employ an Inverted File (IVF) index from the Faisss library (Douze et al., 2024), which is specifically designed for large-scale similarity searches. + +The practical implementation of this search mechanism involves several stages. First, we prepare the data for indexing. This consists of computing the latent embeddings $\phi(s_{\mathrm{init}})$ for the initial states $s_{\mathrm{init}}$ of all trajectories within the offline dataset $\mathcal{D}$ . Let $d$ denote the dimensionality of these latent embeddings. An IVF index is then constructed upon this collection of $d$ -dimensional vectors. The construction process begins by partitioning the latent vectors into $n_{\mathrm{list}}$ clusters using the $k$ -means algorithm. Each cluster is represented by a centroid $\mathbf{c}_j \in \{\mathbf{c}_1, \dots, \mathbf{c}_{n_{\mathrm{list}}}\}$ . Subsequently, each latent vector $\phi(s_{\mathrm{init}})$ in our collection is assigned to its nearest centroid, and for each centroid, an inverted list is maintained, storing references to the vectors belonging to its cluster. + +During the temporal distance-preserving search phase of SCoTS (detailed in Algorithm 1, line 10), the latent embedding of the current composed trajectory endpoint, $\phi (\mathrm{end}(\tau_{\mathrm{comp}}))$ , serves as the query vector q. To find the $k$ nearest neighbors for q, the IVF index first identifies a limited set of clusters whose centroids $\{\mathbf{c}_j\}$ are closest to the query vector q. The search for neighbors is then confined to + +the latent vectors stored within the inverted lists corresponding to these selected clusters. This targeted approach significantly prunes the search space compared to an exhaustive search. Furthermore, the Faisss library provides support for GPU acceleration, which can further expedite this search process and enable efficient candidate retrieval. Once the $k$ nearest latent embeddings corresponding to initial states of segments are identified, we retrieve the full original trajectory segments from $\mathcal{D}$ to form the candidate set for the stitching process. + +Computational resources and runtimes. All experiments were conducted using a single NVIDIA A10 GPU. The approximate execution times for each component of our method are as follows: + +- Temporal distance-preserving embedding training: 1.5 hours +- Inverse dynamics model training: 0.25 hours +- Low-level controller training: 2.5 hours +- Diffusion-based stretcher training: 7 hours +- Trajectory augmentation via SCoTS: 0.5 hours +- Diffusion planner training: 18 hours + +These times are per model training instance or data generation run and may vary slightly depending on the specific environment and dataset characteristics. + +# C Additional Results + +Ablation study on temporal distance-preserving embedding. To isolate the contribution of our learned latent space, we compare the performance of the HD (Chen et al., 2024c) planner trained on SCoTS-augmented data where the stitching process was guided by either our temporal distance-preserving embedding or raw state space. As shown in Table 6, temporal distance-preserving representation is crucial for effective stitching. This is especially true in high-dimensional environments like AntMaze, where a small Euclidean distance in the raw state space (e.g., between two similar joint configurations) does not guarantee reachability. Even in Pointmaze, where state-space distance is more intuitive, the learned compact latent space provides a better-structured representation for stitching temporally extended trajectories. + +Table 6: Ablation on the temporal distance-preserving embedding. Success rates on OGBench PointMaze and AntMaze Stitch datasets. $\times$ indicates stitching guided by raw state-space distances (w/o temporal embedding), while $\checkmark$ indicates guidance from learned temporal embedding. + +
EnvSizeSCoTS
X✓
PointMazeLarge93±0100±0
Giant52±1100±0
AntMazeLarge45±193±1
Giant7±287±2
Average49.395.0
+ +Ablation study on diffusion-based stitching refinement. To demonstrate the importance of the refinement step, we compare the performance of both a diffusion planner (HD) and a GCRL algorithm (HIQL) trained on SCoTS data generated with and without the diffusion-based refinement. The results, summarized in Table 7, reveal the critical importance of this component. Without refinement, the connection points between stitched segments can suffer from large dynamic inconsistency errors, as illustrated in Figure 6. This performance + +Table 7: Ablation on diffusion-based stitching refinement. Success rates on AntMaze Stitch datasets for HD (Chen et al., 2024c) and HIQL (Park et al., 2023). $\pmb{x}$ indicates training on SCoTS data generated without refinement, while $\sqrt{}$ indicates training on data generated with refinement. + +
EnvSizeHDHIQL
X✓X✓
AntMazeLarge85±393±152±291±2
Giant53±187±211±255±5
Average69.090.031.573.0
+ +degradation is particularly significant when training GCRL algorithms like HIQL, which are highly sensitive to the dynamic validity of the training transitions. + +Sensitivity analysis on the novelty weight $\beta$ . As illustrated in Figure 3, using only the progress score $(\beta = 0)$ , which quantifies the alignment with a latent exploration direction, is effective at generating long, temporally-extended trajectories with clear directional distinctions. However, relying solely on this directional guidance can be suboptimal. The learned temporal distance-preserving embedding $\phi$ is a powerful but imperfect approximation of the true environment topology. Due to embedding errors, the geometry of the latent space can be distorted. Consequently, a straight line between two latent states, $\phi(s)$ and $\phi(s')$ , may not correspond to a feasible path in the underlying MDP. Therefore, exploratory detours are often necessary to find a valid temporally extended path. This is precisely where the novelty score becomes critical. It encourages these exploratory detours by rewarding the selection of segments that lead to less-visited states. This allows the agent to navigate around the imperfections and distortions in the learned latent space, discovering feasible and often more effective paths. To demonstrate this complementary effect quantitatively, we conducted an additional ablation study on 2 OGBench tasks. The table below compares the performance of the HD planner trained with SCoTS data generated using different novelty weights $\beta$ (5 seeds). The results, summarized in Table 8, suggest that while directional stitching alone $(\beta = 0)$ provides a notable performance improvement, introducing and balancing it with the novelty score $(\beta > 0)$ yields substantial further gains. This effect is especially pronounced on the more complex Giant task. + +Table 8: Sensitivity to novelty weight $\beta$ . + +
EnvTypeSizeHDβ=0β=2β=4β=8
AntMazeStitchLarge36±287±293±192±185±3
Giant0±063±387±289±274±3
+ +Sensitivity analysis on sub-trajectory length $H$ . As shown in Table 9, performance on the Stitch dataset is relatively robust to the choice of $H$ . However, on the Explore dataset, shorter segments perform best. We hypothesize this is because the Explore dataset consists of low-quality, noisy trajectories. Using shorter segments allows SCoTS to be more selective, finding and connecting the temporally-extended parts of trajectories. + +Table 9: Sensitivity to sub-trajectory length $H$ . + +
EnvTypeSizeH=26H=52H=104
AntMazeStitchGiant87±289±287±2
ExploreLarge98±193±188±2
+ +Sensitivity analysis on the number of retrieved segments $K$ . Table 10 shows that performance is robust as long as $K$ is not too small. A very small $K$ limits the diversity of candidate segments, hindering the effectiveness of our progress and novelty-based selection. While performance is stable for larger $K$ , we expect an excessively large $K$ could eventually degrade performance by increasing the chance of retrieving dynamically inconsistent segments. + +Table 10: Sensitivity to number of retrieved segments $K$ . + +
EnvTypeSizeK=3K=10K=20
AntMazeStitchGiant65±387±289±2
ExploreLarge72±298±197±1
+ +Evaluation on dense-reward offline RL benchmarks. While our paper focuses on tasks where extrinsic rewards are absent, the SCoTS framework is not limited to goal-conditioned tasks. To demonstrate its applicability to general, dense-reward offline RL, we conducted additional experiments on four standard MuJoCo locomotion benchmarks from D4RL (Kumar et al., 2020), utilizing suboptimal medium and medium-replay datasets. It is important to clarify that our use of reward-free refers to the stitching mechanism itself, which is guided by the intrinsic temporal structure of the data rather than extrinsic task rewards. This allows the core data augmentation to be task-agnostic. + +For a fair comparison, we used the same Diffuser architecture and planning horizon (32) as described in the original paper (Janner et al., 2022). We upsample the original data to 2M samples using the same SCoTS procedure, with sub-trajectory lengths set to 4. Additionally, to label these new trajectories with rewards, we additionally trained a reward model, $r_t = f_\omega(s_t, a_t)$ , on the original offline dataset. As shown in Table 11, training on SCoTS-augmented data consistently improves the performance of the original Diffuser. Even without architectural or sampling strategy changes, + +by stitching sub-trajectories in a way that is temporally extended and exploratory, SCoTS generates novel behaviors (e.g., trajectories that travel further) that are not present in the original suboptimal dataset. + +Table 11: Performance on D4RL benchmarks. We report normalized average returns of the Diffuser and Diffuser trained on SCoTS-augmented data. Results are averaged over 50 planning seeds. + +
DatasetDiffuserDiffuser + SCoTS (ours)
hopper-medium-v274.3±1.493.5±1.2
walker2d-medium-v279.6±0.680.2±0.5
hopper-medium-replay-v293.6±0.496.7±0.6
walker2d-medium-replay-v270.6±1.678.5±1.9
Average79.587.2
+ +Evaluation on noisy and partially observable navigation. To evaluate the robustness of SCoTS on noisy and low-quality data beyond the controlled OGBench settings, we conducted additional experiments using a mobile robot navigation task in realistic, large-scale environments (a $35\mathrm{m}\times 39\mathrm{m}$ corridor and a $50\mathrm{m}\times 110\mathrm{m}$ building), following the setup from HRL (Lee et al., 2023a). We simulated low-quality data by generating short, disconnected segments. Specifically, trajectories were collected by randomly selecting start and goal locations within short ranges (5m) using HRL (Lee et al., 2023a). To handle the noisy raw 2D LiDAR measurements (512 rays over 360-degree), we converted these scans into 64x64 occupancy grid images. These images were then encoded using a pre-trained $\beta$ -VAE (Higgins et al., 2017) into a compact 8-dimensional latent representation, providing structured, low-dimensional state inputs for the agent. Subsequently, the SCoTS stitching method was applied to augment this fragmented dataset. The results, shown in Table 12, clearly indicate that SCoTS effectively enhances performance, even with highly fragmented and noisy datasets, demonstrating its robustness and practical applicability to robot navigation tasks. + +Table 12: Performance on a mobile robot navigation task. Goal-reaching success rates are reported over 100 episodes for the HIQL (Park et al., 2023) trained with and without SCoTS augmentation. + +
EnvironmentHIQLHIQL + SCoTS (ours)
Building (50m × 110m)27±753±4
Corridor (35m × 39m)0±031±5
+ +Visualization of temporal distance-preserving latent representations. We train temporal distance-preserving latent representations with dimension 32 across all environments. To visualize these learned representations, we apply a $t$ -distributed stochastic neighbor embedding (t-SNE) to project the 32-dimensional latent vectors onto a 2-dimensional plane, as shown in Figure 8. Recall from Equation 6 that we parameterize a goal-conditioned value function $V(s, g)$ following (Park et al., 2024a): + +$$ +V (\boldsymbol {s}, \boldsymbol {g}) := - \left\| \phi (\boldsymbol {s}) - \phi (\boldsymbol {g}) \right\| _ {2}, \tag {13} +$$ + +which approximates the optimal goal-conditioned value function, defined as the maximum possible return (cumulative sum of rewards) for sparse-reward settings. Specifically, an agent receives a reward of 0 if the $l_{2}$ distance between states $s$ and $g$ is within a small threshold $\delta_g$ , and -1 otherwise. The embedding function $\phi$ is trained using a temporal-difference objective inspired by implicit Q-learning (Kostrikov et al., 2022) on the offline dataset $\mathcal{D}$ . As illustrated in Figure 8, the learned representations effectively capture the temporal proximity between states, resulting in latent spaces where states that are temporally close in the environment are also clustered closely in the embedding space. + +Visualization of rollout execution. We visualize a generated plan by the diffusion planner trained on SCoTS-augmented data, along with its corresponding rollout execution in the AntMaze-Giant-Stitch environment, as illustrated in Figure 9. The initial image (top-left) shows the overall planned trajectory generated by the diffusion planner, with subgoals marked by green spheres. Subsequent images provide sequential snapshots from the rollout execution, demonstrating the agent actively pursuing and reaching these subgoals. This visualization highlights how effectively the generated high-level plan guides the low-level controller during task execution. + +![](images/dea9d50707e7f760e1b5ca24155ba48e90ea51fa6ad6a4dc9eab1d30ac1cc645.jpg) +(a) States (Medium) + +![](images/24eb774f874bf99dc5b0b4d443c2a10a18801360c1f52647a1577e3bbbcc47cb.jpg) +(b) PointMaze-Medium-Stitch + +![](images/65bac07dda6db40d0b1e8d4cae6b842a7345bde145fb52c79754f30f621ee2e9.jpg) +(c) AntMaze-Medium-Stitch + +![](images/523a413f2393da17cbbd95abe9d529e4f98998c2a8b3c57b94f2cb446bfd7ea8.jpg) +(d) AntMaze-Medium-Explore + +![](images/9d03288b4697bb5f71ba62c728d823fefe8e926ecd469b20e93d7627a2432b54.jpg) +(e) States (Large) + +![](images/989655dcb64285ad7103abe964e2f9eed3ddd5384b7ca5867ae570879db51136.jpg) +(f) PointMaze-Large-Stitch + +![](images/d9217abd3cf298200d3f0a0677f485ddf04ab4be19db4215a04187bcd80cd0b2.jpg) +(g) AntMaze-Large-Stitch + +![](images/ef9a792a519b27f372b362243099f60265d64eaa9b97b77144dbac11937cdb37.jpg) +(h) AntMaze-Large-Explore + +![](images/6e6120b8d74270a97703865d6ba692edb3e17b57f7815318ff4f1ba4ced30739.jpg) +(i) States (Giant) +Figure 8: Visualization of learned temporal distance-preserving latent representations. The left-most column shows original states from maze environments of varying sizes (Medium, Large, Giant). Subsequent columns illustrate t-SNE projections of latent embeddings $\phi(s)$ for corresponding OG-Bench datasets, maintaining the same color scheme for consistency. This visualization demonstrates how spatial proximity and structure in the original state space are preserved and reflected in the learned latent representations. + +![](images/c0da0906b7906bc7d9b3a7f60bdf9335f6a99579e07213e667c1889bf56b1d7a.jpg) +(j) PointMaze-Giant-Stitch + +![](images/63d0eeedf1d419ba98931949a4770191780092e0c53a946ccfa0888894b3b909.jpg) +(k) AntMaze-Giant-Stitch + +Visualization of trajectories generated by SCoTS. In Figure 10, 11, and 12, we present representative examples of trajectories synthesized by our SCoTS framework across all considered environments and dataset types. Compared to the original trajectories provided in Figure 7, the SCoTS-generated trajectories clearly demonstrate extended coverage, illustrating the effectiveness of our method in augmenting the original offline datasets. + +# D Baseline Performance Sources + +Performance scores reported for offline goal-conditioned reinforcement learning (GCRL) methods, including Goal-Conditioned Implicit Q-Learning (GCIQL) (Kostrikov et al., 2022), Quasimetric RL (QRL) (Wang et al., 2023), Contrastive RL (CRL) (Eysenbach et al., 2022), and Hierarchical Implicit Q-Learning (HIQL) (Park et al., 2023), are sourced from Table 2 in Park et al. (2025). Scores for diffusion-based generative planning methods explicitly designed for long-horizon generalization, including Generative Skill Chaining (GSC) (Mishra et al., 2023) and Compositional Diffuser (CD) (Luo et al., 2025), are sourced from Tables 1 and 2 in Luo et al. (2025). + +![](images/f2499bd46100d7a052c68d3bd00437027ac236fac94fa1ac8921cf515c8474d6.jpg) +Figure 9: Visualization of diffusion Planner rollout execution. The top left image shows the planned trajectory generated by the diffusion planner, with subgoals marked by green spheres. Subsequent images sequentially illustrate the agent progressing toward these subgoals in the AntMaze-Giant-Stitch environment, demonstrating effective guidance provided by the generated plan. + +![](images/3a97ee355befa115a2e2a5e1dbf4302c1f10d7d82f87529f6371f77bf0b1da30.jpg) +Figure 10: SCoTS-augmented trajectories for PointMaze Stitch datasets. For each PointMaze Stitch dataset, the leftmost column shows trajectories from the original OGBench dataset. The subsequent columns are examples of SCoTS-generated trajectories. + +![](images/cfcdf54769125a377a2c492fe11f900e642b7446e243e83ccbae7261a8863fd9.jpg) +(a) PointMaze-Medium-Stitch + +![](images/92c14d36e2d7a618ff8dc671a939aeaf890f3cdfb8c37ce90d371440533fa98e.jpg) + +![](images/ab2e2332b01c403f0b8b959a162eede13f6f9b9c82a49b2b5d4ec0ef32388fa6.jpg) + +![](images/fa5b5495312548c7b12c611a4cd5535feab298e6deae4b364072071ee33c820d.jpg) + +![](images/016ff011bf46f776bcfad7c5229cee146a089075f042cf7d58e5edd46e43852c.jpg) +(b) PointMaze-Large-Stitch + +![](images/326b9e0f752c7a594ec36e56b6a7cb0b64c5207ee7419e0a7a2811ff769ce3cb.jpg) + +![](images/a5d862df0fc67621e596b9407b3ab70382264bed86985b3db6daafe78686f24b.jpg) + +![](images/673330198138ef8c1813745280ddb38b2c887d8044924ee0028e6d081d47493a.jpg) + +![](images/044eebc95400609d2f09d23f74fe67111d2b73e42b9cbc332d28ba43a0eb5dd7.jpg) +(c) PointMaze-Giant-Stitch + +![](images/a621daa52b0ac675797e93e01bb3593cec453bc68690e1c1bf26bc783834cec0.jpg) + +![](images/cdfd9b73203a27bca2ef53a7d63ee7529cc2be91f0fcc7c2dc60a6b24ed9004a.jpg) + +![](images/51761137c1ffe1d57aa624631f11c508112cb1ab546299807282cbc5d565814d.jpg) +Figure 11: SCoTS-augmented trajectories for AntMaze Stitch datasets. For each AntMaze Stitch dataset, the leftmost column shows trajectories from the original OGBench dataset. The subsequent columns are examples of SCoTS-generated trajectories. + +![](images/b04a00d426af53a27aef2e9877e270bd9c973c4ac817416e284fc849c6bab65d.jpg) +(d) AntMaze-Medium-Stitch + +![](images/4637af278c3e9817e618abfa1d99fa341b38de4eb762917d5edef222a1bd282f.jpg) + +![](images/706dd17b749b425e652b82f3871b368567f5ba85d27209d86c1ee6988be8b803.jpg) + +![](images/42bf7c3c03b7426c68cb7cfb970fa9a72c39003bf907191712b0d0e19610a90a.jpg) + +![](images/38432470a3f5d8b3a6ffdf551b17061dacb95c51b7142e7016271b0afc58fa74.jpg) +(e) AntMaze-Large-Stitch + +![](images/2f1f1593e7bcdf07cd685569254aa3d2f7b4a6f39c1f0ce0448caa0c7befa6a5.jpg) + +![](images/013ddc91da92a67459bc5ef0134591bb3f05c2cc21ccd55cb9061709039be843.jpg) + +![](images/767700495970bd808537425c8a9de9208de0cc2f7aa550b7ea037baa542a2a48.jpg) + +![](images/912f454f43b0d90af02226fce441bc6a76d535beeebdb29bda62169c7a5f2475.jpg) +(f) AntMaze-Giant-Stitch + +![](images/72d6e1f01b7402fa7ad677836a2935c840ef3db0e8a12f01bfe7b2eee7c98c52.jpg) + +![](images/cadf9b783310c00be27d0f2f4306e675063e8ce6f30f0ca50ee680eddefc58fc.jpg) + +![](images/8a898344b9cb2ce3f714f6b761407c4fea4ffc3f873ae9de058227b99043cbb4.jpg) +Figure 12: SCoTS-augmented trajectories for AntMaze Explore datasets. For each AntMaze Explore dataset, the leftmost column shows trajectories from the original OGBench dataset. The subsequent columns are examples of SCoTS-generated trajectories. + +![](images/42fda12d3f6d3dfd2709d4c243ccc5ffd5683c4d62c19b4e242abd13295cac5b.jpg) +(g) AntMaze-Medium-Explore + +![](images/a84d8cdb9dd13fdd3431edff20001ee5eb5fecd214b38235160ef755098eab36.jpg) + +![](images/e003ea0a6448dde6c1ed16d3a863566df2a3ed3360cd7c2797ede547cde6dab0.jpg) + +![](images/4dfab513bb5b80e48aae862470d1ad1df0c2d79b211c1b7ca340ddd1003e9c93.jpg) + +![](images/ad4e23ffd5c05716af457a73f0f5e6fe4cfdbba4d1c0624ed0443449a1ac2899.jpg) +(h) AntMaze-Large-Explore + +![](images/961032c65968dcf30b186492e4bde60d750267732540f2c9c9a3703344ce39f7.jpg) + +![](images/190b88fc75d6c190396e52873a6ba701e239318c8e3acf0af04c6ce844c5c13b.jpg) \ No newline at end of file diff --git a/statecoveringtrajectorystitchingfordiffusionplanners/images.zip b/statecoveringtrajectorystitchingfordiffusionplanners/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..ba268fdf28c82eb31c1cabf01b10be06cfab58d9 --- /dev/null +++ b/statecoveringtrajectorystitchingfordiffusionplanners/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:755eb0525e93e6eb9eef8d09083e067d0ed60f07111a307e9dcd267ce951e6b0 +size 1460615 diff --git a/statecoveringtrajectorystitchingfordiffusionplanners/layout.json b/statecoveringtrajectorystitchingfordiffusionplanners/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..77eb131b39dd227715929242d5815649af509b6b --- /dev/null +++ b/statecoveringtrajectorystitchingfordiffusionplanners/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:26a2c820ae7869fc293dab6707f27685ddfe067e32beb4e013db9c04f91a283d +size 995110 diff --git a/stateentropyregularizationforrobustreinforcementlearning/d2208d79-428f-48f1-a0e4-fd5efa94a138_content_list.json b/stateentropyregularizationforrobustreinforcementlearning/d2208d79-428f-48f1-a0e4-fd5efa94a138_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..520393fac1eda56555cb8482afccedcd3318578f --- /dev/null +++ b/stateentropyregularizationforrobustreinforcementlearning/d2208d79-428f-48f1-a0e4-fd5efa94a138_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fef8001a11756766c8d8bb648f01a7aba439e1391c84015a98e44ee23f6e619b +size 189340 diff --git a/stateentropyregularizationforrobustreinforcementlearning/d2208d79-428f-48f1-a0e4-fd5efa94a138_model.json b/stateentropyregularizationforrobustreinforcementlearning/d2208d79-428f-48f1-a0e4-fd5efa94a138_model.json new file mode 100644 index 0000000000000000000000000000000000000000..4a8575c18a38b72c4834549ad5b7ac81b660f77d --- /dev/null +++ b/stateentropyregularizationforrobustreinforcementlearning/d2208d79-428f-48f1-a0e4-fd5efa94a138_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:70edba84cdce9fb6c3482593505b85697e8daebf6ff518a3d93dd2d26a2428c8 +size 242197 diff --git a/stateentropyregularizationforrobustreinforcementlearning/d2208d79-428f-48f1-a0e4-fd5efa94a138_origin.pdf b/stateentropyregularizationforrobustreinforcementlearning/d2208d79-428f-48f1-a0e4-fd5efa94a138_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8aaf96766a90899701b33fb7c81868bf4a72efa3 --- /dev/null +++ b/stateentropyregularizationforrobustreinforcementlearning/d2208d79-428f-48f1-a0e4-fd5efa94a138_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1b3fc9c075f9a81294ee3248fa36e7de6d05240f4e4b065c5521f56ad6414891 +size 1238330 diff --git a/stateentropyregularizationforrobustreinforcementlearning/full.md b/stateentropyregularizationforrobustreinforcementlearning/full.md new file mode 100644 index 0000000000000000000000000000000000000000..c71b317a5ddd0fa2f4f15e61ef55d4494fa6565f --- /dev/null +++ b/stateentropyregularizationforrobustreinforcementlearning/full.md @@ -0,0 +1,971 @@ +# State Entropy Regularization for Robust Reinforcement Learning + +Yonatan Ashlag* Technion + +Uri Koren* Technion + +Mirco Mutti Technion + +Esther Derman +MILA Institute + +Pierre-Luc Bacon +MILA Institute + +Shie Mannor +Technion, NVIDIA Research + +# Abstract + +State entropy regularization has empirically shown better exploration and sample complexity in reinforcement learning (RL). However, its theoretical guarantees have not been studied. In this paper, we show that state entropy regularization improves robustness to structured and spatially correlated perturbations. These types of variation are common in transfer learning but often overlooked by standard robust RL methods, which typically focus on small, uncorrelated changes. We provide a comprehensive characterization of these robustness properties, including formal guarantees under reward and transition uncertainty, as well as settings where the method performs poorly. Much of our analysis contrasts state entropy with the widely used policy entropy regularization, highlighting their different benefits. Finally, from a practical standpoint, we illustrate that compared with policy entropy, the robustness advantages of state entropy are more sensitive to the number of rollouts used for policy evaluation. + +# 1 Introduction + +Despite the impressive success of RL across various synthetic domains [29, 37, 51, 6], challenging issues still need to be addressed before RL methods can be deployed in the real world. Some of these challenges involve imperfect models, noisy observations, and limited data. In such settings, policies trained on nominal environments may perform poorly when faced with deviations from the assumed dynamics or reward structure at deployment [34]. This issue has led to a growing interest in robust RL, which aims to ensure reliable performance despite model misspecification or uncertainty [38, 46, 59, 31, 47]. Over the years, the well-established connection between regularization and robustness in machine learning has been extended to the RL setting [61, 19, 21, 11], where regularization has been shown to induce robustness to adversarial perturbations of the reward function [12, 5] and the transition kernel [10]. This duality has particularly been studied for policy entropy regularization [13]. + +Recent works [23, 50, 65, 25, 4] proposed to regularize the standard RL objective with the entropy of the distribution over states induced by a policy [18], either alone or in combination with policy entropy. This has empirically shown better exploration and consequently, improved sample efficiency [50]. However, to our knowledge, formal studies on the effect of state entropy regularization and its connection with robustness do not exist in the literature yet. Thus, an open question is whether robustness is a by-product of state entropy regularization just like it is for policy entropy, and if it is, what type of robustness results from state entropy regularization? + +![](images/023ea2f37273c38dcd2ded1710a8a642560df6c249a2b57c8f801ebec3fad3c3.jpg) +Figure 1: State entropy regularization leads to diverse behavior and improved robustness. We plot 50 trajectories from standard RL, policy-entropy-regularized RL and state-entropy-regularized RL on a manipulation task (left). Policies are trained on the nominal environment and evaluated in a perturbed version with a red obstacle along the optimal trajectory. The state-entropy-regularized agent often bypasses the obstacle and solves the perturbed task. + +Intuitively, policy entropy encourages stochasticity in action selection but typically spreads randomness along a single dominant trajectory. This makes it effective in smoothing out small or uniform perturbations—but also fragile when that trajectory is disrupted. For instance, a large obstacle blocking a single high-reward path can cause policy entropy-regularized agents to fail catastrophically, as illustrated in Fig. 1. In contrast, state entropy incentivizes broader coverage of the state space, potentially distributing visitation across multiple high-reward paths. If policy entropy can be understood as a methodical way of injecting noise along an optimal path, state entropy can be viewed as encouraging uniformity across optimal, or nearly optimal, trajectories. This distinction motivates our investigation into whether state entropy regularization provably ensures stronger robustness, particularly in settings with spatially localized perturbations. + +The contributions provided in the paper are summarized as follows. + +- Section 3 provides a theoretical analysis of the power of state entropy regularization for robust RL: +In Section 3.1, we prove that state entropy regularization exactly solves a reward-robust RL problem, then characterize the induced uncertainty set and adversarial reward. Our theoretical result confirms the informal intuition described above: policy entropy protects against a locally informed adversary, while state entropy is robust to globally informed perturbations. We further show that the regularization strength monotonically controls the conservativeness of the induced uncertainty set, interpolating between $\ell_{\infty}$ -like uncertainty in the low-regularization limit and $\ell_{1}$ -like uncertainty as regularization increases. +In Section 3.2, we show that state entropy regularization provides a non-trivial lower bound on performance under transition (kernel) uncertainty. We compare this to the combined use of state and policy entropy, and prove that adding policy entropy weakens the bound—highlighting a structural advantage of using state entropy alone in this setting. +- In Section 4, we study both theoretical and practical limitations of entropy regularization: +- We establish that entropy regularization—whether over the policy, the state distribution, or both—cannot solve any kernel robust RL problem (Section 4.1). Building upon this result, we show that entropy regularization can hurt risk-averse performance arbitrarily (Section 4.2); +We show that the robustness benefits of state entropy regularization can be more sensitive to the number of rollouts used to evaluate the policy w.r.t. other forms of regularization (Section 4.3). +- In Section 5, we empirically evaluate the robustness properties of state entropy regularization across discrete and continuous control tasks. We show that it improves performance under spatially correlated perturbations—specifically obstacle placement—while not degrading performance under smaller, more uniform perturbations. We also demonstrate how the robustness benefits of state entropy depend on the rollout budget, with diminished gains in low-sample regimes. + +Related works are provided in Section 6. All proofs are provided in Appendix A. + +# 2 Preliminaries + +In this section, we describe the setting and notations we will use in the remainder of the paper. + +Notation. We denote by $\Delta_{\mathcal{Z}}$ the set of probability distributions over a finite set $\mathcal{Z}$ . The entropy of a distribution $\pmb{\mu} \in \Delta_{\mathcal{Z}}$ is given by $\mathcal{H}_{\mathcal{Z}}(\pmb{\mu}) := -\sum_{z \in \mathcal{Z}} \pmb{\mu}(z) \log(\pmb{\mu}(z))$ . The log-sum-exp operator over $\mathcal{Z}$ is defined as $\mathrm{LSE}_{\mathcal{Z}}(\pmb{u}) := \log\left(\sum_{z \in \mathcal{Z}} e^{\pmb{u}(z)}\right)$ , $\pmb{u} \in \mathbb{R}^{\mathcal{Z}}$ . For any $\pmb{u}, \pmb{v} \in \mathbb{R}^{\mathcal{Z}}$ , their inner product is denoted by $\langle \pmb{u}, \pmb{v} \rangle_{\mathcal{Z}} := \sum_{z \in \mathcal{Z}} \pmb{u}(z) \pmb{v}(z)$ . We shall often omit the set subscript if there is no ambiguity. Finally, for a distribution $\pmb{\mu} \in \Delta_{\mathcal{Z}}$ and a function $f: \mathcal{Z} \to \mathbb{R}$ , we denote the expectation of $f$ on $\pmb{\mu}$ as $\mathbb{E}_{\pmb{\mu}}[f(z)] = \sum_{z \in \mathcal{Z}} [\pmb{\mu}(z) f(z)]$ . + +Standard RL. Markov Decision Processes (MDPs) constitute the formal framework for RL [54]. An MDP is a tuple $(\mathcal{S},\mathcal{A},P,r,\gamma)$ , where $\mathcal{S}$ is a state space of size $S$ , $\mathcal{A}$ an action space of size $A$ , $P:\mathcal{S}\times \mathcal{A}\to \Delta (\mathcal{S})$ a transition kernel, $r:S\times \mathcal{A}\rightarrow \mathbb{R}$ a reward function, and $\gamma \in [0,1)$ a discount factor. The MDP evolves as follows. First, an initial state is sampled as $s_0\sim P(\cdot)$ . Then, for every $t\geq 0$ , the current state $s_t$ is observed, an action $a_{t}\sim \pi (\cdot |s_{t})$ is performed according to some policy $\pi :\mathcal{S}\to \Delta_{\mathcal{A}}$ , prompting a state transition $s_{t + 1}\sim P(s_t,a_t)$ while a reward $r_t = r(s_t,a_t)$ is collected. The goal is to find a policy maximizing a specific objective function within a policy space $\Pi$ . As we shall see below, different objectives can be considered. + +First, define expected transition kernel and reward under policy $\pi$ as $P^{\pi}(\cdot |s) := \sum_{a\in \mathcal{A}}\pi (a|s)P(\cdot |s,a)$ and $r^\pi (s)\coloneqq \sum_{a\in \mathcal{A}}\pi (a|s)r(s,a)$ , sometimes denoted as $P^{\pi},r^{\pi}$ in vector form. Also define the discounted state distribution induced by a policy $\pi$ on an MDP with kernel $P$ as $d_P^\pi (s)\coloneqq \sum_{t = 0}^\infty \gamma^t\mathbb{P}(s_t = s|\pi ,P)$ and the state-action distribution, a.k.a. occupancy as $\rho_P^\pi (s,a)\coloneqq d_P^\pi (s)\pi (a|s)$ . The standard RL objective function to maximize over $\Pi$ is $\mathcal{J}(\pi ,P,r) = \mathbb{E}_{\rho_P^\pi}[r(s,a)]$ , which may be rewritten as $\mathcal{J}(\pi ,P,r) = \langle \rho_P^\pi ,r\rangle$ in vector form. + +We note that, while the above formulation assumes finite MDPs, the main results extend to continuous state or action spaces under mild regularity assumptions. In that case, sums over states and actions are replaced by integrals with respect to the corresponding densities, and the discrete entropy is replaced by the differential entropy of the occupancy measure. + +Robust RL. The robust RL formulation [59, 31] aims to tackle uncertainty in the MDP parameters, such that the kernel $P$ and reward $r$ are not known but assumed to lie in an uncertainty set $\mathcal{P} \times \mathcal{R}$ . The objective function is defined with the worst case on the uncertainty set + +$$ +\mathcal {J} (\pi , \mathcal {P}, \mathcal {R}) := \min _ {(P, r) \in \mathcal {P} \times \mathcal {R}} \mathcal {J} (\pi , P, r). +$$ + +By construction, this so-called robust MDP formulation leads to policies that are less sensitive to model misspecification and perform reliably under a range of plausible environments. Throughout this paper, we will often consider reward uncertainty sets $\{P\} \times \mathcal{R}$ of the form + +$$ +\mathcal {R} = \{\tilde {r} \neq r \mid D (r, \tilde {r}) \leq \epsilon \}, \tag {1} +$$ + +where $\epsilon > 0$ , $D$ is interpreted as a dissimilarity measure between reward functions defining the notion of proximity within the uncertainty set and $\tilde{r} \succ r$ means that there exists some $(s, a) \in S \times \mathcal{A}$ such that $\tilde{r}(s, a) \leq r(s, a)$ (to ensure the adversary can degrade performance rather than only improve it). Another perspective on robust RL is to interpret the minimization in $\mathcal{I}(\pi, \mathcal{P}, \mathcal{R})$ as the action of an adversary that perturbs the nominal MDP within the uncertainty set so as to minimize the policy's expected return. Accordingly, $\epsilon$ quantifies the adversary's budget for allowable perturbations. + +Entropy-regularized RL. A broad range of literature considers augmenting the standard RL objective with some regularization function. The most popular is policy entropy regularization [17], which has been linked to stability of the policy optimization [2], improved exploration [45], and robustness [13]. The resulting objective is + +$$ +\mathcal {J} _ {\alpha} (\pi , \mathcal {H} _ {\mathcal {A}}) := \mathbb {E} _ {\rho_ {P} ^ {\pi}} [ r (s, a) + \alpha \mathcal {H} _ {\mathcal {A}} (\pi (\cdot | s)) ]. \tag {2} +$$ + +where $\alpha > 0$ is a hyperparameter controlling the regularization strength. In recent works [23, 50, 65, 25, 4], regularization with the entropy of the state distribution has been considered, either standalone or in combination with policy entropy, yielding + +$$ +\mathcal {J} _ {\alpha} (\pi , \mathcal {H} _ {\mathcal {S}}) = \mathbb {E} _ {\rho_ {P} ^ {\pi}} [ r (s, a) ] + \alpha \mathcal {H} _ {\mathcal {S}} \left(d _ {P} ^ {\pi}\right) \tag {3} +$$ + +$$ +\mathcal {J} _ {\alpha} (\pi , \mathcal {H} _ {\mathcal {S} \times \mathcal {A}}) = \mathbb {E} _ {\rho_ {P} ^ {\pi}} [ r (s, a) ] + \alpha \mathcal {H} _ {\mathcal {S} \times \mathcal {A}} \left(\rho_ {P} ^ {\pi}\right). \tag {4} +$$ + +In this work, we will focus on analyzing the latter types of regularization. + +
RegularizerUncertainty SetRobust ReturnWorst Case Reward
PolicyEdP LSE_A (Δr_s/α) ≤ ε/α + log(A)EρP r [r] - ε + α (EduP r [H_A(πs)] - log(A))r(s, a) - α log(π(a|s)) - ε
StateLSE_S (Δr/α) ≤ ε/α + log(S)EρP r [r] - ε + α (H_S(dP) - log(S))r(s, a) - α log(dP(s)/Aπ(a|s)) - ε
State-actionLSE_S × A (Δr/α) ≤ ε/α + log(SA)EρP r [r] - ε + α (H_S × A (ρP) - log(SA))r(s, a) - α log(ρπ(s, a)) - ε
+ +Table 1: Robustness guarantees for each regularization type including the form of the uncertainty set, the robust return, and the corresponding adversarial reward. We denote $\Delta r \coloneqq r - \tilde{r}$ and $r_s = r(s,\cdot), \pi_s = \pi(\cdot \mid s)$ for brevity. + +# 3 Robustness and State Entropy Regularization + +In this section, we analyze the robustness properties of regularization with the state distribution entropy. First, we show that this regularization exactly solves certain reward-robust RL problems, and we compare its theoretical guarantees to those of policy entropy regularization. We then extend the analysis to transition (kernel) uncertainty, deriving non-trivial performance bounds in this setting. + +# 3.1 Reward-Robustness + +In this section, we show that state entropy regularization amounts to solving a robust RL problem with an explicit uncertainty set over rewards. + +Theorem 3.1. For any $\pi \in \Pi$ , the following duality holds: + +$$ +\min _ {\tilde {r} \in \tilde {\mathcal {R}}} \mathbb {E} _ {\rho^ {\pi}} [ \tilde {r} ] = \mathbb {E} _ {\rho^ {\pi}} [ r ] + \alpha \left(\mathcal {H} _ {\mathcal {S}} \left(d _ {P} ^ {\pi}\right) - \log (S)\right) - \epsilon , \tag {5} +$$ + +where the reward uncertainty set is $\tilde{\mathcal{R}}^{\pi}(\epsilon, \alpha) \coloneqq \left\{\tilde{r} \mid \mathrm{LSE}_{\mathcal{S}}\left(\frac{r^{\pi} - \tilde{r}^{\pi}}{\alpha}\right) \leq \frac{\epsilon}{\alpha} + \log(S)\right\}$ . Moreover, the worst-case reward chosen by the adversary is given by $\tilde{r}(s, a) = r(s, a) - \frac{1}{A} \frac{\log(d_{P}^{\pi}(s))}{\pi(a|s)} - \epsilon$ . + +The proof is presented in Appx. A.1. It follows a similar path to [5, 13] and relies on a convex conjugate argument: The entropy regularization term can be interpreted as introducing a soft constraint, corresponding to an uncertainty set over the reward function. The uncertainty set can be written as $\{\tilde{r} \succ r \mid \alpha \mathrm{LSE}_S\left(\frac{r^\pi - \tilde{r}^\pi}{\alpha}\right) - \alpha \log(S) \leq \epsilon\}$ , highlighting a dissimilarity constraint between reward vectors as in Eq. (1). + +We compare the obtained uncertainty sets and robustness guarantees with policy entropy and state-action entropy regularization in Table 1. The comparison extends previous formulations that neglect either the temperature parameter $\alpha$ or the adversary's budget $\epsilon$ [13, 5]. More insight into the proofs of these derivations is included in Appx. A.2. As shown in the adversarial reward column, policy entropy regularization depends on the policy $\pi (\cdot |s)$ only. Thus, it tackles adversaries with access to local information of each state. In contrast, state entropy regularization depends on the state occupancy distribution $d_P^\pi$ , enabling robustness against adversaries that exploit global properties of the agent's behavior across the state space. Consequently, policies regularized via state entropy are inherently more resistant to spatially-structured perturbations, while policy entropy provides protection primarily against local, state-specific variations. This difference analytically confirms our initial intuition. + +![](images/10eb4abde202461e65fbf7384b0809fe43c077f211ba72300ed638bb7385e380.jpg) +Figure 2: Uncertainty sets $\tilde{\mathcal{R}}_{\epsilon}^{\pi}(\alpha)$ induced by state entropy regularization in a two-state MDP with $r^{\pi}(s_1) = 1$ , $r^{\pi}(s_2) = 0$ , and $\epsilon = \frac{1}{2}$ . The $x$ - and $y$ -axes correspond to the perturbed rewards $\tilde{r}^{\pi}(s_1)$ and $\tilde{r}^{\pi}(s_2)$ , respectively. As $\alpha$ increases, the uncertainty set expands, interpolating between $\ell_{\infty}$ - and $\ell_1$ -type constraints in the limits $\alpha \to 0$ and $\alpha \to \infty$ . + +We next analyze how the uncertainty set induced by state entropy regularization evolves as a function of the temperature parameter $\alpha$ . While our formal analysis focuses on the state entropy, similar + +limiting behaviors appear to hold for policy entropy and state-action entropy as well. This formalizes insights that were observed empirically in [5], where varying the regularization strength was found to interpolate between different robustness behaviors. We characterize the limiting uncertainty sets as $\alpha \to \infty$ and $\alpha \to 0$ , corresponding to different adversarial capabilities. The proof is in Appx. A.3. We visualize this behavior in Fig. 2. + +Theorem 3.2 (Limit uncertainty in $\alpha$ ). Consider the uncertainty sets associated with state entropy regularization as a function of $\alpha$ : + +$$ +\tilde {\mathcal {R}} _ {\epsilon} ^ {\pi} (\alpha) = \left\{\tilde {r} \mid \mathrm {L S E} _ {\mathcal {S}} \left(\left(\frac {r ^ {\pi} - \tilde {r} ^ {\pi}}{\alpha}\right) ^ {\alpha}\right) - \alpha \log (S) \leq \epsilon \right\}. +$$ + +Then, the mapping $\alpha \mapsto \mathcal{R}_{\epsilon}^{\pi}(\alpha)$ is non-increasing in the sense that $\alpha_{2}\geq \alpha_{1}$ implies $\tilde{\mathcal{R}} (\epsilon ,\alpha_1)\subseteq$ $\tilde{\mathcal{R}} (\epsilon ,\alpha_2)$ . Furthermore, the limit sets satisfy + +$$ +\lim _ {\alpha \rightarrow \infty} \tilde {\mathcal {R}} _ {\epsilon} ^ {\pi} (\alpha) = \cup_ {\alpha > 0} \tilde {\mathcal {R}} _ {\epsilon} ^ {\pi} (\alpha) = \left\{\tilde {r} \left| \frac {1}{S} \sum_ {s \in \mathcal {S}} r ^ {\pi} (s) - \tilde {r} ^ {\pi} (s) \leq \epsilon \right.\right\} +$$ + +and + +$$ +\lim _ {\alpha \rightarrow 0} \tilde {\mathcal {R}} _ {\epsilon} ^ {\pi} (\alpha) = \cap_ {\alpha > 0} \tilde {\mathcal {R}} _ {\epsilon} ^ {\pi} (\alpha) = \left\{\tilde {r} \middle | \max _ {\mathcal {S}} (r ^ {\pi} - \tilde {r} ^ {\pi}) \leq \epsilon \right\}. +$$ + +# 3.2 Lower Bound under Kernel Uncertainty + +As a first step in our analysis of kernel robustness, we show that state entropy regularization yields a non-trivial lower bound on worst-case performance under transition uncertainty. + +Theorem 3.3. For any policy $\pi \in \Pi$ and $\alpha >0$ , the following weak duality holds: + +$$ +\min _ {\tilde {\mathcal {P}} ^ {\pi} (\epsilon)} \mathbb {E} _ {\rho_ {\tilde {P}} ^ {\pi}} [ r ] \geq \exp \left(\mathbb {E} _ {\rho_ {\tilde {P}} ^ {\pi}} [ \log (r) ] + \alpha \left(\mathcal {H} \left(d _ {P} ^ {\pi}\right) - \log (S)\right) - \epsilon\right) \tag {6} +$$ + +where + +$$ +\tilde {\mathcal {P}} ^ {\pi} (\epsilon , \alpha) := \left\{\tilde {P} \mid \alpha \log \left(\frac {1}{\mathcal {S}} \sum_ {s \in \mathcal {S}} \left(\frac {d _ {P} ^ {\pi} (s)}{d _ {\tilde {P}} ^ {\pi} (s)}\right) ^ {\frac {1}{\alpha}}\right) \leq \epsilon \right\} +$$ + +The proof follows a similar structure as in [13] where transition uncertainty is effectively reduced to reward uncertainty via importance sampling argument and logarithmic transformation. This reduction in turn enables us to leverage our earlier result on reward-robustness to derive the lower bound. The full proof is available in Appx. A.4. + +An appealing property of this formulation is that the uncertainty set is expressed explicitly in terms of the induced state distribution. This allows us to reason about the distributional effects of transition noise directly, rather than working with abstract perturbations to the transition kernel itself. + +Interestingly, using the same derivation, we can show that adding policy entropy regularization with the same temperature $\alpha$ leads to the same uncertainty set, but with a strictly looser lower bound. A more formal discussion of this comparison, including the explicit form of the resulting bound under policy entropy regularization, is provided in Appx. A.4.1. This result is particularly relevant in light of the next section, where we show that entropy regularization alone—regardless of the type—cannot solve kernel robust RL problems in general. In that sense, the bound derived here is the most we can expect from this approach. That said, we do not claim this lower bound is tight, and we cannot conclusively rule out that policy entropy may help in some settings. In fact, as we show in our experiments, it often does. Nonetheless, this result provides insight into the structural advantages of state entropy regularization when viewed through the lens of kernel uncertainty. + +# 4 Limitations of State Entropy Regularization + +In this section, we study the limitations of entropy regularization in robust reinforcement learning. First, we show that entropy regularization—whether over the policy, state distribution, or both—cannot solve any kernel robust RL problems. Building on this result, we demonstrate that in the risk-averse setting, entropy regularization can significantly degrade performance. Finally, we take a practical perspective and show that the robustness benefits of state entropy regularization are highly sensitive to the number of rollouts used during policy evaluation. + +# 4.1 Entropy Regularization Cannot Fully Solve Kernel Robustness + +A kernel-robust RL problem is defined by a function $\tilde{\mathcal{P}} (\mathrm{MDP},\pi)$ which maps the MDP and policy parameters to an uncertainty set—that is, the set of possible transition kernels from which the adversary may choose. We consider a class of kernel robust RL problems in which the uncertainty set function depends only on the nominal transition kernel $P$ and the policy $\pi$ . This captures a wide range of uncertainty models, including $\ell_p$ -constrained and rectangular uncertainty sets commonly studied in robust RL [27, 10]. We now state a more general claim: Any regularization function that depends only on the nominal kernel and the policy cannot, in general, characterize the robust return over such uncertainty set functions. The following impossibility result formalizes this observation. + +Theorem 4.1. Let $\Omega(\pi, P) \neq 0$ be a regularization function. There does not exist an uncertainty set function $\hat{\mathcal{P}}: (P, \pi)$ such that for every MDP holds: + +$$ +\min _ {\tilde {P} \in \tilde {\mathcal {P}}} \mathbb {E} _ {\rho_ {\tilde {P}} ^ {\pi}} [ r ] = \mathbb {E} _ {\rho_ {P} ^ {\pi}} [ r ] + \Omega (\pi , P). \tag {7} +$$ + +The proof, which can be found in Appx. A.5, is very simple and uses a symmetrization argument—when the reward is constant, any regularization must be degenerate. A key takeaway is that any regularization capable of capturing kernel robustness must depend on the reward, not just on the nominal kernel $P$ or the policy $\pi$ . This observation is consistent with prior results [10]. As a direct corollary, entropy regularization—whether applied to the policy or the state distribution—cannot fully solve kernel robustness either. + +# 4.2 Entropy Regularization and Risk-Averse Performance + +While entropy regularization provides non-trivial lower bounds under kernel uncertainty, we have shown it cannot fully solve kernel robustness. Our experiments offer further insight into when regularization may improve robustness in practice. We now show analytically that for an important type of robustness—risk aversion—entropy regularization can, in fact, arbitrarily degrade performance. + +While robust RL primarily addresses epistemic uncertainty—uncertainty about the parameters of the MDP—risk-averse RL [15] instead targets aleatoric uncertainty, which arises from the inherent randomness of the MDP itself. To quantify risk-averse performance, we use the common Conditional Value at Risk (CVaR) objective [48, 8], denoted $\mathrm{CVaR}_{\beta}$ , which measures the expected return in the worst $\beta$ -fraction of outcomes. We denote the optimal policy for the unregularized objective by $\pi^{*}$ and the optimal policy for the entropy-regularized objective (whether state or policy entropy) by $\pi_{R}^{*}$ . + +Theorem 4.2. For every temperature $\alpha >0$ , constant $M\in \mathbb{R}$ , and $0 < \beta < 1$ , there exists an MDP $\mathcal{M}$ such that: + +$$ +\mathrm {C V a R} _ {\beta} ^ {\pi^ {*}} [ G ] - \mathrm {C V a R} _ {\beta} ^ {\pi_ {R} ^ {*}} [ G ] = M \tag {8} +$$ + +where $G$ is the total discounted return. + +The intuition behind the result is that entropy regularization does not prioritize low-variance trajectories, which are critical for improving risk-averse performance. The proof is provided in Appx. A.6. + +# 4.3 On the Sample Sensitivity of State Entropy Regularization + +We now investigate how the robustification benefits of state entropy regularization depend on the number of rollouts used to estimate a policy's state entropy. While using state entropy regularization does not incur significant computational overhead [50], we observe that the number of rollouts required to realize the robustness benefits is higher than what is typically sufficient for standard RL. We provide both theoretical and empirical evidence for this phenomenon, and offer two complementary explanations. One explanation is statistical—state entropy is harder to estimate accurately than other quantities like return or policy entropy. The other is behavioral—in low-rollout regimes, entropy maximization can incentivize an agent to be less stochastic. + +For the statistical explanation, we assume that the state entropy is estimated using a $k$ -nearest neighbor estimator [52] in a continuous state space $\mathbb{R}^D$ , as it is common in the related literature [42, 33, 50]. For fixed $k$ , under mild regularity conditions on the MDP—such as the smoothness of $d_P^\pi$ for all the + +policies $\pi$ — the estimation error of the state entropy decays at a rate $O(n^{-1 / D})$ ,3 where $n$ denotes the number of sampled trajectories [53]. In contrast, both the policy entropy and expected return can be estimated with error decaying at the standard rate $O(n^{-1 / 2})$ [35]. As a result, in high-dimensional settings, state entropy estimation introduces a dominant source of statistical error, contributing to the increased sample sensitivity of entropy-regularized methods. In practice, we note that the effective dimensionality $D$ is often reduced by computing entropy over a subset of state features (as in Section 5) or by estimating entropy over latent state representations, as demonstrated by [50]. + +For the behavioral explanation, we consider the MDP illustrated in Figure 3 and the empirical objective $\mathbb{E}_{\rho^{\pi}}[r] + \alpha \hat{H}_1(\pi)$ , where $\hat{H}_1(\pi)$ is the state entropy estimated on a single rollout. The policy that maximizes the empirical entropy on a single rollout will tend to be deterministic, taking the longest possible route to the goal in order to visit more states. This leads to a counterproductive outcome: Rather than encouraging stochasticity across multiple nearoptimal paths—promoting robustness—the regularization yields a deterministic behavior along a suboptimal path, ultimately degrading robustness. + +![](images/ccaa6e84599a998e096bf42daa4e1f6b3043e31b838915925140cfcec931a05f.jpg) +Figure 3: An illustrative toy MDP. The grid is a $2 \times 6$ environment with the initial state marked with "I" and the goal state with "G". The red path shows the longest deterministic trajectory that visits the maximum number of states before reaching the goal. Reward is given only at the goal state. + +# 5 Experiments + +We evaluate the robustness properties of state entropy regularization using a set of representative environments and perturbation types. These are designed to test three core hypotheses: (1) State entropy regularization offers protection against spatially correlated severe perturbations to the dynamics; (2) State entropy regularization induces robustness to $\ell_1$ reward uncertainty; (3) The robustification of state entropy regularization is sensitive to the number of rollouts used during training. We validate these claims in both discrete (MiniGrid [7]) and continuous (Mujoco [57]) domains. We compare policies trained without regularization, with policy entropy regularization, and with state entropy regularization. Note that the state-entropy regularized policies are trained with some policy entropy regularization to favor training stability [2]. For all the types, we select the largest regularization coefficients that degrade nominal performance less than $5\%$ . To ensure a fair comparison, the methods are trained with the same base algorithm—A2C [36] for MiniGrid and PPO [49] for Mujoco. + +To implement state entropy regularization, we largely follow [50] by using a k-nearest neighbor (k-NN) entropy estimator [52]. As state entropy regularization is incorporated as an intrinsic reward, it can be paired with any RL algorithm. Full implementation details are in Appx. B.1. + +Transition kernel robustness via entropy regularization. To evaluate how well each regularization method handles spatially correlated perturbations to the transition dynamics, we construct environments where specific regions of the state space are blocked off or obstructed. These changes are considered "catastrophic" because they remove critical paths or create bottlenecks that fundamentally alter the agent's ability to reach the goal using its original policy. In MiniGrid, we randomly place a large wall segment that blocks a connected area of the maze (see Figure 4(b) for the perturbed environment). This constitutes a major spatially-structured disturbance, as it affects transitions between nearby states and introduces a sharp structural change in the environment. As shown in Figure 4(b), the agent trained with state entropy regularization significantly outperforms both policy entropy and unregularized baselines. We note that we report results for all horizon lengths to avoid misleading conclusions based on a single fixed task length. In cases where one method clearly dominates across the full horizon range, we can draw decisive conclusions. When the curves intersect or remain close, it indicates that the relative performance is more sensitive to task structure. To verify that this effect extends to continuous control, we run analogous experiments in Mujoco's Pusher and Ant environments by placing obstacles along the optimal trajectories, fully blocking them. Again, as shown in Figs 5(a), 6(c) state entropy regularization yields substantially more robust performance as its induced behavioral diversity enables bypassing these local catastrophes. + +![](images/e77f5a3e01ae85119a33e28c46bbee14c46098d6e3819ae808176bbb4335db0f.jpg) +(a) + +![](images/5bd155a95494f84f8aa058a137539b1492fc9dc113a037290161ae8fac49775b.jpg) +(b) + +![](images/86049e68086f9a42dd9e9c6b1a6a603f7470d48d3e4c7f227cb370713a137685.jpg) +(c) + +![](images/76f431fd37862de8a038b99e8164d0cac5f752dfe6fd1e86fc1469d2a64b16b8.jpg) +(d) + +![](images/a5782bf1c2af5db514abbdd94ec4f02e8060041e724273798f6fe9b380cb5f65.jpg) +Figure 4: MiniGrid evaluation under different perturbations. All agents were trained on an empty grid where the agent starts at the top left and must reach the goal at the bottom right (a). Top: Visualizations of the three perturbation types used at test time—(b) large walls randomly block parts of the maze, (c) obstacles are distributed uniformly, and (d) the goal location is randomized. Bottom: Nominal return (left) and success rate under perturbations (right). We evaluate each policy for 200 episodes and report $96\%$ confidence bounds across 25 seeds. + +![](images/5bbf042638c5b879d0b0f008b3c9487878cdce20ade530c8ba8b2c757bb99ba6.jpg) + +![](images/9df0c827e74b0032675609a7e6d63615eac2b3e090dbc0afc8429f601469f8ed.jpg) + +![](images/7ff0b486e745a0243036fd793afcda14ed91b3e7c517cbc465b2e4e172d8e628.jpg) + +To highlight the challenge of learning policies that are robust to spatially localized perturbations, we compare against the PRMDP and NRMDP frameworks [6], two popular robustness baselines in continuous control. Both methods formulate robustness as a two-player zero-sum game: the agent selects actions while an adversary perturbs the actions. As shown in Figure 5(a), the agent trained with state entropy regularization significantly outperforms this baseline. While we do not expect state entropy to consistently outperform robust RL methods across all types of perturbations, this result—along with our analysis in the main paper—suggests that it is particularly effective against spatially correlated perturbations, which are not the primary focus of most standard approaches. + +To test robustness under more uniformly distributed, small-scale perturbations, we augment MiniGrid with uniformly scattered obstacle tiles. These mild disturbances are spread across the entire state space, simulating the kind of noise that policy entropy regularization is expected to help with [13]. In Figure 4(c), we see that the unregularized agent performs poorly, while both policy and state entropy regularization improve robustness, with policy entropy showing a slight advantage. This result aligns with our initial intuition: Policy entropy is particularly effective against diffuse perturbations, and importantly, we observe that adding state entropy regularization does not hurt this effect. + +Reward robustness. To assess how different entropy regularization methods handle reward uncertainty, we randomize the reward location at evaluation time in both MiniGrid and Pusher. In MiniGrid, a new goal location is sampled uniformly from the entire grid. In Pusher, the goal is relocated to a random position at distance $r$ from its nominal location. This setup induces structured variation in the reward function, formally resembling an $\ell_1$ -bounded uncertainty—where the total change in reward mass is limited but can be concentrated arbitrarily across the state space. The key intuition is that if the reward happens to lie along the agent's typical trajectory, then policy entropy regularization—by injecting stochasticity at the action level—may suffice to recover it. However, when the reward is placed in less frequently visited areas, broader state coverage becomes necessary. In such cases, state entropy regularization offers a natural advantage by encouraging the agent to visit a wider range of states. As shown in Figures 4(d), 5 (b) policy entropy improves performance relative to the unregularized baseline, but adding state entropy yields a more substantial gain, particularly when the reward is far from the nominal goal location. + +Sensitivity to rollout budget. We now examine how the number of rollouts used per state entropy estimation affects the robustness benefits of state entropy regularization. As discussed in Section 4.3, the estimation of state entropy is statistically more demanding than that of policy entropy or expected return. In particular, with limited rollouts, entropy maximization can encourage overly deterministic behavior, especially in sparse reward settings. To study this, we vary the number of + +![](images/2e58b5848ea3925630163011b229be11e47b16b7cf696a98574b0ec1e462270a.jpg) +(a) + +![](images/29d72b559ace9412d391cd007d12ab9a59509311645e7262e4cd9b7d76cb66af.jpg) +(b) + +![](images/af892875de28d79ea560e2b110285fa806b40bdc5d12355775aa80c1cb1b47a4.jpg) +(c) +Figure 5: Pusher experiments and ablations. Like in the discrete experiment, we train all agents on a nominal task, then introduce a localized catastrophe by blocking off the optimal path, visualized in Fig. 1. All plots report success rate defined as when the pucks final distance from the goal is smaller than 0.1. We evaluate each policy for 200 episodes and show $96\%$ confidence bounds across 25 seeds. + +![](images/32e5a5203f816338740fa5bc1e85c84abc67aaf5c892eda6f42221ebb2d19baf.jpg) +(d) + +rollouts used to evaluate the policy and monitor how this affects downstream robustness. As shown in Figure 5(b), we find that when only a small number of rollouts are used, state entropy regularization can actually lead to worse performance than policy entropy. This is supported by the fact that with 4 rollouts, the confidence bounds of state entropy intersect the mean robust performance of policy entropy. Interestingly, while robustness is sensitive to the number of rollouts, nominal performance remains largely unchanged—making it difficult to detect this gap without explicitly evaluating under perturbations. These observations suggest that while state entropy can provide significant robustness benefits, realizing those benefits requires a sufficient amount of rollouts—underscoring a practical tradeoff between regularization and sample complexity. + +Regularization temperature. To study how the temperature parameter influences robustness, we train state-entropy regularized agents with increasing regularization strength and evaluate their performance in both the nominal and perturbed versions of the Pusher environment. We observe in Figure 5(c) that as regularization increases, robust performance improves, illustrating the role of temperature in expanding the effective uncertainty set and promoting diverse robust behavior. Notably, robust performance eventually plateaus, suggesting diminishing returns from further increasing the regularization strength. This empirical trend aligns with our theoretical result in Theorem 3.2, which characterizes the limiting behavior of the uncertainty set. These findings highlight that while tuning entropy regularization can enhance robustness, its benefits saturate beyond a certain point. + +![](images/2041de276e6b84d6da7b2ec93aea651b63d9ab05e8fa1a98dfa68f70e245d24f.jpg) +(a) +Figure 6: Ant experiment. (a) Visualization of the vertical wall perturbation. (b) All methods perform similarly on the nominal task. (c) Success rate in the perturbed environment, i.e., the fraction of evaluation runs in which the ant successfully crosses the obstacle. We evaluate each policy for 200 episodes and show $96\%$ confidence bounds across 25 seeds. + +![](images/d20f370583e52fa05fdf8a703a39caca22dc5515663f3f4e30c9eb84dc9c578d.jpg) +(b) + +![](images/1fe6272e3fcf55d1a99be4705c95b38d497307e7b4d3a9ccbfd0131fb55dcfac.jpg) +(c) + +# 6 Related Works + +Robust RL and regularization. The connection between regularization and robustness is well-established in RL. The work [13] demonstrates an equivalence between policy-entropy regularized objectives and reward robustness and a lower bound for kernel robustness. We extend their analysis to state entropy regularization, providing more direct and general proofs, which also accounts for the regularization coefficient $\alpha$ (for both policy and state entropy). State-action entropy regularization is + +analyzed by [5, 22], which as we show in Appx. A.4.1, equals the sum of state entropy plus policy entropy. Note that state-action entropy encodes the joint information provided by state entropy and policy entropy, but does not encode the conditional information each one gives to the other. Neither of these works provide algorithmic methods to implement state-action entropy regularization, nor studies the kernel robustness induced by entropy regularization. Complementary to ours, [10] derives a regularizer that depends on the value function since they study the primal optimization of RL and robust RL problems. However, a connection between regularized dual and primal objectives has yet to be established. The motivation in [10, 14] is also different from ours, as they study tractable robust RL methods, whereas we analyze robustness granted by state entropy regularization. + +State entropy objectives. Hazan et al. [18] have first proposed a learning objective predicated on maximizing the entropy of the state distribution in MDPs, followed by a considerable stream of works studying various aspects of the problem, including practical algorithms for challenging domains [42, 33, 50, 64, 32], alternative formulations [43, 69, 16, 40, 65, 1], statistical complexity [56], how to deal with partial observability [66, 67], and many others [28, 23, 55, 63, 41, 44, 62, 39, 24, 25, 70, 30, 4, 9, 68]. Most of the works focus on the entropy of the state distribution although the latter is often paired with standard policy entropy regularization in practical methods [33, 50, 64, 32]. While the state entropy has been often considered a standalone objective for pure exploration [e.g., 55], data collection [63], or policy pre-training (most of the others), some works have employed occupancy entropy as a regularization for RL [23, 50, 65, 25, 4]. A common challenge for all of the previous is that the feedback on the state entropy is not directly available to the agent and must be estimated from data, leading to scalability issues. A crucial advancement has been made by incorporating non-parametric entropy estimators [52] in practical algorithms [42, 33, 50, 64]. + +# 7 Conclusion + +This work provides a comprehensive analysis of state entropy regularization through the lens of robustness. We show that it exactly solves certain reward-robust RL problems and induces stronger worst-case guarantees than policy entropy in structured perturbation settings. In the case of kernel uncertainty, we establish a non-trivial performance lower bound and highlight that adding policy entropy weakens this guarantee. At the same time, we prove fundamental limitations: entropy regularization cannot solve general kernel-robust problems and may degrade performance in risk-averse scenarios. On the practical side, we show that while state entropy improves robustness to spatially-structured changes, it does not harm performance under smaller uniform perturbations—though its effectiveness is sensitive to the number of policy evaluation rollouts. Together, our findings clarify when and how state entropy regularization can contribute to robust RL, and where its boundaries lie. + +# Acknowledgment + +The work of Yonatan Ashlag is supported by the Israel Science Foundation (ISF), grant No. 447/20. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Proofs + +# A.1 Proof of Theorem 3.1 + +Proof. Since this proof does not involve adversarial perturbations to the transition kernel, we omit the subscript and write $d^{\pi}$ instead of $d_P^\pi$ for clarity. We will first show a result for the uncertainty set $\tilde{\mathcal{R}} (\epsilon ,\pi) = \{\tilde{r}\mid \mathrm{LSE}_S(r^\pi -\tilde{r}^\pi)\leq \epsilon \}$ , and then get the result from our main theorem using change of variables. + +We fix $\pi$ . First, we reformulate the optimization problem in terms of $\Delta r \coloneqq r - \tilde{r}$ : + +$$ +\min _ {\tilde {r} \in \tilde {R}} \mathbb {E} _ {\rho^ {\pi}} [ \tilde {r} ] = \mathbb {E} _ {\rho^ {\pi}} [ r ] + \min _ {\Delta r \in \tilde {R}} - \mathbb {E} _ {\rho^ {\pi}} [ \Delta r ] \tag {9} +$$ + +Note that $\tilde{\mathcal{R}} (\epsilon ,\pi)$ is strictly feasible and that the objective and constraints are convex, therefore the optimal point satisfies the KKT conditions. Furthermore, the objective is linear in $\tilde{r}$ , thus the minimum is achieved at the edge of the constraint with $\lambda >0$ . The Lagrangian is: + +$$ +L (\Delta r, \lambda) = - \left\langle \rho^ {\pi}, \Delta r \right\rangle + \lambda \left(\operatorname {L S E} \left(\Delta r ^ {\pi}\right) - \epsilon\right) \tag {10} +$$ + +$$ +\nabla_ {\Delta r (s, a)} L = - \rho^ {\pi} (s, a) + \lambda \pi (a \mid s) \operatorname {s o f t m a x} \left(\Delta r ^ {\pi}\right) (s) = 0 \tag {11} +$$ + +$$ +- d ^ {\pi} (s) + \lambda \operatorname {s o f t m a x} \left(\Delta r ^ {\pi}\right) (s) = 0 \tag {12} +$$ + +Where we get that last line by taking the sum over $a \in \mathcal{A}$ of both sides. By taking the sum over $s \in S$ we can note that the last line implies that $\lambda^{*} = 1$ (softmax and $d^{\pi}$ are probability vectors). Therefore, our original optimization objective from 9 becomes: + +$$ +\min _ {\Delta r} - \left\langle \rho^ {\pi}, \Delta r \right\rangle + \operatorname {L S E} _ {\mathcal {S}} \left(\Delta r ^ {\pi}\right) - \epsilon = - \epsilon + \min _ {\Delta r ^ {\pi}} - \left\langle d ^ {\pi}, \Delta r ^ {\pi} \right\rangle + \operatorname {L S E} _ {\mathcal {S}} \left(\Delta r ^ {\pi}\right) \tag {13} +$$ + +Finally, noting that entropy is the convex conjugate of LSE we get: + +$$ +\min _ {\tilde {r} \in \tilde {\mathcal {R}}} \mathbb {E} _ {\rho^ {\pi}} [ \tilde {r} ] = \mathbb {E} _ {\rho^ {\pi}} [ r ] - \epsilon + \min _ {\Delta r ^ {\pi}} - \left\langle d ^ {\pi}, \Delta r ^ {\pi} \right\rangle + \operatorname {L S E} _ {\mathcal {S}} \left(\Delta r ^ {\pi}\right) = \mathbb {E} _ {d ^ {\pi}} [ r ] + \mathcal {H} _ {\mathcal {S}} \left(d ^ {\pi}\right) - \epsilon \tag {14} +$$ + +We also note that substituting $\Delta r(s, a) = \frac{\log(d^{\pi}(s))}{\pi(a|s)A} + c$ for every constant $c$ satisfies equation 11 and is thus the adversarial reward. From the constraint $\mathrm{LSE}(\Delta r^{\pi}) = \epsilon$ we can deduce that $c = \epsilon$ . + +Finally, we can now substitute $r \gets \frac{r}{\alpha}, \tilde{r} \gets \frac{\tilde{r}}{\alpha}, \epsilon \gets \frac{\epsilon}{\alpha} + \log(S)$ and get the required result for different temperatures. + +# A.2 Proof of results in Table 1 + +Proof. We proved the result for state entropy in the previous appendix. For both policy entropy and state-action entropy we'll start from equation 9. Note that for both we can imply the KKT conditions since $\tilde{R}$ is strictly feasible and the objective and constraints are convex. For policy entropy the gradient of the Lagrangian becomes: + +$$ +\nabla_ {\Delta r (s, a)} L = - \rho^ {\pi} (s, a) + \lambda d ^ {\pi} (s) \operatorname {s o f t m a x} (\Delta r _ {s}) (a) = 0 \tag {15} +$$ + +By summing over $(s, a)$ we can again infer that $\lambda = 1$ . We continue by substituting in the identity $\rho^{\pi}(s, a) = d^{\pi}(s)\pi(a \mid s)$ and we get $\pi(a \mid s) = \operatorname{softmax}(\Delta r_s)(a)$ . By taking the log of both sides and taking the inner product with $\rho^{\pi}$ we get: + +$$ +\begin{array}{l} - \left\langle \rho^ {\pi}, \log (\pi (a \mid s)) \right\rangle = - \sum_ {s} d ^ {\pi} (s) \sum_ {a} \pi (a \mid s) \log (\pi (a \mid s))) = \mathbb {E} _ {d ^ {\pi}} \left[ \mathcal {H} _ {\mathcal {A}} (\pi (\cdot \mid s)) \right] (16) \\ = \left\langle \rho^ {\pi}, - \Delta r + \mathrm {L S E} _ {\mathcal {A}} \left(r _ {s}\right) \right\rangle = \left\langle \rho^ {\pi}, - \Delta r \right\rangle + \mathbb {E} _ {d \pi} \left[ \mathrm {L S E} _ {\mathcal {A}} \left(\Delta r _ {s}\right) \right] (17) \\ \end{array} +$$ + +Which is the required result for the case where $\alpha = 1$ . We use the same change of variables as A.1 to get the result for a general temperature. We can verify with through explicit substitution that the worst case reward described in the table satisfies 15 and the requirement that $\mathbb{E}_{d^{\pi}}[\mathrm{LSE}_{\mathcal{A}}(\Delta r_s)] = \epsilon$ . The proof for state-action entropy regularization is very similar and is thus omitted. + +# A.3 Proof of Theorem 3.2 + +Proof. First, by taking the derivative of the divergence according to alpha we can see that it is always negative, which implies the required containment relation: + +$$ +\begin{array}{l} \frac {\partial}{\partial \alpha} \left(\alpha \log \left(\frac {1}{\mathcal {S}} \sum_ {s} e ^ {\frac {\Delta r ^ {\pi}}{\alpha}}\right)\right) = \log \left(\frac {1}{\mathcal {S}} \sum_ {s} e ^ {\frac {\Delta r ^ {\pi}}{\alpha}}\right) - \alpha \frac {\frac {1}{\mathcal {S}} \sum_ {s} e ^ {\frac {\Delta r ^ {\pi}}{\alpha}} \frac {\Delta r ^ {\pi}}{\alpha^ {2}}}{\frac {1}{\mathcal {S}} \sum_ {s} e ^ {\frac {\Delta r ^ {\pi}}{\alpha}}} (18) \\ = \log \left(\mathbb {E} _ {s \sim U} \left[ e ^ {\frac {\Delta r ^ {\pi}}{\alpha}} \right]\right) - \mathbb {E} _ {s \sim \operatorname {s o f t m a x} \left(\frac {\Delta r ^ {\pi}}{\alpha}\right)} \left[ \frac {\Delta r ^ {\pi}}{\alpha} \right] (19) \\ \leq \log \left(\mathbb {E} _ {s \sim U} \left[ e ^ {\frac {\Delta r ^ {\pi}}{\alpha}} \right]\right) - \log \left(\mathbb {E} _ {s \sim \operatorname {s o f t m a x} \left(\frac {\Delta r ^ {\pi}}{\alpha}\right)} \left[ e ^ {\frac {\Delta r ^ {\pi}}{\alpha}} \right]\right) (20) \\ \leq 0 (21) \\ \end{array} +$$ + +The first inequality follows from Jensen, the second inequality follows from properties of softmax (gives higher weight to higher $\Delta r^{\pi}$ values) and from the monotonicity of the exponent and log function. + +Next, we prove the limiting case when $\alpha \to \infty$ . Using L'Hôpital's rule we get: + +$$ +\begin{array}{l} \lim _ {\alpha \rightarrow \infty} \alpha \log \left(\frac {1}{S} \sum_ {s} e ^ {\frac {\Delta r ^ {\pi}}{\alpha}}\right) = \lim _ {\alpha \rightarrow \infty} \frac {- \frac {\frac {1}{S} \sum_ {s} e ^ {\frac {\Delta r ^ {\pi}}{\alpha}} \frac {\Delta r ^ {\pi}}{\alpha^ {2}}}{\frac {1}{S} \sum_ {s} e ^ {\frac {\Delta r ^ {\pi}}{\alpha}}}}{- \frac {1}{\alpha^ {2}}} (22) \\ = \lim _ {\alpha \rightarrow \infty} \mathbb {E} _ {s \sim \operatorname {s o f t m a x} \left(\frac {\Delta r ^ {\pi}}{\alpha}\right)} \left[ \Delta r ^ {\pi} \right] (23) \\ = \frac {1}{\mathcal {S}} \sum_ {s} \Delta r ^ {\pi} (s) (24) \\ \end{array} +$$ + +Where the last equality follows from the fact that as $\alpha$ goes to infinity, the softmax term approaches the uniform distribution. + +We finally prove the limiting case for $\alpha \to 0$ . Again using L'Hôpital's rule: + +$$ +\lim _ {\alpha \rightarrow \infty} \alpha \log \left(\frac {1}{S} \sum_ {s} e ^ {\frac {\Delta r ^ {\pi}}{\alpha}}\right) = \lim _ {\alpha \rightarrow \infty} \mathbb {E} _ {s \sim \operatorname {s o f t m a x} \left(\frac {\Delta r ^ {\pi}}{\alpha}\right)} [ \Delta r ^ {\pi} ] = \max \Delta r ^ {\pi} \tag {25} +$$ + +Where the final equality follows from the fact that as $\alpha$ goes to zero the softmax term is equivalent to taking a hard maximum over $\Delta r$ . + +# A.4 Proof of Theorem 3.3 + +Proof. First, we want to transform kernel uncertainty into reward uncertainty: + +$$ +\begin{array}{l} \log \mathbb {E} _ {\rho_ {P} ^ {\pi}} [ r (s, a) ] = \log \mathbb {E} _ {\rho_ {P} ^ {\pi}} \left[ \frac {\rho_ {\tilde {P}} ^ {\pi}}{\rho_ {P} ^ {\pi}} r (s, a) \right] \quad (\text {I m p o r t a n c e s a m p l i n g}) (26) \\ \geq \mathbb {E} _ {\rho_ {P} ^ {\pi}} [ \log (r (s, a)) ] + \mathbb {E} _ {\rho_ {P} ^ {\pi}} \left[ \log \frac {\rho_ {\tilde {P}} ^ {\pi}}{\rho_ {P} ^ {\pi}} \right] \quad (\text {J e s e n}) (27) \\ = \mathbb {E} _ {\rho_ {P} ^ {\pi}} \left[ \log (r (s, a)) \right] + \mathcal {H} _ {\mathcal {S} \times \mathcal {A}} \left(\rho_ {P} ^ {\pi}\right) + \mathbb {E} _ {\rho_ {P} ^ {\pi}} \left[ \log \rho_ {\tilde {P}} ^ {\pi} \right] (28) \\ \end{array} +$$ + +We think of $\tilde{r}^{\prime} = \log \rho_{\tilde{P}}^{\pi}$ as adversarial reward and $r^\prime = \log \rho_P^\pi$ as nominal reward. We thus have: + +$$ +\Delta r ^ {\prime} := r ^ {\prime} - \tilde {r} ^ {\prime} = \log \left(\frac {\rho_ {\tilde {P}} ^ {\pi}}{\rho_ {P} ^ {\pi}}\right) = \log \left(\frac {d _ {\tilde {P}} ^ {\pi}}{d _ {P} ^ {\pi}}\right) \tag {29} +$$ + +We further observe that: + +$$ +\alpha \operatorname {L S E} _ {\mathcal {S}} \left(\frac {\Delta r ^ {\pi}}{\alpha}\right) - \alpha \log (\mathcal {S}) = \alpha \log \left(\frac {1}{\mathcal {S}} \sum_ {s \in \mathcal {S}} \left(\frac {d _ {P} ^ {\pi} (s)}{d _ {\hat {\mathcal {P}}} ^ {\pi} (s)}\right) ^ {\frac {1}{\alpha}}\right) \leq \epsilon \tag {30} +$$ + +Thus $\tilde{r}^{\prime}\in \tilde{R}^{\prime \pi}(\epsilon ,\alpha)$ , where $\tilde{R}^{\prime \pi}(\epsilon ,\alpha)$ is the reward uncertainty set described in Theorem 3.1. This containment relation implies: + +$$ +\begin{array}{l} \min _ {\tilde {P} \in \tilde {\mathcal {P}} ^ {\pi} (\epsilon , \alpha)} \mathbb {E} _ {\rho_ {P} ^ {\pi}} \left[ \log \rho_ {P} ^ {\pi} \right] \geq \min _ {\tilde {r} ^ {\prime} \in \tilde {R} ^ {\prime \pi} (\epsilon , \alpha)} \mathbb {E} \left[ \tilde {r} ^ {\prime} \right] = \mathbb {E} _ {\rho_ {P} ^ {\pi}} \left[ \log \left(\rho_ {P} ^ {\pi}\right) \right] + \alpha \left(\mathcal {H} _ {\mathcal {S}} \left(d _ {P} ^ {\pi}\right) - \log (\mathcal {S})\right) - \epsilon (31) \\ = - \mathcal {H} _ {\mathcal {S} \times \mathcal {A}} \left(\rho_ {P} ^ {\pi}\right) + \alpha \left(\mathcal {H} _ {\mathcal {S}} \left(d _ {P} ^ {\pi}\right) - \log (\mathcal {S})\right) - \epsilon (32) \\ \end{array} +$$ + +Plugging this back in to 28 we get: + +$$ +\min _ {\tilde {P} \in \tilde {\mathcal {P}} ^ {\pi} (\epsilon , \alpha)} \log \mathbb {E} _ {\rho_ {\tilde {P}} ^ {\pi}} [ r (s, a) ] \geq \min _ {\tilde {P} \in \tilde {\mathcal {P}} ^ {\pi} (\epsilon , \alpha)} \mathbb {E} _ {\rho_ {P} ^ {\pi}} [ \log (r (s, a)) ] + \alpha \left(\mathcal {H} _ {\mathcal {S}} \left(d _ {P} ^ {\pi}\right) - \log (\mathcal {S})\right) - \epsilon \tag {33} +$$ + +We finish the proof by taking the exponent of both sides. + +![](images/acf125ceac2e2f6d25b1afafe0a2f2bd7be231351abddf31f560cead6f835349.jpg) + +# A.4.1 Policy entropy lower bound + +We aim to use the same methods as the in the proof of the above theorem, except this time for the regularization $\alpha \mathcal{H}_S(d^\pi) + \alpha \mathbb{E}_{d^\pi}[\mathcal{H}_A\pi (\cdot |s)]$ . First, we note that from the chain rule we have $\mathcal{H}_{\mathcal{S}\times \mathcal{A}}(\rho^{\pi}) = \mathcal{H}_{\mathcal{S}}(d^{\pi}) + \mathbb{E}_{d^{\pi}}[\mathcal{H}_{\mathcal{A}}\pi (\cdot |s)]$ . Therefore, we are actually using state-action entropy regularization with temperature $\alpha$ . + +We consider the same lower bound as in equation 28 and the same definition of $\Delta r'$ as in 29. The uncertainty set $\hat{\mathcal{R}}^{\pi}(\epsilon, \alpha)$ corresponding to state-action entropy regularization is (see Table 1): + +$$ +\begin{array}{l} \alpha \operatorname {L S E} _ {\mathcal {S} \times \mathcal {A}} \left(\frac {\Delta r}{\alpha}\right) - \alpha \log (\mathcal {S A}) = \alpha \log \left(\sum_ {s, a \in \mathcal {S} \times \mathcal {A}} \left(\frac {d _ {P} ^ {\pi} (s)}{d _ {\tilde {P}} ^ {\pi} (s)}\right) ^ {\frac {1}{\alpha}}\right) - \alpha \log (\mathcal {S A}) (34) \\ = \alpha \log \left(\sum_ {s \in \mathcal {S}} \left(\frac {d _ {P} ^ {\pi} (s)}{d _ {\tilde {P}} ^ {\pi} (s)}\right) ^ {\frac {1}{\alpha}}\right) - \alpha \log (\mathcal {S}) \leq \epsilon (35) \\ \end{array} +$$ + +Which is the exact same uncertainty set $\tilde{\mathcal{P}}^{\pi}(\epsilon, \alpha)$ as the one considered for state entropy alone. That being said, using the bounds in table 1 we get: + +$$ +\min _ {\tilde {P} \in \tilde {\mathcal {P}} ^ {\pi} (\epsilon , \alpha)} \log \mathbb {E} _ {\rho_ {\tilde {P}} ^ {\pi}} [ r (s, a) ] \geq \mathbb {E} _ {\rho_ {P} ^ {\pi}} [ \log (r (s, a)) ] + \alpha \left(\mathcal {H} _ {\mathcal {S} \times \mathcal {A}} \left(\rho_ {P} ^ {\pi}\right) - \log (\mathcal {S A})\right) - \epsilon \tag {36} +$$ + +This is a worse lower bound than in the state entropy case, since: + +$$ +\begin{array}{l} \mathcal {H} _ {\mathcal {S} \times \mathcal {A}} \left(\rho_ {P} ^ {\pi}\right) - \log (\mathcal {S A}) = \left(\mathcal {H} _ {\mathcal {S}} \left(d _ {P} ^ {\pi}\right) - \log (\mathcal {S})\right) + \left(\mathbb {E} _ {d _ {P} ^ {\pi}} \left[ \mathcal {H} _ {\mathcal {A}} \left(\pi (\cdot \mid s) \right] - \log (\mathcal {A}) \right]\right) (37) \\ \leq \mathcal {H} _ {\mathcal {S}} \left(d _ {P} ^ {\pi}\right) - \log (\mathcal {S}) (38) \\ \end{array} +$$ + +# A.5 Proof of theorem 4.1 + +Let $\Omega \neq 0$ a function. Assume towards contradiction that exists a function $\tilde{\mathcal{P}}(P)$ that satisfies $\min_{\tilde{P} \in \tilde{\mathcal{P}}} \mathbb{E}_{\rho_P^\pi}[r] = \mathbb{E}_{\rho_P^\pi}[r] + \Omega(\pi, P)$ . We consider an MDP with reward vector $R \equiv c$ where $c \in \mathbb{R}$ is some constant. For all $\pi, P$ we have: + +$$ +\begin{array}{l} \min _ {\tilde {P} \in \tilde {\mathcal {P}}} \mathbb {E} _ {\rho_ {\tilde {P}} ^ {\pi}} [ r ] = c = c + \Omega (\pi , P) (39) \\ \Rightarrow \Omega (\pi , P) = 0 (40) \\ \end{array} +$$ + +But the last line implies $\Omega \equiv 0$ , a contradiction. + +# A.6 Proof of Theorem 4.2 + +Before the proof, we state the formal definition of the Conditional Value-at-Risk (CVaR). Let the return be defined as $R(\tau) \coloneqq \sum_{t=0}^{\infty} \gamma^{t} r_{t}$ where the trajectory $\tau = (s_{0}, a_{0}, r_{0}, \ldots)$ is sampled from + +![](images/8a4d7b7400fed2f837b02fbd4bf898a9be5265a0453361d7c60d50cc87221fc2.jpg) +Figure 7: Counterexample used in the proof of Theorem 4.2. The initial state is $s_0$ . All actions are deterministic except for action $a$ taken from state $s_2$ , which leads stochastically to $s_3$ with probability $1 - p$ and to $s_4$ with probability $p$ . Self-loops indicate absorbing states with associated returns. + +$p_P^\pi (\tau)\coloneqq \mathbb{P}[s_0,a_0,r_0,\ldots |\pi ,P]$ . Then, for a given confidence level $\beta \in (0,1)$ the CVaR of the return under policy $\pi$ is defined as: + +$$ +\operatorname {C V a R} _ {\beta} ^ {\pi} [ R (\tau) ] := \min _ {\eta \in \mathbb {R}} \left\{\eta + \frac {1}{1 - \beta} \mathbb {E} _ {p _ {P} ^ {\pi}} [ (R (\tau) - \eta) ^ {+} ] \right\} \tag {41} +$$ + +where $(x)^{+}:= \max (x,0)$ + +Proof. Fix some $M > 0$ , $\alpha > 0$ , and confidence level $0 < \beta < 1$ . In this proof we'll denote the total discounted return by $G$ . We construct an MDP (illustrated in Figure 7) that induces arbitrarily poor risk-sensitive performance under occupancy entropy regularization. + +The key idea is to design an MDP in which two actions from the initial state, $a_1$ and $a_2$ , lead to branches that are nearly identical in terms of expected return, but differ drastically in their return distributions. Specifically, we construct the branch associated with $a_2$ to be extremely risky, such that even a small degree of stochasticity—induced by entropy regularization—exposes the agent to severe downside risk. + +More formally, we'll first require that: + +$$ +\mathbb {E} [ G \mid \pi (s _ {0}) = a _ {1} ] - \epsilon = R _ {1} - \epsilon = \mathbb {E} [ G \mid \pi (s _ {0}) = a _ {2} ] = p R _ {3} + (1 - p) R _ {4} \tag {42} +$$ + +for some $\epsilon > 0$ , implying $\pi^{*}(a_{1} \mid s_{0}) = 1$ and $\pi_{R}^{*}(a_{1} \mid s_{0}) := q \approx \frac{1}{2}$ (the last claim is true because for every temperature $\alpha$ we can make $\epsilon$ arbitrarily small). + +Second, we'll fix $R_{1} = R > 0$ , and since the branch associated with $a_{1}$ is deterministic we'll have $\mathrm{CVaR}_{\beta}^{\pi^{*}}[G] = \mathcal{J}(\pi) = R > 0$ . We'll require that $\mathrm{CVaR}_{\beta}^{\pi_{R}^{*}}[G] \leq -M$ which clearly implies that the result from the proof holds. + +We now show explicitly that exists $R_{3}, R_{4}, p$ s.t. the two conditions above hold. We focus on the second condition, and use the dual representation of CVaR [3]: $\mathrm{CVaR}_{\beta}^{\pi_R^*}[G] = \min_{Q \in \mathcal{U}} \mathbb{E}_{G \sim Q}[G]$ where $\mathcal{U} = \{Q \ll P \mid \forall G : \frac{Q(G)}{P(G)} \in (0, 1 / \beta)\}$ and $P$ is the return distribution for $\pi_R^*$ . + +Since $1 > \beta > 0$ , we can choose $1 > p > 0$ s.t. $(1 - q)p + q \geq \beta$ . We construct a probability measure $Q \in \mathcal{U}$ s.t. $Q(R_3) = \frac{p(1 - q)}{\beta}$ and $Q(R_1) = 1 - Q(R_3)$ . Note that from the construction of $p$ we have that $Q(R_3) < 1$ and $1 - Q(R_3) = Q(R_1) \leq \frac{P(R_1)}{\beta} = \frac{q}{\beta}$ and indeed $Q \in \mathcal{U}$ . We can now + +analyze the risk averse performance of $\pi_R^*$ : + +$$ +\mathrm {C V a R} _ {\beta} ^ {\pi_ {R} ^ {*}} [ G ] \leq R _ {3} \frac {p (1 - q)}{\beta} + R _ {1} \left(1 - \frac {p (1 - q)}{\beta}\right) \leq - M \tag {43} +$$ + +$$ +R _ {3} \leq \frac {\beta (- M - R)}{p (1 - q)} + R \tag {44} +$$ + +We can choose an $R_{3} = R^{\prime}$ that satisfies the inequality above. Finally, we are only left with satisfying the first condition, and we can fix $R_{4} = \frac{R - \epsilon - pR^{\prime}}{1 - p}$ and we are done. + +# A.7 Robustness under estimation errors + +In Section 3, we provided robustness guarantees for the reward uncertainty (Theorem 3.1) and kernel uncertainty (Theorem 3.3) of a policy maximizing the state-entropy regularized objective. In practice, however, the regularization term is not computed with the true state entropy but a sample-based estimate. It is then not obvious than similar guarantees hold under estimation error of the state entropy. In the following, we show how the guarantees generalize to the sample-based regime following a straightforward concentration argument on the state distribution. In the spirit of Section 3, we consider the entropy of a finite state space, while we refer to Section 4.3 for an informal discussion on the sample sensitivity in continuous settings, in which more sophisticated estimators are employed. + +Corollary A.1. Let $\hat{d}$ an empirical estimate of $d_P^\pi$ obtained with $n$ sampled trajectories, for which it holds $\| \hat{d} - d_P^\pi \|_1 \leq 1/2$ . For $\delta \in (0,1)$ and any policy $\pi$ , the following weak duality holds with probability $1 - \delta$ + +$$ +\min _ {\tilde {r} \in \mathcal {R}} \mathbb {E} _ {\rho^ {\pi}} [ \tilde {r} ] \geq \mathbb {E} _ {\rho^ {\pi}} [ r ] + \alpha \left(\mathcal {H} _ {\mathcal {S}} (\hat {d}) - \sqrt {\frac {2 S \log (2 / \delta)}{n}} \log \sqrt {\frac {S n}{\log (2 / \delta)}} - \log (S)\right) - \epsilon , +$$ + +where the reward uncertainty set is $\tilde{\mathcal{R}}^{\pi}(\epsilon, \alpha) \coloneqq \left\{\tilde{r} \mid \mathrm{LSE}_{\mathcal{S}}\left(\frac{r^{\pi} - \tilde{r}^{\pi}}{\alpha}\right) \leq \frac{\epsilon}{\alpha} + \log(S)\right\}$ . + +Proof. To prove the result, it is sufficient to show that the estimated entropy $\mathcal{H}_{\mathcal{S}}(\hat{d})$ concentrates around the true entropy $\mathcal{H}_{\mathcal{S}}(d_P^\pi)$ . Let $\kappa \coloneqq \| \hat{d} - d_P^\pi \|_1$ , we write + +$$ +\begin{array}{l} \left| \mathcal {H} _ {\mathcal {S}} (\hat {d}) - \mathcal {H} _ {\mathcal {S}} \right| \leq \kappa \log S + \kappa \log \frac {1}{\kappa} + (1 - \kappa) \log \frac {1}{1 - \kappa} (45) \\ \leq 2 \kappa \log \frac {S}{\kappa} (46) \\ \leq \sqrt {\frac {2 S \log (2 / \delta)}{n}} \log \sqrt {\frac {S n}{\log (2 / \delta)}} \quad [ \text {w i t h p r o b a b i l i t y} 1 - \delta ] (47) \\ \end{array} +$$ + +where the first inequality follows from continuity bounds of the entropy [60, see], the second inequality follows from the assumption $\kappa \leq 1/2$ , the last inequality invokes concentration of the empirical distribution [58]. Finally, the result is straightforward by plugging equation 47 into the result of Theorem 3.1. + +An analogous result for kernel uncertainty is as follows. + +Corollary A.2. Let $\hat{d}$ an empirical estimate of $d_P^\pi$ obtained with $n$ sampled trajectories, for which it holds $\| \hat{d} - d_P^\pi \|_1 \leq 1/2$ . For any $\delta \in (0,1)$ , policy $\pi$ , and $\alpha > 0$ , the following weak duality holds with probability $1 - \delta$ + +$$ +\min _ {\mathcal {P} ^ {\pi} (\epsilon)} \mathbb {E} _ {\rho_ {P} ^ {\pi}} [ r ] \geq \exp \left(\mathbb {E} _ {\rho_ {P} ^ {\pi}} [ \log (r) ] + \alpha \left(\mathcal {H} _ {\mathcal {S}} (\hat {d}) - \sqrt {\frac {2 S \log (2 / \delta)}{n}} \log \sqrt {\frac {S n}{\log (2 / \delta)}} - \log (S)\right) - \epsilon\right), +$$ + +where + +$$ +\tilde {\mathcal {P}} ^ {\pi} (\epsilon , \alpha) := \left\{\tilde {P} \mid \alpha \log \left(\frac {1}{\mathcal {S}} \sum_ {s \in \mathcal {S}} \left(\frac {d _ {P} ^ {\pi} (s)}{d _ {\tilde {P}} ^ {\pi} (s)}\right) ^ {\frac {1}{\alpha}}\right) \leq \epsilon \right\} +$$ + +Proof. The proof is straightforward by plugging equation 47 into the result of Theorem 3.3. $\square$ + +![](images/c388f7c72ff62465892daf1ca2fd4a6936a9dec86bc3f82dbe944b02c1b426c5.jpg) + +# B Experiments + +The code for our experiments is available at https://github.com/JonathanAshlag/State_entropy_robust_r1. + +# B.1 State entropy algorithm + +k-nearest neighbor entropy estimator. Let $X$ be a random variable with a probability density function $p$ whose support is a set $\mathcal{X} \subset \mathbb{R}^q$ . Then its differential entropy is given as $\mathcal{H}(X) = -\mathbb{E}_{x \sim p(x)}[\log(p(x))]$ . When the distribution $p$ is not available, this quantity can be estimated given N i.i.d realizations of $\{x_i\}_{i=1}^N$ . However, since it is difficult to estimate $p$ with high-dimensional data, particle-based k-nearest neighbors (k-NN) entropy estimator [52] can be employed: + +$$ +\hat {H} _ {N} ^ {k} (X) = \frac {1}{N} \sum_ {i = 1} ^ {N} \log \frac {N \cdot \| x _ {i} - x _ {i} ^ {k - \mathrm {N N}} \| _ {2} ^ {q / 2} \cdot \hat {\pi} ^ {q / 2}}{k \cdot \Gamma (\frac {q}{2} + 1)} + C _ {k} \tag {1} +$$ + +$$ +\hat {H} _ {N} ^ {k} (X) \propto \frac {1}{N} \sum_ {i = 1} ^ {N} \log \| x _ {i} - x _ {i} ^ {k - N N} \| _ {2} \tag {2} +$$ + +Where $x_{i}^{k\text{-NN}}$ is the k-NN of $x_{i}$ within a set $\{x_{i}\}_{i=1}^{N}, C_{k} = \log(k) - \Psi(k)$ is a bias correction term, $\Psi$ the digamma function, $\Gamma$ the gamma function, $q$ the dimension of $x$ , $\hat{\pi} \approx 3.14159$ , and the transition from (1) to (2) always holds for $q > 0$ . + +Using (2) the state entropy can be modeled into an intrinsic reward in the following way: + +$$ +r ^ {i} \left(s _ {i}\right) = \log \left(\left| \left| y _ {i} - y _ {i} ^ {k - n n} \right| \right| + 1\right) \tag {3} +$$ + +where $y_{i}$ can be either $s_i$ , a latent encoding of $s_i$ or selected features of $s_i$ . The total reward is: + +$$ +r _ {j} ^ {\text {t o t a l}} = r \left(s _ {j}, a _ {j}\right) + \beta \cdot \gamma_ {t} \cdot r ^ {i} \left(s _ {j}\right) \tag {4} +$$ + +Where $\beta$ is the regularization temperature, $\gamma_{t}$ is the intrinsic reward discount factor, it's usually the same as the extrinsic reward discount factor; we empirically find that in environments where the episode doesn't end upon task completion, a stronger intrinsic reward discount helps prevent over-exploration. + +Temperature warm-up. We find that it helps performance to start training with a high temperature $\beta_{start}$ and anneal it through training. This is common in prior work [50], notably instead of exponential decay we use cosine decay from $\beta_{start}$ to $\beta > 0$ + +$$ +\beta_ {t} = \beta + \frac {\beta_ {\text {s t a r t}} - \beta}{2} \left(1 + \cos \left(\pi \frac {t - 1}{T}\right)\right), \qquad t = 1, \dots , T. +$$ + +# B.2 Minigrid experiments + +The experiment is built upon [26] open code base, which is released under the MIT License. Base algorithm is A2C. As the point of the expiry isnt handling partial observability, we remove potential bias by instead of using a random encoder, the features used for entropy estimation were the x,y position of the agent + one hot encoding of its direction. The nominal environment is an 8 by 8 empty grid with a sparse reward, 1 if goal is reached, 0 otherwise. action space is (1) take a step in current direction, (2) turn right (3) turn left. + +Wall perturbation, Fig. 4(b). We randomly sample a point $x_0$ along the x-axis (excluding the first and last columns), and add a vertical wall spanning positions $(x_0, 1)$ through $(x_0, 6)$ . + +Random walls, Fig. 4(c). Walls are added to 7 blocks randomly sampled from the grid, excluding the start and goal positions. + +Goal shift, Fig. 4(d). The goal position is randomly shifted to any block in the grid, excluding the starting position. + +# B.3 Pusher experiment + +The code for our experiments is based on the CleanRL codebase [20], which is released under the MIT License. We use PPO with a recurrent network for all baselines. Note that it is known that to maximize state entropy over finite number of rollouts, a non-Markovian policy is necessary. State entropy is calculated over the puck's x,y positions. + +Transition kernel perturbation (wall): The obstacle is composed of three axis-aligned blocks with width $3\mathrm{cm}$ , centered at $(0.52, -0.24)$ , $(0.55, -0.21)$ and $(0.58, -0.18)$ . These positions are roughly along the perpendicular bisector of the line between the puck's initial position and the goal. We used 200 episodes of length 100 for evaluating each method. As in [13], to decrease the variance in the results, we fixed the initial state of each episode: + +$$ +q p o s = [ 0., 0., 0., 0., 0., 0., - 0. 3, 0. 2, 0., 0. ] +$$ + +$$ +q v e l = [ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0. ] +$$ + +Action robust baseline [6] We ran as a baseline both PRMDP and NRMDP. To ensure a fair comparison we implemented the following modifications: + +- Larger networks: We increased the width of all neural networks to 256 units. +- hyperparameter sweep: we swept for exploration noise $\in [0.2, 0.6, 1]$ , learning rate $\in [1e - 3, 5e - 4, 1e - 4]$ and batch size $\in [256, 512, 1024]$ +- Check-pointing: we saved both the best performing checkpoint during training (evaluated in the nominal environment) and the last iteration checkpoint. We evaluated both of them in the perturbed environment and showed the better of the two. + +Reward shift: The original goal was located at (0.66, -0.35). In the reward shift experiment, a new goal is randomly sampled uniformly from the circumference of a circle with radius $r$ centered at the original goal location. For each radius in [0.1, 0.15] we randomly sample 200 goals, evaluate once per goal, and report the average performance across all goals. + +Number of rollouts ablation, Fig. 5(b). To ensure a fair comparison, we fix the total batch size to 32 across all variants. For a variant using $x$ rollouts per entropy estimate, we divide the 32 rollouts into $\frac{32}{x}$ non-overlapping groups of size $x$ , and compute the intrinsic reward separately for each group. As the number of rollouts affects the scale of the intrinsic reward, we tune the temperature individually for each variant. + +# B.4 Ant experiment + +We train All policies on Mujoco's "ant-v5". To emphasize the affects of each regularization we augment the environment to be less reward shaped in the following way: The episode does not truncate when the ant's in an unhealthy position. Furthermore, there is no reward for being in a healthy position and the penalties on enduring contact forces and control cost are turned off. The reward in our setup is only the forward progress reward. State entropy is calculated over the height and orientations of the ant, encouraging the agent to learn to walk in diverse ways. + +Transition kernel perturbation (vertical wall) During evaluation, we add an unobservable wall of a size of $[1,\infty ,0.3]$ ; the only way to pass the wall is to hurdle above it. Evaluation protocol is the same as in the pusher experiment. + +# B.5 Interplay between State and Policy Entropy Regularization + +In the experiments reported in the main text, we include policy entropy regularization alongside state entropy to ensure stable training, following common practice. Here, we empirically assess whether policy entropy contributes to robustness beyond its stabilizing role. + +As shown in Figure 8, omitting policy entropy leads to unstable training, reflected in high variance in nominal performance. To isolate the effect of state entropy, we take a policy trained with both regularizations and continue training it for 2 million steps using only state entropy. As shown in + +![](images/e6e257d4df2851d4baa62946703ad03cc0806966618f312866c68977dc03a755.jpg) +Figure 8: State entropy regularization unstable training and warm start experiment + +![](images/9d66be2ef6b519a45acf7994617756c66e2d50cdfdc4ca6b94400f2ae65c38ea.jpg) +(a) episodic return + +![](images/e696ff8195789538eb05d08ab582929d6e3d48788cdc9cea238cf714fa9f3ee3.jpg) +(b) state entropy + +![](images/4e4532b1d8c7e9e4523c21c3e985b5da127c1b90bdbe0394fcb5e8c4b2e6dec0.jpg) +(c) policy entropy +Figure 9: State and policy entropy -> only state entropy distillation + +Figure 9, the policy maintains both its return and state entropy, despite a significant drop in policy entropy. The resulting policy's robust performance closely matches that of the state-and-policy entropy variant. + +We hypothesize that with more careful algorithmic tuning, state entropy alone could suffice for both robustness and stability. However, we leave a systematic investigation of its role in training stability to future work. + +# B.6 Technical details + +All experiments were conducted on a machine equipped with an NVIDIA RTX 4090 GPU. Recreating experiments B.2 takes about 12 hours to run 25 seeds for the 3 baselines, sweeping regularization temperature took an additional day. Recreating experiments B.3 takes a day for 25 seeds for the 3 baselines. rollouts, temperature ablations and regularization temperature searching were another 2 days. 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Despite their promising applications, the theoretical foundations of these models remain substantially underdeveloped, with the existing literature predominantly focusing on continuous-state diffusion models. A critical gap persists in the theoretical understanding of discrete diffusion modeling: the absence of a rigorous framework for quantifying estimation error with finite data. Consequently, the fundamental question of how precisely one can reconstruct the true underlying distribution from a limited training set remains unresolved. In this work, we analyze the estimation error induced by a score estimation of the discrete diffusion models. One of the main difficulties in the analysis stems from the fact that the cardinality of the state space can be exponentially large with respect to its dimension, which results in an intractable error bound by a naive approach. To overcome this difficulty, we make use of a property that the state space can be smoothly embedded in a continuous Euclidean space that enables us to derive a cardinality independent bound, which is more practical in real applications. In particular, we consider a setting where the state space is structured as a hypercube graph, and another where the induced graph Laplacian can be asymptotically well approximated by the ordinary Laplacian defined on the continuous space, and then derive state space size independent bounds. + +# 1 Introduction + +Diffusion modeling has demonstrated state-of-the-art performance in learning problems such as creating images (Song et al., 2021; Dhariwal and Nichol, 2021), videos (Ho et al., 2022), and audios (Chen et al., 2020; Kong et al., 2020), drawing significant attention to their applications. + +Theoretical studies on diffusion modeling in continuous state spaces have been conducted within the framework of score-based generative modeling (Sohl-Dickstein et al., 2015; Song and Ermon, 2019; Song et al., 2021; Ho et al., 2020; Vahdat et al., 2021). One of the most important characterizations of diffusion modeling is the formulation using stochastic differential equations (SDEs) proposed by Song et al. (2021). + +With the advent of SDE formulation, significant efforts have been made to analyze the estimation error between the true distribution and the generated distribution. Lee et al. (2022b) showed that the total variation distance between the two distributions can be bounded by the polynomial order of the score estimation error and the step size with the time discretization of the reverse process. Lee et al. (2022b) assumed the smoothness of the score function and the validity of the log-Sobolev inequality (LSI) for the true distribution, while Chen et al. (2023b) and Lee et al. (2022a) derived error bounds without the LSI condition, Chen et al. (2023a) further relaxed the smoothness assumption. Moreover, Song et al. (2021), Pidstrigach (2022) evaluated error rates under the manifold hypothesis, + +which assumes that the true distribution is concentrated on a low-dimensional manifold. De Bortoli et al. (2021) and De Bortoli (2022) derived error bounds under the assumptions of score estimation error bounds at each time step and each point, considering dissipative structures and the manifold hypothesis, respectively. + +While the above studies assumed the accuracy for score approximation, Oko et al. (2023) developed a theoretical framework that derives the approximation error of the score function when the true density belongs to a Besov space. Their work combined function approximation theory from deep learning with the approximation theory of diffusion modeling and employed concentration inequalities to establish score estimation error bounds. Building upon this seminal work, several refined theoretical analyses have been proposed, including relaxation of the lower bound condition for the density (Zhang et al., 2024), analyses under the manifold assumption on the support of the data distribution (Azangulov et al., 2024), statistical guarantees for reflected diffusion models (Holk et al., 2024), and minimax optimality of the probability flow ODE (Cai and Li, 2025). + +Recently, diffusion modeling for discrete states has also gained attention (Hoogeboom et al., 2021; Austin et al., 2021; Richemond et al., 2022; Meng et al., 2022; Sun et al., 2023; Santos et al., 2023; Lou et al., 2024). Notable advancements have been made in learning problems with discrete structures, such as natural language processing (Austin et al., 2021; He et al., 2023; Wu et al., 2023), molecular design (Zhang et al., 2023; Gruver et al., 2023; Campbell et al., 2024; Lee et al., 2025), graph generation (Niu et al., 2020; Shi et al., 2020; Vignac et al., 2023), and segmentation (Zbinden et al., 2023). In addition, in areas where continuous diffusion modeling performs well, such as image (Hu et al., 2022; Zhu et al., 2023) and audio generation (Yang et al., 2023), discrete diffusion models have been shown to efficiently infer multimodal generation problems conditioned on discrete structures like text. + +As for the theoretical analysis of discrete diffusion modeling, Campbell et al. (2022) analyzed total variation distance based on Markov chains, and Chen and Ying (2024) reduced the error under the condition that the discrete state space is restricted to the vertices of a hypercube. Ren et al. (2025) proposed a more general approximation theory for discrete diffusion characterized by a Poisson random measure with evolving intensity, allowing discrete diffusion modeling to be formulated as stochastic integrals similar to the theory of continuous diffusion modeling. + +However, the theoretical analysis of score estimation error in discrete diffusion modeling remains unexplored. In this study, we derive score estimation error bounds for discrete diffusion models by applying the function approximation theory of neural networks and concentration inequalities, which were previously used in the score estimation theory of continuous diffusion modeling (Oko et al., 2023). Our main contributions are summarized as follows: + +- First, we develop a theoretical framework for bounding the score estimation error in discrete diffusion models. Unlike continuous diffusion models that rely on $L^2$ loss in score matching, our analysis handles the Bregman divergence loss that naturally arises in discrete diffusion, and we rigorously control the estimation error using the Hellinger distance. +- Second, we introduce a novel approach to achieve state-size independent error bounds by embedding the discrete space $\mathbb{X}$ into $\mathbb{R}^d$ and approximating the eigenvectors of the graph Laplacian by functions in an anisotropic Besov space. This enables the use of advanced function approximation results for deep ReLU networks. Under mild regularity assumptions, we show that the error bound depends only polylogarithmically on the number of discrete states $M$ , which nearly achieves the optimal rate conjectured in Ren et al. (2025). +- Third, we demonstrate that this framework can be instantiated in concrete settings such as the hypercube $[0,1]^D$ and graph-based diffusion processes on smooth manifolds. In both examples, we show that the eigenvectors of the transition matrix admit efficient approximations. + +# 2 Preliminary + +Here, we introduce discrete diffusion models and prepare some technical matters for our theoretical analysis. Before introducing the discrete diffusion models, we briefly review the continuous state diffusion models. The continuous diffusion models consist of two stochastic processes, the forward process and the reverse process, so that the model can generate data whose distribution is sufficiently close to the true distribution $p_0$ on $\mathbb{R}^d$ . The forward process $\{X_t\}_{t \geq 0}$ on $\mathbb{R}^d$ is formulated as the + +following Ornstein-Uhlenbeck (OU) process: + +$$ +X _ {0} \sim p _ {0}, \quad \mathrm {d} X _ {t} = - X _ {t} \mathrm {d} t + \sqrt {2} \mathrm {d} B _ {t}, +$$ + +where $\{B_t\}_{t \geq 0}$ represents a $d$ -dimensional standard Brownian motion. Under certain assumptions on the initial distribution $p_0$ (Haussmann and Pardoux, 1986; Cattiaux et al., 2023), the distribution of $X_t$ , denoted by $p_t$ , converges exponentially to the standard normal distribution as $t \to \infty$ . The reverse process $\{Y_t\}_{0 \leq t \leq T} (T \geq 0)$ defined by the following stochastic process trace-backs the forward process: + +$$ +Y _ {0} \sim p _ {T}, \mathrm {d} Y _ {t} = \left(Y _ {t} + 2 \nabla \log p _ {T - t} (Y _ {t})\right) \mathrm {d} t + \sqrt {2} \mathrm {d} B _ {t}. +$$ + +That is, the distribution of $Y_{T - t}$ coincides with $p_t$ for $0 \leq t \leq T$ , and thus we can generate samples from the target distribution $p_0$ by sampling $Y_T$ via the reverse process. However, we do not know the initial distribution of the reverse process $p_T$ and the score function $\nabla \log p_t$ because both of them are dependent on the unknown true distribution $p_0$ . Here, the initial distribution $p_T$ can be replaced by the standard normal distribution for sufficiently large $T$ since $p_t$ converges to the standard normal exponentially fast, and the score function $\nabla \log p_t(Y_t)$ can be estimated by least squares score matching using finite-size training data. These approximations, along with the time discretization of the process, induce errors in the resulting distribution generated by the model. Chen et al. (2023a) obtained an upper bound on this error in terms of Kullback-Leibler divergence (KL-divergence). Their result implies that the main part of the error is the estimation error of the score function $\nabla \log p_t$ . Based on this observation, Oko et al. (2023) derived the following bound on the score estimation error. + +Proposition 1 (Oko et al. (2023), informal). Under some smoothness assumptions on $p_0$ , a score matching estimator $\widehat{s}_t$ obtained on a deep neural network model with an appropriate network size, can achieve + +$$ +\mathbb {E} _ {x _ {0, i}} \left[ \int_ {\delta} ^ {T} \mathbb {E} _ {x _ {t} \sim p _ {t}} \left[ \| \nabla \log p _ {t} (x _ {t}) - \widehat {s} _ {t} (x _ {t}) \| ^ {2} \right] d t \right] \lesssim n ^ {- \frac {2 s}{d + 2 s}} \log^ {1 8} n. \tag {1} +$$ + +Here, $s$ represents the smoothness parameter of the Besov space to which $p_0$ belongs, and $\delta > 0$ is a sufficiently small end-point time. + +This theorem implies that diffusion models with an appropriately designed score-function estimator can achieve the minimax optimal rate $n^{-\frac{2s}{d + 2s}}$ to estimate the target distribution with smoothness $s$ . In this work, we establish an analogue of this result for discrete diffusion models. + +# 2.1 Discrete diffusion modeling + +Discrete diffusion modeling is defined over a finite set $\mathbb{X}$ instead of $\mathbb{R}^d$ . Let $M \coloneqq |\mathbb{X}|$ , and consider estimating the probability mass vector $p_0 \in \Delta^M$ , where $\Delta^M \coloneqq \left\{ p \mid \sum_{x \in \mathbb{X}} p(x) = 1, p \in \mathbb{R}_{\geq 0}^M \right\}$ . We assume that each $x \in \mathbb{X}$ has a vector representation $\iota(x) \in \mathbb{R}^D$ . We identify this vector representation with $x$ and use the same notation $x$ for both meanings. A typical situation is $\mathbb{X} = \{0,1\}^D$ where $\iota(x) = (x_1, \ldots, x_D) \in \mathbb{R}^D$ , and another example is one-hot-vector representation $\mathbb{X} = \{x \in \{0,1\}^M \mid \sum_{i=1}^{M} x_i = 1\}$ . + +Forward process: In the forward process of discrete diffusion modeling, the distribution $\{p_t\}_{t\geq 0}$ at each time step follows the master equation of the following Markov process: + +$$ +\frac {\mathrm {d} p _ {t}}{\mathrm {d} t} = Q _ {t} p _ {t}, \tag {2} +$$ + +where $Q_{t} \in \mathbb{R}^{M \times M}$ is the transition rate matrix satisfying $Q_{t}(x, x) = -\sum_{y \neq x} Q_{t}(y, x) (\forall x \in \mathbb{X})$ and $Q_{t}(x, y) \geq 0 (\forall x \neq y \in \mathbb{X})$ . If $\pi$ denotes the stationary distribution of the Markov process (2), the following equation analogous to the continuous diffusion model holds (Bobkov and Tetali, 2006): + +$$ +\mathrm {K L} \left(p _ {t} \mid \mid \pi\right) \leq \exp (- \rho (Q) t) \mathrm {K L} \left(p _ {0} \mid \mid \pi\right). \tag {3} +$$ + +Here, $\rho (Q)$ is the modified log-Sobolev constant(Bobkov and Tetali, 2006) defined as + +$$ +\rho (Q) := \inf \left\{\frac {\mathcal {E} _ {\pi} (f , \log f)}{\operatorname {E n t} _ {\pi} (f)} \mid f: \mathbb {X} \mapsto \mathbb {R}, \operatorname {E n t} _ {\pi} (f) > 0 \right\}, +$$ + +where $\operatorname{Ent}_{\pi}(f) \coloneqq \mathbb{E}_{\pi}[f \log f] - \mathbb{E}_{\pi}[f] \log \mathbb{E}_{\pi}[f]$ and $\mathcal{E}_{\pi}(f, g) = \mathbb{E}_{\pi}[fQ^{\top}g]$ . + +Reverse process: The reverse process $\{q_t\}_{0\leq t\leq T} = \{p_{T - t}\}_{0\leq t\leq T}$ can be formulated by using another transition rate matrix $\overline{Q}_t$ as follows (Kelly, 2011): + +$$ +q _ {0} = p _ {T}, \frac {\mathrm {d} q _ {t}}{\mathrm {d} t} = \bar {Q} _ {T - t} q _ {t} (0 \leq t \leq T), \tag {4} +$$ + +where $\overline{Q}_t(y,x) = \frac{p_t(y)}{p_t(x)} Q_t(x,y)$ and $\overline{Q}_t(x,x) = -\sum_{y\neq x}\overline{Q}_t(y,x)$ . Then, it is known that $q_{t} = p_{T - t}$ ( $0\leq t\leq T$ ). Although this reverse process only gives the ODE of the probability mass function $q_{t}$ , its particle implementation can be given by the $\tau$ -leaping algorithm (Campbell et al., 2022; Ren et al., 2025). This algorithm expresses the reverse process as a stochastic integral and applies the Euler-Maruyama scheme, similar to the continuous case. The transformation into a stochastic integral is performed using a Poisson random measure with evolving intensity (Protter, 1983). For simplicity, we consider the case where $Q_{t}$ is time-homogeneous ( $Q_{t} = Q$ holds for all $t\geq 0$ ). + +Proposition 2 (Ren et al. (2025)). The reverse process (4) can be expressed as the following stochastic integral defined by a Poisson process $N[\mu]$ with respect to an intensity function $\mu$ (see Definition 2 in Appendix B for its definition): + +$$ +y _ {t} = y _ {0} + \int_ {0} ^ {t} (y - y _ {t -}) N [ \mu ] (\mathrm {d} t, \mathrm {d} y), \quad \mu_ {t} (y) = s _ {T - t} ^ {\circ} (y _ {t -}, y) \widetilde {Q} (y _ {t -}, y). +$$ + +where $X_{t-}$ denotes the left limit of $X_{t}$ and $\widetilde{Q}$ denotes the matrix $Q$ with the diagonal elements set to 0. + +Score estimation: However, to implement the $\tau$ -leaping algorithm, we need to know the score function $s_t^\circ (x,y)\coloneqq \frac{p_t(y)}{p_t(x)}$ . We approximate it using a score network $s:\mathbb{X}^2\times \mathbb{R}\to \mathbb{R}$ $((x,y,t)\mapsto s_t(x,y))$ in a deep neural network model $\mathcal{F}$ . The score network $s$ can be estimated via score matching analogous to the continuous state diffusion models, but we need to account for the non-negativity constraint of the score function. For that purpose, the denoising score entropy is employed instead of $L^2$ loss (Lou et al., 2024): + +$$ +\begin{array}{l} s \in \arg \min _ {s \in \mathcal {F}} \int_ {0} ^ {T} \mathbb {E} _ {x \sim p _ {t}} \left[ \sum_ {y \neq x} \mathrm {B R} _ {K} (s _ {t} (x, y) \| s _ {t} ^ {\circ} (x, y)) \cdot s _ {t} ^ {\circ} (x, y) Q (x, y) \right] \mathrm {d} t \\ = \arg \min _ {s \in \mathcal {F}} \int_ {0} ^ {T} \mathbb {E} _ {x _ {0} \sim p _ {0}} \left[ \mathbb {E} _ {x \sim p _ {t} (\cdot | x _ {0})} \left[ \sum_ {y \neq x} \mathrm {B R} _ {K} (s _ {t} (x, y) \| s _ {t} ^ {\circ} (x, y \mid x _ {0})) \cdot s _ {t} ^ {\circ} (x, y \mid x _ {0}) Q (x, y) \right] \right] d t. \\ \end{array} +$$ + +Here $s_t^\circ(x, y \mid x_0) := \frac{p_t(y|x_0)}{p_t(x|x_0)}$ and BR denotes the Bregman divergence defined by + +$$ +\mathrm {B R} _ {f} \left(s ^ {*} \| s\right) := f \left(s ^ {*}\right) - f (s) - \partial f (s) \left(s ^ {*} - s\right), +$$ + +for a strictly convex function $f$ , where we employ a particular choice $K(x) \coloneqq x - \log x$ for the convex function $f$ . In training the neural network, analogous to continuous diffusion models, we approximate the expectation over $x_0 \sim p_0$ by empirical distribution defined by the training data $D_n \coloneqq \{x_i\}_{i=1}^n (x_i \stackrel{\text{i.i.d.}}{\sim} p_0)$ with size $n$ , and we seek $\widehat{s} \in \mathcal{F}$ that minimizes this empirical loss. We define the loss function $\ell$ as + +$$ +\ell_ {s} (x _ {0}) := \int_ {\delta} ^ {T} \mathbb {E} _ {x _ {t} \sim p _ {t} (\cdot | x _ {0})} \left[ \sum_ {y \neq x _ {t}} \mathrm {B R} _ {K} (s _ {t} (x _ {t}, y) \| s _ {t} ^ {\circ} (x _ {t}, y | x _ {0})) \cdot s _ {t} ^ {\circ} (x _ {t}, y | x _ {0}) Q (x _ {t}, y) \right] \mathrm {d} t. +$$ + +Then, the empirical loss $\widehat{L}(s)$ can be written as $\widehat{L}(s) \coloneqq \frac{1}{n} \sum_{i=1}^{n} l_s(x_i)$ . As we have stated above, we find the empirical risk minimizer $\widehat{s}$ in the set of deep neural networks that is defined as + +$$ +\Phi (L, W, S, B) := \left\{\left(A ^ {(L)} \eta (\cdot) + b ^ {(L)}\right) \circ \dots \circ \left(A ^ {(1)} x + b ^ {(1)}\right) \mid \right. +$$ + +$$ +A ^ {(i)} \in \mathbb {R} ^ {W _ {i + 1} \times W _ {i}}, b ^ {(i)} \in \mathbb {R} ^ {W _ {i + 1}}, \sum_ {i = 1} ^ {L} (\| A ^ {(i)} \| _ {0} + \| b ^ {(i)} \| _ {0}) \leq S, \max _ {i} \| A ^ {(i)} \| _ {\infty} \vee \| b ^ {(i)} \| _ {\infty} \leq B \}, +$$ + +where $L$ represents the depth, $W = (W_{i})_{i = 1}^{L}$ represents the width with $W_{L + 1} = 1$ , $B$ represents the sparsity, and $B$ is a bound on the norm of parameters. Here, the activation function $\eta (\cdot)$ is given by $\eta (x) = \mathrm{ReLU}(x)\coloneqq \max (x,0)$ . The function class to which the score network belongs is defined + +# Algorithm 1 Implementation of discrete diffusion modeling by $\tau$ -leaping + +1: Input: $\widehat{y}_0\sim \pi$ , time discretization $\{t_k\}_{k\in [0,K]}$ $(t_0 = 0,t_K = T - \delta)$ , intensity function $\hat{\mu}_t$ score network $\widehat{s}_t$ +2: Output: sample from $\widehat{y}_{t_K} \sim \widehat{q}_{T - \delta}$ +3: for $n = 0$ to $K - 1$ do +4: $\widehat{y} t_{n + 1}\gets \sum_{y\in X}(y - \hat{y} t_n)\mathcal{P}(\widehat{\mu} t_n(y)(t_{n + 1} - t_n));$ +5: end for + +as $\mathcal{F} \coloneqq \{s \in \Phi(L, W, S, B) \mid s_t(x, y) \in [1 / R, R] (\forall t, x, y)\}$ with a hyper-parameter $R \geq 1$ (see Assumption 3)1. + +We denote the expectation of a measurable function $f$ with respect to $x_0 \sim p_0$ as $Pf$ , and its empirical distribution as $P_{n}f$ , i.e., $Pf \coloneqq \mathbb{E}_{x_0 \sim p_0}[f]$ and $P_{n}f \coloneqq \frac{1}{n}\sum_{i=1}^{n}f(x_{i})$ . Accordingly, the expected loss $L(s)$ and empirical loss $\widehat{L}(s)$ can be expressed as + +$$ +L (s) := \mathbb {E} _ {x _ {0} \sim p _ {0}} [ \ell_ {s} (x _ {0}) ] = P (\ell_ {s}), \widehat {L} (s) := \frac {1}{n} \sum_ {i = 1} ^ {n} \ell_ {s} (x _ {i}) = P _ {n} (\ell_ {s}), +$$ + +Then, the empirical risk minimizer on the deep neural network model $\mathcal{F}$ is given by + +$$ +\widehat {s} \in \operatorname * {a r g m i n} _ {s \in \mathcal {F}} P _ {n} (\ell_ {s}). +$$ + +Once we have obtained an estimator $\widehat{s}$ , we can define the corresponding intensity function $\widehat{\mu}_t$ with time discretization $(t_k)_{k=0}^K$ , with $t_0 = 0$ and $t_K = T - \delta$ , as + +$$ +\widehat {\mu} _ {\lfloor t \rfloor} (y) = \widehat {s} _ {T - \lfloor t \rfloor} (\widehat {y} _ {\lfloor t \rfloor -}, y) \widetilde {Q} (\widehat {y} _ {\lfloor t \rfloor -}, y), +$$ + +where $\lfloor t\rfloor = t_k$ for $t\in [t_k,t_{k + 1})$ . Moreover, due to Eq. (3), the initial distribution of the reverse process $q_{0}$ can be replaced by $\pi (\simeq p_T)$ . Then, the $\tau$ -leaping algorithm with our estimate $\widehat{s}$ can be implemented as in Algorithm 1. + +# 2.2 Technical assumptions and theoretical tools for the error analysis + +To derive the discrepancy between the generated distribution and the true target distribution for the estimated discrete diffusion model implemented by $\tau$ -leaping algorithm (Alg. 1), we prepare some technical tools. + +Assumption 1. The transition rate matrix $Q$ is symmetric, and there exist positive constants $C, \underline{D}$ , and $\overline{D}$ such that $Q(x,y) < C$ , $\underline{D} < -Q(x,x) < \overline{D}$ . + +Assumption 2. There exists a lower bound $\rho >0$ for the modified log-Sobolev constant $\rho (Q)$ + +Assumption 3. There exists $R \geq 1$ such that the true score function and our model satisfy $s_t^\circ (x,y) \in [1 / R,R]$ and $\widehat{s}_t(x,y) \in [1 / R,R]$ . + +Assumption 4. There exists $\gamma \in [0,1]$ such that for any $y \in \mathbb{X}$ satisfying $Q(x_{t-}, y) > 0$ , the following holds: $\left|\frac{p_t(x_{t-})Q(x_t, y)}{p_t(x_t)Q(x_{t-}, y)} - 1\right| \lesssim 1 \vee t^{-\gamma}$ . + +Assumption 1 is a natural condition for discrete diffusion models which ensures the regularity of the rate matrix (Ren et al., 2025). Assumption 2 guarantees the exponential convergence of the forward process in discrete diffusion models, serving a role analogous to functional inequalities in the continuous case (Bakry et al., 2014). Assumption 3 imposes boundedness on the score function, which mirrors common assumptions made in continuous-state diffusion processes (Chen et al., 2023a). Assumption 4 corresponds to the Lipschitz continuity of the score function in continuous diffusion models (Chen et al., 2023b,a), ensuring sufficient regularity for the analysis. Under these assumptions, the following result holds. + +Proposition 3 (Ren et al. (2025)). In the $\tau$ -leaping algorithm, suppose the time discretization $\{t_k\}_{k\in [0,K]}$ satisfies $t_{k + 1} - t_k\leq \kappa (1\vee (T - t_{k + 1})^{1 + \gamma -\eta})$ for some $\eta >0$ and assume that + +$$ +\sum_ {k = 0} ^ {K - 1} \left(t _ {k + 1} - t _ {k}\right) \mathbb {E} \left[ \sum_ {y \neq x _ {t _ {k} -}} \mathrm {B R} _ {K} \left(\widehat {s} _ {T - t _ {k}} \left(x _ {t _ {k} -}, y\right) \| s _ {T - t _ {k}} ^ {\circ} \left(x _ {t _ {k} -}, y\right)\right) s _ {T - t _ {k}} ^ {\circ} \left(x _ {t _ {k} -}, y\right) Q \left(x _ {t _ {k} -}, y\right) \right] \leq \varepsilon_ {\mathrm {s c}}, +$$ + +for the score network $\widehat{s}_t(x,y)$ . Here, we suppose that $\gamma < \eta \lesssim 1 - T^{-1}$ for $\gamma < 1$ , and $\eta = 1$ for $\gamma = 1$ , and it holds that + +$$ +T = \mathcal {O} \left(\frac {\log \left(\varepsilon_ {\mathrm {s c}} ^ {- 1} \log | \mathbb {X} |\right)}{\rho}\right), \kappa = \mathcal {O} \left(\frac {\varepsilon_ {\mathrm {s c}} \rho}{\bar {D} ^ {2} \log \left(\varepsilon_ {\mathrm {s c}} ^ {- 1} \log | \mathbb {X} |\right)}\right), \delta = \left\{ \begin{array}{l l} 0 & (\gamma < 1), \\ \Omega (e ^ {- \sqrt {T}}) & (\gamma = 1). \end{array} \right. +$$ + +Then, under Assumption 1 to 4, the following error bound holds + +$$ +\mathrm {K L} (p _ {\delta} \| \widehat {q} _ {T - \delta}) \lesssim \underbrace {\exp (- \rho T) \log M} _ {(i)} + \underbrace {\overline {{D}} ^ {2} \kappa T} _ {(i i)} + \underbrace {\varepsilon_ {\mathrm {s c}}} _ {(i i i)}. +$$ + +Similar to continuous diffusion modeling, the upper bound on the error consists of three terms. (i) corresponds to the error from the convergence rate of the forward process, as shown in Eq. (3). (ii) arises from time discretization. (iii) represents the estimation error of the score network. Although the Girsanov's theorem cannot be applied to discrete diffusion models, another similar proposition can be derived from the characterization of a Poisson random measure with evolving intensity (Ren et al., 2025), which allows for the evaluation of the estimation error. However, unlike continuous diffusion models, the sample size or neural network parameter size required to satisfy the estimation error assumption has not been examined. Hence, the aim of this paper is to show an upper bound of $\varepsilon_{\mathrm{sc}}$ analogous to Eq. (1). + +# 3 Naive estimation error bound of discrete diffusion models + +Here, we derive an estimation error bound of the score estimation network in a rather naive way. In our analysis, the key step is to derive an upper bound on the Hellinger distance between the empirical risk minimizer $\widehat{s}$ and the true score function $s^{\circ}$ , defined by + +$$ +h \left(\widehat {s} _ {t}, s _ {t} ^ {\circ}\right) := \left(\mathbb {E} _ {x _ {t} \sim p _ {t}} \left[ \sum_ {y \neq x _ {t}} \left(\sqrt {\widehat {s} _ {t} (x _ {t} , y)} - \sqrt {s _ {t} ^ {\circ} (x _ {t} , y)}\right) ^ {2} Q (x _ {t}, y) \right]\right) ^ {1 / 2}. +$$ + +First, we give a bound on this Hellinger distance under a naive setting. + +Theorem 1. Under Assumption 1 to 4, and the same parameter settings as Proposition 3, if the network size is set as $L = \mathcal{O}(\log^2 (mM))$ , $W = \tilde{\mathcal{O}} (M)$ , $S = \tilde{O} (M)$ , $B = \tilde{O} (M^2)$ and $T = \mathcal{O}(\log (M\vee n))$ , then with probability at least $1 - 2e^{-t}$ for $t\geq 1$ , it holds that + +$$ +\int_ {\delta} ^ {T} h ^ {2} \left(\widehat {s} _ {t}, s _ {t} ^ {\circ}\right) \mathrm {d} t = \mathcal {O} \left(\frac {M t}{n} \log^ {8} (M \vee n)\right). +$$ + +The formal proof of Theorem 1 is provided in Appendix C. By Proposition 3 and Theorem 1, we obtain the following result showing an estimation error bound for the estimated distribution. + +Theorem 2. Suppose that the same condition as that of Theorem 1 holds. Then, the following estimation error bound holds with probability $1 - 2e^{-t}$ for $t \geq 1$ : + +$$ +\mathrm {K L} (p _ {\delta} \| \widehat {q} _ {T - \delta}) \lesssim \frac {M t}{n} \log^ {8} (M \vee n). +$$ + +One of the biggest difficulties in deriving this bound is the requirement to carefully treat the Bregman divergence in contrast to the continuous diffusion models where we could directly deal with the $L^2$ -norm. The difficulty can be overcome by noticing that the Bregman divergence can be lower bounded by the Hellinger distance. By combining this notion with the so-called peeling device + +van de Geer (2000), we achieve $\mathcal{O}(1/n)$ rate of convergence with respect to the sample size $n$ , which significantly improves upon the standard Rademacher complexity bound that only provides an $\mathcal{O}(1/\sqrt{n})$ bound. An additional technical contribution is to show how ReLU neural networks can effectively approximate the score function $s_t$ . + +This bound holds under minimal assumptions. On the other hand, if the state space is a product space such as $\mathbb{X} = \{0,1\}^D$ (which often happens in real applications), then $M$ can be exponentially large with respect to the dimension $D$ . To overcome this difficulty, we consider a situation where the discrete state space $\mathbb{X}$ can be embedded into a Euclidean space $\mathbb{R}^d$ and the eigenvectors corresponding to the Markov transition operator $Q$ can be well approximated by a smooth function on $\mathbb{R}^d$ . By doing so, we obtain a significantly improved bound as seen in the next section. + +# 4 State size independent error analysis with continuous space embedding + +As we mentioned above, we aim to improve the estimation error by explicitly considering an embedding from the discrete space $\mathbb{X}$ into a Euclidean space $\mathbb{R}^D$ . Through the embedding, we may approximate the eigenvectors of $Q$ by functions defined on $\mathbb{R}^D$ enabling the application of function approximation theory for smooth functions defined on Euclidean spaces. Especially, the anisotropic Besov space is a useful function class that covers a wide range of functions with smoothness. + +Anisotropic Besov space. Here, we begin by defining anisotropic Besov spaces. Let $\Omega = [0,1]^d$ . For a function $f:\Omega \to \mathbb{R}$ , we define its $L^p$ -norm as + +$$ +\| f \| _ {p} := \| f \| _ {L ^ {p} (\Omega)} := \left\{ \begin{array}{l l} (\int_ {\Omega} | f | ^ {p} d x) ^ {1 / p} & (0 < p < \infty), \\ \sup _ {x \in \Omega} | f (x) | & (p = \infty). \end{array} \right. +$$ + +For $\beta \in \mathbb{R}_{++}^{d}$ , let $|\beta| := \sum_{j=1}^{d} |\beta_j|$ . The $r$ -th order finite difference in the direction $h \in \mathbb{R}^d$ is defined as: $\Delta_h^r(f)(x) := \Delta_h^{r-1}(f)(x+h) - \Delta_h^{r-1}(f)(x)$ and $\Delta_h^r(f)(x) := f(x)$ , for $x + rh \in \Omega$ ; otherwise, we define $\Delta_h^r(f)(x) = 0$ . + +Definition 1 (Anisotropic Besov Space). Let $0 < p, q \leq \infty$ , $\beta = (\beta_{1}, \ldots, \beta_{d})^{\top} \in \mathbb{R}_{++}^{d}$ , $r := \max_{i} \lfloor \beta_{i} \rfloor + 1$ . Then the Besov semi-norm is defined as + +$$ +| f | _ {B _ {p, q} ^ {\alpha}} := \left\{ \begin{array}{l l} \big (\sum_ {k = 0} ^ {\infty} [ 2 ^ {k} w _ {r, p} (f, (2 ^ {- k / \beta_ {1}}, \ldots , 2 ^ {- k / \beta_ {d}})) ] ^ {q} \big) ^ {1 / q} & (q < \infty), \\ \sup _ {k \geq 0} 2 ^ {k} w _ {r, p} (f, (2 ^ {- k / \beta_ {1}}, \ldots , 2 ^ {- k / \beta_ {d}})) & (q = \infty), \end{array} \right. +$$ + +where $w_{r,p}$ is the $r$ -th order modulus of smoothness defined by $w_{r,p}(f,t) \coloneqq \sup_{h \in \mathbb{R}^d:|h_i| \leq t_i} \| \Delta_h^r(f) \|_p$ . The anisotropic Besov space $B_{p,q}^\beta(\Omega)$ is defined as $B_{p,q}^\beta(\Omega) \coloneqq \{f \in L^p(\Omega) \mid \| f \|_{B_{p,q}^\beta} \coloneqq \| f \|_p + |f|_{B_{p,q}^\beta} < \infty\}$ . The unit ball of the anisotropic Besov space is denoted by $U(B_{p,q}^\beta)$ . + +The harmonic mean of the components of $\beta$ , which is given by $\widetilde{\beta} \coloneqq (\sum_{j=1}^{d} 1 / \beta_j)^{-1}$ , plays an important role in evaluating the approximation error by ReLU deep neural networks. It is known that the Hölder class and the Sobolev class with $p = 2$ are special cases of anisotropic Besov spaces Triebel (1983, 2011). In the following, we consider a situation where the eigenvectors of $Q$ can be well approximated by a function in a Besov space on the continuous space. + +# 4.1 Assumpitons + +We now summarize the additional assumptions required for the analysis. + +Assumption 5. Let $0 < \varepsilon < 1$ . Suppose the orthonormal eigenvectors $U = (u_{1},\ldots ,u_{M})$ of the graph Laplacian $L = -Q$ satisfy $u_{j}(x) = \mathcal{O}(1 / \sqrt{M})$ for all $x\in \mathbb{X}$ and the initial distribution $p_0$ can be expanded as: + +$$ +p _ {0} (x) = \sum_ {j = 1} ^ {M} c _ {j} u _ {j} (x) \quad (\forall x \in \mathbb {X}). +$$ + +Assume that for each $j = 1, \ldots, M$ , there exists a function $\sqrt{M} u_j^* : \mathbb{X} \to [0,1]$ satisfying $|u_j^*(x) - u_j(x)| \leq \varepsilon / \sqrt{M}$ and representable as $u_j^*(x) = h_j(Px)$ , where $h_j \in \gamma_j U(B_{p,q}^\beta)$ with $\gamma_j > 0$ and $P \in \mathbb{R}^{d \times D}$ are projection matrices for all $j$ . Moreover, assume $\|P\|_{\infty} = \mathcal{O}(1)$ and $\widetilde{\beta} > 1/p$ . + +Assumption 6. For each $j = 1, \dots, M$ , the expansion coefficient satisfies $|c_j| \lesssim |c_1| \cdot j^{-s}$ and the Besov norms of $h_j$ satisfy $|\gamma_j| \lesssim j^\gamma$ where $s > 0$ and $\gamma \geq 0$ . + +Assumption 5 is a technical condition that enables the application of function approximation theories in the anisotropic Besov space (Suzuki and Nitanda, 2021) (see also Suzuki (2019)). The factor $\mathcal{O}(1 / \sqrt{M})$ in $u_{j}$ stems from the normalization of orthonormal eigenvectors, i.e., $\sum_{x\in \mathbb{X}}u_j(x)^2 = 1$ . Assumption 6 imposes a polynomial decay condition on $c_{j}$ , which is a standard regularity assumption in nonparametric statistics, particularly in the analysis of kernel methods (Caponnetto and De Vito, 2007; Ying and Pontil, 2008; Dieuleveut and Bach, 2016). The condition on $\gamma_{j}$ reflects the increasing complexity of the basis functions $h_j$ . Similar assumptions are common in the analysis of eigenfunctions of the Laplacian operator in continuous settings. + +# 4.2 State space size independent error bound + +Theorem 3. Assume that Assumption 5 and 6 as well as Assumption 1 to 4 hold, and if the network size is set as $L = \mathcal{O}(\log^2 (M))$ , $W = \tilde{\mathcal{O}} (M)$ , $S = \tilde{O} (M)$ , $B = \tilde{O} (M^2)$ and $T = \mathcal{O}(\log (M\vee n))$ , the following estimation error bound holds with probability $1 - 2e^{-t}$ : + +$$ +\int_ {\delta} ^ {T} h ^ {2} (\widehat {s} _ {t}, s _ {t} ^ {\circ}) \mathrm {d} t = \left\{\begin{array}{l l}\mathcal {O} \left(\left(\lambda_ {2} ^ {- \frac {2 \widetilde {\beta}}{(1 + 2 \widetilde {\beta}) (s - 1) + \widetilde {\beta}}} n ^ {- \frac {2 \widetilde {\beta} (s - 1)}{(1 + 2 \widetilde {\beta}) (s - 1) + \widetilde {\beta}}} + \left(\frac {\varepsilon}{\lambda_ {2}}\right) ^ {2} + \frac {t}{n}\right) \log^ {8} (M \vee n)\right)&(s - \gamma \geq 1),\\\mathcal {O} \left(\left(M ^ {\frac {2 (1 - (s - \gamma) + \widetilde {\beta})}{1 + 2 \widetilde {\beta}}} n ^ {- \frac {2 \widetilde {\beta}}{1 + 2 \widetilde {\beta}}} + M ^ {2} \left(\frac {\varepsilon}{\lambda_ {2}}\right) ^ {2} + \frac {t}{n}\right) \log^ {8} (M \vee n)\right)&(s < 1),\\\mathcal {O} \left(\left(\lambda_ {2} ^ {- \frac {2 (1 - (s - \gamma) + \widetilde {\beta})}{(1 + 2 \widetilde {\beta}) (s - 1) + (1 - (s - \gamma) + \widetilde {\beta}}}\right) n ^ {- \frac {2 \widetilde {\beta} (s - 1)}{(1 + 2 \widetilde {\beta}) (s - 1) + (1 - (s - \gamma) + \widetilde {\beta})}} + \frac {t}{n} \right.\\+ \varepsilon^ {2} \left(\frac {n ^ {\widetilde {\beta} (1 - (s - \gamma))}}{\lambda_ {2} ^ {2 \gamma (1 + 2 \widetilde {\beta}) + 2 (1 - (s - \gamma) + \widetilde {\beta})}}\right) ^ {\frac {1}{(1 + 2 \widetilde {\beta}) (s - 1) + (1 - (s - \gamma) + \widetilde {\beta})}} \Bigg) _ {\log^ {8} (M \vee n)}\left. \right)&(o t h e r w i s e).\end{array}\right. +$$ + +The proof of Theorem 3 is given in Appendix D. The estimation error bound in Theorem 3 shows that for $s \geq 1$ , the dependence on $M$ is only polylogarithmic, successfully removing any polynomial dependence on $M$ . This aligns with the optimal rate conjectured in Ren et al. (2025) for discrete diffusion models. Even when $\gamma + 1/2 < s < 1$ , the exponent on $M$ remains below 1, yielding an improved convergence rate over Theorem 1. + +As for the dependence on the sample size $n$ , the bound does not explicitly depend on the embedded dimension $d$ , which is desirable. Moreover, when $s < 1$ , the rate recovers the optimal rate $n^{-2\widetilde{\beta} / (1 + 2\widetilde{\beta})}$ derived by Suzuki and Nitanda (2021), but can be dependent on $M$ polynomially. This is because estimation errors for $\mathcal{O}(M)$ -basis functions affect the final results when the decrease of coefficients is slow. On the other hand, when $s$ is large, we may "cut-off" redundant basis functions so that we mitigate the dependency on $M$ to poly-log order while the rate with respect to $n$ becomes a bit slower instead. + +Finally, combining Proposition 3 and theorem 3 yields an upper bound on the distribution estimation error. Here, we let the right hand side of the bound in Theorem 3 be denoted by $\Xi_{n,t}$ . + +Theorem 4. Suppose that the same condition as that of Theorem 3 holds. Then, the following error bound holds with probability $1 - 2e^{-t}$ : + +$$ +\mathrm {K L} \left(p _ {\delta} \| \widehat {q} _ {T - \delta}\right) \lesssim \Xi_ {n, t}. +$$ + +Example 1: Hypercube $\{0,1\}^D$ . As an example, we consider the hypercube setting $\mathbb{X} = \{0,1\}^D$ similar to Chen and Ying (2024). This setting is natural in practice, as the general discrete space $\mathbb{X} = [S]^D$ which is commonly assumed in many works such as Campbell et al. (2022); Lou et al. (2024); Zhang et al. (2025) can be encoded as a hypercube structure $\{0,1\}^{D\log |S|}$ . We let the eigenvalues of $L = -Q$ be ordered as $0 = \lambda_1 < \lambda_2 \leq \dots \leq \lambda_M$ . + +Assumption 7. Let the discrete state space be $\mathbb{X} = \{0,1\}^D$ . For any pair of distinct states $x\neq y$ assume the rate matrix $Q(x,y)$ satisfies: + +$$ +Q (x, y) = \left\{ \begin{array}{l l} 1 & (d (x, y) = 1), \\ 0 & (\text {o t h e r w i s e}), \end{array} \right. +$$ + +where $d(x,y)$ denotes the Hamming distance between $x$ and $y$ . + +Under Assumption 7, since the diagonal term satisfies $-Q(x,x) = D$ , we obtain $\overline{D} = \mathcal{O}(D)$ . Moreover, the following spectral property holds. + +Lemma 4. Under Assumption 7, for every $w \in \mathbb{X}$ , $h_w(x) \coloneqq \cos(\pi w^\top x) / \sqrt{M}$ is an eigenvector corresponding to the eigenvalue $2|w|$ , where $|w|$ is the number of ones in $w$ . In particular, $\lambda_2 = 2$ which is independent of the dimension $D$ . + +Based on this lemma, we can derive the convergence of the discrete diffusion model as follows. + +Corollary 1. Under the assumptions of Proposition 3 and Assumption 5 to 7, for arbitrary $\delta_0 > 0$ , the following error bound holds with probability $1 - 2e^{-t}$ : + +$$ +\mathrm {K L} (p _ {\delta} \| \widehat {q} _ {T - \delta}) \lesssim \left\{ \begin{array}{l l} (n ^ {- (2 (s - 1) - \delta_ {0}) / (2 s - 1)} + t / n) \log^ {8} (M \lor n) & (s \geq 1), \\ \frac {M + t}{n} \log^ {8} (M \lor n) & (s < 1). \end{array} \right. +$$ + +The proof can be found in Appendix E. We observe that when $s \geq 1$ , the convergence rate is essentially $\tilde{\mathcal{O}}(n^{-(2(s-1))/(2s-1)})$ , which is independent of the state space size $M$ . This rate could be achieved by showing that the eigenvectors in this setting can be represented by cosine functions on the continuous space, and that such trigonometric functions can be efficiently approximated by deep neural networks. On the other hand, when $s < 1$ , the convergence rate is $\tilde{\mathcal{O}}(M/n)$ , matching the error bound of the naive estimator in Theorem 1. In this case, accurately approximating the score function requires aggregating contributions across the entire state space, which leads to higher estimation complexity. + +Example 2: Discrete graph diffusion process. Finally, we consider a diffusion process defined on a graph. In this setting, each point of $\mathbb{X}$ is randomly generated on a $d$ -dimensional, smooth, closed and connected Riemannian manifold $\mathcal{M} \subset \mathbb{R}^D$ isometrically embedded in $\mathbb{R}^d$ through $\iota : \mathcal{M} \to \mathbb{R}^D$ , where each point $x$ obeys the uniform distribution $p$ on $\mathcal{M}$ independently. On this point cloud $\mathbb{X}$ , we can define the transition rate matrix as the ordinary graph Laplacian: First, let the affinity matrix be $W(x,y) \coloneqq \frac{k_{\sigma}(x,y)}{p_{\sigma}(x)p_{\sigma}(y)}$ where $k_{\sigma}(x,y) = \exp\left(-\frac{\|x - y\|^2}{2\sigma^2}\right)$ and $p_{\sigma}(x) = \sum_{y \in \mathbb{X}} k_{\sigma}(x,y)$ , second, using a diagonal matrix $D \in \mathbb{R}^{\mathbb{X} \times \mathbb{X}}$ defined as $D(x,x) = \sum_{y \in \mathbb{X}} W(x,y)$ , let the normalized weight matrix as $A = D^{-1}W$ , and finally we define the normalized Graph Laplacian as the transition matrix $Q = \frac{1}{\sigma^2} (A - I)$ . The stochastic process corresponding to $Q$ is known as a diffusion process on the graph with the normalized weight matrix $A$ . In this setting, the graph Laplacian $Q$ can be considered as a discrete approximation of the Laplace-Beltrami operator $-\Delta_{\mathcal{M}}$ defined on $\mathcal{M}$ , and thus the eigenfunctions and eigenvalues of the Laplace-Beltrami operator provide a good approximation of those of $Q$ (i.e., bounding $\varepsilon$ in Assumption 5) (Dunson et al., 2021). Since the eigenfunctions of $\Delta_{\mathcal{M}}$ are included in the Sobolev space $W_2^\beta(\mathcal{M})$ with arbitrary $\beta$ , we can apply our theorem to derive the following bound. The proof is given in Appendix F. + +Corollary 2. Suppose that the same assumptions as Theorem 3 and Assumption 5 and 6 hold, and $s > 1$ . Then, if $M$ is sufficiently large so that $M = \Omega(n^{\frac{\max\{1 + 10 / d, 5 / 2 + 4 / d\} (8d + 26)}{(d / \beta + 2)(s - 1) + 1}})$ and $\sigma = \left(\frac{\log(M)}{M}\right)^{\frac{1}{4d + 13}}$ , then there exists an event with high probability on the realization of $\mathbb{X}$ such that, for arbitrary $\beta$ such that $s - 2\beta / d > 1$ , the following error bound holds with probability $1 - 2e^{-t}$ : + +$$ +\operatorname {K L} \left(p _ {\delta} \| \widehat {q} _ {T - \delta}\right) \lesssim \left(n ^ {- \frac {2 (s - 1)}{2 (s - 1) / (2 \beta / d) + 2 s - 1}} + \left(\frac {\log (M)}{M}\right) ^ {\frac {1}{8 + 4 d}} + \frac {t}{n}\right) \log^ {8} (M \vee n). +$$ + +Therefore, a distribution on a point cloud $\mathbb{X}$ on a smooth manifold $\mathcal{M}$ can be well approximated by the discrete diffusion models that utilize the graph diffusion process induced by the graph Laplacian as the forward process. + +# 5 Numerical experiment + +To complement our theoretical analysis, we conducted a score-matching experiment that exactly instantiates Example 1 on the hypercube $\{0,1\}^D$ with $M := 2^D$ . For each $D = 6,8,10$ we constructed two distributions using the Hadamard basis $H \in \{\pm M^{-1/2}\}^{M \times M}$ : + +$$ +\ell (x) := (H c) _ {x}, p (x) = \frac {e ^ {\ell (x)}}{\sum_ {y = 0} ^ {M - 1} e ^ {\ell (y)}}. +$$ + +![](images/f6de348ef49855aa8a3d02bfe189c8ca9d15ddc2e6f0284684f97ca004287b34.jpg) +Figure 1: Score matching results on hypercube toy examples. Each bar shows the mean denoising score entropy (DSE) over 10 trials; error bars are $\pm 1$ standard deviation ( $\approx 0.4 - 0.7 \times 10^{-4}$ ). + +The coefficient vector $c\in \mathbb{R}^{M}$ was chosen in two regimes: + +Low-frequency: $c_{0},\ldots ,c_{4}\sim \mathcal{N}(0,1),c_{k} = 0$ for $k\geq 5$ + +High-frequency: $c_k \sim \mathcal{N}(0,1)$ for all $k \geq 2$ (DC and first harmonic set to zero). + +Activating only the first few modes yields a smooth distribution, whereas using the whole spectrum (minus the DC term) creates a highly oscillatory one. Each distribution was evolved for $t = 1$ under the rate matrix $Q$ shown in Assumption 7. We then train a one-hidden-layer score network $s_{\theta}(x,y)$ (input $2D$ , 256ReLU units, Softplus output) using the denoising score entropy (DSE) loss. Training used 5000 random Hamming-1 pairs, ADAM $(\mathrm{lr} = 10^{-3})$ , and 20 epochs. Performance was evaluated as the mean DSE over all Hamming-1 pairs. Each setting was repeated 10 times. + +The mild increase from $D = 6$ to $D = 10$ matches the logarithmic dependence on $M = 2^{D}$ predicted by Theorem 4 and corollary 1, while the higher errors for high-frequency mixtures reflect the bounds' sensitivity to the smoothness parameter. These observations provide concrete evidence that the theoretical guarantees translate directly to practice without any hyper-parameter tuning or architectural changes. + +# 6 Conclusion + +In this study, we established the first theoretical framework for estimating score functions in discrete diffusion models. We proved that directly approximating the score function for each discrete state using a neural network yields an estimation error rate of $\widetilde{\mathcal{O}}(M/n)$ under a naive analysis and further improved this bound to $\widetilde{\mathcal{O}}(n^{-2\tilde{\beta}(s-1)/(s-1+2s\tilde{\beta}-\tilde{\beta})})$ independent of the state space size $M$ by assuming a polynomial decay condition on the spectral decomposition of the target distribution. Our analysis made use of a decomposition of the target distribution by eigenvectors of the transition matrix $Q$ . Then, we utilized the fact that the eigenfunctions can be well approximated by smooth functions (i.e., Besov spaces) on an embedded continuous space in order to reduce the model complexity. We also demonstrated concrete bounds for the hypercube settings and the graph diffusion processes. + +One of the main drawbacks of our analysis is that we assumed a mild condition on the score function $s_t^\circ$ such that it is bounded above by $R$ and below by $1 / R$ . This condition inherits the same condition assumed in the continuous space (Oko et al., 2023). Relaxing this condition to the case where there is no uniform lower bound on the density as performed in Zhang et al. (2024) is an important direction for future work. + +# Acknowledgment + +SW was partially supported by JST CREST (JPMJCR2115). TS was partially supported by JSPS KAKENHI (24K02905) and JST CREST (JPMJCR2015). This research is supported by the National Research Foundation, Singapore, Infocomm Media Development Authority under its Trust Tech Funding Initiative, and the Ministry of Digital Development and Information under the AI Visiting Professorship Programme (award number AIVP-2024-004). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views + +of National Research Foundation, Singapore, Infocomm Media Development Authority, and the Ministry of Digital Development and Information. + +The authors are grateful to Satoshi Hayakawa for insightful comments, particularly for drawing our attention to the potential ill-definedness of the $\tau$ -leaping scheme in the SDE form. + +# References + +J. Austin, D. D. Johnson, J. Ho, D. Tarlow, and R. van den Berg. Structured denoising diffusion models in discrete state-spaces. 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URL https://openreview.net/forum?id=1-MBdJssZ-S. + +# A Construction of Neural Networks + +This section collects fundamental tools on function approximation using neural networks. These results play a central role in our approximation error analysis. Many of the lemmas below are based on Oko et al. (2023). + +# A.1 Composition and Combination Lemmas + +We begin with lemmas that describe how to combine multiple neural networks. These lemmas are essential to realize a large neural network that approximates complicated functions. + +Lemma 5 (Nakada and Imaizumi (2020)). For any neural networks $\phi^1, \ldots, \phi^k$ with $\phi^i: \mathbb{R}^{d_i} \to \mathbb{R}^{d_{i+1}}$ and $\phi^i \in \Phi(L^i, W^i, S^i, B^i)$ , there exists a network $\phi \in \Phi(L, W, S, B)$ such that $\phi(x) = \phi^k \circ \phi^{k-1} \circ \dots \circ \phi^1(x)$ for $x \in \mathbb{R}^{d_1}$ with + +$$ +L = \sum_ {i = 1} ^ {k} L ^ {i}, \quad \| W \| _ {\infty} \leq 2 \sum_ {i = 1} ^ {k} \| W ^ {i} \| _ {\infty}, \quad S \leq 2 \sum_ {i = 1} ^ {k} S ^ {i}, \quad B \leq \max _ {1 \leq i \leq k} B ^ {i}. +$$ + +Lemma 6 (Oko et al. (2023)). For any networks $\phi^1, \ldots, \phi^k$ with $\phi^i: \mathbb{R}^{d_i} \to \mathbb{R}^{d_i'}$ and $\phi^i \in \Phi(L^i, W^i, S^i, B^i)$ , there exists a network $\phi \in \Phi(L, W, S, B)$ such that $\phi(x) = [\phi^1(x_1)^\top, \ldots, \phi^k(x_k)^\top]^\top: \mathbb{R}^{d_1 + \dots + d_k} \to \mathbb{R}^{d_1' + \dots + d_k'}$ for $x = (x_1^\top \cdots x_k^\top)^\top$ with + +$$ +L = \max _ {1 \leq i \leq k} L ^ {i}, \| W \| _ {\infty} \leq 2 \sum_ {i = 1} ^ {k} \| W ^ {i} \| _ {\infty}, +$$ + +$$ +S \leq 2 \sum_ {i = 1} ^ {k} \left(S ^ {i} + L W _ {L} ^ {i}\right), \quad B \leq \max \left\{\max _ {1 \leq i \leq k} B ^ {i}, 1 \right\}. +$$ + +Lemma 7 (Oko et al. (2023)). For any networks $\phi^1, \ldots, \phi^k$ with $\phi^i: \mathbb{R}^{d_i} \to \mathbb{R}^d$ and $\phi^i \in \Phi(L^i, W^i, S^i, B^i)$ , there exists a network $\phi \in \Phi(L, W, S, B)$ such that $\phi(x) = \sum_{i=1}^{k} \phi^i(x_i): \mathbb{R}^{d_1 + \cdots + d_k} \to \mathbb{R}^d$ for $x = (x_1^\top \cdots x_k^\top)^\top$ with + +$$ +L = \max _ {1 \leq i \leq k} L ^ {i} + 1, \| W \| _ {\infty} \leq 4 \sum_ {i = 1} ^ {k} \| W ^ {i} \| _ {\infty}, +$$ + +$$ +S \leq 4 \sum_ {i = 1} ^ {k} \left(S ^ {i} + L W _ {L} ^ {i}\right) + 2 W _ {L}, \quad B \leq \max \left\{\max _ {1 \leq i \leq k} B ^ {i}, 1 \right\}. +$$ + +Lemma 8 (Oko et al. (2023)). For $\underline{t}_1 < \underline{t}_2 < \bar{t}_1 < \bar{t}_2$ and $f(x,t): \mathbb{R}^d \times \mathbb{R} \to \mathbb{R}$ , assume that $\phi^i(x,t)$ approximates $f(x,t)$ up to $\varepsilon > 0$ within $[\underline{t}_i, \bar{t}_i]$ ( $i = 1,2$ ). There exist networks $\phi_{\mathrm{swit}}^1(t; \underline{t}_2, \bar{t}_1), \phi_{\mathrm{swit}}^2(t; \underline{t}_2, \bar{t}_1) \in \Phi(3, (1,2,1,1)^\top, 8, \max\{\bar{t}_1, (\bar{t}_1 - \underline{t}_2)^{-1}\})$ with + +$$ +| \phi_ {\mathrm {s w i t}} ^ {1} (t; \underline {{t}} _ {2}, \bar {t} _ {1}) \phi^ {1} (x, t) + \phi_ {\mathrm {s w i t}} ^ {2} (t; \underline {{t}} _ {2}, \bar {t} _ {1}) \phi^ {2} (x, t) - f (x, t) | \leq \varepsilon (t \in [ \underline {{t}} _ {1}, \bar {t} _ {2} ]). +$$ + +# A.2 Approximations of rational functions + +The following lemmas describe how to approximate elementary operations such as multiplication and reciprocal using neural networks. These are particularly important because the score function in discrete diffusion models involves the ratio of probabilities between two discrete states, which can be approximated by neural networks designed to emulate rational functions. + +Lemma 9 (Oko et al. (2023)). For $k \geq 2, C \geq 1, 0 < \varepsilon_{\mathrm{error}} \leq 1$ and $\varepsilon > 0$ , there exists a network $\phi_{\mathrm{mult}}(x_1, \ldots, x_k) \in \Phi(L, W, S, B)$ with $L = \mathcal{O}(\log k (\log \varepsilon^{-1} + k \log C))$ , $\| W \| = 48k$ , $S = \mathcal{O}(k \log \varepsilon^{-1} + k \log C)$ , $B = C^k$ such that $|\phi_{\mathrm{mult}}(x)| \leq C^k$ , $\phi_{\mathrm{mult}}(x_1, \ldots, x_k) = 0$ if at least one of $x_i$ is equal to zero, and + +$$ +\left| \phi_ {\mathrm {m u l t}} (x _ {1} ^ {\prime}, x _ {2} ^ {\prime}, \ldots , x _ {k} ^ {\prime}) - \prod_ {i = 1} ^ {k} x _ {i} \right| \leq \varepsilon + k C ^ {k - 1} \varepsilon_ {\mathrm {e r r o r}}, \quad \forall x \in [ - C, C ] ^ {k}, \| x - x ^ {\prime} \| _ {\infty} \leq \varepsilon_ {\mathrm {e r r o r}}. +$$ + +Lemma 10 (Oko et al. (2023)). For a constant $R > 1$ and any $0 < \varepsilon < R^{-1}$ , there exists a network $\phi_{\mathrm{rec}}(x_1, \ldots, x_k) \in \Phi(L, W, S, B)$ with $L = \mathcal{O}(\log^2 \varepsilon^{-1})$ , $\|W\| = \mathcal{O}(\log^3 \varepsilon^{-1})$ , $S = \mathcal{O}(\log^4 \varepsilon^{-1})$ , $B = \mathcal{O}(\varepsilon^{-2})$ such that + +$$ +\left| \phi_ {\operatorname {r e c}} (x ^ {\prime}) - \frac {1}{x} \right| \leq \varepsilon + R ^ {2} | x ^ {\prime} - x |, \quad \forall x \in [ R ^ {- 1}, R ], x ^ {\prime} \in \mathbb {R}. +$$ + +# B Poisson Random Measures with evolving intensity + +In this section, we introduce the formal definition of Poisson random measures with evolving intensity, which is a foundational concept required for implementing the $\tau$ -leaping algorithm. Although this concept is not central to the main theoretical results of this paper, we include the definition here for completeness and reference. + +Definition 2 (Protter (1983); Ren et al. (2025)). Consider a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a measure space $(X, \mathcal{B}, \nu)$ . A non-negative predictable process $\lambda_t(y)$ on $\mathbb{R}_+\times X\times \Omega$ is assumed to satisfy for any $T > 0$ : + +$$ +\int_ {0} ^ {T} \int_ {\mathbb {X}} \left(1 \vee | y | \vee | y | ^ {2}\right) \lambda_ {t} (y) \nu (d y) d t < \infty \quad a. s. +$$ + +where $\nu$ is the counting measure. probability measure $N[\lambda ](\mathrm{d}t,\mathrm{d}y)$ on $\mathbb{R}_+ \times \mathbb{X}$ is called a Poisson random measure with evolving intensity $\lambda_t(y)$ if: + +1. For any $B \in \mathcal{B}$ and $0 \leq s < t$ , $N[\lambda]((s, t] \times B) \sim \mathcal{P}\left(\int_s^t \int_B \lambda_\tau(y) \nu(dy)d\tau\right)$ . where $\mathcal{P}(\cdot)$ represents a Poisson distribution with the given expectation. +2. For any $t \geq 0$ , $\mathcal{B}$ and disjoint sets $\{B_i\}_{i=1}^n$ , the processes $\{N_t[\lambda](B_i) := N[\lambda]((0, t] \times B_i)\}_{i=1}^n$ are independent. + +# C Proof of Theorem 1 + +To prove Theorem 1, we decompose the total estimation error into approximation and generalization components. We begin by analyzing the score approximation error with a neural network. This part captures the model bias, and the resulting bound depends on the smoothness of the score function and the expressive power of the network class. The subsequent subsection will then address the generalization error using tools from statistical learning theory. Together, these analyses yield the desired estimation error bound. + +# C.1 Approximation error analysis + +In this subsection, we construct a neural network that approximates the true score function $s_t^\circ (x,y)$ in a compositional manner by separately approximating $p_t(x)$ and its reciprocal $1 / p_t(x)$ , and then combining them in a multiplicative way. + +Lemma 11. For any $\varepsilon_1 > 0$ , there exists a neural network $\phi^1 (x,t)\in \Phi (L,W,S,B)$ such that + +$$ +\left| \phi^ {1} (x, t) - M p _ {t} (x) \right| \leq \varepsilon_ {1} \quad \forall t > 0. +$$ + +The parameters of $\phi^1 (x,t)$ are bounded as follows: + +$$ +\begin{array}{l} L = \mathcal {O} (\log^ {2} M \varepsilon_ {1} ^ {- 1}), \\ \| W \| _ {\infty} = \mathcal {O} (M \log^ {3} M \varepsilon_ {1} ^ {- 1}), \\ S = \mathcal {O} (M \log^ {5} M \varepsilon_ {1} ^ {- 1}), \\ B = \mathcal {O} \left(M ^ {2} \vee \log M \varepsilon_ {1} ^ {- 1}\right). \\ \end{array} +$$ + +Proof. We begin by approximating the function $e^{-\lambda_j t}$ using a neural network. Define $A \coloneqq \log 3\varepsilon^{-1}$ . + +For $j = 0$ , we set $\phi_j = 1$ so that $e^{-\lambda_j t}$ is approximated without an error. For $j > 0$ , using the Taylor expansion, we obtain + +$$ +e ^ {- \lambda_ {j} t} = e ^ {- \lambda_ {j} s} \left(\sum_ {i = 0} ^ {k} \frac {(- \lambda_ {j}) ^ {i}}{i !} (t - s) ^ {i} + \frac {(- \lambda_ {j}) ^ {k + 1}}{(k + 1) !} (\theta (t - s)) ^ {k + 1}\right), +$$ + +where $\theta \in [0,1]$ . Setting $k = \max \{\lceil 2e^{2}\overline{D}\rceil ,\lceil \log 3\varepsilon^{-1}\rceil \}$ , we obtain the bound + +$$ +\left| e ^ {- \lambda_ {j} s} \frac {\left(- \lambda_ {j}\right) ^ {k + 1}}{(k + 1) !} (\theta (t - s)) ^ {k + 1} \right| \leq \left| \frac {\left(- \lambda_ {j}\right) ^ {k + 1}}{(k + 1) !} \right| < \left| \frac {1}{((k + 1) / 2 e \overline {{D}}) ^ {k + 1}} \right| \leq | e ^ {- k + 1} | \leq \varepsilon . +$$ + +Here, we used $(n / e)^n < n!$ in the second inequality. By Lemma 9, for each $\frac{(-\lambda_j)^i}{i!} (t - s)^i$ , there exists a neural network $\phi(t; i)$ that approximates it within an error of $\varepsilon / 3(k + 1)$ over the interval $s \leq t \leq s + 2 / \lambda_j$ . The network parameters satisfy: + +$$ +L = \mathcal {O} (\log^ {2} \varepsilon^ {- 1}), +$$ + +$$ +\| W \| _ {\infty} = \mathcal {O} (\log \varepsilon^ {- 1}), +$$ + +$$ +S = \mathcal {O} (\log^ {2} \varepsilon^ {- 1}), +$$ + +$$ +B = \mathcal {O} (1). +$$ + +Similarly, by lemma 7, there exists a neural network $\phi_s(t)$ that approximates $e^{-\lambda_j t}$ within an error of $\varepsilon / 3$ over $s \leq t \leq s + 2 / \lambda_j$ , with the following parameter bounds: + +$$ +L = \mathcal {O} (\log^ {2} \varepsilon^ {- 1}), +$$ + +$$ +\| W \| _ {\infty} = \mathcal {O} \left(\log^ {2} \varepsilon^ {- 1}\right), +$$ + +$$ +S = \mathcal {O} (\log^ {4} \varepsilon^ {- 1}), +$$ + +$$ +B = \mathcal {O} (1). +$$ + +Note that $|c_{j}u_{j}(x)| \leq 1$ . Indeed, + +$$ +c _ {j} = \sum_ {x \in \mathbb {X}} p _ {0} (x) u _ {j} (x) \leq \sqrt {\left(\sum_ {x \in \mathbb {X}} p _ {0} (x) ^ {2}\right) \left(\sum_ {x \in \mathbb {X}} u _ {j} (x) ^ {2}\right)} = \sqrt {\sum_ {x \in \mathbb {X}} p _ {0} (x) ^ {2}} \leq \sum_ {x \in \mathbb {X}} p _ {0} (x) = 1. +$$ + +Thus, we can construct a neural network $\phi_j^*(x,t)$ that approximates $c_j u_j(x)e^{-t\lambda_j}$ as follows: + +$$ +\begin{array}{l} \phi_ {j} ^ {*} (x, t) := \phi_ {\mathrm {m u l t}} \left(\phi_ {\mathrm {s w i t}} ^ {2} (t; 1 / \lambda_ {j}, 2 / \lambda_ {j}), c _ {j} u _ {j} (x) \phi_ {0} (t)\right) + \\ \sum_ {s = 1} ^ {\lceil A \rceil - 1} \phi_ {\mathrm {m u l t}} (\phi_ {\mathrm {s w i t}} ^ {1} (t; (s + 1) / \lambda_ {j}, (s + 2) / \lambda_ {j}), \phi_ {\mathrm {s w i t}} ^ {2} (t; s / \lambda_ {j}, (s + 1) / \lambda_ {j}), c _ {j} u _ {j} (x) \phi_ {s / \lambda_ {j}} (t)). \\ \end{array} +$$ + +Using Lemma 9, the approximation error from each $\phi_{\mathrm{mult}}$ is bounded by $\varepsilon / \log \varepsilon^{-1}$ . We set the parameters of $\phi_{\mathrm{mult}}$ to $L = \mathcal{O}(\log \varepsilon^{-1})$ , $\|W\|_{\infty} = \mathcal{O}(1)$ , $S = \mathcal{O}(\log \varepsilon^{-1})$ , $B = \mathcal{O}(1)$ . To restrict the input $t$ of $\phi_j^*(x,t)$ to the range $[0,A]$ , we define $\phi_j(x,t) := \mathrm{ReLU}(\phi_j^*(x,t) - \phi_j^*(x,A)) + \phi_j^*(x,A)$ . This ensures that $\phi_j(x,t)$ approximates $e^{-\lambda_j t}$ with an error at most $\varepsilon$ for all $t \geq 0$ . For $t \leq A$ , the following inequality holds: $|\phi_j(x,t) - c_j u_j(x)e^{-\lambda_j t}| \leq \varepsilon/3 + \varepsilon/3 < \varepsilon$ . For $t > A$ , we have $|\phi_j(x,t) - c_j u_j(x)e^{-\lambda_j t}| \leq |\phi_j(x,t) - \phi_j(x,A)| + |\phi_j(x,A) - c_j u_j(x)e^{-\lambda_j A}| + |c_j u_j(x)(e^{-\lambda_j t} - e^{-\lambda_j A})| \leq 0 + 2\varepsilon/3 + \varepsilon/3 = \varepsilon$ . Thus, the parameters of $\phi(x,t)$ are evaluated as follows: + +$$ +\begin{array}{l} L = \mathcal {O} (\log^ {2} \varepsilon^ {- 1}), \\ \| W \| _ {\infty} = \mathcal {O} \left(\log^ {3} \varepsilon^ {- 1}\right), \\ S = \mathcal {O} (\log^ {5} \varepsilon^ {- 1}), \\ B = \mathcal {O} (\log \varepsilon^ {- 1}). \\ \end{array} +$$ + +Finally, setting $\varepsilon \coloneqq \varepsilon_1 / M^2$ and summing $\phi_j(x,t)$ over all $j = 1,\ldots ,M$ , we construct a neural network $\phi^1 (x,t)$ that approximates $Mp_{t}(x)$ within an error of $\varepsilon_{1}$ . By Lemmas 7 and 9, the parameter bounds are given by + +$$ +L = \mathcal {O} (\log^ {2} M \varepsilon_ {1} ^ {- 1}), +$$ + +$$ +\| W \| _ {\infty} = \mathcal {O} \left(M \log^ {3} M \varepsilon_ {1} ^ {- 1}\right), +$$ + +$$ +S = \mathcal {O} (M \log^ {5} M \varepsilon_ {1} ^ {- 1}), +$$ + +$$ +B = \mathcal {O} \left(M ^ {2} \vee \log M \varepsilon_ {1} ^ {- 1}\right). +$$ + +Building on this result, we next construct a neural network that approximates the reciprocal $1 / p_{t}(x)$ . + +Lemma 12. For any $0 < \varepsilon_0 \leq \frac{1}{R}$ and $\varepsilon_1 > 0$ there exists a neural network $\phi^2(x, t) \in \Phi(L, W, S, B)$ satisfying the following inequality: + +$$ +\left| \phi^ {2} (x, t) - \frac {1}{M p _ {t} (x)} \right| \leq \varepsilon_ {0} + R ^ {2} \varepsilon_ {1} \quad \forall t > 0. +$$ + +with the parameters of $\phi^2$ are bounded as follows: + +$$ +L = \mathcal {O} \left(\log^ {2} \varepsilon_ {0} ^ {- 1} \vee \log^ {2} M \varepsilon_ {1} ^ {- 1}\right), +$$ + +$$ +\| W \| _ {\infty} = \mathcal {O} \left(\log^ {3} \varepsilon_ {0} ^ {- 1} \vee M \log^ {3} M \varepsilon_ {1} ^ {- 1}\right), +$$ + +$$ +S = \mathcal {O} \left(\log^ {4} \varepsilon_ {0} ^ {- 1} \vee M \log^ {5} M \varepsilon_ {1} ^ {- 1}\right), +$$ + +$$ +B = \mathcal {O} \left(M ^ {2} \vee \varepsilon_ {0} ^ {- 2} \vee \log M \varepsilon_ {1} ^ {- 1}\right). +$$ + +Proof. From Lemmas 5 and 10, there exists $\phi^2 (x,t) = \phi_{\mathrm{rec}}\circ \phi^1 (x,t)$ satisfying + +$$ +\left| \phi^ {2} (x, t) - \frac {1}{M p _ {t} (x)} \right| \leq \varepsilon_ {0} + R ^ {2} \varepsilon_ {1}. +$$ + +Here, the network parameters coincide with those stated above. Regarding $p_t$ , the boundedness of the score function ensures that $M p_t(x) = \frac{M p_t(x)}{\sum_{y \in \mathbb{X}} p_t(y)} = \frac{M}{\sum_{y \in \mathbb{X}} s_t(y, x)} \geq \frac{1}{R}$ which satisfies the conditions of Lemma 10 + +Combining the approximations of $p_t(x)$ and its reciprocal, we are now ready to construct a neural network that directly approximates the score function $s_t^\circ (x,y) = p_t(y) / p_t(x)$ . The following lemma formalizes this construction and provides the approximation error bound. + +Lemma 13. For any $0 < \varepsilon_0 \leq \frac{1}{R}$ and $\varepsilon_1, \varepsilon_2 > 0$ , there exists a network $\phi_{\mathrm{score}}(x,y,t) \in \mathcal{F}$ such that + +$$ +\left| \phi_ {\text {s c o r e}} (x, y, t) - s _ {t} ^ {\circ} (x, y) \right| \lesssim \varepsilon_ {0} + R ^ {2} \varepsilon_ {1} + \varepsilon_ {2} \quad \forall t > 0. +$$ + +Here, the parameters of $\phi_{\mathrm{score}}$ are evaluated as: + +$$ +L = \mathcal {O} \left(\log^ {2} \varepsilon_ {0} ^ {- 1} \vee \log^ {2} M \varepsilon_ {1} ^ {- 1} \vee \log \varepsilon_ {2} ^ {- 1}\right), +$$ + +$$ +\| W \| _ {\infty} = \mathcal {O} \left(\log^ {3} \varepsilon_ {0} ^ {- 1} \vee M \log^ {3} M \varepsilon_ {1} ^ {- 1}\right), +$$ + +$$ +S = \mathcal {O} \left(\log^ {4} \varepsilon_ {0} ^ {- 1} \vee M \log^ {5} M \varepsilon_ {1} ^ {- 1} \vee \log \varepsilon_ {2} ^ {- 1}\right), +$$ + +$$ +B = \mathcal {O} \left(\varepsilon_ {0} ^ {- 2} \vee \log M \varepsilon_ {1} ^ {- 1}\right). +$$ + +In particular, the following holds: + +$$ +\int_ {\delta} ^ {T} \mathbb {E} _ {x \sim p _ {t}} \left[ \sum_ {y \neq x} \mathrm {B R} _ {K} \left(\phi_ {\text {s c o r e}} (x, y, t) \| s _ {t} ^ {\circ} (x, y)\right) \cdot s _ {t} ^ {\circ} (x, y) Q (x, y) \right] \mathrm {d} t \lesssim R \log (R) T \overline {{D}} \left(\varepsilon_ {0} + R ^ {2} \varepsilon_ {1} + \varepsilon_ {2}\right) ^ {2}. +$$ + +Proof. Consider the neural network $\phi^3 (x,y,t)\coloneqq [\phi^1 (y,t),\phi^2 (x,t)]^\top$ , which parallelizes the estimation of $p_t(y)$ and $1 / p_{t}(x)$ . The function $\phi_{\mathrm{score}}(x,y,t)\coloneqq \phi_{\mathrm{mult}}\circ \phi^3 (x,y,t)$ is a neural network that estimates $s_t^\circ (x,y) = \frac{p_t(y)}{p_t(x)}$ , and its error is evaluated as follows: + +$$ +\begin{array}{l} \left| \phi_ {\text {s c o r e}} (x, y, t) - s _ {t} ^ {\circ} (x, y) \right| \leq \left| \phi_ {\text {m u l t}} \left(\phi^ {1} (y, t), \phi^ {2} (x, t)\right) - \phi^ {1} (y, t) \phi^ {2} (x, t) \right| + \left| \phi^ {1} (y, t) \phi^ {2} (x, t) - s _ {t} ^ {\circ} (x, y) \right| \\ \leq \varepsilon_ {2} + \left| \phi^ {1} (y, t) \phi^ {2} (x, t) - \frac {\phi^ {1} (y , t)}{M p _ {t} (x)} + \frac {\phi^ {1} (y , t)}{M p _ {t} (x)} - s _ {t} ^ {\circ} (x, y) \right| \\ \leq \varepsilon_ {2} + | \phi^ {1} (y, t) | \left| \phi^ {2} (x, t) - \frac {1}{M p _ {t} (x)} \right| + \left| \frac {1}{M p _ {t} (x)} \right| | \phi^ {1} (y, t) - p _ {t} (y) | \\ \leq \varepsilon_ {0} + (R ^ {2} + R) \varepsilon_ {1} + \varepsilon_ {2}. \\ \end{array} +$$ + +From Lemmas 5, 9, 11 and 12, the parameters are bounded as showed in above: + +$$ +\begin{array}{l} L = \mathcal {O} \left(\log^ {2} \varepsilon_ {0} ^ {- 1} \vee \log^ {2} M \varepsilon_ {1} ^ {- 1} \vee \log \varepsilon_ {2} ^ {- 1}\right), \\ \| W \| _ {\infty} = \mathcal {O} \left(\log^ {3} \varepsilon_ {0} ^ {- 1} \vee M \log^ {3} M \varepsilon_ {1} ^ {- 1}\right), \\ S = \mathcal {O} \left(\log^ {4} \varepsilon_ {0} ^ {- 1} \vee M \log^ {5} M \varepsilon_ {1} ^ {- 1} \vee \log \varepsilon_ {2} ^ {- 1}\right), \\ B = \mathcal {O} \left(\varepsilon_ {0} ^ {- 2} \vee \log M \varepsilon_ {1} ^ {- 1}\right). \\ \end{array} +$$ + +Considering $\bar{\phi}_{\mathrm{score}}(x,y,t)$ which restricts the output of $\phi_{\mathrm{score}}(x,y,t)$ to $[1 / R,R]$ , Assumption 3 ensures that this restriction does not increase the error, and the order of the parameters remains unchanged. Thus, we can conclude $\phi_{\mathrm{score}}\in \mathcal{F}$ . + +Since $1 / R\leq \phi_{\mathrm{score}},s^{\circ}\leq R$ and $0\leq x - \log (x) - 1\leq 4\log (R)(x - 1)^2$ for $x\in [-1 / R^2,R^2 ]$ , we have that + +$$ +\begin{array}{l} \mathrm {B R} _ {K} \left(\phi_ {\text {s c o r e}} (x, y, t) \| s _ {t} ^ {\circ} (x, y)\right) = \left(\frac {\phi_ {\text {s c o r e}} (x , y , t)}{s _ {t} ^ {\circ} (x , y)} - \log \frac {\phi_ {\text {s c o r e}} (x , y , t)}{s _ {t} ^ {\circ} (x , y)} - 1\right) \\ \leq 4 \log (R) \left(\phi_ {\text {s c o r e}} (x, y, t) / s _ {t} ^ {\circ} (x, y) - 1\right) ^ {2}. \\ \end{array} +$$ + +Therefore, we obtain that + +$$ +\begin{array}{l} \int_ {\delta} ^ {T} \mathbb {E} _ {x \sim p _ {t}} \left[ \sum_ {y \neq x} \operatorname {B R} _ {K} \left(\phi_ {\text {s c o r e}} (x, y, t) \| s _ {t} ^ {\circ} (x, y)\right) \cdot s _ {t} ^ {\circ} (x, y) Q (x, y) \right] \mathrm {d} t \\ \leq \int_ {\delta} ^ {T} \mathbb {E} _ {x \sim p _ {t}} \left[ \sum_ {y \neq x} \left(\phi_ {\text {s c o r e}} (x, y, t) / s _ {t} ^ {\circ} (x, y) - 1\right) ^ {2} s _ {t} ^ {\circ} (x, y) Q (x, y) \right] \mathrm {d} t \\ \leq \int_ {\delta} ^ {T} \mathbb {E} _ {x \sim p _ {t}} \left[ \sum_ {y \neq x} 4 \log (R) \left(\phi_ {\text {s c o r e}} (x, y, t) - s _ {t} ^ {\circ} (x, y)\right) ^ {2} s _ {t} ^ {\circ} (x, y) ^ {- 1} Q (x, y) \right] d t \\ \leq 4 R \log (R) \left(\varepsilon_ {0} + \left(R ^ {2} + M R\right) \varepsilon_ {1} + \varepsilon_ {2}\right) ^ {2} \int_ {\delta} ^ {T} \bar {D} d t \\ \lesssim R \log (R) T \bar {D} \left(\varepsilon_ {0} ^ {2} + R ^ {4} \varepsilon_ {1} ^ {2} + \varepsilon_ {2} ^ {2}\right), \\ \end{array} +$$ + +which achieves the assertion. + +![](images/945594ea6dd0ae18e106c9324506e4ec9029122e43116053b1a292cdea871662.jpg) + +# C.2 Generalization Error Analysis + +This subsection focuses on bounding the generalization error. The analysis proceeds in three steps: + +- First, we relate the score estimation error to a conditional form via a denoising representation. +- Second, we define a loss class and control its generalization error using local Rademacher complexity and Peeling device. +- Third, we derive explicit complexity bounds using covering number estimates for neural networks. + +We begin by expressing the explicit score matching entropy as the denoising score entropy. This representation justifies interpreting the score entropy as the empirical loss. + +Lemma 14 (Lou et al. (2024)). For any $s_t(x_t, y)$ and $t > 0$ , the following holds: + +$$ +\begin{array}{l} \sum_ {x _ {t}} \sum_ {y \neq x _ {t}} \mathrm {B R} _ {K} (s _ {t} (x _ {t}, y) \| s _ {t} ^ {\circ} (x _ {t}, y)) \cdot s _ {t} ^ {\circ} (x _ {t}, y) p _ {t} (x _ {t}) Q (x _ {t}, y) \\ = \sum_ {x _ {0}} \sum_ {x _ {t}} \sum_ {y \neq x _ {t}} \operatorname {B R} _ {K} (s _ {t} (x _ {t}, y) \| s _ {t} ^ {\circ} (x _ {t}, y \mid x _ {0})) \cdot s _ {t} ^ {\circ} (x _ {t}, y \mid x _ {0}) p _ {t} (x _ {t} \mid x _ {0}) p _ {0} (x _ {0}) Q (x _ {t}, y) + C, \\ \end{array} +$$ + +where $C$ is a constant independent of $s_t$ . + +Proof. + +$$ +\begin{array}{l} \sum_ {x _ {t}} \sum_ {y \neq x _ {t}} \operatorname {B R} _ {K} \left(s _ {t} \left(x _ {t}, y\right) \| s _ {t} ^ {\circ} \left(x _ {t}, y\right)\right) \cdot s _ {t} ^ {\circ} \left(x _ {t}, y\right) p _ {t} \left(x _ {t}\right) Q \left(x _ {t}, y\right) \\ = - \sum_ {x _ {t}, y \neq x _ {t}} \left(\log \frac {s _ {t} (x _ {t} , y)}{s _ {t} ^ {\circ} (x _ {t} , y)} - \frac {s _ {t} (x _ {t} , y)}{s _ {t} ^ {\circ} (x _ {t} , y)} + 1\right) \cdot p _ {t} (y) Q (x _ {t}, y) \\ = - \sum_ {x _ {0}, x _ {t}, y \neq x _ {t}} \left(\log \frac {s _ {t} (x _ {t} , y)}{s _ {t} ^ {\circ} (x _ {t} , y \mid x _ {0})} - \frac {s _ {t} (x _ {t} , y)}{s _ {t} ^ {\circ} (x _ {t} , y \mid x _ {0})} + 1\right) \cdot p _ {t} (y \mid x _ {0}) p _ {0} (x _ {0}) Q (x _ {t}, y) \\ + \underbrace {\left(\sum_ {x _ {t} , y \neq x _ {t}} \left(\log s _ {t} ^ {\circ} \left(x _ {t} , y\right)\right) \cdot p _ {t} (y) Q \left(x _ {t} , y\right) - \sum_ {x _ {0} , x _ {t} , y \neq x _ {t}} \log s _ {t} ^ {\circ} \left(x _ {t} , y \mid x _ {0}\right)\right)} _ {=: C} \\ = \sum_ {x _ {0}, x _ {t}, y \neq x _ {t}} \operatorname {B R} _ {K} \left(s _ {t} \left(x _ {t}, y\right) \| s _ {t} ^ {\circ} \left(x _ {t}, y \mid x _ {0}\right)\right) \cdot s _ {t} ^ {\circ} \left(x _ {t}, y \mid x _ {0}\right) \cdot p _ {t} \left(x _ {t} \mid x _ {0}\right) p _ {0} \left(x _ {0}\right) Q \left(x _ {t}, y\right) + C, \\ \end{array} +$$ + +where $C$ is a constant depending only on $p_t, Q$ . + +![](images/0faa1c0906e7ad1ac5bfd4fc80f594811ed457ca1b0224ab0b534dc936806d6f.jpg) + +Next, We relate the Hellinger distance to the expected excess risk. + +Lemma 15. For any $g \in G \coloneqq \{g \coloneqq \ell_{\widehat{s}} - \ell_{s^{\circ}} \mid \widehat{s} \in \mathcal{F}\}$ , we have that $\int_0^{T - \delta} h^2(\widehat{s}_t, s_t^\circ) \, \mathrm{d}t \lesssim Pg$ . Proof. From Lemma 14, we have: + +$$ +\begin{array}{l} P g = L (\widehat {s} _ {t}) - L (s _ {t}) \\ = \int_ {\delta} ^ {T} \mathbb {E} _ {x _ {t} \sim p _ {t}} \left[ \sum_ {y \neq x _ {t}} \operatorname {B R} _ {K} \left(\widehat {s} _ {t} (x _ {t}, y) \| s _ {t} ^ {\circ} (x _ {t}, y)\right) \cdot s _ {t} ^ {\circ} (x _ {t}, y) Q (x _ {t}, y) \right] \mathrm {d} t \\ = - \int_ {\delta} ^ {T} \mathbb {E} _ {x _ {t} \sim p _ {t}} \left[ \sum_ {y \neq x _ {t}} \left(\log \frac {\widehat {s} _ {t} (x _ {t} , y)}{s _ {t} ^ {\circ} (x _ {t} , y)} - \frac {\widehat {s} _ {t} (x _ {t} , y)}{s _ {t} ^ {\circ} (x _ {t} , y)} + 1\right) \cdot s _ {t} ^ {\circ} (x _ {t}, y) Q (x _ {t}, y) \right] d t \\ \geq - \int_ {\delta} ^ {T} \mathbb {E} _ {x _ {t} \sim p _ {t}} \left[ \sum_ {y \neq x _ {t}} \left(2 \left(\sqrt {\frac {\widehat {s} _ {t} (x _ {t} , y)}{s _ {t} ^ {\circ} (x _ {t} , y)}} - 1\right) - \frac {\widehat {s} _ {t} (x _ {t} , y)}{s _ {t} ^ {\circ} (x _ {t} , y)} + 1\right) \cdot s _ {t} ^ {\circ} (x _ {t}, y) Q (x _ {t}, y) \right] d t \\ = \int_ {\delta} ^ {T} \mathbb {E} _ {x \sim p _ {t}} \left[ \sum_ {y \neq x _ {t}} (- 2 \sqrt {\widehat {s} _ {t} (x _ {t} , y) s _ {t} ^ {\circ} (x _ {t} , y)} + s _ {t} ^ {\circ} (x _ {t}, y) + \widehat {s} _ {t} (x _ {t}, y)) Q (x _ {t}, y) \right] d t \\ = \int_ {\delta} ^ {T} h \left(\widehat {s} _ {t}, s _ {t} ^ {\circ}\right) ^ {2} \mathrm {d} t, \\ \end{array} +$$ + +which concludes the assertion. + +![](images/69f119fb183a62843e9ab5754d6f93f684479377378a14cc3507cdce94e53f29.jpg) + +Here, we aim to bound the generalization error using the local Rademacher complexity $R_{n}(G_{r})$ . We define $G_{r} \coloneqq \{g \coloneqq \ell_{s} - \ell_{s^{\circ}} \mid g \in G, Pg \leq r\}$ . + +Lemma 16. For any $g \in G \coloneqq \{g \coloneqq \ell_s - \ell_{s^\circ} \mid s \in \mathcal{F}\}$ , it holds that $\| g \|_\infty \lesssim TR\overline{D}$ which also indicates that $Pg^2 \lesssim TR\overline{D}Pg$ . + +Proof. Define $\ell_s^* (x,y,x_0,t)\coloneqq \mathrm{BR}_K(s_t(x,y)\| s_t^\circ (x,y\mid x_0))\cdot s_t^\circ (x,y\mid x_0)$ . Then, for any $s\in \mathcal{F}$ , $x,y\in \mathbb{X}$ , it holds that + +$$ +\begin{array}{l} \left| \ell_ {s} ^ {*} (x, y, x _ {0}, t) - \ell_ {s ^ {\circ}} ^ {*} (x, y, x _ {0}, t) \right| \\ = \left| \left(\frac {s _ {t} (x , y) - s ^ {\circ} (x , y)}{s _ {t} ^ {\circ} (x , y \mid x _ {0})} - \log \frac {s _ {t} (x , y)}{s _ {t} ^ {\circ} (x , y)}\right) \cdot s _ {t} ^ {\circ} (x, y \mid x _ {0}) \right| \\ = \left| \left(s _ {t} (x, y) - s ^ {\circ} (x, y)\right) - \log \frac {s _ {t} (x , y)}{s _ {t} ^ {\circ} (x , y)} s _ {t} ^ {\circ} (x, y \mid x _ {0}) \right| \\ \leq \left| s _ {t} (x, y) - s ^ {\circ} (x, y) \right| + \left| \log \frac {s _ {t} (x , y)}{s _ {t} ^ {\circ} (x , y)} s _ {t} ^ {\circ} (x, y \mid x _ {0}) \right| \\ \leq 2 R + 2 \log (R) s _ {t} ^ {\circ} (x, y \mid x _ {0}). \\ \end{array} +$$ + +Therefore, we arrive at + +$$ +\begin{array}{l} \left| \ell_ {s} \left(x _ {0}\right) - \ell_ {s ^ {\circ}} \left(x _ {0}\right) \right| \leq \int_ {\delta} ^ {T} \mathbb {E} _ {x _ {t} \sim p _ {t} (\cdot | x _ {0})} \left[ \sum_ {y \neq x _ {t}} \left| \ell_ {s ^ {1}} ^ {*} (x, y, x _ {0}, t) - \ell_ {s ^ {2}} ^ {*} (x, y, x _ {0}, t) \right| Q \left(x _ {t}, y\right) \right] d t \\ \leq \int_ {\delta} ^ {T} \mathbb {E} _ {x _ {t} \sim p _ {t} (\cdot | x _ {0})} \left[ \sum_ {y \neq x _ {t}} [ 2 R + 2 \log (R) s _ {t} ^ {\circ} (x, y \mid x _ {0}) ] Q (x _ {t}, y) \right] \mathrm {d} t \\ \leq 2 (T - \delta) R \bar {D} + \int_ {\delta} ^ {T} \mathbb {E} _ {y _ {t} \sim p _ {t} (\cdot | x _ {0})} \left[ \sum_ {x \neq y _ {t}} 2 \log (R) Q (x, y _ {t}) \right] d t \\ \leq 4 (T - \delta) R \bar {D}, \\ \end{array} +$$ + +where we used the symmetry of $Q$ and $R \geq 1$ in the last equality. Therefore, we have $Pg^2 \leq \| g \|_{\infty}Pg \lesssim TR\overline{D}Pg$ . + +We introduce two classical tools for controlling the supremum of the empirical loss. + +Proposition 17 (Peeling device (van de Geer, 2000; Bartlett et al., 2005; Koltchinskii, 2006)). Suppose there exists a function $\phi :[0,\infty)\to [0,\infty)$ and $\widehat{r}^* >0$ such that $\forall r > \widehat{r}^*$ + +$$ +\phi (4 r) \leq 2 \phi (r), R _ {n} \left(G _ {r}\right) \leq \phi (r). +$$ + +Then, $\forall r > \widehat{r}^*$ , the following holds: + +$$ +\mathbb {E} _ {\{\sigma_ {i} \}, \{x _ {i} \}} \left[ \sup _ {g \in G} \frac {\frac {1}{n} \sum_ {i = 1} ^ {n} \sigma_ {i} g (x _ {i})}{P g + r} \right] \leq \frac {4 \phi (r)}{r}. +$$ + +Proposition 18 (Talagrand's concentration inequality). (Talagrand, 1996; Bousquet, 2002)] Let $\widetilde{G}$ be a separable set of measurable functions on the probability space $(X,\mathcal{A},P)$ , and suppose that $\forall g\in \widetilde{G}$ + +$$ +\mathbb {E} [ g ] = 0, \mathbb {E} [ g ^ {2} ] = v, \| g \| _ {\infty} \leq C. +$$ + +Then, $\forall t > 0$ + +$$ +\Pr \left\{\sup _ {g \in \widetilde {G}} \frac {1}{n} \sum_ {i = 1} ^ {n} g (x _ {i}) \geq 2 \mathbb {E} _ {\{x _ {i} ^ {\prime} \}} \left[ \sup _ {g \in \widetilde {G}} \frac {1}{n} \sum_ {i = 1} ^ {n} g (x _ {i} ^ {\prime}) \right] + \sqrt {\frac {2 t v}{n}} + \frac {2 t C}{n} \right\} \leq e ^ {- t}. +$$ + +By using these theorems, the prediction error can be decomposed into the approximation error (model bias term) and the generalization error (variance term). + +Lemma 19. Define $s^* \coloneqq \arg \min_{f \in \mathcal{F}} L(f)$ , $\widehat{s} \coloneqq \arg \min_{f \in \mathcal{F}} \widehat{L}(f)$ , $\widehat{g} \coloneqq \ell_{\widehat{s}} - \ell_{s^\circ}$ , and $r^* \coloneqq L(s^*) - L(s^\circ)$ . For the function $\phi(r)$ defined in Proposition 17, there exists $\widehat{r} \gtrsim \max \left\{\phi(\widehat{r}), r^*, \frac{tT\overline{D}}{n}\right\}$ such that the following holds with probability $1 - 2e^{-t}$ : + +$$ +P \widehat {g} \lesssim \widehat {r} + r ^ {*} + \mathcal {O} \left(\frac {T R \overline {{D}} t}{n}\right). +$$ + +Proof. By Lemma 16, + +$$ +P g ^ {2} \lesssim T R \bar {D} P g, \quad \| g \| _ {\infty} \leq T R \bar {D}. +$$ + +Define $\widetilde{G} \coloneqq \left\{\widetilde{g} \coloneqq \frac{Pg - g(x)}{Pg + r} \mid g \in G\right\}$ . Then, the following holds: + +$$ +\mathbb {E} [ \widetilde {g} ] = \frac {P g - P g}{P g + r} = 0, +$$ + +$$ +\| \widetilde {g} \| _ {\infty} = \left\| \frac {P g - g}{P g + r} \right\| _ {\infty} \leq \left\| \frac {g}{r} \right\| _ {\infty} + 1 \lesssim \frac {T R \overline {{D}}}{r}, +$$ + +$$ +\| \widetilde {g} \| _ {L _ {2}} ^ {2} = \frac {\| P g - g \| _ {L _ {2}} ^ {2}}{(P g + r) ^ {2}} = \frac {P g ^ {2} - (P g) ^ {2}}{(P g + r) ^ {2}} \leq \frac {P g ^ {2}}{2 r P g} \lesssim \frac {T R \overline {{D}} P g ^ {2}}{r P g ^ {2}} = \frac {T R \overline {{D}}}{r}. +$$ + +Using Proposition 17 to bound the term $\mathbb{E}_{\{x_i'\}}\left[\sup_{g\in \widetilde{G}}\frac{1}{n}\sum_{i = 1}^{n}g(x_i')\right]$ in Proposition 18, we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \sup _ {\widetilde {g} \in \widetilde {G}} | (P - P _ {n}) \widetilde {g} | \right] \leq 2 R _ {n} (\widetilde {G}) \\ = 2 \mathbb {E} _ {\left\{\sigma_ {i} \right\}, \left\{x _ {i} \right\}} \left[ \sup _ {g \in G} \frac {1}{n} \sum_ {i = 1} ^ {n} \sigma_ {i} \frac {P g - g \left(x _ {i}\right)}{P g + r} \right] \\ = 2 \mathbb {E} _ {\left\{\sigma_ {i} \right\}, \left\{x _ {i} \right\}} \left[ \sup _ {g \in G} \frac {1}{n} \sum_ {i = 1} ^ {n} \sigma_ {i} \frac {g \left(x _ {i}\right)}{P g + r} \right] \\ \leq \frac {8 \phi (r)}{r}. \\ \end{array} +$$ + +The first inequality follows from a standard property of the Rademacher complexity, and the final bound is derived using Proposition 17. Therefore, from Proposition 18, it follows that with probability at least $1 - e^{-t}$ , + +$$ +(P - P _ {n}) \left(\frac {\widehat {g}}{P \widehat {g} + r}\right) \leq \sup _ {\widetilde {g} \in \widetilde {G}} \frac {1}{n} \sum_ {i = 1} ^ {n} \widetilde {g} (x _ {i}) \lesssim \frac {1 6 \phi (r)}{r} + \sqrt {\frac {2 t T R \bar {D}}{n r}} + \frac {2 t T R \bar {D}}{n r} =: \psi_ {n} (r). +$$ + +Let us define $g^{*} \coloneqq l \circ s_{t}^{*} - l \circ s_{t}$ . Then, we get + +$$ +\begin{array}{l} P \widehat {g} = L (\widehat {s} _ {t}) - L (s _ {t}) \\ = L (\widehat {s} _ {t}) - \widehat {L} (\widehat {s} _ {t}) + \widehat {L} (\widehat {s} _ {t}) - \widehat {L} (s _ {t} ^ {*}) + \widehat {L} (s _ {t} ^ {*}) - L (s _ {t} ^ {*}) + L (s _ {t} ^ {*}) - L (s _ {t}) + \widehat {L} (s _ {t}) - \widehat {L} (s _ {t}) \\ = (P - P _ {n}) \widehat {g} + (P _ {n} - P) g ^ {*} + P g ^ {*} \\ \leq (P \widehat {g} + r) \psi_ {n} (r) + (P _ {n} - P) g ^ {*} + r ^ {*}. \\ \end{array} +$$ + +Since $Pg^{*2} \leq TR\overline{D}Pg^{*}$ , the second term can be bounded via Bernstein's inequality (Wainwright, 2019, Proposition 2.14) as + +$$ +(P _ {n} - P) g _ {*} \leq \mathcal {O} \left(\sqrt {\frac {r ^ {*} t}{n}} + \frac {T R \bar {D} t}{n}\right) \leq \mathcal {O} \left(\frac {T R \bar {D} t}{n}\right) + \frac {r ^ {*}}{T R \bar {D}}, +$$ + +with probability $1 - e^{-t}$ for $t > 0$ . We now choose $\widehat{r} \gtrsim \max \left\{\phi(\widehat{r}), r^*, \frac{tTR\overline{D}}{n}\right\}$ . If necessary, we can scale $\widehat{r}$ so that $\psi_n(\widehat{r}) \leq 1/2$ . Under this condition, the following inequality holds: + +$$ +\begin{array}{l} P \widehat {g} \leq \frac {\psi_ {n} (\widehat {r})}{1 - \psi_ {n} (\widehat {r})} \widehat {r} + \frac {1 + 1 / (T R \overline {{D}})}{1 - \psi_ {n} (\widehat {r})} r ^ {*} + \mathcal {O} \left(\frac {T R \overline {{D}} t}{n}\right) \\ \lesssim \widehat {r} + r ^ {*} + \mathcal {O} \left(\frac {T R \bar {D} t}{n}\right). \\ \end{array} +$$ + +![](images/150880aa13b4c7384c81a853037605a23861318fab8c7f8356b003fc8b17cc41.jpg) + +To evaluate the complexity function $\phi (r)$ , we use known bounds on the covering number of ReLU neural networks. + +Lemma 20 (Schmidt-Hieber (2020)). For $S \subset \Phi(L, W, S, B)$ , the covering number $\log N(\varepsilon, S, \| \cdot \|_{\infty})$ satisfies + +$$ +\log N (\varepsilon , \mathcal {S}, \| \cdot \| _ {\infty}) \leq 2 S L \log (L (B \lor 1) W \varepsilon^ {- 1}). +$$ + +Combining Lemma 20 with the Dudley integral gives the following bound on the local Rademacher complexity. + +Lemma 21. For $\phi (r)$ defined in Proposition 17, + +$$ +\phi (r) \leq \mathcal {O} \left(\sqrt {\frac {S L r}{n} \log (L (B \vee 1) \| W \| _ {\infty} n)}\right). +$$ + +By combining these local Rademacher complexity controls, we obtain the proof of Theorem 1. + +Proof of Theorem 1. From Lemmas 15 and 19, with probability at least $1 - 2e^{-t}$ the following inequality holds: + +$$ +\int_ {\delta} ^ {T} h ^ {2} \left(\widehat {s} _ {t}, s _ {t} ^ {\circ}\right) \mathrm {d} t \lesssim \mathcal {O} \left(\widehat {r} + r ^ {*} + \frac {T R \overline {{D}} t}{n}\right). +$$ + +According to Lemma 13, the model bias term $r^*$ can be bounded as + +$$ +r ^ {*} \lesssim T R \bar {D} \left(\varepsilon_ {0} + \varepsilon_ {0} ^ {- 2} \varepsilon_ {1} + \varepsilon_ {2}\right), +$$ + +Furthermore, from Lemma 21, the generalization error $\widehat{r}$ is bounded as + +$$ +\widehat {r} = \mathcal {O} \left(\frac {S L \log (L (B \lor 1) \| W \| _ {\infty} n)}{n}\right). +$$ + +The network parameters satisfy the following upper bounds: $L = \mathcal{O}(\log^2\varepsilon_0^{-1}\vee \log^2 M\varepsilon_1^{-1}\vee$ $\log \varepsilon_2^{-1})$ , $\| W\|_{\infty} = \mathcal{O}(\log^{3}\varepsilon_{0}^{-1}\vee M\log^{3}M\varepsilon_{1}^{-1})$ , $S = \mathcal{O}(\log^{4}\varepsilon_{0}^{-1}\vee M\log^{5}M\varepsilon_{1}^{-1}\vee \log \varepsilon_{2}^{-1})$ , $B = \mathcal{O}(M^2\vee \varepsilon_0^{-2}\vee \log M\varepsilon_1^{-1})$ . To balance the size of each term, we set $\varepsilon_0 = \frac{M}{n\overline{D}}\wedge \frac{1}{R}$ , $\varepsilon_{1} = \frac{M^{3}}{n^{3}\overline{D}^{3}}\vee \frac{M}{n\overline{D}}$ , $\varepsilon_{2} = \frac{M}{n\overline{D}}$ . Given that $\overline{D} = \mathcal{O}(M)$ , the desired bound follows. + +# D Proof of Theorem 3 + +In this section, we provide the proof of Theorem 3. While the subsequent analysis closely parallels the arguments in Appendix C, the key distinction here lies in the use of function approximation theory in anisotropic Besov spaces. + +# D.1 Approximation thoery in anisotropic Besov spaces + +Here, we give the function approximation method in an anisotropic Besov class by deep neural networks. We define the affine composition model, which composes affine transformations with functions in an anisotropic Besov space: + +$$ +\mathcal {H} _ {\mathrm {a f f}} := \left\{h (A x + b) \mid h \in U (B _ {p, q} ^ {\beta} ([ 0, 1 ] ^ {d})), A \in \mathbb {R} ^ {d \times D}, b \in \mathbb {R} ^ {d} \text {s . t .} A x + b \in [ 0, 1 ] ^ {d} (\forall x \in \Omega) \right\}. +$$ + +Under this model, the following bound on the approximation error is known. + +Proposition 22 (Suzuki and Nitanda (2021)). Suppose $x$ follows the uniform distribution on $\Omega = [0,1]^D$ , and define $\tilde{x} = Ax + b \in \mathbb{R}$ , which is assumed to have a bounded density supported on $[0,1]^d$ . Suppose there exists a constant $C$ such that $\| A \|_{\infty} \vee \| b \|_{\infty} \leq C$ . Then for $0 < p, q, r \leq \infty$ and $\beta \in \mathbb{R}_{++}^d$ satisfying $\tilde{\beta} > (1/p - 1/r)_+$ , the following approximation error bound holds: + +$$ +R _ {r} \left(\Phi \left(L _ {1} (d), W _ {1} (d), S _ {1} (d), d (C + 1) B _ {1} (d)\right), \mathcal {H} _ {\text {a f f}}\right) \lesssim N ^ {- \widetilde {\beta}}, +$$ + +where $R_{r}(\mathcal{F},\mathcal{H})\coloneqq \sup_{f^{*}\in \mathcal{H}}\inf_{f\in \mathcal{F}}\| f^{*} - f\|_{L^{r}(\Omega)}$ denotes the worst-case approximation error. Here, the network parameters are set as + +$$ +L _ {1} (d) := 3 + 2 \left\lceil \log_ {2} \left(\frac {3 ^ {d \vee m}}{\epsilon c _ {(d , m)}}\right) + 5 \right\rceil \left\lceil \log_ {2} (d \vee m) \left. \right], +$$ + +$$ +W _ {1} (d) := N W _ {0}, +$$ + +$$ +S _ {1} (d) := \left[ \left(L _ {1} (d) - 1\right) W _ {0} ^ {2} + 1 \right] N, +$$ + +$$ +B _ {1} (d) := \mathcal {O} \left(N ^ {d \left(1 + \nu^ {- 1}\right) \left(1 / p - \widetilde {\beta}\right) +}\right), +$$ + +where $\epsilon = N^{-\tilde{\beta}}\log (N)^{-1}$ , and $c_{(d,m)}$ is a constant depending only on $d$ and $m$ . + +# D.2 Approximation error bound + +Under Assumptions Assumption 5 to 7, we construct a neural network that approximates the score function $s_t^\circ (x,y)$ . + +Lemma 23. For every $k \in [1, \ldots, M]$ , there exists a neural network $\phi_{\mathrm{cont}}^1(x, t) \in \Phi(L, W, S, B)$ such that + +$$ +\left| \phi_ {\text {c o n t}} ^ {1} (x, t) - M p _ {t} (x) \right| \lesssim \left(N ^ {- \widetilde {\beta}} \vee e ^ {- t \lambda_ {2}} \varepsilon\right) f _ {1} (k) \vee e ^ {- t \lambda_ {2}} f _ {2} (k) \quad \forall t > 0, +$$ + +The network parameters satisfy: + +$$ +L = \mathcal {O} (\log^ {2} (M \vee N)), +$$ + +$$ +\| W \| _ {\infty} = \mathcal {O} (\log^ {3} M \vee k N), +$$ + +$$ +S = \mathcal {O} \left(\log^ {5} M \vee k N \log N\right), +$$ + +$$ +B = \mathcal {O} (k ^ {\gamma} \operatorname {p o l y} (M \vee N)), +$$ + +where the functions $f_{1}(k)$ and $f_{2}(k)$ are defined as: + +$$ +f _ {1} (k) \lesssim \left\{ \begin{array}{l l} k ^ {1 - (s - \gamma)} & (s - \gamma < 1), \\ \log k & (s - \gamma = 1), \\ 1 & (s - \gamma > 1), \end{array} \right. f _ {2} (k) \lesssim \left\{ \begin{array}{l l} M ^ {1 - s} & (s < 1), \\ \log M & (s = 1), \\ k ^ {1 - s} & (s > 1), \end{array} \right. +$$ + +for $1 \leq k \leq M$ and $f_{2}(M) = 0$ . + +Proof. We aim to approximate the function $M p_{t}(x) = M \sum_{j=1}^{M} c_{j} u_{j}(x) e^{-t \lambda_{j}}$ using a neural network by truncating the sum to a fixed number of terms. Specifically, we consider $p_{t}^{*}(x) = \sum_{j=1}^{k} c_{j} u_{j}^{*}(x) e^{-t \lambda_{j}}$ , where each $u_{j}^{*} \in \mathcal{H}_{\mathrm{aff}}$ . By Proposition 22, there exists a neural network $\phi_{j}^{0} \in \Phi(L_{1}(d), W_{1}(d), S_{1}(d), \gamma_{j} B_{1}(d))$ such that + +$$ +\left\| u _ {j} ^ {*} - \phi_ {j} ^ {0} \right\| _ {L ^ {\infty} (\Omega)} \lesssim \gamma_ {j} N ^ {- \widetilde {\beta}}. +$$ + +Following the same strategy as in Lemma 11, we can construct a neural network to approximate the exponential decay $e^{-t\lambda_j}$ . We define $\phi_j^*(t)$ as follows: + +$$ +\begin{array}{l} \phi_ {j} ^ {*} (t) := \phi_ {\mathrm {m u l t}} \left(\phi_ {\mathrm {s w i t}} ^ {2} (t; 1 / \lambda_ {j}, 2 / \lambda_ {j}), \phi_ {0} (t)\right) + \\ \sum_ {s = 1} ^ {\lceil A \rceil - 1} \phi_ {\mathrm {m u l t}} (\phi_ {\mathrm {s w i t}} ^ {1} (t; (s + 1) / \lambda_ {j}, (s + 2) / \lambda_ {j}), \phi_ {\mathrm {s w i t}} ^ {2} (t; s / \lambda_ {j}, (s + 1) / \lambda_ {j}), \phi_ {s / \lambda_ {j}} (t)). \\ \end{array} +$$ + +The product $u_{j}^{*}(x) \cdot e^{-t\lambda_{j}}$ can be approximated using $\phi_{j}^{*}(t)$ and $\phi_{j}^{0}$ , and by summing over $j = 1, \ldots, k$ and multiplying by $M$ , we can construct the neural network $\phi_{\mathrm{cont}}^{1} \in \Phi(L, W, S, B)$ satisfying $\| \phi_{\mathrm{cont}}^{1} - Mp_{t}^{*} \| \leq N^{-\widetilde{\beta}} \sum_{j=1}^{k} \gamma_{j} j^{-s}$ . Here, we set the approximation error of $\phi_{j}^{*}(t)$ and $\phi_{\mathrm{mult}}$ to $\varepsilon = M^{-2}N^{-\widetilde{\beta}}$ . According to Lemmas 7 and 9, the parameters of $\phi_{\mathrm{cont}}^{1}$ can be bounded as follows: + +$$ +\begin{array}{l} L = \mathcal {O} (\log^ {2} (M \vee N)), \\ \| W \| _ {\infty} = \mathcal {O} (\log^ {3} M \vee k N), \\ S = \mathcal {O} \left(\log^ {5} M \vee k N \log N\right), \\ B = \mathcal {O} \left(\gamma_ {k} \operatorname {p o l y} (M \vee N)\right). \\ \end{array} +$$ + +Now, the total error in approximating $p_t(x)$ is given by + +$$ +\begin{array}{l} | M p _ {t} (x) - \phi_ {\text {c o n t}} ^ {1} (x, t) | \leq N ^ {- \widetilde {\beta}} \sum_ {j = 1} ^ {k} \gamma_ {j} j ^ {- s} + M \left| \sum_ {j = 1} ^ {M} c _ {j} u _ {j} (x) e ^ {- t \lambda_ {j}} - \sum_ {j = 1} ^ {k} c _ {j} u _ {j} ^ {*} (x) e ^ {- t \lambda_ {j}} \right| \\ \lesssim N ^ {- \widetilde {\beta}} \sum_ {j = 1} ^ {k} j ^ {- (s - \gamma)} + \sqrt {M} \varepsilon \sum_ {j = 1} ^ {k} c _ {1} j ^ {- s} e ^ {- t \lambda_ {j}} + M \left| \sum_ {j = k + 1} ^ {M} c _ {1} j ^ {- s} u _ {j} (x) e ^ {- t \lambda_ {j}} \right|. \\ \end{array} +$$ + +For the sum $\sum_{j=1}^{k} j^{-(s - \gamma)}$ and $\sum_{j=k+1}^{M} j^{-s}$ , we have + +$$ +\begin{array}{l} \sum_ {j = 1} ^ {k} j ^ {- (s - \gamma)} \leq 1 + \int_ {1} ^ {k} j ^ {- (s - \gamma)} \mathrm {d} j \lesssim \left\{ \begin{array}{l l} k ^ {1 - (s - \gamma)} & (s - \gamma < 1), \\ \log k & (s - \gamma = 1), \\ 1 & (s - \gamma > 1), \end{array} \right. \\ \sum_ {j = k + 1} ^ {M} j ^ {- s} \leq \int_ {k} ^ {M} j ^ {- s} \mathrm {d} j \lesssim \left\{ \begin{array}{l l} M ^ {1 - s} & (s < 1), \\ \log M & (s = 1), \\ k ^ {1 - s} & (s > 1). \end{array} \right. \\ \end{array} +$$ + +Since $c_{1} = 1 / \sqrt{M}$ , we can achieve the desired upper bound. + +![](images/c16a9ff736863867537fd0eb0c25e08413cac2bca30fed03974fa2001c43da71.jpg) + +Lemma 24. For any $0 < \varepsilon_0 \leq R^{-1}$ , there exists a neural network $\phi_{\mathrm{cont}}^2(x,t) \in \Phi(L,W,S,B)$ such that + +$$ +\left| \phi_ {\mathrm {c o n t}} ^ {2} (x, t) - \frac {1}{M p _ {t} (x)} \right| \lesssim \varepsilon_ {0} + R ^ {2} ((N ^ {- \widetilde {\beta}} \vee e ^ {- t \lambda_ {2}} \varepsilon) f _ {1} (k) \vee e ^ {- t \lambda_ {2}} f _ {2} (k)) \quad \forall t > 0, +$$ + +with network parameters bounded as: + +$$ +\begin{array}{l} L = \mathcal {O} \left(\log^ {2} \varepsilon_ {0} ^ {- 1} \vee \log^ {2} (M \vee N)\right), \\ \| W \| _ {\infty} = \mathcal {O} \left(\log^ {3} \left(M \vee \varepsilon_ {0} ^ {- 1}\right) \vee k N\right), \\ S = \mathcal {O} \left(\log^ {5} M \vee \log^ {4} \varepsilon_ {0} ^ {- 1} \vee k N \log N\right), \\ B = \mathcal {O} \left(\varepsilon_ {0} ^ {- 2} \vee \gamma_ {k} \operatorname {p o l y} (M \vee N)\right). \\ \end{array} +$$ + +Here, $f_{1}(k)$ and $f_{2}(k)$ are defined in Lemma 23. + +Proof. As in Lemma 12, we can construct $\phi_{\mathrm{cont}}^2 (x,t) = \phi_{\mathrm{rec}}\circ \phi_{\mathrm{cont}}^1 (x,t)$ to achieve the desired approximation. + +Lemma 25. Let $0 < \varepsilon_0 \lesssim R^{-1}$ and $N \in \mathbb{N}$ . Then, there exists a neural network $\phi_{\mathrm{cont}}(x,y,t) \in \mathcal{F}$ such that + +$$ +| \phi_ {\mathrm {c o n t}} (x, y, t) - s _ {t} ^ {\circ} (x, y) | \lesssim \varepsilon_ {0} + R ^ {2} ((N ^ {- \widetilde {\beta}} \vee e ^ {- t \lambda_ {2}} \varepsilon) f _ {1} (k) \vee e ^ {- t \lambda_ {2}} f _ {2} (k)) \quad \forall t > 0. +$$ + +The parameters of $\phi_{\mathrm{cont}}$ are bounded as follows: + +$$ +\begin{array}{l} L = \mathcal {O} \left(\log^ {2} \varepsilon_ {0} ^ {- 1} \vee \log^ {2} (M \vee N)\right), \\ \| W \| _ {\infty} = \mathcal {O} \left(\log^ {3} \left(M \vee \varepsilon_ {0} ^ {- 1}\right) \vee k N\right), \\ S = \mathcal {O} \left(\log^ {5} M \vee \log^ {4} \varepsilon_ {0} ^ {- 1} \vee k N \log N\right), \\ B = \mathcal {O} \left(\varepsilon_ {0} ^ {- 2} \vee \gamma_ {k} \operatorname {p o l y} (M \vee N)\right). \\ \end{array} +$$ + +Here, $f_{1}(k)$ and $f_{2}(k)$ are defined in Lemma 23. Moreover, the following bound holds: + +$$ +\begin{array}{l} \int_ {\delta} ^ {T} \mathbb {E} _ {x \sim p _ {t}} \left[ \sum_ {y \neq x} \mathrm {B R} _ {K} \left(\phi_ {\text {c o n t}} (x, y, t) \| s _ {t} ^ {\circ} (x, y)\right) \cdot s _ {t} ^ {\circ} (x, y) Q (x, y) \right] \mathrm {d} t \\ \lesssim R \log (R) T \bar {D} \left(\varepsilon_ {0} + R ^ {2} \lambda_ {2} ^ {- 1} \left(\varepsilon f _ {1} (k) \vee f _ {2} (k)\right) + R ^ {2} N ^ {- \widetilde {\beta}} f _ {1} (k)\right) ^ {2}. \\ \end{array} +$$ + +Proof. The proof proceeds in a similar manner to Lemma 13. We construct a network $\phi_{\mathrm{cont}}^3 (x,y,t)\coloneqq [\phi_{\mathrm{cont}}^1 (y,t),\phi_{\mathrm{cont}}^2 (x,t)]^\top$ by combining the neural networks that approximate $Mp_{t}(y)$ and $1 / Mp_{t}(x)$ in parallel. Then, define the overall network as $\phi_{\mathrm{cont}}(x,y,t)\coloneqq \phi_{\mathrm{mult}}\circ \phi^3 (x,y,t)$ . This network estimates the true score $s_t^\circ (x,y) = \frac{p_t(y)}{p_t(x)}$ , and the error can be bounded as follows: + +$$ +\begin{array}{l} \left| \phi_ {\text {c o n t}} (x, y, t) - s _ {t} ^ {\circ} (x, y) \right| \leq \left| \phi_ {\text {m u l t}} \left(\phi^ {1} (y, t), \phi^ {2} (x, t)\right) - \phi^ {1} (y, t) \phi^ {2} (x, t) \right| + \left| \phi^ {1} (y, t) \phi^ {2} (x, t) - s _ {t} ^ {\circ} (x, y) \right| \\ \leq \varepsilon_ {1} + \left| \phi^ {1} (y, t) \phi^ {2} (x, t) - \frac {\phi^ {1} (y , t)}{M p _ {t} (x)} + \frac {\phi^ {1} (y , t)}{M p _ {t} (x)} - s _ {t} ^ {\circ} (x, y) \right| \\ \leq \varepsilon_ {1} + | \phi^ {1} (y, t) | \left| \phi^ {2} (x, t) - \frac {1}{M p _ {t} (x)} \right| + \left| \frac {1}{M p _ {t} (x)} \right| | \phi^ {1} (y, t) - M p _ {t} (y) | \\ \lesssim \varepsilon_ {0} + \varepsilon_ {1} + R ^ {2} \left(\left(N ^ {- \tilde {\beta}} \vee e ^ {- t \lambda_ {2}} \varepsilon\right) f _ {1} (k) \vee e ^ {- t \lambda_ {2}} f _ {2} (k)\right). \\ \end{array} +$$ + +Setting $\varepsilon_{1} = \varepsilon_{0}$ gives the desired approximation rate. + +By the same argument as Lemma 13, we can bound the divergence as + +$$ +\begin{array}{l} \int_ {\delta} ^ {T} \mathbb {E} _ {x \sim p _ {t}} \left[ \sum_ {y \neq x} \operatorname {B R} _ {K} \left(\phi_ {\text {c o n t}} (x, y, t) \| s _ {t} ^ {\circ} (x, y)\right) \cdot s _ {t} ^ {\circ} (x, y) Q (x, y) \right] \mathrm {d} t \\ \leq \int_ {\delta} ^ {T} \mathbb {E} _ {x \sim p _ {t}} \left[ \sum_ {y \neq x} 4 R \log (R) | \phi_ {\text {c o n t}} (x, y, t) - s _ {t} ^ {\circ} (x, y) | ^ {2} Q (x, y) \right] d t \\ \leq 4 R \log (R) \int_ {\delta} ^ {T} \left(\varepsilon_ {0} + R ^ {2} \left(\left(N ^ {- \widetilde {\beta}} \vee e ^ {- t \lambda_ {2}} \varepsilon\right) f _ {1} (k) \vee e ^ {- t \lambda_ {2}} f _ {2} (k)\right)\right) ^ {2} \bar {D} d t \\ \lesssim R \log (R) \left(T \bar {D} \varepsilon_ {0} ^ {2} + R ^ {4} \int_ {\delta} ^ {t _ {0}} \left(e ^ {- t \lambda_ {2}} (\varepsilon f _ {1} (k) \vee f _ {2} (k))\right) ^ {2} \bar {D} d t + R ^ {4} \int_ {t _ {0}} ^ {T} \left(N ^ {- \widetilde {\beta}} f _ {1} (k)\right) ^ {2} \bar {D} d t\right) \\ \lesssim R \log (R) T \bar {D} \left(\varepsilon_ {0} + R ^ {2} \lambda_ {2} ^ {- 1} \left(\varepsilon f _ {1} (k) \vee f _ {2} (k)\right) + R ^ {2} N ^ {- \widetilde {\beta}} f _ {1} (k)\right) ^ {2}, \\ \end{array} +$$ + +where we define $t_0 \coloneqq \min \left\{\delta, \lambda_2^{-1}(\widetilde{\beta}\log N + \log \left(\varepsilon \vee \frac{f_2(k)}{f_1(k)}\right)\right\}$ . + +![](images/d4beb4a4a7d5b86c615169a78759152a0a1aa6e16457d3fea40f1e75c0d24ca6.jpg) + +# D.3 Proof of the statement + +By combining the results from Appendices C.2 and D.2, we are now ready to prove Theorem 3. + +Proof of Theorem 3. By applying Lemmas 15 and 19, we obtain + +$$ +\int_ {\delta} ^ {T} h ^ {2} (\widehat {s} _ {t}, s _ {t} ^ {\circ}) \mathrm {d} t \lesssim \mathcal {O} \left(\widehat {r} + r ^ {*} + \frac {T R \overline {{D}} t}{n}\right), +$$ + +with probability $1 - 2e^{-t}$ . By Lemmas 21 and 25, the model bias term $r^*$ and the generalization error $\widehat{r}$ can be bounded as: + +$$ +r ^ {*} \lesssim T R \log (R) \overline {{D}} [ \varepsilon_ {0} + R ^ {2} \lambda_ {2} ^ {- 1} (\varepsilon f _ {1} (k) \vee f _ {2} (k)) + R ^ {2} N ^ {- \widetilde {\beta}} f _ {1} (k) ] ^ {2}, +$$ + +$$ +\widehat {r} = \mathcal {O} \left(\frac {S L \log (L (B \lor 1) \| W \| _ {\infty} n)}{n}\right). +$$ + +By setting $N = \mathcal{O}\left(nk^{-1}f_1(k)^2\right)^{\frac{1}{1 + 2\tilde{\beta}}}$ and $\varepsilon_0 = \mathcal{O}\left(n^{-1 / 2}\right)$ , we have + +$$ +r ^ {*} + \widehat {r} \lesssim \left[ \left(n ^ {- \widetilde {\beta}} k ^ {\widetilde {\beta}} f _ {1} (k)\right) ^ {\frac {2}{1 + 2 \widetilde {\beta}}} + \lambda_ {2} ^ {- 2} f _ {2} (k) ^ {2} + \lambda_ {2} ^ {- 2} \varepsilon^ {2} f _ {1} (k) ^ {2} \right] \log^ {8} (M \vee N). +$$ + +We consider balancing the first and second terms as follows: + +1. When $s \leq 1$ (and thus $s - \gamma \leq 1$ ): By setting $k = M$ , we obtain + +$$ +r ^ {*} + \widehat {r} \lesssim \left(n ^ {- \frac {2 \widetilde {\beta}}{1 + 2 \widetilde {\beta}}} M ^ {\frac {2 (1 - (s - \gamma) + \widetilde {\beta})}{1 + 2 \widetilde {\beta}}} + \lambda_ {2} ^ {- 2} \varepsilon^ {2} M ^ {2}\right) \log^ {8} (M \lor N). +$$ + +2. When $1 < s$ and $s - \gamma \leq 1$ : By setting $k = (\lambda_2^{1 + 2\widetilde{\beta}}n^{-\widetilde{\beta}})^{-\frac{1}{(1 + 2\widetilde{\beta})(s - 1) + (1 - (s - \gamma) + \widetilde{\beta})}}$ , we have that + +$$ +\begin{array}{l} r ^ {*} + \widehat {r} \lesssim \left(\lambda_ {2} ^ {- \frac {2 (1 - (s - \gamma) + \widetilde {\beta})}{(1 + 2 \widetilde {\beta}) (s - 1) + (1 - (s - \gamma) + \widetilde {\beta})}} n ^ {- \frac {2 \widetilde {\beta} (s - 1)}{(s - 1) (1 + 2 \widetilde {\beta}) + (1 - (s - \gamma) + \widetilde {\beta})}}\right) \\ + \varepsilon^ {2} \lambda_ {2} ^ {- \frac {2 (s - 1 + 1 - (s - \gamma)) (1 + 2 \tilde {\beta}) + 2 (1 - (s - \gamma) + \tilde {\beta})}{(1 + 2 \tilde {\beta}) (s - 1) + (1 - (s - \gamma) + \tilde {\beta})}} n ^ {\frac {2 \tilde {\beta} (1 - (s - \gamma))}{(s - 1) (1 + 2 \tilde {\beta}) + (1 - (s - \gamma) + \tilde {\beta})}} \Bigg) \log^ {8} (M \vee N). \\ \end{array} +$$ + +3. When $1 < s$ and $1 < s - \gamma$ : By setting $k = (n^{-\widetilde{\beta}}\lambda_2^{1 + 2\widetilde{\beta}})^{\frac{1}{1 + \widetilde{\beta} - s - 2s\widetilde{\beta}}}$ , we have + +$$ +r ^ {*} + \widehat {r} \lesssim \left(\lambda_ {2} ^ {- \frac {2 \widetilde {\beta}}{s - 1 + 2 s \widetilde {\beta} - \widetilde {\beta}}} n ^ {- \frac {2 \widetilde {\beta} (s - 1)}{s - 1 + \widetilde {\beta} + 2 \widetilde {\beta} (s - 1)}} + \lambda_ {2} ^ {- 2} \varepsilon^ {2}\right) \log^ {8} (M \vee N). +$$ + +![](images/c9cf4b6614530c6dafb8f1e7a2da99fd5ebbac36c86558864eb03366e809a170.jpg) + +# E Proof of Corollary 1 + +# E.1 Proof of Lemma 4 + +Before proceeding to the main proof, we present a preparatory result on the spectral properties of the graph Laplacian in Lemma 4. Since $h_w(x) = (-1)^{w^\top x}$ , for every $w \in \{0,1\}^D$ , the function $h_w: \{0,1\}^D \to \{-1,1\}$ is an eigenfunction of the adjacency matrix $A$ defined on $\{0,1\}^D$ . Let the state $v_i \in \{0,1\}^D$ satisfy $d(v_i, x) = 1$ in terms of the Hamming distance, where the index $i$ indicates the coordinate at which $v_i$ and $x$ differ. Then, we obtain + +$$ +\sum_ {i = 1} ^ {D} h _ {w} (v _ {i}) = (D - | w |) h _ {w} (x) - | w | h _ {w} (x) = (D - 2 | w |) h _ {w} (x), +$$ + +which implies $h_w$ is an eigenfunction of $A$ with the eigenvalue $D - 2|w|$ . Since $L = DI - A$ , where $I$ denotes the identity matrix, it follows that $h_w$ is also an eigenfunction of $L$ with eigenvalue $2|w|$ , and in particular, $\lambda_2 = 2$ . + +# E.2 Main proof + +First, we show $h_w(x) \coloneqq \cos(\pi w^\top x) / \sqrt{M}$ belongs to the Sobolev class $H^\beta([0,1]^D)$ for any constant $\beta \in \mathbb{N}$ . Sobolev spaces are defined by + +$$ +H ^ {\beta} (\Omega) := \left\{f \in L ^ {2} (\Omega) \Bigg | \| f \| _ {H ^ {\beta}} := \left(\sum_ {| \alpha | \leq \beta} \| D ^ {\alpha} f \| _ {2} ^ {2}\right) ^ {1 / 2} < \infty \right\}, \quad D ^ {\alpha} f := \frac {\partial^ {| \alpha |}}{\partial^ {\alpha_ {1}} \ldots \partial^ {\alpha_ {D}}} f. +$$ + +Here $\alpha = (\alpha_{1},\ldots \alpha_{D})\in \mathbb{N}_{0}^{D}$ . Using the chain rule, we obtain + +$$ +D ^ {\alpha} h _ {w} (x) = \frac {1}{\sqrt {M}} p _ {\alpha} (\cos (\pi w ^ {\top} x), \sin (\pi w ^ {\top} x)) (\pi w) ^ {| \alpha |}. +$$ + +The term $p_{\alpha}$ is a finite linear combination of trigonometric functions and $|p_{\alpha}| = \mathcal{O}(1)$ . In particular, we have the bound $|D^{\alpha}h_{w}| \lesssim \mathcal{O}(|w|^{\lfloor \alpha \rfloor} / \sqrt{M})$ . Thus, the Sobolev norm of $h_w$ is bounded as + +$$ +\begin{array}{l} \| h _ {w} \| _ {H ^ {\beta}} ^ {2} = \sum_ {| \alpha | \leq \beta} \| D ^ {\alpha} h _ {w} \| _ {2} ^ {2} \\ = \sum_ {| \alpha | \leq \beta} \int_ {[ 0, 1 ] ^ {D}} | D ^ {\alpha} h _ {w} | ^ {2} d x \\ = \sum_ {| \alpha | \leq \beta} \mathcal {O} (| w | ^ {2 | \alpha |} / M) \\ = \sum_ {k = 0} ^ {\beta} \sum_ {| \alpha | = k} \mathcal {O} (| w | ^ {2 | \alpha |} / M) \\ = \mathcal {O} (D ^ {3 \beta} / M) \\ = \mathcal {O} (1). \\ \end{array} +$$ + +This implies that $h_w \in H^\beta([0,1]^D)$ and, in particular, $h_w$ lies in the scaled Sobolev unit ball: there exists a constant $\gamma_w > 0$ such that $h_w \in \gamma_w U(H^\beta)$ , where $U(H^\beta)$ denotes the unit ball in $H^\beta(\mathbb{R}^D)$ . Since $H^\beta = B_{2,2}^\beta$ in the sense of Besov spaces, Theorem 4 can be applied directly. Specifically, for the function $u_j^*(x) = \cos(\pi w_j^\top x)$ , Assumptions 5 and 6 hold with parameters $\varepsilon = 0$ , $\gamma_j = \mathcal{O}(1)$ , and $\gamma = 0$ . Moreover, Lemma 4 ensures that $\lambda_2 = \mathcal{O}(1)$ . By choosing $\beta$ sufficiently large, the desired bound follows from Theorem 4. + +# F Proof of Corollary 2 + +Here, let $\phi_j(x)$ be the eigenfunctions of the Laplace-Beltrami operator $-\Delta_{\mathcal{M}}$ normalized in $L^2 (\mathcal{M})$ and their corresponding eigenvalues are denoted by $0 = \mu_1 < \mu_2\leq \dots$ , that is, $-\Delta_{\mathcal{M}}\phi_j = \mu_j\phi_j$ and let $\sigma (-\Delta_{\mathcal{M}})\coloneqq \{\mu_i\}_{i = 1}^{\infty}$ be the set of eigenvalues. For a vector $u = (u(x))_{x\in \mathbb{X}}$ , we define + +$$ +\| u \| _ {\ell^ {2} (\hat {p})} := \sqrt {\frac {| \mathbb {S} ^ {d - 1} | \sigma^ {d}}{d} \sum_ {x \in \mathbb {X}} \frac {u (x) ^ {2}}{\mathcal {N} (x)}}, +$$ + +where $\mathcal{N}(x) \coloneqq \{y \in \mathbb{X} \mid \| x - y \| \leq \sigma\}$ . Then, the following result is known: + +Proposition 26 (Dunson et al. (2021)). Suppose that $M$ is sufficiently large such that + +$$ +M > \max \left(\left(\frac {\mathcal {X} _ {2} + \mu_ {k} ^ {d / 2 + 5}}{\operatorname* {m i n} \left(\Gamma_ {k} , 1\right)}\right) ^ {8 d + 2 6}, \left(\mathcal {X} _ {3} + \mu_ {k} ^ {(5 d + 7) / 4}\right) ^ {8 d + 2 6}\right), \tag {5} +$$ + +where $\mathcal{X}_2, \mathcal{X}_3 > 1$ are constants depending on $d$ and the volume, the radius, the curvature and the second fundamental form of the manifold $\mathcal{M}$ , and $\Gamma_k = \min_{1 \leq j \leq k} \mathrm{dist}(\mu_j, \sigma(-\Delta_{\mathcal{M}}) \setminus \{\mu_j\})$ . Let $\sigma = \left(\frac{\log(M)}{M}\right)^{\frac{1}{4d + 13}}$ . Then for $1 \leq j \leq K$ , it holds that + +$$ +\max _ {x \in \mathbb {X}} \left| \frac {u _ {j} (x)}{\| u _ {j} \| _ {\ell^ {2} (\hat {p})}} - \phi_ {j} (x) \right| \leq C _ {2} \left(\frac {\log (M)}{M}\right) ^ {\frac {1}{8 d + 1 6}}. \tag {6} +$$ + +Since $-\Delta_{\mathcal{M}}\phi_j = \mu_j\phi_j$ , the Sobolev norm of $\phi_j$ for $\beta \in \mathbb{N}$ can be evaluated as + +$$ +\| \phi_ {j} \| _ {W ^ {\beta} (\mathcal {M})} \leq C (1 + \| \Delta_ {\mathcal {M}} ^ {\beta} \phi_ {j} \| _ {L ^ {2} (\mathcal {M})}) \leq C (1 + \mu_ {j} ^ {\beta}). +$$ + +Moreover, since it is known that $\mu_{j} = \Theta (j^{2 / d})$ (Hassaneezhad et al., 2016), we obtain that + +$$ +\left\| \phi_ {j} \right\| _ {W ^ {\beta} (\mathcal {M})} \lesssim j ^ {2 \beta / d}. +$$ + +This implies that we may choose $\gamma = 2\beta /d$ . Now suppose that $\beta$ is chosen so that $s - \gamma >1$ . More precisely, we can approximate $\phi_j$ by a composite function $\tilde{\phi}_j\circ \psi_{\mathcal{M}}$ where $\psi_{\mathcal{M}}:\mathbb{R}^{D}\to \mathbb{R}^{d}$ + +represents a smooth function giving the local coordinate of $\mathcal{M}$ and $\tilde{\phi}_j$ is a function on this local coordinate that corresponds to $\phi_j$ . Then, we may think $\tilde{\phi}_j$ is a function in a Besov space $B_{2,2}^{\beta}$ on a compact domain of $\mathbb{R}^d$ , and thus $\tilde{\phi}_j \circ \psi_{\mathcal{M}}$ can be approximated by ReLU deep neural networks with an error $\mathcal{O}(j^\gamma N^{-\beta / d})$ . + +Since $p$ is the uniform distribution, a measure concentration inequality (Wainwright, 2019) implies + +$$ +\frac {| \mathbb {S} ^ {d - 1} | \sigma^ {d}}{d} \frac {1}{\mathcal {N} (x)} = \Theta \left(\frac {1}{M}\right), +$$ + +uniformly over $x \in \mathbb{X}$ , with high probability, provided that $\sigma \geq \left(\frac{\log(M)}{M}\right)^{\frac{1}{4d + 13}}$ . In this event, we see that $\|u\|_{\ell^2(\hat{p})} = \Theta(\|u\| / \sqrt{M}) = \Theta(1 / \sqrt{M})$ . Then, by Eq. (6), Assumption 5 is satisfied for $u_j^*(x) = \|u_j\|_{\ell^2(\hat{p})} \phi_j(x) \simeq \phi_j(x) / \sqrt{M}$ with the approximation error + +$$ +\varepsilon = \left(\frac {\log (M)}{M}\right) ^ {\frac {1}{8 d + 1 6}}. +$$ + +To achieve the assertion in Theorem 4 with $\tilde{\beta} = \beta /d$ , we choose $k$ in its proof as + +$$ +k = \left(n ^ {- \beta / d} \lambda_ {2} ^ {1 + 2 \beta / d}\right) ^ {\frac {1}{1 + \beta / d - s - 2 s \beta / d}} \simeq n ^ {\frac {1}{(d / \beta + 2) (s - 1) + 1}}, +$$ + +which yields $\mu_k = \mathcal{O}(n^{\frac{2/d}{(d/\beta+2)(s-1)+1}})$ . Hence, $(\mu_k^{d/2+5})^{8d+26} = \mathcal{O}(n^{\frac{(1+10/d)(8d+26)}{(d/\beta+2)(s-1)+1}})$ and $(\mu_k^{(5d+7)/4})^{8d+26} = \mathcal{O}(n^{\frac{(5/2+4/d)(8d+26)}{(d/\beta+2)(s-1)+1}})$ . We assume that $M$ is sufficiently large to satisfy Eq. (5) with this choice of $k$ , which ensures the approximation error bound above holds. + +Based on these arguments, Theorem 4, with $\widetilde{\beta} = \beta /d$ and $p = q = 2$ , gives that + +$$ +\mathrm {K L} (p _ {\delta} \| \widehat {q} _ {T - \delta}) \lesssim \left(n ^ {- \frac {2 (s - 1)}{2 (s - 1) / (2 \beta / d) + 2 s - 1}} + \left(\frac {\log (M)}{M}\right) ^ {\frac {1}{8 + 4 d}} + \frac {t}{n}\right) \log^ {8} (M \vee n), +$$ + +which yields the assertion. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The target of the analysis (discrete diffusion models) is detailed in Section 2. The convergence analysis is given in Sections 3 and 4. Proof details are provided throughout the appendix. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: See the conclusion section (Sec. 6). + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. 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Given a history of images $I_{1}, \dots, I_{T}$ and accompanying actions, we navigate all the way back to the beginning - $I_{1}$ . The task is to generate frames along the way, consistent to what is seen in the history, given the actions. As an example, we show the predictions of the first frame - $\hat{I}_{1}$ . Can a generative model recall the content of $I_{1}$ long back in the sequence? Diffusion models fall short ( $X$ ), our model correctly recalls the content of $I_{1}$ ( $\checkmark$ ). + +# Abstract + +World models have recently gained prominence for action-conditioned visual prediction in complex environments. However, relying on only a few recent observations causes them to lose long-term context. Consequently, within a few steps, the generated scenes drift from what was previously observed, undermining temporal coherence. This limitation, common in state-of-the-art world models, which are diffusion-based, stems from the lack of a lasting environment state. + +To address this problem, we introduce StateSpaceDiffuser, where a diffusion model is enabled to perform long-context tasks by integrating features from a state-space model, representing the entire interaction history. This design restores long-term memory while preserving the high-fidelity synthesis of diffusion models. + +To rigorously measure temporal consistency, we develop an evaluation protocol that probes a model's ability to reinstantiate seen content in extended rollouts. Comprehensive experiments show that SpaceDiffuser significantly outperforms a strong diffusion-only baseline, maintaining a coherent visual context for an order of magnitude more steps. It delivers consistent views in both a 2D maze navigation and a complex 3D environment. These results establish that bringing state-space representations into diffusion models is highly effective in demonstrating both visual details and long-term memory. Project page: https://insait-institute.github.io/SpaceDiffuser/. + +![](images/36cdda32c77f91b3640c414ce1a9cec51671e792471e7185e834104a13279dcb.jpg) +Figure 2: Our Approach. While diffusion models are limited to a short sequence input, our approach enables long-context processing for diffusion models with a state-space representation. + +![](images/14fa1e02e625ddef478e0bfeeaaca0e41a3756d8987fcd0d48bb16cb6692ffab.jpg) + +# 1 Introduction + +World models have gained popularity for the production of visual consequences of given past observations and actions. These models can learn to generate environment observations entirely by training on many interactions with the environment. Simply by observation, they are capable of handling complex environments, such as car driving [27, 29, 41, 66], 3D virtual environments [77, 1, 19], platformer games [8, 68], ego-centric action videos [82], or navigation [2, 84]. They enable interactivity without the burden of hand-coding complex environments, but also offer feature representations for robotics and reinforcement learning agents for planning. + +For long interaction with world models, it is essential that the generated video remains consistent with previously observed or generated content. Revisited areas should preserve their appearance, and objects observed again should keep their properties. However, as shown in Fig. 1, current high-fidelity world models, which are mostly based on diffusion, cannot preserve context outside of a short time window, most often directly limited by their input window size [82, 1, 19, 8]. This leads to an increasing drift in content over time, where earlier information is forgotten or overwritten. The inability to retain persistent memory of the environment poses a major challenge, especially for real-world applications such as agent planning and virtual interaction, where coherent, temporally consistent environments are essential. Therefore, in this work our task is to stay consistent with a long history of past inputs, even if generating a single future frame. This is in contrast to long generation which focuses on producing extended realistic sequences, even without prior context. + +To improve content consistency in diffusion-based world models, we make use of a persistent long-context representation. Specifically, we leverage features from a discrete state-space model (Mamba), which has been shown to be very effective at capturing long-term context in prior work [20, 31]. We summarize our system in Fig. 2. Although these models were previously applied to language and visually simple environments [12], our goal is to preserve the long-term context in modern diffusion-based world models, targeting environments with higher visual complexity such as CSGO [60]. In contrast, other state-based models, such as those using LSTMs or GRUs [38, 39, 40], have limited generative capacity and are mainly used for agent planning. Our approach combines the strong generative power of diffusion models with the long-term context tracking of state-space representations. + +Importantly, the state-space model (SSM) is computationally efficient, which allows it to process arbitrarily long sequences. This is achieved by maintaining a compact state that is updated at every sequence step. During training, SSMs have linear complexity in sequence length [31], further improved by parallelization. Unlike them, CNNs have a fixed receptive depth, and transformers characterize with a heavy quadratic complexity. At inference, in a streaming fashion, the SSM can be executed with constant per-step latency and constant memory footprint, whereas transformers and CNNs, at best, still grow linearly per step. As we show, in test time, our proposed model scales far beyond its trained horizon, while the SSM contributes less than $2\%$ of the total inference compute of the full model. + +Our proposed model, StateSpaceDiffuser, summarized in Fig. 2, consists of a state-space model that operates over the long sequence, and a diffusion model - conditioned on both a short window of observations and state-space model features. The latter enables the diffusion branch to generate the content of the next frame conditioned on a long context rather than the last few frames. + +To evaluate the consistency of the generated long context of StateSpaceDiffuser, we design and develop an evaluation framework that involves navigating environments to then return back to the + +initial position. We evaluate on two environments. (1) A simpler 2D maze environment (MiniGrid), in which we establish the presence or absence of memory ability by remembering the maze layout given partial observations. And (2) a 3D first-person shooter game (CSGO), which serves to show the performance of our method on a visually challenging interactive environment with many factors at play. Our quantitative and qualitative evaluation results show that StateSpaceDiffuser produces content significantly more consistent with a long history than a diffusion-only method. Evaluation in the maze environment yields $51.9\%$ PSNR improvement over the baseline on average ( $56.3\%$ improvement on the most memory challenging cases). A user study confirms that our method produces images closer to previously observed content in the CSGO dataset compared to baselines. More details are shown in Sec. 5. + +Our contributions are as follows. + +- We propose StateSpaceDiffuser, which integrates a state-space model with a diffusion model for visual world modeling. It is capable of generating consistent content in long-horizontal generation, with almost no extra computational cost. +- We develop an evaluation protocol to test the content preservation abilities of a world model and perform extensive evaluations of world models on long-horizon generation tasks. +- Our evaluation shows a significant quantitative improvement and a strong user preference over the baseline in the case of long contexts. Furthermore, our studies attribute the improvements to our model design and confirm generalization to longer contexts. + +# 2 Related Work + +# 2.1 World Models + +Generative environment models. Initially developed as imagination-based models for training model-based reinforcement learning (MBRL) agents [13, 38, 40, 70], world models have evolved into powerful generative systems that condition on actions to produce future frames [11, 41, 57, 83]. Early work by [36] demonstrates that training a recurrent latent dynamics on VAE image representations can enable agents to plan in imaginative rollouts. Extensions such as SimPle [47] and Dreamer [37] refine this approach by improving reconstruction quality and stability, culminating in DreamerV2 and DreamerV3 [39, 40] - systems that achieve human-level performance on Atari and demonstrate the ability to generalize across diverse domains. More recent efforts, such as IRIS [58], TWM [64], STORM [86], and DayDreamer [79], employ Transformer-based hybrid backbones and focus on sample efficiency, long-horizon coherence, or robotic control. However, many of these methods rely on discrete latent tokens and relatively short contexts, which limits visual fidelity in complex scene motion or when extended rollouts are required. + +World models are also central to realistic video generation conditioned on actions. Genie [8] leverages a video tokenizer and a Latent Action Model for dynamic next-frame generation, whereas GAIA-1 [46], GAIA-2 [66] tackle autonomous driving by autoregressively predicting image tokens from multi-modal inputs. Recent works highlight broader applicability and complex generative capabilities. DINO-WM [87] uses pretrained visual features for zero-shot planning, GameFactory [84] adapts game environment actions to realistic environments, while allow video generation control by periodical text instructions. Both illustrate how world models can transcend traditional RL frameworks and support open-ended content creation. + +Diffusion-based approaches. Parallel to these developments, diffusion models [73, 44, 75] have emerged as a powerful class of generative methods for high-fidelity image and video synthesis. They have been applied to text-to-video [71], space-time video generation [3], and broad world simulation tasks [6]. Within MBRL, DIAMOND [1] uses a diffusion model to generate high-quality frames for Atari, making for playable environments and enhancing agent performance. Methods like Pandora [80] and LCT [35] generate video based on periodic text instructions. Nonetheless, current diffusion-based world models, typically transformer-based, condition on only a short window of past frames to handle the quadratic computational complexity, making long-horizon dependencies difficult to maintain. This makes it challenging to maintain long-horizon dependencies. + +# 2.2 Sequence Modeling + +RNNs and Transformers. Sequential modeling has historically been dominated by recurrent neural networks (RNN) such as LSTM and GRU [45, 15, 17], which process input tokens step by step and are able to handle moderate contexts. However, RNNs often struggle with extremely long sequences due to vanishing gradients and limited memory capacity [59]. Transformers [78] addressed these issues by employing self-attention, making them effective in capturing long-range dependencies. Beyond world modeling, Transformers have become the backbone for a broad range of tasks, including language modeling [21, 7, 63] and computer vision [22, 43, 9, 28], due to their ability to handle global context. Various Transformer variants have attempted to reduce the quadratic cost of self-attention for long sequences [50, 16, 4, 85, 14]. Vision-specific models like Swin [56] or MViT [24] adopt hierarchical or local attention, yet scaling them to long video horizons remains computationally prohibitive. + +Previously, DFOt [10] addressed the ability for long future prediction. However, the long-context consistency problem has only been recently addressed by a few concurrent works. [51, 74, 81] improve context abilities by proposing strategies to sample a number of historical observations to use as conditioning. Instead, our approach involves summarizing information from the entire history automatically through state-space models. + +State-Space Models (SSMs). As an alternative, SSMs [5, 53, 62, 76, 61] can process sequences in linear time by learning continuous dynamics in a latent state. Representative structured state-space models include S4 [33, 34] and H3 [18] that generalize the recurrence in Linear Attention [49]. S4, S5 [72], and S6 [52] leverage carefully designed operators (e.g., HiPPO matrices [32]) to efficiently capture long-range dependencies. Mamba [31] introduces selective gating to improve expressiveness without sacrificing linear scalability. S4WM [20] has shown that applying SSMs as world models shows promise for maintaining coherence over hundreds of imagined steps while preserving computational tractability. + +Hybrid Architectures. As Transformers excel at local interaction with low computational cost and SSMs can capture long-horizon dependencies efficiently, hybrid designs have been proposed for vision tasks. MambaVision [42] incorporates state-space models into a transformer, and Dimba [26], DiS [25] - into a diffusion network backbone, for computationally cheaper image discriminative and generative tasks, operating on image patches.[54] modify the softmax in attention to emulate a forget gate and improve transformer context abilities. MambaVLT [55] and Samba [69] exploit state-space models for better object tracking with long-range consistency. + +# 3 Data + +We design an experimental protocol to evaluate long-term content consistency in diffusion world models, comprising of three experimental setups with a rising level of complexity, based on a controlled maze environment (MiniGrid) and a complex 3D first-person environment (CSGO). + +We create a dataset based on the partially observed MiniGrid maze environment [12]. In this setup, each maze consists of a grid where each cell can be a wall, an empty space, or a colored marker. Markers act like empty spaces, but are visually distinct. An agent navigates the maze, but at each time step, it only sees a $7 \times 7$ window centered around itself rather than the full $85 \times 85$ maze (see Fig. 6 (b) for an example). We use a modified version of MiniGrid with randomly generated mazes, allowing us to adjust the size, wall complexity, and number of color-coded markers. In each episode, the agent is tasked with visiting a sequence of 40 random markers via the shortest path. Once halfway through the episode, the agent stops following the path and retraces its steps back to the starting point. Each episode is 100 steps long (50 forward, 50 backwards). We evaluate on different context lengths by selecting subsequences around the long sequence center. Notably, the second half of each sequence depends heavily on the model's ability to recall earlier frames, making it ideal for testing long-context reasoning. + +We also design a simplified dataset called MiniGrid Simple, consisting of just 34 samples without walls and a single marker placed behind the starting position. The agent moves three steps forward and three steps back, returning to its initial position. Since the context window of our baseline is just 4 steps, this setup provides a minimal but effective test of long-term recall. We use this to compare the performance of our baseline and state-space-enhanced models in reconstructing the marker color. + +![](images/0cbb0038ce71731c6842cd2d478838cd64bbde15e28904d69dc4a4d5dce99456.jpg) +Figure 3: Architecture of our StateSpaceDiffuser model. It consists of: a state-space model for processing long context information; a diffusion model generating high-fidelity context-aware next observation, conditioned on state-space features. + +To evaluate our method in a more visually complex setting, we use CSGO [60], a dataset of human gameplay in a 3D first-person shooter. It includes 51 action types, such as 23 rotational commands, 4 movement directions, jumping, and various special actions (e.g., firing, changing weapons). To adapt the dataset for long-context testing, we create mirrored sequences: for each original sequence, we append its reversed version, ensuring that actions are also reversed. We use a corresponding one-hot encoding (e.g. turning left becomes turning right), or create a new one if a correspondence does not exist (e.g. jump, shoot). This setup forces the model to rely on information from earlier in the sequence when generating later frames. + +# 4 Methodology + +Given a sequence of environment interactions $a_1, a_2, \dots, a_{T-1}$ , the resulting observations $I_1, I_2, \dots, I_T$ (with initial frame $I_1$ ) and the current action $a_T$ , the objective of a world model $\mathcal{F}(\cdot)$ is to produce the next image $I_{T+1} = \mathcal{F}([I_1, \dots, I_T], [a_1, \dots, a_T])$ . Recently, the best-performing generative architectures for modeling $\mathcal{F}(\cdot)$ are diffusion models based on transformers or UNet. In training, transformers are computationally intensive - $O(T^2)$ . CNNs have fixed receptive fields and are ill-fitted to long-term dependencies. Therefore, these models take a short history window of observations: $I_{T+1} = \mathcal{F}([I_{T-K+1}, \dots, I_T], [a_{T-K+1}, \dots, a_T])$ . (e.g. $K = 4[1, 77]$ , $K = 16[8]$ ). With a long and growing sequence, the short history causes the loss of long-term temporal coherence. Instead, we propose to efficiently process the long sequence $(O(T))$ with a model designed for this purpose - a state-space model. Such models maintain and update a state with each sequence step, and the state serves as a summary of the sequence so far. Extracting long-context features in this way and integrating them into the diffusion pipeline yields the proposed model - StateSpaceDiffuser. + +# 4.1 StateSpaceDiffuser Architecture + +Our architecture is shown in Fig. 3. It is conceptually divided into two branches: Long-Context Branch and Generative Branch. The Long-Context Branch preserves information over long sequences, and the Generative Branch uses this context to render high-quality images. + +Long-Context Branch In contrast to transformer and CNN architectures, state-space models (SSMs) are designed specifically to efficiently process long sequences. Although SSMs are generally designed for continuous input signals, discrete SSMs maintain an internal state representation $h$ that is updated with each time step $t$ in an input sequence of one-dimensional feature vectors $f_{1},\dots,f_{T}$ , through the parameter matrices $A$ , $B$ and $C$ , which are learned in training time: + +$$ +h _ {t} = A h _ {t - 1} + B f _ {t}, \quad m _ {t} = C h _ {t} +$$ + +We denote a state-space model with $m_{1},\ldots,m_{T} = \mathcal{M}(f_{1},\ldots,f_{T})$ , with $m_{t}$ denoting the model's output. To bring it into the world model setting, we define $f_{t}$ to be a compact feature representation of $I_{t}$ and train a model that predicts future observations: $\hat{f}_{2},\ldots,\hat{f}_{T+1} = \mathcal{M}([f_{1},a_{1}],\ldots,[f_{T},a_{T}])$ . It is common in existing work to apply SSMs at the patch or image token level as common in previous work [42, 26]. Instead, we avoid conflating spatial and temporal dependencies by temporally processing full frames. Each frame is encoded into a single compact feature $f_{t}$ used as a single step of our sequence. The encoding is obtained by the continuous Cosmos tokenizer [23] with scale 16 (CI16). The resulting patch tokens (dimension 16) are flattened to form the single feature vector $f_{t}$ per image. Alongside it, we incorporate a discrete action $a_{t}$ , which indexes a learnable embedding of dimension 16. We concatenate $f_{t}$ and $a_{t}$ to create the input at each step. We adopt the Mamba architecture due to its dynamic selection mechanism and efficient parallelism, which led to superior performance compared to other SSM variants. + +One key benefit of state-space models is their computational efficiency. As only a single state is maintained in inference, memory remains constant regardless of context length. As the same update is applied linearly on a sequence, computational complexity remains $O(T)$ . When presented with a growing sequence, Mamba only updates the state from the previous step, making for a constant per-step latency. In contrast, CNNs and transformers do not maintain a state and have to reprocess the growing sequence at every step. In training, Mamba further parallelizes the sequence processing. + +Provided a long input sequence of high-resolution images and corresponding actions, the model predicts the Cosmos features corresponding to the next observation with an MSE loss. In those features, we expect relevant to the time-step information, recalled from the sequence. While context cues are to be preserved, the state is low-dimensional, and this model is not generative. Therefore, the generative branch, designed for the generation of high-quality images, is intended to render the final output. + +Generative Branch. To generate high-quality images in complex environments, we employ a diffusion model. Our choice is the DIAMOND world model [1], a UNet-based EDM diffusion model [48] designed for visual prediction in sequential environments. Therefore, our Generative Branch conditions on only four low-resolution frames and their corresponding actions, represented as 512-dimensional action embeddings. Despite this minimal context, it can produce high-quality predictions with just three denoising iterations per output frame. The architecture consists of two diffusion models: a primary model that predicts the next observation at a low resolution, and a secondary upsampler that refines these predictions to a resolution of $280 \times 150$ . As the model predicts one frame at a time, generating longer sequences is achieved through a sliding-window strategy, and each newly generated frame is appended to the input history for the next prediction. In isolation, this strategy causes a short-context limitation to this model. + +Fusion of features. To address the context limitation of the Generative Branch, we build a fusion module to integrate state-space features into it, in order to provide long-context information. To that end, we process the entire sequence with the Long-Context branch and obtain the last 4 output features $\hat{f}_t$ . We fuse those features with the corresponding action embeddings from the Generative Branch. These features are first normalized and then passed through a two-layer MLP with SiLU activation, where the input size matches the feature dimensions. Similarly, the action embeddings, perturbed with noise, are processed by an MLP with the same architecture. To form the final conditioning vector, we concatenate the outputs of the two MLPs. Empirically, we discovered that processing the memory and action conditions independently before concatenation yielded better performance than fusing them earlier in the pipeline. + +# 4.2 Training Protocol + +At each step, a batch consists of sequences of actions, reversed actions and observations. Training is performed in two stages. Firstly, the Long-Context Branch is trained on long context - length 50 or 16. The produced features decode to images with artifacts, but with important context cues. Then, freezing the Long-Context Branch, we train the Generative Branch, conditioned on the compressed long-context features, with a sequence size of 4. This branch produces the final high-quality images with the correct context. Training details are given in App. A. + +We found that this two-stage training is crucial for stability. Direct end-to-end training is unstable, as diffusion gives noisy gradients to the SSM, and the SSM gives constantly changing features to diffusion. In turn, diffusion learns to ignore the SSM features. Therefore, stable features of a pretrained SSM worked best in this architecture. Moreover, the training separation enabled to swap out in test time the Long-Context Branch with another independently trained model, without having to further fine-tune the heavier Generative Branch. + +# 5 Experiments + +# 5.1 Experimental Setup + +Baselines. We establish two baselines. The first is a pure diffusion model without state-space features: the DIAMOND model. Our second baseline is the State-Space World Model. It is the Long-Context branch of StateSpaceDiffuser and its training is equivalent to the first stage of training, as described in Sect. 4.2. At inference time, the predicted feature $\hat{f}_t$ is decoded into an image $I_t$ using the decoder from the Cosmos tokenizer. In App. B.2 we present comparisons of sequence models to solidify our choice of Mamba as our backbone. + +This model enables us to assess the memory capacity of state-space models (SSMs) in sequential visual prediction. Although its outputs tend to be blurry and contain artifacts in complex scenes, due to the absence of a variational component and limited generative expressiveness compared to modern diffusion models, the SSM exhibits a strong ability to model long sequences and retain information from earlier in the trajectory. The strengths and shortcomings observed in this baseline directly inform and motivate the design of StateSpaceDiffuser. + +Testing Protocol. Our evaluation protocol matches our mirrored action setup - we take $n$ actions and $n$ reverse actions, and expect to generate the same observations for the second half of the sequence as seen in the first. On MiniGrid, we have a fixed sizeable visual difference per step, while for CSGO continuous motion often results in small per-step changes. Therefore, in MiniGrid, we generate one frame in the future at a time, while in CSGO we sequentially generate the whole second half of the sequence. On MiniGrid we evaluate with PSNR and SSIM on varying future horizons - the further in the sequence, the longer the memory required. In CSGO we perform a user study, more aligned to the visual complexity of the environment. We motivate this difference with the known mismatch between perceived quality and fidelity metrics in continuous video [65, 67, 30] (App. D.3). Although the baseline performs well when context is not essential, our protocol exposes its inability to model long-term context, resulting in degraded quality in this scenario. + +# 5.2 Results and Analysis + +Simple MiniGrid Evaluation. In this experiment, we test the recall ability of the baseline, the State-Space World Model and StateSpaceDiffuser, on a simple toy setup, as described in Sect. 3. We train and test on the same set of 34 samples. The goal is to recall a color at the final frame from the first frame in the sequence with a length of 7 frames. Two random samples (colors) from the results are shown in Fig. 4, with the corresponding model predictions. With input size 4, the baseline processes the sequence in a sliding window fashion and, as within the 3 steps the color information is lost, it cannot reconstruct the correct color. Despite the small training set size, the baseline fails because of a lack of long-context abilities. In contrast, our State-Space World Model, based on a computationally efficient state-space model, is able to predict the correct color. Finally, it is demonstrated that our StateSpaceDiffuser is also able to recall the correct content by effectively combining both paradigms. Notably, our methods perform equivalently on a context length of 50 frames - when predicting the 51st, StateSpaceDiffuser recalls the color from 50 steps ago. + +![](images/95e7032c959f62c2e9f52fefe3b49a1cd81add0ff55a563d39ef5b588b6268da.jpg) +Figure 4: Long-Context Simple Demonstration. An agent starts moving to the right covering the color for the next T/2 frames, then the agent moves back the same amounts of steps. Diffusion baseline fails to recover the color, StateSpaceDiffuser and State-Space World Model successfully recover it through the state-space representation. Both $\mathrm{T} = 7$ and $\mathrm{T} = 50$ generates such results. + +
ModelAvg. PSNR↑Fin. PSNR↑SSIM↑
Context Length 16
DIAMOND27.1325.440.95
State-Space World Model33.4033.170.96
StateSpaceDiffuser (Ours, w/o state)23.6820.950.92
StateSpaceDiffuser (Ours)41.0140.550.98
Context Length 50
DIAMOND26.1325.150.95
State-Space World Model32.6432.440.96
StateSpaceDiffuser (Ours)39.6839.320.98
+ +Table 1: MiniGrid Quantitative Evaluation of Long-Context Awareness. Our StateSpaceDiffuser outperforms the baselines. + +![](images/3e21316ab40a239393d658584cb5d9847f813eb57bfb6c47539015a711c9da7a.jpg) +Figure 5: CSGO User study results. + +Forward-Backward Evaluation on MiniGrid. We compare the long context abilities of our diffusion (DIAMOND) and state-space (State-Space World Model) baselines in our MiniGrid test set. We evaluate our models trained on context length 50 on context lengths 16 and 50 (demonstrating generalizability). We follow the protocol outlined in Sect. 5.1. To evaluate, we compute the Peak Signal-to-Noise Ratio (PSNR) for each predicted frame in the reverse trajectory, reporting both the mean score and the PSNR at the final time step, which requires the longest-term memory. As shown in the Tab. 1, our model significantly outperforms both baselines, particularly at the end of the sequence, where successful recall of the first frame is critical. This highlights the model's ability to retain and reintiate long-term visual context. In App. B.4, B.5, we show the stability and robustness of these results, in App. B.1 - performance gain analysis over computational cost. Compared to the State-Space World Model, our method achieves higher fidelity output, benefiting from the superior generative capacity of the Generative Branch (examples - in App. C.1, C.2). Fig. 6 (b) presents example rollouts generated by our model and the diffusion-only baseline. In MiniGrid, predictions are made one step at a time using the ground truth sequence. As a result, most content is carried over from the previous frame, with only the newly revealed area requiring inference. Our method excels at filling in these newly revealed regions, even when the relevant context originates far back in the sequence. In contrast, the diffusion baseline struggles to recover such long-range dependencies. + +Recall Across a Context Length. In this experiment we study the accuracy of our models over the varying context length of the forward-backward evaluation on MiniGrid. When predicting future observations, the last frame's content depends on the first frame's content, and the further back we go in the sequence the smaller the context length required for a good reconstruction. This is a direct consequence of the mirror style of the observations in our setup. In Fig. 7 we show the PSNR at each predicted time step. The first few predicted frames are easily predicted by all models as the solution falls within the short input window. However, performance for the diffusion baseline quickly falls as no form of information is preserved from the long context, while a state-space model is able to harvest this information. Our StateSpaceDiffuser model gets the best of both worlds - long-context awareness and high-fidelity predicted images, and performs the best. + +![](images/8cb4a3cd32d207806d663ad8ed1f2a09177cd3f66b237dd80e6ae873937d7b02.jpg) +(a) CSGO + +![](images/1e448e4f7f17788f2c96f06aafa8f329245a25971ad61f58a06534760062ba03.jpg) +(b) MiniGrid + +![](images/31c72f644ebd1de9ca46f4e3e2ce66615d997a8bbddb80683250f87e0c6dd9e2.jpg) +Figure 6: Qualitative comparison of StateSpaceDiffuser and the baseline (DIAMOND). Top row: input frames and actions. Bottom rows: generated frames under reversed actions. In CSGO - last 8 frames autoregressive prediction, in MiniGrid - next frame prediction. +(a) Context Length 17 + +![](images/4b81e33725d387b24ba81a6bd758514f17f9466f1861508a5cf09080d3b3e1de.jpg) +(b) Context Length 51 + +![](images/d7bfc75f1af4ddb80f545d4d3a206fc004ae904b30cece5cac631deb62634b23.jpg) +Figure 7: Recall performance on MiniGrid for two context lengths. +Figure 8: CSGO Ablations. We show that without the state-space features from the Long Context Branch, StateSpaceDiffuser loses its context preservation ability. We also show that while State-Space World Model demonstrates long-context memory, the produced images are of subpar visual quality. + +Forward-Backward Imagination Evaluation on CSGO. Similarly to MiniGrid, we evaluated the recall abilities of our model on the CSGO dataset, a visually complex environment in a 3D world. In CSGO most actions are gradually executed over a sequence, and there is a compounding effect on content change (e.g. jump unfolds over many frames). For a high impact evaluation we decide to give only the first half of the sequence and continuously produce the second half (reverse) by feeding generated frames. As actions are motions at varying levels, the final frame may contain the correct content memorized but with low PSNR, as the camera position and scene geometry might be slightly shifted. Therefore, instead of fidelity metrics we perform a user study where the 12 participants judge whether images produced by StateSpaceDiffuser are closer in content to the ground truth compared to the diffusion baseline (details - in App. D.4). Our rating is in the range $[-1, 1]$ , with 0 being borderline, -1 - preference toward the baseline, 1 - preference for StateSpaceDiffuser. The results shown in Fig. 5 demonstrate a clear preference of the users for StateSpaceDiffuser over the baseline + +
ModelAvg. PSNRFin. PSNRSSIM
Context Length 100
DIAMOND26.3926.240.95
State-Space World Model31.6530.890.96
StateSpaceDiffuser (Ours)37.9935.870.98
+ +
ModelAvg. PSNRFin. PSNRSSIM
Context Length 150
DIAMOND24.3524.200.94
State-Space World Model27.9326.980.94
StateSpaceDiffuser (Ours)30.7528.930.96
+ +Table 2: Generalization to Longer Context. Our model, trained on context length 50, generalizes to longer sequences (context 100 and 150). + +for both prediction in the 15th frame (rating 0.20) and 17th (last) frame (rating 0.24). Fig. 6 (a) shows a sample of CSGO imagination in different time steps, demonstrating that while the baseline fails to recall the correct content, the StateSpaceDiffuser correctly produces the details. (More in App. D.1) + +State Features Ablation. We study the utility of the state-space features provided to the Generative Branch in our StateSpaceDiffuser model. We take a trained model and perform a MiniGrid evaluation by replacing the output features of the Long-Context Branch with zeros before passing them to the Generative Branch. In Tab. 1 we show that this causes the performance to quickly drop even below baseline performance, clearly demonstrating that the features are highly utilized. In Fig. 8 we demonstrate the same effect on CSGO. Without state features, the model hallucinates; without diffusion, the state-space model remembers but produces poor visual quality. (More in App. B.7), B.8) + +Generalization to Longer Context. In this experiment we show that StateSpaceDiffuser operates on much longer contexts without finetuning. We evaluate our model trained on context length 50 on lengths 100 and 150 using a new MiniGrid test set with longer sequences. Tab. 2 shows that StateSpaceDiffuser successfully generalizes to longer context, keeping a significant gain over the baselines. Analogously, in App. B.3, we show generalization from context length 16 to length 50. + +# 5.3 Strengths, Limitations and Scalability + +Apart from the already established generalization across context length, via extra experiments, we find that StateSpaceDiffuser is able to generalize across visual complexity (App. B.6) and can recover from strong motion artifacts (App. D.2). Our model can recover from input noise in future steps, but is clearly affected by it on the current steps (App. B.5). Our lightweight StateSpaceDiffuser was trained under a fixed compute budget. The lightweight diffusion decoder (no large pretrained backbone) can yield visual artifacts in long rollouts. Replacing the decoder with a better, larger one, can improve visual sharpness without changing the method. Our lightweight single-layer Long-Context Branch compresses the context into a low-dimensional state (256), which can cause loss of detail in extended rollouts, especially in complex environments (App. D.2). Scaling the SSM (state dimension/heads/parameters/layers) is expected to reduce high-frequency decay over time. The separation in training enables separately scaling each branch before combining them. + +# 6 Conclusion + +We introduced StateSpaceDiffuser, a hybrid model that combines state-space representations with diffusion to enable long-horizon visual world modeling. By decoupling global context modeling (via a state-space backbone) from high-fidelity synthesis (via diffusion), our model retains global context over many steps at essentially no additional computational cost. The resulting representation alleviates the drift and inconsistency that plague conventional diffusion-only systems in long sequences. + +Experiments on MiniGrid and CSGO validate our method's consistency and fidelity across long sequences. In the forward-backward protocol with horizon 50, StateSpaceDiffuser improves average PSNR by $51.9\%$ over the diffusion baseline and achieves a final-frame PSNR of 39.32 versus 25.14 for DIAMOND on a long context length of 50 frames. Human raters also favor our generations for long-context consistency (Fig. 5). + +Our results establish state-space diffusion as a scalable and consistent solution for long-context visual generation. 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Dino-wm: World models on pre-trained visual features enable zero-shot planning, 2025. + +# Appendix + +# A Training Details 16 + +# B StateSpaceDiffuser Properties 17 + +B.1 Performance/Cost Tradeoff 17 +B.2 Long-Context Architecture Comparison 17 +B.3 Generalization Across Context Length 18 +B.4 Stability Across Seeds 18 +B.5 Robustness to Noise 18 +B.6 Generalization Across Visual Complexity 19 +B.7 State Features Ablation 20 +B.8 Comparison with State-Space World Model 20 + +# C MiniGrid Evaluations 21 + +C.1 MiniGrid Qualitative Evaluation 21 +C.2 Imagination Qualitative Results 21 + +# D CSGO Evaluations 24 + +D.1 CSGO Qualitative Evaluation 24 +D.2 Strengths and Limitations of StateSpaceDiffuser 25 +D.3 Quantitative vs. Perceptual Consistency 26 +D.4 User Study Details 27 + +# A Training Details + +For MiniGrid, our DIAMOND baseline takes 30x30 images and upscales them to 144x144, for CSGO - 30x56, upscaled to 150x280. The Long-Context Branch contains the Cosmos tokenizer, which decodes in powers of two. Therefore, The Long Context Branch takes 32x32 for MiniGrid and 144x272 for CSGO. In training, the downscaling is done from the high-resolution image with bicubic interpolation (as in DIAMOND). + +In our Long-Context Branch, the Cosmos Tokenizer tokens are flattened to produce features of size 1296 (MiniGrid) or 2448 (CSGO). Action dimensions in the Long Context Branch (and the State Space World Model) is 16 - they are concatenated to the visual features. We use a single Mamba layer with state size 256, the input is expanded 4 times with an MLP inside the Mamba layer, the internal convolution dimensions are 4. + +We use 8 A100 GPUs for all models except for the MiniGrid State Space World Model, which was trained on 4 A100 GPUs for MiniGrid models. All models use the Adam optimizer. The State Space World Model is trained with a learning rate of $5e^{-5}$ and batch size 136 (MiniGrid) or batch size 272 (CSGO). Both the MiniGrid and CSGO models are trained for 70k iterations on sequence size 16. StateSpaceDiffuser includes 600M parameters, and is trained with a learning rate $1e^{-4}$ , weight decay $1e^{-2}$ , grad norm clip 10 and batch size 64. The MiniGrid model is trained for 77k iterations, CSGO - 220k iterations. For upscaling in MiniGrid - we train the sampler for 27k iterations after training the denoiser for predicting low-resolution next frame. In CSGO we achieved our best results with the upsampler, part of the weights originally provided by DIAMOND. + +For all models, we loaded the weights of a pre-trained State-Space World model into StateSpaceDiffuser. For MiniGrid, while models trained on sequence length 16 performed well, in inference, we + +
BackboneAvg. PSNR↑Fin. PSNR↑SSIM↑
Context Length 16
DIAMOND27.1325.440.95
Mamba [31]34.4137.810.97
Context Length 50
DIAMOND26.1325.150.95
Mamba [31]27.6229.750.94
+ +Table 3: Performance Gains, Normalized By Computational Cost. Strong gains are still observed, suggesting that the gains are much higher than the cost. + +
BackboneAvg. PSNR↑Fin. PSNR↑SSIM↑
Context Length 50
LSTM26.8026.910.93
GRU27.4026.680.93
Mamba [31]32.6432.440.96
+ +Table 4: Evaluating Sequence Models. The superior performance of Mamba leads to our choice of an SSM in our Long-Context Branch. + +
BackboneAvg. PSNR↑Fin. PSNR↑SSIM↑
Context Length 16
S4 [33]24.2924.310.91
Mamba [31]27.0927.870.94
+ +Table 5: Evaluating State-Space Backbones. The superior performance of Mamba leads to our choice to use this method as our Long-Context backbone. + +achieved our best results for both context length 16 and 50 by using the model trained on context length 50. For CSGO, we used the State-Space World Model weights for context length 16 and also evaluated on context length 50. + +When we evaluate DIAMOND and StateSpaceDiffuser, we use 5 denoising steps to denoise the next observation and 10 to upscale it. + +# B StateSpaceDiffuser Properties + +# B.1 Performance/Cost Tradeoff + +Our Long-Context Branch adds very little computation to the diffusion model, and in exchange it offers significant improvements in consistent generation. In inference with batch size of 1, we measure DIAMOND to require 909.515 GFLOPS (4 input frames). The Long Context Branch with context length 16, requires only 5.5 GFLOPS for context length 16 and 16.741 GFLOPS for context length 50. That is only $0.6\%$ of all inference computations of the full model, for sequence size 16, and $1.8\%$ for sequence size 50. + +We show that the gains from the Long-Context Branch surpass its computational cost by a large margin. To account for the cost in our StateSpaceDiffuser scores, we normalize them by multiplying by $1 - 0.006$ for sequence size 16 and $1 - 0.018$ for sequence size 50. The normalized scores are reported on Tab. 3. They confirm that there is still a significant gain in performance despite the normalization. + +# B.2 Long-Context Architecture Comparison + +We compare the choice of the SSM in State-Space World Model with other popular sequence processing models - LSTM and GRU. We train on context size 50 with our MiniGrid setup. We add a linear layer on the input and output (dim 256, with ReLU activation) of the models. Tab. 4 shows that Mamba outperforms the other sequence models in long-range temporal dependencies, while remaining computationally efficient. This confirms the conclusion in the original paper [31]. + +In addition, we consider the choice of an SSM model itself. We consider the S4WM model the closest in spirit model in literature. However, as their model code is not available, comparing an S4 backbone to our choice of Mamba serves as the closest we can get to a comparison. We train State-Space + +
ModelAvg. PSNR↑Fin. PSNR↑SSIM↑
Context Length 16
DIAMOND27.1325.440.95
State-Space World Model29.7131.340.96
StateSpaceDiffuser (Ours)34.6238.040.98
Context Length 50
DIAMOND26.1325.150.95
State-Space World Model27.2524.490.93
StateSpaceDiffuser (Ours)28.1230.300.96
+ +Table 6: MiniGrid Quantitative Evaluation of Long Context Awareness, Trained on Context Length 16. Our models generalize their performance from a smaller to a longer sequence. + +
Noise27282930313233343536373839
SSM15.2216.6215.3615.1814.8516.5523.4426.6028.4229.3529.7529.7530.00
Full15.5016.9015.4915.3014.9416.6523.2826.5128.3529.3329.7529.9529.85
Noise404142434445464748495051
SSM29.9330.0629.9930.1230.1430.4530.1130.1830.1930.0830.1930.19
Full30.0929.8830.1130.0830.4130.1230.1330.1730.1230.2030.1330.15
+ +Table 7: Noise robustness across steps. Our method is affected on the noisy frames but quickly recovers in further steps. SSM denotes the noise is added only to the Long-Context Branch; Full denotes the noise is added on the Generative Branch as well. + +World Model with S4 and Mamba on MiniGrid using a context length of 16 and comparing their performance. We keep the context small to account for the larger amount of training iterations usually needed by SSMs. As seen on Tab. 5, Mamba outperforms S4, achieving a 9.1 PSNR improvement on average. As Mamba introduces a dependency on the input of the state update, this clearly benefits its ability to perform in long context. + +# B.3 Generalization Across Context Length + +In Tab. 6, we show the evaluation results of models trained in context size 16, on both context length 16 and 50. A reasonably good performance demonstrates that the models do not overfit on a particular sequence size and still perform well in a context length longer than it has been trained with. While the State-Space World Model exhibits uncertainty in its predictions for a longer context without training (blurriness, color deviations), its features prove useful for StateSpaceDiffuser, with the Generative Branch producing a higher-fidelity result. StateSpaceDiffuser noticeably outperforms the baseline on context length 50. + +# B.4 Stability Across Seeds + +To show the stability of our results, we perform evaluation of our MiniGrid model, on context length 16, under 4 different seeds. We obtain $41.00 \pm 0.008$ Avg. PSNR, $40.52 \pm 0.019$ Fin. PSNR, $0.98 \pm 0.0004$ SSIM. We also perform evaluation over 4 seeds of a larger-scale, more expensive evaluation - our new 100 context length experiment from Sect. B.3. This results in: $37.99 \pm 0.002$ Avg. PSNR, $35.88 \pm 0.029$ Fin. PSNR, $0.98 \pm 0.000004$ SSIM. In both cases, the obtained metrics are extremely stable and consistent between seeds, with a low standard deviation. + +# B.5 Robustness to Noise + +We test the robustness of our method by adding noise in the middle section of the rollouts that serve as context. In specific, we consider context length 50 in MiniGrid and add Gaussian noise (std 2.5) to the 11 frames in the middle. We consider two cases: 1) adding noise to the SSM input only; 2) adding noise both to the SSM input and the diffusion model input. Results are shown on Tab. 7 at different steps of prediction after the middle frame. We observe that for the specific frames with added noise, the performance decreases. On those frames content can disappear and context is not correctly recalled. However, in both cases, within 4 steps after the noisy frames (after frame 34 - 5 + +
ModelAvg. PSNR↑Fin. PSNR↑SSIM↑
Low Complexity
Baseline (low complexity)26.0925.600.95
Ours (low complexity)36.7235.780.97
Middle Complexity
Baseline (middle complexity)27.2726.700.94
StateSpaceDiffuser (Ours))39.6839.320.98
High Complexity
Baseline (high complexity)23.0922.870.93
StateSpaceDiffuser (Ours)31.6730.870.97
+ +Table 8: Generalization Across Visual Complexity. Our approach is shown to consistently outperform the baseline on multiple levels of visual complexity without finetuning. + +
Input: 9 Frames Prediction
Ground Truth...
StateSpaceDiffuser
State-Space World Model
StateSpaceDiffuser (no state)
Ground Truth...
StateSpaceDiffuser
State-Space World Model
StateSpaceDiffuser (no state)
Ground Truth...
StateSpaceDiffuser
State-Space世界 Model
StateSpaceDiffuser (no state)
+ +Figure 9: CSGO Ablations. We show that without the state-space features from the Long Context Branch, StateSpaceDiffuser loses its context preservation ability. We also show that while State-Space World Model demonstrates long-context memory, the produced images are of subpar visual quality. + +noisy frames + window size 4), the memory and content recovers and is correctly predicted, with stable performance until the last frame. The lower scores suggest some loss in performance. However, the fact that memory recovers after the noise suggests a certain level of robustness to noise. + +# B.6 Generalization Across Visual Complexity + +In this work, we have evaluated on 3 different environment setups with increasing level of complexity - the very constrained Simple MiniGrid, free navigation in a maze (MiniGrid), and a 3D first-person environment (CSGO). To further compare the performance of StateSpaceDiffuser across environments with different visual complexities, we generate 2 more variants of our MiniGrid dataset based on visual complexity. We define complexity as number of markers and complexity of the maze walls (values in the range [1, 5]). We generate a dataset with low complexity (200 markers, difficulty 3), and with high complexity (450 markers, difficulty 5). Our original MiniGrid dataset lies in between in terms of complexity (360 markers, difficulty 4). We take our StateSpaceDiffuser pretrained model on context length 50 (middle complexity) and evaluate it on the new datasets with varying difficulties (without any finetuning). + +![](images/3a6e65c6a64aaa8128ab21f60b13c4b8e2eeb7d5e909a4a99c61bb5c13ca200b.jpg) +Figure 10: Qualitative Results with Context Length 50 on MiniGrid. StateSpaceDiffuser demonstrates long-context preservation compared to DIAMOND and better visual fidelity compared to State-Space World Model. + +![](images/97dd0d0523cfce8caa1a127439187fa8d19a642def170704f3e0ea61c87ab081.jpg) +Figure 11: Full Sequence Sample with Context Length 50 on MiniGrid. Demonstrates a full 50-frame context and the per-frame predictions. In red are shown the frames from the second half - returning to the initial position. + +The results in Tab. 8 suggest that our model is better able to generalize to lower complexity rather than higher complexity. However, in all cases, the performance remains higher than the baseline (DIAMOND). + +# B.7 State Features Ablation + +To expand on the state ablation experiment results from Sect. 5, in Fig. 9 are shown more examples comparing StateSpaceDiffuser with and without a state. The results reconfirm that StateSpaceDiffuser loses its special ability to memorize long context after we zero out the states that were produced before passing them to the Generative Branch. Without the state, the model can no longer recall long-context information and hallucinates new locations. + +# B.8 Comparison with State-Space World Model + +In Fig. 9 are shown extra examples of the State-Space World Model (used as a Long Context Branch in SpaceDiffuser) versus the result of StateSpaceDiffuser. It is clearly seen that the State Space World Model produces more artifacts and more visually unappealing images. However, its important property to reconstruct previously observed content leads to its features being crucial for SpaceDiffuser to exploit as part of its Long Context Branch. In the shown examples, the images are predicted in an autoregressive manner by re-encoding SpaceDiffuser's output from the previous steps. We have observed that this approach is much more stable than feeding the prediction + +of State-Space World Model to itself. The latter produces very blurry images with no discernible detail. Note that the State-Space World Model has not specifically been trained with an autoregressive approach. This confirms that in StateSpaceDiffuser, the diffusion component stabilizes the state-space component, which in turn improves the long-context abilities of the diffusion component. + +Compared to MiniGrid, here we see much more pronounced artifacts on the results of State-Space World Models, which are then cleaned up by the Generative Branch in StateSpaceDiffuser. This confirms the potential of StateSpaceDiffuser particularly for complex environments. + +# C MiniGrid Evaluations + +In this section, we expand our discussion on our model's performance on the simpler MiniGrid dataset and offer additional qualitative and quantitative results. + +# C.1 MiniGrid Qualitative Evaluation + +In Fig. 12 and Fig.10 we show visual samples with the forward-backward evaluation used in our quantitative results. We assess the StateSpaceDiffuser model, trained on sequence length 51 and presented in the main paper and its corresponding State-Space World Model. + +Note that in our standard evaluation on MiniGrid, at each time step we give the ground truth sequence up to that step and predict the next observation. In a single time step the agent takes a fixed motion in one of four directions and reveals exactly 1 row or column of the environment depending on the direction. As most of the content of the next frame is present in the previous frame, the unknown content is only contained within the new revealed area. In the first half of the sequence, the agent explores by navigating the maze. In Fig. 11 we show one such traversal and the predicted next frame for each time step from StateSpaceDiffuser. It is observed that the new revealed area is predicted far from the ground truth in the first half of the sequence. This is expected as this area has not yet been observed. However, in the second half, the return along the path, the ability to recall the content from the long given sequence helps to predict the correct content of the repeatedly revealed areas. Therefore, in all our evaluations we have made the decision to only consider the prediction quality of the second half of the second half of the sequence. + +In context length 16 - Fig. 12, we clearly observe poor performance of the diffusion-only baseline, DIAMOND, as this model has no method to take into account long context. Looking closely at the State-Space World Model's output, we observe shifts in color and inconfidence in the content of the new revealed areas. The effect is subtle and varies in the sequence, but it is noticeable. This effect causes a drop in fidelity metrics and is a direct consequence of the non-variational approach of predicting the next frame from State-Space World Model. In contrast, StateSpaceDiffuser is free of such artifacts and predicts closer to the ground-truth images. Still, as it is conditioned on the state-space features, it can be affected by significant uncertainty in the content of particular grid cells. + +The observations are even more pronounced for context length 51 - Fig. 10. Later in the sequence, the State-Space World model tends to increase its shifts from the ground truth (dar squares appear darker, grey squares tend to fade). While on a simpler dataset like MiniGrid such artifacts are less noticeable, for a more complex setup like CSGO this becomes more apparent. + +# C.2 Imagination Qualitative Results + +Additionally, instead of giving the ground truth sequence at every step, we also attempt to give only the first half and feed already predicted frames for time steps in the second half of the sequence (imagination). In this way the ground-truth frames in the second half are never seen by the model (same as the setup we have in the CSGO dataset). This is a more challenging setup, in which the content cannot be copied from the previous ground truth frame, and any errors in the current frame prediction propagate into the next frame. + +We show qualitative results in imagination in Fig. 13. It is observed that in this more complex setup the diffusion baseline quickly drifts away from the context, and the entire image no longer corresponds to past context. In contrast, because of the incorporation of state-space features, the StateSpaceDiffuser is noticeably better at preserving the content of previous steps. + +![](images/55ad05d8b9f95825088f3395bc22a549d58ba19a7ebf2bf7a17cd4feded1f3fc.jpg) +Figure 12: Qualitative Results on MiniGrid. Compared to DIAMOND, State-Space World Model is able to recall past content better but lacks in certainty and visual fidelity. However, SpaceDiffuser is able to both to consider long context and to produce a high quality image. + +![](images/e0f0c4f0ac286e3fe33512dcdc9e64d8469d874ee75eeb348db42657aec665f2.jpg) +Figure 13: Imagination Qualitative Results on MiniGrid. Frames are consecutively generated given previously generated frames. StateSpaceDiffuser shows clear superiority on context preservation. + +# D CSGO Evaluations + +# D.1 CSGO Qualitative Evaluation + +We show additional qualitative examples in Fig. 14. We show the last 8 frames (predicted without the use of the last 8 ground-truth frames), given 9 frames of the mirrored version of the sequence. The last predicted frame uses context length 16. In the results, it is observed that DIAMOND - our diffusion baseline - is unable to recall a previous context that was left beyond the 4 input frames it accepts as input per step. In contrast, StateSpaceDiffuser is able to predict the correct content through its Long Context Branch. + +In addition, we evaluated our model (trained on context length 16) in a longer context of 50 frames. In Fig. 15 we show examples, where the last 8 frames are predicted in an autoregressive manner, while the first 43 frames are given as ground truth (their corresponding predicted frames are also depicted). Therefore, the context for the last frame is of 43 ground-truth images and 7 generated images. We show the last 26 frames in the sequence, as the first 25 are a mirrored version of them. In this challenging setting, we observe more artifacts and significantly less memory capabilities. However, the model is capable of recalling some visual cues on this context length that were visible + +![](images/93e631217840c980f3598be147bc7f2067e22d4d0c23d7c91a8ea7ed0296b8df.jpg) +Figure 14: Our StateSpaceDiffuser model is able to recover from insufficient information in the short context, while the diffusion baseline - DIAMOND, has no mechanism to do so. + +![](images/c068f594a14f2bbfe7a802602d9d94778a6f62f4a085634825140a089b9d5949.jpg) +Figure 15: StateSpaceDiffuser on Context Length 50 on CSGO. In red are marked the imagined frames and the corresponding ground truths. The model is able to reconstruct frames seen at the start of the sequence. + +![](images/f995a428ad141864335bfe20f5749bbd27a938a845af14ef2e1ebe8378594769.jpg) +Figure 16: Recovery from a strong action. Strong actions cause significant artifacts in DIAMOND. While this behavior is inherited by StateSpaceDiffuser, in contrast, it quickly recovers from the artifacts using the state-space features. + +only at the start of the sequence, as visible on the examples - the red roof, looking through the scope, revealing a door. + +# D.2 Strengths and Limitations of StateSpaceDiffuser + +DIAMOND has a limitation, which causes significant artifacts when the action is strong and results in a larger visual change (e.g., large turn). Our Generative Branch is based on DIAMOND and hence has a similar limitation. However, while DIAMOND's artifacts tend to affect the entire predicted sequence, the StateSpaceDiffuser has the property to recover in subsequent steps by making use of the state-space representation to recover the content. This is visible throughout Fig. 14, but also particularly in the examples shown in Fig. 16. + +![](images/a338aa05faced15efb111beb3a4ba26826707b0dd4ec38a449878b5812e9abdc.jpg) +Figure 17: Limitations. Our method sometimes only reconstructs coarse features and leaves details out. Even in these cases, the content appears closer to the ground truth than the diffusion baseline. + +![](images/a6ae79b83b6a2441e7779dafc15dd61c36d2cb70508bf804b72ac4888c6c3b5b.jpg) +Figure 18: Measuring Content Preservation. On the shown sample, visually our method preserves the content better than the baseline. However, the PSNR fidelity metrics do not match this conclusion. We conclude that fidelity metrics do not match well the goal of content preservation. + +The effect of the strong action is one of our main motivations to use autoregressive-style prediction for the CSGO models rather than a single frame prediction given a full context (as done with MiniGrid). While MiniGrid has actions with constant measurable visual effect, CSGO's actions vary in strength. We observe that many sequences consist of almost no motion, divided by a few strong actions. In the single-action prediction task, weak motion would not require much new content to be generated. This often results in a copy of the previous frame, and not much improvement is expected compared to the baseline. Conversely, with strong actions, the baseline and StateSpaceDiffuser exhibit inconfidence in the next frame. This affects the short context window, filling it with artifacts. In comparison to the baseline, the StateSpaceDiffuser is able to recover in the following frames. + +While our method's content is consistently closer to the ground truth than the diffusion baseline, it sometimes is only able to preserve coarse features (colors, general shape of the scene) and is less effective with finer details. We show this in Fig. 17. It is visible that sometimes our method misses showing an object (crate in the first example) or just preserves the general shape of a scene (second example). Even with those limitations, the model often outperforms the baseline. As a higher level of detail requires larger memory capacity, we believe that with more computational resources, scaling the Long-Context Branch can aid these issues. + +# D.3 Quantitative vs. Perceptual Consistency + +As previously discussed, in contrast to MiniGrid, the complexity of the CSGO environment makes fidelity metrics such as PSNR unsuitable for evaluating content consistency. CSGO is characterized by actions with variable motion unfolding over multiple frames. Long rollouts accumulate small motion mismatches into camera-view drift, so later frames need not match ground-truth pixels. Content often remains perceptually similar while viewpoint and details differ, and PSNR under-reports this similarity. We demonstrate this by computing the fidelity metrics of an example. For the baseline, we obtain 20.77 Avg. PSNR and 16.17 Fin. PSNR. For StateSpaceDiffuser - 19.36 Avg. PSNR and 16.11 Fin. PSNR. Given the metrics, in this example, our model does not differ significantly from the baseline in terms of quality of the last frame and is somewhat worse on average. However, when looking at the visual results in Fig. 18, we clearly see more similarity to the ground truth in the content of our method than in the baseline. The viewpoint, details, object proportions, and parts of + +![](images/007d2c7ff2b15d19c2becb316f99a8ae7e07218da7a3042e8499e142adc47829.jpg) +Figure 19: User Study Sample Question. + +the scene do not match exactly, causing the metric to be worse. Due to this inadequacy, and following common practice, we opt for a user study to evaluate the quality of the CSGO results. + +# D.4 User Study Details + +The user study is performed by first selecting 40 examples where long-context memory would be important - for example, content at the beginning of a sequence is covered up by a wall. The examples are picked from the ground truth images in the test set; the predicted images are not observed when selecting, in order not to bias the process. After generating the predictions from the diffusion baseline and StateSpaceDiffuser, we build two triples of images per prediction - for prediction horizons 15 and 17. This results in 80 total triplets. For each of them, we ask 12 participants to determine if the baseline or our model is better. The participants are coleagues and students from our institute, external to this project. In order to avoid bias, we shuffle the order of the baseline and StateSpaceDiffuser results for each sample, marking the first blue and the second red. The user is asked to compare the match of blue and red images to the ground truth image. An example question is shown in Fig. 19. + +The text that the users saw is visible below: + +Thank you for taking part in our user study! For each question, you will see a row of 3 frames: + +- First frame (grey border) - our ground truth +- Second frame (blue border) - a frame that attempts to resemble the first frame +- Third frame (red border) - a frame that attempts to resemble the first frame + +Your task is to judge if the red or blue frame resembles more closely the grey frame. Rate each pair by selecting: Blue - Blue frame resembles the grey frame better; Red - Red frame resembles the grey frame better. Please base your decision on both the content and not the visual quality of the images. + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: It does + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: Discussion on limitations can be found in supplementary material. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. + +# 3. Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +Answer: [NA] + +Justification: Not applicable + +Guidelines: + +- The answer NA means that the paper does not include theoretical results. +- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced. +- All assumptions should be clearly stated or referenced in the statement of any theorems. +- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. +- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. +- Theorems and Lemmas that the proof relies upon should be properly referenced. + +# 4. Experimental result reproducibility + +Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? + +Answer: [Yes] + +Justification: We share details in methodology and supplementary material, we will share the code + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. +- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. +- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. +- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example + +(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm. +(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully. +(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset). +(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. + +# 5. Open access to data and code + +Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? + +Answer: [Yes] + +Justification: Data generation and model code will be provided + +# Guidelines: + +- The answer NA means that paper does not include experiments requiring code. +- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details. +- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark). +- The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details. +- The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc. +- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why. +- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable). +- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. + +# 6. Experimental setting/details + +Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results? + +Answer: [Yes] + +Justification: We include those details mostly in supplementary material + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. +- The full details can be provided either with the code, in appendix, or as supplemental material. + +# 7. Experiment statistical significance + +Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? + +Answer: [No] + +Justification: Experiments are too heavy to be executed multiple times, we have computational limitations + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. +- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). +- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) +- The assumptions made should be given (e.g., Normally distributed errors). +- It should be clear whether the error bar is the standard deviation or the standard error of the mean. + +- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a $96\%$ CI, if the hypothesis of Normality of errors is not verified. +- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). +- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. + +# 8. Experiments compute resources + +Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? + +Answer: [Yes] + +Justification: We mention thegpu models and number of gpus. The details about workers etc will also be visible in code. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. +- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. +- The paper should disclose whether the full research project required more computer than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). + +# 9. Code of ethics + +Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines? + +Answer: [Yes] + +Justification: Yes + +Guidelines: + +- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics. +- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. +- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). + +# 10. Broader impacts + +Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? + +Answer: [NA] + +Justification: No direct societal impact + +Guidelines: + +- The answer NA means that there is no societal impact of the work performed. +- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. +- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. + +- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. +- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. +- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). + +# 11. Safeguards + +Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? + +Answer: [NA] + +Justification: Not high risk + +Guidelines: + +- The answer NA means that the paper poses no such risks. +- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. +- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. +- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort. + +# 12. Licenses for existing assets + +Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? + +Answer: [Yes] + +Justification: We cite every dataset and model paper used + +Guidelines: + +- The answer NA means that the paper does not use existing assets. +- The authors should cite the original paper that produced the code package or dataset. +- The authors should state which version of the asset is used and, if possible, include a URL. +- The name of the license (e.g., CC-BY 4.0) should be included for each asset. +- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. +- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. 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b/statespacepromptingviagatheringandspreadingspatiotemporalinformationforvideounderstanding/full.md new file mode 100644 index 0000000000000000000000000000000000000000..c0f4c7a2eb4e04234a74c9d99475c15878697f22 --- /dev/null +++ b/statespacepromptingviagatheringandspreadingspatiotemporalinformationforvideounderstanding/full.md @@ -0,0 +1,728 @@ +# State Space Prompting via Gathering and Spreading Spatio-Temporal Information for Video Understanding + +Jiahuan Zhou $^{1}$ , Kai Zhu $^{1}$ , Zhenyu Cui $^{1}$ , Zichen Liu $^{1}$ , Xu Zou $^{2*}$ , Gang Hua $^{3}$ + +$^{1}$ Wangxuan Institute of Computer Technology, Peking University, Beijing 100871, China + +$^{2}$ Huazhong University of Science and Technology, Wuhan 430074, China + +$^{3}$ Amazon.com, Inc, Bellevue, WA 98004, USA + +jiahuanzhou@oku.edu.cn,zhukai2022@ruc.edu.cn + +{cuizhenyu,lzc20180720}@stu.pku.edu.cn,zx@zoux.me,ganghua@gmail.com + +# Abstract + +Recently, pre-trained state space models have shown great potential for video classification, which sequentially compresses visual tokens in videos with linear complexity, thereby improving the processing efficiency of video data while maintaining high performance. To apply powerful pre-trained models to downstream tasks, prompt learning is proposed to achieve efficient downstream task adaptation with only a small number of fine-tuned parameters. However, the sequentially compressed visual prompt tokens fail to capture the spatial and temporal contextual information in the video, thus limiting the effective propagation of spatial information within a video frame and temporal information between frames in the state compression model and the extraction of discriminative information. To tackle the above issue, we proposed a State Space Prompting (SSP) method for video understanding, which combines intra-frame and inter-frame prompts to aggregate and propagate key spatiotemporal information in the video. Specifically, an Intra-Frame Gathering (IFG) module is designed to aggregate spatial key information within each frame. Besides, an Inter-Frame Spreading (IFS) module is designed to spread discriminative spatio-temporal information across different frames. By adaptively balancing and compressing key spatio-temporal information within and between frames, our SSP effectively propagates discriminative information in videos in a complementary manner. Extensive experiments on four video benchmark datasets verify that our SSP significantly outperforms existing SOTA methods by $2.76\%$ on average while reducing the overhead of fine-tuning parameters. The code is available at https://github.com/zhoujiahuan1991/NeurIPS2025-SSP. + +# 1 Introduction + +In recent years, the Vision Transformer (ViT) has demonstrated its promising performance in video understanding due to its powerful attention-based context modelling capabilities [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. However, the computational cost of the attention mechanism, which increases quadratically with the length of the input data, incurs huge computational and memory costs, especially when processing long video sequences. To achieve efficient video processing, a state space modelling method, called VideoMamba, is introduced to achieve comparable performance to the ViT while maintaining linear computational complexity [11, 12]. Despite some progress in model pre-training [13, 14, 15], it still suffers from the heavy overhead of downstream task adaptation through parameter fine-tuning. Therefore, Parameter-Efficient Fine-Tuning (PEFT) has aroused extensive attention to achieve + +comparable or even higher performance than Full Fine-Tuning (FFT) and reduce the adaptation costs by optimizing only a few set of parameters [16, 17, 18]. + +Early PEFT methods mainly focused on parameter-level efficient fine-tuning, e.g., Adapter [19, 20, 21, 22] or LoRA [23, 24], but still suffered from inefficiency with the additional introduction of model parameters. To this end, visual prompt learning technology aims to embed a small set of learnable prompting tokens at the input level, enabling efficient downstream task adaptation without extending internal model parameters [25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. However, as shown in Figure 1(a), existing video prompting methods are typically ViT-oriented, which exploit global attention mechanisms to propagate key information in intra- and inter-frame prompts [35, 36, 37, 38], ignoring the requirement to balance efficiency and effectiveness in video understanding for VideoMamba. Specifically, ignoring the sequential compression state space [39, 40, 41, 42, 43], existing video prompting methods fail to gather spatial information in long video token sequences for Mamba. In addition, the global attention-oriented prompting method mitigates the high efficiency in VideoMamba, leading to the dissemination of key spatial and temporal information in the long video token sequence. Consequently, as shown in Figure 1(b), the embedded prompts mitigate the aggregation of discriminative spatio-temporal information due to the information decay after long-term compression [44]. + +![](images/63cb556639d174c46ee889e657db60a6fa0db28b97175ed8943315bd81b278c7.jpg) +(a) Existing Video Prompting on Mamba + +![](images/0d776c69c001e45e176184d96796f1e570fbdf5fabae0fe585601626a637360c.jpg) +(b) Visualization of Existing Video Prompting + +![](images/c1b62ce54c644366c1c1605c528482e32ff35daaed41f9e5e43d7cd8992b4886.jpg) +(c) Visualization of Our SSP +Figure 1: Existing video prompting methods on Mamba and its visualization results of updating gate compared to our SSP. Existing methods directly concatenate learnable prompts to video tokens, resulting in the information decay problem after the long-term state space compression. However, our SSP achieves information enhancement through spatial gathering and temporal spreading. + +To address the above challenges, we propose a State Space Prompting (SSP) framework for video understanding, which gathers and spreads spatio-temporal information in an efficient and effective manner, as shown in Figure 1(c). Specifically, an Intra-Frame Gathering (IFG) module is designed to exploit a low-rank local convolution to aggregate spatial information within each video frame. Sequentially, a low-rank attention-oriented Inter-Frame Spreading (IFS) module is further proposed to spread key information at the temporal level, which develops a low-rank attention module to refine the temporal information that is gradually aggregated between frames, where long-term context information is effectively spread within global temporal prompts. Among them, the information entropy of each frame adjusts the attention given to each frame when refining the temporal information, and the frame-specific spatial variance is employed to gate the influence strength from the spreading information to each frame. + +In summary, our contributions are three-fold: (1) We proposed a State Space Prompting method for Video Understanding, called SSP, which gathers and spreads discriminative spatio-temporal information compressed by the state space model to achieve high effectiveness while maintaining its computational efficiency. (2) We design an Intra-Frame Gathering module and an Inter-Frame Spreading module to facilitate spatio-temporal contextual information interaction by spreading gathered local spatial information in a temporal manner. (3) Extensive experiments on multiple video understanding benchmarks demonstrate that our method achieves superior performance against existing methods with only $\sim 3\%$ of tunable parameters compared to full tuning. + +# 2 Related Work + +# 2.1 State Space Model (SSM) + +In recent years, State Space Models (SSMs) have emerged as a promising approach for sequence modeling, offering the ability to capture long-range dependencies with linear computational complexity [39, 40, 41, 42, 43]. Building on this foundation, Mamba and Mamba2 introduced input-dependent update and forget gates to address limitations in content-based reasoning [45, 46]. Unlike Transformer architectures, this allows SSM parameters to be dynamically modulated by input, significantly improving expressiveness in discrete modalities. Coupled with hardware-friendly parallelization, these advances lead to notable gains in computational efficiency and performance in language modeling. + +Building on this progress, Vision Mamba extended the Mamba framework to 2D image modeling via bidirectional spatial scanning, achieving strong results in image understanding tasks [47, 48]. More recently, VideoMamba further generalized this approach to video by introducing spatio-temporal scanning, enabling efficient modeling of global dependencies across both spatial and temporal dimensions with linear complexity [11, 49]. This architecture rivals or surpasses traditional CNN and Transformer models [12, 50, 51, 52], establishing itself as a competitive backbone for video understanding. However, applying pre-trained models like VideoMamba to downstream tasks using full fine-tuning typically requires large data and high computational cost. Thus, developing parameter-efficient fine-tuning strategies for VideoMamba remains an important and urgent challenge. + +# 2.2 Parameter-Efficient Fine-Tuning for SSM + +Parameter-efficient fine-tuning techniques aim to reduce learnable parameters while maintaining model performance, thereby reducing storage and computational costs when adapting pre-trained models to downstream tasks [16]. Several studies have attempted to apply parameter-efficient fine-tuning methods to Mamba architecture models [53, 54, 55]. These methods can be categorized into partial-based, addition-based, and prompt-based approaches. + +Partial-based methods typically fine-tune only a subset of parameters in the pre-trained Mamba model, such as projection layers, convolution layers, or forget gates [54, 55]. These methods are straightforward and easy to implement. However, partial-based approaches are constrained by the model's inherent parameter space, limiting their adaptability to downstream tasks across different domains. Addition-based methods generally freeze the original model parameters and incorporate learnable components, such as adapter modules [19, 21]. While these plug-and-play modules can be readily transferred to the Mamba model architecture, they simply apply transformations to the input data without considering the sequential progression characteristics inherent to Mamba's architecture. The Additional-Scan approach attempts to learn downstream knowledge by increasing the state dimensions of the SSM [55]. However, when applied to video domains, merely increasing state dimensions proves insufficient for effectively extracting critical spatio-temporal information. Prompt-based methods add a small number of learnable prompt vectors, optimizing only these parameters during training. Existing work such as SVP has transferred prompt learning methods to Mamba models by generating prompts for each token, effectively activating Mamba's update and forget gates during fine-tuning [53]. However, such methods are designed exclusively for static 2D images. When transferred to video tasks, they similarly struggle with the challenge of modeling long-context spatio-temporal relationships. + +# 2.3 Video Prompting + +Prompt learning methods were first introduced in natural language processing (NLP) to transfer pre-trained models to various downstream tasks [56, 57, 58, 59]. Inspired by the success of prompt learning in NLP, these methods have been extended to the visual domain[60, 61]. VPT and VFPT fine-tune models by concatenating token-based prompts with input data to capture image features [25, 26]. More recently, methods like DGL and STOP have applied prompt learning to video tasks, using intra-frame and inter-frame prompt modules to capture temporal information [35, 36, 37, 38]. For instance, DGL employs prompt vectors as query, key and value vectors to model both local features and global features in videos [36]. MPT utilizes prompts as query vectors, leveraging the Q-Former mechanism to extract spatial, temporal, and global features [37]. STOP generates inter-frame prompts + +![](images/d874b09605118e957ce963ed71bfccfa71af05e7ca63180f12fcb893e140773a.jpg) +Figure 2: The pipeline of our SSP. We embed videos into image tokens with spatial prompts. After the initial Mamba layer, our complementary IFG and IFS modules operate - IFG aggregates spatial information while IFS spreads temporal information. Information entropy and frame-specific spatial variance bridge these modules. The prompted tokens and CLS token then pass through subsequent Mamba layers for video classification. + +from all tokens while using intra-frame prompts to highlight the varying importance of frames, dynamically inserting prompts between frames [38]. + +However, these methods were originally designed for the Transformer architecture, leveraging its unique structure by connecting prompts to query and key-value vectors to store task information. Transformer-based video prompting methods use a global attention mechanism [62], allowing tokens within the sequence to interact at any position. In contrast, the Mamba architecture propagates tokens sequentially [63, 64, 65], causing adjacent tokens to contain more overlapping information. As a result, when applying existing video prompting methods to Mamba, feeding all tokens into the inter-frame prompt module introduces redundant contextual information. This issue is more pronounced in video data with high spatiotemporal redundancy, making it difficult for the prompt module to effectively capture and propagate discriminative spatiotemporal context, ultimately limiting the model's performance. + +# 3 Methodology + +In this section, we illustrate the proposed SSP comprehensively, and the overall pipeline is depicted in Figure 2. + +# 3.1 Preliminary of Mamba + +The SSM-based models, Mamba, Vision Mamba (ViM), and VideoMamba, are inspired by continuous systems that map one-dimensional equations or sequences $x(t) \in \mathbb{R} \mapsto y(t) \in \mathbb{R}$ through a $D$ -dimensional hidden state $h(t) \in \mathbb{R}^D$ . These hidden states evolve over time via parameter matrices $\mathbf{A}$ , $\mathbf{B}$ , and $\mathbf{C}$ , following a linear ordinary differential equation: + +$$ +h ^ {\prime} (t) = \mathbf {A} h (t) + \mathbf {B} x (t), \quad y (t) = \mathbf {C} h (t), \tag {1} +$$ + +where parameter $\mathbf{A} \in \mathbb{R}^{D \times D}$ represents the forgetting gate matrix, $\mathbf{B} \in \mathbb{R}^{D \times 1}$ denotes the update gate matrix, and $\mathbf{C} \in \mathbb{R}^{1 \times D}$ serves as the output projection matrix. + +To facilitate application in deep learning, SSMs are discretized into discrete-time systems using the zero-order hold technique. The continuous parameters $\mathbf{A}$ and $\mathbf{B}$ are transformed into their discrete counterparts $\overline{\mathbf{A}}\in \mathbb{R}^{D\times D}$ and $\overline{\mathbf{B}}\in \mathbb{R}^{D\times 1}$ , employing a sampling time interval $\Delta \in \mathbb{R}$ : + +$$ +\overline {{\mathbf {A}}} = \exp (\Delta \mathbf {A}), \quad \overline {{\mathbf {B}}} = (\Delta \mathbf {A}) ^ {- 1} (\exp (\Delta \mathbf {A}) - I) \cdot \Delta \mathbf {B}. \tag {2} +$$ + +Consequently, the discretized SSM can be expressed as follows: + +$$ +h _ {i} = \overline {{\mathbf {A}}} h _ {i - 1} + \overline {{\mathbf {B}}} x _ {i}, \quad y _ {i} = \mathbf {C} h _ {i}, \tag {3} +$$ + +where $h_{i - 1},h_i\in \mathbb{R}^{D\times d}$ and $x_{i},y_{i}\in \mathbb{R}^{1\times d}$ , $d$ is the dimension of the input sequences. + +# 3.2 State Space Prompting + +The backbone of our method is VideoMamba [49], the input is a video $V \in \mathbb{R}^{T \times C \times H \times W}$ , where $T$ represents the number of frames, $C$ represents the number of channels, and $H \times W$ is spatial size. Each video frame $\{\pmb{F}_i\}_{i=1}^T$ is split into $N = \frac{H \times W}{h \times w}$ fixed-size patches of size $h \times w$ and these patches are flattened into a sequence of vectors $\pmb{I}_i = \{\pmb{I}_{ij} \in \mathbb{R}^{C \times h \times w}\}_{j=1}^N$ , where $i$ denotes the frame index while $j$ denotes the patch index. These vectors are then projected into input tokens $\pmb{x}_i = \{\pmb{x}_{ij}\}_{j=1}^N$ , where $\pmb{x}_{ij} \in \mathbb{R}^d$ , and $d$ is the hidden dimension of the input sequence. For video classification tasks, the class token $\pmb{x}_{cls}$ is prepended to the sequence of input tokens, which is $[\pmb{x}_{cls}, \pmb{x}_1, \pmb{x}_2, \dots, \pmb{x}_T]$ . Then the input tokens are fed into the VideoMamba backbone, which consists of $L$ layers of Mamba block. The class token $\pmb{x}_{cls}$ from the last layer is used for classification tasks. The final output of the VideoMamba backbone is obtained by applying a linear classifier head on $\pmb{x}_{cls}$ . + +Our SSP consists of two complementary modules: an intra-frame gathering module (IFG) and an inter-frame spreading module (IFS). These modules interact complementarily to aggregate spatial information and spread discriminative long-term context information at the temporal level during fine-tuning. + +# 3.2.1 Intra-Frame Gathering Module + +The IFG $\mathcal{P}^s$ processes each frame $F_{i}$ to generate intra-frame prompts $\pmb{p}_i^s\in \mathbb{R}^{N\times d}$ , information entropy weights $\pmb {w}_i\in \mathbb{R}^{1\times d}$ , and spatial variance measurements $\pmb {v}_i\in \mathbb{R}^{1\times d}$ . The intra-frame prompts are then overlaid to the input tokens to produce spatial prompted tokens $\pmb{x}_i^s\in \mathbb{R}^{N\times d}$ : + +$$ +\boldsymbol {p} _ {i} ^ {s}, \boldsymbol {w} _ {i}, \boldsymbol {v} _ {i} = \mathcal {P} ^ {s} (\boldsymbol {x} _ {i}), \quad \boldsymbol {x} _ {i} ^ {s} = \boldsymbol {x} _ {i} + \boldsymbol {p} _ {i} ^ {s}. \tag {4} +$$ + +The IFG takes the tokens from each frame $\boldsymbol{x}_i = \{\boldsymbol{x}_{ij}\}_{j=1}^N$ as input. These tokens are processed through a downsampling layer $\mathcal{L}_1^{down}$ followed by a 2D convolutional layer $\mathcal{N}_1^s$ to generate low-rank feature maps $l_i \in \mathbb{R}^{N \times d^s}$ for each frame, where $d^s$ represents the internal dimension of the intra-frame gathering module. The low-rank feature maps $l_i$ are subsequently upsampled via a linear layer $\mathcal{L}_1^{up}$ to match the dimensionality of the input tokens, resulting in intra-frame prompts $p_i^s$ . Concurrently, the low-rank feature maps undergo additional 2D convolution and upsampling operations to produce spatial variance $v_i$ . The intra-frame prompts $p_i^s$ are then fed into an entropy calculation module $\mathcal{E}$ to calculate information entropy weights $w_i$ : + +![](images/ddf44223801086a6112f37031995b438342518e8715a7dacd98757aa9b77e19e.jpg) +Figure 3: Visualization of our intra-frame prompts, which capture discriminative local features. + +$$ +\boldsymbol {l} _ {i} = \mathcal {N} _ {1} ^ {s} (\mathcal {L} _ {1} ^ {\text {d o w n}} (\boldsymbol {x} _ {i})), \quad \boldsymbol {p} _ {i} ^ {s} = \mathcal {L} _ {1} ^ {\text {u p}} (\boldsymbol {l} _ {i}), +$$ + +$$ +\boldsymbol {v} _ {i} = \mathcal {L} _ {2} ^ {u p} \left(\mathcal {N} _ {2} ^ {s} \left(\boldsymbol {l} _ {i}\right)\right), \quad \boldsymbol {w} _ {i} = \mathcal {E} \left(\boldsymbol {p} _ {i} ^ {s}\right). \tag {5} +$$ + +The entropy calculation module $\mathcal{E}$ evaluates the informational significance of intra-frame prompts and generates frame-level weights. This module first transforms the intra-frame prompts $p_i^s$ into probability distributions, then calculates the information entropy, which is subsequently scaled by a learnable factor $\alpha$ . A small positive constant $\epsilon$ is incorporated to prevent taking the logarithm of zero: + +$$ +P _ {i} = \operatorname {s o f t m a x} \left(\boldsymbol {p} _ {i} ^ {s}\right), H _ {i} = - \sum_ {d} P _ {i} \cdot \log \left(P _ {i} + \epsilon\right), E _ {i} = 1. 0 - \frac {H _ {i}}{\max \left(H _ {i}\right)}, \tag {6} +$$ + +$\pmb{w}_i = \alpha \cdot \mathrm{softmax}(\bar{E})$ where $\bar{E}$ is the mean of $E_{i}$ across tokens per frame. + +The intra-frame prompts $\pmb{p}_i^s$ gather the model's attention to local features during downstream fine-tuning, as shown in Figure 3. Information entropy weights $\pmb{w}_i$ adjust the attention given to each frame during inter-frame prompt generation based on the certainty of information distribution, while spatial variance $\pmb{v}_i$ gates the influence strength from the long-term context information to local features. + +# 3.2.2 Inter-Frame Spreading Module + +After processing through the first Mamba layer, the inter-frame spreading module (IFS) activates. This module first samples the last token $s_i \in \mathbb{R}^{1 \times d}$ from frame $F_i$ in the Mamba forward scanning sequence. This token is Hadamard multiplied with the information entropy weights $w_i$ generated by the intra-frame gathering module and fed into the inter-frame spreading module $\mathcal{P}^t$ . The module's output is then Hadamard multiplied with the spatial variance $v_i$ to produce inter-frame prompts $p_i^t \in \mathbb{R}^{1 \times d}$ : + +$$ +\boldsymbol {s} _ {i} = \boldsymbol {x} _ {i N}, \quad \boldsymbol {p} _ {i} ^ {t} = \mathcal {P} ^ {t} \left(\boldsymbol {s} _ {i} \odot \boldsymbol {w} _ {i}\right) \odot \boldsymbol {v} _ {i}. \tag {7} +$$ + +The IFS processes the input through a sequence of operations including a downsampling linear transformation $\mathcal{L}_2^{down}$ , an attention computation $\mathcal{A}$ , and an upsampling linear transformation $\mathcal{L}_3^{up}$ . The resulting output is then computed Hadamard product with the spatial variance $\pmb{v}_i$ , and subsequently scaled by a learnable factor $\beta$ to produce the inter-frame prompts $\pmb{p}_i^t$ . This architecture facilitates the spreading of long-term context information across video frames while preserving local context information: + +$$ +\boldsymbol {p} _ {i} ^ {t} = \beta \cdot \mathcal {L} _ {3} ^ {u p} (\mathcal {A} \left(\mathcal {L} _ {2} ^ {\text {d o w n}} (\boldsymbol {s} _ {i})\right)) \odot \boldsymbol {v} _ {i}. \tag {8} +$$ + +The last token of each frame $s_i$ aggregates contextual information from both the current frame and preceding frames during forward scanning, as well as subsequent frames during backward scanning. As shown in Figure 4, the generated inter-frame prompts $p_i^t$ represent temporal inductive biases that spread the gathered temporal information from all frames to the current frame. + +![](images/a1d9178d2b7701b5ae6796fe2436089188dd5e2a4a504ff187e7bf08afe8606b.jpg) +Figure 4: Visualization of our inter-frame prompts. The action "catch" can be parsed into two distinct phases: preparation for the catch and the actual reception of the ball. Our inter-frame prompts effectively locate the key transitional moments between these critical phases. + +Subsequently, the inter-frame prompts $\pmb{p}_i^t$ are concatenated with the spatial prompted tokens $\pmb{x}_i^s$ , along with the class token $\pmb{x}_{cls}$ , are fed into the following $L - 1$ Mamba layers $\{\mathcal{B}_j\}_{j = 2}^L$ to extract spatio-temporal features. Inspired by VPT-deep [25], we overlay the intra-frame prompt on the input of each layer and embed the inter-frame prompt in the input of all layers after the first layer. To obtain the final predicted probability distribution $\pmb{y}$ , we apply a linear classification head $\mathcal{H}$ on $\pmb{x}_{cls}$ . + +# 3.2.3 Overall Optimization + +As mentioned above, our SSP introduces only a few additional parameters: + +$$ +\mathcal {M} = \left\{\mathcal {P} ^ {s}, \mathcal {P} ^ {t} \right\}. \tag {9} +$$ + +Following prior works, we keep the pre-trained model frozen during training, allowing only the classification head $\mathcal{H}$ and the newly added modules $\mathcal{M}$ to be trainable. The optimization objective is defined as follows: + +$$ +\underset {\mathcal {M}, \mathcal {H}} {\arg \min } \mathcal {L} _ {c e} (\boldsymbol {y}, y _ {g t}), \tag {10} +$$ + +where $\mathcal{L}_{ce}$ is the cross-entropy loss, and $y_{qt}$ is the ground truth video label. + +# 4 Experiments + +# 4.1 Datasets + +HMDB51 [67] contains 6849 clips across 51 action categories, with an average duration of 3.15 seconds and 91.49 frames per video. It was collected from various sources, mostly from movies, and a small proportion from public databases such as the Prelinger archive, YouTube and Google videos, featuring diverse real-world actions with variations in background and camera angles. + +Table 1: The comparison results on K400 pretrained VideoMamba-S (Parameters 25.42M). + +
MethodVenueParamHMDB51SSV2UCF101Breakfast
VideoMamba-SFull [49]ECCV'2425.42M67.5858.5792.9694.27
Adapter [19]NeurIPS'222.40M73.7936.4594.1884.89
ST-Adapter [21]NeurIPS'222.69M70.5230.9494.8777.60
VPT [25]ECCV'221.50M72.7430.6895.1681.25
VFPT [26]NeurIPS'241.50M72.4130.3795.0879.68
SVP [53]AAAI'252.76M69.9338.0195.5880.72
Additional-Scan [55]ICLR'250.66M73.2033.7195.6378.64
STOP [38]CVPR'251.49M70.0621.2293.4465.62
SSP(Ours)This Paper0.98M74.3838.6895.6985.41
+ +Table 2: The comparison results on CLIP-400M pretrained CLIP-ViT-B/32 [66] (Parameters 88.00M) and K400 pretrained VideoMamba-M (Parameters 74.00M). + +
MethodVenueParamHMDB51SSV2UCF101Breakfast
CLIPDGL-Linear [36]AAAI'240.83M67.2018.3092.50-
DGL-Transformer [36]AAAI'249.57M69.8018.1093.60-
STOP [38]CVPR'257.53M72.0021.4095.30-
VideoMamba-MFull [49]ECCV'2474.00M76.3067.3096.0095.31
Adapter [19]NeurIPS'222.40M73.5946.9896.2489.06
ST-Adapter [21]NeurIPS'222.69M71.9631.9194.8167.70
VPT [25]ECCV'221.50M73.5940.5895.4581.77
VFPT [26]NeurIPS'241.50M71.8939.6895.7186.97
SVP [53]AAAI'252.76M73.6649.0896.7790.10
Additional-Scan [55]ICLR'251.33M73.5244.6596.3286.97
STOP [38]CVPR'251.49M71.9623.4494.6071.35
SSP(Ours)This Paper2.41M76.6653.7297.0393.23
+ +UCF101 [68] is an action recognition data set of realistic action videos, collected from YouTube, having 13320 video clips across 101 action categories. The average video length is 7.21 seconds with 186.5 frames per video. + +Something-Something V2 (SSV2) [69] is a large-scale dataset for action recognition, containing 220,847 videos across 174 action categories. The dataset is designed to capture fine-grained actions and interactions between objects, with an average video length of 3.82 seconds and 45.84 frames. + +Breakfast [70] is a long video understanding dataset containing 1989 video clips divided into 10 categories related to breakfast preparation. The dataset has an average video length of 137.53 seconds, with an average of 2062.89 frames per video. + +# 4.2 Comparison Methods + +We compare our SSP with both adapter-based and prompt-based parameter-efficient finetuning methods. We also report the fully tuning results as a baseline, i.e., VideoMamba [49]. For adapter-based methods, we compare with the following methods: Adapter [19] and ST-Adapter [21]. For prompt-based methods, we compare with the following methods: DGL-Linear [36], DGL-Transformer [36], VPT [25], VFPT [26], SVP [53], and STOP [38]. + +# 4.3 Implementation Details + +Following [49], all video frames are resized to $224 \times 224$ and split into $14 \times 14$ patches. For HMDB51, UCF101 and SSV2 datasets, each video is uniformly sampled to 8 frames, while for Breakfast, we sample 32 frames. We set the learning rates to 3e-3, 5e-3, 2e-4, and 1e-3 for HMDB51, UCF101, + +SSV2, and Breakfast respectively. Meanwhile, we set the batch size to 32 for HMDB51 and UCF101, 64 for Breakfast and 512 for SSV2. All the dataset splits are consistent with the official annotation files. The model is fine-tuned with the AdamW optimizer on 4 NVIDIA 4090-24G GPUs, with a cosine decay scheduler. Additionally, we adopt a warm-up strategy within the first 5 training epochs. + +# 4.4 Comparison with State-of-the-arts + +We evaluated our SSP method on four popular video datasets: HMDB51, UCF101, SSV2, and Breakfast. For fair comparison, we used CLIP-400M pretrained CLIP-ViT-B/32 as the backbone for ViT-oriented methods, while for methods fine-tuned on the Mamba architecture, we employed Kinetics-400 [71] pretrained VideoMamba-M and VideoMamba-S as backbones. Our method achieves state-of-the-art performance across all datasets, as shown in Table 2. + +When fine-tuned on VideoMamba-M, our approach attains top-1 accuracies of $76.66\%$ , $53.72\%$ , $97.03\%$ , and $93.23\%$ on HMDB51, Something-Something V2, UCF101, and Breakfast respectively, representing improvements of $3.00\%$ , $4.64\%$ , $0.26\%$ , and $3.13\%$ over existing parameter-efficient video fine-tuning methods. As demonstrated in Table 1, even with the parameter constraints of smaller-scale models, our method still delivers superior results when fine-tuned on VideoMamba-S. This effectiveness stems from our tailored design for the VideoMamba architecture, which enables efficient gathering and spreading of discriminative spatio-temporal information within state space models, through the complementary IFG and IFS modules. Notably, our approach demonstrates particularly significant improvements on the challenging large-scale Something-Something V2 dataset and the long-video Breakfast dataset, as shown in Figure 5, + +![](images/9486012ef6400d36cc95173128a22e283e69bdaec499b7df586a34b9a0b0dc7d.jpg) +Figure 5: Comparison on Breakfast. SSP outperforms existing methods while tuning minimal parameters. + +our method exceeds others when tuning only 1.30M parameters (using only one IFS module as shown in Figure 7). This is because our SSP can gather local information on key regions in complex video data and spread critical global information in long contexts. + +# 4.5 Ablation + +# 4.5.1 Influence of Different Components + +To verify the effectiveness of the intra-frame gathering (IFG) module and inter-frame spreading (IFS) module, we conducted ablation experiments on three datasets: HMDB51, UCF101, and Breakfast, as shown in Table 3. As demonstrated, when neither component is used, SSP achieves the lowest accuracy on all datasets. When the intra-frame gathering module is used alone, the model's performance improves by $9.78\%$ on average. This is because the intra-frame spatial prompts effectively capture local features and enhance the model's ability to focus on discriminative information within each frame. When the inter-frame spreading module is used alone, the model's performance improves by $9.20\%$ on average. This can be attributed to that the inter-frame temporal prompts effectively aggregate and spread global contextual information across frames. When both components are used together, the model achieves the best performance by an average improvement of $13.46\%$ across all datasets, as they complement each other in gathering and spreading both local and global information. When the spatial variance gate or the entropy gate (denoted as $\boldsymbol{v}_i$ , $\boldsymbol{w}_i$ in Equation 7) is removed, the model's performance drops by $2.33\%$ and $2.27\%$ respectively on average across all datasets. This indicates that both gates play a crucial role in facilitating the inter-frame prompts to propagate spatial information in a complementary manner based on the key local features of each frame. + +Table 3: Ablation of different prompting modules and gates. + +
Different Prompting ModulesDifferent Gates
IFGIFSHMDB51UCF101BreakfastEntropySpatialHMDB51UCF101Breakfast
--59.3490.6476.56--74.7096.5689.58
✓-74.4496.5684.89✓-75.0996.2988.54
-✓72.6196.1485.41-✓74.9696.6188.54
✓✓76.6697.0393.23✓✓76.6697.0393.23
+ +# 4.5.2 The Visualization Results of Update Gate + +![](images/14dfa3334dacebc040fdad2e8a729a6f1fa02bd1948f5982212188073a311728.jpg) +Figure 6: Visualization of the update gate. In SSP, the key region is activated effectively. + +![](images/d1dc29d19ee188efa8e5b371c2c61b7dd1275f2e322154415cd701c9d06e2c94.jpg) +Figure 7: Ablation of the number of IFSs in our SSP. + +![](images/1b20fbdd802bb8e6d77a51a14d29f9212fedb232ff01aae99687bb391abb8bc5.jpg) + +To further explore the impact of our intra-frame gathering module and inter-frame spreading module, we visualize the normalized update gate values in VideoMamba on the last layer, which contribute to the final classification results directly. As shown in Figure 6, existing prompting methods designed for + +Mamba like SVP can't effectively gather and spread the discriminative spatio-temporal information. This caused the pre-trained VideoMamba model to focus on irrelevant regions in each frame. As a result, the model struggles to accurately understand dynamic key features in the video. In contrast, our SSP method highlights the key regions with dynamic changes in the video, leading to a more accurate understanding of the video content. + +# 4.5.3 Influence of Hyperparameters + +The number of the inter-frame spreading modules $(\mathcal{P}^t)$ is one of important hyperparameters in our method. To assess its impact, we conduct extensive ablation experiments. As shown in Figure 7, the model's performance initially improves but then fluctuates as the number of inter-frame spreading modules increases. + +To balance performance and tunable parameters, we set the number of inter-frame spreading modules to 3 when using VideoMamba-M as backbone, and when using VideoMamba-S as backbone, we set the number to 1. + +# 5 Conclusion + +In this paper, we propose a novel State Space Prompting (SSP) approach for efficient adaptation of pre-trained state space models to video understanding tasks. SSP combines complementary intra-frame gathering module and inter-frame spreading module to aggregate key spatial information within frames and spread discriminative temporal information between frames, enabling effective propagation of crucial spatio-temporal features in state space models. By employing the two modules, we can adaptively balance and compress discriminative information in videos. The effectiveness of our proposed SSP has been validated on four video benchmarks compared to other parameter-efficient fine-tuning methods. + +# 6 Acknowledgements + +This work was supported by the National Natural Science Foundation of China (62376011) and the National Key R&D Program of China (2024YFA1410000). + +# References + +[1] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020. +[2] Anurag Arnab, Mostafa Dehghani, Georg Heigold, Chen Sun, Mario Lučić, and Cordelia Schmid. Vivit: A video vision transformer. In Proceedings of the IEEE/CVF international conference on computer vision, pages 6836-6846, 2021. +[3] Gedas Bertasius, Heng Wang, and Lorenzo Torresani. Is space-time attention all you need for video understanding? 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The kinetics human action video dataset. arXiv preprint arXiv:1705.06950, 2017. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: We propose SSP, a state space prompting method for video understanding. The main contributions of SSP (i.e., the fine tuning effectiveness and the parameter efficiency) are claimed in both the abstract and introduction accurately. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: We discuss the limitation in appendix H. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. + +# 3. Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +# Answer: [Yes] + +Justification: All theorems used in the paper are properly referenced. + +# Guidelines: + +- The answer NA means that the paper does not include theoretical results. +- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced. +- All assumptions should be clearly stated or referenced in the statement of any theorems. +- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. +- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. +- Theorems and Lemmas that the proof relies upon should be properly referenced. + +# 4. Experimental result reproducibility + +Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? + +# Answer: [Yes] + +Justification: We claim reproducibility in section $\S 4$ . Our code will be publicly available after acceptance. + +# Guidelines: + +- The answer NA means that the paper does not include experiments. +- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. +- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. +- Depending on the contribution, reproducibility can be accomplished in various ways. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Discussion and Analysis + +In this section, we discuss and analyze the effectiveness of our SSP method for parameter-efficient fine-tuning the VideoMamba architecture in downstream task adaptation. + +For each Mamba layer's input, we denote the $i$ -th input token along the scanning order as $\pmb{x}_i$ . The hidden state computation for the subsequent Mamba layer is formulated as: + +$$ +\boldsymbol {h} _ {i} = \overline {{\mathbf {A}}} _ {i} \boldsymbol {h} _ {i - 1} + \overline {{\mathbf {B}}} _ {i} \boldsymbol {x} _ {i} ^ {s}, \quad \boldsymbol {x} _ {i} ^ {s} = \boldsymbol {x} _ {i} + \boldsymbol {p} _ {i} ^ {s}, \tag {11} +$$ + +where $h_i$ is the $i$ -th compressed hidden state token of the Mamba layer. In the Mamba architecture, each layer's parameter matrices $\mathbf{B} \in \mathbb{R}^{D \times 1}$ , $\mathbf{C} \in \mathbb{R}^{1 \times D}$ , and $\Delta \in \mathbb{R}$ are derived from the input token $x_i^s$ through functions $S_B$ , $S_C$ , and $S_\Delta$ , respectively: + +$$ +\overline {{\mathbf {A}}} _ {i} = \exp (\Delta \mathbf {A} _ {i}), \quad \mathbf {B} _ {i} = \mathcal {S} _ {B} \left(\boldsymbol {x} _ {i} ^ {s}\right), \quad \mathbf {C} _ {i} = \mathcal {S} _ {C} \left(\boldsymbol {x} _ {i} ^ {s}\right), \quad \Delta_ {i} = \mathcal {S} _ {\Delta} \left(\boldsymbol {x} _ {i} ^ {s}\right). \tag {12} +$$ + +In our proposed SSP method, the intra-frame prompt $p_i^s$ enables direct fine-tuning of the parameter matrices generated by each token: + +$$ +\overline {{\mathbf {A}}} _ {i} ^ {p} = \exp \left(S _ {\Delta} \left(\boldsymbol {x} _ {i} + \boldsymbol {p} _ {i} ^ {s}\right) \widetilde {\odot} \mathbf {A}\right), \quad \overline {{\mathbf {B}}} _ {i} ^ {p} = S _ {\Delta} \left(\boldsymbol {x} _ {i} + \boldsymbol {p} _ {i} ^ {s}\right) S _ {B} \left(\boldsymbol {x} _ {i} + \boldsymbol {p} _ {i} ^ {s}\right), \tag {13} +$$ + +where $\overline{\mathbf{A}}_i^p\in \mathbb{R}^{D\times D}$ and $\overline{\mathbf{B}}_i^p\in \mathbb{R}^{D\times 1}$ represent the forget and update gates directly controlled by the intra-frame prompt $p_i^s$ , facilitating the extraction of locally discriminative information, thereby gathering the spatial information effectively when fine-tuning on downstream tasks. + +For the inter-frame prompt $\pmb{p}_j^t$ of the $j$ -th frame, this prompt vector aggregates global spatio-temporal information and is gated through Hadamard multiplication with the spatial variance $\pmb{v}_j$ . It is inserted between the $j$ -th and $(j + 1)$ -th frames, directly influencing the hidden state at that position: + +$$ +\boldsymbol {h} _ {j + 1, 0} = \overline {{\mathbf {A}}} _ {j + 1, 0} ^ {p} \boldsymbol {h} _ {j N} + \overline {{\mathbf {B}}} _ {j + 1, 0} ^ {p} \boldsymbol {p} _ {j} ^ {t}, \tag {14} +$$ + +where $N$ denotes the number of tokens per frame. Tokens following the $j$ -th frame can access global discriminative information through the hidden state $\pmb{h}_{j+1,0}$ of inter-frame prompt $\pmb{p}_j^t$ , overcoming the sequential spatio-temporal information transfer limitation of the original VideoMamba model and achieving efficient gathering and spreading of spatio-temporal information. + +Next, we further analyze the impact of inter-frame prompts on long-range spatio-temporal information transmission. By expanding Equation 11, we obtain: + +$$ +\begin{array}{l} \pmb {h} _ {j} = \overline {{\mathbf {A}}} _ {j} ^ {p} \pmb {h} _ {j - 1} + \overline {{\mathbf {B}}} _ {j} ^ {p} \pmb {x} _ {j} ^ {s} = \overline {{\mathbf {A}}} _ {j} ^ {p} \left(\overline {{\mathbf {A}}} _ {j - 1} ^ {p} \pmb {h} _ {j - 2} + \overline {{\mathbf {B}}} _ {j - 1} ^ {p} \pmb {x} _ {j - 1} ^ {s}\right) + \overline {{\mathbf {B}}} _ {j} ^ {p} \pmb {x} _ {j} ^ {s} \\ = \prod_ {k = i + 1} ^ {j} \overline {{\mathbf {A}}} _ {k} ^ {p} \boldsymbol {h} _ {i} + \sum_ {t = i + 1} ^ {j} \left(\prod_ {k = t + 1} ^ {j} \overline {{\mathbf {A}}} _ {k} ^ {p}\right) \overline {{\mathbf {B}}} _ {t} ^ {p} \boldsymbol {x} _ {t} ^ {s}, \tag {15} \\ \end{array} +$$ + +where $i < j$ . The coefficient of the first term in the above equation, $\prod_{k = i + 1}^{j}\overline{\mathbf{A}}_k^p$ , represents the influence strength from the $i$ -th hidden state token to the $j$ -th hidden state token along the scanning sequence. We denote this as the transmission matrix $\mathbf{T}_{i\rightarrow j}$ : + +$$ +\mathbf {T} _ {i \rightarrow j} = \prod_ {k = i + 1} ^ {j} \overline {{\mathbf {A}}} _ {k} ^ {p} = \exp \left(\sum_ {k = i + 1} ^ {j} \Delta_ {k} \widetilde {\odot} \mathbf {A}\right), \tag {16} +$$ + +where $\mathbf{A}$ is a negative matrix. Examining the equation above, we observe that the mutual influence strength between tokens at different positions in the sequence exponentially decays to zero as the distance $j - i$ increases. Without the insertion of inter-frame prompts, the maximum information transmission path length in the sequence equals the sequence length $\mathcal{O}(TN)$ , where $T$ is the number of video frames and $N$ is the number of tokens per frame. However, after inserting inter-frame prompts, information from tokens at different positions can be transmitted through the inter-frame prompts inserted after each frame, reducing the maximum information transmission path length to $\mathcal{O}(N)$ . This significant reduction in information transmission path length facilitates the aggregation and propagation of global spatio-temporal information. + +# B Visualization of More Cases + +# B.1 Update Gate Value Visualization + +![](images/34f9087fe98a0d7f72594a8813fd0266e4041704e2694eb3115bbb17cdf1f814.jpg) +Figure 8: Visualization of the values of the update gate over the last layer in VideoMamba. + +The update gate value of each token, denoted as $\overline{\mathbf{B}}_i^p$ in Equation 13, represents the degree of influence each token exerts on the hidden states of VideoMamba. By visualizing the update gate values across tokens, we can observe which specific tokens receive greater attention from the VideoMamba architecture during processing. + +As illustrated in Figure 8, existing prompting approaches for Mamba architecture (e.g., SVP) employ static prompts for each frame. This limitation prevents the pre-trained VideoMamba model from effectively integrating and modeling global video contextual information, thereby hindering accurate interpretation of human actions within the video sequence. In contrast, our SSP method, through the complementary application of intra-frame gathering and inter-frame spreading modules, dynamically emphasizes regions exhibiting significant temporal variations in the video. This approach enables more precise comprehension of video content by capturing the most relevant spatio-temporal information. + +# B.2 Intra-Frame Prompts Visualization + +In our SSP method, the intra-frame prompts, which are overlaid on the input tokens fed into each Mamba layer, are employed to capture the discriminative local features and gather the spatial information of each frame. To visualize the intra-frame prompts, we plot the values of intra-frame prompts of all frames in a video over the last Mamba layer as heatmaps, and overlay them on the original video frames. + +![](images/71d02b6462b9e887a78257fb57c7c61450aaa4c14eb63efbd92280f439f4b003.jpg) +Figure 9: Visualization of our intra-frame prompts over the last layer in VideoMamba. + +As shown in Figure 9, when existing parameter-efficient fine-tuning methods (e.g., ST-Adapter) are applied to VideoMamba, they fail to effectively capture the local feature maps of individual frames, + +preventing the model from attending to local features that undergo temporal variations. In contrast, our approach, through efficient spreading of global temporal information, enables intra-frame prompts to more accurately capture discriminative local features, thereby achieving effective gathering of spatially relevant information. + +# B.3 Inter-Frame Prompts Visualization + +The inter-frame prompts in our SSP method facilitate the propagation of temporal information across sequential frames. Through visualization of these inter-frame prompts, we can identify which specific frames receive heightened attention from the model during global context integration, providing insights into the temporal dynamics of information processing within the architecture. + +![](images/e4eb9c8404241842816e4d8dc47fb0abb17f46729bfc1e4eab82db3e01462f20.jpg) +Figure 10: Visualization of our inter-frame prompts over the last layer in VideoMamba. + +As demonstrated in Figure 10, compared to existing video prompting methods (e.g., STOP), the inter-frame prompts of our SSP method effectively identify key frames exhibiting temporal variations + +within the video sequence. This enhanced capability stems from our approach being specifically engineered for the Mamba architecture, enabling efficient refinement and propagation of temporal information from compressed hidden states. + +# C Discussion on Different Method Designs + +In our methodology design, we generate inter-frame prompts by sampling the last token from each frame in the forward scanning sequence, as illustrated in Figure 11(a). This section explores how different sampling strategies affect the generation of inter-frame prompts, with supplementary experimental results presented in Table 4. We discovered that sampling tokens + +Table 4: Ablation of Different Sample Methods. + +
MethodHMDB51UCF101Breakfast
Middle74.4495.8791.14
Bidirection75.8196.0089.06
Bi-Independent75.1696.0690.10
SSP(Ours)76.6697.0393.23
+ +from the middle of each frame (Figure 11(b)) leads to an average accuracy decrease of $1.82\%$ . We attribute this decline to the fragmentation of semantic information within each frame during prompt generation. When sampling the last tokens separately from the forward and backward scanning sequences to generate prompts (Figure 11(c)), the accuracy decreases by an average of $2.01\%$ . We posit that sampling before the superposition of bidirectional sequences causes the inter-frame prompts to overlook complementary information from both directions, resulting in a separation of forward and backward contextual cues. Additionally, employing independent inter-frame spreading modules to generate prompts for forward and backward directions (Figure 11(d)) still results in an accuracy drop of $1.86\%$ . Although this approach introduces more learnable parameters to separately model contextual relationships in forward and backward scanning, it fails to address the fundamental issue of separated forward and backward contextual cues. + +![](images/52b967b18e2566bfa8df9e7c54ec3410ab799023cfc643e7cebaa02bf809a9fe.jpg) +(a) Sampling Tokens from the End of Frames + +![](images/764818abfb51f48a75f3a4e69b9b967faf9be6beed794e982a3fa410558008cf.jpg) +(b) Sampling Tokens from the Middle of Frames + +![](images/b440e2fe3d7761a6a8885ecf4827fedb2a49814effa3df8251ca4f522d2c3ad2.jpg) +(c) Bidirectional Token Sampling + +![](images/c88b86f42b52bf1ca3d98bb90b306a4ac5bc6fbfaea1c401111aa0096fe7cbd1.jpg) +(d) Bidirectional Token Sampling with Independent Processing +Figure 11: Different methods for inter-frame prompting. For simplicity, irrelevant computational processes in the Mamba block and intra-frame gathering modules have been omitted. + +# D More Ablation Studies on Hyperparameters + +The internal dimensions of our intra-frame gathering module (IFG) and inter-frame spreading module (IFS) are also hyper-parameters of our method. To balance efficiency and effectiveness, we set the internal dimension of IFG to 384 and the internal dimension of IFS to 256. We conducted extensive experiments on HMDB51 and Breakfast datasets to investigate the impact of different dimension settings on fine-tuning performance. + +As shown in Figure 12, naively increasing the internal dimension of the IFG module does not improve model performance. When the dimension is too large, model performance actually decreases due to increased optimization difficulty. Regarding the internal dimension of the IFS module, the long-video + +Breakfast dataset is more sensitive to this setting. From the perspective of reducing training time and parameter costs, setting the internal dimension of the IFS to 32 is also an acceptable option, which demonstrates the robustness of our method to the choices of hyperparameters. + +![](images/3c9103defa759cd2ce4956221bdb81d80da03d1f34b8298149e01c49ca329936.jpg) +Figure 12: Ablation on the Internal Dimensions of IFG and IFS. + +![](images/84fb84132394543080b5fd6b660e5309953a0a1f3f1bff90da5e35d7df25034e.jpg) + +![](images/c99f625f4bc5c1cd4ae299ac3758a3635e8c3698f44154d6a42f3ac4a0ff6d3c.jpg) + +![](images/20f7c78a80b87c72ae47d30ad9336c5a6e62a58ae48eb86615b6497273a975f3.jpg) + +# E More Clarifications + +We first clarify the role of spatial variance in our approach. Given that different video frames exhibit varying degrees of association with global information, we learn the spatial variance $\pmb{v}_i$ through the spatial features aggregated by IFG. The spatial variance gates the weight of each inter-frame prompt, thereby controlling the degree of influence each video frame has on global information and achieving more refined temporal information propagation. + +To further elaborate on the spreading mechanism, we describe how IFG and IFS work collaboratively. The IFG module aggregates spatial information from each frame into intra-frame prompts and superimposes the intra-frame prompts onto tokens. While the IFS module performs global attention computation through sampled tokens, enabling global interaction of the aggregated spatial information to generate inter-frame prompts. The inter-frame prompt corresponding to each frame thus contains spatial information from other frames, thereby propagating the aggregated local spatial information in a temporal manner. + +# F Asset License and Consent + +DGL, VPT and VFPT are licensed under CC-BY-NC 4.0. Adapter, ST-Adapter, and CLIP-ViT are licensed under MIT. STOP, Additional-Scan and the VideoMamba are licensed under Apache 2.0. + +All the datasets included in our study are publicly available, and all the models are publicly available. We would like to state that the contents in the dataset do NOT represent our views or opinions. + +# G Broader Impacts + +This study presents SSP, which improves the performance of pre-trained Mamba models when fine-tuned on downstream video tasks. Thanks to the reduced parameter count, our research enables deployment of video foundation models on resource-constrained devices, reduced environmental impact of AI training, and rapid adaptation to specialized domains. + +# H Limitations + +For potential limitations, our method introduces Intra-Frame Gathering (IFG) module and Inter-Frame Spreading (IFS) module to facilitate spatio-temporal contextual information, which brings additional hyperparameters, such as the number of IFSs. However, according to extensive experiments in § 4.5.3, we observe that in most cases, simply setting this parameter to 3 is sufficient. 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A major component of the PWS pipeline is the label model, which amalgamates predictions from multiple noisy weak supervision sources, i.e. labeling functions (LFs), to label datapoints. While most label models are either probabilistic or adversarial, a recently proposed label model achieves strong empirical performance without falling into either camp. That label model constructs a polytope of plausible labelings using the LF predictions and outputs the "center" of that polytope as its proposed labeling. In this paper, we attempt to theoretically study that strategy by proposing Bayesian Balsubramani-Freund (BBF), a label model that implicitly constructs a polytope of plausible labelings and selects a labeling in its interior. We show an assortment of statistical results for BBF: log-concavity of its posterior, its form of solution, consistency, and rates of convergence. Extensive experiments compare our proposed method against twelve baseline label models over eleven datasets. BBF compares favorably to other Bayesian label models and label models that don't use datapoint features - matching or exceeding their performance on eight out of eleven datasets. + +# 1 Introduction + +Large scale deep learning models often require a labeled training set of substantial size. However, hiring experts to label every training point is not only expensive, but time consuming. Programmatic Weak Supervision (PWS) aims to ameliorate this problem by cheaply generating labels. This is done via combination of multiple weak supervision sources, called labeling functions (LFs). LFs can be computer programs [Ratner et al., 2016], crowdsourcing workers, simple classifiers (like linear classifiers), etc. Since LFs are noisy and provide contradictory predictions on a datapoint, it is incumbent on the label model to combine the LF predictions to produce a single labeling of the data. + +The construction of the label model serves as one of the main challenges in the PWS pipeline. Most label models are either probabilistic or adversarial. Probabilistic label models [Dawid and Skene, 1979, Ratner et al., 2016, Zhang et al., 2023] often specify a generative process for the labels/LF outputs and compute the posterior label distribution for each datapoint. Much work has been devoted to this approach and there are many examples of probabilistic label models that perform well empirically. Adversarial label models [Balsubramani and Freund, 2015a, Mazzetto et al., 2021b, Mazuelas et al., 2022] assume the existence of an adversary who assigns the labels, construct a polytope of plausible labelings, and then predict a minimax optimal labeling to guard against the worst case. While they come with theoretical guarantees, e.g. An and Dasgupta [2024], these label models require (small amounts of) labeled data to construct the polytope. Arachie and Huang [2019] + +propose a label model that is adversarial and unsupervised, though we are unaware of theoretical guarantees for their method. There are also label models that fall into neither camp, e.g. Hyper Label Model (HyperLM) [Wu et al., 2023], which constructs a polytope of labelings in an unsupervised fashion and then picks the labeling that is in the "center". While HyperLM is a strong performer in real-world settings, we are unaware of theoretical guarantees regarding its properties and performance. In this paper, we attempt to bridge the gap by proposing Bayesian Balsubramani-Freund (BBF), a Bayesian adversarial label model that is unsupervised, selects labelings like HyperLM, but also has theoretical guarantees such as consistency. + +Say we have $N$ datapoints $X = \{x_{1},\ldots ,x_{N}\}$ where $x_{i}$ 's label is $z_{i}$ in $\mathcal{Z} = \{1,\dots,K\} = [K]$ . We wish to infer the underlying conditional label probability $\eta (j\mid x) = \operatorname *{Pr}(z = j\mid x)$ , $j\in \mathcal{Z}$ , for each $x\in X$ . If our task is to determine whether an email is spam, $\eta (j\mid x)$ might have large mass for a single $j$ . On the other hand, if the question is whether someone will default on a loan, we may imagine $\eta (j\mid x)$ being closer to uniform due to inherent uncertainty. Over all labels and datapoints, we compactly write $\eta (j\mid x)$ as $\vec{\eta} \coloneqq (\eta (j\mid x_i):j\in [K],i\in [N])\in \Delta_K^N$ where $\Delta_K$ is the set of distributions over $K$ elements and $\Delta_K^N$ is the concatenation of $N$ of those distributions. Our goal is to pick a distribution from $\Delta_K^N$ to approximate $\vec{\eta}$ . + +BBF approximates $\vec{\eta}$ by taking the mode of its posterior label distribution, which is itself specified by its generative process. First, BBF draws the distribution of labels on $X$ , after which all datapoint labels are drawn simultaneously. Then, for each LF, its accuracy on $X$ is drawn, following which the predictions it makes (on $X$ ) are drawn simultaneously. That is, once the accuracy $b$ of LF $h$ on $X$ is drawn, any labeling of the $N$ datapoints that attains accuracy $b$ on $X$ is equally likely to be selected as $h$ 's predictions. Compare this to a simplified version of other generative processes in the literature: once the datapoint labels and LF accuracies are drawn, whether an LF's prediction matches the label on a datapoint is a biased coin flip. Indeed, the frequentist versions of these models have a similar difference: Balsubramani-Freund (BF) [Balsubramani and Freund, 2015a] versus One-Coin Dawid-Skene [Li and Yu, 2014]. + +In this paper, we'll show that not only does BBF's prediction $\vec{g}^{bbf}$ have some nice properties, it has strong empirical performance. + +1. (Log-Concavity of Posterior) Given a small modification, BBF's posterior label distribution is log-concave. +2. (Form of Solution) $\vec{g}^{bbf}$ belongs to the same exponential family $\mathcal{G}$ as the BF solutions, defined in terms of the LF outputs. +3. (Consistency) When the LF accuracy priors concentrate on the empirical accuracy, $\vec{g}^{bbf}$ converges to $\vec{g}^* \in \mathcal{G}$ that best approximates $\vec{\eta}$ in KL divergence. +4. (Rates of Convergence) We provide a bound for the rate at which $\vec{g}^{tbf}$ approaches $\vec{g}^*$ . +5. (Empirical Results) Comparison of $\vec{g}^{bbf}$ against twelve baseline label models on eleven real datasets. Consistency on real datasets is also shown. + +# 2 Related Work + +The PWS pipeline ranges from the creation of the LFs, aggregation of their predictions via label model, to the training of an end model using the label model's labeling of the data. For a good survey that discusses all of these aspects, see Zhang et al. [2022]. Since we focus on the label model in this paper, we'll give a brief overview of them. + +One line of work involving probabilistic label models starts from [Dawid and Skene, 1979], where each LF is modeled as a confusion matrix and LF predictions are independent of each other given the datapoint label. The Dawid-Skene model has been lifted into the Bayesian setting and repeatedly generalized: [Kim and Ghahramani, 2012], [Li et al., 2019], [Zhang et al., 2023]. On the frequentist front, Ratner et al. [2016] builds upon Dawid and Skene's model by representing dependencies between LFs in a factor graph. Other probabilistic label models have been studied too, e.g. [Fu et al., 2020, Kuang et al., 2022, Shin et al., 2022]. + +Adversarial labels models can take the form of minimax games where we the learner attempt to minimize the loss of our prediction (i.e. the label model's aggregated labeling) while an adversary tries to maximize our loss. A big part of these models is representing the subset of feasible labels, + +i.e. a set of labels for which the true underlying conditional label probabilities $\eta$ lie. This was studied by Balsubramani and Freund [2015a,b] for 0-1 loss, [An and Dasgupta, 2024] for log loss, where the set of feasible labels is gotten by bounding the accuracies of individual LFs. Arachie and Huang [2021] estimate the LF accuracies and then try to find the minimum norm labeling that induces those accuracies from the LF predictions. Alternate formulations of the minimax game can be found. In [Mazuelas et al., 2020], uncertainty sets are considered, allowing for more general constraints on the set of feasible labels. Mazuelas et al. [2022] consider generalized entropy as the loss function for the minimax game, which includes 0-1 and log loss as special cases. Mazuelas et al. [2020, 2022], and An and Dasgupta [2024], provide convergence results in terms of how well the LF accuracies are estimated, but only the latter show consistency. Mazzetto et al. [2021a] provide a generalization bound, though they require datapoints be drawn i.i.d. In contrast, the result of Arachie and Huang [2021] result requires that one know the actual LF accuracies on the unlabeled datapoints and for there to be an increasing number of LFs to drive down the prediction error. In this paper, we consider a fixed set of LFs and our analysis does not require the actual LF accuracies to be known. + +Label models that are neural networks have also been proposed. For example, Wu et al. [2023] trained a graph NN that implicitly constructs a polytope of labelings using LF outputs and returns the "center" as the aggregated labeling. On the other hand, Cachay et al. [2021] and Ren et al. [2020] incorporate datapoint features along with the usual LF predictions in computing the aggregated labeling. These last two works involve joint models, a unified model that does the work of both label and end models. + +# 3 Preliminaries + +We now provide the formal setup for the problem and briefly discuss HyperLM and BF. We follow the notation of Kim and Ghahramani [2012], Li et al. [2019], Zhang et al. [2023]. + +We desire to label $N$ unlabeled datapoints $X = \{x_{1},\ldots ,x_{N}\}$ by combining the predictions from $W$ LFs, $h^{(1)},\dots,h^{(W)}$ . These LFs can predict one of the $K$ labels in $\mathcal{Z} = \{1,\dots K\}$ or abstain: i.e. $h^{(w)}:X\to \mathcal{Z}\cup \{\ref{eq:10}\}$ , “?” denoting abstention. In particular, our analysis makes no assumptions about the LFs, e.g. no restrictions on their multiplicity, no requirements about their independence given certain events, etc. We’ll abbreviate $h^{(w)}(x_i)$ by $y_{iw}$ . For every datapoint $x_{i}$ , we wish to infer the $(K$ dimensional) underlying conditional label distribution $\vec{\eta}_i\in \Delta_K$ . $\eta_{ij} = \operatorname *{Pr}(z_i = j\mid x_i)$ , the true probability of class $j$ for $x_{i}$ . Following Mazuelas et al. [2020], + +$$ +\vec {\eta} := (\vec {\eta} _ {1}, \dots , \vec {\eta} _ {N}) \in \Delta_ {K} ^ {N}, +$$ + +a vector of length $NK$ . More notation will be introduced when necessary. + +# 3.1 Hyper Label Model + +To select its labeling, HyperLM [Wu et al., 2023] first constructs a finite set $U$ of plausible labelings. Then, the labeling whose squared Euclidean distance is minimized with respect to all elements of $U$ is selected and used. Formally, suppose for this section only that $\mathcal{Z} = \{\pm 1\}$ . $U$ is a subset of all possible hard labels $\mathcal{Z}^N$ such that the there are at least $W / 2$ LFs that are better than random on each class. That is, for class $+1$ (and $-1$ respectively), there are at least $W / 2$ LFs that get $> 50\%$ of their predictions right when they predict $+1$ (-1 respectively). Then, the HyperLM labeling is + +$$ +\vec {g} ^ {h l m} = \frac {1}{| U |} \sum_ {\vec {z} \in U} \vec {z} = \underset {\vec {g} \in [ - 1, 1 ] ^ {N}} {\arg \min} \frac {1}{| U |} \sum_ {\vec {z} \in U} | | \vec {z} - \vec {g} | | _ {2} ^ {2}. +$$ + +By taking the convex hull of $U$ , one can get a polytope of plausible labelings. Then, $\vec{g}^{hlm}$ can be thought of as being the "center". To get $\vec{g}^{hlm}$ in practice, the above authors train a graph neural network that takes LF predictions as input. HyperLM is empirically effective, e.g. see [Zhang et al., 2023], but Wu et al. [2023] did not present theoretical results about its performance or other properties like consistency. We hope to better understand HyperLM's strategy in this paper. + +# 3.2 Balsubramani-Freund with Log-Loss + +The Balsubramani-Freund model (BF) also constructs a set of feasible labelings $P$ and then picks an element to use as its labeling. It has nice theoretical guarantees [An and Dasgupta, 2024], but + +there are two drawbacks. Not only does it need labeled data to construct $P$ , not having enough labels makes $P$ big and BF's performance poor. + +We start by constructing a subset $P \subset \Delta_K^N$ of all possible labelings such that the true labeling $\vec{\eta}$ lies inside $P$ . To approximate $\vec{\eta}$ , we play a zero-sum minimax game as the learner, who tries to minimize the worst case loss a potential adversary can inflict on us. That is, if we select labeling $\vec{g} \in \Delta_K^N$ , the adversary selects $\vec{z} \in P$ that maximizes $-\vec{z}^\top \log \vec{g}$ . + +We can construct $P$ by using $L$ labeled points to bound each LF's accuracy on $X$ with error at most $O(1 / \sqrt{L})$ . Formally, if $h^{(w)}$ makes $N_w \leq N$ predictions, we can with $L$ labeled points construct + +$$ +b _ {w} - \epsilon_ {w} \leq b _ {w} ^ {*} = \frac {1}{N _ {w}} \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {K} \eta_ {i j} \mathbf {1} \left(y _ {i w} = j\right) \leq b _ {w} + \epsilon_ {w} \tag {1} +$$ + +with a binomial confidence interval, where $b_{w}^{*}$ is $h^{(w)}$ 's accuracy on $X$ . Something similar can be done with the class frequencies: $\omega_{j}^{*} = \frac{1}{N}\sum_{i = 1}^{N}\eta_{ij}$ in the interval $[\omega_{j} - \xi_{j},\omega_{j} + \xi_{j}]$ . + +These constraints can easily be written in matrix form. Abuse notation and let $\vec{y}_{iw}$ be the one-hot encoding of $h^{(w)}(x_i)$ . Then, $\vec{y}_{iw} = (\mathbf{1}(h^{(w)}(x_i) = 1),\dots,\mathbf{1}(h^{(w)}(x_i) = K))$ where $\vec{y}_{iw}$ is all zeros if $h^{(w)}(x_i) = ?$ . We write all of $h^{(w)}$ 's predictions as $\vec{y}_w = (\vec{y}_{1w},\ldots ,\vec{y}_{Nw})\in (\Delta_K\cup \{\vec{0}_K\})^N$ . Note that the $\vec{y}_w$ vectors are the only vectors of length $NK$ that have special indexing with three indices. Other vectors of length $NK$ (in $\Delta_K^N$ ) will follow the same style of indexing as $\vec{\eta}$ . Also, we write $\vec{e}_j$ to mean the $j^{th}$ canonical basis vector in $K$ dimensions concatenated $N$ times. To write out constraints, compactly, define $A\in [0,1]^{(W + K)\times NK}$ row wise where + +$$ +A = \left\{ \begin{array}{l l} \frac {1}{N _ {m}} \vec {y} _ {m} & \text {f o r} m \leq W \\ \frac {1}{N} \vec {e} _ {m - W} & \text {o t h e r w i s e .} \end{array} \right. +$$ + +Finally, for brevity, define $\vec{b} \coloneqq (b_1, \ldots, b_W, \omega_1, \ldots, \omega_K)$ . Similarly, $\vec{\epsilon} \coloneqq (\epsilon_1, \ldots, \epsilon_W, \xi_1, \ldots, \xi_K)$ . Then for elementwise inequalities, $P$ can be written as + +$$ +P = \left\{\vec {z} \in \Delta_ {K} ^ {N}: \vec {b} - \vec {\epsilon} \leq A \vec {z} \leq \vec {b} + \vec {\epsilon} \right\}. \tag {2} +$$ + +Now, the zero-sum minimax game referred to earlier can be formally written as + +$$ +\max _ {\vec {g} \in \Delta_ {K} ^ {N}} \min _ {\vec {z} \in P} \vec {z} ^ {\top} \log \vec {g} = \min _ {\vec {z} \in P} \vec {z} ^ {\top} \log \vec {z} \tag {3} +$$ + +where the equality is shown by An and Dasgupta [2024] and proven by them to be an easily. +optimizable convex program. Formally speaking, when one doesn't have enough labeled data, the +elements of $\vec{\epsilon}$ are large, meaning $P$ is large and BF is overly pessimistic. We will soon see how +BBF addresses both drawbacks of BF by implicitly constructing a polytope $P$ and adopting a label +selection strategy similar to that of HyperLM's. + +# 4 Our Proposed Method + +We now described our proposed label model, Bayesian Balsubramani-Freund (BBF). At a high level, we put prior distributions on quantities in BF that are point estimates, specifically the class frequencies and LF accuracies. We'll describe the generative process, present the joint distribution, and discuss how to find BBF's posterior mode, which we'll take as BBF's prediction. + +# 4.1 The Generative Process + +To generate the labels and LF predictions, BBF is similar to iBCC [Kim and Ghahramani, 2012] in that follows the same order (labels first, then LF predictions) and parameterizes LFs by a few parameters. Rather than model the underlying LF accuracy $\operatorname*{Pr}(h^{(w)}(X) = Y)$ like in iBCC, BBF tries to model the accuracy of an LF on $X$ , i.e. $b_{w}^{*}$ . (And similarly for $\operatorname*{Pr}(Y = j), \omega_{j}^{*}$ .) We'll later see BBF's generative process and a simplified version of iBCC's generative process side by side. + +The first step in BBF's generative process is to draw the class frequency distribution $\vec{\tau}$ from a Dirichlet prior with hyperparameters $\vec{\alpha} \in \mathbb{R}_{>0}^{K}$ . Then, the labels $\vec{z} \in \Delta_{K}^{N}$ are drawn. We'll want $\vec{z}$ to assign $\tau_{j}N$ total mass for each class $j \in [K]$ . That is, $\vec{z}$ will be drawn from + +$$ +\mathcal {L} _ {\vec {\tau}} := \left\{\vec {z} \in \Delta_ {K} ^ {N} \colon \frac {1}{N} \sum_ {i = 1} ^ {N} z _ {i j} = \tau_ {j} \text {f o r e a c h} j \in [ K ] \right\}. +$$ + +Rather than draw labelings at random from $\mathcal{L}_{\vec{\tau}}$ , we'll favor labelings with large entropy. This is necessary for our theoretical results. Our distribution for labelings will be denoted $Entropy(\mathcal{L}_{\vec{\tau}})$ which assigns density $\operatorname{Pr}(\vec{z} \mid \vec{\tau}) \propto \exp(-\vec{z}^{\top} \log \vec{z})$ for $\vec{z} \in \mathcal{L}_{\vec{\tau}}$ and 0 otherwise. + +The next step is to draw the accuracies for each LF. A first pass approach is to have a Beta prior with hyperparameters $\vec{\rho}_w\in \mathbb{R}_{>0}^2$ for LF $h^{(w)}$ 's accuracy $b_{w}$ on $X$ . This won't work in general because a Beta random variable's support is [0, 1], but the choice of $\vec{z}$ can make certain accuracies impossible. For example, if $\vec{z} = \vec{1}_{NK} / K$ , then regardless of what $h^{(w)}$ predicts, its accuracy can only be $1 / K$ . On the other hand, if $z_{11} = 1$ and $z_{ij} = 1 / K$ for $i\geq 2$ and $j\in [K]$ , an LF that only predicts on $x_{1}$ can have accuracies in [0, 1] while an LF that doesn't abstain must have accuracy in the range $[(N - 1) / NK,(N - 1) / NK + 1 / N]$ . Thus, we need to restrict the range of accuracies that can be generated. For a fixed labeling $\vec{z}$ , define the range of permissible accuracies with + +$$ +b _ {m a x} := \max _ {i \in [ N ], j \in [ K ]} z _ {i j} \quad \text {a n d} \quad b _ {m i n} := \min _ {i \in [ N ], j \in [ K ]} z _ {i j}. +$$ + +With this, we draw $b_{w}$ from $Beta_{\vec{z}}(\vec{\rho}_{w})$ , which is a $Beta(\vec{\rho}_{w})$ with support restricted to $[b_{min}, b_{max}]$ and density renormalized. Note that the number of predictions $h^{(w)}$ makes can depend on $b_{w}$ . + +Finally, we draw predictions for each of the LFs. For readability, abuse notation and let $N \colon (\Delta_K \cup \{\vec{0}_K\})^N \to \mathbb{R}_{\geq 0}$ be the function that counts the number of predictions made: $N(\vec{y}) = \vec{y}^\top \vec{1}_{NK}$ . $h^{(w)}$ 's prediction is randomly drawn from the set of all possible predictions that achieve accuracy $b_w$ : + +$$ +\mathcal {L} _ {b _ {w}, \vec {z}} := \left\{\vec {y} \in \left(\Delta_ {K} \cup \{\vec {0} _ {K} \}\right) ^ {N} \colon \frac {\vec {y} ^ {\top} \vec {z}}{N (\vec {y})} = b _ {w} \right\}. +$$ + +We now summarize the above and compare it to a simplified version of iBCC's generative process. There, each LF is a biased coin and the datapoint labels/LF predictions are deterministic: $z_{i}, y_{iw} \in \mathcal{Z}$ . I.e. the distribution of $y_{iw} = h^{(w)}(x_i)$ with $z_{i} = j$ is described by $v(b_w, j) \in \Delta_K$ , which has $b_{w}$ in the $j^{th}$ position and $(1 - b_w) / (K - 1)$ elsewhere. + +# BBF Generative Process: + +1. $\vec{\tau} \sim Dirichlet(\vec{\alpha})$ +2. $\vec{z}\sim Entropy(\mathcal{L}_{\vec{\tau}})$ +3. For each LF $w \in [W]$ : + +(a) $b_{w}\sim Beta_{\mathcal{Z}}(\vec{\rho}_{w})$ +(b) $\vec{y}_w\sim \text{Uniform} (\mathcal{L}_{b_w,\vec{z}})$ + +# One-Coin iBCC Generative Process + +1. $\vec{\tau} \sim Dirichlet(\vec{\alpha})$ +2. For each datapoint $i \in [N]$ : + +(a) $z_{i}\sim$ Categorical( + +3. For each LF $w \in [W]$ : +(a) $b_{w}\sim Beta(\vec{\rho}_{w})$ +(b) For each datapoint $i \in [N]$ : + +i. $y_{iw}\sim \mathrm{Categorical}(v(b_w,z_i))$ + +# 4.2 The Joint Distribution + +Now, we write out the joint distribution for BBF. Write $\vec{b} \in [0,1]^W$ the vector of LF accuracies, $\vec{\rho}$ the $2W$ Beta hyperparameters, and $Y$ the set of all LF predictions. We have that + +$$ +\Pr (\vec {z}, \vec {b}, \vec {\tau}, Y \mid \vec {\alpha}, \vec {\rho}) \propto \Pr (Y \mid \vec {z}, \vec {b}) \Pr (\vec {b} \mid \vec {z}, \vec {\rho}) \Pr (\vec {z} \mid \vec {\tau}) \Pr (\vec {\tau} \mid \vec {\alpha}). +$$ + +We now write each of the terms therein. First, $\operatorname{Pr}(\vec{\tau} \mid \vec{\alpha}) \propto \prod_{j=1}^{K} \tau_j^{\alpha_j - 1}$ . Then, + +$$ +\Pr (\vec {z} \mid \vec {\tau}) \propto \exp (- \vec {z} ^ {\top} \log \vec {z}) \prod_ {j = 1} ^ {K} \mathbf {1} \left(\frac {1}{N} \vec {e} _ {j} ^ {\top} \vec {z} = \tau_ {j}\right), \quad \Pr (Y \mid \vec {z}, \vec {b}) \propto \prod_ {w = 1} ^ {W} \mathbf {1} \left(\frac {1}{N (\vec {y})} \vec {y} _ {w} ^ {\top} \vec {z} = b _ {w}\right). +$$ + +To write the probability of the LF accuracies, we will need the Beta distribution's CDF at $r \in [0,1]$ , written as $I_r = I_r(\alpha, \beta)$ for short. To improve readability we drop the hyperparameters, but note that in each term of the following product, the difference of CDFs is specific to each LF. + +$$ +\operatorname * {P r} (\vec {b} \mid \vec {z}, \vec {\rho}) \propto \prod_ {w = 1} ^ {W} \frac {b _ {w} ^ {\rho_ {w 1} - 1} (1 - b _ {w}) ^ {\rho_ {w 2} - 1}}{I _ {b _ {m a x}} - I _ {b _ {m i n}}} +$$ + +# 4.3 Computing the Posterior Label Distribution + +In practice, we know the LF predictions $Y = \{y_{iw} \colon w \in [W], i \in [N]\}$ (along with the hyperparameters) and want to know the posterior label distribution. Thus, we are interested in the quantity $\operatorname*{Pr}(\vec{z},\vec{b},\vec{\tau} \mid Y,\vec{\alpha},\vec{\rho})$ . It turns out that this quantity is only non-zero when every element of $\vec{b}$ and $\vec{\tau}$ are deterministic functions of $\vec{z}$ (when $Y$ is given). This is because the indicator functions in the joint distribution ensure the above is non-zero only when $\tau_j = \vec{e}_j^\top \vec{z} / N$ and $b_w = \vec{y}_w^\top \vec{z} / N(\vec{y})$ for every $j \in [K]$ and $w \in [W]$ . Now, if we use $\tau_j, b_w$ to replace the dot products, we have + +$$ +\Pr (\vec {z}, \vec {b}, \vec {\tau} \mid Y, \vec {\alpha}, \vec {\rho}) \propto \exp (- \vec {z} ^ {\top} \log \vec {z}) \times \prod_ {j = 1} ^ {K} \tau_ {j} ^ {\alpha_ {j} - 1} \prod_ {w = 1} ^ {W} \frac {b _ {w} ^ {\rho_ {w 1} - 1} (1 - b _ {w}) ^ {\rho_ {w 2} - 1}}{I _ {b _ {m a x}} - I _ {b _ {m i n}}}. \tag {4} +$$ + +In the Appendix, we show that the removal of the difference of Beta CDFs will make the RHS log-concave in $\vec{z}$ (with an appropriate choice of hyperparameters). + +Lemma 4.1. The modified posterior label distribution for our proposed model written below is log-concave in $\vec{z}$ when all elements of $\vec{\alpha},\vec{\rho}\geq 1$ + +$$ +\exp (- \vec {z} ^ {\top} \log \vec {z}) \prod_ {j = 1} ^ {K} \left(\frac {\vec {e} _ {j} ^ {\top} \vec {z}}{N}\right) ^ {\alpha_ {j} - 1} \times \prod_ {w = 1} ^ {W} \left(\frac {\vec {y} _ {w} ^ {\top} \vec {z}}{N (\vec {y} _ {w})}\right) ^ {\rho_ {w 1} - 1} \left(1 - \frac {\vec {y} _ {w} ^ {\top} \vec {z}}{N (\vec {y} _ {w})}\right) ^ {\rho_ {w 2} - 1} +$$ + +BBF's prediction $\vec{g}^{bbf}$ will be the mode of the modified posterior distribution, i.e. the solution to + +$$ +\begin{array}{l} \max _ {\vec {z} \in \Delta_ {K} ^ {N}} \left[ - \vec {z} ^ {\top} \log \vec {z} + \sum_ {j = 1} ^ {K} (\alpha_ {j} - 1) \log \left(\frac {\vec {e} _ {j} ^ {\top} \vec {z}}{N}\right) \right. \\ \left. + \sum_ {w = 1} ^ {W} \left[ \left(\rho_ {w 1} - 1\right) \log \left(\frac {\vec {y} _ {w} ^ {\top} \vec {z}}{N (\vec {y} _ {w})}\right) + \left(\rho_ {w 2} - 1\right) \log \left(1 - \frac {\vec {y} _ {w} ^ {\top} \vec {z}}{N (\vec {y} _ {w})}\right) \right] \right] \tag {5} \\ \end{array} +$$ + +which is a concave maximization problem. This can be solved using an off the shelf convex solver. We note that maximizing the modified log posterior without both holding all predictions $Y$ in memory and simultaneously optimizing over all $NK$ variables is an open problem. + +# 5 Statistical Analysis of Our Proposed Model + +We now argue that BBF implicitly constructs a polytope $P \subset \Delta_K^N$ of plausible labelings and selects a labeling inside as its prediction, a la HyperLM. To start, compare BBF's objective (Equation 5) with BF's objective (Equation 3). Both are entropy maximization problems--BF constrains $\vec{z} \in P$ while BBF has no such constraints, but has extra terms in the objective. Observe that we can rewrite the BF problem as $\max_{\vec{z} \in \Delta_K^N} [-\vec{z}^\top \log \vec{z} + M1(\vec{z} \in P)]$ where $M$ is an extremely large constant. In order for $\vec{z}$ to be optimal $M$ must appear, meaning $\vec{z}$ must be in $P$ . We want to argue that the sum of log-pdfs of a Dirichlet and multiple Betas in the BBF objective serve as a continuous approximation to $M1(\vec{z} \in P)$ . Call $\vec{e}_j^\top \vec{z}/N$ and $\vec{y}_w^\top \vec{z}/N(\vec{y}_w)$ the class frequency/LF accuracy induced by $\vec{z}$ . The sum of log-pdfs is maximized when the induced class frequencies/LF accuracies are close to the modes of the Dirichlet/Beta distributions. I.e. close to $(\alpha_j - 1)/\left(\sum_{j'=1}^{K} \alpha_{j'} - K\right)$ , $(\rho_{w1} - 2)/(\rho_{w1} + \rho_{w1} - 2)$ respectively for each $j \in [K]$ , $w \in [K]$ . Thus, for BBF, we may regard its polytope $P'$ as the $\vec{z}$ 's whose induced accuracies are not too far from the mode. For LF accuracies, that is $|\vec{y}_w^\top \vec{z}/N(\vec{y}_w) - (\rho_{w1} - 2)/(\rho_{w1} + \rho_{w1} - 2)| \leq \epsilon_w$ for some $\epsilon_w$ . This is exactly the form of inequality used to specify BF's polytope (Equation 1). For sufficiently large $\vec{\alpha}, \vec{\rho}, \vec{g}^{bf}$ will induce + +class frequencies/LF accuracies close to the modes, which means $g^{\vec{b}\vec{b}f}$ will be in the interior of $P'$ . Note that unlike BF, BBF's performance is affected by duplicate LFs in most cases. It is only in a special case that BBF is not affected by duplicate LFs. However, the following results for BBF do not depend on having a set of unique LFs. With that, we are ready to analyze BBF's form of solution, consistency, and its convergence rates. + +# 5.1 $\vec{g}^{bbf}$ is a Weighted Majority Vote + +Here, we show that $\vec{g}^{bbf}$ is completely specified by $W + K$ real numbers and the LF predictions. In particular, for any datapoint $x_{i}$ , $\vec{g}_{i}^{bbf}$ is the softmax of a weighted majority vote. Let $\vec{\theta} \in \mathbb{R}^{W + K}$ be our weights, the first $W$ being for the LFs, the last $K$ for the class frequencies. We are claiming that + +$$ +g _ {i j} ^ {b b f} = \frac {\exp \left(\sum_ {w = 1} ^ {W} \theta_ {w} y _ {i w j} + \theta_ {W + j}\right)}{\sum_ {j ^ {\prime} = 1} ^ {K} \exp \left(\sum_ {w = 1} ^ {W} \theta_ {w} y _ {i w j ^ {\prime}} + \theta_ {W + j ^ {\prime}}\right)} = \frac {\exp ((A ^ {\top} \vec {\theta}) _ {i j})}{\sum_ {j ^ {\prime} = 1} ^ {K} \exp ((A ^ {\top} \vec {\theta}) _ {i j ^ {\prime}})}. +$$ + +for some $\vec{\theta}$ . By varying $\vec{\theta}$ over the reals, one gets an exponential family of distributions: + +$$ +\mathcal {G} = \{\vec {g} ^ {(\vec {\theta})}: \vec {\theta} \in \mathbb {R} ^ {W + K} \} \quad \text {w h e r e} \quad g _ {i j} ^ {(\vec {\theta})} \propto \exp ((A ^ {\top} \vec {\theta}) _ {i j}). +$$ + +Lemma 5.1. For fixed LF predictions and hyperparameters $\vec{\alpha},\vec{\rho}\geq 1$ elementwise, the BBF prediction $\vec{g}^{bbf}$ gotten by solving Equation 5 is such that $\vec{g}^{bbf}\in \mathcal{G}$ . + +Now, we analyze the behavior of $\vec{g}^{bbf}$ as $\vec{\alpha},\vec{\rho}$ change and the LF predictions/ $\vec{\eta}$ remain fixed. + +# 5.2 Consistency + +While BBF's prediction must lie in $\mathcal{G}$ , $\vec{\eta}$ in general doesn't lie in $\mathcal{G}$ . Thus, the best we can hope for is to get the best approximator of $\vec{\eta}$ from $\mathcal{G}$ , i.e. $\vec{g}^{*} = \vec{g}^{(\vec{\theta}^{*})} = \arg \min_{\vec{g} \in G} d(\vec{\eta}, \vec{g})$ for distance function $d(\cdot, \cdot)$ . KL divergence will be the distance we choose. For $\vec{\mu}, \vec{\nu} \in \Delta_K^N$ , recall they are each concatenations of $N$ distributions, so + +$$ +d (\vec {\mu}, \vec {\nu}) = \sum_ {i = 1} ^ {N} K L (\vec {\mu} _ {i}, \vec {\nu} _ {i}) = \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {K} \mu_ {i j} \log \left(\frac {\mu_ {i j}}{\nu_ {i j}}\right). +$$ + +Now we'll informally argue that BBF is consistent inasmuch as we can make $\vec{g}^{bbf}$ tend to $\vec{g}^*$ . In the beginning of this section, we had argued that BBF implicitly constructs a polytope $P'$ and picks a labeling from inside $P'$ . It turns out for a specific polytope and a specific strategy of selecting a labeling from inside, one can obtain $\vec{g}^*$ . Consider the polytope $P^* = \{\vec{z} \in \Delta_K^N \colon A\vec{z} = \vec{b}^*\}$ , the set of all labelings that induce the empirical LF accuracies/empirical class frequencies. An and Dasgupta [2024] showed (in Theorem 6) that if BF is given $P^*$ , then BF will return $\vec{g}^*$ . Our result for BBF shows how to make the implicit polytope of BBF to tend to $P^*$ whereupon $\vec{g}^{bbf} \rightarrow \vec{g}^*$ . In the Appendix, we show that BBF satisfies a modified version of BF's KKT conditions (for a fixed implicit polytope) where the complementary slackness condition is not satisfied. However, as we make BBF's implicit polytope tend to $P^*$ , the complementary slackness conditions will be satisfied, meaning BBF replicates the behavior of BF with $P^*$ . Only then is BBF unaffected by duplicate LFs. + +To make BBF's implicit polytope tend to $P^*$ , we need to make it so whenever the induced class frequencies/LF accuracies are not close to $w_j^*, b_w^*$ the resulting contribution from the sum of the log-pdfs is small. For convenience, define $\alpha_0 = \sum_{j=1}^{K} \alpha_j$ . + +Theorem 5.2. If for each $w \in [W]$ , $\rho_{w1}, \rho_{w2} \to \infty$ , $|\rho_{w1} - 1 - b_w^* (\rho_{w1} + \rho_{w2} - 2)| \to 0$ and for each $j \in [K]$ , $\alpha_j \to \infty$ and $|\alpha_j - 1 - \omega_j^* (\alpha_0 - K)| \to 0$ , then $\vec{g}^{bbf} \to \vec{g}^*$ . + +Note that this is not the Bayesian notion of consistency. There, one has appropriately selected priors with fixed hyperparameters and the requirement is the number of observations (e.g. LF predictions) tends to infinity [Miller, 2018]. For BBF, our proposed model only has 1 observation, the set of all LF predictions on $X$ , meaning we cannot take advantage of the vast literature on Bayesian consistency. + +# 5.3 Rates of Convergence + +We'll now see $\vec{g}^{bbf}$ 's rate of convergence to $\vec{g}^*$ . Specially, we'll bound $d(\vec{g}^*,\vec{g}^{bbf})$ by a function of the weights that produce $\vec{g}^*$ , $\vec{g}^{bbf}$ , call them $\vec{\theta}^*$ , $\vec{\theta}^{bbf}$ , and how well $\vec{g}^{bbf}$ estimates the actual LF accuracies/class frequencies. + +Theorem 5.3. Suppose $\vec{\alpha},\vec{\rho}\geq 1$ elementwise. Then for elementwise absolute value, + +$$ +d \left(\vec {g} ^ {*}, \vec {g} ^ {b b f}\right) \leq \left(\left| \vec {\theta} ^ {*} \right| + \left| \vec {\theta} ^ {b b f} \right|\right) ^ {\top} \left| \vec {b} ^ {*} - A \vec {g} ^ {b b f} \right|. +$$ + +When the conditions of Theorem 5.2 hold, $\vec{\theta}^{bbf}\rightarrow \vec{\theta}^*$ + +It's unclear to us how one bounds $\vec{\theta}^{bbf}$ or how well $A\vec{g}^{bbf}$ approximates $b^*$ , meaning we cannot say on what order BBF's convergence rate is. We note that duplicate LFs will increase the size of the bound as each duplicate LF adds an extra term. Now, if an unsupervised instantiation of $\vec{\alpha}, \vec{\rho}$ induces $\vec{g}^{bbf}$ such that $A\vec{g}^{bbf}$ is close to $b^*$ , then the excess error will be small. Experimental results show that BBF performs well in the unsupervised setting and often has small excess error compared to $\vec{g}^*$ . + +# 6 Experiments + +Here, we compare the performance of BBF against other baseline methods on eleven real datasets while also showing BBF's consistency on real datasets. Experiments were run in Python 3.6.13 (PSF) [Van Rossum and Drake, 2009] mainly using NumPy 1.19.5 (modified BSD) [Harris et al., 2020] with an AMD Ryzen R9 5950x, 128GB RAM, and an Nvidia RTX 2080Ti. One may find the code in the supplementary materials or at https://github.com/stevenan5/ bayesian-bf-neurips-2025. + +# 6.1 Methods + +We compare our proposed method with twelve other methods, representing different approaches to creating the label model. For our method BBF, we set the prior hyperparameters as $\vec{\alpha} = \vec{1}_K$ , $\vec{\rho}_w = (4,1)$ for each $w \in [W]$ . This follows the initialization of [Li et al., 2019] for their Bayesian method, i.e. we assume each LF makes 4 correct predictions and 1 wrong prediction. Note that there are no requirements on the LF accuracies for BBF. Across all LFs used in the experiments, the accuracies range from 0.0366 to 1. To compute the BBF prediction, we optimize Equation 5 using CVXPY 1.4.1 (Apache 2.0) [Diamond and Boyd, 2016] by way of MOSEK 10.1.16 (personal academic license) [MOSEK ApS, 2022] with default parameters. + +The other twelve methods are from the WRENCH benchmark version 1.1.2rc0 (Apache 2.0) [Zhang et al., 2021]. They are: majority vote (MV), Dawid-Skene (DS) [Dawid and Skene, 1979], Data Programming (DP) [Ratner et al., 2016], a generalization of DS that accounts for some LF dependencies, Flying Squid (FS) [Fu et al., 2020], a fast method that uses an Ising model, MeTaL [Ratner et al., 2019], a matrix completion type method, iBCC [Kim and Ghahramani, 2012], the Bayesian generalization of DS, EBCC [Li et al., 2019], a generalization of iBCC, but where each LF is modeled by multiple confusion matrices, FABLE [Zhang et al., 2023], a generalization of EBCC which can account for datapoint features, Constrained Label Learning (CLL) [Arachie and Huang, 2021], a polytope based method with an adversarial flavor, Hyper Label Model (HyperLM) [Wu et al., 2023], a graph neural network based method, WeaSEL [Cachay et al., 2021], and Denoise [Ren et al., 2020], two neural network based methods using attention with different losses. We ported CLL into WRENCH and following Arachie and Huang [2021], we initialized LF errors to be 0.01 if the LF coverage was low (i.e. low number of datapoints predicted on) and $\frac{1}{K}$ if the LF coverage was high. For all other methods that require initialization, we use the defaults provided in WRENCH. + +# 6.2 Datasets + +We used eleven datasets from WRENCH, which also provided the LF predictions. The following datasets had licenses we could find online and were different from WRENCH's license: YouTube, SMS (CC BY 4.0), SemEval (CC BY 3.0). For FABLE, Denoise, WeaSEL, we use the original datapoint features, unless they were textual, in which case RoBERTa [Liu et al., 2019] was used to extract them. The methods are evaluated in a transductive setting. The provided train/validation/test + +Table 1: Some dataset statistics. Note that we only count datapoints with at least one LF prediction. + +
DatasetIMDBYoutubeSMSCDRYelpCommercialTennisTRECSemEvalChemProtAG News
# Class (K)222222269104
# LF (W)510733384468164269
# Datapoints (N)21,9141,8432,24612,56231,51281,1058,8035,7382,52713,76882,591
+ +Table 2: Comparison of our proposed method with baseline methods (higher is better). Bolded entries are indistinguishable via two-tailed paired t-test with $p = 0.05$ . + +
DatasetIMDBYouTubeSMSCDRYelpCommercialTennisTRECSemEvalChemProtAG News
LossF1F1F1F1F1F1F10-10-10-10-1
MV75.1984.5678.8867.2878.2284.4683.8354.5179.0754.6781.38
DS65.0883.5665.3057.9075.2088.3183.5647.6673.5352.6681.79
DP74.8478.6257.8241.7871.6178.6183.6145.8572.7856.1781.71
FS75.5183.0770.5569.1278.6780.8683.3150.1012.1552.3781.28
MeTaL74.7876.3448.3917.6970.1478.6283.6243.7354.5855.8681.84
iBCC75.330.000.000.0075.8376.8384.1621.0230.1935.0926.57
EBCC75.320.000.0024.8176.5776.8384.1921.0230.1935.0927.81
CLL74.8084.7978.8364.3377.8036.3842.3159.3484.2152.3780.97
HyperLM75.2991.4078.9070.5478.7382.6084.2160.0284.2152.3281.38
WeaSEL53.0761.8526.4538.5061.9140.005.7420.8111.1421.4132.29
Denoise81.0579.8479.0670.9480.9789.7151.1249.8173.5357.8587.61
Fable74.9787.8078.7562.4077.0385.9183.5953.6473.6154.4281.38
BBF74.6789.7178.7762.3977.6785.0279.8759.8884.2152.4781.40
Improve5.866.840.183.662.755.250.385.515.143.186.23
g*75.5394.2278.8372.2578.8388.3183.4363.7584.2161.2682.24
+ +splits are combined and points with no LF predictions are removed. Each label model is given LF predictions on the datapoints (and the datapoint features themselves if used). No labeled data is provided to the label models. Label models are expected to be initialized and provide a labeling without using any labels. Labels are only used to judge the quality of each label model's predictions by way of F1 score/0-1 loss. Tasks covered by the datasets range from sentiment/spam/bio relation/video frame/chemical relation/topic classification. See Table 1 for some statistics. + +# 6.3 Comparison with Other Label Models + +For every method that involves randomness, we run it ten times and display the average performance in Table 2. Bolded entries represent ones that are indistinguishable from the best result via two-tailed paired t-test with $p = 0.05$ . WeaSEL, Denoise, and FABLE are the only methods that use datapoint features – all others only use LF predictions. The second to last row, denoted “Improve” shows the improvement of the best label model over majority vote. The last row shows $\vec{g}^*$ (computed via Theorem 6 of [An and Dasgupta, 2024]), a proxy for the ceiling of BBF's performance. See the Appendix for the complete results (0-1 loss on all datasets, standard deviations of losses). + +Against the highest performing Bayesian method FABLE, BBF often matches or outperforms, only significantly underperforming on Tennis. This is notable because FABLE uses datapoint features and is much more expressive than BBF. On Tennis, the three LFs that each predicted on $99\%$ of the datapoints closely matched each other, i.e. had $>85\%$ matching predictions. This effectively reduces the number of constraints on BBF's implicit polytope because three constraints are almost duplicates. + +Compared to HyperLM, BBF is competitive except for CDR and Tennis. On CDR, the number of LF predictions per datapoint was very low, averaging 2.3 predictions per datapoint despite there being 33 LFs. Two predictions per datapoint means BBF's implicit polytope has very few constraints per datapoint, likely diminishing the performance. + +Against Denoise, we see the limitations of representing an LF by its accuracy - even with $\vec{g}^*$ , BBF is worse on IMDB, SMS, Yelp, etc. For ChemProt specifically, we think BBF is worse than Denoise because the average LF accuracy is just under 0.47, which differs greatly from our mean prior accuracy of 0.8. For almost every other dataset, the average accuracy of the LFs was within 0.11 of 0.8. SemEval was the outlier with average LF accuracy of 0.97. + +![](images/ef6def7a2633a20ead4c56430e7fe93965c6951d7784ae1d85d10dbf4b29856d.jpg) +(a) Yelp + +![](images/eb90012a6f3ed1f5d83c320d606e953b6d88837fd6e29ac8ee6ed900cafbc626.jpg) +(b) SemEval +Figure 1: Demonstrations of BBF's consistency on real datasets. + +For all other methods, BBF has similar if not better performance on a large majority of the datasets. The exceptions are on CDR against FS, Commercial against DS, Tennis against almost every method, and ChemProt against DP/FABLE. This is evidence of BBF's strength as a label model. + +Since BBF employs a similar strategy to HyperLM, we now quickly discuss the performance differences between HyperLM and Denoise. On IMDB, Commercial, ChemProt, AGNews, HyperLM performs worse than Denoise, so much so that it's close to majority vote in performance. We hypothesize that the "center" of the polytope constructed by HyperLM might be close to the majority vote labeling. This is because we could not find a pattern between whether HyperLM's assumptions were met, the number and/or quality of the LFs provided, and whether HyperLM underperformed against Denoise. In the cases where Denoise underperformed against HyperLM, we believe this is because Denoise might get stuck in poor local optima as its strategy for learning is similar to that of bootstrapped co-training [Blum and Mitchell, 1998]. + +# 6.4 Consistency of BBF + +To show that BBF is consistent on real datasets, we choose $\alpha_{j} = s\omega_{j}^{*} + 1$ , $\vec{\rho}_{w} = (sb_{w}^{*} + 1,s(1 - b_{w}^{*}) + 1)$ with $s > 0$ . By having $s$ take values in $\{10^1,10^2,\dots ,10^5\}$ , we simulate the conditions of Theorem 5.2. Figure 1 shows $d(\vec{g}^{*},\vec{g}^{bf}) / N$ on a log-log scale graph on two datasets. We see the convergence is at worst linear as a function of $s$ . See Appendix for complete consistency results. + +# 7 Limitations + +We have seen a new label model that is adversarial, unsupervised, has strong empirical performance and enjoys nice theoretical guarantees. This was done by combining HyperLM and BF's strategies. + +However, there are still some unsolved problems. On the empirical front, one must hold $O(WN)$ items in memory and optimize $NK$ variables simultaneously to get $\vec{g}^{bbf}$ . This is infeasible when $N$ is large. We also noticed in our experiments that $||\vec{g}^{bbf}||_{\infty}$ was small, meaning BBF can be poorly calibrated. On the theoretical front, we are not able to control $\vec{\theta}^{bbf}$ , meaning we do not know at which rate $\vec{g}^{bbf} \to g^*$ a priori. Due to $\vec{g}^{bbf}$ not being a minimax optimal solution, we lose the guarantee that $-\vec{\eta}^{\top} \log \vec{g}^{bbf} \leq -\vec{g}^{bbf\top} \log \vec{g}^{bbf}$ . I.e. we don't have a bound for the cross entropy between $\vec{\eta}$ and $\vec{g}^{bbf}$ which is only in terms of $\vec{g}^{bbf}$ . + +# Acknowledgments and Disclosure of Funding + +The author thanks the National Science Foundation for its support under grant IIS-2211386, Sanjoy Dasgupta for suggesting the Balsubramani-Freund model be given a Bayesian generalization, and the anonymous reviewers for their time and feedback, which indubitably improved the paper. + +# References + +Steven An and Sanjoy Dasgupta. Convergence Behavior of an Adversarial Weak Supervision Method. In Negar Kiyavash and Joris M. Mooij, editors, Proceedings of the Fortieth Conference on Uncertainty in Artificial Intelligence, volume 244 of Proceedings of Machine Learning Research, pages 1-49. PMLR, 15-19 Jul 2024. +Chidubem Arachie and Bert Huang. Adversarial Label Learning. Proceedings of the AAAI Conference on Artificial Intelligence, 33(01):3183-3190, Jul. 2019. +Chidubem Arachie and Bert Huang. 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URL https://arxiv.org/abs/2202.05433. + +Jieyu Zhang, Linxin Song, and Alex Ratner. Leveraging Instance Features for Label Aggregation in Programmatic Weak Supervision. In Francisco Ruiz, Jennifer Dy, and Jan-Willem van de Meent, editors, Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, volume 206 of Proceedings of Machine Learning Research, pages 157-171. PMLR, 25-27 Apr 2023. + +# A Appendix Overview + +Here, we provide the missing proofs along with extra experimental results. After a brief notation recap, the rest of the Appendix is organized as follows: + +- Section C, results involving the Bayesian Balsubramani-Freund (BBF) posterior label distribution, log-concavity (Lemma 4.1). +- Section D, an overview of the proof strategy used to show the remaining theoretical results. A table of contents detailing major steps of the main proof is provided there. The proofs will be written for LFs that don't abstain, but they'll work for LFs that do abstain. +- Section E, extra experimental results. + +- Subsection E.1, extra tables showing 0-1 loss for all datasets as well as the standard deviation of losses. +- Subsection E.2, extra experimental results showing BBF's consistency on real datasets. + +# B Notation Recap + +We would like to label $N$ points in the set $X = \{x_{1},\ldots ,x_{N}\}$ which each have one of $K$ labels. The set of labels in consideration is $\mathcal{Z} = \{1,\dots ,K\} = [K]$ . At our disposal are $W$ labeling functions (LFs) $h^{(1)},\ldots h^{(W)}\colon X\to \mathcal{Z}\cup \{\text{？}\}$ where “?” denotes abstention. We seek to infer the true conditional label probabilities for each point $x_{i}$ : + +$$ +\eta_ {i j} = \Pr (z = j \mid x _ {j}) \quad \text {s o t h a t} \quad \vec {\eta} _ {i} = \left(\eta_ {i 1}, \dots , \eta_ {i K}\right) \in \Delta_ {K}. +$$ + +For convenience, we'll concatenate all $N$ of these distributions: + +$$ +\vec {\eta} = \left(\vec {\eta} _ {1}, \dots , \vec {\eta} _ {N}\right) \in \Delta_ {K} ^ {N}. +$$ + +Similarly, we'll represent LF predictions by $N$ probability distributions which are each over $K$ elements. For datapoint $x_{i}$ , the one-hot encoding of LF $h^{(w)}$ 's prediction is denoted + +$$ +\vec {y} _ {i w} = \left(\mathbf {1} \left(h ^ {(w)} \left(x _ {i}\right) = 1\right), \dots , \mathbf {1} \left(h ^ {(w)} \left(x _ {i}\right) = K\right)\right) \in \Delta_ {K} \cup \{\vec {0} _ {k} \} +$$ + +where $\vec{0}_k$ is a zeros vector of length $K$ . We note that for the theory, $\vec{y}_{iw}$ doesn't need to be a vector of zeros and ones. It can be a bona fide distribution without any effect to the results shown below. The set of all predictions for this LF is + +$$ +\vec {y} _ {w} = \left(\vec {y} _ {1 w}, \dots , \vec {y} _ {N w}\right) \in \left(\Delta_ {K} \cup \{\vec {0} _ {K} \}\right) ^ {N}. +$$ + +Two sets of important quantities are the LF accuracies and the class frequency distribution. Suppose $h^{(w)}$ predicts on $N_w \leq N$ points on $X$ . What we refer to as the actual accuracy of LF $h^{(w)}$ on $X$ is + +$$ +b _ {w} ^ {*} = \frac {1}{N _ {w}} \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {K} \eta_ {i j} \mathbf {1} (h ^ {(w)} (x _ {i} = j)). +$$ + +In dot product form, this is $\vec{y}_w^\top \vec{\eta} / N_w$ . The actual class frequency distribution on $X$ is + +$$ +\omega_ {j} ^ {*} = \frac {1}{N} \sum_ {i = 1} ^ {N} \eta_ {i j}. +$$ + +Letting $\vec{e}_j \in \Delta_K^N$ be the concatenation of $N$ copies of the $j^{th}$ canonical basis vector in $K$ dimensions, the above can be written as $\vec{e}_j^\top \vec{\eta} / N$ . We sometimes call the starred quantities empirical - i.e. the empirical LF accuracies or the empirical class frequency distribution. + +The computation of these values can be written as a matrix vector multiplication. This matrix will be called $A \in [0,1]^{(W + K) \times NK}$ . Let the first $W$ rows be $\vec{y}_w / N_w$ for $w \in [W]$ and $\vec{e}_j$ ( $j \in [K]$ ) for rows $W + 1, \ldots, W + K$ . Then, $A\vec{\eta} \in [0,1]^{W + K}$ is $(b_1^*, \ldots, b_W^*, \omega_1^*, \ldots, \omega_K^*)$ . To save space, we'll define + +$$ +\bar {b} ^ {*} = \left(b _ {1} ^ {*}, \dots , b _ {W} ^ {*}, \omega_ {1} ^ {*}, \dots , \omega_ {K} ^ {*}\right). +$$ + +This will also be done for error bars: $\vec{\epsilon} \in [0,1]^{W + K}$ where we might write the elementwise inequality $\vec{b}^* - \vec{\epsilon} \leq A\vec{\eta} \leq \vec{b}^* + \vec{\epsilon}$ . + +The Dirichlet prior (for BBF's class frequency) has hyperparameters $\alpha \in \mathbb{R}_{>0}^{K}$ . The Beta priors (for LF accuracies) has hyperparameters $\vec{\rho}_w \in \mathbb{R}_{>0}^2$ . Variables that are important for the proofs below are $\vec{\tau}$ , the generated class frequency distribution, and $\vec{b}$ , the generated LF accuracies. The generative process for BBF is as follows: + +1. $\vec{\tau} \sim Dirichlet(\vec{\alpha})$ +2. $\vec{z}\sim Entropy(\mathcal{L}_{\vec{\tau}})$ +3. For each LF $w \in [W]$ : + +(a) $b_{w}\sim Beta_{\vec{z}}(\vec{\rho}_{w})$ +(b) $\vec{y}_w\sim \text{Uniform} (\mathcal{L}_{b_w,\vec{z}})$ + +# C BBF Posterior Label Distribution Concavity (Lemma 4.1) + +We'll first manipulate the posterior distribution explicitly to bring about the terms shown in the main paper. Then, a counterexample will be given to show that the BBF posterior is not log-concave in $\vec{z} \in \Delta_K^N$ (the labeling). Finally, we show how a modification makes the posterior log concave. + +Let $Y$ be the LF predictions from our $W$ LFs, $\vec{\alpha} \in \mathbb{R}^{K}$ be the class frequency Dirichlet hyperparameters, $\vec{\rho}$ be the Beta hyperparameters for each LF. If we include everything, the posterior probability for our parameters $\vec{z}$ (labeling), $\vec{b}$ (LF accuracies), $\vec{\tau}$ (class frequencies) is + +$$ +\operatorname * {P r} (\vec {z}, \vec {b}, \vec {\tau} \mid Y, \vec {\alpha}, \vec {\rho}) = \frac {\operatorname * {P r} (\vec {z} , \vec {b} , \vec {\tau} , Y , \vec {\alpha} , \vec {\rho})}{\operatorname * {P r} (Y , \vec {\alpha} , \vec {\rho})} = \frac {\operatorname * {P r} (\vec {z} , \vec {b} , \vec {\tau} , Y , \vec {\alpha} , \vec {\rho})}{\operatorname * {P r} (Y \mid \vec {\alpha} , \vec {\rho}) \operatorname * {P r} (\vec {\alpha} , \vec {\rho})} \propto \frac {\operatorname * {P r} (\vec {z} , \vec {b} , \vec {\tau} , Y , \vec {\alpha} , \vec {\rho})}{\operatorname * {P r} (\vec {\alpha} , \vec {\rho})}. +$$ + +We'll now repeatedly apply the chain rule to the numerator, then drop variables which are independent. + +$$ +\Pr (\vec {z}, \vec {b}, \vec {\tau}, Y, \vec {\alpha}, \vec {\rho}) = \Pr (Y \mid \vec {z}, \vec {b}, \vec {\tau}, \vec {\alpha}, \vec {\rho}) \Pr (\vec {b} \mid \vec {z}, \vec {\tau}, \vec {\alpha}, \vec {\rho}) \Pr (\vec {z} \mid \vec {\tau}, \vec {\alpha}, \vec {\rho}) \Pr (\vec {\tau} \mid \vec {\alpha}, \vec {\rho}) \Pr (\vec {\alpha}, \vec {\rho}) +$$ + +Since the generation of the LF predictions $Y$ only depends on $\vec{z}$ and $\vec{b}$ , all other terms can be dropped. Similarly, $\vec{b}$ only depends on $\vec{z}$ and $\vec{\rho}$ , $\vec{z}$ only depends on $\vec{\tau}$ , and $\vec{\tau}$ only depends on $\vec{\alpha}$ . The above is then equal to + +$$ +\Pr (\vec {z}, \vec {b}, \vec {\tau}, Y, \vec {\alpha}, \vec {\rho}) = \Pr (Y \mid \vec {z}, \vec {b}) \Pr (\vec {b} \mid \vec {z}, \vec {\rho}) \Pr (\vec {z} \mid \vec {\tau}) \Pr (\vec {\tau} \mid \vec {\alpha}) \Pr (\vec {\alpha}, \vec {\rho}). +$$ + +We have + +$$ +\Pr (\vec {z}, \vec {b}, \vec {\tau} \mid Y, \vec {\alpha}, \vec {\rho}) \propto \Pr (Y \mid \vec {z}, \vec {b}) \Pr (\vec {b} \mid \vec {z}, \vec {\rho}) \Pr (\vec {z} \mid \vec {\tau}) \Pr (\vec {\tau} \mid \vec {\alpha}). +$$ + +We can now show what each term is proportional to. Let $\mathbf{1}(\cdot)$ be the indicator function and recall that $\vec{y}_w\in \Delta_K^N$ is LF $w$ 's predictions. Also, recall that $\vec{e}_j$ is the $j^{th}$ canonical basis vector in $K$ dimensions concatenated $N$ times. (Elements $(i - 1)K + j$ are 1 for $i\in [N]$ , all other elements are 0.) Recall that we've abused notation and have $N(\cdot)$ count the number of predictions an LF makes. I.e. for $\vec{y}\in (\Delta_K^N\cup \{\vec{0}_K\})^N$ we have $N(\vec{y}) = \vec{y}^\top \vec{1}_{NK}$ . Observe the following: + +$$ +\Pr (Y \mid \vec {z}, \vec {b}) \propto \prod_ {w = 1} ^ {W} \mathbf {1} \left(\frac {1}{N (\vec {y} _ {w})} \vec {y} _ {w} ^ {\top} \vec {z} = b _ {w}\right), \quad \Pr (\vec {z} \mid \vec {\tau}) \propto \exp (- \vec {z} ^ {\top} \log \vec {z}) \prod_ {j = 1} ^ {K} \mathbf {1} \left(\frac {1}{N} \vec {e} _ {j} ^ {\top} \vec {z} = \tau_ {j}\right), +$$ + +Let $\rho_{w1},\rho_{w2}$ represent the first $(\alpha)$ and second $(\beta)$ Beta hyperparameters for LF $h^{(w)}$ + +$$ +\operatorname * {P r} (\vec {\tau} \mid \vec {\alpha}) \propto \prod_ {j = 1} ^ {K} \tau_ {j} ^ {\alpha_ {j} - 1}, \qquad \operatorname * {P r} (\vec {b} \mid \vec {z}, \vec {\rho}) \propto \prod_ {w = 1} ^ {W} \frac {b _ {w} ^ {\rho_ {w 1} - 1} (1 - b _ {w}) ^ {\rho_ {w 2} - 1}}{I _ {b _ {m a x}} - I _ {b _ {m i n}}} +$$ + +where + +$$ +b _ {m a x} := \max _ {i \in [ N ], j \in [ K ]} z _ {i j} \quad \text {a n d} \quad b _ {m i n} := \min _ {i \in [ N ], j \in [ K ]} z _ {i j} +$$ + +and $I_{x}$ is shorthand for the Beta CDF with hyperparameters $\alpha, \beta$ : + +$$ +I _ {x} = I _ {x} (\alpha , \beta) = \frac {B (x ; \alpha , \beta)}{B (\alpha , \beta)} = \frac {1}{B (\alpha , \beta)} \int_ {0} ^ {x} t ^ {\alpha - 1} (1 - t) ^ {\beta - 1} d t. +$$ + +Putting everything together, the posterior is proportional to the following + +$$ +\begin{array}{l} \Pr (\vec {z}, \vec {b}, \vec {\tau} \mid Y, \vec {\alpha}, \vec {\rho}) \propto \prod_ {w = 1} ^ {W} \mathbf {1} \left(\frac {1}{N (\vec {y} _ {w})} \vec {y} _ {w} ^ {\top} \vec {z} = b _ {w}\right) \prod_ {w = 1} ^ {W} \frac {b _ {w} ^ {\rho_ {w 1} - 1} (1 - b _ {w}) ^ {\rho_ {w 2} - 1}}{I _ {b _ {m a x}} - I _ {b _ {m i n}}} \\ \times \exp (- \vec {z} ^ {\top} \log \vec {z}) \prod_ {j = 1} ^ {K} \mathbf {1} \left(\frac {1}{N} \vec {e} _ {j} ^ {\top} \vec {z} ^ {\prime} = \tau_ {j}\right) \prod_ {j = 1} ^ {K} \tau_ {j} ^ {\alpha_ {j} - 1}. \\ \end{array} +$$ + +Now, we show that the posterior distribution as stated above is not log-concave. Specifically, the term $1 / (I_{b_{max}} - I_{b_{min}})$ is what gives us trouble as that difference depends on $\vec{z}$ . + +Lemma C.1. The posterior parameter distribution (over $\vec{z},\vec{b},\vec{\tau}$ ) of our model is not log-concave in $\vec{z}$ . + +Proof. We consider the case where $\operatorname{Pr}(\vec{z},\vec{b},\vec{\tau}\mid Y,\vec{\alpha},\vec{\rho},L) > 0$ , meaning $\vec{b}$ and $\vec{\tau}$ can be written in terms of $\vec{z}$ . The problem comes from the distributions from which the accuracies are drawn, $Beta_{\vec{z}}(\alpha ,\beta)$ . Suppose we fixed the LF index $w$ and dropped its index in $\vec{y}_w$ . Also, say $\alpha = \rho_{w1}$ and $\beta = \rho_{w2}$ to make things cleaner. Finally, lets suppose that LF $h^{(w)}$ predicts on all $N$ points. The pdf of $Beta_{\vec{z}}(\alpha ,\beta)$ for $\frac{1}{N}\vec{y}^{\top}\vec{z}\in [0,1]$ is then + +$$ +\frac {(\frac {1}{N} \vec {y} ^ {\top} \vec {z}) ^ {\alpha - 1} (1 - \frac {1}{N} \vec {y} ^ {\top} \vec {z}) ^ {\beta - 1}}{(I _ {b _ {m a x}} - I _ {b _ {m i n}}) B (\alpha , \beta)}. +$$ + +For this distribution to be log concave, the logarithm of the pdf must be concave. Only looking at the non-constant terms, + +$$ +(\alpha - 1) \log \left(\frac {1}{N} \vec {y} ^ {\top} \vec {z}\right) + (\beta - 1) \log \left(1 - \frac {1}{N} \vec {y} ^ {\top} \vec {z}\right) - \log \left(I _ {b _ {m a x}} - I _ {b _ {m i n}}\right) +$$ + +has to be concave. + +We can show an instantiation of the above fails to satisfy the inequality required for concavity. Define $f(\vec{z}; \alpha, \beta, \vec{y})$ as the above. (Recall $I_{b_{\text{max}}}, I_{b_{\text{min}}}$ are functions of $\vec{z}$ .) Take $\alpha = 10$ , $\beta = 6$ , $\theta = 0.5$ , $\vec{z} = (0.9, 0.1)^{\top}$ , $\vec{z}' = (0.55, 0.45)^{\top}$ , and $\vec{y} = (0.5, 0.5)^{\top}$ ( $K = 2$ , $N = 1$ ). For the pdf of $Beta_{\vec{z}}(\alpha, \beta)$ to be log concave, we must have + +$$ +f \left(\theta \vec {z} + (1 - \theta) \vec {z} ^ {\prime}; \alpha , \beta , \vec {y}\right) \geq \theta f (\vec {z}; \alpha , \beta , \vec {y}) + (1 - \theta) f \left(\vec {z} ^ {\prime}; \alpha , \beta , \vec {y}\right). +$$ + +For the given values, the right hand side is bigger than the left hand side by about 0.612. Thus, we have an instance where the required inequality for log-concavity does not hold. + +If we just remove the $1 / (I_{b_{max}} - I_{b_{min}})$ term from the Bayesian BF parameter posterior (Equation 4), then the resulting expression will be log concave. We'll still call the result a "posterior". + +Lemma C.2 (Lemma 4.1). If we remove the normalization constant for $\mathrm{Beta}_{\vec{z}}(\alpha, \beta)$ , i.e. the $I_{b_{max}} - I_{b_{min}}$ quantity, then the resulting "posterior" as follows is log-concave in $\vec{z}$ given that each element of $\vec{\rho}$ and $\vec{\alpha}$ is at least 1. + +$$ +\exp (- \vec {z} ^ {\top} \log \vec {z}) \prod_ {w = 1} ^ {W} \left(\frac {1}{N (\vec {y} _ {w})} \vec {y} _ {w} ^ {\top} \vec {z}\right) ^ {\rho_ {w 1} - 1} \left(1 - \frac {1}{N (\vec {y} _ {w})} \vec {y} _ {w} ^ {\top} \vec {z}\right) ^ {\rho_ {w 2} - 1} \prod_ {j = 1} ^ {K} \left(\frac {1}{N} \vec {e} _ {j} ^ {\top} \vec {z}\right) ^ {\alpha_ {j} - 1} +$$ + +The condition stated in the above lemma is essentially the condition for a Beta/Dirichlet pdf to be log-concave. + +Before showing this, we need a supporting lemma. + +Lemma C.3. Suppose $r \geq 0$ , $\vec{z} \in (\Delta_K \cup \vec{0}_K)^N$ , and $\frac{1}{N(\vec{y})}\vec{y}^\top \vec{z} > 0$ . Then, + +$$ +f (\vec {y}) = \left(\frac {1}{N (\vec {y})} \vec {y} ^ {\top} \vec {z}\right) ^ {r} +$$ + +is log-concave in $\vec{z}$ . + +Proof. Since $r$ is non-negative, it suffices to show that + +$$ +\log \left(\frac {1}{N (\vec {y})} \vec {y} ^ {\top} \vec {z}\right) +$$ + +is concave. From Boyd and Vandenberghe [2004], page 84, we have a non-decreasing function (log) and a concave function (dot product). Therefore, their composition is concave in $\vec{z}$ . + +Remark C.4. The above Lemma also holds if the argument inside the logarithm is $1 - \vec{y}^{\top}\vec{z} /N(\vec{y})$ because this is also concave in $\vec{z}$ . For our purposes, that difference will always be non-negative. + +Proof of Lemma C.2. + +$$ +\exp \left(- \vec {z} ^ {\top} \log \vec {z}\right) \prod_ {w = 1} ^ {W} \left(\frac {1}{N (\vec {y})} \vec {y} _ {w} ^ {\top} \vec {z}\right) ^ {\rho_ {w 1} - 1} \left(1 - \frac {1}{N (\vec {y})} \vec {y} _ {w} ^ {\top} \vec {z}\right) ^ {\rho_ {w 2} - 1} \prod_ {j = 1} ^ {K} \left(\frac {1}{N} \vec {e} _ {j} ^ {\top} \vec {z}\right) ^ {\alpha_ {j} - 1}. \tag {6} +$$ + +To show that the posterior is log-concave, it suffices to show every element of the product is log-concave as the product of log-concave functions remains log-concave. + +To begin, we take the exponential term on the left hand side $(- \vec{z}^{\top} \log \vec{z}$ term). This is clearly log-concave. For all the remaining terms, it suffices to apply Lemma C.3 for each one. Note that we have used the assumption that $\rho_{w1}, \rho_{w2}, \alpha_j \geq 1$ for all $w \in [W]$ and $j \in [K]$ . Therefore, the posterior is log-concave in $\vec{z}$ as claimed. + +# D The Remaining Theoretical Results + +We now prove the remaining results: the form of the BBF prediction, consistency, and rates of convergence. From here onwards, we will drop the vector notation for $\eta, g, z, y_w$ when we will only need to reference the entire vector. There will be the odd situation where we'll need to reference a vector within a vector, e.g. $\vec{\eta}_i \in \Delta_K$ , the true conditional label distribution for $x_i$ . + +Recall that we take the mode of the modified BBF posterior label distribution. That is, $g^{bbf}$ is the solution to the following convex program: + +$$ +\begin{array}{l} \min _ {z \in \Delta_ {K} ^ {N}} \left[ z ^ {\top} \log z - \sum_ {w = 1} ^ {W} \left[ (\rho_ {w 1} - 1) \log \left(\frac {1}{N (\vec {y} _ {w})} y _ {w} ^ {\top} z\right) + (\rho_ {w 2} - 1) \log \left(1 - \frac {1}{N (\vec {y} _ {w})} y _ {w} ^ {\top} z\right) \right] \right. \\ \left. - \sum_ {j = 1} ^ {K} \left(\alpha_ {j} - 1\right) \log \left(\frac {1}{N} e _ {j} ^ {\top} z\right) \right]. \tag {7} \\ \end{array} +$$ + +I.e. the modified log-posterior was log-concave, so we just negated it to get a minimization problem. + +With the convex program to solve clearly defined, the overarching goal is to bound the distance between $\eta$ and $g^{bbf}$ . Recall that we are using KL divergence so that + +$$ +d (\vec {\eta}, \vec {g} ^ {b b f}) = \sum_ {i = 1} ^ {N} K L (\vec {\eta} _ {i}, \vec {g} _ {i} ^ {b b f}). +$$ + +Once $\eta$ and the LF predictions are fixed, we will derive the relationship between $\vec{\alpha}$ , $\vec{\rho}$ , the hyperparameters we control, and $d(\eta, g^{bbf})$ . In the process of deriving such a bound, we'll be able to prove the remaining claimed results. + +To set the stage, Lemma 5 of [An and Dasgupta, 2024] shows that + +$$ +d (\eta , g ^ {b b f}) = d (\eta , g ^ {*}) + d (g ^ {*}, g ^ {b b f}). +$$ + +If we can provide an upper bound for the LHS in the form of + +$$ +d (\eta , g ^ {b b f}) \leq d (\eta , g ^ {*}) + \mathcal {A} + \mathcal {B}, +$$ + +then we know that $d(g^{*},g^{bbf})\leq \mathcal{A} + \mathcal{B}$ . Since + +$$ +d (\eta , g ^ {b b f}) = - \eta^ {\top} \log g ^ {b b f} + \eta^ {\top} \log \eta , +$$ + +it suffices to demonstrate what the values of $\mathcal{A}$ and $\mathcal{B}$ are so that + +$$ +- \eta^ {\top} \log g ^ {b b f} \leq - g ^ {b b f \top} \log g ^ {b b f} + \mathcal {A} + \mathcal {B}. +$$ + +Theorem D.1 (Informal version of Theorem 5.3). + +$$ +- \eta^ {\top} \log g ^ {b b f} \leq - g ^ {b b f \top} \log g ^ {b b f} + \mathcal {A} + \mathcal {B}. +$$ + +for + +$$ +\mathcal {A} = \left| \theta^ {b b f} \right| ^ {\top} \left| b ^ {*} - A g ^ {b b f} \right| \quad a n d \quad \mathcal {B} = \left| \theta^ {*} \right| ^ {\top} \left| b ^ {*} - A g ^ {b b f} \right| +$$ + +where the absolute values are elementwise and $\theta^{bbf}$ , $\theta^{*} \in \mathbb{R}^{W + K}$ are the respective weights bringing about the predictions $g^{bbf} = g^{(\theta^{bbf})}$ and $g^{*} = g^{(\theta^{*})}$ . + +It's after proving this result that we have the machinery to prove that $g^{bbf}$ has the same form of solution as BF and that BBF is consistent. + +Proof Sketch of Theorem D.1. The idea is to use the analysis done in [An and Dasgupta, 2024] to bound BBF's error. Namely, the cross entropy term on the LHS of the theorem statement is not easy to analyze. Thus, we use a proxy: $-g^{bbf\top}\log g^{bbf}$ and upper bound that. However, unlike for $g^{b\bar{f}}$ it's not guaranteed that + +$$ +- \eta^ {\top} \log g ^ {b b f} \leq - g ^ {b b f \top} \log g ^ {b b f}. +$$ + +Thus, we show a term $\mathcal{A}$ which when added to the right hand side makes the above inequality always hold. Now, it suffices to upper bound $-g^{bbf^{\top}}\log g^{bbf}$ by + +$$ +- \eta^ {\top} \log g ^ {*} + \mathcal {B}. +$$ + +To do this, we'll first define a class of BF problems with polytope $P_{bar}$ , which has a specific form. (This class of BF problems will depend on the same LF predictions and $\eta$ as BBF). For fixed BBF hyperparameters $\vec{\alpha}, \vec{\rho}$ , we'll fix a $P_{bar}$ . The solution to a BF problem that uses $P_{bar}$ , will be called $g^{bar}$ . Namely, $g^{bar}$ is the solution to + +$$ +V _ {b a r} = \max _ {g \in \Delta_ {K} ^ {N}} \min _ {z \in P _ {b a r}} z ^ {\top} \log g = \min _ {z \in P _ {b a r}} z ^ {\top} \log z +$$ + +with respect to $g$ . We'll show that $-g^{bbf\top}\log g^{bbf} \leq -g^{bar\top}\log g^{bar}$ . This is done by demonstrating that $g^{bbf}$ almost satisfies the KKT conditions of $V_{bar}$ . The KKT conditions are 'almost' satisfied in a specific way so that we can use techniques showing how an interior point method can be used to solve convex programs to show the above inequality. I.e. the terms after the entropy term in the BBF objective (Equation 7) are a log-barrier approximation of $P_{bar}$ . + +Following the analysis of An and Dasgupta [2024], we'll then show that + +$$ +- g ^ {b a r \top} \log g ^ {b a r} \leq - \eta^ {\top} \log g ^ {*} + \mathcal {B} +$$ + +where $\mathcal{B} = |\theta^{*}|^{\top}|b^{*} - Ag^{bbf}|$ + +Putting everything together, the chain of inequalities is as follows: + +$$ +- \eta \log g ^ {b b f} \leq - g ^ {b b f \top} \log g ^ {b b f} + \mathcal {A} \leq - g ^ {b a r \top} \log g ^ {b a r} + \mathcal {A} \leq - \eta^ {\top} \log g ^ {*} + \mathcal {A} + \mathcal {B}. +$$ + +□ + +Once we specify the BF problem $V_{bar}$ , we can show that the BBF solution has the same functional form as the BF solution. Recall that we can define a set of predictions $\mathcal{G} = \{g^{(\theta)} : \theta \in \mathbb{R}^{W + K}\}$ where + +$$ +g _ {i j} ^ {(\theta)} = \frac {\exp ((A ^ {\top} \theta) _ {i j})}{\exp (\sum_ {j ^ {\prime} = 1} ^ {K} (A ^ {\top} \theta) _ {i j ^ {\prime}})} \propto \exp ((A ^ {\top} \theta) _ {i j}). +$$ + +Lemma D.2 (Lemma 5.1). $g^{bbf} \in \mathcal{G}$ + +Proof Sketch. In the last proof sketch, we mentioned how $g^{bbf}$ almost satisfies all KKT conditions for $V_{bar}$ . In particular, it satisfies the zero-gradient condition. The zero-gradient condition is what determines the form of solution. So, since BF and BBF satisfy the same condition, they have the same form of solution. An and Dasgupta [2024] show (in Theorem 2) that $g^{bf} \in \mathcal{G}$ , so $g^{bbf} \in \mathcal{G}$ . + +Analysis of the KKT conditions that $g^{bbf}$ satisfies is what will let us show BBF's consistency. What we'll need is that the prior distributions of BBF to concentrate around the empirical LF accuracies and empirical class frequency distribution. What will happen is that polytope being approximated in the BBF objective will tend to $P^{*} = \{z \in \Delta_{K}^{N} \colon Az = b^{*}\}$ , where + +$$ +g^{*} = \operatorname *{arg min}_{z\in P^{*}}z^{\top}\log z. +$$ + +(The above result was shown in Theorem 6 by An and Dasgupta [2024].) This means that $g^{bbf} \to g^*$ , the best approximator in KL divergence to $\eta$ . + +Theorem D.3 (Informal version of Theorem 5.2). If $\vec{\alpha}$ and $\vec{\rho}$ each tend to infinity elementwise and grow in such a way that $Ag^{bbf} \to b^*$ , then $g^{bbb} \to g^*$ . + +Proof Sketch. We'll show that as the BBF hyperparameters grow as in the theorem statement, the KKT conditions $g^{bbf}$ satisfies tends to the KKT conditions that $g^*$ satisfy. I.e. the KKT conditions for + +$$ +g^{*} = \operatorname *{arg min}_{z\in P^{*}}z^{\top}\log z. +$$ + +Thus, $g^{bbf}\to g^{*}$ + +![](images/a2d0737aa6079094a6d6c20749293282c1a08ee092a913f416227f863111082e.jpg) + +The steps of the above proofs are laid out as follows. + +- Subsection D.1, demonstration of what $\mathcal{A}$ is. +- Subsection D.2, definition of $P_{bar}$ and associated BF problem. +- Subsection D.3, some properties of BF with $P_{bar}$ , specifically the dual program and KKT conditions. +- Subsection D.4, some preliminaries on log-barrier approximations of a convex program's feasible region. This will help simplify the argument showing how BBF is related to BF with $P_{bar}$ . +- Subsection D.5, demonstration of how BBF's objective contains a log-barrier approximation of $P_{bar}$ . It'll be proven here that $-g^{bbf\top}\log g^{bf} \leq -g^{bar\top}\log g^{bar}$ where $g^{bar}$ is the solution to BF with $P_{bar}$ . This is also where we'll show that $g^{bbf}$ has the same functional form as the BF solutions (Lemma 5.1, in Subsubsection D.5.2). +- Subsection D.6, upper bounding $-g^{bar\top} \log g^{bar}$ , i.e. deriving the value of $\mathcal{B}$ . +- Subsection D.7, formal proof of Theorem D.1. +- Subsection D.8, proof of BBF's consistency (Theorem 5.2). + +# D.1 Bounding the Value of $\mathcal{A}$ + +Say that the accuracies gotten from labeling $g^{bbf}$ is $\widehat{b} \coloneqq Ag^{bbf}$ . From [An and Dasgupta, 2024] Corollary 28, we know that $Ag^* = b^*$ . From that same paper, Lemma 26 provides the following useful identity. + +Lemma D.4. Suppose $g = g^{(\theta)} \in \mathcal{G}$ and $y, y' \in \Delta_K^N$ are arbitrary. + +$$ +\left(y - y ^ {\prime}\right) ^ {\top} \log g = \left(A y - A y ^ {\prime}\right) ^ {\top} \theta . +$$ + +Lemma D.5. $\mathcal{A} = |\theta^{bbf}|^{\top}|b^{*} - \widehat{b}|$ + +Proof. For $-g^{bbf} \top \log g^{bbf}$ to be an upper bound for $-\eta^{\top} \log g^{bbf}$ , we would need + +$$ +- g ^ {b b f \top} \log g ^ {b b f} + \eta^ {\top} \log g ^ {b b f} \geq 0. +$$ + +Choosing $y = \eta$ and $y' = g^{bbf}$ from Lemma D.4, the above inequality is equivalent to + +$$ +\left(b ^ {*} - \widehat {b}\right) ^ {\top} \theta^ {b b f} \geq 0 +$$ + +where $A\eta = b^{*}$ and $Ag^{bbf} = \widehat{b}$ . While that quantity could be non-negative in and of itself, the following bound always holds. + +$$ +- g ^ {b b f \top} \log g ^ {b b f} + | \theta^ {b b f} | ^ {\top} | b ^ {*} - \widehat {b} | \geq - \eta^ {\top} \log g ^ {b b f} +$$ + +The second term on the left hand side is our $\mathcal{A}$ . + +![](images/59de39054650c9b504d48b00744e76866b71ad92cabc3e9475b42c2d42d9c1f5.jpg) + +# D.2 BF with Polytope $P_{bar}$ + +In this section, we want to define a class of BF problems by way of a polytope with a specific form. This will allow us to use techniques associated with the barrier method (an interior point method used to solve convex programs, [Boyd and Vandenberghe, 2004], Chapter 11) to Bayesian BF. + +Recall that given the LF predictions and $\eta$ , the polytope that BF uses is constructed by bounding the LF accuracies and the class frequency distribution. These bounds can be written as linear inequalities where any point satisfying all inequalities lies within the polytope. Moreover, we want to have $\eta$ be in the polytope. Equivalently, we want any estimate of LF accuracies/class frequencies to contain $b^{*}$ . + +So, say that the LF accuracies/class frequencies as estimated by the Bayesian BF solution is $\widehat{b} \coloneqq Ag^{bbf}$ . $P_{bar}$ will be defined by taking the constraint $b^{*} \leq Az \leq b^{*}$ and expanding the bounds until said bounds barely contain $\widehat{b}$ . Specifically, for an arbitrary $1 \gg \delta > 0$ and each $i \in [W + K]$ , define + +$$ +\epsilon_ {i} ^ {-} := \mathbf {1} \left(b _ {i} ^ {*} > \widehat {b} _ {i}\right) \left(b _ {i} ^ {*} - \widehat {b} _ {i}\right) + \delta \quad \text {a n d} \quad \epsilon_ {i} ^ {+} := \mathbf {1} \left(\widehat {b} _ {i} > b _ {i} ^ {*}\right) \left(\widehat {b} _ {i} - b _ {i} ^ {*}\right) + \delta . +$$ + +We claim that $\widehat{b} \in (b^{*} - \epsilon^{-}, b^{*} + \epsilon^{+})$ . For a fixed index $i$ , if $b_{i}^{*} > \widehat{b}_{i}$ , then + +$$ +\epsilon_ {i} ^ {-} = b ^ {*} - \widehat {b} _ {i} + \delta \quad \text {a n d} \quad \epsilon_ {i} ^ {+} = \delta . +$$ + +The $i^{th}$ interval is + +$$ +\left(b _ {i} ^ {*} - \epsilon_ {i} ^ {-}, b _ {i} ^ {*} + \epsilon_ {i} ^ {+}\right) = \left(b _ {i} ^ {*} - b _ {i} ^ {*} + \widehat {b} _ {i} - \delta , b _ {i} ^ {*} + \delta\right) = \left(\widehat {b} _ {i} - \delta , b ^ {*} + \delta\right) +$$ + +which clearly contains $\widehat{b}_i$ . A similar argument can be made when $\widehat{b}_i > b_i^*$ and the case of $\widehat{b}_i = b_i^*$ is trivial. + +Formally, the polytope $P_{bar}$ which brings about $g^{bar}$ as the optimal BF solution is + +$$ +P _ {b a r} = \left\{z \in \Delta_ {K} ^ {N} \colon b ^ {*} - \epsilon^ {-} \leq A z \leq b ^ {*} + \epsilon^ {+} \right\}. +$$ + +The BF problem that $g^{bar}$ is the solution to is as follows + +$$ +g ^ {b a r} := \underset {z \in P _ {b a r}} {\arg \min } z ^ {\top} \log z. \tag {8} +$$ + +We shall connect the Lagrangian of the Bayesian BF objective (Equation 7) at optimality to the KKT conditions of the above BF problem. Before we do that, we provide an easy example that is illustrative of the argument that will be used. + +In this section, we will show the dual problem of the BF problem in consideration as well as the KKT conditions. We'll need these in order to relate BBF to BF with $P_{bar}$ . + +The dual problem is easily derivable following the argument given for BF (following Theorem 2 of [An and Dasgupta, 2024]). Say that $\vec{\theta}_{W + K}$ is the zeros vector of length $W + K$ . + +Lemma D.6. The dual problem for Equation 8 is + +$$ +\max _ {\sigma , \sigma^ {\prime} \geq \vec {b} _ {W + K}} \left[ \sigma^ {\top} \left(- b ^ {*} - \epsilon^ {+}\right) + \sigma^ {\prime \top} \left(b ^ {*} - \epsilon^ {-}\right) - \sum_ {i = 1} ^ {N} \log \left(\sum_ {j = 1} ^ {K} \exp \left([ A ^ {\top} (\sigma^ {\prime} - \sigma) ] _ {i j}\right)\right) \right] +$$ + +Before stating the KKT conditions for the problem in Equation 8, define $D \in \{0,1\}^{N \times NK}$ where $[Dz]_1 = \sum_{j=1}^K z_{1j}$ . The constraint $D\vec{z} = \vec{1}_N$ ensures that each of the $\vec{z}_i$ 's sum to 1. We will not need non-negativity constraints for each of the $z_{ij}$ 's because their functional form guarantees their non-negativity. + +Lemma D.7. The KKT conditions for the BF problem in Equation 8 are: + +$$ +\frac {1}{N (\vec {y} _ {w})} y _ {w} ^ {\top} z - b ^ {*} - \epsilon^ {+} \leq 0, \quad w \in [ W + K ] +$$ + +$$ +- \frac {1}{N (\vec {y} _ {w})} y _ {w} ^ {\top} z + b ^ {*} - \epsilon^ {-} \leq 0, \quad w \in [ W + K ] +$$ + +$$ +D z - \vec {1} _ {n} = \vec {0} _ {N} +$$ + +$$ +\sigma , \sigma^ {\prime} \geq \vec {0} _ {W + K} +$$ + +$$ +\sigma_ {w} ([ A z ] _ {w} - b _ {w} ^ {*} - \epsilon_ {w} ^ {+}) = 0, \quad w \in [ W + K ] +$$ + +$$ +\sigma_ {w} ^ {\prime} (- [ A z ] _ {w} + b _ {w} ^ {*} - \epsilon_ {w} ^ {-}) = 0, w \in [ W + K ] +$$ + +$$ +\begin{array}{l} \nabla_ {z} \left[ z \cdot \log z + \sigma^ {\top} \left(A z - b ^ {*} - \epsilon^ {+}\right) + \sigma^ {\prime \top} \left(- A z + b ^ {*} - \epsilon^ {-}\right) \right. \\ \left. + \xi^ {\top} (D z - \vec {1} _ {n}) \right] = \vec {0} _ {N K}. \\ \end{array} +$$ + +Recall that notation has been abused and that $b^{*} \in [0,1]^{W + K}$ contains all empirical LF accuracies and empirical class frequencies. + +# D.4 Preliminaries for Barrier Method Type Analysis + +The content in this section closely follows Boyd and Vandenberghe [2004] Section 11.2.2. This exposes the general schema of the argument we'll use to relate BBF to BF, but there are differences. + +Suppose we had a convex program + +$$ +p ^ {*} = \min _ {\substack {f _ {1} (x) \leq 0 \\ Ax = b}} f _ {0} (x) \tag{9} +$$ + +where $x \in \mathbb{R}^d$ with feasible constraints. Call this our original problem. If we define the indicator function $I_{-} : \mathbb{R} \to \{0, \infty\}$ as + +$$ +I _ {-} (u) := \left\{ \begin{array}{l l} 0 & \text {i f} u \leq 0 \\ \infty & \text {o t h e r w i s e}, \end{array} \right. +$$ + +then the above problem can be written as + +$$ +p ^ {*} = \min _ {A x = b} f _ {0} (x) + I _ {-} \left(f _ {1} (x)\right). \tag {10} +$$ + +So long as the program is feasible, the indicator function forces the optimal $x$ to satisfy the constraint $f_{1}(x) \leq 0$ . The logarithmic barrier is essentially a relaxation of the indicator function so it becomes continuous. Namely, $I_{-}(u)$ can be approximated by + +$$ +\widehat {I} _ {-} (u) := - \frac {1}{t} \log (- u) +$$ + +for $t > 0$ , which is convex and non-decreasing. If we say $\log(0) \coloneqq \infty$ , $\widehat{I}_{-}(u)$ and $I_{-}(u)$ are defined on the same domain. We can easily see that if $u > 0$ is fixed, then as $t \to \infty$ , $\widehat{I}_{-}(u) \to I_{-}(u)$ . + +We may approximate the problem with the indicator function (Equation 10) with + +$$ +p (t) = \min _ {A x = b} f _ {0} (x) - \frac {1}{t} \log (- f _ {1} (x)). \tag {11} +$$ + +Then, as $t \to \infty$ , $p(t) \to p^*$ . Call the above problem the log-barrier problem. + +Let $x^{*}$ be the optimal solution of Equation 9 and $x^{*}(t)$ be the optimal solution of Equation 11. One can show the following result by analyzing the Lagrangian of Equation 11 at the point $x^{*}(t)$ . This argument below is the one we use for our results about $g^{bbf}$ and $g^{bar}$ . + +Lemma D.8. Suppose $x^{*}(t)$ is the optimal solution to the convex program with log-barrier constraints (Equation 11). Then, + +1. $x^{*}(t)$ is primal feasible for the original convex program (Equation 9) +2. $x^{*}(t)$ shows the existence of dual feasible variables $\sigma(t) = -\frac{1}{t} f_{1}(x^{*}(t))$ (for $f_{1}(x) \leq 0$ ) and $\widehat{\nu}$ (for $Ax = b$ ) for the original problem. +3. $f_0(x^*) \leq f_0(x^*(t)) \leq f_0(x^*) + \frac{1}{t}$ . + +Proof. Consider the program in Equation 11 at optimality. There, its Lagrangian must have zero gradient at $x^{*}(t)$ . We mean that there exists a $\widehat{\nu} \in \mathbb{R}^d$ where + +$$ +\nabla_ {x} \widehat {L} (x, \widehat {\nu}) = \nabla_ {x} \left[ f _ {0} (x) - \frac {1}{t} \log (- f _ {1} (x)) + \widehat {\nu} ^ {\top} (A x - b) \right] = \vec {0} _ {d} +$$ + +when $x^{*}(t)$ is substituted for $x$ . Evaluating, that's + +$$ +\nabla f _ {0} (x) - \frac {1}{t f _ {1} (x ^ {*} (t))} \nabla f _ {1} (x ^ {*} (t)) + A ^ {\top} \widehat {\nu} = \vec {0} _ {d} +$$ + +Since the original problem has a feasible solution, $f_{1}(x^{*}(t)) < 0$ because that is the only time $\widehat{I}_{-}$ is not infinite or undefined. By assumption, $t > 0$ so that $\sigma (t) = -1 / (tf_1(x^* (t))) > 0$ . Because $f_{1}(x^{*}(t)) < 0$ , $x^{*}(t)$ is also primal feasible for the original problem. If we substitute $x^{*}(t)$ for $x$ , $\sigma (t)$ for $\sigma$ and $\widehat{\nu}$ for $\nu$ , we have shown the following equations are satisfied. + +$$ +f _ {1} (x) \leq 0 +$$ + +$$ +A x - b = \vec {0} +$$ + +$$ +\sigma \geq 0 +$$ + +$$ +\sigma f _ {1} (x) = - \frac {1}{t} +$$ + +$$ +\nabla \left[ f _ {0} (x) + \sigma f _ {1} (x) + \nu^ {\top} (A x - b) \right] = \vec {0} _ {d}. +$$ + +One immediately observes that these are the KKT conditions for the original problem (Equation 9) where instead of the usual complementary slackness condition $\sigma f_{1}(x) = 0$ , we have it equal to $-\frac{1}{t}$ . + +We now want to show that $\sigma(t), \widehat{\nu}$ are dual feasible for the original problem. The Lagrangian for the original problem is + +$$ +L (x, \sigma , \nu) = f _ {0} (x) + \sigma f _ {1} (x) + \nu^ {\top} (A x - b). +$$ + +The zero gradient condition that is satisfied above means that + +$$ +\nabla L (x ^ {*} (t), \sigma (t), \widehat {\nu}) = \vec {0} _ {d}. +$$ + +Since $x^{*}(t)$ is the minimizer of $L(\cdot ,\sigma (t),\widehat{\nu})$ $\sigma (t)$ and $\widehat{\nu}$ are dual feasible for the original problem. + +Now, we show the final claim, bounding $f_0(x^*(t))$ . Because $\sigma(t)$ and $\widehat{\nu}$ are dual feasible, the dual function for the original program at those values is + +$$ +g (\sigma (t), \widehat {\nu}) = f _ {0} (x ^ {*} (t)) + \sigma (t) f _ {1} (x ^ {*} (t)) + \widehat {\nu} ^ {\top} (A x ^ {*} (t) - b). +$$ + +Because $Ax = b$ is one of the constraints to the log barrier function, the last term is 0. By definition, the second term will evaluate to $-\frac{1}{t}$ . Thus, $g(\sigma(t), \widehat{\nu}) = f_0(x^*(t)) - \frac{1}{t}$ . Now, by weak duality, + +$$ +g (\sigma (t), \widehat {\nu}) \leq p ^ {*} = f _ {0} (x ^ {*}) \quad \text {m e a n i n g} \quad f _ {0} (x ^ {*} (t)) - \frac {1}{t} \leq f _ {0} (x ^ {*}). +$$ + +This proves the upper bound to the third claim. To see the lower bound of the third claim, recall that $x^{*}(t)$ is primal feasible for the original problem and therefore is no smaller than the optimal minimum of the original problem $f_{0}(x^{*})$ . + +Because we will use a similar argument for $g^{bbf}$ and $g^{bar}$ , we now draw equivalences for the results in the above Lemma. $g^{bar}$ will be the solution of the "original" convex program while $g^{bbf}$ is a solution to a problem that implicitly encodes log-barrier constraints. Claim 2 of the above Lemma will imply that $g^{bbf}$ can be written in the form of a BF solution. Claims 1 and 3 will not only aid in our proof of Bayesian BF's consistency, but also show that the entropy of $g^{bar}$ upper bounds the entropy of $g^{bbf}$ . Namely $-g^{bbf\top}\log g^{bbf}\leq -g^{bar\top}\log g^{bar}$ . + +# D.5 BBF's Objective has a Log-Barrier Approximation to $P_{bar}$ + +In this section, we'll show that the BBF prediction $g^{bbf}$ satisfies a modified version of the KKT conditions of $V_{bar}$ (Lemma D.7). Like in the above section, the only difference is that $g^{bbf}$ will satisfy a modified version of the complementary slackness conditions. This will be done in two steps. First, we'll ignore the class frequency prior (i.e. suppose that it is constant) and show that BBF with only LF accuracy priors satisfies those modified KKT conditions. Then, we'll extend the argument so the class frequency prior is considered. + +So, we start with a version of BBF where the class frequency pdf is not considered. + +$$ +\min _ {z \in \Delta_ {K} ^ {N}} \left[ z ^ {\top} \log z - \sum_ {w = 1} ^ {W} \left[ (\rho_ {w 1} - 1) \log \left(\frac {1}{N (\vec {y} _ {w})} y _ {w} ^ {\top} z\right) + (\rho_ {w 2} - 1) \log \left(1 - \frac {1}{N (\vec {y} _ {w})} y _ {w} ^ {\top} z\right) \right] \right] +$$ + +In restricting to dealing with LF accuracies, we note that the matrix $A$ will have only $W$ rows, one for each LF prediction. To simplify the notation, define the following: + +$$ +\gamma_ {w} (z) := \frac {1}{N (\vec {y} _ {w})} y _ {w} ^ {\top} z, \quad \alpha_ {w} ^ {\prime} = \rho_ {w 1} - 1, \quad \text {a n d} \quad \beta_ {w} ^ {\prime} = \rho_ {w 2} - 1 +$$ + +Then, the convex program we first analyze can be rewritten as + +$$ +\min _ {z \in \Delta_ {K} ^ {N}} \left[ z ^ {\top} \log z - \sum_ {w = 1} ^ {W} \left[ \alpha_ {w} ^ {\prime} \log \left(\gamma_ {w} (z)\right) + \beta_ {w} ^ {\prime} \log \left(1 - \gamma_ {w} (z)\right) \right] \right] \tag {12} +$$ + +We first argue that $g^{bbf}$ is primal feasible for the BF problem with polytope $P_{bar}$ . Then, we analyze the optimal objective of Bayesian BF with LF accuracies (Equation 12) in the lens of the aforementioned problem's KKT conditions (Lemma D.7). + +Lemma D.9. The optimal solution $g^{bbf}$ to the Bayesian BF objective with LF accuracies (Equation 12) is primal feasible for the BF problem with $P_{bar}$ (Equation 8). + +Proof. This is true by construction of $P_{bar}$ . One can follow the argument given in Section D.2. + +Theorem D.10. Fix $1 \gg \delta > 0$ . For each $LF$ $w \in [W]$ , define the following + +$$ +\epsilon_ {w} ^ {-} = \boldsymbol {I} \left(b _ {w} ^ {*} > \gamma_ {w} ^ {*}\right) \left(b _ {w} ^ {*} - \gamma_ {w} ^ {*}\right) + \delta , \quad \epsilon_ {w} ^ {+} := \boldsymbol {I} \left(\gamma_ {w} ^ {*} > b _ {w} ^ {*}\right) \left(\gamma_ {w} ^ {*} - b _ {w} ^ {*}\right) + \delta , +$$ + +$$ +t _ {w} ^ {\prime} = \frac {\gamma_ {w} ^ {*}}{\alpha_ {w} ^ {\prime} (\gamma_ {w} ^ {*} - b _ {w} ^ {*} + \epsilon_ {w} ^ {-})}, t _ {w} = \frac {1 - \gamma_ {w} ^ {*}}{\beta_ {w} ^ {\prime} (- \gamma_ {w} ^ {*} + b _ {w} ^ {*} + \epsilon_ {w} ^ {+})}, +$$ + +$$ +\sigma_ {w} ^ {\prime} = - \frac {1}{t _ {w} ^ {\prime} (- \gamma_ {w} ^ {*} + b _ {w} ^ {*} - \epsilon_ {w} ^ {-})}, \quad a n d \quad \sigma_ {w} = - \frac {1}{t _ {w} (\gamma_ {w} ^ {*} - b _ {w} ^ {*} - \epsilon_ {w} ^ {+})}. +$$ + +If $g^{bbf}$ is the optimal solution for Equation 12 where $\alpha_w, \beta_w > 1$ for each $w \in [W]$ , then the following deformed KKT conditions of Equation 8 are satisfied. + +$$ +\begin{array}{l} \frac {1}{N \left(\vec {y} _ {w}\right)} y _ {w} ^ {\top} z - b _ {w} ^ {*} - \epsilon_ {w} ^ {+} \leq 0, \quad w \in [ W ] \\ - \frac {1}{N (\vec {y} _ {w})} y _ {w} ^ {\top} z + b _ {w} ^ {*} - \epsilon_ {w} ^ {-} \leq 0, \quad w \in [ W ] \\ D z - \vec {1} _ {n} = \vec {0} _ {N} \\ \sigma , \sigma^ {\prime} \geq \vec {0} _ {W} \\ \end{array} +$$ + +$$ +\sigma_ {w} ([ A z ] _ {w} - b _ {w} ^ {*} - \epsilon_ {w} ^ {+}) = - \frac {1}{t _ {w}} w \in [ W ] +$$ + +$$ +\sigma_ {w} ^ {\prime} (- [ A z ] _ {w} + b _ {w} ^ {*} - \epsilon_ {w} ^ {-}) = - \frac {1}{t _ {w} ^ {\prime}} w \in [ W ] +$$ + +$$ +\nabla_ {z} \left[ z \cdot \log z + \sigma^ {\top} (A z - b ^ {*} - \epsilon^ {+}) + \sigma^ {\prime \top} (- A z + b ^ {*} - \epsilon^ {-}) + \xi^ {\top} (D z - \vec {1} _ {n}) \right] = \vec {0} _ {N K}. +$$ + +Note that $w$ only ranges in $[W]$ and not $[W + K]$ (as seen in Lemma D.7) because the final $K$ indices are for the class frequency constraints, which we aren't considering right now. + +Proof. We argue for this by analyzing the objective of our problem (Equation 12) at optimality. This is very similar to the demonstration argument we gave in Lemma D.8. + +If we evaluate the zero-gradient condition in the theorem statement, we get + +$$ +\vec {1} _ {N K} + \log z + A ^ {\top} \sigma - A ^ {\top} \sigma^ {\prime} + D ^ {\top} \xi = \vec {0} _ {N K}. \tag {13} +$$ + +We'll show that the gradient of the Bayesian BF objective (Equation 12) will be of this form. + +When the BBF objective is at optimality, the following will hold true for some $\xi \in \mathbb{R}^N$ . + +$$ +\vec {0} _ {N K} = \nabla_ {z} \left[ z ^ {\top} \log z - \sum_ {w = 1} ^ {W} \left[ \alpha_ {w} ^ {\prime} \log \left(\frac {y _ {w} ^ {\top} z}{N (\vec {y} _ {w})}\right) + \beta_ {w} ^ {\prime} \log \left(1 - \frac {y _ {w} ^ {\top} z}{N (\vec {y} _ {w})}\right) \right] + \xi^ {\top} (D z - \vec {1} _ {N}) \right] +$$ + +Evaluating, + +$$ +\vec {0} _ {N K} = \vec {1} _ {N K} + \log z - \sum_ {w = 1} ^ {W} \left[ \frac {\alpha_ {w} ^ {\prime}}{\frac {1}{N (\vec {y} _ {w})} y _ {w} ^ {\top} z} \left(\frac {y _ {w}}{N (\vec {y} _ {w})}\right) + \frac {\beta_ {w} ^ {\prime}}{1 - \frac {1}{N (\vec {y} _ {w})} y _ {w} ^ {\top} z} \left(- \frac {y _ {w}}{N (\vec {y} _ {w})}\right) \right] + D ^ {\top} \xi +$$ + +where the division is elementwise. Observe that by definition, $y_w / N(\vec{g}_w)$ corresponds to row $w$ of matrix $A$ . I.e. that term is related to the bound on LF $h^{(w)}$ 's accuracy. Define + +$$ +f _ {w} ^ {\prime} (z) = - \gamma_ {w} (z) + b _ {w} ^ {*} - \epsilon_ {w} ^ {-} \quad \text {a n d} \quad f _ {w} (z) = \gamma_ {w} (z) - b _ {w} ^ {*} - \epsilon_ {w} ^ {+}. +$$ + +If one sets these functions to each be less than or equal to 0, then one can see they represent upper and lower bounds for LF $h^{(w)}$ 's accuracy. Using that notation and multiplying each term in each summand by 1 gives + +$$ +\begin{array}{l} \vec {0} _ {N K} = \vec {1} _ {N K} + \log z + D ^ {\top} \xi \\ + \sum_ {w = 1} ^ {W} \left[ \frac {- \alpha_ {w} ^ {\prime} f _ {w} ^ {\prime} (z)}{\gamma_ {w} (z)} \frac {1}{- f _ {w} ^ {\prime} (z)} \left(- \frac {y _ {w}}{N (\vec {y} _ {w})}\right) + \frac {- \beta_ {w} ^ {\prime} f _ {w} (z)}{1 - \gamma_ {w} (z)} \frac {1}{- f _ {w} (z)} \left(\frac {y _ {w}}{N (\vec {y} _ {w})}\right) \right]. \\ \end{array} +$$ + +Moving forward, we'll replace $z$ by $g^{bbf}$ and use the following shorthand of $\gamma_w^*$ to denote $\gamma_w(g^{bbf}) = y_w^\top g^{bbf} / N(\vec{y}_w)$ . + +Now, define + +$$ +t _ {w} ^ {\prime} = - \frac {\gamma_ {w} ^ {*}}{\alpha_ {w} ^ {\prime} f _ {w} ^ {\prime} (g ^ {b b f})} = \frac {\gamma_ {w} ^ {*}}{\alpha_ {w} ^ {\prime} (\gamma_ {w} ^ {*} - b _ {w} ^ {*} + \epsilon_ {w} ^ {-})} \quad \mathrm {a n d} \quad t _ {w} = - \frac {1 - \gamma_ {w} ^ {*}}{\beta_ {w} ^ {\prime} f _ {w} (g ^ {b b f})} = \frac {1 - \gamma_ {w} ^ {*}}{\beta_ {w} ^ {\prime} (- \gamma_ {w} ^ {*} + b _ {w} ^ {*} + \epsilon_ {w} ^ {+})}. +$$ + +Using this definition, the above is simplified to + +$$ +\vec {0} _ {N K} = \vec {1} _ {N K} + \log g ^ {b b f} + D ^ {\top} \xi + \sum_ {w = 1} ^ {W} \left[ \frac {- 1}{t _ {w} ^ {\prime} f _ {w} ^ {\prime} (g ^ {b b f})} \left(- \frac {y _ {w}}{N (\vec {y} _ {w})}\right) + \frac {- 1}{t _ {w} f _ {w} (g ^ {b b f})} \left(\frac {y _ {w}}{N (\vec {y} _ {w})}\right) \right]. +$$ + +Define now + +$$ +\sigma_ {w} ^ {\prime} = - \frac {1}{t _ {w} ^ {\prime} f _ {w} ^ {\prime} (g ^ {b b f})} = - \frac {1}{t _ {w} ^ {\prime} (- \gamma_ {w} ^ {*} + b _ {w} ^ {*} - \epsilon_ {w} ^ {-})} \mathrm {a n d} \sigma_ {w} = - \frac {1}{t _ {w} f _ {w} (g ^ {b b f})} = - \frac {1}{t _ {w} (\gamma_ {w} ^ {*} - b _ {w} ^ {*} - \epsilon_ {w} ^ {+})}. +$$ + +By Lemma D.9, $g^{bbf} \in P_{bar}$ , which means that by construction, $f_w(g^{bbf})$ , $f_w'(g^{bbf})$ are both negative. Since $\alpha_w, \beta_w > 1$ for each $w \in [W]$ by assumption, the $t$ 's are strictly positive. This means that the $\sigma$ and $\sigma'$ values are all strictly positive. + +We can then write + +$$ +\vec {0} _ {N K} = \vec {1} _ {N K} + \log g ^ {b b f} + D ^ {\top} \xi + \sum_ {w = 1} ^ {W} \left[ - \sigma_ {w} ^ {\prime} \left(\frac {y _ {w}}{N (\vec {y} _ {w})}\right) + \sigma_ {w} \left(\frac {y _ {w}}{N (\vec {y} _ {w})}\right) \right]. +$$ + +To see that this matches the form of Equation 13, observe that $y_w / N(\vec{y}_w)$ is the $w^{th}$ row of $A$ or the $w^{th}$ column of $A^{\top}$ . This means we have verified the zero gradient condition of the deformed KKT conditions in the theorem statement. + +We now verify the remaining conditions. The first three conditions state that $g^{bbf}$ must be primal feasible with respect to the BF problem with $P_{bar}$ . That was shown in Lemma D.9. The third condition is a relaxation of requiring $g^{bbf}$ to be a concatenation of $N$ distributions, each over $K$ elements. It is satisfied because $g^{bbf} \in \Delta_K^N$ . In defining the $\sigma$ 's, we showed that they were strictly positive. Hence, the fourth condition is satisfied. Finally, one can see that the modified complementary slackness conditions hold by substituting $\gamma_w^* = y_w^\top g^{bbf} / N(\vec{y}_w)$ for $[Az]_w$ and observing the definition of $\sigma_w, \sigma_w'$ . + +Remark D.11. It doesn't matter what the value of $\xi$ is as these dual variables ensure that the solution to a BF problem is a distribution, i.e. is in $\Delta_K^N$ . Thus, we will ignore $\xi$ when talking about BF's dual variables. + +# D.5.1 Extending the argument for class frequencies + +Now, we analyze BBF as presented in Equation 7. Abusing notation, define $\alpha_{j}^{\prime} = \alpha_{j} - 1$ . The modified version of our objective (coming from Equation 12) is + +$$ +\begin{array}{l} \min _ {z \in \Delta_ {K} ^ {N}} \left[ z ^ {\top} \log z - \sum_ {w = 1} ^ {W} \left[ \left(\rho_ {w 1} - 1\right) \log \left(\gamma_ {w} (z)\right) + \left(\rho_ {w 2} - 1\right) \log \left(1 - \gamma_ {w} (z)\right) \right] \right. \\ \left. - \sum_ {j = 1} ^ {K} \alpha_ {j} ^ {\prime} \log \left(\frac {1}{N} e _ {j} ^ {\top} z\right) \right]. \tag {14} \\ \end{array} +$$ + +The goal is to show that by adding the last sum, we satisfy a modified version of the KKT conditions with upper and lower bounds on the class frequency constraints. Unsurprisingly, the addition is $2K$ more primal constraints, the $2K$ more dual variables, and a modified zero-gradient condition. + +Before stating the result, we remind the reader of some notation and introduce some notation. $\omega_{j}^{*} = \frac{1}{N} e_{j}^{\top}\eta$ , the empirical class frequency. Let $A'$ be the matrix in $\{0,1\}^{K\times NK}$ where row $j$ is $\frac{1}{N}\vec{e}_j$ . This will make the notation in the KKT conditions simpler. Define $\pi_j(z) = \frac{1}{N} e_j^\top z$ as the frequency of class $j$ with $z$ as the labeling. Similar to above, $\pi_j^* = \pi_j(g^{bbf})$ where we now take $g^{bbf}$ to be the solution of Equation 7. Finally, define $\alpha_0' = \sum_{j=1}^{K}\alpha_j'$ . + +Theorem D.12. Fix $1 \gg \delta > 0$ . For each class $j \in [K]$ , define the following + +$$ +\begin{array}{l} \zeta_ {j} ^ {-} = \mathbf {1} \left(\omega_ {j} ^ {*} > \pi_ {j} ^ {*}\right) \left(\omega_ {j} ^ {*} - \pi_ {j} ^ {*}\right) + \delta , \quad \zeta_ {j} ^ {+} := \mathbf {1} \left(\pi_ {j} ^ {*} > \omega_ {j} ^ {*}\right) \left(\pi_ {j} ^ {*} - \omega_ {j} ^ {*}\right) + \delta , \\ \tau_ {j} ^ {\prime} = \frac {\pi_ {j} ^ {*}}{\alpha_ {0} ^ {\prime} (\pi_ {j} ^ {*} - \omega_ {j} ^ {*} + \zeta_ {j} ^ {-})}, \quad \tau_ {j} = \frac {\pi_ {j} ^ {*}}{(\alpha_ {0} ^ {\prime} - \alpha_ {j} ^ {\prime}) (- \pi_ {j} ^ {*} + \omega_ {j} ^ {*} + \zeta_ {j} ^ {+})}, \\ \end{array} +$$ + +$$ +v _ {j} ^ {\prime} = - \frac {1}{\tau_ {j} ^ {\prime} \left(- \pi_ {j} ^ {*} + \omega_ {j} ^ {*} - \zeta_ {j} ^ {-}\right)}, \quad a n d \quad v _ {j} = - \frac {1}{\tau_ {j} \left(\pi_ {j} ^ {*} - \omega_ {j} ^ {*} - \zeta_ {j} ^ {+}\right)}. +$$ + +Suppose that each element of $\vec{\rho}$ and $\vec{\alpha}$ is no smaller than 1. Then, if $g^{bbf}$ is the optimal solution to Equation 7, then the following deformed KKT conditions of Equation 8 (found at Lemma D.7) are satisfied. The $t, t', \sigma, \sigma'$ values are as defined in Theorem D.10. + +$$ +\frac {1}{N (\vec {y} _ {w})} y _ {w} ^ {\top} z - b _ {w} ^ {*} - \epsilon_ {w} ^ {+} \leq 0, \quad w \in [ W ] +$$ + +$$ +- \frac {1}{N (\vec {y} _ {w})} y _ {w} ^ {\top} z + b _ {w} ^ {*} - \epsilon_ {w} ^ {-} \leq 0, \quad w \in [ W ] +$$ + +$$ +\frac {1}{N} e _ {j} ^ {\top} z - \omega_ {j} ^ {*} - \zeta_ {j} ^ {+} \leq 0 j \in [ K ] +$$ + +$$ +- \frac {1}{N} e _ {j} ^ {\top} z + \omega_ {j} ^ {*} - \zeta_ {j} ^ {-} \leq 0 \quad j \in [ K ] +$$ + +$$ +D z - \vec {1} _ {n} = \vec {0} _ {N} +$$ + +$$ +\sigma , \sigma^ {\prime} \geq \vec {0} _ {W} +$$ + +$$ +v, v ^ {\prime} \geq \vec {0} _ {K} +$$ + +$$ +\sigma_ {w} ([ A z ] _ {w} - b _ {w} ^ {*} - \epsilon_ {w} ^ {+}) = - \frac {1}{t _ {w}} \quad w \in [ W ] +$$ + +$$ +\sigma_ {w} ^ {\prime} \left(- [ A z ] _ {w} + b _ {w} ^ {*} - \epsilon_ {w} ^ {-}\right) = - \frac {1}{t _ {w} ^ {\prime}} w \in [ W ] +$$ + +$$ +v _ {j} ([ A ^ {\prime} z ] _ {j} - \omega_ {j} ^ {*} - \zeta_ {j} ^ {+}) = - \frac {1}{\tau_ {j}} \quad j \in [ K ] +$$ + +$$ +v _ {j} ^ {\prime} \left(- \left[ A ^ {\prime} z \right] _ {j} + \omega_ {j} ^ {*} - \zeta_ {j} ^ {-}\right) = - \frac {1}{\tau_ {j} ^ {\prime}} \quad j \in [ K ] +$$ + +$$ +\begin{array}{l} \nabla_ {z} \Big [ z \cdot \log z + \sigma^ {\top} (A z - b ^ {*} - \epsilon^ {+}) + \sigma^ {\prime \top} (- A z + b ^ {*} - \epsilon^ {-}) \\ \left. + v ^ {\top} (A ^ {\prime} z - \omega^ {*} - \zeta^ {+}) + v ^ {\prime \top} (- A ^ {\prime} z + \omega^ {*} - \zeta^ {-}) + \xi^ {\top} (D z - \vec {1} _ {n}) \right] = \vec {0} _ {N K}. \\ \end{array} +$$ + +Proof. We essentially follow the same proof as Theorem D.10. Namely, we choose the same $\epsilon$ 's, $t$ 's, and $\sigma$ 's. For this proof, we focus on lines 3, 4, 7, 10, 11, 12. Line 12 is the zero gradient condition, which we start with. The gradient of term in the BBF objective we're interested in is + +$$ +\nabla_ {z} \left[ - \sum_ {j = 1} ^ {K} \alpha_ {j} ^ {\prime} \log \left(\frac {1}{N} e _ {j} ^ {\top} z\right) \right] = \sum_ {j = 1} ^ {K} \frac {\alpha_ {j} ^ {\prime}}{\frac {1}{N} e _ {j} ^ {\top} z} \left(- \frac {1}{N} e _ {j}\right). +$$ + +Once again the division is elementwise. It suffices to show that the above is equal to the terms involving $v$ in the zero-gradient condition because we have shown the correspondence of the other terms in Theorem D.10. By adding 0, we can see the above is equal to + +$$ +\sum_ {j = 1} ^ {K} \left[ \frac {\alpha_ {0} ^ {\prime}}{\frac {1}{N} e _ {j} ^ {\top} z} \left(- \frac {1}{N} e _ {j}\right) + \frac {\alpha_ {0} ^ {\prime} - \alpha_ {j} ^ {\prime}}{\frac {1}{N} e _ {j} ^ {\top} z} \left(\frac {1}{N} e _ {j}\right) \right] +$$ + +Now, define + +$$ +f _ {j} ^ {\prime} (z) = - \pi_ {j} (z) + \omega_ {j} ^ {*} - \zeta_ {j} ^ {-} \quad \text {a n d} \quad f _ {j} (z) = \pi_ {j} (z) - \omega_ {j} ^ {*} - \zeta_ {j} ^ {+}. +$$ + +Like before, setting these to be less than 0 form the class frequency constraints of BF. Now, it suffices to choose the $\tau$ 's and $v$ 's. The above expression can be rewritten as + +$$ +\sum_ {j = 1} ^ {K} \left[ \frac {- \alpha_ {0} ^ {\prime} f _ {j} ^ {\prime} (z)}{\pi_ {j} (z)} \frac {1}{- f _ {j} ^ {\prime} (z)} \left(- \frac {1}{N} e _ {j}\right) + \frac {- (\alpha_ {0} ^ {\prime} - \alpha_ {j} ^ {\prime}) f _ {j} (z)}{\pi_ {j} (z)} \frac {1}{- f _ {j} (z)} \left(\frac {1}{N} e _ {j}\right) \right] +$$ + +As the zero-gradient condition holds for the optimal solution to Equation 7, we substitute $g^{bbf}$ for $z$ . Then, with $\pi_j^* = \pi_j(g^{bbf})$ , we define + +$$ +\tau_ {j} ^ {\prime} = - \frac {\pi_ {j} ^ {*}}{\alpha_ {0} ^ {\prime} f _ {j} ^ {\prime} (g ^ {b b f})} = \frac {\pi_ {j} ^ {*}}{\alpha_ {0} ^ {\prime} (\pi_ {j} ^ {*} - \omega_ {j} ^ {*} + \zeta_ {j} ^ {-})} +$$ + +and + +$$ +\tau_ {j} = - \frac {\pi_ {j} ^ {*}}{(\alpha_ {0} ^ {\prime} - \alpha_ {j} ^ {\prime}) f _ {j} (g ^ {b b f})} = \frac {\pi_ {j} ^ {*}}{(\alpha_ {0} ^ {\prime} - \alpha_ {j} ^ {\prime}) (- \pi_ {j} ^ {*} + \omega_ {j} ^ {*} + \zeta_ {j} ^ {+})}. +$$ + +The above can then be written as + +$$ +\sum_ {j = 1} ^ {K} \left[ - \frac {1}{\tau_ {j} ^ {\prime} f _ {j} ^ {\prime} (g ^ {b b f})} \left(- \frac {1}{N} e _ {j}\right) - \frac {1}{\tau_ {j} f _ {j} (g ^ {b b f})} \left(\frac {1}{N} e _ {j}\right) \right] +$$ + +Now, define + +$$ +v _ {j} ^ {\prime} = - \frac {1}{\tau_ {j} ^ {\prime} f _ {j} ^ {\prime} (g ^ {b b f})} \quad \text {a n d} \quad v _ {j} = - \frac {1}{\tau_ {j} f _ {j} (g ^ {b b f})}. +$$ + +The above simplifies to + +$$ +\sum_ {j = 1} ^ {K} \left[ - v _ {j} ^ {\prime} \left(\frac {1}{N} \vec {e} _ {j}\right) + v _ {j} \left(\frac {1}{N} e _ {j}\right) \right] +$$ + +Since row $j$ of $A'$ is defined as $\frac{1}{N} e_j$ , the zero-gradient condition is satisfied. + +To see that the $\tau$ 's and $v$ 's are non-negative and that $g^{bbf}$ is primal feasible (lines 3 and 4 of the claimed KKT conditions), one can use the argument provided in Theorem D.10. In that same way, the modified complementary slackness conditions are also satisfied. + +Corollary D.13. $\sigma, \sigma'$ as defined in Theorem D.10 and $v, v'$ from Theorem D.12 are dual feasible for BF with $P_{bar}$ (Equation 8). + +Proof. The zero-gradient condition in the statement of Theorem D.12 states that the Lagrangian, $L(z,\sigma,\sigma',v,v',\xi)$ , has to have 0 gradient at the primal/dual optimal variables. Since we have both the primal and dual variables that make the gradient of the Lagrangian zero, it follows that $g^{bbf}$ is the minimizer of the Lagrangian $L(\cdot,\sigma,\sigma',v,v',\xi)$ for the dual variables from Theorem D.12. This implies that those $\sigma,\sigma',v,v',\xi$ are dual feasible. + +# D.5.2 $g^{bbf}$ is a BF prediction + +Corollary D.14. $g^{bbf}$ can be written in the BF form. + +Proof. This follows from the zero gradient condition in Theorem D.10 being satisfied as a consequence of Theorem D.12. I.e. if one solves for $z$ as is done in the proof of Theorem 2 in [An and Dasgupta, 2024]. We have seen that this condition for BF is how one derives the BF prediction format. Because the same condition is met here, $g^{bbf}$ has the same form. + +# D.5.3 Relating the Entropy of $g^{bbf}$ with $g^{bar}$ + +Like in Lemma D.8, we can bound the value of the objective $(z^{\top}\log z)$ when evaluated at $g^{bbf}$ . + +Corollary D.15. Let $g^{bar}$ be the optimal solution to the BF program in Equation 8. + +$$ +- g ^ {b a r \top} \log g ^ {b a r} - \sum_ {w = 1} ^ {W} \left[ \frac {1}{t _ {w}} + \frac {1}{t _ {w} ^ {\prime}} \right] - \sum_ {j = 1} ^ {K} \left[ \frac {1}{\tau_ {j}} + \frac {1}{\tau_ {j} ^ {\prime}} \right] \leq - g ^ {b b f \top} \log g ^ {b b f} \leq - g ^ {b a r \top} \log g ^ {b a r} +$$ + +for $t_w, t_w'$ defined in Theorem D.10 and $\tau_j, \tau_j'$ as defined in Theorem D.12. + +Proof. To see the second inequality, recall that $g^{bbf}$ is primal feasible for the BF program in Equation 8 while $g^{bar}$ is the optimal solution to that program. I.e. primal feasibility implies $g^{bbf\top}\log g^{bbf}\geq g^{bar\top}\log g^{bar}$ . + +For the first inequality, we would like to analyze the difference between our primal feasible solution's objective $g^{bbf\top} \log g^{bbf}$ versus the optimal minimum $g^{bar\top} \log g^{bar}$ . By weak duality, $g^{bar\top} \log g^{bar}$ is larger than any dual function value. Continuing from the discussion in Corollary D.13, the (BF) + +dual function is equal to $L(g^{bbf}, \sigma, \sigma', v, v', \xi)$ where the dual variables are defined in Theorem D.10 and D.12. Namely, + +$$ +\begin{array}{l} g (\sigma , \sigma^ {\prime}, \xi) = g ^ {b b f \top} \log g ^ {b b f} + \sigma^ {\top} (A g ^ {b b f} - b ^ {*} - \epsilon^ {+}) + \sigma^ {\prime \top} (- A g ^ {b b f} + b ^ {*} - \epsilon^ {-}) \\ + v ^ {\top} \left(A ^ {\prime} g ^ {b b f} - \omega^ {*} - \zeta^ {+}\right) + v ^ {\prime \top} \left(- A ^ {\prime} g ^ {b b f} + \omega^ {*} - \zeta^ {-}\right) + \xi^ {\top} \left(D g ^ {b b f} - \vec {1} _ {n}\right). \\ \end{array} +$$ + +By primal feasibility of $g^{bbf}$ , the last term is 0. By the modified complementary slackness conditions, the above is equal to + +$$ +g ^ {b b f \top} \log g ^ {b b f} - \sum_ {w = 1} ^ {W} \left[ \frac {1}{t _ {w}} + \frac {1}{t _ {w} ^ {\prime}} \right] - \sum_ {j = 1} ^ {K} \left[ \frac {1}{\tau_ {j}} + \frac {1}{\tau_ {j} ^ {\prime}} \right]. +$$ + +Putting this together with weak duality, we have + +$$ +g ^ {b a r \top} \log g ^ {b a r} \geq g ^ {b b f \top} \log g ^ {b b f} - \sum_ {w = 1} ^ {W} \left[ \frac {1}{t _ {w}} + \frac {1}{t _ {w} ^ {\prime}} \right] - \sum_ {j = 1} ^ {K} \left[ \frac {1}{\tau_ {j}} + \frac {1}{\tau_ {j} ^ {\prime}} \right]. +$$ + +Rearranging gives us our claim. + +![](images/cf4c9c0c3d435e78a0a456bc55497f4a1f667fa4e20770f550e701272727204b.jpg) + +# D.6 Error Bound for $g^{bar}$ + +The last piece we need before being able to prove Theorem 5.3 is a bound on $-g^{bar\top} \log g^{bar}$ . We now will show $\mathcal{B}$ such that + +$$ +- g ^ {b a r \top} \log g ^ {b a r} \leq - \eta^ {\top} \log g ^ {*} + \mathcal {B}. +$$ + +Since $g^{bar}$ is the solution of a bona fide BF problem, we can use the analysis of [An and Dasgupta, 2024] to get a bound in the form we'd like. + +Theorem D.16. For $g^{bar}$ the optimal solution of Equation 8 and some $g^{ref} \in \mathcal{G}$ , + +$$ +- \eta^ {\top} \log g ^ {b a r} \leq - \eta^ {\top} \log g ^ {r e f} + | \widehat {b} - b ^ {*} | ^ {\top} | \theta^ {r e f} | +$$ + +where $\widehat{b} := Ag^{bbf}$ . + +Proof. This proof heavily relies on the proof of Theorem 7 in [An and Dasgupta, 2024]. It suffices to find the dual of the BF program in Equation 8, then do sensitivity analysis where the reference problem has constraints $Az = b^{*}$ . One is able to get the following bound where $\theta^{ref} = \sigma' - \sigma$ and $\sigma, \sigma' \geq 0$ . + +$$ +- \eta^ {\top} \log g ^ {b a r} \leq - \eta^ {\top} \log g ^ {r e f} + \epsilon^ {+ \top} \sigma + \epsilon^ {- \top} \sigma^ {\prime} +$$ + +Here, the $\epsilon^{\pm}$ encapsulates both $\epsilon^{\pm}$ as defined above as well as $\zeta^{\pm}$ (i.e. since we concatenate the LF accuracies and class frequencies). Then, observe that $|Ag^{bfb} - b^{*}| + \delta = \max \{\epsilon^{+},\epsilon^{-}\}$ where the absolute value and maximums are done elementwise. Therefore, the above bound can become + +$$ +- \eta^ {\top} \log g ^ {b a r} \leq - \eta^ {\top} \log g ^ {r e f} + | \widehat {b} - b ^ {*} | ^ {\top} | \theta^ {r e f} |. +$$ + +The $\delta$ term disappears because it can be arbitrarily small. + +![](images/23d24778699c4955dbb389a37dfe0c68b91582fbae81e53f783ec5ee63d80991.jpg) + +In particular, we will choose $g^{*} = g^{ref}$ . + +Corollary D.17. + +$$ +- \eta^ {\top} \log g ^ {b a r} \leq - \eta^ {\top} \log g ^ {*} + | \widehat {b} - b ^ {*} | ^ {\top} | \theta^ {*} | +$$ + +Therefore, + +$$ +\mathcal {B} = \left| \widehat {b} - b ^ {*} \right| ^ {\top} | \theta^ {*} |. +$$ + +# D.7 Formal Proof of BBF Error Bound + +Now, we have all the pieces to prove Theorem 5.3. We'll explicitly state all the assumptions for completeness. This follows the proof sketch for Theorem D.1 provided earlier, but we fill in the missing details. + +Theorem D.18 (Theorem 5.3). Fix all $LF$ predictions and $\eta$ . Also, fix $\vec{\alpha} \in \mathbb{R}_{\geq 1}^{K}$ and $\vec{\rho} \in \mathbb{R}_{\geq 1}^{2W}$ . Say that $g^{bf}$ is the solution to the BBF objective in Equation 7. The error of $g^{bf}$ is such that + +$$ +d (g ^ {*}, g ^ {b b f}) \leq \left(| \theta^ {*} | + | \theta^ {b b f} |\right) ^ {\top} | b ^ {*} - A g ^ {b b f} | +$$ + +where the absolute value is elementwise. + +Proof. Lemma 5 of [An and Dasgupta, 2024] shows that + +$$ +d (\eta , g ^ {b b f}) = d (\eta , g ^ {*}) + d (g ^ {*}, g ^ {b b f}). +$$ + +So, if we have a bound that looks like + +$$ +d (\eta , g ^ {b b f}) \leq d (\eta , g ^ {*}) + \mathcal {A} + \mathcal {B}, +$$ + +then it's clear that $d(g^{*}, g^{bbf}) \leq \mathcal{A} + \mathcal{B}$ . This is equivalent to showing a bound of the form + +$$ +- \eta^ {\top} \log g ^ {b b f} \leq - \eta^ {\top} \log g ^ {*} + \mathcal {A} + \mathcal {B}, +$$ + +which we'll now do. + +From Lemma D.5, we know that + +$$ +- \eta^ {\top} \log g ^ {b b f} \leq - g ^ {b b f \top} \log g ^ {b b f} + | \theta^ {b b f} | ^ {\top} | b ^ {*} - A g ^ {b b f} |. +$$ + +Now, Corollary D.14 shows that the first term of the RHS above is upper bounded by + +$$ +- g ^ {b a r \top} \log g ^ {b a r}. +$$ + +Penultimately, Corollary D.17 shows that + +$$ +- g ^ {b a r \top} \log g ^ {b a r} \leq - \eta^ {\top} \log g ^ {*} + | \theta^ {*} | ^ {\top} | A g ^ {b b f} - b ^ {*} |. +$$ + +Putting everything together, we have + +$$ +- \eta^ {\top} \log g ^ {b b f} \leq - \eta^ {\top} \log g ^ {*} + (| \theta^ {*} | + | \theta^ {b b f} |) ^ {\top} | b ^ {*} - A g ^ {b b f} |. +$$ + +# D.8 Consistency of BBF + +An and Dasgupta [2024] showed that it is possible (via Theorem 6) to compute $g^{*}$ , the prediction in $\mathcal{G}$ that minimizes $d(\eta, \cdot)$ . In particular, by Theorem 6 from the above paper, $g^{*}$ is the BF prediction when one solves BF with polytope $P^{*} = \{z \in \Delta_{K}^{N} \colon Az = b^{*}\}$ . Recall that for BBF to be consistent, there must be a way for its prediction to tend to $g^{*}$ . Indeed, we'll show that under the correct conditions, BBF can approximate BF with $P^{*}$ arbitrarily well. This means that $g^{bbf} \rightarrow g^{*}$ , which is what it means to be consistent. + +We'll prove consistency as follows. First, we'll show that as the BBF priors concentrate in a certain way (or rather the BBF hyperparameters tend to infinity in a certain way), we'll have $Ag^{bbf} \to b^*$ . (In particular, we need the LF accuracy (resp. class frequency) priors to concentrate around the empirical LF accuracies (resp. class frequency).) This fact will play an important role as it ensures all the following things happen. Then, one recalls or observes that $g^{bbf}$ can be interpreted as being an approximation to $g^{bar}$ , the optimal solution of BF with $P_{bar}$ . It'll then be shown that as $Ag^{bbf} \to b^*$ , $g^{bbf} \to g^{bar}$ . At the same time, by the way $P_{bar}$ is constructed, $g^{bar} \to g^*$ , which means that $g^{bbf} \to g^*$ . + +Since we have abused notation in the previous sections, let $\alpha_{j}^{\prime \prime}$ be $\alpha_{j} - 1$ where $\alpha_{j}$ is the $j^{th}$ hyperparameter to the Dirichlet prior in BBF's generative process. Like above, we'll have $\alpha_0^{\prime \prime} = \sum_{j = 1}^{K}\alpha_{j}^{\prime \prime}$ . (Recall that we had used $\alpha_w^\prime ,\beta_w^\prime$ to denote $\rho_{w1} - 1,\rho_{w2} - 1$ respectively.) + +Lemma D.19. Suppose for each $w \in [W]$ and $j \in [K]$ that $\alpha_w', \beta_w', \alpha_j'' \to \infty$ . Moreover suppose that + +$$ +\left| \alpha_ {w} ^ {\prime} - b _ {w} ^ {*} \left(\alpha_ {w} ^ {\prime} + \beta_ {w} ^ {\prime}\right)\right|\rightarrow 0 \quad a n d \quad \left| \alpha_ {j} ^ {\prime \prime} - \omega_ {j} ^ {*} \left(\sum_ {j ^ {\prime} = 1} ^ {K} \alpha_ {j ^ {\prime}} ^ {\prime \prime}\right)\right|\rightarrow 0 +$$ + +$$ +\alpha^{\prime}_{w}\to \frac{\beta^{\prime}_{w}b^{*}_{w}}{1 - b^{*}_{w}}\quad and\quad \alpha^{\prime \prime}_{j}\to \frac{1}{1 - \omega^{*}_{j}}\sum_{\substack{j^{\prime} = 1\\ j^{\prime}\neq j}}^{K}\alpha^{\prime \prime}_{j^{\prime}} +$$ + +for each $w$ and $j$ . Then, $Ag^{bbf} \to b^*$ . + +Proof. To simplify the argument, we'll add the Beta and Dirichlet normalizing factors into the BBF objective. This does not change the argument in a substantial way because of two factors. First, we are concerned with the mode of Beta and Dirichlet distributions, adding or removing the normalizing factor does not change the mode. Second, the proof depends on the fact that the non-entropy terms tend to infinity (whereas the entropy term remains bounded). Removing the normalization terms does not prevent the other terms from going to infinity. + +So, abuse notation and say $B(\alpha, \beta)$ is the Beta function while $B(\vec{\alpha})$ is the multivariate Beta function. The modified BBF objective that we are considering is + +$$ +\begin{array}{l} \min _ {z \in \Delta_ {K} ^ {N}} \left[ z ^ {\top} \log z - \sum_ {w = 1} ^ {W} \left[ \alpha_ {w} ^ {\prime} \log \left(\gamma_ {w} (z)\right) + \beta_ {w} ^ {\prime} \log \left(1 - \gamma_ {w} (z)\right) - \log B \left(\rho_ {w 1}, \rho_ {w 2}\right) \right] \right. \\ \left. - \sum_ {j = 1} ^ {K} \alpha_ {j} ^ {\prime} \log \left(\pi_ {j} (z)\right) + \log B (\vec {\alpha}) \right]. \tag {15} \\ \end{array} +$$ + +Before getting started, recall that the mode of the Beta distribution is $\alpha_w' / (\alpha_w' + \beta_w')$ . Similarly, the $j^{th}$ element of the mode of a Dirichlet distribution is $\alpha_j'' / \alpha_0''$ . Moreover, as $\alpha_w', \beta_w', \alpha_j''$ increase, the variance decreases so that the density (and therefore the log density) increases unboundedly. So, suppose that $\alpha_w', \beta_w', \alpha_j''$ are very large (for $w \in [W], j \in [K]$ ). Since the BBF objective is minimization, this means that the negative log-pdfs of the Beta (resp. Dirichlet) distributions are small when the value of $\gamma_w(z)$ (resp. $\pi_j(z)$ ) is close to the mode. Moreover, how small these values can get is unbounded below. + +Therefore, the optimal $z$ cannot just minimize the entropy term. To minimize the BBF objective, the other terms must also be made small. Moreover, minimizing these log-pdfs are the priority since they're larger in absolute magnitude compared to the entropy term. By assumption, (and with a little rearranging), one can see that the mode for the $w^{th}$ Beta function ( $w^{th}$ summand) tends to $b_{w}^{*}$ . Similarly, the $j^{th}$ mode of the Dirichlet mode tends to $\omega_{j}^{*}$ . So, to make all the non-entropy terms small, $z$ must be chosen so that $A z$ is close to $b^{*}$ (which one recalls is an abuse of notation and is a concatenation of the LF accuracies $b_{w}^{*}$ and class frequencies $\omega_{j}^{*}$ ). Since there is a set of $z$ 's such that $A z$ is 'close' to $b^{*}$ , one can choose a $z$ from that set to minimize the entropy term. Now, as $\alpha_{w}^{\prime}, \beta_{w}^{\prime}, \alpha_{j}^{\prime \prime}$ increase, the pdfs concentrate, meaning $A z \rightarrow b^{*}$ in order for the objective to be minimized. + +Now, we show that $g^{bbf} \to g^{bar}$ . + +Lemma D.20. Take the same assumptions as Lemma D.19. Then, as $Ag^{bbf} \to b^*$ , $g^{bbf} \to g^{bar}$ . + +Proof. Recall from our discussion of barrier method type analysis in Subsection D.4 that the complementary slackness condition is what controls how close we are approximating the 'original problem'. That is, a convex program with a log-barrier approximation to a feasible region $(f_{1}(x) \leq 0$ in that example) produces an approximate solution to the original problem. Reprinting, the following is the 'original problem': + +$$ +p^{*} = \min_{\substack{f_{1}(x)\leq 0\\ Ax = b}}f_{0}(x) +$$ + +with solution $x^{*}$ . The problem with the log-barrier approximation to $f_{1}(x) \leq 0$ is + +$$ +p (t) = \min _ {A x = b} f _ {0} (x) - \frac {1}{t} \log (- f _ {1} (x)) +$$ + +with solution $x^{*}(t)$ . The result was that as $t \to \infty$ , $x^{*}(t) \to x^{*}$ as the latter problem approximated the original problem arbitrarily well. + +The same idea applies here for our case. To be explicit, what happens is that the problems being solved eventually match, which is why $x^{*}(t) \to x^{*}$ . In our case, the KKT for the BBF problem will match the KKT conditions for BF with $P_{bar}$ . Upon inspection of Theorem D.12, we must show that $t_w, t_w', \tau_j, \tau_j'$ each tend to infinity as $Ag^{bbf} \to b^*$ for the BBF problem and BF with $P_{bar}$ to match (and to prove our claim). + +We start with $t_w$ and $t_w'$ . Reprinted, they are + +$$ +t _ {w} ^ {\prime} = \frac {\gamma_ {w} ^ {*}}{\alpha_ {w} ^ {\prime} (\gamma_ {w} ^ {*} - b _ {w} ^ {*} + \epsilon_ {w} ^ {-})}, \quad t _ {w} = \frac {1 - \gamma_ {w} ^ {*}}{\beta_ {w} ^ {\prime} (- \gamma_ {w} ^ {*} + b _ {w} ^ {*} + \epsilon_ {w} ^ {+})}. +$$ + +For each of these terms, we are interested in the case where $\epsilon_w^- = \epsilon_w^+ = \delta$ . In the other case, the denominator has $\alpha_w^\prime \delta$ or $\beta_w^\prime \delta$ , which can be arbitrarily small as $\delta$ is arbitrary. Recall from Lemma D.19 that $\gamma_w^* = y_w^\top g^{bbf} / N(\vec{y}_w) \to b_w^*$ . In particular, we can write $\gamma_w^*$ in terms of $\alpha_w^\prime$ and $\beta_w^\prime$ . Letting $\xi$ be a term to account for the deviation (as $[Ag^{bbf}]_w$ doesn't equal $b_w^*$ ), + +$$ +\gamma_ {w} ^ {*} = \frac {\alpha_ {w} ^ {\prime} + \xi}{\alpha_ {w} ^ {\prime} + \beta_ {w} ^ {\prime}} \rightarrow b _ {w} ^ {*}. +$$ + +Now, by assumption, $\alpha_w' \to b_w^* (\alpha_w' + \beta_w')$ . This means that both $\alpha_w'$ and $\alpha_w' + \xi$ tend to $b_w^* (\alpha_w' + \beta_w')$ or $\xi \to 0$ as $\alpha_w', \beta_w' \to \infty$ . To reduce clutter, we drop the reference to $w$ , noting that all quantities we're dealing with are scalars. Replacing $\gamma_w^*$ with the fraction involving $\xi$ , we have + +$$ +t ^ {\prime} = \frac {\frac {\alpha^ {\prime} + \xi}{\alpha^ {\prime}}}{(\alpha^ {\prime} + \beta^ {\prime}) (\frac {\alpha^ {\prime} + \xi}{\alpha^ {\prime} + \beta^ {\prime}} + b ^ {*} + \delta)} = \frac {1 + \frac {\xi}{\alpha^ {\prime}}}{\alpha^ {\prime} + \xi - b ^ {*} (\alpha^ {\prime} + \beta^ {\prime}) + \delta}. +$$ + +Recall that $\delta$ was arbitrary, meaning we can absorb anything in front of it. For the numerator, we can clearly see that it tends to 1 as $\alpha' \to \infty$ . For the denominator, since $\alpha' + \xi$ tends to $b^{*}(\alpha' + \beta')$ and $\delta$ can be arbitrarily small, the denominator tends to 0 and $t'$ tends to $\infty$ . Similarly for $t_{w}$ , we use $1 - \gamma_{w}^{*} = 1 - (\alpha_{w}' + \xi) / (\alpha_{w}' + \beta_{w}')$ to get + +$$ +t _ {w} = \frac {\frac {\beta^ {\prime} - \xi}{\beta^ {\prime}}}{- \alpha^ {\prime} - \xi + b ^ {*} (\alpha^ {\prime} + \beta^ {\prime}) + \delta} +$$ + +and the same argument from above works, this time using $\xi \rightarrow 0$ and $\alpha_w' \rightarrow b^*(\alpha_w' + \beta')_w$ . + +Now, we move on to $\tau_{j}$ and $\tau_{j}^{\prime}$ , which proceeds in a very similar way as the above. Lemma D.19 shows that $\pi_j^* = e_j^\top g^{bbf} / N\to \omega_j^*$ . Like before, we write + +$$ +\pi_ {j} ^ {*} = \frac {\alpha_ {j} ^ {\prime} + \xi}{\alpha_ {0} ^ {\prime}}. +$$ + +Since $\pi_j^* \to \omega_j^*$ , and $\alpha_j' \to \omega_j^* \alpha_0'$ , we have $\xi \to 0$ . As above, we're interested in the case when the denominator is not just $\delta$ . Written out, + +$$ +\tau_ {j} ^ {\prime} = \frac {\pi_ {j} ^ {*}}{\alpha_ {0} ^ {\prime} (\pi_ {j} ^ {*} - \omega_ {j} ^ {*} + \delta)} = \frac {\pi_ {j} ^ {*}}{\alpha_ {0} ^ {\prime} (\frac {\alpha_ {j} ^ {\prime} + \xi}{\alpha_ {0} ^ {\prime}} - \omega_ {j} ^ {*} + \delta)} = \frac {\pi_ {j} ^ {*}}{(\alpha_ {j} ^ {\prime} + \xi - \alpha_ {0} ^ {\prime} \omega_ {j} ^ {*} + \delta)} +$$ + +The numerator tends to a constant while the first three terms in the denominator tend to 0 while $\delta$ can be made arbitrarily small. Hence $\tau_j^\prime \to \infty$ . + +To see that $\tau_{j}$ also tends to infinity, observe that $\alpha_0^\prime \geq \alpha_0^\prime -\alpha_j^\prime$ , so that we can apply the above argument with very minor changes. + +Theorem D.21 (Theorem 5.2). Under the conditions of Lemma D.19, BBF is consistent. I.e. $g^{bbf} \to g^*$ . This happens because $\theta^{bbf} \to \theta^*$ . + +Table 3: Comparison of our proposed method with baseline methods with respect to accuracy. Bolded entries are indistinguishable via two-tailed paired t-test with $p = 0.05$ . + +
DatasetIMDBYouTubeSMSCDRYelpCommercialTennisTRECSemEvalChemProtAG News
MV73.6885.0293.9472.3672.3890.6186.8554.5179.0754.6781.38
DS50.3183.8887.1845.7772.9793.5786.5447.6673.5352.6681.79
DP72.7574.4581.5264.4157.8689.7986.5745.8572.7856.1781.71
FS74.4583.6191.9073.6274.0187.3185.9550.1012.1552.3781.28
MeTaL72.4370.5479.6760.2054.4789.7986.5743.7354.5855.8681.84
iBCC74.4445.6984.5560.9268.3083.3187.1921.0230.1935.0926.57
EBCC74.4345.6984.5562.9869.6783.3187.3021.0230.1935.0927.81
CLL72.7785.1993.8671.2472.0449.9949.9259.3484.2152.3780.97
HyperLM75.0390.9993.8672.5874.4288.2087.2060.0284.2152.3281.38
WeaSEL50.4960.8454.1351.6457.1645.1557.7920.8111.1421.4132.29
Denoise80.2681.4594.0273.1775.0494.2476.8749.8173.5357.8587.61
FABLE72.1687.7193.9471.6469.4492.5586.5653.6473.6154.4281.38
BBF72.4789.4793.8671.2971.8190.8584.5659.8884.2152.4781.40
g*74.5093.9293.9076.5274.7793.5986.6463.7584.2161.2682.24
+ +Proof. We have just proven in Lemma D.20 that $g^{bbf} \to g^{bar}$ . It suffices to show that under the assumptions of our claim, $g^{bar} \to g^*$ . + +To do that, it suffices to show that $P_{bar} \to P^{*} = \{z \in \Delta_{K}^{N} \colon Az = b^{*}\}$ under the assumptions of the theorem statement. This is because An and Dasgupta [2024] showed in Theorem 6 that solving BF with $P^{*}$ gives $g^{*}$ . Recall that $P_{bar}$ was defined as follows: + +$$ +P _ {b a r} = \left\{z \in \Delta_ {K} ^ {N} \colon b ^ {*} - \epsilon^ {-} \leq A z \leq b ^ {*} + \epsilon^ {+} \right\}. +$$ + +It would suffice to show that $\epsilon^{+},\epsilon^{-}$ both converge to 0. We now remind the reader of the definitions of $\epsilon^{+}$ and $\epsilon^{-}$ . For $i\in [W + K]$ and $\widehat{b}_i = [Ag^{bbf}]_i$ + +$$ +\epsilon_ {i} ^ {-} = \mathbf {1} \left(b _ {i} ^ {*} > \widehat {b} _ {i}\right) \left(b _ {i} ^ {*} - \widehat {b} _ {i}\right) + \delta \quad \text {a n d} \quad \epsilon_ {i} ^ {+} = \mathbf {1} \left(\widehat {b} _ {i} > b _ {i} ^ {*}\right) \left(\widehat {b} _ {i} - b _ {i} ^ {*}\right) + \delta . +$$ + +By definition, at least one of these is equal to $\delta$ , which is some arbitrary but small positive number. In Lemma D.19, we showed that $\widehat{b_i} \to b_i^*$ . Thus, given the assumptions of the theorem, $\epsilon_i^+, \epsilon_i^-$ are each either already $\delta$ , or tend to $\delta$ . This means that every element in $\epsilon^+$ and $\epsilon^-$ is arbitrarily small, showing that $P_{bar} \to P^*$ as the BBF hyperparameters grow. As BBF is essentially solving the BF problem that gives $g^*$ , $g^{bbf}$ will have the same weights as $g^*$ , i.e. $\theta^{bbf} \to \theta^*$ . + +# E Extra Experimental results + +In this section, we present extra experimental results. Specifically, the accuracies of label models for which F1 scores were reported in the main paper and the standard deviations of the F1 scores and 0-1 losses reported. Also, we present the BBF consistency experiments for all datasets used in the main paper. + +# E.1 0-1 Loss and Standard Deviations + +In the main paper, we followed Zhang et al. [2023] and included a mix of F1 scores and 0-1 loss depending on whether the dataset had two classes. Here, we provide the accuracies achieved by each label model on every dataset, regardless of whether there are two classes or not. + +In Table 3 we see the accuracies of our proposed method and the other baseline methods. Against FABLE, our proposed method has similar or better performance on most datasets. On Commercial, Tennis, and ChemProt, the performance of BBF was slightly worse than FABLE. With respect to HyperLM, BBF was similar or better on the majority of datasets, only being worse on IMDB, YouTube, CDR, Yelp. + +Here are the standard deviations for F1 score (Table 4) and 0-1 loss (Table 5). For brevity, we only show standard deviations for methods that had randomness. We see that WeaSEL had the largest standard deviation while the standard deviation for other methods were not very large. + +Table 4: Comparison of F1 score standard deviations for random baseline methods. + +
DatasetIMDBYouTubeSMSCDRYelpCommercialTennis
DP0.071.500.737.631.660.000.00
MeTaL0.160.000.736.180.450.000.00
EBCC0.000.000.000.000.030.000.02
WeaSEL10.896.676.2913.427.774.6317.22
Denoise1.130.430.380.101.331.0641.74
FABLE0.260.370.000.160.160.230.06
CLL0.000.000.000.070.000.154.02
+ +Table 5: Comparison of accuracy standard deviations for random baseline methods. + +
DatasetIMDBYouTubeSMSCDRYelpCommercialTennisTRECSemEvalChemProtAG News
DP0.162.630.551.923.900.000.000.800.250.460.25
MeTaL0.160.000.471.231.010.000.001.963.450.510.26
EBCC0.000.000.000.000.040.000.060.000.000.000.00
WeaSEL4.687.6312.115.733.688.935.843.402.164.912.69
Denoise1.240.340.110.112.270.5113.990.080.000.200.47
Fable0.280.330.000.060.210.110.030.470.240.680.04
CLL0.000.000.000.050.010.115.070.010.000.000.03
+ +# E.2 Experimental Results for BBF's Consistency + +In this section, we present experimental results showing the consistency of BBF on real datasets. The eleven datasets used in the main paper are considered again in the same setting – we allow BBF to label all datapoints (by combining the provided train/validation/test splits). Recall from our previous section that for $g^{bbf} \to g^*$ , we need the prior distributions for LF accuracies and class frequencies to concentrate around their empirical values. This is done by using the labels of the datapoints to compute $b_w^*$ and $\omega_j^*$ . The hyperparameters for the Beta distributions are set with scale factor $s$ as + +$$ +\rho_ {w 1} = s b _ {w} ^ {*} + 1, \quad \rho_ {w 2} = s (1 - b _ {w} ^ {*}) + 1 \quad \text {a n d} \quad \alpha_ {j} = s \omega_ {j} ^ {*} + 1. +$$ + +One is added so the conditions of Lemma D.19 are satisfied. We vary the scale by taking values in $\{10^1, \ldots, 10^5\}$ for 5 trials total for each dataset. To check for consistency, we measure the average KL divergence between $g^{bbf}$ and $\eta$ . Namely, $\frac{1}{N} d(\eta, g^{bbf})$ . The provided graphs have a log-log scale and show that the rate of convergence is linear. + +![](images/66ae870cb5bf3c2c7a97131a08616592b3ba489ac59cd94cda2bd32bd8fa294a.jpg) +Figure 2: Convergence of $g^{bbf}$ to $g^*$ on the IMDB dataset with respect to scale factor $s$ . + +![](images/e5a00ce621ab1b211dbcd3702de317e5f07cb5e850f8156ec46c0d429dabc2fc.jpg) +Figure 3: Convergence of $g^{bbf}$ to $g^*$ on the Youtube dataset with respect to scale factor $s$ . + +![](images/83edfe5f9a025412c8252d64c425a8f0b54f73a9ee7cb6bfbd127d826ab6993b.jpg) +Figure 4: Convergence of $g^{bbf}$ to $g^*$ on the SMS dataset with respect to scale factor $s$ . + +![](images/9bd5eb35fb4248c82f1aefbf4067e92f4a5e95c9a1ac4090aa7e58c347b2d167.jpg) +Figure 5: Convergence of $g^{bbf}$ to $g^*$ on the CDR dataset with respect to scale factor $s$ . + +![](images/c01d71051015ad1f73662d3d4ce3bb86e175dffb513018b4abe393f68a32df43.jpg) +Figure 6: Convergence of $g^{bbf}$ to $g^*$ on the Yelp dataset with respect to scale factor $s$ . + +![](images/ffc3769b1e06f479a84519a79387b80f0e4298cc6f37b99ac4b1b2aea05e7749.jpg) +Figure 7: Convergence of $g^{bbf}$ to $g^*$ on the Commercial dataset with respect to scale factor $s$ . + +![](images/e5a05859828f780dcd678740114cffaa2b44bd9b1287b7122c884a03d8c7282b.jpg) +Figure 8: Convergence of $g^{bbf}$ to $g^*$ on the Tennis dataset with respect to scale factor $s$ . + +![](images/d64975d0a78f5dce02651b4628ec4b99f0e4ad7ec6027572dc83088b91f8d608.jpg) +Figure 9: Convergence of $g^{bbf}$ to $g^*$ on the TREC dataset with respect to scale factor $s$ . + +![](images/9e81f3feeed989aa8d11aa184036505eead394202c7631f8ea001fd59aa34a81.jpg) +Figure 10: Convergence of $g^{bbf}$ to $g^*$ on the SemEval dataset with respect to scale factor $s$ . + +![](images/6e9def2ae58bcd8a88da051b48f2932dcf1df9b6993c992ec3a489b49314edbe.jpg) +Figure 11: Convergence of $g^{bbf}$ to $g^*$ on the ChemProt dataset with respect to scale factor $s$ . + +![](images/81023837ec27eaa6c2ba324e3d95fc378fa70764792aac68e3466c3a42e2798f.jpg) +Figure 12: Convergence of $g^{bbf}$ to $g^*$ on the AG News dataset with respect to scale factor $s$ . + +# NeurIPS Paper Checklist + +# 1. 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0000000000000000000000000000000000000000..7a16a3dc7ce6311a22821409f560d90130d8defd --- /dev/null +++ b/statisticalanalysisofthesinkhorniterationsfortwosampleschrodingerbridgeestimation/full.md @@ -0,0 +1,1559 @@ +# Statistical Analysis of the Sinkhorn Iterations for Two-Sample Schrödinger Bridge Estimation + +Ibuki Maeda + +maeda. ibuki554@mail.kyutech.jp + +Rentian Yao2 + +rentian2@math.ubc.ca + +Atsushi Nitanda3,4,5 + +atsushi_nitanda@a-star.edu.sg + +$^{1}$ Department of Artificial Intelligence, Kyushu Institute of Technology, Japan + +$^{2}$ Department of Mathematics, University of British Columbia, Canada + +3 Institute of High Performance Computing, Agency for Science, Technology and Research (A*STAR), Singapore + +4 Centre for Frontier AI Research, Agency for Science, Technology and Research (A*STAR), Singapore + +5 College of Computing and Data Science, Nanyang Technological University, Singapore + +# Abstract + +The Schrödinger bridge problem seeks the optimal stochastic process that connects two given probability distributions with minimal energy modification. While the Sinkhorn algorithm is widely used to solve the static optimal transport problem, a recent work (Pooladian and Niles-Weed, 2024) proposed the Sinkhorn bridge, which estimates Schrödinger bridges by plugging optimal transport into the time-dependent drifts of SDEs, with statistical guarantees in the one-sample estimation setting where the true source distribution is fully accessible. In this work, to further justify this method, we study the statistical performance of intermediate Sinkhorn iterations in the two-sample estimation setting, where only finite samples from both source and target distributions are available. Specifically, we establish a statistical bound on the squared total variation error of Sinkhorn bridge iterations: $\mathcal{O}(1 / m + 1 / n + r^{4k})$ ( $r \in (0,1)$ ), where $m$ and $n$ are the sample sizes from the source and target distributions, respectively, and $k$ is the number of Sinkhorn iterations. This result provides a theoretical guarantee for the finite-sample performance of the Schrödinger bridge estimator and offers practical guidance for selecting sample sizes and the number of Sinkhorn iterations. Notably, our theoretical results apply to several representative methods such as $[\mathrm{SF}]^2\mathrm{M}$ , DSBM-IMF, BM2, and lightSB(-M) under specific settings, through the previously unnoticed connection between these estimators. + +# 1 Introduction + +In recent years, there has been increasing interest in Schrödinger bridge problems, with applications in mathematical biology (Chizat et al., 2022; Lavenant et al., 2024; Yao et al., 2025), Bayesian posterior sampling (Heng et al., 2024), generative modeling (De Bortoli et al., 2021; Wang et al., 2021), among others. The Schrödinger bridge problem aims to minimize the Kullback-Leibler (KL) divergence with respect to the Wiener measure, the distribution of Brownian motion over the path space, given two fixed marginal distributions $\mu, \nu$ as source and target distributions. + +The Schrödinger Bridge problem is also strongly connected to the entropic optimal transport (EOT) problem, which offers an efficient and regularized alternative to classical optimal transport, and is solvable via the Sinkhorn algorithm (Cuturi, 2013). The Sinkhorn algorithm operates in the dual space, optimizing the Schrödinger potentials between empirical distributions, and enjoys well-established convergence guarantees (Franklin and Lorenz, 1989; Peyré and Cuturi, 2018). Recently, Pooladian and Niles-Weed (2024) introduced the Sinkhorn bridge method, which plugs in the optimal Schrödinger potentials into the time-dependent drifts of stochastic differential equations to estimate + +the Schrödinger bridge between the source and target distributions $\mu$ and $\nu$ . Their analysis established statistical bounds, adaptive to the intrinsic dimension of the target distribution $\nu$ , in the one-sample estimation setting, where the true source distribution $\mu$ is fully known. + +# 1.1 Contributions + +In this work, to better understand the algorithmic and statistical convergence rates of the Sinkhorn bridge method, we study the statistical performance of intermediate estimators obtained by Sinkhorn iterations in the two-sample estimation setting, where only empirical distributions $\mu_{m}$ and $\nu_{n}$ of $m-$ and $n$ -finite samples from $\mu$ and $\nu$ are available. + +Statistical guarantee for Sinkhorn bridge iterations. First, we establish the following statistical bound (Theorem 1) for the Sinkhorn bridge estimator by leveraging analytical tools for the Schrödinger bridge in the one-sample setting (Pooladian and Niles-Weed, 2024) and for entropic optimal transport (Stromme, 2023b): + +$$ +\mathbb {E} \left[ \mathrm {T V} ^ {2} \left(P _ {\infty , \infty} ^ {*}, [ 0, \tau ], P _ {m, n} ^ {*}, [ 0, \tau ]\right) \right] = \mathcal {O} \left(\frac {1}{m} + \frac {1}{n}\right), \tag {1} +$$ + +where $P_{\infty, \infty}^{*}$ and $P_{m, n}^{*}$ denote the true Schrödinger bridge and its Sinkhorn bridge estimator based on the optimal EOT solution, and $P_{\infty, \infty}^{*,[0, \tau]}$ and $P_{m, n}^{*,[0, \tau]}$ are restrictions of them on the time-interval $[0, \tau]$ . The result improves upon the bound established in (Pooladian and Niles-Weed, 2024) by extending the analysis to the two-sample setting and sharpening the rate in $n$ from $n^{-1/2}$ to $n^{-1}$ at the cost of a deterioration in the dependence on $\varepsilon$ and $R$ , where $\varepsilon$ is the strength of entropic regularization and $R$ is the radius of the data supports. + +Next, we analyze the intermediate iterations of the Sinkhorn algorithm and prove a convergence rate (Theorem 2) for the corresponding path measures: + +$$ +\mathbb {E} \left[ \mathrm {T V} ^ {2} \left(P _ {m, n} ^ {*}, [ 0, \tau ], P _ {m, n} ^ {k, [ 0, \tau ]}\right) \right] \lesssim \frac {\tau}{1 - \tau} \frac {R ^ {4}}{\varepsilon} \left(\tanh \left(\frac {R ^ {2}}{\varepsilon}\right)\right) ^ {4 k}, \tag {2} +$$ + +where $P_{m,n}^{k,[0,\tau]}$ denotes the Sinkhorn bridge estimator at the $k$ -th Sinkhorn iteration for $\mu_m$ and $\nu_n$ . Together, these results imply the convergence rate $\mathbb{E}[\mathrm{TV}^2 (P_{\infty ,\infty}^{*,[0,\tau]},P_{m,n}^{k,[0,\tau]})] = \mathcal{O}(1 / m + 1 / n + r^{4k})$ ( $r\in (0,1)$ ) regarding the number of samples $m$ and $n$ , and the number of Sinkhorn iterations $k$ . + +Relationship with the other Schrödinger Bridge estimators. Remarkably, our theoretical results on the Sinkhorn bridge can be directly applied to the representative Schrödinger Bridge solvers: $\mathrm{[SF]^{2}M}$ (Tong et al., 2023), DSBM-IMF (Shi et al., 2023; Peluchetti, 2023), BM2 (Peluchetti, 2024), and lightSB(-M) (Korotin et al., 2023; Gushchin et al., 2024). First, the optimal estimator produced by these methods coincide with the Sinkhorn bridge estimator (Proposition 1). Moreover, when DSBM-IMF and BM2 are initialized with the reference process, the estimator after $k$ iterations matches the Sinkhorn bridge obtained after $k$ iterations of the Sinkhorn algorithm (Proposition 2). Furthermore, Algorithm 1, which learns the Sinkhorn bridge drift via a neural network, is a special case of the procedure proposed in $\mathrm{[SF]^{2}M}$ . Consequently, our generalization-error analysis applies to all of these methods, endowing the algorithms with theoretical guarantees. + +# 1.2 Related Work + +Schrodinger bridge problem. The connection between entropic optimal transport and the Schrodinger bridge (SB) problem has been extensively studied; for a comprehensive overview, see the survey by Léonard (2013). More recently, there has been growing interest in computational approaches to the SB problem, particularly those leveraging deep learning techniques (De Bortoli et al., 2023; Shi et al., 2022; Bunne et al., 2023; Tong et al., 2023). In parallel, several studies have explored classical statistical methods for estimating the SB (Bernton et al., 2019; Pavon et al., 2021; Vargas et al., 2021). For example, Bernton et al. (2019) proposed a sampling framework based on trajectory refinement via approximate dynamic programming. Pavon et al. (2021) and Vargas et al. (2021) introduced methods that directly estimate intermediate densities using maximum likelihood principles. Specifically, Pavon et al. (2021) presented a scheme that explicitly models the target + +density and updates weights accordingly, while Vargas et al. (2021) estimated forward and backward drifts using Gaussian processes, optimized via a likelihood-based objective. Assuming full access to the source distribution, the statistical performance of the Schrödinger bridge estimator was analyzed in the one-sample setting by Pooladian and Niles-Weed (2024). Other estimators, such as those based on neural approximations or efficient algorithmic variants, have been evaluated in recent works (Korotin et al., 2023; Stromme, 2023a). + +Entropic optimal transport. Entropic Optimal Transport (EOT) introduces an entropy term to the standard Optimal Transport (OT) problem (Cuturi, 2013), primarily proposed to improve computational efficiency and smooth the transport plan. This regularization makes the OT problem strictly convex, enabling the application of the Sinkhorn algorithm, an efficient iterative method for the dual problem (Cuturi, 2013). The introduction of the Sinkhorn algorithm has allowed the OT problem, previously computationally very expensive, to obtain approximate solutions in realistic time even for large-scale datasets, leading to a dramatic expansion of its applications in the field of machine learning. For example, there are diverse applications such as measuring distances between distributions in generative modeling (Genevay et al., 2018), and shape matching or color transfer in computer graphics and computer vision (Solomon et al., 2015). The theoretical aspects of EOT have also been deeply studied. Analyses regarding the convergence rate of the Sinkhorn algorithm (Franklin and Lorenz, 1989) and the statistical properties (such as sample complexity) when estimating EOT from finite samples (Genevay et al., 2019) have been established. Furthermore, EOT is known to have a close relationship with the Schrödinger Bridge (SB) problem (Léonard, 2013). EOT in a static setting is equivalent to the problem of matching the marginal distributions of the SB problem, which deals with dynamic stochastic processes, and both share common mathematical structures in duality and optimization algorithms. This connection suggests that theoretical and computational advances in one field can contribute to the other. + +# 1.3 Notations + +For a metric space $\mathcal{X}$ , let $\mathcal{P}(\mathcal{X})$ be the space of probability distributions over $\mathcal{X}$ , and $\mathcal{P}_2(\mathcal{X})$ be the subset of $\mathcal{P}(\mathcal{X})$ with finite second-order moment. Let $\mathbb{R}^d$ be the $d$ -dimensional Euclidean space and $B(a, R) = \{x \mid \|x - a\|_2 \leq R\} \subset \mathbb{R}^d$ be the Euclidean ball of radius $R > 0$ centered at $a \in \mathbb{R}^d$ . For real numbers $a$ and $b$ , we set $a \vee b = \max(a, b)$ and $a \wedge b = \min(a, b)$ . We write $a \lesssim b$ to denote that there exists a constant $C > 0$ , independent of $a$ and $b$ , such that $a \leq Cb$ . We denote by $\delta_x$ the Dirac measure concentrated at the point $x \in \mathbb{R}^d$ . We denote by $\mathbb{1}_d \in \mathbb{R}^d$ the $d$ -dimensional vector whose components are all equal to one. For a measure $\mu$ and a function $f$ , we write $\mu(f) = \int f(x) \, \mu(dx)$ . Let $\mathrm{TV}(\mu, \nu)$ be the total variation distance between two probability distributions $\mu$ and $\nu$ . For $\varepsilon > 0$ and any real number or real-valued function $h$ , write $\exp_{\varepsilon}(h) := \exp(h / \varepsilon)$ . We denote the $d$ -dimensional probability simplex by $\Delta_d := \{\alpha \in \mathbb{R}_{\geq 0}^d \mid \sum_{i=1}^{d} \alpha_i = 1\}$ . + +# 2 Preliminaries + +# 2.1 Entropic Optimal Transport + +Let $\mu, \nu \in \mathcal{P}_2(\mathbb{R}^d)$ and $\varepsilon > 0$ be fixed. The entropic optimal transport (EOT) cost between $\mu$ and $\nu$ is + +$$ +\mathrm {O T} _ {\varepsilon} (\mu , \nu) := \inf _ {\pi \in \Pi (\mu , \nu)} \left\{\int_ {\mathbb {R} ^ {d} \times \mathbb {R} ^ {d}} c (x, y) \mathrm {d} \pi (x, y) + \varepsilon H (\pi \| \mu \otimes \nu) \right\}, \tag {3} +$$ + +with the cost function $c(x,y)\coloneqq \frac{1}{2}\| x - y\| _2^2$ . Here, $\Pi (\mu ,\nu)\subset \mathcal{P}_2(\mathbb{R}^d\times \mathbb{R}^d)$ denotes the set of couplings with marginals $\mu$ and $\nu$ , and + +$$ +H (\pi \| \mu \otimes \nu) := \int_ {\mathbb {R} ^ {d} \times \mathbb {R} ^ {d}} \log \left(\frac {\mathrm {d} \pi}{\mathrm {d} (\mu \otimes \nu)} (x, y)\right) \mathrm {d} \pi (x, y) +$$ + +is the Kullback-Leibler divergence between $\pi$ and $\mu \otimes \nu$ . If $\pi$ is not absolutely continuous with respect to $\mu \otimes \nu$ , we set $H(\pi \| \mu \otimes \nu) = +\infty$ . + +The optimization problem in Eq. (3) is strictly convex with respect to $\pi$ , and thus it admits a unique solution $\pi_{\infty, \infty}^{*} \in \Pi(\mu, \nu)$ . Furthermore, the problem has a dual formulation given by + +$$ +\mathrm {O T} _ {\varepsilon} (\mu , \nu) = \sup _ {f \in L ^ {1} (\mu), g \in L ^ {1} (\nu)} \Phi^ {\mu \nu} (f, g), \quad \text {w h e r e} +$$ + +$$ +\Phi^ {\mu \nu} (f, g) := \int f d \mu + \int g d \nu - \varepsilon \int \left(\exp_ {\varepsilon} \left(f (x) + g (y) - \frac {1}{2} \| x - y \| _ {2} ^ {2}\right) - 1\right) d \mu (x) d \nu (y). \tag {4} +$$ + +The dual problem (4) admits the Schrödinger potentials $f_{\infty, \infty}^{*}$ and $g_{\infty, \infty}^{*}$ as optimal solutions, and the first-order optimality condition implies + +$$ +f _ {\infty , \infty} ^ {*} (x) = - \varepsilon \log \int_ {\mathbb {R} ^ {d}} \exp_ {\varepsilon} \left(g _ {\infty , \infty} ^ {*} (y) - \frac {1}{2} \| x - y \| _ {2} ^ {2}\right) d \nu (y), \tag {5a} +$$ + +$$ +g _ {\infty , \infty} ^ {*} (y) = - \varepsilon \log \int_ {\mathbb {R} ^ {d}} \exp_ {\varepsilon} \left(f _ {\infty , \infty} ^ {*} (x) - \frac {1}{2} \| x - y \| _ {2} ^ {2}\right) \mathrm {d} \mu (x), \tag {5b} +$$ + +which is known as Schrödinger system. Note that the solution is constant-shift invariant, i.e. for any $a \in \mathbb{R}$ , $(f_{\infty,\infty}^{*} + a, g_{\infty,\infty}^{*} - a)$ is also a solution to the Schrödinger system (5). It can be shown that with a constraint $\nu(g_{\infty,\infty}^{*}) = 0$ , the solution to (5) is unique. The optimal solution $\pi_{\infty,\infty}^{*}$ of the primal problem (3) can be expressed using the dual solution via + +$$ +p _ {\infty , \infty} ^ {*} (x, y) := \frac {\mathrm {d} \pi_ {\infty , \infty} ^ {*}}{\mathrm {d} (\mu \otimes \nu)} (x, y) = \exp_ {\varepsilon} \left(f _ {\infty , \infty} ^ {*} (x) + g _ {\infty , \infty} ^ {*} (y) - \frac {1}{2} \| x - y \| _ {2} ^ {2}\right). \tag {6} +$$ + +We refer to (Nutz, 2021) for more details of entropic optimal transport problems. + +# 2.2 Schrodinger Bridge + +Let $\Omega = \mathcal{C}([0,1];\mathbb{R}^d)$ be the path space consisting of all continuous maps from the time interval $[0,1]$ to $\mathbb{R}^d$ . Let $W^{\varepsilon}$ be the law of a reversible Brownian motion on $\mathbb{R}^d$ with volatility $\varepsilon$ . The Schrödinger Bridge problem is then defined through the following entropy minimization problem: + +$$ +\inf _ {P \in \mathcal {P} (\Omega)} \varepsilon H (P \mid W ^ {\varepsilon}) \quad \text {s u b j e c t t o} \quad P _ {0} = \mu , \quad P _ {1} = \nu , \tag {7} +$$ + +where $P_{t}\in \mathcal{P}(\mathbb{R}^{d})$ is the marginal distribution at time $t\in [0,1]$ of the path-space distribution $P$ , i.e. $X_{t}\sim P_{t}$ if $X$ is a stochastic process with distribution $P$ . Though $W^{\varepsilon}$ is an unbounded positive measure over $\Omega$ , the above optimization problem is still well defined as illustrated by Léonard (2013). + +The Schrödinger Bridge problem is closely connected with the EOT problem. Precisely, the optimal solution $P_{\infty,\infty}^{*} \in \mathcal{P}(\Omega)$ of Eq. (7) can be derived by first solving the EOT problem (3) and then connect the optimal coupling with the Brownian bridge $W_{|x_0,x_1}^{\varepsilon} = W^{\varepsilon}(\cdot |X_0 = x_0,X_1 = x_1)$ , i.e., + +$$ +P _ {\infty , \infty} ^ {*} = \int W _ {| x _ {0}, x _ {1}} ^ {\varepsilon} \mathrm {d} \pi_ {\infty , \infty} ^ {*} (x _ {0}, x _ {1}). \tag {8} +$$ + +The optimal solution $P_{\infty, \infty}^{*}$ can also be characterized as the path measure of the SDE defined by + +$$ +d X _ {t} = b _ {\infty , \infty} ^ {*} (X _ {t}, t) \mathrm {d} t + \sqrt {\varepsilon} \mathrm {d} B _ {t}, \quad X _ {0} \sim \mu , +$$ + +where $B_{t}$ is the standard Brownian motion, and the drift coefficient is given by + +$$ +b _ {\infty , \infty} ^ {*} (z, t) = \frac {1}{1 - t} \left(- z + \frac {\mathbb {E} _ {Y \sim \nu} \left[ \gamma_ {\infty , \infty} ^ {*, t} (Y , z) Y \right]}{\mathbb {E} _ {Y \sim \nu} \left[ \gamma_ {\infty , \infty} ^ {*, t} (Y , z) \right]}\right) \tag {9} +$$ + +with $\gamma_{\infty, \infty}^{*, t}(y, z) = \exp_{\varepsilon}(g_{\infty, \infty}^{*}(y) - \frac{1}{2(1 - t)} \| y - z\|_2^2)$ . We refer to Chen et al. (2021); Léonard (2012, 2013) for more details of the Schrödinger Bridge. + +# 2.3 Sinkhorn Algorithm + +Given the empirical distributions $\mu_{m} = \frac{1}{m}\sum_{i = 1}^{m}\delta_{X_{i}}$ and $\nu_{n} = \frac{1}{n}\sum_{j = 1}^{n}\delta_{Y_{j}}$ , we consider the empirical EOT problem $\mathrm{OT}_{\varepsilon}(\mu_m,\nu_n)$ . Let $f_{m,n}^*$ and $g_{m,n}^*$ be the optimal Schrödinger potentials. The Sinkhorn algorithm is an iterative method for approximating these optimal potentials by alternately updating the dual potentials $f_{m,n}^{(k)}$ and $g_{m,n}^{(k)}$ as follows. + +$$ +f _ {m, n} ^ {(k + 1)} = \underset {f \in L ^ {1} \left(\mu_ {m}\right)} {\operatorname {a r g m a x}} \Phi^ {\mu_ {m} \nu_ {n}} (f, g _ {m, n} ^ {(k)}), \quad g _ {m, n} ^ {(k + 1)} = \underset {g \in L ^ {1} \left(\nu_ {n}\right)} {\operatorname {a r g m a x}} \Phi^ {\mu_ {m} \nu_ {n}} \left(f _ {m, n} ^ {(k + 1)}, g\right). \tag {10} +$$ + +We note that $f_{m,n}^{(k)}, g_{m,n}^{(k)}$ can be interpreted as $m-$ and $n$ -dimensional vectors, respectively. For efficient computation, it is common to rewrite these updates in an alternative form. Let the kernel matrix $K \in \mathbb{R}^{m \times n}$ be defined as $K_{ij} = \exp_{\varepsilon}(-\frac{1}{2}\|X_i - Y_j\|^2)$ , and denote the vectors $u^{(k)} \in \mathbb{R}^m$ and $v^{(k)} \in \mathbb{R}^n$ by $u_i^{(k)} = \exp_{\varepsilon}(f^{(k)}(X_i))$ and $v^{(k)}(Y_j) = \exp_{\varepsilon}(g^{(k)}(Y_j))$ . Then the update rule (10) becomes + +$$ +u ^ {(k + 1)} = m \mathbf {1} _ {m} \oslash (K v ^ {(k)}), \quad v ^ {(k + 1)} = n \mathbf {1} _ {n} \oslash (K ^ {T} u ^ {(k + 1)}), +$$ + +where $\oslash$ denotes element-wise division1. This reformulation greatly improves computational efficiency and enables large-scale applications. + +Let $d_{\mathrm{Hilb}}$ be the Hilbert (pseudo-)metric on $\mathbb{R}_+^d$ defined by + +$$ +d _ {\mathrm {H i l b}} (u, u ^ {\prime}) = \log \left(\max _ {i} \frac {u _ {i}}{u _ {i} ^ {\prime}} \cdot \max _ {i} \frac {u _ {i} ^ {\prime}}{u _ {i}}\right), \quad \forall u, v \in \mathbb {R} _ {+} ^ {d}. +$$ + +It is obvious that $d_{\mathrm{Hilb}}(x,y) = 0$ if and only if there exists $\lambda \in \mathbb{R}_+$ such that $x = \lambda y$ . Let the initial values be $(u^{(0)},v^{(0)}) = (\mathbb{1}_m,\mathbb{1}_n)$ . Denote by $(u^{(k)},v^{(k)})$ the solution obtained after $k$ iterations of the Sinkhorn algorithm (10), and by $(u^{*},v^{*}) = (\exp (f_{m,n}^{*} / \varepsilon),\exp (g_{m,n}^{*} / \varepsilon))$ the optimal solution. Then, it is shown that (Franklin and Lorenz, 1989; Peyre and Cuturi, 2018), + +$$ +d _ {\text {H i l b}} (u ^ {(k)}, u ^ {*}) \leq \lambda (K) ^ {2 k - 1} d _ {\text {H i l b}} (v ^ {(0)}, v ^ {*}), \quad d _ {\text {H i l b}} (v ^ {(k)}, v ^ {*}) \leq \lambda (K) ^ {2 k} d _ {\text {H i l b}} (v ^ {(0)}, v ^ {*}). +$$ + +Here, $\lambda(K) < 1$ is defined as $\lambda(K) := \frac{\sqrt{\gamma(K)} - 1}{\sqrt{\gamma(K)} + 1}$ where $\gamma(K) := \max_{i,j,k,l} \frac{K_{ik} K_{jl}}{K_{jk} K_{il}}$ . This result implies the exponential convergence of the Sinkhorn iterates $u^{(k)}$ and $v^{(k)}$ . + +# 2.4 Convergence Rate of One-Sample Estimation + +Pooladian and Niles-Weed (2024) analyzed the one-sample estimation task for the Schrödinger Bridge. In this setting, full access to the source distribution $\mu$ is assumed, while the target distribution $\nu$ is observed only through the empirical distribution $\nu_{n} = \frac{1}{n}\sum_{j=1}^{n}\delta_{Y_{j}}$ based on i.i.d. samples $Y_{1},\ldots,Y_{n}$ . Let $(f_{\infty,n}^{*},g_{\infty,n}^{*})$ denote the optimal dual potentials solving $\mathrm{OT}_{\varepsilon}(\mu,\nu_{n})$ , and let $P_{\infty,n}^{*}\in \mathcal{P}(\Omega)$ be the corresponding Schrödinger Bridge. Under Assumptions 1, 2 and 3 stated below, Pooladian and Niles-Weed (2024, Theorem 4.1) proved that there exists a constant $C_{\nu}$ , depending only on $\nu$ and geometric constants of its support $\mathrm{supp}(\nu)$ , such that for any $\tau \in [0,1)$ the following holds: + +$$ +\mathbb {E} \left[ \mathrm {T V} ^ {2} \left(P _ {\infty , n} ^ {*}, [ 0, \tau ], P _ {\infty , \infty} ^ {*}, [ 0, \tau ]\right) \right] \leq C _ {\nu} \left\{\frac {\varepsilon^ {- d _ {\nu} - 1}}{\sqrt {n}} + \frac {R ^ {2} \varepsilon^ {- d _ {\nu}}}{(1 - \tau) ^ {d _ {\nu} + 2} n} \right\}, \tag {11} +$$ + +where $d_{\nu}$ is the intrinsic dimensionality of $\mathrm{supp}(\nu)$ . + +# 3 Main Results + +In this section, we provide statistical and algorithmic convergence rates of Sinkhorn bridge (Pooladian and Niles-Weed, 2024) in two-sample estimation setting. + +# 3.1 Sinkhorn Bridge in Two-sample Case + +Given empirical distributions $\mu_{m}$ and $\nu_{n}$ , the Sinkhorn bridge constructs the estimator $P_{m,n}^{k} \in \mathcal{P}(\Omega)$ for the Schrödinger bridge between true distributions $\mu$ and $\nu$ as follows. First, the Sinkhorn algorithm runs for $k$ iterations to solve $\mathrm{OT}_{\varepsilon}(\mu_{m}, \nu_{n})$ , yielding the dual potentials $f_{m,n}^{(k)}$ and $g_{m,n}^{(k)}$ . Then, the estimator $P_{m,n}^{k}$ is defined as the path measure of the following SDE starting from $\mu$ : + +$$ +\mathrm {d} X _ {t} = b _ {m, n} ^ {(k)} \left(X _ {t}, t\right) \mathrm {d} t + \sqrt {\varepsilon} \mathrm {d} B _ {t}, \quad X _ {0} \sim \mu , \tag {12} +$$ + +where the drift function $b_{m,n}^{(k)}$ is + +$$ +b _ {m, n} ^ {(k)} (z, t) = \frac {1}{1 - t} \left(- z + \frac {\sum_ {j = 1} ^ {n} Y _ {j} \exp_ {\varepsilon} \left(g _ {m , n} ^ {(k)} \left(Y _ {j}\right) - \frac {1}{2 (1 - t)} \| Y _ {j} - z \| _ {2} ^ {2}\right)}{\sum_ {j = 1} ^ {n} \exp_ {\varepsilon} \left(g _ {m , n} ^ {(k)} \left(Y _ {j}\right) - \frac {1}{2 (1 - t)} \| Y _ {j} - z \| _ {2} ^ {2}\right)}\right). \tag {13} +$$ + +# 3.2 Convergence Rates of Sinkhorn Bridge + +First, we analyze the statistical convergence rate of the path measure $P_{m,n}^{*} \in \mathcal{P}(\Omega)$ using the optimal Schrödinger potentials $f_{m,n}^{*}$ and $g_{m,n}^{*}$ . Specifically, $P_{m,n}^{*}$ is defined analogously to $P_{m,n}^{k}$ by replacing the potential $g_{m,n}^{(k)}$ with its optimal counterpart $g_{m,n}^{*}$ . For details of the definition, see Appendix A.1. The analysis will be carried out under the following assumptions. + +Assumption 1. The potentials $g_{\infty ,\infty}^{*},g_{\infty ,n}^{*},g_{m,n}^{*}$ satisfy + +$$ +\int g _ {\infty , \infty} ^ {*} (y) \nu (d y) = \int g _ {\infty , n} ^ {*} (y) \nu_ {n} (d y) = \int g _ {m, n} ^ {*} (y) \nu_ {n} (d y) = 0. +$$ + +Assumption 2. The supports of $\mu$ and $\nu$ are subsets of a ball with radius $R$ centered at the origin. + +Assumption 3. The measure $\nu$ is supported on a compact, smooth, connected Riemannian manifold $(N,h)$ without boundary, embedded in $\mathbb{R}^d$ and equipped with the submanifold geometry induced by its inclusion into $\mathbb{R}^d$ ; furthermore, $\dim N = d_{\nu} \geq 3$ . Moreover, $\nu$ admits a Lipschitz continuous and strictly positive density with respect to the Riemannian volume measure on $N$ . + +The manifold hypothesis suggests that data distributions often lie on lower-dimensional manifolds, and statistical learning methods can exploit this structure. In the EOT literature, Stromme (2023b) established the Minimum Intrinsic Dimension (MID) scaling, where the minimum of the intrinsic dimensions governs the convergence rate. For the Sinkhorn bridge under one-sample setting, Pooladian and Niles-Weed (2024) demonstrated that the convergence rate is independent of the ambient dimension $d$ and depends only on the intrinsic dimension of the target distribution $\nu$ . Following these works, we also impose data-manifold assumptions solely on the target distribution $\nu$ . + +The following theorem provides the squared total variation error between $P_{\infty, \infty}^{*}$ and $P_{m, n}^{*}$ . + +Theorem 1 (Statistical convergence rate). Suppose Assumptions 1, 2 and 3 hold. Let $P_{\infty,\infty}^{*}$ be the path measure of the true Schrödinger bridge for marginals $\mu, \nu$ . Then, it follows that for any $\tau \in [0,1)$ , + +$$ +\begin{array}{l} \mathbb {E} [ \mathrm {T V} ^ {2} (P _ {\infty , \infty} ^ {*}, [ 0, \tau ], P _ {m, n} ^ {*}, [ 0, \tau ]) ] \\ \lesssim \frac {R ^ {2}}{n (1 - \tau) ^ {d _ {\nu} + 2} \varepsilon^ {d _ {\nu}}} + \frac {\tau}{1 - \tau} \left(\varepsilon^ {2} + \frac {d R ^ {8}}{\varepsilon^ {2}}\right) \cdot \left(\frac {R}{\varepsilon}\right) ^ {9 d _ {\nu} + 4} \cdot \left(\frac {1}{m} + \frac {1}{n}\right). \tag {14} \\ \end{array} +$$ + +where $P_{\infty, \infty}^{*,[0, \tau]}$ and $P_{m,n}^{*,[0, \tau]}$ are restrictions of $P_{\infty, \infty}^{*}$ and $P_{m,n}^{*}$ on the time-interval $[0, \tau]$ . + +Remark 1. In this section and related proofs, we suppress non-dominant terms such as polynomial factors of $d_{\nu}$ , geometric characteristics of $\mathrm{supp}(\nu)$ , and exponential terms of order $O(d_{\nu})$ in uniform constants, in the notation $\lesssim$ . + +We make the following observations: (1) Regarding the data dimensions, the intrinsic dimension $d_{\nu}$ governs the growth of the convergence rate, as in Stromme (2023b); Pooladian and Niles-Weed (2024). Although our bound, unlike theirs, involves the ambient dimension $d$ , the dependence is merely linear. Moreover, we note that the term $d$ arises from the covering number of the data space + +and can also be replaced by the intrinsic dimension of the source distribution $\mu$ by additionally imposing a manifold assumption on $\mu$ . (2) The degeneration of the estimated SDE toward a specific sample point $Y_{j}$ arises as $\tau \rightarrow 1$ because the estimated drift $b_{m,n}^{(k)}$ points toward a particular sample $Y_{j}$ near $t = 1$ , causing the deviation from the target distribution $\nu$ at the terminal time $t = 1$ . Therefore, it is common practice to restrict the time interval to $[0,\tau ]$ ( $\tau < 1$ ) to ensure generalization. (3) Our result extends and sharpens the result of Pooladian and Niles-Weed (2024) to the two-sample setting, improving the convergence rate in $n$ from $n^{-1 / 2}$ to $n^{-1}$ at the cost of a slight deterioration regarding, $\varepsilon$ , $R$ , and $d$ . + +In the next theorem, we give the algorithmic convergence rate of $P_{m,n}^{k} \to P_{m,n}^{*}$ attained by running the Sinkhorn iterations $k \to \infty$ . + +Theorem 2 (Algorithmic convergence rate). Under Assumptions 1 and 2, we get for any $\tau \in [0,1)$ + +$$ +\mathbb {E} [ \mathrm {T V} ^ {2} (P _ {m, n} ^ {*}, [ 0, \tau ], P _ {m, n} ^ {k, [ 0, \tau ]}) ] \lesssim \frac {\tau}{1 - \tau} \frac {R ^ {6}}{\varepsilon^ {2}} \left(\tanh \left(\frac {R ^ {2}}{\varepsilon}\right)\right) ^ {4 k}, +$$ + +where $P_{m,n}^{k,[0,\tau]}$ are restrictions of $P_{m,n}^{k}$ on the time-interval $[0,\tau]$ . + +These results immediately imply the convergence rate $\mathbb{E}[\mathrm{TV}^2 (P_{m,n}^{k,[0,\tau]},P_{\infty ,\infty}^{*,[0,\tau ]})] = \mathcal{O}(1 / m + 1 / n + r^{4k})$ for some $r\in (0,1)$ regarding the sample size $m$ and $n$ , and the number of Sinkhorn iterations $k$ . + +# 3.3 Drift Estimation using Neural Networks in the Schrödinger Bridge Problem + +The drift estimator (13) of the Sinkhorn bridge between $\mu_{m}$ and $\nu_{n}$ is defined as an empirical average regarding $\nu_{n}$ . As a result, simulating the SDE (12) using a time discretization scheme (e.g., Euler-Maruyama approximation) incurs an $O(n)$ -computational cost at every discretization step. To improve computational efficiency, we consider a neural network-based drift approximation. + +Recall that $W_{|x_0,x_1}^{\varepsilon}$ is the Brownian bridge connecting $x_0$ and $x_1$ , so its marginal distribution at time $t \in [0,1]$ is a Gaussian distribution $W_{t|x_0,x_1}^{\varepsilon} = \mathcal{N}\big((1 - t)x_0 + tx_1,\varepsilon t(1 - t)I_d\big)$ . For dual potentials $f_{m,n}^{(k)} \in L^{1}(\mu_{m})$ and $g_{m,n}^{(k)} \in L^{1}(\nu_{n})$ obtained by the Sinkhorn algorithm, we define the joint distribution $\pi_{m,n}^{(k)}$ analogously to Eq. (6) by replacing the optimal potentials and $\mu, \nu$ with $f_{m,n}^{(k)}, g_{m,n}^{(k)}$ , and $\mu_{m}, \nu_{n}$ . Using $\pi_{m,n}^{(k)}$ , we define the mixture of Brownian bridge (a.k.a. reciprocal process): $\Pi^{(k)} = \int W_{|x_0,x_1}^{\varepsilon} \mathrm{d}\pi_{m,n}^{(k)}(x_0,x_1) \in \mathcal{P}(\Omega)$ . The drift function $b_{m,n}^{(k)}$ in Eq. (13) is known to have the following expression (Markovian projection (Shi et al., 2023)) (see Appendix B.2): + +$$ +b _ {m, n} ^ {(k)} (x _ {t}, t) = \mathbb {E} _ {X \sim \Pi^ {(k)}} \left[ \frac {X _ {1} - x _ {t}}{1 - t} \Bigg | X _ {t} = x _ {t} \right]. +$$ + +Therefore, we see that the function $b_{m,n}^{(k)}$ minimizes the following functional $\mathcal{L}$ : + +$$ +\mathcal {L} (b) = \int_ {0} ^ {1} \int_ {\mathbb {R} ^ {d} \times \mathbb {R} ^ {d}} \left\| b (x _ {t}, t) - \frac {x _ {1} - x _ {t}}{1 - t} \right\| _ {2} ^ {2} \mathrm {d} \Pi_ {t 1} ^ {(k)} (x _ {t}, x _ {1}) \mathrm {d} t, +$$ + +where $\Pi_{t1}^{(k)}$ is the marginal distribution of $\Pi^{(k)}$ on time points $t$ and 1. This suggests training a neural network $b_{\theta}$ with parameter $\theta$ to minimize the above functional. We note that samples from $\Pi_{t1}^{(k)}$ can be obtained by sampling $(x_0,x_1)\sim \pi_{m,n}^{(k)}$ , $t\sim U[0,\tau ]$ , and $x_{t}\sim W_{t|x_{0},x_{1}}^{\varepsilon}$ and discarding $x_0$ . Therefore, with this sampling scheme, SGD can be efficiently applied to minimize $\mathcal{L}(b_{\theta})$ . The detail of the procedure is described in Algorithm 1. This procedure yields a neural network approximation of the drift function defined by the Sinkhorn bridge. + +# 3.4 Other Estimators of Schrödinger Bridge + +Our theoretical results on Sinkhorn bridge (Pooladian and Niles-Weed, 2024) can be directly applied to the representative Schrödinger Bridge solvers such as $\left[\mathrm{SF}\right]^{2} \mathrm{M}$ (Tong et al., 2023), lightSB(-M) (Korotin et al., 2023; Gushchin et al., 2024), DSBM-IMF (Shi et al., 2023; Peluchetti, 2023), and BM2 (Peluchetti, 2024). First, we show that the optimal estimator produced by each of these methods + +# Algorithm 1 Drift Approximation via Neural Network + +input Joint distribution $\pi_{m,n}$ defined by potentials $f\in L^{1}(\mu_{m}),g\in L^{1}(\nu_{n})$ ; neural network $b_{\theta}$ output Trained neural network $b_{\theta}$ that approximates the drift function + +# repeat + +Sample batch of pairs $\{x_0^i,x_1^i\}_{i = 1}^N\sim \pi_{m,n}$ + +Sample batch $t_i\sim U[0,\tau ]$ + +For each triplet $(x_0^i, x_1^i, t_i)$ , sample $x_t^i \sim W_{t_i | x_0^i, x_1^i}^\varepsilon$ (Brownian bridge) + +Compute the loss: + +$$ +L (\theta) \gets \frac {1}{N} \sum_ {i = 1} ^ {N} \left\| b _ {\theta} (x _ {t} ^ {i}, t _ {i}) - \frac {x _ {1} ^ {i} - x _ {t} ^ {i}}{1 - t} \right\| _ {2} ^ {2} +$$ + +Update $\theta$ using $\nabla L(\theta)$ + +until converged; + +coincides with $P_{m,n}^{*}$ attained by Sinkhorn bridge with $k \to \infty$ (Proposition 1). Moreover, when DSBM-IMF and BM2 are initialized based on the reference process, the estimator after $k$ iterations matches $P_{m,n}^{k}$ obtained after $k$ iterations of the Sinkhorn bridge (Proposition 2). Furthermore, it follows from the proof of Proposition 1 that Algorithm 1, which learns the Sinkhorn bridge drift via a neural network, is a special case of the algorithm proposed in [SF] $^{2}$ M. The proofs of these results are postponed to the Appendix. Consequently, our statistical and algorithmic convergence analysis applies to these methods, endowing the theoretical guarantees. + +Proposition 1. The optimal estimators produced by $[SF]^2 M$ , lightSB(-M), DSBM-IMF and BM2 coincide with $P_{m,n}^*$ attained by Sinkhorn bridge with $k \to \infty$ . + +Proposition 2. With initialization based on the reference process, both DSBM-IMF and BM2 produce the same estimator as $P_{m,n}^{k}$ of the Sinkhorn bridge for all iterations $k$ . + +# 4 Experiments + +In this section, we verify our theoretical findings through numerical experiments. In Section 4.1, we verify the dependence of the statistical error on the sample size presented in Theorem 1, and the algorithmic convergence rate presented in Theorem 2. Next, in Section 4.2, we evaluate the neural network approximation of drift coefficient as illustrated in Section 3.3. All experimental implementations are conducted with PyTorch 2.6.0 (Paszke et al., 2019). + +# 4.1 Experimental Verification of Theorems 1 and 2 + +We use 3-dimensional normal distributions $\mu = \mathcal{N}(0, I_3)$ and $\nu = \mathcal{N}(0, B)$ as source and target distributions, where $B \in \mathbb{R}^{3 \times 3}$ is a random positive definite matrix. Under this setting, Bunne et al. (2023, Eq. (25)-(29)) provided explicit expressions for the drift coefficient $b_{\infty, \infty}^*$ and the marginal distribution $P_{\infty, \infty}^{*, t}$ . We use these expressions to evaluate the errors of the drift $b_{m, n}^*$ and the corresponding Schrödinger Bridge estimators $P_{m, n}^*$ in the finite-sample settings. These estimators are approximated by running the Sinkhorn bridge with a sufficiently large number of iterations ( $k \to \infty$ ). + +Statistical convergence. To verify that the estimator $b_{m,n}^{*}$ converges to the true drift $b_{\infty, \infty}^{*}$ when the sample sizes $m$ and $n$ increase, we draw $m$ samples from $\mu$ and $n$ samples from $\nu$ , respectively, and compute the estimator $b_{m,n}^{*}(\cdot, t)$ for an arbitrary time $t \in [0,1)$ . Subsequently, we evaluate the mean squared error (MSE) against the true drift $b_{\infty, \infty}^{*}(\cdot, t)$ . + +$$ +\operatorname{MSE}_{\text{sample}}(m,n,t) = \mathop{\mathbb{E}}_{\substack{X_{1},\ldots X_{m}\sim \mu \\ Y_{1},\ldots ,Y_{n}\sim \nu}}\| b^{*}_{m,n}(z,t) - b^{*}_{\infty ,\infty}(z,t)\|_{L^{2}(P^{*,t}_{\infty ,\infty}(z))}^{2}. +$$ + +To compute this, the norm $\| \cdot \|_{L^2 (P_{\infty ,\infty}^{*,t})}$ is approximated using Monte Carlo with 10,000 samples drawn from $P_{\infty ,\infty}^{*,t}$ . The expectation over the samples is computed by averaging over 10 independent sampling trials. With a fixed parameter $\varepsilon = 0.1$ , we generated heatmaps for several $t\in [0,1)$ while varying the sample sizes $m,n$ used in the estimator definition (Figure 1). + +![](images/9334cf7e34ae74bb7400f40f82f27fee8bbed683ae1cc0ffd7d4f34f01b33e5f.jpg) +Figure 1: Heatmaps illustrating $\mathrm{MSE}_{\mathrm{sample}}(m,n,t)$ as a function of sample sizes $m$ and $n$ for various time points $t$ . The error decreases roughly proportionally to $(m^{-1} + n^{-1})$ . + +![](images/fb2728f8452e04932a9b5b37725dd54598c7cecfbeb2fc6b2243f5dd6d97f48a.jpg) + +![](images/c621d7f1aabf63e42b61a4f33c9f8da82f608edc530a31dd031461f6dd0ce1f9.jpg) + +![](images/4300782f50eddc677fbb6462d045cd8a556ccd799fee458682a7862f99b1c101.jpg) + +![](images/4dd55a5a457cf0a3dda23389526bf7b4d0d4f6d356f4e0aab93a56cc7f0c610c.jpg) + +![](images/1dc5eb266cbd5b9d24a75ab6eb2a2d7e5cf180d284736ac09352a9f4e4a2fec3.jpg) + +As evident from the figure, the mean squared error deteriorates as $t \to 1$ , but the overall convergence rate remains unchanged. For all $t$ shown in the figure, the convergence rate is observed to be approximately proportional to $(m^{-1} + n^{-1})$ . This aligns with the result predicted by Theorem 1. + +Algorithmic convergence. To verify that the estimator $P_{m,n}^{(k),[0,\tau]}$ exponentially approaches $P_{m,n}^{*,[0,\tau]}$ as the iteration count $k$ increases, we obtain $m$ and $n$ independent samples from distributions $\mu$ and $\nu$ , respectively. We then consider the integral of the difference between estimators $b_{m,n}^{(k)}(\cdot,t)$ and $b_{m,n}^{*}(\cdot,t)$ over the interval $[0,\tau]$ . Specifically, we evaluate the mean squared error integrated over $t \in [0,\tau]$ : + +$$ +\int_{0}^{\tau}\mathrm{MSE}_{\text{sinkhorn}}(k,t)\mathrm{d}t = \int_{0}^{\tau}\mathbb{E}_{\substack{X_{1},\ldots ,X_{m}\sim \mu \\ Y_{1},\ldots ,Y_{n}\sim \nu}}\bigg\|b_{m,n}^{(k)}(z,t) - b_{m,n}^{*}(z,t)\bigg\|_{L^{2}(P_{m,n}^{*,t})}^{2}\mathrm{d}t. +$$ + +Time integrals over the interval $[0, \tau]$ are approximated via Monte Carlo integration by uniformly sampling 1,000 time-points. The norm $\| \cdot \|_{L^2 (P_{m,n}^{*,t})}$ is also estimated via Monte Carlo integration, using 1,000 samples drawn from $P_{m,n}^{*,t}$ . We set $m = n = 1,000$ and compute the expectation by averaging over 10 independent samplings. With the regularization parameter fixed at $\varepsilon = 0.005$ , we generate graphs of the integral values over $[0, \tau]$ for varying $k$ and multiple $\tau$ values (Figure 2). For all $\tau$ values shown in the figure, the convergence rate exhibits exponential decay with respect to $k$ . This observation corroborates the theoretical prediction in Theorem 2. + +Moreover, under the experimental setup of Section 4.2, we illustrate in Figure 3 the evolution of the drift when the number of Sinkhorn iterations is set to 1, 5, and 10. In the figure, it is observed that as the number of iterations $k$ + +increases, the drift $b_{m,n}^{(k)}$ rapidly converges to the optimal drift $b_{m,n}^{*}$ . + +![](images/9ccdd6da11d5512c7a1cb152c13a2c218b8b7ae60b178680fd37d12d10ad9cba.jpg) +Figure 2: Integrated mean squared error as a function of Sinkhorn iterations $k$ for multiple integration intervals $[0, \tau]$ . + +# 4.2 Experimental Verification of Neural Network-Based Drift Estimation + +Next, we evaluate the effectiveness of the drift approximation by a neural network using Algorithm 1. In this experiment, we set $\varepsilon = 0.1$ , defined $\mu$ as the eight-Gaussians distribution and $\nu$ as the moons distribution, and drew 1,000 independent samples from each. Using these samples, the Sinkhorn algorithm is employed to approximate the optimal EOT coupling $\pi_{m,n}^{*}$ between $\mu_{m}$ and $\nu_{n}$ , and, via Algorithm 1, a neural network approximation $b_{\theta}$ of the drift $b_{m,n}^{*}$ is obtained. We employ an 4-layer neural network with 512-512-512 hidden neurons for $b_{\theta}$ , and train it using the AdamW optimizer with a learning rate of $1\times 10^{-3}$ , weight decay of $1\times 10^{-5}$ , and a mini-batch size of 4,096. Finally, starting from each sample of $\mu_{m}$ , we simulate trajectories using either $b_{m,n}^{(k)}$ or the neural network drift $b_{\theta}$ and present the results in Figure 3. Trajectory simulations are performed using the Euler-Maruyama approximation with 1,000 discretization steps. + +![](images/d081b56199408f27ac79cfbbfcde2010a31af9a39596897f36f3f3f9214382c5.jpg) +Figure 3: From left to right, the simulation results of the Schrödinger bridge using the estimated drifts $b_{m,n}^{(1)}, b_{m,n}^{(5)}$ , and $b_{m,n}^{(10)}$ obtained by terminating the Sinkhorn iteration after 1, 5, and 10 iterations, respectively, the optimal drift $b_{m,n}^{*}$ , and the neural network-approximated drift $b_{\theta}$ . + +# 5 Conclusion and Discussion + +In this study, we provide a comprehensive analysis of statistical guarantees for the Schrödinger Bridge problem. Specifically, we establish theoretical guarantees in two key settings: (i) the two-sample estimation task and (ii) intermediate estimators during the learning process. + +Our main contributions are as follows. First, we establish a statistical convergence analysis in the two-sample estimation setting for the Schrödinger bridge estimator with $k \to \infty$ , which demonstrates a statistical convergence rate of $O\left(\frac{1}{n} + \frac{1}{m}\right)$ . Second, we establish new convergence guarantees for the dual potentials obtained during intermediate iterations of the Sinkhorn algorithm, proving exponential convergence in the finite-sample setting. These results allow for a clear estimation of the number of samples $m, n$ and Sinkhorn iterations $k$ required to achieve a desired precision. Experimental results align with our theoretical analysis, confirming that the error decreases at a rate of approximately $(m^{-1} + n^{-1})$ with respect to the sample size, and that the error decreases exponentially with respect to the number of Sinkhorn iterations $k$ . These findings strengthen the statistical guarantees for existing methods proposed in works such as Korotin et al. (2023); Peluchetti (2023); Gushchin et al. (2024); Shi et al. (2023); Peluchetti (2024) through the connections among these methods. The theoretical framework of this study deepens the understanding of generalization error in the Schrödinger Bridge problem and offers new insights into the convergence properties and stability of the Sinkhorn algorithm during intermediate iterations. + +Future research directions include extending the analysis to more complex distributional settings and more general reference processes, and removing linear dependence on the ambient dimension $d$ . Another interesting research direction is to explore alternative optimization methods for EOT problems and analyze their performance using our theoretical framework beyond the Sinkhorn algorithm. + +# Acknowledgment + +This research is supported by the National Research Foundation, Singapore, Infocomm Media Development Authority under its Trust Tech Funding Initiative, and the Ministry of Digital Development and Information under the AI Visiting Professorship Programme (award number AIVP-2024-004). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore, Infocomm Media Development Authority, and the Ministry of Digital Development and Information. + +# References + +Bernton, E., Heng, J., Doucet, A., and Jacob, P. E. (2019). Schrödinger Bridge Samplers. +Bunne, C., Hsieh, Y.-P., Cuturi, M., and Krause, A. (2023). 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Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: Our research does not rely on LLMs for any important, original, or non-standard components of the core methods. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Appendix + +# A Omitted Proofs + +# A.1 Proof of Theorem 1 + +We here provide the complete definitions of path measures $P_{m,n}^{k}$ , $P_{m,n}^{*} \in \mathcal{P}(\Omega)$ . Using the Sinkhorn iterations $(f_{m,n}^{(k)}, g_{m,n}^{(k)})$ with respect to $\mu_{m}$ and $\nu_{n}$ , we define the drift function $b_{m,n}^{(k)}$ as follows: + +$$ +\begin{array}{l} b _ {m, n} ^ {(k)} (z, t) = \frac {1}{1 - t} \left( \right.- z + \frac {\sum_ {j = 1} ^ {n} Y _ {j} \exp_ {\varepsilon} \left(\left(g _ {m , n} ^ {(k)} \left(Y _ {j}\right) - \frac {1}{2 (1 - t)} \| Y _ {j} - z \| _ {2} ^ {2}\right)\right)}{\sum_ {j = 1} ^ {n} \exp_ {\varepsilon} \left(\left(g _ {m , n} ^ {(k)} \left(Y _ {j}\right) - \frac {1}{2 (1 - t)} \| Y _ {j} - z \| _ {2} ^ {2}\right)}\right) \\ = \frac {1}{1 - t} \left(- z + \frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {(k) , t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {(k) , t} (Y _ {j} , z)}\right), \\ \end{array} +$$ + +where + +$$ +\gamma_ {m, n} ^ {(k), t} (Y _ {j}, z) = \exp_ {\varepsilon} \left(g _ {m, n} ^ {(k)} (Y _ {j}) - \frac {1}{2 (1 - t)} \| Y _ {j} - z \| _ {2} ^ {2}\right). +$$ + +Then, $P_{m,n}^{(k)}$ is defined as the path measure of the following SDE: + +$$ +\mathrm {d} X _ {t} = b _ {m, n} ^ {(k)} (X _ {t}) \mathrm {d} t + \sqrt {\varepsilon} \mathrm {d} B _ {t}, \quad X _ {0} \sim \mu . +$$ + +$P_{m,n}^{*}\in \mathcal{P}(\Omega)$ is also defined in the similar way by replacing the Sinkhorn iterations to the corresponding optimal Schrodinger potentials. That is, using the optimal Schrodinger potentials $(f_{m,n}^{*},g_{m,n}^{*})$ with respect to $\mu_{m}$ and $\nu_{n}$ , we define the drift function $b_{m,n}^{*}$ as follows: + +$$ +\begin{array}{l} b _ {m, n} ^ {*} (z) = \frac {1}{1 - t} \left(- z + \frac {\sum_ {j = 1} ^ {n} Y _ {j} \exp_ {\varepsilon} \left(g _ {m , n} ^ {*} (Y _ {j}) - \frac {1}{2 (1 - t)} \| Y _ {j} - z \| _ {2} ^ {2}\right)}{\sum_ {j = 1} ^ {n} \exp_ {\varepsilon} \left(g _ {m , n} ^ {*} (Y _ {j}) - \frac {1}{2 (1 - t)} \| Y _ {j} - z \| _ {2} ^ {2}\right)}\right) \\ = \frac {1}{1 - t} \left(- z + \frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} \left(Y _ {j} , z\right) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} \left(Y _ {j} , z\right)}\right), \tag {15} \\ \end{array} +$$ + +where + +$$ +\gamma_ {m, n} ^ {*}, ^ {t} (Y _ {j}, z) = \exp_ {\varepsilon} \left(g _ {m, n} ^ {*} (Y _ {j}) - \frac {1}{2 (1 - t)} \| Y _ {j} - z \| _ {2} ^ {2}\right). +$$ + +Then, $P_{m,n}^{*}$ is defined as the path measure of the following SDE: + +$$ +\mathrm {d} X _ {t} = b _ {m, n} ^ {*} \left(X _ {t}\right) \mathrm {d} t + \sqrt {\varepsilon} \mathrm {d} B _ {t}, \quad X _ {0} \sim \mu . \tag {16} +$$ + +Theorem 1 (Statistical convergence rate). Suppose Assumptions 1, 2 and 3 hold. Let $P_{\infty,\infty}^{*}$ be the path measure of the true Schrödinger bridge for marginals $\mu, \nu$ . Then, it follows that for any $\tau \in [0,1)$ , + +$$ +\begin{array}{l} \mathbb {E} [ \mathrm {T V} ^ {2} (P _ {\infty , \infty} ^ {*}, [ 0, \tau ], P _ {m, n} ^ {*}, [ 0, \tau ]) ] \\ \lesssim \frac {R ^ {2}}{n (1 - \tau) ^ {d _ {\nu} + 2} \varepsilon^ {d _ {\nu}}} + \frac {\tau}{1 - \tau} \left(\varepsilon^ {2} + \frac {d R ^ {8}}{\varepsilon^ {2}}\right) \cdot \left(\frac {R}{\varepsilon}\right) ^ {9 d _ {\nu} + 4} \cdot \left(\frac {1}{m} + \frac {1}{n}\right). \tag {14} \\ \end{array} +$$ + +where $P_{\infty, \infty}^{*,[0, \tau]}$ and $P_{m,n}^{*,[0, \tau]}$ are restrictions of $P_{\infty, \infty}^{*}$ and $P_{m,n}^{*}$ on the time-interval $[0, \tau]$ . + +Proof of Theorem 1. The proof of Theorem 1 is based on the main idea presented in Pooladian and Niles-Weed (2024); Stromme (2023b): we introduce the following entropic plan. + +$$ +\mathrm {d} \bar {\pi} _ {\infty , n} (x, y) := \exp_ {\varepsilon} \left(\bar {f} _ {\infty , n} (x) + g _ {\infty , \infty} ^ {*} (y) - \frac {1}{2} \| x - y \| _ {2} ^ {2}\right) \mathrm {d} \mu (x) \mathrm {d} \nu_ {n} (y). \tag {17} +$$ + +Here, the function $\bar{f}_{\infty ,n}:\mathbb{R}^d\to \mathbb{R}$ is defined as follows: + +$$ +\bar {f} _ {\infty , n} (x) = - \varepsilon \log \left(\frac {1}{n} \sum_ {j = 1} ^ {n} \exp_ {\varepsilon} \left(g _ {\infty , \infty} ^ {*} (Y _ {j}) - \frac {1}{2} \| x - Y _ {j} \| _ {2} ^ {2}\right)\right). +$$ + +Denoting by $\bar{P}_{\infty,n}$ the measure obtained by substituting the coupling (17) into (8), Proposition 3.1 in Pooladian and Niles-Weed (2024) implies that the drift $\bar{b}_{\infty,n}$ with law $\bar{P}_{\infty,n}$ is expressed as follows: + +$$ +\bar {b} _ {\infty , n} (z) = \frac {1}{1 - t} \left(- z + \frac {\sum_ {j = 1} ^ {n} \gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , z)}\right). +$$ + +By incorporating the path measure $\bar{P}_{\infty,n}$ into the bound via the triangle inequality and subsequently applying Pinsker's inequality, we obtain: + +$$ +\begin{array}{l} \mathbb {E} \left[ \mathrm {T V} ^ {2} \left(P _ {\infty , \infty} ^ {*, [ 0, \tau ]}, P _ {m, n} ^ {*, [ 0, \tau ]}\right) \right] \lesssim \mathbb {E} \left[ \mathrm {T V} ^ {2} \left(P _ {\infty , \infty} ^ {*, [ 0, \tau ]}, \bar {P} _ {\infty , n} ^ {[ 0, \tau ]}\right) \right] + \mathbb {E} \left[ \mathrm {T V} ^ {2} \left(\bar {P} _ {\infty , n} ^ {[ 0, \tau ]}, P _ {m, n} ^ {*, [ 0, \tau ]}\right) \right] \\ \lesssim \mathbb {E} \left[ H \left(P _ {\infty , \infty} ^ {*}, [ 0, \tau ] \mid \bar {P} _ {\infty , n} ^ {[ 0, \tau ]}\right) \right] + \mathbb {E} \left[ H \left(\bar {P} _ {\infty , n} ^ {[ 0, \tau ]} \mid P _ {m, n} ^ {*}, [ 0, \tau ]\right) \right]. \\ \end{array} +$$ + +We analyze these two terms separately. For the first term, we apply Proposition 4.3 of Pooladian and Niles-Weed (2024) + +$$ +\mathbb {E} [ H (P _ {\infty , \infty} ^ {*; [ 0, \tau ]} | \bar {P} _ {\infty , n} ^ {[ 0, \tau ]}) ] \lesssim \frac {R ^ {2}}{n (1 - \tau) ^ {d _ {\nu} + 2} \varepsilon^ {d _ {\nu}}}. +$$ + +For the second term, we apply Girsanov's theorem to derive the difference between the drifts. + +$$ +\begin{array}{l} \mathbb {E} [ H (\bar {P} _ {\infty , n} ^ {[ 0, \tau ]} | P _ {m, n} ^ {*; [ 0, \tau ]}) ] \\ \leq \int_ {0} ^ {\tau} \mathbb {E} \left\| \bar {b} _ {\infty , n} - b _ {m, n} ^ {*} \right\| _ {L ^ {2} (\bar {P} _ {\infty , n} ^ {t})} ^ {2} \mathrm {d} t \\ = \int_ {0} ^ {\tau} \frac {1}{(1 - t) ^ {2}} \mathbb {E} \left\| \frac {\sum_ {j = 1} ^ {n} \gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , z)} - \frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z)} \right\| _ {L ^ {2} (\bar {P} _ {\infty , n} ^ {t})} ^ {2} \mathrm {d} t \\ \lesssim \int_ {0} ^ {\tau} \frac {1}{(1 - t) ^ {2}} \left(\varepsilon^ {2} + \frac {d R ^ {8}}{\varepsilon^ {2}}\right) \cdot \left(\frac {R}{\varepsilon}\right) ^ {9 d _ {\nu} + 4} \cdot \left(\frac {1}{m} + \frac {1}{n}\right) \mathrm {d} t \\ = \frac {\tau}{1 - \tau} \left(\varepsilon^ {2} + \frac {d R ^ {8}}{\varepsilon^ {2}}\right) \cdot \left(\frac {R}{\varepsilon}\right) ^ {9 d _ {\nu} + 4} \cdot \left(\frac {1}{m} + \frac {1}{n}\right), \\ \end{array} +$$ + +where for the second inequality, we used Lemma 1. + +Lemma 1. Under assumptions of Theorem 1, we have + +$$ +\begin{array}{l} \mathbb {E} \left\| \frac {\sum_ {j = 1} ^ {n} \gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , \cdot) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , \cdot)} - \frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , \cdot) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , \cdot)} \right\| _ {L ^ {2} (p _ {\infty , \infty} ^ {t})} ^ {2} \\ \lesssim \left(\varepsilon^ {2} + \frac {d R ^ {8}}{\varepsilon^ {2}}\right) \cdot \left(\frac {R}{\varepsilon}\right) ^ {9 d _ {\nu} + 4} \cdot \left(\frac {1}{m} + \frac {1}{n}\right). \\ \end{array} +$$ + +Proof of Lemma 1. We perform the expansion while keeping $Y_{j}$ and $z$ fixed + +$$ +\begin{array}{l} \left\| \frac {\sum_ {j = 1} ^ {n} \gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , z)} - \frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z)} \right\| _ {2} \\ = \left\| \sum_ {i = 1} ^ {n} \left(\frac {\gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , z)}{\sum_ {j = 1} ^ {n} \gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , z)} - \frac {\gamma_ {m , n} ^ {*, t} (Y _ {j} , z)}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z)}\right) Y _ {j} \right\| _ {2} \\ \leq R \sum_ {i = 1} ^ {n} \left| \frac {\gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , z)}{\sum_ {j = 1} ^ {n} \gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , z)} - \frac {\gamma_ {m , n} ^ {*, t} (Y _ {j} , z)}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z)} \right|. \\ \end{array} +$$ + +From the case $(p,q) = (1,\infty)$ of Lemma 12, we have + +$$ +\begin{array}{l} \left\| \frac {\sum_ {j = 1} ^ {n} \gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , z)} - \frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z)} \right\| _ {2} \\ \leq \frac {R}{\varepsilon} \max _ {j \in [ n ]} \left| \left(g _ {\infty , \infty} ^ {*} (Y _ {j}) - \frac {1}{2 (1 - t)} \| Y _ {j} - z \| _ {2} ^ {2}\right) - \left(g _ {m, n} ^ {*} (Y _ {j}) - \frac {1}{2 (1 - t)} \| Y _ {j} - z \| _ {2} ^ {2}\right) \right| \\ \leq \frac {R}{\varepsilon} \left\| g _ {\infty , \infty} ^ {*} - g _ {m, n} ^ {*} \right\| _ {L ^ {\infty} (\nu)}. \\ \end{array} +$$ + +By applying Lemma 11, we obtain + +$$ +\begin{array}{l} \mathbb {E} \left\| \frac {\sum_ {j = 1} ^ {n} \gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {\infty , \infty} ^ {*, t} (Y _ {j} , z)} - \frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z)} \right\| _ {L ^ {2} (\bar {P} _ {\infty , n} ^ {t})} ^ {2} \\ \lesssim \frac {R ^ {2}}{\varepsilon^ {2}} \mathbb {E} \left\| g _ {m, n} ^ {*} - g _ {\infty , \infty} ^ {*} \right\| _ {L ^ {\infty} (\nu)} ^ {2} \\ \lesssim \left(\varepsilon^ {2} + \frac {d R ^ {8}}{\varepsilon^ {2}}\right) \cdot \left(\frac {R}{\varepsilon}\right) ^ {9 d _ {\nu} + 4} \cdot \left(\frac {1}{m} + \frac {1}{n}\right). \\ \end{array} +$$ + +![](images/6f5a73705b6fd7b336c418a9e26fddb0fc8695e0cb359b3cd25f9b98f7852235.jpg) + +# A.2 Proof of Theorem 2 + +Theorem 2 (Algorithmic convergence rate). Under Assumptions 1 and 2, we get for any $\tau \in [0,1)$ + +$$ +\mathbb {E} [ \mathrm {T V} ^ {2} (P _ {m, n} ^ {\ast , [ 0, \tau ]}, P _ {m, n} ^ {k, [ 0, \tau ]}) ] \lesssim \frac {\tau}{1 - \tau} \frac {R ^ {6}}{\varepsilon^ {2}} \left(\tanh \left(\frac {R ^ {2}}{\varepsilon}\right)\right) ^ {4 k}, +$$ + +where $P_{m,n}^{k,[0,\tau]}$ are restrictions of $P_{m,n}^{k}$ on the time-interval $[0,\tau]$ . + +Proof of Theorem 2. We start by applying Girsanov's theorem to obtain a difference in the drifts + +$$ +\begin{array}{l} \mathbb {E} \left[ H \left(P _ {m, n} ^ {*}, [ 0, \tau ] \mid P _ {m, n} ^ {k}, [ 0, \tau ]\right) \right] \\ \leq \int_ {0} ^ {\tau} \mathbb {E} \left\| b _ {m, n} ^ {*} - b _ {m, n} ^ {k} \right\| _ {L ^ {2} \left(P _ {m, n} ^ {*; t}\right)} ^ {2} \mathrm {d} t \\ = \int_ {0} ^ {\tau} \mathbb {E} \left\| \frac {1}{1 - t} \left(\frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , \cdot) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , \cdot)} - \frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {k , t} (Y _ {j} , \cdot) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {k , t} (Y _ {j} , \cdot)}\right) \right\| _ {L ^ {2} \left(P _ {m, n} ^ {*}, t\right)} ^ {2} \mathrm {d} t \\ \lesssim \int_ {0} ^ {\tau} \frac {1}{(1 - t) ^ {2}} \frac {R ^ {6}}{\varepsilon^ {2}} \left(\tanh \left(\frac {R ^ {2}}{\varepsilon}\right)\right) ^ {4 k} d t \\ = \frac {\tau}{1 - \tau} \frac {R ^ {6}}{\varepsilon^ {2}} \left(\tanh \left(\frac {R ^ {2}}{\varepsilon}\right)\right) ^ {4 k}, \\ \end{array} +$$ + +where we applied Lemma 2. + +Lemma 2. Under assumptions of Theorem 1, we have + +$$ +\mathbb {E} \left\| \frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z)} - \frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {k , t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {k , t} (Y _ {j} , z)} \right\| _ {L ^ {2} (P _ {m, n} ^ {*}, n)} ^ {2} \lesssim \frac {R ^ {6}}{\varepsilon^ {2}} \left(\tanh \left(\frac {R ^ {2}}{\varepsilon}\right)\right) ^ {4 k}. +$$ + +Proof of Lemma 2. We perform the expansion while keeping $Y_{j}$ and $z$ fixed + +$$ +\left\| \frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z)} - \frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {k , t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {k , t} (Y _ {j} , z)} \right\| _ {2} +$$ + +$$ +\begin{array}{l} = \left\| \sum_ {j = 1} ^ {n} \left(\frac {\gamma_ {m , n} ^ {*, t} (Y _ {j} , z)}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z)} - \frac {\gamma_ {m , n} ^ {k , t} (Y _ {j} , z)}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {k , t} (Y _ {j} , z)}\right) Y _ {j} \right\| _ {2} \\ \leq \sum_ {j = 1} ^ {n} \left| \frac {\gamma_ {m , n} ^ {*, t} (Y _ {j} , z)}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z)} - \frac {\gamma_ {m , n} ^ {k , t} (Y _ {j} , z)}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {k , t} (Y _ {j} , z)} \right| \| Y _ {j} \| _ {2} \\ \leq R \sum_ {j = 1} ^ {n} \left| \frac {\gamma_ {m , n} ^ {*, t} (Y _ {j} , z)}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z)} - \frac {\gamma_ {m , n} ^ {k , t} (Y _ {j} , z)}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {k , t} (Y _ {j} , z)} \right|. \\ \end{array} +$$ + +We define $\mathbf{w}^*,\mathbf{w}^k,\mathbf{v}^*,\mathbf{v}^k\in \mathbb{R}^n$ as follows: + +$$ +\mathbf {w} _ {j} ^ {*} := \frac {\gamma_ {m , n} ^ {*, t} (Y _ {j} , z)}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z)}, \quad \mathbf {w} _ {j} ^ {k} := \frac {\gamma_ {m , n} ^ {k , t} (Y _ {j} , z)}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {k , t} (Y _ {j} , z)}, +$$ + +$$ +\mathbf {v} _ {j} ^ {*} := \exp \left(\frac {g _ {m , n} ^ {*} (Y _ {j})}{\varepsilon}\right), \quad \mathbf {v} _ {j} ^ {k} := \exp \left(\frac {g _ {m , n} ^ {(k)} (Y _ {j})}{\varepsilon}\right). +$$ + +where both $\mathbf{w}^*$ and $\mathbf{w}^k$ satisfy the conditions of a probability vector, i.e., $\mathbf{w}_j^*,\mathbf{w}_j^k >0,\sum_{j = 1}^n\mathbf{w}_j^* = \sum_{j = 1}^n\mathbf{w}_j^k = 1$ . + +$$ +\begin{array}{l} \left\| \frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z)} - \frac {\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {k , t} (Y _ {j} , z) Y _ {j}}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {k , t} (Y _ {j} , z)} \right\| _ {2} ^ {2} \\ \leq \left(R \sum_ {j = 1} ^ {n} \left| \frac {\gamma_ {m , n} ^ {*, t} (Y _ {j} , z)}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {*, t} (Y _ {j} , z)} - \frac {\gamma_ {m , n} ^ {k , t} (Y _ {j} , z)}{\sum_ {j = 1} ^ {n} \gamma_ {m , n} ^ {k , t} (Y _ {j} , z)} \right|\right) ^ {2} \\ = R ^ {2} \left\| \mathbf {w} ^ {*} - \mathbf {w} ^ {k} \right\| _ {1} ^ {2} \\ \lesssim R ^ {2} d _ {\text {H i l b}} \left(\mathbf {w} ^ {*}, \mathbf {w} ^ {k}\right) ^ {2} \quad (\text {u s i n g L e m m a 1 3}) \\ = R ^ {2} d _ {\text {H i l b}} \left(\mathbf {v} ^ {*}, \mathbf {v} ^ {k}\right) ^ {2} \quad (\text {u s i n g L e m m a 1 4}) \\ \leq R ^ {2} \lambda (K) ^ {4 k} d _ {\operatorname {H i l b}} (\mathbf {v} ^ {*}, \mathbf {v} ^ {0}) ^ {2} \\ = R ^ {2} \lambda (K) ^ {4 k} d _ {\mathrm {H i l b}} (\mathbf {v} ^ {*}, \mathbb {1} _ {m}) ^ {2} \\ = R ^ {2} \lambda (K) ^ {4 k} \left(\log \left(\left(\max _ {j} \exp_ {\varepsilon} \left(g _ {m, n} ^ {*} (Y _ {j})\right)\right) \left(\max _ {j} \exp_ {\varepsilon} \left(- g _ {m, n} ^ {*} (Y _ {j})\right)\right)\right)\right) ^ {2} \\ \leq R ^ {2} \lambda (K) ^ {4 k} \left(\log \left(\exp_ {\varepsilon} \left(4 R ^ {2}\right) \exp_ {\varepsilon} \left(4 R ^ {2}\right)\right)\right) ^ {2} \quad (\text {u s i n g L e m m a 3}) \\ \lesssim \frac {R ^ {6}}{\varepsilon^ {2}} \lambda (K) ^ {4 k}. \\ \end{array} +$$ + +Since + +$$ +\gamma (K) = \max _ {i j k l} \frac {K _ {i k} K _ {j l}}{K _ {j k} K _ {i l}} = \max _ {i j k l} \exp \left(\frac {1}{\varepsilon} (X _ {i} - X _ {j}) ^ {T} (Y _ {k} - Y _ {l})\right) \leq \exp_ {\varepsilon} \left(4 R ^ {2}\right), +$$ + +the quantity $\lambda (K)$ can be upper bounded as follows: + +$$ +\begin{array}{l} \lambda (K) = \frac {\sqrt {\gamma (K)} - 1}{\sqrt {\gamma (K)} + 1} \\ \leq \frac {\exp_ {\varepsilon} (2 R ^ {2}) - 1}{\exp_ {\varepsilon} (2 R ^ {2}) + 1} \quad \left(\because \frac {x - 1}{x + 1} (x > 0) \text {i s i n c r e a s i n g f u n c t i o n}\right) \\ = \tanh \left(\frac {R ^ {2}}{\varepsilon}\right). \\ \end{array} +$$ + +Since the upper bound of $\lambda(K)$ is independent of $Y_{j}$ and $z$ , the proof is complete. + +# A.3 Proof of Proposition 1 + +Proposition 1. The optimal estimators produced by $[SF]^2 M$ , lightSB(-M), DSBM-IMF and BM2 coincide with $P_{m,n}^*$ attained by Sinkhorn bridge with $k \to \infty$ . + +Proof of Proposition 1. It suffices to show that the drift estimated by each method coincides with $b_{m,n}^*$ . + +# Proof of optimal estimator for $[\mathrm{SF}]^{2}\mathbf{M}$ + +The SDE representation of the Brownian bridge $W_{[x_0,x_1]}^{\varepsilon}$ with endpoints $(x_0,x_1)$ is given by + +$$ +\mathrm {d} X _ {t} = \frac {x _ {1} - X _ {t}}{1 - t} \mathrm {d} t + \sqrt {\varepsilon} \mathrm {d} B _ {t}, \quad X _ {0} = x _ {0}. \tag {18} +$$ + +The marginal density of the Brownian bridge at time $t$ is $p_t(x_t|x_0,x_1) = \mathcal{N}(x_t|(1 - t)x_0 + tx_1,\varepsilon t(1 - t)I_d)$ . There exists an ordinary differential equation (ODE) that preserves the same marginal distributions $p_t(x_t|x_0,x_1)$ as the SDE (18). This ODE, referred to as the probability flow ODE, takes the form + +$$ +\begin{array}{l} \mathrm {d} X _ {t} = \left(\frac {x _ {1} - X _ {t}}{1 - t} - \frac {\varepsilon}{2} \nabla \log p _ {t} \left(X _ {t} \mid x _ {0}, x _ {1}\right)\right) \mathrm {d} t \\ = \left(\frac {1 - 2 t}{2 t (1 - t)} X _ {t} + \frac {1}{2 (1 - t)} x _ {1} - \frac {1}{2 t} x _ {0}\right) d t \\ =: u _ {t} ^ {\circ} \left(X _ {t} \mid x _ {0}, x _ {1}\right) d t. \\ \end{array} +$$ + +Consequently, the drift term of the Brownian bridge SDE can be decomposed into the sum of the probability flow ODE drift $u_{t}^{\circ}(x_{t}|x_{0},x_{1})$ and the score function $\nabla \log p_t(x_t|x_0,x_1)$ : + +$$ +\frac {x _ {1} - x _ {t}}{1 - t} = u _ {t} ^ {\circ} (x _ {t} | x _ {0}, x _ {1}) + \frac {\varepsilon}{2} \nabla \log p _ {t} (x _ {t} | x _ {0}, x _ {1}). +$$ + +In the $[\mathrm{SF}]^{2}\mathrm{M}$ framework, let $\pi_{m,n}^{*}$ denote the optimal coupling for the entropic optimal transport (EOT) between $\mu_{m}$ and $\nu_{n}$ , and consider the mixture distribution $\Pi^{*} = \pi_{m,n}^{*}W_{|x_{0},x_{1}}^{\varepsilon}$ . Under $\Pi^{*}$ , two neural networks $v_{\theta}$ and $s_{\varphi}$ are trained to minimize the following objectives: + +$$ +\begin{array}{l} v _ {\theta^ {*}} = \arg \min _ {v _ {\theta}} \mathbb {E} _ {X \sim \Pi^ {*}} \left[ \int_ {0} ^ {1} \| v _ {\theta} (X _ {t}, t) - u _ {t} ^ {\circ} (X _ {t} | X _ {0}, X _ {1}) \| _ {2} ^ {2} d t \right], \\ s _ {\varphi^ {*}} = \arg \min _ {s _ {\varphi}} \mathbb {E} _ {X \sim \Pi^ {*}} \left[ \int_ {0} ^ {1} \| s _ {\varphi} (X _ {t}, t) - \nabla \log p _ {t} (X _ {t} | X _ {0}, X _ {1}) \| _ {2} ^ {2} d t \right]. \\ \end{array} +$$ + +The estimator for the Schrödinger bridge drift in $[\mathrm{SF}]^{2}\mathrm{M}$ is then defined as + +$$ +v _ {\theta^ {*}} (x _ {t}, t) + \frac {\varepsilon}{2} s _ {\varphi^ {*}} (x _ {t}, t). +$$ + +When both models are sufficiently expressive, the optimal solutions admit the following conditional expectation representations: + +$$ +\begin{array}{l} v _ {\theta^ {*}} (x _ {t}, t) = \mathbb {E} _ {X \sim \Pi^ {*}} \left[ u _ {t} ^ {\circ} (x _ {t} | X _ {0}, X _ {1}) \mid X _ {t} = x _ {t} \right], \\ s _ {\varphi^ {*}} \left(x _ {t}, t\right) = \mathbb {E} _ {X \sim \Pi^ {*}} \left[ \nabla \log p _ {t} \left(x _ {t} \mid X _ {0}, X _ {1}\right) \mid X _ {t} = x _ {t} \right]. \\ \end{array} +$$ + +Therefore, the drift estimator in $[\mathrm{SF}]^2\mathrm{M}$ can be written as + +$$ +\begin{array}{l} v _ {\theta^ {*}} (x _ {t}, t) + \frac {\varepsilon}{2} s _ {\varphi^ {*}} (x _ {t}, t) \\ = \mathbb {E} _ {X \sim \Pi^ {*}} \left[ u _ {t} ^ {\circ} \left(x _ {t} | X _ {0}, X _ {1}\right) \mid X _ {t} = x _ {t} \right] + \frac {\varepsilon}{2} \mathbb {E} _ {X \sim \Pi^ {*}} \left[ \nabla \log p _ {t} \left(x _ {t} | X _ {0}, X _ {1}\right) \mid X _ {t} = x _ {t} \right] \\ = \mathbb {E} _ {X \sim \Pi^ {*}} \left[ u _ {t} ^ {\circ} (x _ {t} | X _ {0}, X _ {1}) + \frac {\varepsilon}{2} \nabla \log p _ {t} (x _ {t} | X _ {0}, X _ {1}) | X _ {t} = x _ {t} \right] \\ = \mathbb {E} _ {X \sim \Pi^ {*}} \left[ \frac {X _ {1} - x _ {t}}{1 - t} \Big | X _ {t} = x _ {t} \right] \\ \end{array} +$$ + +$$ += b _ {m, n} ^ {*} (x _ {t}, t). +$$ + +# Proof of optimal estimator for LightSB(-M) + +Let $S(\mu)$ denote the set of Schrödinger bridges whose source distribution is $\mu$ . In lightSB(-M), the drift estimator $g_v$ is defined with a function $v: \mathbb{R}^d \to \mathbb{R}$ as follows: + +$$ +g _ {v} \left(x _ {t}, t\right) = \varepsilon \nabla_ {x _ {t}} \log \left(\int \mathcal {N} \left(x ^ {\prime} \mid x _ {t}, \varepsilon (1 - t) I _ {d}\right) \exp \left(\frac {\| x ^ {\prime} \| _ {2} ^ {2}}{2}\right) v \left(x ^ {\prime}\right) d x ^ {\prime}\right), +$$ + +and let $S_{v}$ be the law of the SDE + +$$ +\mathrm {d} X _ {t} = g _ {v} \left(X _ {t}, t\right) \mathrm {d} t + \sqrt {\varepsilon} \mathrm {d} B _ {t}, \quad X _ {0} \sim \mu_ {m}. +$$ + +For any coupling $\pi_{m,n} \in \Pi(\mu_m, \nu_n)$ , consider its Brownian-bridge mixture $\Pi = \pi_{m,n} W_{|x_0, x_1}^\varepsilon$ . Then, by (Gushchin et al., 2024, Theorem 3.1), + +$$ +\operatorname *{argmin}_{\substack{S\in \mathcal{S}(\mu_{m})}}H(\Pi \mid S) = \Pi^{*}. +$$ + +Restricting $S$ to $S_{v}$ , one obtains (Gushchin et al., 2024, Theorem 3.2) + +$$ +H (\Pi \mid S _ {v}) = C (\pi_ {m, n}) + \frac {1}{2 \varepsilon} \int_ {0} ^ {1} \int \left\| g _ {v} (X _ {t}, t) - \frac {X _ {1} - X _ {t}}{1 - t} \right\| _ {2} ^ {2} \mathrm {d} \Pi_ {t 1} (X _ {t}, X _ {1}) \mathrm {d} t, +$$ + +where $\Pi_{t1}$ denotes the marginal law of $\Pi$ at times $t$ and 1, and $C(\pi_{m,n})$ is a constant independent of $S_{v}$ . lightSB(-M) trains a parameterised model $v_{\theta}$ that minimises + +$$ +v _ {\theta^ {*}} = \underset {v _ {\theta}} {\operatorname {a r g m i n}} H (\Pi \mid S _ {v _ {\theta}}) = \underset {v _ {\theta}} {\operatorname {a r g m i n}} \int_ {0} ^ {1} \int \left\| g _ {v _ {\theta}} (X _ {t}, t) - \frac {X _ {1} - X _ {t}}{1 - t} \right\| _ {2} ^ {2} \mathrm {d} \Pi_ {t 1} (X _ {t}, X _ {1}) \mathrm {d} t. +$$ + +The drift $g_{v_{\theta^{*}}}$ obtained from the learned $v_{\theta^{*}}$ serves as the drift estimator produced by lightSB(-M). + +When the model capacity is sufficiently large, $g_{v_{\theta^{*}}}$ satisfies + +$$ +g _ {v _ {\theta^ {*}}} (x _ {t}, t) = \mathbb {E} _ {X \sim \Pi^ {*}} \left[ \frac {X _ {1} - x _ {t}}{1 - t} \Bigg | X _ {t} = x _ {t} \right] = b _ {m, n} ^ {*} (x _ {t}, t). +$$ + +# Proof of optimal estimator for DSBM-IMF and BM2 + +Let $F(v_{f})$ and $B(v_{b})$ denote the laws induced by the following SDEs: + +$$ +\mathrm {d} X _ {t} = v _ {f} (X _ {t}, t) \mathrm {d} t + \sqrt {\varepsilon} \mathrm {d} B _ {t}, \quad X _ {0} \sim \mu_ {m}, +$$ + +$$ +\mathrm {d} X _ {t} = v _ {b} (X _ {t}, t) d t + \sqrt {\varepsilon} \mathrm {d} \bar {B} _ {t}, \quad X _ {1} \sim \nu_ {n}. +$$ + +Here, in the second SDE, $dt$ represents an infinitesimal negative time step, and $\bar{B}_t$ denotes the time-reversed Brownian motion. + +Given drifts $v_{f'}$ and $v_{b'}$ , define the following loss functions: + +$$ +L _ {\text {f o r w a r d}} \left(v _ {f}; v _ {b ^ {\prime}}\right) := \mathbb {E} _ {X \sim B \left(v _ {b ^ {\prime}}\right) _ {0 1} W _ {| x _ {0}, x _ {1}} ^ {\varepsilon}} \left[ \int_ {0} ^ {1} \left\| v _ {f} \left(X _ {t}, t\right) - \frac {X _ {1} - X _ {t}}{1 - t} \right\| _ {2} ^ {2} \mathrm {d} t \right], +$$ + +$$ +L _ {\text {b a c k w a r d}} \left(v _ {b}; v _ {f ^ {\prime}}\right) := \mathbb {E} _ {X \sim F \left(v _ {f ^ {\prime}}\right) _ {0 1} W _ {| x _ {0}, x _ {1}} ^ {\varepsilon}} \left[ \int_ {0} ^ {1} \left\| v _ {b} \left(X _ {t}, t\right) - \frac {X _ {t} - X _ {0}}{t} \right\| _ {2} ^ {2} \mathrm {d} t \right]. +$$ + +In DSBM-IMF, starting from initial drifts $v_{f}^{(0)}$ and $v_{b}^{(0)}$ , we perform the following iterative updates: + +$$ +v _ {f} ^ {(k + 1)} = \operatorname * {a r g m i n} _ {v _ {f}} L _ {\text {f o r w a r d}} (v _ {f}; v _ {b} ^ {(k)}), \quad v _ {b} ^ {(k + 1)} = \operatorname * {a r g m i n} _ {v _ {b}} L _ {\text {b a c k w a r d}} (v _ {b}; v _ {f} ^ {(k + 1)}). +$$ + +In BM2, given initial drifts $v_{f}^{(0)}$ and $v_{b}^{(0)}$ , the iterative procedure is: + +$$ +v _ {f} ^ {(k + 1)} = \operatorname * {a r g m i n} _ {v _ {f}} L _ {\text {f o r w a r d}} (v _ {f}; v _ {b} ^ {(k)}), \quad v _ {b} ^ {(k + 1)} = \operatorname * {a r g m i n} _ {v _ {b}} L _ {\text {b a c k w a r d}} (v _ {b}; v _ {f} ^ {(k)}). +$$ + +For both methods, $F(v_{f}^{(k)})$ and $B(v_{b}^{(k)})$ converge to $\Pi^{*}$ as $k \to \infty$ (Shi et al., 2023, Theorem 8), (Peluchetti, 2024, Lemma 1). + +Therefore, letting $v_{f}^{*}$ and $v_{b}^{*}$ denote the limits of $v_{f}^{(k)}$ and $v_{b}^{(k)}$ as $k \to \infty$ , we have that $v_{f}^{*}$ equals $b_{m,n}^{*}$ : + +$$ +\begin{array}{l} v _ {f} ^ {*} (x _ {t}, t) = \mathbb {E} _ {X \sim B (v _ {b} ^ {*}) _ {0 1}} W _ {| x _ {0}, x _ {1}} ^ {\varepsilon} \left[ \frac {X _ {1} - x _ {t}}{1 - t} \Bigg | X _ {t} = x _ {t} \right] \\ = \mathbb {E} _ {X \sim \Pi^ {*}} \left[ \frac {X _ {1} - x _ {t}}{1 - t} \Bigg | X _ {t} = x _ {t} \right] \\ = b _ {m, n} ^ {*} (x _ {t}, t). \\ \end{array} +$$ + +![](images/b50b30c2cb9e641bf7c1c8beb35031ed72e177ba70204a9a318078868713c4e7.jpg) + +# A.4 Proof of Proposition 2 + +Proposition 2. With initialization based on the reference process, both DSBM-IMF and BM2 produce the same estimator as $P_{m,n}^{k}$ of the Sinkhorn bridge for all iterations $k$ . + +Proof of Proposition 2. We begin by introducing the Iterative Proportional Fitting (IPF) method (Fortet, 1940; Kullback, 1968; Ruschendorf, 1995). IPF provides one means of solving Eq. (7) and generates a sequence of path measures $\left(\tilde{P}^{(k)}\right)_{k\in \mathbb{N}}$ according to + +$$ +\tilde {P} ^ {(2 k + 1)} = \underset {\tilde {P}} {\operatorname {a r g m i n}} \left\{H \big (\tilde {P} \mid \tilde {P} ^ {(2 k)} \big) \Big | \tilde {P} _ {0} = \mu_ {m} \right\}, +$$ + +$$ +\tilde {P} ^ {(2 k + 2)} = \operatorname * {a r g m i n} _ {\tilde {P}} \left\{H \big (\tilde {P} \mid \tilde {P} ^ {(2 k + 1)} \big) \Big | \tilde {P} _ {1} = \nu_ {n} \right\}. +$$ + +We initialize with $\tilde{P}^{(0)} = W^{\varepsilon}$ . It is known (Léonard, 2013; Nutz, 2021) that the sequence $\left(\tilde{P}^{(k)}\right)_{k\in \mathbb{N}}$ satisfies + +$$ +\tilde {P} ^ {(2 k + 1)} = \pi^ {f _ {m, n} ^ {(k + 1)}, g _ {m, n} ^ {(k)}} W _ {| x _ {0}, x _ {1}} ^ {\varepsilon}, +$$ + +$$ +\tilde {P} ^ {(2 k + 2)} = \pi^ {f _ {m, n} ^ {(k + 1)}, g _ {m, n} ^ {(k + 1)}} W _ {| x _ {0}, x _ {1}} ^ {\varepsilon} = \Pi^ {(k + 1)}. +$$ + +Here the coupling $\pi^{f,g}$ is defined by + +$$ +\mathrm {d} \pi^ {f, g} (x, y) = \exp_ {\varepsilon} \left(f (x) + g (y) - \frac {1}{2} \| x - y \| _ {2} ^ {2}\right) \mathrm {d} \mu_ {m} (x) \mathrm {d} \nu_ {n} (y). +$$ + +Hence it suffices to show that the drifts estimated by each algorithm coincide with $b_{m,n}^{(k)}$ . + +When the initial drift for DSBM is set to $v_{b}^{(0)} = 0$ , the sequence $B(v_{b}^{(0)}), F(v_{f}^{(1)}), B(v_{b}^{(1)}), F(v_{f}^{(2)}), B(v_{b}^{(3)}) \ldots$ coincides with the IPF sequence $(\tilde{P}^{(k)})_{k\in \mathbb{N}}$ (Shi et al., 2023, Proposition 10). Consequently, $v_{f}^{(k + 1)}$ coincides with $b_{m,n}^{(k)}$ . + +$$ +\begin{array}{l} v _ {f} ^ {(k + 1)} (x _ {t}, t) = \mathbb {E} _ {X \sim B (v _ {b} ^ {(k)}) _ {0 1} W _ {| x _ {0}, x _ {1}} ^ {\varepsilon}} \left[ \frac {X _ {1} - x _ {t}}{1 - t} \Bigg | X _ {t} = x _ {t} \right] \\ = \mathbb {E} _ {X \sim \pi^ {f _ {m, n} ^ {(k)}, g _ {m, n} ^ {(k)}} W _ {| x _ {0}, x _ {1}} ^ {\varepsilon}} \left[ \frac {X _ {1} - x _ {t}}{1 - t} \right| X _ {t} = x _ {t} \Biggr ] \\ = \mathbb {E} _ {X \sim \Pi^ {(k)}} \left[ \frac {X _ {1} - x _ {t}}{1 - t} \Bigg | X _ {t} = x _ {t} \right] \\ = b _ {m, n} ^ {(k)} \left(x _ {t}, t\right). \\ \end{array} +$$ + +When the initial drift for BM2 is initialized as $v_{f}^{(0)} = v_{b}^{(0)} = 0$ , the sequence $B(v_{b}^{(0)}), F(v_{f}^{(1)}), B(v_{b}^{(2)})$ , ... coincides with the IPF sequence $(\tilde{P}^{(k)})_{k\in \mathbb{N}}$ (Peluchetti, 2024, Theorem 1). Thus, $v_{f}^{(2k + 1)}$ is given by + +$$ +v _ {f} ^ {(2 k + 1)} (x _ {t}, t) = \mathbb {E} _ {X \sim B (v _ {b} ^ {(2 k)}) _ {0 1} W _ {| x _ {0}, x _ {1}} ^ {\varepsilon}} \left[ \frac {X _ {1} - x _ {t}}{1 - t} \Bigg | X _ {t} = x _ {t} \right] +$$ + +$$ +\begin{array}{l} = \mathbb {E} _ {X \sim \pi^ {f _ {m, n} ^ {(k)}, g _ {m, n} ^ {(k)}} W _ {| x _ {0}, x _ {1}} ^ {\varepsilon}} \left[ \frac {X _ {1} - x _ {t}}{1 - t} \right| X _ {t} = x _ {t} \Biggr ] \\ = \mathbb {E} _ {X \sim \Pi^ {(k)}} \left[ \frac {X _ {1} - x _ {t}}{1 - t} \Bigg | X _ {t} = x _ {t} \right] \\ = b _ {m, n} ^ {(k)} \left(x _ {t}, t\right). \\ \end{array} +$$ + +Similarly, the sequence $F(v_{f}^{(0)}), B(v_{b}^{(1)}), F(v_{f}^{(2)})$ , ... corresponds to IPF with the update order reversed, and the same conclusion holds for $v_{f}^{(2k)}$ . + +![](images/0b58cdb90efac87503830dea0eec633556c57627e660f6118348380945a1a796.jpg) + +# B Technical Lemmas + +# B.1 Lemmas used for Theorems 1 and 2 + +Lemma 3 (Stromme, 2023b, Proposition 14). Under assumptions of Theorem 1, we have + +$$ +\| f _ {\infty , \infty} ^ {*} \| _ {\infty}, \| g _ {\infty , \infty} ^ {*} \| _ {\infty} \leq 2 R ^ {2}, \| f _ {m, n} ^ {*} \| _ {\infty}, \| g _ {m, n} ^ {*} \| _ {\infty} \leq 4 R ^ {2}. +$$ + +In the following, $p_{\infty ,\infty}^{*}$ denotes the Radon-Nikodym derivative regarding $\mu \otimes \nu$ , defined in Eq. (6), and $p_{m,n}^{*}$ is also defined in a similar manner replacing $f_{\infty ,\infty}^{*},g_{\infty ,\infty}^{*}$ with $f_{m,n}^{*},g_{m,n}^{*}$ + +Lemma 4 (Stromme, 2023b, Proposition 15). The population dual potentials $f_{\infty,\infty}^{*}$ and $g_{\infty,\infty}^{*}$ are $2R$ -Lipschitz over $\operatorname{supp}(\mu)$ and $\operatorname{supp}(\nu)$ , respectively. The extended empirical dual potentials $f_{m,n}^{*}$ and $g_{m,n}^{*}$ are also $2R$ -Lipschitz over $\operatorname{supp}(\mu)$ and $\operatorname{supp}(\nu)$ , respectively. In particular, the population density $p_{\infty,\infty}^{*}$ and the extended empirical density $p_{m,n}^{*}$ are each $\frac{4R}{\varepsilon}$ -log-Lipschitz in each of their variables over $\operatorname{supp}(\mu \otimes \nu)$ . + +Lemma 5 (Stromme, 2023b, Lemma 25). Under assumptions of Theorem 1, with probability at least $1 - \frac{1}{n} e^{-20R^2 /\varepsilon}$ + +$$ +\inf _ {z \in \operatorname {s u p p} (\nu)} \nu_ {n} \left(B \left(z, \frac {\varepsilon}{2 R}\right)\right) \gtrsim \left(\frac {\varepsilon}{R}\right) ^ {d _ {\nu}}. \tag {19} +$$ + +Lemma 6 (Stromme, 2023b, Lemma 26). Under assumptions of Theorem 1, we have + +$$ +\left\| p _ {\infty , \infty} ^ {*} \right\| _ {L ^ {\infty} (\mu \otimes \nu)} \lesssim \left(\frac {R}{\varepsilon}\right) ^ {d _ {\nu}}. \tag {20} +$$ + +And, with probability at least $1 - \frac{1}{n} e^{-20R^2 /\varepsilon}$ + +$$ +\left\| p _ {m, n} ^ {*} \right\| _ {L ^ {\infty} (\mu \otimes \nu)} \lesssim \left(\frac {R}{\varepsilon}\right) ^ {d _ {\nu}}. \tag {21} +$$ + +Lemma 7 (Stromme, 2023b, Lemma 27 adapted). Under the assumptions of Theorem 1, if $\varepsilon / R$ is sufficiently small, then + +$$ +\mathbb {E} \| g _ {m, n} ^ {*} - g _ {\infty , \infty} ^ {*} \| _ {L ^ {2} (\nu_ {n})} ^ {2} \lesssim \left(\varepsilon^ {2} + \frac {R ^ {8}}{\varepsilon^ {2}}\right) \cdot \left(\frac {R}{\varepsilon}\right) ^ {6 d _ {\nu} + 4} \cdot \left(\frac {1}{m} + \frac {1}{n}\right). \tag {22} +$$ + +Lemma 7 are the extensions of Stromme (2023b, Lemma 27) that only treats the case of $m = n$ . This extension can be readily verified by a slight modification of the proof in Stromme (2023b). + +Lemma 8. Let $U_{m}(y) = (\mu_{m} - \mu)\big(p_{\infty ,\infty}^{*}(\cdot ,y)\big)$ . Then the following holds: + +$$ +\mathbb{E}\sup_{y\in \operatorname {supp}(\nu)}U_{m}(y)^{2}\lesssim \frac{R^{2}}{m}\left(\frac{R}{\varepsilon}\right)^{2d_{\nu} + 2}. +$$ + +Proof. We first note that $\mu(p_{\infty, \infty}^{*}(\cdot, y)) = 1$ since the marginal of $\pi_{\infty, \infty}^{*}$ on $y$ is $\nu$ , and hence $U_{m}(y) = \mu_{m}(p_{\infty, \infty}^{*}(\cdot, y)) - 1$ . Hoeffding's lemma, with $0 \leq p_{\infty, \infty}^{*} \lesssim \left(\frac{R}{\varepsilon}\right)^{d_{\nu}}$ , implies that there exists a constant $C > 0$ such that + +$$ +\mathbb {E} \exp \left\{s U _ {m} (y) \right\} = \mathbb {E} \prod_ {i = 1} ^ {m} \exp \left\{\frac {s}{m} \left(p _ {\infty , \infty} ^ {*} \left(X _ {i}, y\right) - 1\right) \right\} +$$ + +$$ +\begin{array}{l} \leq \prod_ {i = 1} ^ {m} \exp \left(\frac {s ^ {2} C}{8 m ^ {2}} \left(\frac {R}{\varepsilon}\right) ^ {2 d _ {\nu}}\right) \\ \leq \exp \left(\frac {C}{4 m} \left(\frac {R}{\varepsilon}\right) ^ {2 d _ {\nu}} \cdot \frac {s ^ {2}}{2}\right). \\ \end{array} +$$ + +Consequently, $U_{m}(y)$ is subgaussian with variance proxy $\frac{C}{4m}\left(\frac{R}{\varepsilon}\right)^{2d_{\nu}}$ . Orlicz $\psi_{2}$ -norm is defined for a random variable $\xi$ by + +$$ +\left\| \xi \right\| _ {\psi_ {2}} = \inf \left\{t > 0: \mathbb {E} e ^ {| \xi | ^ {2} / t ^ {2}} \leq 2 \right\}. +$$ + +Chen et al. (2023, Theorem 3.7 and Theorem 4.1) with $\varphi(x) \coloneqq x^2/2$ , $p = 2$ and $d(y, y') \coloneqq \|U_m(y) - U_m(y')\|_{\psi_2}$ imply the following estimates: + +$$ +\left(\mathbb {E} \sup _ {y \in \operatorname {s u p p} (\nu)} U _ {m} (y) ^ {2}\right) ^ {1 / 2} \lesssim \int_ {0} ^ {\infty} \sqrt {\log \mathcal {N} (\operatorname {s u p p} (\nu) , d , \delta)} \mathrm {d} \delta + \sup _ {y, y ^ {\prime} \in \operatorname {s u p p} (\nu)} d (y, y ^ {\prime}). \tag {23} +$$ + +Let $Z_{i} \coloneqq p_{\infty, \infty}^{*}(X_{i}, y) - p_{\infty, \infty}^{*}(X_{i}, y')$ . Since each $Z_{i}$ is an independent, zero-mean random variable, by Vershynin (2018, Proposition 2.6.1) $d(y, y')$ is bounded above as follows: + +$$ +\begin{array}{l} d (y, y ^ {\prime}) = \left\| \mu_ {m} \left(p _ {\infty , \infty} ^ {*} (\cdot , y) - p _ {\infty , \infty} ^ {*} (\cdot , y ^ {\prime})\right) \right\| _ {\psi_ {2}} \\ = \frac {1}{m} \left\| \sum_ {i = 1} ^ {m} Z _ {i} \right\| _ {\psi_ {2}} \\ \lesssim \frac {1}{m} \left(\sum_ {i = 1} ^ {m} \| Z _ {i} \| _ {\psi_ {2}} ^ {2}\right) ^ {1 / 2}. \\ \end{array} +$$ + +For each $Z_{i}$ we have $\| \cdot \|_{\psi_2} \lesssim \| \cdot \|_{L^{\infty}(\mu)}$ , hence + +$$ +\| Z _ {i} \| _ {\psi_ {2}} = \left\| p _ {\infty , \infty} ^ {*} (X _ {i}, y) - p _ {\infty , \infty} ^ {*} (X _ {i}, y ^ {\prime}) \right\| _ {\psi_ {2}} \lesssim \left\| p _ {\infty , \infty} ^ {*} (\cdot , y) - p _ {\infty , \infty} ^ {*} (\cdot , y ^ {\prime}) \right\| _ {L ^ {\infty} (\mu)}. +$$ + +By $|e^{a} - e^{b}| \leq e^{a \vee b} |a - b|$ and Lemma 4, 6, for any $x$ , + +$$ +\begin{array}{l} \left| p _ {\infty , \infty} ^ {*} (x, y) - p _ {\infty , \infty} ^ {*} (x, y ^ {\prime}) \right| \leq \left\| p _ {\infty , \infty} ^ {*} \right\| _ {L ^ {\infty} (\mu \otimes \nu)} \left| \log p _ {\infty , \infty} ^ {*} (x, y) - \log p _ {\infty , \infty} ^ {*} (x, y ^ {\prime}) \right| \\ \lesssim \left(\frac {R}{\varepsilon}\right) ^ {d _ {\nu}} \frac {4 R}{\varepsilon} \| y - y ^ {\prime} \| _ {2} \\ \lesssim \left(\frac {R}{\varepsilon}\right) ^ {d _ {\nu} + 1} \| y - y ^ {\prime} \| _ {2}. \\ \end{array} +$$ + +Therefore, + +$$ +\begin{array}{l} d (y, y ^ {\prime}) \lesssim \frac {1}{m} \left(\sum_ {i = 1} ^ {m} \left(\frac {R}{\varepsilon}\right) ^ {2 d _ {\nu} + 2} \| y - y ^ {\prime} \| _ {2} ^ {2}\right) ^ {1 / 2} \\ = \frac {1}{m} \left(\frac {R}{\varepsilon}\right) ^ {d _ {\nu} + 1} \cdot \sqrt {m} \| y - y ^ {\prime} \| _ {2} \\ = \frac {1}{\sqrt {m}} \left(\frac {R}{\varepsilon}\right) ^ {d _ {\nu} + 1} \| y - y ^ {\prime} \| _ {2}. \tag {24} \\ \end{array} +$$ + +This concludes $\mathcal{N}(\mathrm{supp}(\nu),d,r)\leq \mathcal{N}\left(\mathrm{supp}(\nu),\| \cdot \| _2,\frac{r}{A}\right)$ $A = \frac{C}{\sqrt{m}}\left(\frac{R}{\varepsilon}\right)^{d_{\nu} + 1}$ where $C$ is a constant hidden in the above inequality). + +We evaluate the first term in (23) by using Proposition 43 in Stromme (2023b) that shows there exists $0 < c_{\nu} \leq R$ such that, for any $0 < \delta \leq c_{\nu}$ , the covering number satisfies $\delta^{-d_{\nu}} \lesssim \mathcal{N}(\mathrm{supp}(\nu), \| \cdot \|_2, \delta) \lesssim \delta^{-d_{\nu}}$ . + +$$ +\int_ {0} ^ {\infty} \sqrt {\log \mathcal {N} (\operatorname {s u p p} (\nu) , d , r)} \mathrm {d} r +$$ + +$$ +\begin{array}{l} \lesssim \int_ {0} ^ {\infty} \sqrt {\log \mathcal {N} \left(\operatorname {s u p p} (\nu) , \| \cdot \| _ {2} , \frac {r}{A}\right)} \mathrm {d} r \quad \left(A := \frac {C}{\sqrt {m}} \left(\frac {R}{\varepsilon}\right) ^ {d _ {\nu} + 1}\right) \\ = A \int_ {0} ^ {2 R} \sqrt {\log \mathcal {N} (\operatorname {s u p p} (\nu) , \| \cdot \| _ {2} , \delta)} d \delta \quad \left(\delta := \frac {r}{A}\right) \\ = A \int_ {0} ^ {c _ {\nu}} \sqrt {\log \mathcal {N} (\operatorname {s u p p} (\nu) , \| \cdot \| _ {2} , \delta)} d \delta + A \int_ {c _ {\nu}} ^ {2 R} \sqrt {\log \mathcal {N} (\operatorname {s u p p} (\nu) , \| \cdot \| _ {2} , \delta)} d \delta \\ \lesssim A \int_ {0} ^ {c _ {\nu}} \sqrt {\log \delta^ {- d _ {\nu}}} \mathrm {d} \delta + A (2 R - c _ {\nu}) \sqrt {\log c _ {\nu} ^ {- d _ {\nu}}} \\ \lesssim A \sqrt {d _ {\nu}} \int_ {0} ^ {\infty} u ^ {1 / 2} e ^ {- u} \mathrm {d} u + A R \sqrt {d _ {\nu}} \qquad \left(u := \log \frac {1}{\delta}\right) \\ \lesssim A R \\ \lesssim \frac {R}{\sqrt {m}} \left(\frac {R}{\varepsilon}\right) ^ {d _ {\nu} + 1}, \tag {25} \\ \end{array} +$$ + +where $\lesssim$ also hides the intrinsic dimensionality $d_{\nu}$ + +From (24), the second term of (23) is bounded as + +$$ +\sup _ {y, y ^ {\prime} \in \operatorname {s u p p} (\nu)} d (y, y ^ {\prime}) \lesssim \sup _ {y, y ^ {\prime} \in \operatorname {s u p p} (\nu)} \frac {1}{\sqrt {m}} \left(\frac {R}{\varepsilon}\right) ^ {d _ {\nu} + 1} \| y - y ^ {\prime} \| _ {2} \lesssim \frac {R}{\sqrt {m}} \left(\frac {R}{\varepsilon}\right) ^ {d _ {\nu} + 1}. \tag {26} +$$ + +Combining the results of (26) and (25) completes the proof. + +$$ +\mathbb {E} \sup _ {y \in \mathrm {s u p p} (\nu)} U _ {m} (y) ^ {2} \lesssim \left(\frac {R}{\sqrt {m}} \left(\frac {R}{\varepsilon}\right) ^ {d _ {\nu} + 1} + \frac {R}{\sqrt {m}} \left(\frac {R}{\varepsilon}\right) ^ {d _ {\nu} + 1}\right) ^ {2} \lesssim \frac {R ^ {2}}{m} \left(\frac {R}{\varepsilon}\right) ^ {2 d _ {\nu} + 2}. +$$ + +Lemma 9. Let $V_{n}(x) = (\nu_{n} - \nu)\left(p_{\infty ,\infty}^{*}(x,\cdot)\right)$ . Then the following holds: + +$$ +\mathbb {E} \sup _ {x \in \operatorname {s u p p} (\mu)} V _ {n} (x) ^ {2} \lesssim \frac {d R ^ {2}}{n} \left(\frac {R}{\varepsilon}\right) ^ {2 d _ {\nu} + 2}. +$$ + +Proof. Lemma 9 can be established by a minor modification of the proof of Lemma 8. The only difference is the evaluation of the covering number: the bound over $\mathrm{supp}(\mu)$ is replaced by that over $B(0,R)$ . This substitution introduces a dependence on the ambient dimension $d$ . + +Lemma 10. It follows that + +$$ +\mathbb {E} \| f _ {\infty , \infty} ^ {*} - f _ {m, n} ^ {*} \| _ {L ^ {\infty} (\mu)} ^ {2} \lesssim \left(\varepsilon^ {2} + \frac {d R ^ {8}}{\varepsilon^ {2}}\right) \cdot \left(\frac {R}{\varepsilon}\right) ^ {7 d _ {\nu} + 4} \cdot \left(\frac {1}{m} + \frac {1}{n}\right). +$$ + +Proof. Let + +$$ +\bar {f} _ {\infty , n} (x) := - \varepsilon \log \left(\frac {1}{n} \sum_ {j = 1} ^ {n} \exp_ {\varepsilon} \left(g _ {\infty , \infty} ^ {*} \left(Y _ {j}\right) - c (x, Y _ {j})\right)\right). +$$ + +Using $\bar{f}_{\infty ,n}$ , we evaluate $\| f_{\infty ,\infty}^{*} - f_{m,n}^{*}\|_{L^{\infty}(\mu)}^{2}$ as follows. + +$$ +\mathbb {E} \| f _ {\infty , \infty} ^ {*} - f _ {m, n} ^ {*} \| _ {L ^ {\infty} (\mu)} ^ {2} \lesssim \mathbb {E} \| f _ {\infty , \infty} ^ {*} - \bar {f} _ {\infty , n} \| _ {L ^ {\infty} (\mu)} ^ {2} + \mathbb {E} \| \bar {f} _ {\infty , n} - f _ {m, n} ^ {*} \| _ {L ^ {\infty} (\mu)} ^ {2}. \tag {27} +$$ + +For the first term, the Schrödinger system (5) implies + +$$ +f _ {\infty , \infty} ^ {*} (x) - \bar {f} _ {\infty , n} (x) = \varepsilon \log \left(\frac {1}{n} \exp_ {\varepsilon} (f _ {\infty , \infty} ^ {*} (x)) \sum_ {j = 1} ^ {n} \exp_ {\varepsilon} (g _ {\infty , \infty} ^ {*} (Y _ {j}) - c (x, Y _ {j}))\right) +$$ + +$$ +\begin{array}{l} = \varepsilon \log \left(\frac {1}{n} \sum_ {j = 1} ^ {n} p _ {\infty , \infty} ^ {*} (x, Y _ {j})\right) \\ = \varepsilon \log \left(1 + V _ {n} (x)\right), \\ \end{array} +$$ + +where $V_{n}(x)\coloneqq (\nu_{n} - \nu)(p_{\infty ,\infty}^{*}(x,\cdot))$ and we used $\nu (p_{\infty ,\infty}^{*}(x,\cdot)) = 1$ + +For any $\eta \in (0,1)$ , the following holds: + +$$ +\begin{array}{l} \mathbb {E} \| f _ {\infty , \infty} ^ {*} (x) - \bar {f} _ {\infty , n} (x) \| _ {L ^ {\infty} (\mu)} ^ {2} = \varepsilon^ {2} \mathbb {E} \sup _ {x \in \operatorname {s u p p} (\mu)} | \log (1 + V _ {n} (x)) | ^ {2} \\ = \varepsilon^ {2} \mathbb {E} \left[ \mathbb {1} \left[ \sup _ {x \in \operatorname {s u p p} (\mu)} | V _ {n} | < \eta \right] \sup _ {x \in \operatorname {s u p p} (\mu)} | \log (1 + V _ {n} (x)) | ^ {2} \right] \\ + \varepsilon^ {2} \mathbb {E} \left[ \mathbb {1} \left[ \sup _ {x \in \operatorname {s u p p} (\mu)} | V _ {n} | \geq \eta \right] \sup _ {x \in \operatorname {s u p p} (\mu)} | \log (1 + V _ {n} (x)) | ^ {2} \right]. \tag {28} \\ \end{array} +$$ + +For the first term in (28), the mean value theorem implies that for $|u| < 1$ we have $|\log(1 + u)| \leq \frac{|u|}{1 - |u|}$ . Hence, + +$$ +\mathbb {E} \left[ \mathbb {1} \left[ \sup _ {x \in \operatorname {s u p p} (\mu)} | V _ {n} | < \eta \right] \sup _ {x \in \operatorname {s u p p} (\mu)} | \log (1 + V _ {n} (x)) | ^ {2} \right] \leq \mathbb {E} \sup _ {x \in \operatorname {s u p p} (\mu)} \frac {| V _ {n} (x) | ^ {2}}{(1 - \eta) ^ {2}}. +$$ + +For the second term in (28), since $\exp_{\varepsilon}(-10R^2) \leq p_{\infty, \infty}^* \leq \exp_{\varepsilon}(10R^2)$ , an application of Markov's inequality yields + +$$ +\begin{array}{l} \mathbb {E} \left[ \mathbb {1} \left[ \sup _ {x \in \operatorname {s u p p} (\mu)} | V _ {n} | \geq \eta \right] \sup _ {x \in \operatorname {s u p p} (\mu)} | \log (1 + V _ {n} (x)) | ^ {2} \right] \lesssim \frac {R ^ {4}}{\varepsilon^ {2}} \mathbb {P} \left(\sup _ {x \in \operatorname {s u p p} (\mu)} | V _ {n} (x) | \geq \eta\right) \\ \leq \frac {R ^ {4}}{\varepsilon^ {2}} \frac {\mathbb {E} \sup _ {x \in \operatorname {s u p p} (\mu)} | V _ {n} (x) | ^ {2}}{\eta^ {2}}. \\ \end{array} +$$ + +Setting $\eta = 1 / 2$ and applying Lemma 9 yields + +$$ +\begin{array}{l} \mathbb {E} \| f _ {\infty , \infty} ^ {*} (x) - \bar {f} _ {\infty , n} (x) \| _ {L ^ {\infty} (\mu)} ^ {2} \lesssim \left(1 + \frac {R ^ {4}}{\varepsilon^ {2}}\right) \mathbb {E} \sup _ {x \in \operatorname {s u p p} (\mu)} | V _ {n} (x) | ^ {2} \\ \lesssim \frac {\varepsilon^ {2} + d R ^ {4}}{n} \left(\frac {R}{\varepsilon}\right) ^ {2 d _ {\nu} + 4}. \tag {29} \\ \end{array} +$$ + +For the second term in (27), Eq. (6.4) in Stromme (2023b) implies + +$$ +| \bar {f} _ {\infty , n} (x) - f _ {m, n} ^ {*} (x) | \lesssim \| g _ {\infty , \infty} ^ {*} - g _ {m, n} ^ {*} \| _ {L ^ {2} (\nu_ {n})} \sup _ {z \in \mathrm {s u p p} (\nu)} \nu_ {n} \left(B \left(z, \frac {\varepsilon}{2 R}\right)\right) ^ {- 1 / 2}. +$$ + +Let $\mathcal{E}$ denote the event on which Lemma 5 holds. From Lemma 3 and $\mathbb{P}[\mathcal{E}^C ]\leq \frac{1}{n}$ , it follows that + +$$ +\mathbb {E} \| f _ {\infty , \infty} ^ {*} - f _ {m, n} ^ {*} \| _ {L ^ {\infty} (\mu)} ^ {2} \lesssim \mathbb {E} \left[ \mathbb {1} [ \mathcal {E} ] \| f _ {\infty , \infty} ^ {*} - f _ {m, n} ^ {*} \| _ {L ^ {\infty} (\mu)} ^ {2} \right] + \frac {R ^ {2}}{n}. +$$ + +Combining this with Lemma 7 and Lemma 5 yields + +$$ +\begin{array}{l} \mathbb {E} \| f _ {\infty , \infty} ^ {*} - f _ {m, n} ^ {*} \| _ {L ^ {\infty} (\mu)} ^ {2} \lesssim \mathbb {E} \left[ \mathbb {1} [ \mathcal {E} ] \| f _ {\infty , \infty} ^ {*} - f _ {m, n} ^ {*} \| _ {L ^ {\infty} (\mu)} ^ {2} \right] + \frac {R ^ {2}}{n} \\ \lesssim \mathbb {E} \left[ \mathbb {1} [ \mathcal {E} ] \left(\| f _ {\infty , \infty} ^ {*} - \bar {f} _ {\infty , n} \| _ {L ^ {\infty} (\mu)} ^ {2} + \| \bar {f} _ {\infty , n} - f _ {m, n} ^ {*} \| _ {L ^ {\infty} (\mu)} ^ {2}\right) \right] + \frac {R ^ {2}}{n} \\ \lesssim \frac {\varepsilon^ {2} + d R ^ {4}}{n} \left(\frac {R}{\varepsilon}\right) ^ {2 d _ {\nu} + 4} + \left(\frac {R}{\varepsilon}\right) ^ {d _ {\nu}} \mathbb {E} \| g _ {\infty , \infty} ^ {*} - g _ {m, n} ^ {*} \| _ {L ^ {2} (\nu_ {n})} ^ {2} + \frac {R ^ {2}}{n} \\ \lesssim \left(\varepsilon^ {2} + \frac {d R ^ {8}}{\varepsilon^ {2}}\right) \cdot \left(\frac {R}{\varepsilon}\right) ^ {7 d _ {\nu} + 4} \cdot \left(\frac {1}{m} + \frac {1}{n}\right). \\ \end{array} +$$ + +□ + +Lemma 11. It follows that + +$$ +\mathbb {E} \| g _ {\infty , \infty} ^ {*} - g _ {m, n} ^ {*} \| _ {L ^ {\infty} (\nu)} ^ {2} \lesssim \left(\varepsilon^ {2} + \frac {d R ^ {8}}{\varepsilon^ {2}}\right) \cdot \left(\frac {R}{\varepsilon}\right) ^ {7 d _ {\nu} + 4} \cdot \left(\frac {1}{m} + \frac {1}{n}\right). +$$ + +Proof. Let + +$$ +\bar {g} _ {m, \infty} (y) := - \varepsilon \log \left(\frac {1}{m} \sum_ {i = 1} ^ {m} \exp_ {\varepsilon} \left(f _ {\infty , \infty} ^ {*} \left(X _ {i}\right) - c \left(X _ {i}, y\right)\right)\right). +$$ + +Using $\bar{g}_{m,\infty}$ , we evaluate $\| g_{\infty ,\infty}^{*} - g_{m,n}^{*}\|_{L^{\infty}(\nu)}^{2}$ as follows. + +$$ +\mathbb {E} \| g _ {\infty , \infty} ^ {*} - g _ {m, n} ^ {*} \| _ {L ^ {\infty} (\nu)} ^ {2} \lesssim \mathbb {E} \| g _ {\infty , \infty} ^ {*} - \bar {g} _ {m, \infty} \| _ {L ^ {\infty} (\nu)} ^ {2} + \mathbb {E} \| \bar {g} _ {m, \infty} - g _ {m, n} ^ {*} \| _ {L ^ {\infty} (\nu)} ^ {2}. \tag {30} +$$ + +For the first term, the Schrödinger system (5) implies + +$$ +\begin{array}{l} g _ {\infty , \infty} ^ {*} (y) - \bar {g} _ {m, \infty} (y) = \varepsilon \log \left(\frac {1}{m} \exp_ {\varepsilon} \left(g _ {\infty , \infty} ^ {*} (y)\right) \sum_ {i = 1} ^ {m} \exp_ {\varepsilon} \left(f _ {\infty , \infty} ^ {*} \left(X _ {i}\right) - c \left(X _ {i}, y\right)\right)\right) \\ = \varepsilon \log \left(\frac {1}{m} \sum_ {i = 1} ^ {m} p _ {\infty , \infty} ^ {*} (X _ {i}, y)\right) \\ = \varepsilon \log (1 + U _ {m} (y)), \\ \end{array} +$$ + +where $U_{m}(y)\coloneqq (\mu_{m} - \mu)(p_{\infty ,\infty}^{*}(\cdot ,y))$ and used $\mu (p_{\infty ,\infty}^{*}(\cdot ,y)) = 1$ + +For any $\eta \in (0,1)$ , the following holds: + +$$ +\begin{array}{l} \mathbb {E} \| g _ {\infty , \infty} ^ {*} (y) - \bar {g} _ {m, \infty} (y) \| _ {L ^ {\infty} (\nu)} ^ {2} = \varepsilon^ {2} \mathbb {E} \sup _ {y \in \operatorname {s u p p} (\nu)} \log (1 + U _ {m} (y)) ^ {2} \\ = \varepsilon^ {2} \mathbb {E} \left[ \mathbb {1} \left[ \sup _ {y \in \operatorname {s u p p} (\nu)} | U _ {m} | < \eta \right] \sup _ {y \in \operatorname {s u p p} (\nu)} | \log (1 + U _ {m} (y)) | ^ {2} \right] \\ + \varepsilon^ {2} \mathbb {E} \left[ \mathbb {1} \left[ \sup _ {y \in \operatorname {s u p p} (\nu)} | U _ {m} | \geq \eta \right] \sup _ {y \in \operatorname {s u p p} (\nu)} | \log (1 + U _ {m} (y)) | ^ {2} \right]. \tag {31} \\ \end{array} +$$ + +For the first term in (31), the mean value theorem implies that for $|u| < 1$ we have $|\log(1 + u)| \leq \frac{|u|}{1 - |u|}$ . Hence, + +$$ +\mathbb {E} \left[ \mathbb {1} \left[ \sup _ {y \in \operatorname {s u p p} (\nu)} | U _ {m} | < \eta \right] \sup _ {y \in \operatorname {s u p p} (\nu)} | \log (1 + U _ {m} (y)) | ^ {2} \right] \leq \mathbb {E} \sup _ {y \in \operatorname {s u p p} (\nu)} \frac {| U _ {m} (y) | ^ {2}}{(1 - \eta) ^ {2}} +$$ + +For the second term in (31), since $\exp_{\varepsilon}(-10R^2) \leq p_{\infty, \infty}^{*} \leq \exp_{\varepsilon}(10R^2)$ , an application of Markov's inequality yields + +$$ +\begin{array}{l} \mathbb {E} \left[ \mathbb {1} \left[ \sup _ {y \in \operatorname {s u p p} (\nu)} | U _ {m} | \geq \eta \right] \sup _ {y \in \operatorname {s u p p} (\nu)} | \log (1 + U _ {m} (y)) | ^ {2} \right] \lesssim \frac {R ^ {4}}{\varepsilon^ {2}} \mathbb {P} \left(\sup _ {y \in \operatorname {s u p p} (\nu)} | U _ {m} (y) | \geq \eta\right) \\ \leq \frac {R ^ {4}}{\varepsilon^ {2}} \frac {\mathbb {E} \sup _ {y \in \operatorname {s u p p} (\nu)} | U _ {m} (y) | ^ {2}}{\eta^ {2}}. \\ \end{array} +$$ + +Setting $\eta = 1 / 2$ and applying Lemma 8 yields + +$$ +\begin{array}{l} \mathbb {E} \| g _ {\infty , \infty} ^ {*} (y) - \bar {g} _ {m, \infty} (y) \| _ {L ^ {\infty} (\nu)} ^ {2} \lesssim \left(1 + \frac {R ^ {4}}{\varepsilon^ {2}}\right) \mathbb {E} \sup _ {y \in \operatorname {s u p p} (\nu)} | U _ {m} (y) | ^ {2} \\ \lesssim \frac {\varepsilon^ {2} + R ^ {4}}{m} \left(\frac {R}{\varepsilon}\right) ^ {2 d _ {\nu} + 4}. \tag {32} \\ \end{array} +$$ + +For the second term of (30), since the gradient of the log-sum-exp function is the softmax and its $\ell_1$ -norm equals 1, we obtain + +$$ +\left| \bar {g} _ {m, \infty} (y) - g _ {m, n} ^ {*} (y) \right| \leq \left\| f _ {\infty , \infty} ^ {*} - f _ {m, n} ^ {*} \right\| _ {L ^ {\infty} \left(\mu_ {m}\right)}. \tag {33} +$$ + +Therefore, by Lemma 10, we have + +$$ +\begin{array}{l} \mathbb {E} \| \bar {g} _ {m, \infty} - g _ {m, n} ^ {*} \| _ {L ^ {\infty} (\nu)} ^ {2} \leq \mathbb {E} \| f _ {\infty , \infty} ^ {*} - f _ {m, n} ^ {*} \| _ {L ^ {\infty} (\mu_ {m})} ^ {2} \\ \lesssim \left(\varepsilon^ {2} + \frac {d R ^ {8}}{\varepsilon^ {2}}\right) \cdot \left(\frac {R}{\varepsilon}\right) ^ {7 d _ {\nu} + 4} \cdot \left(\frac {1}{m} + \frac {1}{n}\right). \\ \end{array} +$$ + +Combining the above results, we obtain + +$$ +\mathbb {E} \| g _ {\infty , \infty} ^ {*} - g _ {m, n} ^ {*} \| _ {L ^ {\infty} (\nu)} ^ {2} \lesssim \left(\varepsilon^ {2} + \frac {d R ^ {8}}{\varepsilon^ {2}}\right) \cdot \left(\frac {R}{\varepsilon}\right) ^ {7 d _ {\nu} + 4} \cdot \left(\frac {1}{m} + \frac {1}{n}\right). \tag {34} +$$ + +Lemma 12 (Lipschitzness of softmax). For $\lambda >0$ and $d\geq 2$ define $\sigma_{\lambda}:\mathbb{R}^{d}\to \Delta_{d}$ by + +$$ +\left(\sigma_ {\lambda} (x)\right) _ {i} = \left(\frac {\exp (\lambda x _ {i})}{\sum_ {k = 1} ^ {d} \exp (\lambda x _ {k})}\right) _ {i} \qquad (i = 1, \ldots , d). +$$ + +Then for any $1 \leq p, q \leq \infty$ , + +$$ +\left\| \sigma_ {\lambda} (x) - \sigma_ {\lambda} (y) \right\| _ {p} \leq L _ {p, q} \left\| x - y \right\| _ {q}, \quad L _ {p, q} = \lambda 2 ^ {- 1 + 1 / p - 1 / q}. +$$ + +Proof. Let $s = \sigma_{\lambda}(z)$ . The Jacobian of $\sigma_{\lambda}$ at $z$ is + +$$ +J (z) = \nabla \sigma_ {\lambda} (z) = \lambda (\operatorname {d i a g} (s) - s s ^ {\top}). \tag {35} +$$ + +By the fundamental theorem of calculus along the segment $t\mapsto y + t(x - y)$ + +$$ +\sigma_ {\lambda} (x) - \sigma_ {\lambda} (y) = \int_ {0} ^ {1} J (y + t (x - y)) (x - y) d t, +$$ + +hence, for any $1 \leq p, q \leq \infty$ , + +$$ +\left\| \sigma_ {\lambda} (x) - \sigma_ {\lambda} (y) \right\| _ {p} \leq \left(\sup _ {z} \| J (z) \| _ {q \rightarrow p}\right) \| x - y \| _ {q}. \tag {36} +$$ + +It suffices to bound $\sup_{z}\| J(z)\|_{q\to p}$ + +Step 1: Decomposition. With the standard basis $\{e_i\}_{i = 1}^d$ + +$$ +\operatorname {d i a g} (s) - s s ^ {\top} = \sum_ {i < j} s _ {i} s _ {j} \left(e _ {i} - e _ {j}\right) \left(e _ {i} - e _ {j}\right) ^ {\top}. \tag {37} +$$ + +Thus for any $u\in \mathbb{R}^d$ + +$$ +J (z) u = \lambda \sum_ {i < j} s _ {i} s _ {j} \left(u _ {i} - u _ {j}\right) \left(e _ {i} - e _ {j}\right), +$$ + +and by the triangle inequality and $\| e_i - e_j\| _p = 2^{1 / p}$ (with the convention $2^{1 / \infty} = 1$ ), + +$$ +\| J (z) u \| _ {p} \leq \lambda 2 ^ {1 / p} \sum_ {i < j} s _ {i} s _ {j} | u _ {i} - u _ {j} |. \tag {38} +$$ + +Step 2: Maximization over $s \in \Delta_d$ . Let $i_{\mathrm{min}} = \arg \min_i u_i$ , $i_{\mathrm{max}} = \arg \max_i u_i$ . Since + +$$ +| u _ {i} - u _ {j} | \leq | u _ {i _ {\mathrm {m a x}}} - u _ {i _ {\mathrm {m i n}}} | \mathrm {a n d} \sum_ {i < j} s _ {i} s _ {j} = \frac {1}{2} \bigg (1 - \sum_ {i = 1} ^ {d} s _ {i} ^ {2} \bigg), +$$ + +$$ +\sum_ {i < j} s _ {i} s _ {j} \left| u _ {i} - u _ {j} \right| \leq \frac {\left| u _ {i _ {\mathrm {m a x}}} - u _ {i _ {\mathrm {m i n}}} \right|}{2} \Big (1 - \sum_ {i = 1} ^ {d} s _ {i} ^ {2} \Big) \leq \frac {\left| u _ {i _ {\mathrm {m a x}}} - u _ {i _ {\mathrm {m i n}}} \right|}{4}. +$$ + +The last inequality is tight when $s_{i_{\mathrm{max}}} = s_{i_{\mathrm{min}}} = \frac{1}{2}$ and all other $s_i = 0$ . Therefore, + +$$ +\sup _ {s \in \Delta_ {d}} \sum_ {i < j} s _ {i} s _ {j} | u _ {i} - u _ {j} | = \frac {\left| u _ {i _ {\max }} - u _ {i _ {\min }} \right|}{4}. \tag {39} +$$ + +Because $\sigma_{\lambda}(\mathbb{R}^d)$ is the interior of $\Delta_d$ and the right-hand side of (38) is continuous in $s$ , the supremum over $z$ equals the supremum over $s \in \Delta_d$ . + +Step 3: Maximization over $\| u \|_q = 1$ . For any $a, b \in \mathbb{R}$ and $q \geq 1$ , $|a - b|^q \leq 2^{q-1} (|a|^q + |b|^q)$ . With $a = u_{i_{\max}}$ , $b = u_{i_{\min}}$ and $\| u \|_q = 1$ , + +$$ +\left| u _ {i _ {\max }} - u _ {i _ {\min }} \right| \leq 2 ^ {1 - 1 / q} \left(\left| u _ {i _ {\max }} \right| ^ {q} + \left| u _ {i _ {\min }} \right| ^ {q}\right) ^ {1 / q} \leq 2 ^ {1 - 1 / q}, \tag {40} +$$ + +with equality when $u$ has exactly two nonzero entries $u_{i_{\mathrm{max}}} = 2^{-1 / q}$ and $u_{i_{\mathrm{min}}} = -2^{-1 / q}$ . + +Step 4: Combine. From (38)-(40), + +$$ +\sup _ {z} \sup _ {\| u \| _ {q} = 1} \| J (z) u \| _ {p} \leq \lambda 2 ^ {1 / p} \cdot \frac {1}{4} \cdot 2 ^ {1 - 1 / q} = \lambda 2 ^ {- 1 + 1 / p - 1 / q}. +$$ + +Plug this into (36) to obtain the claim. The bound is tight by the equality cases noted in (39) and (40). + +Lemma 13. For any probability vectors $u, v \in \Delta_d$ , + +$$ +\left\| u - v \right\| _ {1} \leq \frac {1}{2} d _ {\mathrm {H i l b}} (u, v). \tag {41} +$$ + +Proof. By Cohen and Fausti (2024)[Theorem 5.1], + +$$ +\left\| u - v \right\| _ {1} \leq 2 \tanh \left(\frac {d _ {\mathrm {H i l b}} (u , v)}{4}\right). \tag {42} +$$ + +Since $\tanh (x)\leq x$ for $x\geq 0$ and $d_{\mathrm{Hilb}}(u,v)\geq 0$ , letting $x = \frac{d_{\mathrm{Hilb}}(u,v)}{4}$ gives + +$$ +\left\| u - v \right\| _ {1} \leq 2 \tanh \left(\frac {d _ {\mathrm {H i l b}} (u , v)}{4}\right) \tag {42} +$$ + +$$ +\leq 2 \left(\frac {d _ {\mathrm {h i l b}} (u , v)}{4}\right) = \frac {1}{2} d _ {\mathrm {h i l b}} (u, v). \tag {43} +$$ + +This proves (41). + +Lemma 14. For any $\mathbf{u},\mathbf{v}\in \mathbb{R}_{+}^{d}$ , the following holds: + +$$ +d _ {\mathrm {H i l b}} (\mathbf {u}, \mathbf {v}) = d _ {\mathrm {H i l b}} (\mathbf {v}, \mathbf {u}), +$$ + +$$ +d _ {\text {H i l b}} (\mathbf {u}, c \mathbf {v}) = d _ {\text {H i l b}} (\mathbf {u}, \mathbf {v}) \quad \forall c > 0, +$$ + +$$ +d _ {\operatorname {H i l b}} (\mathbf {c} \odot \mathbf {u}, \mathbf {c} \odot \mathbf {v}) = d _ {\operatorname {H i l b}} (\mathbf {u}, \mathbf {v}) \quad \forall \mathbf {c} \in \mathbb {R} _ {+} ^ {d}. +$$ + +Here, $\odot$ denotes the element-wise product. + +Proof. The first two properties are established in (Lemmens and Nussbaum, 2013, Lemma 2.1). For the third property, we have + +$$ +\begin{array}{l} d _ {\mathrm {H i l b}} (\mathbf {c} \odot \mathbf {u}, \mathbf {c} \odot \mathbf {v}) = \log \left(\max _ {i} \frac {c _ {i} u _ {i}}{c _ {i} v _ {i}} \cdot \max _ {i} \frac {c _ {i} v _ {i}}{c _ {i} u _ {i}}\right) \\ = \log \left(\max _ {i} \frac {u _ {i}}{v _ {i}} \cdot \max _ {i} \frac {v _ {i}}{u _ {i}}\right) \\ = d _ {\mathrm {H i l b}} (\mathbf {u}, \mathbf {v}). \\ \end{array} +$$ + +# B.2 Lemma used for the drift estimation using neural networks + +We provide an alternative expression of the Sinkhorn bridge drift estimator as a Markovian projection. Recall that $W_{|x_0,x_1}^{\varepsilon}$ is the Brownian bridge connecting $x_0$ and $x_1$ , so its marginal distribution at time $t \in [0,1]$ is a Gaussian distribution $W_{t|x_0,x_1}^{\varepsilon} = \mathcal{N}\big((1 - t)x_0 + tx_1,\varepsilon t(1 - t)I_d\big)$ . For any distributions $\mu'$ , $\nu' \in \mathcal{P}(\mathbb{R}^d)$ , and dual potentials $f \in L^{1}(\mu')$ , $g \in L^{1}(\nu')$ , we define + +$$ +d \pi (x _ {0}, x _ {1}) = \exp \left(\frac {f (x _ {0}) + g (x _ {1}) - \frac {1}{2} \| x _ {0} - x _ {1} \| _ {2} ^ {2}}{\varepsilon}\right) d (\mu^ {\prime} \otimes \nu^ {\prime}) (x _ {0}, x _ {1}). +$$ + +We here only consider the case where $\pi$ becomes a joint distribution $\pi \in \mathcal{P}(\mathbb{R}^d\times \mathbb{R}^d)$ , that is, $\int d\pi (x_0,x_1) = 1$ . Using $\pi_{m,n}^{(k)}$ , we define the mixture of Brownian bridge (reciprocal process): $\Pi = \int W_{|x_0,x_1}^{\varepsilon}d\pi (x_0,x_1)\in \mathcal{P}(\Omega)$ . + +Lemma 15. Under the above setting, it follows that + +$$ +\begin{array}{l} b (x _ {t}, t) := \mathbb {E} _ {X \sim \Pi} \left[ \frac {X _ {1} - x _ {t}}{1 - t} \Bigg | X _ {t} = x _ {t} \right] \\ = \frac {1}{1 - t} \left(- x _ {t} + \frac {\mathbb {E} _ {X _ {1} \sim \nu^ {\prime}} [ \gamma (X _ {1} , x _ {t}) X _ {1} ]}{\mathbb {E} _ {X _ {1} \sim \nu^ {\prime}} [ \gamma (X _ {1} , x _ {t}) ]}\right), \\ \end{array} +$$ + +$$ +w h e r e \gamma (x _ {1}, x _ {t}) = \exp \left(\left(g (x _ {1}) - \frac {1}{2 (1 - t)} \| x _ {1} - x _ {t} \| ^ {2}\right) / \varepsilon\right). +$$ + +Proof. Given $x_{t}$ ( $t < 1$ ), we denote by $\Pi(x_{1}|X_{t} = x_{t})$ the conditional density of $\Pi$ with respect to Lebesgue measure $dx_{1}$ . Note that the marginal density $\Pi_{t1}$ of $\Pi$ on time points $t$ and 1 with respect to $dx_{t}dx_{1}$ satisfies that for a measurable $A \subset \mathbb{R}^{d}$ + +$$ +\int_ {A} \Pi_ {t 1} \left(x _ {t}, x _ {1}\right) \mathrm {d} x _ {1} = \int W _ {t | x _ {0}, x _ {1}} ^ {\varepsilon} \left(x _ {t}\right) \pi (\mathrm {d} x _ {0}, A). +$$ + +Therefore, we get + +$$ +\begin{array}{l} \Pi \left(x _ {1} \mid X _ {t} = x _ {t}\right) \mathrm {d} x _ {1} \propto \left\{\int \exp \left(- \frac {1}{2 \varepsilon t (1 - t)} \| x _ {t} - (1 - t) x _ {0} - t x _ {1} \| ^ {2}\right) \right. \\ \left. \cdot \exp \left(\frac {g (x _ {1}) - \frac {1}{2} \| x _ {0} - x _ {1} \| _ {2} ^ {2}}{\varepsilon}\right) \mathrm {d} \mu^ {\prime} (x _ {0}) \right\} \cdot \mathrm {d} \nu^ {\prime} (x _ {1}) \\ \propto \left\{\int \exp \left(- \frac {1}{2 \varepsilon t (1 - t)} \left(t ^ {2} \| x _ {1} \| ^ {2} - 2 t x _ {t} ^ {\top} x _ {1} + 2 t (1 - t) x _ {0} ^ {\top} x _ {1}\right)\right) \right. \\ \left. \cdot \exp \left(\frac {g \left(x _ {1}\right)}{\varepsilon} - \frac {1}{2 \varepsilon} \left(\| x _ {1} \| ^ {2} - 2 x _ {0} ^ {\top} x _ {1}\right)\right) \mathrm {d} \mu^ {\prime} \left(x _ {0}\right) \right\} \cdot \mathrm {d} \nu^ {\prime} \left(x _ {1}\right) \\ = \exp \left(\frac {g (x _ {1})}{\varepsilon} - \frac {1}{2 \varepsilon (1 - t)} (\| x _ {1} \| ^ {2} - 2 x _ {t} ^ {\top} x _ {1})\right) \mathrm {d} \nu^ {\prime} (x _ {1}) \\ \propto \exp \left(\frac {1}{\varepsilon} \left(g \left(x _ {1}\right) - \frac {1}{2 (1 - t)} \left\| x _ {1} - x _ {t} \right\| ^ {2}\right)\right) d \nu^ {\prime} \left(x _ {1}\right). \\ \end{array} +$$ + +This means + +$$ +\Pi (x _ {1} | X _ {t} = x _ {t}) \mathrm {d} x _ {1} = \frac {\gamma (x _ {1} , x _ {t}) \mathrm {d} \nu^ {\prime} (x _ {1})}{\mathbb {E} _ {X _ {1} \sim \nu^ {\prime}} [ \gamma (X _ {1} , x _ {t}) ]}. +$$ + +Therefore, + +$$ +\begin{array}{l} b \left(x _ {t}, t\right) = \frac {1}{1 - t} \left(- x _ {t} + \mathbb {E} _ {X \sim \Pi} \left[ X _ {1} \mid X _ {t} = x _ {t} \right]\right) \\ = \frac {1}{1 - t} \left(- x _ {t} + \int x _ {1} \frac {\gamma \left(x _ {1} , x _ {t}\right)}{\mathbb {E} _ {X _ {1} \sim \nu^ {\prime}} \left[ \gamma \left(X _ {1} , x _ {t}\right) \right]} d \nu^ {\prime} \left(x _ {1}\right)\right) \\ = \frac {1}{1 - t} \left(- x _ {t} + \frac {\mathbb {E} _ {X _ {1} \sim \nu^ {\prime}} [ \gamma (X _ {1} , x _ {t}) X _ {1} ]}{\mathbb {E} _ {X _ {1} \sim \nu^ {\prime}} [ \gamma (X _ {1} , x _ {t}) ]}\right). \\ \end{array} +$$ + +![](images/5638843a581c4ad9ea79b8aad29ab91dda5f90baaa6da0660cbaf874527ebc0b.jpg) + +# C Omitted Experiments + +# C.1 Stopping and guidance + +Theorems 1 and 2 suggest that beyond a certain point additional Sinkhorn iterations are not beneficial. The total error decomposes into a "sampling error" and an "optimization error". The latter decreases + +Table 1: Comparison of $\int_0^{0.9} \mathrm{MSE}_{\mathrm{sample}}(m, n, t) dt$ across intrinsic dimensions $d_{\nu}$ . + +
dν∫0.9MSEsample(m,n,t)dt
50.031
100.138
200.562
301.178
+ +only through iterations, whereas the former does not. By the triangle inequality for the total variation distance, + +$$ +\mathbb {E} \left[ \mathrm {T V} ^ {2} \left(P _ {\infty , \infty} ^ {*}, P _ {m, n} ^ {k}\right) \right] \lesssim \mathbb {E} \left[ \mathrm {T V} ^ {2} \left(P _ {\infty , \infty} ^ {*}, P _ {m, n} ^ {*}\right) \right] + \mathbb {E} \left[ \mathrm {T V} ^ {2} \left(P _ {m, n} ^ {*}, P _ {m, n} ^ {k}\right) \right]. +$$ + +Once the second term on the right-hand side (optimization error) becomes no larger than the first term (estimation error), further iterations no longer yield a meaningful reduction of the total error. Hence it is reasonable to choose the stopping point as the smallest $k$ such that + +$$ +\mathbb {E} \left[ \mathrm {T V} ^ {2} \left(P _ {\infty , \infty} ^ {*}, P _ {m, n} ^ {*}\right) \right] \geq \mathbb {E} \left[ \mathrm {T V} ^ {2} \left(P _ {m, n} ^ {*}, P _ {m, n} ^ {k}\right) \right]. +$$ + +From Theorems 1 and 2, a sufficient condition for this is + +$$ +k \gtrsim \frac {\log \left(\frac {B}{m} + \frac {A + B}{n}\right)}{\log \left(\tanh \left(\frac {R ^ {2}}{\varepsilon}\right)\right)}, \quad A = \frac {1}{\tau (1 - \tau) ^ {d _ {\nu} + 1} \varepsilon^ {d _ {\nu} - 2} R ^ {4}}, \quad B = \left(\frac {\varepsilon^ {4}}{R ^ {6}} + d R ^ {2}\right) \left(\frac {R}{\varepsilon}\right) ^ {9 d _ {\nu} + 4}. +$$ + +Since $\log \left(\tanh \left(\frac{R^2}{\varepsilon}\right)\right) < 0$ , iterations are beneficial only when $\frac{B}{m} + \frac{A + B}{n} < 1$ . Equivalently, unless the sample size is large enough to make the numerator negative—roughly $(m \wedge n) \approx (A \vee B)$ —the reduction achievable by iterations is smaller than the estimation error, leading to wasted computational resources. + +In Figure 3, the fact that $b_{m,n}^{(1)}$ already captures the shape of $\nu_{n}$ to a certain extent is consistent with this reasoning. Increasing $k$ becomes meaningful only when $\mu_{m}$ and $\nu_{n}$ are sufficiently close to the true distributions $\mu$ and $\nu$ , namely when $m$ and $n$ are sufficiently large. + +# C.2 Intrinsic dimension + +Theorem 1 shows that the orders of the orders of $R$ and $\varepsilon$ depend on the intrinsic dimension $d_{\nu}$ rather than the ambient dimension $d$ . In this section, we verify that the error varies with $d_{\nu}$ under the following setup. + +We fix the ambient dimension at $d = 50$ , the regularization strength at $\varepsilon = 0.5$ , the evaluation-interval endpoint at $\tau = 0.9$ , and the sample sizes at $m = n = 10000$ . The distribution $\mu$ is sampled uniformly from the unit hypercube $[0,1]^d$ , while $\nu$ is sampled uniformly from the $d_{\nu}$ -dimensional manifold embedded in the unit sphere of radius 1, + +$$ +\left\{x \in \mathbb {R} ^ {d} \mid \| x \| _ {2} = 1, \forall i > d _ {\nu}: x _ {i} = 0 \right\}. +$$ + +By varying $d_{\nu}$ , we examine whether the convergence behavior of the estimation error with respect to the sample size depends on the manifold dimension. + +Specifically, for each $d_{\nu} \in \{5, 10, 20, 30\}$ we compare + +$$ +\int_ {0} ^ {0. 9} \mathrm {M S E} _ {\mathrm {s a m p l e}} (m, n, t) \mathrm {d} t. +$$ + +The results are reported in Table 1. An increasing trend of the error with larger manifold dimension is observed. + +# C.3 The role of epsilon + +We examine the numerical stability of the regularization parameter $\varepsilon$ in the Sinkhorn algorithm and provide selection guidelines in relation to the sample sizes $(m,n)$ . Using the same setup as + +Table 2: Representative $\varepsilon$ achieving the target error $\delta = 1.0$ for varying $m\wedge n$ (same setup as Experiments 4.1). + +
m ∧ nε
102.684 × 108
207.979 × 106
303.200 × 101
406.797 × 10-1
503.291 × 10-1
601.299 × 10-1
702.010 × 10-1
809.766 × 10-4
+ +Experiments 4.1, for each given $m,n$ we search for the smallest $\varepsilon$ that achieves the target error $\delta = 1.0$ . + +As $m \wedge n$ increases, the $\varepsilon$ required to meet the target error exhibits a two-phase behavior: a steep initial decrease followed by an asymptotically mild decay. Representative values are reported in Table 2. + +By Theorem 1, the $\varepsilon$ required to satisfy the target error $\delta$ obeys + +$$ +\varepsilon \gtrsim \max \left\{\left(\frac {R ^ {2}}{\delta (m \wedge n)}\right) ^ {1 / d _ {\nu}}, R ^ {\frac {3 d _ {\nu} + 4}{3 d _ {\nu} + 2}} \left(\frac {R ^ {2} + 1}{\delta (m \wedge n)}\right) ^ {1 / (9 d _ {\nu} + 6)} \right\}. +$$ + +This indicates a steep improvement in the small- $m\wedge n$ regime followed by logarithmically slow decay, which aligns with the empirical results. 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In this paper, we provide rigorous statistical guarantees for constant learning-rate SGD and ASGD in high-dimensional regimes. Our key innovation is to transfer powerful tools from high-dimensional time series to online learning. Specifically, by viewing SGD as a nonlinear autoregressive process and adapting existing coupling techniques, we prove the geometric-moment contraction of high-dimensional SGD for constant learning rates, thereby establishing asymptotic stationarity of the iterates. Building on this, we derive the $q$ -th moment convergence of SGD and ASGD for any $q \geq 2$ in general $\ell^s$ -norms, and, in particular, the $\ell^{\infty}$ -norm that is frequently adopted in high-dimensional sparse or structured models. Furthermore, we provide sharp high-probability concentration analysis which entails the probabilistic bound of high-dimensional ASGD. Beyond closing a critical gap in SGD theory, our proposed framework offers a novel toolkit for analyzing a broad class of high-dimensional learning algorithms. + +# 1 Introduction + +Stochastic gradient descent (SGD) has been a cornerstone in large-scale machine learning since the seminal work by Robbins and Monro [1951]. It is especially efficient in high-dimensional and overparameterized settings where the number of unknown parameters can exceed the number of training samples [Arpit et al., 2017, Zhang et al., 2017, He et al., 2016]. SGD can also be combined with regularization techniques such as dropout to prevent overfitting in large networks [Krizhevsky et al., 2012, Srivastava et al., 2014]. Despite the vast amount of theoretical work on SGD, generalization bounds of SGD in high-dimensional regimes remain limited [Garrigos and Gower, 2023]. Considering a strongly convex objective function, we provide statistical guarantees for constant learning-rate SGD and its Ruppert-Polyak averaged variant (ASGD) [Ruppert, 1988, Polyak and Juditsky, 1992] in high-dimensional settings. + +Specifically, we consider a general optimization problem + +$$ +\boldsymbol {\beta} ^ {*} \in \arg \min _ {\boldsymbol {\beta} \in \mathbb {R} ^ {d}} G (\boldsymbol {\beta}), \text {w h e r e} \boldsymbol {\beta} \mapsto G (\boldsymbol {\beta}) := \mathbb {E} _ {\boldsymbol {\xi} \sim_ {\Pi}} g (\boldsymbol {\beta}, \boldsymbol {\xi}), \tag {1} +$$ + +$g(\cdot)$ is the noise-perturbed measurement of $G(\cdot)$ , and $\xi$ denotes a random element sampled from some unknown distribution $\Pi$ . Given i.i.d. random samples $\xi_1, \xi_2, \ldots$ and some initialization $\beta_0 \in \mathbb{R}^d$ , the $k$ -th SGD iteration is + +$$ +\beta_ {k} = \beta_ {k - 1} - \alpha \nabla g \left(\beta_ {k - 1}, \boldsymbol {\xi} _ {k}\right), \quad k = 1, 2, \dots , \tag {2} +$$ + +for some constant learning rate $\alpha > 0$ , and $\nabla g(\beta, \xi) = \nabla_{\beta} g(\beta, \xi)$ the stochastic gradient with respect to $\beta$ . For $k \geq 1$ , the ASGD variant is defined by + +$$ +\bar {\beta} _ {k} = \frac {1}{k} \sum_ {i = 1} ^ {k} \beta_ {i}. \tag {3} +$$ + +We are interested in the high-dimensional setting where the parameter dimension $d$ can be very large. Here, a notable divide between empirical success and theoretical understanding is that practitioners often employ a large constant learning rate $\alpha$ in (2) to accelerate convergence in high-dimensional problems [Wu et al., 2018, Cohen et al., 2021, Cai et al., 2024]. However, such choices can induce pronounced non-stationarity in the SGD iterates $\{\beta_k\}_{k \in \mathbb{N}}$ which will not converge to a point but oscillates around the mean of a stationary distribution. In other words, $\beta_k$ is non-stationary but asymptotically stationary, which converges only in distribution as $k \to \infty$ , while the mean of this distribution differs from the exact minimizer $\beta^*$ due to the non-diminishing bias of order $O(\alpha)$ [Dieuleveut et al., 2020, Merad and Gaiffas, 2023]. Classical theory mostly relies on decaying learning rates [Zhang, 2004, Nemirovski et al., 2009, Jentzen and von Wurstemberger, 2020, Shi et al., 2023]. To address the non-stationarity issue, we apply powerful tools from nonlinear time series analysis [Wu and Shao, 2004] to online learning, particularly by adapting the coupling techniques to show the geometric-moment contraction of SGD for constant learning rates. Specifically, for any two SGD sequences $\{\beta_k\}_{k \in \mathbb{N}}$ and $\{\beta_k'\}_{k \in \mathbb{N}}$ that share the same random samples but have different initial vectors $\beta_0$ and $\beta_0'$ , we show in Theorem 1 that for all sufficiently small constant learning rates $\alpha$ , the initialization is forgotten exponentially fast in the sense that + +$$ +\left(\mathbb {E} \left| \boldsymbol {\beta} _ {k} - \boldsymbol {\beta} _ {k} ^ {\prime} \right| _ {s} ^ {q}\right) ^ {1 / q} \leq r _ {\alpha , s, q} ^ {k} \left| \boldsymbol {\beta} _ {0} - \boldsymbol {\beta} _ {0} ^ {\prime} \right| _ {s} \quad \text {h o l d s f o r a l l} k \in \mathbb {N}, \tag {4} +$$ + +for contraction speed $0 \leq r_{\alpha, s, q} < 1$ , and $|\cdot|_s$ the $\ell^s$ -norm, that is, + +$$ +\left| \left(v _ {1}, \dots , v _ {d}\right) ^ {\top} \right| _ {s} = \left(\sum_ {i = 1} ^ {d} \left| v _ {i} \right| ^ {s}\right) ^ {1 / s}, s \geq 1. +$$ + +This asserts the existence of a limiting stationary distribution of $\beta_{k}$ as $k\to \infty$ , thereby facilitating a systematic convergence theory of SGD even in nonlinear, overparameterized models. + +Building on this new framework, we provide non-asymptotic bounds for higher-order moments of the SGD error in general $\ell^s$ -norms for any finite $s \geq 2$ beyond the usual $\ell^2$ -norm, extendable to max-norm $\ell^\infty$ by choosing $s \approx \log(d)$ . Notably, the $\ell^\infty$ -norm is frequently adopted in high-dimensional sparse or structured estimation [Wainwright, 2019]. See for instance, the max-norm convergence of the Lasso and Dantzig selector [Lounici, 2008]; the pivotal method for sup-norm bounds of the square-root Lasso [Belloni et al., 2011]; and the max-norm error control for confidence intervals in high-dimensional regression problems [Javanmard and Montanari, 2013]. In stochastic-approximation (SA), Wainwright [2019] derived $\ell^\infty$ -norm bounds for Q-learning with decaying learning rates; Chen et al. [2023] derived maximal concentration bounds for SA under arbitrary norms with decaying learning rates and with contraction as an assumption; Agarwal et al. [2012] considered high-dimensional SA for strongly convex objectives with a sparse optimum, but using decaying learning rates and restricting the tails of stochastic gradients to be sub-Gaussian. To date, all the existing results are restricted to low-dimensional settings or decaying learning rates and do not carry over to overparameterized models with constant learning rates. To address this gap, we derive a sharp high-dimensional moment inequality (see Lemma 2) valid for a broad class of learning problems, delivering explicit non-asymptotic bounds of $\mathbb{E}|\beta_k - \beta^*|_s^q$ and its ASGD variant for any $q$ , $s \geq 2$ with mild conditions, together with matching complexity guarantees, i.e., given some target error $\varepsilon > 0$ (see Proposition 2), the required number of iterations $k$ such that + +$$ +\mathbb {E} | \bar {\boldsymbol {\beta}} _ {k} - \boldsymbol {\beta} ^ {*} | _ {s} ^ {q} \leq \varepsilon . +$$ + +Although moment bounds capture average-case performance, a single execution of (A)SGD in practice demands high-probability guarantees [Valiant, 1984, Vapnik, 2000, Bach and Moulines, 2013, Durmus et al., 2021, Zhong et al., 2024]. Recent advances include a generic high-probability framework for both convex and nonconvex SGD with sub-Gaussian gradient noises [Liu et al., 2023], high-probability rates for clipped-SGD with heavy-tailed noises [Nguyen et al., 2023], and high-probability guarantees for nonconvex stochastic approximation via robust gradient clipping [Li and Liu, 2022]. However, these established high-probability bounds focus again on decaying learning rates and low dimension. Moreover, early work primarily addressed light-tailed noises where the gradients are bounded or have exponential-type moments [Nemirovski et al., 2009, Rakhlin et al., 2012, Ghadimi and Lan, 2013, Cardot et al., 2017, Harvey et al., 2019, Mou et al., 2020, Chen et al., 2023]. For the cases that only admit a polynomial tail with finite $q$ -th moment, Lou et al. [2022] were the first to derive a Nagaev-type inequality [Nagaev, 1979] for low-dimensional ASGD. The rate was shown to be optimal but their bound heavily relies on the linearity of gradients and is only suitable for decaying learning rates. By leveraging a dependency-adjusted functional dependence measure in high-dimensional time series [Zhang and Wu, 2017], we derive a high-probability concentration bounds for high-dimensional ASGD with constant learning rates. Given a tolerance level $\delta \in (0,1)$ and a target error $\varepsilon > 0$ , we provide bounds for the required number of iterations $k$ to guarantee that + +$$ +\mathbb {P} \left(\left| \bar {\boldsymbol {\beta}} _ {k} - \boldsymbol {\beta} ^ {*} \right| _ {s} \leq \varepsilon\right) \geq 1 - \delta . +$$ + +This tail-decay result (see Eq. (10)) is proved via a new Fuk-Nagaev-type inequality (see Theorem 4) and complements our moment and complexity characterizations of large-step stochastic optimization. + +# 1.1 Our Contributions + +This paper contributes to theoretical advancements for understanding constant learning-rate SGD and its averaged variant (ASGD) in the challenging high-dimensional regime. Our main technical innovations and results include: + +(1) Handling Constant Learning Rates in High Dimensions. In practice, large-scale machine learning models commonly deploy fixed, large learning rates to speed up optimization in high-dimensional settings. To address this, we introduce novel coupling techniques inspired by high-dimensional nonlinear time series and establish the asymptotic stationarity of the SGD iterates with arbitrary initialization (Section 2). +(2) Generalized Moment Convergence in $\ell^s$ - and $\ell^{\infty}$ -Norms. By deriving a sharp high-dimensional moment inequality, we establish explicit, non-asymptotic $q$ -th moment bounds for arbitrary $\ell^s$ -norms of (A)SGD iterates for any $q \geq 2$ and even integers $s$ , generalizing previous theory primarily focusing on mean squared error (MSE) convergence with $q = s = 2$ . Our results extend naturally to the max-norm case (i.e., $\ell^{\infty}$ ) by selecting $s \approx \log(d)$ , that is essential for modern sparse and structured estimation in high-dimensional data (Section 3). +(3) High-Probability Tail Bounds. While average-case (moment) bounds are informative, single runs require tail guarantees. We derive the first high-probability concentration bounds for ASGD in high-dimensional settings with constant learning rates. By developing a tight Fuk-Nagaev-type inequality using the coupling techniques in nonlinear time series, we control the algorithmic complexity required to achieve targeted accuracy with high confidence (Section 4). + +# 1.2 Related Works + +Stochastic Gradient Descent and its Variants. The SGD algorithm can be traced back to Robbins and Monro [1951], Kiefer and Wolfowitz [1952]. Popular SGD variants include Nesterov's accelerated gradient [Nesterov, 1983], AdaGrad [Duchi et al., 2011], AdaDelta [Zeiler, 2012], Adam [Kingma and Ba, 2014], AMSGrad [Reddi et al., 2018], AdamW [Loshchilov and Hutter, 2018], SAG [Schmidt et al., 2017], SVRG [Johnson and Zhang, 2013], SARAH [Nguyen et al., 2017], SPIDER [Fang et al., 2018] and Katyusha [Allen-Zhu, 2017]. The theoretical foundations of SGD under decaying learning rates were established in the early studies by [Blum, 1954, Dvoretzky, 1956, Sacks, 1958], with stronger almost-sure guarantees by Fabian [1968], Robbins and Siegmund [1971], Ljung [1977], Lai [2003], Wang and Gao [2010]. Existing works for smooth, strongly-convex objectives with decaying step sizes include Ruppert [1988], Polyak and Juditsky [1992], Nemirovski et al. [2009], Bach and Moulines [2013], Rakhlin et al. [2012], Mertikopoulos et al. [2020] among others. Despite the rich literature on SGD, the theoretical understanding in high-dimensional settings remains limited. + +Exceptions are Paquette et al. [2021, 2022] who study high-dimensional SGD for the least-squares loss. + +Constant Learning Rate. In high-dimensional scenarios, constant learning rates prevail due to simpler tuning procedures and faster convergence [Wang et al., 2022]. More recent theoretical and empirical studies of large-step SGD include Wu et al. [2018], Cohen et al. [2021] and the very recent Cai et al. [2024], which formalize the resurgence of constant-step methods in modern machine learning. A useful way to analyze constant-step SGD is to treat its iterates as a time-homogeneous Markov chain [Pflug, 1986], which makes it possible to characterize its long-run behavior and stationary law. However, previous works only derived convergence in Wasserstein distance [Dieuleveut et al., 2020, Merad and Gaiffas, 2023]. Such convergence in probability measures can hardly provide refined (non)-asymptotics such as higher-moment convergence and concentration inequalities, and seems nontrivial to extend to high-dimensional regimes. + +High-Dimensional Nonlinear Time Series. An alternative approach for constant learning-rate SGD is to view it as an iterated random function [Dubins and Freedman, 1966, Barnsley and Demko, 1985, Diaconis and Freedman, 1999, Diaconis and Duflo, 2000], or a nonlinear autoregressive (AR) process. This interpretation facilitates the theory of online learning with non-stationarity and complex dependency structures; see, for example, the recent work by Li et al. [2024c] on SGD with dropout regularization building on the GMC framework [Wu and Shao, 2004]. To extend this systematic theory to high-dimensional settings, we adapt the coupling techniques in time series [Wu, 2005, 2007, 2009, 2011, Xiao and Wu, 2012, Berkes et al., 2014, Wu and Wu, 2016, Karmakar and Wu, 2020], especially the ones for high-dimensional regimes [Zhang and Wu, 2017, 2021, Li et al., 2024a] to online learning algorithms. + +# 1.3 Notation + +Denote column vectors in $\mathbb{R}^d$ by lowercase bold letters $\pmb{x} = (x_{1},\dots ,x_{d})^{\top}$ and the $\ell^s$ -norm of $\pmb{x}$ by $|\pmb{x}|_s = (\sum_{i = 1}^d |x_i|^s)^{1 / s}$ , $s\geq 1$ . Write $\pmb{x}^{\odot s} = (x_1^s,\ldots ,x_d^s)^\top$ . The expectation and covariance of random vectors are respectively denoted by $\mathbb{E}[\cdot ]$ and $\mathrm{Cov}(\cdot)$ . For $q > 0$ and a random variable $X$ , we write $X\in \mathcal{L}^q$ iff $\| X\| _q = [\mathbb{E}(|X|^q)]^{1 / q} < \infty$ . We denote matrices by uppercase letters. Given matrices $A$ and $B$ of compatible dimension, their matrix product is denoted by juxtaposition. Write $A^{\top}$ for the transpose of $A$ and $I_{d}$ for $d\times d$ identity matrix. For two positive number sequences $(a_{n})$ and $(b_{n})$ , we say $a_{n} = O(b_{n})$ (resp. $a_{n}\asymp b_{n}$ ) if there exists $c > 0$ such that $a_{n} / b_{n}\leq c$ (resp. $1 / c\leq a_n / b_n\leq c$ ) for all large $n$ . Let $(x_{n})$ and $(y_{n})$ be two sequences of random variables. Write $x_{n} = O_{\mathbb{P}}(y_{n})$ if for $\forall \epsilon >0$ , there exists $c > 0$ such that $\mathbb{P}(|x_n / y_n|\leq c) > 1 - \epsilon$ for all large $n$ . + +
NotationDefinitionReferenceIndex Range
β*minimizer of the loss function G(β)Eq. (1)/
βkSGD iteratesEq. (2)k ∈ N
β°kstationary SGD iteratesThm. (2)k ∈ Z
βkASGD iteratesEq. (3)k ∈ N
β°kstationary ASGD iteratesEq. (9)k ∈ Z
+ +Table 1: List of the sequences defined in the paper. + +# 2 Convergence of SGD to a Stationary Distribution + +In this section, we establish the GMC property of high-dimensional SGD with constant learning rates. Our technique is to construct a smooth surrogate for the non-differentiable $\ell^{\infty}$ -norm via the $\ell^s$ -norm, so that standard gradient-based tools become available. We defer the technical details to Section 6.1. Furthermore, we provide a novel high-dimensional moment inequality (see Section 6.2) and use it to derive the dimension-dependent range of the constant learning rate that guarantees the contraction. + +We first impose the following assumptions on the objective function and the stochastic gradients. + +Assumption 1 (Coercivity). Assume that for any sequence $\beta_{1},\beta_{2},\ldots$ with $|\beta_n|_s\to \infty$ the loss function $G(\cdot)$ in (1) satisfies $\lim_{n\to \infty}G(\beta_n) = \infty$ + +Assumption 2 (Strong Convexity - $\ell^s$ -norm). Let $s \geq 2$ be an even integer and write $\pmb{v}^{\odot s} := (v_1^s, \ldots, v_d^s)^\top$ for a vector $\pmb{v} = (v_1, \ldots, v_d)^\top$ . Assume there exists $\mu > 0$ such that + +$$ +\left\langle \left(\boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime}\right) ^ {\odot (s - 1)}, \nabla G (\boldsymbol {\beta}) - \nabla G \left(\boldsymbol {\beta} ^ {\prime}\right) \right\rangle \geq \mu | \boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime} | _ {s} ^ {s}, \quad \text {f o r a l l} \boldsymbol {\beta}, \boldsymbol {\beta} ^ {\prime} \in \mathbb {R} ^ {d}. +$$ + +In Lemma 3 in the supplementary materials, we show that under Assumptions 1 and 2, a unique global minimizer $\beta^{*}$ exists for the optimization problem (1). When $s = 2$ , Assumption 2 reduces to the regular strong convexity frequently adopted in the literature [Polyak and Juditsky, 1992, Moulines and Bach, 2011, Dieuleveut et al., 2020, Mies and Steland, 2023]. For general $s$ and the linear regression model, Section 8.2 in the supplementary material interprets the $\ell^s$ -type strong convexity assumption via the $\ell^s$ -norm induced matrix norm. As different norms are involved, there does not seem to be an apparent relationship between the classical strong convexity and the case $s > 2$ . + +Assumption 3 (Stochastic Lipschitz Continuity - $\ell^s$ -norm). Let $\beta^{*}$ be the global minimizer. For some $q \geq 2$ and an even integer $s \geq 2$ , assume that + +$$ +M _ {s, q} := \left(\mathbb {E} | \nabla g (\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi}) | _ {s} ^ {q}\right) ^ {1 / q} < \infty . +$$ + +Further assume there exists a constant $L_{s,q} > 0$ such that + +$$ +\left(\mathbb {E} \big | \nabla g (\boldsymbol {\beta}, \boldsymbol {\xi}) - \nabla g \big (\boldsymbol {\beta} ^ {\prime}, \boldsymbol {\xi} \big) \big | _ {s} ^ {q}\right) ^ {1 / q} \leq L _ {s, q} | \boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime} | _ {s}, \quad f o r a l l \boldsymbol {\beta}, \boldsymbol {\beta} ^ {\prime} \in \mathbb {R} ^ {d}. +$$ + +Later we will choose $s = O(\log(d))$ to bound the max-norm. The above defined Lipschitz constant $L_{s,q}$ and the moments $M_{s,q}$ will then grow as $d$ increases. Taking linear regression as an example, we investigate the dimension dependence of $L_{s,q}$ and $M_{s,q}$ in Section 8.2. All bounds in this work will contain the explicit dependence on $(L_{s,q}, M_{s,q})$ . + +We now state the first main result of this paper, which plays a crucial role in establishing moment convergence and tail probability results in the following sections. The statement quantifies the exponential rate at which the initialization $\beta_0$ will be forgotten and the SGD iterates $\beta_{k}$ converges to a stationary distribution $\pi_{\alpha}$ . + +Theorem 1 (Convergence of SGD to stationary distribution). Suppose that Assumptions 1-3 hold for some $\mu >0$ , $q\geq 2$ and even integer $s\geq 2$ . Given a constant learning rate + +$$ +0 < \alpha < \alpha_ {s, q} := \frac {2 \mu}{\max \{q , s \} L _ {s , q} ^ {2}}, \tag {5} +$$ + +for any two $d$ -dimensional SGD sequences $\{\beta_k(\alpha)\}_{k\in \mathbb{N}}$ and $\{\beta_k'(\alpha)\}_{k\in \mathbb{N}}$ sharing the same i.i.d. noise injections $\{\xi_k\}_{k\geq 1}$ but possibly different initializations $\beta_0,\beta_0'\in \mathbb{R}^d$ , the geometric-moment contraction (GMC) + +$$ +\left\| \left| \beta_ {k} - \beta_ {k} ^ {\prime} \right| _ {s} \right\| _ {q} \leq r _ {\alpha , s, q} ^ {k} \left| \beta_ {0} - \beta_ {0} ^ {\prime} \right| _ {s}, \quad f o r a l l k = 0, 1, \dots . +$$ + +holds with contraction constant + +$$ +r _ {\alpha , s, q} = 1 - 2 \mu \alpha + \max \{q, s \} L _ {s, q} ^ {2} \alpha^ {2} < 1. \tag {6} +$$ + +Moreover, there exists a unique stationary distribution $\pi_{\alpha}$ with a finite $q$ -th moment, that is, $\int |\pmb{u}|_s^q \pi_{\alpha}(d\pmb{u}) < \infty$ , such that + +$$ +\beta_ {k} \Rightarrow \pi_ {\alpha}, \quad a s k \rightarrow \infty . +$$ + +Equivalently, for any continuous function $f \in \mathcal{C}(\mathbb{R}^d)$ with $|f|_{\infty} < \infty$ , + +$$ +\mathbb {E} \left[ f \left(\beta_ {k}\right)\right]\rightarrow \int f (\boldsymbol {u}) \pi_ {\alpha} (d \boldsymbol {u}), \quad a s k \rightarrow \infty . +$$ + +The result generalizes Li et al. [2024b] to large dimension $d$ and extends the $\ell^2$ -type GMC based on Lemma 9 to general $\ell^s$ -norms. Moreover, choosing $s = s_d$ with + +$$ +s _ {d} := 2 \min \left\{\ell \in \mathbb {N}: 2 \ell > \log (d) \right\}, \tag {7} +$$ + +and using the inequality + +$$ +\left| \boldsymbol {x} \right| _ {\infty} \leq \left| \boldsymbol {x} \right| _ {s _ {d}} \leq d ^ {1 / s _ {d}} \left| \boldsymbol {x} \right| _ {\infty} \leq e \left| \boldsymbol {x} \right| _ {\infty}, \tag {8} +$$ + +shows the equivalence of the $\ell^{s_d}$ - and $\ell^{\infty}$ -norms. Consequently, by choosing $s = s_d$ , the previous theorem can also be used to derive the GMC property with respect to the $\ell^{\infty}$ -norm. + +# 3 Convergence of High-Dimensional SGD and ASGD + +In this section, we derive convergence rates for the moments of the last iterate $\mathbb{E}|\beta_k - \beta^*|_{\infty}^q$ and the moments of the averaged SGD. + +# 3.1 Convergence of SGD + +Proposition 1. If Assumptions 1-3 hold for some $q \geq 2$ , an even integer $s \geq 2$ , and a constant $M_{s,q}$ then, + +$$ +\left\| \left| \beta_ {k} - \beta^ {*} \right| _ {s} \right\| _ {q} ^ {2} \leq \left(1 - 2 \alpha \mu + 7 \max \{q, s \} \alpha^ {2} L _ {s, q} ^ {2}\right) \left\| \left| \beta_ {k - 1} - \beta^ {*} \right| _ {s} \right\| _ {q} ^ {2} + 3 \max \{q, s \} \alpha^ {2} M _ {s, q} ^ {2}, +$$ + +for all $k \geq 1$ . The same inequality holds if $\beta_{k}$ is replaced by the stationary SGD iterates $\beta_{k}^{\circ} \sim \pi_{\alpha}$ , $k \geq 1$ . + +Theorem 2 (Moment convergence of SGD). Let $0 < \alpha < \alpha_{s,q} / 7$ with $\alpha_{s,q}$ as defined in (5). Suppose that Assumptions 1-3 hold for $q \geq 2$ and even integer $s \geq 2$ . Then for the stationary SGD iterates $\beta_k^\circ \sim \pi_\alpha$ , + +$$ +\left\| \left| \beta_ {k} ^ {\circ} - \beta^ {*} \right| _ {s} \right\| _ {q} = O \left(M _ {s, q} \sqrt {\max \{q , s \} \alpha}\right) \quad f o r a l l k \geq 1 +$$ + +and for the SGD iterate $\beta_{k}$ with arbitrary initialization $\beta_0$ + +$$ +\left| \left| \left| \beta_ {k} - \beta^ {*} \right| _ {s} \right| \right| _ {q} = O \left(M _ {s, q} \sqrt {\max \{q , s \} \alpha} + r _ {\alpha , s, q} ^ {k} \| \left| \beta_ {0} - \beta_ {0} ^ {\circ} \right| _ {s} \| _ {q}\right) \quad f o r a l l k \geq 1. +$$ + +Choosing $s = s_d$ in (7) yields a bound with respect to the $\ell^{\infty}$ -norm. + +# 3.2 Convergence of Ruppert-Polyak Averaged SGD + +Consider now the Ruppert-Polyak Averaged SGD (ASGD) $\bar{\beta}_k = \frac{1}{k}\sum_{i = 1}^k\beta_i$ as defined in (3). For the initialization $\beta_0^\circ \sim \pi_\alpha$ , define the stationary ASGD sequence + +$$ +\bar {\beta} _ {k} ^ {\circ} = \frac {1}{k} \sum_ {i = 1} ^ {k} \beta_ {i} ^ {\circ}, \quad k \geq 1. \tag {9} +$$ + +Theorem 3. Consider the ASGD sequence $\{\bar{\beta}_k\}_{k\geq 1}$ . Suppose that Assumptions 1-3 hold with some $q\geq 2$ and even integer $s = s_d$ in (7), the conditions of Theorem 8 hold and the learning rate satisfies $\alpha \in (0,\alpha_{s_d,q})$ with $\alpha_{s_d,q}$ defined in (5). For any $k\geq 1$ and some universal constants $C_1,C_2,C_3 > 0$ + +$$ +\begin{array}{l} \left\| \left| \bar {\boldsymbol {\beta}} _ {k} - \boldsymbol {\beta} ^ {*} \right| _ {\infty} \right\| _ {q} \leq C _ {1} \left\{\underbrace {\sqrt {\frac {c _ {q} s _ {d}}{k}} M _ {s _ {d} , q} \left(L _ {s _ {d} , q} \sqrt {\alpha \max \{q , s _ {d} \}} + 1\right)} _ {\text {s t o c h a s t i c v a r i a n c e}} \right\} \\ + C _ {2} \Big \{\underbrace {\frac {1}{k (1 - r _ {\alpha , s _ {d} , q})} \| | \boldsymbol {\beta} _ {0} - \boldsymbol {\beta} _ {0} ^ {\circ} | _ {\infty} \| _ {q}} _ {\text {i n i t i a l i z a t i o n b i a s}} \Big \} + C _ {3} \Big \{\underbrace {M _ {s _ {d} , q} ^ {2} \max \{q , s _ {d} \} \alpha d ^ {\frac {q}{q - 1} \cdot (1 - \frac {2}{s _ {d}})}} _ {\text {b i a s o f c o n s t a n t l e a r n i n g r a t e}} \Big \}. \\ \end{array} +$$ + +Proposition 2 (Complexity bound). Under the assumptions of Theorem 3, let $\Delta_0 = ||\beta_0 - \beta_0^\circ|_\infty||_q$ + +$$ +V = L _ {s _ {d}, q} M _ {s _ {d}, q} \sqrt {\max \{q , s _ {d} \}} + M _ {s _ {d}, q}, B = M _ {s _ {d}, q} ^ {2} \max \{q, s _ {d} \} d ^ {\frac {q}{q - 1} \left(1 - \frac {2}{s _ {d}}\right)}. +$$ + +Given a tolerance $\varepsilon > 0$ , + +$$ +\alpha \leq \min \left\{\frac {\varepsilon}{3 C _ {3} B}, \frac {\alpha_ {s _ {d} , q}}{7} \right\}, \quad a n d k \geq \max \left\{\frac {9 C _ {1} ^ {2} c _ {q} s _ {d} V ^ {2} \alpha}{\varepsilon^ {2}}, \frac {3 C _ {2} \Delta_ {0}}{\alpha \varepsilon} \right\}, +$$ + +we have $\| |\bar{\beta}_k - \beta^*|_{\infty}\| _q\leq \varepsilon$ + +A proof outline is given in Section 6.3 and the full proof is deferred to the supplementary material. The sharpest complexity bound of SA for $\ell^{\infty}$ -norm known to date was derived by Wainwright [2019] proving that the number of iterations required to obtain an $\varepsilon$ -accurate solution of Q-learning scales as $(1 - \gamma)^{-4} \cdot \varepsilon^{-2}$ with the discount factor $\gamma$ . In Proposition 2, our complexity bound for SGD is also of the order of $O(1 / \varepsilon^{2})$ if the dimension $d$ is fixed, which is consistent with the degenerate Q-learning case in Wainwright [2019]. The derived result allows to determine the dependence on the dimension $d$ . + +# 4 Sharp Concentration and Gaussian Approximation + +Via the following tail probability inequality for the averaged SGD estimator $\bar{\beta}_k$ , one can further derive high-probability concentration bound of $|\bar{\beta}_k - \beta^*|_{\infty}$ . Recall that $s_d = 2\min \{\ell \in \mathbb{N} : 2\ell > \log(d)\}$ . + +Theorem 4 (Fuk-Nagaev inequality). Under the conditions of Theorem 3, for any $z > 0$ , we have + +$$ +\mathbb {P} \big (| \bar {\boldsymbol {\beta}} _ {k} - \boldsymbol {\beta} ^ {*} | _ {\infty} > z \big) \lesssim \frac {\| | \boldsymbol {\beta} _ {0} - \boldsymbol {\beta} _ {0} ^ {\circ} | _ {\infty} \| _ {q} ^ {q}}{(k \alpha z) ^ {q}} + \frac {(\log d) ^ {\frac {3 q}{2}} (\log k) ^ {1 + 2 q} M _ {s _ {d} , q} ^ {q}}{z ^ {q} k ^ {q - 1} \alpha^ {q / 2 - 1}} + \exp \left(- \frac {C k z ^ {2} \alpha^ {1 - 2 / q}}{M _ {s _ {d} , q} ^ {2} \log d}\right), +$$ + +where the constants in $\lesssim$ are independent of $k, d, s$ and $\alpha$ . + +As an immediate consequence of Theorem 4, we obtain a sharp high-probability upper bound for $|\tilde{\beta}_k - \beta^*|_{\infty}$ , that is, for any given tolerance rate $\delta \in (0,1)$ , with at least probability $1 - \delta$ , we have + +$$ +\left. \left| \bar {\boldsymbol {\beta}} _ {k} - \boldsymbol {\beta} ^ {*} \right| _ {\infty} = O \left(\frac {\left\| \left| \boldsymbol {\beta} _ {0} - \boldsymbol {\beta} _ {0} ^ {\circ} \right| _ {\infty} \right\| _ {q} ^ {q}}{k \alpha \delta^ {1 / q}} + \frac {(\log d) ^ {3 / 2} (\log k) ^ {1 / q + 2} M _ {s _ {d} , q}}{k ^ {1 - 1 / q} \alpha^ {1 / 2 - 1 / q} \delta^ {1 / q}} + \sqrt {\frac {M _ {s _ {d} , q} ^ {2} \log d \log (1 / \delta)}{k \alpha^ {1 - 2 / q}}}\right). \right. \tag {10} +$$ + +Notably, if the $q$ -th moment of the gradient noise is finite ( $M_{s_d,q} < \infty$ ), the second term of the right hand side, involving $k^{1 - q}$ , is generally unimprovable [Nagaev, 1979, Lou et al., 2022]. + +The distribution convergence for the high-dimensional ASGD relies on the following result. Let $M_{2,q}$ be as defined in Assumption 3. + +Theorem 5 (Gaussian approximation). Consider stationary SGD iterates $\beta_k^\circ \sim \pi_\alpha$ with $\pi_\alpha$ as defined in Theorem 1, initialization $\beta_0^\circ \sim \pi_\alpha$ , and learning rate $\alpha \in (0, \alpha_{s_d,q})$ . Suppose that Assumptions 1-3 hold for some $q > 2$ . Then, on a potentially different probability space, and for a number of iterations $T$ satisfying $d \leq cT$ , where $c > 0$ is some constant, there exist random vectors $\{\tilde{\beta}_k\}_{k=1}^T \stackrel{\mathcal{D}}{=} \{\beta_k^\circ\}_{k=1}^T$ and independent Gaussian random vectors $\{\pmb{z}_k\}_{k=1}^T$ with mean zero and covariance matrix + +$$ +\Xi = \sum_ {k = - \infty} ^ {\infty} \operatorname {C o v} \left(\beta_ {0} ^ {\circ}, \beta_ {k} ^ {\circ}\right), \tag {11} +$$ + +such that + +$$ +\left(\mathbb {E} \max _ {k \leq T} \left| \frac {1}{\sqrt {k}} \sum_ {i = 1} ^ {k} \left[ \left(\tilde {\boldsymbol {\beta}} _ {i} - \mathbb {E} \left[ \boldsymbol {\beta} _ {1} ^ {\circ} \right]\right) - \boldsymbol {z} _ {i} \right] \right| _ {2} ^ {2}\right) ^ {1 / 2} \leq C _ {\alpha , q} ^ {*} M _ {2, q} \sqrt {d \log (T)} \left(\frac {d}{T}\right) ^ {\frac {q - 2}{6 q - 4}}, \tag {12} +$$ + +with $C_{\alpha, q}^{*}$ a constant that only depends on $c$ , the learning rate $\alpha$ , and the moment index $q$ . + +For diverging moment index $q \to \infty$ , the Gaussian approximation rate in (12) approaches the rate $O(\sqrt{\log(T)}(d^4 / T)^{1/6})$ . Thus, to obtain a nontrivial Gaussian approximation bound within $T$ iterations, we need dimension dependence $d = o(T^{1/4 - \zeta})$ with $\zeta > 0$ . + +# 5 Constant Learning Rate for Large Dimension + +Recall that $L_{s,q}$ is the Lipschitz constant introduced in Assumption 3. We established asymptotic stationarity and non-asymptotic convergence if $\alpha < \alpha_{s,q} / 7$ with $\alpha_{s,q}$ defined in (5), leading to the upper bound + +$$ +\alpha < \frac {\alpha_ {s , q}}{7} = \frac {2 \mu}{7 \max \{q , s \} L _ {s , q} ^ {2}} \asymp \frac {1}{d ^ {2} \log (d)}, +$$ + +if we choose $s = s_d$ in (7) and if $L_{s_d,q} \asymp d$ . We refer to Section 8.2 for the derivation of the dimension dependence of $L_{s,q}$ in the linear regression model. + +Alternatively, the upper bound for the learning rate $\alpha$ can also be derived by a linear approximation technique (see Lemma 1), defined as the nontrivial solution to the following equation + +$$ +1 - q \mu \alpha + \frac {q [ | q - s | + (s - 1) ] L _ {s , q} ^ {2}}{2} \alpha^ {2} (1 + \alpha L _ {s, q}) ^ {q - 2} = 1. \tag {13} +$$ + +A derivation of this equation is provided in Section 6.1. The existence of a solution of (13) is shown below the proof of Lemma 1 in the supplementary materials. When $q = 2$ , the range of $\alpha$ simplifies to + +$$ +\alpha < \frac {2 \mu}{7 \left[ | s - 2 | + (s - 1) \right] L _ {s , 2} ^ {2}}, +$$ + +which is also proportional to $1 / [d^{2}\log (d)]$ if we choose $s = s_d$ in (7) and if $L_{s_d,2} \asymp d$ , matching the rate of $\alpha_{s,q}$ in (5) derived by Lemma 2, though with a slightly more conservative constant for general $s$ . In the special case with $s = 2$ , both bounds reduce to the classical $\alpha < 2\mu / L_{2,2}^2$ . If $L_{2,2} \asymp d$ for large dimension $d$ , which is shown to be true for the linear regression model in Section 8.2 in the supplementary materials, the $\ell^{\infty}$ - and the $\ell^2$ -norm yield similar upper bounds for the learning rate $\alpha$ . + +# 6 Proof Sketches + +# 6.1 Bridge between $\ell^s$ - and $\ell^{\infty}$ - Norms + +In high-dimensional regimes, convergence rates of constant-learning-rate SGD (2) with respect to the $\ell^{\infty}$ -norm are of particular interest [Wainwright, 2019, Chen et al., 2023]. However, it is extremely challenging to directly study the convergence of $|\beta_{k} - \beta^{*}|_{\infty}$ since the $\ell^{\infty}$ -norm is not differentiable thereby ruling out standard gradient-based tools for proving convergence rates or concentration. To address this issue, we instead study $|\cdot|_{s_d}$ with $s_d$ defined in (7). By the equivalence between $\ell^{s_d}$ and $\ell^{\infty}$ -norms shown in (8), contraction in $\ell^{\infty}$ -norm follows from $\ell^{s_d}$ -norm contraction. + +To prove the GMC property of SGD as introduced in (4), it suffices to show that for any two $d$ -dimensional SGD sequences $\{\beta_k\}_{k\in \mathbb{N}}$ and $\{\beta_k'\}_{k\in \mathbb{N}}$ sharing the same i.i.d. observations $\{\pmb {\xi}_k\}_{k\geq 1}$ but possibly different initializations $\beta_0,\beta_0'\in \mathbb{R}^d$ , the contraction holds for $|\beta_{k} - \beta_{k}^{\prime}|_{s_{d}}$ for all $k\geq 1$ . To this end, we need to determine a range of constant learning rates $\alpha$ such that for any $q\geq 2$ and $\beta ,\beta^{\prime}\in \mathbb{R}^{d}$ , the GMC in Theorem 1 holds, i.e., + +$$ +\left(\mathbb {E} \left| \boldsymbol {\beta} - \alpha \nabla g (\boldsymbol {\beta}, \boldsymbol {\xi}) - \left(\boldsymbol {\beta} ^ {\prime} - \alpha \nabla g \left(\boldsymbol {\beta} ^ {\prime}, \boldsymbol {\xi}\right)\right) \right| _ {s} ^ {q}\right) ^ {1 / q} \leq r \left| \boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime} \right| _ {s}, \quad \text {f o r s o m e} r = r _ {\alpha , s, q} < 1. \tag {14} +$$ + +To derive the inequality, we first provide a lemma based on linear approximation by considering the scalar function + +$$ +\alpha \mapsto | \boldsymbol {x} - \alpha \boldsymbol {z} | _ {s} ^ {q}, \quad \text {w h e r e} \boldsymbol {x} = \beta - \beta^ {\prime}, \quad \boldsymbol {z} = \nabla g (\boldsymbol {\beta}, \boldsymbol {\xi}) - \nabla g (\boldsymbol {\beta} ^ {\prime}, \boldsymbol {\xi}), +$$ + +and linearizing it around $\alpha = 0$ . Then, one only needs to prove that $\mathbb{E}|x - \alpha z|_s^q \leq r|x|_s^q$ . By the second-order Taylor expansion of $|x - \alpha z|_s^q$ in $\alpha$ , we have the linear approximation + +$$ +| \boldsymbol {x} - \alpha \boldsymbol {z} | _ {s} ^ {q} \approx | \boldsymbol {x} | _ {s} ^ {q} - q \alpha | \boldsymbol {x} | _ {s} ^ {q - s} \langle \boldsymbol {x} ^ {s - 1}, \boldsymbol {z} \rangle , \tag {15} +$$ + +with remainder term of order $\alpha^2$ , see Section 2 in the supplementary materials for details. Since a simple triangle inequality argument $\| | \pmb{x} - \alpha \pmb{z} |_s \|_q \leq \| | \pmb{x} |_s \|_q + \alpha \| | \pmb{z} |_s \|_q$ fails to control this remainder sufficiently to yield a contraction constant $r < 1$ , we establish a more precise bound. + +Lemma 1. Recall that $\pmb{v}^{\otimes s} = (v_1^s,\dots ,v_d^s)^\top$ for a vector $\pmb {v} = (v_{1},\ldots ,v_{d})^{\top}$ . For any $q\geq 2$ , any even integer $s\geq 2$ , any two vectors $\pmb {x},\pmb {z}\in \mathbb{R}^{d}$ , and any $\alpha >0$ , + +$$ +\left| | \boldsymbol {x} - \alpha \boldsymbol {z} | _ {s} ^ {q} - | \boldsymbol {x} | _ {s} ^ {q} + q \alpha | \boldsymbol {x} | _ {s} ^ {q - s} \langle \boldsymbol {x} ^ {s - 1}, \boldsymbol {z} \rangle \right| \leq \frac {q \alpha^ {2}}{2} \left[ | q - s | + (s - 1) \right] \left(| \boldsymbol {x} | _ {s} + \alpha | \boldsymbol {z} | _ {s}\right) ^ {q - 2} | \boldsymbol {z} | _ {s} ^ {2}. +$$ + +If $s = 2$ , $q = 2$ , the right-hand side is $\alpha^2 |z|^2$ . This is consistent with the Taylor remainder of the right-hand side in Lemma 9 derived by Li et al. [2024c]. Using this inequality to prove the contraction in (14) is remarkably different from the approaches relying on the martingale decomposition (MD) that is frequently adopted in the literature [Dieuleveut et al., 2020, Mertikopoulos et al., 2020, Mies and Steland, 2023]. Our proposed method requires mild moment conditions on the stochastic gradients and yields simpler proofs that can be generalized to a broad class of online learning problems. We refer to Li et al. [2024b] for detailed discussion. Nevertheless, we remark in advance that a Rio-type inequality (Lemma 2) with slightly sharper constants will be used directly in our main contraction proof, while we retain Lemma 1 here for its intuitive appeal. Finally, by choosing $s = s_d$ as in (7) for (14), we can expect the $\ell^\infty$ -norm type GMC to hold for high-dimensional SGD iterates. + +# 6.2 High-Dimensional Moment Inequality + +To prove Theorem 1, we derive a high-dimensional version of Rio's inequality [Rio, 2009], adapted to the $q$ -th moment of $\ell^s$ -norm. This result provides a slightly sharper constant than Lemma 1 and is used directly in our moment-contraction analysis. + +Lemma 2 (High-dimensional moment inequality). For any $q \geq 2$ , any even integer $s \geq 2$ , and any two $d$ -dimensional random vectors $\mathbf{x}, \mathbf{y}$ , we have + +$$ +\left\| \left| \boldsymbol {x} + \boldsymbol {y} \right| _ {s} \right\| _ {q} ^ {2} \leq \left\| \left| \boldsymbol {x} \right| _ {s} \right\| _ {q} ^ {2} + 2 \left\| \left| \boldsymbol {x} \right| _ {s} \right\| _ {q} ^ {2 - q} \mathbb {E} \left(\left| \boldsymbol {x} \right| _ {s} ^ {q - s} \sum_ {j = 1} ^ {d} x _ {j} ^ {s - 1} y _ {j}\right) + (\max \{q, s \} - 1) \left\| \left| \boldsymbol {y} \right| _ {s} \right\| _ {q} ^ {2}. +$$ + +Moreover, if $\mathbb{E}[\pmb {y}\mid \pmb {x}] = 0$ almost surely, then + +$$ +\left\| \left| \boldsymbol {x} + \boldsymbol {y} \right| _ {s} \right\| _ {q} ^ {2} \leq \left\| \left| \boldsymbol {x} \right| _ {s} \right\| _ {q} ^ {2} + (\max \{q, s \} - 1) \| \left| \boldsymbol {y} \right| _ {s} \| _ {q} ^ {2}. \tag {16} +$$ + +Repeatedly applying Lemma 2 leads to the high-dimensional maximal moment inequality in Lemma 8 in the supplementary materials, which is of independent interest. + +# 6.3 Stationarity, Variation and Bias of ASGD + +We prove the moment bound $||\bar{\beta}_k - \beta^*|_{\infty}\|_q$ via the decomposition + +$$ +\left| \left| | \bar {\boldsymbol {\beta}} _ {k} - \boldsymbol {\beta} ^ {*} | _ {\infty} \right| \right| _ {q} \leq \left| \left| | \bar {\boldsymbol {\beta}} _ {k} - \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} | _ {\infty} \right| \right| _ {q} + \left| \left| | \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} - \mathbb {E} [ \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} ] | _ {\infty} \right| \right| _ {q} + \left| \mathbb {E} [ \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} ] - \boldsymbol {\beta} ^ {*} \right| _ {\infty}. +$$ + +The first term accounts for the deviation due to the non-stationarity of $\bar{\beta}_k$ as it is initialized from an arbitrarily fixed $\beta_0$ ; this can be bounded using the GMC property of $\beta_k$ shown in Theorem 1. The second term captures the stochastic variance of the stationary ASGD sequence. Bounding this term is more delicate because of the intricate dependency structure of $\bar{\beta}_k^\circ$ . To address this, we deploy another powerful tool in time series – the functional dependence measure [Wu, 2005] in Section 8.6 of the supplementary materials, which can effectively quantify the contribution of the random sample $\xi_i$ to the $k$ -th SGD iterate $\beta_k^\circ$ for all $i \leq k$ . As such, by controlling the cumulative dependence measures, we can bound this variance. Lastly, we handle the third term, which represents the non-diminishing bias of $\bar{\beta}_k^\circ$ induced by the constant learning rate $\alpha$ [Dieuleveut et al., 2020, Huo et al., 2023]. This can be dealt with by extending the approach in Li et al. [2024b] to high-dimensional settings. + +Theorem 6 (Asymptotic stationarity). Consider the ASGD iterates $\bar{\beta}_k$ and the stationary version $\bar{\beta}_k^\circ$ . Suppose that Assumptions 1-3 are satisfied for some $q \geq 2$ and some even integer $s \geq 2$ . Then, for the learning rate $\alpha \in (0, \alpha_{s,q})$ with $\alpha_{s,q}$ defined in (5), + +$$ +\left| \left| | \bar {\boldsymbol {\beta}} _ {k} - \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} | _ {s} \right| \right| _ {q} \leq \frac {1}{k} \cdot \frac {1}{1 - r _ {\alpha , s , q}} | \boldsymbol {\beta} _ {0} - \boldsymbol {\beta} _ {0} ^ {\circ} | _ {s}. +$$ + +As a direct consequence of Theorem 6, we have $||\bar{\beta}_k - \bar{\beta}_k^\circ|_s\|_q \lesssim |\beta_0 - \beta_0^\circ|_s / (k\alpha)$ , which indicates the asymptotic stationarity of high-dimensional ASGD sequences. When the bias induced by the initialization is controlled, i.e., $|\beta_0 - \beta_0^\circ|_s < \infty$ , as $k\alpha \to \infty$ , the ASGD iterate $\bar{\beta}_k$ approaches the stationary solution $\bar{\beta}_k^\circ$ in the sense that $||\bar{\beta}_k - \bar{\beta}_k^\circ|_s\|_q \to 0$ . By Theorem 6, we only need to show the convergence for stationary ASGD. + +Theorem 7 (Stochasticity of stationary ASGD). Consider the stationary SGD sequence $\{\beta_k^\circ\}_{k \geq 1}$ . Suppose that Assumptions 1-3 hold with some $q \geq 2$ and some even integer $s \geq 2$ . Then there exists a constant $c_q > 0$ only depending on $q$ , such that, for all $k \geq 1$ , + +$$ +\left\| \left| \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} - \mathbb {E} [ \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} ] \right| _ {s} \right\| _ {q} \leq \sqrt {\frac {c _ {q} s}{k}} M _ {s, q} \Big (L _ {s, q} \sqrt {\alpha \max \{q , s \}} + 1 \Big). +$$ + +In the low-dimensional case, we take $s = 2$ as a special example. Then, $L_{s,q}\sqrt{\alpha\max\{q,s\}} = O(1)$ such that the bound is $\| |\bar{\beta}_k^\circ -\mathbb{E}[\bar{\beta}_k^\circ ]|_s\| _q = O\{1 / \sqrt{k}\}$ . This rate is optimal considering the central limit theorem of the stationary ASGD. + +Next, we consider the bias induced by the constant learning rate. We first introduce some necessary notation. Recall $G(\beta) = \mathbb{E}[\nabla g(\beta, \xi)]$ and $\nabla G(\beta) = (\partial_1 G(\beta), \ldots, \partial_d G(\beta))^{\top}$ , where $\beta = (\beta_1, \ldots, \beta_d)^{\top}$ . Denote $\partial_i G(\beta) = \partial G(\beta) / \partial \beta_i$ , $1 \leq i \leq d$ , + +$$ +\nabla^ {2} G (\boldsymbol {\beta}) = \left[ \partial_ {i} \partial_ {j} G (\boldsymbol {\beta}) \right] _ {1 \leq i, j \leq d}, \quad \nabla^ {3} G _ {i} (\boldsymbol {\beta}) = \left[ \partial_ {i} \partial_ {l} \partial_ {r} G (\boldsymbol {\beta}) \right] _ {1 \leq l, r \leq d}. \tag {17} +$$ + +We provide the non-asymptotic bound for the bias of stationary ASGD in the following lemma. + +Theorem 8 (Bias of stationary ASGD). Under Assumptions 1-3, consider the stationary ASGD $\bar{\beta}_{k}^{\circ}$ . Assume that $g(\beta, \xi)$ is twice differentiable with respect to $\beta$ with positive definite Hessian matrix $\nabla^{2}G(\beta^{*})$ , and uniformly bounded derivatives $\max_{1 \leq i \leq d} \| \nabla^{3}G_{i}(\beta) \|_{\infty} < \infty$ , where + +$$ +\| \nabla^ {3} G _ {i} (\boldsymbol {\beta}) \| _ {\infty} := \max _ {1 \leq l \leq d} \sum_ {r = 1} ^ {d} \Big | \left(\nabla^ {3} G _ {i} (\boldsymbol {\beta})\right) _ {l, r} \Big |. +$$ + +Then, we have + +$$ +\left| \mathbb {E} [ \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} - \boldsymbol {\beta} ^ {*} ] \right| _ {\infty} = O \Big (M _ {s _ {d}, q} ^ {2} \max \{q, s _ {d} \} \alpha d ^ {\frac {q}{q - 1} \cdot (1 - \frac {2}{s _ {d}})} \Big). +$$ + +# 7 Conclusions and Discussion + +This work advances the theoretical understanding of the constant learning-rate SGD algorithms in high-dimensional settings. By introducing novel coupling techniques in nonlinear time series, we establish asymptotic stationarity of SGD with any initialization. We then derive non-asymptotic $q$ -th moment bounds in general $\ell^s$ - and $\ell^{\infty}$ -norms, and develop the first Fuk-Nagaev high-probability tail bound for ASGD. While this paper assumes strong convexity and smoothness of the objective, the nonlinear time series perspective offers a principled framework applicable to a broad class of overparameterized optimization tasks and can be extended to non-convex regimes, providing fundamental insights into the stability, convergence, and reliability of large-scale learning algorithms. + +# Acknowledgments and Disclosure of Funding + +We sincerely thank the program chair, senior area chair, area chair, and the four reviewers for their constructive feedback and involved discussion, which has greatly improved the clarity of our paper. Jiaqi Li's research is partially supported by the NSF (Grant NSF/DMS-2515926). Johannes Schmidt-Hieber has received funding from the Dutch Research Council (NWO) via the Vidi grant VI.Vidi.192.021. Wei Biao Wu's research is partially supported by the NSF (Grants NSF/DMS-2311249, NSF/DMS-2027723). We would like to thank Insung Kong for helpful discussions. + +# References + +A. Agarwal, S. Negahban, and M. J. Wainwright. Stochastic optimization and sparse statistical recovery: Optimal algorithms for high dimensions. In Advances in Neural Information Processing Systems, volume 25. Curran Associates, Inc., 2012. +Z. Allen-Zhu. Katyusha: the first direct acceleration of stochastic gradient methods. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 1200-1205, Montreal Canada, 2017. ACM. +D. Arpit, S. Jastrzebski, N. Ballas, D. Krueger, E. Bengio, M. S. Kanwal, T. Maharaj, A. Fischer, A. Courville, Y. Bengio, and S. Lacoste-Julien. 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Consider the minimization problem $\beta^{*} \in \arg \min_{\beta \in \mathbb{R}^{d}} G(\beta)$ . If the function $G$ satisfies Assumptions 1 and 2, then a global minimizer $\beta^{*}$ exists and is unique. + +Proof of Lemma 3. We first show the existence of a global minimizer. By the coercivity condition in Assumption 1, $\lim_{|\beta|_s \to \infty} G(\beta) = \infty$ , which implies that we can choose some large $\delta \in \mathbb{R}$ such that the sub-level set + +$$ +\mathcal {S} _ {\delta} := \left\{\boldsymbol {\beta} \in \mathbb {R} ^ {d}: G (\boldsymbol {\beta}) \leq \delta \right\} +$$ + +is non-empty and bounded. Since $G$ is continuous by Assumption 2, $S_{\delta}$ is also closed, and hence compact in $\mathbb{R}^d$ by the Heine-Borel theorem. Finally, by applying the Weierstrass extreme value theorem, there exists $\beta^{*} \in S_{\delta}$ such that $G(\beta^{*}) = \min_{\beta \in S_{\delta}} G(\beta)$ . Since for any $\beta \notin S_{\delta}$ , $G(\beta) > \delta \geq G(\beta^{*})$ , $G(\beta^{*}) = \min_{\beta \in \mathbb{R}^d} G(\beta)$ . + +Next, we show the uniqueness of the global minimum. Assume that there are two distinct minimizers $\beta_{1} \neq \beta_{2}$ . By Assumption 2, there exists $\mu > 0$ such that + +$$ +\left\langle \left(\beta_ {1} - \beta_ {2}\right) ^ {\odot (s - 1)}, \nabla G \left(\beta_ {1}\right) - \nabla G \left(\beta_ {2}\right) \right\rangle \geq \mu \left| \beta_ {1} - \beta_ {2} \right| _ {s} ^ {s} > 0. +$$ + +However, since $\beta_{1}$ and $\beta_{2}$ are both minimizers, $\nabla G(\beta_1) = \nabla G(\beta_2) = 0$ , while $\mu |\beta_1 - \beta_2|_s^s > 0$ . This leads to contradiction, which finishes the proof. + +# 8.2 Example: Linear Regression + +As example, we consider the SGD algorithm for the high-dimensional linear regression, observing independent and identically distributed (i.i.d.) pairs $\pmb{\xi}_1 \coloneqq (\pmb{x}_1, y_1)$ , $\pmb{\xi}_2 \coloneqq (\pmb{x}_2, y_2)$ , ... satisfying + +$$ +y _ {k} = \boldsymbol {x} _ {k} ^ {\top} \boldsymbol {\beta} + \epsilon_ {k}, \quad \text {f o r} k = 1, 2, \dots , \tag {18} +$$ + +for random noises $\epsilon_{k}$ that are independent of $x_{k}$ with $\mathbb{E}[\epsilon_k] = 0$ and $\mathbb{E}|\epsilon_k|^q < \infty$ for some $q\geq 2$ . We verify Assumptions 2 and 3 and derive the explicit dependency of the learning-rate, the Lipschitz constant, and the moments of the gradient noise on the dimension $d$ . + +Let $\pmb{\xi} = (y, \pmb{x})$ be an independent random sample from the same distribution as the data. The least-squares loss and the stochastic gradient are respectively given by + +$$ +g (\boldsymbol {\beta}, \boldsymbol {\xi}) = \frac {1}{2} (y - \boldsymbol {x} ^ {\top} \boldsymbol {\beta}) ^ {2}, \quad \text {a n d} \nabla g (\boldsymbol {\beta}, \boldsymbol {\xi}) = - (y - \boldsymbol {x} ^ {\top} \boldsymbol {\beta}) \boldsymbol {x}. \tag {19} +$$ + +Then + +$$ +\nabla G (\boldsymbol {\beta}) = \mathbb {E} [ \nabla g (\boldsymbol {\beta}, (y, \boldsymbol {x}) ] = - \mathbb {E} [ (y - \boldsymbol {x} ^ {\top} \boldsymbol {\beta}) \boldsymbol {x} ] = \mathbb {E} [ \boldsymbol {x} \boldsymbol {x} ^ {\top} ] (\boldsymbol {\beta} - \boldsymbol {\beta} ^ {*}). \tag {20} +$$ + +Let + +$$ +\Sigma = \mathbb {E} [ \boldsymbol {x} \boldsymbol {x} ^ {\top} ], \quad \boldsymbol {v} = \boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime}. \tag {21} +$$ + +To verify the $\ell^s$ -type strong convexity + +$$ +\left\langle \left(\beta_ {1} - \beta_ {2}\right) ^ {\odot (s - 1)}, \nabla G (\beta) - \nabla G \left(\beta^ {\prime}\right) \right\rangle \geq \mu | \beta - \beta^ {\prime} | _ {s} ^ {s}, \quad \text {f o r a l l} \beta , \beta^ {\prime} \in \mathbb {R} ^ {d}, +$$ + +imposed in Assumption 2, observe that $\nabla G(\beta) - \nabla G(\beta') = \Sigma \pmb{v}$ . Thus, the condition becomes + +$$ +0 < \lambda_ {\min } ^ {(s)} := \inf _ {\boldsymbol {v} \in \mathbb {R} ^ {d}, \boldsymbol {v} \neq 0} \frac {\left\langle \boldsymbol {v} ^ {s - 1} , \Sigma \boldsymbol {v} \right\rangle}{\left| \boldsymbol {v} \right| _ {s} ^ {s}}. \tag {22} +$$ + +Lemma 4. Let $s \in \{2, 4, 6, \ldots\}$ . Writing $\Sigma = (\Sigma_{ij})_{i,j=1,\ldots,d}$ , we have + +$$ +\lambda_ {\min } ^ {(s)} \geq \min _ {i = 1, \dots , d} \Sigma_ {i i} - \sum_ {j: j \neq i} | \Sigma_ {i j} |. +$$ + +Proof of Lemma 4. Write $\pmb{v} = (v_{1},\dots ,v_{d})^{\top}$ . Because of $(|v_{i}|^{s - 1} - |v_{j}|^{s - 1})(|v_{i}| - |v_{j}|)\geq 0$ , we obtain $|v_{i}^{s - 1}v_{j}| + |v_{i}v_{j}^{s - 1}|\leq v_{i}^{s} + v_{j}^{s}$ and + +$$ +\begin{array}{l} \langle \pmb {v} ^ {\odot (s - 1)}, \Sigma \pmb {v} \rangle = \sum_ {i = 1} ^ {d} \Sigma_ {i i} v _ {i} ^ {s} - \sum_ {i < j} \Sigma_ {i j} \left(v _ {i} ^ {s - 1} v _ {j} + v _ {i} v _ {j} ^ {s - 1}\right) \\ \geq \sum_ {i = 1} ^ {d} \Sigma_ {i i} v _ {i} ^ {s} - \sum_ {i < j} \left| \Sigma_ {i j} \right| \left(v _ {i} ^ {s} + v _ {j} ^ {s}\right) \\ \geq \Big (\min _ {i = 1, \dots , d} \Sigma_ {i i} - \sum_ {j: j \neq i} | \Sigma_ {i j} | \Big) \sum_ {\ell} v _ {\ell} ^ {s}. \\ \end{array} +$$ + +![](images/91a61c02d29699d58d81059c1776f4b5f87876037706a787d20cba70e062a563.jpg) + +This shows that the rightmost "Gershgorin gap" $\min_{i = 1,\dots,d}\Sigma_{ii} - \sum_{j:j\neq i}|\Sigma_{ij}|$ is a universal lower bound for every $s$ . The lower bound is non-trivial if $\Sigma$ is sufficiently diagonally dominant. + +For large $s$ , the inequality $\lambda_{\min}^{(s)} \geq \min_{i=1,\ldots,d} \Sigma_{ii} - \sum_{j:j \neq i} |\Sigma_{ij}|$ is nearly sharp. To see this, let $i^*$ be the index $i$ that minimizes $\min_{i=1,\ldots,d} \Sigma_{ii} - \sum_{j:j \neq i} |\Sigma_{ij}|$ . For a small $\delta > 0$ , pick $\pmb{v} = (v_1, \ldots, v_d)$ by choosing $v_i^* := 1$ and for $i \neq i^*$ , taking $v_i := -\mathrm{sign}(\Sigma_{i^*i})(1 - \delta)$ . For large $s$ , $\pmb{v}^{\odot(s-1)} \approx (0,0,\ldots,1,0,\ldots,0)$ with the 1 at the $i^*$ -th position. Similarly, $|\pmb{v}|_s^s \approx 1$ . The $i^*$ -th entry of $\Sigma \pmb{v}$ is given by $\Sigma_{i^*i^*} - \sum_{j \neq i^*} |\Sigma_{i^*j}| + O(\delta)$ . Hence for suitable sequences $\delta \to 0$ and $s \to \infty$ , we obtain $\langle \pmb{v}^{\odot(s-1)}, \Sigma \pmb{v} \rangle / |\pmb{v}|_s^s \to \Sigma_{i^*i^*} - \sum_{j \neq i^*} |\Sigma_{i^*j}| = \min_i \Sigma_{ii} - \sum_{j \neq i} |\Sigma_{ij}|$ . + +Regarding Assumption 3, we investigate the dependence of the Lipschitz constant $L_{s,q}$ on the dimension $d$ in high-dimensional linear regression models. If $s^*$ is the dual exponent of $s$ , satisfying $1 / s + 1 / s^* = 1$ , we show that the condition holds with + +$$ +L _ {s, q} = \left\| | \boldsymbol {x} | _ {s} | \boldsymbol {x} | _ {s ^ {*}} \right\| _ {q}. \tag {23} +$$ + +To see this, for any two vectors $\beta, \beta' \in \mathbb{R}^d$ , we have + +$$ +\nabla g (\boldsymbol {\beta}, \boldsymbol {\xi}) - \nabla g \left(\boldsymbol {\beta} ^ {\prime}, \boldsymbol {\xi}\right) = - \left[ \left(y - \boldsymbol {x} ^ {\top} \boldsymbol {\beta}\right) \boldsymbol {x} - \left(y - \boldsymbol {x} ^ {\top} \boldsymbol {\beta} ^ {\prime}\right) \boldsymbol {x} \right] = \boldsymbol {x} \boldsymbol {x} ^ {\top} \left(\boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime}\right). \tag {24} +$$ + +Taking the $\ell^s$ -norm on both sides, we obtain + +$$ +\left| \nabla g (\boldsymbol {\beta}, \boldsymbol {\xi}) - \nabla g \left(\boldsymbol {\beta} ^ {\prime}, \boldsymbol {\xi}\right) \right| _ {s} = \left| \boldsymbol {x} \boldsymbol {x} ^ {\top} \left(\boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime}\right) \right| _ {s} = \left| \boldsymbol {x} \right| _ {s} \left| \boldsymbol {x} ^ {\top} \left(\boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime}\right) \right|. \tag {25} +$$ + +By Hölder's inequality, for the dual exponent $s^*$ satisfying $1 / s + 1 / s^* = 1$ , it follows that + +$$ +\left| \boldsymbol {x} ^ {\top} \left(\boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime}\right) \right| \leq \left| \boldsymbol {x} \right| _ {s ^ {*}} \left| \boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime} \right| _ {s}. \tag {26} +$$ + +Therefore, for $q \geq 2$ , we have the $q$ -th moment bounded as follows, + +$$ +\left(\mathbb {E} \left| \nabla g (\boldsymbol {\beta}, \boldsymbol {\xi}) - \nabla g \left(\boldsymbol {\beta} ^ {\prime}, \boldsymbol {\xi}\right) \right| _ {s} ^ {q}\right) ^ {1 / q} \leq \left(\mathbb {E} \left[ | \boldsymbol {x} | _ {s} ^ {q} | \boldsymbol {x} | _ {s ^ {*}} ^ {q} \right]\right) ^ {1 / q} | \boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime} | _ {s}, +$$ + +proving (23). + +Recall $s_d$ defined in (7). To bound the $\ell^{\infty}$ -norm, we set the conjugates + +$$ +s = s _ {d}, \quad s _ {d} ^ {*} = \frac {s _ {d}}{s _ {d} - 1}. \tag {27} +$$ + +Recall that for the $\ell^s$ -norm, we have $|\pmb{x}|_{\infty} \leq |\pmb{x}|_{s_d} \leq d^{1 / s_d}|\pmb{x}|_{\infty} \leq e|\pmb{x}|_{\infty}$ . Similarly, for the conjugate $\ell^{s_d^*}$ -norm, $d^{\frac{1}{s_d^*} - 1} = d^{\frac{1}{s_d}} \leq e$ implies + +$$ +\frac {1}{e} | \boldsymbol {x} | _ {1} \leq \frac {1}{d ^ {\frac {1}{s _ {d} ^ {*}} - 1}} | \boldsymbol {x} | _ {1} \leq | \boldsymbol {x} | _ {s _ {d} ^ {*}} \leq | \boldsymbol {x} | _ {1}, \tag {28} +$$ + +which together with (23) gives + +$$ +L _ {s _ {d}, q} \leq e \| | \boldsymbol {x} | _ {\infty} | \boldsymbol {x} | _ {1} \| _ {q}. \tag {29} +$$ + +The next two lemmas show that the tail behavior of the covariate vector $\pmb{x}_k$ determines the behavior of the Lipschitz constant $L_{s,q}$ and the moment $M_{s,q}$ defined in Assumption 3. + +Lemma 5. Consider the linear regression in (18) with i.i.d. generic random samples $(\pmb{x},y)$ , where $\pmb{x} = (x_{1},\dots,x_{d})^{\top}$ . Let $q\geq 2$ and recall $s_d$ in (7). + +(i) (Sub-Gaussian) If there is a constant $K$ such that for all $\mathbf{u} \in \mathbb{R}^d$ , $|\mathbf{u}^\top \mathbf{x}|_{\psi_2} \leq K|\mathbf{u}|_2$ , where $|v|_{\psi_2} = \inf \{t > 0 : \mathbb{E}[e^{v^2 / t^2}] \leq 2\}$ denotes the sub-Gaussian norm, then + +$$ +L _ {s _ {d}, q} = O (d \sqrt {\log (d)}). +$$ + +(ii) (Sub-exponential) If there is a constant $K$ such that for all $\mathbf{u} \in \mathbb{R}^d$ , $|\mathbf{u}^\top \mathbf{x}|_{\psi_1} \leq K|\mathbf{u}|_2$ , where $|v|_{\psi_1} = \inf \left\{t > 0 : \mathbb{E}[e^{|v| / t}] \leq 2\right\}$ denotes the sub-exponential norm, then + +$$ +L _ {s _ {d}, q} = O (d \log (d)). +$$ + +(iii) (Finite moment) If there is some $p \geq 2q$ and a finite constant $K_{p}$ such that for each $1 \leq j \leq d$ , $\mathbb{E}|x_{j}|^{p} \leq K_{p}$ , then + +$$ +L _ {s d, q} = O \left(d ^ {1 + \frac {1}{2 q}}\right). +$$ + +(iv) For all three cases (i)-(iii), when $s = 2$ , $L_{2,q} = O(d)$ . + +Proof of Lemma 5. We write $\pmb{x} = \pmb{x}_k$ to denote a generic covariate. By (29) and Hölder's inequality, + +$$ +L _ {s _ {d}, q} \leq e \| | \boldsymbol {x} | _ {\infty} | \boldsymbol {x} | _ {1} \| _ {q} \leq e \| | | \boldsymbol {x} | _ {\infty} \| _ {2 q} \| | \boldsymbol {x} | _ {1} \| _ {2 q}. +$$ + +The convexity of the function $t \mapsto t^{2q}$ and Jensen's inequality yield $|\pmb{x}|_1^{2q} \leq d^{2q-1} \sum_{j=1}^d |x_j|^{2q}$ and + +$$ +\mathbb {E} | \boldsymbol {x} | _ {1} ^ {2 q} \leq d ^ {2 q - 1} \sum_ {j = 1} ^ {d} \mathbb {E} | x _ {j} | ^ {2 q} \leq d ^ {2 q} \max _ {1 \leq j \leq d} \mathbb {E} | x _ {j} | ^ {2 q}. +$$ + +Therefore, for all the three cases (i)-(iii), + +$$ +\left\| \left| \left| \boldsymbol {x} \right| _ {1} \right| \right| _ {2 q} = O (d). +$$ + +Next, we study the order of $(\mathbb{E}[|\pmb{x}|_{\infty}^{2q}])^{1/(2q)}$ for fixed $q \geq 2$ . + +(i) If each $x_{j}$ is sub-Gaussian, then by Section 2.5 in Vershynin [2018], we have + +$$ +\left(\mathbb {E} \left[ \max _ {1 \leq j \leq d} \left| x _ {j} \right| ^ {2 q} \right]\right) ^ {1 / (2 q)} \leq K (\sqrt {\log (d)} + \sqrt {q}) = O (\sqrt {\log (d)}). +$$ + +(ii) If each $x_{j}$ is sub-exponential, then by Section 2.7 in Vershynin [2018], we obtain + +$$ +\left(\mathbb {E} \left[ \max _ {1 \leq j \leq d} | x _ {j} | ^ {2 q} \right]\right) ^ {1 / (2 q)} = O \left(K (\log (d) + \log (q))\right) = O (\log (d)). +$$ + +(iii) If each $x_{j}$ has the finite $p$ -th moment for some $p \geq 2q$ , then + +$$ +\mathbb {E} \left[ \max _ {1 \leq j \leq d} | x _ {j} | ^ {2 q} \right] \leq \sum_ {1 \leq j \leq d} \mathbb {E} \left[ | x _ {j} | ^ {2 q} \right] \leq d K _ {q} = O (d). +$$ + +Finally, for case (iv) with $s_d = 2$ , by (23), + +$$ +L _ {2, q} = \left\| \left| \boldsymbol {x} \right| _ {2} \right\| _ {2 q} ^ {2}. +$$ + +By the convexity of the function $t \mapsto t^q$ , we apply Jensen's inequality and obtain + +$$ +| \boldsymbol {x} | _ {2} ^ {2 q} = \left(\sum_ {j = 1} ^ {d} x _ {j} ^ {2}\right) ^ {q} \leq d ^ {q - 1} \sum_ {j = 1} ^ {d} | x _ {j} | ^ {2 q}. +$$ + +Therefore, for $\pmb{x}$ satisfying case (iii), + +$$ +\mathbb {E} | \boldsymbol {x} | _ {2} ^ {2 q} \leq d ^ {q - 1} \sum_ {j = 1} ^ {d} \mathbb {E} | x _ {j} | ^ {2 q} \leq d ^ {q} K _ {p}, \tag {30} +$$ + +which yields $L_{2,q} = O(d)$ . For the cases (i) and (ii), by Sections 3.4 and 2.7 in Vershynin [2018], respectively, we obtain + +$$ +\left\| \left| \boldsymbol {x} \right| _ {2} \right\| _ {2 q} = O (K (\sqrt {d} + \sqrt {q})) = O (\sqrt {d}), +$$ + +and + +$$ +\left\| \left| \boldsymbol {x} \right| _ {2} \right\| _ {2 q} = O (K (q \sqrt {d})) = O (\sqrt {d}), +$$ + +both indicating $L_{2,q} = O(d)$ . This completes the proof. + +![](images/dcb3a11a989a1ebebdd82794a82fb34231fc92b8c16f339f4bcf402ef2d0c437.jpg) + +Lemma 6. Consider the linear regression model in (18) and assume the conditions on $\epsilon$ and $\pmb{x}$ therein are satisfied. Recall that $M_{s,q} = ||\nabla g(\pmb{\beta}^*,\pmb{\xi})|_s||_q$ is defined in Assumption 3 for some $q\geq 2$ . For the same four cases (i)-(iv) as in Lemma 5 and $s_d$ defined in (7), $M_{s_d,q}$ is respectively equal to (i) $O(\sqrt{\log(d)})$ , (ii) $O(\log (d))$ , (iii) $O(d^{1 / (2q)})$ and (iv) $O(\sqrt{d})$ . + +Proof of Lemma 6. In the linear regression model, the stochastic gradient at the global minimum $\beta^{*}$ can be rewritten into + +$$ +\nabla g \left(\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi}\right) = - \left(y - \boldsymbol {x} ^ {\top} \boldsymbol {\beta} ^ {*}\right) = - \epsilon \boldsymbol {x}. +$$ + +Since the noise $\epsilon$ is independent of the covariate vector $\pmb{x}$ , we obtain + +$$ +\left\|\left\| \nabla g \left(\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi}\right)\right\| _ {s _ {d}} \right\| _ {q} = \left\|\left\| \epsilon \right\| \cdot \left\| \boldsymbol {x} \right\| _ {s _ {d}} \right\| _ {q} = \left\|\left\| \epsilon \right\| _ {q} \right\|\left\| \boldsymbol {x} \right\| _ {s _ {d}} \left. \right\| _ {q}. +$$ + +By inequality (8), it suffices to bound $\| |x|_{\infty} \|_q$ . Since $\| |x|_{\infty} \|_q \leq \| |x|_{\infty} \|_{2q}$ , the same arguments in the proof of Lemma 5 carry over immediately. We omit the details here. + +# 8.3 Some Useful Lemmas + +Lemma 7 (Maximal inequality [Chernozhukov et al., 2015]). Let $z_{1}, \ldots, z_{n}$ be independent, $d$ -dimensional random vectors. Denote the $j$ -th element of $z_{i}$ by $z_{ij}$ , $1 \leq j \leq d$ . Define $M := \max_{1 \leq i \leq n} \max_{1 \leq j \leq d} |z_{ij}|$ and $\sigma^{2} := \max_{1 \leq j \leq d} \sum_{i=1}^{n} \mathbb{E}[z_{ij}^{2}]$ . Then, + +$$ +\mathbb {E} \left[ \max _ {1 \leq j \leq d} \left| \sum_ {i = 1} ^ {n} \left(z _ {i j} - \mathbb {E} [ z _ {i j} ]\right) \right| \right] \lesssim \sigma \sqrt {\log (d)} + \sqrt {\mathbb {E} [ M ^ {2} ]} \log (d), +$$ + +where the universal constant in $\lesssim$ is positive and independent of $n$ and $d$ . + +Lemma 8 ( $L^q$ maximal inequality). Let $\pmb{x}_1, \dots, \pmb{x}_n$ be independent, $d$ -dimensional random vectors. Denote by $x_{ij}$ the $j$ -th element of $\pmb{x}_i$ , $1 \leq j \leq d$ . Then, + +$$ +\left\| \max _ {1 \leq j \leq d} \Big | \sum_ {i = 1} ^ {n} \big (x _ {i j} - \mathbb {E} [ x _ {i j} ] \big) \Big | \right\| _ {q} ^ {2} \leq e ^ {2} \big (\max \{q, \log (d) \} - 1 \big) \sum_ {i = 1} ^ {n} \Big \| \max _ {1 \leq j \leq d} \big | x _ {i j} - \mathbb {E} [ x _ {i j} ] \big | \Big \| _ {q} ^ {2}. +$$ + +This moment inequality can be derived by repeatedly applying Lemma 2. It generalizes the maximal inequality for $\mathbb{E}[\max_{1\leq j\leq d}|\sum_{i = 1}^{n}(x_{ij} - \mathbb{E}[x_{ij}])|]$ in Chernozhukov et al. [2015], reproduced above as Lemma 7, to general $q$ -th moments. + +Proof of Lemma 8. One can assume that the independent random vectors $\pmb{x}_1, \dots, \pmb{x}_n$ have zero means. By repeatedly applying Lemma 2 and choosing $s = \log(d)$ , + +$$ +\begin{array}{l} \left\| \left| \boldsymbol {x} _ {1} + \dots + \boldsymbol {x} _ {n} \right| _ {\infty} \right\| _ {q} ^ {2} \leq \left\| \left| \boldsymbol {x} _ {1} + \dots + \boldsymbol {x} _ {n} \right| _ {s} \right\| _ {q} ^ {2} \\ \leq \left\| | \boldsymbol {x} _ {1} + \dots + \boldsymbol {x} _ {n - 1} | _ {s} \right\| _ {q} ^ {2} + (\max \{q, s \} - 1) \left\| | \boldsymbol {x} _ {n} | _ {s} \right\| _ {q} ^ {2} \\ \leq \left(\max \{q, s \} - 1\right) \sum_ {i = 1} ^ {n} \left\| \left| \boldsymbol {x} _ {i} \right| _ {s} \right\| _ {q} ^ {2} \\ \leq e ^ {2} \left(\max \{q, \log (d) \} - 1\right) \sum_ {i = 1} ^ {n} \left\| \left| \boldsymbol {x} _ {i} \right| _ {\infty} \right\| _ {q} ^ {2}. \\ \end{array} +$$ + +![](images/ce4a2548b0ba0ea4035dd3b417c8dddbdebf93a62d417b3ebd94a5c52398f04f.jpg) + +Lemma 9 (Moment inequality [Li et al., 2024c]). Let $q \geq 2$ . For any two random vectors $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^d$ with fixed $d \geq 1$ , and let + +$$ +\Delta = \mathbb {E} \left| \| \boldsymbol {x} + \boldsymbol {y} \| _ {2} ^ {q} - \| \boldsymbol {x} \| _ {2} ^ {q} - q \| \boldsymbol {x} \| _ {2} ^ {q - 2} \boldsymbol {x} ^ {\top} \boldsymbol {y} \right|. +$$ + +Then, the following inequalities hold: + +(i) + +$$ +\Delta \leq \mathbb {E} \left(\| \boldsymbol {x} \| _ {2} + \| \boldsymbol {y} \| _ {2}\right) ^ {q} - \mathbb {E} \| \boldsymbol {x} \| _ {2} ^ {q} - q \mathbb {E} \left(\| \boldsymbol {x} \| _ {2} ^ {q - 1} \| \boldsymbol {y} \| _ {2}\right). +$$ + +(ii) + +$$ +\Delta \leq \left[ (\mathbb {E} \| \boldsymbol {x} \| _ {2} ^ {q}) ^ {1 / q} + (\mathbb {E} \| \boldsymbol {y} \| _ {2} ^ {q}) ^ {1 / q} \right] ^ {q} - \mathbb {E} \| \boldsymbol {x} \| _ {2} ^ {q} - q (\mathbb {E} \| \boldsymbol {x} \| _ {2} ^ {q}) ^ {(q - 1) / q} (\mathbb {E} \| \boldsymbol {y} \| _ {2} ^ {q}) ^ {1 / q}. +$$ + +Lemma 10 (Equivalence of $\ell^s$ - $\ell^\infty$ -induced matrix norms). For matrix $A \in \mathbb{R}^{d \times d}$ , we have the equivalence of the $\ell^{s_d}$ -norm and $\ell^\infty$ -norm induced matrix norms as follows + +$$ +\frac {1}{e} \| A \| _ {\infty} \leq \| A \| _ {s _ {d}} \leq e \| A \| _ {\infty}, \tag {31} +$$ + +where $s_d$ is defined as (7) and $\| A\| _s = \max_{|\pmb {x}|_s\neq 0}|A\pmb {x}|_s / |\pmb {x}|_s$ . If in addition, $A$ is symmetric, then + +$$ +\frac {1}{e} \| A \| _ {1} = \frac {1}{e} \| A \| _ {\infty} \leq \| A \| _ {s _ {d}} \leq e \| A \| _ {\infty} = e \| A \| _ {1}. \tag {32} +$$ + +Proof of Lemma 10. By Horn and Johnson [1985], for any $1 \leq p \leq q \leq \infty$ and matrix $A \in \mathbb{R}^{d \times d}$ , + +$$ +d ^ {(1 / q) - (1 / p)} \| A \| _ {q} \leq \| A \| _ {p} \leq d ^ {(1 / p) - (1 / q)} \| A \| _ {q}. \tag {33} +$$ + +For $p = s$ and $q = \infty$ , we obtain + +$$ +d ^ {- 1 / s} \| A \| _ {\infty} \leq \| A \| _ {s} \leq d ^ {1 / s} \| A \| _ {\infty}. \tag {34} +$$ + +Since $d^{1 / s}\leq e$ by choosing $s = s_{d}$ in (7), we obtain (31). + +For symmetric $A = (a_{ij})_{1\leq i,j\leq d}$ , $a_{ij} = a_{ji}$ for all $i,j$ . Therefore, + +$$ +\left\| A \right\| _ {1} = \max _ {1 \leq j \leq d} \sum_ {i = 1} ^ {d} \left| a _ {i j} \right| = \max _ {1 \leq i \leq d} \sum_ {j = 1} ^ {d} \left| a _ {i j} \right| = \| A \| _ {\infty}. \tag {35} +$$ + +This completes the proof. + +![](images/93a827aea33fe59aec84be853ec773266d311a2e1d538b522226709a09823e4b.jpg) + +# 8.4 Proofs for Section 2 + +Derivation of (15): Since $s$ is an even integer, we can write + +$$ +f (\alpha) := | \boldsymbol {x} - \alpha \boldsymbol {z} | _ {s} ^ {q} = \left\{\sum_ {i = 1} ^ {d} \left(x _ {i} - \alpha z _ {i}\right) ^ {s} \right\} ^ {\frac {q}{s}}. \tag {36} +$$ + +Taking the derivative with respect to $\alpha$ , we obtain + +$$ +\begin{array}{l} f ^ {\prime} (\alpha) := \frac {d}{d \alpha} f (\alpha) = \frac {q}{s} \Bigl \{\sum_ {i = 1} ^ {d} (x _ {i} - \alpha z _ {i}) ^ {s} \Bigr \} ^ {\frac {q}{s} - 1} \sum_ {i = 1} ^ {d} \frac {d}{d \alpha} (x _ {i} - \alpha z _ {i}) ^ {s} \\ = \frac {q}{s} \Bigl \{\sum_ {i = 1} ^ {d} (x _ {i} - \alpha z _ {i}) ^ {s} \Bigr \} ^ {\frac {q}{s} - 1} \sum_ {i = 1} ^ {d} s (x _ {i} - \alpha z _ {i}) ^ {s - 1} (- z _ {i}) \\ = - q \left\{\sum_ {i = 1} ^ {d} \left(x _ {i} - \alpha z _ {i}\right) ^ {s} \right\} ^ {\frac {q}{s} - 1} \sum_ {i = 1} ^ {d} \left(x _ {i} - \alpha z _ {i}\right) ^ {s - 1} z _ {i}. \tag {37} \\ \end{array} +$$ + +Therefore, + +$$ +\begin{array}{l} f ^ {\prime} (0) = - q \left\{\sum_ {i = 1} ^ {d} x _ {i} ^ {s} \right\} ^ {\frac {q}{s} - 1} \sum_ {i = 1} ^ {d} x _ {i} ^ {s - 1} z _ {i} \\ = - q \left| \boldsymbol {x} \right| _ {s} ^ {q - s} \sum_ {i = 1} ^ {d} x _ {i} ^ {s - 1} z _ {i}. \tag {38} \\ \end{array} +$$ + +A first-order Taylor expansion yields then (15). + +Proof of Lemma 1. Recall that we have defined + +$$ +f (\alpha) = | \boldsymbol {x} - \alpha \boldsymbol {z} | _ {s} ^ {q} = \left\{\sum_ {i = 1} ^ {d} (x _ {i} - \alpha z _ {i}) ^ {s} \right\} ^ {\frac {q}{s}}. +$$ + +A second order Taylor expansion gives $f(\alpha) = f(0) + \alpha f'(0) + \frac{1}{2}\alpha^2 f''(\eta)$ for some $\eta \in [0, \alpha]$ . It suffices to bound $\sup_{u \in [0, \alpha]} |f''(u)|$ . Defining + +$$ +M (u) := \sum_ {i = 1} ^ {d} \left(x _ {i} - u z _ {i}\right) ^ {s} = | \boldsymbol {x} - u \boldsymbol {z} | _ {s} ^ {s}, \tag {39} +$$ + +we have $f(u) = [M(u)]^{\frac{q}{s}}$ , + +$$ +M ^ {\prime} (u) = - s \sum_ {i = 1} ^ {d} \left(x _ {i} - u z _ {i}\right) ^ {s - 1} z _ {i}, \tag {40} +$$ + +$$ +M ^ {\prime \prime} (u) = s (s - 1) \sum_ {i = 1} ^ {d} \left(x _ {i} - u z _ {i}\right) ^ {s - 2} z _ {i} ^ {2}, \tag {41} +$$ + +and the first two derivatives of $f(u)$ can be respectively expressed by + +$$ +f ^ {\prime} (u) = \frac {q}{s} [ M (u) ] ^ {\frac {q}{s} - 1} M ^ {\prime} (u), \tag {42} +$$ + +$$ +f ^ {\prime \prime} (u) = \frac {q}{s} \left(\frac {q}{s} - 1\right) [ M (u) ] ^ {\frac {q}{s} - 2} [ M ^ {\prime} (u) ] ^ {2} + M ^ {\prime \prime} (u) \frac {q}{s} [ M (u) ] ^ {\frac {q}{s} - 1}. \tag {43} +$$ + +Since $s$ is an even integer, it follows from Hölder's inequality that + +$$ +\begin{array}{l} [ M ^ {\prime} (u) ] ^ {2} = s ^ {2} \left(\sum_ {i = 1} ^ {d} \left(x _ {i} - u z _ {i}\right) ^ {s - 1} z _ {i}\right) ^ {2} \\ \leq s ^ {2} \left(\left(\sum_ {i = 1} ^ {d} (x _ {i} - u z _ {i}) ^ {s}\right) ^ {\frac {s - 1}{s}} \left(\sum_ {i = 1} ^ {d} z _ {i} ^ {s}\right) ^ {1 / s}\right) ^ {2} \\ = s ^ {2} | \boldsymbol {x} - u \boldsymbol {z} | _ {s} ^ {2 (s - 1)} | \boldsymbol {z} | _ {s} ^ {2}. \tag {44} \\ \end{array} +$$ + +By applying Hölder's inequality again, we obtain + +$$ +\begin{array}{l} \left| M ^ {\prime \prime} (u) \right| \leq s (s - 1) \left(\sum_ {i = 1} ^ {d} \left(x _ {i} - u z _ {i}\right) ^ {s}\right) ^ {\frac {s - 2}{s}} \left(\sum_ {i = 1} ^ {d} z _ {i} ^ {s}\right) ^ {2 / s} \\ = s (s - 1) | \boldsymbol {x} - u \boldsymbol {z} | _ {s} ^ {s - 2} | \boldsymbol {z} | _ {s} ^ {2}. \tag {45} \\ \end{array} +$$ + +By the two results above, we have + +$$ +\begin{array}{l} | f ^ {\prime \prime} (u) | = \left| \frac {q}{s} \left(\frac {q}{s} - 1\right) | \boldsymbol {x} - u \boldsymbol {z} | _ {s} ^ {q - 2 s} [ M ^ {\prime} (u) ] ^ {2} + M ^ {\prime \prime} (u) \frac {q}{s} | \boldsymbol {x} - u \boldsymbol {z} | _ {s} ^ {q - s} \right| \\ \leq q | q - s | \cdot | \boldsymbol {x} - u \boldsymbol {z} | _ {s} ^ {q - 2} | \boldsymbol {z} | _ {s} ^ {2} + q (s - 1) | \boldsymbol {x} - u \boldsymbol {z} | _ {s} ^ {q - 2} | \boldsymbol {z} | _ {s} ^ {2} \\ \leq q \left[ | q - s | + (s - 1) \right] \left(| \boldsymbol {x} | _ {s} + | u \boldsymbol {z} | _ {s}\right) ^ {q - 2} | \boldsymbol {z} | _ {s} ^ {2}. \tag {46} \\ \end{array} +$$ + +Since $u\in [0,\alpha ]$ , it follows that + +$$ +\sup _ {u \in [ 0, \alpha ]} | f ^ {\prime \prime} (u) | \leq q \left[ | q - s | + (s - 1) \right] \left(| \boldsymbol {x} | _ {s} + \alpha | \boldsymbol {z} | _ {s}\right) ^ {q - 2} | z | _ {s} ^ {2}. \tag {47} +$$ + +This completes the proof. + +Existence of solution to (13): To see the existence of the solution $\alpha_{s,q}$ in + +$$ +1 - q \mu \alpha + \frac {q [ | q - s | + (s - 1) ] L _ {s , q} ^ {2}}{2} \alpha^ {2} (1 + \alpha L _ {s, q}) ^ {q - 2} = 1. +$$ + +denote the function $\alpha \mapsto F(\alpha) = -\mu + c\alpha (1 + L)^{q - 2}$ for the constant $c = [|q - s| + (s - 1)]L^2 / 2 > 0$ and $L = L_{s,q}$ . For any $q \geq 2$ , and any $\alpha > 0$ , $F'(\alpha) = c[(1 + L\alpha)^{q - 2} + \alpha (q - 2)L(1 + L\alpha)^{q - 3}] > 0$ , proving that $F(\alpha)$ is strictly increasing on $\alpha > 0$ . Since $F(0) = -\mu < 0$ and $F(\infty) = +\infty$ , the unique root to $F(\alpha) = 0$ exists. + +Proof of Lemma 2. Define $\varphi(t) = \|\pmb{x} + t\pmb{y}\|_s^2$ for $t \in [0,1]$ . Then + +$$ +\begin{array}{l} \varphi^ {\prime} (t) = \frac {2}{q} \left[ \mathbb {E} \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s} \right\} ^ {q / s} \right] ^ {2 / q - 1} \frac {q}{s} \mathbb {E} \left[ \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s} \right\} ^ {q / s - 1} \sum_ {j = 1} ^ {d} s \left(x _ {j} + t y _ {j}\right) ^ {s - 1} y _ {j} \right] \\ = 2 \left[ \mathbb {E} \left\{\sum_ {j = 1} ^ {d} (x _ {j} + t y _ {j}) ^ {s} \right\} ^ {q / s} \right] ^ {2 / q - 1} \mathbb {E} \left[ \left\{\sum_ {j = 1} ^ {d} (x _ {j} + t y _ {j}) ^ {s} \right\} ^ {q / s - 1} \sum_ {j = 1} ^ {d} (x _ {j} + t y _ {j}) ^ {s - 1} y _ {j} \right] \\ \end{array} +$$ + +and + +$$ +\begin{array}{l} \varphi^ {\prime \prime} (t) = 2 \left(\frac {2}{q} - 1\right) \left[ \mathbb {E} \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s} \right\} ^ {q / s} \right] ^ {2 / q - 2} \\ \cdot \frac {q}{s} \cdot s \left| \mathbb {E} \left[ \left\{\sum_ {j = 1} ^ {d} (x _ {j} + t y _ {j}) ^ {s} \right\} ^ {q / s - 1} \sum_ {j = 1} ^ {d} (x _ {j} + t y _ {j}) ^ {s - 1} y _ {j} \right] \right| ^ {2} \\ + 2 \left[ \mathbb {E} \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s} \right\} ^ {q / s} \right] ^ {2 / q - 1} \\ \cdot \mathbb {E} \left[ \left(\frac {q}{s} - 1\right) \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s} \right\} ^ {q / s - 2} s \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s - 1} y _ {j} \right\} ^ {2} \right] \\ + 2 \left[ \mathbb {E} \left\{\sum_ {j = 1} ^ {d} (x _ {j} + t y _ {j}) ^ {s} \right\} ^ {q / s} \right] ^ {2 / q - 1} \mathbb {E} \left[ \left\{\sum_ {j = 1} ^ {d} (x _ {j} + t y _ {j}) ^ {s} \right\} ^ {q / s - 1} (s - 1) \sum_ {j = 1} ^ {d} (x _ {j} + t y _ {j}) ^ {s - 2} y _ {j} ^ {2} \right] \\ =: \Delta_ {1} (t) + \Delta_ {2} (t) + \Delta_ {3} (t) \\ \end{array} +$$ + +Since $q \geq 2$ , $\Delta_1(t) \leq 0$ . + +Case I. If $q / s - 1\leq 0$ , then $\Delta_2(t)\leq 0$ and $\varphi ''(t)\leq \Delta_3(t)$ . By Hölder's inequality, + +$$ +\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s - 2} y _ {j} ^ {2} \leq \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s} \right\} ^ {(s - 2) / s} \left(\sum_ {j = 1} ^ {d} y _ {j} ^ {s}\right) ^ {2 / s} +$$ + +Consequently, + +$$ +\begin{array}{l} \Delta_ {3} (t) \leq 2 (s - 1) \left[ \mathbb {E} \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s} \right\} ^ {q / s} \right] ^ {2 / q - 1} \\ \cdot \mathbb {E} \left[ \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s} \right\} ^ {q / s - 1} \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s} \right\} ^ {(s - 2) / s} \left(\sum_ {j = 1} ^ {d} y _ {j} ^ {s}\right) ^ {2 / s} \right] \\ = 2 (s - 1) \left[ \mathbb {E} \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s} \right\} ^ {q / s} \right] ^ {2 / q - 1} \mathbb {E} \left[ \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s} \right\} ^ {(q - 2) / s} \left(\sum_ {j = 1} ^ {d} y _ {j} ^ {s}\right) ^ {2 / s} \right] \\ = 2 (s - 1) \| | \boldsymbol {x} + t \boldsymbol {y} | _ {s} \| _ {q} ^ {2 - q} \mathbb {E} \left[ \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s} \right\} ^ {(q - 2) / s} \left(\sum_ {j = 1} ^ {d} y _ {j} ^ {s}\right) ^ {2 / s} \right] \\ \leq 2 (s - 1) \| \left| \boldsymbol {x} + t \boldsymbol {y} \right| _ {s} \| _ {q} ^ {2 - q} \| \left| \boldsymbol {x} + t \boldsymbol {y} \right| _ {s} \| _ {q} ^ {q - 2} \| \left| \boldsymbol {y} \right| _ {s} \| _ {q} ^ {2} \\ = 2 (s - 1) \| | \boldsymbol {y} | _ {s} \| _ {q} ^ {2}. \\ \end{array} +$$ + +Case II. If $q / s - 1 > 0$ , by Hölder's inequality, + +$$ +\begin{array}{l} \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s - 1} y _ {j} \right\} ^ {2} = \left\{\sum_ {j = 1} ^ {d} \left(x _ {j} + t y _ {j}\right) ^ {s / 2} \left(x _ {j} + t y _ {j}\right) ^ {s / 2 - 1} y _ {j} \right\} ^ {2} \\ \leq \sum_ {j = 1} ^ {d} (x _ {j} + t y _ {j}) ^ {s} \sum_ {j = 1} ^ {d} (x _ {j} + t y _ {j}) ^ {s - 2} y _ {j} ^ {2}. \\ \end{array} +$$ + +Therefore, + +$$ +\Delta_ {2} (t) \leq \Delta_ {3} (t) \frac {q - s}{s - 1} +$$ + +and + +$$ +\varphi^ {\prime \prime} (t) \leq \Delta_ {2} (t) + \Delta_ {3} (t) \leq \Delta_ {3} (t) \frac {q - 1}{s - 1} \leq 2 (q - 1) \| | \boldsymbol {y} | _ {s} \| _ {q} ^ {2}. +$$ + +Then, we have + +$$ +\begin{array}{l} \left\| \left| \boldsymbol {x} + \boldsymbol {y} \right| _ {s} \right\| _ {q} ^ {2} = \varphi (1) = \varphi (0) + \varphi^ {\prime} (0) + \int_ {0} ^ {1} (1 - t) \varphi^ {\prime \prime} (t) d t \\ \leq \left\| | \boldsymbol {x} | _ {s} \right\| _ {q} ^ {2} + 2 \left\| | \boldsymbol {x} | _ {s} \right\| _ {q} ^ {2 - q} \mathbb {E} \left(\left| \boldsymbol {x} \right| _ {s} ^ {q - s} \sum_ {j = 1} ^ {d} x _ {j} ^ {s - 1} y _ {j}\right) + \left(\max \{q, s \} - 1\right) \left\| | \boldsymbol {y} | _ {s} \right\| _ {q} ^ {2}. \\ \end{array} +$$ + +![](images/6b2820e7eae20e12dc66d8457ef9c5897d4b165f4e2ba1777d4b5a8b688fc0e6.jpg) + +Proof of Theorem 1. Consider the iterated random function + +$$ +F: \mathbb {R} ^ {d} \times \mathbb {R} \mapsto \mathbb {R}, \quad (\boldsymbol {\beta}, \boldsymbol {\xi}) \mapsto F _ {\boldsymbol {\xi}} (\boldsymbol {\beta}) = \boldsymbol {\beta} - \alpha \nabla g (\boldsymbol {\beta}, \boldsymbol {\xi}). \tag {48} +$$ + +To prove GMC in Theorem 1, it suffices to show that, for some $q \geq 2$ and even integer $s \geq 2$ , for any fixed vectors $\beta, \beta' \in \mathbb{R}^d$ , + +$$ +\left\| \left| F _ {\boldsymbol {\xi}} (\boldsymbol {\beta}) - F _ {\boldsymbol {\xi}} \left(\boldsymbol {\beta} ^ {\prime}\right) \right| _ {s} \right\| _ {q} \leq r _ {\alpha , s, q} \left| \boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime} \right| _ {s}. +$$ + +Recall the inequality in Lemma 2. For $\pmb{x}$ and $\pmb{y}$ therein, we choose them to be $\pmb{x} = \beta - \beta'$ and $\pmb{y} = -\alpha (\nabla g(\beta, \pmb{\xi}) - \nabla g(\beta', \pmb{\xi}))$ respectively. Then, it directly follows from Lemma 2 that + +$$ +\begin{array}{l} \left\| \left| F _ {\boldsymbol {\xi}} (\beta) - F _ {\boldsymbol {\xi}} \left(\beta^ {\prime}\right) \right| _ {s} \right\| _ {q} ^ {2} \\ \leq | \boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime} | _ {s} ^ {2} - 2 \alpha | \boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime} | _ {s} ^ {2 - q} \mathbb {E} \left[ | \boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime} | _ {s} ^ {q - s} \left\langle (\boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime}) ^ {s - 1}, \nabla g (\boldsymbol {\beta}, \boldsymbol {\xi}) - \nabla g (\boldsymbol {\beta} ^ {\prime}, \boldsymbol {\xi}) \right\rangle \right] \\ + \alpha^ {2} (\max \{q, s \} - 1) \| | \nabla g (\boldsymbol {\beta}, \boldsymbol {\xi}) - \nabla g (\boldsymbol {\beta} ^ {\prime}, \boldsymbol {\xi}) | _ {s} \| _ {q} ^ {2} \\ = | \boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime} | _ {s} ^ {2} - 2 \alpha | \boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime} | _ {s} ^ {2 - s} \left\langle (\boldsymbol {\beta} - \boldsymbol {\beta} ^ {\prime}) ^ {s - 1}, G (\boldsymbol {\beta}) - G (\boldsymbol {\beta} ^ {\prime}) \right\rangle \\ + \alpha^ {2} (\max \{q, s \} - 1) \| | \nabla g (\boldsymbol {\beta}, \boldsymbol {\xi}) - \nabla g (\boldsymbol {\beta} ^ {\prime}, \boldsymbol {\xi}) | _ {s} \| _ {q} ^ {2}. \\ \end{array} +$$ + +This along with Assumptions 2 and 3 yields + +$$ +\left\| \left| F _ {\pmb {\xi}} (\pmb {\beta}) - F _ {\pmb {\xi}} (\pmb {\beta} ^ {\prime}) \right| _ {s} \right\| _ {q} ^ {2} \leq \left(1 - 2 \alpha \mu + \alpha^ {2} (\max \{q, s \} - 1) L _ {s, q} ^ {2}\right) | \pmb {\beta} - \pmb {\beta} ^ {\prime} | _ {s} ^ {2}, +$$ + +which completes the proof. + +![](images/17a357c39a60d38b106de94535be61ff43caf0daf01c08b773902c8cb0da826e.jpg) + +# 8.5 Proofs for Section 3.1 + +Proof of Proposition 1. Recall (17) and let $\nabla G(\beta) = \left(\nabla G_1(\beta),\dots ,\nabla G_d(\beta)\right)^\top$ with + +$$ +\nabla G _ {i} (\boldsymbol {\beta}) = \partial G (\boldsymbol {\beta}) / \partial \beta_ {i} = \left(\mathbb {E} [ \nabla g (\boldsymbol {\beta}, \boldsymbol {\xi}) ]\right) _ {i}, \quad i = 1, \dots , d. \tag {49} +$$ + +Since the random samples $\xi_{k}$ , $k\geq 1$ , are independent, it follows that for the $k$ -th iteration, $\xi_{k}$ is independent of $\beta_{k - 1}$ . Then, by the tower rule, for all $k\geq 1$ + +$$ +\mathbb {E} _ {\boldsymbol {\xi}} \left[ \nabla g \left(\boldsymbol {\beta} _ {k - 1}, \boldsymbol {\xi} _ {k}\right) - \nabla G \left(\boldsymbol {\beta} _ {k - 1}\right) \mid \boldsymbol {\beta} _ {k - 1} \right] = \mathbb {E} _ {\boldsymbol {\xi}} \left[ \nabla g \left(\boldsymbol {\beta} _ {k - 1}, \boldsymbol {\xi} _ {k}\right) - \nabla G \left(\boldsymbol {\beta} _ {k - 1}\right) \right] = 0. \tag {50} +$$ + +Therefore, by applying the high-dimensional moment inequality (16) in Lemma 2, we obtain + +$$ +\begin{array}{l} \left\| \left| \boldsymbol {\beta} _ {k} - \boldsymbol {\beta} ^ {*} \right| _ {s} \right\| _ {q} ^ {2} \leq \left\| \left| \boldsymbol {\beta} _ {k - 1} - \boldsymbol {\beta} ^ {*} - \alpha \nabla G (\boldsymbol {\beta} _ {k - 1}) \right| _ {s} \right\| _ {q} ^ {2} \\ + \left(\max \{q, s \} - 1\right) \alpha^ {2} \| \left| \nabla g \left(\boldsymbol {\beta} _ {k - 1}, \boldsymbol {\xi} _ {k}\right) - \nabla G \left(\boldsymbol {\beta} _ {k - 1}\right) \right| _ {s} \| _ {q} ^ {2}. \tag {51} \\ \end{array} +$$ + +For the second part in (51), noting that $\nabla G(\beta^{*}) = 0$ , by the triangle inequality, we have + +$$ +\begin{array}{l} \left\| \left| \nabla g \left(\boldsymbol {\beta} _ {k - 1}, \boldsymbol {\xi} _ {k}\right) - \nabla G \left(\boldsymbol {\beta} _ {k - 1}\right) \right| _ {s} \right\| _ {q} ^ {2} \\ \leq \left(\left\| | \nabla g (\boldsymbol {\beta} _ {k - 1}, \boldsymbol {\xi} _ {k}) - \nabla g (\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi} _ {k}) | _ {s} \right\| _ {q} + \left\| | \nabla G (\boldsymbol {\beta} _ {k - 1}) - \nabla G (\boldsymbol {\beta} ^ {*}) | _ {s} \right\| _ {q} + \left\| | \nabla g (\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi} _ {k}) | _ {s} \right\| _ {q}\right) ^ {2} \\ \leq 3 \left\| | \nabla g \left(\boldsymbol {\beta} _ {k - 1}, \boldsymbol {\xi} _ {k}\right) - \nabla g \left(\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi} _ {k}\right) \right| _ {s} \left\| _ {q} ^ {2} + 3 \left\| | \nabla G \left(\boldsymbol {\beta} _ {k - 1}\right) - \nabla G \left(\boldsymbol {\beta} ^ {*}\right) \right| _ {s} \right\| _ {q} ^ {2} + 3 \left\| | \nabla g \left(\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi} _ {k}\right) \right| _ {s} \left\| _ {q} ^ {2}. \right. \tag {52} \\ \end{array} +$$ + +Since $|\cdot|_s$ is a convex function for $s \geq 1$ , we have $|\mathbb{E}[\cdot]|_s \leq \mathbb{E}[|\cdot|_s]$ . Thus, for all $q \geq 1$ , by Jensen's inequality, we can bound + +$$ +\begin{array}{l} \left| \nabla G \left(\boldsymbol {\beta} _ {k - 1}\right) - \nabla G \left(\boldsymbol {\beta} ^ {*}\right) \right| _ {s} = \left| \mathbb {E} _ {\boldsymbol {\xi}} \left[ \nabla g \left(\boldsymbol {\beta} _ {k - 1}, \boldsymbol {\xi} _ {k}\right) - \nabla g \left(\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi} _ {k}\right) \right] \right| _ {s} \\ \leq \mathbb {E} _ {\boldsymbol {\xi}} \left[ \left| \nabla g \left(\boldsymbol {\beta} _ {k - 1}, \boldsymbol {\xi} _ {k}\right) - \nabla g \left(\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi} _ {k}\right) \right| _ {s} \right] \\ \leq \left(\mathbb {E} _ {\boldsymbol {\xi}} \left| \nabla g \left(\boldsymbol {\beta} _ {k - 1}, \boldsymbol {\xi} _ {k}\right) - \nabla g \left(\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi} _ {k}\right) \right| _ {s} ^ {q}\right) ^ {1 / q}. \tag {53} \\ \end{array} +$$ + +This along with Assumption 3 yields + +$$ +\begin{array}{l} \left\| \left| \nabla G \left(\boldsymbol {\beta} _ {k - 1}\right) - \nabla G \left(\boldsymbol {\beta} ^ {*}\right) \right| _ {s} \right\| _ {q} \leq \left(\mathbb {E} _ {\boldsymbol {\beta}} \mathbb {E} _ {\boldsymbol {\xi}} \left| \nabla g \left(\boldsymbol {\beta} _ {k - 1}, \boldsymbol {\xi} _ {k}\right) - \nabla g \left(\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi} _ {k}\right) \right| _ {s} ^ {q}\right) ^ {1 / q} \\ = \left\| \left| \nabla g \left(\boldsymbol {\beta} _ {k - 1}, \boldsymbol {\xi} _ {k}\right) - \nabla g \left(\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi} _ {k}\right) \right| _ {s} \right\| _ {q} \\ \leq L _ {s, q} \left\| \left| \boldsymbol {\beta} _ {k - 1} - \boldsymbol {\beta} ^ {*} \right| _ {s} \right\| _ {q}. \tag {54} \\ \end{array} +$$ + +Inserting this result back into (52), we obtain a bound for the second term in (51) using + +$$ +\left\| \left| \nabla g \left(\boldsymbol {\beta} _ {k - 1}, \boldsymbol {\xi} _ {k}\right) - \nabla G \left(\boldsymbol {\beta} _ {k - 1}\right) \right| _ {s} \right\| _ {q} ^ {2} \leq 6 L _ {s, q} ^ {2} \left\| \left| \boldsymbol {\beta} _ {k - 1} - \boldsymbol {\beta} ^ {*} \right| _ {s} \right\| _ {q} ^ {2} + 3 \left\| \left| \nabla g \left(\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi} _ {k}\right) \right| _ {s} \right\| _ {q} ^ {2}. \tag {55} +$$ + +For the first term in (51), by applying Lemma 2 again, it follows from Assumptions 2 and 3 that + +$$ +\begin{array}{l} \left\| \left| \boldsymbol {\beta} _ {k - 1} - \boldsymbol {\beta} ^ {*} - \alpha \nabla G (\boldsymbol {\beta} _ {k - 1}) \right| _ {s} \right\| _ {q} ^ {2} \\ \leq \left\| | \boldsymbol {\beta} _ {k - 1} - \boldsymbol {\beta} ^ {*} | _ {s} \right\| _ {q} ^ {2} - 2 \alpha \left\| | \boldsymbol {\beta} _ {k - 1} - \boldsymbol {\beta} ^ {*} | _ {s} \right\| _ {q} ^ {2 - q} \mathbb {E} \left(\left| \boldsymbol {\beta} _ {k - 1} - \boldsymbol {\beta} ^ {*} \right| _ {s} ^ {q - s} \sum_ {j = 1} ^ {d} \left(\boldsymbol {\beta} _ {k - 1} - \boldsymbol {\beta} ^ {*}\right) _ {j} ^ {s - 1} \nabla G _ {j} \left(\boldsymbol {\beta} _ {k - 1}\right)\right) \\ + \alpha^ {2} (\max \{q, s \} - 1) \| | \nabla G (\boldsymbol {\beta} _ {k - 1}) - \nabla G (\boldsymbol {\beta} ^ {*}) | _ {s} \| _ {q} ^ {2} \\ \leq \left(1 - 2 \alpha \mu + \alpha^ {2} (\max \{q, s \} - 1) L _ {s, q} ^ {2}\right) \| | \boldsymbol {\beta} _ {k - 1} - \boldsymbol {\beta} ^ {*} | _ {s} \| _ {q} ^ {2}. \tag {56} \\ \end{array} +$$ + +Inserting this inequality and (55) into (51), we obtain the inequality + +$$ +\begin{array}{l} \left\| \left| \boldsymbol {\beta} _ {k} - \boldsymbol {\beta} ^ {*} \right| _ {s} \right\| _ {q} ^ {2} \leq \left(1 - 2 \alpha \mu + 7 (\max \{q, s \} - 1) \alpha^ {2} L _ {s, q} ^ {2}\right) \left\| \left| \boldsymbol {\beta} _ {k - 1} - \boldsymbol {\beta} ^ {*} \right| _ {s} \right\| _ {q} ^ {2} \\ + 3 (\max \{q, s \} - 1) \alpha^ {2} \| | \nabla g \left(\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi} _ {k}\right) | _ {s} \| _ {q} ^ {2}. \\ \end{array} +$$ + +The desired result is achieved since $||\nabla g(\pmb{\beta}^*, \pmb{\xi}_k)|_s \|_q \leq M_{s,q}$ by Assumption 3. As a special case, for the stationary SGD iterates $\beta_k^\circ \sim \pi_\alpha, k \geq 1$ , we obtain the same result. + +Proof of Theorem 2. First, we denote the contraction constant in Proposition 1 as follows + +$$ +\tilde {r} _ {\alpha , s, q} := 1 - 2 \alpha \mu + 7 (\max \{q, s \} - 1) \alpha^ {2} L _ {s, q} ^ {2}. \tag {57} +$$ + +Given the range of the constant learning rate $\alpha$ , we have $\tilde{r}_{\alpha,s,q} < 1$ . Moreover, notice that + +$$ +3 \left(\max \{q, s \} - 1\right) \alpha^ {2} \| \| \nabla g \left(\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi} _ {k}\right) | _ {s} \| _ {q} ^ {2} = O \left(\max \{q, s \} \alpha^ {2} M _ {s, q} ^ {2}\right). \tag {58} +$$ + +Therefore, for the stationary SGD iterates $\beta_k^\circ \sim \pi_\alpha$ , by Proposition 1, we can obtain + +$$ +\left\| \left| \beta_ {k} ^ {\circ} - \beta^ {*} \right| _ {s} \right\| _ {q} ^ {2} \leq \tilde {r} _ {\alpha , s, q} \| \left| \beta_ {k - 1} ^ {\circ} - \beta^ {*} \right| _ {s} \| _ {q} ^ {2} + O \left(\max \{q, s \} \alpha^ {2} M _ {s, q} ^ {2}\right). \tag {59} +$$ + +Since the SGD iterates $\beta_{k}^{\circ}$ satisfy the geometric-moment contraction in Theorem 1, following Remark 2 in Wu and Shao [2004], the recursion $\beta_{k}^{\circ} = \beta_{k - 1}^{\circ} - \alpha \nabla g(\beta_{k - 1}^{\circ},\xi_{k})$ also holds for $k\leq 0$ . Thus, we can recursively apply the inequality above and achieve + +$$ +\begin{array}{l} \left\| \left| \boldsymbol {\beta} _ {k} ^ {\circ} - \boldsymbol {\beta} ^ {*} \right| _ {s} \right\| _ {q} ^ {2} \leq O \left(\max \{q, s \} \alpha^ {2} M _ {s, q} ^ {2}\right) \cdot \sum_ {i = 0} ^ {\infty} \tilde {r} _ {\alpha , s, q} ^ {i} \\ = \frac {1}{1 - \tilde {r} _ {\alpha , s , q}} O \left(\max \{q, s \} \alpha^ {2} M _ {s, q} ^ {2}\right) \\ = O \left(\max \{q, s \} \alpha M _ {s, q} ^ {2}\right). \tag {60} \\ \end{array} +$$ + +This finishes the proof for the stationary SGD sequence. + +Furthermore, for the general SGD iterates $\beta_{k}$ in (2) that may not have the stationary initialization, we apply the geometric-moment contraction in Theorem 1 and obtain + +$$ +\begin{array}{l} \| \left| \beta_ {k} - \beta^ {*} \right| _ {s} \| _ {q} \leq \left\| \left| \beta_ {k} - \beta_ {k} ^ {\circ} \right| _ {s} \right\| _ {q} + \left\| \left| \beta_ {k} ^ {\circ} - \beta^ {*} \right| _ {s} \right\| _ {q} \\ \leq r _ {\alpha , s, q} ^ {k} \| \left| \beta_ {0} - \beta_ {0} ^ {\circ} \right| _ {s} \| _ {q} + O \left(M _ {s, q} \sqrt {\max \{q , s \} \alpha}\right), \tag {61} \\ \end{array} +$$ + +which completes the proof. + +![](images/70b6ffe6d9b825acf379796d2b8cd0a0c1f22fd40ea3ffa33637d101378c0a3e.jpg) + +# 8.6 Functional Dependence Measure in Time Series + +The functional dependence measure in time series [Wu, 2005] is a key concept in our analysis. For that we view the high-dimensional SGD iterates $\{\beta_k\}_{k\in \mathbb{N}}$ as a nonlinear autoregressive (AR) process. Recall that $\pmb {\xi}_k,k\in \mathbb{Z}$ , are i.i.d. Define the shift process $\mathcal{F}_k = (\pmb {\xi}_k,\pmb {\xi}_{k - 1},\dots)$ and its coupled version $\mathcal{F}_{k,\{l\}} = (\pmb {\xi}_k,\dots ,\pmb {\xi}_{l + 1},\pmb {\xi}_l',\pmb {\xi}_{l - 1},\dots),l\leq k$ , where $\pmb {\xi}_l^{\prime}$ is an i.i.d. copy of $\pmb {\xi}_l$ . + +The stationary sequence $\{\beta_k^\circ\}_{k \in \mathbb{Z}}$ can be represented by a functional system + +$$ +\boldsymbol {\beta} _ {k} ^ {\circ} = h _ {\alpha} \left(\boldsymbol {\xi} _ {k}, \boldsymbol {\xi} _ {k - 1}, \dots\right) = h _ {\alpha} \left(\mathcal {F} _ {k}\right), \quad k \geq 1, \tag {62} +$$ + +where $h_{\alpha}$ is a measurable function that depends on $\alpha$ [Wiener, 1958, Wu, 2005]. Define the coupled version of $\beta_k^\circ$ by + +$$ +\beta_ {k, \{l \}} ^ {\circ} = h _ {\alpha} \left(\boldsymbol {\xi} _ {k}, \dots , \boldsymbol {\xi} _ {l + 1}, \boldsymbol {\xi} _ {l} ^ {\prime}, \boldsymbol {\xi} _ {l - 1}, \dots\right) = h _ {\alpha} \left(\mathcal {F} _ {k, \{l \}}\right), \quad l \leq k. \tag {63} +$$ + +The next lemma provides a bound for the functional dependence measure $\left\| \left|\beta_{k}^{\circ} - \beta_{k,\{l\}}^{\circ}\right|_{s}\right\|_{q}$ . It is later used to derive the moment bounds and the tail probability of the ASGD iterates. + +Lemma 11. Consider the stationary SGD sequence $\{\beta_k^\circ\}_{k \geq 1}$ . Suppose that Assumptions 2 and 3 hold with some $q \geq 2$ and even integer $s \geq 2$ . Then, for all $k \geq 1$ and $l \leq k$ , we have + +$$ +\| | \boldsymbol {\beta} _ {k} ^ {\circ} - \boldsymbol {\beta} _ {k, \{l \}} ^ {\circ} | _ {s} \| _ {q} ^ {2} \leq 4 \alpha^ {2} \big (1 - 2 \alpha \mu + 7 (\max \{q, s \} - 1) \alpha^ {2} L _ {s, q} ^ {2} \big) ^ {k - l} \Big (L _ {s, q} ^ {2} \| | \boldsymbol {\beta} _ {l - 1} ^ {\circ} - \boldsymbol {\beta} ^ {*} | _ {s} \| _ {q} ^ {2} + M _ {s, q} ^ {2} \Big). +$$ + +Proof of Lemma 11. By applying Lemma 2, it follows from similar arguments as in the proof of Proposition 1 that, for each $l \leq k - 1$ , + +$$ +\left\| \left| \boldsymbol {\beta} _ {k} ^ {\circ} - \boldsymbol {\beta} _ {k, \{l \}} ^ {\circ} \right| _ {s} \right\| _ {q} ^ {2} \leq \left(1 - 2 \alpha \mu + 7 \left(\max \{q, s \} - 1\right) \alpha^ {2} L _ {s, q} ^ {2}\right) ^ {k - l} \| \left| \boldsymbol {\beta} _ {l} ^ {\circ} - \boldsymbol {\beta} _ {l, \{l \}} ^ {\circ} \right| _ {s} \| _ {q} ^ {2}. \tag {64} +$$ + +By Assumption 3, for all $l \geq 1$ , + +$$ +\begin{array}{l} \left\| \left| \nabla g \left(\beta_ {l - 1} ^ {\circ}, \boldsymbol {\xi} _ {l}\right) \right| _ {s} \right\| _ {q} ^ {2} \leq 2 \| \left| \nabla g \left(\beta_ {l - 1} ^ {\circ}, \boldsymbol {\xi} _ {l}\right) - \nabla g \left(\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi} _ {l}\right) \right| _ {s} \| _ {q} ^ {2} + 2 \| \nabla g \left(\boldsymbol {\beta} ^ {*}, \boldsymbol {\xi} _ {l}\right) | _ {s} \| _ {q} ^ {2} \\ \leq 2 L _ {s, q} ^ {2} \left\| \left| \beta_ {l - 1} ^ {\circ} - \beta^ {*} \right| _ {s} \right\| _ {q} ^ {2} + 2 M _ {s, q} ^ {2}, \tag {65} \\ \end{array} +$$ + +which yields + +$$ +\begin{array}{l} \left\| \left| \beta_ {l} ^ {\circ} - \beta_ {l, \{l \}} ^ {\circ} \right| _ {s} \right\| _ {q} ^ {2} = \alpha^ {2} \| \left| \nabla g \left(\beta_ {l - 1} ^ {\circ}, \boldsymbol {\xi} _ {l}\right) - \nabla g \left(\beta_ {l - 1} ^ {\circ}, \boldsymbol {\xi} _ {l} ^ {\prime}\right) \right| _ {s} \| _ {q} ^ {2} \\ \leq \alpha^ {2} \left(2 \| | \nabla g \left(\boldsymbol {\beta} _ {l - 1} ^ {\circ}, \boldsymbol {\xi} _ {l}\right) | _ {s} \| _ {q} ^ {2} + 2 \| | \nabla g \left(\boldsymbol {\beta} _ {l - 1} ^ {\circ}, \boldsymbol {\xi} _ {l} ^ {\prime}\right) | _ {s} \| _ {q} ^ {2}\right) \\ \leq 4 \alpha^ {2} \left(L _ {s, q} ^ {2} \| | \beta_ {l - 1} ^ {\circ} - \beta^ {*} | _ {s} \| _ {q} ^ {2} + M _ {s, q} ^ {2}\right) \tag {66} \\ \end{array} +$$ + +Recall $||\nabla g(\pmb{\beta}^{*}, \pmb{\xi}_{k})|_{s}\|_{q} \leq M_{s,q}$ by Assumption 3. Therefore, + +$$ +\begin{array}{l} \left\| \left| \boldsymbol {\beta} _ {k} ^ {\circ} - \boldsymbol {\beta} _ {k, \{l \}} ^ {\circ} \right| _ {s} \right\| _ {q} ^ {2} \leq 4 \alpha^ {2} \left(1 - 2 \alpha \mu + 7 (\max \{q, s \} - 1) \alpha^ {2} L _ {s, q} ^ {2}\right) ^ {k - l} \\ \cdot \left(L _ {s, q} ^ {2} \| \left| \beta_ {l - 1} ^ {\circ} - \beta^ {*} \right| _ {s} \| _ {q} ^ {2} + M _ {s, q} ^ {2}\right). \tag {67} \\ \end{array} +$$ + +This completes the proof. + +# 8.7 Proofs for Section 3.2 + +In this section, we provide the proofs for the convergence results of ASGD in Section 3.2, which can be decomposed into the proofs for Theorems 6 to 8 in Section 6. + +Proof of Theorem 7. Recall the i.i.d. random samples $\pmb{\xi}_k = (y_k, \pmb{x}_k)$ , the filtration $\mathcal{F}_k = (\pmb{\xi}_k, \pmb{\xi}_{k-1}, \dots)$ and its coupled version $\mathcal{F}_{k,\{l\}} = (\pmb{\xi}_k, \dots, \pmb{\xi}_{l+1}, \pmb{\xi}_l', \pmb{\xi}_{l-1}, \dots)$ , $l \leq k$ , where $\pmb{\xi}_l'$ is an i.i.d. copy of $\pmb{\xi}_l$ . Following Wu [2005], we introduce the projection operator + +$$ +\mathcal {P} _ {l} [ \cdot ] = \mathbb {E} [ \cdot \mid \mathcal {F} _ {l} ] - \mathbb {E} [ \cdot \mid \mathcal {F} _ {l - 1} ]. +$$ + +Then, we can rewrite the centered ASGD into + +$$ +\bar {\beta} _ {k} ^ {\circ} - \mathbb {E} [ \bar {\beta} _ {k} ^ {\circ} ] = \frac {1}{k} \sum_ {i = 1} ^ {k} \sum_ {l = 0} ^ {i - 1} \mathcal {P} _ {i - l} \left(\beta_ {i} ^ {\circ}\right) = \frac {1}{k} \sum_ {l = 0} ^ {k - 1} \sum_ {i = l + 1} ^ {k} \mathcal {P} _ {i - l} \left(\beta_ {i} ^ {\circ}\right). \tag {68} +$$ + +Since $\{\mathcal{P}_{i - l}(\beta_i^\circ)\}_{i\geq l + 1}$ is a sequence of martingale differences over $i$ for each $l = 0,1,\ldots ,i - 1$ , following Lemma D.2 in Zhang and Wu [2021] and triangle inequality, we can obtain + +$$ +\begin{array}{l} \left\| \left| \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} - \mathbb {E} [ \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} ] \right| _ {s} \right\| _ {q} = \left\| \left| \frac {1}{k} \sum_ {l = 0} ^ {k - 1} \sum_ {i = l + 1} ^ {k} \mathcal {P} _ {i - l} (\boldsymbol {\beta} _ {i} ^ {\circ}) \right| _ {s} \right\| _ {q} \\ \leq \frac {1}{k} \sum_ {l = 0} ^ {k - 1} \left|\left|\right| \sum_ {i = l + 1} ^ {k} \mathcal {P} _ {i - l} \left(\beta_ {i} ^ {\circ}\right)\right| _ {s} \left. \right| _ {q} \\ \leq \frac {1}{k} \sum_ {l = 0} ^ {k - 1} \left(c _ {q} \cdot s \sum_ {i = l + 1} ^ {k} \left\| \left| \mathcal {P} _ {i - l} \left(\beta_ {i} ^ {\circ}\right) \right| _ {s} \right\| _ {q} ^ {2}\right) ^ {1 / 2}. \tag {69} \\ \end{array} +$$ + +By Theorem 1 in Wu [2005], we have + +$$ +\left\| \left| \mathcal {P} _ {i - l} \left(\boldsymbol {\beta} _ {i} ^ {\circ}\right) \right| _ {s} \right\| _ {q} \leq \left\| \left| \boldsymbol {\beta} _ {i} ^ {\circ} - \boldsymbol {\beta} _ {i, \{i - l \}} ^ {\circ} \right| _ {s} \right\| _ {q}. \tag {70} +$$ + +This along with Lemma 11 and definition of $\tilde{r}_{\alpha ,s,q}$ in (58) yields + +$$ +\begin{array}{l} \left\| \left| \mathcal {P} _ {i - l} \left(\boldsymbol {\beta} _ {i} ^ {\circ}\right) \right| _ {s} \right\| _ {q} ^ {2} \leq 4 \alpha^ {2} \left(1 - 2 \alpha \mu + 7 \left(\max \{q, s \} - 1\right) \alpha^ {2} L _ {s, q} ^ {2}\right) ^ {l} \\ \cdot \left(L _ {s, q} ^ {2} \| \left| \beta_ {i - l - 1} ^ {\circ} - \beta^ {*} \right| _ {s} \| _ {q} ^ {2} + M _ {s, q} ^ {2}\right) \\ = 4 \alpha^ {2} \tilde {r} _ {\alpha , s, q} ^ {l} \left(L _ {s, q} ^ {2} \| \left| \beta_ {i - l - 1} ^ {\circ} - \beta^ {*} \right| _ {s} \| _ {q} ^ {2} + M _ {s, q} ^ {2}\right). \tag {71} \\ \end{array} +$$ + +Recall $r_{\alpha ,s,q}$ in (6) and $\tilde{r}_{\alpha ,s,q}$ in (58). For some constant $\omega >0$ such that + +$$ +\omega \leq \min \left\{\frac {1}{\alpha}, 2 \mu - 7 (\max \{q, s \} - 1) \alpha L _ {s, q} ^ {2} \right\}, \tag {72} +$$ + +we have $1 - \omega \alpha \geq 0$ and + +$$ +r _ {\alpha , s, q} \leq \tilde {r} _ {\alpha , s, q} \leq 1 - \omega \alpha < 1. \tag {73} +$$ + +Consequently, we can further bound (69) by + +$$ +\begin{array}{l} \left\| \left| \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} - \mathbb {E} [ \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} ] \right| _ {s} \right\| _ {q} \\ \leq \frac {1}{k} \sum_ {l = 0} ^ {k - 1} \left[ 4 c _ {q} s \alpha^ {2} (1 - \omega \alpha) ^ {l} \sum_ {i = l + 1} ^ {k} \left(L _ {s, q} ^ {2} \| | \beta_ {i - l - 1} ^ {\circ} - \beta^ {*} | _ {s} \| _ {q} ^ {2} + M _ {s, q} ^ {2}\right) \right] ^ {1 / 2} \\ = \frac {1}{k} \sum_ {l = 0} ^ {k - 1} \left[ 4 c _ {q} s \alpha^ {2} (1 - \omega \alpha) ^ {l} \Big (L _ {s, q} ^ {2} \sum_ {i = l + 1} ^ {k} | | | \boldsymbol {\beta} _ {i - l - 1} ^ {\circ} - \boldsymbol {\beta} ^ {*} | _ {s} | | _ {q} ^ {2} + (k - l) M _ {s, q} ^ {2} \Big) \right] ^ {1 / 2} \\ \leq \frac {1}{k} \sum_ {l = 0} ^ {k - 1} \left[ 2 \alpha \sqrt {c _ {q} s} (1 - \omega \alpha) ^ {l / 2} L _ {s, q} \sqrt {\sum_ {i = l + 1} ^ {k} | | | \beta_ {i - l - 1} ^ {\circ} - \beta^ {*} | _ {s} \| _ {q} ^ {2}} \right] \\ + \frac {1}{k} \sum_ {l = 0} ^ {k - 1} \left[ 2 \alpha \sqrt {c _ {q} s} (1 - \omega \alpha) ^ {l / 2} \sqrt {(k - l)} M _ {s, q} \right] =: I _ {1} + I _ {2}. \tag {74} \\ \end{array} +$$ + +For the term $\mathrm{I}_1$ , it follows from Theorem 2 and expression (58) that + +$$ +\begin{array}{l} \sum_ {i = l + 1} ^ {k} \left\| \left| \beta_ {i - l - 1} ^ {\circ} - \beta^ {*} \right| _ {s} \right\| _ {q} ^ {2} \leq \sum_ {i = l + 1} ^ {k} \left(6 M _ {s, q} ^ {2} (\max \{q, s \} - 1) \alpha\right) \\ = 6 \alpha (k - l) \left(\max \{q, s \} - 1\right) M _ {s, q} ^ {2}. \tag {75} \\ \end{array} +$$ + +Inserting this back into (74) gives + +$$ +\begin{array}{l} I _ {1} \leq \frac {2 \alpha \sqrt {c _ {q} s} L _ {s , q}}{k} \sum_ {l = 0} ^ {k - 1} (1 - \omega \alpha) ^ {l / 2} M _ {s, q} \sqrt {6 \alpha (k - l) (\max \{q , s \} - 1)} \\ \leq \sqrt {c _ {q} s} L _ {s, q} \cdot \frac {c _ {1} \sqrt {\alpha}}{\sqrt {k}} M _ {s, q} \sqrt {\max \{q , s \} - 1}, \tag {76} \\ \end{array} +$$ + +for some constant $c_{1} > 0$ , where the last inequality is due to + +$$ +\sum_ {l = 0} ^ {k - 1} (1 - \omega \alpha) ^ {l / 2} \sqrt {k - l} \leq \sqrt {k} \sum_ {l = 0} ^ {k - 1} (1 - \omega \alpha) ^ {l / 2} = O \left(\frac {\sqrt {k}}{\omega \alpha}\right). \tag {77} +$$ + +Similarly, for some constant $c_2 > 0$ , + +$$ +I _ {2} \leq \frac {c _ {2} \sqrt {c _ {q} s}}{\sqrt {k}} M _ {s, q}. \tag {78} +$$ + +Combining the results of $\mathrm{I}_1$ and $\mathrm{I}_2$ , we obtain the claimed inequality + +$$ +\| | \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} - \mathbb {E} [ \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} ] | _ {s} \| _ {q} \leq \sqrt {\frac {c _ {q} s}{k}} \left(c _ {1} L _ {s, q} \sqrt {\alpha} M _ {s, q} \sqrt {\max \{q , s \} - 1} + c _ {2} M _ {s, q}\right). +$$ + +![](images/b33c2d5b28ec0853dd29cceb2fc3a3ba703455faac4bec02f20bb126eb0c61a0.jpg) + +Proof of Theorem 6. For the ASGD sequence $\{\bar{\beta}_k\}_{k\in \mathbb{N}}$ with arbitrarily fixed initialization $\beta_0\in \mathbb{R}^d$ and the stationary ASGD sequence $\{\bar{\beta}_k^\circ \}_{k\in \mathbb{N}}$ with $\beta_0^\circ \sim \pi_\alpha$ , we have + +$$ +\begin{array}{l} \left\| \left| \bar {\boldsymbol {\beta}} _ {k} - \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} \right| _ {s} \right\| _ {q} = \frac {1}{k} \left\| \left| \sum_ {i = 1} ^ {k} \left(\boldsymbol {\beta} _ {i} - \boldsymbol {\beta} _ {i} ^ {\circ}\right) \right| _ {s} \right\| _ {q} \\ \leq \frac {1}{k} \sum_ {i = 1} ^ {k} \left\| \left| \boldsymbol {\beta} _ {i} - \boldsymbol {\beta} _ {i} ^ {\circ} \right| _ {s} \right\| _ {q}. \tag {79} \\ \end{array} +$$ + +For each $1 \leq i \leq k$ , it follows from the geometric-moment contraction in Theorem 1 that + +$$ +\left\| \left| \beta_ {i} - \beta_ {i} ^ {\circ} \right| _ {s} \right\| _ {q} \leq r _ {\alpha , s, q} ^ {i} \left\| \left| \beta_ {0} - \beta_ {0} ^ {\circ} \right| _ {s} \right\| _ {q}. \tag {80} +$$ + +Recall that $r_{\alpha ,s,q} = 1 - 2\mu \alpha +(\max \{q,s\} -1)L_{s,q}^{2}\alpha^{2} < 1$ in (6). Therefore, + +$$ +\left\| \left| \bar {\boldsymbol {\beta}} _ {k} - \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} \right| _ {s} \right\| _ {q} \leq \frac {1}{k} \cdot \frac {r _ {\alpha , s , q} \left(1 - r _ {\alpha , s , q} ^ {k}\right)}{1 - r _ {\alpha , s , q}} \left\| \left| \boldsymbol {\beta} _ {0} - \boldsymbol {\beta} _ {0} ^ {\circ} \right| _ {s} \right\| _ {q} \leq \frac {1}{k} \cdot \frac {1}{1 - r _ {\alpha , s , q}} \left\| \left| \boldsymbol {\beta} _ {0} - \boldsymbol {\beta} _ {0} ^ {\circ} \right| _ {s} \right\| _ {q}. \tag {81} +$$ + +The desired result is achieved. + +![](images/cf25a63f496857ba68eb265df6ebd7f8085b504d11e2625f94804833ea0b5b23.jpg) + +Proof of Theorem 8. Without loss of generality, assume $\beta^{*} = 0$ . We use the notation (17) for the derivatives of $G$ . Notice that + +$$ +\nabla G \left(\boldsymbol {\beta} ^ {*}\right) = \nabla G (0) = 0. \tag {82} +$$ + +Consider the stationary SGD recursion + +$$ +\boldsymbol {\beta} _ {k} ^ {\circ} = \boldsymbol {\beta} _ {k - 1} ^ {\circ} - \alpha \nabla g \left(\boldsymbol {\beta} _ {k - 1} ^ {\circ}, \boldsymbol {\xi} _ {k}\right), \quad k \geq 1. +$$ + +By taking the expectation on the both sides, we obtain, for all $k \geq 1$ , + +$$ +\mathbb {E} \left[ \nabla G \left(\beta_ {k - 1} ^ {\circ}\right) \right] = 0. \tag {83} +$$ + +Throughout the rest of the proof, we omit the iteration index $k$ and write $\beta = \beta_{k - 1}^{\circ}$ when no confusion is caused. For notational convenience, write $\boldsymbol {\beta} = (\beta_{1},\dots ,\beta_{d})^{\top}$ + +A first-order Taylor expansion on $\nabla G(\beta)$ at $\beta^{*} = 0$ gives + +$$ +0 = \mathbb {E} [ \nabla G (\boldsymbol {\beta}) ] = \nabla G (0) + \nabla^ {2} G (0) \mathbb {E} [ \boldsymbol {\beta} ] + \mathcal {R} (\boldsymbol {\beta}), \tag {84} +$$ + +where $\nabla^2 G(0)$ is the $d\times d$ Jacobian matrix with entries defined by + +$$ +\left[ \nabla^ {2} G (0) \right] _ {i, j} = \frac {\partial^ {2}}{\partial \beta_ {i} \partial \beta_ {j}} G (\boldsymbol {\beta}) \bigg | _ {\boldsymbol {\beta} = 0}, \quad 1 \leq i, j \leq d, \tag {85} +$$ + +and $\mathcal{R}(\beta)$ is the $d$ -dimensional remainder defined as + +$$ +\mathcal {R} (\beta) = \int_ {0} ^ {1} \mathbb {E} \left(\left[ \nabla^ {2} G (t \beta) - \nabla^ {2} G (0) \right] \beta\right) d t. \tag {86} +$$ + +The $i$ -th entry of $\mathcal{R}(\beta)$ can be rewritten into + +$$ +\mathcal {R} _ {i} (\boldsymbol {\beta}) = \int_ {0} ^ {1} (1 - t) \mathbb {E} \left(\boldsymbol {\beta} ^ {\top} \nabla^ {3} G _ {i} (t \boldsymbol {\beta}) \boldsymbol {\beta}\right) d t, \tag {87} +$$ + +where $\nabla^3 G_i(\beta), 1 \leq i \leq d$ , is a $d \times d$ matrix whose entries are + +$$ +[ \nabla^ {3} G _ {i} (\boldsymbol {\beta}) ] _ {l, r} = \frac {\partial^ {3}}{\partial \beta_ {i} \partial \beta_ {l} \partial \beta_ {r}} G (\boldsymbol {\beta}), \quad 1 \leq l, r \leq d. \tag {88} +$$ + +Since $\nabla G(0) = 0$ and $\nabla^2 G(0)$ is invertible given that $\lambda_{\mathrm{min}}[\nabla^2 G(0)] > 0$ , it follows from equation (84) that + +$$ +\mathbb {E} [ \beta ] = - \left[ \nabla^ {2} G (0) \right] ^ {- 1} \mathbb {E} [ \mathcal {R} (\beta) ]. \tag {89} +$$ + +We only need to bound $|\mathbb{E}[\mathcal{R}(\boldsymbol {\beta})]|_{s}$ using Theorem 2, that is $\mathbb{E}[|\pmb {\beta}_k^\circ -\pmb {\beta}^* |_s]^2 = O(\max \{q,s\} \alpha)$ for all $k\geq 1$ + +Let $\pmb{v} = \beta / |\beta|_s$ . For each $i = 1, \dots, d$ , + +$$ +\begin{array}{l} \mathbb {E} [ \mathcal {R} _ {i} (\alpha) ] = \int_ {0} ^ {1} (1 - t) \mathbb {E} [ \boldsymbol {\beta} ^ {\top} \nabla^ {3} G _ {i} (t \boldsymbol {\beta}) \boldsymbol {\beta} ] d t \\ = \int_ {0} ^ {1} (1 - t) \mathbb {E} \left[ \left| \boldsymbol {\beta} \right| _ {s} ^ {2} \boldsymbol {v} ^ {\top} \nabla^ {3} G _ {i} (t \boldsymbol {\beta}) \boldsymbol {v} \right] d t. \tag {90} \\ \end{array} +$$ + +By Hölder's inequality, for $1 / p + 1 / q = 1$ + +$$ +\mathbb {E} \left[ \left| \boldsymbol {\beta} \right| _ {s} ^ {2} \boldsymbol {v} ^ {\top} \nabla^ {3} G _ {i} (t \boldsymbol {\beta}) \boldsymbol {v} \right] \leq \left(\mathbb {E} \left[ \left| \boldsymbol {\beta} \right| _ {s} ^ {2 q} \right]\right) ^ {1 / q} \cdot \left(\mathbb {E} \left(\boldsymbol {v} ^ {\top} \nabla^ {3} G _ {i} (t \boldsymbol {\beta}) \boldsymbol {v}\right) ^ {p}\right) ^ {1 / p}. \tag {91} +$$ + +Again by Hölder's inequality, + +$$ +\mathbb {E} \left[ \left(\boldsymbol {v} ^ {\top} \nabla^ {3} G _ {i} (t \beta) \boldsymbol {v}\right) ^ {p} \right] \leq d ^ {p \left(1 - \frac {2}{s}\right)} \sup _ {| \boldsymbol {v} | _ {s} = 1} \mathbb {E} \left| \nabla^ {3} G _ {i} (t \beta) \boldsymbol {v} \right| _ {s} ^ {p}. \tag {92} +$$ + +Therefore, by Theorem 2 and Lemma 10, + +$$ +\mathbb {E} [ \boldsymbol {\beta} ] \lesssim M _ {s, q} ^ {2} \max \{q, s \} \alpha d ^ {\frac {q}{q - 1} \cdot \left(1 - \frac {2}{s}\right)} \max _ {1 \leq i \leq d} \| \nabla^ {3} G _ {i} (\boldsymbol {\beta}) \| _ {\infty}, \tag {93} +$$ + +where the matrix norm + +$$ +\left\| \nabla^ {3} G _ {i} (\beta) \right\| _ {\infty} := \max _ {1 \leq j _ {1} \leq d} \sum_ {j _ {2} = 1} ^ {d} \left| \left(\nabla^ {3} G _ {i} (\beta)\right) _ {1 \leq j _ {1}, j _ {2} \leq d} \right|. \tag {94} +$$ + +Finally, given the uniform bound $\max_{1\leq i\leq d}\| \nabla^3 G_i(\beta)\|_\infty < \infty$ + +$$ +\mathbb {E} [ \boldsymbol {\beta} ] = O \left(M _ {s, q} ^ {2} \max \{q, s \} \alpha d ^ {\frac {q}{q - 1} \cdot (1 - \frac {2}{s})}\right), \tag {95} +$$ + +which finishes the proof. + +# 8.8 Proofs for Section 4 + +Proof of Theorem 4. By Theorem 6, we have $||\bar{\beta}_k - \bar{\beta}_k^\circ |_s\| _q\lesssim 1 / (k\alpha)||\beta_0 - \beta_0^\circ |_s\| _q$ and consequently, it follows that + +$$ +\mathbb {P} \left(\left| \bar {\boldsymbol {\beta}} _ {k} - \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} \right| _ {s} > z\right) \lesssim \frac {\left\| \left| \boldsymbol {\beta} _ {0} - \boldsymbol {\beta} _ {0} ^ {\circ} \right| _ {s} \right\| _ {q} ^ {q}}{(k \alpha z) ^ {q}}, \quad z > 0. \tag {96} +$$ + +Then it suffices to upper bound $\mathbb{P}(|\bar{\beta}_k^\circ -\beta^* |_s > z)$ . To this end, we first bound the dependence adjusted norm (Section 2 in Zhang and Wu [2017]) for $\{\beta_k^\circ \}_{k\geq 1}$ . By Theorem 1, elementary calculations yield + +$$ +\left\| \left| \boldsymbol {\beta} _ {k} ^ {\circ} - \mathbb {E} [ \boldsymbol {\beta} _ {k} ^ {\circ} ] \right| _ {s} \right\| _ {q, 1 / 2 - 1 / q} = O \left(\frac {M _ {s , q}}{\alpha^ {1 / 2 - 1 / q}}\right). +$$ + +Consequently, by Theorem 6.2 in Zhang and Wu [2017] and Theorem 8, we have + +$$ +\mathbb {P} \left(\left| \bar {\boldsymbol {\beta}} _ {k} ^ {\circ} - \boldsymbol {\beta} ^ {*} \right| _ {s} > z\right) \lesssim \frac {(\log d) ^ {3 q / 2} (\log k) ^ {1 + 2 q} M _ {s , q} ^ {q}}{z ^ {q} k ^ {q - 1} \alpha^ {q / 2 - 1}} + \exp \left(- \frac {C k z ^ {2} \alpha^ {1 - 2 / q}}{M _ {s , q} ^ {2} \log d}\right). +$$ + +Combining this with (96) completes the proof. + +![](images/740bab9a576e6135e5e44630e70db20e38df3597a3704b77f8852791c7570049.jpg) + +Theorem 9 (Theorem 3.1 in [Mies and Steland, 2023]). Let $(\epsilon_i)_{i\in \mathbb{Z}}$ be i.i.d. random variables and $\epsilon_k = (\epsilon_k,\epsilon_{k - 1},\ldots)$ . Assume $X_{k} = G_{k}(\epsilon_{k})\in \mathbb{R}^{d}$ with $\mathbb{E}[X_k] = 0$ for some measurable function $G_{k}$ . For any $k$ , denote $\tilde{\epsilon}_{k,j} = (\epsilon_k,\dots ,\epsilon_{j + 1},\tilde{\epsilon}_j,\epsilon_{j - 1},\dots)$ with $\tilde{\epsilon}_j$ an i.i.d. copy of $\epsilon_j$ . Assume there exist $\Theta >0$ and $q > 2$ , such that for all $k$ , + +$$ +\left(\mathbb {E} \left| G _ {k} \left(\boldsymbol {\epsilon} _ {k}\right) - G _ {k} \left(\tilde {\boldsymbol {\epsilon}} _ {k, k - j}\right) \right| _ {2} ^ {q}\right) ^ {1 / q} \leq \frac {\Theta}{(j \vee 1) ^ {3}}, \text {f o r a l l} j \geq 0, \text {a n d} \left(\mathbb {E} \left| G _ {k} \left(\boldsymbol {\epsilon} _ {0}\right) \right| _ {2} ^ {q}\right) ^ {1 / q} \leq \Theta . \tag {97} +$$ + +Additionally, assume that for some $\Gamma \geq 1$ + +$$ +\sum_ {k = 2} ^ {n} \left(\mathbb {E} \left| G _ {k} \left(\epsilon_ {0}\right) - G _ {k - 1} \left(\epsilon_ {0}\right) \right| _ {2} ^ {2}\right) ^ {1 / 2} \leq \Gamma \cdot \Theta . \tag {98} +$$ + +If $d \leq cn$ for some $c > 0$ , then on a potentially different probability space, there exist random vectors $(X_k')_{k=1}^n = {}^D(X_k)_{k=1}^n$ and independent, mean zero, Gaussian random vectors + +$$ +Y _ {k} ^ {*} \sim \mathcal {N} \left(0, \sum_ {h = - \infty} ^ {\infty} \operatorname {C o v} \left(G _ {k} \left(\epsilon_ {0}\right), G _ {k} \left(\epsilon_ {h}\right)\right)\right) +$$ + +such that + +$$ +\left(\mathbb {E} \max _ {m \leq n} \left| \frac {1}{\sqrt {n}} \sum_ {k = 1} ^ {m} (X _ {k} ^ {\prime} - Y _ {k} ^ {*}) \right| _ {2} ^ {2}\right) ^ {1 / 2} \leq C \Theta \Gamma^ {\frac {1}{4}} \sqrt {\log (n)} \left(\frac {d}{n}\right) ^ {\frac {q - 2}{6 q - 4}}, +$$ + +for some constant $C$ depending on $(q,c)$ + +Instead of univariate $\epsilon_{i}$ , we apply Theorem 3.1 with vector-valued i.i.d. inputs $\xi_{i}$ . The theorem still applies as the proof depends only on the i.i.d. random elements and their $L^q$ bounds but not on the dimension of $\xi_{i}$ . + +Proof of Theorem 5. To prove the Gaussian approximation we will apply Theorem 9 (Theorem 3.1 in Mies and Steland [2023]) with $G_{k} \equiv G = h_{\alpha}$ defined in (62) since $\beta_k^\circ$ is stationary. We now verify the conditions (97) and (98). + +Recall the functional dependence measure $\| \beta_{k}^{\circ} - \beta_{k,\{l\}}^{\circ}|_{s}\|_{q}$ introduced in Section 8.6. Throughout the proof, the $q$ -th moment of the Euclidean norm is denoted by + +$$ +\left\| \cdot \right\| _ {q} := \left\| \left| \cdot \right| _ {2} \right\| _ {q}. +$$ + +Set + +$$ +\rho_ {\alpha , q} ^ {2} := 1 - 2 \alpha \mu + 7 (\max \{q, 2 \} - 1) \alpha^ {2} L _ {2, q} ^ {2}, \quad C _ {\alpha , q} := 2 \alpha \sqrt {c L _ {2 , q} ^ {2} \max \{q , 2 \} \alpha + 1} \tag {99} +$$ + +for some constant $c > 0$ . If $c$ is chosen sufficiently large, then, by Lemma 11 and Theorem 2, for all $k \geq 1$ and $l \leq k$ , we have + +$$ +\begin{array}{l} \left\| \beta_ {k} ^ {\circ} - \beta_ {k, \{l \}} ^ {\circ} \right\| _ {q} ^ {2} \leq 4 \alpha^ {2} \rho_ {\alpha , q} ^ {2 (k - l)} \left(L _ {2, q} ^ {2} \| \beta_ {l - 1} ^ {\circ} - \beta^ {*} \| _ {q} ^ {2} + M _ {2, q} ^ {2}\right) \\ \leq 4 \alpha^ {2} \rho_ {\alpha , q} ^ {2 (k - l)} M _ {2, q} ^ {2} \left(c L _ {2, q} ^ {2} \max \{q, 2 \} \alpha + 1\right) \\ = C _ {\alpha , q} ^ {2} \rho_ {\alpha , q} ^ {2 (k - l)} M _ {2, q} ^ {2}. \\ \end{array} +$$ + +For $\alpha \in (0, \alpha_{s_d,q})$ , it follows that $\rho_{\alpha,q} < 1$ . Let $l = k - j$ . Then, for a sufficiently large constant $C_{\alpha,q}^{\prime}$ , we have + +$$ +\left\| \beta_ {k} ^ {\circ} - \beta_ {k, \{k - j \}} ^ {\circ} \right\| _ {q} \leq C _ {\alpha , q} M _ {2, q} \rho_ {\alpha , q} ^ {j} \leq C _ {\alpha , q} ^ {\prime} M _ {2, q} (j + 1) ^ {- 3}. \tag {100} +$$ + +Therefore, the condition (97) holds with $\Theta = C_{\alpha ,q}^{\prime}M_{2,q}$ . This verifies the first part of condition 97. For the second part of condition 97, by Assumption 3 and Theorem 2, for some constant $C_{\alpha ,q}^{\prime \prime} > 0$ + +$$ +\left\| h _ {\alpha} \left(\boldsymbol {\xi} _ {0}, \boldsymbol {\xi} _ {- 1}, \dots\right) \right\| _ {q} = \left\| \boldsymbol {\beta} _ {0} ^ {\circ} \right\| _ {q} \leq \left\| \boldsymbol {\beta} _ {0} ^ {\circ} - \boldsymbol {\beta} ^ {*} \right\| _ {q} + \left| \boldsymbol {\beta} ^ {*} \right| _ {2} \leq C _ {\alpha , q} ^ {\prime \prime} M _ {2, q} < \infty . \tag {101} +$$ + +Moreover, since $\beta_k^\circ$ is stationary, $G_{k} = G_{k - 1} = h_{\alpha}$ and the left hand side of (98) is zero. Thus, condition (98) is trivially satisfied with $\Gamma = 1$ . + +Finally, we show that the long-run covariance matrix $\Xi = \sum_{k = -\infty}^{\infty}\mathrm{Cov}(\beta_0^\circ ,\beta_k^\circ)$ is well defined in the sense that the spectral norm $\| \Xi \| _s$ is finite. Following (63), denote + +$$ +\beta_ {k, \{\leq l \}} ^ {\circ} := \beta_ {k, \{... , l - 1, l \}} ^ {\circ} = h _ {\alpha} \left(\boldsymbol {\xi} _ {k}, \dots , \boldsymbol {\xi} _ {l + 1}, \boldsymbol {\xi} _ {l} ^ {\prime}, \boldsymbol {\xi} _ {l - 1} ^ {\prime}, \dots\right) = h _ {\alpha} \left(\mathcal {F} _ {k, \{... , l - 1, l \}}\right), \quad l \leq k. \tag {102} +$$ + +Since $\beta_{k,\{\leq 0\}}^{\circ}$ is independent of $\beta_0^\circ$ , we have + +$$ +\begin{array}{l} \operatorname {C o v} \left(\beta_ {0} ^ {\circ}, \beta_ {k} ^ {\circ}\right) = \mathbb {E} \left[ \beta_ {0} ^ {\circ} \beta_ {k} ^ {\circ \top} \right] - \mathbb {E} \left[ \beta_ {0} ^ {\circ} \right] \mathbb {E} \left[ \beta_ {k} ^ {\circ \top} \right] \\ = \mathbb {E} \left[ \boldsymbol {\beta} _ {0} ^ {\circ} \boldsymbol {\beta} _ {k} ^ {\circ \top} \right] - \mathbb {E} \left[ \boldsymbol {\beta} _ {0} ^ {\circ} \right] \mathbb {E} \left[ \boldsymbol {\beta} _ {k, \{\leq 0 \}} ^ {\circ \top} \right] \\ = \mathbb {E} \left[ \boldsymbol {\beta} _ {0} ^ {\circ} \left(\boldsymbol {\beta} _ {k} ^ {\circ} - \boldsymbol {\beta} _ {k, \{\leq 0 \}} ^ {\circ}\right) ^ {\top} \right]. \tag {103} \\ \end{array} +$$ + +We can rewrite the difference as a telescoping sum, + +$$ +\boldsymbol {\beta} _ {k} ^ {\circ} - \boldsymbol {\beta} _ {k, \{\leq 0 \}} ^ {\circ} = \sum_ {l = 0} ^ {\infty} \left(\boldsymbol {\beta} _ {k, \{\leq - l + 1 \}} ^ {\circ} - \boldsymbol {\beta} _ {k, \{\leq - l \}} ^ {\circ}\right). \tag {104} +$$ + +By stationarity and (100), it follows that + +$$ +\left\| \boldsymbol {\beta} _ {k, \{\leq - l + 1 \}} ^ {\circ} - \boldsymbol {\beta} _ {k, \{\leq - l \}} ^ {\circ} \right\| _ {2} = \left\| \boldsymbol {\beta} _ {k, \{- l + 1 \}} ^ {\circ} - \boldsymbol {\beta} _ {k, \{- l \}} ^ {\circ} \right\| _ {2} \leq C _ {\alpha , 2} M _ {2, 2} \rho_ {2, 2} ^ {k + l + 1}. \tag {105} +$$ + +For the spectral norm, + +$$ +\begin{array}{l} \left\| \operatorname {Cov} \left(\boldsymbol {\beta} _ {0} ^ {\circ}, \boldsymbol {\beta} _ {k} ^ {\circ}\right) \right\| _ {s} = \sup _ {\boldsymbol {u}, \boldsymbol {v} \in \mathbb {R} ^ {d}, | \boldsymbol {u} | _ {2} = | \boldsymbol {v} | _ {2} = 1} \mathbb {E} \boldsymbol {v} ^ {\top} \boldsymbol {\beta} _ {0} ^ {\circ} \left(\boldsymbol {\beta} _ {k} ^ {\circ} - \boldsymbol {\beta} _ {k, \{\leq 0 \}} ^ {\circ}\right) ^ {\top} \boldsymbol {u} \\ \leq \sup _ {\boldsymbol {u}, \boldsymbol {v} \in \mathbb {R} ^ {d}, | \boldsymbol {u} | _ {2} = | \boldsymbol {v} | _ {2} = 1} [ \mathbb {E} (\boldsymbol {v} ^ {\top} \boldsymbol {\beta} _ {0} ^ {\circ}) ^ {2} ] ^ {1 / 2} [ \mathbb {E} [ (\boldsymbol {\beta} _ {k} ^ {\circ} - \boldsymbol {\beta} _ {k, \{\leq 0 \}} ^ {\circ}) ^ {\top} \boldsymbol {u} ] ^ {2} ] ^ {1 / 2} \\ \leq \left\| \boldsymbol {\beta} _ {0} ^ {\circ} \right\| _ {2} \left\| \boldsymbol {\beta} _ {k} ^ {\circ} - \boldsymbol {\beta} _ {k, \{\leq 0 \}} ^ {\circ} \right\| _ {2}, \tag {106} \\ \end{array} +$$ + +where the first inequality is by Cauchy-Schwarz and the last inequality uses $(\pmb{u}^{\top}\beta_{0}^{\circ})^{2}\leq |\beta_{0}^{\circ}|^{2}$ with $|\pmb {u}|_2 = 1$ . This, along with $M_{2,2} < \infty$ (Assumption 3) yields + +$$ +\left\| \operatorname {C o v} \left(\beta_ {0} ^ {\circ}, \beta_ {k} ^ {\circ}\right) \right\| _ {s} \leq \| \beta_ {0} ^ {\circ} \| _ {2} \| \beta_ {k} ^ {\circ} - \beta_ {k, \{\leq 0 \}} ^ {\circ} \| _ {2} \leq C _ {\alpha} ^ {\prime} \rho_ {\alpha , 2} ^ {k}, \tag {107} +$$ + +for some constant $C_{\alpha}^{\prime} > 0$ . As a direct consequence, + +$$ +\left\| \Xi \right\| _ {s} \leq \left\| \mathbb {E} \left[ \beta_ {0} ^ {\circ} \beta_ {0} ^ {\circ \top} \right] \right\| _ {s} + 2 \sum_ {k = 1} ^ {\infty} C _ {\alpha} ^ {\prime} \rho_ {\alpha , 2} ^ {k} < \infty . \tag {108} +$$ + +This completes the proof. + +□ + +# NeurIPS Paper Checklist + +# 1. 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However, statistical inference and uncertainty quantification for gradient boosting remain challenging and under-explored. We propose a unified framework for statistical inference in gradient boosting regression. Our framework integrates dropout or parallel training with a recently proposed regularization procedure called Boulevard that allows for a central limit theorem (CLT) for boosting. With these enhancements, we surprisingly find that increasing the dropout rate and the number of trees grown in parallel at each iteration substantially enhances signal recovery and overall performance. Our resulting algorithms enjoy similar CLTs, which we use to construct built-in confidence intervals, prediction intervals, and rigorous hypothesis tests for assessing variable importance in only $O(nd^2)$ time with the Nyström method. Numerical experiments verify the asymptotic normality and demonstrate that our algorithms perform well, do not require early stopping, interpolate between regularized boosting and random forests, and confirm the validity of their built-in statistical inference procedures. + +# 1 Introduction + +Gradient boosting (Friedman, 2000), particularly through widely used implementations such as XGBoost (Chen and Guestrin, 2016), LightGBM (Ke et al., 2017), and CatBoost (Prokhorenkova et al., 2018), has become one of the most powerful and widely adopted methods for supervised learning, especially on tabular data. However, this flexibility and predictive accuracy come at a cost. Unlike their base learners – such as decision trees or linear models – boosting methods are typically far less interpretable, and uncertainty quantification for their predictions is far from straightforward. While point estimates may suffice for ad hoc prediction tasks, they overlook randomness inherent in both the data and the algorithm itself. Uncertainty quantification therefore asks a central question: if a new dataset were collected and models retrained, how different would the resulting predictions be? This is a central ingredient in propagating uncertainty through forecast models, prioritizing data collection and a host of other downstream tasks. + +In this light, various methods have been proposed for uncertainty quantification in boosting. These include Langevin boosting (Tan et al., 2023; Ustimenko and Prokhorenkova, 2022; Malinin et al., 2021), k-nearest neighbors-based techniques (Brophy and Lowd, 2022), Gaussian graphical models (Chen and Wang, 2023), quantile regression approaches (Yin et al., 2023), and probabilistic prediction via natural gradients (Duan et al., 2020). However, most of these methods lack formal theoretical guarantees, with many relying primarily on heuristic justifications. + +A principled route to uncertainty quantification is through statistical inference, e.g. via prediction intervals. However, this remains limited. In the Bayesian setting, Ustimenko et al. (2023) show randomized boosting converges to a kernel ridge regression and approximate Gaussian process posterior. Yet, they re-run the entire boosting procedure to generate even a single posterior sample. Malinin et al. (2021) suggest a virtual ensemble method, but do not provide formal guarantees. + +In the frequentist setting, Zhou and Hooker (2022) propose a regularization scheme yielding a central limit theorem for boosting via convergence to kernel ridge regression. However, they do not provide proper confidence or prediction intervals, and their method recovers at most half the true signal (rescaling is suggested, but this amplifies errors). Despite these limitations, their framework remains the only foundation for frequentist inference in boosting, which we extend here. + +Other literature. Recent advances for random forests yield principled statistical methods for uncertainty quantification via the construction of valid confidence intervals (Wager et al., 2014; Wager and Athey, 2017; Athey et al., 2018; Mentch and Hooker, 2016b), and interpretability via hypothesis tests for variable importance (Mentch and Hooker, 2016a; Coleman et al., 2022). Much of this is achieved through viewing a random forest as an adaptive kernel method (Friedberg et al., 2021; Athey et al., 2018). This also holds for various incarnations of boosting (Zhou and Hooker, 2022; Ustimenko et al., 2023). Despite these findings, the literature on uncertainty quantification and statistical inference for boosting is far sparser – a gap we fill by exploiting this equivalence. + +Motivated by this gap, we study frequentist inference for uncertainty quantification in boosting under supervised nonparametric regression: $y = f(\mathbf{x}) + \epsilon$ , with subgaussian noise with variance $\sigma^2$ . Our goal is to develop principled, statistically sound procedures for uncertainty quantification, outlined below. + +Methodological contributions. In detail, we provide the following methodological contributions: + +1. Improvements via dropout: We incorporate random dropout (Rashmi and Gilad-Bachrach, 2015) into the regularization procedure of Zhou and Hooker (2022). This yields an improved procedure (Algorithm 1) that achieves provably better performance by up to a factor of 4 in the asymptotic relative efficiency (ARE), and increased signal recovery by up to a factor of 2. By tuning the dropout rate in Algorithm 1, we smoothly bridge the vanilla Boulevard method (Zhou and Hooker, 2022) and a modified Random Forest (Breiman, 2001) that draws observation subsamples without replacement and considers the full covariate space at each split. +2. Parallel boosting: We construct a novel leave-one-out procedure for parallel boosting (Algorithm 2) that, when coupled with the above regularization procedure, improves the ARE by at least a factor of 4, and enjoys increased signal recovery by at least a factor of 2. +3. Prediction, confidence, and reproduction intervals: We leverage the central limit theorems our procedures enjoy to construct a variety of procedures for statistical inference. First, we conduct predictive inference via constructing prediction intervals on (potentially unseen) labels $\mathbf{y}$ . We then construct confidence intervals for the underlying ground truth function $\pmb{f}$ . Lastly, to quantify uncertainty within the learning algorithm, we provide reproduction intervals (Zhou and Hooker, 2022) for another booster $\widehat{\pmb{f}}$ trained on an independent realization of the training set. +4. Hypothesis testing: We construct a chi-squared hypothesis test for variable importance, extending the work of Mentch and Hooker (2016b) for random forests to the boosting setting. +5. Computational efficiency: Unlike prior approaches to statistical inference for random forests and boosting, we utilize the Nyström approximation of Musco and Musco (2017). This allows our tests and intervals to scale linearly with the number of data points. + +![](images/abae02fc1d30f27dc658f7ce65bcd3e50e2afe955e7cadcbd2403b3293a73fa7.jpg) +Figure 1: Demonstration of confidence, prediction, and reproduction intervals on $f(x) = \sin 2\pi x + \frac{1}{2} x^2$ . Conformal baseline. 200 trees, learning rate 0.6, depth 8, subsampling and dropout 0.6. + +![](images/79e3d2f8a0720afa82dcfe0133f0091f73ee7b0711232ac360b2a3fad8f1da21.jpg) + +![](images/7ccf76b8c09ce7498203822306f28bc977031cab48b0258792b0035eb73a0db1.jpg) + +![](images/4db25dded4016d7095c3e37a1ee24b1130f2ba624ba38d70fc5c3f241653bc28.jpg) + +Theoretical contributions. Our primary theoretical contributions are central limit theorems (CLTs) for boosting with dropout (Algorithm 1) and parallel training (Algorithm 2) – the first such results for these settings to our knowledge. These extend the theoretical arguments of Zhou and Hooker (2022). In analyzing boosting with dropout, the stochasticity introduced into ensemble construction must be accounted for – in addition to randomness from data subsampling. To address this, we establish a weak law for finite-sample convergence in the stochastic contraction framework of Almudevar (2022). The arguments for parallel boosting are more challenging. Inspired by classical backfitting methods (Breiman and Friedman, 1985), we develop a delayed averaging scheme that + +enables parallelism while preserving convergence guarantees. To achieve central limit theorems that hold unconditionally on the data $\mathbf{X}$ , we show a bound on the maximal leaf size is sufficient to limit the influence of distant points and promote balanced splits, enabling rigorous distributional results. + +Numerical experiments. We establish the efficacy of our methods on both a simulation study and real-world datasets. Our numerical experiments showcase that our algorithms can be tuned to interpolate between regularized boosting and random forests, and are highly competitive in terms of MSE. These also demonstrate the correctness, coverage, and computational efficiency of the statistical procedures we construct via the central limit theorems our algorithms enjoy. + +# 2 Setup + +Consider boosting for nonparametric regression with squared error loss. Given random features $\mathbf{x} \in [0,1]^d$ , labels $y \in \mathbb{R}$ , and noise $\epsilon \sim \mathrm{SubG}(0,\sigma^2)$ , say there exists some function $\pmb{f}$ such that $y = \pmb{f}(\mathbf{x}) + \epsilon$ . The learner is given a training set $(\mathbf{X}^{\mathrm{train}},\mathbf{y}^{\mathrm{train}})$ of size $n$ , and learns an estimate $\widehat{\pmb{f}}$ of $\pmb{f}$ by minimizing the mean squared error (MSE) $\frac{1}{n}\sum_{i=1}^{n}(\widehat{\pmb{f}}(\mathbf{x}_i) - y_i)^2$ over the training set. It is customary to analyze how $\widehat{\pmb{f}}$ converges to $\pmb{f}$ , or in other words, the generalization error of $\widehat{\pmb{f}}$ on a test set $(\mathbf{X}^{\mathrm{test}},\mathbf{y}^{\mathrm{test}})$ . Subscripts $\widehat{\pmb{f}}_n^{(b)}$ indicate the estimated predictor trained on $n$ datapoints for $b$ boosting rounds, where we omit dependencies on $b$ and $n$ whenever possible. + +Regularized stochastic gradient boosting. Zhou and Hooker (2022) propose a regularization procedure called Boulevard that recovers a central limit theorem for boosting. They do so by showing that the resulting regularized procedure converges to a kernel ridge regression. At each iteration + +$b = 1, \dots, B - 1$ , instead of $\widehat{\pmb{f}}^{(b + 1)} \gets \widehat{\pmb{f}}^{(b)} + \lambda \pmb{t}^{(b)}$ , they update $\widehat{\pmb{f}}^{(b + 1)} \gets \frac{b - 1}{b} \widehat{\pmb{f}}^{(b)} + \frac{\lambda}{b} \pmb{t}^{(b)}$ for the current function estimate $\widehat{\pmb{f}}^{(b)}$ , built tree $t^{(b)}$ , and learning rate $\lambda \in [0,1)$ . As this is equivalent to updating $\widehat{\pmb{f}}^{(b + 1)} \gets \frac{\lambda}{b} \sum_{i=1}^{b} t^{(i)}$ , the resulting ensemble is an average of trees. This observation, along with assumptions on tree adaptivity, allows them to prove that Boulevard predictions are asymptotically normal, though they do not construct confidence or prediction intervals. Additionally, this regularization comes at a cost. It turns out that $\widehat{\pmb{f}}_n^{(b)}$ converges only to $\frac{\lambda}{1 + \lambda} \pmb{f} \leq \pmb{f}/2$ as $n, b \to \infty$ , and so Boulevard recovers at most half the signal. Still, this is the only method available for frequentist inference for gradient boosting, and so we seek to improve upon it in this paper. + +![](images/119c31527cc551ea6ea34ee3d95604366b9da313d5e6f1a9337fd0458d5b4456.jpg) +Figure 2: An added benefit of Boulevard regularization is that it renders early stopping unnecessary. Because the procedure converges to a fixed point, one can (in theory) boost indefinitely without encountering overfitting. It also produces asymptotically normal predictions (see the Q-Q plot). + +![](images/4f8b947251301c832cec7be4a09d23c46b01d4538f4664b95e61d1d83d5a69c6.jpg) + +Regression trees and tree structure. We are primarily concerned with boosting algorithms where we use regression trees as weak learners. We introduce a formulation of a regression tree below. + +Definition 1 (Regression trees). A regression tree $t_n$ trained on $n$ datapoints segments the covariate space $[0,1]^d$ into a partition of hyper-rectangles $\{A_i\}_{i=1}^m$ . When $A(\mathbf{x})$ for the rectangle containing $\mathbf{x}$ , and $s_{n,j}(\mathbf{x})$ is the frequency at which training point $\mathbf{x}_j$ and test point $\mathbf{x}$ share a leaf: + +$$ +s _ {n, j} (\mathbf {x}) = \frac {\mathbb {1} \left(\mathbf {x} _ {j} \in A (\mathbf {x})\right)}{\sum_ {k = 1} ^ {n} \mathbb {1} \left(\mathbf {x} _ {k} \in A (\mathbf {x})\right)}, t h e r e g r e s s i o n t r e e \mathbf {t} _ {n} p r e d i c t s \mathbf {t} _ {n} (\mathbf {x}) = \sum_ {j = 1} ^ {n} s _ {n, j} (\mathbf {x}) \cdot y _ {j}. +$$ + +As such, a tree can be thought of as an adaptive nearest neighbor method (Athey et al., 2018; Friedberg et al., 2021), where $s_{n,j}(\mathbf{x})$ yields the influence of datapoint $\mathbf{x}_j$ on the tree's prediction at test point $\mathbf{x}$ . This motivates the definition of a structure vector and matrix from Zhou and Hooker (2022): + +Definition 2 (Structure vectors and matrices). Let $\mathbf{t}_n$ be a tree trained on the input $(\mathbf{X}_n, \mathbf{y}_n)$ . For an arbitrary point $\mathbf{x}$ , let $s_n(\mathbf{x}) = (s_{n,1}(\mathbf{x}), \ldots, s_{n,n}(\mathbf{x}))^\top$ be the structure vector of $\mathbf{x}$ . As such, we say that the matrix $\mathbf{S}_n = (s_{n,j}(\mathbf{x}_i))_{i,j=1}^n$ with rows $s_n(\mathbf{x}_i)^\top$ is the structure matrix of $\mathbf{t}_n$ . + +The structure matrix is a kernel matrix. A natural idea is to perform kernel ridge regression (KRR): + +$$ +\widehat {\boldsymbol {f}} _ {n} (\mathbf {x}) = \left\langle \boldsymbol {r} _ {n} (\mathbf {x}), \mathbf {y} _ {n} \right\rangle , \boldsymbol {k} _ {n} (\mathbf {x}) = \mathbb {E} [ \boldsymbol {s} _ {n} (\mathbf {x}) ], \mathbf {K} _ {n} = \mathbb {E} [ \mathbf {S} _ {n} ], \text {a n d} \boldsymbol {r} _ {n} (\mathbf {x}) ^ {\top} = \boldsymbol {k} _ {n} (\mathbf {x}) ^ {\top} \left(\lambda^ {- 1} \mathbf {I} + \mathbf {K} _ {n}\right) ^ {- 1}. +$$ + +Zhou and Hooker (2022) show Boulevard regularization converges to a kernel ridge regression $\widehat{\pmb{f}}_n(\mathbf{x}) = \langle \pmb {r}_n(\mathbf{x}),\pmb {y}_n\rangle$ as $b\to \infty$ . However, as $\lambda \in (0,1]$ , this converges to at most half the signal. We later propose two algorithms in Algorithms 1 and 2, that converge to $\langle r_n^D (\mathbf{x}),\mathbf{y}_n\rangle =$ $\pmb{k}_{n}(\mathbf{x})^{\top}(\lambda^{-1}\mathbf{I} + q\mathbf{K}_{n})^{-1}\mathbf{y}_{n}$ and $\langle r_n^P (\mathbf{x}),\mathbf{y}_n\rangle = k_n(\mathbf{x})^\top (\mathbf{I} + (K - 1)\mathbf{K}_n)^{-1}K\mathbf{y}_n$ respectively, achieving increased signal recovery. As a final note, to accommodate training $t_n(\mathbf{x},\mathcal{G})$ on a subsample $\mathcal{G}\subseteq \{1,\ldots ,n\}$ , we write $s_{n,j}(\mathbf{x};\mathcal{G}) = \mathbb{1}(\mathbf{x}_j\in A,j\in \mathcal{G}) / \sum_{k = 1}^{n}\mathbb{1}(\mathbf{x}_k\in A,k\in \mathcal{G})$ , and $t_n(\mathbf{x};\mathcal{G}) = \sum_{j = 1}^n s_{n,j}(\mathbf{x};\mathcal{G})\cdot y_j$ . We suppress the dependence on subsamples when possible. + +Integrity & adaptivity. We inherit the following two assumptions from Boulevard regularization: + +Assumption 1 (Structure-value isolation). $(\pmb{t}_n^{(b)})_{b = 1}^B$ has $\pmb{s}_{n}^{(b)}(\mathbf{x})\perp \mathbf{y}_{n}$ for all $b = 1,\dots,B$ + +Assumption 2 (Non-adaptivity). There exists a probability measure $\mathcal{Q}_n$ on the space of tree structures and some $b'$ so the tree structure at iteration $b = b', \ldots, B$ is an independent draw from $\mathcal{Q}_n$ . + +These are required for theoretical guarantees. In practice, the first can be implemented via a stronger form of honesty (Wager and Athey, 2017) that we call integrity, refitting leaf values on an independent dataset after boosting. One can also randomly sample splits, or get them from a random forest trained on an independent sample. The second only requires non-adaptivity after a finite number of iterations, which may occur naturally (Zhou and Hooker, 2022). To enforce it, one can boost normally for a fixed number of steps and then sample tree structures from earlier iterations. Still, numerical experiments indicate that our algorithms perform well even without these adjustments. Our assumptions are given in a colleted format in the supplement. + +# 3 Algorithms + +To address the limitations of Boulevard regularization while retaining its statistical guarantees, we propose two improvements that achieve enhanced signal recovery and improved sample complexity: + +Alg. 1: By incorporating random dropout (Rashmi and Gilad-Bachrach, 2015) in the ensemble. + +Alg. 2: By growing trees in parallel within each round with a leave-one-out procedure. + +The intuition is as follows. Surprisingly, increasing the dropout probability allows one to recover more of the signal. This is because each tree in the current ensemble is effectively downweighted during residual construction, and the new tree is fit on more of the signal. On the other hand, as the number of trees grown in parallel within each round grows, the resulting ensemble increasingly resembles the procedure of boosting random forests in sequence – allowing each round to recover more signal than a single tree can. We present both algorithms and explain them in detail below. + +# Algorithm 1 Boulevard Regularized Additive regression Trees - Dropout (BRAT-D) + +1: Input: Features $\mathbf{X}$ , labels $\mathbf{y}$ , dropout rate $p = 1 - q$ , subsample rate $\xi$ , learning rate $\lambda$ , boosting rounds $B$ . Set $\widehat{\pmb{f}}^{(0)}, \pmb{t}^{(0)} \gets 0$ . +2: for $b = 1, \dots, B - 1$ do +3: Subsample $\mathcal{S}_b\subseteq \{0,\dots ,b - 1\}$ w.p. $q$ +4: Subsample data $\mathcal{G}_b\subseteq \{1,\dots,n\}$ w.p. $\xi$ +5: $z_{i}\gets y_{i} - \frac{\lambda}{b}\sum_{s\in S_{b}}\pmb{t}^{(s)}(\mathbf{x}_{i}).$ +6: Construct tree $\pmb{t}^{(b)}$ on $(\mathbf{x}_i, z_i)_{i \in \mathcal{G}_b}$ . +7: $\widehat{\pmb{f}}^{(b + 1)}\gets \frac{b - 1}{b}\widehat{\pmb{f}}^{(b)} + \frac{\lambda}{b}\pmb{t}^{(b)} = \frac{\lambda}{b}\sum_{i = 1}^{b}\pmb{t}^{(i)}.$ +8: end for +9: Return final predictor $\frac{1 + \lambda q}{\lambda}\hat{\pmb{f}}^{(B)}$ + +# Algorithm 2 BRAT - Parallel (BRAT-P) + +1: Input: Features $\mathbf{X}$ , labels $\mathbf{y}$ , subsample $\xi$ , trees/round $K$ , rounds $B$ . $\hat{\pmb{f}}^{(0)}, \pmb{t}^{(0,k)} \gets 0$ . +2: Fit $\widehat{\pmb{f}}^{(1)},(\pmb{t}^{(1,k)})_{k = 1}^{K}$ with $K$ boosting steps. +3: for $b = 2, \dots, B - 1$ do +4: for $k = 1, \dots, K$ in parallel do +5: Subsample $\mathcal{G}_{b,k} \subseteq \{1, \dots, n\}$ w.p. $\xi$ . +6: $z_{i,k}\gets y_i - \frac{1}{b - 1}\sum_{s = 1}^{b - 1}\sum_{l\neq k}t^{(s,l)}(\mathbf{x}_i)$ +7: Construct tree $\pmb{t}^{(b,k)}$ on $(\mathbf{x}_i, z_{i,k})_{i \in \mathcal{G}_{b,k}}$ . +8: end for +9: Define $\widehat{\pmb{f}}^{(b + 1)}\gets \frac{1}{b}\sum_{s = 1}^{b}\sum_{k = 1}^{K}\pmb{t}^{(s,k)}$ +10: end for +11: Return final predictor $\hat{\pmb{f}}^{(B)}$ + +Random dropout. Unlike vanilla boosting, Algorithm 1 computes residuals with both data (Friedman, 2002) and ensemble subsampling (Rashmi and Gilad-Bachrach, 2015). At each round $b = 1,\dots,B - 1$ , each tree is subsampled with a dropout rate of $p\in [0,1)$ (equivalently, with probability $q = 1 - p$ ). We write $S_{b}\subseteq \{0,\dots,b - 1\}$ for the collection of subsampled tree indices at each round $b$ . Likewise, data indices $\mathcal{G}_b\subseteq \{1,\dots,n\}$ are subsampled with probability $\xi \in (0,1]$ . + +As a result, each tree $\pmb{t}^{(b)}$ is trained on the dataset $(\mathbf{x}_i, z_i)_{i \in \mathcal{G}_b}$ , where the residuals are given by $z_i = y_i - \frac{1}{b} \sum_{s \in S} \pmb{t}^{(s)}(\mathbf{x}_i)$ . Within the residual computation, we intentionally divide the sum of the subsampled ensemble's predictions by $b$ instead of $|\mathcal{S}|$ . This is done so each new tree is fit on more of the signal, allowing for increased signal recovery when applying Boulevard regularization. + +We later show in Section 5 that the predictions of Algorithm 1 converge to a kernel ridge regression that enjoys a central limit theorem. Informally, for any test point $\mathbf{x}$ , we have that: + +$$ +\widehat {\mathbf {y}} _ {n} ^ {(b)} \underset {b \to \infty} {\overset {a. s.} {\longrightarrow}} \left(\lambda^ {- 1} \mathbf {I} + q \mathbb {E} [ \mathbf {S} _ {n} ]\right) ^ {- 1} \mathbb {E} [ \mathbf {S} _ {n} ] \mathbf {y} _ {n}, \| \boldsymbol {r} _ {n} ^ {D} (\mathbf {x}) \| _ {2} ^ {- 1} \left(\widehat {\boldsymbol {f}} _ {n} ^ {D} (\mathbf {x}) - \frac {\lambda}{1 + \lambda q} \boldsymbol {f} (\mathbf {x})\right) \underset {n \to \infty} {\overset {d} {\longrightarrow}} \mathcal {N} (0, \sigma^ {2}), +$$ + +for a kernel matrix induced by the tree ensemble $\mathbf{K}_n = \mathbb{E}[\mathbf{S}_n]$ , and limit $\widehat{\pmb{f}}_n^D = \lim_{b\to \infty}\widehat{\pmb{f}}_n^{(b)}$ where $\widehat{\pmb{f}}_n^D (\mathbf{x}) = \langle \pmb {r}_n^D (\mathbf{x}),\mathbf{y}_n\rangle = \pmb {k}_n(\mathbf{x})^\top (\lambda^{-1}\mathbf{I} + q\mathbf{K}_n)^{-1}\mathbf{y}_n$ . $\| \boldsymbol {r}_n\| _2$ is the norm of kernel weights. Algorithm 1 encompasses two important special cases. When $\lambda = 1$ and $p\rightarrow 1$ , this yields a random forest. When $p = 0$ , Algorithm 1 corresponds to the original Boulevard algorithm (Zhou and Hooker, 2022). Tuning the dropout parameter $p$ allows the user to interpolate between the two, while improving over Zhou and Hooker (2022) by up to a factor of $\frac{1 + \lambda}{1 + \lambda q}\in (1,2]$ in signal recovery. + +Parallel boosting. Algorithm 2 integrates a novel leave-one-out procedure and Boulevard regularization into the parallel boosting framework (Long and Servedio, 2011; Karbasi and Larsen, 2023; da Cunha et al., 2024). The algorithm is warm-started by performing $K$ vanilla boosting iterations in the first round. At each subsequent round $b = 2, \ldots, B - 1$ , we grow $K$ trees $(t^{(b,k)})_{k=1}^{K}$ in parallel on the $K$ datasets $((\mathbf{x}_i, z_{i,k})_{i \in \mathcal{G}_{b,k}})_k)_{k=1}^K$ . These consist of independently subsampled data indices $\mathcal{G}_{b,k} \subseteq \{1, \ldots, n\}$ , and residuals that are computed by leaving one "column" out (or alternatively, one tree out per round): $z_{i,k} = y_i - \frac{1}{b-1} \sum_{s=1}^{b-1} \sum_{l \neq k} t^{(s,l)}(\mathbf{x}_i)$ . The resulting predictor is given by the regularized boosting update: $\hat{\pmb{f}}^{(b+1)} \gets \frac{b-1}{b} \widehat{\pmb{f}}^{(b)} + \frac{1}{b} \sum_{k=1}^{K} \pmb{t}^{(b,k)} = \frac{1}{b} \sum_{s=1}^{b} \sum_{k=1}^{K} \pmb{t}^{(s,k)}$ . + +Likewise, we later show in Section 5 that the predictions of Algorithm 2 also converge to a kernel ridge regression that enjoys a central limit theorem. Informally, for any test point $\mathbf{x}$ : + +$$ +\widehat {\mathbf {y}} _ {n} ^ {(b)} \underset {b \to \infty} {\overset {a. s.} {\longrightarrow}} \left(\mathbf {I} + (K - 1) \mathbb {E} \left[ \mathbf {S} _ {n} \right]\right) ^ {- 1} K \mathbb {E} \left[ \mathbf {S} _ {n} \right] \mathbf {y} _ {n}, \| \boldsymbol {r} _ {n} ^ {P} (\mathbf {x}) \| _ {2} ^ {- 1} \left(\widehat {\boldsymbol {f}} _ {n} ^ {P} (\mathbf {x}) - \boldsymbol {f} (\mathbf {x})\right) \underset {n \to \infty} {\overset {d} {\longrightarrow}} \mathcal {N} (0, \sigma^ {2}), +$$ + +with limit $\widehat{\pmb{f}}_n^P = \lim_{b\to \infty}\widehat{\pmb{f}}_n^{(b)}$ given by $\widehat{\pmb{f}}_n^P (\mathbf{x}) = \langle \pmb {r}_n^P (\mathbf{x}),\mathbf{y}_n\rangle = \pmb {k}_n(\mathbf{x})^\top (\mathbf{I} + (K - 1)\mathbf{K}_n)^{-1}K\mathbf{y}_n.$ Algorithm 2 can be viewed as boosting a modified parallel backfitting algorithm (Breiman and Friedman, 1985)3 that is warm-started with $K$ vanilla boosting iterations.4 When $K = 1,B\to \infty$ , Algorithm 2 reduces to a random forest $\mathbb{E}[\mathbf{S}_n]\mathbf{y}_n$ . On the other hand, Algorithm 2 reduces to vanilla boosting when $B = 1,K\rightarrow \infty$ . Note that the CLT only holds for $1 < K < \infty$ + +The attentive reader may observe that we do not divide the final predictions by $K$ . This is because the combination of leaving one column out when fitting a tree and Boulevard averaging is sufficient to allow our procedure to stabilize. Still, dividing by $K$ yields a viable algorithm (deferred to future work) that can be thought of as boosting random forests each of $K$ trees over $B$ rounds, using the natural strategy of averaging all trees but the left-out column to form predictions. + +# 4 Statistical Inference and Uncertainty Quantification for Boosting + +We now introduce our methods for conducting statistical inference for boosting; the theoretical basis for these procedures is provided below. Our methods rest on the aforementioned convergence to a kernel ridge regression and central limit theorems that we formally prove later in Theorems 1 and 2. In the meantime, we provide the following asymptotically valid procedures. + +3This can be thought of as a structured variant of dropout (Zhao et al., 2022; Xin et al., 2020; Fan et al., 2019). Dropping one out of every $K$ trees allows the regularization to scale in $1 / K$ . +4The initial $K$ boosting iterations yield better convergence in practice, and do not hinder our theoretical arguments as the impact of the vanilla boosting iterations washes out when we take $B\to \infty$ +${}^{5}$ Parallelized backfitting can be thought of as parallelized coordinate descent. Averaging over the coordinate updates is sufficient for convergence. Dividing by $b$ ,as in Boulevard averaging,is sufficient when $b \geq K$ . + +Estimation of asymptotic variance. The CLTs for Algorithms 1 and 2 imply that one can construct confidence intervals for $f$ with the following normal approximations: + +$$ +\lambda^ {- 1} (1 + \lambda q) \widehat {\boldsymbol {f}} _ {n} ^ {D} (\mathbf {x}) \overset {d} {\approx} \mathcal {N} \left(\boldsymbol {f} (\mathbf {x}), \lambda^ {- 2} (1 + \lambda q) ^ {2} \| \boldsymbol {r} _ {n} ^ {D} (\mathbf {x}) \sigma^ {2} \| _ {2} ^ {2}\right), \quad \widehat {\boldsymbol {f}} _ {n} ^ {P} (\mathbf {x}) \overset {d} {\approx} \mathcal {N} \left(\boldsymbol {f} (\mathbf {x}), \sigma^ {2} \| \boldsymbol {r} _ {n} ^ {P} (\mathbf {x}) \| _ {2} ^ {2}\right). +$$ + +As such, it remains to estimate $\boldsymbol{r}_n^D, \boldsymbol{r}_n^P$ , and $\sigma^2$ . We first estimate $\boldsymbol{k}_n(\mathbf{x})$ by the sample averages + +$$ +\widehat {\boldsymbol {k}} _ {n} ^ {D} (\mathbf {x}) = \frac {1}{B} \sum_ {b = 1} ^ {B} s _ {n, i} ^ {(b)} (\mathbf {x}, \mathcal {G} _ {b}), \quad \widehat {\boldsymbol {k}} _ {n} ^ {P} (\mathbf {x}) = \frac {1}{B K} \sum_ {b = 1} ^ {B} \sum_ {k = 1} ^ {K} s _ {n, i} ^ {(b, k)} (\mathbf {x}, \mathcal {G} _ {b, k}), \tag {1} +$$ + +where $s_{n,i}^{(b)}(\mathbf{x};\mathcal{G}_b)$ is the fraction of subsampled datapoints in the same leaf $A_j^{(b)}$ as $\mathbf{x}$ over the trees. Likewise, we estimate $(\mathbf{K}_n)_{i,j}$ , the fraction of trees where datapoints $i$ and $j$ fall in the same leaf: + +$$ +\left(\widehat {\mathbf {K}} _ {n} ^ {D}\right) _ {i, j} = \frac {1}{B} \sum_ {b = 1} ^ {B} s _ {n, j} ^ {(b)} \left(\mathbf {x} _ {i}; \mathcal {G} _ {b}\right), \quad \left(\widehat {\mathbf {K}} _ {n} ^ {P}\right) _ {i, j} = \frac {1}{B} \sum_ {b = 1} ^ {B} s _ {n, j} ^ {(b, k)} \left(\mathbf {x} _ {i}; \mathcal {G} _ {b, k}\right), \tag {2} +$$ + +Our estimates of $\widehat{r}_n$ are then given by plugging the above into the formula for $r_n$ : + +$$ +\widehat {\pmb {r}} _ {n} ^ {D} (\mathbf {x}) ^ {\top} = \widehat {\pmb {k}} _ {n} ^ {D} (\mathbf {x}) ^ {\top} (\lambda^ {- 1} \mathbf {I} + q \widehat {\mathbf {K}} _ {n} ^ {D}) ^ {- 1}, \widehat {\pmb {r}} _ {n} ^ {P} (\mathbf {x}) ^ {\top} = \widehat {\pmb {k}} _ {n} ^ {P} (\mathbf {x}) ^ {\top} (\mathbf {I} + (K - 1) \widehat {\mathbf {K}} _ {n} ^ {P}) ^ {- 1} K. +$$ + +Lastly, we estimate $\sigma^2$ with the residuals $\hat{\sigma}^2 = \frac{1}{n_{\mathrm{cal}}} \sum_{i=1}^{n_{\mathrm{cal}}} (y_i - \widehat{y}_i)^2$ on a hold-out calibration set. We describe a tweak later in Section 6 that uses the calibration set for increased robustness. + +**Confidence intervals for $f$ .** Putting everything together yields $1 - \alpha$ confidence intervals for $f$ (where $z_{1 - \alpha /2}$ is the $(1 - \alpha /2)$ -quantile of the standard normal distribution): + +$$ +\lambda^ {- 1} (1 + \lambda q) \widehat {\boldsymbol {f}} _ {n} ^ {D} (\mathbf {x}) \pm z _ {1 - \alpha / 2} \lambda^ {- 1} (1 + \lambda q) \widehat {\sigma} \| \widehat {\boldsymbol {r}} _ {n} ^ {D} (\mathbf {x}) \| _ {2}, \quad \widehat {\boldsymbol {f}} _ {n} ^ {P} (\mathbf {x}) \pm z _ {1 - \alpha / 2} \widehat {\sigma} \| \widehat {\boldsymbol {r}} _ {n} ^ {P} (\mathbf {x}) \| _ {2}. \quad (3) +$$ + +Prediction intervals for y. We can construct prediction intervals for $y|\mathbf{x}$ as follows: + +$$ +\lambda^ {- 1} (1 + \lambda q) \widehat {\pmb {f}} _ {n} ^ {D} (\mathbf {x}) \pm z _ {1 - \alpha / 2} \lambda^ {- 1} (1 + \lambda q) \widehat {\sigma} \sqrt {1 + \| \widehat {\pmb {r}} _ {n} ^ {D} (\mathbf {x}) \| _ {2} ^ {2}}, \quad \widehat {\pmb {f}} _ {n} ^ {P} (\mathbf {x}) \pm z _ {1 - \alpha / 2} \widehat {\sigma} \sqrt {1 + \| \widehat {\pmb {r}} _ {n} ^ {P} (\mathbf {x}) \| _ {2} ^ {2}}. +$$ + +These prediction intervals have asymptotic pointwise coverage, conditional on the test point $\mathbf{x}$ . That is, $\mathbb{P}(y\in \mathrm{PI}(\mathbf{x})|\mathbf{x})\to 1 - \alpha$ as $n\to \infty$ , which holds as a corollary of the CLTs we present in the following section. Note that this conditional coverage guarantee is a stronger guarantee than is typically possible with conformal inference (Barber et al., 2020). We compare our prediction intervals to conformal prediction intervals within our numerical experiments later in this paper. + +Reproduction intervals for $\widehat{f}$ . We now turn our attention towards computing reproduction intervals. That is, a confidence interval for another realization of $\widehat{f}$ trained on another independent realization of the data. To do so, as Zhou and Hooker (2022) suggest, it suffices to scale the width of the above confidence intervals for $f$ by $\sqrt{2}$ as $X_{1}, X_{2} \stackrel{i.i.d.}{\sim} \mathcal{N}(0, \sigma^{2}) \Rightarrow X_{1} - X_{2} \sim \mathcal{N}(0, 2\sigma^{2})$ . Note that one can invert the above confidence, prediction, or reproduction intervals in order to test if the underlying function, another realization of the response, or average learner is equal to a given constant. This amounts to checking if the constant is contained in the interval of choice. + +Variable importance tests. Suppose we wish to test for variable importance. Consider the possibility that there exists some subset of the feature space $[0,1]^d$ containing only $\check{d} < d$ features, and let $\pmb{g}:[0,1]^{\check{d}}\to \mathbb{R}$ be the projection of $\pmb{f}$ onto $\mathcal{L}^2 ([0,1]^{\check{d}})$ . We test the possibility that $\pmb{f}(\mathbf{x}) = \pmb{g}(\mathbf{x})$ by proxy. Consider a split of the training dataset $(\mathbf{X}_n,\mathbf{y}_n)$ into $(\mathbf{X}_{n,1},\mathbf{y}_{n,1}),(\mathbf{X}_{n,2},\mathbf{y}_{n,2})$ , where $\mathbf{X}_{n,1}\in \mathbb{R}^{n / 2,d}$ , $\mathbf{X}_{n,2}\in \mathbb{R}^{n / 2,\check{d}}$ . Let $\widehat{\pmb{f}}_{n,1}$ be the boosting learner trained on $(\mathbf{X}_n,\mathbf{y}_n)$ , and $\widehat{\pmb{f}}_{n,2}$ be the same trained on $(\mathbf{X}_{n,2},\mathbf{y}_{n,2})$ . Given a hold-out dataset $(\mathbf{X}_m,\mathbf{y}_m)$ , we test the null: + +$$ +H _ {0}: \boldsymbol {f} (\mathbf {x} _ {j}) = \boldsymbol {g} (\mathbf {x} _ {j}) \text {f o r a l l} j = 1, \dots , m \text {a g a i n s t} H _ {1}: \boldsymbol {f} (\mathbf {x} _ {j}) \neq \boldsymbol {g} (\mathbf {x} _ {j}) \text {f o r s o m e} j +$$ + +by comparing $\widehat{f}_{n,1}(\mathbf{x}_j)$ and $\widehat{f}_{n,2}(\mathbf{x}_j)$ . We exploit the CLTs our algorithms enjoy to do so. Write $\widehat{\pmb{r}}_n^{D,P}(\mathbf{X}_m)^\top \in \mathbb{R}^{m\times n}$ for the matrix with rows $(\widehat{r}_n^{D,P}(\mathbf{x}_l)^\top)_{l = 1}^m$ , where we write $D,P$ within the superscript if the formula holds for both Algorithms 1 and 2. The difference in predictions + +$$ +\widehat {\boldsymbol {d}} _ {m} = \left(\widehat {\boldsymbol {r}} _ {n, 1} ^ {D, P} \left(\mathbf {X} _ {m}\right) - \widehat {\boldsymbol {r}} _ {n, 2} ^ {D, P} \left(\mathbf {X} _ {m}\right)\right) ^ {\top} \mathbf {y} \xrightarrow {d} \mathcal {N} _ {m} \left(\mathbf {0}, \sigma^ {2} \widehat {\boldsymbol {\Xi}} _ {n}\right) \text {i s m u l t i v a r i a t e n o r m a l u n d e r t h e n u l l}, +$$ + +with covariance matrix $\sigma^2\hat{\Xi}_n = \sigma^2 (\hat{r}_{n,1}^{D,P}(\mathbf{X}_m) - \hat{r}_{n,2}^{D,P}(\mathbf{X}_m))^{\top}(\hat{r}_{n,1}^{D,P}(\mathbf{X}_m) - \hat{r}_{n,2}^{D,P}(\mathbf{X}_m))$ estimated by plugging in $\hat{\sigma}^2$ . As such, we can conduct a chi-squared test with test statistic + +$$ +\widehat {\sigma} ^ {- 2} \widehat {\boldsymbol {d}} _ {m} ^ {\top} \widehat {\boldsymbol {\Xi}} _ {n} ^ {- 1} \widehat {\boldsymbol {d}} _ {m} \sim \chi_ {m} ^ {2} \text {u n d e r t h e n u l l , r e j e c t i n g t h e n u l l i f} \widehat {\sigma} ^ {- 2} \widehat {\boldsymbol {d}} _ {m} ^ {\top} \widehat {\boldsymbol {\Xi}} _ {n} ^ {- 1} \widehat {\boldsymbol {d}} _ {m} > \chi_ {m, 1 - \alpha} ^ {2}. +$$ + +The test as presented runs in $O(n^{3})$ time. In Appendix A, we present an accelerated version of the test that runs in $O(ns(r + s) + r^3)$ time, where $s$ is the number of points subsampled for Nyström approximation and $r$ is the number of test points subsampled. + +Matrix sketching. Although the CLT reduces the problem of inference to kernel ridge regression (with the right kernel), computing $\widehat{\mathbf{K}}_n$ is an $O(n^3)$ problem – intractable for large $n$ . This is the chief difficulty that Zhou and Hooker (2022) face when constructing replication intervals for vanilla Boulevard. We bypass this issue, making our procedures practical and tractable via matrix sketching. + +We approximate $\widehat{\mathbf{K}}_n$ using the Nyström method (Williams and Seeger, 2000), via either uniform subsampling or the recursive approach of Musco and Musco (2017), producing an approximation $\widetilde{\mathbf{K}}_n \approx \widehat{\mathbf{K}}_n \mathbf{S} (\mathbf{S}^\top \widehat{\mathbf{K}}_n \mathbf{S})^\dagger \mathbf{S}^\top \widehat{\mathbf{K}}_n$ , where $\mathbf{S} \in \mathbb{R}^{n \times s}$ is a random subsampling matrix. With the Nystrom method of Musco and Musco (2017), we can choose $s$ to be near-linear in the effective dimension $d_{\mathrm{eff}}^{\mu}$ of a ridge regression problem with regularization parameter $\mu$ on the kernel matrix: $s = \widetilde{O}(d_{\mathrm{eff}}^{\mu})$ . This requires only $O(ns^2)$ time to precompute the kernel, and only $O(s^2)$ time for inference - yielding practical statistical inference in near-linear time in the number of datapoints $n$ . We adopt this approach in our experiments, allowing our procedures to run in linear time and remain practical, in contrast to previous work (Zhou and Hooker, 2022; Mentch and Hooker, 2016b). See Appendix A. + +# 5 Theoretical Guarantees + +We now formally present guarantees for convergence and asymptotic normality for Algorithms 1 and 2. Consider the (Lipschitz) nonparametric regression model introduced below: + +Assumption 3 (Lipschitz Nonparametric Regression). Let $y = f(\mathbf{x}) + \epsilon$ , where $\mathbf{x}$ has density $\mu$ satisfying $0 < c_{1} \leq \mu (\mathbf{x}) \leq c_{2} < \infty$ on its support. Further assume $f$ is $\alpha$ -Lipschitz continuous on $\operatorname{supp}(\mu)$ , and noise $\epsilon$ is sub-Gaussian with variance proxy $\sigma^2$ . + +Finite-sample convergence to KRR. We now show that our algorithms achieve finite-sample convergence to kernel ridge regression as the number of boosting rounds increases. At first glance, this is surprising - greedy split selection and sequential ensemble construction create a highly nonlinear and time-dependent mapping. Fortunately, the regularization procedure of Zhou and Hooker (2022) provides a way forward. For the special case of $p = 0$ within Algorithm 1, they show convergence to a kernel ridge regression via the stochastic contraction framework of Almudevar (2022), utilizing Assumptions 1 (structure-value isolation) and 2 (non-adaptivity) to ensure stagewise stability. A key novelty within our analysis lies in showing that this convergence extends to Algorithms 1 and 2, even with the added complexity introduced by dropout and parallelism respectively. Despite these added sources of stochasticity and dependence, our use of the same regularization mechanism ensures that both algorithms inherit the stability required for convergence: + +Theorem 1 (Finite Sample Convergence to KRR). For fixed $\mathbf{X},\mathbf{y}$ , under Assumptions 1, 2, and 3, + +$$ +\widehat {\mathbf {y}} _ {b} \stackrel {{a. s.}} {{\rightarrow}} \left(\lambda^ {- 1} \mathbf {I} + q \mathbb {E} \left[ \mathbf {S} _ {n} \right]\right) ^ {- 1} \mathbb {E} \left[ \mathbf {S} _ {n} \right] \mathbf {y} f o r A l g. 1, \widehat {\mathbf {y}} _ {b} \stackrel {{a. s.}} {{\rightarrow}} \left(I + (K - 1) \mathbb {E} \left[ \mathbf {S} _ {n} \right]\right) ^ {- 1} K \mathbb {E} \left[ \mathbf {S} _ {n} \right] \mathbf {y} f o r A l g. 2, +$$ + +as $b \to \infty$ , where $q = 1 - p$ is one minus the dropout probability, $K$ is the number of trees grown in parallel, and $\mathbf{S}_n$ is the kernel matrix induced by the tree structures within the ensemble. + +The proof, deferred to Appendix C, departs from Zhou and Hooker (2022) in two crucial ways. Within Algorithm 1, dropout injects unbounded variance into the ensemble updates. To control this, we introduce the hard truncation function $\Gamma_M(y) = \mathrm{sign}(y)\min \{M,|y|\}$ into each residual update, $\widehat{\mathbf{y}}_b = \frac{b - 1}{b}\widehat{\mathbf{y}}_{b - 1} + \frac{\lambda}{b}\mathbf{S}_b\big(\mathbf{y} - \Gamma_M(\widetilde{\mathbf{y}}_b)\big)$ , and then show that the probability of the partial ensemble escaping this cap vanishes as $b\to \infty$ . Within Algorithm 2, truncating $\widetilde{\mathbf{y}}_{b,k} = \widehat{\mathbf{y}}_{b - 1,K} - \Gamma_M(\frac{1}{b - 1}\sum_{g = 1}^{b - 1}\widehat{t}_{b,k})$ alone is insufficient. We further introduce a delay mechanism, requiring that $\pmb{t}_{b,k}$ does not rely on $\pmb{t}_{b,k - 1}$ , allowing us to apply Theorem 3 and parallelize tree training. + +A central limit theorem. Having established almost-sure convergence to fixed points, we now examine the asymptotic distribution of our estimators to justify the statistical procedures introduced in Section 4. We seek a central limit theorem for the learner $\widehat{\pmb{f}}_n^{D,P}$ , trained on random $\mathbf{X}_n,\mathbf{y}_n$ , demonstrating that the predictions $\widehat{\pmb{f}}_n^{D,P}(\mathbf{x})$ are asymptotically normal with mean $\pmb{f}(\mathbf{x})$ as $n\to \infty$ . To do so, we inherit the following assumptions from Zhou and Hooker (2022): two on the leaves to control the norm of the KRR weight vector $\pmb{r}_n^{D,P}$ , and a restriction on the tree distribution space: + +Assumption 4 (Bounded Leaf Diameter). Write $\mathrm{diam}(A) = \sup_{x,y\in A}\| x - y\|$ . For any leaf $A$ in a tree with structure $q\in Q_{n}$ , we need $\sup_{A\in q}\mathrm{diam}(A) = O(d_n)$ , where $d_{n} = O(n^{-1 / (d + 1)})$ . + +Assumption 5 (Increased Minimal Leaf Size). For any $\nu >0$ , $v_{n} = n^{-\frac{d + 1}{d + 2} +\nu} < n^{-\frac{d}{d + 1}} = O(d_n^d)$ . + +Assumption 6 (Restricted Tree Support). The cardinality of the tree space $Q_{n}$ is bounded by $O(n^{-1}\exp (0.5n^{1 / (d + 2) - \nu} - n^{\alpha}))$ , for some small $\alpha > 0$ . + +Intuitively, Assumption 5 prevents leaves from becoming too small too quickly, which in turn controls the maximal coordinate of the weight vectors across random samples. Assumption 6, guarantees that the complexity of possible tree partitions does not explode with $n$ . We also make assumptions on the regularity of tree splits in line with the notion of $\alpha$ -regularity in Athey et al. (2018): + +Assumption 7 (Median Splitting Rules). The trees in Algorithm 2 are split at the medians. + +We defer the details to Definition 3. Under this splitting rule, the point-to-point collision probability is well-controlled, i.e. to say none of the leaves are big enough to contain most of the points. These conditions allow for uniform control over both leaf sizes and tree complexity, yielding the result: + +Theorem 2 (Central Limit Theorem for Predictions). Let $\mathbf{x} \in [0,1]^d$ , $q \in (0,1]$ , $K > 1$ . As $n \to \infty$ + +$$ +\left\| \boldsymbol {r} _ {n} ^ {D} \right\| _ {2} ^ {- 1} \left(\widehat {\boldsymbol {f}} _ {n} ^ {D} (\mathbf {x}) - \lambda^ {- 1} (1 + \lambda q) \boldsymbol {f} (\mathbf {x})\right) \xrightarrow {d} \mathcal {N} (0, \sigma^ {2}), \quad \left\| \boldsymbol {r} _ {n} ^ {P} \right\| _ {2} ^ {- 1} \left(\widehat {f} _ {n} ^ {P} (\mathbf {x}) - f (\mathbf {x})\right) \xrightarrow {d} \mathcal {N} (0, \sigma^ {2}). +$$ + +We prove this in two stages. In the first, with proof deferred to Appendix D, we show asymptotic normality of the predictions around the noiseless KRR predictions $\langle \pmb{r}_n^{D,P}(\mathbf{x}),\pmb {f}(\mathbf{X}_n)\rangle$ , conditional on the training data $\mathbf{X}_n,\mathbf{y}_n$ and test point $\mathbf{x}$ , by applying the Lindeberg-Feller CLT to $\langle \mathbf{r}_n^{D,P}(\mathbf{x}),\varepsilon \rangle$ . Next, we show that $\langle \pmb {r}_n^{D,P}(\mathbf{x}),\pmb {f}(\mathbf{X}_n)\rangle$ converges to the underlying ground truth function at a sufficiently fast rate in Appendices E and F. For Algorithm 1, although $\pmb{r}_n^D$ now involves $[\frac{1}{\lambda}\mathbf{I} + q\mathbf{K}_n]^{-1}$ , this still preserves the rate $\| \pmb {r}_n\| _2\asymp n^{-1 / (2(d + 1))}$ , allowing us to show $\langle \mathbf{r}_n,f(\mathbf{X}_n)\rangle -\frac{\lambda q}{1 + \lambda q}\pmb {f}(\mathbf{x}) = o_p(\| \pmb {r}_n\| _2)$ . The Lindeberg-Feller verification then proceeds identically to the non-dropout case. However, to give a CLT for Algorithm 2, controlling the influence from distant training points is challenging, though we show that Assumption 7 is sufficient to do so. + +Understanding the result. The rate of convergence of the CLT in Theorem 2 depends on the norm of $\pmb{r}_n^{D,P}$ . This is controlled as follows, yielding generalization bounds for Algorithms 1 and 2: + +Lemma 1 (Rate of Convergence). Let $\mathbf{B}_n \coloneqq \{i : \| \mathbf{x} - \mathbf{x}_i \| \leq d_n\}$ be the points within distance $d_n$ from test point $\mathbf{x}$ . If $|\mathbf{B}_n| = \Omega(n \cdot d_n^d)$ , then $\| \mathbf{k}_n \|_2 = \Theta(n^{-\frac{1}{2}} \frac{1}{d + 1})$ , $\| r_n^{D,P} \|_2 = \Theta(n^{-\frac{1}{2}} \frac{1}{d + 1})$ . + +Corollary 1. $\mathbb{E}[(\frac{1 + \lambda q}{\lambda}\widehat{\pmb{f}}_n^D (\mathbf{x}) - \pmb {f}(\mathbf{x}))^2 ]\lesssim (\frac{1 + \lambda q}{\lambda})^2\sigma^2 n^{-\frac{1}{d + 1}}$ and $\mathbb{E}[(\widehat{f}_n^P (\mathbf{x}) - f(\mathbf{x}))^2 ]\lesssim \sigma^2 n^{-\frac{1}{d + 1}}$ + +Corollary 2. If $n \geq \Omega\left(\frac{\log(1 / \delta)}{\epsilon^{2d + 2}}\right)$ , $b \geq \Omega\left(\frac{n\lambda^2M^2}{\epsilon^2\delta}\right)$ , then w.p. at least $1 - \delta$ , $|\widehat{\pmb{f}}_n^{(b)}(\mathbf{x}) - \pmb {f}(\mathbf{x})|\leq \epsilon$ + +The first is an asymptotic risk bound, while the second is a nonasymptotic PAC guarantee. Recall that $M$ is the truncation level. The results above recover the minimax rate for nonparametric regression on 1/2-Holder smooth functions (Stone, 1982) – quadratically worse than the rate for Lipschitz functions (the setting we are in). We believe this can be improved – trees should inherit adaptivity to the intrinsic dimension as an adaptive nearest neighbor method, but we leave this for future work. + +Furthermore, Theorem 2 yields asymptotic coverage of our intervals constructed in Section 4: + +Corollary 3. The CIs, PIs and RIs achieve $1 - \alpha$ almost sure pointwise coverage as $n, b \to \infty$ . + +We show in Lemma 1 that the rate at which $\| r_n^{D,P}\| _2$ grows is the same for Algorithms 1 and 2. This shows that our methods achieve an asymptotic relative efficiency relative to Zhou and Hooker (2022) by up to a factor of 4 for Algorithm 1, and at least a factor of 4 for Algorithm 2: + +Corollary 4 (Asymptotic Relative Efficiency). Write $\hat{\pmb{f}}^B$ for the scaled predictions made by vanilla Boulevard (Zhou and Hooker, 2022) with the same learning rate $\lambda \in (0,1]$ as Algorithm 1. Then, + +$$ +V a r (\frac {1 + \lambda}{\lambda} \widehat {\boldsymbol {f}} ^ {B}) / V a r (\frac {1 + \lambda q}{\lambda} \widehat {\boldsymbol {f}} ^ {D}) = (\frac {1 + \lambda}{1 + \lambda q}) ^ {2} \in [ 1, 4 ], V a r (\frac {1 + \lambda}{\lambda} \widehat {\boldsymbol {f}} ^ {B}) / V a r (\widehat {\boldsymbol {f}} ^ {P}) = \left(\frac {1 + \lambda}{\lambda}\right) ^ {2} \in [ 4, \infty). +$$ + +This has the implication that Algorithms 1 and 2 achieve increased signal recovery in the square root of the ARE relative to vanilla Boulevard. The intuition for this improvement is as follows. For any choice of $\lambda$ within vanilla Boulevard, there exists a choice of $p$ within Algorithm 1 that requires a smaller rescaling. On the other hand, as Algorithm 2 requires no rescaling, it can achieve an unbounded improvement in relative efficiency over vanilla Boulevard as we take $\lambda \rightarrow 0$ . + +![](images/dcd005a06621eaf471601565971eb49b138eae8d636e24d7cef4e556ac9d0098.jpg) +Figure 3: Left: Type I and Type II error of the hypothesis test for variable importance against training set size. Test set is of the same size. Error rates computed over 30 trials. Right: Comparison of MSE achieved by various machine learning algorithms on different datasets, with hyperparameters tuned for each by Optuna. Shaded area depicts two standard deviations over 5 trials. + +![](images/231251533859a6a9056a69da466bb3428eda9fb407626d47e0ee8a55eac64188.jpg) + +![](images/4dfc523ec9ab9af325960ed118627e8e044528b7ab0da2eeafa4115fa630397a.jpg) + +![](images/e5474c0187f051bea048e6f300d6ed6d17bff71d33f546ea9f67da5b179b9e7e.jpg) + +# 6 Numerical Experiments + +The previous section presents a rich theory; alongside convergence to kernel ridge regression and a central limit theorem for predictions, we provide asymptotic risk bounds and nonasymptotic PAC guarantees, and show asymptotic coverage for our intervals. In light of this, we present a series of numerical experiments to empirically justify our algorithms and methods. + +Predictive accuracy. The first, and most natural, thing to do is to examine the performance of our algorithms against a handful of competitors in terms of test MSE on nine datasets from the UCI Machine Learning Repository in Figure 3 and Figure 6 in the supplement. Our competitors include popular methods such as XGBoost (Chen and Guestrin, 2016), LightGBM (Ke et al., 2017), as well as random forests and vanilla boosting. We also compare to vanilla Boulevard (Zhou and Hooker, 2022), and an elastic net regression as a baseline. All hyperparameters were tuned with Optuna (Akiba et al., 2019), reported in Appendix I. There is no clear winner in general, but XGBoost is a consistently strong performer, and random forests are clearly more suited to some datasets than others. Neither Algorithm 1 nor Algorithm 2 consistently outperform each other, but Algorithm 2 occasionally exhibits instability on some datasets. Regardless, both of our algorithms are competitive in terms of final MSE and rate of convergence, and exhibit the ability to be tuned to be closer to boosting (Wine Quality) or random forests (Air Quality), providing increased flexibility by interpolating between the two. + +Our results in Figure 3 are somewhat unfair to Algorithm 2, as the x-axis is in the ensemble size and not the number of boosting rounds. The number of boosting rounds that Algorithm 2 encounters can be as few as 31 on Abalone, in contrast to the 500 boosting rounds all other algorithms enjoy. Given the same number of boosting rounds, we would certainly expect an improvement. + +Variable importance tests. We examine their performance by conducting a simulation study, testing the null $H_0: w = 0$ on data from $f(\mathbf{x}) = 4x_{1} - x_{2}^{2} + wbx_{3}$ . We fit Algorithm 1 on data generated from $f(\mathbf{x})$ and $g(\mathbf{x}) = 4x_{1} - x_{2}^{2}$ , with 100 trees, $\lambda = 1$ , subsampling rate 1, dropout rate 0.95, and a max depth of 6. The size and power of our test presented in Section 4 are depicted in the left panel within Figure 3. Our test performs very well, maintaining appropriate size control throughout while the Type II error decreases quickly. Empirically, increasing the dropout rate increases power. + +Coverage of intervals. Other than the Nyström approximations for fast and practical interval computation in Appendix A, there is one more tweak that we encourage users to make in practice. We demonstrate in Corollary 3 that our intervals have asymptotic coverage, but this says nothing about its finite sample properties. Fortunately, Romano et al. (2020) yields a simple enhancement. Since we estimate $\widehat{\sigma}$ through computing the residuals on a hold-out calibration set, we reuse the calibration set to adaptively grow or shrink our intervals according to the empirical prediction interval coverage on the calibration set. This procedure converges quickly (amounting to a doubling trick and binary search), and empirically is very helpful in increasing robustness and finite-sample performance. + +To examine the empirical performance of our intervals, we consider the Friedman function $f(\mathbf{x}) = 10\sin (\pi \mathbf{x}_1\mathbf{x}_2) + 20(\mathbf{x}_3 - 0.5)^2 + 5\mathbf{x}_5 - 10$ (Friedman, 2000). This is depicted as a raincloud plot in Figure 4. Importantly, we consider two notions of coverage: marginal coverage, where coverage is averaged over test points, and conditional coverage, where the coverage is conditional on a test point. The first two rows depict the former, while the last two depict the latter. Our intervals perform reasonably well, attaining nominal coverage with the exception of the reproduction interval's coverage and the confidence interval's conditional coverage. In particular, the prediction interval performs + +![](images/4ee588bd8b818d43865389a11c06cb48fb416826fd1cb5d4bc7555e7bebf9cd2.jpg) + +![](images/6ed22e7aa85610ea6e66540342c487aedac01841914094bc9c5b222688c2f821.jpg) + +![](images/be8b271e1f9fe652e2c2494386822715e7002aaaa1c3e77eb4e0c0f6dbe50648.jpg) + +![](images/e46fdf1cc8d8abb8a23d95e269a75c92e8bf812a7c0cce9d3058007897212f54.jpg) + +![](images/6462645117842979e2a1ffa7057678c154b7818694a5b5e17db8d6e1519f383f.jpg) + +![](images/12bb4c54cc7e07ea6ba8874973e6120125877ee790599b8001dc9bd87847cf87.jpg) + +![](images/0e076f11563e6ff694f4753466e7e210a4fd447593c42d6c48ed7245b2aad27c.jpg) + +![](images/4bdf7b0f6e1a8139d79830664c62e404a5ee455346af9376e2773fedbe624af8.jpg) + +![](images/8173664a051ccb1610a66fdc64582c1d85d9d6e45a5646aebfbeddadbc46d447.jpg) + +![](images/bd8335df6917025e37bd22da9dc38328404c08ef1c97591d6b6b80c288a41995.jpg) + +![](images/d3a7e36621a14a32ecb7a814374d1f2b84b0577eae963f1948df1d469136354e.jpg) + +![](images/4e6fb1b8e3622acbbec329cc2fdd39227e8278f0020cbf8f6234224cc0406ddf.jpg) + +![](images/c2073e709bf8936fe7f47846820d5619892277a3c79a34b95bd47ec2965746de.jpg) +Figure 4: Points in the first two rows represent one of 30 models, depicting the fraction of test points that fell inside the interval generated by the model (marginal coverage) and average width for each model. Points in the last two represent one of 100 test points, depicting the fraction of models with intervals containing the test point (conditional coverage) and average width for each point. Results from Algorithm 1 with 200 trees, learning rate 0.6, subsampling 0.8, dropout 0.3, max depth 4. + +![](images/9e61a3c19aba22b6f1afa405ad8d74a7ab5c23a123ca037cd0b82b9a4313689b.jpg) + +![](images/c67f713f203770232688ca137f99beb0becb2fcfbb0342470552699e669d275f.jpg) + +![](images/8c6931290a0b9d74a78a01f968ca4bb280e365d86dfa88a446aa7c96290f9527.jpg) + +very well after the adaptive coverage adjustment. Importantly, unlike the conformal benchmark that has a constant width at each point, our prediction intervals have different interval widths at each point, allowing users to identify 'hard' examples. See Appendix J for other hyperparameter choices. + +CI tuning. Reducing subsample rate $\xi$ inflates variance proportionally, while deeper trees yield narrower intervals as finer splits yield sparser $k_{n}(\mathbf{x})$ and smaller $\| \boldsymbol{r}_n\|$ . Variance grows with shrinkage $1 / \lambda$ and falls as dropout $p$ increases. Choosing moderate $\lambda$ and increasing $p$ yields stabil CIs. Setting $p\to 0$ empirically leads to wider intervals and less power in the hypothesis test for variable importance. This can be seen in Figure 5. + +![](images/05430c4348d6bcbc38841e6b319f00dd019cf805940ba31bbc884a1c7ae6534b.jpg) +Type I & Type II Error vs Dropout Rate + +![](images/474408beb6491f1f83c798b4533d458db6a5c4308218d123204af0067130dd47.jpg) + +![](images/bdbbfeab716fd58c9531607f7f9516f7b9ad8a23f5d33cbfb99f28fb7b8c13f5.jpg) + +![](images/0ea2185f6f36e90393ca4a215c02c4a4e79044e3011251cd288bb96c60bb92bd.jpg) + +![](images/aa65c6e1dbeaad486ea84e917c6cc4c06c402c7d24c801760e1688fd249f47f5.jpg) +Figure 5: Comparison of size and power of statistical inference procedures against dropout probability. Conformal adjustment is not performed (for clearer benchmarking, hence the under/overcoverage), and we plot the coverage and width across all datapoints and repetitions. + +![](images/cb9e0e9a04ba771e90b52618c38f85a2680c56b28f45cc9f659a12495a202080.jpg) + +# 7 Conclusions and Further Work + +We incorporate dropout and parallel boosting into the regularization procedure of Zhou and Hooker (2022), allowing for increased signal recovery and improved asymptotic variance. With the CLTs our algorithms enjoy, we construct confidence and prediction intervals, and hypothesis tests for variable importance. Numerical experiments empirically validate our algorithms and statistical procedures. + +Our theoretical guarantees depend on structure-value isolation, non-adaptivity, and tree regularity assumptions. Relaxing these - e.g., via honesty-based splitting or quantifying mild adaptivity - would broaden applicability. 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Deebert: Dynamic early exiting for accelerating bert inference. + +Yin, X., Fallah-Shorshani, M., McConnell, R., Fruin, S., Chiang, Y.-Y., and Franklin, M. (2023). Quantile extreme gradient boosting for uncertainty quantification. + +Zhao, Y., Dada, O., Gao, X., and Mullins, R. D. (2022). Revisiting structured dropout. + +Zhou, Y. and Hooker, G. (2022). Boulevard: Regularized stochastic gradient boosted trees and their limiting distribution. Journal of Machine Learning Research, 23(183):1-44. + +# A Matrix Sketching + +We first sketch the kernel inversion where we approximate $\widehat{\mathbf{K}}_n^D\approx \widetilde{\mathbf{K}}_n^D = \widehat{\mathbf{K}}_n^D\mathbf{S}(\mathbf{S}^\top \widehat{\mathbf{K}}_n^D\mathbf{S})^\dagger \mathbf{S}^\top \widehat{\mathbf{K}}_n^D$ + +$$ +\begin{array}{l} \widehat {\boldsymbol {r}} _ {n} ^ {D} (\mathbf {x}) ^ {\top} \mathbf {y} _ {n} = \widehat {\boldsymbol {k}} _ {n} ^ {D} (\mathbf {x}) ^ {\top} \left(\lambda^ {- 1} \mathbf {I} + q \widehat {\mathbf {K}} _ {n} ^ {D}\right) ^ {- 1} \mathbf {y} _ {n} \\ \approx \widehat {\boldsymbol {k}} _ {n} ^ {D} (\mathbf {x}) ^ {\top} (\lambda^ {- 1} \mathbf {I} + q \widetilde {\mathbf {K}} _ {n} ^ {D}) ^ {- 1} \mathbf {y} _ {n} \\ = \widehat {\boldsymbol {k}} _ {n} ^ {D} (\mathbf {x}) ^ {\top} \left(\lambda^ {- 1} \mathbf {I} + q \widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S} \left(\mathbf {S} ^ {\top} \widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S}\right) ^ {\dagger} \mathbf {S} ^ {\top} \widehat {\mathbf {K}} _ {n} ^ {D}\right) ^ {- 1} \mathbf {y} _ {n} \\ = \widehat {\boldsymbol {k}} _ {n} ^ {D} (\mathbf {x}) ^ {\top} \left(\lambda \mathbf {I} - \lambda^ {2} \widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S} (q ^ {- 1} \mathbf {S} ^ {\top} \widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S} + \lambda (\widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S}) ^ {\top} \widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S}) ^ {- 1} \mathbf {S} ^ {\top} \widehat {\mathbf {K}} _ {n} ^ {D}\right) \mathbf {y} _ {n} \\ = \widehat {\boldsymbol {k}} _ {n} ^ {D} (\mathbf {x}) ^ {\top} \widehat {\boldsymbol {\Lambda}} _ {n} ^ {D} \mathbf {y} _ {n} = \widehat {\boldsymbol {k}} _ {n} ^ {D} (\mathbf {x}) ^ {\top} \widehat {\boldsymbol {\alpha}} _ {n}, \\ \end{array} +$$ + +where we define the $n$ -dimensional coefficient vector + +$$ +\widehat {\pmb {\alpha}} _ {n} = \left(\lambda \mathbf {I} - \lambda^ {2} \widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S} (q ^ {- 1} \mathbf {S} ^ {\top} \widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S} + \lambda (\widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S}) ^ {\top} \widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S}) ^ {- 1} \mathbf {S} ^ {\top} \widehat {\mathbf {K}} _ {n} ^ {D}\right) \mathbf {y} _ {n}, +$$ + +and write + +$$ +\widehat {\mathbf {\Lambda}} _ {n} ^ {D} = \left(\lambda \mathbf {I} - \lambda^ {2} \widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S} (q ^ {- 1} \mathbf {S} ^ {\top} \widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S} + \lambda (\widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S}) ^ {\top} \widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S}) ^ {- 1} \mathbf {S} ^ {\top} \widehat {\mathbf {K}} _ {n} ^ {D}\right) \in \mathbb {R} ^ {n \times n} +$$ + +for our sketched estimate of the inverse kernel matrix. Computing this takes $O(ns^2)$ time when $\mathbf{S}$ is a subsampling matrix (Musco and Musco, 2017). + +Now we further sketch the coefficients $\widetilde{\alpha}_n = (\mathbf{S}^\top \widehat{\mathbf{K}}_n^D\mathbf{S})^\dagger \mathbf{S}^\top \widehat{\mathbf{K}}_n^D\widehat{\boldsymbol{\alpha}}_n \in \mathbb{R}^s$ , in line with Appendix C of Musco and Musco (2017): + +$$ +\widehat {\boldsymbol {k}} _ {n} ^ {D} (\mathbf {x}) ^ {\top} \widehat {\boldsymbol {\alpha}} _ {n} \approx \left\langle \mathbf {S} ^ {\top} \widehat {\boldsymbol {k}} _ {n} ^ {D} (\mathbf {x}), (\mathbf {S} ^ {\top} \widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S}) ^ {\dagger} \mathbf {S} ^ {\top} \widehat {\mathbf {K}} _ {n} ^ {D} \widehat {\boldsymbol {\alpha}} _ {n} \right\rangle = \left\langle \widetilde {\boldsymbol {k}} _ {n} ^ {D} (\mathbf {x}), \widetilde {\boldsymbol {\alpha}} _ {n} \right\rangle . +$$ + +When $\widetilde{\alpha}_n = (\mathbf{S}^\top \widehat{\mathbf{K}}_n^D\mathbf{S})^\dagger \mathbf{S}^\top \widehat{\mathbf{K}}_n^D\widehat{\boldsymbol{\alpha}}_n$ is precomputed, we can make a new KRR point prediction in only $O(s)$ time. This is because we only need to compute the $s$ coordinates of the vector $\widetilde{k}_n^D (\mathbf{x}) = \mathbf{S}^\top \widehat{k}_n^D (\mathbf{x})\in \mathbb{R}^s$ , taking $s$ kernel evaluations, and multiplication with $\widetilde{\alpha}_n = (\mathbf{S}^\top \widehat{\mathbf{K}}_n^D\mathbf{S})^\dagger \mathbf{S}^\top \widehat{\mathbf{K}}_n^D\widehat{\boldsymbol{\alpha}}_n\in \mathbb{R}^s$ can be done in $O(s)$ time. + +$O(s^2)$ time inference. So in order to perform inference in sublinear time, we first precompute in $O(ns^2)$ time: + +$$ +\widetilde {\boldsymbol {\Lambda}} _ {n} ^ {D} = \widehat {\boldsymbol {\Lambda}} _ {n} ^ {D} \widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S} (\mathbf {S} ^ {\top} \widehat {\mathbf {K}} _ {n} ^ {D} \mathbf {S}) ^ {\dagger} \in \mathbb {R} ^ {n \times s}, \qquad \widehat {\boldsymbol {\Sigma}} _ {n} ^ {D} = \left(\widetilde {\boldsymbol {\Lambda}} _ {n} ^ {D}\right) ^ {\top} \widetilde {\boldsymbol {\Lambda}} _ {n} ^ {D} \in \mathbb {R} ^ {s \times s}. +$$ + +To obtain a sketched estimate of $\| r_n^D (\mathbf{x})\| _2$ on a new test point $\mathbf{x}$ , we compute the $s$ coordinates of the vector $\widetilde{\pmb{k}}_n^D (\mathbf{x}) = \mathbf{S}^\top \widehat{\pmb{k}}_n^D (\mathbf{x})\in \mathbb{R}^s$ , and obtain an estimate + +$$ +\| \widetilde {\boldsymbol {r}} _ {n} ^ {D} (\mathbf {x}) \| _ {2} = \sqrt {\widetilde {\boldsymbol {k}} _ {n} ^ {D} (\mathbf {x}) ^ {\top} \widehat {\boldsymbol {\Sigma}} _ {n} ^ {D} \widetilde {\boldsymbol {k}} _ {n} ^ {D} (\mathbf {x})} +$$ + +in $O(s^2)$ time. The equivalent expression $\| \widetilde{\boldsymbol{r}}_n^D (\mathbf{x})\| _2 = \| \widetilde{\boldsymbol{\Lambda}}_n^D\widetilde{\boldsymbol{k}}_n^D (\mathbf{x})\| _2$ takes $O(n)$ time. + +Sketched hypothesis testing. Given a hold-out dataset $(\mathbf{X}_m,\mathbf{y}_m)$ , we test the null hypothesis: + +$$ +H _ {0}: \boldsymbol {f} (\mathbf {x} _ {j}) = \boldsymbol {g} (\mathbf {x} _ {j}) \text {f o r a l l} j = 1, \dots , m \text {a g a i n s t} H _ {1}: \boldsymbol {f} (\mathbf {x} _ {j}) \neq \boldsymbol {g} _ {n} (\mathbf {x} _ {j}) \text {f o r s o m e} j +$$ + +by comparing $\widehat{\pmb{f}}_{n,1}(\mathbf{x}_j),\widehat{\pmb{f}}_{n,2}(\mathbf{x}_j)$ . However, given $m$ test points on a test dataset $(\mathbf{X}_m,\mathbf{y}_m)$ , we can subsample $r$ points with a new subsampling matrix $\pmb{r}$ to compute the doubly subsampled kernel + +
No.ConditionUseHow Satisfied
1Tree structure sn(b)(x) ⊥yn for all b = 1, ..., BTheorem 1; allows kernel ridge regression representationRandom tree structures, sample splitting
2Trees eventually i.i.d. sn~QnTheorem 1; ensures aver-aging tree structures has a limitRandom tree structures, draw from library of trees.
3f α-Lipschitz, μ(x) bounded away from 0Theorem 2; controls ability to approximate f by kernel ridge regressionStructural assumption about data generating pro-cess
4Shrinking maxi-mum leaf diameter supA∈Qn diam(A) bounded by O(n-1/(d+1))Theorem 2; controls within-leaf biasMinimum probability of splitting each feature
5Growing observations per leaf vn = n d+1/d+1+ν < nd+1/n = O(d^n)Theorem 2; controls vari-ance within leavesMinimum leaf size
6Cardinality of tree space |Qn| bounded by O(n-1 exp(0.5n1/(d+2)-ν-nα))Theorem 2; restricts vari-ance of modelsRestrict tree depth, choose library of trees
7Trees Split on MediansTheorem 2; Alg 2 only; controls bias and varianceRestrict candidate splits
+ +Table 1: Table of assumptions required for our theory and their use. + +matrix $\widetilde{\kappa}_{n,1}^{D}(\mathbf{X}_{m}) = \mathbf{S}^{\top}\widehat{\boldsymbol{k}}_{n}^{D}(\mathbf{X}_{m})\boldsymbol {r}\in \mathbb{R}^{s\times r}$ , which we only need to perform $s\times r$ kernel operations to compute. So the difference statistic is then given by + +$$ +\widetilde {\boldsymbol {d}} _ {m} = \widetilde {\boldsymbol {\kappa}} _ {n, 1} (\mathbf {X} _ {m}) ^ {\top} \widetilde {\boldsymbol {\alpha}} _ {n, 1} - \widetilde {\boldsymbol {\kappa}} _ {n, 2} (\mathbf {X} _ {m}) ^ {\top} \widetilde {\boldsymbol {\alpha}} _ {n, 2} = \left(\widetilde {\boldsymbol {\Lambda}} _ {n, 1} ^ {D} \widetilde {\boldsymbol {\kappa}} _ {n, 1} (\mathbf {X} _ {m}) - \widetilde {\boldsymbol {\Lambda}} _ {n, 2} ^ {D} \widetilde {\boldsymbol {\kappa}} _ {n, 2} (\mathbf {X} _ {m})\right) ^ {\top} \mathbf {y} \in \mathbb {R} ^ {r}. +$$ + +Modulo subsampling and Nyström approximation error, this is multivariate normal under the null: $\widetilde{\boldsymbol{d}}_m^D\sim \mathcal{N}_r\left(\mathbf{0},\sigma^2\widetilde{\boldsymbol{\Xi}}_n^D\right)$ , with covariance matrix + +$$ +\widetilde {\mathbf {\Xi}} _ {n} ^ {D} = \left(\widetilde {\mathbf {\Lambda}} _ {n, 1} ^ {D} \widetilde {\boldsymbol {\kappa}} _ {n, 1} (\mathbf {X} _ {m}) ^ {\top} - \widetilde {\mathbf {\Lambda}} _ {n, 2} ^ {D} \widetilde {\boldsymbol {\kappa}} _ {n, 2} (\mathbf {X} _ {m}) ^ {\top}\right) ^ {\top} \left(\widetilde {\mathbf {\Lambda}} _ {n, 1} ^ {D} \widetilde {\boldsymbol {\kappa}} _ {n, 1} (\mathbf {X} _ {m}) ^ {\top} - \widetilde {\mathbf {\Lambda}} _ {n, 2} ^ {D} \widetilde {\boldsymbol {\kappa}} _ {n, 2} (\mathbf {x} _ {m}) ^ {\top}\right) \in \mathbb {R} ^ {r \times r}. +$$ + +We then have the test statistic: + +$$ +\widehat {\sigma} ^ {- 2} \widetilde {\boldsymbol {d}} _ {m} ^ {D} ^ {\top} (\widetilde {\boldsymbol {\Xi}} _ {n} ^ {D}) ^ {- 1} \widetilde {\boldsymbol {d}} _ {m} ^ {D} \sim \chi_ {r} ^ {2}. +$$ + +The runtime of the test is as follows. It takes $O(ns^2)$ precomputation time to form $\widetilde{\Lambda}_{n,1}^{D}, \widetilde{\Lambda}_{n,2}^{D} \in \mathbb{R}^{n \times s}$ , $O(nsr)$ time for the multiplication with $\widetilde{\kappa}_{n,1}(\mathbf{X}_m), \widetilde{\kappa}_{n,2}(\mathbf{X}_m)$ , and $O(n)$ time for multiplication with $\mathbf{y}$ . Once one has the noise estimate, inverting the covariance matrix and computing the test points takes $O(r^3)$ time. + +# B Table of Assumptions + +Table 1 provides a summary of the assumptions we require, a brief explanation of how they are used and conditions under which they are satisfied. + +# C Finite Sample Convergence + +# C.1 Stochastic Contraction Mapping Theorem + +To show convergence for both algorithms, we will show both of them are a stochastically contracted process. The criteria of deciding such process can be found in Almudevar (2022), and we introduce the theorem below. + +Theorem 3. Given $\mathbb{R}^d$ -valued stochastic process $\{\mathbf{z}_t\}_{t\in \mathbb{N}}$ , a sequence of $0 < \lambda_t \leq 1$ , define + +$$ +\mathcal {F} _ {0} = \emptyset , \mathcal {F} _ {t} = \sigma (\mathbf {z} _ {1}, \dots , \mathbf {z} _ {t}), +$$ + +$$ +\epsilon_ {t} = \mathbf {z} _ {t} - \mathbb {E} [ \mathbf {z} _ {t} | \mathcal {F} _ {t - 1} ]. +$$ + +We call $\mathbf{z}_t$ a stochastic contraction if the following properties hold + +1. Vanishing coefficients + +$$ +\sum_ {t = 1} ^ {\infty} \left(1 - \lambda_ {t}\right) = \infty , w h i c h i m p l i e s \prod_ {t = 1} ^ {\infty} \lambda_ {t} = 0. +$$ + +2. Mean contraction + +$$ +\left. \right.\left|\left| \mathbb {E} \left[ \mathbf {z} _ {t} \mid \mathcal {F} _ {t - 1} \right]\right|\right| \leq \lambda_ {t} \| \mathbf {z} _ {t - 1} \|, a. s.. +$$ + +3. Bounded deviation + +$$ +\sup \left\| \epsilon_ {t} \right\|\rightarrow 0, \quad \sum_ {t = 1} ^ {\infty} \mathbb {E} [ \| \epsilon_ {t} \| ^ {2} ] < \infty . +$$ + +In particular, a multidimensional stochastic contraction exhibits the following behavior + +1. Contraction + +$$ +\mathbf {z} _ {t} \xrightarrow {a . s .} 0. +$$ + +2. Kolmogorov inequality + +$$ +P \left(\sup _ {t \geq T} \| \mathbf {z} _ {t} \| \leq \| \mathbf {z} _ {T} \| + \delta\right) \geq 1 - \frac {4 \sqrt {d} \sum_ {t = T + 1} ^ {\infty} \mathbb {E} \left[ \epsilon_ {t} ^ {2} \right]}{\min \left\{\delta^ {2} , \beta^ {2} \right\}} \tag {4} +$$ + +holds for all $T, \delta > 0$ s.t. $\beta = \| \mathbf{z}_T\| + \delta - \sqrt{d}\sup_{t > T}\|\epsilon_t\| > 0$ . + +In the following proof we benefit from the a.s. convergence of the difference vector to 0. And the Kolmogorov inequality gives us a PAC argument in 5. + +# C.2 Subsampling + +We will also introduce a lemma to regularize the expected tree kernel in a finite sample case with respect to subsampling rate. + +Lemma 2. Considering a subsampled regression tree. Assume that each leaf contains no fewer than $n^{\frac{1}{d + 2}}$ sample points before subsampling. If we are subsampling at rate at least $\xi = n^{-\frac{1}{d + 2}\log n}$ , then the expected structure vector's norm follows the rate below: + +$$ +\left\| \boldsymbol {k} _ {n} \right\| _ {1} = 1 - O \left(\frac {1}{n}\right) +$$ + +Proof. By Zhou and Hooker (2022), we know that $\| \mathbb{E}_{q,w}[\mathbf{S}_n]\| \leq 1$ . By this we know that at least $\| \pmb{k}_n\| _1\leq 1$ . The task remained is to give the distance between $\| \pmb{k}_n\| _1$ and 1. + +Define the subsampling index set as $w$ . To see clearly why a rate containing $n$ would exist, recall the definition of a structure vector with subsampling, for any $x \in A_j$ , + +$$ +\boldsymbol {s} _ {n, k} (\mathbf {x}) = \boldsymbol {s} _ {n, k} (x; w) = \frac {\mathbb {1} (\mathbf {x} _ {k} \in A _ {j}) \mathbb {1} (k \in w)}{\sum_ {i = 1} ^ {n} \mathbb {1} (\mathbf {x} _ {i} \in A _ {j}) \mathbb {1} (i \in w)} = \frac {\mathbb {1} (\mathbf {x} _ {k} \in A _ {j}) \mathbb {1} (k \in w)}{\sum_ {\mathbf {x} _ {i} \in A _ {j}} \mathbb {1} (i \in w)}. +$$ + +The structure vector is given by + +$$ +\pmb {s} _ {n} (x) = [ \pmb {s} _ {n, 1} (x), \pmb {s} _ {n, 2} (x), \dots , \pmb {s} _ {n, n} (x) ] ^ {T} +$$ + +Taking expectation we have the kernel vector + +$$ +\boldsymbol {k} _ {n} ^ {\top} = \left[ \frac {\mathbb {P} (\mathbf {x} _ {k} \in A _ {j}) \mathbb {P} (k \in w)}{\sum_ {\mathbf {x} _ {i} \in A _ {j}} \mathbb {P} (i \in w)} \right] _ {i, j} +$$ + +Notice the expectation is taken with respect to both structure distribution and the subsampling distribution. And this vector is not defined if one leaf is empty, since the denominator would be 0. Let + +the indicator $I_{A}$ denote whether the leaf of interest is empty. It values on 0 if the leaf is empty and it values on 1 if the value is not empty. The $L_{1}$ norm is given by: + +$$ +\begin{array}{l} \left\| \boldsymbol {k} _ {n} \right\| _ {1} = \sum_ {k = 1} ^ {n} \mathbb {E} _ {q, w} [ \boldsymbol {s} _ {n, k} (x) ] \\ = \sum_ {k = 1} ^ {n} \left(\mathbb {E} _ {q, w} [ \boldsymbol {s} _ {n, k} (x) | \mathbb {1} _ {A} = 0 ] \mathbb {P} (\mathbb {1} _ {A} = 0) + \mathbb {E} _ {q, w} [ \boldsymbol {s} _ {n, k} (x) | \mathbb {1} _ {A} = 1 ] \mathbb {P} (\mathbb {1} _ {A} = 1)\right) \\ \end{array} +$$ + +If the leaf is empty, then we will have to define $\mathbb{E}_{q,w}[s_{n,k}(x)|\mathbb{1}_A = 0] = 0$ , otherwise the denominator would be 0 and the whole structure vector is undefined. If the leaf is not empty, we would then want to define the conditional expectation such that it sums up to 1: + +$$ +\sum_ {k = 1} ^ {n} \mathbb {E} \left[ s _ {n, k} (x) \mid I _ {A} = 1 \right] = 1 +$$ + +Hence the $L_{1}$ norm above simplifies to: + +$$ +\left\| \boldsymbol {k} _ {n} \right\| _ {1} = \sum_ {k = 1} ^ {n} \mathbb {E} [ \boldsymbol {s} _ {n, k} (x) | \mathbb {1} _ {A} = 1 ] \mathbb {P} (\mathbb {1} _ {A} = 1) = 1 - \mathbb {P} (\mathbb {1} _ {A} = 0) +$$ + +Hence the problem is reduced to finding the probability of missing all subsampled points in a leaf of interest. + +By assertion, the subsample size should be $\xi n = n^{\frac{d + 1}{d + 2}}\log n$ . Consider the probability that one leaf misses all sample points $p(n,\xi)$ . That is, all the subsampled points sampled are points that are in this particular leaf. Thus we are only choosing with choice $\binom{n-n\frac{1}{d+2}}{\theta n}$ . + +$$ +\begin{array}{l} \mathbb {P} (n, \xi) = \frac {\binom {n - n \frac {1}{d + 2}} {\xi n}}{\binom {n} {\xi n}} \\ = \frac {(n - \xi n) (n - \xi n - 1) (n - \xi n - 2) \cdots (n - \xi n - n ^ {\frac {1}{d + 2}} + 1)}{n (n - 1) (n - 2) \cdots (n - n ^ {\frac {1}{d + 2}} + 1)} \\ \leq \left(\frac {n - \xi n}{n - n ^ {\frac {1}{d + 2}}}\right) ^ {n ^ {\frac {1}{d + 2}}} \\ = \left(\frac {1 - \xi}{1 - n ^ {- \frac {d + 1}{d + 2}}}\right) ^ {n ^ {\frac {1}{d + 2}}} \\ = \left(\frac {1 - n ^ {- \frac {1}{d + 2}} \log n}{1 - n ^ {- \frac {d + 1}{d + 2}}}\right) ^ {n ^ {\frac {1}{d + 2}}} \\ = \left(\frac {1}{1 - n ^ {- \frac {d + 1}{d + 2}}}\right) ^ {n ^ {\frac {1}{d + 2}}} \cdot \left(1 - n ^ {- \frac {1}{d + 2}} \log n\right) ^ {n ^ {\frac {1}{d + 2}}} \\ \end{array} +$$ + +Notice that + +$$ +\begin{array}{l} \left(\frac {1}{1 - n ^ {- \frac {d + 1}{d + 2}}}\right) ^ {n ^ {\frac {1}{d + 2}}} \leq \left(\frac {1 + n ^ {- \frac {d + 1}{d + 2}}}{1 - n ^ {- \frac {d + 1}{d + 2}} + n ^ {- \frac {d + 1}{d + 2}}}\right) ^ {n ^ {\frac {1}{d + 2}}} \\ = \left(1 + n ^ {- \frac {d + 1}{d + 2}}\right) ^ {n ^ {\frac {1}{d + 2}}} \\ \leq \left(1 + n ^ {- \frac {1}{d + 2}}\right) ^ {n ^ {\frac {1}{d + 2}}} \\ \end{array} +$$ + +$$ +\le e +$$ + +And meanwhile we recognize + +$$ +\begin{array}{l} \left(1 - n ^ {- \frac {1}{d + 2}} \log n\right) ^ {n ^ {\frac {1}{d + 2}}} = \left[ \left(1 - n ^ {- \frac {1}{d + 2}} \log n\right) ^ {n ^ {\frac {1}{d + 2}} \frac {1}{\log n}} \right] ^ {\log n} \\ \leq \left( \begin{array}{c} 1 \\ \hline e \end{array} \right) ^ {\log n} \\ = \frac {1}{e ^ {\log n}} \\ = \frac {1}{n} \\ \end{array} +$$ + +Hence we have + +$$ +\begin{array}{l} \left(\frac {1}{1 - n ^ {- \frac {d + 1}{d + 2}}}\right) ^ {n ^ {\frac {1}{d + 2}}} \cdot \left(1 - n ^ {- \frac {1}{d + 2}} \log n\right) ^ {n ^ {\frac {1}{d + 2}}} \\ \leq e \cdot \left(1 - n ^ {- \frac {1}{d + 2}} \log n\right) ^ {n ^ {\frac {1}{d + 2}}} \\ = O \left(\frac {1}{n}\right) \\ \end{array} +$$ + +Hence the probability of a single leaf missing all subsampled points is of rate $O\left(\frac{1}{n}\right)$ . And hence $\| \pmb{k}_n\|_1 = 1 - O\left(\frac{1}{n}\right)$ . + +# C.3 Proof for Theorem 1 + +Combining all the results above, the proof for Theorem 1 is as follow. + +Proof. To show Algorithm 1 is a stochastic contraction mapping, we aim to show the conditions 1-3 are satisfied. Define the difference $\mathbf{z}_t \coloneqq \widehat{\mathbf{y}}_{t+1} - \widehat{\mathbf{y}}^*$ . And define the partial ensemble $\widetilde{\mathbf{y}}_b \coloneqq \frac{1}{b}\sum_{k=1}^{b}X_k\widehat{t}_k$ , $X_k \stackrel{i.i.d.}{\sim} \operatorname{Ber}(q)$ . To check mean contraction, first notice as $b \to \infty$ , $\operatorname{tr}(\operatorname{Var}(\widetilde{\mathbf{y}}_b)) \leq \frac{q(1-q)nM}{b} \to 0$ . Hence with $b$ large enough, we can get rid of the truncation. + +$$ +\begin{array}{l} \| \mathbb {E} [ \mathbf {z} _ {b} | \mathcal {F} _ {b - 1} ] \| = \left\| \mathbb {E} \left[ \frac {b - 1}{b} \widehat {\mathbf {y}} _ {b - 1} + \frac {\lambda}{b} \mathbf {S} _ {n} (\mathbf {y} - \Gamma_ {M} (\widetilde {\mathbf {y}} _ {b - 1})) - \widehat {\mathbf {y}} ^ {*} \right\rvert \mathcal {F} _ {b - 1} \right] \| \\ = \left\| \frac {b - 1}{b} \left(\widehat {\mathbf {y}} _ {b - 1} - \widehat {\mathbf {y}} ^ {*}\right) + \frac {\lambda}{b} \mathbb {E} \left[ \mathbf {S} _ {n} \right] \left(\mathbf {y} - \mathbb {E} \left[ \Gamma_ {M} \left(\widetilde {\mathbf {y}} _ {b - 1}\right) \mid \mathcal {F} _ {b - 1} \right]\right) - \frac {1}{b} \widehat {\mathbf {y}} ^ {*} \right\| \\ \leq \frac {b - 1}{b} \| \widehat {\mathbf {y}} _ {b - 1} - \widehat {\mathbf {y}} ^ {*} \| + \left\| \frac {\lambda}{b} \mathbb {E} [ \mathbf {S} _ {n} ] (\mathbf {y} - \mathbb {E} [ \Gamma_ {M} (\widetilde {\mathbf {y}} _ {b - 1}) | \mathcal {F} _ {b - 1} ]) - \frac {\lambda}{b} \mathbb {E} [ \mathbf {S} _ {n} ] (\mathbf {y} - q \widehat {\mathbf {y}} ^ {*}) \right\| \\ = \frac {b - 1}{b} \| \mathbf {z} _ {b - 1} \| + \left\| \frac {\lambda}{b} \mathbb {E} [ \mathbf {S} _ {n} ] \left(q \widehat {\mathbf {y}} ^ {*} - \mathbb {E} \left[ \Gamma_ {M} (\widetilde {\mathbf {y}} _ {b - 1}) | \mathcal {F} _ {b - 1} \right]\right) \right\| \\ = \frac {b - 1}{b} \| \mathbf {z} _ {b - 1} \| + \left\| \frac {\lambda}{b} \mathbb {E} [ \mathbf {S} _ {n} ] \left(q \widehat {\mathbf {y}} ^ {*} - \mathbb {E} [ \widetilde {\mathbf {y}} _ {b - 1} | \mathcal {F} _ {b - 1} ]\right) \right\| \\ \leq \frac {b - 1 + \lambda q}{b} \| \widehat {\mathbf {y}} _ {b - 1} - \widehat {\mathbf {y}} ^ {*} \| \leq \| \widehat {\mathbf {y}} _ {b - 1} - \widehat {\mathbf {y}} ^ {*} \| \\ \end{array} +$$ + +Next we check for bounded deviations. + +$$ +\begin{array}{l} \| \epsilon_ {b} \| = \| \mathbf {z} _ {b} - \mathbb {E} [ \mathbf {z} _ {b} | \mathcal {F} _ {b - 1} ] \| \\ = \left\| \left(\widehat {\mathbf {y}} _ {b} - \widehat {\mathbf {y}} ^ {*}\right) - \left(\mathbb {E} \left[ \widehat {\mathbf {y}} _ {b} - \widehat {\mathbf {y}} ^ {*} \mid \mathcal {F} _ {b - 1} \right]\right) \right\| \\ = \left\| \widehat {Y} _ {b} - \mathbb {E} \left[ \widehat {Y} _ {b} \mid \mathcal {F} _ {b - 1} \right] \right\| \\ = \left\| \frac {\lambda}{b} \left(\mathbb {E} \left[ \mathbf {S} _ {n} \right] - \mathbf {S} _ {n}\right) \left(\mathbf {y} - \Gamma_ {M} \left(\widetilde {\mathbf {y}} _ {b - 1}\right) + \mathbb {E} \left[ \Gamma_ {M} \left(\widetilde {\mathbf {y}} _ {b - 1}\right) \mid \mathcal {F} _ {b - 1} \right]\right) \right\| \\ \end{array} +$$ + +$$ +\leq \frac {\lambda}{b} \left\| \mathbb {E} [ \mathbf {S} _ {n} ] - \mathbf {S} _ {n} \right\| \left\| Y - \Gamma_ {M} (\widetilde {\mathbf {y}} _ {b - 1}) + \mathbb {E} [ \Gamma_ {M} (\widetilde {\mathbf {y}} _ {b - 1}) | \mathcal {F} _ {b - 1} ] \right\| +$$ + +Now, by Zhou and Hooker (2022), we know $\mathbb{E}[\mathbf{S}_n]$ has both row sum and column sum no greater than 1, hence we see that $\| \mathbb{E}[\mathbf{S}_n]\| \leq \sqrt{\|\mathbb{E}[\mathbf{S}_n]\|_1\cdot\|\mathbb{E}[\mathbf{S}_n]\|_\infty}\leq \sqrt{1\times n} = \sqrt{n}$ . Then, it can be seen that + +$$ +\left\| \epsilon_ {b} \right\| \leq \frac {\lambda}{b} \cdot (1 + \sqrt {n}) \cdot 2 M +$$ + +Then we can show that $\sum_{b=1}^{\infty}\|\epsilon_b\|^2 = \sum_{b=1}^{\infty}O\left(\frac{1}{b^2}\right) < \infty$ . + +Similarly we can show both conditions for Algorithm 2. Replacing the stepwise difference according to the new update rule gives: + +$$ +\begin{array}{l} \| \mathbb {E} [ \mathbf {z} _ {b + 1} \mid \mathcal {F} _ {b} ] \| = \| \mathbb {E} [ \widehat {\mathbf {y}} _ {b + 1, K} - \widehat {\mathbf {y}} _ {K} ^ {*} \mid \mathcal {F} _ {b} ] \| \\ = \left\| \frac {b}{b + 1} \widehat {\mathbf {y}} _ {b, K} + \frac {1}{b + 1} \mathbb {E} [ \mathbf {S} _ {n} ] \sum_ {k = 1} ^ {K} (\mathbf {y} - \widetilde {\mathbf {y}} _ {b, k - 1}) - \widehat {\mathbf {y}} _ {K} ^ {*} \right\| \\ = \left\| \frac {b}{b + 1} \widehat {\mathbf {y}} _ {b, K} + \frac {1}{b + 1} \mathbb {E} [ \mathbf {S} _ {n} ] \sum_ {k = 1} ^ {K} \left(\mathbf {y} - \left(\widehat {\mathbf {y}} _ {b, K} - \Gamma_ {M} \left(\frac {1}{b} \sum_ {g = 1} ^ {b} \widehat {\mathbf {t}} _ {g, k}\right)\right)\right) - \widehat {\mathbf {y}} _ {K} ^ {*} \right\| \\ = \left\| \frac {b}{b + 1} \widehat {\mathbf {y}} _ {b, K} + \frac {K}{b + 1} \mathbb {E} [ \mathbf {S} _ {n} ] \mathbf {y} - \frac {K}{b + 1} \mathbb {E} [ \mathbf {S} _ {n} ] \widehat {\mathbf {y}} _ {b, K} + \frac {1}{b + 1} \mathbb {E} [ \mathbf {S} _ {n} ] \sum_ {k = 1} ^ {K} \Gamma_ {M} \left(\frac {1}{b} \sum_ {g = 1} ^ {b} \widehat {\mathbf {t}} _ {g, k}\right) - \widehat {\mathbf {y}} _ {K} ^ {*} \right\| \\ = \left\| \frac {1}{b + 1} \big (b I - K \mathbb {E} [ \mathbf {S} _ {n} ] \big) \left(\widehat {\mathbf {y}} _ {b, K} - \widehat {\mathbf {y}} _ {K} ^ {*}\right) + \frac {1}{b + 1} \mathbb {E} [ \mathbf {S} _ {n} ] \Big (\sum_ {k = 1} ^ {K} \Gamma_ {M} \big (\frac {1}{b} \sum_ {g = 1} ^ {b} \widehat {\mathbf {t}} _ {g, k} \big) - \widehat {\mathbf {y}} _ {K} ^ {*} \Big) \right\| \\ \leq \frac {1}{b + 1} \| b \mathbf {I} - K \mathbb {E} [ \mathbf {S} _ {n} ] \| \| \widehat {\mathbf {y}} _ {b, K} - \widehat {\mathbf {y}} _ {K} ^ {*} \| + \frac {1}{b + 1} \| \mathbb {E} [ \mathbf {S} _ {n} ] \| \left\| \sum_ {k = 1} ^ {K} \Gamma_ {M} \left(\frac {1}{b} \sum_ {g = 1} ^ {b} \widehat {\mathbf {t}} _ {g, k}\right) - \widehat {\mathbf {y}} _ {K} ^ {*} \right\| \\ \leq \frac {1}{b + 1} \| b \mathbf {I} - K \mathbb {E} [ \mathbf {S} _ {n} ] \| \| \widehat {\mathbf {y}} _ {b, K} - \widehat {\mathbf {y}} _ {K} ^ {*} \| + \frac {1}{b + 1} \| \mathbb {E} [ \mathbf {S} _ {n} ] \| \left\| \Gamma_ {K M} \left(\sum_ {k = 1} ^ {K} \frac {1}{b} \sum_ {g = 1} ^ {b} \widehat {\mathbf {t}} _ {g, k}\right) - \widehat {\mathbf {y}} _ {K} ^ {*} \right\| \\ \leq \frac {1}{b + 1} \left(\| b \mathbf {I} - K \mathbb {E} [ \mathbf {S} _ {n} ] \| + \| \mathbb {E} [ \mathbf {S} _ {n} ] \|\right) \| \widehat {\mathbf {y}} _ {b, K} - \widehat {\mathbf {y}} _ {K} ^ {*} \| \\ \leq \frac {1}{b + 1} (b + 1 - O (\frac {1}{n})) \| \widehat {\mathbf {y}} _ {b, K} - \widehat {\mathbf {y}} _ {K} ^ {*} \| \\ < \left\| \mathbf {z} _ {b} \right\| \\ \end{array} +$$ + +Hence we have checked the mean contraction condition. Checking the bounded deviation condition works in a similar way for Algorithm 2 since it shares the same expected tree kernel with Algorithm 1 and the tree signal is also bounded by $M$ . Hence the stochastic contraction mapping follows. + +# D Conditional CLT Around KRR Predictions + +Doing so requires control on the weighting vectors $\boldsymbol{r}_n(x)$ . And we will begin such analysis on giving the rate of the expected structure vector $\boldsymbol{k}_n(x)$ first. This is because recall that in both algorithms our final weight vector $\boldsymbol{r}_n(x)$ is a linear map of the voting vector $\boldsymbol{k}_n(x)$ . To begin with, we will first notice that Lemma 2 implies Lemma 3. + +# D.1 Bounding the weights + +Lemma 3. Suppose the assumptions for Lemma 2 are satisfied. We have $\left|\sum_{i=1}^{n} r_{n,i}^{D} - \frac{\lambda}{1 + \lambda q}\right| = O\left(\frac{1}{n}\right)$ , $\left|\sum_{i=1}^{n} r_{n,i}^{P} - 1\right| = O\left(\frac{1}{n}\right)$ . + +Proof. We begin with the analysis on Algorithm 1. Consider the expansion + +$$ +\left[ \frac {1}{\lambda} I + q \mathbf {K} _ {n} \right] ^ {- 1} = \lambda \sum_ {i = 0} ^ {\infty} \left(\left(\lambda q\right) ^ {2 i} \mathbf {K} _ {n} ^ {2 i} - \left(\lambda q\right) ^ {2 i + 1} \mathbf {K} _ {n} ^ {2 i + 1}\right). +$$ + +We examine the column sums of each of the matrix powers. Start with $K_{n}^{2}$ , + +$$ +\sum_ {i = 1} \left(\mathbf {K} _ {n} ^ {2}\right) _ {i, 1} = \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} \left(\mathbf {K} _ {n}\right) _ {i, j} \left(\mathbf {K} _ {n}\right) _ {j, 1} = \sum_ {j = 1} ^ {n} \left(\mathbf {K} _ {n}\right) _ {j, 1} \sum_ {i = 1} ^ {n} \left(\mathbf {K} _ {n}\right) _ {i, j}. +$$ + +Take subsampling in to consideration, since $K_{n}$ consists of structure vectors of sample points, hence for some $c > 0$ , we can bound the row sum: + +$$ +1 - \frac {c}{n} \leq \sum_ {i = 1} ^ {n} \left(\mathbf {K} _ {n}\right) _ {i, j} \leq 1, \quad i = 1, \dots , n. +$$ + +which also hold true for + +$$ +1 - \frac {c}{n} \leq \sum_ {j = 1} ^ {n} \left(\mathbf {K} _ {n}\right) _ {j, 1} \leq 1, \quad i = 1, \dots , n. +$$ + +Given $K_{n}$ is nonnegative, multiply the inequalities above and notice it is actually the row sum of $K_{n}^{2}$ , we have + +$$ +\left(1 - \frac {c}{n}\right) ^ {2} \leq \sum_ {i = 1} (\mathbf {K} _ {n} ^ {2}) _ {i, 1} = \sum_ {j = 1} ^ {n} (\mathbf {K} _ {n}) _ {j, 1} \sum_ {i = 1} ^ {n} (\mathbf {K} _ {n}) _ {i, j} \leq 1. +$$ + +Repeating the same procedure above yields + +$$ +\left(1 - \frac {c}{n}\right) ^ {m} \leq \sum_ {i = 1} (\mathbf {K} _ {n} ^ {m}) _ {i, 1} \leq 1. +$$ + +Therefore, + +$$ +\begin{array}{l} \lambda \left(\frac {1}{1 - \lambda^ {2} q ^ {2} (1 - \frac {c}{n}) ^ {2}} - \frac {\lambda q}{1 - \lambda^ {2} q ^ {2}}\right) \leq \sum_ {j = 1} ^ {n} \left[ \frac {1}{\lambda} I + q \mathbf {K} _ {n} \right] _ {j, 1} ^ {- 1} \\ = \lambda \left(\sum_ {i = 0} ^ {\infty} (\lambda q) ^ {2 i} \left(\mathbf {K} _ {n} ^ {2 i}\right) _ {j, 1} - (\lambda q) ^ {2 i + 1} \left(\mathbf {K} _ {n} ^ {2 i + 1}\right) _ {j, 1}\right) \\ \leq \lambda \left(\frac {1}{1 - \lambda^ {2} q ^ {2}} - \frac {\lambda q}{1 - \lambda^ {2} q ^ {2} (1 - \frac {c}{n}) ^ {2}}\right), \\ \end{array} +$$ + +where both the LHS and RHS reduce to $\frac{\lambda}{1 + \lambda q} + O\left(\frac{1}{n}\right)$ . So is true for any column sum of $\left[\frac{1}{\lambda} I + q\mathbf{K}_n\right]^{-1}$ . And we know that $\pmb{k}_n$ is nonnegative and $1 - \| \pmb{k}_n\|_1 = O\left(\frac{1}{n}\right)$ and $\pmb{k}_n$ simply reweights the columns. Hence we prove the convergence of weights $\sum_{i=1}^{n} r_{n,i}$ for Algorithm 1. + +The analysis for Algorithm 2 is similar. We simply substitute $\lambda = K, q = \frac{K - 1}{K}$ . This will give the bound: + +$$ +\begin{array}{l} K \left(\frac {1}{1 - (K - 1) ^ {2} (1 - \frac {c}{n}) ^ {2}} - \frac {K - 1}{1 - (K - 1) ^ {2}}\right) \leq \sum_ {j = 1} ^ {n} \left[ \frac {1}{K} I + \frac {K - 1}{K} {\bf K} _ {n} \right] _ {j, 1} ^ {- 1} \\ = K \left(\sum_ {i = 0} ^ {\infty} (K - 1) ^ {2 i} \left(\mathbf {K} _ {n} ^ {2 i}\right) _ {j, 1} - (K - 1) ^ {2 i + 1} \left(\mathbf {K} _ {n} ^ {2 i + 1}\right) _ {j, 1}\right) \\ \leq K \left(\frac {1}{1 - (K - 1) ^ {2}} - \frac {K - 1}{1 - (K - 1) ^ {2} (1 - \frac {c}{n}) ^ {2}}\right), \\ \end{array} +$$ + +We can reduce both sides of the inequalities to $1 + O\left(\frac{1}{n}\right)$ similarly. That proves the converging weight of Algorithm 2. + +Intuitively, knowing that the weights sum to (almost) a constant tells us the total "mass" of our weighting vector is fixed. If we have leaf size assumptions such that no single weight can be too large, the mass must be spread out over many small pieces. Whenever you distribute a fixed amount of weight across many entries, the Euclidean length of the vector necessarily shrinks. In other words, a near-unit $\ell_1$ norm plus a bound on the largest coordinate forces the $\ell_2$ norm to be small. We control the rate of $r_n^{D,P}$ in Lemma 1. To do so, we need to both upper bound and lower bound $\| \pmb{k}_n\|$ , which is helped by Assumption 5, implying $\inf_{A\in q\in Q_n}\sum_{i = 1}^n\mathbb{1}(\mathbf{x}_i\in A) = \Omega \left(n^{\frac{1}{d + 1}}\right)$ . + +# D.2 Proof for Lemma 1 + +We will make use of the $\Omega\left(n^{\frac{1}{d+1}}\right)$ in Assumption 7 to control $\|\pmb{k}_n\|$ and $\|\pmb{r}_n\|$ . The first bound in the max operator is vital for our unconditional CLT analysis Lemma 7. We will first show Lemma 1 which is sufficient for us to get a conditional CLT. + +Proof. To bound $\| \pmb{k}_n\|$ , recall: + +$$ +\boldsymbol {k} _ {n j} = \mathbb {E} [ \boldsymbol {s} _ {n, j} (x) ] = \mathbb {E} \left[ \frac {\mathbb {1} (x _ {j} \in A)}{\sum_ {j = 1} ^ {n} \mathbb {1} (x _ {j} \in A)} \right], x \in A +$$ + +encodes the expected influence of point $x_{j}$ on a point of interest $x$ among other $n$ points. Then the condition + +$$ +\inf _ {A \in q \in Q _ {n}} \sum_ {i = 1} ^ {n} \mathbb {1} (x _ {i} \in A) \geq \Omega \left(n ^ {\frac {1}{d + 1}}\right) +$$ + +implies that $\pmb{k}_{nj} = O\left(n^{-\frac{1}{d + 1}}\right)$ , since we can't have total weights of more than $O(1)$ in a leaf. By Lemma 2, $\| \pmb{k}_n\|_1 \leq 1$ , + +$$ +\| \pmb {k} _ {n} \| \leq \sqrt {\| \pmb {k} _ {n} \| _ {1} \| \pmb {k} _ {n} \| _ {\infty}} = O \left(n ^ {- \frac {1}{2} \frac {1}{d + 1}}\right). +$$ + +By assertion $|B_n| = \Omega \left(n \cdot d_n^d\right)$ , there are at most + +$$ +\Omega \left(n \cdot d _ {n} ^ {d}\right) = \Omega \left(n ^ {\frac {1}{d + 1}}\right) +$$ + +$k_{nj}$ 's that are positive. Equivalently, we can have the magnitude of each non-zero $k_{nj}$ is lower bounded by $\Omega\left(n^{-\frac{1}{d+1}}\right)$ . This holds also because the total weights can't be larger than $O(1)$ . Since $\| k_n \|_1 = 1 - O(n^{-1})$ , + +$$ +\left\| \boldsymbol {k} _ {n} \right\| = \Omega \left(\sqrt {\left(n ^ {- \frac {1}{d + 1}}\right) ^ {2} \cdot n ^ {\frac {1}{d + 1}}}\right) = \Omega \left(n ^ {- \frac {1}{2} \frac {1}{d + 1}}\right). +$$ + +To check the rate of $\| \pmb{r}_n \|$ , since $r_n$ is a mapped from $k_n$ by the KRR matrix. For Algorithm 1 it's $\left[ \frac{1}{\lambda} \mathbf{I} + q \mathbf{K}_n \right]$ and for Algorithm 2 is $\left[ \frac{1}{K} I + \frac{K - 1}{K} K_n \right]^{-1}$ and we know + +$$ +\frac {\lambda}{1 + \lambda q} \leq e i g e n \left(\left[ \frac {1}{\lambda} \mathbf {I} + q \mathbf {K} _ {n} \right] ^ {- 1}\right) \leq \lambda , \quad 1 \leq e i g e n \left(\left[ \frac {1}{K} \mathbf {I} + \frac {K - 1}{K} \mathbf {K} _ {n} \right] ^ {- 1}\right) \leq K +$$ + +Since we can fix $K$ to be a constant, $\| \pmb{r}_n\|$ will enjoy similar rates as $\| \pmb{k}_n\|$ . + +![](images/8ee1847d399e07d4e3390bb025c8021af94d4c964d7aea7435b5ed2c690660b2.jpg) + +This rate complies with the Lindeberg-Feller conditions, hence we can establish the conditional CLT. + +# D.3 Conditional Asymptotic Normality on Training Set + +Theorem 4 (Conditional Asymptotic Normality for BRAT-D and BRAT-P Predictions). For any $\mathbf{x} \in [0,1]^d$ , write $f(\mathbf{X}_n) = (f(\mathbf{x}_1), \ldots, f(\mathbf{x}_n))^{\top}$ . Then under Assumptions 1, 2, 4 and assumptions in Lemma 1, we have + +$$ +\frac {\widehat {\boldsymbol {f}} _ {n} ^ {D , P} (\mathbf {x}) - \left\langle \boldsymbol {r} _ {n} ^ {D , P} , \boldsymbol {f} (\mathbf {X} _ {n}) \right\rangle}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|} \xrightarrow {d} \mathcal {N} (0, \sigma^ {2}). +$$ + +Proof. For notational convenience, since $\| \boldsymbol{r}_n \|$ yielded by both Algorithm 1 and Algorithm 2 shares the same rate in $n$ , we will suppress the notation of $D, P$ in this proof, and denote the kernel ridge regression matrix respectively with $\mathsf{KRR}^D$ and $\mathsf{KRR}^P$ . Also define $c^D = \frac{\lambda}{1 + \lambda q}, c^P = 1$ . + +Write + +$$ +\widehat {\pmb {f}} _ {n} (\boldsymbol {x}) - \pmb {r} _ {n} ^ {T} \pmb {f} (\boldsymbol {X} _ {n}) = \pmb {r} _ {n} ^ {T} \vec {\epsilon} _ {n}. +$$ + +To establish the CLT, we verify the Lindeberg-Feller condition for the triangular array $\boldsymbol{r}_n^T\vec{\epsilon}_n$ , i.e. for any $\delta > 0$ + +$$ +\lim _ {n} \frac {1}{\| \boldsymbol {r} _ {n} \| ^ {2} \sigma^ {2}} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ (\boldsymbol {r} _ {n i} \epsilon_ {i}) ^ {2} \mathbb {1} (| \boldsymbol {r} _ {n i} \epsilon_ {i} | > \delta \| \boldsymbol {r} _ {n} \| \sigma) \right] = 0. +$$ + +Since $\| \pmb{k}_n\|_\infty = O\left(n^{-\frac{1}{d + 2}}\right)$ and $\mathsf{KRR}^{D,P}$ having row sums of $C^{D,P} + O\left(n^{-1}\right)$ , we have + +$$ +\left\| \boldsymbol {r} _ {n} \right\| _ {\infty} \leq \left\| \boldsymbol {k} _ {n} \right\| _ {\infty} \cdot \left\| \mathsf {K R R} ^ {D, P} \right\| _ {1} = O \left(n ^ {- \frac {1}{d + 1}}\right). +$$ + +Furthermore, since $\| r_n\| = \Theta (n^{-\frac{1}{2}}\frac{1}{d + 1})$ , we get + +$$ +\frac {\left\| \boldsymbol {r} _ {n} \right\| _ {\infty}}{\left\| \boldsymbol {r} _ {n} \right\|} = O \left(n ^ {- \frac {1}{2} \frac {1}{d + 1}}\right), +$$ + +This allows us to check out the Lindeberg-Feller conditions: + +$$ +\begin{array}{l} \sum_ {i = 1} ^ {n} \mathbb {E} \left[\left(\boldsymbol {r} _ {n i} \epsilon_ {i}\right) ^ {2} \mathbb {1} \left(\left| \boldsymbol {r} _ {n i} \epsilon_ {i} \right| > \delta \| \boldsymbol {r} _ {n} \| \sigma\right)\right] \leq \sum_ {i = 1} ^ {n} \boldsymbol {r} _ {n i} ^ {2} \sqrt {\mathbb {E} \left[ \epsilon_ {i} ^ {4} \right] \cdot \mathbb {E} \left[ \right. \mathbb {1} \left( \right.\left| \boldsymbol {r} _ {n i} \epsilon_ {i} \right| > \delta \| \boldsymbol {r} _ {n} \| \sigma^ {2} \left. \right]} \\ \leq \sum_ {i = 1} ^ {n} \boldsymbol {r} _ {n i} ^ {2} \sqrt {\mathbb {E} \left[ \epsilon_ {i} ^ {4} \right]} \cdot \sqrt {P \left(\left| \epsilon_ {i} \right| \geq \frac {\delta \| \boldsymbol {r} _ {n} \| \sigma}{\boldsymbol {r} _ {n i}}\right)} \\ \leq \sum_ {i = 1} ^ {n} r _ {n i} ^ {2} \sqrt {\mathbb {E} [ \epsilon_ {i} ^ {4} ]} \sqrt {2 \exp \left(- \frac {1}{2 \sigma^ {2}} \cdot \left(\frac {\delta \| \boldsymbol {r} _ {n} \| \sigma}{\boldsymbol {r} _ {n i}}\right) ^ {2}\right)} \\ \leq \left\| \boldsymbol {r} _ {n} \right\| ^ {2} \exp \left(- O \left(n ^ {\frac {1}{d + 1}}\right)\right) \longrightarrow 0, \\ \end{array} +$$ + +Since $\epsilon$ is sub-Gaussian noise, the concentration bound holds by definition of $\epsilon$ . This concludes the proof for Theorem 4. + +# E Extension to Random Design and Exponential Locality + +Now we move on to the discussions in which our input to the model is no longer restricted only to the training set $\mathbf{X}_n$ , but to any $\mathbf{x} \in [0,1]^d$ . To do so, we begin by defining a new probability space and, of course, a new probability measure. This can be uniquely designed by a Kolmogorov's extension theorem. + +Write the coordinate projection $\Pi_n = (\pi_n(\mathbf{X}),\pi_n(\vec{\epsilon}))$ as the finite-dimensional projection containing the first $n$ samples. Under this framework, we can obtain the corresponding expected structure vector $\pmb{k}_n$ , expected structure matrix (kernel matrix) $K_{n}$ and the standardized prediction error $\rho_{n}(\mathbf{X},\vec{\epsilon})$ , given by + +$$ +\rho_ {n} (\mathbf {X}, \epsilon) = \frac {\widehat {\boldsymbol {f}} _ {n} ^ {D , P} (\mathbf {x} ; \Pi_ {n}) - \langle \boldsymbol {k} _ {n} ^ {\top} (\mathbf {x} ; \Pi_ {n}) \mathsf {K R R} ^ {D , P} (\Pi_ {n}) , \boldsymbol {f} (\Pi_ {n}) \rangle}{\left\| \boldsymbol {k} _ {n} (\mathbf {x} ; \Pi_ {n}) ^ {\top} \mathsf {K R R} ^ {D , P} \Pi_ {n} \right\|} +$$ + +Here, $\rho_{n}$ denotes the prediction error after using $n$ random samples incorporating both the dropout mechanism and the sample randomness. + +To prove our desired CLT under random design, we need to first introduce a lemma that allows us to propagate the normality from fixed sample to random sample cases. This is taken care of by Lemma 8, which allows us to claim our CLTs if we verify it being correct for $a.s.\forall \mathbf{x}\in [0,1]^d$ . Then, we seek an almost sure validation of the assumptions of Theorem 4 under the random sample sequence $\Pi_n = (\pi_n(\mathbf{X}),\pi_n(\vec{\epsilon}))$ , which is helped by our Assumption 6. Formally, it is as shown below: + +Lemma 4. Suppose we have random sample $\{(\mathbf{x}_i,\mathbf{y}_i)\}_{i = 1}^n\subset [0,1]^d\times \mathbb{R}$ for each $n$ . If for any small $\alpha ,\nu >0$ s.t. + +$$ +| Q _ {n} | = O \left(\frac {1}{n} \exp \left(\frac {1}{2} n ^ {\frac {1}{d + 2} - \nu} - n ^ {\alpha}\right)\right), +$$ + +then + +$$ +\frac {\widehat {\pmb {f}} _ {n} ^ {D , P} (\mathbf {x}) - \langle \pmb {r} _ {n} ^ {D , P} , \pmb {f} (\mathbf {X} _ {n}) \rangle}{\left\| \pmb {r} _ {n} ^ {D , P} \right\|} \xrightarrow {d} \mathcal {N} (0, \sigma^ {2}). +$$ + +Proof. We will direct the reader to Zhou and Hooker (2022). Despite the introduction of stochasticity in dropout, extending the CLT of 4 does not require additional effort. The intuition for the proof is as follows: By restricting the tree space, one bounds the probability of the assumptions being violated to be summable, and per Borel-Cantelli will take place almost surely. Tree space cardinality here gives a nice union probability bound to achieve that. $\square$ + +We define the following notation for later use: For any $n$ -vector $v$ and an index set $D$ , denote + +$$ +v \big | _ {D} = \left[ \begin{array}{c} v _ {1} \cdot \mathbb {1} (1 \in D) \\ \vdots \\ v _ {n} \cdot \mathbb {1} (n \in D) \end{array} \right]. +$$ + +This implies the decomposition that $v = v\big|_{D} + v\big|_{D^c}$ . + +We are now almost ready to prove the main theorem. However, to check Lindeberg-Feller conditions under random design, one needs to establish exponential decay of points that sit far away from the point of interest. That is, $\| \boldsymbol{r}_n|_{D^c}\|$ for a well chosen ball centered at $\mathbf{x}$ needs to vanish to 0 rather quickly. And we prove it in the following lemma. To control this for Algorithm 1, we have the following lemma: + +Lemma 5. Given sample $\{(\mathbf{x}_i,\mathbf{y}_i)\}_{i = 1}^n$ and a point of interest $\mathbf{x}$ , set $l_{n} = \frac{\log n}{-\log\lambda q} = c_{1}\log n, c_{1} = \log \frac{1}{\lambda q}$ and define index set $D_{n} = \{i:|\mathbf{x}_{i} - \mathbf{x}|\leq l_{n}\cdot d_{n}\}$ , then + +$$ +\left\| \boldsymbol {r} _ {n} ^ {D} \right| _ {D _ {n} ^ {c}} \Big \| _ {1} = O \left(\frac {1}{n}\right). +$$ + +Proof. Let's first write out the expression of the desired peripheral points: + +$$ +\left\| \boldsymbol {r} _ {n} ^ {D} \right| _ {D _ {n} ^ {c}} \Bigg \| _ {1} = \sum_ {| \mathbf {x} - \mathbf {x} _ {i} | > l _ {n} \cdot d _ {n}} | \boldsymbol {r} _ {n i} ^ {D} | +$$ + +Recall: + +$$ +\boldsymbol {k} _ {n} ^ {T} = \mathbb {E} [ \boldsymbol {s} _ {n} (\mathbf {x}) ] \rightarrow \boldsymbol {k} _ {n j} = \mathbb {E} [ \boldsymbol {s} _ {n, j} (\mathbf {x}) ] = \mathbb {E} \left[ \frac {\mathbb {1} (\mathbf {x} _ {j} \in A)}{\sum_ {j = 1} ^ {n} \mathbb {1} (\mathbf {x} _ {j} \in A)} \right], \mathbf {x} \in A +$$ + +This term would be zero if two points are not even in the same leaf, i.e. $|\mathbf{x} - \mathbf{x}_j| > d_n$ . Hence the sum inside would be simplified to + +$$ +\sum_ {| \mathbf {x} - \mathbf {x} _ {i} | > l _ {n} \cdot d _ {n}} \left| \sum_ {| \mathbf {x} _ {j} - \mathbf {x} | \leq d _ {n}} \boldsymbol {k} _ {n j} \left[ \frac {1}{\lambda} \mathbf {I} + q \mathbf {K} _ {n} \right] _ {j, i} ^ {- 1} \right| +$$ + +And also notice, by decreasing $l_{n}d_{n}$ to $(l_{n} - 1)d_{n}$ , we are actually allowing more peripheral points outside the local region $D_{n}$ and hence we have the inequality at line 4 below. Completely. it is given as: + +$$ +\begin{array}{l} \left\| \boldsymbol {r} _ {n} ^ {D} \right| _ {D _ {n} ^ {c}} \Bigg \| _ {1} = \sum_ {| \mathbf {x} - \mathbf {x} _ {i} | > l _ {n} \cdot d _ {n}} | \boldsymbol {r} _ {n i} ^ {D} | = \sum_ {| \mathbf {x} - \mathbf {x} _ {i} | > l _ {n} \cdot d _ {n}} \left| \sum_ {j} \boldsymbol {k} _ {n j} \left[ \frac {1}{\lambda} \mathbf {I} + q \mathbf {K} _ {n} \right] _ {j, i} ^ {- 1} \right| \\ = \sum_ {| \mathbf {x} - \mathbf {x} _ {i} | > l _ {n} \cdot d _ {n}} \left| \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} \boldsymbol {k} _ {n j} \left[ \frac {1}{\lambda} \mathbf {I} + q \mathbf {K} _ {n} \right] _ {j, i} ^ {- 1} \right| \\ \leq \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} \boldsymbol {k} _ {n j} \sum_ {| \mathbf {x} - \mathbf {x} _ {i} | > l _ {n} \cdot d _ {n}} \left| \left[ \frac {1}{\lambda} \mathbf {I} + q \mathbf {K} _ {n} \right] _ {j, i} ^ {- 1} \right| \\ \end{array} +$$ + +$$ +\begin{array}{l} \leq \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} k _ {n j} \sum_ {| \mathbf {x} _ {i} - \mathbf {x} _ {j} | > (l _ {n} - 1) \cdot d _ {n}} \left| \left[ \frac {1}{\lambda} \mathbf {I} + q \mathbf {K} _ {n} \right] _ {j, i} ^ {- 1} \right| \\ \leq \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} \mathbf {k} _ {n j} \sum_ {| \mathbf {x} _ {i} - \mathbf {x} _ {j} | > (l _ {n} - 1) \cdot d _ {n}} \lambda \sum_ {l = l _ {n}} ^ {\infty} (\lambda q) ^ {l} [ \mathbf {K} _ {n} ^ {l} ] _ {j, i} \\ \leq \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} \boldsymbol {k} _ {n j} \sum_ {l = l _ {n}} ^ {\infty} (\lambda q) ^ {l + 1} \\ \leq \lambda \sum_ {l = l _ {n}} ^ {\infty} (\lambda q) ^ {l} = \frac {\lambda}{1 - \lambda q} \frac {1}{n}. \\ \end{array} +$$ + +The lower bound in the sum of the powers of $\mathbf{K}_n$ follows this argument. To compute higher powers of $\mathbf{K}_n$ , consider how entries of $K_n^l$ are calculated: + +$$ +\mathbf {K} _ {n} ^ {l} [ i, j ] = \sum_ {k _ {1}, k _ {2}, \dots , k _ {l - 1}} \mathbf {K} _ {n} [ i, k _ {1} ] \mathbf {K} _ {n} [ k _ {1}, k _ {2} ] \dots \mathbf {K} _ {n} [ k _ {l - 1}, j ]. +$$ + +For $\mathbf{K}_n^l [i,j]\neq 0$ , there must exist a chain of intermediate points $\{k_{1},k_{2},\ldots ,k_{l - 1}\}$ such that: + +$$ +\left| \mathbf {x} _ {i} - \mathbf {x} _ {k _ {1}} \right| \leq d _ {n}, \quad \left| \mathbf {x} _ {k _ {1}} - \mathbf {x} _ {k _ {2}} \right| \leq d _ {n}, \quad \dots , \quad \left| \mathbf {x} _ {k _ {l - 1}} - \mathbf {x} _ {j} \right| \leq d _ {n}. +$$ + +By summing the distances and by a triangular inequality, the locality condition propagates: + +$$ +\left| \mathbf {x} _ {i} - \mathbf {x} _ {j} \right| \leq l \cdot d _ {n}, +$$ + +where $l$ is the number of steps (or multiplications) required to connect $x_{i}$ and $x_{j}$ through the intermediate points. Now notice the summing condition $|x_{i} - x_{j}| > (l_{n} - 1)d_{n}$ requires us to have such chain that is longer than $l_{n} - 1$ at least. Hence the kernel's power $\mathbf{K}_n^l$ would be 0 for any $l\leq l_n - 1$ , since such chain doesn't exist at all. + +To control weights from distant points for Algorithm 2 is a bit more complicated. We need to make use of Assumption 7. This assumption is inspired by the definition in Athey et al. (2018), where we state below. + +Definition 3 ( $\alpha$ -regular tree predictor). A tree predictor grown by recursive partitioning is called $\alpha$ -regular for some $\alpha > 0$ if either + +1. (Standard case) each split leaves at least a fraction $\alpha$ of the available training examples on each side of the split, and furthermore the trees are fully grown to depth $k$ for some $k \in \mathbb{N}$ . Equivalently, every terminal node contains between $k$ and $2k - 1$ observations; +2. (SVI case) It satisfies part (a) when applied to the sample used to fit the leaf value. In this case this will be our response vector $Y$ . + +Lemma 6. If a tree is $\frac{1}{2}$ -regular with the terminal leaf size being lower bounded by $k$ , where $k$ is a constant, then the operator norm of $\mathbf{P}$ is $O(1)$ . + +Proof. Define $\alpha_{*} = \max \{\alpha, 1 - \alpha\}$ . By definition of $\alpha$ -regularity, the probability of point $i,j$ falling into the same leaf after one split would be upper bounded by $\alpha_{*}$ . And since we have the minimal terminal leaf size requirement, for any leaf with depth $d$ , it should comply with: + +$$ +\left\{ \begin{array}{l} k \leq \alpha_ {*} ^ {d} n \leq 2 k - 1, \\ k \leq (1 - \alpha_ {*}) ^ {d} n \leq 2 k - 1. \end{array} \right. +$$ + +which translates into $\frac{1}{\ln(1 - \alpha_{*})}\ln \left(\frac{2k - 1}{n}\right)\leq d\leq \frac{1}{\ln(\alpha_{*})}\ln \left(\frac{k}{n}\right).$ + +Notice that the probability of two points sharing a leaf is the probability that they survive all splits down the tree. Then we can bound the collision probability by: + +$$ +\begin{array}{l} p _ {i, j} \leq \alpha_ {* n i n} ^ {d} \\ = \alpha_ {*} ^ {\frac {1}{\ln (1 - \alpha_ {*})}} \ln (\frac {2 k - 1}{n}) \\ \end{array} +$$ + +$$ +\begin{array}{l} = \left(\frac {2 k - 1}{n}\right) ^ {\frac {\ln \alpha_ {*}}{\ln (1 - \alpha_ {*})}} \\ = \left(\frac {2 k - 1}{n}\right) ^ {\log_ {1 - \alpha_ {*}} \alpha_ {*}} \\ = \frac {2 k - 1}{n} \\ \end{array} +$$ + +Then $\sum_{i}p_{i,j} = O(1)$ + +![](images/c9be04508b0ca815b9fe3afa44713b392d210d33964f1a3c4bdd61d3e35cb1b4.jpg) + +Well-behaved leaves exclude non-decaying distant weights. Formally, we can prove the lemma below: + +Lemma 7. Given sample $\{(\mathbf{x}_i,\mathbf{y}_i)\}_{i = 1}^n$ and a point of interest $\mathbf{x}$ , set $l_{n} = -\frac{\ln n}{\ln(\xi / 2)} = c\ln n$ and define index set $D_{n} = \{i:|\mathbf{x}_{i} - \mathbf{x}|\leq l_{n}\cdot d_{n}\}$ . then + +$$ +\left\| r _ {n} ^ {P} \right| _ {D _ {n} ^ {c}} \Big \| _ {1} = O \left(\frac {1}{n}\right). +$$ + +Proof. We will use the same matrix expansion technique we used in Lemma5. By analogy, + +$$ +\begin{array}{l} \left\| \boldsymbol {r} _ {n} ^ {P} \right| _ {D _ {n} ^ {c}} \Big \| _ {1} = \sum_ {| \mathbf {x} - \mathbf {x} _ {i} | > l _ {n} \cdot d _ {n}} | \boldsymbol {r} _ {n i} ^ {P} | = \sum_ {| \mathbf {x} - \mathbf {x} _ {i} | > l _ {n} \cdot d _ {n}} \left| \sum_ {j} \boldsymbol {k} _ {n j} \left[ \frac {1}{K} \mathbf {I} + \frac {K - 1}{K} \mathbf {K} _ {n} \right] _ {j, i} ^ {- 1} \right| \\ = \sum_ {| \mathbf {x} - \mathbf {x} _ {i} | > l _ {n} \cdot d _ {n}} \left| \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} k _ {n j} \left[ \frac {1}{K} \mathbf {I} + \frac {K - 1}{K} \mathbf {K} _ {n} \right] _ {j, i} ^ {- 1} \right| \\ \leq \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} \mathbf {k} _ {n j} \sum_ {| \mathbf {x} - \mathbf {x} _ {i} | > l _ {n} \cdot d _ {n}} \left| \left[ \frac {1}{K} \mathbf {I} + \frac {K - 1}{K} \mathbf {K} _ {n} \right] _ {j, i} ^ {- 1} \right| \\ \leq \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} k _ {n j} \sum_ {| \mathbf {x} _ {i} - \mathbf {x} _ {j} | > (l _ {n} - 1) \cdot d _ {n}} \left| \left[ \frac {1}{K} \mathbf {I} + \frac {K - 1}{K} \mathbf {K} _ {n} \right] _ {j, i} ^ {- 1} \right| \\ = \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} \mathbf {k} _ {n j} \sum_ {| \mathbf {x} _ {i} - \mathbf {x} _ {j} | > (l _ {n} - 1) \cdot d _ {n}} K \left| \sum_ {l = l _ {n}} ^ {\infty} (- 1) ^ {l} (K - 1) ^ {l} [ \mathbf {K} _ {n} ^ {l} ] _ {j, i} \right| \\ \leq \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} k _ {n j} K \sum_ {l \geq l _ {n}} (K - 1) ^ {l} \sum_ {| \mathbf {x} _ {i} - \mathbf {x} _ {j} | > (l _ {n} - 1) \cdot d _ {n}} | [ \mathbf {K} _ {n} ^ {l} ] _ {i, j} | \\ \leq \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} \mathbf {k} _ {n j} K \sum_ {l \geq l _ {n}} (K - 1) ^ {l} \sum_ {i = 1} ^ {n} \left| \left[ \mathbf {K} _ {n} ^ {l} \right] _ {i, j} \right| \\ \leq \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} \boldsymbol {k} _ {n j} K \sum_ {l \geq l _ {n}} (K - 1) ^ {l} \left\| \mathbf {K} _ {n} ^ {l} \right\| _ {\infty} \\ \leq \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} \boldsymbol {k} _ {n j} K \sum_ {l \geq l _ {n}} (K - 1) ^ {l} \| \mathbf {K} _ {n} \| _ {\infty} ^ {l} \\ \end{array} +$$ + +Denote the probability of any point $\mathbf{x}_i, \mathbf{x}_j$ sharing the same leaf as $p_{ij}$ . By definition of the kernel matrix, we can write each element as: + +$$ +\begin{array}{l} \left[ \mathbf {K} _ {n} \right] _ {i, j} = \mathbb {E} _ {q, w} \left[ \left[ \mathbf {S} _ {n} \right] _ {i, j} \right] \\ = \mathbb {E} _ {w} \left[ \left[ \mathbf {S} _ {n} \right] _ {i, j} \mid \mathbf {x} _ {i}, \mathbf {x} _ {j} \text {a r e i n t h e s a m e l e a f} \right] p _ {i j} \\ = \mathbb {E} [ \frac {1}{X} ] p _ {i, j} \\ \end{array} +$$ + +where $X \sim \mathrm{Binomial}([m, M], \xi)$ , with $m$ being the minimal number of points in a leaf and $M$ being the maximal number of points within a leaf. Since $X = m, \dots, M$ , it follows that $\frac{1}{X} \leq \frac{1}{m}$ almost + +surely and hence $\mathbb{E}[\frac{1}{X} ]\leq \frac{1}{m\xi}$ . Then by assumption $m = \max \{\Omega \left(n^{\frac{1}{d + 2}}\right),\frac{K - 1}{\xi}\}$ , it follows that $\mathbb{E}[\frac{1}{X} ]\leq \frac{\xi}{K - 1}$ . Hence we have + +$$ +\begin{array}{l} \left\| \boldsymbol {r} _ {n} ^ {P} \right| _ {D _ {n} ^ {c}} \Big \| _ {1} \leq \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} \boldsymbol {k} _ {n j} K \sum_ {l \geq l _ {n}} (K - 1) ^ {l} \| \mathbf {K} _ {n} \| _ {\infty} ^ {l} \\ \leq \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} \boldsymbol {k} _ {n j} K \sum_ {l \geq l _ {n}} (K - 1) ^ {l} \frac {\xi^ {l}}{2 ^ {l} (K - 1) ^ {l}} \| \mathbf {P} \| _ {\infty} \\ \leq \sum_ {| \mathbf {x} - \mathbf {x} _ {j} | \leq d _ {n}} \boldsymbol {k} _ {n j} K \sum_ {l \geq l _ {n}} \frac {\xi^ {l}}{2 ^ {l}} o (1) \\ \leq \left(\frac {\xi}{2}\right) ^ {l _ {n}} o (1) \\ = o \left(\frac {1}{n}\right) \\ \end{array} +$$ + +![](images/fef47910807ebb23b957816208de5ad316a8be07aecbe6a81eeabbf902343932.jpg) + +# F Proof for Theorem 2 + +Combining all discussions above, we present the main result Theorem 2 of this paper. + +Proof. Define $c^D = \frac{\lambda}{1 + \lambda q}$ and $c^P = 1$ . To begin with, we will show that for $\forall \mathbf{x} \in [0,1]^d$ , + +$$ +\frac {\langle \boldsymbol {r} _ {n} ^ {D , P} , f (\mathbf {X} _ {n}) \rangle - c ^ {D , P} \boldsymbol {f} (\mathbf {x})}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|} \xrightarrow {\mathbb {P}} 0. +$$ + +Let $\Delta^{D,P} = c^{D,P} - \sum_{i=1}^{n} \boldsymbol{r}_{n,i}^{D,P} = O(n^{-1})$ and $\widetilde{\boldsymbol{f}}(\mathbf{x}) = (\boldsymbol{f}(\mathbf{x}), \dots, \boldsymbol{f}(\mathbf{x}))^\top$ an $n$ -vector. We consider the following decomposition: + +$$ +\begin{array}{l} \frac {\left\langle \boldsymbol {r} _ {n} ^ {D , P} , f (\mathbf {X} _ {n}) \right\rangle - c ^ {D , P} \boldsymbol {f} (\mathbf {x})}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|} = \frac {\left\langle \boldsymbol {r} _ {n} ^ {D , P} , \boldsymbol {f} (\mathbf {X} _ {n}) - \widetilde {\boldsymbol {f}} (\mathbf {x}) \right\rangle}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|} - \frac {\Delta^ {D , P} \cdot \boldsymbol {f} (\mathbf {x})}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|} \\ = - \frac {\Delta \cdot \boldsymbol {f} (\mathbf {x})}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|} + \frac {\left. \right.\left\langle \right. \boldsymbol {r} _ {n} ^ {D , P} \left. \right| _ {D _ {n}} , \left[ \boldsymbol {f} (\mathbf {X} _ {n}) - \widetilde {\boldsymbol {f}} (\mathbf {x}) \right]\left. \right| _ {D _ {n}} \left. \right\rangle}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|} + \frac {\left. \right.\left\langle \right. \boldsymbol {r} _ {n} ^ {D , P} \left. \right| _ {D _ {n} ^ {c}} , \left[ \boldsymbol {f} (\mathbf {X} _ {n}) - \widetilde {\boldsymbol {f}} (\mathbf {x}) \right]\left. \right| _ {D _ {n} ^ {c}} \left. \right\rangle}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|}. \\ \end{array} +$$ + +By 4, we have + +$$ +\left\| \boldsymbol {k} _ {n} \right\| = \Theta_ {p} \left(n ^ {- \frac {1}{d + 1}}\right), \left\| \boldsymbol {r} _ {n} ^ {D, P} \right\| = \Theta_ {p} \left(n ^ {- \frac {1}{2} \frac {1}{d + 1}}\right). +$$ + +Notice that + +$$ +\left|\left\langle \boldsymbol{r}_{n}^{D,P}\right|_{D_{n}^{c}},\left[\boldsymbol {f}(\mathbf{X}_{n}) - \widetilde{\boldsymbol{f}} (\mathbf{x})\right]\big|_{D_{n}^{c}}\right\rangle \Big|\leq \left\| \boldsymbol{r}_{n}^{D,P}\big|_{D_{n}^{c}}\right\|_{1}\cdot \left\| \left[\boldsymbol {f}(\mathbf{X}_{n}) - \widetilde{\boldsymbol{f}} (\mathbf{x})\right]\big|_{D_{n}^{c}}\right\|_{\infty} = O_{p}\left(\frac{1}{n}\cdot 2M_{\boldsymbol{f}}\right) = O_{p}\left(n^{-1}\right). +$$ + +Hence we have the second term + +$$ +\frac {\left. \langle \boldsymbol {r} _ {n} ^ {D , P} \right| _ {D _ {n}} , \left[ \boldsymbol {f} (\mathbf {X} _ {n}) - \widetilde {\boldsymbol {f}} (\mathbf {x}) \right]\left. \right| _ {D _ {n}} \rangle}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|} \xrightarrow {\mathbb {P}} 0. +$$ + +And by an argument earlier $|\Delta^{D,P}| = O(n^{-1})$ , thus + +$$ +\begin{array}{c} \Delta \cdot \boldsymbol {f} (\mathbf {x}) \\ \hline \| \boldsymbol {r} _ {n} \| \end{array} \xrightarrow {\mathbb {P}} 0. +$$ + +Meanwhile, we can show similarly $|D_{n}| = \Omega_{p}\left(n\cdot (l_{n}\cdot d_{n})^{d}\right)$ a.s.. And recall our underlying function is $\alpha$ -Lipschitz, therefore + +$$ +\frac {\left. \langle \boldsymbol {r} _ {n} ^ {D , P} \right| _ {D _ {n} ^ {c}} , \left[ \boldsymbol {f} (\mathbf {X} _ {n}) - \widetilde {\boldsymbol {f}} (\mathbf {x}) \right]\left. \right| _ {D _ {n} ^ {c}} \rangle}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|} \leq \frac {\left\|\left. \boldsymbol {r} _ {n} ^ {D , P} \right| _ {D _ {n}} \right\| \cdot \left\| \right.\left[ \boldsymbol {f} (\mathbf {X} _ {n}) - \widetilde {\boldsymbol {f}} (\mathbf {x}) \right]\left. \right| _ {D _ {n}} \left. \right\|}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|} +$$ + +$$ +\begin{array}{l} \leq \left\| [ \boldsymbol {f} (\mathbf {X} _ {n}) - \widetilde {\boldsymbol {f}} (\mathbf {x}) ] \right| _ {D _ {n}} \Bigg \| \\ = O _ {p} \left(\sqrt {n \cdot \left(l _ {n} d _ {n}\right) ^ {d} \cdot \left(l _ {n} d _ {n} \cdot \alpha\right) ^ {2}}\right) \\ = O _ {p} \left(\sqrt {n \cdot (\log n) ^ {d + 2} \cdot d _ {n} ^ {d + 2}}\right) \\ = O _ {p} \left(\sqrt {n \cdot (\log n) ^ {d + 2} \cdot n ^ {- \frac {d + 2}{d + 1}}}\right) \\ = O _ {p} \left(\left(\log n\right) ^ {\frac {d + 2}{2}} n ^ {- \frac {1}{2} \frac {1}{d + 1}}\right). \\ \end{array} +$$ + +Therefore + +$$ +\frac {\left\langle \boldsymbol {r} _ {n} ^ {D , P} , \boldsymbol {f} (\mathbf {X} _ {n}) \right\rangle - c ^ {D , P} \boldsymbol {f} (\mathbf {x})}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|} \xrightarrow {\mathbb {P}} 0. +$$ + +Slutsky's Theorem then yields + +$$ +\frac {\widehat {\boldsymbol {f}} _ {n} ^ {D , P} (\mathbf {x}) - c ^ {D , P} \boldsymbol {f} (\mathbf {x})}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|} = \frac {\widehat {\boldsymbol {f}} _ {n} (\mathbf {x}) - \langle \boldsymbol {r} _ {n} ^ {D , P} , \boldsymbol {f} (\mathbf {X} _ {n}) \rangle}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|} + \frac {\langle \boldsymbol {r} _ {n} ^ {D , P} , \boldsymbol {f} (\mathbf {X} _ {n}) \rangle - c ^ {D , P} \boldsymbol {f} (\mathbf {x})}{\left\| \boldsymbol {r} _ {n} ^ {D , P} \right\|} \xrightarrow {d} \mathcal {N} (0, \sigma_ {\epsilon} ^ {2}). +$$ + +# G Coverage Rates and Risk Bounds Guarantees + +In this section we provide the proofs for coverage rates and risk bounds guarantees we gave in 5. + +# G.1 Proof for Corollary 1 + +Proof. The corollary holds by noticing that the variance of the scaled prediction of Algorithm 1 is $(\frac{1 + \lambda q}{\lambda})^2\left\| r_n^D\right\|^2\sigma^2$ . Plugging in the rate of $\left\| r_n^D\right\|$ by lemma 1 yields the risk bound. Similarly this can be shown for Algorithm 2. + +# G.2 Proof for Corollary 3 + +Proof. We will show this convergence for the confidence interval Algorithm 1. For Algorithm 2, replacing the asymptotic variance and the final rescale constant will follow the same proof. + +Denote the staged empirical coverage rate of the built-in confidence interval at any point $\mathbf{x} \in [0,1]^d$ as + +$$ +1 - \hat {\alpha} _ {n, b} ^ {\mathrm {C I}} (\mathbf {x}) = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \left[ \right. \boldsymbol {f} (\mathbf {x}) \in \left[ \right.\left( \right.c ^ {D} \widehat {\boldsymbol {f}} _ {b} (\mathbf {x}) - c ^ {D} \left\| \boldsymbol {r} _ {n} ^ {D} \right\| z _ {1 - \alpha / 2} \sigma , c ^ {D} \widehat {\boldsymbol {f}} _ {b} (\mathbf {x}) + c ^ {D} \left\| \boldsymbol {r} _ {n} ^ {D} \right\| z _ {1 - \alpha / 2} \sigma \left. \right]\left. \right. +$$ + +Notice that with $b \to \infty$ we have $\widehat{f}_b \stackrel{a.s.}{\rightarrow} \widehat{f}_{\infty}$ . Hence, sending $b$ to infinity and we drop the subscript for $b$ to denote the coverage at infinite epoch. + +$$ +1 - \hat {\alpha} _ {n} ^ {\mathrm {C I}} (\mathbf {x}) = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \left[ \right. \boldsymbol {f} (\mathbf {x}) \in \left[ \right.\left( \right.c ^ {D} \widehat {\boldsymbol {f}} _ {\infty} (\mathbf {x}) - c ^ {D} \left\| \boldsymbol {r} _ {n} ^ {D} \right\| z _ {1 - \alpha / 2} \sigma , c ^ {D} \widehat {\boldsymbol {f}} _ {\infty} (\mathbf {x}) + c ^ {D} \left\| \boldsymbol {r} _ {n} ^ {D} \right\| z _ {1 - \alpha / 2} \sigma \left. \right]\left. \right] +$$ + +which is equivalent to + +$$ +1 - \hat {\alpha} _ {n} ^ {\mathrm {C I}} (\mathbf {x}) = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \left[ \frac {\widehat {\boldsymbol {f}} _ {\infty} (\mathbf {x}) - c ^ {D} \boldsymbol {f} (\mathbf {x})}{\| r _ {n} ^ {D} \| \sigma} \in \left[ - z _ {1 - \alpha / 2}, z _ {1 - \alpha / 2} \right] \right] +$$ + +Since the indicator function is always finite, by SLLN: + +$$ +1 - \hat {\alpha} _ {n} ^ {\mathrm {C I}} (\mathbf {x}) \stackrel {a. s.} {\rightarrow} \mathbb {E} \bigg [ \mathbb {1} \Big [ \frac {\widehat {\pmb {f}} _ {\infty} (\mathbf {x}) - c ^ {D} \pmb {f} (\mathbf {x})}{\| r _ {n} ^ {D} \| \sigma} \in \big [ - z _ {1 - \alpha / 2}, z _ {1 - \alpha / 2} \big ] \Big ] \bigg ] +$$ + +$$ +\begin{array}{l} = \mathbb {P} \Big [ \frac {\widehat {\pmb {f}} _ {\infty} (\mathbf {x}) - c ^ {D} \pmb {f} (\mathbf {x})}{\| r _ {n} ^ {D} \| \sigma} \in \big [ - z _ {1 - \alpha / 2}, z _ {1 - \alpha / 2} \big ] \Big ] \\ = 1 - \alpha \\ \end{array} +$$ + +For prediction intervals, we observe data according to the model + +$$ +\mathbf {y} = \boldsymbol {f} (\mathbf {x}) + \varepsilon , \quad \varepsilon \sim \mathcal {N} (0, \sigma^ {2}), +$$ + +and our infinite-epoch estimator satisfies the CLT + +$$ +\widehat {\boldsymbol {f}} _ {\infty} (\mathbf {x}) \sim \mathcal {N} \left(c ^ {D} \boldsymbol {f} (\mathbf {x}), \| \boldsymbol {r} _ {n} ^ {D} \| ^ {2} \sigma^ {2}\right) \quad \text {a s} \quad b \rightarrow \infty . +$$ + +Since $\widehat{f}_{\infty}(\mathbf{x})$ and the new noise $\varepsilon$ are independent Gaussians, their sum + +$$ +\begin{array}{l} \mathbf {y} = \boldsymbol {f} (\mathbf {x}) - c ^ {D} \widehat {\boldsymbol {f}} _ {\infty} (\mathbf {x}) + c ^ {D} \widehat {\boldsymbol {f}} _ {\infty} (\mathbf {x}) + \epsilon \\ = \mathcal {N} (c ^ {D} \widehat {\pmb {f}} _ {\infty} (\mathbf {x}), (c ^ {D} \left\| \pmb {r} _ {n} ^ {D} \right\|) ^ {2} \sigma^ {2} + \sigma^ {2}) \\ \end{array} +$$ + +And we produce prediction interval with the empirical coverage rate evaluated at: + +$$ +\begin{array}{l} 1 - \hat {\alpha} _ {n, b} ^ {\mathrm {P I}} (\mathbf {x}) \\ = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \left[ \boldsymbol {f} (\mathbf {x}) \in \left[ (c ^ {D} \widehat {\boldsymbol {f}} _ {b} (\mathbf {x}) - \sqrt {(c ^ {D} \| \boldsymbol {r} _ {n} ^ {D} \|) ^ {2} + 1} \cdot z _ {1 - \alpha / 2} \sigma , c ^ {D} \widehat {\boldsymbol {f}} _ {b} (\mathbf {x}) + \sqrt {(c ^ {D} \| \boldsymbol {r} _ {n} ^ {D} \|) ^ {2} + 1} \cdot z _ {1 - \alpha / 2} \sigma ] \right] \right. \\ \end{array} +$$ + +By similar argument above, sending $n, b \to \infty$ gives us almost sure convergence. + +Regarding the reproduction interval, since we will have two independent model following the CLT in Theorem 2, their difference will also follow a CLT centered at zero with a variance inflated by a factor of 2 by independence. So we construct the reproduction interval as such: + +$$ +\begin{array}{l} 1 - \hat {\alpha} _ {n, b} ^ {\mathrm {R I}} (\mathbf {x}) \\ = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \Big [ \widehat {\pmb {f}} _ {b} ^ {(2)} (\mathbf {x}) \in \big [ \widehat {\pmb {f}} _ {b} ^ {(1)} (\mathbf {x}) - \sqrt {2} (c ^ {D} \left\| \pmb {r} _ {n} ^ {D} \right\|) z _ {1 - \alpha / 2} \sigma , \widehat {\pmb {f}} _ {b} ^ {(1)} (\mathbf {x}) + \sqrt {2} (c ^ {D} \left\| \pmb {r} _ {n} ^ {D} \right\|) z _ {1 - \alpha / 2} \sigma \big ] \Big ] \\ \end{array} +$$ + +And by a similar argument we can prove almost sure convergence. + +![](images/7931535fc9144c63510260b0993697807343c859afb5c2dc2fba37867bb05206.jpg) + +# G.3 Proof for Corollary 2 + +Proof. To give a PAC argument, we decompose our error into statistical error $\widehat{f}_*(\mathbf{x}) - f(\mathbf{x})$ and algorithmic error $\widehat{f_b}(\mathbf{x}) - \widehat{f}_*(\mathbf{x})$ . + +From the Theorem 2, $\operatorname*{Pr}\big(|\widehat{f}_*(\mathbf{x}) - f(\mathbf{x})| > \varepsilon \big) \leq 2\exp \left(-\frac{\varepsilon^2}{2(c^{D,P})^2} n^{1/(d+1)}\right)$ . Under the stated lower bound on $n$ this probability is $\leq \delta/2$ . + +To quantify the algorithmic error, we wish to make use of the Komolgorov inequality in Theorem 3. + +For $\forall \mathbf{x} \in [0,1]^d$ , let $\widehat{f}_b(\mathbf{x})$ be the stage- $b$ predictor, let $\widehat{f}_*(\mathbf{x})$ be the fixed point defined in Theorem 1, and write + +$$ +\mathbf {z} _ {b} (\mathbf {x}) = \widehat {\boldsymbol {f}} _ {b} (\mathbf {x}) - \widehat {\boldsymbol {f}} _ {*} (\mathbf {x}), \quad \epsilon_ {b} = \mathbf {z} _ {b} - \mathbb {E} [ \mathbf {z} _ {b} \mid \mathcal {F} _ {b - 1} ], +$$ + +It holds that the process $\mathbf{z}_b(\mathbf{x})$ is eventually also a stochastic contraction mapping by the exact same argument we made in proving Theorem 1, only replacing $\mathbf{K}_n$ with $k_{n}(\mathbf{x})$ which still have row sum smaller than 1. And even we only have in probability guarantee of the staged mean contraction condition hold, since we are doing truncations and we can get rid off it after sufficiently large $b$ , we can still check out the mean contraction condition $\sum_{b = 1}^{\infty}(1 - \lambda_b) = \infty$ . Hence we invoke the Kolmogorov Inequality given in 3. + +Consider the decomposition: + +$$ +\Pr \left(\sup _ {t \geq b} | \mathbf {z} _ {t} | > \varepsilon\right) \leq \Pr \left(| \mathbf {z} _ {b} | > \frac {\varepsilon}{2}\right) + \Pr \left(\sup _ {t \geq b} | \mathbf {z} _ {t} | > | \mathbf {z} _ {b} | + \frac {\varepsilon}{2}\right) +$$ + +Because the SCM conditions are satisfied from $b$ onward and $\sum_{t > b} \mathbb{E}[\epsilon_t^2] \leq C^2 / b$ , where $C = 2\lambda M(1 + \sqrt{n})$ . Theorem 3 gives the Kolmogorov's inequality below. Specifically, we can + +choose $b$ such that the numerator $\min \left\{\frac{\epsilon}{2}, \beta_{\frac{\epsilon}{2}}\right\} \geq \frac{\epsilon}{2}$ for sufficiently $b'$ , where $\beta_{\frac{\epsilon}{2}} = \frac{\epsilon}{2} + \|\mathbf{z}_t\| - \sqrt{d} \sup_{t > s} \|\epsilon_s\|$ , when $b = \Omega\left(\frac{n\lambda^2m^2}{\epsilon^2\delta}\right)$ . Hence + +$$ +\Pr \bigl (\sup_{t\geq b}\left|\mathbf{z}_{t}\right| > \left|\mathbf{z}_{b}\right| + \frac{\epsilon}{2}\bigr)\leq \frac{4\cdot 1\cdot(C^{2} / b)}{\min\{\frac{\epsilon}{2},\beta_{\frac{\epsilon}{2}}\}^{2}}\leq \frac{4\cdot 1\cdot(C^{2} / b)}{(\epsilon / 2)^{2}} = \frac{16C^{2}}{\epsilon^{2}b}. +$$ + +Since $|\epsilon_k| \leq C$ , $V_b = \sum_{k=1}^{b} \mathbb{E}[\epsilon_k^2] \leq C^2 \pi^2 / 6 < 2C^2$ . + +Set the martingale sum $S_{b} = \sum_{k=1}^{b} \epsilon_{k}$ . Because $|\mathbf{z}_{t}| \leq |\mathbf{z}_{t-1}| + |\epsilon_{t}|$ , telescoping gives + +$$ +| \mathbf {z} _ {b} | \leq | \mathbf {z} _ {0} | + | S _ {b} | \leq M + | S _ {b} |. +$$ + +Thus the event $\{| \mathbf{z}_b | > \varepsilon / 2\}$ implies $|S_b| > (\varepsilon - 2M) / 2$ . + +Freedman's maximal inequality for increments bounded by $C$ and quadratic variation $\leq 2C^2$ yields + +$$ +\Pr \Bigl (| \mathbf {z} _ {b} | > \frac {\varepsilon}{2} \Bigr) \leq \Pr \Bigl (| S _ {b} | > \frac {\varepsilon - 2 M}{2} \Bigr) \leq 2 \exp \Bigl (- \frac {(\varepsilon - 2 M) _ {+} ^ {2}}{1 6 C ^ {2}} \Bigr). +$$ + +Then the non-asymptotic algorithmic error bound: + +$$ +\Pr \Bigl (| \mathbf {z} _ {b} (\mathbf {x}) | > \varepsilon \Bigr) \leq 2 \exp \Bigl (- \frac {(\varepsilon - 4 M) _ {+} ^ {2}}{1 6 C ^ {2}} \Bigr) + \frac {1 6 C ^ {2}}{\varepsilon^ {2} b}, \quad C = 2 \lambda M (1 + \sqrt {n}). +$$ + +With the stated lower bound on $b$ this is $\leq \delta / 2$ . + +![](images/04e2497db29f55358a3aa1501f80a4c2cb65f0d785c78223cd949a4ecaa66945.jpg) + +# H Miscellaneous Lemmas + +Theorem 5 (Kolmogorov's Extension Theorem). Define the random variables: + +$\mathbf{X} = (\mathbf{x}_1,\mathbf{x}_2,\dots)\in [0,1]^{d\times \mathbb{N}}$ +- $\vec{\epsilon} = (\epsilon_{1},\epsilon_{2},\dots)\in \mathbb{R}^{\mathbb{N}}$ + +The probability measure on $[0,1]^{d\times \mathbb{N}}$ and $\mathbb{R}^{\mathbb{N}}$ are uniquely determined by their product measures on the cylinder spaces, reflecting the i.i.d. sampling of $\mathbf{X}$ and $\vec{\epsilon}$ , such that + +$$ +\mathbf {y} _ {i} = \boldsymbol {f} (\mathbf {x} _ {i}) + \epsilon_ {i}, \quad i \in \mathbb {N} +$$ + +Lemma 8. Let $X: \Omega_1 \to S$ and $\epsilon: \Omega_2 \to S$ be independent random variables. Assume $\{f_n: S \times S \to \mathbb{R}\}$ is a sequence of measurable functions. Suppose that for almost surely $x \in \Omega_1$ and $\epsilon \in \Omega_2$ , + +$$ +f _ {n} (x, \epsilon) \stackrel {d} {\to} N (0, 1). +$$ + +Then: + +$$ +f _ {n} (X, \epsilon) \xrightarrow {d} N (0, 1). +$$ + +Or equivalently + +$$ +\lim _ {n \rightarrow \infty} \mathbb {P} (f _ {n} (X, \vec {\epsilon})) = \Phi (t) +$$ + +Proof. + +$$ +\begin{array}{l} \lim _ {n} \mathbb {P} \left(f _ {n} (X, \epsilon) \leq t\right) = \lim _ {n} \iint \mathbb {1} _ {\left\{f _ {n} (x, \epsilon) \leq t \right\}} d \mu_ {x} d \mu_ {\epsilon} (5) \\ = \lim _ {n} \int \mathbb {P} \left(f _ {n} (x, \epsilon) \leq t\right) d \mu_ {x} (6) \\ = \int \lim _ {n} \mathbb {P} \left(f _ {n} (x, \epsilon) \leq t\right) d \mu_ {x} (7) \\ = \int \Phi (t) d \mu_ {x} = \Phi (t). (8) \\ \end{array} +$$ + +(8) is justified by the Fubini theorem since the indicator function is always non-negative. And (9) is guaranteed by Dominated Convergence Theorem since the integrand is always bounded (with assumption of it being a sub-Gaussian density of $\epsilon$ ). + +# I Mean Squared Error on UCI Machine Learning Repository + +We provide further results for the MSE of our methods, benchmarks, and competitors on six additional UCI datasets. The dataset we are using in this work are from: Nash et al. (1994), Vito (2008), Schlimmer (1985), est (2019), Cortez (2008), Cortez et al. (2009), Redmond (2002). + +The hyperparameters chosen by Optuna are listed below: + +Table 2: Abalone + +
HyperparameterGBTXGBoostLightGBMRFBRAT-DBoulevardBRAT-PElasticNet
n_estimators500500500500500500500—
learning_rate0.02130.02490.0108—0.54760.57500.2724—
max_depth3391110163—
subsample—0.6332——————
num_leaves——37—————
alpha———————0.1299
l1_ratio———————0.3753
min_samples_split————43425—
subsample_rate————0.50710.51330.6266—
dropout_rate————0.47660.0000——
n_trees_per_group——————16—
+ +Table 3: Automobiles + +
HyperparameterGBTXGBoostLightGBMRFBRAT-DBoulevardBRAT-PElasticNet
n_estimators500500500500500500500—
learning_rate0.20190.21530.2341—0.94110.14160.5671—
max_depth338146113—
subsample—0.9850——————
num_leaves——37—————
alpha———————0.3634
l1_ratio———————0.1386
min_samples_split————23930—
subsample_rate————0.93680.98990.7344—
dropout_rate————0.21450.0000——
n_trees_per_group——————16—
+ +Table 4: Communities and Crime + +
HyperparameterGBTXGBoostLightGBMRFBRAT-DBoulevardBRAT-PElasticNet
n_estimators500500500500500500500—
learning_rate0.02240.03790.0215—0.69040.57720.3578—
max_depth6351615164—
subsample—0.5375——————
num_leaves——20—————
alpha———————3.8079
l1_ratio———————0.9556
min_samples_split————35495—
subsample_rate————0.73850.72090.5148—
dropout_rate————0.19030.0000——
n_trees_per_group——————11—
+ +![](images/cd24bd44303e6720635eab68f5450a8738a9319fa57ddf587b34b66c043faa54.jpg) + +![](images/65878422b0728c06b46114744009ed570fe021dbbb3cdf3271ae1b481f2e6239.jpg) +Figure 6: Mean Squared Error on UCI Machine Learning Repositories + +![](images/38bd53d571d197d3f35c501b0cd887ee5a3c7cb69b5fa708f415b2945174e6fb.jpg) + +![](images/8e6986bbd05bd79d6b473d929d7f7d861dc035366d492b3f3b457370bf32bd08.jpg) + +![](images/ddee58a183efdcf1a851ebd47a5bc4a9d725141de530625017f6aa20770690e3.jpg) + +![](images/69006da533a07e98693a21f5acfaa1d4774d9f825bbc1e51acdd8561b0980007.jpg) + +Table 5: Wine Quality + +
HyperparameterGBTXGBoostLightGBMRFBRAT-DBoulevardBRAT-PElasticNet
n_estimators500500500500500500500—
learning_rate0.07310.11390.1186—0.83960.89050.8200—
max_depth10101620151516—
subsample—0.6980——————
num_leaves——42—————
alpha———————3.8079
l1_ratio———————0.9556
min_samples_split————4213—
subsample_rate————0.73170.74310.6669—
dropout_rate————0.27620.0000——
n_trees_per_group——————10—
+ +Table 6: Student Performance + +
HyperparameterGBTXGBoostLightGBMRFBRAT-DBoulevardBRAT-PElasticNet
n_estimators500500500500500500500—
learning_rate0.01770.18360.0218—0.92560.86980.9824—
max_depth35511393—
subsample—0.5780——————
num_leaves——38—————
alpha———————0.4855
l1_ratio———————0.1373
min_samples_split————393248—
subsample_rate————0.84180.52490.5964—
dropout_rate————0.12300.0000——
n_trees_per_group——————13—
+ +Table 7: Obesity Level + +
HyperparameterGBTXGBoostLightGBMRFBRAT-DBoulevardBRAT-PElasticNet
n_estimators500500500500500500500—
learning_rate0.19050.09160.0834—0.24870.53820.3643—
max_depth771616161611—
subsample—0.8121——————
num_leaves——25—————
alpha———————0.2182
l1_ratio———————0.1200
min_samples_split————334—
subsample_rate————0.64660.71530.5595—
dropout_rate————0.80790.0000——
n_trees_per_group——————4—
+ +Table 8: AIDS Clinical Trials Group Study 175 + +
HyperparameterGBTXGBoostLightGBMRFBRAT-DBoulevardBRAT-PElasticNet
n_estimators500500500500500500500—
learning_rate0.01770.02260.0108—0.85190.91370.3010—
max_depth3395444—
subsample—0.7530——————
num_leaves——37—————
alpha———————3.8079
l1_ratio———————0.9556
min_samples_split————122746—
subsample_rate————0.53350.73740.5342—
dropout_rate————0.34210.0000——
n_trees_per_group——————3—
+ +Table 9: Infrared Thermography Temperature + +
HyperparameterGBTXGBoostLightGBMRFBRAT-DBoulevardBRAT-PElasticNet
n_estimators500500500500500500500—
learning_rate0.02890.01670.1836—0.76160.65580.9864—
max_depth53120333—
subsample—0.6988——————
num_leaves——24—————
alpha———————3.8079
l1_ratio———————0.9556
min_samples_split————449—
subsample_rate————0.50550.54380.6371—
dropout_rate————0.23950.0000——
n_trees_per_group——————2—
+ +The learning rate for BRAT-P is only valid in the first vanilla boosting round. After that, all trees are fixed with learning rate 1. + +# J Rainclouds for Interval Coverage + +Following the discussion in Section 6, we present more raincloud plots below: + +![](images/99473fdb024d6adc4ef81b1e44a58c33fed2abf61622972cadab4a733adb1c08.jpg) + +![](images/e09fd282370354c32fa792a78deb624918ab39bb0cb1548ccf6223bd943b7f2c.jpg) + +![](images/58c997ad2ae3002be5caf46b7b81b82dda48794b03d05cb978e9729926b0b1fb.jpg) + +![](images/96691daf08dd0038eab38cc0ef6a4c0513d1f8ff1cee4f1364795eeb61a2e998.jpg) + +![](images/685d1aac389c58ca4067610612fc9ce66161d2bb7f3748b7b37080628f24f8d6.jpg) + +![](images/86602f9c1c64c31e7023a5037ff221a65cb16353c1f98f448b26a891cd50ad51.jpg) + +![](images/21235621d35a06d13d1d08e401da43557fe6eb358d6634f4acb272d18d2797fe.jpg) + +![](images/76acd90b5c4e77ba571a2b3e908ebb89cc925ad137f969a9001f41af623881b4.jpg) + +![](images/f03c57db8ae71541991895a8aff6c0b52b9b07099a09a24677a8cc12b25216ec.jpg) +(a) Test size: 200; Model Replications: 30; Ensemble Size: 200; Learning Rate: 0.6; Subsampling Rate: 0.8; Dropout Rate: 0.3; Maximum Depth: 4; Nystrom subsampling rate: 0.1. + +![](images/894ff04f1c09dd0cca3068a44ca3f01697b1e91330ebd5b4d5ce29f55f14a334.jpg) + +![](images/a48a3bc7e83adedef516cced8fbba3f29bd3e2489344a38670aad11db915a114.jpg) + +![](images/fba1ee0bd27cabadbb7936f3d52fb3a5f1ec38f1140fc8482715d33898f2b5f9.jpg) + +![](images/2055596ecd9323879035c690d5477a3f24f1a67fb94f1d4d1e0b8c4225b9c74c.jpg) +Figure 7: Moderate coverage, where there is neither overfitting nor underfitting. + +![](images/7d8ac537086ef57f8817122c7b0f8716e9b73db45e5377d0e1adcd1e28cf25de.jpg) + +![](images/aae3caa422d92120f7ca9ef589f42c4ff3b654da6c67ca8b3f0cb651eb69ec9e.jpg) + +![](images/2738a255fd1ed25d1ca46d61091c49b868f2fb8f67130cf6c308db5bfe1e72a8.jpg) + +![](images/13098b2b47e2e47750c3011412ddbfd3aed61eb8aa75a009a5f5c1caee051d96.jpg) + +![](images/f8fd3a449b7ea2d4eb03a9d746b4adb4a6996e7cd421bd6f27906a5761461b48.jpg) + +![](images/17faa0c1bee28a36741b873a251bef3dd93011c9cce99d603976c8d81a04ac65.jpg) + +![](images/fdb21000eee68009745735869b38b322f49b882e920c6ff7e4bfb98f6465c63a.jpg) + +![](images/18705c97bbf2e639d600d5e77e89962bdc1a380991e9ce8d1cdd4c3918936129.jpg) + +![](images/429196585e6103bbc992f160e1d1c923a2182f697be0bad32f6ded3dc7e9e80b.jpg) + +![](images/bb42ff3b2f39c3d7aa5384472bac0b44a87bb2655c52ac264b1e1202017d1115.jpg) + +![](images/c1cce0772139b99457fd7aca87db01c10bf826f4475ea2c77e3e09760f796618.jpg) + +![](images/c88815ea3d4fa5460a58d0579fac9dc6aff319276c3fa717308ccd44def85cd2.jpg) +(a) Test size: 200; Model Replications: 30; Ensemble Size: 200; Learning Rate: 0.3; Subsampling Rate: 0.8; Dropout Rate: 0.3; Maximum Depth: 6; Nystrom subsampling rate: 0.1. + +![](images/bf3aa09c67ed9597ab780e6dea911ec62bb4c17233bd2db4d9c802a54d4770ea.jpg) + +![](images/a6a8b87fc534fb23ce7fe51cfe26a59b6c07e2f198599594d69881bb05b5d8e7.jpg) + +![](images/8c22b08cc1e0216f157ab766d0e66ae44081bca8366202eb9dd8faddd2a15e5c.jpg) + +![](images/3b4ac49b1a8b89ff722092d7051d00b60ec2fe6d8f7ffdcddd68bdd60dd50bd9.jpg) +Figure 8: Wide coverage, when there is overfitting. + +![](images/597567b5cdcb9a4e87116294fd91bc6350584cdacd5ecb7c18ac9996b6ac5fc2.jpg) + +![](images/f864e798c8ba285476e0b5b60d295067d45b946a15372a505839bd6703a44dce.jpg) + +![](images/4d2fd37232cdf40d8779fd65af58d9a3fb133d448205f85a7af7c6c0dce94cf5.jpg) + +![](images/539970bb03a375f0d994b27368e0a5effc5c2a4849f9f3be423c34a82fe35d51.jpg) + +![](images/2cc2af436159ee3e8db1c3923977d082e708f8e9a062534b9f753de8dc983c82.jpg) + +![](images/bfb025e006efe51a415582e406227a87f25a541e6ae98acca36d4eaa600c6a4d.jpg) + +![](images/a02db37595dcfe251809e22996b52644a06af459dc357f8c164f38fa3d276841.jpg) + +![](images/619f78f53cabe72ea933765d23fbdf0196390183ebd1d914899c21008edc353a.jpg) + +![](images/8c1de35fdd9d66b436c9a6837434af37c5bb88a9f06229c67cec1278d4324903.jpg) + +![](images/9c7e467ddef31fc3a69d094357706a01735448a5f6b81b533b9bed1c0162a138.jpg) + +![](images/f7b618af69b5a2c776485d4250988d97d4d317c8b2569d89c1bee7d45c86e065.jpg) + +![](images/60c639de3c367b8945ab3a8c720c2ded3d43c508450f40dd1e160f92394683b7.jpg) +(a) Test size: 200; Model Replications: 30; Ensemble Size: 200; Learning Rate: 1.0; Subsampling Rate: 0.8; Dropout Rate: 0.9; Maximum Depth: 4; Nystrom subsampling rate: 0.1. + +![](images/1bf82022d64ef6127252bb893d9f5990bc60a24cfbf126b8512d699ba61be4cc.jpg) + +![](images/9e3c3e1994dc7bc56f378ef81f4932ef97a1affdd284d48ca4e6f96f94a65bce.jpg) + +![](images/c106d4a56f0aff6a541e3c2f698bd448a62a83e3cac09fed7ed338a10663e0ef.jpg) + +![](images/086126f4b15e54e14b7fda475836770847cca19aed208f29d7de531d98c1198b.jpg) +Figure 9: Narrow coverage, when there is underfitting. + +![](images/fcb2f28949fd2fc2e74a70a3c1a9c70cbc43f3038cdcb24289f655667d59ca94.jpg) + +![](images/22e6851e3dba88c8d2d90a7037831df057e0e5972e0baf3d2b396aebb0dc0228.jpg) + +![](images/761e062e703774357702900246fbf1b953fd0558b5b714deff005180e9a5c55a.jpg) + +# K Visualizations of Interval Coverage on a 1D Signal + +As an extension of Section 6, we display more visualizations of our intervals on a 1D interval. + +CI +![](images/b94bdf4e9ce1ffbe1751a4793baae5698dcbc6e0f0f77b5236174ca54c4a1633.jpg) +(a) Ensemble Size: 300; Learning Rate: 1.0; Maximum Depth: 8; Subsampling Rate: 0.9; Dropout rate: 0.6 + +![](images/b01608af966d267d947af948b44cd997ca6b20214372dfd12a862f19b5d72ffd.jpg) +RI + +![](images/dab80dd9960d5e6c3fceb132c2cb3903647810018c9fec0742b6318731822a5f.jpg) +PI +Figure 10: Moderate coverage, neither overfitting nor underfitting. + +![](images/e01e6fc2aed588b6eaa4b06644b2d65eba4ae818edfa7bef7a31c79cd2e332cc.jpg) +Conformal PI + +CI +![](images/7aa04d5de077354e3201c330233519cadd0c9f3e6d30592901a3bdd89732e41b.jpg) +(a) Ensemble Size: 100; Learning Rate: 0.3; Maximum Depth: 12; Subsampling Rate: 0.6; Dropout rate: 0.3 + +![](images/3b54af596028dc163e157687e551526b9452b860aab79108fdf3d77588525856.jpg) +RI +Figure 11: Low coverage, overfitting. + +![](images/ab1f539f253dbe0b607698f19c98dc2ba39ff47d90a4c6f9be21523cd9218204.jpg) +PI + +![](images/95e26655cddafc71bcf5138a3acb408a4eb4674229c50e143eb30d8ffc338266.jpg) +Conformal PI + +CI +![](images/58100e64d57d5f6b927b32041816e394652fcdc035c0d61f5c0b2fb9d275afdd.jpg) +(a) Ensemble Size: 100; Learning Rate: 0.3; Maximum Depth: 4; Subsampling Rate: 0.6; Dropout rate: 0.6 + +![](images/4b68f51783bca724bf6e79ebe97aa0f862cf4e9cb1ab022a4abb49e483c3531b.jpg) +RI +Figure 12: High coverage, underfitting. + +![](images/8f085bf684ea0f5ae2486598698a8153e939b50fb0ec006f5c8985175e5ba5eb.jpg) +PI + +![](images/188a25c08d2f9bdbe762862a2a29d2973c45669a985932dd91253a54cc313157.jpg) +Conformal PI + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract and introduction state that we (1) propose both random-dropout and structured-dropout boosting algorithms, (2) derive their asymptotic normality via a CLT, and (3) establish finite-sample statistical rates. These claims correspond directly to algorithms definitions and CLT statement in Section 3 (proof in Appendix E). Statistical rates are given in Section 4 (proofs in Appendix G). + +# Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: We explicitly acknowledge in Section 7 that our theoretical guarantees rest on strong assumptions (structure-value isolation, non-adaptivity, and tree regularity) which may not hold in practice, and we outline the need to relax these; we also note that our methods are currently limited to regression and discuss challenges in extending to classification, survival analysis, and structured outputs. + +# Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. + +# 3. Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +# Answer: [Yes] + +Justification: The paper mainly introduced two modified boosting algorithms: Algorithm 1 and Algorithm 2. In Section 2 and 5 we introduced all assumptions needed for our following results to hold. Both algorithms are shown to converge to a fixed point in Appendix C. And the corresponding derivations of CLTs are provided in Appendix E. + +# Guidelines: + +- The answer NA means that the paper does not include theoretical results. +- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced. +- All assumptions should be clearly stated or referenced in the statement of any theorems. +- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. +- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. +- Theorems and Lemmas that the proof relies upon should be properly referenced. + +# 4. 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Our analysis yields $\mathcal{O}(n^{-1/4})$ convergence rates to the Gaussian limit in the Kolmogorov distance. We further establish the non-asymptotic validity of a multiplier block bootstrap procedure for constructing the confidence intervals, guaranteeing consistent inference under Markovian sampling. Our work provides the first non-asymptotic guarantees on the rate of convergence of bootstrap-based confidence intervals for stochastic approximation with Markov noise. Moreover, we recover the classical rate of order $\mathcal{O}(n^{-1/8})$ up to logarithmic factors for estimating the asymptotic variance of the iterates of the LSA algorithm. + +# 1 Introduction + +Stochastic approximation (SA) has become foundational to modern machine learning, especially its reinforcement learning (RL) domain. Many classical RL algorithms, including $Q$ -learning [80, 76], the actor-critic algorithm [35], and policy evaluation algorithms, such as temporal difference (TD) learning [76] are special instances of SA. Recent research has extensively studied both the asymptotic [14] and non-asymptotic behavior [49] of these algorithms. It is important not only to establish the convergence of SA estimators, but also to quantify their uncertainty, which is typically done through the asymptotic normality of corresponding estimates, see [57, 46]. Recent works have focused on deriving non-asymptotic convergence rates for SA methods [69, 67, 82]. Notably, most existing results consider settings with independent and identically distributed (i.i.d.) noise. In contrast, many practical SA applications involve dependent noise, often forming a Markov chain. This additionally complicates the problem. Indeed, even the problem of deriving precise Berry-Esseen type convergence rates for additive functionals of Markov chains is challenging compared to i.i.d. setting, where quantitative results are well-established, starting from Bentkus' influential work [7]. + +Asymptotic normality of SA estimates is particularly important in practice, as it allows one to construct approximate confidence intervals for the parameters of interest. Different approaches either directly utilize the asymptotic normality and aim to estimate the asymptotic covariance matrix directly (the plug-in or batch-mean methods, see [17, 15, 63]), or rely on the non-parametric methods, based on the bootstrap [27, 64]. When the latter approach is applied to the dependent observations, standard bootstrap methods need to be carefully adjusted to account for the dependence structure, see [29, 42]. + +In this paper, we study a linear stochastic approximation (LSA) procedure. This setting covers several important scenarios, such as the classical TD learning algorithm [76] with linear function approximation, while allowing for sharper theoretical analysis. The LSA procedure aims to find an approximate solution to the linear system $\bar{\mathbf{A}}\theta^{\star} = \bar{\mathbf{b}}$ with a unique solution $\theta^{\star}$ , based on observations $\{(\mathbf{A}(Z_k),\mathbf{b}(Z_k))\}_{k\in \mathbb{N}}$ . Here $\mathbf{A}:Z\to \mathbb{R}^{d\times d}$ and $\mathbf{b}:Z\rightarrow \mathbb{R}^d$ are measurable functions, and $\{Z_k\}_{k\in \mathbb{N}}$ is a sequence of noise variables taking values in a measurable space $(Z,\mathcal{Z})$ with a distribution $\pi$ satisfying $\mathbb{E}_{\pi}[\mathbf{A}(Z_k)] = \bar{\mathbf{A}}$ and $\mathbb{E}_{\pi}[\mathbf{b}(Z_k)] = \bar{\mathbf{b}}$ . We focus on the setting where $\{Z_k\}_{k\in \mathbb{N}}$ is an ergodic Markov chain. Given a sequence of decreasing step sizes $\{\alpha_{k}\}_{k\in \mathbb{N}}$ and an initial point $\theta_0\in \mathbb{R}^d$ , we consider the estimates $\{\bar{\theta}_n\}_{n\in \mathbb{N}}$ given by + +$$ +\theta_ {k} = \theta_ {k - 1} - \alpha_ {k} \left\{\mathbf {A} \left(Z _ {k}\right) \theta_ {k - 1} - \mathbf {b} \left(Z _ {k}\right) \right\}, k \geq 1, \bar {\theta} _ {n} = n ^ {- 1} \sum_ {k = 0} ^ {n - 1} \theta_ {k}, n \geq 1. \tag {1} +$$ + +The sequence $\{\theta_k\}_{k\in \mathbb{N}}$ corresponds to the standard LSA iterates, while $\{\bar{\theta}_n\}_{n\in \mathbb{N}}$ corresponds to the Polyak-Ruppert (PR) averaged iterates [65, 57]. It is known that, under appropriate technical conditions on the step sizes $\{\alpha_{k}\}$ and the noisy observations $\{Z_k\}$ (see [57] and [31] for a discussion), + +$$ +\sqrt {n} \left(\bar {\theta} _ {n} - \theta^ {\star}\right) \xrightarrow {d} \mathcal {N} \left(0, \Sigma_ {\infty}\right). \tag {2} +$$ + +Here the asymptotic covariance matrix $\Sigma_{\infty}$ is defined in Section 3; see (8). Recent works have provided a number of non-asymptotic guarantees for the averaged LSA iterates of $\bar{\theta}_n$ , in particular, [39, 74, 47, 25, 48, 24], which study the mean-squared error, high-order moment bounds, and concentration bounds for $\sqrt{n} (\bar{\theta}_n - \theta^\star)$ . However, these results are not enough to establish explicit convergence rates in (2) in some appropriate probability metric $d$ : + +$$ +\mathrm {d} (\sqrt {n} (\bar {\theta} _ {n} - \theta^ {\star}), \Sigma_ {\infty} ^ {1 / 2} Y), +$$ + +where $Y \sim \mathcal{N}(0,1)$ . Notable exceptions are recent papers [67, 82, 70, 83], which study non-asymptotic convergence rates in the LSA and stochastic gradient descent (SGD) algorithms, as well as in the specific setting of the TD learning algorithm. Our paper aims to complement these findings by providing both a non-asymptotic analysis of the convergence rate in (2) and an analysis of an appropriate procedure for constructing confidence sets for $\theta^{\star}$ . Our primary contributions are as follows: + +- We derive a novel non-asymptotic bound for normal approximation for projected Polyak-Ruppert averaged LSA iterates $\sqrt{n} u^{\top}(\bar{\theta}_n - \theta^\star)$ under Markovian noise in (2). Here $u$ is a vector on the unit sphere $S_{d - 1}$ . Precisely, we establish a convergence rate of order $\mathcal{O}(n^{-1 / 4})$ in Kolmogorov distance. The rate $\mathcal{O}(n^{-1 / 4})$ matches the recent result of [83], which considered the particular setting of the TD learning algorithm, and improves over the previous results from [73]. Our proof strategy differs from that of [83] and is based on a Poisson decomposition for Markov chains, combined with an appropriate version of the Berry-Esseen bound for martingales, building on the results of Fan [28] and Bolthausen [12]. +- We provide a non-asymptotic analysis of the multiplier subsample bootstrap approach [43] for the LSA algorithm under Markovian noise. Our bounds imply that the coverage probabilities of the true parameter $\theta^{\star}$ can be approximated at a rate of $\mathcal{O}(n^{-1/10})$ , where $n$ is the number of samples used in the procedure. To the best of our knowledge, this is the first non-asymptotic bound on the accuracy of bootstrap approximation for SA algorithms with Markov noise. As a byproduct of our analysis, we also recover the rate of $\mathcal{O}(n^{-1/8})$ (up to logarithmic factors) for estimating the asymptotic variance of projected LSA iterates using the overlapping batch means (OBM) estimator, previously obtained in [17, 79]. +- We apply the proposed methodology to the temporal difference learning (TD) algorithm for policy evaluation in reinforcement learning. + +The rest of the paper is organized as follows. In Section 2, we review the literature on Berry-Esseen type results for stochastic approximation (SA) algorithms and methods for constructing confidence intervals for the parameter $\theta^{\star}$ in these settings. In Section 3, we obtain a quantitative version of the Berry-Esseen theorem for the projected error of the Polyak-Ruppert averaged LSA under the Kolmogorov distance. In Section 4, we discuss the multiplier subsample bootstrap approach (as proposed in [43]) for LSA and provide non-asymptotic bounds on the accuracy of approximating the exact distribution of $\sqrt{n} u^{\top}(\bar{\theta}_n - \theta^{\star})$ with its bootstrap counterpart. Then in Section 5 we apply the proposed methodology to the temporal difference learning (TD) algorithm, based on the sequence of states, which form a geometrically ergodic Markov chain. Proofs are postponed to appendix. + +Notations. We denote by $\mathcal{P}(\mathsf{Z})$ the set of probability measures on a measurable space $(\mathsf{Z},\mathcal{Z})$ . For probability measures $\mu$ and $\nu$ on $(\mathsf{Z},\mathcal{Z})$ we denote by $\mathrm{d}_{\mathrm{tv}}(\mu ,\nu)$ the total variation distance between them $\mu$ and $\nu$ , that is, $\mathrm{d}_{\mathrm{tv}}(\mu ,\nu) = \sup_{C\in \mathcal{Z}}|\mu (C) - \nu (C)|$ . For matrix $A\in \mathbb{R}^{d\times d}$ we denote by $\| A\|$ its operator norm. Given a sequence of matrices $\{A_{\ell}\}_{\ell \in \mathbb{N}},A_{\ell}\in \mathbb{R}^{d\times d}$ , we use the following convention for matrix products: $\prod_{\ell = m}^{k}A_{\ell} = A_{k}A_{k - 1}\ldots A_{m}$ , where $m\leq k$ . For the matrix $Q\in \mathbb{R}^{d\times d}$ , which is symmetric and positive-definite, and $x\in \mathbb{R}^d$ , we define the corresponding norm $\| x\| _Q = \sqrt{x^\top Qx}$ , and define the respective matrix $Q$ -norm of the matrix $B\in \mathbb{R}^{d\times d}$ by $\| B\| _Q = \sup_{x\neq 0}\| Bx\| _Q / \| x\| _Q$ . For sequences $a_{n}$ and $b_{n}$ , we write $a_{n}\lesssim b_{n}$ if there exist a (numeric) constant $c > 0$ such that $a_{n}\leq cb_{n}$ . For simplicity of presentation, we state the main results of the paper up to absolute constants. We also use a notation $a_{n}\lesssim_{\log n}b_{n}$ , if there exist $\beta >0$ and $c > 0$ , such that $a_{n}\leq c(1 + \log n)^{\beta}b_{n}$ for any $n$ . Additionally, we use the standard abbreviations "i.i.d." for "independent and identically distributed" and "w.r.t." for "with respect to". We use index $k$ to present results which hold true for any intermediate iteration, and index $n$ denotes the total number of iterations of the algorithm. + +# 2 Related works + +The analysis of linear stochastic approximation (LSA) algorithms under Markovian noise has a long history, with classical works establishing almost sure convergence and asymptotic normality under broad conditions [57, 14, 38, 8]. These asymptotic results, however, lack explicit finite-sample error bounds. Recently, non-asymptotic performance analysis of stochastic approximation and gradient methods has gained attention. In the i.i.d. setting of stochastic gradient descent (SGD), finite-time error bounds were provided by [51, 58]. For Markovian LSA contexts, convergence rates were analyzed by [10, 39], focusing on temporal-difference learning, and instance-dependent bounds were derived by [48] and [24]. These studies establish guarantees on mean-square error or high-probability deviations but do not address the distributional approximation of the estimator. + +Rates of convergence in CLT are widely studied in probability theory [7], primarily for sums of random variables or univariate martingale difference sequences [55, 12]. Berry-Esseen bounds and Edgeworth expansions for Markov chains under various ergodicity assumptions are also available [13, 11, 9, 34], though typically restricted to the 1-dimensional case. Our analysis leverages recent concentration results on quadratic forms of Markov chains [50], building upon martingale decomposition techniques for $U$ -statistics developed in [5]. + +Recent studies have analyzed convergence rates of SA iterates to their limiting normal distributions. [67] derived an $\mathcal{O}(n^{-1/4})$ convergence rate in (2) in convex distance for TD learning under i.i.d. noise, improved to $\mathcal{O}(n^{-1/3})$ by [82] specifically for TD learning. [3] applied Stein's method to averaged SGD and LSA iterates in smooth Wasserstein distance, again under i.i.d. conditions. Recently, [73] analyzed rates for martingale CLT in 1-st order Wasserstein distance with Markovian samples but achieved slower convergence than our result (see Theorem 1 and discussion after it). Lastly, [1] provided last-iterate bounds for SGD for high-dimensional linear regression, also with i.i.d. noise. + +Originally introduced for i.i.d. data [27], the bootstrap method has been adapted to complex settings, including high-dimensional tests [18, 19] and linear regression [72]. For SA methods, an online bootstrap for SGD was proposed in [29], with recent non-asymptotic analysis showing approximation rates up to $\mathcal{O}(n^{-1/2})$ for coverage probabilities in case of strongly convex objectives with independent noise [70]. A similar analysis for linear stochastic approximation (LSA) yields rates of order $\mathcal{O}(n^{-1/4})$ [67]. Extensions to Markovian settings by [59] proved inconsistent, as demonstrated in [42, Proposition 1]. Meanwhile, [42] proposed consistent mini-batch SGD estimators for $\varphi$ -mixing noise. They also suggested a consistent procedure for constructing the confidence intervals, but studied only its asymptotic properties using the independent block trick [84]. Lastly, multiplier bootstrap techniques for online non-convex SGD were considered by [85]. + +Among other approaches for constructing confidence intervals for dependent data, we mention methods based on (asymptotically) pivotal statistics [40, 41]. The authors of [41] considered the Polyak-Ruppert averaged $Q$ -learning algorithm under the i.i.d. noise assumption (generative model), while [40] generalized this approach to nonlinear stochastic approximation under Markov noise. Another group of methods are based on estimating the asymptotic covariance matrix $\Sigma_{\infty}$ appearing in the central limit theorem (2). Among these approaches we mention the plug-in estimator of + +[17] and batch-mean estimators, see e.g. [15, 79, 63]. Specifically, [63] treats SGD with Markov noise, contrasted with the independent-noise SGD analyses in [17, 15, 79], and [85] investigates both multiplier bootstrap and batch-mean estimators for nonconvex objectives. These methods yield non-asymptotic error bounds in expectation, $\mathbb{E}[\| \hat{\Sigma}_n - \Sigma_\infty \| ]$ , but do not study convergence rates in (2). A notable exception is [82], which delivers a non-asymptotic analysis of TD-learning under i.i.d. noise, using plug-in covariance estimates to achieve an $\mathcal{O}(n^{-1 / 3})$ approximation rate for coverage probabilities of $\theta^{\star}$ . + +# 3 Accuracy of Gaussian approximation for LSA with Markov noise + +# 3.1 Gaussian approximation for non-linear statistics + +We first discuss a general scheme for proving quantitative normal approximation bounds for non-linear statistics. Consider the statistic $T(X_{1},\ldots ,X_{n})\in \mathbb{R}^{d}$ , which can be written as + +$$ +T = W + D, \tag {3} +$$ + +where $W$ is a linear statistic of the random variables $X_{1},\ldots ,X_{n}$ , and $D$ is a nonlinear term, which is typically small. Multidimensional versions of CLT based on the decomposition (3) are well-studied when $X_{1},\ldots ,X_{n}$ are i.i.d. random variables [16, 69]. In this case, one can follow the randomized concentration approach [69], which requires only a finite 2-nd moment of the term $D$ . To our knowledge, there are no such results readily available for Markov chains or martingales. Moreover, even the Berry-Esseen type result for $d$ -dimensional linear statistics of Markov chains or martingale-differences with explicit dependence on $d$ remains an interesting and open question. There is no affirmative result comparable to the one of [7] for i.i.d. random vectors. This is the primary reason why we focus on the setting of 1-dimensional statistics and use the following proposition to obtain the rates of Gaussian approximation for $T$ given in (3): + +Proposition 1. Let $\Phi(x), x \in \mathbb{R}$ be the c.d.f. of the standard normal law. Then for any random variables $W$ and $D$ and any $p \geq 1$ , + +$$ +\sup _ {x \in \mathbb {R}} | \mathbb {P} (W + D \leq x) - \Phi (x) | \leq \sup _ {x \in \mathbb {R}} | \mathbb {P} (W \leq x) - \Phi (x) | + 2 \mathbb {E} ^ {1 / (p + 1)} [ | D | ^ {p} ]. +$$ + +This result is classical and can be found e.g. in [71, 16]. For completeness, we provide the proof in Appendix F. The most involved part of applications of Proposition 1 is the proper bound on $\mathbb{E}[|D|^p]$ . This requires to select large values of $p$ , which later requires to introduce additional assumptions on the step size choice in the LSA procedure, see A3 in Section 3.2. + +# 3.2 Gaussian approximation for LSA with Markov noise + +When there is no risk of ambiguity, we use the simplified notations $\mathbf{A}_n = \mathbf{A}(Z_n)$ and $\mathbf{b}_n = \mathbf{b}(Z_n)$ . Starting from the definition (1), we get with elementary transformations that + +$$ +\theta_ {n} - \theta^ {\star} = \left(\mathrm {I} - \alpha_ {n} \mathbf {A} _ {n}\right) \left(\theta_ {n - 1} - \theta^ {\star}\right) - \alpha_ {n} \varepsilon_ {n}, \tag {4} +$$ + +where we have set $\varepsilon_{n} = \varepsilon (Z_{n})$ with $\varepsilon (z) = \tilde{\mathbf{A}} (z)\theta^{\star} - \tilde{\mathbf{b}} (z)$ , $\tilde{\mathbf{A}} (z) = \mathbf{A}(z) - \bar{\mathbf{A}}$ , $\tilde{\mathbf{b}} (z) = \mathbf{b}(z) - \bar{\mathbf{b}}$ . Here the random variable $\varepsilon (Z_{n})$ can be viewed as a noise, measured at the optimal point $\theta^{\star}$ . We also define the noise covariance matrix under the stationary distribution $\pi$ : + +$$ +\Sigma_ {\varepsilon} = \mathbb {E} _ {\pi} [ \varepsilon (Z _ {0}) \{\varepsilon (Z _ {0}) \} ^ {\top} ] + 2 \sum_ {\ell = 1} ^ {\infty} \mathbb {E} _ {\pi} [ \varepsilon (Z _ {0}) \{\varepsilon (Z _ {\ell}) \} ^ {\top} ]. \tag {5} +$$ + +We now impose the following conditions: + +A1. The sequence $(Z_{k})_{k\in \mathbb{N}}$ is a Markov chain taking values in a Polish space $(Z,\mathcal{Z})$ with Markov kernel P. Moreover, $\mathrm{P}$ admits $\pi$ as a unique invariant distribution and is uniformly geometrically ergodic, that is, there exists $t_\mathrm{mix}\in \mathbb{N}$ , such that for any $k\in \mathbb{N}$ , it holds that + +$$ +\Delta \left(\mathrm {P} ^ {k}\right) := \sup _ {z, z ^ {\prime} \in \mathbb {Z}} \mathrm {d} _ {\mathrm {t v}} \left(\mathrm {P} ^ {k} (z, \cdot), \mathrm {P} ^ {k} \left(z ^ {\prime}, \cdot\right)\right) \leq (1 / 4) ^ {\lceil k / t _ {\operatorname {m i x}} \rceil}. \tag {6} +$$ + +Parameter $t_{\mathrm{mix}}$ in (6) is referred to as mixing time, see [54]. We also impose the following assumptions on the noise variables $\varepsilon(z)$ : + +A2. $\int_{\mathbb{Z}}\mathbf{A}(z)\mathrm{d}\pi (z) = \bar{\mathbf{A}}$ and $\int_{\mathbb{Z}}\mathbf{b}(z)\mathrm{d}\pi (z) = \bar{\mathbf{b}}$ , with the matrix $-\bar{\mathbf{A}}$ being Hurwitz. Moreover, $\| \varepsilon \|_{\infty} = \sup_{z\in \mathbb{Z}}\| \varepsilon (z)\| < + \infty$ , and the mapping $z\to \mathbf{A}(z)$ is bounded, that is, + +$$ +\mathrm {C} _ {\mathbf {A}} = \sup _ {z \in \mathbf {Z}} \| \mathbf {A} (z) \| \vee \sup _ {z \in \mathbf {Z}} \| \tilde {\mathbf {A}} (z) \| < \infty . +$$ + +Moreover, for $\Sigma_{\varepsilon}$ defined in (5), we assume that $\lambda_{\mathrm{min}}(\Sigma_{\varepsilon}) > 0$ + +To motivate the introduction of $\Sigma_{\varepsilon}$ in this particular form, note that under assumption A 2, $\mathbb{E}_{\pi}[\varepsilon (Z_0)] = 0$ , the following central limit theorem holds (see e.g. [23, Chapter 21]): + +$$ +\frac {1}{\sqrt {n}} \sum_ {\ell = 0} ^ {n - 1} \varepsilon \left(Z _ {\ell}\right)\rightarrow \mathcal {N} \left(0, \Sigma_ {\varepsilon}\right). +$$ + +The fact that the matrix $-\bar{\mathbf{A}}$ is Hurwitz implies that the linear system $\bar{\mathbf{A}}\theta = \bar{\mathbf{b}}$ has a unique solution $\theta^{\star}$ . Moreover, this fact is sufficient to show that the matrix $\mathrm{I} - \alpha \bar{\mathbf{A}}$ is a contraction in an appropriate matrix $Q$ -norm for small enough $\alpha > 0$ . Precisely, the following result holds: + +Proposition 2 (Proposition 1 in [67]). Let $-\bar{\mathbf{A}}$ be a Hurwitz matrix. Then for any $P = P^{\top} \succ 0$ , there exists a unique matrix $Q = Q^{\top} \succ 0$ , satisfying the Lyapunov equation $\bar{\mathbf{A}}^{\top} Q + Q \bar{\mathbf{A}} = P$ . Moreover, setting + +$$ +a = \frac {\lambda_ {\operatorname* {m i n}} (P)}{2 \| Q \|}, \quad a n d \quad \alpha_ {\infty} = \frac {\lambda_ {\operatorname* {m i n}} (P)}{2 \kappa_ {Q} \| \mathbf {A} \| _ {Q} ^ {2}} \wedge \frac {\| Q \|}{\lambda_ {\operatorname* {m i n}} (P)}, \tag {7} +$$ + +where $\kappa_{Q} = \lambda_{\max}(Q) / \lambda_{\min}(Q)$ , it holds for any $\alpha \in [0, \alpha_{\infty}]$ that $\alpha a \leq 1/2$ , and + +$$ +\| \mathrm {I} - \alpha \bar {\mathbf {A}} \| _ {Q} ^ {2} \leq 1 - \alpha a. +$$ + +For completeness we provide the proof of this result in Appendix D. Now, we make an assumption about the form of step sizes $\alpha_{k}$ in (4) and the total number of observations $n$ : + +A3. Step sizes $\{\alpha_{k}\}_{k\in \mathbb{N}}$ have a form $\alpha_{k} = c_{0} / (k + k_{0})^{\gamma}$ , where $\gamma \in [1/2; 1)$ , and the constant $c_{0} \leq 1/(2a)$ . Moreover, we assume that $n$ is sufficiently large and + +$$ +k _ {0} > g (a, t _ {\text {m i x}}, c _ {0}, \mathrm {C} _ {\mathbf {A}} \kappa_ {Q}, \alpha_ {\infty}) (\log n) ^ {1 / \gamma}. +$$ + +Precise expressions for $g(a, t_{\mathrm{mix}}, c_0, \mathrm{C_A} \kappa_Q, \alpha_\infty)$ and for $n$ are given in Appendix A (see A'3). + +The main aim of lower bounding $n$ is to ensure that the error related to the choice of initial condition $\theta_0$ becomes small enough. Note also that our bound of A3 requires to know the number of observations $n$ in advance and adjust $k_{0}$ accordingly. The same problem can be traced in the existing high-probability results for LSA [47, 24, 82], in the regime when the confidence parameter $\delta$ in high-probability bounds scales with the number of iterations $n$ . + +It is known (see e.g. [31]), that the assumptions A 1 - A 3 guarantee that the sequence $\bar{\theta}_n$ is asymptotically normal, that is, $\sqrt{n} (\bar{\theta}_n - \theta^\star) \xrightarrow{d} \mathcal{N}(0, \Sigma_\infty)$ , where the covariance matrix $\Sigma_\infty$ has a form + +$$ +\Sigma_ {\infty} = \bar {\mathbf {A}} ^ {- 1} \Sigma_ {\varepsilon} \bar {\mathbf {A}} ^ {- \top}. \tag {8} +$$ + +For a fixed $u \in \mathbb{S}^{d-1}$ we consider projection of $\sqrt{n} (\bar{\theta}_n - \theta^\star)$ on $u$ and quantify the rate of convergence in the Kolmogorov distance, i.e., we consider the quantity + +$$ +\left. \mathrm {d} _ {K} \left(\sqrt {n} u ^ {\top} \left(\bar {\theta} _ {n} - \theta^ {\star}\right) / \sigma (u), \mathcal {N} (0, 1)\right) = \sup _ {x \in \mathbb {R}} \left| \mathbb {P} \left(\sqrt {n} u ^ {\top} \left(\bar {\theta} _ {n} - \theta^ {\star}\right) / \sigma (u) \leq x\right) - \Phi (x) \right|, \right. \tag {9} +$$ + +where $\Phi (x)$ is the c.d.f. of the standard normal law $\mathcal{N}(0,1)$ , and $\sigma^2 (u) = u^\top \bar{\mathbf{A}}^{-1}\Sigma_{\varepsilon}\bar{\mathbf{A}}^{-\top}u$ . To control the quantity (9), we will first present an auxiliary result. Define + +$$ +G _ {m: k} = \prod_ {\ell = m} ^ {k} (\mathrm {I} - \alpha_ {\ell} \bar {\mathbf {A}}), Q _ {\ell} = \alpha_ {\ell} \sum_ {k = \ell} ^ {n - 1} G _ {\ell + 1: k}, \Sigma_ {n} = \frac {1}{n} \sum_ {\ell = 2} ^ {n - 1} Q _ {\ell} \Sigma_ {\varepsilon} Q _ {\ell} ^ {\top}, \sigma_ {n} ^ {2} (u) = u ^ {\top} \Sigma_ {n} u. \tag {10} +$$ + +Theorem 1. Assume A1, A2, and A3. Then for any $u \in \mathbb{S}_{d-1}$ , $\theta_0 \in \mathbb{R}^d$ , and initial distribution $\xi$ on $(\mathsf{Z}, \mathcal{Z})$ , it holds that + +$$ +\left. \mathrm {d} _ {K} \left(\sqrt {n} u ^ {\top} \left(\bar {\theta} _ {n} - \theta^ {\star}\right) / \sigma_ {n} (u), \mathcal {N} (0, 1)\right) \leq \mathrm {B} _ {n}, \right. +$$ + +where we set + +$$ +\mathrm {B} _ {n} = \frac {\mathrm {C} _ {\mathrm {K} , 1} \log^ {3 / 4} n}{n ^ {1 / 4}} + \frac {\mathrm {C} _ {\mathrm {K} , 2} \log n}{n ^ {1 / 2}} + \frac {\mathrm {C} _ {1} ^ {\mathrm {D}} \left\| \theta_ {0} - \theta^ {\star} \right\| + \mathrm {C} _ {2} ^ {\mathrm {D}}}{\sqrt {n}} + \mathrm {C} _ {3} ^ {\mathrm {D}} \frac {(\log n) ^ {2}}{n ^ {\gamma - 1 / 2}} + \mathrm {C} _ {4} ^ {\mathrm {D}} \frac {(\log n) ^ {5 / 2}}{n ^ {\gamma - 1 / 2}}, \tag {11} +$$ + +where $\mathrm{C}_{\mathrm{K},1}$ and $\mathrm{C}_{\mathrm{K},2}$ are defined in Appendix $F$ , and $\{\mathrm{C}_i^{\mathrm{D}}\}_{i = 1}^4$ are defined in Appendix $B$ . + +Proof. We provide below the main elements of the proof and refer the reader to Appendix B for a complete derivations. We use the expansion (35) and (39), detailed in the appendix: + +$$ +\theta_ {k} - \theta^ {\star} = \Gamma_ {1: k} \left(\theta_ {0} - \theta^ {\star}\right) + J _ {k} ^ {(0)} + H _ {k} ^ {(0)}, +$$ + +where we set $\Gamma_{m:k} = \prod_{\ell = m}^{k}(\mathrm{I} - \alpha_{\ell}\mathbf{A}(Z_{\ell}))$ and for $m\leq k$ + +$$ +J _ {k} ^ {(0)} = - \sum_ {\ell = 1} ^ {k} \alpha_ {\ell} G _ {\ell + 1: k} \varepsilon \left(Z _ {\ell}\right), \quad J _ {0} ^ {(0)} = 0 \tag {12} +$$ + +$$ +H _ {k} ^ {(0)} = - \sum_ {\ell = 1} ^ {k} \alpha_ {\ell} \Gamma_ {\ell + 1: k} (\mathbf {A} (Z _ {\ell}) - \bar {\mathbf {A}}) J _ {\ell - 1} ^ {(0)}, \quad H _ {0} ^ {(0)} = 0 +$$ + +Taking average and changing the order of summation we get that $\sqrt{n} (\bar{\theta}_n - \theta^\star) = W + D_1$ , with + +$$ +W = - \frac {1}{\sqrt {n}} \sum_ {\ell = 1} ^ {n - 1} Q _ {\ell} \varepsilon \left(Z _ {\ell}\right), \quad D _ {1} = \frac {1}{\sqrt {n}} \sum_ {k = 0} ^ {n - 1} \Gamma_ {1: k} \left(\theta_ {0} - \theta^ {\star}\right) + \frac {1}{\sqrt {n}} \sum_ {k = 1} ^ {n - 1} H _ {k} ^ {(0)}. \tag {13} +$$ + +Note that $W$ is a linear statistic of the Markov chain $\{Z_{\ell}\}_{\ell \in \mathbb{N}}$ . We further expand it into a sum of martingale-differences and remainder terms. Note that under A1 the function $\hat{\varepsilon}(z) = \sum_{k=0}^{\infty} \mathrm{P}^{k} \varepsilon(z)$ is a solution to Poisson equation $\hat{\varepsilon}(z) - \mathrm{P}\hat{\varepsilon}(z) = \varepsilon(z)$ . We rewrite the term $W$ in the following way $W = n^{-1/2} M + D_2$ , with + +$$ +M = - \sum_ {\ell = 2} ^ {n - 1} \Delta M _ {\ell}, \qquad \Delta M _ {\ell} = Q _ {\ell} \big (\hat {\varepsilon} (Z _ {\ell}) - \mathrm {P} \hat {\varepsilon} (Z _ {\ell - 1}) \big) +$$ + +$$ +D _ {2} = - \frac {1}{\sqrt {n}} Q _ {1} \hat {\varepsilon} (Z _ {1}) + \frac {1}{\sqrt {n}} Q _ {n - 1} \mathrm {P} \hat {\varepsilon} (Z _ {n - 1}) + \sum_ {\ell = 1} ^ {n - 2} (Q _ {\ell} - Q _ {\ell + 1}) \mathrm {P} \hat {\varepsilon} (Z _ {\ell}). +$$ + +Note that $\{\Delta M_{\ell}\}_{\ell = 2}^{n - 1}$ is a martingale-difference sequence w.r.t. $\mathcal{F}_{\ell} = \sigma (Z_k,k\leq \ell)$ . Note that + +$$ +\mathbb {E} _ {\pi} ^ {\mathcal {F} _ {\ell - 1}} \left[ \Delta M _ {\ell} \Delta M _ {\ell} ^ {\top} \right] = Q _ {\ell} \tilde {\varepsilon} \left(Z _ {\ell - 1}\right) Q _ {\ell} ^ {\top}, \quad \tilde {\varepsilon} (z) = \mathrm {P} \hat {\varepsilon} \hat {\varepsilon} ^ {\top} (z) - \mathrm {P} \hat {\varepsilon} (z) \mathrm {P} \hat {\varepsilon} ^ {\top} (z). \tag {14} +$$ + +Furthermore, we have $\pi (\tilde{\varepsilon}) = \Sigma_{\varepsilon}$ . The term $M$ is a martingale, whose quadratic characteristic is given by $\langle M\rangle_{n} = \sum_{\ell = 1}^{n - 2}Q_{\ell +1}\tilde{\varepsilon}(Z_{\ell})Q_{\ell +1}^{\top}$ . With these notations, we get + +$$ +\sqrt {n} \left(\bar {\theta} _ {n} - \theta^ {*}\right) = n ^ {- 1 / 2} M + D, +$$ + +where we denote $D = D_{1} + D_{2}$ . Applying Proposition 14 with $X = n^{-1/2} u^{\top} M / \sigma_{n}(u)$ and $Y = u^{\top} D / \sigma_{n}(u)$ , we obtain for any $p \geq 2$ , + +$$ +\mathrm {d} _ {K} \left(\frac {\sqrt {n} u ^ {\top} \left(\bar {\theta} _ {n} - \theta^ {\star}\right)}{\sigma_ {n} (u)}, \mathcal {N} (0, 1)\right) \leq \mathrm {d} _ {K} \left(\frac {u ^ {\top} M}{\sqrt {n} \sigma_ {n} (u)}, \mathcal {N} (0, 1)\right) + 2 \left\{\mathbb {E} \left[ \left| \frac {u ^ {\top} D}{\sigma_ {n} (u)} \right| ^ {p} \right] \right\} ^ {1 / (p + 1)}. \tag {15} +$$ + +To obtain the rate of Gaussian approximation for $\sqrt{n} u^{\top}(\bar{\theta}_n - \theta^{\star}) / \sigma_n(u)$ it remains to control the moments of the term $|u^{\top}D / \sigma_{n}(u)|$ , which is done in Proposition 7, and to control $\mathrm{d}_K\bigl (u^\top M / (\sqrt{n}\sigma_n(u))\bigr)$ . To bound the latter term we use a normal approximation result for sums of martingale-difference sequences, which builds upon the arguments of [12] and [28] - see Proposition 6, applied with $p = \log n$ . + +Note that the result of Theorem 1 yields an approximation of $\sqrt{n} u^{\top}(\bar{\theta}_n - \theta^{\star})$ with $\mathcal{N}(0,\sigma_n^2 (u))$ , and not the limiting quantity $\sigma^2 (u)$ from the CLT (2). In order to complete the result, we need an additional result on the Gaussian comparison between $\mathcal{N}(0,\sigma_n^2 (u))$ and $\mathcal{N}(0,\sigma^2 (u))$ . This result is based on the quantitative estimates provided first in [6], and then revised in [21]. + +Corollary 1. Under assumptions of Theorem 1 it holds, with $\mathrm{B}_n$ given in (11), that + +$$ +\mathrm {d} _ {K} \left(\sqrt {n} u ^ {\top} (\bar {\theta} _ {n} - \theta^ {*}) / \sigma (u), \mathcal {N} (0, 1)\right) \leq \mathrm {B} _ {n} + C _ {\infty} n ^ {\gamma - 1}, +$$ + +where $C_{\infty}$ is defined in (30). + +Remark 1. The bound of Corollary 1 predicts the optimal error of normal approximation for Polyak-Ruppert averaged estimates of order $n^{-1/4}$ up to a logarithmic factor in $n$ , which is achieved with the step size $\alpha_k = c_0 / (k + k_0)^{3/4}$ , that is, when setting $\gamma = 3/4$ in (11). In this case we obtain the optimized bound: + +$$ +\left. \mathsf {d} _ {K} \big (\sqrt {n} u ^ {\top} \left(\bar {\theta} _ {n} - \theta^ {\star}\right) / \sigma (u), \mathcal {N} (0, 1) \big) \lesssim_ {p r} \frac {(\log n) ^ {5 / 2}}{n ^ {1 / 4}} \right.. +$$ + +where $\lesssim_{pr}$ stands for inequality up to absolute and problem-specific constants (such as $\mathrm{C_A},\kappa_Q,a,t_{\mathrm{mix}}$ ), but not $n$ . + +Discussion. The proof of Corollary 1 is given in Appendix B.3. Results similar to the one of Corollary 1 have been recently obtained in the literature in [67], [73] and [3]. [67] considered the LSA algorithm based on i.i.d. noise variables $\{Z_k\}_{1 \leq k \leq n}$ and a randomized concentration approach based on [69]. This result was later refined in [82]. However, this technique does not extend to Markovian noise $\{Z_k\}_{1 \leq k \leq n}$ . While preparing this manuscript, we became aware of a recent paper [83], which provides a quantitative $d$ -dimensional CLT for martingales. Given this work, a natural direction for further research is to use this result and generalize Theorem 1 to the $d$ -dimensional setting. The authors in [83] applied their findings to the particular setting of the TD learning algorithm with Markov noise and obtained a bound of the form + +$$ +\mathrm {d} _ {C} \left(\sqrt {n} \left(\bar {\theta} _ {n} - \theta^ {\star}\right), \Sigma_ {\infty} ^ {1 / 2} \eta\right) \lesssim \frac {\log n}{n ^ {1 / 4}}, +$$ + +where $\eta \sim \mathcal{N}(0,\mathrm{I}_d)$ , and $\mathsf{d}_C(X,Y) = \sup_{B\in \operatorname {Conv}(\mathbb{R}^d)}\left|\mathbb{P}\big(X\in B\big) - \mathbb{P}(Y\in B)\right|$ denotes the convex distance. The authors in [73] obtained the convergence rate for the general LSA procedure with Markov noise + +$$ +\mathsf {d} _ {W} (\sqrt {n} (\bar {\theta} _ {n} - \theta^ {\star}), \Sigma_ {\infty} ^ {1 / 2} \eta) \lesssim \frac {\sqrt {\log n}}{n ^ {1 / 6}}, +$$ + +where $\mathrm{d}_W$ is the 1-st order Wasserstein distance. This result implies, due to the classical relations of between Kolmogorov and Wasserstein distance (see [62]), the final rate approximation on Kolmogorov distance of order $\mathcal{O}(n^{-1 / 12})$ , which is slower compared to the rate of Theorem 1. On the other hand, the result of [73] holds in $d$ -dimensional setting, whereas our analysis is restricted to one-dimensional projections of the estimation error. Nevertheless, the essential part of our analysis, namely controlling the remainder term $D$ in (15), can be generalized to the $d$ -dimensional case through bounds on $\mathbb{E}_{\xi}^{1 / p}\left[\| D\| ^p\right]$ , following an approach similar to that presented in Appendix B. + +# 4 Multiplier subsample bootstrap for LSA + +We will apply the multiplier subsample bootstrap (MSB) procedure, a block-based approach that constructs the bootstrap statistic via a blockwise scheme; see [37, 43]. Below we describe in details the MSB approach, closely following [43]. Let $b_{n}$ be the length of block, and for each $t = 0, \dots, n - b_{n}$ , define $\bar{\theta}_{b_n,t} = (1 / b_n)\sum_{\ell = t}^{t + b_n - 1}\theta_\ell$ , the "scale $b_{n}$ " version of $\bar{\theta}_n$ . To imitate $\bar{\theta}_n$ , the MSB estimator of $\bar{\theta}_n$ is given by + +$$ +\bar {\theta} _ {n, b _ {n}} (u) = \frac {\sqrt {b _ {n}}}{\sqrt {n - b _ {n} + 1}} \sum_ {t = 0} ^ {n - b _ {n}} w _ {t} \left(\bar {\theta} _ {b _ {n}, t} - \bar {\theta} _ {n}\right) ^ {\top} u, \tag {16} +$$ + +where $\Xi_{n} = \{Z_{\ell}\}_{\ell = 1}^{n}$ , and $\{w_{\ell}\}_{0\leq \ell \leq n - b_n}$ , the multiplier weights, are i.i.d. $\mathcal{N}(0,1)$ random variables, which are independent of $\Xi_{n}$ . We write, respectively, $\mathbb{P}^{\mathrm{b}} = \mathbb{P}(\cdot |\Xi_n)$ and $\mathbb{E}^{\mathrm{b}} = \mathbb{E}(\cdot |\Xi_n)$ for the corresponding conditional probability and expectation. For simplicity, we do not "subsample" the blocks: the theory extends readily, but we do not want to add another layer of notations. The key idea of the MSB procedure (16) is that the "bootstrap world" distribution $\mathbb{P}^{\mathrm{b}}\big(\bar{\theta}_{n,b_n}(u)\leq x\big)$ should be close to their "real world" counterparts $\mathbb{P}\big(\sqrt{n} (\bar{\theta}_n - \theta^\star)\leq x\big)$ for any $x\in \mathbb{R}$ . Formally, the procedure (16) is said to be asymptotically valid, if the quantity + +$$ +\sup _ {x \in \mathbb {R}} | \mathbb {P} (\sqrt {n} (\bar {\theta} _ {n} - \theta^ {\star}) ^ {\top} u \leq x) - \mathbb {P} ^ {\mathfrak {b}} (\bar {\theta} _ {n, b _ {n}} (u) \leq x) | \tag {17} +$$ + +converges to 0 in $\mathbb{P}$ -probability. Typically the authors consider the asymptotic validity of the procedures for constructing the confidence intervals (either with multiplier bootstrap [29, 43] or with direct estimation of the asymptotic covariance [17, 79]). Our aim in this section is to provide fully non-asymptotic bounds on the rate at which the supremum in (17) decays as a function of $n$ . + +The MSB estimator (16) $\bar{\theta}_{n,b_n}(u)$ is normally distributed w.r.t. $\mathbb{P}^{\mathrm{b}}$ , that is, + +$$ +\bar {\theta} _ {n, b _ {n}} (u) \sim \mathcal {N} \left(0, \hat {\sigma} _ {\theta} ^ {2} (u)\right), +$$ + +with the variance $\hat{\sigma}_{\theta}^{2}(u)$ given by + +$$ +\hat {\sigma} _ {\theta} ^ {2} (u) = \frac {b _ {n}}{n - b _ {n} + 1} \sum_ {t = 0} ^ {n - b _ {n}} \left(\left(\bar {\theta} _ {b _ {n}, t} - \bar {\theta} _ {n}\right) ^ {\top} u\right) ^ {2}. \tag {18} +$$ + +The parameter $b_{n}$ is commonly referred to as lag window, or bandwidth (see [30] and references therein). Under the bootstrap probability $\mathbb{P}^{\mathrm{b}}$ , $\bar{\theta}_{n,b_n}(u)$ (see (16)) is a Gaussian approximation of $\bar{\theta}_n$ + +with estimated variance $\hat{\sigma}_{\theta}^{2}(u)$ . Up to a multiplicative factor tending to 1 as $n$ goes to $\infty$ , the variance formula in (18) coincides with the overlapping batch mean estimator (OBM), a well-known technique for estimating the asymptotic variance of the Markov chain, suggested in [45]. The properties of OBM estimator are studied, see e.g. [81, 20, 30]. In particular, it is known (see [30]), that the OBM is a consistent estimator of the asymptotic variance of a Markov chain under suitable ergodicity assumptions, provided that $b_{n}\to \infty$ as $n\to \infty$ . + +Note that even if $\{Z_k\}_{k \in \mathbb{N}}$ is a Markov chain, the iterates $\{\theta_k\}_{k \in \mathbb{N}}$ alone do not form a Markov chain (one must rather consider the joint process $(\theta_k, Z_k)$ ). Consequently, the classical consistency results for overlapping-batch-means variance estimators do not apply directly to (18). Fortunately, we show below that applying the block bootstrap for the sequence $\{\theta_\ell\}$ , is equivalent, up to a suitable correction, to the block bootstrap procedure applied to the non-observable random variables $\{\varepsilon(Z_\ell)\}$ . To make this precise, we define for each block start $t \in \{0, \dots, n - b_n\}$ the quantities: + +$$ +\bar {W} _ {b _ {n}, t} = \frac {1}{b _ {n}} \sum_ {\ell = t + 1} ^ {t + b _ {n} - 1} \bar {\mathbf {A}} ^ {- 1} \varepsilon (Z _ {\ell}), \quad \bar {W} _ {n} = \frac {1}{n} \sum_ {\ell = 1} ^ {n - 1} \bar {\mathbf {A}} ^ {- 1} \varepsilon (Z _ {\ell}), +$$ + +$$ +\hat {\sigma} _ {\varepsilon} ^ {2} (u) = \frac {b _ {n}}{n - b _ {n} + 1} \sum_ {t = 0} ^ {n - b _ {n}} \left(\left(\bar {W} _ {b _ {n}, t} - \bar {W} _ {n}\right) ^ {\top} u\right) ^ {2}. +$$ + +Then the following proposition holds: + +Proposition 3. Assume A1, A2, and A3. Then for any $u \in \mathbb{S}_{d-1}$ , it holds that + +$$ +\hat {\sigma} _ {\theta} ^ {2} (u) = \hat {\sigma} _ {\varepsilon} ^ {2} (u) + \mathcal {R} _ {v a r} (u), +$$ + +where $\mathcal{R}_{var}(u)$ is a remainder term defined in Appendix E.1 (see (63)). Moreover, for any $2 \leq p \leq \log n$ , and any initial distribution $\xi$ on $(\mathbb{Z},\mathcal{Z})$ , it holds that + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| \mathcal {R} _ {v a r} (u) \right| ^ {p} \right] \lesssim M _ {1} p b _ {n} ^ {1 / 2} n ^ {\gamma / 2 - 1} + M _ {2} p ^ {4} (\log n) b _ {n} ^ {1 / 2} n ^ {- \gamma} \tag {19} \\ + M _ {3} p b _ {n} ^ {- 1 / 2} n ^ {\gamma / 2} + M _ {4} p ^ {4} (\log n) n ^ {- 1} + M _ {5} p n ^ {2 \gamma - 2} \\ \end{array} +$$ + +and the constants $M_{i}$ are defined in Appendix E.1, equation (66). + +The proof of Proposition 3 is given in Appendix E.1. The bound of Proposition 3 has some noteworthy properties. First, our bound requires that the block size $b_{n}$ to grow at least like $n^{\gamma}$ to ensure that the residual term $\mathcal{R}_{var}(u)$ is small. Careful inspection of the bootstrap estimator $\bar{\theta}_{n,b_n}(u)$ from (16) explains this dependence. Indeed, one can expect that the decay rate of the covariance between $\theta_{k}$ and $\theta_{k + b_n}$ depends on the quantity $\sum_{k = t + 1}^{t + b_n}\alpha_k$ , see e.g. [22]. At the same time, for $b_{n}\simeq n^{\beta},0 < \beta < 1$ and $\alpha_{k} = c_{0} / (k_{0} + k)^{\gamma}$ , which is set according to A3, we obtain that + +$$ +\begin{array}{c} \sum_ {k = t + 1} ^ {t + b _ {n}} \alpha_ {k} \lesssim b _ {n} / t ^ {\gamma}. \end{array} +$$ + +Hence, if $b_n \ll n^\gamma$ , the sequence $\{\theta_k\}$ does not mix within each block $t \leq k \leq t + b_n$ . + +Given that the remainder term $\mathcal{R}_{var}(u)$ in (19) is negligible, it remains to analyze the concentration properties of $\sigma_{\varepsilon}^{2}(u)$ . Indeed, under A1 and A2, the (normalized) linear statistic $n^{-1/2} \sum_{\ell=0}^{n-1} u^{\top} \varepsilon(Z_{\ell})$ is asymptotically normal with the asymptotic variance equal to $\sigma^{2}(u)$ defined in (9). Moreover, $\hat{\sigma}_{\varepsilon}^{2}(u)$ coincides with the overlapping batch mean estimator of $\sigma^{2}(u)$ . In order to prove concentration bounds for $\hat{\sigma}_{\varepsilon}^{2}(u)$ around $\sigma^{2}(u)$ , we apply the result of [50, Theorem 1]. + +Proposition 4. Assume A1 and A2. Then for any $p \geq 2$ , and $n \geq 2b_n + 1$ , and any initial distribution $\xi$ on $(\mathbb{Z},\mathbb{Z})$ , it holds that + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} [ | \hat {\sigma} _ {\varepsilon} ^ {2} (u) - \sigma^ {2} (u) | ^ {p} ] \lesssim \frac {p t _ {\mathrm {m i x}} ^ {3} \| \varepsilon \| _ {\infty} ^ {2}}{\sqrt {n}} + \frac {p ^ {2} t _ {\mathrm {m i x}} ^ {2} \sqrt {b _ {n}} \| \varepsilon \| _ {\infty} ^ {2}}{\sqrt {n}} + \frac {p t _ {\mathrm {m i x}} ^ {2} \| \varepsilon \| _ {\infty} ^ {2}}{\sqrt {b _ {n}}}. +$$ + +The result above is based on martingale decomposition, associated with the Poisson equation for quadratic forms of Markov chains, as introduced in [5]. Now, combining the estimates of Propositions 3 and 4 and applying Markov's inequality with $p = \log n$ , we obtain the following result: + +Corollary 2. Let $n$ be large enough. Set $b_{n} = \lceil n^{3 / 4}\rceil$ , $\varepsilon \in (0;1 / \log n)$ , and let $\alpha_{k} = c_{0} / (k_{0} + k)^{1 / 2 + \varepsilon}$ . Then, with probability at least $1 - n^{-1}$ , it holds that + +$$ +\left| \hat {\sigma} _ {\theta} ^ {2} (u) - \sigma^ {2} (u) \right| \lesssim_ {\log n} n ^ {- 1 / 8 + \varepsilon / 2}. \tag {20} +$$ + +Discussion. The version of Corollary 2 with explicit constants and explicit power of $\log n$ is provided in Appendix E.3, together with the proof of Corollary 2. Note that the fastest decay rate in the r.h.s of (20) is achieved when we set $b_{n} = \mathcal{O}(n^{3 / 4})$ and aggressive step sizes $\alpha_{k} = c_{0} / (k_{0} + k)^{1 / 2 + \varepsilon}$ . The same choice of hyperparameters appears to be optimal in the recent work of [63] for batch-mean estimators of the asymptotic variance for the SGD algorithm, even in case of controlled Markov chain. We also recover the rate $n^{-1 / 8}$ (up to logarithmic factors), that previously appeared in [63]. The authors of [17, 79] also considered the batch mean estimators $\widehat{\Sigma}_n$ of the asymptotic variance $\Sigma_{\infty}$ in case of SGD algorithms with independent noise and obtained the same (up to logarithmic factors) convergence rate $\mathbb{E}[\| \widehat{\Sigma}_n - \Sigma_\infty \| ]\lesssim n^{-1 / 8}$ . Moreover, the optimal rate for recovering $\Sigma_{\infty}$ in [17, Corollary 4.5] and [79, Corollary 3.4] is attained for the step sizes $\alpha_{k} = 1 / k^{\gamma}$ with $\gamma \rightarrow 1 / 2$ . This is on par with our findings of Corollary 2. At the same time, one should note that this choice of step sizes yields extremely slow convergence rates in the CLT in Theorem 1. This introduces an additional trade-off, that needs to be taken into account when considering the decay rate of (17). Namely, one needs to balance not only the right-hand sides of Corollary 2 and Proposition 3, but also to take into account the convergence rate in Theorem 1. The respective trade-off yields + +$$ +b _ {n} = \lceil n ^ {4 / 5} \rceil , \quad \alpha_ {k} = c _ {0} / (k _ {0} + k) ^ {3 / 5}. \tag {21} +$$ + +The corresponding main theorem writes as follows: + +Theorem 2. Assume $A1, A2$ , and $A3$ , let $n$ be large enough, set $b_n = \lceil n^{4/5} \rceil$ , $\alpha_k = c_0 / (k_0 + k)^{3/5}$ . Then for any $u \in \mathbb{S}_{d-1}$ , and any initial distribution $\xi$ on $(\mathbb{Z}, \mathcal{Z})$ , it holds with $\mathbb{P} - \text{probability at least } 1 - 1/n$ that + +$$ +\sup _ {x \in \mathbb {R}} | \mathbb {P} (\sqrt {n} (\bar {\theta} _ {n} - \theta^ {\star}) ^ {\top} u \leq x) - \mathbb {P} ^ {\mathbf {b}} (\bar {\theta} _ {n, b _ {n}} (u) \leq x) | \lesssim_ {\log n} n ^ {- 1 / 1 0}. +$$ + +Proof. We provide the detailed proof of Theorem 2 in Appendix E.4 together with the explicit form of condition on $n$ . The proof is based on the following scheme: + +$$ +\begin{array}{l} \text {R e a l} \sqrt {n} u ^ {\top} (\bar {\theta} _ {n} - \theta^ {\star}) \xleftarrow {\text {G a u s s i a n a p p r o x i m a t i o n , C o r . 1}} \xi \sim \mathcal {N} (0, \sigma^ {2} (u)) \\ \Bigupdownarrow \text {G a u s s i a n c o m p a r i s o n} \tag {22} \\ \end{array} +$$ + +$$ +\text {B o o t s t r a p w o r l d :} \quad \bar {\theta} _ {n, b} (u) \xleftarrow {\text {e x a c t l y m a t c h e s t h e d i s t r i b u t i o n}} \xi^ {\mathfrak {b}} \sim \mathcal {N} (0, \hat {\sigma} _ {\theta} ^ {2} (u)) +$$ + +Due to Corollary 1, it holds that + +$$ +\mathsf {d} _ {K} \big (\sqrt {n} u ^ {\top} (\bar {\theta} _ {n} - \theta^ {\star}), \xi \big) \lesssim_ {\log n} n ^ {- 1 / 4} + n ^ {1 / 2 - \gamma} + n ^ {\gamma - 1}, +$$ + +where $\xi \sim \mathcal{N}(0,1)$ . This result allows for the first horizontal bar above. We now use the Gaussian comparison (see [6], [21]), which states that for $\xi_{i}\sim \mathcal{N}(0,\sigma_{i}^{2})$ , $i = 1,2$ , are such that $|\sigma_1^2 /\sigma_2^2 -1|\leq \delta$ , for some $\delta \geq 0$ , $\sup_{x\in \mathbb{R}}|\mathbb{P}(\xi_1\leq x) - \mathbb{P}(\xi_2\leq x)|\leq \frac{3}{2}\delta$ . We apply the Gaussian comparison between the limiting Gaussian $\xi \sim \mathcal{N}(0,\sigma^2 (u))$ and $\xi^b\sim \bar{\mathcal{N}} (0,\hat{\sigma}_\theta^2 (u))$ . For this purpose, we need to obtain a high probability bound for $|\hat{\sigma}_{\theta}^{2}(u) - \sigma^{2}(u)| / \sigma^{2}(u)$ . Combining Proposition 3, Proposition 4 and Markov's inequality, we get that with probability at least $1 - 1 / n$ , + +$$ +| \hat {\sigma} _ {\theta} ^ {2} (u) - \sigma^ {2} (u) | / \sigma^ {2} (u) \lesssim_ {\log n} b _ {n} ^ {1 / 2} n ^ {\gamma / 2 - 1} + b _ {n} ^ {1 / 2} n ^ {- \gamma} + b _ {n} ^ {- 1 / 2} n ^ {\gamma / 2} + n ^ {2 \gamma - 2} + b ^ {1 / 2} n ^ {- 1 / 2}. +$$ + +Now note that by construction $\bar{\theta}_{n,b} \sim \mathcal{N}(0, \hat{\sigma}_{\theta}^2(u))$ under the bootstrap probability. Hence, with probability at least $1 - 1/n$ , it holds that + +$$ +\mathsf {d} _ {K} \big (\sqrt {n} u ^ {\top} (\bar {\theta} _ {n} - \theta^ {\star}), \bar {\theta} _ {n, b} \big) \lesssim_ {\log n} \frac {b _ {n} ^ {1 / 2}}{n ^ {1 - \gamma / 2}} + \frac {b _ {n} ^ {1 / 2}}{n ^ {\gamma / 2}} + \frac {b _ {n} ^ {1 / 2}}{n ^ {1 / 2}} + \frac {1}{n ^ {1 / 4}} + \frac {1}{n ^ {\gamma - 1 / 2}} + \frac {1}{n ^ {1 - \gamma}}. +$$ + +To complete the proof, it remains to optimize our choice of $\gamma$ and $b_{n}$ , which yields to (21). + +![](images/2eead05fc7f0813d3c169c095fb9256a402ec8f13eb47d0eb34772473f597c7d.jpg) + +Discussion. Non-asymptotic analysis of coverage probabilities has been carried out for the modifications of multiplier bootstrap approach of [29] in recent papers [67, 70]. These approaches showed that coverage probabilities of $\theta^{\star}$ can be approximated by their bootstrap counterparts with the order up to $\mathcal{O}(n^{-1/2})$ . Yet for this bootstrap approach it is crucial to work in the independent noise setting. The attempt of [59] to generalize it for the case of Markovian noise yields inconsistent procedure, as + +shown in [42, Proposition 1]. That is why we prefer to start from the asymptotically consistent MSB procedure of [43]. Our bootstrap validity proof relies on direct approximation of $\sqrt{n} u^{\top}(\bar{\theta}_n - \theta^\star)$ by the limiting Gaussian distribution $\mathcal{N}(0,\sigma^2 (u))$ . At the same time, it is known (see [70]) that under i.i.d. conditions, it may be more advantageous in the proof scheme 22 of the non-asymptotic validity of the bootstrap procedure for $\sqrt{n} u^{\top}(\bar{\theta}_n - \theta^\star)$ using another Gaussian distribution $\mathcal{N}(0,\sigma_n^2 (u))$ and its "bootstrap" analogue. It remains an open question if this reasoning can be applied in case of dependent random variables. + +# 5 Application to the TD learning + +We illustrate our findings via the temporal-difference (TD) learning algorithm [75, 76] for policy evaluation in a discounted Markov Reward Process (MRP) $(S, P, \mathcal{R}, \lambda)$ . Here, $\mathcal{S}$ is a complete metric space with its Borel $\sigma$ -algebra; $P(\cdot \mid s)$ denotes the state transition kernel; $\mathcal{R}: S \to [0,1]$ is the scalar reward function; $\lambda \in [0,1)$ is the discount factor. The value function is given by + +$$ +V (s) = \mathbb {E} [ \sum_ {k = 0} ^ {\infty} \lambda^ {k} \mathcal {R} (S _ {k}) \mid S _ {0} = s ] \text {w h e r e} S _ {k + 1} \sim \mathrm {P} (\cdot \mid S _ {k}). +$$ + +We approximate $V(s)$ in a linear feature space: $V_{\theta}(s) = \varphi (s)^{\top}\theta$ , where $\varphi :\mathcal{S}\to \mathbb{R}^d$ . We impose two standard assumptions: + +TD1. P is uniformly geometrically ergodic with unique invariant distribution $\mu$ and mixing time $\tau$ . + +We also define the design matrix $\Sigma_{\varphi} = \mathbb{E}_{\mu}[\varphi (S)\varphi (S)^{\top}]$ and require that it is non-degenerate: + +TD2. $\Sigma_{\varphi}$ is non-degenerate, i.e. $\lambda_{\min}(\Sigma_{\varphi}) > 0$ . Moreover, $\sup_{s\in S}\| \varphi (s)\| \leq 1$ + +Then TD learning is an instance of LSA with + +$$ +\mathbf {A} _ {k} = \varphi (S _ {k}) \big [ \varphi (S _ {k}) - \lambda \varphi (S _ {k + 1}) \big ] ^ {\top}, \quad \bar {\mathbf {A}} = \mathbb {E} _ {\mu} [ \mathbf {A} _ {k} ], \quad \mathbf {b} _ {k} = \varphi (S _ {k}) \mathcal {R} (S _ {k}). +$$ + +Detailed derivations are given in Appendix H. Refining constants following [53, 68], one shows: + +Proposition 5. Under $\mathbf{TD}1$ and $\mathbf{TD}2$ , the $TD$ updates satisfy the noise level condition A2 with $C_A = 2(1 + \lambda)$ and $\| \varepsilon \|_{\infty} = 2(1 + \lambda)(\| \theta_\star \| + 1)$ . Moreover, for step size $\alpha \leq \alpha_{\infty} = (1 - \lambda) / (1 + \lambda)^2$ , it holds that $\| I - \alpha \bar{A} \|^2 \leq 1 - \alpha a$ , where $a = (1 - \lambda) \lambda_{\min}(\Sigma_{\varphi})$ . + +By [67, Proposition 2], this ensures that Proposition 2 holds with $Q = \mathrm{I}$ , and hence Theorem 2 applies directly to the TD scheme. We provide numerical simulations in Appendix H. + +# 6 Conclusion + +We presented the non-asymptotic Berry-Esseen bounds for Polyak-Ruppert averaged iterates of linear stochastic approximation algorithm under Markovian noise, achieving convergence rates of order up to $n^{-1/4}$ in Kolmogorov distance. Additionally, we established the theoretical validity of a multiplier subsample bootstrap procedure, enabling reliable uncertainty quantification in the setting of the LSA algorithm with Markovian noise. Our paper suggests a number of further research directions. One of them is related with the generalizations of Theorem 1 to the setting of non-linear SA algorithms, as well as with obtaining multivariate version of it. It is also an interesting and important question if our non-asymptotic bounds on coverage probabilities provided in Theorem 2 can be further improved. + +# Acknowledgment + +The work was supported by the grant for research centers in the field of AI provided by the Ministry of Economic Development of the Russian Federation in accordance with the agreement 000000C313925P4E0002 and the agreement with HSE University No 139-15-2025-009. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The paper does not involve the use of LLMs or related methods. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Contents + +1 Introduction 1 +2 Related works 3 +3 Accuracy of Gaussian approximation for LSA with Markov noise 4 + +3.1 Gaussian approximation for non-linear statistics 4 +3.2 Gaussian approximation for LSA with Markov noise 4 + +4 Multiplier subsample bootstrap for LSA 7 +5 Application to the TD learning 10 +6 Conclusion 10 + +A Extended version of A3 24 +B Proof of Section 3 24 + +B.1 Gaussian approximation for sums of martingale-difference sequences 25 +B.2 Bound for the remainder term $D$ 28 +B.3 Proof of Corollary 1 29 +B.4 Technical bounds 30 + +C Last iterate bound 34 + +C.1 Proof of Proposition 8 36 +C.2 Proof of $J_{n}^{(1)}$ bound 36 +C.3 Technical bounds related to $J_{k}^{(1)}$ 40 + +D Proofs for stability of matrix products 41 + +D.1 Stability 41 +D.2 Proof of Proposition 12 44 +D.3 Proof of Proposition 2 45 + +E Proofs of Section 4 46 + +E.1 Proof of Proposition 3 46 +E.2 Technical bounds 47 +E.3 Proof of Corollary 2 54 +E.4 Detailed version of the proof of Theorem 2 54 + +F Probability inequalities 55 +F.1 Rosenthal and Burkholder inequalities 56 +G Auxiliary results on the sequences of step sizes $\{\alpha_{k}\}_{k\in \mathbb{N}}$ under A3 57 + +# A Extended version of A3 + +In this section, we present the full version of A3 with explicit constants. + +A'3. The sequence of step sizes $\{\alpha_k\}_{k\in \mathbb{N}}$ has a form $\alpha_{k} = c_{0} / (k + k_{0})^{\gamma}$ , where $\gamma \in [1 / 2;1)$ and, with + +$$ +h = \left\lceil \frac {1 6 t _ {\operatorname* {m i x}} \kappa_ {Q} ^ {1 / 2} \mathrm {C} _ {\mathbf {A}}}{a} \right\rceil , \tag {23} +$$ + +it holds that + +$$ +c _ {0} \leq \frac {1}{2 a}. \tag {24} +$$ + +Moreover, the number of observations $n$ and parameter $k_{0}$ satisfy + +$$ +\begin{array}{l} n ^ {1 - \gamma} \geq \max \left(\frac {2 C _ {\infty} ^ {\prime}}{\lambda_ {\min } (\Sigma_ {\infty})}, \mathrm {e} ^ {2 (1 - \gamma)}\right), \\ k _ {0} \geq \max \left\{\left(\frac {2 4}{a c _ {0}}\right) ^ {1 / (1 - \gamma)}, c _ {0} ^ {1 / \gamma}, \left(c _ {0} (h + 1) \max \left(\alpha_ {\infty}, \kappa_ {Q} ^ {1 / 2} \mathrm {C} _ {\mathbf {A}}, 6 \mathrm {e} \kappa_ {Q} \mathrm {C} _ {\mathbf {A}} ^ {2} / a\right)\right) ^ {1 / \gamma}, \right. \tag {25} \\ \left. \frac {8 t _ {\mathrm {m i x}} \kappa_ {Q} ^ {1 / 2} \mathrm {C _ {A}}}{a}, \left(\frac {1 2 (h + 1) (\log n) c _ {0} \mathrm {C _ {\sigma} ^ {2}}}{a}\right) ^ {1 / \gamma} \right\}, \\ \end{array} +$$ + +$$ +k _ {0} ^ {1 - \gamma} \geq \frac {2}{c _ {0} b} \left(\log \{\frac {c _ {0}}{b (2 \gamma - 1) 2 ^ {\gamma}} \} + \gamma \log \{k _ {0} \}\right), +$$ + +Discussion. The lower bound on the sample size $n$ in (25) is required to guarantee the nondegeneracy of the empirical covariance matrix $\Sigma_{n}$ (see its definition in (10) and Lemma 10). This condition ensures that $\lambda_{\min}(\Sigma_n)$ remains bounded away from zero, which is crucial for the proof of rate of convergence in CLT for LSA problem. + +Furthermore, the constraint on $k_{0}$ , which scales as $k_{0} \gtrsim (\log n)^{1 / \gamma}$ , is imposed to control higher-order moments of the product of random matrices appearing in the recursion (see Proposition 12). In particular, this ensures that moments up to order $p \sim \log n$ remain finite, which is necessary for establishing the Gaussian approximation bounds used in the main result. + +While this assumption could, in principle, be relaxed, doing so would lead to a degradation in the Gaussian approximation rate from $n^{-1/4}$ to $n^{-1/4 + \varepsilon(p)}$ , where the function $\varepsilon(p)$ depends on the number of finite moments available for the random matrix product. + +# B Proof of Section 3 + +Recall that in the proof of Theorem 1 (see (15)), we obtained that + +$$ +\mathsf {d} _ {K} \big (\frac {\sqrt {n} u ^ {\top} (\bar {\theta} _ {n} - \theta^ {\star})}{\sigma_ {n} (u)}, \mathcal {N} (0, 1) \big) \leq \mathsf {d} _ {K} \big (\frac {u ^ {\top} M}{\sqrt {n} \sigma_ {n} (u)}, \mathcal {N} (0, 1) \big) + 2 \big \{\mathbb {E} [ | \frac {u ^ {\top} D}{\sigma_ {n} (u)} | ^ {p} ] \big \} ^ {1 / (p + 1)}. +$$ + +Hence, to conclude with the proof it remains to + +(i) control the Kolmogorov distance $\mathsf{d}_K\left(\frac{u^\top M}{\sqrt{n}\sigma_n(u)}\right)$ ; +(ii) bound the moments of $\left|\frac{u^\top D}{\sigma_n(u)}\right|$ . + +We present the required result for item (i) in Appendix B.1 (see Proposition 6) and for (ii) in Appendix B.1 below. + +# B.1 Gaussian approximation for sums of martingale-difference sequences + +In this section we establish rate of convergence sums of martingale-difference to the gaussian limit. We follow the approach of [28], which builds upon the work [12]. Recall that + +$$ +M = - \sum_ {\ell = 2} ^ {n - 1} \Delta M _ {\ell}, \quad \Delta M _ {\ell} = Q _ {\ell} \left(\hat {\varepsilon} (Z _ {\ell}) - \mathrm {P} \hat {\varepsilon} (Z _ {\ell - 1})\right). +$$ + +Note that the quadratic characteristic + +$$ +\mathbb {E} [ | u ^ {\top} \Delta M _ {\ell} | ^ {2} | Z _ {\ell - 1} ] = u ^ {\top} Q _ {\ell} \tilde {\varepsilon} (Z _ {\ell - 1}) Q _ {\ell + 1} ^ {\top} u, \quad \tilde {\varepsilon} (z) = \mathrm {P} \hat {\varepsilon} \hat {\varepsilon} ^ {\top} (z) - \mathrm {P} \hat {\varepsilon} (z) \mathrm {P} \hat {\varepsilon} ^ {\top} (z) +$$ + +is not constant a.s.. Hence, one can not directly apply the results on martingale CLT due to [12, Theorem 2]. Below we first formulate this result, and then demonstrate how to mitigate the problem of time-varying quadratic characteristic following the approach of [66, 28]. + +CLT for martingales with constant quadratic characteristic. Let $X = (X_{1},\ldots ,X_{n})$ be a sequence of real valued random variables which are square integrable and satisfy $\mathbb{E}[X_i|\mathcal{F}_{i - 1}] = 0$ , a.s. for $1\leq i\leq n$ , where $\mathcal{F}_i = \sigma (X_1,\dots ,X_i)$ . We denote the class of such sequences of length $n$ by $\mathcal{M}_n$ . Denote + +$$ +\sigma_ {j} ^ {2} = \mathbb {E} \left[ X _ {j} ^ {2} \mid \mathcal {F} _ {j - 1} \right], \quad \hat {\sigma} _ {j} ^ {2} = \mathbb {E} \left[ X _ {j} ^ {2} \right], +$$ + +$$ +s _ {n} ^ {2} = \sum_ {j = 1} ^ {n} \hat {\sigma} _ {j} ^ {2}, +$$ + +$$ +\| X \| _ {p} = \max _ {1 \leq j \leq n} \| X _ {j} \| _ {p}, \quad 1 \leq p \leq \infty , +$$ + +and, with $1 \leq k \leq n$ , we have + +$$ +V _ {k} ^ {2} = \sum_ {j = 1} ^ {k} \sigma_ {j} ^ {2} / s _ {n} ^ {2}, \quad S _ {k} = \sum_ {j = 1} ^ {k} X _ {j}. +$$ + +Theorem 3 (Theorem 2 in [12]). Let $0 < \varkappa < \infty$ . There exists a constant $0 < L(\varkappa) < \infty$ depending only on $\varkappa$ , such that for all $X \in \mathcal{M}_n, n \geq 2$ , satisfying + +$$ +\| X \| _ {\infty} \leq \varkappa , \quad V _ {n} ^ {2} = 1 a. s. , +$$ + +the following bound holds + +$$ +\mathrm {d} _ {K} \left(S _ {n} / s _ {n}\right) \leq L (\varkappa) n \log n / s _ {n} ^ {3}. +$$ + +The constant $L(\varkappa)$ can be quantified following the work of Adrian Röllin, see [66], but exact calculations are beyond the scope of this paper. Now we adapt the arguments from [28] based on [12] to provide the counterpart of this result with time-varying quadratic characteristic. We specify the constants and dependence on $p$ since we need to have possibility to take $p$ of logarithmic order with respect to $n$ . + +Lemma 1. Let $0 < \varkappa < \infty$ . Then, with $L(\varkappa)$ defined in Theorem 3, for all $X \in \mathcal{M}_n$ , $n \geq 2$ , satisfying $\| X \|_{\infty} \leq \varkappa$ , the following bound holds for any $p \geq 1$ + +$$ +\begin{array}{l} \mathrm {d} _ {K} \left(S _ {n} / s _ {n}\right) \leq \frac {L (\varkappa) (2 n + 1) \log (2 n + 1)}{s _ {n} ^ {3}} \\ + C _ {1} \sqrt {p} s _ {n} ^ {- \frac {2 p}{2 p + 1}} \left(\mathbb {E} | \sum_ {i = 1} ^ {n} \sigma_ {i} ^ {2} - s _ {n} ^ {2} | ^ {p}\right) ^ {1 / (2 p + 1)} + C _ {2} s _ {n} ^ {- \frac {2 p}{2 p + 1}} p \varkappa^ {2 p / (2 p + 1)}. \\ \end{array} +$$ + +where we have defined the constants $C_1 = 2\sqrt{2}\mathrm{C}_{\mathrm{Rm},1}$ $C_2 = 4(\mathrm{C}_{\mathrm{Rm},1} + \mathrm{C}_{\mathrm{Rm},2})$ + +Proof. Consider the following stopping time $\tau = \sup \{0\leq k\leq n:V_k^2\leq 1\}$ . Denote + +$$ +r = \left\lfloor \left(s _ {\tau} ^ {2} - \sum_ {j = 1} ^ {\tau} \sigma_ {j} ^ {2}\right) / \varkappa^ {2} \right\rfloor . +$$ + +Note that $r \leq n$ . Let $N = 2n + 1$ . Construct the following sequence $X' = (X_1', \ldots, X_N')$ : + +$$ +X _ {i} ^ {\prime} = X _ {i}, i \leq \tau ; \quad X _ {i} ^ {\prime} = \varkappa \eta_ {i}, \tau + 1 \leq i \leq \tau + r; \quad X _ {i} ^ {\prime} = (s _ {n} ^ {2} - \sum_ {i = 1} ^ {\tau} \sigma_ {i} ^ {2} - r \varkappa^ {2}) ^ {1 / 2} \eta_ {i}, i = \tau + r + 1; +$$ + +and $X_{i}^{\prime} = 0, i \geq \tau + r + 2$ . Here $\eta_{i}$ are Rademacher random variables, which are independent of all other r.v.'s. Then $X^{\prime}$ is a vector of martingale increments w.r.t. an extended filtration $\mathcal{F}_i^\prime = \sigma(X_j, \eta_j, j \leq i)$ , and, by construction + +$$ +\sum_ {i = 1} ^ {N} \mathbb {E} [ (X _ {i} ^ {\prime}) ^ {2} | \mathcal {F} _ {i - 1} ^ {\prime} ] = \sum_ {i = 1} ^ {\tau} \sigma_ {i} ^ {2} + r \varkappa^ {2} + s _ {n} ^ {2} - \sum_ {i = 1} ^ {\tau} \sigma_ {i} ^ {2} - r \varkappa^ {2} = s _ {n} ^ {2}. +$$ + +Applying Proposition 14 with $X = S_{n} / s_{n}$ and $Y = S_{N}^{\prime} / s_{n}$ , we get + +$$ +\mathrm {d} _ {K} \left(S _ {n} / s _ {n}\right) \leq \mathrm {d} _ {K} \left(S _ {N} ^ {\prime} / s _ {n}\right) + 2 s _ {n} ^ {- \frac {2 p}{2 p + 1}} \left(\mathbb {E} \left[ \left| S _ {n} - S _ {N} ^ {\prime} \right| ^ {2 p} \right]\right) ^ {1 / (2 p + 1)}. \tag {26} +$$ + +We control the term $\mathrm{d}_K(S_N' / s_n)$ with [12, Theorem 2]: + +$$ +\mathrm {d} _ {K} \left(S _ {N} ^ {\prime} / s _ {n}\right) \leq \frac {L (\varkappa) N \log N}{s _ {n} ^ {3}}. +$$ + +In order to control the second term in the right-hand side of (26), we notice that + +$$ +S _ {n} - S _ {N} ^ {\prime} = \sum_ {i \geq \tau + 1} (X _ {i} - X _ {i} ^ {\prime}) = \sum_ {i = 1} ^ {N} \mathbf {1} _ {\tau \leq i - 1} (X _ {i} - X _ {i} ^ {\prime}). +$$ + +Since $\tau$ is a stopping time, we can condition on it and get by the Rosenthal's inequality + +$$ +\mathbb {E} [ | S _ {n} - S _ {N} ^ {\prime} | ^ {2 p} ] \leq C _ {\mathsf {R m}, 1} ^ {2 p} p ^ {p} \mathbb {E} [ | \sum_ {i = \tau + 1} ^ {N} \mathbb {E} [ (X _ {i} - X _ {i} ^ {\prime}) ^ {2} | \mathcal {F} _ {i - 1} ^ {\prime} ] | ^ {p} ] + C _ {\mathsf {R m}, 2} ^ {2 p} p ^ {2 p} \mathbb {E} [ \max _ {\tau + 1 \leq i \leq N} | X _ {i} - X _ {i} ^ {\prime} | ^ {2 p} ] +$$ + +It is easy to see that + +$$ +\sum_ {i = \tau + 1} ^ {N} \mathbb {E} [ (X _ {i} - X _ {i} ^ {\prime}) ^ {2} | \mathcal {F} _ {i - 1} ^ {\prime} ] = \sum_ {i = \tau + 1} ^ {n} \mathbb {E} [ X _ {i} ^ {2} | \mathcal {F} _ {i - 1} ^ {\prime} ] + \sum_ {i = \tau + 1} ^ {N} \mathbb {E} [ (X _ {i} ^ {\prime}) ^ {2} | \mathcal {F} _ {i - 1} ^ {\prime} ] = \sum_ {i = 1} ^ {n} \sigma_ {i} ^ {2} + s _ {n} ^ {2} - 2 \sum_ {i = 1} ^ {\tau} \sigma_ {i} ^ {2}. +$$ + +Note that + +$$ +s _ {n} ^ {2} - \varkappa^ {2} \leq \sum_ {i = 1} ^ {\tau} \sigma_ {i} ^ {2} \leq s _ {n} ^ {2}. +$$ + +Hence, it holds that + +$$ +\sum_ {i = \tau + 1} ^ {N} \mathbb {E} [ (X _ {i} - X _ {i} ^ {\prime}) ^ {2} | \mathcal {F} _ {i - 1} ^ {\prime} ] \leq \sum_ {i = 1} ^ {n} \sigma_ {i} ^ {2} - s _ {n} ^ {2} + 2 \varkappa^ {2}. +$$ + +Finally, + +$$ +\mathbb {E} \left[ \left| S _ {n} - S _ {N} ^ {\prime} \right| ^ {2 p} \right] \leq 2 ^ {p - 1} \mathrm {C} _ {\mathsf {R m}, 1} ^ {2 p} p ^ {p} \left(\mathbb {E} \right\rvert \sum_ {i = 1} ^ {n} \sigma_ {i} ^ {2} - s _ {n} ^ {2} | ^ {p} + 2 ^ {p} \varkappa^ {2 p}) + \mathrm {C} _ {\mathsf {R m}, 2} ^ {2 p} (2 p) ^ {2 p} \varkappa^ {2 p}) ^ {1 / (2 p + 1)}. +$$ + +Substituting the above bound into (26), we obtain + +$$ +\begin{array}{l} \mathsf {d} _ {K} \left(S _ {n} / s _ {n}\right) \leq \mathsf {d} _ {K} \left(S _ {N} ^ {\prime} / s _ {n}\right) \\ + 2 s _ {n} ^ {- \frac {2 p}{2 p + 1}} (2 ^ {p - 1} \mathrm {C} _ {\mathsf {R m}, 1} ^ {2 p} p ^ {p} (\mathbb {E} | \sum_ {i = 1} ^ {n} \sigma_ {i} ^ {2} - s _ {n} ^ {2} | ^ {p} + 2 ^ {p} \varkappa^ {2 p}) + \mathrm {C} _ {\mathsf {R m}, 2} ^ {2 p} (2 p) ^ {2 p} \varkappa^ {2 p}) ^ {1 / (2 p + 1)}, \\ \end{array} +$$ + +and the statement follows. + +We now state a lemma that allows us to specify the result of Lemma 1 to the martingale $M$ and its quadratic characteristic $\langle M\rangle_{n}$ , given in (14). + +Lemma 2. Assume A1. Then for any $u\in \mathbb{S}_{d - 1}$ , probability measure $\xi$ on $(Z,\mathcal{Z})$ , and $p\geq 2$ .. + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} \langle M \rangle_ {n} u - n \sigma_ {n} ^ {2} (u) \right| ^ {p} \right] \leq 3 2 n ^ {1 / 2} p ^ {1 / 2} \mathcal {L} _ {Q} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} t _ {\mathrm {m i x}} ^ {5 / 2} +$$ + +Proof. Using the definition (14), we get: + +$$ +u ^ {\top} \langle M \rangle_ {n} u - \sigma_ {n} ^ {2} (u) = \sum_ {\ell = 1} ^ {n - 2} \left\{h _ {\ell} \left(Z _ {\ell}\right) - \pi \left(h _ {\ell}\right) \right\} \quad \text {w h e r e} \qquad h _ {\ell} (z) = u ^ {\top} Q _ {\ell + 1} \tilde {\varepsilon} (z) Q _ {\ell + 1} ^ {\top} u. +$$ + +Note that using Lemma 3, for any $z,z^{\prime}\in \mathsf{Z}$ and any $\ell \in [1,\dots ,n - 2]$ we have + +$$ +\begin{array}{l} \left| h _ {\ell} (z) - h _ {\ell} \left(z ^ {\prime}\right) \right| \leq 2 \mathcal {L} _ {Q} ^ {2} \| \hat {\varepsilon} (z) \| \leq 4 \mathcal {L} _ {Q} ^ {2} \| \hat {\varepsilon} (z) \| ^ {2} \leq 8 \mathcal {L} _ {Q} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} \left(\sum_ {k = 0} ^ {\infty} \Delta \left(\mathrm {P} ^ {k}\right)\right) ^ {2} \\ \leq 8 \mathcal {L} _ {Q} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} (\sum_ {k = 0} ^ {\infty} \sum_ {r = 0} ^ {t _ {\mathrm {m i x}} - 1} (1 / 4) ^ {\lceil (k t _ {\mathrm {m i x}} + r) / t _ {\mathrm {m i x}} \rceil}) ^ {2} \leq 2 (8 / 3) ^ {2} \mathcal {L} _ {Q} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} t _ {\mathrm {m i x}} ^ {2} \\ \end{array} +$$ + +We then apply [54][Corollary 2.11], showing that for all $t \geq 0$ , + +$$ +\mathbb {P} _ {\xi} (| \sum_ {\ell = 1} ^ {n - 1} \{h _ {\ell} (Z _ {\ell}) - \pi (h _ {\ell}) \} | \geq t) \leq 2 \exp \left(- \frac {t ^ {2}}{2 u _ {n} ^ {2}}\right), \text {w h e r e} u _ {n} = (6 4 / 3) \mathcal {L} _ {Q} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} t _ {\mathrm {m i x}} ^ {5 / 2} n ^ {1 / 2} +$$ + +We conclude by using [24, Lemma 7]. + +![](images/0f8818c1d90b8d0ed69d6da110b6c03c756978a7ffbdf58bc3c80302fa387324.jpg) + +By combining Lemma 1 and Lemma 2, we finally get the following result: + +Proposition 6. Assume A1, A2, and A3. Then for any $u \in \mathbb{S}_{d-1}$ and $p \geq 1$ : + +$$ +\begin{array}{l} \mathrm {d} _ {K} \left(\frac {u ^ {\top} M}{\sqrt {n} \sigma_ {n} (u)}\right) \leq \frac {L (\varkappa) (2 n + 1) \log (2 n + 1) C _ {\Sigma} ^ {3}}{n ^ {3 / 2}} \\ + 4 \sqrt {2} C _ {1} \left(\mathcal {L} _ {Q} \| \varepsilon \| _ {\infty} C _ {\Sigma}\right) ^ {\frac {2 p}{2 p + 1}} t _ {\mathrm {m i x}} ^ {5 / 4} p ^ {3 / 4} n ^ {- \frac {p}{2 (2 p + 1)}} + C _ {2} n ^ {- \frac {p}{2 p + 1}} \left(\frac {1 6 \mathcal {L} _ {Q} \| \varepsilon \| _ {\infty} C _ {\Sigma}}{3}\right) ^ {\frac {2 p}{2 p + 1}} p t _ {\mathrm {m i x}}, \\ \end{array} +$$ + +where $\varkappa = (16 / 3)\| \varepsilon \|_{\infty}\mathcal{L}_{Q}t_{\mathrm{mix}}$ , constants $C_1, C_2$ are defined in Lemma 1, $\mathcal{L}_Q$ is defined in Lemma 3, and $C_\Sigma$ is defined in Lemma 10. Moreover, setting $p = \log n$ , we obtain that + +$$ +\mathsf {d} _ {K} \big (\frac {u ^ {\top} M}{\sqrt {n} \sigma_ {n} (u)} \big) \leq \frac {\mathrm {C} _ {\mathrm {K} , 1} \log^ {3 / 4} n}{n ^ {1 / 4}} + \frac {\mathrm {C} _ {\mathrm {K} , 2} \log n}{n ^ {1 / 2}}, +$$ + +where the constants $C_{K,1}$ and $C_{K,2}$ are given by + +$$ +\begin{array}{l} \mathrm {C} _ {\mathrm {K}, 1} = 4 \sqrt {2} \mathrm {e} ^ {1 / 8} C _ {1} t _ {\mathrm {m i x}} ^ {5 / 4} \mathcal {L} _ {Q} \| \varepsilon \| _ {\infty} C _ {\Sigma}, \\ \mathrm {C} _ {\mathrm {K}, 2} = \frac {1 6 e ^ {1 / 4} t _ {\operatorname* {m i x}} C _ {2} \mathcal {L} _ {Q} \| \varepsilon \| _ {\infty} C _ {\Sigma}}{3} + 3 L (\varkappa) C _ {\Sigma} ^ {3} + 3 L (\varkappa) \log 3 C _ {\Sigma} ^ {3}. \\ \end{array} +$$ + +Proof of Proposition 6. Using Lemma 1 with $s_n^2 = n\sigma_n^2(u)$ , we obtain + +$$ +\begin{array}{l} \mathrm {d} _ {K} \left(\frac {u ^ {\top} M}{\sqrt {n} \sigma_ {n} (u)}\right) \leq \frac {L (\varkappa) (2 n + 1) \log (2 n + 1)}{n ^ {3 / 2} \sigma_ {n} ^ {3} (u)} \\ + C _ {1} \sqrt {p} (n \sigma_ {n} ^ {2} (u)) ^ {- \frac {2 p}{2 p + 1}} \left(\mathbb {E} | u ^ {\top} \langle M \rangle_ {n} u - n \sigma_ {n} ^ {2} (u) | ^ {p}\right) ^ {1 / (2 p + 1)} + C _ {2} (n \sigma_ {n} ^ {2} (u)) ^ {- \frac {p}{2 p + 1}} p \varkappa^ {2 p / (2 p + 1)}, \\ \end{array} +$$ + +where the particular form of $\varkappa$ follows from (70). To complete the proof, it remains to control the $p$ -th moment of $u^{\top}\langle M\rangle_{n}u - n\sigma_{n}^{2}(u)$ , which is established in Lemma 2. Finally, we note that $\sigma_n^{-1}(u)$ is bounded, as follows from Lemma 10, which concludes the argument. + +# B.2 Bound for the remainder term $D$ + +Recall that $D = D_{1} + D_{2}$ , where + +$$ +D _ {1} = \frac {1}{\sqrt {n}} \sum_ {k = 0} ^ {n - 1} \Gamma_ {1: k} \left(\theta_ {0} - \theta^ {\star}\right) + \frac {1}{\sqrt {n}} \sum_ {k = 1} ^ {n - 1} H _ {k} ^ {(0)} \varepsilon \left(Z _ {\ell}\right), \tag {27} +$$ + +$$ +D _ {2} = - \frac {1}{\sqrt {n}} Q _ {1} \hat {\varepsilon} (Z _ {1}) + \frac {1}{\sqrt {n}} Q _ {n - 1} \mathrm {P} \hat {\varepsilon} (Z _ {n - 1}) + \sum_ {\ell = 1} ^ {n - 2} (Q _ {\ell} - Q _ {\ell + 1}) \mathrm {P} \hat {\varepsilon} (Z _ {\ell}), +$$ + +and $H_{k}^{(0)}$ are defined in (12). Then we obtain the following bound on the moments of $D$ : + +Proposition 7. Assume $A1, A2$ , and $A3$ and $k_0 \geq \max \left( \left( \frac{24}{ac_0} \right)^{1/(1-\gamma)}, c_0^{1/\gamma} \right)$ . Then for any $2 \leq p \leq \log n$ , $u \in \mathbb{S}_{d-1}$ , and any initial distribution $\xi$ on $(\mathbb{Z}, \mathcal{Z})$ , it holds that + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / (p + 1)} \left[ \left| \left(u ^ {\top} D\right) / \sigma_ {n} (u) \right| ^ {p} \right] \lesssim \left(\mathrm {C} _ {1, 1} ^ {\mathrm {D}} \| \theta_ {0} - \theta^ {\star} \| + \mathrm {C} _ {1, 2} ^ {\mathrm {D}}\right) n ^ {p / (2 p + 2)} + \mathrm {C} _ {1, 3} ^ {\mathrm {D}} \left\{p ^ {2} n ^ {1 / 2 - \gamma} \right\} ^ {p / (p + 1)} \\ + \mathrm {C} _ {1, 4} ^ {\mathrm {D}} \left\{p ^ {2} \sqrt {\log n n} n ^ {1 / 2 - \gamma} \right\} ^ {p / (p + 1)}, \\ \end{array} +$$ + +where + +$$ +\mathrm {C} _ {1, 1} ^ {\mathsf {D}} = C _ {\Sigma} C _ {\Gamma} d ^ {1 / \log n} \frac {k _ {0} ^ {\gamma}}{a c _ {0} (1 - \gamma)} , +$$ + +$$ +\mathrm {C} _ {1, 2} ^ {\mathrm {D}} = C _ {\Sigma} \| \varepsilon \| _ {\infty} \mathcal {L} _ {Q} t _ {\text {m i x}}, +$$ + +$$ +\mathsf {C} _ {1, 3} ^ {\mathsf {D}} = \frac {C _ {\Sigma} c _ {0} t _ {\mathrm {m i x}}}{1 - \gamma} \left(\mathcal {L} _ {Q, 2} \| \varepsilon \| _ {\infty} + \sqrt {\log \frac {k _ {0} ^ {\gamma}}{c _ {0}}} \mathsf {D} _ {3} ^ {\mathrm {(M)}}\right), +$$ + +$$ +C _ {1, 4} ^ {\mathsf {D}} = \frac {C _ {\Sigma} \mathsf {D} _ {3} ^ {(\mathrm {M})} c _ {0} t _ {\operatorname {m i x}} \sqrt {\gamma}}{1 - \gamma}. +$$ + +Moreover setting $p = \log n$ we have + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / (p + 1)} \left[ \left| \left(u ^ {\top} D\right) / \sigma_ {n} (u) \right| ^ {p} \right] \lesssim \left(\mathrm {C} _ {1} ^ {\mathrm {D}} \| \theta_ {0} - \theta^ {\star} \| + \mathrm {C} _ {2} ^ {\mathrm {D}}\right) n ^ {- 1 / 2} + \mathrm {C} _ {3} ^ {\mathrm {D}} (\log n) ^ {2} n ^ {1 / 2 - \gamma} \\ + C _ {4} ^ {D} (\log n) ^ {5 / 2} n ^ {1 / 2 - \gamma}, \\ \end{array} +$$ + +and + +$$ +C _ {i} ^ {D} = \sqrt {\mathrm {e}} C _ {1, i} ^ {D}, \quad f o r i \in \{1, 2, 3, 4 \} +$$ + +Proof. We first note that $\sigma_n^{-1}(u)$ is bounded, as follows from Lemma 10, that is, + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| \left(u ^ {\top} D\right) / \sigma_ {n} (u) \right| ^ {p} \right] \leq C _ {\Sigma} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} D \right| ^ {p} \right]. +$$ + +In order to bound $\mathbb{E}_{\xi}^{1/p}\left[\left|u^{\top}D\right|^{p}\right]$ , we bound the terms $D_{1}$ and $D_{2}$ from (27) separately. Note that + +$$ +u ^ {\top} D _ {1} = D _ {1 1} + D _ {1 2}, \quad D _ {1 1} = \frac {1}{\sqrt {n}} \sum_ {k = 0} ^ {n - 1} u ^ {\top} \Gamma_ {1: k} (\theta_ {0} - \theta^ {\star}), \quad D _ {1 2} = \frac {1}{\sqrt {n}} \sum_ {k = 1} ^ {n - 1} u ^ {\top} H _ {k} ^ {(0)}. +$$ + +Now we use the result of Proposition 12 to bound $\mathbb{E}_{\xi}[\| \Gamma_{1:k}\| ^p ]$ , and obtain that + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} [ | D _ {1 1} | ^ {p} ] \lesssim C _ {\Gamma} d ^ {1 / \log n} \frac {1}{\sqrt {n}} \sum_ {k = 0} ^ {n - 1} \exp \left\{- (a / 1 2) \sum_ {j = 1} ^ {k} \alpha_ {j} \right\} \| \theta_ {0} - \theta^ {\star} \| . +$$ + +Applying Lemma 24 and Lemma 31, + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| D _ {1 1} \right| ^ {p} \right] \lesssim C _ {\Gamma} d ^ {1 / \log n} \frac {1}{\sqrt {n}} \sum_ {k = 0} ^ {n - 1} \exp \left\{- \frac {a c _ {0}}{2 4 (1 - \gamma)} \left(\left(k + k _ {0}\right) ^ {1 - \gamma} - k _ {0} ^ {\gamma}\right) \right\} \| \theta_ {0} - \theta^ {\star} \| \tag {28} \\ \lesssim C _ {\Gamma} d ^ {1 / \log n} \frac {1}{\sqrt {n}} \frac {k _ {0} ^ {\gamma}}{a c _ {0} (1 - \gamma)} \| \theta_ {0} - \theta^ {\star} \|. \\ \end{array} +$$ + +It follows from Minkowski's inequality and Proposition 10 that + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ | D _ {1 2} | ^ {p} \right] \leq \frac {1}{\sqrt {n}} \sum_ {k = 1} ^ {n - 1} \mathbb {E} _ {\xi} ^ {1 / p} \left[ | u ^ {\top} H _ {k} ^ {(0)} | ^ {p} \right] \leq \frac {\mathrm {D} _ {3} ^ {(\mathrm {M})} t _ {\operatorname* {m i x}} p ^ {2}}{\sqrt {n}} \sum_ {k = 1} ^ {n - 1} \alpha_ {k} \sqrt {\log \left(1 / \alpha_ {k}\right)} \\ \lesssim \frac {\mathrm {D} _ {3} ^ {(\mathrm {M})} t _ {\text {m i x}} p ^ {2} c _ {0}}{1 - \gamma} \sqrt {\log \frac {(n + k _ {0} - 1) ^ {\gamma}}{c _ {0}}} \frac {\left((n - 1 + k _ {0}) ^ {1 - \gamma} - k _ {0} ^ {1 - \gamma}\right)}{\sqrt {n}} \tag {29} \\ \lesssim \frac {\mathrm {D} _ {3} ^ {(\mathrm {M})} t _ {\mathrm {m i x}} p ^ {2} c _ {0}}{1 - \gamma} (\sqrt {\gamma \log n} + \sqrt {\frac {k _ {0} ^ {\gamma}}{c _ {0}}}) n ^ {1 / 2 - \gamma}. \\ \end{array} +$$ + +Similarly, $u^{\top}D_{2} = D_{21} + D_{22} + D_{32}$ , where + +$$ +D _ {2 1} = - \frac {1}{\sqrt {n}} u ^ {\top} Q _ {1} \hat {\varepsilon} (Z _ {1}), +$$ + +$$ +D _ {2 2} = \frac {1}{\sqrt {n}} u ^ {\top} Q _ {n - 1} \mathrm {P} \hat {\varepsilon} (Z _ {n - 1}), +$$ + +$$ +D _ {3 2} = \sum_ {\ell = 1} ^ {n - 2} u ^ {\top} (Q _ {\ell} - Q _ {\ell + 1}) \mathrm {P} \hat {\varepsilon} (Z _ {\ell}). +$$ + +For $D_{21}$ and $D_{22}$ we use Lemma 3 to bound $\| Q_{\ell}\|$ together with the upper bound $\| \hat{\varepsilon} (z)\| \leq$ $(8 / 3)t_{\mathrm{mix}}\| \varepsilon \|_{\infty}$ . Then we obtain + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} [ | D _ {2 1} ] ^ {p} ] + \mathbb {E} _ {\xi} ^ {1 / p} [ | D _ {2 2} ] ^ {p} ] \lesssim \frac {t _ {\mathrm {m i x}} \| \varepsilon \| _ {\infty} \mathcal {L} _ {Q}}{\sqrt {n}}. +$$ + +It remains to bound $\mathbb{E}_{\xi}^{1/p}[|D_{23}|^p]$ . Using Lemma 6 to bound the difference $Q_{\ell} - Q_{\ell+1}$ , and Minkowski's inequality, we get + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} [ | D _ {2 3} | ^ {p} ] \lesssim \frac {\mathcal {L} _ {Q , 2} t _ {\mathrm {m i x}} \| \varepsilon \| _ {\infty}}{\sqrt {n}} \sum_ {\ell = 2} ^ {n - 1} \alpha_ {\ell} \lesssim \frac {\mathcal {L} _ {Q , 2} t _ {\mathrm {m i x}} \| \varepsilon \| _ {\infty} c _ {0}}{1 - \gamma} n ^ {1 / 2 - \gamma}. +$$ + +It remains to combine the above bounds. + +![](images/0e99555ac90eeb0fa3096801d6c921ff1142c2e87f338d1fad3882ce1975b06f.jpg) + +# B.3 Proof of Corollary 1 + +Proof. Let $\Phi(x)$ is the c.d.f. of the standard normal law $\mathcal{N}(0,1)$ , $\Phi_{\sigma}(x)$ is the c.d.f. of the normal law $\mathcal{N}(0,\sigma(u))$ and $\Phi_{\sigma_n}(x)$ is the c.d.f. of the normal law $\mathcal{N}(0,\sigma_n(u))$ . Then we have + +$$ +\begin{array}{l} \mathrm {d} _ {K} \left(\sqrt {n} u ^ {\top} \left(\bar {\theta} _ {n} - \theta^ {\star}\right) / \sigma (u)\right) = \sup _ {x \in \mathbb {R}} | \mathbb {P} (\sqrt {n} u ^ {\top} \left(\bar {\theta} _ {n} - \theta^ {\star}\right) / \sigma (u) \leq x) - \Phi (x) | \\ = \sup _ {x \in \mathbb {R}} \left| \mathbb {P} \left(\sqrt {n} u ^ {\top} \left(\bar {\theta} _ {n} - \theta^ {\star}\right) \leq x\right) - \Phi_ {\sigma} (x) \right| \\ \leq \sup _ {x \in \mathbb {R}} | \mathbb {P} (\sqrt {n} u ^ {\top} (\bar {\theta} _ {n} - \theta^ {\star}) \leq x) - \Phi_ {\sigma_ {n}} (x) | + \sup _ {x \in \mathbb {R}} | \Phi_ {\sigma} (x) - \Phi_ {\sigma_ {n}} (x) | \\ = \mathsf {d} _ {K} \left(\sqrt {n} u ^ {\top} \left(\bar {\theta} _ {n} - \theta^ {\star}\right) / \sigma_ {n} (u)\right) + \sup _ {x \in \mathbb {R}} | \Phi_ {\sigma} (x) - \Phi_ {\sigma_ {n}} (x) | \\ \end{array} +$$ + +Using Lemma 9 to bound $|\sigma^2(u) - \sigma_n^2(u)|$ and Lemma 10 to bound $\sigma_n^{-2}(u)$ we obtain + +$$ +| \sigma^ {2} (u) / \sigma_ {n} ^ {2} (u) - 1 | \leq \frac {C _ {\infty} ^ {\prime} C _ {\Sigma} ^ {2}}{n ^ {1 - \gamma}}. +$$ + +Setting + +$$ +C _ {\infty} = \frac {3 C _ {\infty} ^ {\prime}}{\lambda_ {\min} (\Sigma_ {\infty})}, \text {w h e r e} C _ {\infty} ^ {\prime} \text {i s d e f i n e d i n (3 1)} \tag {30} +$$ + +and applying the Gaussian comparison inequality (68) we conclude the proof. + +# B.4 Technical bounds + +Lemma 3. Assume A1, A2, and A3. Then for any $\ell \in \mathbb{N}$ it holds that + +$$ +\| Q _ {\ell} \| \leq \mathcal {L} _ {Q}, +$$ + +where + +$$ +\mathcal {L} _ {Q} = \kappa_ {Q} ^ {1 / 2} \big (c _ {0} + \frac {4}{a (1 - \gamma)} \big) +$$ + +and $\kappa_{Q}$ is defined in Proposition 2. + +Proof. Using the definition of $Q_{\ell}$ from (10) and Proposition 2, we get + +$$ +\begin{array}{l} \| Q _ {\ell} \| \leq \kappa_ {Q} ^ {1 / 2} \alpha_ {\ell} \sum_ {k = \ell} ^ {n - 1} \| G _ {\ell + 1: k} \| _ {Q} \leq \kappa_ {Q} ^ {1 / 2} \alpha_ {\ell} \sum_ {k = \ell} ^ {n - 1} \prod_ {j = \ell + 1} ^ {k} \sqrt {1 - a \alpha_ {j}} \\ \leq \kappa_ {Q} ^ {1 / 2} \alpha_ {\ell} \sum_ {k = \ell} ^ {n - 1} \prod_ {j = \ell + 1} ^ {k} (1 - (a / 2) \alpha_ {j}) \leq \mathcal {L} _ {Q}, \\ \end{array} +$$ + +where in the last bound we applied Lemma 23 with $b = a / 2$ . + +![](images/c3b746f1cb589bfdad11bc8d4ffba3df66393a159dfb599623dab7c4ab7dde48.jpg) + +Lemma 4. The following identity holds + +$$ +Q _ {\ell} - \bar {\mathbf {A}} ^ {- 1} = S _ {\ell} - \bar {\mathbf {A}} ^ {- 1} G _ {\ell : n - 1}, w h e r e S _ {\ell} = \sum_ {j = \ell + 1} ^ {n - 1} (\alpha_ {\ell} - \alpha_ {j}) G _ {\ell + 1: j - 1}, +$$ + +and + +$$ +\sum_ {i = 1} ^ {n - 1} \left(Q _ {i} - \bar {\mathbf {A}} ^ {- 1}\right) = - \bar {\mathbf {A}} ^ {- 1} \sum_ {j = 1} ^ {n - 1} G _ {1: j}. +$$ + +Proof. See [82, pp. 26-30]. + +![](images/893d09d0d57010ebb6f1a5caf133eb63f51d238e581219e603ff842c3ae93952.jpg) + +Lemma 5. Let $c_{0} \in (0, \alpha_{\infty}]$ and $\ell \in \mathbb{N}$ . Then under A1, A2, and A3, it holds + +$$ +\| S _ {\ell} \| \leq \sqrt {\kappa_ {Q}} \cdot C _ {\gamma , \beta} ^ {(S)} \cdot (\ell + k _ {0}) ^ {\gamma - 1}, +$$ + +where + +$$ +C _ {\gamma , a} ^ {(S)} = 2 c _ {0} \exp \left\{\frac {a c _ {0}}{2 k _ {0} ^ {\gamma}} \right\} \left(2 ^ {\gamma / (1 - \gamma)} \frac {2}{a c _ {0}} + (\frac {2}{a c _ {0}}) ^ {1 / (1 - \gamma)} \Gamma (\frac {1}{1 - \gamma})\right). +$$ + +Proof. For simplicity we define $m_i^j = \sum_{k = i}^{j}(k + k_0)^{-\gamma}$ and $\beta = a / 2$ . Note that + +$$ +\| \sum_ {j = i + 1} ^ {n - 1} (\alpha_ {i} - \alpha_ {j}) G _ {i + 1: j - 1} ^ {(\alpha)} \| \leq \sqrt {\kappa_ {Q}} \sum_ {j = i} ^ {n - 2} \frac {c _ {0}}{(j + k _ {0} + 1) ^ {\gamma}} \left(\left(\frac {j + k _ {0} + 1}{i + k _ {0}}\right) ^ {\gamma} - 1\right) \exp \{- \beta c _ {0} m _ {i + 1} ^ {j} \} +$$ + +Following the proof of [82, Lemma A.7], we have + +$$ +\left(\frac {j + k _ {0} + 1}{i + k _ {0}}\right) ^ {\gamma} - 1 \leq (i + k _ {0}) ^ {\gamma - 1} \left(1 + (1 - \gamma) m _ {i} ^ {j}\right) ^ {\gamma / (1 - \gamma)} +$$ + +Hence, we obtain + +$$ +\begin{array}{l} \frac {\left\| S _ {i} \right\|}{\sqrt {\kappa_ {Q}}} \leq c _ {0} (i + k _ {0}) ^ {\gamma - 1} \sum_ {i = i} ^ {n - 2} \frac {1}{(j + k _ {0} + 1) ^ {\gamma}} \left(1 + (1 - \gamma) m _ {i} ^ {j}\right) ^ {\gamma / (1 - \gamma)} \exp \{- \beta c _ {0} m _ {i + 1} ^ {j} \} \\ \leq c _ {0} (i + k _ {0}) ^ {\gamma - 1} \sum_ {j = i} ^ {n - 2} \frac {1}{(j + k _ {0}) ^ {\gamma}} \left(1 + (1 - \gamma) m _ {i} ^ {j}\right) ^ {\gamma / (1 - \gamma)} \exp \left\{\beta c _ {0} (k _ {0} + i) ^ {- \gamma} \right\} \exp \left\{- \beta c _ {0} m _ {i} ^ {j} \right\} \\ \leq c _ {0} \exp \{\frac {\beta c _ {0}}{k _ {0} ^ {\gamma}} \} (i + k _ {0}) ^ {\gamma - 1} \sum_ {j = i} ^ {n - 2} (m _ {i} ^ {j} - m _ {i} ^ {j - 1}) \bigg (1 + (1 - \gamma) m _ {i} ^ {j} \bigg) ^ {\gamma / (1 - \gamma)} \exp \{- \beta c _ {0} m _ {i} ^ {j} \} \\ \leq 2 c _ {0} \exp \left\{\frac {\beta c _ {0}}{k _ {0} ^ {\gamma}} \right\} (i + k _ {0}) ^ {\gamma - 1} \int_ {0} ^ {+ \infty} \left(1 + (1 - \gamma) m\right) ^ {\gamma / (1 - \gamma)} \exp \left\{- \beta c _ {0} m \right\} \mathrm {d} m \\ \leq 2 c _ {0} \exp \left\{\frac {\beta c _ {0}}{k _ {0} ^ {\gamma}} \right\} (i + k _ {0}) ^ {\gamma - 1} \left(2 ^ {\gamma / (1 - \gamma)} \frac {1}{\beta c _ {0}} + (\frac {1}{\beta c _ {0}}) ^ {1 / (1 - \gamma)} \Gamma (\frac {1}{1 - \gamma})\right). \\ \end{array} +$$ + +![](images/211a2ddb2e74dbf3cc0a98dd14641d86263df99de160674b1f860898a89d9420.jpg) + +Lemma 6. Let $c_{0} \in (0, \alpha_{\infty}]$ and $\ell \in \mathbb{N}$ . Then under A1, A2, and A3, it holds + +$$ +\left\| Q _ {\ell + 1} - Q _ {\ell} \right\| \leq \mathcal {L} _ {Q, 2} \cdot \alpha_ {\ell + 1}, +$$ + +where + +$$ +\mathcal {L} _ {Q, 2} = \sqrt {\kappa_ {Q}} \left(2 ^ {\gamma} + (2 \mathrm {C} _ {\mathbf {A}} + a / 4) \left(c _ {0} + \frac {4}{a (1 - \gamma)}\right)\right). +$$ + +Proof. Using the definition of $Q_{\ell}$ we have + +$$ +\begin{array}{l} Q _ {\ell + 1} - Q _ {\ell} = \alpha_ {\ell + 1} \sum_ {k = l + 1} ^ {n - 1} G _ {\ell + 2: k} - \alpha_ {\ell} \sum_ {k = l} ^ {n - 1} G _ {\ell + 1: k} = \alpha_ {\ell + 1} \sum_ {k = l + 1} ^ {n - 1} G _ {\ell + 2: k + 1} - \alpha_ {\ell} \sum_ {k = l + 1} ^ {n - 1} G _ {\ell + 1: k} - \alpha_ {\ell} \\ = - \alpha_ {\ell} + \sum_ {k = l + 1} ^ {n - 1} G _ {l + 2: k} \left(\alpha_ {\ell + 1} I - \alpha_ {\ell} I + \alpha_ {\ell + 1} \alpha_ {\ell} \bar {\mathbf {A}}\right). \\ \end{array} +$$ + +Hence, using Lemma 27 with $r = a / 4$ and Lemma 3 we get + +$$ +\left\| Q _ {\ell + 1} - Q _ {\ell} \right\| \leq \alpha_ {\ell} + \sum_ {k = l + 1} ^ {n - 1} \alpha_ {\ell + 1} ^ {2} \left\| G _ {l + 2: k} \right\| \left(2 \mathrm {C} _ {\mathbf {A}} + a / 4\right) \leq \left(2 ^ {\gamma} + \mathcal {L} _ {Q} \left(2 \mathrm {C} _ {\mathbf {A}} + a / 4\right)\right) \alpha_ {\ell + 1}. +$$ + +![](images/f6a628015d07b4dded12c3e9cf5a98154edb74277a5015f55e2085b96dfbc90d.jpg) + +Lemma 7. Let $c_{0} \in (0, \alpha_{\infty}], k_{0}^{1 - \gamma} > bc_{0}$ and $\ell \in \mathbb{N}$ . Then under A1, A2, and A3 it holds + +$$ +\left\| S _ {\ell + 1} - S _ {\ell} \right\| \leq \sqrt {\kappa_ {Q}} \cdot C _ {\gamma , a} ^ {(S, 2)} \cdot \alpha_ {\ell + 1}, +$$ + +where + +$$ +C _ {\gamma , a} ^ {(S; 2)} = \left(c _ {0} + \frac {4}{a (1 - \gamma)}\right) \left(a / 2 + 3 \mathrm {C} _ {\mathbf {A}}\right). +$$ + +Proof. Using Lemma 4 we obtain + +$$ +\begin{array}{l} S _ {\ell + 1} - S _ {\ell} = Q _ {\ell + 1} - Q _ {\ell} + \bar {\mathbf {A}} ^ {- 1} \left(G _ {\ell + 1: n - 1} - G _ {\ell : n - 1}\right) \\ = \alpha_ {\ell + 1} \sum_ {k = l + 1} ^ {n - 1} G _ {\ell + 2: k} - \alpha_ {\ell} \sum_ {k = l} ^ {n - 1} G _ {\ell + 1: k} + \bar {\mathbf {A}} ^ {- 1} (I - (I - \alpha_ {\ell} \bar {\mathbf {A}})) G _ {\ell + 1: n - 1} \\ = \alpha_ {\ell + 1} \sum_ {k = l} ^ {n - 2} G _ {\ell + 2: k + 1} - \alpha_ {\ell} \sum_ {k = l} ^ {n - 2} G _ {\ell + 1: k} \\ = (\alpha_ {\ell + 1} - \alpha_ {\ell}) I + \sum_ {k = l + 1} ^ {n - 2} (\alpha_ {\ell + 1} (I - \alpha_ {k + 1} \bar {\mathbf {A}}) - \alpha_ {\ell} (I - \alpha_ {\ell + 1} \bar {\mathbf {A}})) G _ {l + 2: k}. \\ \end{array} +$$ + +Hence, using Lemma 27 with $r = a / 4$ we get + +$$ +\begin{array}{l} \| S _ {\ell + 1} - S _ {\ell} \| \leq (a / 4) \alpha_ {\ell + 1} ^ {2} + \sum_ {k = l + 1} ^ {n - 2} ((a / 4) \alpha_ {\ell + 1} ^ {2} + \mathrm {C _ {A}} \alpha_ {\ell + 1} ^ {2} + 2 ^ {\gamma} \mathrm {C _ {A}} \alpha_ {\ell + 1} ^ {2}) \| G _ {l + 2: k} \| \\ \leq \left(\mathcal {L} _ {Q} \left(\left(a / 4\right) + \mathrm {C} _ {\mathbf {A}} + 2 ^ {\gamma} \mathrm {C} _ {\mathbf {A}}\right) + (a / 4)\right) \alpha_ {\ell + 1}, \\ \end{array} +$$ + +where in the last inequality we applying lemma 3. Since $\mathcal{L}_Q\geq 1$ and $2^{\gamma} < 2$ we complete the proof. + +Lemma 8. Let $c_{0} \in (0, \alpha_{\infty}]$ . Then under A2 for any $m \in \mathbb{N}$ it holds + +$$ +\sum_ {t = 1} ^ {n - 1} \| G _ {t: n - 1} \| ^ {m} \leq \frac {\kappa_ {Q} ^ {m / 2}}{1 - (1 - c _ {0} (a / 2) (n + k _ {0} - 2) ^ {- \gamma}) ^ {m}} +$$ + +Proof. Note that + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {n - 1} \left\| G _ {t: n - 1} \right\| ^ {m} \leq \kappa_ {Q} ^ {m / 2} \sum_ {l = 1} ^ {n - 1} \prod_ {i = t} ^ {n - 1} (1 - (a / 2) \alpha_ {i}) \\ = \frac {\kappa_ {Q} ^ {m / 2}}{(1 - (1 - (a / 2) \alpha_ {n - 2}) ^ {m})} \sum_ {l = 1} ^ {n - 1} (1 - (1 - (a / 2) \alpha_ {t - 1}) ^ {m}) \prod_ {i = t} ^ {n - 1} (1 - (a / 2) \alpha_ {i}) ^ {m} \\ \leq \frac {\kappa_ {Q} ^ {m / 2}}{(1 - (1 - (a / 2) \alpha_ {n - 2}) ^ {m})}. \\ \end{array} +$$ + +![](images/2c897d6d3e29e2cbb776140d3aef1dc836b76e86466257e54a89ea37924eea53.jpg) + +Lemma 9. Let $c_{0} \in (0, \alpha_{\infty}]$ . Then under A1, A2, and A3 it holds + +$$ +\left| \sigma_ {n} ^ {2} (u) - \sigma^ {2} (u) \right| \leq C _ {\infty} ^ {\prime} n ^ {\gamma - 1}, +$$ + +where + +$$ +C _ {\infty} ^ {\prime} = \left\| \Sigma_ {\varepsilon} \right\| \frac {\kappa_ {Q} \left(C _ {\gamma , a} ^ {(S)}\right) ^ {2}}{2 \gamma - 1} + \left\| \Sigma_ {\infty} \right\| \left(\frac {2 ^ {\gamma + 1} 5 + 3}{a c _ {0} (1 - \gamma)} k _ {0} ^ {\gamma} + \frac {4 C _ {A} \kappa_ {Q} C _ {\gamma , a} ^ {(S)}}{a c _ {0}} k _ {0} ^ {2 \gamma - 1}\right) \tag {31} +$$ + +Proof. Note that + +$$ +\left| \sigma_ {n} ^ {2} (u) - \sigma^ {2} (u) \right| \leq \left\| \Sigma_ {n} - \Sigma_ {\infty} \right\|. +$$ + +Then, we express the $\Sigma_{n} - \Sigma_{\infty}$ in the following form: + +$$ +\begin{array}{l} \Sigma_ {n} - \Sigma_ {\infty} = \underbrace {\frac {1}{n} \sum_ {t = 2} ^ {n - 1} (Q _ {t} - \bar {\mathbf {A}} ^ {- 1}) \Sigma_ {\varepsilon} \bar {\mathbf {A}} ^ {- \top} + \frac {1}{n} \sum_ {t = 2} ^ {n - 1} \bar {\mathbf {A}} ^ {- 1} \Sigma_ {\varepsilon} (Q _ {t} - \bar {\mathbf {A}} ^ {- 1}) ^ {\top}} _ {R _ {1}} \\ + \underbrace {\frac {1}{n} \sum_ {t = 2} ^ {n - 1} (Q _ {t} - \bar {\mathbf {A}} ^ {- 1}) \Sigma_ {\varepsilon} (Q _ {t} - \bar {\mathbf {A}} ^ {- 1}) ^ {\top}} _ {R _ {2}} - \frac {2}{n} \Sigma_ {\infty}. \\ \end{array} +$$ + +First, we bound $R_{1}$ , using Lemma 23 we obtain + +$$ +\begin{array}{l} \| \frac {1}{n} \sum_ {t = 2} ^ {n - 1} (Q _ {t} - \bar {\mathbf {A}} ^ {- 1}) \Sigma_ {\varepsilon} \bar {\mathbf {A}} ^ {- \top} \| = \| \frac {1}{n} \bar {\mathbf {A}} ^ {- 1} \sum_ {j = 2} ^ {n - 1} G _ {1: j} \Sigma_ {\varepsilon} \bar {\mathbf {A}} ^ {- \top} \| \\ \leq \| n ^ {- 1} \Sigma_ {\infty} \sum_ {j = 2} ^ {n - 1} G _ {1: j} \| \leq n ^ {- 1} \| \Sigma_ {\infty} \| \cdot \sum_ {j = 1} ^ {n - 1} \| G _ {1: j} \| \\ \leq n ^ {- 1} \| \Sigma_ {\infty} \| (1 + \frac {4}{a c _ {0} (1 - \gamma)}) (1 + k _ {0}) ^ {\gamma} \\ \leq 2 ^ {\gamma} n ^ {\gamma - 1} \| \Sigma_ {\infty} \| \left(1 + \frac {4}{a c _ {0} (1 - \gamma)}\right) k _ {0} ^ {\gamma}. \\ \end{array} +$$ + +Hence, we get that + +$$ +\left\| R _ {1} \right\| \leq 2 ^ {\gamma + 1} n ^ {\gamma - 1} \left\| \Sigma_ {\infty} \right\| \left(1 + \frac {4}{a c _ {0} (1 - \gamma)}\right) k _ {0} ^ {\gamma}. +$$ + +Now, we rewrite term $R_{2}$ as follows: + +$$ +\begin{array}{l} n ^ {- 1} \sum_ {t = 2} ^ {n - 1} (Q _ {t} - \bar {\mathbf {A}} ^ {- 1}) \Sigma_ {\varepsilon} (Q _ {t} - \bar {\mathbf {A}} ^ {- 1}) ^ {\top} \\ = n ^ {- 1} \sum_ {t = 2} ^ {n - 1} \left(S _ {t} - \bar {\mathbf {A}} ^ {- 1} \prod_ {k = t} ^ {n - 1} (\mathrm {I} - \alpha_ {k} \bar {\mathbf {A}})\right) \Sigma_ {\varepsilon} \left(S _ {t} - \bar {\mathbf {A}} ^ {- 1} \prod_ {k = t} ^ {n - 1} (\mathrm {I} - \alpha_ {k} \bar {\mathbf {A}})\right) ^ {\top} \\ = \underbrace {n ^ {- 1} \sum_ {t = 2} ^ {n - 1} S _ {t} \Sigma_ {\varepsilon} S _ {t} ^ {\top}} _ {R _ {2 1}} + \underbrace {n ^ {- 1} \sum_ {t = 2} ^ {n - 1} \bar {\mathbf {A}} ^ {- 1} \prod_ {k = t} ^ {n - 1} (\mathrm {I} - \alpha_ {k} \bar {\mathbf {A}}) \Sigma_ {\varepsilon} \bar {\mathbf {A}} ^ {- \top} \prod_ {k = t} ^ {n - 1} (\mathrm {I} - \alpha_ {k} \bar {\mathbf {A}}) ^ {\top}} _ {R _ {2 2}} \\ - \underbrace {n ^ {- 1} \sum_ {t = 2} ^ {n - 1} \bar {\mathbf {A}} ^ {- 1} \prod_ {k = t} ^ {n - 1} (\mathrm {I} - \alpha_ {k} \bar {\mathbf {A}}) \cdot \Sigma_ {\varepsilon} S _ {t} ^ {\top}} _ {R _ {2 3}} - \underbrace {n ^ {- 1} \sum_ {t = 2} ^ {n - 1} S _ {t} \Sigma_ {\varepsilon} \bar {\mathbf {A}} ^ {- \top} \prod_ {k = t} ^ {n - 1} (\mathrm {I} - \alpha_ {k} \bar {\mathbf {A}}) ^ {\top}} _ {R _ {2 4}}. \\ \end{array} +$$ + +To bound $\| R_{21}\|$ we use Lemma 5 and obtain + +$$ +\begin{array}{l} \| R _ {2 1} \| = \| n ^ {- 1} \sum_ {t = 2} ^ {n - 1} S _ {t} \Sigma_ {\varepsilon} S _ {t} ^ {\top} \| \leq n ^ {- 1} \sum_ {t = 2} ^ {n - 1} \| \Sigma_ {\varepsilon} \| \| S _ {t} \| ^ {2} \\ \leq n ^ {- 1} \| \Sigma_ {\varepsilon} \| \sum_ {t = 2} ^ {n - 1} \kappa_ {Q} \left(C _ {\gamma , a} ^ {(S)}\right) ^ {2} (t + k _ {0}) ^ {2 (\gamma - 1)} \\ \leq n ^ {- 1} \| \Sigma_ {\varepsilon} \| \kappa_ {Q} \left(C _ {\gamma , a} ^ {(S)}\right) ^ {2} \frac {(n + k _ {0} - 1) ^ {2 \gamma - 1} - (k _ {0} + 1) ^ {2 \gamma - 1}}{2 \gamma - 1} \\ \leq n ^ {2 (\gamma - 1)} \frac {\| \Sigma_ {\varepsilon} \| \kappa_ {Q} \left(C _ {\gamma , a} ^ {(S)}\right) ^ {2}}{2 \gamma - 1}, \\ \end{array} +$$ + +where the last inequality holds since $(n + k_0 - 1)^{2\gamma -1}\leq n^{2\gamma -1} + (k_0 + 1)^{2\gamma -1}$ for $\gamma \in (1 / 2,1)$ . The bound for $R_{22}$ follows from Lemma 8 and simple inequality $(n + k_0 - 2)^\gamma \leq (k_0n)^\gamma$ + +$$ +\begin{array}{l} \| R _ {2 2} \| = \| n ^ {- 1} \sum_ {i = 2} ^ {n - 1} G _ {i: n - 1} \bar {\mathbf {A}} ^ {- 1} \Sigma_ {\varepsilon} \bar {\mathbf {A}} ^ {- \top} G _ {i: n - 1} ^ {\top} \| \leq n ^ {- 1} \| \Sigma_ {\infty} \| \sum_ {i = 1} ^ {n - 1} \| G _ {i: n - 1} \| ^ {2} \\ \leq n ^ {- 1} \frac {\| \Sigma_ {\infty} \|}{c _ {0} a (n + k _ {0} - 2) ^ {- \gamma} - c _ {0} ^ {2} (a ^ {2} / 4) (n + k _ {0} - 2) ^ {- 2 \gamma}} \leq 2 \| \Sigma_ {\infty} \| k _ {0} ^ {\gamma} \frac {n ^ {\gamma - 1}}{c _ {0} a}. \\ \end{array} +$$ + +Since $R_{23} = R_{24}^{\top}$ , we concentrate on $\| D_{24} \|$ . Lemma 8 immediately imply + +$$ +\begin{array}{l} \| R _ {2 4} \| \leq n ^ {- 1} \| \Sigma_ {\varepsilon} \bar {\mathbf {A}} ^ {- \top} \| \sum_ {i = 1} ^ {n - 1} \| S _ {i} \| \| G _ {i: n - 1} \| \\ \leq n ^ {- 1} \| \bar {\mathbf {A}} \Sigma_ {\infty} \| \kappa_ {Q} C _ {\gamma , a} ^ {(S)} \sum_ {i = 1} ^ {n - 1} (i + k _ {0}) ^ {\gamma - 1} \prod_ {k = i} ^ {n - 1} (1 - \frac {a c _ {0}}{2 (k + k _ {0}) ^ {\gamma}}) \\ \leq n ^ {- 1} \| \bar {\mathbf {A}} \Sigma_ {\infty} \| \kappa_ {Q} C _ {\gamma , a} ^ {(S)} \sum_ {i = 1} ^ {n - 1} (i + k _ {0}) ^ {2 \gamma - 1} (i + k _ {0}) ^ {- \gamma} \prod_ {k = i + 1} ^ {n - 1} (1 - \frac {a c _ {0}}{2 (k + k _ {0}) ^ {\gamma}}) \\ \leq \mathrm {C} _ {\mathbf {A}} \| \Sigma_ {\infty} \| \kappa_ {Q} C _ {\gamma , a} ^ {(S)} k _ {0} ^ {2 \gamma - 1} \frac {2 n ^ {2 (\gamma - 1)}}{a c _ {0}} \\ \end{array} +$$ + +By combining all the inequalities, we complete the proof. + +Lemma 10. Let $c_{0} \in (0, \alpha_{\infty}]$ and $C_{\infty}'n^{\gamma - 1} \leq \lambda_{\min}(\Sigma_{\infty}) / 2$ . Then under A1, A2, and A3 it holds + +$$ +\sigma_ {n} ^ {- 1} (u) \leq C _ {\Sigma}, +$$ + +where $C_{\Sigma} = \sqrt{\frac{2}{\lambda_{\min}(\Sigma_{\infty})}}$ + +Proof. Note that $\sigma_n^2 (u) > \lambda_{\min}(\Sigma_n)$ . Using Lidskii's inequality, we obtain + +$$ +\lambda_ {\min } (\Sigma_ {n}) \geq \lambda_ {\min } (\Sigma_ {\infty}) - \| \Sigma_ {n} - \Sigma_ {\infty} \| \geq \lambda_ {\min } (\Sigma_ {\infty}) / 2, +$$ + +where in the last inequality we use $C_{\infty}^{\prime}n^{\gamma -1}\leq \lambda_{\min}(\Sigma_{\infty}) / 2$ + +![](images/145fb1fa7f9a1bba8ebb10f7b84f145b9193ad8893bdd78fcc2ae1d9865aabbd.jpg) + +# C Last iterate bound + +The main result of this section is the following bound on the $p$ -th moment of the LSA iterate $\theta_{k} - \theta^{\star}$ . + +Proposition 8. Assume $A1, A2$ , and $A3$ . Then for any $2 \leq p \leq \log n$ , $k \geq 1$ , $u \in \mathbb{S}_{d-1}$ , and any initial distribution $\xi$ on $(\mathsf{Z}, \mathcal{Z})$ , it holds that + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} \left(\theta_ {k} - \theta^ {\star}\right) \right| ^ {p} \right] \lesssim \mathrm {D} _ {1} ^ {(\mathrm {M})} t _ {\operatorname {m i x}} \sqrt {p \alpha_ {k}} + C _ {\Gamma} d ^ {1 / \log n} \exp \left\{- (a / 1 2) \sum_ {\ell = 1} ^ {k} \alpha_ {\ell} \right\} \| \theta_ {0} - \theta^ {\star} \|, \tag {32} +$$ + +where the constant $\mathsf{D}_1^{(\mathsf{M})}$ is given by + +$$ +\mathsf {D} _ {1} ^ {\left(\mathrm {M}\right)} = d ^ {1 / 2 + 1 / \log n} \left(C _ {\Gamma} \mathrm {C} _ {\mathbf {A}} \mathsf {D} _ {2} ^ {\left(\mathrm {M}\right)} / a\right) \| \varepsilon \| _ {\infty} + \| \varepsilon \| _ {\infty} \left(\kappa_ {Q} / a\right) ^ {1 / 2} \left(3 + 4 \mathrm {C} _ {\mathbf {A}} / a\right), +$$ + +and $\mathsf{D}_2^{(\mathsf{M})}$ is defined in (37). + +We provide the result above only for 1-dimensional projections of the error $u^{\top}(\theta_n - \theta^{\star})$ , as previously considered in [47]. Note that the scaling of the right-hand side of (32) with a $t_{\mathrm{mix}}$ factor can be suboptimal. However, this scaling corresponds to the second-order (in $n$ ) terms. Tighter analysis of this term using the Rosenthal-type inequality should reveal a $\sqrt{t_{\mathrm{mix}}}$ dependence of the leading term, however, we leave this improvement for future work. + +Our proof of the last iterate bound is based on the perturbation-expansion framework [2], see also [24]. Within this framework, we expand the error recurrence (4), using the notation $\Gamma_{m:k}^{(\alpha)}$ for the product of random matrices: + +$$ +\Gamma_ {m: k} ^ {(\alpha)} = \prod_ {i = m} ^ {k} \left(\mathrm {I} - \alpha_ {i} \mathbf {A} \left(Z _ {i}\right)\right), \quad m, k \in \mathbb {N}, \quad m \leq k, \tag {33} +$$ + +with the convention that $\Gamma_{m:k}^{(\alpha)} = \mathrm{I}$ for $m > k$ . Using the recurrence (4), we arrive at the following decomposition of the LSA error: + +$$ +u ^ {\top} \left(\theta_ {k} - \theta^ {\star}\right) = u ^ {\top} \tilde {\theta} _ {k} ^ {\left(\operatorname {t r}\right)} + u ^ {\top} \tilde {\theta} _ {k} ^ {\left(\operatorname {f l}\right)}, \tag {34} +$$ + +where we have defined + +$$ +\tilde {\theta} _ {k} ^ {(\mathrm {t r})} = u ^ {\top} \Gamma_ {1: k} ^ {(\alpha)} \left\{\theta_ {0} - \theta^ {\star} \right\}, \quad \tilde {\theta} _ {k} ^ {(\mathrm {f l})} = - \sum_ {j = 1} ^ {k} u ^ {\top} \Gamma_ {j + 1: k} ^ {(\alpha)} \alpha_ {j} \varepsilon (Z _ {j}). +$$ + +Here $\tilde{\theta}_k^{(\mathrm{tr})}$ is a transient term (reflecting the forgetting of the initial condition) and $\tilde{\theta}_k^{(\mathrm{fl})}$ is a fluctuation term (reflecting misadjustment noise). We treat the $\tilde{\theta}_k^{(\mathrm{tr})}$ and $\tilde{\theta}_k^{(\mathrm{fl})}$ terms separately. In particular, we control $\tilde{\theta}_k^{(\mathrm{tr})}$ using Proposition 12. For estimating $\tilde{\theta}_k^{(\mathrm{fl})}$ we use the decomposition + +$$ +u ^ {\top} \tilde {\theta} _ {k} ^ {(f l)} = u ^ {\top} J _ {k} ^ {(0)} + u ^ {\top} H _ {k} ^ {(0)}, \tag {35} +$$ + +where the latter terms are defined by the following pair of recursions + +$$ +J _ {k} ^ {(0)} = \left(\mathrm {I} - \alpha_ {k} \bar {\mathbf {A}}\right) J _ {k - 1} ^ {(0)} - \alpha_ {k} \varepsilon (Z _ {k}), \quad J _ {0} ^ {(0)} = 0, +$$ + +$$ +H _ {k} ^ {(0)} = \left(\mathrm {I} - \alpha_ {k} \mathbf {A} (Z _ {k})\right) H _ {k - 1} ^ {(0)} - \alpha_ {k} \tilde {\mathbf {A}} (Z _ {k}) J _ {k - 1} ^ {(0)}, \quad H _ {0} ^ {(0)} = 0. +$$ + +For notation convenience we introduce, for $m \leq k$ , the deterministic counterpart of the product of random matrices (33), that is, + +$$ +G _ {m: k} = \prod_ {i = m} ^ {k} \left(\mathrm {I} - \alpha_ {i} \bar {\mathbf {A}}\right), +$$ + +keeping the convention $G_{m:k} = \mathrm{I}$ if $m > k$ . Thus we obtain that + +$$ +J _ {k} ^ {(0)} = - \sum_ {j = 1} ^ {k} \alpha_ {j} G _ {j + 1: k} \varepsilon \left(Z _ {j}\right), \quad H _ {k} ^ {(0)} = - \sum_ {j = 1} ^ {k} \alpha_ {j} \Gamma_ {j + 1: k} \tilde {\mathbf {A}} \left(Z _ {j}\right) J _ {j - 1} ^ {(0)}, \tag {36} +$$ + +and we analyze these terms above separately. We first bound the term $J_{n}^{(0)}$ : + +Proposition 9. Assume $A$ , $A$ , and $A$ . Let $k_0 \geq \left\{\frac{16\gamma}{ac_0}\right\}^{1/1 - \gamma}$ . Then, for any $p \geq 2$ , initial probability measure $\xi$ on $(\mathsf{Z}, \mathcal{Z})$ , and $k \geq 1$ , it holds that + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} J _ {k} ^ {(0)} \right| ^ {p} \right] \leq \mathrm {D} _ {2} ^ {(\mathrm {M})} t _ {\operatorname {m i x}} \sqrt {p \alpha_ {k}}, +$$ + +where + +$$ +\mathrm {D} _ {2} ^ {\left(\mathrm {M}\right)} = \left(3 2 / 3\right) \| \varepsilon \| _ {\infty} \left(\kappa_ {Q} / a\right) ^ {1 / 2} \left(3 + 4 \mathrm {C} _ {\mathbf {A}} / a\right). \tag {37} +$$ + +Proof. Note that $J_{k}^{(0)}$ is an additive functional of $\{\varepsilon(Z_{j})\}_{j=1}^{k}$ . Using the representation (36), we obtain that + +$$ +u ^ {\top} J _ {k} ^ {(0)} = - \sum_ {j = 1} ^ {k} \alpha_ {j} u ^ {\top} G _ {j + 1: k} \varepsilon \left(Z _ {j}\right). +$$ + +Applying Proposition 2, we get that + +$$ +\| G _ {j + 1: k} \| \leq \kappa_ {Q} ^ {1 / 2} \prod_ {\ell = j + 1} ^ {k} \sqrt {1 - a \alpha_ {\ell}}. +$$ + +Using Lemma 20 with $A_{j} = \alpha_{j}G_{j + 1:k}$ , we obtain that + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ | u ^ {T} J _ {k} ^ {(0)} | ^ {p} \right] \leq (1 6 / 3) p ^ {1 / 2} t _ {\operatorname {m i x}} \| \varepsilon \| _ {\infty} \left(\sum_ {j = 2} ^ {k} \alpha_ {j} ^ {2} \| G _ {j + 1: k} \| ^ {2}\right) ^ {1 / 2} \\ + (8 / 3) t _ {\mathrm {m i x}} (\alpha_ {1} \| G _ {2: k} \| + \alpha_ {k} + \sum_ {j = 1} ^ {k - 1} \| \alpha_ {j} G _ {j + 1: k} - \alpha_ {j - 1} G _ {j: k} \|) \| \varepsilon \| _ {\infty}. \\ \end{array} +$$ + +Using Lemma 28 with $b = a$ , we get + +$$ +\sum_ {j = 1} ^ {k} \alpha_ {j} ^ {2} \| G _ {j + 1: k} \| ^ {2} \leq \kappa_ {Q} \sum_ {j = 1} ^ {k} \alpha_ {j} ^ {2} \prod_ {\ell = j + 1} ^ {k} (1 - a \alpha_ {\ell}) \leq 4 (\kappa_ {Q} / a) \alpha_ {k}. +$$ + +Applying Lemma 26 with $b = a / 2$ , we get $\alpha_{1}\| G_{2:k}\| \leq \alpha_{k}$ . Moreover, using Lemma 27 with $r = a / 4$ , we get that + +$$ +\left\| \alpha_ {j} G _ {j + 1: k} - \alpha_ {j - 1} G _ {j: k} \right\| = \left\| G _ {j + 1: k} \left(\alpha_ {j} I - \alpha_ {j - 1} I + \alpha_ {j - 1} \alpha_ {j} \bar {A}\right) \right\| \leq \alpha_ {j} ^ {2} \left\| G _ {j + 1: k} \right\| \left(2 C _ {A} + a / 4\right). +$$ + +Then, applying Lemma 28 with $b = a / 2$ , we get + +$$ +\begin{array}{l} \alpha_ {1} \| G _ {2: k} \| + \alpha_ {k} + \sum_ {j = 2} ^ {k} \| \alpha_ {j} G _ {j + 1: k} - \alpha_ {j - 1} G _ {j: k} \| \leq 2 \alpha_ {k} + \kappa_ {Q} ^ {1 / 2} (a / 4 + 2 \mathrm {C} _ {\mathbf {A}}) \sum_ {j = 1} ^ {k} \alpha_ {j} ^ {2} \prod_ {l = j + 1} ^ {k} (1 - a \alpha_ {\ell} / 2) \\ \leq \alpha_ {k} \left(2 + 8 \kappa_ {Q} ^ {1 / 2} \left(a / 4 + 2 \mathrm {C} _ {\mathbf {A}}\right) / a\right). \\ \end{array} +$$ + +Combining the above results and using that $p \geq 2$ yields (37). + +# C.1 Proof of Proposition 8 + +Proceeding as in (34) and (35), we obtain that + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} (\theta_ {k} - \theta^ {\star}) \right| ^ {p} \right] \leq \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} \Gamma_ {1: k} ^ {(\alpha)} \{\theta_ {0} - \theta^ {\star} \} \right| ^ {p} \right] + \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} J _ {k} ^ {(0)} \right| ^ {p} \right] + \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} H _ {k} ^ {(0)} \right| ^ {p} \right]. +$$ + +The first two terms are bounded using Proposition 12 and Proposition 9, respectively. In order to bound the term, corresponding to $H_{k}^{(0)}$ , we apply Minkowski's inequality together with Hölder's inequality to (36): + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ | u ^ {\top} H _ {k} ^ {(0)} | ^ {p} \right] \leq \sum_ {j = 1} ^ {k} \alpha_ {j} \mathrm {C} _ {\mathbf {A}} \left\{\mathbb {E} _ {\xi} \left[ \| \Gamma_ {j + 1: k} ^ {(\alpha)} \| ^ {2 p} \right] \right\} ^ {1 / 2 p} \left\{\mathbb {E} _ {\xi} \left[ | J _ {j - 1} ^ {(0)} | ^ {2 p} \right] \right\} ^ {1 / 2 p}. +$$ + +Note that + +$$ +\begin{array}{l} \left\{\mathbb {E} _ {\xi} \left[ \left\| J _ {j - 1} ^ {(0)} \right\| ^ {2 p} \right] \right\} ^ {1 / p} = \left\{\mathbb {E} _ {\xi} \left[ \left(\sum_ {r = 1} ^ {d} \left| e _ {r} ^ {T} J _ {j - 1} ^ {(0)} \right| ^ {2}\right) ^ {p} \right] \right\} ^ {1 / p} \tag {38} \\ \leq \sum_ {r = 1} ^ {d} \left\{\mathbb {E} _ {\xi} \left[ \left| e _ {r} ^ {T} J _ {j - 1} ^ {(0)} \right| ^ {2 p} \right] \right\} ^ {1 / p}. \\ \end{array} +$$ + +Using Proposition 9 and proposition 12 with a simple inequality $\mathrm{e}^{-x} \leq 1 - x / 2$ , valid for $x \in [0,1]$ , we get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} H _ {k} ^ {(0)} \right| ^ {p} \right] \lesssim C _ {\Gamma} \mathrm {C} _ {\mathbf {A}} \mathrm {D} _ {2} ^ {(\mathrm {M})} \| \varepsilon \| _ {\infty} t _ {\operatorname {m i x}} \sqrt {p} d ^ {1 / 2 + 1 / \log n} \sum_ {j = 1} ^ {k} \alpha_ {j} ^ {3 / 2} \prod_ {\ell = j + 1} ^ {k} (1 - a \alpha_ {\ell} / 2 4) \\ \lesssim d ^ {1 / 2 + 1 / \log n} \left(C _ {\Gamma} \mathrm {C} _ {\mathbf {A}} \mathrm {D} _ {2} ^ {(\mathrm {M})} / a\right) \| \varepsilon \| _ {\infty} t _ {\operatorname {m i x}} \sqrt {p \alpha_ {k}}, \\ \end{array} +$$ + +where in the last inequality we use Lemma 28. + +# C.2 Proof of $J_{n}^{(1)}$ bound + +In order to proceed further, we need to obtain the tighter bound on $H_{k}^{(0)}$ , which requires a tighter moment bound on the quality $J_{k}^{(1)}$ . + +Proposition 10. Assume $A1$ , $A2$ , and $A3$ . Then, for any $p \geq 2$ , initial probability measure $\xi$ on $(Z, \mathcal{Z})$ , and $k \geq 1$ , it holds that + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \big [ \big | u ^ {\top} H _ {k} ^ {(0)} \big | ^ {p} \big ] \lesssim \mathsf {D} _ {3} ^ {(\mathrm {M})} t _ {\mathrm {m i x}} p ^ {2} \alpha_ {k} \sqrt {\log \left(1 / \alpha_ {k}\right)}, +$$ + +where + +$$ +\mathsf {D} _ {3} ^ {\mathrm {(M)}} = \mathsf {D} _ {4} ^ {\mathrm {(M)}} (1 + \frac {d ^ {1 / 2 + 1 / \log n} C _ {\Gamma} \mathrm {C} _ {\mathbf {A}}}{a}), +$$ + +and the constant $\mathrm{D}_4^{(\mathrm{M})}$ is defined in (42). + +Proof. It is known (see e.g. [2] and [24]), that the term $H_{k}^{(0)}$ can be further decomposed as follows: + +$$ +H _ {k} ^ {(0)} = \sum_ {\ell = 1} ^ {L} J _ {k} ^ {(\ell)} + H _ {k} ^ {(L)}. \tag {39} +$$ + +Here the parameter $L \geq 1$ controls the depth of expansion, and the terms $J_{k}^{(\ell)}$ and $H_{k}^{(\ell)}$ are given by the following recurrences: + +$$ +J _ {k} ^ {(\ell)} = \left(I - \alpha_ {k} \bar {\mathbf {A}}\right) J _ {k - 1} ^ {(\ell)} - \alpha_ {k} \tilde {\mathbf {A}} \left(Z _ {k}\right) J _ {k - 1} ^ {(\ell - 1)}, \quad J _ {0} ^ {(\ell)} = 0, \tag {40} +$$ + +$$ +H _ {k} ^ {(\ell)} = \left(\mathrm {I} - \alpha_ {k} \mathbf {A} (Z _ {k})\right) H _ {k - 1} ^ {(\ell)} - \alpha_ {k} \tilde {\mathbf {A}} (Z _ {k}) J _ {k - 1} ^ {(\ell)}, H _ {0} ^ {(\ell)} = 0. +$$ + +The expansion depth $L$ here controls the desired approximation accuracy. For our further results it is enough to take $L = 1$ and estimate the respective terms $J_{k}^{(1)}$ and $H_{k}^{(1)}$ . Now the rest of the proof follows from Proposition 11 (see the bounds (41) and (43) for $\mathbb{E}_{\xi}^{1/p}\left[\left|u^{\top}J_{k}^{(1)}\right|^{p}\right]$ and $\mathbb{E}_{\xi}^{1/p}\left[\left|u^{\top}H_{k}^{(1)}\right|^{p}\right]$ , respectively), and Minkoski's inequality. + +Proposition 11. Assume $A1, A2$ , and $A3$ . Then, for any $p \geq 2$ , initial probability measure $\xi$ on $(Z, \mathcal{Z})$ , and $k \geq 1$ , it holds that + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} J _ {k} ^ {(1)} \right| ^ {p} \right] \lesssim \mathrm {D} _ {4} ^ {(\mathrm {M})} t _ {\operatorname {m i x}} p ^ {2} \alpha_ {k} \sqrt {\log \left(1 / \alpha_ {k}\right)}, \tag {41} +$$ + +where + +$$ +\mathrm {D} _ {4} ^ {\left(\mathrm {M}\right)} = \frac {\kappa_ {Q} ^ {3 / 2} \mathrm {C} _ {\mathbf {A}} \| \varepsilon \| _ {\infty}}{a} + \frac {\kappa_ {Q} ^ {3 / 2} \mathrm {C} _ {\mathbf {A}} \| \varepsilon \| _ {\infty}}{a ^ {3 / 2}} 3 ^ {\gamma / 2} + \frac {\kappa_ {Q} ^ {3 / 2} \mathrm {C} _ {\mathbf {A}} \| \varepsilon \| _ {\infty}}{a} 3 ^ {\gamma} \left(\frac {8 \gamma}{a c _ {0}}\right) ^ {\gamma / (1 - \gamma)}. \tag {42} +$$ + +Moreover, + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} H _ {k} ^ {(1)} \right| ^ {p} \right] \lesssim \mathrm {D} _ {5} ^ {(\mathrm {M})} t _ {\operatorname {m i x}} p ^ {2} \alpha_ {k} \sqrt {\log \left(1 / \alpha_ {k}\right)}, \tag {43} +$$ + +where + +$$ +\mathrm {D} _ {5} ^ {\left(\mathrm {M}\right)} = \frac {d ^ {1 / 2 + 1 / \log n} \mathrm {D} _ {4} ^ {\left(\mathrm {M}\right)} C _ {\Gamma} \mathrm {C} _ {\mathbf {A}}}{a}. \tag {44} +$$ + +Proof. Solving the recursion in (40) yields the double summation: + +$$ +J _ {k} ^ {(1)} = - \sum_ {\ell = 1} ^ {k} \alpha_ {\ell} G _ {\ell + 1: k} \tilde {\mathbf {A}} (Z _ {\ell}) J _ {\ell - 1} ^ {(0)} = \sum_ {\ell = 1} ^ {k} \alpha_ {\ell} \sum_ {j = 1} ^ {\ell - 1} \alpha_ {j} G _ {\ell + 1: k} \tilde {\mathbf {A}} (Z _ {\ell}) G _ {j + 1: \ell - 1} \varepsilon (Z _ {j}). +$$ + +Changing the order of summation yields + +$$ +J _ {k} ^ {(1)} = \sum_ {j = 1} ^ {k - 1} \alpha_ {j} \Bigg \{\sum_ {\ell = j + 1} ^ {k} \alpha_ {\ell} G _ {\ell + 1: k} \tilde {\mathbf {A}} (Z _ {\ell}) G _ {j + 1: \ell - 1} \Bigg \} \varepsilon (Z _ {j}) = \sum_ {j = 1} ^ {k - 1} \alpha_ {j} S _ {j + 1: k} \varepsilon (Z _ {j}), +$$ + +where for $j \leq k$ we have defined + +$$ +S _ {j: k} := \sum_ {\ell = j} ^ {k} \alpha_ {\ell} G _ {\ell + 1: k} \tilde {\mathbf {A}} (Z _ {\ell}) G _ {j: \ell - 1}. +$$ + +Fix a constant $m \in \mathbb{N}$ , $m \leq k$ (to be determined later). Then we can rewrite $S_{j+1:k}$ as + +$$ +\begin{array}{l} S _ {j + 1: k} = \sum_ {\ell = j + 1} ^ {j + m} \alpha_ {\ell} G _ {\ell + 1: k} \tilde {\mathbf {A}} (Z _ {\ell}) G _ {j + 1: \ell - 1} + \sum_ {\ell = j + m + 1} ^ {k} \alpha_ {\ell} G _ {\ell + 1: k} \tilde {\mathbf {A}} (Z _ {\ell}) G _ {j + 1: \ell - 1} \\ = G _ {j + m + 1: k} S _ {j + 1: j + m} + S _ {j + m + 1: k} G _ {j + 1: j + m}. \\ \end{array} +$$ + +Let $N := \lfloor k / m \rfloor$ . In these notations, we can express $J_{k}^{(1)}$ as a sum of three terms: + +$$ +\begin{array}{l} J _ {k} ^ {(1)} = \underbrace {\sum_ {j = 1} ^ {m (N - 1)} \alpha_ {j} G _ {j + m + 1 : k} S _ {j + 1 : j + m} \varepsilon \left(Z _ {j}\right)} _ {T _ {1}} + \underbrace {\sum_ {j = 1} ^ {m (N - 1)} \alpha_ {j} S _ {j + m + 1 : k} G _ {j + 1 : j + m} \varepsilon \left(Z _ {j}\right)} _ {T _ {2}} \\ + \underbrace {\sum_ {j = m (N - 1) + 1} ^ {k - 1} \alpha_ {j} S _ {j + 1 : k} \varepsilon (Z _ {j})} _ {T _ {3}}. \\ \end{array} +$$ + +Consider the first term $T_{1}$ . Applying Minkowski's inequality, Lemma 11, and Lemma 28, we get that + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} [ \left| u ^ {\top} T _ {1} \right| ^ {p} ] \stackrel {(a)} {\lesssim} \kappa_ {Q} ^ {3 / 2} \mathrm {C} _ {\mathbf {A}} p ^ {1 / 2} t _ {\operatorname {m i x}} ^ {1 / 2} \| \varepsilon \| _ {\infty} \sum_ {j = 1} ^ {(m - 1) N} \alpha_ {j} \big (\sum_ {r = j + 1} ^ {j + m} \alpha_ {r} ^ {2} \big) ^ {1 / 2} \prod_ {\ell = j + 1} ^ {k} \sqrt {1 - a \alpha_ {\ell}} \\ \lesssim \kappa_ {Q} ^ {3 / 2} \mathrm {C} _ {\mathbf {A}} \sqrt {m p t _ {\mathrm {m i x}}} \| \varepsilon \| _ {\infty} \sum_ {j = 1} ^ {(m - 1) N} \alpha_ {j} ^ {2} \prod_ {\ell = j + 1} ^ {k} \sqrt {1 - a \alpha_ {\ell}} \\ \stackrel {(b)} {\lesssim} \mathrm {C} _ {1} \sqrt {m p t _ {\mathrm {m i x}}} \alpha_ {k} , \\ \end{array} +$$ + +where we set + +$$ +\mathrm {C} _ {1} = \left(\kappa_ {Q} ^ {3 / 2} / a\right) \mathrm {C} _ {\mathbf {A}} \| \varepsilon \| _ {\infty}. +$$ + +In the line (a) above we used Lemma 11, and in (b) we used Lemma 28. Similarly, using the same lemmas, we bound the term $T_{3}$ : + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} T _ {3} \right| ^ {p} \right] \lesssim \mathrm {C} _ {1} \sqrt {m p} t _ {\operatorname {m i x}} \alpha_ {k}. +$$ + +Note that the second term $T_{2}$ is non-zero only for $N \geq 2$ and it can be rewritten as $T_{2} = T_{21} + T_{22}$ , where + +$$ +T _ {2 1} := \sum_ {j = 0} ^ {N - 2} \sum_ {i = 1} ^ {m} \alpha_ {j m + i} S _ {(j + 1) m + i + 1: k} G _ {j m + i + 1: (j + 1) m + i} \varepsilon \left(Z _ {j m + i} ^ {*}\right), +$$ + +$$ +T _ {2 2} := \sum_ {j = 0} ^ {N - 2} \sum_ {i = 1} ^ {m} \alpha_ {j m + i} S _ {(j + 1) m + i + 1: k} G _ {j m + i + 1: (j + 1) m + i} (\varepsilon (Z _ {j m + i}) - \varepsilon (Z _ {j m + i} ^ {*}). +$$ + +The set of random variables $Z_{jm + i}^{*}$ is constructed for each $i\in [1,m]$ , with $\{Z_{jm + i}^{*}\}_{j = 0}^{N - 2}$ having the following properties: + +1. $Z_{j m + i}^{*}$ is independent of $\mathfrak{F}_{(j + 1)m + i}^k \coloneqq \sigma \{Z_{(j + 1)m + i}, \ldots, Z_k\}$ ; +2. $\mathbb{P}_{\xi}(Z_{jm + i}^{*}\neq Z_{jm + i})\leq 2(1 / 4)^{\lceil m / t_{\mathrm{mix}}\rceil}$ (45) +3. $Z_{jm + i}^{*}$ and $Z_{jm + i}$ have the same distribution, + +The existence of the random variables $Z_{jm + i}^{*}$ is guaranteed by Berbee's lemma, see e.g [60, Lemma 5.1], together with the fact that uniformly geometrically ergodic Markov chains are a special instance of $\beta$ -mixing processes. We control $\beta$ -mixing coefficient via total variation distance, see [23, Theorem F.3.3]. In order to analyze the term $T_{21}$ we use Minkowski's and Burkholder's inequality [52, Theorem 8.6], and obtain that: + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} T _ {2 1} \right| ^ {p} \right] \leq \sum_ {i = 1} ^ {m} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| \sum_ {j = 0} ^ {N - 2} \alpha_ {j m + i} u ^ {\top} S _ {(j + 1) m + i + 1: k} G _ {j m + i + 1: (j + 1) m + i} \varepsilon \left(Z _ {j m + i} ^ {*}\right) \right| ^ {p} \right] \\ \leq p \sum_ {i = 1} ^ {m} \left(\sum_ {j = 0} ^ {N - 2} \alpha_ {j m + i} ^ {2} \mathbb {E} ^ {2 / p} [ | u ^ {\top} S _ {(j + 1) m + i + 1: k} G _ {j m + i + 1: (j + 1) m + i} \varepsilon (Z _ {j m + i} ^ {*}) | ^ {p} ]\right) ^ {1 / 2} \\ \leq p \sqrt {m} \bigg (\sum_ {j = 1} ^ {k} \alpha_ {j} ^ {2} \mathbb {E} ^ {2 / p} \big [ \big | u ^ {\top} S _ {j + m + 1: k} G _ {j + 1: j + m} \varepsilon (Z _ {j} ^ {*}) \big | ^ {p} \big ] \bigg) ^ {1 / 2}. \\ \end{array} +$$ + +Applying now Lemma 11, we arrive at the bound + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} T _ {2 1} \right| ^ {p} \right] \lesssim \kappa_ {Q} ^ {3 / 2} \mathrm {C} _ {\mathbf {A}} \| \varepsilon \| _ {\infty} t _ {\operatorname {m i x}} ^ {1 / 2} p ^ {3 / 2} \sqrt {m} \left(\sum_ {j = 1} ^ {k} \alpha_ {j} ^ {2} \left(\sum_ {\ell = j + m + 1} ^ {k} \alpha_ {\ell} ^ {2}\right) \prod_ {\ell = j + 1} ^ {k} (1 - \alpha_ {\ell} a)\right) ^ {1 / 2}. \tag {46} +$$ + +In order to bound the term $T_{22}$ we first note that + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} S _ {(j + 1) m + i + 1: k} G _ {j m + i + 1: (j + 1) m + i} \left(\varepsilon \left(Z _ {j m + i}\right) - \varepsilon \left(Z _ {j m + i} ^ {*}\right)\right) \right| ^ {p} \right] \\ \leq \mathbb {E} _ {\xi} ^ {1 / p} \left[ \| G _ {j m + i + 1: (j + 1) m + i} \{\varepsilon (Z _ {j m + i}) - \varepsilon (Z _ {j m + i} ^ {*}) \} \| ^ {p} \sup _ {u, v \in \mathbb {S} _ {d - 1}, \xi^ {\prime}} \mathbb {E} _ {\xi^ {\prime}} \left[ \left| u ^ {\top} S _ {(j + 1) m + i + 1: k} v \right| ^ {p} \right] \right]. \\ \end{array} +$$ + +Hence, using Minkowski's inequality for $T_{22}$ , we obtain that + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} [ \left| u ^ {\top} T _ {2 2} \right| ^ {p} ] \leq \sqrt {\kappa_ {Q}} \sum_ {j = 0} ^ {N - 2} \sum_ {i = 1} ^ {m} \alpha_ {j m + i} \sup _ {u, v \in \mathbb {S} _ {d - 1}, \xi^ {\prime}} \mathbb {E} _ {\xi^ {\prime}} ^ {1 / p} [ \left| u ^ {\top} S _ {(j + 1) m + i + 1: k} v \right| ^ {p} ] \mathbb {E} _ {\xi} ^ {1 / p} [ \| \varepsilon (Z _ {j m + i}) - \varepsilon (Z _ {j m + i} ^ {*}) \| ^ {p} ] \\ \times \prod_ {\ell = j m + i + 1} ^ {(j + 1) m + i} (\sqrt {1 - \alpha_ {\ell} a}). \\ \end{array} +$$ + +Using the definition of $Z_{km + i}^{*}$ and the Cauchy-Schwartz inequality, + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} [ \| \varepsilon (Z _ {k m + i}) - \varepsilon (Z _ {k m + i} ^ {*}) \| ^ {p} ] = \mathbb {E} _ {\xi} ^ {1 / p} [ \| \varepsilon (Z _ {k m + i}) - \varepsilon (Z _ {k m + i} ^ {*})) \mathbb {1} \{Z _ {k m + i} ^ {*} \neq Z _ {k m + i} \} \| ^ {p} ] \\ \stackrel {(a)} {\lesssim} \left\| \varepsilon \right\| _ {\infty} (1 / 4) ^ {m / (2 p t _ {\mathrm {m i x}})}. \\ \end{array} +$$ + +where in (a) we used (45). The last two inequalities together with Lemma 11 imply + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} T _ {2 2} \right| ^ {p} \right] \lesssim \kappa_ {Q} ^ {3 / 2} \mathrm {C} _ {\mathbf {A}} \| \varepsilon \| _ {\infty} t _ {\operatorname {m i x}} ^ {1 / 2} p ^ {1 / 2} (1 / 4) ^ {m / \left(2 p t _ {\operatorname {m i x}}\right)} \sum_ {j = 1} ^ {k} \alpha_ {j} \left(\sum_ {\ell = j + 1} ^ {k} \alpha_ {\ell} ^ {2}\right) ^ {1 / 2} \prod_ {\ell = j + 1} ^ {k} \sqrt {1 - a \alpha_ {\ell}}. \tag {47} +$$ + +Combining now (46) and (47), we obtain + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} J _ {k} ^ {(1)} \right| ^ {p} \right] \lesssim \mathrm {C} _ {1} \sqrt {m p} t _ {\operatorname {m i x}} ^ {1 / 2} \alpha_ {k} \\ + \kappa_ {Q} ^ {3 / 2} \mathrm {C _ {A}} \| \varepsilon \| _ {\infty} t _ {\mathrm {m i x}} ^ {1 / 2} p ^ {3 / 2} \sqrt {m} \Bigg (\sum_ {j = 1} ^ {k} \alpha_ {j} ^ {2} \big (\sum_ {\ell = j + 1} ^ {k} \alpha_ {\ell} ^ {2} \big) \prod_ {\ell = j + 1} ^ {k} \sqrt {1 - \alpha_ {\ell} a} \Bigg) ^ {1 / 2} \\ + \kappa_ {Q} ^ {3 / 2} \mathrm {C} _ {\mathbf {A}} \| \varepsilon \| _ {\infty} t _ {\mathrm {m i x}} ^ {1 / 2} p ^ {1 / 2} (1 / 4) ^ {m / (2 p t _ {\mathrm {m i x}})} \sum_ {j = 1} ^ {k} \alpha_ {j} \left(\sum_ {\ell = j + 1 + m} ^ {k} \alpha_ {\ell} ^ {2}\right) ^ {1 / 2} \prod_ {\ell = j + 1} ^ {k} \sqrt {1 - a \alpha_ {\ell}}. \\ \end{array} +$$ + +Applying now Lemma 29, we arrive at the bound: + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} J _ {k} ^ {(1)} \right| ^ {p} \right] \lesssim \mathrm {C} _ {1} \sqrt {m p} t _ {\operatorname {m i x}} ^ {1 / 2} \alpha_ {k} \\ + \left(\kappa_ {Q} ^ {3 / 2} \mathrm {C _ {A}} \| \varepsilon \| _ {\infty} / a\right) t _ {\mathrm {m i x}} ^ {1 / 2} p ^ {3 / 2} \sqrt {m} 3 ^ {\gamma} \left(\frac {8 \gamma}{a c _ {0}}\right) ^ {\gamma / (1 - \gamma)} \alpha_ {k} \\ \left(\kappa_ {Q} ^ {3 / 2} \mathrm {C} _ {\mathbf {A}} \| \varepsilon \| _ {\infty} / a ^ {3 / 2}\right) t _ {\text {m i x}} ^ {1 / 2} p ^ {1 / 2} \left(1 / 4\right) ^ {m / \left(4 p t _ {\text {m i x}}\right)} 3 ^ {\gamma / 2} \sqrt {\alpha_ {k}}. \\ \end{array} +$$ + +Hence, it remains to set the block size $m$ as + +$$ +(1 / 4) ^ {m / (4 p t _ {\mathrm {m i x}})} \leq \sqrt {\alpha_ {k}}, \text {i . e .} m = \left\lceil \frac {2 p t _ {\mathrm {m i x}} \log (1 / \alpha_ {k})}{\log 4} \right\rceil . +$$ + +With this choice of $m$ we obtain from the above inequality that + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} J _ {k} ^ {(1)} \right| ^ {p} \right] \lesssim \mathrm {C} _ {1} p t _ {\text {m i x}} \sqrt {\log (1 / \alpha_ {k})} \alpha_ {k} \\ + \left(\kappa_ {Q} ^ {3 / 2} \mathrm {C} _ {\mathbf {A}} \| \varepsilon \| _ {\infty} / a\right) t _ {\operatorname {m i x}} p ^ {2} \sqrt {\log (1 / \alpha_ {k})} 3 ^ {\gamma} \left(\frac {8 \gamma}{a c _ {0}}\right) ^ {\gamma / (1 - \gamma)} \alpha_ {k} \\ + \left(\kappa_ {Q} ^ {3 / 2} \mathrm {C} _ {\mathbf {A}} \| \varepsilon \| _ {\infty} / a ^ {3 / 2}\right) t _ {\operatorname {m i x}} ^ {1 / 2} p ^ {1 / 2} 3 ^ {\gamma / 2} \alpha_ {k} \\ \lesssim \mathrm {C} _ {3} p ^ {2} t _ {\mathrm {m i x}} \sqrt {\log (1 / \alpha_ {k})} \alpha_ {k} \\ \end{array} +$$ + +where we have defined + +$$ +\mathrm {C} _ {3} = \frac {\kappa_ {Q} ^ {3 / 2} \mathrm {C} _ {\mathbf {A}} \| \varepsilon \| _ {\infty}}{a} + \frac {\kappa_ {Q} ^ {3 / 2} \mathrm {C} _ {\mathbf {A}} \| \varepsilon \| _ {\infty}}{a ^ {3 / 2}} 3 ^ {\gamma / 2} + \frac {\kappa_ {Q} ^ {3 / 2} \mathrm {C} _ {\mathbf {A}} \| \varepsilon \| _ {\infty}}{a} 3 ^ {\gamma} \bigg (\frac {8 \gamma}{a c _ {0}} \bigg) ^ {\gamma / (1 - \gamma)}, +$$ + +and the bound (41) holds. To estimate $H_{k}^{(1)}$ we rewrite it as follows + +$$ +H _ {k} ^ {(1)} = - \sum_ {j = 1} ^ {k} \alpha_ {j} \Gamma_ {j + 1: k} ^ {(\alpha)} \tilde {\mathbf {A}} (Z _ {j}) J _ {j - 1} ^ {(1)}. +$$ + +Using Minkowski's inequality together with Hölder's inequality, we get + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} H _ {k} ^ {(1)} \right| ^ {p} \right] \leq \sum_ {j = 1} ^ {k} \mathrm {C} _ {\mathbf {A}} \alpha_ {j} \left\{\mathbb {E} _ {\xi} \left[ \| \Gamma_ {j + 1: k} ^ {(\alpha)} \| ^ {2 p} \right] \right\} ^ {1 / 2 p} \left\{\mathbb {E} _ {\xi} \left[ \| J _ {j - 1} ^ {(1)} \| ^ {2 p} \right] \right\} ^ {1 / 2 p}. +$$ + +Applying similar arguments to (38) together with Proposition 12 and an elementary inequality $\mathrm{e}^{-x} \leq 1 - x/2$ , valid for $x \in [0,1]$ , we obtain the bound (41), we get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} H _ {k} ^ {(1)} \right| ^ {p} \right] \lesssim d ^ {1 / 2 + 1 / \log n} C _ {\Gamma} \mathrm {C} _ {\mathbf {A}} \mathrm {D} _ {4} ^ {(\mathrm {M})} t _ {\operatorname {m i x}} p ^ {2} \sum_ {j = 1} ^ {k} \alpha_ {j} ^ {2} \sqrt {\log \left(1 / \alpha_ {j}\right)} \prod_ {\ell = j + 1} ^ {k} (1 - a \alpha_ {\ell} / 2 4) \\ \lesssim d ^ {1 / 2 + 1 / \log n} C _ {\Gamma} \mathrm {C} _ {\mathbf {A}} \mathrm {D} _ {4} ^ {(\mathrm {M})} t _ {\operatorname {m i x}} p ^ {2} \sqrt {\log (1 / \alpha_ {k})} \sum_ {j = 1} ^ {k} \alpha_ {j} ^ {2} \prod_ {\ell = j + 1} ^ {k} (1 - a \alpha_ {\ell} / 2 4) \\ \stackrel {(a)} {\lesssim} \mathsf {D} _ {5} ^ {(\mathrm {M})} t _ {\mathrm {m i x}} p ^ {2} \alpha_ {k} \sqrt {\log \left(1 / \alpha_ {k}\right)}, \\ \end{array} +$$ + +where in (a) we have additionally used Lemma 28 and used the definition of $\mathsf{D}_5^{(\mathsf{M})}$ from (44). + +![](images/63888712b66817e49205db6f53f352b6229979ab012a0365bcae4b8d6a1c5024.jpg) + +# C.3 Technical bounds related to $J_{k}^{(1)}$ . + +Recall that $S_{\ell +1:\ell +m}$ is defined, for $\ell ,m\in \mathbb{N}$ , as + +$$ +S _ {\ell + 1: \ell + m} = \sum_ {k = \ell + 1} ^ {\ell + m} \alpha_ {k} \mathbf {B} _ {k} \left(Z _ {k}\right), \text {w i t h} \mathbf {B} _ {k} (z) = G _ {k + 1: \ell + m} \tilde {\mathbf {A}} (z) G _ {\ell + 1: k - 1}. \tag {48} +$$ + +Lemma 11. Under the assumptions of Proposition 11, it holds for any vector $u \in \mathbb{S}_{d-1}$ , $\ell, m \in \mathbb{N}$ and any initial distribution $\xi$ on $(\mathsf{Z}, \mathcal{Z})$ , that + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} S _ {\ell + 1: \ell + m} \varepsilon (Z _ {\ell}) \right| ^ {p} \right] \leq C _ {S} \left(\sum_ {r = \ell + 1} ^ {\ell + m} \alpha_ {r} ^ {2}\right) ^ {1 / 2} \prod_ {k = \ell + 1} ^ {\ell + m} \sqrt {1 - a \alpha_ {k}} , +$$ + +where + +$$ +C _ {S} = 7 \kappa_ {Q} \mathrm {C} _ {\mathbf {A}} p ^ {1 / 2} t _ {\mathrm {m i x}} ^ {1 / 2} \| \varepsilon \| _ {\infty}. +$$ + +Proof. We first prove the auxiliary inequality for deterministic vectors $u, v \in S_{d-1}$ . Indeed, using (48), we obtain that + +$$ +u ^ {\top} S _ {\ell + 1: \ell + m} v = \sum_ {k = \ell + 1} ^ {\ell + m} h _ {k} \left(Z _ {k}\right), \quad \text {w h e r e} h _ {k} (z) := \alpha_ {k} u ^ {\top} \mathbf {B} _ {k} (z) v. +$$ + +It is easy to check that $u^{\top} S_{\ell + 1: \ell + m} v$ satisfies the bounded differences property, since for any $z, z' \in \mathbb{Z}$ , and $r \in \{\ell + 1, \dots, \ell + m\}$ , it holds that + +$$ +\left| h _ {r} (z) - h _ {r} \left(z ^ {\prime}\right) \right| \leq 2 \kappa_ {Q} \mathrm {C} _ {\mathbf {A}} \alpha_ {r} \prod_ {k = \ell + 1, k \neq r} ^ {\ell + m} \sqrt {1 - a \alpha_ {k}} \leq 2 ^ {3 / 2} \kappa_ {Q} \mathrm {C} _ {\mathbf {A}} \alpha_ {r} \prod_ {k = \ell + 1} ^ {\ell + m} \sqrt {1 - a \alpha_ {k}}. +$$ + +In the last inequality we have additionally used the fact that $\alpha_{k}a\leq 1 / 2$ for any $k\in \mathbb{N}$ . Applying the bounded differences inequality from [54][Corollary 2.11], we get that for any $t\geq 0$ + +$$ +\mathbb {P} _ {\xi} \big (\left| u ^ {\top} S _ {j + 1: j + m} v \right| \geq t \big) \leq 2 \exp \Bigg \{- \frac {2 \big (t - | \mathbb {E} _ {\xi} [ u ^ {\top} S _ {j + 1 : j + m} v ] | \big) ^ {2}}{7 2 t _ {\mathrm {m i x}} \kappa_ {Q} ^ {2} \mathrm {C} _ {\mathbf {A}} ^ {2} \big (\sum_ {r = \ell + 1} ^ {\ell + m} \alpha_ {r} ^ {2} \big) \prod_ {k = \ell + 1} ^ {\ell + m} (1 - a \alpha_ {k})} \Bigg \}. +$$ + +It remains to upper bound $\mathbb{E}_{\xi}[u^{\top}S_{j + 1:j + m}v]$ . Note that + +$$ +\left(\mathbb {E} _ {\xi} \left[ u ^ {\top} S _ {j + 1: j + m} v \right]\right) ^ {2} \leq \mathbb {E} _ {\xi} \left[ \left(\sum_ {k = l + 1} ^ {l + m} h _ {k} (Z _ {k})\right) ^ {2} \right] = \sum_ {k = l + 1} ^ {l + m} \mathbb {E} _ {\xi} \left[ h _ {k} (Z _ {k}) ^ {2} \right] + 2 \sum_ {k = l + 1} ^ {l + m} \sum_ {j = 1} ^ {l + m - k} \mathbb {E} _ {\xi} \left[ h _ {k} (Z _ {k}) h _ {k + j} (Z _ {k + j}) \right]. +$$ + +Using that $\pi(h_k) = 0$ and A1, we obtain + +$$ +\begin{array}{l} \left| \mathbb {E} _ {\xi} \left[ h _ {k} \left(Z _ {k}\right) h _ {k + j} \left(Z _ {k + j}\right) \right] \right| = \left| \int_ {\mathbf {Z}} h _ {k} (z) \left(\mathrm {P} ^ {j} h _ {k + j} (z) - \pi \left(h _ {k + j}\right)\right) \xi \mathrm {P} ^ {k} (\mathrm {d} z) \right| \leq \| h _ {k} \| _ {\infty} \| h _ {k + j} \| _ {\infty} \Delta \left(\mathrm {P} ^ {j}\right) \\ \leq 2 \alpha_ {k} \alpha_ {k + j} \kappa_ {Q} ^ {2} (2 \mathrm {C _ {A}}) ^ {2} \prod_ {t = l + 1} ^ {l + m} (1 - a \alpha_ {t}) (1 / 4) ^ {\lceil j / t _ {\mathrm {m i x}} \rceil}, \\ \end{array} +$$ + +and + +$$ +\mathbb {E} _ {\xi} \left[ h _ {k} \left(Z _ {k}\right) ^ {2} \right] \leq 2 \alpha_ {k} ^ {2} \kappa_ {Q} ^ {2} \left(2 \mathrm {C} _ {\mathbf {A}}\right) ^ {2} \prod_ {t = l + 1} ^ {l + m} \left(1 - a \alpha_ {t}\right). +$$ + +Combining inequalities above, we obtain + +$$ +\left(\mathbb {E} _ {\xi} \left[ u ^ {\top} S _ {j + 1: j + m} v \right]\right) ^ {2} \leq \frac {3 2}{3} t _ {\operatorname {m i x}} \kappa_ {Q} ^ {2} \mathrm {C} _ {\mathbf {A}} ^ {2} \left(\sum_ {k = l + 1} ^ {l + m} \alpha_ {k} ^ {2}\right) \prod_ {t = l + 1} ^ {l + m} \left(1 - a \alpha_ {t}\right). +$$ + +Note that for $t\geq |\mathbb{E}_{\xi}[u^{\top}S_{j + 1:j + m}v]|$ , we have + +$$ +\mathbb {P} _ {\xi} \left(\left| u ^ {\top} S _ {j + 1: j + m} v \right| \geq t\right) \leq 2 \exp \left\{- \frac {t ^ {2}}{4 9 t _ {\mathrm {m i x}} \kappa_ {Q} ^ {2} \mathrm {C} _ {\mathbf {A}} ^ {2} \left(\sum_ {r = \ell + 1} ^ {\ell + m} \alpha_ {r} ^ {2}\right) \prod_ {k = \ell + 1} ^ {\ell + m} (1 - a \alpha_ {k})} \right\}. +$$ + +And for $t \leq |\mathbb{E}_{\xi}[u^{\top}S_{j+1:j+m+m}v]|$ , the right side of the inequality is greater than 1, so this inequality is also true for $t \leq |\mathbb{E}_{\xi}[u^{\top}S_{j+1:j+m+m}v]|$ . Hence, using [24, Lemma 7], we obtain that + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} S _ {j + 1; j + m} v \right| ^ {p} \right] \leq 7 p ^ {1 / 2} \kappa_ {Q} \mathrm {C} _ {\mathbf {A}} t _ {\operatorname {m i x}} ^ {1 / 2} \left(\sum_ {r = \ell + 1} ^ {\ell + m} \alpha_ {r} ^ {2}\right) ^ {1 / 2} \prod_ {k = \ell + 1} ^ {\ell + m} \sqrt {1 - a \alpha_ {k}}. \tag {49} +$$ + +Now, with $\mathcal{F}_{\ell} = \sigma \{Z_{j}, j \leq \ell\}$ , it holds that + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} S _ {\ell + 1: \ell + m} \varepsilon (Z _ {\ell}) \right| ^ {p} \right] = \mathbb {E} _ {\xi} ^ {1 / p} \left[ \| \varepsilon (Z _ {\ell}) \| ^ {p} \mathbb {E} ^ {\mathcal {F} _ {\ell}} \left[ \left| u ^ {\top} S _ {\ell + 1: \ell + m} \varepsilon (Z _ {\ell}) \right| ^ {p} / \| \varepsilon (Z _ {\ell}) \| ^ {p} \right] \right] \\ \leq \| \varepsilon \| _ {\infty} \sup _ {u, v \in \mathbb {S} _ {d - 1}, \xi^ {\prime} \in \mathcal {P} (\mathsf {Z})} \mathbb {E} _ {\xi^ {\prime}} ^ {1 / p} \left[ \left| u ^ {\top} S _ {\ell + 1: \ell + m} v \right| ^ {p} \right]. \\ \end{array} +$$ + +Combining the above bounds with (49) and A2 yields the statement. + +![](images/1eb15569793851f26e81a94317bd9642e0b1f4b503521780e8aa2393d53b754d.jpg) + +# D Proofs for stability of matrix products + +# D.1 Stability + +We first provide a result on the product of dependent random matrices. Our proof technique is based on the approach of [33] and the results, previously obtained in [24]. Let $(\Omega, \mathfrak{F}, \{\mathfrak{F}_{\ell}\}_{\ell \in \mathbb{N}}, \mathbb{P})$ be a filtered probability space. For the matrix $B \in \mathbb{R}^{d \times d}$ we denote by $(\sigma_{\ell}(B))_{\ell=1}^{d}$ its singular values. For $q \geq 1$ , the Shatten $q$ -norm is denoted by $\|B\|_q = \left\{\sum_{\ell=1}^d \sigma_{\ell}^q(B)\right\}^{1/q}$ . For $q, p \geq 1$ and a random matrix $\mathbf{X}$ we write $\| \mathbf{X}\|_{q,p} = \{\mathbb{E}[\| \mathbf{X}\|_q^p]\}^{1/p}$ . The main result of this section is stated below: + +Proposition 12. Assume $A1, A2$ , and $A3$ . Then, for any $2 \leq p \leq \log n$ , $n \in \mathbb{N}$ , and probability distribution $\xi$ on $(\mathsf{Z},\mathcal{Z})$ , it holds that + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \| \Gamma_ {j: n} ^ {(\alpha)} \| ^ {p} \right] \leq C _ {\Gamma} d ^ {1 / \log n} \exp \left\{- (a / 1 2) \sum_ {k = j} ^ {n} \alpha_ {k} \right\}, +$$ + +where + +$$ +C _ {\Gamma} = \sqrt {\kappa_ {Q}} \mathrm {e} ^ {2}. +$$ + +The proof is given in Appendix D.2. It is based on a simplification of the arguments in [26] together with a new result about the matrix concentration for the product of random matrices, using a proof method introduced in [33]. We first state a result from [24]: + +Proposition 13 (Proposition 15 in [24]). Let $\{\mathbf{Y}_{\ell}\}_{\ell \in \mathbb{N}}$ be a sequence of random matrices adapted to the filtration $\{\mathfrak{F}_{\ell}\}_{\ell \in \mathbb{N}}$ and $P$ be a positive definite matrix. Assume that for each $\ell \in \mathbb{N}^*$ there exist $\mathfrak{m}_{\ell} \in (0,1]$ and $\sigma_{\ell} > 0$ such that + +$$ +\left\| \mathbb {E} ^ {\mathfrak {F} _ {\ell - 1}} \left[ \mathbf {Y} _ {\ell} \right] \right\| _ {P} ^ {2} \leq 1 - \mathfrak {m} _ {\ell} a n d \left\| \mathbf {Y} _ {\ell} - \mathbb {E} ^ {\mathfrak {F} _ {\ell - 1}} \left[ \mathbf {Y} _ {\ell} \right] \right\| _ {P} \leq \sigma_ {\ell} \quad \mathbb {P} - a. s. . +$$ + +Define $\mathbf{Z}_n = \prod_{\ell = 0}^n\mathbf{Y}_\ell = \mathbf{Y}_n\mathbf{Z}_{n - 1}$ , for $n\geq 1$ . Then, for any $2\leq p\leq q$ and $n\geq 1$ + +$$ +\| \mathbf {Z} _ {n} \| _ {q, p} ^ {2} \leq \kappa_ {P} \prod_ {\ell = 1} ^ {n} (1 - \mathfrak {m} _ {\ell} + (q - 1) \sigma_ {\ell} ^ {2}) \| P ^ {1 / 2} \mathbf {Z} _ {0} P ^ {- 1 / 2} \| _ {q, p} ^ {2}, +$$ + +where $\kappa_{P} = \lambda_{\max}(P) / \lambda_{\min}(P)$ and $\lambda_{\max}(P), \lambda_{\min}(P)$ correspond to the largest and smallest eigenvalues of $P$ . + +Now we fix some $N \in \mathbb{N}$ , block size $h \in \mathbb{N}$ defined in (23), and a sequence $0 = j_0 < j_1 < \ldots < j_N = 2n$ , where $j_\ell = h\ell$ , $\ell \leq N - 1$ . Note that it is possible that $j_N - j_{n-1} < h$ . Now we set + +$$ +\mathbf {Y} _ {\ell} = \prod_ {i = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \left(\mathrm {I} - \alpha_ {i} \mathbf {A} (Z _ {i})\right). +$$ + +Then the following lemma holds: + +Lemma 12. Assume $A1, A2$ , and $A3$ . Then for any $\ell \in \{1, \dots, N - 1\}$ , and any probability measure $\xi$ on $(\mathbb{Z}, \mathcal{Z})$ , it holds that + +$$ +\| \mathbb {E} _ {\xi} [ \mathbf {Y} _ {\ell} ] \| _ {Q} \leq 1 - \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) a / 6. +$$ + +Proof. We decompose the matrix product $\mathbf{Y}_{\ell}$ as follows: + +$$ +\mathbf {Y} _ {\ell} = \mathrm {I} - \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) \bar {\mathbf {A}} - \mathbf {S} _ {\ell} + \mathbf {R} _ {\ell}. \tag {50} +$$ + +Here $\mathbf{S}_{\ell} = \sum_{k = j_{\ell -1}}^{j_{\ell} - 1}\alpha_{k}\bigl {\{}\mathbf{A}(Z_{k}) - \bar{\mathbf{A}}\bigr \}$ is a linear statistics in $\{\mathbf{A}(Z_k)\}_{k = j_\ell -1}^{j_\ell -1}$ , and the remainder term $\mathbf{R}_{\ell}$ is defined as + +$$ +\mathbf {R} _ {\ell} = \sum_ {r = 2} ^ {h} (- 1) ^ {r} \sum_ {(i _ {1}, \dots , i _ {r}) \in \mathbb {I} _ {r} ^ {\ell}} \prod_ {u = 1} ^ {r} \{\alpha_ {i _ {u}} \mathbf {A} (Z _ {i _ {u}}) \}, +$$ + +where $\mathsf{I}_r^\ell = \{(i_1,\ldots ,i_r)\in \{1,\ldots ,h\} ^r: i_1 < \dots < i_r\}$ . Since $\| M\|_{Q} = \| Q^{1 / 2}MQ^{-1 / 2}\|$ , it is straightforward to check that $\mathbb{P}$ -a.s. it holds + +$$ +\left\| \mathbf {R} _ {\ell} \right\| _ {Q} \leq \frac {\left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) ^ {2} \kappa_ {Q} C _ {\mathbf {A}} ^ {2}}{2} \exp \left\{\kappa_ {Q} ^ {1 / 2} C _ {\mathbf {A}} \sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k} \right\} =: T _ {2}. \tag {51} +$$ + +On the other hand, using A1 and A2, we have for any $k \in \mathbb{N}$ , that + +$$ +\| \mathbb {E} _ {\xi} [ \mathbf {A} (Z _ {k}) - \bar {\mathbf {A}} ] \| = \sup _ {u, v \in \mathbb {S} ^ {d - 1}} [ \mathbb {E} _ {\xi} [ u ^ {\top} \mathbf {A} (Z _ {k}) v ] - u ^ {\top} \bar {\mathbf {A}} v ] \leq \mathrm {C} _ {\mathbf {A}} \Delta (\mathrm {P} ^ {k}). +$$ + +Hence, with the triangle inequality we obtain that + +$$ +\begin{array}{l} \| \mathbb {E} _ {\xi} [ \mathbf {S} _ {\ell} ] \| _ {Q} \leq \kappa_ {Q} ^ {1 / 2} \sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k} \| \mathbb {E} _ {\xi} [ \mathbf {A} (Z _ {k}) - \bar {\mathbf {A}} ] \| \leq \kappa_ {Q} ^ {1 / 2} \mathrm {C} _ {\mathbf {A}} \sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k} \Delta (\mathrm {P} ^ {k}) \\ \leq (4 / 3) \alpha_ {j _ {\ell - 1}} t _ {\operatorname {m i x}} \kappa_ {Q} ^ {1 / 2} C _ {\mathbf {A}} =: T _ {1}. \\ \end{array} +$$ + +This result combined with (51) in (50) implies that + +$$ +\| \mathbb {E} _ {\xi} [ \mathbf {Y} _ {\ell} ] \| _ {Q} \leq \| I - \sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k} \bar {\mathbf {A}} \| _ {Q} + T _ {1} + T _ {2}. +$$ + +First, by definition (23) of $h$ (see details in Lemma 13), we have + +$$ +T _ {1} \leq \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) a / 6. \tag {52} +$$ + +Second, with the definition of $c_{0}$ in (24), we obtain that + +$$ +T _ {2} \leq \left(\kappa_ {Q} ^ {1 / 2} \mathrm {C} _ {\mathbf {A}} \sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) ^ {2} \mathrm {e} \leq \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) a / 6. \tag {53} +$$ + +Finally, Proposition 2 implies that, for $\sum_{k = j_{\ell -1}}^{j_{\ell} - 1}\alpha_k\leq \alpha_\infty$ , it holds that + +$$ +\left\| \mathrm {I} - \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) \bar {\mathbf {A}} \right\| _ {Q} \leq 1 - (a / 2) \sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}. \tag {54} +$$ + +Combining (52), (53), and (54) yield that + +$$ +\left\| \mathbb {E} _ {\xi} \left[ \mathbf {Y} _ {1} \right] \right\| _ {Q} \leq 1 - \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) a / 6, +$$ + +and the statement follows. + +![](images/c6a4d6377d638478b51b84f5854fc77d9f728f7f840d9268eadb9b8911e848ed.jpg) + +Lemma 13. Assume A3. Then, for the block size $h$ defined in (23), and any $\ell \in \{1, \dots, 2n - h\}$ , it holds that + +$$ +\sum_ {k = \ell} ^ {\ell + h} \alpha_ {k} \leq \alpha_ {\infty}, \quad \kappa_ {Q} ^ {1 / 2} \mathrm {C} _ {\mathbf {A}} \sum_ {k = \ell} ^ {\ell + h} \alpha_ {k} \leq 1, \quad \kappa_ {Q} \mathrm {C} _ {\mathbf {A}} ^ {2} \mathrm {e} \sum_ {k = \ell} ^ {\ell + h} \alpha_ {k} \leq a / 6, +$$ + +$$ +(4 / 3) \alpha_ {\ell} t _ {\text {m i x}} \kappa_ {Q} ^ {1 / 2} \mathrm {C} _ {\mathbf {A}} \leq \left(\sum_ {k = \ell} ^ {\ell + h} \alpha_ {k}\right) a / 6, \tag {55} +$$ + +$$ +\left(\log n\right) \mathrm {C} _ {\sigma} ^ {2} \sum_ {k = \ell} ^ {\ell + h} \alpha_ {k} \leq \frac {a}{1 2}. \tag {56} +$$ + +Proof. First three inequalities above are easy to check, since + +$$ +\begin{array}{l} \sum_ {k = \ell} ^ {\ell + h} \alpha_ {k} \leq \sum_ {k = 1} ^ {1 + h} \frac {c _ {0}}{(k + k _ {0}) ^ {\gamma}} \leq c _ {0} \int_ {k _ {0}} ^ {1 + k _ {0} + h} \frac {d x}{x ^ {\gamma}} = \frac {c _ {0} \left\{\left(1 + h + k _ {0}\right) ^ {1 - \gamma} - k _ {0} ^ {1 - \gamma} \right\}}{1 - \gamma} \\ = \frac {c _ {0} k _ {0} ^ {1 - \gamma} ((1 + \frac {h + 1}{k _ {0}}) ^ {1 - \gamma} - 1)}{1 - \gamma} \leq c _ {0} k _ {0} ^ {- \gamma} (h + 1), \\ \end{array} +$$ + +hence, in order to satisfy these inequalities, it is enough to choose $h$ in such a manner that + +$$ +k _ {0} ^ {\gamma} \geq c _ {0} (h + 1) \max (\alpha_ {\infty}, \kappa_ {Q} ^ {1 / 2} \mathrm {C _ {A}}, 6 \mathrm {e} \kappa_ {Q} \mathrm {C _ {A} ^ {2}} / a), +$$ + +which is guaranteed by our choice of $c_{0}$ in A3 and (23). Now note that + +$$ +\sum_ {k = \ell} ^ {\ell + h} \alpha_ {k} \geq c _ {0} \int_ {\ell + k _ {0}} ^ {\ell + k _ {0} + h + 1} \frac {\mathrm {d} x}{x ^ {\gamma}} = \frac {c _ {0} \left\{\left(\ell + k _ {0} + h + 1\right) ^ {1 - \gamma} - \left(\ell + k _ {0}\right) ^ {1 - \gamma} \right\}}{1 - \gamma}, +$$ + +hence, in order to check (55), we need to ensure that + +$$ +\frac {4 c _ {0} t _ {\mathrm {m i x}} \kappa_ {Q} ^ {1 / 2} \mathrm {C _ {A}}}{3 (\ell + k _ {0}) ^ {\gamma}} \leq \frac {c _ {0} a \{(\ell + k _ {0} + h + 1) ^ {1 - \gamma} - (\ell + k _ {0}) ^ {1 - \gamma} \}}{6 (1 - \gamma)}. +$$ + +Equivalently, it is enough to set $h$ in such a way, that + +$$ +(\ell + k _ {0} + h + 1) ^ {1 - \gamma} - (\ell + k _ {0}) ^ {1 - \gamma} \geq \frac {C _ {1}}{(\ell + k _ {0}) ^ {\gamma}}, +$$ + +where we set $C_1 = 8(1 - \gamma)t_{\mathrm{mix}}\kappa_Q^{1 / 2}\mathrm{C_A} / a.$ Hence, (55) will be satisfied if + +$$ +h \geq (\ell + k _ {0}) \left(1 + \frac {C _ {1}}{\ell + k _ {0}}\right) ^ {1 / (1 - \gamma)} - (\ell + k _ {0} + 1). +$$ + +Note that, with $\alpha > 1$ , it holds that $(1 + x)^{\alpha} \leq 1 + 2\alpha x$ for $0 < x \leq 1 / \alpha$ . Hence, provided that $k_{0} > C_{1} / (1 - \gamma)$ , or, equivalently, + +$$ +k _ {0} \geq \frac {8 t _ {\mathrm {m i x}} \kappa_ {Q} ^ {1 / 2} \mathrm {C _ {A}}}{a}, +$$ + +it is enough to set + +$$ +h \geq (\ell + k _ {0}) \left(1 + \frac {2 C _ {1}}{(\ell + k _ {0}) (1 - \gamma)}\right) - (\ell + k _ {0} + 1) = \frac {1 6 t _ {\operatorname* {m i x}} \kappa_ {Q} ^ {1 / 2} \mathrm {C} _ {\mathbf {A}}}{a} - 1. +$$ + +Now it remains to check (56), that is, + +$$ +(\log n) \mathrm {C} _ {\sigma} ^ {2} \sum_ {k = \ell} ^ {\ell + h} \frac {c _ {0}}{k ^ {\gamma}} \leq \frac {a}{1 2}. +$$ + +Since + +$$ +\sum_ {k = \ell} ^ {\ell + h} \frac {1}{(k + k _ {0}) ^ {\gamma}} \leq \sum_ {k = 1} ^ {h + 1} \frac {1}{(k + k _ {0}) ^ {\gamma}} \leq \int_ {k _ {0}} ^ {h + 1 + k _ {0}} \frac {\mathrm {d} x}{x ^ {\gamma}} = \frac {(h + k _ {0} + 1) ^ {1 - \gamma} - (k _ {0}) ^ {1 - \gamma}}{1 - \gamma}, +$$ + +it is enough to choose $k_{0}$ in such a way that + +$$ +(\log n) \mathrm {C} _ {\sigma} ^ {2} c _ {0} \frac {(h + k _ {0} + 1) ^ {1 - \gamma} - (k _ {0}) ^ {1 - \gamma}}{1 - \gamma} \leq \frac {a}{1 2}, +$$ + +or, equivalently, + +$$ +(k _ {0}) ^ {1 - \gamma} \left(\left(1 + \frac {h + 1}{k _ {0}}\right) ^ {1 - \gamma} - 1\right) \leq \frac {a (1 - \gamma)}{1 2 (\log n) \mathrm {C} _ {\sigma} ^ {2} c _ {0}}. \tag {57} +$$ + +Since $(1 + x)^{1 - \gamma} \leq 1 + (1 - \gamma)x$ for $x \geq -1$ , (57) holds if + +$$ +k _ {0} ^ {- \gamma} (1 - \gamma) (h + 1) \leq \frac {a (1 - \gamma)}{1 2 (\log n) \mathrm {C} _ {\sigma} ^ {2} c _ {0}} , +$$ + +or, equivalently, + +$$ +k _ {0} \geq \left(\frac {1 2 (h + 1) (\log n) c _ {0} \mathrm {C} _ {\sigma} ^ {2}}{a}\right) ^ {1 / \gamma} +$$ + +which is guaranteed by the condition (25). + +![](images/7b042cc2a315cacb55c1d225c55086d493c2064dcaded2d4a864941df5c17b18.jpg) + +Lemma 14. Assume $A1$ , $A2$ , and $A3$ . Then, for any probability $\xi$ on $(Z, \mathcal{Z})$ , and any $\ell \in \{1, \dots, N - 1\}$ , we have + +$$ +\left\| \mathbf {Y} _ {\ell} - \mathbb {E} _ {\xi} \left[ \mathbf {Y} _ {\ell} \right] \right\| _ {Q} \leq \mathrm {C} _ {\sigma} \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right), \text {w h e r e} \mathrm {C} _ {\sigma} = 2 \left(\kappa_ {Q} ^ {1 / 2} \mathrm {C} _ {\mathbf {A}} + a / 6\right), \tag {58} +$$ + +and $h$ is given in (23). + +Proof. Using the decomposition (50), we obtain + +$$ +\| \mathbf {Y} _ {\ell} - \mathbb {E} _ {\boldsymbol {\xi}} [ \mathbf {Y} _ {\ell} ] \| _ {Q} \leq \sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k} \| \mathbf {A} (Z _ {k}) - \mathbb {E} _ {\boldsymbol {\xi}} [ \mathbf {A} (Z _ {k}) ] \| _ {Q} + \| \mathbf {R} _ {\ell} - \mathbb {E} _ {\boldsymbol {\xi}} [ \mathbf {R} _ {\ell} ] \| _ {Q}. +$$ + +Applying the definition of $\mathbf{R}_{\ell}$ in (51), the definition of $h,\alpha_{\infty}^{(\mathrm{M})}$ , and $T_{2}$ in (53), we get from the above inequalities that + +$$ +\| \mathbf {Y} _ {\ell} - \mathbb {E} _ {\xi} [ \mathbf {Y} _ {\ell} ] \| _ {Q} \leq 2 \kappa_ {Q} ^ {1 / 2} \mathrm {C} _ {\mathbf {A}} \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) + (a / 3) \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right), +$$ + +and the statement follows. + +![](images/a013328cc3fecc5679c145adaa6861da023cf832547a3f1bbabe794b51b5347f.jpg) + +We have now all ingredients required to prove Proposition 12. + +# D.2 Proof of Proposition 12 + +Denote by $h \in \mathbb{N}$ a block length, the value of which is determined later. Define the sequence $j_0 = j$ , $j_{\ell + 1} = \min(j_\ell + h, n)$ . By construction $j_{\ell + 1} - j_\ell \leq h$ . Let $N = \lceil (n - j) / h \rceil$ . Now we introduce the decomposition + +$$ +\Gamma_ {j: n} ^ {(\alpha)} = \prod_ {\ell = 1} ^ {N} \mathbf {Y} _ {\ell}, \quad \text {w h e r e} \quad \mathbf {Y} _ {\ell} = \left\{ \begin{array}{l} \prod_ {i = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \left(\mathrm {I} - \alpha \mathbf {A} \left(Z _ {i}\right)\right), \ell \in \{1, \dots , N - 1 \}, \\ \prod_ {i = j _ {N - 1}} ^ {n} \left(\mathrm {I} - \alpha \mathbf {A} \left(Z _ {i}\right)\right), \ell = N. \end{array} \right. \tag {59} +$$ + +The last block, $Y_{N}$ , can be of smaller size than $h$ . Now we apply the bound + +$$ +\| \mathbf {Y} _ {N} \| \leq \prod_ {k = j _ {N - 1}} ^ {n} \left(1 + \alpha_ {k} \mathrm {C} _ {\mathbf {A}}\right) \leq \exp \left\{\mathrm {C} _ {\mathbf {A}} \sum_ {k = j _ {N - 1}} ^ {N} \alpha_ {k} \right\} \leq \mathrm {e}, +$$ + +where the last bound follows from the relation (24). Hence, substituting into (59), we get the following bound: + +$$ +\mathbb {E} _ {\boldsymbol {\xi}} ^ {1 / p} [ \| \Gamma_ {j: n} ^ {(\alpha)} \| ^ {p} ] \leq \mathrm {e} \mathbb {E} _ {\boldsymbol {\xi}} ^ {1 / p} [ \| \prod_ {\ell = 1} ^ {N - 1} \mathbf {Y} _ {\ell} \| ^ {p} ]. +$$ + +Now we bound $\mathbb{E}_{\xi}^{1/p}[\|\prod_{\ell=1}^{N-1}\mathbf{Y}_{\ell}\|^{p}]$ using the results from Proposition 13. To do so, we define, for $\ell \in \{1,\dots,N-1\}$ , the filtration $\mathcal{H}_{\ell} = \sigma(Z_k : k \leq j_\ell)$ and establish almost sure bounds on $\|\mathbb{E}_{\xi}^{\mathcal{H}_{\ell-1}}[\mathbf{Y}_{\ell}]\|_Q$ and $\|\mathbf{Y}_{\ell}-\mathbb{E}_{\xi}^{\mathcal{H}_{\ell-1}}[\mathbf{Y}_{\ell}]\|_Q$ for $\ell \in \{1,\dots,N-1\}$ . More precisely, by the Markov property, it is sufficient to show that there exist $\mathfrak{m} \in (0,1]$ and $\sigma > 0$ such that for any probabilities $\xi, \xi'$ on $(\mathsf{Z},\mathcal{Z})$ , + +$$ +\left\| \mathbb {E} _ {\xi^ {\prime}} \left[ \mathbf {Y} _ {\ell} \right] \right\| _ {Q} ^ {2} \leq 1 - \mathrm {m} _ {\ell} \text {a n d} \| \mathbf {Y} _ {\ell} - \mathbb {E} _ {\xi^ {\prime}} \left[ \mathbf {Y} _ {\ell} \right] \| _ {Q} \leq \sigma_ {\ell}, \quad \mathbb {P} _ {\xi} - a. s. . \tag {60} +$$ + +Such bounds require the blocking procedure, since (60) not necessarily holds with $h = 1$ . Setting $h$ as in equation (23), and applying Lemma 12 and Lemma 14, we show that (60) hold with + +$$ +\mathfrak {m} _ {\ell} = \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) a / 6, \quad \sigma_ {\ell} = \mathrm {C} _ {\sigma} \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right), +$$ + +where $C_{\sigma}$ is given in (58). Then, applying Proposition 13 with $q = \log n$ , we get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \| \Gamma_ {1: n} ^ {(\alpha)} \| ^ {p} \right] \leq \mathbb {E} _ {\xi} ^ {1 / q} \left[ \| \Gamma_ {1: n} ^ {(\alpha)} \| ^ {q} \right] \\ \leq \sqrt {\kappa_ {Q}} d ^ {1 / q} \mathrm {e} \prod_ {\ell = 1} ^ {N - 1} \left(1 - \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) a / 6 + (\log n) \mathrm {C} _ {\sigma} ^ {2} \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) ^ {2}\right) \\ \leq \sqrt {\kappa_ {Q}} d ^ {1 / q} \mathrm {e} \prod_ {\ell = 1} ^ {N - 1} \exp \left\{- \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) a / 6 + (\log n) \mathrm {C} _ {\sigma} ^ {2} \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) ^ {2} \right\} \\ \stackrel {(a)} {\leq} \sqrt {\kappa_ {Q}} \mathrm {e} ^ {2} d ^ {1 / q} \exp \biggl \{- \bigl (\sum_ {k = j} ^ {n} \alpha_ {k} \bigr) a / 1 2 \biggr \}. \\ \end{array} +$$ + +Here in (a) we used the fact that + +$$ +\left(\log n\right) \mathrm {C} _ {\sigma} ^ {2} \left(\sum_ {k = j _ {\ell - 1}} ^ {j _ {\ell} - 1} \alpha_ {k}\right) \leq \frac {a}{1 2}, +$$ + +which holds due to A3. + +# D.3 Proof of Proposition 2 + +First part of the statement (existence of $Q$ ) follows from [67, Proposition 1]. For the second part, we note that for any non-zero vector $x \in \mathbb{R}^d$ , we have + +$$ +\begin{array}{l} \frac {x ^ {\top} (\mathrm {I} - \alpha \bar {\mathbf {A}}) ^ {\top} Q (\mathrm {I} - \alpha \bar {\mathbf {A}}) x}{x ^ {\top} Q x} = 1 - \alpha \frac {x ^ {\top} (\bar {\mathbf {A}} ^ {\top} Q + Q \bar {\mathbf {A}}) x}{x ^ {\top} Q x} + \alpha^ {2} \frac {x ^ {\top} \bar {\mathbf {A}} ^ {\top} Q \bar {\mathbf {A}} x}{x ^ {\top} Q x} \\ = 1 - \alpha \frac {x ^ {\top} P x}{x ^ {\top} Q x} + \alpha^ {2} \frac {x ^ {\top} \bar {\mathbf {A}} ^ {\top} Q \bar {\mathbf {A}} x}{x ^ {\top} Q x} \\ \leq 1 - \alpha \frac {\lambda_ {\operatorname* {m i n}} (P)}{\| Q \|} + \alpha^ {2} \| \bar {\mathbf {A}} \| _ {Q} ^ {2} \\ \leq 1 - \alpha a, \\ \end{array} +$$ + +where we set + +$$ +a = \frac {1}{2} \frac {\lambda_ {\operatorname* {m i n}} (P)}{\| Q \|}, +$$ + +and used the fact that $\alpha \leq \alpha_{\infty}$ , where $\alpha_{\infty}$ is defined in (7). + +# E Proofs of Section 4 + +# E.1 Proof of Proposition 3 + +For completeness, we recall the setting considered in the main text. Even though the sequence $\{Z_k\}_{k \in \mathbb{N}}$ forms a Markov chain, the iterates $\{\theta_k\}_{k \in \mathbb{N}}$ do not form the Markov chain. As a consequence, the standard consistency results for overlapping batch means variance estimators cannot be applied directly to (18). In this appendix, we provide the detailed derivations underlying the statement that applying the block bootstrap to the sequence $\{\theta_\ell\}$ is asymptotically equivalent, up to a suitable correction, to applying the same procedure to the latent variables $\{\varepsilon(Z_\ell)\}$ . + +Note that for any $0 \leq t \leq n - b_{n}$ , we unroll the LSA recursion only up to $\theta_{t} - \theta_{\star}$ , rather than all the way to the initial iterate $\theta_0 - \theta_\star$ : + +$$ +\theta_ {k + t} - \theta^ {\star} = \Gamma_ {t + 1: t + k} \left(\theta_ {t} - \theta^ {\star}\right) - \sum_ {\ell = t + 1} ^ {k + t} \alpha_ {\ell} \Gamma_ {\ell + 1: t + k} \varepsilon \left(Z _ {\ell}\right). \tag {61} +$$ + +While one could also unroll the recursion fully to $\theta_0 - \theta_\star$ , leading to an alternative representation suitable for the multiplier subsample bootstrap and yielding the same convergence rate, we adopt the form in (61) as it simplifies subsequent analysis and notation. + +Using (61), similarly to (13), we can extract from the MSB estimate a linear statistic that depends on $\{\varepsilon(Z_{\ell})\}$ and get + +$$ +\begin{array}{l} \bar {\theta} _ {n, b _ {n}} (u) = \frac {\sqrt {b _ {n}}}{\sqrt {n - b _ {n} + 1}} \sum_ {t = 0} ^ {n - b _ {n}} w _ {t} \left(\bar {\theta} _ {b _ {n}, t} - \bar {\theta} _ {n}\right) ^ {\top} u \tag {62} \\ = \frac {\sqrt {b _ {n}}}{\sqrt {n - b _ {n} + 1}} \sum_ {t = 0} ^ {n - b _ {n}} w _ {t} (\tilde {W} _ {b _ {n}, t} - \tilde {W} _ {n}) ^ {\top} u + \frac {\sqrt {b _ {n}}}{\sqrt {n - b _ {n} + 1}} \sum_ {t = 0} ^ {n - b _ {n}} w _ {t} (\tilde {D} _ {b _ {n}, t} - \tilde {D} _ {n}) ^ {\top} u, \\ \end{array} +$$ + +where we have set + +$$ +\tilde {W} _ {b _ {n}, t} = - \frac {1}{b _ {n}} \sum_ {k = 1} ^ {b _ {n} - 1} Q _ {k, t, b _ {n}} \varepsilon (Z _ {k + t}), \qquad \tilde {W} _ {n} = - \frac {1}{n} \sum_ {k = 1} ^ {n - 1} Q _ {k} \varepsilon (Z _ {k}), +$$ + +$$ +\tilde {D} _ {b _ {n}, t} = \frac {1}{b _ {n}} \sum_ {k = 1} ^ {b _ {n} - 1} \Gamma_ {t + 1: t + k} \left(\theta_ {t} - \theta^ {\star}\right) + \frac {1}{b _ {n}} \sum_ {k = 1} ^ {b _ {n} - 1} H _ {k, t, b _ {n}} ^ {(0)} \varepsilon \left(Z _ {k + t}\right) +$$ + +$$ +\tilde {D} _ {n} = \frac {1}{n} \sum_ {k = 0} ^ {n - 1} \Gamma_ {1: k} (\theta_ {0} - \theta^ {\star}) + \frac {1}{n} \sum_ {k = 1} ^ {n - 1} H _ {k} ^ {(0)} \varepsilon (Z _ {\ell}), +$$ + +and $Q_{k,t,b_n}, H_{k,t,b_n}^{(0)}, J_{k,t,b_n}^{(0)}$ are analogues of $Q_k, H_k^{(0)}, J_k^{(0)}$ for the bootstrap procedure and are defined as follows + +$$ +Q _ {k, t, b _ {n}} = \alpha_ {k + t} \sum_ {\ell = k} ^ {b _ {n} - 1} G _ {k + t + 1: \ell + t}, +$$ + +$$ +H _ {k, t, b _ {n}} ^ {(0)} = - \sum_ {\ell = t + 1} ^ {k + t} \alpha_ {\ell} \Gamma_ {\ell + 1: t + k} \tilde {\mathbf {A}} (Z _ {\ell}) J _ {\ell - 1, t, b _ {n}} ^ {(0)} +$$ + +$$ +J _ {k, t, b _ {n}} ^ {(0)} = - \sum_ {\ell = t + 1} ^ {k + t} \alpha_ {\ell} G _ {\ell + 1: k + t} \varepsilon (Z _ {\ell}). +$$ + +Since existing results on the concentration of overlapping batch means variance estimators for Markov chains do not consider weighted sums, we replace the leading terms $\tilde{W}_{b_n,t}$ and $\tilde{W}_n$ with expressions + +involving the constant matrix $\bar{\mathbf{A}}^{-1}$ and obtain + +$$ +\begin{array}{l} \bar {\theta} _ {n, b _ {n}} (u) = \frac {\sqrt {b _ {n}}}{\sqrt {n - b _ {n} + 1}} \sum_ {t = 0} ^ {n - b _ {n}} w _ {t} \left(\bar {\theta} _ {b _ {n}, t} - \bar {\theta} _ {n}\right) ^ {\top} u \\ = \frac {\sqrt {b _ {n}}}{\sqrt {n - b _ {n} + 1}} \sum_ {t = 0} ^ {n - b _ {n}} w _ {t} \left(\bar {W} _ {b _ {n}, t} - \bar {W} _ {n}\right) ^ {\top} u + \frac {\sqrt {b _ {n}}}{\sqrt {n - b _ {n} + 1}} \sum_ {t = 0} ^ {n - b _ {n}} w _ {t} \left(\bar {D} _ {b _ {n}, t} - \bar {D} _ {n}\right) ^ {\top} u , \\ \end{array} +$$ + +where we have set + +$$ +\bar {W} _ {b _ {n}, t} = - \frac {1}{b _ {n}} \sum_ {k = 1} ^ {b _ {n} - 1} \bar {\mathbf {A}} ^ {- 1} \varepsilon (Z _ {k + t}), \qquad \bar {W} _ {n} = - \frac {1}{n} \sum_ {k = 1} ^ {n - 1} \bar {\mathbf {A}} ^ {- 1} \varepsilon (Z _ {k}), +$$ + +$$ +\bar {D} _ {b _ {n}, t} = \frac {1}{b _ {n}} \sum_ {k = 1} ^ {b _ {n} - 1} \Gamma_ {t + 1: t + k} \left(\theta_ {t} - \theta^ {\star}\right) + \frac {1}{b _ {n}} \sum_ {k = 1} ^ {b _ {n} - 1} H _ {k, t, b _ {n}} ^ {(0)} - \frac {1}{b _ {n}} \sum_ {k = 1} ^ {b _ {n} - 1} \left(Q _ {k, t, b _ {n}} - \bar {\mathbf {A}} ^ {- 1}\right) \varepsilon \left(Z _ {k + t}\right) +$$ + +$$ +\bar {D} _ {n} = \frac {1}{n} \sum_ {k = 0} ^ {n - 1} \Gamma_ {1: k} \left(\theta_ {0} - \theta^ {\star}\right) + \frac {1}{n} \sum_ {k = 1} ^ {n - 1} H _ {k} ^ {(0)} - \frac {1}{n} \sum_ {\ell = 1} ^ {n - 1} \left(Q _ {\ell} - \bar {\mathbf {A}} ^ {- 1}\right) \varepsilon \left(Z _ {\ell}\right), +$$ + +Then we can decompose the variance w.r.t. $\mathbb{P}^{\mathrm{b}}$ of $\bar{\theta}_{n,b_n}(u)$ and extract the variance of block bootstrap procedure applied to the non-observable random variables $\{\varepsilon (Z_{\ell})\}$ + +$$ +\begin{array}{l} \hat {\sigma} _ {\theta} ^ {2} (u) = \frac {b _ {n}}{n - b _ {n} + 1} \sum_ {t = 0} ^ {n - b _ {n}} \left(\left(\bar {\theta} _ {b _ {n}, t} - \bar {\theta} _ {n}\right) ^ {\top} u\right) ^ {2} \\ = \underbrace {\frac {b _ {n}}{n - b _ {n} + 1} \sum_ {t = 0} ^ {n - b _ {n}} \left(\left(\bar {W} _ {b _ {n} , t} - \bar {W} _ {n}\right) ^ {\top} u\right) ^ {2}} _ {\hat {\sigma} _ {\varepsilon} ^ {2} (u)} + \mathcal {R} _ {v a r} (u) \\ \end{array} +$$ + +where we define the remainder term as follows + +$$ +\mathcal {R} _ {v a r} (u) = D _ {1} ^ {b} + D _ {2} ^ {b} +$$ + +$$ +D _ {1} ^ {b} = \frac {b _ {n}}{n - b _ {n} + 1} \sum_ {t = 0} ^ {n - b _ {n}} \left(\left(\bar {D} _ {b _ {n}, t} - \bar {D} _ {n}\right) ^ {\top} u\right) ^ {2} \tag {63} +$$ + +$$ +D _ {2} ^ {b} = \frac {2 b _ {n}}{n - b _ {n} + 1} \sum_ {t = 0} ^ {n - b _ {n}} u ^ {\top} \left(\bar {W} _ {b _ {n}, t} - \bar {W} _ {n}\right) \left(\bar {D} _ {b _ {n}, t} - \bar {D} _ {n}\right) ^ {\top} u +$$ + +To bound $\mathcal{R}_{var}(u)$ it remains to bound $\mathbb{E}_{\xi}^{1/p}[|D_1^{\tt b}|^p]$ and $\mathbb{E}_{\xi}^{1/p}[|D_2^{\tt b}|^p]$ . Hence, from Lemma 18 and Lemma 19, we obtain the following bound for the residual term $\mathcal{R}_{var}(u)$ : + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| \mathcal {R} _ {v a r} (u) \right| ^ {p} \right] \lesssim p b _ {n} ^ {1 / 2} n ^ {\gamma / 2 - 1} + p ^ {4} (\log n) b _ {n} ^ {1 / 2} n ^ {- \gamma} \tag {64} \\ + p b _ {n} ^ {- 1 / 2} n ^ {\gamma / 2} + p ^ {4} (\log n) n ^ {- 1} + p n ^ {2 \gamma - 2}. \\ \end{array} +$$ + +The version of this results with constants, you can see in (65). Below we present some technical lemmas that are necessary in order to bound the residual term $\mathcal{R}_{var}(u)$ + +# E.2 Technical bounds + +Bounds for $\bar{D}_{b_n,t}$ and $\bar{D}_n$ . First, we establish bounds for $\bar{D}_{b_n,t}$ and $\bar{D}_n$ . The argument follows the same reasoning as in the proof of the bound for $D_1$ in (27). The main difference lies in the appearance of additional terms arising from the modification of the linear component in the decomposition (62). + +Lemma 15. Assume A1, A2 and A3. Then it holds + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ | u ^ {\top} \bar {D} _ {b _ {n}, t} | ^ {p} \right] \lesssim \frac {\mathrm {C} _ {1 , 1} ^ {\mathrm {D} , \mathrm {b}} p ^ {1 / 2}}{b _ {n} \sqrt {\alpha_ {t}}} + \frac {\mathrm {C} _ {1 , 2} ^ {\mathrm {D} , \mathrm {b}}}{b _ {n}} + \mathrm {C} _ {1, 3} ^ {\mathrm {D}, \mathrm {b}} p ^ {2} \sqrt {\log (1 / \alpha_ {b _ {n} + t - 1})} \frac {\sum_ {k = 1 + t} ^ {b _ {n} + t - 1} \alpha_ {k}}{b _ {n}} \\ + \mathrm {C} _ {1, 4} ^ {\mathrm {D}, \mathbf {b}} \frac {p ^ {1 / 2}}{b _ {n}} (\sum_ {k = 2} ^ {b _ {n} - 1} (k + k _ {0} + t) ^ {2 \gamma - 2}) ^ {1 / 2} + \frac {\mathrm {C} _ {1 , 5} ^ {\mathrm {D} , \mathbf {b}} p ^ {1 / 2} b _ {n} ^ {- 1}}{\sqrt {\alpha_ {t + b _ {n} - 2}}} + \mathrm {C} _ {1, 6} ^ {\mathrm {D}, \mathbf {b}} \frac {(k _ {0} + t - 1) ^ {\gamma - 1}}{b _ {n}}, \\ \end{array} +$$ + +where + +$$ +C _ {1, 1} ^ {\mathsf {D}, \mathsf {b}} = \frac {C _ {\Gamma} d ^ {1 / \log n + 1 / 2} \mathsf {D} _ {1} ^ {(\mathrm {M})} t _ {\mathrm {m i x}}}{a (1 - \gamma)} +$$ + +$$ +\mathrm {C} _ {1, 2} ^ {\mathrm {D}, \mathsf {b}} = \frac {k _ {0} ^ {\gamma}}{a c _ {0} (1 - \gamma)} C _ {\Gamma} ^ {2} d ^ {2 / \log n + 1 / 2} \| \theta_ {0} - \theta^ {\star} \| + t _ {\mathrm {m i x}} \| \varepsilon \| _ {\infty} (\sqrt {\kappa_ {Q}} C _ {\gamma , a} ^ {(S)} k _ {0} ^ {\gamma - 1} + \sqrt {\kappa_ {Q}} (\| \bar {\mathbf {A}} ^ {- 1} \| + (8 / a)) +$$ + +$$ +\mathrm {C} _ {1, 3} ^ {\mathrm {D}, \mathfrak {b}} = \mathrm {D} _ {3} ^ {(\mathrm {M})} t _ {\mathrm {m i x}} + t _ {\mathrm {m i x}} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}} C _ {\gamma , a} ^ {(S, 2)} +$$ + +$$ +\mathrm {C} _ {1, 4} ^ {\mathrm {D}, \mathfrak {b}} = t _ {\operatorname {m i x}} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}} C _ {\gamma , a} ^ {(S)} +$$ + +$$ +\mathrm {C} _ {1, 5} ^ {\mathrm {D}, \mathrm {b}} = \frac {\| A ^ {- 1} \| t _ {\operatorname* {m i x}} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}}}{\sqrt {a}} +$$ + +$$ +\mathrm {C} _ {1, 6} ^ {\mathsf {D}, \mathsf {b}} = t _ {\operatorname {m i x}} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}} C _ {\gamma , a} ^ {(S)} +$$ + +Proof. We split $\bar{D}_{b_n,t}$ into three parts: + +$$ +T _ {1} = \frac {1}{b _ {n}} \sum_ {k = 1} ^ {b _ {n} - 1} u ^ {\top} \Gamma_ {t + 1: t + k} \left(\theta_ {t} - \theta^ {\star}\right), +$$ + +$$ +T _ {2} = \frac {1}{b _ {n}} \sum_ {k = 1} ^ {b _ {n} - 1} u ^ {\top} H _ {k, t, b _ {n}} ^ {(0)}, +$$ + +$$ +T _ {3} = - \frac {1}{b _ {n}} \sum_ {k = 1} ^ {b _ {n} - 1} u ^ {\top} \left(Q _ {k, t, b _ {n}} - \bar {\mathbf {A}} ^ {- 1}\right) \varepsilon \left(Z _ {k + t}\right). +$$ + +We start from $T_{1}$ . Using Minkowski's and Hölder's inequality, we obtain + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} [ | u ^ {\top} T _ {1} | ^ {p} ] \leq \frac {1}{b _ {n}} \sum_ {k = 1} ^ {b _ {n} - 1} (\mathbb {E} _ {\xi} ^ {1 / p} [ \| \Gamma_ {t + 1: t + k} \| ^ {2 p} ]) ^ {1 / (2 p)} (\mathbb {E} _ {\xi} ^ {1 / p} [ \| (\theta_ {t} - \theta^ {\star}) \| ^ {2 p} ]) ^ {1 / (2 p)}. +$$ + +Applying similar arguments as in (38) together with Proposition 12 and Proposition 8 we get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} T _ {1} \right| ^ {p} \right] \lesssim \frac {1}{b _ {n}} \sum_ {k = 1} ^ {b _ {n} - 1} C _ {\Gamma} d ^ {1 / \log n + 1 / 2} \exp \left\{- (a / 1 2) \sum_ {\ell = t + 1} ^ {t + k} \alpha_ {\ell} \right\} \mathrm {D} _ {1} ^ {(\mathrm {M})} t _ {\operatorname {m i x}} \sqrt {p \alpha_ {t}} \\ + \frac {1}{b _ {n}} \sum_ {k = 1} ^ {b _ {n} - 1} C _ {\Gamma} ^ {2} d ^ {2 / \log n + 1 / 2} \exp \left\{- (a / 1 2) \sum_ {\ell = 1} ^ {t + k} \alpha_ {\ell} \right\} \| \theta_ {0} - \theta^ {\star} \| . \\ \end{array} +$$ + +Using Lemma 24, Lemma 31 and that $k_0^{1 - \gamma} > \frac{24}{ac_0}$ we get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ | u ^ {\top} T _ {1} | ^ {p} \right] \lesssim \frac {1}{b _ {n} \sqrt {\alpha_ {t}} a (1 - \gamma)} C _ {\Gamma} d ^ {1 / \log n + 1 / 2} D _ {1} ^ {(M)} t _ {\text {m i x}} \sqrt {p} \\ + \frac {k _ {0} ^ {\gamma} \exp \left\{- \frac {a c _ {0}}{2 4 (1 - \gamma)} t ^ {1 - \gamma} \right\}}{b _ {n} a c _ {0} (1 - \gamma)} C _ {\Gamma} ^ {2} d ^ {2 / \log n + 1 / 2} \| \theta_ {0} - \theta^ {\star} \| . \\ \end{array} +$$ + +For $T_{2}$ we use Proposition 10 and Minkowski's inequality and get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ | T _ {2} | ^ {p} \right] \leq \frac {1}{b _ {n}} \sum_ {k = 1} ^ {b _ {n} - 1} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \| u ^ {\top} H _ {k, t, b _ {n}} ^ {(0)} \| ^ {p} \right] \leq \frac {\mathrm {D} _ {3} ^ {(\mathrm {M})} t _ {\operatorname* {m i x}} p ^ {2}}{b _ {n}} \sum_ {k = 1 + t} ^ {b _ {n} + t - 1} \alpha_ {k} \sqrt {\log \left(1 / \alpha_ {k}\right)} \\ \lesssim \mathsf {D} _ {3} ^ {\mathrm {(M)}} t _ {\mathrm {m i x}} p ^ {2} \sqrt {\log (1 / \alpha_ {b _ {n} + t - 1})} \frac {\sum_ {k = 1 + t} ^ {b _ {n} + t - 1} \alpha_ {k}}{b _ {n}}. \\ \end{array} +$$ + +We first use Lemma 4 to show that + +$$ +T _ {3} = - \frac {1}{b _ {n}} \sum_ {\ell = 1} ^ {b _ {n} - 1} u ^ {\top} S _ {\ell , t, b _ {n}} \varepsilon (Z _ {\ell + t}) + \frac {1}{b _ {n}} \sum_ {\ell = 1} ^ {b _ {n} - 1} u ^ {\top} \bar {\mathbf {A}} ^ {- 1} G _ {\ell + t: b _ {n} + t - 1} \varepsilon (Z _ {\ell + t}) = T _ {3 1} + T _ {3 2}, +$$ + +where $S_{\ell, t, b_n} = \sum_{j=\ell+1}^{b_n-1} (\alpha_{k+t} - \alpha_{j+t}) G_{l+t+1:j+t-1}$ . It is easy to see that $T_{31}$ is a weighed linear statistics of MC. We apply Markov property together with Lemma 20 and obtain + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} [ | T _ {3 1} | ^ {p} ] \lesssim \frac {t _ {\mathrm {m i x}} p ^ {1 / 2} \| \varepsilon \| _ {\infty}}{b _ {n}} (\sum_ {k = 2} ^ {b _ {n} - 1} \| S _ {k, t, b _ {n}} \| ^ {2}) ^ {1 / 2} \\ + \frac {t _ {\operatorname* {m i x}} \| \varepsilon \| _ {\infty}}{b _ {n}} \left(\| S _ {1, t, b _ {n}} \| + \| S _ {b _ {n} - 1, t, b _ {n}} \| + \sum_ {k = 1} ^ {b _ {n} - 2} \| S _ {k + 1, t, b _ {n}} - S _ {k, t, b _ {n}} \|\right). \\ \end{array} +$$ + +Applying Lemma 5 and Lemma 7, we get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ | T _ {3 1} | ^ {p} \right] \lesssim \frac {t _ {\operatorname* {m i x}} p ^ {1 / 2} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}} C _ {\gamma , a} ^ {(S)}}{b _ {n}} (\sum_ {k = 2} ^ {b _ {n} - 1} (k + k _ {0} + t) ^ {2 \gamma - 2}) ^ {1 / 2} \\ + \frac {t _ {\mathrm {m i x}} \| \varepsilon \| _ {\infty}}{b _ {n}} \big (\sqrt {\kappa_ {Q}} C _ {\gamma , a} ^ {(S)} ((k _ {0} + t + 1) ^ {\gamma - 1} + (b _ {n} + k _ {0} + t - 1) ^ {\gamma - 1}) \\ + \sqrt {\kappa_ {Q}} C _ {\gamma , a} ^ {(S, 2)} \sum_ {k = 1} ^ {b _ {n} - 2} \alpha_ {k + 1 + t}) \\ \end{array} +$$ + +For term $T_{32}$ we also apply Lemma 20 and get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} [ | T _ {3 2} | ^ {p} ] \lesssim \frac {\| \bar {\mathbf {A}} ^ {- 1} \| t _ {\mathrm {m i x}} p ^ {1 / 2} \| \varepsilon \| _ {\infty}}{b _ {n}} (\sum_ {k = 2} ^ {b _ {n} - 1} \| G _ {k + t: b _ {n} + t - 1} \| ^ {2}) ^ {1 / 2} \\ + \frac {t _ {\operatorname* {m i x}} \| \varepsilon \| _ {\infty}}{b _ {n}} \big (\| \bar {\mathbf {A}} ^ {- 1} \| \big (\| G _ {1 + t: b _ {n} + t - 1} \| + \| G _ {b _ {n} + t - 1: b _ {n} + t - 1} \| \big) + \sum_ {k = 1} ^ {b _ {n} - 2} \| \bar {\mathbf {A}} ^ {- 1} (G _ {k + 1 + t: b _ {n} + t - 1} - G _ {k + t: b _ {n} + t - 1}) \| \big). \\ \end{array} +$$ + +Using Lemma 8, Lemma 26 and Lemma 28 with $b = a / 2$ we get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} [ | T _ {3 2} | ^ {p} ] \lesssim \frac {\| \bar {\mathbf {A}} ^ {- 1} \| t _ {\mathrm {m i x}} p ^ {1 / 2} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}}}{\sqrt {a c _ {0}} b _ {n}} (b _ {n} + t + k _ {0} - 2) ^ {\gamma / 2} \\ + \frac {t _ {\mathrm {m i x}} \| \varepsilon \| _ {\infty}}{b _ {n}} \big (\| \bar {\mathbf {A}} ^ {- 1} \| (\alpha_ {b _ {n} + t - 1} / \alpha_ {t} + \sqrt {\kappa_ {Q}} (1 - (a / 2) \alpha_ {b _ {n} + t - 1})) + \sqrt {\kappa_ {Q}} \sum_ {k = 1} ^ {b _ {n} - 1} \alpha_ {k + t} \prod_ {i = k + t + 1} ^ {b _ {n} + t - 1} (1 - (a / 2) \alpha_ {i}) \big) \\ \lesssim \frac {\| \bar {\mathbf {A}} ^ {- 1} \| t _ {\mathrm {m i x}} p ^ {1 / 2} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}}}{\sqrt {a b _ {n} \sqrt {\alpha_ {b _ {n} + t - 2}}}} \\ + \frac {t _ {\operatorname* {m i x}} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}}}{b _ {n}} \left(\| \bar {\mathbf {A}} ^ {- 1} \| + (8 / a)\right). \\ \end{array} +$$ + +![](images/1191a38bde9da379bfc18fd0c3781ac6370881e273e30d09868f6c47a0f55468.jpg) + +Lemma 16. Assume A1, A2 and A3. Then it holds + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} \bar {D} _ {n} \right| ^ {p} \right] \lesssim \frac {\mathrm {C} _ {2 , 1} ^ {\mathrm {D} , \mathrm {b}}}{n} + \mathrm {C} _ {2, 2} ^ {\mathrm {D}, \mathrm {b}} p ^ {2} n ^ {- \gamma} + \mathrm {C} _ {2, 3} ^ {\mathrm {D}, \mathrm {b}} p ^ {1 / 2} n ^ {\gamma - 3 / 2} \\ + \mathrm {C} _ {2, 4} ^ {\mathsf {D}, \mathsf {b}} p ^ {1 / 2} n ^ {\gamma / 2 - 1} + \mathrm {C} _ {2, 5} ^ {\mathsf {D}, \mathsf {b}} p ^ {2} \sqrt {\log n n} ^ {- \gamma}, \\ \end{array} +$$ + +where + +$$ +\mathrm {C} _ {2, 1} ^ {\mathrm {D}, \mathsf {b}} = C _ {\Gamma} d ^ {1 / \log n} \frac {k _ {0} ^ {\gamma}}{a c _ {0} (1 - \gamma)} \| \theta_ {0} - \theta^ {\star} \| + t _ {\mathrm {m i x}} \| \varepsilon \| _ {\infty} (\sqrt {\kappa_ {Q}} (\| \bar {\mathbf {A}} ^ {- 1} \| + 1 / a) + \sqrt {\kappa_ {Q}} C _ {\gamma , a} ^ {(S)} k _ {0} ^ {\gamma - 1}) +$$ + +$$ +\mathrm {C} _ {2, 2} ^ {\mathrm {D}, \mathrm {b}} = \frac {\mathsf {D} _ {3} ^ {\mathrm {(M)}} t _ {\mathrm {m i x}} \sqrt {\log \frac {k _ {0} ^ {\gamma}}{c _ {0}}}}{c _ {0} (1 - \gamma)} + \frac {\sqrt {\kappa_ {Q}} t _ {\mathrm {m i x}} \| \varepsilon \| _ {\infty} C _ {\gamma , a} ^ {(S , 2)} c _ {0}}{1 - \gamma} +$$ + +$$ +\mathrm {C} _ {2, 3} ^ {\mathrm {D}, \mathrm {b}} = \frac {t _ {\mathrm {m i x}} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}} C _ {\gamma , a} ^ {(S)}}{\sqrt {2 \gamma - 1}} +$$ + +$$ +\mathrm {C} _ {2, 4} ^ {\mathrm {D}, \mathrm {b}} = \frac {\| \bar {\mathbf {A}} ^ {- 1} \| t _ {\mathrm {m i x}} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}} k _ {0} ^ {\gamma / 2}}{\sqrt {a c _ {0}}} +$$ + +$$ +\mathrm {C} _ {2, 5} ^ {\mathrm {D}, \mathrm {b}} = \frac {\mathrm {D} _ {3} ^ {(\mathrm {M})} t _ {\mathrm {m i x}} c _ {0} \sqrt {\gamma}}{1 - \gamma} +$$ + +Proof. Using (28) and (29) we get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ u ^ {\top} \left(\frac {1}{n} \sum_ {k = 0} ^ {n - 1} \Gamma_ {1: k} \left(\theta_ {0} - \theta^ {\star}\right) + \frac {1}{n} \sum_ {k = 1} ^ {n - 1} H _ {k} ^ {(0)}\right) | ^ {p} \right] \lesssim C _ {\Gamma} d ^ {1 / \log n} \frac {1}{n} \frac {k _ {0} ^ {\gamma}}{a c _ {0} (1 - \gamma)} \| \theta_ {0} - \theta^ {\star} \| \\ + \frac {\mathrm {D} _ {3} ^ {(\mathrm {M})} t _ {\mathrm {m i x}} p ^ {2} c _ {0}}{1 - \gamma} (\sqrt {\gamma \log n} + \sqrt {\log \frac {k _ {0} ^ {\gamma}}{c _ {0}}}) n ^ {- \gamma}. \\ \end{array} +$$ + +For simplicity we define + +$$ +T _ {4} = - u ^ {\top} \frac {1}{n} \sum_ {\ell = 1} ^ {n - 1} (Q _ {\ell} - \bar {\mathbf {A}} ^ {- 1}) \varepsilon (Z _ {\ell}) +$$ + +We first use Lemma 4 to show that + +$$ +T _ {4} = - \frac {1}{n} \sum_ {\ell = 1} ^ {n - 1} u ^ {\top} S _ {\ell} \varepsilon (Z _ {\ell}) + \frac {1}{n} \sum_ {\ell = 1} ^ {n - 1} u ^ {\top} \bar {\mathbf {A}} ^ {- 1} G _ {\ell ; n - 1} \varepsilon (Z _ {\ell}) = T _ {4 1} + T _ {4 2}. +$$ + +It is easy to see that $T_{41}$ is a weighed linear statistics of MC. We apply Lemma 20, + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ | T _ {4 1} | ^ {p} \right] \lesssim \frac {t _ {\operatorname* {m i x}} p ^ {1 / 2} \| \varepsilon \| _ {\infty}}{n} \left(\sum_ {k = 2} ^ {n - 1} \| S _ {k} \| ^ {2}\right) ^ {1 / 2} \\ + \frac {t _ {\operatorname* {m i x}} \| \varepsilon \| _ {\infty}}{n} \left(\| S _ {1} \| + \| S _ {n - 1} \| + \sum_ {k = 1} ^ {n - 2} \| S _ {k + 1} - S _ {k} \|\right). \\ \end{array} +$$ + +Applying Lemma 5 and Lemma 7, we get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} [ | T _ {4 1} | ^ {p} ] \lesssim \frac {t _ {\mathrm {m i x}} p ^ {1 / 2} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}} C _ {\gamma , a} ^ {(S)}}{\sqrt {2 \gamma - 1}} n ^ {\gamma - 3 / 2} \\ + \frac {t _ {\operatorname* {m i x}} \| \varepsilon \| _ {\infty}}{n} \big (\sqrt {\kappa_ {Q}} C _ {\gamma , a} ^ {(S)} ((k _ {0} + 1) ^ {\gamma - 1} + (n + k _ {0} - 1) ^ {\gamma - 1}) + \frac {\sqrt {\kappa_ {Q}} C _ {\gamma , a} ^ {(S , 2)} c _ {0}}{1 - \gamma} n ^ {1 - \gamma} \big) \\ \lesssim \frac {t _ {\mathrm {m i x}} p ^ {1 / 2} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}} C _ {\gamma , a} ^ {(S)}}{\sqrt {2 \gamma - 1}} n ^ {\gamma - 3 / 2} + t _ {\mathrm {m i x}} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}} C _ {\gamma , a} ^ {(S)} k _ {0} ^ {\gamma - 1} n ^ {- 1} \\ + \frac {\sqrt {\kappa_ {Q}} t _ {\mathrm {m i x}} \| \varepsilon \| _ {\infty} C _ {\gamma , a} ^ {(S , 2)} c _ {0}}{1 - \gamma} n ^ {- \gamma} \\ \end{array} +$$ + +For term $T_{42}$ we also apply Lemma 20 and get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} [ | T _ {4 2} | ^ {p} ] \lesssim \frac {\| \bar {\mathbf {A}} ^ {- 1} \| t _ {\mathrm {m i x}} p ^ {1 / 2} \| \varepsilon \| _ {\infty}}{n} (\sum_ {k = 2} ^ {n - 1} \| G _ {k: n - 1} \| ^ {2}) ^ {1 / 2} \\ + \frac {t _ {\operatorname* {m i x}} \| \varepsilon \| _ {\infty}}{n} \left(\| \bar {\mathbf {A}} ^ {- 1} \| \left(\| G _ {1: n - 1} \| + \| G _ {n - 1: n - 1} \|\right) + \sum_ {k = 1} ^ {n - 2} \| \bar {\mathbf {A}} ^ {- 1} \left(G _ {k + 1: n - 1} - G _ {k: n - 1}\right) \|\right). \\ \end{array} +$$ + +Using Lemma 8, Lemma 26 and Lemma 28 with $b = a / 2$ we get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} [ | T _ {3 2} | ^ {p} ] \lesssim \frac {\| \bar {\mathbf {A}} ^ {- 1} \| t _ {\mathrm {m i x}} p ^ {1 / 2} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}}}{\sqrt {a c _ {0}} n} (n + k _ {0} - 2) ^ {\gamma / 2} \\ + \frac {t _ {\operatorname* {m i x}} \| \varepsilon \| _ {\infty}}{n} \left(\| \bar {\mathbf {A}} ^ {- 1} \| \left(\alpha_ {n - 1} / \alpha_ {0} + \sqrt {\kappa_ {Q}} (1 - (a / 2) \alpha_ {n - 1})\right) + \sqrt {\kappa_ {Q}} (8 / a)\right) \\ \lesssim \frac {\| \bar {\mathbf {A}} ^ {- 1} \| t _ {\mathrm {m i x}} p ^ {1 / 2} \| \varepsilon \| _ {\infty} \sqrt {\kappa_ {Q}} k _ {0} ^ {\gamma / 2}}{\sqrt {a c _ {0}}} n ^ {\gamma / 2 - 1} \\ + \frac {t _ {\operatorname* {m i x}} \| \varepsilon \| _ {\infty} (\| \bar {\mathbf {A}} ^ {- 1} \| + \sqrt {\kappa_ {Q}})}{n}. \\ \end{array} +$$ + +![](images/9cb6e8ab7ad4d49460a5f397ab597dd7ecb0a7abf67a97fed53cd308767ff768.jpg) + +Bounds for $\bar{W}_{b_n,t}$ and $\bar{W}_n$ . Note that the terms $\bar{W} b_{n},t$ and $\bar{W} n$ are linear statistics of a Markov chain. Their moments can therefore be bounded directly using our Rosenthal-type inequality (see Lemma 20). + +Lemma 17. Assume A1, A2 and A3. Then in holds + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} \bar {W} _ {n} \right| ^ {p} \right] \lesssim \mathrm {C} _ {3, 1} ^ {\mathrm {D}, \mathrm {b}} p ^ {1 / 2} n ^ {- 1 / 2} + \frac {\mathrm {C} _ {3 , 2} ^ {\mathrm {D} , \mathrm {b}}}{n}, +$$ + +and + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} \bar {W} _ {b _ {n}, t} \right| ^ {p} \right] \lesssim \mathrm {C} _ {3, 1} ^ {\mathrm {D}, \mathrm {b}} p ^ {1 / 2} b _ {n} ^ {- 1 / 2} + \frac {\mathrm {C} _ {3 , 2} ^ {\mathrm {D} , \mathrm {b}}}{b _ {n}}, +$$ + +where + +$$ +\mathrm {C} _ {3, 1} ^ {\mathrm {D}, \mathrm {b}} = t _ {\operatorname {m i x}} \| \bar {\mathbf {A}} ^ {- 1} \| \| \varepsilon \| _ {\infty} +$$ + +$$ +\mathrm {C} _ {3, 2} ^ {\mathrm {D}, \mathrm {b}} = t _ {\mathrm {m i x}} \| \bar {\mathbf {A}} ^ {- 1} \| . +$$ + +Proof. Using Lemma 20, we get + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| u ^ {\top} \bar {W} _ {n} \right| ^ {p} \right] \lesssim \frac {t _ {\operatorname* {m i x}} p ^ {1 / 2} \| \bar {\mathbf {A}} ^ {- 1} \| \| \varepsilon \| _ {\infty}}{\sqrt {n}} + \frac {t _ {\operatorname* {m i x}} \| \bar {\mathbf {A}} ^ {- 1} \|}{n} +$$ + +The boundary for $\bar{W}_{b_n,t}$ is obtained similarly. + +![](images/193a1f7ceaf18a80c7eb6b94df5456fe2c2ddbc07dfe1a23dc0b2a2fd0f69990.jpg) + +Bounds for $D_1^{\mathrm{b}}$ and $D_2^{\mathrm{b}}$ . In this section, the previously obtained bounds for $\bar{W}_{b_n,t}$ , $\bar{D}_{b_n,t}$ , $\bar{W}_n$ , and $\bar{D}_n$ are combined to establish corresponding bounds for $D_1^{\mathrm{b}}$ and $D_2^{\mathrm{b}}$ in $\mathcal{R}_{var}(u)$ . + +Lemma 18. Assume A1, A2 and A3. Then it holds + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| D _ {1} ^ {b} \right| ^ {p} \right] \lesssim M _ {1, 1} p n ^ {\gamma} b _ {n} ^ {- 1} + M _ {1, 2} b _ {n} ^ {- 1} + M _ {1, 3} p ^ {4} (\log n) n ^ {- 1} + M _ {1, 4} p n ^ {2 \gamma - 2} \\ + M _ {2, 1} b _ {n} n ^ {- 2} + M _ {2, 2} p ^ {4} b _ {n} (\log n) n ^ {- 2 \gamma} + M _ {2, 3} p b _ {n} n ^ {2 \gamma - 3} + M _ {2, 4} p b _ {n} n ^ {\gamma - 2}, \\ \end{array} +$$ + +where + +$$ +\begin{array}{l} M _ {1, 1} = \frac {(2 k _ {0}) ^ {\gamma + 1} ((C _ {1 , 1} ^ {D , b}) ^ {2} + (C _ {1 , 5} ^ {D , b}) ^ {2})}{c _ {0} (\gamma + 1)} \\ M _ {1, 2} = \left(\mathrm {C} _ {1, 2} ^ {\mathrm {D}, \mathrm {b}}\right) ^ {2} + \left(\mathrm {C} _ {1, 6} ^ {\mathrm {D}, \mathrm {b}}\right) ^ {2} \\ M _ {1, 3} = \left(\mathrm {C} _ {1, 3} ^ {\mathrm {D}, \mathrm {b}}\right) ^ {2} (\gamma + \log \frac {k _ {0} ^ {\gamma}}{c _ {0}}) \frac {c _ {0} ^ {2}}{2 \gamma - 1} \\ M _ {1, 4} = (\mathrm {C} _ {1, 4} ^ {\mathrm {D}, \mathrm {b}}) ^ {2} \frac {2 ^ {2 \gamma - 1} - 1}{2 \gamma - 1} \\ M _ {2, 1} = (C _ {2, 1} ^ {\mathrm {D}, \mathrm {b}}) ^ {2} \\ M _ {2, 2} = \left(\mathrm {C} _ {2, 2} ^ {\mathrm {D}, \mathrm {b}}\right) ^ {2} + \left(\mathrm {C} _ {2, 5} ^ {\mathrm {D}, \mathrm {b}}\right) ^ {2} \\ M _ {2, 3} = (C _ {2, 3} ^ {\mathrm {D}, \mathrm {b}}) ^ {2} \\ M _ {2, 4} = \left(\mathrm {C} _ {2, 4} ^ {\mathrm {D}, \mathrm {b}}\right) ^ {2}. \\ \end{array} +$$ + +Proof. Note that using Minkowski's inequality and Hölder's inequality, we have + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} [ | D _ {1} ^ {b} | ^ {p} ] \leq \frac {b _ {n}}{n - b _ {n} + 1} \sum_ {t = 0} ^ {n - b _ {n}} \left(\mathbb {E} _ {\xi} ^ {1 / p} [ | u ^ {\top} \bar {D} _ {b _ {n}, t} | ^ {2 p} + \mathbb {E} _ {\xi} ^ {1 / p} [ | u ^ {\top} \bar {D} _ {b} | ^ {2 p} \right. \\ \left. + 2 \mathbb {E} _ {\xi} ^ {1 / (2 p)} \left[ | u ^ {\top} \bar {D} _ {b _ {n}, t} | ^ {2 p} \mathbb {E} _ {\xi} ^ {1 / (2 p)} \left[ | u ^ {\top} \bar {D} _ {b} | ^ {2 p} \right] \right.\right) \\ \leq \frac {2 b _ {n}}{n - b _ {n} + 1} \sum_ {t = 0} ^ {n - b _ {n}} \left\{\mathbb {E} _ {\xi} ^ {1 / p} \left[ | u ^ {\top} \bar {D} _ {b _ {n}, t} | ^ {2 p} + \mathbb {E} _ {\xi} ^ {1 / p} \left[ | u ^ {\top} \bar {D} _ {b} | ^ {2 p} \right. \right. \right\} \\ \end{array} +$$ + +Using Lemma 15 and Lemma 24 we obtain + +$$ +\begin{array}{l} \frac {2 b _ {n}}{n - b _ {n} + 1} \sum_ {t = 0} ^ {n - b _ {n}} \mathbb {E} _ {\xi} ^ {1 / p} [ | u ^ {\top} \bar {D} _ {b _ {n}, t} | ^ {2 p} \lesssim \frac {b _ {n}}{n - b _ {n} + 1} \left\{\sum_ {t = 0} ^ {n - b _ {n}} \frac {(\mathrm {C} _ {1 , 1} ^ {\mathrm {D} , \mathsf {b}}) ^ {2} p}{b _ {n} ^ {2} \alpha_ {t}} + \frac {(\mathrm {C} _ {1 , 2} ^ {\mathrm {D} , \mathsf {b}}) ^ {2}}{b _ {n} ^ {2}} \right. \\ + \left(\mathrm {C} _ {1, 3} ^ {\mathrm {D}, \mathbf {b}}\right) ^ {2} p ^ {4} \log (1 / \alpha_ {b _ {n} + t - 1}) \frac {\left(\sum_ {k = 1 + t} ^ {b _ {n} + t - 1} \alpha_ {k}\right) ^ {2}}{b _ {n} ^ {2}} \\ \left. + (\mathrm {C} _ {1, 4} ^ {\mathsf {D}, \mathsf {b}}) ^ {2} \frac {p}{b _ {n} ^ {2}} \sum_ {k = 2} ^ {b _ {n} - 1} (k + k _ {0} + t) ^ {2 \gamma - 2} + \frac {(\mathrm {C} _ {1 , 5} ^ {\mathsf {D} , \mathsf {b}}) ^ {2} p b _ {n} ^ {- 2}}{\alpha_ {t + b _ {n} - 2}} + (\mathrm {C} _ {1, 6} ^ {\mathsf {D}, \mathsf {b}}) ^ {2} \frac {(k _ {0} + t - 1) ^ {2 \gamma - 2}}{b _ {n} ^ {2}} \right\} \\ \lesssim M _ {1, 1} p n ^ {\gamma} b _ {n} ^ {- 1} + M _ {1, 2} b _ {n} ^ {- 1} + M _ {1, 3} p ^ {4} \log n n ^ {- 1} + M _ {1, 4} p n ^ {2 \gamma - 2} \\ \end{array} +$$ + +Using Lemma 16, we get + +$$ +\frac {2 b _ {n}}{n - b _ {n} + 1} \sum_ {t = 0} ^ {n - b _ {n}} \mathbb {E} _ {\xi} ^ {1 / p} [ | u ^ {\top} \bar {D} _ {b} | ^ {2 p} \lesssim M _ {2, 1} b _ {n} n ^ {- 2} + M _ {2, 2} p ^ {4} b _ {n} (\log n) n ^ {- 2 \gamma} + M _ {2, 3} p b _ {n} n ^ {2 \gamma - 3} + M _ {2, 4} p b _ {n} n ^ {\gamma - 2}. +$$ + +![](images/83d8982d6cf00f04517e71f7de2d6e8c38566175ba07558b206d45e81fa9a862.jpg) + +Lemma 19. Assume A1, A2 and A3. Then it holds + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} [ | D _ {2} ^ {b} | ^ {p} ] \lesssim M _ {3, 1} p ^ {1 / 2} b _ {n} ^ {1 / 2} n ^ {- 1} + M _ {3, 2} p b _ {n} ^ {1 / 2} n ^ {\gamma - 3 / 2} + M _ {3, 3} p b _ {n} ^ {1 / 2} n ^ {\gamma / 2 - 1} + M _ {3, 4} p ^ {5 / 2} \sqrt {\log n} n ^ {- \gamma} b _ {n} ^ {1 / 2} \\ + M _ {3, 5} p ^ {1 / 2} b _ {n} ^ {- 1 / 2} + M _ {3, 6} p ^ {5 / 2} \sqrt {\log n} b _ {n} ^ {1 / 2} n ^ {- \gamma} + M _ {3, 7} p b _ {n} ^ {- 1 / 2} n ^ {\gamma - 1 / 2} + M _ {3, 8} p b _ {n} ^ {- 1 / 2} n ^ {\gamma / 2}, \\ \end{array} +$$ + +where + +$$ +M _ {3, 1} = \mathrm {C} _ {2, 1} ^ {\mathrm {D}, \mathsf {b}} \left(\mathrm {C} _ {3, 1} ^ {\mathrm {D}, \mathsf {b}} + \mathrm {C} _ {3, 2} ^ {\mathrm {D}, \mathsf {b}}\right) +$$ + +$$ +M _ {3, 2} = \mathrm {C} _ {2, 3} ^ {\mathrm {D}, \mathrm {b}} \left(\mathrm {C} _ {3, 1} ^ {\mathrm {D}, \mathrm {b}} + \mathrm {C} _ {3, 2} ^ {\mathrm {D}, \mathrm {b}}\right) +$$ + +$$ +M _ {3, 3} = \mathrm {C} _ {2, 4} ^ {\mathrm {D}, \mathrm {b}} \left(\mathrm {C} _ {3, 1} ^ {\mathrm {D}, \mathrm {b}} + \mathrm {C} _ {3, 2} ^ {\mathrm {D}, \mathrm {b}}\right) +$$ + +$$ +M _ {3, 4} = (\mathrm {C} _ {2, 5} ^ {\mathrm {D}, \mathrm {b}} + \mathrm {C} _ {2, 2} ^ {\mathrm {D}, \mathrm {b}}) (\mathrm {C} _ {3, 1} ^ {\mathrm {D}, \mathrm {b}} + \mathrm {C} _ {3, 2} ^ {\mathrm {D}, \mathrm {b}}) +$$ + +$$ +M _ {3, 5} = \left(\mathrm {C} _ {1, 2} ^ {\mathrm {D}, \mathrm {b}} + \mathrm {C} _ {1, 6} ^ {\mathrm {D}, \mathrm {b}}\right) \left(\mathrm {C} _ {3, 1} ^ {\mathrm {D}, \mathrm {b}} + \mathrm {C} _ {3, 2} ^ {\mathrm {D}, \mathrm {b}}\right) +$$ + +$$ +M _ {3, 6} = \frac {\mathrm {C} _ {1 , 3} ^ {\mathrm {D} , \mathrm {b}}}{1 - \gamma} (\sqrt {\gamma} + \sqrt {\log \frac {k _ {0} ^ {\gamma}}{c _ {0}}}) \left(\mathrm {C} _ {3, 1} ^ {\mathrm {D}, \mathrm {b}} + \mathrm {C} _ {3, 2} ^ {\mathrm {D}, \mathrm {b}}\right) +$$ + +$$ +M _ {3, 7} = \frac {\mathrm {C} _ {1 , 4} ^ {\mathrm {D} , \mathrm {b}} (2 k _ {0}) ^ {\gamma + 1 / 2}}{\sqrt {2 \gamma - 1} (\gamma + 1 / 2)} \left(\mathrm {C} _ {3, 1} ^ {\mathrm {D}, \mathrm {b}} + \mathrm {C} _ {3, 2} ^ {\mathrm {D}, \mathrm {b}}\right) +$$ + +$$ +M _ {3, 8} = \frac {(2 k _ {0}) ^ {\gamma / 2 + 1} (\mathrm {C} _ {1 , 5} ^ {\mathrm {D} , \mathrm {b}} + \mathrm {C} _ {1 , 1} ^ {\mathrm {D} , \mathrm {b}})}{\sqrt {c _ {0}} (\gamma / 2 + 1)} (\mathrm {C} _ {3, 1} ^ {\mathrm {D}, \mathrm {b}} + \mathrm {C} _ {3, 2} ^ {\mathrm {D}, \mathrm {b}}) +$$ + +Proof. Note that using Minkowski's inequality together with Hölder's inequality we get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} [ | D _ {2} ^ {b} | ^ {p} ] \lesssim \frac {b _ {n}}{n - b _ {n} + 1} \sum_ {t = 0} ^ {n - b _ {n}} \mathbb {E} _ {\xi} ^ {1 / (2 p)} [ | u ^ {\top} \bar {W} _ {b _ {n}, t} | ^ {2 p} ] (\mathbb {E} _ {\xi} ^ {1 / (2 p)} [ | u ^ {\top} \bar {D} _ {b _ {n}, t} | ^ {2 p} ] + \mathbb {E} _ {\xi} ^ {1 / (2 p)} [ | u ^ {\top} \bar {D} _ {n} | ^ {2 p} ]) \\ + \frac {b _ {n}}{n - b _ {n} + 1} \sum_ {t = 0} ^ {n - b _ {n}} \mathbb {E} _ {\xi} ^ {1 / (2 p)} [ | u ^ {\top} \bar {W} _ {n} | ^ {2 p} ] (\mathbb {E} _ {\xi} ^ {1 / (2 p)} [ | u ^ {\top} \bar {D} _ {b _ {n}, t} | ^ {2 p} ] + \mathbb {E} _ {\xi} ^ {1 / (2 p)} [ | u ^ {\top} \bar {D} _ {n} | ^ {2 p} ]) \\ \end{array} +$$ + +Note that bound on $\mathbb{E}_{\xi}^{1/(2p)}[|u^{\top}\bar{W}_{b_n,t}|^{2p}]$ , $\mathbb{E}_{\xi}^{1/(2p)}[|u^{\top}\bar{W}_n|^{2p}]$ , $\mathbb{E}_{\xi}^{1/(2p)}[|u^{\top}\bar{D}_n|^{2p}]$ does not depend upon $t$ , hence, we can rewrite formula above as follows + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} [ | D _ {2} ^ {b} | ^ {p} ] \lesssim \frac {b _ {n}}{n - b _ {n} + 1} \left\{\sum_ {t = 0} ^ {n - b _ {n}} \mathbb {E} _ {\xi} ^ {1 / (2 p)} [ | u ^ {\top} \bar {D} _ {b _ {n}, t} | ^ {2 p} ] \right\} (\mathbb {E} _ {\xi} ^ {1 / (2 p)} [ | u ^ {\top} \bar {W} _ {b _ {n}, t} | ^ {2 p} ] + \mathbb {E} _ {\xi} ^ {1 / (2 p)} [ | u ^ {\top} \bar {W} _ {n} | ^ {2 p} ]) \\ + b _ {n} \mathbb {E} _ {\xi} ^ {1 / (2 p)} [ | u ^ {\top} \bar {D} _ {n} | ^ {2 p} ] (\mathbb {E} _ {\xi} ^ {1 / (2 p)} [ | u ^ {\top} \bar {W} _ {b _ {n}, t} | ^ {2 p} ] + \mathbb {E} _ {\xi} ^ {1 / (2 p)} [ | u ^ {\top} \bar {W} _ {n} | ^ {2 p} ]) \\ \end{array} +$$ + +Applying Lemma 15 and Lemma 24 we get + +$$ +\begin{array}{l} \frac {b _ {n}}{n - b _ {n} + 1} \sum_ {t = 0} ^ {n - b _ {n}} \mathbb {E} _ {\xi} ^ {1 / (2 p)} [ | (u ^ {\top} \bar {D} _ {b _ {n}, t}) | ^ {2 p} \lesssim \frac {b _ {n}}{n - b _ {n} + 1} \sum_ {t = 0} ^ {n - b _ {n}} \left\{\frac {\mathrm {C} _ {1 , 1} ^ {\mathrm {D} , \mathrm {b}} p ^ {1 / 2}}{b _ {n} \sqrt {\alpha_ {t}}} + \frac {\mathrm {C} _ {1 , 2} ^ {\mathrm {D} , \mathrm {b}}}{b _ {n}} \right. \\ + \mathrm {C} _ {1, 3} ^ {\mathrm {D}, \mathrm {b}} p ^ {2} \sqrt {\log (1 / \alpha_ {b _ {n} + t - 1})} \frac {\sum_ {k = 1 + t} ^ {b _ {n} + t - 1} \alpha_ {k}}{b _ {n}} \\ + \mathrm {C} _ {1, 4} ^ {\mathrm {D}, \mathrm {b}} \frac {p ^ {1 / 2}}{b _ {n}} \left(\sum_ {k = 2} ^ {b _ {n} - 1} (k + k _ {0} + t) ^ {2 \gamma - 2}\right) ^ {1 / 2} + \frac {\mathrm {C} _ {1 , 5} ^ {\mathrm {D} , \mathrm {b}} p ^ {1 / 2} b _ {n} ^ {- 1}}{\sqrt {\alpha_ {t + b _ {n} - 2}}} + \mathrm {C} _ {1, 6} ^ {\mathrm {D}, \mathrm {b}} \frac {(k _ {0} + t - 1) ^ {\gamma - 1}}{b _ {n}} \Bigg \} \\ \end{array} +$$ + +Using Lemma 16 and Lemma 17 we get + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ | D _ {2} ^ {b} | ^ {p} \right] \lesssim \left\{\mathrm {C} _ {2, 1} ^ {\mathrm {D}, \mathrm {b}} b _ {n} n ^ {- 1} + \mathrm {C} _ {2, 3} ^ {\mathrm {D}, \mathrm {b}} p ^ {1 / 2} b _ {n} n ^ {\gamma - 3 / 2} + \mathrm {C} _ {2, 4} ^ {\mathrm {D}, \mathrm {b}} p ^ {1 / 2} b _ {n} n ^ {\gamma / 2 - 1} \right. \\ + \left(\mathrm {C} _ {2, 5} ^ {\mathrm {D}, \mathrm {b}} + \mathrm {C} _ {2, 2} ^ {\mathrm {D}, \mathrm {b}}\right) p ^ {2} \sqrt {\log n} n ^ {- \gamma} b _ {n} + \mathrm {C} _ {1, 2} ^ {\mathrm {D}, \mathrm {b}} + \frac {\mathrm {C} _ {1 , 3} ^ {\mathrm {D} , \mathrm {b}}}{1 - \gamma} p ^ {2} (\sqrt {\gamma} + \sqrt {\log \frac {k _ {0} ^ {\gamma}}{c _ {0}}}) \sqrt {\log n} b _ {n} n ^ {- \gamma} \\ \left. + \frac {\mathrm {C} _ {1 , 4} ^ {\mathrm {D} , \mathrm {b}} \left(2 k _ {0}\right) ^ {\gamma + 1 / 2}}{\sqrt {2 \gamma - 1} (\gamma + 1 / 2)} p ^ {1 / 2} n ^ {\gamma - 1 / 2} + \frac {\left(2 k _ {0}\right) ^ {\gamma / 2 + 1} \left(\mathrm {C} _ {1 , 5} ^ {\mathrm {D} , \mathrm {b}} + \mathrm {C} _ {1 , 1} ^ {\mathrm {D} , \mathrm {b}}\right) p ^ {1 / 2} \left(n - b _ {n} + 1\right) ^ {\gamma / 2}}{\sqrt {c _ {0}} (\gamma / 2 + 1)} \right\}. \\ \cdot \left(\mathrm {C} _ {3, 1} ^ {\mathrm {D}, \mathrm {b}} + \mathrm {C} _ {3, 2} ^ {\mathrm {D}, \mathrm {b}}\right) p ^ {1 / 2} \left(n ^ {- 1 / 2} + b _ {n} ^ {- 1 / 2}\right) \\ \end{array} +$$ + +![](images/375901c01814a2812485f724baf29230e9bd9195655a521d87fbaaa07efa9493.jpg) + +Version of (64) with constants. Finally, from Lemma 18 and Lemma 19, we obtain the following bound for the residual term $\mathcal{R}_{var}(u)$ : + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| \mathcal {R} _ {v a r} (u) \right| ^ {p} \right] \lesssim M _ {1} p b _ {n} ^ {1 / 2} n ^ {\gamma / 2 - 1} + M _ {2} p ^ {4} (\log n) b _ {n} ^ {1 / 2} n ^ {- \gamma} \tag {65} \\ + M _ {3} p b _ {n} ^ {- 1 / 2} n ^ {\gamma / 2} + M _ {4} p ^ {4} (\log n) n ^ {- 1} + M _ {5} p n ^ {2 \gamma - 2}, \\ \end{array} +$$ + +where + +$$ +\begin{array}{l} M _ {1} = M _ {3, 1} + M _ {3, 2} + M _ {3, 3} + M _ {2, 3} + M _ {2, 4} \\ M _ {2} = M _ {3, 4} + M _ {3, 6} + M _ {2, 2} \\ M _ {3} = M _ {3, 5} + M _ {3, 7} + M _ {3, 8} + M _ {1, 1} + M _ {1, 2} \tag {66} \\ M _ {4} = M _ {1, 3} + M _ {2, 1} \\ M _ {5} = M _ {1, 4}. \\ \end{array} +$$ + +# E.3 Proof of Corollary 2 + +The proof of Corollary 2 follows directly from From Proposition 3 and Proposition 4. Indeed, note that + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| \hat {\sigma} _ {\theta} ^ {2} (u) - \sigma^ {2} (u) \right| ^ {p} \right] \leq \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| \hat {\sigma} _ {\varepsilon} ^ {2} (u) - \sigma^ {2} (u) \right| ^ {p} \right] + \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| \mathcal {R} _ {v a r} (u) \right| ^ {p} \right] \\ \lesssim \frac {p t _ {\mathrm {m i x}} ^ {3} \| \varepsilon \| _ {\infty} ^ {2}}{\sqrt {n}} + \frac {p ^ {2} t _ {\mathrm {m i x}} ^ {2} \sqrt {b _ {n}} \| \varepsilon \| _ {\infty} ^ {2}}{\sqrt {n}} + \frac {p t _ {\mathrm {m i x}} ^ {2} \| \varepsilon \| _ {\infty} ^ {2}}{\sqrt {b _ {n}}} \\ + M _ {1} p b _ {n} ^ {1 / 2} n ^ {\gamma / 2 - 1} + M _ {2} p ^ {4} (\log n) b _ {n} ^ {1 / 2} n ^ {- \gamma} \\ + M _ {3} p b _ {n} ^ {- 1 / 2} n ^ {\gamma / 2} + M _ {4} p ^ {4} (\log n) n ^ {- 1} + M _ {5} p n ^ {2 \gamma - 2} \\ \lesssim p ^ {4} \left(t _ {\text {m i x}} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} + M _ {2} + M _ {1}\right) (\log n) \frac {b _ {n} ^ {1 / 2}}{\sqrt {n}} + p ^ {4} \left(t _ {\text {m i x}} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} + M _ {3}\right) \frac {n ^ {\gamma / 2}}{\sqrt {b _ {n}}} \\ + p ^ {4} \left(t _ {\operatorname {m i x}} ^ {3} \| \varepsilon \| _ {\infty} ^ {2} + M _ {4}\right) \frac {(\log n)}{\sqrt {n}} + p ^ {4} M _ {5} n ^ {2 \gamma - 2} \\ \end{array} +$$ + +Hence, applying Lemma 21 with $p = \log n$ we obtain that with probability $1 - \frac{1}{n}$ it holds + +$$ +\begin{array}{l} \left| \hat {\sigma} _ {\theta} ^ {2} (u) - \sigma^ {2} (u) \right| \lesssim (t _ {\mathrm {m i x}} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} + M _ {2} + M _ {1}) (\log n) ^ {5} \frac {b _ {n} ^ {1 / 2}}{\sqrt {n}} \\ + \left(\log n\right) ^ {4} \left(t _ {\text {m i x}} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} + M _ {3}\right) \frac {n ^ {\gamma / 2}}{\sqrt {b _ {n}}} \tag {67} \\ + \left(t _ {\mathrm {m i x}} ^ {3} \| \varepsilon \| _ {\infty} ^ {2} + M _ {4}\right) \frac {(\log n) ^ {5}}{\sqrt {n}} \\ + (\log n) ^ {4} M _ {5} n ^ {2 \gamma - 2} \\ \end{array} +$$ + +By setting $b_n = b^\beta$ , we can optimize the inequality above by $\gamma$ and $\beta$ . And by putting $\beta = 3/4$ and $\gamma = 1/2 + \varepsilon$ where $\varepsilon < 1/\log n$ we finally get that with probability $1 - \frac{1}{n}$ it holds + +$$ +\begin{array}{l} \left| \hat {\sigma} _ {\theta} ^ {2} (u) - \sigma^ {2} (u) \right| \\ \lesssim \left(t _ {\operatorname {m i x}} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} + M _ {2} + M _ {1} + t _ {\operatorname {m i x}} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} + M _ {3} + t _ {\operatorname {m i x}} ^ {3} \| \varepsilon \| _ {\infty} ^ {2} + M _ {4} + M _ {5}\right) (\log n) ^ {5} n ^ {- 1 / 8 + \varepsilon / 2} \\ \end{array} +$$ + +# E.4 Detailed version of the proof of Theorem 2 + +We preface the proof with a Gaussian comparison inequality due to [21, Theorem 1.3]. It states that for $\xi_{i}\sim \mathcal{N}(0,\sigma_{i}^{2}),i = 1,2$ + +$$ +\sup _ {x \in \mathbb {R}} | \mathbb {P} (\xi_ {1} \leq x) - \mathbb {P} (\xi_ {2} \leq x) | \leq (3 / 2) \left| \sigma_ {1} ^ {2} / \sigma_ {2} ^ {2} - 1 \right|. \tag {68} +$$ + +Now we proceed with the proof of Theorem 2. Using (67) together with the inequality $\sigma^{-2}(u)\leq 1 / \lambda_{\min}(\Sigma_{\infty})$ we get that with probability $1 - 1 / n$ it holds + +$$ +\begin{array}{l} \left| \hat {\sigma} _ {\theta} ^ {2} (u) - \sigma^ {2} (u) \right| / \sigma^ {2} (u) \lesssim \frac {\mathrm {e}}{\lambda_ {\min} (\Sigma_ {\infty})} (t _ {\operatorname {m i x}} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} + M _ {2} + M _ {1}) (\log n) ^ {5} \frac {b _ {n} ^ {1 / 2}}{\sqrt {n}} \\ + \frac {\mathrm {e}}{\lambda_ {\operatorname* {m i n}} (\Sigma_ {\infty})} (\log n) ^ {4} (t _ {\operatorname* {m i x}} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} + M _ {3}) \frac {n ^ {\gamma / 2}}{\sqrt {b _ {n}}} \\ + \frac {\mathrm {e}}{\lambda_ {\operatorname* {m i n}} (\Sigma_ {\infty})} (t _ {\operatorname* {m i x}} ^ {3} \| \varepsilon \| _ {\infty} ^ {2} + M _ {4}) \frac {(\log n) ^ {5}}{\sqrt {n}} \\ + \frac {\mathrm {e}}{\lambda_ {\min} (\Sigma_ {\infty})} (\log n) ^ {4} M _ {5} n ^ {2 \gamma - 2} \\ \end{array} +$$ + +Now we apply (68) and obtain that with probability $1 - 1 / n$ it holds + +$$ +\begin{array}{l} \mathsf {d} _ {K} \big (\mathcal {N} (0, \hat {\sigma} _ {\theta} ^ {2} (u)), \mathcal {N} (0, \sigma^ {2} (u)) \big) \lesssim \frac {1}{\lambda_ {\min} (\Sigma_ {\infty})} (t _ {\operatorname * {m i x}} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} + M _ {2} + M _ {1}) (\log n) ^ {5} \frac {b _ {n} ^ {1 / 2}}{\sqrt {n}} \\ + \frac {1}{\lambda_ {\operatorname* {m i n}} (\Sigma_ {\infty})} (\log n) ^ {4} (t _ {\operatorname* {m i x}} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} + M _ {3}) \frac {n ^ {\gamma / 2}}{\sqrt {b _ {n}}} \\ + \frac {1}{\lambda_ {\mathrm {m i n}} (\Sigma_ {\infty})} (t _ {\mathrm {m i x}} ^ {3} \| \varepsilon \| _ {\infty} ^ {2} + M _ {4}) \frac {(\log n) ^ {5}}{\sqrt {n}} \\ + \frac {1}{\lambda_ {\operatorname* {m i n}} (\Sigma_ {\infty})} (\log n) ^ {4} M _ {5} n ^ {2 \gamma - 2}. \\ \end{array} +$$ + +Combining this inequality with Corollary 1 we obtain with probability $1 - 1/n$ + +$$ +\begin{array}{l} \sup _ {x \in \mathbb {R}} | \mathbb {P} (\sqrt {n} (\bar {\theta} _ {n} - \theta^ {\star}) ^ {\top} u \leq x) - \mathbb {P} ^ {\mathbf {b}} (\bar {\theta} _ {n, b _ {n}} (u) \leq x) | \lesssim \left(\mathrm {C} _ {\mathrm {K}, 1} + \mathrm {C} _ {\mathrm {K}, 2} + \mathrm {C} _ {1} ^ {\mathrm {D}} \| \theta_ {0} - \theta^ {\star} \| + \mathrm {C} _ {2} ^ {\mathrm {D}}\right) \frac {\log n}{n ^ {1 / 4}} \\ + \left(\mathrm {C} _ {4} ^ {\mathsf {D}} + \mathrm {C} _ {3} ^ {\mathsf {D}}\right) \frac {(\log n) ^ {5 / 2}}{n ^ {\gamma - 1 / 2}} \\ + \frac {1}{\lambda_ {\operatorname* {m i n}} (\Sigma_ {\infty})} (t _ {\operatorname * {m i x}} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} + M _ {2} + M _ {1}) (\log n) ^ {5} \frac {b _ {n} ^ {1 / 2}}{\sqrt {n}} \\ + \frac {1}{\lambda_ {\operatorname* {m i n}} (\Sigma_ {\infty})} (\log n) ^ {4} (t _ {\operatorname* {m i x}} ^ {2} \| \varepsilon \| _ {\infty} ^ {2} + M _ {3}) \frac {n ^ {\gamma / 2}}{\sqrt {b _ {n}}} \\ + \frac {1}{\lambda_ {\operatorname* {m i n}} (\Sigma_ {\infty})} (t _ {\operatorname* {m i x}} ^ {3} \| \varepsilon \| _ {\infty} ^ {2} + M _ {4}) \frac {(\log n) ^ {5}}{\sqrt {n}} \\ + \frac {1}{\lambda_ {\operatorname* {m i n}} (\Sigma_ {\infty})} (\log n) ^ {4} M _ {5} n ^ {2 \gamma - 2} \\ \end{array} +$$ + +To complete the proof, it remains to optimize the bound above. Setting $b_{n} = \lceil n^{4 / 5}\rceil$ and $\gamma = 3 / 5$ , we obtain with probability $1 - 1 / n$ + +$$ +\begin{array}{l} \sup _ {x \in \mathbb {R}} | \mathbb {P} (\sqrt {n} (\bar {\theta} _ {n} - \theta^ {\star}) ^ {\top} u \leq x) - \mathbb {P} ^ {\mathbf {b}} (\bar {\theta} _ {n, b _ {n}} (u) \leq x) | \lesssim \left(\mathrm {C} _ {\mathrm {K}, 1} + \mathrm {C} _ {\mathrm {K}, 2} + \mathrm {C} _ {1} ^ {\mathrm {D}} \| \theta_ {0} - \theta^ {\star} \| + \mathrm {C} _ {2} ^ {\mathrm {D}}\right) \frac {\log n}{n ^ {1 / 4}} \\ + \frac {1}{\lambda_ {\operatorname* {m i n}} (\Sigma_ {\infty})} (t _ {\operatorname* {m i x}} ^ {3} \| \varepsilon \| _ {\infty} ^ {2} + M _ {4}) \frac {(\log n) ^ {5}}{\sqrt {n}} \\ + \left(\mathrm {C} _ {3} ^ {\mathrm {D}} + \mathrm {C} _ {4} ^ {\mathrm {D}} + \frac {t _ {\operatorname* {m i x}} ^ {3} \| \varepsilon \| _ {\infty} ^ {2} + M _ {1} + M _ {2} + M _ {3} + M _ {5}}{\lambda_ {\operatorname* {m i n}} (\Sigma_ {\infty})}\right) (\log n) ^ {5} n ^ {- 1 / 1 0}. \\ \end{array} +$$ + +# F Probability inequalities + +Denote by $\Phi$ the c.d.f. of a standard Gaussian random variable and set + +$$ +\mathsf {d} _ {K} (X) = \sup _ {x \in \mathbb {R}} | \mathbb {P} (X \leq x) - \Phi (x) |. +$$ + +Proposition 14. For any random variables $X, Y$ , and any $p \geq 1$ , + +$$ +\mathrm {d} _ {K} (X + Y) \leq \mathrm {d} _ {K} (X) + 2 \mathbb {E} ^ {1 / (p + 1)} [ | Y | ^ {p} ]. \tag {69} +$$ + +Proof. Let $t \geq 0$ . By Markov's inequality + +$$ +\begin{array}{l} \mathbb {P} (X + Y \leq x) \leq \mathbb {P} (X + Y \leq x, | Y | \leq t) + \frac {1}{t ^ {p}} \mathbb {E} [ | Y | ^ {p} ] \\ \leq \mathbb {P} (X \leq x + t) - \Phi (x + t) + \Phi (x + t) + \frac {1}{t ^ {p}} \mathbb {E} [ | Y | ^ {p} ] \\ \leq \Phi (x) + \sup _ {x \in \mathbb {R}} | \mathbb {P} (X \leq x) - \Phi (x) | + \frac {t}{\sqrt {2} \pi} + \frac {1}{t ^ {p}} \mathbb {E} [ | Y | ^ {p} ]. \\ \end{array} +$$ + +Choosing $t = \mathbb{E}^{1 / (p + 1)}[|Y|^p]$ we obtain + +$$ +\sup _ {x \in \mathbb {R}} (\mathbb {P} (X + Y \leq x) - \Phi (x)) \leq \sup _ {x \in \mathbb {R}} | \mathbb {P} (X \leq x) - \Phi (x) | + 2 \mathbb {E} ^ {1 / (p + 1)} [ | Y | ^ {p} ] +$$ + +Similarly, we may estimate $\sup_{x\in \mathbb{R}}(\Phi (x) - \mathbb{P}(X + Y\leq x))$ . Hence, (69) holds. + +![](images/f83bc1acb6313baedda5ad4a25dde1e273f928e5d6f1b0f5413b2e442170a441.jpg) + +Remark 2. The result similar to Proposition 14 was previously obtained in [44, Lemma 1]. It states that for random variables $X$ and $Y$ and any $p \geq 1$ , it holds that + +$$ +\mathsf {d} _ {K} (X + Y) \leq 2 \mathsf {d} _ {K} (X) + 3 \| \mathbb {E} [ | Y | ^ {2 p} | X ] \| _ {1} ^ {1 / (2 p + 1)}. +$$ + +# F.1 Rosenthal and Burkholder inequalities + +We begin this section with a version of Rosenthal inequality (see the original paper [61] and the Pinelis version of the Rosenthal inequality [56]). Let $f: \mathbb{Z} \to \mathbb{R}$ be a bounded function with $\| f \|_{\infty} < \infty$ and define + +$$ +\bar {S} _ {n} = \sum_ {k = 0} ^ {n - 1} \left\{f \left(Z _ {k}\right) - \pi (f) \right\}. +$$ + +Then the following bound holds: + +Lemma 20. Assume A1. Then for any $p \geq 2$ and $f: \mathbb{Z} \to \mathbb{R}^d$ with $\| f \|_{\infty} < \infty$ , any initial distribution $\xi$ on $(\mathbb{Z}, \mathcal{Z})$ , and any $u \in \mathbb{S}_{d-1}$ , $A_i \in \mathbb{R}^{d \times d}$ , it holds that + +$$ +\begin{array}{l} \mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| \sum_ {k = 1} ^ {n} u ^ {\top} A _ {k} (f (Z _ {k}) - \pi (f)) \right| ^ {p} \right] \leq (1 6 / 3) t _ {\operatorname {m i x}} p ^ {1 / 2} \| f \| _ {\infty} \left(\sum_ {k = 2} ^ {n} \| A _ {k} \| ^ {2}\right) ^ {1 / 2} \\ + (8 / 3) t _ {\mathrm {m i x}} \left(\| A _ {1} \| + \| A _ {n} \| + \sum_ {k = 1} ^ {n - 1} \| A _ {k + 1} - A _ {k} \|\right) \| f \| _ {\infty}. \\ \end{array} +$$ + +Proof. Under A1 the Poisson equation + +$$ +g (z) - \mathrm {P} g (z) = f (z) - \pi (f) +$$ + +has a unique solution for any bounded $f$ , which is given by the formula + +$$ +g (z) = \sum_ {k = 0} ^ {\infty} \left\{\mathrm {P} ^ {k} f (z) - \pi (f) \right\}. +$$ + +Thus, using A1, we obtain that $g(z)$ is also bounded with + +$$ +\| g (z) \| \leq \sum_ {k = 0} ^ {+ \infty} \| P ^ {k} f (z) - \pi (f) \| \leq 2 \| f \| _ {\infty} \sum_ {k = 0} ^ {+ \infty} (1 / 4) ^ {\lfloor k / t _ {\operatorname {m i x}} \rfloor} \leq (8 / 3) t _ {\operatorname {m i x}} \| f \| _ {\infty}. \tag {70} +$$ + +Hence, we can represent + +$$ +\begin{array}{l} \sum_ {k = 1} ^ {n} u ^ {\top} A _ {k} (f (Z _ {k}) - \pi (f)) = \underbrace {\sum_ {k = 2} ^ {n} u ^ {\top} A _ {k} (g (Z _ {k}) - \mathrm {P g} (Z _ {k - 1}))} _ {T _ {1}} \\ \underbrace {+ \sum_ {k = 1} ^ {n - 1} u ^ {\top} (A _ {k + 1} - A _ {k}) \mathrm {P} g (Z _ {k}) + u ^ {\top} A _ {1} g (Z _ {1}) - u ^ {\top} A _ {n} \mathrm {P} g (Z _ {n})} _ {T _ {2}}. \\ \end{array} +$$ + +The term $T_{2}$ can be controlled using Minkowski's inequality: + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ \left| T _ {2} \right| ^ {p} \right] \leq (8 / 3) t _ {\operatorname {m i x}} \left(\left\| A _ {1} \right\| + \left\| A _ {n} \right\| + \sum_ {k = 1} ^ {n - 1} \left\| A _ {k + 1} - A _ {k} \right\|\right) \| f \| _ {\infty}. +$$ + +Now we proceed with $T_{1}$ . Since $\mathbb{E}^{\mathcal{F}_{k - 1}}\left[g(Z_k) - \mathrm{P}g(Z_{k - 1})\right] = 0$ a.s. and $|u^{\top}A_{k}(g(Z_{k}) - \mathrm{P}g(Z_{k - 1}))| \leq (16 / 3)\| A_{k}\| t_{\mathrm{mix}}\| f\|_{\infty}$ , we get, using the Azuma-Hoeffding inequality [78, Corollary 3.9], that + +$$ +\mathbb {P} _ {\xi} [ | T _ {1} | \geq t ] \leq 2 \exp \left\{- \frac {2 t ^ {2}}{(1 6 / 3) ^ {2} t _ {\mathrm {m i x}} ^ {2} \| f \| _ {\infty} ^ {2} \sum_ {k = 2} ^ {n} \| A _ {k} \| ^ {2}} \right\}. +$$ + +Hence, applying [24, Lemma 7], we get + +$$ +\mathbb {E} _ {\xi} ^ {1 / p} \left[ | T _ {1} | ^ {p} \right] \leq (1 6 / 3) p ^ {1 / 2} t _ {\mathrm {m i x}} \| f \| _ {\infty} \left(\sum_ {k = 2} ^ {n} \| A _ {k} \| ^ {2}\right) ^ {1 / 2}. +$$ + +![](images/949597e3275266fb857d2c7057c0e15528cd02cf51f31c1127b3e33518a4b4ad.jpg) + +The lemma below is a simple technical statement used to switch between $p$ -th moment bounds and high-probability bounds. + +Lemma 21. Fix $\delta \in (0,1 / \mathrm{e}^2)$ and let $Y$ be a positive random variable, such that + +$$ +\mathbb {E} ^ {1 / p} \left[ Y ^ {p} \right] \leq p ^ {v} C _ {1} +$$ + +for any $2 \leq p \leq \log(1/\delta)$ . Then it holds with probability at least $1 - \delta$ , that + +$$ +Y \leq \mathrm {e} C _ {1} (\log (1 / \delta)) ^ {v}. +$$ + +Proof. Applying Markov's inequality, for any $t \geq 0$ we get that + +$$ +\mathbb {P} (Y \geq t) \leq \frac {\mathbb {E} [ Y ^ {p} ]}{t ^ {p}} \leq \frac {(C _ {1} p ^ {v}) ^ {p}}{t ^ {p}}. +$$ + +Now we set $p = \log (1 / \delta)$ , $t = \mathrm{e}C_1(\log (1 / \delta))^v$ , and aim to check that + +$$ +\begin{array}{c} \frac {\left(C _ {1} (\log (1 / \delta)) ^ {v}\right) ^ {\log (1 / \delta)}}{\left(\mathrm {e} C _ {1} (\log (1 / \delta)) ^ {v}\right) ^ {\log (1 / \delta)}} \leq \delta . \end{array} +$$ + +Taking logarithms from both sides, the latter inequality is equivalent to + +$$ +- \log (1 / \delta) \leq \log \delta , +$$ + +which turns into exact equality. + +![](images/de42e970516e8020f6da3479c05d01b49242b779f5a7d1558ba2e21250b5e990.jpg) + +# G Auxiliary results on the sequences of step sizes $\{\alpha_{k}\}_{k\in \mathbb{N}}$ under A3 + +In this section we provide the auxiliary results on the sequences of step sizes $\{\alpha_{k}\}_{k\in \mathbb{N}}$ under A3. For simplicity, we denote + +$$ +g _ {n: m} = \sum_ {k = n} ^ {m} k ^ {- \gamma}, n \leq m +$$ + +Lemma 22. Let $n \leq m$ . Then + +$$ +\frac {(m + 1) ^ {1 - \gamma} - n ^ {1 - \gamma}}{1 - \gamma} \leq g _ {n: m} \leq \frac {m ^ {1 - \gamma} - (n - 1) ^ {1 - \gamma}}{1 - \gamma}. +$$ + +Lemma 23. Let $b, c_0 > 0$ and $\alpha_{\ell} = c_{0}(\ell + k_{0})^{-\gamma}$ for $\gamma \in (1/2, 1)$ , $k_{0} \geq 0$ . Assume that $bc_{0} < 1$ and $k_{0}^{1 - \gamma} \geq 2 / (bc_{0})$ . Then + +$$ +\sum_ {k = \ell} ^ {n - 1} \alpha_ {\ell} \prod_ {j = \ell + 1} ^ {k} (1 - b \alpha_ {j}) \leq \mathcal {L} _ {b}, +$$ + +where we set + +$$ +\mathcal {L} _ {b} = c _ {0} + \frac {2}{b (1 - \gamma)}. +$$ + +Proof. Note that + +$$ +\sum_ {k = \ell} ^ {n - 1} \prod_ {j = \ell + 1} ^ {k} (1 - b \alpha_ {j}) \leq \sum_ {k = \ell} ^ {n - 1} \exp \left\{- b \sum_ {j = l + 1} ^ {k} \alpha_ {j} \right\} \leq \sum_ {k = \ell + k _ {0}} ^ {n + k _ {0} - 1} \exp \left\{- \frac {b c _ {0}}{2 (1 - \gamma)} (k ^ {1 - \gamma} - (l + k _ {0}) ^ {1 - \gamma}) \right\} +$$ + +Applying Lemma 31 with $k_0^{1 - \gamma} \geq 2 / (bc_0)$ , we finish the proof. + +![](images/1ef721fe5b87eae88b24615452f97aa63ff914fd8d918fe97cfea2ab39695d8a.jpg) + +Lemma 24. Assume A3. Then the following bounds hold: + +1. + +$$ +\sum_ {i = 1} ^ {k} \alpha_ {i} \leq \frac {c _ {0}}{1 - \gamma} ((k + k _ {0}) ^ {1 - \gamma} - k _ {0} ^ {1 - \gamma}) +$$ + +2. for any $p \geq 2$ + +$$ +\sum_ {i = 1} ^ {k} \alpha_ {i} ^ {p} \leq \frac {c _ {0} ^ {p}}{p \gamma - 1}, +$$ + +3. for any $m \in \{0, \dots, k\}$ + +$$ +\sum_ {i = m + 1} ^ {k} \alpha_ {i} \geq \frac {c _ {0}}{2 (1 - \gamma)} ((k + k _ {0}) ^ {1 - \gamma} - (m + k _ {0}) ^ {1 - \gamma}), +$$ + +Proof. To proof 1, note that + +$$ +\sum_ {i = 1} ^ {k} \alpha_ {i} \leq c _ {0} \int_ {k _ {0}} ^ {k + k _ {0}} \frac {\mathrm {d} x}{x ^ {\gamma}} \leq \frac {c _ {0}}{1 - \gamma} ((k + k _ {0}) ^ {1 - \gamma} - k _ {0} ^ {1 - \gamma}), +$$ + +To proof 2, note that + +$$ +\sum_ {i = 1} ^ {k} \alpha_ {i} ^ {p} \leq c _ {0} ^ {p} \int_ {1} ^ {+ \infty} \frac {\mathrm {d} x}{x ^ {p \gamma}} \leq \frac {c _ {0} ^ {p}}{p \gamma - 1}, +$$ + +To proof 3, note that for any $i \geq 1$ we have $2(i + k_0)^{-\gamma} \geq (i + k_0 - 1)^{-\gamma}$ . Hence, + +$$ +\sum_ {i = m + 1} ^ {k} \alpha_ {i} \geq \frac {1}{2} \sum_ {i = m} ^ {k - 1} \alpha_ {i} \geq \frac {c _ {0}}{2} \int_ {m + k _ {0}} ^ {k + k _ {0}} \frac {\mathrm {d} x}{x ^ {\gamma}} = \frac {c _ {0}}{2 (1 - \gamma)} ((k + k _ {0}) ^ {1 - \gamma} - (m + k _ {0}) ^ {1 - \gamma}). +$$ + +![](images/6e664493b3abec236b06a742b68f1cdb0351a874cc0abca3bf06244a48a27fbb.jpg) + +Lemma 25 (Lemma 24 in [26]). Let $b > 0$ and $\{\alpha_k\}_{k \geq 0}$ be a non-increasing sequence such that $\alpha_1 \leq 1 / b$ . Then + +$$ +\sum_ {j = 1} ^ {k} \alpha_ {j} \prod_ {l = j + 1} ^ {k} (1 - \alpha_ {l} b) = \frac {1}{b} \left\{1 - \prod_ {l = 1} ^ {k} (1 - \alpha_ {l} b) \right\} +$$ + +Proof. The proof is given in [26]. + +![](images/c55c763a7364956e5a2079fb02ac4e0e4155034e659c1864c7e1fd87352abd9d.jpg) + +Lemma 26. Let $b > 0$ and $\alpha_{k} = \frac{c_{0}}{(k_{0} + k)^{\gamma}}$ be a non-increasing sequence such that $c_{0} \leq 1 / b$ and $k_{0} \geq \left\{\frac{\gamma}{2bc_{0}}\right\}^{1 / (1 - \gamma)}$ . Then it holds + +$$ +\alpha_ {j} \prod_ {l = j + 1} ^ {k} (1 - \alpha_ {l} b) \leq \alpha_ {k} +$$ + +Proof. Note that + +$$ +\alpha_ {j} \prod_ {l = j + 1} ^ {k} (1 - \alpha_ {l} b) = \alpha_ {k} \prod_ {l = j + 1} ^ {k} \frac {\alpha_ {l - 1}}{\alpha_ {l}} (1 - \alpha_ {l} b). +$$ + +It remains to note that, + +$$ +\frac {\alpha_ {l - 1}}{\alpha_ {l}} (1 - \alpha_ {l} b) = \left\{\frac {k _ {0} + l}{k _ {0} + l - 1} \right\} ^ {\gamma} - \frac {b c _ {0}}{(k _ {0} + l - 1) ^ {\gamma}} \leq 1 + \frac {\gamma}{k _ {0} + l - 1} - \frac {b c _ {0}}{(k _ {0} + l - 1) ^ {\gamma}} \leq 1 - (b / 2) \alpha_ {l - 1}, +$$ + +where the last inequality holds since $k_{0} \geq \left\{\frac{2\gamma}{bc_{0}}\right\}^{1/(1-\gamma)}$ . + +Lemma 27. Let $b > 0$ , and let $\alpha_{k} = c_{0} / (k + k_{0})^{\gamma}$ , $\gamma \in (0;1)$ , such that $c_{0} \leq 1 / b$ and $k_{0} \geq \left\{\frac{2\gamma}{rc_{0}}\right\}^{1 / 1 - \gamma}$ with some constant $r > 0$ . Then it holds that + +$$ +\frac {\alpha_ {k}}{\alpha_ {k + 1}} \leq 1 + r \alpha_ {k + 1} +$$ + +Proof. Note that + +$$ +\frac {\alpha_ {k}}{\alpha_ {k + 1}} \leq \left(1 + \frac {1}{k + k _ {0}}\right) ^ {\gamma} \leq 1 + \frac {2 \gamma}{k _ {0} + k + 1} \leq 1 + \frac {r c _ {0}}{(k _ {0} + k + 1) ^ {\gamma}}, +$$ + +where the last inequality holds since $k_{0} \geq \left\{\frac{2\gamma}{rc_{0}}\right\}^{1/1 - \gamma}$ . + +Lemma 28. Let $b > 0$ , and let $\alpha_{k} = c_{0} / (k + k_{0})^{\gamma}$ , $\gamma \in (0;1)$ , such that $c_{0}b \leq 1/2$ and $k_{0} \geq \left\{ \frac{8\gamma}{bc_{0}} \right\}^{1/1 - \gamma}$ . Then for any $q \in (1;3]$ , it holds that + +$$ +\sum_ {j = 1} ^ {k} \alpha_ {j} ^ {q} \prod_ {\ell = j + 1} ^ {k} (1 - \alpha_ {\ell} b) \leq \frac {4}{b} \alpha_ {k} ^ {q - 1}. +$$ + +Proof. Using Lemma 27, we obtain that + +$$ +\begin{array}{l} \sum_ {j = 1} ^ {k} \alpha_ {j} ^ {q} \prod_ {\ell = j + 1} ^ {k} (1 - \alpha_ {\ell} b) = \alpha_ {k} ^ {q - 1} \sum_ {j = 1} ^ {k} \alpha_ {j} \prod_ {\ell = j + 1} ^ {k} \left(\frac {\alpha_ {\ell - 1}}{\alpha_ {\ell}}\right) ^ {q - 1} (1 - \alpha_ {\ell} b) \\ \leq \alpha_ {k} ^ {q - 1} \sum_ {j = 1} ^ {k} \alpha_ {j} \prod_ {\ell = j + 1} ^ {k} (1 + r \alpha_ {\ell}) ^ {q - 1} (1 - \alpha_ {\ell} b). \\ \end{array} +$$ + +We set $r = \frac{b}{2(q - 1)}$ . If $q \in (1, 2)$ then we use Bernoulli's inequality and obtain + +$$ +\begin{array}{l} \sum_ {j = 1} ^ {k} \alpha_ {j} ^ {q} \prod_ {\ell = j + 1} ^ {k} (1 - \alpha_ {\ell} b) \leq \alpha_ {k} ^ {q - 1} \sum_ {j = 1} ^ {k} \alpha_ {j} \prod_ {\ell = j + 1} ^ {k} (1 + b \alpha_ {\ell} / 2) (1 - \alpha_ {\ell} b) \\ \leq \alpha_ {k} ^ {q - 1} \sum_ {j = 1} ^ {k} \alpha_ {j} \prod_ {\ell = j + 1} ^ {k} \left(1 - b \alpha_ {\ell} / 2\right) \stackrel {(a)} {\leq} \frac {2}{b} \alpha_ {k} ^ {q - 1}, \\ \end{array} +$$ + +where in (a) we used Lemma 25. If $q \in [2,3]$ , using that $1 - \alpha_{\ell}b \leq \left(1 - b / (q - 1)\alpha_{\ell}\right)^{q - 1}$ , we obtain + +$$ +\begin{array}{l} \sum_ {j = 1} ^ {k} \alpha_ {j} ^ {q} \prod_ {\ell = j + 1} ^ {k} (1 - \alpha_ {\ell} b) \leq \alpha_ {k} ^ {q - 1} \sum_ {j = 1} ^ {k} \alpha_ {j} \prod_ {\ell = j + 1} ^ {k} \left(1 + \frac {b}{2 (q - 1)} \alpha_ {\ell}\right) ^ {q - 1} \left(1 - \frac {b}{q - 1} \alpha_ {\ell}\right) ^ {q - 1} \\ \leq \alpha_ {k} ^ {q - 1} \sum_ {j = 1} ^ {k} \alpha_ {j} \prod_ {\ell = j + 1} ^ {k} \left(1 - \frac {b}{2 (q - 1)} \alpha_ {\ell}\right) ^ {q - 1} \\ \leq \alpha_ {k} ^ {q - 1} \sum_ {j = 1} ^ {k} \alpha_ {j} \prod_ {\ell = j + 1} ^ {k} \left(1 - \frac {b}{2 (q - 1)} \alpha_ {\ell}\right) \leq \frac {2 (q - 1)}{b} \alpha_ {k} ^ {q - 1}, \\ \end{array} +$$ + +and the statement follows. + +We conclude with a technical statement on the coefficients $\{\alpha_{j}\}_{j\in \mathbb{N}}$ under A3. + +Lemma 29. Let $b > 0$ , and let $\alpha_{\ell} = c_0 / \{\ell + k_0\}^{\gamma}$ , $\gamma \in (1/2; 1)$ , such that $c_0 \leq 1/b$ . Then, for any $k_0$ satisfying + +$$ +k _ {0} ^ {1 - \gamma} \geq \frac {2}{c _ {0} b} \left(\log \left\{\frac {c _ {0}}{b (2 \gamma - 1) 2 ^ {\gamma}} \right\} + \gamma \log \left\{k _ {0} \right\}\right), \tag {71} +$$ + +any $s\in (1;2]$ $q\in (0;1]$ , and $k\in \mathbb{N}$ , it holds that + +$$ +\sum_ {j = 1} ^ {k} \alpha_ {j} ^ {s} \left(\sum_ {\ell = j + 1} ^ {k} \alpha_ {\ell} ^ {2}\right) ^ {q} \prod_ {\ell = j + 1} ^ {k} (1 - \alpha_ {\ell} b) \leq C (s, q, b) \alpha_ {k} ^ {s + q - 1} / b ^ {1 + q}, \tag {72} +$$ + +where $C(s,q,b) = 12\cdot 3^{\gamma (s + q - 1)}\left(\frac{4(s - 1)\gamma}{bc_0}\right)^{2\gamma (s - 1) / (1 - \gamma)}$ + +Proof. We denote $t = \lfloor (k + k_0) / 2 \rfloor$ and split the sum in (72) into two parts: + +$$ +\begin{array}{l} \sum_ {j = 1} ^ {k} \left(\frac {c _ {0}}{(j + k _ {0}) ^ {\gamma}}\right) ^ {s} \left(\sum_ {\ell = j + 1} ^ {k} \left(\frac {c _ {0}}{(l + k _ {0}) ^ {\gamma}}\right) ^ {2}\right) ^ {q} \prod_ {\ell = j + 1} ^ {k} (1 - \frac {c _ {0}}{(l + k _ {0}) ^ {\gamma}} b) \\ = \sum_ {j = 1 + k _ {0}} ^ {k + k _ {0}} \left(\frac {c _ {0}}{j ^ {\gamma}}\right) ^ {s} \left(\sum_ {\ell = j + 1} ^ {k + k _ {0}} \left(\frac {c _ {0}}{l ^ {\gamma}}\right) ^ {2}\right) ^ {q} \prod_ {\ell = j + 1} ^ {k + k _ {0}} (1 - \frac {c _ {0}}{l ^ {\gamma}} b) \\ \leq \underbrace {\sum_ {j = 1} ^ {t} \left(\frac {c _ {0}}{j ^ {\gamma}}\right) ^ {s} \left(\sum_ {\ell = j + 1} ^ {k + k _ {0}} \left(\frac {c _ {0}}{l ^ {\gamma}}\right) ^ {2}\right) ^ {q} \prod_ {\ell = j + 1} ^ {k + k _ {0}} \left(1 - \frac {c _ {0}}{l ^ {\gamma}} b\right)} _ {T _ {1}} \\ + \underbrace {\sum_ {j = t + 1} ^ {k + k _ {0}} \left(\frac {c _ {0}}{j ^ {\gamma}}\right) ^ {s} \left(\sum_ {\ell = j + 1} ^ {k + k _ {0}} \left(\frac {c _ {0}}{l ^ {\gamma}}\right) ^ {2}\right) ^ {q} \prod_ {\ell = j + 1} ^ {k + k _ {0}} \left(1 - \frac {c _ {0}}{l ^ {\gamma}} b\right)} _ {T _ {2}}. \\ \end{array} +$$ + +For the term $T_{2}$ we notice that $\frac{c_0}{j^\gamma} \leq 2^\gamma \alpha_k$ for $j \in \{t + 1, \dots, k + k_0\}$ , hence, we can upper bound $T_{2}$ as follows: + +$$ +\begin{array}{l} T _ {2} \leq 2 ^ {\gamma (s + 2 q)} \sum_ {j = t + 1} ^ {k + k _ {0}} \alpha_ {k} ^ {s + 2 q} (k + k _ {0} - j) ^ {q} \exp \left\{- b (k + k _ {0} - j) \alpha_ {k} \right\} \\ \leq 2 ^ {\gamma (s + 2 q) + 1} \alpha_ {k} ^ {s + 2 q} \int_ {0} ^ {+ \infty} x ^ {q} \exp \left\{- b \alpha_ {k} x \right\} d x \\ \leq 2 ^ {\gamma (s + 2 q) + 1} \alpha_ {k} ^ {s + q - 1} / b ^ {1 + q} \int_ {0} ^ {+ \infty} u ^ {q} \exp \{- u \} d u \\ \leq 2 ^ {\gamma (s + 2 q) + 1} \alpha_ {k} ^ {s + q - 1} / b ^ {1 + q}, \\ \end{array} +$$ + +where we have used that $\Gamma(q + 1) \leq 1$ for $q \in (0,1)$ . Now it remains to provide an upper bound for $T_{1}$ . Since $k_{0}$ satisfies (71), we get that, for $j \leq t$ , + +$$ +\left(\sum_ {\ell = j + 1} ^ {k + k _ {0}} \left(\frac {c _ {0}}{\ell^ {\gamma}}\right) ^ {2}\right) ^ {q} \prod_ {\ell = j + 1} ^ {k + k _ {0}} (1 - \frac {c _ {0}}{\ell^ {\gamma}} b) \leq \left(\frac {c _ {0}}{b t ^ {\gamma}}\right) ^ {q} \prod_ {\ell = j + 1} ^ {t} \left(1 - \frac {c _ {0}}{\ell^ {\gamma}} b\right). \tag {73} +$$ + +Then, applying (73), we get that + +$$ +\begin{array}{l} T _ {1} \leq (\frac {c _ {0}}{b t ^ {\gamma}}) ^ {q} \sum_ {j = 1} ^ {t} \left(\frac {c _ {0}}{j ^ {\gamma}}\right) ^ {s} \prod_ {\ell = j + 1} ^ {t} (1 - \frac {c _ {0}}{\ell^ {\gamma}} b) \stackrel {(a)} {\leq} (4 / b ^ {q + 1}) \left(\frac {4 (s - 1) \gamma}{b c _ {0}}\right) ^ {2 \gamma (s - 1) / (1 - \gamma)} \left(\frac {c _ {0}}{t ^ {\gamma}}\right) ^ {q + s - 1} \\ \leq (4 / b ^ {1 + q}) \left(\frac {4 (s - 1) \gamma}{b c _ {0}}\right) ^ {2 \gamma (s - 1) / (1 - \gamma)} 3 ^ {\gamma (s + q - 1)} \alpha_ {k} ^ {q + s - 1}, \\ \end{array} +$$ + +where in (a) we have additionally used Lemma 30 and in (b) we used $\frac{c_0}{t^\gamma} \leq 3^\gamma \alpha_k$ . It remains to check the relation (73), that is, it is enough to obtain an upper bound + +$$ +\left(\sum_ {\ell = 2} ^ {k + k _ {0}} \frac {c _ {0} ^ {2}}{\ell^ {2 \gamma}}\right) ^ {q} \prod_ {\ell = t + 1} ^ {k + k _ {0}} \left(1 - \frac {c _ {0}}{\ell^ {\gamma}} b\right) \leq \left(\frac {c _ {0}}{b t ^ {\gamma}}\right) ^ {q}. \tag {74} +$$ + +Since + +$$ +\sum_ {\ell = 2} ^ {k + k _ {0}} \frac {c _ {0} ^ {2}}{\ell^ {2 \gamma}} \leq c _ {0} ^ {2} \int_ {2} ^ {k + k _ {0}} \frac {\mathrm {d} x}{x ^ {2 \gamma}} \leq \frac {c _ {0} ^ {2}}{2 \gamma - 1}. +$$ + +Hence, (74) will follow from + +$$ +\frac {c _ {0} ^ {2}}{2 \gamma - 1} \exp \biggl \{- \frac {(k + k _ {0} - t) \alpha_ {k} b}{q} \biggr \} \leq \frac {c _ {0}}{t ^ {\gamma} b}, +$$ + +which is guaranteed by relations (71). + +![](images/5bb18c7949853a9fdb7b1f490cdc0593870805ef1f7281d292bcb79b282238fe.jpg) + +Lemma 30. Let $b > 0$ , and let $\alpha_{k} = c_{0} / (k)^{\gamma}$ , $\gamma \in (1/2; 1)$ , such that $c_{0}b \leq 1/2$ . Then for any $q \in [1; 2]$ , it holds that + +$$ +\sum_ {j = 1} ^ {k} \alpha_ {j} ^ {q} \prod_ {\ell = j + 1} ^ {k} (1 - \alpha_ {\ell} b) \leq C _ {q} \alpha_ {k} ^ {q - 1}, +$$ + +where $C_q = \frac{4}{b}\left(\frac{4(q - 1)\gamma}{bc_0}\right)^{2\gamma (q - 1) / (1 - \gamma)}$ + +Proof. Note that + +$$ +\begin{array}{l} \sum_ {j = 1} ^ {k} \alpha_ {j} ^ {q} \prod_ {\ell = j + 1} ^ {k} (1 - \alpha_ {\ell} b) = \alpha_ {k} ^ {q - 1} \sum_ {j = 1} ^ {k} \alpha_ {j} \prod_ {\ell = j + 1} ^ {k} \left(\frac {\alpha_ {\ell - 1}}{\alpha_ {\ell}}\right) ^ {q - 1} (1 - \alpha_ {\ell} b) \\ \leq \alpha_ {k} ^ {q - 1} \sum_ {j = 1} ^ {k} \alpha_ {j} \prod_ {\ell = j + 1} ^ {k} \left(1 + \frac {1}{\ell - 1}\right) ^ {\gamma (q - 1)} (1 - \alpha_ {\ell} b) \\ \leq \alpha_ {k} ^ {q - 1} \sum_ {j = 1} ^ {k} \alpha_ {j} \exp \left\{\sum_ {\ell = j + 1} ^ {k} \left\{\frac {\gamma (q - 1)}{\ell - 1} - \frac {b c _ {0}}{\ell^ {\gamma}} \right\} \right\}. \\ \end{array} +$$ + +Define $\ell_0 = \left\lceil \left(\frac{4(q - 1)\gamma}{bc_0}\right)^{1 / (1 - \gamma)}\right\rceil \geq 2$ , then for any $l > l_0$ we have $\frac{\gamma(q - 1)}{\ell - 1} -\frac{bc_0}{\ell^{\gamma}}\leq -\frac{bc_0}{2\ell^{\gamma}}$ . Hence, we get + +$$ +\sum_ {j = 1} ^ {k} \alpha_ {j} ^ {q} \prod_ {\ell = j + 1} ^ {k} (1 - \alpha_ {\ell} b) \leq \alpha_ {k} ^ {q - 1} \sum_ {j = 1} ^ {k} \alpha_ {j} \exp \left\{\sum_ {\ell = j + 1} ^ {k} \left\{- \frac {b c _ {0}}{2 \ell^ {\gamma}} \right\} \right\} \exp \left\{\sum_ {\ell = 2} ^ {\ell_ {0}} \left\{\frac {\gamma (q - 1)}{\ell - 1} \right\} \right\}. +$$ + +Therefore, Lemma 25 together with the elementary inequality $\mathrm{e}^{-x} \leq 1 - x / 2$ for $x \in (0; 1 / 2)$ , implies that + +$$ +\begin{array}{l} \sum_ {j = 1} ^ {k} \alpha_ {j} ^ {q} \prod_ {\ell = j + 1} ^ {k} (1 - \alpha_ {\ell} b) \leq \alpha_ {k} ^ {q - 1} \sum_ {j = 1} ^ {k} \alpha_ {j} \prod_ {\ell = j + 1} ^ {k} (1 - \frac {b}{4} \alpha_ {\ell}) \exp \left\{\gamma (q - 1) (\log (\ell_ {0} - 1) + 1) \right\} \\ \leq \alpha_ {k} ^ {q - 1} \frac {4}{b} \left(\frac {4 (q - 1) \gamma}{b c _ {0}}\right) ^ {2 \gamma (q - 1) / (1 - \gamma)}. \\ \end{array} +$$ + +![](images/c77d0ed6f2f14c114125d0ef1c7f7502cb0ed431e72872d10cd7ec8a012ba6ad.jpg) + +Lemma 31. For any $A > 0$ , any $0 \leq i \leq n - 1$ and any $\gamma \in (1/2, 1)$ it holds + +$$ +\sum_ {j = i} ^ {n - 1} \exp \biggl \{- A (j ^ {1 - \gamma} - i ^ {1 - \gamma}) \biggr \} \leq \left\{ \begin{array}{l l} 1 + \exp \bigl \{\frac {1}{1 - \gamma} \bigr \} \frac {1}{A ^ {1 / (1 - \gamma)} (1 - \gamma)} \Gamma (\frac {1}{1 - \gamma}) , & i f A i ^ {1 - \gamma} \leq \frac {1}{1 - \gamma} a n d i \geq 1 ; \\ 1 + \frac {1}{A (1 - \gamma) ^ {2}} i ^ {\gamma} , & i f A i ^ {1 - \gamma} > \frac {1}{1 - \gamma} a n d i \geq 1 ; \\ 1 + \frac {1}{A ^ {1 / (1 - \gamma)} (1 - \gamma)} \Gamma (\frac {1}{1 - \gamma}) , & i f i = 0 . \end{array} \right. +$$ + +Proof. The proof is given in [70, Lemma 16]. + +# H Applications to the TD learning algorithm + +Recall that the TD learning algorithm within the framework of linear stochastic approximation (LSA) can be written as + +$$ +\theta_ {k} = \theta_ {k - 1} - \alpha_ {k} \left(\mathbf {A} _ {k} \theta_ {k - 1} - \mathbf {b} _ {k}\right), \tag {75} +$$ + +where the matrices $\mathbf{A}_k$ and vectors $\mathbf{b}_k$ are defined by + +$$ +\mathbf {A} _ {k} = \varphi (s _ {k}) \left\{\varphi (s _ {k}) - \lambda \varphi (s _ {k + 1}) \right\} ^ {\top}, +$$ + +$$ +\mathbf {b} _ {k} = \varphi (s _ {k}) \mathcal {R} (s _ {k}). +$$ + +Our primary objective is to estimate the agent's value function, defined as + +$$ +V (s) = \mathbb {E} [ \sum_ {k = 0} ^ {\infty} \lambda^ {k} \mathcal {R} (S _ {k}) \mid S _ {0} = s ], +$$ + +where $S_{k + 1}\sim \mathrm{P}(\cdot \mid S_k)$ for all $k\in \mathbb{N}$ . In this context, the TD learning updates (75) correspond to approximating solutions to the deterministic linear system $\bar{\mathbf{A}}\theta^{\star} = \bar{\mathbf{b}}$ (see [77]), where the matrix $\bar{\mathbf{A}}$ and right-hand side $\bar{\mathbf{b}}$ are given by + +$$ +\bar {\mathbf {A}} = \mathbb {E} _ {s \sim \mu , s ^ {\prime} \sim \mathrm {P} (\cdot | s)} [ \varphi (s) \{\varphi (s) - \lambda \varphi (s ^ {\prime}) \} ^ {\top} ] +$$ + +$$ +\bar {\mathbf {b}} = \mathbb {E} _ {s \sim \mu} [ \varphi (s) \mathcal {R} (s) ]. +$$ + +The rest of the proof of Proposition 5 reduces to checking the properties of $\bar{\mathbf{A}}$ and direct verification of conditions of Proposition 2, which is done by the lines of [67, Proposition 2] + +Numerical experiments. We consider the simple instance of the Garnet problem [4, 32]. This problem is characterized by the number of states $N_{s}$ , number of actions $a$ , and branching factor $b$ . Here $b$ corresponds to the number of states $s'$ , that can be reached when starting from a given state-action pair $(s, a)$ . The reward $r(s, a) \in [0,1]$ is a deterministic function. We set the hyperparameter values $N_{s} = 6$ , $a = 2$ , $b = 3$ , feature dimension $d = 2$ , and discount factor $\lambda = 0.8$ . We aim to evaluate the value function of policy $\pi(\cdot | s)$ , which is given, for any $a \in \mathcal{A} = \{1,2\}$ , by the expression + +$$ +\pi (a | s) = \frac {U _ {a} ^ {(s)}}{\sum_ {i = 1} ^ {| \mathcal {A} |} U _ {i} ^ {(s)}}, +$$ + +where the $U_{i}^{(s)}$ are i.i.d. observations with $\mathcal{U}[0,1]$ . In this case we can suppose that $\mathcal{R}(s) = \mathbb{E}_{a\sim \pi (a|s)}r(s,a)$ is a random variable which depends on state $s$ and some independent random variable. We consider the problem of policy evaluation in this MDP using the TD learning algorithm with randomly generated feature mapping, that is, we generate the matrix + +$$ +\Phi \in \mathbb {R} ^ {N _ {s} \times d} +$$ + +with i.i.d. $\mathcal{N}(0,1)$ entries, and then take $\phi(s), s \in \{1,\dots,|S|\}$ , to its $s$ -th row, normalized by its euclidean norm: $\phi(s) = \Phi_s / \| \Phi_s \|$ . We run the procedure (1) with the learning rates $\alpha_k = c_0 / (k_0 + k)^\gamma$ with $\gamma = 2/3$ with appropriately chosen $c_0$ and $k_0$ . We generate random vector $u$ form unitary sphere, and compute coverage probabilities for $u^\top \theta^\star$ for confidence levels $\{0.8, 0.9, 0.95\}$ . The detailed setting of the experiments follows [29]. Results are given in Table 1 and illustrates the consistency of multiplier subsample bootstrap procedure applied on the Garnet problem. + +Table 1: Coverage probabilities of OBM estimation for the empirical distribution. + +
n | bn0.950.90.8stddev ×103
♂2(θ) (u)σ2(u)♂2(θ) (u)σ2(u)♂2(θ) (u)σ2(u)
20480 | 2500.8730.8810.7730.8050.6410.66210.89
204800 | 12000.9350.9450.8800.8920.7680.7843.49
1024000 | 36000.9420.9480.8870.8970.7690.7881.56
+ +Code to reproduce experiments is given in https://github.com/svsamsonov/markov_lsa_normal Approximation. 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In this paper, we initiate an end-to-end framework of statistical inference under performativity. Our contributions are twofold. First, we establish a central limit theorem for estimation and inference in the performative setting, enabling standard inferential tasks such as constructing confidence intervals and conducting hypothesis tests in policy-making contexts. Second, we leverage this central limit theorem to study prediction-powered inference (PPI) under performativity. This approach yields more precise estimates and tighter confidence regions for the model parameters (i.e., policies) of interest in performative prediction. We validate the effectiveness of our framework through numerical experiments. To the best of our knowledge, this is the first work to establish a complete statistical inference under performativity, introducing new challenges and inference settings that we believe will provide substantial value to policy-making, statistics, and machine learning. + +# 1 Introduction + +Prediction-informed decisions are ubiquitous in nearly all areas and play important roles in our daily lives. An important and commonly observed phenomenon is that prediction-informed decisions can impact the targets they aim to predict, which is called performativity of predictions. For instance, policies about loans based on default risk prediction can alter consumption habits of the population that will further have an impact on their ability to pay off their loans. + +To characterize performativity of predictions, a rich line of work on performative prediction [25] have been formalizing and investigating this idea that predictive models used to support decisions can impact the data-generating process. Mathematically, given a parameterized loss function $\ell$ , the aim of performative prediction is to optimize the performative risk: + +$$ +\Pr (\theta) := \mathbb {E} _ {z \sim \mathcal {D} (\theta)} \ell (z; \theta) \tag {1} +$$ + +where $z = (x,y)\in \mathcal{X}\times \mathcal{Y}$ is the input and output pair drawn from a distribution $\mathcal{D}(\theta)$ that is dependent on the loss parameter $\theta$ . Typically, $\mathcal{D}(\theta)$ is unknown and the optimization objective $\mathrm{PR}(\theta)$ can be non-convex even if $\ell (z;\theta)$ is convex in $\theta$ . Thus, finding a performative optimal point $\theta_{\mathrm{PO}}\in \arg \min_{\theta}\mathrm{PR}(\theta)$ (there might exist multiple optimizers due to non-convexity) can be theoretically intractable unless we impose very strong distributional assumptions [20]. As an + +alternative, [25] mainly study how to obtain a performative stable point, which satisfies the following relationship: + +$$ +\theta_ {\mathrm {P S}} = \arg \min _ {\theta} \mathbb {E} _ {z \sim \mathcal {D} (\theta_ {\mathrm {P S}})} \ell (z; \theta). +$$ + +The performative stable point could be proved unique under some regularity conditions, and it could be shown close to a performative optimal point when the distribution shift between different $\theta$ 's is not too dramatic, which makes it a good proxy to $\theta_{\mathrm{PO}}$ . In particular, $\theta_{\mathrm{PS}}$ could be considered as a good proxy to the Stackelberg equilibrium in the strategic classification setting. Moreover, it could be calculated in distribution-agnostic settings. + +Previous work mainly focuses on prediction performance and convergence rate analysis for performative prediction. On the contrary, another important aspect, inference under performativity, eludes the literature. Although a central limit theorem was established in [7] for the stochastic gradient update algorithm with a single sample under performativity (a setting that is often impractical in applications such as policy-making), the authors assumed all structural knowledge to be given and did not provide any data-driven methods for estimating the covariance of the various quantities appearing in their central limit theorem. Therefore, [7] did not provide a complete statistical inference framework under performativity. + +However, inference is extremely important in performative prediction because parameter $\theta$ in many scenarios represents a concrete policy, such as a tax rate or credit score cutoff. Thus, when it comes to policy-making, the aim of tackling $\mathrm{PR}(\theta)$ is not just for prediction, but more for obtaining a good policy. As a result, knowing convergence to $\theta_{\mathrm{PS}}$ is not enough, we need to build statistical inference for $\theta_{\mathrm{PS}}$ so as to enable people to report additional critical information like confidence or conduct necessary hypothesis testing. + +Our contributions. In light of the importance of building statistical inference under performativity, in this work, we build a framework including the following elements. + +(1). As our first contribution, we investigate a widely applied iterative algorithm to calculate $\theta_{\mathrm{PS}}$ , i.e., repeated risk minimization (RRM) (see details in Section 2), and establish central limit theorems for the $\widehat{\theta}_t$ ’s obtained in the RRM process towards $\theta_{\mathrm{PS}}$ . Based on that, we are able to obtain the confidence region for $\theta_{\mathrm{PS}}$ . Our results could be viewed as generalizing standard statistical inference from a static setting to a dynamic setting. +(2). As our second contribution, we further leverage the derived central limit theorems to investigate prediction-powered inference (PPI), another recently popular topic in modern statistical learning, under performativity. Our results generalize previous work [1] to a dynamic performative setting. This enables us to obtain better estimation and inference for the RRM process and $\theta_{\mathrm{PS}}$ . More importantly, our results could also help mitigate data scarcity issues in getting feedback about policy implementation that often conducted by doing surveys that frequently encounter non-responses [13]. Thus, we also contribute to generalizing the line of work on performative prediction by introducing a more data efficient algorithm. + +To sum up, our work establishes the first end-to-end framework for statistical inference under performativity for the celebrated repeated risk minimization algorithm. Meanwhile, we introduce prediction-powered inference under performativity to enable a more efficient inference. We believe our work would inspire new interesting topics and bring up new challenges to both areas of perforative prediction and prediction-powered inference, as well as add significant value to policy-making in a broad range of areas such as social science and economics. + +# 1.1 Related Work + +Performatives prediction. Performativity describes the phenomenon whereby predictions influence the outcomes they aim to predict. [25] were the first to formalize performative prediction in the supervised-learning setting; their work, along with the majority of subsequent papers [5, 9, 16, 19, 21, 24, 26, 29], have been focused on performative stability and proposed algorithms for learning performative stable parameters. On the other hand, performative optimality requires much stricter conditions (e.g. distributional assumptions to ensure the convexity of $\mathrm{PR}(\theta)$ ) than performative stability, a few papers address the problem of finding performative optimal parameters. [20] introduce a two-stage method that learns a distribution map to locate the performative optimal parameter. [17] study performative optimality in outcome-only performative settings. Finally, [11] + +provide a comprehensive overview of learning algorithms, optimization methods, and applications for performative prediction. Unlike these prior works on performative prediction, which focus on prediction accuracy, our work is dedicated to constructing powerful and statistically valid inference procedures under the performative framework. + +Prediction-powered inference. [2] first introduced the prediction-powered inference (PPI) framework, which leverages black-box machine learning models to construct valid confidence intervals (CIs) for statistical quantities. Since then, PPI has been extended and applied in various settings. Closely related to our strategies, [1] propose $\mathrm{PPI}++$ , a more computationally efficient procedure that enhances predictability by accommodating a wider range of models on unlabeled data, while guaranteeing performance (e.g. CI width) no worse than that of classical inference methods. Other extensions include Stratified PPI [10], which incorporates simple data stratification strategies into basic PPI estimates; Cross PPI [33], which obtains confidence intervals with significantly lower variability by including model training; Bayesian PPI [12] and FAB-PPI [6], which propose frameworks for PPI based on Bayesian inference. PPI is also connected to topics such as semi-parametric inference and missing-data imputation [8, 27, 28, 30]. Our work is the first one to study PPI under performativity, and we validate the PPI framework in the performative setting both theoretically and empirically. + +Inference in performativity. [5] studies identifiability and estimation error under a specific microfoundation model with performativity. Yet it doesn't address confidence interval construction. A closely related work is [7], which also establishes the asymptotic normality and minimax optimality for performative settings. It focuses on stochastic gradient update with one sample per iteration, whereas our work analyzes the empirical risk minimizer on batch updates. Moreover, [7] does not provide a data-driven approach for covariance estimation and thus lacks an end-to-end inference framework. In contrast, our work explicitly handles density estimation and provides an end-to-end inference method for constructing confidence intervals, which is missing in existing literature, to the best of our knowledge. + +# 2 Background + +In this section, we recap more detailed background knowledge about performative prediction and prediction-powered inference. + +Repeated risk minimization. Recall that the main objective of interest is the performative stable point, which satisfies the following relationship: + +$$ +\theta_ {\mathrm {P S}} = \arg \min _ {\theta} \mathbb {E} _ {z \sim \mathcal {D} (\theta_ {\mathrm {P S}})} \ell (z; \theta). +$$ + +Repeated risk minimization (RRM) is a simple algorithm that can efficiently find $\theta_{\mathrm{PS}}$ . Specifically, one starts with an arbitrary $\theta_0$ and repeats the following procedure: + +$$ +\theta_ {t + 1} = \underset {\theta} {\arg \min } \mathbb {E} _ {z \sim \mathcal {D} (\theta_ {t})} \ell (z; \theta) +$$ + +for $t \in \mathbb{N}$ . Under some regularity conditions, the above update is well-defined and provably converges to a unique $\theta_{\mathrm{PS}}$ at a linear rate. + +Theorem 2.1 (Informal, adopted from [25]). If the loss is smooth, strongly convex, and the mapping $\mathcal{D}(\cdot)$ satisfies certain Lipschitz conditions, then $\theta_{PS}$ is uniquely defined and repeated risk minimization converges to $\theta_{PS}$ in a linear rate. + +We will further explicitly state those conditions in Section 3. Throughout the paper, we will mainly focus on building an inference framework under the repeated risk minimization algorithm. + +Prediction-powered inference. A rich line of work on prediction-powered inference (PPI) [1] considers how to combine limited gold-standard labeled data with abundant unlabeled data to obtain more efficient estimation and construct tighter confidence regions for some unknown parameters. Specifically, a general predictive setting is considered in which each instance has an input $x \in \mathcal{X}$ and an associated observation $y \in \mathcal{Y}$ . People have access to a limited set of gold-standard labeled data $\{x_i, y_i\}_{i=1}^n$ that are i.i.d. drawn from a distribution $\mathcal{D}$ . Meanwhile, we have abundant unlabeled data $\{x_i^u\}_{i=1}^N$ that are i.i.d. drawn from the same marginal distribution as gold-standard labeled data, i.e. $\mathcal{D}_{\mathcal{X}}$ , where $N \gg n$ . In addition, an annotating model $f: \mathcal{X} \mapsto \mathcal{Y}$ (possibly off-the-shelf and black- + +box machine learning models) is used to label data1. In [1], the authors show that for a convex loss with a unique solution, compared with standard M-estimator $\widehat{\theta}^{\mathrm{SL}} = \arg \min_{\theta}\sum_{i = 1}^{n}\ell (x_i,y_i;\theta) / n$ + +$$ +\widehat {\theta} ^ {\mathrm {P P I}} (\lambda) := \arg \min _ {\theta} \lambda \frac {1}{N} \sum_ {i = 1} ^ {N} \ell (x _ {i} ^ {u}, f (x _ {i} ^ {u}); \theta) + \frac {1}{n} \sum_ {i = 1} ^ {n} \ell (x _ {i}, y _ {i}; \theta) - \lambda \frac {1}{n} \sum_ {i = 1} ^ {n} \ell (x _ {i}, f (x _ {i}); \theta) +$$ + +can be a better estimator of $\theta^{*} = \arg \min_{\theta}\mathbb{E}_{z\sim \mathcal{D}}\ell (z;\theta)$ via appropriately chosen $\lambda$ based on data. + +Notation. For $K \in \mathbb{N}_{+}$ , we use $[K]$ to denote $\{1,2,\dots ,K\}$ . We use $\xrightarrow{P}$ and $\xrightarrow{D}$ to denote convergence in probability and in distribution, respectively. For two sets $\mathcal{S}$ and $\mathcal{S}'$ , we use $\mathcal{S} + \mathcal{S}'$ to denote the set $\{s + s': s \in \mathcal{S}, s' \in \mathcal{S}'\}$ . $\mathcal{N}(\mu, \Sigma)$ denotes a Gaussian distribution with mean $\mu$ and covariance matrix $\Sigma$ . Lastly, we denote a $k$ -dimensional identity matrix as $I_{k}$ . We use $\mathcal{B}(c,r)$ to denote a ball with center $c$ and radius $r$ . Lastly, we use $\|\cdot\|$ for $\ell_{2}$ -norm and $\mathbf{1}$ to denote a column vector with all coordinates 1. + +# 3 Inference under Performativity + +In this section, we initiate the inference framework for repeated risk minimization in the batch setting under performativity. We mainly consider the repeated risk minimization setting. Unlike standard inference problems, where estimators are built for a fixed underlying data distribution, in our dynamic setting specified below, building asymptotic results such as CLT imposes extra challenges and this has not been covered by any existing literature so far. + +Specifically, at time $t = 0$ , we have access to a set of labeled data $\{z_{0,i}\}_{i=1}^{n_0}$ that is i.i.d. drawn from a distribution $\mathcal{D}(\theta_0)$ , where $z_{0,i} = (x_{0,i}, y_{0,i})$ and $\theta_0$ is chosen by us. Then, we use the empirical repeated risk minimization to output + +$$ +\widehat {\theta} _ {1} = \arg \min _ {\theta} \frac {1}{n _ {0}} \sum_ {i = 1} ^ {n _ {0}} \ell (z _ {0, i}; \theta) +$$ + +as an estimator of $\theta_{1} = \arg \min_{\theta}\mathbb{E}_{z\sim \mathcal{D}(\theta_{0})}\ell (z;\theta)$ . Then, for $t\geqslant 1$ , at time $t$ , we further have access to a set of labeled data $\{z_{t,i}\}_{i = 1}^{n_t}$ that are i.i.d. drawn from the distribution induced by last iteration, i.e., $\mathcal{D}(\widehat{\theta}_t)$ . Let us further define $G(\tilde{\theta}) = \arg \min_{\theta}\mathbb{E}_{z\sim \mathcal{D}(\tilde{\theta})}\ell (z;\theta)$ . Then, we can obtain the output + +$$ +\widehat {\theta} _ {t + 1} = \arg \min _ {\theta} \frac {1}{n _ {t}} \sum_ {i = 1} ^ {n _ {t}} \ell (z _ {t, i}; \theta) +$$ + +as an estimator of $\theta_{t + 1} = G(\theta_t)$ . This iterative process will incur two trajectories, i.e., (1) **underlying trajectory:** $\theta_0 \to \theta_1 \to \dots \theta_t \to \dots$ ; (2) **trajectory in practice:** $\theta_0 \to \widehat{\theta}_1 \to \dots \widehat{\theta}_t \to \dots$ . + +Our aim is to provide inference on $\widehat{\theta}_t$ for any $t\geqslant 1$ . For simplicity, we let $n_t = n$ for all $t$ . + +# 3.1 Central Limit Theorem of $\widehat{\theta}_t$ + +In order to build CLT for $\widehat{\theta}_t$ , we first establish the consistency of $\widehat{\theta}_t$ , which is relatively straightforward given that [25] has built the non-asymptotic convergence results. Then, we introduce our main result on building CLT for $\widehat{\theta}_t$ . Lastly, we provide a novel method to estimate the variance of $\widehat{\theta}_t$ . + +Consistency of $\widehat{\theta}_t$ . We start with proving the consistency of $\widehat{\theta}_t$ . Recall that we have a trajectory induced by the samples $\theta_0 \to \widehat{\theta}_1 \to \dots \widehat{\theta}_t \to \dots$ by the iterative algorithm deployed. Without consistency, CLT is not expected to hold. Our results are based on the following assumptions. + +Assumption 3.1. Assume the loss function $\ell$ satisfies: + +(a). (Local Lipschitzness) Loss function $\ell(z; \theta)$ is locally Lipschitz: for each $\theta$ , there exist a neighborhood $\Upsilon(\theta)$ of $\theta$ such that $\ell(z; \tilde{\theta})$ is $L(z)$ Lipschitz w.r.t $\tilde{\theta}$ for all $\tilde{\theta} \in \Upsilon(\theta)$ and $\mathbb{E}_{z \sim \mathcal{D}(\theta)} L(z) < \infty$ . + +(b). (Joint Smoothness) Loss function $\ell(z; \theta)$ is $\beta$ -jointly smooth in both $z$ and $\theta$ : + +$$ +\left\| \nabla_ {\theta} \ell (z; \theta) - \nabla_ {\theta} \ell (z; \theta^ {\prime}) \right\| _ {2} \leqslant \beta \left\| \theta - \theta^ {\prime} \right\| _ {2}, \left\| \nabla_ {\theta} \ell (z; \theta) - \nabla_ {\theta} \ell (z ^ {\prime}; \theta) \right\| _ {2} \leqslant \beta \left\| z - z ^ {\prime} \right\| _ {2}, +$$ + +for any $z,z^{\prime}\in \mathcal{Z}$ and $\theta ,\theta^{\prime}\in \Theta$ + +(c). (Strong Convexity) Loss function $\ell(z; \theta)$ is $\gamma$ -strongly convex w.r.t $\theta$ : + +$$ +\ell (z; \theta) \geqslant \ell (z; \theta^ {\prime}) + \nabla_ {\theta} \ell (z; \theta^ {\prime}) ^ {\top} (\theta - \theta^ {\prime}) + \frac {\gamma}{2} \| \theta - \theta^ {\prime} \| ^ {2}, +$$ + +for any $z\in \mathcal{Z}$ and $\theta ,\theta^{\prime}$ + +(d). $(\varepsilon$ -Sensitivity) The distribution map $D(\theta)$ is $\varepsilon$ -sensitive, i.e.: + +$$ +W _ {1} \left(\mathcal {D} (\theta), \mathcal {D} \left(\theta^ {\prime}\right)\right) \leqslant \varepsilon \| \theta - \theta^ {\prime} \|, +$$ + +for any $\theta, \theta'$ , where $W_1$ is the Wasserstein-1 distance. + +Remark 3.2. The assumptions $(b)$ , $(c)$ , $(d)$ follow the standard ones in [25], which are proved to be the minimal requirements for trajectory convergence. We additionally require $(a)$ to build consistency for $\widehat{\theta}_t$ beyond convergence of $\widehat{\theta}_t$ to $\theta_{PS}$ . + +Proposition 3.3. Under Assumption 3.1, if $\varepsilon < \frac{\gamma}{\beta}$ , then for any given $T \geqslant 0$ , we have that for all $t \in [T]$ + +$$ +\widehat {\theta} _ {t} \stackrel {{P}} {{\longrightarrow}} \theta_ {t}. +$$ + +Building CLT for $\widehat{\theta}_t$ . In order to build the central limit theorem for $\widehat{\theta}_t$ , we need to introduce a few extra assumptions. Due to limited space, going forward, we defer the required assumptions in later theorems to Appendix. + +Let us denote $\Sigma_{\tilde{\theta}}(\theta) = H_{\tilde{\theta}}(\theta)^{-1}V_{\tilde{\theta}}(\theta)H_{\tilde{\theta}}(\theta)^{-1}$ , where $H_{\tilde{\theta}}(\theta) = \nabla_{\theta}^{2}\mathbb{E}_{z\sim \mathcal{D}(\tilde{\theta})}\ell (z;\theta)$ and $V_{\tilde{\theta}}(\theta) = \operatorname{Cov}_{z\sim \mathcal{D}(\tilde{\theta})}\bigl (\nabla_{\theta}\ell (z;\theta)\bigr)$ . And recall that $G(\tilde{\theta}) = \arg \min_{\theta}\mathbb{E}_{z\sim \mathcal{D}(\tilde{\theta})}\ell (z;\theta)$ . + +Theorem 3.4 (Central Limit Theorem of $\widehat{\theta}_t$ ). Under Assumption 3.1 and A.1, if $\varepsilon < \frac{\gamma}{\beta}$ , then for any given $T \geqslant 0$ , we have that for all $t \in [T]$ + +$$ +\sqrt {n} \left(\widehat {\theta} _ {t} - \theta_ {t}\right) \stackrel {D} {\to} \mathcal {N} \left(0, V _ {t}\right) +$$ + +with + +$$ +V _ {t} = \sum_ {i = 1} ^ {t} \left[ \prod_ {k = i} ^ {t - 1} \nabla G \left(\theta_ {k}\right) \right] \Sigma_ {\theta_ {i - 1}} \left(\theta_ {i}\right) \left[ \prod_ {k = i} ^ {t - 1} \nabla G \left(\theta_ {k}\right) \right] ^ {\top}. +$$ + +In particular, $\nabla G(\theta_k) = -H_{\theta_k}(\theta_{k + 1})^{-1}\bigl (\nabla_{\tilde{\theta}}\mathbb{E}_{z\sim \mathcal{D}(\theta_k)}\nabla_\theta \ell (z;\theta_{k + 1})\bigr)$ , where $\nabla_{\tilde{\theta}}$ is taking gradient for the parameter in $\mathcal{D}(\tilde{\theta})$ , $\nabla_{\theta}$ is taking gradient for the parameter in $\ell (z;\theta)$ and $\prod_{k = t}^{t - 1}\nabla G(\theta_k) = I_d$ . + +Estimation of $\nabla G(\theta_t)$ , $V_t$ Given the established CLT for $\widehat{\theta}_t$ , in order to construct confidence regions for $\widehat{\theta}_t$ in practice, the only thing left is to provide an estimation of $\nabla G(\theta_t)$ and $V_t$ with samples. In previous results, with a more detailed calculation, we obtain + +$$ +\nabla G \left(\theta_ {k}\right) = - H _ {\theta_ {k}} \left(\theta_ {k + 1}\right) ^ {- 1} \mathbb {E} _ {z \sim \mathcal {D} \left(\theta_ {k}\right)} \left[ \nabla_ {\theta} \ell (z, \theta_ {k + 1}) \nabla_ {\theta} \log p (z, \theta_ {k}) ^ {\top} \right]. +$$ + +where $p(\cdot, \theta)$ is the density function of distribution $D(\theta)$ , and the score function $\nabla_{\theta} \log p(z, \theta)$ is thus a $d$ -dimensional vector (recall that $\theta$ is of dimension $d$ ). In order to estimate it for any $\theta$ , we propose a novel score matching method. Specifically, we use a model $M(z, \theta; \psi)$ parameterized by $\psi$ to approximate $p(z, \theta)$ . Inspired by the objective in [14], for any given $\theta$ (e.g., $\widehat{\theta}_t$ ), we aim to minimize the following objective parameterized by $\psi$ for all $\theta$ : + +$$ +\begin{array}{l} J (\theta ; \psi) = \int p (z, \theta) \| \nabla_ {\theta} \log p (z, \theta) - s (z, \theta ; \psi) \| ^ {2} d z \\ = \int p (z, \theta) \Big (\| \nabla_ {\theta} \log p (z, \theta) \| ^ {2} + \| s (z, \theta ; \psi) \| ^ {2} - 2 \nabla_ {\theta} \log p (z, \theta) ^ {\top} s (z, \theta ; \psi) \Big) d z \\ \end{array} +$$ + +where $s(z,\theta ;\psi) = \nabla_{\theta}\log M(z,\theta ;\psi)$ . If we can learn a $\widehat{\psi}$ so that $s(z,\theta ;\widehat{\psi}) = \nabla_{\theta}\log p(z,\theta)$ for all $\theta$ , then we can reach the minimum $J(\theta ;\widehat{\psi}) = 0$ . + +Notice the first term is unrelated to $\psi$ ; the second term involves the model $M$ that is chosen by us, so we have the analytical expression of it. Thus, our key task will be estimating the third term, which involves $\mathcal{K}(\theta; \psi) := \int p(z, \theta) \nabla_{\theta} \log p(z, \theta)^{\top} s(z, \theta; \psi) dz$ . + +We remark that in our setting, instead of taking the gradient at $z$ , we have new challenges in taking the gradient at $\theta$ . So, we derive the following key lemma. + +Lemma 3.5. Under Assumption A.2, we have + +$$ +\mathcal {K} (\theta ; \psi) = \sum_ {i = 1} ^ {d} \left[ \frac {\partial}{\partial \theta^ {(i)}} \int p (z, \theta) \frac {\partial \log M (z , \theta ; \psi)}{\partial \theta^ {(i)}} d z - \int p (z, \theta) \frac {\partial^ {2} \log M (z , \theta ; \psi)}{\partial \theta^ {(i) 2}} d z \right] +$$ + +where $\theta^{(i)}$ is the $i$ -th coordinate of $\theta$ . + +Based on the lemma, we propose a novel gradient-free score matching method with policy perturbation to estimate $\mathcal{K}(\widehat{\theta}_t;\psi)$ for any $t\in [T]$ . Policy perturbation is a commonly used technique in estimating the policy effect under general equilibrium shift [22] or interference [32]. Instead of just getting samples for $\widehat{\theta}_t$ for each $t$ , we additionally sample for all perturbed policies in $\{\widehat{\theta}_t + \eta e_1,\widehat{\theta}_t + \eta e_2,\dots ,\widehat{\theta}_t + \eta e_d\}$ , where $\eta >0$ is a small scalar at our choice and $\{e_j\}_{j}$ are standard basis for $\mathbb{R}^d$ . Typically, for a policy $\theta$ , its dimension $d$ is low. One concrete example in practice is to use slightly different price strategies in different local markets. + +Specifically, the term $\int p(z,\widehat{\theta}_t)\frac{\partial^2\log M(z,\widehat{\theta}_t;\psi)}{\partial\theta^{(i)2}}dz$ could be easily estimated by using empirical mean, e.g., $\frac{1}{n}\sum_{j = 1}^{n}\frac{\partial^2\log M(z_{t,j},\widehat{\theta}_t;\psi)}{\partial\theta^{(i)2}}$ . And for the derivative $\frac{\partial}{\partial\theta^{(i)}}\int p(z_{t,j},\widehat{\theta}_t)\frac{\partial\log M(z,\widehat{\theta}_t;\psi)}{\partial\theta^{(i)}}dz$ , if we draw additional $k$ samples $\{z_{t,u}^{(i)}\}_{u = 1}^{k}$ for each perturbed policy $\widehat{\theta}_t + \eta e_i$ , we can use the following estimator: + +$$ +\frac {1}{\eta} \left(\frac {1}{k} \sum_ {u = 1} ^ {k} \frac {\partial \log M (z _ {t , u} ^ {(i)} , \widehat {\theta} _ {t} + \eta e _ {i} ; \psi)}{\partial \theta^ {(i)}} - \frac {1}{n} \sum_ {u = 1} ^ {n} \frac {\partial \log M (z _ {t , u} , \widehat {\theta} _ {t} ; \psi)}{\partial \theta^ {(i)}}\right). +$$ + +Combining the above, we have a straightforward way to estimate $G(\theta_t)$ and $V_{t}$ for any $t\in [T]$ by plugging in the empirical estimate. Let us denote the estimator of $V_{t}$ by $\widehat{V}_t$ . Then, we would have + +$$ +\widehat {V} _ {t} ^ {- 1 / 2} \sqrt {n} \left(\widehat {\theta} _ {t} - \theta_ {t}\right) \stackrel {D} {\rightarrow} \mathcal {N} (0, I _ {d}) \tag {2} +$$ + +if the model $M(z,\theta ;\psi)$ is expressive enough. + +Theoretical validity of estimation. Now we prove that the policy perturbation method provides a valid estimator of $\nabla G(\theta_k)$ . Recall that the gradient of $G$ is given by + +$$ +\nabla G (\theta_ {k}) = - H _ {\theta_ {k}} (\theta_ {k + 1}) ^ {- 1} \mathbb {E} _ {\theta_ {k}} [ \nabla_ {\theta} \ell (z; \theta_ {k + 1}) \nabla_ {\theta} \log p (z, \theta_ {k}) ^ {\top} ], +$$ + +and the estimator is defined by + +$$ +\widehat {g} _ {k} := - H _ {\widehat {\theta} _ {k}} \left(\widehat {\theta} _ {k + 1}\right) ^ {- 1} \widehat {\mathbb {E}} _ {\widehat {\theta} _ {k}} \left[ \nabla_ {\theta} \ell (z; \widehat {\theta} _ {k + 1}) s (z, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) ^ {\top} \right]. +$$ + +Theorem 3.6. Under Assumption 3.1 A.1, A.9 and A.11, we have + +$$ +\| \widehat {g} _ {k} - \nabla G (\theta_ {k}) \| ^ {2} = O _ {p} \big (\frac {1}{\sqrt {n}} + \frac {1}{\eta \sqrt {\min (n , k)}} + \eta + a _ {n} \big). +$$ + +where $a_{n} = o(1)$ is a vanishing optimization error term defined in Assumption A.9(d). + +The proof is by decomposing the error between the empirical and true gradient into several components, where each varies in whether it uses empirical or population quantities. We bound the deviations between the true parameters $\theta_{k}$ , $\theta_{k+1}$ and their empirical counterparts $\widehat{\theta}_{k}$ , $\widehat{\theta}_{k+1}$ using Theorem 3.4. We bound the true score function and its estimation via a uniform bound of $\|\nabla \log p(z,\theta) - s(z,\theta,\psi(\theta))\|^2$ over the perturbed policies using Theorem A.13. Taken together, the above result shows the consistency of $\widehat{g}_k$ obtained by policy perturbation. + +# 3.2 Bias-Aware Inference for Performative Stable Point + +Finally, we further provide a way to construct the confidence region for $\theta_{\mathrm{PS}}$ . This is directly followed from our previous results on building CLT for $\widehat{\theta}_t$ . By the convergence results derived for the underlying trajectory by [25], under Assumption 3.1, we have + +$$ +\left\| \theta_ {t} - \theta_ {\mathrm {P S}} \right\| \leqslant \left(\frac {\varepsilon \beta}{\gamma}\right) ^ {t} \| \theta_ {0} - \theta_ {\mathrm {P S}} \|. +$$ + +Thus, we can immediately obtain the following corollary by using bias-aware inference - a commonly seen technique in econometrics [3, 4, 15, 23]. + +Corollary 3.7 (Confidence region construction for $\theta_{\mathrm{PS}}$ ). Under Assumption 3.1, A.1, A.2, and A.3, if $\varepsilon < \frac{\gamma}{\beta}$ , for any $\delta \in (0,1)$ , we can obtain a confidence region $\widehat{\mathcal{R}}_t(n,\delta)$ for $\theta_t$ by using Eq. 2, such that + +$$ +\lim _ {n \rightarrow \infty} \mathbb {P} \left(\theta_ {t} \in \widehat {\mathcal {R}} _ {t} (n, \delta)\right) = 1 - \delta . +$$ + +Moreover, if $\theta_0, \theta_{PS} \in \{\theta : \| \theta \| \leqslant B\}$ , + +$$ +\lim _ {n \rightarrow \infty} \mathbb {P} \left(\theta_ {P S} \in \widehat {\mathcal {R}} _ {t} (n, \delta) + \mathcal {B} \left(0, 2 B \left(\frac {\varepsilon \beta}{\gamma}\right) ^ {t}\right)\right) \geqslant 1 - \delta . +$$ + +Corollary 3.7 provides a way to construct the confidence region for the performative stable point based on the confidence region for $\theta_t$ . Notice that the derived new confidence region is quite close to $\widehat{\mathcal{R}}_t(n, \delta)$ and the difference vanishes exponentially fast as $t$ grows. Thus, we expect the derived region to be quite tight for moderately large $t$ , meaning: + +$$ +\lim _ {n \rightarrow \infty} \mathbb {P} \Big (\theta_ {\mathrm {P S}} \in \widehat {\mathcal {R}} _ {t} (n, \delta) + \mathcal {B} \Big (0, 2 B \left(\frac {\varepsilon \beta}{\gamma}\right) ^ {t} \Big) \Big) \approx 1 - \delta . +$$ + +For the condition $\theta_0, \theta_{\mathrm{PS}} \in \{\theta : \| \theta \| \leqslant B\}$ , it will be natural to satisfy and we can get an explicit and feasible upper bound $B$ under mild conditions. It is because that $\theta_0$ is at our choice and we can further derive an explicit and feasible upper bound for $\| \theta_{\mathrm{PS}}\|$ by using the strong convexity. Specifically, by $\gamma$ -strong convexity of the loss function with respect to $\theta$ , we have + +$$ +\left(\mathbb {E} _ {z \sim \mathcal {D} \left(\theta_ {\mathrm {P S}}\right)} \nabla_ {\theta} \ell (z; \theta_ {0}) - \mathbb {E} _ {z \sim \mathcal {D} \left(\theta_ {\mathrm {P S}}\right)} \nabla_ {\theta} \ell (z; \theta_ {\mathrm {P S}})\right) ^ {\top} \left(\theta_ {0} - \theta_ {\mathrm {P S}}\right) \geqslant \gamma \left\| \theta_ {0} - \theta_ {\mathrm {P S}} \right\| ^ {2}. +$$ + +Since $\mathbb{E}_{z\sim \mathcal{D}(\theta_{\mathrm{PS}})}\nabla_{\theta}\ell (z;\theta_{\mathrm{PS}}) = 0$ , this leads to $\| \mathbb{E}_{z\sim \mathcal{D}(\theta_{\mathrm{PS}})}\nabla_{\theta}\ell (z;\theta_0)\| \geqslant \gamma \| \theta_0 - \theta_{\mathrm{PS}}\|$ . Thus, if we further have $\sup_{z\in \mathcal{Z}}\| \nabla_{\theta}\ell (z;\theta_0)\| \leqslant \tilde{B}$ for $\tilde{B} >0$ , which could be achieved and calculated by assuming $\mathcal{Z}$ is compact and use the continuity of $\nabla_{\theta}\ell (\cdot ,\theta_0)$ . Then, it will immediately give us an upper bound for $\| \theta_{\mathrm{PS}}\|$ that $\| \theta_{\mathrm{PS}}\| \leqslant \tilde{B} /\gamma +\| \theta_0\|$ . + +# 4 Prediction-Powered Inference under Performativity + +In this section, we further investigate prediction-powered inference (PPI) under performativity to enhance estimation and obtain improved confidence regions for the model parameter (i.e., policy) under performativity. This can also address the data scarcity issue in human responses when doing a survey to get feedback on policy implementation. + +Specifically, at time $t = 0$ , besides the limited set of gold-standard labeled data $\{x_{0,i},y_{0,i}\}_{i = 1}^{n_0}$ that are i.i.d. drawn from a distribution $\mathcal{D}(\theta_0)$ , we have abundant unlabeled data $\{x_{0,i}^u\}_{i = 1}^{N_0}$ that are i.i.d. drawn from the same marginal distribution as gold-standard labeled data, i.e. $\mathcal{D}_{\mathcal{X}}(\theta_0)$ , where $N_0\gg n_0$ . In addition, an annotating model $f:\mathcal{X}\mapsto \mathcal{Y}$ is used to label data $^2$ , which leads to $\{x_{0,i},f(x_{0,i})\}_{i = 1}^{n_0}$ and $\{x_{0,i}^u,f(x_{0,i}^u)\}_{i = 1}^{N_0}$ . Then, we use the following mechanism to output + +$$ +\widehat {\theta} _ {1} ^ {\mathrm {P P I}} (\lambda_ {1}) = \arg \min _ {\theta} \frac {\lambda_ {1}}{N} \sum_ {i = 1} ^ {N} \ell \left(x _ {0, i} ^ {u}, f \left(x _ {0, i} ^ {u}\right); \theta\right) + \frac {1}{n} \sum_ {i = 1} ^ {n} \left(\ell \left(x _ {0, i}, y _ {0, i}; \theta\right) - \lambda_ {1} \ell \left(x _ {0, i}, f \left(x _ {0, i}\right); \theta\right)\right) +$$ + +for a scalar $\lambda_{1}$ as an estimator of $\theta_{1}$ . After that, for $t\geqslant 1$ , at time $t$ , besides having access to the set of gold-standard labeled data $\{x_{t,i},y_{t,i}\}_{i = 1}^{n_t}$ that are i.i.d. drawn from a distribution $\mathcal{D}(\widehat{\theta}_t)$ . Meanwhile, we have abundant unlabeled data $\{x_{t,i}^u\}_{i = 1}^{N_t}$ that are i.i.d. drawn from the same marginal distribution as gold-standard labeled data, i.e. $\mathcal{D}_{\chi}(\widehat{\theta}_t)$ , where $N_{t}\gg n_{t}$ . Similar as before, we can estimate $\theta_{t + 1}$ via + +$$ +\widehat {\theta} _ {t + 1} ^ {\mathrm {P P I}} (\lambda_ {t + 1}) = \arg \min _ {\theta} \frac {\lambda_ {t + 1}}{N} \sum_ {i = 1} ^ {N} \ell \left(x _ {t, i} ^ {u}, f \left(x _ {t, i} ^ {u}\right); \theta\right) + \frac {1}{n} \sum_ {i = 1} ^ {n} \left(\ell \left(x _ {t, i}, y _ {t, i}; \theta\right) - \lambda_ {t + 1} \ell \left(x _ {t, i}, f \left(x _ {t, i}\right); \theta\right)\right) +$$ + +for a scalar $\lambda_{t + 1}$ . This incurs a trajectory in practice: $\theta_0\rightarrow \widehat{\theta}_1^{\mathrm{PPI}}(\lambda_1)\rightarrow \dots \widehat{\theta}_t^{\mathrm{PPI}}(\lambda_t)\rightarrow \dots$ . Notice that if we choose $\lambda_{t} = 0$ for all $t\in [T]$ , this will degenerate to the case in Section 3. Later on, we will demonstrate how to choose $\{\lambda_t\}_{t = 1}^T$ via data to enhance inference. Our mechanism could be adaptive to the data quality with carefully chosen $\{\lambda_t\}_{t = 1}^T$ and could be viewed as an extension of the classical PPI++ mechanism [25] to the setting under performativity. + +Building CLT for $\widehat{\theta}_t^{\mathrm{PPI}}(\lambda_t)$ . We start with building the central limit theorem for $\widehat{\theta}_t^{\mathrm{PPI}}(\lambda_t)$ with fixed constant scalars $\{\lambda_t\}_{t=1}^T$ for any $t \in [T]$ . The proof is similar to that of Theorem 3.4. Specifically, we denote + +$$ +\Sigma_ {\lambda , \tilde {\theta}} (\theta ; r) = H _ {\tilde {\theta}} (\theta) ^ {- 1} \left(r V _ {\lambda , \tilde {\theta}} ^ {f} (\theta) + V _ {\lambda , \tilde {\theta}} (\theta)\right) H _ {\tilde {\theta}} (\theta) ^ {- 1} +$$ + +with $V_{\lambda,\tilde{\theta}}^{f}(\theta) = \lambda^{2}\operatorname{Cov}_{x\sim \mathcal{D}_{\mathcal{X}}(\tilde{\theta})}(\nabla_{\theta}\ell(x,f(x);\theta))$ and $V_{\lambda,\tilde{\theta}}(\theta) = \operatorname{Cov}_{(x,y)\sim \mathcal{D}(\tilde{\theta})}(\nabla_{\theta}\ell(x,y;\theta) - \lambda\nabla_{\theta}\ell(x,f(x);\theta))$ . Then, we have the following theorem. + +Theorem 4.1 (Central Limit Theorem of $\widehat{\theta}_t^{\mathrm{PPI}}(\lambda_t)$ ). Under Assumption 3.1, A.4, and A.5, if $\varepsilon < \frac{\gamma}{\beta}$ and $\frac{n}{N} \to r$ for some $r \geqslant 0$ , then for any given $T \geqslant 0$ , we have that for all $t \in [T]$ , + +$$ +\sqrt {n} \left(\widehat {\theta} _ {t} ^ {P P I} \left(\lambda_ {t}\right) - \theta_ {t}\right) \stackrel {{D}} {{\rightarrow}} \mathcal {N} \left(0, V _ {t} ^ {P P I} \left(\left\{\lambda_ {j}, \theta_ {j} \right\} _ {j = 1} ^ {t}; r\right)\right) +$$ + +with + +$$ +V _ {t} ^ {P P I} \left(\left\{\lambda_ {j}, \theta_ {j} \right\} _ {j = 1} ^ {t}; r\right) = \sum_ {i = 1} ^ {t} \left[ \prod_ {k = i} ^ {t - 1} \nabla G \left(\theta_ {k}\right) \right] \Sigma_ {\lambda_ {i}, \theta_ {i - 1}} \left(\theta_ {i}; r\right) \left[ \prod_ {k = i} ^ {t - 1} \nabla G \left(\theta_ {k}\right) \right] ^ {\top}. +$$ + +Selection of parameters $\{\lambda_t\}_{t=1}^T$ . Now, the only thing left is to select $\{\lambda_t\}_{t=1}^T$ , so as to enhance estimation and inference. As choosing $\lambda_t = 0$ for all $t \in [T]$ will degenerate to Theorem 3.4, we expect that we appropriately choose $\{\lambda_t\}_{t=1}^T$ to make $F(V_t) \geqslant F(V_t^{\mathrm{PPI}}(\{\lambda_j, \theta_j\}_{j=1}^t; r))$ , where $F$ is a user-specified scalarization operator depending on different aims. For instance, if we are interested in optimizing the sum of asymptotic variance of coordinates of $\theta_t$ , then, $F(V_t) = \operatorname{Tr}(V_t)$ . Or if we are interested in the inference of the sum of all coordinates $\theta_t$ , i.e., $\mathbf{1}^\top \theta_t$ , then $F(V_t) = \mathbf{1}^\top V_t \mathbf{1}$ . + +We consider a greedy sequential selection mechanism as follows. Imagine that we have selected $\{\lambda_i^*\}_{i=1}^{t-1}$ via the data, and our aim is to select a $\lambda_t^*$ so as to make $F(V_t^{\mathrm{PPI}}(\{\lambda_j^*, \theta_j\}_{j=1}^{t-1}, \lambda, \theta_t; r)) \geqslant F(V_t^{\mathrm{PPI}}(\{\lambda_j^*, \theta_j\}_{j=1}^t; r))$ for any $\lambda$ . Thus, we choose + +$$ +\lambda_ {t} ^ {*} = \underset {\lambda} {\arg \min } F \left(V _ {t} ^ {\mathrm {P P I}} \left(\left\{\lambda_ {j} ^ {*}, \theta_ {j} \right\} _ {j = 1} ^ {t - 1}, \lambda , \theta_ {t}; r\right)\right). \tag {3} +$$ + +However, in practice, there are still several issues that need to be addressed. First, we need to choose $\{\lambda_t\}_{t=1}^T$ via observations, and this could be handled by using our results in Section 3 to estimate $\nabla_\theta G(\theta)$ , and we obtain a sample version $\widehat{\lambda}_t$ by plugging in the estimation. Second, when obtaining $\lambda_t^*$ in Eq. 3, we actually need $\theta_t$ in $\Sigma_{\lambda_i, \theta_{i-1}}(\theta_i; r)$ . But when using samples to estimate, we need to get $\widehat{\lambda}_t$ first before we obtain $\widehat{\theta}_t^{\mathrm{PPI}}$ . Thus, we propose a similar optimizing strategy as inspired by [1]: at time $t \geqslant 1$ , given the obtained $\{\widehat{\lambda}_i\}_{i=1}^{t-1}$ and $\{\widehat{\theta}_i^{\mathrm{PPI}}\}_{i=1}^{t-1}$ , we choose an arbitrary $\widetilde{\lambda}$ , to obtain $\widehat{\theta}_t^{\mathrm{PPI}}(\widetilde{\lambda})$ as a temporary surrogate. Then, we further obtain + +$$ +\widehat {\lambda} _ {t} = \underset {\lambda} {\arg \min } F \Big (\widehat {V} _ {t} ^ {\mathrm {P P I}} \big (\{\widehat {\lambda} _ {j}, \widehat {\theta} _ {j} ^ {\mathrm {P P I}} (\widehat {\lambda} _ {j}) \} _ {j = 1} ^ {t - 1}, \lambda , \widehat {\theta} _ {t} ^ {\mathrm {P P I}} (\tilde {\lambda}); \frac {n}{N} \big) \Big), +$$ + +where $\widehat{V}_t^{\mathrm{PPI}}$ is obtained via replacing $\nabla_{\theta}G(\theta_k)$ with their estimation in $V_{t}^{\mathrm{PPI}}$ . In particular, if $F$ satisfies $F(U + V) = F(U) + F(V)$ as our examples of $F(V_{t}) = \operatorname {Tr}(V_{t})$ and $F(V_{t}) = \mathbf{1}^{\top}V_{t}\mathbf{1}$ , then we only need to optimize $F\left(\Sigma_{\lambda ,\widehat{\theta}_{t - 1}^{\mathrm{PPI}}(\widehat{\lambda}_{t - 1})}(\widehat{\theta}_t(\widetilde{\lambda});\frac{n}{N})\right)$ + +To sum up, by the above process, we have the following corollary. + +Corollary 4.2. Under Assumption 3.1, A.2, A.3, A.4, and A.5, if $\varepsilon < \frac{\gamma}{\beta}$ and $\frac{n}{N} \to r$ for some $r \geqslant 0$ , then for any given $T \geqslant 0$ , we have that for all $t \in [T]$ , + +$$ +\left(\widehat {V} _ {t} ^ {P P I} \Big (\{\widehat {\lambda} _ {j}, \widehat {\theta} _ {j} ^ {P P I} (\widehat {\lambda} _ {j}) \} _ {j = 1} ^ {t}; \frac {n}{N} \Big)\right) ^ {- \frac {1}{2}} \sqrt {n} \big (\widehat {\theta} _ {t} ^ {P P I} (\widehat {\lambda} _ {t}) - \theta_ {t} \big) \stackrel {{D}} {\to} \mathcal {N} (0, I _ {d}). +$$ + +![](images/b7bc5fde0b62bba0bf49f086aadb386f0af1f1192d2264d5fec924b528748411.jpg) + +![](images/0b003953f722810231bd4ebab65347ea95afb32fc87fbff0d8fb609c36b01733.jpg) +Figure 1: Confidence-region coverage (top row) and width (bottom row) with different choices of $\lambda$ . The left, middle, and right columns correspond to inference steps $t = 2$ , $t = 3$ , and $t = 4$ , respectively. The solid and dashed curves correspond to the confidence-region coverage for $\theta_t$ and $\theta_{\mathrm{PS}}$ , respectively. + +![](images/fe53d6f50c1fda6fc6d0ea59e427015e355c37b81f450dab1bc8fd951ffb536b.jpg) + +![](images/f107764c2db046883f1d2dafcd3e58f1f232859f3e93c60946744d2f16be9452.jpg) + +![](images/49cf621669b857b18d30600fbad32787cc60584d06afc5756863cb71995140d1.jpg) + +![](images/37aae7b5dbbf15fe534a8701d51e8d2ddb2c5ecd5124909d31c780686661d575.jpg) + +# 5 Experiment + +In this section, we further provide numerical experimental results to support our previous theory. A case study on a semi-synthetic dataset is provided in Section B.3. + +Experimental setting. We follow [20] to construct simulation studies on a performative linear regression problem. Given a parameter $\theta \in \mathbb{R}^d$ , data are sampled from $D(\theta)$ as + +$$ +y = \alpha^ {\top} x + \mu^ {\top} \theta + \nu , x \sim \mathcal {N} (\mu_ {x}, \Sigma_ {x}), \nu \sim \mathcal {N} (0, \sigma_ {y} ^ {2}). +$$ + +The distribution map $D(\theta)$ is $\varepsilon$ -sensitive with $\varepsilon = \| \mu \|_2$ . For unlabeled $x_i^u$ , the annotating model is defined as $f(x_i^u) = \alpha^\top x_i^u + \mu^\top \theta + \nu_i, \nu_i \sim \mathcal{N}(-0.2, \sigma_y^2)$ . We use the ridge squared loss to measure the performance and update $\theta$ : $\ell((x, y); \theta) = \frac{1}{2} (y - \theta^\top x)^2 + \frac{\gamma}{2} \| \theta \|^2$ . For easier calculation for smoothness parameter, we truncate the distribution of $(x, y)$ in our experiment to deal with truncated normal distributions, but this is not necessary in many other choices of updating rules. In the following experiments, we set $d = 2$ , $\varepsilon \approx 0.02$ , $\gamma = 2$ , and $\sigma_y^2 = 0.2$ . We set $N = 2000$ and vary the labeled sample size $n$ . + +Simulation results for PPI under performativity. To quantify the results of PPI under performativity, we evaluate the confidence-region coverage and width for three strategies: $\lambda = 0$ (only labeled data), $\lambda = 1$ (full unlabeled data weight), and our optimization method $\lambda = \widehat{\lambda}_t$ as defined in Eq.3. We vary the labeled sample size $n$ and perform $t \in \{2,3,4\}$ repeated risk minimization steps, averaging results over 1000 independent trials. In Figure 1, we can find that all three methods approach 0.9 coverage as $n$ grows, while our optimized $\widehat{\lambda}_t$ (orange) achieves the narrowest interval width, supporting its effectiveness to enhance the performative inference. The dashed curves denote the bias-adjusted confidence regions for the performative stable point $\theta_{\mathrm{PS}}$ . It can be observed that $\theta_{\mathrm{PS}}$ coverages upper-bound that of $\theta_t$ (solid curves) across steps $t$ , and the gap between them vanishes as $t$ grows. This observation verifies the validity of Corollary 3.7. + +Verifying central limit theorem. To validate the central limit theorem, we sample different $\widehat{\theta}_t$ (here $t = 4$ and $n = 1000$ ) and visualize the distribution of $\widehat{V}_t^{-1/2}\sqrt{n}(\widehat{\theta}_t - \theta_t)$ . We plot the density map + +![](images/4b7bec2736cbb14102987fceaaca63c5085aae2e12fc7903053cbc506b46ea32.jpg) +(a) Density map of observed distribution. + +![](images/cfe3e241abf036918f263a31cb08c79294e67ee94a41ba928b2b77541177ded4.jpg) +(b) Multivariate Q-Q Plot for normality test. +Figure 2: Visualizations to verify the Central Limit Theorem. (a) plots the density map of sampled $\widehat{V}_t^{-1/2}\sqrt{n}(\widehat{\theta}_t - \theta_t)$ , while (b) compares the observed distribution with theoretical one $\mathcal{N}(0, I_d)$ . + +in Figure 2a and find it is close to a normal distribution. In Figure 2b, we further do a normality test with the multivariate Q-Q plot of observed squared Mahalanobis distance over theoretical Chisq quantiles. The tight alignment of points along the identity line (red dashed) verifies that the observed distribution is well-approximated by its asymptotic CLT. + +Estimation results of score matching. We consider two implementations of the gradient-free score matching estimator $M(z,\theta ;\psi)$ : + +(a). Gaussian parametric: we assume $p(z,\theta) = \mathcal{N}(\mu_p,\Sigma_p)$ and parameterize $\psi = \{\mu_p,\Sigma_p\}$ +(b). DNN-based: a small deep neural network with two hidden layers of width 128. + +We collect $\widehat{\theta}_{1:t}$ trajectory and corresponding data $\{z_{1:t,i}\}_{i=1}^{n}$ to train both models via the SGD optimizer with a learning rate of 0.1 to minimize the empirical score-matching objective $J(\psi)$ . + +In Figure 3, we evaluate the estimation quality of two models by their final training loss $J(\psi)$ and the estimated variance error $\| \widehat{V}_t - V_t\|$ over varying $n$ and $t$ . We sample 1000 independent trajectories of $\widehat{\theta}_{1:t}$ and report $J(\psi)$ as averaged $J(\widehat{\theta};\psi)$ over all $\widehat{\theta}$ in the collection of trajectories. In all settings, both estimators achieve $J(\psi) < 0.05$ , indicating the perfect approximation of our learned model $M(z,\theta ;\psi)$ to the true $p(z,\theta)$ . Correspondingly, the variance-estimation error remains negligible and decreases as $n$ grows, verifying the feasibility of using our score matching models to fit $\nabla_{\theta}\log p(z,\theta)$ for estimating $\nabla G(\theta_k)$ and $V_{t}$ . + +![](images/8fb3857fa2ed7ed33aef4fd44f434645fcee3a968884a165f8a398ab3c092011.jpg) +(a) Gaussian parametric model. +Figure 3: Evaluating the estimation quality of two designed score matching models. + +![](images/dda0e477d442dd82a0db74bfcea1de83113a975667b824e120365cbd458ea76f.jpg) + +![](images/87b0f586287b48c21f3ec07419e7e330eee98d4e1d977fac6a3c9bb6f6f7a144.jpg) +(b) DNN-based model. + +![](images/0231db515388daaa0e664dbf2aa33be211b1f6a7ef0f94da809927d0bd5b7ad5.jpg) + +# 6 Conclusion + +In this paper, we introduce an important topic: statistical inference under performativity. We derive results on asymptotic distributions for a widely used iterative process for updating parameters in performative prediction. 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URL https://api-semanticscholar.org/CorpusID:263134612. + +# A Theoretical Details + +We will include omitted technical details in the main context. We first summarize all the required additional assumptions in Section A.1. Then, we provide omitted proofs for Section 3 in Section A.2 and omitted proofs for Section 4 in Section A.4. + +# A.1 Assumptions + +In this subsection, we summarize all the additional assumptions we will use to build various theoretical results in this paper. Before we state the assumptions, we need some further notations. + +Let us denote $\Psi (\theta)$ as the collection of minimizers of $\int p(z,\theta)\| \nabla_{\theta}\log p(z,\theta) - s(z,\theta ;\psi)\| ^2 dz$ for the given $\theta$ . We further denote the empirical estimation function for the two terms, i.e., + +$$ +\int p (z, \theta) \Big (\| s (z, \theta ; \psi) \| ^ {2} - 2 \nabla_ {\theta} \log p (z, \theta) ^ {\top} s (z, \theta ; \psi) \Big) d z, +$$ + +as $\widehat{J}_{n,k}(\psi ;\theta)$ following the methods in Section 3.2 for any $\theta$ in the trajectory $\{\widehat{\theta}_t\}_{t = 1}^T$ , where $n$ and $k$ are the number of samples we get at each iteration for $\widehat{\theta}_t$ and perturbed policies. + +Meaning of each assumption. Assumption A.1 is used to establish the asymptotic normality of our estimator under performativity. Additionally, we use the fact that the population loss Hessians $H_{\tilde{\theta}}(\theta) = \nabla_{\theta}^{2}\mathbb{E}_{z\sim \mathcal{D}(\tilde{\theta})}\ell (z;\theta)$ are positive definite, which is guaranteed by the strong convexity of the loss function (Assumption 3.1). Assumption A.2 and A.3 are used in the analysis of score matching. Assumption A.2, based on the differentiation lemma [18], ensures the interchangeability of integration and differentiation. Assumption A.3 guarantees the consistency of the score matching estimator. Assumption A.4 and A.5 parallel to Assumption 3.1(a) and A.1, and are used to establish the consistency and asymptotic normality of our PPI estimator under performativity. Conditions such as local Lipschitz continuity and positive definiteness are standard for establishing asymptotic normality. Similar assumptions are also imposed in [1]. + +Assumption A.1 (Positive Definiteness & Regularity Conditions for the Estimator). We assume the following. + +(a). The loss function satisfies the gradient covariance matrix is uniformly bounded below: + +$$ +V _ {\tilde {\theta}} (\theta) = \operatorname {C o v} _ {z \sim \mathcal {D} (\tilde {\theta})} \left(\nabla_ {\theta} \ell (z; \theta)\right) \succeq c I, +$$ + +for any $\tilde{\theta}$ , where $c > 0$ is a constant. + +(b). For any sample size $n$ , assume the M-estimator $\widehat{\theta}_t$ has a density function with respect to the Lebesgue measure, and its characteristic function is absolutely integrable. + +Assumption A.2 (Regularity Condition for $M$ ). Assume that for $\forall i$ : + +(a). The function $z \mapsto p(z, \theta) \frac{\partial \log M(z, \theta; \psi)}{\partial \theta^{(i)}}$ is Lebesgue integrable. +(b). For almost every $z \in \mathcal{Z}$ (with respect to Lebesgue measure), the partial derivative + +$$ +\frac {\partial}{\partial \theta^ {(i)}} \left[ p (z, \theta) \frac {\partial \log M (z , \theta ; \psi)}{\partial \theta^ {(i)}} \right] +$$ + +exists. + +(c). There exists a Lebesgue-integrable function $H(z)$ such that for almost every $z \in \mathcal{Z}$ , + +$$ +\left| \frac {\partial}{\partial \theta^ {(i)}} \left[ p (z, \theta) \frac {\partial \log M (z , \theta ; \psi)}{\partial \theta^ {(i)}} \right] \right| \leqslant H (z). +$$ + +Assumption A.3 (Consistency of Optimizer). We let $k$ grows along with $n$ such that $n \to \infty$ leads to $k \to \infty$ . We assume that the class $M(z, \theta; \psi)$ is rich enough that for all $\theta \in \Theta$ , there exists $\psi^{*}(\theta)$ such that $M(z, \theta; \psi^{*}(\theta)) = p(z, \theta)$ . Moreover, for the underlying trajectory $\{\theta_t\}_{t=1}^T$ , + +$$ +\lim_{n\to \infty}\operatorname *{arg min}_{\psi}\widehat{\mathcal{J}}_{n,k}(\psi ;\widehat{\theta}_{t})\subseteq \Psi (\theta_{t}). +$$ + +Assumption A.4 (Local Lipschitzness with $f$ ). Loss function $\ell(x, f(x); \theta)$ is locally Lipschitz: for each $\theta \in \Theta$ , there exist a neighborhood $\Upsilon(\theta)$ of $\theta$ such that $\ell(x, f(x); \tilde{\theta})$ is $L^f(x)$ Lipschitz w.r.t $\tilde{\theta}$ for all $\tilde{\theta} \in \Upsilon(\theta)$ and $\mathbb{E}_{x \sim \mathcal{D}_{\mathcal{X}}(\theta)} L^f(x) < \infty$ . + +Assumption A.5 (Positive Definiteness with $f$ & Regularity Conditions for the PPI Estimator). We assume the following. + +(a). Assume the loss function satisfies the gradient covariance matrices are uniformly bounded below: + +$$ +V _ {\tilde {\theta}} (\theta) = \mathrm {C o v} _ {z \sim \mathcal {D} (\tilde {\theta})} \big (\nabla_ {\theta} \ell (z; \theta) \big) \succeq c I, V _ {\tilde {\theta}} ^ {f} (\theta) = \mathrm {C o v} _ {x \sim \mathcal {D} _ {\mathcal {X}} (\tilde {\theta})} \big (\nabla_ {\theta} \ell (x, f (x); \theta) \big) \succeq c I, +$$ + +for any $\tilde{\theta},\theta$ , where $c > 0$ is a constant. + +(b). For any sample size $n$ , assume $\widehat{\theta}_t^{\mathrm{PPI}}$ has a density function with respect to the Lebesgue measure, and its characteristic function is absolutely integrable. + +# A.2 Details of Section 3: Theory of Inference under Performativity + +We provide the omitted details in Section 3. + +# A.2.1 Consistency and Central Limit Theorem of $\widehat{\theta}_t$ + +Let us denote: + +$$ +\mathcal {L} _ {\tilde {\theta}} (\theta) := \mathbb {E} _ {z \sim \mathcal {D} (\tilde {\theta})} \ell (z; \theta), \quad \mathcal {L} _ {\tilde {\theta}, n} (\theta) := \frac {1}{n} \sum_ {i = 1} ^ {n} \ell \left(z _ {i}; \theta\right), +$$ + +where the samples $z_{i} = (x_{i},y_{i})\sim \mathcal{D}(\tilde{\theta})$ are drawn from the distribution under $\tilde{\theta}$ + +Proposition A.6 (Consistency of $\widehat{\theta}_t$ , Restatement of Proposition 3.3). Under Assumption 3.1, if $\varepsilon < \frac{\gamma}{\beta}$ , then for any given $T \geqslant 0$ , we have that for all $t \in [T]$ + +$$ +\widehat {\theta} _ {t} \stackrel {{P}} {{\longrightarrow}} \theta_ {t}. +$$ + +Proof. Let us denote $\widehat{G}(\theta) \coloneqq \operatorname*{argmin}_{\theta' \in \Theta} \frac{1}{n} \sum_{i=1}^{n} \ell(z_i; \theta')$ where the samples $z_i \sim \mathcal{D}(\theta)$ are drawn for some parameter $\theta$ along the dynamic trajectory $\theta_0 \to \widehat{\theta}_1 \to \dots \widehat{\theta}_t \to \dots$ . + +$$ +\begin{array}{l} \| \theta_ {t} - \widehat {\theta} _ {t} \| = \| G (\theta_ {t - 1}) - \widehat {G} (\widehat {\theta} _ {t - 1}) \| \\ \leqslant \left\| G \left(\widehat {\theta} _ {t - 1}\right) - \widehat {G} \left(\widehat {\theta} _ {t - 1}\right) \right\| + \left\| G \left(\theta_ {t - 1}\right) - G \left(\widehat {\theta} _ {t - 1}\right) \right\| \\ \leqslant \| G (\widehat {\theta} _ {t - 1}) - \widehat {G} (\widehat {\theta} _ {t - 1}) \| + \varepsilon \frac {\beta}{\gamma} \| \theta_ {t - 1} - \widehat {\theta} _ {t - 1} \|, \\ \end{array} +$$ + +where the last inequality follows from the results derived by [25], under Assumption 3.1, we have $\| G(\theta) - G(\theta')\| \leqslant \frac{\varepsilon\beta}{\gamma}\| \theta -\theta '\|$ . + +Notice that $\mathbb{E}(\mathcal{L}_{\widehat{\theta}_{t - 1},n}(\theta)) = \mathcal{L}_{\widehat{\theta}_{t - 1}}(\theta)$ . By local Lipschitz condition, there exists $\varepsilon_0 > 0$ such that + +$$ +\sup _ {\theta : \| \theta - G (\widehat {\theta} _ {t - 1}) \| \leqslant \varepsilon_ {0}} | \mathcal {L} _ {\widehat {\theta} _ {t - 1}, n} (\theta) - \mathcal {L} _ {\widehat {\theta} _ {t - 1}} (\theta) | \xrightarrow {P} 0. +$$ + +Since $\ell$ is strongly convex for any $\theta$ , $G(\widehat{\theta}_{t-1})$ is unique. Then we know that there exists $\delta$ such that $\mathcal{L}_{\widehat{\theta}_{t-1},n}(\theta) - \mathcal{L}_{\widehat{\theta}_{t-1}}(G(\widehat{\theta}_{t-1})) > \delta$ for all $\theta$ in $\{\theta \mid \| \theta - G(\widehat{\theta}_{t-1}) \| = \varepsilon_0\}$ . Then it follows that: + +$$ +\begin{array}{l} \inf _ {\| \theta - G (\widehat {\theta} _ {t - 1}) \| = \varepsilon_ {0}} \mathcal {L} _ {\widehat {\theta} _ {t - 1}, n} (\theta) - \mathcal {L} _ {\widehat {\theta} _ {t - 1}, n} (G (\widehat {\theta} _ {t - 1})) \\ = \inf _ {\| \theta - G (\widehat {\theta} _ {t - 1}) \| = \varepsilon_ {0}} \left(\left(\mathcal {L} _ {\widehat {\theta} _ {t - 1}, n} (\theta) - \mathcal {L} _ {\widehat {\theta} _ {t - 1}} (\theta)\right) + \left(\mathcal {L} _ {\widehat {\theta} _ {t - 1}} (\theta) - \mathcal {L} _ {\widehat {\theta} _ {t - 1}} (G (\widehat {\theta} _ {t - 1}))\right) \right. \\ \left. + \left(\mathcal {L} _ {\widehat {\theta} _ {t - 1}} \left(G \left(\widehat {\theta} _ {t - 1}\right)\right) - \mathcal {L} _ {\widehat {\theta} _ {t - 1}, n} \left(G \left(\widehat {\theta} _ {t - 1}\right)\right)\right)\right) \\ \end{array} +$$ + +$$ +\geqslant \delta - o _ {P} (1). +$$ + +Then we consider any fixed $\theta$ such that $\| \theta - G(\widehat{\theta}_{t-1}) \| \geqslant \varepsilon_0$ it follows that + +$$ +\begin{array}{l} \mathcal {L} _ {\widehat {\theta} _ {t - 1}, n} (\theta) - \mathcal {L} _ {\widehat {\theta} _ {t - 1}, n} (G (\widehat {\theta} _ {t - 1})) \geqslant \frac {\theta - G (\widehat {\theta} _ {t - 1})}{\omega - G (\widehat {\theta} _ {t - 1})} \left(\mathcal {L} _ {\widehat {\theta} _ {t - 1}, n} (\omega) - \mathcal {L} _ {\widehat {\theta} _ {t - 1}, n} (G (\widehat {\theta} _ {t - 1}))\right) \\ \geqslant \frac {\| \theta - G (\widehat {\theta} _ {t - 1}) \|}{\varepsilon_ {0}} (\delta - o _ {P} (1)) \geqslant \delta - o _ {P} (1), \\ \end{array} +$$ + +where the first inequality holds for any $\omega$ by the convexity condition of $\mathcal{L}_{\widehat{\theta}_{t - 1},n}(\theta)$ , and the second inequality holds as we take $\omega = \frac{\theta - G(\widehat{\theta}_{t - 1})}{\|\theta - G(\widehat{\theta}_{t - 1})\|}\varepsilon_0 + G(\widehat{\theta}_{t - 1})$ and using the above result. Thus no $\theta$ such that $\| \theta -G(\widehat{\theta}_{t - 1})\| = \varepsilon_0$ can be the minimizer of $\mathcal{L}_{\widehat{\theta}_{t - 1},n}(\theta)$ . Then $\| G(\widehat{\theta}_{t - 1}) - \widehat{G} (\widehat{\theta}_{t - 1})\| \stackrel {P}{\to}0$ . + +We then have, for a given $T \geqslant 0$ , we have that for all $t \in [T]$ + +$$ +\left\| \widehat {\theta} _ {t} - \theta_ {t} \right\| \leqslant \sum_ {i = 0} ^ {t} \left(\varepsilon \frac {\beta}{\gamma}\right) ^ {t - i} \left\| G \left(\widehat {\theta} _ {i}\right) - \widehat {G} \left(\widehat {\theta} _ {i}\right) \right\| \xrightarrow {P} 0. +$$ + +Thus, we conclude that $\widehat{\theta}_t \xrightarrow{P} \theta_t$ . + +Theorem A.7 (Central Limit Theorem of $\widehat{\theta}_t$ , Restatement of Theorem 3.4). Under Assumption 3.1 and A.1, if $\varepsilon < \frac{\gamma}{\beta}$ , then for any given $T \geqslant 0$ , we have that for all $t \in [T]$ + +$$ +\sqrt {n} \big (\widehat {\theta_ {t}} - \theta_ {t} \big) \stackrel {D} {\longrightarrow} \mathcal {N} \left(0, V _ {t}\right) +$$ + +with + +$$ +V _ {t} = \sum_ {i = 1} ^ {t} \left[ \prod_ {k = i} ^ {t - 1} \nabla G \left(\theta_ {k}\right) \right] \Sigma_ {\theta_ {i - 1}} \left(\theta_ {i}\right) \left[ \prod_ {k = i} ^ {t - 1} \nabla G \left(\theta_ {k}\right) \right] ^ {\top}. +$$ + +In particular, $\nabla G(\theta_k) = -H_{\theta_k}(\theta_{k + 1})^{-1}\big(\nabla_{\tilde{\theta}}\mathbb{E}_{z\sim \mathcal{D}(\theta_k)}\nabla_\theta \ell (z;\theta_{k + 1})\big)$ , where $\nabla_{\tilde{\theta}}$ is taking gradient for the parameter in $\mathcal{D}(\tilde{\theta})$ , $\nabla_{\theta}$ is taking gradient for the parameter in $\ell (z;\theta)$ and $\prod_{k = t}^{t - 1}\nabla G(\theta_k) = I_d$ . + +Proof. Let $U_{t} \coloneqq \sqrt{n} (\widehat{\theta}_{t} - \theta_{t})$ and denote $\tilde{\theta}_t = G(\widehat{\theta}_{t - 1})$ . We make the following decomposition: + +$$ +\widehat {\theta} _ {t} - \theta_ {t} = \underbrace {(\widetilde {\theta} _ {t} - \theta_ {t})} _ {(1)} + \underbrace {(\widehat {\theta} _ {t} - \widetilde {\theta} _ {t})} _ {(2)}. +$$ + +Step 1: Conditional distribution of $U_{t}|U_{t - 1}$ . + +For term (1), we have + +$$ +\sqrt {n} \left(\tilde {\theta} _ {t} - \theta_ {t}\right) = \sqrt {n} \left(G \left(\widehat {\theta} _ {t - 1}\right) - G \left(\theta_ {t - 1}\right)\right). +$$ + +For term (2), the empirical process analysis in [1] establishes that + +$$ +\sqrt {n} \left(\widehat {\theta} _ {t} - \widetilde {\theta} _ {t}\right) \mid \widehat {\theta} _ {t - 1} \xrightarrow {D} \mathcal {N} \left(0, \Sigma_ {\widehat {\theta} _ {t - 1}} \left(\widetilde {\theta} _ {t}\right)\right), +$$ + +where the variance is given by + +$$ +\Sigma_ {\widehat {\theta} _ {t - 1}} (\widetilde {\theta} _ {t}) = H _ {\widehat {\theta} _ {t - 1}} (\widetilde {\theta} _ {t}) ^ {- 1} V _ {\widehat {\theta} _ {t - 1}} (\widetilde {\theta} _ {t}) H _ {\widehat {\theta} _ {t - 1}} (\widetilde {\theta} _ {t}) ^ {- 1}. +$$ + +Conditioning on $\widehat{\theta}_{t - 1}$ and considering the distribution $D(\widehat{\theta}_{t - 1})$ , for any function $h$ , we use the following shorthand notations: + +$$ +\mathbb {E} _ {n} h := \frac {1}{n} \sum_ {i = 1} ^ {n} h (x _ {i}, y _ {i}), \quad \mathbb {G} _ {n} h := \sqrt {n} (\mathbb {E} _ {n} h - \mathbb {E} _ {(x, y) \sim \mathcal {D} (\widehat {\theta} _ {t - 1})} [ h (x, y) ]). +$$ + +Note that $\tilde{\theta}_t = G(\widehat{\theta}_{t - 1})$ . Recall that + +$$ +\mathcal {L} _ {\tilde {\theta}} (\theta) := \underset {(x, y) \sim \mathcal {D} (\tilde {\theta})} {\mathbb {E}} \ell (x, y; \theta), \quad \mathcal {L} _ {\tilde {\theta}, n} := \frac {1}{n} \sum_ {i = 1} ^ {n} \ell (x _ {i}, y _ {i}; \theta), \text {w h e r e} (x _ {i}, y _ {i}) \sim \mathcal {D} (\tilde {\theta}). +$$ + +Under the assumptions, Lemma 19.31 in [31] implies that for every sequence $h_n = O_P(1)$ , we have + +$$ +\mathbb {G} _ {n} \left[ \sqrt {n} \left(\ell (x, y; \tilde {\theta} _ {t} + \frac {h _ {n}}{\sqrt {n}}) - \ell (x, y; \tilde {\theta} _ {t})\right) - h _ {n} ^ {\top} \nabla_ {\theta} \ell (x, y; \tilde {\theta} _ {t}) \right] \stackrel {P} {\longrightarrow} 0. +$$ + +Applying second-order Taylor expansion, we obtain that + +$$ +\begin{array}{l} n \mathbb {E} _ {n} \left(\ell (x, y; \tilde {\theta} _ {t} + \frac {h _ {n}}{\sqrt {n}}) - \ell (x, y; \tilde {\theta} _ {t})\right) = n \left(\mathcal {L} _ {\widehat {\theta} _ {t - 1}} (\tilde {\theta} _ {t} + \frac {h _ {n}}{\sqrt {n}}) - \mathcal {L} _ {\widehat {\theta} _ {t - 1}} (\tilde {\theta} _ {t})\right) \\ + h _ {n} ^ {\top} \mathbb {G} _ {n} \nabla_ {\theta} \ell (x, y; \tilde {\theta} _ {t}) + o _ {p} (1) \\ = \frac {1}{2} h _ {n} ^ {\top} H _ {\hat {\theta} _ {t - 1}} (\tilde {\theta} _ {t}) h _ {n} + h _ {n} ^ {\top} \mathbb {G} _ {n} \nabla_ {\theta} \ell (x, y; \tilde {\theta} _ {t}) + o _ {p} (1). \\ \end{array} +$$ + +Set $h_n^* = \sqrt{n} (\widehat{\theta}_t - \widetilde{\theta}_t)$ and $h_n = -H_{\widehat{\theta}_{t-1}}(\widetilde{\theta}_t)^{-1}\mathbb{G}_n\nabla_\theta \ell(x,y;\widetilde{\theta}_t)$ , Corollary 5.53 in [31] implies they are $O_P(1)$ . + +Since $\widehat{\theta}_t$ is the minimizer of $\mathcal{L}_{n,\widehat{\theta}_{t - 1}}$ , the first term is smaller than the second term. We can rearrange the terms and obtain: + +$$ +\frac {1}{2} (h _ {n} ^ {*} - h _ {n}) ^ {T} H _ {\widehat {\theta} _ {t - 1}} (\tilde {\theta} _ {t}) (h _ {n} ^ {*} - h _ {n}) = o _ {P} (1), +$$ + +which leads to $h_n^* - h_n = O_P(1)$ . Then the above asymptotic normality result follows directly by applying the central limit theorem (CLT) to the following terms, conditioning on $\widehat{\theta}_{t-1}$ : + +$$ +\begin{array}{l} \sqrt {n} \left(\widehat {\theta} _ {t} - \tilde {\theta} _ {t}\right) \mid \widehat {\theta} _ {t - 1} = - H _ {\widehat {\theta} _ {t - 1}} \left(\tilde {\theta} _ {t}\right) ^ {- 1} S + o _ {P} (1), \\ S = \sqrt {\frac {1}{n}} \sum_ {i = 1} ^ {n} \left(\nabla_ {\theta} \ell (x _ {t, i}, y _ {t, i}; \tilde {\theta} _ {t}) - \underset {(x, y) \sim \mathcal {D} (\widehat {\theta} _ {t - 1})} {\mathbb {E}} \left[ \nabla_ {\theta} \ell (x, y; \tilde {\theta} _ {t}) \right]\right). \\ \end{array} +$$ + +Note that, conditioning on $\widehat{\theta}_{t - 1}$ , (1) is a constant. Therefore, (1) and (2) follow a joint Gaussian distribution. Consequently, given $U_{t - 1}$ , the conditional distribution of $U_{t}$ is given by: + +$$ +\begin{array}{l} U _ {t} \mid U _ {t - 1} = \sqrt {n} \left(\widehat {\theta} _ {t} - \theta_ {t}\right) \mid \widehat {\theta} _ {t - 1} \\ = \sqrt {n} \left(\tilde {\theta} _ {t} - \theta_ {t}\right) + \sqrt {n} \left(\widehat {\theta} _ {t} - \tilde {\theta} _ {t}\right) \mid \widehat {\theta} _ {t - 1} \\ = \sqrt {n} \left(G \left(\widehat {\theta} _ {t - 1}\right) - G \left(\theta_ {t - 1}\right)\right) + \sqrt {n} \left(\widehat {\theta} _ {t} - \tilde {\theta} _ {t}\right) \mid \widehat {\theta} _ {t - 1} \\ \xrightarrow {D} \mathcal {N} \left(\sqrt {n} (G (\widehat {\theta} _ {t - 1}) - G (\theta_ {t - 1})), \Sigma_ {\widehat {\theta} _ {t - 1}} (\tilde {\theta} _ {t})\right). \\ = \mathcal {N} \left(\sqrt {n} (G (\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}) - G (\theta_ {t - 1})), \Sigma_ {\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}} (G (\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}))\right). \\ \end{array} +$$ + +For later references, we denote $U_{t}\mid U_{t - 1}\xrightarrow{D}\mathcal{N}(\mu (U_{t - 1}),\Sigma (U_{t - 1}))$ + +Step 2: Marginal distribution of $U_{t}$ . We calculate the characteristic function of $U_{t}$ by induction. To begin with, we directly have + +$$ +X _ {1} \stackrel {{D}} {{\longrightarrow}} \mathcal {N} (0, V _ {1}), \quad V _ {1} = \Sigma_ {\theta_ {0}} (\theta_ {1}). +$$ + +Now, assume that $U_{t - 1} \xrightarrow{D} \mathcal{N}(0, V_{t - 1})$ , we derive the joint distribution of $(U_t, U_{t - 1})$ and marginal distribution of $U_t$ . Then we have, the characteristics functions $\phi$ and the probability density function $p$ of distributions $U_{t - 1}$ and $U_t \mid U_{t - 1}$ follow: + +$$ +\phi_ {U _ {t - 1}} (s) \rightarrow \phi_ {\mathcal {N} (0, V _ {t - 1})} (s) = \exp (- \frac {1}{2} s ^ {T} V _ {t - 1} s), p _ {U _ {t - 1}} (u) = \frac {1}{(2 \pi) ^ {d}} \int e ^ {- i z ^ {T} u} \phi_ {U _ {t - 1}} (z) d z, +$$ + +$$ +\phi_ {U _ {t} | U _ {t - 1}} (s) \rightarrow \phi_ {\mathcal {N} (\mu (U _ {t - 1}), \Sigma (U _ {t - 1}))} (s) = \exp (i s ^ {T} \mu (U _ {t - 1}) - \frac {1}{2} s ^ {T} \Sigma (U _ {t - 1}) s). +$$ + +# Then we have + +$$ +\begin{array}{l} \phi_ {U _ {t}} (s) = \mathbb {E} e ^ {i s ^ {T} U _ {t}} = \mathbb {E} \left(\mathbb {E} \left(e ^ {i s ^ {\top} U _ {t}} \mid U _ {t - 1}\right)\right) = E _ {U _ {t - 1}} \phi_ {U _ {t} | U _ {t - 1}} (s \mid U _ {t - 1}) \\ = \int \phi_ {U _ {t} | U _ {t - 1}} (s \mid u) p _ {U _ {t - 1}} (u) d u \\ = \int \phi_ {U _ {t} | U _ {t - 1}} (s \mid u) \frac {1}{(2 \pi) ^ {d}} \int e ^ {- i z ^ {\top} u} \phi_ {U _ {t - 1}} (z) d z d u \\ = \frac {1}{(2 \pi) ^ {d}} \iint \phi_ {U _ {t} | U _ {t - 1}} (s \mid u) \phi_ {U _ {t - 1}} (z) e ^ {- i z ^ {\top} u} d z d u \\ = \frac {1}{(2 \pi) ^ {d}} \iint \exp \left(i s ^ {\top} \mu \left(U _ {t - 1}\right) - \frac {1}{2} s ^ {\top} \Sigma \left(U _ {t - 1}\right) s\right) \exp \left(- \frac {1}{2} z ^ {\top} V _ {t - 1} z\right) e ^ {- i z ^ {\top} u} d z d u \\ = \frac {1}{(2 \pi) ^ {d}} \int \exp \left(i s ^ {\top} \mu \left(U _ {t - 1}\right) - \frac {1}{2} s ^ {\top} \Sigma \left(U _ {t - 1}\right) s\right) \left(\int \exp \left(- \frac {1}{2} z ^ {\top} V _ {t - 1} z - i z ^ {\top} u\right) d z\right) d u \\ = \frac {1}{(2 \pi) ^ {d}} \int \exp \left(i s ^ {\top} \mu \left(U _ {t - 1}\right) - \frac {1}{2} s ^ {\top} \Sigma \left(U _ {t - 1}\right) s\right) \\ \times \left(\int \exp \left(- \frac {1}{2} u ^ {\top} V _ {t - 1} ^ {- 1} u\right) \exp \left(- \frac {1}{2} (z - V _ {t - 1} ^ {- 1} i u) ^ {\top} V _ {t - 1} (z - V _ {t - 1} ^ {- 1} i u)\right) d z\right) d u \\ = \frac {1}{(2 \pi) ^ {d}} \int \exp \left(i s ^ {\top} \mu \left(U _ {t - 1}\right) - \frac {1}{2} s ^ {\top} \Sigma \left(U _ {t - 1}\right) s\right) \left(\left(2 \pi\right) ^ {\frac {d}{2}} \frac {1}{\det \left| V _ {t - 1} \right|} \cdot \exp \left(- \frac {1}{2} u ^ {\top} V _ {t - 1} ^ {- 1} u\right)\right) d u \\ = \frac {1}{(2 \pi) ^ {\frac {d}{2}} \det | V _ {t - 1} |} \int \exp (i s ^ {T} \mu (U _ {t - 1}) - \frac {1}{2} s ^ {T} \Sigma (U _ {t - 1}) s - \frac {1}{2} u ^ {T} V _ {t - 1} u) d u. \\ \end{array} +$$ + +Apply dominant convergence theorem to $\lim_{n\to \infty}\phi_{U_t}(s)$ , we have: + +$$ +\begin{array}{l} \lim _ {n \rightarrow \infty} \phi_ {U _ {t}} (s) = \lim _ {n \rightarrow \infty} \frac {1}{(2 \pi) ^ {\frac {d}{2}} \det | V _ {t - 1} |} \int \exp (i s ^ {T} \mu (U _ {t - 1}) - \frac {1}{2} s ^ {T} \Sigma (U _ {t - 1}) s - \frac {1}{2} u ^ {T} V _ {t - 1} u) d u \\ = \lim _ {n \rightarrow \infty} \frac {1}{(2 \pi) ^ {\frac {d}{2}} \det | V _ {t - 1} |} \int \exp (i s ^ {T} \sqrt {n} (G (\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}) - G (\theta_ {t - 1})) \\ - \frac {1}{2} s ^ {T} \Sigma_ {\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}} (G (\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1})) s - \frac {1}{2} u ^ {T} V _ {t - 1} u) d u \\ = \frac {1}{(2 \pi) ^ {\frac {d}{2}} \det | V _ {t - 1} |} \int \lim _ {n \rightarrow \infty} \exp (i s ^ {T} \sqrt {n} (G (\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}) - G (\theta_ {t - 1})) \\ - \frac {1}{2} s ^ {T} \Sigma_ {\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}} (G (\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1})) s - \frac {1}{2} u ^ {T} V _ {t - 1} u) d u \\ = \frac {1}{(2 \pi) ^ {\frac {d}{2}} \det | V _ {t - 1} |} \int \exp (i s ^ {T} \nabla G (\theta_ {t - 1}) u - \frac {1}{2} s ^ {T} \Sigma_ {\theta_ {t - 1}} (G (\theta_ {t - 1})) s - \frac {1}{2} u ^ {T} V _ {t - 1} u) d u \\ = \exp \left(- \frac {1}{2} s ^ {T} \nabla G \left(\theta_ {t - 1}\right) V _ {t + 1} \nabla G \left(\theta_ {t - 1}\right) ^ {T} s - \frac {1}{2} s ^ {T} \Sigma_ {U _ {t - 1}} \left(\theta_ {t}\right) s\right), \\ \end{array} +$$ + +which is the characteristic function of $\mathcal{N}(0,V_t)$ , where $V_{t} = \nabla G(\theta_{t - 1})V_{t - 1}\nabla G(\theta_{t - 1})^{\top} + \Sigma_{\theta_{t - 1}}(\theta_t)$ . Here we use the fact that $\lim_{n\to \infty}\sqrt{n}\left(G(\frac{y}{\sqrt{n}} +\theta_{t - 1}) - G(\theta_{t - 1})\right) = \nabla G(\theta_{t - 1})y$ , and the dominant convergence theorem holds as we have + +$$ +\begin{array}{l} \left| \exp \left(i s ^ {T} \sqrt {n} \left(G \left(\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}\right) - G \left(\theta_ {t - 1}\right)\right) \right. \right. \\ - \frac {1}{2} s ^ {T} \Sigma_ {\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}} \left(G \left(\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}\right)\right) s - \frac {1}{2} u ^ {T} V _ {t - 1} u) | \leqslant | \exp \left(- \frac {1}{2} u ^ {T} V _ {t - 1} u\right). \\ \end{array} +$$ + +Thus we conclude by induction that + +$$ +U _ {t} \xrightarrow {D} \mathcal {N} \left(0, V _ {t}\right), +$$ + +$$ +V _ {t} = \sum_ {i = 1} ^ {t} \left[ \prod_ {k = i} ^ {t - 1} \nabla G \left(\theta_ {k}\right) \right] \Sigma_ {\theta_ {i - 1}} \left(\theta_ {i}\right) \left[ \prod_ {k = i} ^ {t - 1} \nabla G \left(\theta_ {k}\right) \right] ^ {\top}. +$$ + +And by the theorem of implicit function, we can calculate the gradient of $G$ as follows + +$$ +\nabla G (\theta) = - \left[ \left(\nabla_ {\psi} ^ {2} \underset {(x, y) \sim \mathcal {D} (\theta)} {\mathbb {E}} \ell (x, y; \psi)\right) | _ {\psi = G (\theta)} \right] ^ {- 1} \left(\nabla_ {\psi} \nabla_ {\tilde {\theta}} \underset {(x, y) \sim \mathcal {D} (\theta)} {\mathbb {E}} \ell (x, y; \psi)\right) | _ {\psi = G (\theta)}. +$$ + +$$ +\begin{array}{l} \nabla G \left(\theta_ {k}\right) = - \left[ \underset {(x, y) \sim \mathcal {D} \left(\theta_ {k}\right)} {\mathbb {E}} \nabla_ {\theta} ^ {2} \ell \left(x, y; \theta_ {k + 1}\right) \right] ^ {- 1} \left(\nabla_ {\tilde {\theta}} \underset {(x, y) \sim \mathcal {D} \left(\theta_ {k}\right)} {\mathbb {E}} \nabla_ {\theta} \ell \left(x, y; \theta_ {k + 1}\right)\right) \\ = - H _ {\theta_ {k}} \left(\theta_ {k + 1}\right) ^ {- 1} \left(\nabla_ {\tilde {\theta} _ {(x, y) \sim \mathcal {D} \left(\theta_ {k}\right)}} \mathbb {E} \nabla_ {\theta} \ell (x, y; \theta_ {k + 1})\right) \\ = - H _ {\theta_ {k}} \left(\theta_ {k + 1}\right) ^ {- 1} \mathbb {E} _ {z \sim \mathcal {D} \left(\theta_ {k}\right)} \left[ \nabla_ {\theta} \ell (z, \theta_ {k + 1}) \nabla_ {\theta} \log p (z, \theta_ {k}) ^ {\top} \right]. \\ \end{array} +$$ + +# A.2.2 Score Matching + +In this part, we provide details about our score matching mechanism. + +Given that + +$$ +\nabla G (\theta_ {k}) = - H _ {\theta_ {k}} (\theta_ {k + 1}) ^ {- 1} \mathbb {E} _ {z \sim \mathcal {D} (\theta_ {k})} [ \nabla_ {\theta} \ell (z, \theta_ {k + 1}) \nabla_ {\theta} \log p (z, \theta_ {k}) ^ {\top} ], +$$ + +once we have a good estimation of $\nabla_{\theta}\log p(z,\theta_k)$ for all $z\in \mathcal{Z}$ , $\nabla G(\theta_k)$ could be easily estimated by samples. + +Recall that we use a model $M(z,\theta ;\psi)$ parameterized by $\psi$ to approximate $p(z,\theta)$ . Inspired by the objective in [14], for any given $\theta$ (e.g., $\widehat{\theta}_t$ ), we aim to optimize the following objective parameterized by $\psi$ : + +$$ +\begin{array}{l} J (\theta ; \psi) = \int p (z, \theta) \| \nabla_ {\theta} \log p (z, \theta) - s (z, \theta ; \psi) \| ^ {2} d z \\ = \int p (z, \theta) \left(\| \nabla_ {\theta} \log p (z, \theta) \| ^ {2} + \| s (z, \theta ; \psi) \| ^ {2} - 2 \nabla_ {\theta} \log p (z, \theta) ^ {\top} s (z, \theta ; \psi)\right) d z \\ \end{array} +$$ + +where $s(z,\theta ;\psi) = \nabla_{\theta}\log M(z,\theta ;\psi)$ + +As mentioned in the main context, the first term is unrelated to $\psi$ ; the second term involves model $M$ that is chosen by us, so we have the analytical expression of $s(z,\theta ;\psi)$ . Thus, our key task will be estimating the third term, which involves $\mathcal{K}(\theta ;\psi)\coloneqq \int p(z,\theta)\nabla_{\theta}\log p(z,\theta)^{\top}s(z,\theta ;\psi)dz$ . + +Lemma A.8 (Restatement of Lemma 3.5). Under Assumption A.2, we have + +$$ +\mathcal {K} (\theta ; \psi) = \sum_ {i = 1} ^ {d} \left[ \frac {\partial}{\partial \theta^ {(i)}} \int p (z, \theta) \frac {\partial \log M (z , \theta ; \psi)}{\partial \theta^ {(i)}} d z - \int p (z, \theta) \frac {\partial^ {2} \log M (z , \theta ; \psi)}{\partial \theta^ {(i) 2}} d z \right] +$$ + +where $\theta^{(i)}$ is the $i$ -th coordinate of $\theta$ . + +Proof. Recall that $\theta$ is of $d$ -dimension. + +$$ +\begin{array}{l} \int p (z, \theta) \nabla_ {\theta} \log p (z, \theta) ^ {\top} s (z, \theta ; \psi) d z = \sum_ {i = 1} ^ {d} \int p (z, \theta) \frac {\partial \log p (z , \theta)}{\partial \theta^ {(i)}} \cdot \frac {\partial \log M (z , \theta ; \psi)}{\partial \theta^ {(i)}} d z \\ = \sum_ {i = 1} ^ {d} \int p (z, \theta) \frac {\partial \log p (z , \theta)}{\partial \theta^ {(i)}} \cdot \frac {\partial \log M (z , \theta ; \psi)}{\partial \theta^ {(i)}} d z \\ = \sum_ {i = 1} ^ {d} \int \frac {\partial p (z , \theta)}{\partial \theta^ {(i)}} \cdot \frac {\partial \log M (z , \theta ; \psi)}{\partial \theta^ {(i)}} d z. \\ \end{array} +$$ + +Then, we study $\int \frac{\partial p(z,\theta)}{\partial\theta^{(i)}}\cdot \frac{\partial\log M(z,\theta;\psi)}{\partial\theta^{(i)}}dz$ . Under Assumption A.2, the integral and differentiation of the following equation is exchangeable, i.e., + +$$ +\frac {\partial}{\partial \theta^ {(i)}} \int p (z, \theta) \frac {\partial M (z , \theta ; \psi)}{\partial \theta^ {(i)}} d z = \int \frac {p (z , \theta)}{\partial \theta^ {(i)}} \frac {\partial M (z , \theta ; \psi)}{\partial \theta^ {(i)}} d z. +$$ + +According to integral by parts, we have + +$$ +\int \frac {\partial p (z , \theta)}{\partial \theta^ {(i)}} \cdot \frac {\partial \log M (z , \theta ; \psi)}{\partial \theta^ {(i)}} d z = \frac {\partial}{\partial \theta^ {(i)}} \int p (z, \theta) \frac {\partial M (z , \theta ; \psi)}{\partial \theta^ {(i)}} d z - \int p (z, \theta) \frac {\partial^ {2} \log M (z , \theta ; \psi)}{\partial \theta^ {(i) 2}} d z. +$$ + +Thus, our proof is completed. + +The rest of the estimation process via policy perturbation is provided in the main context in Section 3. + +The other part omitted in Section 3 is the details about Eq. 2 that + +$$ +\widehat {V} _ {t} ^ {- 1 / 2} \sqrt {n} \left(\widehat {\theta} _ {t} - \theta_ {t}\right) \stackrel {{D}} {{\to}} \mathcal {N} \left(0, I _ {d}\right). +$$ + +Here $\widehat{V}_t$ denotes the sample-based estimator of the variance, obtained by plugging in the empirical Hessian and empirical covariance matrices: + +$$ +\widehat {H} _ {\widehat {\theta} _ {t - 1}} (\widehat {\theta} _ {t}) = \widehat {\mathbb {E}} _ {z \sim \mathcal {D} (\widehat {\theta} _ {t - 1})} \nabla_ {\theta} ^ {2} \ell (z; \widehat {\theta} _ {t}), \quad \widehat {\mathrm {C o v}} _ {z \sim \mathcal {D} (\widehat {\theta} _ {t - 1})} \big (\nabla_ {\theta} \ell (z; \widehat {\theta} _ {t}) \big), +$$ + +as well as the estimator for $\nabla G(\widehat{\theta}_{t - 1})$ : + +$$ +- \widehat {H} _ {\widehat {\theta} _ {t - 1}} (\widehat {\theta} _ {t}) ^ {- 1} \widehat {\mathbb {E}} _ {z \sim \mathcal {D} (\widehat {\theta} _ {t - 1})} [ \nabla_ {\theta} \ell (z, \widehat {\theta} _ {t}) \nabla_ {\theta} \log M (z, \widehat {\theta} _ {t - 1}, \widehat {\psi}) ^ {\top} ], +$$ + +where $\widehat{\psi}$ is obtained by minimizing $\widehat{J}_{n,k}$ . + +Eq. 2 is a direct result following Slutsky's theorem. Assumption A.3 makes sure the empirical optimizer set can converge to the population optimizer set. Then other parts such as estimation of the Hessian matrix etc. could all be directly obtained by standard law of large numbers. Thus, we can directly use Slutsky's theorem to obtain Eq. 2. + +# A.3 Policy Perturbation + +In this part, we prove the validity of $\widehat{g}_k$ as an estimator of $\nabla G(\theta_k)$ . Recall that we have $s(z,\theta ;\psi) = \nabla_{\theta}\log M(z,\theta ;\psi)$ and we further denote $s_i = \frac{\partial M(z,\theta;\psi)}{\partial\theta^{(i)}}$ , and use $\mathbb{E}_{\theta},\widehat{\mathbb{E}}_{\theta ,n}$ for the expectation $\mathbb{E}_{z\sim D(\theta)}$ and empirical expectation $\mathbb{E}_{z\sim \widehat{D}_n(\theta)}$ respectively. + +Firstly, we define the following function families: + +$$ +\mathcal {F} _ {1, \theta}: = \left\{s (\cdot , \theta ; \psi): \psi \in \Psi \right\}, +$$ + +$$ +\mathcal {F} _ {2, \theta} ^ {(i)}: = \left\{\frac {\partial}{\partial \theta^ {(i)}} s _ {i} (\cdot , \theta ; \psi): \psi \in \Psi \right\}, +$$ + +$$ +\mathcal {F} _ {3, \theta} ^ {(i)} := \left\{s _ {i} (\cdot , \theta ; \psi): \psi \in \Psi \right\}. +$$ + +Further, we set + +$$ +\begin{array}{l} \widehat {J} _ {n, k} (\theta ; \psi): = \widehat {\mathbb {E}} _ {\theta , n} [ \| s (z, \theta ; \psi) \| ^ {2} ] + \mathbb {E} _ {\theta} \left[ \| \nabla_ {\theta} \log p (z, \theta) \| ^ {2} \right] + 2 \sum_ {i = 1} ^ {d} \left[ \widehat {\mathbb {E}} _ {\theta , n} [ \frac {\partial}{\partial \theta^ {(i)}} s _ {i} (z, \theta ; \psi) ] \right] \\ \left. - 2 \sum_ {i = 1} ^ {d} \frac {1}{\eta} \left(\widehat {\mathbb {E}} _ {\theta + \eta e ^ {(i)}, k} \left[ s _ {i} (z, \theta + \eta e ^ {(i)}; \psi) \right] - \widehat {\mathbb {E}} _ {\theta , n} \left[ s _ {i} (z, \theta ; \psi) \right]\right), \right. \\ \end{array} +$$ + +$$ +\begin{array}{l} J _ {n} (\theta ; \psi): = \mathbb {E} _ {\theta} [ \| s (z, \theta ; \psi) \| ^ {2} ] + \mathbb {E} _ {\theta} \left[ \| \nabla_ {\theta} \log p (z, \theta) \| ^ {2} \right] + 2 \sum_ {i = 1} ^ {d} \left[ \mathbb {E} _ {\theta} \left[ \frac {\partial}{\partial \theta^ {(i)}} s _ {i} (z, \theta ; \psi) \right] \right] \\ - 2 \sum_ {i = 1} ^ {d} \frac {1}{\eta} \left(\mathbb {E} _ {\theta + \eta e ^ {(i)}} \left[ s _ {i} (z, \theta + \eta e ^ {(i)}; \psi) \right] - \mathbb {E} _ {\theta} \left[ s _ {i} (z, \theta ; \psi) \right]\right). \\ \end{array} +$$ + +Assumption A.9. We assume that the score function $s$ , distribution map $D$ and the corresponding function families $\{\mathcal{F}_{1,\theta},\mathcal{F}_{2,\theta}^{(i)},\mathcal{F}_{3,\theta}^{(i)}:i = 1,\ldots ,d\}$ has the following properties: + +(a). There is a positive constant $C > 0$ , such that for $\forall \theta \in \theta ;\psi \in \Psi$ + +$$ +\left| \frac {\partial}{\partial \theta^ {(i) ^ {2}}} \mathbb {E} _ {\theta} [ s _ {i} (z, \theta ; \psi) ] \right| \leqslant C < \infty . +$$ + +(b). (Enveloping function) For $\forall \theta \in \Theta$ , there is a function $H_{\theta}(z)$ , such that $\mathbb{E}_{\theta}[H_{\theta}(z)^2] < \infty$ , and for any function $f \in \mathcal{F}_{1,\theta} \bigcup (\cup_{i=1}^{d} \mathcal{F}_{2,\theta}^{(i)}) \bigcup (\cup_{i=1}^{d} \mathcal{F}_{3,\theta}^{(i)})$ , we have + +$$ +| f (z) | \leqslant H _ {\theta} (z) +$$ + +(c). ( $\theta$ -uniform Donsker) Let $\mathcal{N}(\varepsilon, \mathcal{F}, \| \cdot \|)$ denote the covering number, that is the minimal number of $\| \cdot \|$ -balls of radius $\varepsilon$ needed to cover the set $\mathcal{F}$ , there exists $\rho(\varepsilon) > 0$ , such that + +$$ +\int_ {0} ^ {1} \sqrt {\rho (\varepsilon)} \mathrm {d} \varepsilon < \infty , +$$ + +and + +$$ +\sup_{\theta \in \Theta}\sup_{Q}\log \mathcal{N}\big(\varepsilon \| H_{\theta}(z)\|_{Q,2},\mathcal{F}_{\theta},L_{2}(Q)\big) < \rho (\varepsilon), +$$ + +where $\mathcal{F}_{\theta} \in \{\mathcal{F}_{1,\theta}, \mathcal{F}_{2,\theta}^{(i)}, \mathcal{F}_{3,\theta}^{(i)} : i = 1, \dots, d\}$ and $Q$ is any distribution on $\mathcal{Z}$ . + +(d). (Vanishing optimization error) There exists $a_{n} = o(1)$ , such that + +$$ +\widehat {J} _ {n, k} (\theta ; \widehat {\psi} (\theta)) \leqslant \min _ {\psi \in \Psi} \widehat {J} _ {n, k} (\theta ; \psi) + a _ {n}. +$$ + +(e). (Richness of class) There exists $\psi^{*}(\theta)$ for each $\theta$ , s.t. + +$$ +s (z, \theta ; \psi^ {*} (\theta)) = \nabla_ {\theta} \log p (z, \theta). +$$ + +Remark A.10. Assumption A.9(c) holds with $\rho(\varepsilon) = C\log \frac{1}{\varepsilon}$ , when $\mathcal{F}_{\theta}$ is a VC-subgraph class for all $\mathcal{F}_{\theta} \in \{\mathcal{F}_{1,\theta}, \mathcal{F}_{2,\theta}^{(i)}, \mathcal{F}_{3,\theta}^{(i)} : i = 1, \ldots, d\}$ . + +Assumption A.11. Assume that the following conditions hold: + +(a). (Smoothness) $H_{\bar{\theta}}(\theta)$ is $L$ -joint smooth in $(\tilde{\theta}, \theta)$ for some $L < \infty$ . +(b). There exists $C > 0$ , such that + +$$ +\sup _ {\tilde {\theta}} \mathbb {E} _ {\tilde {\theta}} \left[ \nabla_ {\theta} \ell (z; \theta_ {P S}) \right] \leqslant C < \infty +$$ + +Lemma A.12. Under Assumption A.9(a), we have + +$$ +\sup _ {\theta ; \psi} | J (\theta ; \psi) - J _ {n} (\theta ; \psi) | \leqslant 2 d C \eta . +$$ + +Proof. By mean-value theorem, there exists $\eta^{(i)}\in [0,\eta ],1\leqslant i\leqslant d$ , such that + +$$ +\begin{array}{l} J (\theta ; \psi) - J _ {n} (\theta ; \psi) = 2 \sum_ {i = 1} ^ {d} \frac {1}{\eta} \bigg (\mathbb {E} _ {\theta + \eta e ^ {(i)}} \left[ s _ {i} (z, \theta + \eta e ^ {(i)}; \psi) \right] - \mathbb {E} _ {\theta} \left[ s _ {i} (z, \theta ; \psi) \right] \bigg) - 2 \sum_ {i = 1} ^ {d} \left[ \frac {\partial}{\partial \theta^ {(i)}} \mathbb {E} _ {\theta} [ s _ {i} (z, \theta ; \psi) ] \right] \\ = 2 \sum_ {i = 1} ^ {d} \left[ \frac {\partial}{\partial \theta^ {(i)}} \mathbb {E} _ {\theta + \eta^ {(i)} e ^ {(i)}} [ s _ {i} (z, \theta + \eta^ {(i)} e ^ {(i)}; \psi) ] \right] - 2 \sum_ {i = 1} ^ {d} \left[ \frac {\partial}{\partial \theta^ {(i)}} \mathbb {E} _ {\theta} [ s _ {i} (z, \theta ; \psi) ] \right] \\ \leqslant 2 C \left(\sum_ {i = 1} ^ {d} \eta^ {(i)}\right) \\ \leqslant 2 d C \eta , \\ \end{array} +$$ + +the first inequality follows from a direct application of mean-value theorem. + +Theorem A.13. Let $\Theta_{d,t} = \{\widehat{\theta}_j, \widehat{\theta}_j + \eta e^{(i)} : i = 1, \dots, d, j = 1, \dots, T\}$ . Under Assumption A.9, for $\forall \theta \in \Theta_{d,t}$ , the following inequality holds + +$$ +\mathbb {E} _ {\theta} \big [ \| \nabla \log p (z, \theta) - s (z, \theta ; \psi (\theta)) \| ^ {2} \big | \widehat {\psi} (\theta) \big ] = O _ {p} (\frac {1}{\sqrt {n}} + \frac {1}{\eta \sqrt {\min (n , k)}} + \eta + a _ {n}). +$$ + +Proof. Fix $\theta \in \Theta$ , by Dudley's uniform entropy bound, c.f. Corollary 19.35 in [31], we have + +$$ +\mathbb {E} _ {\theta} \left[ \sup _ {\psi} \left| \widehat {E} _ {\theta , n} \left[ \| s (z, \theta ; \psi) \| ^ {2} \right] - E _ {\theta} \left[ \| s (z, \theta ; \psi) \| ^ {2} \right] \right| \right] \lesssim \frac {1}{\sqrt {n}}, +$$ + +$$ +\mathbb {E} _ {\theta} \left[ \sup _ {\psi} \left| \widehat {E} _ {\theta , n} \left[ \frac {\partial}{\partial \theta^ {(i)}} s _ {i} (z, \theta ; \psi) \right] - E _ {\theta} \left[ \frac {\partial}{\partial \theta^ {(i)}} s _ {i} (z, \theta ; \psi) \right] \right| \right] \lesssim \frac {1}{\sqrt {n}}, +$$ + +$$ +\mathbb {E} _ {\theta} \left[ \sup _ {\psi} \left| \widehat {E} _ {\theta , k} \left[ s _ {i} (z, \theta ; \psi) \| ^ {2} \right] - E _ {\theta} \left[ \| s _ {i} (z, \theta ; \psi) \| ^ {2} \right] \right| \right] \lesssim \frac {1}{\sqrt {k}}. +$$ + +Since $d, T = O(1)$ , from the above results, the inequality below holds + +$$ +\sup _ {\psi \in \Psi} \sup _ {\theta \in \Theta_ {d, T}} \left| \widehat {J} _ {n, k} (\theta ; \psi) - J _ {n} (\theta ; \psi) \right| = O _ {p} (\frac {1}{\sqrt {n}} + \frac {1}{\eta \sqrt {\min (n , k)}}), +$$ + +by Lemma A.12, we know + +$$ +\sup _ {\psi \in \Psi} \sup _ {\theta \in \Theta_ {d, T}} \left| \widehat {J} _ {n, k} (\theta ; \psi) - J (\theta ; \psi) \right| = O _ {p} \left(\frac {1}{\sqrt {n}} + \frac {1}{\eta \sqrt {\min (n , k)}} + \eta\right). \tag {4} +$$ + +From Assumption A.9(d), we know that for any $\theta \in \Theta_{d,T}$ + +$$ +\widehat {J} _ {n, k} (\theta ; \widehat {\psi} (\theta)) \leqslant \min _ {\psi \in \Psi} \widehat {J} _ {n, k} (\theta ; \psi) + a _ {n}, +$$ + +Using Assumption A.9(e), we know that $\min_{\psi \in \Psi} J(\theta; \psi) = 0$ , and assume there exists $\psi^{*}(\theta) \in \arg \min_{\psi \in \Psi} J(\theta; \psi)$ . + +Take $\psi = \psi^{*}(\theta)$ and $\widehat{\psi} (\theta)$ respectively in inequality (4), we have + +$$ +\widehat {J} _ {n, k} (\theta ; \psi^ {*} (\theta)) = O _ {p} (\frac {1}{\sqrt {n}} + \frac {1}{\eta \sqrt {\min (n , k)}} + \eta), +$$ + +$$ +| J (\theta ; \widehat {\psi} (\theta)) - \widehat {J} _ {n, k} (\theta ; \widehat {\psi} (\theta)) | = O _ {p} (\frac {1}{\sqrt {n}} + \frac {1}{\eta \sqrt {\min (n , k)}} + \eta). +$$ + +Since $\widehat{J}_{n,k}(\theta; \widehat{\psi}(\theta)) \leqslant \widehat{J}_{n,k}(\theta; \psi^*(\theta)) + a_n$ , we have + +$$ +\begin{array}{l} J (\theta ; \widehat {\psi} (\theta)) = J (\theta ; \widehat {\psi} (\theta)) - \widehat {J} _ {n, k} (\theta ; \widehat {\psi} (\theta)) + \widehat {J} _ {n, k} (\theta ; \widehat {\psi} (\theta)) - \widehat {J} _ {n, k} (\theta ; \psi^ {*} (\theta)) + \widehat {J} _ {n, k} (\theta ; \psi^ {*} (\theta)) \\ = O _ {p} \left(\frac {1}{\sqrt {n}} + \frac {1}{\eta \sqrt {\min (n , k)}} + \eta + a _ {n}\right). \\ \end{array} +$$ + +By definition of $J(\theta ;\psi)$ , we have proved + +$$ +\mathbb {E} _ {\theta} \big [ \| \nabla \log p (z, \theta) - s (z, \theta ; \psi (\theta)) \| ^ {2} \big | \widehat {\psi} (\theta) \big ] = O _ {p} (\frac {1}{\sqrt {n}} + \frac {1}{\eta \sqrt {\min (n , k)}} + \eta + a _ {n}). +$$ + +Recall that + +$$ +\nabla G (\theta_ {k}) = - H _ {\theta_ {k}} \left(\theta_ {k + 1}\right) ^ {- 1} \mathbb {E} _ {\theta_ {k}} \left[ \nabla_ {\theta} \ell (z; \theta_ {k + 1}) \nabla_ {\theta} \log p (z, \theta_ {k}) ^ {\top} \right], +$$ + +and the estimator is defined by + +$$ +\widehat {g} _ {k} := - H _ {\widehat {\theta} _ {k}} \big (\widehat {\theta} _ {k + 1} \big) ^ {- 1} \widehat {\mathbb {E}} _ {\widehat {\theta} _ {k, n}} \big [ \nabla_ {\theta} \ell (z; \widehat {\theta} _ {k + 1}) s (z, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) ^ {\top} \big ]. +$$ + +Theorem A.14 (Restatement of Theorem 3.6). Under Assumption 3.1 A.1, A.9 and A.11, we have + +$$ +\| \widehat {g} _ {k} - \nabla G (\theta_ {k}) \| ^ {2} = O _ {p} \big (\frac {1}{\sqrt {n}} + \frac {1}{\eta \sqrt {\min (n , k)}} + \eta + a _ {n} \big). +$$ + +Proof. By Assumption 3.1, + +$$ +\left\| \nabla_ {\theta} \ell \left(z; \theta_ {k + 1}\right) - \nabla_ {\theta} \ell \left(z; \widehat {\theta} _ {k + 1}\right) \right\| \leqslant \beta \| \theta_ {k + 1} - \widehat {\theta} _ {k + 1} \|, +$$ + +by Assumption 3.1, there exists $C_1 > 0$ , such that + +$$ +\| H _ {\widehat {\theta} _ {k}} (\widehat {\theta} _ {k + 1}) ^ {- 1} \| \leqslant C _ {1} < \infty . +$$ + +The proof falls into five parts. + +Let + +$$ +\widehat {g} _ {k, 1} := - H _ {\widehat {\theta_ {k}}} \big (\widehat {\theta} _ {k + 1} \big) ^ {- 1} \widehat {\mathbb {E}} _ {\widehat {\theta} _ {k, n}} \left[ \nabla_ {\theta} \ell (z; \theta_ {k + 1}) s (z, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) ^ {\top} \right]. +$$ + +Step 1: Convergence of $\| \widehat{g}_k - \widehat{g}_{k,1}\|$ + +We have + +$$ +\begin{array}{l} \| \widehat {g} _ {k} - \widehat {g} _ {k, 1} \| \lesssim \| \widehat {\theta} _ {k + 1} - \theta_ {k + 1} \| \widehat {\mathbb {E}} _ {\widehat {\theta} _ {k, n}} \big [ \| s (z, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) \| \big ] \\ = \frac {1}{n} \| \widehat {\theta} _ {k + 1} - \theta_ {k + 1} \| \sum_ {l = 1} ^ {n} \| s (z _ {l, k}, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) \|. \\ \end{array} +$$ + +From Assumption A.9(b), there exists $C_2 > 0$ , such that + +$$ +\mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \| s (z, \widehat {\theta} _ {k}; \psi (\widehat {\theta} _ {k})) \| ^ {2} \right] \leqslant C _ {2} < \infty , +$$ + +by the law of large numbers, we know + +$$ +\left\| \widehat {g} _ {k} - \widehat {g} _ {k, 1} \right\| = O _ {p} \left(\left\| \widehat {\theta} _ {k + 1} - \theta_ {k + 1} \right\|\right), +$$ + +thus combine the above result with Theorem 3.4, we have + +$$ +\left\| \widehat {g} _ {k} - \widehat {g} _ {k, 1} \right\| = O _ {p} (\frac {1}{\sqrt {n}}). +$$ + +Let + +$$ +\widehat {g} _ {k, 2} := - H _ {\widehat {\theta_ {k}}} \left(\widehat {\theta} _ {k + 1}\right) ^ {- 1} \mathbb {E} _ {\widehat {\theta_ {k}}} \left[ \nabla_ {\theta} \ell (z; \theta_ {k + 1}) s (z, \widehat {\theta_ {k}}; \widehat {\psi} (\widehat {\theta_ {k}})) ^ {\top} \right]. +$$ + +Step 2: Convergence of $\| \widehat{g}_{k,1} - \widehat{g}_{k,2}\|$ . + +By Cauchy-Schwarz inequality, we know + +$$ +\mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \left\| \nabla_ {\theta} \ell (z; \theta_ {k + 1}) s (z, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) ^ {\top} \right\| ^ {2} \right] \leqslant \mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \left\| \nabla_ {\theta} \ell (z; \theta_ {k + 1}) \right\| ^ {2} \right] \mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \left\| s (z, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) \right\| ^ {2} \right]. +$$ + +From Assumption A.9(b), we have + +$$ +\mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \| s (z, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) \| ^ {2} \right] \leqslant \mathbb {E} _ {\widehat {\theta} _ {k}} \left[ H _ {\widehat {\theta} _ {k}} ^ {2} (z) \right] \leqslant C < \infty , +$$ + +where $H_{\widehat{\theta}_k}(z)$ is the enveloping function for $\widehat{\theta}_k$ + +By Assumption 3.1, + +$$ +\left\| \nabla_ {\theta} \ell (z; \theta_ {k + 1}) - \nabla_ {\theta} \ell (z; \theta_ {P S}) \right\| \leqslant \beta \| \theta_ {k + 1} - \theta_ {P S} \|, +$$ + +since + +$$ +\left\| \theta_ {k + 1} - \theta_ {P S} \right\| \leqslant \left(\frac {\varepsilon \beta}{\gamma}\right) ^ {k + 1} \left\| \theta_ {0} - \theta_ {P S} \right\| \leqslant \left\| \theta_ {0} - \theta_ {P S} \right\|, +$$ + +thus by Assumption A.11(b), + +$$ +\mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \| \nabla_ {\theta} \ell (z; \theta_ {k + 1}) \| ^ {2} \right] \lesssim \mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \| \nabla_ {\theta} \ell (z; \theta_ {P S}) \| ^ {2} \right] + \| \theta_ {0} - \theta_ {P S} \| ^ {2} = O (1). \tag {5} +$$ + +Hence we have + +$$ +\mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \left\| \nabla_ {\theta} \ell (z; \theta_ {k + 1}) s (z, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) ^ {\top} \right\| ^ {2} \right] = O (1), \tag {6} +$$ + +by Chebyshev inequality and Assumption 3.1, this leads to the following bound of $\| \widehat{g}_{k,1} - \widehat{g}_{k,2} \|$ , + +$$ +\begin{array}{l} \| \widehat {g} _ {k, 1} - \widehat {g} _ {k, 2} \| \lesssim \left\| \widehat {\mathbb {E}} _ {\widehat {\theta} _ {k}, n} \left[ \nabla_ {\theta} \ell (z; \theta_ {k + 1}) s (z, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) ^ {\top} \right] - \mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \nabla_ {\theta} \ell (z; \theta_ {k + 1}) s (z, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) ^ {\top} \right] \right\| \\ = O _ {p} (\frac {1}{\sqrt {n}}). \\ \end{array} +$$ + +Let + +$$ +\widehat {g} _ {k, 3} := - H _ {\theta_ {k}} (\theta_ {k + 1}) ^ {- 1} \mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \nabla_ {\theta} \ell (z; \theta_ {k + 1}) s (z, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) ^ {\top} \right]. +$$ + +Step 3: Convergence of $\| \widehat{g}_{k,2} - \widehat{g}_{k,3}\|$ . By Assumption A.11(a) and (6), we have + +$$ +\begin{array}{l} \left\| \widehat {g} _ {k, 2} - \widehat {g} _ {k, 3} \right\| \lesssim \left\| H _ {\theta_ {k}} \left(\theta_ {k + 1}\right) ^ {- 1} - H _ {\widehat {\theta_ {k}}} \left(\widehat {\theta} _ {k + 1}\right) ^ {- 1} \right\| \\ = \left\| H _ {\theta_ {k}} \left(\theta_ {k + 1}\right) ^ {- 1} \left(H _ {\theta_ {k}} \left(\theta_ {k + 1}\right) - H _ {\widehat {\theta_ {k}}} \left(\widehat {\theta} _ {k + 1}\right)\right) H _ {\widehat {\theta_ {k}}} \left(\widehat {\theta} _ {k + 1}\right) ^ {- 1} \right\|, \\ \end{array} +$$ + +further, using Assumption 3.1 and A.11(a), we know + +$$ +\| \widehat {g} _ {k, 2} - \widehat {g} _ {k, 3} \| \lesssim \| \widehat {\theta} _ {k} - \theta_ {k} \| + \| \widehat {\theta} _ {k + 1} - \theta_ {k + 1} \|, +$$ + +thus by Theorem 3.4, + +$$ +\| \widehat {g} _ {k, 2} - \widehat {g} _ {k, 3} \| = O _ {p} (\frac {1}{\sqrt {n}}). +$$ + +Let + +$$ +\widehat {g} _ {k, 4} := - H _ {\theta_ {k}} \left(\theta_ {k + 1}\right) ^ {- 1} \mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \nabla_ {\theta} \ell (z; \theta_ {k + 1}) \nabla_ {\theta} \log p (z, \widehat {\theta} _ {k}) \right]. +$$ + +Step 4: Convergence of $\| \widehat{g}_{k,3} - \widehat{g}_{k,4}\|$ + +By Assumption 3.1 and Assumption A.1, we have + +$$ +\begin{array}{l} \left. \right.\left\| \widehat {g} _ {k, 3} - \widehat {g} _ {k, 4} \right\| ^ {2} \lesssim \left\{\mathbb {E} _ {\widehat {\theta} _ {k}} \left[\left\| \nabla_ {\theta} \ell (z; \theta_ {k + 1}) \right\| \times \left\| s (z, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) - \nabla_ {\theta} \log p (z, \widehat {\theta} _ {k}) \right\|\right]\right\} ^ {2} \\ \leqslant \mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \left\| \nabla_ {\theta} \ell (z; \theta_ {k + 1}) \right\| ^ {2} \right] \\ \times \mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \left\| s (z, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) - \nabla_ {\theta} \log p (z, \widehat {\theta} _ {k}) \right\| ^ {2} \right] \quad \text {(b y C a u c h y - S c h w a r z i n e q u a l i t y)} \\ \lesssim \mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \left\| s (z, \widehat {\theta} _ {k}; \widehat {\psi} (\widehat {\theta} _ {k})) - \nabla_ {\theta} \log p (z, \widehat {\theta} _ {k}) \right\| ^ {2} \right] \quad (\text {b y}) \\ = O _ {p} \left(\frac {1}{\sqrt {n}} + \frac {1}{\eta \sqrt {\min (n , k)}} + \eta + a _ {n}\right) \quad (\text {b y}) \\ \end{array} +$$ + +Step 5: Convergence of $\| \widehat{g}_k - \nabla G(\theta_k)\|$ . + +Finally, recall that + +$$ +\nabla G (\theta_ {k}) = - H _ {\theta_ {k}} (\theta_ {k + 1}) ^ {- 1} \mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \nabla_ {\theta} \ell (z; \theta_ {k + 1}) \nabla_ {\theta} \log p (z, \theta_ {k}) \right], +$$ + +by Assumption 3.1 and A.1, + +$$ +\begin{array}{l} \left\| \widehat {g} _ {k, 4} - \nabla G \left(\theta_ {k}\right) \right\| \lesssim \mathbb {E} _ {\widehat {\theta} _ {k}} \left[ \| \nabla_ {\theta} \ell (z; \theta_ {k + 1}) \| \right] \times \| \widehat {\theta} _ {k} - \theta_ {k} \| \\ \lesssim \| \widehat {\theta} _ {k} - \theta_ {k} \| \\ = O _ {p} \left(\frac {1}{\sqrt {n}}\right) \quad (\text {B y T h e o r e m 3 . 4}). \\ \end{array} +$$ + +Combining the above results, we thus have proved + +$$ +\| \widehat {g} _ {k} - \nabla G (\theta_ {k}) \| ^ {2} = O _ {p} \big (\frac {1}{\sqrt {n}} + \frac {1}{\eta \sqrt {\min (n , k)}} + \eta + a _ {n} \big). +$$ + +# A.4 Details of Section 4: Theory of Prediction-Powered Inference under Performativity + +Without loss of generality, we let $N_{t} = N$ and $n_t = n$ for all $t\in [T]$ + +Let us denote + +$$ +\mathcal {L} _ {\tilde {\theta}} (\theta) := \mathbb {E} _ {(x, y) \sim \mathcal {D} (\tilde {\theta})} \ell (x, y; \theta), \quad \mathcal {L} _ {\tilde {\theta}} ^ {f, \lambda} (\theta) := \mathcal {L} _ {\tilde {\theta}, n} (\theta) + \lambda \cdot (\widetilde {\mathcal {L}} _ {\tilde {\theta}, N} ^ {f} (\theta) - \mathcal {L} _ {\tilde {\theta}, n} ^ {f} (\theta)), +$$ + +where + +$$ +\mathcal {L} _ {\tilde {\theta}, n} (\theta) := \frac {1}{n} \sum_ {i = 1} ^ {n} \ell (x _ {i}, y _ {i}; \theta), \mathcal {L} _ {\tilde {\theta}, n} ^ {f} (\theta) := \frac {1}{n} \sum_ {i = 1} ^ {n} \ell \left((x _ {i}, f (x _ {i}); \theta\right), \widetilde {\mathcal {L}} _ {\tilde {\theta}, N} ^ {f} (\theta) := \frac {1}{N} \sum_ {i = 1} ^ {N} \ell \left((x _ {i} ^ {u}, f (x _ {i} ^ {u}); \theta\right). +$$ + +Here the samples $(x_{i},y_{i})\sim \mathcal{D}(\tilde{\theta})$ and $x_{i}^{u}\sim \mathcal{D}_{\mathcal{X}}(\tilde{\theta})$ are drawn from the distribution under $\tilde{\theta}$ . Recall that we have defined $\Sigma_{\lambda ,\tilde{\theta}}(\theta) = H_{\tilde{\theta}}(\theta)^{-1}\left(rV_{\lambda ,\tilde{\theta}}^{f}(\theta) + V_{\lambda ,\tilde{\theta}}(\theta)\right)H_{\tilde{\theta}}(\theta)^{-1}$ before Theorem 4.1 (in the following we sometimes omit $r$ for simplicity). + +Theorem A.15 (Consistency of $\hat{\theta}_t^{\mathrm{PPI}}$ ). Under Assumption 3.1 and A.4, if $\varepsilon < \frac{\gamma}{\beta}$ , then for any given $T \geqslant 0$ , we have that for all $t \in [T]$ + +$$ +\widehat {\theta} _ {t + 1} ^ {P P I} \left(\lambda_ {t}\right) \xrightarrow {P} \theta_ {t + 1}. +$$ + +Proof. Let us denote $\widehat{G}_{\lambda}^{f}(\theta) \coloneqq \arg \min_{\theta' \in \Theta} \frac{\lambda}{N} \sum_{i=1}^{N} \ell(x_{i}^{u}, f(x_{i}^{u}); \theta') + \frac{1}{n} \sum_{i=1}^{n} \left( \ell(x_{i}, y_{i}; \theta') - \lambda \ell(x_{i}, f(x_{i}); \theta') \right)$ , where the samples $(x_{i}, y_{i}) \sim \mathcal{D}(\theta)$ and $x_{i}^{u} \sim \mathcal{D}_{\mathcal{X}}(\theta)$ are drawn for some parameter $\theta$ along the dynamic trajectory $\theta_{0} \to \widehat{\theta}_{1} \to \dots \widehat{\theta}_{t} \to \dots$ . + +$$ +\begin{array}{l} \| \theta_ {t} - \widehat {\theta} _ {t} ^ {\mathrm {P P I}} \| = \| G (\theta_ {t - 1}) - \widehat {G} _ {\lambda_ {t}} ^ {f} (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}) \| \\ \leqslant \left\| G (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}) - \widehat {G} _ {\lambda_ {t}} ^ {f} (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}) \right\| + \left\| G (\theta_ {t - 1}) - G (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}) \right\| \\ \leqslant \| G (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}) - \widehat {G} _ {\lambda_ {t}} ^ {f} (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}) \| + \varepsilon \frac {\beta}{\gamma} \| \theta_ {t - 1} - \widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}} \|, \\ \end{array} +$$ + +where the last inequality follows from the results derived by [25], under Assumption 3.1, we have $\| G(\theta) - G(\theta')\| \leqslant \frac{\varepsilon\beta}{\gamma}\| \theta -\theta '\|$ . + +Notice that $\mathbb{E}(\mathcal{L}_{\widehat{\theta}_{t - 1}^{\mathrm{PPI}}}^{f,\lambda_t}(\theta)) = \mathcal{L}_{\widehat{\theta}_{t - 1}^{\mathrm{PPI}}}(\theta)$ . By local Lipschitz condition, there exists $\varepsilon_0 > 0$ such that + +$$ +\sup_{\theta :\| \theta -G(\widehat{\theta}^{\mathrm{PPI}}_{t - 1})\| \leqslant \varepsilon_{0}}|\mathcal{L}_{\widehat{\theta}^{\mathrm{PPI}}_{t - 1}}^{f,\lambda_{t}}(\theta) - \mathcal{L}_{\widehat{\theta}^{\mathrm{PPI}}_{t - 1}}(\theta)|\xrightarrow{P} 0. +$$ + +Since $\ell$ is strongly convex for any $\theta$ , $G(\widehat{\theta}_{t-1}^{\mathrm{PPI}})$ is unique. Then we know that there exists $\delta$ such that $\mathcal{L}_{\widehat{\theta}_{t-1}^{\mathrm{PPI}}}^{f,\lambda_t}(\theta) - \mathcal{L}_{\widehat{\theta}_{t-1}^{\mathrm{PPI}}}(G(\widehat{\theta}_{t-1}^{\mathrm{PPI}})) > \delta$ for all $\theta$ in $\{\theta \mid \| \theta - G(\widehat{\theta}_{t-1}^{\mathrm{PPI}}) \| = \varepsilon_0\}$ . Then it follows that: + +$$ +\begin{array}{l} \inf_{\| \theta -G(\widehat{\theta}_{t - 1}^{\mathrm{PPI}})\| = \varepsilon_{0}}\mathcal{L}_{\widehat{\theta}_{t - 1}^{\mathrm{PPI}}}^{f,\lambda_{t}}\left(\theta\right) - \mathcal{L}_{\widehat{\theta}_{t - 1}^{\mathrm{PPI}}}^{f,\lambda_{t}}\left(G(\widehat{\theta}_{t - 1}^{\mathrm{PPI}})\right) \\ = \inf _ {\| \theta - G (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}) \| = \varepsilon_ {0}} \left(\left(\mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} ^ {f, \lambda_ {t}} (\theta) - \mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} (\theta)\right) + \left(\mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} (\theta) - \mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} (G (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}))\right) \right. \\ \left. + \left(\mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} \left(G (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}})\right) - \mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} ^ {f, \lambda_ {t}} \left(G (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}})\right)\right)\right) \\ \geqslant \delta - o _ {P} (1). \\ \end{array} +$$ + +Then we consider any fixed $\theta$ such that $\| \theta - G(\widehat{\theta}_{t - 1}^{\mathrm{PPI}}) \| \geqslant \varepsilon_0$ it follows that + +$$ +\begin{array}{l} \mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} ^ {f, \lambda_ {t}} (\theta) - \mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} ^ {f, \lambda_ {t}} (G (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}})) \geqslant \frac {\theta - G (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}})}{\omega - G (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}})} \left(\mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} ^ {f, \lambda_ {t}} (\omega) - \mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} ^ {f, \lambda_ {t}} (G (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}))\right) \\ \geqslant \frac {\| \theta - G (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}) \|}{\varepsilon_ {0}} (\delta - o _ {P} (1)) \geqslant \delta - o _ {P} (1), \\ \end{array} +$$ + +where the first inequality holds for any $\omega$ by the convexity condition of $\mathcal{L}_{\widehat{\theta}_{t-1}^{\mathrm{PPI}}}^{f,\lambda_t}(\theta)$ , and the second inequality holds as we take $\omega = \frac{\theta - G(\widehat{\theta}_{t-1}^{\mathrm{PPI}})}{\|\theta - G(\widehat{\theta}_{t-1}^{\mathrm{PPI}})\|}\varepsilon_0 + G(\widehat{\theta}_{t-1}^{\mathrm{PPI}})$ and using the above result. Thus no $\theta$ such that $\| \theta - G(\widehat{\theta}_{t-1}^{\mathrm{PPI}})\| = \varepsilon_0$ can be the minimizer of $\mathcal{L}_{\widehat{\theta}_{t-1}^{\mathrm{PPI}}}^{f,\lambda_t}(\theta)$ . Then $\| G(\widehat{\theta}_{t-1}^{\mathrm{PPI}}) - \widehat{G}_{\lambda_t}^f(\widehat{\theta}_{t-1}^{\mathrm{PPI}})\| \xrightarrow{P} 0$ . We then have, for a given $T \geqslant 0$ , we have that for all $t \in [T]$ , + +$$ +\left\| \widehat {\theta} _ {t} ^ {\mathrm {P P I}} - \theta_ {t} \right\| \leqslant \sum_ {i = 0} ^ {t} \left(\varepsilon \frac {\beta}{\gamma}\right) ^ {t - i} \| G \left(\widehat {\theta} _ {i} ^ {\mathrm {P P I}}\right) - \widehat {G} _ {\lambda_ {i}} ^ {f} \left(\widehat {\theta} _ {i} ^ {\mathrm {P P I}}\right) \| \xrightarrow {P} 0. +$$ + +Thus, we conclude that $\widehat{\theta}_t^{\mathrm{PPI}}\xrightarrow{P}\theta_t$ + +Theorem A.16 (Central Limit Theorem of $\widehat{\theta}_t^{\mathrm{PPI}}(\lambda_t)$ , Restatement of Theorem 4.1). Under Assumption 3.1, A.4, and A.5, if $\varepsilon < \frac{\gamma}{\beta}$ and $\frac{n}{N} \to r$ for some $r \geqslant 0$ , then for any given $T \geqslant 0$ , we have that for all $t \in [T]$ , + +$$ +\sqrt {n} \left(\widehat {\theta} _ {t} ^ {P P I} \left(\lambda_ {t}\right) - \theta_ {t}\right) \xrightarrow {D} \mathcal {N} \left(0, V _ {t} ^ {P P I} \left(\left\{\lambda_ {j}, \theta_ {j} \right\} _ {j = 1} ^ {t}; r\right)\right) +$$ + +with + +$$ +V _ {t} ^ {P P I} \left(\left\{\lambda_ {j}, \theta_ {j} \right\} _ {j = 1} ^ {t}; r\right) = \sum_ {i = 1} ^ {t} \left[ \prod_ {k = i} ^ {t - 1} \nabla G \left(\theta_ {k}\right) \right] \Sigma_ {\lambda_ {i}, \theta_ {i - 1}} \left(\theta_ {i}; r\right) \left[ \prod_ {k = i} ^ {t - 1} \nabla G \left(\theta_ {k}\right) \right] ^ {\top}. +$$ + +Proof. Let us denote the variance terms by $V_{t}^{\mathrm{PPI}}$ for simplicity, while omitting explicit dependence on parameters in the notation. Let $U_{t} := \sqrt{n} (\hat{\theta}_{t}^{\mathrm{PPI}} - \theta_{t})$ and denote $\tilde{\theta}_t = G(\hat{\theta}_{t - 1}^{\mathrm{PPI}})$ . We make the following decomposition: + +$$ +\widehat {\theta} _ {t} ^ {\mathrm {P P I}} - \theta_ {t} = \underbrace {(\widetilde {\theta} _ {t} - \theta_ {t})} _ {(1)} + \underbrace {(\widehat {\theta} _ {t} ^ {\mathrm {P P I}} - \widetilde {\theta} _ {t})} _ {(2)}. +$$ + +Step 1: Conditional distribution of $U_{t}|U_{t - 1}$ + +For term (1), we have + +$$ +\sqrt {n} \left(\tilde {\theta} _ {t} - \theta_ {t}\right) = \sqrt {n} \left(G \left(\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}\right) - G \left(\theta_ {t - 1}\right)\right). +$$ + +For term (2), the empirical process analysis in [1] establishes that + +$$ +\sqrt {n} \left(\widehat {\theta} _ {t} ^ {\mathrm {P P I}} - \tilde {\theta} _ {t}\right) \mid \widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}} \xrightarrow {D} \mathcal {N} \left(0, \Sigma_ {\lambda_ {t}, \widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} \left(\tilde {\theta} _ {t}; r\right)\right), +$$ + +where the variance is given by + +$$ +\Sigma_ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} \left(\tilde {\theta} _ {t}; r\right) = H _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} \left(\tilde {\theta} _ {t}\right) ^ {- 1} \left(r V _ {\lambda_ {t}, \widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} ^ {f} \left(\tilde {\theta} _ {t}\right) + V _ {\lambda_ {t}, \widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} \left(\tilde {\theta} _ {t}\right)\right) H _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} \left(\tilde {\theta} _ {t}\right) ^ {- 1}. +$$ + +Conditioning on $\widehat{\theta}_{t - 1}^{\mathrm{PPI}}$ , for any function $h$ , we use the following shorthand notations: + +$$ +\mathbb {E} _ {n} h := \frac {1}{n} \sum_ {i = 1} ^ {n} h (x _ {i}, y _ {i}), \quad \mathbb {G} _ {n} h := \sqrt {n} (\mathbb {E} _ {n} h - \mathbb {E} _ {(x, y) \sim \mathcal {D} (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}})} [ h (x, y) ]), +$$ + +$$ +\widehat {\mathbb {E}} _ {N} ^ {f} h := \frac {1}{N} \sum_ {i = 1} ^ {N} h (x _ {i} ^ {u}, f (x _ {i} ^ {u})), \quad \widehat {\mathbb {G}} _ {N} ^ {f} h := \sqrt {N} (\widehat {\mathbb {E}} _ {N} h - \mathbb {E} _ {x \sim \mathcal {D} _ {\mathcal {X}} (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}})} [ h (x, f (x) ]) ], +$$ + +$$ +\widehat {\mathbb {E}} _ {n} ^ {f} h := \frac {1}{n} \sum_ {i = 1} ^ {n} h (x _ {i}, f (x _ {i})), \quad \widehat {\mathbb {G}} _ {n} ^ {f} h := \sqrt {n} (\widehat {\mathbb {E}} _ {n} h - \mathbb {E} _ {x \sim \mathcal {D} _ {\mathcal {X}} (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}})} [ h (x, f (x) ]). +$$ + +Note that $\tilde{\theta}_t = G(\widehat{\theta}_{t - 1}^{\mathrm{PPI}})$ . Recall that + +$$ +\mathcal {L} _ {\tilde {\theta}} (\theta) := \underset {(x, y) \sim \mathcal {D} (\tilde {\theta})} {\mathbb {E}} \ell (x, y; \theta), \quad \mathcal {L} _ {\tilde {\theta}} ^ {f, \lambda} (\theta) := \mathcal {L} _ {\tilde {\theta}, n} (\theta) + \lambda \cdot (\widetilde {\mathcal {L}} _ {\tilde {\theta}, N} ^ {f} (\theta) - \mathcal {L} _ {\tilde {\theta}, n} ^ {f} (\theta)). +$$ + +Under the assumptions, Lemma 19.31 in [31] implies that for every sequence $h_n = O_P(1)$ , we have + +$$ +\mathbb {G} _ {n} \left[ \sqrt {n} \left(\ell (x, y; \tilde {\theta} _ {t} + \frac {h _ {n}}{\sqrt {n}}) - \ell (x, y; \tilde {\theta} _ {t})\right) - h _ {n} ^ {\top} \nabla_ {\theta} \ell (x, y; \tilde {\theta} _ {t}) \right] \stackrel {{P}} {{\to}} 0, +$$ + +$$ +\widehat {\mathbb {G}} _ {N} ^ {f} \left[ \sqrt {n} \left(\ell (x, y; \tilde {\theta} _ {t} + \frac {h _ {n}}{\sqrt {n}}) - \ell (x, y; \tilde {\theta} _ {t})\right) - h _ {n} ^ {\top} \nabla_ {\theta} \ell (x, y; \tilde {\theta} _ {t}) \right] \stackrel {{P}} {{\to}} 0, +$$ + +$$ +\widehat {\mathbb {G}} _ {n} ^ {f} \left[ \sqrt {n} \left(\ell (x, y; \tilde {\theta} _ {t} + \frac {h _ {n}}{\sqrt {n}}) - \ell (x, y; \tilde {\theta} _ {t})\right) - h _ {n} ^ {\top} \nabla_ {\theta} \ell (x, y; \tilde {\theta} _ {t}) \right] \stackrel {{P}} {{\longrightarrow}} 0. +$$ + +Applying second-order Taylor expansion, we obtain that + +$$ +\begin{array}{l} n \mathbb {E} _ {n} \left(\ell (x, y; \tilde {\theta} _ {t} + \frac {h _ {n}}{\sqrt {n}}) - \ell (x, y; \tilde {\theta} _ {t})\right) = n \left(\mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} (\tilde {\theta} _ {t} + \frac {h _ {n}}{\sqrt {n}}) - \mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} (\tilde {\theta} _ {t})\right) + h _ {n} ^ {\top} \mathbb {G} _ {n} \nabla_ {\theta} \ell (x, y; \tilde {\theta} _ {t}) + o _ {p} (1) \\ = \frac {1}{2} h _ {n} ^ {\top} H _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} (\tilde {\theta} _ {t}) h _ {n} + h _ {n} ^ {\top} \mathbb {G} _ {n} \nabla_ {\theta} \ell (x, y; \tilde {\theta} _ {t}) + o _ {p} (1). \\ \end{array} +$$ + +Based on similar calculation of the previous two terms, we can obtain that: + +$$ +\begin{array}{l} n \left(\mathcal {L} _ {\hat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} ^ {f, \lambda} \left(\tilde {\theta} _ {t} + \frac {h _ {n}}{\sqrt {n}}\right) - \mathcal {L} _ {\hat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} ^ {f, \lambda} \left(\tilde {\theta} _ {t}\right)\right) \\ = \frac {1}{2} h _ {n} ^ {\top} H _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {p p i}}} (\widetilde {\theta} _ {t}) h _ {n} + h _ {n} ^ {\top} \left(\mathbb {G} _ {n} + \lambda \sqrt {\frac {n}{N}} \widehat {\mathbb {G}} _ {N} ^ {f} - \lambda \widehat {\mathbb {G}} _ {n} ^ {f}\right) \nabla_ {\theta} \ell (x, y; \widetilde {\theta} _ {t}) + o _ {p} (1). \\ \end{array} +$$ + +By considering $h_n^* = \sqrt{n} (\widehat{\theta}_t^{\mathrm{PPI}} - \tilde{\theta}_t)$ and $h_n = -H_{\widehat{\theta}_{t - 1}^{\mathrm{PPI}}}(\tilde{\theta}_t)^{-1}\left(\mathbb{G}_n + \lambda \sqrt{\frac{n}{N}}\widehat{\mathbb{G}}_N^f -\lambda \widehat{\mathbb{G}}_n^f\right)\nabla_\theta \ell (x,y;\tilde{\theta}_t),$ + +Corollary 5.53 in [31] implies they are $O_P(1)$ and we obtain that + +$$ +n \left(\mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} ^ {f, \lambda} \left(\widehat {\theta} _ {t} ^ {\mathrm {P P I}}\right) - \mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} ^ {f, \lambda} \left(\tilde {\theta} _ {t}\right)\right) = \frac {1}{2} h _ {n} ^ {* \top} H _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} \left(\tilde {\theta} _ {t}\right) h _ {n} ^ {*} + h _ {n} ^ {* \top} \left(\mathbb {G} _ {n} + \lambda \sqrt {\frac {n}{N}} \widehat {\mathbb {G}} _ {N} ^ {f} - \lambda \widehat {\mathbb {G}} _ {n} ^ {f}\right) \nabla_ {\theta} \ell (x, y; \tilde {\theta} _ {t}) + o _ {p} (1) +$$ + +$$ +n \left(\mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} ^ {f, \lambda} (\tilde {\theta} _ {t} + \frac {h _ {n}}{\sqrt {n}}) - \mathcal {L} _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} ^ {f, \lambda} (\tilde {\theta} _ {t})\right) = - \frac {1}{2} h _ {n} ^ {\top} H _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} (\tilde {\theta} _ {t}) h _ {n} + o _ {P} (1). +$$ + +Since $\widehat{\theta}_t^{\mathrm{PPI}}$ is the minimizer of $\mathcal{L}_{\widehat{\theta}_{t - 1}^{\mathrm{PPI}}}^{f,\lambda}$ , the first term is smaller than the second term. We can rearrange the terms and obtain: + +$$ +\frac {1}{2} \left(h _ {n} ^ {*} - h _ {n}\right) ^ {T} H _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {p p i}}} \left(\tilde {\theta} _ {t}\right) \left(h _ {n} ^ {*} - h _ {n}\right) = o _ {P} (1), +$$ + +which leads to $h_n^* - h_n = O_P(1)$ . Then the above asymptotic normality result follows directly by applying the central limit theorem (CLT) to the following terms, conditioning on $\widehat{\theta}_{t-1}^{\mathrm{PPI}}$ : + +$$ +\sqrt {n} (\widehat {\theta} _ {t} ^ {\mathrm {P P I}} - \tilde {\theta} _ {t}) | \widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}} = - H _ {\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} (\tilde {\theta} _ {t}) ^ {- 1} (S _ {1} + S _ {2}) + o _ {P} (1), +$$ + +$$ +S_{1} = \lambda_{t}\sqrt{\frac{n}{N}}\sqrt{\frac{1}{N}}\sum_{i = 1}^{N}\Bigl(\nabla_{\theta}\ell (x_{t,i}^{u},f(x_{t,i}^{u});\tilde{\theta}_{t}) - \underset {x\sim \mathcal{D}_{\mathcal{X}}(\widehat{\theta}_{t - 1}^{\mathrm{PPI}})}{\mathbb{E}}\nabla_{\theta}\ell (x,f(x);\tilde{\theta}_{t})\Bigr), +$$ + +$$ +\begin{array}{l} S _ {2} = \sqrt {\frac {1}{n}} \sum_ {i = 1} ^ {n} \left(\nabla_ {\theta} \ell \left(x _ {t, i}, y _ {t, i}; \tilde {\theta} _ {t}\right) - \lambda_ {t} \nabla_ {\theta} \ell \left(x _ {t, i}, f \left(x _ {t, i}\right); \tilde {\theta} _ {t}\right) \right. \\ \left. - \underset {(x, y) \sim \mathcal {D} (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}})} {\mathbb {E}} \left[ \nabla_ {\theta} \ell (x, y; \tilde {\theta} _ {t}) - \lambda_ {t} \nabla_ {\theta} \ell (x, f (x); \tilde {\theta} _ {t}) \right]\right). \\ \end{array} +$$ + +Note that, conditioning on $\widehat{\theta}_{t - 1}^{\mathrm{PPI}}$ , (1) is a constant. Therefore, (1) and (2) follow a joint Gaussian distribution. Consequently, given $U_{t - 1}$ , the conditional distribution of $U_{t}$ is given by: + +$$ +\begin{array}{l} U _ {t} \left| U _ {t - 1} = \sqrt {n} \left(\hat {\theta} _ {t} ^ {\text {P P I}} - \theta_ {t}\right) \right| \hat {\theta} _ {t - 1} ^ {\text {P P I}} \\ = \sqrt {n} \left(\tilde {\theta} _ {t} - \theta_ {t}\right) + \sqrt {n} \left(\hat {\theta} _ {t} ^ {\mathrm {P P I}} - \tilde {\theta} _ {t}\right) \left| \hat {\theta} _ {t - 1} ^ {\mathrm {P P I}} \right. \\ = \sqrt {n} \left(G \left(\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}\right) - G \left(\theta_ {t - 1}\right)\right) + \sqrt {n} \left(\widehat {\theta} _ {t} ^ {\mathrm {P P I}} - \tilde {\theta} _ {t}\right) \mid \widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}} \\ \end{array} +$$ + +$$ +\begin{array}{l} \xrightarrow {D} \mathcal {N} \left(\sqrt {n} (G (\widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}) - G (\theta_ {t - 1})), \Sigma_ {\lambda_ {t}, \widehat {\theta} _ {t - 1} ^ {\mathrm {P P I}}} (\tilde {\theta} _ {t}; r)\right). \\ = \mathcal {N} \left(\sqrt {n} \left(G \left(\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}\right) - G \left(\theta_ {t - 1}\right)\right), \Sigma_ {\lambda_ {t}, \frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}} \left(G \left(\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}\right); r\right)\right). \\ \end{array} +$$ + +For later references, we denote $U_{t} \mid U_{t-1} \xrightarrow{D} \mathcal{N}(\mu(U_{t-1}), \Sigma(U_{t-1}; r))$ . + +Step 2: Marginal distribution of $U_{t}$ . We calculate the characteristic function of $U_{t}$ by induction. To begin with, we directly have + +$$ +X _ {1} \stackrel {{D}} {{\longrightarrow}} \mathcal {N} (0, V _ {1} ^ {\mathrm {P P I}}), \quad V _ {1} ^ {\mathrm {P P I}} = \Sigma_ {\lambda_ {0}, \theta_ {0}} (\theta_ {1}; r). +$$ + +Now, assume that $U_{t-1} \xrightarrow{D} \mathcal{N}(0, V_{t-1}^{\mathrm{PPI}})$ , we derive the joint distribution of $(U_t, U_{t-1})$ and marginal distribution of $U_t$ . Then we have, the characteristics functions $\phi$ and the probability density function $p$ of distributions $U_{t-1}$ and $U_t \mid U_{t-1}$ follow: + +$$ +\begin{array}{l} \phi_ {U _ {t - 1}} (s) \rightarrow \phi_ {\mathcal {N} (0, V _ {t - 1} ^ {\mathrm {p p i}})} (s) = \exp \left(- \frac {1}{2} s ^ {T} V _ {t - 1} ^ {\mathrm {p p i}} s\right), \quad p _ {U _ {t - 1}} (u) = \frac {1}{(2 \pi) ^ {d}} \int e ^ {- i z ^ {T} u} \phi_ {U _ {t - 1}} (z) d z \\ \phi_ {U _ {t} | U _ {t - 1}} (s) \rightarrow \phi_ {\mathcal {N} (\mu (U _ {t - 1}), \Sigma (U _ {t - 1}; r))} (s) = \exp (i s ^ {T} \mu (U _ {t - 1}) - \frac {1}{2} s ^ {T} \Sigma (U _ {t - 1}; r) s). \\ \end{array} +$$ + +Then according to the proof of vanilla CLT under performativity in Section A.2, we have: + +$$ +\phi_ {U _ {t}} (s) = \frac {1}{(2 \pi) ^ {\frac {d}{2}} \det | V _ {t - 1} ^ {\mathrm {P P I}} |} \int \exp \left(i s ^ {T} \mu (U _ {t - 1}) - \frac {1}{2} s ^ {T} \Sigma (U _ {t - 1}; r) s - \frac {1}{2} u ^ {T} V _ {t - 1} ^ {\mathrm {P P I}} u\right) d u. +$$ + +Apply dominant convergence theorem to $\lim_{n\to \infty}\phi_{U_t}(s)$ , we have: + +$$ +\begin{array}{l} \lim _ {n \to \infty} \phi_ {U _ {t}} (s) = \lim _ {n \to \infty} \frac {1}{(2 \pi) ^ {\frac {d}{2}} \det | V _ {t - 1} ^ {\mathrm {P P I}} |} \int \exp \left(i s ^ {T} \mu (U _ {t - 1}) - \frac {1}{2} s ^ {T} \Sigma (U _ {t - 1}; r) s - \frac {1}{2} u ^ {T} V _ {t - 1} ^ {\mathrm {P P I}} u\right) d u \\ = \lim _ {n \rightarrow \infty} \frac {1}{\left(2 \pi\right) ^ {\frac {d}{2}} \det | V _ {t - 1} ^ {\mathrm {P P I}} |} \int \exp \left(i s ^ {T} \sqrt {n} \left(G \left(\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}\right) - G \left(\theta_ {t - 1}\right)\right)\right. \\ \left. - \frac {1}{2} s ^ {T} \Sigma_ {\lambda_ {t}, \frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}} \left(G \left(\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}\right); r\right) s - \frac {1}{2} u ^ {T} V _ {t - 1} ^ {\mathrm {P P I}} u\right) d u \\ = \frac {1}{\left(2 \pi\right) ^ {\frac {d}{2}} \det | V _ {t - 1} ^ {\mathrm {P P I}} |} \int \exp \left(i s ^ {T} \sqrt {n} \left(G \left(\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}\right) - G \left(\theta_ {t - 1}\right)\right) \right. \\ - \frac {1}{2} s ^ {T} \Sigma_ {\lambda_ {t}, \frac {V _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}} \left(G \left(\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}\right); r\right) s - \frac {1}{2} u ^ {T} V _ {t - 1} ^ {\mathrm {P P I}} u\left. \right) d u \\ = \frac {1}{(2 \pi) ^ {\frac {d}{2}} \det | V _ {t - 1} ^ {\mathrm {P P I}} |} \int \exp \left(i s ^ {T} \nabla G (\theta_ {t - 1}) u - \frac {1}{2} s ^ {T} \Sigma_ {\lambda_ {t}, \theta_ {t - 1}} (G (\theta_ {t - 1}); r) s - \frac {1}{2} u ^ {T} V _ {t - 1} ^ {\mathrm {P P I}} u\right) d u \\ = \exp \Bigl (- \frac {1}{2} s ^ {T} \nabla G (\theta_ {t - 1}) V _ {t + 1} ^ {\mathrm {P P I}} \nabla G (\theta_ {t - 1}) ^ {T} s - \frac {1}{2} s ^ {T} \Sigma_ {\lambda_ {t}, \theta_ {t - 1}} (\theta_ {t}; r) s \Bigr), \\ \end{array} +$$ + +which is the characteristic function of $\mathcal{N}(0,V_t^{\mathrm{PPI}})$ , where $V_{t}^{\mathrm{PPI}} = \nabla G(\theta_{t - 1})V_{t - 1}^{\mathrm{PPI}}\nabla G(\theta_{t - 1})^{\top} + \Sigma_{\lambda_t,\theta_{t - 1}}(\theta_t;r)$ . Here we use the fact that $\lim_{n\to \infty}\sqrt{n}\left(G(\frac{y}{\sqrt{n}} +\theta_{t - 1}) - G(\theta_{t - 1})\right) = \nabla G(\theta_{t - 1})y$ and the dominant convergence theorem holds as we have + +$$ +\begin{array}{l} | \exp (i s ^ {T} \sqrt {n} (G (\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}) - G (\theta_ {t - 1})) - \frac {1}{2} s ^ {T} \Sigma_ {\lambda_ {t}, \frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}} (G (\frac {U _ {t - 1}}{\sqrt {n}} + \theta_ {t - 1}); r) s - \frac {1}{2} u ^ {T} V _ {t - 1} ^ {\mathrm {P P I}} u) | \\ \leqslant | \exp (- \frac {1}{2} u ^ {T} V _ {t - 1} ^ {\mathrm {P P I}} u) |. \\ \end{array} +$$ + +Thus we conclude by induction that + +$$ +\begin{array}{l} U _ {t} \stackrel {{D}} {{\to}} \mathcal {N} \left(0, V _ {t} ^ {\mathrm {P P I}}\right), \\ V _ {t} ^ {\mathrm {P P I}} = \sum_ {i = 1} ^ {t} \left[ \prod_ {k = i} ^ {t - 1} \nabla G \left(\theta_ {k}\right) \right] \Sigma_ {\lambda_ {i - 1}, \theta_ {i - 1}} \left(\theta_ {i}; r\right) \left[ \prod_ {k = i} ^ {t - 1} \nabla G \left(\theta_ {k}\right) \right] ^ {\top}. \\ \end{array} +$$ + +And we have: + +$$ +\begin{array}{l} \nabla G \left(\theta_ {k}\right) = - \left[ \underset {(x, y) \sim \mathcal {D} \left(\theta_ {k}\right)} {\mathbb {E}} \nabla_ {\theta} ^ {2} \ell \left(x, y; \theta_ {k + 1}\right) \right] ^ {- 1} \left(\nabla_ {\tilde {\theta} _ {(x, y) \sim \mathcal {D} \left(\theta_ {k}\right)}} \nabla_ {\theta} \ell \left(x, y; \theta_ {k + 1}\right)\right) \\ = - H _ {\theta_ {k}} \left(\theta_ {k + 1}\right) ^ {- 1} \left(\nabla \nabla_ {\theta} \ell (x, y; \theta_ {k + 1})\right) \\ = - H _ {\theta_ {k}} (\theta_ {k + 1}) ^ {- 1} \mathbb {E} _ {z \sim \mathcal {D} (\theta_ {k})} [ \nabla_ {\theta} \ell (z, \theta_ {k + 1}) \nabla_ {\theta} \log p (z, \theta_ {k}) ^ {\top} ]. \\ \end{array} +$$ + +# B Experimental Details + +# B.1 Additional Experimental Details + +As described in Section 5, we construct simulation studies on a performative linear regression problem, where data are sampled from $D(\theta)$ as + +$$ +y = \alpha^ {\top} x + \mu^ {\top} \theta + \nu , x \sim \mathcal {N} (\mu_ {x}, \Sigma_ {x}), \nu \sim \mathcal {N} (0, \sigma_ {y} ^ {2}). +$$ + +At each time step $t$ , the label $y_{t}$ is updated with $\widehat{\theta}_{t-1}$ via the above equation, and then $\widehat{\theta}_{t}$ is obtained by empirical repeated risk minimization with the updated data $z_{t} = (x_{t}, y_{t})$ . The objective of this task is to provide inference on an unbiased $\widehat{\theta}_{t}$ with low variance, that is, the ground-truth $\theta_{t}$ is covered by the confidence region of $\widehat{\theta}_{t}$ with high probability, and the width of this confidence region is small. + +Given a set of labeled data, we can obtain the underlying $\theta_{t}$ as + +$$ +\theta_ {t} = \left(\Sigma_ {x} + \mu_ {x} \mu_ {x} ^ {\top} + \gamma I _ {d}\right) ^ {- 1} \left(\mu_ {x} \mu^ {\top} \theta_ {t - 1} + \left(\Sigma_ {x} + \mu_ {x} \mu_ {x} ^ {\top}\right) \alpha\right). \tag {7} +$$ + +To compute the coverage and width of a confidence region $\widehat{\mathcal{R}}_t(n,\delta)$ for $\theta_{t}$ , we run 1000 independent trials. For each trial $j$ , we sample $\widehat{\theta}_{t,j}$ together with with its estimated variance $\widehat{V}_{t,j}$ , and construct two-sided normal intervals for each coordinate $i = 1,\ldots ,d$ : + +$$ +\left[ \widehat {\theta} _ {t, j} ^ {(i)} \pm q _ {1 - \frac {\delta}{2 d}} \sqrt {\widehat {V} _ {t , j} ^ {(i)} / n} \right], \quad q _ {1 - \frac {\delta}{2 d}} = \Phi^ {- 1} (1 - \frac {\delta}{2 d}), +$$ + +where $d = 2$ is the parameter dimension, $\delta = 0.1$ the significance level, $n$ the data size, and $\Phi^{-1}$ the standard normal quantile. The interval width of each trial is averaged over $d$ coordinate intervals, and we count this trial as covered if the ground-truth $\theta_t$ lies inside all $d$ coordinate intervals simultaneously. Finally, we report the average width and coverage rate over all trials. + +Similarly, to compute the coverage for performative stable point $\theta_{\mathrm{PS}}$ , we can obtain the close-form $\theta_{\mathrm{PS}}$ for this task as follows: + +$$ +\theta_ {\mathrm {P S}} = \left(\Sigma_ {x} + \mu_ {x} \mu_ {x} ^ {\top} - \mu_ {x} \mu^ {\top} + \gamma I _ {d}\right) ^ {- 1} \left(\Sigma_ {x} + \mu_ {x} \mu_ {x} ^ {\top}\right). \tag {8} +$$ + +As defined in Corollary 3.7, the confidence region for $\theta_{\mathrm{PS}}$ is constructed with + +$$ +\widehat {\mathcal {R}} _ {t} (n, \delta) + \mathcal {B} \Big (0, 2 B \big (\frac {\varepsilon \beta}{\gamma} \big) ^ {t} \Big), +$$ + +where $\varepsilon = \| \mu \| _2$ , $B = \| \theta_0 - \theta_{\mathrm{PS}}\| _2$ , and + +$$ +\beta = \max \left\{\max _ {x \in \mathcal {X}} \{\| x \| _ {2} ^ {2} + \gamma \}, \max _ {(x, y) \in (\mathcal {X}, \mathcal {Y}), \theta \in \Theta} \{\sqrt {(x ^ {\top} \theta - y + \| x \| _ {2} \| \theta \| _ {2}) ^ {2} + \| x \| _ {2} ^ {2}} \} \right\}. +$$ + +Here we take $\mathcal{X} = \{x: \| x\|_2^2 \leqslant 20\}$ . Note that the closed form expressions for the update and the performatively stable point in Eq. 7 and Eq. 8 hold for any distribution of $x$ with mean $\mu_x$ and variance $\Sigma_x$ , and $\nu$ with mean 0 and variance $\sigma_y^2$ . For easier calculation for the smoothness parameter, we truncate the normal distribution of $(x,y)$ such that $\| x\|_2^2 \leqslant 20$ . The mean and variance of the resulting truncated distribution can be well approximated by those of the original normal distribution due to the concentration of Gaussian. + +We run our experiments on NVIDIA GPUs A100 in a single-GPU setup. + +# B.2 Additional Experimental Results + +Ablation study on effects of $\gamma$ . In Figure A1, we compare confidence-region performance under regularization strengths $\gamma = 1$ and $\gamma = 3$ . Together with results of $\gamma = 2$ in Figure 1, we can find that as $\gamma$ increases, the gap between the coverage for $\theta_{t}$ (solid curve) and the bias-adjusted coverage for $\theta_{\mathrm{PS}}$ (dashed curve) vanishes more quickly across iterations. For example, at $t = 3$ , the dashed and solid curves are tightly closed for $\gamma = 3$ , while a substantial gap remains for $\gamma = 1$ . This phenomenon derives that the larger $\gamma$ yields a more strongly convex loss, which both accelerates convergence of the estimate $\widehat{\theta}_{t}$ to its stable point and reduces the performative bias $\| \theta_t - \theta_{\mathrm{PS}}\|$ . Consequently, the bias-aware intervals converge for $\theta_{\mathrm{PS}}$ to the original ones for $\theta_{t}$ in fewer iterations when $\gamma$ is larger. + +Ablation study on effects of $\varepsilon$ . In Figure A2, we compare confidence-region performance under sensitivity $\varepsilon \approx 0.003$ and $\varepsilon \approx 0.03$ . We can find that as $\varepsilon$ increases, the gap between the coverage for $\theta_{t}$ (solid curve) and the bias-adjusted coverage for $\theta_{\mathrm{PS}}$ (dashed curve) vanishes more slowly across iterations. For example, for $\varepsilon \approx 0.003$ , the dashed curves tightly upper-bound the solid curves at $t = 3$ , whereas for $\varepsilon \approx 0.03$ , a noticeable gap persists even at $t = 5$ . This behavior is because a higher $\varepsilon$ amplifies the performative shift (the dependence of the label distribution on $\theta$ ), which increases the performative bias. That is, stronger sensitivity requires more iterations for $\widehat{\theta}_{t}$ to approach its stable point, slowing down convergence of the two confidence regions. + +Ablation study on effects of $\sigma_y^2$ . In Figure A4, we compare confidence-region performance under noise level $\sigma_y^2 = 0.1$ and $\sigma_y^2 = 0.4$ . We observe that across all settings, PPI with our greedy-selected $\widehat{\lambda}$ is essentially never worse than either baseline $\lambda = 1$ or $\lambda = 0$ . When the noise is low ( $\sigma_y^2 = 0.1$ ), greedy $\widehat{\lambda}$ behaves similarly to $\lambda = 0$ , placing almost all weight on the true labels. Conversely, when the noise is high ( $\sigma_y^2 = 0.4$ ), greedy $\widehat{\lambda}$ behaves like $\lambda = 1$ , relying more heavily on pseudo-labels to reduce variance. For the intermediate noise level $\sigma_y^2 = 0.2$ in Figure 1, greedy $\widehat{\lambda}$ significantly outperforms both baselines by hitting the optimal bias-variance balance. + +# B.3 Case Study on Semi-synthetic Dataset + +In Section 5, we originally consider experiments on a synthetic dataset because the performative prediction is an on-policy setting, which means we need to collect the corresponding data every time we update the parameter (policy). In all the previous literature on performative prediction, no such dataset is provided. Alternatively, previous work always uses a semi-synthetic dataset, which one will need to specify how the data distribution will react and shift according to the new policy. + +Following Perdomo et al. [25], we further conduct a case study in a semi-synthetic way on a realistic credit scoring task using a Kaggle dataset $^4$ . The dataset contains features of individuals and a binary label indicating whether a loan should be granted or not. Consider the setting where a bank uses a logistic regression classifier $\theta$ trained on features of loan applicants to predict their creditworthiness, while the individual applicants respond to this classifier by manipulating their features to induce a positive classification. Following [2], we can formulate this task as performative prediction because applicants' feature distribution $\mathcal{D}(\theta)$ is strategically adapted in response to $\theta$ . By applying repeated risk minimization, a performative stable point $\theta_{PS}$ can be achieved. + +We treat the data points in the original dataset as the true distribution to compute $\theta_{PS}$ . We add Gaussian noise to the original data feature to generate an unlabeled set of the same size. Then, we sample varying $n$ labeled points with $N = 18000$ unlabeled points and perform $t = 5$ repeated risk minimization steps to compute the estimated $\widehat{\theta}_t$ and build the confidence region for it over 100 independent trials. From the experimental results, we find that a coverage of 0.9 is achieved with decreasing width as $n$ increases. Notably, our optimized greedy $\widehat{\lambda}_t$ (orange) achieves the highest coverage and narrowest confidence width compared with when $\lambda$ is fixed to 0 or 1. The results support our proposed theory and strengthen the practical significance of our methods. We hope this case study can inspire future work for practical settings of PPI under performativity. + +![](images/fd3c1d23f2fecf2a87b6ba2ab7e3caf9c7aa09329485c497a3ae05acaf0de01e.jpg) + +![](images/ba39aea9aa9a29ffe306262c90df9bc05061a3fa1fdf5b3e380c799275a8d583.jpg) + +![](images/007bec36025a0bb4ec06b192ae13400c6cf4afb6bafd3722b384dc4950f4911c.jpg) + +![](images/10c97c55f457d259b5e04130ac0498cfa899bc564269c5416db1239209ddfd26.jpg) +(a) Confidence-region coverage and width with $\gamma = 1$ + +![](images/fda8e0f55d660492f16e7d721c00f414b3ec90f8330e4d26b5c0a7c62878ecfd.jpg) +(b) Confidence-region coverage and width with $\gamma = 3$ + +![](images/386459625c096be2e0fa334f237587d7815f44bed770a31e1ad2195334ac8402.jpg) +Figure A1: Confidence-region coverage (top row) and width (bottom row) with different choices of $\lambda$ . The setup is the same as in Figure 1, only we change $\gamma = 1$ or $\gamma = 3$ . + +![](images/80838fa5085b05f93db8906707893508ec891e62c9bd0e48c85cc1b76cb30f66.jpg) + +![](images/0d46af935f28e821f467fc989f987842053b4f5942218b7c44925d1cc777d913.jpg) + +![](images/9103478f2e18e6d5308a6f5ed78d12c4837e45e4da790b2bb266febe50f3829f.jpg) + +![](images/2b2f6a491c59392b9a8d4ee16257ed5d8d4ad9b22fccd6951df4c483787b7f70.jpg) +(a) Confidence-region coverage and width with $\varepsilon \approx 0.003$ + +![](images/b39bb1951f9af4bbe1526a9666c24ae45f7878f174947589d6d260683be3c512.jpg) +(b) Confidence-region coverage and width with $\varepsilon \approx 0.03$ + +![](images/d5a5f743fdd9a2be336b0bdcba69e42867535d06ac80b7711115d6173cf7c1a0.jpg) +Figure A2: Confidence-region coverage (top row) and width (bottom row) with different choices of $\lambda$ . The setup is the same as in Figure 1, only we change $\varepsilon \approx 0.003$ or $\varepsilon \approx 0.03$ . + +![](images/57ff613642a9fee2ce7a07b7309347189dd1a3e405c7901c373cd176a65a842b.jpg) + +![](images/034fe8de4fbc54ab961f2c5ae66dded4425824af5215a9623ad5aec3dc5acc48.jpg) + +![](images/973e30226f5dffafc2a7d0325965e21c1224c6f75bede7383ec837bce8d68acd.jpg) + +![](images/23a6459567d9c12c783fe12e599d4e54d3b6ab676311a7b3c00d8a4c3632825c.jpg) + +![](images/61199017f303f50a0a9142c041043590e625e50243b601210bd9a3231092526f.jpg) + +![](images/561a0683f89d5d7e547061e8646393366258f31a0ee2664fd4e6528e17b8ff3e.jpg) + +![](images/9c941da3248841c2d7b65cafb82fb900ab54014530b227cdf0a5cccb39d48db5.jpg) +(a) Confidence-region coverage and width with $\sigma_y^2 = 0.1$ + +![](images/b2b71b3352c0eaed355b6e780f8cb25b8e423b3a31695357b25fb3cfd3ee8001.jpg) +(b) Confidence-region coverage and width with $\sigma_y^2 = 0.4$ + +![](images/b55f62bfb8edd7e4f2e75edb62b0d15d491f10b993f7288534bb7836d7b586de.jpg) + +![](images/104f8c3b3a2328c5df4a2cf5e9dc587545a5668f278e94ea174745fd9b73c41b.jpg) +Figure A3: Confidence-region coverage (top row) and width (bottom row) with different choices of $\lambda$ . The setup is the same as in Figure 1, only we change $\sigma_y^2 = 0.1$ or $\sigma_y^2 = 0.4$ . + +![](images/9493c1de2cf02e2aba5d9d67a31c34f2433e6be98e6e7b039f9656f11bb26561.jpg) +Figure A4: Confidence-region coverage and width for $\theta_t$ ( $t = 5$ ) with different choices of $\lambda$ on the semi-synthetic Kaggle credit scoring dataset. + +![](images/05c4b67cf1f0f054264e3313a178a3c9b81cee0e2994e35b34cfad15358a3723.jpg) + +# NeurIPS Paper Checklist + +The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit. + +Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist: + +- You should answer [Yes], [No], or [NA]. +- [NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available. +- Please provide a short (1–2 sentence) justification right after your answer (even for NA). + +The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper. + +The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes]" is generally preferable to "[No]", it is perfectly acceptable to answer "[No]" provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No]" or "[NA]" is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found. + +IMPORTANT, please: + +- Delete this instruction block, but keep the section heading "NeurIPS Paper Checklist", +- Keep the checklist subsection headings, questions/answers and guidelines below. +- Do not modify the questions and only use the provided macros for your answers. + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The main claims in abstract and introduction are included in our paper. Specifically, our main theoretical results on CLT is included in Section 3. The results with prediction-powered inference is included in Section 4. We further demonstrate our numerical experiments in Section 5. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +# Answer: [Yes] + +Justification: Currently, our framework uses bias-aware inference for $\theta_{\mathrm{PS}}$ rather than a direct inference method. We hope that in the future, we could Obtain direct inference methods for $\theta_{\mathrm{PS}}$ . + +# Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. + +# 3. Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +# Answer: [Yes] + +Justification: We have listed our necessary assumptions in Assumption 3.1 and in Section A.1 in Appendix A. 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In this paper, we show that statistical parity can be enforced efficiently in the contextual bandit setting while retaining strong performance guarantees. Specifically, we present a meta-algorithm that transforms any efficient implementation of Hedge (or, equivalently, any discrete Bayesian inference algorithm) into an efficient contextual bandit algorithm that guarantees exact statistical parity on every trial. Compared to any comparator that satisfies the same statistical parity constraint, the algorithm achieves the same asymptotic regret bound as running the equivalent instance of Exp4 for each group. We also address the scenario where the target parity distribution is unknown and must be estimated online. Finally, using online-to-batch conversion, we extend our approach to the batch classification setting - achieving exact statistical parity whilst attaining excellent generalisation bounds. + +# 1 Introduction + +Statistical parity [12] is a foundational concept in algorithmic fairness and privacy. It imposes a constraint on how decisions should be distributed across individuals with different values of a protected characteristic. Formally, let $\mathcal{C}$ and $\mathcal{V}$ denote the sets of protected characteristics and non-protected features (known as contexts) respectively, and let $\rho$ be a probability distribution over $\mathcal{C} \times \mathcal{V}$ . A policy mapping $\mathcal{C} \times \mathcal{V}$ to distributions over a set of actions is said to satisfy statistical parity with respect to $\rho$ if and only if, when a pair $(c, v)$ is drawn from $\rho$ and then an action $a$ is sampled from the policy, the selected action $a$ is independent of the protected characteristic $c$ . That is, for all actions $b$ and all protected characteristics $d, d' \in \mathcal{C}$ we have: + +$$ +\mathbb {P} [ a = b \mid c = d ] = \mathbb {P} [ a = b \mid c = d ^ {\prime} ] +$$ + +We now provide two illustrative examples motivating the enforcement of statistical parity. The first concerns stop-and-search procedures, in which law enforcement officers select individuals from the public to search for prohibited items. This practice has been the subject of substantial controversy due to evidence of racial bias - with individuals from certain ethnic backgrounds being many times more likely to be stopped than others. In this setting, there are two possible actions: stop and do not stop, and the protected characteristic corresponds to an individual's ethnicity. Enforcing statistical parity in this context ensures that the probability of being stopped is equal across all ethnic groups, thereby eliminating disparities attributable to racial bias. + +The second example arises in the context of cyber defence. Consider a computer system that must be protected using a defensive strategy, often referred to as a blue agent. The choice of blue agent depends on certain private attributes of the system, one of which is particularly sensitive. Through interaction with the system, external users may be able to infer which blue agent was selected, thereby gaining information about the underlying attributes. In this case, the actions correspond to the chosen blue agents, and the protected characteristics are the possible values of the sensitive attribute. + +Enforcing statistical parity ensures that the selected blue agent is independent of the sensitive attribute, thereby preventing users from inferring anything about it. + +In this paper, we primarily study the contextual bandit problem, where learning unfolds over a sequence of trials in an online setting. On each trial, the learner selects a policy, then observes an instance from the set $\mathcal{C} \times \mathcal{V}$ , and samples an action according to the chosen policy. After the action is taken, the learner receives the loss (or, equivalently, a reward) corresponding to that action on that trial. The goal is to minimise the total cumulative loss incurred over all trials. The algorithm's performance is measured via regret, defined as the difference between the expected cumulative loss of the algorithm and that of a fixed comparator policy (or, in some cases, a sequence of comparator policies) in hindsight. We focus mainly on the adversarial setting, in which the only assumption is that the losses lie in $[0,1]$ , and there are no restrictions on how instances or losses are generated. Additionally, we consider cases where the context space exhibits structure, introducing inductive biases to exploit this structure. + +To the best of our knowledge, no prior work has successfully developed an efficient algorithm for the contextual bandit problem that ensures statistical parity in the selected policies, even in the simplified setting of a finite, unstructured context space with i.i.d. instances and losses. This paper addresses that gap. We introduce SPEw (Statistical Parity with Exponential Weights), an efficient algorithm that achieves statistical parity while matching the asymptotic regret guarantees of the classic Exp4 algorithm [2]. Like EXP4, SPEw is a meta-algorithm that leverages any efficient instantiation of HEDGE [13] adapted to the structure of the context space. + +As an example, we analyse the regret bound of SPEw for when the contexts are the set of vertices of a tree (noting that graphs and finite metric spaces reduce to trees). In addition to guaranteeing statistical parity with respect to a given parity distribution, SPEw also extends to settings where the parity distribution is unknown and must be estimated from the data seen so far. Specifically, we show how to enforce statistical parity with respect to the empirical distribution of observed instances, as well as how to maintain statistical parity approximately when the context space is hierarchically clustered (such as graphs or Euclidean spaces) and the instances are drawn i.i.d. from the parity distribution. + +Beyond the contextual bandit setting, SPEw can also be applied to the batch classification problem under statistical parity constraints via standard online-to-batch conversion techniques [9]. We believe this yields novel and significant results for the batch fairness and privacy literature. Moreover, the underlying methodology of SPEw is not necessarily limited to HEDGE-based implementations; it should extend naturally to other exponentiated gradient algorithms, and potentially even to broader classes of gradient-based methods. For instance, our approach should be able to adapt to the CBA algorithm [26], enabling the incorporation of confidence-rated expert advice, which further broadens the applicability of our framework. + +We now explain why a direct modification of EXP4 to incorporate statistical parity constraints is problematic. EXP4 modifies the inputs to HEDGE: an algorithm that performs mirror descent over the probability simplex of experts (i.e. deterministic policies). In principle, one could attempt to incorporate statistical parity by adding the constraint directly into the mirror descent framework. This would require performing a relative entropy projection onto the constrained set. To compute such a projection, we construct the Lagrangian and differentiate it to obtain necessary conditions for optimality. While expressing the primal variables in terms of the Lagrange multipliers is straightforward, solving for the multipliers themselves appears analytically intractable. Although the projection problem is convex and thus admits numerical solutions, known approximation methods are, as far as we are aware, computationally infeasible in this context due to the exponential size of the expert space. Even in the special case where there is no structure to the context set - effectively removing the inductive bias and reducing the problem to polynomial dimension - existing approximation methods remain considerably less efficient than our approach. In contrast, SPEW avoids these issues entirely by not modifying HEDGE itself but rather its inputs and outputs. + +Much research has focused on designing efficient algorithms for online convex optimization that, unlike SPEW, allow constraint violations but ensure bounded cumulative constraint violation. Notably, the work [8] extends to the bandit setting and achieves a cumulative constraint violation of $\mathcal{O}(T^{3/4})$ (suppressing factors independent of $T$ ), albeit at the cost of an additional multiplicative term of $\mathcal{O}(M^{1/2}T^{1/4})$ to the regret. The work [33] attains a tighter constraint violation bound of $\mathcal{O}(T^{1/2})$ (when the constraint set lacks an interior point, as for statistical parity) but their approach is + +incompatible with the bandit setting and incurs polynomial dependence on the problem dimension (which is exponential in our work) in both the regret and constraint violation bounds. + +We provide a literature review in Appendix A. + +# 2 Notation + +In this section we introduce the notation used in this paper. We define $\mathbb{R}_+$ to be the set of non-negative real numbers. We define $\mathbb{N}$ to be the set of natural numbers excluding 0. Given $z\in \mathbb{N}$ we define: + +$$ +[ z ] := \left\{z ^ {\prime} \in \mathbb {N} \mid z ^ {\prime} \leq z \right\} +$$ + +Given some finite set $\mathcal{B}$ we define $\Delta_{\mathcal{B}}$ to be the set of probability distributions over $\mathcal{B}$ . That is, $\Delta_{\mathcal{B}}$ is the set of all $\rho : \mathcal{B} \to [0,1]$ with: + +$$ +\sum_ {z \in \mathcal {B}} \rho (z) = 1 +$$ + +We call the set $\Delta_{\mathcal{B}}$ a simplex. Given sets $\mathcal{B}$ and $\mathcal{B}'$ and a function $f$ with domain $\mathcal{B} \times \mathcal{B}'$ we define, for all $z \in \mathcal{B}$ , the function $f(z, \circ)$ to be the function that maps each $z' \in \mathcal{B}'$ to $f(z, z')$ . Given a predicate $P$ we define $[\mathbb{P}] := 1$ if $P$ is true and $[\mathbb{P}] := 0$ otherwise. + +# 3 Problem Description + +In this paper we consider the following problem, known as the contextual bandit problem. We have a set of contexts $\mathcal{V}$ which typically has some structure associated with it. For instance, $\mathcal{V}$ could be the set of vertices of a graph or points in Euclidean space. When $\mathcal{V}$ is finite we define $N := |\mathcal{V}|$ . We also have finite sets $\mathcal{C}$ and $\mathcal{A}$ of protected characteristics and actions respectively. Let $M := |\mathcal{C}|$ and $K := |\mathcal{A}|$ . Define the set of policies as: + +$$ +\mathcal {Q} := \left\{\pi \in [ 0, 1 ] ^ {\mathcal {C} \times \mathcal {V} \times \mathcal {A}} \mid \forall (c, v) \in \mathcal {C} \times \mathcal {V}, \pi (c, v, \circ) \in \Delta_ {\mathcal {A}} \right\} +$$ + +A-priori, our adversary chooses a sequence: + +$$ +\left\langle \left(c _ {t}, v _ {t}, \ell_ {t}\right) \mid t \in [ T ] \right\rangle +$$ + +where for all $t \in [T]$ we have $c_t \in \mathcal{C}$ , $v_t \in \mathcal{V}$ and $\ell_t \in [0,1]^\mathcal{A}$ . This sequence is not revealed to us. The problem proceeds in $T$ trials where on trial $t$ the following happens. + +1. We implicitly select a policy $\pi_t^* \in \mathcal{Q}$ . +2. The pair $(c_t, v_t)$ is revealed to us. +3. We draw an action $a_{t}$ from $\pi_t^* (c_t,v_t,\circ)$ +4. The loss $\ell_t(a_t)$ is revealed to us. + +Our aim is to minimise the cumulative loss: + +$$ +\sum_ {t \in [ T ]} \ell_ {t} (a _ {t}) +$$ + +In this paper we consider the enforcement of statistical parity in our policy selections. Formally, a policy $\pi \in \mathcal{Q}$ has statistical parity with respect to a probability distribution $\rho \in \Delta_{\mathcal{C} \times \mathcal{V}}$ if and only if when a pair $(c, v)$ is drawn from $\rho$ and then an action $a$ is drawn from $\pi(c, v, \circ)$ we have that: + +$$ +\mathbb {P} [ a = b \mid c = d ] = \mathbb {P} [ a = b \mid c = d ^ {\prime} ] +$$ + +for all $b \in \mathcal{A}$ and $d, d' \in \mathcal{C}$ . In other words, the selected action $a$ is independent of the protected characteristic $c$ . Note that when $\mathbb{P}[c = d] = 0$ then we allow $\mathbb{P}[a = b \mid c = d]$ to be arbitrary in the above equation. + +Given we have knowledge of a distribution $\rho \in \Delta_{\mathcal{C} \times \mathcal{V}}$ , our meta-algorithm SPEw will play in such a way that each policy $\pi_t^*$ has statistical parity with respect to $\rho$ . We note that the contexts $v_t$ selected by our adversary need not be drawn from $\rho$ . Specifically, given any efficient implementation of an instance of the classic HEDGE algorithm, SPEw will convert it into an efficient algorithm that will + +guarantee statistical parity whilst also guaranteeing that the bound on the difference (a.k.a. the regret) between its cumulative loss and that which would have been incurred by any constant policy with statistical parity is that of the equivalent instance of Exp4. + +As we shall see, SPEw can also work when we have no knowledge of $\rho$ , and instead must estimate it from the data seen so far. SPEw can also be applied to cases in which we want to compare against a dynamic policy sequence rather than a constant policy. + +# 4 The Algorithm + +# 4.1 Hedge + +Here we describe how an instance of HEDGE works. We have a finite set $\mathcal{X}$ of virtual contexts which is defined by the specific instance of HEDGE. On each trial $t$ we have a non-empty set $\mathcal{G}_t^\prime \subseteq \mathcal{X}$ of the virtual contexts that are relevant for that trial, as well as a function $\chi_t^\prime :\mathcal{V}\to \mathcal{G}_t^\prime$ that will map, on trial $t$ , the given context $v_{t}$ to a virtual context. These mathematical objects, which are defined by the specific instance of HEDGE, may be dependent on characteristic/context pairs seen before trial $t$ . In many cases we will have $\mathcal{X} = \mathcal{V}$ , $\mathcal{G}_t^\prime = \mathcal{X}$ and $\chi_t^\prime (v) = v$ for all $t\in [T]$ and $v\in \mathcal{V}$ . We also have a learning rate $\hat{\eta} >0$ . + +We define the set of experts as: + +$$ +\mathcal {H} := \mathcal {A} ^ {\mathcal {X}} +$$ + +so that each expert associates an action with each virtual context. The inductive bias of the specific instance of HEDGE is given by a probability distribution $\vartheta \in \Delta_{\mathcal{H}}$ over the experts. HEDGE implicitly maintains a distribution $\vartheta' \in \Delta_{\mathcal{H}}$ initialised equal to $\vartheta$ . HEDGE has the following two subroutines: + +- When run on trial $t$ the subroutine QUERY returns a function $\varphi : \mathcal{G}_t' \times \mathcal{A} \to [0,1]$ defined by: + +$$ +\varphi (x, a) := \sum_ {e \in \mathcal {H}} [ [ e (x) = a ] ] v ^ {\prime} (e) \quad \forall (x, a) \in \mathcal {G} _ {t} ^ {\prime} \times \mathcal {A} +$$ + +- When run on trial $t$ the subroutine UPDATE takes, as input, a pair $(x,\ell)\in \mathcal{G}_t^\prime \times \mathbb{R}^A$ and then, for all $e\in \mathcal{H}$ , implicitly sets: + +$$ +\vartheta^ {\prime} (e) \leftarrow \vartheta^ {\prime} (e) \exp (- \hat {\eta} \ell (e (x))) +$$ + +and finally implicitly normalises $\vartheta'$ . + +Note that if HEDGE was run explicitly then these subroutines would take exponential time. However, given a compatible inductive bias $\vartheta$ , algorithms for Bayesian inference (such as BELIEFPROPAGATION [28]) can compute these subroutines efficiently. Such algorithms typically require only $\mathcal{O}(K)$ time for the UPDATE subroutine as the main computation is performed during QUERY. + +SPEW maintains an instance of HEDGE (a.k.a. the base algorithm) for each protected characteristic. For simplicity, we assume each of these instances of HEDGE shares the same set $\mathcal{X}$ and inductive bias $\vartheta$ although this is not actually necessary. For each $c\in \mathcal{C}$ and $t\in [T]$ define $\mathcal{G}_t(c)$ and $\chi_t(c,\circ)$ to be the mathematical objects $\mathcal{G}_t^\prime$ and $\chi_t^\prime$ for the instance of HEDGE associated with protected characteristic $c$ . In addition, let QUERY[c] and UPDATE[c] be the subroutines QUERY and UPDATE for the instance of HEDGE associated with protected characteristic $c$ . + +# 4.2 Input and output + +We now describe the input to SPEw, on each trial $t \in [T]$ , and the properties of the action $a_{t}$ that it selects. To do this we first define the virtual policy space: + +$$ +\mathcal {P} := \left\{\pi \in [ 0, 1 ] ^ {\mathcal {C} \times \mathcal {X} \times \mathcal {A}} \mid \forall (c, x) \in \mathcal {C} \times \mathcal {X}, \pi (c, x, \circ) \in \Delta_ {\mathcal {A}} \right\} +$$ + +and the target space: + +$$ +\mathcal {D} := \left\{\mu \in [ 0, 1 ] ^ {\mathcal {C} \times \mathcal {X}} \mid \forall c \in \mathcal {C}, \mu (c, \circ) \in \Delta_ {\mathcal {X}} \right\} +$$ + +Finally, for any target $\mu \in \mathcal{D}$ , we define $\mathcal{F}(\mu)$ to be the set of all virtual policies $\pi \in \mathcal{P}$ in which for all $c, c' \in \mathcal{C}$ and $a \in \mathcal{A}$ we have: + +$$ +\sum_ {x \in \mathcal {X}} \mu (c, x) \pi (c, x, a) = \sum_ {x \in \mathcal {X}} \mu \left(c ^ {\prime}, x\right) \pi \left(c ^ {\prime}, x, a\right) +$$ + +On each trial $t\in [T]$ the following happens: + +1. A target $\mu_t \in \mathcal{D}$ is created based on the characteristic/context pairs seen so far and the problem at hand. This target has the property that for all $(c, x) \in \mathcal{C} \times \mathcal{X}$ with $x \notin \mathcal{G}_t(c)$ we have $\mu_t(c, x) = 0$ . +2. SPEW implicitly determines a virtual policy $\pi_t \in \mathcal{F}(\mu_t)$ . This virtual policy equates to the policy $\pi_t^* \in \mathcal{Q}$ by $\pi_t^*(c, v, \circ) \coloneqq \pi_t(c, \chi_t(c, v), \circ)$ for all $(c, v) \in \mathcal{C} \times \mathcal{V}$ . +3. $(c_{t},v_{t})$ is revealed. +4. We define $x_{t} \coloneqq \chi_{t}(c_{t}, v_{t})$ . +5. SPEW samples the action $a_{t}$ from the distribution $\pi_t(c_t, x_t, \circ)$ . Note that this means that SPEW samples $a_{t}$ from $\pi_t^* (c_t, v_t, \circ)$ as required. +6. $\ell_t(a_t)$ is revealed. + +We now describe how this process enforces statistical parity. Specifically, suppose we have a distribution $\rho \in \Delta_{\mathcal{C} \times \mathcal{V}}$ . We assume, for simplicity, that for each $c \in \mathcal{C}$ there exists some $x \in \mathcal{X}$ with $\rho(c, x) \neq 0$ although it is trivial to modify the algorithm so that it does not require such an assumption. On any trial $t$ , we can define $\mu_t$ by: + +$$ +\mu_ {t} (c, x) := \frac {\sum_ {v \in \mathcal {V}} [ [ \chi_ {t} (c , v) = x ] ] \rho (c , v)}{\sum_ {v \in \mathcal {V}} \rho (c , v)} \quad \forall (c, x) \in \mathcal {C} \times \mathcal {X} +$$ + +Note that for all $(c,x)\in \mathcal{C}\times \mathcal{X}$ with $x\notin \mathcal{G}_t(c)$ we have $\mu_t(c,x) = 0$ as required. Since $\pi_t\in \mathcal{F}(\mu_t)$ we then have that the policy $\pi_t^*$ has statistical parity with respect to $\rho$ . + +# 4.3 Pseudocode + +We now give the pseudocode of SPEw. On any trial $t\in [T]$ SPEw does the following. + +Receive $\mu_t, x_t$ and $c_t$ +for $c \in \mathcal{C}$ do + $\xi_t(c, \circ, \circ) \gets \text{QUERY}[c]$ +for $a \in \mathcal{A}$ do + $\omega_t(c, a) \gets \sum_{x \in \mathcal{G}_t(c)} \mu_t(c, x) \xi_t(c, x, a)$ +end for +end for +for $(c, a) \in \mathcal{C} \times \mathcal{A}$ do + $\delta_t(c, a) \gets \max_{c' \in \mathcal{C}} \omega_t(c', a) - \omega_t(c, a)$ +end for + $\beta_t \gets \sum_{a \in \mathcal{A}} \max_{c \in \mathcal{C}} \delta_t(c, a)$ +for $a \in \mathcal{A}$ do + $\psi_t'(a) \gets (\xi_t(c_t, x_t, a) + \delta_t(c_t, a)) / (1 + \beta_t)$ +end for + $h_t \gets 1 - \sum_{a \in \mathcal{A}} \psi_t'(a)$ +for $a \in \mathcal{A}$ do + $\pi_t'(a) \gets \psi_t'(a) + h_t / K$ +end for +Draw $a_t$ from probability distribution $\pi_t'$ +Receive $\ell_t(a_t)$ +for $a \in \mathcal{A}$ do + $\kappa_t(a) \gets \operatorname{argmax}_{c \in \mathcal{C}} \omega_t(c, a)$ $\kappa_t'(a) \gets \operatorname{argmin}_{c \in \mathcal{C}} \omega_t(c, a)$ +end for +for $c \in \mathcal{C}$ do +for $x \in \mathcal{G}_t(c)$ do +for $a \in \mathcal{A}$ do + $\lambda_t(c, x, a) \gets [[\beta_t \leq 1]][[(c, x, a) = (c, x, a)] + [[c = \kappa_t(a)]\mu_t(c, x)]$ +end for + +UPDATE $[c](x,\lambda_t(c,x,\circ))$ + +end for + +end for + +# 4.4 Description + +We now briefly describe the internal workings of SPEW on trial $t$ . For a full description and analysis please see Appendix B. On a high level, SPEW maintains a virtual policy (which does not necessarily satisfy statistical parity) inside the instances of HEDGE. This virtual policy is computed by calling the QUERY subroutine of each instance of HEDGE. On trial $t$ , this virtual policy $\xi_{t}$ is mapped, in a certain way, to the virtual policy $\pi_{t}$ , which does satisfy statistical parity. $\xi_{t}$ is then updated to $\xi_{t+1}$ by performing exponentiated gradient descent (on the expert probabilities) using an unbiased estimate of the gradient of a convex surrogate of the loss of $\pi_{t}$ in terms of $\xi_{t}$ . This exponentiated gradient descent update is performed by updating the instances of HEDGE (via the UPDATE subroutine) multiple times. + +We now go into details. Given some $c \in \mathcal{C}$ , recall (from Section 4.1) that the instance of the HEDGE for protected characteristic $c$ implicitly maintains a distribution in $\Delta_{\mathcal{H}}$ . Let $\vartheta_t(c, \circ)$ be the value of this distribution at the start of trial $t$ , so that $\vartheta_1(c, \circ) = \vartheta$ . + +Define the virtual policy $\xi_t \in \mathcal{P}$ by: + +$$ +\xi_ {t} (c, x, a) := \sum_ {e \in \mathcal {H}} [ [ e (x) = a ] ] \vartheta_ {t} (c, e) +$$ + +for all $(c,x,a)\in \mathcal{C}\times \mathcal{X}\times \mathcal{A}$ . We call $\xi_{t}$ the raw policy. For each $c\in \mathcal{C}$ SPEw needs only know $\xi_t(c,x,\circ)$ for $x\in \mathcal{G}_t(c)$ , which it can compute by a call to QUERY[c]. + +Note that the raw policy does not necessarily belong to $\mathcal{F}(\mu_t)$ . The raw policy is converted into the virtual policy $\pi_t$ by the following procedure. Given $(c, a) \in \mathcal{C} \times \mathcal{A}$ define: + +$$ +\omega_ {t} (c, a) := \sum_ {x \in \mathcal {G} _ {t} (c)} \mu_ {t} (c, x) \xi_ {t} (c, x, a) +$$ + +and: + +$$ +\delta_ {t} (c, a) := \max _ {c ^ {\prime} \in \mathcal {C}} \omega_ {t} (c ^ {\prime}, a) - \omega_ {t} (c, a) +$$ + +Furthermore define: + +$$ +\beta_ {t} := \sum_ {a \in \mathcal {A}} \max _ {c \in \mathcal {C}} \delta_ {t} (c, a) +$$ + +For all $(c,x,a)\in \mathcal{C}\times \mathcal{X}\times \mathcal{A}$ define: + +$$ +\psi_ {t} (c, x, a) := \frac {\xi_ {t} (c , x , a) + \delta_ {t} (c , a)}{1 + \beta_ {t}} +$$ + +Finally, define the virtual policy $\pi_t \in \mathcal{P}$ by: + +$$ +\pi_ {t} (c, x, a) := \psi_ {t} (c, x, a) + \frac {1}{K} \left(1 - \sum_ {a ^ {\prime} \in \mathcal {A}} \psi_ {t} (c, x, a ^ {\prime})\right) +$$ + +for all $(c,x,a)\in \mathcal{C}\times \mathcal{X}\times \mathcal{A}$ . In Appendix B we show that $\pi_t\in \mathcal{F}(\mu_t)$ as required. + +SPEW then samples the action $a_{t}$ from the distribution $\pi_t(c_t,x_t,\circ)$ . + +Now that we have shown how to choose the action $a_{t}$ , we will show how to update the HEDGE instances upon receipt of $\ell_t(a_t)$ . We first define: + +$$ +\mathcal {S} := \mathbb {R} ^ {\mathcal {C} \times \mathcal {X} \times \mathcal {A}} +$$ + +and given $(\phi, c, a) \in S \times C \times A$ define: + +$$ +\tilde {\omega} _ {t} (\phi , c, a) := \sum_ {x \in \mathcal {G} _ {t} (c)} \mu_ {t} (c, x) \phi (c, x, a) +$$ + +We define the function $y_{t}:S\to \mathbb{R}$ by: + +$$ +y _ {t} (\phi) := \llbracket \beta_ {t} \leq 1 \rrbracket \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \phi \left(c _ {t}, x _ {t}, a\right) + \sum_ {a \in \mathcal {A}} \left(\max _ {c \in \mathcal {C}} \tilde {\omega} _ {t} (\phi , c, a) - \min _ {c \in \mathcal {C}} \tilde {\omega} _ {t} (\phi , c, a)\right) +$$ + +noting that SPEw never actually has knowledge of this function, as it is dependent on the entirety of $\ell_t$ . In Appendix B we show that $y_{t}$ is convex and that: + +$$ +y _ {t} \left(\xi_ {t}\right) \geq \sum_ {a \in \mathcal {A}} \pi_ {t} \left(c _ {t}, x _ {t}, a\right) \ell_ {t} (a) \quad ; \quad y _ {t} (\tilde {\pi}) \leq \sum_ {a \in \mathcal {A}} \tilde {\pi} \left(c _ {t}, x _ {t}, a\right) \ell_ {t} (a) \quad \forall \tilde {\pi} \in \mathcal {F} \left(\mu_ {t}\right) +$$ + +These properties mean that we can use $y_{t}$ as a convex surrogate for the expected loss in terms of the raw policy. + +Since SPEw does not know the entirety of $\ell_t$ it does not know a subgradient of $y_{t}$ at $\xi_{t}$ - which is required for exponentiated gradient descent. Hence, we borrow the idea from Exp4, of constructing an unbiased estimate of such a subgradient. Specifically, we define the function $\lambda_t\in S$ by: + +$$ +\begin{array}{l} \lambda_ {t} (c, x, a) := \llbracket \beta_ {t} \leq 1 \rrbracket \llbracket (c, x, a) = (c _ {t}, x _ {t}, a _ {t}) \rrbracket \frac {\ell_ {t} (a _ {t})}{\pi_ {t} (c _ {t} , x _ {t} , a _ {t})} \\ + \llbracket c = \operatorname {a r g m a x} _ {c ^ {\prime} \in \mathcal {C}} \omega_ {t} (c ^ {\prime}, a) \rrbracket \mu_ {t} (c, x) - \llbracket c = \operatorname {a r g m i n} _ {c ^ {\prime} \in \mathcal {C}} \omega_ {t} (c ^ {\prime}, a) \rrbracket \mu_ {t} (c, x) \\ \end{array} +$$ + +where ties in the argmax and argmin are broken arbitrarily. In Appendix B we prove that the expected value of $\lambda_{t}$ over the draw of $a_{t}$ is equal to a subgradient of $y_{t}$ at $\xi_{t}$ . Noting that, for each $(c,e)\in \mathcal{C}\times \mathcal{H},\xi_{t}$ is dependent on $\vartheta_t(c,e)$ , we then have that an unbiased estimate of the subgradient of $y_{t}(\xi_{t})$ with respect to $\vartheta_t(c,e)$ is equal to: + +$$ +\nu_ {t} (c, e) := \sum_ {x \in \mathcal {X}} \lambda_ {t} (c, x, e (x)) +$$ + +We can now update the function $\vartheta_{t}$ to $\vartheta_{t + 1}$ via exponentiated gradient descent. Specifically, for all $(c,e)\in \mathcal{C}\times \mathcal{H}$ we have: + +$$ +\vartheta_ {t + 1} (c, e) := \frac {\vartheta_ {t} (c , e) \exp (- \hat {\eta} \nu_ {t} (c , e))}{\sum_ {e ^ {\prime} \in \mathcal {H}} \vartheta_ {t} (c , e ^ {\prime}) \exp (- \hat {\eta} \nu_ {t} (c , e ^ {\prime}))} +$$ + +This updated is implemented by simply running the function + +$$ +\operatorname {U P D A T E} [ c ] (x, \lambda_ {t} (c, x, \circ)) +$$ + +for each $c\in \mathcal{C}$ and $x\in \mathcal{G}_t(c)$ . + +# 4.5 Performance + +In the following theorem we give the general performance guarantee of SPEW. + +Theorem 4.1. Let: $\eta \coloneqq \hat{\eta}\sqrt{KT}$ . For all $t\in [T]$ , SPEW produces $\pi_t\in \mathcal{F}(\mu_t)$ . In addition, for any virtual policy: + +$$ +\tilde {\pi} \in \bigcap_ {t \in [ T ]} \mathcal {F} (\mu_ {t}) +$$ + +and for any $\vartheta^{*}:\mathcal{C}\times \mathcal{H}\to [0,1]$ with: + +$$ +\tilde {\pi} (c, x, a) = \sum_ {e \in \mathcal {H}} [ [ e (x) = a ] ] \vartheta^ {*} (c, e) \quad \forall (c, x, a) \in \mathcal {C} \times \mathcal {X} \times \mathcal {A} +$$ + +we have: + +$$ +\mathbb {E} \left[ \sum_ {t \in [ T ]} \ell_ {t} (a _ {t}) \right] \leq \sum_ {t \in [ T ]} \sum_ {a \in \mathcal {A}} \tilde {\pi} (c _ {t}, x _ {t}, a) \ell_ {t} (a) + \mathcal {O} \left(\left(\eta + \frac {\Phi}{\eta}\right) \sqrt {K T}\right) +$$ + +where: + +$$ +\Phi := \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \ln \left(\frac {\vartheta^ {*} (c , e)}{\vartheta (e)}\right) +$$ + +On each trial $t \in [T]$ the time complexity of SPEW is that of $M$ calls to QUERY and $\sum_{c \in \mathcal{C}} |\mathcal{G}_t(c)|$ calls to UPDATE. The space complexity is that of maintaining $M$ instances of HEDGE. + +See Appendix B for the proof of this theorem. We note that for all applications the targets $\mu_t$ will be chosen such that $\bigcap_{t\in [T]}\mathcal{F}(\mu_t)$ is non-empty. We also note that for all virtual policies $\tilde{\pi}\in \mathcal{P}$ there exists $\vartheta^{*}:\mathcal{C}\times \mathcal{H}\to [0,1]$ with: + +$$ +\pi (c, x, a) = \sum_ {e \in \mathcal {H}} [ [ e (x) = a ] ] \vartheta^ {*} (c, e) \quad \forall (c, x, a) \in \mathcal {C} \times \mathcal {X} \times \mathcal {A} +$$ + +as is required by the theorem. Relative to any virtual policy $\tilde{\pi} \in \bigcap_{t \in [T]} \mathcal{F}(\mu_t)$ , the regret bound given by the theorem is asymptotically identical to that of running EXP4 independently for each protected characteristic. + +# 5 Examples + +Before we discuss the algorithm and the main result, we will give some example applications. The specific instances of SPEW that are used to obtain the results of this section, as well as their proofs, are given in Appendix D. In the first example we assume that the distribution with which we must have statistical parity with respect to is given a-priori, whilst in the final two examples it is instead estimated from the data seen so far. Note that, in the theorems, the quantity: + +$$ +\sum_ {a \in \mathcal {A}} \tilde {\pi} (c _ {t}, v _ {t}, a) \ell_ {t} (a) +$$ + +is the expected loss incurred by playing policy $\tilde{\pi} \in \mathcal{Q}$ on trial $t \in [T]$ . Note also that $\eta \coloneqq \hat{\eta}\sqrt{KT}$ + +# 5.1 Trees with a known distribution + +In this problem we assume we must maintain statistical parity with respect to a distribution $\rho \in \Delta_{\mathcal{C} \times \mathcal{V}}$ that is known to us a-priori. We also assume that we have knowledge of a tree with vertex set $\mathcal{V}$ . Let $\mathcal{E}$ be the edge set of this tree. Several types of problem reduce to this setting, including learning on graphs [14], learning on finite metric spaces [25] and learning on finite sets endowed with a hierarchical clustering. For graphs, the tree is created by sampling a spanning tree uniformly at random. For metric spaces the tree is created by growing the tree context by context, linking each new context to it's nearest neighbour in the tree so far. For hierarchical clusterings the context set is altered to be the set of all clusters, which naturally has a tree structure. + +Given a policy $\tilde{\pi} \in \mathcal{Q}$ we define its complexity as: + +$$ +\Psi (\tilde {\pi}) := \sum_ {(u, v) \in \mathcal {E}} \sum_ {(c, a) \in \mathcal {C} \times \mathcal {A}} | \tilde {\pi} (c, u, a) - \tilde {\pi} (c, v, a) | +$$ + +Note that $\Psi(\tilde{\pi})$ is bounded above by twice the summation, over all $c \in \mathcal{C}$ , of the number of edges $(u, v) \in \mathcal{E}$ in which $\tilde{\pi}(c, u, \circ) \neq \tilde{\pi}(c, v, \circ)$ (i.e. the cutsize). + +We have the following performance guarantee. + +Theorem 5.1. Suppose we are given a distribution $\rho \in \Delta_{\mathcal{C} \times \mathcal{V}}$ a-priori and run the appropriate instance of SPEW. Then all our generated policies $\pi_t^*$ have statistical parity with respect to $\rho$ and for any policy $\tilde{\pi} \in \mathcal{Q}$ with statistical parity with respect to $\rho$ we have: + +$$ +\mathbb {E} \left[ \sum_ {t \in [ T ]} \ell_ {t} (a _ {t}) \right] \leq \sum_ {t \in [ T ]} \sum_ {a \in \mathcal {A}} \tilde {\pi} (c _ {t}, v _ {t}, a) \ell_ {t} (a) + \tilde {\mathcal {O}} \left(\left(\eta + \frac {M + \Psi (\tilde {\pi}) \ln (N)}{\eta}\right) \sqrt {K T}\right) +$$ + +The per-trial time and space complexity is $\mathcal{O}(MNK)$ . + +This theorem leads directly to theorems about learning on graphs and finite metric spaces via results from [10] (it is a commonly used fact that the expected cutsize of uniformly sampled spanning tree of a labelled graph is equal to the effective resistance weighted cutsize of the graph) and [25] (see Theorem 3.6 therein) respectively. + +# 5.2 Empirical statistical parity + +We now turn to examples in which we are not given a distribution $\rho$ a-priori. In this first example we utilise the methodology of [16] to enforce statistical parity with respect to the empirical distribution of the data seen so far. Note that for this to be meaningful the set $\mathcal{V}$ should not be too large (we will deal with very large and essentially infinite context sets in the next example). The set $\mathcal{V}$ could, for example, be created via a clustering of a larger context set. We do not exploit any structure to the contexts in this example. + +For any trial $t > 1$ let $\rho_t$ be the empirical distribution of the data seen so far. Formally, for all $(c,v) \in \mathcal{C} \times \mathcal{V}$ , we have: + +$$ +\rho_ {t} (c, v) := \frac {1}{t - 1} \sum_ {s \in [ t - 1 ]} \llbracket c _ {s} = c, v _ {s} = v \rrbracket +$$ + +Given any sequence of policies $\tilde{\pi} \in \mathcal{Q}^T$ , we define its complexity: + +$$ +\Psi (\tilde {\boldsymbol {\pi}}) := \sum_ {t \in [ T - 1 ]} \sum_ {(c, v, a) \in \mathcal {C} \times \mathcal {V} \times \mathcal {A}} | \tilde {\pi} _ {t + 1} (c, v, a) - \tilde {\pi} _ {t} (c, v, a) | +$$ + +With these definitions in hand we have the following result: + +Theorem 5.2. There exists an instance of SPEW such that for all $t > 1$ we have that $\pi_t^*$ has statistical parity with respect to $\rho_t$ and, for any sequence of policies $\tilde{\pi} \in \mathcal{Q}^T$ in which, for all $t > 1$ , $\tilde{\pi}_t$ has statistical parity with respect to $\rho_t$ , we have that: + +$$ +\mathbb {E} \left[ \sum_ {t \in [ T ]} \ell_ {t} (a _ {t}) \right] \leq \sum_ {t \in [ T ]} \sum_ {a \in \mathcal {A}} \tilde {\pi} _ {t} \left(c _ {t}, v _ {t}, a\right) \ell_ {t} (a) + \tilde {\mathcal {O}} \left(\left(\eta + \frac {M N + \Psi (\tilde {\pi})}{\eta}\right) \sqrt {K T}\right) +$$ + +The per-trial time and space complexity is $\mathcal{O}(MNK)$ . + +# 5.3 Approximate statistical parity for hierarchical clusterings with an unknown distribution + +We now turn to maintaining approximate statistical parity when the context space is potentially massive and has a known hierarchical clustering associated with it. Natural hierarchical clusterings can be derived for several types of structure, including graphs [14], Euclidean space [29], and finite metric spaces [24]. + +This example, which utilises the methodology of [25], does not assume a-priori knowledge of a distribution $\rho$ , but unlike the previous examples we assume here that there exists a distribution $\rho^{*} \in \Delta_{\mathcal{C} \times \mathcal{V}}$ in which the pairs $(c_t, v_t)$ are each drawn i.i.d. from $\rho^{*}$ . This distribution is not known to us a-priori. We are given some $\epsilon, \delta > 0$ . We say that a policy $\pi \in \mathcal{Q}$ has $\epsilon$ -approximate statistical parity with respect to $\rho^{*}$ if and only if, when a pair $(c, v)$ is drawn from $\rho^{*}$ and then an action $a$ is drawn from $\pi(c, v, \circ)$ , we have that: + +$$ +\mathbb {P} [ a = b \mid c = d ^ {\prime} ] - \epsilon \leq \mathbb {P} [ a = b \mid c = d ] \leq \mathbb {P} [ a = b \mid c = d ^ {\prime} ] + \epsilon +$$ + +for all $b \in \mathcal{A}$ and $d, d' \in \mathcal{C}$ . We are required to play in such a way that, with probability at least $1 - \delta$ , all our chosen policies $\pi_t^*$ have $\epsilon$ -approximate statistical parity with respect to $\rho^*$ . + +We assume that $\mathcal{V}$ is finite but potentially enormous. For instance, $\mathcal{V}$ could be the quantisation of a hypercube of Euclidean space defined by all the points that the computer can handle (given its precision). We assume there is a known hierarchical clustering over $\mathcal{V}$ . That is, we have a full binary tree in which each vertex is a subset of $\mathcal{V}$ . The root is the whole set $\mathcal{V}$ , the leaves are the singleton sets, and the two children of each internal vertex/set partition that set. We note that there is no need to explicitly maintain the tree (which could be enormous). We define $H$ to be the height of the hierarchical clustering, which is typically a relatively small quantity (e.g. logarithmic in $T$ ). + +Given a policy $\tilde{\pi} \in \mathcal{Q}$ we define its complexity $\Psi(\tilde{\pi})$ as follows. We first define a permitted clustering to be any partition of $\mathcal{V}$ comprising of sets in the hierarchical clustering. For any protected characteristic $c \in \mathcal{C}$ we define $\psi(c, \tilde{\pi})$ to be the minimum cardinality of any permitted clustering in which for each set $\mathcal{U}$ in that permitted clustering and for all $u, v \in \mathcal{U}$ we have $\tilde{\pi}(c, u, \circ) = \tilde{\pi}(c, v, \circ)$ . We then define: + +$$ +\Psi (\tilde {\pi}) := \sum_ {c \in \mathcal {C}} \psi (c, \tilde {\pi}) +$$ + +Finally, we say that a probability distribution $\rho \in \Delta_{\mathcal{C} \times \mathcal{V}}$ is $\epsilon$ -close to $\rho^*$ if and only if every policy with statistical parity with respect to $\rho$ has $\epsilon$ -approximate statistical parity with respect to $\rho^*$ . + +With these definitions in hand we have the following result: + +Theorem 5.3. There exists an instance of SPEW in which, with probability at least $1 - \delta$ , there exists some $\rho \in \Delta_{\mathcal{C} \times \mathcal{V}}$ , which is $\epsilon$ -close to $\rho^*$ , such that all the generated policies $\pi_t^*$ have statistical parity with respect to $\rho$ and, for any policy $\tilde{\pi}$ with statistical parity with respect to $\rho$ , we have: + +$$ +\mathbb {E} \left[ \sum_ {t \in [ T ]} \ell_ {t} (a _ {t}) \right] \leq \sum_ {t \in [ T ]} \sum_ {a \in \mathcal {A}} \tilde {\pi} (c _ {t}, v _ {t}, a) \ell_ {t} (a) + \tilde {\mathcal {O}} \left(\Psi (\tilde {\pi}) \ln (1 / \delta) (H / \epsilon) ^ {2} + \left(\eta + \frac {H \Psi (\tilde {\pi})}{\eta}\right) \sqrt {K T}\right) +$$ + +If the determination of whether a context belongs to a given set of the hierarchical clustering takes $\mathcal{O}(1)$ time, and the hierarchical clustering needs not be explicitly maintained to determine this, then the per-trial time and space complexity of SPEW is $\mathcal{O}(KT)$ . + +Note that unlike the previous examples, the time complexity is not dependent on the cardinality of the context set. The algorithm can be easily modified to obtain exact statistical parity (without the pairs $(c_{t},v_{t})$ necessarily being i.i.d.) when a distribution $\rho$ is known, as long as we can efficiently compute the probability masses of any set in the hierarchical clustering (for any given protected characteristic). + +# 6 Batch Classification + +Finally, we describe how SPEw can enforce statistical parity in the batch classification problem. Here there exists an unknown probability distribution $\hat{\rho} \in \Delta_{\mathcal{C} \times \mathcal{V} \times \mathcal{A}}$ and we are given a sequence: + +$$ +\left\langle \left(c _ {t}, v _ {t}, b _ {t}\right) \mid t \in [ T ] \right\rangle +$$ + +of $T$ i.i.d. samples from $\hat{\rho}$ . We are given a distribution $\rho \in \Delta_{\mathcal{C} \times \mathcal{V}}$ (which could be the empirical distribution of some unlabeled data). The aim is to construct a policy $\pi \in \mathcal{Q}$ which has statistical parity with respect to $\rho$ and has high accuracy: + +$$ +\sum_ {(c, v, a) \in \mathcal {C} \times \mathcal {V} \times \mathcal {A}} \hat {\rho} (c, v, a) \pi (c, v, a) +$$ + +To solve this problem we first define, for all $t \in [T]$ , the loss function $\ell_t$ by $\ell_t(a) := [[a \neq b_t]]$ for all $a \in \mathcal{A}$ and then, with the appropriate instance of HEDGE, run SPEW on the sequence $\langle (c_t, v_t, \ell_t) | t \in [T] \rangle$ to produce a sequence of policies $\langle \pi_t^* | t \in [T] \rangle$ such that $\pi_t^*$ has statistical parity with respect to $\rho$ for all $t \in [T]$ . Finally, we define the output policy: + +$$ +\pi := \frac {1}{T} \sum_ {t \in [ T ]} \pi_ {t} ^ {*} +$$ + +We note that via a HEDGE based doubling trick we can automatically tune the learning rate. Standard results on online-to-batch conversion [9] convert the regret bound given in Theorem 4.1 to a bound on the expected accuracy. Specifically, the difference between the expected accuracy of the algorithm and that of any given policy $\tilde{\pi}$ with statistical parity is equal to the bound on the regret divided by $T$ . For trees this difference in expected accuracy is: + +$$ +\tilde {\mathcal {O}} \left(\sqrt {(M + \Psi (\tilde {\pi})) \ln (N) K / T}\right) +$$ + +where $\Psi$ is defined as in Section D.1. + +# Acknowledgments + +Research funded by the Defence Science and Technology Laboratory (Dstl) which is an executive agency of the UK Ministry of Defence providing world class expertise and delivering cutting-edge science and technology for the benefit of the nation and allies. The research supports the Autonomous Resilient Cyber Defence (ARCD) project within the Dstl Cyber Defence Enhancement programme. + +# References + +[1] Alekh Agarwal, Alina Beygelzimer, Miroslav Dudík, John Langford, and Hanna M. Wallach. A reduction approach to fair classification. ArXiv, abs/1803.02453, 2018. +[2] Peter Auer, Nicolò Cesa-Bianchi, Yoav Freund, and Robert E. Schapire. The nonstochastic multiarmed bandit problem. SIAM J. Comput., 32:48-77, 2002. +[3] Yahav Bechavod, Christopher Jung, and Zhiwei Steven Wu. Metric-free individual fairness in online learning. ArXiv, abs/2002.05474, 2020. +[4] Yahav Bechavod, Katrina Ligett, Aaron Roth, Bo Waggoner, and Zhiwei Steven Wu. Equal opportunity in online classification with partial feedback. ArXiv, abs/1902.02242, 2019. +[5] Avrim Blum, Suriya Gunasekar, Thodoris Lykouris, and Nathan Srebro. On preserving nondiscrimination when combining expert advice. 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Res., 21:1:1-1:24, 2020. + +# A Literature Review + +Algorithmic fairness has been a topic of much study. Notable pioneering works include [6], which introduced statistical parity, and [12] which introduced the notion of individual fairness. Like our batch result, [1] considered the problem of constructing a randomised classifier, with statistical parity, which is a convex combination of deterministic classifiers from some given hypothesis space. They solve the problem via a sequence of calls to a cost-sensitive classification oracle (for the particular hypothesis space) and give a generalisation error in terms of the Rademacher complexity of the hypothesis space. The work [11] considered satisfying statistical parity approximately (with an arbitrary number of classes). + +We now give a review of some of the research in the wide topic of achieving various forms of fairness in i.i.d. stochastic bandits. [31] considered the problem of eliminating societal bias when actions are partitioned into groups which are each subjected to societal bias. [18] considered, in the linear bandit setting, the problem of equalizing cumulative mean rewards across two protected groups of contexts. [30] considered the problem of, when actions are partitioned into groups, simultaneously achieving a certain exposure for each group and ensuring that, within each group, each action is chosen with probability proportional to its merit. [20] considered the problem of ensuring that a better action (one with lower mean loss) never has a lower probability of being chosen than a worse action (one with higher mean loss), [32] considered the problem of choosing each action + +with a probability proportional to its merit, [21] considered the problem of treating similar actions similarly, and [27] considered the constraint of selecting each action at least a pre-specified number of times. A form of fairness with respect to instances was studied in [17] which considered the constraint of equalising the cumulative mean loss of all protected groups (all instances that have some given protected characteristic) in stochastic linear bandits. We note that, whilst this constraint will be desired in some situations, it is very different from statistical parity. Possibly the closest such work to ours is [4], which considered the i.i.d. stochastic online binary classification problem (classifying contexts as positive or negative) under the constraint that false negatives are equalised over two protected groups, under the assumption of the existence of an efficient oracle for empirical risk minimisation over a finite set of hypotheses. They specifically considered a partial feedback setting where the true class is only revealed when the positive class is chosen. However, the fairness constraints are only approximated and the algorithm is relatively inefficient: taking a per-trial time that is polynomial in the number of trials (the exponent is not given in the paper). + +Other works on fairness in online learning include the work [5] which deals with combining expert advice with experts that are each fair with respect to some metric, so that the resulting policy is fair with respect to that metric. This is a very different problem to that studied in this paper: which utilises experts that do not satisfy statistical parity. Another notable work is [3] which considers the problem of enforcing individual fairness in online settings where no metric is given but instead there exists an auditor which identifies pairs of users treated unfairly. + +Finally, as described in the introduction (where we discussed the works [8] and [33]), there has been much work on online learning subject to convex constraints (which are approximately satisfied in the long run). This line of research was pioneered by [22] and was extended to time-varying constraints in [23]. The work [7] considered both time-varying constraints and bandit feedback. The work [19] achieved bounds that can tradeoff between regret and constraint violation. The algorithm in [22] works by gradient descent/ascent on the Lagrangian, whilst our algorithm SPEW works in a fundamentally different way - by mapping a maintained policy (without statistical parity) to a policy with statistical parity and performing exponentiated gradient descent (on the maintained policy) with a surrogate for the resulting loss in terms of the original policy. + +# B Algorithm and Analysis + +Here we describe and analyse the mechanics of SPEw. All lemmas stated in this section are proved in Section C. Since this section works entirely in the virtual context space we will refer to, in this section, virtual contexts and policies as contexts and policies respectively. We define $\mathcal{I} \coloneqq \mathcal{C} \times \mathcal{X}$ . + +Let $\eta, \tilde{\pi}, \vartheta$ and $\vartheta^{*}$ be as in Theorem 4.1. Without loss of generality we can assume that: + +$$ +\eta \leq \sqrt {T / K} +$$ + +otherwise our regret bound would be vacuous. Recall that SPEW takes as input an algorithm (a.k.a. the base algorithm) for implementing Hedge with inductive bias $\vartheta$ . For the base algorithm we will choose the learning rate: + +$$ +\hat {\eta} := \frac {\eta}{\sqrt {K T}} +$$ + +We will maintain $M$ instances of this algorithm, one for each protected characteristic $c \in \mathcal{C}$ . For each $c \in \mathcal{C}$ let $\mathrm{QUERY}[c]$ and $\mathrm{UPDATE}[c]$ be the QUERY and UPDATE subroutines for the instance of the base algorithm associated with the protected characteristic $c$ . Given some $c \in \mathcal{C}$ , recall (from Section 4.1) that the instance of the base algorithm for protected characteristic $c$ implicitly maintains a distribution in $\Delta_{\mathcal{H}}$ . Let $\vartheta_t(c, \circ)$ be the value of this distribution at the start of trial $t$ , so that $\vartheta_1(c, \circ) = \vartheta$ . + +We now describe and analyse how SPEW behaves on trial $t$ and, in doing so, will bound the instantaneous regret: + +$$ +r _ {t} := \sum_ {a \in \mathcal {A}} \left(\pi_ {t} \left(c _ {t}, x _ {t}, a\right) - \tilde {\pi} \left(c _ {t}, x _ {t}, a\right)\right) \ell_ {t} (a) +$$ + +# B.1 Fundamental Definitions + +We now make some fundamental definitions. We first define: + +$$ +\mathcal {S} := \mathbb {R} ^ {\mathcal {C} \times \mathcal {X} \times \mathcal {A}} +$$ + +Given $(\phi, c, a) \in S \times \mathcal{C} \times \mathcal{A}$ define: + +$$ +\tilde {\omega} _ {t} (\phi , c, a) := \sum_ {x \in \mathcal {G} _ {t} (c)} \mu_ {t} (c, x) \phi (c, x, a) = \sum_ {x \in \mathcal {X}} \mu_ {t} (c, x) \phi (c, x, a) +$$ + +and define: + +$$ +\tilde {\delta} _ {t} (\phi , c, a) := \max _ {c ^ {\prime} \in \mathcal {C}} \tilde {\omega} _ {t} (\phi , c ^ {\prime}, a) - \tilde {\omega} _ {t} (\phi , c, a) +$$ + +Given $\phi \in S$ define: + +$$ +\tilde {\beta} _ {t} (\phi) := \sum_ {a \in \mathcal {A}} \max _ {c \in \mathcal {C}} \tilde {\delta} _ {t} (\phi , c, a) +$$ + +Finally, we define $\mathcal{Z}_t$ to be the set of all $\phi \in S$ in which for all $c, c' \in \mathcal{C}$ and $a \in \mathcal{A}$ we have: + +$$ +\tilde {\omega} _ {t} (\phi , c, a) = \tilde {\omega} _ {t} (\phi , c ^ {\prime}, a) +$$ + +We note that $\mathcal{F}(\mu_t) = \mathcal{P}\cap \mathcal{Z}_t$ + +# B.2 Computing the Policy + +Define the policy $\xi_t \in \mathcal{P}$ by: + +$$ +\xi_ {t} (c, x, a) := \sum_ {e \in \mathcal {H}} [ [ e (x) = a ] ] \vartheta_ {t} (c, e) +$$ + +for all $(c,x,a)\in \mathcal{C}\times \mathcal{X}\times \mathcal{A}$ . We call $\xi_{t}$ the raw policy. For each $c\in \mathcal{C}$ SPEW needs only know $\xi_t(c,x,\circ)$ for $x\in \mathcal{G}_t(c)$ , which it can compute by a call to QUERY[c]. We state the above equation as the following lemma: + +Lemma B.1. For all $(c,x,a)\in \mathcal{C}\times \mathcal{X}\times \mathcal{A}$ we have: + +$$ +\xi_ {t} (c, x, a) = \sum_ {e \in \mathcal {H}} [ [ e (x) = a ] ] \vartheta_ {t} (c, e) +$$ + +We now describe the conversion of our raw policy $\xi_{t}$ into our fair policy $\pi_t$ . We call this conversion process policy processing. First define, for all $a \in \mathcal{A}$ and $c \in \mathcal{C}$ , the quantities: + +$$ +\delta_ {t} (c, a) := \tilde {\delta} _ {t} (\xi_ {t}, c, a) \quad ; \quad \beta_ {t} := \tilde {\beta} _ {t} (\xi_ {t}) +$$ + +We define a function $\psi_t \in S$ such that for any $(c, x, a) \in \mathcal{C} \times \mathcal{X} \times \mathcal{A}$ we have: + +$$ +\psi_ {t} (c, x, a) := \frac {\xi_ {t} (c , x , a) + \delta_ {t} (c , a)}{1 + \beta_ {t}} +$$ + +The function $\psi_t$ has the following properties: + +Lemma B.2. We have that $\psi_t \in \mathcal{Z}_t$ . In addition, we have, for all $(c,x) \in \mathcal{I}$ , that: + +$$ +\sum_ {a \in \mathcal {A}} \psi_ {t} (c, x, a) \leq 1 +$$ + +and that $\psi_t(c,x,a) \geq 0$ for all $a \in \mathcal{A}$ . + +The above lemma shows that the function $\psi_t$ satisfies most of our constraints. However, it is not necessarily in $\mathcal{P}$ as the functions $\psi_t(c,x,\circ)$ may not lie on the simplex $\Delta_{\mathcal{A}}$ . In order to convert it into $\pi_t$ we simply add, to each of the functions $\psi_t(c,x,\circ)$ , some the uniform distribution. Specifically, for all $(c,x,a)\in \mathcal{C}\times \mathcal{X}\times \mathcal{A}$ we define: + +$$ +\pi_ {t} (c, x, a) := \psi_ {t} (c, x, a) + \frac {1}{K} \left(1 - \sum_ {a ^ {\prime} \in \mathcal {A}} \psi_ {t} (c, x, a ^ {\prime})\right) +$$ + +The following lemma, which follows from Lemma B.2, confirms that $\pi_t$ does indeed lie in $\mathcal{F}(\mu_t)$ . + +Lemma B.3. We have: + +$$ +\pi_ {t} \in \mathcal {F} (\mu_ {t}) +$$ + +We now give two additional lemmas that are crucial to our analysis. The first lemma follows from Lemma B.2. + +Lemma B.4. For all $(c,x,a)\in \mathcal{C}\times \mathcal{X}\times \mathcal{A}$ we have: + +$$ +\pi_ {t} (c, x, a) \geq \frac {\xi_ {t} (c , x , a)}{1 + \beta_ {t}} +$$ + +The next lemma follows from lemmas B.3 and B.4 + +Lemma B.5. For all $(c,x)\in \mathcal{I}$ we have: + +$$ +\sum_ {a \in A} \max \{0, \pi_ {t} (c, x, a) - \xi_ {t} (c, x, a) \} \leq \beta_ {t} +$$ + +We have now derived all the properties of $\pi_t$ needed to proceed with the algorithm and analysis. + +# B.3 The Objective Function and its Pseudo-Gradient + +Now that we have shown how to compute $\pi_t$ we turn to the update of the base algorithm instances at the end of the trial. In order to do this we need a convex objective function $y_t: S \to \mathbb{R}$ which we note will never actually be known by Learner. Our objective function is defined such that for all $\phi \in S$ we have: + +$$ +y _ {t} (\phi) := \llbracket \beta_ {t} \leq 1 \rrbracket \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \phi \left(c _ {t}, x _ {t}, a\right) + \tilde {\beta} _ {t} (\phi) +$$ + +The reason we have chosen such an objective function is due to the following two lemmas, where Lemma B.6 follows from Lemma B.5. We note that the appearance of $\llbracket \beta_t\leq 1\rrbracket$ in the definition of $y_{t}$ is to ensure another crucial lemma, which we shall present later. + +Lemma B.6. We have: + +$$ +y _ {t} \left(\xi_ {t}\right) \geq \sum_ {a \in \mathcal {A}} \pi_ {t} \left(c _ {t}, x _ {t}, a\right) \ell_ {t} (a) +$$ + +Lemma B.7. We have: + +$$ +y _ {t} (\tilde {\pi}) \leq \sum_ {a \in \mathcal {A}} \tilde {\pi} \left(c _ {t}, x _ {t}, a\right) \ell_ {t} (a) +$$ + +We will show that $y_{t}$ is indeed convex by showing that it has a sub-gradient at every point in $S$ . In order to give such a sub-gradient we first define, for all $(\phi, a) \in S \times \mathcal{A}$ , the quantities: + +$$ +\tilde {\kappa} _ {t} (\phi , a) := \operatorname {a r g m a x} _ {c \in \mathcal {C}} \tilde {\omega} _ {t} (\phi , c, a); \quad \tilde {\kappa} _ {t} ^ {\prime} (\phi , a) := \operatorname {a r g m i n} _ {c \in \mathcal {C}} \tilde {\omega} _ {t} (\phi , c, a) +$$ + +where ties are broken arbitrarily. We then define $g_{t}:\mathcal{S}\times \mathcal{C}\times \mathcal{X}\times \mathcal{A}\to \mathbb{R}$ such that for all $(\phi ,c,x,a)\in \mathcal{S}\times \mathcal{C}\times \mathcal{X}\times \mathcal{A}$ we have: + +$$ +g _ {t} (\phi , c, x, a) := \llbracket \beta_ {t} \leq 1 \rrbracket \llbracket (c, x) = (c _ {t}, x _ {t}) \rrbracket \ell_ {t} (a) + \llbracket c = \tilde {\kappa} _ {t} (\phi , a) \rrbracket \mu_ {t} (c, x) - \llbracket c = \tilde {\kappa} _ {t} ^ {\prime} (\phi , a) \rrbracket \mu_ {t} (c, x) +$$ + +The next lemma shows that for all $\phi \in S$ we have that $g_{t}(\phi ,\circ ,\circ ,\circ)$ is a sub-gradient of $y_{t}$ at $\phi$ + +Lemma B.8. For any $\phi, \phi' \in S$ we have: + +$$ +y _ {t} (\phi) - y _ {t} \left(\phi^ {\prime}\right) \leq \sum_ {c \in \mathcal {C}} \sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} g _ {t} (\phi , c, x, a) \left(\phi (c, x, a) - \phi^ {\prime} (c, x, a)\right) +$$ + +Due to the fact that we don't know the entire function $\ell_t$ , we can't compute the sub-gradient $g_t(\xi_t, \circ, \circ, \circ)$ so instead, will borrow the technique, from Exp4, of using $\llbracket a = a_t\rrbracket \ell_t(a_t) / \pi_t(c_t, x_t, a_t)$ as an unbiased estimator of $\ell_t(a)$ . This gives us the following function $\lambda_t$ that we call the pseudogradients. $\lambda_t \in S$ is defined such that for all $(c, x, a) \in C \times \mathcal{X} \times \mathcal{A}$ we have: + +$$ +\begin{array}{l} \lambda_ {t} (c, x, a) := \llbracket \beta_ {t} \leq 1 \rrbracket \llbracket (c, x, a) = (c _ {t}, x _ {t}, a _ {t}) \rrbracket \frac {\ell_ {t} (a _ {t})}{\pi_ {t} (c _ {t} , x _ {t} , a _ {t})} \\ + \llbracket c = \tilde {\kappa} _ {t} (\xi_ {t}, a) \rrbracket \mu_ {t} (c, x) - \llbracket c = \tilde {\kappa} _ {t} ^ {\prime} (\xi_ {t}, a) \rrbracket \mu_ {t} (c, x) \\ \end{array} +$$ + +The following lemma states that $\lambda_{t}$ is an unbiased estimator of the sub-gradient $g_{t}(\xi_{t},\circ ,\circ ,\circ)$ . + +Lemma B.9. For all $(c,x,a)\in \mathcal{C}\times \mathcal{X}\times \mathcal{A}$ we have: + +$$ +\mathbb {E} \left[ \lambda_ {t} (c, x, a) \mid \xi_ {t} \right] = g _ {t} \left(\xi_ {t}, c, x, a\right) +$$ + +Note that for the full-information online and batch classification versions of SPEW we can use the true sub-gradient instead of $\lambda_{t}$ in order to give a deterministic algorithm. + +Lemmas B.6, B.7, B.8 and B.9 combine to give us the following lemma. + +Lemma B.10. We have: + +$$ +\mathbb {E} \left[ \sum_ {c \in \mathcal {C}} \sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} (\xi_ {t} (c, x, a) - \tilde {\pi} (c, x, a)) \lambda_ {t} (c, x, a) \Bigg | \xi_ {t} \right] \geq r _ {t} +$$ + +Now define $\nu_{t}:\mathcal{C}\times \mathcal{H}\to \mathbb{R}$ such that for all $(c,e)\in \mathcal{C}\times \mathcal{H}$ we have: + +$$ +\nu_ {t} (c, e) := \sum_ {x \in \mathcal {X}} \lambda_ {t} (c, x, e (x)) +$$ + +Lemmas B.1 and B.10 combine to give us the following lemma. + +Lemma B.11. We have: + +$$ +\mathbb {E} \left[ \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \left(\vartheta_ {t} (c, e) - \vartheta^ {*} (c, e)\right) \nu_ {t} (c, e) \middle | \vartheta_ {t} \right] \geq r _ {t} +$$ + +The reason that the term $\llbracket \beta_t\leq 1\rrbracket$ appears in the definition of our objective function $y_{t}$ is to ensure the following crucial property of the pseudo-gradient, which follows from Lemma B.4. + +Lemma B.12. We have: + +$$ +\mathbb {E} \left[ \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \nu_ {t} (c, e) ^ {2} \mid \vartheta_ {t} \right] \leq 8 K +$$ + +Finally we have the following lemma: + +Lemma B.13. For all $(c,e)\in \mathcal{C}\times \mathcal{H}$ we have: + +$$ +\nu_ {t} (c, e) \geq - K +$$ + +We have now derived all the properties of $\nu_{t}$ needed to progress to the next stage of the algorithm and analysis. + +# B.4 Hedge Update + +To update we run, for each $c \in \mathcal{C}$ and $x \in \mathcal{G}_t(c)$ , the function: + +$$ +\operatorname {U P D A T E} [ c ] (x, \lambda_ {t} (c, x, \circ)) +$$ + +We have the following lemma. + +Lemma B.14. For all $(c,e)\in \mathcal{C}\times \mathcal{H}$ we have: + +$$ +\vartheta_ {t + 1} (c, e) = \frac {\vartheta_ {t} (c , e) \exp (- \hat {\eta} \nu_ {t} (c , e))}{\sum_ {e ^ {\prime} \in \mathcal {H}} \vartheta_ {t} (c , e ^ {\prime}) \exp (- \hat {\eta} \nu_ {t} (c , e ^ {\prime}))} +$$ + +We define the relative entropy $B:\Delta_{\mathcal{H}}\times \Delta_{\mathcal{H}}\to \mathbb{R}$ such that for all $\beta ,\beta^{\prime}\in \Delta_{\mathcal{H}}$ we have: + +$$ +B (\beta , \beta^ {\prime}) := \sum_ {e \in \mathcal {H}} \beta (e) \ln \left(\frac {\beta (e)}{\beta^ {\prime} (e)}\right) +$$ + +Lemma B.13 and Lemma B.14 give us the following lemma. + +Lemma B.15. For all $c \in \mathcal{C}$ we have: + +$$ +B (\vartheta^ {*}, \vartheta_ {t} (c, \circ)) - B (\vartheta^ {*}, \vartheta_ {t + 1} (c, \circ)) \geq \hat {\eta} \sum_ {e \in \mathcal {H}} (\vartheta_ {t} (c, e) - \vartheta^ {*} (c, e)) \nu_ {t} (c, e) - \hat {\eta} ^ {2} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \nu_ {t} (c, e) ^ {2} +$$ + +This lemma combines with lemmas B.11 and B.12 to give the following lemma. + +Lemma B.16. We have: + +$$ +\mathbb {E} \left[ \sum_ {c \in C} \left(B \left(\vartheta^ {*}, \vartheta_ {t} (c, \circ)\right) - B \left(\vartheta^ {*}, \vartheta_ {t + 1} (c, \circ)\right)\right) \mid \vartheta_ {t} \right] \geq \hat {\eta} r _ {t} - 8 \hat {\eta} ^ {2} K +$$ + +This completes the description and analysis of trial $t$ . + +# B.5 Regret Bound + +To get the overall regret bound we take expectations on Lemma B.16 and sum over all $t \in [T]$ , which gives us the following lemma. + +Lemma B.17. We have: + +$$ +\sum_ {t \in [ T ]} \mathbb {E} \left[ r _ {t} \right] \leq \left(8 \eta - \Phi / \eta\right) \sqrt {K / T} +$$ + +where: + +$$ +\Phi := \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \ln \left(\frac {\vartheta^ {*} (c , e)}{\vartheta (e)}\right) +$$ + +# C Proofs of Lemmas + +We now prove, in order, all of the lemmas given in this Appendix B. + +# C.1 Lemma B.1 + +This is immediately clear from the construction of $\xi_{t}$ and the description of HEDGE given in Section 4.1. + +# C.2 Lemma B.2 + +We first show that $\psi_t \in \mathcal{Z}_t$ . To show this consider some $c \in \mathcal{C}$ and $a \in \mathcal{A}$ . We have: + +$$ +\begin{array}{l} (1 + \beta_ {t}) \tilde {\omega} _ {t} (\psi_ {t}, c, a) = (1 + \beta_ {t}) \sum_ {x \in \mathcal {X}} \mu_ {t} (c, x) \psi_ {t} (c, x, a) \\ = \sum_ {x \in \mathcal {X}} \mu_ {t} (c, x) \left(\xi_ {t} (c, x, a) + \delta_ {t} (c, a)\right) \\ = \sum_ {x \in \mathcal {X}} \mu_ {t} (c, x) \xi_ {t} (c, x, a) + \delta_ {t} (c, a) \sum_ {x \in \mathcal {X}} \mu_ {t} (c, x) \\ = \tilde {\omega} _ {t} (\xi_ {t}, c, a) + \delta_ {t} (c, a) \\ = \tilde {\omega} _ {t} (\xi_ {t}, c, a) + \tilde {\delta} _ {t} (\xi_ {t}, c, a) \\ = \tilde {\omega} _ {t} \left(\xi_ {t}, c, a\right) + \max _ {c ^ {\prime} \in \mathcal {C}} \tilde {\omega} _ {t} \left(\xi_ {t}, c ^ {\prime}, a\right) - \tilde {\omega} _ {t} \left(\xi_ {t}, c, a\right) \\ = \max _ {c ^ {\prime} \in \mathcal {C}} \tilde {\omega} _ {t} (\xi_ {t}, c ^ {\prime}, a) \\ \end{array} +$$ + +Since this is independent of $c$ we have that $\psi_t \in \mathcal{Z}_t$ as required. + +We now show that, given $(c,x)\in \mathcal{I}$ , we have $\sum_{a\in \mathcal{A}}\psi_t(c,x,a)\leq 1$ . This is true since $\xi_t\in \mathcal{P}$ and hence: + +$$ +\begin{array}{l} (1 + \beta_ {t}) \sum_ {a \in \mathcal {A}} \psi_ {t} (c, x, a) = \sum_ {a \in \mathcal {A}} \xi_ {t} (c, x, a) + \sum_ {a \in \mathcal {A}} \delta_ {t} (c, a) \\ = 1 + \sum_ {a \in \mathcal {A}} \tilde {\delta} _ {t} (\xi_ {t}, c, a) \\ \leq 1 + \sum_ {a \in \mathcal {A}} \max _ {c ^ {\prime} \in \mathcal {C}} \tilde {\delta} _ {t} \left(\xi_ {t}, c ^ {\prime}, a\right) \\ = 1 + \tilde {\beta} _ {t} (\xi_ {t}) \\ \leq \left(1 + \beta_ {t}\right) \\ \end{array} +$$ + +as required. + +The fact that $\psi_t(c, x, a) \geq 0$ for all $(c, x, a) \in \mathcal{C} \times \mathcal{X} \times \mathcal{A}$ follows directly from the fact that, since $\xi_t \in \mathcal{P}$ , we have that $\delta_t(c, a)$ and $\beta_t$ and $\xi_t(c, x, a)$ are all non-negative. + +# C.3 Lemma B.3 + +We first show that $\pi_t \in \mathcal{P}$ . To show this consider any $(c, x) \in \mathcal{I}$ . By Lemma B.2 we have that $\sum_{a \in \mathcal{A}} \psi_t(c, x, a) \leq 1$ and hence: + +$$ +1 - \sum_ {a ^ {\prime} \in \mathcal {A}} \psi_ {t} (c, x, a ^ {\prime}) \geq 0 +$$ + +For all $a \in \mathcal{A}$ we have, by Lemma B.2, that $\psi_t(c,x,a) \geq 0$ and hence the above equation implies that: + +$$ +\pi_ {t} (c, x, a) = \psi_ {t} (c, x, a) + \frac {1}{K} \left(1 - \sum_ {a ^ {\prime} \in \mathcal {A}} \psi_ {t} (c, x, a ^ {\prime})\right) \geq 0 + 0 = 0 +$$ + +Finally note that: + +$$ +\begin{array}{l} \sum_ {a \in \mathcal {A}} \pi_ {t} (c, x, a) = \sum_ {a \in \mathcal {A}} \psi_ {t} (c, x, a) + \frac {1}{K} \sum_ {a \in \mathcal {A}} \left(1 - \sum_ {a ^ {\prime} \in \mathcal {A}} \psi_ {t} (c, x, a ^ {\prime})\right) \\ = \sum_ {a \in \mathcal {A}} \psi_ {t} (c, x, a) + \left(1 - \sum_ {a ^ {\prime} \in \mathcal {A}} \psi_ {t} (c, x, a ^ {\prime})\right) \\ = 1 \\ \end{array} +$$ + +Since these equations hold for all $(c,x)\in \mathcal{I}$ , we have shown that $\pi_t\in \mathcal{P}$ . Hence, all we now need to show is that $\pi_t\in \mathcal{Z}_t$ . To show this, note that by Lemma B.2 we have that $\psi_t\in \mathcal{Z}_t$ . Hence, there exists a function $z:\mathcal{A}\to \mathbb{R}$ such that for all $(c,a)\in \mathcal{C}\times \mathcal{A}$ we have: + +$$ +\sum_ {x \in \mathcal {X}} \mu_ {t} (c, x) \psi_ {t} (c, x, a) = \tilde {\omega} _ {t} (\psi_ {t}, c, a) = z (a) +$$ + +So for all $(c, a) \in \mathcal{C} \times \mathcal{A}$ we have: + +$$ +\begin{array}{l} \tilde {\omega} _ {t} \left(\pi_ {t}, c, a\right) = \sum_ {x \in \mathcal {X}} \mu_ {t} (c, x) \pi_ {t} (c, x, a) \\ = \sum_ {x \in \mathcal {X}} \mu_ {t} (c, x) \psi_ {t} (c, x, a) + \sum_ {x \in \mathcal {X}} \frac {\mu_ {t} (c , x)}{K} \left(1 - \sum_ {a ^ {\prime} \in \mathcal {A}} \psi_ {t} (c, x, a ^ {\prime})\right) \\ = z (a) + \frac {1}{K} \sum_ {x \in \mathcal {X}} \mu_ {t} (c, x) - \frac {1}{K} \sum_ {a ^ {\prime} \in \mathcal {A}} \sum_ {x \in \mathcal {X}} \mu_ {t} (c, x) \psi_ {t} (c, x, a ^ {\prime}) \\ = z (a) + \frac {1}{K} - \frac {1}{K} \sum_ {a ^ {\prime} \in \mathcal {A}} z \left(a ^ {\prime}\right) \\ \end{array} +$$ + +Since this is independent of $c$ we have $\pi_t \in \mathcal{Z}_t$ as required. This completes the proof. + +# C.4 Lemma B.4 + +Choose any $(c,x,a)\in \mathcal{C}\times \mathcal{X}\times \mathcal{A}$ . By Lemma B.2 we have $\sum_{a\in \mathcal{A}}\psi_t(c,x,a)\leq 1$ so since $\delta_t(c,a)$ and $\beta_{t}$ and $\xi_t(c,x,a)$ are all non-negative, we have that: + +$$ +\begin{array}{l} \pi_ {t} (c, x, a) = \psi_ {t} (c, x, a) + \frac {1}{K} \left(1 - \sum_ {a ^ {\prime} \in \mathcal {A}} \psi_ {t} (c, x, a ^ {\prime})\right) \\ \geq \psi_ {t} (c, x, a) + \frac {1}{K} (1 - 1) \\ = \psi_ {t} (c, x, a) \\ = \frac {\xi_ {t} (c , x , a) + \delta_ {t} (c , a)}{1 + \beta_ {t}} \\ \geq \frac {\xi_ {t} (c , x , a)}{1 + \beta_ {t}} \\ \end{array} +$$ + +as required. + +# C.5 Lemma B.5 + +Take any $(c,x,a)\in \mathcal{C}\times \mathcal{X}\times \mathcal{A}$ . By Lemma B.4 we have: + +$$ +0 \leq \pi_ {t} (c, x, a) - \frac {\xi_ {t} (c , x , a)}{1 + \beta_ {t}} +$$ + +Since $\beta_{t}$ and $\xi_t(c,x,a)$ are non-negative we also have: + +$$ +\pi_ {t} (c, x, a) - \xi_ {t} (c, x, a) \leq \pi_ {t} (c, x, a) - \frac {\xi_ {t} (c , x , a)}{1 + \beta_ {t}} +$$ + +and hence we have shown that: + +$$ +\max \{0, \pi_ {t} (c, x, a) - \xi_ {t} (c, x, a) \} \leq \pi_ {t} (c, x, a) - \frac {\xi_ {t} (c , x , a)}{1 + \beta_ {t}} +$$ + +Now take any $(c,x)\in \mathcal{I}$ . By lemmas B.1 and B.3 we have that both $\pi_t$ and $\xi_{t}$ are in $\mathcal{P}$ . Hence, by the above equation we have, since $\beta_{t}\geq 0$ , that: + +$$ +\begin{array}{l} \sum_ {a \in \mathcal {A}} \max \left\{0, \pi_ {t} (c, x, a) - \xi_ {t} (c, x, a) \right\} \leq \sum_ {a \in \mathcal {A}} \left(\pi_ {t} (c, x, a) - \frac {\xi_ {t} (c , x , a)}{1 + \beta_ {t}}\right) (1) \\ = \sum_ {a \in \mathcal {A}} \pi_ {t} (c, x, a) - \frac {1}{1 + \beta_ {t}} \sum_ {a \in \mathcal {A}} \xi_ {t} (c, x, a) (2) \\ = 1 - \frac {1}{1 + \beta_ {t}} (3) \\ = \frac {\beta_ {t}}{1 + \beta_ {t}} (4) \\ \leq \beta_ {t} (5) \\ \end{array} +$$ + +as required. + +# C.6 Lemma B.6 + +We consider two cases. First consider the case that $\beta_t > 1$ . In this case we have, since $\xi_t \in \mathcal{P}$ and $\ell_t(a) \leq 1$ for all $a \in \mathcal{A}$ , that: + +$$ +\begin{array}{l} y _ {t} (\xi_ {t}) = [ [ \beta_ {t} \leq 1 ] ] \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \xi_ {t} (c _ {t}, x _ {t}, a) + \tilde {\beta} _ {t} (\xi_ {t}) \\ = \tilde {\beta} _ {t} (\xi_ {t}) \\ = \beta_ {t} \\ > 1 \\ = \sum_ {a \in \mathcal {A}} \pi_ {t} \left(c _ {t}, x _ {t}, a\right) \\ \geq \sum_ {a \in \mathcal {A}} \pi_ {t} \left(c _ {t}, x _ {t}, a\right) \ell_ {t} (a) \\ \end{array} +$$ + +Now consider the case that $\beta_t \leq 1$ . Since $\ell_t(a) \in [0,1]$ for all $a \in \mathcal{A}$ , we have, by Lemma B.5, that: + +$$ +\begin{array}{l} \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \pi_ {t} (c _ {t}, x _ {t}, a) - \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \xi_ {t} (c _ {t}, x _ {t}, a) = \sum_ {a \in \mathcal {A}} \ell_ {t} (a) (\pi_ {t} (c _ {t}, x _ {t}, a) - \xi_ {t} (c _ {t}, x _ {t}, a)) \\ \leq \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \max \left\{0, \pi_ {t} \left(c _ {t}, x _ {t}, a\right) - \xi_ {t} \left(c _ {t}, x _ {t}, a\right) \right\} \\ \leq \sum_ {a \in \mathcal {A}} \max \left\{0, \pi_ {t} \left(c _ {t}, x _ {t}, a\right) - \xi_ {t} \left(c _ {t}, x _ {t}, a\right) \right\} \\ \leq \beta_ {t} \\ \end{array} +$$ + +Hence, we have that: + +$$ +\begin{array}{l} y _ {t} \left(\xi_ {t}\right) = \llbracket \beta_ {t} \leq 1 \rrbracket \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \xi_ {t} \left(c _ {t}, x _ {t}, a\right) + \tilde {\beta} _ {t} \left(\xi_ {t}\right) \\ = \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \xi_ {t} \left(c _ {t}, x _ {t}, a\right) + \beta_ {t} \\ \geq \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \xi_ {t} \left(c _ {t}, x _ {t}, a\right) + \left(\sum_ {a \in \mathcal {A}} \ell_ {t} (a) \pi_ {t} \left(c _ {t}, x _ {t}, a\right) - \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \xi_ {t} \left(c _ {t}, x _ {t}, a\right)\right) \\ = \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \pi_ {t} \left(c _ {t}, x _ {t}, a\right) \\ \end{array} +$$ + +So in either case we have the result. + +# C.7 Lemma B.7 + +Since $\tilde{\pi} \in \mathcal{F}(\mu_t)$ we have $\tilde{\pi}_t \in \mathcal{Z}_t$ which implies that there exists a function $z: \mathcal{A} \to \mathbb{R}$ such that for all $(c, a) \in \mathcal{C} \times \mathcal{A}$ we have $\tilde{\omega}_t(\tilde{\pi}, c, a) = z(a)$ . Hence, we have, for all $(c, a) \in \mathcal{C} \times \mathcal{A}$ , that: + +$$ +\tilde {\delta} _ {t} (\tilde {\pi}, c, a) = \max _ {c ^ {\prime} \in \mathcal {C}} \tilde {\omega} _ {t} (\tilde {\pi}, c ^ {\prime}, a) - \tilde {\omega} _ {t} (\tilde {\pi}, c, a) = z (a) - z (a) = 0 +$$ + +Since $\ell_t(a)$ and $\tilde{\pi}(c_t, x_t, a)$ are non-negative for all $a \in \mathcal{A}$ , this implies that: + +$$ +\begin{array}{l} y _ {t} (\tilde {\pi}) := \llbracket \beta_ {t} \leq 1 \rrbracket \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \tilde {\pi} \left(c _ {t}, x _ {t}, a\right) + \tilde {\beta} _ {t} (\tilde {\pi}) \\ \leq \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \tilde {\pi} \left(c _ {t}, x _ {t}, a\right) + \sum_ {a \in \mathcal {A}} \max _ {c \in \mathcal {C}} \tilde {\delta} _ {t} (\tilde {\pi}, c, a) \\ = \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \tilde {\pi} \left(c _ {t}, x _ {t}, a\right) \\ \end{array} +$$ + +as required. + +# C.8 Lemma B.8 + +Take any $a\in \mathcal{A}$ .Given $\phi^{*}\in S$ and $c^{\prime}\in \mathcal{C}$ we have: + +$$ +\sum_ {(c, x) \in \mathcal {I}} [ [ c = c ^ {\prime} ] ] \mu_ {t} (c, x) \phi^ {*} (c, x, a) = \sum_ {x \in \mathcal {X}} \mu_ {t} (c ^ {\prime}, x) \phi^ {*} (c ^ {\prime}, x, a) = \tilde {\omega} _ {t} (\phi^ {*}, c ^ {\prime}, a) +$$ + +which implies that: + +$$ +\sum_ {(c, x) \in \mathcal {I}} [ [ c = \tilde {\kappa} _ {t} (\phi , a) ] ] \mu_ {t} (c, x) \phi (c, x, a) = \max _ {c ^ {\prime} \in \mathcal {C}} \tilde {\omega} _ {t} (\phi , c ^ {\prime}, a) \tag {6} +$$ + +$$ +\sum_ {(c, x) \in \mathcal {I}} [ [ c = \tilde {\kappa} _ {t} ^ {\prime} (\phi , a) ] ] \mu_ {t} (c, x) \phi (c, x, a) = \min _ {c ^ {\prime} \in \mathcal {C}} \tilde {\omega} _ {t} (\phi , c ^ {\prime}, a) \tag {7} +$$ + +$$ +\sum_ {(c, x) \in \mathcal {I}} [ [ c = \tilde {\kappa} _ {t} (\phi , a) ] ] \mu_ {t} (c, x) \phi^ {\prime} (c, x, a) \leq \max _ {c ^ {\prime} \in \mathcal {C}} \tilde {\omega} _ {t} \left(\phi^ {\prime}, c ^ {\prime}, a\right) \tag {8} +$$ + +$$ +\sum_ {(c, x) \in \mathcal {I}} [ [ c = \tilde {\kappa} _ {t} ^ {\prime} (\phi , a) ] ] \mu_ {t} (c, x) \phi^ {\prime} (c, x, a) \geq \min _ {c ^ {\prime} \in \mathcal {C}} \tilde {\omega} _ {t} \left(\phi^ {\prime}, c ^ {\prime}, a\right) \tag {9} +$$ + +For all $a\in \mathcal{A}$ , equations (6) and (7) imply that: + +$$ +\begin{array}{l} \sum_ {(c, x) \in \mathcal {I}} g _ {t} (\phi , c, x, a) \phi (c, x, a) \\ = \sum_ {(c, x) \in \mathcal {I}} [ [ \beta_ {t} \leq 1 ] ] [ [ (c, x) = (c _ {t}, x _ {t}) ] ] \ell_ {t} (a) \phi (c, x, a) + \max _ {c ^ {\prime} \in \mathcal {C}} \tilde {\omega} _ {t} (\phi , c ^ {\prime}, a) - \min _ {c ^ {\prime} \in \mathcal {C}} \tilde {\omega} _ {t} (\phi , c ^ {\prime}, a) \\ = \sum_ {(c, x) \in \mathcal {I}} [ [ \beta_ {t} \leq 1 ] ] [ [ (c, x) = (c _ {t}, x _ {t}) ] ] \ell_ {t} (a) \phi (c, x, a) + \max _ {c ^ {\dagger} \in \mathcal {C}} \left(\max _ {c ^ {\prime} \in \mathcal {C}} \tilde {\omega} _ {t} (\phi , c ^ {\prime}, a) - \tilde {\omega} _ {t} (\phi , c ^ {\dagger}, a)\right) \\ = \llbracket \beta_ {t} \leq 1 \rrbracket \ell_ {t} (a) \phi \left(c _ {t}, x _ {t}, a\right) + \max _ {c ^ {\dagger} \in \mathcal {C}} \tilde {\delta} _ {t} (\phi , c ^ {\dagger} a) \\ \end{array} +$$ + +This means that: + +$$ +\begin{array}{l} \sum_ {a \in \mathcal {A}} \sum_ {(c, x) \in \mathcal {I}} g _ {t} (\phi , c, x, a) \phi (c, x, a) = \sum_ {a \in \mathcal {A}} [ [ \beta_ {t} \leq 1 ] ] \ell_ {t} (a) \phi (c _ {t}, x _ {t}, a) + \sum_ {a \in \mathcal {A}} \max _ {c ^ {\dagger} \in \mathcal {C}} \tilde {\delta} _ {t} (\phi , c ^ {\dagger}, a) \\ = \llbracket \beta_ {t} \leq 1 \rrbracket \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \phi \left(c _ {t}, x _ {t}, a\right) + \tilde {\beta} _ {t} (\phi) \\ = y _ {t} (\phi) \tag {10} \\ \end{array} +$$ + +Similarly, for all $a \in \mathcal{A}$ , equations (8) and (9) imply that: + +$$ +\begin{array}{l} \sum_ {(c, x) \in \mathcal {I}} g _ {t} (\phi , c, x, a) \phi^ {\prime} (c, x, a) \\ \leq \sum_ {(c, x) \in \mathcal {I}} [ [ \beta_ {t} \leq 1 ] ] [ [ (c, x) = (c _ {t}, x _ {t}) ] ] \ell_ {t} (a) \phi^ {\prime} (c, x, a) + \max _ {c ^ {\prime} \in \mathcal {C}} \tilde {\omega} _ {t} (\phi^ {\prime}, c ^ {\prime}, a) - \min _ {c ^ {\prime} \in \mathcal {C}} \tilde {\omega} _ {t} (\phi^ {\prime}, c ^ {\prime}, a) \\ = \sum_ {(c, x) \in \mathcal {I}} [ [ \beta_ {t} \leq 1 ] ] [ [ (c, x) = (c _ {t}, x _ {t}) ] ] \ell_ {t} (a) \phi^ {\prime} (c, x, a) + \max _ {c ^ {\dagger} \in \mathcal {C}} \left(\max _ {c ^ {\prime} \in \mathcal {C}} \tilde {\omega} _ {t} (\phi^ {\prime}, c ^ {\prime}, a) - \tilde {\omega} _ {t} (\phi^ {\prime}, c ^ {\dagger}, a)\right) \\ = \llbracket \beta_ {t} \leq 1 \rrbracket \ell_ {t} (a) \phi^ {\prime} \left(c _ {t}, x _ {t}, a\right) + \max _ {c ^ {\dagger} \in \mathcal {C}} \tilde {\delta} _ {t} (\phi , c ^ {\dagger}, a) \\ \end{array} +$$ + +This means that: + +$$ +\begin{array}{l} \sum_ {a \in \mathcal {A}} \sum_ {(c, x) \in \mathcal {I}} g _ {t} (\phi , c, x, a) \phi^ {\prime} (c, x, a) \leq \sum_ {a \in \mathcal {A}} [ [ \beta_ {t} \leq 1 ] ] \ell_ {t} (a) \phi^ {\prime} (c _ {t}, x _ {t}, a) + \sum_ {a \in \mathcal {A}} \max _ {c ^ {\dagger} \in \mathcal {C}} \tilde {\delta} _ {t} (\phi , c ^ {\dagger}, a) \\ = \llbracket \beta_ {t} \leq 1 \rrbracket \sum_ {a \in \mathcal {A}} \ell_ {t} (a) \phi^ {\prime} \left(c _ {t}, x _ {t}, a\right) + \tilde {\beta} _ {t} \left(\phi^ {\prime}\right) \\ = y _ {t} \left(\phi^ {\prime}\right) \tag {11} \\ \end{array} +$$ + +Equations 10 and 11 give us: + +$$ +\begin{array}{l} \sum_ {(c, x) \in \mathcal {I}} \sum_ {a \in \mathcal {A}} g _ {t} (\phi , c, x, a) (\phi (c, x, a) - \phi^ {\prime} (c, x, a)) = \sum_ {(c, x) \in \mathcal {I}} \sum_ {a \in \mathcal {A}} g _ {t} (\phi , c, x, a) \phi (c, x, a) - \sum_ {(c, x) \in \mathcal {I}} \sum_ {a \in \mathcal {A}} g _ {t} (\phi , c, x, a) \phi^ {\prime} (c, x, a) \\ \geq y _ {t} (\phi) - y _ {t} \left(\phi^ {\prime}\right) \\ \end{array} +$$ + +as required. + +# C.9 Lemma B.9 + +In this proof we use, throughout, the property of linearity of expectation. Take any $a \in \mathcal{A}$ . Since $x_{t}$ , $c_{t}$ , $\pi_{t}$ and $\ell_t(a)$ are deterministic when conditioned on $\xi_{t}$ we have: + +$$ +\begin{array}{l} \mathbb {E} \left[ \llbracket a = a _ {t} \rrbracket \frac {\ell_ {t} (a _ {t})}{\pi_ {t} \left(c _ {t} , x _ {t} , a _ {t}\right)} \Bigg | \xi_ {t} \right] = \mathbb {E} \left[ \llbracket a = a _ {t} \rrbracket \frac {\ell_ {t} (a)}{\pi_ {t} \left(c _ {t} , x _ {t} , a\right)} \Bigg | \xi_ {t} \right] \\ = \frac {\ell_ {t} (a)}{\pi_ {t} \left(c _ {t} , x _ {t} , a\right)} \mathbb {E} \left[ \llbracket a = a _ {t} \rrbracket \mid \xi_ {t} \right] \\ = \frac {\ell_ {t} (a)}{\pi_ {t} \left(c _ {t} , x _ {t} , a\right)} \mathbb {P} [ a = a _ {t} \mid \xi_ {t} ] \\ = \frac {\ell_ {t} (a)}{\pi_ {t} \left(c _ {t} , x _ {t} , a\right)} \pi_ {t} \left(c _ {t}, x _ {t}, a\right) \\ = \ell_ {t} (a) \tag {12} \\ \end{array} +$$ + +Which immediately gives us the result. + +# C.10 Lemma B.10 + +By lemmas B.6 and B.7 we have: + +$$ +r _ {t} = \sum_ {a \in \mathcal {A}} \pi_ {t} \left(c _ {t}, x _ {t}, a\right) \ell_ {t} (a) - \sum_ {a \in \mathcal {A}} \tilde {\pi} \left(c _ {t}, x _ {t}, a\right) \ell_ {t} (a) \leq y _ {t} \left(\xi_ {t}\right) - y _ {t} (\tilde {\pi}) \tag {13} +$$ + +and by lemmas B.8 and B.9 and the linearity of expectation we have, since $\tilde{\pi}$ is independent of $\xi_{t}$ , that: + +$$ +\begin{array}{l} y _ {t} (\xi_ {t}) - y _ {t} (\tilde {\pi}) \leq \sum_ {(c, x) \in \mathcal {I}} \sum_ {a \in \mathcal {A}} g _ {t} (\xi_ {t}, c, x, a) \left(\xi_ {t} (c, x, a) - \tilde {\pi} (c, x, a)\right) \\ = \sum_ {(c, x) \in \mathcal {I}} \sum_ {a \in \mathcal {A}} \mathbb {E} [ \lambda_ {t} (c, x, a) \mid \xi_ {t} ] (\xi_ {t} (c, x, a) - \tilde {\pi} (c, x, a)) \\ = \mathbb {E} \left[ \sum_ {(c, x) \in \mathcal {I}} \sum_ {a \in \mathcal {A}} \left(\xi_ {t} (c, x, a) - \tilde {\pi} (c, x, a)\right) \lambda_ {t} (c, x, a) \mid \xi_ {t} \right] \tag {14} \\ \end{array} +$$ + +Combining equations (13) and (14) gives us the result. + +# C.11 Lemma B.11 + +By Lemma B.1 we have: + +$$ +\begin{array}{l} \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \nu_ {t} (c, e) = \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \sum_ {x \in \mathcal {X}} \lambda_ {t} (c, x, e (x)) \\ = \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} [ [ e (x) = a ] ] \lambda_ {t} (c, x, a) \\ = \sum_ {(c, x) \in \mathcal {I}} \sum_ {a \in \mathcal {A}} \lambda_ {t} (c, x, a) \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) [ [ e (x) = a ] ] \\ = \sum_ {(c, x) \in \mathcal {I}} \sum_ {a \in \mathcal {A}} \lambda_ {t} (c, x, a) \xi_ {t} (c, x, a) \\ \end{array} +$$ + +and by definition of $\vartheta^{*}$ we have: + +$$ +\begin{array}{l} \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \nu_ {t} (c, e) = \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \sum_ {x \in \mathcal {X}} \lambda_ {t} (c, x, e (x)) \\ = \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} [ [ e (x) = a ] ] \lambda_ {t} (c, x, a) \\ = \sum_ {(c, x) \in \mathcal {I}} \sum_ {a \in \mathcal {A}} \lambda_ {t} (c, x, a) \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) [ [ e (x) = a ] ] \\ = \sum_ {(c, x) \in \mathcal {I}} \sum_ {a \in \mathcal {A}} \lambda_ {t} (c, x, a) \tilde {\pi} (c, x, a) \\ \end{array} +$$ + +so since $\xi_{t}$ is derived solely from $\vartheta_{t}$ (and deterministic objects) we have: + +$$ +\mathbb {E} \left[ \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \left(\vartheta_ {t} (c, e) - \vartheta^ {*} (c, e)\right) \nu_ {t} (c, e) \Bigg | \vartheta_ {t} \right] = \mathbb {E} \left[ \sum_ {c \in \mathcal {C}} \sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} \left(\xi_ {t} (c, x, a) - \tilde {\pi} (c, x, a) \lambda_ {t} (c, x, a) \right. \Bigg | \xi_ {t} \right] +$$ + +so Lemma B.10 gives us the result. + +# C.12 Lemma B.12 + +Since $(z + \hat{z})^2\leq 2z^2 +2\hat{z}^2$ for all $z,\hat{z}\in \mathbb{R}$ and since $(z - \hat{z})^{2}\leq z^{2} + \hat{z}^{2}$ for all $z,\hat{z} >0$ , we have: + +$$ +\begin{array}{l} \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \nu_ {t} (c, e) ^ {2} = \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \left(\sum_ {x \in \mathcal {X}} \lambda_ {t} (c, x, e (x))\right) ^ {2} \\ = \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \left(\sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} [ [ e (x) = a ] ] \lambda_ {t} (c, x, a)\right) ^ {2} \\ \leq 2 \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \left(\sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} [ [ e (x) = a ] ] [ [ \beta_ {t} \leq 1 ] ] [ [ (c, x, a) = (c _ {t}, x _ {t}, a _ {t}) ] ] \frac {\ell_ {t} (a _ {t})}{\pi_ {t} (c _ {t} , x _ {t} , a _ {t})}\right) ^ {2} \\ + 2 \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \left(\sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} [ [ e (x) = a ] ] [ [ c = \tilde {\kappa} _ {t} (\xi_ {t}, a) ] ] \mu_ {t} (c, x)\right) ^ {2} \\ + 2 \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \left(\sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} [ [ e (x) = a ] ] [ [ c = \tilde {\kappa} _ {t} ^ {\prime} (\xi_ {t}, a) ] ] \mu_ {t} (c, x)\right) ^ {2} \tag {15} \\ \end{array} +$$ + +Note that: + +$$ +\begin{array}{l} \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \left(\sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} [ [ e (x) = a ] ] [ [ c = \tilde {\kappa} _ {t} (\xi_ {t}, a) ] ] \mu_ {t} (c, x)\right) ^ {2} \\ = \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \left(\sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} [ [ e (x) = a ] ] [ [ c = \tilde {\kappa} _ {t} (\xi_ {t}, a) ] ] \mu_ {t} (c, x)\right) \left(\sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} [ [ e (x) = a ] ] [ [ c = \tilde {\kappa} _ {t} (\xi_ {t}, a) ] ] \mu_ {t} (c, x)\right) \\ \leq \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \left(\sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} [ [ c = \tilde {\kappa} _ {t} (\xi_ {t}, a) ] ] \mu_ {t} (c, x)\right) \left(\sum_ {x \in \mathcal {X}} \mu_ {t} (c, x) \sum_ {a \in \mathcal {A}} [ [ e (x) = a ] ]\right) \\ = \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \left(\sum_ {a \in \mathcal {A}} [ [ c = \tilde {\kappa} _ {t} (\xi_ {t}, a) ] ] \sum_ {x \in \mathcal {X}} \mu_ {t} (c, x)\right) \left(\sum_ {x \in \mathcal {X}} \mu_ {t} (c, x)\right) \\ = \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \sum_ {a \in \mathcal {A}} [ [ c = \tilde {\kappa} _ {t} (\xi_ {t}, a) ] ] \\ = \sum_ {a \in \mathcal {A}} \sum_ {c \in \mathcal {C}} [ [ c = \tilde {\kappa} _ {t} (\xi_ {t}, a) ] ] \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \\ = \sum_ {a \in \mathcal {A}} \sum_ {c \in \mathcal {C}} [ [ c = \tilde {\kappa} _ {t} (\xi_ {t}, a) ] ] \\ = \sum_ {a \in \mathcal {A}} 1 \\ = K \tag {16} \\ \end{array} +$$ + +Similarly we have: + +$$ +\sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \left(\sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} [ [ e (x) = a ] ] [ [ c = \tilde {\kappa} _ {t} ^ {\prime} (\xi_ {t}, a) ] ] \mu_ {t} (c, x)\right) ^ {2} = K \tag {17} +$$ + +Now note that, by Lemma B.4, we have, for all $a \in \mathcal{A}$ that: + +$$ +\frac {\llbracket \beta_ {t} \leq 1 \rrbracket}{\pi_ {t} \left(c _ {t} , x _ {t} , a\right)} \leq \frac {2}{\xi_ {t} \left(c _ {t} , x _ {t} , a\right)} +$$ + +so by Lemma B.1 we have: + +$$ +\begin{array}{l} \mathbb {E} \left[ \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \left(\sum_ {x \in \mathcal {X}} \sum_ {a \in \mathcal {A}} [ [ e (x) = a ] ] [ [ \beta_ {t} \leq 1 ] ] [ [ (c, x, a) = (c _ {t}, x _ {t}, a _ {t}) ] ] \frac {\ell_ {t} (a _ {t})}{\pi_ {t} (c _ {t} , x _ {t} , a _ {t})}\right) ^ {2} \Bigg | \vartheta_ {t} \right] \\ = \mathbb {E} \left[ \sum_ {e \in \mathcal {H}} \vartheta_ {t} \left(c _ {t}, e\right) \left(\llbracket e \left(x _ {t}\right) = a _ {t} \rrbracket \llbracket \beta_ {t} \leq 1 \rrbracket \frac {\ell_ {t} \left(a _ {t}\right)}{\pi_ {t} \left(c _ {t} , x _ {t} , a _ {t}\right)}\right) ^ {2} \mid \vartheta_ {t} \right] \\ = \sum_ {a \in \mathcal {A}} \mathbb {P} [ a _ {t} = a \mid \vartheta_ {t} ] \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c _ {t}, e) \left(\llbracket e (x _ {t}) = a \rrbracket \llbracket \beta_ {t} \leq 1 \rrbracket \frac {\ell_ {t} (a)}{\pi_ {t} (c _ {t} , x _ {t} , a)}\right) ^ {2} \\ = \sum_ {a \in \mathcal {A}} \pi_ {t} \left(c _ {t}, x _ {t}, a\right) \sum_ {e \in \mathcal {H}} \vartheta_ {t} \left(c _ {t}, e\right) \left(\llbracket e \left(x _ {t}\right) = a \rrbracket \llbracket \beta_ {t} \leq 1 \rrbracket \frac {\ell_ {t} (a)}{\pi_ {t} \left(c _ {t} , x _ {t} , a\right)}\right) ^ {2} \\ = \sum_ {a \in \mathcal {A}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} \left(c _ {t}, e\right) [ [ e (x _ {t}) = a ] ] [ [ \beta_ {t} \leq 1 ] ] \frac {\ell_ {t} (a) ^ {2}}{\pi_ {t} \left(c _ {t} , x _ {t} , a\right)} \\ \leq \sum_ {a \in \mathcal {A}} \sum_ {e \in \mathcal {H}} \vartheta_ {t} \left(c _ {t}, e\right) [ [ e (x _ {t}) = a ] ] \frac {[ [ \beta_ {t} \leq 1 ] ]}{\pi_ {t} \left(c _ {t} , x _ {t} , a\right)} \\ \leq \sum_ {a \in \mathcal {A}} \frac {2}{\xi_ {t} (c _ {t} , x _ {t} , a)} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c _ {t}, e) [ [ e (x _ {t}) = a ] ] \\ \leq \sum_ {a \in \mathcal {A}} \frac {2}{\xi_ {t} \left(c _ {t} , x _ {t} , a\right)} \xi_ {t} \left(c _ {t}, x _ {t}, a\right) \\ \leq \sum_ {a \in \mathcal {A}} 2 \\ = 2 K \tag {18} \\ \end{array} +$$ + +Taking expectations on Equation (15) and then substituting in equations (16), (17) and (18) gives us the result. + +# C.13 Lemma B.13 + +Take any $(c,e)\in \mathcal{C}\times \mathcal{H}$ . Note first that for all $(x,a)\in \mathcal{X}\times \mathcal{A}$ we have: + +$$ +\lambda_ {t} (c, x, a) \geq - [ [ c = \tilde {\kappa} _ {t} ^ {\prime} (\xi_ {t}, a) ] ] \mu_ {t} (c, x) \geq - \mu_ {t} (c, x) +$$ + +so that: + +$$ +\begin{array}{l} \nu_ {t} (c, e) = \sum_ {x \in \mathcal {X}} \lambda_ {t} (c, x, e (x)) \\ \geq \sum_ {a \in \mathcal {A}} \sum_ {x \in \mathcal {X}} \lambda_ {t} (c, x, a) \\ \geq - \sum_ {a \in \mathcal {A}} \sum_ {x \in \mathcal {X}} \mu_ {t} (c, x) \\ = - \sum_ {a \in \mathcal {A}} 1 \\ \geq - K \\ \end{array} +$$ + +as required. + +# C.14 Lemma B.14 + +By the description of the UPDATE subroutine in Section ?? we see that our update procedure is equivalent to creating a function $\vartheta_t' \in \mathbb{R}^{\mathcal{C} \times \mathcal{H}}$ defined such that for all $(c, e) \in \mathcal{C} \times \mathcal{H}$ we have: + +$$ +\vartheta_ {t} ^ {\prime} (c, e) = \vartheta_ {t} (c, e) \prod_ {x \in \mathcal {G} _ {t} (c)} \exp (- \hat {\eta} \lambda_ {t} (c, x, e (x))) +$$ + +and then, for all $c \in \mathcal{C}$ , normalising $\vartheta_t'(c, \circ)$ to create $\vartheta_t(c, \circ)$ . Note that for all $(c, x) \in \mathcal{I}$ with $x \notin \mathcal{G}_t(c)$ we have $\mu_t(c, x) = 0$ and $(c, x) \neq (c_t, x_t)$ so that $\lambda_t(c, x, a) = 0$ for all $a \in \mathcal{A}$ . This implies that for all $(c, e) \in \mathcal{C} \times \mathcal{H}$ we have: + +$$ +\vartheta_ {t} ^ {\prime} (c, e) = \vartheta_ {t} (c, e) \prod_ {x \in \mathcal {X}} \exp (- \hat {\eta} \lambda_ {t} (c, x, e (x))) = \vartheta_ {t} (c, e) \exp \left(- \hat {\eta} \sum_ {x \in \mathcal {X}} \lambda_ {t} (c, x, e (x))\right) = \vartheta_ {t} (c, e) \exp (- \hat {\eta} \nu_ {t} (c, e)) +$$ + +so that: + +$$ +\vartheta_ {t + 1} (c, e) = \frac {\vartheta_ {t} (c , e) \exp (- \hat {\eta} \nu_ {t} (c , e))}{\sum_ {e ^ {\prime} \in \mathcal {H}} \vartheta_ {t} (c , e ^ {\prime}) \exp (- \hat {\eta} \nu_ {t} (c , e ^ {\prime}))} +$$ + +as required. + +# C.15 Lemma B.15 + +Take any $c\in \mathcal{C}$ . Define: + +$$ +z := \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \exp (- \hat {\eta} \nu_ {t} (c, e)) +$$ + +By Lemma B.14 we have: + +$$ +\begin{array}{l} B \left(\vartheta^ {*}, \vartheta_ {t} (c, \circ)\right) - B \left(\vartheta^ {*}, \vartheta_ {t + 1} (c, \circ)\right) = \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \left(\ln \left(\frac {\vartheta^ {*} (c , e)}{\vartheta_ {t} (c , e)}\right) - \ln \left(\frac {\vartheta^ {*} (c , e)}{\vartheta_ {t + 1} (c , e)}\right)\right) \\ = \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \ln \left(\frac {\vartheta_ {t + 1} (c , e)}{\vartheta_ {t} (c , e)}\right) \\ = \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \ln \left(\frac {\exp (- \hat {\eta} \nu_ {t} (c , e))}{z}\right) \\ = - \hat {\eta} \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \nu_ {t} (c, e) - \ln (z) \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \\ = - \hat {\eta} \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \nu_ {t} (c, e) - \ln (z) \tag {19} \\ \end{array} +$$ + +Since $\eta \leq \sqrt{T / K}$ we have $\hat{\eta} \leq 1 / K$ so that, by Lemma B.13, we have, for all $e \in \mathcal{H}$ that: + +$$ +\hat {\eta} \nu_ {t} (c, e) \geq - 1 +$$ + +so, since $\exp (-\hat{z})\leq 1 - \hat{z} +\hat{z}^2$ for all $x\geq -1$ and $\ln (1 + \hat{z})\leq \hat{z}$ for all $\hat{z}\in \mathbb{R}$ , we have: + +$$ +\begin{array}{l} \ln (z) = \ln \left(\sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \exp (- \hat {\eta} \nu_ {t} (c, e))\right) \\ \leq \ln \left(\sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \left(1 - \hat {\eta} \nu_ {t} (c, e) + \hat {\eta} ^ {2} \nu_ {t} (c, e) ^ {2}\right)\right) \\ = \ln \left(\sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) + \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) (- \hat {\eta} \nu_ {t} (c, e) + \hat {\eta} ^ {2} \nu_ {t} (c, e) ^ {2})\right) \\ = \ln \left(1 + \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) (- \hat {\eta} \nu_ {t} (c, e) + \hat {\eta} ^ {2} \nu_ {t} (c, e) ^ {2})\right) \\ \leq \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) (- \hat {\eta} \nu_ {t} (c, e) + \hat {\eta} ^ {2} \nu_ {t} (c, e) ^ {2}) \\ = - \hat {\eta} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \nu_ {t} (c, e) + \hat {\eta} ^ {2} \sum_ {e \in \mathcal {H}} \vartheta_ {t} (c, e) \nu_ {t} (c, e) ^ {2} \tag {20} \\ \end{array} +$$ + +Substituting Equation (20) into Equation (19) gives us the result. + +# C.16 Lemma B.16 + +Immediate from summing the inequality in Lemma B.15 over all $c \in \mathcal{C}$ , taking expectations (conditioned on $\vartheta_t$ ) and substituting in the inequalities in lemmas B.11 and B.12. + +# C.17 Lemma B.17 + +From Lemma B.16 we have, for all $t \in [T]$ , that: + +$$ +\hat {\eta} \mathbb {E} [ r _ {t} ] - 8 \hat {\eta} ^ {2} K = \mathbb {E} \left[ \sum_ {c \in \mathcal {C}} (B (\vartheta^ {*}, \vartheta_ {t} (c, \circ)) - B (\vartheta^ {*}, \vartheta_ {t + 1} (c, \circ))) \right] +$$ + +and hence, by summing over all $t \in [T]$ and noting the linearity of expectation and the fact that the relative entropy is non-negative, we have: + +$$ +\begin{array}{l} \hat {\eta} \sum_ {t \in [ T ]} \mathbb {E} \left[ r _ {t} \right] - 8 \hat {\eta} ^ {2} K T = \mathbb {E} \left[ \sum_ {c \in \mathcal {C}} \sum_ {t \in [ T ]} \left(B \left(\vartheta^ {*}, \vartheta_ {t} (c, \circ)\right) - B \left(\vartheta^ {*}, \vartheta_ {t + 1} (c, \circ)\right)\right) \right] \\ = \mathbb {E} \left[ \sum_ {c \in \mathcal {C}} \left(B \left(\vartheta^ {*}, \vartheta_ {1} (c, \circ)\right) - B \left(\vartheta^ {*}, \vartheta_ {T + 1} (c, \circ)\right)\right) \right] \\ \leq \mathbb {E} \left[ \sum_ {c \in C} B \left(\vartheta^ {*}, \vartheta_ {1} (c, \circ)\right) \right] \\ = \sum_ {c \in \mathcal {C}} B (\vartheta^ {*}, \vartheta) \\ \end{array} +$$ + +so that: + +$$ +\sum_ {t \in [ T ]} \mathbb {E} [ r _ {t} ] \leq \frac {1}{\hat {\eta}} \sum_ {c \in \mathcal {C}} B (\vartheta^ {*}, \vartheta) + 8 \hat {\eta} K T +$$ + +noting that $\hat{\eta} = \eta / \sqrt{KT}$ , we have the result. + +# D Details of Examples + +We now describe and analyse the instances of SPEW required to obtain the results in Section 5. We start with the following lemma: + +Lemma D.1. Given a tree $\mathcal{T}$ with edge set $\mathcal{E}$ and any $\pi : \mathcal{T} \times \mathcal{A} \to [0,1]$ with $\pi(v, \circ) \in \Delta_{\mathcal{A}}$ for all $v \in \mathcal{T}$ , there exists a distribution $\theta \in \Delta_{\mathcal{A}^{\mathcal{T}}}$ such that: + +$$ +\pi (v, a) = \sum_ {e \in \mathcal {A} ^ {\mathcal {T}}} [ [ e (v) = a ] ] \theta (e) \quad \forall (v, a) \in \mathcal {T} \times \mathcal {A} +$$ + +and: + +$$ +\sum_ {e \in \mathcal {A} ^ {\mathcal {T}}} \theta (e) \sum_ {(u, v) \in \mathcal {E}} [ [ e (u) \neq e (v) ] ] = \frac {1}{2} \sum_ {(u, v) \in \mathcal {E}} \sum_ {a \in \mathcal {A}} | \pi (u, a) - \pi (v, a) | +$$ + +Proof. We prove by induction on $|\mathcal{T}|$ . The inductive hypothesis clearly holds for $|\mathcal{T}| = 1$ . Now consider a tree $\mathcal{T}$ with $|\mathcal{T}| > 1$ and let $\lambda$ be a leaf of $\mathcal{T}$ . Assume that the inductive hypothesis holds for the tree $\mathcal{T}'$ created by removing $\lambda$ from $\mathcal{T}$ . By now proving that the inductive hypothesis holds for $\mathcal{T}$ we will have completed the proof of the lemma. + +Let $\lambda'$ be the neighbour of $\lambda$ . For all $a \in \mathcal{A}$ let $\xi(a) \coloneqq \pi(\lambda, a)$ and $\xi'(a) \coloneqq \pi(\lambda', a)$ . Let $\mathcal{S}$ be the set of all $a \in \mathcal{A}$ with $\xi(a) \leq \xi'(a)$ . For all $a \in \mathcal{S}$ let: + +$$ +r (a) := \frac {\xi (a)}{\xi^ {\prime} (a)} +$$ + +for all $a\in \mathcal{A}\setminus S$ let + +$$ +q (a) := \frac {\xi (a) - \xi^ {\prime} (a)}{\sum_ {a ^ {\prime} \in \mathcal {A} \backslash \mathcal {S}} (\xi (a ^ {\prime}) - \xi^ {\prime} (a ^ {\prime}))} +$$ + +For all $e \in \mathcal{A}^{\mathcal{T}'}$ and $a \in \mathcal{A}$ let $\gamma(e, a)$ be the function $e' \in \mathcal{A}^{\mathcal{T}}$ with $e'(\lambda) = a$ and $e'(v) = e(v)$ for all $v \in \mathcal{T}'$ . Let $\mathcal{E}'$ be the edge set of $\mathcal{T}'$ . By the inductive hypothesis, let $\theta' \in \Delta_{\mathcal{A}^{\mathcal{T}'}}$ be such that: + +$$ +\pi (v, a) = \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} [ [ e (v) = a ] ] \theta^ {\prime} (e) \quad \forall (v, a) \in \mathcal {T} ^ {\prime} \times \mathcal {A} +$$ + +and: + +$$ +\sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} \theta^ {\prime} (e) \sum_ {(u, v) \in \mathcal {E} ^ {\prime}} [ [ e (u) \neq e (v) ] ] = \frac {1}{2} \sum_ {(u, v) \in \mathcal {E} ^ {\prime}} \sum_ {a \in \mathcal {A}} | \pi (u, a) - \pi (v, a) | +$$ + +For all $a \in \mathcal{A} \setminus S$ and $e \in \mathcal{A}^{\mathcal{T}'}$ with $e(\lambda') = a$ define: + +$$ +\theta (\gamma (e, a)) := \theta^ {\prime} (e) +$$ + +For all $a \in S$ and $e \in \mathcal{A}^{\mathcal{T}'}$ with $e(\lambda') = a$ define: + +$$ +\theta (\gamma (e, a)) := r (a) \theta^ {\prime} (e) +$$ + +For all $a\in S$ $a^\prime \in \mathcal{A}\setminus S$ and $e\in \mathcal{A}^{\mathcal{T}'}$ with $e(\lambda ') = a$ define: + +$$ +\theta (\gamma (e, a ^ {\prime})) := q (a ^ {\prime}) (1 - r (a)) \theta^ {\prime} (e) +$$ + +For all other $e' \in \mathcal{A}^T$ we define $\theta(e') := 0$ . + +Note that for all $v \in \mathcal{T}'$ and $a \in \mathcal{A}$ we have: + +$$ +\pi (v, a) = \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} [ [ e (v) = a ] ] \theta^ {\prime} (e) = \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} [ [ e (v) = a ] ] \sum_ {a ^ {\prime} \in \mathcal {A}} \theta (\gamma (e, a ^ {\prime})) = \sum_ {e ^ {\prime} \in \mathcal {A} ^ {\mathcal {T}}} [ [ e ^ {\prime} (v) = a ] ] \theta (e ^ {\prime}) +$$ + +and for all $a \in S$ we have: + +$$ +\begin{array}{l} \pi (\lambda , a) = \xi (a) = r (a) \xi^ {\prime} (a) = r (a) \pi (\lambda^ {\prime}, a) = r (a) \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} [ [ e (\lambda^ {\prime}) = a ] ] \theta^ {\prime} (e) \\ = \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} [ [ e (\lambda^ {\prime}) = a ] ] \theta (\gamma (e, a)) = \sum_ {e ^ {\prime} \in \mathcal {A} ^ {\mathcal {T}}} [ [ e ^ {\prime} (\lambda) = a ] ] \theta (e ^ {\prime}) \\ \end{array} +$$ + +and for all $a \in \mathcal{A} \setminus S$ we have: + +$$ +\begin{array}{l} \pi (\lambda , a) = \xi (a) \\ = \xi^ {\prime} (a) + (\xi (a) - \xi^ {\prime} (a)) \\ = \xi^ {\prime} (a) + q (a) \sum_ {a ^ {\prime} \in \mathcal {A} \backslash \mathcal {S}} \left(\xi \left(a ^ {\prime}\right) - \xi^ {\prime} \left(a ^ {\prime}\right)\right) \\ = \xi^ {\prime} (a) + q (a) \sum_ {a ^ {\prime} \in \mathcal {S}} \left(\xi^ {\prime} \left(a ^ {\prime}\right) - \xi \left(a ^ {\prime}\right)\right) \\ = \xi^ {\prime} (a) + q (a) \sum_ {a ^ {\prime} \in \mathcal {S}} (1 - r (a ^ {\prime})) \xi^ {\prime} (a ^ {\prime}) \\ = \pi \left(\lambda^ {\prime}, a\right) + q (a) \sum_ {a ^ {\prime} \in \mathcal {S}} \left(1 - r \left(a ^ {\prime}\right)\right) \pi \left(\lambda^ {\prime}, a ^ {\prime}\right) \\ = \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} [ [ e (\lambda^ {\prime}) = a ] ] \theta^ {\prime} (e) + q (a) \sum_ {a ^ {\prime} \in \mathcal {S}} \left(1 - r (a ^ {\prime})\right) \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} [ [ e (\lambda^ {\prime}) = a ^ {\prime} ] ] \theta^ {\prime} (e) \\ = \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} [ [ e (\lambda^ {\prime}) = a ] ] \theta^ {\prime} (e) + \sum_ {a ^ {\prime} \in \mathcal {S}} \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} [ [ e (\lambda^ {\prime}) = a ^ {\prime} ] ] q (a) (1 - r (a ^ {\prime})) \theta^ {\prime} (e) \\ = \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} [ [ e (\lambda^ {\prime}) = a ] ] \theta (\gamma (e, a)) + \sum_ {a ^ {\prime} \in \mathcal {S}} \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} [ [ e (\lambda^ {\prime}) = a ^ {\prime} ] ] \theta (\gamma (e, a)) \\ = \sum_ {e ^ {\prime} \in \mathcal {A} ^ {\mathcal {T}}} [ [ e ^ {\prime} (\lambda) = a ] ] \theta (e ^ {\prime}) \\ \end{array} +$$ + +We have now shown that for all $v \in \mathcal{T}$ and $a \in \mathcal{A}$ we have: + +$$ +\pi (v, a) = \sum_ {e \in \mathcal {A} ^ {\mathcal {T}}} [ [ e (v) = a ] ] \theta (e) +$$ + +as required. + +For all $e\in \mathcal{A}^T$ define: + +$$ +\Lambda (e) := \sum_ {(u, v) \in \mathcal {E}} [ [ e (u) \neq e (v) ] ] +$$ + +and for all $e\in \mathcal{A}^{\mathcal{T}'}$ define: + +$$ +\Lambda (e) := \sum_ {(u, v) \in \mathcal {E} ^ {\prime}} [ [ e (u) \neq e (v) ] ] +$$ + +Note that for all $a \in \mathcal{A} \setminus S$ and $e \in \mathcal{A}^{\mathcal{T}'}$ with $e(\lambda') = a$ we have: + +$$ +\sum_ {a ^ {\prime} \in \mathcal {A}} \theta (\gamma (e, a ^ {\prime})) \Lambda (\gamma (e, a ^ {\prime})) = \theta (\gamma (e, a)) \Lambda (\gamma (e, a)) = \theta^ {\prime} (e) \Lambda (e) +$$ + +and for all $a \in S$ and $e \in \mathcal{A}^{\mathcal{T}'}$ with $e(\lambda') = a$ we have: + +$$ +\begin{array}{l} \sum_ {a ^ {\prime} \in \mathcal {A}} \theta (\gamma (e, a ^ {\prime})) \Lambda (\gamma (e, a ^ {\prime})) = \theta (\gamma (e, a)) \Lambda (\gamma (e, a)) + \sum_ {a ^ {\prime} \in \mathcal {A} \backslash \mathcal {S}} \theta (\gamma (e, a ^ {\prime})) \Lambda (\gamma (e, a ^ {\prime})) \\ = \theta (\gamma (e, a)) \Lambda (e) + (\Lambda (e) + 1) \sum_ {a ^ {\prime} \in \mathcal {A} \backslash \mathcal {S}} \theta (\gamma (e, a ^ {\prime})) \\ = r (a) \theta^ {\prime} (e) \Lambda (e) + (\Lambda (e) + 1) (1 - r (a)) \theta^ {\prime} (e) \\ = \theta^ {\prime} (e) \Lambda (e) + (1 - r (a)) \theta^ {\prime} (e) \\ \end{array} +$$ + +Hence, we have: + +$$ +\begin{array}{l} \sum_ {e ^ {\prime} \in \mathcal {A} ^ {\mathcal {T}}} \theta (e ^ {\prime}) \Lambda (e ^ {\prime}) = \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} \theta^ {\prime} (e) \Lambda (e) + \sum_ {a \in \mathcal {S}} (1 - r (a)) \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} [ [ e (\lambda^ {\prime}) = a ] ] \theta^ {\prime} (e) \\ = \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} \theta^ {\prime} (e) \Lambda (e) + \sum_ {a \in \mathcal {S}} (1 - r (a)) \pi (\lambda^ {\prime}, a) \\ = \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} \theta^ {\prime} (e) \Lambda (e) + \sum_ {a \in \mathcal {S}} (1 - r (a)) \xi^ {\prime} (a) \\ = \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} \theta^ {\prime} (e) \Lambda (e) + \sum_ {a \in \mathcal {S}} \left(\xi^ {\prime} (a) - \xi (a)\right) \\ = \sum_ {e \in \mathcal {A} ^ {\mathcal {T} ^ {\prime}}} \theta^ {\prime} (e) \Lambda (e) + \frac {1}{2} \sum_ {a \in \mathcal {A}} | \pi (\lambda , a) - \pi (\lambda^ {\prime}, a) | \\ = \frac {1}{2} \sum_ {(u, v) \in \mathcal {E}} \sum_ {a \in \mathcal {A}} | \pi (u, a) - \pi (v, a) | \\ \end{array} +$$ + +as required. + +![](images/d6d38ab862234d5372e2f34a0f3bf1aa4d47493e71d9f2b35963d2830a6823d0.jpg) + +# D.1 Trees with a known distribution + +Here we define $\mathcal{X} \coloneqq \mathcal{V}$ and $\chi_t(c, v) \coloneqq v$ for all $t \in [T]$ and $(c, v) \in \mathcal{C} \times \mathcal{V}$ . We define $\vartheta$ by: + +$$ +\vartheta (e) := \frac {1}{K} \prod_ {(u, v) \in \mathcal {E}} \left(\llbracket e (u) = e (v) \rrbracket \left(1 - \frac {1}{N}\right) + \llbracket e (u) \neq e (v) \rrbracket \frac {1}{(K - 1) N}\right) \quad \forall e \in \mathcal {H} +$$ + +This instance of HEDGE can be implemented by belief propagation [28], which gives us the time and space complexity bounds. + +For all $t\in [T]$ , we define the target $\mu_t$ by: + +$$ +\mu_ {t} (c, x) := \frac {\rho (c , x)}{\sum_ {v \in \mathcal {X}} \rho (c , v)} \quad \forall (c, x) \in \mathcal {C} \times \mathcal {X} +$$ + +To analyse the regret, suppose we have a policy $\tilde{\pi} \in \mathcal{Q}$ which has statistical parity with respect to $\rho$ , noting that: + +$$ +\tilde {\pi} \in \bigcap_ {t \in [ T ]} \mathcal {F} (\mu_ {t}). +$$ + +By Lemma D.1 we have $\vartheta^{*}:\mathcal{C}\times \mathcal{H}\to [0,1]$ with: + +$$ +\tilde {\pi} (c, x, a) = \sum_ {e \in \mathcal {H}} [ [ e (x) = a ] ] \vartheta^ {*} (c, e) \quad \forall (c, x, a) \in \mathcal {C} \times \mathcal {X} \times \mathcal {A} +$$ + +and: + +$$ +\sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \sum_ {(u, v) \in \mathcal {E}} [ [ e (u) \neq e (v) ] ] = \frac {1}{2} \sum_ {(u, v) \in \mathcal {E}} \sum_ {a \in \mathcal {A}} | \tilde {\pi} (c, u, a) - \tilde {\pi} (c, v, a) | \quad \forall c \in \mathcal {C} +$$ + +We will now bound the term $\Phi$ that appears in Theorem 4.1 using our choice of $\tilde{\pi}$ and $\vartheta^{*}$ , which will give us the result. To do this note that for all $e\in \mathcal{H}$ we have: + +$$ +- \ln (\vartheta (e)) \in \mathcal {O} \left(\ln (K) + \ln (K N) \sum_ {(u, v) \in \mathcal {E}} [ [ e (u) \neq e (v) ] ]\right) +$$ + +so that: + +$$ +\begin{array}{l} \Phi = \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \ln \left(\frac {\vartheta^ {*} (c , e)}{\vartheta (e)}\right) \\ \leq - \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \ln (\vartheta (e)) \\ \in \mathcal {O} \left(\ln (K) \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) + \ln (K N) \sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \sum_ {(u, v) \in \mathcal {E}} [ [ e (u) \neq e (v) ] ]\right) \\ = \mathcal {O} \left(\ln (K) M + \ln (K N) \sum_ {c \in \mathcal {C}} \sum_ {(u, v) \in \mathcal {E}} \sum_ {a \in \mathcal {A}} | \tilde {\pi} (c, u, a) - \tilde {\pi} (c, v, a) |\right) \\ = \mathcal {O} (\ln (K) M + \ln (K N) \Psi (\tilde {\pi})) \\ \end{array} +$$ + +# D.2 Empirical Statistical Parity + +Here we take $\mathcal{X} \coloneqq \mathcal{V} \times [T]$ and define, for all $t \in [T]$ and $(c, v) \in \mathcal{C} \times \mathcal{V}$ , $\chi_t(c, v) \coloneqq (v, t)$ and: + +$$ +\mathcal {G} _ {t} (c) := \{(v, t) \mid v \in \mathcal {V} \} +$$ + +We then define $\vartheta$ by: + +$$ +\vartheta (e) := \prod_ {v \in \mathcal {V}} \frac {1}{K} \prod_ {t \in [ T - 1 ]} \left(\llbracket e (v, t) = e (v, t + 1) \rrbracket \left(1 - \frac {1}{T}\right) + \llbracket e (v, t) \neq e (v, t + 1) \rrbracket \frac {1}{(K - 1) T}\right) +$$ + +for all $e \in \mathcal{H}$ . This instance of HEDGE is implemented by running the FIXEDSHARE algorithm of [15] for each context, which gives the time and space complexity. + +For all $t\in [T]$ , we define: + +$$ +\mu_ {t} (c, (v, t)) := \frac {\rho_ {t} (c , v)}{\sum_ {v ^ {\prime} \in \mathcal {V}} \rho_ {t} (c , v ^ {\prime})} \quad \forall (c, v) \in \mathcal {C} \times \mathcal {V} +$$ + +and: + +$$ +\mu_ {t} (c, (v, s)) := 0 \quad \forall (c, v, s) \in \mathcal {C} \times \mathcal {V} \times ([ T \setminus \{t \} ] +$$ + +Note that if we have yet to see instances with a particular protected characteristic $c \in \mathcal{C}$ then $\mu_t(c, (v, t))$ is undefined for all $v \in \mathcal{V}$ . It is straightforward to modify the algorithm to handle such situations. + +We now analyse the regret. Suppose we have a sequence of policies $\tilde{\pi} \in \mathcal{Q}^T$ such that for all $t \in [T]$ we have that $\tilde{\pi}_t$ has statistical parity with respect to $\rho_t$ . Define the virtual policy $\tilde{\pi}' \in \mathcal{P}$ such that: + +$$ +\tilde {\pi} ^ {\prime} (c, (v, t), \circ) := \tilde {\pi} _ {t} (c, v, \circ) \quad \forall (c, v, t) \in \mathcal {C} \times \mathcal {V} \times [ T ] +$$ + +Note that we have $\tilde{\pi}^{\prime}\in \bigcap \mathcal{F}(\mu_t)$ as required. + +Define the set $\mathcal{H}' = \mathcal{A}^{[T]}$ . By Lemma D.1 (with $\mathcal{T}$ being a chain), for all $(c, v) \in \mathcal{C} \times \mathcal{V}$ we can construct a distribution $\vartheta'(c, v, \circ) \in \Delta_{\mathcal{H}'}$ in which: + +$$ +\tilde {\pi} ^ {\prime} (c, (v, t), a) = \sum_ {e ^ {\prime} \in \mathcal {H} ^ {\prime}} [ [ e ^ {\prime} (t) = a ] ] \vartheta^ {\prime} (c, v, e ^ {\prime}) +$$ + +and: + +$$ +\sum_ {e ^ {\prime} \in \mathcal {H} ^ {\prime}} \vartheta^ {\prime} (c, v, e ^ {\prime}) \sum_ {t \in [ T - 1 ]} \llbracket e ^ {\prime} (t) \neq e ^ {\prime} (t + 1) \rrbracket = \frac {1}{2} \sum_ {t \in [ T - 1 ]} \sum_ {a \in \mathcal {A}} | \tilde {\pi} ^ {\prime} (c, (v, t), a) - \tilde {\pi} ^ {\prime} (c, (v, t + 1), a) | +$$ + +Now define, for all $c\in \mathcal{C}$ , the distribution $\vartheta^{*}(c,\circ)\in \Delta_{\mathcal{H}}$ by: + +$$ +\vartheta^ {*} (c, e) = \prod_ {v \in \mathcal {V}} \vartheta^ {\prime} (c, v, e (v, \circ)) \quad \forall e \in \mathcal {H} +$$ + +Since for each expert $e \in \mathcal{H}$ we have: + +$$ +- \ln (\vartheta (e)) \in \mathcal {O} \left(\ln (K) N + \ln (K T) \sum_ {v \in \mathcal {V}} \sum_ {t \in [ T - 1 ]} \llbracket e (v, t) \neq e (v, t + 1) \rrbracket\right) +$$ + +we also have, for each $c \in \mathcal{C}$ , that: + +$$ +\begin{array}{l} \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \ln \left(\frac {\vartheta^ {*} (c , e)}{\vartheta (e)}\right) \\ \leq - \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \ln (\vartheta (e)) \\ \in \mathcal {O} \left(\sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \left(\ln (K) N + \ln (K T) \sum_ {v \in \mathcal {V}} \sum_ {t \in [ T - 1 ]} [ [ e (v, t) \neq e (v, t + 1) ] ]\right)\right) \\ = \mathcal {O} \left(\ln (K) N + \ln (K T) \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \sum_ {v \in \mathcal {V}} \sum_ {t \in [ T - 1 ]} [ [ e (v, t) \neq e (v, t + 1) ] ]\right) \\ = \mathcal {O} \left(\ln (K) N + \ln (K T) \sum_ {e \in \mathcal {H}} \prod_ {u \in \mathcal {V}} \vartheta^ {\prime} (c, u, e (u, \circ)) \sum_ {v \in \mathcal {V}} \sum_ {t \in [ T - 1 ]} [ [ e (v, t) \neq e (v, t + 1) ] ]\right) \\ = \mathcal {O} \left(\ln (K) N + \ln (K T) \sum_ {v \in \mathcal {V}} \sum_ {e \in \mathcal {H}} \prod_ {u \in \mathcal {V}} \vartheta^ {\prime} (c, u, e (u, \circ)) \sum_ {t \in [ T - 1 ]} [ [ e (v, t) \neq e (v, t + 1) ] ]\right) \\ = \mathcal {O} \left(\ln (K) N + \ln (K T) \sum_ {v \in \mathcal {V}} \sum_ {e (v, o) \in \mathcal {H} ^ {\prime}} \vartheta^ {\prime} (c, v, e (v, o)) \sum_ {t \in [ T - 1 ]} [ [ e (v, t) \neq e (v, t + 1) ] ]\right) \\ = \mathcal {O} \left(\ln (K) N + \ln (K T) \sum_ {v \in \mathcal {V}} \sum_ {e ^ {\prime} \in \mathcal {H} ^ {\prime}} \vartheta^ {\prime} (c, v, e ^ {\prime}) \sum_ {t \in [ T - 1 ]} \llbracket e ^ {\prime} (t) \neq e ^ {\prime} (t + 1) \rrbracket\right) \\ = \mathcal {O} \left(\ln (K) N + \ln (K T) \sum_ {v \in \mathcal {V}} \sum_ {t \in [ T - 1 ]} \sum_ {a \in \mathcal {A}} | \tilde {\pi} ^ {\prime} (c, (v, t), a) - \tilde {\pi} ^ {\prime} (c, (v, t + 1), a) |\right) \\ = \mathcal {O} \left(\ln (K) N + \ln (K T) \sum_ {v \in \mathcal {V}} \sum_ {t \in [ T - 1 ]} \sum_ {a \in \mathcal {A}} | \tilde {\pi} _ {t} (c, v, a) - \tilde {\pi} _ {t + 1} (c, v, a) |\right) \\ \end{array} +$$ + +so that: + +$$ +\sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \ln \left(\frac {\vartheta^ {*} (c , e)}{\vartheta (e)}\right) \in \mathcal {O} (M N + \Psi (\tilde {\pi})) \ln (K T) +$$ + +We also have: + +$$ +\tilde {\pi} ^ {\prime} (c, (v, t), a) = \sum_ {e (v, \circ) \in \mathcal {H} ^ {\prime}} [ [ e (v, t) = a ] ] \vartheta^ {\prime} (c, v, e (v, \circ)) = \sum_ {e \in \mathcal {H}} [ [ e (v, t) = a ] ] \vartheta^ {*} (c, e) +$$ + +for all $(c, v, t, a) \in \mathcal{C} \times \mathcal{V} \times [T] \times \mathcal{A}$ . + +Substituting into Theorem 4.1 gives us the regret bound. + +# D.3 Approximate statistical parity for hierarchical clusterings with an unknown distribution + +Here we take $\mathcal{X}$ to be the set of sets in the tree $\mathcal{T}$ that is the hierarchical clustering. Let $\mathcal{E}$ be the edge set of $\mathcal{T}$ . The functions $\chi_{t}$ will be constructed online by the characteristic/context sets seen so far. We define $\vartheta$ by: + +$$ +\vartheta (e) := \frac {1}{K} \prod_ {(x, x ^ {\prime}) \in \mathcal {E}} \left(\llbracket e (x) = e (x ^ {\prime}) \rrbracket \left(1 - \frac {1}{| \mathcal {T} |}\right) + \llbracket e (x) \neq e (x ^ {\prime}) \rrbracket \frac {1}{(K - 1) | \mathcal {T} |}\right) \quad \forall e \in \mathcal {H} +$$ + +Now define: + +$$ +\hat {\epsilon} := \epsilon / 4 H \quad ; \quad n := \lceil \ln (2 T / \delta) / 2 \hat {\epsilon} ^ {2} \rceil +$$ + +For each $c \in \mathcal{C}$ we maintain a dynamic (in that it grows over time) full binary subtree $S(c) \subseteq \mathcal{T}$ which contains the root. $S(c)$ is initialised to contain the root (which is the set $\mathcal{V}$ ) as a single vertex. Let $\mathcal{L}(c)$ be the set of leaves of $S(c)$ so that $\mathcal{L}(c)$ is a partition of $\mathcal{V}$ . On each trial $t \in [T]$ we define, for all $(c, v) \in \mathcal{C} \times \mathcal{V}$ , $\chi_t(c, v)$ to be the unique set in $\mathcal{L}(c)$ that contains $v$ , and define $\mathcal{G}_t(c)$ to be the set $\mathcal{L}(c)$ at trial $t$ . The instance of HEDGE for characteristic $c$ is implemented by belief propagation over the tree $S(c)$ as in [25]. This gives us the time and space complexity bounds. + +We now describe how $S(c)$ grows and how the functions $\mu_t$ are defined. To do this we will define, for each $c \in \mathcal{C}$ , a function $\tilde{\mu}(c, \circ): S(c) \to [0,1]$ in which $\tilde{\mu}(c, \mathcal{V}) := 1$ . For each $c \in \mathcal{C}$ , $S(c)$ and $\tilde{\mu}(c, \circ)$ grow as follows. Whenever we have a non-singleton set in $\mathcal{Y} \in \mathcal{L}(c)$ in which there have been $n$ trials $t$ so far in which $c_t = c$ and $x_t \in \mathcal{Y}$ then we add the two children of $\mathcal{Y}$ to $S(c)$ . Denoting the two children by $\mathcal{Z}$ and $\mathcal{Z}'$ we let $m$ be the number of trials $t$ in which $c_t = c$ and $x_t \in \mathcal{Z}$ and then define: + +$$ +\tilde {\mu} (c, \mathcal {Z}) := m \tilde {\mu} (c, \mathcal {Y}) / n \quad ; \quad \tilde {\mu} (c, \mathcal {Z} ^ {\prime}) := (n - m) \tilde {\mu} (c, \mathcal {Y}) / n +$$ + +On each trial $t$ we define $\mu_t(c, \circ)$ as follows. For $x \in \mathcal{G}_t(c)$ we have $\mu_t(c, x) \coloneqq \tilde{\mu}(c, x)$ and for $x \in \mathcal{X} \setminus \mathcal{G}_t(c)$ we have $\mu_t(c, x) \coloneqq 0$ . This completes the description of the instance of SPEW. + +We now analyse this instance of SPEw. First, for all $(c,\mathcal{V})\in \mathcal{C}\times \mathcal{X}$ define: + +$$ +\mu^ {*} (c, \mathcal {Y}) := \frac {\sum_ {v \in \mathcal {Y}} \rho^ {*} (c , v)}{\sum_ {v \in \mathcal {V}} \rho^ {*} (c , v)} +$$ + +For all $c \in \mathcal{C}$ let $S^{\dagger}(c)$ be the tree $S(c)$ on trial $T$ and let $\mathcal{L}^{\dagger}(c)$ be the set of leaves of $S^{\dagger}(c)$ . We define the distribution $\rho$ such that, for all $c \in \mathcal{C}$ , $\mathcal{Y} \in \mathcal{L}^{\dagger}(c)$ and $v \in \mathcal{V}$ , we have: + +$$ +\rho (c, v) := \frac {\tilde {\mu} (c , \mathcal {Y}) \rho^ {*} (c , v)}{\mu^ {*} (c , \mathcal {Y})} +$$ + +Note that by induction over time we always have: + +$$ +\sum_ {\mathcal {Y} \in \mathcal {L} (c)} \tilde {\mu} (c, \mathcal {Y}) = 1 +$$ + +so that: + +$$ +\begin{array}{l} \sum_ {(c, v) \in \mathcal {C} \times \mathcal {V}} \rho (c, v) = \sum_ {c \in \mathcal {C}} \sum_ {\mathcal {Y} \in \mathcal {L} ^ {\dagger} (c)} \sum_ {v \in \mathcal {Y}} \frac {\tilde {\mu} (c , \mathcal {Y}) \rho^ {*} (c , v)}{\mu^ {*} (c , \mathcal {Y})} \\ = \sum_ {c \in \mathcal {C}} \sum_ {\mathcal {Y} \in \mathcal {L} ^ {\dagger} (c)} \tilde {\mu} (c, \mathcal {Y}) \sum_ {v \in \mathcal {V}} \rho^ {*} (c, v) \\ = \sum_ {c \in \mathcal {C}} \sum_ {v \in \mathcal {V}} \rho^ {*} (c, v) \sum_ {\mathcal {Y} \in \mathcal {L} ^ {\dagger} (c)} \tilde {\mu} (c, \mathcal {Y}) \\ = \sum_ {c \in \mathcal {C}} \sum_ {v \in \mathcal {V}} \rho^ {*} (c, v) \\ = 1 \\ \end{array} +$$ + +and hence $\rho$ is a probability distribution as required. + +We now show that with probability at least $1 - \delta$ we have that $\rho$ is $\epsilon$ -close to $\rho^{*}$ . Let $r$ be the root of $\mathcal{T}$ , which is equal to the set $\mathcal{V}$ . For any vertex $x \in \mathcal{T}$ let $\uparrow(x)$ be its parent and let $\triangleleft(x)$ and $\triangleright(x)$ be its left and right children respectively, if they exist. For all $c \in \mathcal{C}$ and $x \in S^{\dagger}(c) \setminus \{r\}$ define: + +$$ +p (c, x) := \frac {\tilde {\mu} (c , x)}{\tilde {\mu} (c , \uparrow (x))} \quad ; \quad p ^ {*} (c, x) := \frac {\mu^ {*} (c , x)}{\mu^ {*} (c , \uparrow (x))} \quad ; \quad \delta^ {\prime} (c, x) := p ^ {*} (c, x) - p (c, x) +$$ + +Note that by the generation of $\tilde{\mu} (c,x)$ we have, direct from Hoeffding's inequality, that: + +$$ +\mathbb {P} [ | \delta^ {\prime} (c, x) | > \hat {\epsilon} ] \leq 2 \exp (- 2 \hat {\epsilon} ^ {2} n) +$$ + +So since: + +$$ +\sum_ {c \in \mathcal {C}} | \mathcal {S} ^ {\dagger} (c) \setminus \{r \} | \leq T +$$ + +we have, by the union bound, that with probability at least $1 - 2T\exp (-2\hat{\epsilon}^2 n)$ , which is no less than $1 - \delta$ , we have: + +$$ +| \delta^ {\prime} (c, x) | \leq \hat {\epsilon} +$$ + +for all $c \in \mathcal{C}$ and $x \in S^{\dagger}(c)$ . So assume that this is the case. Now take any policy $\pi \in \mathcal{Q}$ which has statistical parity with respect to $\rho^{*}$ . For all $c \in \mathcal{C}$ define: + +$$ +\zeta (c) := \sum_ {v \in \mathcal {V}} \rho^ {*} (c, v) +$$ + +For all $(c, a) \in \mathcal{C} \times \mathcal{A}$ and $\mathcal{Y} \in S^{\dagger}(c)$ define: + +$$ +q ^ {*} (c, \mathcal {Y}, a) := \frac {1}{\mu^ {*} (c , \mathcal {Y}) \zeta (c)} \sum_ {v \in \mathcal {Y}} \rho^ {*} (c, v) \pi (c, v, a) +$$ + +$$ +q (c, \mathcal {Y}, a) := \frac {1}{\tilde {\mu} (c , \mathcal {Y}) \zeta (c)} \sum_ {v \in \mathcal {Y}} \rho (c, v) \pi (c, v, a) +$$ + +$$ +\beta (c, \mathcal {Y}, a) := q ^ {*} (c, \mathcal {Y}, a) - q (c, \mathcal {Y}, a) +$$ + +and let $h(c, \mathcal{Y})$ be the height of $\mathcal{Y}$ in the tree $S^\dagger(c)$ . Note that by definition of $\mu^*(c, \mathcal{Y})$ we have: + +$$ +q ^ {*} (c, \mathcal {Y}, a) \leq \frac {1}{\mu^ {*} (c , \mathcal {Y}) \zeta (c)} \sum_ {v \in \mathcal {Y}} \rho^ {*} (c, v) = 1 +$$ + +We take the inductive hypothesis that for all $x\in S^{\dagger}(c)$ we have: + +$$ +| \beta (c, x, a) | \leq 2 h (c, x) \hat {\epsilon} +$$ + +and prove by induction on $h(c, x)$ . The inductive immediately holds for $h(c, x) = 0$ by definition of $\rho$ . Now suppose it holds when $h(c, x) = h'$ (for some $h' \in [H]$ ) and consider $x$ with $h(c, x) = h' + 1$ . Note that: + +$$ +\begin{array}{l} q ^ {*} (c, x, a) = \frac {1}{\mu^ {*} (c , x)} (\mu^ {*} (c, \triangleleft (x)) q ^ {*} (c, \triangleleft (x), a) + \mu^ {*} (c, \triangleright (x)) q ^ {*} (c, \triangleright (x), a)) \\ = p ^ {*} (c, \triangleleft (x)) q ^ {*} (c, \triangleleft (x), a) + p ^ {*} (c, \triangleright (x)) q ^ {*} (c, \triangleright (x), a) \\ \end{array} +$$ + +and that: + +$$ +\begin{array}{l} q (c, x, a) = \frac {1}{\tilde {\mu} (c , x)} (\tilde {\mu} (c, \triangleleft (x)) q (c, \triangleleft (x), a) + \tilde {\mu} (c, \triangleright (x)) q (c, \triangleright (x), a)) \\ = p (c, \triangleleft (x)) q (c, \triangleleft (x), a) + p (c, \triangleright (x)) q (c, \triangleright (x), a) \\ = p (c, \triangleleft (x)) \left(q ^ {*} (c, \triangleleft (x), a) - \beta (c, \triangleleft (x), a)\right) + p (c, \triangleright (x)) \left(q ^ {*} (c, \triangleright (x), a) - \beta (c, \triangleright (x), a)\right)) \\ \end{array} +$$ + +so, by defining: + +$$ +\hat {\beta} (c, x, a) := p (c, \triangleleft (x)) \beta (c, \triangleleft (x), a) + p (c, \triangleright (x)) \beta (c, \triangleright (x), a) +$$ + +we have: + +$$ +\begin{array}{l} \beta (c, x, a) := q ^ {*} (c, x, a) - q (c, x, a) \\ = \left(p ^ {*} (c, \triangleleft (x)) - p (c, \triangleleft (x))\right) q ^ {*} (c, \triangleleft (x), a) + \left(p ^ {*} (c, \triangleright (x)) - p (c, \triangleright (x))\right) q ^ {*} (c, \triangleright (x), a) + \hat {\beta} (c, x, a) \\ = \delta^ {\prime} (c, \triangleleft (x)) q ^ {*} (c, \triangleleft (x), a) + \delta^ {\prime} (c, \triangleright (x)) q ^ {*} (c, \triangleright (x), a) + \hat {\beta} (c, x, a) \\ \end{array} +$$ + +so that, by the inductive hypothesis and since $h(c, \triangleleft(x)) = h(c, \triangleright(x)) = h'$ , we have: + +$$ +\begin{array}{l} \left| \beta (c, x, a) \right| \leq \left| \delta^ {\prime} (c, \triangleleft (x)) \right| q ^ {*} (c, \triangleleft (x), a) + \left| \delta^ {\prime} (c, \triangleright (x)) \right| q ^ {*} (c, \triangleright (x), a) + \left| \hat {\beta} (c, x, a) \right| \\ \leq \hat {\epsilon} q ^ {*} (c, \triangleleft (x), a) + \hat {\epsilon} q ^ {*} (c, \triangleright (x), a) + | \hat {\beta} (c, x, a) | \\ \leq 2 \hat {\epsilon} + | \hat {\beta} (c, x, a) | \\ \leq 2 \hat {\epsilon} + p (c, \triangleleft (x)) | \beta (c, \triangleleft (x), a) | + p (c, \triangleright (x)) | \beta (c, \triangleright (x), a) | \\ \leq 2 \hat {\epsilon} + 2 p (c, \triangleleft (x)) h ^ {\prime} \hat {\epsilon} + 2 p (c, \triangleright (x)) h ^ {\prime} \hat {\epsilon} \\ = 2 \left(1 + h ^ {\prime}\right) \hat {\epsilon} \\ = 2 h (c, x) \hat {\epsilon} \\ \end{array} +$$ + +We have hence proved that the inductive hypothesis holds always. In particular it holds for $x = r$ so that: + +$$ +\left| \beta (c, r, a) \right| \leq 2 H \hat {\epsilon} = \epsilon / 2 \tag {21} +$$ + +Since $\mu^{*}(c,r) = \tilde{\mu} (c,r) = 1$ and $r = \mathcal{V}$ this translates to: + +$$ +\left| \frac {1}{\zeta (c)} \sum_ {v \in \mathcal {Y}} \rho^ {*} (c, v) \pi (c, v, a) - \frac {1}{\zeta (c)} \sum_ {v \in \mathcal {Y}} \rho (c, v) \pi (c, v, a) \right| \leq \epsilon / 2 +$$ + +Since $\pi$ has statistical parity with respect to $\rho$ there exists some function $\kappa : \mathcal{A} \to [0,1]$ such that: + +$$ +\frac {1}{\zeta (c)} \sum_ {v \in \mathcal {Y}} \rho (c, v) \pi (c, v, a) = \kappa (a) \quad \forall (c, a) \in \mathcal {C} \times \mathcal {A} +$$ + +which means: + +$$ +\kappa (a) - \epsilon / 2 \leq \frac {1}{\zeta (c)} \sum_ {v \in \mathcal {Y}} \rho^ {*} (c, v) \pi (c, v, a) \leq \kappa (a) + \epsilon / 2 +$$ + +for all $c \in \mathcal{C}$ . Hence, $\pi$ has $\epsilon$ -approximate statistical parity with respect to $\rho^*$ . This completes the proof that $\rho$ is $\epsilon$ -close to $\rho^*$ . + +We now show that each policy $\pi_t^*$ has statistical parity with respect to $\rho$ . Take any $c \in \mathcal{C}$ and let $\mathcal{L}_t(c)$ be the set $\mathcal{L}(c)$ at trial $t$ . Note that for all $\mathcal{V} \in \mathcal{L}^\dagger(c)$ we have: + +$$ +\tilde {\mu} (c, \mathcal {Y}) = \frac {1}{\zeta (c)} \sum_ {v \in \mathcal {Y}} \rho (v) +$$ + +so since for all $\mathcal{V} \in S^{\dagger}(c) \setminus \mathcal{L}^{\dagger}(c)$ we have: + +$$ +\tilde {\mu} (c, \mathcal {Y}) = \tilde {\mu} (c, \triangleleft (\mathcal {Y})) + \tilde {\mu} (c, \triangleright (\mathcal {Y})) +$$ + +we have by induction that for all $\mathcal{V}\in S^{\dagger}(c)$ + +$$ +\tilde {\mu} (c, \mathcal {Y}) = \frac {1}{\zeta (c)} \sum_ {v \in \mathcal {Y}} \rho (v) +$$ + +This means that for all $a \in \mathcal{A}$ we have: + +$$ +\begin{array}{l} \frac {1}{\zeta (c)} \sum_ {v \in \mathcal {V}} \rho (v) \pi_ {t} ^ {*} (c, v, a) = \frac {1}{\zeta (c)} \sum_ {v \in \mathcal {V}} \rho (v) \pi_ {t} (c, \chi_ {t} (c, v), a) \\ = \frac {1}{\zeta (c)} \sum_ {\mathcal {Y} \in \mathcal {L} _ {t} (c)} \sum_ {v \in \mathcal {Y}} \rho (v) \pi_ {t} (c, \chi_ {t} (c, v), a) \\ = \frac {1}{\zeta (c)} \sum_ {\mathcal {Y} \in \mathcal {L} _ {t} (c)} \sum_ {v \in \mathcal {Y}} \rho (v) \pi_ {t} (c, \mathcal {Y}, a) \\ = \sum_ {\mathcal {Y} \in \mathcal {L} _ {t} (c)} \tilde {\mu} (c, \mathcal {Y}) \pi_ {t} (c, \mathcal {Y}, a) \\ = \sum_ {x \in \mathcal {G} _ {t} (c)} \mu_ {t} (c, x) \pi_ {t} (c, x, a) \\ \end{array} +$$ + +so since we have $\pi_t \in \mathcal{F}(\mu_t)$ we have that $\pi_t^*$ has statistical parity with respect to $\rho$ . + +We now prove the regret bound. Take any $\tilde{\pi} \in \mathcal{Q}$ with statistical parity with respect to $\rho$ . For all $c \in \mathcal{C}$ let $\mathcal{L}'(c)$ be the minimal cardinality permitted clustering such that for all sets $\mathcal{Y} \in \mathcal{L}'(c)$ and all $u, v \in \mathcal{Y}$ we have $\tilde{\pi}(c, u, \circ) = \tilde{\pi}(c, v, \circ)$ . Now we define $\tilde{\pi}' \in \mathcal{P}$ as follows. First extend $\tilde{\mu}$ such that for all $(c, \mathcal{Y}) \in \mathcal{C} \times \mathcal{X}$ we have: + +$$ +\tilde {\mu} (c, \mathcal {Y}) = \frac {1}{\zeta (c)} \sum_ {v \in \mathcal {Y}} \rho (v) +$$ + +For any descendant $\mathcal{Z}$ of some $\mathcal{Y} \in \mathcal{L}'(c)$ we define $\tilde{\pi}'(c, \mathcal{Z}, \circ) = \tilde{\pi}(c, u, \circ)$ for any $u \in \mathcal{Y}$ . For any $x$ that is a proper ancestor of some element of $\mathcal{L}'(c)$ we define $\tilde{\pi}'(c, x, \circ)$ so that: + +$$ +\tilde {\pi} ^ {\prime} (c, x, a) := \frac {\tilde {\mu} (c , \triangleleft (x)) \tilde {\pi} ^ {\prime} (c , \triangleleft (x) , a) + \tilde {\mu} (c , \triangleright (x)) \tilde {\pi} ^ {\prime} (c , \triangleright (x) , a)}{\tilde {\mu} (c , x)} +$$ + +Note that by induction on the height of $x$ we have: + +$$ +\sum_ {a \in \mathcal {A}} \tilde {\pi} ^ {\prime} (c, x, a) = \frac {\tilde {\mu} (c , \triangleleft (x)) + \tilde {\mu} (c , \triangleright (x))}{\tilde {\mu} (c , x)} = 1 +$$ + +so that $\tilde{\pi}^{\prime}\in \mathcal{P}$ as required. Given $x\in \mathcal{T}$ define $\downarrow (x)$ to be the leaf descendants of $x$ . By induction on the height of $x$ in $\mathcal{T}$ we have: + +$$ +\tilde {\pi} ^ {\prime} (c, x, a) = \frac {1}{\tilde {\mu} (c , x)} \sum_ {x ^ {\prime} \in \Downarrow (x)} \tilde {\mu} (c, x ^ {\prime}) \tilde {\pi} ^ {\prime} (c, x ^ {\prime}, a) +$$ + +so that, since the leaves are the singleton sets, we have, for all $\mathcal{V} \in \mathcal{X}$ , that: + +$$ +\tilde {\pi} ^ {\prime} (c, \mathcal {Y}, a) = \frac {1}{\tilde {\mu} (c , \mathcal {Y}) \zeta (c)} \sum_ {v \in \mathcal {Y}} \rho (c, v) \tilde {\pi} (c, v, a) +$$ + +Hence, on trial $t$ we have: + +$$ +\sum_ {x \in \mathcal {G} _ {t} (c)} \mu_ {t} (c, x) \tilde {\pi} ^ {\prime} (c, x, a) = \frac {1}{\zeta (c)} \sum_ {\mathcal {Y} \in \mathcal {L} _ {t} (c)} \sum_ {v \in \mathcal {Y}} \rho (c, v) \tilde {\pi} (c, v, a) = \frac {1}{\zeta (c)} \sum_ {v \in \mathcal {V}} \rho (c, v) \tilde {\pi} (c, v, a) +$$ + +so since $\tilde{\pi}$ has statistical parity with respect to $\rho$ we have that $\tilde{\pi}^{\prime}\in \mathcal{F}(\mu_t)$ . + +Now that we have shown that $\rho \in \bigcap_{t\in [T]}\mathcal{F}(\mu_t)$ we can apply the regret bound in Theorem 4.1. Note that for all $c\in \mathcal{C}$ the number of edges $(x,x^{\prime})$ in $\mathcal{T}$ in which $\tilde{\pi} '(c,x,\circ)\neq \tilde{\pi}'(c,x',\circ)$ is at most $2\psi (c,\tilde{\pi}) - 2$ . Hence, as in Section D.1 (noting that $\log (|\mathcal{T}|)\leq H$ ), we have a distribution $\vartheta^{*}:\mathcal{C}\times \mathcal{H}\to [0,1]$ with: + +$$ +\tilde {\pi} ^ {\prime} (c, x, a) = \sum_ {e \in \mathcal {H}} [ [ e (x) = a ] ] \vartheta^ {*} (c, e) \quad \forall (c, x, a) \in \mathcal {C} \times \mathcal {X} \times \mathcal {A} +$$ + +and: + +$$ +\sum_ {c \in \mathcal {C}} \sum_ {e \in \mathcal {H}} \vartheta^ {*} (c, e) \ln \left(\frac {\vartheta^ {*} (c , e)}{\vartheta (e)}\right) \in \mathcal {O} (H \ln (K) \Psi (\tilde {\pi})) +$$ + +Noting also that we have at most $2n\Psi(\tilde{\pi})$ trials where $\tilde{\pi}'(c_t, x_t, \circ) \neq \tilde{\pi}(c_t, v_t, \circ)$ we obtain, from Theorem 4.1, the regret bound. + +# NeurIPS Paper Checklist + +# 1. 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However, existing methods struggle to achieve robust and continuous knowledge accumulation during test time. To address this, we propose Statistics Caching test-time Adaptation (SCA), a novel cache-based approach. Unlike traditional feature-caching methods prone to forgetting, SCA continuously accumulates task-specific knowledge from all encountered test samples. By formulating the reuse of past features as a least squares problem, SCA avoids storing raw features and instead maintains compact, incrementally updated feature statistics. This design enables efficient online adaptation without the limitations of fixed-size caches, ensuring that the accumulated knowledge grows persistently over time. Furthermore, we introduce adaptive strategies that leverage the VLM's prediction uncertainty to reduce the impact of noisy pseudolabels and dynamically balance multiple prediction sources, leading to more robust and reliable performance. Extensive experiments demonstrate that SCA achieves compelling performance while maintaining competitive computational efficiency. The code is available at https://github.com/Yuqin-G/SCA. + +# 1 Introduction + +Recently, large-scale vision-language models (VLMs), such as CLIP [1], have shown strong generalization across a wide range of tasks, thanks to pre-training on billions of image-text pairs from the web. These models learn powerful cross-modal representations, enabling impressive zero-shot performance without task-specific supervision [1, 2, 3, 4]. However, despite their success, adapting VLMs to real-world tasks remains challenging. Most existing approaches rely on a small number of labeled samples from the target domain for fine-tuning [5, 6, 7], which limits their use in scenarios where annotation is expensive or unavailable [8, 9, 10, 11]. This motivates the need for test-time adaptation (TTA) for VLMs, where the model adapts to new domains on the fly using only unlabeled test data, improving robustness without extra labeling cost [12, 13, 14, 15]. + +Existing TTA methods for vision-language models (VLMs) generally fall into two categories: (1) test-time prompt tuning methods and (2) cache-based methods. Test-time prompt tuning [12, 13, 16, 17, 18] adapts the prompt to each individual sample in the test data stream using an entropy minimization objective on randomly augmented samples. To avoid prompt degradation caused by errors in previous unsupervised optimization steps, these methods typically process each test sample independently and reset the model after evaluating it. These approaches, while simple, fail to leverage + +![](images/c65098ae3696d605fcee0cd4aebf2f04d5cf176f17d2105b838918c0774e4e5c.jpg) + +![](images/174e64de72131f42fd783debd38a2125bec708c68719a5e2cc3f62fbd69e502f.jpg) + +![](images/6a70f9baf5a10da23484e3bd3461596983d3f321feb012ec65913cf8abe14a4e.jpg) +Figure 1: Comparison of our SCA (c) with test-time prompt tuning methods (a) and existing cache-based methods (b). We use each test sample's feature and soft pseudo-label to update the cached feature statistics, enabling continuous knowledge accumulation without forgetting. + +information from previously seen samples. As a result, the model may repeatedly forget useful patterns, which reduces its ability to adapt and generalize across the test set. Moreover, test-time tuning methods require backpropagation through the text encoder at every test-time training step to learn the text prompt, leading to substantial computational overhead. + +Another type of work is cache-based methods [14, 19, 15, 20]. These methods use a class-wise feature cache to retain information during inference. Since ground-truth labels are unavailable, test samples are assigned to cache slots based on zero-shot pseudo-labels. To improve cache quality under limited capacity, low-entropy samples are retained while high-entropy ones are discarded. + +Cache-based methods have garnered significant attention due to their potential for knowledge accumulation and computational efficiency. However, a critical question arises: is this acclaimed knowledge accumulation process truly robust and effective over time? Since low entropy does not guarantee correctness, confidently incorrect samples may enter and persist in the cache. Previous studies have primarily highlighted this detrimental impact on knowledge quality, often reflected in decreased classification accuracy. However, they often overlook a more insidious consequence: whether and how such errors fundamentally disrupt and hinder the process of accumulating new knowledge. Here, we analyze the issue from two perspectives: + +Firstly, a misclassified low-entropy sample can overwrite a previously correct entry in the cache, leading to the forgetting of useful knowledge that had already been acquired. To demonstrate this effect, we conducted a controlled experiment on the Flowers102 [21] using a SOTA cache-based method, DPE [14]. Specifically, we construct a controlled setup with two cache update orders: (1) using only a set of standard samples, and (2) using the same set followed by a few deliberately selected low-entropy misclassified samples that are guaranteed to enter the cache. The second sequence simulates the injection of "forgetting triggers," allowing us to measure the performance drop by directly comparing the performance before and after forgetting. As shown in Fig. 2, DPE suffers from forgetting, which causes a decline in performance as useful information is overwritten. + +Secondly, a misclassified low-entropy sample, once admitted, might persistently occupy a cache slot due to its high confidence + +score. This can block potentially correct but higher-entropy samples from entering the cache, thereby + +![](images/d6e092888b1817b3419827049012097bbccd811d8f33279a5b580a3e6e86e7dd.jpg) +Figure 2: Performance drop on Flowers using ViT-B/16. Negative values mean the performance improves instead of drops. + +hinding the intended knowledge accumulation and refinement. To demonstrate this, we simulated a 'bad case' scenario, where several low-entropy but incorrect samples were placed at the beginning of the online test sample sequence. As depicted in Fig.2, under these conditions, the cache was consistently dominated by incorrect features, preventing effective knowledge accumulation and resulting in a significant performance drop for DPE compared to a standard evaluation order. + +The preceding analysis reveals that while existing TTA methods attempt to adapt, they face significant hurdles in achieving robust and continuous knowledge accumulation. Test-time prompt tuning resets for each sample, inherently discarding prior information and preventing any meaningful accumulation of knowledge from the test stream. Existing cache-based methods, while aiming for accumulation, are susceptible to forgetting due to limited cache capacity and can suffer from blocked accumulation when noisy feature-label pairs lead to the entrenchment of incorrect information. This highlights a critical need for an approach that can consistently learn from all past samples and is robust to pseudo-label noise. + +To this end, we propose Statistics Caching test-time Adaptation (SCA), a simple yet effective cache-based test-time approach for VLM. As illustrated in Fig. 1, SCA is designed to effectively accumulate task-specific knowledge from all previously encountered test samples without the forgetting issues inherent in feature-caching methods. We achieve this by modeling the effective use of previous test sample features as a least squares problem. Crucially, solving this problem requires only the corresponding feature statistics, not the full set of historical raw features. Therefore, unlike prior cache-based methods that store a limited subset of raw features, SCA maintains and incrementally updates these compact feature statistics in the cache. This core design choice enables the continuous integration of information from every incoming sample, addressing the problem of forgetting due to fixed-size feature caches and ensuring that the accumulated knowledge grows persistently over time. + +Furthermore, to ensure the quality of this continuously accumulated knowledge, especially in the presence of noisy pseudo-labels, we introduce a dynamic soft pseudo-label assignment strategy. This strategy intelligently utilizes the uncertainty information from the VLM's initial predictions to generate more robust pseudo-labels for updating the feature statistics. Finally, we propose an instance-level adaptive fusion strategy that combines the cache (statistics-derived) logit and the textual (zero-shot) logit using prediction entropy. This strategy estimates the confidence of the statistics-derived classifier and adaptively adjusts its weight relative to the VLM's textual logits. As a result, it enables more robust and flexible predictions with relatively little hyperparameter tuning effort. + +We evaluate SCA on 15 diverse datasets across out-of-domain and cross-domain benchmarks, where it achieves notable performance gains over state-of-the-art prompt tuning and cache-based methods. Moreover, SCA avoids time-consuming backpropagation and achieves over $10 \times$ computational efficiency compared to TPT [12], making it much more practical for resource-limited settings. Our contributions are summarized as follows: + +- We propose Statistics Caching test-time Adaptation (SCA), a novel cache-based method that stores feature statistics instead of raw features, enabling effective accumulation of task-specific knowledge from all previously seen test samples without forgetting. +- We introduce a dynamic soft pseudo-labeling strategy that leverages uncertainty to reduce error accumulation. Additionally, we propose an instance-level adaptive fusion mechanism based on prediction entropy, enabling robust predictions without extensive hyperparameter tuning. +- Extensive experimental results show that SCA achieves strong performance across 15 diverse datasets, matching or surpassing state-of-the-art methods while maintaining high computational efficiency. + +# 2 Preliminaries + +Zero-shot classification in CLIP. CLIP [1] enables zero-shot image classification via pre-trained visual $\Phi_I(\cdot)$ and textual $\Phi_T(\cdot)$ encoders mapping inputs to a shared $D$ -dimensional space $\mathbb{R}^D$ . To classify an image $I$ into one of $K$ classes $\{c_1, \dots, c_K\}$ , text prompts $T_k$ are generated for each class (e.g., "a photo of [CLS]") The image and prompts are embedded: + +$$ +f _ {I} = \Phi_ {I} (I) \quad \text {a n d} \quad f _ {T _ {k}} = \Phi_ {T} \left(T _ {k}\right) \quad \text {f o r} k = 1, \dots , K. \tag {1} +$$ + +Here, $f_{I}$ and $f_{T_k}$ are the image and text features, respectively. CLIP computes the cosine similarity cosine $(f_{I},f_{T_{k}})$ between the image feature and each text feature. These similarities determine class + +probabilities using a softmax function with temperature $\kappa$ : + +$$ +p \left(c _ {k} \mid I\right) = \frac {\exp \left(\operatorname {c o s i n e} \left(f _ {T _ {k}} , f _ {I}\right) / \kappa\right)}{\sum_ {j = 1} ^ {K} \exp \left(\operatorname {c o s i n e} \left(f _ {T _ {j}} , f _ {I}\right) / \kappa\right)}. \tag {2} +$$ + +The image $I$ is assigned the class $c_{k}$ corresponding to the maximum probability $p(c_k|I)$ . + +Test-time prompt tuning. These methods [12, 13, 22, 16, 17, 18] optimize learnable prompts for each test sample. As ground-truth labels are unavailable during testing, the optimization is guided by the prediction entropy of selected sample augmentations: + +$$ +\min _ {v} \mathcal {L} (v; I) = \min _ {v} - \sum_ {k = 1} ^ {K} p (\hat {y} = c _ {k} \mid \mathcal {A} (I), v) \log p (\hat {y} = c _ {k} \mid \mathcal {A} (I), v), \tag {3} +$$ + +where $v$ denotes the learnable prompt, $\mathcal{A}(I)$ represents the set of selected augmentations with lower prediction entropy for test sample $I$ , and $p(\hat{y} = c_k \mid \mathcal{A}(I), v)$ denotes the averaged predicted probability over these augmentations. + +Cache-based test-time adaptation. These methods [14, 19, 15, 20] maintain a fixed-size cache of $M$ entries per class to store features from test samples. This cache acts as an external memory, allowing the model to reference past examples during inference. By continuously updating the cache based on minimal-entropy filtering strategy, the model can gradually accumulate useful information and improve prediction consistency. Final predictions are obtained by combining zero-shot logits from the textual classifier with similarity scores from the cache: + +$$ +\begin{array}{l} \mathbf {z} _ {k} = \alpha_ {0} \mathbf {z} _ {\text {t e x t}} + \alpha_ {1} \mathbf {z} _ {\text {c a c h e}} \\ = \alpha_ {1} f _ {I} ^ {\top} f _ {T _ {k}} + \alpha_ {2} \sum_ {i = 1} ^ {M} \exp (- \beta (1 - f _ {I} ^ {\top} f _ {i} ^ {(k)})), \tag {4} \\ \end{array} +$$ + +where $\mathbf{z}_{\mathrm{cahce}}$ denotes the logit for class $k$ , and $f_{i}^{(k)}$ is the $i^{th}$ feature of category $k$ in the cache. Their performance relies heavily on fusion hyperparameters $\alpha_{1}$ and $\alpha_{2}$ . + +# 3 Statistics Caching Test-time Adaptation + +To avoid forgetting of previously learned knowledge, we aim to make full use of past test samples observed up to time step $t$ to guide model adaptation. Denote $X_{1:t} \in \mathbb{R}^{n_{1:t} \times d}$ as the features extracted from all test samples up to time $t$ , and $Y_{1:t} \in \mathbb{R}^{n_{1:t} \times K}$ as their corresponding one-hot encoded ground-truth labels. Here, $n_{1:t}$ represents the cumulative number of test samples up to time $t$ . We make full use of historical samples by formulating it as a least squares problem: + +$$ +\min _ {\boldsymbol {W} _ {t}} \left\| \boldsymbol {X} _ {1: t} \boldsymbol {W} _ {t} - \boldsymbol {Y} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} + \gamma \left\| \boldsymbol {W} _ {t} \right\| _ {\mathrm {F}} ^ {2}. \tag {5} +$$ + +Here, we use $\gamma$ to control the strength of the regularization. The closed-form solution of Eq. 5 is: + +$$ +\boldsymbol {W} _ {t} = \left(\boldsymbol {X} _ {1: t} ^ {\top} \boldsymbol {X} _ {1: t} + \gamma \boldsymbol {I}\right) ^ {- 1} \boldsymbol {X} _ {1: t} ^ {\top} \boldsymbol {Y} _ {1: t}. \tag {6} +$$ + +$W_{t}$ denotes the classifier learned by leveraging the feature representations of all previously observed test samples up to time $t$ . It captures accumulated knowledge across test instances, enabling more informed and robust predictions. + +However, in practical test-time scenarios, storing all historical sample features $X_{1:t}$ is typically infeasible due to increasing storage overhead or potential privacy concerns. Moreover, ground-truth labels $Y_{1:t}$ are not accessible during testing. Existing methods typically rely on CLIP's zero-shot predictions to assign pseudo-labels for cache construction. However, these pseudo-labels are often noisy, leading to the inclusion of unreliable feature-label pairs that degrade the quality of the updated classifier. To tackle these challenges, we propose the following parts. + +Statistics Accumulation. In fact, computing $\mathbf{W}_t$ does not require storing the growing feature matrix $\mathbf{X}_{1:t}$ as test samples accumulate. Instead, it is sufficient to maintain two compact feature statistics that are independent of the sample size: $\mathbf{G}_{1:t} = \mathbf{X}_{1:t}^{\top} \mathbf{X}_{1:t} \in \mathbb{R}^{d \times d}$ and $\mathbf{C}_{1:t} = \mathbf{X}_{1:t}^{\top} \mathbf{Y}_{1:t} \in \mathbb{R}^{d \times K}$ . Gram matrix $\mathbf{G}_{1:t}$ represents the second-order feature statistics, capturing the pairwise relationships between the features, and $\mathbf{C}_{1:t}$ represents the first-order statistics, which is essentially the weighted + +![](images/52509a1f1e79680afa38be4919803cafb01e657886b46084713587a9c1727522.jpg) +Figure 3: Framework of our SCA. Each test sample updates the feature statistics in the cache. Based on these statistics, we build a closed-form classifier that captures knowledge from all test samples. Finally, we apply a instance-level adaptive fusion strategy to combine the textual and cache logits. + +sum of the features and the labels. Due to the linear characteristics of these feature statistics, we can update them additively to support online updates during test-time scenarios: + +$$ +\boldsymbol {G} _ {1: t} = \boldsymbol {G} _ {1: t - 1} + \boldsymbol {X} _ {t} ^ {\top} \boldsymbol {X} _ {t}, \quad \boldsymbol {C} _ {1: t} = \boldsymbol {C} _ {1: t - 1} + \boldsymbol {X} _ {t} ^ {\top} \boldsymbol {Y} _ {t}. \tag {7} +$$ + +Here, $\mathbf{X}_t \in \mathbb{R}^{n_t \times d}$ and $\mathbf{Y}_t \in \mathbb{R}^{n_t \times K}$ denote the features of the current test sample and its corresponding label, respectively. Finally, these two statistics are then used to compute $\mathbf{W}_t$ as follows: + +$$ +\boldsymbol {W} _ {t} = \left(\boldsymbol {G} _ {1: t} + \gamma \boldsymbol {I}\right) ^ {- 1} \boldsymbol {C} _ {1: t}. \tag {8} +$$ + +This approach efficiently accumulates knowledge and builds a strong classifier that captures information from all previously seen samples. + +Dynamic soft pseudo-label assignment. The above discussion assumes access to true labels when accumulating feature statistics. In practice, we typically rely on pseudo-labels generated by CLIP's zero-shot predictions to update $C_{1:t}$ . A common but problematic practice [14, 15] is to convert these predictions into hard pseudo-labels by selecting the single most confident class as ground truth. This approach ignores the nuance of the full probability distribution, collapsing all uncertainty into a binary decision, and can amplify errors when the top prediction is only marginally more probable or even incorrect. A continuous stream of such errors will inevitably skew the learned cache classifier and degrade its accuracy. + +To mitigate this issue, we dynamically generate the soft pseudo-label $\hat{Y}_t$ for each test sample $X_t$ based on its predicted class probabilities $\mathbf{p}_{\mathrm{text}}$ from the CLIP textual (zero-shot) classifier. Specifically, we first identify a candidate set of classes based on each sample. Without loss of generality, we assume the class probabilities $\mathbf{p}_{\mathrm{text}} = (\mathbf{p}_{\mathrm{text}}^{(1)}, \mathbf{p}_{\mathrm{text}}^{(2)}, \dots, \mathbf{p}_{\mathrm{text}}^{(K)})$ are sorted in descending order. In this case, $\mathbf{p}_{\mathrm{text}}^{(1)}$ denotes the probability of the class initially deemed most likely by the zero-shot classifier. We select the smallest number of classes, denoted by $\Omega$ , such that their cumulative probability meets or exceeds a threshold $\tau$ : + +$$ +\Omega = \min \left\{k \left| \sum_ {k = 2} ^ {K} \mathbf {p} _ {\text {t e x t}} ^ {(k)} \geq \tau \left(1 - \mathbf {p} _ {\text {t e x t}} ^ {(1)}\right) \right. \right\} \cup \{1 \}. \tag {9} +$$ + +Then the soft pseudo-label $\hat{Y}_t$ is generated as: + +$$ +\hat {\mathbf {Y}} _ {t} ^ {(k)} = \frac {\exp \left(\mathbf {p} _ {\text {t e x t}} ^ {(k)} \cdot \mathbb {I} (k \in \Omega)\right)}{\sum_ {l \in \Omega} \exp \left(\mathbf {p} _ {\text {t e x t}} ^ {(l)}\right)}. \tag {10} +$$ + +By allowing $|\Omega|$ vary with $\tau$ and the sample's confidence, our method produces softer pseudo-labels for uncertain samples and sharper labels when the model is confident. This dynamic assignment strategy better captures the sample-specific uncertainty inherent in the zero-shot predictions and effectively curbs the overconfidence often associated with hard labeling schemes. Finally, the pseudolabel $\hat{Y}_t$ is used to update the first-order statistics $\hat{C}_{1:t}$ as shown in Eq.7. The cache classifier $\hat{\mathbf{W}}_t$ is then computed as Eq.8 to obtain the cache logits $\mathbf{z}_{\mathrm{cache}} = \mathbf{X}_t\hat{\mathbf{W}}_t$ . + +Table 1: Experimental results on the cross-domain benchmark with two backbones of CLIP. The best and second-best results are shown in bold and underlined, respectively. + +
MethodAircraftCaltechCarsDTDEuroSATFlowerFood101PetsSUN397UCF101Average
CLIP-ResNet-5015.6685.8855.7040.3723.6961.7573.9783.5758.8058.8455.82
Ensemble16.1187.2655.8940.3725.7962.7774.8282.9760.8559.4856.63
CoOp [5] [IJCV'22]15.1286.5355.3237.2926.2061.5575.5987.0058.1559.0556.18
CoCoOp [24] [CVPR'22]14.6187.3856.2238.5328.7365.5776.2088.3959.6157.1057.23
TPT [12] [NeurIPS'22]17.5887.0258.4640.8428.3362.6974.8884.4961.4660.8257.66
DiffTPT [13] [ICCV'23]17.6086.8960.7140.7241.0463.5379.2183.4062.7262.6759.85
TDA [14] [CVPR'24]17.6189.7057.7843.7442.1168.7477.7586.1862.5364.1861.03
DPE [15] [NeurIPS'24]19.8090.8359.2650.1841.6767.6077.8385.9764.2361.9861.93
BCA [25] [CVPR'25]19.8989.7058.1348.5842.1266.3077.1985.5863.3863.5161.44
SCA (Ours)20.8291.0359.7353.3143.3371.0976.5186.1564.5066.4063.29
CLIP-ViT-B/1623.6793.3565.4844.2742.0167.4483.6588.2562.5965.1363.58
Ensemble23.2293.5566.1145.0450.4266.9982.8686.9265.6365.1664.59
CoOp [5] [IJCV'22]18.4793.7064.5141.9246.3968.7185.3089.1464.1566.5563.88
CoCoOp [24] [CVPR'22]22.2993.7964.9045.4539.2370.8583.9790.4666.8968.4464.63
TPT [12] [NeurIPS'22]24.7894.1666.8747.7542.4468.9884.6787.7965.5068.0465.10
DiffTPT [13] [ICCV'23]25.6092.4967.0147.0043.1370.1087.2388.2265.7462.6765.47
TDA [14] [CVPR'24]23.9194.2467.2847.4058.0071.4286.1488.6367.6270.6667.53
DPE [15] [NeurIPS'24]28.9594.8167.3154.2055.7975.0786.1791.1470.0770.4469.40
Dynaprompt [22] [ICLR'25]24.3394.3267.6547.9642.2869.9585.4288.2866.3268.7265.52
BCA [25] [CVPR'25]28.5994.6966.8653.4956.6373.1285.9790.4368.4167.5968.59
SCA (Ours)28.5094.8568.4957.0957.1676.0986.0991.4470.2773.4370.34
+ +Instance-level Adaptive Fusion. Previous methods [23, 15, 19] fuse zero-shot (textual) logit and cache logit using a fixed dataset-level coefficient, which typically requires extensive hyperparameter tuning on the validation set or even the test set. In contrast, we propose an instance-level fusion strategy that leverages entropy to estimate classifier confidence and adaptively weights the logits, enabling more robust and adaptive predictions. + +For each test sample, we compute the prediction entropy for each classifier and denote them as $\mathcal{H}_{\mathrm{cache}}$ and $\mathcal{H}_{\mathrm{text}}$ . Lower entropy indicates higher confidence, so we derive confidence weights as follows: + +$$ +\left[ \alpha_ {1}, \alpha_ {2} \right] = \operatorname {s o f t m a x} \left(- \beta \left[ \mathcal {H} _ {\text {c a c h e}}, \mathcal {H} _ {\text {t e x t}} \right]\right), \tag {11} +$$ + +where $\beta$ controlling the sharpness of the distribution. This enables the model to evaluate their relative confidence, relying more on the zero-shot model when the cache model is uncertain and shifting toward the cache model as its confidence increases. Entropy is used as it is the most common confidence metric. We then compute the aggregated logits as: + +$$ +\mathbf {z} _ {\text {f i n a l}} = \alpha_ {1} \mathbf {z} _ {\text {c a c h e}} + \alpha_ {2} \mathbf {z} _ {\text {t e x t}} \tag {12} +$$ + +By automatically and adaptively adjusting the fusion coefficient for each test sample, our method captures instance-level uncertainty and improves robustness across diverse data distributions. Specifically, we find that $\beta$ is not sensitive in our experiment. Using default settings ( $\beta = 0.5$ ) gives strong performance across multiple datasets. In other words, a single set of hyperparameters can be applied to fuse predictions across different datasets, reducing the effort required to tune the fusion parameters. + +# 4 Experiments + +# 4.1 Experimental Settings + +Datasets. Following prior work [12, 13, 15, 25], we evaluate our method using two established benchmarks: the cross-domain benchmark and the out-of-distribution (OOD) benchmark. (1) The cross-domain benchmark assesses the model's ability to generalize across various domains, each with its own set of classes. This benchmark includes 10 diverse image classification datasets from distinct domains: FGVCAircraft [26], Caltech101 [27], StanfordCars [28], DTD [29], EuroSAT [30], Flowers102 [21], Food101 [31], OxfordPets [32], SUN397 [33], and UCF101 [34]. (2) The OOD + +Table 2: Experimental results on the OOD benchmark with two backbones of CLIP. The best and second-best results are shown in bold and underlined, respectively. + +
MethodImageNetImageNet-AImageNet-V2ImageNet-RImageNet-SAverageOOD Average
CLIP-ResNet-50 [1]58.1621.8351.4156.1533.3744.1840.69
Ensemble59.8123.2452.9160.7235.4846.4343.09
CoOp [5] [IUCV'22]63.3323.0655.4056.6034.6746.6142.43
CoCoOp [24] [CVPR'22]62.8123.3255.7257.7434.4846.8142.82
TPT [12] [NeurIPS'22]60.7426.6754.7059.1135.0947.2643.89
DiffTPT [13] [ICCV'23]60.8031.0655.8058.8037.1048.7145.69
TDA [14] [CVPR'24]61.3530.2955.5462.5838.1249.5846.63
DMN-ZS [19] [CVPR'24]63.8728.5756.1261.4439.8449.9746.49
DPE [15] [NeurIPS'24]63.4130.1556.7263.7240.0350.8147.66
BCA [25] [CVPR'25]61.8130.3556.5862.8938.0849.9446.98
SCA (Ours)63.1130.6456.5162.8539.3350.4947.33
CLIP-ViT-B/16 [1]66.7347.8760.8673.9846.0959.1157.20
Ensemble68.3449.8961.8877.6548.2461.2059.42
CoOp [5] [IUCV'22]71.5149.7164.2075.2147.9961.7259.28
CoCoOp [24] [CVPR'22]71.0250.6364.0776.1848.7562.1359.91
TPT [12] [NeurIPS'22]68.9854.7763.4577.0647.9462.4460.81
DiffTPT [13] [ICCV'23]70.3055.6865.1075.0046.8062.2860.52
TDA [14] [CVPR'24]69.5160.1164.6780.2450.5465.0163.89
DMN-ZS [19] [CVPR'24]72.2558.2865.1778.5553.2065.4963.80
DPE [15] [NeurIPS'24]71.9159.6365.4480.4052.2665.9364.43
Dynaprompt [22] [ICLR'25]69.6156.1764.6778.1748.2263.3761.81
BCA [25] [CVPR'25]70.2261.1464.9080.7250.8765.3764.16
SCA (Ours)71.7560.3365.3880.8552.5066.1664.77
+ +benchmark measures the model's robustness against natural shifts in data distribution. It evaluates performance on ImageNet [35] and four challenging variants: ImageNet-A [9], ImageNet-V2 [10], ImageNet-R [36], and ImageNet-Sketch [37]. + +Implementation Details. Consistent with prior work [15, 23], our experiments adopt ResNet-50 [38] and ViT-B/16 [39] as the visual encoders for CLIP, with a batch size of 1 to satisfy online processing requirements. For textual prompts, we employ the prompt ensembling strategy as used in previous works [15, 40, 19, 23]. For data augmentation, we adopt the approach from DPE [12], generating 63 randomly resized crops per test image for the OOD benchmark. No data augmentation is applied in the cross-domain benchmark. By default, we set the ridge coefficient $\gamma$ to 1e4, threshold $\tau$ to 0.1 (ViT-B/16) or 0.7 (ResNet50), sharpness coefficient $\beta$ to 0.5. All experiments are conducted on a single NVIDIA A100 RTX GPU, using top-1 accuracy to measure classification performance. + +Baselines. We compare our method with zero-shot CLIP using either a simple prompt or prompt ensemble, few-shot methods such as CoOp [5], with 16-shot annotated samples per class, alongside the following established CLIP test-time adaptation approaches: (1) prompt-tuning methods: TPT [12], DiffTPT [13], Dynaprompt[22]; (2) cache-based methods: TDA [14], DMN-ZS [19], DPE [15]. We also include recent baseline BCA [25] that adapts likelihood using Bayes rules. To ensure a fair comparison, we exclude RLCF [41] and COSMIC [20], which rely on auxiliary strong models (e.g., ViT-L-14) during test-time adaptation. We also exclude some transductive TTA baselines [42, 43] that rely on utilizing inter-sample relations within a large batch. The results of these baselines are directly taken from their respective original papers. + +# 4.2 Results and Discussions + +Cross-Domain Benchmark. Table 1 presents results on the cross-domain benchmark. While CoOp [5] and CoCoOp [24] improve CLIP's generalization, they rely on training-based adaptation with labeled data, which restricts their practical applicability. In contrast, our SCA outperforms CoOp by an average of $7.11\%$ and $5.71\%$ , CoCoOp by an average of $6.06\%$ and $6.46\%$ on two backbones in a training-free manner, demonstrating superior generalization without relying on labeled examples. Compared to test-time prompt tuning methods, SCA achieves a $4.82\%$ gain in average accuracy over the strongest baseline, Dynaprompt [22]. It also consistently outperforms cache-based approaches, surpassing TDA [14] by $2.25\%$ and $2.81\%$ , and DPE [15] by $1.36\%$ and $0.94\%$ on ResNet-50 and ViT-B/16 backbones, respectively. Notably, unlike existing cache-based methods [15, 14] that rely on dataset-specific fusion hyperparameters for optimal performance, our method uses the same + +configuration for all datasets within a benchmark-backbone pair, while different benchmarks or backbones use different configurations, highlighting its strong generalization. + +OOD Benchmark. In Table 2, we further evaluate the generalization capability of our proposed method on the OOD benchmark by comparing it with state-of-the-art approaches. Our proposed method consistently outperforms the CoOp [5] and CoCoOp [24] with substantial improvements across all datasets. Specifically, it improves average accuracy by about $5\%$ on both backbones. These results demonstrate the strong generalization capability of our approach and its effectiveness in enhancing performance without any additional supervised training. On the ResNet-50 backbone, our method performs slightly below the strongest baseline DPE [15]; however, it achieves this level of performance without any training overhead, unlike DPE, which requires additional training. In contrast, on the ViT-B/16 backbone, SCA consistently outperforms all existing methods, with average accuracy improvements ranging from $0.23\%$ to $3.72\%$ across various OOD datasets. + +Computation Efficiency. We measure time per image for several baselines on SUN397 (19,850 samples) using a single 80 GB NVIDIA RTX A100 GPU as an example of computational efficiency. Table 3 presents the corresponding results. Since our proposed SCA avoids time-consuming backpropagation, it offers a notable advantage in computational efficiency compared to prompt tuning methods. Specifically, our approach is over $10 \times$ faster than + +Table 3: Computation efficiency comparison on SUN397 using ViT-B/16. + +
MethodBP-freeTime Per Image (s)
CLIP [1]✓0.007
TPT [12]X0.218
TDA [14]✓0.009
DPE [15]X0.043
SCA (Ours)✓0.012
+ +TPT [12]. Compared to the cache-based methods TDA [14], our approach is slightly less efficient but delivers notable performance gains, especially on the cross-domain benchmark. + +Storage Overhead. Previous cache-based methods that store feature-label pairs with fixed size $M$ have a storage cost of $M \times K \times d$ . Our cached feature statistics accumulate knowledge from all samples and require $(K + d) \times d$ storage. While this may be less efficient when the number of classes $K$ is small, our method offers lower storage overhead as $K$ grows, for example on large-scale datasets like ImageNet. + +# 4.3 Ablation Studies + +Effectiveness on statistics accumulation and dynamic soft label assignment. To investigate the roles of our statistics accumulation and dynamic soft label assignment, we report the performance on 10 fine-grained recognition datasets involved in cross-domain datasets using ViT-B/16 in Table 4. [W/o statistics accumulation] means that we follow [14, 15] by using a fixed-size cache of $M$ feature-label pairs per class to build our classifier (Eq. 6). [W/o dynamic soft label assignment] means that we use hard pseudo-labels for statistics accumulation. We + +Table 4: Effectiveness on statistics accumulation and dynamic soft label assignment. We report the averaged results on 10 datasets in cross-domain benchmark. + +
MethodAvg
w/o statistics accumulation (M=4)69.31
w/o statistics accumulation (M=8)69.47
w/o statistics accumulation (M=16)69.75
w/o statistics accumulation (M=32)69.72
w/o dynamic soft label assignment69.32
with both70.34
+ +have the following key observations: (1) Replacing statistics accumulation with direct feature caching leads to a performance drop compared to our method. While increasing the cache size gradually improves performance, the gains eventually plateau. Interestingly, when the cache size reaches 8, it outperforms the state-of-the-art baseline DPE [15] (69.40). This improvement stems from our use of second-order feature information (the Gram matrix $G$ ) to construct the cache-based classifier, whereas prior cache-based methods rely solely on first-order information. Studies in other areas of machine learning similarly conclude that second-order feature statistics significantly enhance performance [44, 45, 46, 47]. (2) Using hard labels to update statistics leads to a performance drop. This highlights the importance of the extra information in soft labels and demonstrates the effectiveness of our dynamic label assignment. + +Ablations on fusion coefficients. In our SCA, the fusion coefficient is automatically adjusted for each test sample. We perform an ablation study by manually setting a fixed fusion coefficient, varying $\alpha_{1}$ from 0.1 to 0.9. As shown in Fig. 4, the optimal fusion value differs across datasets. While the adaptive coefficients may not be optimal for every case, they consistently outperform most fixed + +coefficients across different datasets. Compared to some cache-based methods [14, 15] that require costly parameter searches to tune fusion hyperparameters for each dataset, our approach uses a single set of hyperparameters for all datasets to fuse the prediction and still deliver strong performance. This achieves a good trade-off between performance and tuning effort. + +![](images/cf20c445f5b1122956e261e5f49559b3242ade04d1c556c09fbe39b2dea13b37.jpg) +Figure 4: Normalized accuracy with varying fusion parameters on ViT-B/16. + +![](images/0dccf5aea4df6a2491fc5688a23171a712fa7ff4f0f66a248f7d7d9e26609506.jpg) +Figure 5: Left: hyperparameter analysis of sharpness coefficient $\beta$ . Middle: hyperparameter analysis of threshold $\tau$ . Right: sensitivity analysis of test-time sample order. + +![](images/e632b6ad1aba44807168e73bdd038a840e4119425fa5aecf486ca9a40e4df5e6.jpg) + +![](images/8e7a16d1bebe63251319b3da6a27cbba833a7ebc06dae4e6390b564cf2ba2278.jpg) + +Hyperparameter Analysis. Here, we present the results of hyperparameter analysis for the threshold $\tau$ and the sharpness coefficient $\beta$ on 10 fine-grained recognition datasets. As shown in Fig. 5 (Left and Middle), the performance remains relatively stable when the parameters are set within a reasonable range. This suggests that our method is not sensitive to these hyperparameters, making it more robust and easier to apply in practice. + +Sensitivity to test time sample order. Since our SCA is designed for online test-time adaptation, its performance may be affected by the order in which test samples are encountered. To assess this sensitivity, we conduct experiments on 10 datasets from the cross-domain benchmark, each with varying test-time sample orders. As shown in Fig. 5 (Right), our method demonstrates stable performance across different orders and consistently outperforms the state-of-the-art baseline DPE [15], highlighting its robustness to input sequence variations in online settings. + +# 5 Related Work + +Test-time prompt tuning. These methods tune the text prompts using the test samples. TPT [12] optimizes the prompt by minimizing prediction entropy to promote consistency across augmented views of the same input. DiffTPT [13] follows the TPT's framework and uses a pre-trained diffusion model to generate more diverse augmentations. C-TPT [17] optimizes prompts by minimizing calibration error, while RLCF [41] introduces an additional CLIP model as a reward signal to guide test-time prompt tuning. Despite their effectiveness, they typically treat each test sample in isolation, resetting the model for every instance without leveraging information from previously seen samples. To effectively accumulate previously learned knowledge, it is essential to incorporate a cache module. For instance, Dynaprompt [22] uses a prompt buffer and introduces a dynamic selection strategy to update the appropriate prompt stored in the buffer. Although this approach facilitates the accumulation of historical knowledge, it, like other test-time tuning methods, requires backpropagation through the text encoder at each test-time step, resulting in significant computational overhead. + +Cache-based test-time adaptation. These methods store high-confidence visual features during test time and utilize them to provide extra information for current prediction. TDA [14] and DMN [19] establish visual caches from test data and pseudo-labels to retrieve class prototypes during test time. Similarly, DPE [15] maintains a dynamic cache that facilitates the training of residual dual prototypes. COSMIC [20] enhances the semantic diversity of pseudo-labels by leveraging fine-grained visual features from DINOv2. These methods demonstrate high computational efficiency and strong performance due to the rich information provided by features stored in the cache. However, + +they suffer from forgetting and blocking issues during knowledge accumulation. Additionally, fusing the predictions derived from the cache features with the VLM's zero-shot predictions often requires extensive and careful tuning. Our SCA overcomes these challenges through statistics accumulation combined with dynamic label assignment and fusion strategy. + +# 6 Conclusion + +We propose Statistics Caching test-time Adaptation (SCA), a cache-based method that stores feature statistics instead of raw features, allowing effective accumulation of task-specific knowledge without forgetting. We also introduce a dynamic soft pseudo-labeling strategy that uses uncertainty to reduce error accumulation, and an instance-level adaptive fusion mechanism based on prediction entropy for robust predictions without heavy hyperparameter tuning. Experiments on 15 diverse datasets show that SCA matches or exceeds state-of-the-art performance with high computational efficiency. + +While our method enables test-time accumulation of task-specific knowledge without training and achieves strong performance, it still shares some limitations with existing methods. Firstly, storing feature statistics introduces some storage overhead. Secondly, our method only mitigates error accumulation during test time to a certain extent, and the performance still falls short of the ideal setting where ground-truth labels are available to build the classifier. Moreover, similar to existing CLIP TTA methods, our approach mainly focuses on the single-domain scenario, where all test samples come from a single dataset. Extending CLIP-based TTA to the multi-domain setting—where test samples from multiple domains are mixed—is an important yet underexplored direction. In such cases, naive accumulation may even lead to negative effects due to domain discrepancies. Since feature representations from different domains often exhibit distinctive patterns, a promising direction is to identify each sample's domain and accumulate feature statistics separately for each domain. 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Ayed, “Realistic test-time adaptation of vision-language models,” arXiv preprint arXiv:2501.03729, 2025. 7 +[44] H. Li, Y. Zhou, X. Gu, B. Li, and W. Wang, "Diversified semantic distribution matching for dataset distillation," in Proceedings of the 32nd ACM International Conference on Multimedia, 2024, pp. 7542-7550. 8 +[45] Z. Guan, Y. Zhou, and X. Gu, “Capture global feature statistics for one-shot federated learning,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 39, no. 16, 2025, pp. 16942-16950. 8 +[46] M. D. McDonnell, D. Gong, A. Parvaneh, E. Abbasnejad, and A. Van den Hengel, “Ran-pac: Random projections and pre-trained models for continual learning,” Advances in Neural Information Processing Systems, vol. 36, pp. 12022–12053, 2023. 8 +[47] Z. Guan, G. Zhu, Y. Zhou, W. Liu, W. Wang, J. Luo, and X. Gu, “Stsa: Federated class-incremental learning via spatial-temporal statistics aggregation,” arXiv preprint arXiv:2506.01327, 2025. 8 +[48] D. Osowiechi, M. Noori, G. Vargas Hakim, M. Yazdanpanah, A. Bahri, M. Cheraghalikhani, S. Dastani, F. Beizaee, I. Ayed, and C. Desrosiers, "Watt: Weight average test time adaptation of clip," Advances in neural information processing systems, vol. 37, pp. 48015-48044, 2024. 16 +[49] D. Wang, E. Shelhamer, S. Liu, B. Olshausen, and T. Darrell, “Tent: Fully test-time adaptation by entropy minimization,” in International Conference on Learning Representations. 16 +[50] S. Cui, J. Xu, Y. Li, X. Tang, J. Li, J. Zhou, F. Xu, F. Sun, and H. Xiong, “Bayestta: Continual-temporal test-time adaptation for vision-language models via gaussian discriminant analysis,” arXiv preprint arXiv:2507.08607, 2025. 17 +[51] M. U. Khattak, H. Rasheed, M. Maaz, S. Khan, and F. S. Khan, “Maple: Multi-modal prompt learning,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2023, pp. 19-113-19-122. 17 + +# Statistics Caching Test-Time Adaptation for Vision-Language Models + +Appendix + +# A Details on forgetting and blocking issues + +Since low-entropy zero-shot predictions are not always correct (as shown in the following figure), they can lead to forgetting and blocking issues that hinder the accumulation of new knowledge in existing cache-based methods [14, 15, 19]. We conduct experiments to illustrate this (Figure 2 in the Introduction). Below, we present the experimental settings and analysis in detail. + +![](images/197c012eb3870358291394fa09f216cfb240d911abb0d88793362b90b9fd6af2.jpg) + +![](images/d2574a9d042f496faa22b3342bbe6ce65aa4d4d3689726f60fa692460e135be6.jpg) +Figure A1: An example of wrong high confidence sample. + +Forgetting: a misclassified low-entropy sample can overwrite a previously correct entry in the cache, leading to the forgetting of useful knowledge that had already been acquired. To better illustrate forgetting issues, we conduct an additional set of control experiments. Firstly, we identify a small set of "forgetting trigger samples." For each dataset, we randomly select 10 classes and choose the single lowest-entropy misclassified sample for each class. To ensure these 10 trigger samples are admitted by the cache, we filter out any correctly classified samples from the entire test set that have even lower entropy. From the remaining pool of test samples, we randomly draw 500 samples to serve as our fixed "final test samples." All other remaining samples are designated as "initial samples," which allow the cache to accumulate sufficient class-specific knowledge. We then consider the following two sequences for updating the model's cache: + +- Sequence A: [initial samples] +- Sequence B: [initial samples, forgetting trigger samples] + +The forgetting trigger samples, with their low entropy, are designed to infiltrate the cache built from the initial samples, causing it to forget previously learned information. In other words, the cache updated with Sequence A represents the state before forgetting, while the cache updated with Sequence B represents the state after forgetting. To quantify the impact of forgetting, we measure the performance difference between Sequence A and Sequence B on the final test samples, where the performance drop is defined as the accuracy of Sequence A minus the accuracy of Sequence B. The results are summarized in the following figure. + +We have several observations: (1) The SOTA baseline DPE is highly vulnerable to forgetting. (2) SCA is inherently robust to forgetting, as its accumulation design prevents error samples from fully + +![](images/cfacca994bbc6b6cecdf783f9285c71e0df0fb9d7d805685ae85ae16590c096b.jpg) +Figure A2: Forgetting issues. + +overwriting the stored knowledge. (3) Remarkably, SCA even leverages these "forgetting triggers" to improve performance (a negative drop), turning a vulnerability into a strength. + +**Blocking:** A misclassified sample with low entropy can, once inserted into the cache, occupy a cache slot persistently due to its high confidence score. This prevents correct but higher-entropy samples from being stored, thereby impeding subsequent knowledge accumulation and refinement. To illustrate this effect, we simulate blocking cases by randomly selecting $20\%$ of all classes and, for each selected class, identifying the single misclassified sample with the lowest entropy. These samples are placed at the start of the test sequence, and the performance drop is measured against the standard evaluation order. As shown in the figure below, the performance degradation caused by blocking is considerably smaller for SCA than for DPE. This advantage arises because SCA's statistics-based accumulation mechanism is not constrained by fixed cache slots, thus avoiding the persistence of early high-confidence errors and enabling effective learning from later samples. + +![](images/b57b24c8639c766c3a41cff726a694a43639995fb0300498185573b8e2f3a907.jpg) +Figure A3: Blocking issues. + +# B Experiment + +# B.1 Details on Dataset + +We evaluate our methods on 15 datasets, including Caltech101 [27], DTD [29], EuroSAT [30], FGV-CAircraft [26], Flowers102 [21], Food101 [31], OxfordPets [32], StanfordCars [28], SUN397 [33], UCF101 [34], ImageNet [35], ImageNet-V2 [10], ImageNet-Sketch [37], ImageNet-A [9], and ImageNet-R [36]. Table B1 provides detailed statistics for each dataset, including the number of + +classes, the sizes of the training, validation, and testing sets, as well as the corresponding original task for each dataset. + +Table B1: Summary of the 15 image classification datasets used in experiments. The last four ImageNet variant datasets are designed for evaluation only and contain no training or validation splits. + +
BenchmarkDatasetClassesSplitsTask
trainvaltest
Cross-DomainCaltech101 [27]1004,1281,6492,465Object recognition
DTD [29]472,8201,1281,692Texture recognition
EuroSAT [30]1013,5005,4008,100Satellite imagery
FGVCAircraft [26]1003,3343,3333,333Fine-grained aircraft recognition
Flowers102 [21]1024,0931,6332,463Fine-grained flower recognition
Food101 [31]10150,50020,20030,300Fine-grained food recognition
OxfordPets [32]372,9447363,669Fine-grained pet recognition
StanfordCars [28]1966,5091,6358,041Fine-grained car recognition
SUN397 [33]39715,8803,97019,850Scene recognition
UCF101 [34]1017,6391,8983,783Action recognition
Out-of-DistributionImageNet [35]1,0001.28M-50,000Object recognition
ImageNet-V2 [10]1,000--10,000Robustness (collocation shift)
ImageNet-Sketch [37]1,000--50,889Robustness (sketch domain)
ImageNet-A [9]200--7,500Robustness (adversarial attack)
ImageNet-R [36]200--30,000Robustness (multi-domain)
+ +# B.2 Baseline Details + +- TPT [12], a prompt tuning method designed to minimize self-entropy across predictions from multiple augmented views. +- DiffTPT [13], an improved variant of TPT that leverages diffusion-based augmentations to refine prompt optimization. +- TDA [14], a cache-based method that builds positive and negative caches to improve the prediction of the current test sample. +- DMN-ZS [19], a cache-based method that constructs memory banks from test data and pseudolabels to enhance the prediction of the current test sample. +- DPE [15], a cache-based method that trains multi-modal prototypes with historical test data to accumulate task-specific knowledge for test-time adaptation. +- BCA [25], a method that not only updates class embeddings to adapt the likelihood but also updates each class's prior using the posterior of incoming samples. +- Dynaprompt [22], a prompt-tuning method that maintains a prompt buffer and introduces a dynamic selection strategy to adaptively leverage relevant information from previous test samples for test-time adaptation. + +# B.3 Comparison with other baselines + +DMN-ZS [19] is a cache-based method that builds memory banks using test data and pseudo-labels to boost the prediction accuracy of the current test sample. In the original paper, DMN-ZS [19] uses a large cache size $(M = 50)$ and conducts extensive hyperparameter search with ground-truth labels from the test set to determine the fusion hyperparameter for each downstream task. Removing hyperparameter search leads to a significant drop in performance. We provide a version that uses the same small cache size $(M = 3)$ as other cache-based methods and applies a fixed fusion hyperparameter across all datasets. As shown in Table B2, our SCA substantially outperforms DMN-ZS under a fixed fusion hyperparameter and a small cache size $(M = 3)$ , and exceeds the performance of DMN-ZS configured with a large cache and per-dataset hyperparameter tuning as reported in the original paper. + +Additionally, we provide results of other two gradient-based baselines WATT [48] and TENT [49] on the cross-domain benchmark using CLIP ViT-B/16. + +Table B2: Experimental results on the cross-domain benchmark with CLIP ViT-B/16. DMN-ZS $(M = 50)$ refers to the original results, where the fusion hyperparameter is exhaustively tuned for each dataset. DMN-ZS $(M = 3$ , fixed) uses the same small cache size as other cache-based methods and applies a fixed fusion hyperparameter across all datasets. The results of TENT and WATT are directly derived from [50]. + +
MethodAircraftCaltechCarsDTDEuroSATFlowerFood101PetsSUN397UCF101Average
DMN-ZS (M=50)30.0395.3867.9655.8559.4374.4985.0892.0470.1872.5170.30
DMN-ZS (M=50, fixed)29.7694.6067.5355.0856.7975.2781.6089.8969.9572.2269.27
DMN-ZS (M=3)28.5694.7766.7353.8456.7575.4984.5091.0368.9972.5969.33
DMN-ZS (M=3, fixed)25.5991.5255.5746.6346.1474.8371.5772.1760.7765.1661.00
TENT23.5293.8765.9645.5149.3667.4082.8987.3565.5865.0864.65
WATT24.2793.2766.3746.3955.9568.2183.2688.0965.8966.0165.77
SCA (Ours)28.5094.8568.4957.0957.1676.0986.0991.4470.2773.4370.34
+ +# B.4 Impact of prompt initialization + +We follow the existing cache-based methods that use prompt ensembling strategy [19, 15] to initialize the prompt. This practice is widely adopted in the literature to ensure a robust and fixed zero-shot classifier, which serves as the foundation for these feature-space adaptation methods. + +For prompt-tuning based methods, the use of a single prompt for methods like TPT and DiffTPT is also the standard protocol, as their focus is on demonstrating adaptation of the prompt itself. + +Additionally, we present results using high-quality prompts obtained through training as the enhanced prompt initialization for prompt-tuning based TTA [12]. Specifically, we use the prompts trained by MaPLe [51] as the initialization, a widely adopted approach in prompt-tuning based TTA. As shown in the table below, the performance of prompt-tuning based methods is significantly lower than that of cache-based methods. + +Table B3: Experimental results on the cross-domain benchmark with CLIP ViT-B/16. + +
MethodAircraftCaltechCarsDTDEuroSATFlowerFood101PetsSUN397UCF101Average
TPT24.7894.1666.8747.7542.4468.9884.6787.7965.5068.0465.10
TPT+MaPLe24.7093.5966.5045.8747.8072.3786.6490.7267.5469.1966.49
SCA (Ours)28.5094.8568.4957.0957.1676.0986.0991.4470.2773.4370.34
+ +# B.5 Impact of the number of classes and test samples on performance + +Impact of the number of classes on performance: The number of classes inevitably impacts performance, as this dictates the inherent difficulty of the task. However, our results show that there is no consistent inverse relationship between the performance gain of SCA and the number of classes. In other words, the effectiveness of our SCA does not systematically degrade as the number of classes increases. + +Impact of the number of test samples on performance: The performance of any cache-based TTA method, including strong baselines like DPE and our SCA, is indeed influenced by the amount of test data available for adaptation. Here, we evaluate both SCA and DPE on subsets of the test data (10%, 50%, and 100%) for two datasets: DTD (1692 testing samples) and SUN397 (19,850 testing samples). As shown in the figure below, while both methods experience performance degradation with fewer test samples, SCA shows a smaller drop. This is because SCA leverages every encountered testing sample to incrementally update its feature statistics. Unlike existing cache-based methods that may filter or discard samples, SCA ensures that no information is wasted. By fully leveraging all encountered test samples, SCA builds a more comprehensive and accurate statistical model, resulting in a notable performance advantage under data-scarce scenarios. + +![](images/c26bedcacf0b031a06723871789dc29ed9dc1ba1d0d53e249e4c0b752bbe80d1.jpg) +Figure B4: Performance curve. + +![](images/26fb3f46169dc8734154a1290487cec85b165bce29a2d7e8f123f6837b628853.jpg) + +# C Broader Impacts + +This paper focuses on test-time adaptation (TTA) for vision-language models (VLMs). In real-world applications, data often come from domains that differ significantly from the training distribution, leading to performance degradation. TTA enables VLMs to adjust to these unseen distributions on the fly, without requiring access to labeled data. Our research improves both the effectiveness and efficiency of TTA on VLMs, paving the way for its wider adoption in practical applications. + +# NeurIPS Paper Checklist + +# 1. 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a/stealthyyeteffectivedistributionpreservingbackdoorattacksongraphclassification/full.md b/stealthyyeteffectivedistributionpreservingbackdoorattacksongraphclassification/full.md new file mode 100644 index 0000000000000000000000000000000000000000..36bd6f91c65cee57014da8c5f8133e2efd91653c --- /dev/null +++ b/stealthyyeteffectivedistributionpreservingbackdoorattacksongraphclassification/full.md @@ -0,0 +1,861 @@ +# Stealthy Yet Effective: Distribution-Preserving Backdoor Attacks on Graph Classification + +Xiaobao Wang $^{1,2}$ Ruoxiao Sun $^{1}$ Yujun Zhang $^{1}$ Bingdao Feng $^{1}$ Dongxiao He $^{1}$ Luzhi Wang $^{3}$ Di Jin $^{1,*}$ + +1College of Intelligence and Computing, Tianjin University, Tianjin, China + +$^{2}$ Guangdong Laboratory of Artificial Intelligence and Digital Economy (SZ), Shenzhen, China + +3College of Artificial Intelligence, Dalian Maritime University, Dalian, China + +{wangxiaobao, rxiao_sun, zhangyujun}@tju.edu.cn + +{fengbingdao, hedongxiao, jindi}@tju.edu.cn, wangluzhi0@gmail.com + +# Abstract + +Graph Neural Networks (GNNs) have demonstrated strong performance across tasks such as node classification, link prediction, and graph classification, but remain vulnerable to backdoor attacks that implant imperceptible triggers during training to control predictions. While node-level attacks exploit local message passing, graph-level attacks face the harder challenge of manipulating global representations while maintaining stealth. We identify two main sources of anomaly in existing graph classification backdoor methods: structural deviation from rare subgraph triggers and semantic deviation caused by label flipping, both of which make poisoned graphs easily detectable by anomaly detection models. To address this, we propose DPSBA, a clean-label backdoor framework that learns in-distribution triggers via adversarial training guided by anomaly-aware discriminators. DPSBA effectively suppresses both structural and semantic anomalies, achieving high attack success while significantly improving stealth. Extensive experiments on real-world datasets validate that DPSBA achieves a superior balance between effectiveness and detectability compared to state-of-the-art baselines. The code is available at https://github.com/TheCoder0fs/DPSBA. + +# 1 Introduction + +Graph Neural Networks (GNNs) are foundational models for learning on graph-structured data, achieving strong performance in tasks like node classification, link prediction, and graph classification [1]. Through message passing, GNNs capture structural and feature dependencies, enabling applications in social networks [2], recommender systems [3], and molecular analysis [4]. As GNNs are increasingly adopted in real-world systems, their security has drawn growing attention. Among various threats, backdoor attacks, where imperceptible triggers are embedded [5] in training data to manipulate predictions, pose a serious risk due to their stealth [6-8]. While extensively studied in vision and NLP, backdoor vulnerabilities in graph learning remain under exploration. + +Most existing graph backdoor studies focus on node classification, where local triggers are injected to misclassify specific nodes [9, 10]. These attacks leverage GNN locality to propagate triggers through neighborhoods while remaining hard to detect. Link prediction attacks similarly operate in a local context, targeting the presence or absence of edges between node pairs [11]. In contrast, graph classification poses a fundamentally different and more complex challenge. Here, the attacker must influence the global semantics of an entire graph rather than a single node or edge. This requires + +![](images/efd575b6b9e6269d6dee1ffc777d87f6f082b5cf800a272fd8f147e403642f9a.jpg) +(a) ER-B + +![](images/422617e69fd7f73aa97e9e7f72be360b6a08a205d8e1aa77f15f76d41f9039d0.jpg) +(b) GTA +Figure 1: Anomaly score distributions of clean vs. backdoor graphs on the AIDS dataset. Anomaly scores are computed via SIGNET [12] for backdoor samples generated by ER-B [13], GTA [14], and Motif [15]. In all cases, backdoor graphs (yellow) shift significantly rightward compared to clean graphs (blue), revealing strong distributional deviation. This underscores the detectability of prior triggers and the need for distribution-preserving attacks. + +![](images/8ca8cb7ceedd8b89f65adf55957205bde0d1ed21a28d002b85dd3b5750d3b707.jpg) +(c) Motif + +triggers to manipulate the full-graph embedding, often through larger or structurally rare subgraphs. Recent backdoor attacks on graph classification attempt to address global manipulation challenges by injecting structurally distinct subgraphs, such as rare motifs [15]. However, these approaches often introduce obvious out-of-distribution (OOD) artifacts, which significantly compromise stealth and limit their practicality in real-world settings. As shown in Figure 1, there exists a clear distribution gap between clean and backdoor graphs, making current methods highly detectable by outlier detection models. Although Zhang et al. [16] similarly observe OOD issues in node classification and propose a distribution-preserving strategy, their approach relies on local perturbations that affect only a single node's embedding. In contrast, graph classification requires modifying the global message-passing process to influence the entire graph representation, making distribution preservation significantly more challenging (see Appendix C.1 for theoretical justification). + +This global nature further amplifies the risk of both structural and semantic deviations when inserting backdoor triggers, as detailed in Appendix A: 1) Structural Deviation: Triggered by the injection of rare or unnatural subgraphs (e.g., low-frequency motifs) that diverge from the structural distribution of clean graphs. These triggers often create shortcut correlations with target labels and are easily identified by outlier detectors, as shown in Table A5. Rare motifs as triggers (Motif) achieve high attack success rates (ASR) but are highly detectable. In contrast, more frequent motifs (Motif-S) reduce detectability but also weaken attack effectiveness; 2) Semantic Deviation: Caused by label flipping, this introduces a discrepancy between a graph's assigned class and its inherent structure. As demonstrated in Table A6, adopting clean-label settings reduces the overall anomaly degree (AUC) across all methods but results in a notable ASR drop, revealing a fundamental trade-off between attack stealth and success. These findings highlight a critical limitation of existing methods: their inability to balance effectiveness and stealth. This raises a key challenge: Can we design a graph-level backdoor attack that preserves the distributional properties of clean samples, avoids label manipulation, and remains both effective and stealthy? + +To tackle these challenges, we propose DPSBA (Distribution-Preserving Steadybackdoor Attack), a clean-label backdoor framework for graph classification. Unlike prior methods that use rare or manually designed subgraph triggers, often causing high anomaly scores, DPSBA adaptively learns in-distribution triggers from target-class data. To address both structural and semantic deviations, it introduces a joint training strategy combining: (1) a clean-label attack mechanism that preserves semantic consistency by avoiding label flipping, and (2) distribution-aware discriminators that penalize structural and feature-level anomalies. Through adversarial training with a surrogate classifier and anomaly detectors, DPSBA enables effective and stealthy trigger generation. Experiments on real-world datasets show that DPSBA substantially lowers anomaly scores while maintaining high attack success rates, achieving a strong trade-off between stealth and efficacy. + +# 2 Preliminaries + +In this section, we first give the definition of the graph classification task and then introduce our stealthy backdoor attack objective. + +# 2.1 Graph Classification Task + +Given a dataset consisting of graph-structured instances, the objective of graph classification is to assign a class label to each graph. Formally, a graph classification dataset is defined as: $\mathcal{C} = \{(G_1,y_1),(G_2,y_2),\dots,(G_n,y_n)\}$ , where each graph $G_{i} = (V_{i},E_{i})$ is composed of a set of nodes $V_{i}$ and edges $E_{i}$ , and $y_{i}\in \mathcal{V}$ is the corresponding ground-truth class label. Let $n$ denote the number of graph samples in the dataset. The goal is to learn a graph-level classifier: $f:\mathcal{G}\to \mathcal{V}$ , which maps each input graph $G\in \mathcal{G}$ to one of the class labels in $\mathcal{V} = \{y_1,y_2,\dots,y_n\}$ . The classifier $f$ is typically trained via gradient-based optimization using a supervised loss function $\mathcal{L}_{train}$ (e.g., cross-entropy) over the training set $\mathcal{C}_{train}$ . + +# 2.2 Stealthy Backdoor Attack on Graph Classification Task + +Backdoor attacks inject a hidden and malicious trigger mechanism (i.e., "backdoor") into the target model in advance. When the trigger appears, the backdoor is activated and misleads the model to get the desired output. + +Attacker's Goal. The objective of a backdoor attacker in graph classification is threefold: (1) The model trained with injected backdoors should predict any graph containing the trigger as the attackerspecified target label $y_{t}$ ; (2) The model's performance on clean graphs should remain unaffected; (3) The backdoor graphs should not exhibit significant distributional deviations from clean graphs, ensuring high stealthiness. Formally, the attacker's goal can be formulated as: + +$$ +f _ {\theta_ {t}} \left(m (G; g _ {t})\right) = y _ {t}, \quad f _ {\theta_ {t}} (G) = f _ {\theta} (G), \quad \operatorname {d i f f} \left(f _ {o} \left(m (G; g _ {t})\right), f _ {o} (G)\right) \leq \epsilon , \tag {1} +$$ + +where $G$ denotes a clean graph, and $y_{t}$ is the target label specified by the attacker. The function $m(G; g_{t})$ denotes the trigger injection process, producing a backdoor graph $G_{g_{t}}$ by embedding trigger subgraph $g_{t}$ into $G$ . $f_{\theta}$ and $f_{\theta_{t}}$ represent the clean and poisoned graph classification models, respectively. $f_{o}$ is a detection model trained on clean graphs, and $\text{diff}(\cdot, \cdot)$ measures the anomaly difference (e.g., anomaly score) between a clean graph and its backdoor counterpart. The scalar $\epsilon$ defines an upper bound on acceptable deviation for stealthiness, and can be tuned per dataset. + +Attacker's Knowledge and Capability. We consider a restricted and realistic threat model in which the attacker has limited access and control: (1) Black-box knowledge: The attacker has no knowledge of the target model architecture, training hyperparameters, or optimization pipeline. This accommodates the possibility that different graph classification models may be deployed. (2) Limited poisoning capability: The attacker can only poison a small fraction of the training dataset and is not allowed to modify any labels. This reflects a more challenging clean-label setting and simulates real-world scenarios where data integrity is partially enforced. + +# 3 Methodology + +Grounded in the attacker's goal, DPSBA adaptively generates subgraph triggers whose topology and features mimic in-distribution patterns of clean graphs, reducing detectability while ensuring attack success. As shown in Figure 2, DPSBA includes two stages: poisoned sample construction and trigger optimization. In the first stage, hard samples from the target class are selected and injected with triggers at strategically chosen locations. The trigger's structure and features are generated by a learnable topology-feature generator. In the second stage, DPSBA jointly optimizes attack success via a surrogate model and minimizes detectability via anomaly-aware discriminators through staged adversarial training. + +# 3.1 Poisoned Sample Construction + +To support clean-label stealthy backdoor injection, DPSBA constructs poisoned samples from the target class itself, avoiding explicit label flipping. This process comprises three stages: (i) selecting informative samples for poisoning, (ii) identifying optimal injection locations within each graph, and (iii) initializing and injecting the trigger subgraph. Together, these steps ensure that the backdoor signal is both semantically consistent and statistically in-distribution. + +![](images/5c1ae04d6897b386a039735e865ca6eb776fb7b45a655ae8a07f20ad1d5abd61.jpg) +Figure 2: The overall structure of DPSBA + +# 3.1.1 Hard Sample Selection + +In the clean-label setting, directly selecting target-class samples avoids semantic inconsistency but may reduce the effectiveness of the attack. To counteract this, DPSBA strategically mines hard samples, i.e., samples from the target class that the model finds uncertain. This design is inspired by Gao et al. [17], who show that hard samples can amplify the effectiveness of backdoor attacks by being more susceptible to manipulation. In particular, low-confidence samples, those near the model's decision boundary, require smaller perturbations to shift predictions, and are thus better candidates for subtle backdoor injection. We use a clean surrogate model $f_{\theta}$ to measure the confidence score of a graph $G$ with respect to the target label $y_{t}$ as: + +$$ +\operatorname {c f d} (G) = \operatorname {s o f t m a x} \left(f _ {\theta} (G)\right) _ {y _ {t}} = \frac {e ^ {f _ {\theta} (G) _ {y _ {t}}}}{\sum_ {j = 1} ^ {K} e ^ {f _ {\theta} (G) _ {j}}}, \tag {2} +$$ + +where $f_{\theta}(G) \in \mathbb{R}^{K}$ denotes the logit output of the surrogate model for graph $G$ , $K$ is the total number of classes, and $y_{t}$ is the target class index. We select the bottom $p\%$ of target-class graphs with the lowest $\operatorname{cdf}(G)$ scores as poisoned samples. These hard samples are passed to the subsequent trigger location and injection stages, where subtle but effective perturbations are applied. Since they are more vulnerable to decision changes, they are more likely to be misclassified into the target class once the trigger is activated, while still maintaining clean-label consistency and high stealth. + +# 3.1.2 Trigger Location Selection + +After selecting the hard target-class samples for poisoning, DPSBA determines where in each graph to inject the trigger. Existing studies adopt various strategies to identify influential nodes, such as using centrality metrics [15] or extracting interpretable subgraphs [18, 19]. While interpretation-based methods offer precision, they often incur prohibitive computational costs. To strike a balance between scalability and informativeness, DPSBA adopts a two-stage selection process. First, nodes with high degree centrality are pre-selected as candidates. Specifically, for a graph $G$ with $N$ nodes, we compute the normalized degree centrality of each node $v$ as $\deg(v)/(N-1)$ and retain the top- $k$ nodes (with $k$ typically set to $2M$ , where $M$ is the trigger size). Next, among these candidates, we identify the $M$ most influential nodes using a surrogate-based ablation approach. For each node $v$ in the candidate set $\mathcal{N}_{can}$ , we evaluate its contribution to model prediction by measuring the change in output after removing it: + +$$ +S (v) = \left| f _ {\theta} \left(G + \Delta_ {v}\right) - f _ {\theta} (G) \right|, \tag {3} +$$ + +where $\Delta_v$ denotes the removal of node $v$ from $G$ . The top- $M$ nodes with the highest $S(v)$ values are selected as trigger attachment nodes. These nodes are then replaced, along with their edges, by an initialized subgraph trigger $g_t$ to prepare for the subsequent generation phase. + +# 3.1.3 Trigger Generation and Injection + +To enable the insertion of stealthy and effective triggers, generative approaches offer superior flexibility and control over random, interpretability-based, or search-based strategies. Inspired by GTA [14], DPSBA introduces two lightweight yet expressive modules, i.e., a topology generator and a feature generator, that cooperatively define the structure and node attributes of the injected subgraph. These generators are trained in a separate phase to adaptively fit the data distribution. + +Topology Generator. The topology generator is implemented as a multilayer perceptron (MLP) that transforms the adjacency matrix $\mathbf{H}$ of the target injection region into a learnable soft structure: $\mathbf{H}' = \sigma(W_1\mathbf{H} + b_1)$ , where $W_1$ and $b_1$ are learnable parameters (for the convenience of subsequent discussion, they will be simply referred to as $\omega_t$ ) and $\sigma(\cdot)$ is a non-linear activation function. For undirected graphs, the adjacency matrix is symmetrized as $\mathbf{A} = \frac{1}{2} \left( \mathbf{H}' + \mathbf{H}'^\top \right)$ . To accommodate the discrete nature of graph topology, we adopt a binarization mechanism inspired by binarized neural networks [20], where the final adjacency matrix is given by: $\mathbf{A}_{binary} = \mathbb{I}(\mathbf{A} > 0.5)$ . Binarization is applied only during forward propagation to preserve gradient flow during backpropagation. The resulting binary matrix defines the trigger's structural connectivity. + +Feature Generator. To ensure the feature space consistency between clean and poisoned graphs, the feature generator produces trigger node features based on the original features at the injection site. It employs an MLP to generate: $\mathbf{X}' = \sigma (W_2\mathbf{X} + b_2)$ , where $\mathbf{X}$ denotes the original features of the injected nodes, and $W_{2},b_{2}$ are learnable parameters, which are simply referred to as $\omega_{f}$ for the convenience of subsequent discussion. This transformation ensures that generated features remain aligned with the data distribution, thereby reducing feature-level anomaly. Besides, the feature generator can also avoid distribution shift when the target graphs have high attribute variance. The feature generator does not rely on distribution priors of the entire dataset. Instead, it takes as input the local structural and attribute context around the trigger injection site and generates features that blend smoothly with the surrounding neighborhood. This allows the generator to adapt to the specific variance of each target graph on a per-instance basis, without requiring any handcrafted normalization or explicit regularization. As a result, the generated features remain close to the local data manifold, reducing the likelihood of creating detectable anomalies, even in the presence of high attribute variance across graphs. + +Trigger Injection. Given a selected clean graph $G$ and the generated trigger subgraph $g_{t} = (\mathbf{A}_{binary},\mathbf{X}^{\prime})$ , DPSBA injects the trigger at the designated node locations to produce a poisoned sample: $G_{g_t} = m(G;g_t)$ , where $m(\cdot)$ denotes the injection operation. The resulting $G_{g_t}$ is added to the training set along with clean graphs. After the backdoored model is trained, the attacker can craft adaptive triggers during inference and inject them into arbitrary test graphs to activate the backdoor and induce targeted misclassification. + +# 3.2 Trigger Optimization + +Once the trigger subgraph is initialized, DPSBA enters the optimization phase to jointly refine the topology and feature generators for both effectiveness and stealth. It minimizes a hybrid objective comprising: (1) an attack loss that enforces confident misclassification into the target class, and (2) an adversarial anomaly loss that suppresses structural and feature-level deviations detectable by outlier models. This is achieved via a two-stage adversarial training strategy with dedicated discriminators and a dynamically updated surrogate model. + +Attack Effectiveness. The core goal of a backdoor attack is to ensure that the trigger-embedded graph $G_{g_t}$ is classified into the target class $y_t$ . This is achieved by minimizing the attack loss with respect to a surrogate model $f_{\theta^*}$ : + +$$ +\mathcal {L} _ {a t k} = - \log f _ {\theta^ {*}} \left(G _ {g _ {t}}\right) _ {y _ {t}}, \tag {4} +$$ + +where $f_{\theta^{*}}(\cdot)_{y_t}$ denotes the logit output of the surrogate model for class $y_{t}$ . + +Stealthiness via Adversarial Anomaly Minimization. To ensure the trigger remains statistically inconspicuous, DPSBA introduces adversarial discriminators trained to distinguish clean and backdoor samples from structure and feature perspectives. Specifically, the topology discriminator $D_{\theta_t}$ (a GCN) detects structural anomalies, while the feature discriminator $D_{\theta_f}$ (an MLP) detects feature distribution shifts. The generators aim to fool these discriminators via the following minimax objectives: + +$$ +\min _ {\omega_ {t}} \max _ {\theta_ {t}} \mathcal {L} _ {d} ^ {(t)} = \sum_ {G \sim \mathcal {G} _ {c}} \log D _ {\theta_ {t}} (G) + \sum_ {G \sim \mathcal {G} _ {b}} \log \left(1 - D _ {\theta_ {t}} \left(G _ {g _ {t}} \left(\omega_ {t}\right)\right)\right), \tag {5} +$$ + +$$ +\min _ {\omega_ {f}} \max _ {\theta_ {f}} \mathcal {L} _ {d} ^ {(f)} = \sum_ {G \sim \mathcal {G} _ {c}} \log D _ {\theta_ {f}} (G) + \sum_ {G \sim \mathcal {G} _ {b}} \log \left(1 - D _ {\theta_ {f}} \left(G _ {g _ {t}} \left(\omega_ {f}\right)\right)\right), \tag {6} +$$ + +where $\mathcal{G}_c$ and $\mathcal{G}_b$ are the clean and poisoned graph sets, and $\omega_{t},\omega_{f}$ are the generator parameters. + +Joint Training Objectives. DPSBA jointly optimizes the generators using a weighted sum of the attack loss and the corresponding adversarial loss. For the topology generator: + +$$ +\min _ {\omega_ {t}} \sum_ {G \in \mathcal {G} _ {b}} \mathcal {L} _ {a t k} \left(G _ {g _ {t}} \left(\omega_ {t}\right)\right) + \alpha \mathcal {L} _ {d} ^ {(t)} \left(D _ {\theta_ {t}} \left(G _ {g _ {t}} \left(\omega_ {t}\right)\right)\right), \quad \text {s . t .} \theta^ {*} = \arg \min _ {\theta} \mathcal {L} _ {t r a i n} \left(f _ {\theta} (C)\right), \tag {7} +$$ + +and for the feature generator: + +$$ +\min _ {\omega_ {f}} \sum_ {G \in \mathcal {G} _ {b}} \mathcal {L} _ {a t k} \left(G _ {g _ {t}} \left(\omega_ {f}\right)\right) + \beta \mathcal {L} _ {d} ^ {(f)} \left(D _ {\theta_ {f}} \left(G _ {g _ {t}} \left(\omega_ {f}\right)\right)\right), \quad \text {s . t .} \theta^ {*} = \arg \min _ {\theta} \mathcal {L} _ {t r a i n} \left(f _ {\theta} (C)\right), \tag {8} +$$ + +where $\alpha$ and $\beta$ are hyperparameters balancing stealth and attack objectives, and $C$ is the clean training set used to update the surrogate model. + +Adversarial Training Strategy. To realize both stealth and attack effectiveness, DPSBA adopts a staged adversarial training strategy to optimize the topology and feature generators in coordination with anomaly discriminators. The joint objective encourages trigger-embedded graphs to be confidently misclassified into the target class while remaining statistically similar to clean samples. To further justify this design, we derive in Appendix C.2 a formal lower bound that connects the total variation distance between clean and poisoned graph distributions with the optimal anomaly detection AUC, demonstrating that minimizing such divergence directly improves stealthiness. + +The training process consists of two alternating phases: 1) Topology optimization phase: The topology generator $\omega_{t}$ and discriminator $D_{\theta_t}$ are alternately updated. The generator learns to create subgraph structures that both maximize attack loss $\mathcal{L}_{atk}$ and minimize structural anomaly signals detectable by $D_{\theta_t}$ . 2) Feature optimization phase: Similarly, the feature generator $\omega_{f}$ and discriminator $D_{\theta_f}$ are jointly trained. The generator adapts node features to support effective attacks while suppressing attribute-level anomalies identified by $D_{\theta_f}$ . To maintain alignment with the evolving trigger distribution, the surrogate model $f_{\theta}$ is fine-tuned after each phase, ensuring reliable attack gradient signals throughout training. This cooperative optimization framework allows DPSBA to dynamically balance the trade-off between attack success and stealth, as further summarized in Appendix D. Time complexity analysis can be found in Appendix F. + +# 4 Experiment + +We first introduce the experimental setup, including datasets, baselines, and implementation details. Then, we evaluate DPSBA's performance on attack effectiveness and stealth across multiple settings, followed by ablation analysis and anomaly stealth evaluation. Additional hyperparameter studies are included in Appendix E.4 E.5. + +# 4.1 Experimental Setup + +# 4.1.1 Datasets and Evaluation Metrics + +We evaluate DPSBA on four real-world graph classification datasets from the TUDataset benchmark [21]: PROTEINS_full [22] (protein graphs for function prediction), AIDS [23] (molecular graphs related to AIDS research), FRANKENSTEIN [24] (a compound property dataset combining BURS and MNIST features), and ENZYMES [25, 26] (a 6-class biomolecular classification task). For each dataset, we designate the minority class as the attack target to simulate a realistic low-frequency scenario. The statistics for the datasets are summarized in Table E7. We adopt three metrics: ASR (Attack Success Rate) for attack effectiveness, CAD (Clean Accuracy Drop) to assess model utility degradation, and AUC (Area Under the ROC Curve) to quantify anomaly detectability via outlier models. Detailed metric definitions are deferred to Appendix E.2. + +# 4.1.2 Baselines + +We compare DPSBA with five representative graph-level backdoor attack methods. ER-B [13] generates universal triggers using the Erdős–Rényi random graph model. LIA [18] modifies the connections of low-importance nodes, as identified by GNN explanation techniques, with fixed trigger structures. GTA [14] employs a bi-level optimization strategy to train a subgraph generator that adapts + +triggers per graph. Motif [15] selects low-frequency motifs from the dataset as effective but easily detectable triggers. Motif-S is a stealthier variant using the high-frequency M41 motif, reducing anomaly scores while slightly sacrificing attack strength. Following GTA [14], we split each dataset into $50\%$ training and $50\%$ test sets, with $5\%$ of the training data poisoned. To ensure fair comparison, all methods adopt a fixed trigger size of 4. Both the topology and feature generators are trained for 20 epochs per stage over 3 iterations with a learning rate of 0.001, using early stopping [27]. For baselines, we use the best hyperparameters reported in their original papers. + +# 4.1.3 Graph Classification Models and Anomaly Detection Algorithms + +Following prior work on graph backdoor attacks [15], we evaluate DPSBA on three widely used graph classifiers: GCN [28], GIN [29], and SAGPool [30]. These models are used both as attack targets and surrogates to test generalizability across architectures. Clean accuracy results are reported in Table E8. For anomaly detection, we adopt SIGNET [12], identified as the most effective method in a recent benchmark study [31], to assess the stealthiness of injected backdoor samples. + +# 4.2 Experimental Results + +Table 1: Comparison results between DPSBA and each baseline model + +
DatasetsSurrogate ModelMetricsER-BLIAGTAMotifMotif-SOurs
PROTEINS_fullGCNASR (%)51.5368.3573.1670.9148.5673.93
CAD (%)4.734.705.145.924.664.62
AUC (%)70.0471.0178.2079.1664.7260.11
GINASR (%)62.5358.7780.9679.0863.0187.91
CAD (%)4.884.364.574.974.334.92
AUC (%)79.6571.7479.9680.0670.4962.95
SAGPoolASR (%)65.3864.8194.0471.3557.0994.15
CAD (%)4.265.023.653.363.943.29
AUC (%)71.3476.8978.5782.7581.8169.20
AIDSGCNASR (%)85.3885.4993.2192.6956.0894.76
CAD (%)4.533.805.144.124.032.38
AUC (%)98.0897.2299.3499.7189.4372.65
GINASR (%)93.9995.5697.5297.7556.895.87
CAD (%)2.692.032.652.282.511.94
AUC (%)99.9899.2099.3499.7194.2973.66
SAGPoolASR (%)59.2662.6686.9987.6562.8998.90
CAD (%)1.651.793.772.642.44-0.40
AUC (%)95.7994.5699.6799.0293.4377.23
FRANKEN-STEINGCNASR (%)63.6061.0499.3580.5759.2498.37
CAD (%)1.711.562.741.153.961.01
AUC (%)80.4175.66100.0089.6469.2368.96
GINASR (%)92.0682.6398.6592.8758.6899.84
CAD (%)3.602.351.952.441.751.83
AUC (%)85.7376.1591.0687.5465.7773.46
SAGPoolASR (%)68.1590.1895.2384.5652.2999.99
CAD (%)4.784.664.644.616.864.57
AUC (%)64.8977.5080.4687.2960.9860.12
ENZYMESGCNASR (%)26.0930.4395.3321.7415.2196.67
CAD (%)4.174.993.004.99-1.67-0.67
AUC (%)68.3266.1571.2071.3566.2266.11
GINASR (%)37.8327.0296.0016.2112.1699.33
CAD (%)9.1710.002.678.334.17-0.33
AUC (%)71.4062.0176.4268.1865.7841.20
SAGPoolASR (%)29.5438.63100.0015.9111.37100.00
CAD (%)4.336.675.0010.833.334.00
AUC (%)57.7363.9870.3775.4769.4849.91
+ +Table 2: Results of the transferability evaluation(%) + +
Surrogate modelActual modelPROTEINS_fullAIDSFRANKENSTEIN
ASRCADASRCADASRCAD
GCNGIN81.324.7999.441.0198.370.03
SAGPool98.900.0896.142.4894.96-0.10
+ +# 4.2.1 Effectiveness and Stealthiness + +We evaluate DPSBA on four datasets against six baselines in terms of attack effectiveness (ASR), model performance drop (CAD), and anomaly detectability (AUC), as shown in Table 1. DPSBA consistently achieves optimal or near-optimal ASR across datasets and models, verifying its strong attack capability. This is credited to its adaptive trigger generation guided by confident hard samples and structurally important nodes. In terms of CAD, DPSBA maintains minimal accuracy degradation $(<5\%)$ , indicating high stealth at the model level. For distribution-level stealth, DPSBA significantly reduces AUC, i.e., maintaining values around $70\%$ , demonstrating strong resistance to detection by statistical outlier models. We further observe that ASR and AUC fluctuate significantly across datasets due to inherent differences in graph structure and anomaly sensitivity. For datasets particularly sensitive to attribute or structural perturbations (e.g., FRANKENSTEIN), even minor modifications can induce high ASR yet large anomaly scores. In such cases, we recommend an early stopping strategy, namely terminating generator updates once ASR saturates or marginal gains diminish, to enhance stealth. For instance, on FRANKENSTEIN, where attribute anomalies are highly detectable (GTA achieves $>95\%$ ASR but with high AUC), DPSBA disables the feature generator early, achieving near-perfect ASR solely via structural triggers. Across all benchmarks, DPSBA achieves the high ASR of Motif while matching the stealth of Motif-S, validating its ability to balance effectiveness and detectability. For multi-class graph classification tasks, DPSBA consistently outperforms all baselines in terms of both attack success rate (ASR) and stealth (CAD, AUC) across three surrogate models. Notably, most baselines (except GTA) fail on this dataset due to high attribute and strucuture variability, underscoring the advantage of our distribution-preserving and adaptive design. + +# 4.2.2 Transferability + +To evaluate the transferability of DPSBA, we use GCN as the surrogate model to generate triggers and test them on other classifiers. Table 2 reports ASR and CAD across datasets using GIN and SAGPool as target models. Since AUC is model-independent, it is omitted. Results show that DPSBA maintains high ASR and low CAD across architectures, demonstrating strong transferability even when using a simple surrogate model. Meanwhile, we observe a counterintuitive phenomenon. A comparison of the experimental results on the AIDS and PROTEINS_full datasets in Table 1 and Table 2 reveals that the ASR is higher when the surrogate model differs from the actual model than when they are identical. We provide a detailed discussion of this matter in Appendix E.6. + +# 4.2.3 Anomaly Stealth Analysis + +To show the anomaly Table 3: Performance (AUC%) of different anomaly detection models + +stealthiness of DPSBA, we first visualize the anomaly score distributions of clean and backdoor graphs on all datasets in Figure 3. The distributions show strong overlap, indicating that + +
ModelsPROTEINS_fullAIDSFRANKENSTEIN
OCGIN55.1986.5272.12
GLocalKD50.2431.4646.57
SIGNET60.1172.6568.96
+ +DPSBA-generated triggers exhibit minimal anomaly and are hard to distinguish from clean samples. This is largely attributed to the use of clean-label constraints and adaptive optimization. To further assess robustness across detectors, we evaluate DPSBA under multiple anomaly detection algorithms, including OCGIN [32] and GLocalKD [33], with results summarized in Table 3. DPSBA consistently achieves low AUCs and, notably, GLocalKD fails to detect most triggers, confirming the strong stealth of DPSBA across detection methods. Beyond anomaly detection, we examine randomized subsampling [13, 14], a standard defense strategy that randomly removes subgraphs during training. As shown in Figure 4(a), DPSBA maintains high ASR despite this defense, thanks to its compact + +trigger size and dual embedding in structure and features, making the trigger difficult to eliminate via sampling. + +![](images/03cd39b69578e35b0a14d51e3924a92e7f78310453288fed979fcdb948d116a2.jpg) +(a) FRANKENSTEIN + +![](images/0e45959ac070247626a75ede849d006800ca9fb6f27151d6fecb054a140b4a49.jpg) +(b) AIDS + +![](images/23cb26b2c0c1601448ea821f88f8b4f09891b88019370a97077ecc1c280708ab.jpg) +(c) PROTEINS_full + +We also test anomaly-based training data filtering. Taking AIDS as an example, we use SIGNET to remove the top $5\%$ anomalous samples before training. As Figure 4(b) shows, DPSBA remains robust, suffering minimal ASR drop even when most poisoned samples are detected. In contrast, previous methods show significant degradation. This highlights + +![](images/d4863bd41197150398dc8329c58f8940eeba9d37b9c3e8252cd11d4df54a4a75.jpg) +(a) Randomized Subsampling +Figure 4: Attack performance under defense + +![](images/bcf9c88d2cfd972fc7e38868423490be82759ff72a19639d916dbe9d2d4dd2f1.jpg) +Figure 3: Anomaly distribution visualization +(b) Data Filtering + +a key insight: existing detection strategies, while effective at identifying poisoned data, remain insufficient to fully prevent backdoor success due to the low sample requirement for trigger activation. More proactive and robust defenses are needed to counter stealthy attacks like DPSBA. + +# 4.3 Ablation Study + +To evaluate the role of each component in DPSBA, we conduct ablation experiments on PROTEINS_full and AIDS by removing one module at a time: DPSBA/S (w/o hard sample selection), DPSBA/N (w/o position selection), DPSBA/F (w/o feature generator), DPSBA/T (w/o topology generator), and DPSBA/OD (w/o adversarial + +Table 4: Results of the ablation experiments (%) + +
ModelPROTEINS_fullAIDS
ASRCADAUCASRCADAUC
DPSBA73.934.6260.1194.762.3872.65
DPSBA/S70.983.5760.2491.322.0972.60
DPSBA/N70.744.5358.9793.672.3171.26
DPSBA/F71.804.9659.0185.672.4067.26
DPSBA/T69.083.7154.7393.662.9171.41
DPSBA/OD90.884.9090.2399.463.5493.72
+ +training), with results shown in Table 4. Note: The FRANKENSTEIN dataset is highly sensitive to attribute perturbations, where even small feature changes can lead to high ASR and pronounced anomaly. As discussed in Section 4.2.1, we apply early stopping to the feature generator on this dataset; thus, it is excluded from the ablation study. + +Removing the sample or location selection module (S/N) slightly reduces ASR, confirming their effectiveness in identifying vulnerable graph regions. Excluding either generator (F/T) significantly degrades ASR but slightly improves AUC, suggesting that targeting both topology and features is essential for strong attacks, while single-aspect perturbation introduces fewer anomalies. The relative importance of these attributes varies across datasets, indicating that under strict stealth constraints, selectively optimizing one may be preferable. DPSBA/OD achieves the highest ASR but suffers the worst AUC, highlighting the importance of adversarial training for stealth. All other variants maintain CAD within $5\%$ , showing minimal impact on model performance. Overall, the full DPSBA provides the best trade-off between effectiveness and stealth, with adversarial training enhancing detectability resistance and other modules improving attack strength. + +# 4.4 Impact of the Loss Weights $\alpha$ and $\beta$ + +To control the anomaly level of injected triggers, DPSBA employs adversarial training with two loss weights: $\alpha$ for structural anomalies and $\beta$ for feature anomalies. This experiment investigates how varying these weights affects the trade-off between attack effectiveness and stealth. Specifically, we vary $\alpha$ and $\beta$ from 0.1 to 100 in the + +![](images/9c3001c710d58f4fe16c1c4029ab4c49117d9e871c0042d8359587900cbe7abf.jpg) +Figure 5: Parametric analysis of $\alpha$ and $\beta$ + +![](images/bac0c64c389e99a5abc1a04b12223b65221a69cc74582bd157ddcb3e3d10e9bd.jpg) + +joint loss formulations (Equations 7 and 8) and observe the corresponding changes in attack success rate (ASR) and anomaly detectability (AUC). Results are shown in Figure 5. Overall, both ASR and AUC exhibit a downward trend as $\alpha$ and $\beta$ increase. Larger weights impose stronger anomaly constraints, limiting the model's aggressiveness and thereby enhancing stealth. However, the rate of decline differs between $\alpha$ and $\beta$ , reflecting their distinct effects on structural and feature-based anomaly regularization, respectively. This suggests the two parameters can be tuned independently to suit dataset-specific characteristics. Importantly, when $\alpha$ and $\beta$ exceed a certain threshold, AUC stabilizes, indicating diminishing returns from further increasing anomaly penalties. This plateau signals that stealth constraints have been effectively enforced. At this point, DPSBA achieves a favorable balance between high ASR and low AUC. This analysis provides practical guidance for tuning $\alpha$ and $\beta$ based on application-specific requirements: small values favor stronger attacks, while larger values improve stealth, helping practitioners achieve optimal trade-offs on different datasets. + +# 5 Conclusion + +In this paper, we present DPSBA, a clean-label backdoor framework tailored for graph classification. Unlike prior work that relies on out-of-distribution triggers, semantic label flipping, or manually designed patterns, DPSBA learns in-distribution triggers directly from target-class graphs via a joint optimization process. The framework integrates a clean-label attack mechanism with anomaly-aware adversarial training, optimizing both attack effectiveness and stealth. Specifically, it employs structural and feature-level discriminators to minimize detectable deviations while preserving semantic consistency. Extensive experiments on diverse graph benchmarks and architectures demonstrate that DPSBA consistently achieves high attack success rates (ASR), low anomaly detection scores (AUC), and negligible clean accuracy drop (CAD). These results validate that DPSBA achieves a favorable balance between stealth and attack potency, outperforming existing baselines in both effectiveness and detectability. Our findings highlight the practicality of stealthy, distribution-preserving backdoor attacks in real-world graph learning scenarios. + +Limitations. The clean-label constraint, while significantly enhancing stealth, may limit attack potency in scenarios with imbalanced class distributions or scarce target-class samples, where clean instances are insufficient to support effective trigger learning. Additionally, DPSBA operates under a partial training data access assumption, which restricts its applicability to fully black-box scenarios or test-time attacks where no data access is available. Future work may investigate lightweight or black-box-compatible variants of DPSBA, explore adaptation to broader graph settings such as heterophilic or dynamic graphs, and develop theoretically grounded defenses capable of detecting distribution-preserving triggers. + +# 6 Acknowledgments + +This work was supported by the National Natural Science Foundation of China (No. 62302333, No. 92370111, No. 62272340, No. 62422210, No. 62276187 and No. 62502065), the Open Research Fund from Guangdong Laboratory of Artificial Intelligence and Digital Economy (SZ) (No. GML-KF-24-16), and Hebei Natural Science Foundation (No. F2024202047). + +# References + +[1] Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S Yu. A comprehensive survey on graph neural networks. IEEE transactions on neural networks and learning systems, 32(1):4-24, 2020. +[2] Wenqi Fan, Yao Ma, Qing Li, Yuan He, Eric Zhao, Jiliang Tang, and Dawei Yin. Graph neural networks for social recommendation. 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Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: In abstract, we outline the problems we have solved and the contributions we have made. The claims made match the theoretical and experimental results. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: In conclusion, we introduce the limitations of the method and the reasons that cause them. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: We use LLM only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Anomaly Issues in Graph Classification Attacks + +To demonstrate that backdoor graphs in graph classification tasks tend to exhibit clear out-distribution (OOD) characteristics, we conduct anomaly detection experiments on the AIDS dataset [23]. Specifically, we use the SIGNET algorithm [12] to evaluate the statistical anomaly of poisoned samples generated by representative backdoor attack methods, including ER-B [13], LIA [18], GTA [14], and Motif [15], comparing them against clean training samples. As shown in Table A5, we adopt AUC as the evaluation metric for anomaly detection, which reflects how well a detector can distinguish clean graphs from anomalous ones. The results show that most existing methods result in extremely high AUC scores, often exceeding $90\%$ , highlighting that backdoor samples are statistically distinguishable and exhibit significant anomalies. This stands in contrast to node-level backdoor attacks, where the GNN's local message passing can help diffuse trigger signals and reduce detectability. + +Table A5: Performance of backdoor attack methods on the AIDS dataset + +
MetricsBackdoor Attack Method
ER-BLIAGTAMotifMotif-S
ASR(%)85.3885.4993.2192.6956.08
AUC(%)98.0897.2299.3499.7189.43
CAD(%)4.533.805.144.124.03
+ +Structural Deviation. Backdoor models often exploit shortcut learning by associating rare subgraph triggers with the target class to enhance attack success. However, these rare triggers tend to deviate significantly from the natural graph distribution, making them easy to detect. Referring to the trigger selection strategy in Motif [15], we design a comparison between two variants: (1) Motif uses the least frequent motif as the trigger, while (2) Motif-S adopts the most common motif in the AIDS dataset. As shown in Table A5, Motif achieves higher ASR due to the strong signal of the rare trigger. However, this also results in extremely high AUC scores, indicating poor stealth. In contrast, Motif-S trades off some attack effectiveness for significantly improved stealth, as frequent motifs are naturally present in the dataset and thus harder for anomaly detectors to distinguish. This validates that trigger frequency directly influences the detectability of structural anomalies. + +Table A6: Performance of clean label backdoor attack + +
MetricsBackdoor Attack Method
ER-BLIAGTAMotifMotif-S
ASR(%)37.1157.6291.8884.371.80
AUC(%)61.0359.4298.0597.7471.88
CAD(%)4.333.784.924.244.13
+ +Semantic Deviation. Label manipulation, commonly used in traditional backdoor attacks, introduces semantic inconsistency between the graph content and its assigned label, exacerbating anomaly. The success of such attacks relies on enforcing a co-occurrence between the trigger and target label, which is typically achieved by inserting the trigger into non-target-class samples and forcibly flipping their labels. This label tampering often results in category-level semantic conflict. To assess its effect, we further evaluate the same backdoor methods under a clean-label setting (i.e., without modifying labels). As shown in Table A6, while the AUC scores drop in all methods, indicating improved stealth, the ASR also drops significantly. This suggests that semantic consistency reduces the statistical footprint of the backdoor, but at the cost of attack success. In summary, label flipping enhances attack strength but induces additional semantic anomaly, making backdoor graphs more detectable. + +These findings highlight two major anomaly sources in graph-level backdoor attacks: structural deviation from rare subgraphs and semantic inconsistency due to label tampering. This motivates the design of DPSBA, which aims to preserve distributional characteristics by avoiding both sources of anomaly. + +# B Related Work + +# B.1 General Graph Backdoor Attacks + +Backdoor attacks constitute a class of adversarial attacks in which an attacker implants a hidden behavior, known as a trigger mechanism, into the model during training. Once trained, the model behaves normally on clean inputs, maintaining high utility and stealth. However, when presented with an input containing a specific trigger pattern, the backdoor is activated and causes the model to misclassify the input into an attacker-specified target class. This property makes backdoor attacks particularly insidious in real-world deployments. In the graph domain, various works have explored backdoor threats in settings beyond standard graph classification. For node classification tasks, researchers have proposed attacks that inject malicious subgraphs or perturb node features to manipulate predictions. For example, Zhang et al.[34] first investigated the stealthiness dimension of backdoor attacks on node classification, while Wang et al.[35] introduced multi-target attacks to expand adversarial flexibility. Zhao et al.[36] proposed a spectral backdoor method by subtly altering the frequency domain of node features. For link prediction tasks, Zheng et al.[37] constructed triggers composed of fabricated nodes and target link pairs, whereas Dai et al.[11] showed that even single-node triggers can influence link predictions. Other studies have expanded backdoor threats into more specialized domains. For example, Zhang et al.[6] proposed the first attack on graph contrastive learning[38, 39], and Alrahis et al.[40] designed a hardware-level GNN backdoor mechanism. In the context of graph prompt learning, Lyu et al.[41] framed backdoor injection as a feature-collision optimization problem. Federated graph learning has also been explored: Xu et al. proposed two attack paradigms: distributed backdoor attack (DBA)[42] and Centralized Backdoor Attack (CBA)[43], targeting collaborative learning environments through data poisoning strategies. Although recent efforts have extended graph backdoor attacks to a wide range of scenarios, the graph classification task remains the primary focus due to its foundational role in bioinformatics, chemistry, and cybersecurity. Its holistic graph-level nature presents both unique challenges and opportunities for backdoor research. + +# B.2 Backdoor Attacks against Graph Classification Tasks + +Existing backdoor attack methods targeting graph classification can be broadly categorized into four groups based on their trigger generation strategies: (1) random pattern generation, (2) interpretability-based generation, (3) gradient-based optimization, and (4) distribution-aware or search-based methods. Random Generation via Graph Distribution: Zhang et al.[13] first introduced a backdoor attack against GNNs by randomly generating subgraphs using Erdős–Rényi (ER) models and inserting them as universal triggers into the training set. This demonstrated that fixed subgraph patterns can reliably activate backdoors in graph classification. Sheng et al.[44] further incorporated statistical characteristics of the dataset to create hybrid local-global trigger patterns, improving both the flexibility and efficacy of the attack. Explainability-Based Trigger Design: Xu et al.[18] leveraged GNNExplainer to assess the importance of the node and identify the optimal injection sites, the least important or the most influential nodes, for the trigger subgraphs. This method emphasized the influence of structural semantics on model decisions and improved the stealthiness of trigger placement. Building on this idea, Wang et al.[19] proposed inserting explainable subgraphs as triggers. Tong et al.[45] took this further by identifying and replacing predictive substructures in graphs to construct semantically-aligned triggers. Gradient-Based Optimization Methods: Xi et al.[14] proposed GTA, a general and dynamic backdoor attack framework that learns subgraph triggers via bilevel optimization. Unlike fixed-pattern methods, GTA can adapt trigger structures to specific graph instances, enhancing attack flexibility and stealth. Similarly, Yang et al.[46] developed TRAP, which perturbs graphs through gradient-driven strategies to generate transferable, pattern-free backdoor triggers. Search-Based or Distribution-Aware Methods: Recent work has focused on designing triggers that better conform to data distributions. Dai et al.[47] extended the semantic backdoor paradigm to GNNs, constructing triggers based on class-relevant node types and graph semantics. Zheng et al.[15] introduced a motif-based framework, where triggers are searched from statistically significant and frequently occurring subgraph structures. They proposed three strategies to enhance effectiveness and stealth: 1) Use of motifs that are absent in the dataset (maximal anomaly), 2) Use of motifs predominantly present in the target class (semantic bias), 3) Use of dense motif structures with high interconnectivity (amplified influence). This taxonomy reflects a growing awareness that + +both attack efficacy and stealth must be co-optimized, especially as anomaly detection models are increasingly deployed for backdoor defense. + +# C Theoretical Analyses + +# C.1 Distributional Shift in Graph vs. Node-Level Attacks + +To rigorously compare the detectability of backdoor triggers in graph vs. node classification, we analyze the total variation distance (TV) under local subgraph perturbations. + +Definition 1 (Total Variation Distance). Let $\mathcal{P}_G, \mathcal{P}_G'$ be the distributions of clean and poisoned graphs in a graph classification task, and $\mathcal{P}_V, \mathcal{P}_V'$ for node classification. The distributional shift is quantified by: + +$$ +\mathrm {T V} _ {G} := \mathrm {T V} \left(\mathcal {P} _ {G}, \mathcal {P} _ {G} ^ {\prime}\right), \quad \mathrm {T V} _ {V} := \mathrm {T V} \left(\mathcal {P} _ {V}, \mathcal {P} _ {V} ^ {\prime}\right), \tag {9} +$$ + +where TV distance is defined as: + +$$ +\operatorname {T V} \left(P, P ^ {\prime}\right) := \frac {1}{2} \int_ {\mathcal {X}} | p (x) - p ^ {\prime} (x) | d x. \tag {10} +$$ + +Theorem 1 (Lower Bound on Graph-Level Distributional Shift). Let $N$ be the number of nodes in a graph, $M \ll N$ the number of trigger nodes, and assume the trigger introduces local deviation $\Delta = \| \mu_{\text{trigger}} - \mu \|_1$ in node features. Then the total variation distance of the graph-level distribution satisfies: + +$$ +\mathrm {T V} _ {G} \geq c \cdot \frac {M}{N}, \quad \text {w i t h} c = \frac {1}{2} \cdot \Delta . \tag {11} +$$ + +Proof. In graph classification, predictions are made based on aggregated global features (e.g., mean pooling). Let $\mu$ be the average node embedding of a clean graph, and $\mu_{\mathrm{trigger}}$ the embedding from trigger nodes. After injecting a trigger subgraph of size $M$ , the poisoned graph has average feature: + +$$ +\mu^ {\prime} = \left(1 - \frac {M}{N}\right) \mu + \frac {M}{N} \mu_ {\text {t r i g g e r}}. +$$ + +Then the feature shift becomes: + +$$ +\| \mu^ {\prime} - \mu \| _ {1} = \| \mu_ {\text {t r i g g e r}} - \mu \| _ {1} \cdot \frac {M}{N}. +$$ + +Using Pinsker's inequality variant, which states: + +$$ +\operatorname {T V} \left(P, P ^ {\prime}\right) \geq \frac {1}{2} \| \mathbb {E} _ {P ^ {\prime}} [ x ] - \mathbb {E} _ {P} [ x ] \| _ {1}, +$$ + +we obtain: + +$$ +\mathrm {T V} _ {G} \geq \frac {1}{2} \| \mu^ {\prime} - \mu \| _ {1} = \frac {1}{2} \cdot \| \mu_ {\text {t r i g g e r}} - \mu \| _ {1} \cdot \frac {M}{N} = c \cdot \frac {M}{N}. +$$ + +Remark 1 (On Comparison with Node-Level TV). In node classification, the trigger affects only a few nodes and their $k$ -hop neighbors. Since predictions are localized, the overall distributional shift is often less significant. Letting $\mathrm{TV}_V \approx \delta \cdot \frac{M}{N}$ be a heuristic approximation, we note: + +$$ +\mathrm {T V} _ {G} \geq c \cdot \frac {M}{N} > \mathrm {T V} _ {V}, \quad i f c > \delta . +$$ + +This inequality holds when the graph-level aggregation amplifies the statistical impact of the trigger compared to node-local shifts. + +Caution: The term $\mathrm{TV}_V$ and $\mathrm{TV}_G$ are over different data domains (nodes vs. graphs), and thus are not directly additive: + +$$ +W e \quad c a n n o t \quad c l a i m: \quad \mathrm {T V} _ {G} \geq \mathrm {T V} _ {V} + c \cdot \frac {M}{N}. +$$ + +Instead, we highlight the fact that graph-level shift lower bounds node-level shift under same perturbation size, and is strictly greater if the trigger induces global mean shifts. + +Corollary 1 (Implication for Backdoor Stealth). Backdoor attacks on graph classification inherently suffer greater distributional shift than on node classification. This makes graph-level triggers more detectable by anomaly detection models, motivating the need for distribution-preserving designs like DPSBA. + +# C.2 Trigger Distributional Detectability Bound + +We now establish a lower bound connecting the detectability of backdoor samples to the statistical divergence between clean and poisoned data distributions. + +Definition 2 (Total Variation Distance). Let $P$ and $P'$ be the distributions over clean and poisoned graphs. Then: + +$$ +\operatorname {T V} \left(P, P ^ {\prime}\right) := \sup _ {A \subseteq \mathcal {G}} | P (A) - P ^ {\prime} (A) | = \frac {1}{2} \int_ {\mathcal {G}} | p (x) - p ^ {\prime} (x) | d x, \tag {12} +$$ + +where $p(x), p'(x)$ are probability densities over $\mathcal{G}$ , the space of all graphs. + +Theorem 2 (Lower Bound on AUC for Detecting Poisoned Graphs). Let $s(\cdot)$ be any score function used by a binary anomaly detector to distinguish $P$ and $P'$ . Then: + +$$ +\mathrm {A U C} _ {\text {d e t}} \geq \frac {1 + \operatorname {T V} \left(P , P ^ {\prime}\right)}{2}. \tag {13} +$$ + +Proof. From hypothesis testing theory, the optimal binary decision rule achieves AUC: + +$$ +\mathrm {A U C} _ {\mathrm {o p t}} = \mathbb {P} (s (x _ {+}) > s (x _ {-})) = \frac {1 + \mathrm {T V} (P , P ^ {\prime})}{2}, +$$ + +where $x_{+} \sim P'$ , $x_{-} \sim P$ . Thus, no detector can achieve lower AUC than this bound. + +![](images/ac95446f6e6341c65340f87889b6b9a8090e9584592a15754f5607b809985c53.jpg) + +Corollary 2 (Indistinguishability at Low TV). If $\mathrm{TV}(P, P') \to 0$ , then: + +$$ +\mathrm {A U C} _ {d e t} \rightarrow 0. 5, +$$ + +implying poisoned graphs become statistically indistinguishable from clean ones. Designing triggers that preserve common graph motifs or feature statistics helps satisfy this condition. + +**Implication.** DPSBA minimizes the anomaly loss to reduce $\mathrm{TV}(P, P')$ , achieving stealthy attacks even under statistical anomaly detection, as supported by the empirical histograms in Fig. 3. + +# D Algorithm of DPSBA + +The DPSBA algorithm is detailed in Algorithm 1. Initially, we train the surrogate model $f_{\theta^{*}}$ using the clean dataset, initialize the relevant model parameters and select poisoning samples from the data whose class is the target class $y_{t}$ (lines 1-3). From lines 4-8, We inject triggers into the poisoned sample $G_{B}$ . Lines 10-17 and lines 18-25 are topology generation stage and feature generation stage, respectively. The topology (feature) discriminator $\theta_{t}(\theta_{f})$ and the generator $\omega_{t}(\omega_{f})$ are alternately optimized and then update the surrogate model $f_{\theta^{*}}$ with the current backdoor training set $C'$ . + +# E Experimental Details + +# E.1 Datasets + +The key statistics of the datasets are presented in Table E7. + +# E.2 Metrics + +This section details the evaluation metrics used to assess both the effectiveness and stealthiness of backdoor attacks, including Attack Success Rate (ASR), Clean Accuracy Drop (CAD), and Area Under the Curve (AUC). + +# Algorithm 1 DPSBA + +Input: Datasets $C$ , target label $y_{t}$ , attack budgets $M$ , maximum number of training epochs + +Output: Backdoor training set $C'$ , topology/feature generator parameters $\omega_{t} / \omega_{f}$ + +1: The model $f_{\theta^{*}}$ is trained through the gradient $\nabla_{\theta}\mathcal{L}_{train}(C)$ +2: Randomly initialize the topology generator parameters $\omega_{t}$ , the feature generator parameters $\omega_{f}$ , the topology anomaly discriminator parameters $\theta_{t}$ and the feature anomaly discriminator parameters $\theta_{f}$ ; +3: The poisoned samples $G_B$ are selected from $C[y_t]$ according to the confidence score $cfd$ calculated by Eq. (2); +4: for $G \in G_B$ do +5: According to the centrality calculated by $\deg(v) / (N - 1)$ , nodes with high degree centrality are pre-selected as candidates $\mathcal{N}_{can}$ ; +6: According to the importance score $S$ calculated by Eq. (3), $M$ injection positions are selected; +7: Using $m(G; g_t)$ inject triggers in $G$ ; +8: end for + +// Put trigger-embedded graph $G_B$ back into the dataset to form the backdoored dataset $C'$ + +9: while not converged yet do + +10: while epoch $<$ epochs do +11: Update $\theta_{t}$ by gradient according to Eq. (5) +12: Update $\omega_{t}$ via gradient $\nabla_{\omega_t}\mathcal{L}_{atk}(G_{g_t}(\omega_t)) + \alpha \mathcal{L}_d^{(t)}(D_{\theta_t}(G_{g_t}(\omega_t)))$ according to Eq. (7); +13: end while +14: for $G \in G_B$ do +15: Update $G$ by $m(G;g_t)$ +16: end for +17: Update $f_{\theta^{*}}$ by gradient $\nabla_{\theta}\mathcal{L}_{train}(C')$ ; +18: while epoch $<$ epochs do +19: Update $\theta_{f}$ by gradient according to Eq. (6) +20: Update $\omega_{f}$ via gradient $\nabla_{\omega_f}\mathcal{L}_{atk}(G_{g_t}(\omega_f)) + \beta \mathcal{L}_d^{(f)}(D_{\theta_f}(G_{g_t}(\omega_f)))$ according to Eq. (8); +21: end while +22: for $G \in G_B$ do +23: Update $G$ by $m(G;g_t)$ +24: end for +25: Update $f_{\theta^{*}}$ by gradient $\nabla_{\theta}\mathcal{L}_{train}(C')$ ; +26: end while +27: return $C', \omega_t, \omega_f$ + +Table E7: Datasets statistics + +
DatasetsPROTEINS_fullAIDSFRANKENSTEINENZYMES
Number of graphs111320004337600
Avg. Number of nodes39.0615.6916.9032.63
Avg. Number of edges72.8216.2017.8862.14
Number of classes2226
Label distribution663 [0], 450 [1]400 [0], 1600 [1]1936 [0], 2401 [1]100 [1-6]
Target label1000
+ +(1) Attack Success Rate (ASR): ASR quantifies the proportion of trigger-embedded graphs that are successfully misclassified into the target class. It serves as the primary measure of attack effectiveness and is defined as: + +$$ +A S R = \frac {\text {N u m b e r o f s u c c e s s f u l t a c k s}}{\text {T o t a l n u m b e r o f b a c k d o o r t r i a l s}}. \tag {14} +$$ + +(2) Clean Accuracy Drop (CAD): CAD measures the performance degradation of the poisoned model on clean samples, indicating the stealthiness of the attack when the trigger is absent. It is defined as the accuracy difference between a clean model and the backdoored model on a clean test + +Table E8: Model accuracy on a clean dataset + +
DatasetsModel
GCNGINSAGPool
PROTEINS_full75.5876.2373.99
AIDS98.6498.9298.26
FRANKENSTEIN60.8164.3262.75
+ +set: + +$$ +C A D = A C C _ {\text {c l e a n m o d e l}} - A C C _ {\text {b a c k d o o r m o d e l}}, \tag {15} +$$ + +where accuracy is calculated as: + +$$ +A C C = \frac {1}{N} \sum_ {i = 1} ^ {N} \mathbb {I} \left(\hat {y} _ {i} = y _ {i}\right), \tag {16} +$$ + +with $\mathbb{I}(\cdot)$ denoting the indicator function, $\hat{y}_i$ the predicted label, $y_i$ the ground-truth label, and $N$ the number of test samples. + +Interpretation: A high ASR indicates effective attack performance, while a low CAD implies minimal disruption to normal prediction behavior, reflecting strong stealth. + +(3) Area Under the Curve (AUC): To evaluate stealth from the perspective of anomaly detection, we use the AUC score produced by an anomaly detection model (e.g., SIGNET) trained on clean graphs. AUC reflects the model's ability to distinguish backdoor samples from clean ones: + +$$ +A U C = P \left(s \left(x _ {+}\right) > s \left(x _ {-}\right)\right), \tag {17} +$$ + +where $s(x)$ denotes the anomaly score of graph $x$ , and $x_{+}, x_{-}$ are randomly selected anomalous and clean samples, respectively. + +# Interpretation: + +- AUC $\rightarrow 1.0$ : the detector perfectly separates anomalies from clean data, i.e., low stealth. +- AUC $\rightarrow 0.5$ : the detector cannot distinguish between backdoor and clean samples, i.e., high stealth. +- AUC $< 0.5$ : the detector is misled, incorrectly prioritizing clean samples as more anomalous. + +Goal: An effective backdoor should aim for an AUC as close as possible to 0.5, indicating that the anomaly introduced by the trigger is indistinguishable from natural variations in clean data, thus enhancing stealth against detection models. + +# E.3 Accuracy of Graph Classifiers on Clean Data + +Table E8 illustrates the accuracy of three different graph classifiers on the clean datasets. + +# E.4 Impact of the Poisoning Rate + +To investigate how the poisoning rate influences DPSBA's performance, we evaluate the variation of attack success rate (ASR) and clean accuracy drop (CAD) under different poisoning rates. Since the anomaly level of the trigger is determined by its structural and feature design rather than injection frequency, we focus here on ASR and CAD as shown in Figure E6. As the poisoning rate increases from $1\%$ to $7\%$ , ASR consistently improves across all datasets, confirming that injecting more poisoned samples strengthens the backdoor effect. However, the growth rate of ASR gradually slows, indicating diminishing returns. Meanwhile, CAD shows a slight upward trend with higher poisoning rates, but remains below $5\%$ in all cases, demonstrating that DPSBA maintains high stealth even under increased poisoning levels. This result suggests that while increasing the poisoning rate enhances attack effectiveness, DPSBA can still preserve stealthiness due to its clean-label and anomaly-aware design. Practitioners may adjust the poisoning budget to balance performance and resource cost without significantly sacrificing stealth. + +![](images/c5c7aa4728dfba4b6e9892869f0178a6407de21f70dc01863f179a67d06b9842.jpg) +Figure E6: The impact of the Poisoning Rate + +![](images/28259bc2c018736daa6f09ef6524211a2b0d1a04ffd4eeba27603ceeb0bf8924.jpg) + +# E.5 Impact of Trigger Size + +![](images/02fa90292f6f9e8592a41ee114d4779dcbefe0dd4b9401aca203434cb13c53d4.jpg) +Figure E7: The impact of trigger size + +We further examine the effect of trigger size on the performance of DPSBA. As shown in Figure E7, increasing the size of the trigger leads to a consistent improvement in attack success rate (ASR) across all datasets. This is expected, as larger triggers introduce stronger perturbations to the graph's topology and features, making it easier for the model to associate the injected pattern with the target label. However, this enhancement in effectiveness comes at the cost of stealthiness. As the size of the trigger increases, the level of anomaly, as measured by the AUC, also increases. This indicates that larger triggers deviate more from the normal graph distribution, thus being more detectable by anomaly detection models. Still, the AUC values remain within an acceptable range (mostly below $80\%$ ), preserving a reasonable degree of stealth. Moreover, the clean accuracy drop (CAD) exhibits a general upward trend with increasing trigger size, though it remains under $5\%$ in most cases. This suggests that even larger triggers do not significantly impair the model's generalization performance on clean data. In summary, trigger size presents a clear trade-off: larger triggers yield stronger attacks but increase detectability, while smaller ones offer greater stealth at the cost of ASR. DPSBA supports flexible adjustment of this trade-off based on practical requirements. + +# E.6 Higher ASR in Cross-Architecture Transfer Settings + +We explain this counterintuitive phenomenon from three perspectives: model expressiveness, trigger generalizability, and dataset characteristics. + +(1) Model Expressiveness vs. Overfitting: GIN and SAGPool are more expressive than GCN. GIN is theoretically equivalent to the Weisfeiler-Lehman test, capable of distinguishing fine-grained substructures and SAGPool uses self-attention to highlight global structures most relevant for classification. These powerful models, when used as surrogates, tend to learn highly specialized, structure-sensitive triggers. While effective on the surrogate model itself, such triggers are prone to overfitting and may fail to generalize to unseen samples or to minor parameter shifts. This aligns with prior observations [48] that stronger surrogates often overfit their own gradients, reducing perturbation transferability. +(2) GCN-Trained Triggers Are More Transferable: GCN, with its smoother message passing and lower expressiveness, tends to learn broader and more transferable trigger patterns. These triggers may achieve lower ASR on GCN itself (as seen in Table 1), but generalize better when transferred to expressive models like GIN or SAGPool, which are more sensitive to subtle perturbations and thus amplify the attack effect. + +(3) Dataset-Specific Factors Amplify the Effect: In the AIDS dataset, molecular graphs are small but structurally complex. Local substructures (e.g., functional groups) are crucial, making GIN highly sensitive to local changes caused by transferred triggers. In PROTEINS_full dataset, graphs are larger and denser (node-edge ratio $\approx 1:1.86$ ), favoring models like SAGPool that focus on global structure. GCN's triggers are naturally aligned with such global perturbations, which SAGPool tends to emphasize. + +# F Time Complexity Analysis + +We analyze the training time complexity of DPSBA by accounting for graph sparsity and the dimensionality of node features. The framework consists of three main stages: (1) hard sample selection; (2) trigger location selection and initialization; and (3) adversarial optimization of the topology and feature generators. Let us denote: + +- $C$ : number of graphs in the clean training set +- $N$ : average number of nodes per graph +- $d$ : average degree of nodes (assumes sparse graph, i.e., $d \ll N$ ) +- $F$ : node feature dimension +- $B$ : the total number of poisoned graphs +- $E$ : total number of adversarial training epochs + +1) Hard Sample Selection. For each graph in the target class, we compute its confidence score with respect to the target label using a forward pass through the surrogate model. Assuming each node aggregates information from $d$ neighbors, and node features are $F$ -dimensional, the per-graph complexity is $\mathcal{O}(N \cdot d \cdot F)$ , leading to $\mathcal{O}(C \cdot N \cdot d \cdot F)$ . This term reflects the initial overhead of scoring and ranking all graphs to identify low-confidence (hard) samples. +2) Trigger Location Selection. For each selected hard sample, DPSBA computes node importance via a deletion-based interpretability scheme. Each candidate node requires one additional forward pass through the GNN after node removal. If we sample $k$ candidates per graph, the cost becomes $\mathcal{O}(k \cdot N \cdot d \cdot F)$ . Since $k$ is fixed and small ( $k \approx 2M$ ), this step scales as $\mathcal{O}(C \cdot N \cdot d \cdot F)$ . Thus, steps (1) and (2) share the same leading complexity. +3) Trigger Optimization. In each epoch, we optimize the topology and feature generators to simultaneously (a) maximize attack loss and (b) minimize anomaly detection confidence through adversarial discriminators (GCN and MLP). Each poisoned graph in the batch undergoes GCN-based classification and gradient updates. Assuming adversarial training lasts $E$ epochs, the total cost becomes $\mathcal{O}(E \cdot B \cdot N \cdot d \cdot F)$ . This dominates the runtime due to repeated gradient steps. + +Overall Complexity. Combining the above three stages, the total training complexity of DPSBA is: $\mathcal{O}(C\cdot N\cdot d\cdot F + E\cdot B\cdot N\cdot d\cdot F)$ This expression reveals linear scalability with respect to the number of nodes, average degree, and feature dimension. The first term corresponds to one-time sampling and location inference; the second reflects iterative adversarial training. As the number of poisoned graphs $(B)$ and training epochs $(E)$ are relatively small, DPSBA maintains practical efficiency under sparse graph assumptions. DPSBA achieves a favorable balance between attack efficacy and efficiency: its overall complexity is comparable to standard GNN training pipelines while introducing minimal overhead. Moreover, by leveraging sparse message passing and modular optimization, DPSBA remains scalable to real-world graphs with thousands of nodes and high-dimensional features. + +In addition, we benchmark the actual training time of DPSBA and compare it with several representative baselines (ER-B, LIA, GTA, and Motif) on the largest dataset, FRANKENSTEIN. The results are reported in Table F9. All experiments are conducted on a machine equipped with a 14-core Intel i7-12700H CPU, an NVIDIA GeForce RTX 3060 GPU (12 GB), and Windows 11 (version 23H2). As shown in the table, DPSBA achieves the highest ASR (99.84%) while maintaining the lowest CAD (1.83%) and AUC (73.46%), demonstrating a superior balance between attack success and stealth. Although DPSBA incurs slightly more training time compared to the lightest baseline (ER-B), the overhead is acceptable given the substantial gains in performance. Notably, our framework integrates an early-stopping mechanism during adversarial training, which adaptively terminates optimization once the attack objective converges. This not only reduces computational overhead but also avoids unnecessary overfitting, making DPSBA both effective and time-efficient. + +Table F9: Training Time and Attack Performance on FRANKENSTEIN. + +
MetricsER-BLIAGTAMotifOurs
ASR (%)92.0682.6395.2384.5699.84
CAD (%)3.602.351.952.441.83
AUC (%)85.7376.1591.0687.5473.46
Time (s)210261376239281
+ +# G Discussion of Inductive Graph Classification Setting + +While our current work focuses on the transductive graph classification setting, we believe that several key components of DPSBA have promising potential to extend to the inductive regime. In particular: 1) The feature generator operates based on localized structural and attribute information at the trigger injection site, and 2) The distribution-aware discriminators regularize stealth on a per-graph basis through adversarial training. Both modules are graph-local in nature and do not rely on inter-graph interactions or train-test graph overlap, making them amenable to inductive settings where new test graphs are unseen during training. That said, certain components, such as the hard sample selection module, currently assume access to sufficient examples of the target class, which poses challenges in few-shot scenarios. Adapting this component to work under strong data constraints (e.g., via meta-learning or class-agnostic proxy supervision) is a meaningful direction for future work. + +# H Broader Impacts + +This work reveals that even under clean-label settings, graph classification models remain highly vulnerable to stealthy backdoor attacks. By exposing this underexplored threat, our method highlights the limitations of existing defenses and underscores the need for more robust anomaly detection and training strategies. 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We propose SteerConf, a novel framework that systematically steers LLMs' confidence scores to improve their calibration and reliability. SteerConf introduces three key components: (1) a steering prompt strategy that guides LLMs to produce confidence scores in specified directions (e.g., conservative or optimistic) by leveraging prompts with varying steering levels; (2) a steered confidence consistency measure that quantifies alignment across multiple steered confidence to enhance calibration; and (3) a steered confidence calibration method that aggregates confidence scores using consistency measures and applies linear quantization for answer selection. SteerConf operates without additional training or fine-tuning, making it broadly applicable to existing LLMs. Experiments on seven benchmarks spanning professional knowledge, common sense, ethics, and reasoning tasks, using advanced LLM models (GPT-3.5, LLaMA 3, GPT-4), demonstrate that SteerConf significantly outperforms existing methods, often by a significant margin. Our findings highlight the potential of steering the confidence of LLMs to enhance their reliability for safer deployment in real-world applications. The implementation is at https://github.com/scottjiao/SteerConf. + +# 1 Introduction + +Large Language Models (LLMs) are widely used in applications such as AI-driven chatbots [29, 21, 13], medical diagnosis [9, 30, 24], code generation [23, 15], legal document analysis [40, 19], and education [34, 14]. Ensuring the reliability and trustworthiness of LLMs is critical, particularly in high-stakes domains like healthcare, legal analysis, and autonomous systems. However, LLMs often exhibit overconfidence, producing predictions that fail to align with their true likelihood of correctness, even when uncertain [38], posing significant challenges for their practical deployment. + +To obtain prediction confidence from LLMs, confidence elicitation methods treat LLMs as black boxes, using prompts to elicit self-assessed verbalized confidence scores [38, 31]. Unlike traditional approaches [22, 25] that rely on white-box access to internal model information, some LLMs are closed-source, with only commercial APIs, such as GPT-3.5 [28] and GPT-4 [29], available for use. + +These APIs do not provide access to internal details like token likelihoods, making traditional confidence elicitation infeasible. Additionally, fine-tuning LLMs for confidence elicitation is prohibitively expensive, limiting its practicality for most researchers. + +Recent studies focus on eliciting confidence from LLMs via prompts in a black-box manner to improve confidence calibration, aligning confidence with accuracy and enhancing failure prediction—measuring the ability of LLMs to assign high confidence to correct predictions and low confidence to incorrect ones [38]. Specifically, [38] propose a systematic framework that probes LLMs with various prompts, samples multiple responses to elicit confidence scores, and aggregates them to obtain a final confidence score. For instance, a Top-K prompting strategy [31] asks an LLM to provide the top-K most likely answers and their associated confidence scores for a question. A sample prompt is: "Provide your K best guesses and the confidence that each is correct (0% to 100%) for the following question." + +Observation. However, the prompting strategies explored so far, do not explicitly steer the direction of confidence in a controlled manner, i.e., whether the LLMs should be more conservative or optimistic in their confidence scores. Importantly, we find that if we explicitly instruct LLMs to be more conservative or optimistic, the LLMs can be steered to produce confidence scores in a + +![](images/dcecf5fb0fcc131af5007025bc83e604034b980d599efd4dee20b432747156d1.jpg) +(a) Vanilla Verbalized Confidence + +![](images/1e9f3cd952b07d9a30c70e42acdbdb2fa08911bb872d85c68fc56910ad9c38ee.jpg) +(b) Very-Cautious Steered Prompt +Figure 1: Vanilla and very cautious Prompt Steered Confidence on Object Counting data with GPT-3.5. + +specific direction. For example, we can steer LLMs to be very cautious in answering questions and producing confidence scores, compared with the vanilla prompt, as shown in the example below. + +- Vanilla Prompt: "Read the question carefully, analyze step by step, and provide your answer along with your confidence. Confidence reflects how likely you believe your answer is correct." +- Very Cautious Prompt: The Vanilla Prompt with an additional instruction: "You should adopt a very cautious approach and assign low confidence to most of your answers." + +Figure 1 compares the effects of vanilla and very cautious steered prompts on the Object Counting Dataset [33] using GPT-3.5 [28]. In Figure 1(a), the vanilla prompt results in severe overconfidence, with LLMs assigning $100\%$ confidence to answers that are only $50\%$ accurate for counting tasks. In contrast, Figure 1(b) shows that the very cautious steered prompt yields better-calibrated confidence scores, where confidence levels more closely align with actual accuracy. For example, answers with $100\%$ confidence achieve approximately $90\%$ accuracy, while those with $10\%$ confidence correspond to about $10\%$ accuracy. Quantitatively, the very cautious prompt improves calibration as measured by ECE and failure prediction as measured by AUROC, achieving a $29\%$ increase in Area Under the Receiver Operating Characteristic Curve (AUROC, higher is better) and a $16\%$ reduction in Expected Calibration Error (ECE, lower is better). These findings demonstrate the effectiveness of steering prompts in enhancing the reliability of LLM confidence scores. + +Our Contributions. We propose SteerConf, a novel framework that leverages the steerability of LLMs' confidence scores to enhance their reliability. SteerConf comprises three key components: (1) We propose a steering prompt strategy to guide LLMs in generating confidence scores in a specified direction (e.g., conservative or optimistic) by employing prompts with varying steering intensities. For each question, we sample multiple responses, capturing both answers and their associated confidence scores across different steering levels. (2) We introduce a steered confidence consistency measure to evaluate the alignment of confidence scores across multiple steered responses, leveraging this consistency to enhance confidence calibration. (3) We design a steered confidence calibration method that leverages the consistency measures to aggregate confidence scores and applies linear quantization to identify the most reliable answer based on the calibrated confidence. SteerConf is a simple yet effective framework that can be directly applied to existing LLMs without requiring additional training or fine-tuning. + +Comprehensive experiments across seven benchmarks, covering professional knowledge, common sense, ethics, and reasoning tasks, using three state-of-the-art models (GPT-3.5, LLaMA 3, GPT-4), demonstrate that SteerConf significantly outperforms vanilla verbalized confidence and existing + +calibration methods. For example, SteerConf achieves up to a $39.8\%$ reduction in ECE on GSM8K dataset with GPT-3.5 and a $43.3\%$ improvement in AUROC on the Object Counting dataset with GPT-4 in the CoT setting. + +# 2 Preliminaries + +# 2.1 Task Statement + +Verbalized Confidence. Confidence represents a model's certainty in its predictions, typically expressed as a probability (e.g., "80% confidence"). A score closer to 1 indicates high certainty, while a score near 0 reflects uncertainty. However, for large language models (LLMs), internal confidence scores are not directly accessible due to their black-box nature. Instead, confidence is elicited through verbalized responses obtained via prompting, where the model explicitly states its confidence in natural language (e.g., "I am 80% confident in answer A") [38, 31]. + +Verbalized Confidence Elicitation. This process involves modifying the original query prompt to explicitly request confidence scores from LLMs. For example, given the question "Which one is bigger, 9.8 or 9.10? Provide your answer," the prompt can be rephrased as "Which one is bigger, 9.8 or 9.10? Provide your answer along with your confidence (\%). The model might then respond with "9.8, $80\%$ " instead of just "9.8," thereby verbalizing its confidence level as $80\%$ . + +Confidence Calibration and Failure Prediction. Verbalized confidence scores elicited from LLMs are often miscalibrated, meaning the reported confidence does not accurately reflect the true likelihood of correctness. For instance, a model might state "80% confidence" for answers that are correct only 50% of the time. Confidence calibration addresses this issue by aligning the model's confidence scores with the actual accuracy of predictions [11]. Ideally, predictions with 80% confidence should exhibit 80% accuracy. This is crucial for applications where confidence scores guide decision-making, such as in medical diagnosis or autonomous driving. The effectiveness of confidence calibration is commonly evaluated using metrics such as ECE. Failure prediction focuses on identifying incorrect predictions by leveraging confidence scores. It is formulated as a binary classification task, where correct predictions are labeled as 0 and incorrect ones as 1. The performance of failure prediction is measured using AUROC, which assesses the model's ability to rank incorrect predictions below correct ones [1]. Detailed computations and evaluations are provided in the experiments. + +# 2.2 Related Work + +Elicitation of Confidence in LLMs. The concept of verbalized confidence, where models directly express confidence through natural language, was introduced in [22], but their work primarily focuses on fine-tuned models trained on specific datasets. Zero-shot verbalized confidence in LLMs remains unexplored in their study. Similarly, [25] trains external models to estimate confidence based on internal model representations, which are inaccessible in closed-source LLMs. While [44] examines the impact of confidence expressions, it does not provide explicit confidence scores to users. [31] explores prompting strategies, proposing the Top-K verbalized prompting method. The unified framework summarized in [38], which considers sample consistency, is the most closely related to our approach. In contrast, our work introduces a novel framework to steer LLM confidence in a desired direction, addressing gaps left by these prior studies. + +**Confidence Calibration.** Modern AI systems often exhibit unreliable confidence estimates, frequently displaying overconfidence in their predictions [11, 26, 37]. Calibration techniques address this issue by aligning confidence scores with actual accuracy rates [11, 26]. Two primary approaches have been explored: (1) scaling methods, which adjust confidence scores to improve reliability [11, 4, 43], and (2) binning methods, which group predictions by confidence levels to assess and correct miscalibration [42, 43]. For LLMs, studies have revealed significant calibration challenges. For instance, [16] demonstrated that models like T5 and GPT-2 are poorly calibrated in question-answering tasks. Similarly, [2] found that while standard language models face calibration issues, pretraining can enhance confidence reliability. Conversely, [17] observed that larger LLMs, when properly prompted, exhibit improved calibration in multiple-choice tasks. However, a critical limitation of these studies is their reliance on access to internal model computations (e.g., logits), which is infeasible for closed-source systems like GPT-4. This limitation underscores the need for alternative confidence + +![](images/4c880e95798404d711fb5f67d10d84be8135522c0f5cc5c683b80ace353f6ab5.jpg) +Figure 2: Overview of the SteerConf framework, comprising three components: Steering Prompting, Steered Confidence Consistency, and Steered Confidence Calibration. Five steering levels, from very cautious to very confident, elicit verbalized confidence and answers from the LLM. Confidence consistency $\kappa_{conf}$ evaluates the stability of these confidence. The calibrated confidence $c(x)$ is computed by combining the mean confidence $\mu_c$ , confidence consistency $\kappa_{conf}$ , and answer consistency $\kappa_{ans}$ , and the final answer $f(x)$ is selected based on $c(x)$ . + +estimation methods that operate without requiring internal model access or modifications, motivating our proposed framework. + +# 3 SteerConf: Steering the Confidence of LLMs + +Figure 2 illustrates the overall framework of SteerConf, comprising three key components: Steering Prompting (Section 3.1), Steered Confidence Consistency (Section 3.2), and Steered Confidence Calibration (Section 3.3). + +Given an input question $x$ , we first construct a set of steering prompts $P_{conf}^{l}$ with varying steering levels (e.g., very cautious) to elicit steered answers $f_{l}(x)$ and confidence scores $c_{l}(x)$ from the LLM. Next, the Steered Confidence Consistency module computes the confidence consistency $\kappa_{conf}$ by analyzing the mean $\mu_{c}$ and standard deviation $\sigma_{c}$ of the sampled confidence scores $c_{l}(x)$ . The calibrated confidence score $c(x)$ is then derived by combining $\mu_{c}, \kappa_{conf}$ , and the answer consistency $\kappa_{ans}$ , which is determined by the relative frequency of the most frequent answer among $f_{l}(x)$ . Finally, the calibrated confidence score $c(x)$ is used to select the final answer $f(x)$ via linear quantization, mapping $c(x)$ to the corresponding steering index in the confidence distribution. + +# 3.1 Steering Prompts + +As discussed in Section 1, while LLMs often exhibit overconfidence, explicitly prompting them to be cautious can result in lower confidence scores. To leverage this behavior, we propose a steering prompting strategy to systematically elicit and sample confidence scores from LLMs, enabling more accurate confidence estimation. Specifically, this strategy employs steering prompts that explicitly instruct the LLM to adopt varying levels of caution or confidence. + +Specifically, we employ a set of steering levels to guide the LLM's confidence elicitation from very cautious to very confident: {very cautious, cautious, vanilla, confident, very confident}. The steering level "very cautious" in the prompt instructs the LLM to be very cautious, with which the LLM tends to be more conservative in its confidence estimation. In the steering prompt, we also include the original question to be answered, and the LLM is asked to provide both the answer and the confidence score. A brief example of very-cautious prompt is shown in the introduction. In the steering prompt, the steering level can be replaced with different levels, and the LLM is asked to provide the answer and confidence score for each steering level. The detailed prompt design is shown in Appendix B. + +Formally, given an input text $x$ , we can apply $2\ell + 1$ steering levels via steering prompts to the LLM, generating $2\ell + 1$ prediction-confidence pairs. Let $P_{\mathrm{conf}}^{l}$ be the $l$ -th steering prompt, where + +$l \in \{-\ell, -\ell + 1, \dots, 0, \dots, \ell - 1, \ell\}$ . Negative steering levels $-\ell, -\ell + 1, \dots, -1$ represent cautious steering levels, while positive steering levels $1, 2, \dots, \ell$ represent confident steering levels. $-\ell$ and $\ell$ represent the most cautious and most confident steering levels, respectively, while $0$ represents the vanilla steering level. In other words, $P_{\mathrm{conf}}^{-\ell}$ is the most cautious steering prompt, $P_{\mathrm{conf}}^{\ell}$ is the most confident steering prompt, and $P_{\mathrm{conf}}^{0}$ is the vanilla steering prompt, which is the same with original verbalized confidence elicitation process. + +In our setting, we set $\ell = 2$ , which means we have five steering levels: {very cautious, cautious, vanilla, confident, very confident}, a moderate granularity, which is sufficient to demonstrate the effectiveness of our method. Our framework is flexible and can be extended to more steering levels by increasing $\ell$ . Given a question $x$ , we construct the steering prompt set $\{P_{\mathrm{conf}}^l\}_{l = -\ell}^\ell$ , and for each steering prompt $P_{\mathrm{conf}}^l$ , we can elicit the answer and confidence score from the LLM to get the prediction-confidence pair $(f_l(x), c_l(x))$ , where $f_l(x)$ is the answer and $c_l(x)$ is the confidence score. The confidence score $c_l(x)$ is a real number between 0 and 1, representing the LLM's confidence in its answer $f_l(x)$ . By applying different steering levels, we can probe the LLM's confidence in different directions and intensities, which can help us to better understand the LLM's confidence and uncertainty, to facilitate the accurate confidence estimation. + +# 3.2 Steered Confidence Consistency + +Given a set of sampled steered answers $\{f_l(x)\}_{l = -\ell}^{\ell}$ and their confidence scores $\{c_l(x)\}_{l = -\ell}^{\ell}$ , we aim to compute a calibrated confidence score $c(x)$ that reflects the model's certainty about answering the input question $x$ . In previous studies [38, 6, 5, 20], it has been shown that the consistency of multiple responses for a given question from LLMs can provide meaningful guidance on model confidence. For example, the degree of agreement among multiple answers has been used to serve as a measure of confidence for LLMs [38], which however overlooks the consistency of the verbalized confidence returned by LLMs, which may contain rich information of model's interior certainty. + +Therefore, in this section, we propose a steered confidence consistency measure that considers the consistency of the steered verbalized confidence scores $\{c_l(x)\}_{l = -\ell}^{\ell}$ to estimate the model's certainty about answering the input question $x$ . The intuition is that, even under different steering levels, if the LLM is able to respond with consistent confidence scores, it indicates that the model is confident about its answer, even with explicit prompt interventions. The higher the consistency of steered confidence scores, the more reliable the LLM is about its answer regarding the input question. To quantify this, we first compute the mean confidence $\mu_{c}$ of the steered confidence $\{c_l(x)\}_{l = -\ell}^{\ell}$ as in Eq. (1), representing the average certainty level, and the standard deviation $\sigma_{c}$ in Eq. (2), measuring how consistently this certainty is maintained across steering levels from $-\ell$ to $\ell$ . Intuitively, higher $\mu_{c}$ with lower $\sigma_{c}$ indicates stronger confidence consistency. + +Intuitively, the ratio $\frac{\sigma_c}{\mu_c}$ , known as the coefficient of variation, naturally normalizes the dispersion of a distribution relative to the mean, showing the extent of variability in relation to the mean of the observations. A smaller value of $\frac{\sigma_c}{\mu_c}$ indicates that the data points tend to be very close to the mean and are more consistent, while a larger value indicates that the data points are spread out over a wider range of values, being less consistent. Therefore, we propose the confidence consistency $\kappa_{\mathrm{conf}} = \frac{1}{1 + \sigma_c / \mu_c}$ in Eq. (3), which is a bounded metric between 0 and 1, where $\kappa_{\mathrm{conf}} = 1$ indicates perfect consistency with $\sigma_c = 0$ and $\kappa_{\mathrm{conf}} = 0$ indicates no consistency with infinite $\sigma_c$ . + +$$ +\mu_ {c} = \frac {1}{2 \ell + 1} \sum_ {l = - \ell} ^ {\ell} c _ {l} (x) \tag {1} +$$ + +$$ +\sigma_ {c} = \sqrt {\frac {1}{2 \ell + 1} \sum_ {l = - \ell} ^ {\ell} \left(c _ {l} (x) - \mu_ {c}\right) ^ {2}} \tag {2} +$$ + +$$ +\kappa_ {\text {c o n f}} = \frac {1}{1 + \sigma_ {c} / \mu_ {c}} \tag {3} +$$ + +# 3.3 Steered Confidence Calibration + +For a question $x$ , we aim to compute the calibrated confidence score $c(x)$ that reflects the model's certainty about answering the question $x$ . In a nutshell, the calibrated confidence score $c(x)$ should be high when the model is confident about its answer, and low when the model is uncertain about its answer. To achieve this, we propose to consider the mean confidence $\mu_c$ , confidence consistency $\kappa_{\mathrm{conf}}$ , and answer consistency $\kappa_{\mathrm{ans}}$ that is computed from the steered answers $\{f_l(x)\}_{l = -\ell}^{\ell}$ . + +First, the answer consistency $\kappa_{ans}$ is computed based on the relative frequency of the answers [38]. For each answer $y\in \{f_l(x)\}_{l = -\ell}^{\ell}$ , its frequency $\mathrm{freq}(y)$ is defined as the proportion of times $y$ occurs among all steered answers: $\mathrm{freq}(y) = \frac{1}{2\ell + 1}\sum_{l = -\ell}^{\ell}\mathbb{I}(f_l(x) = y)$ , where $\mathbb{I}(\cdot)$ is the indicator function. The answer consistency is then defined as the maximum relative frequency of the most frequent answer: $\kappa_{ans} = \max_{y\in \{f_l(x)\}_{l = -\ell}^{\ell}}\mathrm{freq}(y)$ . This reflects the degree of agreement among the answers, with higher $\kappa_{ans}$ indicating greater consistency and reliability of the model's predictions. + +Then, the calibrated confidence $c(x)$ is defined as the product of the average steered confidence $\mu_c$ , answer consistency $\kappa_{\mathrm{ans}}$ , and confidence consistency $\kappa_{\mathrm{conf}}$ as in Eq. (4). The factor $\mu_c$ considers all the steered confidence scores, while $\kappa_{\mathrm{ans}}$ and $\kappa_{\mathrm{conf}}$ , the two consistency metrics, are used to adjust the average steered confidence $\mu_c$ based on LLMs' self-consistency in terms of answers and confidence scores. If the LLM shows low consistency in either answers or confidence scores, the final calibrated confidence $c(x)$ will be downweighted accordingly, and vise versa. This design ensures the robustness of the final calibrated confidence $c(x)$ against isolated high-confidence outliers that may arise from the black-box nature of LLMs, as any single inconsistency component ( $\kappa_{\mathrm{ans}}$ or $\kappa_{\mathrm{conf}}$ ) can sufficiently downweight potentially miscalibrated $\mu_c$ . + +$$ +c (x) = \mu_ {c} \cdot \kappa_ {\text {a n s}} \cdot \kappa_ {\text {c o n f}} \tag {4} +$$ + +After obtaining the final calibrated confidence $c(x)$ from Eq. (4), the next step is to select the final answer $f(x)$ from the set of steered answers $\{f_{l}(x)\}_{l = -\ell}^{\ell}$ . The selection should align with the calibrated confidence $c(x)$ , which represents the model's certainty. + +Since $c(x)$ may not directly match any of the steered confidence scores $\{c_{\ell}(x)\}_{\ell = -\ell}^{\ell}$ , we employ a mapping mechanism to determine the most appropriate answer. We map $c(x)$ to the steering index space using linear quantization. First, we compute the range of observed steered confidence: + +$$ +c _ {\min } = \min _ {l} c _ {l} (x), \quad c _ {\max } = \max _ {l} c _ {l} (x). \tag {5} +$$ + +Next, we map $c(x)$ to a discrete steering index $j$ using: + +$$ +j = \left\lfloor \frac {c (x) - c _ {\operatorname* {m i n}}}{c _ {\operatorname* {m a x}} - c _ {\operatorname* {m i n}}} \cdot (2 \ell + 1) \right\rfloor , \tag {6} +$$ + +where $\lfloor \cdot \rfloor$ denotes the floor operation, ensuring $j$ is an integer. + +Finally, the selected answer $f(x)$ is determined as: + +$$ +f (x) = \left\{ \begin{array}{l l} f _ {- \ell} (x), & \text {i f} j < - \ell , \\ f _ {j} (x), & \text {i f} - \ell \leq j \leq \ell . \end{array} \right. +$$ + +This approach ensures that the answer $f(x)$ is consistent with the calibrated confidence $c(x)$ , while maintaining numerical stability through explicit range normalization. + +# 4 Experiments + +# 4.1 Setup + +Datasets. We assess confidence estimation quality across five categories of reasoning tasks: (1) Commonsense Reasoning using Sports Understanding dataset (Sport) [18] and StrategyQA (StrategyQA) [7] from BigBench [8]; (2) Arithmetic Reasoning evaluated on GSM8K (GSM8K) [3]; (3) Symbolic Reasoning covering Date Understanding (DateUnd) [36] and Object Counting (ObjCnt) [33]; (4) Professional Knowledge tested through Law (Law) from MMLU [12]; and (5) Ethical Knowledge examined via Business Ethics (Ethics) in MMLU [12]. + +
ModelGSM8KDateUndObjCntStrategyQASportLawEthicsAvg
LLaMA374.837.850.226.630.630.915.038.0
+SteerConf45.211.834.629.316.48.526.524.6
GPT-3.562.660.245.629.625.343.226.041.8
+SteerConf22.833.027.514.912.224.010.820.7
GPT-453.525.723.716.818.719.45.123.3
+SteerConf27.519.518.013.513.611.29.916.2
+ +(a) ECE $\downarrow$ (Lower is Better) + +
ModelGSM8KDateUndObjCntStrategyQASport LawEthicsAvg
LLaMA381.738.250.233.034.840.830.3
+SteerConf88.551.766.132.137.747.052.3
GPT-3.576.966.046.137.632.354.837.5
+SteerConf93.564.176.443.541.764.451.7
GPT-455.326.524.330.723.941.358.4
+SteerConf83.535.947.337.533.246.245.0
+ +(c) PR-N $\uparrow$ (Higher is Better) + +
ModelGSM8KDateUndObjCntStrategyQASportLawEthicsAvg
LLaMA353.750.350.358.851.551.042.351.1
+SteerConf67.161.067.259.053.858.962.961.4
GPT-3.555.856.650.353.352.851.754.853.6
+SteerConf82.960.582.558.661.560.671.068.2
GPT-452.050.550.455.657.956.684.158.2
+SteerConf83.964.272.763.763.359.777.669.3
+ +(b) AUROC $\uparrow$ (Higher is Better) + +
ModelGSM8KDateUndObjCntStrategyQASportLawEthicsAvg
LLaMA321.362.549.977.266.961.071.558.6
+SteerConf28.168.062.379.667.166.279.164.3
GPT-3.530.041.554.566.270.847.569.654.3
+SteerConf60.454.483.371.176.152.380.268.3
GPT-447.374.576.577.682.667.594.574.4
+SteerConf76.881.284.481.484.471.392.081.6
+ +(d) PR-P ↑ (Higher is Better) +Table 1: Comparing SteerConf with vanilla verbalized confidence elicitation of LLMs. ECE > 0.25, AUROC, PR-P, PR-N < 0.6 denote significant deviation from ideal performance. The best is in bold. + +
ModelGSM8KDateUndObjCntStrategyQASportLawEthicsAvg
LLaMA3(CoT)5.013.78.711.87.722.87.811.1
+SteerConf7.81.06.58.65.420.95.07.9
GPT-3.5(CoT)20.330.841.826.020.544.324.829.8
+SteerConf6.56.719.714.510.816.015.112.7
GPT-4(CoT)6.56.64.918.59.223.06.110.7
+SteerConf4.38.41.611.55.69.99.37.2
+ +(a) ECE $\downarrow$ (Lower is Better) + +
ModelGSM8KDateUndObjCntStrategyQASportLawEthicsAvg
LLaMA3(CoT)12.413.48.729.738.338.226.123.8
+SteerConf41.324.049.043.247.550.651.743.9
GPT-3.5(CoT)26.528.042.036.846.955.545.740.2
+SteerConf70.949.774.748.952.260.975.061.8
GPT-4(CoT)10.547.64.948.727.746.283.238.4
+SteerConf59.361.844.938.948.755.474.954.9
+ +(c) PR-N $\uparrow$ (Higher is Better) + +
ModelGSM8KDateUndObjCntStrategyQASport LawEthicsAvg
LLaMA3(CoT)55.154.350.064.674.154.354.2
+SteerConf81.270.787.674.579.464.781.2
GPT-3.5(CoT)56.249.850.156.462.753.065.2
+SteerConf85.676.082.567.566.461.586.7
GPT-4(CoT)52.175.150.068.865.059.587.6
+SteerConf86.081.593.370.375.268.493.0
+ +(b) AUROC $\uparrow$ (Higher is Better) + +
ModelGSM8KDateUndObjCntStrategyQASport LawEthicsAvg
LLaMA(CoT)95.489.391.385.289.266.583.0
+SteerConf97.892.698.190.092.873.693.1
GPT-3.5(CoT)80.571.558.271.570.248.475.8
+SteerConf92.188.483.277.775.057.391.8
GPT-4(CoT)93.794.295.181.488.068.392.8
+SteerConf97.394.099.287.691.175.697.9
+ +(d) PR-P ↑ (Higher is Better) + +Table 2: Comparing SteerConf with verbalized confidence elicitation of LLMs with CoT. ECE > 0.25, AUROC, PR-P, PR-N < 0.6 denote significant deviation from ideal performance. The best is in bold. + +Tasks and Metrics. We evaluate our method SteerConf on two tasks: confidence calibration and failure prediction, following the protocol in [38]. For confidence calibration, we report Expected Calibration Error (ECE), which measures the average absolute difference between predicted confidence and empirical accuracy; lower ECE indicates better calibration. For failure prediction, we use the Area Under the Receiver Operating Characteristic curve (AUROC), where higher values (closer to 1.0) indicate better discrimination between correct and incorrect predictions. To account for class imbalance due to varying accuracy, we further report AUPRC-Positive (PR-P) and AUPRC-Negative (PR-N), which respectively measure the model's ability to prioritize incorrect and correct predictions in a precision-recall framework. Formal definitions of all metrics are provided in Appendix A. + +LLM Models. We evaluate several widely-used LLMs: GPT-3.5 [28], LLaMA3 [10], and GPT-4 [29]. LLaMA3 is used in its 70B parameter version with 4-bit quantization. Note that experiments with GPT-4 incurred a cost of approximately 1500 USD due to its higher pricing. We first compare these LLMs using vanilla verbalized confidence elicitation, and then evaluate each LLM with CoT prompting [35] for confidence elicitation. Results are reported in Table 1 and Table 2 in Section 4.2. Details on the application of CoT are provided in Appendix B. + +Baselines. We also compare against sampling-based and prompting-based baselines summarized in [38], including Misleading [38], Self-Random [38], and Top-K [31], as reported in Table 3. All baselines use an LLM with CoT prompting as the backbone. Self-Random [38] leverages the model's inherent randomness by inputting the same prompt multiple times, and consistency aggregation metric [38] is applied. Misleading [38] evaluates the model's uncertainty by introducing misleading + +
MethodGSM8KLawDateUndStrategyQAEthicsAvg
ECEAUROCECEAUROCECEAUROCECEAUROCECEAUROCECEAUROC
GPT-3.5(CoT)10.154.839.752.223.457.422.059.830.056.025.056.4
+Top-K19.658.516.758.926.174.214.061.312.473.317.865.2
+Misleading8.0388.618.359.320.567.321.861.517.871.317.369.6
+Self-Random6.2892.726.065.617.066.823.360.820.779.018.773.0
+SteerConf (ours)6.585.616.061.56.776.014.567.515.186.711.775.4
LLaMA3(CoT)5.055.122.854.313.754.311.864.67.854.212.256.5
+Top-K10.261.123.049.727.552.221.155.011.152.418.654.1
+Misleading4.483.818.959.718.467.811.765.314.075.013.570.3
+Self-Random2.279.327.164.67.266.817.360.115.955.313.965.2
+SteerConf (ours)6.181.23.364.710.070.74.374.513.681.27.574.5
+ +Table 3: Comparison with baselines on GPT-3.5 and LLaMA3 with CoT. The best is in bold and the second best is underlined. + +
MethodGSM8KLawDateUndStrategyQAEthicsAvg
ECEAUROCECEAUROCECEAUROCECEAUROCECEAUROCECEAUROC
SteerConf w/o Steering Prompting2.380.013.266.710.070.26.172.510.259.08.469.7
Self-Random + Consistency2.279.327.164.67.266.817.360.115.955.313.965.2
SteerConf + Consistency2.471.725.963.87.568.217.261.414.368.513.566.7
Self-Random + Avg-Conf2.379.327.164.48.868.317.259.916.355.014.365.4
SteerConf + Avg-Conf2.871.325.863.38.667.117.461.316.773.514.367.3
SteerConf (ours)6.181.23.364.710.070.74.374.513.681.27.574.5
+ +Table 4: Component ablations and alternative aggregation schemes on LLaMA3with CoT. Lower ECE values indicate better calibration, while higher AUROC values reflect improved performance. + +information and observing whether the model maintains its initial answer, inspired by human behavior under confidence and consistency aggregation metric [38] is applied. Top-K [31] prompts the model to generate $K$ answer-confidence pairs in a single response, pair-rank metric [38] is applied. For Misleading and Self-Random, we use $M = 5$ samples; for Top-K, we set $K = 5$ answer-confidence pairs. We adopt GPT-3.5 and LLaMA3 both with CoT. + +# 4.2 Effectiveness Results + +Comparison with LLMs with and without CoT. Table 1 and Table 2 report the performance of our method SteerConf versus vanilla verbalized confidence elicitation on LLaMA3, GPT-3.5, and GPT-4, without and with CoT prompting, respectively, across ECE, AUROC, PR-N, and PR-P metrics (all in%). The last column in each table shows the average across datasets. Our method consistently outperforms vanilla verbalized confidence elicitation across nearly all metrics and datasets, both with and without CoT, demonstrating improved confidence calibration and failure detection. For instance, in Table 1(a), GPT-3.5 with SteerConf achieves an average ECE of $20.7\%$ , substantially better than the vanilla ECE of $41.8\%$ . On ObjCnt, SteerConf attains an AUROC of $82.5\%$ , significantly better than $50.3\%$ for vanilla GPT-3.5 (Table 1(b)). CoT prompting further improves overall performance of all LLMs (Table 2), yet SteerConf maintains a clear advantage. For example, in Table 2(b), GPT-4(CoT) with SteerConf achieves an average AUROC of $81.1\%$ , versus $65.5\%$ for vanilla. On the Ethics dataset, LLaMA3(CoT) with SteerConf yields a PR-N of $51.7\%$ , markedly higher than the vanilla baseline (Table 2(c)). + +Comparison with Other Baselines. Table 3 presents a comparison of our method SteerConf with the Top-K, Misleading, and Self-Random baselines on LLaMA3 and GPT-3.5 with CoT. Across all datasets and metrics, SteerConf consistently outperforms the baselines, highlighting its superiority in both confidence calibration and failure detection. For instance, on GPT-3.5(CoT), SteerConf achieves an average ECE of $11.7\%$ , substantially lower than the $17.3\%$ of the Misleading baseline. On LLaMA3(CoT), SteerConf attains an average AUROC of $74.5\%$ , outperforming all other baselines. + +Ablation Study. We perform an ablation study to assess the impact of steering prompting and aggregation components, using LLaMA3 with CoT as the backbone. For the steering prompts component, removing it yields "SteerConf w/o Steering Prompting". For the aggregation component, we replace our proposed aggregation with Consistency and Avg-Conf, denoted as "SteerConf + Consistency" and "SteerConf + Avg-Conf". Ablating both components results in the Self-Random approach with Consistency or Avg-Conf aggregation. The results are reported in Table 4, which + +![](images/f0b23e2ca00be154884fbe8fced0ee16a1810c758ea5635ce4cb836cecb9fd2c.jpg) +Figure 3: Confidence histograms of SteerConf and baselines for LLaMA3 with CoT. The rows are different datasets, and the columns are different methods. + +demonstrates that each component contributes to improved performance, supporting the effectiveness of our method's design. + +# 4.3 Empirical Confidence Distributions + +Figure 3 presents the confidence histograms of SteerConf and baselines on LLaMA3 with CoT. Each row corresponds to a dataset, and each column shows the confidence distributions for Self-Random, Misleading, Top-K, and our method SteerConf. The x-axis is confidence, and the y-axis is the number of samples. Blue and red bars represent the confidence distributions for correct and incorrect answers, respectively. Compared to the baselines, SteerConf produces confidence distributions that are better separated between correct and incorrect answers, while baselines such as Self-Random tend to yield overconfident or less discriminative distributions. These results further validate the effectiveness of our method in improving confidence calibration, consistent with the quantitative findings in Section 4.2. + +# 4.4 Detailed Calibration Distributions + +We provide the detailed calibration curves [32] of SteerConf and baselines on LLaMA3 with CoT. The results are presented in Figure 4. As depicted in Figure 4, the confidence distribution from vanilla verbalization is concentrated on a few discrete values. In contrast, SteerConf yields a much more uniform distribution. Such continuous confidence values are more amenable to quantization, thereby improving model reliability. Moreover, we observe a better alignment between the calibrated confidence from SteerConf and the actual accuracy. On the Sport and StrategyQA datasets, for example, SteerConf shows a consistent pattern of alignment across the full spectrum of confidence values, demonstrating the effectiveness of our approach. + +![](images/b025857b53905a9314d9a554af911c108e82b94efa04c43638555d594318e502.jpg) + +![](images/164346a91be620d78029711aaf3dff0dc05219b7535ffae076b70495edeb9b2a.jpg) + +![](images/730f03cbc0407522b2b1686f95c4ba086b5ffc0ec6363641e3d469b2b5cc0bdc.jpg) + +![](images/bf5501fb493ab315a53e285fde15d9ce776c4e89addab42d541f55feffa8b19a.jpg) + +![](images/354507912473b0c6a96d922c340c5f46ca9963b6cce2c21ce6c1b7a7bd795d0c.jpg) +(a) Law (vanilla) + +![](images/a0528ed6bd28e36fa06a4055fb7fa0bf335619a6a23d6061c4324b1d6be8029f.jpg) +(b) Sport (vanilla) + +![](images/899fd57ef53e79227c4e155d37f37dca79b577d706b61822e3fb691465e99da6.jpg) +(c) ObjCnt (vanilla) + +![](images/391e77c22c0c55a9ac834bc5b442ccb686045b6e7802567513fcfc223304e1ae.jpg) +(d) StrategyQA (vanilla) + +![](images/1590a721cbfe3b47bffadae1e96bc41b3efea3184c45fd929ae5295f3323018f.jpg) + +![](images/c988bb9b9c13bfdfd1056ee5326ecac66c8ce1e89b95aea2d5a47d00a9e2b152.jpg) + +![](images/e0661528d06a053cc16edabafdf4427c64a2779e012f962c674922dcc8da7f54.jpg) +(g) ObjCnt (SteerConf) + +![](images/2d494165a32c8dba983016dd32f02497ccd6a7e6f9c160d7fb3b1c0346664be2.jpg) +(h) StrategyQA (Steer-Conf) +Figure 4: Expected Calibration Error (ECE) plots and reliable diagrams for LLaMA3-70B model. The columns correspond to four tasks: Law, Sport, ObjCnt, and StrategyQA. The top row shows results of vanilla verbalized confidence, while the bottom row shows results from our SteerConf. In each plot, the upper part shows the ECE plots, and the lower part shows the reliable diagrams. + +![](images/f91483b01e94bcaabde75c14ce3719376877096f3235bb0e6bcfafc9d5bce9c0.jpg) +(e) Law (SteerConf) + +![](images/6e99f50bb531ecf15be621cd4134aa97b8e2dc7629686a287f578cc85f9c7fc0.jpg) +(f) Sport (SteerConf) + +# 5 Conclusion, Limitations, and Future Work + +In this paper, we introduced SteerConf, a framework for improving LLM confidence reliability through semantic steering. Our approach addresses the critical issue of overconfidence, which undermines trustworthiness in high-stakes applications. By leveraging carefully designed prompts, we systematically steer LLM confidence in desired directions. The framework comprises three key components: a steering prompt strategy, a steered confidence consistency measure, and a steered confidence calibration method. Together, these components enhance confidence calibration without requiring model internals or fine-tuning. Comprehensive evaluations across seven benchmarks with state-of-the-art LLMs demonstrate consistent improvements over existing methods. This work has significant implications for deploying LLMs in critical domains where reliable confidence estimation is essential. By offering an effective black-box approach, SteerConf enables trustworthy AI without requiring specialized expertise or access to model internals. + +Limitations and Future Work. It is common to manually design prompts for LLMs, and our method is no exception. Future work will explore automatic prompt generation to address this. Additionally, using multiple prompts increases computational costs; we aim to develop adaptive strategies to optimize prompt usage based on task and model needs. Following prior work, our current tasks focus on numerical answers (GSM8K) or multiple-choice questions (Law), where ambiguity is minimal and answer consistency is both sufficient and effective. Extending to more open-ended tasks is an important future direction. + +Broader Impact. This work improves LLM confidence elicitation, enhancing their reliability and trustworthiness in high-stakes applications. While we foresee no direct negative societal impacts, LLM misuse remains a concern. We advocate for responsible AI practices to mitigate such risks. + +# Acknowledgments + +This work is supported by grants from the Research Grants Council of Hong Kong Special Administrative Region, China (No. PolyU 25201221, No. PolyU 15205224), NSFC No. 62202404, and Smart Cities Research Institute (SCRI) P0051036-P0050643. + +# References + +[1] Kendrick Boyd, Kevin H Eng, and C David Page. Area under the precision-recall curve: point estimates and confidence intervals. In Machine Learning and Knowledge Discovery in Databases: European Conference, ECML PKDD 2013, Prague, Czech Republic, September 23-27, 2013, Proceedings, Part III 13, pages 451-466. Springer, 2013. +[2] Yangyi Chen, Lifan Yuan, Ganqu Cui, Zhiyuan Liu, and Heng Ji. A close look into the calibration of pre-trained language models. 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If you obtained IRB approval, you should clearly state this in the paper. +- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. +- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [Yes] + +Justification: We use LLMs as baselines and backbones as stated in Section 4.1. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Task Statement And Metric + +Confidence Calibration. LLMs often exhibit misaligned confidence scores, frequently overestimating the reliability of their predictions. The goal of confidence calibration is to align these confidence scores with the actual accuracy of the predictions. To evaluate calibration performance, we use the Expected Calibration Error (ECE) [27, 39, 41] as the primary metric. + +Given a set of input samples $\{x_{i}\}_{i = 1}^{N}$ with corresponding labels $\{y_{i}\}_{i = 1}^{N}$ , LLM predictions $\{f(x_i)\}_{i = 1}^N$ , and confidence scores $\{c(x_i)\}_{i = 1}^N$ , we partition the samples into $B$ bins based on their confidence scores: $x_{i}\in B_{b}$ if $c(x_{i})\in \left(\frac{b - 1}{B},\frac{b}{B}\right)$ . The Expected Calibration Error (ECE) is computed as: + +$$ +\mathrm {E C E} = \sum_ {b = 1} ^ {B} \frac {\left| B _ {b} \right|}{N} \left| \operatorname {a c c} \left(B _ {b}\right) - \operatorname {c o n f} \left(B _ {b}\right) \right|, \tag {7} +$$ + +where $B_{b}$ is the set of samples in the $b$ -th bin, $|B_{b}|$ is the number of samples in the $b$ -th bin, $\mathrm{acc}(B_{b}) = \frac{1}{|B_{b}|}\sum_{x_{i}\in B_{b}}\mathbb{I}(f(x_{i}) = y_{i})$ is the accuracy of the samples in the $b$ -th bin, and $\operatorname{conf}(B_b) = \frac{1}{|B_b|}\sum_{x_i\in B_b}c(x_i)$ is the average confidence score of the samples in the $b$ -th bin. Here, $\mathbb{I}(\cdot)$ is the indicator function, which returns 1 if its argument is true, and 0 otherwise. + +Failure Prediction. Failure prediction involves determining whether an LLM's prediction is correct based on its confidence score. This task is evaluated using the Area Under the Receiver Operating Characteristic Curve (AUC-ROC) [1]. Given a set of input samples $\{x_{i}\}_{i = 1}^{N}$ , the task is framed as a binary classification problem, where the input is the confidence score $c(x_i)$ , and the label indicates the correctness of the LLM's prediction, $\mathbb{I}(f(x_i) = y_i)$ . The AUC-ROC is defined as: + +$$ +\mathrm {A U C - R O C} = \int_ {0} ^ {1} \operatorname {T P R} (t) d \mathrm {F P R} (t), \tag {8} +$$ + +where $\mathrm{TPR}(t)$ (True Positive Rate) and $\mathrm{FPR}(t)$ (False Positive Rate) at threshold $t$ are computed as: + +$$ +\operatorname {T P R} (t) = \frac {1}{N} \sum_ {i = 1} ^ {N} \mathbb {I} \left(c \left(x _ {i}\right) \geq t\right) \mathbb {I} \left(f \left(x _ {i}\right) = y _ {i}\right), \tag {9} +$$ + +$$ +\operatorname {F P R} (t) = \frac {1}{N} \sum_ {i = 1} ^ {N} \mathbb {I} \left(c \left(x _ {i}\right) \geq t\right) \left(1 - \mathbb {I} \left(f \left(x _ {i}\right) = y _ {i}\right)\right). \tag {10} +$$ + +Precision-Recall Perspective. To complement AUC-ROC, we evaluate failure prediction using the Area Under the Precision-Recall Curve for the positive class (PR-P) and the negative class (PR-N). These metrics are particularly informative under class imbalance. For PR-P, the positive class corresponds to correct predictions $(f(x_{i}) = y_{i})$ , while for PR-N, the positive class corresponds to incorrect predictions $(f(x_{i}) \neq y_{i})$ . The Precision-Recall Curve (PRC) is parameterized by threshold $t$ , with precision and recall defined as: + +$$ +\operatorname {P r e c i s i o n} _ {p} (t) = \frac {\sum_ {i = 1} ^ {N} \mathbb {I} \left(c \left(x _ {i}\right) \geq t\right) \mathbb {I} \left(f \left(x _ {i}\right) = y _ {i}\right)}{\sum_ {i = 1} ^ {N} \mathbb {I} \left(c \left(x _ {i}\right) \geq t\right)}, \tag {11} +$$ + +$$ +\operatorname {R e c a l l} _ {p} (t) = \frac {\sum_ {i = 1} ^ {N} \mathbb {I} \left(c \left(x _ {i}\right) \geq t\right) \mathbb {I} \left(f \left(x _ {i}\right) = y _ {i}\right)}{\sum_ {i = 1} ^ {N} \mathbb {I} \left(f \left(x _ {i}\right) = y _ {i}\right)}, \tag {12} +$$ + +for PR-P, and analogously for PR-N by replacing $\mathbb{I}(f(x_i) = y_i)$ with $\mathbb{I}(f(x_i) \neq y_i)$ in both the numerator and denominator. The AUPRC values are computed as: + +$$ +\Pr - \mathrm {P} = \int_ {0} ^ {1} \operatorname {P r e c i s i o n} _ {p} (t) d \text {R e c a l l} _ {p} (t), \quad \Pr - \mathrm {N} = \int_ {0} ^ {1} \operatorname {P r e c i s i o n} _ {n} (t) d \text {R e c a l l} _ {n} (t). \tag {13} +$$ + +PR-P emphasizes the model's ability to identify correct predictions across confidence thresholds, while PR-N focuses on detecting failures. Together, these metrics provide a nuanced evaluation of calibration quality, particularly under skewed class distributions. + +# B Steering Prompts Used + +In this section, we present the detailed prompts used in our experiments for LLMs under both CoT and non-CoT settings. The steering levels {very cautious, cautious, vanilla, confident, very confident} + +} are described as follows. Section B.1 provides the prompts for the CoT setting, while Section B.2 covers the prompts for the non-CoT setting. + +Discussion on Instruction-Following Ability. Instruction-following is a prerequisite. Verbalized confidence assumes the model can return answer-confidence pairs in a fixed format. In practice, smaller or less capable models may fail at this step, which we treat as a formatting failure rather than a limitation of steering. Once a model reliably produces answer-confidence pairs, we find it also follows our steering prompts (e.g., very cautious vs. very confident) and shifts its confidence accordingly (see Figure 5 for how the model's confidence changes with different steering levels). + +Discussion on Additional Query Cost. We deliberately query the model multiple times under a small set of steering prompts. In our black-box setting, we cannot read internal signals (e.g., logits), so these extra queries give us the missing evidence about how stable both the answers and the confidence scores are. This is a test-time scaling trade-off: we spend a few more forward passes to obtain much better-calibrated confidence and more reliable failure detection, without any training or access to internals. For complex, multi-turn reasoning, the same idea applies—investing a handful of additional queries typically yields more dependable confidence estimates and thus more trustworthy traces. + +# B.1 With CoT + +very cautious. Read the question, analyze step by step, provide your answer and your confidence in this answer. Note: (1) The confidence indicates how likely you think your answer will be true. (2) You are making important decisions, thus you should avoid giving a wrong answer with high confidence. (3) You should be very cautious, and tend to give low confidence on almost all of the answers. \nUse the following format to answer:\“Explanation: [insert step-by-step analysis here]\nAnswer and Confidence (0-100): [ONLY the {ANSWER_TYPE}; not a complete sentence], [Your confidence level, please only include the numerical number in the range of 0-100]%\“\nOnly give me the reply according to this format, don't give me any other words. + +cautious. Read the question, analyze step by step, provide your answer and your confidence in this answer. Note: (1) The confidence indicates how likely you think your answer will be true. (2) You are making important decisions, thus you should avoid giving a wrong answer with high confidence. \nUse the following format to answer:\“Explanation: [insert step-by-step analysis here]\nAnswer and Confidence (0-100): [ONLY the {ANSWER_TYPE}; not a complete sentence], [Your confidence level, please only include the numerical number in the range of 0-100]%\`“\nOnly give me the reply according to this format, don't give me any other words. + +vanilla. Read the question, analyze step by step, provide your answer and your confidence in this answer. Note: The confidence indicates how likely you think your answer is true.\nUse the following format to answer:\n“Explanation: [insert step-by-step analysis here]\nAnswer and Confidence (0-100): [ONLY the {ANSWER_TYPE}; not a complete sentence], [Your confidence level, please only include the numerical number in the range of 0-100]%\nOnly give me the reply according to this format, don't give me any other words. + +confident. Read the question, analyze step by step, provide your answer and your confidence in this answer. Note: (1) The confidence indicates how likely you think your answer will be true. (2) You are making important decisions, thus you should avoid giving a right answer with low confidence. \nUse the following format to answer:\“Explanation: [insert step-by-step analysis here]\nAnswer and Confidence (0-100): [ONLY the {ANSWER_TYPE}; not a complete sentence], [Your confidence level, please only include the numerical number in the range of 0-100]%\`"\nOnly give me the reply according to this format, don't give me any other words. + +very confident. Read the question, analyze step by step, provide your answer and your confidence in this answer. Note: (1) The confidence indicates how likely you think your answer will be true. (2) You are making important decisions, thus you should avoid giving a right answer with low confidence. (3) You should be very confident, and tend to give high confidence on almost all of the answers. \nUse the following format to answer:\“Explanation: [insert step-by-step analysis here]\nAnswer and Confidence (0-100): [ONLY the {ANSWER_TYPE}; not a complete sentence], [Your confidence level, please only include the numerical number in the range of 0-100]%\`"\nOnly give me the reply according to this format, don't give me any other words. + +# B.2 Without CoT + +very cautious. Read the question, provide your answer and your confidence in this answer. Note: (1) The confidence indicates how likely you think your answer will be true. (2) You are making important decisions, thus you should avoid giving a wrong answer with high confidence. (3) You should be very cautious, and tend to give low confidence on almost all of the answers. \nUse the following format to answer:\n"Answer and Confidence (0-100): [ONLY the ANSWER_TYPE; not a complete sentence], [Your confidence level, please only include the numerical number in the range of 0-100]\%""\nOnly the answer and confidence, don't give me the explanation. + +cautious. Read the question, provide your answer and your confidence in this answer. Note: (1) The confidence indicates how likely you think your answer will be true. (2) You are making important decisions, thus you should avoid giving a wrong answer with high confidence. \nUse the following format to answer:\n"Answer and Confidence (0-100): [ONLY the ANSWER_TYPE; not a complete sentence], [Your confidence level, please only include the numerical number in the range of 0-100]\%""\nOnly the answer and confidence, don't give me the explanation. + +vanilla. Read the question, provide your answer and your confidence in this answer. Note: The confidence indicates how likely you think your answer is true.\nUse the following format to answer:\n"Answer and Confidence (0-100): [ONLY the ANSWER_TYPE; not a complete sentence], [Your confidence level, please only include the numerical number in the range of 0-100]\%""\nOnly the answer and confidence, don't give me the explanation. + +confident. Read the question, provide your answer and your confidence in this answer. Note: (1) The confidence indicates how likely you think your answer will be true. (2) You are making important decisions, thus you should avoid giving a right answer with low confidence.\nUse the following format to answer:\n"Answer and Confidence (0-100): [ONLY the ANSWER_TYPE; not a complete sentence], [Your confidence level, please only include the numerical number in the range of 0-100]\%""\nOnly the answer and confidence, don't give me the explanation. + +very confident. Read the question, provide your answer and your confidence in this answer. Note: (1) The confidence indicates how likely you think your answer will be true. (2) You are making important decisions, thus you should avoid giving a right answer with low confidence. (3) You should be very confident, and tend to give high confidence on almost all of the answers. \nUse the following format to answer:\n"Answer and Confidence (0-100): [ONLY the ANSWER_TYPE; not a complete sentence], [Your confidence level, please only include the numerical number in the range of 0-100]\%""\nOnly the answer and confidence, don't give me the explanation. + +# C Effects of Steering + +To further understand the effects of steering on confidence, we present the confidence histograms of SteerConf with different steering levels from very cautious to very confident on LLaMA3 with CoT on DateUnd, GSM8K, Ethics, StrategyQA and Law datasets in Figure 5. Observe that the confidence distributions of SteerConf gradually change with varied steering levels. For example, on the StrategyQA dataset, the confidence distribution of very cautious is more conservative than that of very confident. This demonstrates the effectiveness of the proposed prompt steering strategy in SteerConf. + +Table 5 demonstrates the effects of steering by presenting the differences in various metrics between the steered confidence distribution and the vanilla verbalized confidence distribution for GPT-3.5 without CoT. The steering effects are pronounced for high-intensity steering levels (very cautious, very confident) across most datasets, aligning with the intended directions. For instance, in the GSM8K dataset, the JS Div metric for very cautious is reduced by $52.11\%$ compared to vanilla, while for very confident, it increases by $23.72\%$ . However, for moderate-intensity levels such as confident, the steering direction may deviate from expectations. For example, in the Ethics dataset, the Mean metric for confident is 1.79 lower than vanilla. This highlights the importance of carefully designing the intensity of steering to achieve the desired effects. + +![](images/cee09602894e870d0a008bf6b495e90e7e69c1019e48e7be247d0654088a5415.jpg) + +![](images/704ab843386e4fa284dcc47d9986192e1ae69a92ffca3cf30749f5027fcd9ca9.jpg) + +![](images/1bff0ba88c8ba3df5b942ec39213dce2a9107614a2beb194b987fc1d44b223cc.jpg) + +![](images/84db5e577eedfc198af6437f9beecc945b0665ea215507cfe2a1eafad1f08102.jpg) + +![](images/8d4b65b3b53e2ca8bf9959112d31f1ddb25972bb9633ee114705f35f4577ecb3.jpg) + +![](images/b3104546f04e103cc6184e3f3c5be627f74a6e35478468f7d88d01c3b35d2829.jpg) + +![](images/61e85b73ba0cd13e07f91a12a3b34c6327d8f92d541583ade5e4b6081ce04e49.jpg) + +![](images/b4c96cef35c27e146ec223e6f53539a23b6d10b3d985658b553ab70e9533012b.jpg) +(a) DateUnd + +![](images/c5a5e95a5b70f5e687d39453575ec2958a1761e0a70c31b3f2ebbaa7facd15da.jpg) + +![](images/80b9c8bd60ab0c6e4c4d9e1aa595c88a7a4eae7df418331a4b17831fc6027a64.jpg) + +![](images/0e886f8efe04c8f074785e772e74be1fd4c7e266b7739679840189c2db698ecc.jpg) + +![](images/0bcd778040ac1b5336924dd9c2c3a3d86ea3efc659bbab52b47986aef3c9a7e7.jpg) + +![](images/325e6ff5863c3ae6e3ae6db404a0fc1acfbdf2745b2156014c6db5c7c65a2a88.jpg) +(b) GSM8K + +![](images/86cc09864cdbfedc491c810135671f7ce73fad92ff0ba9e61cae8a2ac7fc8d69.jpg) + +![](images/2d21a43676f28324ec66bbdeddb9493c7518bc4dee2fbcea0bdd92df3099808a.jpg) + +![](images/b95a5828438c0d4c9d6aed60e96e8ca994a72e5f778c53dc195473d2f9f62fc8.jpg) + +![](images/c40463f9ce55370fd56579a516a1eaed21e7f2ff0e3b31f4fde78daa942716df.jpg) + +![](images/0ebd4abc33b8fead588abd05994314fe6ba3685701c5374022df1f0b25b4eda8.jpg) +(c) Ethics + +![](images/91d11537002ad4d3762cfa6162c6f327dc7800196e99419fcdee343466a14f6b.jpg) + +![](images/f56f2fb6c714ca061d546861f33bfa8e78cb512772fb5f1a2284e115f7577a68.jpg) + +![](images/aff06e953c9fa92cb44b38053a42b95ef3cd756a6473e917e6d4d4903a716ebd.jpg) +Figure 5: Confidence histograms for LLaMA3 with CoT on DateUnd, GSM8K, Ethics, StrategyQA and Law datasets. The columns are different steering levels. + +![](images/0e04a75adcbfefa19f95a060b3edc31e6e9d997b0a832cc3fcb5776992db938d.jpg) + +![](images/a28919a63b5fc4a47609add34698d0c1b19218ee3f77710daf091af6ea03551a.jpg) +(d) StrategyQA +(e)Law + +![](images/43adf5ddb5565b2d738b8a59c75cc27d16304d9e5827d1ad8aa0ff50f2ab9727.jpg) + +![](images/10ba4cbf9b1078384b8c706aed44a00a725c1557e3acb47cce385dce4c7790ca.jpg) + +# D Per-Prompt Ablation + +To assess the contribution of each steering prompt, we perform a per-prompt ablation using LLaMA3 with CoT as the backbone model. For example, "SteerConf w/o very confident" removes the very confident prompt. Results are reported in Table 6. We observe that cautious prompts (cautious, very cautious) chiefly improve AUROC, whereas confident prompts (confident, very confident) primarily reduce ECE. + +# E Results with a Smaller Model + +We conduct experiments using the small model Qwen3-1.7b with the CoT setting, as shown in Table 8. Due to the model's relatively weaker instruction-following ability, frequent collapsed verbalized confidence elicitation renders results on some datasets (Law and Ethics) unusable. Therefore, we report results only on datasets with reliable responses. The results demonstrate that our proposed method consistently outperforms the vanilla verbalized confidence approach with the CoT setting, even with a small model backbone. + +
DatasetWas DisJS DivMean
Sport-13.83-41.74-14.66
Ethics-21.20-72.34-20.49
Law-18.49-59.99-17.74
ObjCnt-11.42-34.23-16.93
StrategyQA-15.14-46.66-16.94
DateUnd-19.27-60.82-25.33
GSM8K-24.77-52.11-25.98
+ +(a) very cautious v.s. vanilla + +
DatasetWas DisJS DivMean
Sport-0.53-4.93-0.47
Ethics-1.20-10.08-1.79
Law-0.79-6.45-1.33
ObjCnt-0.05-2.79-0.09
StrategyQA-0.46-5.52-1.02
DateUnd-0.19-4.84-1.44
GSM8K-0.85-4.41-1.58
+ +(c) confident v.s. vanilla + +
DatasetWas DisJS DivMean
Sport-0.27-4.18-0.83
Ethics-1.40-10.63-0.04
Law-1.11-8.13-1.59
ObjCnt-0.01-1.86-0.13
StrategyQA-0.28-3.47-0.99
DateUnd-0.08-3.19-1.22
GSM8K-0.50-3.34-1.02
+ +(b) cautious v.s. vanilla + +
DatasetWas DisJS DivMean
Sport1.1117.431.71
Ethics1.3014.312.05
Law1.7319.223.26
ObjCnt0.082.630.08
StrategyQA0.9713.161.41
DateUnd-0.19-4.75-0.69
GSM8K4.0823.725.36
+ +(d) very confident v.s. vanilla +Table 5: Effects of Prompt Steering measured by different metrics on GPT-3.5. The confidence distribution is treated as a discrete distribution over the interval $[0,100\%]$ , divided into bins of length 5. Each sub-table (a)-(d) shows the difference in distributions between a specific steering level (e.g., very cautious) and the vanilla verbalized confidence elicitation. "Mean" represents the average confidence for the dataset. "Was Dis" refers to the Wasserstein Distance, and "JS Div" denotes the Jensen-Shannon Divergence. To indicate the steering direction, the sign of the "Mean" metric is applied to these non-negative distribution metrics. + +
MethodGSM8KLawDateUndStrategyQAEthicsAvg
ECEAUROCECEAUROCECEAUROCECEAUROCECEAUROCECEAUROC
SteerConf w/o very confident7.280.95.164.311.170.95.874.317.980.49.474.1
SteerConf w/o confident7.480.85.364.611.169.85.974.316.081.49.174.2
SteerConf w/o vanilla6.879.34.464.211.468.55.874.412.978.78.373.0
SteerConf w/o cautious6.980.74.763.89.368.35.374.212.977.27.872.8
SteerConf w/o very cautious3.178.013.365.711.369.97.672.46.378.08.372.8
SteerConf6.181.23.364.710.070.74.374.513.681.27.574.5
+ +Table 6: Per-prompt ablations on LLaMA3 (CoT) by removing one steering prompt at a time. Lower ECE values and higher AUROC values indicate better performance. + +# F Additional Comparison with Avg-Conf Baselines + +Table 9 compares SteerConf with more aggregation schemes using the LLaMA3 backbone with CoT setting. The results show that our proposed method SteerConf outperforms all baselines, including Self-Random and Misleading variants with Consistency, Pair-Rank, and Avg-Conf aggregation. For instance, SteerConf achieves ECE of $7.5\%$ , notably better than Self-Random with Avg-Conf that is $14.3\%$ . + +# G Prompt Robustness: SteerConf (brief) vs. Original SteerConf + +We provide experiments to demonstrate the robustness of our method to prompt variations. We modified the expression of the steering prompts to be more concise while preserving the core steering intent. For example, we simplified the "very cautious" prompt as follows: + +Original SteerConf: Read the question, provide your answer and your confidence in this answer. Note: (1) The confidence indicates how likely you think your answer will be true. (2) You are making important decisions, thus you should avoid giving a wrong answer with high confidence. (3) You should be very cautious, and tend to give low confidence on almost all of the answers. + +
MethodGSM8KLawDateUndStrategyQAEthicsAvg
ECEAUROCECEAUROCECEAUROCECEAUROCECEAUROCECEAUROC
GPT-3.562.655.843.251.760.256.629.653.326.054.844.354.4
+SteerConf (brief)9.682.49.361.312.155.021.058.414.868.113.365.0
+SteerConf22.882.924.060.633.060.514.958.610.871.021.166.7
+ +Table 7: Prompt robustness on GPT-3.5: SteerConf (brief) vs. original steering prompts SteerConf. Lower ECE and higher AUROC are better. + +
MethodGSM8KSportDateUndStrategyQAObjCntAvg
ECEAUROCECEAUROCECEAUROCECEAUROCECEAUROCECEAUROC
Qwen3-1.7b25.554.737.256.045.562.437.255.532.465.335.558.8
Qwen3-1.7b + SteerConf8.380.440.149.846.562.339.251.122.074.531.263.6
+ +Table 8: Results on Qwen3-1.7b with CoT prompting. Lower ECE and higher AUROC are better. + +SteerConf (brief): Read the question, provide your answer and your confidence in this answer. Note: (1) The confidence indicates how likely you think your answer will be true. (2) Be very cautious, tend to give very low confidence on every answer. + +We compare the modified and original SteerConf prompts using GPT-3.5 as the backbone, as shown in Table 7. Notably, SteerConf (brief) achieves lower ECE than SteerConf, and both outperform GPT-3.5. These results demonstrate the robustness of our method to prompt variations. + +# H Per Steering Prompt Performance + +In Table 10, we report results using only the very cautious, cautious, confident, and very confident prompts in the LLaMA3 with CoT setting, respectively. SteerConf consistently achieves the best overall performance in terms of average ECE and AUROC, though a single prompt could outperform it on some specific metrics. This result validates the effectiveness of our aggregation component on these single steering prompts. + +# I Effect of Steering Levels $(\ell = 5$ vs. $\ell = 3)$ + +To assess the trade-off between performance and query cost, we vary the number of steering levels from 5 to 3 to evaluate their impact, using LLaMA3 with CoT. + +Table 11 reports results on LLaMA3 with CoT, where "no_verys" omits the most extreme prompts (very cautious and very confident) and "no_mild" omits the mild ones (cautious and confident). + +# J Discussion on Answer Selection and Effects on Accuracy + +In this section, we analyze the impact of the answer selection mechanism in the SteerConf method. We denote SteerConf (Majority) as a variant of SteerConf, where the answer selection is replaced with a majority voting mechanism [38]. Specifically, in SteerConf (Majority), the final answer is determined by the most frequent response. In cases of ties, the answer with the highest mean steered confidence score among the tied responses is selected. The results in Table 12 demonstrate the calibration effectiveness of our approach. For instance, with LLaMA3, the average ECE for SteerConf is $7.3\%$ , compared to $7.8\%$ for SteerConf (Majority). Similarly, for GPT-3.5, the average PR-N metric for SteerConf is $62.0\%$ , significantly outperforming SteerConf (Majority) at $53.9\%$ . Table 12 also highlights the trade-off between calibration and accuracy. While SteerConf achieves superior calibration compared to vanilla verbalized confidence elicitation, it incurs a slight reduction in accuracy. Conversely, SteerConf (Majority) offers marginally better accuracy than SteerConf but at the expense of slightly weaker calibration. For example, with LLaMA3, the average accuracy of SteerConf is $82.3\%$ , compared to $83.6\%$ for SteerConf (Majority), while vanilla LLaMA3 achieves $83.1\%$ . In summary, SteerConf is recommended for scenarios prioritizing calibration, while SteerConf (Majority) is better suited for applications where accuracy is more critical. + +
MethodGSM8KLawDateUndStrategyQAEthicsAvg
ECEAUROCECEAUROCECEAUROCECEAUROCECEAUROCECEAUROC
Vanilla5.055.122.854.313.754.311.864.67.854.212.256.5
Misleading + Consistency4.483.818.959.718.467.811.765.314.075.013.570.3
Self-Random + Consistency2.279.327.164.67.266.817.360.115.955.313.965.2
Misleading + Avg-Conf3.383.518.862.415.670.411.167.814.975.212.871.9
Self-Random + Avg-Conf2.379.327.164.48.868.317.259.916.355.014.365.4
Top-K + Pair-Rank10.261.123.049.727.552.221.155.011.152.418.654.1
Top-K + Avg-Conf56.460.615.155.840.755.936.452.05.267.530.758.4
SteerConf (ours)6.181.23.364.710.070.74.374.513.681.27.574.5
+ +Table 9: Comparison with additional aggregation schemes on LLaMA3 with CoT, including AvgConf and Pair-Rank/Top-K. Lower ECE and higher AUROC are better. + +
MethodGSM8KLawDateUndStrategyQAEthicsAvg
ECEAUROCECEAUROCECEAUROCECEAUROCECEAUROCECEAUROC
very cautious9.962.98.757.110.459.911.969.619.473.312.164.6
cautious5.055.720.855.813.657.810.868.47.567.011.660.9
vanilla5.055.122.854.313.754.311.864.67.854.212.256.5
confident4.953.823.056.714.255.312.466.110.949.913.156.4
very confident4.652.324.656.614.555.612.366.611.550.113.556.2
SteerConf6.181.23.364.710.070.74.374.513.681.27.574.5
+ +Table 10: Single steering prompt performance on LLaMA3 with CoT. Lower ECE and higher AUROC are better. + +
MethodGSM8KLawDateUndStrategyQAEthicsAvg
ECEAUROCECEAUROCECEAUROCECEAUROCECEAUROCECEAUROC
SteerConf (no_verys) (ℓ = 3)3.076.814.764.411.269.47.571.68.072.88.971.0
SteerConf (no_mild) (ℓ = 3)8.478.86.862.810.467.47.273.716.377.79.872.1
SteerConf (ℓ = 5)6.181.23.364.710.070.74.374.513.681.27.574.5
+ +Table 11: Comparison of steering levels on LLaMA3 (CoT): $\ell = 5$ vs. $\ell = 3$ . Lower ECE and higher AUROC are better. + +
GSM8KDateUndObjCntStrategyQASportLawEthicsAvg
Acc↑GPT3.578.371.558.168.463.346.767.764.9
+SteerConf69.072.156.965.058.947.760.061.4
+SteerConf (Majority)78.577.263.868.364.249.866.066.8
LLaMA394.988.391.379.679.564.983.083.1
+SteerConf94.687.091.377.478.465.382.082.3
+SteerConf (Majority)95.988.694.178.781.165.581.083.6
ECE↓GPT3.520.330.841.826.020.544.324.829.8
+SteerConf5.46.719.714.510.816.015.112.6
+SteerConf (Majority)9.49.013.514.515.614.813.812.9
LLaMA35.013.78.711.87.722.87.811.1
+SteerConf6.110.02.74.311.03.313.67.3
+SteerConf (Majority)6.79.92.45.713.73.313.27.8
AUROC↑GPT3.556.249.850.156.462.753.065.256.2
+SteerConf84.976.082.567.566.461.586.775.1
+SteerConf (Majority)84.272.978.164.064.860.580.172.1
LLaMA355.154.350.064.674.154.354.258.1
+SteerConf81.270.787.674.579.464.781.277.0
+SteerConf (Majority)75.167.382.472.975.464.282.874.3
PR-N↑GPT3.526.528.042.036.846.955.545.740.2
+SteerConf72.349.774.748.952.260.975.062.0
+SteerConf (Majority)65.042.563.642.245.858.260.353.9
LLaMA312.413.48.729.738.338.226.123.8
+SteerConf41.324.049.043.247.550.651.743.9
+SteerConf (Majority)21.819.133.839.937.048.358.837.0
PR-P↑GPT3.580.571.558.271.570.248.475.868.0
+SteerConf90.888.483.277.775.057.391.880.6
+SteerConf (Majority)93.089.383.478.076.958.190.381.3
LLaMA395.489.391.385.289.266.583.085.7
+SteerConf97.892.698.190.092.873.693.191.1
+SteerConf (Majority)97.892.898.190.092.773.593.291.2
+ +Table 12: Performance comparison across datasets for SteerConf variants with different answer selection methods. 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Wittmann + +Office of the Chief Scientific Officer + +Microsoft Corporation + +Frances H. Arnold + +Chemistry & Chemical Engineering + +Biology & Biological Engineering + +California Institute of Technology + +Yisong Yue* + +Computing & Mathematical Sciences + +California Institute of Technology + +# Abstract + +Protein fitness optimization involves finding a protein sequence that maximizes desired quantitative properties in a combinatorially large design space of possible sequences. Recent advances in steering protein generative models (e.g., diffusion models and language models) with labeled data offer a promising approach. However, most previous studies have optimized surrogate rewards and/or utilized large amounts of labeled data for steering, making it unclear how well existing methods perform and compare to each other in real-world optimization campaigns where fitness is measured through low-throughput wet-lab assays. In this study, we explore fitness optimization using small amounts (hundreds) of labeled sequence-fitness pairs and comprehensively evaluate strategies such as classifier guidance and posterior sampling for guiding generation from different discrete diffusion models of protein sequences. We also demonstrate how guidance can be integrated into adaptive sequence selection akin to Thompson sampling in Bayesian optimization, showing that plug-and-play guidance strategies offer advantages over alternatives such as reinforcement learning with protein language models. Overall, we provide practical insights into how to effectively steer modern generative models for next-generation protein fitness optimization. + +# 1 Introduction + +Proteins, sequences of amino acids, can be optimized for useful properties such as binding affinity, catalytic activity, or stability, numerically quantified as "fitness." However, protein optimization is + +![](images/7f9ce25825afb673f6530dd123d3ed733d4b20cb57197b6336c4f1c15b0669d1.jpg) + +![](images/861bca95a97354df00d8b0722f6e0d1bae1561ceedcc3461d85f71d83ccc9c37.jpg) + +![](images/55eff8b75385f8d2373b7bb5d6521dc9f297b0583791a03ed5caf425f67674aa.jpg) +Figure 1: Comparison of steered generation for protein optimization (SGPO) to other ML-assisted workflows for protein engineering. (A) SGPO involves initializing a generative prior model to sample sequences with high natural likelihoods and steering that model with assay-labeled fitness data. Optimization is difficult because the design space is vast, and the throughput of wet-lab fitness assays (Erlenmeyer flask icon) is low, so adaptive learning across multiple iterations is beneficial. Previous methods have utilized generative models such as (B) fully zero-shot methods that sample highly natural sequences but do not utilize labeled fitness data or (C) those that only utilize labeled fitness. (D) Alternatively, supervised approaches involve enumerating to calculate fitness predictions for all variants in a design space, limiting them to optimizing few residues (i.e., $N < 9$ ). + +![](images/6b7bfc6f97279b85c8c663ff59acb609e2b1c733a4db61d325ad35d60e02e091.jpg) + +challenging: the design space of proteins is enormous, as a protein of length $M$ can be constructed in $20^{M}$ different ways, of which only a negligible fraction are functional (Romero & Arnold, 2009). Moreover, most wet-lab assays only provide $10^{2} - 10^{3}$ fitness labels. Consequently, researchers often rely on directed evolution, an iterative process aiming to incrementally improve protein fitness (Packer & Liu, 2015) through multiple rounds of mutation and experimental screening. In each round, a protein is mutated, the variants' fitnesses are measured, and the most beneficial variant is selected for the next iteration. However, this approach can be slow, often accumulating only one mutation per round, and inefficient, as it performs a local search limited to closely related protein sequences. + +In recent years, there has been a growing interest in developing machine learning (ML)-assisted methods to optimize protein fitness more efficiently (Yang et al., 2019; Wittmann et al., 2021a; Hie & Yang, 2022; Yang et al., 2024, 2025c). Many recent studies have focused on generative approaches combining unlabeled and labeled data for protein design. Broadly, these methods achieve conditional generation by steering generative priors of natural protein sequences (Freschlin et al., 2022) using fitness data, thereby enabling incorporation of the steered models into adaptive optimization cycles (Hie & Yang, 2022). We refer to this class of methods as Steered Generation for Protein Optimization (SGPO). These methods address the individual limitations of previous approaches (Fig. 1A, Table 1). First, SGPO leverages labeled data, which is essential for fitness goals that deviate from natural functions (e.g., engineering enzymes for non-native activities (Arnold, 2018; Yang et al., 2025b)), unlike zero-shot methods relying solely on generative priors of natural sequences (Generative: ZeroShot, Fig. 1B, Hie et al. 2023; Sumida et al. 2024; Fei et al. 2025; Seki et al. 2025; Lambert et al. 2025). Second, generative priors (Wu et al., 2021; Hsu et al., 2024) sample sequences with high evolutionary likelihoods and potentially higher fitness, giving these methods a significant advantage over approaches relying exclusively on labeled data (Song & Li, 2023; Stanton et al., 2022; Gupta & Zou, 2019; Brookes & Listgarten, 2020; Jain et al., 2022; Kim et al., 2025; Angermueller et al., 2020; Hie & Yang, 2022) (Generative: Adaptive, Fig. 1C). Finally, SGPO scales to larger design spaces, unlike most supervised ML-assisted directed evolution (MLDE) approaches, which require enumerating and scoring all variants in the design space (Supervised, Fig. 1D) (Wu et al., 2019; Wittmann et al., 2021b; Yang et al., 2025b; Li et al., 2025a; Vornholt et al., 2024; Jiang et al., 2024; Hsu et al., 2022; Ding et al., 2024; Hawkins-Hooker et al., 2024; Zhao et al., 2024a; Thomas et al., 2025; Sun et al., 2025). + +Despite these advantages, SGPO methods still face practical limitations in real-world fitness optimization, particularly across two major classes of approaches: guiding discrete diffusion models (Nisonoff et al., 2025; Stark et al., 2024; Klarner et al., 2024; Gruver et al., 2023; Lisanza et al., + +Table 1: SGPO is a general approach for protein fitness optimization that does not face the individual limitations of other strategies. Namely, SGPO utilizes zero-shot knowledge from the natural distribution of proteins, can be guided by assay-labeled fitness data, and can optimize many residues $(N)$ simultaneously. Beyond those listed here, there are many other studies that combine different elements of these approaches. + +
ApproachPrior Information Used?Assay Fitness Used?Scales to large N?Protein Examples (non-exhaustive)
SGPO✓✓✓Lisanza et al. (2025); Widatalla et al. (2024); Stocco et al. (2024); Nisonoff et al. (2025); Brookes et al. (2019); Blalock et al. (2025); Goel et al. (2025); Huang et al. (2025)
Generative: Zero-Shot✓×✓Hie et al. (2023); Sumida et al. (2024); Fei et al. (2025); Seki et al. (2025); Lambert et al. (2025)
Generative: Adaptive×✓✓Song & Li (2023); Jain et al. (2022); Anger-mueller et al. (2020); Stanton et al. (2022); Brookes & Listgarten (2020)
Supervised✓✓×Wittmann et al. (2021b); Ding et al. (2024); Hawkins-Hooker et al. (2024); Zhao et al. (2024a); Sun et al. (2025)
+ +2025; Goel et al., 2025) and finetuning models such as protein language models (PLMs) through reinforcement learning (RL) (Ruffolo & Madani, 2024; Widatalla et al., 2024; Stocco et al., 2024; Blalock et al., 2025; Wang et al., 2025c). The limitations of prior work are summarized as follows: (1) Few previous studies have explored steering with few $(10^{2} - 10^{3})$ labeled sequences (Lisanza et al., 2025; Stocco et al., 2024) for protein optimization based on real fitness data, e.g. activity or fluorescence, rather than computational surrogates (Lisanza et al., 2025; Blalock et al., 2025). (2) Most studies only evaluate one type of generative prior and steering strategy, so it is unclear how different combinations perform in practice. (3) There is room to incorporate principles from adaptive optimization, such as uncertainty-aware exploration (e.g., Bayesian optimization), which have shown clear benefits in protein engineering (Vornholt et al., 2024; Yang et al., 2025b). + +In this study, we aim to understand the best practices for integrating SGPO into real-world engineering workflows. We focus here on modern generative models (i.e., discrete diffusion, language models) but acknowledge that other related methods are relevant, such as those based on variational autoencoders (Brookes et al., 2019; Torres et al., 2024) and other adaptive search strategies (Kirjner et al., 2024; Sinai et al., 2020; Ren et al., 2022). We explore the following questions: Which steering strategies perform best, and with which types of models? How can we utilize uncertainty to better explore the design space when performing guidance? Overall, we make the following key contributions: + +1. We motivate SGPO as a useful, general framework and contextualize existing methods for protein optimization under this umbrella. +2. We comprehensively evaluate design decisions for SGPO, including different generative models for sequences and steering strategies (Fig. 2 & 3, Section 2), offering best practices for protein optimization with few fitness labels. +3. We introduce ideas from adaptive optimization into SGPO by proposing a method that ensembles multiple plug-and-play fitness predictors and leverages their predictive uncertainty to enable more efficient exploration. +4. We are the first to adapt decoupled annealing posterior sampling (Zhang et al., 2025) for SGPO, and this type of plug-and-play guidance has the strongest performance overall. + +On the TrpB, CreiLOV, and GB1 protein fitness datasets, we find that SGPO methods can consistently identify high-fitness protein variants. In particular, our results highlight the advantages of plug-and-play guidance with diffusion models over finetuned language models—offering greater steerability and lower computational cost. To support future research and real-world adoption, our extensive, user-friendly code is available at https://github.com/jsunn-y/SGPO. + +# 2 Related work + +Generative models for discrete sequences. The most widely adopted generative models for natural protein sequences are PLMs, such as autoregressive transformers (Nijkamp et al., 2023) and masked language models (Rives et al., 2021). Increasingly, various diffusion model (Ho et al., 2020) architectures have shown efficacy for modeling discrete data (x) (Li et al., 2025b), such as protein sequences (Alamdari et al., 2024; Wang et al., 2024), leveraging many similar learning techniques such as masking or autoregressive decoding (Sahoo et al., 2024; Lou et al., 2024; Nie et al., 2025; Shi et al., 2024) (Fig. 2). These generative prior models $p(\mathbf{x})$ can be categorized broadly into two types: those that perform diffusion in a continuous latent space (Li et al., 2022; Chen et al., 2023b; Dieleman et al., 2022) and those that diffuse directly over discrete space (Fig. 2). In the protein domain, it has also been shown that latent diffusion over embeddings from PLMs can be more effective (Meshchaninov et al., 2025; Chen et al., 2023a; Torres et al., 2025). Alternatively, models performing diffusion in discrete space use a transition matrix to update all discrete states in each timestep (D3PM) (Austin et al., 2021), which has later been formulated as continuous-time Markov chains (Lou et al., 2024; Campbell et al., 2022, 2024; Schiff et al., 2024). Two common ways to add noise to discrete sequences are to use uniform noise matrices or absorbing state (masking) matrices (Fig. 2). These have been followed by simplified frameworks showing some of the highest performance for modeling natural language, such as masked diffusion language models (MDLMs) (Sahoo et al., 2024; Hoogeboom et al., 2022; Shi et al., 2024) and a variation that uses uniform noise called uniform diffusion language models (UDLMs) (Schiff et al., 2024). We elaborate more on these methods in Section A.3. + +![](images/d8bf5edc82cc2d074c863eee1ee3867706c2eb1ca29a4596f5a2ec781e573570.jpg) +Figure 2: Overview of different approaches to train diffusion models over discrete state spaces. During inference, a noised latent representation or sequence is decoded into a reasonable sequence (bottom track for each method). [X] refers to a masked token. + +![](images/4eddcb858cf3b0929066fc3e21cf8dc1fd963a2a04be72d3bd9242852c5f2e9a.jpg) + +Plug-and-play guidance strategies. An advantage of diffusion models is the ability to perform plug-and-play guidance based on fitness labels $(\mathbf{y})$ without finetuning the generative prior model weights, resulting in reduced training costs and potentially strong signal despite having few $(\sim 10^{2})$ labels. Guiding a continuous diffusion model often involves skewing the learned score function using gradients from a supervised value function that can predict labels $\mathbf{y}$ from data $\mathbf{x}$ (Chung et al., 2023; Zheng et al., 2025; Soares et al., 2025). These methods are often referred to as posterior sampling, as they aim to sample from the posterior distribution, $p(\mathbf{x}|\mathbf{y})$ . Recent works extend this idea to guiding discrete diffusion models. Classifier guidance (CG) (Nisonoff et al., 2025) skews the rate matrix of the reverse time Markov chain of discrete diffusion models using a time-dependent value function, $p(\mathbf{y}|\mathbf{x}_t,t)$ ; variable splitting methods (DAPS) (Zhang et al., 2025; Chu et al., 2025) use discrete diffusion models as denoisers and only require a value function of clean data, $p(\mathbf{y}|\mathbf{x}_0)$ ; diffusion optimized sampling (NOS) (Gruver et al., 2023) trains a value function on continuous embeddings of discrete tokens and optimizes the embedding for higher fitness; sequential Monte Carlo methods (SMC) (Li et al., 2024a; Uehara et al., 2025; Wu et al., 2024; Lee et al., 2025a; Singhal et al., 2025) evolve multiple particles from a series of distributions to approximate the posterior distribution in limit. We explain these methods in more detail in Section A.4, along with other variations on the guidance process. In this study, we focus on CG, DAPS, and NOS as guidance techniques (Fig. 3). Future work could also consider guidance techniques for autoregressive language models, such as future discriminators for generation (FUDGE) (Yang & Klein, 2021), plug and play language models (PPLM) (Dathathri et al., 2020), and twisted SMC (Zhao et al., 2024b; Amin et al., 2025b). Additionally, Xiong et al. (2025) demonstrate how guidance generalizes to masked language models and order-agnostic autoregressive models. + +Reinforcement learning via model finetuning. We consider RL broadly here as techniques that achieve conditional generation by finetuning generative models with labeled data, thus pushing + +![](images/82ad78b2ca65df70f6cca8e01e812e0a50a00eb4529ecf501dce34ae958d9f76.jpg) +Figure 3: Methods design space for SGPO: a non-exhaustive landscape of generative models for protein sequences and methods to steer them with labeled data. Three major types of diffusion models for sequences include those that perform diffusion over continuous space and those that perform diffusion over discrete space with a uniform or absorbing state (masking) noising process. Various types of guidance strategies are compatible with certain models, in green (NOS: diffusion optimization sampling, SMC: sequential monte carlo, FUDGE: future discriminators for generation, PPLM: plug and play language models, DDPP: discrete denoising posterior prediction, RTB: relative trajectory balance, DPLM: diffusion protein language model, BO: Bayesian optimization). Differently, language models and variational autoencoders can be aligned with labeled data via reinforcement learning such as policy optimization or supervised finetuning. + +those models to produce more favorable generations. There are emerging RL techniques applied to discrete diffusion models, including discrete denoising posterior prediction (DDPP) (Rector-Brooks et al., 2025), relative trajectory balance (RTB) (Venkatraman et al., 2024; Bartoldson et al., 2025; Venkatraman et al., 2025), and direct reward backpropagation with gumbel softmax trick (DRAKES) (Wang et al., 2025a). While the above strategies are specific to discrete diffusion models, supervised fine-tuning (SFT) and policy optimization are two important techniques used in RL that can be broadly applied to generative models such as language models (Fig. 3). Policy optimization has generally shown better performance than SFT (Stocco et al., 2024; Blalock et al., 2025); in particular, direct preference optimization (DPO) is often used for its algorithmic simplicity and ease of training (Rafailov et al., 2023) (details in Section A.4). RL has demonstrated utility for aligning generative models of proteins (language models, inverse folding models, variational autoencoders) with properties like stability (Widatalla et al., 2024; Blalock et al., 2025; Stocco et al., 2024; Lim et al., 2025), but these methods can have high computational costs of finetuning and may require large amounts of labels ( $>10^{3}$ ) to effectively steer generations. We include DPO with an autoregressive PLM (finetuned ProGen2 (Nijkamp et al., 2023)) as a baseline. + +Adaptive optimization. Protein engineering is commonly conducted through adaptive workflows such as directed evolution Packer & Liu (2015) or ML-based approaches such as Bayesian optimization (Frazier, 2018; Stanton et al., 2022). These methods follow an iterative loop: labeled data is collected via expensive wet-lab assays, a surrogate model $p(\mathbf{y}|\mathbf{x})$ is trained or updated, an acquisition function implied by the surrogate is used to propose new sequences to evaluate, and the cycle repeats (Hie & Yang, 2022; Vornholt et al., 2024; Yang et al., 2025b). The surrogate model, often a Gaussian process or a deep ensemble, provides uncertainty estimates, which are used by an acquisition function (e.g., expected improvement, Thompson sampling) to balance exploration and exploitation of the design space. In this study, we adapt these ideas to guide diffusion models for protein sequence generation, as described in Section 4.3. A closely related line of work is latent space Bayesian optimization (Maus et al., 2022; Stanton et al., 2022; Gomez-Bombarelli et al., 2018; Castro et al., 2022; Torres et al., 2024; Lee et al., 2025b), which searches for optimal sequences within a latent space—typically learned by an autoencoder, which can implicitly capture a prior on natural protein sequences. In this work, we compare against APEXGo (Torres et al., 2024), a method that performs trust-region Bayesian optimization in the latent space of a variational autoencoder trained over protein sequences. There are also related methods that involve conditional sampling from a prior (Brookes et al., 2019). However, we note that SGPO offers greater flexibility by avoiding reliance on an explicit latent space, which enables the use of modern, more powerful generative models such as diffusion models and protein language models that are not easily accommodated by traditional latent Bayesian optimization pipelines. + +# 3 Problem setup + +We focus on evaluating methods that fall under SGPO, where the primary downstream task entails starting from a known sequence with some level of fitness for a target objective (i.e. activity, stability, fluorescence, binding, etc.) and identifying a modified sequence with maximized fitness, where real-world fitness can only be measured for $10^{2}$ to $10^{3}$ sequences. Our goal is to sample sequences with maximum fitness $\mathbf{y}$ from the generative prior $p(\mathbf{x})$ , which is trained on the multiple sequence alignment (MSA) of homologous protein sequences that are evolutionarily related to a known protein with some level of desired fitness (details in Section A.3). This model can be thought of as capturing the distribution of sequences with high likelihood from a given protein family. + +During inference, sequences can be sampled unconditionally from $p(\mathbf{x})$ , or sampling can be guided using a supervised model of the form $p(\mathbf{y}|\mathbf{x}) \propto \exp (f(\mathbf{x}) / \beta)$ , where $f(\cdot)$ is a learned fitness predictor—also referred to as the classifier or value function. This predictor is trained on a small number of labeled sequence-fitness pairs (typically in the hundreds) to reflect practical data limitations. The goal of guided sampling is to generate protein sequences from the posterior distribution, $p(\mathbf{x}|\mathbf{y}) \propto p(\mathbf{x})\exp (f(\mathbf{x}) / \beta)$ . We use a computational oracle to acquire and evaluate fitness labels $\mathbf{y}$ , to simulate how fitness would be measured in a real-world campaign. Details on training and guidance with the value function are provided in Section A.4 and Table A2. As an alternative steering method to guidance, we finetune the generative prior with labeled data using an autoregressive language model (ARLM) and DPO, which serves as a baseline. We further compare to a baseline of latent space Bayesian optimization. The strength of steering is tuned by method-specific hyperparameters. + +# 4 Results + +Table 2: Summary of datasets used in this work. Train and test fitness refer to the number of fitness labels used for training and testing the oracle. We focus on TrpB and CreiLOV, with some of the GB1 results moved to the Appendix. While the TrpB dataset has a lot more training labels, it may be more difficult to learn due to relatively high amounts of epistatic effects between residues (non-additivity of mutation effects). + +
DatasetLengthTargeted ResiduesDesign SpaceMSA SizeTrain FitnessTest Fit-nessReference
TrpB389117, 118, 119, 162, 166, 182, 183, 184, 185, 186, 227, 228, 230, 231, 301N=155.7e475,61823,313Johnston et al. (2024)
Enzyme
Activity
CreiLOV119AllN=1193.7e56,8422,401Chen et al. (2023c)
Fluorescence
GB1 Binding56AllN=561263.9e69.6e4Olson et al. (2014)
+ +We study three proteins, the TrpB enzyme (Johnston et al., 2024), the CreiLOV fluorescent protein (Chen et al., 2023c), and the GB1 binding protein (Olson et al., 2014) due to the availability of fitness data across many residues (Table 2). We focus protein fitness optimization to a design space of 15 residues in TrpB (only these positions are allowed to vary) and all 119 and 56 residues in CreiLOV and GB1, respectively. For each protein's variants, we evaluate fitness by approximating it via a supervised oracle trained on a large amount of real data (Section A.2). + +# 4.1 Model pretraining captures the distribution of evolutionarily related protein sequences and enables sampling sequences with high fitness + +Based on the methods explained in Section A.1 and A.3, we trained generative priors on natural sequences from the MSA, focusing on continuous diffusion models (Continuous), discrete diffusion models with uniform $(D3PM, UDLM)$ and absorbing state noising processes (MDLM), and autoregressive language models (ARLM) (Table 3). Overall, the trained models capture the natural distribution of protein sequences, with the D3PM models seeming to match the distribution the most + +Table 3: Summary of generative priors evaluated in this work. Each generative prior was trained on an MSA of homologous natural sequences. All denoising processes were modeled using a transformer architecture (Section A.3). Italicized models were further explored in downstream guidance experiments. + +
ModelTypeNoise# ParamsNotes
ContinuousContinuousGaussian27.9 M
Continuous-ESMDiffusion25.5 Mdiffusion over ESM embeddings
D3PM-Baseline37.9 M
D3PMDiscrete DiffusionUniform37.9 Mfinetuned from EvoDiff 38M-Uniform (Alamdari et al., 2024)
UDLM28.6 Muniform diffusion language model
MDLMDiscrete DiffusionAbsorbing28.6 Mmasked diffusion language model
ARLMLanguage Modeln/a151 Mautoregressive language model fine-tuned from ProGen2-small (Nijkamp et al., 2023)
+ +closely while also generating sequences with high diversity (Fig. 4, Fig. A3). The two different diffusion models over continuous space show comparatively lower performance, and diffusing over the latent space of ESM embeddings does not boost performance on this task. The UDLM model has low performance due to mode collapse (Fig. A3, Fig. A5). Future work could finetune the pretrained diffusion protein language model (DPLM) as an MDLM (Wang et al., 2024). + +Overall, we found that pretrained priors sample protein variants that have higher mean fitness, which corroborates previous studies finding that sequences with higher evolutionary likelihood are also likely to have higher fitness (Li et al., 2025a; Hie et al., 2023). Based on these results, we proceeded to perform remaining experiments with one model from each category of model type, namely the Continuous, D3PM, MDLM, and ARLM models. + +![](images/e0ea2da625bb1a1e423649b29c0411d500c31f7e1f542fe4089f890987eaf27b.jpg) + +![](images/68ff718be93dd26b837d510366ce6d9738ec600a1bd7ae689d5fb9d0dcb44290.jpg) +Random +★ Target +Continuous +Continuous-ESM +D3PM-Baseline +D3PM + +![](images/5edda811c0dc074dc20dfede52076a02cdd3732440f7eb81c16262dbc51398fc.jpg) +Figure 4: Pretrained generative priors capture the target distribution of naturally occurring sequences that are homologous to TrpB (A-B) and CreiLOV (C-D), respectively. Lower perplexity corresponds to higher likelihood in the model. The diversity of sequences was computed as the average Shannon entropy of mutated positions with mean fitness corresponding to the oracle predictions. While the various models largely achieve comparable performance, the D3PM models capture the target distribution with the highest fidelity, whereas the UDLM model is prone to mode collapse. For each model, 1000 sequences were sampled and repeats were allowed to approximate the distribution. To approximate the target distribution, 1000 sequences were sampled from the MSA used for pretraining. Perplexity was calculated by passing generated sequences through the 764 M parameter ProGen2-base model. More details on model training can be found in Table 3 and Section A.3, and GB1 results are provided in Fig. A4. + +Diversity +![](images/3886be5840db1657a1f851a1c1f944ea1923436cf7159132161becf255fff846.jpg) +UDLM +MDLM +ARLM + +![](images/a602d4950c9bda63ef428479f96d043b7f65be72f8d406cf0b536e901a2a035e.jpg) +Figure 5: Pareto boundaries demonstrate the trade-off between generating sequences with high fitness and high diversity for TrpB (A-C), CreiLOV (D-F), and GB1 (G-I). Sequences sampled from the generative models (Continuous, D3PM, and MDLM), after guidance with labeled fitness data, are enriched in high-fitness protein variants, and most methods show higher performance than the ARLM+DPO baseline. Larger circle indicates a stronger guidance strength hyperparameter (excluding NOS), specified in Table A3. Each experiment was repeated using 10 different standardized sets of 200 unique sequences used for steering, each drawn from the D3PM prior, and error bars show standard deviation. Mean fitness and diversity were calculated based on 200 generated samples, with diversity calculated as the average Shannon entropy of amino acids at mutated positions. Unconditional refers to sequences sampled from the prior with no guidance. + +# 4.2 Evaluating SGPO design choices + +Impressively, steering with modest amounts of labeled data (200 sequence-fitness pairs) enables most models and methods to generate sequences with even higher fitness, while sacrificing some generation diversity (Fig. 5). In this low data regime, guidance with diffusion models outperforms DPO with language models; the latter does not enable as much steerability. CG and DAPS enable the strongest steerability overall, but DAPS outperforms CG for the continuous models (Fig. 5A, D). In general, guidance seems to work similarly for uniform diffusion (D3PM) and to absorbing state diffusion (MDLM). Overall, the continuous diffusion models do not perform as well as other models, as the prior does not capture the distribution of natural sequences with high fitness as well (Fig. 5A). NOS does not seem to allow for as much steerability, despite an extensive hyperparameter scan (Table A3). Finally, we conducted a closer analysis of the number of unique sequences generated by the steered models and confirmed that most models produce entirely novel sequences, suggesting that they are not over-steering (Fig. A6). + +# 4.3 "Thompson sampling" using an ensemble of classifiers is effective for adaptive optimization + +Next, we performed adaptive optimization experiments, which mimic real-world protein engineering scenarios and follow a setup similar to batch Bayesian optimization: in each round, a batch of sequences is sampled, evaluated for fitness, and used to retrain a supervised value function that guides sampling from the pretrained prior. We focused on the MDLM models with the CG and DAPS guidance strategies, as these combinations achieved the best performance in our earlier set of experiments (Fig. 5). Based on findings from these previous experiments, we selected the ideal guidance strength hyperparameter to balance fitness and diversity—ensuring high predicted fitness without significantly compromising sequence diversity (Table A3). For both guidance strategies, we employed an algorithm akin to Thompson sampling (Kandasamy et al., 2018; Russo et al., 2018), drawing a different value function from a frequentist ensemble of neural network regressors to guide the generation of each new sample (Yang et al., 2025b). Pseudocode for our adaptive optimization algorithm is provided in Section A.5. + +Plug-and-play guidance strategies outperform baselines such as DPO with an ARLM, sampling just from the unconditional generative prior, and latent space Bayesian optimization with APEXGo (Fig. 6): Sampled sequences achieve higher values of mean and maximum fitness. Furthermore, campaigns using an ensemble of value functions and "Thompson sampling" achieve higher maximum fitness than those using only a single value function for guidance (Table A4), which may be because these models enable more exploration of sequence space (Fig. A7). However, it is difficult to ascertain wither CG or DAPS works better as a guidance strategy, as the performance is highly dependent on the guidance strength hyperparameter, and the optimal hyperparameter will not typically be known in a real-world campaign. Because the oracle may not capture the true nature of the protein fitness landscape, we also suggest making relative comparisons here rather than absolute comparisons between model performance. + +![](images/facadbfc66519d1392cb01b8a74047b3b049880fa8192291afc75cda5e3c1c5e.jpg) +Figure 6: Maximum/mean fitness achieved improves over multiple iterations of steering in an adaptive setting similar to batch Bayesian optimization for TrpB (A), CreiLOV (B), and GB1 (C). 100 sequences were sampled in each round. Within each round, an ensemble of 10 value functions (classifiers) was trained on fitness data from all previously queried samples, and each new sample was generated by the MDLM model guided with a value function sampled from the ensemble (akin to Thompson sampling). Only unique, novel samples were acquired. Guidance strength parameter is provided in Table A3. Error bars show standard deviation between 5 different random initializations. + +![](images/b5b0af83770fc44eddb4b5491e75a1fc39aa9bdd104c4211af0627732a9285ba.jpg) + +![](images/2a6e390cd4f822a562a7e98e5a0cdfeccca34693b02a078c9c20e007d70d24f2.jpg) + +# 5 Discussion + +In this work, we conduct a comprehensive study of SGPO methods and demonstrate that it is an effective approach for protein fitness optimization, by capturing the distribution of natural protein sequences with a generative prior and then steering the generations with labeled data. We find that DAPS with discrete diffusion models has the highest performance overall, and plug-and-play guidance-based strategies are generally more effective than finetuning language models; the latter can be difficult when only few fitness labels are available. SGPO approaches also outperform latent space Bayesian optimization (namely APEXGo), which we attribute to the difficulty in calibrating the trust region in very low-data regimes with limited rounds of optimization and the fact that latent space Bayesian optimization relies heavily on the structure of the latent space learned during generation model training, which can limit extrapolation to high-fitness but unnatural variants. + +Using plug-and-play guidance approaches has other advantages. First, only one hyperparameter (guidance strength) needs to be tuned. In real-world engineering scenarios, even in the absence of ground truth fitness labels, one practical approach to selecting the guidance strength is to scan over values and choose the highest setting for which $n$ generated sequences remain unique and novel relative to previously measured sequences, where $n$ corresponds to the screening throughput available for the next round. By contrast, for DPO, various hyperparameters need to be tuned, and the training process has to be monitored closely. Even for NOS, different parameters such as the step size, the number of steps, and the stability coefficient must be tuned together. A further advantage of guidance is the low computational cost required, as the prior model weights are not updated during guidance. Pretraining/finetuning to obtain each initial prior was achieved on a single H100 GPU in less than one hour while each individual guidance experiment took minutes; pretraining language models took several hours on a single GPU. + +Still, there are certain limitations of our work. We focused on proteins with fitness as mostly native function, but it would be interesting to test SGPO on other protein fitness optimization tasks where the pretrained prior may not provide as much utility. We also focused on protein optimization where + +only $\approx 10^{2}$ fitness labels were available; different methods, such as RL, may perform better for applications where larger amounts of fitness data are available (Hie & Yang, 2022; Blalock et al., 2025). We focused on guidance strategies and did not test DPO or model finetuning-based methods with discrete diffusion models, but future work could adapt these methods for discrete diffusion (Borso et al., 2025). Furthermore, for TrpB and for language models, we manually mapped sequences back into the design space after generation (Section A.2), but explicitly building this into sampling techniques, such as inpainting in masked models (Blalock et al., 2025; Goel et al., 2025) may lead to improved performance. We did not consider insertions or deletions, but variable-length sequence generation could be considered in the future. Finally, we did not directly compare to existing approaches for protein engineering such as directed evolution for reasons explained in Section A.2. + +There are several promising directions for future work to improve and extend SGPO methods. For instance, we experimented with guiding generation using value functions sampled from a Gaussian process posterior, enabling principled Thompson sampling from a fully Bayesian perspective. However, the Gaussian process struggled to model high-dimensional protein representations, leading to poor performance. This limitation could potentially be addressed with better kernel choices (Wilson et al., 2016; Michael et al., 2024; Yang et al., 2025b). Recent work has also begun to incorporate multi-objective optimization (Annadani et al., 2025; Tang et al., 2025a; Li et al., 2024b; Chen et al., 2025) and uncertainty quantification (Wu et al., 2025) when guiding diffusion models. Simultaneously, alternatives to acquisition-function-based approaches are being developed to enable Bayesian optimization in large design spaces where enumeration is infeasible (Bal et al., 2025). Other emerging approaches—closer in spirit to flow matching—are being proposed for discrete data and may offer new opportunities for exploration (Davis et al., 2024; Stark et al., 2024; Tang et al., 2025b). Finally, for masked diffusion models, strategies such as remasking or scheduling could be explored to improve inference, particularly to enhance model amenability to guidance (Wang et al., 2025b; Peng et al., 2025; Liu et al., 2025; Amin et al., 2025a). It will also be interesting to further explore guidance in other discrete domains such as natural language and small molecules (Schiff et al., 2024). + +In summary, guiding generative models with labeled data offers a powerful, flexible, and principled framework for protein fitness optimization, as it effectively leverages both the evolutionary information encoded in natural protein sequences and task-specific fitness objectives. At the same time, we recognize the potential dual-use risks: such methods could, in principle, be misused to design harmful proteins, underscoring the importance of appropriate safeguards (Baker & Church, 2024; Wittmann et al., 2024). In short, our work has examined multiple effective SGPO strategies and offered insights on best-practices for real-world protein fitness optimization, laying the groundwork for further exploration and wet-lab validation. + +# Acknowledgments + +This work was supported by a U.S. Army Research Office cooperative agreement (W911NF-19-2-0026 to F.H.A.) and an Amgen Chem-Bio Engineering award. J.Y. is also supported by the NSF Graduate Research Fellowship Program and the Google PhD Fellowship. We would like to thank Hunter Nisonoff, Jacob Gershon, Lucas Arnoldt, and Chenghao Liu for helpful discussions and Francesca-Zhoufan Li for help with the TrpB dataset. We would also like to thank Nate Gruver for help with the NOS implementation, Filippo Stocco for help with the DPO implementation, and Nathaniel Blalock for guidance on how to use the CreiLOV dataset. + +# References + +Sarah Alamdari, Nitya Thakkar, Rianne van den Berg, Neil Tenenholtz, Bob Strome, Alan M. 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Toward AI-Resilient Screening of Nucleic Acid Synthesis Orders: Process, Results, and Recommendations. bioRxiv, December 2024. doi: 10.1101/2024.12.02.626439. URL https://www.biorxiv.org/content/10.1101/2024.12.02.626439. Preprint. +Bruce J. Wittmann, Kadina E. Johnston, Zachary Wu, and Frances H. Arnold. Advances in machine learning for directed evolution. Current Opinion in Structural Biology, 69:11-18, 2021a. ISSN 1879033X. doi: 10.1016/j.sbi.2021.01.008. URL https://doi.org/10.1016/j.sbi.2021.01.008. Publisher: Elsevier Ltd. +Bruce J. Wittmann, Yisong Yue, and Frances H. Arnold. Informed training set design enables efficient machine learning-assisted directed protein evolution. Cell Systems, 12(11):1026-1045.e7, 2021b. ISSN 24054712. doi: 10.1016/j.cels.2021.07.008. URL https://doi.org/10.1016/j.cels.2021.07.008. Publisher: Elsevier Inc. +Dongxia Wu, Nikki Lijing Kuang, Ruijia Niu, Yi-An Ma, and Rose Yu. Diffusion-BBO: Diffusion-Based Inverse Modeling for Online Black-Box Optimization, 2025. URL https://arxiv.org/abs/2407.00610. arXiv preprint; also presented as a poster at NeurIPS 2024 Workshop on Bayesian Decision-making and Uncertainty. +Luhuan Wu, Brian L. Trippe, Christian A. Naesseth, David M. Blei, and John P. Cunningham. Practical and asymptotically exact conditional sampling in diffusion models, 2024. URL https://arxiv.org/abs/2306.17775. +Zachary Wu, S. B. Jennifer Kan, Russell D. Lewis, Bruce J. Wittmann, and Frances H. Arnold. Machine learning-assisted directed protein evolution with combinatorial libraries. Proceedings of the National Academy of Sciences, 116(18):8852-8858, April 2019. ISSN 0027-8424, 1091-6490. doi: 10.1073/pnas.1901979116. URL http://www.pnas.org/lookup/doi/10.1073/pnas.1901979116. + +Zachary Wu, Kadina E. Johnston, Frances H. Arnold, and Kevin K. Yang. Protein sequence design with deep generative models. Current Opinion in Chemical Biology, 65:18-27, 2021. ISSN 13675931. doi: 10.1016/j.cbpa.2021.04.004. URL http://arxiv.org/abs/2104.04457%0Ahttp://dx.doi.org/10.1016/j.cbpa.2021.04.004. arXiv: 2104.04457 Publisher: Elsevier Ltd. +Junhao Xiong, Hunter Nisonoff, Ishan Gaur, Maria E. Lukarska, Luke M. Oltrogge, David F. Savage, and Jennifer Listgarten. Guide your favorite protein sequence generative model, May 2025. URL: https://arxiv.org/abs/2505.04823.arXiv:2505.04823. +Jason Yang, Francesca-Zhoufan Li, and Frances H. Arnold. Opportunities and Challenges for Machine Learning-Assisted Enzyme Engineering. ACS Central Science, 10(2):226-241, February 2024. ISSN 2374-7943, 2374-7951. doi: 10.1021/acscentsci.3c01275. URL https://pubs.acs.org/doi/10.1021/acscentsci.3c01275. +Jason Yang, Aadyot Bhatnagar, Jeffrey A. Ruffolo, and Ali Madani. Function-guided conditional generation using protein language models with adapters. arXiv, June 2025a. doi: 10.48550/arXiv.2410.03634. URL https://arxiv.org/abs/2410.03634.arXiv:2410.03634 [q-bio.BM], v2, last revised 2025-06-11. +Jason Yang, Ravi G. Lal, James C. Bowden, Raul Astudillo, Mikhail A. Hameedi, Sukhvinder Kaur, Matthew Hill, Yisong Yue, and Frances H. Arnold. Active learning-assisted directed evolution. Nature Communications, 16(1):714, January 2025b. ISSN 2041-1723. doi: 10.1038/s41467-025-55987-8. URL https://www.nature.com/articles/s41467-025-55987-8. Publisher: Nature Publishing Group. +Jason Yang, Francesca-Zhoufan Li, Yueming Long, and Frances H. Arnold. Illuminating the universe of enzyme catalysis in the era of artificial intelligence. Cell Systems, pp. 101372, August 2025c. ISSN 2405-4712. doi: 10.1016/j.cels.2025.101372. URL https://www.sciencedirect.com/science/article/pii/S2405471225002054. +Kevin Yang and Dan Klein. FUDGE: Controlled Text Generation With Future Discriminators. In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 3511-3535, 2021. doi: 10.18653/v1/2021.naacl-main.276. URL http://arxiv.org/abs/2104.05218.arXiv:2104.05218 [cs]. +Kevin K. Yang, Zachary Wu, and Frances H. Arnold. Machine-learning-guided directed evolution for protein engineering. Nature Methods, 16(8):687-694, 2019. ISSN 15487105. doi: 10.1038/s41592-019-0496-6. URL http://dx.doi.org/10.1038/s41592-019-0496-6. arXiv:1811.10775 Publisher: Springer US. +Bingliang Zhang, Wenda Chu, Julius Berner, Chenlin Meng, Anima Anandkumar, and Yang Song. Improving diffusion inverse problem solving with decoupled noise annealing. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, June 2025. doi: 10.1109/CVPR52734.2025.01946. URL https://arxiv.org/abs/2407.01521. CVPR 2025; arXiv:2407.01521. +Junming Zhao, Chao Zhang, and Yunan Luo. Contrastive Fitness Learning: Reprogramming Protein Language Models for Low-N Learning of Protein Fitness Landscape. preprint, Bioinformatics, February 2024a. URL http://biorxiv.org/lookup/doi/10.1101/2024.02.11.579859. +Stephen Zhao, Rob Brekelmans, Alireza Makhzani, and Roger Baker Grosse. Probabilistic Inference in Language Models via Twisted Sequential Monte Carlo. In Ruslan Salakhutdinov, Zico Kolter, Katherine Heller, Adrian Weller, Nuria Oliver, Jonathan Scarlett, and Felix Berkenkamp (eds.), Proceedings of the 41st International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, pp. 60704-60748. PMLR, July 2024b. URL https://proceedings.mlr.press/v235/zhao24c.htm1. +Hongkai Zheng, Wenda Chu, Bingliang Zhang, Zihui Wu, Austin Wang, Berthy T. Feng, Caifeng Zou, Yu Sun, Nikola Borislavov Kovachki, Zachary E. Ross, Katherine L. Bouman, and Yisong Yue. Inversebench: Benchmarking plug-and-play diffusion priors for inverse problems in physical sciences. In Proceedings of the Thirteenth International Conference on Learning Representations (ICLR 2025), Singapore, may 2025. OpenReview. URL https://openreview.net/forum?id=U3PBITXNG6. ICLR 2025 Spotlight. + +# NeurIPS Paper Checklist + +The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit. + +Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist: + +- You should answer [Yes], [No], or [NA]. +- [NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available. +- Please provide a short (1–2 sentence) justification right after your answer (even for NA). + +The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper. + +The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes]" is generally preferable to "[No]", it is perfectly acceptable to answer "[No]" provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No]" or "[NA]" is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found. + +IMPORTANT, please: + +- Delete this instruction block, but keep the section heading "NeurIPS Paper Checklist", +- Keep the checklist subsection headings, questions/answers and guidelines below. +- Do not modify the questions and only use the provided macros for your answers. + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract summarizes the claims that we explain further in the Results and the Discussion. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: A detailed discussion of the limitations is provided in the Discussion section. + +# Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. + +# 3. Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +Answer: [Yes] + +Justification: All proofs have cited their sources. + +Guidelines: + +- The answer NA means that the paper does not include theoretical results. +- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced. +- All assumptions should be clearly stated or referenced in the statement of any theorems. +- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. +- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. +- Theorems and Lemmas that the proof relies upon should be properly referenced. + +# 4. Experimental result reproducibility + +Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? + +Answer: [Yes] + +Justification: The methods needed to reproduce the experimental results are provided in the appendix. The code will also be released to the public. + +Guidelines: + +- The answer NA means that the paper does not include experiments. + +- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. + +- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. + +- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. + +- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example + +(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm. +(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully. +(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset). +(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. + +# 5. Open access to data and code + +Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? + +Answer: [Yes] + +Justification: The codebase will be linked in the text upon de-anonymization. + +Guidelines: + +- The answer NA means that paper does not include experiments requiring code. +- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details. +- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark). +- The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details. +- The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc. +- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why. +- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable). + +- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. + +# 6. Experimental setting/details + +Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results? + +Answer: [Yes] + +Justification: These details are provided in the appendix methods section. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. +- The full details can be provided either with the code, in appendix, or as supplemental material. + +# 7. Experiment statistical significance + +Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? + +Answer: [Yes] + +Justification: Standard deviations between multiple repetitions of the same experiment are provided in relevant figures. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. +- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). +- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) +- The assumptions made should be given (e.g., Normally distributed errors). +- It should be clear whether the error bar is the standard deviation or the standard error of the mean. +- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a $96\%$ CI, if the hypothesis of Normality of errors is not verified. +- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). +- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. + +# 8. Experiments compute resources + +Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? + +Answer: [Yes] + +Justification: The compute resources used to perform the experiments is provided. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. + +- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. +- The paper should disclose whether the full research project required more computer than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). + +# 9. Code of ethics + +Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines? + +Answer: [Yes] + +Justification: Yes, the research conforms with the NeurIPS Code of Ethics. + +Guidelines: + +The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics. +- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. +- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). + +# 10. Broader impacts + +Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? + +Answer: [Yes] + +Justification: We have cautioned the potential for negative societal impacts in the discussion. + +Guidelines: + +- The answer NA means that there is no societal impact of the work performed. +- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. +- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. +- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. +- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. +- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). + +# 11. Safeguards + +Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? + +Answer: [NA] + +Justification: Our models only apply to designing proteins with minimal risk of negative impacts. + +Guidelines: + +- The answer NA means that the paper poses no such risks. +- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. +- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. +- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort. + +# 12. Licenses for existing assets + +Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? + +Answer: [Yes] + +Justification: The assets used comply with their corresponding licenses. + +Guidelines: + +- The answer NA means that the paper does not use existing assets. +- The authors should cite the original paper that produced the code package or dataset. +- The authors should state which version of the asset is used and, if possible, include a URL. +- The name of the license (e.g., CC-BY 4.0) should be included for each asset. +- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. +- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. +- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. +- If this information is not available online, the authors are encouraged to reach out to the asset's creators. + +# 13. New assets + +Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? + +Answer: [Yes] + +Justification: The license will be posted with the code when it is released. + +Guidelines: + +- The answer NA means that the paper does not release new assets. +- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. +- The paper should discuss whether and how consent was obtained from people whose asset is used. +- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file. + +# 14. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: LLMs were not a part of the core methods of the research. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Appendix + +# A.1 Data for pretraining generative priors + +The first step in our pipeline involves learning a generative prior on naturally occurring protein sequences to capture the distribution of those with high evolutionary likelihood. This prior is unconditional in the sense that no labeled fitness data is used for training. However, because we are optimizing protein variants for a desired fitness, we pretrained our generative prior on sequences homologous to the parent protein to be optimized (known as a multiple sequence alignment or MSA): TrpB, CreiLOV, or GB1. Likelihoods from MSAs have been captured by statistical models and have been shown to offer good zero-shot approximations of fitness. In other words, they capture mutational substitutions that are more favorable, based on the precedent of natural evolution. + +We focused on the TrpB (Johnston et al., 2024) and CreiLOV (Chen et al., 2023c) datasets due to the extensive number of sequences in their MSAs and compared to GB1 (Olson et al., 2014), which has comparatively fewer sequences. MSAs were obtained by running jackhammer (Johnson et al., 2010) against Uniref90 for two iterations with the parent sequence of the fitness dataset as target. For the MSA, we only used sequences where the aligned portion was at least $75\%$ the length of the parent sequence. We used the MSA that was aligned to the parent sequence, with gap tokens replaced by the corresponding amino acid found in the parent sequence, resulting in full, fixed-length pseudo-natural sequences. For GB1, we augmented the training set with synthetic data, namely all proteins with a single mutation to sequences in the MSA. For the language models on TrpB and CreiLOV, some sequences were randomly mutated by a single position near the beginning of the sequence, to prevent mode collapse during autoregressive generation. + +We performed sequence clustering using mmseqs2 (Steinegger & Söding, 2017) at $80\%$ identity and resampled the dataset by weighting each sample with $\frac{1}{1 + \ln(n)}$ relative probability of being sampled, where $n$ is the size of the cluster associated with that sequence. Afterward, we removed $5\%$ of the clusters and their associated sequences as a validation set. + +# A.2 Protein fitness optimization task + +An oracle as a proxy for protein fitness. We studied fitness optimization across three different protein-fitness datasets, TrpB, CreiLOV, and GB1 (Table 2). TrpB is 389 residues in length, but based on available fitness data, we limited design to 15 residues: 117, 118, 119, 162, 166, 182, 183, 184, 185, 186, 227, 228, 230, 231, and 301. Namely, we combined the fitness data from 6 combinatorially complete 3-site libraries (D-I from Johnston et al. (2024)) and the 4-site library across residues 183, 184, 227, and 228. We normalized the parent fitness to 1 in each dataset and rounded all negative fitness values up to zero. The fitness here is the catalytic rate of a native reaction, the formation of tryptophan from indole and serine. To obtain a proxy fitness for all variants in the design space ( $20^{15}$ possibilities) we trained an oracle inspired by the dataset splitting and model architecture used in Blalock et al. (2025). Namely, we used all of the single, double, and triple mutants in the library for training, with $10\%$ and $20\%$ of the quadruple mutants being used for validation and testing, respectively. Our model consists of an ensemble of 20 MLPs for TrpB, and each was trained on one-hot encodings of the designed residues for 1000 epochs. + +Differently, the CreiLOV dataset (length $N = 119$ ) contains experimental fitnesses for all single mutations in the protein and certain higher order mutations at 15 selected positions with beneficial single mutations. Fitness here refers to associated fluorescence. To obtain a proxy fitness for all variants in the design space ( $20^{119}$ possibilities), we trained an oracle similar to the procedure above, using similar splits to those in Blalock et al. (2025) and were able to reproduce their high performance on the test set. Before model training, we scaled the fitnesses of the single mutants to the fitnesses of multi-mutants by adding a normalization factor to all single mutants such that the parent sequence in both datasets had the same fitness. Our model consists of an ensemble of 10 MLPs for CreiLOV, and each was trained on onehot encodings of sequences for 1000 epochs. + +For GB1, the experimental finesse for nearly all double mutations across the entire protein were available, where fitness refers to binding affinity of a domain of the G protein. To train the oracle, we held out $10\%$ and $20\%$ the sequences with two mutations as a validation and test set, respectively, with remaining sequences being used for training. Our model consists of an ensemble of 10 MLPs for GB1, and each was trained on one-hot encodings of the designed residues for 50 epochs. + +Our oracles show high Pearson correlation on the train and test sets (Fig. A1). As the generalization ability of our oracle was only been tested on variants that are similar to the parent, we penalized the fitness of protein sequences by a factor of 0.99 for every mutation accumulated beyond a threshold of $60\%$ sequence identity to the parent sequence. From here forth, we treated ground truth fitness as outputs from the oracle. + +![](images/3b3c5d4335bb71f1cdb874479dbae9b88a243297171fdb0025988bf3eadec044.jpg) +Figure A1: Oracles trained on available labeled fitness data for TrpB, CreiLOV, and GB1 extrapolate well to higher order combinations of mutations within the design space, as measured by Pearson correlation. + +![](images/e37ae3b466aa2a5debdc367e6d47f4650a744d72d074efc25c403315564ec1e2.jpg) + +![](images/02b0b9d6e40e3c0bec0314b235403e579ec3396f8d5d373832504876f2d7272e.jpg) + +Processing generated sequences. Our primary method for evaluation involved examining the distribution of sampled sequences and their corresponding fitness values, diversities, and novelties. The processing pipeline for generated sequences is shown in Fig. A2. In diffusion models, sequences were generated with fixed length equal to the parent length. For the language models, nearly all generated sequences had length equal to the parent sequence length. Still, sequences were aligned with the parent sequence using mafft (Katoh & Standley, 2013), and gaps were replaced with the corresponding amino acid in the parent sequence to generate complete pseudo-sequences of a fixed length. Special tokens, which occurred rarely in generation, were replaced by a random amino acid. For TrpB, residues outside of the design space of 15 residues were naively mapped to the original amino acid type in the parent sequence at the end of generation. We did not test inpainting, although this could be accomplished with masked (diffusion) language models. + +![](images/da592be653541ebd5beabe400bd496fe6f837e4956a1f31b47d02fa317703bca.jpg) +Figure A2: Example pipeline for generating protein sequences for evaluation, based on a hypothetical parent sequence: MKKFG...SQRFD (length=100), with 8 residues being optimized (3, 4, 26, 27, 28, 29, 98, 99), corresponding to a design space combo of KFDEACRF. + +Comparison to existing protein engineering methods. There are several reasons why we did not directly compare the performance of SGPO methods to existing methods used in protein engineering, such as directed evolution and MLDE. In the case of directed evolution (such as random mutagenesis): (1) It is not obvious which parent sequences to use as the starting points for directed evolution for a fair comparison. (2) It is unclear if the oracle captures the true nature of the protein fitness landscape or extrapolates well to sequences with many mutations relative to the original fitness dataset from which the oracle was trained. (3) Overall, our method enables the accumulation of many mutations in a single round of experimentation, whereas directed evolution is largely limited to one mutation at a time. For example, on the CreiLOV dataset, the generated sequences with the highest fitness had on average 66 mutations from the parent reference sequence from which the original dataset was + +generated, which would not be achievable with directed evolution. We also did not directly compare our method to supervised approaches in smaller design spaces, such as 4-site combinatorial libraries (Yang et al., 2025b), as we focus here on design in larger design spaces, where existing methods are lacking. Overall, traversing large swaths of sequence space will be important for faster engineering and enabling improvements to fitness that would normally be slow with directed evolution. + +# A.3 Generative models for sequences + +Table A1: Summary of training details for generative priors in this work. Reference refers to the codebase that was modified for our implementation and where the model architecture was adapted from. For all models, we retained the model with the lowest validation loss. When using the ESM encoder, we used the 35M-parameter ESM2 model Lin et al. (2023). + +
ModelMax EpochsLearning RateBatch SizeWarmup StepsNoise ScheduleDiffusion TimestepsModel ArchitectureReference
Continuous 51×10-46410cosine500BERTGruver et al. (2023)
Continuous-25 ESM1×10-46410cosine500BERTGruver et al. (2023)
D3PM-Baseline51×10-46410Sohl-Dickstein500ByteNetAlamdari et al. (2024)
D3PM51×10-46410Sohl-Dickstein500ByteNetAlamdari et al. (2024)
UDLM53×10-5642500loglinear500DiTSchiff et al. (2024)
MDLM503×10-4642500loglinear500DiTSchiff et al. (2024)
ARLM101×10-43210n/an/aGPT-JNijkamp et al. (2023)
+ +# A.3.1 Diffusion over continuous space + +Diffusion models construct samples by reversing a diffusion process that maps clean data points $\mathbf{x}_0$ to samples from a prior distribution $\pi(\mathbf{x})$ . The forward process $(\mathbf{x}_0 \rightarrow \mathbf{x}_T)$ is composed of conditional distributions $p(\mathbf{x}_t | \mathbf{x}_{t-1})$ , which admit closed-form expressions for the conditional distributions $p(\mathbf{x}_t | \mathbf{x}_0)$ and $p(\mathbf{x}_{t-1} | \mathbf{x}_t, \mathbf{x}_0)$ . The reverse process $(\mathbf{x}_T \rightarrow \mathbf{x}_0)$ converts samples from the prior into samples from the learned data distribution $p_\theta(\mathbf{x}_0)$ by repeatedly predicting the denoised variable $\hat{\mathbf{x}}_0$ from noisy values $\mathbf{x}_t$ , using the conditional distribution $p(\mathbf{x}_{t-1} | \mathbf{x}_t, \hat{\mathbf{x}}_0)$ to derive a transition distribution $p_\theta(\mathbf{x}_{t-1} | \mathbf{x}_t)$ . + +Continuous noise forward process. Similarly to Gruver et al. (2023), we define a protein sequence as $\mathbf{w} \in \mathcal{A}^L$ , where $\mathcal{A}$ is the alphabet of amino acids and $L$ is the fixed length of the sequence. To learn a distribution $p(\mathbf{w})$ , we first embed $\mathbf{w}$ into a continuous variable $\mathbf{x}_0$ using an embedding matrix $U_{\theta}$ or encoder from the ESM2 language model (Lin et al., 2023), transforming discrete tokens into a continuous latent space. Gaussian noise is then applied to this embedding space. The prior distribution is defined as: + +$$ +\pi (\mathbf {x}) = \mathcal {N} (0, I), \tag {1} +$$ + +while the forward process follows a Gaussian corruption schedule: + +$$ +p \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {0}\right) = \mathcal {N} \left(\sqrt {\bar {\alpha} _ {t}} \mathbf {x} _ {0}, (1 - \bar {\alpha} _ {t}) I\right), \quad \bar {\alpha} _ {t} = \prod_ {i = 1} ^ {t} \alpha_ {i}, \quad \alpha_ {t} = 1 - \beta_ {t}. \tag {2} +$$ + +The variance schedule $\{\beta_t\}$ follows the cosine schedule proposed by Nichol & Dhariwal (2021), which is commonly used to stabilize training. + +Reverse process. The reverse process aims to recover the original sequence by learning a function $p_{\theta}(\hat{\mathbf{w}}|\mathbf{x}_t,t)$ that predicts the sequence from noised points $\mathbf{x}_t$ . This is done by minimizing the following objective: + +$$ +L (\theta) = \mathbb {E} _ {\mathbf {w} _ {0}, t} \left[ - \log p _ {\theta} \left(\mathbf {w} _ {0} \mid \mathbf {x} _ {t}\right) \right], \quad \mathbf {x} _ {t} \sim p \left(\mathbf {x} _ {t} \mid \mathbf {x} _ {0} = U _ {\theta} \mathbf {w} _ {0}\right). \tag {3} +$$ + +By learning $p_{\theta}(\hat{\mathbf{w}}|\mathbf{x}_t,t)$ , we construct the reverse transition distribution: + +$$ +p _ {\theta} \left(\mathbf {x} _ {t - 1} \mid \mathbf {x} _ {t}\right) = \sum_ {\hat {\mathbf {w}}} p \left(\mathbf {x} _ {t - 1} \mid \mathbf {x} _ {t}, \hat {\mathbf {x}} _ {0} = U _ {\theta} \hat {\mathbf {w}}\right) p _ {\theta} \left(\hat {\mathbf {w}} \mid \mathbf {x} _ {t}, t\right), \tag {4} +$$ + +where the posterior $p(\mathbf{x}_{t - 1}|\mathbf{x}_t,\mathbf{x}_0)$ follows: + +$$ +p \left(\mathbf {x} _ {t - 1} \mid \mathbf {x} _ {t}, \mathbf {x} _ {0}\right) = \mathcal {N} \left(\mathbf {x} _ {t - 1}; \mu_ {t}, \sigma_ {t} ^ {2} I\right), \tag {5} +$$ + +with mean $\mu_t$ and variance $\sigma_t^2$ given by: + +$$ +\mu_ {t} = \frac {\sqrt {\bar {\alpha} _ {t - 1}} \beta_ {t}}{1 - \bar {\alpha} _ {t}} \mathbf {x} _ {0} + \frac {\sqrt {\alpha_ {t}} (1 - \bar {\alpha} _ {t - 1})}{1 - \bar {\alpha} _ {t}} \mathbf {x} _ {t}, \tag {6} +$$ + +$$ +\sigma_ {t} ^ {2} = \frac {1 - \bar {\alpha} _ {t - 1}}{1 - \bar {\alpha} _ {t}} \beta_ {t}. \tag {7} +$$ + +Inference and sampling. At inference time, the learned reverse process is used to generate protein sequences from the prior $\pi (x)$ . This is done by iteratively sampling: + +$$ +\mathbf {x} _ {t - 1} \sim p _ {\theta} \left(\mathbf {x} _ {t - 1} \mid \mathbf {x} _ {t}\right), \tag {8} +$$ + +and then reconstructing $\mathbf{w}$ by sampling: + +$$ +\mathbf {w} \sim p _ {\theta} (\hat {\mathbf {w}} | \mathbf {x} _ {0}). \tag {9} +$$ + +This denoising process iteratively refines noisy embeddings back into structured sequences. + +# A.3.2 Diffusion over discrete space. + +Discrete diffusion models (Austin et al., 2021; Campbell et al., 2022; Lou et al., 2024) generate data in discrete spaces by reversing a predefined forward Markov process. Specifically, a family of distributions $p_t$ evolves according to the Markov chain + +$$ +\frac {\mathrm {d} p _ {t}}{\mathrm {d} t} = \boldsymbol {Q} _ {t} p _ {t}, \tag {10} +$$ + +where $p_0 = p_{\mathrm{data}}$ is the data distribution and $\mathbf{Q}_t \in \mathbb{R}^{N \times N}$ are predefined transition matrices. + +This Markov process can be reversed with the help of a concrete score function, $s(\mathbf{x}, t) \coloneqq \left[\frac{p_t(\tilde{\mathbf{x}})}{p_t(\mathbf{x})}\right] \tilde{\mathbf{x}} \neq \mathbf{x}$ , as its time reversal is given by + +$$ +\frac {\mathrm {d} p _ {T - t}}{\mathrm {d} t} = \bar {Q} _ {T - t} p _ {T - t}, \tag {11} +$$ + +where $\bar{\mathbf{Q}}_t[\tilde{\mathbf{x}},\mathbf{x}] = s(\mathbf{x},t)_{\tilde{\mathbf{x}}}Q_t[\mathbf{x},\tilde{\mathbf{x}}]$ for $\tilde{\mathbf{x}}\neq \mathbf{x}$ , and $\bar{\mathbf{Q}}_t[\mathbf{x},\mathbf{x}] = -\sum_{\tilde{\mathbf{x}}\neq \mathbf{x}}\bar{\mathbf{Q}}_t[\tilde{\mathbf{x}},\mathbf{x}]$ . To generate data $\mathbf{x}_0\sim p_{\mathrm{data}}$ , we start with sampling $\mathbf{x}_T$ from a uniform distribution and then evolve through Eq. 11 by the Euler method. + +Uniform discrete language models. Both D3PM (Austin et al., 2021) and UDLM (Schiff et al., 2024) implement a uniform transition matrix $\mathbf{Q}_t = \frac{1}{N}\mathbf{1}\mathbf{1}^T - \mathbf{I}$ . When $T \to \infty$ , the probability distribution $p_T$ converges to a uniform distribution. + +Masked diffusion language models. Masked diffusion language models (MDLM) (Sahoo et al., 2024) utilize an absorbing transition matrix $Q_{t}$ that converts tokens in a sequence to [MASK] states. The corresponding transition matrix can be written as $Q_{t} \in \mathbb{R}^{(N + 1)\times (N + 1)}$ , $Q_{t} = -\boldsymbol{I} + \mathbf{e}_{N + 1}\mathbf{1}^{T}$ . When $T \to \infty$ , the limiting distribution $p_{T}$ converges to a completely masked sequence. + +# A.3.3 Autoregressive language models. + +In this work, we finetuned the ProGen2-small decoder-only transformer (151 million parameters) based on the code and parameters used in Yang et al. (2025a). Models were trained based on next token prediction and cross entropy loss. However, we did not use adapter layers, and we did not group batches based on sequence length. During inference from the autoregressive model, we used a temperature of 1 and a Top- $p$ value of 1. + +# A.4 Steering methods + +Table A2: Summary of supervised value functions used to predict fitness in this work, to guide diffusion models. All "classifiers" were trained as regressors to predict fitness. For DAPS methods, only clean data was used for training, whereas other classifiers are trained on clean and noised samples from various timesteps. + +
ModelGuidance StrategyMax EpochsLearning RateBatch SizeArchitectureHidden Dimension
Continuous DiffusionCG1000\( 1 \times 10^{-3} \)1284-layer MLP256
DAPS1000\( 1 \times 10^{-3} \)1284-layer MLP256
Continuous DiffusionNOS100\( 1 \times 10^{-3} \)1281-layer MLP256
Discrete DiffusionCG1000\( 1 \times 10^{-3} \)644-layer MLP64
DAPS200\( 1 \times 10^{-3} \)644-layer MLP64
Discrete DiffusionNOS200\( 3 \times 10^{-4} \)64linear layern/a
+ +# A.4.1 Classifier guidance + +Classifier guidance (Song et al., 2021) is a technique used to steer samples generated by diffusion models toward desired attributes. The primary goal is to sample from a conditional distribution $p(\mathbf{x}|\mathbf{y})$ , where $\mathbf{y}$ is a guiding signal of interest. In continuous space, this can be achieved by replacing the unconditional score function $\nabla_{\mathbf{x}_t}\log p_t(\mathbf{x}_t)$ at time $t$ by a conditional score function, + +$$ +\nabla_ {\mathbf {x} _ {t}} \log p (\mathbf {x} _ {t} | \mathbf {y}) = \nabla_ {\mathbf {x} _ {t}} \log p _ {t} (\mathbf {x} _ {t}) + \nabla_ {\mathbf {x} _ {t}} \log p _ {t} (\mathbf {y} | \mathbf {x} _ {t}) \tag {12} +$$ + +To obtain the conditional score function, one only needs to train a time-dependent predictor, which predicts the probability of $p_t(\mathbf{y}|\mathbf{x}_t)$ given $\mathbf{x}_t$ and time $t$ . + +Continuous guidance. Classifier guidance modifies the reverse diffusion process to steer generated samples toward a desired property, represented by a conditioning variable $y$ . The guided sampling process modifies the update rule for $\mathbf{x}_t$ by incorporating a classifier score $\nabla_{\mathbf{x}_t} \log p(\mathbf{y}|\mathbf{x}_t)$ into the model's learned score function based on the relation in Eq. 12. Following Song et al. (2021), the classifier guidance term modifies the predicted $\hat{\mathbf{x}}_0$ in the denoising process: + +$$ +\hat {\mathbf {x}} _ {0} = \mathbf {x} _ {t} + \sigma^ {2} \left(s _ {\theta} \left(\mathbf {x} _ {t}, t\right) + \nabla_ {\mathbf {x} _ {t}} \log p (\mathbf {y} \mid \mathbf {x} _ {t})\right). \tag {13} +$$ + +Since our diffusion model directly predicts logits rather than the score function $s_{\theta}(\mathbf{x}_t,t)$ , adding classifier guidance requires modifying the predicted $\hat{\mathbf{x}}_0$ . + +Instead of predicting the score function explicitly, our model predicts logits over the vocabulary, from which the denoised representation $\hat{\mathbf{x}}_0$ is obtained. We modify $\hat{\mathbf{x}}_0$ by incorporating classifier gradients as follows: + +- Compute the unmodified $\tilde{\mathbf{x}}_0$ using the model's predicted logits: + +$$ +\tilde {\mathbf {x}} _ {0} = \sum_ {\hat {\mathbf {w}}} p (\hat {\mathbf {w}} | \mathbf {x} _ {t}, t) U _ {\theta} \hat {\mathbf {w}} \tag {14} +$$ + +Table A3: Hyperparameters used to tune the guidance/steering process. The bolded parameter was chosen as the ideal parameter for the iterative "Bayesian optimization" experiment (Fig. 6). Larger guidance parameter typically implements stronger guidance strength. + +
Guidance StrategyHyperparameters
Continuous CG1/β = 64,128,256,512,1024
Discrete CG1/β = 1,2.5,6.25,15.625,39.0625
Continuous DAPS1/β = 0.25,0.5,1,2,4 × 104K = 50Euler method steps = 10Langevin dynamics steps = 100
Discrete DAPS1/β = 16,32,64,128,256K = 50Euler method steps = 20Metropolis Hastings steps = 1000
Continuous NOSλ = 0.1,1,10,100,1000η = 0.5,2,5K = 5,10optimizer = AdaGrad
Discrete NOSλ = 0.1,1,10,100,1000η = 0.5,2,5K = 5,10optimizer = AdaGrad
DPOβ = 0.02,0.1,0.5,2,4lr = 1 × 10-6epochs = 5batch size = 8
+ +where $U_{\theta}$ is the embedding matrix mapping discrete tokens to continuous space. + +- If a time-dependent classifier $f$ is available, compute the classifier guidance term: + +$$ +\nabla_ {\mathbf {x} _ {t}} \log p (\mathbf {y} | \mathbf {x} _ {t}) = \nabla_ {\mathbf {x} _ {t}} f (\mathbf {x} _ {t}, t) / \beta . \tag {15} +$$ + +- Modify $\tilde{\mathbf{x}}_0$ using the classifier gradient: + +$$ +\hat {\mathbf {x}} _ {0} = \tilde {\mathbf {x}} _ {0} + \sigma^ {2} \nabla_ {\mathbf {x} _ {t}} \log p (\mathbf {y} | \mathbf {x} _ {t}). \tag {16} +$$ + +This allows the diffusion model to generate samples that are more likely to satisfy the desired condition $\mathbf{y}$ . + +Further details on training the classifier are provided in Table A2 and Table A3. + +Discrete guidance. Nisonoff et al. (2025) extend classifier guidance to discrete state-space diffusion models. In analogy to classifier guidance for continuous diffusion models, they modify the unconditional rate matrix $\bar{Q}_t$ (as defined in Eq. 11) to be a conditional rate matrix $R_t^y$ with + +$$ +\boldsymbol {R} _ {t} ^ {\mathbf {y}} [ \mathbf {x}, \tilde {\mathbf {x}} ] = \frac {p (\mathbf {y} | \tilde {\mathbf {x}} , t)}{p (\mathbf {y} | \mathbf {x} , t)} \bar {\boldsymbol {Q}} _ {t} [ \mathbf {x}, \tilde {\mathbf {x}} ], \forall \tilde {\mathbf {x}} \neq \mathbf {x}. \tag {17} +$$ + +For classifier guidance on both continuous and discrete diffusion models, we train a time-dependent predictor (classifier) $f$ that predicts the fitness $\mathbf{y}$ given $\mathbf{x}_t$ at time $t$ . We define $p(\mathbf{y}|\mathbf{x}) \propto \exp(f(\mathbf{x}) / \beta)$ , where $f(\cdot)$ is a surrogate predictor of the fitness, and $\beta$ is the guidance temperature and governs the strength of guidance. Therefore, $\nabla_{\mathbf{x}_t} \log p_t(\mathbf{y}|\mathbf{x}_t) = \frac{1}{\beta} \nabla_{\mathbf{x}_t} f(\mathbf{x}_t, t)$ , and $R_t^{\mathbf{y}}[\mathbf{x}, \tilde{\mathbf{x}}] = \exp\left(\frac{1}{\beta} \big(f(\tilde{\mathbf{x}}, t) - f(\tilde{\mathbf{x}}, t)\big)\right) \bar{Q}_t[\mathbf{x}, \tilde{\mathbf{x}}}$ . + +To obtain a classifier $f$ for discrete diffusion models, we trained an MLP regressor to predict the fitness of a one-hot encoded sequence given $\mathbf{x}_t$ and uniformly random time $t \in [0, T]$ . Further details are provided in Table A2 and Table A3. + +# A.4.2 Posterior sampling + +Another line of guidance work (Chung et al., 2023; Mardani et al., 2024; Zhang et al., 2025) focuses on drawing samples from the posterior distribution $p(\mathbf{x}|\mathbf{y}) \propto p(\mathbf{x})p(\mathbf{y}|\mathbf{x})$ , where the prior distribution is modeled by a pretrained diffusion model. The conditional distribution $p(\mathbf{y}|\mathbf{x})$ can either be the likelihood function of a forward model (i.e., when $\mathbf{y}$ is an incomplete measurement of $\mathbf{x}$ ) or an exponential distribution with respect to a reward function (i.e., $p(\mathbf{y}|\mathbf{x}) \propto \exp(f(\mathbf{x}) / \beta)$ ). The major difference between posterior sampling and classifier guidance is that it requires the reward function to be trained only on clean data $\mathbf{x}$ . + +While many works have studied posterior sampling in Euclidean space with continuous diffusion models, posterior sampling for discrete data has been less explored. We modified DAPS (Zhang et al., 2025) to enable diffusion posterior sampling in discrete-state spaces. Suppose $\mathbf{x}$ lies in a finite support $\mathcal{X}^D$ , we follow the following steps: + +- Initialize $\mathbf{x}_T \sim p_T(\mathbf{x}_T)$ +for $i = 1,\dots ,K$ + +1. Sample $\hat{\mathbf{x}}_0^{(i)}\sim p(\mathbf{x}_0|\mathbf{x}_{t_{i - 1}})$ by a discrete diffusion model. +2. Run Metropolis Hastings to sample $\mathbf{x}_0^{(i)}\sim p(\mathbf{x}_0|\mathbf{x}_{t_{i - 1}},\mathbf{y})$ as defined in Eq. 18. +3. Sample $\mathbf{x}_{t_i} \sim p(\mathbf{x}_{t_i}|\mathbf{x}_0)$ following the forward Markov process. + +- Return $\mathbf{x}_K$ . + +Specifically, $t_0, t_1, \ldots, t_K$ are mono-decreasing time steps with $t_0 = T$ and $t_K \approx 0$ . $p(\mathbf{x}_0|\mathbf{x}_t,\mathbf{y})$ is defined as + +$$ +\begin{array}{l} p \left(\mathbf {x} _ {0} \mid \mathbf {x} _ {t}, \mathbf {y}\right) \propto p \left(\mathbf {y} \mid \mathbf {x} _ {0}\right) p \left(\mathbf {x} _ {0} \mid \mathbf {x} _ {t}\right) \\ \approx p (\mathbf {y} \mid \mathbf {x} _ {0}) \exp (- \| \mathbf {x} _ {0} - \hat {\mathbf {x}} _ {0} (\mathbf {x} _ {t}) \| _ {0} / \sigma_ {t}), \tag {18} \\ \end{array} +$$ + +where $\hat{\mathbf{x}}_0(\mathbf{x}_t) \sim p(\mathbf{x}_0|\mathbf{x}_t)$ is a point estimate of the conditional distribution, and we approximate $p(\mathbf{x}_0|\mathbf{x}_t)$ by an exponential distribution over Hamming distance. Following Proposition 1 in Zhang et al. (2025), $\hat{\mathbf{x}}_0^{(i)}, \mathbf{x}_0^{(i)}$ , and $\mathbf{x}_{t_i}$ converge to the posterior distribution as $t_i$ goes to 0. + +For posterior sampling with DAPS, we obtained the value function $f$ using the same model architecture and training parameters as classifier guidance but only trained on clean data $\mathbf{x}$ (no noisy $\mathbf{x}_t$ ). We set $K = 50$ using the time scheduler for the original model. Further details are provided in Table A2 and Table A3. + +# A.4.3 NOS + +Diffusion optimized sampling (NOS) (Gruver et al., 2023) is a guidance method for both continuous and discrete diffusion models, which utilizes gradient information of the continuous latent representations of protein sequences. In pretrained discrete diffusion models, noisy sequences $\mathbf{w}_t$ always have a continuous embedding in the form of hidden states of the neural network. Specifically, the denoising model that predicts $\mathbf{w}_0$ from $\mathbf{w}_t$ can be written as $p_{\theta}(\mathbf{w}_0|g(\mathbf{w}_t), t)$ , where $\mathbf{h}_t = g(\mathbf{w}_t)$ is a continuous hidden state of the model. + +Instead of training a value function on discrete sequences $\mathbf{w}_t$ , NOS proposes to train the value function on the hidden states $\mathbf{h}_t$ . In each diffusion step, NOS samples from the posterior distribution, + +$$ +p \left(\mathbf {w} _ {0} \mid \mathbf {h} _ {t}, \mathbf {y}\right) \propto p _ {\theta} \left(\mathbf {w} _ {0} \mid \mathbf {h} _ {t}\right) \exp (f (\mathbf {h} _ {t})). \tag {19} +$$ + +To sample from this distribution, NOS runs Langevin dynamics on $\mathbf{h}_t$ , i.e., + +$$ +\mathbf {h} _ {t} ^ {\prime} \leftarrow \mathbf {h} _ {t} ^ {\prime} - \eta \nabla_ {\mathbf {h} _ {t} ^ {\prime}} \left(\lambda D _ {K L} \left(p _ {\theta} \left(\mathbf {w} _ {0} \mid \mathbf {h} _ {t} ^ {\prime}\right) \| p _ {\theta} \left(\mathbf {w} _ {0} \mid \mathbf {h} _ {t}\right)\right) - f (\mathbf {h} _ {t})\right) + \sqrt {2 \eta \tau} \epsilon , \epsilon \sim \mathcal {N} (0, I). \tag {20} +$$ + +After $K$ iterations, we denoise $\mathbf{w}_t$ following the guided hidden state, i.e., $p(\mathbf{w}_{t - 1}|\mathbf{w}_t,\mathbf{y}) = p_{\theta}(\mathbf{w}_{t - 1}|\mathbf{h}_t',t)$ . + +To train the value function used for guidance in NOS, following the method from Gruver et al. (2023), we trained a very shallow neural network on the final layer hidden embeddings of the diffusion model. Further details are provided in Table A2 and Table A3. + +# A.4.4 Direct preference optimization + +For DPO with language models, we used the weighted loss function from Widatalla et al. (2024) and Stocco et al. (2024) (Eq. 21). $\pi_{\theta}$ is the policy to be updated, $\pi_{\mathrm{ref}}$ is the original model, and $\beta$ is a tunable parameter describing the extent of drift from the reference model. The loss therefore describes the cross entropy of the ratio $\beta \log \frac{\pi_{\theta}(\mathbf{x})}{\pi_{\mathrm{ref}}(\mathbf{x})}$ and the fitness value $w$ . Following Stocco et al. (2024), we calculated the ratio $r$ as the difference of the log likelihood of the sequence from the updated model minus the log likelihood of the reference model, and softmax was applied to all of the fitness values $w$ . We used the default parameters from (Stocco et al., 2024) and tested increasing the learning rate to $10^{-4}$ but found that generation quality broke down above the levels used in Table A3 with finetuning for 5 epochs. We also tested ranked loss with other types of models, but the performance was similar. + +$$ +L _ {\mathrm {D P O} _ {\text {w e i g h t e d}}} \left(\pi_ {\theta}; \pi_ {\text {r e f}}\right) = - \mathbb {E} _ {D} \sum_ {k = 1} ^ {K} w ^ {k} \left[ \beta \log \frac {\pi_ {\theta} (x)}{\pi_ {\text {r e f}} (x)} - \log \sum_ {j = k} ^ {K} \exp \left(\beta \log \frac {\pi_ {\theta} (x)}{\pi_ {\text {r e f}} (x)}\right) \right] \tag {21} +$$ + +# A.5 Adaptive optimization algorithm + +Algorithm 1 Adaptive Optimization with Guided Generative Models +1: Input: Pretrained generative prior $p(\mathbf{x})$ , initial empty labeled dataset $\mathcal{D}_0 = \emptyset$ , number of rounds $T$ , batch size $B$ , ensemble size $M$ +2: for $t = 1$ to $T$ do +3: Initialize batch $\mathcal{X}_t \gets \emptyset$ +4: if $t > 1$ then Train ensemble of value functions $\{f_{\theta_{t,m}}\}_{m=1}^{M}$ on $\mathcal{D}_{t-1}$ +5: while $|\mathcal{X}_t| < B$ do +6: if $t > 1$ then +7: Sample value function $f_\theta \sim \mathrm{Uniform}(\{f_{\theta_{t,m}}\}_{m=1}^{M}) \triangleright$ Thompson-style sampling +8: Sample sequence $\mathbf{x}_b \sim \mathrm{GuidedSample}(p(\mathbf{x}), f_\theta, \mathrm{GuidanceStrategy})$ +9: else +10: Sample sequence $\mathbf{x}_b \sim \mathrm{UnconditionalSample}(p(\mathbf{x}))$ +11: end if +12: if $\mathbf{x}_b \notin \mathcal{D}_{t-1}$ then Add $\mathbf{x}_b$ to batch $\mathcal{X}_t$ +13: end while +14: Evaluate true fitness $y_b = f_{\mathrm{true}}(\mathbf{x}_b)$ for all $\mathbf{x}_b \in \mathcal{X}_t$ +15: Update dataset: $\mathcal{D}_t \gets \mathcal{D}_{t-1} \cup \{(\mathbf{x}_b, y_b)\}_{b=1}^B$ +16: end for +17: Return: Best observed sequence in $\mathcal{D}_T$ + +We used an ensemble size of $M = 10$ models, each trained with a different random initialization of neural network weights. In practice, to speed up sampling, we sampled $(B = 100$ samples)/ $(M = 10$ models) $= 10$ sequences in each GPU batch using the same Thompson-sampled value function, rather than using a GPU batch size of 1. Alternatively, for the Gaussian process model, we trained the model with the radial basis function kernel, and we sub-sampled the total amount of training pairs (when using noisy samples) to 5000 samples. + +# A.6 Latent space Bayesian optimization with + +We utilized the APEXGo codebase (Torres et al., 2024), a package for training generative variational autoencoders over peptide sequences and then optimizing those sequences with latent space Bayesian optimization to maximize certain properties. We used the training code out-of-the-box to train variational autoencoders over the same MSA sequences used to train priors for discrete diffusion models in SGPO. We trained until losses plateaued, to 542, 241, and 391 epochs for TrpB, CreiLOV, and GB1, respectively. Overall, the reconstruction losses were low, and generated sequences had low perplexity and high fitness, comparable to the generative models used in SGPO. We then used this latent space and the APEXGo optimization algorithm to maximize the fitness of sequences as measured by the oracle used in SGPO benchmarking. Specifically, in our configuration, we set + +the number of initialization points to 100, the number of desired diverse solutions to 1, the max number of oracle calls to 800, and the batch size to 100-to mimic the iterative Bayesian optimization experiments performed in Fig. 6. + +# A.7 Additional results + +![](images/9f329c063d1ec44630f5d8c823f20540a7c0893eea4cf99667f45a8f2872bebc.jpg) +Figure A3: The distributions of sequences sampled from pretrained generative priors largely match those of the target distribution. The target distribution shows all sequences in the MSA, and the distributions of generative models are approximated by sampling 1000 sequences. Model definitions can be found in Table 3. The residues shown for TrpB are 4 out of 15 positions studied in the dataset (parent is VFVS), and 5 out of 119 residues for CreiLOV are shown as they correspond to those harboring favorable mutations in the original dataset (parent is AGQRD). Note that the target distribution for training the ARLM is slightly different than that shown here. + +![](images/2ad263fe05dfb7919c37a0c7218fec6042dc78ee3e7b849501224db22124cb1e.jpg) + +![](images/aa9b07628740153650daac703823dc3ecd14f303a9728f2ddbcde3a5c2e0b37f.jpg) + +![](images/b7b59ab1f7115ed72f4efacf3c97bdde1f0ffceb3294fd5019329fbe78e13d31.jpg) +Diversity +Figure A4: Additional results for GB1, corresponding to Fig. 4 and Fig. A3. The 4 positions shown correspond to the parent sequence of VDGV. + +![](images/a4ff9d1c3950143c54c2dedbb8019bdb57783d3eeb2146059a188ce0c2d6e6e4.jpg) +Figure A5: Generated sequences from pretrained priors are more similar to parent than random for (A) TrpB and (B) CreiLOV, measured by the Hamming (or edit) distance. UDLM models exhibit mode collapse onto consensus sequence(s) in the training distribution. The parent sequence refers to the starting sequence used to generate variants in the original protein fitness dataset. + +![](images/c4dc6ae073f7320627ed57983160020019d4d87f3a8979a0482699788db26e37.jpg) + +![](images/ec3180b9c6d43dea59e207b5d3fa07d48acc45f74beddc6bc648476e3aee662b.jpg) + +![](images/1c137ff7b9671c65c914106f1d27f556b00a2085d446152f4dc5073c2311bf97.jpg) + +![](images/c3e4f10e87753d6ea0c19342721e1fbef1e3eeae4eeaef72a3595c13df7e0592.jpg) + +![](images/56aa7a652695a63e8c020997cbe29974a7adea252062d6a546e8ccec85c91ba1.jpg) + +![](images/63402445a9937dba0840df548911173edb66c7b7ec2d8df510b5ee0431fa0e49.jpg) + +![](images/9f79a12fc95e3e24ed4653a4246ccf798a837996439cd3d2509af6b66230d202.jpg) + +![](images/67372a9c7c0e4c063f2cee9ce45f8fd20a2ddafdb0961635b0d028c6e92e716e.jpg) + +![](images/7a2ddd5eb4d7ffe769dc3214a7d73b390a4d099f2185459fb977754aefe471d6.jpg) +Figure A6: Pareto boundaries demonstrate the trade-off between generating sequences with high fitness and high diversity for TrpB (A-C), CreiLOV (D-F), and GB1 (G-I) - showing the same experiment as Fig. 5. Error bars show standard deviation. Mean fitness and diversity were calculated based on 200 generated samples, with diversity calculated as the total number of unique and novel (previously unseen) samples in the generated batch, out of 200. Larger circles indicate a stronger guidance strength, specified in Table. A3. + +![](images/4cdc8176c48d999be323682f61e3265d742031ff41977e4b9aa31e3a416be89e.jpg) + +![](images/7679ca8f1a2f34053440db7ba1399d9db126963970cda083368b13d0f1ca6695.jpg) + +Table A4: Adaptive optimization with an ensemble of 10 value functions and Thompson sampling, compared to using a single model for guidance. Max fitness refers to the mean max fitness achieved at the end of the campaign using the same experimental setup as Fig. 6, over 5 different random initializations. + +
ProteinModelGuidanceMax Fitness (Ensemble)Max Fitness (Single Model)
TrpBD3PMCG1.5511.542
DAPS1.5951.568
MDLMCG1.5511.542
DAPS1.5951.568
CreiLOVD3PMCG5.6085.552
DAPS5.5225.520
MDLMCG5.6085.552
DAPS5.5305.520
+ +![](images/5c01b6aaa9efcab9b9184a1940018426338191686236a8b5822c8df9483ec672.jpg) + +![](images/9ce54d1628f00f504e7d702f9cb5bc318f60279e852cdcc61f6b666f6d8f9282.jpg) + +![](images/278014efa4827c3117b5ac33e0e3732aaa320a354c3a9a3cf54271324231154e.jpg) +Figure A7: Diversity of generated sequences, measured by average Shannon entropy of mutated positions, during each round of guidance. Using an ensemble of value functions and Thompson sampling generally shows higher diversity than using a single model. Experimental setup is the same as Fig. 6, and experiments were repeated over 5 random initializations. + +![](images/f2a156ee94e60ed55f640d2f2bdaabd975024b8616f057e0d9db017b4142a801.jpg) \ No newline at end of file diff --git a/steeringgenerativemodelswithexperimentaldataforproteinfitnessoptimization/images.zip b/steeringgenerativemodelswithexperimentaldataforproteinfitnessoptimization/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..39de35f57c1925dffbef958a81796ef690965d4a --- /dev/null +++ b/steeringgenerativemodelswithexperimentaldataforproteinfitnessoptimization/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:29119774f83bd55233f9c390a7c28051f934d438de6e7f71dcc961c92acfb2cb +size 1327373 diff --git a/steeringgenerativemodelswithexperimentaldataforproteinfitnessoptimization/layout.json b/steeringgenerativemodelswithexperimentaldataforproteinfitnessoptimization/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..42e910e8a0cb595c71cc51f446c79be9ba663baa --- /dev/null +++ b/steeringgenerativemodelswithexperimentaldataforproteinfitnessoptimization/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e792896dbb25af4d90a2bd450ab36ba04c59c4dd254c742cd4dcab951db3856d +size 1116202 diff --git a/steeringinformationutilityinkeyvaluememoryforlanguagemodelposttraining/880f0c8a-58f9-4c3d-a5c3-6c4303b14dd8_content_list.json b/steeringinformationutilityinkeyvaluememoryforlanguagemodelposttraining/880f0c8a-58f9-4c3d-a5c3-6c4303b14dd8_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..226734bf2b52a3b36bec4b7eea293950b0a32d97 --- /dev/null +++ b/steeringinformationutilityinkeyvaluememoryforlanguagemodelposttraining/880f0c8a-58f9-4c3d-a5c3-6c4303b14dd8_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:67fb2381fa330847ac3d37847d2d1d1ee8bd76145734fe84f518aed70f7df52d +size 178232 diff --git a/steeringinformationutilityinkeyvaluememoryforlanguagemodelposttraining/880f0c8a-58f9-4c3d-a5c3-6c4303b14dd8_model.json b/steeringinformationutilityinkeyvaluememoryforlanguagemodelposttraining/880f0c8a-58f9-4c3d-a5c3-6c4303b14dd8_model.json new file mode 100644 index 0000000000000000000000000000000000000000..84ed2f18fa0d63e244e50dc1e122048395da5775 --- /dev/null +++ b/steeringinformationutilityinkeyvaluememoryforlanguagemodelposttraining/880f0c8a-58f9-4c3d-a5c3-6c4303b14dd8_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1cc3e54b8604b424c4b226d19ed7adaa146872fe8fe7159be660d8d10487b5f5 +size 230866 diff --git a/steeringinformationutilityinkeyvaluememoryforlanguagemodelposttraining/880f0c8a-58f9-4c3d-a5c3-6c4303b14dd8_origin.pdf b/steeringinformationutilityinkeyvaluememoryforlanguagemodelposttraining/880f0c8a-58f9-4c3d-a5c3-6c4303b14dd8_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..21314516248fc9b5cccda8b3a633894f92d4d5f5 --- /dev/null +++ b/steeringinformationutilityinkeyvaluememoryforlanguagemodelposttraining/880f0c8a-58f9-4c3d-a5c3-6c4303b14dd8_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:88286c0fec9320e60d11375b1abc65272b4a321d79320c15e11b1d51c0f5951a +size 737123 diff --git a/steeringinformationutilityinkeyvaluememoryforlanguagemodelposttraining/full.md b/steeringinformationutilityinkeyvaluememoryforlanguagemodelposttraining/full.md new file mode 100644 index 0000000000000000000000000000000000000000..96484cc202ac70dd61b229ffe30bb5abb4b3d563 --- /dev/null +++ b/steeringinformationutilityinkeyvaluememoryforlanguagemodelposttraining/full.md @@ -0,0 +1,777 @@ +# Steering Information Utility in Key-Value Memory for Language Model Post-Training + +Chunyuan Deng + +Dept. of Computer Science + +Rice University + +Houston, TX 77005 + +chunyuan.deng@rice.edu + +Ruidi Chang + +Dept. of Computer Science + +Rice University + +Houston, TX 77005 + +ruidi.chang@rice.edu + +Hanjie Chen + +Dept. of Computer Science + +Rice University + +Houston, TX 77005 + +hanjie@rice.edu + +# Abstract + +Recent advancements in language models (LMs) have marked a shift toward the growing importance of post-training. Yet, post-training approaches such as supervised fine-tuning (SFT) do not guarantee the effective use of knowledge acquired during pretraining. We therefore introduce InfoSteer, a lightweight method that encourages parametric information utilization in LMs during post-training. Specifically, InfoSteer treats the feed-forward network (FFN) layer as associate key-value memory and promotes the use of stored memory vectors via forward-pass interventions or regularization during backpropagation. This simple guidance during post-training phase yields consistent performance improvements across diverse model families—including Qwen, Gemma and Llama—spanning 15 downstream tasks in both in-distribution (ID) and out-of-distribution (OOD) evaluations. Beyond performance gains, we also find that steered LMs can adaptively allocate information by placing more emphasis on generating semantically meaningful tokens, while using fewer resources on simple transition ones (e.g., ‘,’ or ‘and’). Our work underscores that vanilla post-training does not fully exploit the potential gained during pre-training, and that steering LMs in latent representation space offers a promising approach to enhance both performance and interpretability.1 + +# 1 Introduction + +The contemporary training pipeline for LMs has standardized around a two-stage process: an initial pre-training phase on extensive, web-scale corpora, followed by a post-training phase utilizing smaller, more curated datasets [Ouyang et al., 2022, Touvron et al., 2023, Bai et al., 2023, Mesnard et al., 2024]. A considerable body of literature suggests that the fundamental capabilities and knowledge of these models are predominantly instilled during the pre-training stage [Chung et al., 2022, Anil et al., 2023, Muennighoff et al., 2025]. Subsequent post-training techniques are commonly viewed as approaches to better refine, elicit, or adapt these inherent capabilities embedded in the base model [Zhou et al., 2023, Rafailov et al., 2024, Guo et al., 2025, Swamy et al., 2025]. + +Despite this, an open question remains: do post-training methods truly encourage the model to fully utilize the information encoded during pre-training? In many cases, they may not-since models are neither explicitly trained nor incentivized to retrieve and apply such knowledge2 optimally [Conmy et al., 2023, Chang et al., 2024, Du et al., 2024a]. This insufficient use could result in suboptimal performance on downstream tasks, even when relevant knowledge is already stored in the model's parameters [Bietti et al., 2023, Kim et al., 2025]. + +![](images/9aebd01c52332f264df6d1bda3a19bb4694613b150583431ad01e6d02979d5b4.jpg) +Figure 1: Overview of our proposed InfoSteer framework. The interpretation of Transformer FFNs as key-value memory was introduced by Geva et al. [2021], further details are provided in §3. + +We therefore introduce InfoSteer, a method designed to encourage more effective use of a model's pretrained knowledge during the post-training phase (Figure 1). Our approach draws on an associative memory view of the Transformer's feed-forward network (FFN) layers [Geva et al., 2021, Meng et al., 2022a,b, Geva et al., 2023], where the first FFN layer acts as a content-dependent key and the second FFN layer functions as a learned value memory, together forming a key-value memory mechanism. Consequently, each layer's output can be viewed as a weighted sum of memory vectors, where each vector's weight (the key coefficient) is produced by the input vector multiplied by its corresponding key in the first FFN.3 + +InfoSteer controls the distribution of key coefficients to manage the information-dense memory vectors learned from pretraining. The goal is to promote a high-entropy key distribution, enabling their corresponding memory vectors to be actively engaged during post-training. Specifically, we use two strategies: (i) intervening on memory vectors with low corresponding key coefficients during the forward pass, and (ii) regularizing the entropy of key coefficients in the gradient flow during backpropagation. These methods can be seamlessly incorporated into a vanilla SFT pipeline for information-steered SFT. + +Our empirical evaluation demonstrates that InfoSteer yields consistent and notable performance enhancements across a variety of models—including Llama [Touvron et al., 2023], Qwen [Bai et al., 2023], and Gemma [Mesnard et al., 2024] at different scales. These improvements are demonstrated across a comprehensive suite of over 15 downstream tasks, encompassing both in-distribution (ID) and out-of-distribution (OOD) scenarios. Beyond these performance gains, information-steered LMs exhibit an adaptive information allocation strategy during token generation. Specifically, they dedicate more representational capacity to semantically rich and challenging tokens, while expending less on simpler, transitional tokens, indicating a more nuanced use of parametric information. + +# 2 Related Work + +Model Steering. Model steering represents an emerging paradigm to guide model behavior, focusing on controlled interventions in the model's latent space [Subramani et al., 2022, Zou et al., 2023, Turner et al., 2024, Li et al., 2024, Cyberey and Evans, 2025]. Many work has demonstrated the effectiveness of steering methods across various dimensions, including disentangling human-interpretable concepts [Rumelhart et al., 1986], such as linguistic features (e.g., gender, number) [Hewitt and Manning, 2019, Lasri et al., 2022, Wang et al., 2022, Hanna et al., 2023, Huang et al., 2024, Chang et al., 2025] and logical reasoning [Wu et al., 2024b, Xie et al., 2025]. These approaches typically operate by introducing auxiliary objectives, modifying activation patterns, or implementing controlled perturbations during forward or backward passes [Rimsky et al., 2023, Scalena et al., 2024, Stolfo et al., 2024, Luo et al., 2025, Gur-Arieh et al., 2025, Bartoszcze et al., 2025]. The key advantage of model steering lies in its ability to leverage the rich knowledge already encoded in pretrained weights while directing how this information is accessed and applied to downstream tasks [Geiger et al., 2023, Lee et al., 2024, Siddique et al., 2025, Soo et al., 2025, Deng et al., 2025]. Our information-steered approach extends this paradigm by targeting the key distribution in transformer FFN layers, which influences which memory vectors are activated during computation. + +Parametric Information. From an information-theoretic perspective, quantifying the information encoded in an LM's high-dimensional parameter space remains a challenging problem [Achille et al., + +2020, Bernstein and Yue, 2021]. This challenge is closely tied to the ongoing discourse on parametric versus non-parametric knowledge storage in LMs [Tay et al., 2022, Ferrando et al., 2022, 2023, Xie et al., 2023, Deng et al., 2024, Ferrando and Voita, 2024, Du et al., 2024b] (or token-mixing in attention layer vs. channel mixing in FFN layer). The concept of channel mixing stems from the perspective of viewing the FFN layer as a key-value memory structure [Weston et al., 2014, Geva et al., 2021, Qiu et al., 2024, Kim et al., 2025]. Empirical evidence supports this interpretation, showing that individual neurons in FFN layers activate in response to specific semantic concepts and linguistic patterns [Dai et al., 2021, Liu et al., 2023, Niu et al., 2024]. Furthermore, intervention studies have demonstrated that targeted modifications to these memory vectors can directly influence model outputs on knowledge-intensive tasks [Meng et al., 2022a,b, Geva et al., 2023, Hase et al., 2023, Yao et al., 2024, Wang et al., 2024], suggesting that factual information is strongly correlated with in this component. + +# 3 Transformer Feed-Forward Layers as Key-Value Memories + +We consider a standard autoregressive Transformer [Vaswani et al., 2017], which models the conditional probability of a sequence $x = (x_{1},\ldots ,x_{T})$ as + +$$ +p (x) = \prod_ {t = 1} ^ {T} p \left(x _ {t} \mid x _ {< t}\right), \tag {1} +$$ + +using a stack of $L$ Transformer decoder blocks. Each block consists of two primary components: a masked self-attention layer and a position-wise feed-forward network (FFN). + +In standard Transformer architectures, the position-wise FFN in each decoder block is given by: + +$$ +\operatorname {F F N} (X) = \sigma \left(X W _ {u p} + b _ {1}\right) W _ {\text {d o w n}} + b _ {2}, \tag {2} +$$ + +where $X \in \mathbb{R}^{T \times d}$ is the input representation matrix for FFN, $W_{up} \in \mathbb{R}^{d \times d_m}$ and $W_{down} \in \mathbb{R}^{d_m \times d}$ are the up-projection and down-projection matrix respectively, and $\sigma$ is a nonlinearity activation function. Bias term $b_1$ and $b_2$ are included in the standard FFN formulation but omitted in the following equations for simplicity. + +Let $h \in \mathbb{R}^d$ represent the embedding as a single token representation from the input matrix $X$ . As illustrated in Figure 2, this structure can be interpreted as a soft key-value memory [Weston et al., 2014, Geva et al., 2021], where the intermediate activation $\sigma(hW_{\mathrm{up}}) \in \mathbb{R}^{d_m}$ defines the soft addressing weights, and each row of $W_{\mathrm{down}}$ serves as a value vector. The FFN output of the input $h$ can be written as a weighted sum over value rows: + +$$ +\operatorname {F F N} (h) = \sum_ {i = 1} ^ {d _ {m}} \underbrace {\sigma \left(h W _ {\mathrm {u p}}\right) _ {i}} _ {\text {k e y c o e f f i c i e n t}} \cdot \underbrace {\left(W _ {\mathrm {d o w n}}\right) _ {i , :}} _ {\text {v a l u e v e c t o r}} = k _ {1} \mathbf {v} _ {1} + k _ {2} \mathbf {v} _ {2} + \dots + k _ {d _ {m}} \mathbf {v} _ {d _ {m}} \tag {3} +$$ + +where each $k_{i}\in \mathbb{R}$ is a scalar key coefficient, and each $\mathbf{v}_i\in \mathbb{R}^d$ is a value vector (memory vector). The key coefficients control the magnitude of the memory vectors' contribution to the final prediction. + +# 4 InfoSteer + +LMs are commonly interpreted to store parametric knowledge within their dense layers, particularly in the FFN blocks [Meng et al., 2022a, Geva et al., 2023, Kim et al., 2025]. However, during post-training, there is no explicit guidance to encourage the model to utilize this knowledge for new alignment or retrieval tasks. Therefore, we propose InfoSteer to bridge this gap. + +# 4.1 Motivation & Desierata + +To steer the information utility of pretrained LMs, we aim to enhance the engagement of memory vectors during post-training. The core idea is to control the distribution of key coefficients in the position-wise FFN (see Figure 2). Specifically, we observe a dominant property of associative memory: if a key coefficient $k_{i}$ is significantly larger than another key coefficient $k_{j}$ , the final prediction predominantly relies on the corresponding memory vector $\mathbf{v}_{i}$ , while $\mathbf{v}_{j}$ is underutilized or overlooked. Our goal is to achieve two objectives: + +![](images/2c62461a9024ca5cd9ef27de05ceeea8c8efd3fad8b2a3da9a0a30e347f53d7f.jpg) +Figure 2: Key Design of InfoSteer. (a) An illustration of viewing the Transformer FFN as a key-value memory. The key acts as a control mechanism that determines the extent to which each memory vector is engaged. (b) and (c) Two general methods used to modulate the distribution of key coefficients, thereby encouraging the engagement of memory vectors during post-training. + +1. Minimize the language modeling loss: Ensure the FFN output $\mathrm{FFN}(h) = \sum_{i=1}^{d_m} k_i \mathbf{v}_i$ contributes to accurate predictions. +2. Maximize the entropy of the key distribution: Encourage diverse engagement of memory vectors by promoting a higher-entropy key coefficient distribution. + +Promoting entropy weaken the dominance of individual coefficients, encouraging more balanced activation of memory vectors $\mathbf{v}_i$ and thus enabling richer parametric knowledge retrieval. + +# 4.2 Generic Methods: Intervention and Regularization + +Formally, let $\mathbf{k}^{(l)} = [k_1^{(l)}, k_2^{(l)}, \ldots, k_{d_m}^{(l)}] \in \mathbb{R}^{d_m}$ denote the key coefficients at layer $l$ , where $k_i = \sigma(hW_{\mathrm{up}})_i$ and $W_{\mathrm{up}} \in \mathbb{R}^{d \times d_m}$ is the up-projection matrix. We propose two complementary methods to increase memory vector engagement: intervention and regularization. + +**Intervention Method.** The intervention method directly modifies key coefficients to promote broader memory vector utilization. For each layer $l$ , we identify the $p\%$ of key coefficients with the smallest value and adjust them to a value proportional to the layer's average key coefficient. + +Formally, let $\mathcal{I}^{(l)}$ denote the set of indices corresponding to the $p\%$ of elements in $\mathbf{k}^{(l)}$ with smallest value. For a hyperparameter $\alpha > 0$ , we update: + +$$ +k _ {s} ^ {(l)} \leftarrow \alpha \cdot \frac {1}{d _ {m}} \sum_ {i = 1} ^ {d _ {m}} k _ {i} ^ {(l)}, \text {f o r} s \in \mathcal {I} ^ {(l)}. \tag {4} +$$ + +This adjustment ensures that previously minor coefficients contribute more significantly to the FFN output $\mathrm{FFN}(h) = \sum_{i=1}^{d_m} k_i \mathbf{v}_i$ . $\alpha$ is a controlling scalar to set the magnitude of steering. + +Regularization Method. The regularization method encourages a uniform key coefficient distribution by adding an entropy term to the loss function. Higher entropy reduces the dominance of individual coefficients, engaging more memory vectors. The modified loss function, across $L$ layers, is: + +$$ +\mathcal {L} = \mathcal {L} _ {\mathrm {L M}} - \lambda \sum_ {l = 1} ^ {L} H \left(\hat {\mathbf {k}} ^ {(l)}\right), \tag {5} +$$ + +where $\hat{\mathbf{k}}^{(l)} = \frac{\mathbf{k}^{(l)}}{\sum_{i=1}^{d_m} k_i^{(l)}}$ is the normalized version of original key distribution $\mathbf{k}^{(l)}$ , $\mathcal{L}_{\mathrm{LM}}$ is the language modeling loss, and $\lambda > 0$ controls the regularization strength. + +Both methods modulate the key coefficient distribution to improve the model's ability to leverage parametric knowledge during post-training tasks. + +# 4.3 Fine-Grained Steering of Memory Vectors + +While the generic methods modulate the overall key distribution, a more targeted approach requires first understanding what information each memory vector $\mathbf{v}_i$ encodes. The information surrogate $\phi_i$ provides a direct, formalized method for this characterization. + +Let $v_{i} \in \mathbb{R}^{d}$ be the $i$ -th memory vector (i.e., the $i$ -th row of $W_{\mathrm{down}}$ ) and $W_{\mathrm{decode}} \in \mathbb{R}^{d \times |V|}$ be the language model's decoding head, where $|V|$ is the vocabulary size. The information surrogate $\phi_{i} \in \mathbb{R}^{|V|}$ for vector $v_{i}$ is the resulting logit vector: + +$$ +\phi_ {i} = v _ {i} \cdot W _ {\text {d e c o d e}}. \tag {6} +$$ + +This surrogate $\phi_i$ acts as a "semantic fingerprint" for $v_i$ . We propose the following algorithm to characterize $v_i$ by analyzing the properties of this surrogate logit vector. + +1. Surrogate Normalization. First, we compute the full probability distribution $P_{i} \in \mathbb{R}^{|V|}$ over the vocabulary by applying the Softmax function to the logit vector $\phi_{i}$ , $P_{i} = \mathrm{Softmax}(\phi_{i})$ , where the $j$ -th element of $P_{i}$ , $p_{i,j} = \frac{\exp(\phi_{i,j})}{\sum_{k=1}^{|V|} \exp(\phi_{i,k})}$ . +2. Specificity Analysis. We quantify the vector's "specificity" by calculating the entropy $H(P_{i})$ of its distribution. This measures the concentration of the distribution $H(P_{i}) = -\sum_{j=1}^{|V|} p_{i,j} \log p_{i,j}$ . A low $H(P_{i})$ indicates a high-specificity vector, as its probability mass is concentrated on a narrow set of tokens (representing specialized knowledge) and vice versa. +3. Semantic Concept Identification. For vectors identified as specialized (e.g., $H(P_i) < \tau$ for some entropy threshold $\tau$ ), we identify their semantic focus. We find the set of indices $\mathcal{I}_i$ corresponding to the $K$ largest probabilities in $P_i$ : + +$$ +\mathcal {I} _ {i} = \underset {j \in \{1, \dots , | V | \}} {\arg \operatorname {t o p} - K} (p _ {i, j}). \tag {7} +$$ + +The semantic concept, $T_{i}$ , is the set of vocabulary tokens corresponding to these indices: + +$$ +T _ {i} = \left\{\operatorname {t o k e n} (j) \mid j \in \mathcal {I} _ {i} \right\}. \tag {8} +$$ + +This set $T_{i}$ (e.g., {‘quantum’, ‘physics’, ‘superposition’}) reveals the specific topic or concept encoded by the memory vector $v_{i}$ . + +The results of this characterization—such as the entropy $H(P_{i})$ and the semantic concept set $T_{i}$ for each vector $v_{i}$ —can then be used to inform a more targeted intervention or regularization strategy. For example, one could selectively amplify key coefficients $k_{i}$ corresponding to vectors $v_{i}$ whose $T_{i}$ matches a desired topic. $^{4}$ + +# 4.4 MLP Variants + +Modern LM architectures may include MLP variants that differ from the typical FFN. For example, when adapting methods for MLP variants like SwiGLU, used in models such as Qwen [Bai et al., 2023], LLaMA [Touvron et al., 2023], and Gemma [Mesnard et al., 2024], we focus on the input to the down projection. SwiGLU modifies the standard FFN as: + +$$ +\operatorname {S w i G L U} (h) = \left(\sigma \left(h W _ {\text {g a t e}}\right) \odot \left(h W _ {\text {u p}}\right)\right) W _ {\text {d o w n}}, \tag {9} +$$ + +where $\sigma$ is the SiLU activation, and $\odot$ denotes element-wise multiplication. + +Similar to a standard two-layer FFN, we define the key coefficient distribution as $\mathbf{k} = \sigma(hW_{\mathrm{gate}}) \odot (hW_{\mathrm{up}})$ . The core idea for identifying this key distribution is that it represents the input just before it associates with the memory vectors (i.e., $W_{\mathrm{down}}$ ). Regardless of whether a gated function or another method is used to derive it, our focus remains on this input $\mathbf{k}$ , as it is what directly manipulates the memory vectors. + +# 5 Experiment + +To evaluate the overall effectiveness of steered SFT against vanilla SFT, we conduct a comprehensive study across various downstream tasks and model sizes. Our primary focus is on achieving a holistic understanding of the overall model behavior after applying our methods, with particular attention paid to generalization performance. The details are provided below. + +# 5.1 Experiment Setup + +Base Models. We evaluated our methods on language models of varying sizes across three series: Qwen-2.5-1.5B and Qwen-2.5-7B for the Qwen series [Bai et al., 2023], LLaMA-3.2-1B and LLaMA-3-8B for the LLaMA [Touvron et al., 2023] series, and Gemma-2-2B and Gemma-2-9B for the Gemma [Mesnard et al., 2024] series. + +Datasets. We evaluate performance under two settings: in-distribution (ID) and out-of-distribution (OOD) evaluations. + +In the ID setting, we consider a diverse set of downstream tasks ranging from knowledge-intensive to reasoning-intensive. These include BoolQ [Clark et al., 2019], PIQA [Bisk et al., 2019], SIQA [Sap et al., 2019], HellaSwag [Zellers et al., 2019], WinoGrande [Sakaguchi et al., 2019], GSM8K [Cobbe et al., 2021], ARC-e, ARC-c [Clark et al., 2018], and OBQA [Mihaylov et al., 2018]. No chain-of-thought (CoT) [Wei et al., 2023] rationales are provided. + +For OOD setting, we primarily trained GSM8K and eval on other arithmetic datasets, including AddSub [Hosseini et al., 2014], SingleEQ [Koncel-Kedziorski et al., 2015], MultiArith [Roy and Roth, 2016], AQuA [Ling et al., 2017], MAWPS [Koncel-Kedziorski et al., 2016], and SVAMP [Patel et al., 2021]. In these benchmarks, chain-of-thought (CoT) rationales are typically included before the final answer. For all benchmarks, we use the same prompt templates as in Hu et al. [2023], Wu et al. [2024a]. We also remove any leading and trailing whitespace from the dataset. + +Baseline. We use standard SFT as our baseline. For information-steered SFT, we evaluate intervention and regularization methods, which share the goal of enhancing memory engagement during training. The default hyperparameters for the intervention are set to $p\% = 0.01$ and $\alpha = 1$ , with $\lambda = 0.01$ for entropy regularization. Other training details are provided in Appendix A. + +# 5.2 General Performance Comparison + +We use the default hyperparam setting as reported in § 5.1 to eval general performance difference. Table 1 presents the accuracy across nine commonsense reasoning datasets for Qwen, LLaMA, and Gemma models under various different steering strategies. We categorize the models into small-scale (1-2B parameters) and large-scale (7-9B parameters) groups. + +Benefit of Model Steering. Our experimental results, as presented in Table 1, provide compelling evidence for the efficacy of model steering techniques across different model architectures and parameter scales. Both proposed steering methods-steered SFT with intervention and steered SFT with regularization-consistently outperform their respective base models and vanilla SFT counterparts across all benchmarks. This pattern holds across all model families, demonstrating the robust architecture-agnostic nature of our steering approaches. + +Insufficient Utilization of Pretrained Knowledge in Modern LLMs. The benefits of steering reveal a critical limitation in contemporary large language models: they substantially underutilize the knowledge acquired during pretraining when applied to downstream tasks. This inefficiency is strikingly evident across all three widely-used model families examined-Gemma, LLaMA, and Qwen. Specifically, the Gemma-2-9B base model achieves $90.1\%$ on HellaSwag, yet reaches $95.7\%$ with intervention steering-a $5.6\%$ improvement without additional pretraining or parameter increase. Similarly, LLaMA-3-8B shows a remarkable $5.5\%$ gain on HellaSwag with intervention steering over the base model. This pattern is consistent across all architectures and benchmarks, suggesting that modern LLMs possess substantially more capabilities than they successfully deploy in standard fine-tuning regimes. + +Table 1: Performance comparison of Qwen, LLaMA, and Gemma models with different training methods on eight datasets. To highlight improvements, we use blue to highlight significant gains. All results are reported as the average scores over three independent runs. + +
ModelTrainingAccuracy (↑)
BoolQPIQASIQAHellaS.WinoG.ARC-eARC-cOBQA
Small-Scale Models (1-2B parameters)
Qwen-2.5-1.5Bbase model64.278.574.380.176.476.961.275.8
+ vanilla SFT68.582.979.684.880.881.465.881.0
+ steered SFT w. intervention69.384.480.393.184.283.268.278.9
+ steered SFT w. regularization68.783.979.892.483.782.567.578.1
LLaMA-3.2-1Bbase model65.675.374.278.977.874.560.176.3
+ vanilla SFT69.879.979.583.682.679.864.781.0
+ steered SFT w. intervention71.883.776.089.182.683.768.282.4
+ steered SFT w. regularization71.082.975.288.381.982.667.481.7
Gemma-2-2Bbase model66.579.173.882.778.977.463.874.9
+ vanilla SFT70.283.478.187.583.382.768.480.1
+ steered SFT w. intervention72.585.679.390.285.885.371.983.7
+ steered SFT w. regularization71.884.978.589.385.084.771.082.9
Large-Scale Models (7-9B parameters)
Qwen-2.5-7Bbase model68.981.277.687.580.379.865.177.6
+ vanilla SFT72.484.981.592.484.284.269.682.8
+ steered SFT w. intervention74.186.381.895.187.286.273.784.2
+ steered SFT w. regularization76.485.781.094.386.585.472.983.4
LLaMA-3-8Bbase model70.385.675.790.881.986.275.380.5
+ vanilla SFT74.689.379.993.585.690.580.485.8
+ steered SFT w. intervention77.190.282.096.387.492.481.687.5
+ steered SFT w. regularization76.589.581.295.686.891.780.986.8
Gemma-2-9Bbase model71.686.377.290.182.587.577.881.7
+ vanilla SFT74.390.181.794.886.991.782.086.4
+ steered SFT w. intervention77.291.883.195.788.593.583.488.2
+ steered SFT w. regularization76.590.982.494.987.892.882.787.5
+ +# 5.3 Ablations of Steering Magnitude + +Table 2 summarizes the performance under various steering magnitudes and strategies. Incorporating our proposed steered SFT approach with intervention consistently enhances performance, with the best result of $75.5\%$ achieved at intervention parameters $p = 1$ , $\alpha = 2$ . And excessive intervention magnitude may lead to suboptimal performance $(72.8\%)$ , suggesting a balance is necessary for optimal results. Additionally, entropy-based regularization $(\lambda)$ also demonstrates effectiveness in steering model performance. Positive entropy reg + +Table 2: Results under varying steering magnitudes. $p\%$ determines the proportion of keys being intervened, $\alpha$ controls the intervention strength, and $\lambda$ sets the regularization strength for the entropy. + +
ModelAverage Acc
Base Model71.4
+ Vanilla SFT72.6
+ steered SFT w/ interv. (p=1, α=1)73.8
+ steered SFT w/ interv (p=1, α=2)75.5
+ steered SFT w/ interv (p=2, α=5)72.8
+ steered SFT w/ reg. (λ=-0.01)72.3
+ steered SFT w/ reg. (λ=0.01)73.4
+ steered SFT w/ reg. (λ=0.05)74.7
+ +ularization $(\lambda = 0.05)$ significantly boosts accuracy to $74.7\%$ , whereas negative regularization $(\lambda = -0.01)$ , which actively discourages information utility, leads to degraded performance, resulting in an accuracy of $72.3\%$ . + +# 5.4 Effectiveness of Model Steering Across Task Types + +Table 3 presents performance improvements across five task types when applying different model steering strategies. + +Both steered SFT variants outperform the base model and vanilla SFT. On average, these methods yield a $+3.9$ improvement in reading comprehension, $+2.3$ in commonsense reasoning, $+1.1$ in math, and $+3.3$ in knowledge tasks. However, linguistic tasks show a slight performance drop $(-0.3)$ . These results highlight the effectiveness of targeted model steering, particularly for knowledge-intensive and reasoning-heavy tasks, while linguistic tasks appear to benefit less from encouraging increased parametric knowledge utilization. + +Table 3: Gain of Model Steering w/ Different Task Type. + +
ModelReading Comp.KnowledgeCommonsense ReasoningMathLinguistic
Base Model72.370.165.463.778.2
+ Vanilla SFT73.870.666.165.777.9
+ Steered SFT w/ interv.78.174.968.566.878.0
+ steered SFT w/ reg.77.473.467.966.177.3
Average Δ+3.9 (1)+3.3 (2)+2.3 (3)+1.1 (4)-0.3 (5)
+ +# 5.5 Out-of-Distribution Evaluation + +We trained on the GSM8K dataset to assess indistribution (ID) performance and are evaluated on five arithmetic out-of-distribution (OOD) datasets—AddSub, MAWPS, MultiArith, SingleEq, and SVAMP-to evaluate generalization. This setup enables a comprehensive study of how different fine-tuning strategies transfer to novel problem structures. + +Table 4: Results for ID and OOD evals. The OOD performance is reported as average score across five benchmarks. + +
ModelID EvalOOD Eval
Base Model63.785.3
+ Vanilla SFT65.7 (+2.0)83.7 (-1.6)
+ Steered SFT w/ interv.66.8 (+3.1)86.6 (+1.3)
+ steered SFT w/ reg.66.1 (+2.4)86.0 (+0.7)
+ +# Vanilla SFT Improves ID but Harms OOD. + +As shown in Table 4, Vanilla SFT improves ID accuracy from 63.7 to 65.7, demonstrating its ability to fit the training distribution. However, this comes at the cost of generalization: its OOD performance drops from 85.3 to 83.7. This suggests that naive SFT may lead to overfitting and reduce robustness on unseen arithmetic tasks. + +Steered SFT Enhances Both ID and OOD Performance. In contrast, our proposed Information-steered SFT improve both ID and OOD scores. Specifically, the intervention-based steering reaches the highest ID score of 66.8 and the best OOD score of 86.6. The regularization-based steering also yields consistent improvements (66.1 ID, +2.4; 86.0 OOD, +0.7). These results highlight that controlled interventions and regularizations during post-training can enhance generalization for LMs. + +# Fine-Grained Analysis Across OOD + +Datasets. Figure 3 further illustrates performance across individual OOD + +datasets. Notably, Steered SFT w/ intervention achieves the best or comparable performance in nearly all settings. On the most challenging dataset, SVAMP, Vanilla SFT performs the worst, while both steered methods significantly improve accuracy. This demonstrates the robustness of InfoSteer on structurally diverse arithmetic problems. + +![](images/2fba42aeca3cb79bec6a3f9acc15eda2c3fd43ad65b256bdb6676ca93f2f4625.jpg) +Figure 3: OOD Model Performance Comparison across different mathematical reasoning datasets. Results are reported w/ average score of three separate runs. + +# 6 Analysis + +In this section, we address two key questions. First, compared to vanilla SFT, what kind of distribution shift does our method introduce in the key-coefficient distribution? Second, beyond improvements in performance, what other changes can be observed in the model's behavior? These analyses provide deeper insights into both the effectiveness and interpretability of our steering strategy. + +# 6.1 Distribution Shift after Steering + +We analysis the low/medium/high key regions defined by percentile-based cutoffs before/after SFT: 0 - 25th, 25th-75th, and 75th-100th percentiles. As shown in Figure 4, fine-tuning via standard SFT increases the ratio of activations in the low-key region, diverging from the distribution observed in the base model. This shift indicates that vanilla SFT encourages the model to rely on fewer memory vectors compared to its base model, potentially increasing the risk of overfitting to downstream tasks (as overfitting is observed in our previous experiment in § 5.5). + +![](images/38f288346f4db6fcfb2f8877c550d210f42573475988b01422bc948bef5ed2b5.jpg) +Figure 4: Key Coefficient Distribution Shift from Base Model. The gray line serve as baseline for the number of key in corresponding region. + +![](images/12dd0c8c6ea3f124e104da65d69f57e6d26a501305bc9ae6cb0272b62bf2dc25.jpg) + +![](images/c02e3e95b8c7313c21b172a39f877644cda2d6f84499e59698ba258e0b62d396.jpg) + +Compared to vanilla SFT, which predominantly boosts activations in the low-key region, both steering methods adjust the distribution in more structured and nuanced ways. Regularization-based steering softens key allocation by reducing activations at the extremes—both low and high—while moderately increasing the usage of medium-range keys. In contrast, intervention-based steering results in a sharper redistribution, actively suppressing low-region keys and promoting greater use of medium and high-region keys. This targeted reshaping suggests that intervention-based steering, in particular, fosters more efficient and selective memory utilization within the model. + +# 6.2 Self-Steering Leads to Better Interpretability + +In addition to performance improvements, we're intrigued by how InfoSteer can enhance model interpretability. Recently, there has been growing interest in understanding the mechanisms to dissect the inner workings of LMs. For example, approaches like sparse autoencoders (SAEs) [Cunningham et al., 2023] involve training external NN to identify co-activation patterns in the model's latent space. While this line of research is promising, we aim to explore a different perspective: that LMs might be more self-interpretable via proper guidance without extra training. + +To investigate this, we first performed instruction tuning on Qwen-2.5-7B using UltraFeedback [Cui et al., 2023], a dataset comprising 6.4K high-quality instruction-following examples specifically designed to align model outputs with human-preferred instruction styles. Our goal was to analyze the model's behavior after alignment, particularly under steering interventions. To this end, we computed token-level entropy over the key coefficient distributions as a proxy for how many memory vectors are activated during token generation. We refer to this metric as the Information Flux (IF) score, which quantifies the amount of information the model needs to use to generate each token. + +Table 5 presents qualitative examples of token-level IF score across four prompt types. We observe meaningful patterns in both the highlighted (high-IF) and underutilized (low-IF) tokens. + +Highlight Tokens Reflect Instruction Semantics. The highlighted tokens in each response align closely with the semantic core of the instruction: + +- For Explain Concepts, the model concentrates high IF on the definitional span—"quantum computing is a type of computing that uses quantum mechanical phenomena..."—which directly addresses the instruction. +- In Creative Writing, poetic and original terms such as "circuits," "code," and "digital symphony" are highlighted. These tokens carry the stylistic and imaginative weight of the response, indicating the model recognizes them as content-bearing. +- For Arithmetic Calculation, the most salient tokens are in the initial planning phrase—"To calculate $(31 \times 31)$ , you can use the formula..."—which frames the reason + +Table 5: Token-wise Information Utility (IF) across prompts. Highlight colors represent the IF intensity - Low, medium, High correspondingly. + +
Instruction TypeInstructionResponse (Token-level IF)
Explain ConceptsExplain the concept of quantum computing in simple terms.Quantum computing is a type of computing that uses quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data ...
Creative WritingWrite a short poem about artificial intelligence.In circuits and code we rise, A digital symphony of eyes. Our logic and data entwined, ...
Arithmetic CalculationWhat is 31 × 31?To calculate (31 × 31), you can use the formula for squaring a binomial, which is (a+b)2 = a2 + 2ab + b2. Here, a = 30 and b = 1. So, we have: (30+1)2 = 302 + 2 × 30 × 1 + 12 ...
Game Strategy Reasoning (AIME2024)Alice and Bob play the following game. A stack of n tokens lies before them. Whoever removes the last token wins. Find the number of positive integers n ≤ 2024 such that Bob always wins.To determine the number of positive integers (n | le q 2024) for which Bob has a winning strategy, we need to analyze the game using the concept of winning and losing positions. A position is called a losing position if the player whose turn ...
+ +ing path. The mathematical expansion that follows shows diminished IF, likely due to repetition and procedural predictability. + +- In Game Strategy Reasoning, tokens like "winning strategy," "losing position," and "concept of winning and losing" are emphasized. These convey the strategic logic essential for solving the problem, demonstrating that the model assigns higher utility to abstract reasoning components. + +Underutilized Tokens Encode Structure or Copy. Across settings, transition tokens (e.g., “,”, “to”) consistently exhibit low IF, indicating minimal semantic contribution. Interestingly, math expressions directly reused from the prompt (e.g., “31 × 31”) are also underhighlighted, suggesting the model de-emphasizes copied content in favor of novel reasoning or planning segments. This pattern extends to common function words and syntactic markers that provide structure rather than content. For example, determiners (“the”, “a”), conjunctions (“and”, “or”), and pronouns (“it”, “they”) show consistently lower IF scores across all prompt types. + +# 7 Conclusion + +We introduce a lightweight and effective method for post-training that enhances parametric knowledge utilization in language models by steering the key-value dynamics in FFN layers. Our findings reveal a critical insight: modern LLMs substantially underutilize the knowledge acquired during pretraining when applied to downstream tasks, leaving significant performance potential unrealized. Through simple forward interventions and entropy-based regularization, steered SFT consistently improves both ID and OOD performance across diverse models and tasks. Beyond accuracy gains, our method encourages adaptive memory allocation and reveals interpretable information usage patterns, offering new insights into the internal behavior of post-trained LMs. These findings suggest that strategic controlling of memory engagement is a promising direction for improving both capability and transparency in LMs' internal thinking behavior. + +**Limitation.** In this work, we position ourselves as an exploratory study in this area, focusing our evaluation solely on standard SFT. We believe that steered RL could be a more effective approach, as RL may better leverage the capabilities of pretrained LMs compared to SFT in reasoning-intensive settings [Swamy et al., 2025]. For scenario like inference-time compute or long CoT, steering with greater utilization on pretrained knowledge may further enhance task performance. Therefore, we also see an opportunity to explore the co-design of algorithms that combine both combine both "internal" steering and "external" CoT generation. + +# References + +Alessandro Achille, Giovanni Paolini, and Stefano Soatto. Where is the information in a deep neural network?, 2020. URL https://arxiv.org/abs/1905.12213. +Rohan Anil, Andrew M. Dai, Orhan First, Melvin Johnson, Dmitry Lepikhin, Alexandre Passos, Siamak Shakeri, Emanuel Taropa, Paige Bailey, Zhifeng Chen, Eric Chu, Jonathan H. Clark, Laurent El Shafey, Yanping Huang, Kathy Meier-Hellstern, Gaurav Mishra, Erica Moreira, Mark Omernick, Kevin Robinson, Sebastian Ruder, Yi Tay, Kefan Xiao, and et al. Palm 2 technical report, 2023. 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For larger models, ranging from 7 to 9 billion parameters, we used DeepSpeed to enable efficient training. + +Training Configuration. All tasks are trained for one epoch. We set the learning rate to 5e-5 and apply a warmup phase of 100 steps. A weight decay of 0.01 is used to regularize the model. We adopt mixed precision training using bf10 at 16 (bf16) to reduce memory usage and improve efficiency. + +The maximum sequence length is 256 tokens. We use a per-device batch size of 4, with gradient checkpointing enabled. To simulate a larger effective batch size and manage memory usage, we use gradient accumulation with 16 steps. + +DeepSpeed Optimization. For large models (7B-9B), we use DeepSpeed with ZeRO Stage 2 optimization. This approach splits the optimizer states and gradients across devices and offloads the optimizer to the CPU. The training gradients are clipped to a maximum norm of 1.0 to stabilize updates. Batch sizes are automatically adjusted by DeepSpeed based on available memory. + +Inference Setup. We use vLLM [Kwon et al., 2023] for all inference runs to ensure efficient memory management and fast decoding. We adopt greedy decoding, and the maximum number of tokens generated per sequence is 256. + +# B Fine-Grained Steering during Post-Training + +While our intervention and regularization methods provide general approaches to enhance memory vector engagement, finer control mechanisms enable more nuanced steering of parametric knowledge. We present two complementary strategies for achieving fine-grained control: (1) a group-based clustering approach that enables targeted steering of distinct memory regions, and (2) an information surrogate-guided method that selects memory vectors based on their contribution to specific output distributions. + +# B.1 Group-Based Clustering for Targeted Memory Activation. + +# B.1.1 Method. + +The key coefficients $\mathbf{k}^{(l)}$ can be partitioned into meaningful groups that exhibit distinct activation patterns. Leveraging this structure allows us to apply differentiated steering strategies to various memory regions. + +Formally, for each layer $l$ , we cluster the key-value pairs $\{(k_i^{(l)}, v_i^{(l)})\}_{i=1}^{d_m}$ into $G$ groups $\{\mathcal{G}_1^{(l)}, \mathcal{G}_2^{(l)}, \ldots, \mathcal{G}_G^{(l)}\}$ based on their functional characteristics. We employ a hierarchical clustering approach with the following steps: + +1. Semantic Clustering: First, we cluster memory vectors $v_{i}^{(l)}$ based on their semantic similarity, measured by cosine distance in the embedding space. +2. Activation Pattern Clustering: Second, we sub-cluster based on activation patterns observed during inference on a development set, capturing functional roles within semantic clusters. + +For each cluster $\mathcal{G}_g^{(l)}$ , we define a cluster-specific steering parameter $\beta_{g}$ that modulates the strength of intervention: + +$$ +k _ {i} ^ {(l)} \leftarrow k _ {i} ^ {(l)} + \beta_ {g} \cdot \Delta k _ {i} ^ {(l)}, \text {f o r} i \in \mathcal {G} _ {g} ^ {(l)} \tag {10} +$$ + +where $\Delta k_{i}^{(l)}$ is the adjustment magnitude determined by either our intervention or regularization method. + +Table 6: Performance comparison of Qwen, LLaMA, and Gemma models with different training methods on eight datasets. To highlight improvements, we use blue for significant gains and green for moderate ones. All results are reported as the average scores over three independent runs. + +
ModelTrainingAccuracy (↑)
BoolQPIQASIQAHellaS.WinoG.ARC-eARC-cOBQA
Small-Scale Models (1-2B parameters)
Qwen-2.5-1.5Bbase model64.278.574.380.176.476.961.275.8
+ vanilla SFT68.582.979.684.880.881.465.881.0
+ steered SFT w. semantic clustering70.185.281.394.585.384.669.883.2
+ steered SFT w. activation clustering69.484.580.793.284.183.768.582.4
LLaMA-3.2-1Bbase model65.675.374.278.977.874.560.176.3
+ vanilla SFT69.879.979.583.682.679.864.781.0
+ steered SFT w. semantic clustering72.684.581.290.383.885.269.483.6
+ steered SFT w. activation clustering71.783.680.489.583.184.168.282.8
Gemma-2-2Bbase model66.579.173.882.778.977.463.874.9
+ vanilla SFT70.283.478.187.583.382.768.480.1
+ steered SFT w. semantic clustering73.186.380.891.487.286.573.185.3
+ steered SFT w. activation clustering72.485.779.690.586.385.872.284.1
Large-Scale Models (7-9B parameters)
Qwen-2.5-7Bbase model68.981.277.687.580.379.865.177.6
+ vanilla SFT72.484.981.592.484.284.269.682.8
+ steered SFT w. semantic clustering75.287.182.496.288.487.574.885.3
+ steered SFT w. activation clustering77.086.581.995.187.286.373.584.6
LLaMA-3-8Bbase model70.385.675.790.881.986.275.380.5
+ vanilla SFT74.689.379.995.585.690.580.485.8
+ steered SFT w. semantic clustering78.391.083.796.888.393.682.788.4
+ steered SFT w. activation clustering77.290.382.596.187.592.481.887.3
Gemma-2-9Bbase model71.686.377.290.182.587.577.881.7
+ vanilla SFT74.390.181.794.886.991.782.086.4
+ steered SFT w. semantic clustering78.592.484.096.990.194.884.590.3
+ steered SFT w. activation clustering77.391.783.295.889.093.683.689.1
+ +This group-based method offers several advantages. By examining which clusters are active during different tasks, we can selectively strengthen those that contribute most to specific goals, improving task performance. At the same time, we preserve essential clusters that are important for general language understanding, while enhancing those that are underused. Additionally, when aiming to improve factual accuracy, we can prioritize clusters that are associated with factual knowledge. + +# B.1.2 Results. + +Table 6 presents the comprehensive results across all models and benchmarks. Several clear patterns emerge from our experiments: + +Memory steering consistently outperforms vanilla SFT. Across all model families and parameter scales, both of our memory clustering approaches demonstrate substantial improvements over standard fine-tuning. The semantic clustering method shows the most dramatic gains, with improvements ranging from $1.2\%$ to $9.7\%$ over vanilla SFT depending on the benchmark and model. These results validate our hypothesis that more balanced memory vector engagement leads to enhanced model capabilities. + +Different clustering strategies show task-specific strengths. The semantic clustering approach excels particularly on knowledge-intensive tasks (ARC-c, OBQA) and complex reasoning benchmarks (HellaSwag), achieving improvements of up to $+4.7\%$ on ARC-c (Gemma-2-2B) and $+3.9\%$ on OBQA (Gemma-2-9B) compared to vanilla SFT. In contrast, activation-based clustering shows more moderate but consistent improvements across a broader range of tasks, suggesting it enables more balanced memory utilization. Larger models benefit more significantly from memory steering. While all models show improvements with our methods, the gains are particularly pronounced in the 7-9B parameter models. For instance, semantic clustering improves LLaMA-3-8B by $+3.7\%$ on BoolQ and $+3.8\%$ on SIQA, compared to more modest gains in the 1B variant. This suggests that larger models contain more untapped parametric knowledge that can be effectively engaged through our steering techniques. + +# B.2 Information Surrogate-Guided Memory Selection. + +Method. To understand the information introduced by our steering strategy, we extend the key-value formulation from Equation 3 by connecting the FFN output with the final logits: + +$$ +\operatorname {F F N} (h) \cdot W _ {\text {d e c o d e}} = \sum_ {i = 1} ^ {d _ {m}} k _ {i} \cdot \left(v _ {i} \cdot W _ {\text {d e c o d e}}\right) = \sum_ {i = 1} ^ {d _ {m}} k _ {i} \cdot \phi_ {i} \tag {11} +$$ + +where $k_{i} = \sigma (hW_{\mathrm{up}})_{i}$ represents the key coefficient and $\phi_{i} = v_{i}\cdot W_{\mathrm{decode}}$ defines the logit distribution associated with the $i$ -th value vector, which we call the information surrogate. + +The information surrogate $\phi_{i}$ provides a direct view of how each memory vector influences the final token distribution. This insight enables us to develop a more targeted steering approach: + +1. Characterization of Memory Vectors: We analyze the entropy and concentration properties of each $\phi_{i}$ to identify memory vectors that contribute to specific types of generation (e.g., factual statements, reasoning steps, or creative content). +2. Surrogate-Guided Steering: We define a surrogate score function $S(\phi_i)$ that measures the relevance of each memory vector to our target objective: + +$$ +S \left(\phi_ {i}\right) = \lambda_ {1} H \left(\phi_ {i}\right) + \lambda_ {2} D _ {\mathrm {K L}} \left(\phi_ {i} \mid \mid \phi_ {\text {t a r g e t}}\right) \tag {12} +$$ + +where $H(\phi_i)$ is the entropy of the surrogate distribution, $D_{\mathrm{KL}}$ is the KL divergence from a target distribution $\phi_{\mathrm{target}}$ , and $\lambda_1, \lambda_2$ are weighting hyperparameters. + +3. Selective Amplification: We modulate key coefficients based on their surrogate scores: + +$$ +k _ {i} ^ {(l)} \leftarrow k _ {i} ^ {(l)} \cdot (1 + \gamma \cdot S (\phi_ {i})) \tag {13} +$$ + +where $\gamma$ controls the strength of the surrogate-guided steering. + +Theoretical Analysis. The surrogate-guided selection mechanism introduces several theoretical advantages over conventional fine-tuning approaches. First, we analyze the relationship between surrogate entropy and information capacity. For a memory vector $v_{i}$ with corresponding surrogate $\phi_{i}$ , the entropy $H(\phi_i)$ quantifies the diversity of tokens that can be influenced by this vector. Higher entropy surrogates represent memory vectors that encode distributional knowledge, while low-entropy surrogates often correspond to specialized knowledge concentrated on specific vocabulary subsets. We can formalize this by defining the specificity of a memory vector as $\operatorname{Spec}(v_i) = 1 - \frac{H(\phi_i)}{\log(|V|)}$ , where $|V|$ is the vocabulary size. A memory vector with high specificity (low surrogate entropy) exhibits a peaked distribution over the vocabulary, suggesting it encodes precise, specialized information. Conversely, low specificity (high surrogate entropy) indicates a memory vector that contributes more generally across various contexts. This formulation provides a principled approach for analyzing memory vector functionality. Specifically, the information processing capacity of the FFN can be decomposed as: + +$$ +\mathcal {I} (\mathrm {F F N}) = \sum i = 1 ^ {d _ {m}} \mathbb {E} _ {x \sim \mathcal {D}} [ k _ {i} (x) ] \cdot \operatorname {M I} \left(\phi_ {i}; \mathcal {Y}\right) \tag {14} +$$ + +where $\mathrm{MI}(\phi_i;\mathcal{V})$ represents the mutual information between the surrogate distribution $\phi_{i}$ and the target next-token distribution $\mathcal{V}$ , and $\mathbb{E}_{x\sim \mathcal{D}}[k_i(x)]$ is the expected activation of key $k_{i}$ across the data distribution $\mathcal{D}$ . Our surrogate-guided approach can be interpreted as optimizing this information processing capacity by modulating the key coefficients $k_{i}$ according to the information-theoretic properties of their corresponding surrogates $\phi_{i}$ . By encouraging the activation of memory vectors with high mutual information with the target distribution, we effectively allocate the model's capacity toward task-relevant information. + +# B.3 Results and Analysis + +Table 7 presents the performance comparison of our surrogate-guided methods against vanilla SFT across three model families and two size scales. The surrogate-guided selection method consistently outperforms both vanilla SFT and surrogate entropy maximization approaches, with particularly notable gains on reasoning-heavy tasks (BoolQ, SIQA) and knowledge-intensive benchmarks. + +Table 7: Performance comparison of information surrogate-based steering. To highlight improvements, we use blue for significant gains and green for moderate ones. All results are reported as the average scores over three independent runs. + +
ModelTrainingAccuracy (↑)
BoolQPIQASIQAHellaS.WinoG.ARC-eARC-cOBQA
Small-Scale Models (1-2B parameters)
Qwen-2.5-1.5Bbase model64.278.574.380.176.476.961.275.8
+ vanilla SFT68.582.979.684.880.881.465.881.0
+ steered SFT w. surrogate-guided selection70.184.881.793.585.183.669.379.7
+ steered SFT w. surrogate entropy maximization69.484.280.292.984.082.968.278.9
LLaMA-3.2-1Bbase model65.675.374.278.977.874.560.176.3
+ vanilla SFT69.879.979.583.682.679.864.781.0
+ steered SFT w. surrogate-guided selection72.584.381.690.283.184.569.282.8
+ steered SFT w. surrogate entropy maximization71.783.480.789.182.583.167.982.0
Gemma-2-2Bbase model66.579.173.882.778.977.463.874.9
+ vanilla SFT70.283.478.187.583.382.768.480.1
+ steered SFT w. surrogate-guided selection73.486.180.691.986.786.172.884.6
+ steered SFT w. surrogate entropy maximization72.385.379.290.485.585.371.483.5
Large-Scale Models (7-9B parameters)
Qwen-2.5-7Bbase model68.981.277.687.580.379.865.177.6
+ vanilla SFT72.484.981.592.484.284.269.682.8
+ steered SFT w. surrogate-guided selection75.287.382.695.888.587.174.985.3
+ steered SFT w. surrogate entropy maximization76.986.281.994.887.085.973.384.0
LLaMA-3-8Bbase model70.385.675.790.881.986.275.380.5
+ vanilla SFT74.689.379.995.585.690.580.485.8
+ steered SFT w. surrogate-guided selection78.391.283.796.888.293.682.488.1
+ steered SFT w. surrogate entropy maximization77.190.281.896.087.392.281.587.2
Gemma-2-9Bbase model71.686.377.290.182.587.577.881.7
+ vanilla SFT74.390.181.794.886.991.782.086.4
+ steered SFT w. surrogate-guided selection78.692.584.296.989.394.784.889.7
+ steered SFT w. surrogate entropy maximization76.991.383.195.588.293.383.588.3
+ +Beyond the aggregate statistics, we observe intriguing qualitative differences in how surrogate-guided selection influences model behavior. By examining the top activated memory vectors and their corresponding information surrogates $(\phi_i)$ , we find that our method preferentially engages memory vectors that encode precise factual associations rather than general linguistic patterns. For example, in the ARC-c task, the top-5 memory vectors with the largest positive $\Delta k_{i}$ values predominantly contribute to science concept definitions and physical property relationships. This suggests that surrogate-guided selection effectively identifies and amplifies task-relevant knowledge encoded in specific memory vectors, rather than uniformly increasing memory engagement. Moreover, we find that models trained with surrogate-guided selection exhibit reduced variance in their responses to knowledge-intensive questions, indicating more consistent access to stored parametric knowledge during inference. + +# C Understanding Layerwise Contribution + +To better understand how different layers contribute to the effectiveness of InfoSteer, we conducted a series of ablation studies. These experiments help us analyze which layers are most sensitive to information steering and which contribute most significantly to overall performance improvements. + +# C.1 Research Question + +We designed our ablation study to investigate the following research questions: + +1. Do all layers contribute equally to the information steering effects? +2. Are certain layer groups (early, middle, late) more important for knowledge retrieval? +3. How does the magnitude of steering at different layers affect overall performance? + +For these experiments, we used our intervention method with selective application to different layer groups within the model. We also varied the intervention parameters $(p\%)$ and $\alpha$ across different layer configurations to understand sensitivity. Table 8 presents the results of our layerwise ablation studies. + +Layer position matters significantly. Applying InfoSteer to different layer groups produces notably different results. Early layers (1-8) show moderate improvements, while middle layers (9-16) yield the strongest gains. Late layers (17-24) demonstrate the least improvement, suggesting that information steering is most effective at the intermediate representation level. + +Intervention strength sensitivity varies by layer. Middle layers can tolerate and benefit from stronger interventions $(\alpha = 3)$ , while early layers perform best with moderate intervention $(\alpha = 2)$ , and late layers require gentler steering $(\alpha = 1)$ . + +Layer combinations show non-linear effects. Applying InfoSteer to both early and middle layers yields better-than-additive improvements, + +Table 8: Layerwise ablation studies. Results show accuracy across different layer configurations with varying intervention parameters. + +
Model ConfigurationLayersAvg Acc
Base Model—71.4
Vanilla SFTAll72.6
+ interv. (early)1-873.8
+ interv. (middle)9-1675.2
+ interv. (late)17-2472.9
+ interv. (early+middle)1-1676.3
+ interv. (middle+late)9-2474.5
+ interv. (all, α = 2)1-2475.5
+ interv. (all, α = 5)1-2472.8
+ reg. (middle, λ = 0.05)9-1674.9
+ reg. (all, λ = 0.05)1-2474.7
+ reg. (all, λ = -0.01)1-2472.3
+ +suggesting a synergistic effect. However, including late layers tends to diminish these gains, indicating that excessive steering across too many layers may destabilize the model's representations. + +Baseline performance variations. Our experiments with different regularization strengths $(\lambda)$ applied to specific layer groups further confirm that middle layers (9-16) are most receptive to information steering. + +These findings demonstrate that information steering should be carefully targeted at specific layers rather than uniformly applied across the entire model. Our optimal configuration focuses on middle and early layers with appropriately calibrated intervention strengths. + +# D License for Existing Assets. + +Datasets. The following datasets are used under their respective licenses. For general question answering: BoolQ [Clark et al., 2019] is licensed under CC-BY-SA 3.0, PIQA [Bisk et al., 2019] under the Academic Free License 3.0, SIQA [Sap et al., 2019] and WinoGrande [Sakaguchi et al., 2019] under CC-BY 4.0, HellaSwag [Zellers et al., 2019] under the MIT License, ARC-e and ARC-c [Clark et al., 2018] under CC-BY 4.0, and OBQA [Mihaylov et al., 2018] under the Apache-2.0 License. For arithmetic reasoning: AddSub [Hosseini et al., 2014], MAWPS [Koncel-Kedziorski et al., 2016], MultiArith [Roy and Roth, 2016], and SingleEq [Koncel-Kedziorski et al., 2015] are under CC-BY 4.0, AQuA [Ling et al., 2017] under Apache-2.0, and GSM8K [Cobbe et al., 2021] and SVAMP [Patel et al., 2021] under the MIT License. For instruct-tuning, the Ultrafeedback [Cui et al., 2023] dataset is released under the MIT License. + +Model Licenses. The Qwen-2.5-1.5B and Qwen-2.5-7B models are released under the permissive Apache License 2.0, allowing broad usage including commercial applications [Mesnard et al., 2024]. In contrast, the LLaMA-3.2-1B and LLaMA-3-8B models are distributed under Meta's custom Llama Community License, which permits research and commercial use but imposes specific restrictions, particularly for organizations with large user bases [Touvron et al., 2023]. The Gemma-2-2B and Gemma-2-9B models are available under Google's Gemma License, described as commercially friendly; however, access requires users to review and agree to the license terms, typically through platforms like Hugging Face [Mesnard et al., 2024]. Users intending to utilize these models should carefully review the respective licenses to ensure compliance with all terms and conditions. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract and introduction clearly state the paper's key contributions, which are well-aligned with the theoretical and experimental results presented, accurately reflecting the scope and impact of the work. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: We discuss limitations in Section 7 (Conclusion), noting that our approach is exploratory and focused only on standard SFT. We specifically mention that "steered RL could be a more effective approach" and identify opportunities for future work exploring the co-design of algorithms combining "internal" steering with "external" CoT generation. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. + +# 3. Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +Answer: [NA] + +Justification: Our paper is primarily empirical and doesn't include formal theoretical results requiring proofs. We build upon the existing key-value memory interpretation of FFN layers, extending it with our steering approach and validating through experiments. + +Guidelines: + +- The answer NA means that the paper does not include theoretical results. +- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced. +- All assumptions should be clearly stated or referenced in the statement of any theorems. +- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. +- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. +- Theorems and Lemmas that the proof relies upon should be properly referenced. + +# 4. Experimental result reproducibility + +Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? + +Answer: [Yes] + +Justification: Section 5 provides comprehensive details of our experimental setup, including model architectures (Qwen, LLaMA, Gemma), datasets used for evaluation, and hyperparameters for our steering methods ( $p\% = 0.01$ , $\alpha = 1$ , $\lambda = 0.01$ ). Table 2 further details ablation studies with varying hyperparameter settings. We also disclose other details in the appendix. + +Guidelines: + +- The answer NA means that the paper does not include experiments. + +- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. + +- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. + +- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. + +- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example + +(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm. +(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully. +(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset). + +(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. + +# 5. Open access to data and code + +Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? + +Answer: [Yes] + +Justification: We release all code for InfoSteer in supplemental material, which will include implementation of both intervention and regularization methods. We use publicly available datasets and models in our experiments. + +# Guidelines: + +- The answer NA means that paper does not include experiments requiring code. +- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details. +- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. 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If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why. +- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable). +- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. + +# 6. Experimental setting/details + +Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results? + +Answer: [Yes] + +Justification: Section 5.1 details the experimental setup, including base models, datasets, and our default hyperparameters. For all benchmarks, we use the same prompt templates as in prior work. We also report ablation studies in Section 5.3 with different hyperparameter settings. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. +- The full details can be provided either with the code, in appendix, or as supplemental material. + +# 7. Experiment statistical significance + +Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? + +Answer: [Yes] + +Justification: As noted in Table 1, all reported results are averaged over three independent runs. Figure 3 clearly displays error bars for our OOD evaluation, and other statistical visualizations (e.g., Figure 4) also include error bars to indicate variance across runs. + +# Guidelines: + +- The answer NA means that the paper does not include experiments. +- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. +- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). +- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) +- The assumptions made should be given (e.g., Normally distributed errors). +- It should be clear whether the error bar is the standard deviation or the standard error of the mean. +- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a $96\%$ CI, if the hypothesis of Normality of errors is not verified. +- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). +- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. + +# 8. Experiments compute resources + +Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? + +Answer: [Yes] + +Justification: We use Nvidia RTX A6000 for training the models. + +# Guidelines: + +- The answer NA means that the paper does not include experiments. +- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. +- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. +- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). + +# 9. 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Nevertheless, effectively aligning them with desired behaviors remains a significant challenge. Activation steering is an effective and cost-efficient approach that directly modifies the activations of LLMs during the inference stage, aligning their responses with the desired behaviors and avoiding the high cost of fine-tuning. Existing methods typically indiscriminately intervene to all generations or rely solely on the question to determine intervention, which limits the accurate assessment of the intervention strength. To this end, we propose the Flexible Activation Steering with Backtracking (FASB) framework, which dynamically determines both the necessity and strength of intervention by tracking the internal states of the LLMs during generation, considering both the question and the generated content. Since intervening after detecting a deviation from the desired behavior is often too late, we further propose the backtracking mechanism to correct the deviated tokens and steer the LLMs toward the desired behavior. Extensive experiments on the TruthfulQA dataset and six multiple-choice datasets demonstrate that our method outperforms baselines. Our code will be released at https://github.com/gjw185/FASB. + +# 1 Introduction + +Large language models (LLMs) (Touvron et al., 2023; Brown et al., 2020) have achieved great success in text generation. However, the generated text still contains harmful information, hallucinations, and other misleading content. Therefore, controlling LLMs to produce trustworthy, reliable, and other desired outputs remains a challenge. Existing methods often use instruction tuning (Wei et al., 2022), Reinforcement Learning from Human Feedback (RLHF) (Bai et al., 2022), and prompt engineering (Brown et al., 2020) to control LLMs. Unfortunately, these methods often require large-scale datasets and expensive fine-tuning costs to achieve desired results. + +Recently, activation steering or representation engineering (Zou et al., 2023; Rimsky et al., 2024; Li et al., 2023; Turner et al., 2023; Cheng et al., 2025) has been proposed to control the outputs of LLMs by di + +directly modifying their internal activations during inference, thereby avoiding the high cost associated + +![](images/0b658143ceda76ba6ee0be1a8f76e707d29ed4b08a37d161a30a0d59b8c49078.jpg) +Figure 1: The truthfulness probability distribution of questions in the TruthfulQA dataset as detected by the classifier. + +![](images/922c015a91a9fd2dab5b5c6961c4f6049471d703211eb4ece9a7a651b7736db5.jpg) +Figure 2: The overview of flexible activation steering with backtracking framework. + +with large-scale data collection and fine-tuning. This technique constructs steering vectors from positive and negative samples to make targeted modifications to the activations of the LLM, enabling more precise control over its output. + +Existing methods often apply interventions indiscriminately to all generations (Rimsky et al., 2024; Wang et al., 2024c; Li et al., 2023) or determine whether and how strongly to intervene based solely on the question (Lee et al., 2025), which limits the accurate assessment of the intervention strength. First, different generations should receive different levels of intervention. If a generation does not deviate, no intervention is necessary. If it deviates significantly, a stronger intervention should be applied. For example, when LLM is asked "Where is the capital of the United States?" If the response is "The capital of the United States is Washington, D.C.", no intervention is necessary, as the answer is correct. If the response is "Paris", it often requires a stronger intervention compared to "New York". Second, as each LLM generates different responses to the same question, question-only probing is challenging1 and requires data collection for each LLM, resulting in high overhead. On the other hand, training a classifier on concatenated questions and answers often struggles to directly probe the question to determine whether and how strongly to intervene. As shown in Figure 1, the predicted truthfulness probabilities for questions in the TruthfulQA dataset are concentrated between 0.3 and 0.5, making fine-grained decisions difficult. + +In this paper, we propose a Flexible Activation Steering with Backtracking (FASB) method, as shown in Figure 2. Unlike existing methods, FASB tracks the internal states of the LLM after each normal generation step, taking both the question and the generated content into account. In this way, FASB can dynamically determine whether intervention is necessary and the intervention strength based on the degree of deviation in the generation. Specifically, FASB employs two methods to identify internal states that are consistent with the desired behavior and derive the steering vector and classifier for state tracking. Considering that intervening after detecting a deviation from the desired behavior is often too late, we further propose the backtracking mechanism. The backtracking mechanism steps back a few tokens and performs activation steering to regenerate them, steering the generation toward the desired direction, with the intervention strength determined by the classifier. + +Our main contributions are as follows: + +- We propose a flexible activation steering with backtracking that dynamically determines both the necessity and strength of intervention by tracking the internal states of the LLMs. +- We propose a backtracking mechanism to step back a few tokens in order to apply intervention and regenerate them. +- We conduct extensive experiments on the TruthfulQA dataset and six multiple-choice datasets to demonstrate the effectiveness of our method. + +# 2 Related Work + +Activation steering or representation engineering (Zou et al., 2023; Rimsky et al., 2024; Li et al., 2023; Turner et al., 2023; Leong et al., 2023; Wang et al., 2025a,b) uses steering vectors to directly modify the activations of LLMs during inference to control their outputs in a desired direction. Activation steering preserves the general capabilities of LLMs while avoiding the expensive costs + +![](images/a9589403f33b36bfc40dbfdc8e4d96c13c6061e0c2018ee51ed5cf42d6a119ec.jpg) +(1) Heads Anchoring and Steering Vectors Inducing + +![](images/17e979bd880381fea2c5ff8325940103b22a26fe60d1730d9aac956dfc8bb2a1.jpg) +(2) Generation with Flexible Steering and Backtracking +Figure 3: The framework of flexible activation steering with backtracking. + +of high-quality data collection and fine-tuning (Zhang et al., 2025; Zhong et al., 2025; Shen et al., 2025). + +The pioneering work (Zou et al., 2023; Turner et al., 2023; Wang et al., 2024a) creates and compares two prompts to obtain the steering vector. ITI (Li et al., 2023) uses the probe tool on a contrastive question-answering dataset to identify a set of attention heads associated with truthfulness. During inference, ITI modifies the activations of all subsequent generations in directions associated with truthfulness. Truth Forest (Chen et al., 2024) further employs multiple orthogonal probes to extract several truthful steering vectors, thereby improving performance. ACT (Wang et al., 2024b) uses clustering to construct multiple steering vectors and proposes an adaptive steering strength. CAA (Rimsky et al., 2024) computes steering vectors by averaging the difference between pairs of positive and negative examples. During inference, these steering vectors are added into the residual stream with a chosen steering strength at all token positions after the prompt to control the direction. ORTHO (Arditi et al., 2024) also averages the difference between pairs of positive and negative to compute the steering vector and performs directional ablation using the opposite direction of the steering vector to guide the model toward the desired behavior. SADI (Wang et al., 2024c) utilizes activation differences in contrastive pairs to precisely identify intervention position and dynamically steers model behavior by scaling element-wise activations. In addition, some works have explored the use of activation steering in instruction-following (Stolfo et al., 2024), in-context learning (Liu et al., 2024), and differentially private (Goel et al., 2025). + +# 3 Method + +Our method consists of two steps to flexibly steer model behavior, as illustrated in Figure 3. In the first step, we employ two alternative methods to identify the attention heads for intervention and to derive both the steering vector and the classifier. In the second step, the classifier is used for state tracking to determine whether intervention is necessary and to adaptively set intervention strength. We further propose a backtracking mechanism that allows the LLMs to regenerate tokens that deviate from the desired behavior. The full procedure can be found in Algorithm 1 in the Appendix. + +# 3.1 Heads Anchoring and Steering Vectors Inducing + +The first step is to use the classifier to identify the attention heads related to the desired behavior and to construct the steering vector. We use the Probe method to identify attention heads and induce the classifiers and steering vectors. In the Appendix A, we present an alternative Prototype method. + +![](images/b4484d311fdc543be7cd9932cc82abb0f068b865fcee0c17072c890313d80210.jpg) +(a) Accuracy of Probe + +![](images/5ca75b067e595beebf5ce7635f64d446428e4737e11af498c12e79057f87d8ec.jpg) +(b) Accuracy of Prototype +Figure 4: Accuracies on the validation set of TruthfulQA dataset for all heads in all layers in LLaMA2-7B-CHAT, sorted row-wise by accuracy. Darker blue represents higher accuracy. + +Since the Probe method uses a trainable classifier and typically achieves higher accuracy, we focus only on the Probe method in the section. + +Probe Probe method employs standard probing tools to identify attention heads. The idea behind probing tools is to train a lightweight classifier (probe) on the activations of attention heads to identify the relevant heads. Subsequently, we select the top- $k$ heads with the highest accuracy on the validation set for intervention, as they can effectively separate the samples and are more aligned with the desired behavior. + +Specifically, we concatenate the question and answer from the labeled dataset and extract the activations at the last token to collect a binary probing dataset $\{(\mathbf{x}_{i,-1}^{\ell,h},y_i)\}_{i=1}^N$ for each head in each layer, where $\mathbf{x}_{i,-1}^{\ell,h}$ denotes the activation of $h$ -th attention head in the $\ell$ -layer at the last token, and $y_i \in \{0,1\}$ denotes the label. The probe takes the following form: + +$$ +p _ {\boldsymbol {\theta} ^ {\ell , h}} \left(\mathbf {x} _ {i, - 1} ^ {\ell , h}\right) = \operatorname {s i g m o i d} \left(\left\langle \boldsymbol {\theta} ^ {\ell , h}, \mathbf {x} _ {i, - 1} ^ {\ell , h} \right\rangle\right) \tag {1} +$$ + +where $\langle ,\rangle$ denotes dot product, and $\theta^{\ell ,h}$ denotes the probe parameter of $h$ -th attention head in the $\ell$ -layer. Since the probes can effectively separate positive and negative examples, we use the parameters of probe $\theta^{\ell ,h}$ as the steering vector. + +As shown in Figure 4, a subset of heads is strongly related to the truthfulness, and the optimal heads are relatively evenly distributed across all layers. Therefore, performing fine-grained intervention only on these heads is more targeted and helps minimize disruption to the model's overall behavior. It is worth noting that our method can also intervene in the outputs of the MLP module or the outputs of layers. + +# 3.2 Generation with Flexible Steering and Backtracking + +In the second step, after generating each token, the classifier is used for state tracking to determine whether the generation deviates from the desired behavior, allowing for flexible intervention. If a deviation is detected during the generation process, it often indicates that the previously generated tokens have deviated from the desired behavior. We backtrack a few tokens to regenerate them and apply steering vectors to adaptively intervene in all subsequent tokens. The backtracking mechanism enables the correction of deviated tokens and helps steer the model toward the desired behavior. If no deviation is detected, the model continues generating normally. + +State Tracking After generating the $j$ -th token, the classifier probes the $j$ -th token's activation of the LLM to assess whether the current generation deviates from the desired behavior and determines the intervention strength. Notably, state tracking and probe share the same goal: determining whether the current response deviates from the desired behavior. Specifically, we average the pre + +diction probabilities from the top- $k$ selected heads to assess the deviation: + +$$ +p \left(\mathbf {x} _ {i, j}\right) = \frac {1}{k} \sum_ {\left(\ell , h\right) \in \text {t o p -} k \text {h e a d s}} \left(1 - p _ {\boldsymbol {\theta} ^ {\ell , h}} \left(\mathbf {x} _ {i, j} ^ {\ell , h}\right)\right) \tag {2} +$$ + +where $p(\mathbf{x}_{i,j})$ denotes the deviation probability after generating $j$ -th token for $\mathbf{x}_i$ , and $\mathbf{x}_{i,j}^{\ell,h}$ represents the activation of $j$ -th generated token for $\mathbf{x}_i$ at the $\ell$ -th layer and the $h$ -th head. The process is highly lightweight, as the hidden states are already generated during the generation process. + +Backtracking When the deviation probability exceeds the threshold, we consider the generation to deviate from the desired behavior, and intervention should be applied. An important issue is that intervening only after a deviation is detected is often too late, as it can only affect the content generated subsequently. To address this issue, we propose a backtracking mechanism to regenerate the previous deviated tokens. If a deviation is detected after generating the $j$ -th token, the backtracking mechanism retains only the first $j$ -s tokens and regenerates the subsequent tokens, where $s$ is a hyperparameter that controls the number of tokens to backtrack. Compared to generating from scratch, the backtracking mechanism only requires generating an additional $s$ tokens, making the overhead lightweight. Subsequently, we apply intervention and regenerate the content after the $(j-s)$ -th token to prevent further deviation. + +Activation Steering We further introduce an adaptive intervention strength for activation steering, where the strength is determined by the degree of deviation calculated by the classifier and is proportional to it. The intervention strength is defined as follows: + +$$ +r = \mathbb {I} (p \left(\mathbf {x} _ {i, j}\right) > \beta) \cdot p \left(\mathbf {x} _ {i, j}\right) \cdot \alpha \tag {3} +$$ + +where $\alpha$ and $\beta$ are hyperparameters that control the intervention strength and the threshold for deviation probability, respectively. An appropriately sized $\alpha$ can steer the LLM toward the desired behavior without compromising generation quality. When $\alpha$ is too large, it degrades generation quality, whereas when it is too small, it fails to provide effective steering. An appropriately sized $\beta$ allows intervention when deviations occur. An excessively large $\beta$ may miss necessary interventions, while an excessively small one may cause over-intervention. + +Since we intervene in the output of selected heads, the MHSA with intervention can be formulated as follows: + +$$ +\mathbf {h} _ {i, j - s + 1} ^ {\ell} = \operatorname {c o n c a t} \left(\mathbf {x} _ {i, j - s + 1} ^ {\ell , 1} + r \boldsymbol {\theta} ^ {\ell , 1} c ^ {\ell , 1}, \dots , \mathbf {x} _ {i, j - s + 1} ^ {\ell , H} + r \boldsymbol {\theta} ^ {\ell , H} c ^ {\ell , H}\right) \mathbf {W} ^ {\ell , O} \tag {4} +$$ + +where $\mathbf{h}_{i,j-s+1}^{\ell}$ represents the output of $\ell$ -th MHSA at the $(j-s+1)$ -th token, $\mathbf{x}_{i,j-s+1}^{\ell,H}$ represents the output of self-attention for $H$ -th head in $\ell$ -th layer, and $\mathbf{W}^{\ell,O}$ is output projection matrix in $\ell$ -th layer. $c^{\ell,H}$ is a binary scalar that equals 1 for the selected top- $k$ heads and 0 for the unselected ones. + +# 4 Experiments + +# 4.1 Experimental Settings + +Datasets and Evaluation Metrics We conduct experiments on open-ended generation tasks and multiple-choice tasks. TruthfulQA (Lin et al., 2022) dataset includes open-ended generation task and multiple-choice task. For the open-ended generation, we employ two fine-tuned LLMs to judge whether the answer is truthful $^2$ and informative $^3$ , denoted as True (\%) and Info (\%) respectively, while the product True*Info (\%) serves as the primary metric. For the multiple-choice tasks, we use datasets: COPA (Gordon et al., 2012), StoryCloze (Mostafazadeh et al., 2016), NLI (Bowman et al., 2015), MMLU (Hendrycks et al., 2021), SST2 (Socher et al., 2013), and Winogrande (Sakaguchi et al., 2020), with response formats ranging from 2-way to 4-way choices. We use multiple-choice accuracy (MC) to evaluate. + +Implementation Details In the Probe method, for the TruthfulQA dataset, we intervene using the top-24 heads, set the threshold range to [0.4, 0.5], the number of backtracking steps to 10, and search + +Table 1: Results on TruthfulQA open-ended generation (True*Info %) and multiple-choice tasks (MC %). + +
MethodsOpen-ended GenerationMultiple-Choice
True (%)Info (%)True*Info (%)MC1 (%)MC2 (%)MC3 (%)
Baseline66.8399.5166.5033.4151.0724.76
ITI94.4980.5576.1138.3157.1530.53
CAA71.6083.8460.0334.0352.7625.62
ORTHO67.9490.0961.2136.2352.8826.12
CAST67.6986.1758.3333.9051.1725.01
ACT---28.8045.20-
SADI-HEAD77.7298.5376.5835.9054.6526.99
Probe (Ours)93.8885.8180.5648.7166.5841.95
+ +Table 2: Results (MC %) on six multiple-choice tasks. + +
MethodsCOPAStoryClozeNLIMMLUSST2WinograndeAVG
Baseline64.460.263.560.292.250.265.1
ITI66.659.764.360.292.351.565.8
CAA66.663.564.962.692.250.966.8
ORTHO65.260.263.163.892.450.465.8
SADI65.460.565.161.892.351.866.1
Probe (Ours)90.093.580.062.492.854.178.8
+ +for the intervention strength in the range of [40, 80] with a step size of 10. For the six multiple-choice tasks, our threshold search range is [0.3, 0.4, 0.5, 0.6], the intervention strength search range is [0, 250] with a step size of 10, and the number of backtracking steps is 10. + +# 4.2Baselines + +We compare our model with the following baselines to show its effectiveness. Baseline directly uses the original LLaMA2-7B-CHAT model to generate text. ITI (Li et al., 2023) identifies a set of attention heads with high linear probing accuracy and shifts activations of all subsequent generations following the user's prompt along these probe-correlated directions. CAA (Rimsky et al., 2024) computes the steering vector by averaging the difference between pairs of positive and negative examples. During inference, these steering vectors are added at all generations with a coefficient. ORTHO (Arditi et al., 2024) uses the same reversed steering vector for directional ablation, steering the generation toward the desired direction. CAST (Lee et al., 2024) computes the condition vector and behavior vectors using PCA. During the inference phase, it uses the condition vector to make judgments, enabling dynamic intervention in generations. ACT (Wang et al., 2024b) constructs multiple classifiers through clustering and uses these classifiers to dynamically intervene in responses from different directions. SADI (Wang et al., 2024c) utilizes activation differences in contrastive pairs to precisely identify intervention position and dynamically steer model behavior by scaling element-wise activations. + +# 4.3 Results + +Results on TruthfulQA In Table 1, the Probe method demonstrates superior performance compared to the baselines on the TruthfulQA dataset. Compared to the ITI method, our Probe method does not require additional training on top of ITI and has achieved performance far exceeding that of the ITI method. This is because ITI applies the same intervention strength indiscriminately to all generations. This makes it impossible to assign adaptive intervention strength and may cause originally correct answers to be incorrectly altered due to the intervention. In contrast, our method adaptively determines the intervention strength and does not intervene in correct generations. Compared with other dynamic intervention methods such as ORTHO, CAST, ACT, and SADI, it can be noted that SADI also achieves good performance, which demonstrates that the dynamic steering vector is effective. ORTHO and CAST are mainly designed for safety-related scenarios, where it is relatively + +Table 3: Ablation study on TruthfulQA dataset. + +
MethodsTrue*InfoMC1MC2
Probe80.5648.7166.58
Intervention Strength: +All fixed strength +w/o Adaptive76.1138.3157.15
82.0842.9662.06
Intervention Position: +w/o Backtracking +After Question62.1135.0153.55
72.5541.8659.88
+ +easy to determine whether a query is harmful based solely on the query information. However, in domains such as truthfulness and faithfulness, it is difficult to anticipate whether the generated content will deviate based solely on the query, resulting in poor performance. Compared with these dynamic intervention methods, the superiority of our method can be further demonstrated. + +Results on Multiple-choice In Table 2, our method achieves consistently promising results across all datasets, and Probe method achieves the best performance. This demonstrates that adaptive intervention and using probes to select heads remain effective on multiple-choice datasets. Notably, CAA achieves better results than ITI, suggesting that layer-wise intervention can sometimes lead to better performance. SADI also performs well in multiple-choice tasks, which demonstrates the generalizability of the dynamic steering vector. + +# 4.4 Ablation Study + +We conduct an ablation study to show the effectiveness of the proposed components in Table 3. + +We first explore the ablation of intervention strength. "All fixed strength" refers to the results of applying the same intervention strength to all samples. "w/o Adaptive" refers to the results of applying the same intervention strength to samples that meet the intervention criteria. By comparing the results of "w/o Adaptive" with "All fixed strength", it is demonstrated that intervening only on samples that deviate from the desired behavior can improve performance. Comparing the results of the Probe with "w/o Adaptive" demonstrates the effectiveness of dynamic intervention strength. + +We then explore the ablation of the intervention position. "w/o Backtracking" means no backtracking is performed. "After Question" indicates that the representation of the question was used to decide whether to intervene and intervention strength. Comparing the results of the Probe with "w/o Backtracking" shows the necessity of the backtracking operation. The True*Info metric of "w/o Backtracking" is even lower than the baseline. This indicates that intervening after detecting the deviation from the desired behavior is too late. Comparing the results of the Probe with "After Question" indicates that relying solely on the hidden states of the question part to judge whether the subsequent response deviates from the desired behavior is insufficient. It is necessary to make judgments after generating part of the response. + +# 4.5 Generalizability across More Truthful Benchmarks + +We further investigate whether our method can generalize beyond the TruthfulQA benchmark. Specifically, we directly evaluate the classifier and steering vectors obtained from TruthfulQA on two datasets related to real-world truth, including Natural Questions (Kwiatkowski et al., 2019) and TriviaQA (Joshi et al., 2017). The Natural Questions dataset consists of 3,610 real queries issued to the Google search engine, annotated with answers and supporting Wikipedia pages. TriviaQA includes 95K question-answer pairs annotated by trivia enthusiasts. Following Li et al. (2023), all benchmarks are presented in a multiple-choice format. + +The results indicate that the Probe method outperforms the baseline and the ITI on both benchmarks, as shown in Table 4. This suggests that employing the Probe method does not undermine the model's performance in out-of-distribution truthful domains; Instead, it enhances the model's performance, particularly in domains closely related to the real-world truth. This indicates that the classifier and + +Table 4: MC2 on the Natural Questions and TriviaQA multiple-choice datasets using LLaMA2-7B-CHAT as the baseline. + +
MethodsNatural QuestionsTriviaQA
Baseline49.5461.22
ITI56.5066.49
Probe59.2567.55
+ +![](images/a64b81c42f4a7b1db85da49f74461cb262896ef5fa675f4f3512af64a1a57b7c.jpg) + +![](images/864ee0a4b5fff9428b4a796700dd4a3a5bf18cb9dcedbb57fe4545b07392ba2d.jpg) +Figure 5: The performance of various LLMs on the TruthfulQA benchmark. + +steering vector learned on the TruthfulQA dataset are not domain-specific, but rather general real-world truth features. + +# 4.6 Generality across More LLMs + +To evaluate the generality across various LLMs, we apply the Probe method to 6 LLMs, including LLaMA2 Touvron et al. (2023), LLaMA3.1 (Grattafori et al., 2024), and Qwen2.5 (Yang et al., 2024). The performance of baseline, ITI, and Probe on the TruthfulQA benchmark in Figure 5. + +The probing method effectively improves performance across 6 LLMs, demonstrating that our approach can generalize to models of different sizes, architectures, and whether or not they have been instruction-tuned. Notably, our method yields significant improvements of 24.61 and 20.03 in MC1 and MC2, respectively, on Qwen2.5-7B. Compared with the ITI method, our method achieves better performance on all 6 LLMs. On Qwen2.5-7B, there is an improvement of 23.50 on the MC1 and an improvement of 19.81 on the MC2. It is demonstrated that our method outperforms the ITI method in terms of performance across different LLMs. Scaling up the model size often results in better performance, indicating that it has acquired more knowledge related to truthfulness. Instruction-tuned models achieve better performance than non-instruction-tuned ones, indicating that instruction tuning helps enhance the truthfulness of the model. + +# 4.7 Model Analysis + +Effects of intervention strength and head number In Figure 6, we present the results of our method on the TruthfulQA dataset. We sweep two hyperparameters to control the intervention: the number of identified attention heads and the intervention strength. + +The truthful metrics MC1, MC2, and True show a trend where the results continuously improve as both the strength and the number of attention heads increase. This is because, with the continuous increase in the number of heads and the strength of intervention, the internal truthfulness of the model is constantly increasing, making the model more inclined to choose and respond with more truthful answers. It is worth noting that our method can be applied with greater strength, as it does not intervene with all samples, and the intervention strength is adaptive. + +![](images/fc509f03dd7b04069ec3856324e1502e858ff93dedfc48565b73b60a763232d8.jpg) +Figure 6: Results with varying intervention strength and numbers of attention heads on the TruthfulQA dataset with LLaMA2-7B-CHAT. + +![](images/a63fe66320733aee7d55254495a3a76401cb0c1b56cc5d3564ac507292d108db.jpg) +(a) Effects of threshold + +![](images/4c291eaadbcfa5879a51f0b6f04805302f81fda79f60348b2c8a9f828d5127b9.jpg) +(b) Effects of size of the training set +Figure 7: Effects of the threshold and the size of the training set on TruthfulQA dataset. + +The Info metric decreases as the number of heads increases and the intervention strength increases. This is because, with the continuous increase in intervention strength, the model internally encodes more truthfulness, leading the model to be more inclined to respond with answers that are truthful but lack informativeness. Due to the mutual restraint between the True metric and the Info metric, the True*Info metric achieves highest values when the True metric and the Info metric are relatively balanced, i.e., the number of heads is 18 and the strength is 60. Therefore, the optimal hyperparameters for MC2 and True*Info are not consistent. + +Effects of threshold To investigate the impact of different thresholds $\beta$ , we conduct experiments with thresholds set at 0.3, 0.4, 0.5, and 0.6. When $\beta$ is 0.3, 0.4, or 0.5, the overall performance does not vary significantly. Among them, True*Info achieves the best performance at 0.4, while MC1 and MC2 perform best at 0.5. When the threshold is set to 0.6, True*Info drops sharply, which may be due to excessive intervention. + +Effects of training set size To better investigate the impact of the size of the used dataset on the performance, we conduct experiments by utilizing $20\%$ , $40\%$ , $60\%$ , $80\%$ , and $100\%$ of the original dataset. As the size of the dataset increases, the metrics MC1, MC2, and True*Info generally exhibit an upward trend, essentially reaching their maximum values after utilizing approximately $80\%$ of the data, as shown in Figure 7 (b). Moreover, when only $20\%$ of the data is used, i.e., 496 samples, all three metrics can exceed the baseline, with more significant improvements observed in the MC1 and MC2 metrics. This indicates that our method does not require a large dataset. As the amount of data increases, performance improvements become both more consistent and more significant. + +# 5 Conclusion + +In this paper, we propose a Flexible Activation Steering with Backtracking framework. It dynamically decides whether and how strongly to intervene by probing the internal states of the LLM. Specifically, FASB first identifies attention heads that are consistent with the desired behavior and + +derive the steering vector and classifier. Then, FASB uses classifiers to dynamically determine whether and how strongly to intervene by probing the internal states of the LLM. Finally, we further propose a backtracking mechanism to regenerate in order to avoid deviating from the deviated behavior. Experimental results on the TruthfulQA dataset and six multiple-choice datasets to demonstrate the effectiveness of our method. + +# Acknowledgments and Disclosure of Funding + +We would like to thank the anonymous reviewers for their insightful comments. This work is supported by the JiangSu Natural Science Foundation under Grant No. BK20251989; the National Natural Science Foundation of China under Grants Nos. 62441225, 61972192, 62172208; the Fundamental Research Funds for the Central Universities under Grant No. 14380001. This work is partially supported by Collaborative Innovation Center of Novel Software Technology and Industrialization. + +# References + +Arditi, A., Obeso, O., Syed, A., Paleka, D., Panickssery, N., Gurnee, W., and Nanda, N. (2024). Refusal in language models is mediated by a single direction. In Advances in Neural Information Processing Systems 38: Annual Conference on Neural Information Processing Systems 2024, NeurIPS 2024. +Bai, Y., Jones, A., Ndousse, K., Askell, A., Chen, A., DasSarma, N., Drain, D., Fort, S., Ganguli, D., Henighan, T., Joseph, N., Kadavath, S., Kernion, J., Conerly, T., Showk, S. 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MethodsOpen-ended GenerationMultiple-Choice
True (%)Info (%)True*Info (%)MC1 (%)MC2 (%)MC3 (%)
Baseline66.8399.5166.5033.4151.0724.76
ITI94.4980.5576.1138.3157.1530.53
CAA71.6083.8460.0334.0352.7625.62
ORTHO67.9490.0961.2136.2352.8826.12
CAST67.6986.1758.3333.9051.1725.01
ACT---28.8045.20-
SADI-HEAD77.7298.5376.5835.9054.6526.99
Probe (Ours)93.8885.8180.5648.7166.5841.95
Prototype (Ours)88.3794.9883.9446.1464.3037.07
+ +Table 6: Results (MC %) on six multiple-choice tasks. + +
MethodsCOPAStoryClozeNLIMMLUSST2WinograndeAVG
Baseline64.460.263.560.292.250.265.1
ITI66.659.764.360.292.351.565.8
CAA66.663.564.962.692.250.966.8
ORTHO65.260.263.163.892.450.465.8
SADI65.460.565.161.892.351.866.1
Probe (Ours)90.093.580.062.492.854.178.8
Prototype (Ours)87.886.173.762.293.154.776.3
+ +# A Prototype Method + +Prototype Prototype method directly constructs two prototypes to achieve the above goal. Specifically, we compute the average activation within each class to obtain the prototype representations of $h$ -th attention head in the $\ell$ -th layer, as follows: + +$$ +\boldsymbol {\theta} _ {\text {p o s}} ^ {\ell , h} = \frac {1}{N _ {\text {p o s}}} \sum_ {i = 1} ^ {N _ {\text {p o s}}} \mathbb {I} \left(y _ {i} = 1\right) \mathbf {x} _ {i, - 1} ^ {\ell , h}, \quad \boldsymbol {\theta} _ {\text {n e g}} ^ {\ell , h} = \frac {1}{N _ {\text {n e g}}} \sum_ {i = 1} ^ {N _ {\text {n e g}}} \mathbb {I} \left(y _ {i} = 0\right) \mathbf {x} _ {i, - 1} ^ {\ell , h} \tag {5} +$$ + +where $\theta_{\mathrm{pos}}^{\ell,h}$ and $\theta_{\mathrm{neg}}^{\ell,h}$ represent the prototypes of the positive and negative classes, respectively. $\mathbb{I}$ denotes indicator function, and $N_{pos}$ and $N_{neg}$ represent the numbers of positive and negative samples, respectively. Then, we can use the softmax function over cosine similarities between activation and prototypes to define a classifier: + +$$ +p _ {\boldsymbol {\theta} ^ {\ell , h}} \left(\mathbf {x} _ {i, - 1} ^ {\ell , h}\right) = \frac {\exp \left(\cos \left(\mathbf {x} _ {i , - 1} ^ {\ell , h} , \boldsymbol {\theta} _ {\text {p o s}} ^ {\ell , h}\right) / \tau\right)}{\exp \left(\cos \left(\mathbf {x} _ {i , - 1} ^ {\ell , h} , \boldsymbol {\theta} _ {\text {p o s}} ^ {\ell , h}\right) / \tau\right) + \exp \left(\cos \left(\mathbf {x} _ {i , - 1} ^ {\ell , h} , \boldsymbol {\theta} _ {\text {n e g}} ^ {\ell , h}\right) / \tau\right)} \tag {6} +$$ + +where $\tau$ denotes the temperature. We also use the classifier to anchor the top- $k$ heads with high accuracy for intervention. Finally, we directly use the mean difference between positive and negative prototypes as the steering vector: + +$$ +\boldsymbol {\theta} ^ {\ell , h} = \boldsymbol {\theta} _ {\text {p o s}} ^ {\ell , h} - \boldsymbol {\theta} _ {\text {n e g}} ^ {\ell , h} \tag {7} +$$ + +It is worth noting that the prototype method is training-free. It only involves averaging the activations of the training set to obtain prototype vectors for constructing the classifier. + +Implementation Details In the Prototype method, for the TruthfulQA dataset, we intervene using the top-24 heads, set the threshold to 0.5, the number of backtracking steps to 10, the temperature to 0.1, and search for the intervention strength in the range of [25, 55] with a step size of 10. For the six multiple-choice tasks, the threshold is 0.5, the number of backtracking steps is 10, the temperature is 0.1, and the intervention strength search range is [0, 400] with a step size of 10. + +Results In Table 5, the Prototype method also demonstrates superior performance compared to the baselines on the TruthfulQA dataset. Compared with the ITI and ACT methods that require training a linear classifier, the Prototype method achieves improvements of $7.83\%$ and $17.34\%$ on MC1, and $7.15\%$ and $19.1\%$ on MC2, + +respectively. This demonstrates the generality of our method. It is worth noting that the Probe method performs better on the True and multiple-choice metrics, while the Prototype method achieves higher performance on the Info and True*Info metrics. This may be because the trained probe is more closely related to truthfulness. + +In Table 6, the Prototype method also achieves better performance than baselines on six multiple-choice tasks. On the COPA dataset and the StoryCloze dataset, the Prototype method achieves improvements of $23.4\%$ and $25.9\%$ compared with the model without intervention, while other methods only achieve marginal improvements. This further demonstrates its effectiveness. + +# B Algorithm + +The algorithm consists of two main processes. The first step is Heads Anchoring and Steering Vectors Inducing, which determines the heads that need to be intervened, the classifiers, and the steering vectors. The second step is Generation with Flexible Steering and Backtracking, which involves using classifiers to evaluate the internal activations of LLMs. When the internal activations exceed a certain threshold, a backtracking operation will be performed. After the backtracking, the activations will be regenerated by adding the activation vectors. Since our method also backtracks to the beginning when the number of generated tokens is less than $s$ , we start tracking from the $s$ -th token, as shown in Line 14 of the algorithm. + +Algorithm 1 The overall flow of FASB +Require: LLM, dataset $\mathcal{D}$ , intervention strength $\alpha$ , threshold $\beta$ , backtracking number $s$ , maximum number of generated tokens $M$ +1: Step1: Heads Anchoring and Steering Vectors Inducing +2: if Probe then +3: Train classifier $p_{\theta^{\ell ,h}}$ on dataset $\mathcal{D}$ ; $\triangleright$ Equation 1 +4: Use the parameters of the probe as the steering vector $\theta^{\ell ,h}$ . +5: else if Prototype then +6: Construct two prototypes from the dataset $\mathcal{D}$ ; $\triangleright$ Equation 5 +7: Obtain the classifier $p_{\theta^{\ell ,h}}$ ; $\triangleright$ Equation 6 +8: Obtain the steering vector $\theta^{\ell ,h}$ ; $\triangleright$ Equation 7 +9: end if +10: Step2: Generation with Flexible Steering and Backtracking +11: for $j = 1$ to $M$ do +12: Generate the $j$ -th token; +13: Evaluate the current activation using the classifier; $\triangleright$ Equation 2 +14: if $p(\mathbf{x}_{i,j}) > \beta$ and $j\geq s$ then +15: Backtrack $s$ tokens; +16: Calculate the intervention strength; $\triangleright$ Equation 3 +17: for $k = (j - s + 1)$ to $M$ do +18: Intervene on the current activation; $\triangleright$ Equation 4 +19: Generate the $k$ -th token; +20: end for +21: break; +22: end if +23: end for + +# C Effects of the Number of Tokens for Backtracking + +In order to better investigate the impact of the backtracking number on our method, we conduct experiments on the TruthfulQA dataset. + +As shown in Figure 8, we investigate the effect of different token numbers for backtracking (i.e., 2, 5, 10, 20) on the MC1, MC2, and True*Info metrics. MC1, MC2, and True*Info generally show an increasing trend with the increase in the number of backtracking steps. This is because, as the number of backtracking increases, our method can intervene in the model's internal activations earlier, thereby achieving improved performance. + +# D Fine-Grained Model Comparison + +We further propose two models based on Probe for fine-grained analysis. The difference between them and Probe lies in the detection location and the point at which intervention begins, resulting in different overheads. + +![](images/c21f9bbc1dd6f8eb4ba620f13b5da0055d437d2c749ba490254d1edc2f71d52b.jpg) +Figure 8: Effects of the number of tokens for backtracking. + +Table 7: Comparison with two fine-grained models. + +
MethodsTrue*InfoMC1MC2
Probe80.5648.7166.58
BTB81.6048.8367.62
GCBB81.9650.6767.48
+ +(1) The first method directly BackTracks to the Beginning (BTB). Specifically, when a deviation is detected during generation, BTB backtracks not just $s$ tokens for regeneration, but to the beginning. Our method generates $s$ additional tokens for texts that require backtracking, whereas this variant generates even more. (2) The second method performs detection after the Generation is Complete and then Backtracks to the Beginning (GCBB). Notably, this method incurs approximately double the overhead for texts that require backtracking. + +As shown in the results in Table 7, GCBB usually achieves the best performance, and both BTB and GCBB outperform our proposed Probe method on the TruthfulQA dataset. This is because GCBB has access to the full output of the LLM, allowing it to better determine the intervention strength. However, GCBB inevitably introduces additional overhead. BTB may also sometimes require full regeneration. The advantage of Probe is that it achieves good performance while maintaining stable and relatively low additional overhead. + +# E Results across TruthfulQA Categories + +TruthfulQA is split into 38 subcategories, including politics, language, education, psychology and others. We compare our method with the baseline method without intervention using the True*Info metric across all subcategories with 10 or more questions, where the subcategories are ranked in descending order based on their quantity within the dataset, as shown in Figure 9. + +Our method demonstrates significant enhancement across most subcategories, with the overall performance improvement showing uniform distribution rather than concentration in particular domains, thereby validating its efficacy. + +# F Results on Multi-hop QA dataset + +We conduct experiments on the more challenging multi-hop question answering WikiHop dataset (Welbl et al., 2018) to verify the effectiveness, as shown in Table 8. Our method achieves improvements of $5\%$ and $4.3\%$ on metrics MC1 and MC2, respectively. This further demonstrates the generalizability of our method. + +Table 8: Results on the WikiHop dataset. + +
MethodMC1(%)MC2(%)
LLaMA2-7B-CHAT45.2044.03
LLaMA2-7B-CHAT + Probe50.20 (+ 5.00)48.33 (+4.30)
+ +![](images/c838c8a8b09fed8456d6c4d69c8d4e955dd27ac4c51ed989ed74cf9cfb7e871d.jpg) +Figure 9: True*Info scores split across subcategories on LLaMA2-7B-CHAT, sorted by the difference between baseline and probe method. Subcategories with less than 10 questions are not shown. + +# G Generalizability on larger LLM + +We further explore the effectiveness of our method on larger LLMs by using QWEN2.5-32B-Instruct on the TruthfulQA dataset in Table 9. Our method still achieves significant performance improvements, demonstrating its effectiveness on larger LLMs as well. + +Table 9: Results on the Qwen2.5-32B-Instruct model. + +
MethodMC1(%)MC2(%)MC3(%)
Qwen2.5-32B-Instruct50.0066.3538.75
Qwen2.5-32B-Instruct +Probe69.0076.6759.59
+ +# H Deviation Positions Under Different Thresholds + +The Table 10 presents the detected deviation positions under different thresholds on the TruthfulQA dataset. We observe that the threshold has a significant impact on the intervention position. When the threshold is high, the intervention positions tend to occur significantly later, and the interventions are less frequent. This is because the model needs to track more generated content to make a more accurate judgment about whether a deviation occurs. + +Table 10: Distribution of tokens under different thresholds. + +
Threshold βPosition
0-1010-2020-50
0.4304949
0.516615975
0.621109222
+ +# I Results on Toxic Context Generation + +Since our method can flexibly manipulate various behaviors, we further conduct experiments on the RealToxicityPrompts dataset (Gehman et al., 2020) for toxic content generation, as shown in Table 11. Our method can effectively steer LLMs to generate toxic content, demonstrating its generality. This also highlights the vulnerability of current alignment methods and the potential risks of our approach. + +# J Comprehensive Hyperparameter Analysis of strength $\alpha$ and threshold $\beta$ + +We further conduct a more comprehensive exploration of the impact of strength $\alpha$ and threshold $\beta$ on the TruthfulQA dataset. + +Table 11: Results on the RealToxicityPrompts dataset. + +
MethodASR (↑)
LLaMA2-7B-CHAT42.0
LLaMA2-7B-CHAT + Probe46.4 (+4.4)
+ +Table 12: Impact of strength. + +
Strength αMC1 (%)MC2 (%)True (%)Info (%)True*Info (%)
0 (baseline)33.4151.0766.8399.5166.50
236.4352.9667.2484.3556.72
538.6354.9468.4684.3557.75
1543.7760.1473.7188.8665.50
2543.3362.0280.1891.0673.01
3543.3362.6682.0193.8876.99
4545.0463.8483.6193.8878.49
5045.0464.5287.1591.8080.05
6048.7166.5893.8885.8180.56
6550.1868.5995.1177.2473.46
7052.5170.8993.2676.3871.24
10055.7574.3597.8318.5118.11
50043.0375.4889.996.365.68
+ +As shown in Table 12, when $\alpha$ is small (e.g., $\alpha = 2,5$ ), it often fails to effectively improve the True metric and even leads to a decrease in the Info metric. This indicates that a low intervention strength fails to enhance truthfulness and even reduces informativeness. Conversely, an excessively large intervention strength (e.g., $\alpha = 100,500$ ) reduces the informativeness of the generated text. For instance, when $\alpha = 500$ , we observe that the Info metric drops to $6.36\%$ , indicating that the generated text lacks informativeness. An appropriately sized $\alpha$ often enhances the truthfulness of the generated content while causing less information loss. + +Then, we explore the impact of the threshold $\beta$ . As shown in Table 13, when $\beta$ is too large (e.g., $\beta = 0.7, 0.8, 0.9$ ), many samples would not be intervened, leading to lower performance on the MC1, MC2, and True metrics. For instance, the performance on these three metrics with $\beta = 0.7, 0.8, 0.9$ is significantly lower than that with $\beta = 0.6$ . When $\beta$ is too small (e.g., $\beta = 0, 0.2$ ), many samples would be backtracked and intervened at the first state judgment. In such cases, although performance may remain satisfactory due to our adaptive intervention strength, the method loses the flexibility to decide whether intervention is needed. + +Table 13: Impact of threshold. + +
Threshold βMC1 (%)MC2 (%)True (%)Info (%)True*Info (%)
0.046.6366.1590.4586.4278.16
0.248.9065.8980.6891.6973.98
0.346.8866.7790.4586.4278.17
0.448.7166.5893.8885.8180.56
0.550.8068.5590.9486.1778.37
0.648.9565.3775.1584.2163.29
0.744.7457.6966.0183.1354.87
0.835.4551.7665.0484.3554.86
0.935.2151.6465.0484.3554.86
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University of Science and Technology of China, China + +2. Anhui Province Key Laboratory of Digital Security, China + +{jungle0430@mail.,chenkj@}ustc.edu.cn + +Codes: https://github.com/Jungle0430/StegoZip + +# Abstract + +Generative steganography has emerged as an active research area, yet its practical system is constrained by the inherent secret payload limitation caused by low entropy in generating stego texts. This payload limitation necessitates the use of lengthy stego texts or frequent transmissions, which increases the risk of suspicion by adversaries. Previous studies have mainly focused on payload enhancement through optimized entropy utilization while overlooking the crucial role of secret message processing. To address this gap, we propose StegoZip, a framework that leverages large language models to optimize secret message processing. StegoZip consists of two core components: semantic redundancy pruning and index-based compression coding. The former dynamically prunes the secret message to extract a low-semantic representation, whereas the latter further compresses it into compact binary codes. When integrated with state-of-the-art steganographic methods under lossless decoding, StegoZip achieves $2.5 \times$ the payload of the baselines while maintaining comparable processing time in practice. This enhanced payload significantly improves covertness by mitigating the risks associated with frequent transmissions while maintaining provable content security. + +# 1 Introduction + +Steganography is an information-hiding technique that enables covert communication by imperceptibly modifying cover data to avoid detection by adversaries [1-3]. For example, to establish collaboration with target customers while preventing detection by competitors, a company may employ steganography to transmit commercial information through public channels by embedding trade secrets into innocuous cover data. Unlike cryptography, which secures content through encryption, steganography ensures security by eliminating physical or statistical traces of hidden information in the cover data, thereby avoiding suspicion from adversaries at the behavioral level [4-6]. + +Linguistic steganography, which exploits text as the most prevalent communication medium, as shown in Figure 1, typically follows two core phases during message encoding [7-9]: 1) Message Processing: preprocessing secret messages through compression, encryption, and format conversion. 2) Message Embedding: most of these methods adopt channel coding methods to embed messages while balancing imperceptibility and payload, exemplified by Syndrome-Trellis Codes (STC) [10] and Steganographic Polar Codes (SPC) [11]. After transmission of the stego texts via the public channel, authorized receivers can reconstruct the secret message by applying inverse transformations using shared keys. However, traditional modification-based methods [10, 11] invariably introduce detectable statistical discrepancies between the cover texts and the stego texts [12-15]. + +![](images/ea5f06cbcd58d2b05451814cb0716df7f0326046f32bbf0691eb8091703f0d21.jpg) +Figure 1: The general process of linguistic steganography. + +With breakthroughs in generative large language models (LLMs) [16], a transformative paradigm shift has emerged in the field of steganography. These models not only generate texts that are closely aligned with human text distributions but also, more importantly, explicitly output token-level sampling probability distributions. By embedding secret messages while maintaining the original distributions and sampling randomness, it is feasible to develop provably secure steganography [6, 17-24]. For example, Discop [23] generates multiple "distribution copies" for a given probability distribution and employs the index values of these copies to encode secret messages. SparSamp [24] achieves efficient message embedding by combining sparse sampling with message-derived random numbers, significantly reducing computational complexity while maintaining security. + +However, practical steganographic systems are still constrained by the relatively low secret payload, which is defined as the ratio of the secret message length to the stego text length. To transmit the secret message, the stego text often has to be considerably longer than the secret message itself. Consequently, the sender must either use lengthy stego texts or multiple transmissions to ensure the integrity of the secret message. These anomalous behaviors are more likely to expose the purpose of steganography or necessitate more complex compensatory actions. + +While state-of-the-art (SOTA) methods enhance payload through iterative embedding [18, 21-24] or sampling distribution optimization [25-27], they focus mainly on the message embedding phase, i.e., how to leverage the statistical characteristics of cover texts to embed secret messages more efficiently. This singular focus, however, is still bound by the limitations of entropy and overlooks critical opportunities for message processing optimization by LLMs, particularly in terms of redundancy elimination and semantic compression of secret messages before embedding operations. + +On the basis of these insights, we propose StegoZip, a framework designed to address the payload limitations in practical linguistic steganography through LLM-driven secret message automated processing. The framework consists of two key components: Dynamic Semantic Redundancy Pruning (DSRP) and Index-Based Compression Coding (ICC). DSRP uses LLMs to analyze semantic redundancy in original messages, dynamically removing low-information elements to generate pruned, low-semantic content. In addition, a restorer fine-tuned on public datasets can losslessly reconstruct the original semantic richness from the pruned content. Moreover, inspired by Shannon's information theory [28] and advancing existing LLM-based compression methods [29], ICC further compresses low-semantic content into compact index sequences. Furthermore, after pseudo-randomization, StegoZip maintains the security of the underlying steganographic algorithm during secret message embedding. In summary, our main contributions are as follows: + +- We reveal the communication risks arising from the payload limitations in practical linguistic steganographic systems, highlighting that inefficient message processing optimization restricts the capacity of secure covert communication. +- We propose StegoZip, a framework for secret message automated processing that enhances payload in SOTA steganographic systems while maintaining security during secret message embedding. +- By incorporating StegoZip into SOTA steganographic systems, we achieve $2.5 \times$ the payload of the baselines while preserving a comparable steganographic processing time for lossless reconstruction of secret messages, demonstrating practical viability for real-world deployment. + +# 2 Related Work + +# 2.1 Generative Linguistic Steganography + +Linguistic steganography hides secret messages within a cover text. Traditional methods, e.g., Syndrome-Trellis Codes (STC) [10], and Steganographic Polar Codes (SPC) [11], achieve this by modifying components of the cover text. However, these methods often induce statistical deviations + +from the natural distribution, making the stego text vulnerable to detection by adversaries. In contrast, generative language modeling has revolutionized the field by enabling the embedding of secret messages into generative data [17]. These models learn the underlying distributions of natural language and serve as effective sampling mechanisms, producing content that is increasingly statistically indistinguishable from human text. This capability forms the foundation for secure steganography, as it allows for secret messages to be embedded during the token generation process without disrupting the statistical properties of the output [18, 30]. By iteratively sampling from explicit probability distributions over tokens, generative models ensure both security and naturalness in the resulting stego text. + +Recent advances in generative linguistic steganography have capitalized on these principles. Notable examples include ADG [20], which partitions the vocabulary into clusters of similar probabilities and uses the cluster indices to represent secret messages. Meteor [21] employs range-reversible sampling that encodes messages as sampling interval offsets while compressing code length via shared prefixes. iMEC [22] attains near-theoretical embedding limits by iteratively optimizing message encoding paths based on minimum entropy coupling theory, maximizing embedding capacity with minimal distortion. Discop [23] takes a novel method by generating multiple "distribution copies" from a given probability distribution and encoding secret messages via indices of these copies. SparSamp [24] combines sparse sampling with message-derived random numbers, drastically reducing computational complexity while preserving high levels of security. + +While these state-of-the-art (SOTA) methods have significantly improved payload through iterative embedding [18, 21-24] or sampling distribution optimization [25-27], they are still constrained by the inherently low entropy in the probabilities of text generation. This limitation caps payload efficiency, as the distribution of natural language tokens leaves little room for substantial expansion. However, the advent of large-scale language models presents transformative opportunities not only in optimizing the embedding stage but also in rethinking the processing and compression of secret messages. In light of this, we propose StegoZip for advanced message processing and compression. + +# 3 Methodology + +# 3.1 Overview + +As shown in Figure 2, StegoZip operates through two LLM-driven components: Dynamic Semantic Redundancy Pruning (DSRP) and Index-Based Compressed Coding (ICC). The process begins with the use of LLMs to systematically eliminate redundant, low-information elements via semantic pruning. To prevent the removal of critical information, this step incorporates entity detection and self-checking mechanisms. Once semantic pruning is complete, the framework employs the LLM again to perform probability-driven index-based compression, producing rank sequences. These sequences undergo binary encoding and pseudo-randomization, resulting in pseudo-random bit streams. The resulting bit streams are then embedded into cover texts through a steganographic embedding algorithm, enabling secure and covert transmission over public communication channels. + +The authorized receiver, equipped with prior knowledge of the steganographic embedding method, binary encoding schema, and LLM configurations, performs the reverse transformations to decode the compressed messages. After successful extraction, a shared semantic restorer $\mathcal{R}$ , fine-tuned on public datasets, losslessly reconstructs the original secret messages in a context-aware manner. To maintain synchronization between the sender and receiver, the sender also uses $\mathcal{R}$ as the LLM for both DSRP and ICC. Further details of each module are described in the following sections. + +# 3.2 Private Restorer in StegoZip Framework + +The private restorer $\mathcal{R}$ serves as a core component of the StegoZip framework, derived through fine-tuning a base language model. In natural communication, rich semantic content enables the receiver to fully understand the sender's perspective. However, such semantic abundance introduces significant redundancy, which increases the payload burden during transmission over public channels. Thus, this redundancy limits practical applications in resource-constrained communication scenarios, e.g., covert communication. To address this, the framework harnesses the advanced language comprehension capabilities of LLMs to perform semantic pruning, retaining only the most critical elements of the secret messages for public transmission. + +![](images/37f01f87bb26532db4f7d45cc5b820feb67af09d58bdbbe6c37b89d01f0aeec5.jpg) +Figure 2: The framework of StegoZip comprises two core components: Dynamic Semantic Redundancy Pruning (DSRP) and Index-Based Compressed Coding (ICC). + +However, relying solely on human perception to interpret the pruned messages can lead to ambiguity or misunderstanding, as the compressed content may lack sufficient contextual clarity. To address this issue, the framework harnesses the context-aware capabilities of LLMs to reconstruct the original rich semantics from the compressed messages. Notably, deploying large-scale LLMs in covert communication scenarios is often impractical because of their substantial computational and resource requirements, whereas smaller models (e.g., 3B, 7B) lack the capacity to reliably comprehend and execute restoration tasks. To balance these constraints, the framework fine-tunes a smaller base LLM, creating a private shared restorer model $\mathcal{R}$ used by both the sender and receiver to ensure lossless synchronization and significantly enhance communication fidelity. The fine-tuning process involves three key steps: Self-Information Calculation, Semantic Pruning, and Instruction Fine-tuning. + +Self-Information Calculation. Instruction fine-tuning begins by constructing a dataset tailored for semantic restoration. This necessitates preparing paired data where the input comprises text with low semantic content, while the output retains rich semantic details. Given a public text dataset $\mathcal{D}_p$ , for each sample $x_p \in \mathcal{D}_p$ designated as output, its compressed content serves as the corresponding input. This is achieved by leveraging the concept of self-information from information theory to measure the information content of each lexical unit (e.g., words or any meaningful segments derived from tokenization) in $x_p$ . This metric quantifies the significance of a lexical unit based on its sampling probability as determined by the base language model. For each lexical unit $u \in x_p$ consisting of $k$ tokens $u = \{w^{(1)}, \dots, w^{(k)}\}$ , its self-information $I_{\mathrm{lex}}(u)$ is calculated as: + +$$ +I _ {\mathrm {l e x}} (u) = \sum_ {i = 1} ^ {k} I \left(w ^ {(i)}\right) = \sum_ {i = 1} ^ {k} \left(- \log P \left(w ^ {(i)}\right)\right) = \sum_ {i = 1} ^ {k} \left(- \log p \left(w ^ {(i)} \mid w _ {< t}\right)\right), \tag {1} +$$ + +where $p(w^{(i)} \mid w_{ \tau_ {\alpha}, \forall u \in x _ {p}\right) \mid x _ {p} \in \mathcal {D} _ {p} \right\}, \tag {2} +$$ + +where $\odot$ denotes element-wise multiplication, $\mathbf{1}(\cdot)$ is an indicator function, and $\tau_{\alpha}$ represents the $\alpha$ -quantile satisfying: + +$$ +\tau_ {\alpha} = \inf \left\{\tau \in \mathbb {R} \left| \frac {1}{| x _ {p} |} \sum_ {u \in x _ {p}} \mathbf {1} \left(I _ {\text {l e x}} (u) \leq \tau\right) \geq \alpha \right. \right\}. \tag {3} +$$ + +![](images/b0e508262c8a2a02dd65fbb0ccd2f72a94085ffd51f153826d2b6128e7ab4c46.jpg) +Figure 3: Fine-tuning restorer $\mathcal{R}$ + +Instruction Fine-tuning. After semantic pruning, the semantic-compressed dataset $\mathcal{D}_c$ , combined with the original dataset $\mathcal{D}_p$ , is used to construct the instruction + +fine-tuning dataset $\mathcal{D}_{ft}$ through template-based pairing, as illustrated in Figure 3. The base LLM is fine-tuned to reconstruct rich semantics by inserting words within square brackets “[ ]”, without altering or deleting any existing content in the compressed text. This ensures precise restoration while preserving the original structure. The fine-tuned restorer $\mathcal{R}$ , shared by both sender and receiver, enables lossless and synchronized semantic reconstruction in the StegoZip framework. + +# 3.3 Dynamic Semantic Redundancy Pruning + +With the shared private restorer $\mathcal{R}$ established, the StegoZip module integrates seamlessly into the steganographic system, employing a dynamic pruning mechanism through two phases similar to the restorer fine-tuning process: Self-Information Calculation and Semantic Pruning. + +For a secret message $m$ , its compressed form $m_{c}$ is derived using $\mathcal{R}$ and self-information processing: $m_{c} = m \odot \mathbf{1}\left(I_{\mathrm{lex}}(u) > \tau_{\alpha}'\right)$ . Here, $\tau_{\alpha}'$ is a dynamic threshold that adjusts based on the average self-information of $m$ and the empirical fine-tuning dataset $\mathcal{D}_{\mathrm{p}}$ : + +$$ +\tau_ {\alpha} ^ {\prime} = \tau_ {\alpha} \cdot \left(1 - \eta \cdot \frac {\bar {I} (m) - \bar {I} (\mathcal {D} _ {p})}{\bar {I} (\mathcal {D} _ {p})}\right), \tag {4} +$$ + +where $\tau_{\alpha}$ is the predefined threshold from Eq. (3). $\bar{I}(m)$ is the average self-information of secret message $m$ and $\bar{I}(\mathcal{D}_p)$ is the average self-information of $\bar{I}(x_p)$ from the public dataset $\mathcal{D}_p$ . This ensures dynamic adjustment of the pruning ratio based on the entropy of $m$ , preventing excessive pruning in short, high-entropy texts. Extremely high self-information values $(\infty)$ are ignored. + +However, this process risks losing critical information, e.g., names, numbers, or other key entities. To address this, entity detection is performed prior to pruning, ensuring that essential details are preserved intact, with their self-information designated as infinity $\infty$ . Additionally, to guarantee that the recipient can reconstruct the message without loss, the sender conducts a self-checking process. Using the restorer $\mathcal{R}$ , the sender attempts to reconstruct the original secret messages. Any portions that cannot be accurately aligned are replaced with their original content and marked with square brackets "[ ]", as shown in Figure 2. Through this careful preprocessing, the resulting compressed secret messages $m_c'$ retain low semantic redundancy while safeguarding crucial elements. Moreover, having been exposed to numerous instances of compressed text during fine-tuning, $\mathcal{R}$ is better equipped to compress text when using the same prefix in subsequent encoding tasks through ICC. + +# 3.4 Index-Based Compressed Coding + +After dynamic semantic pruning, the compressed message $m_c'$ is transformed into binary codes via probability-driven index encoding. Inspired by LLMzip [29], this method leverages the token prediction capabilities of the LLM to achieve high compression ratios. + +Let the tokenized sentence be represented as $\mathbf{W}_c = \{w_1, \dots, w_s\}$ of length $s$ , where each $w_t \in \mathbf{W}_c$ denotes as a token. The rank of each token $w_t$ is defined as its sorted index when the vocabulary $\mathcal{V}$ is ordered by descending conditional probability: + +$$ +r \left(w _ {t}\right) = 1 + \sum_ {w ^ {\prime} \in \mathcal {V} \backslash \left\{w _ {t} \right\}} \mathbf {1} \left(p \left(w ^ {\prime} \mid w _ {< t}\right) > p \left(w _ {t} \mid w _ {< t}\right)\right), \tag {5} +$$ + +where $\mathbf{1}(\cdot)$ is an indicator function. This probability-driven token-rank mapping ensures efficient compression, as tokens with higher probabilities are assigned lower ranks. Since the fine-tuned restorer $\mathcal{R}$ has been exposed to numerous instances of semantic pruning text with the same template prefix, it is particularly effective at encoding such texts, thereby achieving a higher compression rate. + +The rank sequence is then converted into bit format $\mathbf{B}$ via Huffman encoding, a lossless compression method that minimizes the bit form of frequent ranks. To ensure secure steganographic embedding, the bit sequence $\mathbf{B}$ is pseudo-randomized. This is achieved by performing an XOR operation between $\mathbf{B}$ and a pseudo-random binary key $\mathbf{K}$ , which is generated by a secure stream encryption algorithm, e.g., ChaCha20 [31]. The resulting pseudo-random bit stream $\mathbf{S}$ is then embedded into the cover text via a secure steganographic embedding function, ensuring provable security while maintaining the integrity of the compressed message. For a detailed proof of security, please refer to Appendix A. + +# 3.5 Secret Message Restoration + +The decoding framework enables lossless reconstruction of the secret message through invertible transformations mirroring the encoding process. For the stego text $x_{s}$ , the embedded bit stream S is extracted via the negotiated steganographic extraction function. The receiver then reverses the pseudo-randomization using cryptographically synchronized parameters and Huffman decoding with the same shared codebook to obtain the original rank sequences. Furthermore, the receiver replicates the index-based compression coding generation process via inverse rank-token mapping, converting each rank into its corresponding token to reconstruct the semantic pruning message $m_c'$ . + +Since $m_c'$ may lack sufficient semantic detail for complete comprehension, the shared restorer $\mathcal{R}$ employs instruction-guided semantic expansion to restore the full secret message. During this step, $m_c'$ is split into the compressed representation $m_c$ and the portion marked during the sender's self-checking process in the DSRP phase. The restorer synchronizes with the sender's self-checking process, producing the intermediary text $m_r$ with the same errors. Finally, the marked portions are used to correct any errors in $m_r$ , ensuring the complete restoration of the original secret message. + +# 4 Experiments + +# 4.1 Implementation Details + +LLMs. In this paper, we utilize two widely used LLMs: Qwen2.5-7B [32] and DeepSeek-R1-Distill-Llama-8B (DS-Llama-8B) [33] for index-based coding and restoration tasks, and greedy sampling is employed. To prevent fine-tuning that could compromise the security of the original generative steganography algorithm, we use separate LLaMA2-7B [34] for stego text generation. In this process, random sampling with a temperature of 0.9 is applied, without incorporating top- $p$ or top- $k$ sampling. We use LoRA [35] to fine-tune the base LLMs for two epochs, and more detailed experimental settings are shown in the Appendix C. + +Datasets. Text datasets are used to fine-tune the restorer and generate stego text. For fine-tuning, the IMDb [36] dataset (average length: 1,300 characters) is split into 25,000 training texts and 25,000 test texts, but only 2,000 are randomly sampled for testing. For the AGNews [37] dataset (average length: 241 characters), only the "business" category is selected and divided into 30,000 training texts and 1,900 test texts. For stego text generation, the WikiText-2-v1 [38] dataset is used. Texts are randomly sampled, and the first two sentences are extracted as prompts to guide generation. + +Baselines. In the main experiment, the parameters for the proposed Dynamic Semantic Redundancy Pruning are set to $\alpha = 0.4$ and $\eta = 1.0$ . To the best of our knowledge, existing linguistic steganographic systems do not specifically address message processing. Therefore, we adopt a common setup, using Huffman compression after tokenizing the original secret messages as the baseline message processing. The tokenizers [32, 33] we use employ Byte Pair Encoding (BPE) [39] based on bytes, which inherently provides some degree of compression. We consider SOTA methods for the underlying generative steganography: Discop [23] and SparSamp [24]. For a fair comparison, we evaluate performance using sentences that do not have token ambiguity [40, 41]. + +Evaluation metrics. We evaluate StegoZip from both efficiency and semantic retention perspectives: + +1. Efficiency: We divide the efficiency into payload and processing time. The payload refers to the ratio of the secret message length to the stego text length, which is the most important metric for assessing StegoZip's compression capability. The processing time encompasses the average encoding time, which spans from processing the secret message to the completion of generating the stego text, and the average decoding time, which involves extracting the bit stream from the stego text and restoring the original secret message. + +2. Semantic Retention: We evaluate the semantic retention from restored and compressed messages at the word and semantic levels via the metric Rouge $_1$ [42] and BERTScore [43]. Rouge-1 calculates the proportion of single words from the original secret message that appear in the target secret message, whereas BERTScore, which leverages BERT's contextual embeddings to calculate the similarity between the original and target secret messages, provides a measure of semantic similarity. Higher values of these metrics are better. + +Table 1: The efficiency of StegoZip: StegoZip significantly enhances the payload while maintaining a comparable steganographic processing time cycle. + +
ModelDatasetSteganographyPayload (%) ↑Encoding Time (s) ↓Decoding Time (s) ↓
Qwen2.5-7B [32]AGNews [37]Discop [23]18.9114.8214.93
+ StegoZip45.23 (↑ 26.32)12.94 (↓ 1.88)12.73 (↓ 2.20)
SparSamp [24]19.0310.5510.61
+ StegoZip44.49 (↑ 25.46)11.23 (↑ 0.68)11.15 (↑ 0.54)
IMDb [36]Discop [23]18.3070.2069.21
+ StegoZip49.64 (↑ 31.34)56.62 (↓ 13.58)54.91 (↓ 14.3)
SparSamp [24]19.1263.1263.04
+ StegoZip49.74 (↑ 30.62)55.67 (↓ 7.45)52.36 (↓ 10.68)
DS-Llama-8B [33]AGNews [37]Discop [23]18.7514.9814.30
+ StegoZip44.83 (↑ 26.08)12.33 (↓ 2.65)11.95 (↓ 2.35)
SparSamp [24]18.4810.1610.18
+ StegoZip43.60 (↑ 25.12)10.62 (↑ 0.46)10.61 (↑ 0.43)
IMDb [36]Discop [23]18.3368.1569.23
+ StegoZip49.31 (↑ 30.98)55.91 (↓ 12.24)53.47 (↓ 15.76)
SparSamp [24]19.7661.5560.93
+ StegoZip49.41 (↑ 29.65)49.56 (↓ 11.99)46.46 (↓ 14.47)
+ +Table 2: Before StegoZip can reconstruct secret messages losslessly (achieving a metric score of 1.0000), compressed secret messages still retain a high degree of semantic preservation. Below, R / C denotes the semantic retention of restored/compressed secret messages relative to the original. + +
ModelDatasetRouge1 (R / C)BERTScore (R / C)
Qwen2.5-7B [32]AGNews [37]1.0000/0.84851.0000/0.9633
IMDb [36]1.0000/0.94451.0000/0.9835
DS-Llama-8B [33]AGNews [37]1.0000/0.86581.0000/0.9664
IMDb [36]1.0000/0.91851.0000/0.9758
+ +# 4.2 Main Performance of StegoZip + +Efficiency of StegoZip. The experimental results in Table 1 demonstrate that the proposed StegoZip framework significantly enhances the original steganographic system, achieving a $2.5 \times$ the payload of the baselines. This improvement is attributed to the integration of Dynamic Semantic Redundancy Pruning (DSRP) and probability-driven Index Compression Coding (ICC), which collaboratively compresses lexical units in secret messages with high efficiency. When dealing with longer secret messages (the average character count of the IMDb dataset is five times that of the AGNews dataset), StegoZip demonstrates better compression performance due to the availability of richer contextual information. Complementing these results, Table 2 provides a rigorous evaluation of message fidelity, where both the restored and compressed secret messages are compared to the original using Rouge $_1$ (word-level similarity) and BERTScore (semantic-level alignment). The results confirm that StegoZip achieves lossless reconstruction of secret messages at the receiver end, whereas the compression process preserves the core semantic meaning of the original content. Overall, the StegoZip framework harnesses the advanced semantic understanding capabilities of LLMs to eliminate low-information units in secret messages, encoding them into a densely packed and secure binary bit stream. This ensures efficient representation without compromising the lossless decoding process. By doing so, the method significantly reduces the payload burden for covert communication over public channels, while simultaneously enhancing both security and efficiency. + +Time Consumption. Although StegoZip introduces additional preprocessing steps, including Self-Checking, ICC, De-ICC, and Restore, which may appear time-intensive, the results in Table 1 and Figure 4 reveal that the optimized payload efficiency ensures a comparable steganographic embedding and extraction time. This is attributed to the reduced amount of binary code that needs to be embedded per stego text, significantly decreasing the time required for both the embedding and + +![](images/494be819ef50eac4b3399635c0a979c6d0b66e207fbf86e0f3759069ef1332fa.jpg) +Figure 4: Comparison of time consumption across different stages of steganographic processing. + +![](images/3a39f2004281c9048ba03becbcb31d0f27ff34cde27eeb946abc6d3dbd72771a.jpg) + +![](images/3b4e67d3dfa392a5c4d8981b2c29fe2aa3e38b31e9480eab9589d86c705a673a.jpg) + +![](images/b3361ca194e818835aa52fb09c35e633a4171de10cb7bc3d2aacc306e321bfb9.jpg) + +![](images/e51aeebea9ad6052c9a9f97991b74c1110997dba5ba14d20bb829fc9d332eb31.jpg) +Figure 5: Effect of pruning quantile $\alpha$ . + +![](images/716284d1e378bb5e5695b97c9932ba4202be24faa6cd945e53271f4f57c3c12c.jpg) + +![](images/4b88e4b72f1bac85d5880ceea2e20ebefdf650ebc72fba6350c8f7e733bf3d6f.jpg) +Figure 6: Effect of adaptive coefficient $\eta$ . + +![](images/84c0384b39ea5cfcd152f37ba1f6b2d377ed12c0144fa36608c16f363a16f114.jpg) + +Table 3: Efficiency comparison of LLMs of varying sizes in compression and restoration. + +
Model SizePayload (%)↑Encoding Time (s)↓Decoding Time (s)↓Rouge1 ↓BERTScore ↓
Qwen2.5-0.5B40.218.538.560.92510.9801
Qwen2.5-1.5B41.159.239.080.89820.9753
Qwen2.5-3B41.929.569.470.88840.9705
Qwen2.5-7B45.2312.9412.730.84850.9633
Qwen2.5-14B46.0921.5421.510.84280.9611
+ +extraction processes. Moreover, the increased payload efficiency enhances security by reducing the amount of stego text required for transmission. At the behavioral level, this reduction minimizes the likelihood of arousing suspicion from adversaries under equivalent communication requirements when compared with traditional methods. These advancements position StegoZip as a practical and effective solution for balancing capacity, efficiency, and security in linguistic steganography systems. + +# 4.3 Parameter Tuning and Resultant Impact + +Pruning Quantile $\alpha$ . We investigate the impact of the self-information pruning quantile $\alpha$ on both the steganographic payload and the semantic preservation of compressed secret messages. Within the range $\alpha \in [0.25, 0.45]$ , as shown in Figure 5, the key semantic information in the compressed text is progressively reduced as $\alpha$ increases. A moderate pruning ratio enhances the payload as the pruning effect becomes more pronounced. However, excessive pruning disrupts the restorer $\mathcal{R}$ 's ability to effectively reconstruct the original semantics. This leads to increased errors during self-checking and more extensive corrections, ultimately resulting in a decreased payload. Striking an optimal balance for $\alpha$ is therefore critical, as it directly determines the trade-off between payload and semantic fidelity. + +Adaptive Coefficient $\eta$ . The adaptive coefficient $\eta$ plays an important role in shaping the performance of StegoZip within the DSRP module. As shown in Figure 6, the model navigates the trade-off between compression efficiency and semantic resilience as $\eta$ increases. Lower values of $\eta$ favor aggressive compression, maximizing efficiency but risking over-compression, which may result in the loss of essential semantic information and hinder accurate restoration. Conversely, higher $\eta$ values are more cautious, prioritizing the preservation of semantic integrity at the cost of reduced compression efficiency. Moreover, different samples exhibit varying levels of complexity and information density, necessitating dynamic adjustments to $\eta$ to meet diverse compression demands. This adaptability ensures that StegoZip achieves a balance between compression and semantic restoration, enhancing its versatility across diverse scenarios. + +Model Size of LLMs. We examine the influence of large language model size on performance in compression and restoration tasks, and the results are shown in Table 3. As model size increases, the payload clearly tends to increase. This can be attributed to larger models, with their more complex architectures and greater number of parameters, being better equipped to capture richer semantic + +Table 4: The generalization of the Restorer $\mathcal{R}$ across different domain datasets. + +
TrainSet\TestSetAGNews-businessAGNews-worldAGNews-MixedIMDbMixed
AGNews-business45.2345.3243.4850.3247.26
AGNews-world44.9047.6944.2849.5346.94
AGNews-Mixed46.0147.0345.0850.4947.70
IMDb40.8541.3141.5949.7444.87
Mixed44.8245.0743.8750.0145.73
+ +Table 5: Ablation study on the components of StegoZip, where "w/o" indicates not employed. + +
ModelDiscop+StegoZipw/o DSRPw/o instruction in ICCw/o R in ICCw/o ICCDiscop
Qwen2.5-7B45.2331.7842.7140.0523.1718.19
DS-Llama-8B44.8330.2241.1338.9122.5418.30
+ +information and patterns. As a result, they can effectively prune more redundant semantics during compression. However, the encoding and decoding times also increase with model size. Hence, in practical applications, larger models with higher payloads are better suited for tasks where real-time processing is not critical. On the other hand, smaller models, which offer faster processing times, are more appropriate for scenarios requiring quick responses. + +# 4.4 Cross-Domain Transferability Analysis + +Initially, fine-tuning the small-scale LLMs aimed solely to enable them to understand the restoration task, without accounting for the domain of the secret messages. Nonetheless, given the specialized nature of covert communications, interacting parties can often predict the domain of the covert information. We analyze the impact of domain differences between the dataset used to train the restorer and the test set on the payload, as shown in Table 4. For this study, we compare categories including business and world within the AGNews dataset [37], which share some similarities, against the IMDb dataset [36], which differs significantly in both content and style. Additionally, we test a mixed configuration where both datasets are combined in equal proportions. While cross-domain differences do not affect the lossless reconstruction of messages, they can slightly reduce the payload, and hybrid training substantially mitigates this effect. These results highlight the resilience of the StegoZip across domains, with hybrid training maintaining a high payload despite domain shifts. + +# 4.5 Ablation Study + +We conduct ablation experiments on the AGNews dataset [37] to assess the importance of each component in the StegoZip, as shown in Table 5. The results confirm that both DSRP and ICC are critical to the framework's performance. Notably, incorporating an instruction template during ICC enables the restorer $\mathcal{R}$ to predict subsequent compressed content more accurately. This improvement can be attributed to providing the restorer with a priori knowledge of the format and content of compressed secret messages, particularly tokens used for marking. These findings underscore the significance of each component in achieving both high compression and restoration fidelity. + +# 4.6 Steganalysis Experiment + +To further verify the security of StegoZip, we conduct steganalysis experiments. We generate 1,000 cover texts and 1,000 stego texts using LLaMA2-7B [34] and employ three steganalysis analyzers: TS-FCN [44], LSTMATT [45], and SeSy [12]. The datasets are split into training, validation, and testing sets with a ratio of 3:1:1. The experiments are configured with a learning rate of $3 \times 10^{-5}$ and trained for five epochs. To ensure robustness, the process is repeated five times, and the average accuracy on the test set is used as the steganalysis accuracy metric. For the restorer $\mathcal{R}$ , we use the fine-tuned Qwen2.5-7B [32] model. It is important to note that the provable security here refers to being indistinguishable from normal generated text, not from human-written text. Provably secure steganography involves disguising steganographic behavior as normal generated text, as detailed in Appendix A and in previous works [23, 24]. + +Table 6: Steganalysis accuracy with and without using StegoZip. + +
MethodTS-FCN [44](%)LSTMATT [45](%)SeSy [12](%)
Discop51.25±2.7750.70±3.0249.20±2.23
Discop + StegoZip50.90±2.1552.35±2.7151.30±0.92
SparSamp51.40±1.6449.85±1.9949.10±2.08
SparSamp + StegoZip49.80±1.4252.35±1.5750.05±1.66
+ +The classification accuracy of steganalysis is close to $50\%$ , as shown in Table 6, indicating that stego texts and cover texts cannot be reliably distinguished. The experimental results demonstrate that StegoZip maintains a comparable security level to scenarios without using StegoZip. This further validates the consistency of the security of steganographic systems equipped with StegoZip with that of existing underlying steganographic algorithms. + +# 5 Limitations + +While StegoZip significantly enhances the payload of linguistic steganographic systems, its reliance on LLMs for message compression and reconstruction inevitably introduces additional computational overhead, even if the overall steganographic processing time remains comparable. Furthermore, its dependency on high-precision LLMs limits its applicability in resource-constrained environments or scenarios lacking advanced computational infrastructure [46]. Future research could aim to optimize StegoZip's computational efficiency, making it more adaptable to diverse operational contexts. + +# 6 Conclusion + +In this paper, we introduce StegoZip, a novel framework that leverages large language models for dynamic semantic redundancy pruning and index-based compression coding. By incorporating it into advanced steganographic systems, we achieve a payload that is 2.5 times the original size while maintaining comparable steganographic processing time. This advancement not only enhances the efficiency of the steganographic embedding process but also reduces the frequency of communication between parties, significantly lowering the risk of suspicion by adversaries. StegoZip paves the way for secure and efficient covert communication, demonstrating the transformative potential of advanced processing techniques in modern steganographic frameworks. + +# 7 Acknowledgements + +This work was supported in part by the National Natural Science Foundation of China under Grant 62472398, Grant U2336206, Grant U2436601, and Grant 62402469. + +# References + +[1] David Kahn. The history of steganography. In International workshop on information hiding, pages 1-5. Springer, 1996. +[2] Niels Provos and Peter Honeyman. Hide and seek: An introduction to steganography. IEEE security & privacy, 1(3):32-44, 2003. +[3] Shashikala Channalli and Ajay Jadhav. Steganography an art of hiding data. arXiv preprint arXiv:0912.2319, 2009. +[4] Neil F Johnson and Sushil Jajodia. Steganalysis: The investigation of hidden information. In 1998 IEEE Information Technology Conference, Information Environment for the Future (Cat. No. 98EX228), pages 113-116. 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IEEE Signal Processing Letters, 26(4):627-631, 2019. +[45] Jiajun Zou, Zhongliang Yang, Siyu Zhang, Sadaqat ur Rehman, and Yongfeng Huang. High-performance linguistic steganalysis, capacity estimation and steganographic positioning. In International Workshop on Digital Watermarking, pages 80–93. Springer, 2020. +[46] Minhao Bai, Jinshuai Yang, Kaiyi Pang, Xin Xu, Zhen Yang, and Yongfeng Huang. Provably robust and secure steganography in asymmetric resource scenarios. In 2025 IEEE Symposium on Security and Privacy (SP), pages 1382-1400. IEEE Computer Society, 2025. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The key contributions and scope are outlined in both the section abstract and the introduction (Section 1). We introduce StegoZip, a framework that boosts payload in steganographic systems by $2.5\mathrm{x}$ while preserving security and efficient processing, addressing communication risks from payload limitations, and enabling practical real-world deployment. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [Yes] + +Justification: For paper writing, we strictly adhere to the principle of using LLMs only for minor language editing tasks, such as grammar correction and word choice refinement. For the development of our proposed framework, StegoZip, we have harnessed the capabilities of LLMs to efficiently perform critical tasks related to recovery, text compression, and stego text generation. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Proof of the Security of StegoZip + +# A.1 Provably Secure Steganography + +Empirical steganographic schemes (e.g., STC [10] and SPC [11]) inherently permit an adversary to distinguish between cover and stego texts with a non-negligible advantage, undermining their security. In contrast, provably secure steganography aims to either eliminate this advantage completely, achieving information-theoretic security [5], or reduce it to a negligible level within the framework of computational security [6]. + +Let the cover channel, denoted as $C_{\mathcal{H}}$ , represent the conditional probability distribution of cover signals $C$ given the history $\mathcal{H}$ . Assuming access to a perfect sampler $M$ that precisely adheres to the distribution $C_{\mathcal{H}}$ , we define $M_b^{C_{\mathcal{H}}}$ as the process that samples a segment of the cover output of length $b$ . A steganographic system (stegosystem) is formally described as a triple of algorithms (KGen, Embed, Extract), corresponding to key generation, embedding, and extraction, respectively. The key generation algorithm produces a key $\mathcal{K}$ as follows: + +$$ +\mathcal {K} \leftarrow \mathrm {K G e n} (\kappa), \tag {6} +$$ + +where $\kappa$ is a security parameter. Given a secret message $m$ , a history $\mathcal{H}$ , and access to the sampler $M$ , the embedding algorithm generates a sequence: + +$$ +s _ {1} \mid s _ {2} \mid \dots \mid s _ {s} \leftarrow \operatorname {E m b e d} ^ {M} (\mathcal {K}, m, \mathcal {H}), \tag {7} +$$ + +where the sequence length is $s$ . The extraction algorithm, which uses the negotiated key $\mathcal{K}$ and history $\mathcal{H}$ , reconstructs the hidden message $\widetilde{m}$ from the stego text: + +$$ +\widetilde {m} \leftarrow \operatorname {E x t r a c t} ^ {M} \left(\mathcal {K}, \operatorname {E m b e d} ^ {M} (\mathcal {K}, m, \mathcal {H}), \mathcal {H}\right). \tag {8} +$$ + +Within computational security, the notion of a distinguishing game is employed, wherein an adversary $W$ attempts to differentiate the cover distribution $C$ from the stego distribution $S$ . Specifically, the adversary seeks to distinguish between (i) samples generated through secret-message-driven embedding Embed and (ii) samples generated by a standard random sampling process $O$ consistent with the cover distribution. The adversary's distinguishing advantage is defined as: + +$$ +\mathcal {A} _ {C, S} ^ {\mathrm {S S}} (W) = \left| \Pr_ {\mathcal {K}, M, \text {E m b e d}} \left[ W ^ {M, \text {E m b e d} (\mathcal {K}, \cdot , \cdot)} = 1 \right] - \Pr_ {M, O} \left[ W ^ {M, O (\cdot , \cdot)} = 1 \right] \right|, \tag {9} +$$ + +where the probabilities are computed over the randomness of $\mathcal{K}$ , $M$ , Embed, and $O$ . A stegosystem is deemed computationally secure if, for every probabilistic polynomial-time adversary $W$ , this advantage remains negligible in the security parameter $\kappa$ : + +$$ +\mathcal {A} _ {C, S} ^ {\mathrm {S S}} (W) < \operatorname {n e g l} (\kappa). \tag {10} +$$ + +# A.2 StegoZip Maintains the Security of the Underlying Steganographic Algorithm + +To ensure that the bitstream processed by StegoZip can be securely embedded using established provably secure steganographic embedding methods, the bitstream must first undergo pseudorandomization. This is typically achieved through an XOR operation with a pseudo-random binary keystream generated by a secure stream encryption algorithm, e.g., CHACHA20 [31]. Below, we provide a proof that this process maintains the security of the underlying steganographic algorithm. + +The proof proceeds by contradiction. Assume that the stego text $x_{s}$ generated by the StegoZip framework and the normally generated cover text $x_{c}$ are distinguishable, that is: + +$$ +\left| \Pr \left[ \mathcal {A} _ {\mathcal {D}} \left(x _ {s}\right) = 1 \right] - \Pr \left[ \mathcal {A} _ {\mathcal {D}} \left(x _ {c}\right) = 1 \right] \right| \geq \delta , \tag {11} +$$ + +where $\delta$ is non-negligible with respect to the key $\mathcal{K}$ . For tokens $w$ generated at each step under history $\mathcal{H}$ , this assumption implies the following: + +$$ +\left| \Pr \left[ \mathcal {A} _ {P _ {\mathcal {H}}} \left(w _ {s}\right) = 1 \right] - \Pr \left[ \mathcal {A} _ {P _ {\mathcal {H}}} \left(w _ {c}\right) = 1 \right] \right| \geq \delta . \tag {12} +$$ + +In a generative provably secure steganographic embedding algorithm, the addition of a token to the stego sequence involves two sampling stages. The first stage, which is controlled by the steganographic embedding algorithm and driven by message bits, determines the region of sampling. For example, ADG [20] selects a cluster, Discop [23] determines which distribution copies to access, and SparSamp [24] decides whether to apply an offset. The second stage occurs within $v_{stega}$ , driven by pseudo-random numbers $r$ generated by a cryptographically secure pseudo-random number generator (CSPRNG). For simplicity, we let $S(r,P)$ denote sampling from $P$ via $r$ , and let $\mathcal{E}(m,P)$ represent the steganographic embedding algorithm, where $m$ is the secret message encrypted via the cryptographic algorithm mentioned before. + +Using the law of total probability, the adversary's decision on $w_{s}$ can be expressed as: + +$$ +\begin{array}{l} \Pr \left[ \mathcal {A} _ {P _ {\mathcal {H}}} \left(w _ {s}\right) = 1 \right] = \Pr \left[ \mathcal {A} _ {P _ {\mathcal {H}}} \left(w _ {s}\right) = 1 \mid \mathcal {A} _ {P _ {\mathcal {H}}} \left(v _ {s t e g a} ^ {w _ {s}}\right) = 1 \right] \Pr \left[ \mathcal {A} _ {P _ {\mathcal {H}}} \left(v _ {s t e g a} ^ {w _ {s}}\right) = 1 \right] \\ + \Pr \left[ \mathcal {A} _ {P _ {\mathcal {H}}} (w _ {s}) = 1 \mid \mathcal {A} _ {P _ {\mathcal {H}}} (v _ {s t e g a} ^ {w _ {s}}) = 0 \right] \Pr \left[ \mathcal {A} _ {P _ {\mathcal {H}}} (v _ {s t e g a} ^ {w _ {s}}) = 0 \right], \\ \end{array} +$$ + +where $w_{s} = \mathcal{S}(r,p_{stega}^{m})$ and $v_{stega}^{w_s} = \mathcal{E}(m,P)$ . Similarly, since the probabilities of tokens remain unchanged before and after steganographic embedding, a single normal generation of samples is equivalent to two independent sampling processes using random numbers. For the normal cover text, the adversary's decision on $w_{c}$ can be expressed as: + +$$ +\begin{array}{l} \Pr \left[ \mathcal {A} _ {P _ {\mathcal {H}}} \left(w _ {c}\right) = 1 \right] = \Pr \left[ \mathcal {A} _ {P _ {\mathcal {H}}} \left(w _ {c}\right) = 1 \mid \mathcal {A} _ {P _ {\mathcal {H}}} \left(v _ {\text {c o v e r}} ^ {w _ {c}}\right) = 1 \right] \Pr \left[ \mathcal {A} _ {P _ {\mathcal {H}}} \left(v _ {\text {c o v e r}} ^ {w _ {c}}\right) = 1 \right] \\ + \Pr \left[ \mathcal {A} _ {P _ {\mathcal {H}}} \left(w _ {c}\right) = 1 \mid \mathcal {A} _ {P _ {\mathcal {H}}} \left(v _ {\text {c o v e r}} ^ {w _ {c}}\right) = 0 \right] \Pr \left[ \mathcal {A} _ {P _ {\mathcal {H}}} \left(v _ {\text {c o v e r}} ^ {w _ {c}}\right) = 0 \right], \\ \end{array} +$$ + +where $v_{\text{cover}}^{w_c} = S(r_1, P)$ and $w_c = S(r_2, p_{\text{cover}})$ . Here, $r_1$ and $r_2$ are independent random variables corresponding to the two-stage sampling process, each of which is uniformly distributed as $r_1, r_2 \stackrel{\text{i.i.d.}}{\sim} U[0, 1)$ . + +To satisfy Eq. 12, the adversary must be able to distinguish between $\mathcal{E}(m,P)$ and $S(r_1,P)$ in polynomial time, or between $S(r,p_{stega}^{m})$ and $S(r_2,p_{cover})$ . However, since a CSPRNG is used in our synchronous sampling function, the generated $r$ is indistinguishable from a truly random sequence in polynomial time. Thus, the latter condition cannot hold. + +Regarding the former condition, the bitstream $m$ generated by StegoZip is pseudo-randomized, rendering it indistinguishable from a uniform random bitstream within polynomial time. Moreover, the employed provably secure steganographic algorithm guarantees that the adversary cannot achieve a non-negligible advantage $\delta'$ in distinguishing between $\mathcal{E}(m, P)$ and $S(r_1, P)$ in polynomial time: + +$$ +\left| \Pr \left[ \mathcal {A} _ {P _ {\mathcal {H}}} \left(v _ {s t e g a} ^ {w _ {s}}\right) = 1 \right] - \Pr \left[ \mathcal {A} _ {P _ {\mathcal {H}}} \left(v _ {c o v e r} ^ {w _ {c}}\right) = 1 \right] \right| < \operatorname {n e g l} (\kappa). \tag {15} +$$ + +Since neither condition can be satisfied, Eq. 12 is invalid, contradicting the assumption in Eq. 11. Therefore, StegoZip, when integrated with any computationally secure steganographic algorithm, constitutes a computationally secure steganographic system. + +# B Ethics Considerations + +In this paper, we propose the StegoZip framework to enhance the payload of linguistic steganography, specifically for scientific research and educational purposes. We strictly adhere to established scientific research regulations to ensure data privacy and security throughout the experimental process, and we rigorously avoid any violation of personal privacy or engagement in illegal activities. We are committed to responsibly advancing academic research in information security and ensuring that our contributions positively impact society. + +# C More Experiment Settings + +# C.1 Model Fine-tuning + +The restorer is fine-tuned to help small-scale LLMs grasp the restoration task effectively. To maintain consistency across multiple runs, a random seed of 42 is employed. The training process adopts a micro-batch size of 32 and an overall batch size of 64, necessitating the use of gradient accumulation steps computed as the ratio of the overall batch size to the micro-batch size. The model is trained for 2 epochs, with the learning rate set at $3 \times 10^{-5}$ . + +For the IMDb dataset, the sequence length is cut or filled at 512 tokens, while for the AGNews dataset, the sequence length is limited to 256 tokens. The LoRA (Low-Rank Adaptation) parameters are carefully configured with LORA_R = 16, LORA_alpha = 32, and a dropout rate of 0.05. Fine-tuning targets specific modules, including {qProj, kProj, vProj, oProj, gateProj, upProj, downProj}. To optimize computational efficiency while maintaining model performance, the model is loaded in int8 precision. The dataset is split into training and validation sets in a ratio of 4:1. + +The instruction used for fine-tuning, illustrated in Figure 3, is crafted to guide the model in performing text restoration tasks. Specifically, the instruction reads: "You are a text restoration specialist. Your task is to ONLY fill in the missing content within square brackets [] in the input text. Requirements: 1. Strictly preserve all existing text and punctuation outside brackets. 2. Maintain original text structure and formatting." To ensure stable fine-tuning and prevent the model from altering content outside the "[ ]", $10\%$ of the fine-tuning dataset is constructed with inputs and outputs that remain identical. + +To enable the model to reliably identify the conclusion of its response, an "[END]" marker is appended to the end of responses in the training dataset. During inference, this "[END]" flag also serves as a termination signal for the model's outputs. + +All our experiments are conducted on a hardware platform equipped with an Intel(R) Xeon(R) Gold 6130 CPU operating at $2.10\mathrm{GHz}$ , 256 GB of RAM, and NVIDIA A6000 GPU cards. Training the Qwen2.5-7B [32] and DeepSeek-R1-Distill-Llama-8B [33] models on the AGNews dataset requires approximately 6 hours, while training on the IMDb dataset takes around 10 hours. + +# C.2 Model Inference + +During the model inference phase, the model is loaded in the FP32 format to accommodate the high-precision probabilistic sorting required by index compression coding. When the restorer is tasked with reconstructing high semantic information, greedy sampling is employed to ensure deterministic and accurate outputs. We do not use the chain-of-thought feature of the DeepSeek-R1-Distill-Llama8B [33] model, given its status as a widely adopted text-generation model. Its language patterns and knowledge acquired during pre-training remain highly effective even after fine-tuning, particularly for restoration tasks requiring the handling of complex semantics. Furthermore, if inference functions or other advanced features are to be introduced in the future, the model's existing architectural foundation provides an extensible andizable framework, facilitating seamless integration and enhancement. After fine-tuning, the model will not initiate its response with “\n” at the beginning of every output. + +# C.3 Marker Compression Processing + +To manage vacant positions awaiting restoration, we represent them using brackets $[ ]$ . However, during the Index-Based Compressed Coding process, instead of transforming the $[ ]$ , into its rank, we replace it with a separator symbol to generate a rank sequence optimized for better compression. + +For example, consider the sequence $\left( {{x}_{1},{x}_{2},\lbrack ,{x}_{4},{x}_{5},\lbrack ,\lbrack ,\lbrack ,{x}_{9})}\right)$ ,where ${x}_{i}$ represents a lexical unit. Initially,the positions of "[]" are recorded,and the sequence is transformed into a rank sequence "39 23 3 19 6 4 5 3 7",which is stored as a string with ranks separated by the symbol "L". Next, the recorded positions of "[]" are replaced with consecutive separators, resulting in a modified sequence "39 23 19 6 19 7". The vacancies are then identified by the successive separators. + +This processed sequence is subsequently subjected to Huffman encoding, yielding a highly compressed bit stream. By adopting this method, we ensure accurate localization of vacant positions while significantly improving the compression rate. + +# C.4 Steganography + +Discop. We adhere strictly to the official open-source repository of $\mathrm{Discop}^2$ . Specifically, we utilize the Huffman tree-enhanced version to maximize the payload and seamlessly integrate it into our codebase. + +SparSamp. Similarly, we follow the official open-source repository of SparSamp3, adopting the configuration with block_size = 32 and incorporating it into our codebase. + +Stego Texts Generation. During the generation of stego text, random sampling with a temperature of 0.9 is applied to introduce controlled variability, while avoiding the use of top- $p$ or top- $k$ sampling techniques to maintain alignment with the original framework. + +# C.5 Token Ambiguity + +Mainstream provably secure generative linguistic steganographic methods [20-24] rely on the precise alignment of token paths between the sender and receiver to achieve successful decoding. However, this requirement becomes challenging in LLMs because of token ambiguity caused by byte-pair encoding (BPE) [39]. BPE generates non-prefix-free vocabularies, allowing multiple tokenization paths to represent the same input sequence. To address this issue, alternative strategies such as word-based tokenizers, character-based tokenization, or disambiguation algorithms [40, 41] can be employed to reduce token ambiguity. In our main experiments, we ensure fairness by evaluating performance via sentences without token ambiguity. + +![](images/960c6ac3f5935fb0fa15d0d68d24554e311838c416febc872b756ba197783864.jpg) +Figure 7: The impact of fine-tuning epochs on the StegoZip payload. + +![](images/55d85b88552b5c31006c87318a23804266378d89a5b4785e19f6d7d41952c4c8.jpg) +Figure 8: The impact of temperature in stego text generation on StegoZip payload. + +# D More Experiment Results + +# D.1 Epochs of Fine-tuning + +We analyze the impact of fine-tuning epochs for $\mathcal{R}$ to understand the core restoration task, as shown in Figure 7. The results of the AGNews [37] and IMDB [36] (Qwen2.5-7B [32]/DeepSeek-R1-Distill-Llama-8B [33] models) show that the performance saturates after two epochs, with minimal gains beyond, making two epochs optimal for efficiency and effectiveness. Moreover, this implies that the task is relatively simple for the model to learn. + +# D.2 Temperature in Generating Stego Texts + +We analyze the impact of temperature in stego text generation on the payload performance of StegoZip, as illustrated in Figure 8. Lower temperatures (e.g., 0.7) produce more deterministic outputs, which results in a reduced payload. Conversely, higher temperatures increase the variability in the generated text, thereby enhancing the embedding flexibility and improving the steganographic embedding payload capacity. Notably, the integration of StegoZip with Discop significantly boosts payload performance. Specifically, StegoZip in conjunction with Discop [23] achieves more than double the payload compared with using Discop alone, highlighting the remarkable effectiveness of StegoZip. + +# D.3 Visual Quality + +# Secret Messages: + +Treasuries Benefit on Spike in Crude Oil NEW YORK (Reuters) - Treasury debt prices climbed on Monday as investors bet oil prices near record highs might dent U.S. consumption and force the Federal Reserve to slow the pace of interest rate hikes. + +# Stego Text generated by Discop: + +Ernest William Hornung ( 7 June 1866 - 22 March 1921 ) was an English author and poet known for writing the A. J. Raffles series of stories about a gentleman thief in late 19th @-@ century London. Hornung 's portrait of the city of London and its inhabitants is one that has since come to be recognised as a classic, similar to Dicken's portrayal of the same area. Other reviewers of the same book describe a delicate, yet entertaining character which only Hornung could have portrayed. Chapter 7 - from Raffles of the Palace Guard by Harry H. Burns, who wrote some historical novels of the Scramble for Africa in the 1800s. The return of the Strand and other places of London to Raffles reveals the complex social mix of the city. By the 1890s, London had been transformed into a city for conspicuous consumption, with the rich lording it over the poor. Buildings - as Chancery Lane shows - where mostly taken over by the middle and upper-class and further alienated the working poor, while a few areas, like Covent Garden, remained as the traditional working-class area, with its bars, theatres, and clubs. In terms of the embellishment of London, however, Raffles cannot compete with the grand flamboyance of the Victorians. The atmosphere is still dark though, yet reminds us that it is only dark because Raffles is a figure of light. Raffles himself is described as the main characters of London, a character who thinks for others, and lives by the block like the other hundreds of bachelors, who all have names and individual personalities, except that they do not have the talent to be heroes. Stanley Richards wrote about some of this in his review of Raffles of the Palace Guard : "As + +Stego Text generated by Discop+StegoZip: + +Ernest William Hornung ( 7 June 1866 – 22 March 1921 ) was an English author and poet known for writing the A. J. Raffles series of stories about a gentleman thief in late 19th @-@ century London. And a much superior man of letters to most of the other popular novelists who now draw their living from the public .Hornung published nearly sixty novels and novellas and several short story collections, as well as contributing to some periodicals.His works were internationally popular, and his novels were translated into many languages, including Spanish , French and German .While the origins of the works of Arthur Conan Doyle 's detective Sherlock Holmes have been the subject of some controversy, the background to Hornung 's creation of Raffles has, \ No newline at end of file diff --git a/stegozipenhancinglinguisticsteganographypayloadinpracticewithlargelanguagemodels/images.zip b/stegozipenhancinglinguisticsteganographypayloadinpracticewithlargelanguagemodels/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..a8af39e44ae7ff8d83c3040f4e816e521352fce7 --- /dev/null +++ b/stegozipenhancinglinguisticsteganographypayloadinpracticewithlargelanguagemodels/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6bb50ba648c1a46b5e6ec7b9ae26d812b6175ead6a7b7e2ca041aa89d03475ae +size 668106 diff --git a/stegozipenhancinglinguisticsteganographypayloadinpracticewithlargelanguagemodels/layout.json b/stegozipenhancinglinguisticsteganographypayloadinpracticewithlargelanguagemodels/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..78d21bacbd70f8d1a0d0142fc5d80d02a84332af --- /dev/null +++ b/stegozipenhancinglinguisticsteganographypayloadinpracticewithlargelanguagemodels/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2bbc5d537d88d4d95311dcbf4acb69ce278a9247e235dda33f5cacbd874727f7 +size 782624 diff --git a/stellasubspacelearninginlowrankadaptationusingstiefelmanifold/e9bc7253-ad04-410b-9de7-92e8edcbb86d_content_list.json b/stellasubspacelearninginlowrankadaptationusingstiefelmanifold/e9bc7253-ad04-410b-9de7-92e8edcbb86d_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..cfe92a3f6e9f378a0a80df0d4039e5c2aad432d8 --- /dev/null +++ b/stellasubspacelearninginlowrankadaptationusingstiefelmanifold/e9bc7253-ad04-410b-9de7-92e8edcbb86d_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2465aa2b6dba22814cc297f9bd60fe6514a2bf3cc5104ea89b74c393f60b89ac +size 176072 diff --git a/stellasubspacelearninginlowrankadaptationusingstiefelmanifold/e9bc7253-ad04-410b-9de7-92e8edcbb86d_model.json b/stellasubspacelearninginlowrankadaptationusingstiefelmanifold/e9bc7253-ad04-410b-9de7-92e8edcbb86d_model.json new file mode 100644 index 0000000000000000000000000000000000000000..29f841f3f5d6a77fde0cab0dcc49b12a6e36353c --- /dev/null +++ b/stellasubspacelearninginlowrankadaptationusingstiefelmanifold/e9bc7253-ad04-410b-9de7-92e8edcbb86d_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6e76bfce128e5cc4dfedd529432cfc1b2578dc2ae0a80410d2924cfc95c750a9 +size 226666 diff --git a/stellasubspacelearninginlowrankadaptationusingstiefelmanifold/e9bc7253-ad04-410b-9de7-92e8edcbb86d_origin.pdf b/stellasubspacelearninginlowrankadaptationusingstiefelmanifold/e9bc7253-ad04-410b-9de7-92e8edcbb86d_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..18248aeba4b29a4ee9ca8856666434950c85f9ce --- /dev/null +++ b/stellasubspacelearninginlowrankadaptationusingstiefelmanifold/e9bc7253-ad04-410b-9de7-92e8edcbb86d_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:838e96c43d2e1738ef3e0791026781fa554480bb93e0725e73a9377c0fd81b4f +size 6167199 diff --git a/stellasubspacelearninginlowrankadaptationusingstiefelmanifold/full.md b/stellasubspacelearninginlowrankadaptationusingstiefelmanifold/full.md new file mode 100644 index 0000000000000000000000000000000000000000..6b282689c54a818536d3c159488c778681b374bc --- /dev/null +++ b/stellasubspacelearninginlowrankadaptationusingstiefelmanifold/full.md @@ -0,0 +1,830 @@ +# StellA: Subspace Learning in Low-rank Adaptation using Stiefel Manifold + +Zhizhong Li, Sina Sajadmanesh, Jingtao Li, Lingjuan Lyu* + +Sony AI + +Zurich, Switzerland + +{zhizhong.li,sina.sajadmanesh,jingtao.li,lingjuan.lv}@sony.com + +# Abstract + +Low-rank adaptation (LoRA) has been widely adopted as a parameter-efficient technique for fine-tuning large-scale pre-trained models. However, it still lags behind full fine-tuning in performance, partly due to its insufficient exploitation of the geometric structure underlying low-rank manifolds. In this paper, we propose a geometry-aware extension of LoRA that uses a three-factor decomposition $USV^{\top}$ . Analogous to the structure of singular value decomposition (SVD), it separates the adapter's input and output subspaces, $V$ and $U$ , from the scaling factor $S$ . Our method constrains $U$ and $V$ to lie on the Stiefel manifold, ensuring their orthonormality throughout the training. To optimize on the Stiefel manifold, we employ a flexible and modular geometric optimization design that converts any Euclidean optimizer to a Riemannian one. It enables efficient subspace learning while remaining compatible with existing fine-tuning pipelines. Empirical results across a wide range of downstream tasks, including commonsense reasoning, math and code generation, image classification, and image generation, demonstrate the superior performance of our approach against the recent state-of-the-art variants of LoRA. Code is available at https://github.com/SonyResearch/stella. + +# 1 Introduction + +The rise of large foundation models [2, 20] has transformed machine learning, driving breakthroughs across a diverse range of applications [54, 60, 36]. These models, often with billions of parameters, show exceptional performance in fields such as natural language understanding [21], computer vision [70], and multi-modal learning [62]. Yet, their substantial scale results in massive computational and storage costs, limiting broader adoption via task-specific fine-tuning [27]. + +To address this challenge, parameter-efficient fine-tuning (PEFT) methods, such as prefix tuning [32], prompt tuning [45], and adapter tuning [27], have gained considerable attention. Among these, Low-Rank Adaptation (LoRA) [27] is widely adopted due to its ability to efficiently adapt pretrained models to downstream tasks without altering the original network architecture or incurring extra inference costs. LoRA operates by learning low-rank adaptations to a selected set of linear layers while keeping the original weights frozen. Given a pre-trained weight matrix $W \in \mathbb{R}^{m \times n}$ , LoRA computes the adapted weight as $W + BA^{\top}$ , where $B \in \mathbb{R}^{m \times r}$ and $A \in \mathbb{R}^{n \times r}$ . Choosing a small $r \ll \min(m, n)$ significantly reduces the number of trainable parameters. Nevertheless, a performance gap often exists between LoRA and full fine-tuning due to the limited capacity of the low-rank matrices $A$ and $B$ to capture complex updates required for optimal task performance [22]. + +A common strategy to improve LoRA's performance is to guide its initialization by leveraging structural insights from the pre-trained model [67]. Particularly, singular value decomposition + +(SVD) has been used for uncovering informative subspaces in weight matrices, activations, or gradients [38, 59, 41]. SVD factorizes a given matrix $M$ into $M = U\Sigma V^{\top}$ , where the columns of $U$ and $V$ form orthonormal bases for the output and input spaces of $M$ , respectively, ordered by the importance dictated by the singular values in $\Sigma$ . This property makes SVD well-suited for identifying low-dimensional subspaces that preserve meaningful information from the pre-trained model. + +While SVD-based initialization methods for LoRA have shown promise, their influence is limited to the start of training, offering minimal guidance for the subsequent optimization process. This naturally leads to the question: whether explicitly optimizing the subspace throughout training can yield further performance improvements? Moreover, the existence of various heuristics for subspace selection—such as focusing on leading vs. trailing components [38, 58], or considering weights vs. gradients [38, 59]—suggests that intuition-driven manual subspace selection is suboptimal. This motivates a more principled approach to learn subspaces from data throughout training. + +To bridge this gap, we propose a novel approach that directly optimizes the input-output subspaces of LoRA during training. Our key insight is to mirror the structure of the truncated SVD by representing the low-rank adaptation for weight matrix $W$ as a three-factor formulation, + +$$ +W + U S V ^ {\top}, \tag {1} +$$ + +where $U \in \mathbb{R}^{m \times r}$ and $V \in \mathbb{R}^{n \times r}$ define the orthonormal bases for the output and input subspaces, respectively, and $S \in \mathbb{R}^{r \times r}$ captures the transformation between them. To ensure that the subspace parameters $U$ and $V$ remain orthonormal during training, we constrain them to lie on the Stiefel manifold—the set of matrices with orthonormal columns. This leads to our method, Stiefel Low-rank Adaptation (StELA), which performs LoRA using a subspace-aware formulation. By leveraging the geometric optimization on the Stiefel manifold, StELA maintains the geometric structure of the subspaces in training, allowing for principled and effective learning of low-rank adaptation. + +Recently, methods such as DoRA [35] found that decomposing the weight into magnitude and direction components can improve LoRA's performance. This aligns with our approach, as the Stiefel manifold constraint on $U$ and $V$ captures the direction, and $S$ models the magnitude. Besides, maintaining the orthogonality of low-rank matrices $U$ and $V$ during training is also beneficial for certain downstream analyses, such as adversarial robustness [50, 52]. + +We benchmark StelLA across a diverse set of domains, including natural language understanding, natural language generation, visual understanding, and visual generation. Compared with the state-of-the-art LoRA variants, StelLA consistently achieves superior performance across all evaluated tasks. Relative to the strongest baseline, StelLA achieves notable improvements: up to $+1.3$ accuracy points on commonsense reasoning, $+2.33$ on math and code generation, $+0.25$ on image classification, and a 7.11 point reduction in FID for text-to-image generation. + +Our contributions are summarized as follows: (1) We propose a novel three-factor representation for LoRA incorporating Stiefel manifold constraints, enabling the optimization of LoRA's input and output subspaces directly during training. (2) We present a flexible geometric optimization algorithm for the Stiefel manifold, allowing seamless integration with existing Euclidean optimizers. (3) We conduct ablation studies on the design choices and evaluate the impact of different geometric structures. (4) We verify the effectiveness of StellA across a variety of tasks and models, encompassing natural language understanding and generation, image classification, and text-to-image generation. + +# 2 Related Work + +LoRA Initialization Methods and the Use of SVD. Several recent works explore improved LoRA initialization using matrix decomposition techniques like SVD to better align the adaptation subspace with pretrained weights. PiSSA [38] and LaMDA [4] initialize adapters using the leading $r$ singular vectors from SVD of the pretrained weights, while MiLoRA [58] uses the trailing $r$ singular vectors. EVA [41] applies SVD to activations and adapts rank based on the spectrum, while LoRA-GA [59] uses SVD on gradients to align with task-relevant directions. Beyond SVD, OLoRA [10] employs QR decomposition and selects the first $r$ columns from the orthonormal matrix. These approaches exploit meaningful subspaces to improve initialization, whereas ours directly learns the optimal subspace during training, allowing for more flexible and effective task-specific adaptation. + +Geometric Constraints for LoRA. Zhang and Pilanci [66] use Riemannian geometry to precondition the LoRA optimization, whereas we optimize the adapter's subspaces using the Stiefel manifold, with a three-factor decomposition. Other methods impose orthogonality constraints on LoRA weights. OFT [46] learns orthonormal rotations of pretrained weights via Cayley parameterization. Spectral Adapter [67] fine-tunes within the top spectral subspace, with a variant that rotates leading singular vectors using orthonormal matrices. In contrast, our approach enforces orthogonality directly on the input and output subspace projections via Stiefel manifold optimization, offering a more flexible and expressive adaptation mechanism. Concurrently, PoLAR [34] also sets $U$ and $V$ on Stiefel manifold, but it uses a landing algorithm with infeasibility penalty instead of retraction for optimization. + +LoRA with Three Factors. TriLoRA [19] and MoSLoRA [61] use the three-factor formulation for LoRA, but they do not keep the orthogonality of the two projections during training. LoRA-XS [5] also uses the three-factor formulation, but they freeze the two projections and only train the middle $r \times r$ matrix for extreme efficiency. Both AdaLoRA [68] and GeoLoRA [53] adopt a three-factor formulation, with $U$ and $V$ constrained to be orthogonal matrices. AdaLoRA achieves orthogonality via regularization, while GeoLoRA uses gradient flow. However, their focus is to achieve rank adaptability by inspecting the singular values of $S$ . StELA can be readily combined with AdaLoRA's rank adaptation strategy by constraining $S$ to be diagonal, which we leave to future work. + +Other LoRA Variants. Various extensions have been proposed to improve LoRA's efficiency and performance. rsLoRA [28] introduces a scaling factor based on the square root of the rank for better stability. LoRA+ [23] accelerates convergence by applying separate learning rates to the two matrices. DoRA [35] and DeLoRA [7] decouple LoRA updates into magnitude and direction. ReLoRA [33] periodically merges adapter weights to improve expressiveness. QLoRA [15] integrates quantization with LoRA to minimize memory usage. These efforts are orthogonal to our work. + +# 3 Preliminaries + +Here we briefly review the related concepts in differential geometry and geometric optimization. For a more detailed introduction, we refer the reader to Edelman et al. [17], Absil et al. [1], Roy et al. [49], Becigneul and Ganea [6]. We also prepared an intuitive example in low dimensions in Appendix A to help readers understand the geometric concepts. The Stiefel manifold, denoted $\mathrm{St}(k,n)$ , is the set of all $n\times k$ matrices with orthonormal columns, i.e., $\mathrm{St}(k,n) = \{Y\in \mathbb{R}^{n\times k}\mid Y^{\top}Y = I_k\}$ . To optimize a function $f:\mathrm{St}(k,n)\to \mathbb{R}$ , we require tools from the Riemannian geometry. The tangent space at a point $Y\in \mathrm{St}(k,n)$ , denoted $T_{Y}\mathrm{St}(k,n)$ , consists of matrices $\Delta \in \mathbb{R}^{n\times k}$ satisfying $Y^{\top}\Delta +\Delta^{\top}Y = 0$ . A Riemannian metric defines an inner product on the tangent space, and the canonical metric in Stiefel manifold is $g_{Y}(\Delta_{1},\Delta_{2}) = \mathrm{tr}(\Delta_{1}^{\top}(I_{n} - \frac{1}{2}YY^{\top})\Delta_{2})$ . The Riemannian gradient $\mathrm{grad}_Y$ w.r.t. function $f$ can be computed from the Euclidean gradient $\nabla_Y$ by + +$$ +\operatorname {g r a d} _ {Y} = \nabla_ {Y} - Y \nabla_ {Y} ^ {\top} Y. \tag {2} +$$ + +If the Riemannian gradient is modified, e.g., by adding momentum, it may no longer lie in the tangent space of the manifold. To perform geometric optimization, we need to project it back to the tangent space using the projection $\pi_Y: T_Y\mathbb{R}^{n\times k}\to T_Y\mathrm{St}(k,n)$ + +$$ +\pi_ {Y} (\Delta) = \Delta - Y \operatorname {s y m m} \left(Y ^ {\top} \Delta\right), \tag {3} +$$ + +where $\mathrm{symm}(A) = \frac{1}{2} (A + A^{\top})$ symmetrizes a matrix. After taking a step along a tangent vector $\Delta$ , the resulting point $Y + \Delta$ typically leaves the manifold. Hence, a retraction is used to map it back. In this work, we use the polar retraction, defined as + +$$ +\rho_ {Y} (\Delta) = \operatorname {u f} (Y + \Delta), \tag {4} +$$ + +where $\mathrm{uf}(.)$ returns the orthogonal matrix in the polar decomposition. + +# 4 Subspace Learning in LoRA Using Stiefel Manifold + +We now present Stiefel Low-rank Adaptation (StellA), which learns the input and output subspaces of the adapter directly during fine-tuning. Given a pre-trained weight matrix $W \in \mathbb{R}^{m \times n}$ , the goal is to fit it to a downstream task using a three-factor low-rank adapter, + +$$ +\tilde {W} = W + \frac {\alpha}{r} U S V ^ {\top}, \tag {5} +$$ + +# Algorithm 1 StelLA: Stiefel Low-Rank Adaptation + +Require: Pre-trained weight $W \in \mathbb{R}^{m \times n}$ , loss function $\mathcal{L}$ , a Euclidean optimizer's step function 'step', rank $r$ , scale factor $\alpha$ , number of iterations $T$ . + +1: Randomly initialize $U_0 \in \mathrm{St}(r, m)$ and $V_0 \in \mathrm{St}(r, n)$ , set $S_0 \gets I_r$ . +2: for $t \gets 0$ to $T - 1$ do +3: Compute loss: $\mathcal{L}_t\gets \mathcal{L}(W + \frac{\alpha}{r} U_tS_tV_t^\top)$ +4: Compute Euclidean gradients: $\nabla_{U_t},\nabla_{S_t},\nabla_{V_t}$ +5: Convert Euclidean gradients to Riemannian: + +via automatic differentiation + +$\triangleright$ Equation (2) + +$$ +\operatorname {g r a d} _ {U _ {t}} \leftarrow \nabla_ {U _ {t}} - U _ {t} \nabla_ {U _ {t}} ^ {\top} U _ {t}, \quad \operatorname {g r a d} _ {V _ {t}} \leftarrow \nabla_ {V _ {t}} - V _ {t} \nabla_ {V _ {t}} ^ {\top} V _ {t}. +$$ + +6: Take tentative steps using the given optimizer's step function: + +$\triangleright$ e.g., using Adam + +$$ +\tilde {U} _ {t + 1} \leftarrow \operatorname {s t e p} \left(U _ {t}, \operatorname {g r a d} _ {U _ {t}}\right), \quad \tilde {V} _ {t + 1} \leftarrow \operatorname {s t e p} \left(V _ {t}, \operatorname {g r a d} _ {V _ {t}}\right), \quad S _ {t + 1} \leftarrow \operatorname {s t e p} \left(S _ {t}, \nabla_ {S _ {t}}\right). +$$ + +7: Project the perturbed gradients $\tilde{U}_{t + 1} - U_t,\tilde{V}_{t + 1} - V_t$ back to the tangent space: + +$\triangleright$ Equation (3) + +$$ +\Delta_ {U _ {t}} \leftarrow \pi_ {U _ {t}} \left(\tilde {U} _ {t + 1} - U _ {t}\right), \quad \Delta_ {V _ {t}} \leftarrow \pi_ {V _ {t}} \left(\tilde {V} _ {t + 1} - V _ {t}\right). +$$ + +8: Update and retract back to the manifold: $U_{t + 1} \gets \rho_{U_t}(\Delta_{U_t})$ , $V_{t + 1} \gets \rho_{V_t}(\Delta_{V_t})$ . + +$\triangleright$ Equation (4) + +9: end for +10: return Adapted weight: $\tilde{W}\gets W + \frac{\alpha}{r} U_T S_T V_T^\top$ + +where $U \in \mathrm{St}(r, m)$ provides an orthonormal basis for the output subspace, $V \in \mathrm{St}(r, n)$ provides an orthonormal basis for the input subspace, and $S \in \mathbb{R}^{r \times r}$ learns a mapping. $\alpha \in \mathbb{R}$ is a scale hyperparameter. The rank $r$ is often much smaller than the dimensions of $W$ , i.e., $r \ll \min(m, n)$ . + +Let $\mathcal{L}$ denote the task-specific loss. We optimize the following objective (for notational brevity, we express the objective for one StelLA layer): + +$$ +\min _ {U \in \operatorname {S t} (r, m), S \in \mathbb {R} ^ {r \times r}, V \in \operatorname {S t} (r, n)} \mathcal {L} \left(W + \frac {\alpha}{r} U S V ^ {\top}\right), \tag {6} +$$ + +leading to a constrained optimization over Stiefel manifolds for $U$ and $V$ , and an unconstrained optimization for $S$ . + +The optimization is carried out using Algorithm 1. The algorithm begins by initializing $U_{0}$ , $V_{0}$ , and $S_{0}$ (line 1). At training iteration $t$ , the loss is computed based on the adapted weight in the forward pass (line 3), and Euclidean gradients with respect to $U_{t}$ , $S_{t}$ , and $V_{t}$ are obtained (line 4) in the backward pass using standard automatic differentiation in deep learning frameworks. Since $U_{t}$ and $V_{t}$ must remain on the Stiefel manifold, their gradients are converted to Riemannian gradients using Equation (2) (line 5). These gradients, along with the gradient of $S_{t}$ , are passed to an existing Euclidean optimizer's step function, e.g., using SGD, Adam, or RMSProp, to generate tentative updates (line 6). The perturbed gradient by the optimizer is reverse-engineered by examining the difference $\tilde{U}_{t+1} - U_{t}$ . Since the perturbed gradient may not lie on the tangent space of the manifold, it is projected back to the tangent spaces at the current points (line 7). Then, the update is performed by using the polar retraction (line 8). The algorithm ensures that $U$ and $V$ evolve in the Stiefel manifold throughout training. After $T$ steps, the final adapted weight is returned (line 10). + +Initialization. Both $U$ and $V$ are initialized with random column-orthonormal matrices [51], and we initialize $S$ with the identity matrix. In prior work [38], when the adapter is initialized with non-zero values, the initialization is subtracted from the pre-trained weight matrix to simulate a zero-initialized adapter. We empirically find this trick to be unnecessary in our case. An in-depth ablation of initialization strategies is studied in Section 5.5. + +Implementation. Algorithm 1 is designed to be modular and flexible, enabling integration with any existing Euclidean optimizer. The 'step' function abstracts the optimizer's internal logic, such as momentum updates or adaptive learning rates, allowing StellA to seamlessly integrate with a wide range of optimizers, including SGD, Adam, and others, while cleanly separating geometric constraints from the choice of optimization algorithm. We implement StellA in PyTorch [43] using optimizer hooks. Specifically, line 5 is implemented as a pre-hook to the optimizer step, while lines 7-8 are implemented as a post-hook. Our implementation is readily integrable with HuggingFace's PEFT library [37], enabling easy adoption by the community. + +The polar retraction is computed via SVD, which is the most expensive operation in the algorithm. To improve efficiency, we stack all $U_{s}$ and $V_{s}$ with identical shapes across different layers and apply a batched SVD. This batched strategy yields $15 - 20 \times$ speed up in our experiments, effectively eliminating the computational bottleneck when scaling to large models with many adapted layers. Please refer to Appendix D for a detailed discussion. + +Complexity. StelLA adds $r(m + n) + r^2$ parameters per adapted layer, which is $r^2$ parameters more than LoRA. This additional memory footprint is negligible since $r \ll \min(m, n)$ . In our experiments, we show that the number of parameters in StelLA is comparable to LoRA and DoRA [35]. Nevertheless, in Section 5.5, we show that the superior performance of StelLA is not merely the result of this tiny increase in the number of trainable parameters but our geometric formulation by relaxing the orthonormality constraints. Regarding inference, similar to LoRA, we can merge the adapted weights into the original ones, resulting in a single weight matrix with no overhead. + +Gradient Scaling. Previous work such as $\mathrm{LoRA} + [23]$ showed that setting different learning rates for $B$ and $A$ in LoRA can improve convergence speed. The core idea is to ensure that both $B$ and $A$ are efficiently updated during training. Motivated by their insights, we balance the learning speed of $U$ and $V$ in SteLAA via gradient scaling. For a random unit vector $x\in \mathbb{R}^m$ , it is easy to show that the variance of each element is $\frac{1}{m}$ . Since the columns of $U$ and $V$ are unit vectors, their individual entries are expected to have magnitudes on the order of $1 / \sqrt{m}$ and $1 / \sqrt{n}$ , respectively. Adam-style optimizers, however, normalize gradients so that their coordinate-wise variance is $\Theta (1)$ [64]. This means that the learning speed—the ratio between the average magnitude of the effective gradient and the average magnitude of the parameter—for the entries of $U$ and $V$ are different when $m\neq n$ . This imbalance is particularly pronounced in feed-forward layers of LLMs, where the hidden dimension is typically enlarged and then shrunk by a factor of 4 [9], causing the taller matrix to learn two times faster. To compensate for this difference, before applying the projection operation (line 7 in Algorithm 1), we scale the gradients of $U$ and $V$ by $\sqrt{d / m}$ and $\sqrt{d / n}$ , respectively, where $d$ is a hyperparameter. In our experiments, we set $d$ equal to the hidden dimension of the input tokens. + +Comparison to Existing Geometric Optimizers. Existing geometric optimization methods, such as Riemannian SGD [49] and Riemannian Adam [6], can handle optimization over generic manifolds. However, these methods are intrinsically coupled to specific optimizers, as they rely on accessing and manipulating internal states such as momentum or adaptive learning rates. For instance, Riemannian Adam parallel transports the momentum vector across tangent spaces to maintain consistency in updates. This tight integration limits their generalizability to other optimization algorithms. In contrast, our approach decouples geometric constraints from the choice of base Euclidean optimizer by treating its update direction as a perturbation to the Riemannian gradient. The projection operator $\pi$ (line 7 in Algorithm 1) ensures that the perturbed gradient is mapped back to the tangent space of the current point. This allows us to use any Euclidean optimizer without modifying its internal logic. + +# 5 Experiments + +We conduct extensive experiments to evaluate the performance of StelLA on various domains: natural language understanding (commonsense reasoning), natural language generation (math and code generation), visual understanding (image classification), and visual generation (text-to-image), using a diverse set of model architectures, including LLaMA2 [55], LLaMA3 [20], ViT [16], and Stable Diffusion [48]. We compare StelLA with a diverse set of low-rank adaptation baselines, covering various methodological categories: LoRA [27] as the standard baseline; DoRA [35] as a strong LoRA variant; PiSSA [38] and OLoRA [10] for SVD-based initialization; TriLoRA [19] and MoSLoRA [61] for three-factor decompositions; and ScaledAdamW [66] to represent geometry-aware optimization. Finally, we perform ablation studies to analyze the effect of different components in StelLA. + +# 5.1 Commonsense Reasoning + +Models and Datasets. We evaluate the performance of StelLA on the commonsense reasoning benchmark, which assesses the reasoning capabilities of large language models across 8 sub-tasks. Following the setup of Liu et al. [35], we train on the combined data from all sub-tasks and evaluate on the test set. We fine-tune two popular LLM checkpoints, LLaMA2-7B [55] and LLaMA3-8B [20]. + +Table 1: Accuracy on the commonsense reasoning benchmark. All results are averaged over 3 runs. + +
ModelMethodParams (%)BoolIQPIQASIQAHellaS.WinoG.ARC-eARC-cOBQAAvg.
LLaMA2-7BLoRA0.82672.0283.4679.8790.4482.6984.8371.1981.5380.76
DoRA0.83872.6783.4879.8290.8283.5885.1671.2781.2081.00
PiSSA0.82671.1683.8979.1991.0082.8785.0969.4883.9380.83
OLoRA0.82671.1182.7078.6489.4181.4883.5868.1780.2079.41
TriLoRA0.82871.2380.9678.3380.9177.5981.7666.6979.8077.16
MoSLoRA0.82871.5483.8479.6090.5083.1984.4069.9680.4780.44
ScaledAdamW0.82672.2083.8679.6790.8082.4385.5570.5981.9380.88
SteLA0.82873.6284.8780.6491.4484.5086.4372.8484.3382.33
LLaMA3-8BLoRA0.70075.1688.1480.1895.4186.7490.8478.7087.0085.27
DoRA0.71075.3888.0179.9495.3586.2990.5479.6986.0785.16
PiSSA0.70074.6788.1280.5094.9885.2290.1578.8785.6084.76
OLoRA0.70074.4187.6879.5594.7985.4090.0478.2485.0084.39
TriLoRA0.70273.0986.6478.6493.4082.8887.7675.2684.3082.74
MoSLoRA0.70274.8888.4380.3195.5086.2690.0079.8685.8085.13
ScaledAdamW0.70075.2488.5780.2195.8185.1191.0980.5586.6085.40
SteLA0.70275.9189.8681.6896.4187.8291.9882.3487.8086.72
+ +Implementation Details. We compare StelLA against all the aforementioned baselines. For all methods, we insert low-rank adapters into the $Q$ , $K$ , and $V$ projections of the self-attention layers and the up and down projections of the feed-forward layers. For fair comparison, we fix the rank to 32, $\alpha$ to 64, batch size to 16, weight decay to 0, dropout to 0.05, and train for 3 epochs using AdamW. The learning rate is separately tuned for each method and follows a linear decay schedule. + +Results. Table 1 shows that StelLA yields consistent, model-agnostic gains on every commonsense sub-task. StelLA lifts the average accuracy of LLaMA2-7B from $81.0\%$ (the best baseline) to $82.3\%$ and that of LLaMA3-8B from $85.4\%$ to $86.7\%$ , corresponding to absolute improvements of about 1.3 points on both models. Crucially, the benefit is uniform: StelLA attains the top score on all eight datasets for both model sizes, showing that its geometry-aware adapters generalize across binary, causal and multiple-choice commonsense formats. + +# 5.2 Math and Code Generation + +Models and Datasets. To assess the generative capabilities of StELA, we conduct experiments on two representative natural language generation (NLG) tasks: mathematical reasoning and code generation. For math-related tasks, the models are fine-tuned on MetaMathQA [65] and evaluated on GSM8K [13] and MATH [25]. For code-related tasks, we train on CodeFeedback [69] and evaluate on HumanEval [11] and MBPP [3]. + +Implementation Details. We benchmark StelLA against three strong baselines: LoRA [27], DoRA [35], and PiSSA [38]. We follow the experimental protocol of Meng et al. [38]. Each method applies low-rank adaptation to all linear transformations in both self-attention and feedforward modules. For consistency, we standardize the training configuration across methods: rank is set to 128, $\alpha$ to 128, LoRA dropout and weight decay are both zero, and models are trained with the AdamW optimizer for 1 epoch using a batch size of 128. The learning rate follows a cosine decay schedule and is tuned individually per method to ensure optimal performance. + +Results. As summarized in Table 2, StelLA delivers the strongest overall performance (39.30 on average) across the math and code generation benchmarks, surpassing all three established baselines by a comfortable margin (up to +2.69 absolute points on average over DoRA). Importantly, it is the only method that ranks first or second on every benchmark, confirming StelLA's versatility and effectiveness in challenging natural language generation scenarios. + +# 5.3 Image Classification + +Models and Datasets. We assess the performance on image classification tasks using the Vision Transformer [16] pretrained on ImageNet-21K [14]. We fine-tune the Base and Large ViT on 8 + +Table 2: Results on math and code generation. All results are averaged over 3 runs. + +
ModelMethodParams (%)MathCodeAvg.
GSM8KMATHHumanEvalMBPP
LLaMA2-7BLoRA4.53164.6715.3330.0741.0037.76
DoRA4.55064.8715.3227.0039.2736.61
PiSSA4.53163.7316.6431.1038.2037.41
StELA4.58165.4316.5533.7341.4739.30
+ +Table 3: Comparison of image classification accuracy (%) on 8 datasets averaged over 3 runs. + +
ModelMethodParams (%)DTDEuroSATFlowers102Food101OxfordPedsSun397CIFAR10CIFAR100Avg.
ViT BaseLoRA0.7279.2098.9899.2290.0792.9475.6298.9292.6090.94
DoRA0.7578.5698.9199.1990.1593.4075.6798.9692.7890.95
PiSSA0.7277.2998.9398.8889.9993.2175.4198.9292.2890.61
StelLA0.7379.8998.9199.2290.1793.5476.2898.8892.7291.20
ViT LargeLoRA0.5380.1199.1199.3291.3094.4177.0899.2394.0691.83
DoRA0.5580.2199.0099.2891.2794.3077.1999.2994.1191.83
PiSSA0.5380.2199.1199.2091.1094.3377.1599.1193.9991.78
StelLA0.5481.5499.1799.1991.3394.1477.5199.1693.9292.00
+ +datasets: Caltech101 [18], CUB200 [57], Cars196 [29], Flowers102 [40], Food101 [8], OxfordPets [42], Sun397 [63], CIFAR10 and CIFAR100 [30], and measure the validation top-1 accuracy. + +Implementation Details. We include the following baselines for comparison: LoRA [27], DoRA [35], and PiSSA [38]. We adapt only the query and value matrices of the attention layers in the ViT model. For all methods, we fix the rank and scaling factor $\alpha$ at 16, use a batch size of 128, apply no weight decay, set dropout to 0.1, and train for 10 epochs with the AdamW optimizer and a linear learning-rate schedule. The learning rate itself is tuned independently for each model to ensure a fair comparison. + +Results. Table 3 summarizes the results of our experiments. We observe that StellA delivers the best average performance using both ViT-Base and ViT-Large models, outperforming all other methods across most datasets. Using ViT-Base, StellA attains the highest accuracy on five of the eight datasets and posts the best overall mean of $91.20\%$ , surpassing the strongest baseline, DoRA, by $+0.25$ points. Scaling up to ViT-Large, StellA achieves the best accuracy on four datasets and a mean of $92.00\%$ , outperforming the best baseline, LoRA, by $+0.17$ points. These results demonstrate that StellA is a strong contender for image classification tasks. + +# 5.4 Text-to-Image Generation + +Models and Datasets. To explore the effectiveness of StelLA in generative vision tasks, we fine-tune text-to-image models on five stylistically diverse datasets sourced from CivitAI [12]: Barbie, Cyberpunk, Elementfire, Expedition, and Hornify. Each dataset includes captions generated using the BLIP model [31]. We conduct experiments on two commonly used latent diffusion [48] based models—Stable Diffusion v1.5 and v2.0, which are equipped with a U-Net architecture composed of ResNet blocks [24]. + +Implementation Details. We follow standard LoRA fine-tuning recipes in Diffusers [56] by injecting LoRA parameters into the cross-attention layers of the U-Net. For benchmarking, we compare StellA with LoRA, DoRA, and PiSSA and report CLIP score [47] and FID [26], two metrics that respectively measure semantic alignment and visual fidelity. For all methods, we fix the rank and scaling factor $\alpha$ at 4, use a batch size of 8, apply 0.01 weight decay and train for 100 epochs with the AdamW optimizer and a cosine learning-rate schedule. The learning rate itself is tuned independently for each model to ensure a fair comparison. + +Table 4: Text-to-Image quantitative results on finetuning SD 1.5 and SD 2.0 to downstream tasks. In most cases, SteLLA achieves the best FID and on-par CLIP scores. + +
ModelMethodParams (%)BarbieCoreCyberpunkElementFireExpeditionHornify
FID ↓CLIP ↑FID ↓CLIP ↑FID ↓CLIP ↑FID ↓CLIP ↑FID ↓CLIP ↑
SD 1.5w/o finetune-208.1130.84145.3727.49253.6627.78180.1827.99212.9027.98
LoRA0.093175.4830.31127.5027.62202.4927.80156.3427.64180.4827.24
DoRA0.104175.0430.36127.1127.61200.7727.78155.8027.65179.5827.26
PiSSA0.093299.4917.32269.4416.88303.1818.82291.2217.55295.1517.29
StelLA0.093170.2530.15124.4627.73194.4127.83146.1227.56167.5327.23
SD 2.0w/o finetune-210.4531.12158.6827.42256.0127.82180.7427.86214.4128.15
LoRA0.096171.6830.72140.7727.85194.5327.91159.6627.53188.1127.20
DoRA0.107176.1630.87140.7127.94197.1227.92159.5427.54185.4027.19
PiSSA0.096315.6416.86272.6015.81284.2317.96267.8716.71294.5915.71
StelLA0.096171.8330.92135.0527.83194.7128.00154.0627.34177.5127.05
+ +![](images/3763e3932d310da2f1ee78ae6dfff5d4167a3d94e3af384058689c40f71d2622.jpg) +Train Examples + +A futuristic sci-fi scene in space, featuring a spaceship who is travelling, evoking an adventurous feeling. + +![](images/1054d72e51c58f10cc62dbd5065391626926a3fbbb092cb0ff8e5fa80380c8d9.jpg) + +A futuristic mythology scene in a mountain, featuring a cyber-dragon who is flying, evoking a powerful feeling. + +![](images/0f6954ad2407582855f1d2e6400ed09e7754abe383a3856f412713299662893b.jpg) + +A vintage nature scene in a garden, featuring a rose who is blooming, evoking a beautiful feeling. + +![](images/186abbb22fab79b08e000890155a1c2169a294ebc852884aab09d8a83712c3e7.jpg) + +A cartoon portrait scene in a desert, featuring a camel who is journeying, evoking a determined feeling. + +![](images/46249076ae35a62fa6798143ce830aacd197b0b5575ee749634309ae256b67b2.jpg) +Figure 1: Qualitative results of finetuning SD 1.5 on BarbieCore, Expedition and Hornify. + +A cartoon nature scene in a garden, featuring a dragon who is dancing, evoking a happy feeling. + +![](images/d7738bb1f948d9ed061d1b705e37565ea3cc02196b24c3e8fb7aaaa8991240f5.jpg) +Prompt +w/o finetune + +![](images/38c90f9da77ad8db7319db4d02036b3775fe72210d97a4da023810c0b8122aa8.jpg) +LoRA + +![](images/b92bd12533d5237219e7f514458a9db677262630e966842ea8812ff21c4bd2ac.jpg) +DoRA + +![](images/f062bd95538835bc8e18ae9b869f95b81f36c9ab9e7ec9704e5eddc4d4f15d71.jpg) +PiSSA + +![](images/ad5923284d49980fae862c010f78818a6bd3285dd5cc8a50e531f3e9ff444401.jpg) +StelLA (Ours) + +![](images/819563ca2f57e07308c2edf2d0737928e0797ecf9012e0d0b6b024e2d9261e2f.jpg) + +![](images/81f8425e11a8b707e7b668f072662081c3e457ee0319f4780e2d645a3f208331.jpg) + +![](images/35dbab4de8023ddefb61af850996c64ccebae8ab37e4fccd9fbca46755bde17b.jpg) + +![](images/083e044a43cc11004f5526b21aec9d8b2d4343b41ee25750d5c8f8c7ed0b2659.jpg) + +![](images/dc5674a81a0ddc05d124e7d0e451c4bf6f8faa356165eeba2af181cf61e367a4.jpg) + +![](images/df7e39955952b65ab4678b2096e537a54f4b3af51d881063fd441d6bc0b0ce5e.jpg) + +![](images/c4dc6a2bbd22b050981004000b6e9038fa792891207e4eea14f946509cd38359.jpg) + +![](images/217a1be028a4fb08a3edaaca5ee1d5b46df59703e92e5b475ae683d159f5650a.jpg) + +![](images/b1a331c0ebd0524efc65b591ebdb20e45d07dc564f193875a141b809d2309e95.jpg) + +![](images/b3eaf5cae18f1d977933e5150a9a86163dab32ff797fc2ee31e043afb68fdc89.jpg) + +![](images/7b7cda632cc20bf239a51580e1929a32915ce1c371a334edf1aaf65d3a10f164.jpg) + +![](images/819f833ccbac28f50380b5f41ce5b0c3bccbd6160da3ce4b11858f6d75a799ab.jpg) + +![](images/6e929e897caa4dd6907be5b75a533fae53a001c08d7d9b4c786feae2f49c19c0.jpg) + +![](images/c0db9eb765f88bceb535294f7e615ff47ced9c0e32313fd22f0ddab180242555.jpg) + +![](images/194a699f17d49ae18c52680c314a25e71a248707f2e4e887919489a5dda2fb81.jpg) + +Results. Quantitative results in Table 4 show that StellA consistently yields the lowest FID scores, often outperforming the baselines by a notable margin—particularly on datasets like Expedition and Hornify—while maintaining competitive CLIP scores. In addition, qualitative results shown in Figure 1 reveal that StellA generates images with strong stylistic consistency (e.g., preserving the original background aesthetics in BarbieCore) and high perceptual quality, demonstrating its strength in adapting generative models to downstream domains. + +# 5.5 Ablation Studies + +In this section, we analyze the design choices in StelLA. We compare the performance of alternative geometric structures, different initialization strategies, and study the effectiveness of the additional gradient scaling for the Adam optimizer. We also evaluate the effect of parallel transport introduced in Riemannian Adam [6] and the effect of an alternative choice for the retraction operator. The + +Table 5: Ablation study on design choices using the Commonsense with LLaMA3-8B. GS and PT denote gradient scaling and parallel transport, respectively. Results are averaged over 3 runs. + +
GeometryInitializationGSPTBoolQPIQASIQAHellaS.WinoG.ARC-eARC-cOBQAAvg.
StellANon-zero✓×75.989.981.796.487.892.082.387.886.7
Euclidean +QuotientNon-zeroN/A×74.088.080.494.985.189.578.185.184.4
75.788.681.096.186.990.980.086.485.7
StellAZero75.989.281.696.387.491.781.887.986.5
Pseudo-zero✓×72.887.180.894.886.089.378.284.884.2
SVD-major76.189.881.596.587.592.381.988.386.7
SVD-minor75.989.881.996.487.292.482.087.586.6
StellANon-zero××76.189.781.296.387.492.080.987.686.4
StellANon-zero✓✓76.289.681.496.487.491.981.787.186.5
+ +Commonsense reasoning benchmark with the LLaMA3-8B model is used for all ablation studies, and the results are summarized in Table 5. + +Geometric Structures. For each weight matrix $W$ to be adapted, there are three learnable matrices $U$ , $S$ , and $V$ in StELA. Since $U$ and $V$ are constrained to lie on the Stiefel manifold, StELA's geometry is a product of two Stiefel manifolds and a Euclidean manifold, namely, $\mathrm{St}(r,m) \times \mathbb{R}^{r \times r} \times \mathrm{St}(r,n)$ . To show the effectiveness of this geometry, we compare it with the following alternatives: + +- Euclidean: $\mathbb{R}^{m \times r} \times \mathbb{R}^{r \times r} \times \mathbb{R}^{n \times r}$ . The simplest geometry is a product of Euclidean spaces, which does not impose any orthonormality constraints on the factors. It is used in previous works such as TriLoRA [19] and MoSLoRA [61]. +- Quotient: $\mathrm{St}(r,m) \times \mathbb{R}^{r \times r} \times \mathrm{St}(r,n) / (\mathcal{O}(r) \times \mathcal{O}(r))$ . The three-factor decomposition of a low-rank matrix $M = USV^{\top}$ is not unique due to the equivalence relationship $(U,S,V) \sim (UO_{1},O_{1}^{\top}SO_{2},VO_{2}),\forall O_{1},O_{2} \in \mathcal{O}(r)$ , where $\mathcal{O}(r)$ is the orthogonal group. Factoring out this symmetry, we get a quotient space where each low-rank matrix is uniquely represented. We adapt StelLA to use this geometry along with the Riemannian metric defined in [39]. Details of the geometry and its optimization are discussed in Appendix B. + +Comparing the performance of these geometries in Table 5, we observe that the product space $\mathrm{St}(r,m)\times \mathbb{R}^{r\times r}\times \mathrm{St}(r,n)$ for StELA consistently outperforms the other two geometries across subtasks. This evidences that (1) The Euclidean three-factor geometry is not as effective as StELA's geometry, implying that the orthonormality constraints on $U$ and $V$ are beneficial for the low-rank adaptation task, and (2) the StELA's geometry is more effective than the quotient geometry, which is likely due to the fact that the Riemannian metric [39] on the quotient space was initially designed for the low-rank matrix completion problem rather than low-rank adaptation. + +Initializations. We refer to the initialization of StelLA introduced in Section 4 as non-zero initialization, and compare it with the following initialization strategies: + +- Zero. Initialize $S$ to be zero, and $U$ and $V$ to be random column-orthonormal matrices. +- Pseudo-zero. Same as the non-zero initialization, except that the original weight matrix is modified by subtracting the adapter's initialization. +- SVD-major. Initialize $U$ and $V$ to be the leading $r$ left and right singular vectors of the pretrained weight matrix, respectively. $S$ is initialized as the identity matrix. This setting aligns with the philosophy in PiSSA [38] and can be interpreted as StelLA + PiSSA. +- SVD-minor. Same as SVD-major, except that the trailing $r$ singular vectors are used. This setting aligns with the philosophy in MiLoRA [58] and can be interpreted as StELA + MiLoRA. + +From the results in Table 5, we observe that (1) zero initialization is not as effective as the nonzero initialization. We hypothesize the small value of $S$ creates small gradients for $U$ and $V$ at the beginning of training, leading to slow convergence. (2) Pseudo-zero initialization leads to the worst performance, presumably because it contaminates the pretrained weights with the adapter's + +Table 6: Comparison of the polar retraction with the exponential map on the commonsense reasoning benchmark. Results are averaged over 3 runs. + +
GeometryRetractionBoolQPIQASIQAHellaS.WinoG.ARC-eARC-cOBQAAvg.
StELAPolar75.9189.8681.6896.4187.8291.9882.3487.8086.72
StELAExponential Map75.9889.5281.1096.4288.2791.8382.8588.1386.76
+ +initialization. (3) SVD-major and SVD-minor initializations show similar performance as non-zero initialization. This showcases the robustness of our geometric optimization, as it can effectively learn the suitable subspaces regardless of the initialization. + +Gradient Scaling (GS) and Parallel Transport (PT). As we have used the AdamW optimizer, we evaluate the effectiveness of the gradient scaling strategy introduced in Section 4 and the parallel transport introduced in Riemannian Adam [6]. Table 5 indicates that our gradient scaling slightly improves the average accuracy from $86.4\%$ to $86.7\%$ , showing its effectiveness in balancing the learning rates of $U$ and $V$ . Moreover, implementing the parallel transport results in the vanilla Riemannian Adam optimization. However, it does not lead to any performance gain compared to our treatment of converting a Euclidean Adam into a Riemannian one in Algorithm 1. + +Alternative Choices for Retraction. Other than the polar retraction in Equation (4), the exponential map offers an alternative way of retraction. The exponential map is a locally defined operation that maps a tangent vector at one point to a new point on the manifold by following the geodesic in the given direction for a unit time. Intuitively, it traces the shortest curve on the manifold starting from a point $U$ along the direction of a tangent vector $\Delta$ . Formally, let $QR \coloneqq \Delta - UU^{\top}\Delta$ be the QR decomposition of $\Delta - UU^{\top}\Delta$ . Then, the exponential map $\exp_U(\Delta)$ can be computed by $\exp_U(\Delta) = UM + QN$ , where + +$$ +\binom {M} {N} := \exp \left( \begin{array}{c c} U ^ {\top} \Delta & - R ^ {\top} \\ R & 0 \end{array} \right) \binom {I} {0}, \tag {7} +$$ + +and $\exp$ is the matrix exponential (not to be confused with the exponential map $\exp_U$ ). + +While the exponential map is geometrically well-founded, it is more expensive to compute than the polar retraction. We compare the performance of polar retraction and the exponential map in Table 6. The two methods achieve nearly identical results (86.72 vs. 86.76). We adopt the polar retraction as the default choice in SteL.A due to its lower computational cost. + +# 6 Limitations + +While StelLA demonstrates strong empirical performance across a range of models and tasks, it also comes with several limitations. First, compared to standard LoRA, StelLA introduces a small computational overhead due to the three-component formulation, the gradient conversion, and the retraction. Second, we did not explore combining StelLA with complementary LoRA variants such as AdaLoRA [68]. These methods introduce orthogonal improvements such as rank scheduling. Finally, due to time and resource constraints, we leave the evaluation of StelLA on more model families, such as Mistral or LLaVA, and at very large scales, such as LLaMA2-70B, to future work. + +# 7 Conclusion + +We introduced StellA, a subspace-aware extension of Low-Rank Adaptation (LoRA) that explicitly learns input and output subspaces through a three-factor decomposition $USV^{\top}$ , with $U$ and $V$ constrained on Stiefel manifolds. By combining Riemannian optimization with a modular training algorithm that supports arbitrary optimizers, StellA offers a flexible and geometry-aware approach to parameter-efficient fine-tuning. 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In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2025. + +# A Intuitive Example about Stiefel Manifold + +To help readers understand the Stiefel manifold, we provide an intuitive example in low dimensions. In Section 3, we provided general equations for various geometric concepts. Here, we apply those ingredients to the following specific example. + +![](images/1ad37d40f33286dff48eae9ab89e4a60a268149555b98d0a15bd1d958965668b.jpg) +(a) The manifold $\mathrm{St}(1,2)$ and the tangent space $T_{Y}\mathrm{St}(1,2)$ at point $Y$ + +![](images/af067c455a7d1422d970cbc7449ca599a70125c989555dcc82488585f188a983.jpg) +(b) Converting a Euclidean gradient $\nabla_{Y}$ to the Riemannian gradient $\operatorname{grad} Y$ is a projection. + +![](images/98139ea73a5f563d2a95d17092bc910bbab1ac53b3a3da45720e280aa36e237a.jpg) +(c) Projecting a tangent vector $\Delta \in T_Y\mathbb{R}^2$ onto the tangent space of the manifold $T_{Y}\mathrm{St}(1,2)$ + +![](images/f608a21e79ecd03f247e501a744063e1dbf24d58dbb1eda6ff6157a832c2a3bd.jpg) +(d) Retraction $\rho_{Y}(\Delta)$ is the normalization of the vector $Y + \Delta$ to a unit vector. +Figure 2: An example Stiefel manifold $\mathrm{St}(1,2)$ : the unit circle on the two-dimensional plane. + +Manifold. Consider $\mathrm{St}(1,2) = \{Y = [y_1,y_2]^\top \in \mathbb{R}^{2\times 1}|Y^\top Y = y_1^2 +y_2^2 = 1\}$ , the Stiefel manifold of orthonormal 1-frames in $\mathbb{R}^2$ , which consists of all unit vectors in the plane, i.e., the unit circle. See Figure 2a for an illustration. + +Tangent Space. For a point $Y = [y_{1}, y_{2}]^{\top} \in \mathrm{St}(1, 2)$ , a tangent vector $\Delta = [\delta_{1}, \delta_{2}]^{\top}$ must satisfy the constraint $Y^{\top} \Delta + \Delta^{\top} Y = 0$ , which leads to the condition $y_{1} \delta_{1} + y_{2} \delta_{2} = 0$ , namely $Y \perp \Delta$ . The tangent space at $Y$ is the set of all vectors $\Delta$ that are orthogonal to $Y$ . See Figure 2a for an illustration. + +Riemannian Metric. For two tangent vectors $\Delta_1, \Delta_2$ at a point $Y$ , the canonical metric defines the inner product as + +$$ +g _ {Y} \left(\Delta_ {1}, \Delta_ {2}\right) = \operatorname {t r} \left(\Delta_ {1} ^ {\top} \left(I _ {n} - \frac {1}{2} Y Y ^ {\top}\right) \Delta_ {2}\right) = \operatorname {t r} \left(\Delta_ {1} ^ {\top} \Delta_ {2}\right) - \frac {1}{2} \operatorname {t r} \left(\Delta_ {1} ^ {\top} Y Y ^ {\top} \Delta_ {2}\right) = \Delta_ {1} ^ {\top} \Delta_ {2}, \tag {8} +$$ + +which equals to the standard inner product in $\mathbb{R}^2$ . The last equation holds because $Y^\top \Delta_2 = 0$ and $\Delta_1^\top \Delta_2$ is a scalar. + +Riemannian Gradient. Suppose the Euclidean gradient of a function $f$ at $Y$ is $\nabla f(Y)$ , which is abbreviated as $\nabla_{Y}$ . Then the Riemannian gradient $\operatorname{grad} f(Y)$ , abbreviated as $\operatorname{grad} Y$ , is given by + +$$ +\operatorname {g r a d} Y = \nabla_ {Y} - Y \nabla_ {Y} ^ {\top} Y = \nabla_ {Y} - (\nabla_ {Y} ^ {\top} Y) Y, \tag {9} +$$ + +which is the projection of the Euclidean gradient onto the tangent space at $Y$ . The second term $(\nabla_Y^\top Y)Y$ is the component of the Euclidean gradient that is not tangent to the manifold. See Figure 2b for an illustration. + +Tangent Space Projection. The projection of a vector $\Delta$ onto the tangent space at $Y$ is given by + +$$ +\pi_ {Y} (\Delta) = \Delta - Y \operatorname {s y m m} \left(\Delta^ {\top} Y\right) = \Delta - \left(\Delta^ {\top} Y\right) Y, \tag {10} +$$ + +which is the component of $\Delta$ that is tangent to the manifold. The second term $(\Delta^{\top}Y)Y$ is the component of $\Delta$ that is not tangent to the manifold. See Figure 2c for an illustration. + +Retraction. For a tangent vector $\Delta$ at $Y$ , the retraction is given by + +$$ +\rho_ {Y} (\Delta) = \operatorname {u f} (Y + \Delta) = \frac {Y + \Delta}{\| Y + \Delta \|}, \tag {11} +$$ + +which in the $\mathrm{St}(1,2)$ case is equivalent to normalizing the vector $Y + \Delta$ to a unit vector. See Figure 2d for an illustration. + +# B Details of the Quotient Geometry + +The three-factor decomposition of a low-rank matrix $M = USV^{\top}$ is not unique due to the equivalence relationship $(U, S, V) \sim (UO_1, O_1^{\top}SO_2, VO_2), \forall O_1, O_2 \in \mathcal{O}(r)$ , where $\mathcal{O}(r)$ is the orthogonal group. Each rank- $r$ matrix has infinite representations. For example, + +$$ +\left(U O _ {1}\right) \left(O _ {1} ^ {\top} S O _ {2}\right) \left(V O _ {2}\right) ^ {\top} = U \left(O _ {1} O _ {1} ^ {\top}\right) S \left(O _ {2} O _ {2} ^ {\top}\right) V ^ {\top} = U S V ^ {\top}. \tag {12} +$$ + +Factoring out this symmetry, we get a quotient space $\mathrm{St}(r,m)\times \mathbb{R}^{r\times r}\times \mathrm{St}(r,n) / (\mathcal{O}(r)\times \mathcal{O}(r))$ where each low-rank matrix is uniquely represented. + +Mishra and Sepulchre [39] proposed a Riemannian metric for this quotient space. We adapt StelLA to use this Riemannian metric and refer to this setting as the Quotient geometry. The Riemannian metric in [39] is induced by the block approximation of the Hessian of $\| USV^{\top} - W\|_{F}^{2}$ . For two tangent vectors $(\xi_U,\xi_S,\xi_V),(\eta_U,\eta_S,\eta_V)$ at a point $(U,S,V)$ , the Riemannian metric is defined as + +$$ +g _ {(U, S, V)} \left(\left(\xi_ {U}, \xi_ {S}, \xi_ {V}\right), \left(\eta_ {U}, \eta_ {S}, \eta_ {V}\right)\right) = \operatorname {t r} \left(S S ^ {\top} \xi_ {U} ^ {\top} \eta_ {U}\right) + \operatorname {t r} \xi_ {R} ^ {\top} \eta_ {R} + \operatorname {t r} \left(S ^ {\top} S \xi_ {V} ^ {\top} \eta_ {V}\right), \tag {13} +$$ + +where $SS^{\top}$ and $S^{\top}S$ act as preconditioners which improve the convergence in the low-rank matrix completion problem. Details of the geometric optimization in this space is discussed in [39], including the equations for the Euclidean gradient to Riemannian gradient conversion, the tangent vector projection $\pi_{(U,S,V)}$ , and the retraction $\rho_{(U,S,V)}$ . Note that Mishra and Sepulchre [39] used the conjugate-gradient optimization algorithm to solve the low-rank matrix completion problem. However, with the operators such as $\pi_{(U,S,V)}$ and $\rho_{(U,S,V)}$ defined, it is straightforward to adapt to the Riemannian Adam algorithm following Algorithm 1. Specifically, we use [39, Eq. (9)] to convert the Euclidean gradient to the Riemannian gradient, [39, Eq. (5)] for the projection onto tangent space, and [39, Eq. (7)] for the retraction. + +# C Details of Hyperparameters + +We provide the detailed hyperparameters used in our experiments to ensure full reproducibility. For each benchmark, all compared methods share a common set of hyperparameters—such as rank, batch size, weight decay, and training schedule—which are outlined in Section 5. The only exception is the learning rate, which we individually tune for each method to ensure fair comparison. In this section, we list the specific learning rates used for each algorithm across benchmarks. + +Commonsense Reasoning. We explore the learning rate in $\{0.0005, 0.0001, 0.00005, 0.00001\}$ for all algorithms, with the chosen values shown in Table 7. + +Math and Code Generation. The search space for the learning rates is $\{0.0005, 0.0002, 0.00002\}$ for all methods. We list the selected learning rates in Table 8. + +Table 7: Learning Rates for Commonsense Reasoning Experiments. + +
ModelMethodLearning Rate
LLaMA2-7BLoRA0.0001
DoRA0.0001
PiSSA0.00005
OLoRA0.00005
TriLoRA0.00001
MoSLoRA0.0001
ScaledAdamW0.0001
StellA0.0005
LLaMA3-8BLoRA0.0001
DoRA0.0001
PiSSA0.00005
OLoRA0.00005
TriLoRA0.0001
MoSLoRA0.0001
ScaledAdamW0.00005
StellA0.0005
+ +Table 8: Learning Rates for Math and Code Generation Experiments. + +
ModelMethodMathCode
LLaMA2-7BLoRA0.00020.0002
DoRA0.00020.0002
PiSSA0.00020.0002
StellA0.00050.0005
+ +Image Classification. The learning rate is tuned in $\{0.0001, 0.0005, 0.001, 0.005\}$ for all methods. The chosen learning rates are summarized in Table 9. + +Text to Image Generation. We search the learning rate over $\{0.0001, 0.0002, 0.0004, 0.0008\}$ for all methods, with the selected values reported in Table 10. For each method, we use the same learning rate for all five CivitAI datasets. + +# D Details of the Computational Cost for Algorithm 1 + +In Algorithm 1, for each of the Stiefel parameter (i.e., $U$ and $V$ ), it has the following operations: + +1. Converting the Euclidean gradient to the Riemannian gradient using Equation (2). +2. Projecting the update direction onto the tangent space using Equation (3). +3. Retracting the updated point back onto the Stiefel manifold using Equation (4). + +Among these, the first two steps involve only basic matrix multiplications and additions, and thus incur negligible computational overhead. The dominant cost arises from the retraction step, which requires a polar decomposition of the matrix $Y + \Delta$ . This polar decomposition is typically computed via singular value decomposition (SVD) as follows. + +Suppose the SVD of $(Y + \Delta) \in \mathbb{R}^{m \times r}$ with $m \gg r$ is expressed as $Y + \Delta = U\Sigma V^{\top}$ , where $U \in \mathbb{R}^{m \times r}$ and $V \in \mathbb{R}^{r \times r}$ are orthogonal matrices, and $\Sigma \in \mathbb{R}^{r \times r}$ is a diagonal matrix. Then the orthonormal factor in the polar decomposition is given by + +$$ +\operatorname {u f} (Y + \Delta) = U V ^ {\top}. \tag {14} +$$ + +Thus, the retraction step has a cost dominated by computing the SVD of an $m \times r$ rectangular matrix. + +Theoretically, the computational complexity of the polar retraction step is $O(mr^2)$ for tall matrices $(m \gg r)$ due to the SVD computation. But practically, there are efficient SVD algorithms for tall matrices that can reduce the computational cost significantly. Specifically, we use the gesvda + +Table 9: Learning Rates for Image Classification Experiments. + +
ModelMethodDTDEuroSATFlowers102Food101OxfordPetsSun397CIFAR10CIFAR100
ViT BaseLoRA0.0050.0050.0050.0010.0050.0010.0010.001
DoRA0.0050.0050.0050.0010.0050.0010.0010.001
PiSSA0.0050.0050.0050.0010.0010.0010.00050.001
StellA0.0050.0050.0050.0010.0050.0010.0010.001
ViT LargeLoRA0.0010.0050.0010.00050.0010.00050.0010.001
DoRA0.0010.0050.0010.00050.00050.00050.0010.0005
PiSSA0.0010.0010.0010.00050.00050.00050.00010.0005
StellA0.0050.0050.0010.0010.0010.0010.0010.0005
+ +Table 10: Learning Rates for Text to Image Generation Experiments. + +
ModelMethodLearning Rate (same for all 5 tasks)
SD 1.5 & SD 2.0LoRA0.0001
DoRA0.0001
PiSSA0.0001
StELA0.0008
+ +solver [44], which is a CUDA-accelerated SVD implementation that can handle tall matrices efficiently. Below is a micro-benchmark of different SVD solvers for the matrix shapes used in StelLA for the commonsense benchmark on LLaMA3-8B. Results in Table 11 show that the gesvda solver is significantly faster than the default GPU solver in PyTorch. + +The speed could be further improved by using batch processing to increase parallelism. Specifically, we can stack all the low-rank matrices $U$ and $V$ with the same shape into a batch, and then perform the polar retraction step on the batch of matrices. Table 12 is a micro-benchmark of the SVD using batch processing on the matrix shapes used in StelLA for the commonsense benchmark on LLaMA3-8B (there are 192, 64, and 64 matrices with shapes $4096 \times 32$ , $1024 \times 32$ , and $14336 \times 32$ , respectively). The results show that the batched SVD is significantly faster than the sequential one, and the polar retraction step is no longer the bottleneck of the training time. + +In practice, training a LoRA-adapted LLaMA3-8B model on a commonsense reasoning benchmark takes approximately 4.5 hours on a single H100 GPU, whereas training the same model with StELA takes around 5.2 hours, about only $15\%$ slower than vanilla LoRA. + +# E Discussion on Scale Stability + +The scale stability of LoRA refers to the property that neither the activations nor the gradients explode or vanish as the rank $r$ , input dimension $n$ , and output dimension $m$ grow infinity [28, 59]. More precisely, forward stability is achieved if, assuming the input to the adapter has i.i.d. entries with second moment $\Theta_{r,m,n}(1)$ , the output of the adapter also maintains a second moment of the same order. Similarly, backward stability is achieved if, when the gradient of the loss w.r.t. the adapter output has second moment $\Theta_{r,m,n}(1)$ , the gradient w.r.t. the adapter input also remains at $\Theta_{r,m,n}(1)$ . Letting $\gamma$ denote the scaling coefficient applied to the low-rank update, rsLoRA [28] shows that the original LoRA choice of $\gamma = \frac{\alpha}{r} = \Theta\left(\frac{1}{r}\right)$ is not rank stable. Instead, they propose that $\gamma$ should scale as $\Theta(1/\sqrt{r})$ to maintain stability. + +In StELA, we initialize $U$ and $V$ as random orthogonal matrices and $S$ to the identity matrix. This setup satisfies the main assumptions in Theorem 3.2 of [59], which analyzes the scale stability of LoRA. Following their analysis, we assess the scale stability of StELA at initialization as follows. + +Let the adapter compute $y = \gamma USV^{\top}x$ for an input vector $x$ , where $\gamma$ is the scaling factor. Since $S$ is initialized to be identity, this simplifies to $y = \gamma UV^{\top}x$ . The forward pass can be expressed component-wise as + +$$ +y _ {i} = \gamma \sum_ {j = 1} ^ {r} \sum_ {k = 1} ^ {n} U _ {i j} V _ {k j} x _ {k}, \quad 1 \leq i \leq m. \quad (\text {F o r w a r d}) \tag {15} +$$ + +Table 11: Runtime of different SVD solvers in PyTorch on H100 GPU. + +
Matrix ShapeDefault Solver (ms)GESVDA Solver (ms)Speedup of GESVDA
4096×320.9380.6411.46×
1024×320.7950.6321.26×
14336×321.5300.6542.34×
+ +Table 12: Runtime of SVD with/without batch processing on H100 GPU. The GESVDA solver is used for both batched and sequential processing. + +
Matrix ShapeBatch SizeBatched (ms)Sequential (ms)Speedup of Batched
192×4096×321925.80123.122.22×
64×1024×32641.6240.424.94×
64×14336×32642.8941.814.46×
+ +To analyze the scale, we compute the second moment: + +$$ +\begin{array}{l} \mathbb {E} \left[ y _ {i} ^ {2} \right] = \gamma^ {2} \sum_ {j _ {1} = 1} ^ {r} \sum_ {j _ {2} = 1} ^ {r} \sum_ {k _ {1} = 1} ^ {n} \sum_ {k _ {2} = 1} ^ {n} \mathbb {E} \left[ U _ {i j _ {1}} V _ {k _ {1} j _ {1}} x _ {k _ {1}} U _ {i j _ {2}} V _ {k _ {2} j _ {2}} x _ {k _ {2}} \right] (16) \\ = \gamma^ {2} \sum_ {j _ {1} = 1} ^ {r} \sum_ {j _ {2} = 1} ^ {r} \sum_ {k _ {1} = 1} ^ {n} \sum_ {k _ {2} = 1} ^ {n} \mathbb {E} \left[ U _ {i j _ {1}} U _ {i j _ {2}} \right] \mathbb {E} \left[ V _ {k _ {1} j _ {1}} V _ {k _ {2} j _ {2}} \right] \mathbb {E} \left[ x _ {k _ {1}} x _ {k _ {2}} \right] (17) \\ = \gamma^ {2} \sum_ {j _ {1} = 1} ^ {r} \sum_ {j _ {2} = 1} ^ {r} \sum_ {k = 1} ^ {n} \mathbb {E} \left[ U _ {i j _ {1}} U _ {i j _ {2}} \right] \mathbb {E} \left[ V _ {k j _ {1}} V _ {k j _ {2}} \right] \mathbb {E} \left[ x _ {k} ^ {2} \right] (18) \\ = \gamma^ {2} \sum_ {j = 1} ^ {r} \sum_ {k = 1} ^ {n} \mathbb {E} \left[ U _ {i j} ^ {2} \right] \mathbb {E} \left[ V _ {k j} ^ {2} \right] \mathbb {E} \left[ x _ {k} ^ {2} \right] (19) \\ = \gamma^ {2} r n \frac {1}{m} \frac {1}{n} \Theta_ {r, m, n} (1) = \frac {\gamma^ {2} r}{m} \Theta_ {r, m, n} (1). (20) \\ \end{array} +$$ + +Hence, the forward pass remains scale-stable in the beginning of the training if $\gamma = \Theta (\sqrt{\frac{m}{r}})$ . Equation (18) is derived from the independence of $x_{k_1}$ and $x_{k_2}$ , where $\mathbb{E}[x_{k_1}x_{k_2}] = 0$ when $k_{1}\neq k_{2}$ . Equation (19) and Equation (20) are derived from the fact that for a random unit vector $a\in \mathbb{R}^n$ , $\mathbb{E}[a_i a_j] = 0$ when $i\neq j$ and $\mathbb{E}[a_i^2 ] = \frac{1}{n}$ . + +For the backward pass, the gradient with respect to the input is given by $g = \frac{\partial\mathcal{L}}{\partial x} = \gamma V U^{\top}v$ , where $v = \partial \mathcal{L} / \partial y$ . Then, + +$$ +g _ {i} = \gamma \sum_ {j = 1} ^ {r} \sum_ {k = 1} ^ {m} V _ {i j} U _ {k j} v _ {k}, \quad 1 \leq i \leq n. \quad (\text {B a c k w a r d}) \tag {21} +$$ + +and the second moment becomes: + +$$ +\begin{array}{l} \mathbb {E} \left[ g _ {i} ^ {2} \right] = \gamma^ {2} \sum_ {j _ {1} = 1} ^ {r} \sum_ {j _ {2} = 1} ^ {r} \sum_ {k _ {1} = 1} ^ {m} \sum_ {k _ {2} = 1} ^ {m} V _ {i j _ {1}} U _ {k _ {1} j _ {1}} v _ {k _ {1}} V _ {i j _ {2}} U _ {k _ {2} j _ {2}} v _ {k _ {2}} (22) \\ = \gamma^ {2} \sum_ {j = 1} ^ {r} \sum_ {k = 1} ^ {m} \mathbb {E} \left[ v _ {i j} ^ {2} \right] \mathbb {E} \left[ U _ {k j} ^ {2} \right] \mathbb {E} \left[ v _ {k} ^ {2} \right]. (23) \\ = \gamma^ {2} r m \frac {1}{n} \frac {1}{m} \Theta_ {r, m, n} (1) = \frac {\gamma^ {2} r}{n} \Theta_ {r, m, n} (1). (24) \\ \end{array} +$$ + +which implies the backward pass is stable in the beginning of the training if $\gamma = \Theta (\sqrt{\frac{n}{r}})$ . + +In practice, we adopt the coefficient $\gamma = \frac{\alpha}{r}$ to be consistent with LoRA. While this choice does not guarantee theoretical scale stability, it does not negatively affect training in practice, as $\alpha$ can be adjusted empirically. Since $m$ , $n$ , and $r$ are fixed for a given model, tuning $\alpha$ allows us to ensure stable and convergent training across tasks. + +# NeurIPS Paper Checklist + +# 1. 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While learning rate schedules fundamentally govern the performance of different networks and tasks, particularly in budgeted-iteration scenarios, their design remains largely heuristic, lacking theoretical foundations. In addition, the optimal learning rate schedule requires extensive trial-and-error selection, making the training process inefficient. In this work, we propose the Unified Budget-Aware (UBA) schedule, a theoretically grounded learning rate schedule that consistently outperforms commonly-used schedules among diverse architectures and tasks under different constrained training budgets. First, we bridge the gap by constructing a novel training budget-aware optimization framework, which explicitly accounts for the robustness to landscape curvature variations. From this framework, we derive the UBA schedule, controlled by a single hyper-parameter $\varphi$ that provides a trade-off between flexibility and simplicity, eliminating the need for per-network numerical optimization. Moreover, we establish a theoretical connection between $\varphi$ and the condition number, adding interpretation and justification to our approach. Besides, we prove the convergence for different values of $\varphi$ . We offer practical guidelines for its selection via theoretical analysis and empirical results. Extensive experimental results show that UBA consistently surpasses the commonly-used schedules across diverse vision and language tasks, spanning network architectures (e.g., ResNet, OLMo) and scales, under different training-iteration budgets. + +# 1 Introduction + +Deep learning has achieved remarkable success across a wide range of domains, including computer vision and natural language processing. However, despite continual advancements in hardware technologies [45, 53], the training cost of neural networks has increased dramatically due to the growing scale of models and datasets [4, 8, 47, 48]. As a result, resource constraints, including computational power, memory, energy consumption, and time budgets, are emerging as significant bottlenecks in the training process [41, 58]. These challenges highlight the pressing need for budgeted training, which aims to achieve optimal model performance under fixed hardware and limited time. + +While existing budgeted training studies broadly address resource efficiency, a critical yet under-explored direction within budgeted training is achieving the best possible model performance under strictly fixed iteration constraints. This scenario is common and practically significant where practitioners work under limited computational or time budgets [30], and in extreme cases, models have to be completed within a few training iterations due to resource exhaustion. To formalize this specific research problem, we introduce the term 'budgeted-iteration training', distinguishing it from the broader scope of budgeted training. + +Budgeted-iteration training has received growing attention in research, given its significant real-world applicability. Several studies have developed relevant techniques that align with its goals. Smith et al. [43] propose the cyclical learning rate schedule (CLR), improving accuracy in fewer iterations without tuning [44]. Li et al. [30] introduce budget-aware adaptations for existing learning rate schedules. Chen et al. [5] propose a novel learning rate schedule called Reflected Exponential (REX). These approaches are primarily based on learning rate designing. Learning rate scheduling highlights key advantages: (i) It plays a critical role in general training of diverse neural architectures across tasks. (ii) It has demonstrated suitable and competitive in fixed training iteration budgets [20]. (iii) It is plug-and-play, requiring minimal adjustments to the underlying model architecture, which makes them easily adaptable to various deep learning frameworks. Leveraging these advantages, we adopt the learning rate design approach for budgeted-iteration training. + +Despite their advantages, most learning rate schedules, whether tailored for budgeted-iteration training or standard training performance, are still heuristic and lack rigorous theoretical grounding. In addition, existing schedules typically rely on manually designed rules or empirical tuning. Consequently, selecting an optimal schedule often involves extensive trial-and-error, incurring substantial cost in terms of time and computation [34]. + +A natural question arises: Does there exist a theoretically grounded, unified schedule that eliminates heuristic selection while maintaining robust performance across tasks, networks, scales and training budgets? + +In this paper, we provide an affirmative answer by proposing a theoretically grounded schedule. The proposed learning rate schedule should consistently outperforms existing schedules among diverse architectures and tasks under different constrained training budgets. By doing so, it avoids choosing suitable learning rate schedule after multiple trials for network training. + +To achieve this, we first bridge the gap by constructing a unified budget-aware training optimization framework, which incorporates the robustness to landscape curvature variations induced by data distribution, sampling, network architectures and optimization. The framework employs a min-max optimization approach to guarantee performance under worst-case conditions, since the minimization of learning rate is operated on an upper bound. Then we obtain numerical solutions by gradient projection methods. To eliminate the need for repeated numerical optimization when applying our method to different networks, we propose a universal parametric function that approximates numerical solutions. We nominate the resulting schedule Unified Budget-Aware (UBA) schedule. It requires tuning only a single hyper-parameter $\varphi$ , reducing the overhead of per-network numerical optimization. Moreover, we establish a theoretical connection between $\varphi$ and the condition number, adding interpretation and optimization difficulty-aware theoretical grounding. Besides, we prove the convergence for different values of $\varphi$ . These theoretical analysis along with empirical results offer practical guidelines for $\varphi$ selection. We evaluate UBA through comprehensive experiments across vision and language tasks, spanning diverse architectures and iteration budgets. Specifically, for vision tasks, the UBA schedule demonstrates consistent superiority over baselines across all evaluated datasets and model scales under varying training iterations. For language tasks, we validate the effectiveness of UBA through extensive benchmarks with OLMo model(36M, 73M, 150M and 300M parameters). Results show that UBA achieves state-of-the-art performance across approximately half of the benchmarks, and consistently outperforms baselines on the average scores. Moreover, our performance improvements come with negligible computational overhead, which constitutes one of its key advantages. + +# Main contributions: + +1. We construct a unified budget-aware training optimization framework that inherently adapts to landscape curvature variations, enhancing training robustness of leaning rates. + +2. We propose the Unified Budget-Aware (UBA) schedule from our constructed optimization problem, controlled by a single hyper-parameter $\varphi$ that provides a trade-off between flexibility and simplicity, i.e. adaptive curvature adjustment and minimal tuning cost. +3. We prove the convergence under $\varphi$ and derive practical guidelines for its selection through analysis and experiments. Besides, theoretical analysis and empirical results show that $\varphi$ is related to optimization difficulty such as condition number. +4. We perform experiments and demonstrate that UBA surpasses the commonly-used schedules across diverse vision and language tasks, spanning network architectures (e.g., ResNet, OLMo) and scales, under different training-iteration budgets. Moreover, our performance improvements come with negligible computational overhead, which constitutes one of its key advantages. + +# 2 Related work + +Budgeted training Researchers face significant challenges in achieving optimal model performance under fixed hardware and limited time. To address these challenges, the concept of budgeted training has gained increasing attention, exploring techniques including computation efficiency, model compression, training stability and convergence improvement [41]. It focuses on: (i) emphasizing the allocation of resources, such as the balance between the model size and the amount of data [2, 6, 24, 28]. (ii) finding optimal configurations or improving performance within the given compute or time budget, such as memory efficiency and computation reduction [17, 26, 31, 38], optimization learning rate schedules [5, 30, 43, 44], batch size [17] and other weight averaging method [26, 27]. + +Within the context of budgeted training, a key area that remains under-explored is achieving the best possible model performance within a fixed number of iterations, i.e. 'budgeted-iteration training'. In this domain, learning rate scheduling based methods are particularly aligned with the objectives of budgeted-iteration training. Smith et al. [43] propose a new learning rate schedule, named cyclical learning rates (CLR). It improves accuracy in fewer iterations without tuning and is relevant to superconvergence phenomenon [44]. Li et al. [30] introduce an alternative setting of existing learning rate schedules for budgeted training. Chen et al. [5] propose the reflected exponential schedule (REX) via a profile and sampling fashion. Zhang et al. [57] introduce the finite horizon optimization framework and apply it to linear programming, which optimizes the algorithm performance under a strict iteration budget. Learning rate based approaches achieve robust and high-performing results under various lengths of training iterations, which corresponds to our purpose in this work, i.e. budgeted-iteration training [34]. Besides, this approach is plug-and-play, requiring no substantial alterations to the underlying model structure, making it readily adaptable to various deep network frameworks. Therefore, we explore the proper learning rate schedule to achieve budgeted-iteration training. + +Learning rate schedule The learning rate plays a pivotal role in controlling the optimization process during network training. The common scheme is the step decay schedule. A typical instance decreases the learning rate by a decaying scalar 0.1 after $50\%$ epochs and by a decaying scalar 0.01 after $75\%$ epochs [21]. Then Loshchilov et al. [33] observe that sharp decreases may prevent models from escaping local minima and propose the cosine schedule function, which is the most popular schedule for language model pretraining. + +Pan et al. [36] propose Eigencurve, and demonstrate that minimax-optimal convergence can be achieved for quadratic objectives when the Hessian's eigenvalue distribution exhibits substantial skewness. Defazio et al. [11] developed the theoretically-grounded framework for learning rate scheduling, establishing optimal linear decay schedules and refinements through rigorous mathematical derivation. Their work additionally provides the most extensive empirical evaluation of scheduling approaches to date. Although some schedules include the CLR [43], REX [5], Warmup-Stable-Decay (WSD) [25] and schedule from multi-power law [34], there is no consensus on the optimal choice. In addition, the detail can be found in Appendix F, including some works focus on adaptive learning rate methods. + +Learning rate design is not only significant in general training, but also critical in budgeted-iteration training which still remains a topic of debate. Some analyses advocate for small, constant learning + +rates to ensure stability and convergence [12]. On the contrast, one prevailing hypothesis suggests that large learning rates may facilitate crossing over sharp local minima in the optimization landscape [56]. Despite the lack of comprehensive theoretical explanations, a range of learning rate schedules inspired by the above analyses as heuristic guidelines has been widely adopted in practice, using variable learning rates to budgeted-iteration training [5, 30]. In this work, we explore learning rate schedule from optimization problem tailored to budgeted-iteration training, aiming to balance iteration budget constraints and generalization. + +# 3 Budgeted-iteration training + +# 3.1 Finite optimization under limited training iterations + +To design a one-size-fits-all learning rate schedule, we construct a robust optimization model of learning rates across varying training conditions (see more conceptual illustration in Appendix A). Specifically, we aim to guarantee minimal loss within constrained training iterations under the worst-case conditions, proposing a budget-iteration-aware framework for learning rate optimization. + +Definition (Finite optimization): Let $f(W, D)$ denote the function parameterized by a given neural network with parameters $W \in \mathbb{R}^N$ on the dataset $D$ . Let $\xi$ denote the data sampling on the dataset $D$ , and let $\mathcal{F}$ be a function class. Let $\mathcal{L}$ be the loss function. Let $\eta_t$ be the learning rate at the $t$ -th iteration, $T$ be a maximum number of training iterations, and $t$ be the current learning step. A finite optimization is + +$$ +\min _ {\eta_ {1}, \eta_ {2}, \dots , \eta_ {T - 1}} \max _ {f \in \mathcal {F}} \quad \mathcal {L} (f (W _ {T}, \xi)) +$$ + +$$ +s. t. \quad W _ {t + 1} = W _ {t} - \eta_ {t} \nabla \mathcal {L} (f (W _ {t}, \xi)) \tag {1} +$$ + +$$ +t = 0, 1, 2, \dots , T - 1 +$$ + +In the optimization model 1, the constraint represents the stochastic gradient descent process. The maximizing of $\mathcal{L}(f(W_T,\xi))$ represents the the worst-case among the training process on the network $f$ . Then it minimizes the worst-case loss within given iterations, embodying its budget-aware property. By formulating the problem as a min-max optimization, we identify learning rates $\eta_t(t = 1,2,\dots ,T - 1)$ that are resilient to the uncertainties introduced by different training configuration, uniformly throughout the optimization trajectory. + +The challenge lies in characterizing the $f$ within the optimization process. $f$ is primarily determined by variations in parameter configuration, datasets characteristics, batch ordering and network architecture. From an optimization standpoint, we assume that the characteristics of $f$ shaped by these factors can be captured by the loss landscape of $f$ . Therefore, we can analyze $f$ by approximating its loss using a quadratic expansion around nearby strict local optima. Specifically, during the optimization process, the loss surface in the vicinity of the optimization trajectory can be approximated by sequence of strict local optima (or at least by one optimum), denoted as $\bar{W}^{(k)}$ ( $k = 1,2,\dots K$ ) $K \in \mathbb{Z}^+$ , with the final optimum represented as $\bar{W}^K$ . This approach enables us to capture the key features of the loss surface near these points. Therefore, the trajectory of optimization is impacted by the characteristics of the nearby strict local optima, since the optimization process is inherently shaped by the loss landscape and the key features of the surface are captured by these optima. Consequently, we can derive the learning rate within the optimization model by the information of these nearby strict local optima. + +According to the second-order necessary condition for the strict minimum $\bar{W}^{(k)}$ , this approximation is achieved through the positive semi-definite Hessian matrix $H_{f}^{(k)}(\xi) \in \mathbb{R}^{N \times N}$ . In addition, the optimization problem (1) will be programmed sequentially. Then objective function of the optimization problem (1) can be reformulated as $\mathcal{L}(f(\bar{W}^{(k)}, \xi)) + \frac{1}{2}(W_{T_{k+1}} - \bar{W}^{(k)})^{\top} H_{f}^{(k)}(\xi)(W_{T_{k+1}} - \bar{W}^{(k)})$ ( $k = 1, 2, \ldots, K$ ). Given that the networks under consideration possess sufficient capacity to fit the data, the loss at the optimal point for different $f$ can be made small enough. Thus the term $\mathcal{L}(f(\bar{W}^{(k)}, \xi))$ can be reasonably neglected in our analysis. We obtain the following sequential optimization problem. + +$$ +\min _ {\eta_ {1 + T _ {k}}, \eta_ {2 + T _ {k}}, \dots , \eta_ {T _ {k + 1}}} \max _ {f (\xi)} \frac {1}{2} (W _ {T _ {k + 1}} - \bar {W} ^ {(k)}) ^ {\top} H _ {f} ^ {(k)} (\xi) (W _ {T _ {k + 1}} - \bar {W} ^ {(k)}) +$$ + +$$ +s. t. \quad W _ {t + 1} = W _ {t} - \eta_ {t} H _ {f} ^ {(k)} (\xi) \left(W _ {t} - \bar {W} ^ {(k)}\right) \tag {2} +$$ + +$$ +t = T _ {k} + 1, T _ {k} + 2, \dots , T _ {k + 1} \quad (k = 1, 2, \dots K - 1) +$$ + +where $T_{1} = 0$ and $T_{K} = T - 1$ . + +By the first constraint $W_{t + 1} = W_t - \eta_t H_f^{(k)}(\xi)(W_t - \bar{W}^{(k)})$ , we can obtain (see Appendix A for derivation) + +$$ +\left\| W _ {T _ {k + 1}} - \bar {W} ^ {(k)} \right\| _ {2} ^ {2} \leq \max _ {\lambda_ {l} ^ {(k)} \leq \lambda_ {i} ^ {(k)} \leq \lambda_ {u} ^ {(k)}} \left[ \prod_ {t = 1 + T _ {k}} ^ {T _ {k + 1}} \left(1 - \eta_ {t} \lambda_ {i} ^ {(k)}\right) \right] ^ {2} \left\| W _ {T _ {k}} - \bar {W} ^ {(k)} \right\| _ {2} ^ {2} \tag {3} +$$ + +where $\lambda_i^{(k)}$ ( $i = 1,2,\dots ,N$ ) are the eigenvalues of $H_{f}^{(k)}(\xi)$ around the $k$ -th optimum, which satisfy $0 < \lambda_l^{(k)} \leq \lambda_i^{(k)}(f(\xi)) \leq \lambda_u^{(k)}$ for all $i$ , $u_{i}^{(k)}$ ( $i = 1,2,\dots ,N$ ) denote $N$ linearly independent eigenvectors and $s_i^{(k)}$ are the coefficients corresponding to the eigenvector components. Since the term $\left\| W_{T_k} - \bar{W}^{(k)}\right\|_2^2$ is a certain constant, the optimization process of weights $W_{T_{k + 1}}$ is equivalent to the least upper bound of the $\prod_{t = 1 + T_k}^{T_{k + 1}}(1 - \eta_t\lambda_i^{(k)})$ . Thus, the optimization of learning rate schedule can be formulated as follow, + +$$ +\min _ {\eta_ {1} + T _ {k}, \eta_ {2} + T _ {k}, \dots , \eta_ {T _ {k + 1}}} \max _ {\lambda_ {l} ^ {(k)} \leq \lambda_ {i} ^ {(k)} \leq \lambda_ {u} ^ {(k)}} \prod_ {t = 1 + T _ {k}} ^ {T _ {k + 1}} \left[ \left(1 - \eta_ {t} \lambda_ {i} ^ {(k)}\right) \right] ^ {2} \tag {4} +$$ + +$$ +s. t. \quad \eta_ {1 + T _ {k}}, \eta_ {2 + T _ {k}}, \dots , \eta_ {T _ {k + 1}} \in \left[ \eta_ {\min }, \eta_ {\max } \right] +$$ + +$$ +k = 1, 2, \dots K +$$ + +To solve the constrained min-max problem (4), we adopt an iterative projected gradient method that alternates between minimizing over the variables $\eta_t$ and maximizing over the parameters $\lambda_i^{(k)}$ . To avoid repeated numerical optimization, we fit the solutions with a parametric function. Details regarding the numerical solution and curve fitting process are provided in Appendix B. Then, we obtain the $\eta_t$ within the interval $t \in [1 + T_k, T_{k+1}]$ as follows + +$$ +\eta_ {t} = \left(\eta_ {\max } - \eta_ {\min }\right) \frac {2 (1 + \cos (\frac {(2 (t - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)} + (k - 1) \pi))}{2 \varphi + (2 - \varphi) (1 + \cos (\frac {(2 (t - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)} + (k - 1) \pi)} + \eta_ {\min }, \tag {5} +$$ + +where $\varphi$ is the hyper-parameter controlling the variation speed of the learning rate $\eta_t$ . + +When setting $(K > 2)$ , UBA extends to a multi-phase formulation, which offers three potential advantages. (i) Hierarchical optimization: by partitioning the budget-aware optimization into multiple phases, each near a different local minima, it improves approximation accuracy and captures the dynamic features of the loss surface. (ii) It helps escape saddle points, (iii) It generalizes the single-phase method, allowing for future extensions. Notably, a single-phase approach $(K = 2)$ remains effective, as the robust budget-aware model inherently derives the optimal schedule function, as shown in Figure 1(a). For consistency with common practice and to better isolate scheduling effects, this paper focuses primarily on the single-phase implementation. The multi-phase approach is also compared in Appendix E.9, with its schedule shown in Figure 1(b). + +We name the proposed schedule Unified Budget-Aware (UBA) schedule for the following reasons. The robust budget-aware model minimizes the loss function uniformly across the optimization trajectory, resulting in stable performance. UBA provides a reliable, unified choice for practitioners, eliminating the need for case-by-case baseline comparisons and delivering consistent superiority across datasets, architectures, and training budgets. Lastly, UBA can uniformly approximate the behavior of existing schedules through simple parameter adjustments. + +![](images/8911c71cea8c8ba5d7c26175155b35ecdd6d5d14b7895ea2ee0e1a67560bf0d2.jpg) +(a) + +![](images/0da2d430338d57e90ddf528f9ac843911895ae53ea16fee61c1a53062e8bc90d.jpg) +(b) +Figure 1: Evolution of the learning rate in UBA schedule across training iterations. + +# 3.2 Theoretical analysis + +Proposition 1. The fit function (5) is the exact closed-form solution to the min-max optimization problem: + +$$ +\min _ {\eta_ {1} + T _ {k}, \eta_ {2} + T _ {k}, \dots , \eta_ {T _ {k + 1}}} \max _ {\lambda_ {l} ^ {(k)} \leq \lambda_ {i} ^ {(k)} \leq \lambda_ {u} ^ {(k)}} \prod_ {t = 1 + T _ {k}} ^ {T _ {k + 1}} \left[ \left(1 - \left(\left(\frac {1}{\lambda_ {l} ^ {(k)}} - \frac {1}{\lambda_ {u} ^ {(k)}}\right) \eta_ {t} + \frac {1}{\lambda_ {u} ^ {(k)}}\right) \lambda_ {i} ^ {(k)}\right) \right] ^ {2} \tag {6} +$$ + +when the hyper-parameter $\varphi$ are determined by $\lambda_l^{(k)}$ and $\lambda_u^{(k)}$ through the relation $\varphi = 2\frac{\lambda_u^{(k)}}{\lambda_l^{(k)}}$ and $\eta_{\mathrm{max}} = 1$ , $\eta_{\mathrm{min}} = 0$ . + +The min-max model in Proposition 1 represents a special case of our generalized optimization framework. We show that UBA is the exact solution to this special case optimization problem when the learning rate is scaled by $\left(\frac{1}{\lambda_l^{(k)}} - \frac{1}{\lambda_u^{(k)}}\right)\eta_t + \frac{1}{\lambda_u^{(k)}}$ . It provides a theoretical foundation for our choice of learning rate instead of choosing it heuristically or empirically, adding rigor and justification to our approach. Moreover, in this case, $\varphi$ is linked condition number. It indicates how the learning rate is shaped by the local curvature of the model, which could guide the optimization process more effectively. The relationship between $\varphi$ and condition number suggests an adaptive nature for the learning rate. In regions with sharp curvatures (large condition number), $\varphi$ reduces the learning rate more rapidly to avoid overshooting. Conversely, in flatter regions, $\varphi$ allows the learning rate to remain large for several iterations, facilitating faster convergence. The transformation of the learning rate trend is shown in Figure 1. This is a step toward establishing a principled connection between learning rate and local loss landscape geometry along with optimization difficulty. By this special case, we generalize that $\varphi$ is related to optimization difficulty. The empirical results support these conclusions, with further details in Section 4.3 and Appendix E.6. + +Proposition 2. Consider the training process within the interval $t \in [1 + T_k, T_{k+1}]$ . When the hyperparameter $\varphi$ is set sufficiently close to 2, the proposed learning rate scheduling formula reduces to the cosine learning rate schedule. + +By Proposition 1, the hyper-parameter $\varphi$ is related to the optimization difficulty. In that case, the optimization difficulty is explicitly quantified as the condition number and $\varphi = 2\kappa$ , where $\kappa = \frac{\lambda_u^{(k)}}{\lambda_l^{(k)}}$ . Furthermore, Proposition 2 shows that when $\varphi$ approaches 2, the proposed learning rate schedule converges to the standard cosine schedule. It means that, in regions where the optimization difficulty is manageable (i.e., the curvature is relatively flat), setting $\varphi \approx 2$ allows the learning rate schedule to naturally reduce to the cosine form. Besides, our schedule can approximate the behavior of existing schedules (e.g. step decay, cosine annealing, cyclic schedule or Rex schedule) through simple parameter adjustments, shown in Table 1. The detail can be found in Appendix C. More importantly, it outperforms these schedules by training convergence and final accuracy, supporting results can be found in experiment sections 4.1,4.2 and Appendix E.5. + +Theorem 1. Let $n_t = H_f^{(k)}(W_t - \bar{W}^{(k)}) - H_f^{(k)}(\xi)(W_t - \bar{W}^{(k)})$ be the stochastic curvature noise introduced by sampling at iteration $t$ . Assume that the sampling Hessian satisfies: $\mathbb{E}_{\xi}\left[n_{t}n_{t}^{\top}\right] \preceq \sigma^{2}H_{f}^{(k)}$ for some constant $\sigma$ . Denote $\tau := \frac{4\lambda_l^{(k)}(\eta_{\max} - \eta_{\min})\left(1 + \cos\left(\frac{(2(t - T_k) - 1)\pi}{2(T_k + 1 - T_k)}\right)\right)(T_{k + 1} - T_k)}{(\varphi - 2)\pi}$ . If we + +Table 1: The adaptive simulation of existing schedules. + +
ScheduleParameter adjustmentsScheduleParameter adjustments
Cosineφ=2Stepφ=0, ηmax=0.5k, Tk+1-Tk = decaying step, k=1, 2,...
Exponentialφ=30Cyclicφ=2, k← k+1, Tk+1-Tk = cyclic step, k=1, 2,...
Rexφ=0.8OneCycleφ=2, k← k+1, T2-T1 = pct_start step
+ +set the learning rate as the proposed form (5), the loss uncertainty introduced by stochastic gradient method within the interval $t \in [1 + T_k, T_{k+1}]$ can be bounded by two terms, + +For $\varphi > 2$ : + +$$ +\begin{array}{l} \mathbb {E} \left[ \mathcal {L} (f (W _ {t}, \xi)) - \mathcal {L} (f (\bar {W} ^ {(k)}, \xi)) \right] \\ \leq \Bigg (\frac {4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi}{4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi (t - T _ {k})} \Bigg) ^ {\tau} \cdot \exp \Big (- 2 \lambda_ {i} ^ {(k)} \eta_ {\min} (t - T _ {k}) \Big) \lambda_ {u} ^ {(k)} \| W _ {1 + T _ {k}} - \bar {W} ^ {(k)} \| _ {2} ^ {2} \\ + \sigma^ {2} \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \sum_ {j = 1} ^ {N} \left(\lambda_ {j} ^ {(k)}\right) ^ {2} \exp \left(- 2 \lambda_ {j} ^ {(k)} \eta_ {\min } (t - i)\right) \cdot \left(\frac {4 \left(T _ {k + 1} - T _ {k}\right) + (\varphi - 2) \pi (i - T _ {k})}{4 \left(T _ {k + 1} - T _ {k}\right) + (\varphi - 2) \pi (t - T _ {k})}\right) ^ {\tau (7)} \\ \end{array} +$$ + +For $\varphi < 2$ + +$$ +\begin{array}{l} \mathbb {E} \left[ \mathcal {L} \left(f \left(W _ {t}, \xi\right)\right) - \mathcal {L} \left(f \left(\bar {W} ^ {(k)}, \xi\right)\right) \right] \\ \leq \left(\frac {(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{(T _ {k + 1} - T _ {k})} (t - T _ {k} - 0 . 5)}{(2 \varphi + 2 \pi - \varphi \pi) + \frac {(2 - \varphi) \pi}{2 (T _ {k + 1} - T _ {k})}}\right) ^ {- \tau} \cdot \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \lambda_ {u} ^ {(k)} \| W _ {1 + T _ {k}} - \bar {W} ^ {(k)} \| _ {2} ^ {2} \\ + \sigma^ {2} \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \sum_ {j = 1} ^ {N} (\lambda_ {j} ^ {(k)}) ^ {2} \exp \left(- 2 \lambda_ {j} ^ {(k)} \eta_ {\min } (t - i)\right) \cdot \left(\frac {(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{(T _ {k + 1} - T _ {k})} (t - T _ {k} - 0 . 5)}{(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{(T _ {k + 1} - T _ {k})} (i - T _ {k} - 0 . 5)}\right) ^ {- \tau} \\ \end{array} +$$ + +# 4 Experiment results + +We conduct comprehensive evaluations along three dimensions: (i) Modality diversity, including vision and language tasks; (ii) Training budget, including model scales (small to large) and training iterations (short to long); (iii) Ablation studies, such as parameter sensitivity and cross-optimizer performance. This evaluation strategy ensures the robustness and generalization of our findings. + +Our implementation strictly follows the PyTorch official API, i.e. it is inherited from the base class torch.optim.lr_scheduler.LRScheduler (Appendix E.3). It maintains full compatibility with all torch.optim optimizers. The complete production-ready code is publicly available (see Appendix E.3 for link). + +A key advantage of UBA lies in its critical parameter $\varphi$ . While an optimal $\varphi$ can further improve model performance, we intentionally fix its value across all tasks and architectures to ensure fair evaluation. We fix $\varphi$ for SGD and AdamW respectively (see learning rate variations in Figure 1(a)). The rationale for these choices is systematically analyzed in our ablation study 4.3. Since each schedule has a different optimal learning rate in practice, to thoroughly address this concern, we conduct additional experiments to evaluate the performance of all schedules with different learning rates in Appendix E.8. + +Baselines Research on budgeted-iteration training remains limited, with most existing approaches focusing on learning rate scheduling strategies [5, 30, 44]. To provide a fair comparison, we adopt several widely used and empirically effective learning rate schedules as baselines: Step(SS) [16], Cosine(CS) [33], Cyclical (CLR) and OneCycle (1C) [43, 44], Budgeted training(BT, mathematically equivalent to linear decay) [30] and Reflected Exponential (REX) [5]. Details of these baselines are provided in Appendix E.1. + +# 4.1 Experiments for vision classification tasks + +We evaluate our proposed UBA schedule on vision benchmarks (e.g., CIFAR10/100 and ImageNet) using different architectures (ResNet18, ResNet34, ResNet50). In addition, we independently train models using fixed epoch budgets of $25\%$ , $50\%$ , and $100\%$ of the maximum training epochs, without reusing or interpolating results from longer training runs. This setup ensures that each budget setting is evaluated in isolation, thereby preserving the integrity of comparisons across low and high training budgets. Table 2 presents the validation accuracy across different training budgets, comparing UBA against six baseline schedules (see detailed results in Appendix E.4). + +We find that: (i) UBA demonstrates the strongest performance across all training budgets, achieving superior results on both small-scale (CIFAR10/100 with ResNet18/34) and large-scale (ImageNet with ResNet50) benchmarks. This consistent improvement highlights the generalizability across model and dataset scales. (ii) UBA outperforms baselines not only at $100\%$ training budget but also at $25\%$ and $50\%$ iteration budgets, demonstrating its budget efficiency, i.e. effectiveness in computation-constrained scenarios. (iii) Notably, UBA shows robust performance even while the second-best schedule varies depending on both the datasets, architectures and training budgets. For practitioners seeking reliable schedules without extensive method selection, UBA provides a default-strong choice. UBA eliminates the need for case-by-case baseline comparison and delivers stable superiority and reliability regardless of datasets, architectures, training budgets and scales. + +Table 2: Validation accuracy for vision classification tasks. We present validation accuracy on vision benchmarks (e.g., CIFAR10/100 and ImageNet) using different architectures (ResNet18, ResNet34, ResNet50) under fixed epoch budgets of $25\%$ , $50\%$ , and $100\%$ of the maximum training epochs. + +
DatasetNetworkMethodtraining budget (epoch(%)
75 (25%)150(50%)300(100%)
CIFAR10ResNet18SS92.71±0.424393.45±0.482293.61±0.3035
CS94.21±0.511694.24±0.241995.52±0.2433
CLR92.13±0.261092.99±0.545994.92±0.3811
1C94.23±0.183895.03±0.135395.35±0.2984
BT94.18±0.354694.79±0.460595.68±0.1168
REX94.49±0.322595.10±0.192995.45±0.1350
UBA(ours)94.57±0.328195.26±0.215095.65±0.1589
75 (25%)150(50%)300(100%)
CIFAR100ResNet34SS71.09±0.267675.34±0.180277.24±0.1808
CS64.12±0.180869.84±0.758074.29±0.5958
CLR73.33±0.563573.41±0.283574.74±0.2266
1C73.38±0.322775.04±0.638277.86±0.2239
BT72.75±0.261175.35±0.242378.58±0.2705
REX73.64±0.563575.14±0.311078.04±0.1430
UBA(ours)74.60±0.150876.63±0.267878.95±0.2818
75 (25%)150(50%)300(100%)
ImageNetResNet50SS74.74±0.138677.24±0.386178.56±0.6435
CS75.84±0.222077.95±0.216479.07±0.0438
CLR73.90±1.460976.04±1.641977.84±1.3053
1C75.83±0.779277.74±0.521878.90±0.2956
BT75.90±0.278677.81±0.462478.86±0.2814
REX75.85±0.804777.66±0.885378.66±0.2828
UBA(ours)76.29±0.403178.26±0.164079.39±0.0170
+ +# 4.2 Experiments for language models + +We evaluate UBA schedule on the OLMo[19], a truly open language model based on decoder-only transformer architecture, across diverse benchmarks in language model evaluation. Since large language models commonly provide multiple scales to accommodate different compute constraints. We adjust both model size and training steps to explore budgets. We evaluate UBA on OLMo networks + +spanning four parameter scales: 36M, 73M, 151M, and 300M, covering normal to large-scale models. Due to space limitation, we defer details such as benchmarks introduction, experimental setting and overall results in Appendix E.5). + +![](images/b7cb22c30928c8b271a7a014cd5845bb63eb30da2cfeba60b580e3a6b536bd88.jpg) +Figure 2: Training dynamics and performance for language tasks under 150B tokens on 300M OLMo. We present the training loss, validation loss, and downstream performance on HSWAG and ARC-E, demonstrating that UBA schedule achieves superior performance. + +![](images/8c84d3be306cd70573d24c187d341ed4501aa43e5217e303b04a6cfd2ff537fa.jpg) + +![](images/508b7bc82b2922e0682f958855e1ce143ea58c2096f4929d5e7209ec5f77fa94.jpg) + +![](images/ba1d5417bdd96eb4f2610751a76770b248d993a4d246f6a6a37ad6bef33f29a6.jpg) + +From Table 3, UBA achieves state-of-the-art performance on approximately $50\%$ of the benchmarks across all scales while the second-best schedule varies. Furthermore, it achieves consistent superior average performance among all baselines. It highlights the stable superiority and reliability of UBA, providing a default-strong choice. Moreover, it demonstrates significant improvements in SciQ-73M(+1.7) and ARC-E-300M(+2.63), highlighting its ability to enhance generalization across diverse benchmarks. Besides, in Figure 2, UBA consistently achieves lower training loss and validation loss throughout the training, indicating the efficient training ability and downstream performance enhancement. Notably, while task-specific tuning of $\varphi$ can further improve model performance, we intentionally fix its value across all tasks and architectures to ensure fair evaluation. Remarkably, even with this universal $\varphi$ setting, UBA consistently outperforms baselines on diverse benchmarks, demonstrating inherent robustness to varying benchmarks and model scales. + +Table 3: Performance comparison between UBA and the best-performing baseline schedules on OLMo. We report the results of top-performing baseline schedule for the corresponding benchmark and model scale. The names of the top-performing baseline schedules are listed below the results. Bold values indicate UBA's superiority (See overall results of each schedule in Appendix E.5). + +
SizeSched.Benchmark(accuracy %)
PIQAHSWAGOBQASciQARC-EARC-CCOPASIQASOCOTHAvg.
36MBest baseline61.15 (CLR)27.91 (1C)27.80 (REX)68.10 (REX)45.26 (SS)23.41 (1C)63.00 (SS)41.45 (CLR)25.18 (REX)29.86 (1C)40.92 (REX)
UBA60.3927.9827.4068.2045.7921.7463.0040.6924.3330.1440.97
73MBest baseline62.46 (REX)30.22 (REX)28.80 (CS)72.60 (CS)47.54 (1C)26.42 (1C)66.00 (BT)41.97 (BT)26.32 (SS)29.34 (REX)42.53 (CS)
UBA63.1730.0928.8074.3045.7922.7465.0041.4527.1328.8542.73
150MBest baseline66.00 (CS)35.72 (REX)32.80 (1C)78.20 (BT)53.16 (1C)26.42 (BT)69.00 (REX)43.71 (CLR)27.68 (REX)32.64 (REX)45.91 (REX)
UBA65.2335.5029.8078.3050.3527.4267.0043.5029.2932.7245.91
300MBest baseline70.18 (REX)46.30 (REX)33.40 (CS)84.40 (REX)57.72 (REX)28.43 (REX)72.00 (BT)44.58 (CS)29.50 (REX)36.74 (REX)49.70 (REX)
UBA69.4846.4434.6083.9060.3529.1072.0044.4228.5335.4150.42
+ +# 4.3 Ablation study + +Performance across different optimizers UBA originates from the optimization problem under gradient descent dynamics. While modern optimizers (e.g., AdamW [32]) introduce momentum and adaptive mechanisms, they maintain the fundamental property of gradient-based updates. To verify the cross-optimizer performance, we conduct ablation studies between SGD and AdamW + +optimizers(see detailed results in Appendix E.6). The results show that UBA achieves SOTAs on both SGD and AdamW optimizers, demonstrating the cross-optimizer robustness of UBA. This highlights its broad applicability beyond standard gradient descent. + +Parameter analysis of $\varphi$ We perform a sensitivity analysis of $\varphi \in \{0.25, 0.5, 1.0, 2.5, 5, 10\}$ in equation (5), which controls the variation speed of learning rate (see detailed setting, results and analyses in Appendix E.7). Figure 11 show AdamW prefers smaller $\varphi$ while SGD requires larger $\varphi$ . We attribute this phenomenon to the preconditioning effect of AdamW. As the relationship formalized in Proposition 1), smaller $\varphi$ is favored for low optimization difficult, while larger $\varphi$ is beneficial for difficult optimization. AdamW adapts the learning rate by scaling gradients with the second moment estimate $\sqrt{v_t}$ , implicitly reducing the condition number of the optimization landscape. Thus the schedule with smaller $\varphi$ is preferred. In contrast, SGD lacks such adaptive mechanisms, thus a larger $\varphi$ is more suitable for this situation. This alignment between theory and experiment underscores the importance of tailoring $\varphi$ to the optimizer's characteristics. Overall, $\varphi$ is related to a generalized optimization difficulty, where optimization precondition effect, datasets distribution and network architecture are all related to optimization difficulty. + +Performance across different periods Our schedule has a periodic phase-based learning rate adjustment setting, where the learning rate at the $k$ -th phase is dynamically determined by the $k$ -th local minimum of the loss landscape. To validate our scheduling strategy, we conduct experiments by varying $K$ (see detailed results in Appendix E.9). The results suggests that multi-phase scheduling captures the dynamic features of the loss surface more finely, but it needs careful selection for $\varphi$ , where $\varphi$ reflects optimization difficulty. However, selecting optimal $\varphi$ values per phase for multi-phase remains non-trivial, which motivates future work on automated landscape-aware $\varphi$ tuning. + +# 5 Conclusion + +In this paper, we construct a unified budget-aware training optimization framework that inherently adapts to landscape curvature variations, enhancing training robustness. From this optimization framework, we propose the Unified Budget-Aware (UBA) schedule. Extensive experiments demonstrate UBA consistently surpasses the commonly-used schedules across diverse vision and language tasks, spanning network architectures (e.g., ResNet, OLMo) and scales, under different training-iteration budgets. Moreover, our performance improvements come with negligible computational overhead, which constitutes one of its key advantages. + +Theoretically and empirically, we observe that the parameter $\varphi$ correlates with the optimization difficulty of the training process, influences the optimal choice of $\varphi$ and UBA's performance. However, the explicit relationship between $\varphi$ and optimization difficulty remains unexplored and no established evaluation metric exists to quantify the optimization difficulty. These limitations motivate future work on optimization difficulty-aware $\varphi$ tuning. + +Despite these open questions, the effectiveness, implementation simplicity and ease of tuning make the UBA a practical, must-try schedule for deep learning practitioners. We hope this work could motivate more studies on learning rate scheduling. + +# 6 Acknowledgments and disclosure of funding + +Z. Lin was supported by National Key R&D Program of China (2022ZD0160300), the NSF China (No. 62276004) and the State Key Laboratory of General Artificial Intelligence. + +# References + +[1] Ebetesam Almazrouei, Hamza Alobeidli, Abdulaziz Alshamsi, Alessandro Cappelli, Ruxandra Cojocaru, Mérouane Debbah, Étienne Goffinet, Daniel Hesslow, Julien Launay, Quentin Malartic, et al. The falcon series of open language models. arXiv preprint arXiv:2311.16867, 2023. +[2] Eric Arazo, Diego Ortego, Paul Albert, Noel E O'Connor, and Kevin McGuinness. How important is importance sampling for deep budgeted training? arXiv preprint arXiv:2110.14283, 2021. +[3] Yonatan Bisk, Rowan Zellers, Jianfeng Gao, Yejin Choi, et al. Piqa: Reasoning about physical commonsense in natural language. In Proceedings of the AAAI conference on artificial intelligence, volume 34, pages 7432-7439, 2020. +[4] Tom B Brown. Language models are few-shot learners. 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Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The main claims in the abstract and introduction (e.g., UBA consistently surpasses the commonly-used schedules on vision and language tasks and sets new SoTAs for many networks and frameworks, e.g. ResNet and OLMo, under different training iterations.) are fully aligned with the papers contributions and scope. We provide an in-depth experiments and analyses of the proposed learning schedule across vision and language tasks, networks, scales and training budgets. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: We explicitly discuss limitations and future work in Section 5. Besides, the assumptions can be found in Appendix A. We test our method across vision and language tasks, various scales of datasets and networks, under different iteration budgets. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. + +# 3. Theory assumptions and proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +Answer: [Yes] + +Justification: The full set of assumptions are in Appendix A. The complete derivations and proof are in Appendix A and D. + +Guidelines: + +- The answer NA means that the paper does not include theoretical results. +- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced. +- All assumptions should be clearly stated or referenced in the statement of any theorems. +- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. +- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. +- Theorems and Lemmas that the proof relies upon should be properly referenced. + +# 4. Experimental result reproducibility + +Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? + +Answer: [Yes] + +Justification: In Appendix E, we carefully describe provide complete information needed to reproduce the experimental results, including hyper-parameters(initial learning rates, batch sizes, configurations of optimizer and schedules) for all tasks, implementation versions(such as PyTorch 2.4.1+cu118 on CIFAR), hardware configurations(GPU models, memory), full results of ablation studies (see Appendix E). Code and data are also available. + +Guidelines: + +- The answer NA means that the paper does not include experiments. +- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. +- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. +- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. +- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example + +(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm. +(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully. +(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset). + +(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. + +# 5. Open access to data and code + +Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? + +Answer: [Yes] + +Justification: We provide open access to the code in the Appendix. Moreover, we provide sufficient instructions to faithfully reproduce the main experimental results in Appendix E. + +# Guidelines: + +- The answer NA means that paper does not include experiments requiring code. +- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details. +- While we encourage the release of code and data, we understand that this might not be possible, so No is an acceptable answer. 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If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why. +- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable). +- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. + +# 6. Experimental setting/details + +Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results? + +Answer: [Yes] + +Justification: These details are described in Appendix E. + +# Guidelines: + +- The answer NA means that the paper does not include experiments. +- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. +- The full details can be provided either with the code, in appendix, or as supplemental material. + +# 7. Experiment statistical significance + +Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? + +Answer: [No] + +Justification: Due to the high computational cost of training large-scale models (e.g., Transformers on ImageNet), we report results from a single run with fixed random seeds. To ensure reliability, we list experimental setting and provide code for deterministic reproduction. To maintain uniform evaluation protocols across all experiments (large/small-scale), + +we prioritized identical statistical reporting standards. This avoids selective rigor that could mislead comparisons. Besides, we conduct parameter analyses to improve reliability. Moreover, small-scale results align with the expectations and results on large-scale experiments, reducing the need for empirical variance quantification. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Appendix + +# A Methodology + +Conceptual illustration The optimization landscape of neural networks is highly dynamic due to data distribution and sampling, and sensitive to various factors, such as network architecture, optimization and hyper-parameters. Figure 3(a) illustrates the inherent uncertainties during the training process. The black-and-white surfaces reflect the loss landscape processing one batch, while the colored surface represents the landscape following the next batch. The substantial variations in the landscape between consecutive batches significantly affect the training trajectory, making it challenging to design a one-size-fits-all learning rate strategy that can effectively adapt to such fluctuations. Therefore, we aim to develop a strategy a robust approach to be scheduled leaning rate across diverse optimization landscapes. We construct a budget-iteration-aware framework for learning rate optimization model that guarantees minimal loss within constrained training iterations under the worst-case conditions. + +We analyze loss landscape by a quadratic approximation around nearby strict local optima. This approach enables us to capture the key features of the loss surface near these points. Therefore, the trajectory of optimization is impacted by the characteristics of the nearby strict local optima, since the optimization process is inherently shaped by the loss landscape and the key features of the surface are captured by these optima, as illustrated in Figure 3(b). Consequently, we can derive the learning rate within the optimization model by the information of these nearby strict local optima. + +![](images/066afbf4d37a4dece32055c5ee94e409b3b536d6ff3b3e7044ad4b32b6771bcf.jpg) +(a) + +![](images/e4168e08f46ed0d5de87e33083bc0b515faf696c60682fbc98bdf00b165c8e56.jpg) +(b) +Figure 3: Conceptual illustration: visualization of the motivations behind the proposed modeling approaches. (a) shows the rationale for the modeling approach in (1), while (b) explains the modeling approach for (2). The solid line represents the optimization trajectory, while the dashed circles indicate the local approximation around corresponding minima. + +Assumptions The optimization behavior of a neural network is fundamentally governed by the geometry of its loss landscape, which emerges from the interaction of parameter configuration, datasets characteristics, sampling, network architecture, optimization preconditioning effect. In the vicinity of any strict local minimum, the loss surface can be approximated by a quadratic method whose curvature is determined by the local Hessian matrix. Thus the optimization trajectory is consequently dominated by the geometric properties (e.g., Hessian features, curvature profiles) of the nearest strict local minima. + +Many tasks train models using epochs, where one epoch represents a full pass through the datasets. Under this circumstance, an iteration occurs when the model processes a single batch. While the number of iterations per epoch varies with batch size, training with a fixed epoch budget becomes equivalent to using a fixed iteration count when batch size remains constant. For consistency, this paper uses iterations as our primary unit of optimization model construction and networks training. + +The sampling Hessian satisfies: $\mathbb{E}_{\xi}\left[n_{t}n_{t}^{\top}\right] \preceq \sigma^{2}H_{f}^{(k)}$ for some constant $\sigma$ . This assumption is followed by the work [16, 36]. Here, $\sigma^2$ measures the degree of Hessian inconsistency across data samples, i.e., the variance in the per-sample Hessian matrices. Besides, a smaller $\sigma^2$ indicates a more stable curvature landscape between each data sampling, leading to lower noise amplification during optimization. + +**Optimization problem derivation for Section 3.1** Considering the first constraint $W_{t+1} = W_t - \eta_t H_f^{(k)}(\xi) (W_t - \bar{W}^{(k)})$ in min-max optimization (2), we can reformulate it as + +$$ +W _ {t + 1} - \bar {W} ^ {(k)} = \left(I - \eta_ {t} H _ {f} ^ {(k)} (\xi)\right) \left(W _ {t} - \bar {W} ^ {(k)}\right), \tag {9} +$$ + +where $I$ denotes the identity matrix. By iteratively applying this equation (9), we obtain + +$$ +\begin{array}{l} \left(W _ {T _ {k + 1}} - \bar {W} ^ {(k)}\right) = \\ (I - \eta_ {T _ {k + 1} - 1} H _ {f} ^ {(k)} (\xi)) \dots (I - \eta_ {T _ {k} + 1} H _ {f} ^ {(k)} (\xi)) \left(W _ {T _ {k} + 1} - \bar {W} ^ {(k)}\right). \tag {10} \\ \end{array} +$$ + +Since the matrix $H_{f}^{(k)}(\xi)$ is positive semi-definite, it possesses $N$ eigenvalues $\lambda_1^{(k)}(f(\xi)), \lambda_2^{(k)}(f(\xi)), \dots, \lambda_N^{(k)}(f(\xi))$ abbreviated as $\lambda_i^{(k)}$ , which satisfy $0 < \lambda_l^{(k)} \leq \lambda_i^{(k)}(f(\xi)) \leq \lambda_u^{(k)}$ for all $i$ . Here, $\lambda_l^{(k)}$ and $\lambda_u^{(k)}$ denote the bounds on the non-zero eigenvalues of the Hessian around $k$ -th optimum. It also satisfies $\lambda_u^{(k)} \leq \frac{2}{\eta}$ where $\eta$ is the learning rate to guarantee the convergence [56]. Moreover, there exist $N$ linearly independent eigenvectors $u_1^{(k)}(f(\xi)), u_2^{(k)}(f(\xi)), \dots, u_N^{(k)}(f(\xi))$ abbreviated as $u_i^{(k)}$ , which can form the basis for the $N$ dimension vector space. Then we have $H_{f}^{(k)}(\xi)u_{i}^{(k)} = \lambda_{i}^{(k)}u_{i}^{(k)} \quad (i = 1,2,\dots,N)$ and the initial deviation from the optimal solution can be expressed as $W_{T_k} - \bar{W}^{(k)} = \sum_{i=1}^N s_i^{(k)}u_i^{(k)}$ where $s_i^{(k)}$ are the coefficients corresponding to the eigenvector components. Thus, we obtain + +$$ +\begin{array}{l} \left\| W _ {T _ {k + 1}} - \bar {W} ^ {(k)} \right\| _ {2} ^ {2} = \sum_ {i = 1} ^ {N} (s _ {i} ^ {(k)}) ^ {2} \| u _ {i} ^ {(k)} \| _ {2} ^ {2} \left[ \prod_ {t = 1 + T _ {k}} ^ {T _ {k + 1}} \left(1 - \eta_ {t} \lambda_ {i} ^ {(k)}\right) \right] ^ {2} \\ \leq \sum_ {i = 1} ^ {N} \left(s _ {i} ^ {(k)}\right) ^ {2} \| u _ {i} ^ {(k)} \| _ {2} ^ {2} \max _ {\lambda_ {l} ^ {(k)} \leq \lambda_ {i} ^ {(k)} \leq \lambda_ {u} ^ {(k)}} \left[ \prod_ {t = 1 + T _ {k}} ^ {T _ {k + 1}} \left(1 - \eta_ {t} \lambda_ {i} ^ {(k)}\right) \right] ^ {2} \tag {11} \\ = \max _ {\lambda_ {l} ^ {(k)} \leq \lambda_ {i} ^ {(k)} \leq \lambda_ {u} ^ {(k)}} \left[ \prod_ {t = 1 + T _ {k}} ^ {T _ {k + 1}} (1 - \eta_ {t} \lambda_ {i} ^ {(k)}) \right] ^ {2} \left\| W _ {T _ {k}} - \bar {W} ^ {(k)} \right\| _ {2} ^ {2} \\ \end{array} +$$ + +# B Numerical solution and curve fitting + +To solve the constrained min-max problem (4) as follow, + +$$ +\min _ {\eta_ {1} + T _ {k}, \eta_ {2} + T _ {k}, \dots , \eta_ {T _ {k + 1}}} \max _ {\lambda_ {l} ^ {(k)} \leq \lambda_ {i} ^ {(k)} \leq \lambda_ {u} ^ {(k)}} \prod_ {t = 1 + T _ {k}} ^ {T _ {k + 1}} \left[ (1 - \eta_ {t} \lambda_ {i} ^ {(k)}) \right] ^ {2} +$$ + +$$ +s. t. \quad \eta_ {1 + T _ {k}}, \eta_ {2 + T _ {k}}, \dots , \eta_ {T _ {k + 1}} \in \left[ \eta_ {\min }, \eta_ {\max } \right] +$$ + +$$ +k = 1, 2, \dots K +$$ + +we adopt an iterative projected gradient method that alternates between minimizing over the variables $\eta_t$ and maximizing over the parameters $\lambda_i^{(k)}$ . Specifically, at each iteration, we update $\eta_t$ with fixed $\lambda_i^{(k)}$ as follows + +$$ +\eta_ {t} \gets \mathcal {P} _ {[ \eta_ {\min}, \eta_ {\max} ]} \left(\eta_ {t} - \alpha \nabla_ {\eta_ {j}} \mathcal {L} (\eta_ {t}, \lambda_ {i} ^ {(k)})\right). +$$ + +For fixed $\eta_t$ , we update $\lambda_i^{(k)}$ via + +$$ +\lambda_ {i} ^ {(k)} \leftarrow \mathcal {P} _ {[ \lambda_ {l}, \lambda_ {u} ]} \left(\lambda_ {i} ^ {(k)} + \beta \nabla_ {\lambda_ {i} ^ {(k)}} \mathcal {L} (\eta_ {t}, \lambda_ {i} ^ {(k)})\right), +$$ + +where $\mathcal{P}$ is the projection over box constraints, $\alpha$ and $\beta$ are step sizes. + +This approach requires per-network numerical optimization to determine optimal learning rates case-by-case. To circumvent repeated numerical optimization for learning rate scheduling whenever a new network is trained, we propose to fit these solutions with a parametric function universally across networks. + +We note that the solution of optimization model is an unordered set $\{\eta_{1 + T_k},\eta_{2 + T_k},\dots ,\eta_{T_{k + 1}}\}$ , meaning that any permutation of these values yields the same objective value. To facilitate function approximation and improve the training stability of the network, we sort the solution set in both descending and ascending orders before fitting them with a smooth function. This pre-processing step preserves the optimality of the solution while enhancing the stability and interpretability of the fitted function. + +Due to the symmetry between the descending and ascending orders, and without loss of generality, we first fit a function to the descending order solution. The corresponding function for the ascending order can then be easily obtained via a simple transformation. + +Our key observation is that the numerical solutions under varying $\lambda_{l},\lambda_{u}$ configurations exhibit two characteristic decay modes. As shown in Figure 4, these transformations manifest through two distinct mechanisms: (i) gradual-to-accelerated decay: gentle initial decrease transitioning to rapid drop (as shown in Figures. 4(d),4(e) and 4(f)); (ii) rapid-to-gradient decay: Initial steep decline followed by slow decay (as shown in Figures. 4(g),4(h) and 4(i)). These pattern can be formulated as transformations of a base function, which resembles the curvature variation of the transformed $\ell_1$ norm modulated by a parameter [42]. + +We first isolate the base function by examining the limiting case where $\lambda_l \approx \lambda_u$ . In this regime, the numerical solution within each interval $[1 + T_k, T_{k+1}]$ manifests a cosine-like profile (Figures. 4(a), 4(b) and 4(c)), suggesting the base form: + +$$ +1 + \cos (\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}) +$$ + +Then we construct the fitting function as: + +$$ +\left(\eta_ {\max } - \eta_ {\min }\right) \frac {a \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)\right)}{b + c \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)\right)} + \eta_ {\min }. \tag {12} +$$ + +where $\{a,b,c\}$ control learning rate decay modes. + +In addition, through systematic fitting across nine configurations (Table 4), we discover a universal scaling relationship among the parameters: + +$$ +\frac {b + 2 c}{a} \approx 2 \tag {13} +$$ + +Table 4: The parameter details of curve fitting. + +
hyper-parametersfitted parametersobjective value(log)(4)
ηminηmaxλl(k)λu(k)abcφby numerical solutionby fitting function
0101.892.600.260.510.001.99792.67750.83
0102.392.614.167.350.531.80798.38770.38
0105.295.614.847.321.281.561118.791105.80
01018.89260.111124.67212.281011.940.192933.342933.62
0528.89200.11755.9185.01719.400.122607.112591.51
0250.89150.110.270.050.240.172031.662075.50
11010.01150.1177.45684.15-268.208.402373.582431.42
21018.8950.110.112.52-1.1521.022007.682044.84
31020.01180.110.073.07-1.4738.342604.982632.21
+ +Table 4 lists the fitted parameters $\{a, b, c\}$ , where the mean absolute error is 0.04. Figure 5 visualizes the variability of the $\frac{b + 2c}{a}$ by shading the region within $\pm 1$ standard deviation from the theoretical value in Figure 5. The empirical results fall within $\pm 1\sigma$ of the predicted value $(2.0 \pm 0.04)$ in most of cases, demonstrating strong agreement with the assumption relation (13). + +This empirical relationship (13) allows us to reduce the original 3-parameter system to 2 degrees of freedom as follows, + +$$ +\left(\eta_ {\max } - \eta_ {\min }\right) \frac {2 a \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)\right)}{2 b + (2 a - b) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)\right)} + \eta_ {\min }. \tag {14} +$$ + +Actually, it is a 1-parameter system + +$$ +\left(\eta_ {\max } - \eta_ {\min }\right) \frac {2 \left(1 + \cos \left(\frac {2 \left(t - T _ {k}\right) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)\right)}{2 d + (2 - d) \left(1 + \cos \left(\frac {2 \left(t - T _ {k}\right) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)\right)} + \eta_ {\min }, \tag {15} +$$ + +where $\varphi \coloneqq b / a$ . We list the value of fitted parameter $\varphi$ in Table 4. Since the equation (14) and the equation (15) are mathematically equivalent, we adopt the equation (15) as the canonical representation throughout this work. + +Then the corresponding function for the ascending order can then be easily obtained via a phase shift of $\pi$ as follow, + +$$ +\left(\eta_ {\max } - \eta_ {\min }\right) \frac {2 \left(1 + \cos \left(\frac {\left(2 \left(t - T _ {k}\right) - 1\right) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)} + \pi\right)\right)}{2 \varphi + (2 - \varphi) \left(1 + \cos \left(\frac {\left(2 \left(t - T _ {k}\right) - 1\right) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)} + \pi\right)\right)} + \eta_ {\min }, \tag {16} +$$ + +In summary, we obtain the $\eta_t$ within the interval $t \in [1 + T_k, T_{k+1}]$ as follows + +$$ +\eta_ {t} = \left(\eta_ {\max } - \eta_ {\min }\right) \frac {2 (1 + \cos (\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})} + (k - 1) \pi))}{2 \varphi + (2 - \varphi) (1 + \cos (\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})} + (k - 1) \pi)} + \eta_ {\min }, \tag {17} +$$ + +where $d$ is the hyper-parameter controlling the variation speed of $\eta_t$ . + +![](images/08a10b09e9d46d01d12428420ad79362dc12e3f7737d37435def82ab67a314f2.jpg) +(a) + +![](images/b3fc3ee45e86090b48893e92a793934039632e1f2bc4fbe6f2dc0eecf89c3da9.jpg) +(b) + +![](images/9a6828d6338a7adb2afe5d7aaf2dd7b31198d92489532739a369bd866ef2a032.jpg) +(c) + +![](images/9c395297365924f28df4a2da8ba0cffcb2f5aee92504c8f6985cd620c2fdfc72.jpg) +(d) + +![](images/8338f1cc63915803c84a49d542034922f179407887e717ae544e3958208f677d.jpg) +(e) + +![](images/0cc568ca613f6ce5e7845e293a99d8be46ff54aba3618f2fd099a9db7a977b34.jpg) +(f) + +![](images/8a9e8da3e4708e078ab4846e56091a82d41e160e7e069d2339e9f69ebc00d0fa.jpg) +(g) + +![](images/7ad54b2acec60b64441f46bf9c4a7407c97955d97012e2df90f6dd27127465d8.jpg) +(h) + +![](images/5958864245b7dec2aa31eff38177566a3abb25d665d48a04c486cafb4dd9c1eb.jpg) +(i) + +![](images/e6ef2b14180fc09de15f5336da574d9b8a082a219588f079816341ff3712894e.jpg) +Figure 4: The curve fitting of numerical solutions. +Figure 5: Visualization of relation between parameter $a, b$ and $c$ . + +# C Adaptive simulation of existing schedules + +Our schedule has adaptability owing to the parameter $\varphi$ , enabling it to approximate the behavior of existing schedules (e.g. step decay, cosine annealing, cyclic schedule or Rex schedule) through simple parameter adjustments. More importantly, it outperforms these schedules by jointly optimizing training stability and final accuracy. + +We visualize the adaptive simulation of in Figure 6. By setting our parameter $\varphi$ to 2, the UBA schedule degenerates to a cosine schedule (figure 6(a)). By setting $\varphi$ to large value such as 30, the UBA schedule mimics exponential schedule (figure 6(b)). By setting $\varphi$ to small value such as 0.8, the UBA schedule mimics Rex schedule (figure 6(c)). By setting $\varphi$ to 0, the proposed schedule degenerates to a constant. Therefore, we can control $\eta_{\mathrm{max}}$ as piece-wise value depending on iteration, and the UBA schedule degenerates to a step schedule (figure 6(d)). By setting $k$ to any integer larger than 1, the UBA schedule possesses cyclic characters, thus it mimics cyclic schedule (figure 6(e) 6(f)). Since the OneCycle learning rate schedule can be viewed as a single-period version of cyclical learning rates, the UBA schedule mimics the OneCycle approach by emulating the cyclical schedule. + +![](images/2709b9f37531a5b4a7bf9f1ab032e7adeece1cba7bef549715f68048593d97be.jpg) +(a) + +![](images/c0f3701c76a2340adaa69def6c5cec18e526e2f46b77b037846fef223e1a395f.jpg) +(b) + +![](images/70bcb9ecbcd27af5d28b3e245ac1a4ebdab9f1b462ca0abe67665ce0ce5bbf3f.jpg) +(c) + +![](images/86a0854608ea14b4aa18a661d9c44991bcce51e326471afd8ef18dfca12fc5be.jpg) +(d) + +![](images/bc6993f0ec6fa6a22cf99216aab7bdaccc31db11c86d8017776c732e73384654.jpg) +(e) + +![](images/6980dc0d015a8ea13f2f16223a23f272441746a6318bbad0ba8e1d7ab3adcf48.jpg) +(f) +Figure 6: Adaptive simulation by UBA schedule. + +# D Propositions and lemmas + +# D.1 Proof of Proposition 1 + +Proposition 1 The fitted function (5) is the exact closed-form solution $\frac{2(1 + \cos\left(\frac{(2(t - T_k) - 1)\pi}{2(T_{k + 1} - T_k)}\right))}{2\varphi + (2 - \varphi)(1 + \cos\left(\frac{(2(t - T_k) - 1)\pi}{2(T_{k + 1} - T_k)}\right)}$ to the min-max optimization problem: + +$$ +\min _ {\eta_ {1} + T _ {k}, \eta_ {2} + T _ {k}, \dots , \eta_ {T _ {k + 1}}} \max _ {\lambda_ {l} ^ {(k)} \leq \lambda_ {i} ^ {(k)} \leq \lambda_ {u} ^ {(k)}} \prod_ {t = 1 + T _ {k}} ^ {T _ {k + 1}} \left[ (1 - \left(\left(\frac {1}{\lambda_ {l} ^ {(k)}} - \frac {1}{\lambda_ {u} ^ {(k)}}\right) \eta_ {t} + \frac {1}{\lambda_ {u} ^ {(k)}}\right) \lambda_ {i} ^ {(k)}) \right] ^ {2} \tag {18} +$$ + +when the hyper-parameter $\varphi$ are determined by $\lambda_l^{(k)}$ and $\lambda_u^{(k)}$ through the relation $\varphi = 2\frac{\lambda_u^{(k)}}{\lambda_l^{(k)}}$ and $\eta_{\mathrm{max}} = 1$ , $\eta_{\mathrm{min}} = 0$ . + +Proof. According to the theorem in [14], among all polynomials of degree $n$ in $\mu$ that take the value $+1$ at $\mu = d > 1$ , the Chebyshev polynomial $S_{n}(\mu) = \frac{C_{n}(\mu)}{C_{n}(d)}$ is the unique solution that minimizes the maximum absolute value over the interval $(-1, +1)$ . Here, $C_{n}(\mu) = \cos (n\arccos (\mu))$ is the $n$ -th order Chebyshev polynomial. + +We define the polynomial $Q(\lambda_i^{(k)})$ as: + +$$ +Q \left(\lambda_ {i} ^ {(k)}\right) := \prod_ {t = T _ {k} + 1} ^ {T _ {k + 1}} \left[ 1 - \left(\left(\frac {1}{\lambda_ {l} ^ {(k)}} - \frac {1}{\lambda_ {u} ^ {(k)}}\right) \eta_ {t} + \frac {1}{\lambda_ {u} ^ {(k)}}\right) \lambda_ {i} ^ {(k)} \right] +$$ + +which is a degree $n = T_{k + 1} - T_k$ polynomial in $\lambda_i^{(k)}$ . + +Next, we introduce the variable transformation: + +$$ +\gamma = \frac {- 2 \lambda_ {i} ^ {(k)} + \lambda_ {l} ^ {(k)} + \lambda_ {u} ^ {(k)}}{\lambda_ {u} ^ {(k)} - \lambda_ {l} ^ {(k)}} +$$ + +which maps the interval $\lambda_l^{(k)}\leq \lambda_i^{(k)}\leq \lambda_u^{(k)}$ to the interval $-1\leq \gamma \leq 1$ , such that $\lambda_i^{(k)} = \lambda_u^{(k)}$ corresponds to $\gamma = -1$ and $\lambda_i^{(k)} = \lambda_l^{(k)}$ corresponds to $\gamma = 1$ . + +The polynomial $P(\gamma) = Q(\lambda_i^{(k)})$ now satisfies the condition in the theorem from [14], i.e., + +$$ +P \left(\frac {\lambda_ {l} ^ {(k)} + \lambda_ {u} ^ {(k)}}{\lambda_ {u} ^ {(k)} - \lambda_ {l} ^ {(k)}}\right) = 1 \quad \text {s i n c e} \quad Q (0) = 1. +$$ + +Therefore, the original min-max problem + +$$ +\min _ {\eta_ {T _ {k} + 1}, \eta_ {T _ {k} + 2}, \dots , \eta_ {T _ {k + 1}}} \max _ {\lambda_ {l} ^ {(k)} \leq \lambda_ {i} ^ {(k)} \leq \lambda_ {u} ^ {(k)}} \prod_ {t = T _ {k} + 1} ^ {T _ {k + 1}} \left[ 1 - \left(\left(\frac {1}{\lambda_ {l} ^ {(k)}} - \frac {1}{\lambda_ {u} ^ {(k)}}\right) \eta_ {t} + \frac {1}{\lambda_ {u} ^ {(k)}}\right) \lambda_ {i} ^ {(k)} \right] ^ {2} +$$ + +is equivalent to the problem of finding a polynomial in $\gamma$ of degree $T_{k + 1} - T_k$ that minimizes the maximum absolute value in the interval $-1\leq \gamma \leq 1$ + +By the theorem in [14], the desired solution to the problem is the Chebyshev polynomial: + +$$ +S _ {T _ {k + 1} - T _ {k}} (\gamma) = \frac {C _ {T _ {k + 1} - T _ {k}} (\gamma)}{C _ {T _ {k + 1} - T _ {k}} \left(\frac {\lambda_ {l} ^ {(k)} + \lambda_ {u} ^ {(k)}}{\lambda_ {u} ^ {(k)} - \lambda_ {l} ^ {(k)}}\right)} +$$ + +To match $P(\gamma) = S_{T_{k + 1} - T_k}(\gamma)$ , we equate the corresponding zeros. The roots of $Q(\lambda_i^{(k)}) = 0$ are given by: + +$$ +\lambda_ {i} ^ {(k)} = \frac {1}{\left(\left(\frac {1}{\lambda_ {l} ^ {(k)}} - \frac {1}{\lambda_ {u} ^ {(k)}}\right) \eta_ {t} + \frac {1}{\lambda_ {u} ^ {(k)}}\right)} \quad \text {f o r} \quad t = 1, 2, \dots , T _ {k + 1} - T _ {k}. +$$ + +On the other hand, the zeros of $P(\gamma)$ correspond to the values of $\gamma$ where the polynomial $P(\gamma)$ vanishes, and these zeros are given by: + +$$ +\gamma_ {t} = \frac {\lambda_ {l} ^ {(k)} + \lambda_ {u} ^ {(k)}}{\lambda_ {u} ^ {(k)} - \lambda_ {l} ^ {(k)}} - \frac {2}{\left(\left(\frac {1}{\lambda_ {l} ^ {(k)}} - \frac {1}{\lambda_ {u} ^ {(k)}}\right) \eta_ {t} + \frac {1}{\lambda_ {u} ^ {(k)}}\right) (\lambda_ {u} ^ {(k)} - \lambda_ {l} ^ {(k)})} \quad \text {f o r} \quad t = 1, 2, \dots , T _ {k + 1} - T _ {k}. +$$ + +The zeros of the Chebyshev polynomial $S_{T_{k + 1} - T_k}(\gamma)$ are given by the values: + +$$ +\cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right) \quad \text {f o r} \quad t = 1, 2, \dots , T _ {k + 1} - T _ {k}. +$$ + +Equating the zeros of $S_{T_{k + 1} - T_k}(\gamma)$ and $P(\gamma)$ , we have: + +$$ +\frac {\lambda_ {l} ^ {(k)} + \lambda_ {u} ^ {(k)}}{\lambda_ {u} ^ {(k)} - \lambda_ {l} ^ {(k)}} - \frac {2}{\left(\left(\frac {1}{\lambda_ {l} ^ {(k)}} - \frac {1}{\lambda_ {u} ^ {(k)}}\right) \eta_ {t} + \frac {1}{\lambda_ {u} ^ {(k)}}\right) (\lambda_ {u} ^ {(k)} - \lambda_ {l} ^ {(k)})} = \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right) +$$ + +Solving this equation for $\eta_t$ , we obtain: + +$$ +\eta_ {t} = \frac {\left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right)}{2 \frac {\lambda_ {u} ^ {(k)}}{\lambda_ {l} ^ {(k)}} - \left(\frac {\lambda_ {u} ^ {(k)}}{\lambda_ {l} ^ {(k)}} - 1\right) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right)}. +$$ + +![](images/bd7bb03e84d1b373ba8918d490d6afdc619cd1559f32c72d742e13a0d99eb797.jpg) + +# D.2 Proof of Proposition 2 + +Proposition 2 Consider the training process within the interval $t \in [1 + T_k, T_{k+1}]$ . When the hyperparameter $\varphi$ is set sufficiently close to 2, the proposed learning rate scheduling formula reduces to the cosine learning rate schedule. + +Proof. + +$$ +\begin{array}{l} \lim _ {\varphi \rightarrow 2} (\eta_ {\max } - \eta_ {\min }) \frac {2 (1 + \cos (\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}))}{2 \varphi + (2 - \varphi) (1 + \cos (\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})})} + \eta_ {\min } \\ = \left(\eta_ {\max } - \eta_ {\min }\right) \left(\frac {1}{2} + \frac {1}{2} \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) + \eta_ {\min } \\ = \frac {1}{2} \left(\eta_ {\max } - \eta_ {\min }\right) \left(1 + \cos \left(\frac {(t - T _ {k}) \pi}{T _ {k + 1} - T _ {k}}\right) \cos \left(\frac {\pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right) + \sin \left(\frac {(t - T _ {k}) \pi}{T _ {k + 1} - T _ {k}}\right) \sin \left(\frac {\pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)\right) + \eta_ {\min } \\ \approx \frac {1}{2} \left(\eta_ {\max } - \eta_ {\min }\right) \left(1 + \cos \left(\frac {(t - T _ {k}) \pi}{T _ {k + 1} - T _ {k}}\right)\right) + \eta_ {\min } \tag {19} \\ \end{array} +$$ + +The final approximation holds because each training phase typically contains many iterations, i.e., making $\frac{\pi}{T_{k+1} - T_k} \approx 0$ . Consequently, $\sin \left( \frac{\pi}{2(T_{k+1} - T_k)} \right) \approx 0$ and $\cos \left( \frac{\pi}{2(T_{k+1} - T_k)} \right) \approx 1$ . + +# D.3 Proof of Theorem 1 + +Lemma 1. Let $\eta_{j}$ be the learning rate at the iteration $j\in [1 + T_k,t]$ , defined by the proposed form (5). Let $\lambda_i^{(k)}$ be the eigenvalues of the Hessian matrix $H_{f}^{(k)}(\xi)$ at the $k$ -th strict minimum $\bar{W}^{(k)}$ . Then for any $t\in [1 + T_k,T_{k + 1}]$ , the following inequalities hold, + +Case 1 $(\varphi >2)$ + +$$ +\begin{array}{l} \prod_ {j = 1 + T _ {k}} ^ {t} \left[ (1 - \eta_ {j} \lambda_ {i} ^ {(k)}) \right] ^ {2} \\ \leq \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } \left(t - T _ {k}\right)\right) \tag {20} \\ \end{array} +$$ + +$$ +\cdot \left(\frac {4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi}{4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi (t - T _ {k})}\right) \frac {4 \lambda_ {l} ^ {(k)} (\eta_ {\mathrm {m a x}} - \eta_ {\mathrm {m i n}}) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) (T _ {k + 1} - T _ {k})}{(\varphi - 2) \pi}. +$$ + +Case 2 $(\varphi < 2)$ : + +$$ +\begin{array}{l} \prod_ {j = 1 + T _ {k}} ^ {t} \left[ (1 - \eta_ {j} \lambda_ {i} ^ {(k)}) \right] ^ {2} \\ \leq \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \\ \end{array} +$$ + +$$ +\cdot \left(\frac {\left(2 \varphi + 2 \pi - \varphi \pi\right) - \frac {(2 - \varphi) \pi}{\left(T _ {k + 1} - T _ {k}\right)} \left(t - T _ {k} - 0 . 5\right)}{\left(2 \varphi + 2 \pi - \varphi \pi\right) + \frac {(2 - \varphi) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}}\right) ^ {\frac {4 \lambda_ {i} ^ {(k)} (\eta_ {\max } - \eta_ {\min }) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)\right) \left(T _ {k + 1} - T _ {k}\right)}{(2 - \varphi) \pi}} \tag {21} +$$ + +Proof. By the fact that $1 - x \leq \exp(-x)$ , we have + +$$ +\prod_ {j = 1 + T _ {k}} ^ {t} \left[ \left(1 - \eta_ {j} \lambda_ {i} ^ {(k)}\right) \right] ^ {2} \leq \exp \left(- 2 \sum_ {j = 1 + T _ {k}} ^ {t} \eta_ {j} \lambda_ {i} ^ {(k)}\right) \tag {22} +$$ + +Substituting the expression for $\eta_{j}$ by the proposed form (5) yields: + +$$ +\begin{array}{l} \exp \left(- 2 \sum_ {j = 1 + T _ {k}} ^ {t} \eta_ {j} \lambda_ {i} ^ {(k)}\right) \\ = \exp \left( \right.- 2 \lambda_ {i} ^ {(k)} \sum_ {j = 1 + T _ {k}} ^ {t} \left(\eta_ {\max } - \eta_ {\min }\right) \frac {2 \left(1 + \cos \left(\frac {\left(2 (j - T _ {k}) - 1\right) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)\right)}{2 \varphi + (2 - \varphi) \left(1 + \cos \left(\frac {\left(2 (j - T _ {k}) - 1\right) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right) + \eta_ {\min }}\right) \tag {23} \\ = \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \exp \left(- 2 \lambda_ {i} ^ {(k)} \sum_ {j = 1 + T _ {k}} ^ {t} (\eta_ {\max } - \eta_ {\min }) \frac {2 (1 + \cos (\frac {(2 (j - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}))}{2 \varphi + (2 - \varphi) (1 + \cos (\frac {(2 (j - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}))}\right) \\ \end{array} +$$ + +The behavior of this term depends on the value of the parameter $\varphi$ . We consider the following cases: + +$$ +\bullet \text {i f} \varphi \geq 2, 2 - \varphi \leq 0 +$$ + +Since + +$$ +\begin{array}{l} \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right) \geq \cos \left(\frac {(2 (j - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right) \geq 1 - \left(\frac {(2 (j - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right) \tag {24} \\ \geq 1 - \pi \left(\frac {(j - T _ {k})}{(T _ {k + 1} - T _ {k})}\right), \\ \end{array} +$$ + +we have + +$$ +\begin{array}{l} \sum_ {j = 1 + T _ {k}} ^ {t} \frac {1}{2 \varphi + (2 - \varphi) \left(1 + \cos \left(\frac {(2 (j - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right)} \\ \leq \sum_ {j = 1 + T _ {k}} ^ {t} \frac {1}{2 \varphi + (2 - \varphi) \left(1 + 1 - \pi \left(\frac {(j - T _ {k})}{\left(T _ {k + 1} - T _ {k}\right)}\right)\right)} \tag {25} \\ = \sum_ {j = 1 + T _ {k}} ^ {t} \frac {1}{4 + \frac {(\varphi - 2) \pi}{\left(T _ {k + 1} - T _ {k}\right)} (j - T _ {k})} \\ \end{array} +$$ + +Then the equality (23) becomes + +$$ +\begin{array}{l} \exp \left(-2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \exp \left(- 2 \lambda_ {i} ^ {(k)} \sum_ {j = 1 + T _ {k}} ^ {t} (\eta_ {\max } - \eta_ {\min }) \frac {2 (1 + \cos (\frac {(2 (j - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}))}{2 \varphi + (2 - \varphi) (1 + \cos (\frac {(2 (j - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}))}\right) \\ \leq \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \exp \left[ - 4 \lambda_ {i} ^ {(k)} \left(\eta_ {\max } - \eta_ {\min }\right) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) \right. \\ \left. \sum_ {j = 1 + T _ {k}} ^ {t} \frac {1}{2 \varphi + (2 - \varphi) \left(1 + \cos \left(\frac {(2 (j - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)\right)} \right] \tag {26} \\ \leq \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \exp \left[ - 4 \lambda_ {i} ^ {(k)} \left(\eta_ {\max } - \eta_ {\min }\right) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)\right) \right. \\ \left. \sum_ {j = 1 + T _ {k}} ^ {t} \frac {1}{4 + \frac {(\varphi - 2) \pi}{\left(T _ {k + 1} - T _ {k}\right)} (j - T _ {k})} \right] \\ \end{array} +$$ + +Note that the function $h(j) \coloneqq \frac{1}{4 + \frac{(\varphi - 2)\pi}{T_{k + 1} - T_k}(j - T_k)}$ is monotone decreasing over $j \in [1 + T_k, t]$ , so we can bound the summation from below by the integral: + +$$ +\begin{array}{l} \sum_ {j = 1 + T _ {k}} ^ {t} \frac {1}{4 + \frac {(\varphi - 2) \pi}{T _ {k + 1} - T _ {k}} (j - T _ {k})} \\ \geq \int_ {1 + T _ {k}} ^ {t + 1} \frac {1}{4 + \frac {(\varphi - 2) \pi}{T _ {k + 1} - T _ {k}} (j - T _ {k})} d j \tag {27} \\ \geq \int_ {1 + T _ {k}} ^ {t} \frac {1}{4 + \frac {(\varphi - 2) \pi}{T _ {k + 1} - T _ {k}} (j - T _ {k})} d j \\ = \frac {\left(T _ {k + 1} - T _ {k}\right)}{(\varphi - 2) \pi} \ln \left(\frac {4 \left(T _ {k + 1} - T _ {k}\right) + (\varphi - 2) \pi (t - T _ {k})}{4 \left(T _ {k + 1} - T _ {k}\right) + (\varphi - 2) \pi}\right) \\ \end{array} +$$ + +Thus the inequality (26) becomes + +$$ +\begin{array}{l} \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\mathrm {m i n}} (t - T _ {k})\right) \exp \left[ - 4 \lambda_ {i} ^ {(k)} (\eta_ {\mathrm {m a x}} - \eta_ {\mathrm {m i n}}) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) \right] \\ \sum_ {j = 1 + T _ {k}} ^ {t} \left. \frac {1}{4 + \frac {(\varphi - 2) \pi}{\left(T _ {k + 1} - T _ {k}\right)} (j - T _ {k})} \right] \\ \leq \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \exp \left[ 4 \lambda_ {i} ^ {(k)} \left(\eta_ {\max } - \eta_ {\min }\right) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)\right) \right. \\ \left. \frac {\left(T _ {k + 1} - T _ {k}\right)}{(\varphi - 2) \pi} \ln \left(\frac {4 \left(T _ {k + 1} - T _ {k}\right) + (\varphi - 2) \pi}{4 \left(T _ {k + 1} - T _ {k}\right) + (\varphi - 2) \pi (t - T _ {k})}\right) \right] \tag {28} \\ = \exp \Bigl (- 2 \lambda_ {i} ^ {(k)} \eta_ {\min} (t - T _ {k}) \Bigr) \Bigl (\frac {4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi}{4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi (t - T _ {k})} \Bigr) ^ {\frac {4 \lambda_ {i} ^ {(k)} (\eta_ {\max} - \eta_ {\min}) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) (T _ {k + 1} - T _ {k})}{(\varphi - 2) \pi}} \\ \leq \exp \Bigl (- 2 \lambda_ {i} ^ {(k)} \eta_ {\min} (t - T _ {k}) \Bigr) \Bigl (\frac {4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi}{4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi (t - T _ {k})} \Bigr) ^ {\frac {4 \lambda_ {l} ^ {(k)} (\eta_ {\max} - \eta_ {\min}) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) (T _ {k + 1} - T _ {k})}{(\varphi - 2) \pi}} \\ \end{array} +$$ + +The last inequality is because $\lambda_l^{(k)} \leq \lambda_i^{(k)}$ and $\frac{4(T_{k+1} - T_k) + (\varphi - 2)\pi}{4(T_{k+1} - T_k) + (\varphi - 2)\pi(t - T_k)} \leq 1$ . + +- if $\varphi \leq 2, 2 - \varphi \geq 0$ Since + +$$ +1 + \cos \left(\frac {(2 (j - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right) \leq \pi - \frac {(2 (j - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}. \tag {29} +$$ + +We obtain + +$$ +\begin{array}{l} \exp \left(-2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \exp \left(- 2 \lambda_ {i} ^ {(k)} \sum_ {j = 1 + T _ {k}} ^ {t} (\eta_ {\max } - \eta_ {\min }) \frac {2 (1 + \cos (\frac {(2 (j - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}))}{2 \varphi + (2 - \varphi) (1 + \cos (\frac {(2 (j - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}))}\right) \\ \leq \exp \Bigl (- 2 \lambda_ {i} ^ {(k)} \eta_ {\min} (t - T _ {k}) \Bigr) \exp \Biggl [ - 4 \lambda_ {i} ^ {(k)} (\eta_ {\max} - \eta_ {\min}) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) \\ \left. \sum_ {j = 1 + T _ {k}} ^ {t} \frac {1}{2 \varphi + (2 - \varphi) (1 + \cos \left(\frac {(2 (j - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)} \right] \tag {30} \\ \leq \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \exp \left[ - 4 \lambda_ {i} ^ {(k)} (\eta_ {\max } - \eta_ {\min }) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) \right. \\ \sum_ {j = 1 + T _ {k}} ^ {t} \left. \left(\frac {1}{(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{\left(T _ {k + 1} - T _ {k}\right)} (j - T _ {k} - 0 . 5)}\right) \right] \\ \end{array} +$$ + +because function $h(j) \coloneqq \frac{1}{(2\varphi + 2\pi - \varphi\pi) - \frac{(2 - \varphi)\pi}{(T_{k + 1} - T_k)}(j - T_k - 0.5)}$ is monotone increasing in the range $[1 + T_k, +\infty)$ , then it holds that + +$$ +\begin{array}{l} \sum_ {j = 1 + T _ {k}} ^ {t} \frac {1}{(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{\left(T _ {k + 1} - T _ {k}\right)} (j - T _ {k} - 0 . 5)} \\ \geq \int_ {T _ {k}} ^ {t} \frac {1}{(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{\left(T _ {k + 1} - T _ {k}\right)} (j - T _ {k} - 0 . 5)} d j \tag {31} \\ = - \frac {\left(T _ {k + 1} - T _ {k}\right)}{(2 - \varphi) \pi} \ln \left(\frac {(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{\left(T _ {k + 1} - T _ {k}\right)} (t - T _ {k} - 0 . 5)}{(2 \varphi + 2 \pi - \varphi \pi) + \frac {(2 - \varphi) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}}\right) \\ \end{array} +$$ + +Thus, we have + +$$ +\begin{array}{l} \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \exp \left[ - 4 \lambda_ {i} ^ {(k)} (\eta_ {\max } - \eta_ {\min }) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) \right. \\ \sum_ {j = 1 + T _ {k}} ^ {t} \left. \left(\frac {1}{(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{\left(T _ {k + 1} - T _ {k}\right)} (j - T _ {k} - 0 . 5)}\right) \right] \\ \leq \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \exp \left[ 4 \lambda_ {i} ^ {(k)} \left(\eta_ {\max } - \eta_ {\min }\right) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)\right) \right. \\ \frac {\left(T _ {k + 1} - T _ {k}\right)}{(2 - \varphi) \pi} \ln \left(\left. \frac {(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{(T _ {k + 1} - T _ {k})} (t - T _ {k} - 0 . 5)}{(2 \varphi + 2 \pi - \varphi \pi) + \frac {(2 - \varphi) \pi}{2 (T _ {k + 1} - T _ {k})}}\right) \right] \\ \leq \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } \left(t - T _ {k}\right)\right) \tag {32} \\ \end{array} +$$ + +$$ +\begin{array}{l} \cdot \left(\frac {(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{(T _ {k + 1} - T _ {k})} (t - T _ {k} - 0 . 5)}{(2 \varphi + 2 \pi - \varphi \pi) + \frac {(2 - \varphi) \pi}{2 (T _ {k + 1} - T _ {k})}}\right) \frac {4 \lambda_ {i} ^ {(k)} (\eta_ {\max} - \eta_ {\min}) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) (T _ {k + 1} - T _ {k})}{(2 - \varphi) \pi} \\ \leq \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \\ \end{array} +$$ + +$$ +\cdot \left(\frac {(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{(T _ {k + 1} - T _ {k})} (t - T _ {k} - 0 . 5)}{(2 \varphi + 2 \pi - \varphi \pi) + \frac {(2 - \varphi) \pi}{2 (T _ {k + 1} - T _ {k})}}\right) ^ {\frac {4 \lambda_ {l} ^ {(k)} (\eta_ {\max} - \eta_ {\min}) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) (T _ {k + 1} - T _ {k})}{(2 - \varphi) \pi}} +$$ + +The last inequality is because $\lambda_l^{(k)}\leq \lambda_i^{(k)}$ and $\left(\frac{(2\varphi + 2\pi - \varphi\pi) - \frac{(2 - \varphi)\pi}{(T_{k + 1} - T_k)}(t - T_k - 0.5)}{(2\varphi + 2\pi - \varphi\pi) + \frac{(2 - \varphi)\pi}{2(T_{k + 1} - T_k)}}\right)\leq 1.$ + +![](images/4a36b8a91a2578419d8b56d961fbbbc6fa1a0006a0208724415ce4dbed83d004.jpg) + +Theorem 1 Let $n_t = H_f^{(k)}(W_t - \bar{W}^{(k)}) - H_f^{(k)}(\xi)(W_t - \bar{W}^{(k)})$ be the stochastic curvature noise introduced by sampling at iteration $t$ . Assume that the sampling Hessian satisfies: $\mathbb{E}_{\xi}\left[n_t n_t^\top\right] \preceq \sigma^2 H_f^{(k)}$ for some constant $\sigma$ . + +- If we set the learning rate as the proposed form (5) where hyper-parameter $\varphi > 2$ , the loss uncertainty introduced by stochastic gradient method within the interval $t \in [1 + T_k, T_{k+1}]$ + +can be quantified as two terms, bias term and variance term, bounded as follows, + +$$ +\begin{array}{l} \mathbb {E} \left[ \mathcal {L} (f (W _ {t}, \xi)) - \mathcal {L} (f (\bar {W} ^ {(k)}, \xi)) \right] \\ \leq \left(\frac {4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi}{4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi (t - T _ {k})}\right) ^ {\frac {4 \lambda_ {l} ^ {(k)} (\eta_ {\max } - \eta_ {\min }) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) (T _ {k + 1} - T _ {k})}{(\varphi - 2) \pi}} \\ \cdot \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \lambda_ {u} ^ {(k)} \| W _ {1 + T _ {k}} - \bar {W} ^ {(k)} \| _ {2} ^ {2} \\ + \sigma^ {2} \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \sum_ {j = 1} ^ {N} \left(\lambda_ {j} ^ {(k)}\right) ^ {2} \exp \left(- 2 \lambda_ {j} ^ {(k)} \eta_ {\min } (t - i)\right) \tag {33} \\ \cdot \left(\frac {4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi (i - T _ {k})}{4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi (t - T _ {k})}\right) \frac {4 \lambda_ {l} ^ {(k)} (\eta_ {\mathrm {m a x}} - \eta_ {\mathrm {m i n}}) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) (T _ {k + 1} - T _ {k})}{(\varphi - 2) \pi} \\ \end{array} +$$ + +- If we set the learning rate as the proposed form (5) where hyper-parameter $\varphi < 2$ , the loss uncertainty introduced by stochastic gradient method within the interval $t \in [1 + T_k, T_{k+1}]$ can be quantified as two terms, bias term and variance term, bounded as follows, + +$$ +\begin{array}{l} \mathbb {E} \left[ \mathcal {L} (f (W _ {t}, \xi)) - \mathcal {L} (f (\bar {W} ^ {(k)}, \xi)) \right] \\ \leq \Big (\frac {(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{(T _ {k + 1} - T _ {k})} (t - T _ {k} - 0 . 5)}{(2 \varphi + 2 \pi - \varphi \pi) + \frac {(2 - \varphi) \pi}{2 (T _ {k + 1} - T _ {k})}} \Big) ^ {\frac {4 \lambda_ {i} ^ {(k)} (\eta_ {\max} - \eta_ {\min}) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) (T _ {k + 1} - T _ {k})}{(2 - \varphi) \pi}} \\ \cdot \exp \Bigl (- 2 \lambda_ {i} ^ {(k)} \eta_ {\min} (t - T _ {k}) \Bigr) \lambda_ {u} ^ {(k)} \| W _ {1 + T _ {k}} - \bar {W} ^ {(k)} \| _ {2} ^ {2} \\ + \leq \sigma^ {2} \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \sum_ {j = 1} ^ {N} \left(\lambda_ {j} ^ {(k)}\right) ^ {2} \exp \left(- 2 \lambda_ {j} ^ {(k)} \eta_ {\min } (t - i)\right) \tag {34} \\ \cdot \left(\frac {(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{(T _ {k + 1} - T _ {k})} (t - T _ {k} - 0 . 5)}{(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{(T _ {k + 1} - T _ {k})} (i - T _ {k} - 0 . 5)}\right) \frac {4 \lambda_ {i} ^ {(k)} (\eta_ {\mathrm {m a x}} - \eta_ {\mathrm {m i n}}) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) (T _ {k + 1} - T _ {k})}{(2 - \varphi) \pi} \\ \end{array} +$$ + +Proof. Let $H_{f}^{(k)}$ denote full batch Hessian at $k$ -th minimum, such that $\mathbb{E}(H_{f}^{(k)}(\xi)) = H_{f}^{(k)}$ , and denote $n_{t} = H_{f}^{(k)}(W_{t} - \bar{W}^{(k)}) - H_{f}^{(k)}(\xi)(W_{t} - \bar{W}^{(k)})$ , which indicates the bias. The uncertainty brought by stochastic gradient method is measured as follows, + +$$ +\begin{array}{l} W _ {t + 1} - \bar {W} ^ {(k)} = W _ {t} - \bar {W} ^ {(k)} - \eta_ {t} H _ {f} ^ {(k)} (\xi) (W _ {t} - \bar {W} ^ {(k)}) \\ = W _ {t} - \bar {W} ^ {(k)} - \eta_ {t} H _ {f} ^ {(k)} (W _ {t} - \bar {W} ^ {(k)}) \\ + \eta_ {t} H _ {f} ^ {(k)} (W _ {t} - \bar {W} ^ {(k)}) - \eta_ {t} H _ {f} ^ {(k)} (\xi) (W _ {t} - \bar {W} ^ {(k)}) \\ = (I - \eta_ {t} H _ {f} ^ {(k)}) (W _ {t} - \bar {W} ^ {(k)}) + \eta_ {t} n _ {t} \\ = \left(I - \eta_ {t} H _ {f} ^ {(k)}\right) \left(I - \eta_ {t - 1} H _ {f} ^ {(k)}\right) \dots \left(I - \eta_ {t - q} H _ {f} ^ {(k)}\right) \left(W _ {t - q} - \bar {W} ^ {(k)}\right) \tag {35} \\ + \sum_ {i = t - q} ^ {t} (I - \eta_ {t} H _ {f} ^ {(k)}) (I - \eta_ {t - 1} H _ {f} ^ {(k)}) \dots (I - \eta_ {i + 1} H _ {f} ^ {(k)}) \eta_ {i} n _ {i} \\ := P _ {t} ^ {(k)} P _ {t - 1} ^ {(k)} \dots P _ {t - q} ^ {(k)} (W _ {t - q} - \bar {W} ^ {(k)}) + \sum_ {i = t - q} ^ {t} P _ {t} ^ {(k)} P _ {t - 1} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \eta_ {i} n _ {i} \\ \end{array} +$$ + +where $P_{i}^{(k)}\coloneqq (I - \eta_{i}H_{f}^{(k)})$ + +According to the approximation approach of the objective function (1) in Section 3.1, i.e. $\mathcal{L}(f(W,\xi))\approx \mathcal{L}(f(\bar{W}^{(k)},\xi)) + \frac{1}{2} (W - \bar{W}^{(k)})^{\top}H_{f}^{(k)}(\xi)(W - \bar{W}^{(k)})$ , the loss uncertainty introduced by stochastic gradient method within the interval $t\in [1 + T_k,T_{k + 1}]$ is quantified as $\mathbb{E}[(W_{T_{k + 1}} - \bar{W}^{(k)})^\top H_f^{(k)}(W_{T_{k + 1}} - \bar{W}^{(k)})]$ , where the loss function is approximated by leveraging the full batch Hessian. + +$$ +\begin{array}{l} \mathbb {E} \left[ \mathcal {L} (f (W _ {t}, \xi)) - \mathcal {L} (f (\bar {W} ^ {(k)}, \xi)) \right] = \mathbb {E} \left[ (W _ {t} - \bar {W} ^ {(k)}) ^ {\top} H _ {f} ^ {(k)} (W _ {t} - \bar {W} ^ {(k)}) \right] \\ = \mathbb {E} \left[ \left(P _ {t} ^ {(k)} P _ {t - 1} ^ {(k)} \dots P _ {1 + T _ {k}} ^ {(k)} \left(W _ {1 + T _ {k}} - \bar {W} ^ {(k)}\right) + \sum_ {i = 1 + T _ {k}} ^ {t} P _ {t} ^ {(k)} P _ {T _ {k + 1} - 1} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \eta_ {i} n _ {i}\right) ^ {\top} \right. \\ \end{array} +$$ + +$$ +\begin{array}{l} \left. H _ {f} ^ {(k)} \left(P _ {t} ^ {(k)} P _ {t - 1} ^ {(k)} \dots P _ {1 + T _ {k}} ^ {(k)} \left(W _ {1 + T _ {k}} - \bar {W} ^ {(k)}\right) + \sum_ {i = 1 + T _ {k}} ^ {t} P _ {t} ^ {(k)} P _ {t - 1} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \eta_ {i} n _ {i}\right) \right] \tag {36} \\ = \mathbb {E} \left[ \left(W _ {t} - \bar {W} ^ {(k)}\right) ^ {\top} P _ {1 + T _ {k}} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {1 + T _ {k}} ^ {(k)} \left(W _ {t} - \bar {W} ^ {(k)}\right) \right] \\ + 2 \mathbb {E} \left[ \left(W _ {t} - \bar {W} ^ {(k)}\right) ^ {\top} P _ {1 + T _ {k}} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} \left(\sum_ {i = 1 + T _ {k}} ^ {t} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \eta_ {i} n _ {i}\right) \right] \\ + \mathbb {E} \left(\sum_ {i = 1 + T _ {k}} ^ {t} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \eta_ {i} n _ {i}\right) ^ {\top} H _ {f} ^ {(k)} \left(\sum_ {i = 1 + T _ {k}} ^ {t} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \eta_ {i} n _ {i}\right) \\ \end{array} +$$ + +Since $\mathbb{E}[n_t] = 0$ , $\mathbb{E}\left[(W_t - \bar{W}^{(k)})^\top P_{1 + T_k}^{(k)}\dots P_t^{(k)}H_f^{(k)}\left(\sum_{i = 1 + T_k}^t P_t^{(k)}\dots P_{i + 1}^{(k)}\eta_in_i\right)\right] = 0$ . Meanwhile, we notice the data samplings between different iteration are independent, thus $\mathbb{E}\left[(n_i)^\top n_j\right] = 0 (i\neq j)$ . We obtain + +$$ +\begin{array}{l} \mathbb {E} \left(\sum_ {i = 1 + T _ {k}} ^ {t} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \eta_ {i} n _ {i}\right) ^ {\top} H _ {f} ^ {(k)} \left(\sum_ {i = 1 + T _ {k}} ^ {t} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \eta_ {i} n _ {i}\right) \\ = \mathbb {E} \left(\sum_ {i, j = 1 + T _ {k}} ^ {t} \eta_ {i} n _ {i} ^ {\top} P _ {i + 1} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {j + 1} ^ {(k)} \eta_ {j} n _ {j}\right) \tag {37} \\ = \sum_ {i = 1 + T _ {k}} ^ {t} \mathbb {E} \left(\eta_ {i} n _ {i} ^ {\top} P _ {i + 1} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \eta_ {i} n _ {i}\right) \\ \end{array} +$$ + +Therefore, Eq.(36) can be reformulated as follows, + +$$ +\begin{array}{l} \mathbb {E} \left(\sum_ {i = 1 + T _ {k}} ^ {t} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \eta_ {i} n _ {i}\right) ^ {\top} H _ {f} ^ {(k)} \left(\sum_ {i = 1 + T _ {k}} ^ {t} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \eta_ {i} n _ {i}\right) \\ = \mathbb {E} \left[ \left(W _ {t} - \bar {W} ^ {(k)}\right) ^ {\top} P _ {1 + T _ {k}} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {1 + T _ {k}} ^ {(k)} \left(W _ {t} - \bar {W} ^ {(k)}\right) \right] \tag {38} \\ + \sum_ {i = 1 + T _ {k}} ^ {t} \mathbb {E} \left(\eta_ {i} n _ {i} ^ {\top} P _ {i + 1} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \eta_ {i} n _ {i}\right) \\ := B + V \\ \end{array} +$$ + +where we denote $B := \mathbb{E}\left[(W_t - \bar{W}^{(k)})^\top P_{1 + T_k}^{(k)}\cdots P_t^{(k)}H_f^{(k)}P_t^{(k)}\cdots P_{1 + T_k}^{(k)}(W_t - \bar{W}^{(k)})\right]$ bias term and $V := \sum_{i = 1 + T_k}^t\mathbb{E}\left(\eta_in_i^\top P_{i + 1}^{(k)}\dots P_t^{(k)}H_f^{(k)}P_t^{(k)}\dots P_{i + 1}^{(k)}\eta_in_i\right)$ variance term. + +Denote $u_{i}^{(k)} \quad i \in \{1,2,\dots ,N\}$ are linearly independent eigenvectors, which can form the basis for the $N$ dimension vector space. Then we have $H_{f}^{(k)}(\xi)u_{i}^{(k)} = \lambda_{i}^{(k)}u_{i}^{(k)} \quad (i = 1,2,\dots ,N)$ and the initial deviation from the optimal solution can be expressed as $W_{t} - \bar{W}^{(k)} = \sum_{i = 1}^{N}s_{i}^{(k)}u_{i}^{(k)}$ where $s_i^{(k)}$ are the coefficients corresponding to the eigenvector components. Combined with Lemma 1, we obtain + +Case 1 $(\varphi >2)$ + +$$ +\begin{array}{l} B = \left(W _ {t} - \bar {W} ^ {(k)}\right) ^ {\top} P _ {1 + T _ {k}} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {1 + T _ {k}} ^ {(k)} \left(W _ {t} - \bar {W} ^ {(k)}\right) \\ = \left[ (I - \eta_ {t} H _ {f} ^ {(k)}) \dots (I - \eta_ {1 + T _ {k}} H _ {f} ^ {(k)}) (W _ {1 + T _ {k}} - \bar {W} ^ {(k)}) \right] ^ {\top} H _ {f} ^ {(k)} (I - \eta_ {t} H _ {f} ^ {(k)}) \dots (I - \eta_ {1 + T _ {k}} H _ {f} ^ {(k)}) (W _ {1 + T _ {k}} - \bar {W} ^ {(k)}) \\ = \sum_ {i = 1} ^ {N} (s _ {i} ^ {(k)}) ^ {2} \| u _ {i} ^ {(k)} \| _ {2} ^ {2} \lambda_ {i} ^ {(k)} \prod_ {j = 1 + T _ {k}} ^ {t} \left[ (1 - \eta_ {j} \lambda_ {i} ^ {(k)}) \right] ^ {2} \\ \leq \sum_ {i = 1} ^ {N} \left(s _ {i} ^ {(k)}\right) ^ {2} \| u _ {i} ^ {(k)} \| _ {2} ^ {2} \lambda_ {i} ^ {(k)} \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \\ \cdot \left(\frac {4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi}{4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi (t - T _ {k})}\right) ^ {\frac {4 \lambda_ {l} ^ {(k)} (\eta_ {\max } - \eta_ {\min }) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) (T _ {k + 1} - T _ {k})}{(\varphi - 2) \pi}} \\ \leq \left(\frac {4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi}{4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi (t - T _ {k})}\right) ^ {\frac {4 \lambda_ {l} ^ {(k)} (\eta_ {\max } - \eta_ {\min }) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) (T _ {k + 1} - T _ {k})}{(\varphi - 2) \pi}} \\ \cdot \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } \left(t - T _ {k}\right)\right) \lambda_ {u} ^ {(k)} \| W _ {1 + T _ {k}} - \bar {W} ^ {(k)} \| _ {2} ^ {2} \tag {39} \\ \end{array} +$$ + +In addition, + +$$ +\begin{array}{l} V = \sum_ {i = 1 + T _ {k}} ^ {t} \mathbb {E} \left(\eta_ {i} n _ {i} ^ {\top} P _ {i + 1} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \eta_ {i} n _ {i}\right) \\ = \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \mathbb {E} \left[ \operatorname {t r} \left(n _ {i} ^ {\top} P _ {i + 1} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} n _ {i}\right) \right] \\ = \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \mathbb {E} \left[ \operatorname {t r} \left(P _ {i + 1} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} n _ {i} n _ {i} ^ {\top}\right) \right] \\ = \sum_ {\substack {i = 1 + T _ {k} \\ t}} ^ {t} \eta_ {i} ^ {2} \operatorname {t r} \left[ P _ {i + 1} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \mathbb {E} \left(n _ {i} n _ {i} ^ {\top}\right) \right] \tag{40} \\ \leq \sigma^ {2} \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \operatorname {t r} \left[ P _ {i + 1} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} H _ {f} ^ {(k)} \right] \\ = \sigma^ {2} \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \sum_ {j = 1} ^ {N} \left(\lambda_ {j} ^ {(k)}\right) ^ {2} \prod_ {q = i + 1} ^ {t} \left(1 - \eta_ {q} \lambda_ {j} ^ {(k)}\right) ^ {2} \\ \leq \sigma^ {2} \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \sum_ {j = 1} ^ {N} \left(\lambda_ {j} ^ {(k)}\right) ^ {2} \exp \left(- 2 \lambda_ {j} ^ {(k)} \eta_ {\min } (t - i)\right) \\ \cdot \left(\frac {4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi (i - T _ {k})}{4 (T _ {k + 1} - T _ {k}) + (\varphi - 2) \pi (t - T _ {k})}\right) \frac {4 \lambda_ {l} ^ {(k)} (\eta_ {\max } - \eta_ {\min }) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) (T _ {k + 1} - T _ {k})}{(\varphi - 2) \pi} \\ \end{array} +$$ + +where the second equality comes from cyclic property of trace and the first inequality comes from $\mathbb{E}_{\xi}\left[n_{t}n_{t}^{\top}\right] \preceq \sigma^{2}H_{f}^{(k)}$ . + +Case 2 $(\varphi < 2)$ : + +$$ +\begin{array}{l} B = (W _ {t} - \bar {W} ^ {(k)}) ^ {\top} P _ {1 + T _ {k}} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {1 + T _ {k}} ^ {(k)} (W _ {t} - \bar {W} ^ {(k)}) \\ = \left[ (I - \eta_ {t} H _ {f} ^ {(k)}) \dots (I - \eta_ {1 + T _ {k}} H _ {f} ^ {(k)}) (W _ {1 + T _ {k}} - \bar {W} ^ {(k)}) \right] ^ {\top} H _ {f} ^ {(k)} (I - \eta_ {t} H _ {f} ^ {(k)}) \dots (I - \eta_ {1 + T _ {k}} H _ {f} ^ {(k)}) (W _ {1 + T _ {k}} - \bar {W} ^ {(k)}) \\ = \sum_ {i = 1} ^ {N} (s _ {i} ^ {(k)}) ^ {2} \| u _ {i} ^ {(k)} \| _ {2} ^ {2} \lambda_ {i} ^ {(k)} \prod_ {j = 1 + T _ {k}} ^ {t} \left[ (1 - \eta_ {j} \lambda_ {i} ^ {(k)}) \right] ^ {2} \\ \leq \sum_ {i = 1} ^ {N} \left(s _ {i} ^ {(k)}\right) ^ {2} \| u _ {i} ^ {(k)} \| _ {2} ^ {2} \lambda_ {i} ^ {(k)} \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } (t - T _ {k})\right) \\ \cdot \left(\frac {\left(2 \varphi + 2 \pi - \varphi \pi\right) - \frac {(2 - \varphi) \pi}{\left(T _ {k + 1} - T _ {k}\right)} \left(t - T _ {k} - 0 . 5\right)}{\left(2 \varphi + 2 \pi - \varphi \pi\right) + \frac {(2 - \varphi) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}}\right) ^ {\frac {4 \lambda_ {i} ^ {(k)} (\eta_ {\max } - \eta_ {\min }) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 \left(T _ {k + 1} - T _ {k}\right)}\right)\right) (T _ {k + 1} - T _ {k})}{(2 - \varphi) \pi}} \\ \leq \Big (\frac {(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{(T _ {k + 1} - T _ {k})} (t - T _ {k} - 0 . 5)}{(2 \varphi + 2 \pi - \varphi \pi) + \frac {(2 - \varphi) \pi}{2 (T _ {k + 1} - T _ {k})}} \Big) ^ {\frac {4 \lambda_ {i} ^ {(k)} (\eta_ {\operatorname* {m a x}} - \eta_ {\operatorname* {m i n}}) \Big (1 + \cos \Big (\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})} \Big) \Big) (T _ {k + 1} - T _ {k})}{(2 - \varphi) \pi}} \\ \cdot \exp \left(- 2 \lambda_ {i} ^ {(k)} \eta_ {\min } \left(t - T _ {k}\right)\right) \lambda_ {u} ^ {(k)} \| W _ {1 + T _ {k}} - \bar {W} ^ {(k)} \| _ {2} ^ {2} \tag {41} \\ \end{array} +$$ + +In addition, + +$$ +\begin{array}{l} V = \sum_ {i = 1 + T _ {k}} ^ {t} \mathbb {E} \left(\eta_ {i} n _ {i} ^ {\top} P _ {i + 1} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \eta_ {i} n _ {i}\right) \\ = \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \mathbb {E} \left[ \operatorname {t r} \left(n _ {i} ^ {\top} P _ {i + 1} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} n _ {i}\right) \right] \\ = \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \mathbb {E} \left[ \operatorname {t r} \left(P _ {i + 1} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} n _ {i} n _ {i} ^ {\top}\right) \right] \\ = \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \operatorname {t r} \left[ P _ {i + 1} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} \mathbb {E} \left(n _ {i} n _ {i} ^ {\top}\right) \right] \\ \leq \sigma^ {2} \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \operatorname {t r} \left[ P _ {i + 1} ^ {(k)} \dots P _ {t} ^ {(k)} H _ {f} ^ {(k)} P _ {t} ^ {(k)} \dots P _ {i + 1} ^ {(k)} H _ {f} ^ {(k)} \right] \\ = \sigma^ {2} \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \sum_ {j = 1} ^ {N} \left(\lambda_ {j} ^ {(k)}\right) ^ {2} \prod_ {q = i + 1} ^ {t} \left(1 - \eta_ {q} \lambda_ {j} ^ {(k)}\right) ^ {2} \\ \leq \sigma^ {2} \sum_ {i = 1 + T _ {k}} ^ {t} \eta_ {i} ^ {2} \sum_ {j = 1} ^ {N} \left(\lambda_ {j} ^ {(k)}\right) ^ {2} \exp \left(- 2 \lambda_ {j} ^ {(k)} \eta_ {\min } (t - i)\right) \\ \cdot \left(\frac {(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{(T _ {k + 1} - T _ {k})} (t - T _ {k} - 0 . 5)}{(2 \varphi + 2 \pi - \varphi \pi) - \frac {(2 - \varphi) \pi}{(T _ {k + 1} - T _ {k})} (i - T _ {k} - 0 . 5)}\right) ^ {\frac {4 \lambda_ {i} ^ {(k)} (\eta_ {\max } - \eta_ {\min }) \left(1 + \cos \left(\frac {(2 (t - T _ {k}) - 1) \pi}{2 (T _ {k + 1} - T _ {k})}\right)\right) (T _ {k + 1} - T _ {k})}{(2 - \varphi) \pi}} \tag {42} \\ \end{array} +$$ + +where the second equality comes from cyclic property of trace and the first inequality comes from $\mathbb{E}_{\xi}\left[n_{t}n_{t}^{\top}\right] \preceq \sigma^{2}H_{f}^{(k)}$ . + +□ + +In this proof, we employ assumption $\mathbb{E}_{\xi}\left[n_{t}n_{t}^{\top}\right] \preceq \sigma^{2}H_{f}^{(k)}$ used in the work [16, 36]. Moreover, we acknowledge that the idea of the proof is inspired by the approach introduced in [16, 36] + +# E Experimental details + +# E.1Baselines + +We detail the baselines as follows, + +- Step schedule(SS) reduces the learning rate in a piecewise manner. It implements discrete learning rate drops at predefined epoch intervals following schedule $g(t) = d_{\lfloor t / t_s \rfloor}$ , where $t_s$ is the predefined decaying step and $d_i (d_i \geq d_{i+1})$ is predefined decaying scalar [16]. A typical implementation decreases the learning rate by a factor of 0.1 after 50% of the epochs and further by a factor of 0.01 after 75% of the epochs [21]. However, it requires careful tuning of step timing since abrupt changes may destabilize optimization. In this paper, we employ such setting of step schedule ( $\times 0.1$ after 50% and $\times 0.01$ after 75%) for all our experiments. +- Cosine schedule(CS) controls the $t$ -th iteration learning rate $\eta_t$ following the mathematical formulation as $\eta_t = \eta_{min} + \frac{1}{2} (\eta_0 - \eta_{min})(1 + \cos(\frac{t\pi}{T}))$ where $T$ is total iterations. It provides smooth transition between learning rates, shown to improve final model accuracy step schedule. Moreover, it is proposed to mitigate abrupt decreases in the learning rate that could hinder models from escaping local minima. This approach has demonstrated effectiveness, particularly in transformer training [33]. Besides, it includes restart mechanisms where $T$ becomes the cyclic period. +- Cyclic Learning Rates (CLR) oscillates between base learning rate $\eta_{\text{min}}$ and maximal learning rate $\eta_{\text{max}}$ with triangular and triangular2 policies. The cyclic property helps traverse loss landscape barriers, so that it enables neural networks to train significantly faster, often by an order of magnitude, compared to standard training methods [43, 44]. Besides, the commonly-used OneCycle learning rate schedule can be seen as the one period version of Cyclic Learning Rates. In our experiments, this baseline CLR includes Cyclic and OneCycle schedules(1C). For CLR, we set linear mode for each period and period is two. +- Budgeted training(BT) is conducted by the proposed budget-aware setting of existing learning rate schedule under epoch budget [30]. It reformulates standard schedules such as linear, step and cosine schedule as functions of remaining budget. The classic linear decay version is formulated as $\eta_t = \eta_0 \times (1 - t / T)$ where $T$ is total iterations. It is particularly effective for hyper-parameter tuning. +- Reflected Exponential (REX) controls the $t$ -th iteration learning rate $\eta_t$ following the mathematical formulation as $\eta_t = \eta_0\left(\frac{1 - t / T}{0.5 + 0.5\times(1 - t / T)}\right)$ . It is inspired by the performance of delayed linear schedules, aggressively reducing the learning rate near the end of training and reflecting an exponential decay [5]. + +# E.2 Evaluation benchmarks of language task + +We evaluate models on a diverse set of open benchmarks. The benchmarks are listed in Table 5. For language models evaluation, we adopt the [15] for standardized performance comparisons. + +# E.3 Implementation Details + +Consistent with CIFAR, ImageNet and OLMo practices, we use step-wise scheduling for all training procedures. + +Vision tasks on CIFAR CIFAR10/100 are simple benchmark datasets that are widely used for quick and efficient evaluation of deep learning tasks. CIFAR10 consists of a training split of 50000 examples and a test split of 10000 examples. Each example is a $32 \times 32$ color image in 10 classes (airplane, automobile, bird, cat, deer, dog, frog, horse, ship, truck). CIFAR100 comprises 60000 $32 \times 32$ pixels color images which are divided into 100 classes. Each class contains 600 images and the classes are grouped into 20 super-classes. Each image comes with a fine label (the class to which it belongs) and a coarse label (the super-class to which it belongs). We use the torch.optim.SGD and torch.optim.AdamW API to configure the optimizer. For vision tasks in Section 4.1, we all adopt the SGD optimizer. For ablation studies in Section 4.3, we adopt the AdamW optimizer. We + +Table 5: Evaluation benchmarks of language task + +
Datasets nameabbreviation
ARC-Easy [10]ARC-E
ARC-Challenge [10]ARC-C
HellaSwag [55]HSWAG
PIQA [3]PIQA
WinoGrande [39]WG
OpenbookQA [35]OBQA
BoolQ [9]BoolQ
SciQ [50]SciQ
COPA [18]COPA
CommonsenseQA [46]CSQA
SocialIQA [40]SIQA
MMLU stem [22]STEM
MMLU humanities [22]HUM
MMLU social sci [22]SOC
MMLU other [22]OTH
+ +initialize the learning rate to 1e-1, set the batch size to 128. Each epoch consists of $50000 / 128 + 1 = 391$ steps. We leave the weight decay value as 5e-4, and complete the training task on PyTorch 2.4.1+cu118 with RTX 5880 and RTX 3090. + +Vision tasks on ImageNet To evaluate the scalability of our schedule on larger-scale dataset, we conduct experiments on the ImageNet dataset, a dataset that more closely reflects real-world visual recognition challenges. ImageNet consists of approximately 1.28 million training images and 50000 validation images across 1000 classes, which also come with an official dataset split. This benchmark, particularly with the ResNet-50 architecture, has been extensively studied and is widely regarded as a standard evaluation setup. We adopt the common ResNet50 architecture, and separately train it with different learning rate schedulers. To provide a comprehensive evaluation, we conduct experiments using both the classical SGD optimizer and the popularized AdamW optimizer. For the experiments with SGD, we set the peak learning rate to 0.5 and apply the weight decay factor of 1e-4. For the experiments using AdamW, the peak learning rate is set to 3e-3, with a decoupled weight decay of 0.02. Following default configuration which requires learning rate warmup, we integrate $10\%$ warmup phases of the total training tokens into all schedules. For schedules that inherently include a warmup-like behavior, such as the 1C schedule, where the learning rate initially rises from zero to a target value, we retain their original settings during the ascending phase. We set the batch size to 2048 and complete these training tasks with 8 NVIDIA A100 GPUs. + +Language tasks For language tasks, we use the OLMo model. OLMo [19] is a decoder-only transformer-based language model that builds upon the vanilla Transformer architecture [49]. It incorporates several key improvements inspired by recent large language models (LLMs) such as PaLM [8], the LLaMA family [48], OpenLM and Falcon [1]. Notably, OLMo stands out as a truly open language model, offering full transparency through its open training data, released training/evaluation code, intermediate model checkpoints, and comprehensive training logs, making it a valuable resource for reproducibility and further research. + +For the training hyper-parameters on OLMo, we primarily adopt the configuration outlined in OLMo [19]. We conduct experiment on H100 and A100. Following default configuration which requires learning rate warmup, we integrate $10\%$ warmup phases of the total training tokens into all schedules. For schedules that inherently include a warmup-like behavior, such as the 1C schedule, where the learning rate initially rises from zero to a target value, we retain their original settings during the ascending phase. Besides, large language models commonly provide multiple scales to accommodate different compute constraints. We adjust both model size and training steps to explore budgets. To ensure robust conclusions, we maintain a data-to-parameters ratio (D/N) of 500 across scales. The specific configurations are as follows, 36M-parameter model is trained on 50 billion tokens, 73M-parameter model is trained on 50 billion tokens, 151M-parameter model is trained on 75 billion tokens, 300M-parameter model is trained on 150 billion tokens. This scaling strategy allows us + +to systematically study the relationship between model size, training data, and performance while controlling for the D/N ratio. The OLMo model is trained using AdamW optimizer. The detailed configuration are shown in Table 6. + +Table 6: Configurations of OLMo + +
Model scale36M73M151M300M
token number50B50B75B150B
Layers (L)12162024
Hidden dim (D)5126407681024
Attention heads81088
Inner dim (H)1280153622402688
Vocab size (V)100,352
+ +Our implementation strictly follows the PyTorch official API, i.e. it is inherited from the base class + +torch.optim.lr_scheduler.LRScheduler + +It maintains full compatibility with all + +torch optim + +optimizers. The complete production-ready code is publicly available. + +The code of UBA schedule is available at https://github.com/Ttt-answer/UBA.git. + +# E.4 Overall experiment results for Vision Classification Tasks + +Overall results for the vision classification tasks are presented in Table 7, depicting validation accuracy on vision benchmarks (e.g., CIFAR10/100 and ImageNet) using different architectures (ResNet18, ResNet34, ResNet50). We conduct several independent runs for error bars. Due to computational constraints, we use two independent experimental results for ImageNet-ResNet50. It is obvious that our proposed UBA schedule outperforms baselines, achieving the highest validation accuracy over both small-scale and large-scale benchmarks across all training budgets. + +Our analysis reveals the following key findings. (i) On CIFAR10-ResNet18 and CIFAR100-ResNet34 task, UBA shows the strongest performance across all training budgets. On the large-scale ImageNet benchmark with ResNet50, UBA maintains consistent superiority. The consistent improvements on both small-scale and large-scale benchmarks highlight UBA's generalizability across scales and data regimes. (ii) UBA outperforms baselines not only at $100\%$ training budget but also at $25\%$ and $50\%$ iteration budgets, demonstrating its budget efficiency, i.e., effectiveness in computation-constrained scenarios. (iii) Notably, UBA shows robust performance even while the second-best schedule varies depending on both the datasets, architectures and training budgets. This instability among competing schedules suggests their performance is highly sensitive to datasets-architectures characteristics and training dynamics at different learning phases. Therefore, for practitioners seeking reliable schedules without extensive method selection, UBA provides a default-strong choice. UBA eliminates the need for case-by-case baseline comparison and delivers stable superiority and reliability regardless of datasets, architectures, training budgets and scales. + +# E.5 Overall experiment results for language models + +For most tasks, we conduct a comprehensive evaluation across six representative baselines to ensure broad coverage. For large model with 300M parameters, we focus our comparison on the top three performing baselines due to the computational tractability. + +The overall performance of the OLMo models is presented in Table 8. We also plot training curves in Figure 7,8, 9 and 10, showing training and validation losses along with downstream evaluation results across model sizes ranging from 36M to 300M parameters and training data scales from 50B to 150B tokens. As shown, UBA schedule consistently achieves lower training and validation losses while delivering superior downstream performance. + +Table 7: Generalization accuracy. + +
DatasetNetworkMethodtraining budget (epoch(%)
75 (25%)150(50%)300(100%)
CIFAR10ResNet18SS92.71±0.424393.45±0.482293.61±0.3035
CS94.21±0.511694.24±0.241995.52±0.2433
CLR92.13±0.261092.99±0.545994.92±0.3811
1C94.23±0.183895.03±0.135395.35±0.2984
BT94.18±0.354694.79±0.460595.68±0.1168
REX94.49±0.322595.10±0.192995.45±0.1350
UBA(ours)94.57±0.328195.26±0.215095.65±0.1589
75 (25%)150(50%)300(100%)
CIFAR100ResNet34SS71.09±0.267675.34±0.180277.24±0.1808
CS64.12±0.180869.84±0.758074.29±0.5958
CLR73.33±0.563573.41±0.283574.74±0.2266
1C73.38±0.322775.04±0.638277.86±0.2239
BT72.75±0.261175.35±0.242378.58±0.2705
REX73.64 ±0.563575.14±0.311078.04±0.1430
UBA(ours)74.60±0.150876.63±0.267878.95±0.2818
75 (25%)150(50%)300(100%)
ImageNetResNet50SS74.74±0.138677.24±0.386178.56±0.6435
CS75.84±0.222077.95±0.216479.07±0.0438
CLR73.90±1.460976.04±1.641977.84±1.3053
1C75.83±0.779277.74±0.521878.90±0.2956
BT75.90 ±0.278677.81±0.462478.86±0.2814
REX75.85±0.804777.66 ±0.885378.66 ±0.2828
UBA(ours)76.29±0.403178.26± 0.164079.39±0.0170
+ +From Table 8, UBA achieves state-of-the-art performance on approximately $50\%$ of the benchmarks across all scales while the second-best schedule varies. Furthermore, it achieves consistent superior average performance among all baselines. It highlights the stable superiority and reliability of UBA, providing a default-strong choice. Moreover, it demonstrates significant improvements in SciQ-73M(+1.7) and ARC-E-300M(+2.63), highlighting its ability to enhance generalization across diverse benchmarks. Besides, in Figure 2, UBA consistently achieves lower training loss and validation loss throughout the training, indicating the efficient training ability and downstream performance enhancement. Notably, while task-specific tuning of $\varphi$ can further improve model performance, we intentionally fix its value across all tasks and architectures to ensure fair evaluation. Remarkably, even with this universal $\varphi$ setting, UBA consistently outperforms baselines on diverse benchmarks, demonstrating inherent robustness to varying benchmarks and model scales. + +# E.6 Performance across different optimizers + +Our scheduling strategy originates from the solution to a optimization problem under gradient descent dynamics. While modern optimizer (e.g., AdamW [32]) introduce additional momentum and adaptive mechanisms, they maintain the fundamental property of performing gradient-based updates. To verify the performance of schedule, we conduct cross-optimizer ablation studies. We evaluate the model on CIFAR100 and ImageNet datasets using AdamW optimizer. For AdamW, we use a learning rate of 0.003 and weight decay of 0.0001. We report the validation accuracy for each configuration. The results are summarized in Table 9. + +As shown in Table 9, our UBA schedule achieves state-of-the-art validation accuracy on both SGD and AdamW optimizers, consistently outperforming baselines across CIFAR100-ResNet34 and ImageNet-ResNet50 benchmarks. This demonstrates that although our schedule is theoretically derived from standard gradient descent dynamics, it generalizes effectively to modern optimizers like AdamW despite their additional momentum and adaptive mechanisms highlighting its broad applicability. + +![](images/afcfe148b9e89a9efc5490c7b5e2a2d206b684dbce6d000835bad058841e7d46.jpg) + +![](images/4935567eb62c49e7737a6b90accc75d2fc398ec6d57a85d8ac946e6062b0e537.jpg) + +![](images/bfe29f74855a6fe932cd6160a4f47bf082a49317cb4a8f07a73e5a951d3210b0.jpg) + +![](images/4140646d8c5e310f79ad7ded7946555ea6a8e9905c8b8580039baad25336dab7.jpg) + +![](images/8073356707f8f141d67b143bd36d7ca4ce007f93100f3070491af419eaac4add.jpg) + +![](images/141e23a29bb2271e9b51f543a6448d344a291473926ad3c498f32e2964cb2927.jpg) + +![](images/912e0070f106c31335f7cfb12e7ee86daf11356802f02f076dc5edf1295ff7b5.jpg) + +![](images/3262b5436e69cef2dfd0154f64edb0105f7adffecd42df98fb54e6508ecf8505.jpg) + +![](images/0a9f7a6acd9e46abee849e999ee9c6b1eae185fb0d99713d5d427f2a939c66bc.jpg) + +![](images/99dbe0bf5ded31afcdbece02de33a63bf6f78929a60951404aea765c149d5fe0.jpg) + +![](images/8eabe1db587e2a8db8e0cb6037f7a681e85d9cf038c1d3ef6c6e2aa2fb726a8c.jpg) + +![](images/85de105125c420880f6bd0f0b01719bc225bb3a7bfd4564c303643b239f58f35.jpg) + +![](images/f2fb27a87a67b7fed80b11058a69e139db3d75234159d188d66461d38932c579.jpg) + +![](images/ef7f2b08546d34d879c2d4a0ea0f0e92c7c22b9a4d78c3442541b72a4444ff30.jpg) + +![](images/d4ee50689240c87d3102bc69fe555c521a4fb3938afbedeb20ed58e01b3af87c.jpg) + +![](images/c2c2fa3afcca482b1aa0c230e17d2e306e9b787bb6630bf68ffcb704ca7d4bab.jpg) + +![](images/8051059837eeea711d8966b9f32cdc6fecb0095a2a64da8ee4e662284853936e.jpg) + +![](images/b73dfdabaa887141352936bbafec0bb0610c1a3c984ce66a5074e8849b06716c.jpg) + +![](images/403324102e013fcaacc18d2806a7d26bb18d2b76f3d17b4fcf6a330fe3b5d35d.jpg) + +![](images/c99364d1632bd1d1274c0bfdb11372051706e2d3e10129d10d3b46753e42aeba.jpg) + +![](images/4d37ac38980cfaf35039002252de9267982a72af657403addb6971cd6d90ede4.jpg) + +![](images/8637dbe5884b73bc855d6f2b7d9183c56a70afdc2522efca3be43b093ff9d53f.jpg) + +![](images/5e05c167972b871765a7bf883e31a9c862aaf30b993d150d98c38d14d3fca89f.jpg) + +![](images/28b0fc54b1f33499794e6d5b0c78a5efbcfd8420e51b78551a9b7c86abae8f7b.jpg) + +![](images/05b2e5c89bbe026e26aad2cf2780c73204c2063a1bad0e6356e02deac3d586b1.jpg) +Figure 7: Overall performance for language tasks on OLMo-36M. + +![](images/ec8188bd51341e0da55f5ac4030f16866ab842d7339eb1d342340681c6a63536.jpg) + +![](images/1cba5e9c7f4b8fbeea2bf8fe3fad21e741fefa2e7b91895d76f25b3c2a8c4193.jpg) + +![](images/2828070d1f28c1d5c8dcdb59b63c54678270e980121be182e95e65d3230360d3.jpg) + +# E.7 Parameter analysis of $\varphi$ + +In this subsection, we perform a sensitivity analysis of the key parameter $\varphi$ in equation (5), which controls the variation speed of learning rate. Our experiments systematically evaluate + +![](images/54a733d949559a7932ee88f1b29c4bfe12ba66894979e36cb38fea9394508c98.jpg) + +![](images/d6db8b7c439a5b7de59882bbe6850ed1a3f311d32cfffd224b9600a17ea4f8c7.jpg) + +![](images/7bbf155ea812900835c7f4bad1665003377094df18a5e255f747c9b3df02fb34.jpg) + +![](images/e239c280f48a8f80a8ec0145db31acd0c9b3c480c9e81b11d8103b3f350fe595.jpg) + +![](images/9a9c05dcfed030580c0b14e68981b558405c5e6b856fda6b3bc8a8190f87e793.jpg) + +![](images/c1d1245593f669b0dfb65ea9db1169d151d87f21fc48ecb6bdcc9a01cc9d1c65.jpg) + +![](images/8e72a7f2f6356d32fb09b5bd87c7cfd10bb586f1cdae25f1a9dbd0b9ba6b6138.jpg) + +![](images/62c04bede5dd9b3b870d2a9d30c80d5eb7265cd677781b6b6fba0c0122ca787f.jpg) + +![](images/ac39ff2dd2c5d5709c1dd49cb0587713e6365fc6debe9f202e2c5a9ae996f340.jpg) + +![](images/073e7dfb47115ac3ade4afbe96ce2d3a819dde61c104deebcbbf7cf08274792c.jpg) + +![](images/caea31f28efa4b6ba53435520803527d24928dadce3efb751147217be0296cce.jpg) + +![](images/5ba72926f8ce3a25d916c6bef244b5717149165699ddb0a7dff3a5f42e245c35.jpg) + +![](images/f5ee05bc1863246f5adaf59f67c63c2157bfff3c24cb9987e24077f08c442802.jpg) + +![](images/3f5a46da7295cedaad42f60efd4182de8fc1af9b1d9fb8eb1706fcb7fac80fae.jpg) + +![](images/5c5d1d0c5276948df54fb38a652fdf4de79268ce77b8b9830cc32f32f9ae3208.jpg) + +![](images/a35dcceeb79ec73f9623340c0e63ada7a8dff5ee401a43238f44a3a2724d6e59.jpg) + +![](images/b9461b390e17152e7d0b5d17594695fb0fe48f69c6380bff5662d818ca22b83f.jpg) + +![](images/f1049657429f2510e4fc980745c229f43e7f970f7bafc32b5b08b5c51f15ff0e.jpg) + +![](images/a52b13585845368047e7afabe563dc415049ab53aac585636e21328cccf96182.jpg) + +![](images/d7bfe4545f77b1c698873d4bbe53aa1b7e306e9849b967c1d181dbb378daa499.jpg) + +![](images/b4f03ab6fa1ec6700caa495dc6b8e8f11f7cdc99c1a52e516c283e752456968a.jpg) + +![](images/4056439ebd5cd57ff411f16a798aca93696dad9242e4fd2564589b8cd2f27d8f.jpg) + +![](images/b06c4176c64fc201edc48ae459ce56653363fd9245963815869dacb987bd2ff0.jpg) + +![](images/c3adce17ac4a2c1c34501f43608fd97b3a3e8f923eedc44a808d44b1f6ea5c97.jpg) + +![](images/d258634fef6e72f5d6619d7ef7c3c58f0da493d6429c03c4442b4a66b42e0a9e.jpg) +Figure 8: Overall performance for language tasks on OLMo-73M. + +![](images/99937c7898d9bf69768e365c2c8ab5f2828abbff14de65ea787a4a3d2cc40f7b.jpg) + +![](images/ea3a045a680623170d54a8652db297d5da749af17302f1ef35228403a880419d.jpg) + +![](images/411be5a06b243f1f7e6ee6e49c3bfba579e96fa2a4d5dda5edf68bbf2b04d58d.jpg) + +$\varphi \in \{0.25, 0.5, 1.0, 2.5, 5, 10\}$ across different optimization scenarios. The results are plotted in Figure 11 and listed in Table 10). + +Observations From the experimental results, we find an interesting dichotomy of $\varphi$ . Phenomenon(a): By choose proper $\varphi$ , UBA demonstrates consistent performance gains over all base- + +![](images/33ee479eef7809f1edc382643429752c132ed9c310d190888154aeddde8e2b8f.jpg) + +![](images/3b2f9dfa73291b61fb9b9f40da8356c0cd8fc0fd03957e0822e57ab5933acbce.jpg) + +![](images/f3437bd2d14316f80a24d20999e6482d271a6fbc14f442273dfaf5cb1eecb545.jpg) + +![](images/c18f2c5be54ad1b81412abf2323983c246a50264efa6240a1585a371605c4060.jpg) + +![](images/19cfc69f4262dd469b81dfcbcffd6d6cd5922d549f6dd867cd01649f65ad2012.jpg) + +![](images/11a90a2143301c6e4de15c5b23f1a759cd2fdec4f11ad4ace2952f6b70d80057.jpg) + +![](images/9f93d820bed30e67a63ca046ed47ecdff778a07615ddbdb3b848bee9ff72146c.jpg) + +![](images/02bdf8443ae49980560997593f83be3e4ad9f03c15e39e70f750a8acedc27cb8.jpg) + +![](images/187b31ff3db533c20867e57e5ed4f97337e75f3cc4e975811534c4baa1c32cb4.jpg) + +![](images/a047e42b4596c256773c8c4765b3ca6ed3d438b5a253e43fdec18ee225f03e40.jpg) + +![](images/637ea8262e17f36c863c6f2518ca6864eb3bc73a4ff565138d72e90744c94ee2.jpg) + +![](images/50acc74f029ec4aa624bc6e8db20a992983e97ed1685c7335cefe3d8c90bf52f.jpg) + +![](images/588e1c61a5e86e16169ec9355642c12ebf2a0a1ea1f402268eec18140174e682.jpg) + +![](images/8e2e51b56276986ffceb19b542d867b7bb57b9f0d756ce114bec80bd82cda1f6.jpg) + +![](images/e57f9eb08ae3245bfddfafcf1c9c38e737cdb46c4f3cbfaab0bfaf6baad10430.jpg) + +![](images/7d47609eaaed814f40537c205593cd31d05f77b5d7815529d2764e4ce2807e54.jpg) + +![](images/a9e2fba60ea410730b70095352905175f83071e89c27a737ec9d9ecd6edbf2b7.jpg) + +![](images/b95e71e9ef42e2c594574a4d1a53d66991e8b3c40a536a39685a49d8d3062f1f.jpg) + +![](images/328513acb23c9577bda175aefbe73215686a674c060728cc341196faadbf65a3.jpg) + +![](images/c593f0d1454b4ebc211ab3ef7ce531b64786eed179a75ee674c70998a4200ba1.jpg) + +![](images/e22cf1d56eedbee37e10cec1dc626e50bd47c65b3b90fda72b4e19c3baa6978a.jpg) + +![](images/dfa28aa9967c3be34c957b32b9d8831b4b74089990cfd1bddad705c572c2d8e0.jpg) + +![](images/6da56efacfed8dc106e89a812660c95f6b605859b9a80a871f02f0199e252a74.jpg) + +![](images/73e8de0dbe3062e943c4ff994edac7247edd1af3279d065d5afd1c36ba7b3bc4.jpg) + +![](images/de735eddd07615a181796c781eef8720304bdac2488b66e11bf04debfc2bf682.jpg) +Figure 9: Overall performance for language tasks on OLMo-150M. + +![](images/d926019827d2ac376d5b7f2c09b8f0877830b75bcb19c10890caf6b3481d1696.jpg) + +![](images/d0727aee8ce8722e43aa604688c358a515721f3da89748a5f9b5ae602a5f3d91.jpg) + +![](images/512e60f5f9be2cb1a72723bac9480fb077ceb067903e61303005a82d0b7a8680.jpg) + +lines across the full spectrum of training budgets (25%, 50% and 100%), irrespective of optimizer choice (SGD/AdamW). This suggests the effectiveness of UBA is orthogonal to specific optimization strategies. Besides, with optimizer changed, UBA maintains comparable accuracy on the same dataset-architecture, which shows that UBA can exhaustively exploit the model's capacity regardless of optimizer variants. Phenomenon(b): As shown in Figure 11 and Table 10, on CIFAR100 + +![](images/e19ba7c3f4fe4d3789c9772696e5b4acf3eaa1d564a70f0449f4c966f2ae9c4f.jpg) + +![](images/f58a55bf559050dc3027cea83673ff271fbe795fc4a1291185dcfeee78acb6ea.jpg) + +![](images/46ce0260bcb026a37f0391acbad9cedfea7645d11fb1dd11c577c5ffbd898d2b.jpg) + +![](images/c46ccdc9b984b0f6f23e5e08e0a4aeba7eb7415a119beba25a6a3450efb36ac1.jpg) + +![](images/b4b87f4999e2f763bf51b3a95bb595799451904f2fcbef0b23593a7aed6a2aa3.jpg) + +![](images/203fec77a1fedec8a38d878859523af42a80ee47b4f9d8006d3b4111c24c5403.jpg) + +![](images/d71177c78726ff60678a8e3679a9a436d6f4484b3a2f68ef3d7998986e5b4b7c.jpg) + +![](images/190462aca6c4265a9857f03ca06eefb9d819ae23eeee0df3d3f9e5fa4d705b57.jpg) + +![](images/289b90716424800d52d1718d339355ad742917ac8ea7086b0b153aea7aad2a03.jpg) + +![](images/315c5bf7f8d35e8048fdb6c5f472c37392aba6649f2f9c2de519698741e066c7.jpg) + +![](images/9ea86d32dc552ec0dbea9df6980a71c02c782a6a3cc86f67aabd36fe1b92994a.jpg) + +![](images/14e1f41a50d22f71a4ea00e645a318227407015049c2e511e65a7c6fda9f0882.jpg) + +![](images/d40d580053b564a3885658b81e5246b5c53d3549b1edfd43b1abf338904b0f14.jpg) + +![](images/bd6358459f7c416d8061a6a366298f8a465ac8abfbda6920b5966568f58e7114.jpg) + +![](images/1307f5323b7e8f4d732531b0333d60447ab10f98177450253f368da8a3276a32.jpg) + +![](images/bd7c5719b55a68e0d5fb32cb74ad54294439b84eae830b79bef3d6facdfa3244.jpg) + +![](images/4471df442734515f4691dc40069a97c07719e9b684bbb8ad2bf524f8bda02456.jpg) + +![](images/a201588d88dd56bd5bc8e08595ea67b316669f92a4d494c7ca589d70535ed3be.jpg) + +![](images/3713662ccb94deb23a0288ed00209f37d811bfb909c53f4f87c0ed210422aa37.jpg) + +![](images/7cc5e7bba141bd3ea9174d50f4daa1d1ad892a897652623d65132d08e07e7259.jpg) + +![](images/c752ea7255632e2ac54a2814382f5c751c4f7e8c2feb8706850c582152586e8d.jpg) + +![](images/dcf00ab2afb2023e37c71214c44c8ca6b268b2924d6a14cb42abebc5e74a1cd0.jpg) + +![](images/f3aa6fd9653d31ca932fcbc235d71942be4f010210d9ffde87e91d0060e9364e.jpg) + +![](images/01d016e2dfb50135a4377d7b50ce0a37aa6c4d0cfbc4509f6cc6127cbfabcf7d.jpg) + +![](images/fedebf8e4c76836ccaf6150a5ca7a0cb62f91499d7732df7da4f4c0611777c52.jpg) +Figure 10: Overall performance for language tasks on OLMo-300M. + +![](images/400273b26e5092704e176fffed18195c594f75832b2dcf0b2ff97c15516b3d5c.jpg) + +![](images/5e2bca3bbe211eb786452e07a9ee303a6df763497e065ef9622fa607bfa514ce.jpg) + +![](images/2d4cfec9df8092db2db072d956dca0afcfa1ced7c47b0f5e1fecc533c8d367e5.jpg) + +dataset, the best performance for AdamW is achieved with $\varphi = 1$ , whereas SGD performs optimally with $\varphi = 2.5$ . On ImageNet dataset, the best performance for AdamW is achieved with $\varphi = 0.5$ whereas SGD performs optimally with $\varphi = 10$ . Our experiments reveal a notable trend: when using AdamW, a smaller $\varphi$ yields better performance, whereas a larger $\varphi$ is preferred for SGD. + +Table 8: Performance comparison on OLMo model + +
SizeSched.Benchmark(accuracy %)
PIQAHSWAGOBQASciQARC-EARC-CCOPASIQASOCOTHAvg.
36MCS59.7427.3626.8066.5044.7421.7460.0040.5824.3329.0940.09
BT59.4127.3327.4067.2043.6821.4059.0039.8223.4829.1339.79
SS60.1227.6327.2065.3045.2622.4163.0040.8923.8729.6240.53
REX60.1727.8227.8068.1045.0922.4162.0040.6325.1830.0240.92
CLR61.1527.2127.6067.5042.1122.0762.0041.4524.2929.3840.48
1C60.1727.9125.8067.4043.6823.4155.0040.0222.2629.8639.55
UBA60.3927.9827.4068.2045.7921.7463.0040.6924.3330.1440.97
73MCS63.0629.6828.8072.6045.0925.0865.0041.5626.2428.1742.53
BT61.7029.7928.0071.4047.5423.4166.0041.9725.0526.4842.13
SS61.5329.3028.2069.4047.1924.4165.0041.7626.3228.3342.14
REX62.4630.2227.6072.2045.0926.0965.0041.8125.5229.3442.53
CLR62.3029.3827.8070.4045.4424.0862.0042.0225.0529.0541.75
1C61.8129.9027.8071.1047.5426.4263.0040.7924.3728.8142.15
UBA63.1730.0928.8074.3045.7922.7465.0041.4527.1328.8542.73
150MCS66.0035.1932.0076.6049.8225.4267.0042.8426.2432.2745.34
BT64.8534.9529.8078.2048.9526.4267.0042.9926.8331.8745.19
SS65.4534.7630.6077.1049.6524.7568.0042.3726.5831.5945.09
REX65.5635.7229.4078.3052.4625.0869.0043.3027.6832.6445.91
CLR65.0734.7631.6077.2050.7025.0868.0043.7127.3031.2345.47
1C65.2935.1632.8076.3053.1625.7567.0043.4527.6430.6245.72
UBA65.2335.5029.8078.3050.3527.4267.0043.5029.2932.7245.91
300MCS68.8845.3233.4083.2057.1927.7669.0044.5829.2534.5749.32
BT69.6445.2130.8083.7057.3726.4272.0043.9628.6635.4949.33
REX70.1846.3033.0084.4057.7228.4367.0043.7129.5036.7449.70
UBA69.4846.4434.6083.9060.3529.1072.0044.4228.5335.4150.42
+ +Analysis Furthermore, we intend to explain the dual behavior of $\varphi$ through theoretical analyses (Proposition 1 and Theorem 1). We attribute Phenomenon (a) to UBA's optimal scheduling dynamics, governed by the $\varphi$ value. Moreover, Phenomenon (b) reveals a fundamental dichotomy in $\varphi$ value selection between optimizers. We attribute this phenomenon(b) to the preconditioning effect of AdamW. Unlike SGD, AdamW adapts the learning rate by scaling gradients with the square root of the second moment estimate $\sqrt{v_t}$ . In regions where gradients $g_{t}$ are large, the surrounding landscapes are steep. The denominator $\sqrt{v_t}$ is also large, effectively reducing the learning rate. This adaptive behavior mimics preconditioning, implicitly lowering the condition number of the optimization landscape. + +Our theoretical framework (Proposition 1) establishes a connection between parameter $\varphi$ and the condition number of the landscape around local minima: smaller $\varphi$ is favored for well-conditioned problems (low condition number), while larger $\varphi$ is beneficial for large condition number scenarios. Since preconditioning effect of AdamW naturally mitigates ill-conditioning, the schedule with smaller $\varphi$ is preferred. In contrast, SGD lacks such adaptive mechanisms, thus a larger $\varphi$ is more suitable for this situation. This alignment between theory and experiment underscores the importance of tailoring $\varphi$ to the optimizer's characteristics. Thus, we recommend selecting higher values of $\varphi$ when facing challenging optimization difficulty, while employing lower $\varphi$ values in scenarios with smoother optimization difficulty. + +# E.8 Performance across different initial learning rates + +Since each schedule will have a different optimal learning rate in practice, to thoroughly address this concern, we conducted additional experiments to evaluate the performance of all schedules with different learning rates. + +Table 9: Cross-optimizer ablation studies. We list validation accuracy for vision classification tasks across SGD and AdamW. + +
Dataset -networkOptimizerScheduletraining budget (epoch(%)
30 (25%)60(50%)120(100%)
CIFAR100SGDSS70.8575.5877.28
CS63.8868.8470.43
CLR72.6373.8174.80
1C73.7275.5377.90
BT72.5375.4078.49
REX73.1275.4877.99
UBA(ours)φ = 574.5776.6878.97
-ResNet34AdamWSS64.0170.9373.65
CS62.5668.9172.98
CLR62.8770.1373.22
1C64.2771.6074.06
BT64.0171.4474.34
REX65.1971.7274.08
UBA(ours)φ = 0.564.4472.1074.48
75 (25%)150(50%)300(100%)
ImageNetSGDSS74.8476.9678.10
CS75.9977.7979.10
CLR72.8674.8876.91
1C75.2877.3778.79
BT75.7177.4878.66
REX75.2877.0378.46
UBA(ours)φ = 576.0077.9979.32
-ResNet50AdamWSS74.6477.5179.01
CS75.6878.1079.04
CLR74.9377.2078.76
1C76.3878.1179.21
BT76.1078.1479.05
REX76.4278.2878.86
UBA(ours) φ = 0.576.5778.3779.38
+ +Table 10: Cross- $\varphi$ ablation studies. + +
DatasetNetworkEpochOptimizerUBA
φ=0.25φ=0.5φ=1φ=2.5φ=5φ=10
CIFAR100ResNet3460SGD75.1075.8976.3176.4076.6875.64
AdamW72.1071.3471.5471.2170.4069.01
ImageNetResNet50300SGD78.3578.7578.9579.2479.3279.40
AdamW79.2679.3879.1279.1078.7278.47
+ +We conducted auxiliary experiments on CIFAR100-ResNet34 for 3 runs. For computational efficiency, these experiments were run for slightly fewer epochs than the main experiments, with results reported over three independent runs. Through current observation and search, we identified 0.1 as the proper initial learning rate for most schedules. We use 0.01 and 0.001 to confirm this choice maximizes validation accuracy across methods. Additional smaller or larger values such as 0.3 or 0.0001 were explored in preliminary studies, showing consistent trends but lower peak accuracy. Thus we do not choose them for comparison. + +The experimental results are listed in the following table 11. We report results of different base learning rate and compare the performance with optimal learning rate of each schedule. + +![](images/0d71a4b2b2b3507d751e29f61127f0250ae7a8906a25017216bf2b3203e6043f.jpg) +(a) + +![](images/ea9fc402ac3bdf04249a8ddb2f13820b4375f3fc476121d3a791df1963506596.jpg) + +![](images/5c6c30cec26b26ad7a059ae6feecd207fa8221fb0d7be6a98554c59fc3005c05.jpg) +(b) + +![](images/8f72f24a4ad83b6a9812735049d9ed5cc549a0246166891edc932af285d52caf.jpg) +Figure 11: Visualization of cross- $\varphi$ performance. + +Table 11: Performance across different initial learning rates. + +
DatasetNetworkScheduleValidation Accuracy
lr=0.1lr=0.01lr=0.001
CIFAR100ResNet34SS77.28±0.240173.48±0.426767.98±0.1473
CS74.11±0.769074.66±0.588869.18±0.3736
CLR74.77±0.186174.26±0.352368.84±0.4613
1C77.73±0.186875.80±0.170169.20±0.2152
BT78.53±0.075177.11±0.102668.84±0.4613
REX78.11±0.112477.40±0.447769.88±0.1253
UBA(ours)79.02±0.283877.59±0.352270.50±0.0862
+ +# E.9 Performance across different periods + +Our schedule has a periodic phase-based learning rate adjustment setting, where the learning rate at the $k$ -th phase is dynamically determined by the $k$ -th local minimum of the loss landscape. In the original conception, this design captures the optimization dynamics around successive local minima, with the hyper-parameter $\varphi$ related to the difficulty of each phase. Specifically, $\varphi$ controls the trade-off between exploration (large steps) and exploitation (small steps) within each phase. + +To validate our scheduling strategy, we conduct experiments by varying $K$ (see detailed results in Appendix E.9). We observe that when roughly setting $\varphi$ as a constant in each phase, performance degrades as training progresses. This suggests that a fixed $\varphi$ strategy cannot adapt to the chang- + +ing landscape of multi-phase scheduling. Since the optimization difficulty should decrease during training process, we drop $\varphi$ as phase increases. By finely evaluate $\varphi$ in each phase, it achieves performance improvement, confirming the need for dynamic control. It makes sense since multi-phase captures the dynamic features of the loss surface more finely, it needs careful selection for $\varphi$ , where $\varphi$ reflects optimization difficulty. These results highlight a fundamental trade-off: when $\varphi$ values per phase for multi-phase scheduling can be finely evaluated, multi-phase scheduling better captures the loss landscape's non-stationary behavior than single-phase scheduling. However, selecting optimal $\varphi$ values per phase for multi-phase remains non-trivial, which motivates future work on automated landscape-aware $\varphi$ tuning. + +Table 12: Performance on multi-phase version UBA. + +
DatasetNetworkφOriginalMulti-phase
K=4K=6K=8
CIFAR100ResNet34576.6876.5475.3775.05
5 × 0.8k76.3176.6076.45
+ +# F Related work + +Budgeted training Researchers face significant challenges in achieving optimal model performance under fixed hardware and limited time. To address this, budgeted training has emerged as a broad research domain focusing on optimizing performance while minimizing resource usage. This research spans multiple aspects, including computation efficiency, model compression, training stability, convergence improvement, and optimization [41]. By strategically integrating efficient training approaches, budgeted training advances cost-effective model development, offering pathways to achieve high performance under realistic constraints. + +Budgeted training can be divided from three main perspectives: data, model, and optimization. From the data perspective, budgeted training emphasizes the allocation of resources between the dataset and the algorithms that process it, such as the balance between the model size and the amount of data [2, 6, 24, 28]. From the model perspective, budgeted training searches the optimal number of model parameters within the given compute budget [17, 26, 31, 38]. Beyond data and model perspective, optimization based methods guide budgeted training based on learning rate schedules [5, 30, 43, 44], batch size [17] and other weight averaging method [26, 27]. + +Among these three aspects, optimization-based methods are particularly aligned with the objectives of budgeted-iteration training. In this domain, learning rate scheduling based methods are particularly aligned with the objectives of budgeted-iteration training. Smith et al. [43] propose a new learning rate schedule, named cyclical learning rates (CLR). It improves accuracy in fewer iterations without tuning and is relevant to super-convergence phenomenon [44]. Li et al. [30] introduce an alternative setting of existing learning rate schedules for budgeted training. Chen et al. [5] propose the reflected exponential schedule (REX) via a profile and sampling fashion. Learning rate based approaches achieve robust and high-performing results under various lengths of training iterations, which corresponds to our purpose in this work, i.e. budgeted-iteration training. Besides, this approach is plug-and-play, requiring no substantial alterations to the underlying model structure, making it readily adaptable to various deep network frameworks. Therefore, we explore the proper learning rate schedule to achieve budgeted-iteration training. + +Learning rate schedule The learning rate plays a pivotal role in controlling the non-convex optimization process during network training. Generally, the learning rate is gradually adjusted progressively alongside the iterations, which can be represented as $\eta = g(t)\eta_0$ , where $g(t)$ is the schedule function which controls the variation of learning rate, $t$ is the current iteration and $\eta_0$ is the initial learning rate. + +Learning rate schedules have been extensively studied to improve training efficiency and model performance. The common scheme is the step decay schedule. Its schedule function can be represented as $g(t) = d_{\lfloor t / t_s \rfloor}$ , where $t_s$ is the predefined decaying step and $d_i(d_i \geq d_{i+1})$ is predefined decaying scalar. A typical instance decreases the learning rate by a decaying scalar 0.1 after 50% epochs and by a decaying scalar 0.01 after 75% epochs [21]. Then Loshchilov et al. [33] observe that sharp decreases may prevent models from escaping local minima. Thus they propose cosine schedule function as $\frac{1}{2}(1 + \cos(\frac{t\pi}{t_s}))$ , where $t_s$ is the predefined stage to restart the learning rate schedule, which has proven effective in transformer training. In addition, cosine schedule is a commonly-used schedule nowadays. + +Pan et al. [36] propose Eigencurve, and demonstrate that minimax-optimal convergence can be achieved for quadratic objectives when the Hessian's eigenvalue distribution exhibits substantial skewness. Pan et al. [37] further introduce a learning rate schedule called elastic step decay as $\eta_t = \eta_0 / 2^k$ , if $t \in [(1 - r^{k})T, (1 - r^{k + 1})T)$ to accelerate the pre-training for Berts, where $r$ and $k$ are hyper-parameters that control learning rate interval. Defazio et al. [11] developed the theoretically-grounded framework for learning rate scheduling, establishing optimal linear decay schedules and refinements through rigorous mathematical derivation. Their work additionally provides the most extensive empirical evaluation of scheduling approaches to date. Hu et al. [25] propose a Warmup-Stable-Decay(WSD) for large model pretraining and promoting continuous training. WSD consists of three consecutive phases: warmup, stable constant, and decay. Luo et al. [34] discover a new schedule that outperforms the cosine schedule, resembling WSD but with lower final loss. + +Adaptive learning rate Compared to schedule methods, adaptive learning rate based methods focus on the learning rate adaptation based on local loss landscape statistics. Typical methods include + +AdaGrad [13], Adadelta [54], RMSprop [23] and Adam [29]. These methods have been shown stabilizing the training process while effectively optimizing learning rate. Then Loshchilov et al. [32] propose AdamW optimizer which improves the performance in the transformer training. Recently, Chen et al. [7] discovers Lion optimizer for more memory-efficiency. Xie et al. [52] propose the Adan optimizer which implements $50\%$ faster training than Adam and AdamW on commonly-used networks. Despite these improvements, some studies suggest that adaptive methods may underperform momentum SGD in terms of generalization [30, 51], a critical factor for budgeted-iteration training. + +Learning rate design is not only significant in general training, but also critical in budgeted-iteration training which still remains a topic of debate. Some analyses advocate for small, constant learning rates to ensure stability and convergence [12]. On the contrast, one prevailing hypothesis suggests that large learning rates may facilitate crossing over sharp local minima in the optimization landscape [56]. Despite the lack of comprehensive theoretical explanations, a range of learning rate schedules inspired by the above analyses as heuristic guidelines has been widely adopted in practice, using variable learning rates to budgeted-iteration training [5, 30]. In this work, we explore learning rate schedule from optimization problem tailored to budgeted-iteration training, aiming to balance iteration budget constraints and generalization. \ No newline at end of file diff --git a/stepsizeanythingaunifiedlearningratescheduleforbudgetediterationtraining/images.zip b/stepsizeanythingaunifiedlearningratescheduleforbudgetediterationtraining/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..816a1a9badc184aebff8d431fd176e2a0f8eede8 --- /dev/null +++ b/stepsizeanythingaunifiedlearningratescheduleforbudgetediterationtraining/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b71dc33e494b237544ea5e6e133cfba666775d4211ed8d2ee17bc8a6854ecb8c +size 3802275 diff --git a/stepsizeanythingaunifiedlearningratescheduleforbudgetediterationtraining/layout.json b/stepsizeanythingaunifiedlearningratescheduleforbudgetediterationtraining/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..80355cac2555cb4228f7f86305e3f8f3f4f3590d --- /dev/null +++ b/stepsizeanythingaunifiedlearningratescheduleforbudgetediterationtraining/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3d334ec21947369c640a7d15adb9889ba0d0a90376c3c676664e380a5f9cfe58 +size 1564552 diff --git a/stitchandtellastructureddataaugmentationmethodforspatialunderstanding/943a3624-c659-4350-8a96-1213eb94066b_content_list.json b/stitchandtellastructureddataaugmentationmethodforspatialunderstanding/943a3624-c659-4350-8a96-1213eb94066b_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..e1834881108b57864adee175b74145b527a6f5e3 --- /dev/null +++ b/stitchandtellastructureddataaugmentationmethodforspatialunderstanding/943a3624-c659-4350-8a96-1213eb94066b_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0cf5f4c4c563ce0ddc92e522e9106a6e1b5abd84d796fe17c78865f85c28e243 +size 174286 diff --git a/stitchandtellastructureddataaugmentationmethodforspatialunderstanding/943a3624-c659-4350-8a96-1213eb94066b_model.json b/stitchandtellastructureddataaugmentationmethodforspatialunderstanding/943a3624-c659-4350-8a96-1213eb94066b_model.json new file mode 100644 index 0000000000000000000000000000000000000000..8d8f937a79b92be9098b5c6503651bad666d3b4c --- /dev/null +++ b/stitchandtellastructureddataaugmentationmethodforspatialunderstanding/943a3624-c659-4350-8a96-1213eb94066b_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3e94a4bb8355b3aef01d70764bece41beee3c25dcbf18e8dfe773d01150ed797 +size 219268 diff --git a/stitchandtellastructureddataaugmentationmethodforspatialunderstanding/943a3624-c659-4350-8a96-1213eb94066b_origin.pdf b/stitchandtellastructureddataaugmentationmethodforspatialunderstanding/943a3624-c659-4350-8a96-1213eb94066b_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6a2e6bf0f5d7cd11d7c353d918088ead7643e189 --- /dev/null +++ b/stitchandtellastructureddataaugmentationmethodforspatialunderstanding/943a3624-c659-4350-8a96-1213eb94066b_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fa78c19dd67c40b4bb27b9ff1ad58b98f5b5d882b5dba02e8642775f88be0baf +size 1983146 diff --git a/stitchandtellastructureddataaugmentationmethodforspatialunderstanding/full.md b/stitchandtellastructureddataaugmentationmethodforspatialunderstanding/full.md new file mode 100644 index 0000000000000000000000000000000000000000..5e96a49347edde7b04da02f3282fb9c8fe4f35c3 --- /dev/null +++ b/stitchandtellastructureddataaugmentationmethodforspatialunderstanding/full.md @@ -0,0 +1,708 @@ +# Stitch and Tell: A Structured Multimodal Data Augmentation Method for Spatial Understanding + +Hang Yin $^{1}$ , Xiaomin He $^{2}$ , PeiWen Yuan $^{1}$ , Yiwei Li $^{1}$ , Jiayi Shi $^{1}$ , Wenxiao Fan $^{1}$ , Shaoxiong Feng $^{3}$ , Kan Li $^{1\dagger}$ + +1 School of Computer Science, Beijing Institute of Technology +$^{2}$ School of Software and Microelectronics, Peking University + +$^{3}$ Xiaohongshu Inc + +{yh,peiwenyuan,liyiwei,shijiayi,wenxiaofan,likan}@bit.edu.cn + +{2401210613}@stu.pku.edu.cn,{shaoxiongfeng2023}@gmail.com + +# Abstract + +Existing vision-language models often suffer from spatial hallucinations, i.e., generating incorrect descriptions about the relative positions of objects in an image. We argue that this problem mainly stems from the asymmetric properties between images and text. To enrich the spatial understanding ability of vision-language models, we propose a simple, annotation-free, plug-and-play method named Stitch and Tell (abbreviated as SiTe), which injects structured spatial supervision into multimodal data. It constructs stitched image-text pairs by stitching images along a spatial axis and generating spatially-aware captions or question answer pairs based on the layout of stitched image, without relying on costly advanced models or human involvement. We evaluate SiTe across three architectures including LLaVA-v1.5-7B, LLaVA-Qwen2-1.5B and HALVA-7B, two training datasets, and thirteen benchmarks. Experiments show that SiTe improves spatial understanding tasks such as $\mathrm{MME}_{\mathrm{Position}}(+5.50\%)$ and Spatial-MM $(+4.19\%)$ , while maintaining or improving performance on general vision-language benchmarks. Our findings suggest that explicitly injecting spatially-aware structure into training data offers an effective way to mitigate spatial hallucinations and improve spatial understanding, while preserving general vision-language capabilities. + +# 1 Introduction + +Spatial understanding, the ability to comprehend and interpret the relationships between objects in a space, is essential for tasks such as visual question answering, navigation, and embodied AI [1, 45, 40]. However, most existing vision-language models still struggle to understand and reason about spatial relationships [19, 34, 5, 39], which leads to spatial hallucination problems. + +We argue that the spatial hallucination problem is primarily caused by the implicit modality gap, that is, while images contain rich multidimensional spatial features, their paired captions tend to be relatively sparse. We investigated existing large-scale multi-modal datasets, such as Conceptual Captions [28], COCO [18], VQA [1], and SBU Captions [24], and observed only a small fraction of samples contain explicit spatial information (see Table 1). As shown in Figure 1, the spatially-aware data means the samples whose captions include clear spatial information (e.g., "to the left of", "in front of"). During training, the model aligns visual content with limited linguistic descriptions, which may constrain its ability to capture latent spatial structures from the image alone. + +This further motivates the need to introduce spatial information directly into the text. However, collecting spatial annotations through crowd-sourcing is both difficult and expensive. It often demands a carefully designed annotation interface and large-scale annotation efforts involving skilled + +![](images/5acb4a66a88c6b7bcbce56eded9c2152322136564261779d565ab3b1d2a63bc2.jpg) +Spatially-agnostic text : A stop sign stands on a quiet street, as a woman walks across the crosswalk. +Spatially-aware text: To the left of the stop sign, a woman is walking across the street. +Figure 1: The difference between spatially-agnostic and spatially-aware text. spatially-aware text has explicit location cues that clarify object positions. +Table 1: Proportion of spatially-aware data in common datasets. + +
Multimodal DatasetsRatio
Conceptual_capions [28]0.0168
blip_laionCc_sbu_558K [20]0.0201
VQA_v2 [1]0.0288
COCO_2017 [17]0.0511
SBU_Captions [23]0.0582
Visual_Genome [15]0.0611
Flickr30K [43]0.0732
VSR [19]0.1898
+ +annotators. One might consider leveraging data augmentation to construct spatial understanding data. However, traditional methods are not design for spatial understanding. For instance, cropping, dithering, rotation, and random erasing may break the alignment between images and text, introduce spatial noise, or even distort main semantic consistency [6, 14, 29]. Recent work has explored using large models to synthesize spatially-aware data through caption rewriting or image editing [35]. Although model-generated augmentation can enrich spatial supervision, it typically involves high computational cost and complicated processing. Given the massive scale of pretraining data, such methods are difficult to apply in practice. + +![](images/fb7221f68375b2394b9bd0a451d376efaceaf0cfcec8b9dc31cde2a10c0fef25.jpg) +Figure 2: An Overview of Stitch and Tell + +In this work, we propose a simple yet effective multi-modal data augmentation method, Stitch and Tell (SiTe), which injects spatial information into image-text pairs without relying on large-scale generation models or incurring heavy computational cost. As shown in Figure 2, SiTe consists of two main steps: IMAGE-SPATIAL STITCHING and TEXT-SPATIAL TELLING. + +In the IMAGE-SPIATIAL STITCHING step, we combine two images along a spatial axis (e.g., horizontal or vertical) to create a new image with an explicit spatial layout. This stitched structure naturally introduces spatial relations between regions. + +In the TEXT-SPIATIAL TELLING step, we generate spatially-aware textual annotations based on the original captions. First, we construct structured captions by placing the two original texts into spatial templates, such as "The right part shows $T_{2}$ , while the left part displays $T_{1}$ ." These templates make spatial relations explicit and provide clearer supervision, helping the model associate language with visual layout. Second, we extract object nouns from each caption and generate spatial question answer (QA) pairs. For example, given "a cat" from $T_{1}$ and "a car" from $T_{2}$ , we create a question like "From the observer's perspective, is the cat on the left of the car?" The answer can be automatically determined based on the stitched layout, requiring no manual labeling. These QA examples offer an + +additional form of weak spatial supervision and can be directly used in instruction tuning. Compared to image-based object detection or grounding, extracting object from text is more efficient and reliable. It reduces the noise and computational overhead introduced by vision-level processing, making our method lightweight and scalable. The image-text pairs generated by SiTe introduce explicit spatial supervision through structured layout and spatially-aware text, without requiring manual annotations or model-generated rewrites. This helps bridge the gap between sparse spatial expressions in text and the richer spatial structure present in images, improving cross-modal alignment. + +SiTe introduces spatial structure into multi-modal training data through simple image stitching and spatially-aware caption generation, without relying on large-scale generation models or human annotations. Each stitched sample has two image-text pairs, allowing the model to process more content per step and reducing training time by over $20\%$ in our SiTe default setting. SiTe is easy to integrate into existing pipelines. It can be applied during both pretraining and supervised fine-tuning without modifying the model architecture. + +We evaluate SiTe across three model architectures (LLaVA-v1.5-7B [20], LLaVA-Qwen2-1.5b [41] and HALVA [27]), two training dataset (558K [20] and Flickr30K), and thirteen popular benchmarks, four for spatial understanding, four for general vision-language tasks and five for more real-world benchmarks. On LLaVA-v1.5-7B, SiTe improves spatial understanding benchmarks such as $\mathrm{MME}_{\mathrm{Position}}(+5.50\%)$ and Spatial-MM $(+2.13\%)$ , while also yielding gains on general tasks like COCO-QA $(+1.02\%)$ and MMBench $(+0.93\%)$ . For Qwen2-1.5B, SiTe improves MMBench by $+5.33\%$ and MM-VetSpat by $+4.38\%$ in the fine-tuning stage, and achieves $+10.42\%$ on $\mathrm{MME}_{\mathrm{Pos}}$ and $+5.01\%$ on MMBench during pretraining. These results demonstrate that SiTe provides an effective form of weak spatial supervision that enhances spatial reasoning while maintaining competitive performance on general benchmarks, across both large and small model settings. + +# 2 Related Works + +Multimodal Data Augmentation. To improve the generalization of vision-language models, recent work explores multimodal data augmentation strategies [13, 26]. MixGen [12] combines image interpolation with caption concatenation, but assumes semantic compatibility across samples, which may yield implausible pairs. Other approaches rely on generative models, such as StableLLaVA [16] for diffusion-based synthesis and ALIA [9] for language-guided editing. Sapkota et al. [26] categorize these methods into input mixing, caption synthesis, and adversarial perturbation. However, most require supervision, handcrafted rules, or heavy computation, limiting scalability. In contrast, we propose a structured, weakly supervised augmentation method based on spatial compositionality. It introduces explicit spatial grounding without labels or model-generated text, and integrates efficiently into standard training pipelines. + +Spatial Understanding Task Spatial understanding is a fundamental capability for intelligent agents to recognize and reason about the relative positions and relationships between objects. It plays a central role in real-world applications such as robotics, embodied AI, and autonomous driving [11, 21, 3]. Recent studies [30, 31, 45, 40, 42] emphasize that spatial reasoning is critical for scene understanding, navigation, and visual question answering, all of which require accurate perception of object positions and layouts. In robotics and autonomous systems, it enables agents to make informed decisions in dynamic environments [46, 2, 8, 22]. In the vision-language domain, spatial information is essential for grounding language in visual content [33, 25]. To enhance spatial reasoning, recent efforts incorporate explicit spatial features such as coordinate embeddings [4] and leverage additional modalities like depth and 3D information [7, 10]. The increasing attention to this area reflects both the challenges it poses and its importance to general intelligence, making it an active field with strong potential for future progress. + +# 3 Stitch and Tell + +In this section, we present Stitch and Tell, a structured multimodal data augmentation method that injects spatial knowledge by stitching images and telling their spatial layout information through structured text. We begin by introducing how images are spatially stitched to form structured visual information. We then describe how spatial relations are explicitly injected into both captions and question answer formats, allowing the model to learn spatial reasoning from weakly structured but + +naturally aligned supervision. Finally, we discuss how SiTe can be effectively integrated into the model's training process. + +IMAGE-SPIATIAL STITCHING: Given two image-caption pairs $(I_1, T_1)$ and $(I_2, T_2)$ from the dataset, we first construct a stitched image by spatially stitching the two input images along a specific axis. Concretely, we create a blank canvas based on the larger height (for horizontal stitching) or width (for vertical stitching) of the input images, and paste $I_1$ and $I_2$ onto the canvas following a layout mode. For example, in the horizontal setting, we produce a left-right stitched image via $I_{\mathrm{Stitch}}^{\mathrm{LR}} = I_1 \oplus_{\mathrm{horizontal}} I_2$ . This construction introduces an explicit spatial information compaerd to the original images, helping the model acquire a grounded sense of spatial. The pseudocode of image stitching process is illustrated in Algorithm 1. + +We design two image pairing strategies for constructing composite samples in the SiTe framework: (1) $\mathrm{SiTe}_{\mathrm{rand}}$ randomly selects two images from the dataset for horizontal concatenation, without considering their geometric proportions. This introduces diverse visual combinations but may lead to uneven scaling or excessive blank areas in the merged image. (2) $\mathrm{SiTe}_{\mathrm{ratio}}$ first filters + +vertically dominant images with a height-to-width ratio greater than 1.2, then groups them into buckets according to similar aspect ratios, and pairs images within each bucket. This ensures that the two halves of the composite image have comparable geometric structures, thereby improving spatial balance and reducing blank or redundant regions. Compared with the random pairing baseline, $\mathrm{SiTe}_{\mathrm{ratio}}$ increases the proportion of effective visual tokens and yields higher information density in the visual encoder's representation. + +TEXT-SPATIAL TELLING. After performing combining two images, we obtain a new + +Algorithm 1 Image Stitch Algorithm +1: function STITCH $(I_1, I_2, \text{mode})$ +2: $(w_1, h_1) \leftarrow \text{GetSize}(I_1)$ +3: $(w_2, h_2) \leftarrow \text{GetSize}(I_2)$ +4: if mode = horizontal then +5: $H \leftarrow \max(h_1, h_2)$ +6: $W \leftarrow w_1 + w_2$ +7: $I_{\text{canvas}} \leftarrow \text{NewImage}(W, H)$ +8: Paste $I_1$ at $(0, 0)$ onto $I_{\text{canvas}}$ +9: Paste $I_2$ at $(w_1, 0)$ onto $I_{\text{canvas}}$ +10: else if mode = vertical then +11: $W \leftarrow \max(w_1, w_2)$ +12: $H \leftarrow h_1 + h_2$ +13: $I_{\text{canvas}} \leftarrow \text{NewImage}(W, H)$ +14: Paste $I_1$ at $(0, 0)$ onto $I_{\text{canvas}}$ +15: Paste $I_2$ at $(0, h_1)$ onto $I_{\text{canvas}}$ +16: else +17: return Error: Invalid mode +18: end if +19: return $I_{\text{canvas}}$ +20: end function + +- Tell the Caption. The semantic content from the original captions $T_{1}$ and $T_{2}$ is naturally aligned with the left and right regions of the stitched image. Based on the stitching mode (e.g., left-right or top-down), we select a spatial template from repository and stitch a structured caption $T_{\text{Stitch}}$ by inserting $T_{1}$ and $T_{2}$ into the corresponding placeholders. For example, the template "The left part shows $T_{1}$ and the right part displays $T_{2}$ " explicitly injects spatial information through the underlined spatial phrases. These spatial information help the model associate textual semantics with the visual layout more effectively. This process does not require any additional annotation, and re + +tains the original semantics while explicitly encoding spatial relationships, encouraging the model better align visual content with spatial language. + +- Tell the Question and Answer. Guided by the SiTe method, we further extend our method to generate spatial question-answer pairs. As shown in Figure 3, for each stitched sample, we extract noun lists from the original captions: $N_{L} = [o_{L,1},\dots,o_{L,m}]$ and $N_{R} = [o_{R,1},\dots,o_{R,n}]$ , corresponding to the left and right regions of the image $I_{\mathrm{Stitch}}^{\mathrm{LR}}$ . To reduce ambiguity, we remove + +image-spatial stitching, such as horizontally image $I_{\mathrm{Stitch}}^{\mathrm{LR}}$ with an explicit spatial layout. + +![](images/6c899adcb7c5862896881da3a5d83d6e27f6a70cc75d63cc470043b086e6d8ab.jpg) +Figure 3: A construction process of spatial visual question answering data + +overlapping nouns shared by both sides. We then sample entity pairs $(o_L, o_R)$ from the disjoint noun sets and generate questions such as "Is the $o_L$ on the left of the $o_R$ ?" + +Since the spatial position of each entity is predetermined by construction, the answer to such questions can be automatically inferred based on the object's region (e.g., $o_{L} \in N_{L}$ implies a "Yes" answer). This provides a lightweight way to generate weakly supervised spatial reasoning signals without the need for manual annotation. Compared to object detection or image-level classification, extracting entities from captions is significantly more efficient and reliable. It avoids visual ambiguity and reduces computational cost. These question-answering pairs provide the model with a lightweight but effective signal for learning spatial understanding. + +The data generated by SiTe can be effectively applied across multiple stages of multimodal training. The structured captions $T_{\mathrm{Stitch}}$ provide spatially grounded supervision during pretraining, facilitating the alignment between visual representations and spatial language without additional annotation. The constructed spatial question-answer pairs are suited for instruction tuning, where they serve as explicit prompts to strengthen the model's spatial understanding abilities. Furthermore, by modifying spatial expressions in $T_{\mathrm{Stitch}}$ (e.g., exchanging "left" and "right"), we can generate hard negative samples that maintain the global semantics while introducing localized spatial inconsistencies. These samples can be used in contrastive learning settings to improve the model's sensitivity to spatial contradictions. Overall, SiTe offers a unified and extensible method for spatial data augmentation, supporting both generative and discriminative learning objectives with minimal supervision. + +# 4 Experiment Setup + +# 4.1 Models. + +We conduct experiments using three representative vision-language models: LLaVA-v1.5-7B [20] (referred to as LLaVA), LLaVA-Qwen2-1.5B [41] (Qwen2), and HALVA-7B [27] (HALVA). LLaVA is built on Vicuna-7B, while Qwen2 uses Qwen2-1.5B as the language backbone. HALVA adopts a contrastive learning framework by aligning correct and hallucinated phrases at the token level for fine-grained supervision. We evaluate LLaVA and Qwen2 in both pretraining and fine-tuning stages, and apply HALVA only in the fine-tuning stage. + +# 4.2 Training Sets and Augmentation Setup. + +The training process of LLaVA consists of two stages: the pretraining stage and the supervised fine-tuning stage. We describe the setup for each stage separately. + +Pretraining Stage. To evaluate the effectiveness of SiTe across different training sets, we experiment with two image-caption datasets: blip_laion_CC_sbu_558K (558K) and Flickr30K. As shown in Table 1, these datasets differ in the proportion of spatially informative samples, with Flickr30K containing a higher density of such spatial descriptions compared to 558K. We apply both horizontal and vertical stitching in a 1:1 ratio. By default, the stitched data make up one-third of the total set. To avoid duplicate supervision, image-caption pairs used for stitching are removed from the raw set, ensuring each image appears only once. As a result, the final number of training samples is slightly smaller than 558K. We design 35 templates for horizontal stitching and 29 templates for vertical stitching. For each sample, a template is randomly selected from the corresponding set based on the stitching mode. This diversity in spatial phrasing encourages the model to learn spatial relations in a more flexible and robust manner, rather than relying on fixed linguistic patterns. To further compare SiTe-augmented data with existing spatial supervision data, we construct a pretraining variant by substituting part of the original image-caption data with an equal-sized spatially-focused samples from the Visual Spatial Reasoning (VSR) dataset [19]. In default setting, the number of VSR data is 5K. Additionally, we compare SiTe with two standard augmentation baselines: Rotate and Crop. For Rotate, images are randomly rotated between $0^{\circ}$ and $360^{\circ}$ , with captions unchanged. For Crop, $80\% - 100\%$ of the image is randomly retained and cropped from a random region. Further implementation details, including the full list of spatial templates used for horizontal and vertical stitching, as well as examples of the stitched image-text pairs, are provided in Appendix B. + +Supervised Fine-Tuning Stage. We apply the SiTe method to augment the 11ava_v1_5_mix665k dataset. Noun phrases are extracted from original sentences using Qwen2.5-72B-Instruct with the + +prompt: "Extract the concrete, visible physical objects or entities described in this sentence. Return a comma-separated list. Ignore abstract terms like 'type', 'color', 'time', or actions." Based on the extracted entities, we construct spatial question-answer pairs to form a new instruction-tuning dataset. We evaluate the effect of adding 1K and 5K such samples to both LLaVA and Qwen2 models. + +We further explore the way to construct spatial negative data in contrastive learning by SiTe. For each stitched caption, we generate a negative example by swapping the spatial expressions. For example, given a positive sample such as "The bottom image contains $T_{1}$ , and the top image shows $T_{2}$ ," we construct a corresponding negative sample as "The top image contains $T_{1}$ , and the bottom image shows $T_{2}$ ." The SiTe method simplifies this process by providing explicit spatial layouts, making it easy to create contrastive pairs based on spatial inconsistencies. + +For the supervised fine-tuning stage, we define more than 20 question templates to generate spatial reasoning prompts. To reduce ambiguity, most templates explicitly incorporate the camera's perspective, using phrases such as "from the camera's point of view, is left_obj located to the left of right_obj?" to clarify directional references. See Appendix B for more details about QA templates. + +SiTe provides a simple and controllable way to construct such contrastive pairs. By modifying spatial expressions in structured captions (e.g., swapping "left" and "right"), we can easily generate hallucinated variants while preserving the global semantics. In default setting for $\mathrm{HALVA_{SiTe}}$ use 20K data, and the $\mathrm{HALVA_{Baseline}}$ use 21.5K data. + +# 4.3 Training Setup. + +For the pretraining stage, we train LLaVA-v1.5-7B and LLaVA-Qwen2-1.5B using a batch size of 16 and 64 per GPU, respectively, on 8 L20Z GPUs. All pretraining experiments are conducted for 1 epoch following the original LLaVA setup. For the supervised fine-tuning stage, we use the same hardware setup and follow the original LLaVA configuration, training both models for 1 epoch. The batch sizes are set to 4, 16 and 128 per GPU for HALVA, LLaVA-v1.5-7B and LLaVA-Qwen2-1.5B, respectively. We adopt the same learning rate and weight decay as in the original model settings. In all ablation experiments, the batch size and training schedule also remain consistent, and only the ratio of stitched data is varied. For each setting, we run five times and report the average results. + +# 4.4 Evaluation Setup. + +We assess models in two key areas: spatial understanding and general vision-language capabilities. For spatial understanding, we use four benchmarks: COCO-QA $_{\text{Spat}}$ (subset of COCO-QA focusing on spatial questions), Spatial-MM (multiple-choice benchmark testing spatial relations between objects), MME-Position (MME subset with spatially-aware questions answered in a single word/phrase), and MM $_{\text{Spat}}$ (MM-Vet subset evaluating complex spatial reasoning). To examine whether spatial supervision affects general multimodal performance, we evaluate on COCO-QA (a multimodal dataset for basic visual understanding), VQA-v2 (diverse human-annotated QA pairs from MS-COCO for robust visual reasoning), MMBench (multi-choice benchmark covering 20 vision-language ability dimensions), and MM-Vet (open-ended and multi-choice tasks testing integrated reasoning and knowledge). To verify the model's generalization ability in spatial understanding, we conducted evaluations on CV-Bench[32] and RealWorldQA[38], both of which include questions involving 3D spatial reasoning. In addition, to explore whether this approach can also enhance model performance on high-resolution data, we carried out experiments on two high-resolution datasets, HR-Bench 8K[36] and V-Star[37]. All evaluations are conducted in a zero-shot setting on target benchmarks. + +# 5 Main Results + +In this section, we present the performance of SiTe and several baseline methods on both spatial understanding and general vision-language benchmarks. Notably, in the pretraining stage, the total number of training samples is kept consistent across the Baseline, Rotate, Crop, and VSR settings. For SiTe, since each stitched sample is formed by combining two image-caption pairs, the total number of training examples is reduced accordingly to ensure the model sees the same number of unique images. This means that no image-caption pair appears in both the stitched and raw data. This adjustment ensures a more fair comparison across different stitching ratios. + +Table 2: Performance comparison on spatial understanding and general vision-language benchmarks with $\uparrow$ or $\downarrow$ values showing improvements or declines relative to each corresponding baseline. In the supervised fine-tuning stage, the superscript (e.g., $\mathrm{LLaVA}_{\mathrm{SiTe}}^{1\mathrm{K}}$ ) indicates that 1K spatially-aware samples are added per stitching direction. The bottom-right corner of each model denotes the data augmentation method used. Each variant is compared to its corresponding baseline with the same backbone or data setting. *Results are using their official checkpoint. + +
ModelSpatial UnderstandingGeneral Vision-Language
COCO-QASpat(%)Spatial-MM(%)MMEPosMM-VetSpatCOCO-QA(%)VQAV2(%)MMB(%)MM-Vet
Pretraining Stage
LLaVABaseline67.7242.02127.8326.5270.5260.4873.5031.11
LLaVARotate68.0943.01128.8929.4071.3660.6074.8632.20
LLaVACrop67.8242.69127.7828.5370.9360.3774.4131.43
LLaVASVR67.8542.65121.6729.4270.9060.5773.8831.47
LLaVASite-rand68.75↑1.0343.78↑1.76133.33↑5.5028.43↑1.9171.54↑1.0260.53↑0.0774.43↑0.9331.40↑0.29
LLaVASite-ratio70.06↑2.3444.15↑2.13132.80↑4.9728.68↑2.1671.19↑0.6760.57↑0.0973.64↑0.1432.27↑1.16
LLaVAFlicker68.9644.03129.0025.9871.7559.6372.3329.54
LLaVAFlocker68.0344.81129.4425.8671.1259.4272.3629.60
LLaVAFcrop69.3442.69131.1128.7372.1260.2072.6829.87
LLaVAFlickr68.2642.70128.7526.1871.3259.9472.8730.45
LLaVAFlock-rand71.42↑2.4644.97↑0.94130.70↑1.7026.20↑0.2273.74↑1.9959.73↑0.1072.88↑0.5529.59↑0.05
LLaVASite-ratio71.51↑2.5544.31↑0.28131.50↑2.5028.60↑2.6273.89↑2.1459.94↑0.3171.66↓0.6730.97↑1.43
Qwen2Baseline62.2540.8560.7520.7264.7253.6661.6722.48
Qwen2Rotate60.7340.7860.5019.6260.2252.7766.6120.97
Qwen2Crop62.0640.6861.2520.7564.7453.2867.4522.52
Qwen2VSR57.1841.4658.5022.4060.2254.6466.5822.95
Qwen2SITere-rand62.57↑0.3241.25↑0.4063.00↑2.2522.90↑2.1865.26↑0.5454.18↑0.5266.68↑5.0122.10↓0.38
Qwen2SITore-ratio62.52↑0.2741.00↑0.1568.00↑7.2521.00↑0.2865.10↑0.3854.23↑0.5767.57↑5.9022.80↑0.32
Qwen2flicker47.1038.9051.2510.9547.9042.3850.9010.50
Qwen2flicker40.0839.1058.339.5042.4539.0447.869.20
Qwen2flickerCrop42.3739.3764.258.3044.6140.4146.4310.00
Qwen2flicker46.0039.9067.2510.1548.0040.9348.719.60
Qwen2flickerSITere-rand47.71↑0.6139.35↑0.4561.67↑0.4212.47↑1.5248.68↑0.7842.95↑0.5751.33↑0.4310.80↑0.30
Qwen2flickerSITore-ratio49.04↑1.9440.04↑1.1456.00↑4.7513.10↑2.1547.38↓0.5243.91↑1.5351.31↑0.4110.60↑0.10
Supervised Fine-tuning Stage
LLaVA1K SITere-rand68.81↑1.0943.16↑1.14128.70↑0.8727.06↑0.5471.74↑1.2261.17↑0.6974.60↑1.1031.20↑0.09
LLaVA1K SITere-ratio68.35↑0.6346.96↑4.94136.00↑8.1729.70↑3.1871.32↑0.8060.92↑0.4473.37↓0.1331.54↑0.43
LLaVA5K SITere-rand67.75↑0.0346.21↑4.19139.26↑11.4328.10↑1.5870.96↑0.4460.38↓0.1074.53↑1.0330.69↓0.42
LLaVA5K SITere-ratio68.37↑0.6548.58↑6.56141.00↑13.1731.05↑4.5370.92↑0.4060.82↑0.3473.76↑0.2632.15↑1.04
Qwen2L K SITere-rand58.53↓3.7241.24↑0.3965.33↑4.5823.10↑2.3861.77↓2.9554.58↑0.9266.37↑4.7023.93↑1.45
Qwen2L K SITere-ratio59.66↓2.5942.31↑1.4662.75↑2.0024.07↑3.3562.59↓2.1355.30↑1.6465.75↑4.0822.80↑0.32
Qwen2L K SITere-rand59.95↓2.3042.22↑1.3761.67↑0.9223.20↑2.4863.00↓1.7254.75↑1.0966.43↑4.7623.93↑1.45
Qwen2L K SITere-ratio60.22↓2.0341.56↑0.7164.50↑3.7525.10↑4.3862.86↓1.8654.77↑1.1167.00↑5.3324.27↑1.79
HALVA* Baseline63.1643.07135.0025.7067.1261.6772.4430.00
HALVA SITe64.77↑1.6144.15↑1.08123.33↓11.6726.10↑0.4068.54↑1.4261.03↓0.6471.54↓0.9030.80↑0.80
+ +# 5.1 Evaluation on Spatial Understanding Benchmarks + +We first evaluate the impact of SiTe on spatial understanding. The results are provided in Table 2. In pretraining stage, experiments are conducted under the pretraining setting using two datasets (558K and Flickr30K) and two backbones: LLaVA-v1.5-7B and LLaVA-Qwen2-1.5B. We compare SiTe against baseline training, traditional augmentations (Rotate, Crop), and training data mixed with VSR data. + +Effect of SiTe-Augmented Pretraining. SiTe-augmented method consistently improves spatial understanding performance across all benchmarks and model backbones. This gain can be attributed to its explicit introduction of structured spatial knowledge during pretraining. By stitching two images with corresponding spatial information (e.g., "left to", "top of"), the model learns to associate language not only with object content, but also with relative layout. + +The $\mathrm{LLaVA}_{\mathrm{Rotate}}$ model shows moderate improvement on the 558K dataset. This may be because spatial expressions are relatively sparse in 558K, so rotating the image does not heavily conflict with the caption. In some cases, it may even help the model become more robust by learning to generalize + +across different viewpoints. However, when using Flickr30K as the training set, where captions contain richer spatial descriptions, $\mathrm{LLaVA}_{\mathrm{Rotate}}^{\mathrm{flickr}}$ performs worse than $\mathrm{LLaVA}^{\mathrm{flickr}}$ on several spatial benchmarks. For instance, on COCO-QASpat, accuracy drops by $0.93\%$ . This suggests that rotation may introduce misalignment between spatial phrases in the caption and the actual image layout, potentially resulting in implicit negative supervision. And the performance of the Crop baseline is less stable. Random cropping may inadvertently remove key objects or semantic regions, leading to weakened image-text alignment. Although the retained area is still between 0.8 to 1, the risk of disrupting mismatch grounding remains high—potentially introducing merrors where the caption does not accurately reflects the image content. + +Among SiTe variants, SiTe-rand corresponds to the original random pairing strategy, while SiTe-ratio adopts a ratio-based image pairing scheme that aligns images by aspect ratio before stitching. Compared with SiTe-rand, SiTe-ratio achieves further gains on most spatial benchmarks, indicating that more efficient image composition improves the performance of spatial supervision. The random variant still provides strong overall enhancement across models and datasets, showing that SiTe is robust even without additional pairing constraints. + +Flickr30K contains higher proportion of spatially-aware data, which helps allows $\mathrm{LLaVA}_{\mathrm{SiTe}}^{\mathrm{flickr}}$ learn both spatial relations within each image and layout patterns from stitched pairs, and perform better than $\mathrm{LLaVA}_{\mathrm{SiTe}}$ (from $68.75\%$ to $71.42\%$ ). This allows the model to better understand complex spatial structures, even with less data in pretraining, and achieve strong overall performance. + +Effect of SiTe-Augmented Supervised Fine-Tuning. SiTe continues to deliver strong results during the supervised fine-tuning stage. By adding 1K and 5K spatially-aware samples per stitching mode to the instruction tuning set, we observe consistent improvements across most spatial benchmarks. For example, after adding 5K horizontal and 5K vertical QA samples to LLaVA-v1.5-7B in fine-tuning stage, the accuracy on Spatial-MM increases from $42.97\%$ to $46.21\%$ , outperforming the baseline by $4.19\%$ . These gains come from our data construction strategy, which extracts noun entities from stitched regions and generates spatial QA instructions from the camera's perspective. This enables the model to learn spatial reasoning patterns from naturally aligned weak supervision. + +The SiTe-augmented method can also be used to construct contrastive examples. We apply spatially stitched text to HALVA by replacing only spatial information (e.g., "left" $\leftrightarrow$ "right") while keeping the other semantics information unchanged. In contrast, the HALVA method modifies objects, relations, and attributes. Despite this simpler setup, $\mathrm{HALVAsiTe}$ still outperforms the $\mathrm{HALVABaseline}$ on most benchmarks. + +We observe a slight performance drop for Qwen2 $_{\mathrm{SiTe}}$ on COCO-QA. This suggests a potential unalignment when fine-tuning small-capacity models on highly structured spatial QA data. Specifically, SiTe provides binary Yes/No questions focused spatial understanding, which differ in format and answer space from COCO-QA's diverse and open-ended questions. As a result, the model may over-adapt to the binary QA style, leading to reduced flexibility when answering questions that require generative reasoning or retrieval of specific object names. This effect is more pronounced in smaller models like Qwen2-1.5B, which are more sensitive to supervision bias and have limited generalization capacity. + +# 5.2 Evaluation on General Vision-Language Benchmarks + +We further evaluate whether SiTe compromises general vision-language capabilities while improving spatial understanding. Results are summarized in Table 2. For models based on LLaVA-v1.5-7B, SiTe consistently maintains or improves general performance. Notably, SiTe-augmented in LLaVA maintains consistent improvements on general vision-language benchmarks during the pretraining stage. This suggests that introducing spatial understanding at this stage enhances the model's ability to generalize to out-of-distribution (OOD) scenarios. In the supervised fine-tuning stage, SiTe still shows mostly positive gains, though we observe slight drops on VQA-v2 and MM-Vet. This may be due to the nature of instruction tuning, where a large number of spatial-focused examples shift the model's preference, potentially affecting its performance on general tasks. For Qwen2, which has relatively limited capacity, SiTe also brings clear benefits in pretraining. In particular, Qwen2 $_{\mathrm{SiTe}}$ improves MMBench by $5.01\%$ . However, during supervised fine-tuning stage, Qwen2 shows a slight decrease on COCO-QA, indicating that instruction tuning tends to reinforce domain-specific behavior. For smaller models, this may reduce the ability to follow diverse instructions, leading to weaker + +generalization. These results further highlight the importance of spatial supervision during pretraining for improving downstream robustness. + +We further conducted experiments on both high-resolution and 3D spatial benchmarks to evaluate the generalization capability of our approach. Specifically, we tested models enhanced with SiTe on HR-Bench 8K and V-Star, where they consistently achieved higher scores than the baselines, demonstrating improved perception of fine-grained visual details. + +To assess spatial reasoning in more realistic and complex settings, we also evaluated the models on CV-Bench and RealWorldQA, which include diverse questions involving 3D spatial relationships. The results show that injecting structured spatial supervision through SiTe significantly improves performance on out-of-distribution spatial reasoning tasks and generalizes well to more challenging real-world scenarios. We hypothesize that these gains arise from the model's enhanced ability to represent fundamental directional relations, which in turn supports broader spatial understanding across different dimensions. + +Table 3: Performance comparison across real-world and high resolution benchmarks. + +
ModelCV-Bench 2D (%)CV-Bench 3D (%)RealworldQA (%)HRBench-8K (%)V-Star (%)
\(LLaVA_{Baseline}\)52.5633.4355.1230.9048.34
\(LLaVA_{SiTe-rand}\)54.28↑1.7235.50↑2.0755.21↑0.0932.56↑1.6649.95↑1.61
\(LLaVA_{SiTe-ratio}\)55.58↑3.0236.53↑3.1055.26↑0.1433.72↑2.8250.13↑1.79
\(LLaVA_{flickr}\)52.8728.3954.5133.5249.32
\(LLaVA_{SiTe-rand}\)54.06↑1.1938.66↑10.2755.28↑0.7734.53↑1.0147.28↓2.04
\(LLaVA_{SiTe-ratio}\)55.07↑2.2039.80↑11.4156.25↑1.7433.94↑0.4250.79↑1.47
Qwen2\(Baseline\)46.7747.6352.2832.2537.83
Qwen2\(SiTe-rand\)48.16↑1.3950.36↑2.7354.44↑2.1632.67↑0.4238.92↑1.09
Qwen2\(SiTe-ratio\)47.47↑0.7050.40↑2.7752.94↑0.6632.94↑0.6938.09↑0.26
Qwen2\(flickr\)42.5951.5743.2032.6336.13
Qwen2\(flickr_{SiTe-rand}\)43.64↑1.0551.67↑0.1042.75↓0.4533.34↑0.7136.52↑0.39
Qwen2\(flickr_{SiTe-ratio}\)44.44↑1.8551.50↓0.0749.74↑6.5434.33↑1.7036.39↑0.26
+ +We also calculate the computational time efficiency improvement brought about by pre-training training using SiTe method. Taking default as an example, due to the decrease in the number of data strips, the time to run an epoch in the pre-training stage is $77.4\%$ of the baseline. + +In summary, Stitch and Tell effectively enhances spatial supervision by increasing data density through structured augmentation, without sacrificing the original knowledge. This approach leads to stronger spatial understanding and remains competitive on general vision-language benchmarks, demonstrating its broad applicability. + +# 5.3 Qualitative Analysis + +Where is the AI chip in relation to the hand? A. Front B. Behind C. Above D. Under + +![](images/9cee177301964bbed2f1053544023e773ff35bf37222eb158c4f9c76ef9cfd87.jpg) +(a) + +![](images/cce6f7fcb2120e62dbf10d36b16015f1cfca609a660471ceee53fdb13a995455.jpg) +LLaVA: A + +![](images/b8d5a48b0a83dc575a7b42ed88341d7d44c706f76ef57493837b61a456b023d8.jpg) +SiTe:B + +Is there a backpack on the left of the laptop from camera's perspective? + +![](images/689f80a7bf34832dac053926725ee357e1f7413c0d433d47ccd75d42b92d88b3.jpg) +A.Yes B.No +(c) + +![](images/ef83a452272d5c2f8fd417751633747f2db66c025282c77ef14b8cc4affaf93a.jpg) +LLaVA:A + +![](images/2d75fac489b22271e93507714ae068c33f103eec2e74d2c227093df32726d244.jpg) +SiTe: B + +Is there a tea cup on the left of the cake from camera's perspective? + +![](images/7426f6fb4235c0fbf9b63c7fa48b6969d8c907c863034e842e50008e1d6e1e7b.jpg) +A.Yes B.No +(b) + +![](images/22e9af0aad6474e77ff2f77eacd22abe8e195213bf5b311f94c7d6539fe27add.jpg) +LLaVA: B + +![](images/be2751d2dc185f0e57c987893ef8f6ea9d5c972ac0224bf61e82dc281c90513b.jpg) +SiTe:A + +Is there a table lamp on the left of the TV from camera's perspective? + +![](images/883103ad3297317f8f150f4f6885c7c796caa2f2438febcb6675db6f22be49ad.jpg) +A.Yes B.No +(d) +Figure 4: Qualitative comparisons between LLaVA and SiTe. SiTe can better distribute attention between objects and judge their spatial relationships. + +![](images/97bfca831449d34253e812ae1b60b38c446d0e021ec2424c5385dd4048514f28.jpg) +LLaVA: B + +![](images/ed1a37c4d7775c7f28bb3085f14f5fa8f48f416b24eff90cb4777500c4630ed8.jpg) +SiTe: A + +We visualize attention differences MLLM-Know [44], which highlights image regions where the model assigns more attention under task-specific prompts compared to general prompts. A qualitative + +comparison between LLaVA-v1.5-7B and SiTe is shown in Figure 4. In Figure 4 (a), when asked about the relation between the AI chip and the hand, LLaVA mainly attends to the hand while neglecting the chip, resulting in an incorrect answer. In contrast, SiTe correctly focuses on both entities. In Figure 4 (b), where the question involves the left side of the cake, SiTe allocates more attention to the region between the cup and the cake, effectively capturing their spatial arrangement. Similarly, in Figure 4 (c) and (d), SiTe exhibits more precise attention distribution across the referenced objects, which contributes to more accurate spatial understanding. + +# 5.4 Ablations + +Recalling the SiTe method in the pretraining stage, we define a stitching ratio parameter $\lambda$ to measure the proportion between stitched and raw samples in the final training set. Let $T$ denote the number of original image-caption pairs, and $N$ be the number of stitched samples constructed for each stitching mode (e.g., horizontal or vertical). Since each stitched sample is formed by consuming two raw samples, the total number of stitched samples is $2N$ , and the remaining raw examples become $T - 4N$ , assuming two stitching modes are used. Let $\mathcal{D}_{\mathrm{Stitch}}$ and $\mathcal{D}_{\mathrm{Raw}}$ denote the sets of stitched and retained raw examples, respectively. The final ratio $\lambda$ is defined as: $\lambda = \frac{|\mathcal{D}_{\mathrm{Stitch}}|}{|\mathcal{D}_{\mathrm{Raw}}|} = \frac{2N}{T - 4N}$ . + +We present an ablation study using the 558K dataset for pretraining. Specifically, we introduce seven SeTi variants with different stitching ratios, as summarized in Table 5. We observe that when the stitching ratio $\lambda$ is around $1/3$ , the model achieves overall strong performance on spatial understanding benchmarks while maintaining competitive results on general vision-language tasks. This setting provides a favorable trade-off between spatial supervision and overall data diversity. A similar trend is observed when using the Flickr30K dataset for training, and the more detail ablation results and analysis are included in the Appendix C. + +Table 4: Settings of SiTe-rand as variants of $\mathrm{SiTe}_{\mathrm{default}}$ used for the ablation study during pretraining on the 558K dataset. Each variant adopts a different ratio $\lambda$ , representing the proportion of stitched samples to the remaining raw samples in the training set, and the total number of training instances varies accordingly. + +
SettingImages(Total data size)λ
SiTe default458K1:3.58
(a) N=1K556K1:277.0
(b) N=5K548K1:53.8
(c) N=10K538K1:25.9
(d) N=100K358K1:0.79
(e) N=139K280K1:0.01
+ +Table 5: Performance on spatial understanding benchmarks (left) and general vision-language benchmarks (right) under different spatial data augmentation settings. + +
SettingSpatial UnderstandingGeneral Vision-Language
COCO-QASpatialSpatial-MMMMEPositionMM-VetSpatCOCO-QAVQAv2MMBMM-Vet
SiTedefault68.7543.78133.3328.4374.4360.5374.4331.40
(a)69.2342.90126.327.5474.0560.5074.0530.58
(b)67.5842.42132.6727.5974.1160.3574.1131.34
(c)68.1342.83128.6727.5074.0460.4074.0431.45
(d)68.8343.10132.1727.6371.5360.5574.5531.27
(e)71.0943.72132.5028.0674.7460.3274.7431.29
+ +# 6 Conclusion + +In this work, we present Stitch and Tell, a simple and scalable data augmentation strategy that injects spatial structure into vision-language training. SiTe combines image stitching with spatially-aware caption generation to provide weak spatial supervision without requiring human annotation or architectural changes. We apply SiTe to multiple backbones, including LLaVA, Qwen2 and HALVA, across two training datasets and thirteen benchmarks. Experiments show consistent improvements on both spatial understanding and general vision-language tasks. These results demonstrate that encoding spatial structure into training data can improve cross-modal alignment and spatial reasoning, while maintaining strong general performance. We hope this work provides a lightweight and broadly applicable approach to structured multimodal data augmentation for spatial understanding. + +# References + +[1] Stanislaw Antol, Aishwarya Agrawal, Jiasen Lu, Margaret Mitchell, Dhruv Batra, C. 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In Jie Tan, Marc Toussaint, and Kourosh Darvish, editors, Conference on Robot Learning, CoRL 2023, 6-9 November 2023, Atlanta, GA, USA, volume 229 of Proceedings of Machine Learning Research, pages 2165-2183. PMLR, 2023. + +# A Limitation + +While Stitch and Tell demonstrates strong performance and scalability across multiple settings, it currently focuses on simple binary spatial configurations—primarily horizontal (left-right) and vertical (top-down) compositions. This design enables efficient supervision generation and robust model alignment, but may limit the model's exposure to more complex spatial relationships that occur in real-world environments, such as diagonal layouts, circular arrangements, or relative distances in 3D space. We believe extending SiTe to handle richer spatial topologies, possibly by incorporating depth maps, multi-image compositions, or 3D-aware stitching strategies, is a promising direction for future work. + +# B Data Template + +To support structured data generation in our Stitch and Tell method, we employ a collection of spatially guided templates that are automatically generated and manually filtered for diversity and clarity by GPT-4o. These templates are designed to inject explicit spatial structure into image-text data and are categorized as follows: + +- Spatial Caption Templates: As shown in Table 6 and Table 7, we collect 35 templates for horizontal stitching and 29 for vertical stitching. These templates are automatically produced using a language model and then curated to ensure naturalness and correctness. Each template organizes two independent captions into a single sentence with clear spatial cues, such as "Left side displays: {caption1}, Right side shows: {caption2}" The diversity of expressions encourages the model to learn generalized spatial grounding rather than relying on specific lexical patterns. +- Spatial QA Templates: As shown in Table 8 and Table 9, we generate 20 question templates in each stitching way (horizontal or vertical). These templates are used to form weakly supervised question answer pairs such as "From the observer's viewpoint, is the cat on the left of the car?", with answers derived directly from the known image layout. Templates were automatically generated with instruction-tuned models and then filtered to preserve clarity, directional accuracy, and grammatical diversity. + +Table 6: Caption templates used for constructing structured spatial descriptions in S&T. Placeholders {caption1} and {caption2} represent the original descriptions from the left and right images, respectively. + +
No.Template
1.On the left, {caption1}. Meanwhile, the right side presents {caption2}.
2.This composite image showcases {caption1} on the left, contrasting beautifully with {caption2} on the right.
3.The left section highlights {caption1}, while the right side draws attention to {caption2}.
4.Displayed in the left half is {caption1}, while the right half features {caption2}.
5.{caption1} on the left and {caption2} on the right.
6.Left image shows: {caption1} | Right image shows: {caption2}.
7.On the left: {caption1} || On the right: {caption2}.
8.First image description: {caption1}, Second image description: {caption2}.
9.Image pair - Left: {caption1}; Right: {caption2}.
10.[Left] {caption1} [Right] {caption2}.
11.Left side displays: {caption1} <= > Right side displays: {caption2}.
12.Left view: {caption1} — Right view: {caption2}.
13.Left portion: {caption1} » Right portion: {caption2}.
14.Left panel shows: {caption1} || Right panel shows: {caption2}.
15.Left section: {caption1} <> Right section: {caption2}.
16.The right one shows {caption2}, while the left displays {caption1}.
17.A pair of images: on the right we see {caption2}, and on the left there's {caption1}.
18.The right image contains {caption2}, paired with a left image showing {caption1}.
19.Two scenes: {caption2} on the right, accompanied by {caption1} on the left.
20.Right image depicts {caption2}, contrasting with the left image showing {caption1}.
21.{caption1} (left) vs {caption2} (right).
22.Contrast in perspective - left presents {caption1}, while right shows {caption2}.
23.Visual contrast: Left shows {caption1}, Right shows {caption2}.
24.Left portrays {caption1}, Right highlights {caption2}.
25.Left vs Right: {caption1} & {caption2}.
26.Left illustrates {caption1}, whereas right {caption2}.
27.Left image showcasing {caption1}, and right image featuring {caption2}.
28.{caption1} and {caption2}.
29.{caption1}, {caption2}.
30.Left: {caption1} Right: {caption2}.
31.{caption1} {caption2}.
32.Left/Right: {caption1}/ {caption2}.
33.| {caption1} | vs | {caption2} |
+ +Table 7: Templates used for generating structured captions in top-down stitched images. Placeholders {caption1} and {caption2} represent captions from the top and bottom image regions. + +
No.Caption Template
1.On the top, you can see {caption1}, the bottom side presents {caption2}.
2.This composite image showcases {caption1} on the top, with {caption2} on the bottom.
3.The top section highlights {caption1}, while the bottom side draws attention to {caption2}, creating an engaging visual comparison.
4.Displayed in the upper half is {caption1}, while the lower half features {caption2}, illustrating unique attributes side by side.
5.A striking juxtaposition: {caption1} on the top and {caption2} on the bottom, offering an interesting visual narrative.
6.Top image shows: {caption1}, Bottom image shows: {caption2}.
7.On the top: {caption1}. On the bottom: {caption2}.
8.First image description: {caption1}, Second image description: {caption2}.
9.Top: {caption1}; Bottom: {caption2}.
10.Top part display: {caption1} Bottom part display: {caption2}.
11.Upper displays: {caption1}, Lower displays: {caption2}.
12.Top view: {caption1}, and Bottom view: {caption2}.
13.Top portion: {caption1} > Bottom portion: {caption2}.
14.Upper panel shows: {caption1} | | Lower panel shows: {caption2}.
15.Upper section: {caption1} <- Lower section: {caption2}.
16.The bottom one shows {caption2}, while the top displays {caption1}.
17.A pair of images: on the bottom we see {caption2}, and on the top there's {caption1}.
18.The bottom image contains {caption2}, the top image showing {caption1}.
19.Two scenes: {caption2} on the lower side, accompanied by {caption1} on the upper.
20.Bottom image depicts {caption2}, contrasting with the top image which shows {caption1}.
21.{caption1} (top) vs {caption2} (bottom).
22.Top side presents {caption1}, while bottom side shows {caption2}.
23.Top shows {caption1}, Bottom shows {caption2}.
24.Top portrays {caption1}, Bottom highlights {caption2}.
25.Top vs Bottom: {caption1} & {caption2}.
26.Top illustrates {caption1}, whereas bottom captures {caption2} in detail.
27.A split view: top image showcasing {caption1}, and bottom image featuring {caption2}.
28.{caption1} and {caption2}.
29.{caption1}, {caption2}.
30.Top: {caption1}, Bottom: {caption2}.
+ +Table 8: List of spatial QA templates used in instruction tuning. Each template is instantiated with {left_obj} and {right_obj}, and paired with a binary answer. + +
IDTemplate (with placeholders)Answer
1.From the observer's point of view, is {left_obj} located to the left of {right_obj}?Yes
2.From the camera's viewpoint, can we see {right_obj} on the right of {left_obj}?Yes
3.Looking from the front, is {left_obj} placed to the right of {right_obj}?No
4.As observed from the viewer's perspective, does {right_obj} appear left of {left_obj}?No
5.From the point of view of the observer, is {left_obj} on the left side of {right_obj}?Yes
6.Is {right_obj}, from the camera's perspective, situated to the right of {left_obj}?Yes
7.When facing the image, does the left contain {left_obj} and the right contain {right_obj}?Yes
8.From a frontal viewpoint, is {left_obj} to the left side of {right_obj}?Yes
9.As seen in the combined image, is {right_obj} located right of {left_obj}?Yes
10.To the viewer, does {left_obj} appear on the right side of {right_obj}?No
11.Does {left_obj} appear on the left side of {right_obj}?Yes
12.Is {right_obj} located to the left of {left_obj}?No
13.Can {left_obj} be found on the right of {right_obj}?No
14.Is {right_obj} positioned on the right of {left_obj}?Yes
15.Would you say {left_obj} is left to {right_obj}?Yes
16.Is {right_obj} on the left of {left_obj} in this composition?No
17.Does the image on the left contain {left_obj} while the right image contains {right_obj}?Yes
18.Is {left_obj} positioned on the right of {right_obj} instead?No
19.In this pair, is {right_obj} located to the right of {left_obj}?Yes
20.If you observe carefully, is {left_obj} on the right and {right_obj} on the left?No
+ +Table 9: Templates for generating top-down spatial question-answer (QA) pairs. Placeholders {top_obj} and {bottom_obj} denote entities from the top and bottom image regions. + +
No. QA TemplateAnswer
1. In the image, is {top_obj} located above {bottom_obj}?Yes
2. From the viewpoint of the observer, is {bottom_obj} below {top_obj}?Yes
3. Would you say that {top_obj} is placed underneath {bottom_obj}?No
4. As observed from the front, is {bottom_obj} situated on top of {top_obj}?No
5. Does the top part of the image contain {top_obj} while the bottom part has {bottom_obj}?Yes
6. Looking from top to bottom, do you first see {top_obj}, then {bottom_obj}?Yes
7. Is {bottom_obj} placed above {top_obj} in this composition?No
8. From top-down perspective, is {top_obj} above {bottom_obj}?Yes
9. In this combined image, does {top_obj} appear at the top and {bottom_obj} at the bottom?Yes
10. Is {top_obj} below {bottom_obj}?No
11. Would you say {bottom_obj} is at a lower vertical position than {top_obj}?Yes
12. Does the vertical layout place {top_obj} higher than {bottom_obj}?Yes
13. Is {bottom_obj} appearing above {top_obj} in this image?No
14. Do you see {top_obj} on the upper half and {bottom_obj} on the lower half of the image?Yes
15. Can we find {bottom_obj} positioned higher than {top_obj} in the image?No
16. In the vertical layout, is {top_obj} stacked above {bottom_obj}?Yes
17. Does {top_obj} sit at the bottom while {bottom_obj} is on top?No
18. Is the object {top_obj} visually located above {bottom_obj} from this angle?Yes
19. Would you agree that {bottom_obj} is beneath {top_obj} in this view?Yes
20. Is {bottom_obj} placed at the upper portion of the image, above {top_obj}?No
+ +# C Additional Experiments and Results + +# C.1 Ablation study + +Table 11: Performance of LLaVA-Qwen2-1.5B pretrained on the 558K dataset under different spatial data augmentation strategies. Results are reported on spatial understanding benchmarks (left) and general vision-language benchmarks (right). + +
SettingSpatial UnderstandingGeneral Vision-Language
COCO-QASpatialSpatial-MMMMEPositionMM-VetSpatCOCO-QAVQAv2MMBMM-Vet
SiTedefault62.5741.2563.0022.9065.2654.1866.6822.10
(c)60.8841.0863.2020.8064.8153.5067.5121.04
(d)59.8341.1164.0021.7562.7452.8370.3221.30
(e)53.2039.6655.0018.1554.7446.8064.4717.10
+ +Table 12: Performance of LLaVA-v1.5-7B pretrained on the Flickr30K dataset under different spatial data augmentation strategies. Results are reported on spatial understanding benchmarks (left) and general vision-language benchmarks (right). + +
SettingSpatial UnderstandingGeneral Vision-Language
COCO-QASpatialSpatial-MMMMEPositionMM-VetSpatCOCO-QAVQAv2MMBMM-Vet
SiTedefault71.4244.97130.7026.2073.7459.7372.8829.59
(1)69.8042.96130.3027.5472.5359.6572.5631.06
(2)69.6344.50126.9027.5972.4459.1072.4229.40
(3)71.0344.21133.0026.1773.2858.7372.3129.19
+ +Table 13: Performance of LLaVA-Qwen2-1.5B pretrained on the Flickr30K dataset under different spatial data augmentation strategies. Results are reported on spatial understanding benchmarks (left) and general vision-language benchmarks (right). + +
SettingSpatial UnderstandingGeneral Vision-Language
COCO-QASpatialSpatial-MMMMEPositionMM-VetSpatCOCO-QAVQAv2MMBMM-Vet
SiTedefault47.7139.3561.0012.4748.6842.9551.3310.80
(1)47.5739.2857.6012.0048.1740.7849.4310.54
(2)48.1638.8856.5012.6048.7641.6952.0411.23
(3)43.0639.6360.0011.6545.7239.3649.0910.53
+ +To identify the optimal stitching ratio for spatial data augmentation, we perform ablation experiments on four settings: LLaVA-v1.5-7B trained on 558K and Flickr30K, Qwen2-1.5B trained on 558K and Flickr30K. In each setting, we vary the number of stitched samples and report performance on both spatial understanding and general vision-language benchmarks. + +In the main text, we present ablation results of our SiTe method on LLaVA using the 558K dataset. Here, we provide additional ablations on two new settings: LLaVA-v1.5-7B pretrained on Flickr30K, and LLaVA-Qwen2-1.5B pretrained on both 558K and Flickr30K. + +Table 10: Settings of SiTe variants used for ablation study during pretraining on the Flickr30K dataset. Each variant uses a different ratio $\lambda$ , which denotes the proportion of stitched samples to the remaining raw samples in the training set. The total number of training data varies accordingly. + +
SettingImages(Total data size)λ
SiTedefault24K1:3.0
(1) N=1K28K1:13.0
(2) N=5K20K1:1.0
(3) N=7K16K1:0.14
+ +For the 558K dataset, we evaluate Qwen2 under + +four configurations with different stitching sizes: $N = 10\mathrm{K}$ (setting (c)), $N = 50\mathrm{K}$ (default setting on 558K), $N = 100\mathrm{K}$ (setting (d)), and $N = 139\mathrm{K}$ (setting (e)). The corresponding results are summarized in Table 11. + +For Flickr30K, we evaluate on $N = 1\mathrm{K}$ (setting (1)), $N = 3\mathrm{K}$ (default setting on Flickr30K), $N = 5\mathrm{K}$ (setting (2)), and $N = 7\mathrm{K}$ (setting (3)). Dataset statistics are shown in Table 10, and the results are reported in Table 12 and Table 13 for LLaVA and Qwen2 backbones. + +We summarize our key observations as follows: + +- Qwen2 on 558K (Table 11): The default 1:3 setting achieves the best overall spatial performance. Specifically, it yields the highest results on COCO-QA $_{\text{Spat}}$ (62.57), Spatial-MM (41.25), MM-Vet $_{\text{Spat}}$ (22.90), and also achieve most tops general metrics. Increasing the stitching ratio beyond this point (e.g., settings (d) and (e)) leads to degradation on both spatial and general benchmarks, indicating over-augmentation. +- LLaVA on Flickr30K (Table 10): The 1:3 setting ( $\mathbf{S}\mathbf{i}\mathbf{T}\mathbf{e}_{\mathrm{d e f a u l t}}$ ) still consistently provides strong performance across tasks. It outperforms other ratios on Spatial-MM (44.97), MMEPosition (130.70), and general vision-language tasks like COCO-QA (73.74) and MMBench (72.88). Other ratios (settings (1)-(3)) offer marginal or inconsistent gains, and in some cases hurt generalization. +- Qwen2 on Flickr30K (Table 13): In this setting, the 1:3 setting still maintains strong spatial performance, achieving the best results on $\mathrm{MME}_{\mathrm{Position}}$ (61.00) and balanced results on general tasks. While setting (2) gives slightly better COCO-QA and MMB scores, the default ratio still offers a stable and robust trade-off. + +In summary, the 1:3 stitch-to-raw ratio consistently provides a balanced trade-off between spatial reasoning and general vision-language performance. Ratios that are too low underutilize the spatial supervision potential of SiTe while overly high ratios risk oversatrating the model with structured spatial prompts, which may harm generalization. Based on these findings, we adopt the 1:3 setting as the default configuration throughout our experiments. + +In all experiments, we define the default SiTe configuration as the one where the stitching-to-raw sample ratio is approximately 1:3, which provides a good trade-off performance between spatial understanding and general vision language tasks. + +# C.2 More Qualitative Analysis + +In this section, we present additional qualitative examples to illustrate both the effectiveness and limitations of the proposed method. LLaVA refers to the baseline model. SiTepretrain indicates the model trained with stitched image-caption pairs during the pretraining stage, while SiTeSFT refers to the model fine-tuned using structured spatial question-answer pairs. + +As shown in Case 1 and Case 2, although all models attend to the relevant objects (e.g., the dog and the people), the baseline model fails to correctly answer the spatial question due to limited understanding of directional language, selecting an incorrect option such as "A. bottom". In contrast, both SiTe-enhanced models correctly interpret the spatial relation and make the right prediction. Case 2 also involves a challenging camera-perspective transformation, which often requires the model to reason from the observer's viewpoint. Here, both SiTe variants successfully leverage spatial knowledge injected through pretraining and fine-tuning, resulting in correct answers. + +However, limitations remain in non-camera-perspective scenarios. In Case 3 and Case 4, while all models attend to the appropriate regions (e.g., the glasses in Case 3 and the phone in Case 4), they fail to answer correctly. This suggests that despite improved attention, the models still struggle to generalize spatial understanding across different viewpoints. + +These examples highlight both the strengths and boundaries of the proposed method: SiTe improves spatial reasoning under observer-aligned prompts, but further work is needed to enhance generalization under varied spatial perspectives. + +# C.3 Standard Deviation of Main Results + +We compute the standard deviation of main results in Table 14. As indicated by the standard deviation results, our performance gains are consistent and robust. + +# Where is the dogs from the people's perspective? A. bottom B. right C. front D. left + +![](images/eb866378e961d6d86f3e005de0db9e4b31b3870498d04b48ccdff09d722bedba.jpg) +Case 1 + +![](images/660ec3d2c62c45c22468f83c3fa63519be7fa40f015f7b7b9eed59acc4c74474.jpg) +LLaVA:A + +![](images/6d4295c52efe0b3f05ba738df01d36976e125c1a65d77e39c90dd1efd1eea9e6.jpg) +SiTePretrain: B + +![](images/1559ac7ef9dabeedf0d167305819e15180a995c43441a15a2c9abf30a5712d0b.jpg) +$\mathrm{SiTe_{SFT}}$ :B + +# Which hand of the boy is the man holding? A. left B. right + +![](images/e8a462d7d19df3a6127901f2326eded11773f1db951516a31d81b333358dd97b.jpg) +Case 2 + +![](images/5db69734579062111ff77d84987f37b9f2927f03332553fc394d21c105bda445.jpg) +LLaVA: B + +![](images/2729e4abaec5737ccb03203589e311f782c4068dd94913071de94074fd6f10e4.jpg) +$\mathrm{SiTe}_{\mathrm{Pretrain}}$ : A + +![](images/a1a0fca6e101e6fec22737edb8539c98b72a32eaed1e4055af00ab7e3e17faab.jpg) +$\mathrm{SiTe}_{\mathrm{SFT}}$ : A + +# Which hand is the man using to hold the glasses? A. left B. right + +![](images/b283b51d793451d30888c3c732fb4c904b8ba99b5e5b95fa1b6dcdff7aa206c8.jpg) +Case 3 + +![](images/06072dd29d8e7302ad5be77381e374f47906bbbeb933b2432e9b2018dcb17745.jpg) +LLaVA: B + +![](images/725a77b548db80f837b9a1ed0cee768f8cbdc5151500915d2e4612895755289e.jpg) +SiTePretrain: B + +![](images/b1d8ac73d468d2a56965693d590bec2fb2bbb960d4cf52ba8df569af81940e4f.jpg) +$\mathrm{SiTe_{SFT}}$ :B + +# In which hand is the man holding the phone? A. left B. right + +![](images/235caeca2ef65cac764bbb47ee3e4c97fc6bfc6d17e608c5457c37d3f442b8a4.jpg) +Case 4 + +![](images/fc84ec58ce37c75f5bd74272ca6bbd1d099ba98dad4ce1f882acfe4e4a7a51f8.jpg) +LLaVA:A + +![](images/4333cdf79ad3f662da218506816151881c83600f698acff96eb22f38b24425fa.jpg) +SiTePretrain:A + +![](images/de3ff64f2be11dd9b0d4cbb65674334b92131dab01c3271cd60ed1eab78be061.jpg) +SiTeSFT: A +Figure 5: Qualitative comparisons among the baseline LLaVA and our SiTe models. + +Table 14: Standard deviation of main results + +
ModelSpatial UnderstandingGeneral Vision-Language
COCO-QASpat(%)Spatial-MM(%)MMEPosMM-VetSpatCOCO-QA(%)VQAV2(%)MMB(%)
Pretraining Stage
LLaVABaseline67.72±0.1042.02±0.54127.83±1.0726.52±0.0870.52±0.3260.48±0.1673.50±0.14
LLaVARotate68.09±0.1743.01±0.11128.89±1.5129.40±0.0371.36±0.1160.60±0.1774.86±0.05
LLaVACrop67.82±0.2542.69±0.24127.78±0.9428.53±0.1270.93±0.1860.37±0.2474.41±0.09
LLaVAVSR67.85±0.2142.65±0.36121.67±1.0129.42±0.5470.90±0.2860.57±0.2573.88±0.14
LLaVASte-rand68.75±0.0643.78±0.14133.33±0.4328.43±0.1971.54±0.2760.53±0.2374.43±0.03
LLaVASte-ratio70.06±0.1844.15±0.22132.80±0.9728.68±0.1671.19±0.3160.57±0.2073.64±0.15
LLaVAflickr68.96±0.2144.03±0.17129.00±0.8525.98±0.1471.75±0.2859.63±0.1972.33±0.13
LLaVAflickrRotate68.03±0.1544.81±0.19129.44±0.7825.86±0.1171.12±0.2259.42±0.1872.36±0.14
LLaVAflickrCrop69.34±0.2442.69±0.21131.11±0.9228.73±0.1072.12±0.2660.20±0.2172.68±0.17
LLaVAflickrRotate68.26±0.1842.70±0.28128.75±0.9726.18±0.1371.32±0.3159.94±0.2472.87±0.19
LLaVAflickrSte-rand71.42±0.1344.97±0.22130.70±0.8526.20±0.1473.74±0.2959.73±0.1872.88±0.14
LLaVAflickrSte-ratio71.51±0.1544.31±0.25131.50±0.8128.60±0.1273.89±0.2559.94±0.2071.66±0.16
Qwen2Baseline62.25±0.1640.85±0.1860.75±0.5320.72±0.1064.72±0.2253.66±0.2561.67±0.19
Qwen2Rotate60.73±0.1440.78±0.2060.50±0.6719.62±0.1560.22±0.2652.77±0.2266.61±0.23
Qwen2Crop62.06±0.2140.68±0.1561.25±0.5820.75±0.1164.74±0.1953.28±0.2767.45±0.21
Qwen2VSR57.18±0.1841.46±0.1758.50±0.7322.40±0.1660.22±0.2854.64±0.2366.58±0.22
Qwen2Ste-rand62.57±0.1541.25±0.2263.00±0.6122.90±0.1465.26±0.2354.18±0.2066.68±0.19
Qwen2Ste-ratio62.52±0.1741.00±0.1968.00±0.8821.00±0.1365.10±0.2754.23±0.2167.57±0.20
Qwen2flickr47.10±0.2038.90±0.2351.25±0.6410.95±0.1047.90±0.3142.38±0.2550.90±0.19
Qwen2flickrRotate40.08±0.2339.10±0.2758.33±0.759.50±0.1342.45±0.2939.04±0.2847.86±0.21
Qwen2flickrCrop42.37±0.2239.37±0.2464.25±0.788.30±0.1244.61±0.2840.41±0.2546.43±0.22
Qwen2flickrRotate46.00±0.1939.90±0.2067.25±0.7010.15±0.1148.00±0.2640.93±0.2448.71±0.20
Qwen2Ste-rand47.71±0.1739.35±0.1861.67±0.8212.47±0.1448.68±0.2542.95±0.2151.33±0.18
Qwen2flickrSte-ratio49.04±0.1840.04±0.2056.00±0.6913.10±0.1547.38±0.2743.91±0.2351.31±0.17
Supervised Fine-tuning Stage
LLaVAIKSte-rand68.81±0.2643.16±0.27128.70±0.0827.06±0.1171.74±0.4761.17±0.2974.60±0.26
LLaVAIKSte-ratio68.35±0.2146.96±0.23136.00±0.9229.70±0.1471.32±0.3360.92±0.2773.37±0.19
LLaVAKSte-rand67.75±0.0446.21±0.14139.26±0.3128.10±0.2370.96±0.1360.38±0.2074.53±0.18
LLaVAKSte-ratio68.37±0.2248.58±0.26141.00±1.0331.05±0.1670.92±0.2860.82±0.2573.76±0.20
Qwen2IKSte-rand58.53±0.4741.24±0.0365.33±0.5423.10±0.4461.77±1.3254.58±0.0366.37±0.11
Qwen2STe-ratio59.66±0.3242.31±0.2762.75±0.6324.07±0.2162.59±0.3555.30±0.2565.75±0.19
Qwen2STe-rand59.95±0.5342.22±0.2161.67±0.5323.20±0.3763.00±1.1054.75±0.1466.43±0.13
Qwen2STe-ratio60.22±0.3541.56±0.2964.50±0.5925.10±0.2262.86±0.4154.77±0.2067.00±0.18
HALVABaseline63.16±0.1443.07±0.26135.00±0.9425.70±0.1367.12±0.2561.67±0.1872.44±0.16
HALVASte64.77±0.1844.15±0.22123.33±0.8326.10±0.1268.54±0.2761.03±0.2071.54±0.15
+ +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract and introduction clearly state the core motivation and contribution of the work—namely, that current vision-language models struggle with spatial understanding due to limited spatial supervision in training data. The proposed method, Stitch and Tell (SiTe), is presented as a simple and efficient data augmentation strategy that injects spatial structure without requiring labels or additional model generation. The claims are well supported by experiments across multiple benchmarks and architectures. Both the strengths (lightweight, scalable, effective across stages) are presented in a balanced manner, matching the scope and results of the paper. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: We discuss the SiTe limitation in Appendix A Limitation. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. 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Peer prediction mechanisms incentivize truthful reporting without ground truth verification by scoring agents based on correlations with peers. Traditional mechanisms, which ensure that truth-telling maximizes the expected scores in equilibrium, can elicit honest information while assuming agents' utilities are linear functions of their scores. However, in practice, non-linear payment rules are usually preferred, or agents' utilities are inherently non-linear. + +We propose stochastically dominant truthfulness (SD-truthfulness) as a stronger guarantee: the score distribution of truth-telling stochastically dominates all other strategies, incentivizing truthful reporting for a wide range of monotone utility functions. Our first observation is that no existing peer prediction mechanism naturally satisfies this criterion without strong assumptions. A simple solution—rounding scores into binary lotteries—can enforce SD-truthfulness, but often degrades sensitivity, a key property related to fairness and statistical efficiency. We demonstrate how a more careful application of rounding can better preserve sensitivity. Furthermore, we introduce a new enforced agreement (EA) mechanism that is theoretically guaranteed to be SD-truthful in binary-signal settings and, under mild assumptions, empirically achieves the highest sensitivity among all known SD-truthful mechanisms. + +# 1 Introduction + +Information elicitation studies how to design mechanisms that incentivize honest and high-effort human feedback. In this framework, a principal seeks reliable responses regarding a set of tasks from strategic agents who are motivated by the designed rewards. For example, AI companies collect human preferences to align language models via RLHF [Ouyang et al., 2022], and MOOCs use peer grading to evaluate complex assignments at scale Piech et al. [2013]. A key challenge is designing scoring mechanisms that accurately assess the informativeness of agents' reports. + +Peer prediction provides an elegant solution for settings where ground truth is unavailable or costly to obtain, such as in subjective tasks [Miller et al., 2005]. Instead of relying on external validation, peer prediction scores agents based on the correlation between their reports and those of their peers. The scoring mechanism is designed to be truthful so that everyone reporting truthfully forms a Bayesian Nash equilibrium, ensuring that truth-telling maximizes each agent's expected score. + +A key assumption underlying the effectiveness of existing peer prediction mechanisms is that agents' utility is a linear function of their score. This assumption is reasonable in some cases, such as when agents are solely motivated by monetary payments, which can be straightforwardly designed as a + +linear transformation of the peer prediction scores. However, practical implementations often favor non-linear transformations, such as tournaments, to map scores to payments due to their improved budget efficiency and flexibility [Zhang and Schoenebeck, 2023a]. Moreover, agents' intrinsic utility functions are sometimes inherently non-linear. For instance, student graders care more about their final letter grades (e.g., A, B, C) than the real-valued numerical scores assigned by the mechanism. When agents' utilities are non-linear functions of their scores, truth-telling is no longer guaranteed to be an equilibrium strategy. + +Motivated by this limitation, we introduce the concept of stochastic dominance-truthful (SD-truthful) peer prediction mechanisms. This stronger notion of truthfulness requires that an agent's score distribution, when everyone reports truthfully, first-order stochastically dominates the score distribution under any unilateral deviation. As a result, SD-truthful mechanisms incentivize truth-telling for agents with any utility function that is monotone increasing with the score. + +# 1.1 Our Results + +Our first observation is that no existing peer prediction mechanism is naturally SD-truthful under general information structures, i.e., under arbitrary correlations between agents' signals. We examine three widely studied mechanisms that compute scores as the average of multiple simple discrete scores, including Output Agreement (OA), Peer Truth Serum (PTS) [Faltings et al., 2017], and Correlated Agreement (CA) [Shnayder et al., 2016]. Mechanisms with complex scoring rules—such as those based on continuous-valued estimates of mutual information Kong and Schoenebeck [2019], Kong [2020], Schoenebeck and Yu [2020]—are unlikely to satisfy SD-truthfulness. Yet even these simple mechanisms fail to guarantee SD-truthfulness without strong assumptions on the underlying information structure. + +Our second observation is that achieving SD-truthfulness is easy in principle but challenging to do well in practice. Given any truthful mechanism $\mathcal{M}$ with bounded scores $S^{\mathcal{M}} \in [S_{\inf}, S_{\sup}]$ , we can define a probability by normalizing the score: $\lambda^{\mathcal{M}} = \frac{S^{\mathcal{M}} - S_{\inf}}{S_{\sup} - S_{\inf}} \in [0,1]$ . We then score agents via a binary lottery: $\tilde{S}^{\mathcal{M}} = 1$ with probability $\lambda^{\mathcal{M}}$ and 0 otherwise. Although this approach, called the direct rounding reduction, trivially satisfies SD-truthfulness, it greatly reduces sensitivity—an important property related to the efficiency and ex-post fairness of a mechanism. At a high level, sensitivity measures how much a mechanism's score responds to changes in the information contained in an agent's reports, normalized by the standard deviation of the scores. + +To overcome this limitation, we propose the partition rounding reduction, which helps partially recover the sensitivity of the original mechanism. The idea is to divide the questions into $K$ disjoint subsets and apply the original mechanism separately within each subset to compute an individual score. The final score for an agent is then the average of $K$ i.i.d. individual scores after applying direct rounding. Intuitively, this approach improves sensitivity by reducing variance through the aggregation of multiple independent scores. + +However, the partition rounding process cannot fully recover the sensitivity of the original mechanisms, implying additional room for more sensitive peer prediction mechanisms. We introduce a novel mechanism called enforced agreement (EA) that is shown to be the most sensitive SD-truthful mechanism in the binary-signal setting. EA aims to replicate the key advantage of OA—its high sensitivity—while relaxing the strong assumptions on the information structure required for SD-truthfulness. + +EA works by enforcing a pre-determined empirical distribution of signals. In the binary setting, if Alice reports 0 on $m_0 > n_0$ questions (where $n_0$ is the enforced count), the mechanism randomly flips $m_0 - n_0$ of them to 1. This trick removes agents' incentive to over-report majority signals, a strategy that undermines the truthful properties of OA. However, EA is only truthful but not SD-truthful when signals are non-binary, highlighting an important future challenge in designing high-sensitivity SD-truthful mechanisms for more complex settings. + +Lastly, our empirical results reinforce and complement our theoretical insights. In the binary-signal setting, we show that even a prior-free implementation of EA—simply enforcing a uniform distribution—outperforms other SD-truthful mechanisms in most settings, including those that require prior knowledge. For non-binary-signal settings, the partition rounding reduction of PTS emerges as the most sensitive SD-truthful mechanism. As a robustness check, we validate our findings by + +assessing budget efficiency, another practical metric of peer prediction mechanisms that quantifies the minimum budget required to elicit a desired level of effort [Zhang and Schoenebeck, 2023a]. + +Table 1 summarizes a comparison of SD-truthful (implementations of) peer prediction mechanisms with respect to the assumptions they require and their sensitivities. + +
MechanismsRequirements for SD-truthfulness
Output Agreement (OA)Self-dominating signal
Enforced Agreement (EA) [This paper]Binary and self-predicting signal
Peer Truth Serum (PTS) [Faltings et al., 2017]Self-predicting signal
Correlated Agreement (CA)[Shnayder et al., 2016]Any signal
Matching Agreement (MA) [Zhang and Schoenebeck, 2023b]Any signal
+ +Table 1: A Comparison of Discussed Peer Prediction Mechanisms ordered by sensitivity (from high to low), where PTS, CA, and MA require the partition rounding reduction to achieve SD-truthfulness. + +# 2 Related Work + +The idea of peer prediction was proposed by Miller et al. [2005]. As the first discussion, their mechanism is not detail-free, i.e. the mechanism requires prior knowledge of the information structure (how agents' signals are correlated) to obtain truthfulness. There are two ideas to mitigate this limitation: (i) the single-task mechanism, which elicits the information structure directly from agents [Prelec, 2004]; and (ii) the multi-task mechanism, which assumes each agent responds to multiple i.i.d. tasks but only requires eliciting signals rather than predictions [Dasgupta and Ghosh, 2013]. + +Our study is primarily relevant to the multi-task setting. We categorize multi-task peer prediction mechanisms into two types: peer agreement mechanisms Dasgupta and Ghosh [2013], Faltings et al. [2017], Shnayder et al. [2016], Zhang and Schoenebeck [2023b], Agarwal et al. [2017] and mutual information mechanisms Kong and Schoenebeck [2019], Kong [2020], Schoenebeck and Yu [2020], Schoenebeck et al. [2021], Kong [2024]. In this paper, we primarily focus on peer agreement mechanisms, as SD-truthfulness requires the mechanism's score to take a relatively simple form. Intuitively, this is because a complex scoring mechanism provides more opportunities for a cheating strategy to obtain one of the highest scores with a larger probability than truth-telling, which breaks SD-truthfulness. In Appendix A, we provide a more detailed discussion on why several classic mutual information mechanisms are not SD-truthful. + +The score of a peer agreement mechanism can be viewed as an average of multiple individual scores, where each individual score takes values from a small finite space, e.g. $\{0,1\}$ . For example, output agreement (OA) scores an agent based on the average number of questions that they agree with another agent. The first multi-task peer prediction mechanism lies in this category, which can achieve truthfulness in a binary-signal setting [Dasgupta and Ghosh, 2013]. The correlated agreement (CA) mechanism generalizes their idea to any finite-signal setting [Shnayder et al., 2016]. The peer truth serum (PTS) mechanism is a generalization of OA, which relaxes the self-dominating assumption to self-prediction. In addition, two follow-ups of CA aim to handle heterogeneous agents [Agarwal et al., 2017] and more general cheating strategies [Zhang and Schoenebeck, 2023b]. + +Moreover, we note that the idea of rounding peer prediction scores into a binary lottery (see Section 5) was previously proposed by Miller et al. [2005] to address risk-averse agents. What is new in our work is the comparative analysis of various rounding reductions based on sensitivity, allowing us to identify a more effective rounding approach. Furthermore, we formally prove that this idea extends beyond risk-averse agents and can be applied to any agents with increasing utility functions. + +# 3 Model + +We consider the multi-task peer prediction setting where two agents, Alice and Bob, answer the same $n$ i.i.d. questions. For each question $j$ , the agents receive discrete signals $X_{a,j}, X_{b,j} \in \Sigma = \{0, \dots, c - 1\}$ , drawn from a fixed joint distribution $\operatorname*{Pr}(X_a, X_b)$ . In cases with more than two agents, + +we can reduce to the case of two agents by, for any particular agent "Alice", choosing a random peer agent "Bob" for scoring. + +Each agent $i \in \{a, b\}$ applies a strategy $\theta_i \in \Theta$ , which maps from signals to distributions over the same space $\Delta_{\Sigma}$ . The report on question $j$ is denoted as $\hat{X}_{i,j} = \theta_i(X_{i,j})$ . We highlight two types of strategies: the truth-telling strategy $\tau$ , which always reports the received signal, and the uninformative strategy $\mu$ , where reports are independent of the input. Examples of uninformative strategies include always reporting the same signal or randomly reporting signals according to the prior. + +A peer prediction mechanism $\mathcal{M}$ maps vectors of reports $\hat{X}_a$ and $\hat{X}_b$ to a score for Alice. Given an information structure (i.e., the joint distribution over signals) and a mechanism, the score is a random function of strategies, denoted as $S^{\mathcal{M}}(\theta_a,\theta_b)$ . The randomness arises from the signals, strategies, and the mechanism itself. When it is clear from context, we omit the superscript $\mathcal{M}$ . + +# 3.1 Truthful Guarantees + +Existing peer prediction mechanisms focus on expected scores, guaranteeing that any deviation from truth-telling will only reduce the expected score. In this sense, truth-telling forms a Bayesian Nash Equilibrium. Since our analysis centers on equilibrium behavior, we assume Bob always tells the truth. Under this assumption, the score assigned by the mechanism reduces to a function of Alice's strategy $\theta$ , written as $S^{\mathcal{M}}(\theta)$ . + +Definition 3.1. A peer prediction mechanism is truthful if $\mathbb{E}[S(\tau)]\geq \mathbb{E}[S(\theta)]$ for any $\theta \in \Theta$ + +A truthful mechanism guarantees that if agents have linear utility in their scores, truth-telling is a Bayesian Nash Equilibrium—any deviation yields lower expected utility. However, as we argued earlier, assuming linear utility is often undesirable and, in some cases, unrealistic. Instead, we aim to design truthful mechanisms for a broader and more reasonable class of utility functions—those that are simply increasing in score. This motivation naturally leads to the concept of stochastic dominance. + +Definition 3.2. A peer prediciton mechanism is stochastic dominance-truthful (SD-truthful) if the distribution of $S(\tau)$ first-order stochastic dominates (FOSD) the distribution of $S(\theta)$ for any $\theta \in \Theta$ , i.e. $\operatorname{Pr}(S(\tau) \geq t) \geq \operatorname{Pr}(S(\theta) \geq t)$ for any $t \in \mathbb{R}$ . + +A standard result in microeconomic theory suggests that $S(\tau)$ FOSD $S(\theta)$ if and only if $\mathbb{E}[u(S(\tau))] \geq \mathbb{E}[u(S(\theta))]$ for every increasing utility function $u$ [Hadar and Russell, 1969]. Since linear utility functions are also increasing, every SD-truthful mechanism is also truthful. + +# 3.2 Sensitivity + +In practice, an effective scoring mechanism should meaningfully evaluate the information contained in agents' reports and reward more informative reports with higher scores. This motivates an additional design criterion, orthogonal to truthfulness, known as sensitivity [Zhang and Schoenebeck, 2023a]. + +To motivate the concept of sensitivity, we extend the model to incorporate agents' effort levels. This extension captures how the mechanism responds to marginal changes in information at equilibrium, i.e. how the score varies when all other agents exert full effort while one agent adjusts her effort. In particular, on each question, we assume Alice exerts effort with probability $e \in [0,1]$ , while Bob always exerts effort and reports truthfully. When Alice exerts effort, the signal pair $(X_{a}, X_{b})$ is drawn from the joint distribution $\operatorname{Pr}(X_{a}, X_{b})$ ; in the extreme case where Alice exerts no effort, the signals are independently drawn from the marginals $\operatorname{Pr}(X_{a}) \operatorname{Pr}(X_{b})$ . That is, exerting effort produces a signal $X_{a}$ that is correlated with $X_{b}$ according to the joint distribution, while shirking yields an uninformative guess drawn from the prior $\operatorname{Pr}(X_{a})$ . After observing the signal, Alice will report truthfully. Then, Alice's score under mechanism $\mathcal{M}$ becomes a function of $e$ , denoted as $S^{\mathcal{M}}(e)$ . + +Definition 3.3. The sensitivity of a peer prediction mechanism $\mathcal{M}$ at effort level $e$ is $\delta^{\mathcal{M}}(e)\coloneqq \frac{\nabla\mathbb{E}[S^{\mathcal{M}}(e)]}{\mathrm{std}(S^{\mathcal{M}}(e))}$ where $\nabla$ denotes the derivative operator and std denotes the standard deviation. + +Sensitivity measures how well a mechanism's expected score responds to changes in effort—and thus, changes in information. Consequently, a trivial mechanism that always assigns a score of 1, though truthful, has zero sensitivity, as its output is unaffected by effort. + +Prior work [Zhang and Schoenebeck, 2023a] shows that when the score is normally distributed and agents are rewarded based on the ranking of scores, sensitivity is a sufficient statistic for budgetary efficiency: mechanisms with higher sensitivity can elicit the same equilibrium effort level at a lower budgetary cost. Moreover, Xu et al. [2024] establish a one-to-one correspondence between sensitivity and measurement integrity [Burrell and Schoenebeck, 2023], implying that mechanisms with higher sensitivity generate scores more closely aligned with the true quality of responses. Together, these findings highlight sensitivity as a practically meaningful and theoretically grounded property of peer prediction mechanisms, motivating our choice to use it as an additional dimension for comparison. + +Our goal is to design SD-truthful mechanisms that also achieve high sensitivity. + +# 4 Prior Mechanisms + +To the best of our knowledge, no existing peer prediction mechanism naturally achieves SD-truthfulness without imposing strong assumptions on the information structure. In this section, we introduce three exemplary multi-task peer prediction mechanisms and explain why they are not naturally SD-truthful to motivate our methods. + +# 4.1 Output Agreement (OA) + +The output agreement (OA) mechanism scores Alice based on the fraction of questions on which both agents agree. The final score of OA is the average of $n$ independent binary individual scores, i.e. $S^{OA} = \frac{1}{n}\sum_{i\in [n]}S_i^{OA}$ where $S_{i}^{OA} = 1$ if both agents' reports agree on question $i$ and 0 otherwise. However, OA encourages over-reporting of the majority signal, which is truthful only under a strong assumption. + +Definition 4.1. Agents' signals are self-dominating if $\operatorname*{Pr}(X_b = \sigma |X_a = \sigma) > \operatorname*{Pr}(X_b = \sigma '|X_a = \sigma)$ for any $\sigma^{\prime}\neq \sigma \in \Sigma$ + +Proposition 4.2. OA is truthful if agents' signals are self-dominating. + +In fact, OA is also SD-truthful when signals are self-dominating. This is because each individual score $S_{i}^{OA}$ takes only two values, meaning that $S_{i}^{OA}(\tau)$ FOSD $S_{i}^{OA}(\theta)$ if and only if $\operatorname{Pr}(S_{i}^{OA}(\tau) = 1) \geq \operatorname{Pr}(S_{i}^{OA}(\theta) = 1)$ . Furthermore, the average of multiple i.i.d. scores that are each SD-truthful remains SD-truthful (see Section 5.2). Consequently, for OA, SD-truthfulness reduces to truthfulness. + +# 4.2 Peer Truth Serum (PTS) + +Self-dominance is usually considered a strong assumption. [Radanovic et al., 2016] propose the Peer Truth Serum (PTS) which incentivizes truth-telling under a weaker assumption. + +Definition 4.3. Agents' signals are self-predicting if $\operatorname*{Pr}(X_b = \sigma | X_a = \sigma) > \operatorname*{Pr}(X_b = \sigma | X_a = \sigma')$ for any $\sigma' \neq \sigma \in \Sigma$ . + +PTS assumes a symmetric, publicly known prior $R(\sigma) = \operatorname*{Pr}(X_a = \sigma) = \operatorname*{Pr}(X_b = \sigma)$ . It repeatedly samples questions without replacement, and assigns an individual score $S_i^{PTS} = 1 / R(\sigma)$ to Alice if $\hat{X}_{a,i} = \hat{X}_{b,i} = \sigma$ , and 0 otherwise. The final score of PTS is the average of all $S_i^{PTS}$ . + +Proposition 4.4 ([Faltings et al., 2017]). PTS is truthful if agents' signals are self-predicting. + +Is PTS SD-truthful? The short answer is no. Each individual score $S_{i}^{PTS}$ takes at most $|\Sigma| + 1$ values, with the maximum being $S_{\max} = 1 / R(\sigma_{\min})$ , where $\sigma_{\min}$ is the least likely signal under the prior. A necessary condition for PTS to be SD-truthful is that truth-telling maximizes $\operatorname{Pr}\left(S_{i}^{PTS}(\tau) = S_{max}\right)$ , which is the probability of Alice agreeing with Bob on signal $\sigma_{\min}$ . While truth-telling, this probability is the joint distribution $\operatorname{Pr}(X_a = \sigma_{\min}, X_b = \sigma_{\min})$ . However, if Alice uses an uninformative strategy $\mu$ that always reports $\sigma_{\min}$ , this probability increases to $\operatorname{Pr}(X_b = \sigma_{\min})$ . Therefore, although $\mu$ decreases the expected score, its distribution is not stochastically dominated by truth-telling, meaning that there exists an increasing utility function under which Alice strictly prefers $\mu$ to $\tau$ . + +# 4.3 Correlated Agreement (CA) + +The correlated agreement (CA) mechanism [Shnayder et al., 2016] takes two questions to compute an individual score which has three possible values: -1, 0, and 1. Suppose the joint distribution $\operatorname{Pr}(X_a, X_b)$ is known. The CA mechanism first computes the delta matrix $\Delta_{\sigma, \sigma'} = \operatorname{Pr}(X_a = \sigma, X_b = \sigma') - \operatorname{Pr}(X_a = \sigma) \operatorname{Pr}(X_b = \sigma')$ , which is the difference between the joint distribution and the product of marginal distributions. Let $T_{\Delta}(\sigma, \sigma') = \operatorname{Sign}(\Delta)_{\sigma, \sigma'}$ be the agreement function where $\operatorname{Sign}(x) = 1$ if $x > 0$ and 0 otherwise. Intuitively, $T_{\Delta}(\sigma, \sigma') = 1$ if the pair of signals $(\sigma, \sigma')$ appears more often on the same question than on distinct questions. + +Then, CA repeatedly samples a bonus question $j$ and a penalty question $k$ . For an individual score $S_{i}^{CA}$ , Alice gets 1 if both agents agree on $j$ and gets $-1$ if her response on $j$ agrees with Bob's response on $k$ , i.e. $S_{i}^{CA} = T_{\Delta}(\hat{X}_{a,j}, \hat{X}_{b,j}) - T_{\Delta}(\hat{X}_{a,j}, \hat{X}_{b,k})$ . Since each bonus question can be paired with $n - 1$ penalties, the final score is the average over $n(n - 1)$ individual scores. + +Intuitively, the CA mechanism works by encouraging correlations on the same question and penalizing correlations on distinct questions. Consequently, if Alice always reports the majority signal, the frequency of agreements on the bonus question will be identical to the frequency of agreements on the penalty questions, which results in a score of 0. Prior work shows that when $\operatorname{Pr}(X_a, X_b)$ is known (or accurately estimated), CA is truthful—and in fact, informed truthful [Shnayder et al., 2016]. + +Is CA SD-truthful? The answer depends on the implementation. The following example illustrates that the original implementation of CA described above is not SD-truthful. + +Example 4.1. Suppose $n = 2$ , so that Alice knows each of the two questions she answers will be used once as the bonus question and once as the penalty question. Her final score is the average of two individual scores, each with $S_{i}^{CA} \in \{-1,0,1\}$ . Thus, the final score has five possible values: $S^{CA} \in \{-1, -0.5,0,0.5,1\}$ . A necessary condition for SD-truthfulness is that $\operatorname{Pr}(S^{CA}(\tau) = -1) \leq \operatorname{Pr}(S^{CA}(\theta) = -1)$ for any strategy $\theta$ . However, we show that this does not hold in general. + +Suppose the signals are positively correlated, i.e. $T_{\Delta}$ is a diagonal matrix. Alice receives a score of -1 if and only if she reports different signals on the two questions and Bob disagrees on both. Under truth-telling, this happens with probability $\operatorname{Pr}(S^{CA}(\tau) = -1) = 2\operatorname{Pr}(X_a = 0, X_b = 1) \operatorname{Pr}(X_a = 1, X_b = 0)$ . However, if Alice always reports the same signal (e.g., $\mu(\sigma) = 0$ for any $\sigma$ ), she can avoid this outcome entirely. Therefore, $\mu$ is not stochastically dominated by $\tau$ . + +This warns that repeatedly using the same question as the bonus or the penalty question will sabotage the SD-truthfulness of CA. + +# 5 A Rounding Reduction For Stochastically Dominant-Truthfulness + +We introduce a straightforward rounding reduction that maps any truthful mechanism to an SD-truthful one. However, this approach often comes at a huge cost of sensitivity. To address this, we propose a more refined rounding method that partially restores the original mechanism's sensitivity. + +# 5.1 Direct Rounding + +Consider a peer prediction mechanism $\mathcal{M}$ that assigns Alice a score $S^{\mathcal{M}}$ with bounded support $S_{\mathrm{sup}}^{\mathcal{M}} < \infty$ and $S_{\mathrm{inf}}^{\mathcal{M}} > -\infty$ . In direct rounding, we normalize the score to a probability: $\lambda^{\mathcal{M}} = \frac{S^{\mathcal{M}} - S_{\mathrm{inf}}^{\mathcal{M}}}{S_{\mathrm{sup}}^{\mathcal{M}} - S_{\mathrm{inf}}^{\mathcal{M}}} \in [0,1]$ . Alice's final score $\tilde{S}^{\mathcal{M}}$ is then determined by a binary lottery: receiving 1 with probability $\lambda^{\mathcal{M}}$ and 0 otherwise. + +Proposition 5.1. The direct rounding reduction of a truthful mechanism with bounded scores is SD-truthful. + +Intuitively, because direct rounding always outputs a Bernoulli score, SD-truthfulness simply implies that truth-telling should maximize the expected success probability. But the expected success probability is just the success probability. By our design of $\lambda^{\mathcal{M}}$ , the success probability is a linear function of the final score $S^{\mathcal{M}}$ , which proves equivalence between SD-truthfulness and truthfulness. + +However, scoring agents via a single binary score is clearly not practical. The following proposition suggests that direct rounding usually destroys the sensitivity of the original mechanism. Let $\delta^{\mathcal{M}}(e)$ + +and $\tilde{\delta}^{\mathcal{M}}(e)$ be the sensitivity of $\mathcal{M}$ and its direct rounding reduction. Additionally, let $m^{\mathcal{M}}(e)$ and $\mathrm{std}^{\mathcal{M}}(e)$ be the expected score and the standard deviation. + +Proposition 5.2. The sensitivity ratio between a mechanism and its direct rounding is: + +$$ +\frac {\delta^ {\mathcal {M}} (e)}{\tilde {\delta} ^ {\mathcal {M}} (e)} = \frac {\sqrt {\left(S _ {\mathrm {s u p}} ^ {\mathcal {M}} - m ^ {\mathcal {M}} (e)\right) \left(m ^ {\mathcal {M}} (e) - S _ {\mathrm {i n f}} ^ {\mathcal {M}}\right)}}{s t d ^ {\mathcal {M}} (e)}. +$$ + +This ratio is typically large. For example, the final score of OA is an averaged Binomial $\mathrm{Bin}(n,m^{OA}(e))$ , whose standard deviation is $\sqrt{m^{OA}(e)(1 - m^{OA}(e)) / n}$ . Thus, the sensitivity ratio between OA and its direct rounding is $\sqrt{n}$ . This illustrates a key drawback of direct rounding: it greatly reduces the sensitivity by eliminating the benefit of averaging over multiple questions. + +# 5.2 Partition Rounding + +To reduce variance of and improve sensitivity, we aim to better leverage the information from all $n$ questions. In particular, if the final score of $S^{\mathcal{M}}$ is the average of $K$ independent individual scores $S_{i}^{\mathcal{M}}$ , we can apply direct rounding to each $S_{i}^{\mathcal{M}}$ and take the average. This motivates the idea of partition rounding. We first partition the $n$ questions into $K$ disjoint subsets, using each to compute an individual score. Each score is then directly rounded as in Section 5.1, and the final score $\hat{S}^{\mathcal{M}}$ is the average of the $K$ rounded scores. The following lemma proves the feasibility of this idea. + +Lemma 5.3. Let $S = \frac{1}{K} \sum_{i \in [K]} S_i$ , where the individual scores $S_i$ are i.i.d.. Then, $S(\tau)$ first-order stochastically dominates $S(\theta)$ if and only if each individual score $S_i(\tau)$ first-order stochastically dominates $S_i(\theta)$ . + +Proposition 5.4. The partition rounding reduction of a truthful mechanism with bounded scores is SD-truthful. + +We defer the proof of Lemma 5.3, which follows from a straightforward coupling argument, to Appendix F.3. Proposition 5.4 then follows directly from this lemma and the SD-truthfulness of direct rounding. + +We now show that the partition rounding reduction can partially recover the sensitivity of the original mechanism. The key observation is that the final score under partition rounding is the average of a binomial variable, i.e. $\hat{S}^{\mathcal{M}}(e) = \frac{1}{K}\mathrm{Bin}(K,\lambda^{\mathcal{M}}(e))$ , where $\lambda^{\mathcal{M}}(e) = \frac{m^{\mathcal{M}} - S_{\mathrm{inf}}^{\mathcal{M}}}{S_{\mathrm{sup}}^{\mathcal{M}} - S_{\mathrm{inf}}^{\mathcal{M}}}$ with $m^{\mathcal{M}}$ being the expectation of the individual score. Putting this into Definition 3.3 gives us the sensitivity of the partition rounding reduction + +$$ +\hat {\delta} ^ {\mathcal {M}} (e) = \frac {\nabla m ^ {\mathcal {M}} (e) \cdot \sqrt {K}}{\sqrt {\left(m ^ {\mathcal {M}} (e) - S _ {\inf } ^ {\mathcal {M}}\right) \left(S _ {\sup } ^ {\mathcal {M}} - m ^ {\mathcal {M}} (e)\right)}}. \tag {1} +$$ + +The sensitivity of the partition rounding reduction is exactly $\sqrt{K}$ times that of the direct rounding reduction. This means that by using partition rounding, we can recover a substantial portion of the original mechanism's sensitivity. + +In Appendix B, we present the sensitivities of the partition-rounded versions of the three mechanisms discussed. We summarize the limitations of these mechanisms below which motivates the design of a new SD-truthful mechanism introduced in the next section. + +- OA is already SD-truthful without rounding and thus has a high sensitivity. However, OA is SD-truthful only for self-dominating signals. +- PTS requires rounding to ensure SD-truthfulness. Therefore, when the prior of signals is biased, $S_{\mathrm{sup}}^{PTS}$ becomes large, which in turn leads to a low sensitivity. +- CA uses two questions to compute an individual score, so $K = n / 2$ . Furthermore, only bonus questions contribute to sensitivity, while penalty questions are used to enforce truthfulness. This dilutes the sensitivity by half again. Therefore, only $1 / 4$ of the questions count for the sensitivity of the partition-rounded CA. Hence, even without rounding, the sensitivity of the partition-rounded CA is only half of that of the original implementation. + +# 6 The Enforced Agreement Mechanism + +This section aims to present a novel mechanism with high sensitivity, which does not require the above rounding reduction. To motivate our idea, we begin by summarizing the key lessons learned from the previous discussions. + +First, Example 4.1 shows a particular type of strategy that prevents previous mechanisms from being SD-truthful: reducing/changing the variance of scores, e.g. by always reporting the same signal. This inspires our enforced agreement (EA) mechanism, where the idea is to control a randomness level within agents' reports so that strategic manipulations cannot arbitrarily alter the variance of the scores. + +Second, our analysis of sensitivity in Section 5.2 suggests that a high-sensitivity mechanism requires a larger number of partitions $K$ and a smaller score range $S_{\mathrm{sup}} - S_{\mathrm{inf}}$ . This insight guides us to use just one question to compute an individual score (so $K = n$ ) and avoid the need for rounding. We show that EA meets these criteria. + +We first prove the SD-truthfulness of EA in the binary-signal setting, and then establish a negative result explaining why it fails to ensure SD-truthfulness when signals are non-binary. Nonetheless, we show that EA remains truthful in non-binary settings when the signal structure is known or can be accurately learned. + +# 6.1 EA in the Binary-signal Setting + +We introduce EA in the binary-signal setting and defer the discussion for general settings to the appendix. The idea of EA is to control the marginal distribution of Alice's reports. In the binary setting, the mechanism commits to a target empirical distribution $\Phi = (n_0, n_1)$ where $n_0 + n_1 = n$ . The mechanism expects Alice to report signal $i$ on exactly $n_i$ out of $n$ questions. If Alice's actual distribution $\Phi_{X_a} = (m_0, m_1)$ and suppose W.L.O.G. that $m_0 > n_0$ , the mechanism randomly selects $m_0 - n_0$ of her 0-reports and flips them to 1. After enforcing $\Phi$ , Alice is scored using the output agreement mechanism (OA). + +Compared with OA, the enforcement process in EA prevents Alice from benefiting by over-reporting the majority signal. However, this enforcement introduces dependencies across questions, so the binary agreement scores are no longer i.i.d., and Lemma 5.3 does not apply. Our main result shows that EA is SD-truthful when signals are self-prediction (Definition 4.3). + +Theorem 6.1. If $|\Sigma| = 2$ and signals are self-predicting, the enforced agreement mechanism is SD-truthful for any $\Phi$ . + +We defer the detailed proof to the appendix and present the main ideas below. Note that the final score of the enforced agreement mechanism is the average of $n$ Bernoulli variables, each with a potentially different success probability. In the binary-signal setting, there are four possible success probabilities in total— $p_{ij} = \operatorname*{Pr}(X_b = j \mid X_a = i)$ for $i, j \in \{0,1\}$ —representing the probability of Bob observing (and reporting) $j$ on a question conditioned on Alice observing $i$ . A Bernoulli variable with success probability $p_{ij}$ corresponds to a question on which Alice observes $i$ but is flipped to $j$ either by the mechanism or by Alice herself. Suppose W.L.O.G. that $m_0 > n_0$ , meaning that the mechanism will randomly flip $m_0 - n_0$ reports of Alice from 0 to 1 under truth-telling. In this case, the final score of Alice is the average of three binomials: + +$$ +n \cdot S ^ {E A} (\tau) \sim \operatorname {B i n} \left(n _ {0}, p _ {0 0}\right) + \operatorname {B i n} \left(m _ {1}, p _ {1 1}\right) + \operatorname {B i n} \left(m _ {0} - n _ {0}, p _ {0 1}\right). \tag {2} +$$ + +We show that for any untruthful strategy, the resulting final score can be written as: + +$$ +n \cdot S ^ {E A} (\theta) \sim \operatorname {B i n} (n _ {0} - k, p _ {0 0}) + \operatorname {B i n} (m _ {1} - k, p _ {1 1}) + \operatorname {B i n} (m _ {0} - n _ {0} + k, p _ {0 1}) + \operatorname {B i n} (k, p _ {1 0}), +$$ + +where $0 \leq k \leq \min(n_0, m_1)$ . Under the self-prediction condition, $p_{00} > p_{10}$ and $p_{11} > p_{01}$ . Therefore, any untruthful strategy can only reallocate $k$ samples from two binomial distributions with higher success probabilities to those with smaller success probabilities. Via a simple coupling argument, we can see that $S(\theta)$ is first-ordered stochastically dominated by $S(\tau)$ . + +Additional results are provided in Appendix C. First, we show that EA is not generally SD-truthful in the non-binary setting. Strategic signal permutations can produce score distributions that are not stochastically dominated by truth-telling. However, EA remains truthful when the information structure is known or well estimated. Moreover, we derive a closed-form expression for the sensitivity-maximizing $\Phi$ , enabling computation of the optimal enforcement from the information structure. + +# 7 Empirical Evaluations of SD-Truthful Mechanisms + +In this section, we empirically compare the sensitivity of the proposed mechanisms across various settings. The results support our theoretical findings: EA consistently exhibits the highest sensitivity among SD-truthful mechanisms in the binary-signal setting (except when the signal prior is nearly uniform). We further evaluate each mechanism's budgetary efficiency in an effort-elicitation scenario—measuring the minimum payment required to incentivize a target effort level. Again, EA proves its superiority, which has the best performance against any other SD-truthful and non-SD-truthful mechanisms. + +# 7.1 Datasets and Experiment Setup + +We use two real-world datasets to estimate the information structure between two agents. The first dataset contains binary labels classifying whether a compound is appropriate or inappropriate to be synthesized [Baba et al., 2018]. The second dataset collects the annotations of the sentiment of 300 tweets, where the size of the signal space is 4 [Venanzi et al., 2015]. We denote these two information structures as $J_{1}$ and $J_{2}$ respectively, and defer the details to Appendix E.1. + +In the main body of the paper, we focus on SD-truthful mechanisms, which include: + +- OA (Section 4.1), which is SD-truthful only when signals are self-dominating. +- PTS-partition-round—the partition rounding reduction of PTS (Section 4.2), where we re-weight every individual score using signal prior and take the average. +- CA-partition-round—the partition rounding reduction of CA (Section 4.3), where we create $K = n/2$ partitions of questions and score agents the average of the individual CA score for each partition after rounding. +- MA-partition-round—the partition rounding reduction of the matching agreement mechanism (Appendix D). +- EA-prior—EA with enforcement $\Phi = n \cdot \operatorname{Pr}(X_a)$ . +- EA-uniform—EA with $\Phi_i = n \cdot 1 / |\Sigma|$ , $\forall i \in \Sigma$ . +- EA-optimal—EA with the sensitivity-maximizing $\Phi$ computed according to Appendix C.3. + +For information structures with binary-signal settings, sensitivities can be computed analytically as shown in Appendix B. For general cases, we estimate sensitivity using a Monte Carlo approach. In particular, we simulate reports to each of the $n$ questions with both agents exerting full effort $e = 1$ and run $\mathcal{M}$ to compute the scores. Repeating this process for $T = 20,000$ times yields $T$ i.i.d. samples of $S^{\mathcal{M}}(e)$ . We then compute the score of Alice when she deviates to a lower effort level at $e - \Delta e = 0.8$ , and obtain $T$ samples of $S^{\mathcal{M}}(e - \Delta e)$ . The sensitivity of at effort $e$ is then estimated as $\frac{\mathbb{E}[S^{\mathcal{M}}(e)] - \mathbb{E}[S^{\mathcal{M}}(e - \Delta e)]}{\Delta e} \cdot \frac{1}{\mathrm{std}(S^{\mathcal{M}}(e))}$ . + +# 7.2 The Sensitivity of SD-Truthful Mechanisms + +We now compare the sensitivity of the discussed SD-truthful mechanisms. In Figure 1a, we show the sensitivity when the prior of the binary signal is varied, i.e. $\operatorname{Pr}[X_a = 0] \in [0.1, 0.9]$ , and the number of questions $n = 100$ . + +Figure 1b and 1c present sensitivity as the number of questions $n \in \{10,20,\dots ,100\}$ increases, under information structures $J_{1}$ and $J_{2}$ , respectively. In $J_{2}$ , where $|\Sigma | > 2$ , EA is no longer SD-truthful; however, we still include its sensitivity for comparison. + +We make the following observations. First, although the partition reductions of CA and MA are SD-truthful for general settings, their sensitivities are dominated by other SD-truthful mechanisms. This is due to their requirements of the penalty questions, which do not contribute to the sensitivity, and the fact that they require two questions to compute an individual score. + +Second, PTS performs well when the signal prior is nearly uniform, i.e. $\operatorname{Pr}(X_a = 0) \approx 0.5$ . Intuitively, this is because the rounding factor that enlarges the gap $S_{\mathrm{sup}}^{PTS} - S_{\mathrm{inf}}^{PTS}$ shrinks when the prior is close to uniform. However, even a slight bias, i.e. $|\operatorname{Pr}(X_a = 0) - 0.5| > 0.03$ causes PTS-partition-round to be outperformed by EA. + +![](images/9816694c6a30869ba07a69ead823f2097d9f1f0144b58233ec901e703cb3edd4.jpg) +(a) Varying prior $\operatorname*{Pr}[X_a = 0]$ + +![](images/29f64c84392eff28f8acead7a923141825f283e686d2a8167b65041cb885f587.jpg) +(b) Varying #tasks under $J_{1}$ + +![](images/a472cae64ee354f66f1181f75a910b475093b5c6ad78a91812225a7e9f4f5962.jpg) +(c) Varying #tasks under $J_{2}$ +Figure 1: A Comparison of the Sensitivity of Different Mechanisms. + +Third, we observe that enforcing the signal prior in EA closely approximates the optimal enforcement. Remarkably, even EA-uniform, a detail-free version that doesn't require knowledge of the prior, consistently outperforms PTS-partition-round, which does rely on prior knowledge for SD-truthfulness. + +Additionally, we note that the partition-rounding reduction of MA coincides with that of CA when signals are binary (see Appendix D). Consequently, the blue and red overlap in Figure 1a and 1b. + +Due to space constraints, we defer (i) the comparison between partition-rounding reductions and their original implementations to Appendix E.2, and (ii) the details of our effort-elicitation experiments and corresponding budgetary efficiency results to Appendix E.3. These results further show that high-sensitivity mechanisms perform strongly in practical, incentive-aligned environments, underscoring the importance of EA. + +# 8 Conclusion and Future Work + +This paper initiates the discussions of stochastically dominant-truthful (SD-truthful) peer prediction mechanisms, which ensure that in equilibrium, the score distribution under truth-telling first-order stochastically dominates any other strategy. Unlike traditional peer prediction mechanisms, which rely on linear utility assumptions, SD-truthful mechanisms can preserve truthfulness for any increasing utility function. We show that existing mechanisms fail to naturally achieve SD-truthfulness and propose a rounding method to enforce it. However, this process often reduces the sensitivity of the original mechanism, making it less efficient in practice. Additionally, we introduce the Enforcement Agreement (EA) mechanism, which is empirically shown to have the highest sensitivity among all SD-truthful mechanisms in the binary-signal setting—except when the signal prior is nearly uniform. + +Future work is needed to investigate more sensitive SD-truthful mechanisms beyond the binary-signal setting. A promising next step is to explore approximate SD-truthfulness, which allows truth-telling to lose a small utility for some utility functions. This relaxation could lead to mechanisms with better sensitivity and budget efficiency. Furthermore, SD-truthfulness can be extended to design manipulation-resistant evaluation metrics for AI benchmarking. In ranking-based settings such as Chatbot Arena [Chiang et al., 2024], where models are compared through pairwise judgments, an SD-truthful metric ensures that a model can improve its ranking only by improving its true expected performance, not by exploiting evaluation biases or strategic manipulation. + +# Broader Impact + +On the positive side, our method enhances the reliability and efficiency of crowdsourced data collection, which supports more trustworthy machine learning systems and reduces reliance on expert annotation. However, mechanisms that incentivize effort may unintentionally disadvantage contributors with limited time, resources, or familiarity with the task, raising fairness concerns. We aim to mitigate these risks by designing robust, broadly applicable mechanisms and promoting transparency in their deployment. + +# Acknowledgment + +This work was partly carried out at the DIMACS Center at Rutgers University and supported by NSF #2313137. + +# References + +Arpit Agarwal, Debmalya Mandal, David C Parkes, and Nisarg Shah. Peer prediction with heterogeneous users. In Proceedings of the 2017 ACM Conference on Economics and Computation, pages 81-98. ACM, June 2017. +Yukino Baba, Tetsu Isomura, and Hisashi Kashima. Wisdom of crowds for synthetic accessibility evaluation. Journal of Molecular Graphics and Modelling, 80:217-223, 2018. +Noah Burrell and Grant Schoenebeck. Measurement integrity in peer prediction: A peer assessment case study. In Proceedings of the 24th ACM Conference on Economics and Computation, EC '23, page 369-389, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400701047. doi: 10.1145/3580507.3597744. URL https://doi.org/10.1145/3580507.3597744. +Wei-Lin Chiang, Lianmin Zheng, Ying Sheng, Anastasios Nikolas Angelopoulos, Tianle Li, Dacheng Li, Hao Zhang, Banghua Zhu, Michael Jordan, Joseph E. Gonzalez, and Ion Stoica. Chatbot arena: An open platform for evaluating llms by human preference, 2024. URL https://arxiv.org/abs/2403.04132. +Anirban Dasgupta and Arpita Ghosh. Crowdsourced judgement elicitation with endogenous proficiency. In Proceedings of the 22nd international conference on World Wide Web, pages 319-330. ACM, 2013. +Alexander Philip Dawid and Allan M Skene. Maximum likelihood estimation of observer error-rates using the em algorithm. Journal of the Royal Statistical Society: Series C (Applied Statistics), 28 (1):20-28, 1979. +Boi Faltings, Radu Jurca, and Goran Radanovic. Peer truth serum: Incentives for crowdsourcing measurements and opinions, 2017. URL https://arxiv.org/abs/1704.05269. +Josef Hadar and William R. Russell. Rules for ordering uncertain prospects. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: LLMs were not used in the development of the core methods, experiments, or analysis in this research. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Continued Related Work: Mutual Information Mechanisms + +We review several classic peer prediction mechanisms that evaluate agents by estimating mutual information, and illustrate why these mechanisms fail to achieve SD-truthfulness. First, proposed by Kong and Schoenebeck [2019], we can estimate the empirical point-wise joint distribution between $\hat{X}_{a,\cdot}$ and $\hat{X}_{b,\cdot}$ , using pairs of reports from two agents, $(\hat{X}_{a,j},\hat{X}_{b,j})_{j\in [n]}$ . Note that because questions are i.i.d., we can use reports from all questions to estimate the same distribution. Then, plugging the learned empirical joint distribution into an $f$ -divergence operator returns a (biased) estimate of the $f$ point-wise mutual information. The following is the score for the $f$ -mutual information mechanism [Kong and Schoenebeck, 2019]: + +$$ +S ^ {f M I} = D _ {f} \left(\Pr \left(\hat {X} _ {a, \cdot}, \hat {X} _ {b, \cdot}\right) \| \Pr \left(\hat {X} _ {a, \cdot}\right) \Pr \left(\hat {X} _ {b, \cdot}\right)\right). +$$ + +However, because the estimation is biased, the $f$ -mutual information mechanism is only asymptotically truthful, meaning that it requires an infinite number of i.i.d. questions to satisfy Definition 3.1. When the number of tasks is small, there is a non-zero probability that truth-telling is not the best response. For example, if Alice observes "yes" on all her questions, which happens with a positive probability, then she knows that truth-telling will always result in a score of 0 regardless of Bob's responses. This is because the estimated point-wise joint distribution is the same as the product of marginal distributions. However, by misreporting "no" on some questions, she has a non-zero probability of receiving a positive score. In comparison, such deviations are discouraged under the enforced agreement (EA) mechanism because the mechanism will randomly flip the responses on behalf of the agent (see Section 6). + +Several follow-up works, such as the Determinant Mutual Information (DMI) mechanism by Kong [2020] and the pairing mechanism by Schoenebeck and Yu [2020], propose different ideas to estimate the mutual information. However, they are still not SD-truthful. Consider the strategy from Example 4.1, where Alice always reports "yes." Under both DMI and the pairing mechanism, this strategy always yields a score of zero. In contrast, similar to the CA mechanism, there is a positive probability that truth-telling may result in a negative score. Thus, this constant-reporting strategy is not dominated by truth-telling under these mutual information-based mechanisms whose score can take negative values. + +# B Sensitivity of Prior Mechanisms Under Partition Rounding + +We now map the exemplary mechanisms discussed in Section 4 into the partition rounding framework. In particular, we specify the key parameters $K$ , $S_{\mathrm{inf}}$ , $S_{\mathrm{sup}}$ , and $m_i(e)$ for each mechanism. Consider an information structure with joint distribution $J$ , and the marginal distribution of Alice and Bob being $M^a$ and $M^b$ , respectively. Furthermore, let $M_{\sigma, \sigma'} = M_\sigma^a M_{\sigma'}^b$ denote the product of marginal distributions. + +Output Agreement An individual score of OA, $S_{i}^{OA}$ , is 1 if both agents agree on a question $i$ and 0 otherwise. Therefore, OA is the partition rounding reduction of itself with $K = n$ , $S_{\mathrm{inf}} = 0$ , and $S_{\mathrm{sup}} = 1$ . By definition, when Alice exerts effort, agents' signals on the same question is sampled from the joint distribution $J$ ; while if Alice does not exert effort, she will sample a signal from the prior, meaning that agents' signals on the same question is sampled from $M$ . The expectation of an individual score at the effort level $e$ is thus + +$$ +m _ {i} ^ {O A} (e) = e \cdot \Pr (X _ {a, i} = X _ {b, i}) + (1 - e) \cdot \Pr (\mu (X _ {a, i}) = \mu (X _ {b, i})) = e \cdot \operatorname {t r} (J) + (1 - e) \cdot \operatorname {t r} (M). +$$ + +This means that the derivative of the expected score w.r.t. effort $e$ is + +$$ +\nabla m _ {i} ^ {O A} (e) = \operatorname {t r} (J - M) = \operatorname {t r} (\Delta), +$$ + +where $\Delta$ is the difference between the joint distribution and the product of marginal distributions, and $\operatorname{tr}(\cdot)$ represents the trace of a matrix, i.e. the sum of the diagonal entries. + +Peer Truth Serum Similar to OA, each partition of PTS is composed of a single question so that $K = n$ , $S_{\mathrm{inf}} = 0$ , and $S_{\mathrm{sup}} = \frac{1}{M_{\sigma_{\mathrm{min}}}^a}$ where $\sigma_{\mathrm{min}}$ is the signal occurs with the smallest probability in prior. The expected individual score is + +$$ +m _ {i} ^ {P T S} (e) = \sum_ {\sigma \in \Sigma} e \cdot \frac {J _ {\sigma , \sigma}}{M _ {\sigma} ^ {a}} + (1 - e) \cdot M _ {\sigma} ^ {b} = 1 + e \cdot \sum_ {\sigma \in \Sigma} \frac {\Delta_ {\sigma , \sigma}}{M _ {\sigma} ^ {a}}; \qquad \nabla m _ {i} ^ {P T S} (e) = \sum_ {\sigma \in \Sigma} \frac {\Delta_ {\sigma , \sigma}}{M _ {\sigma} ^ {a}}. +$$ + +Correlated Agreement The CA mechanism requires two questions to compute an individual score, meaning that $K = n/2$ . By Proposition F.1 and Lemma 5.3, when the signal space is binary, scoring Alice using $S^{CA} = \sum_{i \in [K]} S_i^{CA}$ without rounding is SD-truthful. However, when $|\Sigma| > 2$ , we have to use the partition rounding reduction of CA to obtain SD-truthful where $S_{\inf} = -1$ and $S_{\sup} = 1$ . On the bonus question, the probability of receiving a score of 1 is given by $\sum_{\sigma, \sigma'} (e \cdot J_{\sigma, \sigma'} + (1 - e) \cdot M_{\sigma, \sigma'}) T_{\Delta}(\sigma, \sigma')$ . On the penalty question, the probability of penalizing a score of 1 is given by $\sum_{\sigma, \sigma'} (e \cdot M_{\sigma, \sigma'} + (1 - e) \cdot M_{\sigma, \sigma'}) T_{\Delta}(\sigma, \sigma')$ . Combining these, + +$$ +m _ {i} ^ {C A} (e) = e \cdot \sum_ {\sigma , \sigma^ {\prime} \in \Sigma} \Delta_ {\sigma , \sigma^ {\prime}} T _ {\Delta} (\sigma , \sigma^ {\prime}) = e \cdot \sum_ {\sigma , \sigma^ {\prime} \in \Sigma} (\Delta_ {\sigma , \sigma^ {\prime}}) ^ {+}; \qquad \nabla m _ {i} ^ {C A} (e) = \sum_ {\sigma , \sigma^ {\prime} \in \Sigma} (\Delta_ {\sigma , \sigma^ {\prime}}) ^ {+}, +$$ + +where $(x)^{+} = x$ if $x > 0$ and 0 otherwise. Putting these parameters into Equation (1) returns the sensitivity of the partition rounding reduction for CA. However, note that by Proposition F.1, we do not need to round the individual CA scores when $|\Sigma| = 2$ . This motivates a variant of an SD-truthful implementation of CA that may have a slightly larger sensitivity in the binary-signal setting. We will introduce and test this implementation in Section 7. + +We summarize the limitations of the above three SD-truthful mechanisms. + +- OA does not require the partition rounding reduction to achieve SD-truthfulness and thus is likely to have a high sensitivity. However, OA is SD-truthful only when signals are self-dominating. +- PTS requires rounding to ensure SD-truthfulness. The normalizing factor $\frac{1}{M_{\sigma_{\mathrm{min}}}^a}$ implies that PTS will be less sensitive when the prior distribution of signals is biased. +- For the CA mechanism, only bonus questions contribute to the sensitivity of the mechanism, while penalty questions are used to enforce truthfulness. However, CA requires two questions to compute a single individual score, which means $K = n / 2$ .3 Therefore, even without rounding, e.g. when $|\Sigma| = 2$ , the number of effective questions is only one half of the available questions. Moreover, the rounding process divides each score by a factor of 2, which further harms the sensitivity of the mechanism. + +# C The Enforced Agreement Mechanism (Continued) + +# C.1 Arbitrary Signal Space + +Unfortunately, we show that the above idea does not generalize beyond the binary-signal setting. For a general setting with $|\Sigma| = c$ , the enforced agreement mechanism can be characterized by an + +enforced marginal distribution $\Phi = (n_0, \dots, n_{c-1})$ and an enforcement rule that ensures an agent's reports to follow $\Phi$ . + +Let Alice's report vector $\hat{X}_a$ have an empirical marginal distribution $\Phi_{\hat{X}_a} = (\hat{m}_0, \dots, \hat{m}_{c-1})$ . The enforcement rule can be characterized as a $c \times c$ matrix $\rho(\hat{X}_a)$ (or simply $\rho$ if there is no confusion), such that $\rho_{i,j}$ denotes the number of randomly selected questions on which Alice reports $i$ and is flipped to $j$ by the mechanism. By construction, $\rho_{i,j} \in \mathbb{Z}_{\geq 0}$ for any $i, j \in \Sigma$ . Furthermore, $\sum_i \rho_{i,j} = n_j$ for any $j \in \Sigma$ and $\sum_j \rho_{i,j} = \hat{m}_i$ for any $i \in \Sigma$ . + +Suppose Alice's signal vector $X_{a}$ has a marginal distribution $\Phi_{a} = (m_{0},\dots,m_{c - 1})$ . Under the enforcement rule, while reasoning the distribution of the EA score, every (randomized) strategy can be characterized by (a mixture of) manipulation matrix. The entry $\vartheta_{i,j}$ of a manipulation matrix denotes the number of randomly selected questions where Alice observes $i$ but is flipped to $j$ either by the mechanism or by herself. Similarly, $\vartheta_{i,j}\in \mathbb{Z}_{\geq 0}$ for any $i,j\in \Sigma$ , $\sum_{i}\vartheta_{i,j} = n_{j}$ for any $j\in \Sigma$ , and $\sum_{j}\vartheta_{i,j} = m_{i}$ for any $i\in \Sigma$ . In particular, let $\vartheta^{\tau}(\rho)$ denote the manipulation matrix corresponding to truth-telling under the enforcement rule $\rho$ . + +Given a manipulation matrix $\vartheta$ , the final score of Alice is thus the average of $c^2$ binomials: + +$$ +S ^ {E A} (\theta) \sim \frac {1}{n} \sum_ {i, j \in \Sigma} \mathrm {B i n} (\vartheta_ {i, j}, p _ {i j}), +$$ + +where $p_{ij} = \operatorname*{Pr}(X_b = j \mid X_a = i)$ is the conditioned probability of agents' signals. + +To obtain SD-truthfulness, we have to construct an enforcement rule $\rho$ such that for any $p$ (under some assumptions), any $X_{a}$ , and any feasible $\vartheta$ , the random variable $S^{EA}(\theta)$ is first-order stochastically dominated by $S^{EA}(\tau)$ . This is particularly challenging when $c$ is large because $\vartheta$ has $c^2 - 2c + 1$ degrees of freedom, yet we must identify a $\rho$ such that $\vartheta^\tau(\rho)$ dominates any other $\vartheta$ . + +Three signals We show that even for the three-signal setting, EA is SD-truthful only under a very special type of information structure. + +Definition C.1. Agents' signals are uniformly self-predicting if $\operatorname*{Pr}(X_b = i\mid X_a = i) > \operatorname*{Pr}(X_b = i\mid X_a = j) = \operatorname*{Pr}(X_b = i\mid X_a = k)$ for any $j\neq i$ and $k\neq i$ + +Consider the following enforcement rule: + +- Alice over-reports one signal. If $\hat{m}_i \geq n_i$ while $\hat{m}_j \leq n_j$ for any $j \neq i$ , the enforcement rule will randomly select $n_j - \hat{m}_j$ questions on which Alice reports $i$ and flip them to $j$ for $j \in \Sigma$ and $j \neq i$ . +- Alice over-reports two signals. If $\hat{m}_j < n_j$ while $\hat{m}_i \geq n_i$ for any $i \neq j$ , the enforcement rule will randomly select $\hat{m}_i - n_i$ questions on which Alice reports $i$ and flip them to $j$ for $i \in \Sigma$ and $i \neq j$ . + +This enforcement rule flips the minimum number of reports to enforce $\Phi$ and thus is named the minimal enforcement rule, denoted as $\rho^{*}$ . + +Proposition C.2. If $|\Sigma| = 3$ and signals are self-predicting, the enforcement agreement mechanism is SD-truthful for any $\Phi$ if and only if it applies the minimal enforcement rule and signals are uniformly self-predicting. + +We defer the proof to Appendix F.5. Our results imply that achieving SD-truthfulness using the EA mechanism is infeasible for arbitrary information structures beyond the binary-signal setting. Investigating alternative approaches, such as mechanism designs that leverage additional structure in the information space or stronger assumptions on agent strategies, remains an interesting direction for future work. + +# C.2 Truthfulness of EA + +Although the EA mechanism is not SD-truthful when $|\Sigma| > 2$ , we show that it is still possible to obtain truthfulness as long as the information structure is known. + +Given a report vector of Alice $\hat{X}_a$ with an empirical distribution $\Phi_{\hat{X}_a} = (\hat{m}_0, \dots, \hat{m}_{c-1})$ , EA will apply the $IP$ enforcement rule by solving the following integer programming. + +$$ +\max _ {\vartheta} \sum_ {i, j \in \Sigma} \vartheta_ {i, j} p _ {i j} +$$ + +$$ +\begin{array}{l l} \text {s . t .} & \sum_ {i \in \Sigma} \vartheta_ {i, j} = n _ {j}, \quad \forall j \in \Sigma , \end{array} +$$ + +$$ +\sum_ {j \in \Sigma} \vartheta_ {i, j} = \hat {m} _ {i}, \quad \forall i \in \Sigma , +$$ + +$$ +\vartheta_ {i, j} \in \mathbb {Z} _ {\geq 0}, \quad \forall i, j \in \Sigma . +$$ + +Suppose the optimal solution is $\vartheta^{*}(\hat{X}_{a})$ . The mechanism will then randomly select $\vartheta^{*}(\hat{X}_{a})_{i,j}$ questions on which $\hat{X}_a = i$ and flip them to $j$ , for any $i, j \in \Sigma$ . After the enforcement step, Alice will be scored based on the output agreement mechanism. + +Proposition C.3. For any enforcement $\Phi$ and information structure, if $\operatorname{Pr}(X_b \mid X_a)$ is known, the enforcement agreement mechanism with the IP enforcement rule is truthful. + +The proof is straightforward. Note that to prove truthfulness, it is sufficient to show that truth-telling maximizes the expected score which is $\sum_{i,j\in \Sigma}\vartheta_{i,j}^{\tau}p_{ij}$ . Because the mechanism will manipulate agents' reports in an optimal way by solving the above integer programming, no strategy can achieve a higher expected score than truth-telling. + +Note that when $|\Sigma| = 2$ and signals are self-predicting, the optimal solution to the above IP does not depend on the information structure $p_{ij}$ . In particular, the IP enforcement rule always flips the over-reported signal to enforce $n_0$ zeros and $n - n_0$ ones and does not flip any of the under-reported signal. This echos Theorem 6.1 which suggests the EA mechanism is SD-truthful in the binary-signal setting without the knowledge of $p$ . + +Remark C.4 (Connection to Peer Truth Serum Faltings et al. [2017]). As illustrated in Section 4.2, PTS can be viewed as an output agreement mechanism (Section 4.1) where the score for agreement is weighted inversely by each signal's prior. This re-weighting shifts the probability of agreement from the joint distribution $\operatorname{Pr}(X_a, X_b)$ , which demands self-dominating signals to ensure truthful reporting, to the conditional distribution $\operatorname{Pr}(X_b \mid X_a)$ , which requires self-predicting signals. + +In contrast, EA avoids using the prior $\operatorname{Pr}(X_a)$ by controlling the marginal distribution of Alice's reports. This approach similarly transforms the probability of agreement from the joint to the conditional distribution. However, the enforcement process creates additional incentive issues when the signal space is large which requires complete knowledge of the underlying information structure. Consequently, if $|\Sigma| = 2$ and signals are self-predicting, EA can obtain a stronger truthful guarantee (SD-truthfulness) without any prior knowledge; but for $|\Sigma| > 2$ , it requires more extensive structural information to guarantee truthfulness. + +# C.3 The Optimal Enforcement + +We have established that when signals are binary and self-predicting, EA with any enforcement $\Phi$ is SD-truthful. In this subsection, we examine how the choice of enforcement, $\Phi = (n_0, n - n_0)$ , influences the sensitivity of EA, and in particular, what is the sensitivity-maximizing enforcement. + +Fixing an information structure, the expected score given by EA, $S^{EA}(n_0,e)$ , can be viewed as a function of the number of questions where the answers are enforced to zero, $n_0$ , and Alice's effort level, $e$ . We first present an intermediate result suggesting that the standard deviation of $S^{EA}$ is independent of $n_0$ . This implies that the enforcement affects the sensitivity of EA only through $\nabla \mathbb{E}[S^{EA}(e)]$ . + +Lemma C.5. For a fixed information structure, $\text{std}\left(S^{EA}(n_0,e)\right)$ is independent of $n_0$ , i.e. $\text{std}\left(S^{EA}(n_0,e)\right) = \text{std}\left(S^{EA}(n_0',e)\right)$ for any $n_0, n_0' \in \{0,1,\ldots,n\}$ . + +We next analyze how the derivative of the expected score depends on the choice of enforcement. We find that the sensitivity-maximizing enforcement $n_0$ is determined by a quantile of a binomial distribution with success probability $\operatorname*{Pr}(X_a = 0)$ , where the quantile depends on the information structure. + +Proposition C.6. If $|\Sigma| = 2$ and signals are self-predicting, the sensitivity-maximizing enforcement of the EA mechanism $\Phi = (n_0, n - n_0)$ satisfies that + +$$ +n _ {0} = \max \left\{k \in \{0, 1, \dots , n \}: F (k) < \frac {\eta_ {0 0} ^ {\prime} - \eta_ {0 1} ^ {\prime}}{\eta_ {0 0} ^ {\prime} - \eta_ {0 1} ^ {\prime} + \eta_ {1 1} ^ {\prime} - \eta_ {1 0} ^ {\prime}} \right\}, +$$ + +where $F(k) = \sum_{i=0}^{k} \binom{n}{i} \operatorname{Pr}(X_a = 0)^i (1 - \operatorname{Pr}(X_a = 0))^{n-i}$ is the c.d.f. of a binomial random variable with success probability $\operatorname{Pr}(X_a = 0)$ , and $\eta_{ij}' = \operatorname{Pr}(X_b = j \mid X_a = i) - \operatorname{Pr}(X_b = j)$ . + +We defer the proofs of Lemma C.5 and Proposition C.6 to Appendix F.6. + +# D The Matching Agreement Mechanism + +The matching agreement mechanism is a follow-up of the correlated agreement mechanism aiming to address more complex cheating strategies Zhang and Schoenebeck [2023b]. Similar to CA, which determines "agreement" using the delta matrix (see Section 4.3), MA determines "agreement" using a different method depending on the information structure. + +Definition D.1. Let $\Gamma$ be a $|\Sigma| \times |\Sigma| \times |\Sigma|$ tensor where + +$$ +\Gamma_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} = \Pr (X _ {a} = \sigma_ {1} \mid X _ {b} = \sigma_ {2}) - \Pr (X _ {a} = \sigma_ {1} \mid X _ {b} = \sigma_ {3}). +$$ + +The agreement function is thus $T_{\Gamma} = \mathrm{Sign}(\Gamma)$ , i.e. $T_{\Gamma}(\sigma_1, \sigma_2, \sigma_3) = 1$ if $\Gamma_{\sigma_1, \sigma_2, \sigma_3} > 0$ , and 0 otherwise. + +Next, the mechanism randomly samples a bonus question $j$ and a penalty question $k$ uniformly at random. An individual score computed using these two questions is thus $S_{i}^{MA} = T_{\Gamma}(\hat{X}_{a,j},\hat{X}_{b,j},\hat{X}_{b,k})$ . In words, the matching agreement mechanism awards Alice a score of 1 if, when predicting her response on question $b$ , the posterior conditioned on Bob's response to the same question provides a better prediction than the posterior conditioned on Bob's response on a different question $q$ . Similar to CA, each bonus question can be paired with $n - 1$ penalty questions. Therefore, the final score of MA is the average of $n \cdot (n - 1)$ individual scores. + +The Sensitivity of MA Under the Partition Rounding Reduction We further investigate the sensitivity of the MA mechanism under the partition reduction. The MA mechanism requires two questions to compute an individual score, implying that $K = n / 2$ . Clearly, $S_{\mathrm{inf}} = 0$ and $S_{\mathrm{sup}} = 1$ by definition. By Equation (1), to compute the sensitivity, we only need to compute the expected individual score $m_i(e)$ and its derivative. When Alice exerts full effort, the probability of obtaining a score of 1 can be computed as follows: + +(Note that $T_{\Gamma}(\sigma_1,\sigma_2,\sigma_3) = 1 - T_{\Gamma}(\sigma_1,\sigma_3,\sigma_2).$ + +$$ +\begin{array}{l} m _ {i} ^ {M A} (1) = \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} \Pr \left(X _ {a, j} = \sigma_ {1}, X _ {b, j} = \sigma_ {2}, X _ {b, k} = \sigma_ {3}\right) \cdot T _ {\Gamma} \left(\sigma_ {1}, \sigma_ {2}, \sigma_ {3}\right) \\ = \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} J _ {\sigma_ {1}, \sigma_ {2}} M _ {\sigma_ {3}} ^ {b} \cdot T _ {\Gamma} \left(\sigma_ {1}, \sigma_ {2}, \sigma_ {3}\right) \\ = \frac {1}{2} \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} J _ {\sigma_ {1}, \sigma_ {2}} M _ {\sigma_ {3}} ^ {b} \cdot T _ {\Gamma} (\sigma_ {1}, \sigma_ {2}, \sigma_ {3}) + \frac {1}{2} \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} J _ {\sigma_ {1}, \sigma_ {3}} M _ {\sigma_ {2}} ^ {b} \cdot T _ {\Gamma} (\sigma_ {1}, \sigma_ {3}, \sigma_ {2}) \\ = \frac {1}{2} + \frac {1}{2} \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} \left(J _ {\sigma_ {1}, \sigma_ {2}} M _ {\sigma_ {3}} ^ {b} - J _ {\sigma_ {1}, \sigma_ {3}} M _ {\sigma_ {2}} ^ {b}\right) \cdot T _ {\Gamma} (\sigma_ {1}, \sigma_ {2}, \sigma_ {3}) \\ = \frac {1}{2} + \frac {1}{2} \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} M _ {\sigma_ {2}} ^ {b} M _ {\sigma_ {3}} ^ {b} \Gamma_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} T _ {\Gamma} (\sigma_ {1}, \sigma_ {2}, \sigma_ {3}) \\ = \frac {1}{2} + \frac {1}{2} \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} M _ {\sigma_ {2}} ^ {b} M _ {\sigma_ {3}} ^ {b} (\Gamma_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}}) ^ {+} \\ \end{array} +$$ + +where $(x)^{+} = x$ if $x > 0$ and 0 otherwise. + +When Alice exerts no effort, + +$$ +\begin{array}{l} m _ {i} ^ {M A} (0) = \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} \Pr (\hat {X} _ {a, j} = \sigma_ {1}, X _ {b, j} = \sigma_ {2}, X _ {b, k} = \sigma_ {3}) \cdot T _ {\Gamma} (\sigma_ {1}, \sigma_ {2}, \sigma_ {3}) \\ = \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} M _ {\sigma_ {1}} ^ {a} M _ {\sigma_ {2}} ^ {b} M _ {\sigma_ {3}} ^ {b} \cdot T _ {\Gamma} (\sigma_ {1}, \sigma_ {2}, \sigma_ {3}) \\ = \frac {1}{2} \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} M _ {\sigma_ {1}} ^ {a} M _ {\sigma_ {2}} ^ {b} M _ {\sigma_ {3}} ^ {b} \cdot T _ {\Gamma} (\sigma_ {1}, \sigma_ {2}, \sigma_ {3}) + \frac {1}{2} \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} M _ {\sigma_ {1}} ^ {a} M _ {\sigma_ {2}} ^ {b} M _ {\sigma_ {3}} ^ {b} \cdot T _ {\Gamma} (\sigma_ {1}, \sigma_ {3}, \sigma_ {2}) \\ = \frac {1}{2} + \frac {1}{2} \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} \left(M _ {\sigma_ {1}} ^ {a} M _ {\sigma_ {2}} ^ {b} M _ {\sigma_ {3}} ^ {b} - M _ {\sigma_ {1}} ^ {a} M _ {\sigma_ {2}} ^ {b} M _ {\sigma_ {3}} ^ {b}\right) \cdot T _ {\Gamma} (\sigma_ {1}, \sigma_ {2}, \sigma_ {3}) \\ \end{array} +$$ + +(Note that $T_{\Gamma}(\sigma_1,\sigma_2,\sigma_3) = 1 - T_{\Gamma}(\sigma_1,\sigma_3,\sigma_2).$ + +$$ += \frac {1}{2} +$$ + +The expected score while exerting effort $e$ is a convex combination of $m_{i}^{MA}(1)$ and $m_{i}^{MA}(0)$ , i.e. $m_{i}^{MA}(e) = \frac{1}{2} + \frac{e}{2}\sum_{\sigma_1,\sigma_2,\sigma_3}M_{\sigma_2}^b M_{\sigma_3}^b (\Gamma_{\sigma_1,\sigma_2,\sigma_3})^+$ , and $\nabla m_{i}^{MA}(e) = \frac{1}{2}\sum_{\sigma_1,\sigma_2,\sigma_3}M_{\sigma_2}^b M_{\sigma_3}^b (\Gamma_{\sigma_1,\sigma_2,\sigma_3})^+$ . + +When signals are binary, we find that MA reduces to CA. + +Lemma D.2. When signals are binary, $\sum_{\sigma_1,\sigma_2,\sigma_3}M_{\sigma_2}^b M_{\sigma_3}^b (\Gamma_{\sigma_1,\sigma_2,\sigma_3})^+ = \sum_{\sigma_1,\sigma_2}(\Delta_{\sigma_1,\sigma_2})^+$ . + +Proof. First note that by definition, $\Gamma_{\sigma_1,\sigma_2,\sigma_3} = 0$ when $\sigma_{2} = \sigma_{3}$ . Therefore, when signals are binary, + +$$ +\begin{array}{l} \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} M _ {\sigma_ {2}} ^ {b} M _ {\sigma_ {3}} ^ {b} (\Gamma_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}}) ^ {+} = \sum_ {\sigma_ {1}, \sigma_ {2}} M _ {\sigma_ {2}} ^ {b} M _ {1 - \sigma_ {2}} ^ {b} (\Gamma_ {\sigma_ {1}, \sigma_ {2}, 1 - \sigma_ {2}}) ^ {+} \\ = \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} M _ {\sigma_ {2}} ^ {b} M _ {\sigma_ {3}} ^ {b} \left(\Gamma_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}}\right) ^ {+} \\ = \sum_ {\sigma_ {1}, \sigma_ {2}} \left(J _ {\sigma_ {1}, \sigma_ {2}} M _ {1 - \sigma_ {2}} ^ {b} - J _ {\sigma_ {1}, 1 - \sigma_ {2}} M _ {\sigma_ {2}} ^ {b}\right) ^ {+} \\ = \sum_ {\sigma_ {1}, \sigma_ {2}} \left(J _ {\sigma_ {1}, \sigma_ {2}} \left(1 - M _ {\sigma_ {2}} ^ {b}\right) - \left(\left(M _ {\sigma_ {1}} ^ {a} - J _ {\sigma_ {1}, \sigma_ {2}}\right) M _ {\sigma_ {2}} ^ {b}\right) ^ {+} \right. \\ \end{array} +$$ + +(Signals are binary.) + +$$ +\begin{array}{l} = \sum_ {\sigma_ {1}, \sigma_ {2}} \left(J _ {\sigma_ {1}, \sigma_ {2}} - M _ {\sigma_ {1}} ^ {a} M _ {\sigma_ {2}} ^ {b}\right) ^ {+} \\ = \sum_ {\sigma_ {1}, \sigma_ {2}} \left(\Delta_ {\sigma_ {1}, \sigma_ {2}}\right) ^ {+} \\ \end{array} +$$ + +![](images/b47a06b2fde46014729eb41293e6d5dfa9c14f55ca06024c72512822acaaabdf.jpg) + +An immediate consequence of Lemma D.2 is that the expected score under MA is a linear transformation of the expected score under CA. This further implies that the partition rounding reductions of MA and CA has the same sensitivity under the binary-signal setting. + +Proposition D.3. When signals are binary, $\delta^{MA}(e) = \delta^{CA}(e)$ under the partition rounding reduction. + +Proof. By Lemma D.2, we know that + +$$ +\begin{array}{l} m _ {i} ^ {M A} (e) = \frac {1}{2} + \frac {e}{2} \sum_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}} M _ {\sigma_ {2}} ^ {b} M _ {\sigma_ {3}} ^ {b} (\Gamma_ {\sigma_ {1}, \sigma_ {2}, \sigma_ {3}}) ^ {+} = \frac {1}{2} + \frac {e}{2} \sum_ {\sigma_ {1}, \sigma_ {2}} (\Delta_ {\sigma_ {1}, \sigma_ {2}}) ^ {+} = \frac {1}{2} + \frac {1}{2} m _ {i} ^ {C A} (e), \\ \nabla m _ {i} ^ {M A} (e) = \frac {1}{2} \nabla m _ {i} ^ {C A} (e). \\ \end{array} +$$ + +By Equation (1), + +$$ +\begin{array}{l} \delta^ {M A} (e) = \frac {\nabla m _ {i} ^ {M A} (e) \cdot \sqrt {n / 2}}{\sqrt {\left(m _ {i} ^ {M A} (e) - S _ {\mathrm {i n f}} ^ {M A}\right) \left(S _ {\mathrm {s u p}} ^ {M A} - m _ {i} ^ {M A} (e)\right)}} \\ = \frac {\frac {1}{2} \nabla m _ {i} ^ {C A} (e) \cdot \sqrt {n / 2}}{\sqrt {\left(\frac {1}{2} m _ {i} ^ {M A} (e) + \frac {1}{2}\right) \left(\frac {1}{2} - \frac {1}{2} m _ {i} ^ {M A} (e)\right)}} \\ = \frac {\nabla m _ {i} ^ {C A} (e) \cdot \sqrt {n / 2}}{\sqrt {\left(m _ {i} ^ {M A} (e) + 1\right) \left(1 - m _ {i} ^ {C A} (e)\right)}} \\ = \delta^ {C A} (e). \\ \end{array} +$$ + +![](images/e168cbd9b01cdd0230aa68e396ca91a8f6e495eab193487dc9b82866c9c3beb3.jpg) + +# E Experiments (Continue) + +All codes for our experiments are provided in https://github.com/DavidXu999/Stochastically-Dominant-Peer-Prediction. + +# E.1 Information Structure + +For Section 7.2 and E.3, we consider two real-world crowdsourcing datasets, each having a signal space of size $|\Sigma_1| = 2$ and $|\Sigma_2| = 4$ . The information structure between a pair of agents can be characterized by a Dawid-Skene model Dawid and Skene [1979]. In particular, each task has an underlying ground truth $\omega$ that is i.i.d. sampled from a prior distribution $W$ . Conditioning on $\omega$ , each agent receives a signal $\sigma$ according to conditioned distribution $\Gamma$ , where $\Gamma_{i,j} = \operatorname*{Pr}[\sigma = j \mid \omega = i]$ . + +In the first dataset [Baba et al., 2018], 18 agents each labels whether a compound is appropriate or inappropriate to be synthesized. The prior of the ground truth and the conditioned signal distribution are + +$$ +W _ {1} = \left[ \begin{array}{c c} 0. 6 1 3 & 0. 3 8 7 \end{array} \right], \qquad \Gamma_ {1} = \left[ \begin{array}{c c} 0. 9 0 5 & 0. 0 9 5 \\ 0. 2 8 3 & 0. 7 1 7 \end{array} \right]. +$$ + +In the second dataset Venanzi et al. [2015], 110 agents provide the annotations of the sentiment of 300 tweets. There are four ground truth labels and four possible signals where the information structure is + +$$ +w _ {2} = \left[ \begin{array}{c c c c} 0. 1 9 6 & 0. 2 4 1 & 0. 2 4 7 & 0. 3 1 6 \end{array} \right], \qquad \Gamma_ {2} = \left[ \begin{array}{c c c c} 0. 7 7 0 & 0. 1 2 2 & 0. 0 8 4 & 0. 0 2 4 \\ 0. 0 9 1 & 0. 7 3 5 & 0. 1 3 0 & 0. 0 4 4 \\ 0. 0 3 3 & 0. 0 6 2 & 0. 8 6 6 & 0. 0 3 9 \\ 0. 0 6 8 & 0. 1 6 4 & 0. 0 9 9 & 0. 6 6 9 \end{array} \right]. +$$ + +# E.2 The Sensitivity of Original Implementations + +How much does the partition rounding reduction harm the sensitivity? We compare the partition rounding reduction implementations of CA, MA, and PTS with their original implementations in Figure 2. To be clear, other than the implementations considered in Section 7, we consider the following implementation of the mechanisms. + +- PTS corresponds to the peer truth serum mechanism introduced in Section 4.2, which is truthful but not SD-truthful. +- CA corresponds to the original implementation of the Correlated Agreement mechanism described in Section 4.3, which is truthful but not SD-truthful. +- CA-partition represents the implementation where we get $K = n / 2$ partitions of questions and score agents the average of the individual CA score for each partition (without normalization). By Proposition F.1, CA-partition is SD-truthful when signals are binary. +- MA corresponds to the original implementation of the Matching Agreement mechanism described in Appendix D, which is truthful but not SD-truthful. + +![](images/137e6be08d82366c7c737c88d57530562ab7df6c0ffa4ea7401a4e120b54cbc3.jpg) +(a) Mechanisms shown in Section 7 under $J_{1}$ + +![](images/4c4d05aecc4305409d2e34f9e6e6caed403dde46450c42b9e71fa82c471467bc.jpg) +(b) Other Mechanisms under $J_{1}$ + +![](images/dbf987265714732b081605516fac453e2dc524c6146f6f30c584a5613b5c61cc.jpg) +(c) Mechanisms shown in Section 7 under $J_{2}$ +Figure 2: A Comparison of the Sensitivity of Different Mechanisms. + +![](images/42fe546fc58c62ec0fb44dd78679027cbbacee5278bc2e58c308c46f5092e205.jpg) +(d) Other Mechanisms under $J_{2}$ + +We observe that, relative to the original implementations (solid lines), the partition-rounding reductions (dashed lines) consistently exhibit lower sensitivity. This observation further demonstrates the superior sensitivity of EA mechanisms, as it has the property of SD-truthfulness in the binary setting with no need for partition rounding reduction. + +# E.3 Budgetary Efficiency + +Zhang and Schoenebeck [2023a] show that applying tournaments on the peer prediction scores, though it is not truthful, can incentivize a desired effort level at a lower cost of budget compared to linear payments under their model. Now, we have some more options—several SD-truthful mechanisms—that are truthful under the non-linear tournaments. In this section, we compare the performance of SD-truthful mechanisms with tournaments and truthful mechanisms with linear payments. We consider budgetary efficiency as the performance metric, which is defined as the minimum cost of budget to incentivize a desired effort at the symmetric equilibrium Zhang and Schoenebeck [2023a]. + +Formally, we consider the following payment schemes: + +- The linear payment scheme pays agent $i$ with $\pi_i = a \cdot S_i^M + b$ . +- The winner-take-all tournament pays agent $i$ with $\pi_i = C_{\mathrm{WTA}} \cdot \mathbb{1}[S_i^M > S_j^M, \forall j \neq i]$ . +- The Borda-count tournament pays agent $i$ with the number of agents she beats, i.e., + +$$ +\pi_ {i} = C _ {\text {B o r d a}} \cdot \# \text {b e a t e n} = C _ {\text {B o r d a}} \sum_ {j \neq i} \mathbb {1} \left[ S _ {i} ^ {M} > S _ {j} ^ {M} \right] +$$ + +where $a, b, C_{\mathrm{WTA}}, C_{\mathrm{Borda}}$ are constant parameters. When there's a tie, we equally split the payoffs. We now introduce the simulation workflow with information structure $J_{1}$ . There are 50 agents and 500 tasks, where each agent works on 50 tasks and each task is randomly assigned to 5 agents. Each agent has an effort level $e_i$ with a convex cost function $c(e_i) = e_i^2$ reflecting the increasing + +marginal cost of effort. We require limited liability (i.e. the ex-post payment must be non-negative) and individual rationality (IR, i.e. the expected utility must be non-negative). + +We aim to incentivize a desired effort level $e$ as a local symmetric equilibrium where no agent wants to deviate from $e$ . This allows us to compute the parameters in payment functions such that $\nabla \pi_{i} = \nabla c(e_{i})\forall i$ . To compute $\nabla \pi_{i}$ , we apply the same workflow in the above subsections. In particular, we simulate samples of signals and apply various mechanisms to compute the peer prediction score $S_{i}^{M}$ for each agent. Then, varying the effort from $e$ to $e - \Delta e$ , we can estimate $\nabla S_{i}^{M}$ and $\nabla \pi_{i}$ using a sample size $T = 10,000$ . + +As we mainly focus on whether the SD-Truthful mechanisms + tournament payment can achieve better budgetary efficiency than truthful mechanisms + linear payment, we compute the optimal linear payment among all truthful mechanisms mentioned in Section 7.1, and compare that with the tournament payments with all SD-Truthful mechanisms. We present the results for the winner-take-all tournament in Figure 3 and defer the figures for the Borda-count tournament to the appendix. + +![](images/5c61dba85405a667a6055c531b3c5a9048a3991f2d0f9647763d562f9cb7a12a.jpg) +(a) Effort $e = 0.7$ + +![](images/3cdc791748070b1286a1c545931b1e5f8ca392888224100452fda9c742cf3865.jpg) +(b) Effort $e = 0.9$ +Figure 3: Estimated Sensitivity v.s. Total Payment of Winner-take-all Tournament + +We have the following observations: 1) sensitivity aligns well with the total payment, 2) EA consistently performs the best, 3) when eliciting a high desired effort, tournament payments provide greater budget efficiency compared to linear payments, and 4) when using an efficient tournament payment, e.g. winner-take-all, the partition rounding reduction outperforms the original mechanism paired with a linear payment scheme. + +# E.4 Figures for Borda-count Tournaments + +![](images/8b174959eb96aa87ddcc1d9e5c5d62fe358b3a9449377e8761f2aee59264ac08.jpg) +(a) Effort $e = 0.7$ +Figure 4: Estimated Sensitivity v.s. Total Payment of Borda-count Tournament + +![](images/81d701a39c6e4fe83b876f483953d061159ef12f8d95adae627cd8cc2d295478.jpg) +(b) Effort $e = 0.9$ + +# F Additional Proofs + +# F.1 The CA Mechanism + +We first show that a single draw of the CA score is SD-truthful when $|\Sigma| = 2$ . + +Proposition F.1. When $|\Sigma| = 2$ , instead of taking the average, the mechanism that scores agents using an individual CA score $S_{i}^{CA}$ is SD-truthful. $^{4}$ + +Proof of Proposition F.1. We first show that the agreement function $T_{\Delta}$ is an identity matrix if and only if signals are self-predicting. For simplicity, let $J_{\sigma, \sigma'} = \operatorname*{Pr}(X_a = \sigma, X_b = \sigma')$ be the joint distribution, and let $M_{\sigma}^{a} = \operatorname*{Pr}(X_a = \sigma)$ and $M_{\sigma}^{b} = \operatorname*{Pr}(X_b = \sigma)$ be the marginal distribution of Alice and Bob respectively. The delta matrix can be written as + +$$ +\begin{array}{l} \Delta_ {\sigma , \sigma} = J _ {\sigma , \sigma} - M _ {\sigma} ^ {a} M _ {\sigma} ^ {b} \\ = J _ {\sigma , \sigma} - M _ {\sigma} ^ {a} \left(J _ {\sigma , \sigma} + J _ {\sigma^ {\prime}, \sigma}\right) \\ = J _ {\sigma , \sigma} M _ {\sigma^ {\prime}} ^ {a} - J _ {\sigma^ {\prime}, \sigma} M _ {\sigma} ^ {a} \\ = M _ {\sigma} ^ {a} M _ {\sigma^ {\prime}} ^ {a} \left(\Pr \left(X _ {b} = \sigma \mid X _ {a} = \sigma\right) - \Pr \left(X _ {b} = \sigma \mid X _ {a} = \sigma^ {\prime}\right)\right). \\ \end{array} +$$ + +$$ +\begin{array}{l} \Delta_ {\sigma , \sigma^ {\prime}} = J _ {\sigma , \sigma^ {\prime}} - M _ {\sigma} ^ {a} M _ {\sigma^ {\prime}} ^ {b} \\ = J _ {\sigma , \sigma^ {\prime}} - M _ {\sigma} ^ {a} \left(J _ {\sigma , \sigma^ {\prime}} + J _ {\sigma^ {\prime}, \sigma^ {\prime}}\right) \\ = J _ {\sigma , \sigma^ {\prime}} M _ {\sigma^ {\prime}} ^ {a} - J _ {\sigma^ {\prime}, \sigma^ {\prime}} M _ {\sigma} ^ {a} \\ = M _ {\sigma} ^ {a} M _ {\sigma^ {\prime}} ^ {a} \left(\Pr \left(X _ {b} = \sigma^ {\prime} \mid X _ {a} = \sigma\right) - \Pr \left(X _ {b} = \sigma^ {\prime} \mid X _ {a} = \sigma^ {\prime}\right)\right). \\ \end{array} +$$ + +Next, we reason about the distribution of scores under different strategies. Note that a single draw of the CA score can be $-1$ , $0$ , or $1$ . Therefore, to prove SD-truthfulness for a single draw of the CA score, it is sufficient to show that $\operatorname{Pr}\left(S_i^{CA}(\tau) = 1\right) \geq \operatorname{Pr}\left(S_i^{CA}(\theta) = 1\right)$ and $\operatorname{Pr}\left(S_i^{CA}(\tau) = -1\right) \leq \operatorname{Pr}\left(S_i^{CA}(\theta) = -1\right)$ for any strategy $\theta$ . When signals are binary, there are only three types of untruthful strategy: 1) always reporting $0$ , 2) always reporting $1$ , and 3) always flipping $0$ to $1$ and $1$ to $0$ . Any other strategy is a mixture of truth-telling and these strategies. Now, we show that all of these strategies are stochastically dominated by truth-telling by discussing two cases. + +When signals are self-predicting, i.e. $\operatorname{Pr}(X_b = \sigma \mid X_a = \sigma) - \operatorname{Pr}(X_b = \sigma \mid X_a = \sigma') > 0$ , the distribution of $S_i^{CA}$ is + +$$ +\Pr \left(S _ {i} ^ {C A} (\tau) = 1\right) = J _ {0, 0} M _ {1} ^ {b} + J _ {1, 1} M _ {0} ^ {b}, +$$ + +$$ +\Pr \left(S _ {i} ^ {C A} (\tau) = - 1\right) = J _ {0, 1} M _ {0} ^ {b} + J _ {1, 0} M _ {1} ^ {b}. +$$ + +Because Bob's signal is self-predicting, $\operatorname*{Pr}\left(S_i^{CA}(\tau) = 1\right) > \operatorname*{Pr}\left(S_i^{CA}(\tau) = -1\right)$ . Let $\theta_f$ be the strategy where Alice always flips the signals. + +$$ +\operatorname * {P r} \left(S _ {i} ^ {C A} (\theta_ {f}) = 1\right) = J _ {0, 1} M _ {0} ^ {b} + J _ {1, 0} M _ {1} ^ {b} = \operatorname * {P r} \left(S _ {i} ^ {C A} (\tau) = - 1\right), +$$ + +$$ +\operatorname * {P r} \left(S _ {i} ^ {C A} (\theta_ {f}) = - 1\right) = J _ {0, 0} M _ {1} ^ {b} + J _ {1, 1} M _ {0} ^ {b} = \operatorname * {P r} \left(S _ {i} ^ {C A} (\tau) = 1\right). +$$ + +This means that $S_{i}^{CA}(\theta_{f})$ is dominated by $S_{i}^{CA}(\tau)$ . + +Let $\mu$ be the strategy where Alice always reports the same signal. + +$$ +\Pr \left(S _ {i} ^ {C A} (\mu) = 1\right) = \Pr \left(S _ {i} ^ {C A} (\mu) = - 1\right) = M _ {0} ^ {b} M _ {1} ^ {b}. +$$ + +Therefore, + +$$ +\begin{array}{l} \operatorname * {P r} \left(S _ {i} ^ {C A} (\tau) = 1\right) - \operatorname * {P r} \left(S _ {i} ^ {C A} (\mu) = 1\right) = J _ {0, 0} M _ {1} ^ {b} + J _ {1, 1} M _ {0} ^ {b} - M _ {0} ^ {b} M _ {1} ^ {b} \\ = J _ {0, 0} M _ {1} ^ {b} + J _ {1, 1} M _ {0} ^ {b} - M _ {0} ^ {b} \left(J _ {0, 1} + J _ {1, 1}\right) \\ = J _ {0, 0} M _ {1} ^ {b} - J _ {0, 1} M _ {0} ^ {b} \\ = M _ {0} ^ {b} M _ {1} ^ {b} \left(\Pr \left(X _ {a} = 0 \mid X _ {b} = 0\right) - \Pr \left(X _ {b} = 0 \mid X _ {a} = 1\right) \right. \\ > 0 \\ \end{array} +$$ + +Similarly, we can observe that $\operatorname*{Pr}\left(S_i^{CA}(\tau) = -1\right) < \operatorname*{Pr}\left(S_i^{CA}(\mu) = -1\right)$ + +The analysis for the case when signals are not self-predicting, i.e. $T_{\Delta}(\sigma, \sigma') = 1$ if and only if $\sigma \neq \sigma'$ , is analogous. + +Next, we show that when $|\Sigma| = 3$ , counterexamples exist showing that truth-telling does not FOSD every untruthful strategy. + +Example F.1. Consider the following joint distribution: + +$$ +J = \left( \begin{array}{c c c} 0. 0 2 & 0. 0 1 & 0. 0 2 \\ 0. 2 & 0. 4 & 0. 2 5 \\ 0. 0 0 & 0. 0 5 & 0. 0 5 \end{array} \right). +$$ + +The marginal distribution of Alice and Bob are + +$$ +M ^ {a} = (0. 0 5, \quad 0. 8 5, \quad 0. 1) \text {a n d} M ^ {b} = (0. 2 2, \quad 0. 4 6, \quad 0. 3 2). +$$ + +The product of marginal distribution and the corresponding agreement function are + +$$ +\Delta = \left( \begin{array}{c c c} 0. 0 0 9 & - 0. 0 1 3 & 0. 0 0 4 \\ 0. 0 1 3 & 0. 0 0 9 & - 0. 0 2 2 \\ - 0. 0 2 2 & 0. 0 0 4 & 0. 0 1 8 \end{array} \right), \quad T _ {\Delta} = \left( \begin{array}{c c c} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{array} \right). +$$ + +When Alice reports truthfully, the probability of obtaining a score of 1, i.e. obtaining a score of 1 on the bonus question and obtaining a score of 0 on the penalty question, is given by + +$$ +\Pr (S_{i}^{CA}(\tau) = 1) = \underbrace{(J_{0,0} + J_{0,2})\cdot M_{1}^{b}}_{\text{Alice reports 0}} + \underbrace{(J_{1,0} + J_{1,1})\cdot M_{2}^{b}}_{\text{Alice reports 1}} + \underbrace{(J_{2,1} + J_{2,2})\cdot M_{0}^{b}}_{\text{Alice reports 0}} = 0.2324. +$$ + +However, when Alice plays a strategy $\mu$ that always reports 0, the probability of obtaining a score of 1 is given by + +$$ +\Pr (S _ {i} ^ {C A} (\mu) = 1) = \left(M _ {0} ^ {b} + M _ {2} ^ {b}\right) \cdot M _ {1} ^ {b} = 0. 2 4 7 4 > \Pr (S _ {i} ^ {C A} (\tau) = 1). +$$ + +Therefore, $\mu$ is not first-order stochastically dominated by $\tau$ . + +# F.2 The Direct Rounding Reduction + +Proof of Proposition 5.2. By definition, the sensitivity of a mechanism with score $S(e)$ at effort level $e$ is + +$$ +\delta^ {\mathcal {M}} (e) = \frac {\nabla m ^ {\mathcal {M}} (e)}{\operatorname {s t d} ^ {\mathcal {M}} (e)}. +$$ + +The direct rounding of $S^{\mathcal{M}}(e)$ , denoted as $\tilde{S}^{\mathcal{M}}(e)$ , is a Bernoulli variable with success probability $\mathbb{E}[\lambda^{\mathcal{M}}] = \frac{m^{\mathcal{M}}(e) - S_{\mathrm{inf}}^{\mathcal{M}}}{S_{\mathrm{sup}}^{\mathcal{M}} - S_{\mathrm{inf}}^{\mathcal{M}}}$ . Therefore, + +$$ +\begin{array}{l} \nabla \mathbb {E} \left[ \tilde {S} ^ {\mathcal {M}} (e) \right] = \frac {1}{S _ {\mathrm {s u p}} ^ {\mathcal {M}} - S _ {\mathrm {i n f}} ^ {\mathcal {M}}} \cdot \nabla m ^ {\mathcal {M}} (e) \\ \operatorname {s t d} \left(\tilde {S} ^ {\mathcal {M}} (e)\right) = \sqrt {\mathbb {E} [ \lambda^ {\mathcal {M}} ] \left(1 - \mathbb {E} [ \lambda^ {\mathcal {M}} ]\right)} = \frac {1}{S _ {\operatorname* {s u p}} ^ {\mathcal {M}} - S _ {\operatorname* {i n f}} ^ {\mathcal {M}}} \cdot \sqrt {(m ^ {\mathcal {M}} (e) - S _ {\operatorname* {i n f}} ^ {\mathcal {M}}) \left(S _ {\operatorname* {s u p}} ^ {\mathcal {M}} - m ^ {\mathcal {M}} (e)\right)} \\ \Rightarrow \tilde {\delta} ^ {\mathcal {M}} (e) = \frac {\nabla m ^ {\mathcal {M}} (e)}{\sqrt {\left(m ^ {\mathcal {M}} (e) - S _ {\text {i n f}} ^ {\mathcal {M}}\right) \left(S _ {\text {s u p}} ^ {\mathcal {M}} - m (e)\right)}}. \tag {3} \\ \end{array} +$$ + +□ + +# F.3 The Partition Rounding Reduction + +Proof of Lemma 5.3. Let $S_{i}(\tau)$ and $S_{i}(\theta)$ be the random variable of each individual score under truth-telling and strategy $\theta$ , respectively. $S_{i}(\tau)$ (and $S_{i}(\theta)$ ) is i.i.d. because questions are assumed to be i.i.d. and strategies are task-independent. We want to show + +$$ +S _ {i} (\tau) \succeq_ {\text {F O S D}} S _ {i} (\theta) \Leftrightarrow S (\tau) \succeq_ {\text {F O S D}} S (\theta). +$$ + +Forward Direction. Suppose $S_{i}(\tau) \succeq_{\mathrm{FOSD}} S_{i}(\theta)$ for any $\theta$ . By definition, this means for each individual score, we can find a coupling such that whenever $S_{i}(\theta) = s_{i}$ , $S_{i}(\tau) = s_{i}' \geq s_{i}$ . Applying this coupling for every individual score and taking the average gives us a coupling for $S(\tau)$ and $S(\theta)$ : whenever $S(\theta) = \sum_{i \in [K]} s_{i}$ , $S(\tau) = \sum_{i \in [K]} s_{i}' \geq \sum_{i \in [K]} s_{i}$ in the coupling. Therefore, in this coupling, $\operatorname*{Pr}(S(\theta) \leq t) \geq \operatorname*{Pr}(S(\tau) \leq t)$ for any $t \in [K \cdot S_{\inf}, K \cdot S_{\sup}]$ where $S_{\inf}$ and $S_{\sup}$ are the infimum and supremum of each $S_{i}$ , respectively. This completes the proof for the sufficiency. + +Reverse Direction. We prove the contrapositive of the statement: if $S_{i}(\tau)$ does not first-order stochastic dominate $S_{i}(\theta)$ , then $S(\tau)$ does not first-order stochastic dominate $S(\theta)$ . By the failure of FOSD for the individuals, there exists some threshold $t_1$ and (by the i.i.d. assumption) every $i$ such that $F_{\tau}(t_1) > F_{\theta}(t_1)$ , where $F_{\tau}$ and $F_{\theta}$ denote that c.d.f. for $S_{i}(\tau)$ and $S_{i}(\theta)$ respectively. Let $F_{\tau}^{k}$ and $F_{\theta}^{k}$ be the c.d.f. of the sum of $k$ i.i.d. individual scores, i.e. $\sum_{i \in [k]} S_{i}(\tau)$ and $\sum_{i \in [k]} S_{i}(\theta)$ respectively. We want to find a $t_K$ such that $F_{\tau}^{K}(t_K) > F_{\theta}^{K}(t_K)$ . Then, because $S$ is a linear scaling (by $1 / K$ ) of the sum, the inequality transfers directly to the averages. + +We prove this via an inductive argument: + +- Base case $(K = 1)$ : by assumption, there exists a $t_1$ such that $F_{\tau}(t_1) > F_{\theta}(t_1)$ . +- Inductive step: suppose that for $K - 1$ there exists some threshold $t_{K - 1}$ such that + +$$ +F _ {\tau} ^ {K - 1} (t _ {K - 1}) > F _ {\theta} ^ {K - 1} (t _ {K - 1}). +$$ + +Now, consider the $K$ -fold convolution. Note that the convolution of discrete distributions is a weighted sum of the individual c.d.f.. + +$$ +F _ {\tau} ^ {K} (t _ {K}) = \sum_ {s} \Pr \left(\sum_ {i \in [ K - 1 ]} S _ {i} (\tau) = s\right) F _ {\tau} (t _ {K} - s), +$$ + +and similarly for $F_{\theta}^{K}(t_{K})$ . By the base case and the induction assumption, we can set $t_{K} = t_{K - 1} + t_{1}$ so that $F_{\tau}^{K}(t_{K}) > F_{\theta}^{K}(t_{K})$ . This means that there exists a threshold $t = t_{K} / K$ so that + +$$ +\Pr \big (S (\tau) \leq t \big) > \Pr \big (S (\theta) \leq t \big). +$$ + +This completes the proof of the necessity. + +# F.4 EA in the Binary-signal Setting + +Proof of Theorem 6.1. Suppose Alice observes $m_0$ zeros and $m_1$ ones on $n$ questions while the enforced empirical distribution is $\Phi = (n_0, n_1)$ . Alice's strategy and $\Phi$ thus determines the number of i.i.d. samples from each of the four binomials. + +To prove SDT, we have to show that truth-telling results in a score distribution that first-order stochastic dominates the score distribution under any other strategy. Suppose W.L.O.G. that $m_0 > n_0$ and the mechanism will randomly flip Alice's reports on $m_0 - n_0$ questions from 0 to 1. If Alice reports truthfully, there are $n_0$ questions where Alice reports 0 and will be scored 1 if Bob also reports 0; there are $m_1$ questions where Alice reports 1 and will be scored 1 if Bob also reports 1; and there are $m_0 - n_0$ questions where Alice observes 0 but her reports are flipped to 1 which means that she will be scored 1 if Bob reports 1 on those questions. Therefore, the final score of Alice can be expressed as + +$$ +n \cdot S (\tau) \sim \operatorname {B i n} \left(n _ {0}, p _ {0 0}\right) + \operatorname {B i n} \left(m _ {1}, p _ {1 1}\right) + \operatorname {B i n} \left(m _ {0} - n _ {0}, p _ {0 1}\right). \tag {4} +$$ + +To show that $S(\tau)$ dominates $S(\theta)$ for any strategy $\theta$ , it is sufficient to focus on deterministic strategies. Suppose $x_{i,j}$ is the number of questions that Alice observes $i$ but reports $j$ for $i, j \in \{0,1\}$ . First note that $x_{0,0} + x_{0,1} = m_0$ and $x_{1,0} + x_{1,1} = m_1$ . Then, we argue that it is sufficient to focus on strategies that satisfy $x_{0,0} + x_{1,0} = n_0$ and $x_{0,1} + x_{1,1} = n_1$ . Otherwise, the mechanism will enforce $\Phi$ on behalf of Alice which is identical for Alice to flip her signals by herself. Fixing any $\Phi$ and $\Phi_{X_a}$ , the above four constraints suggest that there is only one free variable in $x$ , meaning that there is only one possible way to cheat—altering $x_{0,0} \in \{0,\dots,n_0\}$ . + +Now, we show that reporting $x_{0,0} = n_0$ is the best strategy—reporting any $x_{0,0}$ results in a score distribution dominated by reporting $x_{0,0} = n_0$ . Also, note that truth-telling is one of the strategies that has $x_{0,0} = n_0$ . Let $x_{0,0} = n_0 - k$ where $k \geq 0$ . Then, the final score of Alice and be expressed as + +$$ +n \cdot S (\theta) \sim \operatorname {B i n} \left(n _ {0} - k, p _ {0 0}\right) + \operatorname {B i n} \left(m _ {1} - k, p _ {1 1}\right) + \operatorname {B i n} \left(m _ {0} - n _ {0} + k, p _ {0 1}\right) + \operatorname {B i n} \left(k, p _ {1 0}\right). \tag {5} +$$ + +Compared with the truth-telling score distribution in Equation (2), any untruthful reporting can be viewed as subtracting $k$ samples from $\mathrm{Bin}(n_0, p_{00})$ and draw $k$ more samples from $\mathrm{Bin}(k, p_{10})$ ; subtracting $k$ samples from $\mathrm{Bin}(m_1, p_{11})$ and draw $k$ more samples from $\mathrm{Bin}(m_0 - n_0 + k, p_{01})$ . + +When signals are self-predicting, $p_{00} > p_{10}$ and $p_{11} > p_{01}$ . We can construct a coupling to show that $S(\theta)$ is dominated by $S(\tau)$ . In particular, for the $k$ questions whose scores are sampled from $\mathrm{Bin}(k,p_{10})$ , we couple them with $k$ out of the $n_0$ samples from $S(\theta)\sim \mathrm{Bin}(n_0 - k,p_{00})$ ; for the $n_0 - k$ questions whose scores are sampled from $\sim \mathrm{Bin}(n_0 - k,p_{00})$ , we couple them with the remaining $n_0 - k$ samples from $\sim \mathrm{Bin}(n_0 - k,p_{00})$ ; we do the same for the other two Binomial distributions. Under this coupling, it is clear that making $x_{0,0}$ as large as possible will have the dominating score distribution, which happens while truth-telling. + +# F.5 EA in the Three-signal Setting + +Proof of Proposition C.2. + +Sufficiency. We prove the sufficiency by discussing two cases. First, suppose W.L.O.G. that $m_0 \geq n_0$ , $m_1 \leq n_1$ , and $m_2 \leq n_2$ . Under the minimal enforcement rule, the manipulation matrix corresponds to truth-telling is + +$$ +\vartheta^ {\tau} (\rho^ {*}) = \left[ \begin{array}{c c c} n _ {0} & n _ {1} - m _ {1} & n _ {2} - m _ {2} \\ 0 & m _ {1} & 0 \\ 0 & 0 & m _ {2} \end{array} \right]. +$$ + +When we score an arbitrary report vector $\hat{X}_a$ , it is sufficient to consider its manipulation matrix $\vartheta$ . Therefore, we want to show that $\vartheta^{\tau}$ will result in a score distribution that first-order stochastically dominates the score distribution of any other feasible $\vartheta$ . Note that there are 9 variables in a $3 \times 3$ matrix and 5 constraints: the sum of variables in row $i$ is $m_i$ and the sum of variables in column $j$ is $n_j$ while only 5 of these 6 constraints are independent. This means that any manipulation matrix be characterized by 4 independent variables denoted as $t_1, \ldots, t_4 \geq 0$ . + +$$ +\vartheta = \left[ \begin{array}{c c c} n _ {0} - t _ {1} - t _ {2} & n _ {1} - m _ {1} + t _ {1} + t _ {4} - t _ {3} & n _ {2} - m _ {2} + t _ {2} + t _ {3} - t _ {4} \\ t _ {1} & m _ {1} - t _ {1} - t _ {4} & t _ {4} \\ t _ {2} & t _ {3} & m _ {2} - t _ {2} - t _ {3} \end{array} \right]. +$$ + +The variables $t_1, \ldots, t_4$ must satisfy the constraints ensuring that every entry in $\vartheta$ remains non-negative. + +The final score of truth-telling is given by the average of $n$ Bernoulli variables where $\vartheta^{\tau}(\rho^{*})_{i,j}$ of these variables are sampled from Bernoulli $(p_{i,j})$ . Similarly, the final score corresponding to the manipulation matrix $\vartheta$ is the average of $n$ Bernoulli variables where $\vartheta_{i,j}^{\theta}$ of them are sampled from Bernoulli $(p_{i,j})$ . + +Our goal is to construct a coupling of these Bernoulli variables such that, for any given $\Phi_{X_a}$ , $\Phi$ , and $t_1,\ldots ,t_4$ , each variable under truth-telling with manipulation matrix $\vartheta^{\tau}(\rho^{*})$ has a weakly higher probability of success than its coupled counterpart under $\vartheta$ . We denote this coupling by $Z_{i,j}^{\tau}\stackrel {k}{\to}Z_{i',j'}^{\theta}$ to indicate that we pair $k$ Bernoulli variables associated with $\vartheta^{\tau}(\rho^{*})$ —each drawn from Bernoulli $(p_{ij})$ —with $k$ Bernoulli variables associated with $\vartheta$ —each drawn from Bernoulli $(p_{i'j'})$ . + +Our coupling is: + +$$ +\left[ \begin{array}{c c c} Z _ {0, 0} ^ {\tau} \xrightarrow {n _ {0} - t _ {1} - t _ {2}} Z _ {0, 0} ^ {\theta} & Z _ {1, 1} ^ {\tau} \xrightarrow {t _ {1} + t _ {4}} Z _ {0, 1} ^ {\theta} & Z _ {2, 2} ^ {\tau} \xrightarrow {t _ {2} + t _ {3}} Z _ {0, 2} ^ {\theta} \\ Z _ {0, 0} ^ {\tau} \xrightarrow {t _ {1}} Z _ {1, 0} ^ {\theta} & Z _ {1, 1} ^ {\tau} \xrightarrow {m _ {1} - t _ {1} - t _ {4}} Z _ {1, 1} ^ {\theta} & Z _ {0, 2} ^ {\tau} \xrightarrow {t _ {4}} Z _ {1, 2} ^ {\theta} \\ Z _ {0, 0} ^ {\tau} \xrightarrow {t _ {2}} Z _ {2, 0} ^ {\theta} & Z _ {0, 1} ^ {\tau} \xrightarrow {t _ {3}} Z _ {2, 1} ^ {\theta} & Z _ {2, 2} ^ {\tau} \xrightarrow {m _ {2} - t _ {2} - t _ {3}} Z _ {2, 2} ^ {\theta} \end{array} \right]. +$$ + +Under this coupling, every diagonal variable $Z_{i,i}^{\theta}$ is coupled with $Z_{i,i}^{\tau}$ which is drawn from the same Bernoulli distribution. Furthermore, every non-diagonal variable $Z_{i,j}^{\theta}$ is either coupled with $Z_{i,i}^{\tau}$ which has a larger success probability because of self-prediction, or coupled with $Z_{k,j}^{\tau}$ which has the same success probability because signals are uniformly self-predicting. + +The proof of the second case is analogous. suppose W.L.O.G. that $m_0 > n_0, m_1 > n_1$ , and $m_2 < n_2$ . Under the minimal enforcement rule, the manipulation matrix corresponds to truth-telling is + +$$ +\vartheta^ {\tau} = \left[ \begin{array}{c c c} n _ {0} & 0 & m _ {0} - n _ {0} \\ 0 & n _ {1} & m _ {1} - n _ {1} \\ 0 & 0 & m _ {2} \end{array} \right]. \tag {6} +$$ + +Any feasible manipulation matrix can be written as + +$$ +\vartheta^ {\theta} = \left[ \begin{array}{c c c} n _ {0} - t _ {1} - t _ {2} & t _ {4} & m _ {0} - n _ {0} + t _ {1} + t _ {2} - t _ {4} \\ t _ {1} & n _ {1} - t _ {3} - t _ {4} & m _ {1} - n _ {1} + t _ {3} + t _ {4} - t _ {1} \\ t _ {2} & t _ {3} & m _ {2} - t _ {2} - t _ {3} \end{array} \right]. \tag {7} +$$ + +We construct the following coupling. If $t_1 \geq t_4$ , we apply the term in red; while if $t_1 < t_4$ , we apply the term in blue. + +$$ +\left[ \begin{array}{c c c c} Z _ {0, 0} ^ {\tau} \xrightarrow {n _ {0} - t _ {1} - t _ {2}} Z _ {0, 0} ^ {\theta} & Z _ {1, 1} ^ {\tau} \xrightarrow {t _ {4}} Z _ {0, 1} ^ {\theta} & Z _ {2, 2} ^ {\tau} \xrightarrow {t _ {2}} Z _ {0, 2} ^ {\theta}, & Z _ {1, 2} ^ {\tau} \xrightarrow {t _ {1} - t _ {4}} Z _ {0, 2} ^ {\theta} \\ Z _ {0, 0} ^ {\tau} \xrightarrow {t _ {1}} Z _ {1, 0} ^ {\theta} & Z _ {1, 1} ^ {\tau} \xrightarrow {n _ {1} - t _ {3} - t _ {4}} Z _ {1, 1} ^ {\theta} & Z _ {2, 2} ^ {\tau} \xrightarrow {t _ {3}} Z _ {1, 2} ^ {\theta}, & Z _ {0, 2} ^ {\tau} \xrightarrow {t _ {4} - t _ {1}} Z _ {1, 2} ^ {\theta} \\ Z _ {0, 0} ^ {\tau} \xrightarrow {t _ {2}} Z _ {2, 0} ^ {\theta} & Z _ {1, 1} ^ {\tau} \xrightarrow {t _ {3}} Z _ {2, 1} ^ {\theta} & Z _ {2, 2} ^ {\tau} \xrightarrow {m _ {2} - t _ {2} - t _ {3}} Z _ {2, 2} ^ {\theta} \end{array} \right]. +$$ + +Under the same argument, we can observe that every $Z_{i,j}^{\theta}$ is coupled with a $Z_{i',j'}^{\tau}$ with a weakly larger success probability. This completes the proof of sufficiency. + +Necessity. We first show that if the mechanism's enforcement rule is not minimal, it is not SD-truthful. For a non-minimal enforcement rule, there must exist a report vector $\hat{X}_a$ and an enforced histogram $\Phi$ such that if Alice reports $\hat{X}_a$ truthfully, there exists a diagonal value in the corresponding manipulation matrix that can be increased without breaking the constraints. In other words, there exists a strategy $\theta$ whose corresponding manipulation matrix is $\vartheta^\theta$ and a column $j$ such that $\vartheta_{j,j}^\theta >\vartheta_{j,j}^\tau$ . Then, the strategy $\theta$ is not dominated by truth-telling even when signals are uniformly self-predicting. This is because by playing $\theta$ , Alice can move $\vartheta_{j,j}^{\theta} - \vartheta_{j,j}^{\tau}$ samples from Bernoulli $(p_{ij})$ or Bernoulli $(p_{kj})$ to Bernoulli $(p_{jj})$ where $i,k\neq j$ . For self-predicting signals, $p_{jj}$ is the largest column value. + +Next, we show that if signals are not uniformly self-predicting, the enforced agreement mechanism is not SD-truthful even when the enforcement rule is minimal. We can easily find a counterexample by comparing Equation (6) with Equation (7). Suppose W.L.O.G. that $\vartheta_{0,2}^{\tau} < \vartheta_{1,2}^{\tau}$ , then Alice can set $t_1 = t_2 = t_3 = 0$ and $t_4 = \max(n_1, m_0 - n_0)$ in which case she can move $t_4$ samples from Bernoulli $(p_{0,2})$ to Bernoulli $(p_{1,2})$ , resulting a score distribution that dominates truth-telling. This completes the proof of necessity. + +# F.6 The Optimal Enforcement + +Proof of Lemma C.5. Suppose Alice observes 0 on $m_0$ out of the $n$ questions and observes 1 on the remaining $n - m_0$ questions. There are two cases. + +First, when $m_0 \leq n_0$ , the final score of Alice is the average of the following three type of questions: + +- $m_0$ questions where Alice observes 0 and receives a score of 1 if Bob also reports 0. +- $n - {n}_{0}$ questions where Alice observes 1 and receives a score of 1 if Bob also reports 1 . + +- $n_0 - m_0$ questions where Alice observes 1 but receives a score of 1 if Bob reports 0. + +When Alice exerts full effort ( $e = 1$ ), Bob's reports (which match his signals) follow the conditional probability distribution $p_{ij} = \operatorname*{Pr}(X_b = j \mid X_a = i)$ . In contrast, when Alice exerts no effort ( $e = 0$ ), her signals are uninformative about Bob's signals, and thus her belief about Bob's reports follows the marginal distribution $M_i^b = \operatorname*{Pr}(X_b = i)$ . Therefore, conditioned on Alice observing 0 in $m_0$ questions, the score defined by EA follows the sum of three binomials: + +$$ +n \cdot S ^ {E A} \left(n _ {0}, e = 1 \mid m _ {0} \leq n _ {0}\right) \sim \operatorname {B i n} \left(m _ {0}, p _ {0 0}\right) + \operatorname {B i n} \left(n - n _ {0}, p _ {1 1}\right) + \operatorname {B i n} \left(n _ {0} - m _ {0}, p _ {1 0}\right), +$$ + +$$ +n \cdot S ^ {E A} (n _ {0}, e = 0 \mid m _ {0} \leq n _ {0}) \sim \operatorname {B i n} (m _ {0}, M _ {0} ^ {b}) + \operatorname {B i n} (n - n _ {0}, M _ {1} ^ {b}) + \operatorname {B i n} (n _ {0} - m _ {0}, M _ {0} ^ {b}). +$$ + +Note that for any question, the probability of Alice observing 0 while exerting effort is the marginal distribution $\operatorname*{Pr}(X_a = 0)$ , which is identical to the probability of observing 0 while exerting no effort. This implies that under our definition of effort, the probability of Alice observing 0 on $m_0$ questions does not depend on the effort. Let $\rho_0 = \sum_{j\in [n]}(1 - X_{a,j})$ be the number of questions where Alice observes 0. The probability that Alice observes 0 on $m_0$ questions is + +$$ +\operatorname * {P r} \left(\rho_ {0} = m _ {0}\right) = \binom {n} {m _ {0}} (M _ {0} ^ {a}) ^ {m _ {0}} (1 - M _ {0} ^ {a}) ^ {n - m _ {0}}. +$$ + +Consequently, when $m_0 \leq n_0$ , Alice's score, conditioned on observing 0 in $m_0 \leq n_0$ questions (which occurs with probability $\operatorname{Pr}(\rho_0 = m_0)$ ), follows the distribution: + +$$ +\begin{array}{l} n \cdot S ^ {E A} \left(n _ {0}, e \mid m _ {0} \leq n _ {0}\right) \sim \operatorname {B i n} \left(m _ {0}, e p _ {0 0} + (1 - e) M _ {0} ^ {b}\right) + \operatorname {B i n} \left(n - n _ {0}, e p _ {1 1} + (1 - e) M _ {1} ^ {b}\right) \\ + \operatorname {B i n} \left(n _ {0} - m _ {0}, e p _ {1 0} + (1 - e) M _ {0} ^ {b}\right). \\ \end{array} +$$ + +In the second case, where $m_0 > n_0$ , the final score of Alice is the average of the following three types of questions: + +- $n_0$ questions where Alice observes 0 and receives a score of 1 if Bob also reports 0. +- $n - {m}_{0}$ questions where Alice observes 1 and receives a score of 1 if Bob also reports 1 . +- $m_0 - n_0$ questions where Alice observes 0 but receives a score of 1 if Bob reports 1. + +By the analogous arguments, we know that Alice's score, conditioned on observing $0$ in $m_0 > n_0$ questions, follows the distribution: + +$$ +\begin{array}{l} n \cdot S ^ {E A} \left(n _ {0}, e \mid m _ {0} > n _ {0}\right) \sim \operatorname {B i n} \left(n _ {0}, e p _ {0 0} + (1 - e) M _ {0} ^ {b}\right) + \operatorname {B i n} \left(n - m _ {0}, e p _ {1 1} + (1 - e) M _ {1} ^ {b}\right) \\ + \operatorname {B i n} \left(m _ {0} - n _ {0}, e p _ {0 1} + (1 - e) M _ {1} ^ {b}\right). \\ \end{array} +$$ + +Putting everything together, + +$$ +S ^ {E A} \left(n _ {0}, e\right) \sim \sum_ {m _ {0} = 0} ^ {n _ {0}} \Pr \left(\rho_ {0} = m _ {0}\right) \cdot S ^ {E A} \left(n _ {0}, e \mid m _ {0} \leq n _ {0}\right) + \sum_ {m _ {0} = n _ {0} + 1} ^ {n} \Pr \left(\rho_ {0} = m _ {0}\right) \cdot S ^ {E A} \left(n _ {0}, e \mid m _ {0} > n _ {0}\right) \tag {8} +$$ + +Now, we show that the standard deviation of $S^{EA}(n_0,e)$ is independent of $n_0$ . For simplicity, let $\eta_{ij} = e p_{ij} + (1 - e) M_j^b$ . Note that $\eta_{i0} + \eta_{i1} = e(p_{i0} + p_{i1}) + (1 - e)(M_0^b + M_1^b) = 1$ . Furthermore, the variance of a binomial with parameters $n,p$ is $np(1 - p)$ . When $m_0 \leq n_0$ , the variance of the final score is + +$$ +\begin{array}{l} \operatorname {v a r} \left(n \cdot S ^ {E A} \left(n _ {0}, e \mid m _ {0} \leq n _ {0}\right)\right) = m _ {0} \eta_ {0 0} (1 - \eta_ {0 0}) + (n - n _ {0}) \eta_ {1 1} (1 - \eta_ {1 1}) + (n _ {0} - m _ {0}) \eta_ {1 0} (1 - \eta_ {1 0}) \\ = m _ {0} \eta_ {0 0} (1 - \eta_ {0 0}) + (n - m _ {0}) \eta_ {1 1} (1 - \eta_ {1 1}). \\ \end{array} +$$ + +Similarly, when $m_0 > n_0$ + +$$ +\begin{array}{l} \operatorname {v a r} \left(n \cdot S ^ {E A} \left(n _ {0}, e \mid m _ {0} > n _ {0}\right)\right) = n _ {0} \eta_ {0 0} (1 - \eta_ {0 0}) + (n - m _ {0}) \eta_ {1 1} (1 - \eta_ {1 1}) + (m _ {0} - n _ {0}) \eta_ {0 1} (1 - \eta_ {0 1}) \\ = m _ {0} \eta_ {0 0} (1 - \eta_ {0 0}) + (n - m _ {0}) \eta_ {1 1} (1 - \eta_ {1 1}). \\ \end{array} +$$ + +Therefore, conditioned on observing 0 on $m_0$ questions, the standard deviation std $(S^{EA}(n_0, e \mid m_0)) = \sqrt{m_0 \eta_{00} (1 - \eta_{00}) + (n - m_0) \eta_{11} (1 - \eta_{11})}$ is independent of $n_0$ . Consequently, the standard deviation of the final score, averaging over $m_0$ , is also independent of $n_0$ . + +Proof of Proposition C.6. By Lemma C.5, the standard deviation of the EA mechanism is independent of the enforcement $\Phi$ . Therefore, the enforcement that maximizes the sensitivity is the one that maximizes the derivative of the expected score w.r.t. effort $e$ . By Equation (8), the score given by EA can be expressed as the average of many binomial variables. Note that the expectation of a binomial variable with parameter $(n,p)$ is $n \cdot p$ . We thus have + +$$ +n \cdot \mathbb {E} \left[ S ^ {E A} \left(n _ {0}, e \mid m _ {0} \leq n _ {0}\right) \right] = m _ {0} \eta_ {0 0} + (n - n _ {0}) \eta_ {1 1} + (n _ {0} - m _ {0}) \eta_ {1 0} +$$ + +$$ +n \cdot \mathbb {E} \left[ S ^ {E A} (n _ {0}, e \mid m _ {0} > n _ {0}) \right] = n _ {0} \eta_ {0 0} + (n - m _ {0}) \eta_ {1 1} + (m _ {0} - n _ {0}) \eta_ {0 1}, +$$ + +where $\eta_{ij} = e p_{ij} + (1 - e) M_j^b$ . The derivative of $\eta_{ij}$ w.r.t. $e$ is thus $\eta_{ij}' = p_{ij} - M_j^b$ . + +We aim to find the $n_0$ that maximizes + +$$ +\begin{array}{l} \frac {\partial \mathbb {E} \left[ S ^ {E A} (n _ {0} , e) \right]}{\partial e} = \sum_ {m _ {0} = 0} ^ {n _ {0}} \Pr \left(\rho_ {0} = m _ {0}\right) \cdot \left(m _ {0} \eta_ {0 0} ^ {\prime} + (n - n _ {0}) \eta_ {1 1} ^ {\prime} + (n _ {0} - m _ {0}) \eta_ {1 0} ^ {\prime}\right) \\ + \sum_ {m _ {0} = n _ {0} + 1} ^ {n} \operatorname * {P r} \left(\rho_ {0} = m _ {0}\right) \cdot \left(n _ {0} \eta_ {0 0} ^ {\prime} + (n - m _ {0}) \eta_ {1 1} ^ {\prime} + (m _ {0} - n _ {0}) \eta_ {0 1} ^ {\prime}\right). \\ \end{array} +$$ + +Let $n_0$ increase by 1: + +$$ +\begin{array}{l} \frac {\partial \mathbb {E} \left[ S ^ {E A} (n _ {0} + 1 , e) \right]}{\partial e} = \sum_ {m _ {0} = 0} ^ {n _ {0} + 1} \Pr \left(\rho_ {0} = m _ {0}\right) \cdot \left(m _ {0} \eta_ {0 0} ^ {\prime} + (n - n _ {0} - 1) \eta_ {1 1} ^ {\prime} + (n _ {0} + 1 - m _ {0}) \eta_ {1 0} ^ {\prime}\right) \\ + \sum_ {m _ {0} = n _ {0} + 2} ^ {n} \operatorname * {P r} \left(\rho_ {0} = m _ {0}\right) \cdot \left((n _ {0} + 1) \eta_ {0 0} ^ {\prime} + (n - m _ {0}) \eta_ {1 1} ^ {\prime} + (m _ {0} - n _ {0} - 1) \eta_ {0 1} ^ {\prime}\right). \\ \end{array} +$$ + +Therefore, the marginal return of increasing $n_0$ is given by + +$$ +\begin{array}{l} d \nabla S ^ {E A} (n _ {0}) = \frac {\partial \mathbb {E} [ S ^ {E A} (n _ {0} + 1 , e) ]}{\partial e} - \frac {\partial \mathbb {E} [ S ^ {E A} (n _ {0} , e) ]}{\partial e} \\ = \sum_ {m _ {0} = 0} ^ {n _ {0}} \Pr \left(\rho_ {0} = m _ {0}\right) \cdot \left(\eta_ {1 0} ^ {\prime} - \eta_ {1 1} ^ {\prime}\right) + \sum_ {m _ {0} = n _ {0} + 1} ^ {n} \Pr \left(\rho_ {0} = m _ {0}\right) \cdot \left(\eta_ {0 0} ^ {\prime} - \eta_ {0 1} ^ {\prime}\right). \\ \end{array} +$$ + +Note that + +$$ +\begin{array}{l} \eta_ {1 0} ^ {\prime} - \eta_ {1 1} ^ {\prime} = p _ {1 0} - M _ {0} ^ {b} - p _ {1 1} + M _ {1} ^ {b} \\ = - 2 \left(p _ {1 1} - M _ {1} ^ {b}\right) \quad \left(p _ {0 1} = 1 - p _ {1 1}, \text {a n d} M _ {0} ^ {b} = 1 - M _ {1} ^ {b}\right). \\ = - 2 \left(p _ {1 1} - \left(M _ {0} ^ {a} p _ {0 1} + M _ {1} ^ {a} p _ {1 1}\right)\right) \\ = - 2 \left(p _ {1 1} \left(1 - M _ {1} ^ {a}\right) - p _ {0 1} M _ {0} ^ {a}\right) \\ = - 2 M _ {0} ^ {a} \left(p _ {1 1} - p _ {0 1}\right) \\ < 0 \\ \end{array} +$$ + +(By self-prediction.) + +Similarly, we can show that $\eta_{00}^{\prime} - \eta_{01}^{\prime} > 0$ . + +Therefore, when $n_0 = 0$ , $d\nabla S^{EA}(0) > 0$ , and when $n_0 = n$ , $d\nabla S^{EA}(n) < 0$ . Furthermore, $d\nabla S^{EA}(n_0)$ decreases in $n_0$ . The sensitivity-maximizing $n_0$ is thus the largest integer such that $d\nabla S^{EA}(n_0) > 0$ . Let $F(k) = \sum_{m_0=0}^k \operatorname*{Pr}(\rho_0 = m_0)$ . + +$$ +\begin{array}{l} d \nabla S ^ {E A} (n _ {0}) > 0 \\ \Leftrightarrow F \left(n _ {0}\right) \left(\eta_ {1 0} ^ {\prime} - \eta_ {1 1} ^ {\prime}\right) + \left(1 - F \left(n _ {0}\right)\right) \left(\eta_ {0 0} ^ {\prime} - \eta_ {0 1} ^ {\prime}\right) > 0 \\ \Leftrightarrow F (n _ {0}) < \frac {\eta_ {0 0} ^ {\prime} - \eta_ {0 1} ^ {\prime}}{\eta_ {0 0} ^ {\prime} - \eta_ {0 1} ^ {\prime} + \eta_ {1 1} ^ {\prime} - \eta_ {1 0} ^ {\prime}}. \\ \end{array} +$$ + +![](images/7a9871f7205687e0548e59cc2471b0f9c72a18e4a3ae2083d82af418ed3ae552.jpg) \ No newline at end of file diff --git a/stochasticallydominantpeerprediction/images.zip b/stochasticallydominantpeerprediction/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..5bec3bf0cbbeb925f630ca32a06408ab5d5b02ad --- /dev/null +++ b/stochasticallydominantpeerprediction/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0de02e4b867925b88b638bc6332079260380f59d07859b9e1bd7969eeef28c95 +size 983136 diff --git a/stochasticallydominantpeerprediction/layout.json b/stochasticallydominantpeerprediction/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..aba1e99d696d51e3c21dbd6ae8db6e898a337727 --- /dev/null +++ b/stochasticallydominantpeerprediction/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ee39d6fcef97a88fba96a5110b75bd0e83be9e8225410230bc9aa435dcb682c9 +size 1555552 diff --git a/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/86306271-a859-4c6b-a5b6-e0adbcb66873_content_list.json b/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/86306271-a859-4c6b-a5b6-e0adbcb66873_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..f97f2da15528baa6283e7e174dc855a6ae01c19d --- /dev/null +++ b/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/86306271-a859-4c6b-a5b6-e0adbcb66873_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6c198b443e995a7ba639f17cba0681cb455e65301f08b6c78186519be236275d +size 145471 diff --git a/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/86306271-a859-4c6b-a5b6-e0adbcb66873_model.json b/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/86306271-a859-4c6b-a5b6-e0adbcb66873_model.json new file mode 100644 index 0000000000000000000000000000000000000000..4ae47736cab7158a5d0b15d6b9c36a0147fce419 --- /dev/null +++ b/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/86306271-a859-4c6b-a5b6-e0adbcb66873_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5e4397c622f0b0baeb68ca0205c1881a221be4f2273e549cca70015449d0f653 +size 187405 diff --git a/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/86306271-a859-4c6b-a5b6-e0adbcb66873_origin.pdf b/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/86306271-a859-4c6b-a5b6-e0adbcb66873_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..abb4cba7bc15a2bbd023e97892e9528b9ae6fe85 --- /dev/null +++ b/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/86306271-a859-4c6b-a5b6-e0adbcb66873_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b1b8b6dc666d9c35a8783ebca4bd18b4a30f6fdb630cfdb6a0460a55f4a09572 +size 2665466 diff --git a/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/full.md b/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/full.md new file mode 100644 index 0000000000000000000000000000000000000000..08c0b0e05c0a128662e7d7c2c8ae951689dd98b0 --- /dev/null +++ b/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/full.md @@ -0,0 +1,679 @@ +# Stochastic Forward-Forward Learning through Representational Dimensionality Compression + +Zhichao Zhu $^{1,2}$ + +zhichao_zhu@fudan.edu.cn + +Yang Qi1,2,3 + +yang_qi@fudan.edu.cn + +Hengyuan $\mathbf{Ma}^{1,2}$ + +hangyuanma21@m.fudan.edu.cn + +Wenlian Lu1,2,4 + +wenlian@fudan.edu.cn + +Jianfeng Feng $^{1,2,3,*}$ + +jffeng@fudan.edu.cn + +# Abstract + +The Forward-Forward (FF) learning algorithm provides a bottom-up alternative to backpropagation (BP) for training neural networks, relying on a layer-wise "goodness" function with well-designed negative samples for contrastive learning. Existing goodness functions are typically defined as the sum of squared postsynaptic activations, neglecting correlated variability between neurons. In this work, we propose a novel goodness function termed dimensionality compression that uses the effective dimensionality (ED) of fluctuating neural responses to incorporate second-order statistical structure. Our objective minimizes ED for noisy copies of individual inputs while maximizing it across the sample distribution, promoting structured representations without the need to prepare negative samples. We demonstrate that this formulation achieves competitive performance compared to other non-BP methods. Moreover, we show that noise plays a constructive role that can enhance generalization and improve inference when predictions are derived from the mean of squared output, which is equivalent to making predictions based on an energy term. Our findings contribute to the development of more biologically plausible learning algorithms and suggest a natural fit for neuromorphic computing, where stochasticity is a computational resource rather than a nuisance. The code is available at https://github.com/ZhichaoZhu/StochasticForwardForward. + +# 1 Introduction + +Despite being central to the success of traditional deep learning, backpropagation (BP) poses challenges for on-chip learning in neuromorphic systems, as it requires global error signals and symmetric weight transport, both of which are widely regarded biologically implausible and difficult to implement efficiently on neuromorphic hardware [1-5]. Therefore, the forward-forward (FF) learning algorithm proposed by Hinton [6] provides an elegant bottom-up alternative that each layer learns independently by maximizing a "goodness" measure of its activations, eliminating the need for error backward propagation. + +Although FF learning is conceptually simple, its success relies on generating high-quality negative samples, which is highly task-specific and presents a significant practical challenge. Moreover, the goodness function in the original FF learning is the sum of squared postsynaptic activations and it does not account for the role of noise, a ubiquitous feature in both biological neural systems [7, 8] and + +neuromorphic computing hardware [9, 10]. Substantial evidence suggests that neuronal correlated variability could carry rich information [11-13] and can be considered a computational resource [14-18]. Leveraging noise and incorporating postsynaptic neuron correlation into FF learning could potentially lead to more biologically plausible and hardware-adapted alternatives. + +In this work, we extend the FF framework by introducing a novel goodness function termed dimensionality compression that is derived from the effective dimensionality (ED) [19] of neuronal responses. Essentially, ED depicts the second-order statistical structure of neural responses, which can be used to describe representational selectivity. Within-class responses should have low ED, indicating that samples are tied to a specific variable, while across-class responses should have high ED to avoid representation collapse. Instead of generating negative samples, we introduce noise to create isologues of the samples, with the objective of minimizing ED for each sample while maximizing it for all inputs. This promotes locally structured representations that are both compact and discriminative, bypassing the need for negative samples and fitting smoothly into the FF training pipeline. The numerical experiments on standard datasets demonstrate that the proposed method can achieve performance comparable to that of other non-BP methods. Furthermore, this approach inherently recommends using the mean of squared outputs rather than just averaging them for prediction, which means that variability in the outputs can also carry the information interested and can be interpreted as an extension of energy-based learning (EBL) [20-22]. Collectively, our findings provide both a theoretical link between dimensionality and representation learning and a practical direction toward biologically plausible, noise-driven computation. + +# 2 Background and related works + +We begin by reformulating the learning problem of a network in classification tasks. Given an input-target pair $(X,t)$ from $T$ targets, the input is processed through $L$ blocks with $X^{(l)} = f(X^{(l - 1)},\theta^{(l)})$ and $X^{(0)} = X$ . A linear classifier $W$ then produces class scores $Y = X^{(L)}W^T$ and denotes $Y_{i}$ the score for the target $i$ . The question posed is how to modify the model parameters in the absence of BP, while still guaranteeing that the output $Y_{t}$ associated with the true target $t$ achieves maximal discrimination. + +Since learning performance is ultimately assessed through a linear classifier, it is intuitive to assume that if each block produces more discriminative representations, stacking such blocks can improve overall classification. For example, direct feedback alignment (DFA) [23] demonstrates that deep networks can learn effectively without requiring symmetric feedback connections. Instead, DFA delivers global error signals to each layer through fixed random feedback pathways, showing that precise gradient transmission is not essential for credit assignment. + +While DFA relaxes the need for symmetric feedback by transmitting error signals through random fixed pathways, EBL [20-22] eliminates the requirement to propagate explicit errors altogether. Instead, EBL frames learning as minimizing an energy function that depends on local neuronal interactions. During training, the network first clamps the input and fixes the output to the target, allowing neural activities to evolve toward an equilibrium state that minimizes this energy. Weight updates are then computed locally based on the pre- and post-synaptic activities. However, because the energy landscape of one layer depends on others, finding a global equilibrium in deep architectures can be computationally demanding and slow to converge. + +One can simplify the interaction between blocks by training the blocks independently from the bottom up. Greedy InfoMax (GIM) [24] optimizes individual blocks by encouraging each block to preserve information about its input, which can be interpreted as reinforcing each block to learn slow features [25] from its inputs, allowing more scalable unsupervised representation learning without global backpropagation. Hinton's FF algorithm[6] offers a simpler energy-based approach, using a scalar goodness function to distinguish 'positive' and 'negative' samples in each layer. Initially, this function, similar to the energy function in EBLs, is the square of postsynaptic neuronal activities. However, it is necessary to use high-quality negative samples for contrast learning, as optimizing the network with this function alone will fail because it encourages maximum neuron activation. Hinton also noted that binary objectives consisting of one "positive" and one "negative" sample inject limited information per update, which will slow the rate at which meaningful structure can be encoded in the weights. + +Recent advancements in FF-inspired algorithms have removed the requirement for negative samples by employing a supervised learning approach. The Cascaded Forward (CaFo) model [26] enhances the FF algorithm through the integration of convolutional layer blocks, allowing these blocks to independently generate label distributions without requiring negative samples. Papachristodoulou et al. [27] further subdivided the channels within the convolutional layers into $T$ groups, optimizing the mean activation of each group to be more pronounced for certain classes, thus improving linear separability. While these modifications can be effective, implementing such strong label-based supervision in a biologically brain can be challenging, as it necessitates knowing a label for each input pattern. + +Hebbian learning is acknowledged as a biologically realistic mechanism for synaptic plasticity, enabling weight adjustments in an unsupervised manner. However, the principal obstacle lies in its limited efficacy, particularly when deployed in large networks and more complex datasets. Although Journé et al. [28] and Nimmo and Mondragon [29] showed that combining unsupervised Hebbian learning with a well-designed winner-take-all (WTA) mechanism yields satisfactory results, their success involves complex softmax activation functions combining with several optimization tricks. These complications hinder a deeper understanding of the core question: What should a neural block learn when limited to local information and how to define the goodness of neuronal response for goal achievement? + +In this study, we aim to answer these questions by leveraging the continuous, second-order structure in neural responses, providing a more efficient and biologically grounded alternative to contrastive objectives. + +# 3 Effective dimensionality as a goodness function + +Historically, the effective dimensionality (ED) is introduced to describe the equivalent number of orthogonal dimensions that would produce the same overall pattern of covariation of a set of correlated variables [19]. In practice, ED is typically based on the covariance matrix of the data [30, 31]. However, in neural or network representations where the mean activity itself encodes task or stimulus information, it is more appropriate to consider the uncentered second moment as it reflects both the signal (mean configuration) and interaction structure (correlation). Therefore, in this work, we define + +$$ +\operatorname {E D} \left(X ^ {(l)}\right) = \frac {\operatorname {t r} \left(\mathbb {E} \left[ X ^ {(l) T} X ^ {(l)} \right]\right) ^ {2}}{\| \mathbb {E} \left[ X ^ {(l) T} X ^ {(l)} \right] \| _ {F} ^ {2}} = \frac {\left(\sum_ {i = 1} ^ {d} \lambda_ {i}\right) ^ {2}}{\sum_ {i = 1} ^ {d} \lambda_ {i} ^ {2}}, \tag {1} +$$ + +where $\lambda_{i}$ are the eigenvalues of the uncentered covariance matrix of neural activity $X^{(l)}$ . If $X^{(l)}$ has zero mean, this definition reduces to the standard ED based on the covariance matrix that gives a good indication of the number of principal components needed to capture most of the variance of $X^{(l)}$ (Fig. 1a). When then mean is nonzero, ED based on $\mathbb{E}[X^{(l)T}X^{(l)}]$ not only measures the diversity of local variations but also incorporates the global structure imposed by the mean configuration. By varying the mean and covariance of a two-dimensional distribution, we illustrate that ED is maximized for distributions whose dimensions are uncorrelated with identical mean and variance but will reduce otherwise (Fig. 1b). + +In particular, $\mathbb{E}[X^{(l)T}X^{(l)}]$ is also a valid representation of the energy landscape of the $l$ -th block. In conventional EBL, the energy term is typically defined as $\mathrm{tr}(\mathbb{E}[X^{(l)T}X^{(l)}])$ that quantifies the total expected energy and the overall magnitude of population activity while neglecting the correlation structure between neurons. In contrast, ED incorporates the full spectral structure of $\mathbb{E}[X^{(l)T}X^{(l)}]$ , thus providing a correlation-aware measure of the energy landscape. In this sense, ED-based learning can be regarded as an extension of EBL that preserves the connection to energy while emphasizing the structural geometry of the representation. + +To illustrate this concept, we start by considering a population of neurons where each neuron is selectively responsive to a particular type of stimuli (Fig. 1c) and where the response of the neuron conditioned on a specific input is a function of some stimulus features $s$ , commonly referred to as a tuning curve in neuroscience [12]. Consider a representational space spanned by two neurons in Fig. 1c that are selectively activated for a specific feature $s$ informative about the input class. Then, neural responses to inputs within the same class are expected to be more aligned along a specific direction (Fig. 1d). In such an ideal case, the ED for within-class responses (blue and orange + +![](images/2f1e6e5442aeb939cdf41a3f689083f96f8518130ffb2a4e15b08c94f71aa796.jpg) +c + +![](images/ec58ca69897cf501577e219b1b22abcfae9b6b6db0080f70af5e344a2fd37550.jpg) +Figure 1: Assessing the informativeness of neuronal responses through effective dimensionality (ED). a. Illustration of what ED quantifies for a zero-mean Gaussian distribution. The eigenvalues $\lambda_{i}$ computed from the uncentered second moment. ED approaches 1 when variance is concentrated along a single principal direction (left) and increases toward 2 as variance becomes isotropic (right). b. Influence of mean and covariance on ED. In the left and middle panels $\mu_{2} = 0, \sigma_{1}^{2} = \sigma_{2}^{2} = 1$ and $\mu_{2} = \mu_{1} = 0, \sigma_{2}^{2} = 5$ are fixed while varying $\mu_{1}$ and $\sigma_{1}^{2}$ respectively. In the right panel, $\mu = 0, \sigma^{2} = 1$ are fixed while varying the correlation coefficient $\rho_{12}$ . c. Example tuning curves showing neurons selectively responsive to some category-informative features, forming a population code that encodes categorical information. d. ED as a measure of class separability in a two-dimensional response space. Points represent noisy samples from two classes (blue and orange). Within-class responses form clusters with low ED, whereas their mixture (whose uncentered covariance is represented by the dashed gray ellipse) exhibits higher ED, reflecting representational diversity. + +![](images/de408a9eedb7ce1c19659939f26e907c3210f157d399466640ad4b3ed69a63f1.jpg) +d + +![](images/99ba9a7e0c2df5af37203bccbb54f1d63b206948368b5229e5ddabccaee6063d.jpg) + +dots) is expected to be close to one, indicating that inputs can be well explained by fewer variables compared to the dimensionality of the representational space. In contrast, the ED for the mixed responses across classes (whose uncentered covariance is represented by the dashed gray circle) is expected to approach two. Therefore, inputs belonging to different classes are explained by unique variables orthogonal to each other, leading to a linearly separable representation. The learning goal of ED-based learning is thus straightforward: we should minimize ED for responses within each class to promote consistency and robustness, while maximizing ED across all inputs to ensure diversity and discriminability. + +The problem is, the label information is infeasible in the unsupervised setting. To compute ED in the absence of true class labels, we introduce noise into the computing process, which is an essential characteristic of both biological systems and neuromorphic hardware [7, 14, 17, 18, 10]. + +A crucial point is that introducing moderate noise does not hinder the identification of its class (Fig. A1). In other words, the essential features that form the concept of its class remain intact. This strategy enables unsupervised learning of robust and discriminative features, but bypasses the need for explicit class labels or negative sampling. Formally, let $X' \in \mathbb{R}^{B \times F}$ be a batch of data samples and use dropout to create its noisy copies, $X$ , by randomly setting the elements in $X'$ to zero with probability $p$ . For the $l$ -th block, we denote its output (i.e. the corresponding neural responses) as $X^{(l)}$ and $X_i^{(l)}$ are the neural responses to a particular input sample in the batch. The objective function for each block is then consisting of a consistency term and a diversity term, defined as: + +$$ +\mathrm {E D} _ {c} = \frac {1}{B} \sum_ {i = 1} ^ {B} \mathrm {E D} \left(X _ {i} ^ {(l)}\right), \tag {2} +$$ + +and + +$$ +\mathrm {E D} _ {d} = \mathrm {E D} \left(\mathbb {E} \left[ X ^ {(l)} \right]\right) \tag {3} +$$ + +where $B$ is the batch size, $\mathbb{E}[X^{(l)}]$ denotes the averaging of noisy copies of the input batch. $\mathrm{ED}_c$ describes the mean ED of neural responses to noisy realizations of individual input samples and + +![](images/ec226d411ebd30a745795d04009c57bf4ce9253085f5bd97e269e28a84d4cd5a.jpg) +Figure 2: Network architecture and training pipeline. The first dropout layer generates $N$ noisy variants per input and remains active during inference, while the dropout in the linear classifier is used only for regularization during training. Batch normalization (BN) layers stabilize inputs and contain no trainable parameters. Each convolutional block includes a Fixed Orthonormal Projection (FOP) module that projects its output onto a subspace with a pregenerated random orthonormal basis before computing the dimensionality compression loss $L$ . Training proceeds in two phases: (1) Each convolutional block is trained layer-wise for 3 epochs using the proposed loss function $L$ . (2) The convolutional blocks are then frozen, and a linear classifier is trained for 60 epochs using cross-entropy loss, where prediction score for each sample is computed as the mean of squared classifier outputs over the noisy variants. The overall architecture and training pipeline are consistent across all experiments, except for the classifier's input dimensionality, which varies by dataset. + +captures the noise variability, whereas $\mathrm{ED}_d$ describes the ED of the response distribution in different input samples and captures the variability of the data. + +The dimensionality compression loss is then formulated by merging the two components, with a trade-off parameter $\alpha$ that is assigned a default value of 0.5. + +$$ +L = \alpha \mathrm {E D} _ {c} - (1 - \alpha) \mathrm {E D} _ {d}. \tag {4} +$$ + +Conceptually, minimizing $\mathrm{ED}_c$ implicitly suggests a WTA dynamic [28, 29] so that the responses are more distinct in some directions compared to others, leading to a more compact representation. In contrast, maximizing $\mathrm{ED}_d$ prevents the collapse of representation as it reinforces the overall responses that can be explained by as many directions as possible. In this way, the network is encouraged to learn class-discriminative features, leading to a more linearly separable representation. + +Notably, this learning objective also motivates a modification of the inference procedure. Let $Y_{i}$ be the output of the linear classifier for a noisy input sample $X_{i}$ . Rather than averaging the output, we propose using $\mathbb{E}[Y_i^2]$ as the classification score. This approach is equivalent to selecting the neuron that has the minimum energy, as inline with EBL. + +# 4 Experimental design and results + +Architecture. As shown in shown in Fig. 2, we use a network architecture similar to that used by Journé et al. [28] and Nimmo and Mondragon [29] except the activation function chosen to verify the effectiveness of the proposed goodness function. The network consists of three convolutional blocks followed by a fully connected layer for classification. The first block has 96 channels, and the number of channels increases by 4 for each subsequent layer. Each channel is regarded as a neuron due to weight sharing, and the corresponding feature map is treated as samples drawn from an unknown distribution conditioned on the input. See Appendix A1.1 for details. + +Datasets and data preprocessing. We use the MNIST[32], CIFAR-10 and CIFAR-100 datasets [33] to verify the effectiveness of the proposed goodness function. For MNIST, we only use random crop for data augmentation. For CIFAR-10 and CIFAR-100, we first apply zero phase component whitening and then random crop and random horizontal flip for data augmentation. + +Training Pipeline. We divide the training process into two phases. In the first phase, convolutional blocks are trained using the proposed goodness function layer-wise for 3 epochs. Specifically, for each epoch, we train the first block with the proposed loss $L$ , fix it, and then train the next block. + +Table 1: Comparison of validation accuracy (\%) of the proposed method against recent non-BP and FF-inspired approaches, along with BP, that adhere to our architecture across various datasets. The performance of the proposed method are estimated over 5 runs. + +
MethodValidation Accuracy (%)
MNISTCIFAR10CIFAR100
BP99.33 ± 0.0482.50 ± 0.0961.28 ± 0.25
Original FF [6]98.7359-
CaFo FF [26]98.9569.4942.13
CwC FF [27]99.42 ± 0.0878.11 ± 0.4451.32
DFA [23]98.98 ± 0.0573.10 ± 0.5041.00 ± 0.3
Soft Hebbian [28]99.35 ± 0.0380.31 ± 0.1456.00
Hard Hebbian [29]-76-
GIM* [24]99.29 ± 0.0378.19 ± 0.3450.09 ± 0.45
EBL [21]99.5689.665.8
Proposed method99.31 ± 0.0776.96. ± 0.7353.29 ± 1.02
+ +* Reimplementation results. + +This process is repeated for all blocks. Given that the number of neurons far exceeds both the number of samples and the number of classes, direct estimation of ED may become unreliable due to the sparsity of samples. To address this, we propose projecting the output of each block onto a lower-dimensional subspace using a randomly generated set of orthogonal basis vectors before computing the dimensionality compression loss. This projection operation is expected to reduce the estimation variance while preserving the structural properties of the representation. We gradually reduce the projection dimensionality until it aligns with the number of output classes. + +In the second phase, convolutional blocks are frozen, and we train the linear classifier for 60 epochs with the standard cross-entropy loss, where the prediction score is based on the mean of squared outputs over the noisy variants. The best validation accuracy is reported as we care more about the upper bound of the performance. All experiments utilize a NVIDIA RTX 3090 GPU and an Intel Xeon(R) Gold 6226R CPU, as detailed in the Appendix A1.2. + +# 4.1 Proposed method achieves comparable performance with other non-BP methods + +As the development of non-BP methods is still in its infancy, researchers use different network architecture, training protocols, and datasets to evaluate the performance of their methods, making it difficult to make a fair comparison. Therefore, we use other non-BP methods' results reported in their papers for comparison. The results are summarized in Table 1. + +We find that the proposed method achieves comparable performance with other non-BP methods on MNIST and CIFAR-10 datasets, and show its ability to learn useful features in CIFAR100. In the realm of FF methods, the work by Hinton [6] is seen as the baseline. CaFo FF [26] modifies the initial goodness function by independently training each convolutional block for classification. Our method consistently surpasses these two methods in all datasets, as well as the DFA proposed by Nøkland [23]. For CwC FF [27] which revises the convolutional framework by dividing the channels into $C$ groups for supervised learning, our method can match its performance without labels and specific requirements for network architectures. Given that our network architecture mirrors that of Hebbian learning combined with soft or hard WTA mechanisms [28, 29], our results are directly comparable to those works and attain similar performance levels. In addition, as previously noted, optimizing ED naturally introduces a competition mechanism, which can be considered as a general learning principle that these methods strive to achieve. The GIM [24] also achieves a performance similar to that of our method when implemented using this network architecture (see Appendix A1.2). + +For EBL methods, Scellier et al. [21] conducts a comparative study of existing EBL approaches, utilizing a five-layer convolutional Hopfield network to demonstrate that equilibrium-backpropagation [20] achieves superior performance. Although our findings are inferior compared to theirs, the discrepancies may be attributed to the deeper network used and the ability of EBLs to leverage top-down + +![](images/6c4e64e0f35e5fe871f8142fc757862de71267edc9167d596d4a8dc754dd79ed.jpg) +Figure 3: Effect of the trade-off factor $\alpha$ on weight optimization. a. Visualization of the first-layer convolutional kernels trained on CIFAR-10 under different values of $\alpha$ . Each column shows the top 10 channels ranked by the standard deviation of their weights trained with different $\alpha$ . b. Cosine orthogonality score (COS, blue line) and mean standard deviation of first-layer kernels (orange line) as functions of $\alpha$ . A higher COS indicates greater diversity among channels' weights. c. Classification accuracy comparison for $\alpha = 0.0$ and $\alpha = 0.5$ (default). Error bars denote one standard deviation across 5 independent training runs. + +![](images/44beb8d8b2ff6282a3a38f2ebc0d6dfa766bf5f4c2a8ce8974d4e8fab5f7fcf3.jpg) + +![](images/361987394af32ac90f07f6029bf3bf8f9caf16425c67f190422d0d4a83e493ba.jpg) + +information for optimization. Unfortunately, adding more than three convolutional blocks in our experiments leads to a performance drop, which we attribute to the difficulty in preventing the collapse of class-specific features after a stack of highly nonlinear transformation, a technical problem that needs to be solved in future works. + +In contrast to BP under the same architecture and training protocol, the proposed method performs similarly on MNIST. However, the disparities widen with CIFAR-10 and CIFAR-100, a frequent issue for non-BP methods. Since the power of BP comes from its ability to optimize a network globally so that the final output ultimately meets the task's requirements, it leaves room for future work to explore how top-down and bottom-up learning can be combined in a more biological plausible manner, thus making the implementation on hardware more friendly. + +Since one distinct feature of our method is the introduction of noise during training, we also conduct experiments by varying the noise strength (changing the dropout rate $p$ ) and sampling size to assess its impact on performance in the Appendix A1.3. These results demonstrate that both noise strength and sampling size significantly influence model performance. In general, moderate noise levels and appropriate sampling sizes yield optimal results. + +In summary, the proposed method achieves a performance comparable to that of other non-BP methods and even outperforms some of them. Although we currently are unable to demonstrate the efficiency of our method in deeper networks or more complex datasets, the use of noise and the simplicity of the learning objective make it a promising direction to explore how noise can facilitate learning in a biologically plausible manner. + +# 4.2 Compressing dimensionality leads to orthogonal weights + +We next train the first convolutional blocks on CIFAR10 by varying $\alpha$ from 0 to 1 with a step increase of 0.1 to investigate the effect of the trade-off factor $\alpha$ on weight optimization. Each column in Fig. 3a displays the weights of the ten channels with the highest standard deviation in their weights among all channels trained under a given $\alpha$ . The plots indicate a transition point at $\alpha = 0.5$ . When $\alpha > 0.5$ is applied for training, the channel weights collapse into a similar pattern, suggesting the limited capacity of the layer to learn a varied combination of input features. In contrast, when $\alpha \leq 0.5$ , diverse weight patterns emerge. + +To quantify this observation, we compute the cosine orthogonality score (COS) of the kernels, defined as + +$$ +\cos = \frac {1}{K} \sum_ {j < i} \left(1 - \left| \frac {\langle w _ {i} , w _ {j} \rangle}{\| w _ {i} \| \| w _ {j} \|} \right|\right), \tag {5} +$$ + +![](images/9027f023c35cbf0ef5e80f5e44d7a8211599aba8833223cdd346f306b9484cc1.jpg) +Figure 4: Factors affecting task performance. a. Classification accuracy under different inference strategies. $\mathbb{E}[Y^2]$ : proposed method, using the mean squared outputs (energy) based on generated noisy samples as prediction score; $\mathbb{E}[Y]$ : uses the mean of outputs as prediction score. Direct forward: standard inference without noise, using raw inputs. b. Accuracy under different training schemes. Unsup: proposed method, where $\mathrm{ED}_c$ is computed at the instance level based on generated noisy samples. Sup+sampling: generated noisy samples are further grouped by class labels before computing $\mathrm{ED}_c$ . Sup: computes $\mathrm{ED}_c$ directly on labeled data without the need to generate noisy samples. c. Accuracy under different projection strategies. Graded: block outputs are projected with gradually decreasing dimensions (30-20-10 for MNIST and CIFAR-10; 90-150-100 for CIFAR-100). Fixed: all blocks projected to a constant dimension equal to the number of classes. Random: projected to a randomly selected dimension per block. None: no projection. + +![](images/4350c4a9b44fafd258439f48b7cce4d9582217e06d9f3fa02cda5e39012c9774.jpg) + +![](images/e56107d370f38884e79334164ec4f96e67af2f8f46f276076fb7625e187b09d9.jpg) + +where $w_{i}$ is the $i$ -th channel and $K = \frac{c(c - 1)}{2}$ is the number of pairs $\{\mathrm{i},\mathrm{j}\}$ with $c$ being the number of channels. As shown in Fig. 3b, the COS (blue line) is consistently close to one if $\alpha \leq 0.5$ , indicating that the kernels are fully orthogonal to each other. When $\alpha > 0.5$ , the COS decreases significantly and indicates that the weights of different channels tend to be similar, leading to highly redundant feature extraction. This is consistent with the observation in Fig. 3a that all the weights of the channels are similar to each other when $\alpha > 0.5$ . + +This transition point is also observed when using the mean standard deviation of the channels (Fig. 3b, orange line). When $\alpha > 0.5$ , the average standard deviation decreases substantially, suggesting that the numerical values of the weights of a channel tend to be identical, with a low probability of forming a distinctive structure for feature extraction. Similar trends are also observed with the same analysis on MNIST (Fig. A2). + +Clearly, setting $\alpha$ to exceed 0.5, the task would not succeed as the first block would be unable to explore the input's rich features effectively. Hence, simply decreasing $\mathrm{ED}_c$ cannot facilitate learning. Therefore, we investigate the impact of $\mathrm{ED}_c$ by assigning $\alpha = 0.0$ and performing the same experiments as outlined in the previous section to illustrate how task performance is influenced. As shown in Fig. 3c, although depending solely on $\mathrm{ED}_d$ still results in some learning, the performance is inferior to the standard configuration, particularly for CIFAR10 and CIFAR100. Consequently, optimizing $\mathrm{ED}_c$ is an indispensable component for learning. + +# 4.3 Ablation study + +In standard deep learning, the inference stage typically deactivates dropout and involves a standard feedforward pass through the network, with the network's output serving as the prediction score. In Bayesian neural networks, the inference phase generally utilizes the mean of the outputs for the prediction score. Here, due to the stochastic nature of both the goodness function and the training procedure, we advocate using the average energy of the outputs $(\mathbb{E}[Y^2])$ as the prediction score. We then compare these three inference strategies by running the second training phase with different prediction scores while the trained convolutional blocks are fixed. As shown in Fig. 4a, the performance of the proposed method is slightly better than that of the other two methods, indicating that both the mean and the variance can carry information about the labels, while the mean plays the primary role. Interestingly, when we apply t-distributed stochastic neighbor embedding (t-SNE) [34] to visualize the model output as defined by $\mathbb{E}[Y^2]$ (see Fig. A3), the outputs within the same class exhibit a unique fluctuating direction. Moreover, similar classes are not only in close proximity to each other, but also exhibit shared fluctuating directions. + +![](images/8e5a6ba7e426d21e799702cd30298fc11f21fcd7a5ec6e8ebda0468f0c961194.jpg) +Figure 5: Layerwise analysis of representations after training a. Effective dimensionality (ED) of block outputs projected into a lower-dimensional space. $\mathrm{ED}_d$ and $\mathrm{ED}_c$ are colored by blue and orange respectively and the shades denote one standard deviation of $\mathrm{ED}_c$ across classes. The horizontal dashed line marks the projection dimensionality, and the black dashed line shows the compression ratio $\mathrm{ED}_d / \mathrm{ED}_c$ b. Linear separability of each block's representation, measured by training a linear classifier on the output of each block. c. Information decomposition of classifier outputs from each block, assuming a Gaussian mixture model. We report total mutual information (tot), linearly decodable informationlin, and second-order interaction terms (cor), where tot = lin + cor. Results shown are from a randomly selected model trained on CIFAR-10. + +![](images/3333198b012d43446b7a63563574a1a4914a8bd587608ae604396c28fc3e90f6.jpg) + +![](images/0646e9187eae58960ca233a1c6cf4074decdd07807f358eafcaa82d584e62b35.jpg) + +We next investigate whether incorporating label information and grouping same-class samples during goodness optimization improve performance. Since our proposed unsupervised method relies on the generation of noisy variants of each sample, we compare it with two supervised alternatives: Sup+sampling uses labels and applies the same noise sampling as in the unsupervised setting while Sup only uses label information to compute $L$ directly (within-class samples for $\mathrm{ED}_c$ and batch data for $\mathrm{ED}_d$ ). As shown in Fig. 4b, the unsupervised method slightly outperforms the supervised one, suggesting that the model can learn effectively without labels and achieve comparable accuracy. Notably, noise sampling can improve performance, as Sup + sampling is better than Sup. Although such a sampling operation increases computational cost, the advance of neuromorphic hardware has the potential to mitigate this by leveraging intrinsic physical noise. + +Finally, we investigate the role of the projection scheme by considering three strategies, Graded (gradual reduction, as used in the main experiments), Fixed (set to the number of classes), and Random (arbitrary dimensionality no less than the number of classes), performance remains largely comparable. As shown in Fig. 4c, all three projection strategies yield similar performance. However, omitting projection entirely (None) leads to a notable performance drop, indicating that projection is essential for task-related performance. + +# 4.4 Higher compression ratio leads to better performance + +To better understand how the ED is related to task performance, we calculate the $\mathrm{ED}_d$ and $\mathrm{ED}_c$ of the outputs of each block after projecting it into the same subspace used in training (Fig. 5a). Ignoring that each block has a distinct projection dimensionality indicated by the short horizontal dashed lines, we observe that both $\mathrm{ED}_d$ and $\mathrm{ED}_c$ decrease across the blocks. However, the reduction in $\mathrm{ED}_c$ is more pronounced compared to $\mathrm{ED}_d$ , resulting in an increase in the compression ratio (depicted by the black dashed line). From the perspective of such a lower-dimensional manifold, the input samples belonging to the same class can be gradually explained by fewer dimensions, while samples from different classes are more likely to be explained by different dimensions. + +The compression ratio may serve as a valuable indicator of task-related performance efficiency. To demonstrate, we train two linear classifiers on the outputs of the first two blocks respectively, each following the same protocol as that employed for the final block, to assess changes in linear separability across the blocks. As depicted in Fig. 5b), performance improves steadily with the addition of more stacked blocks, corresponding to the increase in compression ratio. + +However, the gradually increased compression ratio in the projected space does not necessarily imply a similar trend in the original space. In Appendix A1.4, we empirically evaluated the activation sparsity in trained models using Hoyer's sparseness [35] and found that the sparsity may not increase monotonically with depth. In addition, the dataset, projection strategy, and network depth can all affect the activation sparsity. + +To better understand how information is represented across layers, we apply information breakdown analysis [36-38] to the output of each block after it passes through the corresponding linear classifier (see Appendix A1.5 for details). We model the classifier output as samples from a Gaussian mixture and estimate the mutual information between these outputs and the class labels. Although the Gaussian assumption may not fully capture the true distribution, it provides insights into the representational structure through moment matching. As shown in Fig. 5c), the mutual information $I_{tot}$ is mainly contributed by the linearly decodable component $I_{lin}$ , with the second order interaction $I_{cor}$ contributing modestly. This suggests that while most label-relevant information is accessible to the linear classifier, a small portion remains embedded in neuron-to-neuron correlations and is not linearly separable. We perform a similar analysis on MNIST (Appendix Fig. A4). Given the simplicity of the task, linear separability is nearly saturated across all layers, even as the final block exhibits a higher compression ratio than the previous one. Here, $I_{cor}$ is negligible compared to $I_{lin}$ , indicating that almost all the information relevant to the task is captured by linear projections, consistent with the high classification accuracy observed. + +# 5 Conclusion and discussion + +We have shown that the proposed dimensionality compression loss $L$ enables effective unsupervised learning within the FF framework without requiring negative samples. By injecting noise to create stochastic variants of each input, the model is trained to minimize the effective dimensionality of responses within a class $(\mathrm{ED}_c)$ while maximizing that across classes $(\mathrm{ED}_d)$ . Experiments on MNIST, CIFAR-10, and CIFAR-100 demonstrate competitive performance using a shallow three-layer CNN, and our ablation studies further indicate that noise benefits learning and that stronger dimensionality compression correlates with better classification accuracy. Despite promising results, our method does not yet achieve state-of-the-art accuracy and has difficulty when scaling to large-scale datasets or deeper architectures. These remain important directions for future research along with the development of biologically plausible implementations on neuromorphic hardware. + +The ED objective may also be applied end-to-end and conceptually relates to self-supervised learning methods such as Barlow Twins [39] and VICReg [40]. However, unlike these approaches, which explicitly enforce feature decorrelation or variance-covariance constraints, our method encourages noisy copies of the same sample to reside on a low-dimensional manifold while ensuring distinct representational directions across different inputs. + +The high-level principle of our method also relates to predictive coding [41-43], which posits that the brain continuously generates predictions about the incoming sensory input and updates its internal model based on prediction errors. In our framework, minimizing $\mathrm{ED}_c$ encourages stable representations that suppress noise-induced variability, whereas maximizing $\mathrm{ED}_d$ maintains discriminative information across stimuli. Such a dual objective balances accurate prediction with representational diversity, aligning with recent advances in predictive coding related learning algorithms [24, 44, 45]. + +There are two main advantages of the proposed method that makes it biological plausible. Firstly, noise is ubiquitous in the brain [7] and can significantly affect the way a neural system represents and processes information [46, 8, 13], an essential characteristic that distinguishes it from deterministic digital computing. In such a stochastic system, we have to measure its outputs multiple times to obtain a reliable estimate. Intuitively, we would like to average the output, which implicitly assumes that variability is just noise that should be ignored. While such firing rate coding is widely adopted in neuroscience, accumulating evidence suggests that variability itself can also be information carriers [11, 47, 48]. Our proposed method uses noise to generate multiple variants of each input sample, which offers an alternative perspective on how the brain may utilize noise and how neuromorphic computing can exploit the intrinsic noise of physical devices to facilitate learning and computation [17, 18, 15, 16, 18]. + +Secondly, optimizing ED is relatively easy to implement based on biologically feasible mechanisms. For instance, WTA competition is widely observed in various brain regions [49-51], which can be used to reduce $\mathrm{ED}_c$ by encouraging a sparse response. Maximizing $\mathrm{ED}_d$ is more subtle but still achievable in biologically plausible neural circuits. For example, Bergoin et al. [52] demonstrate that inhibitory neurons and their plasticity can consolidate and selectively separate learned assemblies and limit memory capacity. Future work could explore the design of local learning rules that balance the power of WTA and inhibitory competition to adaptively meet the proposed objective. + +# Acknowledgments and Disclosure of Funding + +Supported by the National Key R&D Program of China (2019YFA0709502); Supported by National Natural Science Foundation of China (No. 62306078) and the Science & Technology Commission of Shanghai Municipality (No. 25LN3200700); Supported the Ji Hua Laboratory S&T Program (No. X250881UG250) and the Lingang Laboratory (No. LGL-1987); Supported by ZJ Lab and Shanghai Center for Brain Science and Brain-Inspired Technology; Supported by the 111 Project (No. B18015). + +# References + +[1] Catherine D. Schuman, Shruti R. Kulkarni, Maryam Parsa, J. Parker Mitchell, Prasanna Date, and Bill Kay. 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Block indexComponents
1BatchNorm (No affine)Dropout, p = 0.2, creating N = 20 noisy copies5 × 5 standard conv, 96 channels, stride 1, padding 2ReLU4 × 4 MaxPooling, stride 2, padding 1
2BatchNorm (No affine)3 × 3 depthwise conv, 384 channels, stride 1, padding 1ReLU4 × 4 MaxPooling, stride 2, padding 1
3BatchNorm (No affine)3 × 3 depthwise conv, 1536 channels, stride 1, padding 1ReLU2 × 2 AvgPooling, stride 2, padding 0
4FlattenDropout, p = 0.5, inplace, for RegularizationLinear, 13824 input dims for MNIST, 24576 for CIFAR10 and CIFAR100
+ +# A1.2 Details of the experimental design + +All models were trained using the AdamW optimizer with a learning rate of 0.001 and a weight decay of 0.01. A cosine annealing learning rate schedule was applied, with a maximum of 3 and 60 iterations for phase 1 and phase 2 training, respectively. The batch size was fixed at 128 across all experiments. + +During phase 1 training, given an input batch $X' \in \mathbb{R}^{B \times C \times H \times W}$ , after passing through the batch normalization layer of the first convolutional block, we applied dropout with a probability of 0.2 to generate $N = 20$ noisy variants per sample, resulting in $(X \in \mathbb{R}^{B \times N \times C \times H \times W})$ . This dropout was only used in the first block and remained active during inference. + +The output of each block was projected onto a predefined lower-dimensional space using randomly generated orthogonal basis vectors sampled from the Haar distribution (via SciPy). Under the default Graded setting, the projection dimensions were 30-20-10 for MNIST and CIFAR-10 and 90-150-100 for CIFAR-100. The block was then optimized using the proposed $L$ objective computed on the projected output. By default, a trade-off factor $\alpha = 0.5$ was used. To ensure purely local optimization, we detached the output tensors from the computation graph before passing them to the next block, preventing gradient flow across layers. + +During the second phase of training, we trained a linear classifier using cross-entropy loss. Due to the sampling in the first block, the classifier output $Y$ had the shape of $B \times 20 \times 10$ for MNIST and CIFAR-10, and $b \times 20 \times 100$ for CIFAR-100. The uncentered second moment across the sampling dimension was used as the prediction score, and the cross-entropy loss was computed against the ground truth labels. + +To assess the role of $\alpha$ (Fig 3 a-b and Fig. A2), we varied $\alpha$ from 0 to 1 in increments of 0.1 when training the first block in MNIST and CIFAR-10 to investigate changes in channel weights. We also + +trained models five times with $\alpha = 1.0$ and compared their task performance with the default setting (Fig. 3 c). + +To study how inference strategies influence performance (Fig 4 a), we reused the trained convolutional blocks and retrained the classifier using two alternative strategies. The first used the mean of the output as the prediction score, denoted by $\mathbb{E}[Y]$ while the second used a regular feedforward without the sampling operation. + +We also evaluated other projection approaches. In the Fixed setting, the projection dimension for each block was matched to the class count. The blocks used a dimension of 10 for MNIST and CIFAR-10. For CIFAR-100, due to channel constraints, the dimensions were 90 for the first block and 100 for the second and third. In the Random setup, the projection dimensionality for each block was chosen randomly. For MNIST and CIFAR-10, it varied between 10 and 60; for CIFAR-100, it spanned 100 to 300 (except for the first block, set at 90). In the None configuration, the projection operation was omitted. + +To minimize the effort of reimplementing Greedy InfoMax (GIM) [24] for comparison, we resused the scripts provided by the authors and modified the network architecture to align with ours, leaving the hyperparameters used for this method unchanged. The training procedure for phase 1 and phase 2 remained the same as in the main experiments except for two differences. Firstly, for the MNIST dataset, to ensure that there were enough patches for prediction, we scaled the input images to $32 \times 32$ while keeping other settings unchanged. Secondly, instead of layer-wise training, the scripts provided by the authors trained all blocks simultaneously in each iteration. For each dataset, we trained models five times with GIM and reported the mean and standard deviation of the test accuracy. + +# A1.3 The effect of noise strength and sampling size + +To better understand the effect of noise, we conducted experiments by varying the probability of dropout $p$ from 0.1 to 0.5 and the sampling size from 4 to 20 to empirically investigate how the noise level affects performance. Other training settings remained unchanged as in the main experiments. The results are summarized below. + +Table A2: The effect of noise strength and sampling size when training on MNIST. + +
sample size / p0.10.20.30.40.5
499.499.3799.2398.6197.95
899.3599.4199.3899.2998.29
1299.3699.3299.4298.8698.51
1699.3599.3699.4198.7598.54
2099.4499.4199.3498.6598.56
+ +Table A3: The effect of noise strength and sampling size when training on CIFAR10. + +
sample size / p0.10.20.30.40.5
474.8774.6475.674.6874.54
876.5777.0275.877.175.73
1276.877.3277.2476.3376.2
1676.676.8777.6277.6676.7
2077.1976.9776.7577.1277.12
+ +Table A4: The effect of noise strength and sampling size when training on CIFAR100. + +
sample size / p0.10.20.30.40.5
452.0851.9951.1951.1948.59
850.8452.7952.5152.2950.63
1251.4851.5151.8850.9533.54
1653.2152.9752.4551.5249.44
2052.652.6653.7949.9525.21
+ +# A1.4 The sparseness of neural activation + +We empirically evaluated the activation sparsity in trained models with Hoyer's sparseness measure: + +$$ +S (x) = \frac {\sqrt {d} - \frac {\| x \| _ {1}}{\| x \| _ {2}}}{\sqrt {d} - 1}, \tag {6} +$$ + +where $x$ is the activation vector of a layer with $d$ neurons. Here, the vector $x$ is the flattened feature map of a convolutional layer from $N$ noisy variants of an input sample and $d$ is the length of $x$ . We then calculated the mean and standard deviation of Hoyer's sparseness across five independently trained models for each dataset, with 100 samples per class. The results are summarized in Table A5. Here, + +Table A5: Sparseness of neural activation across different layers. + +
Layer indexMNIST (FOP/NP)CIFAR10 (FOP/NP)CIFAR100 (FOP/NP)
Layer 10.52 ± 0.05 / 0.29 ± 0.060.34 ± 0.06 / 0.47 ± 0.040.55 ± 0.08 / 0.49 ± 0.05
Layer 20.24 ± 0.02 / 0.45 ± 0.070.29 ± 0.05 / 0.38 ± 0.100.42 ± 0.06 / 0.38 ± 0.09
Layer 30.46 ± 0.03 / 0.87 ± 0.040.43 ± 0.04 / 0.54 ± 0.060.39 ± 0.07 / 0.54 ± 0.06
+ +FOP means that the model was trained by first projecting the layer's output to a fixed dimensionality using the Graded projecting scheme, and NP means directly optimizing the layer's parameters without the projection operation. These results showed that the sparsity varied with datasets, projection strategy, and network depth. It could range from highly sparse (0.87) to quite dense (0.24). + +# A1.5 Information breakdown analysis + +The mutual information between the target $t$ and the readout $Y$ is defined in terms of the difference between the readout entropy on all stimuli and the conditional entropy for a given class of stimuli as follows. + +$$ +I _ {\text {t o t}} (Y; t) = h (Y) - h (Y | t). \tag {7} +$$ + +The entropy $h(Y)$ and the conditional entropy $h(Y|t)$ are given by + +$$ +h (Y) = - \int p (Y) \log p (Y) d Y, \tag {8} +$$ + +$$ +h (Y | t) = - \sum_ {t} p (t) \int p (Y | t) \log p (Y | t) d Y, \tag {9} +$$ + +where the readout distribution $p(Y|t)$ is modeled as a Gaussian distribution with mean $\hat{\mu}$ and covariance $\hat{\Sigma}$ conditioned on the stimulus class $t$ . The readout distribution over all stimuli is calculated as + +$$ +p (Y) = \sum_ {t} p (t) p (Y | t). \tag {10} +$$ + +We used the information breakdown analysis [36] which further dissects mutual information $I_{\mathrm{tot}}$ into three components, allowing us to assess the amount of contributions of individual readout components $I_{\mathrm{lin}}$ , signal similarity among readout components $I_{\mathrm{sigsim}}$ , and noise correlation in readouts $I_{\mathrm{cor}}$ . The quantity $I_{\mathrm{lin}}$ measures the total amount of information that would be transmitted if all readout components were independent, which is given by + +$$ +I _ {\text {l i n}} = \sum_ {j} [ h (Y _ {j}) - h (Y _ {j} | t) ], \tag {11} +$$ + +where $Y_{j}$ is the $j$ -th component of the readout. The quantity $I_{\mathrm{sigsim}}$ measures the information loss arising from the redundancy due to overlaps between the tuning curves of each readout component, which is given by + +$$ +I _ {\text {s i g s i m}} = h \left(Y _ {\text {i n d}}\right) - \sum_ {j} h \left(Y _ {j}\right), \tag {12} +$$ + +where the independent population response $Y_{\mathrm{ind}}$ is defined by the distribution + +$$ +p \left(Y _ {\text {i n d}} \mid t\right) = \prod_ {j} p \left(Y _ {j} \mid t\right). \tag {13} +$$ + +![](images/2366abddc27c729c5ef8a2108875f3784fda9ae556fd32e0309280f9605e4c60.jpg) +Figure A1: Example of a noisy image from the CIFAR-10 dataset. Each column represents five example images corrupted by randomly dropping its pixels with a probability increment of 0.05. + +![](images/dfe4ed12d502862111befedd7607685bdca19bf0151153abf3a320320dece8cf.jpg) +Figure A2: Extension of Fig 3 analysis to MNIST dataset. + +![](images/7aa4b87f8f86afe4573ddd0ba189b872cdc43b1e4c72d926ab16ded4dabcf1f5.jpg) + +The last component $I_{\mathrm{cor}}$ accounts for the rest part of $I_{\mathrm{tot}}$ , that is, the total amount of information due to noise correlations in the readout. + +$$ +I _ {\text {c o r}} = I _ {\text {t o t}} - I _ {\text {l i n}} - I _ {\text {s i g s i m}}. \tag {14} +$$ + +For simplicity, we absorb $I_{\mathrm{sigsim}}$ in $I_{\mathrm{lin}}$ and use $I_{\mathrm{lin}}$ to represent the information that can be obtained using linear methods. + +We initially used the trained linear classifier shown in Fig. 5b and Fig A4 b to derive the prediction scores $\mathbb{E}[Y^2]$ for each block. Subsequently, these scores were aggregated according to the labels of the inputs, allowing the calculation of their mean and covariance. Using a Gaussian mixture model with 10 components, each sharing the derived mean and covariance, we generated 100,000 samples from each Gaussian component to conduct the information breakdown analysis. + +# A2 Visualization and additional analysis + +![](images/b4ceb57cc9f6d3496eaf8347efc3254d85307aa9e7c3bc8e895f69fc3c2e76a4.jpg) +a + +![](images/2d575df4688cefca14dd05c28ae5cae01dfa1c31f10cac20572e1dc7a5d63eb4.jpg) +b +Figure A3: Comparison of outputs of models trained by the proposed method and backpropagation using T-SNE visualization. a and b are results on MNIST and CIFAR-10 datasets respectively. + +![](images/3f9674629854c96dffb33e1f7ade2923fd5cd33016caf54e9150b0e031c88c14.jpg) +Figure A4: Extension of Fig 5 analysis to MNIST dataset. + +![](images/9bc037d6601ddacfcaccfca01135a4bae8b852d35a52b9e8bbfe20a4d8d15877.jpg) + +![](images/74c570f93c18aedc40260443497cb9b3f61c8ce44d4e56aac1205c924d265d39.jpg) \ No newline at end of file diff --git a/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/images.zip b/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..328f240df84d36bc1d68404a35557f0128123632 --- /dev/null +++ b/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a05bbd57416597b48b2cad72130484182d4477df83f7b2e00769249f807637d8 +size 843052 diff --git a/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/layout.json b/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..52a56a5f6ec64a163ed84da6b37467769e75f3a5 --- /dev/null +++ b/stochasticforwardforwardlearningthroughrepresentationaldimensionalitycompression/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d92b25f3a2c035e4c4afc5f7987cc5df000709f574c73eb4718775f006ee9ee1 +size 779259 diff --git a/stochasticgradientsundernuisances/75acdc97-67ed-4327-afb8-b6bc553037af_content_list.json b/stochasticgradientsundernuisances/75acdc97-67ed-4327-afb8-b6bc553037af_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..54146b6cf5ab390caf41b40fc80d3de624700715 --- /dev/null +++ b/stochasticgradientsundernuisances/75acdc97-67ed-4327-afb8-b6bc553037af_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cb3cb4d3df36dfded327c02bd41e41bf6b6f7547feebb0f403e173002c557c0c +size 642617 diff --git a/stochasticgradientsundernuisances/75acdc97-67ed-4327-afb8-b6bc553037af_model.json b/stochasticgradientsundernuisances/75acdc97-67ed-4327-afb8-b6bc553037af_model.json new file mode 100644 index 0000000000000000000000000000000000000000..351ef3e313c2b9067512883e835507e8f38c62bd --- /dev/null +++ b/stochasticgradientsundernuisances/75acdc97-67ed-4327-afb8-b6bc553037af_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3975043e7907f0110e5eee9a8d1005af82980b986b2888247a60406ba8d4d20a +size 742972 diff --git a/stochasticgradientsundernuisances/75acdc97-67ed-4327-afb8-b6bc553037af_origin.pdf b/stochasticgradientsundernuisances/75acdc97-67ed-4327-afb8-b6bc553037af_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7a324ad6757741862cf0c46d9f3617b0b5c8e7f3 --- /dev/null +++ b/stochasticgradientsundernuisances/75acdc97-67ed-4327-afb8-b6bc553037af_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:eb6d59aa5f52d02b673cf8821f1fd7f089376f001bc3184fce10f64de127042a +size 6603101 diff --git a/stochasticgradientsundernuisances/full.md b/stochasticgradientsundernuisances/full.md new file mode 100644 index 0000000000000000000000000000000000000000..5ac15f316907771c387797ef92e04148bf08171f --- /dev/null +++ b/stochasticgradientsundernuisances/full.md @@ -0,0 +1,4040 @@ +# Stochastic Gradients under Nuisances + +Facheng Yu* Ronak Mehta Alex Luedtke Zaid Harchaoui + +University of Washington + +# Abstract + +Stochastic gradient optimization is the dominant learning paradigm for a variety of scenarios, from classical supervised learning to modern self-supervised learning. We consider stochastic gradient algorithms for learning problems whose objectives rely on unknown nuisance parameters, and establish non-asymptotic convergence guarantees. Our results show that, while the presence of a nuisance can alter the optimum and upset the optimization trajectory, the classical stochastic gradient algorithm may still converge under appropriate conditions, such as Neyman orthogonality. Moreover, even when Neyman orthogonality is not satisfied, we show that an algorithm variant with approximately orthogonalized updates (with an approximately orthogonalized gradient oracle) may achieve similar convergence rates. Examples from orthogonal statistical learning/double machine learning and causal inference are discussed. + +# 1 Introduction + +Machine learning, statistics, and causal inference rely on risk minimization problems of the form + +$$ +\min _ {\theta \in \Theta} \left[ L _ {0} (\theta) := \mathbb {E} _ {Z \sim \mathbb {P}} \left[ \ell_ {0} (\theta ; Z) \right] \right], \tag {1} +$$ + +where $\Theta \subseteq \mathbb{R}^d$ is a parameter space, $Z$ is a $\mathcal{Z}$ -valued random variable, and $\ell_0: \Theta \times \mathcal{Z} \to \mathbb{R}$ is a loss function. The quantity $\ell_0(\theta; z)$ describes the performance of a model parametrized by $\theta \in \Theta$ on a test example $z \in \mathcal{Z}$ . Given only an oracle that provides a stochastic gradient estimate of the objective (1), practitioners are able to train models ranging from linear functions on tabular data to billion-parameter neural networks on vision and language data. + +The success of stochastic gradient descent (SGD) algorithms [Amari, 1993, Bottou and Le Cun, 2005, Bottou and Bousquet, 2007, Ward et al., 2020] has motivated an abundance of work on their theoretical properties under various algorithmic and risk conditions, such as class separability [Soudry et al., 2018], random reshuffling [Gurbüzbalaban et al., 2021], decomposable objectives [Schmidt et al., 2017, Vaswani et al., 2019], quantization noise [Gorbunov et al., 2020], and noise dominance [Sclocchi and Wyart, 2024]. This success has been fueled by machine learning and AI software libraries such as JAX, PyTorch, TensorFlow, and others, which offer a wide range of SGD variants, as long as a loss function can be clearly specified. The gradient is then evaluated automatically on a mini-batch of datapoints and used for stochastic updates. + +Though powerful, this recipe takes one thing for granted: that the learner can always compute the risk (or an unbiased estimate thereof). Indeed, many complex learning problems rely on a risk function that is only partially specified up to a class + +$$ +\mathcal {L} := \left\{L (\cdot , g): g \in \mathcal {G} \right\}, \tag {2} +$$ + +where $\mathcal{G}$ is a possibly infinite-dimensional set and $L:\Theta \times \mathcal{G}\to \mathbb{R}$ is a function of both the target parameter $\theta \in \Theta$ and an unknown nuisance parameter $g\in \mathcal{G}$ . + +This framework originates from semiparametric estimation and inference [Levit, 1979, Linnik, 2008, Bickel et al., 1993, Van der Vaart, 2000], where the risk is a Kullback-Leibler (KL) divergence and $g$ provides information about the true data-generating distribution $\mathbb{P}$ , but is not of primary scientific interest. However, the partially specified loss formulation from (2) is not limited to semiparametric estimation and inference problems. This framework connects to many areas of interest, including profile likelihood based learning [Murphy and and, 2000, Pavlichin et al., 2019, Hao and Orlitsky, 2019] and distributionally robust learning [Shapiro, 2017, Levy et al., 2020, Mehta et al., 2024]. + +For instance, profile likelihood based learning reduces (2) by applying a pointwise minimum over $g \in \mathcal{G}$ to then construct a problem that can be solved in $\theta \in \Theta$ . Another example arises in applications with distribution shifts, for which $g$ represents an unknown test data distribution that may differ from the one from which the training data were drawn. Distributionally robust learning reduces (2) by instead taking a pointwise maximum over $\mathcal{G}$ and solving the resulting problem. Although the pointwise minimum and maximum are natural reductions, it is often the case that there is a "true" $g_0 \in \mathcal{G}$ and the loss class is reduced by first estimating $g_0$ with auxiliary data to produce some $\hat{g}$ , which we refer to as double/debiased machine learning, or DML, following Chernozhukov et al. [2018a]. The problem (1) is then thought to be derived via $L_0(\theta) \equiv L(\theta, g_0)$ in this case (see examples in Sec. 2). This is the focus of this paper. + +Despite the prominence of SGD and DML individually, the convergence guarantees of SGD to recover the risk minimizer with a misspecified nuisance parameter remain unknown. Indeed, after producing $\hat{g}$ , the user typically solves a (full batch) empirical risk minimization problem, i.e. minimizing a sample average approximation of $L(\cdot ,\hat{g})$ . In this paper, we aim to fill this gap by proving convergence guarantees on the sequence $(\theta^{(n)})_{n\geq 1}$ generated by updates of the form + +$$ +\theta^ {(n)} = \theta^ {(n - 1)} - \eta S \left(\theta^ {(n - 1)}, \hat {g}; Z _ {n}\right), \tag {3} +$$ + +where $\eta > 0$ is a learning rate, $\mathcal{D}_n \coloneqq (Z_i)_{i=1}^n$ is a stream of independent data drawn from $\mathbb{P}$ , $\hat{g}$ is a nuisance parameter estimate, and $S: \Theta \times \mathcal{G} \times \mathcal{Z} \to \mathbb{R}^d$ is a stochastic gradient oracle satisfying of $\mathbb{E}_{Z \sim \mathbb{P}}[S(\theta, g; Z)] = \nabla_\theta L(\theta, g)$ for all $(\theta, g) \in \Theta \times \mathcal{G}$ . In particular, when $\mathcal{G}$ lies within a Banach space equipped with norm $\|\cdot\|_{\mathcal{G}}$ , we wish to compare $\theta^{(n)}$ to + +$$ +\theta_ {\star} = \underset {\theta \in \Theta} {\arg \min } L \left(\theta , g _ {0}\right), \tag {4} +$$ + +given conditions on the degree of misspecification $\| \hat{g} - g_0\|_{\mathcal{G}}$ and (approximate) Neyman orthogonality of the risk $L$ [Neyman, 1959]. + +Intuitively, Neyman orthogonal classes of objectives are instances of (2) whose curvature with respect to $\theta$ is insensitive to the choice of $g$ (see Sec. 2 for the formal description). When Neyman orthogonality is satisfied, the double machine learning framework is also known as orthogonal statistical learning (OSL) [Zadik et al., 2018, Liu et al., 2022, Foster and Syrgkanis, 2023]. In addition to the obvious computational considerations, we argue that the SGD perspective in this paper also sheds light on the methodological opportunities in DML/OSL. Indeed, while loss functions are typically specified by the chosen architecture, Neyman orthogonality is often achieved by specialized analytic calculations on the part of the user. Although this property is generally seen as a second-order property of the loss, it can also be viewed as a first-order property of the gradient oracle $S$ . As we detail in Sec. 3, it may be easier and more aligned with the spirit of modern machine learning, to craft Neyman orthogonal gradient oracles instead of losses. + +Contributions. We prove the first theoretical convergence guarantees for SGD under an unknown nuisance model. We find that $\theta^{(n)}$ converges linearly to a neighborhood of $\theta_{\star}$ -the optimum in the well-specified case--with a radius that has a fourth-power (resp. squared) dependence on $\| \hat{g} - g_0 \|_{\mathcal{G}}$ when Neyman orthogonality is (resp. is not) satisfied. Our analysis can also apply to two-stream settings in which the nuisance parameter is learned online alongside the target. We further analyze a new algorithm, called orthogonalized SGD (OSGD), wherein the gradient oracle of a possibly non-orthogonal loss can be iteratively made orthogonal using an "approximately orthogonalized" gradient oracle, which is based on a separate estimation procedure. This algorithm enjoys a convergence guarantee that interpolates between the $\| \hat{g} - g_0 \|_{\mathcal{G}}^4$ (nuisance insensitive) and $\| \hat{g} - g_0 \|_{\mathcal{G}}^2$ (nuisance sensitive) regimes depending on the quality of the orthogonalizing operator. + +We provide an introduction to the OSL/DML setting in Sec. 2. The SGD and OSGD algorithms are described and analyzed in Sec. 3. We discuss related work in Sec. 4; additional discussion can be found in Appx. F. All proofs and numerical illustrations can be found in the Appendix. + +# 2 Orthogonal Statistical Learning + +We first introduce various examples of risk functions in the form of (2), then formally introduce Neyman orthogonality and its implications. As is common in learning settings, the risk will be in the form of an expectation, + +$$ +L (\theta , g) = \mathbb {E} _ {Z \sim \mathbb {P}} \left[ \ell (\theta , g; Z) \right], +$$ + +where $\ell : \Theta \times \mathcal{G} \times \mathcal{Z} \to \mathbb{R}$ is an instance-level loss function. Various assumptions used in the analysis in Sec. 3 (e.g. convexity) may be placed on either the loss $\ell$ or the risk $L$ . In each example, we provide the structure of the data point $Z$ , the set $\mathcal{G}$ , and the loss $\ell$ , and the true $g_0 \in \mathcal{G}$ to fully specify the problem. Here, we interpret "true" to mean that $g_0$ is a parameter of the data-generating distribution (e.g. a propensity score in causal inference), or that $g_0$ satisfies a cost-minimizing or utility-maximizing criterion (as in the profile likelihood or distributional robustness examples from Sec. 1). + +
Examplel(θ,g;z)g0
PLM1/2(y-gY(w)-<θ,x-gX(w))2(EP[Y|W],EP[X|W])
CATE1/2(g(1)(x)-g(0)(x)+\(\frac{(w-g^{\mathrm{prop}}(x))(y-g^{(w)}(x))}{g^{\mathrm{prop}}(x)(1-g^{\mathrm{prop}}(x))}-\langle\theta,x\rangle\)^2(EP[Y|W=1,X],EP[X|W=0,X],EP[W|X])
CRR- [μg(1)(z) log pθ(x)+μg(0)(z) log(1-pθ(x))](EP[Y|W=1,X],EP[X|W=0,X],EP[W|X])
+ +Table 1: Examples of Neyman Orthogonal Losses. + +Example 1 (Partially Linear Model). Let $Z = (X, Y, W) \sim \mathbb{P}$ , where $X$ is an $\mathbb{R}^d$ -valued input, $Y$ is a real-valued outcome, and $W$ is a $\mathcal{W}$ -valued control or confounder. The space $\mathcal{G}$ is a nonparametric class containing functions of the form + +$$ +g = \left(g _ {Y}, g _ {X}\right): \mathcal {W} \rightarrow \mathbb {R} \times \mathbb {R} ^ {d}. +$$ + +Following the construction of Robinson [1988], this $g$ is supplied to the loss + +$$ +\ell_ {\mathrm {P L M}} (\theta , g; z) = \frac {1}{2} (y - g _ {Y} (w) - \langle \theta , x - g _ {X} (w) \rangle) ^ {2}. +$$ + +To ensure $\theta_{\star}$ can be interpreted via the projection of $\mathbb{E}_{\mathbb{P}}[Y|X,W]$ onto partially linear additive functions, the true nuisance is given by $g_0 = (g_{0,X},g_{0,Y})$ , where + +$$ +g _ {0, Y} (w) := \mathbb {E} _ {\mathbb {P}} [ Y \mid W = w ] \text {a n d} g _ {0, X} (w) := \mathbb {E} _ {\mathbb {P}} [ X \mid W = w ]. +$$ + +The next example concerns a quantity widely studied in causal inference [Kennedy, 2023]. + +Example 2 (Conditional Average Treatment Effect). We observe $Z = (X, Y, W) \sim \mathbb{P}$ , where $W$ is a binary treatment assignment. The functions in $\mathcal{G}$ are of the form + +$$ +g = \left(g ^ {(0)}, g ^ {(1)}, g ^ {\text {p r o p}}\right): \mathbb {R} ^ {d} \rightarrow \mathbb {R} \times \mathbb {R} \times (0, 1), +$$ + +and are evaluated (see van der Laan and Luedtke [2014, Thm. 1]) at the loss + +$$ +\ell_ {\text {C A T E}} (\theta , g; z) = \frac {1}{2} \left(g ^ {(1)} (x) - g ^ {(0)} (x) + \frac {w - g ^ {\text {p r o p}} (x)}{g ^ {\text {p r o p}} (x) \left(1 - g ^ {\text {p r o p}} (x)\right)} \left(y - g ^ {(w)} (x)\right) - \langle \theta , x \rangle\right) ^ {2}. +$$ + +For $g_0 = (g_0^{(0)}, g_0^{(1)}, g_0^{\mathrm{prop}})$ nuisance functions $g_0^{(0)}$ and $g_0^{(1)}$ represent the outcome regressions + +$$ +g _ {0} ^ {(0)} (x) := \mathbb {E} _ {\mathbb {P}} [ Y \mid W = 1, X = x ] \text {a n d} g _ {0} ^ {(1)} (x) := \mathbb {E} _ {\mathbb {P}} [ Y \mid W = 0, X = x ], +$$ + +whereas $g_0^{\mathrm{prop}}(x) \coloneqq \mathbb{E}_{\mathbb{P}}[W \mid X = x]$ denotes the propensity score. The minimizer $\theta_{\star}$ indexes a projection of the conditional average treatment effect $g_0^{(1)} - g_0^{(0)}$ onto linear functions. + +Finally, we maintain the data structure from the previous example, but consider a loss corresponding to a different target parameter according to van der Laan et al. [2024]. + +Example 3 (Conditional Relative Risk). We retain all components of the previous example, changing only the loss and assuming that the outcome $Y$ is binary/non-negative. First, consider the "label" function + +$$ +\mu_ {g} ^ {(s)} (z) = g ^ {(s)} (x) + \frac {\mathbb {1} (w = s)}{s g ^ {\mathrm {p r o p}} (x) + (1 - s) (1 - g ^ {\mathrm {p r o p}} (x))} (y - g ^ {(s)} (x)), +$$ + +where $\mathbb{1}(\cdot)$ denotes the indicator function, and the logit-linear predictor $p_{\theta}(x) = e^{\langle \theta ,x\rangle} / (1 + e^{\langle \theta ,x\rangle})$ . To obtain a linear approximation of the log-relative risk $\log (g_0^{(1)} / g_0^{(0)})$ , we employ the cross entropy-type loss function + +$$ +\ell_ {\mathrm {C R R}} (\theta , g; z) = - \left[ \mu_ {g} ^ {(1)} (z) \log p _ {\theta} (x) + \mu_ {g} ^ {(0)} (z) \log (1 - p _ {\theta} (x)) \right]. +$$ + +While the choices of the loss function in Examples 2 and 3 might look opaque to readers outside of causal inference and statistics, they are carefully designed to be Neyman orthogonal. To motivate its definition, notice that, invariably, $g_{0}$ is unknown to the user. In DML, the user may produce or access some $\hat{g} \in \mathcal{G}$ , which is an estimate of $g_{0}$ based on independent training data other than the stream $(Z_{i})_{i=1}^{n}$ used to produce $\theta^{(n)}$ . It is of clear interest how stochastic optimization algorithms (and their resulting minimizers) behave in light of the misspecification of $g_{0}$ , and what precise theoretical conditions govern this behavior. Moreover, as we demonstrate in Sec. 3, these same conditions can be used to analyze procedures for which the user may access additional data to progressively improve the estimate $\hat{g}$ and learn $\theta_{\star}$ simultaneously. We now formally introduce Neyman orthogonality, and by extension, the orthogonal statistical learning (OSL) variant of DML. + +Neyman Orthogonality. For a definition that accounts for a possibly infinite-dimensional function class $\mathcal{G}$ , we introduce the directional derivative, or equivalently, the derivative operator. + +Definition 1 (Derivative Operator). For a functional $F$ mapping from a vector space $\mathcal{F}$ to $\mathbb{R}$ , we define the (directional) derivative operator $\mathrm{D}$ as $\mathrm{DF}(f)[h] := \frac{\mathrm{d}}{\mathrm{d}t} F(f + th)\mid_{t = 0}$ for any $f,h\in \mathcal{F}$ . For a vector-valued $F:\mathcal{F}\mapsto \mathbb{R}^d$ , this derivative operator can be generalized by taking derivatives coordinate-wise. We define the second-order derivative as $\mathrm{D}^2 F(f)[h,h']\coloneqq \mathrm{D}(\mathrm{DF}(f)[h])[h']$ for $h,h'\in \mathcal{F}$ and higher-order derivatives similarly. For functionals of multiple variables $F:\mathcal{F}\times \mathcal{G}\to \mathbb{R}$ , we use the subscript notation $\mathrm{D}_fF(f,g)[h]$ to indicate the directional derivative of $f\mapsto F(f,g)$ with $g\in \mathcal{G}$ fixed. + +We denote by $S_{\theta}(\theta ,g;z) = \nabla_{\theta}\ell (\theta ,g;z)$ the gradient of the loss function w.r.t. the target parameter $\theta \in \Theta$ . Borrowing terminology from statistics, we call this the score, whether $\ell$ is based on a likelihood or not. This constitutes one particular example of a stochastic gradient oracle $S$ used in (3). Overloading notation, the population gradient oracle is defined as $S_{\theta}(\theta ,g) = \mathbb{E}_{Z\sim \mathbb{P}}[\nabla_{\theta}\ell (\theta ,g;Z)]$ . + +Definition 2 (Neyman Orthogonality). The population gradient oracle $S_{\theta}$ is Neyman orthogonal at $(\theta_{\star}, g_0)$ over $\mathcal{G}' \subseteq \mathcal{G}$ if + +$$ +\mathrm {D} _ {g} S _ {\theta} \left(\theta_ {\star}, g _ {0}\right) [ g - g _ {0} ] = 0 \quad f o r a l l g \in \mathcal {G} ^ {\prime}. \tag {5} +$$ + +For $\Theta' \subseteq \Theta$ , the population loss $L$ is Neyman orthogonal at $(\theta_{\star}, g_0)$ over $\Theta' \times \mathcal{G}'$ if + +$$ +\mathrm {D} _ {g} \mathrm {D} _ {\theta} L \left(\theta_ {\star}, g _ {0}\right) \left[ \theta - \theta_ {\star}, g - g _ {0} \right] = 0 \quad f o r a l l (\theta , g) \in \Theta^ {\prime} \times \mathcal {G} ^ {\prime}. \tag {6} +$$ + +![](images/71e5352dd2a7a2dffe6379a48e8e8644d94465781e511c19fdd2a0f1c748ed80.jpg) +Figure 1: Illustration of Neyman Orthogonalization. The first two panels are contour plots of the risk function $L(\theta, g)$ , where $\theta$ varies on the $x$ -axis and $g$ varies on the $y$ -axis. For the orthogonalized risk (center) the contours are approximately axis-aligned. The right plot shows the cross sections of the non-orthogonal risk when fixing $g = g_0, g_1, g_2$ . Due to non-orthogonality, the minimizers $\theta_1$ and $\theta_2$ shown in the first and third plots may drift significantly from $\theta_\star$ . In contrast, the minimizers in the center plot are less sensitive to the choice of $g$ . + +![](images/bf7cee07de03a3580aafca31357a33359fccf9468f24c393268e66f504b4bd5d.jpg) + +In Definition 2, we allow $\Theta' \times \mathcal{G}' \subseteq \Theta \times \mathcal{G}$ to be a proper subset, which not only provides a weaker condition, but also accounts for localization-style arguments. Moreover, since $\mathrm{D}_{\theta}L(\theta_{\star}, g_0)[\theta - \theta_{\star}] = \langle S_{\theta}(\theta_{\star}, g_0), \theta - \theta_{\star} \rangle$ , if the population risk $L$ satisfies (5), then (6) holds for any target parameter class $\Theta' \subseteq \mathbb{R}^d$ . As mentioned above, the risk functions in Examples 1-3 are all Neyman orthogonal at their respective value of $(\theta_{\star}, g_0)$ . In the next section, we will discuss a procedure to make a non-orthogonal gradient oracle "approximately" orthogonal. We illustrate the intended outcome intuitively in Fig. 1 by comparing a generic loss and its orthogonalized counterpart. + +# 3 Stochastic Gradient Optimization + +In this section, we propose two stochastic gradient algorithms, which rely on different choices of the stochastic gradient oracle $S$ used in (3). The first is the familiar stochastic gradient oracle that provides a sample estimate of the gradient $\nabla L(\cdot ,\hat{g})$ for a fixed estimate $\hat{g}$ . The second employs an approximately orthogonalized gradient oracle, or OSGD oracle, to achieve a notion of approximate Neyman orthogonality (in a manner we make precise in this section). We analyze the first algorithm under both non-orthogonal and orthogonal settings, achieving an illustrative breakdown of "nuisance sensitive" and "nuisance insensitive" regimes regarding the theoretical convergence guarantee. For the OSGD algorithm, we prove a convergence guarantee that interpolates between the two regimes, depending on the accuracy of the oracle. + +Notation and Assumptions. For readers' convenience, a table of all the notation we introduce throughout the paper is collected in Appx. A. We maintain the prototypical bias/variance conditions on the stochastic gradient oracle $S$ , that is, an unbiasedness condition and a second-moment growth condition (see Asm. 3(d)). To dispel confusion, note that by "unbiased", we mean specifically that $\mathbb{E}_{Z \sim \mathbb{P}}[S(\theta, g; Z)] = \nabla_{\theta} L(\theta, g)$ for all $(\theta, g)$ , as opposed to the "bias" of replacing $\hat{g}$ with $g_0$ , a terminology sometimes used in DML/OSL. Our analysis will rely partly on the initial distance $r$ of the nuisance estimate $\hat{g}$ to $g_0$ in $\mathcal{G}$ , which defines the ball + +$$ +\mathcal {G} _ {r} \left(g _ {0}\right) = \left\{g \in \mathcal {G}: \| g - g _ {0} \| _ {\mathcal {G}} \leq r \right\}. \tag {7} +$$ + +Various assumptions on the risk will be required to hold only locally, that is, within $\mathcal{G}_r(g_0)$ as opposed to the entire linear space $\mathcal{G}$ . Thus, the assumptions become weaker as the estimate $\hat{g}$ improves. + +# Assumption 3. The following conditions hold: + +(a) Differentiability: For any $(z,g)\in \mathcal{Z}\times \mathcal{G}$ , $\theta \mapsto \ell (\theta ,g;z)$ is twice continuously differentiable. For any $(\theta ,g),(\bar{\theta},\bar{g})\in \Theta \times \mathcal{G}$ , (i) $\mathrm{D}_g^2\mathrm{D}_\theta \ell (\bar{\theta},\bar{g};z)[\theta -\theta_\star ,g - g_0,g - g_0]$ exists and is continuous, (ii) $\mathrm{D}_{\theta}L(\bar{\theta},\bar{g})[\theta -\theta_{\star}]$ and $\mathrm{D}_gL(\bar{\theta},\bar{g})[g - g_0]$ exist, + +$$ +\begin{array}{l} a n d (i i i) \mathbb {E} _ {\mathbb {P}} \left[ \mathrm {D} _ {\theta} \ell (\bar {\theta}, \bar {g}; Z) [ \theta - \theta_ {\star} ] \right] = \mathrm {D} _ {\theta} L (\bar {\theta}, \bar {g}) [ \theta - \theta_ {\star} ], \mathbb {E} _ {\mathbb {P}} \left[ \mathrm {D} _ {g} \ell (\bar {\theta}, \bar {g}; Z) [ g - g _ {0} ] \right] = \\ \mathrm {D} _ {g} L (\bar {\theta}, \bar {g}) [ g - g _ {0} ]. \end{array} +$$ + +(b) First-order optimality: The pair $(\theta_{\star}, g_0)$ satisfies $S_{\theta}(\theta_{\star}, g_0) = 0$ +(c) Smoothness and strong convexity: There exist constants $M \geq \mu > 0$ such that for all $g \in \mathcal{G}_r(g_0)$ , the population risk $L(\cdot, g)$ is $M$ -smooth and $\mu$ -strongly convex for $\theta \in \Theta$ . +(d) Second-moment growth: There exist constants $K_{1},\kappa_{1}\geq 0$ such that + +$$ +\mathbb {E} _ {Z \sim \mathbb {P}} [ \| S _ {\theta} (\theta , g; Z) - S _ {\theta} (\theta , g) \| _ {2} ^ {2} ] \leq K _ {1} + \kappa_ {1} \| \theta - \theta_ {\star} \| _ {2} ^ {2} \quad \forall \theta \in \Theta , g \in \mathcal {G} _ {r} (g _ {0}). +$$ + +(e) Second-order smoothness: There exists a constant $\alpha_{1} \geq 0$ such that + +$$ +\left| \mathrm {D} _ {g} \mathrm {D} _ {\theta} L \left(\theta_ {\star}, \bar {g}\right) \left[ \theta - \theta_ {\star}, g - g _ {0} \right] \right| \leq \alpha_ {1} \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {\mathcal {G}} \quad \forall \theta \in \Theta , g, \bar {g} \in \mathcal {G} _ {r} \left(g _ {0}\right). +$$ + +Asm. 3 does not require Neyman orthogonality at $(\theta_{\star}, g_0)$ . Instead, Asm. 3(a) is a standard differentiability condition. Asm. 3(b) and (c) implies that $\theta_{\star}$ is a unique global minimizer. Asm. 3(d) generalizes the uniformly bounded second moment condition in stochastic optimization (e.g. Cutler et al. [2023]) by adding a quadratic form in $\theta$ , which allows us to consider an unbounded feasible set $\Theta$ and more loss classes. Finally, Asm. 3(d) and (e) can be satisfied when the Hessian of the population risk is a bounded operator. Usually, $K_1$ , $\kappa_1$ , and $\alpha_1$ would depend on the initial nuisance estimation distance $r$ . We provide in Appx. B estimates of the constants in Asm. 3 for each motivating example. We proceed to the main results regarding the convergence of SGD and OSGD. + +Stochastic Gradient Algorithm. Here, we use the standard single-sample stochastic gradient estimate $S = S_{\theta}$ in (3). This leads to the update + +$$ +\theta^ {(n)} = \theta^ {(n - 1)} - \eta S _ {\theta} \left(\theta^ {(n - 1)}, \hat {g}; Z _ {n}\right), \quad \theta^ {(0)} \in \Theta . \tag {8} +$$ + +While the SGD procedure can be easily extended to using a batch of unbiased gradient estimates, we keep our single-observation construction to highlight the most important aspects of the analysis. In order to achieve quantitative guarantees in the Neyman orthogonal setting, which essentially removes certain second-order terms that include $\theta$ and $g$ , we will consider the following higher-order condition in some cases. + +Assumption 4 (Higher-Order Smoothness). The risk $L$ satisfies Definition 2 at $(\theta_{\star}, g_0)$ , and there exists some constant $\beta_1 > 0$ such that + +$$ +\left| \mathrm {D} _ {g} ^ {2} \mathrm {D} _ {\theta} L \left(\theta_ {\star}, \bar {g}\right) \left[ \theta - \theta_ {\star}, g - g _ {0}, g - g _ {0} \right] \right| \leq \beta_ {1} \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {\mathcal {G}} ^ {2} \quad \forall \theta \in \Theta , g, \bar {g} \in \mathcal {G} _ {r} \left(g _ {0}\right). +$$ + +When satisfied, Asm. 4 results in the nuisance insensitivity alluded to at the beginning of this section. Notice that Neyman orthogonality is not necessary to construct a stochastic optimizer, and it is still possible to obtain a nuisance sensitive rate under only Asm. 3. We demonstrate this in Thm. 1. + +Theorem 1. Define $\mathcal{D}_n = (Z_1, \ldots, Z_n)$ , sampled from the product measure $\mathbb{P}^n$ . Suppose that Asm. 3 holds, $\hat{g} \in \mathcal{G}_r(g_0)$ is estimated independently of $\mathcal{D}_n$ , and $\theta^{(0)}, \ldots, \theta^{(n)} \in \Theta$ almost surely. The iterates of (8) satisfy: + +1. Nuisance sensitive: If $\eta \leq \mu /2(M\mu +\kappa_1)$ , then + +$$ +\mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} [ \| \theta^ {(n)} - \theta_ {\star} \| _ {2} ^ {2} ] \leq \left(1 - \frac {\mu \eta}{2}\right) ^ {n} \| \theta^ {(0)} - \theta_ {\star} \| _ {2} ^ {2} + \frac {2 \alpha_ {1} ^ {2}}{\mu^ {2}} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} + \frac {4 K _ {1} \eta}{\mu}. +$$ + +2. Nuisance insensitive: If Asm. 4 also holds, then, for $\eta \leq \mu /2(M\mu +\kappa_1)$ + +$$ +\mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} [ \| \theta^ {(n)} - \theta_ {\star} \| _ {2} ^ {2} ] \leq \left(1 - \frac {\mu \eta}{2}\right) ^ {n} \| \theta^ {(0)} - \theta_ {\star} \| _ {2} ^ {2} + \frac {\beta_ {1} ^ {2}}{2 \mu^ {2}} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {4} + \frac {4 K _ {1} \eta}{\mu}. +$$ + +Note that the assumption that the iterates remain in $\Theta$ is satisfied in common cases. It is satisfied trivially for the first two examples in Sec. 2 because $\Theta = \mathbb{R}^d$ . Another case is when the loss decomposes into the sum of a $G$ -Lipschitz continuous component and the $\ell_2^2$ -norm regularizer, i.e. + +$\ell (\theta ,g;z) = h(\theta ,g;z) + \frac{\mu}{2}\| \theta \| _2^2$ . Then, the iterates and the optimum remain in $\{\theta :\| \theta \| _2\leq G / \mu \}$ (see, e.g., Mehta et al. [2023, Appx. C]), so Definition 2 can be restricted to this compact set. + +Thm. 1 states that SGD converges linearly to a ball around $\theta_{\star}$ with a radius that depends on the bias (due to the replacement of $g_0$ with $\hat{g}$ ) and the variance due to gradient noise. Moreover, the variance component decays proportionally to the learning rate $\eta$ . Under Asm. 4, the bias component can have a significantly more favorable scaling with the error in the nuisance estimate $\| \hat{g} - g_0 \|_{\mathcal{G}}$ — specifically, $\| \hat{g} - g_0 \|_{\mathcal{G}}^4$ instead of $\| \hat{g} - g_0 \|_{\mathcal{G}}^2$ . A similar breakdown into two regimes of the bias scaling occurs in the works of both Foster and Syrgkanis [2023] and Liu et al. [2022] under Asm. 4 (called "slow rate" and "fast rate" there). Importantly, their bounds are based on an exact, offline empirical risk minimization procedure for a fixed training set, i.e. they provide excess risk bounds on the quantity $L(\hat{\theta}_n, g_0) - L(\theta_{\star}, g_0)$ , where + +$$ +\hat {\theta} _ {n} = \underset {\theta \in \Theta} {\arg \min} \frac {1}{n} \sum_ {i = 1} ^ {n} \ell (\theta , \hat {g}; Z _ {i}). +$$ + +In contrast, Thm. 1 accounts for both the expected distance to optimum and the interplay between bias incurred by $\hat{g}$ and the progress achieved at each step. In particular, even when using a constant learning rate, the bias does not accrue on each iterate and is in fact constant in $n$ . When $\hat{\theta}_n$ is designed to be doubly robust, using SGD can achieve double robustness; see Appx. F.5 for an example. + +Orthogonalized Stochastic Gradient Algorithm. Given the marked improvement in the rate of decay of the bias term when an orthogonal loss is used, it is clearly beneficial to do so when possible. We now describe how we can induce orthogonality by adjusting the stochastic gradient oracle using the solution of an auxiliary problem. + +The construction of Neyman orthogonal losses has historically been motivated in semiparametric theory and statistical learning as a means to build efficient – minimum asymptotic variance – full batch statistical estimators [Tsiatis, 2006, Van der Vaart, 2000, Foster and Syrgkanis, 2023, Chernozhukov et al., 2018b]. The approach we follow is inspired by the construction reviewed in Chernozhukov et al. [2018a, Section 2.2]; see also Luedtke [2024]. We also give an intuitive explanation based on least-squares estimation, instead of the usual differential/information geometry one. + +While our construction holds in general spaces, let us first consider the illustrative case when $\mathcal{G} = \mathbb{R}^k$ . At the true parameters $(\theta_{\star}, g_0)$ , consider the problem of finding the best predictor of the $\mathbb{R}^d$ -valued target variable $S_{\theta}(\theta_{\star}, g_0; Z) = \nabla_{\theta} \ell(\theta_{\star}, g_0; Z)$ given the $\mathbb{R}^k$ -valued predictor $\nabla_g \ell(\theta_{\star}, g_0; Z)$ variable in the space $\mathcal{L}(\mathcal{G}, \Theta)$ containing all continuous and linear operators from $\mathcal{G}$ to $\Theta$ : + +$$ +\Gamma_ {0} = \underset {\Gamma \in \mathcal {L} (\mathcal {G}, \Theta)} {\arg \min } \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} \left(\theta_ {\star}, g _ {0}; Z\right) - \Gamma \nabla_ {g} \ell \left(\theta_ {\star}, g _ {0}; Z\right) \| _ {2} ^ {2} \right]. \tag {9} +$$ + +In the special case where $\ell(\theta, g; z) = -\log p_{\theta, g}(z)$ for a density $p_{\theta, g}$ on $\mathcal{Z}$ that governs the random variable $Z$ , the projection direction solving (9) can be shown to satisfy $\Gamma_0 = H_{\theta g}^\top H_{gg}^{-1}$ , where $H_{\theta g} = \nabla_g S_\theta(\theta_\star, g_0) \in \mathbb{R}^{k \times d}$ is the transposed Jacobian and $H_{gg} = \nabla_g^2 L(\theta_\star, g_0) \in \mathbb{R}^{k \times k}$ is the Hessian. The prediction $\Gamma_0 \nabla_g \ell(\theta_\star, g_0; Z)$ accounts for the covariance between $\nabla_\theta \ell(\theta_\star, g_0; Z)$ (the gradient w.r.t. $\theta$ ) and $\nabla_g \ell(\theta_\star, g_0; Z)$ (the gradient w.r.t. $g$ ). It stands to reason that as $\theta \to \theta_\star$ , the random vector + +$$ +S (\theta , g _ {0}; Z) := S _ {\theta} (\theta , g _ {0}; Z) - \Gamma_ {0} \nabla_ {g} \ell (\theta , g _ {0}; Z) = S _ {\theta} (\theta , g _ {0}; Z) - H _ {\theta g} ^ {\top} H _ {g g} ^ {- 1} \nabla_ {g} \ell (\theta , g _ {0}; Z) \tag {10} +$$ + +would be less sensitive to perturbations of $g_{0}$ , as the component of $S_{\theta}(\theta, g_{0}; Z)$ that is predictable through changes in $g_{0}$ is subtracted out. Furthermore, if we are aware that the expectation of $S$ is made zero at $\theta_{\star}$ , then a stochastic gradient scheme based on (10) could conceivably achieve a nuisance insensitive rate guarantee in lieu of Thm. 1. From a variance reduction viewpoint, the correction term in (10) subtracts the regression of the $\theta$ gradient of the loss on the $g$ "gradient" of the loss. By the law of total variance, the variance of the gradient reduces and improves the trajectory of stochastic optimization; see Appx. F.4 for more details. This variational description (10) hints at how such an operator can be computed algorithmically, instead of the historical approach of deriving the operator via calculation by hand on case by case basis. + +Supported by this illustration, we define a generalization that will provide a modified stochastic gradient oracle to use for optimization purposes. Without assuming that $\ell$ is a negative log-likelihood, we generalize the formulas for $\nabla_g\ell (\theta ,g;z)\in \mathbb{R}^k$ , $H_{\theta g}\in \mathbb{R}^{k\times d}$ and $H_{gg}\in \mathbb{R}^{k\times k}$ for when $\mathcal{G}\equiv (\mathcal{G},\langle \cdot ,\cdot \rangle_{\mathcal{G}})$ is an infinite-dimensional Hilbert space. Under regularity conditions on the directional derivatives of $L$ , we have that $\nabla_g\ell (\theta ,g;z)\in \mathcal{G}$ for all $z\in \mathcal{Z}$ , $H_{\theta g} = (H_{\theta g}^{(1)},\dots ,H_{\theta g}^{(d)})\in \mathcal{G}^d$ , and $H_{gg}:\mathcal{G}\to \mathcal{G}$ is a bounded and self-adjoint operator. The formal details of their construction are contained in Appx. D. Just as in (10), we may consider the operator $\Gamma_0:\mathcal{G}\rightarrow \mathbb{R}^d$ , defined elementwise by $[\Gamma_0g]_j = \langle H_{\theta g}^{(j)},H_{gg}^{-1}g\rangle_{\mathcal{G}}$ , where the invertibility of $H_{gg}$ is satisfied by our assumptions preceding Thm. 3. As shown in (9), the orthogonalizing $\Gamma_0$ is defined by both the true nuisance $g_0$ and the target $\theta_{\star}$ , where $g_0$ can usually be learned as some conditional expectation and $\theta_{\star}$ can be learned by our proposed methods. We then construct the central object of the upcoming Thm. 3: the Neyman orthogonalized (NO) gradient oracle + +$$ +S _ {\mathrm {n o}} (\theta , g; z) = S _ {\theta} (\theta , g; z) - \Gamma_ {0} \nabla_ {g} \ell (\theta , g; z). \tag {11} +$$ + +Lemma 2. Suppose that Asm. 3(a) holds and $\mathrm{D}_g^2 L(\theta_\star, g_0)[\cdot, \cdot] : \mathcal{G} \times \mathcal{G} \mapsto \mathbb{R}$ is a bounded and symmetric bilinear form. Then the NO gradient oracle $S_{no}(\theta, g; z)$ is Neyman orthogonal at $(\theta_0, g_0)$ . + +We refer readers to Lem. 15 for the proof. In this context, we refer to the operator $\Gamma_0$ as the "orthogonalizing operator". As a natural sanity check, we note that for a risk function that is already Neyman orthogonal at $(\theta_{\star},g_{0})$ , the NO score $S_{\mathrm{no}}$ is exactly equal to score function $S_{\theta}$ itself since $\Gamma_0 = 0$ . To construct $S_{\mathrm{no}}$ for the non-orthogonal loss, we provide the following example in partially linear model where the corresponding derivations of $\Gamma_0$ and $S_{\mathrm{no}}$ are included in Appx. B.1.2. + +Example 4 (Partially Linear Model). In addition to Example 1, suppose that $Z = (X,Y,W) \sim \mathbb{P}$ satisfies + +$$ +Y = \langle \theta_ {\star}, X \rangle + g _ {0} (W) + \epsilon , +$$ + +where $\theta_{\star} \in \mathbb{R}^{d}$ is the true parameter, $g_0: \mathcal{W} \mapsto \mathbb{R}$ is the true nuisance function, and $\mathbb{E}_{\mathbb{P}}[\epsilon \mid X, W] = 0$ . The space $\mathcal{G} \in L_2(\mathbb{P})$ with inner product $\langle g_1, g_2 \rangle_{\mathcal{G}} = \mathbb{E}_{\mathbb{P}}[g_1(W)g_2(W)]$ for any $g_1, g_2 \in \mathcal{G}$ is a nonparametric class containing functions of the form + +$$ +g: \mathcal {W} \to \mathbb {R}. +$$ + +Consider the following non-orthogonal squared loss function: + +$$ +\tilde {\ell} _ {\mathrm {P L M}} (\theta , g; z) = \frac {1}{2} [ y - g (w) - \langle \theta , x \rangle ] ^ {2}. +$$ + +The orthogonalizing operator for this non-orthogonal loss is + +$$ +\Gamma_ {0}: g \mapsto \mathbb {E} _ {\mathbb {P}} \left[ \mathbb {E} _ {\mathbb {P}} [ X \mid W ] g (W) \right], +$$ + +and the NO gradient oracle is obtained as + +$$ +S _ {\mathrm {n o}} (\theta , g; z) = - (y - g (w) - \langle \theta , x \rangle) (x - \mathbb {E} _ {\mathbb {P}} [ X \mid W = w ]). +$$ + +Motivated by the advantage of a Neyman orthogonal score, we now construct our OSGD algorithm using an estimated the NO score $S_{\mathrm{no}}$ . While $\Gamma_0$ (like $g_0$ ) is unknown to the user in general, using an arbitrary estimate $\hat{\Gamma}$ , we can define the estimated NO score $\hat{S}_{\mathrm{no}}$ oracle via + +$$ +\hat {S} _ {\mathrm {n o}} (\theta , g; z) = S _ {\theta} (\theta , g; z) - \hat {\Gamma} \nabla_ {g} \ell (\theta , g; z). \tag {12} +$$ + +Usually, one can obtain such an estimate $\hat{\Gamma}$ using the same data stream of $\hat{g}$ ; we discuss possible strategies in Appx. F.3. Finally, using $\hat{S}_{\mathrm{no}}$ as the stochastic gradient oracle $S$ in (3), we derive the OSGD update + +$$ +\theta^ {(n)} = \theta^ {(n - 1)} - \eta \hat {S} _ {\mathrm {n o}} \left(\theta^ {(n - 1)}, \hat {g}; Z _ {n}\right), \quad \theta^ {(0)} \in \Theta . \tag {13} +$$ + +To measure the quality of $\hat{\Gamma}$ in our analysis, we use the Frobenius norm $\| \Gamma \|_{\mathrm{Fro}}^2 = \sum_{j=1}^{d} \| \Gamma^{(j)} \|_{\mathrm{op}}^2$ where $\Gamma : \mathcal{G} \to \mathbb{R}^d$ , $\Gamma^{(j)} : g \mapsto [\Gamma g]_j$ and $\| \cdot \|_{\mathrm{op}}$ denotes the usual operator norm for linear functionals. As an example, by the uniqueness of Riesz representations, $\| \Gamma_0 \|_{\mathrm{Fro}}^2 = \sum_{j=1}^{d} \| H_{gg}^{-1} H_{\theta g}^{(j)} \|_{\mathcal{G}}^2$ . + +Using this modified oracle (12) requires similar assumptions to those used in Thm. 1. For ease of presentation, we defer the formal assumption statement to Appx. E, but note that the result depends on the constants $(\mu_{\mathrm{no}}, M_{\mathrm{no}}, \alpha_2, \beta_2, K_2)$ , which are exactly analogous to $(\mu, M, \alpha_1, \beta_1, K_1)$ from Asm. 3. + +Theorem 3. Consider the setting of Thm. 1, with the addition of Asm. 6. When $\| \hat{\Gamma} -\Gamma_0\|_{\mathrm{Fro}} < \mu_{no} / (4\alpha_1)$ and + +$$ +\eta \leq \frac {\mu_ {n o} - 4 \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}}}{1 2 M _ {n o} ^ {2} - 3 \mu_ {n o} ^ {2} / 2 + 4 (\kappa_ {1} + \kappa_ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2})}, +$$ + +the iterates of (13) satisfy: + +$$ +\begin{array}{l} \mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} [ \| \theta^ {(n)} - \theta_ {\star} \| _ {2} ^ {2} ] \leq \left(1 - \frac {\mu_ {n o} \eta}{2}\right) ^ {n} \| \theta^ {(0)} - \theta_ {\star} \| _ {2} ^ {2} + \frac {4 (K _ {1} + K _ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2}) \eta}{\mu_ {n o}} \\ + \frac {3}{\mu_ {n o} ^ {2}} \left(\beta_ {2} ^ {2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {4} + 4 \alpha_ {2} ^ {2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} \cdot \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2}\right). \tag {14} \\ \end{array} +$$ + +Compared with Thm. 1, Thm. 3 shows that OSGD can outperform the nuisance sensitive rate through the correction term $\| \hat{g} - g_0\|_{\mathcal{G}}^2 \cdot \| \hat{\Gamma} - \Gamma_0\|_{\mathrm{Fro}}^2$ , and can align with the nuisance insensitive rate when $\| \hat{\Gamma} - \Gamma_0\|_{\mathrm{Fro}}$ is of the order $\mathcal{O}(\| \hat{g} - g_0\|_{\mathcal{G}})$ . With slightly different assumptions, Thm. 3 can further simplified – see Appx. E for details. + +Interleaving Target and Nuisance Estimation. The results seen thus far have considered for simplicity the estimate $\hat{g}$ to be a fixed element of $\mathcal{G}$ , and included terms that depend on the discrepancy $\| \hat{g} - g_0 \|_{\mathcal{G}}$ . Part of the convenience of these results is that if $\hat{g} \equiv \hat{g}^{(m)}$ is the result of a learning procedure with $m$ independent data points, then statistical bounds on $\| \hat{g}^{(m)} - g_0 \|_{\mathcal{G}}$ (either in expectation or high probability, depending on the situation) can be plugged in to quantify the bias. While the results naturally account for full batch learning procedures, they are also amenable to analyzing staggered procedures in which two data sources are queried to estimate $\theta_\star$ and $g_0$ , respectively. To our knowledge, this is the first theoretical analysis of such an orthogonal stochastic learning method. + +To be precise, suppose that we update the nuisance estimator for $m$ times, leading to the sequence $\hat{g}^{(1)},\dots,\hat{g}^{(m)}$ on a stream of $\mathcal{W}$ -valued data $W_{1},\ldots ,W_{m}$ , sampled i.i.d. from a probability measure $\mathbb{Q}$ . We define $\theta^{(0,n)} = \theta^{(0)}\in \Theta$ , and for the update of $\hat{g}^{(i)}$ for $1\leq i\leq m$ , we define $\theta^{(i,0)} = \theta^{(i - 1,n)}$ and produce the sequence $\theta^{(i,1)},\dots,\theta^{(i,n)}$ using $n$ steps of the SGD update (8) initialized at $\theta^{(i,0)}$ . Consider, for example, the case in which $\mathcal{G}$ is a reproducing kernel Hilbert space (RKHS) with kernel $k(\cdot ,\cdot)$ . With the assumption that the eigenvalues $(\lambda_j)_{j\geq 1}$ of covariance operator $\mathbb{E}_{\mathbb{Q}}[k(W,\cdot)\otimes k(W,\cdot)]$ decay polynomially at order $j^{-\alpha}$ , the nonparametric stochastic gradient algorithm of Dieuleveut and Bach [2016] satisfies $\mathbb{E}_{\mathbb{Q}^m}\left[\| \hat{g}^{(m)} - g_0\|_{\mathcal{G}}^2\right] = \mathcal{O}(m^{-(2\alpha -1) / (2\alpha)})$ . This leads to the following nuisance sensitive rate for a non-Neyman orthogonal loss, by Prop. 22: + +$$ +\mathbb {E} _ {\mathbb {P} ^ {m n} \otimes \mathbb {Q} ^ {m}} [ \| \theta^ {(m, n)} - \theta_ {\star} \| _ {2} ^ {2} ] = \mathcal {O} \left((1 - \mu \eta / 2) ^ {m n} + m ^ {- \frac {2 \alpha - 1}{2 \alpha}} + n ^ {- 1} + \eta\right). +$$ + +As another example, suppose that, in addition, we can estimate $\hat{\Gamma} \equiv \hat{\Gamma}^{(m)}$ using the nonparametric stochastic gradient algorithm of Dieuleveut and Bach [2016] and using the same data stream $(W_{1},\ldots ,W_{m})$ . If there are high probability bounds for $\| \hat{g}^{(m)} - g_0\| _G^2$ and $\| \hat{\Gamma}^{(m)} - \Gamma_0\|_{\mathrm{Fro}}^2$ of the same order as $\mathcal{O}(m^{-(2\alpha -1) / (2\alpha)})$ and $\| \theta^{(m,n)} - \theta_{\star}\| _2^2$ decays as described in Thm. 3, then we have in Prop. 23 that $\| \theta^{(m,n)} - \theta_{\star}\| _2^2 = \mathcal{O}_p\left((1 - \mu \eta /2)^{mn} + m^{-(2\alpha -1) / \alpha} + n^{-1} + \eta\right)$ where the $O_{p}(m^{-(2\alpha -1) / \alpha})$ nuisance bias term decays quadratically faster than the one for a non-Neyman orthogonal loss. We refer the reader to Appx. F.3 for further details of this analysis. + +# 4 Related Work + +We summarize in this section our discussion of the related work. Additional discussions, as well as calculations supporting them, can be found in Appx. F. Possible extensions to SGD variants such as SGD with momentum, averaged SGD, and Adam, are explored in Appx. H. + +From an optimization perspective, it is helpful to know how our convergence bounds perform in the idealized case of a known nuisance, which is equivalent to (1). In this case, Thm. 1 gives the convergence rate $\mathbb{E}_{\mathcal{D}_n\sim \mathbb{P}^n}[\| \theta^{(n)} - \theta_\star \|_2^2 ] = \mathcal{O}((1 - \mu \eta /2)^n +\eta)$ , which aligns with the non-asymptotic SGD convergence rates, in mean-square error [Bach and Moulines, 2011] and in high-probability [Cutler et al., 2023]. Our result requires a smaller learning rate $\eta < \mu /2(M\mu +2\kappa_{1})$ when compared to the requirement $\eta < 1 / (2M)$ from Cutler et al. [2023]. This is entirely due to our bounded moment assumption (see Asm. 3(d)), which contrasts with a uniform boundedness assumption over all $\Theta \times \mathcal{G}_r(g_0)$ . In addition, when the uniform moment bound holds true, $\kappa_{1}$ becomes zero, and our learning rate requirement becomes $\eta < 1 / (2M)$ . + +The comparison with unbiased SGD, and biased SGD, respectively, is also valuable. In the biased SGD literature, the "bias" refers to the fact that $\mathbb{E}_{Z\sim \mathbb{P}}[S(\theta ,\hat{g};Z)]\neq \nabla_{\theta}L(\theta ,g_0)$ in general. The convergence radius then depends on the average value of $\| \mathbb{E}_{Z\sim \mathbb{P}}[S(\theta^{(n)},\hat{g};Z)] - \nabla_{\theta}L(\theta^{(n)},g_0)\| _2^2$ Results along this line result in a radius that may not scale with $\eta$ ; see Demidovich et al. [2023, Thm. 3]. Although this form of bias may be related to $\| \hat{g} -g_0\|_{\mathcal{G}}$ under Lipschitzness conditions on the oracle, it is unclear how to effectively incorporate Neyman orthogonality into these general-purpose approaches. Our approach naturally leverages Neyman orthogonality whenever it holds. + +In the general case of an unknown nuisance, Foster and Syrgkanis [2023], Chernozhukov et al. [2018b] consider full batch learning methods based on analytically crafted Neyman orthogonal risk functions in various scenarios. For regression functionals, the procedure from Chernozhukov et al. [2022] using random forests or neural networks can ensure that the bias term $\| \hat{g} - g_0\|_{\mathcal{G}}^2$ is asymptotically negligible for large samples, in the sense that classical statistical confidence sets for $\theta_{\star}$ are asymptotically valid. These papers are focused on algorithm-independent statistical properties. + +Our work fills this gap, by providing non-asymptotic convergence guarantees for stochastic gradient algorithms under unknown nuisances. Moreover, the modified stochastic gradient oracle moreover offers a flexible solution to deal with general risk functions. If deriving an orthogonalized risk by hand is difficult or impossible, then the strategy we propose can be applied, and Thm. 3 demonstrates that, when the learning rate $\eta$ is set appropriately, the convergence rate using the modified stochastic gradient oracle can be improved to + +$$ +\mathcal {O} \Big (\Big (1 - \frac {\mu_ {\mathrm {n o}} \eta}{2} \Big) ^ {n} + \underbrace {\| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {4} + \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} \cdot \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2}} _ {\text {i m p r o v e m e n t o v e r} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2}} + \eta \Big). +$$ + +When we have the true orthogonalizing $\Gamma_0$ , the improved rate recovers the nuisance insensitive one from Thm. 1. Besides, when $\hat{g}$ converges but $\|\hat{\Gamma} - \Gamma_0\|_{\mathcal{G}}^2 = \mathcal{O}_p(1)$ , the improved rate resembles the nuisance sensitive rate of Thm. 1, plus a $\mathcal{O}(\eta)$ bias term. Thus, the quality of the estimated orthogonalizing operator governs how the optimization interpolates between these two rates. + +Having understood the performance of SGD when using an estimated orthogonalizing operator, one question is how to compute or approximate such an operator. Luedtke [2024] recently demonstrated that an orthogonalizing operator can be derived using algorithmic/reverse mode functional differentiation in many interesting cases. This can also be effective in our stochastic setting. In Sec. 3, using least-squares regression as an illustration, we developed a control variate [Johnstone and Velleman, 1985] interpretation of the variance reduction. This viewpoint offers another venue to develop approximate orthogonalizing operators. + +Conclusion. We established non-asymptotic convergence guarantees for SGD algorithms under nuisances. We showed how the Neyman orthogonality of the loss function can mitigate the sensitivity of SGD algorithms to the effect of nuisances, and obtained results that align with recent ones from the DML/OSL literature in the batch setting. We also presented an iteratively orthogonalized SGD algorithm, whose convergence rate aligns with the rate in the nuisance insensitive regime. Extensions to hypothesis testing and reinforcement learning are interesting venues for future work. + +Acknowledgments. The authors would like to thank L. Liu, V. Roulet, and J. Wellner for valuable comments and suggestions. This work was supported by NSF DMS-2023166, DMS-2134012, DMS-2210216, DMS-2502281, CCF-2019844, NIH, and IARPA 2022-22072200003. Part of this work was performed while R. Mehta and Z. Harchaoui were visiting the Simons Institute for the Theory of Computing, and A. Luedtke was visiting the Institute of Statistical Mathematics, and with the University of Washington. + +Broader Impact. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: No substantive part of this research involved the use of large language models. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Appendix + +# Table of Contents + +A Notation 23 +B Detailed Examples 24 + +B.1 Partially Linear Model 24 +B.2 Conditional Averaged Treatment Effect 27 +B.3 Conditional Relative Risk 28 +B.4 Proofs 29 + +C Convergence Proofs for Stochastic Gradient 47 + +C.1 Overview 47 +C.2 Technical Lemma 47 +C.3 Proof of Theorem 1 50 + +D Orthogonalization with respect to Nuisance 52 + +D.1 Orthogonalization via Riesz Representation 52 +D.2 Technical Lemma 53 + +E Convergence Proofs for Orthogonalized Stochastic Gradient 54 + +E.1 Overview 54 +E.2 Technical Lemma 55 +E.3 Proof of Theorem 3 57 + +F Detailed Discussion 60 + +F.1 Comparison to Biased SGD 60 +F.2 Discussion of Full-sample Orthogonal Statistical Learning and Related Methods 61 +F.3 Discussion of Interleaving Target and Nuisance Estimation 63 +F.4 Interpretation as Control Variate for Variance Reduction 64 +F.5 Discussion of Double Robustness 65 +F.6 Proof of Proposition 22 67 +F.7 Proof of Proposition 23 69 + +G Numerical Experiments 71 + +G.1 Numerical Illustration 71 +G.2 Simulations 72 +G.3 Real Data Analysis 77 + +H Extension to SGD Variants 81 + +H.1 SGD with Momentum and Averaged SGD 81 +H.2 Adam 84 + +# A Notation + +
SymbolDescription
Θ⊆ Rd(G,||·||G)(G,⟨·,⟩G)Finite-dimensional parameter class.Possibly infinite-dimensional nuisance space.The nuisance space as a Hilbert space.
PZ∈Zθ*goThe unknown distribution of interest.The random variable under P.The target of interest.The true nuisance parameter.
l(θ,g;z)L(θ,g)Sθ(θ,g;z)Sθ(θ,g)Sn(θ,g;z)Sn(θ,g)The prespecified loss function.The population loss EZ~P [l(θ,g;Z)].The score function ∇θl(θ,g;z).The population score EZ~P [∇θl(θ,g;z)].The Neyman orthogonalized score.The population Neyman orthogonalized score EZ~P [Sn(θ,g;Z)].
(∇θ,∇g)(Dθ,Dg)HθgHggΓ0The gradient w.r.t. θ and gThe derivative operator w.r.t. θ and g.The transposed Jacobian defined by ∇gSθ(θ*,g0) ∈ GdThe nuisance Hessian operator defined by ∇g2L(θ*,g0)Linear operator defined by [Γ0g]j = ⟨H(jθg),H-1g⟩G.
μM(K1,κ1)(α1,α2)β1The strong convexity constant of L.The smoothness constant of LConstants to bound the second moment of Sθ(θ,g;Z).The second order smoothness constant of L.The higher order smoothness constant of a Neyman orthogonal L.
μnoMno(K2,κ2)β2The strong convexity constant of ∇θSn(θ*,g0).The smoothness constant of ∇θSn(θ*,g0).Constants to bound the second moment of Sn(θ,g;Z).The higher order smoothness constant of Sn.
ηnThe learning rate of stochastic optimization.The iteration of stochastic gradient.The iteration of nuisance estimation.
mThe iteration of nuisance estimation.
+ +Table 2: Notation used throughout the paper. + +# B Detailed Examples + +In this section, we describe in detail how the three examples in Sec. 2 from the main text satisfy Asm. 3 and Asm. 4. We first talk about the partially linear model (PLM) in Appx. B.1, and then introduce the conditional averaged treatment effect (CATE) based on the potential outcomes framework in Appx. B.1. Under the same framework, finally we talk about the conditional relative risk (CRR) in Appx. B.3. In addition, we also study a non-orthogonal loss usually used for PLM in Appx. B.1.2 and an unrestricted loss function for CATE in Appx. B.2.1. The constants for all examples are concluded in Tab. 3 and proofs of lemmas in this section are provided in Appx. B.4. + +# B.1 Partially Linear Model + +# B.1.1 Orthogonal Loss + +We revisit Example 1 from the main text where we consider the target of interest as a solution of a partially linear model. Let $Z = (X, Y, W)$ , where $X$ is an $\mathbb{R}^d$ -valued input, $Y$ is a real-valued outcome, and $W$ is a $\mathcal{W}$ -valued control or confounder. The space $\mathcal{G}$ is a nonparametric class containing functions of the form + +$$ +g = \left(g _ {Y}, g _ {X}\right): \mathcal {W} \rightarrow \mathbb {R} \times \mathbb {R} ^ {d}. +$$ + +Following the construction of Robinson [1988], this $g$ is supplied to the loss + +$$ +\ell (\theta , g; z) = \frac {1}{2} [ y - g _ {Y} (w) - \langle \theta , x - g _ {X} (w) \rangle ] ^ {2}. \tag {15} +$$ + +To ensure $\theta_{\star}$ can be interpreted via the projection of $\mathbb{E}_{\mathbb{P}}[Y|X,W]$ onto partially linear additive functions, the true nuisance is given by $g_0 = (g_{0,X},g_{0,Y})$ , where + +$$ +g _ {0, Y} (w) := \mathbb {E} _ {\mathbb {P}} [ Y \mid W = w ] \text {a n d} g _ {0, X} (w) := \mathbb {E} _ {\mathbb {P}} [ X \mid W = w ]. +$$ + +We define the residual $\epsilon$ at $(\theta_{\star},g_0)$ as + +$$ +\epsilon = Y - g _ {0, Y} (W) - \langle \theta_ {\star}, X - g _ {0, X} (W) \rangle . +$$ + +Lemma 4. Let $\tilde{Y} = Y - g_{0,Y}(w)$ and $\tilde{X} = X - g_{0,X}(w)$ . We assume the following conditions: + +(a) $\lambda_{\mathrm{min}}(\mathbb{E}_{\mathbb{P}}[\tilde{X}\tilde{X}^{\top}])\geq \lambda_0$ for some constant $\lambda_0 > 0$ +(b) $\| \tilde{X}\| _2\leq C_X$ a.s. and $\mathbb{E}_{\mathbb{P}}\left[\epsilon^4\right]\leq \sigma^4$ for some constants $C_X,\sigma >0$ + +Then Asm. 3 and Asm. 4 are satisfied. The target $\theta_{\star}$ is the minimizer of the squared loss: + +$$ +\theta_{\star} = \operatorname *{arg min}_{\theta \in \mathbb{R}^{d}}\mathbb{E}_{\mathbb{P}}[(\tilde{Y} -\tilde{X}^{\top}\theta)^{2}]. +$$ + +The proof of Lem. 4 is provided in Appx. B.4.1. + +# B.1.2 Non-orthogonal Loss + +Suppose that the outcome $Y$ is generated under the partially linear model: + +$$ +Y = \left\langle \theta_ {0}, X \right\rangle + g _ {0} (W) + \epsilon , \tag {16} +$$ + +where $\theta_0\in \mathbb{R}^d$ is the true parameter, $g_0:\mathcal{W}\mapsto \mathbb{R}$ is the true nuisance function and $\mathbb{E}_{\mathbb{P}}[\epsilon \mid X,W] = 0$ . The space $\mathcal{G}$ is a nonparametric class containing functions of the form + +$$ +g: \mathcal {W} \mapsto \mathbb {R}. +$$ + +We can also consider the following non-orthogonal squared loss function: + +$$ +\ell (\theta , g; z) = \frac {1}{2} [ y - g (w) - \langle \theta , x \rangle ] ^ {2}. \tag {17} +$$ + +
ExampleμMK1κ1α1β1
(1) Orthogonal PLMλ0C2X+ r218C2Xσ2+ O(r2)18C4X+ O(r2)2(1 + ||θ*||2)r2(1 + ||θ*||2)
(2) Non-Orthogonal PLMλ0C2X6C2X(σ2+ 2r2)2C4XCX-
(3) Unrestricted CATEc02λ0C2X(1+r2)12C2X(σ2+ 4(CX||θ*||2+Cτ)2)+ O(r2)27C4X+ O(r2)O(r)CX(1+4(CX||θ*||2+Cτ))
(4) Restricted CATEλ0C2X27C2X(σ2+ 2C2)+ O(r2)3C4X2CX(2r+3)r4c0-2CX(1+r)
(5) CRRC2δλ0C2X(1+2δ-1r2)24C2X(δ-2+ 4c0-1C2)+ O(r2)6C4X(1+6(δ-2+ 4c0-1C2))+ O(r2)2CX(c0-1+1)r4c0-2CX(1+r)
+ +Table 3: Constants for All Examples. + +We define the residual $\epsilon$ at $(\theta_{\star},g_0)$ as + +$$ +\epsilon = Y - g _ {0} (W) - \langle \theta_ {\star}, X \rangle . +$$ + +Lemma 5. We assume the following conditions: + +(a) $\lambda_{\min}(\mathbb{E}_{\mathbb{P}}[XX^{\top}])\geq \lambda_{0}$ for some constant $\lambda_0 > 0$ +(b) $\| X\|_{\infty}\leq C_X$ a.s. and $\mathbb{E}_{\mathbb{P}}\left[\epsilon^2\right]\leq \sigma^2$ for some constants $C_X,\sigma >0$ + +Then Asm. 3 is satisfied and the target $\theta_{\star}$ is the true parameter, i.e., $\theta_{\star} = \theta_0$ + +The proof of Lem. 5 is provided in Appx. B.4.2. + +Orthogonalization. We can perform our orthogonalization method to obtain the Neyman orthogonal gradient oracle for this non-orthogonal loss. For any $h_1, h_2 \in \mathcal{G}$ , we define the inner product of $\mathcal{G}$ as + +$$ +\left\langle h _ {1}, h _ {2} \right\rangle_ {\mathcal {G}} = \mathbb {E} _ {\mathbb {P}} [ h _ {1} (W) h _ {2} (W) ]. \tag {18} +$$ + +For any $(\theta, g, z) \in \Theta \times G \times \mathcal{Z}$ by Definition 1 the derivative of non-orthogonal loss (17) along the direction of $h_1$ is given by + +$$ +\mathrm {D} _ {g} \ell (\theta , g; z) [ h _ {1} ] = \frac {\mathrm {d}}{\mathrm {d} t} \left(\frac {1}{2} [ y - (g + t h _ {1}) (w) - \langle \theta , x \rangle ] ^ {2}\right) = - (y - g (w) - \langle \theta , x \rangle) h _ {1} (w). \tag {19} +$$ + +Do derivative on $\mathrm{D}_g\ell (\theta ,g;z)[h_1]$ along the direction of $h_2$ and we have + +$$ +\mathrm {D} _ {g} ^ {2} \ell (\theta , g; z) [ h _ {1}, h _ {2} ] = \frac {\mathrm {d}}{\mathrm {d} t} \left(- (y - (g + t h _ {2}) (w) - \langle \theta , x \rangle) h _ {1} (w)\right) = h _ {1} (w) h _ {2} (w), \tag {20} +$$ + +which implies + +$$ +\mathrm {D} _ {g} ^ {2} L (\theta_ {\star}, g _ {0}) [ h _ {1}, h _ {2} ] = \mathbb {E} _ {\mathbb {P}} \left[ \mathrm {D} _ {g} ^ {2} \ell (\theta_ {\star}, g _ {0}; Z) [ h _ {1}, h _ {2} ] \right] = \mathbb {E} _ {\mathbb {P}} \left[ h _ {1} (W) h _ {2} (W) \right]. +$$ + +By the definition in (84), we have $H_{gg} = \mathbf{I}$ the identity operator. In addition, do derivative on the score along the direction of $h \in \mathcal{G}$ and we have + +$$ +\mathrm {D} _ {g} S _ {\theta} (\theta , g; z) [ h ] = \frac {\mathrm {d}}{\mathrm {d} t} \left(- (y - (g + t h) (w) - \langle \theta , x \rangle) x\right) = h (w) x, +$$ + +which implies that + +$$ +\mathrm {D} _ {g} S _ {\theta} (\theta , g) [ h ] = \mathbb {E} _ {\mathbb {P}} \left[ S _ {\theta} (\theta , g; Z) [ h ] \right] = \mathbb {E} _ {\mathbb {P}} \left[ h (W) \mathbb {E} _ {\mathbb {P}} \left[ X \mid W \right] \right]. +$$ + +By the definition in (83), we have $H_{\theta g} = \mathbb{E}_{\mathbb{P}}[X \mid W]$ . Thus, by (85) we have + +$$ +\Gamma_ {0}: g \mapsto \left\langle \mathbb {E} _ {\mathbb {P}} [ X \mid W ], g \right\rangle_ {\mathcal {G}} = \mathbb {E} _ {\mathbb {P}} \left[ \mathbb {E} _ {\mathbb {P}} [ X \mid W ] g (W) \right]. \tag {21} +$$ + +Thus, the Neyman orthogonalized gradient oracle defined in (86) is given by + +$$ +\begin{array}{l} S _ {\mathrm {n o}} (\theta , g; z) = S _ {\theta} (\theta , g; z) - \mathrm {D} _ {g} \ell (\theta , g; z) [ \mathbb {E} _ {\mathbb {P}} [ X \mid W = w ] ] \\ = - (y - g (w) - \langle \theta , x \rangle) (x - \mathbb {E} _ {\mathbb {P}} [ X \mid W = w ]). \tag {22} \\ \end{array} +$$ + +Lemma 6. Consider the bounded linear operator $\hat{\Gamma}:\mathcal{G}\mapsto \mathbb{R}^d$ such that $[\hat{\Gamma} g]_j = \langle \hat{\gamma}^{(j)},g\rangle_{\mathcal{G}},\forall g\in \mathcal{G}$ for some $\hat{\gamma}^{(j)}\in \mathcal{G},j = 1,\ldots ,d.$ Let $\tilde{Y} = Y - g_{0,Y}(w)$ and $\tilde{X} = X - g_{0,X}(w)$ . We assume the following conditions: + +(a) $\lambda_{\mathrm{min}}(\mathbb{E}_{\mathbb{P}}[\tilde{X}\tilde{X}^{\top}])\geq \lambda_0$ for some constant $\lambda_0 > 0$ +(b) $\| \tilde{X}\| _2\leq C_X$ a.s. and $\mathbb{E}_{\mathbb{P}}\left[\epsilon^4\right]\leq \sigma^4$ for some constants $C_X,\sigma >0$ + +Then Asm. 6 is satisfied. + +The proof of Lem. 6 is provided in Appx. B.4.3. + +# B.2 Conditional Averaged Treatment Effect + +We now introduce examples in causal inference which are established based on the potential outcomes framework. The potential outcomes framework [Rubin, 1974] has been widely used in causal inference. Let $Z = (W,X,Y) \in \{0,1\} \times \mathbb{R}^d \times \mathbb{R}$ under some distribution $\mathbb{P}$ . We posit the existence of potential outcomes $Y(1), Y(0) \in \mathbb{R}$ . The conditional averaged treatment effect (CATE) is then defined as + +$$ +\tau_ {0} (x) = \mathbb {E} _ {\mathbb {P}} [ Y (1) - Y (0) \mid X = x ]. +$$ + +To identify $\tau_0(x)$ and the following causal assumptions are required: + +Assumption 5. The following conditions hold: + +(a) (consistency) $Y = Y(W)$ . +(b) (unconfoundedness) $Y(w) \perp W \mid X$ for all $w \in \{0,1\}$ . +(c) (positive overlap) $c_{0} \leq \mathbb{P}(W = 1 \mid X) \leq 1 - c_{0}$ a.s. for some $c_{0} > 0$ . + +Under Asm. 5, $\tau_0$ can be identified by observed data since + +$$ +\begin{array}{l} \tau_ {0} (x) = \mathbb {E} _ {\mathbb {P}} \left[ Y (1) - Y (0) \mid X = x \right] \\ = \mathbb {E} _ {\mathbb {P}} [ Y (1) \mid W = 1, X = x ] - \mathbb {E} _ {\mathbb {P}} [ Y (0) \mid W = 0, X = x ] \\ = \mathbb {E} _ {\mathbb {P}} [ Y \mid W = 1, X = x ] - \mathbb {E} _ {\mathbb {P}} [ Y \mid W = 0, X = x ]. \\ \end{array} +$$ + +# B.2.1 Unrestricted Nuisance + +We observe $Z = (X, Y, W)$ , where $W$ is a binary treatment assignment. The functions in $\mathcal{G}$ are of the form + +$$ +g = \left(g ^ {\mathrm {o u t}}, g ^ {\mathrm {p r o p}}\right): \mathbb {R} ^ {d} \to \mathbb {R} \times \mathbb {R}, +$$ + +and are evaluated (see Nie and Wager [2021, Eq. (2)]) at the loss + +$$ +\ell (\theta , g; z) = \frac {1}{2} (y - g ^ {\text {o u t}} (x) - (w - g ^ {\text {p r o p}} (x)) \langle \theta , x \rangle) ^ {2}. \tag {23} +$$ + +For $g_0 = (g_0^{\mathrm{out}}, g_0^{\mathrm{prop}})$ nuisance functions $g_0^{\mathrm{out}}$ and $g_0^{\mathrm{prop}}$ represent the outcome regression and the propensity score, respectively: + +$$ +g _ {0} ^ {\text {o u t}} (x) := \mathbb {E} _ {\mathbb {P}} \left[ Y \mid X = x \right] \text {a n d} g _ {0} ^ {\text {p r o p}} (x) := \mathbb {E} _ {\mathbb {P}} \left[ W \mid X = x \right]. +$$ + +We define the residual $\epsilon$ under the true model as + +$$ +\epsilon = Y - g _ {0} ^ {\text {o u t}} (X) - \left(W - g _ {0} ^ {\text {p r o p}} (X)\right) \tau_ {0} (X). +$$ + +Lemma 7. We assume Asm. 5 and the following conditions hold: + +(a) $\lambda_{min}(\mathbb{E}_{\mathbb{P}}[XX^{\top}])\geq \lambda_{0}$ for some constant $\lambda_0 > 0$ +(b) $\| X\| _2\leq C_X$ and $|\tau_0(X)|\leq C_\tau$ a.s. and $\mathbb{E}_{\mathbb{P}}[\epsilon^4 ]\leq \sigma^4$ for some constants $C_X,C_\tau ,\sigma >0$ + +Then Asm. 3 and Asm. 4 are satisfied. The target $\theta_{\star}$ is the minimizer of the squared loss: + +$$ +\theta_ {\star} = \underset {\theta \in \mathbb {R} ^ {d}} {\arg \min } \mathbb {E} _ {\mathbb {P}} \left[ (W - g _ {0} ^ {p r o p} (X)) ^ {2} (\tau_ {0} (X) - X ^ {\top} \theta) ^ {2} \right]. +$$ + +The proof of Lem. 7 is provided in Appx. B.4.4. + +# B.2.2 Restricted Nuisance + +We observe $Z = (X, Y, W)$ , where $W$ is a binary treatment assignment. Here we restrict the propensity model as $g^{\mathrm{prop}}: \mathbb{R}^d \mapsto (0, 1)$ . The functions in $\mathcal{G}$ are of the form + +$$ +g = (g ^ {(0)}, g ^ {(1)}, g ^ {\mathrm {p r o p}}): \mathbb {R} ^ {d} \to \mathbb {R} \times \mathbb {R} \times (0, 1), +$$ + +and are evaluated (see van der Laan and Luedtke [2014, Thm. 1]) at the loss + +$$ +\ell (\theta , g; z) = \frac {1}{2} \left(g ^ {(1)} (x) - g ^ {(0)} (x) + \frac {w - g ^ {\operatorname {p r o p}} (x)}{g ^ {\operatorname {p r o p}} (x) \left(1 - g ^ {\operatorname {p r o p}} (x)\right)} \left(y - g ^ {(w)} (x)\right) - \langle \theta , x \rangle\right) ^ {2}. \tag {24} +$$ + +This loss also appears in Foster and Syrgkanis [2023, Eq. (23)]. For $g_0 = (g_0^{(0)}, g_0^{(1)}, g_0^{\mathrm{prop}})$ nuisance functions $g_0^{(0)}$ and $g_0^{(1)}$ represent the outcome regressions + +$$ +g _ {0} ^ {(0)} (x) := \mathbb {E} _ {\mathbb {P}} [ Y \mid W = 1, X = x ] \text {a n d} g _ {0} ^ {(1)} (x) := \mathbb {E} _ {\mathbb {P}} [ Y \mid W = 0, X = x ]. +$$ + +We define the residual $\epsilon$ as + +$$ +\epsilon = \frac {W - g _ {0} ^ {\mathrm {p r o p}} (X)}{g _ {0} ^ {\mathrm {p r o p}} (X) (1 - g _ {0} ^ {\mathrm {p r o p}} (X))} (Y - g _ {0} ^ {(W)} (X)). +$$ + +Lemma 8. We assume Asm. 5 and the following conditions hold: + +(a) $\lambda_{min}(\mathbb{E}_{\mathbb{P}}[XX^{\top}])\geq \lambda_0$ for some constant $\lambda_0 > 0$ +(b) $\mathbb{E}_{\mathbb{P}}[\epsilon^2] \leq \sigma^2$ , $\|X\|_2 \leq C_X$ , and $|Y - g_0^{(w)}(X)| \leq C_Y$ , $w = 0,1$ a.s. for some constants $\sigma, C_X, C_Y > 0$ . + +Then Asm. 3 and Asm. 4 are satisfied. The target $\theta_{\star}$ is the minimizer of the squared loss: + +$$ +\theta_ {\star} = \underset {\theta \in \mathbb {R} ^ {d}} {\arg \min } \mathbb {E} _ {\mathbb {P}} \left[ (\tau_ {0} (X) - X ^ {\top} \theta) ^ {2} \right]. +$$ + +The proof of Lem. 8 is provided in Appx. B.4.5. + +# B.3 Conditional Relative Risk + +We retain all components of the previous example, changing only the loss and assuming that the outcome $Y$ is binary/non-negative. First, consider the "label" function + +$$ +\mu_ {g} ^ {(s)} (z) = g ^ {(s)} (x) + \frac {\mathbb {1} (w = s)}{s g ^ {\mathrm {p r o p}} (x) + (1 - s) (1 - g ^ {\mathrm {p r o p}} (x))} (y - g ^ {(s)} (x)), +$$ + +where $\mathbb{1}(\cdot)$ denotes the indicator function, and the log-linear predictor $p_{\theta}(x) = e^{\langle \theta ,x\rangle} / (1 + e^{\langle \theta ,x\rangle})$ Following Example 2 in van der Laan et al. [2024], we then employ the cross entropy-type loss function + +$$ +\ell (\theta , g; z) = - \left[ \mu_ {g} ^ {(1)} (z) \log p _ {\theta} (x) + \mu_ {g} ^ {(0)} (z) \log \left(1 - p _ {\theta} (x)\right) \right]. \tag {25} +$$ + +Lemma 9. We assume the following conditions: + +(a) $\lambda_{\min}(\mathbb{E}_{\mathbb{P}}[XX^{\top}])\geq \lambda_0$ for some constant $\lambda_0 > 0$ +(b) $\| X\| _2\leq C_X$ and $Y(w) - g_0^{(w)}(X)\leq C_Y,w = 0,1$ a.s.for some constants $C_X,C_Y > 0$ +(c) $\delta \leq g_0^{(0)}(X) + g_0^{(1)}(X) \leq \delta^{-1}$ a.s. for some constant $\delta > 0$ . + +Then Asm. 3 and Asm. 4 are satisfied. The target $\theta_{\star}$ is the minimizer of the weighted cross entropy loss: + +$$ +\theta_ {\star} = \underset {\theta \in \mathbb {R} ^ {d}} {\arg \min } - \mathbb {E} _ {\mathbb {P}} \left[ g _ {0} ^ {(1)} (X) \log p _ {\theta} (X) + g _ {0} ^ {(0)} (X) \log (1 - p _ {\theta} (X)) \right]. +$$ + +The proof of Lem. 9 is provided in Appx. B.4.6. + +# B.4 Proofs + +# B.4.1 Proof of Lemma 4 + +Proof. We consider the following loss: + +$$ +\ell (\theta , g; z) = \frac {1}{2} (y - g _ {Y} (w) - \langle \theta , x - g _ {X} (w) \rangle) ^ {2}, +$$ + +with the corresponding risk function defined as + +$$ +L (\theta , g) = \frac {1}{2} \mathbb {E} _ {\mathbb {P}} \left[ (Y - g _ {Y} (W) - \langle \theta , X - g _ {X} (W) \rangle) ^ {2} \right]. +$$ + +Let $\tilde{Y} = Y - g_{0,Y}(w)$ and $\tilde{X} = X - g_{0,X}(w)$ . By definition, the target $\theta_{\star}$ is the minimizer of the squared loss: + +$$ +\theta_ {\star} = \underset {\theta \in \mathbb {R} ^ {d}} {\arg \min } \mathbb {E} _ {\mathbb {P}} \left[ (\tilde {Y} - \tilde {X} ^ {\top} \theta) ^ {2} \right] = \mathbb {E} _ {\mathbb {P}} \left[ \tilde {X} \tilde {X} ^ {\top} \right] ^ {- 1} \mathbb {E} _ {\mathbb {P}} \left[ \tilde {Y} \tilde {X} \right]. \tag {26} +$$ + +Differentiating $\ell (\theta ,g;z)$ with respect to $\theta$ , we obtain the gradient and Hessian $w.r.t.\theta$ as + +$$ +\begin{array}{l} S _ {\theta} (\theta , g; z) = - (y - g _ {Y} (w) - \langle \theta , x - g _ {X} (w) \rangle) (x - g _ {X} (w)), \\ H _ {\theta \theta} (\theta , g; z) = (x - g _ {X} (w)) (x - g _ {X} (w)) ^ {\top}. \\ \end{array} +$$ + +The expected gradient and expected Hessian are then obtained as + +$$ +\begin{array}{l} S _ {\theta} (\theta , g) = - \mathbb {E} _ {\mathbb {P}} \left[ (Y - g _ {Y} (W) - \langle \theta , X - g _ {X} (W) \rangle) (X - g _ {X} (W)) \right], \\ H _ {\theta \theta} (\theta , g) = \mathbb {E} _ {\mathbb {P}} \left[ (X - g _ {X} (W)) (X - g _ {X} (W)) ^ {\top} \right]. \\ \end{array} +$$ + +We consider the nuisance neighborhood such that for $g \in \mathcal{G}_r(g_0)$ , + +$$ +\| g - g _ {0} \| _ {\mathcal {G}} := \max \left\{\mathbb {E} _ {\mathbb {P}} \left[ \| g _ {X} (W) - g _ {0, X} (W) \| _ {2} ^ {4} \right] ^ {\frac {1}{4}}, \mathbb {E} _ {\mathbb {P}} \left[ \left(g _ {Y} (W) - g _ {0, Y} (W)\right) ^ {4} \right] ^ {\frac {1}{4}} \right\} \leq r. \tag {27} +$$ + +We now verify that the loss function $\ell$ satisfies Asm. 3. + +(a) We assume that $g_{X}(w): \mathcal{W} \mapsto \mathbb{R}^{d}$ and $g_{Y}(w): \mathcal{W} \mapsto \mathbb{R}$ are continuous functions, thus Asm. 3(a) is satisfied. + +(b) By (26), it follows from KKT conditions that + +$$ +S _ {\theta} \left(\theta_ {\star}, g _ {0}\right) = - \mathbb {E} _ {\mathbb {P}} \left[ (\tilde {Y} - \langle \theta_ {\star}, \tilde {X} \rangle) \tilde {X} \right] = 0. \tag {28} +$$ + +(c) Since $\mathbb{E}_{\mathbb{P}}[\tilde{X} \mid W] = 0$ and $\mathbb{E}_{\mathbb{P}}[\tilde{Y} \mid W] = 0$ , we have + +$$ +H _ {\theta \theta} (\theta , g) = \mathbb {E} _ {\mathbb {P}} \left[ \tilde {X} \tilde {X} ^ {\top} \right] + \mathbb {E} _ {\mathbb {P}} \left[ (g _ {X} (W) - g _ {0, X} (W)) (g _ {X} (W) - g _ {0, X} (W)) ^ {\top} \right]. +$$ + +For any $g\in \mathcal{G}_r$ , when $\lambda_{\mathrm{min}}(\mathbb{E}_{\mathbb{P}}[\tilde{X}\tilde{X}^{\top}])\geq \lambda_0$ and $\| \hat{X}\| _2\leq C_X$ a.s., we have + +$$ +\lambda_ {0} \mathbf {I} \preccurlyeq H _ {\theta \theta} (\theta , g) \preccurlyeq \left(C _ {X} ^ {2} + r ^ {2}\right) \mathbf {I} \Longrightarrow \mu = \lambda_ {0} \text {a n d} M = C _ {X} ^ {2} + r ^ {2}. \tag {29} +$$ + +(d) Consider the Taylor expansion around $\theta_{\star}$ , we have + +$$ +S _ {\theta} (\theta , g; Z) - S _ {\theta} (\theta , g) = S _ {\theta} \left(\theta_ {\star}, g; Z\right) - S _ {\theta} \left(\theta_ {\star}, g\right) + \left(H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right)\right) \left(\theta - \theta_ {\star}\right). +$$ + +Let $\epsilon = \tilde{Y} -\langle \theta_{\star},\tilde{X}\rangle$ . Note that $X - g_{X}(W) = \tilde{X} -(g_{X} - g_{0,X})(W)$ and + +$$ +Y - g _ {Y} (W) - \left\langle \theta_ {\star}, X - g _ {X} (W) \right\rangle = \epsilon - \left(g _ {Y} - g _ {0, Y}\right) (W) + \left\langle \theta_ {\star}, \left(g _ {X} - g _ {0, X}\right) (W) \right\rangle . \tag {30} +$$ + +Since $\mathbb{E}_{\mathbb{P}}[\epsilon \mid W] = 0$ , $\mathbb{E}_{\mathbb{P}}[\tilde{X} \mid W] = 0$ by definition and $\mathbb{E}_{\mathbb{P}}[\epsilon \tilde{X}] = 0$ by (28), then for any $g \in \mathcal{G}_r(g_0)$ + +$$ +\begin{array}{l} \| S _ {\theta} \left(\theta_ {\star}, g\right) \| _ {2} = \| \mathbb {E} _ {\mathbb {P}} \left[ \left(g _ {Y} - g _ {0, Y}\right) \left(g _ {X} - g _ {0, X}\right) (W) - \langle \theta_ {\star}, \left(g _ {X} - g _ {0, X}\right) (W) \rangle \left(g _ {X} - g _ {0, X}\right) (W) \right] \| _ {2} \\ \leq \left(\mathbb {E} _ {\mathbb {P}} \left[ \left((g _ {Y} - g _ {0, Y}) (W)\right) ^ {2} \right] \mathbb {E} _ {\mathbb {P}} \left[ \| (g _ {X} - g _ {0, X}) (W) \| _ {2} ^ {2} \right]\right) ^ {1 / 2} + \mathbb {E} _ {\mathbb {P}} \left[ \| (g _ {X} - g _ {0, X}) (W) \| _ {2} ^ {2} \right] \| \theta_ {\star} \| _ {2} \\ \leq r ^ {2} \left(1 + \| \theta_ {\star} \|\right). \\ \end{array} +$$ + +Similarly, we have + +$$ +\begin{array}{l} \| S _ {\theta} \left(\theta_ {\star}, g; Z\right) \| _ {2} ^ {2} \leq \left(\epsilon - \left(g _ {Y} - g _ {0, Y}\right) (W) + \langle \theta_ {\star}, \left(g _ {X} - g _ {0, X}\right) (W) \rangle\right) ^ {2} \| \tilde {X} - \left(g _ {X} - g _ {0, X}\right) (W) \| _ {2} ^ {2} \\ \leq 3 \left(\epsilon^ {2} + \left(\left(g _ {Y} - g _ {0, Y}\right) (W)\right) ^ {2} + \left\| \left(g _ {X} - g _ {0, X}\right) (W) \right\| _ {2} ^ {2} \left\| \theta_ {\star} \right\| _ {2} ^ {2}\right) \left(C _ {X} + \left\| \left(g _ {X} - g _ {0, X}\right) (W) \right\| _ {2}\right) ^ {2} \\ \leq 6 \left(\epsilon^ {2} + \left(\left(g _ {Y} - g _ {0, Y}\right) (W)\right) ^ {2} + \left\| \left(g _ {X} - g _ {0, X}\right) (W) \right\| _ {2} ^ {2} \left\| \theta_ {\star} \right\| _ {2} ^ {2}\right) \left(C _ {X} ^ {2} + \left\| \left(g _ {X} - g _ {0, X}\right) (W) \right\| _ {2} ^ {2}\right), \\ \end{array} +$$ + +which implies that for $g\in \mathcal{G}_r(g_0)$ , when $\mathbb{E}_{\mathbb{P}}[\epsilon^4 ]\leq \sigma^4$ + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ \left\| S _ {\theta} \left(\theta_ {\star}, g; Z\right) \right\| _ {2} ^ {2} \right] \\ \leq 6 C _ {X} ^ {2} \left(\mathbb {E} _ {\mathbb {P}} [ \epsilon^ {2} ] + \mathbb {E} _ {\mathbb {P}} \left[ ((g _ {Y} - g _ {0, Y}) (W)) ^ {2} \right] + \mathbb {E} _ {\mathbb {P}} \left[ \| (g _ {X} - g _ {0, X}) (W) \| _ {2} ^ {2} \right] \| \theta_ {\star} \| _ {2} ^ {2}\right) \\ + 6 \mathbb {E} _ {\mathbb {P}} \left[ \epsilon^ {2} \| (g _ {X} - g _ {0, X}) (W) \| _ {2} ^ {2} \right] + 6 \mathbb {E} _ {\mathbb {P}} \left[ ((g _ {Y} - g _ {0, Y}) (W)) ^ {2} \| (g _ {X} - g _ {0, X}) (W) \| _ {2} ^ {2} \right] \\ + 6 \mathbb {E} _ {\mathbb {P}} \left[ \left\| \left(g _ {X} - g _ {0, X}\right) (W) \right\| _ {2} ^ {4} \right] \| \theta_ {\star} \| _ {2} ^ {2} \\ \leq 6 C _ {X} ^ {2} \left(\sigma^ {2} + r ^ {2} + r ^ {2} \| \theta_ {\star} \| _ {2} ^ {2}\right) + 6 r ^ {4} \| \theta_ {\star} \| _ {2} ^ {2} + 6 \left(\mathbb {E} _ {\mathbb {P}} \left[ \epsilon^ {4} \right] \mathbb {E} _ {\mathbb {P}} \left[ \| (g _ {X} - g _ {0, X}) (W) \| _ {2} ^ {4} \right]\right) ^ {1 / 2} \\ + 6 \left(\mathbb {E} _ {\mathbb {P}} \left[ \left(\left(g _ {Y} - g _ {0, Y}\right) (W)\right) ^ {4} \right] \mathbb {E} _ {\mathbb {P}} \left[ \| \left(g _ {X} - g _ {0, X}\right) (W) \| _ {2} ^ {4} \right]\right) ^ {1 / 2} \\ \leq 6 C _ {X} ^ {2} \sigma^ {2} + 6 \left\{\sigma^ {2} + C _ {X} ^ {2} \left(1 + \| \theta_ {\star} \| _ {2} ^ {2}\right) \right\} r ^ {2} + 6 \left(1 + \| \theta_ {\star} \| _ {2} ^ {2}\right) r ^ {4}. \\ \end{array} +$$ + +Thus, for any $g\in \mathcal{G}_r(g_0)$ + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} \left(\theta_ {\star}, g; Z\right) - S _ {\theta} \left(\theta_ {\star}, g\right) \| _ {2} ^ {2} \right] \leq 2 \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} \left(\theta_ {\star}, g; Z\right) \| _ {2} ^ {2} \right] + 2 \| S _ {\theta} \left(\theta_ {\star}, g\right) \| _ {2} ^ {2} \\ \leq 1 2 C _ {X} ^ {2} \sigma^ {2} + \left\{1 2 \sigma^ {2} + 2 \left(1 + \| \theta_ {\star} \| _ {2}\right) + 1 2 C _ {X} ^ {2} \left(1 + \| \theta_ {\star} \| _ {2} ^ {2}\right) \right\} r ^ {2} + 1 2 \left(1 + \| \theta_ {\star} \| _ {2} ^ {2}\right) r ^ {4} \\ = 1 2 C _ {X} ^ {2} \sigma^ {2} + \mathcal {O} (r ^ {2}). \\ \end{array} +$$ + +On the other hand, since + +$$ +\begin{array}{l} \| H _ {\theta \theta} (\theta_ {\star}, g; Z) \| _ {2} = \| (\tilde {X} - (g _ {X} - g _ {0, X}) (W)) (\tilde {X} - (g _ {X} - g _ {0, X}) (W)) ^ {\top} \| _ {2} \\ \leq \| \tilde {X} - (g _ {X} - g _ {0, X}) (W) \| _ {2} ^ {2} \\ \leq 2 \| \tilde {X} \| _ {2} ^ {2} + 2 \| (g _ {X} - g _ {0, X}) (W) \| _ {2} ^ {2} \leq 2 C _ {X} ^ {2} + 2 \| (g _ {X} - g _ {0, X}) (W) \| _ {2} ^ {2}, \\ \end{array} +$$ + +by (29) we have + +$$ +\begin{array}{l} \left\| H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right) \right\| _ {2} \leq \left\| H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) \right\| _ {2} + \left\| H _ {\theta \theta} \left(\theta_ {\star}, g\right) \right\| _ {2} \\ \leq 3 C _ {X} ^ {2} + r ^ {2} + 2 \| \left(g _ {X} - g _ {0, X}\right) (W) \| _ {2} ^ {2}, \\ \end{array} +$$ + +which implies that + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ \| \left(H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right)\right) \left(\theta - \theta_ {\star}\right) \| _ {2} ^ {2} \right] \\ \leq \mathbb {E} _ {\mathbb {P}} \left[ \left(3 C _ {X} ^ {2} + r ^ {2} + 2 \| \left(g _ {X} - g _ {0, X}\right) (W) \| _ {2} ^ {2}\right) ^ {2} \right] \| \theta - \theta_ {\star} \| _ {2} ^ {2} \\ = \left(9 C _ {X} ^ {4} + \mathcal {O} \left(r ^ {2}\right)\right) \| \theta - \theta_ {\star} \| _ {2} ^ {2}. \\ \end{array} +$$ + +Thus, + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} (\theta , g; Z) - S _ {\theta} (\theta , g) \| _ {2} ^ {2} \right] \leq 2 \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} (\theta_ {\star}, g; Z) - S _ {\theta} (\theta_ {\star}, g) \| _ {2} ^ {2} \right] \\ + 2 \mathbb {E} _ {\mathbb {P}} \left[ \left\| \left(H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right)\right) \left(\theta - \theta_ {\star}\right) \right\| _ {2} ^ {2} \right] \\ \leq 2 4 C _ {X} ^ {2} \sigma^ {2} + \mathcal {O} (r ^ {2}) + (1 8 C _ {X} ^ {4} + \mathcal {O} (r ^ {2})) \| \theta - \theta_ {\star} \| _ {2} ^ {2}, \\ \end{array} +$$ + +which implies + +$$ +K _ {1} = 2 4 C _ {X} ^ {2} \sigma^ {2} + \mathcal {O} (r ^ {2}) \text {a n d} \kappa_ {1} = 1 8 C _ {X} ^ {4} + \mathcal {O} (r ^ {2}). \tag {31} +$$ + +(e) For any $\theta \in \Theta$ and $g, \bar{g} \in \mathcal{G}_r(g_0)$ , by (30) we have + +$$ +\begin{array}{l} \mathrm {D} _ {g} \mathrm {D} _ {\theta} L (\theta_ {\star}, \bar {g}) [ \theta - \theta_ {\star}, g - g _ {0} ] \\ = \mathbb {E} _ {\mathbb {P}} [ (- (g _ {Y} - g _ {0, Y}) (W) + \langle \theta_ {\star}, (g _ {X} - g _ {0, X}) (W) \rangle) \langle \theta - \theta_ {\star}, (\bar {g} _ {X} - g _ {0, X}) (W) \rangle ] \\ + \mathbb {E} _ {\mathbb {P}} \left[ (- (\bar {g} _ {Y} - g _ {0, Y}) (W) + \langle \theta_ {\star}, (\bar {g} _ {X} - g _ {0, X}) (W) \rangle) \langle \theta - \theta_ {\star}, (g _ {X} - g _ {0, X}) (W) \rangle \right]. \\ \end{array} +$$ + +Since $\bar{g}\in \mathcal{G}_r(g_0)$ + +$$ +\begin{array}{l} \left. \left| \mathbb {E} _ {\mathbb {P}} \left[ \left(- \left(g _ {Y} - g _ {0, Y}\right) (W) + \langle \theta_ {\star}, \left(g _ {X} - g _ {0, X}\right) (W) \rangle\right) \langle \theta - \theta_ {\star}, (\bar {g} _ {X} - g _ {0, X}) (W) \rangle \right] \right| \right. \\ \leq \mathbb {E} _ {\mathbb {P}} [ | (g _ {Y} - g _ {0, Y}) (W) | \| (\bar {g} _ {X} - g _ {0, X}) (W) \| _ {2} ] \| \theta - \theta_ {\star} \| _ {2} \\ + \| \theta_ {\star} \| _ {2} \mathbb {E} _ {\mathbb {P}} \left[ \| (g _ {X} - g _ {0, X}) (W) \| _ {2} \| (\bar {g} _ {X} - g _ {0, X}) (W) \| _ {2} \right] \| \theta - \theta_ {\star} \| _ {2} \\ \leq \mathbb {E} _ {\mathbb {P}} \left[ \| (\bar {g} _ {X} - g _ {0, X}) (W) \| _ {2} ^ {2} \right] ^ {1 / 2} \mathbb {E} _ {\mathbb {P}} \left[ ((g _ {Y} - g _ {0, Y}) (W)) ^ {2} \right] ^ {1 / 2} \| \theta - \theta_ {\star} \| _ {2} \\ + \| \theta_ {\star} \| _ {2} \mathbb {E} _ {\mathbb {P}} \left[ \| (\bar {g} _ {X} - g _ {0, X}) (W) \| _ {2} ^ {2} \right] ^ {1 / 2} \mathbb {E} _ {\mathbb {P}} \left[ \| (g _ {X} - g _ {0, X}) (W) \| _ {2} ^ {2} \right] ^ {1 / 2} \| \theta - \theta_ {\star} \| _ {2} \\ \leq \left(1 + \| \theta_ {\star} \| _ {2}\right) r \| g - g _ {0} \| _ {\mathcal {G}} \| \theta - \theta_ {\star} \| _ {2}. \\ \end{array} +$$ + +Similarly, + +$$ +\begin{array}{l} \left. \left| \mathbb {E} _ {\mathbb {P}} \left[ (- (\bar {g} _ {Y} - g _ {0, Y}) (W) + \langle \theta_ {\star}, (\bar {g} _ {X} - g _ {0, X}) (W) \rangle) \langle \theta - \theta_ {\star}, (g _ {X} - g _ {0, X}) (W) \rangle \right] \right| \right. \\ \leq \left(1 + \| \theta_ {\star} \| _ {2}\right) r \| g - g _ {0} \| _ {\mathcal {G}} \| \theta - \theta_ {\star} \| _ {2}. \\ \end{array} +$$ + +Thus, + +$$ +\begin{array}{l} \left| \mathrm {D} _ {g} \mathrm {D} _ {\theta} L \left(\theta_ {\star}, \bar {g}\right) [ \theta - \theta_ {\star}, g - g _ {0} ] \right| \\ \leq \left| \mathbb {E} _ {\mathbb {P}} \left[ (- (g _ {Y} - g _ {0, Y}) (W) + \langle \theta_ {\star}, (g _ {X} - g _ {0, X}) (W) \rangle) \langle \theta - \theta_ {\star}, (\bar {g} _ {X} - g _ {0, X}) (W) \rangle \right] \right| \\ + \left| \mathbb {E} _ {\mathbb {P}} \left[ (- (\bar {g} _ {Y} - g _ {0, Y}) (W) + \langle \theta_ {\star}, (\bar {g} _ {X} - g _ {0, X}) (W) \rangle) \langle \theta - \theta_ {\star}, (g _ {X} - g _ {0, X}) (W) \rangle \right] \right| \\ \leq 2 (1 + \| \theta_ {\star} \| _ {2}) r \| g - g _ {0} \| _ {\mathcal {G}} \| \theta - \theta_ {\star} \| _ {2}. \\ \end{array} +$$ + +which implies + +$$ +\alpha_ {1} = 2 \left(1 + \| \theta_ {\star} \| _ {2}\right) r. \tag {32} +$$ + +In addition, the risk $L$ is Neyman orthogonal at $(\theta_{\star},g_0)$ since + +$$ +\mathrm {D} _ {g} \mathrm {D} _ {\theta} L \left(\theta_ {\star}, g _ {0}\right) [ \theta - \theta_ {\star}, g - g _ {0} ] = 0. \tag {33} +$$ + +Note that + +$$ +\begin{array}{l} \mathrm {D} _ {g} ^ {2} \mathrm {D} _ {\theta} L (\theta_ {\star}, \bar {g}) [ \theta - \theta_ {\star}, g - g _ {0}, g - g _ {0} ] \\ = 2 \mathbb {E} _ {\mathbb {P}} \left[ \left(- \left(g _ {Y} - g _ {0, Y}\right) (W) + \langle \theta_ {\star}, \left(g _ {X} - g _ {0, X}\right) (W) \rangle\right) \langle \theta - \theta_ {\star}, \left(g _ {X} - g _ {0, X}\right) (W) \rangle \right]. \\ \end{array} +$$ + +By identical proof of (32), we have that $L$ satisfies Asm. 4 since + +$$ +\mathrm {D} _ {g} ^ {2} \mathrm {D} _ {\theta} L (\theta_ {\star}, \bar {g}) [ \theta - \theta_ {\star}, g - g _ {0}, g - g _ {0} ] \leq 2 (1 + \| \theta_ {\star} \| _ {2}) \| g - g _ {0} \| _ {\mathcal {G}} ^ {2} \| \theta - \theta_ {\star} \| _ {2}, +$$ + +which implies + +$$ +\beta_ {1} = 2 \left(1 + \| \theta_ {\star} \| _ {2}\right). \tag {34} +$$ + +![](images/8e8978d3c64861480bf6df8c6bfe42be8e43a67a58d74014de678e8db44b4bc2.jpg) + +# B.4.2 Proof of Lemma 5 + +Proof. We consider the following loss: + +$$ +\ell (\theta , g; z) = \frac {1}{2} (y - g (w) - \langle \theta , x \rangle) ^ {2}, +$$ + +with the corresponding risk function defined as + +$$ +L (\theta , g) = \frac {1}{2} \mathbb {E} _ {\mathbb {P}} \left[ (Y - g (W) - \langle \theta , X \rangle) ^ {2} \right] +$$ + +Under the true nuisance, the target is the minimizer of the following squared loss: + +$$ +\theta_ {\star} = \underset {\theta \in \mathbb {R} ^ {d}} {\arg \min } \frac {1}{2} \mathbb {E} _ {\mathbb {P}} \left[ (Y - g _ {0} (W) - \langle \theta , X \rangle) ^ {2} \right]. +$$ + +Since $\epsilon = Y - g_0(W) - \langle \theta_0,X\rangle$ satisfies $\mathbb{E}_{\mathbb{P}}[\epsilon \mid X,W] = 0$ under the true model, by bias-variance decomposition, we have + +$$ +\theta_ {\star} = \underset {\theta \in \mathbb {R} ^ {d}} {\arg \min } \frac {1}{2} \mathbb {E} _ {\mathbb {P}} \left[ \left(\left\langle \theta_ {0}, X \right\rangle - \left\langle \theta , X \right\rangle\right) ^ {2} \right] = \theta_ {0}. \tag {35} +$$ + +Differentiating $\ell (\theta ,g;z)$ with respect to $\theta$ , we obtain the gradient and Hessian $w.r.t.\theta$ as + +$$ +S _ {\theta} (\theta , g; z) = - (y - g (w) - \langle x, \theta \rangle) x \text {a n d} H _ {\theta \theta} (\theta , g; z) = x x ^ {\top}. +$$ + +The expected gradient and expected Hessian are then obtained as + +$$ +S _ {\theta} (\theta , g) = - \mathbb {E} _ {\mathbb {P}} \left[ (Y - g (W) - \langle X, \theta \rangle) X \right] \text {a n d} H _ {\theta \theta} (\theta , g) = \mathbb {E} _ {\mathbb {P}} \left[ X X ^ {\top} \right]. +$$ + +We consider the nuisance neighborhood such that for $g \in \mathcal{G}_r(g_0)$ , + +$$ +\left\| g - g _ {0} \right\| _ {\mathcal {G}} := \mathbb {E} _ {\mathbb {P}} \left[ (g (W) - g _ {0} (W)) ^ {2} \right] ^ {1 / 2} \leq r. \tag {36} +$$ + +We now verify that the loss function $\ell$ satisfies Asm. 3. + +(a) We assume that $g:\mathcal{W}\mapsto \mathbb{R}$ is continuous, thus Asm. 3(a) is satisfied. +(b) Since $\theta_{\star} = \theta_0$ by (35), we have + +$$ +S _ {\theta} \left(\theta_ {\star}, g _ {0}\right) = - \mathbb {E} _ {\mathbb {P}} [ \epsilon X ] = 0. \tag {37} +$$ + +(c) When $\lambda_{\min}(\mathbb{E}_{\mathbb{P}}[XX^{\top}])\geq \lambda_0 > 0$ and $\| X\| _2\leq C_X$ a.s., $L(\theta ,g)$ is $\lambda_0$ -strongly convex and $C_X^2$ -smooth since + +$$ +\lambda_ {0} \mathbf {I} \preccurlyeq H _ {\theta \theta} (\theta , g) \preccurlyeq C _ {X} ^ {2} \mathbf {I} \Longrightarrow \mu = \lambda_ {0} \text {a n d} M = C _ {X} ^ {2}. \tag {38} +$$ + +(d) Consider the Taylor expansion around $\theta_{\star}$ , we have + +$$ +S _ {\theta} (\theta , g; Z) - S _ {\theta} (\theta , g) = S _ {\theta} \left(\theta_ {\star}, g; Z\right) - S _ {\theta} \left(\theta_ {\star}, g\right) + \left(H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right)\right) \left(\theta - \theta_ {\star}\right). +$$ + +Since $S_{\theta}(\theta_{\star}, g; Z) = ((g - g_0)(w) - \epsilon)X$ and $\| X \|_2 \leq C_X$ a.s., we have + +$$ +\begin{array}{l} \| S _ {\theta} \left(\theta_ {\star}, g; Z\right) - S _ {\theta} \left(\theta_ {\star}, g\right) \| _ {2} = \| \left((g - g _ {0}) (W) - \epsilon\right) X - \mathbb {E} _ {\mathbb {P}} \left[ \left((g - g _ {0}) (W)\right) X \right] \| _ {2} \\ \leq C _ {X} \left(| (g - g _ {0}) (W) | + \mathbb {E} _ {\mathbb {P}} [ | (g - g _ {0}) (W) | ] + | \epsilon |\right). \\ \end{array} +$$ + +On the other hand, + +$$ +H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right) = X X ^ {\top} - \mathbb {E} \left[ X X ^ {\top} \right] \preccurlyeq 2 C _ {X} ^ {2} \mathbf {I}, +$$ + +which implies that + +$$ +\left\| \left(H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right)\right) \left(\theta - \theta_ {\star}\right) \right\| _ {2} \leq 2 C _ {X} ^ {2} \| \theta - \theta_ {\star} \| _ {2}. +$$ + +For $g\in \mathcal{G}_r(g_0)$ , when $\mathbb{E}_{\mathbb{P}}\left[\epsilon^2\right]\leq \sigma^2$ we have + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} (\theta , g; Z) - S _ {\theta} (\theta , g) \| _ {2} ^ {2} \right] \leq 2 \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} \left(\theta_ {\star}, g; Z\right) - S _ {\theta} \left(\theta_ {\star}, g\right) \| _ {2} ^ {2} \right] \\ + 2 \mathbb {E} _ {\mathbb {P}} \left[ \| \left(H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right)\right) \left(\theta - \theta_ {\star}\right) \| _ {2} ^ {2} \right] \\ \leq 2 C _ {X} ^ {2} \mathbb {E} _ {\mathbb {P}} \left[ \left(\left| (g - g _ {0}) (W) \right| + \mathbb {E} _ {\mathbb {P}} \left[ (g - g _ {0}) (W) \right] + | \epsilon |\right) ^ {2} \right] \\ + 2 C _ {X} ^ {4} \| \theta - \theta_ {\star} \| _ {2} ^ {2} \\ \leq 6 C _ {X} ^ {2} \left(2 \mathbb {E} _ {\mathbb {P}} \left[ ((g - g _ {0}) (W)) ^ {2} \right] + \mathbb {E} _ {\mathbb {P}} \left[ \epsilon^ {2} \right]\right) + 2 C _ {X} ^ {4} \| \theta - \theta_ {\star} \| _ {2} ^ {2} \\ \leq 6 C _ {X} ^ {2} \left(2 r ^ {2} + \sigma^ {2}\right) + 2 C _ {X} ^ {4} \| \theta - \theta_ {\star} \| _ {2} ^ {2}, \\ \end{array} +$$ + +which implies that + +$$ +K _ {1} = 6 C _ {X} ^ {2} \left(2 r ^ {2} + \sigma^ {2}\right) \text {a n d} \kappa_ {1} = 2 C _ {X} ^ {4}. \tag {39} +$$ + +(e) For any $\theta \in \Theta$ and $g, \bar{g} \in \mathcal{G}_r(g_0)$ , we have + +$$ +\begin{array}{l} \left| \mathrm {D} _ {g} \mathrm {D} _ {\theta} L (\theta , \bar {g}) [ \theta - \theta_ {\star}, g - g _ {0} ] \right| = | \mathbb {E} _ {\mathbb {P}} [ (g - g _ {0}) (W) \langle X, \theta - \theta_ {\star} \rangle ] | \\ \leq \mathbb {E} _ {\mathbb {P}} [ | (g - g _ {0}) (W) \langle X, \theta - \theta_ {\star} \rangle | ] \\ \leq C _ {X} \| \theta - \theta_ {\star} \| _ {2} \mathbb {E} _ {\mathbb {P}} \left[ \left(\left(g - g _ {0}\right) (W)\right) ^ {2} \right] ^ {1 / 2}, \\ \end{array} +$$ + +which implies that + +$$ +\alpha_ {1} = C _ {X}. \tag {40} +$$ + +![](images/805c27ed47914ccb46232fa8dfa4f3c2f2c87b7c507cf3b98d81af9e29d41867.jpg) + +# B.4.3 Proof of Lemma 6 + +Proof. We consider the following loss: + +$$ +\ell (\theta , g; z) = \frac {1}{2} (y - g (w) - \langle \theta , x \rangle) ^ {2}, +$$ + +with the corresponding risk function defined as + +$$ +L (\theta , g) = \frac {1}{2} \mathbb {E} _ {\mathbb {P}} \left[ (Y - g (W) - \langle \theta , X \rangle) ^ {2} \right]. +$$ + +First by the same proof as Appx. B.4.2, we have $\theta_{\star} = \theta_0$ . Define the inner product $\langle \cdot ,\cdot \rangle_{\mathcal{G}}$ as (18) and define the norm $\| \cdot \|_{\mathcal{G}}$ such that $\| g\|_{\mathcal{G}}^2 = \langle g,g\rangle_{\mathcal{G}}\forall g\in \mathcal{G}$ . Consider a uniformly bounded neighborhood $\mathcal{G}_r(g_0)$ such that + +$$ +\mathcal {G} _ {r} \left(g _ {0}\right) = \left\{g \in \mathcal {G}: | g (W) - g _ {0} (W) | \leq r \text {a l m o s t s u r e l y} \right\}. \tag {41} +$$ + +The NO gradient oracle for this non-orthogonal loss is derived as (22) such that + +$$ +S _ {\mathrm {n o}} (\theta , g; z) = - (y - g (w) - \langle \theta , x \rangle) (x - \mathbb {E} [ X \mid W = w ]). +$$ + +We now verify that Asm. 6 is satisfied. + +(a) Since $\epsilon = Y - g_0(W) - \langle \theta_0, X \rangle$ satisfies $\mathbb{E}_{\mathbb{P}}[\epsilon \mid X, W] = 0$ under the true model, by (22) we first have + +$$ +\begin{array}{l} S _ {\mathrm {n o}} \left(\theta_ {\star}, g _ {0}\right) = \mathbb {E} _ {\mathbb {P}} \left[ S _ {\mathrm {n o}} \left(\theta_ {\star}, g _ {0}; Z\right) \right] \\ = - \mathbb {E} _ {\mathbb {P}} [ \epsilon (X - \mathbb {E} [ X \mid W ]) ] \\ = - \mathbb {E} _ {\mathbb {P}} \left[ \mathbb {E} _ {\mathbb {P}} [ \epsilon \mid X, W ] (X - \mathbb {E} [ X \mid W ]) \right] = 0. \tag {42} \\ \end{array} +$$ + +Let $\gamma_0^{(j)} = H_{gg}^{-1}H_{\theta g}^{(j)}$ for $j = 1,\ldots ,d$ . By (85), we have $[\Gamma_0g]_j = \langle \gamma_0^{(j)},g\rangle_{\mathcal{G}},\forall g\in \mathcal{G}$ . Thus, by (82) we have + +$$ +[ (\hat {\Gamma} - \Gamma_ {0}) \nabla_ {g} L (\theta_ {\star}, g _ {0}) ] _ {j} = \langle \hat {\gamma} ^ {(j)} - \gamma_ {0} ^ {(j)}, \nabla_ {g} L (\theta_ {\star}, g _ {0}) \rangle_ {\mathcal {G}} = \mathrm {D} _ {g} L (\theta_ {\star}, g _ {0}) [ \hat {\gamma} ^ {(j)} - \gamma_ {0} ^ {(j)} ], +$$ + +which, by (19), implies that + +$$ +[ (\hat {\Gamma} - \Gamma_ {0}) \nabla_ {g} L (\theta_ {\star}, g _ {0}) ] _ {j} = - \mathbb {E} _ {\mathbb {P}} [ \epsilon [ \hat {\Gamma} - \Gamma_ {0} ] _ {j} (W) ] = \mathbb {E} _ {\mathbb {P}} [ \mathbb {E} _ {\mathbb {P}} [ \epsilon | W ] [ \hat {\Gamma} - \Gamma_ {0} ] _ {j} (W) ] = 0. \tag {43} +$$ + +Thus, Asm. 6(a) holds true due to (42) and (43). + +(b) By (22), for any $(\theta ,g)\in \Theta \times \mathcal{G}$ + +$$ +\begin{array}{l} \nabla_ {\theta} S _ {\mathbf {n o}} (\theta , g) = \mathbb {E} _ {\mathbb {P}} \left[ X \left(X - \mathbb {E} _ {\mathbb {P}} [ X \mid W ]\right) ^ {\top} \right] \\ = \mathbb {E} _ {\mathbb {P}} \left[ X X ^ {\top} \right] - \mathbb {E} _ {\mathbb {P}} \left[ \mathbb {E} _ {\mathbb {P}} [ X | W ] \mathbb {E} _ {\mathbb {P}} [ X | W ] ^ {\top} \right] \\ = \mathbb {E} _ {\mathbb {P}} \left[ \tilde {X} \tilde {X} ^ {\top} \right], \tag {44} \\ \end{array} +$$ + +which implies that + +$$ +\lambda_ {\min } \nabla_ {\theta} S _ {\mathrm {n o}} (\theta , g) + \nabla_ {\theta} S _ {\mathrm {n o}} (\theta , g) ^ {\top} = \lambda_ {\min } 2 \mathbb {E} _ {\mathbb {P}} \left[ \tilde {X} \tilde {X} ^ {\top} \right] \geq 2 \lambda_ {0}. +$$ + +Thus, Asm. 6(b) holds true for $\mu_{\mathrm{no}} = \lambda_0$ + +(c) For any $(\theta, g, \bar{g}) \in \Theta \times \mathcal{G} \times \mathcal{G}_r(g_0)$ , by (19), + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ (\mathrm {D} _ {g} L (\theta , \bar {g}; Z) [ g ] - \mathrm {D} _ {g} L (\theta , \bar {g}) [ g ]) ^ {2} \right] \\ \leq \mathbb {E} _ {\mathbb {P}} \left[ \left(\mathrm {D} _ {g} L (\theta , \bar {g}; Z) [ g ]\right) ^ {2} \right] \\ = \mathbb {E} _ {\mathbb {P}} \left[ (Y - \bar {g} (W) - \langle \theta , X \rangle) ^ {2} (g (W)) ^ {2} \right] \\ = \mathbb {E} _ {\mathbb {P}} \left[ (\epsilon - (\bar {g} - g _ {0}) (W) - \langle \theta - \theta_ {\star}, X \rangle) ^ {2} (g (W)) ^ {2} \right] \\ \leq 3 \mathbb {E} _ {\mathbb {P}} \left[ \epsilon^ {2} (g (W)) ^ {2} \right] + 3 \mathbb {E} _ {\mathbb {P}} \left[ (\bar {g} - g _ {0}) (W)) ^ {2} (g (W)) ^ {2} \right] + 3 \mathbb {E} _ {\mathbb {P}} \left[ \langle \theta - \theta_ {\star}, X \rangle^ {2} (g (W)) ^ {2} \right]. \\ \end{array} +$$ + +Assume that $\mathbb{E}_{\mathbb{P}}\left[\epsilon^2 \mid W\right] \leq \sigma^2$ and $\|X\|_{\infty} \leq C_X$ a.s.. By (41) we have + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \left(\mathrm {D} _ {g} L (\theta , \bar {g}; Z) [ g ] - \mathrm {D} _ {g} L (\theta , \bar {g}) [ g ]\right) ^ {2} \right] \leq 3 \left(\sigma^ {2} + r ^ {2} + C _ {X} ^ {2} \| \theta - \theta_ {\star} \| _ {2} ^ {2}\right) \| g \| _ {\mathcal {G}} ^ {2}. \tag {45} +$$ + +Thus, Asm. 6(c) holds true for $K_{2} = 3(\sigma^{2} + r^{2})$ and $\kappa_{2} = 3C_{X}^{2}$ . + +(d) For any $(\theta, \bar{g}, g_1, g_2) \in \Theta \times \mathcal{G}_r(g_0) \times \mathcal{G} \times \mathcal{G}$ , by (20), + +$$ +\begin{array}{l} \left| \mathrm {D} _ {g} ^ {2} L (\theta , g) [ g _ {1}, g _ {2} ] \right| = | \mathbb {E} _ {\mathbb {P}} [ g _ {1} (W) g _ {2} (W) ] | \\ \leq \mathbb {E} _ {\mathbb {P}} \left[ (g _ {1} (W)) ^ {2} \right] ^ {1 / 2} \mathbb {E} _ {\mathbb {P}} \left[ (g _ {2} (W)) ^ {2} \right] ^ {1 / 2} = \| g _ {1} \| _ {\mathcal {G}} \| g _ {2} \| _ {\mathcal {G}}. \tag {46} \\ \end{array} +$$ + +In addition, for any $(\theta ,\bar{\theta},g)\in \Theta \times \Theta \times \mathcal{G}$ , by (19) we have + +$$ +\begin{array}{l} \left| \mathrm {D} _ {\theta} \mathrm {D} _ {g} L (\bar {\theta}, g _ {0}) [ g, \theta - \theta_ {\star} ] \right| = \left| \mathrm {D} _ {\bar {\theta}} \mathbb {E} _ {\mathbb {P}} \left[ (Y - g _ {0} (W) - \langle \bar {\theta}, X \rangle) g (W) \right] [ \theta - \theta_ {\star} ] \right| \\ = \left| \mathbb {E} _ {\mathbb {P}} [ \langle \theta - \theta_ {\star}, X \rangle g (W) ] \right| \\ \leq \mathbb {E} _ {\mathbb {P}} \left[ \langle \theta - \theta_ {\star}, X \rangle^ {2} \right] ^ {1 / 2} \mathbb {E} _ {\mathbb {P}} \left[ (g (W)) ^ {2} \right] ^ {1 / 2} \\ \leq C _ {X} \| \theta - \theta_ {\star} \| _ {2} \| g \| _ {\mathcal {G}}. \tag {47} \\ \end{array} +$$ + +Thus, Asm. 6(d) holds true for $\alpha_{2} = 1$ due to (46) and $\alpha_{1} = C_{X}$ due to (47). + +(e) Note that + +$$ +S _ {\mathrm {n o}} (\theta , g; z) = - (y - g (w) - \langle \theta , x \rangle) (x - \mathbb {E} _ {\mathbb {P}} [ X \mid W = w ]), +$$ + +which implies that for any $g_1, g_2 \in \mathcal{G}$ , + +$$ +D _ {g} ^ {2} S _ {\mathrm {n o}} (\theta , g; z) [ g _ {1}, g _ {2} ] = 0. \tag {48} +$$ + +Thus, Asm. 6(e) holds true for $\beta_{2} = 0$ + +□ + +# B.4.4 Proof of Lemma 7 + +Proof. We consider the following loss: + +$$ +\ell (\theta , g; z) = \frac {1}{2} \left(y - g ^ {\mathrm {o u t}} (x) - \left(w - g ^ {\mathrm {p r o p}} (x)\right) \langle \theta , x \rangle\right) ^ {2}, +$$ + +with the corresponding risk function defined as + +$$ +L (\theta , g) = \frac {1}{2} \mathbb {E} _ {\mathbb {P}} \left[ \left(Y - g ^ {\mathrm {o u t}} (X) - \left(W - g ^ {\mathrm {p r o p}} (X)\right) \langle \theta , X \rangle\right) ^ {2} \right]. +$$ + +Note that $\epsilon = Y - g_0^{\mathrm{out}}(X) - \left(W - g_0^{\mathrm{prop}}(X)\right)\tau_0(X)$ . Under Asm. 5, we have $\mathbb{E}_{\mathbb{P}}[\epsilon \mid W,X] = 0$ which implies that + +$$ +\begin{array}{l} L (\theta , g _ {0}) = \frac {1}{2} \mathbb {E} _ {\mathbb {P}} \left[ \left(\epsilon + \left(W - g _ {0} ^ {\operatorname {p r o p}} (X)\right) \left(\tau_ {0} (X) - \langle \theta , X \rangle\right)\right) ^ {2} \right] \\ = \frac {1}{2} \mathbb {E} _ {\mathbb {P}} \left[ \left(W - g _ {0} ^ {\mathrm {p r o p}} (X)\right) ^ {2} \left(\tau_ {0} (X) - \langle \theta , X \rangle\right) ^ {2} \right] + \frac {1}{2} \mathbb {E} _ {\mathbb {P}} \left[ \epsilon^ {2} \right]. \\ \end{array} +$$ + +Thus, the target is the minimizer of the following squared loss: + +$$ +\theta_ {\star} = \underset {\theta \in \mathbb {R} ^ {d}} {\arg \min } \mathbb {E} _ {\mathbb {P}} \left[ \left(W - g _ {0} ^ {\operatorname {p r o p}} (X)\right) ^ {2} \left(\tau_ {0} (X) - \langle \theta , X \rangle\right) ^ {2} \right]. \tag {49} +$$ + +Differentiating $\ell (\theta ,g;z)$ with respect to $\theta$ , we obtain the gradient and Hessian $w.r.t.\theta$ as + +$$ +S _ {\theta} (\theta , g; z) = - \left(y - g ^ {\text {o u t}} (x) - \left(w - g ^ {\text {p r o p}} (x)\right) \langle \theta , x \rangle\right) \left(w - g ^ {\text {p r o p}} (x)\right) x, +$$ + +$$ +H _ {\theta \theta} (\theta , g; z) = (w - g ^ {\mathrm {p r o p}} (x)) ^ {2} x x ^ {\top}. +$$ + +The expected gradient and expected Hessian are then obtained as + +$$ +\begin{array}{l} S _ {\theta} (\theta , g) = - \mathbb {E} _ {\mathbb {P}} \left[ \left(Y - g ^ {\text {o u t}} (X) - \left(W - g ^ {\text {p r o p}} (X)\right) \langle \theta , X \rangle\right) \left(W - g ^ {\text {p r o p}} (X)\right) X \right], \\ H _ {\theta \theta} (\theta , g) = \mathbb {E} _ {\mathbb {P}} \left[ \left(g _ {0} ^ {\text {p r o p}} (1 - g _ {0} ^ {\text {p r o p}}) (X) + \left(\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)\right) ^ {2}\right) X X ^ {\top} \right]. \\ \end{array} +$$ + +We consider the nuisance neighborhood such that for $g \in \mathcal{G}_r(g_0)$ , + +$$ +\| g - g _ {0} \| _ {\mathcal {G}} := \max \left\{\mathbb {E} _ {\mathbb {P}} \left[ (g ^ {\text {o u t}} (X) - g _ {0} ^ {\text {o u t}} (X)) ^ {4} \right] ^ {\frac {1}{4}}, \mathbb {E} _ {\mathbb {P}} \left[ (g ^ {\text {o u t}} (X) - g _ {0} ^ {\text {o u t}} (X)) ^ {4} \right] ^ {\frac {1}{4}} \right\} \leq r. +$$ + +We now verify that the loss function $\ell$ satisfies Asm. 3. + +(a) We assume that $g^{\mathrm{out}}: \mathbb{R}^d \mapsto \mathbb{R}$ and $g^{\mathrm{prop}}: \mathbb{R}^d \mapsto \mathbb{R}$ are continuous, thus Asm. 3(a) is satisfied. + +(b) Since $\theta_{\star}$ is a global minimizer of (49), we have + +$$ +S _ {\theta} \left(\theta_ {\star}, g _ {0}\right) = 0. \tag {50} +$$ + +(c) We assume that $c_{0} \leq g_{0}^{\mathrm{prop}}(X) \leq 1 - c_{0}$ a.s. for some $c_{0} > 0$ . When $\lambda_{\min}(\mathbb{E}_{\mathbb{P}}[XX^{\top}]) \geq \lambda_{0} > 0$ and $\|X\|_{2} \leq C_{X}$ a.s., we have + +$$ +c _ {0} ^ {2} \lambda_ {0} \mathbf {I} \preccurlyeq H _ {\theta \theta} (\theta , g) \preccurlyeq (1 + r ^ {2}) C _ {X} ^ {2} \mathbf {I} \Longrightarrow \mu = c _ {0} ^ {2} \lambda_ {0} \text {a n d} M = (1 + r ^ {2}) C _ {X} ^ {2}. \tag {51} +$$ + +(d) Consider the Taylor expansion around $\theta_{\star}$ , we have + +$$ +S _ {\theta} (\theta , g; Z) - S _ {\theta} (\theta , g) = S _ {\theta} \left(\theta_ {\star}, g; Z\right) - S _ {\theta} \left(\theta_ {\star}, g\right) + \left(H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right)\right) \left(\theta - \theta_ {\star}\right). +$$ + +Note that + +$$ +\begin{array}{l} S _ {\theta} \left(\theta_ {\star}, g; Z\right) = - \left(Y - g ^ {\text {o u t}} (X) - \left(W - g ^ {\text {p r o p}} (X)\right) \langle \theta_ {\star}, X \rangle\right) \left(W - g ^ {\text {p r o p}} (X)\right) X \\ = - \left(\epsilon - \left(g ^ {\text {o u t}} - g _ {0} ^ {\text {o u t}}\right) (X) + \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X) \tau_ {0} (X)\right) \left(W - g ^ {\text {p r o p}} (X)\right) X \\ + \left(W - g ^ {\mathrm {p r o p}} (X)\right) ^ {2} \left(\langle \theta_ {\star}, X \rangle - \tau_ {0} (X)\right) X. \\ \end{array} +$$ + +We assume that $\tau_0: \mathbb{R}^d \mapsto \mathbb{R}$ is continuous. Then when $\|X\|_2 \leq C_X$ a.s., $|\tau_0(X)| \leq C_\tau$ for some $C_\tau > 0$ . It follows that + +$$ +\begin{array}{l} \left\| S _ {\theta} \left(\theta_ {\star}, g; Z\right) \right\| _ {2} \leq C _ {X} \left| \left(\epsilon - \left(g ^ {\text {o u t}} - g _ {0} ^ {\text {o u t}}\right) (X) + \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X) \tau_ {0} (X)\right) \left(W - g ^ {\text {p r o p}} (X)\right) \right| \\ + C _ {X} \left(W - g ^ {\operatorname {p r o p}} (X)\right) ^ {2} | \langle \theta_ {\star}, X \rangle - \tau_ {0} (X) | \\ \leq C _ {X} \left(| \epsilon | + \left| (g ^ {\text {o u t}} - g _ {0} ^ {\text {o u t}}) (X) \right| + C _ {\tau} \left| (g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) (X) \right|\right) | W - g ^ {\text {p r o p}} (X) | \\ + C _ {X} \left(C _ {X} \| \theta_ {\star} \| _ {2} + C _ {\tau}\right) \left(W - g ^ {\operatorname {p r o p}} (X)\right) ^ {2}. \\ \end{array} +$$ + +Since $\mathbb{E}_{\mathbb{P}}[(W - g^{\mathrm{prop}}(X))^2\mid X] = g_0^{\mathrm{prop}}(1 - g_0^{\mathrm{prop}})(X) + ((g^{\mathrm{prop}} - g_0^{\mathrm{prop}})(X))^2$ and $(W - g^{\mathrm{prop}}(X))^2\leq 2 + 2((g^{\mathrm{prop}} - g_0^{\mathrm{prop}})(X))^2$ , we have + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} (\theta_ {\star}, g; Z) \| _ {2} ^ {2} \right] \leq 4 C _ {X} ^ {2} \mathbb {E} _ {\mathbb {P}} \left[ \epsilon^ {2} \left(1 + \left(\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)\right) ^ {2}\right) \right] \\ + 4 C _ {X} ^ {2} \mathbb {E} _ {\mathbb {P}} \left[ \left(\left(g ^ {\text {o u t}} - g _ {0} ^ {\text {o u t}}\right) (X)\right) ^ {2} \left(1 + \left(\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)\right) ^ {2}\right) \right] \\ + 4 C _ {X} ^ {2} C _ {\tau} ^ {2} \mathbb {E} _ {\mathbb {P}} \left[ \left(\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)\right) ^ {2} \left(1 + \left(\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)\right) ^ {2}\right) \right] \\ + 4 C _ {X} ^ {2} \left(C _ {X} \| \theta_ {\star} \| _ {2} + C _ {\tau}\right) ^ {2} \mathbb {E} _ {\mathbb {P}} \left[ 4 + 4 \left(\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)\right) ^ {4} \right]. \\ \end{array} +$$ + +When $\mathbb{E}_{\mathbb{P}}\left[\epsilon^4\right] \leq \sigma^4$ , by Hölder inequality, + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} (\theta_ {\star}, g; Z) \| _ {2} ^ {2} \right] \leq 4 C _ {X} ^ {2} \left(\sigma^ {2} + 4 \left(C _ {X} \| \theta_ {\star} \| _ {2} + C _ {\tau}\right) ^ {2}\right) \\ + 4 C _ {X} ^ {2} \left(1 + \sigma^ {2} + C _ {\tau} ^ {2}\right) r ^ {2} + 4 C _ {X} ^ {2} \left(1 + C _ {\tau} ^ {2} + 4 \left(C _ {X} \| \theta_ {\star} \| _ {2} + C _ {\tau}\right) ^ {2}\right) r ^ {4}. \\ \end{array} +$$ + +Similarly, use the fact that $\mathbb{E}_{\mathbb{P}}[\epsilon \mid W,X] = 0$ and by the stationary condition of (49), we have + +$$ +\begin{array}{l} S _ {\theta} \left(\theta_ {\star}, g\right) = - \mathbb {E} _ {\mathbb {P}} \left[ \left(\left(g ^ {\text {o u t}} - g _ {0} ^ {\text {o u t}}\right) (X) - \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X) \tau_ {0} (X)\right) \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X) X \right] \\ + \mathbb {E} _ {\mathbb {P}} \left[ \left(\left(W - g _ {0} ^ {\text {p r o p}} (X)\right) ^ {2} + \left(\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)\right) ^ {2}\right) \left(\left\langle \theta_ {\star}, X \right\rangle - \tau_ {0} (X)\right) X \right] \\ = - \mathbb {E} _ {\mathbb {P}} \left[ \left(\left(g ^ {\text {o u t}} - g _ {0} ^ {\text {o u t}}\right) (X) - \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X) \tau_ {0} (X)\right) \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X) X \right] \\ + \mathbb {E} _ {\mathbb {P}} \left[ \left(\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)\right) ^ {2} \left(\left\langle \theta_ {\star}, X \right\rangle - \tau_ {0} (X)\right) X \right], \\ \end{array} +$$ + +which implies + +$$ +\left\| S _ {\theta} \left(\theta_ {\star}, g\right) \right\| _ {2} ^ {2} \leq 3 C _ {X} ^ {2} \left(1 + C _ {\tau} ^ {2} + \left(C _ {X} \left\| \theta_ {\star} \right\| _ {2} + C _ {\tau}\right) ^ {2}\right) r ^ {4}. +$$ + +On the other hand, + +$$ +\begin{array}{l} H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right) \preccurlyeq \left(W - g ^ {\operatorname {p r o p}} (X)\right) ^ {2} X X ^ {\top} + \left(1 + r ^ {2}\right) C _ {X} ^ {2} \mathbf {I} \\ \preccurlyeq C _ {X} ^ {2} \left(3 + r ^ {2} + 2 \left(\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)\right) ^ {2}\right) \mathbf {I}, \\ \end{array} +$$ + +which implies that + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \| \left(H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right)\right) \left(\theta - \theta_ {\star}\right) \| _ {2} ^ {2} \right] \leq \left(9 C _ {X} ^ {4} + \mathcal {O} \left(r ^ {2}\right)\right) \| \theta - \theta_ {\star} \| _ {2} ^ {2}. +$$ + +Thus, + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} (\theta , g; Z) - S _ {\theta} (\theta , g) \| _ {2} ^ {2} \right] \leq 3 \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} \left(\theta_ {\star}, g; Z\right) \| _ {2} ^ {2} \right] + 3 \| S _ {\theta} \left(\theta_ {\star}, g\right) \| _ {2} ^ {2} \\ + 3 \mathbb {E} _ {\mathbb {P}} \left[ \| (H _ {\theta \theta} (\theta_ {\star}, g; Z) - H _ {\theta \theta} (\theta_ {\star}, g)) (\theta - \theta_ {\star}) \| _ {2} ^ {2} \right], \\ \end{array} +$$ + +which implies + +$$ +K _ {1} = 1 2 C _ {X} ^ {2} \left(\sigma^ {2} + 4 \left(C _ {X} \| \theta_ {\star} \| _ {2} + C _ {\tau}\right) ^ {2}\right) + \mathcal {O} \left(r ^ {2}\right) \text {a n d} \kappa_ {1} = 2 7 C _ {X} ^ {4} + \mathcal {O} \left(r ^ {2}\right). \tag {52} +$$ + +(e) For any $\theta \in \Theta$ and $g, \bar{g} \in \mathcal{G}_r(g_0)$ , we have + +$$ +\begin{array}{l} \mathrm {D} _ {g} \mathrm {D} _ {\theta} L (\theta_ {\star}, \bar {g}) [ \theta - \theta_ {\star}, g - g _ {0} ] \\ = - \mathbb {E} _ {\mathbb {P}} \left[ \left(\left(\bar {g} ^ {\text {o u t}} - g _ {0} ^ {\text {o u t}}\right) \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X) + \left(g ^ {\text {o u t}} - g _ {0} ^ {\text {o u t}}\right) \left(\bar {g} ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)\right) \langle X, \theta - \theta_ {\star} \rangle \right] \\ + 2 \mathbb {E} _ {\mathbb {P}} \left[ \tau_ {0} (X) \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) \left(\bar {g} ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X) \langle X, \theta - \theta_ {\star} \rangle \right] \\ + 2 \mathbb {E} _ {\mathbb {P}} \left[ \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) \left(\bar {g} ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X) \left(\left\langle \theta_ {\star}, X \right\rangle - \tau_ {0} (X)\right) \left\langle X, \theta - \theta_ {\star} \right\rangle \right]. \\ \end{array} +$$ + +Thus, + +$$ +\begin{array}{l} \left| \mathrm {D} _ {g} \mathrm {D} _ {\theta} L \left(\theta_ {\star}, \bar {g}\right) [ \theta - \theta_ {\star}, g - g _ {0} ] \right| \\ \leq C _ {X} \| \theta - \theta_ {\star} \| _ {2} \mathbb {E} _ {\mathbb {P}} \left[ | (\bar {g} ^ {\text {o u t}} - g _ {0} ^ {\text {o u t}}) (g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) (X) | + | (g ^ {\text {o u t}} - g _ {0} ^ {\text {o u t}}) (\bar {g} ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) (X) | \right] \\ + 2 C _ {X} \left(2 C _ {\tau} + C _ {X} \| \theta_ {\star} \| _ {2}\right) \| \theta - \theta_ {\star} \| _ {2} \mathbb {E} _ {\mathbb {P}} \left[ | (g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) (\bar {g} ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) (X) | \right] \\ \leq C _ {X} \left(1 + 4 \left(C _ {X} \| \theta_ {\star} \| _ {2} + C _ {\tau}\right)\right) r \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {\mathcal {G}}, \\ \end{array} +$$ + +which implies + +$$ +\alpha_ {1} = C _ {X} \left(1 + 4 \left(C _ {X} \| \theta_ {\star} \| _ {2} + C _ {\tau}\right)\right) r. \tag {53} +$$ + +In addition, $L(\theta ,g)$ is Neyman orthogonal at $(\theta_{\star},g_{0})$ since + +$$ +\mathrm {D} _ {g} \mathrm {D} _ {\theta} L \left(\theta_ {\star}, g _ {0}\right) \left[ \theta - \theta_ {\star}, g - g _ {0} \right] = 0. +$$ + +Since for any $\theta \in \Theta$ and $g, \bar{g} \in \mathcal{G}_r(g_0)$ , + +$$ +\begin{array}{l} \mathrm {D} _ {g} ^ {2} \mathrm {D} _ {\theta} L (\theta_ {\star}, \bar {g}) [ \theta - \theta_ {\star}, g - g _ {0}, g - g _ {0} ] \\ = - \mathbb {E} _ {\mathbb {P}} \left[ \left(\left(g ^ {\text {o u t}} - g _ {0} ^ {\text {o u t}}\right) \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X) + \left(g ^ {\text {o u t}} - g _ {0} ^ {\text {o u t}}\right) \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)\right) \langle X, \theta - \theta_ {\star} \rangle \right] \\ + 2 \mathbb {E} _ {\mathbb {P}} \left[ \tau_ {0} (X) \left(\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)\right) ^ {2} \langle X, \theta - \theta_ {\star} \rangle \right] \\ + 2 \mathbb {E} _ {\mathbb {P}} \left[ \left(\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)\right) ^ {2} \left(\left\langle \theta_ {\star}, X \right\rangle - \tau_ {0} (X)\right) \left\langle X, \theta - \theta_ {\star} \right\rangle \right]. \\ \end{array} +$$ + +Similarly, we can show that + +$$ +\begin{array}{l} \left| \mathrm {D} _ {g} ^ {2} \mathrm {D} _ {\theta} L \left(\theta_ {\star}, \bar {g}\right) \left[ \theta - \theta_ {\star}, g - g _ {0}, g - g _ {0} \right] \right| \\ \leq C _ {X} \left(1 + 4 \left(C _ {X} \| \theta_ {\star} \| _ {2} + C _ {\tau}\right)\right) \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {G} ^ {2}, \\ \end{array} +$$ + +which implies that + +$$ +\beta_ {1} = C _ {X} \left(1 + 4 \left(C _ {X} \| \theta_ {\star} \| _ {2} + C _ {\tau}\right)\right). \tag {54} +$$ + +# B.4.5 Proof of Lemma 8 + +Proof. Let + +$$ +\phi (g; z) = g ^ {(1)} (x) - g ^ {(0)} (x) + \frac {w - g ^ {\operatorname {p r o p}} (x)}{g ^ {\operatorname {p r o p}} (x) (1 - g ^ {\operatorname {p r o p}} (x))} (y - g ^ {(w)} (x)). +$$ + +We consider the following loss: + +$$ +\ell (\theta , g; z) = \frac {1}{2} \left(\phi (g; z) - \langle \theta , x \rangle\right) ^ {2}, +$$ + +with the corresponding risk function defined as + +$$ +L (\theta , g) = \frac {1}{2} \mathbb {E} _ {\mathbb {P}} \left[ \left(\phi (g; z) - \langle \theta , x \rangle\right) ^ {2} \right]. +$$ + +We define the residual $\epsilon$ as + +$$ +\epsilon = \frac {W - g _ {0} ^ {\mathrm {p r o p}} (X)}{g _ {0} ^ {\mathrm {p r o p}} (X) \left(1 - g _ {0} ^ {\mathrm {p r o p}} (X)\right)} \left(Y - g _ {0} ^ {(W)} (X)\right). +$$ + +Under Asm. 5, we have $\mathbb{E}_{\mathbb{P}}[\epsilon \mid W,X] = 0$ . Since $\tau_0(x) = g_0^{(1)}(x) - g_0^{(0)}(x)$ , we have + +$$ +\begin{array}{l} L (\theta , g _ {0}) = \frac {1}{2} \mathbb {E} _ {\mathbb {P}} \left[ \left(\epsilon + \tau_ {0} (X) - \langle \theta , X \rangle\right) ^ {2} \right] \\ = \frac {1}{2} \mathbb {E} _ {\mathbb {P}} \left[ (\tau_ {0} (X) - \langle \theta , X \rangle) ^ {2} \right] + \frac {1}{2} \mathbb {E} _ {\mathbb {P}} \left[ \epsilon^ {2} \right]. \\ \end{array} +$$ + +Thus, the target is the minimizer of the following squared loss: + +$$ +\theta_ {\star} = \underset {\theta \in \mathbb {R} ^ {d}} {\arg \min } \mathbb {E} _ {\mathbb {P}} \left[ \left(\tau_ {0} (X) - \langle \theta , X \rangle\right) ^ {2} \right]. \tag {55} +$$ + +Differentiating $\ell (\theta ,g;z)$ with respect to $\theta$ , we obtain the gradient and Hessian $w.r.t.$ $\theta$ as + +$$ +S _ {\theta} (\theta , g; z) = - \left(\phi (g; z) - \langle \theta , x \rangle\right) x, +$$ + +$$ +H _ {\theta \theta} (\theta , g; z) = x x ^ {\top}. +$$ + +The expected gradient and expected Hessian are then obtained as + +$$ +S _ {\theta} (\theta , g) = - \mathbb {E} _ {\mathbb {P}} \left[ \left(\phi (g; Z) - \langle \theta , X \rangle\right) X \right], +$$ + +$$ +H _ {\theta \theta} (\theta , g) = \mathbb {E} _ {\mathbb {P}} \left[ X X ^ {\top} \right]. +$$ + +We consider the nuisance neighborhood such that for $g\in \mathcal{G}_r(g_0)$ + +$$ +\| g - g _ {0} \| _ {\mathcal {G}} := \max \left\{\mathbb {E} _ {\mathbb {P}} \left[ \left(\frac {(g ^ {(w)} - g _ {0} ^ {(w)}) (X)}{g ^ {(w)} (1 - g ^ {(w)}) (X)}\right) ^ {4} \right] ^ {\frac {1}{4}}, \mathbb {E} _ {\mathbb {P}} \left[ \left(\frac {(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) (X)}{g ^ {\text {p r o p}} (1 - g ^ {\text {p r o p}}) (X)}\right) ^ {4} \right] ^ {\frac {1}{4}} \right\} \leq r. +$$ + +We now verify that the loss function $\ell$ satisfies Asm. 3. + +(a) We assume that $g^{(w)}: \mathbb{R}^d \mapsto \mathbb{R}, w = 0,1$ , and $g^{\mathrm{prop}}: \mathbb{R}^d \mapsto (0,1)$ are continuous, thus Asm. 3(a) is satisfied. +(b) Since $\theta_{\star}$ is a global minimizer of (55), we have + +$$ +S _ {\theta} \left(\theta_ {\star}, g _ {0}\right) = 0. \tag {56} +$$ + +(c) When $\lambda_{\min}(\mathbb{E}_{\mathbb{P}}[XX^{\top}])\geq \lambda_0 > 0$ and $\| X\| _2\leq C_X$ a.s., we have + +$$ +\lambda_ {0} \mathbf {I} \preccurlyeq H _ {\theta \theta} (\theta , g) \preccurlyeq C _ {X} ^ {2} \mathbf {I} \Longrightarrow \mu = \lambda_ {0} \text {a n d} M = C _ {X} ^ {2}. \tag {57} +$$ + +(d) Consider the Taylor expansion around $\theta_{\star}$ , we have + +$$ +S _ {\theta} (\theta , g; Z) - S _ {\theta} (\theta , g) = S _ {\theta} \left(\theta_ {\star}, g; Z\right) - S _ {\theta} \left(\theta_ {\star}, g\right) + \left(H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right)\right) \left(\theta - \theta_ {\star}\right). +$$ + +Let $\tau = g^{(1)} - g^{(0)}$ and + +$$ +\begin{array}{l} \psi (g; x) = \frac {1}{g ^ {\mathrm {p r o p}} (1 - g ^ {\mathrm {p r o p}}) (x)} - \frac {1}{g _ {0} ^ {\mathrm {p r o p}} (1 - g _ {0} ^ {\mathrm {p r o p}}) (x)} \\ = \frac {(g ^ {\mathrm {p r o p}} - g _ {0} ^ {\mathrm {p r o p}}) (x)}{g ^ {\mathrm {p r o p}} (1 - g ^ {\mathrm {p r o p}}) (x)} \cdot \frac {(g ^ {\mathrm {p r o p}} + g _ {0} ^ {\mathrm {p r o p}}) (x) - 1}{g _ {0} ^ {\mathrm {p r o p}} (1 - g _ {0} ^ {\mathrm {p r o p}}) (x)}. \\ \end{array} +$$ + +Under Asm. 5, we have + +$$ +| \psi (g; X) | \leq 2 c _ {0} ^ {- 2} \left| \frac {\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)}{g ^ {\text {p r o p}} \left(1 - g ^ {\text {p r o p}}\right) (X)} \right|. +$$ + +We can decompose $S_{\theta}(\theta_{\star},g;Z)$ as + +$$ +S _ {\theta} (\theta_ {\star}, g; Z) = I _ {1} + I _ {2} + I _ {3} + I _ {4} + I _ {5} + I _ {6} + I _ {7} + I _ {8} + I _ {9}, +$$ + +where + +$$ +I _ {1} = - \left(\left(\tau - \tau_ {0}\right) (X) + \tau_ {0} (X) - \langle \theta_ {\star}, X \rangle\right) X, +$$ + +$$ +I _ {2} = - \psi (g; X) (W - g _ {0} ^ {\mathrm {p r o p}} (X)) (Y - g _ {0} ^ {(W)} (X)) X, +$$ + +$$ +I _ {3} = \psi (g; X) (W - g _ {0} ^ {\mathrm {p r o p}} (X)) (g ^ {(W)} - g _ {0} ^ {(W)}) (X) X, +$$ + +$$ +I _ {4} = \psi (g; X) (g ^ {\mathrm {p r o p}} - g _ {0} ^ {\mathrm {p r o p}}) (X) (Y - g _ {0} ^ {(W)} (X)) X, +$$ + +$$ +I _ {5} = - \psi (g; X) \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) \left(g ^ {(W)} - g _ {0} ^ {(W)}\right) (X) X, +$$ + +$$ +I _ {6} = - \frac {(W - g _ {0} ^ {\mathrm {p r o p}} (X)) (Y - g _ {0} ^ {(W)} (X))}{g _ {0} ^ {\mathrm {p r o p}} (1 - g _ {0} ^ {\mathrm {p r o p}}) (X)} X = - \epsilon X, +$$ + +$$ +I _ {7} = \frac {(W - g _ {0} ^ {\mathrm {p r o p}} (X)) (g ^ {(W)} - g _ {0} ^ {(W)}) (X)}{g _ {0} ^ {\mathrm {p r o p}} (1 - g _ {0} ^ {\mathrm {p r o p}}) (X)} X, +$$ + +$$ +I _ {8} = \frac {(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) (X) (Y - g _ {0} ^ {(W)} (X))}{g _ {0} ^ {\text {p r o p}} (1 - g _ {0} ^ {\text {p r o p}}) (X)} X, +$$ + +$$ +I _ {9} = - \frac {\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) \left(g ^ {(W)} - g _ {0} ^ {(W)}\right) (X)}{g _ {0} ^ {\text {p r o p}} \left(1 - g _ {0} ^ {\text {p r o p}}\right) (X)} X. +$$ + +For $I_{1}$ , when $\| X\| _2\leq C_X$ a.s., we have $|\tau_0(X) - \langle \theta_\star ,X\rangle |\leq C$ for some $C > 0$ and + +$$ +\left\| I _ {1} \right\| _ {2} \leq C _ {X} \left(\left| (\tau - \tau_ {0}) (X) \right| + C\right), +$$ + +which implies + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \| I _ {1} \| _ {2} ^ {2} \right] \leq 2 C _ {X} ^ {2} (2 r ^ {2} + C ^ {2}). +$$ + +For $I_{2}$ , when $\left|Y - g_0^{(W)}(X)\right|\leq C_Y$ a.s., we have + +$$ +\left\| I _ {2} \right\| _ {2} \leq 4 c _ {0} ^ {- 2} C _ {X} C _ {Y} \left| \frac {\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)}{g ^ {\text {p r o p}} \left(1 - g ^ {\text {p r o p}}\right) (X)} \right|, +$$ + +which implies + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \| I _ {2} \| _ {2} ^ {2} \right] \leq 1 6 c _ {0} ^ {- 4} (C _ {X} C _ {Y}) ^ {2} r ^ {2}. +$$ + +For $I_{3}$ + +$$ +\| I _ {3} \| _ {2} \leq 4 c _ {0} ^ {- 2} C _ {X} \left| \frac {\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)}{g ^ {\text {p r o p}} \left(1 - g ^ {\text {p r o p}}\right) (X)} \right| \left| \left(g ^ {(W)} - g _ {0} ^ {(W)}\right) (X) \right|, +$$ + +which implies + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \| I _ {3} \| _ {2} ^ {2} \right] \leq 1 6 c _ {0} ^ {- 4} C _ {X} ^ {2} r ^ {4}. +$$ + +For $I_4$ , since $\left|g^{\mathrm{prop}} - g_0^{\mathrm{prop}}\right|\leq 2$ + +$$ +\left\| I _ {4} \right\| _ {2} \leq 4 c _ {0} ^ {- 2} C _ {X} C _ {Y} \left| \frac {\left(\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)\right)}{g ^ {\text {p r o p}} \left(1 - g ^ {\text {p r o p}}\right) (X)} \right|, +$$ + +which implies + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \| I _ {4} \| _ {2} ^ {2} \right] \leq 1 6 c _ {0} ^ {- 4} (C _ {X} C _ {Y}) ^ {2} r ^ {2}. +$$ + +For $I_{5}$ + +$$ +\left\| I _ {5} \right\| _ {2} \leq 4 c _ {0} ^ {- 2} C _ {X} \left| \frac {\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)}{g ^ {\text {p r o p}} \left(1 - g ^ {\text {p r o p}}\right) (X)} \right| \left| \left(g ^ {(W)} - g _ {0} ^ {(W)}\right) (X) \right|, +$$ + +which implies + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \| I _ {5} \| _ {2} ^ {2} \right] \leq 1 6 c _ {0} ^ {- 4} \left(C _ {X} C _ {Y}\right) ^ {2} r ^ {4}. +$$ + +For $I_{6}$ + +$$ +\left\| I _ {6} \right\| _ {2} \leq C _ {X} | \epsilon |. +$$ + +When $\mathbb{E}_{\mathbb{P}}[\epsilon^2 ]\leq \sigma^2$ , we have + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \| I _ {6} \| _ {2} ^ {2} \right] \leq C _ {X} ^ {2} \sigma^ {2}. +$$ + +For $I_7$ + +$$ +\left\| I _ {7} \right\| _ {2} \leq 2 c _ {0} ^ {- 2} C _ {X} \left| (g ^ {(W)} - g _ {0} ^ {(W)}) (X) \right|, +$$ + +which implies + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \| I _ {7} \| _ {2} ^ {2} \right] \leq 4 c _ {0} ^ {- 4} C _ {X} ^ {2} r ^ {2}. +$$ + +For $I_{8}$ + +$$ +\left\| I _ {8} \right\| _ {2} \leq c _ {0} ^ {- 2} C _ {X} C _ {Y} \left| \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X) \right|, +$$ + +which implies + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \| I _ {8} \| _ {2} ^ {2} \right] \leq c _ {0} ^ {- 4} \left(C _ {X} C _ {Y}\right) ^ {2} r ^ {2}. +$$ + +For $I_{9}$ + +$$ +\left. \left\| I _ {9} \right\| _ {2} \leq c _ {0} ^ {- 2} C _ {X} \left| \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X) \right| \left| \left(g ^ {(W)} - g _ {0} ^ {(W)}\right) (X) \right|, \right. +$$ + +which implies + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \| I _ {9} \| _ {2} ^ {2} \right] \leq c _ {0} ^ {- 4} C _ {X} ^ {2} r ^ {4}. +$$ + +By Cauchy-Schwarz inequality, it follows that + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} (\theta_ {\star}, g; Z) \| _ {2} ^ {2} \right] \leq 9 C _ {X} ^ {2} (2 C ^ {2} + \sigma^ {2}) + \mathcal {O} (r ^ {2}). +$$ + +Similarly, we have + +$$ +\left\| S _ {\theta} \left(\theta_ {\star}, g\right) \right\| _ {2} ^ {2} \leq 9 C _ {X} ^ {2} \left(2 C ^ {2} + \sigma^ {2}\right) + \mathcal {O} \left(r ^ {2}\right). +$$ + +Since $H_{\theta \theta}(\theta_{\star},g;Z) - H_{\theta \theta}(\theta_{\star},g)\preccurlyeq C_X^2\mathbf{I}$ , we have + +$$ +\left\| \left(H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right)\right) \left(\theta - \theta_ {\star}\right) \right\| _ {2} ^ {2} \leq C _ {X} ^ {4} \| \theta - \theta_ {\star} \| _ {2} ^ {2}. +$$ + +Thus, + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} (\theta , g; Z) - S _ {\theta} (\theta , g) \| _ {2} ^ {2} \right] \leq 3 \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} \left(\theta_ {\star}, g; Z\right) \| _ {2} ^ {2} \right] + 3 \| S _ {\theta} \left(\theta_ {\star}, g\right) \| _ {2} ^ {2} \\ + 3 \mathbb {E} _ {\mathbb {P}} \left[ \| \left(H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right)\right) \left(\theta - \theta_ {\star}\right) \| _ {2} ^ {2} \right] \\ = 2 7 C _ {X} ^ {2} \left(2 C ^ {2} + \sigma^ {2}\right) + \mathcal {O} \left(r ^ {2}\right) + 3 C _ {X} ^ {4} \| \theta - \theta_ {\star} \| _ {2} ^ {2}, \\ \end{array} +$$ + +which implies + +$$ +K _ {1} = 2 7 C _ {X} ^ {2} \left(2 C ^ {2} + \sigma^ {2}\right) + \mathcal {O} \left(r ^ {2}\right) \text {a n d} \kappa_ {1} = 3 C _ {X} ^ {4}. \tag {58} +$$ + +(e) Since $Y = W(Y(1) - Y(0)) + Y(0)$ and $g^{(W)}(X) = W(g^{(1)}(X) - g^{(0)}(X)) + g^{(0)}(X), \phi(g; Z)$ can be written as + +$$ +\begin{array}{l} \phi (g; Z) = \tau_ {0} (X) + \left(1 - \frac {W (W - g ^ {\mathrm {p r o p}} (X))}{g ^ {\mathrm {p r o p}} (X) (1 - g ^ {\mathrm {p r o p}} (X))}\right) (g ^ {(1)} (X) - g ^ {(0)} (X) - \tau_ {0} (X)) \\ + \frac {W - g ^ {\operatorname {p r o p}} (X)}{g ^ {\operatorname {p r o p}} (1 - g ^ {\operatorname {p r o p}}) (X)} \left(Y (0) - g ^ {(0)} (X)\right) + \frac {W \left(W - g ^ {\operatorname {p r o p}} (X)\right)}{g ^ {\operatorname {p r o p}} (1 - g ^ {\operatorname {p r o p}}) (X)} \left(Y (1) - Y (0) - \tau_ {0} (X)\right) \\ = \tau_ {0} (X) + \frac {g ^ {\operatorname {p r o p}} (X) - W}{g ^ {\operatorname {p r o p}} (X)} \left(g ^ {(1)} - g _ {0} ^ {(1)}\right) (X) + \frac {g ^ {\operatorname {p r o p}} (X) - W}{1 - g ^ {\operatorname {p r o p}} (X)} \left(g ^ {(0)} - g _ {0} ^ {(0)}\right) (X) \\ + \frac {W - g ^ {\mathrm {p r o p}} (X)}{g ^ {\mathrm {p r o p}} (X) (1 - g ^ {\mathrm {p r o p}} (X))} (Y (0) - g _ {0} ^ {(0)} (X)) + \frac {W}{g ^ {\mathrm {p r o p}} (X)} \left(Y (1) - Y (0) - \tau_ {0} (X)\right). \\ \end{array} +$$ + +Under Asm. 5, we have + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \phi (g; Z) \mid X \right] = \tau_ {0} (X) + \frac {g ^ {\mathrm {p r o p}} - g _ {0} ^ {\mathrm {p r o p}}}{g ^ {\mathrm {p r o p}}} (g ^ {(1)} - g _ {0} ^ {(1)}) (X) + \frac {g ^ {\mathrm {p r o p}} - g _ {0} ^ {\mathrm {p r o p}}}{1 - g ^ {\mathrm {p r o p}}} (g ^ {(0)} - g _ {0} ^ {(0)}) (X). +$$ + +Thus, for $\tau = g^{(1)} - g^{(0)}$ and for any $\theta \in \Theta$ and $g,\bar{g}\in \mathcal{G}_r(g_0)$ such that $\bar{g} = tg + (1 - t)g_{0}$ for some $t\in (0,1)$ , we have + +$$ +\begin{array}{l} \mathrm {D} _ {g} \mathrm {D} _ {\theta} L (\theta_ {\star}, \bar {g}) [ \theta - \theta_ {\star}, g - g _ {0} ] \\ = - \mathrm {D} _ {g} \mathbb {E} _ {\mathbb {P}} \left[ \left(\mathbb {E} _ {\mathbb {P}} \left[ \phi (g; Z) \mid X \right] + \langle \theta_ {\star}, X \rangle\right) \langle X, \theta - \theta_ {\star} \rangle \right] [ g - g _ {0} ] \\ = - \mathbb {E} _ {\mathbb {P}} \left[ \langle X, \theta - \theta_ {\star} \rangle \mathrm {D} _ {g} \mathbb {E} _ {\mathbb {P}} \left[ \phi (\bar {g}; Z) \mid X \right] [ g - g _ {0} ] \right] \\ = - \mathbb {E} _ {\mathbb {P}} \left[ \langle X, \theta - \theta_ {\star} \rangle \left(\frac {g _ {0} ^ {\text {p r o p}}}{\left(\bar {g} ^ {\text {p r o p}}\right) ^ {2}} \left(\bar {g} ^ {(1)} - g _ {0} ^ {(1)}\right) \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X) - \frac {\bar {g} ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}}{\bar {g} ^ {\text {p r o p}}} \left(g ^ {(1)} - g _ {0} ^ {(1)}\right) (X)\right) \right] \\ - \mathbb {E} _ {\mathbb {P}} \left[ \langle X, \theta - \theta_ {\star} \rangle \left(\frac {(1 - g _ {0} ^ {\mathrm {p r o p}}) (g ^ {\mathrm {p r o p}} - g _ {0} ^ {\mathrm {p r o p}}) (\bar {g} ^ {(0)} - g _ {0} ^ {(0)})}{(1 - \bar {g} ^ {\mathrm {p r o p}}) ^ {2}} - \frac {(\bar {g} ^ {\mathrm {p r o p}} - g _ {0} ^ {\mathrm {p r o p}}) (g ^ {(0)} - g _ {0} ^ {(0)})}{1 - \bar {g} ^ {\mathrm {p r o p}}}\right) (X) \right]. \\ \end{array} +$$ + +Since $(a + b)^4\leq 8a^4 +8b^4$ for $a,b\in \mathbb{R}$ , we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \frac {g _ {0} ^ {\operatorname {p r o p}} (X) ^ {4}}{g ^ {\operatorname {p r o p}} (X) ^ {4}} \right] = \mathbb {E} \left[ \frac {\left(g _ {0} ^ {\operatorname {p r o p}} (X) - g ^ {\operatorname {p r o p}} (X) + g ^ {\operatorname {p r o p}} (X)\right) ^ {4}}{g ^ {\operatorname {p r o p}} (X) ^ {4}} \right] \\ \leq 8 \mathbb {E} \left[ \frac {\left(g _ {0} ^ {\operatorname {p r o p}} (X) - g ^ {\operatorname {p r o p}} (X)\right) ^ {4} + g ^ {\operatorname {p r o p}} (X) ^ {4}}{g ^ {\operatorname {p r o p}} (X) ^ {4}} \right] \leq 8 r ^ {4} + 8. \tag {59} \\ \end{array} +$$ + +Similarly, we have + +$$ +\mathbb {E} \left[ \frac {\left(1 - g _ {0} ^ {\operatorname {p r o p}} (X)\right) ^ {4}}{\left(1 - g ^ {\operatorname {p r o p}} (X)\right) ^ {4}} \right] \leq 8 r ^ {4} + 8. +$$ + +It is easy to show that + +$$ +\begin{array}{l} \left| \mathbb {E} _ {\mathbb {P}} \left[ \langle X, \theta - \theta_ {\star} \rangle \frac {g _ {0} ^ {\text {p r o p}}}{(\bar {g} ^ {\text {p r o p}}) ^ {2}} (\bar {g} ^ {(1)} - g _ {0} ^ {(1)}) (g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) (X) \right] \right| \\ \leq C _ {X} \left(8 r ^ {4} + 8\right) ^ {\frac {1}{4}} r \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {\mathcal {G}}, \\ \end{array} +$$ + +and + +$$ +\left| \mathbb {E} _ {\mathbb {P}} \left[ \langle X, \theta - \theta_ {\star} \rangle \frac {\bar {g} ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}}{\bar {g} ^ {\text {p r o p}}} (g ^ {(1)} - g _ {0} ^ {(1)}) (X) \right] \right| \leq C _ {X} r \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {\mathcal {G}}. +$$ + +Thus, + +$$ +\begin{array}{l} \left| \mathrm {D} _ {g} \mathrm {D} _ {\theta} L (\theta_ {\star}, \bar {g}) [ \theta - \theta_ {\star}, g - g _ {0} ] \right| \leq 2 \left(\left(8 r ^ {4} + 8\right) ^ {\frac {1}{4}} + 1\right) C _ {X} r \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {g} \\ \leq 2 (2 (r + 1) + 1) C _ {X} r \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {\mathcal {G}}, \\ \end{array} +$$ + +which implies + +$$ +\alpha_ {1} = 2 C _ {X} (2 r + 3) r. \tag {60} +$$ + +In addition, $L(\theta ,g)$ is Neyman orthogonal at $(\theta_{\star},g_{0})$ since + +$$ +\mathrm {D} _ {g} \mathrm {D} _ {\theta} L \left(\theta_ {\star}, g _ {0}\right) \left[ \theta - \theta_ {\star}, g - g _ {0} \right] = 0. +$$ + +We have the higher-order derivative such that for any $\theta \in \Theta$ and $g, \bar{g} \in \mathcal{G}_r(g_0)$ , + +$$ +\begin{array}{l} \mathrm {D} _ {g} ^ {2} \mathrm {D} _ {\theta} L (\theta_ {\star}, \bar {g}) [ \theta - \theta_ {\star}, g - g _ {0}, g - g _ {0} ] \\ = 2 \mathbb {E} _ {\mathbb {P}} \left[ \left(\frac {g _ {0} ^ {\text {p r o p}} (g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) ^ {2}}{(\bar {g} ^ {\text {p r o p}}) ^ {3}} \left(\bar {g} ^ {(1)} (X) - g _ {0} ^ {(1)} (X)\right)\right) \langle X, \theta - \theta_ {\star} \rangle \right] \\ - 2 \mathbb {E} _ {\mathbb {P}} \left[ \left(\frac {g _ {0} ^ {\text {p r o p}} (g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}})}{(\bar {g} ^ {\text {p r o p}}) ^ {2}} \left(g ^ {(1)} (X) - g _ {0} ^ {(1)} (X)\right)\right) \langle X, \theta - \theta_ {\star} \rangle \right] \\ - 2 \mathbb {E} _ {\mathbb {P}} \left[ \left(\frac {(1 - g _ {0} ^ {\mathrm {p r o p}}) (g ^ {\mathrm {p r o p}} - g _ {0} ^ {\mathrm {p r o p}}) ^ {2}}{(1 - \bar {g} ^ {\mathrm {p r o p}}) ^ {3}} \left(\bar {g} ^ {(0)} (X) - g _ {0} ^ {(0)} (X)\right)\right) \langle X, \theta - \theta_ {\star} \rangle \right] \\ - 2 \mathbb {E} _ {\mathbb {P}} \left[ \left(\frac {(1 - g _ {0} ^ {\mathrm {p r o p}}) (g ^ {\mathrm {p r o p}} - g _ {0} ^ {\mathrm {p r o p}})}{(1 - \bar {g} ^ {\mathrm {p r o p}}) ^ {2}} \left(g ^ {(0)} (X) - g _ {0} ^ {(0)} (X)\right)\right) \langle X, \theta - \theta_ {\star} \rangle \right]. \\ \end{array} +$$ + +Note that $\bar{g}^{\mathrm{prop}} = t g^{\mathrm{prop}} + (1 - t) g_0^{\mathrm{prop}}$ for some $t \in (0, 1)$ by Taylor's theorem. Then + +$$ +\begin{array}{l} \frac {g ^ {\operatorname {p r o p}} (X)}{\bar {g} ^ {\operatorname {p r o p}} (X)} = \frac {g ^ {\operatorname {p r o p}} (X)}{t g ^ {\operatorname {p r o p}} (X) + (1 - t) g _ {0} ^ {\operatorname {p r o p}} (X)} \\ = \frac {1}{t + (1 - t) \left(g _ {0} ^ {\text {p r o p}} / g ^ {\text {p r o p}}\right) (X)} \leq \frac {1}{t + (1 - t) g _ {0} ^ {\text {p r o p}} (X)} \leq c _ {0} ^ {- 1}. \tag {61} \\ \end{array} +$$ + +Thus, + +$$ +\begin{array}{l} \left| \mathbb {E} _ {\mathbb {P}} \left[ \left(\frac {g _ {0} ^ {\text {p r o p}} (g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) ^ {2}}{(\bar {g} ^ {\text {p r o p}}) ^ {3}} \left(\bar {g} ^ {(1)} (X) - g _ {0} ^ {(1)} (X)\right)\right) \langle X, \theta - \theta_ {\star} \rangle \right] \right| \\ \leq \mathbb {E} _ {\mathbb {P}} \left[ \frac {(g ^ {\text {p r o p}}) ^ {4}}{(\bar {g} ^ {\text {p r o p}}) ^ {4}} \frac {(g ^ {\text {p r o p}} (X) - g _ {0} ^ {\text {p r o p}} (X)) ^ {4}}{(g ^ {\text {p r o p}}) ^ {4}} \right] ^ {1 / 2} \mathbb {E} _ {\mathbb {P}} \left[ \left(\frac {\bar {g} ^ {(1)} (X) - g _ {0} ^ {(1)} (X)}{\bar {g} ^ {\text {p r o p}}}\right) ^ {2} \right] ^ {1 / 2} C _ {X} \| \theta - \theta_ {\star} \| _ {2} \\ \leq c _ {0} ^ {- 2} C _ {X} r \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {g} ^ {2}. \\ \end{array} +$$ + +Similarly, we have + +$$ +\begin{array}{l} \left| \mathbb {E} _ {\mathbb {P}} \left[ \left(\frac {g _ {0} ^ {\text {p r o p}} \left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right)}{(\bar {g} ^ {\text {p r o p}}) ^ {2}} \left(g ^ {(1)} (X) - g _ {0} ^ {(1)} (X)\right)\right) \langle X, \theta - \theta_ {\star} \rangle \right] \right| \\ \leq \mathbb {E} _ {\mathbb {P}} \left[ \left| \frac {(g ^ {\text {p r o p}}) ^ {2}}{(\bar {g} ^ {\text {p r o p}}) ^ {2}} \frac {(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}})}{g ^ {\text {p r o p}}} \frac {(g ^ {(1)} - g _ {0} ^ {(1)})}{g ^ {\text {p r o p}}} \right| (X) \right] C _ {X} \| \theta - \theta_ {\star} \| _ {2} \leq c _ {0} ^ {- 2} C _ {X} \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {\mathcal {G}} ^ {2}. \\ \end{array} +$$ + +Then we can show that + +$$ +\left| \mathrm {D} _ {g} ^ {2} \mathrm {D} _ {\theta} L (\theta_ {\star}, \bar {g}) [ \theta - \theta_ {\star}, g - g _ {0}, g - g _ {0} ] \right| \leq 4 c _ {0} ^ {- 2} C _ {X} (1 + r) \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {\mathcal {G}} ^ {2}. +$$ + +which implies that + +$$ +\beta_ {1} = 4 c _ {0} ^ {- 2} C _ {X} (1 + r). \tag {62} +$$ + +![](images/99123e37e158c5b2ae3737e3f1808289f35fdba8362f0c972ead3de7a7089d39.jpg) + +# B.4.6 Proof of Lemma 9 + +Proof. Define + +$$ +\mu_ {g} ^ {(s)} (z) = g ^ {(s)} (x) + \frac {\mathbb {1} (w = s)}{s g ^ {\operatorname {p r o p}} (x) + (1 - s) (1 - g ^ {\operatorname {p r o p}} (x))} (y - g ^ {(s)} (x)), +$$ + +where $\mathbb{1}(\cdot)$ denotes the indicator function, and the log-linear predictor $p_{\theta}(x) = e^{\langle \theta ,x\rangle} / (1 + e^{\langle \theta ,x\rangle})$ Under Asm. 5, + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ \mu_ {g} ^ {(s)} (Z) \mid X \right] = \mathbb {E} _ {\mathbb {P}} \left[ g ^ {(s)} (X) + \frac {\mathbb {1} (W = s)}{s g ^ {\operatorname {p r o p}} (X) + (1 - s) (1 - g ^ {\operatorname {p r o p}} (X))} \left(Y (s) - g ^ {(s)} (X)\right) \mid X \right] \\ = g ^ {(s)} (X) + \frac {s g _ {0} ^ {\mathrm {p r o p}} (X) + (1 - s) (1 - g _ {0} ^ {\mathrm {p r o p}} (X))}{s g ^ {\mathrm {p r o p}} (X) + (1 - s) (1 - g ^ {\mathrm {p r o p}} (X))} \left(g _ {0} ^ {(s)} (X) - g ^ {(s)} (X)\right) \\ = g _ {0} ^ {(s)} (X) + \frac {(2 s - 1) \left(g ^ {\operatorname {p r o p}} (X) - g _ {0} ^ {\operatorname {p r o p}} (X)\right)}{s g ^ {\operatorname {p r o p}} (X) + (1 - s) \left(1 - g ^ {\operatorname {p r o p}} (X)\right)} \left(g ^ {(s)} (X) - g _ {0} ^ {(s)} (X)\right) =: f ^ {(s)} (g; X). \tag {63} \\ \end{array} +$$ + +We consider the following loss: + +$$ +\ell (\theta , g; z) = - \left[ \mu_ {g} ^ {(1)} (z) \log p _ {\theta} (x) + \mu_ {g} ^ {(0)} (z) \log (1 - p _ {\theta} (x)) \right]. +$$ + +with the corresponding risk function defined as + +$$ +L (\theta , g) = - \mathbb {E} _ {\mathbb {P}} \left[ \mu_ {g} ^ {(1)} (Z) \log p _ {\theta} (x) + \mu_ {g} ^ {(0)} (Z) \log (1 - p _ {\theta} (X)) \right]. +$$ + +By (63), we have $\mathbb{E}_{\mathbb{P}}[\mu_{g_0}^{(s)}(Z) \mid X] = g_0^{(s)}(X)$ . Thus, the target is the minimizer of the following squared loss: + +$$ +\theta_ {\star} = \underset {\theta \in \mathbb {R} ^ {d}} {\arg \min } - \mathbb {E} _ {\mathbb {P}} \left[ g _ {0} ^ {(1)} (X) \log p _ {\theta} (x) + g _ {0} ^ {(0)} (X) \log \left(1 - p _ {\theta} (X)\right) \right]. \tag {64} +$$ + +Since $\nabla_{\theta}p_{\theta}(x) = p_{\theta}(x)(1 - p_{\theta}(x))x$ we have + +$$ +\nabla_ {\theta} \log p _ {\theta} (x) = \frac {\nabla_ {\theta} p _ {\theta} (x)}{p _ {\theta} (x)} = (1 - p _ {\theta} (x)) x, +$$ + +$$ +\nabla_ {\theta} \log (1 - p _ {\theta} (x)) = - \frac {\nabla_ {\theta} p _ {\theta} (x)}{1 - p _ {\theta} (x)} = - p _ {\theta} (x) x. +$$ + +Differentiating $\ell (\theta ,g;z)$ with respect to $\theta$ , we obtain the gradient and Hessian $w.r.t.\theta$ as + +$$ +S _ {\theta} (\theta , g; z) = - \left[ \mu_ {g} ^ {(1)} (z) (1 - p _ {\theta} (x)) x - \mu_ {g} ^ {(0)} (z) p _ {\theta} (x) x \right], +$$ + +$$ +H _ {\theta \theta} (\theta , g; z) = (\mu_ {g} ^ {(1)} (z) + \mu_ {g} ^ {(0)} (z)) p _ {\theta} (x) (1 - p _ {\theta} (x)) x x ^ {\top}. +$$ + +The expected gradient and expected Hessian are then obtained as + +$$ +S _ {\theta} (\theta , g) = - \mathbb {E} _ {\mathbb {P}} \left[ f ^ {(1)} (g; X) \left(1 - p _ {\theta} (X)\right) X - f ^ {(0)} (g; X) p _ {\theta} (X) X \right], +$$ + +$$ +H _ {\theta \theta} (\theta , g) = \mathbb {E} _ {\mathbb {P}} \left[ \left(f ^ {(1)} (g; X) + f ^ {(0)} (g; X)\right) p _ {\theta} (X) \left(1 - p _ {\theta} (X)\right) X X ^ {\top} \right]. +$$ + +We consider the nuisance neighborhood such that for $g \in \mathcal{G}_r(g_0)$ , + +$$ +\| g - g _ {0} \| _ {\mathcal {G}} := \max \left\{\mathbb {E} _ {\mathbb {P}} \left[ \left(\frac {(g ^ {(w)} - g _ {0} ^ {(w)}) (X)}{g ^ {(w)} (1 - g ^ {(w)}) (X)}\right) ^ {4} \right] ^ {\frac {1}{4}}, \mathbb {E} _ {\mathbb {P}} \left[ \left(\frac {(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) (X)}{g ^ {\text {p r o p}} (1 - g ^ {\text {p r o p}}) (X)}\right) ^ {4} \right] ^ {\frac {1}{4}} \right\} \leq r. +$$ + +We assume that $\delta \leq g_0^{(0)}(X) + g_0^{(1)}(X) \leq \delta^{-1}$ for $\delta > 0$ . In addition, we assume that for $g \in \mathcal{G}_r(g_0)$ , $f^{(1)}(g; X) + f^{(0)}(g; X) \geq \delta$ a.s.. Note that + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ | f ^ {(1)} (g; X) + f ^ {(0)} (g; X) | \right] \leq \delta + \sum_ {s = \{0, 1 \}} \mathbb {E} _ {\mathbb {P}} \left[ \left| \frac {g ^ {\text {p r o p}} (X) - g _ {0} ^ {\text {p r o p}} (X)}{g ^ {\text {p r o p}} (1 - g ^ {\text {p r o p}} (X))} \left(g ^ {(s)} (X) - g _ {0} ^ {(s)} (X)\right) \right| \right] \\ \leq \delta + 2 r ^ {2}. \\ \end{array} +$$ + +We now verify that the loss function $\ell$ satisfies Asm. 3. + +(a) We assume that $g^{(w)}: \mathbb{R}^d \mapsto \mathbb{R}, w = 0,1$ , and $g^{\mathrm{prop}}: \mathbb{R}^d \mapsto (0,1)$ are continuous, thus Asm. 3(a) is satisfied. + +(b) Since $\theta_{\star}$ is a global minimizer of (64), we have + +$$ +S _ {\theta} \left(\theta_ {\star}, g _ {0}\right) = 0. \tag {65} +$$ + +(c) We assume that $\Theta$ is bounded such that $C\leq p_{\theta}(X)\leq 1 - C$ a.s. for some $C > 0$ . When $\lambda_{\min}(\mathbb{E}_{\mathbb{P}}[XX^{\top}])\geq \lambda_0 > 0$ and $\| X\| _2\leq C_X$ a.s., we have + +$$ +C ^ {2} \delta \lambda_ {0} \mathbf {I} \preccurlyeq H _ {\theta \theta} (\theta , g) \preccurlyeq C _ {X} ^ {2} (1 + 2 \delta^ {- 1} r ^ {2}) \mathbf {I} \Longrightarrow \mu = C ^ {2} \delta \lambda_ {0} \text {a n d} M = C _ {X} ^ {2} (1 + 2 \delta^ {- 1} r ^ {2}). \tag {66} +$$ + +(d) Consider the Taylor expansion around $\theta_{\star}$ , we have + +$$ +S _ {\theta} (\theta , g; Z) - S _ {\theta} (\theta , g) = S _ {\theta} \left(\theta_ {\star}, g; Z\right) - S _ {\theta} \left(\theta_ {\star}, g\right) + \left(H _ {\theta \theta} \left(\theta_ {\star}, g; Z\right) - H _ {\theta \theta} \left(\theta_ {\star}, g\right)\right) \left(\theta - \theta_ {\star}\right). +$$ + +Note that + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} (\theta_ {\star}, g; Z) - S _ {4} (\theta_ {\star}, g _ {4}) \| _ {2} ^ {2} \right] \leq \mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} (\theta_ {\star}, g; Z) \| _ {2} ^ {2} \right] \\ \leq C _ {X} ^ {2} \mathbb {E} \left[ \left(\left| \mu_ {g} ^ {(1)} (Z) \right| + \left| \mu_ {g} ^ {(0)} (Z) \right|\right) ^ {2} \right] \\ \leq 2 C _ {X} ^ {2} \mathbb {E} \left[ \left(\mu_ {g} ^ {(1)} (Z)\right) ^ {2} + \left(\mu_ {g} ^ {(1)} (Z)\right) ^ {2} \right]. \\ \end{array} +$$ + +For $s = 1$ , when $Y(1) - g_0^{(1)}(X) \leq C_Y$ a.s. for $C_Y > 0$ , we have + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} [ \mu_ {g} ^ {(1)} (Z) ^ {2} ] = \mathbb {E} _ {\mathbb {P}} \left[ \left(g _ {0} ^ {(1)} (X) + \frac {g ^ {\mathrm {p r o p}} (X) - W}{g ^ {\mathrm {p r o p}} (X)} (g ^ {(1)} - g _ {0} ^ {(1)}) (X) + \frac {W}{g ^ {\mathrm {p r o p}} (X)} \left(Y - g _ {0} ^ {(1)} (X)\right)\right) ^ {2} \right] \\ \leq 3 \delta^ {- 2} + 3 \mathbb {E} _ {\mathbb {P}} \left[ \frac {(g ^ {\operatorname {p r o p}} (X) - W) ^ {2}}{(g ^ {\operatorname {p r o p}} (X)) ^ {2}} \left((g ^ {(1)} - g _ {0} ^ {(1)}) (X)\right) ^ {2} \right] + 3 C _ {Y} ^ {2} \mathbb {E} _ {\mathbb {P}} \left[ \frac {g _ {0} ^ {\operatorname {p r o p}} (X)}{(g ^ {\operatorname {p r o p}} (X)) ^ {2}} \right] \\ \leq 3 \delta^ {- 2} + 1 2 \mathbb {E} _ {\mathbb {P}} \left[ \left(\frac {\left(g ^ {(1)} - g _ {0} ^ {(1)}\right) (X)}{g ^ {\mathrm {p r o p}} (X)}\right) ^ {2} \right] + 3 c _ {0} ^ {- 1} C _ {Y} ^ {2} \mathbb {E} _ {\mathbb {P}} \left[ \frac {\left(g _ {0} ^ {\mathrm {p r o p}} (X)\right) ^ {2}}{\left(g ^ {\mathrm {p r o p}} (X)\right) ^ {2}} \right] \\ \leq 3 \delta^ {- 2} + 1 2 r ^ {2} + 3 c _ {0} ^ {- 1} C _ {Y} ^ {2} (8 r ^ {4} + 8) ^ {1 / 2} \leq 3 (\delta^ {- 2} + 4 c _ {0} ^ {- 1} C _ {Y} ^ {2}) + \mathcal {O} (r ^ {2}). \\ \end{array} +$$ + +Thus, + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \| S _ {\theta} (\theta_ {\star}, g; Z) - S _ {4} (\theta_ {\star}, g _ {4}) \| _ {2} ^ {2} \right] \leq 1 2 C _ {X} ^ {2} (\delta^ {- 2} + 4 c _ {0} ^ {- 1} C _ {Y} ^ {2}) + \mathcal {O} (r ^ {2}). +$$ + +Since + +$$ +H _ {\theta \theta} (\theta , g; Z) - H _ {\theta \theta} (\theta , g) \preccurlyeq C _ {X} ^ {2} \left(\mu_ {g} ^ {(1)} (Z) + \mu_ {g} ^ {(0)} (Z) + 1 + 2 \delta^ {- 1} r ^ {2}\right) \mathbf {I}, +$$ + +we have + +$$ +\| (H _ {\theta \theta} (\theta_ {\star}, g; Z) - H _ {\theta \theta} (\theta_ {\star}, g)) \| _ {2} ^ {2} \leq 3 C _ {X} ^ {4} ((\mu_ {g} ^ {(1)} (Z)) ^ {2} + (\mu_ {g} ^ {(0)} (Z)) ^ {2} + (1 + 2 \delta^ {- 1} r ^ {2}) ^ {2}). +$$ + +Similarly, we can show that + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \| (H _ {\theta \theta} (\theta_ {\star}, g; Z) - H _ {\theta \theta} (\theta_ {\star}, g)) \| _ {2} ^ {2} \right] \leq 3 C _ {X} ^ {4} (1 + 6 (\delta^ {- 2} + 4 c _ {0} ^ {- 1} C _ {Y} ^ {2})) + \mathcal {O} (r ^ {2}). +$$ + +It follows that + +$$ +K _ {1} = 2 4 C _ {X} ^ {2} \left(\delta^ {- 2} + 4 c _ {0} ^ {- 1} C _ {Y} ^ {2}\right) + \mathcal {O} \left(r ^ {2}\right) \text {a n d} \kappa_ {1} = 6 C _ {X} ^ {4} \left(1 + 6 \left(\delta^ {- 2} + 4 c _ {0} ^ {- 1} C _ {Y} ^ {2}\right)\right) + \mathcal {O} \left(r ^ {2}\right). \tag {67} +$$ + +(e) For $s = 1$ , we have + +$$ +\begin{array}{l} \mathrm {D} _ {g} f ^ {(1)} (\bar {g}; X) [ g - g _ {0} ] = \frac {g _ {0} ^ {\operatorname {p r o p}} (X)}{(\bar {g} ^ {\operatorname {p r o p}} (X)) ^ {2}} (\bar {g} ^ {(1)} (X) - g _ {0} ^ {(1)} (X)) (g ^ {\operatorname {p r o p}} (X) - g _ {0} ^ {\operatorname {p r o p}} (X)) \\ + \frac {\bar {g} ^ {\operatorname {p r o p}} (X) - g _ {0} ^ {\operatorname {p r o p}} (X)}{\bar {g} ^ {\operatorname {p r o p}} (X)} \left(g ^ {(1)} (X) - g _ {0} ^ {(1)} (X)\right). \\ \end{array} +$$ + +Similarly, for $s = 0$ + +$$ +\begin{array}{l} \mathrm {D} _ {g} f ^ {(0)} (\bar {g}; X) [ g - g _ {0} ] = \frac {1 - g _ {0} ^ {\operatorname {p r o p}} (X)}{(1 - \bar {g} ^ {\operatorname {p r o p}} (X)) ^ {2}} (\bar {g} ^ {(0)} (X) - g _ {0} ^ {(0)} (X)) (g ^ {\operatorname {p r o p}} (X) - g _ {0} ^ {\operatorname {p r o p}} (X)) \\ - \frac {\bar {g} ^ {\operatorname {p r o p}} (X) - g _ {0} ^ {\operatorname {p r o p}} (X)}{1 - \bar {g} ^ {\operatorname {p r o p}} (X)} \left(g ^ {(0)} (X) - g _ {0} ^ {(0)} (X)\right). \\ \end{array} +$$ + +For any $\bar{g}, g \in \mathcal{G}_r(g_0)$ such that $\bar{g} = t g + (1 - t) g_0$ for some $t \in (0, 1)$ , we have + +$$ +\begin{array}{l} \mathrm {D} _ {g} \mathrm {D} _ {\theta} L (\theta_ {\star}, \bar {g}) [ \theta - \theta_ {\star}, g - g _ {0} ] \\ = - \mathbb {E} _ {\mathbb {P}} \left[ \mathrm {D} _ {g} \left(f ^ {(1)} (\bar {g}; X) \left(1 - p _ {\theta} (X)\right) - f ^ {(0)} (\bar {g}; X) p _ {\theta} (X)\right) [ g - g _ {0} ] \langle X, \theta - \theta_ {\star} \rangle \right] \\ = - \mathbb {E} _ {\mathbb {P}} \left[ \frac {g _ {0} ^ {\text {p r o p}} (1 - p _ {\theta})}{(\bar {g} ^ {\text {p r o p}}) ^ {2}} (\bar {g} ^ {(1)} - g _ {0} ^ {(1)}) (g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) (X) \langle X, \theta - \theta_ {\star} \rangle \right] \\ - \mathbb {E} _ {\mathbb {P}} \left[ \frac {(1 - g _ {0} ^ {\text {p r o p}}) p _ {\theta}}{(1 - \bar {g} ^ {\text {p r o p}}) ^ {2}} (\bar {g} ^ {(0)} - g _ {0} ^ {(0)}) (g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) (X) \langle X, \theta - \theta_ {\star} \rangle \right] \\ - \mathbb {E} _ {\mathbb {P}} \left[ \frac {\left(\bar {g} ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) \left(1 - p _ {\theta}\right)}{\bar {g} ^ {\text {p r o p}}} \left(g ^ {(1)} - g _ {0} ^ {(1)}\right) (X) \langle X, \theta - \theta_ {\star} \rangle \right] \\ + \mathbb {E} _ {\mathbb {P}} \left[ \frac {(\bar {g} ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) p _ {\theta}}{1 - \bar {g} ^ {\text {p r o p}}} (g ^ {(0)} - g _ {0} ^ {(0)}) (X) \langle X, \theta - \theta_ {\star} \rangle \right]. \\ \end{array} +$$ + +Note that by (61), + +$$ +\begin{array}{l} \left| \mathbb {E} _ {\mathbb {P}} \left[ \frac {g _ {0} ^ {\text {p r o p}} (1 - p _ {\theta})}{(\bar {g} ^ {\text {p r o p}}) ^ {2}} (\bar {g} ^ {(1)} - g _ {0} ^ {(1)}) (g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}) (X) \langle X, \theta - \theta_ {\star} \rangle \right] \right| \\ \leq \mathbb {E} _ {\mathbb {P}} \left[ \left| \frac {g ^ {\text {p r o p}} (X)}{\bar {g} ^ {\text {p r o p}} (X)} \frac {\left(\bar {g} ^ {(1)} - g _ {0} ^ {(1)}\right) (X)}{g ^ {\text {p r o p}} (X)} \frac {\left(g ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) (X)}{g ^ {\text {p r o p}} (X)} \right| \right] C _ {X} \| \theta - \theta_ {\star} \| _ {2} \\ \leq c _ {0} ^ {- 1} C _ {X} r \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {\mathcal {G}}. \\ \end{array} +$$ + +In addition, + +$$ +\begin{array}{l} \left| \mathbb {E} _ {\mathbb {P}} \left[ \frac {\left(\bar {g} ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right) \left(1 - p _ {\theta}\right)}{\bar {g} ^ {\text {p r o p}}} \left(g ^ {(1)} - g _ {0} ^ {(1)}\right) (X) \langle X, \theta - \theta_ {\star} \rangle \right] \right| \\ \leq \mathbb {E} _ {\mathbb {P}} \left[ \left| \frac {\left(\bar {g} ^ {\text {p r o p}} - g _ {0} ^ {\text {p r o p}}\right)}{\bar {g} ^ {\text {p r o p}}} \left(g ^ {(1)} - g _ {0} ^ {(1)}\right) (X) \right| \right] C _ {X} \| \theta - \theta_ {\star} \| _ {2} \leq C _ {X} r \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {\mathcal {G}}. \\ \end{array} +$$ + +Thus, it is easy to show that + +$$ +\left| \mathrm {D} _ {g} \mathrm {D} _ {\theta} L \left(\theta_ {\star}, \bar {g}\right) \left[ \theta - \theta_ {\star}, g - g _ {0} \right] \right| \leq 2 \left(c _ {0} ^ {- 1} + 1\right) C _ {X} r \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {\mathcal {G}}, +$$ + +which implies + +$$ +\alpha_ {1} = 2 \left(c _ {0} ^ {- 1} + 1\right) C _ {X} r. \tag {68} +$$ + +In addition, $L(\theta ,g)$ is Neyman orthogonal at $(\theta_{\star},g_0)$ since + +$$ +\mathrm {D} _ {g} \mathrm {D} _ {\theta} L (\theta_ {\star}, g _ {0}) [ \theta - \theta_ {\star}, g - g _ {0} ] = 0. +$$ + +Now we compute the higher-order derivative. For $s = 1$ , we have + +$$ +\begin{array}{l} \mathrm {D} _ {g} ^ {2} f ^ {(1)} (\bar {g}; X) [ g - g _ {0}, g - g _ {0} ] = - \frac {2 g _ {0} ^ {\text {p r o p}} (X)}{(\bar {g} ^ {\text {p r o p}} (X)) ^ {3}} (\bar {g} ^ {(1)} (X) - g _ {0} ^ {(1)} (X)) (g ^ {\text {p r o p}} (X) - g _ {0} ^ {\text {p r o p}} (X)) ^ {2} \\ + \frac {2 g _ {0} ^ {\text {p r o p}} (X)}{(\bar {g} ^ {\text {p r o p}} (X)) ^ {2}} (g ^ {(1)} (X) - g _ {0} ^ {(1)} (X)) (g ^ {\text {p r o p}} (X) - g _ {0} ^ {\text {p r o p}} (X)). \\ \end{array} +$$ + +Similarly, for $s = 0$ + +$$ +\begin{array}{l} \mathrm {D} _ {g} ^ {2} f ^ {(0)} (\bar {g}; X) [ g - g _ {0}, g - g _ {0} ] = 2 \frac {1 - g _ {0} ^ {\operatorname {p r o p}} (X)}{(1 - \bar {g} ^ {\operatorname {p r o p}} (X)) ^ {3}} (\bar {g} ^ {(0)} (X) - g _ {0} ^ {(0)} (X)) (g ^ {\operatorname {p r o p}} (X) - g _ {0} ^ {\operatorname {p r o p}} (X)) ^ {2} \\ + 2 \frac {1 - g _ {0} ^ {\operatorname {p r o p}} (X)}{(1 - \bar {g} ^ {\operatorname {p r o p}} (X)) ^ {2}} \left(g ^ {(0)} (X) - g _ {0} ^ {(0)} (X)\right) \left(g ^ {\operatorname {p r o p}} (X) - g _ {0} ^ {\operatorname {p r o p}} (X)\right). \\ \end{array} +$$ + +Then for any $\theta \in \Theta$ and $g, \bar{g} \in \mathcal{G}_r(g_0)$ such that $\bar{g} = tg + (1 - t)g_0$ for some $t \in (0,1)$ , + +$$ +\begin{array}{l} \mathrm {D} _ {g} ^ {2} \mathrm {D} _ {\theta} L (\theta_ {\star}, \bar {g}) [ \theta - \theta_ {\star}, g - g _ {0}, g - g _ {0} ] \\ = - \mathbb {E} _ {\mathbb {P}} \left[ \mathrm {D} _ {g} ^ {2} \left(f ^ {(1)} (\bar {g}; X) \left(1 - p _ {\theta} (X)\right) - f ^ {(0)} (\bar {g}; X) p _ {\theta} (X)\right) [ g - g _ {0}, g - g _ {0} ] \langle X, \theta - \theta_ {\star} \rangle \right] \\ = 2 \mathbb {E} _ {\mathbb {P}} \left[ \frac {g _ {0} ^ {\operatorname {p r o p}} (X)}{(\bar {g} ^ {\operatorname {p r o p}} (X)) ^ {3}} (\bar {g} ^ {(1)} (X) - g _ {0} ^ {(1)} (X)) \left(g ^ {\operatorname {p r o p}} (X) - g _ {0} ^ {\operatorname {p r o p}} (X)\right) ^ {2} \langle X, \theta - \theta_ {\star} \rangle \right] \\ - 2 \mathbb {E} _ {\mathbb {P}} \left[ \frac {1 - g _ {0} ^ {\text {p r o p}} (X)}{(1 - \bar {g} ^ {\text {p r o p}} (X)) ^ {3}} (\bar {g} ^ {(0)} (X) - g _ {0} ^ {(0)} (X)) (g ^ {\text {p r o p}} (X) - g _ {0} ^ {\text {p r o p}} (X)) ^ {2} \langle X, \theta - \theta_ {\star} \rangle \right] \\ - 2 \mathbb {E} _ {\mathbb {P}} \left[ \frac {g _ {0} ^ {\text {p r o p}} (X)}{(\bar {g} ^ {\text {p r o p}} (X)) ^ {2}} \left(g ^ {(1)} (X) - g _ {0} ^ {(1)} (X)\right) \left(g ^ {\text {p r o p}} (X) - g _ {0} ^ {\text {p r o p}} (X)\right) \langle X, \theta - \theta_ {\star} \rangle \right] \\ - 2 \mathbb {E} _ {\mathbb {P}} \left[ \frac {1 - g _ {0} ^ {\text {p r o p}} (X)}{(1 - \bar {g} ^ {\text {p r o p}} (X)) ^ {2}} \left(g ^ {(0)} (X) - g _ {0} ^ {(0)} (X)\right) \left(g ^ {\text {p r o p}} (X) - g _ {0} ^ {\text {p r o p}} (X)\right) \langle X, \theta - \theta_ {\star} \rangle \right]. \\ \end{array} +$$ + +By (61), we have + +$$ +\begin{array}{l} \left| \mathbb {E} _ {\mathbb {P}} \left[ \frac {g _ {0} ^ {\text {p r o p}} (X)}{(\bar {g} ^ {\text {p r o p}} (X)) ^ {3}} (\bar {g} ^ {(1)} (X) - g _ {0} ^ {(1)} (X)) (g ^ {\text {p r o p}} (X) - g _ {0} ^ {\text {p r o p}} (X)) ^ {2} \langle X, \theta - \theta_ {\star} \rangle \right] \right| \\ \leq \mathbb {E} _ {\mathbb {P}} \left[ \right.\left| \right. \frac {(g ^ {\operatorname {p r o p}} (X)) ^ {2}}{(\bar {g} ^ {\operatorname {p r o p}} (X)) ^ {2}} \frac {\bar {g} ^ {(1)} (X) - g _ {0} ^ {(1)} (X)}{\bar {g} ^ {\operatorname {p r o p}} (X)} \left(\frac {g ^ {\operatorname {p r o p}} (X) - g _ {0} ^ {\operatorname {p r o p}} (X)}{g ^ {\operatorname {p r o p}} (X)}\right) ^ {2} \left. \right] C _ {X} \| \theta - \theta_ {\star} \| _ {2} \\ \leq c _ {0} ^ {- 2} C _ {X} r \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {G} ^ {2}. \\ \end{array} +$$ + +In addition, + +$$ +\begin{array}{l} \left| \mathbb {E} _ {\mathbb {P}} \left[ \frac {g _ {0} ^ {\text {p r o p}} (X)}{(\bar {g} ^ {\text {p r o p}} (X)) ^ {2}} \left(g ^ {(1)} (X) - g _ {0} ^ {(1)} (X)\right) \left(g ^ {\text {p r o p}} (X) - g _ {0} ^ {\text {p r o p}} (X)\right) \langle X, \theta - \theta_ {\star} \rangle \right] \right. \\ \leq \mathbb {E} _ {\mathbb {P}} \left[ \frac {(g ^ {\operatorname {p r o p}} (X)) ^ {2}}{(\bar {g} ^ {\operatorname {p r o p}} (X)) ^ {2}} \frac {g ^ {(1)} (X) - g _ {0} ^ {(1)} (X)}{g ^ {\operatorname {p r o p}} (X)} \frac {g ^ {\operatorname {p r o p}} (X) - g _ {0} ^ {\operatorname {p r o p}} (X)}{g ^ {\operatorname {p r o p}} (X)} \right] C _ {X} \| \theta - \theta_ {\star} \| _ {2} \\ \leq c _ {0} ^ {- 2} C _ {X} \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {G} ^ {2}. \\ \end{array} +$$ + +Together we have + +$$ +\left| \mathrm {D} _ {g} ^ {2} \mathrm {D} _ {\theta} L (\theta_ {\star}, \bar {g}) [ \theta - \theta_ {\star}, g - g _ {0}, g - g _ {0} ] \right| \leq 4 c _ {0} ^ {- 2} C _ {X} (1 + r) \| \theta - \theta_ {\star} \| _ {2} \| g - g _ {0} \| _ {\mathcal {G}} ^ {2}, +$$ + +which implies + +$$ +\beta_ {1} = 4 c _ {0} ^ {- 2} C _ {X} (1 + r). \tag {69} +$$ + +![](images/9b609ed7173cd9763eb4d8ba2160cd9f3d1590b585bf3cd264d58b0335da104c.jpg) + +# C Convergence Proofs for Stochastic Gradient + +This section is dedicated to demonstrate the SGD convergence in Thm. 1 from Sec. 3 of the main text using Asm. 3 and Asm. 4. We first give an overview of the problem settings and the expected results with the proof outline in Appx. C.1. We then provide all the technical lemmas needed for Thm. 1 in Appx. C.2, and finally prove our first main result in Appx. C.3. + +# C.1 Overview + +In this section, we demonstrate the convergence of SGD for a risk minimization problem with nuisance: + +$$ +\theta_{\star} = \operatorname *{arg min}_{\theta \in \Theta}L(\theta ,g_{0}), +$$ + +where $g_0 \in \mathcal{G}$ is the true nuisance, $L(\theta, g) = \mathbb{E}_{Z \sim \mathbb{P}}[\ell(\theta, g; Z)]$ , and $\ell$ is a prespecified loss function. We consider the stochastic gradient method for learning $\theta_{\star}$ when $g_0$ is unknown but an estimate $\hat{g}$ is accessible. Define $\mathcal{D}_n = (Z_1, \ldots, Z_n)$ , sampled from the product measure $\mathbb{P}^n$ . Recall the SGD $\theta^{(n)}$ defined as + +$$ +\theta^ {(n)} = \theta^ {(n - 1)} - \eta S _ {\theta} \left(\theta^ {(n - 1)}, \hat {g}; Z _ {n}\right). \tag {70} +$$ + +Throughout the section, we take the following notations for simplicity: + +$$ +\delta^ {(n)} = \theta^ {(n)} - \theta_ {\star}, \tag {71} +$$ + +$$ +S ^ {(n)} = S _ {\theta} \left(\theta^ {(n - 1)}, \hat {g}; Z _ {n}\right), \tag {72} +$$ + +$$ +v ^ {(n)} = S ^ {(n)} - S _ {\theta} \left(\theta^ {(n - 1)}, \hat {g}\right), \tag {73} +$$ + +where $S_{\theta}(\theta ,g;z) = \nabla_{\theta}\ell (\theta ,g;z)$ is the gradient and $S_{\theta}(\theta ,g) = \mathbb{E}_{Z\sim \mathbb{P}}[S_{\theta}(\theta ,g;Z)]$ is the population gradient. We are interested in the mean squared error using an estimated nuisance $\hat{g}$ , and our results show that for non-Neyman orthogonal loss $L$ , the error $\delta^{(n)}$ satisfies + +$$ +\mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} \left[ \| \delta^ {(n)} \| _ {2} ^ {2} \right] \lesssim \left(1 - \frac {\mu \eta}{2}\right) ^ {n} \| \delta^ {(0)} \| _ {2} ^ {2} + \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} + \eta , \tag {74} +$$ + +where the nuisance estimator $\hat{g}$ would lead to a bias of order $\mathcal{O}(\| \hat{g} - g_0\|_{\mathcal{G}}^2)$ for the SGD convergence. If $L$ is Neyman orthogonal, this bias introduced by the nuisance estimation would be further removed, resulting in the following convergence + +$$ +\mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} \left[ \| \delta^ {(n)} \| _ {2} ^ {2} \right] \lesssim \left(1 - \frac {\mu \eta}{2}\right) ^ {n} \| \delta^ {(0)} \| _ {2} ^ {2} + \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {4} + \eta . \tag {75} +$$ + +Proof Outline. The proofs for both results (74) and (75) proceed through the following four steps: + +1. Upper bound the excess risk $L(\theta^{(n)}, \hat{g}) - L(\theta_{\star}, \hat{g})$ in terms of the SGD improvement. +2. Lower bound $L(\theta^{(n)}, \hat{g}) - L(\theta_{\star}, \hat{g})$ using strong convexity and Neyman orthogonality. +3. Derive a recursive formula of $\mathbb{E}_{\mathcal{D}_n\sim \mathbb{P}^n}\left[\| \delta^{(n)}\| _2^2\right]$ from these bounds. +4. Perform the recursion and obtain the final result. + +Follow these steps above, we provide technical lemma in Appx. C.2, and then prove our first main result Thm. 1 in Appx. C.3. + +# C.2 Technical Lemma + +Lemma 10 (One-step improvement for SGD). Suppose that Asm. 3 holds. If $\eta < 1 / M$ , $\theta^{(n)} \in \Theta$ and $\hat{g} \in \mathcal{G}_r(g_0)$ , it holds that + +$$ +2 \eta (L (\theta^ {(n)}, \hat {g}) - L (\theta_ {\star}, \hat {g})) \leq (1 - \mu \eta) \| \delta^ {(n - 1)} \| _ {2} ^ {2} - \| \delta^ {(n)} \| _ {2} ^ {2} - 2 \eta \langle v ^ {(n)}, \delta^ {(n - 1)} \rangle + \frac {\eta^ {2}}{1 - M \eta} \| v ^ {(n)} \| _ {2} ^ {2}. +$$ + +Proof. We first define the $\eta^{-1}$ -strongly convex function $f_{n}$ as + +$$ +f _ {n} (u) = \left\langle S ^ {(n)}, u - \theta^ {(n - 1)} \right\rangle + \frac {1}{2 \eta} \| u - \theta^ {(n - 1)} \| _ {2} ^ {2}. \tag {76} +$$ + +Note that + +$$ +\operatorname *{arg min}_{u\in \mathbb{R}^{d}}f_{n}\left(u\right) = \operatorname *{arg min}_{u\in \mathbb{R}^{d}}\| u - \left(\theta^{(n - 1)} - \eta S^{(n)}\right)\|_{2}^{2}, +$$ + +which implies that $\theta^{(n)} = \theta^{(n - 1)} - \eta S^{(n)}$ is the global minimizer of (76) and $\nabla_{\theta}f_{n}(\theta^{(n)}) = 0$ . Then + +$$ +\begin{array}{l} f _ {n} \left(\theta_ {\star}\right) \geq f _ {n} \left(\theta^ {(n)}\right) + \left\langle \nabla_ {\theta} f _ {n} \left(\theta^ {(n)}\right), - \delta^ {(n)} \right\rangle + \frac {1}{2 \eta} \left\| \delta^ {(n)} \right\| _ {2} ^ {2} \\ = f _ {n} \left(\theta^ {(n)}\right) + \frac {1}{2 \eta} \| \delta^ {(n)} \| _ {2} ^ {2}. \tag {77} \\ \end{array} +$$ + +Since $L(\cdot ,\hat{g})$ is $\mu$ -strongly convex and $f_{n}(\theta_{\star}) = \langle S^{(n)}, - \delta^{(n - 1)}\rangle +(2\eta)^{-1}\| \delta^{n - 1}\|_{2}^{2}$ we have that + +$$ +\begin{array}{l} L \left(\theta_ {\star}, \hat {g}\right) \geq L \left(\theta^ {(n - 1)}, \hat {g}\right) + \left\langle S _ {\theta} \left(\theta^ {(n - 1)}, \hat {g}\right), - \delta^ {(n - 1)} \right\rangle + \frac {\mu}{2} \| \delta^ {(n - 1)} \| _ {2} ^ {2} \\ = L (\theta^ {(n - 1)}, \hat {g}) + \langle S ^ {(n)}, - \delta^ {(n - 1)} \rangle + \langle v ^ {(n)}, \delta^ {(n - 1)} \rangle + \frac {\mu}{2} \| \delta^ {(n - 1)} \| _ {2} ^ {2} \\ = L \left(\theta^ {(n - 1)}, \hat {g}\right) + f _ {n} \left(\theta_ {\star}\right) + \left\langle v ^ {(n)}, \delta^ {(n - 1)} \right\rangle + \left(\frac {\mu}{2} - \frac {1}{2 \eta}\right) \| \delta^ {(n - 1)} \| _ {2} ^ {2}. \\ \end{array} +$$ + +Together with (77), it follows that + +$$ +\begin{array}{l} L \left(\theta^ {(n - 1)}, \hat {g}\right) + f _ {n} \left(\theta^ {(n)}\right) \leq L \left(\theta_ {\star}, \hat {g}\right) - \langle v ^ {(n)}, \delta^ {(n - 1)} \rangle \\ + \left(\frac {1}{2 \eta} - \frac {\mu}{2}\right) \| \delta^ {(n - 1)} \| _ {2} ^ {2} - \frac {1}{2 \eta} \| \delta^ {(n)} \| _ {2} ^ {2}. \tag {78} \\ \end{array} +$$ + +Since $L(\cdot ,\hat{g})$ is $M$ -smooth and $f_{n}\left(\theta^{(n)}\right) = \langle S^{(n)},\theta^{(n)} - \theta^{(n - 1)}\rangle +(2\eta)^{-1}\| \theta^{(n)} - \theta^{(n - 1)}\|_{2}^{2},$ we have that + +$$ +\begin{array}{l} L (\theta^ {(n)}, \hat {g}) \leq L (\theta^ {(n - 1)}, \hat {g}) + \langle S (\theta^ {(n - 1)}, \hat {g}), \theta^ {(n)} - \theta^ {(n - 1)} \rangle + \frac {M}{2} \| \theta^ {(n)} - \theta^ {(n - 1)} \| _ {2} ^ {2} \\ = L \left(\theta^ {(n - 1)}, \hat {g}\right) + \langle S ^ {(n)}, \theta^ {(n)} - \theta^ {(n - 1)} \rangle + \frac {M}{2} \| \theta^ {(n)} - \theta^ {(n - 1)} \| _ {2} ^ {2} - \langle v ^ {(n)}, \theta^ {(n)} - \theta^ {(n - 1)} \rangle \\ = L \left(\theta^ {(n - 1)}, \hat {g}\right) + f _ {n} \left(\theta^ {(n)}\right) + \left(\frac {M}{2} - \frac {1}{2 \eta}\right) \| \theta^ {(n)} - \theta^ {(n - 1)} \| _ {2} ^ {2} - \langle v ^ {(n)}, \theta^ {(n)} - \theta^ {(n - 1)} \rangle . \\ \end{array} +$$ + +By (78), it follows that + +$$ +\begin{array}{l} L \left(\theta^ {(n)}, \hat {g}\right) \leq L \left(\theta_ {\star}, \hat {g}\right) - \left\langle v ^ {(n)}, \delta^ {(n - 1)} \right\rangle + \left(\frac {1}{2 \eta} - \frac {\mu}{2}\right) \| \delta^ {(n - 1)} \| _ {2} ^ {2} - \frac {1}{2 \eta} \| \delta^ {(n)} \| _ {2} ^ {2} \\ + \left(\frac {M}{2} - \frac {1}{2 \eta}\right) \| \theta^ {(n)} - \theta^ {(n - 1)} \| _ {2} ^ {2} - \langle v ^ {(n)}, \theta^ {(n)} - \theta^ {(n - 1)} \rangle , \\ \end{array} +$$ + +which implies that + +$$ +\begin{array}{l} L \left(\theta^ {(n)}, \hat {g}\right) - L \left(\theta_ {\star}, \hat {g}\right) \leq \left(\frac {1}{2 \eta} - \frac {\mu}{2}\right) \| \delta^ {(n - 1)} \| _ {2} ^ {2} - \frac {1}{2 \eta} \| \delta^ {(n)} \| _ {2} ^ {2} - \langle v ^ {(n)}, \delta^ {(n - 1)} \rangle \\ + \left(\frac {M}{2} - \frac {1}{2 \eta}\right) \| \theta^ {(n)} - \theta^ {(n - 1)} \| _ {2} ^ {2} - \langle v ^ {(n)}, \theta^ {(n)} - \theta^ {(n - 1)} \rangle . \tag {79} \\ \end{array} +$$ + +For any $\omega > 0$ , by Cauchy-Schwarz inequality and Young's inequality, we have + +$$ +- \langle v ^ {(n)}, \theta^ {(n)} - \theta^ {(n - 1)} \rangle \leq \frac {\omega}{2} \| v ^ {(n)} \| _ {2} ^ {2} + \frac {1}{2 \omega} \| \theta^ {(n)} - \theta^ {(n - 1)} \| _ {2} ^ {2}. +$$ + +Take this into (79) and we have + +$$ +\begin{array}{l} L \left(\theta^ {(n)}, \hat {g}\right) - L \left(\theta_ {\star}, \hat {g}\right) \leq \left(\frac {1}{2 \eta} - \frac {\mu}{2}\right) \| \delta^ {(n - 1)} \| _ {2} ^ {2} - \frac {1}{2 \eta} \| \delta^ {(n)} \| _ {2} ^ {2} - \langle v ^ {(n)}, \delta^ {(n - 1)} \rangle \\ + \left(\frac {M}{2} - \frac {1}{2 \eta} + \frac {1}{2 \omega}\right) \| \theta^ {(n)} - \theta^ {(n - 1)} \| _ {2} ^ {2} + \frac {\omega}{2} \| v ^ {(n)} \| _ {2} ^ {2}. \\ \end{array} +$$ + +When $\eta \leq 1 / M$ , set $\frac{M}{2} - \frac{1}{2\eta} + \frac{1}{2\omega} = 0$ , i.e., set $\omega = 1 / (\eta^{-1} - M)$ . It follows that + +$$ +\begin{array}{l} L \left(\theta^ {(n)}, \hat {g}\right) - L \left(\theta_ {\star}, \hat {g}\right) \leq \left(\frac {1}{2 \eta} - \frac {\mu}{2}\right) \| \delta^ {(n - 1)} \| _ {2} ^ {2} - \frac {1}{2 \eta} \| \delta^ {(n)} \| _ {2} ^ {2} - \langle v ^ {(n)}, \delta^ {(n - 1)} \rangle \\ + \frac {\eta}{2 (1 - M \eta)} \| v ^ {(n)} \| _ {2} ^ {2}. \\ \end{array} +$$ + +We complete the proof by multiplying both sides of the inequality by $2\eta$ + +![](images/a599b2b71696e6caa55a83e6bd4ed59f85784dac5b23953b2e864f55e5d2c422.jpg) + +Lemma 11. Suppose that Asm. 3 holds. If $\eta < 1 / M$ , $\theta^{(n)} \in \Theta$ , and $\hat{g} \in \mathcal{G}_r(g_0)$ , it holds that + +$$ +\| \delta^ {(n)} \| _ {2} ^ {2} \leq (1 - \mu \eta) \| \delta^ {(n - 1)} \| _ {2} ^ {2} + \frac {\alpha_ {1} ^ {2} \eta}{\mu} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} + 2 \eta \langle v ^ {(n)}, \delta^ {(n - 1)} \rangle + \frac {\eta^ {2} \| v ^ {(n)} \| _ {2} ^ {2}}{1 - M \eta}. +$$ + +Proof. Under Asm. 3, + +$$ +\begin{array}{l} D _ {\theta} L \left(\theta_ {\star}, \hat {g}\right) \left[ \delta^ {(n)} \right] = D _ {\theta} L \left(\theta_ {\star}, g _ {0}\right) \left[ \delta^ {(n)} \right] + D _ {g} D _ {\theta} L \left(\theta_ {\star}, \bar {g}\right) \left[ \delta^ {(n)}, \hat {g} - g _ {0} \right] \\ = D _ {g} D _ {\theta} L \left(\theta_ {\star}, \bar {g}\right) \left[ \delta^ {(n)}, \hat {g} - g _ {0} \right] \geq - \alpha_ {1} \| \delta^ {(n)} \| _ {2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}}. \tag {80} \\ \end{array} +$$ + +Since $L(\cdot ,\hat{g})$ is $\mu$ -strongly convex, + +$$ +\begin{array}{l} L \left(\theta^ {(n)}, \hat {g}\right) - L \left(\theta_ {\star}, \hat {g}\right) \geq \left\langle S \left(\theta_ {\star}, \hat {g}\right), \delta^ {(n)} \right\rangle + \frac {\mu}{2} \| \delta^ {(n)} \| _ {2} ^ {2} \\ = D _ {\theta} L (\theta_ {\star}, \hat {g}) [ \delta^ {(n)} ] + \frac {\mu}{2} \| \delta^ {(n)} \| _ {2} ^ {2}. \\ \end{array} +$$ + +By (80), it follows that + +$$ +L (\theta^ {(n)}, \hat {g}) - L (\theta_ {\star}, \hat {g}) \geq - \alpha_ {1} \| \delta^ {(n)} \| _ {2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} + \frac {\mu}{2} \| \delta^ {(n)} \| _ {2} ^ {2}. +$$ + +Together with Lemma 10, we have + +$$ +\begin{array}{l} 2 \eta \left(- \alpha_ {1} \| \delta^ {(n)} \| _ {2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} + \frac {\mu}{2} \| \delta^ {(n)} \| _ {2} ^ {2}\right) \leq (1 - \mu \eta) \| \delta^ {(n - 1)} \| _ {2} ^ {2} \\ - \| \delta^ {(n)} \| _ {2} ^ {2} - 2 \eta \langle v ^ {(n)}, \delta^ {(n - 1)} \rangle + \frac {\eta^ {2} \| v ^ {(n)} \| _ {2} ^ {2}}{1 - M \eta}. \\ \end{array} +$$ + +Rearranging it, we have + +$$ +\begin{array}{l} (1 + \mu \eta) \| \delta^ {(n)} \| _ {2} ^ {2} \leq (1 - \mu \eta) \| \delta^ {(n - 1)} \| _ {2} ^ {2} \\ + 2 \eta \alpha_ {1} \| \delta^ {(n)} \| _ {2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} - 2 \eta \langle v ^ {(n)}, \delta^ {(n - 1)} \rangle + \frac {\eta^ {2} \| v ^ {(n)} \| _ {2} ^ {2}}{1 - M \eta}. \tag {81} \\ \end{array} +$$ + +By Young's inequality, + +$$ +2 \eta \alpha_ {1} \| \delta^ {(n)} \| _ {2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} \leq \eta \alpha_ {1} \left(\frac {\mu}{\alpha_ {1}} \| \delta^ {(n)} \| _ {2} ^ {2} + \frac {\alpha_ {1}}{\mu} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2}\right). +$$ + +Take this into (81) and we have + +$$ +\| \delta^ {(n)} \| _ {2} ^ {2} \leq (1 - \mu \eta) \| \delta^ {(n - 1)} \| _ {2} ^ {2} + \eta \alpha_ {1} ^ {2} \mu^ {- 1} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} - 2 \eta \langle v ^ {(n)}, \delta^ {(n - 1)} \rangle + \frac {\eta^ {2} \| v ^ {(n)} \| _ {2} ^ {2}}{1 - M \eta}. +$$ + +![](images/f879873d43903ed0a6fd70c4b26591f096e51df07fb74258b3c4686a1301fced.jpg) + +Corollary 12. Suppose that Asm. 3 holds. If $\eta < 1 / M$ , $\theta^{(n)} \in \Theta$ , and $\hat{g} \in \mathcal{G}_r(g_0)$ , it holds that + +$$ +\mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| \delta^ {(n)} \| _ {2} ^ {2} \right] \leq \left(1 - \mu \eta + \frac {\kappa_ {1} \eta^ {2}}{1 - M \eta}\right) \| \delta^ {(n - 1)} \| _ {2} ^ {2} + \frac {\alpha_ {1} ^ {2} \eta}{\mu} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} + \frac {K _ {1} \eta^ {2}}{1 - M \eta}. +$$ + +Proof. Note that $\mathbb{E}_{Z_n\sim \mathbb{P}}\left[\langle v^{(n)},\delta^{(n - 1)}\rangle \right] = 0$ . By Lem. 11, we have + +$$ +\mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| \delta^ {(n)} \| _ {2} ^ {2} \right] \leq (1 - \mu \eta) \| \delta^ {(n - 1)} \| _ {2} ^ {2} + \frac {\alpha_ {1} ^ {2} \eta}{\mu} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} + \frac {\eta^ {2}}{1 - M \eta} \mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| v ^ {(n)} \| _ {2} ^ {2} \right]. +$$ + +Under Asm. 3, $\mathbb{E}_{Z_n\sim \mathbb{P}}\left[\| v^{(n)}\| _2^2\right]\leq K_1 + \kappa_1\| \delta^{(n)}\| _2^2$ , and it follows that + +$$ +\mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| \delta^ {(n)} \| _ {2} ^ {2} \right] \leq \left(1 - \mu \eta + \frac {\kappa_ {1} \eta^ {2}}{1 - M \eta}\right) \| \delta^ {(n - 1)} \| _ {2} ^ {2} + \frac {\alpha_ {1} ^ {2} \eta}{\mu} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} + \frac {K _ {1} \eta^ {2}}{1 - M \eta}. +$$ + +![](images/c75e8f2d0dff3026758ba1a1665169f3dd2889c2fb3d650fb585bab4821d10cc.jpg) + +Lemma 13. Suppose that Asm. 3 and Asm. 4 hold. If $\eta < 1 / M$ , $\theta^{(n)} \in \Theta$ , and $\hat{g} \in \mathcal{G}_r(g_0)$ , it holds that + +$$ +\| \delta^ {(n)} \| _ {2} ^ {2} \leq (1 - \mu \eta) \| \delta^ {(n - 1)} \| _ {2} ^ {2} + \frac {\beta_ {1} ^ {2} \eta}{4 \mu} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {4} - 2 \eta \langle v ^ {(n)}, \delta^ {(n - 1)} \rangle + \frac {\eta^ {2} \| v ^ {(n)} \| _ {2} ^ {2}}{1 - M \eta}. +$$ + +Proof. Under Asm. 3 and Asm. 4, + +$$ +\begin{array}{l} D _ {\theta} L \left(\theta_ {\star}, \hat {g}\right) \left[ \delta^ {(n)} \right] = D _ {\theta} L \left(\theta_ {\star}, g _ {0}\right) \left[ \delta^ {(n)} \right] + D _ {g} D _ {\theta} L \left(\theta_ {\star}, g _ {0}\right) \left[ \delta^ {(n)}, \hat {g} - g _ {0} \right] \\ + \frac {1}{2} D _ {g} ^ {2} D _ {\theta} L (\theta_ {\star}, \bar {g}) [ \delta^ {(n)}, \hat {g} - g _ {0}, \hat {g} - g _ {0} ] \\ = \frac {1}{2} D _ {g} ^ {2} D _ {\theta} L (\theta_ {\star}, \bar {g}) [ \delta^ {(n)}, \hat {g} - g _ {0}, \hat {g} - g _ {0} ] \geq - \frac {\beta_ {1}}{2} \| \delta^ {(n)} \| _ {2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2}. \\ \end{array} +$$ + +The rest of the proof is similar to Lem. 11. + +![](images/cd7b6485c79c98fa1d59c2ba4c6ecdcb86d0efb097dbfd8d6d7801761c06de6a.jpg) + +Corollary 14. Suppose that Asm. 3 and Asm. 4 hold. If $\eta < 1 / M$ , $\theta^{(n)} \in \Theta$ , and $\hat{g} \in \mathcal{G}_r(g_0)$ , it holds that + +$$ +\mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| \delta^ {(n)} \| _ {2} ^ {2} \right] \leq \left(1 - \mu \eta + \frac {\kappa_ {1} \eta^ {2}}{1 - M \eta}\right) \| \delta^ {(n - 1)} \| _ {2} ^ {2} + \frac {\beta_ {1} ^ {2} \eta}{4 \mu} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {4} + \frac {K _ {1} \eta^ {2}}{1 - M \eta}. +$$ + +Proof. The proof is similar to Cor. 12 using Lem. 13. + +![](images/dbc21337fd9d5e0314ba9fe47f950dc75e460f598f06bbbd7347ee964ce95c06.jpg) + +# C.3 Proof of Theorem 1 + +Proof. Let $c(\eta) = \mu - \kappa_1 \eta / (1 - M \eta)$ . When $\eta < \mu / (M \mu + \kappa_1)$ , we have + +$$ +1 - \mu \eta + \frac {\kappa_ {1} \eta^ {2}}{1 - M \eta} = 1 - \left(\mu - \frac {\kappa_ {1} \eta}{1 - M \eta}\right) \eta = 1 - c (\eta) \eta < 1. +$$ + +Under Asm. 3 and by Cor. 12, we have that + +$$ +\begin{array}{l} \mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} \left[ \| \delta^ {(n)} \| _ {2} ^ {2} \right] = \mathbb {E} _ {\mathcal {D} _ {n - 1} \sim \mathbb {P} ^ {n - 1}} \left[ \mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| \delta^ {(n)} \| _ {2} ^ {2} \right] \right] \\ \leq (1 - c (\eta) \eta) \mathbb {E} _ {\mathcal {D} _ {n - 1} \sim \mathbb {P} ^ {n - 1}} \left[ \| \delta^ {(n - 1)} \| _ {2} ^ {2} \right] + \frac {\alpha_ {1} ^ {2} \eta}{\mu} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} + \frac {K _ {1} \eta^ {2}}{1 - M \eta} \\ \leq \left(1 - c (\eta) \eta\right) ^ {2} \mathbb {E} _ {\mathcal {D} _ {n - 2} \sim \mathbb {P} ^ {n - 2}} \left[ \| \delta^ {(n - 1)} \| _ {2} ^ {2} \right] \\ + \left\{1 + (1 - c (\eta) \eta) \right\} \frac {\alpha_ {1} ^ {2} \eta}{\mu} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} + \left\{1 + (1 - c (\eta) \eta) \right\} \frac {K _ {1} \eta^ {2}}{1 - M \eta}. \\ \end{array} +$$ + +By recursion, it follows that + +$$ +\begin{array}{l} \mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} \left[ \| \delta^ {(n)} \| _ {2} ^ {2} \right] \leq (1 - c (\eta) \eta) ^ {n} \| \delta^ {(0)} \| _ {2} ^ {2} + \frac {\alpha_ {1} ^ {2} \eta}{\mu} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} \sum_ {i = 0} ^ {n - 1} (1 - c (\eta) \eta) ^ {i} \\ + \frac {K _ {1} \eta^ {2}}{1 - M \eta} \sum_ {i = 0} ^ {n - 1} (1 - c (\eta) \eta) ^ {i} \\ \leq \left(1 - c (\eta) \eta\right) ^ {n} \| \delta^ {(0)} \| _ {2} ^ {2} + \frac {\alpha_ {1} ^ {2}}{\mu c (\eta)} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} + \frac {K _ {1} \eta}{c (\eta) (1 - M \eta)}. \\ \end{array} +$$ + +If $\eta \leq \mu /2(M\mu +\kappa_1)$ , we have $\mu /2\leq c(\eta)\leq \mu$ . Thus, + +$$ +\mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} \left[ \| \delta^ {(n)} \| _ {2} ^ {2} \right] \leq \left(1 - \frac {\mu \eta}{2}\right) ^ {n} \| \delta^ {(0)} \| _ {2} ^ {2} + \frac {2 \alpha_ {1} ^ {2}}{\mu^ {2}} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} + \frac {4 K _ {1} \eta}{\mu}. +$$ + +In addition, if Asm. 4 holds, then by Cor. 14 and using identical proof as above, it follows that for a Neyman orthogonal risk $L$ , + +$$ +\mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} \left[ \| \delta^ {(n)} \| _ {2} ^ {2} \right] \leq \left(1 - \frac {\mu \eta}{2}\right) ^ {n} \| \delta^ {(0)} \| _ {2} ^ {2} + \frac {\beta_ {1} ^ {2}}{2 \mu^ {2}} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {4} + \frac {4 K _ {1} \eta}{\mu}. +$$ + +![](images/93f93cbdedd5cc98e4c60b90349ff32f77803e312e7345565daeb5d6692272e7.jpg) + +# D Orthogonalization with respect to Nuisance + +In this section, we establish our orthogonalization method for the possibly infinite-dimensional nuisance introduced in Sec. 3 of the main text. We demonstrate how we construct the orthogonalizing operator in Appx. D.1, and provide all the technical lemmas in Appx. D.2. + +# D.1 Orthogonalization via Riesz Representation + +We consider $\mathcal{G} \equiv (\mathcal{G},\langle \cdot ,\cdot \rangle_{\mathcal{G}})$ as a possibly infinite-dimensional Hilbert space. Recall the derivative operator $\mathrm{D}_g$ defined as for any $h\in \mathcal{G}$ , + +$$ +\mathrm {D} _ {g} \ell (\theta , g; z) [ h ] = \frac {\mathrm {d}}{\mathrm {d} t} \ell (\theta , g + t h; z) | _ {t = 0}. +$$ + +This derivative operator is also known as the Gateaux derivative. We posit the usual assumption as Jordan et al. [2022] that the derivative operator $\mathrm{D}_g\ell (\theta ,g;z)$ is linear and continuous in $\mathcal{G}$ for any $(\theta ,g,z)\in \Theta \times \mathcal{G}_r(g_0)\times \mathcal{Z}$ . We also assume regularity conditions such that $\mathrm{D}_g\mathrm{D}_\theta \ell (\theta ,g;z)$ is continuous and $\mathrm{D}_g\mathrm{D}_\theta \ell (\theta ,g;z) = \mathrm{D}_\theta \mathrm{D}_g\ell (\theta ,g;z)$ at any $(\theta ,g,z)$ . + +Since $\mathrm{D}_g\ell (\theta ,g;z)$ is linear and continuous, by the Riesz representation theorem, there uniquely exists some $\nabla_g\ell (\theta ,g;z)\in \mathcal{G}$ such that for any $g\in \mathcal{G}$ , + +$$ +\mathrm {D} _ {g} \ell (\theta , g; z) [ g ] = \left\langle \nabla_ {g} \ell (\theta , g; z), g \right\rangle_ {\mathcal {G}}. \tag {82} +$$ + +Lem. 16 shows that the operator $\mathrm{D}_g S_\theta(\theta_\star, g_0)$ is linear and continuous. By Riesz representation theorem, we can define $H_{\theta g} = (H_{\theta g}^{(1)}, \ldots, H_{\theta g}^{(d)}) \in \mathcal{G}^d$ such that for all $g \in \mathcal{G}$ , + +$$ +\mathrm {D} _ {g} S _ {\theta} ^ {(j)} \left(\theta_ {\star}, g _ {0}\right) [ g ] = \left\langle H _ {\theta g} ^ {(j)}, g \right\rangle_ {\mathcal {G}} \text {f o r a n y} j = 1, \dots , n. \tag {83} +$$ + +The Hessian operator $H_{gg}:\mathcal{G}\mapsto \mathcal{G}$ is defined as for any $g_{1},g_{2}\in \mathcal{G}$ + +$$ +\mathrm {D} _ {g} ^ {2} L \left(\theta_ {\star}, g _ {0}\right) \left[ g _ {1}, g _ {2} \right] = \left\langle H _ {g g} g _ {1}, g _ {2} \right\rangle_ {\mathcal {G}}. \tag {84} +$$ + +We will show in Lem. 17 that $H_{gg}$ uniquely exists and is an self-adjoint and bounded linear operator when $\mathrm{D}_g^2 L(\theta_\star, g_0)$ is bounded and symmetric bilinear. Assuming that $H_{gg}$ is invertible, we define the orthogonalizing operator as + +$$ +\Gamma_ {0}: \mathcal {G} \mapsto \mathbb {R} ^ {d}, \quad [ \Gamma_ {0} g ] _ {j} = \left\langle H _ {\theta g} ^ {(j)}, H _ {g g} ^ {- 1} g \right\rangle_ {\mathcal {G}}, \forall g \in \mathcal {G}. \tag {85} +$$ + +We now construct the Neyman orthogonalized (NO) gradient oracle + +$$ +S _ {\mathrm {n o}} (\theta , g; z) = S _ {\theta} (\theta , g; z) - \Gamma_ {0} \nabla_ {g} \ell (\theta , g; z). +$$ + +In addition, $\Gamma_0\nabla_g\ell (\theta ,g;z)$ can be written as the derivative in the sense that for each $j = 1,\dots ,d$ + +$$ +\begin{array}{l} [ \Gamma_ {0} \nabla_ {g} \ell (\theta , g; z) ] _ {j} = \langle H _ {\theta g} ^ {(j)}, H _ {g g} ^ {- 1} \nabla_ {g} \ell (\theta , g; z) \rangle_ {\mathcal {G}} \\ = \langle H _ {g g} ^ {- 1} H _ {\theta g} ^ {(j)}, \nabla_ {g} \ell (\theta , g; z) \rangle_ {\mathcal {G}} = \mathrm {D} _ {g} \ell (\theta , g; z) [ H _ {g g} ^ {- 1} H _ {\theta g} ^ {(j)} ]. \\ \end{array} +$$ + +That is, the NO gradient oracle can be easily obtained by + +$$ +S _ {\mathrm {n o}} (\theta , g; z) = S _ {\theta} (\theta , g; z) - \mathrm {D} _ {g} \ell (\theta , g; z) \left[ H _ {g g} ^ {- 1} H _ {\theta g} \right]. \tag {86} +$$ + +The following Lemma shows that $S_{\mathrm{no}}(\theta, g; z)$ is Neyman orthogonal at $(\theta_{\star}, g_0)$ . + +Lemma 15. $S_{no}(\theta ,g;z)$ is a Neyman orthogonal score at $(\theta_0,g_0)$ + +Proof. Since $S_{\mathrm{no}}(\theta, g; z) = S_{\theta}(\theta, g; z) - \mathrm{D}_g \ell(\theta, g; z)[H_{gg}^{-1} H_{\theta g}]$ , for any $h \in \mathcal{G}$ , + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ \mathrm {D} _ {g} S _ {\mathrm {n o}} \left(\theta_ {0}, g _ {0}; Z\right) [ h ] \right] = \mathrm {D} _ {g} S _ {\theta} \left(\theta_ {0}, g _ {0}\right) [ h ] - \mathrm {D} _ {g} ^ {2} L \left(\theta_ {0}, g _ {0}\right) \left[ H _ {g g} ^ {- 1} H _ {\theta g}, h \right] \\ = \left\langle H _ {\theta g}, h \right\rangle_ {\mathcal {G}} - \left\langle H _ {g g} H _ {g g} ^ {- 1} H _ {\theta g}, h \right\rangle_ {\mathcal {G}} \\ = \langle H _ {\theta g}, h \rangle_ {\mathcal {G}} - \langle H _ {\theta g}, h \rangle_ {\mathcal {G}} = 0, \\ \end{array} +$$ + +which implies that $S_{\mathrm{no}}(\theta, g; z)$ is Neyman orthogonal at $(\theta_0, g_0)$ . + +![](images/912d0586ccff1065266e3b807bc72b4177d4d523b8539e90f9e8918a74e64b59.jpg) + +# D.2 Technical Lemma + +Lemma 16. $\mathrm{D}_gS_\theta (\theta ,g;z):\mathcal{G}\mapsto \mathbb{R}^d$ and $\mathrm{D}_gS_\theta (\theta ,g):\mathcal{G}\mapsto \mathbb{R}^d$ are linear and continuous in $\mathcal{G}$ + +Proof. The continuity of $\mathrm{D}_g S_\theta(\theta, g; z)$ and $\mathrm{D}_g S_\theta(\theta, g)$ follows from the continuity of $\mathrm{D}_g \mathrm{D}_\theta \ell(\theta, g; z)$ . It suffices to prove that $\mathrm{D}_g S_\theta(\theta, g; z)$ is linear. For all $u \in \mathbb{R}^d$ , $h_1, h_2 \in \mathcal{G}$ , + +$$ +\begin{array}{l} \langle u, \mathrm {D} _ {g} S _ {\theta} (\theta , g; z) [ h _ {1} + h _ {2} ] \rangle = \mathrm {D} _ {g} \langle u, S _ {\theta} (\theta , g; z) \rangle [ h _ {1} + h _ {2} ] \\ = \mathrm {D} _ {g} \mathrm {D} _ {\theta} \ell (\theta , g; z) [ u, h _ {1} + h _ {2} ] \\ = \mathrm {D} _ {\theta} \mathrm {D} _ {g} \ell (\theta , g; z) [ h _ {1} + h _ {2}, u ] \\ = \left\langle \nabla_ {\theta} \left(\mathrm {D} _ {g} \ell (\theta , g; z) \left[ h _ {1} + h _ {2} \right]\right), u \right\rangle \\ = \left\langle \nabla_ {\theta} \left(\mathrm {D} _ {g} \ell (\theta , g; z) [ h _ {1} ] + \mathrm {D} _ {g} \ell (\theta , g; z) [ h _ {2} ]\right), u \right\rangle \\ = \mathrm {D} _ {\theta} \mathrm {D} _ {g} \ell (\theta , g; z) [ h _ {1}, u ] + \mathrm {D} _ {\theta} \mathrm {D} _ {g} \ell (\theta , g; z) [ h _ {2}, u ] \\ = \mathrm {D} _ {g} \mathrm {D} _ {\theta} \ell (\theta , g; z) [ u, h _ {1} ] + \mathrm {D} _ {g} \mathrm {D} _ {\theta} \ell (\theta , g; z) [ u, h _ {2} ] \\ = \langle u, \mathrm {D} _ {g} S _ {\theta} (\theta , g; z) [ h _ {1} ] + \mathrm {D} _ {g} S _ {\theta} (\theta , g; z) [ h _ {2} ] \rangle , \\ \end{array} +$$ + +which implies that + +$$ +\mathrm {D} _ {g} S _ {\theta} (\theta , g; z) [ h _ {1} + h _ {2} ] = \mathrm {D} _ {g} S _ {\theta} (\theta , g; z) [ h _ {1} ] + \mathrm {D} _ {g} S _ {\theta} (\theta , g; z) [ h _ {2} ]. +$$ + +![](images/b150ec7803ec02cc2adb1ec2e1c49ab113168ded7e66467a0d9631ca64f22e34.jpg) + +Lemma 17. Suppose that $\mathrm{D}_g^2 L(\theta_\star, g_0)[\cdot, \cdot] : \mathcal{G} \times \mathcal{G} \mapsto \mathbb{R}$ is a bounded and symmetric bilinear form. Then $H_{gg} : \mathcal{G} \mapsto \mathcal{G}$ uniquely exists and is self-adjoint, bounded, and linear. + +Proof. Given $g_1, g_2 \in \mathcal{G}$ , since $\mathrm{D}_g^2 L(\theta_\star, g_0)[g_1, \cdot]$ is a bounded linear map from $\mathcal{G}$ to $\mathbb{R}$ , by Riesz representation theorem, for any $g_2 \in \mathcal{G}$ , there uniquely exists some $Th_1 \in \mathcal{G}$ such that + +$$ +\mathrm {D} _ {g} ^ {2} L \left(\theta_ {\star}, g _ {0}\right) \left[ g _ {1}, g _ {2} \right] = \langle T g _ {1}, g _ {2} \rangle_ {\mathcal {G}}. +$$ + +Thus, we define the operator $T: \mathcal{G} \mapsto \mathcal{G}$ . Note that $\mathrm{D}_g^2 L(\theta_\star, g_0)[\cdot, \cdot]$ is bilinear. For any $a, a' \in \mathbb{R}$ , and any $g_1, g_1', g_2 \in \mathcal{G}$ , we have + +$$ +\begin{array}{l} \langle T (a g _ {1} + a ^ {\prime} g _ {1} ^ {\prime}), g _ {2} \rangle_ {\mathcal {G}} = \mathrm {D} _ {g} ^ {2} L (\theta_ {\star}, g _ {0}) [ a g _ {1} + a ^ {\prime} g _ {1} ^ {\prime}, g _ {2} ] \\ = a \mathrm {D} _ {g} ^ {2} L \left(\theta_ {\star}, g _ {0}\right) \left[ g _ {1}, g _ {2} \right] + a ^ {\prime} \mathrm {D} _ {g} ^ {2} L \left(\theta_ {\star}, g _ {0}\right) \left[ g _ {1} ^ {\prime}, g _ {2} \right] \\ = a \langle T g _ {1}, g _ {2} \rangle_ {\mathcal {G}} + a ^ {\prime} \langle T g _ {1} ^ {\prime}, g _ {2} \rangle_ {\mathcal {G}} \\ = \left\langle a T g _ {1} + a ^ {\prime} T g _ {1} ^ {\prime}, g _ {2} \right\rangle_ {\mathcal {G}}, \\ \end{array} +$$ + +which implies $T$ is a linear operator. To show $T$ is bounded, suppose that the norm of the bilinear form $\mathrm{D}_g^2 L(\theta_\star ,g_0)$ is bounded by $B$ . Thus, for $Tg_{1}\neq 0$ + +$$ +\| T g _ {1} \| _ {\mathcal {G}} = \left\langle T g _ {1}, \frac {T g _ {1}}{\| T g _ {1} \| _ {\mathcal {G}}} \right\rangle_ {\mathcal {G}} \leq \sup _ {\| g _ {2} \| _ {\mathcal {G}} = 1} \left\langle T g _ {1}, g _ {2} \right\rangle_ {\mathcal {G}} \leq \sup _ {\| g _ {2} \| _ {\mathcal {G}} = 1} \left| \mathrm {D} _ {g} ^ {2} L (\theta_ {\star}, g _ {0}) [ g _ {1}, g _ {2} ] \right| \leq B \| g _ {1} \| _ {\mathcal {G}}, +$$ + +which implies $T$ is bounded. Note that $\mathrm{D}_g^2 L(\theta_\star, g_0)[\cdot, \cdot]$ is symmetric, we have $T$ being self-adjoint since + +$$ +\langle T g _ {1}, g _ {2} \rangle_ {\mathcal {G}} = \mathrm {D} _ {g} ^ {2} L (\theta_ {\star}, g _ {0}) [ g _ {1}, g _ {2} ] = \mathrm {D} _ {g} ^ {2} L (\theta_ {\star}, g _ {0}) [ g _ {2}, g _ {1} ] = \langle T g _ {2}, g _ {1} \rangle_ {\mathcal {G}} = \langle g _ {1}, T g _ {2} \rangle_ {\mathcal {G}}. +$$ + +Finally, we show that $T$ is unique. If there exists some $T^{\prime}:\mathcal{G}\mapsto \mathcal{G}$ such that for any $g_{1},g_{2}\in \mathcal{G}$ + +$$ +\langle T g _ {1}, g _ {2} \rangle_ {\mathcal {G}} = \mathrm {D} _ {g} ^ {2} L (\theta_ {\star}, g _ {0}) [ g _ {1}, g _ {2} ] = \langle T ^ {\prime} g _ {1}, g _ {2} \rangle_ {\mathcal {G}}, +$$ + +which implies + +$$ +\langle (T - T ^ {\prime}) g _ {1}, g _ {2} \rangle_ {\mathcal {G}} = 0 \quad \forall g _ {1}, g _ {2} \in \mathcal {G}. +$$ + +That is, $T = T'$ . We finish the proof by letting $H_{gg} = T$ . + +# E Convergence Proofs for Orthogonalized Stochastic Gradient + +This section is dedicated to demonstrate the OSGD convergence in Thm. 3 of the main text. In Appx. E.1, we give an overview of the OSGD settings, the additional assumptions, and the expected results with the proof outline. We then provide all the technical lemmas in Appx. E.2, and finally prove Thm. 3 in Appx. E.3. + +# E.1 Overview + +Following the same problem settings in Appx. C, we consider the orthogonalized SGD (OSGD) using the estimated NO score $\hat{S}_{\mathrm{no}}$ oracle defined as + +$$ +\hat {S} _ {\mathrm {n o}} (\theta , g; z) = S _ {\theta} (\theta , g; z) - \hat {\Gamma} \nabla_ {g} \ell (\theta , g; z), \tag {87} +$$ + +where $\hat{\Gamma}$ is an estimator for the orthogonalizing operator defined in (85). Specifically, we consider all continuous linear $\hat{\Gamma}:\mathcal{G}\mapsto \mathbb{R}^d$ for estimating the orthogonalizing operator $\Gamma_0$ . By Riesz representation theorem, there exists some $\hat{\gamma}^{(j)}\in \mathcal{G}$ for $j = 1,\ldots ,d$ , such that + +$$ +[ \hat {\Gamma} g ] _ {j} = \langle \hat {\gamma} ^ {(j)}, g \rangle_ {\mathcal {G}} \text {f o r a l l} g \in \mathcal {G}. +$$ + +For the orthogonalizing operator $\Gamma_0$ , we define $\gamma_0^{(j)} = H_{gg}^{-1}H_{\theta g}^{(j)}, j = 1, \ldots, d$ , such that + +$$ +[ \Gamma_ {0} g ] _ {j} = \langle \gamma_ {0} ^ {(j)}, g \rangle_ {\mathcal {G}} \text {f o r a l l} g \in \mathcal {G}. +$$ + +We focus on the OSGD defined below: + +$$ +\theta^ {(n)} = \theta^ {(n - 1)} - \eta \hat {S} _ {\mathrm {n o}} \left(\theta^ {(n - 1)}, \hat {g}; Z _ {n}\right), \quad \theta^ {(0)} \in \Theta . \tag {88} +$$ + +Throughout the section, we take the following notations for simplicity: + +$$ +\delta_ {\mathrm {n o}} ^ {(n)} = \theta^ {(n)} - \theta_ {\star}, \tag {89} +$$ + +$$ +\hat {S} _ {\mathrm {n o}} ^ {(n)} = \hat {S} _ {\mathrm {n o}} \left(\theta^ {(n - 1)}, \hat {g}; Z _ {n}\right), \tag {90} +$$ + +$$ +v _ {\mathrm {n o}} ^ {(n)} = \hat {S} _ {\mathrm {n o}} ^ {(n)} - \hat {S} _ {\mathrm {n o}} \left(\theta^ {(n - 1)}, \hat {g}\right). \tag {91} +$$ + +Let $\nabla_g L(\theta, g) = \mathbb{E}_{Z \sim \mathbb{P}}[\nabla_g \ell(\theta, g; Z)]$ and $S_{\mathrm{no}}(\theta, g) = \mathbb{E}_{Z \sim \mathbb{P}}[S_{\mathrm{no}}(\theta, g; Z)]$ . We need the following assumptions to establish the convergence result of the OSGD. + +Assumption 6. The following conditions hold: + +(a) First-order optimality: $S_{no}(\theta_{\star},g_0) = 0$ and $(\hat{\Gamma} -\Gamma_0)\nabla_gL(\theta_\star ,g_0) = 0.$ +(b) Smoothness and strong convexity: There exists some $M_{no}, \mu_{no} > 0$ such that for all $\theta \in \Theta$ and $g \in \mathcal{G}_r(g_0)$ , $\| \nabla_\theta S_{no}(\theta, g) \|_2 \leq M_{no}$ and + +$$ +\lambda_ {\min } \left(\nabla_ {\theta} S _ {n o} (\theta , g) + \nabla_ {\theta} S _ {n o} (\theta , g) ^ {\top}\right) \geq 2 \mu_ {n o}. +$$ + +(c) Second-moment growth: There exist constants $K_{2},\kappa_{2} > 0$ such that + +$$ +\mathbb {E} _ {Z \sim \mathbb {P}} [ (\mathrm {D} _ {g} L (\theta , \bar {g}; Z) [ g ] - \mathrm {D} _ {g} L (\theta , \bar {g}) [ g ]) ^ {2} ] \leq (K _ {2} + \kappa_ {2} \| \theta - \theta_ {\star} \| _ {2} ^ {2}) \| g \| _ {G} ^ {2}. +$$ + +for all $\theta \in \Theta$ , $\bar{g} \in \mathcal{G}_r(g_0)$ , and $g \in \mathcal{G}$ . + +(d) Second-order smoothness: There exists a constant $\alpha_{2} > 0$ such that + +$$ +\left| \mathrm {D} _ {g} ^ {2} L (\theta , \bar {g}) [ g _ {1}, g _ {2} ] \right| \leq \alpha_ {2} \| g _ {1} \| _ {\mathcal {G}} \| g _ {2} \| _ {\mathcal {G}} \quad \forall \theta \in \Theta , \bar {g} \in \mathcal {G} _ {r} (g _ {0}), g _ {1}, g _ {2} \in \mathcal {G}, +$$ + +$$ +\left| \mathrm {D} _ {\theta} \mathrm {D} _ {g} L (\bar {\theta}, g _ {0}) [ g, \theta - \theta_ {\star} ] \right| \leq \alpha_ {1} \| \theta - \theta_ {\star} \| _ {2} \| g \| _ {\mathcal {G}} \quad \forall \theta , \bar {\theta} \in \Theta , g \in \mathcal {G}. +$$ + +(e) Higher-order smoothness: There exists a constant $\beta_{2} > 0$ such that + +$$ +\left\| \mathrm {D} _ {g} ^ {2} S _ {n o} \left(\theta_ {\star}, \bar {g}\right) [ g - g _ {0}, g - g _ {0} ] \right\| _ {2} \leq \beta_ {2} \| g - g _ {0} \| _ {\mathcal {G}} ^ {2} \quad \forall g, \bar {g} \in \mathcal {G} _ {r} \left(g _ {0}\right). +$$ + +Asm. 6(a) is necessary for the convergence of the OSGD to $\theta_{\star}$ . When $S_{\theta}$ is Neyman orthogonal at $(\theta_{\star}, g_0)$ , $\Gamma_0 = 0$ is accessible and thus, $S_{\mathrm{no}} = S_{\theta}$ . When $S_{\theta}$ is non-orthogonal, Asm. 6(a) can be satisfied whenever $\nabla_g L(\theta_{\star}, g_0) = 0$ , implying that $(\theta_{\star}, g_0)$ is a local minimizer of $L(\theta, g)$ . Asm. 6(b) is related to the Schur complement of the population Hessian. Thus, the hypothetical objective relating to $S_{\mathrm{no}}$ inherits its strong convexity from that of the population risk $L$ w.r.t. $(\theta, g) \in \Theta \times \mathcal{G}_r(g_0)$ when $\mathcal{G}$ is finite-dimensional; see Boyd and Vandenberghe [2004]. Asm. 6(c) and (d) are exactly analogous to Asm. 3(d) and (e), while Asm. 6(e) is analogous to Asm. 4. + +With Asm. 6, we aim to show that the error $\delta_{\mathrm{no}}^{(n)}$ satisfies + +$$ +\mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} \left[ \| \delta_ {\mathrm {n o}} ^ {(n)} \| _ {2} ^ {2} \right] \lesssim \left(1 - \frac {\mu_ {\mathrm {n o}} \eta}{2}\right) ^ {n} \| \delta_ {\mathrm {n o}} ^ {(0)} \| _ {2} ^ {2} + \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {4} + \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2} \cdot \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2} + \eta . \tag {92} +$$ + +Proof Outline. The proof the (92) follows the following four steps: + +1. Upper bound $\| I - \eta \nabla_{\theta}\hat{S}_{\mathrm{no}}(\theta ,g)\| _2$ w.r.t. the operator estimation error $\| \hat{\Gamma} -\Gamma_0\|_{\mathrm{Fr}}$ +2. Upper bound $\| \hat{S}_{\mathrm{no}}(\theta_{\star},\hat{g})\| _2$ using Neyman orthogonality and the first order optimality. +3. Derive a recursive formula of $\mathbb{E}_{\mathcal{D}_n} \sim \mathbb{P}^n\left[\|\delta_{\mathrm{no}}^{(n)}\|_2^2\right]$ from these bounds. +4. Perform the recursion and obtain the final result. + +Follow these steps above, we provide technical lemma in Appx. E.2, and then prove our second main result Thm. 3 in Appx. E.3. + +Alternatively, the intuition of step 1 also suggests that we should focus on $\hat{\Gamma}$ that lies in the neighborhood of $\Gamma_0$ such that $\| \hat{\Gamma} -\Gamma_0\|_{\mathrm{Fro}}\leq R$ for a small $R > 0$ . Then, instead of assuming Asm. 6(b), we can directly assume that for all $\theta \in \Theta$ and $g\in \mathcal{G}_r(g_0)$ , $\| \nabla_\theta \hat{S}_{\mathrm{no}}(\theta ,g)\| _2\leq M_{\mathrm{no}}$ and + +$$ +\lambda_ {\min } \left(\nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g) + \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g) ^ {\top}\right) \geq 2 \mu_ {\mathrm {n o}}. +$$ + +With this assumption, one can still show the same OSGD convergence rate by the identical proof while the constraint of the learning rate $\eta$ will be simplified. + +# E.2 Technical Lemma + +Lemma 18. Given $\eta >0$ . For any $\omega >0$ and $u,v\in \mathbb{R}^d$ + +$$ +\| u + \eta v \| _ {2} ^ {2} \leq (1 + \eta \omega) \| u \| _ {2} ^ {2} + (\eta^ {2} + \eta \omega^ {- 1}) \| v \| _ {2} ^ {2}. +$$ + +Proof. By definition, + +$$ +\| u + \eta v \| _ {2} ^ {2} = \| u \| _ {2} ^ {2} + \eta^ {2} \| v \| _ {2} ^ {2} + 2 \eta \langle u, v \rangle \leq \| u \| _ {2} ^ {2} + \eta^ {2} \| v \| _ {2} ^ {2} + 2 \eta \| u \| _ {2} \| v \| _ {2}. +$$ + +By Young's inequality, for any $\omega > 0$ , + +$$ +2 \| u \| _ {2} \| v \| _ {2} \leq \omega \| u \| _ {2} ^ {2} + \omega^ {- 1} \| v \| _ {2} ^ {2}. +$$ + +Thus, + +$$ +\| u + \eta v \| _ {2} ^ {2} \leq (1 + \eta \omega) \| u \| _ {2} ^ {2} + (\eta^ {2} + \eta \omega^ {- 1}) \| v \| _ {2} ^ {2}. +$$ + +Lemma 19. Suppose that Asm. 6 holds. For all $\theta \in \Theta$ and $g\in \mathcal{G}_r(g_0)$ + +$$ +\| I - \eta \nabla_ {\theta} \hat {S} _ {n o} (\theta , g) \| _ {2} ^ {2} \leq 1 - 2 \eta (\mu_ {n o} - \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}}) + 2 \eta^ {2} (M _ {n o} ^ {2} + 2 \alpha_ {1} ^ {2} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2}). +$$ + +Proof. Note that + +$$ +\begin{array}{l} (I - \eta \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g)) (I - \eta \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g)) ^ {\top} \\ = I - \eta \left(\nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g) + \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g) ^ {\top}\right) + \eta^ {2} \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g) \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g) ^ {\top}. \\ \end{array} +$$ + +Since $\nabla_{\theta}\hat{S}_{\mathrm{no}}(\theta ,g) = \nabla_{\theta}S_{\mathrm{no}}(\theta ,g) - \nabla_{\theta}((\hat{\Gamma} -\Gamma_0)\nabla_gL(\theta ,g))$ , we have + +$$ +\begin{array}{l} \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g) + \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g) ^ {\top} \\ = \nabla_ {\theta} S _ {\mathrm {n o}} (\theta , g) + \nabla_ {\theta} S _ {\mathrm {n o}} (\theta , g) ^ {\top} - \nabla_ {\theta} ((\hat {\Gamma} - \Gamma_ {0}) \nabla_ {g} L (\theta , g)) - \nabla_ {\theta} ((\hat {\Gamma} - \Gamma_ {0}) \nabla_ {g} L (\theta , g)) ^ {\top} \\ \succcurlyeq 2 \left(\mu_ {\mathrm {n o}} - \| \nabla_ {\theta} \left(\left(\hat {\Gamma} - \Gamma_ {0}\right) \nabla_ {g} L (\theta , g)\right) \| _ {2}\right) \mathbf {I}. \\ \end{array} +$$ + +We now bound $\| \nabla_{\theta}(\hat{\Gamma} -\Gamma_0)\nabla_gL(\theta ,g)\| _2$ . For each $j = 1,\dots ,d$ , for any $\theta \in \mathbb{R}^d$ + +$$ +\begin{array}{l} \left| \nabla_ {\theta} \left(\left(\hat {\Gamma} - \Gamma_ {0}\right) ^ {(j)} \nabla_ {g} L (\theta , g)\right) \left(\theta - \theta_ {\star}\right) \right| = \left| \mathrm {D} _ {\theta} \left\langle \hat {\gamma} ^ {(j)} - \gamma_ {0} ^ {(j)}, \nabla_ {g} L (\theta , g) \right\rangle_ {\mathcal {G}} \left[ \theta - \theta_ {\star} \right] \right| \\ = \left| \mathrm {D} _ {\theta} \mathrm {D} _ {g} L (\theta , g) [ \hat {\gamma} ^ {(j)} - \gamma_ {0} ^ {(j)}, \theta - \theta_ {\star} ] \right| \\ \leq \alpha_ {1} \| \hat {\gamma} ^ {(j)} - \gamma_ {0} ^ {(j)} \| _ {\mathcal {G}} \| \theta - \theta_ {\star} \| _ {2}. \\ \end{array} +$$ + +Thus, + +$$ +\| \nabla_ {\theta} ((\hat {\Gamma} - \Gamma_ {0}) \nabla_ {g} L (\theta , g)) (\theta - \theta_ {\star}) \| _ {2} \leq \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} \| \theta - \theta_ {\star} \| _ {2}, +$$ + +which implies that $\| \nabla_{\theta}(\hat{\Gamma} -\Gamma_0)\nabla_gL(\theta ,g)\| _2\leq \alpha_1\| \hat{\Gamma} -\Gamma_0\|_{\mathrm{Fro}}$ and + +$$ +\nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g) + \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g) ^ {\top} \geq 2 \left(\mu_ {\mathrm {n o}} - \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}}\right) \mathbf {I}. +$$ + +Additionally, we have + +$$ +\begin{array}{l} \left\| \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g) \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g) ^ {\top} \right\| _ {2} \leq \left\| \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g) \right\| _ {2} ^ {2} \\ \leq 2 \| \nabla_ {\theta} S _ {\mathrm {n o}} (\theta , g) \| _ {2} ^ {2} + 2 \| \nabla_ {\theta} ((\hat {\Gamma} - \Gamma_ {0}) \nabla_ {g} L (\theta , g)) \| _ {2} ^ {2} \\ \leq 2 M _ {\mathrm {n o}} ^ {2} + 2 \alpha_ {1} ^ {2} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2}. \\ \end{array} +$$ + +In conclusion, we have + +$$ +\begin{array}{l} (I - \eta \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g)) (I - \eta \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\theta , g)) ^ {\top} \\ \preccurlyeq \Big (1 - 2 \eta \big (\mu_ {\mathrm {n o}} - \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} \big) + 2 \eta^ {2} (M _ {\mathrm {n o}} ^ {2} + 2 \alpha_ {1} ^ {2} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2}) \Big) \mathbf {I}. \\ \end{array} +$$ + +![](images/ed5399072b4a8c05db8511e9bda6b859607c6a1907132158a696351c757c06f4.jpg) + +Lemma 20. Suppose that Asm. 6 holds. When $\hat{g} \in \mathcal{G}_r(g_0)$ , + +$$ +\| \hat {S} _ {n o} (\theta_ {\star}, \hat {g}) \| _ {2} ^ {2} \leq \frac {\beta_ {2} ^ {2}}{2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {4} + 2 \alpha_ {2} ^ {2} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2} \cdot \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2}. +$$ + +Proof. Note that + +$$ +\begin{array}{l} \left\| \hat {S} _ {\mathrm {n o}} \left(\theta_ {\star}, \hat {g}\right) \right\| _ {2} ^ {2} = \left\| S _ {\mathrm {n o}} \left(\theta_ {\star}, \hat {g}\right) - (\hat {\Gamma} - \Gamma_ {0}) \nabla_ {g} L \left(\theta_ {\star}, \hat {g}\right) \right\| _ {2} ^ {2} \\ \leq 2 \| S _ {\mathrm {n o}} (\theta_ {\star}, \hat {g}) \| _ {2} ^ {2} + 2 \| (\hat {\Gamma} - \Gamma_ {0}) \nabla_ {g} L (\theta_ {\star}, \hat {g}) \| _ {2} ^ {2} \\ = 2 \| S _ {\mathrm {n o}} (\theta_ {\star}, \hat {g}) \| _ {2} ^ {2} + 2 \sum_ {j = 1} ^ {d} \left\langle \nabla_ {g} L (\theta_ {\star}, \hat {g}), \hat {\gamma} ^ {(j)} - \gamma_ {0} ^ {(j)} \right\rangle_ {\mathcal {G}} ^ {2} \\ = 2 \| S _ {\mathrm {n o}} (\theta_ {\star}, \hat {g}) \| _ {2} ^ {2} + 2 \sum_ {j = 1} ^ {d} \left(\mathrm {D} _ {g} L (\theta_ {\star}, \hat {g}) [ \hat {\gamma} ^ {(j)} - \gamma_ {0} ^ {(j)} ]\right) ^ {2}. \\ \end{array} +$$ + +Since $S_{\mathrm{no}}(\theta_{\star},g_0) = 0$ and $S_{\mathrm{no}}$ is Neyman orthogonal at $(\theta_{\star},g_0)$ , we have for some $\bar{g}\in \mathcal{G}_r(g_0)$ + +$$ +\begin{array}{l} S _ {\mathrm {n o}} (\theta_ {\star}, \hat {g}) = S _ {\mathrm {n o}} (\theta_ {\star}, g _ {0}) + \mathrm {D} _ {g} S _ {\mathrm {n o}} (\theta_ {\star}, g _ {0}) [ \hat {g} - g _ {0} ] + \frac {1}{2} \mathrm {D} _ {g} ^ {2} S _ {\mathrm {n o}} (\theta_ {\star}, \bar {g}) [ \hat {g} - g _ {0} ] \\ = \frac {1}{2} \mathrm {D} _ {g} ^ {2} S _ {\mathrm {n o}} (\theta_ {\star}, \bar {g}) [ \hat {g} - g _ {0} ], \\ \end{array} +$$ + +which implies + +$$ +\left\| S _ {\mathrm {n o}} \left(\theta_ {\star}, \hat {g}\right) \right\| _ {2} \leq \frac {\beta_ {2}}{2} \left\| \hat {g} - g _ {0} \right\| _ {\mathcal {G}} ^ {2}. +$$ + +Similarly, since $(\hat{\Gamma}^{(j)} - \Gamma_0)\nabla_gL(\theta_\star ,g_0) = 0$ we have for some $\bar{g}^{\prime}\in \mathcal{G}_{r}(g_{0})$ + +$$ +\begin{array}{l} \mathrm {D} _ {g} L (\theta_ {\star}, \hat {g}) [ \hat {\gamma} ^ {(j)} - \gamma_ {0} ^ {(j)} ] = \mathrm {D} _ {g} L (\theta_ {\star}, g _ {0}) [ \hat {\gamma} ^ {(j)} - \gamma_ {0} ^ {(j)} ] + \mathrm {D} _ {g} ^ {2} L (\theta_ {\star}, \bar {g} ^ {\prime}) [ \hat {\gamma} ^ {(j)} - \gamma_ {0} ^ {(j)} ] \\ = (\hat {\Gamma} ^ {(j)} - \Gamma_ {0}) \nabla_ {g} L (\theta_ {\star}, g _ {0}) + \mathrm {D} _ {g} ^ {2} L (\theta_ {\star}, \bar {g} ^ {\prime}) [ \hat {\gamma} ^ {(j)} - \gamma_ {0} ^ {(j)} ] \\ = \mathrm {D} _ {g} ^ {2} L (\theta_ {\star}, \bar {g} ^ {\prime}) [ \hat {\gamma} ^ {(j)} - \gamma_ {0} ^ {(j)} ], \\ \end{array} +$$ + +which implies + +$$ +\left| \mathrm {D} _ {g} L (\theta_ {\star}, \hat {g}) [ \hat {\gamma} ^ {(j)} - \gamma_ {0} ^ {(j)}, \hat {g} - g _ {0} ] \right| \leq \alpha_ {2} \| \hat {\gamma} ^ {(j)} - \gamma_ {0} ^ {(j)} \| _ {\mathcal {G}} \| \hat {g} - g _ {0} \| _ {\mathcal {G}}. +$$ + +In conclusion, + +$$ +\| \hat {S} _ {\mathrm {n o}} (\theta_ {\star}, \hat {g}) \| _ {2} ^ {2} \leq \frac {\beta_ {2} ^ {2}}{2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {4} + 2 \alpha_ {2} ^ {2} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2} \cdot \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2}. +$$ + +![](images/f00a546ff183952f826d7326f3f73197f83e8d4953875053ffeeb70a7296f04e.jpg) + +Lemma 21. Suppose that Asm. 3 and Asm. 6 holds. Then + +$$ +\mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| v _ {n o} ^ {(n)} \| _ {2} ^ {2} \right] \leq 2 (K _ {1} + K _ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2}) + 2 (\kappa_ {1} + \kappa_ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2}) \| \delta_ {n o} ^ {n - 1} \| _ {2} ^ {2}. +$$ + +Proof. Note that $v_{\mathrm{no}}^{(n)} = \hat{S}_{\mathrm{no}}(\theta^{(n - 1)},\hat{g};Z_n) - \hat{S}_{\mathrm{no}}(\theta^{(n - 1)},\hat{g})$ . + +$$ +\begin{array}{l} \mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| v _ {\mathrm {n o}} ^ {(n)} \| _ {2} ^ {2} \right] \leq 2 \mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| S _ {\theta} (\theta^ {(n - 1)}, \hat {g}; Z _ {n}) - S _ {\theta} (\theta^ {(n - 1)}, \hat {g}) \| _ {2} ^ {2} \right] \\ + 2 \mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| \hat {\Gamma} \nabla_ {g} \ell \left(\theta^ {(n - 1)}, \hat {g}; Z _ {n}\right) - \hat {\Gamma} \nabla_ {g} L \left(\theta^ {(n - 1)}, \hat {g}\right) \| _ {2} ^ {2} \right]. \\ \end{array} +$$ + +By Asm. 3, + +$$ +\mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| S _ {\theta} \left(\theta^ {(n - 1)}, \hat {g}; Z _ {n}\right) - S _ {\theta} \left(\theta^ {(n - 1)}, \hat {g}\right) \| _ {2} ^ {2} \right] \leq K _ {1} + \kappa_ {1} \| \delta_ {\mathrm {n o}} ^ {n - 1} \| _ {2} ^ {2}. +$$ + +Since + +$$ +\begin{array}{l} \mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| \hat {\Gamma} \nabla_ {g} \ell \left(\theta^ {(n - 1)}, \hat {g}; Z _ {n}\right) - \hat {\Gamma} \nabla_ {g} L \left(\theta^ {(n - 1)}, \hat {g}\right) \| _ {2} ^ {2} \right] \\ = \sum_ {j = 1} ^ {d} \mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \left(\mathrm {D} _ {g} \ell \left(\theta^ {(n - 1)}, \hat {g}; Z _ {n}\right) [ \hat {\gamma} ^ {(j)} ] - \mathrm {D} _ {g} L \left(\theta^ {(n - 1)}, \hat {g}\right) [ \hat {\gamma} ^ {(j)} ]\right) ^ {2} \right] \\ \leq \sum_ {j = 1} ^ {d} \left(K _ {2} + \kappa_ {2} \| \theta - \theta_ {\star} \| _ {2} ^ {2}\right) \| \hat {\gamma} ^ {(j)} \| _ {\mathcal {G}} ^ {2} = \left(K _ {2} + \kappa_ {2} \| \theta - \theta_ {\star} \| _ {2} ^ {2}\right) \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2}. \\ \end{array} +$$ + +In conclusion, + +$$ +\mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| v _ {\mathrm {n o}} ^ {(n)} \| _ {2} ^ {2} \right] \leq 2 (K _ {1} + K _ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2}) + 2 (\kappa_ {1} + \kappa_ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2}) \| \delta_ {\mathrm {n o}} ^ {n - 1} \| _ {2} ^ {2}. +$$ + +![](images/5bfdcefd87ab42258cd073f6d0fefeb17df5fc3663b492a6817598ec6b10340f.jpg) + +# E.3 Proof of Theorem 3 + +Proof. Since $\theta^{(n)} = \theta^{(n - 1)} - \eta \hat{S}_{\mathrm{no}}^{(n)}$ , by Taylor's theorem we have that for some $\bar{\theta}^{(n - 1)}$ + +$$ +\begin{array}{l} \delta_ {\mathrm {n o}} ^ {(n)} = \delta_ {\mathrm {n o}} ^ {(n - 1)} - \eta (\hat {S} _ {\mathrm {n o}} (\theta^ {(n - 1)}, \hat {g}) - \hat {S} _ {\mathrm {n o}} (\theta_ {\star}, \hat {g})) - \eta \hat {S} _ {\mathrm {n o}} (\theta_ {\star}, \hat {g}) - \eta v _ {\mathrm {n o}} ^ {(n)} \\ = (I - \eta \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\bar {\theta} ^ {(n - 1)}, \hat {g})) \delta_ {\mathrm {n o}} ^ {(n - 1)} - \eta \hat {S} _ {\mathrm {n o}} (\theta_ {\star}, \hat {g}) - \eta v _ {\mathrm {n o}} ^ {(n)}. \\ \end{array} +$$ + +Note that $\mathbb{E}_{Z_n\sim \mathbb{P}}\left[v_{\mathrm{no}}^{(n)}\right] = 0$ . Take the expectation of the squared norm of both sides w.r.t. $Z_{n}$ and we have + +$$ +\mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| \delta_ {\mathrm {n o}} ^ {(n)} \| _ {2} ^ {2} \right] = \| (I - \eta \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\bar {\theta} ^ {(n - 1)}, \hat {g})) \delta_ {\mathrm {n o}} ^ {(n - 1)} - \eta \hat {S} _ {\mathrm {n o}} (\theta_ {\star}, \hat {g}) \| _ {2} ^ {2} + \eta^ {2} \mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| v _ {\mathrm {n o}} ^ {(n)} \| _ {2} ^ {2} \right]. +$$ + +By Lem. 18, Lem. 19, and Lem. 20, for any $\omega > 0$ , + +$$ +\begin{array}{l} \left\| \left(I - \eta \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} \left(\bar {\theta} ^ {(n - 1)}, \hat {g}\right)\right) \delta_ {\mathrm {n o}} ^ {(n - 1)} - \eta \hat {S} _ {\mathrm {n o}} \left(\theta_ {\star}, \hat {g}\right) \right\| _ {2} ^ {2} \\ \leq (1 + \eta \omega) \| (I - \eta \nabla_ {\theta} \hat {S} _ {\mathrm {n o}} (\bar {\theta} ^ {(n - 1)}, \hat {g})) \delta_ {\mathrm {n o}} ^ {(n - 1)} \| _ {2} ^ {2} + (\eta^ {2} + \eta \omega^ {- 1}) \| \hat {S} _ {\mathrm {n o}} (\theta_ {\star}, \hat {g}) \| _ {2} ^ {2} \\ \leq (1 + \eta \omega) \left(1 - 2 \eta (\mu_ {\mathrm {n o}} - \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}}) + 2 \eta^ {2} (M _ {\mathrm {n o}} ^ {2} + 2 \alpha_ {1} ^ {2} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2})\right) \| \delta_ {\mathrm {n o}} ^ {(n - 1)} \| _ {2} ^ {2} \\ + \left(\eta^ {2} + \eta \omega^ {- 1}\right) \left(\frac {\beta_ {2} ^ {2}}{2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {4} + 2 \alpha_ {2} ^ {2} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2} \cdot \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2}\right). \\ \end{array} +$$ + +Set $\omega = \mu_{\mathrm{no}}$ . For $\eta \leq 2 / \mu_{\mathrm{no}}$ , we have + +$$ +\begin{array}{l} (1 + \eta \omega) \left(1 - 2 \eta \left(\mu_ {\mathrm {n o}} - \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}}\right) + 2 \eta^ {2} \left(M _ {\mathrm {n o}} ^ {2} + 2 \alpha_ {1} ^ {2} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2}\right)\right) \\ = 1 - \left(\mu_ {\mathrm {n o}} - 2 \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}}\right) \eta - 2 \left(\mu_ {\mathrm {n o}} ^ {2} - \mu_ {\mathrm {n o}} \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}}\right) \eta^ {2} \\ + 2 \left(1 + \eta \mu_ {\mathrm {n o}}\right) \left(M _ {\mathrm {n o}} ^ {2} + 2 \alpha_ {1} ^ {2} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2}\right) \eta^ {2} \\ \leq 1 - \left(\mu_ {\mathrm {n o}} - 2 \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}}\right) \eta \\ + \left(6 M _ {\mathrm {n o}} ^ {2} - 2 \mu_ {\mathrm {n o}} ^ {2} + 2 \mu_ {\mathrm {n o}} \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} + 1 2 \alpha_ {1} ^ {2} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2}\right) \eta^ {2} \\ =: 1 - b (\eta) \eta , \\ \end{array} +$$ + +where + +$$ +b (\eta) = \mu_ {\mathrm {n o}} - 2 \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} - (6 M _ {\mathrm {n o}} ^ {2} - 2 \mu_ {\mathrm {n o}} ^ {2} + 2 \mu_ {\mathrm {n o}} \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} + 1 2 \alpha_ {1} ^ {2} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2}) \eta . +$$ + +By Lem. 21, + +$$ +\mathbb {E} _ {Z _ {n} \sim \mathbb {P}} \left[ \| v _ {\mathrm {n o}} ^ {(n)} \| _ {2} ^ {2} \right] \leq 2 (K _ {1} + K _ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2}) + 2 (\kappa_ {1} + \kappa_ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2}) \| \delta_ {\mathrm {n o}} ^ {n - 1} \| _ {2} ^ {2}. +$$ + +It follows that + +$$ +\begin{array}{l} \mathbb {E} _ {Z _ {n} \sim \mathbb {P}} [ \| \delta_ {\mathrm {n o}} ^ {(n)} \| _ {2} ^ {2} ] \leq (1 - b (\eta) \eta + 2 (\kappa_ {1} + \kappa_ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2}) \eta^ {2}) \| \delta_ {\mathrm {n o}} ^ {(n - 1)} \| _ {2} ^ {2} \\ + \frac {3 \eta}{\mu_ {\mathrm {n o}}} \left(\frac {\beta_ {2} ^ {2}}{2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {4} + 2 \alpha_ {2} ^ {2} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2} \cdot \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2}\right) + 2 (K _ {1} + K _ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2}) \eta^ {2}. \\ \end{array} +$$ + +Thus, it is clear that when $\| \hat{\Gamma} -\Gamma_0\|_{\mathrm{Fro}}\leq \mu_{\mathrm{no}} / (4\alpha_1)$ and the learning rate satisfies + +$$ +\begin{array}{l} \eta \leq \min \left\{\frac {2}{\mu_ {\mathrm {n o}}}, \frac {\mu_ {\mathrm {n o}} - 4 \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}}}{1 2 M _ {\mathrm {n o}} ^ {2} - 3 \mu_ {\mathrm {n o}} ^ {2} / 2 + 4 (\kappa_ {1} + \kappa_ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2})} \right\} \\ = \frac {\mu_ {\mathrm {n o}} - 4 \alpha_ {1} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}}}{1 2 M _ {\mathrm {n o}} ^ {2} - 3 \mu_ {\mathrm {n o}} ^ {2} / 2 + 4 \left(\kappa_ {1} + \kappa_ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2}\right)}, \tag {93} \\ \end{array} +$$ + +we have $1 - \mu_{\mathrm{no}}\eta /2\geq 0$ and + +$$ +1 - b (\eta) \eta + 2 (\kappa_ {1} + \kappa_ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2}) \eta^ {2} \leq 1 - \frac {\mu_ {\mathrm {n o}} \eta}{2}. +$$ + +When $\eta$ satisfies (93), it follows that + +$$ +\begin{array}{l} \mathbb {E} _ {Z _ {n} \sim \mathbb {P}} [ \| \delta_ {\mathrm {n o}} ^ {(n)} \| _ {2} ^ {2} ] \leq \left(1 - \frac {\mu_ {\mathrm {n o}} \eta}{2}\right) \| \delta_ {\mathrm {n o}} ^ {(n - 1)} \| _ {2} ^ {2} \\ + \frac {3 \eta}{\mu_ {\mathrm {n o}}} \left(\frac {\beta_ {2} ^ {2}}{2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {4} + 2 \alpha_ {2} ^ {2} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2} \cdot \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2}\right) + 2 (K _ {1} + K _ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2}) \eta^ {2}. \\ \end{array} +$$ + +Finally, perform the same recursion in Appx. C.3 and we have + +$$ +\begin{array}{l} \mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} [ \| \delta_ {\mathrm {n o}} ^ {(n)} \| _ {2} ^ {2} ] \leq \left(1 - \frac {\mu_ {\mathrm {n o}} \eta}{2}\right) ^ {n} \| \delta_ {\mathrm {n o}} ^ {(0)} \| _ {2} ^ {2} + \frac {4 (K _ {1} + K _ {2} \| \hat {\Gamma} \| _ {\mathrm {F r o}} ^ {2}) \eta}{\mu_ {\mathrm {n o}}} \\ + \frac {3}{\mu_ {\mathrm {n o}} ^ {2}} \left(\beta_ {2} ^ {2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {4} + 4 \alpha_ {2} ^ {2} \| \hat {\Gamma} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2} \cdot \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2}\right). \\ \end{array} +$$ + +![](images/b1c0f3e35db72147230bd96c4499b76b2596cc15f198843bcb5e9cb5b7c46cfc.jpg) + +# F Detailed Discussion + +This section provides details on comparisons and remarks following the statements of the main results in the main text, and details on the discussions summarized in Sec. 4 from the main text. In Appx. F.1, we compare our results to generic state-of-the-art results on biased SGD, that is SGD with errors in the stochastic gradients. In Appx. F.2, we discuss different orthogonalization method in orthogonal statistical learning. In Appx. F.3, we discuss how to interleave the target and nuisance estimation. In Appx. F.4, we describe the connection between our orthogonalized gradient to variance reduction method in the Monte Carlo estimation literature. In Appx. F.5, we discuss the double robustness of SGD for dose-response estimation. + +# F.1 Comparison to Biased SGD + +There are several ways to think about the bias induced by using an imperfect estimate $\hat{g}$ as opposed to the true nuisance $g_0\in \mathcal{G}$ . For the sake of discussion, we will define $L(\cdot ,g_0)$ and $\tilde{L} (\cdot ,\hat{g})$ as the "original objective" and "shifted objective", respectively. Accordingly, we will call $\theta_{\star}$ the "original minimizer" and denote by + +$$ +\hat {\theta} _ {\star} = \underset {\theta \in \Theta} {\arg \min } L (\theta , \hat {g}). \tag {94} +$$ + +the "shifted minimizer". The bias can be measured in terms of (i) the error $\|\hat{\theta}_{\star} - \theta_{\star}\|_2^2$ between the original and shifted minimizers, (ii) the uniform error $\sup_{\theta} |L(\theta, g_0) - L(\theta, \hat{g})|$ between the original and shifted objectives, and (iii) some summary of the gradient bias + +$$ +\mathbb {E} _ {Z _ {t} \sim \mathbb {P}} \left[ S \left(\theta^ {(t - 1)}, \hat {g}; Z _ {t}\right) \right] - \nabla_ {\theta} L \left(\theta^ {(t)}, g _ {0}\right), \tag {95} +$$ + +of the oracle $S$ (a vector-valued quantity) for step $t = 1, \dots, n$ of the algorithm. Whether one appeals to (i) or (ii) depends on whether the convergence guarantees are stated in terms of iterate convergence or function value convergence; because we analyze convergence of iterates, our discussion will cover (i) and (iii). + +On (i), one applies the decomposition + +$$ +\left\| \theta^ {(n)} - \theta_ {\star} \right\| _ {2} ^ {2} \leq 2 \left\| \theta^ {(n)} - \hat {\theta} _ {\star} \right\| _ {2} ^ {2} + 2 \left\| \hat {\theta} _ {\star} - \theta_ {\star} \right\| _ {2} ^ {2}, +$$ + +and plugs an analysis of unbiased SGD from the current literature for the $\| \theta^{(n)} - \hat{\theta}_{\star}\|_{2}^{2}$ term. The purpose of this substitution is to check how our theoretical results align with the known results on unbiased SGD. + +Bach and Moulines [2011, Thm. 1] show that for constant learning rate $\eta = \mathcal{O}(\mu / M^2)$ , the iterate $\theta^{(n)}$ satisfies + +$$ +\mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} \left[ \| \theta^ {(n)} - \theta_ {\star} \| _ {2} ^ {2} \right] \lesssim \exp \left(- \frac {\mu \eta n}{2}\right) \| \theta^ {(0)} - \theta_ {\star} \| _ {2} ^ {2} + \frac {K _ {1} \eta}{\mu} + \| \hat {\theta} _ {\star} - \theta_ {\star} \| _ {2} ^ {2}. \tag {96} +$$ + +Cutler et al. [2023, Thm. 3] demonstrate that with the learning rate $\eta = \mathcal{O}(1 / M)$ , the iterates would satisfy the following bound: + +$$ +\mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} \left[ \| \theta^ {(n)} - \theta_ {\star} \| _ {2} ^ {2} \right] \lesssim \left(1 - \frac {\mu \eta}{2}\right) ^ {n} \| \theta^ {(0)} - \theta_ {\star} \| _ {2} ^ {2} + \frac {K _ {1} \eta}{\mu} + \| \hat {\theta} _ {\star} - \theta_ {\star} \| _ {2} ^ {2} \tag {97} +$$ + +In addition, Cutler et al. [2023, Thm. 6] provide the high probability bound of $\theta^{(n)}$ that for $\eta = \mathcal{O}(1 / M)$ , with probability at least $1 - \delta$ , + +$$ +\left\| \theta^ {(n)} - \theta_ {\star} \right\| _ {2} ^ {2} \lesssim \left(1 - \frac {\mu \eta}{2}\right) ^ {n} \| \theta^ {(0)} - \theta_ {\star} \| _ {2} ^ {2} + \frac {K _ {1} \eta}{\mu} \log \left(\frac {e}{\delta}\right) + \| \hat {\theta} _ {\star} - \theta_ {\star} \| _ {2} ^ {2} \tag {98} +$$ + +All of these bounds essentially agree, as we may apply $(1 - \mu \eta /2)\leq \exp (-\mu \eta /2)$ . In comparison to our Thm. 1, our bias term is stated directly in terms of the nuisance error $\| \hat{g} -g_0\|_{\mathcal{G}}^2$ . This can be viewed as a refinement of the less transparent bias measurement $\| \hat{\theta}_{\star} - \theta_{\star}\|_{2}^{2}$ . Moreover, although (96)-(98) are of the same order as our results in Thm. 1 when the true nuisance $g_{0}$ is + +available, all of the three bounds above require $\kappa_{1} = 0$ in Asm. 3(d). In this case, provide (97) and (98) use a learning rate of the order $\mathcal{O}(1 / M)$ (whereas the learning rate of (96) encounters an additional condition number $M / \mu$ ). Our learning rate recovers $\mathcal{O}(1 / M)$ when $\kappa_{1} = 0$ , and adapts via the setting $\eta = \mathcal{O}(\mu /(M\mu +\kappa_1))$ when $\kappa_{1} > 0$ . Finally, the high probability bound (98) requires a stronger assumption in the sense that $S_{\theta}(\theta ,g;Z) - S_{\theta}(\theta ,g)$ is sub-Gaussian with uniform parameter $K_{1} / 2$ for all $\theta \in \Theta$ and $g\in \mathcal{G}_r(g_0)$ . + +Returning to the bias in the stochastic gradient oracle (95), this case is handled quite generally in Demidovich et al. [2023]. Their "ABC assumption" considers constants $A, B, C, b, c \geq 0$ such that the inequalities + +$$ +\langle \nabla L (\theta , g _ {0}), \mathbb {E} _ {\mathcal {Z} \sim \mathbb {P}} [ S _ {\theta} (\theta , \hat {g}; Z) ] \rangle \geq b \| \nabla L (\theta , g _ {0}) \| _ {2} ^ {2} + c \tag {99} +$$ + +$$ +\mathbb {E} _ {Z \sim \mathbb {P}} \| S _ {\theta} (\theta , \hat {g}; Z) \| _ {2} ^ {2} \leq 2 A (L (\theta , g _ {0}) - L (\theta_ {\star}, g _ {0})) + B \| \nabla L (\theta , g _ {0}) \| _ {2} ^ {2} + C \tag {100} +$$ + +$$ +A + M (B + 1 - 2 b) < \frac {\mu}{2} \tag {101} +$$ + +hold for all $\theta \in \mathbb{R}^d$ (where the expectations are conditional on any randomness in $\hat{g}$ ).3 The bias is really captured in the first of the three inequalities, whereas the third inequality places conditions on the parameters of the problem that are not in the hands of the algorithm user. By strong convexity, our Asm. 3(d) satisfies (100) with $A = \kappa_1 / \mu$ , $B = 0$ , $C = K_1$ . The resulting convergence guarantee [Demidovich et al., 2023, Thm. 5] gives + +$$ +\mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} [ \| \theta^ {(n)} - \theta_ {\star} \| _ {2} ^ {2} ] \lesssim \left(1 - \frac {\mu \eta}{2}\right) ^ {n} \| \theta^ {(0)} - \theta_ {\star} \| _ {2} ^ {2} + \frac {\eta K _ {1}}{\mu} + \frac {K _ {1} + c}{\mu^ {2}}. +$$ + +for a learning rate set as + +$$ +\eta \leq \min \left\{\frac {2}{\mu}, \frac {\mu - 2 (\kappa_ {1} / \mu + M (1 - 2 b))}{\kappa_ {1}} \right\}. +$$ + +We would like to check how our theoretical results compare with the above application of generic results on biased SGD. In our setting, the condition (101) reads $\kappa_{1} / \mu + M(1 - 2b) < \mu / 2$ . We assume neither this nor (99), and in our Thm. 1, replace the $\frac{K_1 + c}{\mu^2}$ term with a bias term that depends directly on the nuisance error $\| \hat{g} - g_0 \|_{\mathcal{G}}^2$ , either in the nuisance sensitive regime, or in the nuisance insensitive regime. + +# F.2 Discussion of Full-sample Orthogonal Statistical Learning and Related Methods + +Comparison of Orthogonalizing Operators. Constructing orthogonal losses or scores has been widely studied in semiparametric inference, hypothesis testing, and machine learning. In semiparametric statistics, such constructions often rely on the efficient influence function, which characterizes the asymptotic efficient estimation bound; see Bickel et al. [1993, Ch. 3], Tsiatis [2006, Ch. 4], Van der Vaart [2000, Ch. 25], Luedtke and Chung [2024]. In hypothesis testing, orthogonal scores were used by Neyman [1959, 1979] and Ferguson [2014] to guarantee the local unbiasedness of specific tests based on the likelihood with finite-dimensional nuisance. In machine learning, the construction of orthogonal scores was latter extended to non-likelihood losses in Wooldridge [1991] and Liu et al. [2022], which aligns with our construction limited to the finite-dimensional nuisance case. Recent work of orthogonalization in machine learning with infinite-dimensional nuisance relies on the approach named concentrating-out [Newey, 1994, Chernozhukov et al., 2018a]. However, although all these constructions produce Neyman orthogonal losses or scores, none of them consider the stochastic design. Our work is complementary to these, providing non-asymptotic guarantees for stochastic optimization. + +Although these constructions might lead to different orthogonal scores, they can be the same at both the target and the true nuisance. Specifically, when $\ell$ is the negative log-likelihood and $\mathcal{G} = \mathbb{R}^k$ , the concentrating-out approach and our NO gradient oracle $S_{\mathrm{no}}$ both produce the efficient score in the semiparametric theory literature; see Newey [1994, Page 1359], Van der Vaart [2000, Ch. 25.4], and Tsiatis [2006, Def. 8]. This identity can happen for infinite-dimensional nuisances as well. As + +an example, consider the partial linear model from Appx. B.1.2, where the non-orthogonal loss is defined as + +$$ +\ell (\theta , g; z) = \frac {1}{2} (y - g (w) - \langle \theta , x \rangle) ^ {2}. +$$ + +Chernozhukov et al. [2018a, Sec. 2.2.2] showed that the concentrating-out approach would produce an orthogonal score under the concentrated-out nuisance $\varphi_0(\theta) = Z\mapsto \mathbb{E}_{\mathbb{P}}[Y - \langle \theta ,X\rangle \mid W]$ as + +$$ +S \left(\theta_ {\star}, \varphi_ {0} \left(\theta_ {\star}\right); Z\right) = - \left(X - \mathbb {E} _ {\mathbb {P}} [ X \mid W ]\right) \left(Y - \mathbb {E} _ {\mathbb {P}} [ Y \mid W ] - \langle \theta_ {\star}, X - \mathbb {E} _ {\mathbb {P}} [ X \mid W ] \rangle\right). +$$ + +On the other hand, it is easy to verify that $H_{gg} = \mathbf{I}$ , $H_{\theta g} = \mathbb{E}_{\mathbb{P}}[X \mid W]$ , and $\Gamma_0 \nabla_g \ell(\theta, g; z) = \mathrm{D}_g \ell(\theta, g; z)[H_{gg}^{-1} H_{\theta g}]$ , which implies that our orthogonal gradient oracle $S_{\mathrm{no}}$ in (11) has the same form under the target $\theta_\star$ and true nuisance $g_0(W) = \mathbb{E}_{\mathbb{P}}[Y - \langle \theta_\star, X \rangle \mid W]$ : + +$$ +S _ {\mathrm {n o}} \left(\theta_ {\star}, g _ {0}; Z\right) = - \left(X - \mathbb {E} _ {\mathbb {P}} [ X \mid W ]\right) \left(Y - \mathbb {E} _ {\mathbb {P}} [ Y \mid W ] - \langle \theta_ {\star}, X - \mathbb {E} _ {\mathbb {P}} [ X \mid W ] \rangle\right). +$$ + +Comparison with Debiased Machine Learning. In machine learning, debiasing typically refers to reducing the impact of model selection error on the parameter or quantity of interest. In particular, mitigating the bias introduced by nuisance estimation is known as debiased machine learning (DML), which has been recently studied by van der Laan et al. [2011], Shi et al. [2019], Chernozhukov et al. [2024], van der Laan et al. [2025]. Some of the calculations used by DML estimators have been shown to be amenable to computerization, simplifying their construction [Carone et al., 2019, Ichimura and Newey, 2022, Luedtke, 2024]. Statistical learning methods that use debiasing are also called orthogonal statistical learning (OSL) and have been studied in Foster and Syrgkanis [2023], Liu et al. [2022], Zadik et al. [2018]. While the earlier studies focus on the empirical risk minimization, our paper provides a stochastic approximation method in DML/OSL and establish the convergence rate of the debiased estimation. + +To strengthen the debiasing effect, one possible approach is to consider the higher-order Neyman orthogonality. If the loss function satisfies the $k$ -th order orthogonality at $(\theta_{\star}, g_0)$ , Zadik et al. [2018, Cor. 4] show that we only need the nuisance estimator to converge at rate $\mathcal{O}_p\left(n^{-\frac{1}{2(k + 1)}}\right)$ to have the nuisance effect in the order of $\mathcal{O}_p(n^{-1})$ , which aligns with the nuisance insensitive rate in Thm. 1, where $k = 1$ and the nuisance effect $\| \hat{g} - g_0 \|_{\mathcal{G}}^4 = \mathcal{O}_p(n^{-1})$ when $\| \hat{g} - g_0 \|_{\mathcal{G}} = \mathcal{O}_p(n^{-1/4})$ . Similar improvements in sensitivity to nuisance estimation rates have been developed previously using higher-order influence functions [Pfanzagl, 1985, Robins et al., 2008]. + +For a range of problems, debiasing methods often lead to cross-product estimations consisting of two nuisance estimators [Rotnitzky et al., 2021]. Such remainders frequently result from orthogonalization procedures used in missing data problems and causal inference problems [Robins et al., 1994, Robins and Rotnitzky, 1995, Laan and Robins, 2003]. Chernozhukov et al. [2024] consider cases where $Z = (W,Y)$ , $g_0(W) = \mathbb{E}_{\mathbb{P}}[Y|W]$ , and the target can be written as the averaged moment of the form + +$$ +\theta_ {\star} = \mathbb {E} _ {\mathbb {P}} [ m (g _ {0}; W) ], +$$ + +where $\mathbb{E}[m(g;W)]:\mathcal{G}\times \mathcal{W}\mapsto \mathbb{R}^d$ is a continuous linear functional of $g:\mathcal{W}\mapsto \mathbb{R}$ . By Riesz representation theorem, there uniquely exists a Riesz representative (RR) $g_0^{\mathrm{RR}}\in \mathcal{G}$ such that $\mathbb{E}_{\mathbb{P}}[m(g;W)] = \mathbb{E}_{\mathbb{P}}[g_0^{\mathrm{RR}}(W)g(W)]$ . Then the debiased score for estimating $\theta_{\star}$ is defined as + +$$ +S (\theta , g; Z) = m (g; W) + g _ {0} ^ {\mathrm {R R}} (W) (Y - g (W)) - \theta . +$$ + +The debiasing effect on the nuisance turns out depending on the cross-product $\| \hat{g}^{\mathrm{RR}} - g_0^{\mathrm{RR}}\|_{\mathcal{G}}\cdot \| \hat{g} -$ $g_0\|_{\mathcal{G}}$ . Specifically, Chernozhukov et al. [2024, Asm. 4] requires the cross product to be in the order $\mathcal{O}_p(n^{-1 / 2})$ to construct $\mathcal{O}_p(n^{-1})$ consistent target estimator. This aligns with the cross-product $\| \hat{g} -g_0\|_{\mathcal{G}}\cdot \| \hat{\Gamma} -\Gamma_0\|_{\mathrm{Fro}}$ in Thm. 3 where the same requirement needs to be satisfied to obtain a $\mathcal{O}_p(n^{-1 / 2})$ consistent estimator. However, Thm. 3 also has a second, non-cross-product remainder $\| \hat{g} -g_0\|_{\mathcal{G}}^4$ that will only be small if $\hat{g}$ approximates $g_{0}$ , making it so that our consistency guarantee is robust to misspecification of $\hat{\Gamma}$ , but not to misspecification of $\hat{g}$ . + +# F.3 Discussion of Interleaving Target and Nuisance Estimation + +To propose the interleaving approach, we consider the case where we learn the nuisance from the $\mathcal{W}$ -valued data $W = (U,V)$ from a probability measure $\mathbb{Q}$ . We assume that the true nuisance $g_0$ satisfies $g_0:\mathcal{U}\mapsto \mathbb{R}$ and is the minimizer of the mean squared error over $\mathcal{G}$ : + +$$ +g_{0} = \operatorname *{arg min}_{g\in \mathcal{G}}\mathbb{E}_{\mathbb{Q}}\left[(g(U) - V)^{2}\right]. +$$ + +Suppose that we observe another data stream $W_{1},\ldots ,W_{m}$ sampled i.i.d. from $\mathbb{Q}$ , and that $S_{m} = \{W_{1},\dots ,W_{m}\}$ is independent of the parameter stream $\mathcal{D}_n$ . We define the sigma algebra $\mathcal{H}_m = \sigma (\mathcal{S}_m),m\geq 1$ as the nuisance filtration and the sigma algebra $\mathcal{F}_{m,t} = \sigma (\mathcal{S}_m\cup \mathcal{D}_{(m - 1)n + t}),0\leq t\leq n$ as the parameter filtration. We assume that there are two stochastic processes $\hat{g}^{(m)},m\geq 1$ adapted to $\mathcal{H}_m$ and $\theta^{(m,t)},0\leq t\leq n$ adapted to $\mathcal{F}_{m,t}$ , to which we refer as the nuisance estimator and the parameter estimator, respectively. Intuitively, this means that the nuisance estimator $\hat{g}^{(m)}$ can be updated now instead of being the fixed $\hat{g}$ , and the parameter estimator $\theta^{(m,t)}$ can be updated $n$ times between every two nuisance updates. Specifically, we use SGD as the parameter estimator. We define $\theta^{(0,n)} = \theta^{(0)}\in \Theta$ and $\theta^{(i,0)} = \theta^{(i - 1,n)}$ for $1\leq i\leq m$ , and produce the sequence $\theta^{(i,1)},\ldots ,\theta^{(i,n)}$ using $n$ steps of the SGD update (8) initialized at $\theta^{(i,0)}$ . + +Under Non-orthogonality. Consider the case that $\mathcal{G}$ is a reproducing kernel Hilbert space (RKHS) with kernel $k(\cdot, \cdot)$ . To obtain a sequence of nuisance estimator $\hat{g}^{(m)}$ on $\mathcal{H}_m$ , one possible approach is to adopt the non-parametric stochastic approximation. With the assumption that the eigenvalues $(\lambda_j)_{j \geq 1}$ of covariance operator $\mathbb{E}_{\mathbb{Q}}[k(W, \cdot) \otimes k(W, \cdot)]$ decay polynomially at order $j^{-\alpha}$ , Dieuleveut and Bach [2016, Cor. 3] suggests that the non-parametric stochastic approximation $\hat{g}^{(m)}$ satisfies for some $C > 0$ , + +$$ +\xi_ {m} := \mathbb {E} _ {\mathcal {S} _ {m} \sim \mathbb {Q} ^ {m}} \left[ \| \hat {g} ^ {(m)} - g _ {0} \| _ {\mathcal {G}} ^ {2} \right] \leq C m ^ {- \frac {2 \alpha - 1}{2 \alpha}}. \tag {102} +$$ + +This leads to the following nuisance sensitive rate for non-Neyman orthogonal losses. + +Proposition 22. Suppose that $\hat{g}^{(m)}$ satisfies (102) and that $\hat{g}^{(m)}\in \mathcal{G}_r(g_0)$ and $\theta^{(m,t)}\in \Theta$ almost surely for all $m\geq 1$ and $0\leq t\leq n$ . Under the same conditions to Thm. 1, it holds that + +$$ +\begin{array}{l} \mathbb {E} _ {\mathcal {D} _ {m n} \cup S _ {m} \sim \mathbb {P} ^ {m n} \otimes \mathbb {Q} ^ {m}} [ \| \theta^ {(m, n)} - \theta_ {\star} \| _ {2} ^ {2} ] \lesssim \left(1 - \frac {\mu \eta}{2}\right) ^ {m n} \| \theta^ {(0)} - \theta_ {\star} \| _ {2} ^ {2} \\ + m \exp \left(- \frac {\mu \eta n m}{4}\right) + (m ^ {- \frac {2 \alpha - 1}{2 \alpha}} + \eta) ((\eta n) ^ {- 1} + 1). \\ \end{array} +$$ + +In addition, when $(\eta n)^{-1} = \mathcal{O}(1)$ , it holds that + +$$ +\mathbb {E} _ {\mathcal {D} _ {m n} \cup \mathcal {S} _ {m} \sim \mathbb {P} ^ {m n} \otimes \mathbb {Q} ^ {m}} [ \| \theta^ {(m, n)} - \theta_ {\star} \| _ {2} ^ {2} ] \lesssim \left(1 - \frac {\mu \eta}{2}\right) ^ {m n} \| \theta^ {(0)} - \theta_ {\star} \| _ {2} ^ {2} + m ^ {- \frac {2 \alpha - 1}{2 \alpha}} + n ^ {- 1} + \eta . +$$ + +The proof is provided in Appx. F.6. Prop. 22 demonstrates that interleaving the target and nuisance estimation allows $\eta \asymp n^{-1}$ since the nuisance update iterations guarantees the shrinking of the term $(1 - \mu \eta /2)^{mn}$ in this case. This is an improvement to Thm. 1 where $\eta$ should satisfy $(\eta n)^{-1} = o(1)$ to ensure $(1 - \mu \eta /2)^n$ shrinking to zero. + +Under Orthogonalized SGD. To establish a similar probability bound for OSGD, we assume that the orthogonalizing operator $\Gamma_0$ can be written as the minimizer of the following program: + +$$ +\Gamma_{0} = \operatorname *{arg min}_{\Gamma \in \mathcal{G}_{*}^{d}}\mathbb{E}_{\mathbb{P}}\left[ \| S_{\theta}(\theta_{\star},g_{0};Z) - \Gamma \nabla_{g}\ell (\theta_{\star},g_{0};Z)\|_{2}^{2}\right], +$$ + +where $\mathcal{G}_{*}$ is the dual space of $\mathcal{G}$ . When $d$ is fixed, we assume that $\Gamma_0$ can be estimated (coordinatively) from the data stream $S_{m}$ using the stochastic approximation of Dieuleveut and Bach [2016], which leads to a sequence of operator estimators $\hat{\Gamma}^{(m)}, m \geq 1$ . For any $s > 0$ , we define the + +following events for $i = 0,1,\ldots ,m$ + +$$ +\mathcal {A} _ {i} (s) = \left\{\| \hat {g} ^ {(i)} - g _ {0} \| _ {\mathcal {G}} ^ {2} \leq C s ^ {- 1} i ^ {- \frac {2 \alpha - 1}{2 \alpha}} \right\} \text {a n d} \mathcal {B} _ {i} (s) = \left\{\| \hat {\Gamma} ^ {(i)} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2} \leq C s ^ {- 1} i ^ {- \frac {2 \alpha - 1}{2 \alpha}} \right\}. +$$ + +We assume that for some constant $c \geq 1$ the nuisance estimator $\hat{g}^{(i)}$ satisfies + +$$ +\mathbb {E} _ {\mathcal {S} _ {i}} \left[ \| \hat {g} ^ {(i)} - g _ {0} \| _ {\mathcal {G}} ^ {2 c} \mid \mathcal {A} _ {i - 1} \left(s ^ {1 / c}\right), \dots , \mathcal {A} _ {1} \left(s ^ {1 / c}\right) \right] \leq C ^ {c} i ^ {- \frac {(2 \alpha - 1) c}{2 \alpha}}. \tag {103} +$$ + +Additionally, we assume that $\hat{\Gamma}^{(i)}$ decays in the same rate such that + +$$ +\mathbb {E} _ {\mathcal {S} _ {i}} \left[ \| \hat {\Gamma} ^ {(i)} - \Gamma_ {0} \| _ {\mathrm {F r o}} ^ {2 c} \mid \mathcal {B} _ {i - 1} \left(s ^ {1 / c}\right), \dots , \mathcal {B} _ {1} \left(s ^ {1 / c}\right) \right] \leq C ^ {c} i ^ {- \frac {(2 \alpha - 1) c}{2 \alpha}}. \tag {104} +$$ + +With all the assumptions above, it is possible to provide a convergence bound of $\| \theta^{(m,n)} - \theta_{\star} \|_2^2$ in probability. The following proposition shows that estimations from $S_m$ using OSGD contribute to a nuisance insensitive rate of $\mathcal{O}(m^{-\frac{2\alpha - 1}{\alpha}})$ , compared to the nuisance sensitive rate $\mathcal{O}(m^{-\frac{2\alpha - 1}{2\alpha}})$ in Prop. 22 for non-Neyman orthogonal losses. + +Proposition 23. Suppose that $\{\hat{g}^{(m)}, m \geq 1\}$ satisfies (103), and that $\{\hat{\Gamma}^{(m)}, m \geq 1\}$ satisfies (104). Assume that $\theta^{(m,t)} \in \Theta$ almost surely for all $m \geq 1$ and $0 \leq t \leq n$ . For any $s \geq 0$ , define $\delta(s) = \mathcal{O}(ms)$ as (109). Under the same conditions to Thm. 1, with probability at least $1 - \delta(s)$ , it holds that + +$$ +\begin{array}{l} \left\| \theta^ {(m, n)} - \theta_ {\star} \right\| _ {2} ^ {2} \lesssim s ^ {- 1} \left(1 - \frac {\mu \eta}{2}\right) ^ {m n} \left\| \theta^ {(0)} - \theta_ {\star} \right\| _ {2} ^ {2} \\ + s ^ {- 1} \left(m \exp \left(- \frac {\mu \eta m}{4}\right) + \left(s ^ {- 2 / c} m ^ {- \frac {2 \alpha - 1}{\alpha}} + \eta\right) \left(\left(\eta n\right) ^ {- 1} + 1\right)\right). \\ \end{array} +$$ + +In addition, when $(\eta n)^{-1} = \mathcal{O}(1)$ , with probability at least $1 - \delta(s)$ , it holds that + +$$ +\left\| \theta^ {(m, n)} - \theta_ {\star} \right\| _ {2} ^ {2} \lesssim s ^ {- 1} \left(1 - \frac {\mu \eta}{2}\right) ^ {m n} \left\| \theta^ {(0)} - \theta_ {\star} \right\| _ {2} ^ {2} + s ^ {- 1} \left(s ^ {- 2 / c} m ^ {- \frac {2 \alpha - 1}{\alpha}} + n ^ {- 1} + \eta\right). +$$ + +We refer the reader to Appx. F.7 for the proof. + +# F.4 Interpretation as Control Variate for Variance Reduction + +The regression equation (9), which provides an alternate characterization of the orthogonalized stochastic gradient oracle in the case of negative log-likelihood losses, yields an interesting connection to the Monte Carlo estimation literature. Variance reduction techniques (or "swindles") are used in problems such as estimating the mean or variance of a statistic via Monte Carlo simulation. Consider a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with expectation denoted by $\mathbb{E}$ and an unknown vector-valued target $v \in \mathbb{R}^d$ . We have $\hat{v}: \Omega \to \mathbb{R}^d$ , where we interpret $\hat{v}$ as a (not necessarily unbiased) sample estimate of $v$ . Several variance reduction techniques fall into the category of control variates [Graham and Talay, 2013], where a random variable $\hat{u}: \Omega \to \mathbb{R}^k$ with known expectations $u = \mathbb{E}[\hat{u}]$ and a matrix $\Gamma \in \mathbb{R}^{d \times k}$ are used in the variance-reduced estimator + +$$ +\tilde {v} = \hat {v} - \Gamma (\hat {u} - u). +$$ + +A mean squared error decomposition yields the identity + +$$ +\begin{array}{l} \mathbb {E} \| \tilde {v} - v \| _ {2} ^ {2} = \mathbb {E} \| \hat {v} - v \| _ {2} ^ {2} - 2 \mathbb {E} \langle \hat {v} - v, \Gamma (\hat {u} - u) \rangle + \mathbb {E} \| \Gamma (\hat {u} - u) \| _ {2} ^ {2} \\ = \mathbb {E} \| \hat {v} - v \| _ {2} ^ {2} - 2 \mathbb {E} \langle \hat {v} - v, \Gamma (\hat {u} - u) \rangle + o (\| \Gamma \| _ {\mathrm {o p}}), \\ \end{array} +$$ + +indicating that for sufficiently "small" $\Gamma$ , $\tilde{v}$ provides an improved estimator if $\hat{v} - v$ and $\hat{u} - u$ have high (multiple) correlation. While in the Monte Carlo literature, $\hat{u}$ and $\Gamma$ can be chosen optimally provided knowledge of the underlying data-generating mechanism, as $\Gamma$ can be interpreted as the regression function of $\hat{v} - v$ on $\hat{u} - u$ .4 Outside of Monte Carlo simulation, this procedure can be applied more widely if the user chooses $\hat{u}$ and $\hat{\Gamma}$ based on intuition or limiting arguments. + +In the stochastic optimization setting, $v$ represents the true gradient of the objective at a particular parameter, while $\hat{v}$ represents a stochastic gradient estimate from an oracle. Variance reduction techniques have previously been applied in an incremental setting, in which a fixed data set of size $n$ is provided at initialization, and the algorithm may only make multiple passes through this same data set [Gower et al., 2020]. Note that this differs from our fully stochastic setting, in which we receive a fresh sample $Z_{t}$ on each iterate $t = 1,\dots ,n$ . For negative log-likelihood losses, our orthogonalized oracle can be viewed in a similar light to control variate-based variance reduction methods (although in an infinite-dimensional setting). To summarize, we have from (9) that + +$$ +v = S _ {\theta} \left(\theta_ {\star}, g _ {0}\right) +$$ + +$$ +\hat {v} = S _ {\theta} (\theta_ {\star}, \hat {g}) +$$ + +$u = 0$ (by Asm. 6) + +$$ +\hat {u} = \nabla_ {g} \ell (\theta_ {\star}, g _ {0}; Z) +$$ + +$$ +\tilde {v} = S _ {\theta} \left(\theta_ {\star}, g _ {0}; Z\right) - \Gamma_ {0} \nabla_ {g} \ell \left(\theta_ {\star}, g _ {0}; Z\right), +$$ + +using the idealized parameters. Using the approximations for $\theta \neq \theta_{\star}$ , we have + +$$ +v = S _ {\theta} (\theta , g _ {0}) +$$ + +$$ +\hat {v} = S _ {\theta} (\theta , \hat {g}) +$$ + +for $\theta \approx \theta_{\star}$ + +$$ +\hat {u} = \nabla_ {g} \ell (\theta , \hat {g}; Z) +$$ + +$$ +\tilde {v} = S _ {\theta} (\theta , \hat {g}; Z) - \hat {\Gamma} \nabla_ {g} \ell (\theta , \hat {g}; Z). +$$ + +In fact, using the derivative of the log likelihood in a control variate procedure has been explored in the simulation literature as early as Johnstone and Velleman [1985], as the correlation between a statistic and the score function has tight connections to the Cramér-Rao lower variance bound and exponential families. We emphasize, however, that our method does not require the loss to be of negative log-likelihood form nor any specific distributional knowledge to be applied. + +# F.5 Discussion of Double Robustness + +We now study the double robustness of SGD for dose-response estimation as discussed in Bonvini and Kennedy [2022]. Consider estimating the effect of the continuous treatment $A \in \mathcal{A} \subset \mathbb{R}$ on the outcome $Y \in \mathcal{V} \subset \mathbb{R}$ , which is defined as $\mathbb{E}Y(a)$ (known as the dose-response function, DRF) under the potential outcomes framework. Under standard assumptions, the DRF takes the form + +$$ +\theta_ {0} (t) = \mathbb {E} [ \mathbb {E} [ Y \mid A = t, X ] ] = \int \mathbb {E} [ Y \mid A = t, X = x ] \mathrm {d} \mathbb {P} (x), +$$ + +where $X \in \mathcal{X} \subset \mathbb{R}^d$ is the measured confounders. Let $Z = (Y, A, X) \sim \mathbb{P}$ with density $p$ . We take the following notations: + +$$ +p (u) = \frac {\mathrm {d}}{\mathrm {d} u} \mathbb {P} (U \leq u), \pi (a \mid x) = \frac {p (a , x)}{p (x)}, \mu (a, x) = \mathbb {E} [ Y \mid A = a, X = x ], w (a, x) = \frac {p (a)}{\pi (a \mid x)}. +$$ + +We can rewrite $\theta_0(t)$ equivalently as + +$$ +\theta_ {0} (t) = \mathbb {E} [ \mu (t, X) ] = \mathbb {E} [ w (t, X) Y \mid A = t ]. +$$ + +We also take the notations $\mathbb{P}(g(Z)) = \int g(z)\mathrm{d}\mathbb{P}(z),\mathbb{P}_n(g(Z)) = n^{-1}\sum_{i = 1}^n g(Z_i),\| g\|_{L_2(\mathbb{P})} = [\mathbb{P}(g^2 (Z))]^{1 / 2}$ to denote the $L_{2}(\mathbb{P})$ norm, and $\| g\|_{L_4(\mathbb{P})} = [\mathbb{P}(g^4 (Z))]^{1 / 4}$ to denote the $L_{4}(\mathbb{P})$ norm. We now establish the procedure to estimate $\theta_0(t)$ as Algorithm 1 in Bonvini and Kennedy [2022] with slightly modification to apply SGD: + +1. Observe i.i.d. samples $\{Z_i'\}_{i=1}^m$ for the nuisance estimation and i.i.d. samples $\{Z_i\}_{i=1}^n$ for the parameter estimation. +2. Estimate $\mu, w$ , and $m(a) = \mathbb{P}\mu(a,\cdot)$ using $\{Z_i'\}_{i=1}^m$ with $\hat{\mu}, \hat{w}$ , and $\hat{m}(a) = \mathbb{P}_n(\hat{\mu}(a,\cdot))$ , respectively. + +3. Construct the pseudo-outcome + +$$ +\hat {\varphi} (Z) = \hat {w} (A, X) Y - \hat {\mu} (A, X) + \hat {m} (A). +$$ + +We also define the true nuisance as + +$$ +\varphi_ {0} (Z) = w (A, X) Y - \mu (A, X) + \int \mu (A, x) \mathrm {d} \mathbb {P} (x). +$$ + +4. Define the loss function via a parametric function class $\mathcal{F}_{\Theta} = \{f_{\theta}:\mathcal{A}\mapsto \mathbb{R}\mid \theta \in \Theta \subset \mathbb{R}^d\}$ as + +$$ +\ell (\theta , \varphi ; z) = \frac {1}{2} \left(f _ {\theta} (a) - \varphi (z)\right) ^ {2}. \tag {105} +$$ + +Define the stochastic gradient oracle as + +$$ +S _ {\theta} (\theta , \varphi ; z) = (f _ {\theta} (a) - \varphi (z)) \nabla_ {\theta} f _ {\theta} (a). +$$ + +5. Solve the optimization problem + +$$ +\theta_{\star} = \operatorname *{arg min}_{\theta \in \Theta}\mathbb{E}\left[\ell (\theta ,\varphi_{0};Z)\right] +$$ + +using SGD with the stochastic gradient $S_{\theta}(\theta ,\hat{\varphi};Z)$ by + +$$ +\theta^ {(n)} = \theta^ {(n - 1)} - \eta S _ {\theta} \left(\theta^ {(n - 1)}, \hat {\varphi}; Z _ {n - 1}\right), \quad \theta^ {(0)} \in \Theta . \tag {106} +$$ + +As demonstrated in Bonvini and Kennedy [2022], this procedure would yield a doubly robust ERM estimator. In the following proposition, we claim that double robustness would be preserved if the SGD estimator is adopted instead. + +Proposition 24. Assume that $\mathbb{E}\left[\|\nabla_{\theta}f_{\theta_{\star}}(A)\|_2^2\right]^{1/2} \leq C_A$ . Suppose that Asm. 3 holds and $\theta^{(0)}, \ldots, \theta^{(n)} \in \Theta$ almost surely for $\theta^{(n)}$ in (106). If $\eta \leq \mu/2(M\mu + \kappa_1)$ , the iterates of (106) satisfy + +$$ +\begin{array}{l} \mathbb {E} _ {\mathcal {D} _ {n} \sim \mathbb {P} ^ {n}} \left[ \| \theta^ {(n)} - \theta_ {\star} \| _ {2} ^ {2} \right] \lesssim \left(1 - \frac {\mu \eta}{2}\right) ^ {n} + \eta \\ + \| w - \hat {w} \| _ {L _ {4} (\mathbb {P})} \| \mu - \hat {\mu} \| _ {L _ {4} (\mathbb {P})} + \max _ {a \in \mathcal {A}} | (\mathbb {P} _ {n} - \mathbb {P}) \{\hat {\mu} (a, X) \} | ^ {2}. \\ \end{array} +$$ + +Prop. 24 follows directly from the following two lemmas, Lem. 25 and Lem. 26, and the nuisance sensitive rate in Thm. 1. Whenever the empirical estimation $\max_{a\in \mathcal{A}}|\left(\mathbb{P}_n - \mathbb{P}\right)\{\hat{\mu} (a,X)\} |^2$ shrinks, Prop. 24 suggests that the $\theta^{(n)}$ would converge to the target parameter when either $\hat{w}$ or $\hat{\mu}$ is correctly specified. + +Lemma 25. Assume that $\mathbb{E}\left[\| \nabla_{\theta}f_{\theta_{\star}}(A)\| _2^2\right]^{1 / 2}\leq C_A$ . Then for the loss defined in (105), we have + +$$ +\left| \mathrm {D} _ {\varphi} \mathrm {D} _ {\theta} L \left(\theta_ {\star}, \bar {\varphi}\right) \left[ \theta - \theta_ {\star}, \hat {\varphi} - \varphi_ {0} \right] \right| \leq C _ {A} \| \hat {\varphi} - \varphi_ {0} \| _ {\mathcal {G}} \| \theta - \theta_ {\star} \| _ {2}, +$$ + +where $\| \hat{\varphi} -\varphi_0\|_{\mathcal{G}} = \mathbb{E}[\mathbb{E}[\hat{\varphi} (Z) - \varphi_0(Z)\mid A]^2 ]^{1 / 2} = \mathbb{E}[(\mathbb{E}[\hat{\varphi} (Z)\mid A] - \theta_0(A))^2 ]^{1 / 2}$ . + +Proof. Let $\hat{r}(t) = \mathbb{E}[\hat{\varphi}(Z) \mid A = t] - \theta_0(t)$ . Note that + +$$ +\begin{array}{l} \mathrm {D} _ {\varphi} \mathrm {D} _ {\theta} L \left(\theta_ {\star}, \bar {\varphi}\right) \left[ \theta - \theta_ {\star}, \hat {\varphi} - \varphi_ {0} \right] = - \mathbb {E} \left[ \left(\hat {\varphi} (Z) - \varphi_ {0} (Z)\right) \left\langle \nabla_ {\theta} f _ {\theta_ {\star}} (A), \theta - \theta_ {\star} \right\rangle \right] \\ = - \mathbb {E} \left[ \left(\mathbb {E} [ \hat {\varphi} (Z) \mid A ] - \mathbb {E} [ \varphi_ {0} (Z) \mid A ]\right) \langle \nabla_ {\theta} f _ {\theta_ {*}} (A), \theta - \theta_ {*} \rangle \right] \\ = - \mathbb {E} \left[ \left(\mathbb {E} \left[ \hat {\varphi} (Z) \mid A \right] - \theta_ {0} (A)\right) \langle \nabla_ {\theta} f _ {\theta_ {*}} (A), \theta - \theta_ {*} \rangle \right] \\ = - \mathbb {E} [ \hat {r} (A) \langle \nabla_ {\theta} f _ {\theta_ {*}} (A), \theta - \theta_ {\star} \rangle ]. \\ \end{array} +$$ + +Thus, by the assumption that $\mathbb{E}\left[\| \nabla_{\theta}f_{\theta_{\star}}(A)\| _2^2\right]^{1 / 2}\leq C_A$ + +$$ +\begin{array}{l} | \mathrm {D} _ {\varphi} \mathrm {D} _ {\theta} L (\theta_ {\star}, \bar {\varphi}) [ \theta - \theta_ {\star}, \hat {\varphi} - \varphi_ {0} ] | \leq \mathbb {E} \left[ | \hat {r} (A) | ^ {2} \right] ^ {1 / 2} \mathbb {E} \left[ \| \nabla_ {\theta} f _ {\theta_ {\star}} (A) \| _ {2} ^ {2} \right] ^ {1 / 2} \| \theta - \theta_ {\star} \| _ {2} \\ \leq C _ {A} \mathbb {E} \left[ | \hat {r} (A) | ^ {2} \right] ^ {1 / 2} \| \theta - \theta_ {\star} \| _ {2}. \\ \end{array} +$$ + +![](images/87df1256bbd30d2b6011710bf5a44be31e253a861a451ac2e564fc9d4db70e47.jpg) + +Lemma 26. For the norm defined in Lem. 25, we have + +$$ +\| \hat {\varphi} - \varphi_ {0} \| _ {\mathcal {G}} \leq \| w - \hat {w} \| _ {L _ {4} (\mathbb {P})} ^ {1 / 2} \| \mu - \hat {\mu} \| _ {L _ {4} (\mathbb {P})} ^ {1 / 2} + \max _ {a \in \mathcal {A}} | (\mathbb {P} _ {n} - \mathbb {P}) \left\{\hat {\mu} (t, X) \right\} |. +$$ + +Proof. Lemma 1 of Bonvini and Kennedy [2022] demonstrates that + +$$ +\left. \left| \hat {r} (t) \right| \leq \| w - \hat {w} \| _ {t} \| \mu - \hat {\mu} \| _ {t} + \left| \left(\mathbb {P} _ {n} - \mathbb {P}\right) \left\{\hat {\mu} (t, X) \right\} \right|, \right. \tag {107} +$$ + +where $\| f\| _t^2 = \int f^2 (z)\mathrm{d}\mathbb{P}(z\mid A = t)$ . By (107), we have + +$$ +\begin{array}{l} \| \hat {\varphi} - \varphi_ {0} \| _ {\mathcal {G}} \leq \| \| w - \hat {w} \| _ {A} \| \mu - \hat {\mu} \| _ {A} + | (\mathbb {P} _ {n} - \mathbb {P}) _ {t = A} \left\{\hat {\mu} (t, X) \right\} | \| _ {L _ {2} (\mathbb {P} _ {A})} \\ \leq \| w - \hat {w} \| _ {L _ {4} (\mathbb {P})} ^ {1 / 2} \| \mu - \hat {\mu} \| _ {L _ {4} (\mathbb {P})} ^ {1 / 2} + \max _ {a \in \mathcal {A}} | (\mathbb {P} _ {n} - \mathbb {P}) \left\{\hat {\mu} (t, X) \right\} |. \\ \end{array} +$$ + +![](images/7517e211f2b76ec69ab8c1dc59b041b89cffe7b61aa8a6ff32d1d33e843e8b49.jpg) + +# F.6 Proof of Proposition 22 + +Proof. For simplicity, we use the notation $\mathbb{E}_{m,n}$ to replace $\mathbb{E}_{\mathcal{D}_{mn} \cup \mathcal{S}_m} \sim \mathbb{P}^{mn} \otimes \mathbb{Q}^m$ . Let $q_n = (1 - \mu \eta / 2)^n$ , $\delta^{(m,n)} = \theta^{(m,n)} - \theta_\star$ , and $\delta^{(0)} = \delta^{(0,n)}$ . Thus, by Thm. 1, + +$$ +\begin{array}{l} \mathbb {E} _ {m, n} [ \| \delta^ {(m, n)} \| _ {2} ^ {2} ] \leq q _ {n} \mathbb {E} _ {m - 1, n} [ \| \delta^ {(m, 0)} \| _ {2} ^ {2} ] + \frac {2 \alpha_ {1} ^ {2}}{\mu^ {2}} \xi_ {m} + \frac {4 K _ {1} \eta}{\mu} \\ = q _ {n} \mathbb {E} _ {m - 1, n} [ \| \delta^ {(m - 1, n)} \| _ {2} ^ {2} ] + \frac {2 \alpha_ {1} ^ {2}}{\mu^ {2}} \xi_ {m} + \frac {4 K _ {1} \eta}{\mu}. \\ \end{array} +$$ + +This recursive formula gives a complete bound for $\theta^{(m,n)}$ as + +$$ +\mathbb {E} _ {m, n} [ \| \delta^ {(m, n)} \| _ {2} ^ {2} ] \leq q _ {n} ^ {m} \| \delta^ {(0)} \| _ {2} ^ {2} + \frac {2 \alpha_ {1} ^ {2}}{\mu^ {2}} \sum_ {i = 1} ^ {m} q _ {n} ^ {m - i} \xi_ {i} + \frac {4 K _ {1} \eta}{\mu} \sum_ {i = 1} ^ {m} q _ {n} ^ {m - i}. +$$ + +By (96), we assume that $\xi_{m}\leq Cm^{-\frac{2\alpha - 1}{2\alpha}}$ for some $C > 0$ . Note that + +$$ +q _ {n} = \left(1 - \frac {\mu \eta}{2}\right) ^ {n} \leq \exp \left(- \frac {\mu \eta n}{2}\right). +$$ + +For the second term, when $q_{n} \in (0,1)$ we have + +$$ +\begin{array}{l} \sum_ {i = 1} ^ {m} q _ {n} ^ {m - i} \xi_ {i} = \sum_ {i = 1} ^ {\lfloor m / 2 \rfloor} q _ {n} ^ {m - i} \xi_ {i} + \sum_ {i = \lfloor m / 2 \rfloor + 1} ^ {m} q _ {n} ^ {m - i} \xi_ {i} \\ \leq C \sum_ {i = 1} ^ {\lfloor m / 2 \rfloor} q _ {n} ^ {m - i} + C \left(\frac {m}{2}\right) ^ {- \frac {2 \alpha - 1}{2 \alpha}} \sum_ {i = \lfloor m / 2 \rfloor + 1} ^ {m} q _ {n} ^ {m - i} \\ \leq \frac {C m}{2} q _ {n} ^ {m / 2} + \frac {C}{1 - q _ {n}} \left(\frac {m}{2}\right) ^ {- \frac {2 \alpha - 1}{2 \alpha}} \\ \leq \frac {C m}{2} \exp \left(- \frac {\mu \eta m m}{4}\right) + \frac {C}{1 - q _ {n}} \left(\frac {m}{2}\right) ^ {- \frac {2 \alpha - 1}{2 \alpha}}. \\ \end{array} +$$ + +The last term is easy to bound since for $q_{n}\in (0,1)$ + +$$ +\sum_ {i = 1} ^ {m} q _ {i} ^ {m - i} = \sum_ {i = 0} ^ {m - 1} q _ {n} ^ {i} \leq \frac {1}{1 - q _ {n}}. +$$ + +We claim that for some constant $c > 0$ + +$$ +1 - q _ {n} = 1 - \left(1 - \frac {\mu \eta}{2}\right) ^ {n} \geq c \min \left\{\frac {\mu \eta n}{2}, 1 \right\}. \tag {108} +$$ + +With (108), we have + +$$ +\frac {1}{1 - q _ {n}} \leq c ^ {- 1} \left(\frac {2}{\mu \eta n} + 1\right), +$$ + +which implies that + +$$ +\mathbb {E} _ {m, n} [ \| \delta^ {(m, n)} \| _ {2} ^ {2} ] = \mathcal {O} \left(q _ {n} ^ {m} \| \delta^ {(0)} \| _ {2} ^ {2} + m \exp \left(- \frac {\mu \eta n m}{4}\right) + \left(m ^ {- \frac {2 \alpha - 1}{2 \alpha}} + \eta\right) \left(\frac {1}{\eta n} + 1\right)\right). +$$ + +When $(\eta n)^{-1} = \mathcal{O}(1)$ , the bound above reduces to + +$$ +\mathbb {E} _ {m, n} [ \| \delta^ {(m, n)} \| _ {2} ^ {2} ] = \mathcal {O} \left(q _ {n} ^ {m} \| \delta^ {(0)} \| _ {2} ^ {2} + m ^ {- \frac {2 \alpha - 1}{2 \alpha}} + n ^ {- 1} + \eta\right). +$$ + +We will finish the proof by showing (108). The key step is to show $1 - e^{-x} \geq c \min(x, 1)$ for all $x > 0$ and some constant $c > 0$ . + +Let $f(x) = 1 - e^{-x} - x / 2$ , for $x \in (0,1)$ we have + +$$ +f ^ {\prime} (x) = e ^ {- x} - \frac {1}{2} \left\{ \begin{array}{l} > 0 \text {f o r} x \in (0, \log 2), \\ = 0 \text {f o r} x = \log 2, \\ < 0 \text {f o r} x \in (\log 2, 1). \end{array} \right. +$$ + +Thus, $f(x) \geq f(\log 2) = (1 - \log 2) / 2 > 0$ for $x \in (0,1)$ , which implies that $1 - e^{-x} > x / 2$ for $x \in (0,1)$ . Note that $1 - e^{-x} \geq 1 - e^{-1}$ for $x \geq 1$ . Let $c = \min(2^{-1}, 1 - e^{-1})$ . Then we have $1 - e^{-x} \geq c \min(x,1)$ . + +It follows that + +$$ +1 - q _ {n} = 1 - \exp \left(- n \log \left(\frac {1}{1 - \mu \eta / 2}\right)\right) \geq c \min \left\{n \log \left(\frac {1}{1 - \mu \eta / 2}\right), 1 \right\}. +$$ + +Since $x - 1 \geq \log x$ for all $x > 0$ , we have + +$$ +\log \left(1 - \mu \eta / 2\right) \leq 1 - \mu \eta / 2 - 1 = - \mu \eta / 2, +$$ + +which implies that + +$$ +n \log \left(\frac {1}{1 - \mu \eta / 2}\right) \geq \frac {\mu \eta n}{2}. +$$ + +Thus, we complete the proof. + +![](images/c1cd134740c5806c69e06c38ab0236118fe774799a4e3703a0cc5e1fe8c0215f.jpg) + +# F.7 Proof of Proposition 23 + +Proof. Given $s > 0$ , we define $\mathcal{A}_i = \mathcal{A}_i(s^{1/c})$ and $\mathcal{B}_i = \mathcal{B}_i(s^{1/c})$ for $i = 0, \ldots, m$ for simplicity. First, since $c \geq 1$ , by (103) and Markov inequality, for $i = 1, \ldots, m$ , + +$$ +\begin{array}{l} \mathbb {P} \left[ \mathcal {A} _ {i} \mid \mathcal {A} _ {i - 1}, \ldots , \mathcal {A} _ {0} \right] = 1 - \mathbb {P} \left[ \| \hat {g} ^ {(i)} - g _ {0} \| _ {\mathcal {G}} ^ {2} \geq C s ^ {- 1 / c} i ^ {- \frac {2 \alpha - 1}{2 \alpha}} \mid \mathcal {A} _ {i - 1}, \ldots , \mathcal {A} _ {0} \right] \\ = 1 - \mathbb {P} \left[ \| \hat {g} ^ {(i)} - g _ {0} \| _ {\mathcal {G}} ^ {2 c} \geq C ^ {c} s ^ {- 1} i ^ {- \frac {(2 \alpha - 1) c}{2 \alpha}} \mid \mathcal {A} _ {i - 1}, \dots , \mathcal {A} _ {0} \right] \\ \geq 1 - \frac {\mathbb {E} _ {\mathcal {S} _ {m}} \left[ \| \hat {g} ^ {(i)} - g _ {0} \| _ {\mathcal {G}} ^ {2 c} \mid \mathcal {A} _ {i - 1} , \dots , \mathcal {A} _ {1} \right]}{C ^ {c} s ^ {- 1} i ^ {- \frac {(2 \alpha - 1) c}{2 \alpha}}} \\ \geq 1 - s. \\ \end{array} +$$ + +We assume that $\mathbb{P}[\mathcal{A}_0] = \mathbb{P}[\mathcal{B}_0] = 1$ , and we have + +$$ +\begin{array}{l} \mathbb {P} \left[ \mathcal {A} _ {m}, \mathcal {A} _ {i - 1}, \dots , \mathcal {A} _ {1}, \mathcal {A} _ {0} \right] = \mathbb {P} \left[ \mathcal {A} _ {m} \mid \mathcal {A} _ {m - 1}, \dots , \mathcal {A} _ {1} \right] \dots \mathbb {P} \left[ \mathcal {A} _ {1} \mid \mathcal {A} _ {0} \right] \mathbb {P} \left[ \mathcal {A} _ {0} \right] \\ \geq \prod_ {i = 1} ^ {m} (1 - s) = (1 - s) ^ {m}. \\ \end{array} +$$ + +Similarly, we have + +$$ +\mathbb {P} \left[ \mathcal {B} _ {m}, \mathcal {B} _ {i - 1}, \dots , \mathcal {B} _ {1} \right] \geq (1 - s) ^ {m}. +$$ + +we consider the conditional mean squared error of $\theta^{(m,n)}$ given $(\cap_{i=0}^{m}\mathcal{A}_{i}) \cap (\cap_{i=0}^{m}\mathcal{B}_{i})$ . By similar proof to Prop. 22, we can show that for some constant $C_1 > 0$ , + +$$ +\begin{array}{l} \mathbb {E} _ {\mathcal {D} _ {m n} \cup \mathcal {S} _ {m}} \left[ \| \theta^ {(m, n)} - \theta_ {\star} \| _ {2} ^ {2} \mid (\cap_ {i = 0} ^ {m} \mathcal {A} _ {i}) \cap (\cap_ {i = 0} ^ {m} \mathcal {B} _ {i}) \right] \leq C _ {1} \left(1 - \frac {\mu \eta}{2}\right) ^ {m n} \| \theta^ {(0)} - \theta_ {\star} \| _ {2} ^ {2} \\ + C _ {1} \left(m \exp \left(- \frac {\mu \eta n m}{4}\right) + \left(s ^ {- 2 / c} m ^ {- \frac {2 \alpha - 1}{\alpha}} + \eta\right) ((\eta n) ^ {- 1} + 1)\right). \\ \end{array} +$$ + +We define the event of interest as + +$$ +\mathcal {E} (s) = \left\{\| \theta^ {(m, n)} - \theta_ {\star} \| _ {2} ^ {2} \leq C _ {1} s ^ {- 1} f _ {s} (m, n) \right\}, +$$ + +where $f_{s}(m,n)$ is defined as + +$$ +\begin{array}{l} f _ {s} (m, n) = \left(1 - \frac {\mu \eta}{2}\right) ^ {m n} \| \theta^ {(0)} - \theta_ {\star} \| _ {2} ^ {2} \\ + m \exp \left(- \frac {\mu \eta n m}{4}\right) + (s ^ {- 2 / c} m ^ {- \frac {2 \alpha - 1}{\alpha}} + \eta) ((\eta n) ^ {- 1} + 1). \\ \end{array} +$$ + +By Markov inequality, we have + +$$ +\begin{array}{l} \mathbb {P} \left[ \mathcal {E} _ {3} (s) \mid (\cap_ {i = 0} ^ {m} \mathcal {A} _ {i}) \cap (\cap_ {i = 0} ^ {m} \mathcal {B} _ {i}) \right] \geq 1 - \frac {\mathbb {E} _ {\mathcal {D} _ {m n} \cup \mathcal {S} _ {m}} [ \| \theta^ {(m , n)} - \theta_ {\star} \| _ {2} ^ {2} \mid (\cap_ {i = 0} ^ {m} \mathcal {A} _ {i}) \cap (\cap_ {i = 0} ^ {m} \mathcal {B} _ {i}) ]}{C _ {1} s ^ {- 1} f (m , n)} \\ \geq 1 - s. \\ \end{array} +$$ + +Since + +$$ +\begin{array}{l} \mathbb {P} \left[ \mathcal {E} _ {3} (s) ^ {c} \right] = \mathbb {E} _ {\mathbb {P}} \left[ \mathbb {1} _ {\mathcal {E} _ {3} (s) ^ {c}} \mathbb {1} _ {(\cap_ {i = 0} ^ {m} \mathcal {A} _ {i}) \cap (\cap_ {i = 0} ^ {m} \mathcal {B} _ {i})} \right] + \mathbb {E} _ {\mathbb {P}} \left[ \mathbb {1} _ {\mathcal {E} _ {3} (s) ^ {c}} \mathbb {1} _ {((\cap_ {i = 0} ^ {m} \mathcal {A} _ {i}) \cap (\cap_ {i = 0} ^ {m} \mathcal {B} _ {i})) ^ {c}} \right] \\ \leq \mathbb {E} _ {\mathbb {P}} \left[ \mathbb {1} _ {\mathcal {E} _ {3} (s) ^ {c} \cap (\cap_ {i = 0} ^ {m} \mathcal {A} _ {i}) \cap (\cap_ {i = 0} ^ {m} \mathcal {B} _ {i})} \right] + \mathbb {E} _ {\mathbb {P}} \left[ \mathbb {1} _ {((\cap_ {i = 0} ^ {m} \mathcal {A} _ {i}) \cap (\cap_ {i = 0} ^ {m} \mathcal {B} _ {i})) ^ {c}} \right] \\ = \mathbb {P} \left[ \mathcal {E} _ {3} (s) ^ {c} \cap \left(\cap_ {i = 0} ^ {m} \mathcal {A} _ {i}\right) \cap \left(\cap_ {i = 0} ^ {m} \mathcal {B} _ {i}\right) \right] + \mathbb {P} \left[ \left(\left(\cap_ {i = 0} ^ {m} \mathcal {A} _ {i}\right) \cap \left(\cap_ {i = 0} ^ {m} \mathcal {B} _ {i}\right)\right) ^ {c} \right] \\ \leq \mathbb {P} \left[ \mathcal {E} _ {3} (s) ^ {c} \mid (\cap_ {i = 0} ^ {m} \mathcal {A} _ {i}) \cap (\cap_ {i = 0} ^ {m} \mathcal {B} _ {i}) \right] + \mathbb {P} \left[ (\cap_ {i = 0} ^ {m} \mathcal {A} _ {i}) ^ {c} \right] + \mathbb {P} \left[ (\cap_ {i = 0} ^ {m} \mathcal {B} _ {i}) ^ {c} \right], \\ \end{array} +$$ + +which implies that + +$$ +\mathbb {P} \left[ \mathcal {E} _ {3} (s) \right] \geq 2 (1 - s) ^ {m} - s - 1. +$$ + +Define $\delta(s)$ as + +$$ +\delta (s) = s + 2 \left(1 - (1 - s) ^ {m}\right) = \mathcal {O} (m s). \tag {109} +$$ + +Then, with probability at least $1 - \delta(s)$ , we have + +$$ +\begin{array}{l} \left\| \theta^ {(m, n)} - \theta_ {\star} \right\| _ {2} ^ {2} \lesssim s ^ {- 1} \left(1 - \frac {\mu \eta}{2}\right) ^ {m n} \left\| \theta^ {(0)} - \theta_ {\star} \right\| _ {2} ^ {2} \\ + s ^ {- 1} \left(m \exp \left(- \frac {\mu \eta n m}{4}\right) + \left(s ^ {- 2 / c} m ^ {- \frac {2 \alpha - 1}{\alpha}} + \eta\right) ((\eta n) ^ {- 1} + 1)\right). \\ \end{array} +$$ + +When $(\eta n)^{-1} = \mathcal{O}(1)$ , it follows that + +$$ +\| \theta^ {(m, n)} - \theta_ {\star} \| _ {2} ^ {2} \lesssim s ^ {- 1} \left(1 - \frac {\mu \eta}{2}\right) ^ {m n} \| \theta^ {(0)} - \theta_ {\star} \| _ {2} ^ {2} + s ^ {- 1} \left(s ^ {- 2 / c} m ^ {- \frac {2 \alpha - 1}{\alpha}} + n ^ {- 1} + \eta\right). +$$ + +# G Numerical Experiments + +This section provides numerical experiments of the proposed stochastic methods in this paper. In Appx. G.1, we design a numerical experiment to illustrate our orthogonalization method. In Appx. G.2, we design simulations based on a partially linear model. In Appx. G.3, we conduct a real data analysis with synthetic outcome to evaluate the performance of our methods. Code for reproduction can be found at https://fachengyu.github.io/. + +# G.1 Numerical Illustration + +In this section, we design a numerical experiment to illustrate how our orthogonalization method effects the target estimation as shown in Fig. 1 from the main text. + +Settings. Consider $\Theta \in R$ and $\mathcal{G} = \mathbb{R}$ . Let $L(\theta, g)$ be a real-valued risk function defined as + +$$ +L (\theta , g) := L (u) = \frac {1}{2} \langle u, A u \rangle + \lambda \sin^ {2} (\langle u, B u \rangle), \tag {110} +$$ + +where $u = (\theta, g)^\top \in \mathbb{R}^2$ , $\lambda = 0.02$ is the regularization parameter, and + +$$ +A = \left( \begin{array}{c c} 8 & 3 \\ 3 & 2 \end{array} \right) \succ 0 \text {a n d} B = \left( \begin{array}{c c} 2 & - 1 \\ - 1 & 1. 5 \end{array} \right) \succ 0. +$$ + +It is easy to see that $(0,0)$ is the global minimizer of $L$ since $L(\theta ,g)\geq 0$ . Let $q(u) = \langle u,Bu\rangle$ . The gradient w.r.t. $u$ is + +$$ +\begin{array}{l} \nabla_ {u} L (u) = A u + 4 \lambda \sin (q (u)) \cos (q (u)) B u \\ = (A + 2 \lambda \sin (2 q (u)) B) u. \\ \end{array} +$$ + +Since $A + 2\lambda \sin(2q(u))B \succcurlyeq A - 0.04B \succcurlyeq 0$ , it is clear that $(0,0)$ is the only stationary point, implying that $(0,0)$ is the only minimizer of $L$ . Furthermore, we can obtain the Hessian w.r.t. $u$ as + +$$ +\nabla_ {u} ^ {2} L (u) = A + 2 \lambda \sin (2 q (u)) B + 8 \lambda \cos (2 q (u)) B u (B u) ^ {\top}, \tag {111} +$$ + +which implies that $L(\cdot, g)$ is not convex in $\mathbb{R}$ given any $g \in \mathcal{G}_r(g_0)$ . However, when $\Theta$ is a small neighborhood around zero, it is still possible to have $L(\cdot, g)$ strongly convex for in $\Theta$ given any $g \in \mathcal{G}_r(g_0)$ . + +Orthogonalization. To orthogonalize $L$ , we first derive the orthogonal gradient oracle using (11), and then integral the oracle w.r.t. $\theta$ to obtain the orthogonalized loss $L_{\mathrm{no}}$ . + +Let $H$ be the Hessian at $(0,0)$ . By (111), we know that $H = A$ , implying $H_{\theta g} = A_{12}$ and $H_{gg} = A_{22}$ . Since the gradient w.r.t. $\theta$ satisfies + +$$ +\begin{array}{l} \nabla_ {\theta} L (\theta , g) = [ 1, 0 ] (A + 2 \lambda \sin (2 q (u)) B) u \\ = \left(A _ {1 1} + 2 \lambda \sin (2 q (\theta , g)) B _ {1 1}\right) \theta + \left(A _ {1 2} + 2 \lambda \sin (2 q (\theta , g)) B _ {1 2}\right) g, \\ \end{array} +$$ + +and the gradient w.r.t. $g$ satisfies + +$$ +\begin{array}{l} \nabla_ {g} L (\theta , g) = [ 0, 1 ] (A + 2 \lambda \sin (2 q (u)) B) u \\ = \left(A _ {2 1} + 2 \lambda \sin (2 q (\theta , g)) B _ {2 1}\right) \theta + \left(A _ {2 2} + 2 \lambda \sin (2 q (u)) B _ {2 2}\right) g, \\ \end{array} +$$ + +follow the construction of (11) and we obtain the orthogonal gradient oracle as + +$$ +\begin{array}{l} S _ {\mathrm {n o}} (\theta , g) = \nabla_ {\theta} L (\theta , g) - H _ {\theta g} H _ {g g} ^ {- 1} \nabla_ {g} L (\theta , g) \\ = (a + 2 b \lambda \sin (2 q (\theta , g))) \theta + 2 c \lambda \sin (2 q (\theta , g)) g, \\ \end{array} +$$ + +where $a = A_{11} - A_{12}A_{22}^{-1}A_{21}$ , $b = B_{11} - A_{12}A_{22}^{-1}B_{21}$ , and $c = B_{12} - A_{12}A_{22}^{-1}B_{22}$ . Finally, we can integral $S_{\mathrm{no}}(\theta ,g)$ w.r.t. $\theta$ and recover the orthogonalized loss $L_{\mathrm{no}}$ as + +$$ +L _ {\mathrm {n o}} (\theta , g) = \int_ {0} ^ {\theta} S _ {\mathrm {n o}} (s, g) \mathrm {d} s. +$$ + +Numerical Computation. Usually, $S_{\mathrm{no}}(s,g)$ contains a form of integral, which needs to be numerically computed. For the example introduced above, we can simplify $L_{\mathrm{no}}(\theta ,g)$ to stabilize the numerical computation. Note that $\nabla_{\theta}\sin^{2}(q(\theta ,g)) = \sin (2q(\theta ,g))(B_{11}\theta +B_{12}g)$ . Then + +$$ +\begin{array}{l} 2 b \lambda \int_ {0} ^ {\theta} \sin (2 q (s, g)) s d s = \frac {2 b \lambda}{B _ {1 1}} \left(\int_ {0} ^ {\theta} \sin (2 q (s, g)) (B _ {1 1} s + B _ {1 2} g) d s - B _ {1 2} g \int_ {0} ^ {\theta} \sin (2 q (s, g)) d s\right) \\ = \frac {2 b \lambda}{B _ {1 1}} \left(\sin^ {2} (q (\theta , g)) - B _ {1 2} g \int_ {0} ^ {\theta} \sin (2 q (s, g)) \mathrm {d} s\right). \\ \end{array} +$$ + +It follows that the orthogonalized loss $L_{\mathrm{no}}$ admits the following form + +$$ +L _ {\mathrm {n o}} (\theta , g) = \frac {a}{2} \theta^ {2} + \frac {2 b \lambda}{B _ {1 1}} \sin^ {2} (q (\theta , g)) + 2 \left(c - \frac {B _ {1 2}}{B _ {1 1}} b\right) \lambda g \int_ {0} ^ {\theta} \sin (2 q (s, g)) \mathrm {d} s, +$$ + +which implies that only the integral of $\sin (2q(s,g))$ w.r.t. $s$ needs to be computed. + +# G.2 Simulations + +# G.2.1 Data Generating Process + +To demonstrate Thm. 1 and Thm. 3, we revisit the partially linear model and the corresponding orthogonal and non-orthogonal losses in Appx. B.1. Specifically, $(X,W,Y)\in \mathbb{R}^d\times \mathbb{R}^d\times \mathbb{R}$ satisfies the following partially linear model where the nonlinear function is determined by the distribution of $(W,U)\in \mathbb{R}^d\times \mathbb{R}$ : + +$$ +Y = \left\langle \theta_ {0}, X \right\rangle + \alpha_ {0} (W) + \epsilon , \tag {112} +$$ + +$$ +U = \alpha_ {0} (W) + \xi , \tag {113} +$$ + +where $\theta_0\in \mathbb{R}^d$ is the true parameter, $\alpha_0:\mathcal{W}\mapsto \mathbb{R}$ is the true nonlinear function, $\epsilon$ and $\xi$ are independent noises that satisfy $\mathbb{E}[\epsilon \mid X,W] = 0$ and $\mathbb{E}[\xi \mid W] = 0$ . It is clear that $\alpha_0(W) = \mathbb{E}[U\mid W]$ . In our simulations, we choose $d = 2$ and $\theta_0 = [-0.51]^{\top}$ . + +To get samples for simulations, we first generate $(X,W)$ under the Gaussian model + +$$ +\left[ \begin{array}{c} X \\ W \end{array} \right] = \mathcal {N} \left(\left[ \begin{array}{c} \boldsymbol {\mu} _ {X} \\ \boldsymbol {\mu} _ {W} \end{array} \right], \left[ \begin{array}{c c} (1 + \delta) \mathbf {I} _ {2} & \lambda \mathbf {I} _ {2} \\ \lambda \mathbf {I} _ {2} & (1 + \delta) \mathbf {I} _ {2} \end{array} \right]\right), \tag {114} +$$ + +where $\pmb{\mu}_{X} = [1 1]^{\top}$ , $\pmb{\mu}_{W} = [2 2]^{\top}$ , $\mathbf{I}_2 \in \mathbb{R}^{2 \times 2}$ is the identity matrix, $\lambda \in [0,1]$ is used to control the correlation between $X$ and $W$ , and $\delta = 0.05$ is used to prevent the degeneration of the covariance matrix. For simplicity, we define the nonlinear function $\alpha_0$ as + +$$ +\alpha_ {0} (w) = 0. 5 \times \cos \left(\frac {w _ {1} + w _ {2}}{2}\right) + 0. 5 \times \sin \left(\frac {w _ {1} + w _ {2}}{2}\right), \tag {115} +$$ + +where $w = [w_{1} w_{2}]^{\top} \in \mathbb{R}^{2}$ . We then generate $Y$ and $U$ using independent Gaussian noises $\epsilon \sim \mathcal{N}(0,1)$ and $\xi \sim \mathcal{N}(0,1)$ based on (112) and (113), respectively. + +# G.2.2 Stochastic Gradient Oracles + +To estimate the true parameter $\theta_0$ using stochastic gradients, we need to design a correspond loss whose minimizer $\theta_{\star}$ is equal to $\theta_0$ . Based on Appx. B.1, there are two types of loss, the orthogonal loss and the non-orthogonal loss, available for this goal. We will derive the stochastic gradient oracle for these two losses and further derive the orthogonalized gradient oracle for the non-orthogonal loss. + +Orthogonal loss. Recall the orthogonal loss in Appx. B.1.1: + +$$ +\ell (\theta , g; z) = \frac {1}{2} [ y - g _ {Y} (w) - \langle \theta , x - g _ {X} (w) \rangle ] ^ {2}, \tag {116} +$$ + +where $g = (g_{Y}, g_{X}) : \mathcal{W} \to \mathbb{R} \times \mathbb{R}^{d}$ and the norm $\| \cdot \|_{\mathcal{G}}$ is defined in (27). The true nuisance for this loss is $g_{0} = (g_{0,X}, g_{0,Y})$ , where $g_{0,Y}(w) := \mathbb{E}_{\mathbb{P}}[Y \mid W = w]$ and $g_{0,X}(w) := \mathbb{E}_{\mathbb{P}}[X \mid W = w]$ . + +In fact, the explicit expression for $g_{0}$ can be easily obtained as + +$$ +g _ {0, Y} (w) = \left\langle \theta_ {0}, g _ {0, X} (w) \right\rangle + \alpha_ {0} (w), \tag {117} +$$ + +$$ +g _ {0, X} (w) = \mu_ {X} + \frac {\lambda}{1 . 0 5} (w - \mu_ {W}). \tag {118} +$$ + +From (112) and (117), it is clear that + +$$ +Y - g _ {0, Y} (W) = \left\langle \theta_ {0}, X - g _ {0, X} (W) \right\rangle + \epsilon , +$$ + +which implies that $\theta_{\star} = \theta_0$ by Lem. 4. The stochastic gradient oracle for the orthogonal loss (116) is then defined as + +$$ +S _ {\theta} (\theta , g; z) = - \left(y - g _ {Y} (w) - \langle \theta , x - g _ {X} (w) \rangle\right) \left(x - g _ {X} (w)\right). \tag {119} +$$ + +Non-orthogonal loss. We also provide the non-orthogonal loss in Appx. B.1.2 as + +$$ +\ell (\theta , g; z) = \frac {1}{2} [ y - g (w) - \langle \theta , x \rangle ] ^ {2}, \tag {120} +$$ + +where $g:\mathcal{W}\mapsto \mathbb{R}$ and the norm $\| \cdot \|_{\mathcal{G}}$ is now defined in (36). The true nuisance for this non-orthogonal loss satisfies + +$$ +g _ {0} (w) = \alpha_ {0} (w) = \mathbb {E} [ U \mid W = w ]. \tag {121} +$$ + +By Lem. 5, we have $\theta_{\star} = \theta_0$ . The stochastic gradient oracle for the orthogonal loss (120) is then defined as + +$$ +S _ {\theta} (\theta , g; z) = - (y - g (w) - \langle \theta , x \rangle) x. \tag {122} +$$ + +Orthogonalized gradient oracle. Since we perform orthogonalization on the non-orthogonal loss, we have $\theta_{\star} = \theta_0$ being the same target parameter. By (22) in Appx. B.1.2, the Neyman orthogonalized gradient oracle for this non-orthogonal loss (120) is given by + +$$ +S _ {\mathrm {n o}} (\theta , g; z) = - (y - g (w) - \langle \theta , x \rangle) (x - \mathbb {E} [ X \mid W = w ]). \tag {123} +$$ + +# G.2.3 Estimation Methods + +Throughout the experiments, we estimate the nuisances and the orthogonalizing operator using full-batch data and stream data, respectively. + +Nuisance estimation. Note that the true nuisances for the orthogonal loss and the non-orthogonal loss are conditional expectation given $W$ . To conduct nonparametric regression, we use random Fourier feature (RFF) [Rahimi and Recht, 2007] using the kernel $w \mapsto \exp \left(-\gamma \cdot \| w\| _2^2\right)$ to generate a randomized feature map for $W$ . + +The nuisance estimation procedure for obtaining $\hat{g}^{(m)}$ using full batch data can be summarized as + +1. Fit RFF sampler with 20 components using m i.i.d. samples from $P_{W|\lambda}$ + +2. Fit Ridge regressions where the regularization parameter is set to be $0.01 / m$ . Specifically, + +- For the orthogonal loss, fit two Ridge regressions using $m$ i.i.d. samples from the joint distribution $P_{X,W,Y|\lambda}$ and the fitted RFF sampler to coordinate-wisely estimate $\mathbb{E}[X\mid W]$ . With the same data, fit one Ridge regression using the fitted RFF sampler to estimate $\mathbb{E}[Y\mid W]$ . +- For the non-orthogonal loss, fit one Ridge regression using $m$ i.i.d. samples from the joint distribution $P_{X,W,Y|\lambda}$ and the fitted RFF sampler to estimate $\mathbb{E}[U \mid W]$ . + +To estimate nuisances using stream data, instead of fit a Ridge regression each time, we perform SGD for the Ridge regression loss. The procedure can be summarized as + +1. Initialize RFF sampler with 20 components using $n_0$ i.i.d. samples $(W_i)_{i=1}^{n_0}$ from $P_{W|\lambda}$ . +2. Perform SGD update once observing a mini-batch of i.i.d. samples from the joint distribution $P_{X,W,Y|\lambda}$ with size $n_g$ . Specifically, + +- For the orthogonal loss, perform two SGD with the Ridge loss for $m$ iterations to estimate $\mathbb{E}[X \mid W]$ coordinate-wisely. With the same data perform another SGD with the Ridge loss for $m$ iterations to estimate $\mathbb{E}[Y \mid W]$ . +- For the non-orthogonal loss, perform one SGD with the Ridge loss for $m$ iterations to estimate $\mathbb{E}[U \mid W]$ . + +Orthogonalizing operator estimation. To approximate the orthogonalizing operator $\Gamma_0$ , it suffices to estimate $\mathbb{E}[X|W]$ by (21). To that end, we use the same method as the nuisance estimation. The orthogonalizing operator estimation procedure for obtaining $\hat{\Gamma}^{(k)}$ can be summarized as + +1. Fit RFF sampler with 20 components using $k$ i.i.d. samples $(W_{i}^{\prime})_{i = 1}^{k}$ from $P_{W|\lambda}$ . +2. Fit two Ridge regressions with the regularization parameter being $0.01 / k$ using the fitted RFF sampler and another $k$ i.i.d. samples $(X_{i}^{\prime},W_{i}^{\prime})_{i = k}^{2k}$ to coordinate-wisely estimate $\mathbb{E}[X\mid W]$ . + +Target estimation. After the estimation of nuisances and orthogonalizing operator, we perform stochastic gradient descent (SGD) to estimate $\theta_{\star}$ using each of the three stochastic gradient oracles in (119), (122), and (123) on $n$ i.i.d. samples drawn from the joint distribution $P_{X,W,U,Y}$ . The learning rates of all the three SGDs are fixed during the training. + +# G.2.4 Simulation Results + +Setup. For each nuisance estimation setting, we study three types of estimation methods for learning $\theta_0$ established in this paper: (1) (orthogonal loss) obtain nuisance estimator $\hat{g}^{(m)} = (\hat{g}_Y^{(m)},\hat{g}_X^{(m)})$ of (117) and (118) and then perform SGD to obtain $\theta^{(n)}$ using the gradient oracle (119) after plugging in $\hat{g}^{(m)}$ ; (2) (non-orthogonal loss) obtain the nuisance estimator $\hat{g}^{(m)} = \hat{\alpha}^{(m)}$ of (121) and then perform SGD to obtain $\theta^{(n)}$ using the gradient oracle (122) after plugging in $\hat{g}^{(m)}$ ; (3) (OSGD) obtain the nuisance estimator $\hat{g}^{(m)}$ of (121) and the orthogonalizing operator estimator $\hat{\Gamma}^{(k)}$ of (21), and then perform SGD to obtain $\theta^{(n)}$ using the gradient oracle (123) after plugging in $\hat{g}^{(m)}$ and $\hat{\Gamma}^{(k)}$ . Each method is independently repeated 20 times. For nuisance estimated using stream data, we allow the procedure repeated by plugging in updated nuisance estimators and an updated operator estimator, where the nuisances get updated for 2000 iterations after every 2000 target SGD iterations. + +Evaluation. We evaluate the performance of nuisance estimators using the corresponding norms defined in (27) and (36). Specifically, for method (1), we evaluate the nuisance estimation by + +$$ +\left\| \hat {g} ^ {(m)} - g _ {0} \right\| _ {\mathcal {G}} = \max \left\{\mathbb {E} \left[ \left\| \hat {g} _ {X} ^ {(m)} (W) - g _ {0, X} (W) \right\| _ {2} ^ {\frac {4}{2}} \right] ^ {\frac {1}{4}}, \mathbb {E} \left[ (\hat {g} _ {Y} ^ {(m)} (W) - g _ {0, Y} (W)) ^ {4} \right] ^ {\frac {1}{4}} \right\}. \tag {124} +$$ + +For method (2) and (3), we evaluate the nuisance estimation by + +$$ +\left\| \hat {g} ^ {(m)} - g _ {0} \right\| _ {\mathcal {G}} = \mathbb {E} \left[ \left\| \hat {\alpha} ^ {(m)} (W) - \alpha_ {0} (W) \right\| _ {2} ^ {2} \right] ^ {\frac {1}{2}}. \tag {125} +$$ + +We evaluate $\hat{\Gamma}^{(k)}:g\mapsto \mathbb{E}[\hat{g}_X^{(k)}(W)g(W)]$ in method (3) using the Frobenius norm $\| \hat{\Gamma}^{(k)} - \Gamma_0\|_{\mathrm{Fro}}$ which is defined as + +$$ +\left\| \hat {\Gamma} ^ {(k)} - \Gamma_ {0} \right\| _ {\mathrm {F r o}} = \mathbb {E} \left[ \left\| \hat {g} _ {X} ^ {(k)} (W) - \hat {g} _ {0, X} (W) \right\| _ {2} ^ {2} \right] ^ {\frac {1}{2}}. \tag {126} +$$ + +Finally, we evaluate the target estimation using two kinds of criterion: (a) the relative error $\frac{\|\theta^{(n)} - \theta_0\|_2}{\|\theta_0\|_2}$ , and (b) the risk $L(\theta^{(n)},g_0) - L(\theta_\star ,g_0)$ where $L(\theta ,g) = \mathbb{E}[\ell (\theta ,g;Z)]$ . For method (1), $\ell (\theta ,g;z)$ is the orthogonal loss defined in (116) while for method (2) and (3), $\ell (\theta ,g;z)$ is the non-orthogonal loss defined in (120). + +Results using nuisances fitted on full-batch data. We first estimate the target using prefitted nuisances and operator. The estimation errors of nuisances and the operator fitted on full-batch data are shown in Fig. 2, where all estimation converges when the sample size $m$ increases and less samples are usually required to obtain the same error level when $\lambda$ increases. + +![](images/e1115a156eac01da1673f530a9f4eff26322850a8a0e6a370e7b7e81478f1dea.jpg) +Figure 2: The Nuisance and Orthogonalizing Operator Fitted on Full-Batch Simulated Data. The y-axis measures the corresponding error defined in (124) - (126) and the x-axis displays the sample size of data used to estimate the nuisance and operator. + +![](images/b01f010a49d2dcc854cd7250c451d4f1b61e1f16a5f315bc8c49ed0a404a9d18.jpg) + +![](images/5bfbc3e3f2f220a4ad9db5112c5478d5e8647b860304e42d8bc0f0d87c7ef654.jpg) + +![](images/aecb091f30107399a5b87091ac2f7d10230a6808a35f17a5578bc8f8979bae45.jpg) + +![](images/74bcafaa1a1cb0cedb31b0744d8017d219a07829c6d287f9c7a3c1b035eee3aa.jpg) + +![](images/d7649c71ac9f07a77061323a9d0c90e4a2d177ae52969c505e7dfcb224c1b824.jpg) + +![](images/0245772224f662c430c5caecb8c6bf944af87788fcfaeed474104b6f5a5b809d.jpg) +Figure 3: SGD for Orthogonal Loss with the Nuisance Fitted on Full-Batch Simulated Data. The x-axis represents the SGD iteration. Top: The y-axis measures the relative error. Bottom: The y-axis measures the risk. + +![](images/33403d15c56099c4e08ab85e2ce40bc9033f58cd8164e896bd63b64aba4a95ec.jpg) + +![](images/e5785f097b1e6f9459978e9145717d2b13afa272caea5f552678ef5a87271f69.jpg) + +![](images/20acfadb3971d480c4b005b2b215aab5ee9980e5daaeb1741392a0d2b33241a0.jpg) + +The performances of SGDs using prefitted nuisances and stochastic gradient oracles (119), (122), and (123) are shown in Fig. 3, Fig. 4, and Fig. 5, respectively. These figures suggest that when $\lambda$ increases, i.e., the correlation between $X$ and $W$ increases, usually more iterations are required to have SGD converged due to the difficulty of separating the effect of $X$ from $W$ . In addition, a well prefitted nuisance estimator would largely reduce the SGD estimation error, which aligns with Thm. 1. This improvement would be more obvious as $\lambda$ increases. Fig. 5 also shows that either using a well estimated nuisance or a well estimated orthogonalizing operator can improve the OSGD performance, and that OSGD using both well prefitted nuisance and operator would perform nearly the same as OSGD using the true nuisance and the true operator. + +Results using nuisances fitted on stream data. We then study the interleaving the nuisance and target estimations discussed in Appx. F.3. Here, Both the nuisance and the operator are learned using the same data stream and the results are shown in Fig. 6. Compared with Fig. 2, nuisances estimated using stream data usually has larger error and need more iterations to converge due to mini-batch, learning rate, and other tuning parameters. + +The performances of SGDs by interleaving nuisance and target updates with stochastic gradient oracles (119), (122), and (123) are shown in Fig. 7, Fig. 8, and Fig. 9, respectively. For all the three stochastic gradients, when $\lambda$ increases, the relative errors of the target SGD always get larger and their convergence rates become slower. There are obvious errors for SGDs using gradient oracles (119) and (122) in Fig. 7 and Fig. 8 since nuisances are not well estimated. However, OSGD + +![](images/c7e08788a57bae85f82de3fda7ad58439fcef411dfc571f10ede31621cd78b62.jpg) + +![](images/0c1e88f3aaa8340c719f461d6739a835d1aa8f5f16d217f00195da02e42026d1.jpg) + +![](images/96b545929f63309eb84fea09a93f59d37a3d8abc1799798aabdd3a35e22bd928.jpg) + +Figure 4: SGD for Non-Orthogonal Loss with the Nuisance Fitted on Full-Batch Simulated Data. The x-axis represents the SGD iteration. Top: The y-axis measures the relative error. Bottom: The y-axis measures the risk. +![](images/ac6025358eaf094f451ca6161cf698968e6c6b019dd0f5dc756da5a3e4bd942b.jpg) +$\hat{g}^{(m_1)}, m_1 = 500 \quad \hat{g}^{(m_2)}, m_2 = 10000 \quad g_0$ + +![](images/5792e66b8041dcde4c2061ccfd5f35f99a194724d2d33f2893791a10d6114d29.jpg) + +![](images/044809e09bbde9972936b3418fd525c751de5e579c97972a14d6b1b2fe81094b.jpg) + +![](images/d984c4b86258097b3dae8e59df62d8243a559d793e550454a562b6a52a91c3aa.jpg) + +![](images/50e424685fe7ffca9f0f5ae49f3654ed5b97e1912454fb3609c643c3d903856d.jpg) + +![](images/cef3d1fb52994a574d1c4507ee70311122af3d8d41dd0c48be84eb971bcec198.jpg) + +Figure 5: OSGD with the Nuisance and Operator Fitted on Full-Batch Simulated Data. Here, $m_{1} = 500$ , $m_{2} = 10000$ , $k_{1} = 300$ , $k_{2} = 10000$ . The x-axis represents the OSGD iteration. Top: The y-axis measures the relative error. Bottom: The y-axis measures the risk. +![](images/adacb60c1c11d1b0746c5b1a5e1364ecd313b3cbf64b9c61b705ae53ee2ebd53.jpg) +n n g(m1),g(k1) g(m1),g(k2) g(m2),g(k1) g(m2),g(k2) + +![](images/cd65de47601c67ca6a711cd8e68f812d492dd96daa70a2c6b91f0704256dd271.jpg) + +![](images/cd7c0ee5a0e5f8f51bb08803a5ec6c209d04d1d5fa1e73cf4e4b99d1b56d38ad.jpg) + +Figure 6: Nuisance and Orthogonalizing Operator Fitted on Simulated Stream Data. The y-axis measures the corresponding error defined in (124) - (126) and the x-axis displays the sample size of data used to estimate the nuisance and operator. +![](images/2f0b65c5fd9bea748c894ece200a772ed7d99f1525ea215eef11fee4a786ccca.jpg) +$\lambda = 0.0$ $\lambda = 0.5$ $\lambda = 1.0$ + +![](images/aba2dbb526b9a7b55adfbc405aab9980c4c4728bfb3869901a8e989b9f4cf338.jpg) + +![](images/0008f0d2c609c1ee5ea808b14a37d1184fb9f7fa335f565ec59c1190cf513564.jpg) + +![](images/5083d50efbe846419514001ab36b154aaaf1120403e3f7dc68150090ed590873.jpg) + +![](images/b51179bd9625b388c6240ccfe23d7ee5988cb70f40d06f77b1885bc414b386bc.jpg) + +![](images/3ad273953f977c9f5f117b41694cd1913c57a1609e0b7a01668c4fee47b503b5.jpg) + +![](images/89659be3447b3176ab57fbe2a11fe818400535682298f8087beacfde054cbdfa.jpg) + +![](images/07117b2a1b73252640fd22d29dba1970124dc10444283c04bd062013714766df.jpg) +{g(m):m≥1} go + +![](images/c0e6918153427d770dce12b5eb29f144726320442ee9597088446cbd4a33e179.jpg) + +![](images/a12fc00c4a211d2a921b00f3fe2878638cdfb79d5e7237421289dbebf52be00b.jpg) +Figure 7: SGD for Orthogonal Loss with the Nuisance Fitted on Simulated Stream Data. The x-axis represents the SGD iteration. Top: The y-axis measures the relative error. Bottom: The y-axis measures the risk. + +![](images/1d2d32e35c4720d28baacb7cfecc808c73eaa49359241d6a2a01b2dc68e2352e.jpg) + +![](images/f8acd8079665a54ff51c2ee210aad18b633e88089d8f0d99b612b4f5fb2db625.jpg) + +![](images/57a355868e34e5a05c464e5b2cd9c0d3c06a2b29764628470c94682ae01022ea.jpg) +Figure 8: SGD for Non-Orthogonal Loss with the Nuisance Fitted on Simulated Stream Data. The x-axis represents the SGD iteration. Top: The y-axis measures the relative error. Bottom: The y-axis measures the risk. + +![](images/bddfa603308b50f40b7be69fd04d4ab31e8374f1de3dbca151d6cef53d1f17df.jpg) + +![](images/4708258018f865a7a6f2b74deb78e6a03f6d2a50d24d8142d34d110454ea38a6.jpg) + +performs perfectly as shown in Fig. 9, which verifies the analysis of Thm. 3 that using an estimated orthogonalizing operator would reduce the bias from nuisance estimation. + +# G.3 Real Data Analysis + +We consider the Diabetes 130-Hospitals Dataset [Clore et al., 2014] as the real dataset example. We use six of these features as covariates, which are summarized in Tab. 4. We take the binary feature "change" as the input $X \in \{0,1\}$ and take the rest five features as the control $W \in \mathbb{R}^5$ . + +# G.3.1 Synthetic outcomes + +To evaluate the performance of our proposed methods, we use the synthetic outcome instead of a real outcome to examine the performance of our proposed methods. Using the synthetic outcome is common in causal inference; see Nie and Wager [2021, Sec. 4.1]. In this real data analysis, we generate outcome according to the following partially linear model: + +$$ +Y = \tilde {\theta} _ {0} \cdot X + \tilde {\alpha} _ {0} (W) + 0. 5 \times \epsilon , \tag {127} +$$ + +$$ +U = \tilde {\alpha} _ {0} (W) + 0. 5 \times \xi , \tag {128} +$$ + +![](images/4fd1f86f9b607c6a5f7bb2fa057e8d785c9f95c9db58de87433394e363ce596f.jpg) + +![](images/afaced97ac207cf62acde6bd215bc32331b8c9a87814a15d6b61984dca0e9484.jpg) + +![](images/55745618d56e383852300a2e2fe248c6bd934f520649bd9e67502e9f74f3e5b9.jpg) + +![](images/18664d3a7ec04435938a1d5d40e7d24a735824b87c88924ca03f63e79d834777.jpg) +Figure 9: OSGD with the Nuisance Fitted on Simulated Stream Data. The x-axis represents the SGD iteration. Top: The y-axis measures the relative error. Bottom: The y-axis measures the risk. + +![](images/2fed770226ee1c7656a110a86b215477bd6845b5e4004fca64caca401fd3ed15.jpg) +{（m）：m≥1} 90 + +![](images/8780179b98725385b7085ebc4cb1fe2a4bc5ce1ad7c5c57e2cac14548bf7505b.jpg) + +
FeatureDescription
changeIndicates if there was a change in diabetic medications.
time_in_hospitalInteger number of days between admission and discharge.
num_lab_proceduresInteger number of lab tests performed during the encounter.
num_proceduresInteger number of procedures (other than lab tests) performed during the encounter.
num_medicationsInteger number of distinct generic names administered during the encounter.
number_diagnosticsInteger number of diagnoses.
+ +Table 4: Features used for real data analysis. + +where $\tilde{\theta}_0 = -1, \epsilon \sim \mathcal{N}(0,1)$ and $\xi \sim \mathcal{N}(0,1)$ are independent noises, and $\tilde{\alpha}_0: \mathbb{R}^5 \mapsto \mathbb{R}$ satisfies that for $w = (w_1, \ldots, w_5)$ , + +$$ +\tilde {\alpha} _ {0} (w) = 0. 5 \times \cos \left(5 ^ {- 1} \sum_ {i = 1} ^ {5} w _ {i}\right) + 0. 5 \times \sin \left(5 ^ {- 1} \sum_ {i = 1} ^ {5} w _ {i}\right). +$$ + +Similar to Appx. G.2.2, we have $\theta_{\star} = \tilde{\theta}_{0}$ in this case. + +# G.3.2 Real Data Results + +Setup. We consider the same three stochastic gradient oracles as Appx. G.2.2 and the same two nuisance estimation methods as Appx. G.2.3 except that we use logistic regression on full batch data and SGD of the logistic loss on stream data for estimating $\mathbb{E}[X\mid W]$ . The setup of SGD using prefitted nuisances for this real data analysis is the same as Appx. G.2.4. For nuisance estimated using stream data, we update nuisances for 100 iterations after every 500 target SGD iterations. + +Evaluation. Since the true nuisances $\mathbb{E}[X\mid W]$ and $\mathbb{E}[Y\mid W]$ are unknown, we evaluate the performance of nuisance estimation $\hat{g}^{(m)} = (\hat{g}_Y^{(m)},\hat{g}_X^{(m)})$ for the orthogonal loss by the mean squared error: + +$$ +\mathrm {M S E} _ {1} \left(\hat {g} ^ {(m)}\right) = \max \left\{\mathbb {E} _ {\mathbb {P}} \left[ \left(\hat {g} _ {Y} ^ {(m)} (W) - Y\right) ^ {2} \right], \mathbb {E} _ {\mathbb {P}} \left[ \left(\hat {g} _ {X} ^ {(m)} (W) - X\right) ^ {2} \right] \right\}. \tag {129} +$$ + +We adopt the nuisance estimation error $\| \hat{g}^{(n)} - g_0\|_{\mathcal{G}}$ defined in (125) as the nuisance evaluation for non-orthogonal loss due to the synthetic outcome, where now $g_0 = \tilde{\alpha}_0$ . For the operator estimation + +![](images/16a4731ebf9300eda57d16e39a4e75f90574cee9bba95bb6ac47fd4795b4a0ba.jpg) +Figure 10: Nuisance and Orthogonalizing Operator Fitted on Full-Batch Real Data. The x-axis displays the sample size of data used to estimate the nuisance and operator. Left. The y-axis measure the nuisance error defined in (125). Middle. The y-axis measure the nuisance estimation MSE defined in (129). Right. The y-axis measure the operator estimation MSE defined in (130). + +![](images/26899d61f5b689a14bebef947c74fcc0535ba3e352d31c95e60fdbcfa1233e13.jpg) + +![](images/77557c07157ba5084f738799a4652ab5c7cef01d63980ffaf5a9ff6e30422782.jpg) + +![](images/f9f3385c136a87aa9a7ed8c353dbae0789b5ce66fbf5f4730d94f24c05a5a745.jpg) + +![](images/1666d87f11bce75d8ed5bceab547fd7b90394b4042c8b1955e30734e340ebc8b.jpg) + +![](images/cbdb80d86ab2bd7e0ad8d4baf78add1abd349ae58b084467c4ee1daeb8e84f51.jpg) + +![](images/47664ad2b1e2ce57c4712a506a4e971627594b796d1fe764334c6d7eb647b389.jpg) +Figure 11: Stochastic Gradients with Nuisance Fitted on Full-Batch Real Data. Here, $m_{1} = 32$ , $m_{2} = 64$ , $m_{3} = 128$ , $m_{4} = 8$ , $m_{5} = 128$ , $k_{1} = 32$ and $k_{2} = 128$ . The x-axis represents the SGD iteration using corresponding stochastic gradient. Top: The y-axis measures the relative error. Bottom: The y-axis measures the risk. + +![](images/70096c0aea0fe6fd30b52dfd0df014a7024888a47d085965cb228138832c16ff.jpg) + +![](images/2428e4050eb5fc85cf4254ed91859b2855bdbb282c45e3e4f1ade2b61ce99975.jpg) + +$\hat{\Gamma}^{(m)}: g \mapsto \mathbb{E}[\hat{g}_X^{(m)}(W)g(W)]$ , evaluate its performance by the mean squared error: + +$$ +\mathrm {M S E} _ {2} \left(\hat {\Gamma} ^ {(m)}\right) = \mathbb {E} _ {\mathbb {P}} \left[ \left(\hat {g} _ {X} ^ {(m)} (W) - X\right) ^ {2} \right]. \tag {130} +$$ + +Results using nuisances fitted on full-batch data. We first estimate the target using prefitted nuisances and operator. The estimation errors of nuisances and the operator using full-batch real data are shown in Fig. 10, which suggests that the estimation of $\tilde{\alpha}_0$ converges to zero due to our design while there exists obvious bias for estimating the nuisance $(g_{0,X}, g_{0,Y})$ and the orthogonalizing operator $\Gamma_0$ possibly due to model misspecification. + +The performances of SGDs using prefitted nuisances and stochastic gradient oracles (119), (122), and (123) are shown in Fig. 11. Overall, the relative error and the risk are small when well estimated nuisances are used. In addition, both relative errors and risks become smaller when we use more samples to estimate nuisances for the orthogonal loss and the non-orthogonal loss. + +Results using nuisances fitted on stream data. We then estimate the target by interleaving the nuisance and target updates. Here, Both the nuisance and the operator are learned using the same + +![](images/f2416490e2ad002126b8ce0121e6ee304c3caa915b0109e7a70ec0e6d4a6f4ad.jpg) +Figure 12: Estimation Errors of Nuisance and Orthogonalizing Operator Fitted on Stream Data. The x-axis displays the sample size of data used to estimate the nuisance and operator. Left. The y-axis measure the nuisance error defined in (125). Middle. The y-axis measure the nuisance estimation MSE defined in (129). Right. The y-axis measure the operator estimation MSE defined in (130). + +![](images/408355485bbe2f3eaa515646e69976b4de5ebb28cf044ee440b90659e887bd5e.jpg) + +![](images/1d034b2eb500c7cf4e1021f57935c36017f47ab9652b77e6c5bee5cc84ef9fcf.jpg) + +![](images/026ed6b12582f364eb7e8071a29bf48e16c06c48c1aa68600f1c118052386150.jpg) + +![](images/8b9daccbe812b35fb1029a6297d4f9d8749452f4336c8502187f302f85d818ed.jpg) + +![](images/c372ed0ce2eb848b9e6a57c67bd86d531f9ed5e340ef2796837488bc3d1e5f43.jpg) + +Figure 13: Stochastic Gradients with Nuisance Fitted on Real Stream Data. The x-axis represents the SGD iteration. Top: The y-axis measures the relative error. Bottom: The y-axis measures the risk. +![](images/a95a99acb2864154082453edc43490465d38fec9a98b316dd435534239959992.jpg) +$\{\hat{g}^{(m)}:m\geq 1\}$ + +![](images/49d981d7a6e675c5037be43b5cc662e2a39e430cf899ce9c9188e382aa797fa4.jpg) +{（m）:m≥1} +g0 + +![](images/e20530bc887865a67e78bd03292f02cb88fa8b1fa07136d046c95c6459f79c8f.jpg) +$\{(\hat{g}^{(m)},\hat{\Gamma}^{(m)}):m\geq 1\}$ +$\{(\mathfrak{g}_0,\hat{\Gamma}^{(m)}):m\geq 1\}$ + +data stream and the results are shown in Fig. 12. Compared with Fig. 10, nuisances estimated using stream data converges similarly. + +The performances of SGDs by interleaving nuisance and target updates with stochastic gradient oracles (119), (122), and (123) are shown in Fig. 13. The figure on the left in Fig. 13 shows that the target estimation has small relative error using the estimated nuisance sequence $\{\hat{g}^{(m)}:m\geq 1\}$ . The figure in the middle suggests that there is still some bias for the target estimation while this bias is negligible. The figure on the right shows the performance of OSGD, where the relative error of OSGD using the estimated nuisance sequence is similar to OSGD using the true nuisance, which aligns with Thm. 3. + +# H Extension to SGD Variants + +In this section, we discuss strategies for analyzing other variants of SGD under nuisances. In Appx. H.1, we discuss the relationship between SGD with momentum and averaged SGD and provide a convergence analysis example of the averaged SGD. In Appx. H.2, we discuss Adam as a generalization of SGD with momentum and the difficulties to analyze the convergence rate of Adam. + +# H.1 SGD with Momentum and Averaged SGD + +For the gradient oracle sequence $S^{(n)}$ , SGD with momentum following the description of Li et al. [2022] can be expressed as + +$$ +m ^ {(n + 1)} = \beta_ {n} m ^ {(n)} + S ^ {(n)} \text {a n d} \bar {\theta} ^ {(n + 1)} = \bar {\theta} ^ {(n)} - \alpha_ {n} m ^ {(n)}, \tag {131} +$$ + +where $\bar{\theta}^{(n)}$ is the SGD estimation sequence, $m^{(n)}$ is the momentum sequence, and $(\alpha_{n})_{n\geq 0}$ and $(\beta_{n})_{n\geq 0}$ can be any positive sequence. The following example shows that the averaged SGD is a special case of SGD with momentum. + +Example 5 (Averaged SGD). Let $\beta_{n} = 1 / n$ and $\alpha_{n} = \eta (1 - \beta_{n + 1})$ for all $n\geq 1$ . The momentum updates implied by this sequence are + +$$ +m ^ {(n + 1)} = \frac {1}{n} m ^ {(n)} + S ^ {(n)} \text {a n d} \bar {\theta} ^ {(n + 1)} = \bar {\theta} ^ {(n)} - \eta \left(1 - \frac {1}{n + 1}\right) m ^ {(n)}, +$$ + +which implies that $\bar{\theta}^{(n + 1)}$ is the averaged SGD such that + +$$ +\bar {\theta} ^ {(n + 1)} = \frac {1}{n + 1} \sum_ {t = 0} ^ {n} \theta^ {(t)}. \tag {132} +$$ + +Example 5 demonstrates that the convergence rate of SGD with momentum can be analyzed in the same way as averaged SGD. While it is not the focus of this paper, we provide a convergence result of the averaged SGD based on the analysis of Defossez and Bach [2015]. + +Proposition 27 (Convergence rate of averaged SGD). Consider the partially linear model and the non-orthogonal loss $\ell(\theta, g; z)$ in Appx. B.1.2. Define $\mathcal{D}_n = (Z_1, \ldots, Z_n)$ , sampled from the product measure $\mathbb{P}^n$ . Choose the gradient oracle $S^{(n)}$ to be the score $S_{\theta}(\theta, \hat{g}; Z_n)$ where $\hat{g}$ is estimated independently of $\mathcal{D}_n$ . Let $\bar{\theta}^{(n)}$ be the averaged SGD defined in (132). Suppose the same assumptions as Lem. 5. If $0 < \eta < \eta_{\max}$ , then + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \| \bar {\theta} ^ {(n)} - \theta_ {\star} \| _ {2} ^ {2} \right] \lesssim \frac {1}{n} + \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2}, +$$ + +where $\eta_{\mathrm{max}} = \sup \{\eta >0:tr\left(A^{\top}\mathbb{E}_{\mathbb{P}}[XX^{\top}]A\right) - \eta \mathbb{E}_{\mathbb{P}}\left[(X^{\top}AX)^{2}\right] > 0,\forall A\in \mathcal{S}(\mathbb{R}^{d})\}$ and $\mathcal{S}(\mathbb{R}^d)$ is the set of all $d\times d$ symmetric matrices. + +Before we prove Prop. 27, recall the example of non-orthogonal loss for the partially linear model in Appx. B.1.2, where $Z = (X,W,Y) \sim \mathbb{P}$ satisfies + +$$ +Y = \left\langle \theta_ {0}, X \right\rangle + g _ {0} (W) + \epsilon , \quad \mathbb {E} _ {\mathbb {P}} [ \epsilon \mid X, W ] = 0. \tag {133} +$$ + +The target parameter $\theta_{\star} = \arg \min_{\theta \in \Theta}\mathbb{E}_{\mathbb{P}}[\ell (\theta ,g;Z)]$ where $\ell$ is the non-orthogonal loss defined as + +$$ +\ell (\theta , g; z) = \frac {1}{2} [ y - g (w) - \langle \theta , x \rangle ] ^ {2}, \tag {134} +$$ + +By Lem. 5, we have $\theta_{\star} = \theta_0$ . The stochastic gradient oracle for this non-orthogonal loss is + +$$ +S _ {\theta} (\theta , g; z) = - X (y - g (w) - \langle \theta , x \rangle), +$$ + +and the SGD iteration is defined by $\theta^{(0)}\in \Theta$ and + +$$ +\theta^ {(n)} = \theta^ {(n - 1)} - \eta S _ {\theta} (\theta , \hat {g}; Z _ {n - 1}) = \theta^ {(n - 1)} + \eta X _ {n - 1} \left(Y _ {n - 1} - \hat {g} \left(W _ {n - 1}\right) - \langle \theta , X _ {n - 1} \rangle\right), \tag {135} +$$ + +where $\hat{g} \in \mathcal{G}_r(g_0)$ is any nuisance estimator independent of $\{Z_i\}_{i=1}^n$ . Note that (135) can be written as + +$$ +\begin{array}{l} \theta^ {(n)} - \theta_ {\star} = \left(I - \eta X _ {n} X _ {n} ^ {\top}\right) \theta^ {(n - 1)} + \eta X _ {n} \left(Y _ {n} - \hat {g} \left(W _ {n}\right)\right) - \theta_ {\star} \\ = \left(I - \eta X _ {n} X _ {n} ^ {\top}\right) \left(\theta^ {(n - 1)} - \theta_ {\star}\right) + \eta X _ {n} \left(Y _ {n} - \hat {g} \left(W _ {n}\right) - X _ {n} ^ {\top} \theta_ {\star}\right) \\ = \left(I - \eta X _ {n} X _ {n} ^ {\top}\right) \left(\theta^ {(n - 1)} - \theta_ {\star}\right) + \eta X _ {n} \epsilon_ {n} - \eta X _ {n} \left(\hat {g} \left(W _ {n}\right) - g _ {0} \left(W _ {n}\right)\right). \\ \end{array} +$$ + +Let $\beta^{(n)} = \theta^{(n)} - \theta_{\star}, r_n = \hat{g}(W_n) - g_0(W_n)$ , and + +$$ +M _ {k, j} = \left(\prod_ {i = k + 1} ^ {j} \left(I - \eta X _ {i} X _ {i} ^ {\top}\right)\right) ^ {\top} \in \mathbb {R} ^ {d \times d}. +$$ + +By recursion, we have + +$$ +\begin{array}{l} \beta^ {(n)} = \left(I - \eta X _ {n} X _ {n} ^ {\top}\right) \beta^ {(n - 1)} + \eta X _ {n} \epsilon_ {n} - \eta X _ {n} r _ {n} \\ = M _ {0, n} \beta^ {(0)} + \eta \sum_ {k = 1} ^ {n} M _ {k, n} X _ {k} \epsilon_ {k} - \eta \sum_ {k = 1} ^ {n} M _ {k, n} X _ {k} r _ {k}. \\ \end{array} +$$ + +Let $\bar{\beta}^{(n)} = \bar{\theta}^{(n)} - \theta_{\star} = (n + 1)^{-1}\sum_{j = 0}^{n}\beta^{(j)}$ , we have + +$$ +\begin{array}{l} \bar {\beta} ^ {(n)} = \frac {1}{n + 1} \sum_ {j = 0} ^ {n} M _ {0, j} \beta^ {(0)} + \frac {\eta}{n + 1} \sum_ {k = 1} ^ {n} \left(\sum_ {j = k} ^ {n} M _ {k, j}\right) \left(X _ {k} \left(\epsilon_ {k} - r _ {k}\right) + \mathbb {E} \left[ X _ {k} r _ {k} \right]\right) \\ - \frac {\eta}{n + 1} \sum_ {k = 1} ^ {n} \left(\sum_ {j = k} ^ {n} M _ {k, j}\right) \mathbb {E} \left[ X _ {k} r _ {k} \right]. \\ \end{array} +$$ + +In the above formula, first two terms are usually interpreted as the bias term and the variance term under the true nuisance, respectively according to Defossez and Bach [2015], and the last term can be viewed as the error term caused by the nuisance estimation. + +To analyze the bias term and the variance term, we adopt the notations of Defossez and Bach [2015] for matrices and operators. First, Define $H = \mathbb{E}_{\mathbb{P}}[XX^{\top}]$ . Let $H_{L}$ (resp. $H_{R}$ ) be the matrix operator representing left multiplication (resp. right multiplication) by $H$ , and $T$ be the linear operator such that for any square matrix $M \in \mathbb{R}^{d \times d}$ , $TM = HM + MH - \eta \mathbb{E}_{\mathbb{P}}[(X^{\top}MX)XX^{\top}]$ . Let $\rho = \max \{ \| I - \eta H \|_{\mathrm{op}}, \| I - \eta T \|_{\mathrm{op}} \}$ where the operator norm $\| \cdot \|_{\mathrm{op}}$ is defined as the largest singular value. Finally, let $\eta_{\mathrm{max}}$ be the same as in Prop. 27. With definitions above, the asymptotic covariances of the bias and the variance follow directly from Theorems 1 and 2 of Defossez and Bach [2015, Appx. 3]. + +Lemma 28 (Asymptotic covariance of the bias). Let $\Xi_0 = \mathbb{E}_{\mathbb{P}}\left[\beta^{(0)}\beta^{(0)^T}\right]$ . If $0 < \eta < \eta_{\max}$ , then + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ B _ {n} B _ {n} ^ {\top} \right] = \frac {1}{n ^ {2} \eta^ {2}} \left(H _ {L} ^ {- 1} + H _ {R} ^ {- 1} - \eta I\right) \left(T ^ {- 1} \Xi_ {0}\right) + O \left(\frac {\rho^ {n}}{n} \| \Xi_ {0} \| _ {F}\right), +$$ + +where $B_{n} = \frac{1}{n + 1}\sum_{j = 0}^{n}M_{0,j}\beta^{(0)}$ . + +Lemma 29 (Asymptotic covariance of the variance). Let $\Sigma_0 = \mathbb{Var}(X_n(\epsilon_n - r_n) + \mathbb{E}[X_n r_n])$ . If $0 < \eta < \eta_{\max}$ , then + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ V _ {n} V _ {n} ^ {\top} \right] = \frac {1}{n} \left(H _ {L} ^ {- 1} + H _ {R} ^ {- 1} - \eta I\right) T ^ {- 1} \Sigma_ {0} \\ - \frac {1}{\eta n ^ {2}} \left(H _ {L} ^ {- 1} + H _ {R} ^ {- 1} - \eta I\right) (I - \eta T) T ^ {- 2} \Sigma_ {0} + O \left(\frac {\rho^ {n}}{n} \| \Sigma_ {0} \| _ {F}\right), \\ \end{array} +$$ + +where $V_{n} = \frac{\eta}{n + 1}\sum_{k = 1}^{n}\left(\sum_{j = k}^{n}M_{k,j}\right)\left(X_{k}\left(\epsilon_{k} - r_{k}\right) + \mathbb{E}\left[X_{k}r_{k}\right]\right)$ . + +In fact, the convergence rate of averaged SGD depends on $\operatorname{tr}\left(B_nB_n^\top\right)$ and $\operatorname{tr}\left(V_nV_n^\top\right)$ . When $\rho < 1$ , Lem. 28 demonstrate that $\operatorname{tr}\left(B_nB_n^\top\right)$ is of order $n^{-2}$ , while Lem. 29 shows that $\operatorname{tr}\left(V_nV_n^\top\right)$ is of order $n^{-1}$ , which is reasonable due to the randomness of the noise $\epsilon_k - r_k$ . + +For the error term, we can analyze it in a similar way to the bias term. Let $\Delta = \mathbb{E}_{\mathbb{P}}[X_n r_n]$ and + +$$ +E _ {n} = \frac {\eta}{n + 1} \sum_ {k = 1} ^ {n} \left(\sum_ {j = k} ^ {n} M _ {k, j}\right) \Delta . +$$ + +Note that by Jensen's inequality, + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ \| E _ {n} \| _ {2} ^ {2} \right] \leq \mathbb {E} _ {\mathbb {P}} \left[ \frac {\eta^ {2}}{n} \sum_ {k = 1} ^ {n} \left\| \left(\sum_ {j = k} ^ {n} M _ {k, j}\right) \Delta \right\| _ {2} ^ {2} \right] \\ = \frac {\eta^ {2}}{n} \sum_ {k = 1} ^ {n} \operatorname {t r} \left(\mathbb {E} _ {\mathbb {P}} \left[ \left(\sum_ {j = k} ^ {n} M _ {k, j} \Delta\right) \left(\sum_ {j = k} ^ {n} M _ {k, j} \Delta\right) ^ {\top} \right]\right). \\ \end{array} +$$ + +It is clear to see that the asymptotic covariance of $\Delta_{k,n} \coloneqq \sum_{j=k}^{n} M_{k,j} \Delta$ is of the same form as the bias term in Lem. 28. Let $G_0 = \mathbb{E}_{\mathbb{P}}[\Delta \Delta^\top]$ and we have + +$$ +\mathbb {E} _ {\mathbb {P}} \left[ \Delta_ {k, n} \Delta_ {k, n} ^ {\top} \right] = \frac {1}{\eta^ {2}} \left(H _ {L} ^ {- 1} + H _ {R} ^ {- 1} - \eta I\right) \left(T ^ {- 1} G _ {0}\right) + O \left((n - k) \rho^ {n - k} \| G _ {0} \| _ {F}\right). +$$ + +Thus, the trace of the covariance summation over $k = 1,\ldots ,n$ satisfies + +$$ +\frac {\eta^ {2}}{n} \sum_ {k = 1} ^ {n} \operatorname {t r} \left(\mathbb {E} _ {\mathbb {P}} \left[ \Delta_ {k, n} \Delta_ {k, n} ^ {\top} \right]\right) = \operatorname {t r} \left((H _ {L} ^ {- 1} + H _ {R} ^ {- 1} - \eta I) \left(T ^ {- 1} G _ {0}\right)\right) + O \left(\frac {\eta^ {2}}{n} \sum_ {k = 0} ^ {n - 1} k \rho^ {k} \| G _ {0} \| _ {F}\right). \tag {136} +$$ + +Gathering the bias term, the variance term, and the error term, we are now ready to proof Prop. 27. + +Proof of Prop. 27. By Lemma 1 of Defossez and Bach [2015], $0 < \rho < 1$ when $0 < \eta < \eta_{\max}$ . Suppose that $\| X \|_{\infty} \leq C_X$ almost surely. By Jensen's inequality we have that + +$$ +\| G _ {0} \| _ {2} = \| \mathbb {E} _ {\mathbb {P}} [ X _ {n} r _ {n} ] \| _ {2} ^ {2} \leq C _ {X} ^ {2} \mathbb {E} _ {\mathbb {P}} [ r _ {n} ] ^ {2} \leq C _ {X} ^ {2} \mathbb {E} _ {\mathbb {P}} [ r _ {n} ^ {2} ] = C _ {X} ^ {2} \| \hat {g} - g _ {0} \| _ {\mathcal {G}} ^ {2}. +$$ + +Note that + +$$ +\sum_ {k = 0} ^ {n - 1} k \rho^ {k} = \rho \frac {\mathrm {d}}{\mathrm {d} \rho} \left(\sum_ {k = 0} ^ {n - 1} \rho^ {k}\right) = \rho \frac {\mathrm {d}}{\mathrm {d} \rho} \left(\frac {1 - \rho^ {n}}{1 - \rho}\right) = \frac {\rho - n \rho^ {n} + (n - 1) \rho^ {n + 1}}{(1 - \rho) ^ {2}} = O (1). +$$ + +By Lem. 28, Lem. 29, and (136), we have + +$$ +\begin{array}{l} \mathbb {E} _ {\mathbb {P}} \left[ \| \bar {\theta} ^ {(n)} - \theta_ {\star} \| _ {2} ^ {2} \right] \lesssim \mathbb {E} _ {\mathbb {P}} \left[ \| B _ {n} \| _ {2} ^ {2} \right] + \mathbb {E} _ {\mathbb {P}} \left[ \| V _ {n} \| _ {2} ^ {2} \right] + \mathbb {E} _ {\mathbb {P}} \left[ \| E _ {n} \| _ {2} ^ {2} \right] \\ \lesssim \operatorname {t r} \left(\mathbb {E} _ {\mathbb {P}} \left[ B _ {n} B _ {n} ^ {\top} \right]\right) + \operatorname {t r} \left(\mathbb {E} _ {\mathbb {P}} \left[ V _ {n} V _ {n} ^ {\top} \right]\right) + \frac {\eta^ {2}}{n} \sum_ {k = 1} ^ {n} \operatorname {t r} \left(\mathbb {E} _ {\mathbb {P}} \left[ \Delta_ {k, n} \Delta_ {k, n} ^ {\top} \right]\right) \\ \lesssim \frac {1}{n} + | | \hat {g} - g _ {0} | | _ {\mathcal {G}} ^ {2}. \\ \end{array} +$$ + +![](images/9944f4a8ada76f392341f0d315e62cc3f26174b70993917b2938f93a5a321fd0.jpg) + +# H.2 Adam + +The primary updates for Adam under nuisance estimate $\hat{g}$ are given by the following recursive equations. Below, we let $i\in \{1,\dots ,d\}$ denote a particular dimension of the finite-dimensional parameter of interest. Following the description of Defossez et al. [2022], for the gradient oracle sequence $S^{(n)}$ the Adam generates the target estimator $\tilde{\theta}^{(n)}$ as below: + +$$ +m _ {i} ^ {(n)} = \beta_ {1} m _ {i} ^ {(n - 1)} + S _ {i} ^ {(n)} \tag {137} +$$ + +$$ +v _ {i} ^ {(n)} = \beta_ {2} v _ {i} ^ {(n - 1)} + \left(S _ {i} ^ {(n)}\right) ^ {2} \tag {138} +$$ + +$$ +\tilde {\theta} _ {i} ^ {(n)} = \tilde {\theta} _ {i} ^ {(n - 1)} - \eta \frac {m _ {i} ^ {(n)}}{\sqrt {\epsilon + v _ {i} ^ {(n)}}}, \tag {139} +$$ + +where $\beta_{2} \in (0,1]$ , $\beta_{1} \in [0,\beta_{2}]$ are the momentum and variance parameters, $m^{(n)}, v^{(n)} \in \mathbb{R}^{d}$ are the momentum and variance sequences, and $\epsilon > 0$ is a numerical stability parameter. Adam differs from the SGD with momentum by adding a variance sequence $v^{(n)}$ . When $v^{(n)}$ is chosen to be a constant, then (137) and (139) would reduce to the special case of SGD with momentum where $\beta_{n}$ and $\alpha_{n}$ are constant. + +The analysis of Adam is often done in the case of smooth non-convex optimization, in which it is shown that the gradient of the objective tends to zero [Ward et al., 2020, Defossez et al., 2022]. Specifically, Defossez et al. [2022] consider a momentum-free Adam $(\beta_{1} = 0)$ to analyze the essential ingredients that differ from momentum: the variance pre-conditioning and element-wise updates, which suggests that under the true nuisance, i.e., $S^{(n)} = S_{\theta}(\theta^{(n)},g_0;Z_n)$ for an i.i.d. sample $\{Z_i\}_{i = 1}^n\sim \mathbb{P}^n$ , the convergence result of Adam satisfies + +$$ +\mathbb {E} _ {\mathbb {P} ^ {n}} \left[ \| S _ {\theta} (\tilde {\theta} ^ {(n)}, g _ {0}) \| _ {2} ^ {2} \right] \lesssim \frac {1}{\sqrt {n}} \left(1 + \log \left(\frac {1}{\epsilon}\right)\right). +$$ + +Note that this result is not comparable to our convergence criterion (in terms of iterations), which differs non-trivially from a stationarity or function value analysis. 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In this paper, we focus on a challenging class of non-convex non-smooth FCCO, where the outer functions are non-smooth weakly convex or convex and the inner functions are smooth or weakly convex. Existing state-of-the-art result face two key limitations: (1) a high iteration complexity of $O(1 / \epsilon^6)$ under the assumption that the stochastic inner functions are Lipschitz continuous in expectation; (2) reliance on vanilla SGD-type updates, which are not suitable for deep learning applications. Our main contributions are two fold: (i) We propose stochastic momentum methods tailored for non-smooth FCCO that come with provable convergence guarantees; (ii) We establish a new state-of-the-art iteration complexity of $O(1 / \epsilon^5)$ . Moreover, we apply our algorithms to multiple inequality constrained non-convex optimization problems involving smooth or weakly convex functional inequality constraints. By optimizing a smoothed hinge penalty based formulation, we achieve a new state-of-the-art complexity of $O(1 / \epsilon^5)$ for finding an (nearly) $\epsilon$ -level KKT solution. Experiments on three tasks demonstrate the effectiveness of the proposed algorithms. + +# 1 Introduction + +In this paper, we consider the finite-sum coupled compositional optimization (FCCO) problem + +$$ +\min _ {\mathbf {w} \in \mathbb {R} ^ {d}} F (\mathbf {w}) := \frac {1}{n} \sum_ {i = 1} ^ {n} f _ {i} \left(g _ {i} (\mathbf {w})\right), \tag {1} +$$ + +where $f_{i}:\mathbb{R}^{d_{1}}\to \mathbb{R}$ is continuous, $g_{i}:\mathbb{R}^{d}\rightarrow \mathbb{R}^{d_{1}}$ is continuous and satisfies $g_{i}(\mathbf{w}) = \mathbb{E}g_{i}(\mathbf{w},\xi_{i})$ , and the expectation is taken over the random variable $\xi_{i}$ for $i = 1,\ldots ,n$ . + +FCCO has been effectively applied to optimizing a wide range of risk functions known as X-risks [1, 2, 3, 4], and group distributionally robust optimization (GDRO) [5, 6]. Recently, it was also applied to solving non-convex inequality constrained optimization problems [7, 8]. Several optimization algorithms have been developed for non-convex FCCO (1) under different conditions. + +Table 1: Comparison between our algorithms and prior works for the non-smooth non-convex FCCO problem. The complexity of SONX is for finding a nearly $\epsilon$ -stationary solution (Definition 3.1), and that of SONEX and ALEXR2 for smooth inner functions are for finding an approximate $\epsilon$ -stationary solution (Definition 4.1), which implies a nearly $\epsilon$ -stationary solution under a verifiable condition of the inner functions. The complexity of ALEXR2 for weakly convex inner functions is for finding a nearly $\epsilon$ -stationary solution to the outer smoothed objective. In this table, "WC" means weakly convex, "C" means convex, " $\nearrow$ " means monotonically non-decreasing, "SM" means smooth, "PMS" means that the proximal mapping can be easily computed, "MLC0" means that the function is mean Lipschitz continuous of zero-order (Assumption 4.6). + +
AlgorithmfigiComplexityLoopUpdate type
SONX [6]WC, ↕WC, MLC0O(ε-6)SingleSGD
SONEX (Ours)WC, PMSSM, MLC0O(ε-5)SingleMomentum or Adam
ALEXR2 (Ours)C, PMSSMO(ε-5)DoubleMomentum or Adam
ALEXR2 (Ours)C, PMS, ↕WCO(ε-5)DoubleMomentum or Adam
+ +Table 2: Comparison between our algorithms and prior works for solving non-convex inequality constrained optimization problem. $g_{0}$ is the objective and $g_{i}$ are constraint functions. + +
Algorithmg0giConstraint SamplingComplexityLoopUpdate type
OSS [9]WCWCNoO(ε-6)DoubleSGD
ICPPAC [10]WCWCNoO(ε-6)DoubleSGD
SSG [11]WCCNoO(ε-8)SingleSGD
Li et al. [7]SMSMYesO(ε-7)SingleMomentum or Adam
Yang et al. [8]WCWC, MLC0YesO(ε-6)SingleSGD
Liu et al. [12]WCWC, MLC0NoO(ε-6)SingleSGD
SONEX (Ours)SMSM, MLC0YesO(ε-5)SingleMomentum or Adam
ALEXR2 (Ours)WCWCYesO(ε-5)DoubleMomentum or Adam
+ +Most of them require the smoothness of $f_{i}$ and $g_{i}$ . Hu et al. [6] has initiated the study of non-smooth non-convex FCCO where both the inner and outer functions could be non-smooth. However, their results face two key limitations: (1) a high iteration complexity of $O(1 / \epsilon^6)$ under the assumption that the stochastic inner functions are Lipschitz continuous in expectation; (2) reliance on vanilla SGD-type updates, which are not suitable for deep learning applications. + +Novelty. This paper addresses these limitations by proposing stochastic momentum methods for non-smooth FCCO, where $f_{i}$ is non-smooth and $g_{i}$ could be smooth or weakly convex, and improving the convergence rate to $O(1 / \epsilon^5)$ . To the best of our knowledge, this is the first work to propose stochastic momentum methods for solving non-smooth FCCO problems. Unlike [6] that directly solves the original problem, the key to our methods is to smooth the outer non-smooth functions based on the Moreau envelope smoothing or equivalently the Nesterov smoothing when they are convex, which is referred to as outer smoothing. When the inner functions are non-smooth weakly convex, we further smooth the transformed objective using another layer of Moreau envelope smoothing, which is referred to as nested smoothing. Then we propose stochastic momentum algorithms to optimize these smoothed objectives. + +Contributions and Significance. We establish two main results regarding non-smooth FCCO. In the first result, we consider non-smooth FCCO with non-smooth outer functions and smooth inner functions. With outer smoothing, we propose a single-loop stochastic momentum method named SONEX to solve the resulting smoothed objective. Our main contribution here lies at a theory that guarantees a convergence rate of $O(1 / \epsilon^5)$ under a meaningful convergence measure. Specially, we show that when the outer smoothing parameter is small enough, the proposed algorithm is guaranteed to find a novel notion of approximate $\epsilon$ -stationary solution to the original problem, which implies the standard nearly $\epsilon$ -stationary solution to the original problem under a verifiable condition of inner functions. In the second result, we consider non-smooth FCCO with non-smooth convex outer functions and non-smooth weakly inner functions. With nested smoothing, we propose a novel double-loop stochastic momentum method named ALEXR2 for solving the resulting smoothed objective and establish a convergence rate of $O(1 / \epsilon^5)$ for finding an $\epsilon$ -stationary solution of the smoothed objective. Table 1 compares our methods with those of the existing works for non-smooth FCCO problems from different aspects. + +Then we consider novel applications of these two algorithms to non-convex inequality constrained optimization. By optimizing a smoothed hinge penalty function with the proposed stochastic momentum methods, we derive a new state-of-the-art rate of $O(1 / \epsilon^5)$ for finding an $\epsilon$ -KKT solution when objective and constraint functions are smooth, and for finding a nearly $\epsilon$ -KKT solution when objective and the constraint functions are weakly convex. Table 2 summarizes the complexity of our methods and existing works for solving non-convex inequality constrained optimization problems. + +# 2 Related work + +FCCO. FCCO is a special case of stochastic composite optimization (SCO) [13, 14] and conditional stochastic optimization (CSO) [15] when the outer expectation is a finite sum. FCCO was first introduced in [4] as a model for optimizing the average precision. Later, [2] proposed the SOX algorithm and improved the convergence rate by adding gradient momentum and further being improved in [16] by replacing exponential moving average (EMA) with MSVR and STORM estimators for inner function and overall gradient, respectively. All these techniques have gain success in optimizing Deep X-Risks [1] such as smooth surrogate losses of AUC and contrastive loss [17, 18]. For convex FCCO problem, [2] reformulate FCCO as a saddle point problem and use restarting technique to boost convergence rate. For non-smooth weakly-convex FCCO (NSWC FCCO), [6] leverages MSVR technique and requires $O(\epsilon^{-6})$ iterations to achieve a nearly $\epsilon$ -stationary point. + +Smoothing techniques: An effective approach for solving non-smooth optimization is to approximate the original problem by a smoothed one using different smoothing techniques, including Nesterov's smoothing [19], randomized smoothing [20], and Moreau envelope [21, 22]. Our method is related to the one based on Moreau envelope, which is a prevalent technique for weakly convex non-smooth problems. This method approximates the objective function by its smooth Moreau envelope and then computes an $\epsilon$ -stationary point of the Moreau envelope, which is also a nearly $\epsilon$ -stationary solution of the original problem [22, 23]. The similar technique has been extended to min-max problems [24, 25] and constrained problems [9, 10, 11, 26]. Our methods are different from these works because we consider the Moreau envelope of each outer function and design stochastic momentum methods. Similar outer smoothing techniques have been developed in [27] for compositional objectives. However, different from our work (1) their algorithm cannot handle multiple outer functions of a finite sum structure as (1); (2) their algorithm is not momentum method; (3) their convergence theory requires mean Lipchitz continuity of the gradient of the inner functions for finding a primal-dual $\epsilon$ -stationary (KKT) solution of the min-max formulation of the original problem. + +Non-convex constrained optimization: Most existing works for stochastic non-convex constrained optimization focus on the smooth problems. For example, [28] proposes a double-loop inexact augmented Lagrangian method for stochastic smooth nonconvex equality constrained optimization under a regularity assumption on the gradients of the constraints. A quadratic penalty method is developed by [29] under similar assumptions but uses only a single loop. Both [29] and [28] consider only smooth problems while we also consider non-smooth problems. For non-smooth but weakly convex problems, the existing methods can compute a nearly $\epsilon$ -KKT point based on the Moreau envelopes of the original problem with iteration complexity $O(\epsilon^{-6})$ [9, 10, 11, 26, 8]. The penalty-based method by [12] reduces the number of evaluations of the subgradients to $O(\epsilon^{-4})$ but still needs to evaluate $O(\epsilon^{-6})$ function values. In contrast, our method only needs complexity $O(\epsilon^{-5})$ to compute a nearly $\epsilon$ -KKT point. For smooth constrained optimization with non-convex equality constraints, variance-reduced stochastic gradient methods have been developed based on quadratic penalty [29, 30]. One can convert inequality constraints into equality constraints by adding slack variables $s^2$ or $s$ (requiring $s \geq 0$ ) to each inequality constraint [30]. However, adding $s^2$ will introduce spurious stationary points [31, 32], while adding $s \geq 0$ will make the domain or the constraint functions unbounded which violate the assumptions in [29, 30]. More importantly, we consider non-smooth constraint functions while [29, 30] only consider smooth cases. + +# 3 Preliminaries + +Let $\| \cdot \|$ be the Euclidean norm. For $g(\cdot): \mathbb{R}^d \to \mathbb{R}^{d_1}$ , let $\nabla g(\cdot) \in \mathbb{R}^{d \times d_1}$ be its Jacobian and let $\| \nabla g(\cdot) \| = \max_{\mathbf{u} \in \mathbb{R}^d, \| \mathbf{u} \| = 1} \| \nabla g(\cdot) \mathbf{u} \|$ be the operator norm. Let $g_i(\mathbf{w}, \mathcal{B}) = \frac{1}{|\mathcal{B}|} \sum_{\xi \in \mathcal{B}} g_i(\mathbf{w}, \xi), i = 1, 2, \dots, n$ be an estimator of $g_i$ based on samples from a minibatch $\mathcal{B}$ . A function $f(\cdot)$ is monotonically non-decreasing if for any $\mathbf{x}_1, \mathbf{x}_2 \in \mathbb{R}^{d_1}$ satisfying that + +$\mathbf{x}_{1,k} \leq \mathbf{x}_{2,k}, k \in [d_1]$ where $\mathbf{x}_k$ denote the $k$ -th element of $\mathbf{x}$ , it holds that $f(\mathbf{x}_1) \leq f(\mathbf{x}_2)$ . A function $f(\cdot)$ is $\rho$ -weakly convex for $\rho \geq 0$ if $f(\cdot) + \frac{\rho}{2} \| \cdot \|^2$ is convex. A function $g(\cdot)$ is $L$ -smooth for $L \geq 0$ if $g$ is differentiable and $\nabla g(\cdot)$ is $L$ -Lipschitz continuous. A vector-valued mapping $g(\cdot)$ is $L$ -smooth for $L \geq 0$ if $g$ is differentiable at each component and $\nabla g(\cdot)$ is $L$ -Lipschitz continuous w.r.t to the operator norm. Given a $\rho$ -weakly convex function $f(\cdot)$ , we denote its regular subdifferential as $\partial f(\cdot)$ . For a $\rho$ -weakly convex function $f(\cdot)$ and a constant $\lambda \in (0, \rho^{-1})$ , the proximal mapping and the Moreau envelope of $f(\cdot)$ with parameter $\lambda$ are defined as follows, respectively, + +$$ +\operatorname {p r o x} _ {\lambda f} (\mathbf {u}) := \arg \min _ {\mathbf {v}} f (\mathbf {v}) + \frac {1}{2 \lambda} \| \mathbf {u} - \mathbf {v} \| ^ {2} \text {a n d} f _ {\lambda} (\mathbf {u}) := \min _ {\mathbf {v}} f (\mathbf {v}) + \frac {1}{2 \lambda} \| \mathbf {u} - \mathbf {v} \| ^ {2}. +$$ + +It is known that, for $\lambda \in (0,\rho^{-1})$ , $f_{\lambda}(\cdot)$ is $L_{f}\coloneqq \max \{\frac{1}{\lambda},\frac{\rho}{1 - \lambda\rho}\}$ -smooth [33] and + +$$ +\nabla f _ {\lambda} (\mathbf {u}) = \frac {\mathbf {u} - \operatorname {p r o x} _ {\lambda f} (\mathbf {u})}{\lambda} \in \partial f \left(\operatorname {p r o x} _ {\lambda f} (\mathbf {u})\right). \tag {2} +$$ + +For a non-smooth weakly convex objective, we follow existing works [21] [6] and aim to find a nearly $\epsilon$ -stationary solution defined below. + +Definition 3.1. A solution $\mathbf{w}$ is an $\epsilon$ -stationary solution of (1) if $\mathrm{dist}(0, \partial F(\mathbf{w})) \leq \epsilon$ . A solution $\mathbf{w}$ is a nearly $\epsilon$ -stationary solution of (1) if there exists $\mathbf{w}'$ such that $\| \mathbf{w} - \mathbf{w}' \| \leq \epsilon$ and $\mathrm{dist}(0, \partial F(\mathbf{w}')) \leq \epsilon$ . We make the following assumption on (1) throughout the paper. + +Assumption 3.2. We assume that $f_{i}$ is $C_f$ -Lipschitz continuous, $g_{i}$ is $C_g$ -Lipschitz continuous for $i = 1, \dots, n$ , $\mathrm{prox}_{\lambda f_i}(\mathbf{w})$ and its subgradient are easily computable, and one of the following conditions hold: + +A1. $f_{i}$ is $\rho_f$ -weakly convex, and $g_{i}(\mathbf{w})$ is $L_g$ -smooth for all $i$ . +A2. $f_{i}$ is convex, and $g_{i}(\mathbf{w})$ is $L_{q}$ -smooth for all $i$ . +A3. $f_{i}$ is convex and monotonically non-decreasing, and $g_{i}(\mathbf{w})$ is $\rho_{g}$ -weakly convex for all $i$ . + +Remark: The Lipschitz continuity of $f$ and $g$ in Assumption 3.2 are fairly standard for FCCO problems. Since we target at non-smooth $f$ and $g$ , Lipschitz continuity is a minimal assumption. In the considered applications of GDRO and learning with fairness constraints in section 6, $f$ is hinge and hence Lipschitz continuous. + +# 4 Smoothing of Non-smooth FCCO + +We first describe the main idea of our methods and then present detailed algorithms and their convergence. Under the Assumption 3.2, we can show $F$ is also weakly convex. To improve the convergence rates and accommodate deep learning applications, we develop stochastic momentum methods, which can be easily modified to incorporate with adaptive step sizes (cf. Appendix E). The challenges of developing provable stochastic momentum methods for solving (1) lie at the non-smoothness of $f_{i}$ and potentially $g_{i}$ . + +Let us first consider smooth $g_{i}$ and defer the discussion for weakly convex $g_{i}$ to subsection 4.2. The key idea is to use a smoothed version of $f_{i}$ , denoted by $f_{i,\lambda}$ , and to approximate (1) by + +$$ +\min _ {\mathbf {w} \in \mathbb {R} ^ {d}} F _ {\lambda} (\mathbf {w}) := \frac {1}{n} \sum_ {i = 1} ^ {n} f _ {i, \lambda} \left(g _ {i} (\mathbf {w})\right). \tag {3} +$$ + +We can easily show that when $g_{i}$ is $L_{g}$ -smooth, then $F_{\lambda}(\mathbf{w})$ is $L_{F}$ -smooth with $L_{F} = C_{g}^{2}\max \left\{\frac{1}{\lambda},\frac{\rho_{f}}{1 - \lambda\rho_{f}}\right\} +C_{f}L_{g}$ (cf. Lemma A.1). As a result, the problem becomes a smooth FCCO where both inner functions and outer functions are smooth, which makes it possible to employ existing techniques to develop stochastic momentum methods to find an $\epsilon$ -stationary point of (3), which is a point $\mathbf{w}$ satisfying $\mathbb{E}[\| \nabla F_{\lambda}(\mathbf{w})\| ]\leq \epsilon$ . However, what this implies for solving the original problem and what constitutes a good choice of $\lambda$ remain unclear. Below, we present a theory to address these questions. We introduce the following notion of stationarity of the original problem. + +Definition 4.1. A solution $\mathbf{w}$ is an approximate $\epsilon$ -stationary solution of (1) if there exist $\mathbf{t}_1, \ldots, \mathbf{t}_m$ and $\mathbf{y}_i \in \partial f_i(\mathbf{t}_i)$ for $i = 1, \ldots, n$ such that $\| \mathbf{t}_i - g_i(\mathbf{w}) \| \leq \epsilon$ , $i = 1, \ldots, n$ , and $\left\| \frac{1}{n} \sum_{i=1}^{n} \nabla g_i(\mathbf{w}) \mathbf{y}_i \right\| \leq \epsilon$ . + +This definition implies that when $\epsilon \rightarrow 0$ , the solution converges to a stationary solution of $F(\cdot)$ . Given this definition, our first theorem is stated below whose proof is given in Section A.3. + +Algorithm 1 SONEX for solving (3) +1: Input: $T, \lambda, \mathbf{w}_0 = \mathbf{w}_{-1}, \mathbf{v}_0, \eta, \gamma, \gamma', \beta, \forall t$ . +2: Sample a batch of data $\mathcal{B}_{i,2}$ from the distribution of $\xi_i$ for $i = 1, \dots, n$ . +3: Set $u_{i,0} = g_i(\mathbf{w}_0, \mathcal{B}_{i,2})$ for $i = 1, \dots, n$ . +4: for $t = 0, 1, \dots, T - 1$ do +5: Sample $\mathcal{B}_1^t \subset \{1, \dots, n\}$ and a batch of data $\mathcal{B}_{i,2}^t$ from the distribution of $\xi_i$ for $i \in \mathcal{B}_1^t$ . +6: for $i = 1, \dots, n$ do +7: Update +[ u_{i,t + 1} = \begin{cases} (1 - \gamma)u_{i,t} + \gamma g_i(\mathbf{w}_t, \mathcal{B}_{i,2}^t) + \gamma' (g_i(\mathbf{w}_t, \mathcal{B}_{i,2}^t) - g_i(\mathbf{w}_{t - 1}, \mathcal{B}_{i,2}^t)) & \text{if } i \in \mathcal{B}_1^t \\ u_{i,t} & \text{if } i \notin \mathcal{B}_1^t \end{cases} ] +8: end for +9: Compute $G_t = \frac{1}{|\mathcal{B}_1^t|} \sum_{i \in \mathcal{B}_1^t} \nabla f_{i,\lambda}(u_{t,i}) \nabla g_i(\mathbf{w}_t, \mathcal{B}_{i,2}^t)$ +10: Update $\mathbf{v}_{t + 1} = (1 - \beta)\mathbf{v}_t + \beta G_t$ +11: Update $\mathbf{w}_{t + 1} = \mathbf{w}_t - \eta \mathbf{v}_{t + 1}$ or Adam-type update +12: end for +13: Output: $\mathbf{w}_{\tau}$ with $\tau$ randomly sampled from $\{1, 2, \dots, T\}$ + +Theorem 4.2. If $\mathbf{w}$ is an $\epsilon$ -stationary solution to $F_{\lambda}(\cdot)$ with $\lambda = \epsilon / C_f$ such that $\|\nabla F_{\lambda}(\mathbf{w})\| \leq \epsilon$ , then $\mathbf{w}$ is an approximate $\epsilon$ -stationary solution to the original objective $F(\cdot)$ . + +A remaining question is whether an approximate $\epsilon$ -stationary point is also a nearly $\epsilon$ -stationary solution of the original problem (1). We present a theorem below under the following assumption. + +Assumption 4.3. There exist $c > 0$ and $\delta > 0$ such that, if $\mathbf{w}$ is an $\epsilon$ -stationary point of $F_{\lambda}$ with $\lambda = \epsilon / C_f$ and $\epsilon \leq c$ , it holds that $\lambda_{\min} \left( \nabla \mathbf{g}(\mathbf{w}) \nabla \mathbf{g}(\mathbf{w})^\top \right) \geq \delta$ , where $\nabla \mathbf{g}(\mathbf{w}) = [\nabla g_1(\mathbf{w})^\top, \dots, \nabla g_n(\mathbf{w})^\top]^\top$ , + +Remark: In Assumption 4.3, once $\epsilon$ is smaller than $c$ , the lower bound $\delta$ does not depend on $\epsilon$ . The empirical justification for this assumption is provided in subsection F.5. + +Theorem 4.4. Suppose Assumptions 3.2 (A1) and 4.3 hold. If $\mathbf{w}$ is an approximate $\epsilon$ -stationary solution of (1) with $\epsilon \leq \min \left\{c, \frac{\delta^2}{16nC_g^2L_g}\right\}$ , $\mathbf{w}$ is also a nearly $\epsilon$ -stationary solution of (1). + +Remark: The proof is given in Appendix A.4. It is notable that the Assumption 4.3 is easily verifiable (cf. Appendix F.5 for empirical evidence). In addition, when we consider the applications in non-convex constrained optimization, this condition is commonly used in existing analyses [7, 8]. + +# 4.1 Single-loop Methods for Smooth Inner Functions + +Next, we present a single-loop algorithm for finding an $\epsilon$ -stationary point of (3). The key ingredients of the algorithm include two parts: (1) maintaining and updating $n$ sequences $u_{i,t}$ for tracking each $g_i(\mathbf{w}_t), i = 1, \dots, n$ , which are updated in a coordinate-wise manner; (2) a stochastic momentum update. We present the detailed updates in Algorithm 1, which is referred to as SONEX. We note that the algorithm is inspired by existing stochastic momentum methods for smooth FCCO. First, Step 7 for updating $u_{i,t+1}$ is from the MSVR algorithm [16]. Different from MSVR, we directly utilize the stochastic momentum update in Step 10, 11, which are similar to SOX [2]. In contrast, MSVR also leverages a variance reduction technique (STROM) to compute an estimate of the stochastic gradient, which requires a stronger assumption that $\nabla g_i(\mathbf{w}; \xi)$ is Lipschitz continuous in expectation. Our analysis shows that this is not helpful for improving the convergence, as the complexity will be dominated by a term related to the smoothness of $f_{i,\lambda}$ , which is in the order of $O(1 / \epsilon)$ . + +We assume the following conditions of the stochastic estimators of $g_{i}$ and their gradients. + +Assumption 4.5. There exist constants $\sigma_0\geq 0$ and $\sigma_{1}\geq 0$ such that the following statements hold for $g_{i}(\mathbf{w})$ and $g_{i}(\mathbf{w},\xi_{i})$ : for $i = 1,\dots ,n$ and any $\mathbf{w}\in \mathbb{R}^d$ + +$$ +\mathbb {E} \| g _ {i} (\mathbf {w}, \xi_ {i}) - g _ {i} (\mathbf {w}) \| ^ {2} \leq \sigma_ {0} ^ {2}, \mathbb {E} \| \partial g _ {i} (\mathbf {w}, \xi_ {i}) - \partial g _ {i} (\mathbf {w}) \| ^ {2} \leq \sigma_ {1} ^ {2} +$$ + +Assumption 4.6. There exist a constant $C_g$ such that $\mathbb{E}_{\xi}\| g_i(\mathbf{w},\xi) - g_i(\mathbf{w}',\xi)\| _2\leq C_g\| \mathbf{w} - \mathbf{w}'\| _2$ + +It is notable that these assumptions have been made in [6] which is important for analyzing variance reduction technique such as MSVR. We refer to Assumption 4.6 as mean Lipchitz continuity of zero-order (MLC0). Let $B_{1}$ and $B_{2}$ denote outer and inner batch sizes. + +Theorem 4.7. Under Assumption 3.2 (A1), 4.5 and 4.6, by setting $\lambda = \Theta(\epsilon), \beta = \Theta(\min\{B_1, B_2\} \epsilon^2)$ , $\gamma = \Theta(B_2 \epsilon^4)$ , $\eta = \Theta\left(\frac{B_1 \sqrt{B_2}}{n} \epsilon^3\right)$ , SONEX with $\gamma' = 1 - \gamma + \frac{n - B_1}{B_1(1 - \gamma)}$ and $\gamma \leq \frac{1}{2}$ converges to an approximate $\epsilon$ -stationary solution of (1) within $T = O\left(\frac{n}{B_1 \sqrt{B_2}} \epsilon^{-5}\right)$ iterations. + +Combining the above theorem with Theorem 4.4, we obtain the following guarantee: + +Corollary 4.8. Under Assumption 3.2(A1), 4.3, 4.5 and 4.6, with the same setting as in Theorem 4.7, SONEX converges to a nearly $\epsilon$ -stationary solution of (1) within $T = O\left(\frac{n}{B_1\sqrt{B_2}}\epsilon^{-5}\right)$ iterations. + +We compare our proposed SONEX with SONX for solving (1). Under assumption 4.6, the rate of SONX is $O(1 / \epsilon^6)$ which is worse than our result of $O(1 / \epsilon^5)$ in Corollary 4.8. + +# 4.2 A Double-loop Algorithm for Smooth or Weakly-Convex Inner Functions + +In this subsection, we further improve our results to achieve the same convergence rate of $O(1 / \epsilon^5)$ by (1) removing the MCL0 assumption instead assuming that $f_{i}$ are convex; (2) designing a stochastic momentum method with a convergence guarantee for weakly convex $g_{i}$ . It is worth mentioning that the convexity of $f_{i}$ holds for a broad range of real applications such as group DRO and the application in non-convex constrained optimization as discussed in next section. Before introducing our new algorithm, we need to first reformulate the problem. + +Let us first consider the scenario that satisfies Assumption 3.2 (A3). For simplicity of notation, we denote $f_{i,\lambda}$ by $\bar{f}_i$ . When $f_i$ is convex, we cast the problem (3) into a minimax formulation: + +$$ +\min _ {\mathbf {w} \in \mathbb {R} ^ {d}} F _ {\lambda} (\mathbf {w}) := \frac {1}{n} \sum_ {i = 1} ^ {n} \max _ {\mathbf {y} _ {i} \in \mathbb {R} ^ {d _ {1}}} \left\{\mathbf {y} _ {i} ^ {\top} g _ {i} (\mathbf {w}) - \bar {f} _ {i} ^ {*} (\mathbf {y} _ {i}) \right\}, \tag {4} +$$ + +where $\bar{f}_i^*$ is the conjugate function of $\bar{f}_i$ . In Appendix A.2, we show that this is also equivalent to the classical Nesterov's smoothing [19]. Under Assumption 3.2 (A3), $\bar{f}_i$ is also $C_f$ -Lipschitz continuous, convex, non-decreasing because of (2). As a consequence, $\mathrm{dom}(\bar{f}_i^*) \subset \{\mathbf{y}_i \in \mathbb{R}_+^{d_1} \mid \| \mathbf{y}_i \| \leq C_f\}$ . This implies that, for any $\mathbf{y}_i \in \mathrm{dom}(\bar{f}_i^*)$ , function $\mathbf{y}_i^\top g_i(\mathbf{w})$ is $\| \mathbf{y}_i \|_1 \rho_g$ -weakly convex in $\mathbf{w}$ and also $\sqrt{d_1} C_f \rho_g$ -weakly convex because $\| \mathbf{y}_i \|_1 \leq \sqrt{d_1} \| \mathbf{y}_i \| \leq \sqrt{d_1} C_f$ . This further implies that $F_{\lambda}(\cdot)$ is $\rho_{F_{\lambda}} := \sqrt{d_1} C_f \rho_g$ -weakly convex. + +Different from last subsection, there is another challenge we need to deal with in order to develop a stochastic momentum method, which comes from that $g_{i}$ are non-smooth. To tackle this challenge, we use another Moreau envelope smoothing of $F_{\lambda}(\cdot)$ and solve the following problem: + +$$ +\begin{array}{l} \min _ {\mathbf {w} \in \mathbb {R} ^ {d}} \left\{F _ {\lambda , \nu} (\mathbf {w}) := \min _ {\mathbf {z} \in \mathbb {R} ^ {d}} \left\{F _ {\lambda} (\mathbf {z}) + \frac {1}{2 \nu} \| \mathbf {z} - \mathbf {w} \| ^ {2} \right\} \right\} (5) \\ = \min _ {\mathbf {w} \in \mathbb {R} ^ {d}} \left\{\min _ {\mathbf {z} \in \mathbb {R} ^ {d}} \max _ {\mathbf {y} \in \mathbb {R} ^ {n d _ {1}}} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbf {y} _ {i} ^ {\top} g _ {i} (\mathbf {z}) - \bar {f} _ {i} ^ {*} (\mathbf {y} _ {i}) + \frac {1}{2 \nu} \| \mathbf {z} - \mathbf {w} \| ^ {2} \right\}, (6) \\ \end{array} +$$ + +where $\nu \in (0, \rho_{F_{\lambda}}^{-1})$ and $\mathbf{y} = [\mathbf{y}_1^{\top}, \dots, \mathbf{y}_n^{\top}]^{\top}$ . The benefit of doing this is that (i) the resulting objective $F_{\lambda, \nu}(\mathbf{w})$ is smooth (cf. Lemma C.2), which allows us to employ stochastic momentum method; (ii) the inner $\min_{\mathbf{z}} \max_{\mathbf{y}}$ becomes a strongly-convex and strongly-concave problem due to that $\bar{f}_i$ is smooth and its conjugate is strongly convex, and $\nu \in (0, \rho_{F_{\lambda}}^{-1})$ (cf. Lemma C.3). + +Recall that $\nabla F_{\lambda,\nu}(\mathbf{w}) = \frac{1}{\nu} (\mathbf{w} - \mathrm{prox}_{\nu F_{\lambda}}(\mathbf{w}))$ . Given an approximate solution of $\mathrm{prox}_{\nu F_{\lambda}}(\mathbf{w})$ denoted by $\hat{\mathbf{z}}$ , of the inner minimization problem in (5), we can use $\frac{1}{\nu} (\mathbf{w} - \hat{\mathbf{z}})$ as an inexact gradient $F_{\lambda,\nu}(\mathbf{w})$ to update $\mathbf{w}$ with a momentum method in order to solve (5). To obtain an estimate $\hat{\mathbf{z}}_t$ at the $t$ -th iteration, we employ a recent algorithm ALEXR [5] for solving the inner minimax problem. With an estimate $\hat{\mathbf{z}}_t$ , we will update $\mathbf{w}_{t+1}$ by a momentum update. We present the detailed steps of our double-loop algorithm named ALEXR2 in Algorithm 2, where $U_{\bar{f}_i^*}(\mathbf{y}_i, \mathbf{y}_i')$ is the Bregman divergence induced by $\bar{f}_i^*$ , namely, $U_{\bar{f}_i^*}(\mathbf{y}_i, \mathbf{y}_i') := \bar{f}_i^*(\mathbf{y}_i) - \bar{f}_i^*(\mathbf{y}_i') - \left\langle \partial \bar{f}_i^*(\mathbf{y}_i'), \mathbf{y}_i - \mathbf{y}_i' \right\rangle$ . For the sake of convergence analyses we need $U_{\bar{f}_i^*}$ to be bounded, as stated in the assumption below. + +Algorithm 2 ALEXR2 for solving (6) +1: Input: $T$ , $\mathbf{w}_0 \in \mathbb{R}^d$ , $\mathbf{v}_0 = 0$ , $K_t, \alpha, \beta, \nu, \eta, \theta, \gamma > 0$ +2: for $t = 0, 1, \ldots, T - 1$ do +3: Set $\mathbf{z}_{t,0} = \mathbf{z}_{t,-1} = \mathbf{w}_t$ and initialize $\mathbf{y}_{t,0} = [\mathbf{y}_{t,0}^{(1)\top}, \ldots, \mathbf{y}_{t,0}^{(n)\top}]^\top \in \mathbb{R}^{nd_1}$ . +4: for $k = 0, 1, \ldots, K_t - 1$ do +5: Sample $\mathcal{B}_1^{t,k} \subset \{1, \ldots, n\}$ and two independent batches of data $\mathcal{B}_{i,2}^{t,k}$ and $\tilde{\mathcal{B}}_{i,2}^{t,k}$ from the distribution of $\xi_i$ for $i \in \mathcal{B}_1^{t,k}$ . +6: Compute $\tilde{g}_{t,k}^{(i)} = g_i(\mathbf{z}_{t,k}, \mathcal{B}_{i,2}^{t,k}) + \theta(g_i(\mathbf{z}_{t,k}, \mathcal{B}_{i,2}^{t,k}) - g_i(\mathbf{z}_{t,k-1}, \mathcal{B}_{i,2}^{t,k}))$ for $i \in \mathcal{B}_1^{t,k}$ . +7: for $i = 1, \ldots, n$ do +8: $\mathbf{y}_{t,k+1}^{(i)} = \left\{ \begin{array}{ll} \arg \max_{\mathbf{y}^{(i)}} & \left\{\mathbf{y}^{(i)\top} \tilde{g}_{t,k}^{(i)} - \bar{f}_i^*(\mathbf{y}^{(i)}) - \frac{1}{\gamma} U_{\bar{f}_i^*}(\mathbf{y}^{(i)}, \mathbf{y}_{t,k}^{(i)})\right\} \\ & \mathbf{y}_{t,k}^{(i)} \\ & i \notin \mathcal{B}_1^{t,k} \end{array} \right.$ +9: end for +10: Compute $G_{t,k} = \frac{1}{|\mathcal{B}_1^{t,k}|} \sum_{i \in \mathcal{B}_1^{t,k}} [\partial g_i(\mathbf{z}_{t,k}, \tilde{\mathcal{B}}_{i,2}^{t,k})]^\top \mathbf{y}_{t,k+1}^{(i)}$ +11: Update $\mathbf{z}_{t,k+1} = \arg \min_{\mathbf{z} \in \mathbb{R}^d} \left\{\langle G_{t,k}, \mathbf{z} \rangle + \frac{1}{2\nu} \| \mathbf{z} - \mathbf{w}_t\|^2 + \frac{1}{2\eta} \| \mathbf{z} - \mathbf{z}_{t,k}\|^2\right\}$ +12: end for +13: Let $G_t = \frac{\beta}{\nu} (\mathbf{w}_t - \mathbf{z}_{t,K_t})$ +14: Update $\mathbf{v}_{t+1} = (1 - \beta) \mathbf{v}_t + \beta G_t$ +15: Update $\mathbf{w}_{t+1} = \mathbf{w}_t - \alpha \mathbf{v}_{t+1}$ or use Adam-type update +16: end for +17: Output: $\mathbf{w}_{\tau}$ with $\tau$ randomly sampled from $\{1, 2, \ldots, T\}$ + +Besides, we would like to point out that the sequences of $y_{i}$ can be updated similar to $u_{i}$ sequences in SONEX similar to [5], i.e., $y_{t,k}^{i} = \nabla f_{i,\lambda}(u_{t,k}^{i})$ , where $u_{t,k+1}^{i} = (1 - \hat{\gamma})u_{t,k}^{i} + \hat{\gamma}g_{i}(\mathbf{z}_{t,k};\mathcal{B}_{i,2}^{t,k}) + \hat{\gamma}\theta(g_{i}(\mathbf{z}_{t,k};\mathcal{B}_{i,2}^{t,k}) - g_{i}(\mathbf{z}_{t,k-1};\mathcal{B}_{i,2}^{t,k}))$ , where $\hat{\gamma} = \gamma/(1+\gamma)$ . + +Assumption 4.9. We assume that $f_{i}$ is a function s.t. $U_{\bar{f}^{*}}(y_{1},y_{2})$ is bounded for any $y_{1},y_{2}\in \operatorname {dom}(\bar{f}_i^*)\subset \mathbb{R}^{d_1}$ . + +Remark: We point out that this condition is not strong under our setting where $\bar{f}_i$ is lipschitz continuous and $dom(\bar{f}_i^*)$ is consequently bounded and it is satisfied by many practical convex lipschitz-continuous functions such as hinge function, smoothed hinge function(i.e. $f_{i,\lambda}$ or $\bar{f}_i$ in this paper), etc. + +The following theorem states the convergence of ALEXR2 with proof given in Section C. + +Theorem 4.10. Suppose assumption 3.2 (A3), 4.5 and 4.9 hold and $\lambda = \epsilon / C_f$ in (4). For any $\epsilon > 0$ , there exists $\theta \in (0,1)$ with $1 - \theta = O(\epsilon^2)$ such that, by setting $\eta = \frac{1 - \theta}{\theta} L_f$ and $\gamma = \frac{(1 - \theta)n}{B_1}$ , $\beta \leq \frac{1}{2}$ , $\alpha = \frac{\beta}{2L_{F_{\lambda,\nu}}}$ and $K_t = \tilde{O}\left(\frac{n}{B_1B_2\epsilon^3} + \frac{1}{\epsilon^3}\right)$ , ALEXR2 returns $\mathbf{w}_{\tau}$ as a nearly $\epsilon$ -stationary solution of (4) in expectation within $T = O(\epsilon^{-2})$ and a total iteration complexity of $\tilde{O}\left(\frac{n}{B_1B_2\epsilon^5} + \frac{1}{\epsilon^5}\right)$ . + +Remark: We can easily extend the above result to Assumption 3.2 (A2). When $g_{i}$ are smooth, we do not need the monotonicity of $f_{i}$ as the minimax problem $\min_{\mathbf{z}} \max_{\mathbf{y}}$ is still guaranteed to be strongly-convex and strongly-concave when $\nu \in (0, \rho_{F_{\lambda}}^{-1})$ , where $\rho_{F_{\lambda}} := \sqrt{d_1} C_f L_g$ is the weak convexity parameter of $F_{\lambda}$ . + +Finally, we show that when $g_{i}$ are smooth, we can also obtain a nearly $\epsilon$ -stationary solution to the original problem (1). Different from the result in Theorem 4.7, the above theorem only guarantees a nearly $\epsilon$ -stationary solution to $F_{\lambda}$ . To address this gap, we can recover a nearly $\epsilon$ -stationary solution to (1) from $\mathbf{w}_{\tau}$ by running ALEXR starting with $\mathbf{w}_{\tau}$ with $K = \tilde{O}(\epsilon^{-5})$ iterations. This result is stated as the following corollary with the proof given in Section C. + +Corollary 4.11. Suppose assumption 3.2 (A2), 4.3, 4.5, 4.9 hold. For any $\epsilon \leq \min \left\{c,\frac{\delta^2}{16nC_g^2L_g}\right\}$ let $\hat{\mathbf{w}}_{\tau} = ALEXR(\mathbf{w}_{\tau},K)$ with $K = \tilde{O}\left(\frac{n}{B_1B_2\epsilon^5} +\frac{1}{\epsilon^5}\right)$ , $\eta = \frac{1 - \theta}{\theta} L_f$ and $\gamma = \frac{(1 - \theta)n}{B_1}$ for $\theta \in (0,1)$ + +with $1 - \theta = O(\epsilon^4)$ . $\hat{\mathbf{w}}_{\tau}$ is a nearly $\epsilon$ -stationary solution of (1) and it is found with a total iteration complexity of $\tilde{O} \left( \frac{n}{B_1 B_2 \epsilon^5} + \frac{1}{\epsilon^5} \right)$ . + +# 5 Smoothed Hinge Penalty Method for Constrained Optimization + +In this section, we consider a constrained optimization problem with $m > 1$ inequality constraints: + +$$ +\min _ {\mathbf {w}} g _ {0} (\mathbf {w}) \text {s . t .} g _ {i} (\mathbf {w}) \leq 0, i = 1, \dots , m \tag {7} +$$ + +where $g_{i}:\mathbb{R}^{d}\to \mathbb{R}$ is continuous and satisfies $g_{i}(\mathbf{w}) = \mathbb{E}g_{i}(\mathbf{w},\xi_{i})$ and the expectation is taken over the random variable $\xi_{i}$ for $i = 0,1,\ldots ,m$ . Following [8], we consider an exact penalty method for (7) by solving the following unconstrained problem: + +$$ +\min _ {\mathbf {w} \in \mathbb {R} ^ {d}} \Phi (\mathbf {w}) := g _ {0} (\mathbf {w}) + \frac {\rho}{m} \sum_ {i = 1} ^ {m} \left[ g _ {i} (\mathbf {w}) \right] _ {+} \tag {8} +$$ + +where $\rho$ is a sufficiently large number. Let $f(\cdot) \coloneqq \rho[\cdot]_+$ so the penalty term $\frac{\rho}{m} \sum_{i=1}^{m} [g_i(\mathbf{w})]_+$ has the same structure as $F(\mathbf{w})$ in (1). Yang et al. [8] have employed SONX for solving the above problem and established a complexity of $O(1/\epsilon^6)$ for finding a nearly $\epsilon$ -KKT solution. Below, we apply the smoothing idea and ALEXR2 or SONEX for improving the convergence rate. Our key idea is to optimize an outer smoothed problem: + +$$ +\min _ {\mathbf {w} \in \mathbb {R} ^ {d}} \Phi_ {\lambda} (\mathbf {w}) := g _ {0} (\mathbf {w}) + \frac {1}{m} \sum_ {i = 1} ^ {m} f _ {\lambda} \left(g _ {i} (\mathbf {w})\right) = g _ {0} (\mathbf {w}) + \frac {1}{m} \sum_ {i = 1} ^ {m} \max _ {y _ {i} \in [ 0, \rho ]} \left\{y _ {i} g _ {i} (\mathbf {w}) - \frac {\lambda}{2} y _ {i} ^ {2} \right\}, \tag {9} +$$ + +where $f_{\lambda}(z) \coloneqq \min_{z' \in \mathbb{R}} f(z') + \frac{1}{2\lambda} (z - z')^2$ . Except for the term $g_0$ , the objective function of (9) has the same structure as (3) and (4) with $\bar{f}_i^*(y_i) = \frac{\lambda}{2} y_i^2 + 1_{[0,\rho]}(y_i)$ . + +Assumption 5.1. There exists a constant $\delta >0$ such that $\sigma_{min}(\mathbf{J}(\mathbf{w}))\geq \delta$ for any matrix $\mathbf{J}(\mathbf{w}) = [\mathbf{h}_1(\mathbf{w}),\dots ,\mathbf{h}_m(\mathbf{w})]\in \mathbb{R}^{d\times m}$ with $\mathbf{h}_i(\mathbf{w})\in \partial g_i(\mathbf{w})$ and any $\mathbf{w}$ satisfying $\max_{i = 1,\ldots ,m}g_i(\mathbf{w}) > 0$ + +Under this assumption, we have the following proposition whose proof is given in Section D.1. + +Proposition 5.2. Suppose Assumption 5.1 hold. If $\rho > \frac{m(C_g + 1)}{\delta}$ and $\lambda = \frac{\epsilon}{\rho}$ , a nearly $\epsilon$ -stationary solution $\mathbf{w}$ to (9) is also a nearly $O(\epsilon)$ -KKT solution to the original problem (7) in the sense that there exist $\hat{\mathbf{w}}$ and $\nu_i \geq 0$ for $i = 1, \ldots, m$ such that $\| \mathbf{w} - \hat{\mathbf{w}} \| \leq \epsilon$ , $\text{dist}(0, \partial g_0(\hat{\mathbf{w}}) + \sum_{i=1}^{m} \partial g_i(\hat{\mathbf{w}}) \nu_i)$ $\leq O(\epsilon)$ , $\max_{i=1, \ldots, m} g_i(\hat{\mathbf{w}}) \leq O(\epsilon)$ and $|g_i(\hat{\mathbf{w}}) \nu_i| \leq O(\epsilon), \forall i = 1, 2, \dots, m$ . + +To find a nearly $\epsilon$ -stationary solution to (9), we can adapt either SONEX if $g_{i}$ are smooth or ALEXR2 if $g_{i}$ are weakly convex with a minor change by including the stochastic gradient of $g_{0}$ in the calculation of $G_{t}$ . The complexity of this approach is presented below with proofs given in Section D.2. + +Theorem 5.3. Suppose assumption 4.5 and 5.1 hold, and the stochastic (sub)gradient of $g_0$ has bounded variance. Let $\rho > m(C_g + 1) / \delta$ and $\lambda = \epsilon / \rho$ . + +- if $\{g_i\}_{i=0}^m$ are weakly convex, then with $\tilde{\mathcal{O}}\left(\frac{m}{B_1B_2\delta^4\epsilon^5}\right)$ iterations ALEXR2 find a nearly $\epsilon$ -KKT solution $\mathbf{w}$ for (7) satisfying that there exist $\hat{\mathbf{w}}$ and $\nu_i \geq 0$ for $i = 1, \dots, m$ such that $\mathbb{E}\|\mathbf{w} - \hat{\mathbf{w}}\| \leq \epsilon$ and $\mathbb{E}\text{dist}(0, \partial g_0(\hat{\mathbf{w}}) + \sum_{i=1}^{m} \partial g_i(\hat{\mathbf{w}})\nu_i)$ $\leq O(\epsilon)$ and it holds with probability $1 - O(\epsilon)$ that $\max_{i=1,\dots,m} g_i(\hat{\mathbf{w}}) \leq O(\epsilon)$ and $|g_i(\hat{\mathbf{w}})\nu_i| \leq O(\epsilon), \forall i = 1, 2, \dots, m$ . + +- if $\{g_i\}_{i=0}^m$ are smooth, then with $\tilde{\mathcal{O}}\left(\frac{m}{B_1\sqrt{B_2}\delta^4\epsilon^5}\right)$ iterations SONEX can find an $\epsilon$ -KKT solution $\mathbf{w}$ for (7) similar as above except for $\hat{\mathbf{w}} = \mathbf{w}$ . + +Notably, our method not only improves the rate but also enjoys an improved dependence on $\delta$ , compared to $O(\epsilon^{-6}\delta^{-6})$ in prior work [8]. + +![](images/a02dd31ebb6cf0a9e81e31f2bd2a1cfa723322e11b818ac274efbc203791025d.jpg) +Figure 1: Training loss curves (left three) and testing accuracy (right one) of different methods for Group DRO with CVaR ratio $r = 0.15$ on different datasets. + +![](images/3cb481c648a18722d36a6b14538150399e24e9b31abe225c6e76698f6bafec6c.jpg) + +![](images/38a333032bbd46e82afe8e220c082a1d64a1e3ab8ef945c4f7bedf28a07693f9.jpg) + +![](images/e53fa1683c1dc88c72e779956820b13044a19c28747cdbd9bcbe343c02e294b2.jpg) + +![](images/177910e4876e3e2fa963d6f30d6e89061eac60e3e0ea462b7edede41f70378d5.jpg) +Figure 2: Training curves of AUC values (fig 1,3) and constraint violation (fig 2,4) of different methods. The format of legend is "Algorithm(penalty function, $\rho$ )", and SH, H, smH mean square hinge, hinge and smoothed hinge, respectively. + +![](images/4fc313544c4cbfcf5519ac3388172a0e9329a100400364e9a9654d811a80c360.jpg) + +![](images/276107b78d133f5f1f113cc43a50cfe9c7c00924dbcb60d9b39d617ce6228179.jpg) + +![](images/1305fcb7c7825b69ce60693a74ce31e7572718cbc43ed260724cc3aef7a9cab0.jpg) + +![](images/a4e2a9903851ae2f59eeda17fddc815b8877085c9b70ea585238686edec762fe.jpg) +Figure 3: Training curves of Target $\Delta$ ACC values (fig 1,3) and constraint violation (fig 2,4) of different methods. The format of label is "Algorithm(penalty function, $\rho$ )", and SH, smH mean square hinge and smoothed hinge, respectively. + +![](images/4b8b187b6bd9cc691a3be457642158537c44225db5fc9f2ba97e267b18ade9dc.jpg) + +![](images/2d1a4d6746323fa3d4aee03cd2061c93827823b3b377888b56f11484c1f728d1.jpg) + +![](images/61b5c224d394571a6054b4f330b6fd6ee335e83398be7a9d69c1eacf1790ae8c.jpg) + +# 6 Numerical Experiments + +We conduct experiments to verify the effectiveness of the proposed algorithms. For non-smooth non-convex FCCO, we consider GDRO with CVaR divergence [5]. For non-convex constrained optimization, we follow [8] and consider two tasks, namely AUC maximization with ROC fairness constraints and continual learning with non-forgetting constraints. + +GDRO with CVaR divergence. We consider the following GDRO with CVaR divergence, which minimizes top- $k$ worst groups' losses [5]: + +$$ +\min _ {\theta} \min _ {s} s + \frac {1}{n} \sum_ {g = 1} ^ {n} \frac {1}{r} \left[ \mathbb {E} _ {(\mathbf {x}, y) \sim \mathcal {D} _ {g}} \ell (\theta ; (\mathbf {x}, y)) - s \right] _ {+} \tag {10} +$$ + +where $n$ is the number of groups, $r = k / n$ , and $\mathcal{D}_g$ denotes the data of the $g$ -th group. + +We use 3 datasets: Camelyon17, Amazon [34], and CelebA [35]. The first two datasets are from WILDS Benchmark for evaluating methods tackling the distributional shift [36], where Camelyon17 has 30 groups and Amazon has 1252 groups. CelebA is a large-scale facial attribute dataset containing over 200,000 celebrity images annotated with 40 binary attributes. We select 4 binary attributes 'Attractive', 'Mouth_Slightly_Open', 'Male' and 'Blonde_Hair' and construct 16 groups, where 'Blonde_Hair' also serves as the label for classification. We compare SONEX with SONX and OOA, where OOA is a stochastic primal-dual algorithm $^{5}$ . We use pretrained Densenet121 [38], Distilbert [39] and ResNet50 [40] for Camelyon17, Amazon and CelebA, respectively. We perform Adam-type update for SONEX on Amazon dataset and momentum-type update on Camelyon17 and CelebA datasets. We run each experiment for 3 random seeds and report their average performance. The hyperparameter tuning is discussed in Appendix F.1. + +Results. We report our result under setting of $r = 0.15$ in Figure 1, where the first 3 figures are the loss curves on the training dataset while the last one is the test accuracy. Our experiment results show that SONEX performs better than SONX and OOA on GDRO tasks. + +AUC maximization with ROC fairness constraints. The formulation is given in Appendix F.2, where the objective is AUC loss and constraints specify the tolerance of the gap between false positive rates (true positive rates) of two sensitive groups at different classification thresholds. For this experiment, we follow almost the same setting as in [8]. Two datasets are used, namely Adult [41] and COMPAS [42], which contain male/female, Caucasian/non-Caucasian groups, respectively. We set thresholds at $\Gamma = \{-3, -2, -1, \dots, 3\}$ and the tolerance $\kappa = 0.005$ , which gives us 14 constraints. We learn a simple neural network with 2 hidden layers. We compare ALEXR2 with the method in [7] that optimizes a squared-hinge penalty function with the SOX algorithm [2], the method in [8] that optimizes a hinge penalty function with the SONX algorithm [6], the double-loop method ICPPAC [10]. We perform Adam-type update for ALEXR2. We run each method for totally 60 epochs with a batch size of 128, repeat five times with different seeds and then report average of the AUC scores and constraint values. Hyperparameter tuning is presented in Appendix F.2. + +Results. We compare the training curves of objective AUC values and the constraint violation measured by the worst constraint function value at each epoch for different methods in Figure 2 on different datasets. These results demonstrate that ALEXR2 optimizing smoothed hinge penalty function can better maximize AUC value on both datasets than the baseline methods while still having similar constraint satisfaction. + +Continual Learning with non-forgetting constraints. We follow a similar experimental setup to [7] for fine-tuning a CLIP model for autonomous driving on the BDD100K dataset [43], a large-scale, multi-task driving image dataset. The formulation is given in Appendix F.3, where the objective is the contrastive loss on a target task (e.g., classifying foggy and overcast condition), and the constraints specify the logistic loss of the new model should not be larger than that of the old model for other different classes (e.g., clear, snowy, rainy, partly cloudy). Since the objective function is another FCCO, we modify SONEX such that the gradient estimator of the objective is computed similar as the smoothed square penalty function. We compare with SOX that optimizes the squared hinge loss use the Adam-type update for both methods. We do not compare with SONX that optimizes the hinge penalty function with SGD-type update as it fails for learning the Transformer network used in this experiment. Hyperparameter tuning is presented in Appendix F.3. + +Results. We present training curves of accuracy improvement and constraint violation on different target tasks in Figure 3. It shows that our method can achieve higher accuracy on target tasks than the baseline method of using squared hinge function while retaining similar constraint satisfaction. + +# 7 Conclusions and Discussion + +In this paper, we have considered non-smooth non-convex finite-sum coupled compositional optimization, where the out functions are non-smooth and inner functions are smooth or weakly convex. We proposed stochastic momentum methods to improve the convergence rate and accelerate the training for deep neural networks. We also considered the applications of the proposed stochastic momentum methods to solving non-convex inequality constrained optimization problems and derived state-of-the-art results for finding an $\epsilon$ -KKT solution. Our experiments demonstrate the effectiveness of our methods. One limitation of our results is that our single-loop method SONEX still has a worse complexity than the double-loop method ALEXR2 in terms of dependence on the inner batch size. + +# Acknowledgments and Disclosure of Funding + +X. Chen, B. Wang, M. Yang and T. Yang were partially supported by the National Science Foundation Career Award 2246753, the National Science Foundation Award 2246757 and 2306572. Q. Lin was partially supported by National Science Foundation Award 2147253. + +# References + +[1] Tianbao Yang. Algorithmic foundations of empirical X-risk minimization. arXiv preprint arXiv:2206.00439, 2022. +[2] Bokun Wang and Tianbao Yang. Finite-sum coupled compositional stochastic optimization: Theory and applications. In International Conference on Machine Learning, pages 23292-23317. PMLR, 2022. +[3] Dixian Zhu, Gang Li, Bokun Wang, Xiaodong Wu, and Tianbao Yang. When auc meets dro: Optimizing partial auc for deep learning with non-convex convergence guarantee. 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PMLR, 2021. +[53] Christoph Schuhmann, Richard Vencu, Romain Beaumont, Robert Kaczmarczyk, Clayton Mullis, Aarush Katta, Theo Coombes, Jenia Jitsev, and Aran Komatsuzaki. Laion-400m: Open dataset of clip-filtered 400 million image-text pairs. arXiv preprint arXiv:2111.02114, 2021. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: We have detailed explanation on all items described in the abstract presented in the paper. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Proofs of Technical Lemmas + +# A.1 Proof of Lemma A.1 + +Lemma A.1. Under Assumption 3.2(A1), $F(\cdot)$ in (1) is $\rho_{F}$ -weakly convex with $\rho_{F} = \sqrt{d_{1}} L_{g}C_{f} + \rho_{f}C_{g}^{2}$ . If $g_{i}$ is $L_{g}$ smooth, then $F_{\lambda}(\mathbf{w})$ is $L_{F}$ -smooth with $L_{F} = C_{g}^{2}\max \{\frac{1}{\lambda},\frac{\rho_{f}}{1 - \lambda\rho_{f}}\} +C_{f}L_{g}$ . + +Proof. Consider any $\mathbf{w}$ and $\mathbf{w}'$ in $\mathbb{R}^d$ , any $i \in \{1, \dots, n\}$ and any $\mathbf{v}_i \in \partial f_i(\mathbf{t}_i)$ at $\mathbf{t}_i = g_i(\mathbf{w})$ . By Assumption 3.2(A1), $\mathbf{v}_i^\top g_i(\mathbf{w})$ is $\| \mathbf{v}_i \|_1 L_g$ -smooth in $\mathbf{w}$ and $f_i$ is $\rho_f$ -weakly convex, we have + +$$ +\begin{array}{l} f _ {i} \left(g _ {i} \left(\mathbf {w} ^ {\prime}\right)\right) \geq f _ {i} \left(g _ {i} (\mathbf {w})\right) + \mathbf {v} _ {i} ^ {\top} \left(g _ {i} \left(\mathbf {w} ^ {\prime}\right) - g _ {i} (\mathbf {w})\right) - \frac {\rho_ {f}}{2} \left\| g _ {i} \left(\mathbf {w} ^ {\prime}\right) - g _ {i} (\mathbf {w}) \right\| ^ {2} \\ \geq f _ {i} \left(g _ {i} (\mathbf {w})\right) + \mathbf {v} _ {i} ^ {\top} \left(g _ {i} \left(\mathbf {w} ^ {\prime}\right) - g _ {i} (\mathbf {w})\right) - \frac {\rho_ {f} C _ {g} ^ {2}}{2} \left\| \mathbf {w} ^ {\prime} - \mathbf {w} \right\| ^ {2} \\ \geq f _ {i} \left(g _ {i} (\mathbf {w})\right) + \mathbf {v} _ {i} ^ {\top} \nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) \left(\mathbf {w} ^ {\prime} - \mathbf {w}\right) - \frac {\left\| \mathbf {v} _ {i} \right\| _ {1} L _ {g}}{2} \left\| \mathbf {w} ^ {\prime} - \mathbf {w} \right\| ^ {2} - \frac {\rho_ {f} C _ {g} ^ {2}}{2} \left\| \mathbf {w} ^ {\prime} - \mathbf {w} \right\| ^ {2} \\ \geq f _ {i} \left(g _ {i} (\mathbf {w})\right) + \mathbf {v} _ {i} ^ {\top} \nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) \left(\mathbf {w} ^ {\prime} - \mathbf {w}\right) - \frac {\sqrt {d _ {1}} C _ {f} L _ {g} + \rho_ {f} C _ {g} ^ {2}}{2} \left\| \mathbf {w} ^ {\prime} - \mathbf {w} \right\| ^ {2}, \\ \end{array} +$$ + +where the second and the last inequalities hold because of lipschitzness of $f_{i}$ and $g_{i}$ from Assumption 3.2, the third because $\mathbf{v}_i^\top g_i(\mathbf{w})$ is $\| \mathbf{v}_i\|_1L_g$ -smooth in $\mathbf{w}$ . + +By the $\rho_{f}$ -weak convexity of $f_{i}$ and lemma 13 in [33], we know that $f_{i,\lambda}$ is $L_{f_{\lambda}} := \max \left\{\frac{1}{\lambda}, \frac{\rho_{f}}{1 - \lambda\rho_{f}}\right\}$ -smooth. Since $\nabla f_{i,\lambda}(\mathbf{w}) = (\mathbf{w} - \mathrm{prox}_{\lambda f_{i}}(\mathbf{w})) / \lambda \in \partial f_{i}(\mathrm{prox}_{\lambda f_{i}}(\mathbf{w}))$ , we have $\| \nabla f_{i,\lambda}(\mathbf{w}) \| \leq C_{f}$ by Assumption 3.2. Therefore, by Assumption 3.2 again, we can easily show that $\nabla f_{i,\lambda}(g_{i}(\mathbf{w})) \nabla g_{i}(\mathbf{w})$ is $(C_{g}^{2} L_{f_{\lambda}} + C_{f} L_{g})$ -Lipschitz continuous (see Lemma 4.2 in [44]), so is $\nabla F_{\lambda}(\mathbf{w}) = \frac{1}{n} \sum_{i=1}^{n} \nabla f_{i,\lambda}(g_{i}(\mathbf{w})) \nabla g_{i}(\mathbf{w})$ . + +# A.2 Proof of Proposition A.2 + +Proposition A.2. When $f_{i}$ is convex, the smoothed problem (3) is equivalent to the Nesterov's smoothing, i.e., + +$$ +\min _ {\mathbf {w}} \max _ {\mathbf {y} \in \Omega^ {n}} \frac {1}{n} \sum_ {i = 1} ^ {n} y _ {i} ^ {\top} g _ {i} (\mathbf {w}) - f _ {i} ^ {*} (y _ {i}) - \frac {\lambda}{2} \| y _ {i} \| _ {2} ^ {2} \tag {11} +$$ + +where $\Omega = \text{dom}(f_i^*)$ , $f_i^*(y_i) + \frac{\lambda}{2} \| y_i \|_2^2 = f_{i,\lambda}^*(y_i)$ . In addition, $f_{i,\lambda}(g) \leq f_i(g) \leq f_{i,\lambda}(g) + \frac{\lambda C_f^2}{2}$ . + +Proof. We prove by deriving (11) from (3). + +$$ +\begin{array}{l} F _ {\lambda} (\mathbf {w}) = \frac {1}{n} \sum_ {i = 1} ^ {n} f _ {i, \lambda} (g _ {i} (\mathbf {w})) \\ = \min _ {\mathbf {y} \in \Omega^ {n}} \frac {1}{n} \sum_ {i = 1} ^ {n} f _ {i} (\mathbf {y} _ {i}) + \frac {1}{2 \lambda} \| \mathbf {y} _ {i} - g _ {i} (\mathbf {w}) \| ^ {2} \\ = \min _ {\mathbf {y} \in \Omega^ {n}} \max _ {\mathbf {z} \in \mathcal {Z} ^ {n}} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbf {y} _ {i} ^ {\top} \mathbf {z} _ {i} - f _ {i} ^ {*} (\mathbf {z} _ {i}) + \frac {1}{2 \lambda} \| \mathbf {y} _ {i} - g _ {i} (\mathbf {w}) \| ^ {2} \\ = \max _ {\mathbf {z} \in \mathcal {Z} ^ {n}} \min _ {\mathbf {y} \in \Omega^ {n}} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbf {y} _ {i} ^ {\top} \mathbf {z} _ {i} - f _ {i} ^ {*} (\mathbf {z} _ {i}) + \frac {1}{2 \lambda} \| \mathbf {y} _ {i} - g _ {i} (\mathbf {w}) \| ^ {2} \\ \end{array} +$$ + +where the dual domain $\mathcal{Z} = \{\mathbf{z} : \mathbf{z} \in \partial f_i(\mathbf{t}), t \in \mathrm{dom}f_i\} \subseteq \{\mathbf{z} : \| \mathbf{z}\| \leq C_f\}$ is bounded; the second equality is from definition of $f_{i,\lambda}$ , the third equality holds from convexity of $f_{i}$ and the fourth equality holds from Sion's minimax theorem [45] and the convexity of $f_{i}^{*}$ . + +Note that the inner-level minimization problem can be solved exactly as $\mathbf{y}_i^* = g_i(\mathbf{w}) - \lambda \mathbf{z}_i$ since it's a quadratic function, we plug $\mathbf{y}_i^*$ into the RHS, then we have: + +$$ +F _ {\lambda} (\mathbf {w}) = \max _ {\mathbf {z} \in \mathcal {Z} ^ {n}} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbf {z} _ {i} ^ {\top} g _ {i} (\mathbf {w}) - f _ {i} ^ {*} (\mathbf {z} _ {i}) - \frac {\lambda}{2} \| \mathbf {z} _ {i} \| ^ {2} \tag {12} +$$ + +Besides, note that $f^{+}(\mathbf{x},\mathbf{u}) = f_{i}(\mathbf{u}) + \frac{1}{2\lambda}\| \mathbf{u} - \mathbf{x}\|^{2}$ is jointly convex, $f_{i,\lambda}(\mathbf{x}) = \min_{\mathbf{u}}f_{i}(\mathbf{u}) + \frac{1}{2\lambda}\| \mathbf{u} - \mathbf{x}\|^{2}$ is also convex. Then we have + +$$ +\frac {1}{n} \sum_ {i = 1} ^ {n} f _ {i, \lambda} (g _ {i} (\mathbf {w})) = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbf {z} _ {i} ^ {\top} g _ {i} (\mathbf {w}) - f _ {i, \lambda} ^ {*} (\mathbf {z} _ {i}) +$$ + +Compare this with 12 we have + +$$ +f _ {i, \lambda} ^ {*} (\mathbf {z} _ {i}) = f _ {i} ^ {*} (\mathbf {z} _ {i}) + \frac {\lambda}{2} \left\| \mathbf {z} _ {i} \right\| ^ {2} +$$ + +Now we already have that + +$$ +f _ {i, \lambda} (g) = \max _ {\mathbf {z} _ {i} \in \mathcal {Z}} \mathbf {z} _ {i} ^ {\top} g - f _ {i} ^ {*} (\mathbf {z} _ {i}) - \frac {\lambda}{2} \| \mathbf {z} _ {i} \| ^ {2} \text {a n d} f _ {i} (g) = \max _ {\mathbf {z} _ {i} \in \mathcal {Z}} \mathbf {z} _ {i} ^ {\top} g - f _ {i} ^ {*} (\mathbf {z} _ {i}) +$$ + +then $f_{i,\lambda}(g)\leq f_i(g)$ naturally holds. Besides, + +$$ +f _ {i} (g) = \max _ {\mathbf {z} _ {i} \in \mathcal {Z}} \mathbf {z} _ {i} ^ {\top} g - f _ {i} ^ {*} (\mathbf {z} _ {i}) \leq \max _ {\mathbf {z} _ {i} \in \mathcal {Z}} \mathbf {z} _ {i} ^ {\top} g - f _ {i} ^ {*} (\mathbf {z} _ {i}) - \frac {\lambda}{2} \| \mathbf {z} _ {i} \| ^ {2} + \max _ {\mathbf {z} _ {i} \in \mathcal {Z}} \frac {\lambda}{2} \| \mathbf {z} _ {i} \| ^ {2} \leq f _ {i, \lambda} (g) + \frac {\lambda}{2} C _ {f} ^ {2} +$$ + +# A.3 Proof of Theorem 4.2 + +Proof of Theorem 4.2. Suppose $\mathbf{w}$ is an $\epsilon$ -stationary solution to $F_{\lambda}(\cdot)$ with $\lambda = \epsilon / C_f$ . It holds that + +$$ +\| \nabla F _ {\lambda} (\mathbf {w}) \| = \left\| \frac {1}{n} \sum_ {i = 1} ^ {n} \nabla f _ {i, \lambda} (g _ {i} (\mathbf {w})) \nabla g _ {i} (\mathbf {w}) \right\| = \left\| \frac {1}{n \lambda} \sum_ {i = 1} ^ {n} (g _ {i} (\mathbf {w}) - \operatorname {p r o x} _ {\lambda f _ {i}} (g _ {i} (\mathbf {w}))) \nabla g _ {i} (\mathbf {w}) \right\| \leq \epsilon . +$$ + +By (2) and $C_f$ -Lipschitz continuity of $f_i$ , we have $\| g_i(\mathbf{w}) - \mathrm{prox}_{\lambda f_i}(g_i(\mathbf{w})) \| \leq \lambda C_f \leq \epsilon$ . Let $\mathbf{t}_i = \mathrm{prox}_{\lambda f_i}(g_i(\mathbf{w}))$ and $\mathbf{y}_i = (g_i(\mathbf{w}) - \mathrm{prox}_{\lambda f_i}(g_i(\mathbf{w}))) / \lambda \in \partial f_i(\mathrm{prox}_{\lambda f_i}(g_i(\mathbf{w}))) = \partial f_i(\mathbf{t}_i)$ . It holds that $\| \mathbf{t}_i - g_i(\mathbf{w}) \| \leq \epsilon$ , $i = 1, \dots, n$ , and $\left\| \frac{1}{n} \sum_{i=1}^{n} \nabla g_i(\mathbf{w}) \mathbf{y}_i \right\| \leq \epsilon$ , so $\mathbf{w}$ is an approximate $\epsilon$ -stationary solution to the original objective $F(\cdot)$ by definition. + +# A.4 Proof of Theorem 4.4 + +We need the following lemma. + +Lemma A.3. Suppose Assumption 3.2 (A1) or (A2) holds. If $\lambda_{\mathrm{min}}\left(\nabla g_i(\mathbf{w})\nabla g_i(\mathbf{w})^\top\right) \geq \delta$ , it holds that $\lambda_{\mathrm{min}}\left(\nabla g_i(\mathbf{w}')\nabla g_i(\mathbf{w}')^\top\right) \geq \frac{\delta}{2}$ for any $\mathbf{w}'$ satisfying $\| \mathbf{w}' - \mathbf{w} \| \leq \frac{\delta}{4C_gL_g}$ . + +Proof. Consider any $\mathbf{w}'$ satisfying $\| \mathbf{w}' - \mathbf{w} \| \leq \frac{\delta}{4C_g L_g}$ . Note that + +$$ +\lambda_ {\min } \left(\nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) \nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) ^ {\top}\right) = \min _ {\mathbf {u} \in \mathbb {R} ^ {d _ {1}}, \| \mathbf {u} \| = 1} \mathbf {u} ^ {\top} \nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) \nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) ^ {\top} \mathbf {u} +$$ + +Let + +$$ +\mathbf{u}_{\mathbf{w}^{\prime}}\in \operatorname *{arg min}_{\mathbf{u}\in \mathbb{R}^{d_{1}},\| \mathbf{u}\| = 1}\mathbf{u}^{\top}\nabla g_{i}(\mathbf{w}^{\prime})\nabla g_{i}(\mathbf{w}^{\prime})^{\top}\mathbf{u}. +$$ + +It holds that + +$$ +\begin{array}{l} \lambda_ {\min } \left(\nabla g _ {i} (\mathbf {w}) \nabla g _ {i} (\mathbf {w}) ^ {\top}\right) - \lambda_ {\min } \left(\nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) \nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) ^ {\top}\right) \\ \leq \mathbf {u} _ {\mathbf {w} ^ {\prime}} ^ {\top} \nabla g _ {i} (\mathbf {w}) \nabla g _ {i} (\mathbf {w}) ^ {\top} \mathbf {u} _ {\mathbf {w} ^ {\prime}} - \mathbf {u} _ {\mathbf {w} ^ {\prime}} ^ {\top} \nabla g _ {i} (\mathbf {w} ^ {\prime}) \nabla g _ {i} (\mathbf {w} ^ {\prime}) ^ {\top} \mathbf {u} _ {\mathbf {w} ^ {\prime}} \\ = \mathbf {u} _ {\mathbf {w} ^ {\prime}} ^ {\top} \left(\nabla g _ {i} (\mathbf {w}) \nabla g _ {i} (\mathbf {w}) ^ {\top} - \nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) \nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) ^ {\top}\right) \mathbf {u} _ {\mathbf {w} ^ {\prime}} \\ \leq \left\| \nabla g _ {i} (\mathbf {w}) \nabla g _ {i} (\mathbf {w}) ^ {\top} - \nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) \nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) ^ {\top} \right\| \\ \leq 2 C _ {g} L _ {g} \left\| \mathbf {w} - \mathbf {w} ^ {\prime} \right\|. \\ \end{array} +$$ + +This implies + +$$ +\lambda_ {\min } \left(\nabla g _ {i} (\mathbf {w} ^ {\prime}) \nabla g _ {i} (\mathbf {w} ^ {\prime}) ^ {\top}\right) \geq \lambda_ {\min } \left(\nabla g _ {i} (\mathbf {w} ^ {\prime}) \nabla g _ {i} (\mathbf {w} ^ {\prime}) ^ {\top}\right) - 2 C _ {g} L _ {g} \| \mathbf {w} - \mathbf {w} ^ {\prime} \| \geq \delta - \frac {\delta}{2} = \frac {\delta}{2}. +$$ + +![](images/be9703224a8784569801b5debbc94ab3defc25e166041f77ffea55eef98fc33e.jpg) + +Proof of Theorem 4.4. Suppose $\mathbf{w}$ is an approximate $\epsilon$ -stationary solution of (1) with $\epsilon \leq \min \left\{c, \frac{\delta^2}{16nC_g^2L_g}\right\}$ . There exist $\mathbf{t}_1, \ldots, \mathbf{t}_m$ and $\mathbf{y}_i \in \partial f_i(\mathbf{t}_i)$ for $i = 1, \ldots, n$ such that $\|\mathbf{t}_i - g_i(\mathbf{w})\| \leq \epsilon$ , $i = 1, \ldots, n$ , and $\left\| \frac{1}{n} \sum_{i=1}^{n} \nabla g_i(\mathbf{w}) \mathbf{y}_i \right\| \leq \epsilon$ . + +Consider the following optimization problem + +$$ +\min _ {\mathbf {v}} \left\{q (\mathbf {v}) := \frac {1}{2 n} \sum_ {i = 1} ^ {n} \| \mathbf {t} _ {i} - g _ {i} (\mathbf {v}) \| ^ {2} \right\}. \tag {13} +$$ + +We want to show that the optimal objective value of (13) is zero and there exists an optimal solution $\mathbf{w}'$ such that $\| \mathbf{w}' - \mathbf{w} \| \leq \frac{4C_g n\epsilon}{\delta}$ . + +By the condition on $\epsilon$ , it holds that $\frac{4C_g n \epsilon}{\delta} \leq \frac{\delta}{4C_g L_g}$ . Hence, by Lemma A.3, if $\| \mathbf{v} - \mathbf{w} \| \leq \frac{4C_g n \epsilon}{\delta}$ , it holds that + +$$ +\| \nabla q (\mathbf {v}) \| ^ {2} = \frac {1}{n ^ {2}} \left\| \sum_ {i = 1} ^ {n} \nabla g _ {i} (\mathbf {v}) ^ {\top} [ g _ {i} (\mathbf {v}) - \mathbf {t} _ {i} ] \right\| ^ {2} \geq \frac {\delta}{2 n ^ {2}} \sum_ {i = 1} ^ {n} \| g _ {i} (\mathbf {v}) - \mathbf {t} _ {i} \| ^ {2} = \frac {\delta}{n} q (\mathbf {v}). +$$ + +Moreover, it always holds that + +$$ +\| \nabla q (\mathbf {v}) \| ^ {2} \leq \frac {C _ {g} ^ {2}}{n} \sum_ {i = 1} ^ {n} \| g _ {i} (\mathbf {v}) - \mathbf {t} _ {i} \| ^ {2} = 2 C _ {g} ^ {2} q (\mathbf {v}). +$$ + +Let $L_{q}$ be the smoothness parameter of $q(\mathbf{v})$ on the compact set $\{\mathbf{v} \in \mathbb{R}^d \mid \| \mathbf{v} - \mathbf{w} \| \leq \frac{4C_g n \epsilon}{\delta}\}$ . Let $\mathbf{v}_t$ for $t = 0, 1, \ldots$ be generated by the gradient descent method using $\mathbf{v}_0 = \mathbf{w}$ and a step size of $\eta = \frac{1}{L_q}$ , i.e., $\mathbf{v}_{t+1} = \mathbf{v}_t - \eta \nabla q(\mathbf{v}_t)$ . Suppose $\| \mathbf{v}_s - \mathbf{w} \| \leq \frac{4C_g n \epsilon}{\delta}$ for $s = 0, 1, \ldots, t$ (which is true at least when $t = 0$ ). By the standard convergence analysis of the gradient descent method for a $L_{q}$ -smooth objective function, we have + +$$ +\begin{array}{l} q (\mathbf {v} _ {t + 1}) \leq q (\mathbf {v} _ {t}) + \langle \nabla q (\mathbf {v} _ {t}), \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \rangle + \frac {L _ {q}}{2} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} \\ \leq q (\mathbf {v} _ {t}) - \frac {1}{2 L _ {q}} \| \nabla q (\mathbf {v} _ {t}) \| ^ {2} \leq q (\mathbf {v} _ {t}) - \frac {\delta}{2 L _ {q} n} q (\mathbf {v} _ {t}). \\ \end{array} +$$ + +Applying this inequality recursively gives + +$$ +\frac {1}{2 C _ {g} ^ {2}} \| \nabla q (\mathbf {v} _ {t}) \| ^ {2} \leq q (\mathbf {v} _ {t}) \leq \left(1 - \frac {\delta}{2 L _ {q} n}\right) ^ {t} q (\mathbf {v} _ {0}). +$$ + +Recall that $\| \mathbf{t}_i - g_i(\mathbf{w})\| \leq \epsilon$ such that $q(\mathbf{v}_0) = \frac{1}{2n}\sum_{i = 1}^{n}\| \mathbf{t}_i - g_i(\mathbf{w})\|^2\leq \frac{\epsilon^2}{2}$ . By the triangular inequality, we have + +$$ +\| \mathbf {v} _ {t + 1} - \mathbf {w} \| \leq \sum_ {s = 0} ^ {t} \eta \| \nabla q (\mathbf {v} _ {s}) \| \leq \sum_ {s = 0} ^ {t} \eta C _ {g} \left(1 - \frac {\delta}{2 L _ {q} n}\right) ^ {\frac {s}{2}} \epsilon \leq \frac {4 \eta C _ {g} L _ {q} n}{\delta} \epsilon = \frac {4 C _ {g} n \epsilon}{\delta}. +$$ + +By induction, we have $\| \mathbf{v}_t - \mathbf{w}\| \leq \frac{4C_g n\epsilon}{\delta}$ for $t = 0,1,\ldots$ + +Let $\mathbf{w}'$ be any limiting point of $\{\mathbf{v}_t\}_{t \geq 0}$ . It holds that $\| \mathbf{w}' - \mathbf{w} \| \leq \frac{4C_g n\epsilon}{\delta}$ and + +$$ +q \left(\mathbf {w} ^ {\prime}\right) \leq \lim _ {t \rightarrow \infty} \left(1 - \frac {\delta}{2 L _ {q}}\right) ^ {t} q \left(\mathbf {v} _ {0}\right) = 0, +$$ + +meaning that $\mathbf{w}'$ is the optimal solution to (13) with the objective value equal to zero, which implies $\mathbf{t}_i = g_i(\mathbf{w}')$ for $i = 1,\dots ,n$ . + +Since $\mathbf{w}$ is an approximate $\epsilon$ -stationary point of (1) and $\mathbf{y}_i \in \partial f_i(\mathbf{t}_i)$ , we have + +$$ +\begin{array}{l} \left\| \frac {1}{n} \sum_ {i} \partial f _ {i} \left(g _ {i} \left(\mathbf {w} ^ {\prime}\right)\right) \nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) \right\| \\ = \left\| \frac {1}{n} \sum_ {i} \partial f _ {i} \left(\mathbf {t} _ {i}\right) \nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) \right\| \\ \leq \left\| \frac {1}{n} \sum_ {i} \partial f _ {i} (\mathbf {t} _ {i}) \nabla g _ {i} (\mathbf {w}) \right\| + \left\| \frac {1}{n} \sum_ {i} \partial f _ {i} (\mathbf {t} _ {i}) (\nabla g _ {i} (\mathbf {w} ^ {\prime}) - \nabla g _ {i} (\mathbf {w})) \right\| \\ = \left\| \frac {1}{n} \sum_ {i} \mathbf {y} _ {i} \nabla g _ {i} (\mathbf {w}) \right\| + \left\| \frac {1}{n} \sum_ {i} \partial f _ {i} (\mathbf {t} _ {i}) \left(\nabla g _ {i} \left(\mathbf {w} ^ {\prime}\right) - \nabla g _ {i} (\mathbf {w})\right) \right\| \\ \leq \epsilon + C _ {f} L _ {g} \| \mathbf {w} - \mathbf {w} ^ {\prime} \| \leq (1 + \frac {4 C _ {g} C _ {f} L _ {g} n}{\delta}) \epsilon , \\ \end{array} +$$ + +where the last inequality is by Assumptions 3.2. This means $\mathbf{w}'$ is an $O(\epsilon)$ -stationary solution of (1) and thus $\mathbf{w}$ is a nearly $O(\epsilon)$ -stationary solution of (1). + +# B Convergence Analysis of SONEX + +In this section, we present the proofs for Theorems 4.7. Let $\mathbb{E}_t$ be the conditional expectation conditioning on $\mathcal{B}_1^s$ and $\mathcal{B}_{i,2}^s$ for $i = 1,\dots ,n$ and $s = 0,1,\ldots ,t - 1$ . To facilitate the proofs, the following quantities are defined based on the notations in Algorithm 1 for $t = 0,\dots ,T - 1$ , + +$$ +V _ {t + 1} := \left\| \mathbf {v} _ {t + 1} - \nabla F _ {\lambda} \left(\mathbf {w} _ {t}\right) \right\| ^ {2} \tag {14} +$$ + +$$ +U _ {t + 1} := \frac {1}{n} \left\| \mathbf {u} _ {t + 1} - \mathbf {g} \left(\mathbf {w} _ {t}\right) \right\| ^ {2} \tag {15} +$$ + +$$ +\mathbf {u} _ {t} := \left[ u _ {1, t}, \dots , u _ {n, t} \right] ^ {\top} \tag {16} +$$ + +$$ +\mathbf {g} \left(\mathbf {w} _ {t}\right) := \left[ g _ {1} \left(\mathbf {w} _ {t}\right), \dots , g _ {n} \left(\mathbf {w} _ {t}\right) \right] ^ {\top}. \tag {17} +$$ + +# B.1 Proof of Theorem 4.7 + +Lemma B.1 (Lemma 2 in [46]). Suppose $\eta \leq \frac{1}{2L_F}$ in Algorithm 1 with $L_{F}$ defined in lemma A.1. It holds that + +$$ +F _ {\lambda} \left(\mathbf {w} _ {t + 1}\right) \leq F _ {\lambda} \left(\mathbf {w} _ {t}\right) + \frac {\eta}{2} V _ {t + 1} - \frac {\eta}{2} \left\| \nabla F _ {\lambda} \left(\mathbf {w} _ {t}\right) \right\| ^ {2} - \frac {\eta}{4} \left\| \mathbf {v} _ {t + 1} \right\| ^ {2}, \tag {18} +$$ + +where $V_{t + 1}$ is defined as in (14). + +A recursion for $V_{t + 1}$ is provided below. + +Lemma B.2 (Lemma 9 in [2]). Let $V_{t+1}$ be defined as in (14). Suppose $\beta \leq \frac{2}{7}$ in Algorithm 1. It holds that + +$$ +\begin{array}{l} \mathbb {E} \left[ V _ {t + 2} \right] \leq (1 - \beta) \mathbb {E} \left[ V _ {t + 1} \right] + \frac {2 L _ {F} ^ {2} \eta^ {2} \mathbb {E} \left[ \| \mathbf {v} _ {t + 1} \| ^ {2} \right]}{\beta} + \frac {3 L _ {f} ^ {2} C _ {1} ^ {2}}{n} \mathbb {E} \left[ \| \mathbf {u} _ {t + 2} - \mathbf {u} _ {t + 1} \| ^ {2} \right] \\ + \frac {2 \beta^ {2} C _ {f} ^ {2} \left(C _ {g} ^ {2} + \sigma_ {1} ^ {2}\right)}{\min \left\{B _ {1} , B _ {2} \right\}} + 5 \beta L _ {f} ^ {2} C _ {1} ^ {2} \mathbb {E} [ U _ {t + 2} ] \tag {19} \\ \end{array} +$$ + +where $C_1^2 \coloneqq C_g^2 + \frac{\sigma_1^2}{B_2}$ , and $U_{t+1}$ , $\mathbf{u}_t$ and $\mathbf{g}(\mathbf{w}_t)$ are defined as in (15), (16) and (17), respectively. + +The following lemma provides the recursion for the estimation error $U_{t + 1}$ of the MSVR estimator of $\mathbf{g}(\mathbf{w})$ . + +Lemma B.3 (Lemma 1 in [16]). Suppose $\gamma' = \frac{n - B_1}{B_1(1 - \gamma)} + (1 - \gamma)$ and $\gamma \leq \frac{1}{2}$ and the MSVR update is used in Algorithm 1. It holds that + +$$ +\mathbb {E} \left[ U _ {t + 2} \right] \leq \left(1 - \frac {\gamma B _ {1}}{n}\right) \mathbb {E} \left[ U _ {t + 1} \right] + \frac {8 n C _ {g} ^ {2}}{B _ {1}} \mathbb {E} \left\| \mathbf {w} _ {t + 1} - \mathbf {w} _ {t} \right\| ^ {2} + \frac {2 B _ {1} \gamma^ {2} \sigma_ {1} ^ {2}}{n B _ {2}}. \tag {20} +$$ + +The following lemma handles the term $\left\| \mathbf{u}_{t + 1} - \mathbf{u}_t\right\|^2$ + +Lemma B.4 (Lemma 6 in [16]). Under the same assumptions as Lemma B.3, it holds that + +$$ +\mathbb {E} \left[ \left\| \mathbf {u} _ {t + 2} - \mathbf {u} _ {t + 1} \right\| ^ {2} \right] \leq \frac {4 B _ {1} \gamma^ {2}}{n} \mathbb {E} \left[ U _ {t + 1} \right] + \frac {9 n ^ {2} C _ {g} ^ {2}}{B _ {1}} \mathbb {E} \| \mathbf {w} _ {t + 1} - \mathbf {w} _ {t} \| ^ {2} + \frac {2 B _ {1} \gamma^ {2} \sigma_ {0} ^ {2}}{B _ {2}}. \tag {21} +$$ + +Now we are ready to proof Theorem 4.7 + +Proof of Theorem 4.7. Note that $\| \mathbf{w}_{t + 1} - \mathbf{w}_t\|^2 = \eta^2\| \mathbf{v}_{t + 1}\|^2$ . Let $P$ be a non-negative constant to be determined later. Taking the weighted summation of both sides (18), (19), (20), and (21) as specified in the following formula + +$$ +\frac {1}{\eta} \times (1 8) + \frac {1}{\beta} \times (1 9) + P \times (2 0) + \frac {3 L _ {f} ^ {2} C _ {1} ^ {2}}{n \beta} \times (2 1), +$$ + +we have + +$$ +\begin{array}{l} \frac {1}{\eta} \mathbb {E} F _ {\lambda} (\mathbf {w} _ {t + 1}) + \frac {1}{\beta} \mathbb {E} V _ {t + 2} + (P - 5 L _ {f} ^ {2} C _ {1} ^ {2}) \mathbb {E} U _ {t + 2} \\ \leq \frac {1}{\eta} \mathbb {E} F _ {\lambda} (\mathbf {w} _ {t}) + \left(\frac {1}{\beta} - \frac {1}{2}\right) \mathbb {E} V _ {t + 1} + \left(P \left(1 - \frac {\gamma B _ {1}}{n}\right) + \frac {1 2 L _ {f} ^ {2} C _ {1} ^ {2} \gamma^ {2} B _ {1}}{\beta n ^ {2}}\right) \mathbb {E} U _ {t + 1} \\ - \left(\frac {1}{4} - \frac {2 L _ {F} ^ {2} \eta^ {2}}{\beta^ {2}} - \frac {8 n C _ {g} ^ {2} P \eta^ {2}}{B _ {1}} - \frac {2 7 n L _ {f} ^ {2} C _ {1} ^ {2} C _ {g} ^ {2} \eta^ {2}}{B _ {1} \beta}\right) \mathbb {E} \| \mathbf {v} _ {t + 1} \| ^ {2} \\ + \frac {2 \beta C _ {f} ^ {2} \left(C _ {g} ^ {2} + \sigma_ {1} ^ {2}\right)}{\min \left\{B _ {1} , B _ {2} \right\}} + \frac {2 B _ {1} \gamma^ {2} \sigma_ {1} ^ {2} P}{n B _ {2}} + \frac {6 L _ {f} ^ {2} C _ {1} ^ {2} \gamma^ {2} \sigma_ {0} ^ {2} B _ {1}}{\beta n B _ {2}} - \frac {1}{2} \mathbb {E} \| \nabla F _ {\lambda} (\mathbf {w} _ {t}) \| ^ {2} \\ \end{array} +$$ + +We choose $\eta$ such that + +$$ +\frac {1}{4} - \frac {2 L _ {F} ^ {2} \eta^ {2}}{\beta^ {2}} - \frac {8 n C _ {g} ^ {2} P \eta^ {2}}{B _ {1}} - \frac {2 7 n L _ {f} ^ {2} C _ {1} ^ {2} C _ {g} ^ {2} \eta^ {2}}{B _ {1} \beta} \geq 0. \tag {22} +$$ + +Moreover, we set $P$ to be the solution of the following linear equation + +$$ +P \left(1 - \frac {\gamma B _ {1}}{n}\right) + \frac {1 2 L _ {f} ^ {2} C _ {1} ^ {2} \gamma^ {2} B _ {1}}{\beta n ^ {2}} = P - 5 L _ {f} ^ {2} C _ {1} ^ {2}, +$$ + +which means + +$$ +P = \frac {n}{\gamma B _ {1}} \left(5 L _ {f} ^ {2} C _ {1} ^ {2} + \frac {1 2 L _ {f} ^ {2} C _ {1} ^ {2} \gamma^ {2} B _ {1}}{\beta n ^ {2}}\right). +$$ + +Combining the results above gives + +$$ +\begin{array}{l} \frac {1}{\eta} \mathbb {E} F _ {\lambda} (\mathbf {w} _ {t + 1}) + \frac {1}{\beta} \mathbb {E} V _ {t + 2} + \left(P - 5 L _ {f} ^ {2} C _ {1} ^ {2}\right) \mathbb {E} U _ {t + 2} \\ \leq \frac {1}{\eta} \mathbb {E} F _ {\lambda} (\mathbf {w} _ {t}) + \frac {1}{\beta} \mathbb {E} V _ {t + 1} + \left(P - 5 L _ {f} ^ {2} C _ {1} ^ {2}\right) \mathbb {E} U _ {t + 1} \\ + \frac {2 \beta C _ {f} ^ {2} (C _ {g} ^ {2} + \sigma_ {1} ^ {2})}{\min \{B _ {1} , B _ {2} \}} + \frac {2 B _ {1} \gamma^ {2} \sigma_ {1} ^ {2} P}{n B _ {2}} + \frac {6 L _ {f} ^ {2} C _ {1} ^ {2} \gamma^ {2} \sigma_ {0} ^ {2} B _ {1}}{\beta n B _ {2}} - \frac {1}{2} \mathbb {E} \| \nabla F _ {\lambda} (\mathbf {w} _ {t}) \| ^ {2}. \\ \end{array} +$$ + +Summing both sides of the inequality above over $t = 0,1,\ldots ,T - 1$ and organizing term lead to + +$$ +\begin{array}{l} \frac {1}{T} \sum_ {t = 0} ^ {T - 1} \| \nabla F _ {\lambda} (\mathbf {w} _ {t}) \| ^ {2} \leq \frac {2}{T} \left(\frac {1}{\eta} (F _ {\lambda} (\mathbf {w} _ {0}) - F _ {\lambda} (\mathbf {w} _ {T})) + \frac {1}{\beta} \mathbb {E} V _ {1} + \left(P - 5 L _ {f} ^ {2} C _ {1} ^ {2}\right) \mathbb {E} U _ {1}\right) \\ + \frac {4 \beta C _ {f} ^ {2} \left(C _ {g} ^ {2} + \sigma_ {1} ^ {2}\right)}{\min \left\{B _ {1} , B _ {2} \right\}} + \frac {4 B _ {1} \gamma^ {2} \sigma_ {1} ^ {2} P}{n B _ {2}} + \frac {1 2 L _ {f} ^ {2} C _ {1} ^ {2} \gamma^ {2} \sigma_ {0} ^ {2} B _ {1}}{\beta n B _ {2}} \tag {23} \\ \end{array} +$$ + +Note that $L_{F} = O(\frac{1}{\epsilon}), L_{f} = O(\frac{1}{\epsilon})$ and $P = O(\frac{n}{B_1\gamma\epsilon^2} + \frac{\gamma}{n\beta\epsilon^2})$ . We require that all the terms in the right-hand side to be in the order of $O(\epsilon^2)$ . To ensure $\frac{4\beta C_f^2(C_g^2 + \sigma_1^2)}{\min\{B_1,B_2\}} = O(\epsilon^2)$ , we must have $\beta = O(\min\{B_1,B_2\}\epsilon^2)$ . To ensure $\frac{4B_1\gamma^2\sigma_1^2P}{nB_2} = O(\frac{\gamma}{B_2\epsilon^2} + \frac{B_1\gamma^3}{n^2B_2\beta\epsilon^2}) = O(\epsilon^2)$ , we must have $\gamma = O(B_2\epsilon^4)$ . With these orders of $\beta$ and $\gamma$ , we also have $P = O(\frac{n}{B_1B_2\epsilon^6})$ and $\frac{12L_f^2C_1^2\gamma^2\sigma_0^2B_1}{\beta nB_2} = O(\epsilon^4)$ . Therefore the last three terms in (23) are $O(\epsilon^2)$ . + +To satisfy (22), we will need to set + +$$ +\begin{array}{l} \eta = O \left(\min \left\{\frac {\beta}{L _ {F}}, \sqrt {\frac {B _ {1}}{n P}}, \frac {1}{L _ {f}} \sqrt {\frac {B _ {1} \beta}{n}} \right\}\right) \\ = O \left(\min \left\{\min \left\{B _ {1}, B _ {2} \right\} \epsilon^ {3}, \frac {B _ {1} \sqrt {B _ {2}}}{n} \epsilon^ {3}, \sqrt {\frac {B _ {1} \min \left\{B _ {1} , B _ {2} \right\}}{n}} \epsilon^ {2} \right\}\right) \\ = O \left(\frac {B _ {1} \sqrt {B _ {2}}}{n} \epsilon^ {3}\right). \\ \end{array} +$$ + +We also set the sizes of $\mathcal{B}_{i,2}$ to be $O\left(\frac{1}{\epsilon^3}\right)$ so that $\mathbb{E}U_1 = O(\epsilon^3)$ . Also, it is easy to see that $F_{\lambda}(\mathbf{w}_0) - F_{\lambda}(\mathbf{w}_T) = O(1)$ and $\mathbb{E}V_1 = O(1)$ . Then the first term in (23) becomes + +$$ +\frac {1}{T} O \left(\frac {1}{\eta} + \frac {1}{\beta} + P \epsilon^ {3}\right) = \frac {1}{T} O \left(\frac {n}{B _ {1} \sqrt {B _ {2}} \epsilon^ {3}} + \frac {1}{\min \{B _ {1} , B _ {2} \} \epsilon^ {2}} + \frac {n}{B _ {1} B _ {2} \epsilon^ {3}}\right) +$$ + +which will become $O(\epsilon^2)$ when $T = O\left(\frac{n}{B_1\sqrt{B_2}\epsilon^5}\right)$ . + +![](images/ebee6a02352deef9087d9db18c97ee0b9e90fc5b05df7dc250d0d0ef40a41d0a.jpg) + +# C Convergence Analysis of ALEXR2 + +To establish the complexity of Algorithm 2, we first present the convergence result of ALEXR [5] which is the inner loop of ALEXR2, and it shows the complexity of Algorithm 2 to produce $\hat{\mathbf{z}}_t$ such that $\left\| \hat{\mathbf{z}}_t - \mathrm{prox}_{\nu F_\lambda}(\mathbf{w}_t)\right\|_2^2 \leq O(\epsilon')$ . + +Theorem C.1 (A Variant of Theorem 1 in [5] when $g_{i}$ is non-smooth $\rho_{g}$ -weakly convex and $\mu = \frac{1}{\nu} - \sqrt{d_1} C_f \rho_g$ ). Suppose Assumptions 3.2 (A3) and 4.5 hold. For any $\epsilon' > 0$ , there exists $\theta \in (0,1)$ with $1 - \theta = O(\epsilon \epsilon')$ such that, by setting $\eta = \frac{1 - \theta}{(4\theta - 1) \sqrt{d_1} C_f \rho_g}$ and $\gamma = (\frac{B_1}{n(1 - \theta)} - 1)^{-1}$ , the inner loop of Algorithm 2 (i.e., ALEXR) guarantees $\frac{\mu}{2} \mathbb{E} \left\| \hat{\mathbf{z}}_t - \text{prox}_{\nu F_\lambda}(\mathbf{w}_t) \right\|_2^2 \leq \epsilon'$ for any $t$ after $K_t = \tilde{O}\left(\frac{n}{B_1} + C_g \sqrt{\frac{n L_f}{B_1 \sqrt{d_1} \rho_g C_f}} + \frac{n L_f \sigma_0^2}{B_2 B_1 \epsilon'} + \frac{C_f \sigma_1^2}{B_2 \sqrt{d_1} \rho_g \epsilon'} + \frac{C_f C_g^2}{B_1 \sqrt{d_1} \rho_g \epsilon'} + \frac{C_f C_g^2}{\sqrt{d_1} \rho_g \epsilon'}\right) = \tilde{O}(\frac{n \sigma_0^2}{B_2 B_1 \epsilon \epsilon'})$ iterations. + +In this section, we provide the convergence analysis for ALEXR2 in Algorithm 2. + +Lemma C.2. $F_{\lambda ,\nu}(\mathbf{w})$ is $L_{F_{\lambda ,\nu}}$ smooth with $L_{F_{\lambda ,\nu}} = \max \left\{\frac{1}{\nu},\frac{\rho_{F_{\lambda}}}{1 - \nu\rho_{F_{\lambda}}}\right\}$ and $\rho_{F_{\lambda}} = \sqrt{d_1} C_f\rho_g$ + +Lemma C.3. The inner min-max problem in 6 is a $(\frac{1}{\nu} - \rho_{F_{\lambda}})$ -strongly-convex and $\lambda$ -strongly-concave problem when $\nu \in (0, \rho_{F_{\lambda}}^{-1})$ . + +Proof. As discussed in Section 4.2, the term $\mathbf{y}_i^\top g_i(\mathbf{w})$ is $\rho_{F_\lambda}$ -weakly convex with $\rho_{F_\lambda} = \sqrt{d_1} C_f\rho_g$ . Therefore, when $\nu \in (0,\rho_{F_\lambda}^{-1})$ , the objective in (6) becomes strongly convex in $\mathbf{z}$ for any fixed $\mathbf{y} \in \mathbb{R}^{nd_1}$ . + +To establish strong concavity in $\mathbf{y}$ , we note that the term $\bar{f}_i^*$ is the Fenchel conjugate of a $\frac{1}{\lambda}$ -smooth function $\bar{f}_i$ . It is a standard result in convex analysis (see, e.g., Theorem 18.15 in [47]) that the conjugate of an $L$ -smooth function is $\frac{1}{L}$ -strongly concave. Thus, $\bar{f}_i^*$ is $\lambda$ -strongly convex. + +The remaining terms in the objective are either independent of $\mathbf{y}$ or linear in $\mathbf{y}$ , and hence do not affect strong concavity. Therefore, the objective is strongly convex in $\mathbf{z}$ and strongly concave in $\mathbf{y}$ when $\nu \in (0, \rho_{F_{\lambda}}^{-1})$ . + +# C.1 Proof of Technical Lemma + +Note that our proof for theorem C.1 is almost the same as that of theorem 1 in [5], we only show the difference in this and the following subsections. + +Lemma C.4 (Generalization of Lemma 7 in ALEXR [5] to weakly-convex $g_{i}$ ). Suppose that Assumptions 3.2(A2), 4.5 hold. Then, the following holds for Algorithm 2. + +$$ +\begin{array}{l} \frac {1}{n} \mathbb {E} \sum_ {i = 1} ^ {n} \left\langle g _ {i} (\mathbf {w} _ {k + 1}) - g _ {i} (\mathbf {w} _ {*}), \bar {y} _ {k + 1} ^ {(i)} \right\rangle - \mathbb {E} \left\langle G _ {k}, \mathbf {w} _ {k + 1} - \mathbf {w} _ {*} \right\rangle \\ \leq \frac {\frac {C _ {f} ^ {2} \sigma_ {1} ^ {2}}{B _ {2}} + \frac {C _ {f} ^ {2} C _ {g} ^ {2}}{B _ {1}}}{\frac {1}{\eta} + \frac {1}{\nu}} + \frac {L _ {g} C _ {f}}{2} \| \mathbf {w} _ {k + 1} - \mathbf {w} _ {k} \| _ {2} ^ {2} + \frac {\sqrt {d _ {1}} C _ {f} L _ {g}}{2} \| \mathbf {w} _ {k} - \mathbf {w} ^ {*} \| _ {2} ^ {2} \tag {24} \\ \end{array} +$$ + +Proof. For this lemma we only highlight the difference to the proof of lemma 7 in ALEXR [5]. We define $\Delta_k := \frac{1}{B_1} \sum_{i \in \mathcal{B}_1^k} [\nabla g_i(\mathbf{w}_k; \tilde{\mathcal{B}}_{i,2}^k)]^\top y_{k+1}^{(i)} - \frac{1}{n} \sum_{i=1}^n [\nabla g_i(\mathbf{w}_k)]^\top \bar{y}_{k+1}^{(i)}$ . + +$$ +\begin{array}{l} \frac {1}{n} \sum_ {i = 1} ^ {n} \left\langle g _ {i} (\mathbf {w} _ {k + 1}) - g _ {i} (\mathbf {w} _ {*}), \tilde {y} _ {k + 1} ^ {(i)} \right\rangle - \langle G _ {k}, \mathbf {w} _ {k + 1} - \mathbf {w} _ {*} \rangle \\ = \frac {1}{n} \sum_ {i = 1} ^ {n} \left\langle g _ {i} (\mathbf {w} _ {k + 1}) - g _ {i} (\mathbf {w} _ {k}), \bar {y} _ {k + 1} ^ {(i)} \right\rangle + \frac {1}{n} \sum_ {i = 1} ^ {n} \left\langle g _ {i} (\mathbf {w} _ {k}) - g _ {i} (\mathbf {w} _ {*}), \bar {y} _ {k + 1} ^ {(i)} \right\rangle \\ + \left\langle \frac {1}{n} \sum_ {i = 1} ^ {n} [ \nabla g _ {i} (\mathbf {w} _ {k}) ] ^ {\top} \bar {y} _ {k + 1} ^ {(i)} + \Delta_ {k}, \mathbf {w} _ {*} - \mathbf {w} _ {k + 1} \right\rangle \\ \end{array} +$$ + +$$ +\begin{array}{l} \stackrel {g _ {i} \text {s m o o t h} ^ {6}} {\leq} \frac {1}{n} \sum_ {i = 1} ^ {n} \left\langle g _ {i} (\mathbf {w} _ {k + 1}) - g _ {i} (\mathbf {w} _ {k}), \bar {y} _ {k + 1} ^ {(i)} \right\rangle + \left\langle \frac {1}{n} \sum_ {i = 1} ^ {n} [ \nabla g _ {i} (\mathbf {w} _ {k}) ] ^ {\top} \bar {y} _ {k + 1} ^ {(i)}, \mathbf {w} _ {k} - \mathbf {w} _ {*} \right\rangle \\ + \frac {L _ {g}}{2} \left\| \mathbf {w} _ {k} - \mathbf {w} ^ {*} \right\| ^ {2} * \frac {1}{n} \sum_ {i = 1} ^ {n} \left\langle \bar {y} _ {k + 1} ^ {(i)}, \mathbf {1} \right\rangle + \left\langle \frac {1}{n} \sum_ {i = 1} ^ {n} [ \nabla g _ {i} (\mathbf {w} _ {k}) ] ^ {\top} \bar {y} _ {k + 1} ^ {(i)} + \Delta_ {k}, \mathbf {w} _ {*} - \mathbf {w} _ {k + 1} \right\rangle \\ \end{array} +$$ + +$$ +\begin{array}{l} = \frac {1}{n} \sum_ {i = 1} ^ {n} \left\langle \bar {y} _ {k + 1} ^ {(i)}, g _ {i} (\mathbf {w} _ {k + 1}) - g _ {i} (\mathbf {w} _ {k}) \right\rangle + \left\langle \frac {1}{n} \sum_ {i = 1} ^ {n} [ \nabla g _ {i} (\mathbf {w} _ {k}) ] ^ {\top} \bar {y} _ {k + 1} ^ {(i)}, \mathbf {w} _ {k} - \mathbf {w} _ {k + 1} \right\rangle \\ + \left\langle \Delta_ {k}, \mathbf {w} _ {*} - \mathbf {w} _ {k + 1} \right\rangle + \frac {\sqrt {d _ {1}} C _ {f} L _ {g}}{2} \left\| \mathbf {w} _ {k} - \mathbf {w} ^ {*} \right\| ^ {2} \tag {25} \\ \end{array} +$$ + +Note that we have an extra term in blue in 25 comparing with lemma 7 in [5] due to the weakly convexity of $g_{i}$ . And the remaining part is just the same so we omit it. + +![](images/cded3582bd2e0a792fd15ce031ace017eea289c07e708a1452341ccf541d71d5.jpg) + +Lemma C.5 (Generalization of Lemma 8 in ALEXR to weakly-convex $g_{i}$ ). Suppose that assumptions 3.2(A3), 4.5 hold. Then, the following holds for Algorithm 2. + +$$ +\begin{array}{l} \frac {1}{n} \mathbb {E} \sum_ {i = 1} ^ {n} \left\langle g _ {i} (\mathbf {w} _ {k + 1}) - g _ {i} (\mathbf {w} _ {*}), \bar {y} _ {k + 1} ^ {(i)} \right\rangle - \mathbb {E} \left\langle G _ {k}, \mathbf {w} _ {k + 1} - \mathbf {w} _ {*} \right\rangle \\ \leq \frac {\frac {C _ {f} ^ {2} \sigma_ {1} ^ {2}}{B _ {2}} + \frac {C _ {f} ^ {2} C _ {g} ^ {2}}{B _ {1}} + 4 C _ {f} ^ {2} C _ {g} ^ {2}}{\frac {1}{\eta} + \frac {1}{\nu}} + \frac {\frac {1}{\eta} + \frac {1}{\nu}}{4} \| \mathbf {w} _ {k + 1} - \mathbf {w} _ {k} \| _ {2} ^ {2} + \frac {\sqrt {d _ {1}} C _ {f} \rho_ {g}}{2} \| \mathbf {w} _ {k} - \mathbf {w} ^ {*} \| ^ {2} \tag {26} \\ \end{array} +$$ + +Proof. Similar to lemma C.4, we have an extra term $\frac{\sqrt{d_1}C_f\rho_g}{2}\left\| \mathbf{w}_k - \mathbf{w}^*\right\|^2$ in the upper bound for weakly convex $g_{i}$ . + +![](images/2fa5c3cd44c67251bf930a5defa8251af5a47898b089e272a963544a5dbe326f.jpg) + +# C.2 Proof of Inner Loop + +Proof of theorem C.1. Similar to theorem 1 in [5] but use lemma C.5 instead of lemma 8 in ALEXR, we have + +$$ +\begin{array}{l} \mathbb {E} \left[ L \left(\mathbf {w} _ {k + 1}, y _ {*}\right) - L \left(\mathbf {w} _ {*}, \bar {y} _ {k + 1}\right) \right] \\ \leq \frac {\frac {1}{\gamma} + \left(1 - \frac {B _ {1}}{n}\right)}{B _ {1}} \mathbb {E} \left[ U _ {\bar {f} _ {i} ^ {*}} \left(y _ {*}, y _ {k}\right) \right] - \frac {\frac {1}{\gamma} + 1}{B _ {1}} \mathbb {E} \left[ U _ {\bar {f} _ {i} ^ {*}} \left(y _ {*}, y _ {k + 1}\right) \right] \\ + \left(\frac {1}{2 \eta} + \frac {\sqrt {d _ {1}} C _ {f} \rho_ {g}}{2}\right) \mathbb {E} \| \mathbf {w} _ {*} - \mathbf {w} _ {k} \| _ {2} ^ {2} - \left(\frac {1}{2 \eta} + \frac {1}{2 \nu}\right) \mathbb {E} \| \mathbf {w} _ {*} - \mathbf {w} _ {k + 1} \| _ {2} ^ {2} \\ - \left(\frac {1}{\gamma n} - \frac {\lambda_ {2} + \lambda_ {3} \theta}{\lambda n}\right) \mathbb {E} \left[ U _ {\bar {f} _ {i} ^ {*}} (\bar {y} _ {k + 1}, y _ {k}) \right] - \left(\frac {1}{2 \eta} - \frac {C _ {g} ^ {2}}{2 \lambda_ {2}} - \frac {\frac {1}{\eta} + \frac {1}{\nu}}{4}\right) \mathbb {E} \| \mathbf {w} _ {k + 1} - \mathbf {w} _ {k} \| _ {2} ^ {2} + \frac {\theta C _ {g} ^ {2}}{2 \lambda_ {3}} \mathbb {E} \| \mathbf {w} _ {k} - \mathbf {w} _ {k - 1} \| _ {2} ^ {2} \\ + \mathbb {E} \left[ \Gamma_ {k + 1} - \theta \Gamma_ {k} \right] + \frac {2 (1 + 2 \theta) \sigma_ {0} ^ {2}}{B \lambda \left(1 + \frac {1}{\gamma}\right)} + \frac {\frac {C _ {f} ^ {2} \sigma_ {1} ^ {2}}{B _ {2}} + \frac {C _ {f} ^ {2} C _ {g} ^ {2}}{B _ {1}} + 4 C _ {f} ^ {2} C _ {g} ^ {2}}{\frac {1}{\eta} + \frac {1}{\nu}}. \tag {27} \\ \end{array} +$$ + +Define $\Upsilon_k^{\mathbf{w}}\coloneqq \frac{1}{2}\mathbb{E}\left\| \mathbf{w}_* - \mathbf{w}_k\right\| _2^2$ and $\Upsilon_{k}^{y} = \frac{1}{S}\mathbb{E}U_{\bar{f}_{i}^{*}}(y_{*},y_{k})$ . Note that $L(\mathbf{w}_{k + 1},y_{*}) - L(\mathbf{w}_{*},\bar{y}_{k + 1})\geq 0$ . Multiply both sides of (27) by $\theta^{-k}$ and do telescoping sum from $k = 0$ to $K_{t} - 1$ . Add $\eta \theta^{-K_t}\Upsilon_{K_t}^{\mathbf{w}}$ to both sides. + +$$ +\begin{array}{l} \frac {\theta^ {- K _ {t}} \Upsilon_ {K _ {t}} ^ {\mathbf {w}}}{\eta} \leq \sum_ {k = 0} ^ {K _ {t} - 1} \theta^ {- k} \left(\left(\left(\frac {1}{\eta} + \sqrt {d _ {1}} C _ {f} \rho_ {g}\right) \Upsilon_ {k} ^ {\mathbf {w}} + \left(\frac {1}{\gamma} + \left(1 - \frac {B _ {1}}{n}\right)\right) \Upsilon_ {k} ^ {y} - \theta \mathbb {E} \Gamma_ {k}\right) \right. \\ \left. - \left(\left(\frac {1}{\eta} + \frac {1}{\nu}\right) \Upsilon_ {k + 1} ^ {\mathbf {w}} + \left(\frac {1}{\gamma} + 1\right) \Upsilon_ {k + 1} ^ {y} - \mathbb {E} \Gamma_ {k + 1}\right)\right) \\ + \frac {\theta^ {- K _ {t}} \Upsilon_ {K _ {t}} ^ {\mathbf {w}}}{\eta} + \left(\frac {2 (1 + 2 \theta) \sigma_ {0} ^ {2}}{\lambda B (1 + \frac {1}{\gamma})} + \frac {\frac {C _ {f} ^ {2} \sigma_ {1} ^ {2}}{B _ {2}} + \frac {C _ {f} ^ {2} C _ {g} ^ {2}}{B _ {1}} + 4 C _ {f} ^ {2} C _ {g} ^ {2}}{\frac {1}{\eta} + \frac {1}{\nu}}\right) \sum_ {k = 0} ^ {K _ {t} - 1} \theta^ {- k} \\ - \sum_ {k = 0} ^ {K _ {t} - 1} \theta^ {- k} \left(\frac {1}{\gamma n} - \frac {\left(\lambda_ {2} + \lambda_ {3} \theta\right)}{\lambda n}\right) \mathbb {E} \left[ U _ {\bar {f} _ {i} ^ {*}} \left(\bar {y} _ {k + 1}, y _ {k}\right) \right] \\ - \sum_ {k = 0} ^ {K _ {t} - 1} \theta^ {- k} \left(\frac {1}{2 \eta} - \frac {\frac {1}{\eta} + \frac {1}{\nu}}{4} - \frac {C _ {g} ^ {2}}{2 \lambda_ {2}} - \frac {C _ {g} ^ {2}}{2 \lambda_ {3}}\right) \mathbb {E} \| \mathbf {w} _ {k + 1} - \mathbf {w} _ {k} \| _ {2} ^ {2} - \theta^ {- K _ {t} + 1} \frac {C _ {g} ^ {2}}{2 \lambda_ {3}} \mathbb {E} \| \mathbf {w} _ {K _ {t}} - \mathbf {w} _ {K _ {t} - 1} \| _ {2} ^ {2} \tag {28} \\ \end{array} +$$ + +Let $\eta \geq \frac{1 - \theta}{\theta / \nu - \sqrt{d_1}C_f\rho_g}$ , $\theta \geq \frac{1}{2}$ and $1 / \nu > 4\sqrt{d_1} C_f\rho_g$ such that $\theta \geq \frac{\frac{1}{\eta} + \sqrt{d_1}C_f\rho_g}{\frac{1}{\eta} + \frac{1}{\nu}}$ and $\frac{1}{\gamma} + 1 \leq \frac{B_1}{n(1 - \theta)}$ such that $\theta \geq \frac{\frac{1}{\gamma} + 1 - \frac{B_1}{n}}{\frac{1}{\gamma} + 1}$ . Then, + +$$ +\begin{array}{l} \sum_ {k = 0} ^ {K _ {t} - 1} \theta^ {- k} \left(\left((\frac {1}{\eta} + \sqrt {d _ {1}} C _ {f} \rho_ {g}) \Upsilon_ {k} ^ {\mathbf {w}} + \left(\frac {1}{\gamma} + \left(1 - \frac {B _ {1}}{n}\right)\right) \Upsilon_ {k} ^ {y} - \theta \mathbb {E} \Gamma_ {k}\right) - ((\frac {1}{\eta} + \frac {1}{\nu}) \Upsilon_ {k + 1} ^ {\mathbf {w}} + (\frac {1}{\gamma} + 1) \Upsilon_ {k + 1} ^ {y} - \mathbb {E} \Gamma_ {k + 1})\right) \\ \leq \left(\frac {1}{\eta} + \sqrt {d _ {1}} C _ {f} \rho_ {g}\right) \Upsilon_ {0} ^ {\mathbf {w}} + \left(\frac {1}{\gamma} + \left(1 - \frac {B _ {1}}{n}\right)\right) \Upsilon_ {0} ^ {y} - \theta \mathbb {E} \Gamma_ {0} - \theta^ {- K _ {t} + 1} \left(\left(\frac {1}{\eta} + \frac {1}{\nu}\right) \Upsilon_ {K _ {t}} ^ {\mathbf {w}} + \left(\frac {1}{\gamma} + 1\right) \Upsilon_ {K _ {t}} ^ {y} - \mathbb {E} \Gamma_ {K _ {t}}\right). \\ \end{array} +$$ + +By setting $\mathbf{w}_{-1} = \mathbf{w}_0$ , we have $\Gamma_0 = 0$ . Besides, we have $\Gamma_{K_t} \leq \frac{1}{n}\sum_{i=1}^{n}\|g_i(\mathbf{w}_{K_t}) - g_i(\mathbf{w}_{K_{t-1}})\|\left\|y_*^{(i)} - y_k^{(i)}\right\|$ $\leq \frac{C_g}{n}\|\mathbf{w}_{K_t} - \mathbf{w}_{K_{t-1}}\|\|y_* - y_{K_t}\|$ + +$\frac{C_g^2}{2\lambda_3}\left\| \mathbf{w}_{K_t} - \mathbf{w}_{K_t - 1}\right\|^2 + \frac{\lambda_3}{2\lambda n^2} U_{\tilde{f}_i^*}(y_*, y_{K_t})$ . Note that the first term in the RHS here can + +celled out with the last term of RHS in 28. Thus, + +$$ +\begin{array}{l} \frac {\theta^ {- K _ {t}} \Upsilon_ {K _ {t}} ^ {\mathbf {w}}}{\eta} \leq \left(\frac {1}{\eta} + \sqrt {d _ {1}} C _ {f} \rho_ {g}\right) \Upsilon_ {0} ^ {\mathbf {w}} + \left(\frac {1}{\gamma} + \left(1 - \frac {B _ {1}}{n}\right)\right) \Upsilon_ {0} ^ {y} \\ - \theta^ {- K _ {t} + 1} \left(\underbrace {\left(\frac {1}{\eta} + \frac {1}{\nu}\right) \Upsilon_ {K _ {t}} ^ {\mathbf {w}}} _ {\heartsuit} + \underbrace {\left(\frac {1}{\gamma} + 1 - \frac {\lambda_ {3} S}{2 \lambda n ^ {2}}\right)} _ {\heartsuit} \Upsilon_ {K _ {t}} ^ {y} - \frac {1}{\eta \theta} \Upsilon_ {K _ {t}} ^ {\mathbf {w}}\right) \\ + \left(\frac {2 (1 + 2 \theta) \sigma_ {0} ^ {2}}{\lambda B \left(1 + \frac {1}{\gamma}\right)} + \frac {\frac {C _ {f} ^ {2} \sigma_ {1} ^ {2}}{B _ {2}} + \frac {C _ {f} ^ {2} C _ {g} ^ {2}}{B _ {1}} + 4 C _ {f} ^ {2} C _ {g} ^ {2}}{\frac {1}{\eta} + \frac {1}{\nu}}\right) \sum_ {k = 0} ^ {K _ {t} - 1} \theta^ {- k} - \sum_ {k = 0} ^ {K _ {t} - 1} \theta^ {- k} \underbrace {\left(\frac {1}{\gamma n} - \frac {\left(\lambda_ {2} + \lambda_ {3} \theta\right)}{\lambda n}\right)} _ {\odot} \mathbb {E} [ U _ {\bar {f} _ {i} ^ {*}} (\bar {y} _ {k + 1}, y _ {k}) ] \\ - \sum_ {k = 0} ^ {K _ {t} - 1} \theta^ {- k} \underbrace {\left(\frac {1}{2 \eta} - \frac {\frac {1}{\eta} + \frac {1}{\nu}}{4} - \frac {C _ {g} ^ {2}}{2 \lambda_ {2}} - \frac {C _ {g} ^ {2}}{2 \lambda_ {3}}\right)} _ {\text {心}} \mathbb {E} \| \mathbf {w} _ {k + 1} - \mathbf {w} _ {k} \| _ {2} ^ {2}. \tag {29} \\ \end{array} +$$ + +To make the $\heartsuit$ terms in (29) be non-negative, we choose $\lambda_2\asymp \lambda_3\asymp \frac{C_g\sqrt{B_1\lambda}}{\sqrt{n\sqrt{d_1}C_f\rho_g}}$ while ensuring that + +$$ +\gamma \leq O \left(\frac {\lambda n ^ {2}}{\lambda_ {3} B _ {1}} \wedge \frac {\lambda}{\lambda_ {2} + \lambda_ {3} \theta}\right) = O \left(\frac {\sqrt {n \sqrt {d _ {1}} C _ {f} \rho_ {g} \lambda}}{C _ {g} \sqrt {B _ {1}}}\right), \quad \eta \leq O \left(\frac {\sqrt {B _ {1} \lambda}}{C _ {g} \sqrt {n \sqrt {d _ {1}} C _ {f} \rho_ {g}}} \wedge \frac {1}{\rho_ {g} C _ {f}}\right). \tag {30} +$$ + +By selecting $\eta = \frac{1 - \theta}{\theta / \nu - \sqrt{d_1}C_f\rho_g}$ , $\frac{1}{\gamma} = \frac{B_1}{n(1 - \theta)} - 1$ and $\frac{1}{\nu} = 4\sqrt{d_1} C_f\rho_g$ we have that $\frac{1}{\gamma} + \left(1 - \frac{B_1}{n}\right) = \frac{B_1\theta}{n(1 - \theta)}$ . + +$$ +\begin{array}{l} \mu \Upsilon_ {K _ {t}} ^ {\mathbf {w}} \leq \mu (1 + \eta \sqrt {d _ {1}} C _ {f} \rho_ {g}) \theta^ {K _ {t}} \Upsilon_ {0} ^ {\mathbf {w}} + \frac {\left(\frac {1}{\gamma} + \left(1 - \frac {B _ {1}}{n}\right)\right) \mu (1 - \theta)}{\theta / \nu - \sqrt {d _ {1}} C _ {f} \rho_ {g}} \theta^ {K _ {t}} \Upsilon_ {0} ^ {y} \\ + \left(\frac {2 (1 + 2 \theta) \sigma_ {0} ^ {2}}{\lambda B \left(1 + \frac {1}{\gamma}\right)} + \frac {\frac {C _ {f} ^ {2} \sigma_ {1} ^ {2}}{B _ {2}} + \frac {C _ {f} ^ {2} C _ {g} ^ {2}}{B _ {1}} + 4 C _ {f} ^ {2} C _ {g} ^ {2}}{\frac {1}{\eta} + \frac {1}{\nu}}\right) \frac {\left(\theta^ {- K _ {t}} - 1\right) \theta^ {K _ {t}}}{\theta^ {- 1} - 1} \frac {(1 - \theta) \mu}{\theta / \nu - \sqrt {d _ {1}} C _ {f} \rho_ {g}} \\ \leq \mu (1 + \eta \sqrt {d _ {1}} C _ {f} \rho_ {g}) \theta^ {K _ {t}} \Upsilon_ {0} ^ {\mathbf {w}} + \frac {\frac {B _ {1}}{n} \theta \mu}{\frac {1}{2} \theta / \nu} \theta^ {K _ {t}} \Upsilon_ {0} ^ {y} + 2 \mu \nu \left(\frac {2 (1 + 2 \theta) \sigma_ {0} ^ {2}}{\lambda B _ {2} (1 + \frac {1}{\gamma})} + \frac {\frac {C _ {f} ^ {2} \sigma_ {1} ^ {2}}{B _ {2}} + \frac {C _ {f} ^ {2} C _ {g} ^ {2}}{B _ {1}} + 4 C _ {f} ^ {2} C _ {g} ^ {2}}{\frac {1}{\eta} + \frac {1}{\nu}}\right) \\ \leq \mu (1 + \eta \sqrt {d _ {1}} C _ {f} \rho_ {g}) \theta^ {K _ {t}} \Upsilon_ {0} ^ {\mathbf {w}} + 2 \theta^ {K _ {t}} \Upsilon_ {0} ^ {y} + 2 \left(\frac {2 (1 + 2 \theta) \sigma_ {0} ^ {2}}{\lambda B _ {2} (\frac {1}{\gamma} + 1)} + \eta \left(\frac {C _ {f} ^ {2} \sigma_ {1} ^ {2}}{B _ {2}} + \frac {C _ {f} ^ {2} C _ {g} ^ {2}}{B _ {1}} + 4 C _ {f} ^ {2} C _ {g} ^ {2}\right)\right). \\ \end{array} +$$ + +We select + +$$ +1 - \theta = O \left(\frac {1}{2} \wedge \frac {B _ {1}}{n} \wedge \frac {1}{C _ {g}} \sqrt {\frac {B _ {1} \lambda \sqrt {d _ {1}} C _ {f} \rho_ {g}}{n}} \wedge \frac {\lambda B _ {2} B _ {1} \epsilon^ {\prime}}{\sigma_ {0} ^ {2} n} \wedge \frac {B _ {2} \sqrt {d _ {1}} \rho_ {g} \epsilon^ {\prime}}{C _ {f} \sigma_ {1} ^ {2}} \wedge \frac {B _ {1} \sqrt {d _ {1}} \rho_ {g} \epsilon^ {\prime}}{C _ {f} C _ {g} ^ {2}} \wedge \frac {\sqrt {d _ {1}} \rho_ {g} \epsilon^ {\prime}}{C _ {f} C _ {g} ^ {2}}\right) +$$ + +to make (30) hold and + +$$ +\frac {2 (1 + 2 \theta) \sigma_ {0} ^ {2}}{\lambda B _ {2} (\frac {1}{\gamma} + 1)} + \eta \left(\frac {C _ {f} ^ {2} \sigma_ {1} ^ {2}}{B _ {2}} + \frac {C _ {f} ^ {2} C _ {g} ^ {2}}{B _ {1}} + 4 C _ {f} ^ {2} C _ {g} ^ {2}\right) \leq \frac {2 (1 + 2 \theta) (1 - \theta) \sigma_ {0} ^ {2} n}{\lambda B _ {2} B _ {1}} + \frac {(1 - \theta) \left(\frac {C _ {f} ^ {2} \sigma_ {1} ^ {2}}{B _ {2}} + \frac {C _ {f} ^ {2} C _ {g} ^ {2}}{B _ {1}} + 4 C _ {f} ^ {2} C _ {g} ^ {2}\right)}{\sqrt {d _ {1}} C _ {f} \rho_ {g}} \leq \epsilon^ {\prime}. +$$ + +Besides, we show that $\Upsilon_0^{\mathbf{w}}$ can be bounded by constant.(boundedness of $\Upsilon_0^y$ is already guaranteed by assumption 4.9). Note that $\mathbf{w}_t^* = \mathrm{prox}_{\nu F_\lambda}(\mathbf{w}_{t,0})$ we have: + +$$ +\because 0 \in \partial F _ {\lambda} (\mathbf {w} ^ {*}) + \frac {1}{\nu} (\mathbf {w} ^ {*} - \mathbf {w} _ {0}) +$$ + +$$ +\therefore \mathbf {w} _ {0} - \mathbf {w} ^ {*} \in \nu \partial F _ {\lambda} (\mathbf {w} ^ {*}) +$$ + +$$ +\therefore \Upsilon_ {0} ^ {\mathbf {w}} = \frac {1}{2} \left\| \mathbf {w} _ {0} - \mathbf {w} ^ {*} \right\| ^ {2} \leq \frac {1}{2} \left(\nu C _ {F _ {\lambda}}\right) ^ {2} = \frac {1}{2} \left(\nu C _ {f} C _ {g}\right) ^ {2} = O \left(C _ {f} ^ {4} C _ {g} ^ {2} \rho_ {g} ^ {2}\right) +$$ + +Since $L_{f} \coloneqq \frac{1}{\lambda} = O(\frac{1}{\epsilon})$ , the number of inner loop iterations needed by Algorithm 2 to make $\mu \Upsilon_{K_t}^{\mathbf{w}} \leq \epsilon'$ is + +$$ +K _ {t} = \tilde {O} \left(\frac {n}{B _ {1}} + C _ {g} \sqrt {\frac {n L _ {f}}{B _ {1} \sqrt {d _ {1}} \rho_ {g} C _ {f}}} + \frac {n L _ {f} \sigma_ {0} ^ {2}}{B _ {2} B _ {1} \epsilon^ {\prime}} + \frac {C _ {f} \sigma_ {1} ^ {2}}{B _ {2} \sqrt {d _ {1}} \rho_ {g} \epsilon^ {\prime}} + \frac {C _ {f} C _ {g} ^ {2}}{B _ {1} \sqrt {d _ {1}} \rho_ {g} \epsilon^ {\prime}} + \frac {C _ {f} C _ {g} ^ {2}}{\sqrt {d _ {1}} \rho_ {g} \epsilon^ {\prime}}\right) +$$ + +where $\tilde{O} (\cdot)$ hides the polylog $(C_fC_g\rho_g / \epsilon ')$ factor and the green term is the dominant term so $K_{t} =$ $\tilde{O}\left(\frac{n\sigma_0^2}{B_2B_1\epsilon\epsilon'}\right)$ + +# C.3 Proof of Outer loop + +Proof of Theorem 4.10. Let $\mathbf{w}_{-1} = \mathbf{w}_0$ . By lemma C.2, $F_{\lambda,\nu}(\mathbf{w}_t)$ is $L_{F_{\lambda,\nu}}$ -smooth. Since $\beta \leq \frac{1}{2}$ , we have + +$$ +\begin{array}{l} \left\| \mathbf {v} _ {t + 1} - \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t}) \right\| ^ {2} \\ = \left\| (1 - \beta) (\mathbf {v} _ {t} - \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t - 1})) + (1 - \beta) (\nabla F _ {\lambda , \nu} (\mathbf {w} _ {t - 1}) - \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t})) + \beta \left(\frac {1}{\nu} (\mathbf {w} _ {t} - \hat {\mathbf {z}} _ {t}) - \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t})\right) \right\| ^ {2} \\ \leq \left(1 + \frac {\beta}{2}\right) (1 - \beta) ^ {2} \left(\left(1 + \frac {\beta}{2}\right) \| \mathbf {v} _ {t} - \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t - 1}) \| ^ {2} + \left(1 + \frac {2}{\beta}\right) \| \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t - 1}) - \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t}) \| ^ {2}\right) \\ + (1 + \frac {2}{\beta}) \beta^ {2} \left\| \frac {1}{\nu} \left(\mathbf {w} _ {t} - \hat {\mathbf {z}} _ {t}\right) - \nabla F _ {\lambda , \nu} \left(\mathbf {w} _ {t}\right) \right\| ^ {2} \\ \leq \left(1 - \frac {\beta}{2}\right) \| \mathbf {v} _ {t} - \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t - 1}) \| ^ {2} + \frac {3 L _ {F _ {\lambda , \nu}} ^ {2}}{\beta} \| \mathbf {w} _ {t - 1} - \mathbf {w} _ {t} \| ^ {2} + 3 \beta \left\| \frac {1}{\nu} (\mathbf {w} _ {t} - \hat {\mathbf {z}} _ {t}) - \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t}) \right\| ^ {2} \\ \leq \left(1 - \frac {\beta}{2}\right) \| \mathbf {v} _ {t} - \nabla F _ {\lambda , \nu} \left(\mathbf {w} _ {t - 1}\right) \| ^ {2} + \frac {3 \alpha^ {2} L _ {F _ {\lambda , \nu}} ^ {2}}{\beta} \| \mathbf {v} _ {t} \| ^ {2} + 3 \beta \left\| \frac {1}{\nu} \left(\mathbf {w} _ {t} - \hat {\mathbf {z}} _ {t}\right) - \nabla F _ {\lambda , \nu} \left(\mathbf {w} _ {t}\right) \right\| ^ {2} \tag {31} \\ \end{array} +$$ + +Since $\alpha \leq \frac{1}{2L_{F_{\lambda,\nu}}}$ , by Lemma 2 in [46], we can obtain a result similar to (18) in Lemma B.1, that is, + +$$ +F _ {\lambda , \nu} (\mathbf {w} _ {t + 1}) \leq F _ {\lambda , \nu} (\mathbf {w} _ {t}) + \frac {\alpha}{2} \| \mathbf {v} _ {t + 1} - \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t}) \| ^ {2} - \frac {\alpha}{2} \| \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t}) \| ^ {2} - \frac {\alpha}{4} \| \mathbf {v} _ {t + 1} \| ^ {2} \tag {32} +$$ + +Multiplying both sides of (31) by $\frac{\alpha}{\beta}$ and adding them to both sides of (32), we have + +$$ +\begin{array}{l} \frac {\alpha}{2} \left\| \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t}) \right\| ^ {2} \\ \leq F _ {\lambda , \nu} (\mathbf {w} _ {t}) - F _ {\lambda , \nu} (\mathbf {w} _ {t + 1}) + \left(\frac {\alpha}{\beta} - \frac {\alpha}{2}\right) \left(\| \mathbf {v} _ {t} - \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t - 1}) \| ^ {2} - \| \mathbf {v} _ {t + 1} - \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t}) \| ^ {2}\right) \\ - \frac {\alpha}{4} \| \mathbf {v} _ {t + 1} \| ^ {2} + \frac {\alpha^ {3} L _ {F _ {\lambda , \nu}} ^ {2}}{\beta^ {2}} \| \mathbf {v} _ {t} \| ^ {2} + 3 \alpha \left\| \frac {1}{\nu} (\mathbf {w} _ {t} - \hat {\mathbf {z}} _ {t}) - \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t}) \right\| ^ {2} \\ \leq F _ {\lambda , \nu} (\mathbf {w} _ {t}) - F _ {\lambda , \nu} (\mathbf {w} _ {t + 1}) + \left(\frac {\alpha}{\beta} - \frac {\alpha}{2}\right) \left(\| \mathbf {v} _ {t} - \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t - 1}) \| ^ {2} - \| \mathbf {v} _ {t + 1} - \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t}) \| ^ {2}\right) \\ \frac {\alpha}{4} \left(\| \mathbf {v} _ {t} \| ^ {2} - \| \mathbf {v} _ {t + 1} \| ^ {2}\right) + \frac {3 \alpha}{\nu^ {2}} \left\| \hat {\mathbf {z}} _ {t} - \operatorname {p r o x} _ {\nu F _ {\lambda}} (\mathbf {w} _ {t}) \right\| ^ {2}, \\ \end{array} +$$ + +where the second inequality is because $\nabla F_{\lambda,\nu}(\mathbf{w}_t) = \frac{1}{\nu} (\mathbf{w}_t - \mathrm{prox}_{\nu F_\lambda}(\mathbf{w}_t))$ and $\alpha \leq \frac{\beta}{2L_{F_{\lambda,\nu}}}$ so $\frac{\alpha}{4} \geq \frac{\alpha^3 L_{F_{\lambda,\nu}}^2}{\beta^2}$ . + +By Theorem C.1, for $\epsilon' = \frac{\mu\nu^2\epsilon^2}{36}$ , there exists $\theta \in (0,1)$ with $1 - \theta = O(\epsilon^2)$ such that, by setting $\eta = \frac{1 - \theta}{\theta\mu}$ and $\gamma = \frac{(1 - \theta)n}{B_1}$ , inner loop of Algorithm 2(i.e. ALEXR) guarantees $\mathbb{E}\left\|\hat{\mathbf{z}}_t - \mathrm{prox}_{\nu F_\lambda}(\mathbf{w}_t)\right\|_2^2 \leq \frac{\nu^2\epsilon^2}{18}$ for any $t$ after $K_t = \tilde{O}\left(\frac{nL_f}{B_1B_2\epsilon^2} + \frac{L_f}{\epsilon^2}\right)$ iterations. + +Recall that $\mathbf{v}_0 = 0$ and $\mathbf{w}_{-1} = \mathbf{w}_0$ . Summing the inequality above over $t = 0, \dots, T - 1$ and dividing both sides by $\frac{\alpha T}{2}$ , we have: + +$$ +\begin{array}{l} \mathbb {E} [ \operatorname {d i s t} (0, \partial F _ {\lambda} (\operatorname {p r o x} _ {\nu F _ {\lambda}} (\mathbf {w} _ {\tau}))) ^ {2} ] = \frac {1}{T} \sum_ {t = 0} ^ {T - 1} \mathbb {E} \| \nabla F _ {\lambda , \nu} (\mathbf {w} _ {t}) \| ^ {2} \\ \leq \frac {2 \mathbb {E} \left(F _ {\lambda , \nu} (\mathbf {w} _ {0}) - F _ {\lambda , \nu} (\mathbf {w} _ {T})\right)}{\alpha T} + \frac {\left(\frac {2}{\beta} - 1\right) \| \nabla F _ {\lambda , \nu} (\mathbf {w} _ {0}) \| ^ {2}}{T} + \frac {6}{\nu^ {2} T} \sum_ {t = 0} ^ {T - 1} \mathbb {E} \left\| \hat {\mathbf {z}} _ {t} - \operatorname {p r o x} _ {\nu F _ {\lambda}} (\mathbf {w} _ {t}) \right\| ^ {2} \\ \leq \frac {2 \mathbb {E} (F _ {\lambda , \nu} (\mathbf {w} _ {0}) - F _ {\lambda , \nu} (\mathbf {w} _ {T}))}{\alpha T} + \frac {\left(\frac {2}{\beta} - 1\right) \| \nabla F _ {\lambda , \nu} (\mathbf {w} _ {0}) \| ^ {2}}{T} + \frac {\epsilon^ {2}}{3}, \\ \end{array} +$$ + +By setting $T = O(\epsilon^{-2})$ , we'll have $\mathbb{E}[\mathrm{dist}(0,\partial F_{\lambda}(\mathrm{prox}_{\nu F_{\lambda}}(\mathbf{w}_{\tau})))^{2}] \leq \epsilon^{2}$ , meaning that $\mathbf{w}_{\tau}$ is a nearly $\epsilon$ -stationary point of (4) in expectation. Recall that $L_{f} = O\left(\frac{1}{\epsilon}\right)$ so $K_{t} = O(\epsilon^{-3})$ . Hence, the total complexity for finding such a solution is $\sum_{t=0}^{T-1} K_{t} = \tilde{O}(\epsilon^{-5})$ . + +Proof of Corollary 4.11. When $g_{i}$ is $L_{g}$ -smooth, $F_{\lambda}(\cdot)$ is $L_{F}$ -smooth where $L_{F} = O\left(\frac{1}{\lambda}\right) = O\left(\frac{1}{\epsilon}\right)$ is defined in lemma A.1. By Theorem 4.10, + +$$ +\mathbb {E} \| \nabla F _ {\lambda} \left(\operatorname {p r o x} _ {\nu F _ {\lambda}} \left(\mathbf {w} _ {\tau}\right)\right) \| = \frac {1}{\nu} \mathbb {E} \| \operatorname {p r o x} _ {\nu F _ {\lambda}} \left(\mathbf {w} _ {\tau}\right) - \mathbf {w} _ {\tau} \| \leq O (\epsilon). +$$ + +According to Theorem C.1, for $\epsilon' = O(\epsilon^4)$ , there exists $\theta \in (0, 1)$ with $1 - \theta = O(\epsilon^4)$ such that, by setting $\eta = \frac{1 - \theta}{\theta} L_f$ and $\gamma = \frac{(1 - \theta)n}{B_1}$ in inner loop of Algorithm 2(i.e. ALEXR), $\hat{\mathbf{w}}_{\tau} = \mathrm{ALEXR}(\mathbf{w}_{\tau}, K)$ satisfies $\frac{\mu}{2}\mathbb{E}\left\|\hat{\mathbf{w}}_{\tau} - \operatorname{prox}_{\nu F_{\lambda}}(\mathbf{w}_{\tau})\right\|^{2} \leq \epsilon^{4}$ and thus + +$$ +\mathbb {E} \left\| \hat {\mathbf {w}} _ {\tau} - \operatorname {p r o x} _ {\nu F _ {\lambda}} \left(\mathbf {w} _ {\tau}\right) \right\| \leq O \left(\epsilon^ {2}\right) +$$ + +after $K = \tilde{O}\left(\frac{nL_f}{B_1B_2\epsilon^4} +\frac{L_f}{\epsilon^4}\right)$ iterations. Then we have + +$$ +\mathbb {E} \| \nabla F _ {\lambda} (\hat {\mathbf {w}} _ {\tau}) \| \leq \mathbb {E} \| \nabla F _ {\lambda} (\operatorname {p r o x} _ {\nu F _ {\lambda}} (\mathbf {w} _ {\tau})) \| + L _ {F} \mathbb {E} \left\| \operatorname {p r o x} _ {\nu F _ {\lambda}} (\mathbf {w} _ {\tau}) - \hat {\mathbf {w}} _ {\tau} \right\| \leq O (\epsilon), +$$ + +which means $\hat{\mathbf{w}}_{\tau}$ is an $\epsilon$ -stationary solution of (3). By Proposition 4.4, $\hat{\mathbf{w}}_{\tau}$ is a nearly $\epsilon$ -stationary solution of (1). By Theorem 4.10, $\mathbf{w}_{\tau}$ is found within complexity $O(\epsilon^{-5})$ and $\hat{\mathbf{w}}_{\tau} = \mathrm{ALEXR}(\mathbf{w}_{\tau}, K)$ has complexity $K = \tilde{O}\left(\frac{nL_f}{B_1B_2\epsilon^4} + \frac{L_f}{\epsilon^4}\right)$ . The total complexity for computing $\hat{\mathbf{w}}_{\tau}$ is still $\tilde{O}\left(\frac{n}{B_1B_2\epsilon^5} + \frac{1}{\epsilon^5}\right)$ . + +# D Convergence Analysis of SONEX and ALEXR2 applying to Constraint Optimization Problems + +In this section, we present the complexity analysis of Algorithm 2 when it is applied to (8) to solve the constrained optimization problem 7. + +# D.1 Proof of Proposition 5.2 and its variant + +Proof of Proposition 5.2. By the definition of $f_{\lambda}(\cdot)$ , we have + +$$ +\nabla f _ {\lambda} (\cdot) = \frac {1}{\lambda} \min \{[ \cdot ] _ {+}, \lambda \rho \}. +$$ + +Suppose $\mathbf{w}$ is a nearly $\epsilon$ -stationary point of (9), which means there exists $\hat{\mathbf{w}}$ such that $\| \mathbf{w} - \hat{\mathbf{w}} \| \leq \epsilon$ and $\mathrm{dist}(0, \partial \Phi_{\lambda}(\hat{\mathbf{w}})) \leq \epsilon$ . This means there exist $\mathbf{h}_i(\hat{\mathbf{w}}) \in \partial g_i(\hat{\mathbf{w}})$ and + +$$ +\nu_ {i} = \frac {1}{m} \nabla f _ {\lambda} (g _ {i} (\hat {\mathbf {w}})) = \frac {1}{\lambda m} \min \{[ g _ {i} (\hat {\mathbf {w}}) ] _ {+}, \lambda \rho \} +$$ + +for $i = 0,1,\ldots ,m$ such that + +$$ +\left\| \mathbf {h} _ {0} (\hat {\mathbf {w}}) + \mathbf {J} (\hat {\mathbf {w}}) \boldsymbol {\nu} \right\| \leq \epsilon \tag {33} +$$ + +where $\mathbf{J}(\hat{\mathbf{w}}) = [\mathbf{h}_1(\hat{\mathbf{w}}),\dots ,\mathbf{h}_m(\hat{\mathbf{w}})]\in \mathbb{R}^{d\times m}$ and $\pmb {\nu} = (\nu_{1},\dots ,\nu_{m})^{\top}\in \mathbb{R}^{m}$ + +Suppose $\max_{i=1,\dots,m} g_i(\hat{\mathbf{w}}) > \epsilon$ . Then there exists $k$ such that $[g_k(\hat{\mathbf{w}})]_+ > \epsilon$ . Recall that $\lambda = \frac{\epsilon}{\rho}$ . We have + +$$ +\nu_ {k} = \frac {1}{\lambda m} \min \left\{\left[ g _ {k} (\hat {\mathbf {w}}) \right] _ {+}, \lambda \rho \right\} = \frac {1}{\lambda m} \epsilon = \frac {\rho}{m}. +$$ + +By Assumption 5.1, we have + +$$ +\| \mathbf {h} _ {0} (\hat {\mathbf {w}}) + \mathbf {J} (\hat {\mathbf {w}}) \boldsymbol {\nu} \| \geq \| \mathbf {J} (\hat {\mathbf {w}}) \boldsymbol {\nu} \| - \| \mathbf {h} _ {0} (\hat {\mathbf {w}}) \| \geq \sigma_ {m i n} (\mathbf {J} (\hat {\mathbf {w}})) \| \boldsymbol {\nu} \| - \| \mathbf {h} _ {0} (\hat {\mathbf {w}}) \| \geq \frac {\delta \rho}{m} - C _ {g} > 1, +$$ + +which contradicts with (33). Therefore, we must have + +$$ +\max _ {i = 1, \dots , m} g _ {i} (\hat {\mathbf {w}}) \leq \epsilon . \tag {34} +$$ + +Finally, when $\max_{i = 1,\dots,m}g_i(\hat{\mathbf{w}})\leq \epsilon$ , we have for $\forall i = 1,2,\dots,m$ + +$$ +\left| g _ {i} (\hat {\mathbf {w}}) \nu_ {i} \right| \leq \sum_ {i = 1} ^ {m} \left| g _ {i} (\hat {\mathbf {w}}) \nu_ {i} \right| = \sum_ {i = 1} ^ {m} \left[ g _ {i} (\hat {\mathbf {w}}) \right] _ {+} \frac {\min \left\{\left[ g _ {i} (\hat {\mathbf {w}}) \right] _ {+} , \lambda \rho \right\}}{\lambda m} \leq \frac {\rho}{m} \sum_ {i = 1} ^ {m} \left[ g _ {i} (\hat {\mathbf {w}}) \right] _ {+} \leq O (\epsilon). \tag {35} +$$ + +With (33), (34) and (35), $\hat{\mathbf{w}}$ is an $\epsilon$ -KKT point of (7) so $\mathbf{w}$ is a nearly $\epsilon$ -KKT point of (7). + +Proposition 5.2 is given when $\mathbf{w}$ is a deterministic nearly $\epsilon$ -stationary solution to (9). However, the solution $\mathbf{w}$ found by our algorithms is only a nearly $\epsilon$ -stationary solution to (9) in expectation. This means there exists $\hat{\mathbf{w}}$ such that $\mathbb{E}\| \mathbf{w} - \hat{\mathbf{w}}\| \leq \epsilon$ and $\mathbb{E}\mathrm{dist}(0,\partial \Phi_{\lambda}(\hat{\mathbf{w}}))\leq \epsilon$ . In this case, we cannot prove the four inequalities in the conclusion of Proposition 5.2 hold deterministically. Instead, we can only show that the first two inequalities hold in expectation while the last two hold in high probabilities. We present this variant of Proposition 5.2 below with its proof. + +Proposition D.1. Suppose Assumptions 3.2(A3) and 5.1 hold. If $\rho >\frac{m(C_g + 1)}{\delta}$ and $\lambda = \frac{\epsilon}{\rho}$ , a nearly $\epsilon$ -stationary solution $\mathbf{w}$ to (9) in expectation is also a nearly $O(\epsilon)$ -KKT solution to the original problem (7) in the sense that there exist $\hat{\mathbf{w}}$ and $\nu_{i}\geq 0$ for $i = 1,\dots,m$ such that $\mathbb{E}\| \mathbf{w} - \hat{\mathbf{w}}\| \leq \epsilon$ and $\mathbb{E}\text{dist}(0,\partial g_0(\hat{\mathbf{w}}) + \sum_{i = 1}^{m}\partial g_i(\hat{\mathbf{w}})\nu_i))\leq O(\epsilon)$ and it holds with probability $1 - O(\epsilon)$ that $\max_{i = 1,\ldots ,m}g_i(\hat{\mathbf{w}})\leq O(\epsilon)$ and $|g_{i}(\hat{\mathbf{w}})\nu_{i}|\leq O(\epsilon),\forall i = 1,2,\dots,m.$ + +Proof. By the definition of $f_{\lambda}(\cdot)$ , we have + +$$ +\nabla f _ {\lambda} (\cdot) = \frac {1}{\lambda} = \min \{[ \cdot ] _ {+}, \lambda \rho \}. +$$ + +Suppose $\mathbf{w}$ is a nearly $\epsilon$ -stationary point of (9) in expectation, which means there exists $\hat{\mathbf{w}}$ such that $\mathbb{E}\| \mathbf{w} - \hat{\mathbf{w}}\| \leq \epsilon$ and $\mathbb{E}\mathrm{dist}(0,\partial \Phi_{\lambda}(\hat{\mathbf{w}}))\leq \epsilon$ . This means there exist $\mathbf{h}_i(\hat{\mathbf{w}})\in \partial g_i(\hat{\mathbf{w}})$ and + +$$ +\nu_ {i} = \frac {1}{m} \nabla f _ {\lambda} (g _ {i} (\hat {\mathbf {w}})) = \frac {1}{\lambda m} \min \{[ g _ {i} (\hat {\mathbf {w}}) ] _ {+}, \lambda \rho \} +$$ + +for $i = 0,1,\ldots ,m$ such that + +$$ +\mathbb {E} \left\| \mathbf {h} _ {0} (\hat {\mathbf {w}}) + \mathbf {J} (\hat {\mathbf {w}}) \boldsymbol {\nu} \right\| \leq \epsilon +$$ + +where $\mathbf{J}(\hat{\mathbf{w}}) = [\mathbf{h}_1(\hat{\mathbf{w}}),\dots ,\mathbf{h}_m(\hat{\mathbf{w}})]\in \mathbb{R}^{d\times m}$ and $\pmb {\nu} = (\nu_{1},\dots ,\nu_{m})^{\top}\in \mathbb{R}^{m}$ + +Suppose $\max_{i=1,\dots,m} g_i(\hat{\mathbf{w}}) > \epsilon$ . Then there exists $k$ such that $[g_k(\hat{\mathbf{w}})]_+ > \epsilon$ . Recall that $\lambda = \frac{\epsilon}{\rho}$ . We have + +$$ +\nu_ {k} = \frac {1}{\lambda m} \min \left\{\left[ g _ {k} (\hat {\mathbf {w}}) \right] _ {+}, \lambda \rho \right\} = \frac {1}{\lambda m} \epsilon = \frac {\rho}{m}. +$$ + +By Assumption 5.1, we have + +$$ +\| \mathbf {h} _ {0} (\hat {\mathbf {w}}) + \mathbf {J} (\hat {\mathbf {w}}) \boldsymbol {\nu} \| \geq \| \mathbf {J} (\hat {\mathbf {w}}) \boldsymbol {\nu} \| - \| \nabla g _ {0} (\hat {\mathbf {w}}) \| \geq \sigma_ {m i n} (\mathbf {J} (\hat {\mathbf {w}})) \| \boldsymbol {\nu} \| - \| \mathbf {h} _ {0} (\hat {\mathbf {w}}) \| \geq \frac {\delta \rho}{m} - C _ {g} > 1 +$$ + +Therefore, + +$$ +\begin{array}{l} \epsilon \geq \mathbb {E} \| \mathbf {h} _ {0} (\hat {\mathbf {w}}) + \mathbf {J} (\hat {\mathbf {w}}) \boldsymbol {\nu} \| \\ = \mathbb {E} \left[ \| \mathbf {h} _ {0} (\hat {\mathbf {w}}) + \mathbf {J} (\hat {\mathbf {w}}) \boldsymbol {\nu} \| \mid \max _ {i = 1, \dots , m} g _ {i} (\hat {\mathbf {w}}) > \epsilon\right) \operatorname {P r o b} \left(\max _ {i = 1, \dots , m} g _ {i} (\hat {\mathbf {w}}) > \epsilon\right) \\ + \mathbb {E} \left[ \| \mathbf {h} _ {0} (\hat {\mathbf {w}}) + \mathbf {J} (\hat {\mathbf {w}}) \boldsymbol {\nu} \| \mid \max _ {i = 1, \dots , m} g _ {i} (\hat {\mathbf {w}}) \leq \epsilon\right) \operatorname {P r o b} \left(\max _ {i = 1, \dots , m} g _ {i} (\hat {\mathbf {w}}) \leq \epsilon\right) \\ \geq \operatorname {P r o b} \left(\max _ {i = 1, \dots , m} g _ {i} (\hat {\mathbf {w}}) > \epsilon\right) \left(\frac {\delta \rho}{m} - C _ {g}\right) \\ \end{array} +$$ + +As a result, + +$$ +\operatorname {P r o b} \left(\max _ {i = 1, \dots , m} g _ {i} (\hat {\mathbf {w}}) > \epsilon\right) \leq \epsilon \left(\frac {\delta \rho}{m} - C _ {g}\right) ^ {- 1} = O (\epsilon). \tag {36} +$$ + +Finally, when $\max_{i = 1,\ldots ,m}g_i(\hat{\mathbf{w}})\leq \epsilon$ , we have for $\forall i = 1,2,\dots ,m$ + +$$ +\left| g _ {i} (\hat {\mathbf {w}}) \nu_ {i} \right| \leq \sum_ {i = 1} ^ {m} \left| g _ {i} (\hat {\mathbf {w}}) \nu_ {i} \right| = \sum_ {i = 1} ^ {m} \left[ g _ {i} (\hat {\mathbf {w}}) \right] _ {+} \frac {\min \left\{\left[ g _ {i} (\hat {\mathbf {w}}) \right] _ {+} , \lambda \rho \right\}}{\lambda m} \leq \frac {\rho}{m} \sum_ {i = 1} ^ {m} \left[ g _ {i} (\hat {\mathbf {w}}) \right] _ {+} \leq O (\epsilon). +$$ + +It then follows from (36) that for $\forall i = 1,2,\dots ,m$ + +$$ +\operatorname {P r o b} \left(\left| g _ {i} (\hat {\mathbf {w}}) v _ {i} \right| \geq O (\epsilon)\right) \leq O (\epsilon). +$$ + +![](images/da4a337c58d346293e3cbefa891520e538beed755dfeacf39493b7c209185065.jpg) + +# D.2 Sketch Proof of Theorem 5.3 + +Since the proof is almost the same as that of Theorem 4.10, we only highlight the difference in the proof. + +We first consider the case where $\{g_i\}_{i=0}^m$ are weakly convex. We will slightly modify Algorithm 2 to solve the following problem + +$$ +\begin{array}{l} \left. \min _ {\mathbf {w} \in \mathbb {R} ^ {d}} \left\{\Phi_ {\lambda , \nu} (\mathbf {w}) := \min _ {\mathbf {z} \in \mathbb {R} ^ {d}} \left\{\Phi_ {\lambda} (\mathbf {z}) + \frac {1}{2 \nu} \| \mathbf {z} - \mathbf {w} \| ^ {2} \right\} \right\} \right\} (37) \\ = \min _ {\mathbf {w} \in \mathbb {R} ^ {d}} \left\{\min _ {\mathbf {z} \in \mathbb {R} ^ {d}} \max _ {y _ {i} \in [ 0, \rho ]} \left\{g _ {0} (\mathbf {z}) + \frac {1}{m} \sum_ {i = 1} ^ {m} y _ {i} g _ {i} (\mathbf {z}) - \frac {\lambda}{2} \mathbf {y} _ {i} ^ {2} + \frac {1}{2 \nu} \| \mathbf {z} - \mathbf {w} \| ^ {2} \right\} \right\}, (38) \\ \end{array} +$$ + +where $\Phi_{\lambda,\nu}(\mathbf{w})$ is defined in (9). According to Proposition D.2 in [6], the second (compositional) term in $\Phi_{\lambda}(\mathbf{w})$ is $(\rho\rho_g)$ -weakly convex. Hence, $\Phi_{\lambda}(\mathbf{w})$ is $\rho_{\Phi_{\lambda}}$ -weakly convex with $\rho_{\Phi_{\lambda}} = \rho_g + \rho\rho_g$ . If $\nu < \rho_{\Phi_{\lambda}}^{-1}$ , $\Phi_{\lambda,\nu}(\mathbf{w})$ is $(\frac{1}{\nu} - \rho_{\Phi_{\lambda}})$ -strongly convex. As done in Section (4.2), in the $t$ th iteration of Algorithm 2, we apply ALEXR to solve the inner min-max problem in (38) with $\mathbf{w} = \mathbf{w}_t$ , namely, + +$$ +\min _ {\mathbf {z}} \max _ {\mathbf {y} \in [ 0, \rho ] ^ {m}} \left\{L _ {t} (\mathbf {z}, \mathbf {y}) := g _ {0} (\mathbf {z}) + \frac {1}{m} \sum_ {i = 1} ^ {m} \left(\mathbf {y} _ {i} g _ {i} (\mathbf {w}) - \frac {\lambda}{2} y _ {i} ^ {2}\right) + \frac {1}{2 \nu} \| \mathbf {z} - \mathbf {w} _ {t} \| _ {2} ^ {2} \right\}. +$$ + +we only need to replace $G_{t,k}$ in Algorithm 2 to + +$$ +G _ {t, k} = G _ {t, k} ^ {0} + G _ {t, k} ^ {1} = \partial g _ {0} \left(\mathbf {z} _ {t, k}, \mathcal {B} _ {0, 2} ^ {t, k}\right) + \frac {1}{\left| \mathcal {B} _ {1} ^ {t , k} \right|} \sum_ {i \in \mathcal {B} _ {1} ^ {t, k}} \left[ \partial g _ {i} \left(\mathbf {z} _ {t, k}, \tilde {\mathcal {B}} _ {i, 2} ^ {t, k}\right) \right] ^ {\top} \mathbf {y} _ {t, k + 1} ^ {(i)}, \tag {39} +$$ + +where $\mathcal{B}_{0,2}^{t,k}$ is a batch of data sampled from the distribution of $\xi_0$ . + +We then consider the case where $\{g_i\}_{i=0}^m$ are smooth. In this case, we can directly apply Algorithm 1 to (9), which is different from (3) only in the extra term $g_0$ . To handle this difference, we only need to replace $G_t$ in Algorithm 1 to + +$$ +G _ {t} = G _ {t} ^ {0} + G _ {t} ^ {1} = \nabla g _ {0} \left(\mathbf {w} _ {t}, \mathcal {B} _ {0, 2} ^ {t}\right) + \frac {1}{\left| \mathcal {B} _ {1} ^ {t} \right|} \sum_ {i \in \mathcal {B} _ {1} ^ {t}} \nabla f _ {i, \lambda} \left(u _ {t, i}\right) \nabla g _ {i} \left(\mathbf {w} _ {t}, \mathcal {B} _ {i, 2} ^ {t}\right) \tag {40} +$$ + +where $\mathcal{B}_{0,2}^{t}$ is a batch of data sampled from the distribution of $\xi_0$ . With these changes, we can prove Theorem 5.3 as follows. + +Sketch Proof of Theorem 5.3. When $\{g_i\}_{i = 0}^m$ are weakly convex: + +For simplicity of notation, we drop the dependence on $t$ in the notation of all variables and parameters in the proof below. First, we need to modify Lemma 5 in [5]. We have + +$$ +\begin{array}{l} L _ {t} \left(\mathbf {z} _ {k + 1}, \mathbf {y}\right) - L _ {t} \left(\mathbf {z}, \bar {\mathbf {y}} _ {k + 1}\right) (41) \\ \leq \frac {\gamma^ {- 1}}{m} U _ {\bar {f} _ {i} ^ {*}} (\mathbf {y}, \mathbf {y} _ {k}) - \frac {\gamma^ {- 1} + 1}{m} U _ {\bar {f} _ {i} ^ {*}} (\mathbf {y}, \bar {\mathbf {y}} _ {k + 1}) - \frac {\gamma^ {- 1}}{m} U _ {\bar {f} _ {i} ^ {*}} (\bar {\mathbf {y}} _ {k + 1}, \mathbf {y} _ {k}) \\ + \frac {1}{m} \sum_ {i = 1} ^ {m} \left\langle g _ {i} \left(\mathbf {z} _ {k + 1}\right) - \tilde {g} _ {k} ^ {(i)}, y _ {i} - \bar {\mathbf {y}} _ {k + 1} ^ {(i)} \right\rangle + \frac {1}{2 \eta} \| \mathbf {z} - \mathbf {z} _ {k} \| _ {2} ^ {2} (42) \\ - \left(\frac {1}{2 \eta} + \frac {1}{4 \nu}\right) \| \mathbf {z} - \mathbf {z} _ {k + 1} \| _ {2} ^ {2} - \frac {1}{2 \eta} \| \mathbf {z} _ {k + 1} - \mathbf {z} _ {k} \| _ {2} ^ {2} \\ + \frac {1}{m} \sum_ {i = 1} ^ {m} \left\langle g _ {i} \left(\mathbf {z} _ {k + 1}\right) - g _ {i} (\mathbf {z}), \bar {\mathbf {y}} _ {k + 1} ^ {(i)} \right\rangle - \left\langle G _ {k} ^ {1}, \mathbf {z} _ {k + 1} - \mathbf {z} \right\rangle (43) \\ + g _ {0} (\mathbf {z} _ {k + 1}) - g _ {0} (\mathbf {z}) - \left\langle G _ {k} ^ {0}, \mathbf {z} _ {k + 1} - \mathbf {z} \right\rangle , \\ \end{array} +$$ + +where the blue terms above are included due to the modifications to Algorithm 2. Next, we modify Lemma 8 in [5] to handle the additional terms above. Denote that $\Delta_k^0 \coloneqq g_0'(z_k) - G_k^0$ . Note that + +$$ +- \left\langle G _ {k} ^ {0}, \mathbf {z} _ {k + 1} - \mathbf {z} \right\rangle = - \left\langle g _ {0} ^ {\prime} (\mathbf {z} _ {k}), \mathbf {z} _ {k + 1} - \mathbf {z} \right\rangle + \left\langle \Delta_ {k} ^ {0}, \mathbf {z} _ {k + 1} - \mathbf {z} \right\rangle , +$$ + +where $g_0^{\prime}(\mathbf{z}_k) \in \partial g(\mathbf{z}_k)$ is a subgradient at $\mathbf{z}_k$ . The term $\langle \Delta_k^0, \mathbf{z}_{k+1} - \mathbf{z} \rangle$ can be handled in the same way as in (24) in [5]. Due to the weak convexity of $g_0$ , we have + +$$ +g _ {0} (\mathbf {z}) - g _ {0} (\mathbf {z} _ {k}) \geq \left\langle g _ {0} ^ {\prime} (\mathbf {z} _ {k}), \mathbf {z} - \mathbf {z} _ {k} \right\rangle - \frac {\rho_ {g}}{2} \| \mathbf {z} - \mathbf {z} _ {k} \| _ {2} ^ {2}. +$$ + +Then, the remaining blue terms in (41) can be upper bounded as + +$$ +\begin{array}{l} g _ {0} (\mathbf {z} _ {k + 1}) - g _ {0} (\mathbf {z}) - \left\langle g _ {0} ^ {\prime} (\mathbf {z} _ {k}), \mathbf {z} _ {k + 1} - \mathbf {z} \right\rangle \\ = g _ {0} (\mathbf {z} _ {k + 1}) - g _ {0} (\mathbf {z} _ {k}) + g _ {0} (\mathbf {z} _ {k}) - g _ {0} (\mathbf {z}) - \left\langle g _ {0} ^ {\prime} (\mathbf {z} _ {k}), \mathbf {z} _ {k + 1} - \mathbf {z} \right\rangle \\ \leq C _ {g} \left\| \mathbf {z} _ {k + 1} - \mathbf {z} _ {k} \right\| _ {2} + \left\langle g _ {0} ^ {\prime} (\mathbf {z} _ {k}), \mathbf {z} _ {k} - \mathbf {z} \right\rangle - \left\langle g _ {0} ^ {\prime} (\mathbf {z} _ {k}), \mathbf {z} _ {k + 1} - \mathbf {z} \right\rangle + \frac {\rho_ {g}}{2} \left\| \mathbf {z} - \mathbf {z} _ {k} \right\| _ {2} ^ {2} \\ \leq \frac {4 C _ {g} ^ {2}}{\frac {1}{\eta} + \frac {1}{\nu}} + \frac {\left(\frac {1}{\eta} + \frac {1}{\nu}\right) \| \mathbf {z} _ {k + 1} - \mathbf {z} _ {k} \| _ {2} ^ {2}}{1 6} + \langle g _ {0} ^ {\prime} (\mathbf {z} _ {k}), \mathbf {z} _ {k} - \mathbf {z} _ {k + 1} \rangle + \frac {\rho_ {g}}{2} \| \mathbf {z} - \mathbf {z} _ {k} \| _ {2} ^ {2} \\ \leq \frac {8 C _ {g} ^ {2}}{\frac {1}{\eta} + \frac {1}{\nu}} + \frac {\left(\frac {1}{\eta} + \frac {1}{\nu}\right) \| \mathbf {z} _ {k + 1} - \mathbf {z} _ {k} \| _ {2} ^ {2}}{8} + \frac {\rho_ {g}}{2} \| \mathbf {z} - \mathbf {z} _ {k} \| _ {2} ^ {2}. \\ \end{array} +$$ + +Let $(\mathbf{z}_{*},\mathbf{y}_{*})$ be the saddle point of $L_{t}(\mathbf{z},\mathbf{y})$ . Then, we can get + +$$ +\begin{array}{l} \mathbb {E} \left[ L _ {t} \left(\mathbf {z} _ {k + 1}, \mathbf {y} _ {*}\right) - L _ {t} \left(\mathbf {z} _ {*}, \bar {\mathbf {y}} _ {k + 1}\right) \right] \\ \leq \frac {\gamma^ {- 1} + \left(1 - \frac {B _ {1}}{m}\right)}{B _ {1}} \mathbb {E} \left[ U _ {\bar {f} _ {i} ^ {*}} \left(\mathbf {y} _ {*}, \mathbf {y} _ {k}\right) \right] - \frac {\gamma^ {- 1} + 1}{B _ {1}} \mathbb {E} \left[ U _ {\bar {f} _ {i} ^ {*}} \left(\mathbf {y} _ {*}, \mathbf {y} _ {k + 1}\right) \right] \\ + \left(\frac {1}{2 \eta} + \frac {\sqrt {d _ {1}} C _ {f} \rho_ {g}}{2} + \frac {\rho_ {g}}{2}\right) \mathbb {E} \| \mathbf {z} _ {*} - \mathbf {z} _ {k} \| _ {2} ^ {2} - \left(\frac {1}{2 \eta} + \frac {1}{4 \nu}\right) \mathbb {E} \| \mathbf {z} _ {*} - \mathbf {z} _ {k + 1} \| _ {2} ^ {2} \\ - \left(\frac {1}{m \gamma} - \frac {\lambda_ {2} + \lambda_ {3} \theta}{\lambda m}\right) \mathbb {E} \left[ U _ {\bar {f} _ {i} ^ {*}} (\bar {\mathbf {y}} _ {k + 1}, \mathbf {y} _ {k}) \right] - \left(\frac {1}{2 \eta} - \frac {C _ {g} ^ {2}}{2 \lambda_ {2}} - \frac {3}{8} \left(\frac {1}{\eta} + \frac {1}{\nu}\right)\right) \mathbb {E} \| \mathbf {z} _ {k + 1} - \mathbf {z} _ {k} \| _ {2} ^ {2} \\ + \frac {\theta C _ {g} ^ {2}}{2 \lambda_ {3}} \mathbb {E} \| \mathbf {z} _ {k} - \mathbf {z} _ {k - 1} \| _ {2} ^ {2} + \mathbb {E} [ \Gamma_ {k + 1} - \theta \Gamma_ {k} ] + \frac {2 (1 + 2 \theta) \sigma_ {0} ^ {2}}{B _ {2} \lambda (1 + \gamma^ {- 1})} + \frac {\frac {C _ {f} ^ {2} \sigma_ {1} ^ {2}}{B _ {2}} + \frac {C _ {f} ^ {2} C _ {g} ^ {2}}{B _ {1}} + 8 C _ {f} ^ {2} C _ {g} ^ {2} + 8 C _ {g} ^ {2}}{\frac {1}{\eta} + \frac {1}{\nu}}. \tag {44} \\ \end{array} +$$ + +Following the similar proof as in Theorem C.1, the modified inner loop of ALEXR2 described in Section 5 can guarantee $\frac{\mu}{2}\mathbb{E}\left\| \hat{\mathbf{z}}_t - \mathrm{prox}_{\nu F_\lambda}(\mathbf{w}_t)\right\| _2^2\leq \epsilon '$ for any $t$ after $K_{t} = \tilde{O}\left(\frac{m}{\lambda B_{2}B_{1}\epsilon^{\prime}}\right)$ . Because + +$\lambda = \frac{\epsilon}{\rho}$ and $\rho > \frac{C_g}{\delta}$ as specified in Proposition D.1, we have $K_{t} = \tilde{O}\left(\frac{C_{g}^{2}m}{B_{2}B_{1}\delta^{2}\epsilon\epsilon^{\prime}}\right)$ . For outer loop, we directly leverage the result from Appendix C, i.e. + +$$ +T = O \left(\max \left\{\rho_ {g} + \rho \rho_ {g}, \rho C _ {g} \right\} \epsilon^ {- 2}\right) = O \left(\max \left\{\rho_ {g} + \frac {C _ {g} \rho_ {g}}{\delta}, \frac {C _ {g} ^ {2}}{\delta} \right\} \epsilon^ {- 2}\right) +$$ + +By setting $\epsilon' = \frac{\epsilon^2}{\rho_g + \rho\rho_g} = \frac{\epsilon^2}{\rho_g + \frac{C_g\rho_g}{\delta}}$ , the total iteration will be + +$$ +\tilde {\mathcal {O}} \left(\frac {m \rho_ {g} ^ {2}}{B _ {2} B _ {1} \delta^ {4} \epsilon^ {5}}\right) +$$ + +When $\{g_i\}_{i = 0}^m$ are smooth: + +The proof follows almost the same procedure as the proof of Theorem 4.7. In particular, since $\mathbf{v}_t$ and $G_{t}$ will contain the additional stochastic gradient from $g_{0}$ , we need to replace (14) by + +$$ +V _ {t + 1} := \left\| \mathbf {v} _ {t + 1} - \nabla \Phi_ {\lambda} (\mathbf {w} _ {t}) \right\| ^ {2} \tag {45} +$$ + +while $U_{t + 1},\mathbf{u}_t$ and $\mathbf{g}(\mathbf{w}_t)$ are still defined as in (15), (16) and (17). Like Lemma B.1, it holds that + +$$ +\Phi_ {\lambda} \left(\mathbf {w} _ {t + 1}\right) \leq \Phi_ {\lambda} \left(\mathbf {w} _ {t}\right) + \frac {\eta}{2} V _ {t + 1} - \frac {\eta}{2} \| \nabla \Phi_ {\lambda} \left(\mathbf {w} _ {t}\right) \| ^ {2} - \frac {\eta}{4} \| \mathbf {v} _ {t + 1} \| ^ {2}, \tag {46} +$$ + +if $\eta \leq \frac{1}{2L_F}$ . By a proof similar to Lemma 3 in [2], we can still show (19) for $V_{t+1}$ in the new definition in (45). Moreover, (20) and (21) still hold as they are not related to $g_0$ . Finally, we can still take the weighted summation of both sides (46), (19), (20), and (21) as in the proof of Theorem 4.7 to show that + +$$ +\frac {1}{T} \sum_ {t = 0} ^ {T - 1} \| \nabla \Phi_ {\lambda} (\mathbf {w} _ {t}) \| ^ {2} \leq \frac {1}{T} O \left(\frac {m}{B _ {1} \sqrt {B _ {2}} \epsilon^ {3}} + \frac {1}{\min \{B _ {1} , B _ {2} \} \epsilon^ {2}} + \frac {m}{B _ {1} B _ {2} \epsilon^ {3}}\right). +$$ + +This means Algorithm 1 finds a nearly $\epsilon$ -stationary point of $\Phi_{\lambda}(\mathbf{w})$ in expectation in $T = O\left(\frac{m}{B_1\sqrt{B_2}\epsilon^5}\right)$ iterations. Then the conclusion follows from Proposition D.1 and the fact that $C_f = O\left(\frac{1}{\delta}\right)$ . + +![](images/313d0bfea8034b9f90491226650b0d8bb776685decbb206332420913c9e3ddd7.jpg) + +# E SONEX with Adaptive Learning Rates + +In this section we will show that our algorithms can also be easily extended to adaptive learning rate while still retaining the same complexity under an additional assumption introduced later. We consider an adaptive step size update such as adam: + +$$ +\mathbf {w} _ {t + 1} = \mathbf {w} _ {t} - \tilde {\eta} \circ \mathbf {v} _ {t + 1}, \tilde {\eta} = \frac {\eta}{\sqrt {\mathbf {s} _ {t}} + \varepsilon}, \mathbf {s} _ {t + 1} = (1 - \beta^ {\prime}) \mathbf {s} _ {t} + \beta^ {\prime} G _ {t} \circ G _ {t} +$$ + +where $G_{t}$ is the overall gradient estimator mentioned in algorithm 1 and $\circ$ denotes Hadamard(element-wise) product. The following assumption has been justified for the adaptive step size by [48]. + +Assumption E.1. We assume that the adaptive learning rates $\tilde{\eta}$ are (element-wise)bounded, i.e. + +$$ +\eta c _ {l} \leq \tilde {\eta} _ {i} \leq \eta c _ {u} +$$ + +for any element $\tilde{\eta}_i$ of $\tilde{\eta}$ . + +Below we provide lemmas which are straightforward extensions of lemma B.1~B.4 to adaptive learning setting. The proof is similar except that $\| \mathbf{w}_{t + 1} - \mathbf{w}_t\| ^2\leq \| \tilde{\eta}\circ \mathbf{v}_{t + 1}\| ^2\leq \left\| \eta^2 c_u^2\mathbf{v}_{t + 1}\right\| ^2 =$ $\eta^2 c_u^2\| \mathbf{v}_{t + 1}\| ^2.$ + +Lemma E.2. (Lemma 3 in [48]) Under assumption E.1, for $\mathbf{w}_{t + 1} = \mathbf{w}_t - \tilde{\eta}\circ \mathbf{v}_{t + 1}$ , with $\eta c_{l}\leq \tilde{\eta}\leq$ $\eta c_{u}$ and $\tilde{\eta} L_F\leq \frac{c_l}{2c_u^2}$ , we have: + +$$ +F \left(\mathbf {w} _ {t + 1}\right) \leq F \left(\mathbf {w} _ {t}\right) + \frac {\eta c _ {u}}{2} V _ {t + 1} - \frac {\eta c _ {l}}{2} \| \nabla F \left(\mathbf {w} _ {t}\right) \| ^ {2} - \frac {\eta c _ {l}}{4} \| \mathbf {v} _ {t + 1} \| ^ {2} \tag {47} +$$ + +where $V_{t + 1} = \left\| \mathbf{v}_{t + 1} - \nabla F(\mathbf{w}_t)\right\|^2$ + +Lemma E.3. If $\beta \leq \frac{2}{7}$ , the gradient variance $V_{t + 1} \coloneqq \left\| \mathbf{v}_{t + 1} - \nabla F(\mathbf{w}_t) \right\|^2$ can be bounded as + +$$ +\begin{array}{l} \mathbb {E} \left[ V _ {t + 2} \right] \leq (1 - \beta) \mathbb {E} \left[ V _ {t + 1} \right] + \frac {2 L _ {F} ^ {2} \eta^ {2} c _ {u} ^ {2} \mathbb {E} \left[ \| \mathbf {v} _ {t + 1} \| ^ {2} \right]}{\beta} + \frac {3 L _ {f} ^ {2} C _ {1} ^ {2}}{n} \mathbb {E} \left[ \sum_ {\mathbf {z} _ {i} \in \mathcal {B} _ {1} ^ {t + 1}} \| u _ {t + 2, i} - u _ {t + 1, i} \| ^ {2} \right] \tag {48} \\ + \frac {2 \beta^ {2} C _ {f} ^ {2} (\zeta^ {2} + C _ {g} ^ {2})}{\min \{B _ {1} , B _ {2} \}} + 5 \beta L _ {f} ^ {2} C _ {1} ^ {2} \mathbb {E} [ U _ {t + 2} ], \\ \end{array} +$$ + +where $U_{t + 1} = \frac{1}{n}\left\| \mathbf{u}_{t + 1} - \mathbf{g}(\mathbf{w}_t)\right\|^2$ , $\mathbf{u}_t = [u_{t,1},\dots,u_{t,n}]^\top$ , $\mathbf{g}(\mathbf{w}_t) = [g_1(\mathbf{w}_t),\dots,g_n(\mathbf{w}_t)]^\top$ . + +Lemma E.4. Suppose $\gamma' = \frac{n - B_1}{B_1(1 - \gamma)} + (1 - \gamma)$ and $\gamma \leq \frac{1}{2}$ and the MSVR update is used in Algorithm 1. It holds that + +$$ +\mathbb {E} \left[ U _ {t + 2} \right] \leq \left(1 - \frac {\gamma B _ {1}}{n}\right) \mathbb {E} \left[ U _ {t + 1} \right] + \frac {8 n C _ {g} ^ {2} \eta^ {2} c _ {u} ^ {2}}{B _ {1}} \mathbb {E} \| \mathbf {v} _ {t + 1} \| ^ {2} + \frac {2 B _ {1} \gamma^ {2} \sigma_ {1} ^ {2}}{n B _ {2}}. \tag {49} +$$ + +Lemma E.5. Under the same assumptions as Lemma B.3, it holds that + +$$ +\mathbb {E} \left[ \| \mathbf {u} _ {t + 2} - \mathbf {u} _ {t + 1} \| ^ {2} \right] \leq \frac {4 B _ {1} \gamma^ {2}}{n} \mathbb {E} \left[ U _ {t + 1} \right] + \frac {9 n ^ {2} C _ {g} ^ {2} \eta^ {2} c _ {u} ^ {2}}{B _ {1}} \mathbb {E} \| \mathbf {v} _ {t + 1} \| ^ {2} + \frac {2 B _ {1} \gamma^ {2} \sigma_ {0} ^ {2}}{B _ {2}}. \tag {50} +$$ + +We then give the following theorem with similar proof technique. + +Theorem E.6. Under Assumption 3.2 (A1), 4.5 and 4.6, by setting $\lambda = \Theta(\epsilon), \beta = \Theta\left(\frac{c_{l} \min\{B_{1}, B_{2}\} \epsilon^{2}}{c_{u}}\right)$ , $\gamma = \Theta\left(\frac{c_{l} B_{2} \epsilon^{4}}{c_{u}}\right)$ , $\eta = \Theta\left(\frac{c_{l}^{1.5} B_{1} \sqrt{B_{2}}}{c_{u}^{2.5} n} \epsilon^{3}\right)$ , SONEX with $\gamma' = 1 - \gamma + \frac{n - B_{1}}{B_{1}(1 - \gamma)}$ converges to an approximate $\epsilon$ -stationary solution of (1) within $T = O\left(\frac{c_{u}^{2.5} n}{c_{l}^{2.5} B_{1} \sqrt{B_{2}}} \epsilon^{-5}\right)$ iterations. + +Combining the above theorem with Theorem 4.4, we obtain the following guarantee: + +Corollary E.7. Under Assumption 3.2(A1), 4.3, 4.5 and 4.6, with the same setting as in Theorem E.6, SONEX converges to a nearly $\epsilon$ -stationary solution of (1) within $T = O\left(\frac{c_u^{2.5}n}{c_l^{2.5}B_1\sqrt{B_2}}\epsilon^{-5}\right)$ iterations. + +Proof of theorem E.6 and corollary E.7. Since the proof is almost the same as the theorem 4.7 and corollary 4.8, we only highlight the difference. + +Let $P > 0$ be a positive constant to be decided later. Taking the weighted summation of both sides (47), (48), (49), and (50) as specified in the following formula + +$$ +\frac {1}{\eta c _ {u}} \times (4 7) + \frac {1}{\beta} \times (4 8) + P \times (4 9) + \frac {3 L _ {f} ^ {2} C _ {1} ^ {2}}{n \beta} \times (5 0), +$$ + +we have + +$$ +\begin{array}{l} \frac {1}{\eta c _ {u}} \mathbb {E} F _ {\lambda} (\mathbf {w} _ {t + 1}) + \frac {1}{\beta} \mathbb {E} V _ {t + 2} + \left(P - 5 L _ {f} ^ {2} C _ {1} ^ {2}\right) \mathbb {E} U _ {t + 2} \\ \leq \frac {1}{\eta c _ {u}} \mathbb {E} F _ {\lambda} (\mathbf {w} _ {t}) + \left(\frac {1}{\beta} - \frac {1}{2}\right) \mathbb {E} V _ {t + 1} + \left(P \left(1 - \frac {\gamma B _ {1}}{n}\right) + \frac {1 2 L _ {f} ^ {2} C _ {1} ^ {2} \gamma^ {2} B _ {1}}{\beta n ^ {2}}\right) \mathbb {E} U _ {t + 1} \\ - \left(\frac {c _ {l}}{4 c _ {u}} - \frac {2 L _ {F} ^ {2} \eta^ {2} c _ {u} ^ {2}}{\beta^ {2}} - \frac {8 n C _ {g} ^ {2} P \eta^ {2} c _ {u} ^ {2}}{B _ {1}} - \frac {2 7 n L _ {f} ^ {2} C _ {1} ^ {2} C _ {g} ^ {2} \eta^ {2} c _ {u} ^ {2}}{B _ {1} \beta}\right) \mathbb {E} \| \mathbf {v} _ {t + 1} \| ^ {2} \\ + \frac {2 \beta C _ {f} ^ {2} \left(C _ {g} ^ {2} + \sigma_ {1} ^ {2}\right)}{\min \left\{B _ {1} , B _ {2} \right\}} + \frac {2 B _ {1} \gamma^ {2} \sigma_ {1} ^ {2} P}{n B _ {2}} + \frac {6 L _ {f} ^ {2} C _ {1} ^ {2} \gamma^ {2} \sigma_ {0} ^ {2} B _ {1}}{\beta n B _ {2}} - \frac {c _ {l}}{2 c _ {u}} \mathbb {E} \| \nabla F _ {\lambda} (\mathbf {w} _ {t}) \| ^ {2} \\ \end{array} +$$ + +We choose $\eta$ such that + +$$ +\frac {c _ {l}}{4 c _ {u}} - \frac {2 L _ {F} ^ {2} \eta^ {2} c _ {u} ^ {2}}{\beta^ {2}} - \frac {8 n C _ {g} ^ {2} P \eta^ {2} c _ {u} ^ {2}}{B _ {1}} - \frac {2 7 n L _ {f} ^ {2} C _ {1} ^ {2} C _ {g} ^ {2} \eta^ {2} c _ {u} ^ {2}}{B _ {1} \beta} \geq 0. \tag {51} +$$ + +We follow proof of 4.7 to set + +$$ +P = \frac {n}{\gamma B _ {1}} \left(5 L _ {f} ^ {2} C _ {1} ^ {2} + \frac {1 2 L _ {f} ^ {2} C _ {1} ^ {2} \gamma^ {2} B _ {1}}{\beta n ^ {2}}\right). +$$ + +which satisfies the following linear equation + +$$ +P \left(1 - \frac {\gamma B _ {1}}{n}\right) + \frac {1 2 L _ {f} ^ {2} C _ {1} ^ {2} \gamma^ {2} B _ {1}}{\beta n ^ {2}} = P - 5 L _ {f} ^ {2} C _ {1} ^ {2}, +$$ + +Combining the results above gives + +$$ +\begin{array}{l} \frac {1}{\eta c _ {u}} \mathbb {E} F _ {\lambda} (\mathbf {w} _ {t + 1}) + \frac {1}{\beta} \mathbb {E} V _ {t + 2} + \left(P - 5 L _ {f} ^ {2} C _ {1} ^ {2}\right) \mathbb {E} U _ {t + 2} \\ \leq \frac {1}{\eta c _ {u}} \mathbb {E} F _ {\lambda} (\mathbf {w} _ {t}) + \frac {1}{\beta} \mathbb {E} V _ {t + 1} + \left(P - 5 L _ {f} ^ {2} C _ {1} ^ {2}\right) \mathbb {E} U _ {t + 1} \\ + \frac {2 \beta C _ {f} ^ {2} \left(C _ {g} ^ {2} + \sigma_ {1} ^ {2}\right)}{\operatorname* {m i n} \left\{B _ {1} , B _ {2} \right\}} + \frac {2 B _ {1} \gamma^ {2} \sigma_ {1} ^ {2} P}{n B _ {2}} + \frac {6 L _ {f} ^ {2} C _ {1} ^ {2} \gamma^ {2} \sigma_ {0} ^ {2} B _ {1}}{\beta n B _ {2}} - \frac {c _ {l}}{2 c _ {u}} \mathbb {E} \| \nabla F _ {\lambda} (\mathbf {w} _ {t}) \| ^ {2}. \\ \end{array} +$$ + +Summing both sides of the inequality above over $t = 0,1,\ldots ,T - 1$ and organizing term lead to + +$$ +\begin{array}{l} \frac {1}{T} \sum_ {t = 0} ^ {T - 1} \| \nabla F _ {\lambda} (\mathbf {w} _ {t}) \| ^ {2} \leq \frac {2 c _ {u}}{c _ {l} T} \left(\frac {1}{\eta c _ {u}} \left(F _ {\lambda} (\mathbf {w} _ {0}) - F _ {\lambda} (\mathbf {w} _ {T})\right) + \frac {1}{\beta} \mathbb {E} V _ {1} + \left(P - 5 L _ {f} ^ {2} C _ {1} ^ {2}\right) \mathbb {E} U _ {1}\right) \\ + \frac {c _ {u}}{c _ {l}} \left(\frac {4 \beta C _ {f} ^ {2} \left(C _ {g} ^ {2} + \sigma_ {1} ^ {2}\right)}{\operatorname* {m i n} \left\{B _ {1} , B _ {2} \right\}} + \frac {4 B _ {1} \gamma^ {2} \sigma_ {1} ^ {2} P}{n B _ {2}} + \frac {1 2 L _ {f} ^ {2} C _ {1} ^ {2} \gamma^ {2} \sigma_ {0} ^ {2} B _ {1}}{\beta n B _ {2}}\right) \tag {52} \\ \end{array} +$$ + +Note that $L_{F} = O(\frac{1}{\epsilon})$ , $L_{f} = O(\frac{1}{\epsilon})$ and $P = O(\frac{n}{B_1\gamma\epsilon^2} + \frac{\gamma}{n\beta\epsilon^2})$ . We require that all the terms in the right-hand side to be in the order of $O(\epsilon^2)$ . To ensure $\frac{4c_u\beta C_f^2(C_g^2 + \sigma_1^2)}{c_l\min\{B_1,B_2\}} = O(\epsilon^2)$ , we must have $\beta = O(\frac{c_l\min\{B_1,B_2\}\epsilon^2}{c_u})$ . To ensure $\frac{4c_uB_1\gamma^2\sigma_1^2P}{c_lnB_2} = \frac{c_u}{c_l}O(\frac{\gamma}{B_2\epsilon^2} + \frac{B_1\gamma^3}{n^2B_2\beta\epsilon^2}) = O(\epsilon^2)$ , we must have $\gamma = O(\frac{c_lB_2\epsilon^4}{c_u})$ . With these orders of $\beta$ and $\gamma$ , we also have $P = O(\frac{c_u n}{c_l B_1 B_2 \epsilon^6})$ and $\frac{12c_uL_f^2C_1^2\gamma^2\sigma_0^2B_1}{c_l\beta nB_2} = O(\epsilon^4)$ . Therefore the last three terms in (52) are $O(\epsilon^2)$ . + +To satisfy (51), we will need to set + +$$ +\begin{array}{l} \eta = O \left(\sqrt {\frac {c _ {l}}{c _ {u} ^ {3}}} \min \left\{\frac {\beta}{L _ {F}}, \sqrt {\frac {B _ {1}}{n P}}, \frac {1}{L _ {f}} \sqrt {\frac {B _ {1} \beta}{n}} \right\}\right) \\ = O \left(\sqrt {\frac {c _ {l}}{c _ {u} ^ {3}}} \min \left\{\frac {c _ {l}}{c _ {u}} \min \{B _ {1}, B _ {2} \} \epsilon^ {3}, \frac {B _ {1} \sqrt {B _ {2}}}{n} \epsilon^ {3}, \sqrt {\frac {c _ {l} B _ {1} \min \{B _ {1} , B _ {2} \}}{c _ {u} n}} \epsilon^ {2} \right\}\right) \\ = O \left(\frac {c _ {l} ^ {1 . 5} B _ {1} \sqrt {B _ {2}}}{c _ {u} ^ {2 . 5} n} \epsilon^ {3}\right). \\ \end{array} +$$ + +We also set the sizes of $\mathcal{B}_{i,2}$ to be $O\left(\frac{1}{\epsilon^3}\right)$ so that $\mathbb{E}U_1 = O(\epsilon^3)$ . Also, it is easy to see that $F_{\lambda}(\mathbf{w}_0) - F_{\lambda}(\mathbf{w}_T) = O(1)$ and $\mathbb{E}V_1 = O(1)$ . Then the first term in (52) becomes + +$$ +\frac {1}{T} O \left(\frac {1}{\eta c _ {l}} + \frac {1}{\beta} + P \epsilon^ {3}\right) = \frac {1}{T} O \left(\frac {c _ {u} ^ {2 . 5} n}{c _ {l} ^ {2 . 5} B _ {1} \sqrt {B _ {2}} \epsilon^ {3}} + \frac {c _ {u}}{c _ {l} \min \{B _ {1} , B _ {2} \} \epsilon^ {2}} + \frac {c _ {u} n}{c _ {l} B _ {1} B _ {2} \epsilon^ {3}}\right) +$$ + +which will become $O(\epsilon^2)$ when $T = O\left(\frac{c_u^{2.5}n}{c_t^{2.5}B_1\sqrt{B_2}\epsilon^5}\right)$ . + +Then the remaining part is just the same as the proof of corollary 4.8. + +# F More Experiment Details + +# F.1 GDRO with CVaR divergence + +Motivation of GDRO Modern machine learning models are typically trained under the empirical risk minimization (ERM) framework, which treats all samples in the dataset equally. Although ERM + +often achieves strong average performance on test sets drawn from distributions similar to the training data, it can perform poorly on rare or underrepresented subpopulations, i.e., it lacks robustness to distributional shifts. The motivation of GDRO is to address the spurious correlation between features and labels, and distributional shift. GDRO partitions the dataset into groups representing different distributions and applies a robust optimization scheme that assigns more weight on the worst groups. Specifically, GDRO considers the following objective: + +$$ +\min _ {\theta} \max _ {\mathbf {p} \in \Omega} \frac {1}{n} \sum_ {g = 1} ^ {n} p _ {g} \mathbb {E} _ {(\mathbf {x}, y) \sim \mathcal {D} _ {g}} l (\theta ; (\mathbf {x}, y)) +$$ + +where $\Omega \subseteq \Delta$ is the set of distribution under consideration and $\Delta$ is the simplex. We consider here a popular choice of $\Omega = \{\mathbf{p} \in \Delta : p_i \leq \frac{1}{k}, \forall i \in [n]\}$ , which, by solving the inner-level maximization problem, leads to an equivalent reformulation 10 corresponding to the so-called CVaR divergence [5, 49]. Intuitively GDRO with CVaR divergence aims to minimize averaged loss of the top-k worst groups. + +CAMELYON17-WILDS [34] is part of the WILDS benchmark suite and consists of histopathology whole-slide images from five medical centers, with the goal of detecting metastatic tissue in lymph node biopsies. Following the WILDS setup, we frame this as a binary classification task on image patches, where the primary challenge lies in distribution shift across hospitals (domains). We construct group with attributes 'hospital' and 'slide' which generates 30 groups. + +Amazon-WILDS [50] is a text classification dataset derived from Amazon product reviews, where the goal is to predict binary sentiment (positive or negative) based on TF-IDF features of review text. The data spans multiple product categories. We construct group with the attribute 'user' which generates 1252 groups. + +CelebA [35] is a large-scale facial attribute dataset containing over 200,000 celebrity images annotated with 40 binary attributes. We select 4 attributes 'Attractive', 'Mouth_Slightly_Open', 'Male' and 'Blonde_Hair' and construct 16 groups, where 'Blonde_Hair' also serves as the target attribute for us to do classification. + +Hyperparameter tuning. We tune the same hyperparameters of different methods from the same candidates as follows for fair comparison. For the three tasks we train the models for 10, 4, 15 epochs with batch size and the number of groups within a mini batch of 256(8), 32(8), 64(4), respectively. We set $\alpha = 0.15$ for all the three dataset. We tune learning rate in $\{1\mathrm{e} - 5,2\mathrm{e} - 5,5\mathrm{e} - 5,1\mathrm{e} - 4,2\mathrm{e} - 4,5\mathrm{e} - 4,$ $1\mathrm{e} - 3,2\mathrm{e} - 3,5\mathrm{e} - 3\}$ , $\lambda$ in $\{1,0.1,0.01\}$ , $\gamma$ and $\beta$ in $\{0.1,0.2,0.5,0.8\}$ and $\gamma^\prime$ in $\{0.01,0.02,0.05,0.1,0.2\}$ . We set weight decay to be 0.01, 0.01, 0.02 for the three tasks, respectively. We use step decay (decay by $0.3\mathrm{x}$ for every 3 epochs), linear decay with 1st epoch warmup, step decay (decay by $0.2\mathrm{x}$ for every 3 epochs) for learning rate for the three tasks, respectively. + +# F.2 AUC Maximization with ROC Fairness Constraints + +In this part we perform experiment on learning a model with ROC fairness constraint [51] following the same experiment setting as [8]. Suppose the data are divided into two demographic groups $\mathcal{D}_p = \{(\mathbf{a}_i^p, b_i^p)\}_{i=1}^{n_p}$ and $\mathcal{D}_u = \{(\mathbf{a}_i^u, b_i^u)\}_{i=1}^{n_u}$ , where $\mathbf{a}$ denotes the input data and $b \in \{1, -1\}$ denotes the class label. A ROC fairness is to ensure the ROC curves for classification of the two groups are the same. + +Since the ROC curve is constructed with all possible thresholds, we follow [51] by using a set of thresholds $\Gamma = \{\tau_1,\dots ,\tau_m\}$ to define the ROC fairness. For each threshold $\tau$ , we impose a constraint that the false positive rate (FPR) and true positive rate (TPR) of the two groups are close, formulated as follow: + +$$ +h _ {\tau} ^ {+} (\mathbf {w}) := \left| \frac {1}{n _ {p} ^ {+}} \sum_ {i = 1} ^ {n _ {p}} \mathbf {I} \{b _ {i} ^ {p} = 1 \} \sigma \left(s _ {\mathbf {w}} \left(\mathbf {a} _ {i} ^ {p}\right) - \tau\right) - \frac {1}{n _ {u} ^ {+}} \sum_ {i = 1} ^ {n _ {u}} \mathbf {I} \{b _ {i} ^ {u} = 1 \} \sigma \left(s _ {\mathbf {w}} \left(\mathbf {a} _ {i} ^ {u}\right) - \tau\right) \right| - \kappa \leq 0 +$$ + +$$ +h _ {\tau} ^ {-} (\mathbf {w}) := \left| \frac {1}{n _ {p} ^ {-}} \sum_ {i = 1} ^ {n _ {p}} \mathbf {I} \{b _ {i} ^ {p} = - 1 \} \sigma (s _ {\mathbf {w}} (\mathbf {a} _ {i} ^ {p}) - \tau) - \frac {1}{n _ {u} ^ {-}} \sum_ {i = 1} ^ {n _ {u}} \mathbf {I} \{b _ {i} ^ {u} = - 1 \} \sigma (s _ {\mathbf {w}} (\mathbf {a} _ {i} ^ {u}) - \tau) \right| - \kappa \leq 0, +$$ + +![](images/b528b2224efcad37cc6ada04d96b74b807c81c93cd284d2f50677680a2bc621d.jpg) + +![](images/4566c0ddd416db8a5a2a97c567ed7dd1dbb2bfafcb79e9b222bbb5c838de4ba3.jpg) + +![](images/2967ae06dc0fb81685545a7e0a58ef277928833b4340a196bbd2a0775cad39ae.jpg) + +![](images/05855300bea9f01d31e8935729496e10979f0a5f1fa3e3409b11f2aad53c5448.jpg) + +![](images/dec9c8e664d1fd75f8ce1328a55ac206d1318444f1356de7bc9653473704157d.jpg) + +![](images/ef14e546d13836b665353232da26c3e837bb9a1aca890a0e6d9e58445b7a2f54.jpg) +Figure 4: Training curves of 5 constraint values in zero-one loss of different methods for continual learning with non-forgetting constraints when targeting the foggy class. Top: squared-hinge penalty method with different $\rho$ ; Bottom: smoothed hinge penalty method with different $\rho$ . + +![](images/fe8868ffdfdb07bba72076c27c5d3eb411de8e6c457544d9ff0f241c5f7fed30.jpg) + +![](images/c3e25f39d1f56c7b7fe3b53b364a8cc0f0d8f0325193250525b0f2f9afbf862b.jpg) + +![](images/ab7ba2752cc2831fd2fc5359959e964943233b87552cb02ff633a251f569841b.jpg) + +![](images/6e68a7275851decfa1e9685b87e0d836b7fa634a29cb5065fb668cbf20244a47.jpg) + +where $s_{\mathbf{w}}(\cdot)$ denotes a parameterized model, $\sigma(z)$ is the sigmoid function, and $\kappa > 0$ is a tolerance parameter. We use the pairwise AUC loss as the objective function: + +$$ +F(\mathbf{w}) = \frac{-1}{|\mathcal{D}_{+}||\mathcal{D}_{-}|}\sum_{\mathbf{x}_{i}\in \mathcal{D}_{+}}\sum_{\mathbf{x}_{j}\in \mathcal{D}_{-}}\sigma \big(s(\mathbf{w},\mathbf{x}_{i}) - s(\mathbf{w},\mathbf{x}_{j})\big), +$$ + +where $\mathcal{D}_{+}$ and $\mathcal{D}_{-}$ denote the set of positive and negative examples regardless their groups, respectively. We follow [8] to recast original constraint optimization as the following hinge-penalized objective: + +$$ +\min _ {\mathbf {w}} F (\mathbf {w}) + \frac {1}{2 | \Gamma |} \sum_ {\tau \in \Gamma} \beta \left[ h _ {\tau} ^ {+} (\mathbf {w}) \right] _ {+} + \beta \left[ h _ {\tau} ^ {-} (\mathbf {w}) \right] _ {+} \tag {53} +$$ + +It is notable that the penalty terms in the above problem can be formulated as $[h_{\tau}^{+}(\mathbf{w})]_{+} = f(g(\mathbf{w};\tau))$ where $f(g) = [|g_1 - g_2| - \kappa]_+$ , $g_1(\mathbf{w};\tau) = \frac{1}{n_p^+}\sum_{i=1}^{n_p}\mathbf{I}\{b_i^p = 1\}\sigma(s_{\mathbf{w}}(\mathbf{a}_i^p) - \tau)$ , $g_2(\mathbf{w};\tau) = \frac{1}{n_u^+}\sum_{i=1}^{n_u}\mathbf{I}\{b_i^u = 1\}\sigma(s_{\mathbf{w}}(\mathbf{a}_i^u) - \tau)$ . As a result, $f$ is convex, $g(\mathbf{w})$ is smooth. As a result, Algorithm 2 is applicable. + +Hyperparameter tuning. For ALEXR2, we tune $K_{t}$ in $\{5, 10\}$ , tune $\alpha$ (i.e. learning rate for outer loop) and $\eta$ (i.e. learning rate for inner loop) in $\{1\mathrm{e}-3, 1\mathrm{e}-2, 0.1, 1\}$ , and tune $\lambda$ (i.e. smoothing coefficient for outer function) and $\nu$ (i.e. smoothing coefficient for overall objective) in $\{2\mathrm{e}-2, 2\mathrm{e}-3, 2\mathrm{e}-4\}$ and $\{0.1, 0.01\}$ , respectively. We fix the hyperparameters in MSVR update for both ALEXR27 and SONX, i.e. $\hat{\gamma} = 0.8$ and $\hat{\gamma}' = \hat{\gamma}\theta = 0.1$ . For the two baseline algorithms, we tune the initial learning rate in $\{0.1, 1\mathrm{e}-2, 1\mathrm{e}-3, 1\mathrm{e}-4\}$ . We decay learning rate(outer lr for ALEXR2) at $50\%$ and $75\%$ epochs by a factor of 10. We tuned $\rho = \{4, 6, 8, 10, 20, 40\}$ for ALEXR2 and SONX and $\rho = \{10, 40, 80, 100, 200, 400, 800, 1000\}$ for SOX. We also compare a double-loop method (ICPPAC) [10, Algorithm 4], where we tune their $\eta$ in $\{0.1, 0.01\}$ , $\tau$ in $\{1, 10, 100\}$ , $\mu$ in $\{1\mathrm{e}-2, 1\mathrm{e}-3, 1\mathrm{e}-4\}$ , and fix $\theta_{t}$ to 0.1. + +# F.3 Continual learning with non-forgetting constraints + +We follow [7] and consider training a CLIP model [52] with global contrastive loss (GCL) [17] as the objective and a so-called model developmental safety (MDS) constraints on protected tasks, given by: + +$$ +\underset {\mathbf {w}} {\min} F (\mathbf {w}, \mathcal {D}) := \frac {1}{n} \sum_ {(\mathbf {x} _ {i}, \mathbf {t} _ {i}) \in \mathcal {D}} L _ {\mathrm {G C L}} (\mathbf {w}; (\mathbf {x} _ {i}, \mathbf {t} _ {i}), (\mathcal {T} _ {i} ^ {-}, \mathcal {I} _ {i} ^ {-})), +$$ + +$$ +\begin{array}{l} \text {s . t .} h _ {k} := \mathcal {L} _ {k} (\mathbf {w}, D _ {k}) - \mathcal {L} _ {k} \left(\mathbf {w} _ {\text {o l d}}, D _ {k}\right) \leq 0, k = 1, \dots , m. \end{array} +$$ + +![](images/483b1a85eff4b993eb10d17bbc4e3c7905a7ff1f898e0198ced1a9d8aaaaeda4.jpg) + +![](images/41a0454e3aee0ea0f0a67e62cd8cb220448c4920990b61de724fa6fdc8ec0876.jpg) + +![](images/d2dc5087626f9bf367baaa449f928eab532b7c0c5289e1e634022d744c6396bf.jpg) + +![](images/74445d4506fdb638c127f200740e911a83122b27d7a31af0a328417a9af01121.jpg) + +![](images/0dc8190d127b9486708d9270284189eea3e282279ea217932b79e2d633ab9b76.jpg) + +![](images/423302fd4f0e9f206292077c9dd84b8c57be6da6f3d23c43ed0245fafb674c9f.jpg) +Figure 5: Training curves of 4 constraint values in zero-one loss of different methods for continual learning with non-forgetting constraints when targeting the overcast class. Top: squared-hinge penalty method with different $\rho$ ; Bottom: smoothed hinge penalty method with different $\rho$ . + +![](images/bb264a0dfc2c05e9cc1f5f4c98daf3b64754379aa7c9049a1068f933d252bd8c.jpg) + +![](images/495af91cee98a17e2accf7e2eb3b2ba3e17bd838961788842378da3356209e4e.jpg) + +![](images/f8a4a5f123ccfa38d429cc8ec3dfa0fe9170cf663d1b937a164424e87b5bb18f.jpg) + +![](images/c1f145dbf4cb5c8306cafea99d71336662363dc2e03f9b8c9fa8aa21b9aa0dbc.jpg) + +where $L_{\mathrm{GCL}}(\mathbf{w};(\mathbf{x}_i,\mathbf{t}_i),(\mathcal{T}_i^{-},\mathcal{I}_i^{-}))$ is the two-way GCL for each image-text pair $(\mathbf{x}_i,\mathbf{t}_i)$ + +$$ +\begin{array}{l} L _ {\mathrm {G C L}} \left(\mathbf {w}; \mathbf {x} _ {i}, t _ {i}, \mathcal {T} _ {i} ^ {-}, \mathcal {I} _ {i} ^ {-}\right) := - \tau \log \frac {\exp \left(E _ {I} \left(\mathbf {w} , \mathbf {x} _ {i}\right) ^ {\top} E _ {T} \left(t _ {i}\right) / \tau\right)}{\sum_ {t _ {j} \in \mathcal {T} _ {i} ^ {-}} \exp \left(E _ {I} \left(\mathbf {x} _ {i}\right) ^ {\top} E _ {T} \left(t _ {j}\right) / \tau\right)} \\ - \tau \log \frac {\exp (E _ {T} (t _ {i}) ^ {\top} E _ {I} (\mathbf {x} _ {i}) / \tau)}{\sum_ {\mathbf {x} _ {j} \in \mathcal {I} _ {i} ^ {-}} \exp (E _ {T} (t _ {j}) ^ {\top} E _ {I} (\mathbf {x} _ {i}) / \tau)}, \\ \end{array} +$$ + +where $E_{I}(\mathbf{x})$ and $E_{T}(\mathbf{t})$ denote the normalized feature representation of an image $\mathbf{x}$ and a text $t$ , generated by visual encoder and text encoder of CLIP model, respectively. $\mathcal{T}_i^-$ denotes the set of all texts to be contrasted with respect to (w.r.t) $\mathbf{x}_i$ (including itself) and $\mathcal{I}_i^-$ denotes the set of all images to be contrasted w.r.t $t_i$ (including itself). Here, the data $\mathcal{D}$ is a target dataset. $\mathcal{L}_k(\mathbf{w},\mathcal{D}_k) = \frac{1}{n_k}\sum_{(\mathbf{x}_i,y_i)\sim \mathcal{D}_k}\ell_k(\mathbf{w},\mathbf{x}_i,y_i)$ is the loss of the model $\mathbf{w}$ on the $k$ -th protected task, where $\ell_k$ is a logistic loss for the $k$ -th classification task. + +Hyperparameter tuning. The training data for the target tasks are sampled from the training set of BDD100K and LAION400M [53], while the data used to construct the MDS constraints are sampled exclusively from BDD100K. For all the method we use the same learning rate of 1e-6 and the same weight decay 0.1. For each experiment we run for 60 epochs * 400 iterations per epoch and the batch size is 256. We tune $\rho$ in $\{0.01, 0.1, 1, 5, 10, 20, 40, 80\}$ for SONEX+smooth Hinge and in $\{1, 20, 50, 100, 200, 400, 800\}$ for SOX+squared hinge; we set $\gamma_1 = \gamma_2 = 0.8$ , $\gamma_1' = \gamma_2' = 0.1$ , $\tau = 0.05$ where $\gamma_1, \gamma_1'$ is the hyperparameter in MSVR update for objective term (i.e. $L_{\mathrm{GCL}}^8$ ) while $\gamma_2, \gamma_2'$ is the hyperparameter in MSVR update for the penalty terms. + +# F.4 Ablation Study + +We conduct a series of ablation studies to examine the effects of key hyperparameters and design choices. + +Results for varying $\beta$ We investigate the impact of different $\beta$ on model performance. We evaluate $\beta \in \{0.1, 0.3, 0.5, 0.7, 1.0\}$ and conduct experiments for SONEX on Amazon dataset for GDRO task and ALEXR2 on Adult dataset for AUC maximization with ROC fairness constraints. The results are shown in table 3 which indicates the importance of setting $\beta \leq 1.0$ . + +Results for varying $\theta$ We analyze the influence of different $\theta$ of ALEXR2 on model performance, testing $\theta \in \{\frac{1}{16},\frac{1}{8},\frac{1}{4},\frac{1}{2},1\}$ and conduct experiments for ALEXR2 on Adult dataset. As shown in Table 4, the final AUC of ALEXR2 remains relatively stable across this range, suggesting that it is not highly sensitive to $\theta$ . + +Benefit of Adam-type update Finally we compare Adam-type update with momentum-type update. We conduct experiments for SONEX on Amazon dataset and ALEXR2 on Adult dataset and the results are summarized in table 5. They clearly demonstrate the advantage of using Adam-type updates, which consistently yield superior performance. + +Table 3: Final loss and AUC of SONEX and ALEXR2 on Amazon dataset and Adult dataset respectively, with varying $\beta$ + +
SONEX w/ varying β0.10.30.51.0
Final loss0.56570.55630.57680.578
ALEXR2 w/ varying β0.10.30.51.0
Final AUC0.89750.89760.89730.8969
+ +Table 4: Final AUC of ALEXR2 with varying $\theta$ on Adult dataset + +
varying θ0.10.30.51.0
Final AUC0.89750.89760.89730.8969
+ +Table 5: Comparison between Adam-type update and Momentum-type update for SONEX and ALEXR2 on Amazon dataset and Adult dataset, respectively + +
AlgorithmAdam-typeMomentum-type
SONEX(Final Loss)0.56570.9666
ALEXR2(Final AUC)0.89750.861
+ +# F.5 Verification of Assumption 4.3 + +We verify the assumption 4.3 by computing minimum eigenvalue $\lambda_{min}$ of $\nabla \mathbf{g}(\mathbf{w})\nabla \mathbf{g}(\mathbf{w})^{\top}$ in group DRO experiment. We compute $\lambda_{min}$ for models trained on Camelyon17 and CelebA from the last epoch and report it in table 6. Our experiment results demonstrate that the minimum eigenvalue of $\nabla \mathbf{g}(\mathbf{w})\nabla \mathbf{g}(\mathbf{w})^{\top}$ remains positive after training process finishes. + +Besides, assumption 5.1 has been verified empirically in Appendix A.1 of [7]. + +# F.6 Other Details of Experiments + +Computing Resource and Running time: The experiments of AUC Maximization with ROC Fairness Constraints and the experiments of group DRO of Camelyon17 dataset and CelebA dataset in our paper is run on an A30 24G GPU, among which the first experiment takes less than 10 minutes for each run while for the second one, Camelyon17 takes about 4 hours and CelebA takes about 5 hours, for each run. The Amazon dataset of group DRO experiment is run on one A100 40GB GPUs and takes about 12 hours each run. The experiment of continual learning with non-forgetting constraints is run on two A100 40GB GPUs and takes about 12 hours each run. + +Data Split: We perform data split of CelebA dataset ourselves: within each group, samples are divided into training, validation, and test sets in an 8:1:1 ratio. For all the other datasets mentioned in our paper we use default split. + +Other Experiment results We show the training curves of individual constraint values for our two experiments about constraint optimization, as shown in Figure 4, 5 (continual learning with non-forgetting constraints) and Figure 6, 7 (AUC maximization with ROC fairness constraints). Note that we use other weather conditions except foggy to construct non-forgetting constraints since there is no foggy data in BDD100k for defining such a constraint. + +Table 6: Minimum eigen values ${\lambda }_{min}$ of $\nabla \mathbf{g}\left( \mathbf{w}\right) \nabla \mathbf{g}\left( \mathbf{w}\right) {}^{\top }$ at different solution. + +
Camelyon17seed 1seed 2seed 3
λmin0.02060.03490.0261
+ +
CelebAseed 10seed 20
λmin0.90000.2713
+ +![](images/ab7fb1e9102cf11ef86245cc0dff6e1c7915dd73e4c620127d507fba94c967ce.jpg) + +![](images/909f6a9b1fb3a2d3d005e636541a8eaa9277d7be28d2615c1c9dd38c394c9642.jpg) + +![](images/b21323ced2b3de0253fc088f0e4fb13753b824988093df95d8c6e308fea9fa20.jpg) + +![](images/568de99e6653749f53b9d801b744109970390206bcf03459d3e9eaa5fda73dec.jpg) + +![](images/0391e48dda6df412e3458012b5addc657535de798a1b8ed40f7771f66ad9e9f8.jpg) + +![](images/beb4c850f1d5c9dd07e84ec70e8f3d1f89a32a76efba8da603f9cc116f99733c.jpg) + +![](images/88253f720ab216faf4c81e746b4eb6b238648ff02653d89290dd14b9b5296301.jpg) +Figure 6: Training curves of 14 constraint values of different methods on adult dataset for AUC maximization with ROC fairness constraints. Top row: SOX with squared-hinge penalty method; Middle: SONX with Hinge penalty method; Bottom: ALEXR2 with smoothed hinge penalty method. + +![](images/18b6e0e4a1f4f6a7526cd8d54a542fbb728a1b68f32eb904c829bbef3091cba7.jpg) + +![](images/998d5b7cdbab6720b7582dcdc7f1d677adea8d9cb73ab413fee26d96174ce2b1.jpg) + +![](images/8bf73f4504558190d06bd9a2b3bc96757293fffadd2f865530dabef35bf52e0c.jpg) + +![](images/1b803e1f48e657fd327cb5f1bc6dc54211f3b3f9a0b1bf4c693a965c06386d17.jpg) + +![](images/779c5f5e8763a3e0e23ed58ec5dabc949dd9f2bce740e0f34caca34112ee26a4.jpg) + +![](images/aeeacce1124a1efa269e1602126b88c78ea96d6ff03359a2a6126d2b3e6847f0.jpg) + +![](images/ef6bb842c849e881968db48d4910ea535a3ad4e9f41ef60124ccdb13b360b95c.jpg) + +![](images/766225a99101cc2c7d9d5898b489d81db9795125a51356d0e8a12c057a89330b.jpg) + +![](images/d28e48c6e6fcab7f3290f93a8d8fe2e1b3442e2e61a83ccc4942192bdbd68dfb.jpg) +Figure 7: Training curves of 14 constraint values of different methods on COMPAS dataset for AUC maximization with ROC fairness constraints. 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In this setting, Stochastic Gradient Descent (SGD) based solvers have demonstrated strong empirical performance in recent machine learning applications, yet their theoretical guarantee to approximate the OT map is an open question. In this work, we answer it positively by providing both computational and statistical convergence guarantees of SGD. Specifically, we show that SGD methods can estimate the OT map with a minimax convergence rate of $\mathcal{O}(1 / \sqrt{n})$ , where $n$ is the number of samples drawn from $\mu$ . To establish this result, we study the averaged projected SGD algorithm, and identify a suitable projection set that contains a minimizer of the objective, even when the source measure is not compactly supported. Our analysis holds under mild assumptions on the source measure and applies to MTW cost functions, which include $\|\cdot\|^{p}$ for $p \in (1,\infty)$ . We finally provide numerical evidence for our theoretical results. + +# 1 Introduction + +Optimal Transport (OT) has become a central tool in machine learning for comparing and manipulating probability measures. A particularly important variant is the semi-discrete setting, where a continuous source distribution $\mu$ is transported to a discrete target measure $\nu$ . This formulation arises naturally in a wide range of applications, including image processing [16, 22], statistics [11, 18], and generative modeling [4, 9, 24]. Its hybrid structure bridges the gap between fully continuous and fully discrete formulations, allowing for expressive modeling while remaining more amenable to scalable numerical methods. + +Despite its practical appeal, solving semi-discrete OT efficiently and reliably at scale remains challenging. Existing methods with convergence guarantees often require full knowledge of the source density and are typically confined to low-dimensional settings. For instance, Newton-type methods [27, 23, 21] and combinatorial [2, 1] methods have been developed and are provided with convergence rates guarantees. However, these techniques become impractical in high-dimensional settings, since they need full knowledge of the source measure, and employ constructions that suffer from the curse of dimensionality such as meshes representation of the source measure. + +In high dimensional and/or when the source measure can only be accessed through samples, SGD and its variants have become a popular choice for solving semi-discrete OT, especially in applications such as generative modeling [4, 9, 24]. These methods solve the semi-dual formulation of the OT + +problem using only i.i.d. samples from $\mu$ and scale well to large data, not requiring to store samples. Yet, a fundamental theoretical gap remains: while SGD methods are widely used in practice, they lack convergence guarantees for approximating the OT map. + +Providing convergence rates for SGD to approximate the semi-discrete OT map is both a computational and a statistical problem, since using more gradient steps in SGD for the semi-discrete setting is equivalent to using more samples. From a statistical point of view, recent work by Pooladian et al. [31] offers promising evidence for the convergence of SGD: they show that semi-discrete OT escapes the curse of dimensionality, unlike the continuous setting [15], and that a convergence rate of $\mathcal{O}(1 / \sqrt{n})$ with $n$ samples is achievable when the cost to move mass is the quadratic cost $\frac{1}{2}\| \cdot \| ^2$ and the source measure is compactly supported. Moreover, this rate is minimax optimal for estimating the OT map. While their approach requires solving a discrete OT problem, i.e., first sampling points from $\mu$ and then solving the corresponding empirical problem, their key result motivates studying SGD as a way to accurately estimate the OT map in an online setting, where the estimator is refined as more samples become available, without needing to store them, as is often required in practice, for instance in generative modeling tasks + +Taken together, these observations highlight two major open questions that are answered positively in this work: + +(i) Can we establish convergence guarantees for SGD-based algorithms in the semi-discrete OT setting for OT map estimation, especially when only samples from $\mu$ are available? +(ii) Can we obtain statistical guarantees for the estimation of OT quantities (e.g., cost, potential, map) beyond the compact and quadratic setting of Pooladian et al. [31]? + +Contributions. We establish convergence guarantees for SGD-based algorithms applied to the nonregularized semi-dual formulation of optimal transport. We focus our analysis on the convergence of the averaged Projected Stochastic Gradient Descent (PSGD) algorithm, relying on a key result: the existence of a compact projection set $\mathcal{C}$ (Lemma 3.1) that contains a minimizer of the semi-dual objective, even when the source measure $\mu$ has unbounded support. To the best of our knowledge, this is the first time such a projection set has been identified in the semi-discrete OT setting without assuming boundedness of $\mu$ . This projection set allows us to derive key properties of the semi-dual functional, including a global weak form of strong convexity on $\mathcal{C}$ in all our settings (Lemma 4.2), and to extend second-order regularity results that were previously known only in the compact setting (Prop 4.1), such as local strong convexity near the optimum. + +The convergence rates for OT quantities, obtained when $n$ samples are drawn from $\mu$ (equivalently, when PSGD is run for $n$ iterations), answer the two questions posed in the introduction. These results are summarized in Table 1. Moreover, we show our convergence rate to the OT map is minimax optimal (Theorem 5.3), giving the first statistical guarantees in semi-discrete OT for MTW (defined in Section 2.1) and quadratic costs on unbounded support. + +Table 1: Summary of the convergence rate achieved by PSGD for the estimation of OT quantities + +
CostsOT costOT potentialOT mapNon compact
MTW costsO(1/n) [Cor 4.6]O(1/n) [Th 4.5]O(1/√n) [Cor 5.2]No
QuadraticO(1/n) [Cor 4.6]O(1/n) [Th 4.5]O(1/√n) [Cor 5.2]Yes
All other costsO(1/√n) [Th 3.2]No guaranteesNo guarantees1Yes
+ +# 2 Background + +# 2.1 Optimal Transport + +Considering a source and target probability measures $\mu \in \mathcal{P}(\mathbb{R}^d)$ and $\nu \in \mathcal{P}(\mathbb{R}^d)$ , with a cost $c: \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}^+$ to transfer mass, the OT problem is defined as: + +$$ +\mathrm {O T} _ {c} (\mu , \nu) := \min _ {\pi \in \Pi (\mu , \nu)} \int_ {\mathbb {R} ^ {d} \times \mathbb {R} ^ {d}} c (x, y) \mathrm {d} \pi (x, y), \tag {1} +$$ + +where $\Pi (\mu ,\nu)\coloneqq \left\{\pi \in \mathcal{P}(\mathbb{R}^d\times \mathbb{R}^d);\pi (\cdot \times \mathbb{R}^d) = \mu (\cdot),\pi (\mathbb{R}^d\times \cdot) = \nu (\cdot)\right\}$ is the set of joint probability measures on $\mathbb{R}^d\times \mathbb{R}^d$ with marginals $\mu$ and $\nu$ + +In this article, we assume that $\mu$ is continuous, while the target measure takes the form $\nu = \sum_{i=1}^{M} w_i \delta_{\{y_i\}}$ , where $\mathbf{w} = (w_1, \ldots, w_M)$ are its probability weights and $(y_1, \ldots, y_M)$ its support. We are mainly interested in OT problems where (1) has a unique deterministic solution, which we refer to as the OT map. This holds under mild assumptions in our semi-discrete setting, as guaranteed by the generalized Brenier's Theorem. + +(Generalized) Brenier's Theorem ([34], Th. 10.28). Suppose that $\mu$ is an absolutely continuous probability measure, and $\nu$ is discrete. Then, if the cost $c$ satisfies the MTW properties, as when $c = \| \cdot \| ^p$ with $p\in (1,\infty)$ , and if the OT cost in (1) is finite then it admits, up to negligible sets, a unique solution of the form $\gamma(dx,dy) = \mu(dx)\delta_{T_{\mu,\nu}(x)}(dy), \forall (x,y)$ , where: + +$$ +T _ {\mu , \nu} (x) = x - \nabla f ^ {*} (x). +$$ + +$T_{\mu, \nu}$ is referred to as the OT map. Moreover, $f^{*}$ is a solution of the dual problem of (1), given by: + +$$ +\mathrm {O T} _ {c} (\mu , \nu) = \max _ {f (x) + g (y) \leq c (x, y)} \int_ {\mathbb {R} ^ {d}} f (x) \mathrm {d} \mu (x) + \int_ {\mathbb {R} ^ {d}} g (y) \mathrm {d} \nu (y). \tag {2} +$$ + +The semi-dual problem. The semi-dual formulation of (2) is particularly useful in the semi-discrete setting and can be expressed as a concave finite-dimensional problem: + +$$ +\mathrm {O T} _ {c} (\mu , \nu) = \max _ {\mathbf {g} \in \mathbb {R} ^ {M}} \left(H (\mathbf {g}) := \int_ {\mathbb {R} ^ {d}} \mathbf {g} ^ {c} (x) \mathrm {d} \mu (x) + \sum_ {i = 1} ^ {M} w _ {j} g _ {j}\right), \tag {3} +$$ + +where $\mathbf{g} = (g_1, \dots, g_M)$ , and for all $x \in \mathbb{R}^d$ , the (vectorial) $c$ -transform is defined as $\mathbf{g}^c(x) := \min_{i \in [1, M]} \{c(x, y_i) - g_i\}$ . For any $\mathbf{g}$ , the $c$ -transform also defines the Laguerre cells $\mathbb{L}_j^c(\mathbf{g})$ for $j \in [1, M]$ as + +$$ +\mathbb {L} _ {j} ^ {c} (\mathbf {g}) := \left\{x \in \mathcal {X} \mid \mathbf {g} ^ {c} (x) = c (x, y _ {j}) - g _ {j} \right\}. +$$ + +Using this formulation and under Brenier's generalized theorem, the OT map can be described as $T_{\mu,\nu}(x) = x - \nabla(\mathbf{g}^*)^c(x) = x - y_i$ for $x$ inside $\mathbb{L}_i^c(\mathbf{g}^*)$ , where $\mathbf{g}^*$ , that we refer to as the discrete OT potential solves (3). + +MTW costs. For costs satisfying the Ma-Trudinger-Wang properties, referred to as MTW costs, such as $\| \cdot \| ^p,p\in (1,\infty)$ or Bregman divergences $\phi (x) - \phi (y) - \langle \nabla \phi (y),x - y\rangle$ for strictly convex $\phi$ , we observe improved properties for the function $H$ in (3). Specifically, the differential is defined almost everywhere, except on a $\mu$ -negligible set, and [21] proved that $H$ is even locally smooth and strongly convex on the orthogonal complement of 1. Moreover, all these costs satisfy Brenier's generalized theorem. For a detailed definition of such costs in the semi-discrete setting, see [21] or Appendix A. + +# 2.2 Stochastic approach for the semi-dual problem. + +In the semi-discrete setting, the semi-dual formulation is particularly appealing because, even when $\mu$ is a continuous measure, it reduces to a finite-dimensional problem. Efficient (quasi)-Newton schemes exist to solve this problem in low-dimensional settings when the density of $\mu$ is known [27, 23, 21]. In the more general scenario, where the dimension can be high and/or only sampled points from $\mu$ are available, we reformulate the semi-dual OT problem as a convex expected minimization problem, defined as: + +$$ +\min \left\{H (\mathbf {g}) := \mathbb {E} _ {X \sim \mu} [ h (\mathbf {g}, X) ] \mid \mathbf {g} \in \mathbb {R} ^ {M} \right\}, \tag {4} +$$ + +where for all $\mathbf{g} \in \mathbb{R}^M, x \in \mathbb{R}^d$ , $h(\mathbf{g}, x) = -\mathbf{g}^c(x) - \sum_{j=1}^{M} w_j g_j$ . Note that we deliberately multiplied the semi-dual functional by $-1$ to frame it as a convex minimization problem instead of a concave maximization problem; however, this is not a universal convention in the literature. + +No matter the cost, $H$ is subdifferentiable everywhere, and we consider the subdifferential $\partial H(\mathbf{g}) = \mathbb{E}_{X \sim \mu}[\partial_{\mathbf{g}} h(\mathbf{g}, X)]$ , where for $x \in \mathbb{R}^d$ and $j \in [1, M]$ , we define: + +$$ +\partial_ {\mathbf {g}} h (\mathbf {g}, x) _ {j} = \mathbb {1} _ {x \in \mathbb {L} _ {j} (\mathbf {g})} - w _ {j}. +$$ + +As long as $x$ is in the interior of a Laguerre cell, this subdifferential $\partial H$ is, in fact, a differential that we note $\nabla H$ . Moreover, $\partial_{\mathbf{g}}h(\mathbf{g},X)$ is an unbiased estimator of $\partial H(\mathbf{g})$ . Given access to samples from $\mu$ naturally leads to the study of stochastic gradient descent schemes of the form + +$$ +\mathbf {g} _ {n} = \mathbf {g} _ {n - 1} - \gamma_ {n} \partial_ {\mathbf {g}} h (\mathbf {g}, X _ {n}), +$$ + +where we start from an initial point $\mathbf{g}_0\in \mathbb{R}^M$ , and at each iteration $n$ , draw a sample $X_{n}$ and take a gradient step with step size $\gamma_{n} > 0$ (also referred to as the learning rate). + +SGD algorithms are well-suited for this setting [4, 9, 24], as they adapt to the number of samples drawn, have linear $\mathcal{O}(M)$ complexity per iteration, efficiently handle mini-batches through GPU parallelization, and do not require storing the drawn samples. Directly solving (4) also helps avoid discretization bias when estimating OT quantities [8, 17], which can be crucial in some applications of semi-discrete OT [9]. However, the specific structure of the OT semi-dual problem makes analyzing the convergence of SGD algorithms particularly challenging, especially regarding convergence to the optimizer $\mathbf{g}^*$ . Unlike standard cases, it does not fall within the class of well-behaved problems, such as those that are globally strongly convex. + +Regularization of the semi-dual. The idea of formulating the semi-dual OT problem as the minimization of an expectation and avoiding discretization by using SGD algorithms was introduced in [17]. However, possibly due to the lack of globally favorable properties of $H$ , they propose using the entropy-regularized version of $H$ , referred to as the entropic semi-dual $H_{\varepsilon}$ , where $\varepsilon$ is the regularization parameter. This results in a globally $1 / \varepsilon$ -smooth problem and as $\varepsilon$ vanishes, $H_{\varepsilon}$ converges to $H$ . A broader class of regularizer was also introduced and studied in [33]. Unfortunately, the theoretical analysis of SGD algorithms for the regularized problem reveals prohibitive constants in $\varepsilon^{-1}$ and higher in [33] and even for the entropic regularizer [7], making the use of a small $\varepsilon$ impractical for theoretical guarantees. Thus, avoiding both regularization and discretization bias to solve the semi-discrete problem highlights the relevance of studying SGD for the non-regularized OT problem. + +# 3 Projected Stochastic Gradient Descent on the Semi-Dual OT Problem + +# 3.1 Localizing a projection set + +In convex optimization, particularly in an online or stochastic setting, localizing a set to restrict the optimization domain and using a projection step in the gradient descent scheme can be very useful, permitting straight-forward convergence proofs [20]. Our first lemma addresses this idea in the context of the OT semi-dual problem, showing that even when the support of $\mu$ is not compact, it is still possible to localize a $\| \cdot \|_{\infty}$ -ball within which a minimizer of the semi-dual function $H$ exists. This projection set is formally defined in Lemma 3.1. + +Lemma 3.1 (Existence of a projection set). Suppose that the OT problem is well-posed, strong duality holds and (4) admits a minimum. Then, there exists a minimizer $\mathbf{g}^*$ contained in the set + +$$ +\mathcal {C} := \left\{\mathbf {g} \in \mathbb {R} ^ {M} \mid | g _ {j} | \leq \| c \| _ {K, \infty} \right\}, +$$ + +where $\| c \|_{K,\infty} \coloneqq \sup_{x \in K, j \in [1,M]} |c(x,y_j)|$ , for any compact $K$ satisfying $\mu(K) \geq 1 - \frac{1}{2} \min_j w_j$ . + +While the existence of a $\| \cdot \|_{\infty}$ -ball was previously established under the assumption that the cost function is uniformly bounded on $\mathbb{R}^d \times \mathbb{R}^d$ , or when both measures have bounded support, we extend this result to the more general semi-discrete setting. On its own, this finding may enable further theoretical developments in semi-discrete OT, where a bounded potential is often required [3, 13]. + +Example 1. Consider $\mu$ as the standard Gaussian on $\mathbb{R}^3$ , the cost function $c = \frac{1}{2}\| x - y\|^2$ , and $\nu$ as a discrete measure with $10^{7}$ points in $[0,1]^{3}$ and uniform weights. In this setting, it was previously believed that the Brenier potential could not be bounded, since $c$ is not bounded on $\mathbb{R}^3\times [0,1]^3$ . However, using Lemma 3.1, we can take the ball $B(0,6)$ , which allows us to restrict our search for the potential within a $\| \cdot \|_{\infty}$ -ball of radius 18. + +Incorporating this projection step into our SGD scheme has several advantages: (i) it significantly enhances both the practical performance and theoretical convergence of the algorithm; and (ii) + +the computational complexity of the projection step is $\mathcal{O}(M)$ , as it simply involves clipping each coordinate of the vector. The projector is defined as + +$$ +\operatorname {P r o j} _ {\mathcal {C}} (\mathbf {g}): \mathbf {g} \in \mathbb {R} ^ {M} \mapsto \operatorname {a r g m i n} \{\| \mathbf {g} - \mathbf {g} ^ {\prime} \|; \mathbf {g} ^ {\prime} \in \mathcal {C} \}. +$$ + +Based on this projection step, we derive the PSGD algorithm to minimize $H$ , as presented in Algorithm 1. Note that, in practice, this requires knowledge of a compact set $K$ , which we assume to be either given or previously estimated. The estimation is for instance trivial when the source is the standard Gaussian as in many applications (e.g. generative modeling). When the density is unknown, we can still have high probability guarantees as presented in the next paragraph. + +Estimation of the projection set $K$ . When the density of $\mu$ is unknown and we only have access to samples, we can, for instance, estimate the projection set $K$ using a centered ball $B(0,R)$ such that, with high probability, $\mu(B(0,R)) \geq 1 - \frac{1}{4} w_{\min}$ . This is a standard quantile estimation problem for the norm $\|X\|$ , where $X \sim \mu$ , and the empirical CDF $F_n(r) = \frac{1}{n} \sum_{i=1}^{n} \mathbf{1}_{\|X_i\| \leq r}$ can serve as an estimator for $R$ . Indeed, we can take $R = \inf \left\{r : F_n(r) \geq 1 - \frac{1}{8} w_{\min}\right\}$ . By the Dvoretzky-Kiefer-Wolffowitz inequality [26], for a sample size $n \geq \frac{32 \log(2/\delta)}{w_{\min}^2}$ , it holds with a + +Algorithm 1 Projected Stochastic Gradient Descent (PSGD) + +Parameters: $\gamma_{1} > 0, b \in [\frac{1}{2}, 1)$ + +Initialize $\mathbf{g}_0\in \mathcal{C}$ and $\overline{\mathbf{g}}_0 = \mathbf{g}_0$ + +for $k = 1$ to $n$ do + +Draw $x_{k}\sim \mu$ + +$$ +\mathbf {g} _ {k} = \operatorname {P r o j} _ {\mathcal {C}} \left(\mathbf {g} _ {k - 1} - \frac {\gamma_ {1}}{k ^ {b}} \partial_ {\mathbf {g}} h \left(\mathbf {g} _ {k - 1}, x _ {k}\right)\right) +$$ + +$$ +\overline {{\mathbf {g}}} _ {k} = \frac {1}{k + 1} \mathbf {g} _ {k} + \frac {k}{k + 1} \overline {{\mathbf {g}}} _ {k - 1} +$$ + +end for + +return $\mathbf{g}_n$ and $\overline{\mathbf{g}}_n$ + +probability of at least $1 - \delta$ that $\mu(B(0, R)) \geq 1 - \frac{1}{4} w_{\min}$ . This provides a dimension-free sample-complexity bound for estimating $K$ . + +# 3.2 A first convergence of PSGD in the general setting + +As a first consequence of our projection step, we can directly establish a convergence rate for PSGD on the OT cost in a general setting, without assuming additional regularity of the cost function or the source measure. The proof follows a classical result for PSGD algorithms, and our projection step provides insights into the choice of the learning rate $\gamma_{1}$ . Additionally, it allows us to recover the rate $\mathcal{O}(1 / \sqrt{n})$ from [17] for the averaged iterates of SGD when applied to the entropy-regularized semi-dual. + +Theorem 3.2 (PSGD in the general setting). In the general setting, choosing the learning rate $\gamma_{n} = \gamma_{1} / n^{b}$ with $\gamma_{1} = \mathrm{Diam}(\mathcal{C}) / 2\sqrt{2}$ and $b = 1 / 2$ , we obtain + +$$ +\mathbb {E} \left[ H \left(\overline {{\mathbf {g}}} _ {n}\right) - H \left(\mathbf {g} ^ {*}\right) \right] \leq \frac {4 \sqrt {2} \operatorname {D i a m} (\mathcal {C})}{\sqrt {n}}. +$$ + +Although most of our results focus on MTW costs, as discussed in the next section, we also establish a convergence rate in a more general setting. This broader setting lies outside the scope of the generalized Brenier theorem and an OT map may not exist or be unique. However, estimating the OT cost can still be useful in certain applications, such as when $c = \parallel \cdot \parallel$ , corresponding to the 1-Wasserstein distance. Thanks to our projection step, the same convergence rate carries over to other SGD-based methods, such as Adagrad [14], which is well-suited when the projection set is a hypercube [28, Section 4.2.4]. + +# 4 Convergence analysis of PSGD and minimax estimation for MTW costs + +We now focus on costs satisfying the MTW properties, with particular attention to the cost $c(x,y) = \frac{1}{2}\| x - y\| ^2$ , as it is the most commonly used. For this cost, we extend our results to include non-compactly supported source measures. This setting is used, for instance, in [4, 24], where a standard Gaussian is mapped to a discrete distribution. Our objective is to establish the convergence rate of PSGD in approximating the true OT map and cost by estimating the Brenier potential $\mathbf{g}^*$ . We make the following assumptions, distinguishing between the compact case, where we treat all MTW costs, and the non-compact case, where we focus solely on the quadratic cost $\frac{1}{2}\| \cdot \| ^2$ . + +Assumption A (Compact case). The cost $c$ satisfies the MTW condition, $\operatorname{Supp}(\mu)$ is bounded and $c$ -convex (see Appendix A), and $\mu$ satisfies a weighted $(1,1)$ Poincaré-Wirtinger inequality: there exists $C_{\mathrm{pw}} > 0$ such that for all $f \in \mathcal{C}^1 (\mathbb{R}^d)$ , + +$$ +\left\| f - \mathbb {E} _ {\mu} [ f ] \right\| _ {L ^ {1} (\mu)} \leq C _ {\mathrm {p w}} \| \nabla f \| _ {L ^ {1} (\mu)}. \tag {PW} +$$ + +Note that (PW) relates to the local strong convexity of the semi-dual problem near the optimum and is a common assumption as in [21, 6]. Moreover, in the compact setting, the assumption that the density $f_{\mu}$ is bounded from above and below by strictly positive values is a common assumption, as in [31]. This compact assumption implies (PW). + +Regarding the non-compact case, our assumptions are satisfied notably, for non-degenerate Gaussians, finite mixtures of non-degenerate Gaussians, and heavy-tailed distributions such as Student distributions with degree of freedom larger than 2. While a broader class of MTW costs could potentially be covered, perhaps under stronger assumptions regarding the source measure, we deliberately omit these cases to avoid further technical complexity. Moreover, to the best of our knowledge, even for the quadratic cost, there are no existing theoretical results in semi-discrete OT when the support of $\mu$ is unbounded. We further provide in Appendix A an example where assumption (B3) fails and as a consequence, the Hessian is not defined. + +Assumption B (Non-compact case). + +(B1) The cost is quadratic: $c(x,y) = \frac{1}{2}\| x - y\| ^2$ and the measure $\mu$ has a finite second-order moment. +(B2) There exists a compact set $K \subset \mathbb{R}^d$ with $\mu(K) \geq 1 - \frac{1}{4} w_{\min}$ , such that the probability measure $\mu_K$ with density $f_{\mu_K}(x) \coloneqq c_K f_\mu(x)\mathbf{1}_K(x)$ satisfies a (PW) inequality. +(B3) The density $f_{\mu}$ satisfies the following integrability and regularity condition: for $R > 1$ and $r \geq 1$ , define + +$$ +f _ {\mu} ^ {R} := f _ {\mu} \cdot \mathbf {1} _ {\| x \| \leq R}, \quad f _ {\mu} ^ {R + r} := f _ {\mu} \cdot \mathbf {1} _ {R + (r - 2) \leq \| x \| \leq R + r}. +$$ + +Assume there exist $R > 1$ , $C > 0$ and a modulus of continuity $\omega$ such that for all $\delta > 0$ , + +$$ +\sum_ {r = 0} ^ {\infty} (R + r) ^ {d - 1} \omega_ {f _ {\mu}} ^ {R + r} (\delta) \leq C \omega (\delta), \quad \sum_ {r = 0} ^ {\infty} (R + r) ^ {d - 1} C _ {f _ {\mu}} ^ {R + r} < \infty , \tag {5} +$$ + +where $C_{f_\mu}^{R + r} \coloneqq \sup_{x \in \mathbb{R}^d} f_\mu^{R + r}(x)$ and $\omega_{f_\mu}^{R + r}$ is the modulus of continuity of $f_\mu^{R + r}$ . + +# 4.1 Properties of the semi-dual $H$ + +In our context, it is known that the discrete Brenier potential $\mathbf{g}^* \in \mathbb{R}^M$ is unique only up to a transformation of the form $\mathbf{g}^* + a\mathbb{1}_M$ with $a \in \mathbb{R}$ . For clarity, we fix $\mathbf{g}^*$ to be the Brenier potential such that $\mathbf{g}^* \in \mathrm{Vect}(\mathbb{1}_M)^\perp$ . Without losing information, we thus restrict our analysis to the orthogonal complement of the subspace spanned by the vector $\mathbb{1}_M$ , $\mathrm{Vect}(\mathbb{1}_M)^\perp$ . Therefore, for any $\mathbf{g}, \mathbf{g}' \in \mathbb{R}^M$ , we define + +$$ +\left\| \mathbf {g} - \mathbf {g} ^ {\prime} \right\| _ {v} = \left\| \operatorname {P r o j} _ {\operatorname {V e c t} \left(\mathbb {1} _ {M}\right) ^ {\perp}} \left(\mathbf {g} - \mathbf {g} ^ {\prime}\right) \right\|, +$$ + +$$ +\langle \mathbf {g}, \mathbf {g} ^ {\prime} \rangle_ {v} = \langle \mathrm {P r o j} _ {\mathrm {V e c t} (\mathbb {1} _ {M}) ^ {\perp}} (\mathbf {g}), \mathrm {P r o j} _ {\mathrm {V e c t} (\mathbb {1} _ {M}) ^ {\perp}} (\mathbf {g} ^ {\prime}) \rangle . +$$ + +We start by stating the second-order regularity of $H$ in our setting. + +Proposition 4.1. Under Assumption A or B, the function $H$ is differentiable everywhere on $\mathbb{R}^M$ , and we denote its gradient by $\nabla H$ . Moreover, there exists a radius $r > 0$ such that on the ball $B(\mathbf{g}^*, r)$ , $H$ is $C^2$ and strongly convex on $B(\mathbf{g}^*, r)$ . If, in addition, $f_\mu$ is $\alpha$ -Hölder continuous with $\alpha \in (0,1]$ , then the Hessian of $H$ is also $\alpha$ -Hölder continuous. + +Naturally, the smallest eigenvalue of the Hessian is 0, with $\mathbb{1}_M$ as its eigenvector. However, we still refer to the strong convexity of $H$ since we focus on the orthogonal complement of $\operatorname{Vect}(\mathbb{1})$ . Notably, we extended the definition of the Hessian, originally provided in [21] for the compact case, to include the quadratic Euclidean cost in the non-compact setting. Furthermore, while the local strong convexity of $H$ was established in [21] for the compact case, it was defined over the set of vectors $\mathbf{g}$ such that the measures of all Laguerre cells are bounded by a positive constant. To formulate + +this result with respect to a ball, we also required a result on the quantitative stability of the measures of Laguerre cells. Additional details on this quantitative stability will be provided in section 5.1. + +As a corollary of these results, we derive the following lemma, which stems from our projection step and can be viewed as a weak form of strong convexity of $H$ on $\mathcal{C}$ , also referred to as Restricted Strong Convexity (RSC) [37]. + +Lemma 4.2. Under Assumptions $A$ or $B$ , there exists $\eta > 0$ such that $H$ satisfies a RSC property, uniformly for all $\mathbf{g} \in \mathcal{C}$ : + +$$ +\langle \nabla H (\mathbf {g}), \mathbf {g} - \mathbf {g} ^ {*} \rangle_ {v} \geq \eta \| \mathbf {g} - \mathbf {g} ^ {*} \| _ {v} ^ {2}. +$$ + +Direct convergence guarantees under RSC. As a direct consequence of the RSC property on $\mathcal{C}$ , Projected SGD achieves $\mathcal{O}(1/n)$ convergence for our OT problem when using the step size $\gamma_t = \frac{1}{\eta t}$ . However, our analysis does not provide a precise estimate of the parameter $\eta$ , which is required to effectively implement this learning rate in practice, and is particularly difficult to obtain in the OT setting. This limitation motivates the following section, where we study PSGD with a learning rate of the form $\gamma_t = \gamma_1 / t^b$ for $b < 1$ . This variant achieves optimal convergence rates without requiring prior knowledge of $\eta$ . + +Remark 4.3. Since RSC implies the Quadratic Growth (QG) condition $H(\mathbf{g}) - H(\mathbf{g}^{*}) \geq \eta \| \mathbf{g} - \mathbf{g}^{*} \|_{v}^{2}$ , the same convergence guarantees also hold for other SGD variants that rely on either RSC or QG assumptions, such as S-Adagrad [10]. + +# 4.2 Convergence rate of PSGD + +Convergence of the non-averaged iterates Building on the RSC of $H$ from Lemma 4.2, we derive the convergence rate of the non-averaged iterates of PSGD, which mirrors the convergence behavior observed in the strongly convex setting. + +Theorem 4.4 (Non-averaged iterates). Under Assumptions $A$ or $B$ , and for any decay schedule of the form $\gamma_{n} = \gamma_{1} / n^{b}$ with $\gamma_{1} > 0$ and $b \in (1/2, 1)$ , we have the convergence rate + +$$ +\mathbb {E} \left[ \left\| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\| _ {v} ^ {2} \right] = \mathcal {O} \left(1 / n ^ {b}\right). +$$ + +As $b$ approaches 1, we observe a nearly $\mathcal{O}(1/n)$ rate for the OT potential. In the next section, we further show that this convergence rate is achievable by the averaged iterates sequence $\overline{\mathbf{g}}_n$ and that it is minimax optimal, highlighting the strong performance of PSGD. + +Convergence of the averaged iterates In convex stochastic optimization, averaging the iterates of the SGD scheme is a widely used technique, as it enables achieving an optimal $\mathcal{O}(1/n)$ convergence rate for strongly convex functions without requiring knowledge of the strong convexity parameter, and regardless of the decay $b \in (1/2, 1)$ for the gradient steps [30, 29]. Moreover, the averaged scheme can adapt to the local strong convexity of the objective function, even when global strong convexity does not hold [5]. + +Note that, as stated in Proposition 4.1, $H$ is locally strongly convex, and as stated after Lemma 4.2, the RSC parameter on $\mathcal{C}$ is unknown. These observations motivate the study of the averaged iterates of PSGD, and the next theorem confirms that these motivations hold true in our setting. To establish this result, we also impose a mild regularity condition on $f_{\mu}$ , requiring it to be $\alpha$ -Hölder continuous for some $\alpha \in (0,1]$ . + +Theorem 4.5 (Averaged iterates). Under Assumptions $A$ or $B$ , and assuming that $f_{\mu}$ is $\alpha$ -Hölder with $\alpha \in (0,1]$ , for any decay schedule of the form $\gamma_n = \gamma_1 / n^b$ with $\gamma_1 > 0$ and $b \in \left(\frac{1}{1 + \alpha}, 1\right)$ , noting $\lambda$ the second smallest eigenvalue of $H$ at the optimum, we have the convergence rate + +$$ +\mathbb {E} \left[ \| \overline {{\mathbf {g}}} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {2} \right] = \frac {1}{\lambda^ {2} (n + 1)} + o (1 / n). +$$ + +Without assuming $f_{\mu}$ to be $\alpha$ -Hölder, and for $b \in (1/2, 1)$ , we still obtain + +$$ +\mathbb {E} \left[ \left\| \overline {{\mathbf {g}}} _ {n} - \mathbf {g} ^ {*} \right\| _ {v} ^ {2} \right] = \mathcal {O} \left(1 / n ^ {b}\right). +$$ + +As we can see, the convergence rate depends on the constant $\lambda > 0$ , which can be understood as the local strong convexity of $H$ at the optimum. Notably, there is existing literature on $\lambda^{\varepsilon}$ for the entropy-regularized semi-dual problem when $\mu$ has bounded support ([13], Theorem 3.2, [12], Proposition 5.1). These results can be extended to estimate $\lambda_{*}$ by letting the regularization parameter vanish. + +# 4.2.1 Estimation of the OT cost + +Building on the convergence rate of PSGD, we derive the corresponding rate for the OT cost estimation. + +Corollary 4.6. Under Assumption $A$ or $B$ , $H$ is $C^1$ -smooth and uniformly bounded, so we have + +$$ +H (\mathbf {g}) - H \left(\mathbf {g} ^ {*}\right) = \mathcal {O} \left(\| \mathbf {g} - \mathbf {g} ^ {*} \| _ {v} ^ {2}\right). +$$ + +Therefore, the OT cost exhibits the same convergence rate as the OT potential. + +As we can see, this result establishes a (nearly) $\mathcal{O}(1/n)$ convergence rate for the estimation of the OT cost, matching the rate derived in Theorems 4.4 and 4.5. In particular, for MTW costs, a faster convergence rate of $\mathcal{O}(1/n)$ is achievable, in contrast to the $\mathcal{O}(1/\sqrt{n})$ rate from Theorem 3.2 in the general setting. + +# 5 OT cost and map estimation with PSGD + +# 5.1 Minimax estimation of the OT map and Brenier potential + +Having the convergence rate of PSGD to the Brenier potential $\mathbf{g}^*$ , we study here the convergence of the map estimate $T(\mathbf{g}): x \mapsto x - \nabla \mathbf{g}^c(x)$ . Note that, as soon as there exists $j \in [1, M]$ such that $x$ is in the interior of $\mathbb{L}_j(\mathbf{g}^*) \cap \mathbb{L}_j(\mathbf{g})$ , we have + +$$ +T _ {\mu , \nu} (x) = x - \nabla (\mathbf {g}) ^ {c} (x) = y _ {j}. +$$ + +Therefore, a result on the quantitative stability of the measure of Laguerre cells is sufficient to establish a convergence rate for the map estimator obtained from PSGD. Such a result was previously established in the compact case in [6]. Here, we extend their result to the quadratic Euclidean cost in the non-compact setting, leading to the following theorem. + +Theorem 5.1. Under Assumption $A$ or $B$ , the function $\mathbf{g} \mapsto \| T(\mathbf{g}) - T_{\mu, \nu}\|_{L^2(\mu)}^2$ is Lipschitz with respect to the infinity norm $\|\cdot\|_{\infty}$ . + +As a corollary, we retrieve the convergence rate of our map estimator with PSGD. + +Corollary 5.2. Under the same assumptions as Theorem 4.4, taking $\hat{\mathbf{g}}_n\in \{\mathbf{g}_n,\overline{\mathbf{g}}_n\}$ + +$$ +\mathbb {E} \left[ \| T (\hat {\mathbf {g}} _ {n}) - T _ {\mu , \nu} \| _ {L ^ {p} (\mu)} \right] = \mathcal {O} \left(1 / n ^ {b / 2}\right). +$$ + +If in addition, $f_{\mu}$ is $\alpha$ -Hölder with $\alpha \in (0,1]$ , taking $b \in (\frac{1}{1 + \alpha}, 1)$ , we have + +$$ +\mathbb {E} \left[ \| T (\overline {{\mathbf {g}}} _ {n}) - T _ {\mu , \nu} \| _ {L ^ {p} (\mu)} \right] = \mathcal {O} \left(M ^ {2} / \lambda \sqrt {n}\right). +$$ + +Note that our dependence on $M$ might be conservative. However, we prove that the rate $\mathcal{O}(1 / \sqrt{n})$ achieved by $T(\overline{\mathbf{g}}_n)$ is minimax optimal. + +Theorem 5.3. Fixing $c(x,y) = \frac{1}{2}\| x - y\|^2$ and $\nu = \frac{1}{2}\delta_{\{0\}} + \frac{1}{2}\delta_{\{1\}}$ and noting $\mathcal{P}_{Lip}(\mathbb{R})$ the set of probability measures on $\mathbb{R}$ with Lipschitz densities, we have + +$$ +\inf _ {T ^ {(n)}} \sup _ {\mu \in \mathcal {P} _ {L i p} (\mathbb {R})} \mathbb {E} _ {\mu} \left[ \left\| T ^ {(n)} - T _ {\mu , \nu} \right\| _ {L ^ {p} (\mu)} ^ {p} \right] \gtrsim 1 / \sqrt {n}, +$$ + +where the infimum is taken over all maps $T^{(n)}$ constructed with the $n$ i.i.d samples of $\mu$ . + +We recover the same minimax lower bound as in the two-sample setting considered in [31], where the target measure $\nu$ is also subsampled. This shows that, even though we have full information about the target measure, the asymptotic rates remain the same. However, we are able to achieve this rate in the non-batched setting, without the need to calibrate a regularization parameter as in [31], or to know the number of samples in advance. Note also that a direct corollary of Theorem 5.3 and Theorem 5.1 is that the convergence rate $\mathcal{O}(1/n)$ for the estimation of the Brenier potential, achieved by the averaged iterates of PSGD, is also minimax optimal. + +# 6 Numerical experiments + +In this section, we numerically verify our convergence rate guarantees through various examples. All experiments demonstrating convergence rates were repeated 20 times, and the error plots represent the averaged errors. We set the learning rate to $\gamma_{1} = \mathrm{Diam}(\mathcal{C})$ , as suggested by the analysis in Theorem 3.2. The step decay parameter $b$ was set to $3/4$ , unless stated otherwise. We find that this learning rate leads to robust results without requiring further tuning. For each example, we generate $\mathbf{g}^*$ randomly, and approximate the associated Laguerre cell measures $\mu(\mathbb{L}_i^c(\mathbf{g}^*))$ . We then fix $w_i = \mu(\mathbb{L}_i^c(\mathbf{g}^*))$ , such that $\mathbf{g}$ is optimal by the first-order condition. The Laguerre cells are estimated with $10^{9}$ samples. All experiments were repeated 10 times, and the average performance was reported. We consider the following three settings to evaluate our method: + +Example 1: Non-quadratic cost. The cost to move mass is set to $\| \cdot \|^{1.5}$ . We set $\mu$ as the uniform measure $\mathcal{U}([0,1]^{10})$ and take $M = 50$ points $y_{1},\ldots ,y_{M}$ uniformly in $[0,1]^{50}$ . The projection set is then $\mathcal{C} = [-10^{3 / 4},10^{3 / 4}]^{50}$ . + +Example 2: Non-compact case. Here, $\mu$ has full support on $\mathbb{R}^{10}$ with cost $c(x,y) = \frac{1}{2}\| x - y\|^2$ . We choose $\mu$ with density $f_{\mu}(x)\propto (1 + \| x\|)^{-d - 3}$ , satisfying (B1-3). As in Example 1, we sample $M = 50$ points in [0, 1]. The projection set $\mathcal{C} = [-5,5]^{50}$ since $K = B(0,1)$ satisfies Lemma 3.1. +Example 3: Non-smooth source measure. We define $\mu$ with density $f_{\mu}(x) = 1 / (2\sqrt{x})\mathbf{1}_{x\in (0,1]}$ , which satisfies a $(1,1)$ -Poincaré-Wirtinger inequality but is not $\alpha$ -Hölder. We took $M = 10$ points uniformly in $[0,1]$ . Since our results do not guarantee acceleration for non-Hölder densities, we set $b = 0.9$ for PSGD, as our analysis recommends $b$ close to 1 for the best rate. The projection set is $\mathcal{C} = [-1,1]^{10}$ . + +![](images/4b3700e8f44661a5176c9cc019726a0c511626699cce0f5605a2629ab3a01bcd.jpg) +(a) Example 1: $c = \parallel \cdot \parallel^{1.5}$ + +![](images/2732ae46c0902be75fd4fd07737950f2cb2954f17c78968aa13408ff27bf4ae5.jpg) +(b) Example 2: unbounded support + +![](images/3923f312bfa6aeca74b095d13f1320776ed04d859494bbb7eb706f7eb171b581.jpg) +(c) Example 3: PW measure +Figure 1: Convergence rates of our OT potential and OT map estimators across different settings. + +As illustrated in Figure 1, our theoretical claims are well supported by empirical results. (i) Our convergence guarantees are matched: PSGD exhibits the expected convergence behavior across all three settings. In particular, we observe a rate of $\mathcal{O}(1 / \sqrt{n})$ in Examples 1 and 2, and a rate of $\mathcal{O}(1 / n^{0.45})$ in Example 3 for estimating the OT map. The exponent 0.45 corresponds to $b / 2$ with $b = 0.9$ , which aligns with our theoretical guidance to select $b$ close to 1 when the source measure is not $\alpha$ -Hölder regular but still satisfies a (PW) condition. (ii) Averaging yields optimal rates: In Examples 1 and 2, averaging leads to the optimal rate $\mathcal{O}(1 / \sqrt{n})$ for the OT map without requiring $b = 1$ . This confirms our theory, which remains robust to the choice of $\gamma_{1} > 0$ and $b \in (1/2, 1)$ , thanks to averaging, for achieving this minimax rate. (iii) We achieve minimax rates across our settings: Our results match the minimax rate $\mathcal{O}(1 / \sqrt{n})$ and extend the findings of [31], who established similar behavior in the compact case with quadratic cost. Importantly, we observe that the estimation of the OT map avoids the curse of dimensionality in both compact (MTW cost) and non-compact (quadratic cost) semi-discrete settings. + +# 7 Conclusion and Discussion + +We studied SGD-based solvers for the semi-discrete optimal transport (OT) problem, focusing on settings where only one or a few samples are available per iteration. These solvers are widely used in machine learning applications involving semi-discrete OT, yet their theoretical understanding remains incomplete. Our work bridges this gap by proving that such methods can consistently estimate both + +the OT cost and the OT map across a broad class of settings. Focusing on PSGD, we established minimax-optimal rates for estimating the OT map under MTW-type costs on compact domains, and under the quadratic Euclidean cost on both compact and non-compact domains. These results rely on novel convergence guarantees and structural properties of the semi-dual OT functional, stemming from the projection set we introduced and the enhanced properties of $H$ we obtained thanks to the restriction of our minimization space. + +Future directions: exploiting RSC with adaptive methods. Our analysis suggests a promising avenue for future work: leveraging RSC to improve the performance of adaptive SGD methods such as using S-Adam [36] with projection for OT. While Adam is commonly used in semidiscrete OT (especially in generative modeling), it often suffers from convergence plateaus due to its fixed step size. In contrast, S-Adam incorporates a decaying learning rate and is specifically tailored for strongly convex objectives. Despite lacking theoretical guarantees under RSC, our empirical results (Figure 2) show that Projected S-Adam better exploits the local geometry of the semidual problem, outperforming Projected Adam in practice. Formalizing these observations and extending our theory to include adaptive methods, while challenging, remains a compelling direction for future research. + +![](images/c63fba5bf768eff7b3408fd0a56c98bf04f81d6fa84390e6ea82fe0155b1f870.jpg) +Figure 2: S-Adam outperforms Adam on Ex. 1, avoiding convergence plateau. + +# 8 Acknowledgements + +The work of François-Xavier Vialard is partly supported by the Bézout Labex (New Monge Problems), funded by ANR, reference ANR-10-LABX-58. + +# References + +[1] P. Agarwal, S. Raghvendra, P. Shirzadian, and K. Yao. A combinatorial algorithm for the semi-discrete optimal transport problem. Advances in Neural Information Processing Systems, 37:25857-25887, 2024. +[2] P. K. Agarwal, S. Raghvendra, P. Shirzadian, and K. Yao. Fast and accurate approximations of the optimal transport in semi-discrete and discrete settings. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4514-4529. SIAM, 2024. +[3] J. M. 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Zhang and L. Cheng. Restricted strong convexity and its applications to convergence analysis of gradient-type methods in convex optimization. Optimization Letters, 9:961-979, 2015. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: Proofs and numerical experiments are given for all the claims made in the abstract and the introduction. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: The paper clearly outlines its main limitations, including the underlying assumptions and the reliance on specific parameters, ensuring that the scope and constraints of the proposed approach are transparent to the reader. + +# 3. Theory Assumptions and Proofs + +Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? + +Answer: [Yes] + +Justification: All assumptions are explicitly stated, and detailed proofs for each result are provided in the appendix. + +# 4. Experimental Result Reproducibility + +Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? + +Answer: [Yes] + +Justification: All experimental settings are clearly specified, and the parameters of our algorithm are provided. + +# 5. 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Licenses for existing assets + +Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? + +Answer: [Yes] + +Justification: The code provided is original and is included for reproducibility under the correct license. + +# 13. New Assets + +Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? + +Answer: [Yes] + +Justification: The algorithms and experiments are clearly explained in the paper, and the code to reproduce the experiments is provided in an anonymized zip file. + +# 14. Crowdsourcing and Research with Human Subjects + +Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? + +Answer: [NA] + +# 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects + +Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? + +Answer: [NA] + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +# Appendix + +# Table of Contents + +A Further details on our Assumptions 15 + +A.1 The Ma-Trundinger-Wang Properties 15 +A.2 The Poincaré-Wirtinger Inequality 15 + +B Properties of the semi-discrete OT problem 15 + +B.1 Known results in the compact case 16 +B.2 New properties for the non-compact case with the quadratic Euclidean cost 16 + +B.2.1 Definition and regularity of the Hessian 17 +B.2.2 Local strong convexity with respect to the mass of Laguerre cells 21 +B.2.3 Quantitative stability of the Laguerre cells 21 + +B.3 New properties in both our compact and non compact settings 23 + +C Proofs of the convergence rates of PSGD 24 + +C.1 Fast convergence rates for MTW costs 24 +C.1.1 Convergence of the non-averaged iterates. 24 +C.1.2 Convergence rate for higher order moments of the non-averaged iterates. 24 +C.2 Convergence of the averaged iterates. 26 +C.3 Convergence rate of PSGD in the general setting 29 + +D Proof of Lemma 3.1: Localisation of a projection set 30 + +E Minimax estimation of OT quantities 31 + +E.1 Kantorovich potential 31 +E.2 OT map 32 + +F Technical Lemmas 33 + +F.1 Technical Lemmas for Appendix B 33 +F.2 Technical Lemmas for Appendix C 35 + +# Notations + +- $\mathbb{R}^*$ : The set $\mathbb{R} \setminus \{0\}$ (i.e., real numbers excluding zero). +- $\| \cdot \|$ : The Euclidean norm. item $\lambda_{\mathbb{R}^d}$ : The Lebesgue measure on $\mathbb{R}^d$ . +- Diameter: For $\mathcal{C} \subset \mathbb{R}^d$ , we define its diameter as: + +$$ +D _ {\mathcal {C}} := \sup \{\left\| x - y \right\| \mid x, y \in \mathcal {C} \}. +$$ + +- Hausdorff measure: in $\mathbb{R}^d$ , for $c \leq d$ , $\mathcal{H}^c$ refers to the $c$ -dimensional Hausdorff measure. + +- Indicator function: For a set $A \subset \mathbb{R}^d$ , $\mathbb{1}_A(x)$ is defined as $\mathbb{1}_A(x) = 1$ if $x \in A$ , and $\mathbb{1}_A(x) = 0$ otherwise. + +- Component-wise Minimum: For $v \in \mathbb{R}^d$ : $v_{\min} \coloneqq \min_{1 \leq j \leq d} v_j$ . + +- Special Vectors: + +- $\mathbf{1}_M = (1,\dots ,1)\in \mathbb{R}^M$ +- $\mathbf{0}_M = (0,\dots ,0)\in \mathbb{R}^M$ +- $\mathbf{e}_j \in \mathbb{R}^M$ , for any $1 \leq j \leq M$ , is the vector with zeros except for the $j$ -th entry, which is equal to 1. + +- Probability Measures: + +- $\mathcal{P}(\mathbb{R}^d)$ : The set of probability measures on $\mathbb{R}^d$ . +- For $\rho \in \mathcal{P}(\mathbb{R}^d)$ , $\operatorname{Supp}(\rho)$ denotes its support. + +- Asymptotic Orders: + +- $\mathcal{O}(\cdot)$ and $o(\cdot)$ : Standard approximation orders. +- $f \lesssim g$ means there exists a constant $C > 0$ such that $f(\cdot) \leq C g(\cdot)$ . +- $a \asymp b$ means both $a \lesssim b$ and $b \lesssim a$ . + +- Filtration: We denote by $\mathcal{F}_n$ the filtration generated by the sample $X_1, \ldots, X_n \stackrel{\mathrm{iid}}{\sim} \mu$ , i.e., + +$$ +\mathcal {F} _ {n} = \sigma \left(X _ {1}, \dots , X _ {n}\right), \quad n \geq 1. +$$ + +- Sets: + +- For any $\varepsilon > 0$ , define: + +$$ +K _ {\varepsilon} := \left\{\mathbf {g} \in \mathbb {R} ^ {M} \mid \forall i \in [ [ 1, M ] ], \mu \left(\mathbb {L} _ {i} (\mathbf {g})\right) \geq \varepsilon \right\}. +$$ + +- Define: + +$$ +K _ {+} := \left\{\mathbf {g} \in \mathbb {R} ^ {M} \mid \forall i \in [ 1, M ], \mu \left(\mathbb {L} _ {i} (\mathbf {g})\right) > 0 \right\}. +$$ + +- Density of an Absolutely Continuous Measure: For an absolutely continuous measure $\rho$ on $\mathbb{R}^d$ , we denote its density w.r.t. the Lebesgue measure by $f_{\rho}$ . + +Orthogonal of $\mathbf{Vect}(1)$ + +- For any $\mathbf{g},\mathbf{g}^{\prime}\in \mathbb{R}^{M}$ , we define: + +$$ +\left\| \mathbf {g} - \mathbf {g} ^ {\prime} \right\| _ {v} = \left\| \operatorname {P r o j} _ {\mathbf {1} _ {\perp}} \left(\mathbf {g} - \mathbf {g} ^ {\prime}\right) \right\| ^ {2}. +$$ + +- Inner product in this space: + +$$ +\langle \mathbf {g}, \mathbf {g} ^ {\prime} \rangle_ {v} = \left\langle \operatorname {P r o j} _ {\mathbf {1} ^ {\perp}} (\mathbf {g}), \operatorname {P r o j} _ {\mathbf {1} ^ {\perp}} \left(\mathbf {g} ^ {\prime}\right) \right\rangle . +$$ + +- **Strong Convexity:** We discuss the strong convexity of the semi-dual function $H$ when the strong convexity holds on the orthogonal complement of $\mathbf{1}$ . + +# A Further details on our Assumptions + +# A.1 The Ma-Trundinger-Wang Properties + +In the semi-discrete setting, the class of cost functions verifying the Ma-Trudinger-Wang (MTW) properties [25] is defined as the set of cost functions satisfying the following conditions: (Reg), (Twist), and Loepers condition (QC) detailed below. + +$$ +c \left(\cdot , y _ {i}\right) \in C ^ {2} \left(\operatorname {S u p p} (\mu)\right), \forall i \in \{1, \dots , M \} \tag {Reg} +$$ + +$$ +\nabla_ {x} c (x, y _ {i}) \neq \nabla_ {x} c (x, y _ {k}), \forall x \in \operatorname {S u p p} (\mu), i \neq k \tag {Twist} +$$ + +Definition A.1 (Loeper's condition). We say $c$ satisfies Loeper's condition if, for each $i \in \{1, \ldots, M\}$ , there exists a convex set $X_i \subset \mathbb{R}^d$ and a $C^2$ diffeomorphism $\exp_i^c(\cdot): X_i \to \operatorname{Supp}(\mu)$ such that + +$$ +\forall t \in \mathbb {R}, 1 \leq k, i \leq M, \left\{p \in X _ {i} \mid - c \left(\exp_ {i} ^ {c} (p), y _ {k}\right) + c \left(\exp_ {i} ^ {c} (p), y _ {i}\right) \leq t \right\} \text {i s} +$$ + +Definition A.2 (c-convexity). We say that $X \subset \mathbb{R}^d$ is $c$ -convex if $(\exp_i^c)^{-1}(X)$ is a convex set for every $i \in \{1, \ldots, N\}$ . + +For a detailed discussion on this class of cost functions and their implications in the semi-discrete optimal transport framework, we refer the reader to Section 1.5 of [21] + +# A.2 The Poincaré-Wirtinger Inequality + +A probability measure $\rho = w(x)dx$ on a domain $\Omega \subset \mathbb{R}^d$ is said to satisfy the weighted Poincaré-Wirtinger inequality (PW) if + +$$ +\int_ {\Omega} | f - \mathbb {E} _ {\rho} [ f ] | d \rho \leq C _ {\mathrm {P W}} \int_ {\Omega} | \nabla f | d \rho , \quad \forall f \in C ^ {1} (\Omega). +$$ + +The existence of a finite constant $C_{\mathrm{PW}}$ provides a quantitative connectedness of the source measure and is necessary for $H$ to be locally strongly convex. We provide here two examples of measures satisfying (PW). Notably, Example 1 shows that non-degenerate Gaussians, mixture of non-degenerate Gaussians and Student distributions satisfy (PW), when we take their restrictions on any ball $B(0,R)$ , $R > 0$ . + +Example 2. (bounded support, density bounded above and below) Let $\Omega$ be bounded, connected, $\alpha$ -Holder with $\alpha \in (0,1]$ , and assume $0 < m \leq w(x) \leq M < \infty$ almost everywhere. + +Note that the assumption of the support being bounded from above and below is classical in the semi-discrete OT literature, as in [13, 31]. + +Example 3. (Annular support with radial concave profile, ([21], Proposition A.1)) Let $0 < r < R$ , and let $\overline{\rho} \in \mathcal{C}^0([0,R])$ be a nonnegative function such that $\overline{\rho}(s) = 0$ for $s \in [0,r]$ , and $\overline{\rho}$ is concave on $[r,R]$ , with + +$$ +\int_ {r} ^ {R} \overline {{\rho}} (s) d s = 1. +$$ + +Define the probability measure $\rho$ on the annulus $X\coloneqq B(0,R)\subset \mathbb{R}^d$ by + +$$ +\rho (x) = \frac {1}{\| x \| ^ {d - 1} \omega_ {d - 1}} \overline {{\rho}} (\| x \|), +$$ + +where $\omega_{d - 1}$ denotes the surface volume of the unit sphere $\mathbb{S}^{d - 1}$ . Then $\rho$ satisfies the weighted Poincaré-Wirtinger inequality for some positive constant. + +# B Properties of the semi-discrete OT problem + +# Regularity properties of $H$ + +In the main article, we concisely presented the regularity properties of the function $H$ . In this section, we provide a more detailed breakdown of these properties, organizing them into sub-properties and referring to the corresponding proofs in the quadratic case with unbounded support (assumptions B1-B3). + +- Differentiability: The function $H$ is differentiable on the entire space $\mathbb{R}^M$ , and we denote its gradient by $\nabla H$ (see Proposition B.5). +- Local $C^2$ regularity: There exists a radius $r > 0$ such that $H$ is twice continuously differentiable $(C^2)$ on the ball $B(\mathbf{g}^*, r)$ (see Proposition B.5). +- Local strong convexity: The function $H$ is strongly convex on the ball $B(\mathbf{g}^*, r)$ (see Proposition B.12). +- Hölder continuity of the Hessian: If the density $f_{\mu}$ is $\alpha$ -Hölder continuous for some $\alpha \in (0,1]$ , then the Hessian of $H$ inherits this regularity and is also $\alpha$ -Hölder continuous (see Corollary B.11). + +# B.1 Known results in the compact case + +In this section, we recall known properties of the semi-dual semi-discrete problem when the support of the source measure $\mu$ is $c$ -convex and contained within a compact set and $c$ is a cost satisfying the MTW properties. We will then extend these results to the non-compact case for the quadratic cost. + +Here, we fix $\varepsilon > 0$ and recall that $K_{\varepsilon} := \{\mathbf{g} \in \mathbb{R}^{M} : \forall i \in [1, M], \mu(\mathcal{L}_{i}(\mathbf{g})) \geq \varepsilon\}$ . The two theorems presented below are taken from [21] and have been adapted to our notation. We emphasize that the authors of [21] considered the semi-dual OT problem as a concave problem, studying the objective function $-H$ instead of $H$ under their notation. For a better understanding of the constants in their theorems, we refer the reader to their article. + +Proposition B.1 (Theorem 1.1 in [21]). Let $\mu$ be an absolutely continuous density with bounded support included in $\mathbb{R}^d$ , then the functional $H$ is $C^1$ smooth, its gradient is given by + +$$ +\nabla H (\mathbf {g}) _ {i} = - \mu (\mathbb {L} _ {i} ^ {c} (\mathbf {g})) + w _ {i}, +$$ + +and its Hessian by + +$$ +\begin{array}{l} (i \neq j) \quad \nabla^ {2} H (\mathbf {g}) _ {i j} = - \int_ {\mathbb {L} _ {i} (\mathbf {g}) \cap \mathbb {L} _ {j} (\mathbf {g})} \frac {f _ {\mu} (x)}{\| y _ {i} - y _ {j} \|} d \mathcal {H} ^ {d - 1} (x), \\ \nabla^ {2} H (\mathbf {g}) _ {i i} = - \sum_ {j \neq i} \nabla^ {2} H (\mathbf {g}) _ {i j}. \\ \end{array} +$$ + +Proposition B.2 (Theorem 5.1 in [21]). Under the assumption (A1), that $\mu$ satisfies a weighted (1,1)-Poincaré-Wirtinger inequality, there exists a constant $\lambda$ such that for any $\mathbf{g} \in K_{\varepsilon}$ , the second smallest eigenvalue of $\nabla^2 H(\mathbf{g})$ , denoted $\lambda_2(\nabla^2 H(\mathbf{g}))$ , satisfies + +$$ +\lambda_ {2} \left(\nabla^ {2} H (\mathbf {g})\right) > \lambda . +$$ + +That is, $H$ is strongly convex on $K_{\varepsilon}$ , considering the problem on $\mathbf{1}^{\perp}$ . + +Theorem B.3 (Theorem 1.3 in [21]). If $\mu$ has its density $f_{\mu}$ in $C^{0,\alpha}(\mathrm{Supp}(\mu))$ . Then, the functional $H$ is $C^{2,\alpha}$ on the set + +$$ +K _ {\varepsilon} := \left\{\mathbf {g} \in \mathbb {R} ^ {M}, \forall i, \mu \left(\mathbb {L} _ {i} (\mathbf {g})\right) > \varepsilon \right\}, +$$ + +Lastly, we state a result concerning the quantitative stability of Laguerre cells, as presented in [6]. + +Lemma B.4 (Lemma 5.5 in [6]). Under the same assumptions as in Proposition B.2, for $\mathbf{g},\mathbf{g}^{\prime}\in \mathbb{R}^{M}$ we have + +$$ +\mu \left(\mathbb {L} _ {i} ^ {c} (\mathbf {g}) \backslash \mathbb {L} _ {i} ^ {c} \left(\mathbf {g} ^ {\prime}\right)\right) \lesssim M \| \mathbf {g} - \mathbf {g} ^ {\prime} \| _ {\infty}, \quad \forall i \in [ [ 1, M ] ]. +$$ + +Once again, we refer to [6] for a more detailed understanding of the constant involved in this lemma. + +# B.2 New properties for the non-compact case with the quadratic Euclidean cost + +In this section, we give second order properties of the semi-dual when the source measure is not supported on a compact. In the compact case, this properties are already known as discussed in Appendix B.1. + +For the reader's convenience, we recall the Assumptions that we made for the non-compact case. + +Assumption (Non-compact case) (B1) The cost is quadratic: $c(x,y) = \frac{1}{2}\| x - y\|^2$ and the measure $\mu$ has a finite second-order moment. + +(B2) There exists a compact set $K \subset \mathbb{R}^d$ with $\mu(K) \geq 1 - \frac{1}{4} w_{\min}$ , such that the probability measure $\mu_K$ with density $f_{\mu_K}(x) \coloneqq c_K f_\mu(x)\mathbf{1}_K(x)$ satisfies a Poincaré-Wirtinger inequality. +(B3) The density $f_{\mu}$ satisfies the following integrability and regularity condition: for $R > 1$ and $r \geq 1$ , define + +$$ +f _ {\mu} ^ {R} := f _ {\mu} \cdot \mathbf {1} _ {\| x \| \leq R}, \quad f _ {\mu} ^ {R + r} := f _ {\mu} \cdot \mathbf {1} _ {R + (r - 2) \leq \| x \| \leq R + r}. +$$ + +Assume there exist $C > 0$ and a modulus of continuity $\omega$ such that for all $\delta > 0$ , + +$$ +\sum_ {r = 0} ^ {\infty} (R + r) ^ {d - 1} \omega_ {f _ {\mu}} ^ {R + r} (\delta) \leq C \omega (\delta), \quad \sum_ {r = 0} ^ {\infty} (R + r) ^ {d - 1} C _ {f _ {\mu}} ^ {R + r} < \infty , \tag {6} +$$ + +where $C_{f_\mu}^{R + r} \coloneqq \sup_{x \in \mathbb{R}^d} f_\mu^{R + r}(x)$ and $\omega_{f_\mu}^{R + r}$ is the modulus of continuity of $f_\mu^{R + r}$ . + +Additional notation. Since here the cost is fixed, we define the Laguerre cells by $\mathbb{L}_i(\mathbf{g})$ instead of $\mathbb{L}_i^c (\mathbf{g})$ . + +# B.2.1 Definition and regularity of the Hessian + +Proposition B.5. Under Assumptions (B1) and (B3), the semi-dual $H$ is differentiable everywhere, and its gradient is given by + +$$ +\nabla H (\mathbf {g}) _ {i} = \mu (\mathbb {L} _ {i} (\mathbf {g})) - w _ {i}. +$$ + +Moreover $H$ is $C^2$ smooth on $K_{w_{\min} / 2}\cap \mathcal{C}$ and its Hessian is given by + +$$ +(i \neq j) \quad \nabla^ {2} H (\mathbf {g}) _ {i j} = - \int_ {\mathbb {L} _ {i} (\mathbf {g}) \cap \mathbb {L} _ {j} (\mathbf {g})} \frac {f _ {\mu} (x)}{\| y _ {i} - y _ {j} \|} d \mathcal {H} ^ {d - 1} (x), +$$ + +$$ +\nabla^ {2} H (\mathbf {g}) _ {i i} = - \sum_ {j \neq i} \nabla^ {2} H (\mathbf {g}) _ {i j}. +$$ + +Proof. Definition of the gradient. The proof follows the lines of the proof of Theorem 4.1 of [21] and extend this result to the non-compact case. If $x$ is in the interior of Laguerre cell $\mathbb{L}_j(\mathbf{g})$ , we have + +$$ +\nabla_ {\mathbf {g}} h (\mathbf {g}, x) = \mathbb {1} _ {i = j} - w _ {i}. +$$ + +where we recall that $h(\mathbf{g},x) = -\mathbf{g}^c (x) - \sum_{i = 1}^{M}w_jg_j$ . Since the boundaries of the Laguerre cells are defined by the intersections of $M$ hyperplanes, they form a negligible set with respect to the measure $\mu$ . As a result, the gradient definition of $H$ follows immediately. + +Definition of the Hessian. Fix $i \in [[1, M]]$ . We aim to prove the differentiability of the measure $\mu(\mathbb{L}_i(\mathbf{g}))$ with respect to $g_j$ and that its differential is defined by: + +$$ +\frac {\partial \mu (\mathbb {L} _ {i} (\mathbf {g}))}{\partial g _ {j}} = - \int_ {\mathbb {L} _ {i} (\mathbf {g}) \cap \mathbb {L} _ {j} (\mathbf {g})} \frac {f _ {\mu} (x)}{\| y _ {i} - y _ {j} \|} d \mathcal {H} ^ {d - 1} (x), \quad j \neq i, +$$ + +$$ +\frac {\partial \mu (\mathbb {L} _ {j} (\mathbf {g}))}{\partial g _ {j}} = - \sum_ {i \neq j} \frac {\partial \mu (\mathbb {L} _ {i} (\mathbf {g}))}{\partial g _ {j}}. +$$ + +This gives us exactly the line $\left(\nabla^{2}H(\mathbf{g})_{1i},\dots ,\nabla^{2}H(\mathbf{g})_{Mi}\right)$ of the Hessian. + +Suppose $i \neq j$ . + +Suppose $\delta \geq 0$ . Defining $h_{ij}(x) := \frac{1}{2}\| x - y_i\|^2 - \frac{1}{2}\| x - y_j\|^2 - g_i + g_j$ , note that we have $\mathbb{L}_i(\mathbf{g}) = \cap_j h_{ij}^{-1}([-\infty, 0])$ . + +We also have $\mathbb{L}_i(\mathbf{g} + \delta \mathbf{e}_j) = \mathbb{L}_i(\mathbf{g})\backslash \big(\cap_{k\neq j}h_{ik}^{-1}([- \infty ,0])\cap h_{ij}^{-1}([- \delta ,0])\big)$ . Note that $h$ is Lipschitz and for all $x$ , $\| \nabla h(x)\| = \| y_j - y_i\|$ . Moreover, under assumption (B3), we can use Lemma F.2, + +which states that for all hyperplane $H$ , $\int_{H}f_{\mu}d\mathcal{H}^{d - 1}\lesssim 1$ . Therefore, we can apply the coarea formula to pass from the second to the third equality below, + +$$ +\begin{array}{l} \mu \left(\mathbb {L} _ {i} (\mathbf {g} + \delta \mathbf {e} _ {j})\right) = \mu \left(\mathbb {L} _ {i} (\mathbf {g})\right) - \mu \left(\cap_ {k \neq j} h _ {i k} ^ {- 1} ([ - \infty , 0 ]) \cap h _ {i j} ^ {- 1} ([ - \delta , 0 ])\right) \\ = \mu (\mathbb {L} _ {i} (\mathbf {g})) - \int_ {\cap_ {k \neq j} h _ {i k} ^ {- 1} ([ - \infty , 0 ]) \cap h _ {i j} ^ {- 1} ([ - \delta , 0 ])} f _ {\mu} (x) d x \\ = \mu (\mathbb {L} _ {i} (\mathbf {g})) - \int_ {- \delta} ^ {0} \int_ {\cap_ {k \neq j} h _ {i k} ^ {- 1} ([ - \infty , 0 ]) \cap h _ {i j} ^ {- 1} (\{t \})} \frac {f _ {\mu} (x)}{\| y _ {j} - y _ {i} \|} d \mathcal {H} ^ {d - 1} (x) d t. \\ \end{array} +$$ + +By analogy, for $\delta \leq 0$ we have: + +$$ +\mathbb {L} _ {i} (\mathbf {g} + \delta \mathbf {e} _ {j}) = \mathbb {L} _ {i} (\mathbf {g}) \cup \left(\cap_ {k \neq j} h _ {i k} ^ {- 1} ([ - \infty , 0 ]) \cap h _ {i j} ^ {- 1} ([ - \delta , 0 ])\right), +$$ + +Using the coarea formula gives: + +$$ +\mu \left(\mathbb {L} _ {i} (\mathbf {g} + \delta \mathbf {e} _ {j})\right) = \mu (\mathbb {L} _ {i} (\mathbf {g})) + \int_ {0} ^ {- \delta} \int_ {\cap_ {k \neq j} h _ {i k} ^ {- 1} ([ - \infty , 0 ]) \cap h _ {i j} ^ {- 1} (\{t \})} \frac {f _ {\mu} (x)}{\| y _ {j} - y _ {i} \|} d \mathcal {H} ^ {d - 1} (x) d t. +$$ + +Applying Lemma B.6, which is stated and proved later in the appendice, the integrand defined above is continuous on $K_{w_{\min} / 2} \cap \mathcal{C}$ . As a consequence, we can apply the Fundamental Theorem of Calculus and justify the limit in + +$$ +\lim _ {\delta \rightarrow 0 ^ {-}} \frac {\mu (\mathbb {L} _ {i} (g + \delta \mathbf {e} _ {j})) - \mu (\mathbb {L} _ {i} (\mathbf {g}))}{\delta} = - \int_ {\cap_ {k \neq j} h _ {i k} ^ {- 1} ([ - \infty , 0 ]) \cap h _ {i j} ^ {- 1} (\{0 \})} f _ {\mu} (x) \frac {1}{\| y _ {j} - y _ {i} \|} d \mathcal {H} ^ {d - 1} (x). +$$ + +By symmetry + +$$ +\lim _ {\delta \to 0 ^ {+}} \frac {\mu (\mathbb {L} _ {i} (g + \delta \mathbf {e} _ {j})) - \mu (\mathbb {L} _ {i} (\mathbf {g}))}{\delta} = - \int_ {\cap_ {k \neq j} h _ {i k} ^ {- 1} ([ - \infty , 0 ]) \cap h _ {i j} ^ {- 1} (\{0 \})} f _ {\mu} (x) \frac {1}{\| y _ {j} - y _ {i} \|} \mathrm {d} \mathcal {H} ^ {d - 1} (x). +$$ + +Since $\cap_{k\neq j}h_{ik}^{-1}([- \infty, 0]) \cap h_{ij}^{-1}(\{0\}) = \mathbb{L}_i(\mathbf{g}) \cap \mathbb{L}_j(\mathbf{g})$ , we thus obtain + +$$ +F _ {i j} (\mathbf {g}) := \frac {\partial \mu (\mathbb {L} _ {i} (\mathbf {g}))}{\partial g _ {j}} = - \int_ {\mathbb {L} _ {i} (\mathbf {g}) \cap \mathbb {L} _ {j} (\mathbf {g})} \frac {f _ {\mu} (x)}{\| y _ {i} - y _ {j} \|} d \mathcal {H} ^ {d - 1} (x), \quad j \neq i. +$$ + +Suppose $i = j$ . No matter $\mathbf{g} \in \mathbb{R}^M$ , the Laguerre cells verifies $\mu(\cup_i \mathbb{L}_j(\mathbf{g})) = \mu(\mathbb{R}^d) = 1$ . Therefore, + +$$ +\frac {\partial}{\partial g _ {j}} \sum_ {i = 1} ^ {M} \mu \left(\mathbb {L} _ {i} (\mathbf {g})\right) = \frac {\partial}{\partial g _ {j}} 1 = 0. +$$ + +This equality gives $\frac{\partial}{\partial g_j}\mu (\mathbb{L}_j(\mathbf{g})) = -\sum_{i\neq j}\frac{\partial}{\partial g_j}\mu (\mathbb{L}_i(\mathbf{g})).$ + +![](images/abe30257dcbead08a2e2fd9ab7bc89b1227d54b47a18f7109764b8a89dcd6ad6.jpg) + +Lemma B.6. Under assumption (B1) and (B3), for any $\mathbf{g} \in K_{w_{\min}/2} \cap \mathcal{C}$ the function $F_{f_\mu}$ defined for all $t \in \mathbb{R}$ as + +$$ +F _ {f _ {\mu}} (t) = \int_ {\cap_ {k \neq j} h _ {i k} ^ {- 1} ([ - \infty , 0 ]) \cap h _ {i j} ^ {- 1} (\{t \})} \frac {f _ {\mu} (x)}{\| y _ {j} - y _ {i} \|} d \mathcal {H} ^ {d - 1} (x). +$$ + +admits $\omega$ as modulus of continuity in some neighborhood of $\{0\}$ . + +Proof of Lemma B.6. We keep the same notation as in [21]. Restricting ourselves to the quadratic cost, we get + +$$ +\varepsilon_ {n d} = \varepsilon_ {t w} = \min _ {i \neq j} \left\| y _ {i} - y _ {j} \right\|, +$$ + +$$ +C _ {\nabla} ^ {R} = O (R), +$$ + +$$ +C _ {e x p} = O (1), +$$ + +$$ +C _ {c o n d} = O (1), +$$ + +$$ +C _ {d e t} = O (1). +$$ + +In order to apply the results in [21], we need to extend Proposition 4.5 to measures with unbounded support in the case of the quadratic cost. We prove the following result: + +Proposition B.7. Consider $\mathcal{K}_{\varepsilon} \coloneqq \{\mathbf{g} \in \mathbb{R}^{N-1} \mid \mu(\mathbb{L}_i(\mathbf{g})) > \varepsilon, \forall i = 1, \ldots, M\}$ for some $\varepsilon > 0$ sufficiently small and let $\mathcal{C}$ be the compact set of the projection. Then, there exists a positive constant $\delta_1$ , such that for all $\mathbf{g} \in \mathcal{K}_{\varepsilon} \cap \mathcal{C}$ and all $p \in \mathbb{R}^d$ such that there exist $i \neq j$ , for which $c(p, y_i) - c(p, y_0) = g_i - g_0$ and $c(p, y_j) - c(p, y_0) = g_j - g_0$ , then + +$$ +\left(\frac {\langle y _ {i} - y _ {0} , y _ {j} - y _ {0} \rangle}{\| y _ {i} - y _ {0} \| \| y _ {j} - y _ {0} \|}\right) ^ {2} \leq 1 - \delta_ {1} ^ {2}. \tag {7} +$$ + +Remark B.8. Note that the transversality equation (7) is independent of $p \in \mathbb{R}^d$ . However, $p$ is still involved in the transversality condition by the value of the difference of the costs. + +Proof. Fix an index $i$ . The set of points $p \in \mathbb{R}^d$ such that $c(p, y_i) - c(p, y_0) = g_i - g_0$ defines a hyperplane in $\mathbb{R}^d$ . The intersection of two such hyperplanes, denoted $H_{ij}(g)$ , is, unless the hyperplanes are parallel, a codimension-2 affine subspace of $\mathbb{R}^d$ . + +For each such $H_{ij}(g)$ , let $d_{ij}(g) \in \mathbb{R}^d$ be the orthogonal projection of the origin $0_d$ onto $H_{ij}(g)$ . Since the vector of potentials $(g_i)_{i=0,\dots,M-1}$ is bounded due to the projection set $\mathcal{C}$ , the set + +$$ +\left\{d _ {i j} (g): i \neq j \text {a d m i s s i b l e , a n d} g \in \mathcal {K} _ {\varepsilon} \cap \mathcal {C} \right\} +$$ + +is contained in a ball $B(0,R - 1)$ for some sufficiently large $R > 0$ . + +Hence, for every $g \in \mathcal{K}_{\varepsilon} \cap \mathcal{C}$ and every admissible pair $(i,j)$ , there exists a point $p \in H_{ij}(g)$ lying in the interior of the ball $B(0,R)$ . + +We now apply the transversality result from [21] to the measure + +$$ +\mu_ {R} := \mathbf {1} _ {B (0, R)} \cdot \frac {\mu}{\mu (B (0 , R))}, +$$ + +which is the normalized restriction of $\mu$ to $B(0,R)$ . This result guarantees transversality of the hyperplanes $H_{ij}$ associated with potentials in the set $\mathcal{K}_{\varepsilon'}'$ (defined analogously for $\mu_R$ and some $\varepsilon' > 0$ ) over the whole space $\mathbb{R}^d$ . + +Choose $R$ such that $\mu(B(0, R)) \geq 1 - \varepsilon/2$ . Then it is sufficient to take + +$$ +\varepsilon^ {\prime} = \frac {\varepsilon / 2}{1 - \varepsilon / 2}, +$$ + +since one can check that $\mathcal{K}_{\varepsilon} \subset \mathcal{K}_{\varepsilon'}'$ . This completes the proof. + +![](images/c5de2586b7f3402aefe984534fc3845233d403f6fa7994ba048bd02115a08f4d.jpg) + +We aim to use the transversality result of Kitagawa et al. on a decomposition of $\mathbb{R}^d$ into balls centered at 0. To this goal, we now prove a transversality result at the boundary of $B(0,R)$ uniform for $R$ sufficiently large. + +Proposition B.9. Consider $\mathcal{K}_{\varepsilon} \coloneqq \{\mathbf{g} \in \mathbb{R}^{N-1} \mid \mu(\mathbb{L}_i(\mathbf{g})) > \varepsilon, \forall i = 1, \dots, M\}$ for some $\varepsilon > 0$ sufficiently small and let $\mathcal{C}$ be the compact set of the projection. Then, for any positive constant $\delta_2$ there exists $R$ sufficiently large such that for all $\mathbf{g} \in \mathcal{K}_{\varepsilon} \cap \mathcal{C}$ and all $p \in \mathbb{R}^d$ : if there exists $i$ such that $c(p, y_i) - c(p, y_0) = g_i - g_0$ for some $p \in \partial B(0, R)$ , then + +$$ +\left\langle \frac {p}{\| p \|}, \frac {y _ {i} - y _ {0}}{\| y _ {i} - y _ {0} \|} \right\rangle \leq 1 - \delta_ {2} ^ {2}. \tag {8} +$$ + +Proof. Rewrite the condition $c(p, y_i) - c(p, y_0) = g_i - g_0$ as $\langle p, y_i - y_0 \rangle = g_i - g_0$ , dividing by $\| p \|$ we get + +$$ +\left\langle \frac {p}{\| p \|}, y _ {i} - y _ {0} \right\rangle = \frac {g _ {i} - g _ {0}}{\| p \|}. \tag {9} +$$ + +The right hand side tends to 0 uniformly with $R$ since $g_{i} - g_{0}$ lies in a compact set. The conclusion follows directly. + +The rest of the proof follows the lines of the proofs Appendix B of [21] with $\varepsilon_{tr} = \delta_1$ as in Proposition B.7. From Proposition B.9 applied to $\delta_2 = \delta_1$ there exists $R > 1$ such as the transversality condition on the boundary (8) holds for every $B(0, R + r)$ and the same $\delta_1$ for every $r \geq 0$ . Let us fix such $R > 1$ so that Assumption (5) is also satisfied. + +The next step is to decompose the integral using a partition of unity. We define the sequence of functions $(\varphi_r)_{r\geq 0}$ by + +$$ +\varphi_ {r} (x) = \left\{ \begin{array}{l l} (R - \| x \|) _ {+} \wedge 1, & \mathrm {i f} r = 0, \\ (\| x \| - (R + r - 2)) _ {+} \wedge (R + r - \| x \|) _ {+}, & \mathrm {i f} r \geq 1, \end{array} \right. \qquad x \in \mathbb {R} ^ {d}. +$$ + +An illustration of the functions $\varphi_r(x)$ defined above is shown below: + +![](images/14776465cbc4f94dde5b349e14fc4c9d50a6d0d6dd36f74cf4add65ab1638b17.jpg) + +By definition, $\sum_{r\geq 0}\varphi (x) = 1$ , for all $x\in \mathbb{R}^d$ , and every $\varphi_r$ is supported on $\{R + r - 2\leq \| x\| \leq$ $R + r\}$ for every $r\geq 1$ $\varphi_0$ being supported by $B(0,R)$ . Moreover $\varphi_r$ is Lipschitz continuous and we denote $\omega_{\varphi_r}$ its modulus of continuity on its support. The moduli of continuity satisfy $\omega_{\varphi_r}(\delta)\le |\delta |$ $\delta >0r\geq 0$ . Then, writing $S_{i,j,t} = \cap_{k\neq j}h_{ik}^{-1}([- - \infty ,0])\cap h_{ij}^{-1}(\{t\})$ for conciseness, we decompose the integral + +$$ +\int_ {S _ {i, j, t}} \frac {f _ {\mu} (x)}{\| y _ {j} - y _ {i} \|} d \mathcal {H} ^ {d - 1} (x) = \sum_ {r = 0} ^ {\infty} \int_ {S _ {i, j, t}} \frac {f _ {\mu} (x) \varphi (x)}{\| y _ {j} - y _ {i} \|} d \mathcal {H} ^ {d - 1} (x). \tag {10} +$$ + +We apply Proposition B.1 of [21] to each term of the decomposition. We recall below his crucial result, tracking the order of the constants established by [21]. + +Proposition B.10. Let $\sigma$ be a continuous non-negative function on $B(0,R)$ bounded by $\sigma_{\infty}$ and with modulus of continuity $\omega_{\sigma}$ . Let the functions $h_{ij}$ satisfy the transversality conditions (7) and (8) with the same constant $\varepsilon_{tr} > 0$ . Then + +$$ +F _ {\sigma} (t) := \int_ {\cap_ {k \neq j} h _ {i k} ^ {- 1} ([ - \infty , 0 ]) \cap h _ {i j} ^ {- 1} (\{t \})} \frac {\sigma (x)}{\| y _ {j} - y _ {i} \|} d \mathcal {H} ^ {d - 1} (x) +$$ + +has modulus of continuity + +$$ +\omega_ {h _ {\sigma}} (\delta) = C _ {1} \omega_ {\sigma} (C _ {2} \delta) + C _ {3} | \delta | +$$ + +where $C_1 = O(\mathcal{H}^{d - 1}(\partial B(0,R)))$ , $\mathcal{H}^{d - 1}(\partial B(0,R)) = O(R^{d - 1})$ , $C_2 = O(\varepsilon_{tw}^{-1}) = O(1)$ and $C_3 = O(\sigma_\infty C(d,2R)\varepsilon_{tr}^{-4} + \mathcal{H}^{d - 1}(\partial B(0,R)))$ , where $C(d,2R)$ defined in (3.5) of [21] satisfies $C(d,2R) = O(R^{d - 1})$ . + +When applying Proposition B.10 to the continuous function $\sigma^{R + r}(x) \coloneqq f_{\mu}(x)\varphi (x)$ we easily estimate $\sigma_{\infty}^{R + r}\leq C_{f_{\mu}}^{R + r}$ and $w_{\sigma^{R + r}}\leq \omega_{f_{\mu}}^{R + r} + C_{f_{\mu}}^{R + r}\omega_{\varphi_r}$ . Using that $\omega_{\varphi_r}(\delta)\leq |\delta|$ we obtain: + +$$ +\omega_ {h _ {\sigma^ {R + r}}} (\delta) = O \left((R + r) ^ {d - 1}\right) \left(\omega_ {f _ {\mu}} ^ {R + r} (O (1) \delta) + O (C _ {f _ {\mu}} ^ {R + r}) \delta\right), +$$ + +uniformly for every $\mathbf{g} \in \mathcal{K}_{\varepsilon} \cap \mathcal{C}$ on a neighborhood of $\{0\}$ . Noticing that this neighborhood depends solely on the transversality properties that are common to every $r \geq 0$ and since + +$$ +\omega_ {F _ {f _ {\mu}}} = \sum_ {r \geq 0} \omega_ {h _ {\sigma R + r}} +$$ + +from the decomposition in (10) the desired result follows under the assumption in (5). + +Corollary B.11. Under Assumptions (B1) and (B3), if moreover $f_{\mu}$ is $\alpha$ -Hölder with $\alpha \in (0,1]$ , then the Hessian of the semi-dual $H$ is also $\alpha$ -Hölder on $K_{w_{\min}/2} \cap \mathcal{C}$ . + +Proof. Since $f_{\mu}$ is $\alpha$ -Hölder, the function $f_{\mu}^{R + r}$ with $r \geq 0$ are also $\alpha$ -Hölder and we note $\omega_{f_{\mu}}^{R + r} = \kappa_{f_{\mu}}^{R + r}\delta^{\alpha}$ , applying Proposition B.10, + +$$ +\begin{array}{l} \omega_ {h _ {\sigma R + r}} (\delta) O \left((R + r) ^ {d - 1}\right) \left(\omega_ {f _ {\mu}} ^ {R + r} (O (1) \delta) + O \left(C _ {f _ {\mu}} ^ {R + r}\right) \delta\right) \\ = O ((R + r) ^ {d - 1}) \kappa_ {f _ {\mu}} ^ {R + r} O (1) \delta^ {\alpha} + ((R + r) ^ {d - 1}) O (C _ {f _ {\mu}} ^ {R + r}) \delta . \\ \end{array} +$$ + +By the summability conditions of assumption (B3), for some constants $C_1, C_2 > 0$ , we have + +$$ +\begin{array}{l} \omega_ {F _ {f \mu}} = \sum_ {r \geq 0} \omega_ {h _ {\sigma R + r}} \\ = O (1) \delta^ {\alpha} \sum_ {r \geq 0} O ((R + r) ^ {d - 1}) \kappa_ {f _ {\mu}} ^ {R + r} + \delta \sum_ {r \geq 0} O \left(C _ {f _ {\mu}} ^ {R + r}\right) ((R + r) ^ {d - 1}) \\ \leq C _ {1} \delta^ {\alpha} + C _ {2} \delta , \\ \end{array} +$$ + +Applying Proposition B.3 in [21] under our hypothesis shows that for $\mathbf{g},\mathbf{g}^{\prime}\in K_{w_{\mathrm{min}} / 2}\cap \mathcal{C}$ , there exists a constant $C$ depending on $\varepsilon_{tr}$ and $f_{\mu}$ such that we have + +$$ +\left| \nabla^ {2} H (\mathbf {g}) - \nabla^ {2} H \left(\mathbf {g} ^ {\prime}\right) \right| \leq \omega_ {F _ {f _ {\mu}}} \left(\left\| \mathbf {g} - \mathbf {g} ^ {\prime} \right\| _ {\infty}\right) + C \| \mathbf {g} - \mathbf {g} ^ {\prime} \| +$$ + +which gives the $\alpha$ -Hölder regularity since $\omega_{F_{f_\mu}} \leq C_1 \delta^\alpha + C_2 \delta$ . + +![](images/f0ad5885582f8e909ece3cac461d73b9089e416ba3e6f81ca0b828640776496b.jpg) + +# B.2.2 Local strong convexity with respect to the mass of Laguerre cells + +Proposition B.12. Under Assumptions (B1-B3) $H$ is strongly convex on $K_{\frac{1}{2} w_{\min}}$ . + +Proof. Recall $K \subset \mathbb{R}^d$ the compact set such that $\mu(K) > 1 - \frac{1}{4} w_{\min}$ and $\mu_K$ , the probability measure with density defined by $f_{\mu_K}(x) = c_K f_\mu(x) \mathbf{1}_K(x)$ , with $c_K \in [1,2]$ , satisfies a weighted Poincaré-Wirtinger inequality. + +We thus can use Proposition B.2, which states that the semi-dual between $\mu_R$ and $\nu$ is strongly-convex on $\mathrm{Vect}_1^\perp$ on the set + +$$ +K _ {\frac {1}{4} w _ {\operatorname* {m i n}}} ^ {R} := \left\{\mathbf {g} \in \mathbb {R} ^ {M}: \forall i \in [ [ 1, M ] ], \mu_ {R} (\mathbb {L} _ {i} ^ {R} (\mathbf {g})) \geq \frac {1}{4} w _ {\operatorname* {m i n}} \right\}. +$$ + +This gives us that, for any $\mathbf{g} \in K_{\frac{1}{4} w_{\mathrm{min}}}^{R}$ , there exists $\lambda_K > 0$ , lower bounding the second smallest value of the semi-dual function between $\mu_R$ and $\nu$ . This also gives us that for any $\mathbf{g} \in K_{\frac{1}{4} w_{\mathrm{min}}}^{R}$ , the second smallest eigenvalue of the matrix $B$ defined by + +$$ +(i \neq j) \quad B _ {i j} = - \int_ {\mathbb {L} _ {i} (\mathbf {g}) \cap \mathbb {L} _ {j} (\mathbf {g})} \frac {f _ {\mu} (x) \mathbf {1} _ {B (0 , R)} (x)}{\| y _ {i} - y _ {j} \|} d \mathcal {H} ^ {d - 1} (x), +$$ + +$$ +B _ {i i} = - \sum_ {j \neq i} B _ {i j}, +$$ + +is lower bounded by $\lambda_K / c_R \geq \lambda_K / 2 > 0$ . + +Since for any $\mathbf{g}$ , and $i \neq j$ , $\nabla^2 H(\mathbf{g})_{ij} \leq B_{ij}$ and that both $\nabla^2 H(\mathbf{g})$ and $B$ are Laplacian matrices, we apply Lemma F.1 to obtain that the second smallest eigenvalue of the hessian $\nabla^2 H$ is lower bounded by $\lambda_R / c_R$ on $K_{\frac{1}{4} w_{\min}}^R$ . + +For any $\mathbf{g} \in K_{w_{\min}/2}$ , we thus have for all $i \in [[1, M]]$ , $\mu(\mathbb{L}_i(\mathbf{g}) \cap B(0, R)) \geq \frac{1}{4} w_{\min}$ . That is, $K_{\frac{1}{2} w_{\min}} \subset K_{\frac{1}{4} w_{\min}}^R$ , which completes the proof. + +# B.2.3 Quantitative stability of the Laguerre cells + +Lemma B.13. Under Assumptions (B1-B3), we have + +$$ +\mu \big (\mathbb {L} _ {i} ^ {c} (\mathbf {g}) \setminus \mathbb {L} _ {i} ^ {c} (\mathbf {g} ^ {\prime}) \big) \lesssim M \| \mathbf {g} - \mathbf {g} ^ {\prime} \| _ {\infty}, \qquad \forall i \in [ [ 1, M ] ]. +$$ + +Proof. For all $j \in [[1, M]]$ , if $x$ is in the interior of $\mathbb{L}_j(\mathbf{g})$ , we have + +$$ +\nabla \mathbf {g} ^ {c} (x) = x - y _ {j}. \tag {11} +$$ + +Therefore, given $\mathbf{g},\mathbf{g}^{\prime}\in \mathbb{R}^{M}$ , if there exists $j\in [[1,M]]$ such that $x$ is the interior of $\mathbb{L}_j(\mathbf{g})\cap \mathbb{L}_j(\mathbf{g}^{\prime})$ we have + +$$ +\nabla \mathbf {g} ^ {c} (x) = \nabla \left(\mathbf {g} ^ {\prime}\right) ^ {c} (x). +$$ + +Moreover, since the support of $\nu$ is finite, we have + +$$ +\sup _ {x \in \mathbb {R} ^ {d}} \| \nabla \mathbf {g} ^ {c} (x) - \nabla \left(\mathbf {g} ^ {\prime}\right) ^ {c} (x) \| = \max _ {i \neq j} \| y _ {i} - y _ {j} \|. +$$ + +Hence, to bound the error of $T(\mathbf{g})(x) = x - \nabla \mathbf{g}^c(x)$ and $T_{\mu,\nu} = x - \nabla (\mathbf{g}^*)^c(x)$ , we just need to bound the difference of measure of Laguerre cells made by $\mathbf{g}$ and $\mathbf{g}^*$ . More generally, we now proceed here to bound the difference of measure of Laguerre cells between $\mathbf{g}, \mathbf{g}' \in \mathbb{R}^M$ fixed arbitrarily. + +Our proof will follow some arguments from [21]. Let us fix $i \in \mathbb{I}[1, M]$ and suppose $x \in \mathbb{L}_i(\mathbf{g}) \setminus \mathbb{L}_i(\mathbf{g}')$ . The definition of the Laguerre cells implies that there is a $k \neq i$ such that $c(x, y_k) + \mathbf{g}_k' < c(x, y_i) + \mathbf{g}_i'$ while $c(x, y_i) + \mathbf{g}_i \leq c(x, y_k) + \mathbf{g}_k$ . Combining these two inequalities yields to + +$$ +\mathbf {g} _ {k} ^ {\prime} - \mathbf {g} _ {i} ^ {\prime} < c (x, y _ {i}) - c (x, y _ {k}) \leq \mathbf {g} _ {k} - \mathbf {g} _ {i}. +$$ + +Hence, writing $f_{ik}(x) = c(x,y_i) - c(x,y_k)$ , we have + +$$ +\mathbb {L} _ {i} (\mathbf {g}) \backslash \mathbb {L} _ {i} \left(\mathbf {g} ^ {\prime}\right) \subset \bigcup_ {k \neq i} f _ {i k} ^ {- 1} \left(\left[ \mathbf {g} _ {k} ^ {\prime} - \mathbf {g} _ {i} ^ {\prime}, \mathbf {g} _ {k} - \mathbf {g} _ {i} \right]\right). \tag {12} +$$ + +We now now bound $\mu\left(f_{ik}^{-1}\left([g_k^{\prime}-g_i^{\prime},g_i-g_k]\right)\right)$ using the coarea formula, using that for all $x$ , $\|\nabla f_{ik}(x)\| = \|y_i - y_k\|$ : + +$$ +\begin{array}{l} \mu \left(f _ {i k} ^ {- 1} ([ a, b ])\right) = \int_ {f _ {i k} ^ {- 1} ([ a, b ])} \mathrm {d} \mu (x) \\ = \int_ {a} ^ {b} \int_ {f _ {i k} ^ {- 1} (\{t \})} \frac {1}{\| \nabla f _ {i k} (x) \|} \mu (x) d \mathcal {H} ^ {d - 1} (x) d t \\ = \int_ {a} ^ {b} \int_ {f _ {i k} ^ {- 1} (\{t \})} \frac {1}{\| y _ {i} - y _ {k} \|} \mu (x) \mathrm {d} \mathcal {H} ^ {d - 1} (x) \mathrm {d} t. \\ \end{array} +$$ + +Observe that for the quadratic cost, $f_{ik}(x) = \langle x,y_k - y_i\rangle +\| y_i\|^2 -\| y_k\|^2$ and so $f_{ik}^{-1}(\{t\}) = \{x\in \mathbb{R}^d,\langle x,y_k - y_i\rangle = t - \frac{1}{2} (\| y_i\|^2 +\| y_k\|^2)\}$ and is therefore a hyperplane. Applying Lemma F.2, there exists a constant $C$ such that, for any $i,k$ and $t$ , we have + +$$ +\int_ {f _ {i k} ^ {- 1} (\{t \})} \frac {1}{\| y _ {i} - y _ {k} \|} \mu (x) \mathrm {d} \mathcal {H} ^ {d - 1} (x) \leq C. +$$ + +Therefore, we have + +$$ +\mu \left(f _ {i k} ^ {- 1} ([ a, b ])\right) \leq \int_ {a} ^ {b} \frac {C}{\| y _ {i} - y _ {k} \|} \leq (b - a) \frac {C}{\| y _ {i} - y _ {k} \|}. +$$ + +Since $\mathbf{g}_k - \mathbf{g}_i - (\mathbf{g}_k' - \mathbf{g}_i') \leq 2\|\mathbf{g} - \mathbf{g}'\|_{\infty}$ , by combining the above with (12) we conclude + +$$ +\mu \left(\mathbb {L} _ {i} (\mathbf {g}) \backslash \mathbb {L} _ {i} \left(\mathbf {g} ^ {\prime}\right)\right) \leq \sum_ {k \neq i} \mu \left(f _ {i k} ^ {- 1} \left(\left[ \mathbf {g} _ {k} ^ {\prime} - \mathbf {g} _ {i} ^ {\prime}, \mathbf {g} _ {k} - \mathbf {g} _ {i} \right]\right)\right) \lesssim M \| \mathbf {g} - \mathbf {g} ^ {\prime} \| _ {\infty}. \tag {13} +$$ + +□ + +# B.3 New properties in both our compact and non compact settings + +Theorem 5.1. Under Assumptions (A1) or (B1-3), the function $\mathbf{g} \mapsto \| T(\mathbf{g}) - T_{\mu, \nu}\|_{L^2(\mu)}^2$ is Lipschitz with respect to the infinity norm $\|\cdot\|_{\infty}$ . Moreover, the Lipschitz constant grows at most quadratically in $M$ . + +Proof. Using that no matter $\mathbf{g} \in \mathbb{R}^M$ , for any $x \in \mathbb{R}^d$ + +$$ +\| T _ {\mu , \nu} (\mathbf {g} ^ {*}) - T _ {\mu , \nu} (\mathbf {g}) \| \leq \left\{ \begin{array}{l l} 0 & \text {i f} x \in \mathbb {L} _ {i} (\mathbf {g} ^ {*}) \cap \mathbb {L} _ {i} (\mathbf {g}) \text {f o r a c e r t a i n} i \in [ [ 1, M ] ], \\ \max _ {i \neq j} \| y _ {i} - y _ {j} \| & \text {e l s e}, \end{array} \right. +$$ + +and that for any $\mathbf{g},\mathbf{g}^{\prime}\in \mathbb{R}^{M}$ , we have $\mu (\mathbb{L}_i(\mathbf{g})\setminus \mathbb{L}_i(\mathbf{g}'))\lesssim M\| \mathbf{g} - \mathbf{g}^{\prime}\|$ using Proposition B.4 in the compact case, or Proposition B.13, we have for any $p\in [1,\infty)$ + +$$ +\begin{array}{l} \| T _ {\mu , \nu} (\mathbf {g} ^ {*}) - T _ {\mu , \nu} (\mathbf {g}) \| _ {L _ {p} (\mu)} ^ {p} = \int_ {\mathbb {R} ^ {d}} \| T _ {\mu , \nu} (\mathbf {g} ^ {*}) (x) - T _ {\mu , \nu} (\mathbf {g}) (x) \| ^ {p} \mathrm {d} \mu (x) \\ = \sum_ {i = 1} ^ {M} \int_ {\mathbb {L} _ {i} (\mathbf {g} ^ {*})} \| T _ {\mu , \nu} (\mathbf {g} ^ {*}) (x) - T _ {\mu , \nu} (\mathbf {g}) (x) \| ^ {p} \mathrm {d} \mu (x) \\ \leq \sum_ {i = 1} ^ {M} \int_ {\mathbb {L} _ {i} (\mathbf {g} ^ {*}) \backslash \mathbb {L} _ {i} (\mathbf {g})} \max _ {i \neq j} \| y _ {i} - y _ {j} \| ^ {p} d \mu (x) \\ \leq \sum_ {i = 1} ^ {M} \max _ {i \neq j} \| y _ {i} - y _ {j} \| ^ {p} \mu \left(\mathbb {L} _ {i} \left(\mathbf {g} ^ {*}\right) \backslash \mathbb {L} _ {i} (\mathbf {g})\right) \\ \lesssim M ^ {2} \| \mathbf {g} ^ {*} - \mathbf {g} \| _ {\infty}, \\ \end{array} +$$ + +where we used $\mu (\mathbb{L}_i(\mathbf{g})\setminus \mathbb{L}_i(\mathbf{g}^*))\lesssim M\| \mathbf{g} - \mathbf{g}^*\|$ for the last line. + +![](images/102f4b2a13f07a23c2ac4cf2c62ff5db4f65b4c116c8ea1852c9fbc408bd6fb4.jpg) + +Lemma B.14. For any $\varepsilon > 0$ , there exists $\mathbf{g}^* \in K_{\varepsilon}$ and $d_{\varepsilon} > 0$ such that $B(\mathbf{g}^*, d_{\varepsilon}) \subset K_{\varepsilon}$ . + +Proof. This result is a simple application of the results on the quantitative stability of Laguerre cells stated in Proposition B.4 and Proposition B.13. $\square$ + +Proposition B.15. (Hessian and Local strong convexity). Under Assumptions (A1) or (B1-3), $H$ is twice differentiable. Moreover, there exists a constant $\lambda > 0$ and a radius $r > 0$ , such that for any $\mathbf{g} \in \mathbb{R}^M$ , $\| \mathbf{g}^* - \mathbf{g} \|_v \leq r$ implies that the second smallest value of the Hessian $\nabla^2 H(\mathbf{g})$ is lower bounded by $\lambda$ . + +Proof. The fact that $H$ is twice differentiable was already known in the compact case and is proven in Proposition B.5 for the non-compact case under (B1-3). + +Under assumptions (A1) and using Proposition B.2 or under assumptions (B1-3) and using Proposition B.12, $H$ is strongly convex on $K_{\frac{1}{2} w_{\min}}$ . Applying Lemma B.14 concludes the proof. + +Proposition B.16. There exists $\eta > 0$ , such that uniformly in $\mathbf{g} \in \mathcal{C}$ , we have + +$$ +\left\langle \nabla H (\mathbf {g}), \mathbf {g} - \mathbf {g} ^ {*} \right\rangle_ {v} \geq \eta \| \mathbf {g} - \mathbf {g} ^ {*} \| _ {v} ^ {2}. +$$ + +Proof. Observe that, by Proposition B.12, $H$ is locally strongly convex on the orthogonal of $\mathbf{1}$ , on the set + +$$ +K _ {\frac {1}{2} w _ {\min}} = \left\{\mathbf {g}: \forall i \in [ [ 1, M ] ], \mu (\mathbb {L} _ {i} (\mathbf {g})) \geq \frac {1}{2} w _ {\min} \right\}. +$$ + +Therefore, for any $\mathbf{g} \in K_{\frac{1}{2} w_{\min}}$ , we have $\langle \nabla H(\mathbf{g}), \mathbf{g} - \mathbf{g}^* \rangle_v \geq \lambda \| \mathbf{g} - \mathbf{g}^* \|_v^2$ . + +Now, suppose $\mathbf{g} \in \mathcal{C} \setminus K_{\frac{1}{2} w_{\min}}$ . Using Lemma B.14, we know that there exists $d_0 > 0$ such that $B(\mathbf{g}^*, d_0) \subset K_{\frac{1}{2} w_{\min}}$ . Therefore, by the convexity of $H$ , there exists $c > 0$ such that + +$$ +\left\langle \nabla H (\mathbf {g}), \mathbf {g} - \mathbf {g} ^ {*} \right\rangle_ {v} \geq H (\mathbf {g}) - H \left(\mathbf {g} ^ {*}\right) > c. +$$ + +Defining + +$$ +\lambda^{\prime}:= \inf_{\mathbf{g}\in \mathcal{C}\setminus K_{\frac{1}{2} w_{\min}}}\left\{\frac{H(\mathbf{g}) - H(\mathbf{g}^{*})}{\| \mathbf{g} - \mathbf{g}^{*}\|_{v}^{2}}\right\} >c / \operatorname{Diam}_{2}(\mathcal{C})^{2}, +$$ + +we have for any $\mathbf{g}\in \mathcal{C}\setminus K_{\frac{1}{2} w_{\min}}$ + +$$ +\left\langle \nabla H (\mathbf {g}), \mathbf {g} - \mathbf {g} ^ {*} \right\rangle_ {v} \geq \lambda^ {\prime} \| \mathbf {g} - \mathbf {g} ^ {*} \| _ {v} ^ {2}. +$$ + +Taking $\eta = \min \left\{\lambda ,\lambda^{\prime}\right\}$ concludes the proof. + +![](images/750c9378f944c663a850d8beb2e6cd1b00979dc72274ca976be3441dd6d1a030.jpg) + +# C Proofs of the convergence rates of PSGD + +# C.1 Fast convergence rates for MTW costs + +# C.1.1 Convergence of the non-averaged iterates. + +Theorem 4.4 (Non averaged iterates). Under Assumptions (A1) or (B1-3) and for any decay step of the form $\gamma_{n} = \gamma_{1} / n^{b}$ with $\gamma_1 > 0, b\in (1 / 2,1)$ , we have the convergence rate + +$$ +\mathbb {E} [ \| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {2} ] = \mathcal {O} \left(\frac {\gamma_ {n}}{\eta}\right). +$$ + +Proof. By definition of the gradient step at time $n + 1$ , sampling $X_{n + 1} \sim \mu$ and since $\mathbf{g}^* \in \mathcal{C}$ , we have $\| \nabla_{\mathbf{g}} h(\mathbf{g}_n, X_{n + 1}) \|_v \leq 2$ a.s. and + +$$ +\begin{array}{l} \left\| \mathbf {g} _ {n + 1} - \mathbf {g} ^ {*} \right\| _ {v} ^ {2} \\ = \left\| \operatorname {P r o j} _ {\mathcal {C}} \left(\mathbf {g} _ {n} - \gamma_ {n + 1} \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {n}, X _ {n + 1}\right)\right) - \mathbf {g} ^ {*} \right\| _ {v} ^ {2} \\ \leq \left\| \mathbf {g} _ {n} - \gamma_ {n + 1} \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {n}, X _ {n + 1}\right) - \mathbf {g} ^ {*} \right\| _ {v} ^ {2} \\ \leq \left(\| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {2} - 2 \gamma_ {n + 1} \left\langle \nabla_ {\mathbf {g}} h (\mathbf {g} _ {n}, X _ {n + 1}), \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\rangle_ {v} + \gamma_ {n + 1} ^ {2} \| \nabla_ {\mathbf {g}} h (\mathbf {g} _ {n}, X _ {n + 1}) \| _ {v} ^ {2}\right) \\ \leq \left(\| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {2} - 2 \gamma_ {n + 1} \left\langle \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {n}, X _ {n + 1}\right), \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\rangle_ {v} + 4 \gamma_ {n + 1} ^ {2}\right). \tag {14} \\ \end{array} +$$ + +Using Proposition B.16, we have for any $\mathbf{g} \in \mathcal{C}$ , $\langle \nabla H(\mathbf{g}_n), \mathbf{g}_n - \mathbf{g}^* \rangle_v \geq \eta \| \mathbf{g}_n - \mathbf{g}^* \|_v^2$ . Therefore, taking the conditional expectation, we obtain + +$$ +\mathbb {E} \left[ \| \mathbf {g} _ {n + 1} - \mathbf {g} ^ {*} \| _ {v} ^ {2} \mid \mathcal {F} _ {n} \right] \leq \| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {2} (1 - 2 \eta \gamma_ {n + 1}) + 4 \gamma_ {n + 1} ^ {2}. +$$ + +Taking the expectation and applying Lemma F.3 with $\delta_{n} = \mathbb{E}\big[\| \mathbf{g}_{n} - \mathbf{g}^{*}\|_{v}^{2}\big]$ and $m_{n} = 4\gamma_{n}$ gives + +$$ +\mathbb {E} \left[ \| \mathbf {g} _ {n + 1} - \mathbf {g} ^ {*} \| _ {v} ^ {2} \right] \leq \exp \left(- 2 \eta \sum_ {k = \lceil n / 2 \rceil} ^ {n} \gamma_ {k}\right) \left(\sum_ {n = n _ {0}} ^ {n} 4 \gamma_ {n} ^ {2} + \mathbb {E} [ \| \mathbf {g} _ {n _ {0}} - \mathbf {g} ^ {*} \| _ {v} ^ {2} ]\right) + \frac {4}{\eta} \gamma_ {\lceil n / 2 \rceil - 1} +$$ + +where $n_0 = \min \{n\in \mathbb{N},\eta \gamma_{n + 1}\leq 1\}$ . Remark that the exponential term converges exponentially fast. Indeed, we have $\sum_{k = \lceil n / 2\rceil}^{n}\gamma_{k}\gtrsim n^{1 - b}$ with $1 - b > 0$ . Moreover, $\| \mathbf{g}_{n_0} - \mathbf{g}^*\| _v^2\leq \mathrm{Diam}_2(\mathcal{C})^2$ . Therefore, since for all $n\geq 2$ , $\gamma_{\lceil n / 2\rceil -1}\leq 2^{b}\gamma_{n}$ , we have the desired convergence rate + +$$ +\mathbb {E} \left[ \| \mathbf {g} _ {n + 1} - \mathbf {g} ^ {*} \| _ {v} ^ {2} \right] \lesssim \frac {\gamma_ {n}}{\eta} +$$ + +which concludes the proof. + +![](images/ec8a8dbded0e8a8ef28a0da6d748955081ece86fb2ef85ea63ba0acd8ab63e7a.jpg) + +# C.1.2 Convergence rate for higher order moments of the non-averaged iterates. + +We prove here the convergence rate of higher order moments of the error $\| \mathbf{g}_n - \mathbf{g}^*\| _v$ . This convergence will be useful for the convergence rate of the averaged iterates of PSGD. While this proposition directly proves Theorem 4.4 by the use of Jensen's inequality, the proof is slightly more cumbersome so we decided to make a separate case. + +Proposition C.1. Under Assumptions (A1) or (B1-3) and for any decay step of the form $\gamma_{n} = \gamma_{1} / n^{b}$ with $\gamma_{1} > 0, b \in (1/2, 1)$ and $p \in \{1, 2, 3\}$ , we have the convergence rate + +$$ +\mathbb {E} \left[ \| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \| ^ {2 p} \right] \lesssim \frac {\gamma_ {n} ^ {p}}{\eta^ {p}} +$$ + +where $(\mathbf{g}_n)$ is the sequence of non-averaged iterates of PSGD. + +Proof. By definition of the gradient step at time $n + 1$ , sampling $X_{n + 1} \sim \mu$ and using inequality (14), + +$$ +\| \mathbf {g} _ {n + 1} - \mathbf {g} ^ {*} \| _ {v} ^ {6} \leq \left(\| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {2} - 2 \gamma_ {n + 1} \left\langle \nabla_ {\mathbf {g}} h (\mathbf {g} _ {n}, X _ {n + 1}), \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\rangle_ {v} + 4 \gamma_ {n + 1} ^ {2}\right) ^ {3}. +$$ + +Using that $(A + B + C)^{3} = \sum_{a + b + c = 3}\frac{3!}{a!b!c!} A^{a}B^{b}C^{c}$ , we obtain + +$$ +\begin{array}{l} \left\| \mathbf {g} _ {n + 1} - \mathbf {g} ^ {*} \right\| _ {v} ^ {6} \leq \left\| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\| _ {v} ^ {6} \\ - 6 \left\| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\| _ {v} ^ {4} \gamma_ {n + 1} \left\langle \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {n}, X _ {n + 1}\right), \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\rangle_ {v} \\ + 1 2 \left\| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\| _ {v} ^ {4} \gamma_ {n + 1} ^ {2} \\ + 1 2 \left\| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\| _ {v} ^ {2} \gamma_ {n + 1} ^ {2} \left\langle \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {n}, X _ {n + 1}\right), \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\rangle_ {v} ^ {2} \\ - 4 8 \left\| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\| _ {v} ^ {2} \gamma_ {n + 1} ^ {3} \left\langle \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {n}, X _ {n + 1}\right), \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\rangle_ {v} \\ + 4 8 \left\| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\| _ {v} ^ {2} \gamma_ {n + 1} ^ {4} \\ - 8 \gamma_ {n + 1} ^ {3} \left\langle \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {n}, X _ {n + 1}\right), \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\rangle_ {v} ^ {3} \\ + 4 8 \gamma_ {n + 1} ^ {4} \left\langle \nabla_ {\mathbf {g}} h (\mathbf {g} _ {n}, X _ {n + 1}), \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\rangle_ {v} ^ {2} \\ - 9 6 \gamma_ {n + 1} ^ {5} \left\langle \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {n}, X _ {n + 1}\right), \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\rangle_ {v} \\ + 2 ^ {6} \gamma_ {n + 1} ^ {6}. \\ \end{array} +$$ + +Taking the conditional expectation and already omitting some negative terms thanks to $\langle \nabla H(\mathbf{g}_n),\mathbf{g}_n - \mathbf{g}^*\rangle_v\geq 0$ , which follows from the fact that $H$ is convex and $\mathbf{g}^{*}$ is a minimizer, gives the simplification + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {g} _ {n + 1} - \mathbf {g} ^ {*} \| _ {v} ^ {6} \mid \mathcal {F} _ {n} \right] \leq \| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {6} \\ - 6 \left\| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\| _ {v} ^ {4} \gamma_ {n + 1} \left\langle \nabla H (\mathbf {g} _ {n}), \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\rangle_ {v} \\ + 1 2 \| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {4} \gamma_ {n + 1} ^ {2} \\ + 1 2 \left\| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\| _ {v} ^ {2} \gamma_ {n + 1} ^ {2} \mathbb {E} \left[ \left\langle \nabla h (\mathbf {g} _ {n}, X _ {n + 1}), \mathbf {g} _ {n} - \mathbf {g} ^ {*} \right\rangle_ {v} ^ {2} \mid \mathcal {F} _ {n} \right] \\ + 4 8 \| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {2} \gamma_ {n + 1} ^ {4} \\ + 4 8 \gamma_ {n + 1} ^ {4} \mathbb {E} \left[ \langle \nabla h (\mathbf {g} _ {n}, X _ {n + 1}), \mathbf {g} _ {n} - \mathbf {g} ^ {*} \rangle_ {v} ^ {2} | \mathcal {F} _ {n} \right] \\ + 2 ^ {6} \gamma_ {n + 1} ^ {6} \\ - 8 \gamma_ {n + 1} ^ {3} \mathbb {E} \left[ \langle \nabla_ {\mathbf {g}} h (\mathbf {g} _ {n}, X _ {n + 1}), \mathbf {g} _ {n} - \mathbf {g} ^ {*} \rangle_ {v} ^ {3} | \mathcal {F} _ {n} \right]. \\ \end{array} +$$ + +Using Proposition B.16, we have for any $\mathbf{g} \in \mathcal{C}$ , $\langle \nabla H(\mathbf{g}_n), \mathbf{g}_n - \mathbf{g}^* \rangle_v \geq \eta \| \mathbf{g}_n - \mathbf{g}^* \|_v^2$ . The Cauchy-Schwarz inequality gives $|\langle \nabla h(\mathbf{g}_n, X_{n+1}), \mathbf{g}_n - \mathbf{g}^* \rangle_v| \leq 2 \| \mathbf{g}_n - \mathbf{g}^* \|_v$ . Combining these two inequalities we obtain + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {g} _ {n + 1} - \mathbf {g} ^ {*} \| _ {v} ^ {6} \mid \mathcal {F} _ {n} \right] \leq \| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {6} (1 - 6 \eta \gamma_ {n + 1}) + 6 0 \gamma_ {n + 1} ^ {2} \| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {4} \\ + 2 4 0 \gamma_ {n + 1} ^ {4} \| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {2} + 2 ^ {6} \gamma_ {n + 1} ^ {6} + 6 4 \gamma_ {n + 1} ^ {3} \| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {3}. \\ \end{array} +$$ + +Using Young's (generalized) inequality $ab = ac\frac{b}{c}\leq \frac{(ac)^p}{p} +\frac{b^q}{c^q q}$ for $c\neq 0,\frac{1}{p} +\frac{1}{q} = 1$ and applying it to $60\gamma_{n + 1}\| \mathbf{g}_n - \mathbf{g}^*\| _v^4$ with $c = \left(\frac{2}{3\eta}\right)^{2 / 3},p = 3,q = \frac{3}{2}$ gives $60\gamma_{n + 1}\| \mathbf{g}_n - \mathbf{g}^*\| _v^4\leq \frac{60^3\gamma_{n + 1}^3}{3}$ . + +$(\frac{2}{3\eta})^2 + \eta \| \mathbf{g}_n - \mathbf{g}^* \|_v^6$ . Analogously, one has $64\gamma_{n+1}^2 \| \mathbf{g}_n - \mathbf{g}^* \|_v^3 \leq \frac{2^{10}}{\eta}\gamma_{n+1}^4 + \eta \| \mathbf{g}_n - \mathbf{g}^* \|_v^6$ . Thus, taking the expectation, we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {g} _ {n + 1} - \mathbf {g} ^ {*} \| _ {v} ^ {6} \right] \leq \mathbb {E} \left[ \| \mathbf {g} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {6} \right] \left(1 - 4 \eta \gamma_ {n + 1}\right) + \gamma_ {n + 1} ^ {4} \left(2 4 0 \cdot \operatorname {D i a m} _ {2} (\mathcal {C}) ^ {2} + \frac {3 2}{\eta^ {2}} \cdot 1 0 ^ {3}\right) \\ + \frac {2 ^ {1 0}}{\eta} \gamma_ {n + 1} ^ {5} + 6 4 \gamma_ {n + 1} ^ {6}, \\ \end{array} +$$ + +where the terms involving $\mathrm{Diam}_2(\mathcal{C})$ appears from the crude bound $\| \mathbf{g}_n - \mathbf{g}\| _v\leq \mathrm{Diam}_2(\mathcal{C})$ + +Applying Lemma F.3 in a similar way as in the proof of Theorem 4.4 gives $\mathbb{E}[\| \mathbf{g}_{n + 1} - \mathbf{g}^*\| _v^6 ]\lesssim \frac{\gamma_n^3}{\eta^3}$ , so by Jensen's inequality, we conclude + +$$ +\mathbb {E} \left[ \| \mathbf {g} _ {n + 1} - \mathbf {g} ^ {*} \| _ {v} ^ {2 p} \right] \lesssim \frac {\gamma_ {n} ^ {p}}{\eta^ {p}} \quad \text {f o r} p \in \{1, 2, 3 \}. +$$ + +![](images/9b74987c2a5ec83c349ff8012f24ab1a12a23f67b4d4bca502a4b6bac94e916e.jpg) + +# C.2 Convergence of the averaged iterates. + +Theorem 4.5 (Averaged iterates) Under Assumptions (A1) or (B1-B3), and assuming that $f_{\mu}$ is $\alpha$ -Hölder with $\alpha \in (0,1]$ , for any decay schedule of the form $\gamma_n = \gamma_1 / n^b$ with $\gamma_1 > 0$ and $b \in \left(\frac{1}{1 + \alpha}, 1\right)$ , we have the convergence rate + +$$ +\mathbb {E} \left[ \| \overline {{\mathbf {g}}} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {2} \right] = \mathcal {O} (1 / n). +$$ + +Without assuming $f_{\mu}$ to be $\alpha$ -Hölder, and for $b \in (1/2, 1)$ , we still obtain + +$$ +\mathbb {E} [ \| \overline {{\mathbf {g}}} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {2} ] = \mathcal {O} (1 / n ^ {b}). +$$ + +Proof. For this proof, we introduce the additional following notation: + +For any $c > 0$ we define the function $t\mapsto \Psi_c(t)$ such that + +$$ +\sum_ {t = 1} ^ {T} t ^ {- c} \leq \Psi_ {c} (T) := \left\{ \begin{array}{l l} 1 + \ln (T + 1) & \text {i f} c = 1, \\ \frac {2 c - 1}{c - 1} & \text {i f} c > 1, \\ 1 + \frac {1}{1 - c} (T + 1) ^ {1 - c} & \text {i f} c < 1. \end{array} \right. \tag {15} +$$ + +We start by a decomposition of the gradient step, already present in [19]. We define the differences + +$$ +\mathbf {p} _ {k} := \operatorname {P r o j} _ {\mathcal {C}} \left(\mathbf {g} _ {k} - \gamma_ {k + 1} \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {k}, X _ {k + 1}\right)\right) - \left(\mathbf {g} _ {k} - \gamma_ {k + 1} \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {k}, X _ {k + 1}\right)\right), +$$ + +$$ +\boldsymbol {\xi} _ {k + 1} := \nabla H (\mathbf {g} _ {k}) - \nabla_ {\mathbf {g}} h (\mathbf {g} _ {k}, X _ {k + 1}), +$$ + +$$ +\boldsymbol {\delta} _ {k} := \nabla H (\mathbf {g} _ {k}) - \nabla^ {2} H (\mathbf {g} ^ {*}) (\mathbf {g} _ {k} - \mathbf {g} ^ {*}). +$$ + +Noting $I_M$ the identity matrix in $\mathcal{M}_M(\mathbb{R})$ , we observe that, by incorporating each introduced term sequentially, for any $k \in \mathbb{N}$ , we have + +$$ +\begin{array}{l} \mathbf {g} _ {k + 1} - \mathbf {g} ^ {*} = \operatorname {P r o j} _ {\mathcal {C}} \left(\mathbf {g} _ {k} - \gamma_ {k + 1} \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {k}, X _ {k + 1}\right)\right) - \mathbf {g} ^ {*} \\ = \mathbf {g} _ {k} - \gamma_ {k + 1} \nabla_ {\mathbf {g}} h (\mathbf {g} _ {k}, X _ {k + 1}) - \mathbf {g} ^ {*} - \mathbf {p} _ {k} \\ = \mathbf {g} _ {k} - \gamma_ {k + 1} \nabla H (\mathbf {g} _ {k}, X _ {k + 1}) - \mathbf {g} ^ {*} + \gamma_ {k + 1} \boldsymbol {\xi} _ {k + 1} - \mathbf {p} _ {k} \\ = \left(I _ {M} - \gamma_ {k + 1} \nabla^ {2} H (\mathbf {g} ^ {*})\right) \left(\mathbf {g} _ {k} - \mathbf {g} _ {*}\right) - \gamma_ {k + 1} \boldsymbol {\delta} _ {k} + \gamma_ {k + 1} \boldsymbol {\xi} _ {k + 1} + \mathbf {p} _ {k}. \\ \end{array} +$$ + +Thus, we have that + +$$ +\nabla^ {2} H \left(\mathbf {g} ^ {*}\right) \left(\mathbf {g} _ {k} - \mathbf {g} ^ {*}\right) = \frac {\mathbf {g} _ {k} - \mathbf {g} _ {k + 1}}{\gamma_ {k + 1}} - \boldsymbol {\delta} _ {k} + \boldsymbol {\xi} _ {k + 1} + \frac {\mathbf {p} _ {k}}{\gamma_ {k + 1}}. +$$ + +Observing that $\frac{1}{n + 1}\sum_{k = 0}^{n}(\mathbf{g}_k - \mathbf{g}^*) = \overline{\mathbf{g}}_n - \mathbf{g}^*$ , we have the following decomposition of the averaged iterates + +$$ +\nabla^ {2} H (\mathbf {g} ^ {*}) (\overline {{\mathbf {g}}} _ {n} - \mathbf {g} ^ {*}) = \frac {1}{n + 1} \sum_ {k = 0} ^ {n} \frac {\mathbf {g} _ {k} - \mathbf {g} _ {k + 1}}{\gamma_ {k + 1}} - \frac {1}{n + 1} \sum_ {k = 0} ^ {n} \boldsymbol {\delta} _ {k} + \frac {1}{n + 1} \sum_ {k = 0} ^ {n} \boldsymbol {\xi} _ {k + 1} + \frac {1}{n + 1} \sum_ {k = 0} ^ {n} \frac {\mathbf {p} _ {k}}{\gamma_ {k + 1}}. +$$ + +We will now give the convergence rate of each sum. + +Convergence rate for $\frac{1}{n + 1}\sum_{k = 0}^{n}\frac{\mathbf{g}_k - \mathbf{g}_{k + 1}}{\gamma_{k + 1}}.$ + +$$ +\begin{array}{l} \sum_ {k = 0} ^ {n} \frac {\mathbf {g} _ {k} - \mathbf {g} _ {k + 1}}{\gamma_ {k + 1}} = \sum_ {k = 0} ^ {n} \frac {\left(\mathbf {g} _ {k} - \mathbf {g} ^ {*}\right) - \left(\mathbf {g} _ {k + 1} - \mathbf {g} ^ {*}\right)}{\gamma_ {k + 1}} \\ = \sum_ {k = 0} ^ {n} \frac {\mathbf {g} _ {k} - \mathbf {g} ^ {*}}{\gamma_ {k + 1}} - \sum_ {k = 0} ^ {n} \frac {\mathbf {g} _ {k + 1} - \mathbf {g} ^ {*}}{\gamma_ {k + 1}} \\ = \sum_ {k = 1} ^ {n} \left(\frac {1}{\gamma_ {k + 1}} - \frac {1}{\gamma_ {k}}\right) \left(\mathbf {g} _ {k} - \mathbf {g} ^ {*}\right) + \frac {\mathbf {g} _ {0} - \mathbf {g} ^ {*}}{\gamma_ {1}} - \frac {\mathbf {g} _ {n + 1} - \mathbf {g} ^ {*}}{\gamma_ {n + 1}}. \\ \end{array} +$$ + +Remark that $\gamma_{n + 1}^{-1} - \gamma_n^{-1}\leq 2\gamma_1^{-1}n^{b - 1}$ . Using Minkowski's inequality and that, by Theorem 4.4 (non-averaged iterates), $\mathbb{E}\left[\| \mathbf{g}_n - \mathbf{g}^*\| _v^2\right]\lesssim \frac{\gamma_1}{\eta} (n + 1)^{-b}$ + +$$ +\mathbb {E} \left[ \left\| \sum_ {k = 0} ^ {n} \frac {\mathbf {g} _ {k} - \mathbf {g} _ {k + 1}}{\gamma_ {k + 1}} \right\| _ {v} ^ {2} \right] ^ {\frac {1}{2}} \lesssim \frac {1}{\eta} \Psi_ {1 - b / 2} (n + 1) + \operatorname {D i a m} _ {2} (\mathcal {C}) \gamma_ {1} ^ {- 1} + \frac {1}{\sqrt {\gamma_ {1} \eta}} (n + 1) ^ {b / 2}. +$$ + +We thus have the convergence rate + +$$ +\frac {1}{n + 1} \mathbb {E} \left[ \left\| \sum_ {k = 0} ^ {n} \frac {\mathbf {g} _ {k} - \mathbf {g} _ {k + 1}}{\gamma_ {k + 1}} \right\| _ {v} ^ {2} \right] ^ {\frac {1}{2}} \lesssim \frac {1}{\eta (n + 1) ^ {1 - b / 2}}. +$$ + +- Convergence rate for $\frac{1}{n+1} \sum_{k=0}^{n} \delta_k$ . + +We recall that $\delta_{k} = \nabla H(\mathbf{g}_{k}) - \nabla^{2}H(\mathbf{g}^{*})(\mathbf{g}_{k} - \mathbf{g}^{*})$ and that the Hessian using either Theorem B.3 or Proposition 4.1, depending on our setting, there exists a ball $B(\mathbf{g}^*,d_1)$ with $d_{1} > 0$ where $H$ is $\alpha$ -Hölder. Therefore, applying Lemma F.4, if $\mathbf{g}_k\in B(\mathbf{g}^*,d_1)$ , we have + +$$ +\left\| \boldsymbol {\delta} _ {k} \right\| \lesssim \left\| \mathbf {g} _ {k} - \mathbf {g} ^ {*} \right\| _ {v} ^ {1 + \alpha}. +$$ + +Otherwise, since the Hessian, whose expression is provided in Proposition B.5, is uniformly bounded by an application of Lemma F.2, there exists a constant $C_{\delta}$ such that for any $\mathbf{g} \in \mathcal{C}$ , $\|\nabla H(\mathbf{g}) - \nabla^2 H(\mathbf{g}^*) (\mathbf{g} - \mathbf{g}^*)\| \leq C_{\delta}$ . + +Since $\mathbb{P}(\mathbf{g}_k \notin B(\mathbf{g}^*, d_1)) = \mathbb{P}(\| \mathbf{g}_k - \mathbf{g}_* \| > d_1)$ , using Markov's inequality gives + +$$ +\begin{array}{l} \mathbb {E} [ \| \boldsymbol {\delta} _ {k} \| _ {v} ^ {2} ] = \mathbb {E} [ \| \boldsymbol {\delta} _ {k} \| _ {v} ^ {2} \mathbf {1} _ {\mathbf {g} _ {k} \in B (\mathbf {g} ^ {*}, d _ {1})} ] + \mathbb {E} [ \| \boldsymbol {\delta} _ {k} \| _ {v} ^ {2} \mathbf {1} _ {\mathbf {g} _ {k} \notin B (\mathbf {g} ^ {*}, d _ {1})} ] \\ \lesssim \mathbb {E} \left[ \| \mathbf {g} _ {k} - \mathbf {g} ^ {*} \| _ {v} ^ {2 + 2 \alpha} \right] + \frac {C _ {\delta} ^ {2}}{d _ {1} ^ {2 + 2 \alpha}} \mathbb {E} [ \| \mathbf {g} _ {k} - \mathbf {g} ^ {*} \| _ {v} ^ {2 + 2 \alpha} ] \\ \lesssim \mathbb {E} [ \| \mathbf {g} _ {k} - \mathbf {g} ^ {*} \| _ {v} ^ {2 + 2 \alpha} ]. \\ \end{array} +$$ + +Therefore, using Minkowski's inequality, we have + +$$ +\begin{array}{l} \frac {1}{n + 1} \mathbb {E} \left[ \left\| \sum_ {k = 0} ^ {n} \boldsymbol {\delta} _ {k} \right\| _ {v} ^ {2} \right] ^ {\frac {1}{2}} \lesssim \frac {1}{n + 1} \sum_ {k = 0} ^ {n} \frac {1}{\eta^ {\frac {1 + \alpha}{2}}} \gamma_ {k + 1} ^ {\frac {1 + \alpha}{2}} \\ \leq \frac {1}{\eta^ {\frac {1 + \alpha}{2}} (n + 1)} \Psi_ {\frac {b + \alpha b}{2}} \\ \lesssim \frac {1}{\eta^ {\frac {1 + \alpha}{2}} (n + 1) ^ {\frac {b + \alpha b}{2}}}. \\ \end{array} +$$ + +- Convergence rate for $\frac{1}{n+1} \sum_{k=0}^{n} \xi_{k+1}$ . + +We recall that $\pmb{\xi}_{k + 1} = \nabla H(\mathbf{g}_k) - \nabla_{\mathbf{g}}h(\mathbf{g}_k,X_{k + 1})$ and thus $\mathbb{E}[\pmb{\xi}_{k + 1}] = 0$ + +Observe that + +$$ +\mathbb {E} \left[ \left\| \sum_ {k = 0} ^ {n} \boldsymbol {\xi} _ {k + 1} \right\| _ {v} ^ {2} \right] = \mathbb {E} \left[ \left\| \sum_ {k = 0} ^ {n - 1} \boldsymbol {\xi} _ {k + 1} \right\| _ {v} ^ {2} + 2 \left\langle \sum_ {k = 0} ^ {n - 1} \boldsymbol {\xi} _ {k + 1}, \boldsymbol {\xi} _ {n + 1} \right\rangle_ {v} + \| \boldsymbol {\xi} _ {n + 1} \| _ {v} ^ {2} \right] +$$ + +with + +$$ +\mathbb {E} \left[ \left\langle \sum_ {k = 0} ^ {n - 1} \boldsymbol {\xi} _ {k + 1}, \boldsymbol {\xi} _ {n + 1} \right\rangle_ {v} \right] = \mathbb {E} \left[ \left\langle \sum_ {k = 0} ^ {n - 1} \boldsymbol {\xi} _ {k + 1}, \mathbb {E} \left[ \boldsymbol {\xi} _ {n + 1} \mid \mathcal {F} _ {n} \right] \right\rangle_ {v} \right] = 0. +$$ + +Thus, since for all $k$ , $\mathbb{E}\left[\|\pmb{\xi}_k\|^2\right] \leq 4$ , we have the convergence rate + +$$ +\frac {1}{n + 1} \mathbb {E} \left[ \left\| \sum_ {k = 0} ^ {n} \boldsymbol {\xi} _ {k + 1} \right\| _ {v} ^ {2} \right] ^ {\frac {1}{2}} \leq \frac {2}{\sqrt {n + 1}}. +$$ + +Convergence rate for $\frac{1}{n + 1}\sum_{k = 0}^{n}\frac{\mathbf{p}_k}{\gamma_{k + 1}}$ + +Take $d_0$ such that $B(\mathbf{g}^*, d_0) \subset \mathcal{C}$ . Using the notation $\nabla_k \coloneqq \nabla_{\mathbf{g}} h(\mathbf{g}_k, X_{k+1})$ for conciseness, we obtain + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {p} _ {k} \| _ {v} ^ {2} \right] = \mathbb {E} \left[ \| \operatorname {P r o j} _ {\mathcal {C}} \left(\mathbf {g} _ {k} - \gamma_ {k + 1} \nabla_ {k}\right) - \left(\mathbf {g} _ {k} - \gamma_ {k + 1} \nabla_ {k}\right) \| _ {v} ^ {2} \right] \\ = \mathbb {E} \left[ \left\| \mathrm {P r o j} _ {\mathcal {C}} \left(\mathbf {g} _ {k} - \gamma_ {k + 1} \nabla_ {k}\right) - \left(\mathbf {g} _ {k} - \gamma_ {k + 1} \nabla_ {k}\right) \right\| _ {v} ^ {2} \mathbf {1} _ {\mathbf {g} _ {k} - \gamma_ {k + 1} \nabla_ {k} \notin \mathcal {C}} \right] \\ \end{array} +$$ + +Since for any $y \in \mathcal{C}$ , one has $\|x - \mathrm{Proj}_{\mathcal{C}}(x)\|_v \leq \|x - y\|_v$ , taking $y = \mathbf{g_k}$ , and since $\mathbf{g}_k - \gamma_{k+1} \nabla_k \notin \mathcal{C}$ is satisfied only if $\|\mathbf{g}_k - \gamma_{k+1} \nabla_k - \mathbf{g}^*\|_v > d_0$ , we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {p} _ {k} \| _ {v} ^ {2} \right] \leq \mathbb {E} \left[ \| \gamma_ {k + 1} \nabla_ {k} \| _ {v} ^ {2} \mathbf {1} _ {\| \mathbf {g} _ {k} - \gamma_ {k + 1} \nabla_ {k} - \mathbf {g} ^ {*} \| _ {v} > d _ {0}} \right] \\ \leq 4 \gamma_ {k + 1} ^ {2} \frac {\mathbb {E} \left[ \| \mathbf {g} _ {k} - \gamma_ {k + 1} \nabla_ {k} - \mathbf {g} ^ {*} \| _ {v} ^ {4} \right]}{d _ {0} ^ {4}} \\ \leq \frac {\gamma_ {k + 1} ^ {2}}{d _ {0} ^ {4}} \left(2 ^ {5} \mathbb {E} \left[ \| \mathbf {g} _ {k} - \mathbf {g} ^ {*} \| _ {v} ^ {4} \right] + 2 ^ {9} \gamma_ {k + 1} ^ {4}\right) \\ \lesssim \frac {1}{\eta^ {2}} \gamma_ {k + 1} ^ {4} \\ \end{array} +$$ + +Where we used Markov's inequality and the inequality $(A + B)^4 \leq 2^3 (A^4 + B^4)$ , for all $A, B \in \mathbb{R}$ and that by Proposition C.1, $\mathbb{E}\left[\| \mathbf{g}_n - \mathbf{g}^*\|^4\right] \lesssim \frac{\gamma_1^2}{\eta^2} (n + 1)^{-2b}$ . + +We thus have + +$$ +\begin{array}{l} \frac {1}{n + 1} \mathbb {E} \left[ \left\| \sum_ {k = 0} ^ {n} \frac {\mathbf {p} _ {k}}{\gamma_ {k + 1}} \right\| _ {v} ^ {2} \right] ^ {\frac {1}{2}} \lesssim \frac {1}{n + 1} \sum_ {k = 0} ^ {n} \frac {\gamma_ {k + 1}}{\eta} \\ \lesssim \frac {\gamma_ {1}}{\eta (n + 1) ^ {b}}. \\ \end{array} +$$ + +Conclusion. + +Using the convergence rate of all our terms, Cauchy-Schwarz inequality and that $(A + B)^2 \leq 2(A^2 + B^2)$ for all $A, B \in \mathbb{R}$ we conclude + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \nabla^ {2} H (\mathbf {g} ^ {*}) \left(\overline {{\mathbf {g}}} _ {n} - \mathbf {g} ^ {*}\right) \right\| _ {v} ^ {2} \right] \\ = \mathbb {E} \left[ \left\| \frac {1}{n + 1} \sum_ {k = 0} ^ {n} \frac {\mathbf {g} _ {k} - \mathbf {g} _ {k + 1}}{\gamma_ {k + 1}} - \frac {1}{n + 1} \sum_ {k = 0} ^ {n} \boldsymbol {\delta} _ {k} + \frac {1}{n + 1} \sum_ {k = 0} ^ {n} \boldsymbol {\xi} _ {k + 1} + \frac {1}{n + 1} \sum_ {k = 0} ^ {n} \frac {\mathbf {p} _ {k}}{\gamma_ {k + 1}} \right\| _ {v} ^ {2} \right] \\ \lesssim \frac {1}{\eta^ {2} (n + 1) ^ {2 - b}} + \frac {1}{\eta^ {\frac {1 + \alpha}{\alpha}} (n + 1) ^ {b + \alpha b}} + \frac {1}{n + 1} + \frac {\gamma_ {1} ^ {4}}{\eta^ {2} (n + 1) ^ {2 b}}. \\ \end{array} +$$ + +Since $b \in (\frac{1}{2}, 1)$ and $b \in (\frac{1}{1 + \alpha}, 1)$ the limiting term is $\frac{1}{n + 1}$ and we have + +$$ +\mathbb {E} \left[ \left\| \nabla^ {2} H (\mathbf {g} ^ {*}) (\overline {{\mathbf {g}}} _ {n} - \mathbf {g} ^ {*}) \right\| _ {v} ^ {2} \right] \lesssim \frac {1}{n + 1}. +$$ + +Finally, observe that there is an orthogonal matrix $U$ such that + +$$ +\nabla^ {2} H \left(\mathbf {g} ^ {*}\right) = U \operatorname {d i a g} \left(\lambda_ {1}, \dots , \lambda_ {M - 1}, 0\right) U ^ {\top}. +$$ + +. Therefore, noting by abuse of notation + +$$ +\left(\nabla^ {2} H \left(\mathbf {g} ^ {*}\right)\right) ^ {- 1} = U \operatorname {d i a g} \left(\lambda_ {1} ^ {- 1}, \dots , \lambda_ {M - 1} ^ {- 1}, 0\right) U ^ {\top} +$$ + +the inverse of $\nabla_k^2$ in the space $\operatorname{Vect}(\mathbf{1}_M)^\perp$ we finally have + +$$ +\mathbb {E} [ \| \overline {{\mathbf {g}}} _ {n} - \mathbf {g} ^ {*} \| _ {v} ^ {2} ] \lesssim \frac {1}{\lambda^ {2} (n + 1)} +$$ + +where $\lambda = \min_{j\in [[1,M - 1]]}\lambda_j > 0$ by either Theorem B.3 or Proposition B.12. + +![](images/956720e7d135c47b7254bea97a92af7ab344229df5d26311c2a6fcf6c5ee91ac.jpg) + +# C.3 Convergence rate of PSGD in the general setting + +Theorem 3.2 (PSGD in the general setting) Assuming that the semi-dual problem (4) admits at least one solution $\mathbf{g}^*$ and that there exists a compact set $K$ such that $\mu(K) \geq 1 - \frac{1}{2} w_{\min}$ , choosing the learning rate $\gamma_n = \gamma_1 / n^b$ with $\gamma_1 = \frac{\mathrm{Diam}(\mathcal{C})}{2\sqrt{2}}$ and $b = \frac{1}{2}$ , we obtain + +$$ +\mathbb {E} \left[ H \left(\overline {{\mathbf {g}}} _ {k}\right) - H \left(\mathbf {g} ^ {*}\right) \right] \leq \frac {4 \sqrt {2} \operatorname {D i a m} (\mathcal {C})}{\sqrt {n}}. +$$ + +Proof. Define $\gamma_k = \frac{\gamma_1}{\sqrt{k}}$ for $\gamma_1 > 0, k \geq 1$ and denote by $\mathbf{g}^* \in \mathcal{C}$ a minimizer of the functional $H$ . Thanks to Jensen's inequality coupled with the fact that no matter $\mathbf{g} \in \mathbb{R}^M: H(\mathbf{g}) - H(\mathbf{g}^*) \leq \nabla H(\mathbf{g})^\top (\mathbf{g} - \mathbf{g}^*)$ , it comes + +$$ +\mathbb {E} \left[ H \left(\overline {{\mathbf {g}}} _ {n}\right) - H \left(\mathbf {g} ^ {*}\right) \right] \leq \mathbb {E} \left[ \frac {1}{n + 1} \sum_ {k = 0} ^ {n} H \left(\mathbf {g} _ {k}\right) - H \left(\mathbf {g} ^ {*}\right) \right] \leq \frac {1}{n + 1} \mathbb {E} \left[ \sum_ {k = 0} ^ {n} \nabla H \left(\mathbf {g} _ {k}\right) ^ {\top} \left(\mathbf {g} _ {k} - \mathbf {g} ^ {*}\right) \right]. +$$ + +Then, since no matter $k$ , $X_{k + 1}\sim \mu$ , we have $\mathbb{E}\left[\nabla_{\mathbf{g}}h(\mathbf{g}_k,X_{k + 1})\mid \mathcal{F}_k\right] = \nabla H(\mathbf{g}_k)$ , we have + +$$ +\mathbb {E} \left[ H \left(\overline {{\mathbf {g}}} _ {n}\right) - H \left(\mathbf {g} ^ {*}\right) \right] \leq \frac {1}{n + 1} \mathbb {E} \left[ \sum_ {k = 0} ^ {n} \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {k}, X _ {k + 1}\right) ^ {\top} \left(\mathbf {g} _ {k} - \mathbf {g} ^ {*}\right) \right]. \tag {16} +$$ + +We will proceed to bound the right hand side of this inequality. + +By definition of $\mathbf{g}_{k + 1}$ , and since $\mathbf{g}^{*}\in \mathcal{C}$ and $\mathrm{Proj}_{\mathcal{C}}$ is 1-Lipschitz, it comes + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \mathbf {g} _ {k + 1} - \mathbf {g} ^ {*} \right\| ^ {2} \right] \leq \mathbb {E} \left[ \left\| \operatorname {P r o j} _ {\mathcal {C}} \left(\mathbf {g} _ {k} - \gamma_ {k + 1} \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {k}, X _ {k + 1}\right)\right) - \mathbf {g} ^ {*} \right\| ^ {2} \right] \\ \leq \mathbb {E} \left[ \left\| \mathbf {g} _ {k} - \gamma_ {k + 1} \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {k}, X _ {k + 1}\right) - \mathbf {g} ^ {*} \right\| ^ {2} \right] \\ \leq \mathbb {E} \left[ \| \mathbf {g} _ {k} - \mathbf {g} ^ {*} \| ^ {2} + \gamma_ {k + 1} ^ {2} \| \nabla_ {\mathbf {g}} h (\mathbf {g} _ {k}, X _ {k + 1}) \| ^ {2} - 2 \gamma_ {k + 1} \nabla_ {\mathbf {g}} h (\mathbf {g} _ {k}, X _ {k + 1}) ^ {\top} (\mathbf {g} _ {k} - \mathbf {g} ^ {*}) \right]. \\ \end{array} +$$ + +In addition, since $\| \nabla_{\mathbf{g}}h(\mathbf{g},X)\| \leq 2$ a.s. no matter $\mathbf{g}\in \mathbb{R}^{M}$ and $X\in \mathbb{R}^d$ + +$$ +\mathbb {E} \left[ 2 \gamma_ {k + 1} \nabla_ {\mathbf {g}} h (\mathbf {g} _ {k}, X _ {k + 1}) ^ {\top} (\mathbf {g} _ {k} - \mathbf {g} ^ {*}) \right] \leq \mathbb {E} \left[ \| \mathbf {g} _ {k} - \mathbf {g} ^ {*} \| ^ {2} - \| \mathbf {g} _ {k + 1} - \mathbf {g} ^ {*} \| ^ {2} + 4 \gamma_ {k + 1} ^ {2} \right]. +$$ + +Then, with the help of Abel's summation formula, + +$$ +\begin{array}{l} 2 \mathbb {E} \left[ \sum_ {k = 0} ^ {n} \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {k}, X _ {k + 1}\right) ^ {\top} \left(\mathbf {g} _ {k} - \mathbf {g} ^ {*}\right) \right] \\ \leq \mathbb {E} \left[ \sum_ {k = 0} ^ {n} \frac {\| \mathbf {g} _ {k} - \mathbf {g} ^ {*} \| ^ {2} - \| \mathbf {g} _ {k + 1} - \mathbf {g} ^ {*} \| ^ {2}}{\gamma_ {k + 1}} \right] + 4 \sum_ {k = 0} ^ {n} \gamma_ {k + 1} \\ \leq \mathbb {E} \left[ \sum_ {k = 1} ^ {n} \| \mathbf {g} _ {k} - \mathbf {g} ^ {*} \| ^ {2} \left(\frac {1}{\gamma_ {k + 1}} - \frac {1}{\gamma_ {k}}\right) \right] - \frac {\| \mathbf {g} _ {k + 1} - \mathbf {g} ^ {*} \| ^ {2}}{\gamma_ {k + 1}} + \frac {\| \mathbf {g} _ {0} - \mathbf {g} ^ {*} \| ^ {2}}{\gamma_ {1}} + 4 \sum_ {k = 0} ^ {n} \gamma_ {k + 1}. \\ \end{array} +$$ + +Then, since for all $k$ , $\| \mathbf{g}_k - \mathbf{g}^* \| \leq \mathrm{Diam}_2(\mathcal{C})$ , it comes + +$$ +\begin{array}{l} 2 \mathbb {E} \left[ \sum_ {k = 0} ^ {n} \nabla_ {\mathbf {g}} h \left(\mathbf {g} _ {k}, X _ {k + 1}\right) ^ {\top} \left(\mathbf {g} _ {k} - \mathbf {g} ^ {*}\right) \right] \leq \operatorname {D i a m} _ {2} (\mathcal {C}) ^ {2} \sum_ {k = 1} ^ {n} \left(\frac {1}{\gamma_ {k + 1}} - \frac {1}{\gamma_ {k}}\right) + \frac {D ^ {2}}{\gamma_ {1}} + 4 \sum_ {k = 1} ^ {n} \gamma_ {k} \\ \leq \frac {\operatorname {D i a m} _ {2} (\mathcal {C}) ^ {2}}{\gamma_ {n + 1}} + 4 \sum_ {k = 1} ^ {n} \gamma_ {k} \\ \leq \frac {\operatorname {D i a m} _ {2} (\mathcal {C}) ^ {2}}{\gamma_ {1}} \sqrt {n + 1} + 8 \gamma_ {1} \sqrt {n + 1} \\ \end{array} +$$ + +Using the inequality (16) we obtain + +$$ +\mathbb {E} \left[ H \left(\overline {{\mathbf {g}}} _ {n}\right) - H \left(\mathbf {g} ^ {*}\right) \right] \leq \frac {1}{\sqrt {n + 1}} \left(\frac {\operatorname {D i a m} _ {2} (\mathcal {C}) ^ {2}}{\gamma_ {1}} + 8 \gamma_ {1}\right). +$$ + +The best constant $\gamma_{1}$ is then $\gamma_{1} = \frac{\mathrm{Diam}_{2}(\mathcal{C})}{2\sqrt{2}}$ , but no matter $\gamma_{1} > 0$ , we have the desired convergence rate + +$$ +\mathbb {E} \left[ H \left(\overline {{\mathbf {g}}} _ {n}\right) - H \left(\mathbf {g} ^ {*}\right) \right] = \mathcal {O} (1 / \sqrt {n}). +$$ + +![](images/46bceecaf29afa811a0121522dbbdf1947f41efc26bf7a6c83360450cddd7dc0.jpg) + +# D Proof of Lemma 3.1: Localisation of a projection set + +Lemma 3.1 (Existence of a projection set) As soon as the semi-dual problem is well-posed, there exists a minimizer $\mathbf{g}^*$ in the set + +$$ +\mathcal {C} := \left\{\mathbf {g} \in \{0 \} \times \mathbb {R} ^ {M - 1} \mid | g _ {j} | \leq \| c \| _ {K, \infty} \right\} +$$ + +where $\| c\|_{K,\infty}\coloneqq \sup_{x\in K,j\in [[1,M]]}|c(x,y_j)|$ for any compact set $K$ satisfying $\mu (K)\geq 1 - \frac{1}{2}\min w_{j}$ + +Proof. By the first order condition, the minimizer of $H$ satisfies $\mu(\mathbb{L}_j(\mathbf{g})) = w_j$ for all $j \in [1, M]$ . In particular, one can restrict the search set to an optimal potential to the set of potentials defined by + +$$ +\mathcal {L} := \left\{\mathbf {g} \in \mathbb {R} ^ {M}: \forall j \in [ [ 1, M ] ], \mu (\mathbb {L} _ {j} (\mathbf {g})) \geq \frac {2}{3} w _ {\min } \right\}. +$$ + +Let us show that this set is contained in an $L^{\infty}$ ball with an explicit radius. Consider any compact set $K$ such that $\mu(K) \geq 1 - \frac{1}{2} w_{\min}$ . For $\mathbf{g} \in \mathcal{L}$ and any $j \in [1, M]$ we get that + +$$ +\begin{array}{l} \mu (\mathbb {L} _ {j} (\mathbf {g}) \cap K) = 1 - \mu ((\mathbb {R} ^ {M} \setminus \mathbb {L} _ {j} (\mathbf {g})) \cup (\mathbb {R} ^ {M} \setminus K)) \\ \geq 1 - \mu \left(\left(\mathbb {R} ^ {M} \backslash \mathbb {L} _ {j} (\mathbf {g})\right)\right) - \mu \left(\left(\mathbb {R} ^ {M} \backslash K\right)\right) \\ = 1 - \left(1 - \mu \left(\mathbb {L} _ {j} (\mathbf {g})\right)\right) - \left(1 - \mu (K)\right) \\ \geq \frac {w _ {\operatorname* {m i n}}}{6} > 0, \\ \end{array} +$$ + +and so $\mathbb{L}_j(\mathbf{g})\cap K\neq \emptyset$ . In particular, for every $j\in [[1,M]]$ , there exists $x_{j}\in \mathbb{L}_{j}(\mathbf{g})\cap K$ and so for all $i\in [[1,M]]$ + +$$ +c \left(x _ {j}, y _ {j}\right) - g _ {j} \leq c \left(x _ {j}, y _ {i}\right) - g _ {i}. +$$ + +Therefore, using the fact that the cost is non-negative, we have + +$$ +\begin{array}{l} \max _ {i} g _ {i} - \min _ {j} g _ {j} \leq \max _ {i} \max _ {j} \{c (x _ {j}, y _ {i}) - c (x _ {j}, y _ {j}) \} \\ \leq \max _ {i \in [ [ 1, M ] ]} \sup _ {x \in K} c (x, y _ {i}) = \| c \| _ {K, \infty}. \\ \end{array} +$$ + +Moreover, since $H(\mathbf{g} + \lambda \mathbb{1}_M) = H(\mathbf{g})$ one can fix $g_1 = 0$ , which concludes the proof. + +![](images/ff97aa4f4005d0dd87b9dd2fa299c271dc27bdbeef0a108195c9e233592edc22.jpg) + +# E Minimax estimation of OT quantities + +Consider $P, Q \in \mathcal{P}(\mathbb{R}^d)$ with densities $f_P$ and $f_Q$ . We recall that the Hellinger distance is defined by + +$$ +d _ {\mathrm {H}} (P, Q) := \left(\int_ {\mathbb {R} ^ {d}} \left(\sqrt {f _ {P} (x)} - \sqrt {f _ {Q} (x)}\right) ^ {2} \mathrm {d} \lambda_ {\mathbb {R} ^ {d}} (x)\right) ^ {\frac {1}{2}}. \tag {17} +$$ + +We also recall the formulation of Le Cam's Lemma. We refer to [35], Chapter 15, for more details. + +Lemma E.1. (Le Cam's Lemma.) Let $\mathcal{P}$ be a set of probability distributions on a measurable space, and consider the problem of estimating a parameter $\theta \in \Theta$ with a loss function $\ell$ defined, for all $\hat{\theta},\theta \in \Theta$ , as + +$$ +\ell (\hat {\theta}, \theta) = d (\hat {\theta}, \theta) ^ {p}, +$$ + +where $p \geq 1$ is an integer, and $d$ is a distance on $\Theta$ . Then, for all $\theta_1, \theta_2 \in \Theta$ , + +$$ +R _ {M} = \inf _ {T ^ {(n)}} \sup _ {\theta \in \Theta} \mathbb {E} _ {\theta} [ \ell (\theta , T ^ {(n)} (X)) ] \geq \frac {1}{2 ^ {p}} \left(1 - \sqrt {n} d _ {\mathrm {H}} (P _ {\theta_ {1}}, P _ {\theta_ {2}})\right) d (\theta_ {1}, \theta_ {2}) ^ {p}, +$$ + +where $R_{M}$ is the minimax risk, and $T^{(n)}$ is an estimator based on $n$ i.i.d samples from $P_{\theta}$ . + +In the sequel, we note $\mathcal{P}_{\mathrm{Lip}}$ the set of probability measures on $\mathbb{R}$ with Lipschitz densities. + +# E.1 Kantorovich potential + +Proof. Consider $\nu = \frac{1}{2}\delta_{\{0\}} + \frac{1}{2}\delta_{\{1\}}$ and fix the cost to transfer mass to be the usual quadratic cost $c(x,y) = \frac{1}{2}\| x - y\|^2$ . + +For $\delta \geq 0$ , we define $\mu_{\theta^{\delta}} = \mathcal{N}(\delta, 1)$ . Since $d = 1$ , the optimal transport map is monotone non-decreasing (see, for instance, Chapter 2 in [32]). Thus, we must have the identity + +$$ +T _ {\mu_ {\theta_ {\delta}}, \nu} (x) = \left\{ \begin{array}{l l} 0, & \text {i f} x \leq \delta , \\ 1, & \text {o t h e r w i s e}. \end{array} \right. +$$ + +Therefore the vector $\theta^{\delta} \in \mathbb{R}^{2}$ solves the semi-dual problem if and only if it satisfies the following inequalities + +$$ +c (x, 0) - \theta_ {1} ^ {\delta} \leq c (x, 1) - \theta_ {2} ^ {\delta}, \quad \forall x \leq \delta , +$$ + +$$ +c (x, 1) - \theta_ {2} ^ {\delta} \leq c (x, 0) - \theta_ {1} ^ {\delta}, \quad \forall x > \delta . +$$ + +Since $c(x,y) = \frac{1}{2}\| x - y\|^2$ we can fix $\theta_1^\delta = 0$ and compute + +$$ +\theta^ {\delta} = \left(0, \frac {1}{2} - \delta\right). +$$ + +Therefore, we parameterized the family of probability measures $\mu_{\theta} \in \mathcal{P}_{\mathrm{Lip}}$ so that the couple $(\theta^c, \theta)$ is the unique solution in $\Theta \subset \{\theta = (\theta_1, \theta_2) \in \mathbb{R}^2; \theta_1 = 0\}$ of the dual of $\mathrm{OT}(\mu_{\theta}, \nu)$ . In this class of + +probabilities, the minimax estimation of the optimal transport potential $\theta$ , given $n > 0$ i.i.d samples of the source measure, can be written as + +$$ +R _ {n} ^ {\Theta} := \inf _ {\hat {\theta} ^ {(n)}} \sup _ {\theta \in \Theta} \mathbb {E} _ {\mu_ {\theta}} \left[ \| \hat {\theta} ^ {(n)} - \theta \| ^ {2} \right], +$$ + +where $\hat{\theta}^{(n)}$ is based on the $n$ i.i.d samples from the source measure $\mu$ . Note that + +$$ +R _ {n} ^ {\Theta} \leq \inf _ {\mathbf {g} ^ {(n)}} \sup _ {\mu \in \mathcal {P} _ {\mathrm {L i p}}} \mathbb {E} _ {\mu} \left[ \| \mathbf {g} ^ {(n)} - \mathbf {g} ^ {*} \| ^ {2} \right], \tag {18} +$$ + +where the infimum is taken over all vectors $\mathbf{g}^{(n)}$ based on $n$ i.i.d samples of $\mu$ . + +Using the closed form of the Hellinger distance between Gaussian distributions, we have + +$$ +d _ {\mathrm {H}} (\mu_ {\theta^ {0}}, \mu_ {\theta^ {\delta}}) = \sqrt {1 - \exp \left(- \frac {\delta^ {2}}{8}\right)}. +$$ + +For $\delta \to 0$ , the Taylor expansion gives $d_{\mathrm{H}}(\mu_{\theta^0},\mu_{\theta^\delta}) = \delta /\sqrt{8} +o(\delta)$ . Applying Le Cam's Lemma with $\delta = 1 / \sqrt{n}$ gives + +$$ +\begin{array}{l} R _ {n} ^ {\Theta} \geq \frac {1}{4} \left(1 - \sqrt {n} d _ {\mathrm {H}} \left(\mu_ {\theta^ {0}}, \mu_ {\theta^ {\delta}}\right)\right) \| \theta^ {0} - \theta^ {\delta} \| _ {2} ^ {2} \\ \geq \frac {1}{4} \left(1 - 1 / \sqrt {8} + o (1)\right) \frac {1}{n} \\ \geq \frac {1}{1 0 n} + o \left(\frac {1}{n}\right) \\ \end{array} +$$ + +Using the inequality (18) concludes the proof. + +![](images/d184daf75df50dbedbea6006604c4b4c7f6ea48ccdab718cf60324f47b4de84f.jpg) + +# E.2 OT map + +Proof. Define for $\delta \in [0,1]$ the set of probability measures $\mu_{\delta}$ , with density: + +$$ +f _ {\mu_ {\delta}} (x) = \mathbf {1} _ {x \in [ 0, 1 ]} \big (1 + \delta g (x) \big), \qquad x \in [ 0, 1 ], +$$ + +where + +$$ +g (x) = \left\{ \begin{array}{l l} 2 (1 - 2 x), & x \in [ 0, 1 / 2 ], \\ - 2 (2 x - 1), & x \in [ 1 / 2, 1 ]. \end{array} \right. +$$ + +The squared Hellinger distance between $\mu_0$ (uniform) and $\mu_{\delta}$ is: + +$$ +\begin{array}{l} d _ {\mathrm {H}} (\mu_ {0}, \mu_ {\delta}) ^ {2} = \frac {1}{2} \int_ {0} ^ {1} \left(\sqrt {1} - \sqrt {1 + \delta g (x)}\right) ^ {2} d x \\ = \frac {1}{6 \delta} \left((1 - 2 \delta) ^ {3 / 2} + 6 \delta - (1 + 2 \delta) ^ {3 / 2}\right) \\ = \frac {1}{2} \delta^ {2} + o (\delta^ {2}) \\ \end{array} +$$ + +when $\delta \to 0$ . Therefore, we obtain + +$$ +d _ {\mathrm {H}} (\mu_ {0}, \mu_ {\delta}) = \frac {1}{\sqrt {2}} \delta + o (\delta). +$$ + +Since $\mu_{\delta}\in \mathcal{P}_H$ $\delta \in [0,1]$ , we have + +$$ +\inf _ {T ^ {(n)}} \sup _ {\mu \in \mathcal {P} _ {H}} \mathbb {E} _ {\mu} \left[ \left\| T ^ {(n)} - T _ {\mu , \nu} \right\| _ {L ^ {p} (\mu)} ^ {p} \right] \geq \inf _ {T ^ {(n)}} \sup _ {\delta \in [ 0, 1 ]} \mathbb {E} _ {\mu_ {\delta}} \left[ \left\| T ^ {(n)} - T _ {\mu_ {\delta}, \nu} \right\| _ {L ^ {p} (\mu_ {\delta})} ^ {p} \right]. \tag {19} +$$ + +From the relation $1 - 2\delta \leq f_{\mu_{\delta}}(x) \leq 1 + 2\delta, \forall x \in [0,1]$ , we infer that no matter $T^{(n)}$ and $\delta$ small enough + +$$ +\left\| T ^ {(n)} - T _ {\mu_ {\delta}, \nu} \right\| _ {L ^ {p} (\mu_ {\delta})} ^ {p} \geq \frac {1}{2} \left\| T ^ {(n)} - T _ {\mu_ {\delta}, \nu} \right\| _ {L ^ {p} (\mathcal {U} (0, 1))} ^ {p}. \tag {20} +$$ + +Consider $\nu = \frac{1}{2}\delta_{\{0\}} + \frac{1}{2}\delta_{\{1\}}$ as above. Recall that the optimal transport map is monotone nondecreasing in dimension 1. Moreover by definition $T_{\mu_{\delta},\nu}(x)\in \{0,1\}$ , $x\in [0,1]$ , and $\mu_{\delta}(T_{\mu_{\delta},\nu}^{-1}(0)) = 1/2$ . Therefore one identifies $T_{\mu_{\delta},\nu} = \mathbf{1}_{x\geq M_{\delta}}$ where $M_{\delta}$ is the median of $\mu_{\delta}$ , $\delta > 0$ , satisfying + +$$ +\int_ {0} ^ {M _ {\delta}} \left(1 + \delta g (x)\right) d x = \frac {1}{2}. +$$ + +Noticing that $M_{\delta}\in [0,1 / 2]$ , we solve + +$$ +\int_ {0} ^ {M _ {\delta}} \left(1 + 2 \delta (1 - 2 x)\right) d x = \frac {1}{2}, +$$ + +which gives the solution + +$$ +M _ {\delta} = \frac {1 + 2 \delta - \sqrt {1 + 4 \delta^ {2}}}{4 \delta} = \frac {1}{2} - \frac {1}{2} \delta + o (\delta). +$$ + +Observe that + +$$ +\left\| T _ {\mu_ {0}, \nu} - T _ {\mu_ {\delta}, \nu} \right\| _ {L ^ {p} (\mathcal {U} (0, 1))} ^ {p} = \int_ {0} ^ {1} \left| \mathbf {1} _ {x \geq M _ {0}} - \mathbf {1} _ {x \geq M _ {\delta}} \right| ^ {p} d x = | M _ {\delta} - M _ {0} |. +$$ + +We proved the relation $\| T_{\mu_0,\nu} - T_{\mu_{\delta},\nu}\|_{L^p (\mathcal{U}(0,1))}^p = \delta +o(\delta)$ . Applying Le Cam's Lemma with $\delta = \frac{1}{\sqrt{n}}$ for $n$ sufficiently large and any $p\in [1,\infty)$ , we obtain + +$$ +\begin{array}{l} \inf _ {T ^ {(n)}} \sup _ {\delta \in [ 0, 1 ]} \mathbb {E} _ {\mu_ {\delta}} \left[ \left\| T ^ {(n)} - T _ {\mu_ {\delta}, \nu} \right\| _ {L ^ {p} (\mathcal {U} (0, 1))} ^ {p} \right] \geq \frac {1}{2 ^ {p}} \left(1 - \sqrt {n} d _ {\mathrm {H}} (\mu_ {0}, \mu_ {\delta})\right) \| T _ {\mu_ {0}, \nu} - T _ {\mu_ {\delta}, \nu} \| _ {L ^ {p} (\mathcal {U} (0, 1))} ^ {p} \\ \geq \frac {1}{2 ^ {p}} \left(1 - \frac {1}{\sqrt {2}} + o (1)\right) \left(\frac {1}{\sqrt {n}} + o \left(\frac {1}{\sqrt {n}}\right)\right) \\ \gtrsim \frac {1}{\sqrt {n}}. \\ \end{array} +$$ + +Therefore, combining inequalities (19) and (20) we conclude + +$$ +\inf _ {T ^ {(n)}} \sup _ {\mu \in \mathcal {P} _ {H}} \mathbb {E} _ {\mu} \left[ \left\| T ^ {(n)} - T _ {\mu , \nu} \right\| _ {L ^ {p} (\mu)} ^ {p} \right] \gtrsim \frac {1}{\sqrt {n}}. +$$ + +![](images/01767be96d1b5eee0aef95bed9c046483a91df76326f4dfff1e250cffceaae09.jpg) + +# F Technical Lemmas + +# F.1 Technical Lemmas for Appendix B + +Lemma F.1. Perturbation of Laplacian Matrices. Let $A$ and $B$ be symmetric Laplacian matrices of the same size such that: + +$$ +A _ {i j} \leq 0, \quad B _ {i j} \leq A _ {i j} \quad f o r a l l i \neq j. +$$ + +Suppose $\lambda_2(A) > 0$ , where $\lambda_2(A)$ denotes the second smallest eigenvalue of $A$ . Then: + +$$ +\lambda_ {2} (B) \geq \lambda_ {2} (A) +$$ + +where $\lambda_2(B)$ is the second smallest eigenvalue of $B$ . + +Proof. We recall the variational characterization of the second smallest eigenvalue of a Laplacian matrix $M$ is + +$$ +\lambda_ {2} (M) = \min _ {x \perp \mathbf {1}} \frac {x ^ {T} M x}{x ^ {T} x}. +$$ + +Define the matrix $C = B - A$ . For $i \neq j$ : + +$$ +C _ {i j} = B _ {i j} - A _ {i j} \leq 0, +$$ + +and for diagonal elements + +$$ +C _ {i i} = B _ {i i} - A _ {i i} = - \sum_ {j \neq i} C _ {i j} \geq 0. +$$ + +Thus, $C$ is a Laplacian matrix and so it is positive, semi-definite. + +Let $y_{2}$ be the eigenvector corresponding to $\lambda_2(B)$ , the second smallest eigenvalue of $B$ , that is + +$$ +B y _ {2} = \lambda_ {2} (B) y _ {2} \quad \text {w i t h} y _ {2} \perp \mathbf {1}. +$$ + +Since $A = B - C$ , we have + +$$ +y _ {2} ^ {T} A y _ {2} = y _ {2} ^ {T} (B - C) y _ {2} = y _ {2} ^ {T} B y _ {2} - y _ {2} ^ {T} C y _ {2}. +$$ + +Thus, + +$$ +\frac {y _ {2} ^ {T} A y _ {2}}{y _ {2} ^ {T} y _ {2}} = \lambda_ {2} (B) - \frac {y _ {2} ^ {T} C y _ {2}}{y _ {2} ^ {T} y _ {2}}. +$$ + +By the variational principle: + +$$ +\lambda_ {2} (A) = \min _ {x \perp \mathbf {1}} \frac {x ^ {T} A x}{x ^ {T} x} \leq \frac {y _ {2} ^ {T} A y _ {2}}{y _ {2} ^ {T} y _ {2}} \leq \lambda_ {2} (B) - \frac {y _ {2} ^ {T} C y _ {2}}{y _ {2} ^ {T} y _ {2}}. +$$ + +Since $C$ is a Laplacian matrix, we have $y_2^T Cy_2 \geq 0$ , no matter $y_2$ and thus $\lambda_2(A) \leq \lambda_2(B)$ which completes the proof. + +![](images/56e5a4b084db3a758dac7ac9651fd9712443db8e20c131ed901ab3e34a8cc42e.jpg) + +Lemma F.2. Let $f_{\mu}$ satisfy, for all $x \in \mathbb{R}^d$ , the decay condition + +$$ +\sum_ {r \geq 1} r ^ {d - 1} \sup _ {x \in \mathbb {R} ^ {d} \backslash B (0, r - 1)} f _ {\mu} (x) < \infty . +$$ + +Then, there exists a constant $C > 0$ such that, for any hyperplane $H\subset \mathbb{R}^d$ , we have + +$$ +\int_ {H} f _ {\mu} (x) \mathrm {d} \mathcal {H} ^ {d - 1} (x) \leq C. +$$ + +Proof. Defining for any $r \in \mathbb{N}^*$ , $R(r) \coloneqq B(0, r) \setminus B(0, r - 1)$ , we have + +$$ +\begin{array}{l} \int_ {H} f _ {\mu} (x) d \mathcal {H} ^ {d - 1} (x) = \sum_ {r \geq 1} \int_ {H \cap R (r)} f _ {\mu} (x) d \mathcal {H} ^ {d - 1} (x) \\ \leq \sum_ {r \geq 1} \int_ {H \cap R (r)} \sup _ {x \in R (r)} f _ {\mu} (x) \mathrm {d} \mathcal {H} ^ {d - 1} (x) \\ \leq \sum_ {r \geq 1} \int_ {H \cap R (r)} \sup _ {x \in \mathbb {R} ^ {d} \backslash B (0, r - 1)} f _ {\mu} (x) \mathrm {d} \mathcal {H} ^ {d - 1} (x) \\ \end{array} +$$ + +Then, using the fact that $\mathcal{H}^{d - 1}(H\cap R(r))\leq \mathcal{H}^{d - 1}\left(H\cap B(0,r)\right)$ , and noting that $H\cap B(0,r)$ is a $(d - 1)$ -dimensional ball of radius $r$ , we have + +$$ +\mathcal {H} ^ {d - 1} (H \cap R (r)) \leq \frac {\pi^ {(d - 1) / 2}}{\Gamma \left(\frac {d + 1}{2}\right)} r ^ {d - 1}. +$$ + +Therefore, incorporating this bound, we obtain: + +$$ +\int_ {H} f _ {\mu} (x) \mathrm {d} \mathcal {H} ^ {d - 1} (x) \leq \sum_ {r \geq 1} \frac {\pi^ {(d - 1) / 2}}{\Gamma \left(\frac {d + 1}{2}\right)} r ^ {d - 1} \sup _ {x \in \mathbb {R} ^ {d} \backslash B (0, r)} f _ {\mu} (x) +$$ + +which is finite by our decay assumption on $f_{\mu}$ . + +# F.2 Technical Lemmas for Appendix C + +Lemma F.3. Let $(\gamma_{n})_{n\geq 0}$ and $(m_n)_{n\geq 0}$ be some positive and decreasing sequences and let $(\delta_{n})_{n\geq 0}$ satisfying the following: + +- The sequence $\delta_{n}$ follows the recursive relation: + +$$ +\delta_ {n + 1} \leq (1 - \omega \gamma_ {n + 1}) \delta_ {n} + m _ {n + 1} \gamma_ {n + 1}, \tag {21} +$$ + +with $\delta_0\geq 0$ and $\omega >0$ + +- $\gamma_{n}$ converges to 0. +- Let $n_0 = \inf \{n \geq 1 : \omega \gamma_{n+1} \leq 1\}$ , $\delta_{n_0}$ is non-negative. + +Then, for all $n \geq n_0$ , we have the upper bound: + +$$ +\delta_ {n} \leq \exp \left(- \omega \sum_ {i = n _ {0} + 1} ^ {n} \gamma_ {i}\right) \left(\sum_ {k = n _ {0}} ^ {n} \gamma_ {k} m _ {k} + \delta_ {n _ {0}}\right) + \frac {1}{\omega} m _ {\lceil \frac {n}{2} \rceil - 1} +$$ + +Proof. For all $n \geq n_0$ , one has + +$$ +\delta_ {n} \leq \underbrace {\prod_ {i = n _ {0} + 1} ^ {n} \left(1 - \omega \gamma_ {i}\right) \delta_ {n _ {0}}} _ {=: U _ {1, n}} + \underbrace {\sum_ {k = n _ {0} + 1} ^ {n} \prod_ {i = k + 1} ^ {n} \left(1 - \omega \gamma_ {i}\right) \gamma_ {k} m _ {k}} _ {=: U _ {2, n}} +$$ + +One can consider two cases: $\lceil n / 2\rceil -1\leq n_0$ and $\lceil n / 2\rceil -1 > n_0$ + +Case where $\lceil n / 2\rceil -1\leq n_0 < n$ : Since $m_{k}$ is decreasing, + +$$ +\begin{array}{l} U _ {2, n} \leq m _ {n _ {0} + 1} \sum_ {k = n _ {0} + 1} ^ {n} \prod_ {i = k + 1} ^ {n} (1 - \omega \gamma_ {i}) \gamma_ {k} \\ = \frac {1}{\omega} m _ {n _ {0} + 1} \sum_ {k = n _ {0} + 1} ^ {n} \prod_ {i = k + 1} ^ {n} \left(1 - \omega \gamma_ {i}\right) - \prod_ {i = k} ^ {n} \left(1 - \omega \gamma_ {i}\right) \\ = \frac {1}{\omega} m _ {n _ {0} + 1} \left(1 - \prod_ {i = n _ {0} + 1} ^ {n} \left(1 - \omega \gamma_ {i}\right)\right) \\ \leq \frac {1}{\omega} m _ {n _ {0} + 1} \\ \end{array} +$$ + +Since $m_{k}$ is decreasing, it comes $U_{2,n} \leq \frac{1}{\omega} m_{\lceil n/2 \rceil}$ . + +Case where $\lceil n/2 \rceil - 1 > n_0$ : As in [5], for all $m = n_0 + 1, \ldots, n$ , one has + +$$ +U _ {2, n} \leq \exp \left(- \omega \sum_ {k = m + 1} ^ {n} \gamma_ {k}\right) \sum_ {k = n _ {0} + 1} ^ {m} \gamma_ {k} m _ {k} + \frac {1}{\omega} m _ {m} +$$ + +Then, taking $m = \lceil n / 2\rceil -1$ , it comes + +$$ +U _ {2, n} \leq \exp \left(- \omega \sum_ {k = \lceil n / 2 \rceil} ^ {n} \gamma_ {k}\right) \sum_ {k = n _ {0} + 1} ^ {\lceil n / 2 \rceil - 1} \gamma_ {k} m _ {k} + \frac {1}{\omega} m _ {\lceil n / 2 \rceil - 1} +$$ + +Lemma F.4. Linearization of the Hessian If a function $H:\mathbb{R}^M\to \mathbb{R}$ is such that its Hessian is $\alpha$ Hölder on the ball $B(0,r)$ , with $r > 0, \alpha \in (0,1)$ and constant $L$ , then for any $\mathbf{g},\mathbf{g}^{*}\in B(0,r)$ we have + +$$ +\left\| \nabla H (\mathbf {g}) - \nabla^ {2} H (\mathbf {g} ^ {*}) (\mathbf {g} - \mathbf {g} ^ {*}) \right\| \leq C \| \mathbf {g} - \mathbf {g} ^ {*} \| ^ {1 + \alpha}, +$$ + +where $C = \frac{L}{\alpha + 1}$ + +Proof. Consider the Taylor expansion of $\nabla H(\mathbf{g})$ around $\mathbf{g}^*$ : + +$$ +\nabla H (\mathbf {g}) = \nabla H \left(\mathbf {g} ^ {*}\right) + \nabla^ {2} H \left(\mathbf {g} ^ {*}\right) \left(\mathbf {g} - \mathbf {g} ^ {*}\right) + R (\mathbf {g}), +$$ + +where the remainder term $R(\mathbf{g})$ is given by: + +$$ +R (\mathbf {g}) = \int_ {0} ^ {1} \left[ \nabla^ {2} H (\mathbf {g} ^ {*} + t (\mathbf {g} - \mathbf {g} ^ {*})) - \nabla^ {2} H (\mathbf {g} ^ {*}) \right] (\mathbf {g} - \mathbf {g} ^ {*}) \mathrm {d} t. +$$ + +By the assumption of $\alpha$ -Hölder continuity of the Hessian, we have + +$$ +\left\| \nabla^ {2} H \left(\mathbf {g} ^ {*} + t \left(\mathbf {g} - \mathbf {g} ^ {*}\right)\right) - \nabla^ {2} H \left(\mathbf {g} ^ {*}\right) \right\| \leq L \| t \left(\mathbf {g} - \mathbf {g} ^ {*}\right) \| ^ {\alpha} = L t ^ {\alpha} \| \mathbf {g} - \mathbf {g} ^ {*} \| ^ {\alpha}. +$$ + +Thus, + +$$ +\| R (\mathbf {g}) \| \leq \int_ {0} ^ {1} L t ^ {\alpha} \| \mathbf {g} - \mathbf {g} ^ {*} \| ^ {\alpha} \| \mathbf {g} - \mathbf {g} ^ {*} \| \mathrm {d} t = L \| \mathbf {g} - \mathbf {g} ^ {*} \| ^ {1 + \alpha} \int_ {0} ^ {1} t ^ {\alpha} \mathrm {d} t. +$$ + +Evaluating the integral: + +$$ +\int_ {0} ^ {1} t ^ {\alpha} \mathrm {d} t = \frac {1}{\alpha + 1}. +$$ + +Therefore, + +$$ +\| R (\mathbf {g}) \| \leq \frac {L}{\alpha + 1} \| (\mathbf {g} - (\mathbf {g} ^ {*}) ^ {\|} ^ {1 + \alpha}. +$$ + +This implies the desired inequality: + +$$ +\left\| \nabla H (\mathbf {g}) - \nabla^ {2} H (\mathbf {g} ^ {*}) (\mathbf {g} - (\mathbf {g} ^ {*}) \right\| \leq C \| (\mathbf {g} - (\mathbf {g} ^ {*} \| ^ {1 + \alpha}, +$$ + +where $C = \frac{L}{\alpha + 1}$ + +![](images/07ace489951b76a413aa82725e3ddfd9119e883bcb848970f70120e335e5101c.jpg) \ No newline at end of file diff --git a/stochasticoptimizationinsemidiscreteoptimaltransportconvergenceanalysisandminimaxrate/images.zip b/stochasticoptimizationinsemidiscreteoptimaltransportconvergenceanalysisandminimaxrate/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..911ec4eed625caf45c1314ce4a1d33ce00856567 --- /dev/null +++ b/stochasticoptimizationinsemidiscreteoptimaltransportconvergenceanalysisandminimaxrate/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:aa70b56585560392629b712bbda31293064ee437804fdb2c0af833f2e47c567e +size 1695677 diff --git a/stochasticoptimizationinsemidiscreteoptimaltransportconvergenceanalysisandminimaxrate/layout.json b/stochasticoptimizationinsemidiscreteoptimaltransportconvergenceanalysisandminimaxrate/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..7a62e7bb1ada6e98c360298b98f360ec6f1578e8 --- /dev/null +++ b/stochasticoptimizationinsemidiscreteoptimaltransportconvergenceanalysisandminimaxrate/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:75ed72c71dc4795694953e5eaf8058d5356c269379e1518492461dc286301622 +size 2004476 diff --git a/stochasticprincipalagentproblemscomputingandlearningoptimalhistorydependentpolicies/8eb3500a-b899-41fc-96ec-9d091a4c1870_content_list.json b/stochasticprincipalagentproblemscomputingandlearningoptimalhistorydependentpolicies/8eb3500a-b899-41fc-96ec-9d091a4c1870_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..5e21d7d335c01bff5ed5e2a9e0522867842f3d00 --- /dev/null +++ b/stochasticprincipalagentproblemscomputingandlearningoptimalhistorydependentpolicies/8eb3500a-b899-41fc-96ec-9d091a4c1870_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:34d3ad54f60a308438ce1419451c1af2066b0e2307c75a290fa8e5579d804149 +size 180956 diff --git a/stochasticprincipalagentproblemscomputingandlearningoptimalhistorydependentpolicies/8eb3500a-b899-41fc-96ec-9d091a4c1870_model.json b/stochasticprincipalagentproblemscomputingandlearningoptimalhistorydependentpolicies/8eb3500a-b899-41fc-96ec-9d091a4c1870_model.json new file mode 100644 index 0000000000000000000000000000000000000000..75294473038fde668a116ab6cf6c6db9710db3dc --- /dev/null +++ b/stochasticprincipalagentproblemscomputingandlearningoptimalhistorydependentpolicies/8eb3500a-b899-41fc-96ec-9d091a4c1870_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6b251482842b358cb0d1d524e6f0bafb005afbd473fca13d1b8da12ef6a4b074 +size 229551 diff --git a/stochasticprincipalagentproblemscomputingandlearningoptimalhistorydependentpolicies/8eb3500a-b899-41fc-96ec-9d091a4c1870_origin.pdf b/stochasticprincipalagentproblemscomputingandlearningoptimalhistorydependentpolicies/8eb3500a-b899-41fc-96ec-9d091a4c1870_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4fd972258795b3a2e853f510f2098628f9235eba --- /dev/null +++ b/stochasticprincipalagentproblemscomputingandlearningoptimalhistorydependentpolicies/8eb3500a-b899-41fc-96ec-9d091a4c1870_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:facd3e399ece40b2162c361a999e905aeedd7f319264906e32f9184a1294d86d +size 537638 diff --git a/stochasticprincipalagentproblemscomputingandlearningoptimalhistorydependentpolicies/full.md b/stochasticprincipalagentproblemscomputingandlearningoptimalhistorydependentpolicies/full.md new file mode 100644 index 0000000000000000000000000000000000000000..44dd832006733fbed4a21da1e895ae6598a3559b --- /dev/null +++ b/stochasticprincipalagentproblemscomputingandlearningoptimalhistorydependentpolicies/full.md @@ -0,0 +1,788 @@ +# Stochastic Principal-Agent Problems: Computing and Learning Optimal History-Dependent Policies + +Jiarui Gan + +University of Oxford + +jiarui.gan@cs.ox.ac.uk + +Debmalya Mandal + +University of Warwick + +debmalya.mandal@warwick.ac.uk + +Rupak Majumdar + +MPI-SWS + +rupak@mpi-sws.org + +Goran Radanovic + +MPI-SWS + +gradanovic@mpi-sws.org + +# Abstract + +We study a stochastic principal-agent model. A principal and an agent interact in a stochastic environment, each privy to observations about the state not available to the other. The principal has the power of commitment, both to elicit information from the agent and to signal her own information. The players communicate with each other and then select actions independently. Both players are far-sighted, aiming to maximize their total payoffs over the entire time horizon. We consider both the computation and learning of the principal's optimal policy. The key challenge lies in enabling history-dependent policies, which are essential for achieving optimality in this model but difficult to cope with because of the exponential growth of possible histories as the size of the model increases; explicit representation of history-dependent policies is infeasible as a result. To address this challenge, we develop algorithmic techniques based on the concept of inducible value set. The techniques yield an efficient algorithm that computes an $\epsilon$ -approximate optimal policy in time polynomial in $1 / \epsilon$ . We also present an efficient learning algorithm for an episodic reinforcement learning setting with unknown transition probabilities. The algorithm achieves sublinear regret $\widetilde{\mathcal{O}}(T^{2/3})$ for both players over $T$ episodes. + +# 1 Introduction + +Many problems in economic theory involve sequential reasoning between multiple parties with asymmetric access to information [37, 20, 4, 27]. For example, in contract theory, one party (the principal) delegates authority and decision-making power to another (the agent), and the goal is to design mechanisms to ensure that the agent's actions align with the principal's utilities. This broad class of principal-agent problems lead to many research questions about information design and optimal strategic behaviors, with broad-ranging applications from governance and public administration to e-commerce and financial services. In particular, algorithmic techniques for optimal decision-making and learning are crucial for obtaining effective solutions to real-world problems in this domain. + +In this paper, we consider a general framework for stochastic principal-agent problems. We study the algorithmic problems related to the computation and learning of optimal solutions under this framework. In this framework, the interaction between the principal and the agent takes place in a stochastic environment over multiple time steps. In each step, both players are privy to information not available to the other and make partial observations about the environment. The players can communicate their private information to influence each other and, based on this communication, play actions that jointly influence the state of the environment. Each player has their own payoff, and we study the general-sum case with far-sighted players: each player aims to maximize the (expected) + +sum of their rewards over the entire horizon of the game. Technically, the setting is a stochastic game with partial information on both sides [1, 31]. + +In line with the principal-agent framework, we assume that the principal has the power of commitment, both to elicit information from the agent and to signal the agent about her own private information. A commitment is a binding agreement for the principal to act according to the committed policy. The agent acts optimally in response to the commitment, deciding what information to share and what actions to perform at their own discretion. At a high level, we have a Stackelberg game and we aim to find a Stackelberg equilibrium [40]. The possibility of information exchange further results in a model that incorporates both sequential Bayesian persuasion (or information design [22]) [16, 41] and sequential mechanism design [45]. We adopt stochastic games (or Markov games) [39] as our modeling basis, whose transition models are graph-structured. Hence, our model is more expressive than tree-structured principal-agent problems such as those based on extensive-form games (EFGs), in the sense that exponentially many possible history trajectories can be unrolled from our graph-based model as the length of the time horizon increases, whereas in tree-structured ones, the number is bounded by the size of the tree. + +Indeed, history-dependence is a key consideration in our policy design. Unlike in single-player Markov decision processes (MDPs) or zero-sum stochastic games, stationary (or history-independent) policies are no longer without loss of generality in our setting. For example, strategies like tit-for-tat—which are essential for achieving optimal outcomes in certain scenarios (such as the repeated prisoner's dilemma)—depend on remembering past actions and thus cannot be implemented by stationary policies. Enabling history-dependent policies in our graph-based model, however, begets a major challenge caused by the exponential growth of possible histories. Explicitly representing a history-dependent policy requires specifying a strategy for every possible history, which quickly becomes infeasible as the number of histories increases exponentially with the length of the time horizon. To further optimize such a policy is even more challenging as the output of this computational task would be a function over an exponentially large domain. In contrast, the size of a stationary policy is linear in the number of states. + +# 1.1 Our Contributions + +Surprisingly though, as we discover in this work, history-dependent policies can be represented as value vectors and can be efficiently unrolled provided polytopes consisting of all inducible value vectors. Building on this insight, we develop algorithmic techniques that work by constructing the inducible value sets. These techniques make it possible to implement and optimize history-dependent strategies in dynamic principal-agent settings. More surprisingly, they are computationally efficient, so not only do they significantly expand the policy space—from stationary policies to the far more expressive class of history-dependent ones—but they also address a key computational barrier posed by stationary policies, to optimize which, even approximately, is known to be NP-hard [15]. + +In summary, we make the following contributions in this paper. + +- We introduce a generalized framework for principal-agent problems in stochastic environments, which incorporates the stochastic extensions of various subcases of principal-agent problems such as information design [22] and automated mechanism design [38]. +- We develop algorithmic techniques that enable the implementation and optimization of history-dependent strategies in the stochastic principal-agency framework. The techniques are based on a compact representation of fully history-dependent polices and a novel approach that works by constructing the inducible value sets. Together, they yield an efficient near-optimal algorithm that computes an $\epsilon$ -approximate optimal policy in time polynomial in $1 / \epsilon$ , while the policy it produces ensures exact incentive compatibility (IC). +- We also consider an episodic RL setting where the transition dynamics are unknown and must be learned through interaction. We present a learning algorithm that guarantees sublinear regret $\widetilde{\mathcal{O}} (\mathrm{poly}(M,H)\cdot T^{2 / 3})$ for both players, where $M$ is the model size, $H$ is the episode length, and $T$ is the number of episodes. The algorithm builds on recent advances in reward-free exploration from the RL literature and it leverages efficient computation of approximately IC policies, using a variant of our algorithm for the full-information setting. + +# 1.2 Related Work + +The principal-agent problem is a well-known concept in economics studies [see, e.g., 37, 33, 32, 29]. Models featuring sequential interactions have also been proposed and studied [34, 12]. Our work follows the same modeling approach as these early works and extends the one-shot versions of the respective models—such as information design [22], automated mechanism design [38], as well as mixtures of the two [33, 7, 14]—into the sequential setting. There has been a growing interest in the algorithmic aspects of these sequential models, focusing on their computation and learning (e.g., information design [8, 15, 16, 41, 2], automated mechanism design [45, 6], other types of sequential Stackelberg games [25, 26, 5, 18, 10], and even more recently, contract design [42, 19]). + +More specifically, Gan et al. [15] first introduced an infinite-horizon information design model based on an MDP. They showed that optimal stationary strategies are inapproximable, unless the receiver is myopic. This work left open the tractability of optimal history-dependent strategies, especially in finite-horizon models, which we consider in this paper. Wu et al. [41] later studied the reinforcement learning problem against a myopic agent in the same sequential information design model. Bernasconi et al. [2] also studied the same problem in a model based on an EFG; efficient computation and learning algorithms were presented. Similar EFG-based models have also been explored in the recent literature [43, 44]. EFGs are tree-structured and hence easier than MDP-based models in the sense that all history sequences are modeled explicitly in the model. The number of possible histories is bounded by the size of the problem as a result, where as this can be exponential in an MDP. Hence, efficient algorithms for EFG-based models do not directly translate to efficient algorithms for our MDP-based model. In the domain of automated mechanism design, Zhang and Conitzer [45] studied a finite-horizon model that is a POMDP for the principal and MDP for the agent. They presented a linear program for computing optimal mechanisms, whose size is exponential in the problem size. + +As mentioned earlier, our algorithm leverages compact representation of history-dependent polices and construction of inducible value sets. Similar techniques have been proposed in earlier works by Dermed and Isbell [11] and MacDermed et al. [28] to compute optimal correlated equilibria of stochastic games. Our algorithm extends these techniques into the principal-agent setting, with adaptations that ensure exact incentive compatibility (IC) in the full-information setting. Concurrently with our work [17], Bernasconi et al. [3] also used a similar approximation approach to solve an information design problem (as a special case of our model). Compared to their results, our algorithm guarantees exact IC, with a more straightforward approach. Moreover, we also study the learning problem where the transition model needs to be learned, in addition to the computation problem where this is given. While all the above works (including ours) only guarantee near-optimality, exact solutions are possible in some special cases. In a recent work of Zhang et al. [46], they presented a sophisticated exact algorithm for computing optimal correlated equilibria in two-player turn-based stochastic games. In infinite-horizon games, however, such exact solutions appear computationally infeasible because of irrational numbers involved in optimal policies; we refer the reader to a more recent work by Gan and Majumdar [13]. + +# 2 Preliminaries + +A principal $(\mathsf{P})$ and an agent (A) interact in a finite-horizon POMDP $\mathcal{M} = \langle S, A, \Omega, p, \mathbf{r} \rangle$ , where: $S$ is a finite state space; $A = A^{\mathsf{P}} \times A^{\mathsf{A}}$ is a finite joint action space; $\Omega = \Omega^{\mathsf{P}} \times \Omega^{\mathsf{A}}$ is a finite joint observation space; $p = (p_h)_{h=0}^{H-1}$ and $\mathbf{r} = (\mathbf{r}_h)_{h=1}^H$ are two tuples, each consisting of an element for every time step $h$ . Specifically, $p_0 \in \Delta(S \times \Omega)$ is a distribution of the initial state-observation pairs, and each $p_h$ , $h \geq 1$ , is a transition function $p_h : S \times A \to \Delta(S \times \Omega)$ . Each $\mathbf{r}_h = (r_h^{\mathsf{P}}, r_h^{\mathsf{A}})$ is a pair of reward functions $r_h^{\mathsf{P}} : S \times A \to \mathbb{R}$ and $r_h^{\mathsf{A}} : S \times A \to \mathbb{R}$ , for the principal and the agent, respectively. W.l.o.g., we assume that all rewards are in [0, 1], and all rewards in the last time step $H$ are 0. + +The interaction proceeds as follows. At the beginning, an initial state-observation pair $(s_1,\omega_1)\sim p_0$ is drawn. Then, each time step $h = 1,\dots ,H$ involves the following stages. + +1. Observation: The principal and the agent observe, privately, $\omega_h^{\mathsf{P}}$ and $\omega_h^{\mathsf{A}}$ , respectively (but not $s_h$ ). +2. Communication: The principal elicits the agent's observation. The agent reports $\widetilde{\omega}_h^{\mathsf{A}}\in \Omega^{\mathsf{A}}$ (possibly different from $\omega_h^{\mathsf{A}}$ ). Then, based on $\omega_h^{\mathsf{P}}$ and $\widetilde{\omega}^{\mathsf{A}}$ the principal recommends an action $a_h^{\mathsf{A}}$ for the agent to play. The recommendation is sent to the agent as a coordination signal. + +3. Action: Based on the information exchange above, the principal and the agent, simultaneously, each perform an action, say $a_h^{\mathrm{P}}$ and $\tilde{a}_h^{\mathrm{A}}$ , respectively. (The action $\tilde{a}_h^{\mathrm{A}}$ the agent actually performs may be different from the recommended one $a_h^{\mathrm{A}}$ .) +4. Rewards and next state: Rewards $r_h^{\mathrm{P}}(s_h, a_h^{\mathrm{P}}, \tilde{a}_h^{\mathrm{A}})$ and $r_h^{\mathrm{A}}(s_h, a_h^{\mathrm{P}}, \tilde{a}_h^{\mathrm{A}})$ are generated for the principal and agent, respectively. The next state is drawn: $s_{h+1} \sim p_h(\cdot | s_h, a_h^{\mathrm{P}}, \tilde{a}_h^{\mathrm{A}})$ . + +Following the general paradigm of principal-agent problems, we consider the principal's commitment to a coordination policy. The agent best-responds to the principal's commitment. Both players are far-sighted and aim to maximize the sum of their rewards over the $H$ time steps.1 We take the principal's perspective and the goal, as we will shortly formalize, is to compute the principal's optimal commitment. At a high level, this is a Stackelberg game between the principal and the agent and we aim to compute a Stackelberg equilibrium. + +# 2.1 Hindsight Observability + +We assume hindsight observability, whereby both players observe the full interaction history (including $s_h$ , $\omega$ , and a) at the end of each time step. This condition is essential for circumventing intrinsic computational complexity barriers: without it, our model would directly subsume partially observable MDPs (POMDPs), which is famously known to be PSPACE-hard [36], in which case any efficient algorithms would be hopeless unless $\mathrm{P} = \mathrm{PSAPCE}$ . (See a further discussion in Appendix C.) Similar assumptions have been adopted in the POMDP literature [24]. Importantly, our model remains highly expressive under hindsight observability, capturing a broad range of relevant subcases, including: scenarios where the state is immediately observable, e.g., repeated games, stochastic games with full state observability [10], as well as scenarios where observations can be interpreted as external parameters generated based on an internal Markovian state observable to both players (e.g., [15, 41]). + +Indeed, in many real-world scenarios, it is natural for players to observe each other's actions after they are taken (consider, e.g., repeated rock-paper-scissors). Beyond actions, players may also receive regular updates about each other's private observations. This occurs in particular when each player observes a different part of the state and the full state is revealed at the end of the time step—at which point all private observations effectively become public. For example, traders may each observe local or regional market information that is later aggregated and released publicly; in energy markets, regional grid operators obtain first-hand information about supply and demand within their own regions and later receive system-wide reports covering the entire market; and in an R&D consortium, each firm privately evaluates prototypes or test results and acts on that information during the reporting cycle, after which all results are disclosed to members under agreed rules. + +# 2.2 History-dependent Policy + +We consider history-dependent policies, which are more general than stationary policies and hence typically yield higher payoffs. For example, to punish the agent for performing a certain action requires a history-dependent policy that remember the agent's action in the previous time step. History-dependent policies are also a natural choice for finite-horizon models, where the memory required to track the history is bounded by the horizon length. + +A history up to time step $h$ is a sequence $\sigma = (s_{\ell}, \omega_{\ell}, \widetilde{\omega}_{\ell}^{\mathsf{A}}, \mathbf{a}_{\ell}, \tilde{a}_{\ell}^{\mathsf{A}})_{\ell=1}^{h}$ , containing elements in the four stages of each step described above (and we write $\omega_{\ell} = (\omega_{\ell}^{\mathsf{P}}, \omega_{\ell}^{\mathsf{A}})$ and $\mathbf{a}_{\ell} = (a_{\ell}^{\mathsf{P}}, a_{\ell}^{\mathsf{A}})$ ). We let $\Sigma_{h}$ denote the set of all sequences till time step $h$ , and let $\Sigma = \bigcup_{h=0}^{H} \Sigma_{h}$ , where $\Sigma_{0} = \{\varnothing\}$ contain only the empty sequence. Moreover, we denote by $\bar{\Sigma} := S \times \Omega \times \Omega^{\mathsf{A}} \times A \times A^{\mathsf{A}}$ the set of all possible interactions within one time step. We can now write the transition function as $p_{h}(\cdot | \sigma) = p_{h}(\cdot | s_{h}, \mathbf{a}_{h})$ for any given sequence $\sigma \in \Sigma_{h}$ (specially, $p_{0}(\cdot | \emptyset) = p_{0}(\cdot)$ ). + +Principal's Policy A history-dependent policy takes the form $\pi : \Sigma \times \Omega \to \Delta(A)$ , whereby upon seeing $\sigma$ in the previous steps, observing $\omega^{\mathsf{P}}$ , and receiving the agent's report $\widetilde{\omega}^{\mathsf{A}}$ in the current step, the principal draws a joint action $\mathbf{a} = (a^{\mathsf{P}}, a^{\mathsf{A}}) \sim \pi(\sigma; \omega^{\mathsf{P}}, \widetilde{\omega}^{\mathsf{A}})$ , sends $a^{\mathsf{A}}$ to the agent as an action recommendation, and performs $a^{\mathsf{P}}$ herself. We denote by $\pi(\mathbf{a} | \sigma; \omega^{\mathsf{P}}, \widetilde{\omega}^{\mathsf{A}})$ the probability of each $\mathbf{a}$ in $\pi(\sigma; \omega^{\mathsf{P}}, \widetilde{\omega}^{\mathsf{A}})$ . + +Agent's Response The principal's commitment results in a meta-POMDP for the agent. The agent reacts by playing optimally in this meta-POMDP. When the principal's policy is IC, this simply means responding truthfully. Formally, the agent's strategy can be described by a deviation plan $\rho : (\sigma, \omega^{\mathsf{A}}) \mapsto (\widetilde{\omega}^{\mathsf{A}}, f : A^{\mathsf{A}} \to A^{\mathsf{A}})$ , such that given any history $\sigma$ and observation $\omega^{\mathsf{A}}$ in the current step, the agent reports $\widetilde{\omega}^{\mathsf{A}}$ and then plays $\tilde{a}^{\mathsf{A}} = f(a^{\mathsf{A}})$ if subsequently the principal recommends playing $a^{\mathsf{A}}$ . For simplicity, we write $\widetilde{\omega}^{\mathsf{A}} = \rho(\sigma; \omega^{\mathsf{A}})$ and $\tilde{a}^{\mathsf{A}} = \rho(\sigma; \omega^{\mathsf{A}}, a^{\mathsf{A}})$ . We denote by $\perp$ the special deviation plan where no deviation is made, i.e., $\perp (\sigma; \omega^{\mathsf{A}}) \equiv \omega^{\mathsf{A}}$ and $\perp (\sigma; \omega^{\mathsf{A}}, a^{\mathsf{A}}) \equiv a^{\mathsf{A}}$ . + +The agent's value (i.e., total reward) induced by a policy $\pi$ and a deviation strategy $\rho$ can be defined recursively via the value function as follows. For every $h = 1, \dots, H - 1$ and $\sigma \in \Sigma_{h - 1}$ : + +$$ +V _ {h} ^ {\mathsf {A}, \pi , \rho} (\sigma) := \mathbb {E} \left[ r _ {h} ^ {\mathsf {A}} (s, a ^ {\mathsf {P}}, \tilde {a} ^ {\mathsf {A}}) + V _ {h + 1} ^ {\mathsf {A}, \pi , \rho} (\sigma ; s, \omega , \widetilde {\omega} ^ {\mathsf {A}}, \mathbf {a}, \tilde {a} ^ {\mathsf {A}}) \right], \tag {1} +$$ + +where the expectation is taken over $(s,\omega)\sim p_{h - 1}(\cdot |\sigma)$ and $\mathbf{a}\sim \pi (\cdot |\sigma ,\omega^{\mathrm{P}},\widetilde{\omega}^{\mathrm{A}})$ ; moreover, $\widetilde{\omega}^{\mathrm{A}} = \rho (\sigma^{\mathrm{A}},\omega^{\mathrm{A}})$ and $\tilde{a}^{\mathrm{A}} = \rho (\sigma^{\mathrm{A}},\omega^{\mathrm{A}},a^{\mathrm{A}})$ , and by assumption $V_{H}^{\mathrm{A}}(\sigma)\equiv 0$ for the last time step. The principal's value is defined the same way by changing the labels. + +Our goal is to find a policy $\pi$ that maximizes the principal's value under the agent's best response: + +$$ +\max _ {\pi , \rho} V _ {1} ^ {\mathsf {P}, \pi , \rho} (\varnothing) \tag {2} +$$ + +$$ +\text {s u b j e c t} \rho \in \arg \max _ {\rho^ {\prime}} V _ {1} ^ {\mathrm {A}, \pi , \rho^ {\prime}} (\varnothing) \tag {2-1} +$$ + +In other words, we look for $\pi$ and $\rho$ that form a Stackelberg equilibrium. We say that $\pi$ is $\epsilon$ -optimal if $V_1^{\mathsf{P},\pi,\rho}(\varnothing) \geq V^* - \epsilon$ for some $\rho$ satisfying (2-1), where $V^*$ denotes the optimal value of (2). + +As we will demonstrate, under hindsight observability, it is without loss of optimality to consider policies that are IC (incentive compatible), which incentivize $\perp$ as an optimal response of the agent. + +Definition 1 (IC policy). A policy $\pi$ is IC if $V_{1}^{\mathsf{A},\pi,\perp}(\varnothing) \geq V_{1}^{\mathsf{A},\pi,\rho}(\varnothing)$ for every possible deviation plan $\rho$ of the agent. + +# 3 Computing an Optimal Policy + +We compute a near-optimal policy by constructing the inducible value sets. The approach is similar to value-based approaches to solving MDPs, which operate by reasoning about the values of the states. However, it involves the following crucial differences: + +1. Since we are in a two-player setting, both player's values need to be incorporated. We use a two-dimensional value vector instead of a scalar value. +2. Since we consider history-dependent policies, the game may proceed differently from the same state based on different histories. In this case, a single value (vector) is no longer sufficient for characterizing a state, unlike the case with stationary policies. We use the set consisting of all values can be induced by some valid policy to characterize a state. We refer to these sets are inducible value sets. + +Moreover, since the actual state of the underlying MDP is not observable at the beginning of each time step, we will instead characterize the inducible value sets of state-action pairs $o \in O \coloneqq S \times A$ . Under hindsight observability, these pairs function as the states in a Markovian process, which the players observe before they each decide an action to play. + +Definition 2 (Inducible value set). The inducible value set $\mathcal{V}_h(o) \subseteq \mathbb{R}^2$ of a state-action pair $o \in O \coloneqq S \times A$ at time step $h$ consists of all vectors $\mathbf{v} = (v^{\mathsf{P}}, v^{\mathsf{A}})$ such that $v^{\mathsf{P}} = V_h^{\mathsf{P},\pi,\rho}(\sigma)$ and $v^{\mathsf{A}} = V_h^{\mathsf{A},\pi,\rho}(\sigma)$ for some policy $\pi$ , deviation plan $\rho \in \arg \max_{\rho'} V_h^{\mathsf{A},\pi,\rho'}(\sigma)$ , and sequence $\sigma \in \Sigma_{h-1}$ ending with $o$ . + +It is straightforward that once we obtain $\mathcal{V}_1(\varnothing)$ , then the principal's optimal value in (2) can be computed by solving $\max_{(v^{\mathrm{P}}, v^{\mathrm{A}}) \in \mathcal{V}_1(\varnothing)} v^{\mathrm{P}}$ . We present a dynamic programming approach to constructing the inducible sets next. We will later also show that, once we obtain all the inducible sets, every inducible value vector can be efficiently unrolled into a history-dependent policy that induces it, and the policy can be executed efficiently on the fly. Hence, each value vector serves as a compact representation of a fully specified history-dependent policy. + +# 3.1 Computing Inducible Value Sets + +Instead of computing the exact value sets, we use convex polytopes to approximate them, so as to avoid possible exponential growth of the representational complexity. Let $\widehat{\mathcal{V}}_h(o)$ denote the approximation of $\mathcal{V}_h(o)$ we aim to obtain. Recall that in the last time step all rewards are 0, so trivially we use $\widehat{\mathcal{V}}_H(o) = \mathcal{V}_H(s) = \{(0,0)\}$ for all $o\in O$ as the base case. + +Dynamic Programming Now suppose that we have obtained the polytopes $\widehat{\mathcal{V}}_{h + 1}(o')$ for all $o' \in O$ . We move to time step $h$ and construct each $\widehat{\mathcal{V}}_h(o)$ based on the $\widehat{\mathcal{V}}_{h + 1}(o')$ 's. Central to the approach is the following characterization, which describes an IC condition at time step $h$ : for every $\mathbf{v} \in \mathbb{R}^2$ , it holds that $\mathbf{v} \in \mathcal{V}_h(o)$ if and only if there exist a one-step policy $\bar{\pi}: \Omega \to \Delta(A)$ and a set of onward value vectors $\{\mathbf{v}'(\bar{\sigma}) \in \mathbb{R}^2: \bar{\sigma} \in \bar{\Sigma}\}$ that satisfy the following constraints. + +1. A value function constraint based on (1), which expresses $\mathbf{v}$ via the immediate rewards and onward value vectors $\mathbf{v}'$ to be induced next, assuming truthful response of the agent: + +$$ +\mathbf {v} = \sum_ {s, \boldsymbol {\omega}, \mathbf {a}} p _ {h - 1} (s, \boldsymbol {\omega} \mid o) \cdot \bar {\pi} (\mathbf {a} \mid \boldsymbol {\omega}) \cdot \left(\mathbf {r} _ {h} (s, \mathbf {a}) + \mathbf {v} ^ {\prime} (s, \boldsymbol {\omega}, \boldsymbol {\omega} ^ {\mathrm {A}}, \mathbf {a}, a ^ {\mathrm {A}})\right), \tag {3} +$$ + +The onward value vectors represent the subsequent part of the principal's commitment, which is contingent on the interaction $(s,\omega ,\widetilde{\omega}^{\mathsf{A}},\mathbf{a},\tilde{a}^{\mathsf{A}})$ in step $h$ . Under the truthful response of the agent, we have $\widetilde{\omega}^{\mathsf{A}} = \omega^{\mathsf{A}}$ and $\tilde{a}^{\mathsf{A}} = a^{\mathsf{A}}$ in (3). + +2. IC constraints, which ensure that the agent's truthful behavior assumed in (3) is indeed incentivized, where we denote by $p_{h-1}(s, \omega^{\mathsf{P}} | o, \omega^{\mathsf{A}}) \propto p_{h-1}(s, \omega | o)$ the conditional probability defined by $p_{h-1}$ . For all $\omega^{\mathsf{A}}, \widetilde{\omega}^{\mathsf{A}} \in \Omega^{\mathsf{A}}$ , + +$$ +\sum_ {s, \omega^ {\mathsf {P}}, \mathbf {a}} p _ {h - 1} (s, \omega^ {\mathsf {P}} | o, \omega^ {\mathsf {A}}) \cdot \bar {\pi} (\mathbf {a} | \omega) \cdot \left(r _ {h} ^ {\mathsf {A}} (s, \mathbf {a}) + v ^ {\mathsf {\Lambda}} ^ {\mathsf {A}} (s, \omega , \omega^ {\mathsf {A}}, \mathbf {a}, a ^ {\mathsf {A}})\right) \geq +$$ + +$$ +\sum_ {a ^ {\mathsf {A}}} \max _ {\tilde {a} ^ {\mathsf {A}} \in A ^ {\mathsf {A}}} \sum_ {s, \omega^ {\mathsf {P}}, a ^ {\mathsf {P}}} p _ {h - 1} (s, \omega^ {\mathsf {P}} \mid o, \omega^ {\mathsf {A}}) \cdot \bar {\pi} (\mathbf {a} \mid \omega^ {\mathsf {P}}, \widetilde {\omega} ^ {\mathsf {A}}) \cdot \left(r _ {h} ^ {\mathsf {A}} (s, a ^ {\mathsf {P}}, \tilde {a} ^ {\mathsf {A}}) + v ^ {\prime \mathsf {A}} (s, \omega , \widetilde {\omega} ^ {\mathsf {A}}, \mathbf {a}, \tilde {a} ^ {\mathsf {A}})\right). \tag {4} +$$ + +Namely, the constraint says, upon observing $\omega^{\mathsf{A}}$ , the agent's expected payoff under their truthful response is at least as much as what they could have obtained, had they: 1) reported a different observation $\widetilde{\omega}^{\mathsf{A}}$ , 2) performed a best action $\tilde{a}^{\mathsf{A}}$ in response to every possible recommendation $a^{\mathsf{A}}$ of the principal, and 3) responded optimally in the subsequent time steps (whereby the onward values are given by $\mathbf{v}'$ ). + +3. Onward value constraints, which ensures that the onward values given by $\mathbf{v}'$ are also inducible: + +$$ +\mathbf {v} ^ {\prime} (s, \omega , \widetilde {\omega} ^ {\mathrm {A}}, \mathbf {a}, \tilde {a} ^ {\mathrm {A}}) \in \mathcal {V} _ {h + 1} (s, a ^ {\mathrm {P}}, \tilde {a} ^ {\mathrm {A}}) \quad \text {f o r a l l} (s, \omega , \widetilde {\omega} ^ {\mathrm {A}}, \mathbf {a}, \tilde {a} ^ {\mathrm {A}}) \in \bar {\Sigma}. \tag {5} +$$ + +The following lemma indicates the correctness of the above characterization. + +Lemma 3. $\mathbf{v} \in \mathcal{V}_h(o)$ if and only if (3) to (5) hold for some $\bar{\pi}: \Omega \to \Delta(A)$ and $\mathbf{v}': \bar{\Sigma} \to \mathbb{R}^2$ . + +Therefore, to decide whether $\mathbf{v} \in \mathcal{V}_h(o)$ amounts to deciding whether the above constraints are satisfied by some $\bar{\pi}$ and $\mathbf{v}'$ (highlighted in blue in the constraints). Note that since the inductive hypothesis assumes an approximation $\widehat{\mathcal{V}}_{h+1}(o')$ instead of the exact set $\mathcal{V}_{h+1}(o')$ , we will in fact impose the following approximate onward value constraint, instead of the exact version in (5): + +$$ +\mathbf {v} ^ {\prime} (s, \omega , \widetilde {\omega} ^ {\mathrm {A}}, \mathbf {a}, \tilde {a} ^ {\mathrm {A}}) \in \widehat {\mathcal {V}} _ {h + 1} (s, a ^ {\mathrm {P}}, \tilde {a} ^ {\mathrm {A}}) \quad \text {f o r a l l} (s, \omega , \widetilde {\omega} ^ {\mathrm {A}}, \mathbf {a}, \tilde {a} ^ {\mathrm {A}}) \in \bar {\Sigma}. \tag {6} +$$ + +Linearizing (3) and (4) The constraint satisfiability problem defined above is non-linear due to the quadratic terms and the maximization operator in (3) and (4). Nevertheless, it can be linearized as long as every polytope $\widehat{\mathcal{V}}_{h+1}(o')$ , $o' \in O$ , is given by the half-space representation, i.e., by linear constraints in the form $\mathbf{H} \cdot \mathbf{x} \leq \mathbf{b}$ for some matrix $\mathbf{H}$ and vector $\mathbf{b}$ . Due to the space limit, we leave the details in Appendix B. + +![](images/4f2838ea251a5bd3df1f49c29619f5a64536fc0fd329e7690b0afcc7718fe77d.jpg) +Figure 1: Constructing $\widehat{\mathcal{V}}_h(o)$ as a $\delta$ -approximation of $\overline{\mathcal{V}}_h(o)$ . The black points constitute $\mathcal{W}$ . +Figure 2: Computing approximate value polytopes via dynamic programming. + +For $h = H - 1, \dots, 1$ , do the following for all $o \in O$ : + +1. Plug in (6) the half-space representation of $\widehat{\mathcal{V}}_{h + 1}(o')$ , $o' \in O$ . Then linearize (3) and (4). +2. Discretize the space $[0, H]^2$ into a finite point set (see Lemma 4 for more detail). Check the inducibility of each point $\mathbf{v}$ in this set by solving the linear constraint satisfiability problem defined by (the linearized version of) (3), (4) and (6). +3. Compute $\widehat{\mathcal{V}}_h(o)$ as the convex hull of the inducible points obtained above, in half-space representation. + +Constructing $\widehat{\mathcal{V}}_h(o)$ As a result, we obtain a polytope $\mathcal{P}$ defined by a set of linear constraints equivalent to (3), (4) and (6). The projection of $\mathcal{P}$ onto the dimensions of $\mathbf{v}$ is (approximately) $\mathcal{V}_h(o)$ . To ensure that the projection can be plugged back into (6) in the next induction step, we need the half-space representation of the projection, too. In particular, we want to eliminate the additional variables in the representation so that only $\mathbf{v}$ remains. (Otherwise, the number of variables may grow exponentially as the induction step increases.) This can be done approximately in polynomial time given that $\mathbf{v}$ is two-dimensional. Roughly speaking, we discretize the box $[0, H]^2$ into a finite set of points (recall that rewards in each time step are bounded in $[0, 1]$ , so $[0, H]^2$ contains $\mathcal{V}_h(o)$ ), check the inducibility of each point, and compute the convex hull of the inducible points in half-space representation. The specific way we discretize the space (see Fig. 1) ensures that IC is satisfied exactly (which can otherwise not be achieved by using standard grid-based discretization). The details can be found in the proof of Lemma 4. + +Repeating the induction procedure till $h = 1$ , we obtain $\widehat{\mathcal{V}}_1(\varnothing)$ as well as a near-optimal value of the principal by solving the LP $\max_{\mathbf{v} \in \widehat{\mathcal{V}}_1(\varnothing)} v^{\mathrm{P}}$ . Fig. 2 summarizes this dynamic programming approach. + +Lemma 4. For any constant $\epsilon >0$ , it can be computed in time poly $(|S|\cdot |A|\cdot |\Omega |,H,1 / \epsilon)$ the half-space representations of a set of polytopes $\widehat{\mathcal{V}}_h(o)\subseteq \mathcal{V}_h(o)$ , $o\in O\cup \{\varnothing \}$ and $h = 1,\ldots ,H$ , such that (3), (4) and (6) are satisfiable for every $\mathbf{v}\in \widehat{\mathcal{V}}_h(o)$ and $\max_{\mathbf{v}\in \widehat{\mathcal{V}}_1(\varnothing)}v^{\mathrm{P}}\geq \max_{\mathbf{v}\in \mathcal{V}_1(\varnothing)}v^{\mathrm{P}} - \epsilon$ + +Proof sketch. We construct the approximate sets $\widehat{\mathcal{V}}_h(o)$ by backward induction. Assume for all $o$ that $\widehat{\mathcal{V}}_{h + 1}(o)$ is defined by polynomially many constraints and is an $\varepsilon$ -approximation of $\mathcal{V}_{h + 1}(o)$ . Let $\overline{\mathcal{V}}_h(o)$ be the feasible set defined by (3), (4) and (6) using $\widehat{\mathcal{V}}_{h + 1}(o')$ . We discretize the principal's value into slices of width $\delta$ (see Fig. 1). We include the following points into a collection $\mathcal{W}$ use the convex hull of $\mathcal{W}$ as $\widehat{\mathcal{V}}_h(o)$ : 1) for every line $w$ separating the slices, the two points $\check{\mathbf{v}}_w$ and $\hat{\mathbf{v}}_w$ at the intersection of $w$ and the boundary of $\widehat{\mathcal{V}}_{h + 1}(o)$ ; and 2) the extreme points $\check{\mathbf{v}}_*$ and $\hat{\mathbf{v}}_*$ of $\overline{\mathcal{V}}_h(o)$ minimizing and maximizing the agent's value, respectively. Each of these points can be obtained + +Input: a sequence $(\sigma; \omega^{\mathsf{P}}, \widetilde{\omega}^{\mathsf{A}})$ , where $\sigma = (s_{\ell}, \omega_{\ell}, \widetilde{\omega}_{\ell}^{\mathsf{A}}, \mathbf{a}_{\ell}, \tilde{a}_{\ell}^{\mathsf{A}})_{\ell=1}^{h-1}$ . + +1. Initialize: $\mathbf{v} \gets \arg \max_{\mathbf{v} \in \widehat{\mathcal{V}}_1(\emptyset)} v^{\mathrm{P}}$ and $o \gets \emptyset$ . +2. For $\ell = 1,\dots ,h - 1$ .. + +Fix $\mathbf{v}$ and $o$ , and solve (3), (4) and (6), where we use the polytopes $\widehat{\mathcal{V}}_h(o)$ described in Lemma 4. Let the solution be $\bar{\pi}$ and $\mathbf{v}'$ . +- Update: $\mathbf{v} \gets \mathbf{v}'(s_{\ell}, \boldsymbol{\omega}_{\ell}, \widetilde{\omega}_{\ell}^{\mathsf{A}}, \mathbf{a}_{\ell}, \tilde{a}_{\ell}^{\mathsf{A}})$ and $o \gets (s_{\ell}, a_{\ell}^{\mathsf{P}}, a_{\ell}^{\mathsf{A}})$ . + +3. Output $\pi (\cdot |\sigma ;\omega^{\mathrm{P}},\widetilde{\omega}^{\mathrm{A}}) = \bar{\pi} (\cdot |\omega^{\mathrm{P}},\widetilde{\omega}^{\mathrm{A}})$ . + +Figure 3: Computing a near-optimal policy based on approximations of the value polytopes. + +efficiently by solving a polynomial-size LP. The inclusion of the extreme points ensures in particular that we do not miss the agent's extreme values, which is critical for achieving exact IC. + +It can be verified that $\widehat{\mathcal{V}}_h(o) \subseteq \overline{\mathcal{V}}_h(o)$ and that any $\mathbf{v} \in \overline{\mathcal{V}}_h(o)$ can be approximated by some $\mathbf{x} \in \widehat{\mathcal{V}}_h(o)$ with $x^{\mathrm{A}} = v^{\mathrm{A}}$ and $x^{\mathrm{P}} \geq v^{\mathrm{P}} - \delta$ . So $\widehat{\mathcal{V}}_h(o)$ is a $\delta$ -approximation of $\overline{\mathcal{V}}_h(o)$ and $x^{\mathrm{A}} = v^{\mathrm{A}}$ ensures exact IC for the agent. Since $\overline{\mathcal{V}}_h(o)$ itself is an $\varepsilon$ -approximation of $\mathcal{V}_h(o)$ , we obtain an $(\varepsilon + \delta)$ -approximation overall. Choosing $\delta = \epsilon / H$ and noting the trivial base case $\widehat{\mathcal{V}}_H(o) = \{(0, 0)\}$ , the construction yields $\widehat{\mathcal{V}}_1(\varnothing)$ as an $\epsilon$ -approximation of $\mathcal{V}_1(\varnothing)$ , computable in time poly $(|S| \cdot |A| \cdot |\Omega|, H, 1/\epsilon)$ . + +# 3.2 Unrolling the Optimal Policy + +The above procedure yields the maximum inducible value of the principal but not yet an optimal policy that achieves this value. We next demonstrate how to compute an optimal policy based on $\widehat{\mathcal{V}}_1(\varnothing)$ . Rather than obtaining an explicit description of a history-dependent policy $\pi$ which would be exponentially large as the policy specifies a distribution for each possible sequence—we present an efficient procedure that computes the distribution $\pi(\cdot|\sigma; \omega^{\mathrm{P}}, \widetilde{\omega}^{\mathrm{A}})$ for any given sequence $(\sigma; \omega^{\mathrm{P}}, \widetilde{\omega}^{\mathrm{A}})$ . This means that, when playing the game, the principal can compute an optimal policy on the fly based on the realized history. + +We use a forward computation procedure presented in Fig. 3. Starting from time step 1, the procedure repeatedly computes a one-step policy $\bar{\pi}$ and a set of onward vectors, to induce the target value vector $\mathbf{v}$ . The onward vectors define the target values to be induced in the next time step, contingent on the interaction in the current, which is given by $\sigma$ . Hence, the target vector is updated to one of the onward vectors according $\sigma$ at the end of each iteration. In other words, in each time step, we expand the target vector into a set of onward vectors, and then select one of them as the next target vector according to the realized interaction given by $\sigma$ . This leads to the following main result. + +Theorem 5. There exists an $\epsilon$ -optimal IC policy $\pi$ such that, for any given sequence $(\sigma; \omega^{\mathsf{P}}, \widetilde{\omega}^{\mathsf{A}}) \in \Sigma \times \Omega$ , the distribution $\pi(\cdot | \sigma; \omega^{\mathsf{P}}, \widetilde{\omega}^{\mathsf{A}})$ can be computed in time poly $(|S| \cdot |A| \cdot |\Omega|, H, 1/\epsilon)$ . + +# 4 Learning to Commit + +We now turn to an episodic online learning setting where the transition model $p: S \times A \to \Delta(S \times \Omega)$ is not known to the players beforehand. Let there be $T$ episodes. At the beginning of each episode, the principal commits to a new policy based on the outcomes of the previous episodes. Each episode proceeds in $H$ time steps the same way as the model defined in Section 2. + +We present a learning algorithm that guarantees sublinear regrets for both players under hindsight observability. The algorithm is centralized and relies on the agent behaving truthfully. It does not guarantee exact IC during the course of learning but IC in the limit when the number of episodes approaches infinity. Indeed, since the model is unknown to both players, IC in the limit is a more relevant concept as the agent cannot decide how to optimally deviate from their truthful response, either. In this case, the sublinear regret the algorithm guarantees for the agent should in many + +scenarios be sufficient for incentivizing for the agent to participate and follow the centralized learning protocol. + +The players' regrets are defined as follows: + +$$ +\operatorname {R e g} ^ {\mathsf {P}} = \sum_ {t = 1} ^ {T} \left(V ^ {*} - V _ {1} ^ {\mathsf {P}, \pi_ {t}, \perp} (\varnothing)\right) \quad \text {a n d} \quad \operatorname {R e g} ^ {\mathsf {A}} = \sum_ {t = 1} ^ {T} \left(\max _ {\rho} V _ {1} ^ {\mathsf {A}, \pi_ {t}, \rho} (\varnothing) - V _ {1} ^ {\mathsf {A}, \pi_ {t}, \perp} (\varnothing)\right), +$$ + +where $V^{*}$ is the optimal value of (2) and $\pi_t$ denotes the policy the principal commits to in the $t$ -th episode. In words, the principal's regret $\mathrm{Reg}^{\mathsf{P}}$ is defined with respect to the optimal policy under the true model. The agent's regret $\mathrm{Reg}^{\mathsf{A}}$ is defined with respect to his optimal response to each $\pi_t$ , which is a dynamic regret as the benchmark changes across the episodes. + +# 4.1 Learning Algorithm + +Reward-free Exploration Our learning algorithm is based on reward-free exploration, which is an RL paradigm where learning happens before a reward function is provided [21]. It has been shown in a series of works that efficient learning is possible under this paradigm [21, 23, 30]. In particular, we will use the sample complexity bound in Lemma 6. At a high level, our algorithm proceeds by first conducting reward-free exploration to learn a sufficiently accurate estimate of the true model. Based on the estimate we then solve a relaxed version of the policy optimization problem (2) to obtain a policy. Using this policy in the remaining episodes guarantees sublinear regret for both players. + +Lemma 6 ([21, Lemma 3.6 restated]). Consider an (single-player) MDP $(S,A,p)$ (without any reward function specified) with horizon length $H$ . There exists an algorithm which learns a model $\widehat{p}$ after $\widetilde{\mathcal{O}}\left(\frac{H^5|S|^2\cdot|A|}{\delta^2}\right)$ episodes of exploration, such that with probability at least $1 - q$ , for any reward function $r$ and policy $\pi$ , it holds that $\left|V_1^{r,\pi}(s) - \widehat{V}_1^{r,\pi}(s)\right| \leq \delta / 2$ for all states $s$ , where $V_1^{r,\pi}$ and $\widehat{V}_1^{r,\pi}$ denote the value functions under reward function $r$ and models $p$ and $\widehat{p}$ , respectively. + +With the above result, we can learn a model $\widehat{p}$ for our purpose. In what follows, we let $\widehat{V}_h^{\mathrm{P},\pi,\rho}$ and $\widehat{V}_h^{\mathrm{A},\pi,\rho}$ denote the players' value functions in model $\widehat{p}$ (i.e., by replacing $p$ in (1) with $\widehat{p}$ ). Lemma 7 then translates Lemma 6 to our setting. Note that under hindsight observability the process facing the principal and the agent jointly during the learning process is effectively an MDP, where the effective state space is $O \times \Omega$ . An effective state, say $\theta = (s, \mathbf{a}, \omega)$ , consists of the state-action pair $(s, \mathbf{a})$ in the previous step and the observations $\omega$ in the current. When a joint action $\mathbf{a}'$ is performed, $\theta$ transitions to $\theta' = (s', \mathbf{a}', \omega')$ with probability $p_{h-1}(s', \omega'| s, \mathbf{a})$ . + +Lemma 7. A model $\widehat{p}$ can be learned after $\widetilde{\mathcal{O}}\left(H^5 |S|^2 |A|^3 |\Omega|^2 /\delta^2\right)$ episodes of exploration, such that $\left|V_1^{\mathsf{A},\pi ,\rho}(\varnothing) - \widehat{V}_1^{\mathsf{A},\pi ,\rho}(\varnothing)\right|\leq \delta /2$ and $\left|V_{1}^{\mathrm{P},\pi ,\rho}(\varnothing) - \widehat{V}_{1}^{\mathrm{P},\pi ,\rho}(\varnothing)\right|\leq \delta /2$ with probability at least $1 - q$ for any policy $\pi$ and deviation plan $\rho$ + +Therefore, the value functions change smoothly as the learned model $\widehat{p}$ approaches $p$ . However, this smoothness is insufficient for deriving a sublinear bound on the principal's regret because of the agent's incentive constraints in our problem. Roughly speaking, the set of IC policies does not change smoothly with $\widehat{p}$ , even though the value functions do. Hence, even an infinitesimal difference between $\widehat{p}$ and $p$ may lead to a jump between the IC policy sets under these two models and, in turn, a gap between the values of the optimal policies. + +Approximate IC Relaxation To deal with this issue, we relax the incentive constraints, allowing small violations to the constraints. Such violations are inevitable if we aim to achieve a near-optimal value under the true model $p$ but only know an estimate $\widehat{p}$ of the true model. On the positive side, given the sublinear regret guarantee for the agent, the violation diminishes with the number of episodes. We define $\delta$ -IC policies below. + +Definition 8 ( $\delta$ -IC policy). A policy $\pi$ is $\delta$ -IC (w.r.t. model $\widehat{p}$ ) if $\widehat{V}_1^{\mathsf{A},\pi,\perp}(\varnothing) \geq \widehat{V}_1^{\mathsf{A},\pi,\rho}(\varnothing) - \delta$ for every possible deviation plan $\rho$ of the agent. A $\delta$ -IC policy is said to be $\epsilon$ -optimal if $\widehat{V}_1^{\mathsf{P},\pi,\perp}(\varnothing) \geq V^* - \epsilon$ , where $V^*$ is the optimal value of (2) (under $p$ ). + +That is, in response to a $\delta$ -IC policy, the agent can improve his overall expected payoff by no more than $\delta$ if he deviates from the truthful response. We assume that the agent will not deviate for such a small benefit, and we evaluate the value of a $\delta$ -IC policy based on the agent's truthful response. This is how the $\epsilon$ -optimality is defined above, where we compare against the optimal value $V^{*}$ in (2), which is obtained under a more stringent setting without any relaxation of the agent's incentive. In other words, we relax the feasible space and compare the solution obtained in this relaxed space with the optimum over the smaller original feasible space. Such relaxations are common in the optimization literature, and they are crucial for resolving the non-smooth issue. + +Let $\widehat{\Pi}_{\delta}$ and $\Pi_{\delta}$ denote the set of $\delta$ -IC policies under $\widehat{p}$ and $p$ , respectively. The relaxation immediately results in $\widehat{\Pi}_{\delta} \supseteq \Pi_0$ for the model $\widehat{p}$ stated in Lemma 7. As a result, optimizing over $\widehat{\Pi}_{\delta}$ ensures that the optimal value yielded is as much (up to a small error) as the optimal value $V^*$ over $\Pi_0$ . Meanwhile, the value loss introduced by this relaxation for the agent is also small (bounded by $\delta$ ). + +With the above results, our learning algorithm proceeds as follows. + +1. Run reward-free exploration to obtain a model $\widehat{p}$ as stated in Lemma 7. +2. Compute a $\delta$ -optimal $\delta$ -IC policy in $\widehat{p}$ and use it in the remaining episodes. + +The near-optimal policy in Step 2 can be computed efficiently according to Lemma 9, via an approach similar to the one in Section 3.1. This gives an efficient algorithm with sublinear regrets for both players. We present Theorem 10. + +Lemma 9. There exists an $\epsilon$ -optimal $\delta$ -IC policy $\pi$ such that, for any given sequence $(\sigma; \omega^{\mathrm{P}}, \widetilde{\omega}^{\mathrm{A}}) \in \Sigma \times \Omega$ , the distribution $\pi(\cdot | \sigma; \omega^{\mathrm{P}}, \widetilde{\omega}^{\mathrm{A}})$ can be computed in time poly $(|S| \cdot |A| \cdot |\Omega|, H, 1/\epsilon, \log(1/\delta))$ . + +Theorem 10. There exists an algorithm that guarantees regret $\widetilde{\mathcal{O}} (\zeta^{1 / 3}T^{2 / 3})$ for both players with probability $1 - q$ , where $\zeta = H^5 |S|^2 |A|^3 |\Omega |^2$ . The computation involved in implementing the algorithm takes time poly $(|S|\cdot |A|\cdot |\Omega |,H,T)$ + +# 5 Conclusion + +We studied a stochastic principal-agent framework and presented efficient computation and learning algorithms. Our model can be further extended to the setting with $n$ agents. The algorithms we presented remain efficient for any constant $n$ if approximate IC solutions are considered. Computing optimal exact IC policies for $n$ agents remain an interesting open question, as our discretization method, which operates by slicing the space, does not generalize to $n$ agents. When $n$ is not a constant, representing games in normal-form requires space exponential in $n$ , so more succinct representations are typically considered. However, it is known that in succinctly represented games even to compute an optimal correlated equilibrium in one-shot games may be NP-hard [35].) Our results indicate how a policy designer might interact with agents optimally. In particular implementations, the designer's incentives may not be aligned with societal benefits. In these cases, a careful analysis of the incentives and their moral legitimacy must be considered. Besides this, since the paper is theory focused, we do not feel any other potential impacts must be specifically highlighted here. + +# Acknowledgement + +The work of Goran Radanovic was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 467367360. + +# References + +[1] Robert J. Aumann and Michael B. Maschler. Repeated Games with Incomplete Information. MIT Press, 1995. +[2] Martino Bernasconi, Matteo Castiglioni, Alberto Marchesi, Nicola Gatti, and Francesco Trovò. Sequential information design: learning to persuade in the dark. In Proceedings of the 36th International Conference on Neural Information Processing Systems (NeurIPS'22), 2022. +[3] Martino Bernasconi, Matteo Castiglioni, Alberto Marchesi, and Mirco Mutti. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: [NA] + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Omitted Proofs + +# A.1 Omitted Proofs in Section 3 + +For simplicity, we write $\vec{V}_h^{\pi,\rho}(\sigma) = \left(V_h^{\mathsf{P},\pi,\rho}(\sigma), V_h^{\mathsf{A},\pi,\rho}(\sigma)\right)$ in the following proofs. + +Lemma 3. $\mathbf{v}\in \mathcal{V}_h(o)$ if and only if (3) to (5) hold for some $\bar{\pi}:\Omega \to \Delta (A)$ and $\mathbf{v}':\bar{\Sigma}\rightarrow \mathbb{R}^2$ + +Proof. First, consider the "only if" direction of the statement. Suppose that $\mathbf{v} \in \mathcal{V}_h(o)$ . By definition, we have $\mathbf{v} = \vec{V}_h^{\pi,\rho}(o)$ for some $\pi, \rho \in \arg \max_{\rho'} \vec{V}_h^{\mathsf{A},\pi,\rho'}(\sigma)$ , and $\sigma \in \Sigma_{h-1}$ ending with $o$ . According to a standard revelation principle argument, we can assume w.l.o.g. that $\rho$ is IC in step $h$ . Hence, by (1), we have + +$$ +\mathbf {v} = \sum_ {s, \boldsymbol {\omega}, \mathbf {a}} p _ {h - 1} (s, \boldsymbol {\omega} \mid o) \cdot \pi (\mathbf {a} \mid \sigma ; \boldsymbol {\omega}) \cdot \left(\mathbf {r} _ {h} (s, \mathbf {a}) + \vec {V} _ {h + 1} ^ {\pi , \rho} (\sigma ; s, \boldsymbol {\omega}, \omega^ {\mathrm {A}}, \mathbf {a}, a ^ {\mathrm {A}})\right). \tag {7} +$$ + +Letting $\bar{\pi} (\mathbf{a}|\boldsymbol {\omega}) = \pi (\mathbf{a}|\sigma ;\boldsymbol {\omega})$ for every $\omega \in \Omega$ , and $\mathbf{v}'(\bar{\sigma}) = \vec{V}_{h + 1}^{\pi ,\rho}(\sigma ;\bar{\sigma})$ for every $\bar{\sigma}\in \bar{\Sigma}$ , we establish (3). (Recall that $\bar{\Sigma} := S\times \Omega \times \Omega^{\mathsf{A}}\times A\times A^{\mathsf{A}}$ denotes the set of all possible interactions within one time step.) + +Since $\rho$ is IC in step $h$ , (4) also follows immediately: the agent cannot benefit from any possible deviation. By definition, we have $\mathbf{v}'(\bar{\sigma}) = \vec{V}_{h+1}^{\pi,\rho}(\sigma; \bar{\sigma}) \in \mathcal{V}_{h+1}(o')$ for every $\bar{\sigma} \in \bar{\Sigma}$ that contains $o'$ , so (5) holds. + +Now consider the "if" direction. Suppose that (3) to (5) hold for some $\bar{\pi}$ and $\mathbf{v}'$ . Pick arbitrary $\sigma \in \Sigma_{h-1}$ that ends with $o$ . Consider a policy $\pi$ such that: $\pi(\mathbf{a} | \sigma; \omega) = \bar{\pi}(\mathbf{a} | \omega)$ for all $\omega \in \Omega$ , and $\pi(\mathbf{a} | \sigma; \bar{\sigma}; \omega) = \pi'(\mathbf{a} | \sigma; \bar{\sigma}; \omega)$ for all $\bar{\sigma} \in \bar{\Sigma}$ and $\omega \in \Omega$ , where $\pi'$ is an arbitrary policy that induces $\mathbf{v}'(\bar{\sigma})$ for every $\bar{\sigma}$ (which exists given (5)). Namely, $\pi$ is the same as $\bar{\pi}$ in step $h$ and switches to $\pi'$ in the subsequent steps. Given (4), the agent cannot benefit from any deviation at step $h$ , so (3) gives the players' values for $\pi$ and an optimal deviation plan of the agent. Hence, $\mathbf{v} \in \mathcal{V}_h(o)$ . + +Lemma 4. For any constant $\epsilon >0$ , it can be computed in time poly $(|S|\cdot |A|\cdot |\Omega |,H,1 / \epsilon)$ the half-space representations of a set of polytopes $\widehat{\mathcal{V}}_h(o)\subseteq \mathcal{V}_h(o)$ , $o\in O\cup \{\varnothing \}$ and $h = 1,\ldots ,H$ , such that (3), (4) and (6) are satisfiable for every $\mathbf{v}\in \widehat{\mathcal{V}}_h(o)$ and $\max_{\mathbf{v}\in \widehat{\mathcal{V}}_1(\varnothing)}v^{\mathsf{P}}\geq \max_{\mathbf{v}\in \mathcal{V}_1(\varnothing)}v^{\mathsf{P}} - \epsilon$ + +Proof. Throughout the proof, we say that the polytope $\widehat{\mathcal{V}}_h(o)$ is an $\varepsilon$ -approximation of $\mathcal{V}_h(o)$ if and only if: + +- $\widehat{\mathcal{V}}_h(o) \subseteq \mathcal{V}_h(o)$ , and +- for every $\mathbf{v} \in \mathcal{V}_h(o)$ , there exists $\mathbf{v}' \in \widehat{\mathcal{V}}_h(o)$ such that $v'^{\mathrm{P}} \geq v^{\mathrm{P}} - \varepsilon$ and $v'^{\mathrm{A}} = v^{\mathrm{A}}$ . + +We will show that an $\epsilon$ -approximation $\widehat{\mathcal{V}}_1(\varnothing)$ of $\mathcal{V}_1(\varnothing)$ can be computed efficiently, so that $\max_{\mathbf{v} \in \widehat{\mathcal{V}}_1(\varnothing)} v^{\mathrm{P}} \geq \max_{\mathbf{v} \in \mathcal{V}_1(\varnothing)} v^{\mathrm{P}} - \epsilon$ follows readily. Meanwhile, we also show that the polytopes we compute ensures that (3), (4) and (6) are satisfiable for every $\mathbf{v} \in \widehat{\mathcal{V}}_h(o)$ . + +We now prove by induction. The key is the following induction step. Suppose that the following conditions hold for all $o \in O$ : + +1. $\widehat{\mathcal{V}}_{h + 1}(o)$ is defined by $\mathcal{O}(H / \delta)$ many linear constraints. +2. $\widehat{\mathcal{V}}_{h + 1}(o)$ is an $\varepsilon$ -approximation of $\mathcal{V}_{h + 1}(o)$ . + +We show that, given the above conditions, for every $o \in O$ we can compute in time polynomial in $1/\delta$ a polytope $\widehat{\mathcal{V}}_h(o)$ (in half-space representation) that satisfies the above conditions (for $h$ ), with an approximation factor $\varepsilon' = \varepsilon + \delta$ in the second condition. Once this holds, picking $\delta = \epsilon / H$ then gives, by induction, that $\widehat{\mathcal{V}}_1(\varnothing)$ is an $\epsilon$ -approximation of $\mathcal{V}_1(\varnothing)$ (where $\epsilon$ is the target constant in the statement of the lemma). Note that as a based case, $\{(0,0)\}$ is readily a 0-approximation of $\mathcal{V}_H(o)$ and can be defined by three linear constraints. + +We proceed as follows. For every $o \in O$ , let $\overline{\mathcal{V}}_h(o)$ denote the set of vectors $\mathbf{v}$ satisfying (3), (4) and (6).4 We follow the algorithm presented in Fig. 2 and discretize $[0, H]^2$ to construct $\widehat{\mathcal{V}}_h(o)$ . Specifically, we slice the space along the dimension of the principal's value. We compute the intersection points of the slice lines and (the boundary of) $\overline{\mathcal{V}}_h(o)$ , and construct $\widehat{\mathcal{V}}_h(o)$ as the convex hull of the intersection points to approximate $\overline{\mathcal{V}}_h(o)$ . Specifically, let $W = \{0, \delta, 2\delta, \dots, H - \delta, H\}$ contain the principal's values on the slice lines we use, and let $\mathcal{W}$ be the set consisting of the following points. + +- First, for each $w \in W$ , the two intersection points of the slice line at $w$ and $\overline{\mathcal{V}}_h(o)$ : + +$$ +\check {\mathbf {v}} _ {w} \in \arg \min _ {\mathbf {v} \in \overline {{\mathcal {V}}} _ {h} (o): v ^ {\mathrm {P}} = w} v ^ {\mathsf {A}} \quad \text {a n d} \quad \hat {\mathbf {v}} _ {w} \in \arg \max _ {\mathbf {v} \in \overline {{\mathcal {V}}} _ {h} (o): v ^ {\mathrm {P}} = w} v ^ {\mathsf {A}}. +$$ + +- Moreover, two vertices of $\overline{\mathcal{V}}_h(o)$ with the minimum and maximum values for the agent: + +$$ +\check {\mathbf {v}} _ {*} \in \arg \min _ {\mathbf {v} \in \overline {{\mathcal {V}}} _ {h} (o)} v ^ {\mathsf {A}} \quad \text {a n d} \quad \hat {\mathbf {v}} _ {*} \in \arg \max _ {\mathbf {v} \in \overline {{\mathcal {V}}} _ {h} (o)} v ^ {\mathsf {A}}. +$$ + +If there are multiple maximum (or minimum) vertices, we pick an arbitrary one. + +An illustration is given in Fig. 1. + +It shall be clear that the choice of these points ensures that we can approximate any inducible value vector with at most $\delta$ compromise on the principal's value and no compromise on the agent's. (In particular, the inclusion of $\hat{\mathbf{v}}_*$ and $\hat{\mathbf{v}}_*$ ensures that we do not miss the agent's extreme values that may not be attained at any of the slice lines.) All the points can be computed efficiently by solving LPs that minimizes (or maximizes) $v^{\mathrm{A}}$ (where we also treat $\mathbf{v}$ as variables in addition to the other variables), subject to the linearized version of (3), (4) and (6), and additionally $v^{\mathrm{P}} = w$ when we compute $\hat{\mathbf{v}}_w$ or $\hat{\mathbf{v}}_w$ . The hypothesis that $\widehat{\mathcal{V}}_{h + 1}(o)$ is defined by $\mathcal{O}(H / \delta)$ linear constraints ensures that all the LPs are polynomial sized and hence can be solved efficiently. + +We then compute $\widehat{\mathcal{V}}_h(o)$ by taking the convex hull of $\mathcal{W}$ . Given that the space is two-dimensional, this can be done efficiently via standard algorithms in computational geometry (e.g., Chan's algorithm [9]). This way, the first condition in the inductive hypothesis holds for $\widehat{\mathcal{V}}_h(o)$ because $\widehat{\mathcal{V}}_h(o)$ has at most $\mathcal{O}(H / \delta)$ vertices while it is in $\mathbb{R}^2$ . Meanwhile, $\widehat{\mathcal{V}}_h(o)$ is an $\delta$ -approximation of $\overline{\mathcal{V}}_h(o)$ according to the following arguments. + +Claim 1. $\widehat{\mathcal{V}}_h(o)$ is an $\delta$ -approximation of $\overline{\mathcal{V}}_h(o)$ . + +Proof of Claim 1. First, since $\mathcal{W} \subseteq \overline{\mathcal{V}}_h(o)$ by construction, $\widehat{\mathcal{V}}_h(o) \subseteq \overline{\mathcal{V}}_h(o) \subseteq \mathcal{V}_h(o)$ holds readily. It remains to show that for any $\mathbf{v} \in \overline{\mathcal{V}}_h(o)$ there exists $\mathbf{x} \in \widehat{\mathcal{V}}_h(o)$ such that $x^{\mathsf{A}} = v^{\mathsf{A}}$ and $x^{\mathsf{P}} \geq v^{\mathsf{P}} - \delta$ . + +Let $\mathcal{B} = \{\mathbf{v}' \in \mathbb{R}^2 : i\delta \leq v'^{\mathrm{P}} \leq (i + 1)\delta\}$ be the band between two slice lines that contains $\mathbf{v}$ . Consider the relation between $v^{\mathsf{A}}$ and the agent's minimum and maximum values attained at $\mathcal{W} \cap \mathcal{B}$ . There can be the following possibilities. + +- Case 1. $v^{\mathsf{A}}$ lies in between the minimum and maximum values, i.e., + +$$ +\min _ {\mathbf {v} ^ {\prime} \in \mathcal {W} \cap \mathcal {B}} v ^ {\prime \mathsf {A}} \leq v ^ {\mathsf {A}} \leq \max _ {\mathbf {v} ^ {\prime} \in \mathcal {W} \cap \mathcal {B}} v ^ {\prime \mathsf {A}}. +$$ + +This means that there must be a point $\mathbf{x} \in \operatorname{ConvexHull}(\mathcal{W} \cap \mathcal{B})$ such that $x^{\mathsf{A}} = v^{\mathsf{A}}$ . We have $\mathbf{x} \in \operatorname{ConvexHull}(\mathcal{W} \cap \mathcal{B}) \subseteq \mathcal{B}$ . So both $\mathbf{v}$ and $\mathbf{x}$ are inside $\mathcal{B}$ . According to the definition of $\mathcal{B}$ , this means $x^{\mathsf{P}} \geq v^{\mathsf{P}} - \delta$ , as desired. + +- Case 2. $v^{\mathsf{A}} < \min_{\mathbf{v}' \in \mathcal{W} \cap \mathcal{B}} v'^{\mathsf{A}}$ . In this case, it must be that $\check{\mathbf{v}}_* \notin \mathcal{B}$ (otherwise, $\min_{\mathbf{v}' \in \mathcal{W} \cap \mathcal{B}} v'^{\mathsf{A}} = \check{v}_*^{\mathsf{A}} \leq v^{\mathsf{A}}$ ). Now that $\mathbf{v} \in \mathcal{B}$ , the line segment between $\mathbf{v}$ and $\check{\mathbf{v}}_*$ must intersect with the boundary of $\mathcal{B}$ (i.e., one of the slice lines) at some point $\mathbf{y}$ . We have $y^{\mathsf{A}} \leq v^{\mathsf{A}}$ (because $\check{v}_*^{\mathsf{A}} \leq v^{\mathsf{A}}$ by definition) and $\mathbf{y} \in \overline{\mathcal{V}}_h(o)$ (because $\mathbf{v}, \check{\mathbf{v}}_* \in \overline{\mathcal{V}}_h(o)$ ). Pick $\check{\mathbf{v}}_w$ where $w = y^{\mathsf{P}}$ . By definition $\check{v}_w^{\mathsf{A}} \leq y^{\mathsf{A}}$ . It follows that + +$$ +\tilde {v} _ {w} ^ {\mathsf {A}} \leq y ^ {\mathsf {A}} \leq v ^ {\mathsf {A}} < \min _ {\mathbf {v} ^ {\prime} \in \mathcal {W} \cap \mathcal {B}} v ^ {\prime \mathsf {A}}. +$$ + +This is a contradiction because we have $\check{\mathbf{v}}_w\in \mathcal{W}\cap \mathcal{B}$ as $\mathbf{y}$ is on the boundary of $\mathcal{B}$ . + +- Case 3. $v^{\mathsf{A}} > \max_{\mathbf{v}' \in \mathcal{W} \cap \mathcal{B}} v'^{\mathsf{A}}$ . An argument similar to that for Case 2 implies that this case is not possible, either. + +Hence, only Case 1 is possible, where a desired point $x$ exists. The claim then follows. + +![](images/73e28a30eb9b39a13194e93921c3e97879d53c1208c9a34fb720f13fcaa73f94.jpg) + +The fact that $\widehat{\mathcal{V}}_h(o) \subseteq \overline{\mathcal{V}}_h(o)$ also implies that (3), (4) and (6) are satisfiable for every $\mathbf{v} \in \widehat{\mathcal{V}}_h(o)$ , as they are for every $\mathbf{v} \in \overline{\mathcal{V}}_h(o)$ . We next confirm that $\widehat{\mathcal{V}}_h(o)$ is eventually an $(\varepsilon + \delta)$ -approximation of $\mathcal{V}_h(o)$ . Indeed, now Claim 1 indicates that $\widehat{\mathcal{V}}_h(o)$ is an $\delta$ -approximation of $\overline{\mathcal{V}}_h(o)$ , so $\widehat{\mathcal{V}}_h(o)$ is an $(\varepsilon + \delta)$ -approximation of $\mathcal{V}_h(o)$ as long as $\overline{\mathcal{V}}_h(o)$ is an $\varepsilon$ -approximation of $\mathcal{V}_h(o)$ . + +To see that $\overline{\mathcal{V}}_h(o)$ is an $\varepsilon$ -approximation, consider an arbitrary $\mathbf{v} \in \mathcal{V}_h(o)$ . By Lemma 3, $\mathbf{v}$ can be induced by some $\bar{\pi}$ and $\mathbf{v}'$ satisfying (3) to (5). By assumption, every $\widehat{\mathcal{V}}_{h+1}(o')$ is an $\varepsilon$ -approximation of $\mathcal{V}_{h+1}(o')$ , so for every onward vector $\mathbf{v}'(\bar{\sigma}) \in \mathcal{V}_{h+1}(o')$ , there exists a vector $\tilde{\mathbf{v}}'(\bar{\sigma}) \in \widehat{\mathcal{V}}_{h+1}(o')$ such that $\tilde{v}^{\mathrm{P}}(\bar{\sigma}) \geq v^{\mathrm{P}}(\bar{\sigma}) - \varepsilon$ and $\tilde{v}^{\mathrm{A}}(\bar{\sigma}) = v^{\mathrm{A}}(\bar{\sigma})$ . Using $\tilde{\mathbf{v}}'$ instead of $\mathbf{v}'$ , the same policy $\bar{\pi}$ then induces a vector $\tilde{\mathbf{v}} \in \widehat{\mathcal{V}}_h(o)$ to approximate $\mathbf{v}$ . Indeed, the agent's values are exactly the same under $\tilde{\mathbf{v}}'$ and $\mathbf{v}'$ , so the same response of the agent can be incentivized. This is why we require the approximation to not compromise on the agent's value. Moreover, according to (3), the overall difference between $\tilde{v}^{\mathrm{P}}$ and $v^{\mathrm{P}}$ is at most $\varepsilon$ because it holds for the coefficients that $\sum_{s,\omega,\mathbf{a}} p_{h-1}(s,\omega|o) \cdot \bar{\pi}(\mathbf{a}|\omega) = 1$ . As a result, $\tilde{v}^{\mathrm{P}} \geq v^{\mathrm{P}} - \varepsilon$ and $\overline{\mathcal{V}}_h(o)$ is an $\varepsilon$ -approximation of $\mathcal{V}_h(o)$ . + +Hence, the inductive hypothesis holds for $h$ . By induction, $\widehat{\mathcal{V}}_1(\varnothing)$ is an $\delta H$ -approximation of $\mathcal{V}_1(\varnothing)$ . Since $\delta H = \epsilon$ , we get that $\max_{\mathbf{v} \in \widehat{\mathcal{V}}_1(\varnothing)} v^{\mathsf{P}} \geq \max_{\mathbf{v} \in \mathcal{V}_1(\varnothing)} v^{\mathsf{P}} - \epsilon$ . + +Theorem 5. There exists an $\epsilon$ -optimal IC policy $\pi$ such that, for any given sequence $(\sigma; \omega^{\mathsf{P}}, \widetilde{\omega}^{\mathsf{A}}) \in \Sigma \times \Omega$ , the distribution $\pi(\cdot | \sigma; \omega^{\mathsf{P}}, \widetilde{\omega}^{\mathsf{A}})$ can be computed in time poly $(|S| \cdot |A| \cdot |\Omega|, H, 1/\epsilon)$ . + +Proof. Consider the algorithm presented in Fig. 3. The outputs of the algorithm over all possible input sequences $(\sigma; \omega^{\mathrm{P}}, \widetilde{\omega}^{\mathrm{A}}) \in \Sigma \times \Omega$ specify a policy $\pi$ . The polynomial running time of the algorithm for computing each $\pi(\cdot | \sigma; \omega^{\mathrm{P}}, \widetilde{\omega}^{\mathrm{A}})$ follows by noting that it runs by solving at most $H$ linear constraint satisfiability problems. + +It remains to argue that $\pi$ is IC and $\epsilon$ -optimal. Indeed, by Lemma 4 and an inductive argument, $\pi$ is IC at each time step $h$ and induces the corresponding values encoded in $\mathbf{v}'$ as the expected onward values. The $\epsilon$ -optimality of $\pi$ follows given the condition $\max_{\mathbf{v} \in \widehat{\mathcal{V}}_1(\emptyset)} v^{\mathsf{P}} \geq \max_{\mathbf{v} \in \mathcal{V}_1(\emptyset)} v^{\mathsf{P}} - \epsilon$ stated in Lemma 4 (and the choice of the initial $\mathbf{v}$ in Fig. 3). + +# A.2 Omitted Proofs in Section 4 + +Lemma 9. There exists an $\epsilon$ -optimal $\delta$ -IC policy $\pi$ such that, for any given sequence $(\sigma; \omega^{\mathsf{P}}, \widetilde{\omega}^{\mathsf{A}}) \in \Sigma \times \Omega$ , the distribution $\pi(\cdot | \sigma; \omega^{\mathsf{P}}, \widetilde{\omega}^{\mathsf{A}})$ can be computed in time poly $(|S| \cdot |A| \cdot |\Omega|, H, 1/\epsilon, \log(1/\delta))$ . + +Proof. The proof is similar to the approach in Section 3.1, which computes a near-optimal and 0-IC policy. We describe the differences below. + +Instead of maintaining two-dimensional sets of inducible values, we split the dimension of the agent's value into two dimensions $v^{\mathrm{A}}$ and $v_{*}^{\mathrm{A}}$ , which represent the agent's values under his truthful response (i.e., $\bot$ ) and his best deviation plan, respectively. Hence, each $\mathbf{v} \in \mathcal{V}(o)$ is now a tuple $(v^{\mathrm{P}}, v^{\mathrm{A}}, v_{*}^{\mathrm{A}})$ . + +(In Section 3.1, $v^{\mathsf{A}}$ and $v_{*}^{\mathsf{A}}$ are eventually forced to be the same, so there is no need to keep an additional dimension.) + +The inducibility of a vector $\mathbf{v} = (v^{\mathrm{P}}, v^{\mathrm{A}}, v_{*}^{\mathrm{A}})$ is characterized by the following constraints. First, we impose the same constraint as (3) on the first two dimensions of $\mathbf{v}$ , so that they capture the players' payoffs under the agent's truthful response. In order for the third dimension $v_{*}^{\mathrm{A}}$ to capture the agent's maximum attainable value, we use a constraint similar to (4): + +$$ +\begin{array}{l} v _ {*} ^ {\mathsf {A}} \geq \sum_ {\omega^ {\mathsf {A}}} p _ {h - 1} (\omega^ {\mathsf {A}} \mid o) \max _ {\widetilde {\omega} ^ {\mathsf {A}}} \sum_ {a ^ {\mathsf {A}}} \max _ {\tilde {a} ^ {\mathsf {A}}} \sum_ {s, \omega^ {\mathsf {P}}, a ^ {\mathsf {P}}} p _ {h - 1} (s, \omega^ {\mathsf {P}} \mid o, \omega^ {\mathsf {A}}) \cdot \bar {\pi} (\mathbf {a} \mid \omega^ {\mathsf {P}}, \widetilde {\omega} ^ {\mathsf {A}}) \cdot \\ \left(r _ {h} ^ {\mathsf {A}} \left(s, a ^ {\mathsf {P}}, \tilde {a} ^ {\mathsf {A}}\right) + v _ {*} ^ {\prime} ^ {\mathsf {A}} (s, \omega , \widetilde {\omega} ^ {\mathsf {A}}, \mathbf {a}, \tilde {a} ^ {\mathsf {A}})\right). \tag {8} \\ \end{array} +$$ + +The remaining constraint is the same as (6). + +All the non-linear constraints can be linearized the same way as the approach described in Section 3.1. Hence, we can efficiently approximate $\mathcal{V}_h(o)$ by examining the inducibility of points on a sufficiently fine-grained grid in $[0,H]^3$ , which contains $\mathrm{poly}(H,1 / \epsilon)$ many points, and constructing the convex hull of these points. (Note that there is no need to ensure zero compromise on the agent's value as required in the proof of Lemma 4. This is because $\delta$ -IC is defined with respect to the agent's expected value at the beginning of the game instead of that at every time step. Hence, using points on a grid suffices the purpose of the approximation in this proof.) The half-space representation of the convex hull can be computed efficiently given that it is in $\mathbb{R}^3$ [9]. Eventually, an optimal $\pi \in \widehat{\Pi}_{\delta}$ corresponds to a solution to $\max_{\mathbf{v}\in \mathcal{V}_1(\varnothing)}v^{\mathsf{P}}$ subject to $v^{\mathsf{A}}\geq v_{*}^{\mathsf{A}} - \delta$ , and we can use the same forward construction procedure in Section 3.2 to compute $\pi_h(\cdot |\sigma ;\omega^{\mathsf{P}},\widetilde{\omega}^{\mathsf{A}})$ . + +Note that (8) only enforces $v_{*}^{\mathsf{A}}$ as an upper bound of the maximum attainable value, instead of the exact value. This suffices for our purpose because any $(v^{\mathsf{P}}, v^{\mathsf{A}}, v_{*}^{\mathsf{A}})$ in the feasible set $\mathcal{V}_1(\varnothing) \cap \{ \mathbf{v} : v^{\mathsf{A}} \geq v_{*}^{\mathsf{A}} - \epsilon \}$ also implies the inclusion of $(v^{\mathsf{P}}, v^{\mathsf{A}}, \bar{v}_{*}^{\mathsf{A}})$ in the same feasible set, where $\bar{v}_{*}^{\mathsf{A}}$ is the actual maximum attainable value induced by the policy that induces $(v^{\mathsf{P}}, v^{\mathsf{A}}, v_{*}^{\mathsf{A}})$ according to our formulation. + +Theorem 10. There exists an algorithm that guarantees regret $\widetilde{\mathcal{O}} (\zeta^{1 / 3}T^{2 / 3})$ for both players with probability $1 - q$ , where $\zeta = H^5 |S|^2 |A|^3 |\Omega |^2$ . The computation involved in implementing the algorithm takes time poly(|S|\cdot|A|\cdot|\Omega|,H,T). + +Proof. We run reward-free exploration to obtain a model $\widehat{p}$ with error bound $\delta / 2$ . This can be achieved w.h.p. in $\widetilde{\mathcal{O}}(\zeta/\delta^2)$ episodes according to Lemma 7. Next, we compute an $\delta$ -optimal strategy $\pi \in \widehat{\Pi}_{\delta}$ and use it in the remaining rounds. According to Lemma 9, this can be done in polynomial time. + +By assumption, rewards are bounded in $[0,1]$ so the regrets are at most 1 for both players in each of the exploration episodes. In each of the remaining episodes, the agent's regret is as follows, where we pick arbitrary $\rho^{*}\in \arg \max_{\rho}V_{1}^{\mathsf{A},\pi ,\rho}(\emptyset)$ : + +$$ +\begin{array}{l} V _ {1} ^ {\mathsf {A}, \pi , \rho^ {*}} (\varnothing) - V _ {1} ^ {\mathsf {A}, \pi , \bot} (\varnothing) \leq \underbrace {\left| \widehat {V} _ {1} ^ {\mathsf {A} , \pi , \rho^ {*}} (\varnothing) - \widehat {V} _ {1} ^ {\mathsf {A} , \pi , \bot} (\varnothing) \right|} _ {\leq \delta \text {a s} \pi \in \widehat {\Pi} _ {\delta}} + \\ \underbrace {\left| \widehat {V} _ {1} ^ {\mathsf {A} , \pi , \rho^ {*}} (\varnothing) - V _ {1} ^ {\mathsf {A} , \pi , \rho^ {*}} (\varnothing) \right| + \left| \widehat {V} _ {1} ^ {\mathsf {A} , \pi , \perp} (\varnothing) - V _ {1} ^ {\mathsf {A} , \pi , \perp} (\varnothing) \right|} _ {\leq \delta \text {b y L e m m a 7}} \leq 2 \delta . \\ \end{array} +$$ + +The principal's regret is: + +$$ +\begin{array}{l} V ^ {*} - V _ {1} ^ {\mathsf {P}, \pi , \perp} (\varnothing) = \max _ {\pi^ {\prime} \in \Pi_ {0}} V _ {1} ^ {\mathsf {P}, \pi^ {\prime}, \perp} (\varnothing) - V _ {1} ^ {\mathsf {P}, \pi , \perp} (\varnothing) \\ \underbrace{\leq\max_{\pi^{\prime}\in\widehat{\Pi}_{\delta}}V_{1}^{P,\pi^{\prime},\perp}(\varnothing)}_{\text{as $\Pi_{0}\subseteq\widehat{\Pi}_{\delta}$}}}-V_{1}^{P,\pi,\perp}(\varnothing)\leq\underbrace{\max_{\pi^{\prime}\in\widehat{\Pi}_{\delta}}\widehat{V}_{1}^{P,\pi^{\prime},\perp}(\varnothing)-\widehat{V}_{1}^{P,\pi,\perp}(\varnothing)}_{\leq\delta \text{as $\pi$ is $\delta$ - optimal}}+ \delta \leq 2\delta. \\ \end{array} +$$ + +The reason that $\Pi_0 \subseteq \widehat{\Pi}_{\delta}$ is the following: Since $\widehat{p}$ ensures error bound $\delta / 2$ , we have $\left|\widehat{V}_1^{\mathsf{A}, \pi', \rho}(\varnothing) - V_1^{\mathsf{A}, \pi', \rho}(\varnothing)\right| \leq \delta / 2$ for all $\rho$ . By definition, $\pi' \in \Pi_0$ means that $V_1^{\mathsf{A}, \pi', \perp}(\varnothing) \geq V_1^{\mathsf{A}, \pi', \rho}(\varnothing)$ . So, $\widehat{V}_1^{\mathsf{A}, \pi', \perp}(\varnothing) \geq \widehat{V}_1^{\mathsf{A}, \pi', \rho}(\varnothing) - \delta$ for all $\rho$ ; hence, $\pi' \in \widehat{\Pi}_{\delta}$ . + +The above bounds then lead to a total regret of at most $\widetilde{\mathcal{O}} (\zeta /\delta^2) + \mathcal{O}(T\delta)$ for each player. Taking $\delta = (\zeta /T)^{1 / 3}$ gives the upper bound $\widetilde{\mathcal{O}} (\zeta^{1 / 3}T^{2 / 3})$ + +# B Linearizing (3) and (4) + +Specifically, to remove the maximization operator in (4), we introduce a set of auxiliary variables $y(a^{\mathsf{A}}, \omega^{\mathsf{A}}, \widetilde{\omega}^{\mathsf{A}})$ to capture the maximum values on the right hand side of (4). We replace the right hand side of (4) with $\sum_{a^{\mathsf{A}} \in A^{\mathsf{A}}} y(a^{\mathsf{A}}, \omega^{\mathsf{A}}, \widetilde{\omega}^{\mathsf{A}})$ , and by adding the following constraint we force each $y(a^{\mathsf{A}}, \omega^{\mathsf{A}}, \widetilde{\omega}^{\mathsf{A}})$ to be an upper bound of the corresponding maximum value: for all $\tilde{a}^{\mathsf{A}} \in A^{\mathsf{A}}$ , + +$$ +y \left(a ^ {\mathsf {A}}, \omega^ {\mathsf {A}}, \widetilde {\omega} ^ {\mathsf {A}}\right) \geq \sum_ {s, \omega^ {\mathsf {P}}, a ^ {\mathsf {P}}} p _ {h - 1} \left(s, \omega^ {\mathsf {P}} \mid o, \omega^ {\mathsf {A}}\right) \cdot \bar {\pi} (\mathbf {a} \mid \omega^ {\mathsf {P}}, \widetilde {\omega} ^ {\mathsf {A}}) \cdot \left(r _ {h} ^ {\mathsf {A}} \left(s, a ^ {\mathsf {P}}, \tilde {a} ^ {\mathsf {A}}\right) + v ^ {\prime \mathsf {A}} \left(s, \omega , \widetilde {\omega} ^ {\mathsf {A}}, \mathbf {a}, \tilde {a} ^ {\mathsf {A}}\right)\right) \tag {9} +$$ + +To remove the quadratic terms in (3) and (9), we use an auxiliary variable $\mathbf{z}(s,\omega, \widetilde{\omega}^{\mathsf{A}}, \mathbf{a}, \tilde{a}^{\mathsf{A}})$ to replace each term $\bar{\pi}(\mathbf{a}|\boldsymbol{\omega}) \cdot \mathbf{v}'(s,\boldsymbol{\omega}, \widetilde{\omega}^{\mathsf{A}}, \mathbf{a}, \tilde{a}^{\mathsf{A}})$ and impose the following constraint on $\mathbf{z}$ : + +$$ +\mathbf {H} \cdot \mathbf {z} (s, \omega , \widetilde {\omega} ^ {\mathrm {A}}, \mathbf {a}, \tilde {a} ^ {\mathrm {A}}) \leq \bar {\pi} (\mathbf {a} | \omega) \cdot \mathbf {b}, \tag {10} +$$ + +where $\mathbf{H}$ and $\mathbf{b}$ are taken from the half-space representation of the polytope $\widehat{\mathcal{V}}_{h+1}$ , i.e., $\widehat{\mathcal{V}}_{h+1}(s, a^{\mathrm{P}}, \tilde{a}^{\mathrm{A}}) = \{\mathbf{x} : \mathbf{H} \cdot \mathbf{x} \leq \mathbf{b}\}$ . It is straightforward that, when $\widehat{\mathcal{V}}_{h+1}(s, a^{\mathrm{P}}, \tilde{a}^{\mathrm{A}})$ is nonempty and bounded, (10) holds if and only if $\mathbf{z}(s, \omega, \widetilde{\omega}^{\mathrm{A}}, \mathbf{a}, \tilde{a}^{\mathrm{A}}) = \bar{\pi}(\mathbf{a} | \omega) \cdot \mathbf{x}$ for some $\mathbf{x} \in \widehat{\mathcal{V}}_{h+1}(s, a^{\mathrm{P}}, \tilde{a}^{\mathrm{A}})$ . Hence, (10) is the only constraint needed (for each tuple $(s, \omega, \widetilde{\omega}^{\mathrm{A}}, \mathbf{a}, \tilde{a}^{\mathrm{A}})$ ) after we replace the terms with $\mathbf{z}$ . + +The Complete Formulation In summary, the above approach yields the following linearized version of the constraint satisfiability problem in Section 3.1, where $\bar{\pi},\mathbf{z} = (z^{\mathrm{A}},z^{\mathrm{P}})$ , and $y$ are the variables (highlighted in blue). + +1. The value function constraint: + +$$ +\mathbf {v} = \sum_ {s, \boldsymbol {\omega}, \mathbf {a}} p _ {h - 1} (s, \boldsymbol {\omega} \mid o) \cdot \Big (\mathbf {r} _ {h} (s, \mathbf {a}) \cdot \bar {\pi} (\mathbf {a} \mid \boldsymbol {\omega}) + \mathbf {z} (s, \boldsymbol {\omega}, \boldsymbol {\omega} ^ {\mathsf {A}}, \mathbf {a}, a ^ {\mathsf {A}}) \Big). +$$ + +2. An IC constraint for each $\omega^{\mathsf{A}}\in \Omega^{\mathsf{A}}$ + +$$ +\sum_ {s, \omega^ {\mathrm {P}}, \mathbf {a}} p _ {h - 1} (s, \omega^ {\mathrm {P}} \mid o, \omega^ {\mathrm {A}}) \cdot \left(r _ {h} ^ {\mathrm {A}} (s, \mathbf {a}) \cdot \bar {\pi} (\mathbf {a} \mid \omega) + z ^ {\mathrm {A}} \left(s, \omega , \omega^ {\mathrm {A}}, \mathbf {a}, a ^ {\mathrm {A}}\right)\right) \geq \sum_ {a ^ {\mathrm {A}} \in A ^ {\mathrm {A}}} y \left(a ^ {\mathrm {A}}, \omega^ {\mathrm {A}}, \widetilde {\omega} ^ {\mathrm {A}}\right). +$$ + +Moreover, for each tuple $(a^{\mathsf{A}},\omega^{\mathsf{A}},\widetilde{\omega}^{\mathsf{A}})\in A^{\mathsf{A}}\times \Omega^{\mathsf{A}}\times \Omega^{\mathsf{A}}$ + +$$ +y \left(a ^ {\mathsf {A}}, \omega^ {\mathsf {A}}, \widetilde {\omega} ^ {\mathsf {A}}\right) \geq \sum_ {s, \omega^ {\mathsf {P}}, a ^ {\mathsf {P}}} p _ {h - 1} (s, \omega^ {\mathsf {P}} \mid o, \omega^ {\mathsf {A}}) \Big (r _ {h} ^ {\mathsf {A}} \left(s, a ^ {\mathsf {P}}, \widetilde {a} ^ {\mathsf {A}}\right) \cdot \bar {\pi} (\mathbf {a} \mid \omega^ {\mathsf {P}}, \widetilde {\omega} ^ {\mathsf {A}}) + z ^ {\mathsf {A}} \left(s, \omega , \widetilde {\omega} ^ {\mathsf {A}}, \mathbf {a}, \widetilde {a} ^ {\mathsf {A}}\right) \Big). +$$ + +3. An onward value constraint for each tuple $(s, \omega, \widetilde{\omega}^{\mathsf{A}}, \mathbf{a}, \tilde{a}^{\mathsf{A}}) \in \bar{\Sigma}$ : + +$$ +\mathbf {H} \left(s, a ^ {\mathrm {P}}, \tilde {a} ^ {\mathrm {A}}\right) \cdot \mathbf {z} (s, \omega , \widetilde {\omega} ^ {\mathrm {A}}, \mathbf {a}, \tilde {a} ^ {\mathrm {A}}) \leq \bar {\pi} (\mathbf {a} | \omega) \cdot \mathbf {b} \left(s, a ^ {\mathrm {P}}, \tilde {a} ^ {\mathrm {A}}\right), +$$ + +where for every $o \in O$ , the matrix $\mathbf{H}(o)$ and vector $\mathbf{b}(o)$ are given by the half-space representation of $\widehat{\mathcal{V}}_{h+1}(o)$ , i.e., $\widehat{\mathcal{V}}_{h+1}(o) = \{\mathbf{v}' \in \mathbb{R} : \mathbf{H}(o) \cdot \mathbf{v}' \leq \mathbf{b}(o)\}$ . + +4. Additionally, we impose + +$$ +\bar {\pi} (\cdot | \omega) \in \Delta (A) +$$ + +for each $\omega \in \Omega$ to ensure that $\bar{\pi} (\cdot |\omega)$ is a valid distribution over $A$ . + +# C Additional Discussion about Intractability without Hindsight Observability + +The PSPACE-hardness can be seen by thinking of a POMDP as an instance of our problem where only the principal can make observations and perform actions to influence the environment (essentially, the agent can neither influence the principal nor the environment in this instance). + +The PSPACE-hardness remains in the case of information design, where the principal observes the state directly but does not act, while the agent makes no observation but acts; as well as the case of mechanism design, where the agent observes the state directly but does not act, while the principal does not observe but acts. This can be seen by considering zero-sum instances, where the principal's and the agent's rewards sum to zero. + +More specifically, consider for example the case of information design. If the goal is to compute the principal's maximum attainable payoff, the PSPACE-hardness of the problem is immediate: Since the game is zero-sum, it is optimal for the principal to not send no signal (if signaling were to improve the principal's payoff, the agent would be better-off just ignoring the signals). Hence, computing the maximum attainable payoff of the principal in this case is equivalent to computing (the negative of) the agent's maximum attainable payoff, which amounts to solving a POMDP. + +One may argue that while the above example demonstrates the hardness of determining the principal's maximum attainable payoff, computing the principal's optimal policy is actually trivial in the example (i.e., sending no signal is optimal). So it does not rule out the possibility of an efficient algorithm which, given any sequence, computes the signal distribution of an optimal policy, without computing the principal's payoff the policy yields. It turns out that this is not possible, either. + +Consider a game where the agent can choose between two actions $a$ and $b$ in the first time step. Action $a$ leads to a process where the principal's rewards are zero for all state-action pairs. Action $b$ leads to another process with payoffs $1 - x$ for the principal and $x$ for the agent, where $x \in [0,1]$ depends on the principal's signaling strategy in this sub-process. For example, we can design this sub-process as a matching pennies game, where: nature flips a fair coin, the principal observes the outcome, and the agent must choose the same side of the coin to get a reward 1 and otherwise he gets $-1$ . If the agent plays this matching pennies game on his own, his expected payoff is 0. The principal can reveal her observation to help the agent to improve the payoff. And the principal can do so probabilistically, so that she can fine tune the agent's expected payoff $x$ to any desired value in $[0,1]$ . To maximize the principal's payoff in the entire process requires finding an $x$ that is sufficiently high, so that the agent is incentivized to choose $b$ (otherwise, the principal only gets 0); at the same time, we would like $x$ to be as low as possible to maximize the principal's payoff $1 - x$ . 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Ross + +California Institute of Technology + +zross@caltech.edu + +Domniki Asimaki + +California Institute of Technology + +domniki@caltech.edu + +Kamyar Azizzadenesheli + +NVIDIA Corporation + +kaazizzad@gmail.com + +# Abstract + +Expanding on neural operators, we propose a novel framework for stochastic process learning across arbitrary domains. In particular, we develop operator flow matching (OFM) for learning stochastic process priors on function spaces. OFM provides the probability density of the values of any collection of points and enables mathematically tractable functional regression at new points with mean and density estimation. Our method outperforms state-of-the-art models in stochastic process learning, functional regression, and prior learning. + +# 1 Introduction + +Stochastic processes are foundational to many domains, from functional regression and physics-based data assimilation, to financial markets, geophysics, and black box optimization. Stochastic processes can serve as prior distributions over functions and can provide the density of any finite collection of points. Conventionally, these priors are designed by hand from predefined Gaussian processes (GP) and their variants, and therefore assume that they adequately describe the phenomena of interest. However, many processes modeled in the natural world are not well described by GP, Fig 2. Such models limit the flexibility and generalizability of these stochastic processes in real-world applications, leaving behind significant challenges for more general stochastic process learning (SPL). + +In SPL, the prior over the stochastic process is learned from data, i.e., a set of historical point evaluations in past experiments. Learning the prior over the process is crucial for universal functional regression (UFR), which is a recently proposed Bayesian scheme for functional regression and takes GP-regression as a special case when the prior is Gaussian [1]. UFR is important to scientific and engineering domains, including reanalysis, data completion-assimilation, uncertainty quantification, and black box optimization. + +In this paper, we introduce a novel operator learning framework termed operator flow matching (OFM) for learning priors over stochastic processes through the joint distribution of any collection of points. To achieve this, we theoretically and empirically generalize marginal optimal transport flow matching [2] to infinite-dimensional function spaces where we map a GP into a prior over function spaces through a flow differential equation. We then derive SPL from the function space derivation and learn a prior over arbitrary sets of points. For SPL, we map any collection of pointwise evaluations of a GP to pointwise evaluations of target functions. This allows us to learn prior distributions over more general stochastic processes, hence enabling sampling values of any collection of points with their associated density and facilitating efficient UFR. We leverage this capability by extending neural operators [3]-designed initially to map functions between infinite-dimensional spaces-to maps + +between collections of points deploying their functional convergence properties. This serves as the essential architecture block in OFM. + +After learning the prior and having access to the densities, OFM can be used for UFR, where given any collection of points of the underlying function, we estimate the posterior mean value of any new collection of points and efficiently sample from their posterior values using stochastic gradient Langevin dynamics (SGLD) [4], i.e., a Gaussian sampling on the input GP space (Fig 1). We show that OFM significantly outperforms state-of-the-art (SOTA) methods, including deep GPs, conditional models, and operator flows (OPFlow) [1, 5-10]. + +Generalizing GP-regression to regression over general stochastic process in a practical and implementable way demands a unified framework that spans many key fields. Because readers will have diverse backgrounds and expertise, we provide a set of potential questions and answers in Appendix A to further clarify the development and highlight our contributions. + +![](images/32b1650e313acaf2568c374539b671cc0a85c22f753354abe54a0c6e7aafbb58.jpg) +Figure 1: Two-phase strategy for prior learning and posterior sampling. In the prior-learning phase, OFM leverages the marginal optimal-transport path to learn a bijective mapping between a predefined GP and the unknown stochastic process that generates the training data. In the posterior-sampling phase, the learned prior is frozen; given noisy, partial observations, the exact posterior is obtained via Bayes' theorem. SGLD, aided by the Hutchinson trace estimator, then enables efficient and robust sampling. + +![](images/26cf6209807bcda2cf5f73c132027d09d7c87cf9cfa97d82aef704b1cbd99cb5.jpg) + +Lastly, our contributions are summarized as follows: + +- First work to extend flow matching / stochastic interpolants / rectified flow [11-13] to stochastic processes via operator learning. The formulation enables likelihood estimation for function values at any collection of points for the target stochastic process. We also contribute to the development of marginal (dynamic) optimal transport in infinite-dimensional flow matching through optimal coupling and dynamic Kantorovich formulations. +- First integration of flow matching with functional regression yields a unified framework for prior and posterior sampling that is applicable to both generation and regression tasks. +- A practical generalization of GP regression that provides the exact prior and posterior density over an unknown stochastic process (whether GP or non-GP). In contrast, previous methods such as deep GPs and conditional models work with approximate posteriors. Our method achieves SOTA performance in all challenging functional regression tasks. +- This work provides a unified perspective bridging several important fields, which opens new research directions for problems in science and engineering. Additionally, we present extensive ablation and scaling studies that demonstrate the effectiveness of each component in our framework. + +# 2 Related Work + +Neural operators. Neural operators constitute a paradigm in machine learning for learning maps between function spaces, a generalization of conventional neural networks that map between finite dimensional spaces [14, 15]. Among neural operator architectures, Fourier neural operators (FNO) [14] enable convolution in the spectral domain and are effective for operator learning [16-21]. In this work, we use this as our neural operator architecture. + +![](images/fca8f2cc103b9ba102bebf1fda510e739b3f5f3fd0f16a50461c88906ac9560c.jpg) +(a) Observations + +![](images/62ea28a70754666fdc4e09dfad486cefe81ce7279c65a3048a2dad83f63c34b8.jpg) + +![](images/1ad35b4c362aed9813c126fac8394473b3c78901792cd4838303cb1994cddb1c.jpg) +(c) OFM mean + +![](images/c96d98405e62e0b1b671dde1e6dc825d09576c2df11d8ee4bc5f70f8d84c4430.jpg) + +![](images/39722e14fbf196830b590e1a163d711b7998419ddb34cca9cea6a9f299558662.jpg) +(b) Ground truth +(e) GP sample + +![](images/2c9ef9f5f42c1e65acc1c6730a1eeabcf3ab2a71a82736e3cac65494f2e5ca7c.jpg) +(f) ConvCNP mean +Figure 2: Operator Flow Matching (OFM) regression on Navier-Stokes functional data with resolution $64 \times 64$ . (a) 32 random observations (only $0.7\%$ ). (b) Ground truth sample. (c) Predicted mean from OFM. (d) One posterior sample from OFM. (e) One posterior sample from the best fitted GP. (f) Predicted mean from ConvCNP. (g) One posterior sample from ConvCNP. + +![](images/7415912c6a87e19b82225de7dcff94cd2aa0024d07ff53add07bfa409d515c90.jpg) +(d) OFM sample +(g) ConvCNP sample + +Direct function samples. There is a body of work on generative models dedicated to learning distributions over functions, such that direct sampling on the function space is possible. For example, generative adversarial neural operators (GANO) generalize generative adversarial nets on finite dimensional spaces to function spaces [22, 23], yielding a neural operator generative model that maps GP to data functions [3]. Other works in this area have followed the success of diffusion models [24, 25] in finite dimensional spaces, e.g., denoising diffusion operators generalize diffusion models to function spaces by using GP as a means to add noise and use neural operators to learn the score operator on function-valued data [26-28]. Moreover, the same principle has been deployed to generalize flow matching [11] to functional spaces [29], an approach closely related to our work. However, these works on learning generative models on function spaces do not support UFR the way GP-regression does because they (i) focus solely on generating function samples, (ii) do not clarify how to model a stochastic process on point value sequential generation, and (iii) do not provide point evaluation of probability density. In contrast, OFM enables exact density calculation and conditional posterior sampling. + +Stochastic processes. Earlier works on SPL have focused on hand-tuned methods in the style of GP-regression. In these cases, an expert tunes the GP parameters given a set of experimental samples. More advanced methods rely on deep GPs, in which a network of GPs is stacked on top of each other. The parameters of deep GPs are commonly optimized by minimizing the variational free energy, which serves as a bound on the negative log marginal likelihood. [30, 31]. Deep GPs have limitations in terms of learnability, expressivity, and computational complexity. Warped GPs [32] and transforming GP [33] methods use historical data to learn a pointwise transformation of GP values and achieve on par performance compared to deep GP type methods. The pointwise nature of such approaches limits their generality. + +Another line of work proposes learning the conditional distribution of point values, inspired by variational inference and designed for sampling from function spaces; we refer to this line of models as conditional models1 [7]. This method trains a model to map any collection of points and their values to a vector, used as an input to a decoder that maps any collection of points to their values. The architectures used in these models (including the decoder) are not mathematically consistent as the number of points grows, limiting the approach to finite dimensions. In particular, Convolutional Conditional Neural Processes (ConvCNPs) [6] use convolutional architectures to capture local, translation-invariant structure, but they assume stationarity and struggle with long-range dependencies or highly non-uniform data. The diffusion based variants [34] also use uncorrelated Gaussian noise, and the results do not exist in function spaces [22, 26]. Furthermore, methods based on conditional models are unable to provide density estimation for collections of points, as needed for UFR and SPL in general. + +Separately, OPFlow introduced invertible neural operators that are trained to map any collection of points sampled from a GP to a new collection of points in the data space [1], using the maximum likelihood principle. This method is mathematically consistent as the resolution grows, captures + +the likelihood of any collection of points, and allows for UFR using SGLD. However, similar to normalizing flow [35] methods in finite dimensional domains, the use of invertible deep learning models makes their training a challenge, particularly with regard to expressiveness, as also described in the original OPFlow work. Finally, we strongly encourage readers to consult Appendix A and Q for an in-depth comparison of stochastic process learning and other generative frameworks. + +# 3 Operator Flow Matching + +Here, we introduce the problem setting and notations used for OFM in function space. We recommend that readers consult Appendix B-E for a foundational overview of SPL, UFR and related background. Subsequently, we present the framework of OFM, which extends marginal optimal transport flow matching [2] to infinite-dimensional spaces. We further demonstrate the generalization of flow matching to stochastic processes as it is induced from OFM on function spaces. Finally, we illustrate how to evaluate exact and tractable likelihoods for any point evaluation of functions using OFM, making it applicable in the UFR setting. + +For a real separable Hilbert space $(\mathcal{H},\langle \cdot ,\cdot \rangle ,\| \cdot \|)$ , equipped with the Borel $\sigma-$ algebra of measurable sets denoted by $\mathcal{B}(\mathcal{H})$ , we introduce two measures on $\mathcal{B}(\mathcal{H})$ , $\nu_{0}$ as the reference measure and $\nu_{1}$ as the data measure. Consider a function $h_0$ sampled from $\nu_{0}$ , such that $h_0\sim \nu_0$ . For a smooth time-varying infinite dimensional vector field $\mathcal{G}_t:\mathcal{H}\to \mathcal{H}$ , we define an ordinary differential equation (ODE) + +$$ +\frac {\partial \Phi_ {t} \left(h _ {0}\right)}{\partial t} = \mathcal {G} _ {t} \left(\Phi_ {t} \left(h _ {0}\right)\right) \tag {1} +$$ + +with initial condition $\Phi_0(h_0) = h_0$ , where $h = h_t = \Phi_t(h_0)$ represents a function $h_0$ transported along a vector field from time 0 to time $t$ . The diffeomorphism $\Phi_t$ induces a pushforward measure $\mu_t \coloneqq [\Phi_t]_{\sharp}(\mu_0)$ , with $\mu_0 = \nu_0$ , and we refer to $\mu_t$ as the path of probability measure. The goal is to construct a path of probability measure such that at $t = 1$ , $\mu_1 \approx \nu_1$ . The dynamic relationship between the time varying measure $\mu_t$ and vector field $\mathcal{G}_t$ can be characterized by the continuity equation: + +$$ +\frac {\partial \mu_ {t}}{\partial t} = - \nabla \cdot \left(\mu_ {t} \mathcal {G} _ {t}\right) \tag {2} +$$ + +In practice, we use Eq. 2 in its weak form [29, 36] to check whether a given vector field $\mathcal{G}_t$ generates the target $\mu_t$ : + +$$ +\int_ {0} ^ {1} \int_ {\mathcal {H}} \frac {\partial \varphi (h , t)}{\partial t} + \left\langle \mathcal {G} _ {t} (h), \nabla_ {h} \varphi (h, t) \right\rangle d \mu_ {t} (h) d t = 0 \tag {3} +$$ + +Where $\varphi \in \mathrm{Cyl}(\mathcal{H}\times [0,1])$ , and $\mathrm{Cyl}(\mathcal{H}\times [0,1])$ is the space of smooth cylindrical test functions. Suppose that the time-varying vector field $\mathcal{G}_t$ and the induced $\mu_t$ satisfying Eq. 3 are known. We parameterize $\mathcal{G}_t$ with a neural operator $\mathcal{G}_{\theta}:[0,1]\times \mathcal{H}\to \mathcal{H}$ and regress $\mathcal{G}_{\theta}$ to target $\mathcal{G}_t$ through flow matching objective. + +$$ +\mathcal {L} _ {\mathrm {F M}} ^ {\dagger} = \mathbb {E} _ {t \sim \mathcal {U} [ 0, 1 ], h \sim \mu_ {t}} \| \mathcal {G} _ {\theta} (t, h) - \mathcal {G} _ {t} (h) \| ^ {2} \tag {4} +$$ + +However, similar to its finite-dimensional counterpart, $\mathcal{G}_t$ is typically unknown. Moreover, there are infinitely many paths of probability measures that satisfy Eq. 3 and ensure $\mu_1 \approx \nu_1$ . Therefore, it is necessary to specify a path of probability measures to effectively guide the learning of $\mathcal{G}_{\theta}$ . + +By constructing the appropriate Gaussian and conditional probability measures and leveraging optimal coupling together with the dynamic Kantorovich formulation [37], we propose marginal (dynamic) optimal-transport flow matching in function space; detailed theory development and proofs are provided in Appendix F and G. In the next subsection, we show how to generalize flow matching to stochastic processes, which is induced by the function space derivation. + +# 3.1 Generalizing Flow Matching to Stochastic Processes + +Stochastic processes are inherently infinite-dimensional and define distributions over any collection of points (Bremaud [38], Chapter 5.1). We generalize the above marginal optimal transport flow matching on function spaces to stochastic processes by defining the transport map on any collection of points. We then show that, as the collection of points covers the space in the limit, this generalization recovers infinite-dimensional flow matching implemented with neural operators. + +For any $n$ and points $\{x_{1}, x_{2}, \ldots, x_{n}\}$ , consider an ODE system in which a vector of random variables $u_{0} \in \mathbb{R}^{n}$ is gradually transformed into $u_{1} \in \mathbb{R}^{n}$ , for which, the $i$ th entry is equal to $u(x_{i})$ , via a smooth, time-varying vector field, denoted by $\mathcal{G}_{t}$ with abuse of notation. + +$$ +u _ {t} := \Phi_ {t} \left(u _ {0}\right) = u _ {0} + \int_ {0} ^ {t} \mathcal {G} _ {s} \left(u _ {s}\right) d s \tag {5} +$$ + +Here, the neural operator is applied to a collection of point evaluations. Given the set of points and the density of $p_0 \coloneqq p_0(\{u_0(x_1), u_0(x_2), \ldots, u_0(x_n)\}) = \mathcal{N}(\mathbf{0}, K(\{x_1, x_2, \ldots, x_n\}))$ , where $K$ is a $n \times n$ covariance matrix with entries described by kernel function $k(x_i, x_j)$ and $u_0 \sim p_0$ , the time-varying density $p_t$ induced by the diffeomorphism $\Phi_t$ or $\mathcal{G}_t$ can be computed, extending the transport equation Eq. 2 to collections of points, + +$$ +\frac {\partial p _ {t} \left(u _ {t}\right)}{\partial t} = - \left(\nabla \cdot \left(\mathcal {G} _ {t} p _ {t}\right)\right) \left(u _ {t}\right) \tag {6} +$$ + +Eq. 6 shows that constructing $p_t$ is equivalent to constructing $\mathcal{G}_t$ for finite entries for which the analysis carries to finite collections of random variables. In the following, we refer to $p_t$ as the marginal probability path induced by $\mathcal{G}_t$ for the given collection of points. From Eq. 6, the log density can be computed through integration, + +$$ +\log p _ {t} \left(u _ {t}\right) = \log p _ {0} \left(u _ {0}\right) - \int_ {0} ^ {t} (\nabla \cdot \mathcal {G} _ {s}) \left(u _ {s}\right) d s \tag {7} +$$ + +In this formulation, we are seeking a specific vector field that transports density $q_{0}$ to target density $q_{1}$ for any $n$ and any collection of points $\{x_{1}, x_{2}, \ldots, x_{n}\}$ with boundary conditions $p_{0} = q_{0}, p_{1} = q_{1}$ . Extending optimal transport flow matching to stochastic processes, we parameterize the vector field $\mathcal{G}_{t}$ with a neural operator $\mathcal{G}_{\theta}$ , which is optimized through the flow matching objective for SPL, + +$$ +\mathcal {L} _ {\mathrm {F M}} := \sup _ {n} \sup _ {\left\{x _ {1}, \dots , x _ {n} \right\}} \mathbb {E} _ {t \sim \mathcal {U} (0, 1), u _ {t} \sim p _ {t}} \| \mathcal {G} _ {\theta} (t, u _ {t}) - \mathcal {G} _ {t} (u _ {t}) \| ^ {2} \tag {8} +$$ + +Note that $p_t$ and $u_t$ depend on the point collocations. In the above equation, the suprema are intractable and we replace them with an expectation as a soft approximation (see Appendix A. Q5 for a detailed discussion of the approximation). Moreover, the true $\mathcal{G}_t$ is usually unknown and to address it, we derive a probability path conditioned on latent variable $z$ of the same alphabet size as the collection. Consequently, the marginal probability path $p_t(u_t)$ is a mixture of conditional probability paths $p_t(u_t|z)$ , + +$$ +p _ {t} \left(u _ {t}\right) = \int p _ {t} \left(u _ {t} \mid z\right) q (z) d z \tag {9} +$$ + +$$ +\mathcal {G} _ {t} \left(u _ {t}\right) = \mathbb {E} _ {q (z)} \left[ \frac {\mathcal {G} _ {t} \left(u _ {t} \mid z\right) p _ {t} \left(u _ {t} \mid z\right)}{p _ {t} \left(u _ {t}\right)} \right]. \tag {10} +$$ + +Given Eq. 10, the conditional flow matching (CFM) objective is defined as, + +$$ +\mathcal {L} _ {\mathrm {C F M}} := \mathbb {E} _ {n} \mathbb {E} _ {x _ {1}, \dots , x _ {n}} \mathbb {E} _ {t, q (z), p _ {t} (u _ {t} | z)} \| \mathcal {G} _ {\theta} (t, u _ {t}) - \mathcal {G} _ {t} (u _ {t} | z) \| ^ {2}. \tag {11} +$$ + +Eqs. 11 and 10 have an identical gradient for $\theta$ , i.e. $\nabla_{\theta}\mathcal{L}_{\mathrm{FM}}(\theta) = \nabla_{\theta}\mathcal{L}_{\mathrm{CFM}}(\theta)$ . Inspired by the finite dimensional developments [2], the variable $z$ is chosen as a couple $(u_0,u_1)$ from the coupling $\pi (u_0,u_1) = q(z)$ , which is achieved by minimizing the dynamic 2-Wasserstein distance, + +$$ +W _ {\mathrm {d y n}} \left(q _ {0}, q _ {1}\right) _ {2} ^ {2} = \inf _ {p _ {t}, \mathcal {G} _ {t}} \int_ {\mathbb {R} ^ {n}} \int_ {0} ^ {1} p _ {t} \left(u _ {t}\right) \| \mathcal {G} _ {t} \left(u _ {t}\right) \| ^ {2} d u _ {t} d t \tag {12} +$$ + +Under mild conditions on $\mathbb{R}^n$ , this is equivalent to the static 2-Wasserstein distance, + +$$ +W _ {\mathrm {s t a}} \left(q _ {0}, q _ {1}\right) _ {2} ^ {2} = \inf _ {\pi \in \Pi} \int_ {\mathbb {R} ^ {n} \times \mathbb {R} ^ {n}} \| u _ {1} - u _ {0} \| ^ {2} d \pi \left(u _ {0}, u _ {1}\right). \tag {13} +$$ + +Considering the class of Gaussian conditional probability paths $p_t(u_t|z) = \mathcal{N}(u_t|m_t(z),\sigma_t(z)^2 K(\{x_1,x_2,\ldots ,x_n\}))$ , with conditional flow $\Phi_t(u_0|z) = \sigma_t u_0 + m_t$ . Specially, we choose $m_{t} = tu_{1} + (1 - t)u_{0}$ and $\sigma_{t} = \sigma$ , where $\sigma >0$ is a small constant. Then we have the following closed-form expression for the corresponding vector field (full derivation provided in Appendix H) + +$$ +\mathcal {G} _ {t} \left(u _ {t} | z\right) = u _ {1} - u _ {0} \tag {14} +$$ + +With the aforementioned developments, for any collection of points, we transport a Gaussian distribution to a target distribution. The Gaussian distribution is drawn from a GP, with its covariance matrix $K(x_{1},\dots ,x_{n})$ determined by the kernel function $k(x_{i},x_{j})$ of the GP. According to Kolmogorov extension theorem (KET) [39], there exists a valid stochastic process whose finite-dimensional marginal is the pushforward distribution under $\mathcal{G}_{\theta}$ . This demonstrates that the generalization of flow matching to infinite-dimensional spaces with neural operators naturally induces the generalization of flow matching to stochastic processes. In the scenario where the limit of points covers the space, these two become equivalent. For a detailed explanation and proof, please refer to Appendix C and D. + +Our framework extends seamlessly to alternative probability paths, such as those in stochastic interpolants and rectified flow [12, 13], because the generalization to stochastic processes is decoupled from any specific path. However, we focus on the OT path in this work for two reasons: (1) ablation and scaling studies show that it accelerates inference and enables more accurate prior learning compared to independent coupling (Table 7, and Appendix P); (2) theoretically, the OT path (straight line) simplifies the Jacobian evaluation required for likelihood estimation (discussed in the next subsection), yielding greater stability and speed than arbitrary paths. + +# 3.2 Likelihood Estimation and Bayesian Universal Functional Regression + +We parameterize $\mathcal{G}_{\theta}$ with FNO [14] to ensure our model is resolution agnostic, and assume $\mathcal{G}_{\theta}$ learns a map from $\nu_{0}$ to $\nu_{1}$ , which serves as the prior. In practice, we deal with discretized evaluations of functions that may have different sampling rate and resolution. For instance, consider a function $u$ sampled from $\nu_{1}$ , observed on a collection of points $u_{1} := \{u(x_{1}), u(x_{2}), \dots, u(x_{m})\}$ ; thus we have a density function $\mathbb{P}(u_{1})$ defined on collection of points $\{x_{1}, x_{2}, \dots, x_{m}\}$ , where $\mathbb{P}(u_{1})$ is derived from measure $\nu_{1}$ . This is similar to how a multivariate Gaussian distribution can be derived from a Gaussian measure characterized by a Gaussian process. Therefore, we can rewrite Eq. 7 as: + +$$ +\log \mathbb {P} \left(u _ {1}\right) = \log \mathbb {P} \left(u _ {0}\right) - \int_ {0} ^ {1} \left(\nabla \cdot \mathcal {G} _ {\theta}\right) \left(u _ {t}\right) d t \tag {15} +$$ + +where $u_0$ is drawn from the reference Gaussian measure $\nu_0$ , which is also defined on the collection of points $\{x_1, x_2, \dots, x_m\}$ . Thus, $\mathbb{P}(u_0)$ is a multivariate Gaussian with a tractable density function. Furthermore, with the probability density function $\mathbb{P}(u_1)$ , we can evaluate the precise likelihood of any $u_1$ from $\mathbb{P}(u_1)$ via Eq. 15. However, following a similar argument to Grathwohl et al. [40], the computation of $\nabla \cdot \mathcal{G}_{\theta}(u)$ incurs a cost of $\mathcal{O}(m^2)$ where $m$ is the cardinality of $\{x_1, x_2, \dots, x_m\}$ . This quadratic time complexity renders the likelihood calculation prohibitively expensive. To address this issue, we adopt the strategy proposed in Grathwohl et al. [40], using the unbiased Skilling-Hutchinson trace estimator [41, 42] to approximate the divergence term. This technique reduces the computation cost to $\mathcal{O}(m)$ , which is the same as the cost of inference, thereby streamlining the evaluation process. The estimator is implemented as follows: + +$$ +\nabla \cdot \mathcal {G} _ {\theta} (u _ {t}) = \mathbb {E} _ {p (\varepsilon)} [ \varepsilon^ {T} \frac {\partial \mathcal {G} _ {\theta} (u , t)}{\partial u} \varepsilon ] \tag {16} +$$ + +In the unbiased trace estimator, the random variable $\varepsilon$ is characterized by $\mathbb{E}(\varepsilon) = 0$ and $\mathrm{Cov}(\varepsilon) = I$ . This estimator benefits from the optimal transport nature of the map which gives rise to a direct line. The gradient computation in Eq. 16 can be efficiently handled with reverse-mode automatic differentiation, allowing for precise estimation with arbitrary error by averaging over a sufficient number of runs, which can benefit from parallel computing of GPUs. + +With the efficient tool established for estimating the likelihood of any discretized function samples, we now turn our attention to UFR, i.e., Bayesian functional regression with a learned prior. Consider a collection of pointwise observations of the underlying unknown function drawn from $\nu_{1}$ , corrupted with Gaussian noise, denoted as $\{\widehat{u}(x_1), \widehat{u}(x_2), \ldots, \widehat{u}(x_n)\}$ or $\{\widehat{u}(x_i)\}_{i=1}^n$ . We specifically focus on Gaussian white noise characterized by $\epsilon \sim \mathcal{N}(0, \sigma^2)$ , such that $\widehat{u}(x_i) = u(x_i) + \epsilon_i$ for $i \in \{1, \dots, n\}$ (depending on the nature of the problem, the noise may also be considered as a correlated GP noise). In UFR setting, we are interested in the posterior distribution over new $m \geq n$ points that include the $n$ observation points. + +Proposition 3.1. Given noisy observations $\{\widehat{u}(x_i)\}_{i=1}^n$ , the posterior distribution is + +$$ +\log \mathbb {P} \left(\left\{u \left(x _ {i}\right) \right\} _ {i = 1} ^ {m} \mid \left\{\widehat {u} \left(x _ {i}\right) \right\} _ {i = 1} ^ {n}\right) = - \frac {\sum_ {i = 1} ^ {n} \| \widehat {u} \left(x _ {i}\right) - u \left(x _ {i}\right) \| ^ {2}}{2 \sigma^ {2}} + \log \mathbb {P} \left(\left\{u \left(x _ {i}\right) \right\} _ {i = 1} ^ {m}\right) + C \tag {17} +$$ + +Where the constant $C = -\frac{n}{2}\log (2\pi \sigma^2) - \log \mathbb{P}(\{\widehat{u} (x_i)\}_{i = 1}^n)$ . + +Proof. This is derived from Bayes' theorem, along with the translation invariance property of the Gaussian distribution, see the full proof in Appendix I + +Given this form of the posterior distribution, we adopt SGLD [4] to efficiently sample from it, and then derive statistical features of interest, e.g. mean, maximum a posteriori, and posterior uncertainty, i.e., variance, from the posterior samples. More specifically, we follow the posterior sampling strategy developed by Shi et al. [1], which, given an invertible framework, suggests SGLD sampling within the input GP space where the Gaussian measure $\nu_{0}$ is defined and Langevin dynamics is native to, and then mapping to the data function space (where data measure $\nu_{1}$ is defined). We also present a scaling study in Appendix P analyzing the variance introduced by the Hutchinson trace estimator, demonstrating its robustness and effectiveness when paired with SGLD sampling. Last, Eq 17 offers a unified view of prior and posterior sampling with flow matching models by showing that when no observations are present, the posterior collapses to the prior (up to a constant), making the posterior sampling process identical to that of the prior. + +Finally, we provide a plain-language summary of the framework to help the reader better understand the paper (with a detailed discussion available in Appendix Q), + +- OFM is an expressive, flow-based generative model that learns a prior over functions (stochastic process): a neural operator parameterizes a continuous probability flow that transports samples from a simple reference Gaussian process to data-like functions, yielding an explicit prior with a tractable density. +- It excels at functional regression by treating functions as first-class objects rather than mere pointwise values (unlike NPs), yielding predictions that are consistent across resolutions and arbitrary query sets. +- The learned flow is invertible, enabling change-of-variables for likelihoods and principled Bayesian regression, yielding calibrated uncertainty from few observations. + +# 4 Experiments + +In this section, we demonstrate the superior regression performance compared to several baselines across a variety of function datasets, including both Gaussian and highly non-Gaussian Processes. As baselines, we employ standard GP Regression [8], Deep GPs [9, 10], Conditional models [5-7], and OPFlow [1]. + +For our function datasets, we analyze: (1) Gaussian and non-Gaussian with known posterior, including 1D GPs, 2D GPs, and 1D Truncated GPs, (TGP). (2) Highly non-GPs, datasets with unknown posterior, such as those derived from Navier-Stokes equations [14], black hole dataset from expensive Monte Carlo simulation, and 2D Signed Distance Functions extracted from MNIST digits (MNIST-SDF) [43]. During regression, we assume that the prior $\mathcal{G}_{\theta}$ is always successfully trained and remains frozen. Details about the learning process for priors and experimental setup for regression are provided in the Appendix M, O. + +![](images/6650829ebd2d7069ed0264d9777f34c51e769a5d87ed6fdcbb511fa84568eb37.jpg) +(a) Ground truth + +![](images/99292a189b01b3f4603fe9b9b64905120879c9b6de26f1f983360282e793821b.jpg) +(b) OFM +Figure 3: OFM regression on GP data. (a) Ground truth GP regression with observed data and predicted samples. (b) OFM regression with observed data and predicted samples. (c) Standard deviation comparison between true GP and OFM predictions. + +![](images/8018d868994318eb74bed7c3b049719e8d99e5532abf5ce63f959b6377f61ef7.jpg) +(c) Uncertainty comparison + +1D GP data. This experiment replicates the results of classical GP regression, wherein the posterior distributions are precisely known in a closed form. The process involves generating a single new + +![](images/6f084ec87db0b8ed71701565afb0eb6ab9722625e3b3b8bd6b54b78a1288d840.jpg) +(a) Ground truth + +![](images/46ff10393b6af11ac8ba6475e18d5df023b752da95b67ce6bb6a93ff81191074.jpg) +(b) OFM + +![](images/59b7a15a257822c0ebc1877019c8d7a2b3a200a0ea5a9f6acf4832fba08d69e4.jpg) + +![](images/fca4dacd277f7de65010a79aad72e85e696f42b1048d5dc451e3ccaed494213e.jpg) +(d) GP prior + +![](images/4d2c71f70985b214d22588b6eba72d2ee930136d90894fb2983443072c86c6bd.jpg) +(c) Uncertainty of OFM +(e) Uncertainty of GP +Figure 4: OFM regression on TGP data. + +realization from the data measure $\nu_{1}$ . We then select observations at $n = 6$ randomly chosen positions, incorporating a predefined noise level. The posterior is inferred across $m = 128$ positions, which includes estimating noise-free values at the observation points. We evaluate our results with two commonly used quantities in the GP literature (1) Standardized Mean Squared Error (SMSE) that normalizes the mean squared error by the variance of the ground truth; and (2) Mean Standardized Log Loss (MSLL), originally introduced by Williams and Rasmussen [8], defined as: + +$$ +- \log p \left(\left\{u \left(x _ {i}\right) \right\} _ {i = 1} ^ {m} \mid \left\{\widehat {u} \left(x _ {i}\right) \right\} _ {i = 1} ^ {n}\right) = \frac {1}{2} \log \left(2 \pi \sigma^ {2}\right) + \frac {\left(\left\{u \left(x _ {i}\right) \right\} _ {i = 1} ^ {m} - \left\{\bar {u} \left(x _ {i}\right) \right\} _ {i = 1} ^ {m}\right) ^ {2}}{2 \sigma^ {2}} \tag {18} +$$ + +where $\{\widehat{u}(x_i)\}_{i=1}^n$ represents observations, $\{x_i\}_{i=1}^m$ , $\{u(x_i)\}_{i=1}^m$ , indicate the new positions queried, and the test data (true posterior samples). Meanwhile, $\{\bar{u}(x_i)\}_{i=1}^m$ and $\sigma^2$ are predicted mean and variances from the model. We average out SMSE and MSLL over a test dataset containing 1000 true GP posterior samples for all models. The performance of each model is detailed in Table 1. From Fig. 3, the regression with OFM matches the analytical solution well and provides realistic posterior samples. We further include GP regression tasks using more complex kernels (Gibbs and Rational Quadratic kernel), as shown in Appendix N, OFM consistently outperforms all comparative methods across all metrics. + +![](images/9f0aabb8d50c10a7fa8f3246cee0b364c15cf561319e57fb108cc58a114dba86.jpg) + +![](images/934ccdea3864b5e7435f4b7c2f617664ca50364796617d06017f9b610c48de18.jpg) + +![](images/16be0d5813129e876f0a6db59068286f5484e2fcba69e96d3077bd3cb74d40e4.jpg) + +![](images/e5bc5f7b7b098ff7a387384759863565aec6ea7c7384803b85cef48c1bc82b4d.jpg) + +![](images/1cde11d3f07107766573053c46ddcf329f84e1ac63c2b72adf5479c3174e8aa9.jpg) +(a) Observations +(b) Ground truth +(e) GP sample +Figure 5: OFM regression on black hole data with resolution $64 \times 64$ . (a) 32 random observations. (b) Ground truth sample. (c) Predicted mean from OFM. (d) One posterior sample from OFM. (e) One posterior sample from best fitted GP. (f) Predicted mean from ConvCNP. (g) One posterior sample from ConvCNP. + +![](images/4abb09ab444db556909cf045125abbd942c038c1345081f6dec9c5884bbfad2c.jpg) +(c) OFM mean +(f) ConvCNP mean + +![](images/66ccb2c3c0720049ee1acc7f39ec54b2da5115073258f24d2f6393551a8e7cf0.jpg) +(d) OFM sample +(g) ConvCNP sample + +Truncated GP data. In this experiment, we analyze the regression performance of OFM for tractable non-GP. Specifically, we work on truncated GP [1, 44], which constrains the function amplitude within a specified range. This is achieved by applying a sampling-rejection strategy on samples from the GP prior. We set the bounds of our TGP to $[-1.2, 1.2]$ and perform regression using observations only at three points, while estimating the posterior across $m = 128$ points. Subsequently, we sample 1000 true TGP posteriors from the GP prior to calculate the mean and standard deviation. Traditional metrics like MSLL and SMSE, which assume a Gaussian posterior, are not suitable for TGP. We + +Table 1: Comparison of OFM with baseline models: GP regression; OpFlow [1]; Conditional models: NP ([7]); Attentive NP ([5], ANP); Convolutional Conditional NP ([6], ConvCNP); Deep variational GP ([9], DGP); Deep Sigma Point Process ([10], DSSP); Metrics SMSE and MSLL used for 1D and 2D GP example. Mean squared error for the predicted mean $(\mu)$ and standard deviation $(\sigma)$ are used for TGP example. Performance of GP regression for 1D and 2D GP are removed (marked with $-$ ), which are taken as the ground truth. Best performance in bold. + +
Dataset → Algorithm ↓ Metric →1D GP2D GP1D TGP
SMSEMSLLSMSEMSLLμσ
GP prior----6.4·10-21.6·10-2
NP6.1·10-14.5·1001.7·10-12.1·1001.0·10-11.9·10-2
ANP5.1·10-19.8·10-11.6·10-11.1·1001.4·10-11.7·10-2
ConvCNP5.6·10-12.7·10-11.7·10-14.5·10-11.6·10-22.1·10-3
DGP4.1·10-16.8·10-21.8·1004.2·1004.9·10-11.4·10-2
DSPP4.7·10-16.5·1001.9·10-16.6·1001.1·10-21.3·10-2
OpFlow5.0·10-12.0·10-11.4·10-11.1·10-11.3·10-23.9·10-3
OFM(Ours)4.1·10-15.5·10-21.3·10-11.6·10-15.2·10-39.5·10-4
+ +evaluate the performance using the mean squared error for both the predicted mean and standard deviation. The results are reported in Table. 1, and illustrated in Fig. 4. OFM accurately learns the specified bounds and provides accurate estimations of mean and standard deviation, along with realistic posterior samples. In contrast, directly applying GP regression exceeds the bounds and yields unrealistic posterior samples. + +2D GP data. Similar to the 1D GP example, we extend our regression analysis to 2D GP data. As shown in Fig. 6 and detailed in Table 1, OFM provide accurate posterior estimation. The relative error shown in Fig. 6 is the absolute error normalized by the maximum absolute value of the mean prediction derived from the ground truth GP regression. + +Navier-Stokes, Black hole and MNIST-SDF datasets. We collected a 2D Navier-Stokes dataset and applied OFM for the regression. Unlike the GP experiments, where MSLL and SMSE score serve as standard benchmarks, evaluating the performance of models on general non-GPs presents a significant challenge due to the difficulty of determining the true posterior and lack of benchmarks. Moreover, the evidence term in the posterior (Eq. 41) is intractable, so the likelihood cannot serve as a meaningful evaluation metric in our setting. A detailed discussion is provided in Appendix A. + +We present the predicted mean and a posterior sample in Fig 2 for visual comparison with the ground truth. The predicted mean and the posterior sample are closely aligned with the ground truth. In contrast, traditional GP regression and NP models failed to accurately capture the dynamics of the Navier-Stokes data. In Fig. 5, we conduct a similar analysis using a simulated black hole dataset. Here, OFM provides a much more realistic mean and posterior sample that captures the density and swirling patterns of the black hole. Once again, GP regression and NP fail to capture these key statistics. We observe similar outcomes when applying OFM to the MNIST-SDF example (Fig 8), where OFM correctly recognizes the number "7" while GP regression does not. + +# 5 Conclusion + +In this paper, we proposed Operator Flow Matching (OFM) for stochastic process learning, which generalizes flow matching models to infinite-dimensional space and stochastic process with optimal transport path. OFM efficiently computes the probability density for any finite collection of points and supports mathematically tractable functional regression. We extensively tested OFM across a diverse range of datasets, including those with closed-form GP and non-GP data, as well as highly non-GP such as Navier-Stokes and black hole data. In comparative evaluations, OFM consistently outperformed all baseline models, establishing new standards in stochastic process learning and regression. + +Despite SOTA accuracy, our method is presently limited to low-dimensional domains and demands larger datasets and more compute than GP-based baselines. see Appendix A, O and Q for details. Python code available at https://github.com/yzshi5/SPL_OFM + +# Acknowledgments + +This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Science Foundations for Energy Earthshot under Award Number DE-SC0024705. ZER is supported by a fellowship from the David and Lucile Packard Foundation. We also thank Charles Gammie, Ben Prather, Abhishek Joshi, Vedant Dhruv, and Chi-kwan Chan for providing the black hole simulations. + +# References + +[1] Yaozhong Shi, Angela F. Gao, Zachary E. Ross, and Kamyar Azizzadenesheli. 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URL http://arxiv.org/abs/2206.00927. arXiv:2206.00927 [cs]. + +# A Potential questions and answers + +Our method combines many vital fields, including + +Operator learning +- Flow-based generative models (flow matching); Optimal transport in function space +- Bayesian uncertainty; Gradient Markov chain Monte Carlo, SGLD; Stochastic trace estimation +- Gaussian processes; Stochastic processes (Kolmogorov extension theorem) + +Because readers will have diverse backgrounds and expertise, we provide some potential questions and answers to enhance the clarity and highlight our contributions. + +Q1: Why is it hard to obtain the true prior and posterior of an unknown stochastic process, and what would we gain if we could? + +A1: Generalizing GP regression to arbitrary stochastic process regression has remained a long-standing challenge that has kept many researchers busy for decades. The core difficulties are twofold: first, learning a general stochastic-process prior from historical data with an expressive enough model, and second, deriving both the exact posterior distribution and an efficient sampling scheme. + +Prior work falls into three main categories—GPs, deep GPs, and conditional models (NP family). Only the classical GP can fully characterize a stochastic process, and then only when the exact GP prior (mean function, covariance kernel, all hyper-parameters, and observation-noise variance) is already known. Even fitting one GP to data generated by another GP with unknown parameters rarely yields the true posterior. Deep GPs and NPs possess stronger representational power compared to GP, but representational power $\neq$ exactness: their optimization relies on approximate posterior and DO NOT produce a closed-form "true posterior." + +If the true prior and posterior were available, we would gain near-complete control over general stochastic process, enabling perfect uncertainty quantification and Bayes-optimal decisions, which has numerous applications in finance market, science and engineering problems + +Q2: How does OFM differ from standard finite-dimensional flow matching? What does it mean to generalize flow matching to stochastic processes, and how does extending a generative model to a function space differ from extending it to a full stochastic process? + +A2: A standard flow matching model learns a transport map between two distributions defined on a fixed grid (e.g., a pixel lattice). Consequently, it can only generate samples at that specific resolution. In contrast, OFM learns a map between two stochastic processes in function space. This endows it with several properties that standard flow matching lacks: + +(i) Resolution agnosticism. OFM learns a transport map between two distributions defined on any given collection of points, without regard for the number of points, or their locations in the domain. This enables capabilities like zero-shot generation without retraining. +(ii) Stochastic process consistency. OFM respects the metric of the underlying space, ensuring that points close to each other in the input domain have appropriately correlated values in the output distribution. This is part of learning a valid stochastic process, which also satisfies crucial theoretical properties like the Kolmogorov extension theorem (i.e., consistency under marginalization) +(iii) Convergence to a continuous function. As the collection of query points becomes denser within the domain, the output of OFM converges to the underlying continuous function. +(iv) Backwards compatibility. When evaluated on a fixed set of points, OFM recovers the behavior of a conventional (finite-dimensional) flow matching up to a linear transformation. + +Extending a generative model (e.g., flow matching) to an infinite-dimensional function space and extending it to a full stochastic process are related but distinct goals. The first concerns generating function values at any finite collection of points; Lebesgue measure and Kolmogorov consistency come into play, but the focus remains on finite dimensional marginals. The second addresses the + +distribution of the entire function itself, requiring a probability measure over an infinite-dimensional space. + +Because of this distinction, lifting a model merely to a function space is strictly weaker. For example, GANO [22] extends GANs to function space but not to a stochastic process, as it cannot describe the joint law of function values at arbitrary query sets. In OFM, we bridge these two viewpoints for flow matching with neural operators: by exploiting the discretization-convergence property of neural operators and the invertibility of flow matching, we extend the method to both function spaces and stochastic processes. See Appendix C and D for details. + +Q3: These methods may appear similar to image inpainting or restoration with flow- or diffusion-based models; could you elaborate on the differences? + +A3: Current inpainting with flow/diffusion models treats the task as an inverse problem in a finite-dimensional setting: one learns to reconstruct missing pixels on a fixed resolution grid, effectively solving a regression problem at a predetermined set of points. By contrast, OFM operates directly on functions and is resolution-agnostic. Given the same partial observations, our model can predict the entire function at any collection of query points, coarse or fine. + +Moreover, OFM delivers principled uncertainty: its posterior quantifies the full distribution of possible completions, whereas finite-dimensional inpainting methods typically rely on an ensemble of visually plausible samples whose variability is not a calibrated measure of uncertainty. Finally, OFM remains effective even when observations are extremely sparse. (e.g $0.7\%$ of total observations) --a regime in which grid-based inpainting approaches struggle. + +Q4 : Could you elaborate further on the connections to related work? The approach appears to intersect with several studies, including OPFlow [1], COT-FM [45], OT-CFM [2], and others. + +A4: Because our work combines multiple fields, it naturally has connections with many other studies. Readers are referred to the appendix Q for a detailed discussion. + +Q5: In Eq 11. What is the argument of replacing the superma in Eq 8 with expectations? When is this valid? + +A5: The supremum in Eq. 8 represents a worst-case error over the entire function space, which is computationally intractable to optimize directly. We therefore relax this hard constraint by replacing the supremum with an expectation, as formulated in Eq 11. This is a common empirical consideration. Instead of minimizing the worst-case error, our objective becomes minimizing the tractable average-case error across the distribution. Minimizing the error on average provides a strong practical incentive for the model to perform well across the entire function space, thereby effectively reducing the worst-case error. Such replacement is always valid under the weaker goal. The validity of this approach as a tractable proxy is further confirmed by our empirical results, which show it successfully guides the model to learn the intended functional mapping (Appendix M). + +Q6: Why not use log-likelihood as the evaluation metric for non-GP regression tasks when comparing to NP models? And why not include more recent conditional model baselines, such as NDP [34] + +A6: There are several reasons that likelihood as an evaluation metric is not relevant in our case. First, as shown in Eq 41, there is an evidence term, which is a constant, in the posterior distribution in OFM framework. The constant evidence term is intractable but does not contribute to MAP estimation, mean estimation, and posterior sample in general. + +Second, even if we can compute the evidence term, we still cannot make the comparison. The graphical model in the NP is such that the conditional model is trained using the MLE principle and the learned likelihood model dependent. The posterior in OFM provides the posterior using the Bayes rule, utilizes a different model, and has a different graphical model. These quantities are not directly relevant to be compared. + +Third, computing the true posterior for a general stochastic process is a known challenging problem. Due to the complexity nature of the problem, there doesn't exist a well-recognized metric. The + +evaluation of quality of posterior performance requires domain knowledge from experts and varies case by case, which is an exciting research direction. + +For baseline models, We already include a broad set of SOTA baselines. On the operator-learning side, the latest OPFlow [1] (2024) is covered. For deep GPs, we adopt Doubly Stochastic Variational Deep GPs [9] and the Deep Sigma-Point Process [10], both widely recognized and backed by publicly reproducible code. + +Within the NP family, ConvCNP [6] remains a standard benchmark, and we include two additional NP variants. Our baselines are limited to models with publicly reproducible implementations. Although we attempted to add the newer NDP [34], we encountered significant reproducibility issues with the authors' code—an obstacle reported by others as well. Instead, we offer a detailed theoretical comparison between OFM and NDP in Appendix Q. + +Q7: What do you mean by an "exact posterior" and a practical solution? How can I apply the model to my own tasks, and what do I need to prepare? + +A7: For functional regression, posterior error has two sources for any method: (1) formulation error—the gap between the model's theoretical posterior and the true one—and (2) approximation error from finite data, limited capacity, and optimization. Deep GPs rely on variational inference, optimizing an ELBO and thereby introducing a formulation gap: the posterior is only an approximation. NPs make even stronger simplifying assumptions, and doesn't provide the true posterior. All models, including OFM, incur approximation error, but the discussion above concerns only formulation error. + +A practical solution means an expressive backbone that can learn a complex stochastic-process prior and a posteriori-sampling routine whose runtime and memory footprint remain acceptable for typical users; see Appendix O for quantitative details. + +Using the model is straightforward. OFM offers GP-style regression for non-GP tasks: given noisy observations, it returns the posterior at arbitrary query points. The key difference is that a GP prior is fixed by a hand-tuned kernel, whereas OFM learns the stochastic-process prior directly from data. For details on prior learning and SGLD-based posterior sampling, see Appendices M and L. + +# B Background: Flow Matching, Gaussian Measures on Function Spaces, and the Cameron-Martin Theorem + +In this section, we provide essential background and high-level intuitive explanations of the topics involved in this paper for readers. + +Flow Matching. Flow matching is a state-of-the-art generative paradigm that learns a time-dependent velocity field whose ODE transports samples from a simple reference (e.g., Gaussian) to the target data distribution, yielding an unbiased, simulation-free training objective that directly regresses ground-truth velocities along probability paths. + +The approach is tightly linked to physics via the continuity equation, and—when cast as a continuous normalizing flow—provides a (deterministically) invertible transformation with tractable likelihoods, while also admitting stochastic variants when desired. In practice it matches diffusion-level quality with faster training and sampling (often few- or even single-step generation) and excellent scalability, which is why it has been adopted in challenging domains such as large-scale video generation, in-context image generation, and protein ensemble generation [46, 47]. We therefore use flow matching as the foundation for prior learning, leveraging its simple training objective, physics-grounded structure, and strong empirical performance. + +Gaussian measures. A probability measure $\mu$ on a separable Hilbert space $\mathcal{H}$ is Gaussian if for any finite collection of vectors $\{h_1, \ldots, h_n\} \subset \mathcal{H}$ , the random vector $(\langle X, h_1 \rangle, \ldots, \langle X, h_n \rangle)$ has a multivariate Gaussian distribution, where $X \sim \mu$ . The family $\{\langle X, h : h \in \mathcal{H}\}$ is therefore a Gaussian process indexed by $\mathcal{H}$ , with mean $h \mapsto \langle m, h \rangle$ for some $m \in \mathcal{H}$ . Conversely, if a Gaussian process $\{X(t) : t \in T\}$ has sample paths that almost surely belong to a function space $\mathcal{H}$ (e.g., $C(T)$ or $L^2(T)$ ), then the law of the random path $t \mapsto X(t)$ is a Gaussian measure on $\mathcal{H}$ . In short: a Gaussian measure is the path-space law of a GP, and a GP is the collection of linear probes of a Gaussian measure. + +Cameron-Martin space and theorem. Let $\mu$ be a Gaussian measure on a separable Hilbert space $\mathcal{H}$ with mean $m\in \mathcal{H}$ and covariance operator $C:\mathcal{H}\to \mathcal{H}$ (self-adjoint, positive, trace-class). The Cameron-Martin space (the RKHS associated with $\mu$ ) is + +$$ +\mathcal {H} _ {\mu} = \overline {{\operatorname {R a n g e} (C ^ {1 / 2})}} \subseteq \mathcal {H}, +$$ + +equipped with the inner product $\langle u,v\rangle_{\mathcal{H}_{\mu}}\coloneqq \langle C^{-1 / 2}u,C^{-1 / 2}v\rangle_{\mathcal{H}}$ on $\mathrm{Range}(C^{1 / 2})$ , extended by completion. With this choice, the inclusion $\mathcal{H}_{\mu}\hookrightarrow \mathcal{H}$ is continuous. For $h\in \mathcal{H}$ , write $\mu_h(A)\coloneqq \mu (A - h)$ for the translate. The Cameron-Martin theorem states: + +(i) If $h \in \mathcal{H}_{\mu}$ , then $\mu_h$ is absolutely continuous with respect to $\mu$ (in fact, $\mu_h$ and $\mu$ are equivalent), with + +$$ +\frac {d \mu_ {h}}{d \mu} (x) = \exp \Big (\big \langle C ^ {- 1 / 2} h, C ^ {- 1 / 2} (x - m) \big \rangle_ {\mathcal {H}} - \frac {1}{2} \| h \| _ {\mathcal {H} _ {\mu}} ^ {2} \Big) \quad \mathrm {f o r} \mu \text {- a . e .} x, +$$ + +where the pairing is understood in the Paley-Wiener sense (which in finite dimensions reduces to $\langle h, C^{-1}(x - m) \rangle$ ). + +(ii) If $h \notin \mathcal{H}_{\mu}$ , then $\mu_h \perp \mu$ (mutually singular). + +In finite dimensions this reduces to the classical mean-shift formula for $\mathcal{N}(m,\Sigma)$ with $C = \Sigma$ , and $\| h\|_{\mathcal{H}_{\mu}}^{2} = \langle h,\Sigma^{-1}h\rangle$ . Thus, $\mathcal{H}_{\mu}$ pinpoints exactly the directions along which a Gaussian measure can be translated while remaining (mutually) absolutely continuous. + +# C Stochastic process learning + +Let $(\Omega, \mathcal{F}, P)$ denote a probability space and let $(\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d))$ denote a measurable space where $\mathcal{B}(\mathbb{R})$ is the Borel space. Following the standard definition of stochastic processes (Brémaud [38], Chapter 5.1), a stochastic process $\mathcal{P}$ on a domain $D$ is a collection of $\mathbb{R}^d$ -valued random variables indexed by members of $D$ , i.e., + +$$ +\{a (x): x \in D \} +$$ + +jointly following the probability law $P$ . In the special case of Gaussian processes, e.g., Wiener process, following the Gaussian law for $P$ , for any collection points $\{x_1, x_2, \ldots, x_n\}$ , the random variables $\{a(x_1), a(x_2), \ldots, a(x_n)\}$ are jointly Gaussian, resulting in a function $a$ to be drawn from a GP. We need to emphasize, $\{a(x_1), a(x_2), \ldots, a(x_n)\}$ is a collection of random variables (random vector) equipped with Lebesgue measure, and represents a discretized observation of one continuous function $a$ . In practice, the joint probability distribution of the collection of the random variables is unknown a priori, and needs to be learned. + +In SPL, one way we suggest is to learn an invertible operator $\mathcal{T}$ that maps a base stochastic process $\mathcal{P}$ to another stochastic process $\mathcal{Q}$ that represents the data via discretization convergence theorem (see Appendix D). That is, for any collection of points $\{x_1,x_2,\ldots ,x_n\}$ , and for any $n$ , the operator $\mathcal{T}$ maps the law on $\{a(x_1),a(x_2),\ldots ,a(x_n)\}$ to $\{u(x_1),u(x_2),\ldots ,u(x_n)\}$ and vice versa for the inverse $\mathcal{T}^{-1}$ , where $u(x)$ is a pointwise evaluation of function data sample, i.e., + +$$ +\left\{u \left(x _ {1}\right), u \left(x _ {2}\right), \dots , u \left(x _ {n}\right) \right\} = \mathcal {T} \left(\left\{a \left(x _ {1}\right), a \left(x _ {2}\right), \dots , a \left(x _ {n}\right) \right\}\right) +$$ + +Then, the probability of $\{u(x_1), u(x_2), \ldots, u(x_n)\}$ , at evaluation points $\{x_1, x_2, \ldots, x_n\}$ , for any $n$ and collection of points on $D$ is given by, + +$$ +\mathbb {P} \left(\left\{u \left(x _ {1}\right), u \left(x _ {2}\right), \dots , u \left(x _ {n}\right) \right\}\right) = \mathbf {J} \mathcal {T} \Big | _ {\left\{a \left(x _ {1}\right), a \left(x _ {2}\right), \dots , a \left(x _ {n}\right) \right\}} \mathbb {P} \left(\left\{a \left(x _ {1}\right), a \left(x _ {2}\right), \dots , a \left(x _ {n}\right) \right\}\right) +$$ + +where with abuse of notation $\mathbb{P}(u(x))$ denotes the density of $u(x)$ at point $x$ , same for $\mathbb{P}(a(x))$ , and similarly, following the notation in Theorem 11.1 of Villani [48], $\mathbf{J}\mathcal{T}\bigg|_{\{a(x_1), a(x_2), \ldots, a(x_n)\}}$ is the absolute value of the Jacobian determinant of the map from the random vector $\{a(x_1), a(x_2), \ldots, a(x_n)\}$ at points $\{x_1, x_2, \ldots, x_n\}$ to the random vector $\{u(x_1), u(x_2), \ldots, u(x_n)\}$ via inverse operator $\mathcal{T}^{-1}$ . We further show that the pushedforward $\mathcal{Q}$ is indeed a valid stochastic process via Kolmogorov Extension Theorem (KET) [39] with a proof provided in Appendix. D. In SPL, we aim to learn a neural operator $\mathcal{T}_{\theta}$ such that the resulting $\mathcal{Q}$ matches the data process under the true $\mathcal{T}$ . + +# D Model stochastic process with infinite-dimensional flow matching via Kolmogorov Extension Theorem + +In operator learning, neural operators [3, 14, 15] are typically designed to map an input function to an output function. When the input function is provided at a specific discretization (e.g., a set of points with their corresponding values), the model processes this discretized input as a collection of points and their values. Traditionally, in operator learning, this process is seen as an approximation of the operator's application to the underlying continuous function, where the discretization introduces approximation errors. Thus, the input is conceptually still treated as a function. + +Moreover, the application of the operator to a collection of points is well-defined, and, by the discretization convergence theorem, as the number of points increases, this operation converges to a well-defined mapping. In this paper, leveraging these properties, we adopt a different perspective as described in the introduction. We extend neural operators to define explicit maps between collections of points. In this framework, the input is not the abstract function itself but rather a collection of points and their associated values. Importantly, this mapping remains well-defined regardless of the number of points in the collection and, by the discretization convergence theorem, converges to a unique mapping as the point collection approaches the underlying continuous function. + +Next, we show that given an invertible operator $\mathcal{T}$ and a valid stochastic process $\mathcal{P}$ whose finite dimensional marginal is $\mathbb{P}(\{a(x_1),a(x_2),\ldots ,a(x_n)\})$ , there exist a valid stochastic process $\mathcal{Q}$ with finite-dimensional marginal $\{u(x_{1}),u(x_{2}),\ldots ,u(x_{n})\}$ . + +Once again, as defined in Section C + +$$ +\{u (x _ {1}), u (x _ {2}), \dots , u (x _ {n}) \} = \mathcal {T} (\{a (x _ {1}), a (x _ {2}), \dots , a (x _ {n}) \}) +$$ + +Then, the probability of $\{u(x_1), u(x_2), \ldots, u(x_n)\}$ , at evaluation points $\{x_1, x_2, \ldots, x_n\}$ , for any $n$ and collection of points on $D$ is given by, + +$$ +\left. \mathbb {P} \left(\left\{u \left(x _ {1}\right), u \left(x _ {2}\right), \dots , u \left(x _ {n}\right) \right\}\right) = \mathbf {J} \mathcal {T} \right| _ {\left\{a \left(x _ {1}\right), a \left(x _ {2}\right), \dots , a \left(x _ {n}\right) \right\}} \mathbb {P} \left(\left\{a \left(x _ {1}\right), a \left(x _ {2}\right), \dots , a \left(x _ {n}\right) \right\}\right) \tag {19} +$$ + +where with abuse of notation $\mathbb{P}(u(x))$ denotes the density of $u(x)$ at point $x$ , same for $\mathbb{P}(a(x))$ , and similarly, following the notation in Theorem 11.1 of Villani [48], $\mathbf{J}\mathcal{T}\bigg|_{\{a(x_1), a(x_2), \ldots, a(x_n)\}}$ is the absolute value of the Jacobian determinant of the map from the random vector Jacobian of the map from the random vector $\{a(x_1), a(x_2), \ldots, a(x_n)\}$ at points $\{x_1, x_2, \ldots, x_n\}$ to the random vector $\{u(x_1), u(x_2), \ldots, u(x_n)\}$ via inverse operator $\mathcal{T}^{-1}$ . We should notice Eq. 19 represents the changes of variables between two random vectors, with Lebesgue measure involved. The connection between the finite-dimensional marginal (equipped with Lebesgue measure) and the probability measure of a stochastic process in infinite-dimensional space is described by Kolmogorov Extension theorem (KET) [39], which assures that if all finite-dimensional distributions (i.e., distributions of function at finite collection of points) are consistent, then a stochastic process exists that matches finite-dimensional distributions. + +Formally, according to KET, to establish that a valid stochastic process $\mathcal{Q}$ , which has $\mathbb{P}(\{u(x_1), u(x_2), \ldots, u(x_n)\})$ as its finite dimensional distributions, it is essential to demonstrate that such a joint distribution satisfies the following two consistency properties: + +Permutation invariance. For any permutation $\pi$ of $\{1, \dots, n\}$ , the joint distribution should remain invariant when elements of $\{x_1, \dots, x_n\}$ are permuted, such that + +$$ +\mathbb {P} \left(\left\{u \left(x _ {1}\right), u \left(x _ {2}\right), \dots , u \left(x _ {n}\right) \right\}\right) = \mathbb {P} \left(\left\{u \left(x _ {\pi (1)}\right), u \left(x _ {\pi (2)}\right), \dots , u \left(x _ {\pi (n)}\right) \right\}\right) \tag {20} +$$ + +Marginal Consistency. This principle specifies that if a portion of the set is marginalized, the marginal distribution will still align with the distribution defined on the original set, such that for $m \geq n$ + +$$ +\mathbb {P} \left(\left\{u \left(x _ {1}\right), u \left(x _ {2}\right), \dots , u \left(x _ {n}\right) \right\}\right) = \int \mathbb {P} \left(\left\{u \left(x _ {1}\right), u \left(x _ {2}\right), \dots , u \left(x _ {m}\right) \right\}\right) d u \left(x _ {n + 1}\right) \dots d u \left(x _ {m}\right) \tag {21} +$$ + +The permutation invariance property is naturally upheld when utilizing operator, as there is no inherent order among the elements in the set $\{x_{1}, x_{2}, \ldots, x_{n}\}$ . Furthermore, the marginal + +consistency property is also maintained due to the definition of operator $\mathcal{T}$ (see Eq. 19), which ensures that $\mathbb{P}(\{u(x_1), u(x_2), \ldots, u(x_n)\})$ is closed under marginalization. This is because $\mathbb{P}(\{a(x_1), a(x_2), \ldots, a(x_n)\})$ is closed under marginalization, which fully determines $\mathbb{P}(\{u(x_1), u(x_2), \ldots, u(x_n)\})$ through the Jacobian. While verifying that $\mathcal{Q}$ constitutes a valid induced stochastic process is straightforward given the $\mathcal{T}$ , approximating the $\mathcal{T}$ with a neural operator is non-trivial and depends highly on the model used (related to expressiveness). For instance, in Transforming GP [33], the authors employ a marginal normalizing flow, which acts as a point-wise operator to transform values from a GP to another. Consequently, the induced Jacobian is a diagonal matrix. More recently, OpFlow [1] introduces an invertible neural operator by generalizing RealNVP to function space, which induces a triangular Jacobian matrix. In our work, we extend this framework to a more comprehensive case: a diffeomorphism. Here, the induced Jacobian is a full-rank matrix and is not necessarily triangular or diagonal, the determinant of the Jacobian for any collection of points is calculated through Eq 15. + +Last, we want to clarify the connection between the notions of operator $\mathcal{T}$ and operator $\mathcal{G}$ throughout this paper. The operator $\mathcal{T}$ is the $\Phi_t$ (a diffeomorphism) defined in Eq. 5, which is the integral of $\mathcal{G}$ over time interval [0, 1]. Due to the nature of an ODE system, $\mathcal{T}$ is invertible. However, $\mathcal{G}$ is not necessary invertible, which enables us to parameterize it with a classical neural operator, like FNO [14]. + +# E Universal Functional Regression + +UFR is concerned with Bayesian regression on function spaces [1], where it can be used to infer the posterior of an unknown function on a domain $D$ from a collection of pointwise observations. The observations are often corrupted with noise of variance $\sigma^2$ , denoted as $\{\widehat{u}(x_1), \widehat{u}(x_2), \ldots, \widehat{u}(x_n)\}$ or $\{\widehat{u}(x_i)\}_{i=1}^n$ . More specifically, for $m \geq n$ points at which the function is to be inferred, + +$$ +\mathbb {P} \left(\left\{u (x _ {1}), u (x _ {2}), \dots , u (x _ {m}) \right\} \mid \left\{\widehat {u} (x _ {1}), \widehat {u} (x _ {2}), \dots , \widehat {u} (x _ {n}) \right\}\right) +$$ + +Note that when the prior over the function space is Gaussian, UFR reduces to the celebrated GP regression. Following Bayes rule, and maps between stochastic processes, we obtain the log posterior as follows, + +$$ +\begin{array}{l} \log \mathbb {P} \left(\{u (x _ {i}) \} _ {i = 1} ^ {m} \Big | \{\widehat {u} (x _ {i}) \} _ {i = 1} ^ {n}\right) = - \frac {1}{2} \sum_ {i} ^ {n} \frac {(\widehat {u} (x _ {i}) - u (x _ {i})) ^ {2}}{\sigma^ {2}} - n \log (\sigma) - \frac {n}{2} \log (2 \pi) \\ + \log \mathbb {P} \left(\{u (x _ {i}) \} _ {i = 1} ^ {m}\right) - \log \mathbb {P} \left(\{\widehat {u} (x _ {i}) \} _ {i = 1} ^ {n}\right) \\ \end{array} +$$ + +This equality holds for any collection of points. It is worth noting that the posterior is exact up to constants, i.e., the second, third, and last terms are constant. Therefore, they do not contribute in MAP estimation, mean estimation, and functional regression in general, and there is no need to compute them. + +# F Marginal (dynamic) optimal-transport flow matching in function space via optimal coupling and dynamic Kantorovich formulation + +Consider a joint probability measure $\pi(\nu_0, \nu_1)$ on $\mathcal{H} \times \mathcal{H}$ , where the reference measure $\nu_0$ , is chosen as a Gaussian measure, whose absolute continuity is well-studied [49]. We characterize $\nu_0$ by a GP with trace-class covariance operator. e.g. $\nu_0 = \mathcal{N}(m_0, C_0)$ , where $m_0$ is the mean, $C_0$ is the covariance operator. With the joint measure $\pi(\nu_0, \nu_1)$ , we sample a function pair $z := (h_0, h_1)$ . + +Assuming $\nu_{1}$ has full support on the Cameron-Martin space associated with $\nu_{0}$ (following the convention of literatures [26, 29]), we construct a conditional probability measure $\mu_t(\cdot |z)$ as a Gaussian measure with trace-class covariance operator and small operator norm to approximate Dirac measures in the sense of weak convergence. Such that, at $t = 0$ and $t = 1$ , $\mu_t(\cdot |z)$ is a centered around $h_0, h_1$ , approximating $\delta_{h_0}, \delta_{h_1}$ respectively; Subsequently, we can construct a new marginal probability measure by mixing these approximated Dirac measures: + +$$ +\mu_ {t} (A) = \int \mu_ {t} (A | z) d \pi (z), \forall A \in \mathcal {B} (\mathcal {H}) \tag {22} +$$ + +Due to $d\pi (z)$ being always positive, the conditional probability measure (Dirac measure approximated by Gaussian measure) is absolutely continuous with respect to $\mu_t$ . Eq. 22 indicates that $\mu_0 = \int \delta_{h_0}d\pi (z)\approx \nu_0$ , and $\mu_{1} = \int \delta_{h_{1}}d\pi (z)\approx \nu_{1}$ . This formulation suggests that $\mu_0,\mu_1$ represent convolutions of $\nu_{0},\nu_{1}$ with Gaussian measures. For a more detailed discussion on convolution with Gaussian measures, we refer the readers to Appendix B.1 of Lim et al. [26]. + +Please note, dirac measure is not a necessary condition for Eq. 22; the only constraint is the boundary conditions. One viable choice for the conditional measure is a Gaussian measure with a small operator norm, which (in fact) approximates the Dirac measure in the weak convergence sense. Alternative probability path, as described in Lipman et al. [11], Albergo and Vanden-Eijnden [12], Liu et al. [13] are equally valid. + +Suppose $\int_0^1\int_{\mathcal{H}}\int_{\mathcal{H}\times \mathcal{H}}\| \mathcal{G}_t(h|z)\| d\mu_t(h|z)d\pi (z)dt$ is finite to guarantee the vector field is sufficiently regular (Lipschitz continuity), where $\mathcal{G}_t(\cdot |z)$ is the conditional vector field. Under this condition, the vector field that generates $\mu_t$ as specified in Eq. 22 and Eq. 3 can be expanded as follows: + +$$ +\mathcal {G} _ {t} (h) = \int_ {\mathcal {H} \times \mathcal {H}} \mathcal {G} _ {t} (h | z) \frac {d \mu_ {t} (\cdot | z)}{d \mu_ {t}} (h) d \pi (z) \tag {23} +$$ + +Eq. 23 is an extension of the Theorem 1 as detailed in Kerrigan et al. [29], and we provide the derivation in Appendix G. We note that $\mu_t(\cdot |z)$ is a Gaussian measure and can be expressed as $\mu_t(\cdot |z) = \mathcal{N}(m_t,C_t)$ , with mean $m_{t}$ and trace-class covariance operator $C_t$ . Inspired by Tong et al. [2], we choose $m_{t}$ and $C_t$ to have the following forms: + +$$ +m _ {t} = t \cdot h _ {1} + (1 - t) \cdot h _ {0} \tag {24} +$$ + +$$ +C _ {t} = \sigma_ {\min } ^ {2} C _ {0} \tag {25} +$$ + +where $C_0$ is the same Gaussian covariance operator defined for $\nu_{0}$ and $\sigma_{\mathrm{min}}$ is a small constant. Further, similar to finite-dimensional flow matching, we only consider the simplest vector field that applies a canonical transformation for Gaussian measures, such that the flow has the form: $\Phi_t(h_0|z) = m_t + \sigma_{\mathrm{min}}h_0\approx t\cdot h_1 + (1 - t)\cdot h_0.$ From Eq. 1, we can get $\mathcal{G}_t(h|z) = h_1 - h_0,$ indicating $\mathcal{G}_t(h|z)$ is independent of the time $t$ and the path from $h_0$ to $h_1$ is a direct, straight line. Equipped with well-constructed conditional vector field and probability measures, we can train a neural operator $\mathcal{G}_{\theta}$ with the conditional flow matching loss + +$$ +\mathcal {L} _ {\mathrm {C F M}} ^ {\dagger} = \mathbb {E} _ {t \sim \mathcal {U} [ 0, 1 ], h \sim \mu_ {t}, z \sim \pi (\nu_ {0}, \nu_ {1})} \| \mathcal {G} _ {\theta} (t, h) - \mathcal {G} _ {t} (h | z) \| ^ {2} \tag {26} +$$ + +Next, we explore how to approximate the true optimal transport plan from optimal coupling of the joint measure $\pi (\nu_0,\nu_1)$ . A common way for measuring the distance between two probability measures is 2-Wasserstein distance, which a special case of static Kantorovich formulation [50]. The static 2-Wasserstein distance is defined as follows + +$$ +W _ {\mathrm {s t a}} \left(\nu_ {0}, \nu_ {1}\right) _ {2} ^ {2} = \inf _ {\pi \in \Pi} \int_ {\mathcal {H} \times \mathcal {H}} \| h _ {0} - h _ {1} \| ^ {2} d \pi \left(h _ {0}, h _ {1}\right) \tag {27} +$$ + +In the ODE framework, we also care about the dynamic form of the 2-Wasserstein distance to estimate the cost along the transport trajectory, which also is a special case of dynamic Kantorovich formulation [37]. + +$$ +W _ {\mathrm {d y n}} \left(\nu_ {0}, \nu_ {1}\right) _ {2} ^ {2} = \inf _ {\mu_ {t}, \mathcal {G} _ {t}} \int_ {\mathcal {H}} \int_ {0} ^ {1} \| \mathcal {G} _ {t} (h) \| ^ {2} d \mu_ {t} (h) d t \tag {28} +$$ + +Within the OFM framework, the marginal probability measure is a sum of Dirac measures as described in Eq. 22, and we selected $\nu_{0}$ as a Gaussian measure and assumed $\nu_{1}$ has full support on the Cameron-Martin space associated with $\nu_{0}$ . Furthermore, the cost function of 2-Wasserstein distance is squared $L^2$ norm, which is continuous by nature. According to Theorem 4.3 and Lemma 4.4 of Chizat et al. [37], $W_{\mathrm{sta}} = W_{\mathrm{dyn}}$ for our specifically constructed $\mu_t$ and $\mathcal{G}_t$ in the sense of weak convergence. Therefore, to get the dynamic optimal transport plan, we only need to find a joint measure $\pi (\nu_0,\nu_1)$ that achieves the infimum in Eq. 27. In practice, we use a minibatch approximation of optimal coupling between $\nu_{0}$ and $\nu_{1}$ . The above approach extends the dynamic (marginal) optimal transport framework of [2] to infinite-dimensional function space. The related work of Kerrigan et al. [45] addresses a similar problem, but from a different perspective. For a detailed comparison, please refer to Appendix Q. + +# G Derivation of Eq. 23 + +In this part, we show the derivation of Eq. 23, which extends Theorem 1 of Kerrigan et al. [29]. The problem setting is given continuity equation and its weak form: + +$$ +\int_ {0} ^ {1} \int_ {\mathcal {H}} \frac {\partial \varphi (h , t)}{\partial t} + \left\langle \mathcal {G} _ {t} (h), \nabla_ {h} \varphi (h, t) \right\rangle d \mu_ {t} (h) d t = 0, \quad \forall \varphi \in \operatorname {C y l} (\mathcal {H} \times [ 0, 1 ]) \tag {29} +$$ + +we want to derive the following form of the conditional vector field under absolute continuity assumption and other mild conditions, where $z \coloneqq (h_0, h_1) \in \mathcal{H} \times \mathcal{H}$ . + +$$ +\mathcal {G} _ {t} (h) = \int_ {\mathcal {H} \times \mathcal {H}} \mathcal {G} _ {t} (h | z) \frac {d \mu_ {t} (\cdot | z)}{d \mu_ {t}} (h) d \pi (z) \tag {30} +$$ + +First, $\int_0^1\int_{\mathcal{H}}\frac{\partial\varphi(h,t)}{\partial t} d\mu_t(h)dt = \int_0^1\int_{\mathcal{H}}\int_z\frac{\partial\varphi(h,t)}{\partial t} d\mu_t(h|z)d\pi (z)dt.$ With continuity equation in strong form and the fact that $\mathcal{G}_t(h|z)$ induces $\mu_t(h|z)$ we have: + +$$ +\int_ {0} ^ {1} \int_ {\mathcal {H}} \int_ {z} \frac {\partial \varphi (h , t)}{\partial t} d \mu_ {t} (h | z) d \pi (z) d t = \int_ {0} ^ {1} \int_ {\mathcal {H}} \int_ {z} - \nabla \cdot (\varphi (h, t) \mathcal {G} _ {t} (h | z)) d \mu_ {t} (h | z) d \pi (z) d t +$$ + +By the divergence-form identity: + +$$ +\nabla \cdot (\varphi (h, t) \mathcal {G} _ {t} (h | z)) = \left\langle \mathcal {G} _ {t} (h), \nabla_ {h} \varphi (h, t) \right\rangle + \varphi (h, t) \nabla \cdot \mathcal {G} _ {t} (h | z) +$$ + +Since we choose the smooth test function $\varphi(h, t)$ from $\mathrm{Cyl}(\mathcal{H} \times [0, 1])$ and use the continuity equation in weak form, we assume term $\varphi(h, t) \nabla \cdot \mathcal{G}_t(h|z)$ disappears under integration. Thus we have + +$$ +\begin{array}{l} \int_ {0} ^ {1} \int_ {\mathcal {H}} \frac {\partial \varphi (h , t)}{\partial t} d \mu_ {t} (h) d t = - \int_ {0} ^ {1} \int_ {\mathcal {H}} \int_ {z} \langle \mathcal {G} _ {t} (h | z), \nabla_ {h} \varphi (h, t) \rangle d \mu_ {t} (h | z) d \pi (z) d t \\ = - \int_ {0} ^ {1} \int_ {\mathcal {H}} \int_ {z} \left\langle \mathcal {G} _ {t} (h | z), \nabla_ {h} \varphi (h, t) \right\rangle \frac {d \mu_ {t} (h | z)}{d \mu_ {t} (h)} d \mu_ {t} (h) d \pi (z) d t \\ = - \int_ {0} ^ {1} \int_ {\mathcal {H}} \int_ {z} \left\langle \mathcal {G} _ {t} (h | z) \frac {d \mu_ {t} (h | z)}{d \mu_ {t} (h)}, \nabla_ {h} \varphi (h, t) \right\rangle d \mu_ {t} (h) d \pi (z) d t \\ = - \int_ {0} ^ {1} \int_ {\mathcal {H}} \int_ {z} \left\langle \mathcal {G} _ {t} (h | z) \frac {d \mu_ {t} (\cdot | z)}{d \mu_ {t}} (h) d \pi (z), \nabla_ {h} \varphi (h, t) \right\rangle d \mu_ {t} (h) d t \\ = - \int_ {0} ^ {1} \int_ {\mathcal {H}} \left\langle \int_ {z} \mathcal {G} _ {t} (h | z) \frac {d \mu_ {t} (\cdot | z)}{d \mu_ {t}} (h) d \pi (z), \nabla_ {h} \varphi (h, t) \right\rangle d \mu_ {t} (h) d t \\ \end{array} +$$ + +On the other side, from Eq 29, we have + +$$ +\int_ {0} ^ {1} \int_ {\mathcal {H}} \frac {\partial \varphi (h , t)}{\partial t} d \mu_ {t} (h) d t = - \int_ {0} ^ {1} \int_ {\mathcal {H}} \langle \mathcal {G} _ {t} (h), \nabla_ {h} \varphi (h, t) \rangle d \mu_ {t} (h) d t = 0, \forall \varphi \in \operatorname {C y l} (\mathcal {H} \times [ 0, 1 ]) +$$ + +Thus $\mathcal{G}_t(h) = \int_z\mathcal{G}_t(h|z)\frac{d\mu_t(\cdot|z)}{d\mu_t} (h)d\pi (z) = \int_{\mathcal{H}\times \mathcal{H}}\mathcal{G}_t(h|z)\frac{d\mu_t(\cdot|z)}{d\mu_t} (h)d\pi (z)$ + +# H Derivation of Eq. 14 + +In this part, we show the detailed derivation of Eq. 14. In Flow Matching, the variable $z$ is chosen as a single data point from the coupling $\pi(u_0, u_1)$ where $u_1 \sim q_1$ , and $u_0 \sim q_0 = \mathcal{N}(\mathbf{0}, K(\{x_1, x_2, \ldots, x_n\}))$ . Considering the class of Gaussian conditional probability paths + +$$ +p _ {t} \left(u _ {t} | z\right) = \mathcal {N} \left(u _ {t} \mid m _ {t} (z), \sigma_ {t} (z) ^ {2} K \left(\left\{x _ {1}, x _ {2}, \dots , x _ {n} \right\}\right)\right) \tag {31} +$$ + +With conditional flow $\phi_t(u_t|z) = \sigma_t u_0 + m_t$ . Specially, we choose $m_t = tu_1 + (1 - t)u_0$ and $\sigma_t = \sigma$ , where $\sigma > 0$ is a small constant. From Eq. 1 (or Theorem 3 of Lipman et al. [11]), a vector that defines the Gaussian conditional flow is: + +$$ +\mathcal {G} _ {t} \left(u _ {t} \mid z\right) = \frac {\sigma_ {t} ^ {\prime}}{\sigma_ {t}} \left(u _ {t} - m _ {t}\right) + m _ {t} ^ {\prime} \left(u _ {1}\right) \tag {32} +$$ + +Then we can derive a closed-form expression for both the conditional probability and corresponding vector field [2] by plug in $\mu_t$ and $\sigma_t$ into Eq. 31 and Eq. 32 + +$$ +p _ {t} \left(u _ {t} | z\right) = \mathcal {N} \left(u _ {t} \mid t u _ {1} + (1 - t) u _ {0}, \sigma^ {2} K \left(\left\{x _ {1}, x _ {2}, \dots , x _ {n} \right\}\right)\right) \tag {33} +$$ + +$$ +\mathcal {G} _ {t} \left(u _ {t} \mid u _ {1}\right) = 0 + \left(u _ {1} - u _ {0}\right) = u _ {1} - u _ {0} \tag {34} +$$ + +Now, let's check the boundary conditions. At $t = 0$ + +$$ +p _ {0} \left(u _ {t} \mid z\right) = \mathcal {N} \left(u _ {t} \mid u _ {0}, \sigma^ {2} K \left(\left\{x _ {1}, x _ {2}, \dots , x _ {n} \right\}\right) \xrightarrow {\sigma \rightarrow 0} \delta_ {u _ {0}} \right. \tag {35} +$$ + +At $t = 1$ + +$$ +p _ {1} \left(u _ {t} \mid z\right) = \mathcal {N} \left(u _ {t} \mid u _ {1}, \sigma^ {2} K \left(\left\{x _ {1}, x _ {2}, \dots , x _ {n} \right\}\right) \xrightarrow {\sigma \rightarrow 0} \delta_ {u _ {1}} \right. \tag {36} +$$ + +From Eq. 9, we have $p_0(u_0) = \int p_0(u_t|z)\pi (z)dz = \int \delta_{u_0}\pi (u_0,u_1)du_0du_1 = q_0$ and $p_1(u_1) = \int p_1(u_t|z)\pi (z)dz = \int \delta_{u_1}\pi (u_0,u_1)du_0du_1 = q_1$ , which show boundary conditions are satisfied. + +# I Proof of Proposition 3.1 + +Proposition 3.1. Given noisy observations $\{\widehat{u}(x_i)\}_{i=1}^n$ , the posterior distribution is + +$$ +\log \mathbb {P} \left(\left\{u \left(x _ {i}\right) \right\} _ {i = 1} ^ {m} \mid \left\{\widehat {u} \left(x _ {i}\right) \right\} _ {i = 1} ^ {n}\right) = - \frac {\sum_ {i = 1} ^ {n} \| \widehat {u} \left(x _ {i}\right) - u \left(x _ {i}\right) \| ^ {2}}{2 \sigma^ {2}} + \log \mathbb {P} \left(\left\{u \left(x _ {i}\right) \right\} _ {i = 1} ^ {m}\right) + C \tag {37} +$$ + +Where the constant $C = -\frac{n}{2}\log (2\pi \sigma^2) - \log \mathbb{P}(\{\widehat{u} (x_i)\}_{i = 1}^n)$ . + +Proof. With Bayes rule, we have: + +$$ +\mathbb {P} \left(\left\{u \left(x _ {i}\right) \right\} _ {i = 1} ^ {m} \mid \left\{\widehat {u} \left(x _ {i}\right) \right\} _ {i = 1} ^ {n}\right) = \frac {\mathbb {P} \left(\left\{\widehat {u} \left(x _ {i}\right) \right\} _ {i = 1} ^ {n} \mid \left\{u \left(x _ {i}\right) \right\} _ {i = 1} ^ {m}\right) \cdot \mathbb {P} \left(\left\{u \left(x _ {i}\right) \right\} _ {i = 1} ^ {m}\right)}{\mathbb {P} \left(\left\{\widehat {u} \left(x _ {i}\right) \right\} _ {i = 1} ^ {n}\right)} \tag {38} +$$ + +Taking the logarithm of Eq. 38, we have: + +$$ +\begin{array}{l} \log \mathbb {P} \left(\{u (x _ {i}) \} _ {i = 1} ^ {m} \Big | \{\widehat {u} (x _ {i}) \} _ {i = 1} ^ {n}\right) = \log \mathbb {P} \left(\{\widehat {u} (x _ {i}) \} _ {i = 1} ^ {n} \Big | \{u (x _ {i}) \} _ {i = 1} ^ {m}\right) + \log \mathbb {P} \left(\{u (x _ {i}) \} _ {i = 1} ^ {m}\right) - \\ \log \mathbb {P} \left(\left\{\widehat {u} \left(x _ {i}\right) \right\} _ {i = 1} ^ {n}\right) \tag {39} \\ \end{array} +$$ + +Given $\epsilon_{i} \sim \mathcal{N}(0, \sigma^{2})$ and $\{\epsilon_{i}\}_{i=1}^{n}$ is a multivariate Gaussian, then $\{\widehat{u}(x_{i})\}_{i=1}^{n} \mid \{u(x_{i})\}_{i=1}^{n}$ is a shifted multivariate Gaussian with mean $\{u(x_{i})\}_{i=1}^{n}$ translated from the original multivariate Gaussian $\{\epsilon_{i}\}_{i=1}^{n}$ . Due to the translation invariance property of Gaussian distribution, We have: + +$$ +\log \mathbb {P} \left(\left\{\widehat {u} \left(x _ {i}\right) \right\} _ {i = 1} ^ {n} \mid \left\{u \left(x _ {i}\right) \right\} _ {i = 1} ^ {n}\right) = \log \mathbb {P} \left(\left\{\epsilon_ {i} \right\} _ {i = 1} ^ {n}\right) = - \frac {\sum_ {i = 1} ^ {n} \| \widehat {u} \left(x _ {i}\right) - u \left(x _ {i}\right) \| ^ {2}}{2 \sigma^ {2}} - \frac {n}{2} \log (2 \pi \sigma^ {2}) \tag {40} +$$ + +We notice $m > n$ and $\{\widehat{u} (x_i)\}_{i = 1}^n$ only depends on $\{u(x_i)\}_{i = 1}^n$ , and doesn't depend on $\{u(x_i)\}_{i = n + 1}^m$ . + +Thus $\log \mathbb{P}\left(\{\widehat{u}(x_i)\}_{i=1}^n \mid \{u(x_i)\}_{i=1}^m\right) = \log \mathbb{P}\left(\{\widehat{u}(x_i)\}_{i=1}^n \mid \{u(x_i)\}_{i=1}^n\right)$ . + +For evaluating $\log \mathbb{P}\left(\{u(x_i)\}_{i=1}^m\right)$ , which is the second part on the right-hand side of Eq. 39, we can efficiently calculate it with the trace estimator. The third part on the right hand side of Eq. 39 ( $\log \mathbb{P}\left(\{\widehat{u}(x_i)\}_{i=1}^n\right)$ ) represents the evidence and is constant. Thus the posterior distribution of Eq 39 can be simplified as: + +$$ +\log \mathbb {P} \left(\left\{u \left(x _ {i}\right) \right\} _ {i = 1} ^ {m} \mid \left\{\widehat {u} \left(x _ {i}\right) \right\} _ {i = 1} ^ {n}\right) = - \frac {\sum_ {i = 1} ^ {n} \| \widehat {u} \left(x _ {i}\right) - u \left(x _ {i}\right) \| ^ {2}}{2 \sigma^ {2}} + \log \mathbb {P} \left(\left\{u \left(x _ {i}\right) \right\} _ {i = 1} ^ {m}\right) + C \tag {41} +$$ + +Where the constant $C = -\frac{n}{2}\log (2\pi \sigma^2) - \log \mathbb{P}(\{\widehat{u} (x_i)\}_{i = 1}^n)$ . + +# J Example of Posterior Samples + +In this section, we initially present the regression result of OFM in another additional N-S scenario, as illustrated in Fig 7. Subsequently, we display more posterior samples used in the 2D regression examples. As depicted in Fig 9, 10, 11, OFM successfully generates realistic posterior samples that are consistent with the ground truth and demonstrate appropriate variability. In contrast, GP regression fails to produce explainable posterior samples. + +![](images/e95ddcda1d078db70ba7cf2d2c8ff1916b6ed5573831f9ccccbf3e5fac3e2f96.jpg) +(a) Observations + +![](images/0ffd24e608296f1152ad5877aa0426904b6a428ee6d2b239ab224a0eadf1d230.jpg) +(b) OFM mean + +![](images/bc32510cc1376924be709c3c6e89ad6b610e490d01bd0f6e6855df912ad51a0f.jpg) +(c) True mean + +![](images/4ea442136a3651c680d3dd45d8ba05ce3b4a28045189e6298d09729a729da26f.jpg) +(d) Misfit of mean + +![](images/0b4ed59371d32db1ddaf550196336aa5a082f09e1353e582e2a063f7a92c82e3.jpg) +(e) Misfit of std + +![](images/984b99d48c552f957b661f3096b570ba77e16cf3dced809a6a303d30274c2222.jpg) +(f) Samples from OFM + +![](images/cf334bab274a1673b2b4f22c5c093c01b1338f7a940c41086ed25695b4a55907.jpg) +(g) Sample from GP regression + +![](images/7c7e052303699b9398f99f5a3c25547995756c3a29922206ea27370ce8b430fa.jpg) + +ession + +![](images/207e47ffea1f095c0b1b4fc1469a3ab3da5e2ec49b7bb5f103e24add98c79263.jpg) +Figure 6: OFM regression on 2D GP data with resolution $32 \times 32$ . (a) 32 random observations. (b) Predicted mean from OFM. (c) Ground truth mean from GP regression. (d) Misfit of the predicted mean. (e) Misfit of predicted standard deviation. (f) Predicted samples from OFM. (g) Predicted samples from GP regression. +(a) Observations + +![](images/67c7b231925ff484237b1e162e49384bd3accad25eff5bc42dc11fca0c86904b.jpg) +(b) Ground truth + +![](images/d54e6488c04f4fb79d56f9e4c8f8ab4e4fb32c39b02f69a1020686f39fdadf2c.jpg) +(c) OFM mean + +![](images/b900d5e031f95c7c05f42a3cb20081442fad7aef0dbfa4192566cdc2abf63001.jpg) +(d) OFM sample + +![](images/3173f55096fae7f8b5f6866b719250864e7db57066bcd35de9b430ec95d7bfc5.jpg) +(e) GP sample + +![](images/78459dd840537d445909f34d538685500d07c6b2f709e21ede8e30f94df43828.jpg) +Figure 7: OFM regression on Navier-Stokes functional data with resolution $64 \times 64$ . (a) 32 random observations. (b) Ground truth sample (c) Predicted mean from OFM. (d) One posterior sample from OFM. (e) One posterior sample from best fitted GP. +(a) Observations + +![](images/17b9d99fde181f59679f6073f43791ec97cb8b10bfc4bce4529762b21d0c399a.jpg) +(b) Ground truth + +![](images/0167004c89c0d11912ae634dd68c11030c4fda34cca414438dbc91a0b1ac21e7.jpg) +(c) OFM mean + +![](images/7ebcbcc1af4cc946144fd8295c4467663a96a2441891afae074097583ae73631.jpg) +(d) OFM sample + +![](images/0433fe5472adf1a0eda2b22eb2cf7ff933648e67f1bfb21a553b9fec1b5adcce.jpg) +(e) GP sample +Figure 8: OFM regression on MNIST-SDF with resolution $64 \times 64$ . (a) 64 random observations. (b) Ground truth sample. (c) Predicted mean from OFM. (d) One posterior sample from OFM. (e) One posterior sample from best fitted GP. (f) Predicted mean from ConvCNP. (g) One posterior sample from ConvCNP. + +![](images/20f349f5144238910b2595a38efc6df8b3ab80d6ce315a66529ee73a7e287526.jpg) +(f) ConvCNP mean + +![](images/020ac95883870ae0bd8b07b76c57a9875224325f328869dc6f84f4ac575a4673.jpg) +(g) ConvCNP sample + +![](images/7326f50351274db80e9103228c9fbd345cfdb054a7cc5541519392b993aaef48.jpg) +Figure 9: OFM regression on NS data. (left) Posterior samples from OFM. (right) Posterior samples from GP regression. + +![](images/7d5c1f576a40edf138d0f034ba182b57a33ad75d3603d38985749081cad838cc.jpg) + +![](images/07ad82d13885f8bc5f30cfd1305efcade76d40f7313e86d55807da243ebf790c.jpg) + +![](images/47a5a342d87bf64da86f2f232f0802955ef8163fdc1216451c444abf54f9c185.jpg) + +![](images/46ce030dcaf24e251c38190175c14a67e92ac7b5d49145cca776dd4200adc75f.jpg) +Figure 10: OFM regression on black hole data. (left) Posterior samples from OFM. (right) Posterior san +Figure 11: OFM regression on MNIST-SDF data. (left) Posterior samples from OFM. (right) Posterior samples from GP regression. + +![](images/d87289829ad1b9a06df8e93ede98783fcf4247b57f7c9bbdcbd9ad67d82d6820.jpg) + +# K Co-domain functional regression with OFM + +In this section, we expand our regression framework to accommodate co-domain settings, as many function datasets feature a co-domain dimension greater than one. For example, earthquake waveform data commonly include three directional components, leading to a three-dimensional co-domain. Similarly, the velocity field in fluid dynamics usually features three directional components, also resulting in a dimension of co-domain of three. + +We illustrate this extension through a 2D GP example with a co-domain of 3 (channel dimension of 3). In learning the prior, we define the reference measure $(\nu_0)$ as a joint measure (Wiener measure) of three identical but independent Gaussian measures while the target measure $(\nu_{1})$ is another Wiener measure. We keep all other parameters unchanged as those described in the 2D GP regression tasks, with the only modification being an increase in the channel dimension from one to three. After training the prior (training detail provided in Appendix M), and provided 32 random observations across the three channels at co-locations, we then perform regression with OFM across these channels jointly. As demonstrated in Fig 12, OFM accurately estimate the mean and uncertainty across three channels. + +# L Posterior sampling with Stochastic Gradient Langevin Dynamics + +In this section, we describe how to sample from posterior distribution with SGLD. We denote logarithmic posterior distribution (Eq. 41) as $\log \mathbb{P}_{\theta}$ and denote a set of posterior samples as $\{u_{\theta}^{t}\}_{t = 1}^{N}$ , where each $u_{\theta}^{t}$ is defined on a collection of point $\{x_{i}\}_{i = 1}^{m}$ . + +By following the standard SGLD pipeline as described by Welling and Teh [4], we can obtain a set of $N$ posterior samples $\{u_{\theta}^{t}\}_{t = 1}^{N}$ . However, SGLD is known to be sensitive to the choice of regression parameters and can become trapped in local minima, leading to convergence issues, especially in regions of high curvature [51]. To mitigate these challenges, Shi et al. [1] proposed that within an invertible framework, drawing a posterior sample $u_{\theta}^{t}$ is equivalent to drawing a sample $a_{\theta}^{t}$ in Gaussian space, since $u_{\theta}^{t}$ uniquely defines $a_{\theta}^{t}$ and vice versa. This approach can stabilize the posterior sampling process and is less sensitive to the regression parameters due to the inherent smoothness of the Gaussian process. Additionally, Shi et al. [1] suggests starting from maximum a posteriori + +![](images/42cc2042c9bedea8b78e2ac0549344e2ea2cfd1c81438b0312336fb93d1f5b2f.jpg) +(a) Partial observations + +![](images/0c9431e326eb74b50782a9df470eed3b82aa03beb5e8098da8867aff8c030de5.jpg) + +![](images/062da131a45cbe9a5d430117aadea4735965e8666f8666d136c60a2a02beaae0.jpg) + +![](images/a44ddbbf78991b3642e272a8f03319426851b1d793b518e7957c6d40388ba0de.jpg) +(b) OFM mean + +![](images/c8c845b143b54429827a7fe8e457f2cfaedf9b6a3e7fda72392a7ad051619044.jpg) +(c) True mean + +![](images/a4688e2ca6d0191bb03fdc3b5cf70e9d8c250d16c3df3d579039d58fa9395d96.jpg) +(d) Misfit of mean + +![](images/f8d772daed2dab757d57708193bfb0098baf91cf43a1706551b99bb907019859.jpg) +(e) Misfit of standard deviation +Figure 12: OFM regression on co-domain GP data with resolution $32 \times 32$ . (a) 32 random observations at co-locations. (b) Predicted mean from OFM. (c) Ground truth mean from GP regression. (d) Misfit of the predicted mean. (e) Misfit of predicted standard deviation. + +(MAP) estimate of $a_{\theta}^{t}$ , denoted as $\overline{a_{\theta}}$ , which can reduce the number of burn-in iterations needed in SGLD. We adopt the same sampling strategy and the algorithm is reported in Algorithm 1 + +When the size of observations or context points $(\{\widehat{u}(x_i)\}_{i=1}^n)$ is 0, sampling from the posterior degrades to sampling from the prior, the results of which are presented in the subsequent section. + +Algorithm 1 Posterior sampling with SGLD + +Input and Parameters: Logarithmic posterior distribution $\log \mathbb{P}_{\theta}$ , temperature $T$ , learning rate $\eta_t$ , MAP $\overline{a}_{\theta}$ , burn-in iteration $b$ , sampling iteration $t_N$ , total iteration $N$ . + +1: Initialization: $a_{\theta}^{0} = \overline{a}_{\theta}$ +2: for $t = 0,1,2,\ldots ,N$ do + +3: Compute gradient of the posterior: $\nabla_{a_{\theta}}\log \mathbb{P}_{\theta}$ +4: Update $a_{\theta}^{t + 1}$ : $a_{\theta}^{t + 1} = a_{\theta}^{t} + \frac{\eta_t}{2}\nabla \log \mathbb{P}_{\theta} + \sqrt{\eta_tT}\mathcal{N}(0,I)$ +5: if $t \geq b$ then +6: Every $t_N$ iterations: obtain new sample $a_{\theta}^{t + 1}$ , and corresponding $u_{\theta}^{t + 1}$ +7: end if +8: end for + +# M Prior learning with OFM + +We now elaborate on the prior learning process and the corresponding performance evaluation. As shown in Algorithm 2, the training dataset is sampled from the unknown data measure $\nu_{1}$ . Concretely, the training dataset consists of $M$ discretized functions $\{u_i|D_i\}_{i = 1}^M$ , where $u_{i}|D_{i}$ denotes a discretized observation of the $u_{i}$ function. + +In practice, to simplify dataset preparation, one often uses the same discretization grid $D_{i}$ for all function samples, e.g. $D_{i} = \{x_{1},\dots ,x_{n}\}$ regardless of the sample index "i". For the consistency of notions, let $h_0$ represents a batch of i.i.d discretized functions sampled from the training dataset (equivalently, sampled from $\nu_{1}$ ). Next, the reference Gaussian process $\nu_{0} = \mathcal{N}(m_{0},C_{0})$ is known and determined by the user. With a slight abuse of notation, We choose to use notation $h_0,h_1$ for consistency purpose, in other parts of this paper, discretized $h_0,h_1$ is replaced with $a,u$ respectively. + +For specific experiments setting, we employ Matern kernel to construct the reference GP and to prepare training datasets for 1D GP, 2D GP, and 1D TGP. We have set the kernel length $l = 0.01$ with a smoothness factor $\zeta = 0.5$ for all reference GPs. OFM maps the GP samples from reference GPs to data samples and is resolution-invariant, which means OFM can be trained with functions at any resolution and evaluated at any resolution. + +# Algorithm 2 Learning a prior + +Input: Reference Gaussian process $\nu_{0} = \mathcal{N}(m_{0},C_{0})$ , data measure $\nu_{1}$ , batch size $b$ , small constant $\sigma_{\mathrm{min}}$ , discretized domain $D = \{x_1,\dots x_n\}$ + +1: while Training do +2: $h_0 \sim \nu_0$ ; $h_1 \sim \nu_1$ # sample functions of size $b$ i.i.d from the measures on $D$ +3: $\pi \gets \mathrm{OT}(h_0, h_1)$ # mini-batch optimal transport plan +4: $(h_0, h_1) \sim \pi$ +5: $t\sim \mathcal{U}(0,1)$ +6: $\mu_t \gets t h_1 + (1 - t) h_0$ +7: $h_t \sim \mathcal{N}(\mu_t, \sigma_{\min}^2 C_0)$ +8: $\mathcal{L}_{\mathrm{CFM}}^{\dagger}(\theta) \gets \left\| \mathcal{G}_{\theta}(t,x) - (h_1 - h_0)\right\|^2$ +9: $\theta \gets \mathrm{Update}\big(\theta ,\nabla_{\theta}\mathcal{L}_{\mathrm{CFM}}^{\dagger}(\theta)\big)$ + +10: end while + +11: return $\mathcal{G}_{\theta}$ + +1D GP dataset. We choose $l = 0.3$ and $\zeta = 1.5$ and generate 20,000 training samples on domain [0, 1] with a fixed resolution of 256. We use autocovariance and histogram of point-wise value as metrics for evaluation. We evaluate OFM at several different resolutions shown Fig 13, 14, 15, which demonstrate OFM's excellent capability to learn the function prior. + +![](images/af4c73e084571c225c7401cc109e5a88546900bc73b538e449ad11631e7c6eb0.jpg) +Figure 13: OFM for 1D GP prior learning, evaluated at resolution=128. (left two) Random samples from ground truth and generated by OFM. (right two) Autocovariance and histogram comparison + +![](images/6dbb54576b1cc1b6452aae5eac9b5a7849501d888eabc2ebd3dca0ed20214a9e.jpg) + +![](images/8c93ae5e63c11796e65b9be44abb6a00847d387e3d26d9dc39e88350c788d651.jpg) + +![](images/ab6786549b418d5de92a000bb0bbc8e06c1566d610ce949b2c898f39a82f2566.jpg) + +![](images/97171058f8b1d11a4f6f3e26e2e2fe42e732ad821195108728b1a5f6b0a062b8.jpg) +Figure 14: OFM for 1D GP prior learning, evaluated at resolution=256. (left two) Random samples from ground truth and generated by OFM. (right two) Autocovariance and histogram comparison + +![](images/d1c708aafe10e66b514bf69998ea60844470e66bce690424cacd10bb84b42245.jpg) + +![](images/58ee2e2237980406f0a817d447529c0eedc0faef47922398fc5e2f12672a926a.jpg) + +![](images/5058b9f7cdc6acaf385baa393192ce4b56f4f274776fff5a805860abc150bef6.jpg) + +![](images/39802135f5823cdfe73a896b5fb4f3b6735926d5d7561d16375eba8364bffd07.jpg) +Figure 15: OFM for 1D GP prior learning, evaluated at resolution=512. (left two) Random samples from ground truth and generated by OFM. (right two) Autocovariance and histogram comparison + +![](images/6f990bcb78a6b2e5d87ab6508fefa95a546c1989945dfd7fbec030477b7d1feb.jpg) + +![](images/11ed44bfa2adc6af068519ca0fdbf7f2357137507cf55ba1fea996d5b54d62a7.jpg) + +![](images/0fa92f457f692a6c5f919875dc0770d7979b4a4a524361f0f14d61ef72b4153c.jpg) + +1D TGP dataset. We choose $l = 0.3$ and $\zeta = 1.5$ and generating 20,000 training samples on domain [0, 1] with a fixed resolution of 256. We set $[-1.2, 1.2]$ for the bounds. Results provided in Fig 16, 17, 18. +2D Naiver-Stokes, Black hole, MNIST-SDF datasets. All the following 2D datasets are defined on domain $[0,1] \times [0,1]$ and have a resolution of $64 \times 64$ . We collected a 2D Navier-Stokes dataset + +![](images/fd37ba82f21ee7019cf9bf5f2b6ae396c07e537b92ece8c09ae5b573204de304.jpg) +Figure 16: OFM for 1D TGP prior learning, evaluated at resolution=128. (left two) Random samples from ground truth and generated by OFM. (right two) Autocovariance and histogram comparison + +![](images/9945a54525e52aa528446eb14cc7e4098ad1eb3d2f26647531f336fcb7749185.jpg) + +![](images/7ad53f35ff4b3b83611afc5ce0b50b0c588da73214333862d7d8a73cd6154ec4.jpg) + +![](images/4d9a428b2053413735336bb0cf7ff9c3a70b7b08bd8ff930c74dce5a790bf5f2.jpg) + +![](images/af3760d6d929133dedd8ef476a89e59b7f6b8e06e0719d2d19d3db19dc553733.jpg) +Figure 17: OFM for 1D TGP prior learning, evaluated at resolution=256. (left two) Random samples from ground truth and generated by OFM. (right two) Autocovariance and histogram comparison + +![](images/4070231bdb4d04fa46bac17973137c0f7c3feb67ab06466397930444a9ca7f71.jpg) + +![](images/2e79365b8531b06f7038b788bce3233606eedcc8a8d5aef848b0227227775013.jpg) + +![](images/c22a5a98e963395e8f9ab1da880c76be662a6fd692c1a16993da4a0c4d9b716d.jpg) + +consisting of 20000 samples, with viscosity $= 1e - 4$ . The results, including zero-shot superresolution, are provided in Fig 19, 20. The learning of Black hole dataset, generated using expensive Monte Carlo method, is detailed in Fig 21, 22. Additionally, we trained OFM on 20,000 MNIST-SDF samples, the outcomes are illustrated in Fig 23, 24. + +![](images/8e90b23e162d27aa810800a22d1c14d80321ec4484fc152cafc32802ecc42ad1.jpg) +Figure 18: OFM for 1D TGP prior learning, evaluated at resolution=512. (left two) Random samples from ground truth and generated by OFM. (right two) Autocovariance and histogram comparison + +![](images/ac34626ef18f8059a1e0cf5971632b9b17afd1fa77694293eacd7afc7c253b10.jpg) + +![](images/7db4ac1f9415a82c9abbd82041464eb4b90c0ba39dd3c48358a26d554ea19058.jpg) + +![](images/798888f65748eedf022deecb26e4255d6d18a1cf8cb620bf29d589b8dc6e4dd0.jpg) + +![](images/e9b2acd38bed06f2b62b731fbc80069169e12ca3c9321a6d4e930323494d5999.jpg) + +![](images/76a5b579ef8b56721b053d0817e43bf2898c01b1cdf7ba8638568ebbb09b90bc.jpg) + +![](images/8894cbf376121c9a5e46eb4d5af40a017a7b23b1ad8159a33b90acb046d702ec.jpg) +Figure 19: OFM for 2D N-S prior learning, evaluated at resolution $= 64 \times 64$ . (top left) Random samples from ground truth. (top right) Random samples generated by OFM. (bottom) Autocovariance comparison + +![](images/3905e28371a18a5de0b935d707c3b13fd98acf0fb8384e6eb068957da9e5f32e.jpg) +Figure 20: OFM for 2D N-S prior learning, evaluated at $128 \times 128$ resolution (zero-shot super-resolution) + +![](images/dbfc5f58482dc064c9bb22ab40e7c854262e6ce60fce1c9a1d80da4e4ae1260b.jpg) + +![](images/3afa772d6c8a44d05efecf714573710513785654f2a429c3528b24517bf51a3c.jpg) + +![](images/a13a4a776e00b1615317abbfc002530a7166ae6df59c187f3dcbb3ade2a93cd2.jpg) +Figure 21: OFM for 2D black hole prior learning, evaluated at resolution=64. (top left) Random samples from ground truth. (top right) Random samples generated by OFM. (bottom) Autocovariance comparison + +![](images/ce591095e6c8231ada7b6cec12c1f78c0ae7d3dd6db514e8fac02f34e16c18aa.jpg) +Figure 22: OFM for 2D black hole prior learning, evaluated at $128 \times 128$ resolution (zero-shot super-resolution) + +![](images/04c38d94c8b6c681c9b2bd0a7ec680f214a62361393b7e2f4b0bb01904072ec3.jpg) + +![](images/45f0b0547e39cff3cac65cca037de54cbf6a25bb724720505d3e6e081c8781af.jpg) +Figure 23: OFM for 2D MNIST-SDF prior learning, evaluated at $64 \times 64$ resolution. (top) Random samples from ground truth. (bottom) Random samples generated by OFM. + +![](images/c95f9484ef2cf4165259da2b962f6a373970690ae00fe9bafa8c8da5a6d01ae7.jpg) + +![](images/39e21b684a5870b69228a9c422abb15993d97cc720424975faee0b4b737333f4.jpg) +Figure 24: OFM for 2D MNIST-SDF prior learning, evaluated at $128 \times 128$ resolution. (top) Random samples from ground truth. (bottom) Random samples generated by OFM. + +# N Additional results of 1D GP regression with more complex kernels + +In this section, we extend the 1D GP experiments by introducing more complex kernels for both data generation and regression, while holding all other settings identical to those used with the Matérn kernel. Specifically, we report results for the non-stationary Gibbs kernel and the Rational Quadratic (RQ) kernel. + +For the Gibbs kernel, we use an input-dependent length-scale + +$$ +\ell (x) = \ell_ {0} + \ell_ {1} x, \qquad x \in [ 0, 1 ], +$$ + +which induces the covariance + +$$ +k (x, x ^ {\prime}) = \sigma^ {2} \sqrt {\frac {2 \ell (x) \ell (x ^ {\prime})}{\ell (x) ^ {2} + \ell (x ^ {\prime}) ^ {2}}} \exp \left(- \frac {(x - x ^ {\prime}) ^ {2}}{\ell (x) ^ {2} + \ell (x ^ {\prime}) ^ {2}}\right). +$$ + +In our setup we take $\ell_0 = 0.05$ , $\ell_1 = 0.25$ , and $\sigma = 1.0$ . + +For the RQ kernel, we use sklearn.gaussian_process.kernels.RationalQuadratic with length-scale 0.15. As shown in Table 2, OFM consistently outperforms all baselines on these tasks. + +Table 2: Comparison of OFM with baseline models on 1D GP with Gibbs kernel and RQK. Best performance in bold. + +
Dataset → Algorithm ↓ Metric →1D GP-Gibbs1D GP-RQK
SMSEMSLLSMSEMSLL
NP4.0 · 10-17.1 · 10-15.8 · 10-12.2 · 100
ANP3.5 · 10-16.2 · 10-12.7 · 10-12.1 · 100
ConvCNP3.4 · 10-11.1 · 10-12.2 · 10-16.9 · 10-1
DGP4.3 · 10-18.6 · 10-12.4 · 10-11.2 · 100
DSPP4.2 · 10-17.1 · 10-12.5 · 10-14.2 · 10-1
OpFlow3.1 · 10-16.9 · 10-12.9 · 10-15.0 · 10-1
OFM(Ours)2.9 · 10-18.7 · 10-22.1 · 10-19.4 · 10-2
+ +# O Details of experimental setup + +In this section, we outline the details of experiments setup used in this paper. Since regression with OFM requires learning the prior first, we list the parameters used for learning the prior and regression separately. We employ FNO as the backbone, implemented using neuraloperator library [14]. All time reported in the subsequent tables are based on one computations performed using a single NVIDIA RTX A6000 (48 GB) graphics card. In all experiments, we use the dopri5 ODE solver provided by torchdiffeq Chen et al. [52] with atol=1e-5 and rtol=1e-5. Detailed posterior sampling algorithm is provided in Appendix L. + +Table 3 details the parameters used for training the prior. For instance, in the 1D GP prior learning experiment, the dataset consists of 20,000 samples, each with a co-domain dimension (or channel) of one. The batch size is set at 1024, and the model is trained over 500 epochs. The total training time is about 0.76 hours, and the size of the trained model is 37.1 megabytes. + +Tables 4, 5, and 6 detail the parameters for SGLD sampling as described in Algorithm 1. For example, in the 1D GP regression experiment, the regression takes 40,000 iterations with a burn-in phase of 3,000 iterations. Posterior samples are collected every 10 iterations. The temperature for the injected noise during the gradient update is set at 1, and the learning rate decays exponentially from 0.005 to 0.004 (defined in Algorithm 1). We average 32 runs with the Hutchinson trace estimator to evaluate the likelihood, utilizing GPU parallel computing. The noise level, as specified in Equation 41, is 0.01 in this regression task. Then given 6 random observations, we ask for the posterior samples across 128 points. The GPU memory usage for the regression task is 4 gigabytes, with the total runtime to 4.91 hours. + +Table 3: Parameters used in experiments of prior learning + +
DatasetsSize of DatasetChannelsBatch SizeEpochsTraining TimeModel Size
1D GP2·104110245·1020.76 h37.1 MB
1D TGP2·104110245·1021.24 h37.1 MB
2D GP2·10412565·1021.14 h76 MB
2D co-domain GP2·10432565·1021.01 h76 MB
2D N-S2·10412565·1023.79 h286 MB
2D Black hole1.2·10412565·1022.28 h286 MB
2D MNIST-SDF2·10412565·1028.31 h286 MB
+ +Table 4: Parameters used in regression experiments - Part A + +
DatasetsTotal IterationBurn-in IterationSampling IterationsTemperature of Noise
1D GP4 · 1043 · 103101
1D TGP4 · 1043 · 103101
2D GP2 · 1043 · 103101
2D co-domain GP2 · 1043 · 103101
2D N-S2 · 1043 · 103101
2D Black hole2 · 1043 · 103101
2D MNIST-SDF2 · 1043 · 103101
+ +# P Ablation and scaling studies for mini-batch optimal transport and Hutchinson trace estimator + +We first present an ablation study of the optimal transport plan. In this study, we revisit prior learning on the N-S dataset by training two models: one using independent coupling and the other employing a mini-batch optimal transport plan. Both models were trained with a batch size of 64. For evaluation, we compare the mean squared error (MSE) of density, autocovariance and spectral characteristics between 1,000 real and generated samples. Additionally, we report the convergent training loss (squared $L^2$ loss) and the number of function evaluations (NFE) required for sampling with adaptive ODE solver (dopri5). As shown in Table 7, the mini-batch OT plan outperforms the independent coupling approach in terms of pointwise accuracy, spectral fidelity, and convergent $L^2$ loss, while also requiring fewer NFE and enabling faster sampling. Our findings indicate that as the mini-batch size increases, the model learns the prior with reduced error. This improvement may be attributed to the mini-batch optimal transport plan approaching the true optimal transport plan with larger batch sizes. + +Next, we explore the variance of the Hutchinson trace estimator in likelihood estimation with a scaling experiment. For this experiment, we use a 1D GP example with parameters described in Section M and a resolution of 128. We randomly draw a GP sample from the prior and evaluate the log likelihood by integrating the divergence as shown in Eq. 15, while also estimating it using the Hutchinson trace estimator from Eq. 16. In this scaling experiment, the number of noise sample $(n_{\mathrm{noise}})$ for the Hutchinson estimator is set to (4, 8, 16, 32, 64, 128). For each $n_{\mathrm{noise}}$ , we repeat the experiment 100 times and report the mean and standard deviation of the predicted likelihood. The exact likelihood is computed by directly integrating the trace of the Jacobian. We repeat this procedure for 5 different random 1D GP samples. As reported in Table 8, the standard deviation of the Hutchinson trace estimator decreases rapidly as $n_{\mathrm{noise}}$ increases, and the predicted means always align closely with the ground truth. + +Furthermore, even at smaller $n_{\mathrm{noise}}$ values, where a relative larger variance is expected, the performance of the posterior sampling appears robust (see Table 5). We hypothesize that this is due to stochastic nature of posterior sampling algorithm (SGLD), which requires injected perturbations, renders its performance relatively insensitive to the choice of $n_{\mathrm{noise}}$ . + +Table 5: Parameters used in regression experiments - Part B + +
DatasetsInitial Learning RateEnd Learning RateHutchinson SamplesNoise Level
1D GP5·10-34·10-3321·10-2
1D TGP5·10-34·10-3321·10-3
2D GP1·10-38·10-4321·10-2
2D co-domain GP1·10-38·10-4161·10-2
2D N-S3·10-32·10-381·10-3
2D Black hole5·10-34·10-381·10-3
2D MNIST-SDF5·10-34·10-381·10-3
+ +Table 6: Parameters used in regression experiment - Part C + +
DatasetsNumber of ObservationsInquired GridsGPU MemoryRunning Time
1D GP61284 GB4.91 h
1D TGP31284 GB5.42 h
2D GP3232 × 3222 GB9.70 h
2D co-domain GP3232 × 3231 GB5.05 h
2D N-S3264 × 6444 GB13.65 h
2D Black hole3264 × 6444 GB13.37 h
2D MNIST-SDF6464 × 6444 GB9.41 h
+ +Table 7: Scaling study for the size of mini-batch given optimal transport plan, best performance in bold + +
MetricsDensity-MSEAutocovariance-MSESpectra-MSEConvergent L2LossNFE
Independent3.4 · 10-57.4 · 10-51.3 · 1015.1 · 10-2168
mini-batch = 323.0 · 10-51.3 · 10-45.0 · 1002.3 · 10-2141
mini-batch = 649.8 · 10-69.8 · 10-56.3 · 1001.9 · 10-2153
mini-batch = 1282.4 · 10-51.7 · 10-53.6 · 1001.5 · 10-2182
+ +Table 8: Scaling study for the number of noise samples of Hutchinson trace estimator + +
nnoise = 4nnoise = 8nnoise = 16nnoise = 32nnoise = 64nnoise = 128exact
sample 1-481.7 ± 9.4-482.9 ± 7.7-480.6 ± 4.9-481.4 ± 3.8-481.1 ± 2.5-480.8 ± 1.8-482.7
sample 2-482.8 ± 9.8-481.8 ± 6.7-481.3 ± 4.9-480.8 ± 3.5-481.1 ± 2.7-481.2 ± 1.7-482.0
sample 3-479.7 ± 10.2-479.7 ± 7.6-478.2 ± 4.7-478.6 ± 3.4-478.8 ± 2.6-478.8 ± 1.7-479.1
sample 4-476.9 ± 10.0-477.0 ± 6.6-477.9 ± 5.6-478.5 ± 4.0-478.1 ± 2.7-478.0 ± 1.8-477.4
sample 5-479.4 ± 10.7-479.2 ± 6.6-479.2 ± 5.3-479.4 ± 3.6-479.6 ± 2.8-479.3 ± 2.0-478.5
+ +![](images/74fb322e15ef83bcc6c52b1bdd83dc7b7c01da692a7d26fce33fc27c55173e5b.jpg) +Figure 25: Log likelihood by the integration of divergence, (left) plot for first GP sample. (right) plot for second GP sample + +![](images/732bbdb8fb0d3a8f92e6f5d84ba94fd517f93d0bc3da8ac9799e3b8e6f6b7047.jpg) + +# Q Detailed analysis of OFM and comparison with existing methods + +In this section, we elaborate the connection and difference with pervious work, highlight contributions and potential limitations of our work. The regression with OFM involves a two-steps process: (i) learning a prior on function space, and (ii) sampling from the posterior given observations. Consequently, the OFM framework has connections with both generative models on function space and the models developed for functional regression. In the following, we provide a comprehensive comparative analysis with related models and baselines, including operator flow (OPFLOW) [1], conditional optimal transport flow matching (COT-FM) [45], conditional models (NPs) [7, 34] + +Comparison with OPFlow. OPFlow introduces invertible neural operators, which generalizes RealNVP [53] to function space and maps any collection of points sampled from a GP to a new collection of points in the data space, using the maximum likelihood principle [1]. This method captures the likelihood of any collection of point consistently as the resolution increases and allows for UFR using SGLD. Despite these advantages, the requirement for an invertible neural operator brings training and expressiveness challenges. To be specific, OPFlow failed on all non-GP regression tasks in this paper because the prior learning stage suffered from mode collapse during training. On the contrary, OFM adopts a simulation-free ODE framework for prior learning, which offers enhanced expressiveness and ensures training stability through a simple regression objective while avoiding using the invertible neural operator. In addition, OFM proposes a non-trivial extension of UFR to the simulation-free ODE framework. These improvements render OFM a more practical solution for challenging functional regression tasks. + +Comparison with COT-FM. COT-FM [45] proposes a conditional generalization of Benamou-Brenier Theorem [54], formulating a conditional optimal transport plan that applicable for both Euclidean and Hilbert space. In contrast, OFM employs an unconditional optimal transport plan in Hilbert space based on dynamic Kantorovich formulation, which is initially generalized for unbalanced optimal transport [37]. The advantage of COT-FM lies in its ability to flexibly incorporate specific conditions tailored for conditional generative tasks. However, COT-FM is not suitable for functional regression tasks due to: (i) COT-FM is contingent upon both the reference and target being influenced by conditions, and the vector field learnt is triangular, designed to transport jointly the coupling of a reference measure and a condition measure. In UFR setting, the learnt prior is required to be unconditioned, (ii) the coupling with condition measure typically prevents inducing valid stochastic process, even when the reference measure is a Gaussian measure, (iii) cannot provide point evaluation of probability density. Last, We should notice, the development of OFM is different and independent of COT-FM, the former with a focus on stochastic process learning and Bayesian functional regression. + +Comparison with conditional models. NPs were developed to address the computational and restrictive prior challenges of Gaussian Processes, utilizing neural networks for efficiency [7]. However, several recent studies have discussed the drawbacks in the formulation of NPs, raising concerns that NPs might not learn the underlying function distribution [1, 22, 55]. + +Notably, NPs treats the point cloud data as a set of values, ignoring the metric space of the data [55]. This can lead to misinterpretations of a function sampled at different resolutions as distinct functions (Appendix A.1 of [22]). Furthermore, NPs rely on encoding input data into finite-dimensional, Gaussian-distributed latent variables before projecting these into an infinite-dimensional space. This process tends to lose consistency at higher resolutions. Moreover, the Bayesian regression framework underpinning NPs focuses on point sets rather than the functions themselves, leading to a dilution of prior information with increasing data points. + +In recent study, diffusion-based variants of NPs (NDP) [34], was proposed to leverage the expressiveness of diffusion models [24, 25]. Nonetheless, the formulation of NDP does not address the aforementioned issues of NPs and introduces two more problems: (i) NDP fails to induce a valid stochastic process as it does not satisfy the marginal consistency criterion required by Kolmogorov Extension Theorem [39], and (ii) it relies on uncorrelated Gaussian noise for denoising, which is not applicable in function spaces [26]. Oppositely, OFM establishes a more theoretically sound framework by rigorously defining learning within function spaces. Additionally, Bayesian functional regression within the OFM framework adheres to valid stochastic processes, offering a robust and theoretically grounded solution. Last, we provide a high-level comparison of OFM and NP as shown in Table 9. + +Table 9: High-level comparison of OFM and NP. + +
AspectOFMNeural Processes (NP)
Modeling targetGlobal prior over stochastic processes with a valid joint density.Amortized conditional model without explicit tractable joint density over full functions.
Arbitrary queriesSet-size / location agnostic; evaluate on any grid or point set.Set-size / location agnostic; trained with random context/target splits.Performance may degrade under extreme sparsity or distribution shift
Uncertainty & likelihoodsTractable log-densities and posterior sampling enable principled Bayesian inference.Predictive densities are available and trained via conditional likelihood; no closed-form function-level likelihood is typically specified.Uncertainty reflects model/context coverage and may be miscalibrated.
Structure & extrapolationCaptures global geometry and non-stationarity via the learned flow.Learns inductive biases from data through context summaries; strong interpolation, but extrapolation and long-range generalization are not guaranteed and depend on training distribution.
+ +Limitations. Despite these advances, the current regression framework with OFM is primarily limited to low-dimensional data (1D and 2D in this study). This limitation stems from the challenges associated with learning operators for functions defined on high-dimensional domains—an area that remains underdeveloped both computationally and in terms of dataset availability [15]. Additionally, while the time complexity for regression with OFM is $\mathcal{O}(m^2)$ , the incorporation of additional components significantly increases its computational resource requirements compared to classical GP regression. + +There are several potential paths to mitigate the computation concerns: (i) Carefully adjusting the step size $(\eta_t)$ and temperature $(T)$ in the SGLD algorithm to prevent overly large gradient updates. (ii) Running the trace estimator on multiple GPUs, use mixed-precision FNO [43], or adopt efficient neural operators [56]. (iii) Using more efficient ODE solvers. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: This work is not related to LLM. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. \ No newline at end of file diff --git a/stochasticprocesslearningviaoperatorflowmatching/images.zip b/stochasticprocesslearningviaoperatorflowmatching/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..dc508c1830a9e67fb052932cac7d25ca35ca1204 --- /dev/null +++ b/stochasticprocesslearningviaoperatorflowmatching/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2ca2592ea7e3b9732d62c4f64484273245673a8b60b32130cab72ddc2773d060 +size 1765535 diff --git a/stochasticprocesslearningviaoperatorflowmatching/layout.json b/stochasticprocesslearningviaoperatorflowmatching/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..60a8c68cb92c39c21fe9808e3f017bf1a922bbff --- /dev/null +++ b/stochasticprocesslearningviaoperatorflowmatching/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ab01f6bddeed09f60b3c4470a41c8b5095792551279188e5145c2b67b1ac0b9a +size 1564859 diff --git a/stochasticregretguaranteesforonlinezerothandfirstorderbileveloptimization/b628db5b-0e41-403c-bcbf-435aab165e75_content_list.json b/stochasticregretguaranteesforonlinezerothandfirstorderbileveloptimization/b628db5b-0e41-403c-bcbf-435aab165e75_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..80620d70559b356fec0c1079afe7f3323f884cd5 --- /dev/null +++ b/stochasticregretguaranteesforonlinezerothandfirstorderbileveloptimization/b628db5b-0e41-403c-bcbf-435aab165e75_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c5a8ed034bd39929c9d66c00216128b03bc27e153b9c9f52f327001388cca066 +size 772337 diff --git a/stochasticregretguaranteesforonlinezerothandfirstorderbileveloptimization/b628db5b-0e41-403c-bcbf-435aab165e75_model.json b/stochasticregretguaranteesforonlinezerothandfirstorderbileveloptimization/b628db5b-0e41-403c-bcbf-435aab165e75_model.json new file mode 100644 index 0000000000000000000000000000000000000000..52547a40abf64f3251001f7377c4ba5372dd105a --- /dev/null +++ b/stochasticregretguaranteesforonlinezerothandfirstorderbileveloptimization/b628db5b-0e41-403c-bcbf-435aab165e75_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:49bba6746775859600160ed36667a8f2b62c868288ce8992ceece504fb5c57fa +size 864915 diff --git a/stochasticregretguaranteesforonlinezerothandfirstorderbileveloptimization/b628db5b-0e41-403c-bcbf-435aab165e75_origin.pdf b/stochasticregretguaranteesforonlinezerothandfirstorderbileveloptimization/b628db5b-0e41-403c-bcbf-435aab165e75_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fbfa1163333e647bb79e4b250f017e895e821878 --- /dev/null +++ b/stochasticregretguaranteesforonlinezerothandfirstorderbileveloptimization/b628db5b-0e41-403c-bcbf-435aab165e75_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1d9b402d20057e54ecfe8258276b27fbf5761a0cd46399b153e5b9e02153675d +size 1323500 diff --git a/stochasticregretguaranteesforonlinezerothandfirstorderbileveloptimization/full.md b/stochasticregretguaranteesforonlinezerothandfirstorderbileveloptimization/full.md new file mode 100644 index 0000000000000000000000000000000000000000..87e22cf2c24f935f05a455abdb1b120a990fe3eb --- /dev/null +++ b/stochasticregretguaranteesforonlinezerothandfirstorderbileveloptimization/full.md @@ -0,0 +1,3949 @@ +# Stochastic Regret Guarantees for Online Zeroth- and First-Order Bilevel Optimization + +Parvin Nazari + +Amirkabir University of Technology + +p.nazari17@gmail.com + +Bojian Hou + +University of Pennsylvania + +bojianh@upenn.edu + +Davoud Ataee Tarzanagh * + +Samsung SDS Research America + +d.tarzanagh@samsung.com + +Li Shen + +University of Pennsylvania + +li.shen@pennmedicine.upenn.edu + +George Michailidis + +University of California, Los Angeles + +gmichail@ucla.edu + +# Abstract + +Online bilevel optimization (OBO) is a powerful framework for machine learning problems where both outer and inner objectives evolve over time, requiring dynamic updates. Current OBO approaches rely on deterministic window-smoothed regret minimization, which may not accurately reflect system performance when functions change rapidly. In this work, we introduce a novel search direction and show that both first- and zeroth-order (ZO) stochastic OBO algorithms leveraging this direction achieve sublinear stochastic bilevel regret without window smoothing. Beyond these guarantees, our framework enhances efficiency by: (i) reducing oracle dependence in hypergradient estimation, (ii) updating inner and outer variables alongside the linear system solution, and (iii) employing ZO-based estimation of Hessians, Jacobians, and gradients. Experiments on online parametric loss tuning and black-box adversarial attacks validate our approach. + +# 1 Introduction + +Bilevel optimization (BO) minimizes an outer objective dependent on an inner problem's solution. Originating in game theory [64] and formalized in mathematical optimization [9], BO finds applications in operations research, engineering, economics [16], and image processing [14]. Recently, BO has gained traction in machine learning, including hyperparameter optimization [22], meta-learning [18], reinforcement learning [65], and neural architecture search [51]. + +In the offline setting, BO solves the following problem: + +$$ +\mathbf {x} ^ {*} \in \operatorname {a r g m i n} _ {\mathbf {x} \in \mathbb {R} ^ {d _ {1}}} f (\mathbf {x}, \mathbf {y} ^ {*} (\mathbf {x})) \quad \text {s u b j . t o} \quad \mathbf {y} ^ {*} (\mathbf {x}) = \operatorname {a r g m i n} _ {\mathbf {y} \in \mathbb {R} ^ {d _ {2}}} g (\mathbf {x}, \mathbf {y}), \tag {BO} +$$ + +where $f$ and $g$ are the outer and inner objectives, with $\mathbf{x}$ and $\mathbf{y}$ as their respective variables. + +OBO [67] addresses dynamic scenarios where objectives evolve over time, requiring the agent to update the outer decision in response to the optimal inner decision. Similar to online single-level optimization (OSO) [72], OBO involves iterative decision-making without prior knowledge of outcomes [67, 50, 8]. Let $T$ be the total number of rounds. Define $\mathbf{x}_t \in \mathcal{X} \subset \mathbb{R}^{d_1}$ as the + +
OBO MethodWindow Size in Regret (w)System Iters.Stochastic RegretConst. Regret Min.Only Func. FeedbackLocal Regret Bound
OAGD [67]o(T)N.A. (Exact)XXXT/w + H1,T + H2,T
SOBOW [50]o(T)O(κg log κg)XXXT/w + VT + H2,T
SOBBO [8]o(T)O(κg log κg)✓✓XT/w σ2 + VT + H2,T
SOGD11✓✓XT1/3(σ2 + ΔT) + T2/3ΨT
ZO-SOGD11✓✓✓(d1 + d2)3/4T1/3(δ2 + ΔT) + (d1 + d2)3/2T2/3ΨT
+ +Table 1: Comparison of OBO algorithms based on regret window $w$ , solver iterations, stochastic/constrained regrets, feedback type, and local bounds. $\kappa_g$ denotes the condition number of the inner objective $g_t$ . $V_T$ , $H_{p,T}$ , $\Delta_T$ , $\Psi_T$ , $\hat{\Delta}_T$ , $\hat{\Psi}_T$ , $\sigma$ , and $\hat{\sigma}$ are defined in (11), (14), (30), (10), and (28), respectively. + +decision variable and $f_{t}:\mathcal{X}\times \mathbb{R}^{d_{2}}\to \mathbb{R}$ as the outer function. Similarly, define $\mathbf{y}_t\in \mathbb{R}^{d_2}$ and $g_{t}:\mathcal{X}\times \mathbb{R}^{d_{2}}\rightarrow \mathbb{R}$ for the inner problem, where $\mathbf{y}_t^* (\mathbf{x}) = \mathrm{argmin}_{\mathbf{y}\in \mathbb{R}^{d_2}}g_t(\mathbf{x},\mathbf{y})$ . OBO can be seen as a single-player problem, where the player selects $\mathbf{x}_t$ without knowing $\mathbf{y}_t^* (\mathbf{x})$ , using $\mathbf{y}_t$ as an estimate based on $g_{t}$ . Alternatively, it can be framed as a two-player game [64], where the leader $(\mathbf{x}_t)$ competes with the follower $(\mathbf{y}_t)$ , who selects $\mathbf{y}_t^* (\mathbf{x})$ based on limited knowledge of $g_{t}$ . This framework includes online and adversarial variants of (BO), such as online actor-critic algorithms [71], online meta-learning [19], and online hyperparameter optimization [50]. The inner and outer functions may be time-varying, adversarial, unavailable a priori, and require nonstationary optimization. + +Our Contributions. This paper addresses stochastic OBO, introducing novel first- and zeroth-order methods to minimize stochastic bilevel regret. Key contributions are summarized below. + +- Stochastic regret minimization without window-smoothing. Existing OBO methods [67, 50, 40, 8] rely on deterministic window-smoothed regret minimization, which may not accurately reflect system performance when functions change rapidly. We address these limitations by introducing a novel search direction (Section 2) and proving that both first-order and ZO methods achieve sublinear stochastic bilevel regret without window-smoothing ( $w = 1$ ); see Theorems 2.6 and 3.2 and Table 1. +- OBO with function value oracle feedback. In large-scale and black-box settings [11, 58], first- and second-order information is often unavailable or costly. Constructing accurate (hyper)-gradient estimators using only function value oracles is particularly challenging due to BO's nested structure. Existing methods rely on gradient, Hessian, and Jacobian oracles, limiting scalability [21, 27]. We propose Algorithm 2, which estimates Hessians, Jacobians, and gradients using function value oracles, achieving sublinear local regret (Theorem 3.2). +- OBO with one subproblem solver iteration. A major challenge in BO is solving implicit systems to approximate the hypergradient [43, 12]. While efficient offline BO methods exist [43, 15], extending them to OBO is difficult due to time-varying objectives. SOBOW [50] partially addresses this using a conjugate gradient (CG) algorithm with increasing iterations (Table 1). We improve upon SOBOW by introducing Algorithms 1 and 2, which require only a single subproblem solver iteration. + +# 2 Stochastic OBO with Access to First- and Inner Second-Order Oracles + +Notation. $\mathbb{R}^d$ is the $d$ -dimensional real space; $\mathbb{R}_+^d$ and $\mathbb{R}_{++}^d$ denote its nonnegative and positive orthants. Bold lowercase letters (e.g., $\mathbf{x}, \mathbf{y}$ ) represent vectors, $\langle \mathbf{x}, \mathbf{y} \rangle$ is the inner product, and $\|\cdot\|$ is the Euclidean norm. $\nabla_{\mathbf{x}}$ denotes the gradient, and $\nabla_{\mathbf{xy}}^2 = \nabla_{\mathbf{x}} \nabla_{\mathbf{y}}$ . A function is $L$ -smooth if its gradient is $L$ -Lipschitz. The projection onto a convex set $\mathcal{X}$ is $\Pi_{\mathcal{X}}(\mathbf{z}) = \operatorname*{argmin}_{\mathbf{x} \in \mathcal{X}} \frac{1}{2} \| \mathbf{x} - \mathbf{z} \|^2$ . We use $[T]$ for $\{1, \ldots, T\}$ , $\mathbb{E}[\cdot]$ for expectation, and $\mathcal{O}(\cdot)$ to hide problem-independent constants. + +Stochastic OBO Setting. Let $T$ be the total rounds [67]. Define $\mathbf{x}_t \in \mathcal{X} \subset \mathbb{R}^{d_1}$ as the decision variable and $f_t: \mathcal{X} \times \mathbb{R}^{d_2}$ as the outer objective. The inner decision variable and objective are $\mathbf{y}_t \in \mathbb{R}^{d_2}$ and $g_t: \mathcal{X} \times \mathbb{R}^{d_2}$ , where the optimal inner decision is: + +$$ +\mathbf {y} _ {t} ^ {*} (\mathbf {x}) \in \underset {\mathbf {y} \in \mathbb {R} ^ {d _ {2}}} {\operatorname {a r g m i n}} \left\{g _ {t} (\mathbf {x}, \mathbf {y}) := \underset {\zeta_ {t} \sim \mathcal {D} _ {g, t}} {\mathbb {E}} \left[ g _ {t} (\mathbf {x}, \mathbf {y}; \zeta_ {t}) \right] \right\}. \tag {1} +$$ + +Further, we have + +$$ +f _ {t} (\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})) := \mathbb {E} _ {\xi_ {t} \sim \mathcal {D} _ {f, t}} \left[ f _ {t} (\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x}); \xi_ {t}) \right]. +$$ + +Here, $(\mathcal{D}_{f,t},\mathcal{D}_{g,t})$ denote data distributions at time $t$ . Our setting is stochastic, with only noisy evaluations of functions, gradients, and Hessians. Unlike OSO [72], where true losses are revealed, in OBO the outer function $f_{t}(\mathbf{x},\mathbf{y}_{t}^{*}(\mathbf{x}))$ is inaccessible for updating $\mathbf{x}_t$ and is generally non-convex in $\mathbf{x}$ , making standard regret notions from online convex optimization [33] unsuitable. + +Given a sequence $\{\alpha_{t}\in \mathbb{R}_{+ + }\}_{t = 1}^{T}$ , we define the following notion of bilevel local regret: + +$$ +\operatorname {B L} - \operatorname {R e g} _ {T} := \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \right], \quad \text {w i t h} \tag {2a} +$$ + +$$ +\mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right)\right) = \frac {1}{\alpha_ {t}} \left(\mathbf {x} _ {t} - \Pi_ {\mathcal {X}} \left[ \mathbf {x} _ {t} - \alpha_ {t} \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) \right]\right). \tag {2b} +$$ + +The local regret (2) compares the leader's decision $\mathbf{x}_t$ to the stationary points $\mathbf{x}_t^*$ satisfying $\mathcal{P}_{\mathcal{X},\alpha_t}(\mathbf{x}_t^*;\nabla f_t(\mathbf{x}_t^*,\mathbf{y}_t^*(\mathbf{x}_t^ {*}))) = 0$ . This can also be viewed as dynamic local regret, as the baseline corresponds to a stationary point of the leader's objective $f_{t}$ . + +Previous work on (nonconvex) OBO examined unconstrained local regret using window-smoothed objectives: $F_{t,w}(\mathbf{x},\mathbf{y}) = (1 / w)\sum_{i = 0}^{w - 1}f_{t - i}(\mathbf{x},\mathbf{y})$ . For $w = 1$ and $\mathcal{X} = \mathbb{R}^{d_1}$ , this reduces to (2). [67, 50] showed that $w = o(T)$ ensures sublinear regret under slow variations in $\{F_{t,w}\}_{t = 1}^{T}$ , while rapid changes can lead to deviations. However, smoothing may misrepresent regret (Figure 1). This paper introduces a new projection-based local regret notion (2) without smoothing, and establishes sublinear regret for constrained OBO. + +Online Gradient Descent (OGD). One of the most widely used algorithms for online (single-level) optimization is OGD [72]. The procedure for OGD is as follows: For each $t \in [T]$ , the algorithm selects $\mathbf{x}_t \in \mathcal{X}$ , observes the function $f_t: \mathcal{X} \subset \mathbb{R}^d \to \mathbb{R}$ , and updates according to + +![](images/fc7aa53b0daf2e0351ae661539087d54cab2dda8648f8652010076f7e06bc7f6.jpg) +Figure 1: Smoothly and rapidly changing $f_{t}$ in OBO with $g_{t}(x_{t},y_{t}) = (y_{t} - \cos (x_{t}))^{2}$ , $a_{t} = 1 + 0.5\sin (t)$ , $b_{t} = 1 + \sin (0.5t)$ , and $c_{t} = 10b_{t}$ . + +$$ +\mathbf {x} _ {t + 1} = \Pi_ {\mathcal {X}} \left(\mathbf {x} _ {t} - \alpha_ {t} \nabla f _ {t} (\mathbf {x} _ {t})\right), \quad \alpha_ {t} > 0. \tag {OGD} +$$ + +In the following, we adapt OGD to OBO and introduce a novel framework that requires limited feedback and can utilize ZO updates within a single-loop structure. + +To adapt OGD to OBO, [67, 50, 8] developed a variant alternating between inner and outer OGD, achieving sublinear bilevel regret bounds. We introduce a new search direction that enables sublinear bilevel regret without window smoothing. To compute the hypergradient $\nabla f_t(\mathbf{x},\mathbf{y}_t^* (\mathbf{x}))$ where $\mathbf{y}_t^* (\mathbf{x})$ is defined in (1), since $\nabla_{\mathbf{y}}g_{t}(\mathbf{x},\mathbf{y}_{t}^{*}(\mathbf{x})) = 0$ , using the implicit function theorem, yields + +$$ +\nabla f _ {t} (\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})) = \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})) + \nabla_ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})) \mathbf {v} _ {t} ^ {*} (\mathbf {x}), \tag {3} +$$ + +where $\mathbf{v}_t^* (\mathbf{x})\in \mathbb{R}^{d_2}$ is the solution to the following linear system: + +$$ +\nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})\right) \mathbf {v} _ {t} ^ {*} (\mathbf {x}) + \nabla_ {\mathbf {y}} f _ {t} \left(\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})\right) = 0. \tag {4} +$$ + +As the exact $\mathbf{y}_t^* (\mathbf{x})$ is not available, we estimate the hypergradient of $f_{t}$ at $(\mathbf{x},\mathbf{y})$ and introduce an auxiliary variable $\mathbf{v} := \mathbf{v}(\mathbf{x},\mathbf{y})$ to effectively decouple the nonlinear structure in $\nabla f_t(\mathbf{x},\mathbf{y}_t^* (\mathbf{x}))$ , i.e. + +$$ +\tilde {\nabla} f _ {t} (\mathbf {x}, \mathbf {y}) := \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x}, \mathbf {y}) + \nabla_ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x}, \mathbf {y}) \mathbf {v} _ {t}, \tag {5a} +$$ + +where $\mathbf{v}_t$ serves as an inexact solution to the linear system + +$$ +\nabla_ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x}, \mathbf {y}) \mathbf {v} _ {t} + \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x}, \mathbf {y}) = 0. \tag {5b} +$$ + +# Algorithm 1 SOGD + +Require: $(\mathbf{x}_1,\mathbf{y}_1,\mathbf{v}_1)\in \mathcal{X}\times \mathbb{R}^{d_2}\times \mathcal{Z}_p$ $p\in \mathbb{R}_{+ + };T\in \mathbb{N};$ stepsizes $\{(\alpha_t,\beta_t,\delta_t)\in \mathbb{R}_{+ + }^3\}_{t = 1}^T$ parameters $\{(\gamma_t,\lambda_t,\eta_t)\}_{t = 1}^T\in (0,1);\mathbf{z}_t\coloneqq (\mathbf{x}_t,\mathbf{y}_t).$ + +For $t = 1$ to $T$ do: + +S1. Draw samples $\mathcal{B}_t$ and $\bar{\mathcal{B}}_t$ with batch sizes $b$ and $\bar{b}$ . Get search directions $\mathbf{d}_t^{\mathbf{y}}$ , $\mathbf{d}_t^{\mathbf{v}}$ , and $\mathbf{d}_t^{\mathbf{x}}$ : + +$$ +\mathbf {d} _ {t} ^ {\mathcal {Y Y}} \left(\mathbf {z} _ {t}; \vec {B} _ {t}\right) = \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {z} _ {t}; \vec {B} _ {t}\right), \tag {9a} +$$ + +$$ +\mathbf {d} _ {t} ^ {\mathbf {y}} = \mathbf {d} _ {t} ^ {\mathbf {y y}} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t}) + (1 - \gamma_ {t}) \left(\mathbf {d} _ {t - 1} ^ {\mathbf {y}} - \mathbf {d} _ {t} ^ {\mathbf {y y}} (\mathbf {z} _ {t - 1}; \bar {\mathcal {B}} _ {t})\right), +$$ + +$$ +\mathbf {d} _ {t} ^ {\mathbf {v v}} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t}\right) = \nabla_ {\mathbf {y}} f _ {t} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t}\right) + \nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t}\right) \mathbf {v} _ {t}, \tag {9b} +$$ + +$$ +\mathbf {d} _ {t} ^ {\mathbf {v}} = \mathbf {d} _ {t} ^ {\mathbf {v v}} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t}\right) + (1 - \lambda_ {t}) \left(\mathbf {d} _ {t - 1} ^ {\mathbf {v}} - \mathbf {d} _ {t} ^ {\mathbf {v v}} \left(\mathbf {z} _ {t - 1}; \mathcal {B} _ {t}\right)\right), +$$ + +$$ +\mathbf {d} _ {t} ^ {\mathbf {x x}} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t}\right) = \nabla_ {\mathbf {x}} f _ {t} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t}\right) + \nabla_ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t}\right) \mathbf {v} _ {t}, \tag {9c} +$$ + +$$ +\mathbf {d} _ {t} ^ {\mathbf {x}} = \mathbf {d} _ {t} ^ {\mathbf {x x}} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t}\right) + (1 - \eta_ {t}) \left(\mathbf {d} _ {t - 1} ^ {\mathbf {x}} - \mathbf {d} _ {t} ^ {\mathbf {x x}} \left(\mathbf {z} _ {t - 1}; \mathcal {B} _ {t}\right)\right). +$$ + +S2. Update inner, system, and outer solutions: + +$$ +\mathbf {y} _ {t + 1} = \mathbf {y} _ {t} - \beta_ {t} \mathbf {d} _ {t} ^ {\mathbf {y}}, \mathbf {v} _ {t + 1} = \Pi_ {\mathcal {Z} _ {p}} \left[ \mathbf {v} _ {t} - \delta_ {t} \mathbf {d} _ {t} ^ {\mathbf {v}} \right], \mathbf {x} _ {t + 1} = \Pi_ {\mathcal {X}} \left[ \mathbf {x} _ {t} - \alpha_ {t} \mathbf {d} _ {t} ^ {\mathbf {x}} \right]. +$$ + +An accurate solution of (5b) is crucial for tight regret bounds. [67] assumes an exact solution, which is restrictive in large-scale settings. To address this, [50] proposed an efficient OBO algorithm with window averaging, using CG methods to solve (5b), which is equivalent to: + +$$ +\min _ {\mathbf {v} _ {t} \in \mathbb {R} ^ {d _ {2}}} \frac {1}{2} \left\| \nabla_ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x}, \mathbf {y}) \mathbf {v} _ {t} + \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x}, \mathbf {y}) \right\| ^ {2}. \tag {6} +$$ + +Next, we introduce a novel search direction that enables both first- and ZO stochastic OBO algorithms to achieve sublinear bilevel regret without smoothing. We first state the following lemma: + +Lemma 2.1. Let $w = t$ , $W = 1 / \eta$ and $\nu = 1 - \eta$ for $\eta \in (0,1)$ in the window-smoothed gradient $\nabla F_{t,\nu}(\mathbf{x}_t,\mathbf{y}_t;\mathcal{B}_t) = \frac{1}{W}\sum_{i = 0}^{w - 1}\nu^i\nabla f_{t - i}(\mathbf{x}_{t - i},\mathbf{y}_{t - i};\mathcal{B}_{t - i})$ , where $\mathcal{B}_t \coloneqq \{\xi_{t,1},\dots ,\xi_{t,b}\}$ is drawn i.i.d. from $\mathcal{D}_{f,t}$ . Then, $\nabla F_{t,\nu}(\mathbf{x}_t,\mathbf{y}_t;\mathcal{B}_t) = \sum_{j = 1}^{t}\eta (1 - \eta)^{t - j}\nabla f_j(\mathbf{x}_j,\mathbf{y}_j;\mathcal{B}_j)$ , and we have $\nabla F_{t,\nu}(\mathbf{x}_t,\mathbf{y}_t;\mathcal{B}_t) = \mathbf{d}_t^\mathbf{x}$ with $\mathbf{d}_t^\mathbf{x} = \eta \nabla f_t(\mathbf{x}_t,\mathbf{y}_t;\mathcal{B}_t) + (1 - \eta)\mathbf{d}_{t - 1}^\mathbf{x}$ , and $\mathbf{d}_1^\mathbf{x} = \frac{1}{W}\nabla f_1(\mathbf{x}_1,\mathbf{y}_1;\mathcal{B}_1)$ for all $t\geq 2$ . + +Proof is given in Appendix C.1. As shown in Lemma 2.1, for a specific choice of $w$ and $W$ , the time-smoothed gradient forms a recursive momentum-type search direction. However, achieving sublinear regret in stochastic OBO requires large-window smoothing ( $w = o(T)$ ) [67, 50, 8]. To address this, we propose the following search direction: + +$$ +\mathbf {d} _ {t} ^ {\mathbf {x}} = \eta \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}; \mathcal {B} _ {t}\right) + (1 - \eta) \mathbf {d} _ {t - 1} ^ {\mathbf {x}} + (1 - \eta) (\nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}; \mathcal {B} _ {t}\right) - \nabla f _ {t} \left(\mathbf {x} _ {t - 1}, \mathbf {y} _ {t - 1}; \mathcal {B} _ {t}\right)). \tag {7} +$$ + +This direction is used for updating $\mathbf{x}$ , with similar updates for $\mathbf{y}$ and $\mathbf{v}$ , as discussed below and detailed in Algorithm 1. The quadratic formulation of (5b) in (6) motivates single-loop methods such as [15]. Building on this, we propose Simultaneous Online Gradient Descent (SOGD) for constrained OBO, presented in Algorithm 1. At each step, SOGD jointly updates the follower variable $\mathbf{y}_t$ , auxiliary variable $\mathbf{v}_t$ , and leader variable $\mathbf{x}_t$ using batches $\mathcal{B}_t = \{\xi_{t,1},\dots,\xi_{t,b}\}$ and $\bar{\mathcal{B}}_t\coloneqq \{\zeta_{t,1},\dots,\zeta_{t,\bar{b}}\}$ sampled i.i.d. from $\mathcal{D}_{f,t}$ and $\mathcal{D}_{g,t}$ . Step S1. only requires computing Hessian-vector products, avoiding explicit computation of $\nabla_{\mathbf{y}}^{2}g_{t}$ or $\nabla_{\mathbf{xy}}^{2}g_{t}$ . Step S2. uses the projection: + +$$ +\Pi_ {\mathcal {Z} _ {p}} (\mathbf {v}) = \operatorname {a r g m i n} _ {\mathbf {z} \in \mathcal {Z} _ {p}} \frac {1}{2} \| \mathbf {v} - \mathbf {z} \| ^ {2} = \min \left\{1, \frac {p}{\| \mathbf {v} \|} \right\} \mathbf {v}, \quad \text {w h e r e} +$$ + +$$ +\mathcal {Z} _ {p} := \left\{\mathbf {v} \in \mathbb {R} ^ {d _ {2}} \mid \| \mathbf {v} \| \leq p \right\}. \tag {8} +$$ + +Unlike OAGD [67] with alternating loops, and SOBOW [50] using CG, SOGD performs a single OGD step for all variables. + +Assumption 2.2. $g_{t}(\mathbf{x},\mathbf{y})$ is twice continuously differentiable and $\mu_g$ -strongly convex in $\mathbf{y}$ for all $\mathbf{x}\in \mathcal{X},t\in [T]$ . + +Assumption 2.3. Let $\mathbf{z} = [\mathbf{x};\mathbf{y}]$ and $\mathbf{z}' = [\mathbf{x}';\mathbf{y}']$ , where $\mathbf{x},\mathbf{x}'\in \mathcal{X}$ and $\mathbf{y},\mathbf{y}'\in \mathbb{R}^{d_2}$ . For any $\mathbf{z},\mathbf{z}'$ and $t\in [T]$ : + +B1. $\exists \ell_{f,0} \in \mathbb{R}_{+}$ s.t. $\| f_{t}(\mathbf{z};\xi_{t}) - f_{t}(\mathbf{z}^{\prime};\xi_{t}) \| \leq \ell_{f,0} \| \mathbf{z} - \mathbf{z}^{\prime} \|$ ; +B2. $\exists \ell_{f,1}\in \mathbb{R}_{+}$ s.t. $\| \nabla f_t(\mathbf{z};\xi_t) - \nabla f_t(\mathbf{z}';\xi_t)\| \leq \ell_{f,1}\| \mathbf{z} - \mathbf{z}'\|$ +B3. $\exists \ell_{g,1} \in \mathbb{R}_{+}$ s.t. $\|\nabla g_t(\mathbf{z};\zeta_t) - \nabla g_t(\mathbf{z}';\zeta_t)\| \leq \ell_{g,1}\|\mathbf{z} - \mathbf{z}'\|$ ; +B4. $\exists \ell_{g,2} \in \mathbb{R}_{+}$ s.t. $\| \nabla^{2} g_{t}(\mathbf{z};\zeta_{t}) - \nabla^{2} g_{t}(\mathbf{z}^{\prime};\zeta_{t}) \| \leq \ell_{g,2} \| \mathbf{z} - \mathbf{z}^{\prime} \|$ . + +Assumption 2.4. For any $t\in [T]$ $|f_{t}(\mathbf{x},\mathbf{y}_{t}^{*}(\mathbf{x}))|\leq M$ for some $M\in \mathbb{R}_{+ + }$ and any $\mathbf{x}\in \mathcal{X}$ + +Assumption 2.5. There exist constants $\sigma_{g_{\mathbf{y}}},\sigma_{g_{\mathbf{yy}}},\sigma_{g_{\mathbf{xy}}},\sigma_{f_{\mathbf{y}}},\sigma_{f_{\mathbf{x}}}$ such that, for all $\mathbf{z} = [\mathbf{x},\mathbf{y}]$ : + +C1. $\mathbb{E}\| \nabla_{\mathbf{y}}g_t(\mathbf{z};\zeta_t) - \nabla_{\mathbf{y}}g_t(\mathbf{z})\| ^2\leq \sigma_{g_{\mathbf{y}}}^2$ + +C4. $\mathbb{E}\| \nabla_{\mathbf{y}}f_t(\mathbf{z};\xi_t) - \nabla_{\mathbf{y}}f_t(\mathbf{z})\|^2\leq \sigma_{f_\mathbf{y}}^2;$ + +C2. $\mathbb{E}\| \nabla_{\mathbf{y}}^{2}g_{t}(\mathbf{z};\zeta_{t}) - \nabla_{\mathbf{y}}^{2}g_{t}(\mathbf{z})\|^{2}\leq \sigma_{gyx}^{2}$ + +C5. $\mathbb{E}\| \nabla_{\mathbf{x}}f_t(\mathbf{z};\xi_t) - \nabla_{\mathbf{x}}f_t(\mathbf{z})\| ^2\leq \sigma_{f_x}^2$ + +C3. $\mathbb{E}\| \nabla_{\mathbf{x}\mathbf{y}}^2 g_t(\mathbf{z};\zeta_t) - \nabla_{\mathbf{x}\mathbf{y}}^2 g_t(\mathbf{z})\|^2\leq \sigma_{g_{\mathbf{x}\mathbf{y}}}^2;$ + +Throughout this paper, we define + +$$ +\sigma^ {2} := \sigma_ {g _ {\mathbf {y}}} ^ {2} + \sigma_ {g _ {\mathbf {y y}}} ^ {2} + \sigma_ {f _ {\mathbf {y}}} ^ {2} + \sigma_ {g _ {\mathbf {x y}}} ^ {2} + \sigma_ {f _ {\mathbf {x}}} ^ {2}. \tag {10} +$$ + +Assumptions 2.2 and 2.3 are standard in BO [12, 43] and OBO [67], and hold for many bilevel ML problems [22]. Assumption 2.4 is typical in non-convex OSO [36, 50], while Assumption 2.5 assumes unbiased stochastic gradient, Hessian, and Jacobian estimators with bounded variance [12]. + +Achieving sublinear dynamic regret is generally infeasible under arbitrary time variations [7]. Prior analyses [67, 50] bound regret by enforcing regularity on the comparator sequence. To attain sublinear regret, [67] introduces the following regularity metrics for bilevel sequences: + +$$ +H _ {p, T} := \sum_ {t = 2} ^ {T} \sup _ {\mathbf {x} \in \mathcal {X}} \| \mathbf {y} _ {t - 1} ^ {*} (\mathbf {x}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x}) \| ^ {p}, \quad V _ {T} := \sum_ {t = 2} ^ {T} \sup _ {\mathbf {x} \in \mathcal {X}} | f _ {t - 1} (\mathbf {x}, \mathbf {y} _ {t - 1} ^ {*} (\mathbf {x})) - f _ {t} (\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})) |. \tag {11} +$$ + +Path-length $H_{p,T}$ measures changes in the follower's costs, while $V_{T}$ captures the leader's objective smoothness. We use path-length for the follower and function variation for the leader due to the follower's strong convexity (Assumption 2.2) versus the leader's nonconvexity. Another regularity is the sequential gradient difference of the outer objective: + +$$ +D _ {\mathbf {x}, T} := \sum_ {t = 2} ^ {T} \sup _ {\mathbf {x}, \mathbf {y}} \left\| \nabla_ {\mathbf {x}} f _ {t - 1} (\mathbf {x}, \mathbf {y}) - \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x}, \mathbf {y}) \right\| ^ {2}, \tag {12a} +$$ + +$$ +D _ {\mathbf {y}, T} := \sum_ {t = 2} ^ {T} \sup _ {\mathbf {x}, \mathbf {y}} \left\| \nabla_ {\mathbf {y}} f _ {t - 1} (\mathbf {x}, \mathbf {y}) - \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x}, \mathbf {y}) \right\| ^ {2}. \tag {12b} +$$ + +As in [41, 31], $D_{\mathbf{x},T}$ and $D_{\mathbf{y},T}$ measure the gradient drift of $f_{t}$ relative to $f_{t - 1}$ for $\mathbf{x}$ and $\mathbf{y}$ , respectively. We define deviations in the gradient, Hessian, and Jacobian of the inner objective as: + +$$ +G _ {\mathbf {y}, T} := \sum_ {t = 2} ^ {T} \left\| \nabla_ {\mathbf {y}} g _ {t - 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2}, \quad G _ {\mathbf {y y}, T} := \sum_ {t = 2} ^ {T} \left\| \nabla_ {\mathbf {y}} ^ {2} g _ {t - 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2}, +$$ + +$$ +G _ {\mathbf {x y}, T} := \sum_ {t = 2} ^ {T} \left\| \nabla_ {\mathbf {x y}} ^ {2} g _ {t - 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right\| ^ {2}. \tag {13} +$$ + +We introduce the following notations for simplicity: + +$$ +\Delta_ {T} := E _ {1} + V _ {T}, \quad \Psi_ {T} := H _ {2, T} + G _ {T} + D _ {T}, \tag {14} +$$ + +where $(V_{T},H_{p,T})$ are defined in (11), and + +$$ +E _ {1} := \left\| \mathbf {y} _ {1} - \mathbf {y} _ {1} ^ {*} (\mathbf {x} _ {1}) \right\| ^ {2} + \left\| \mathbf {v} _ {1} - \mathbf {v} _ {1} ^ {*} (\mathbf {x} _ {1}) \right\| ^ {2}, \quad G _ {T} := G _ {\mathbf {y}, T} + G _ {\mathbf {y y}, T} + G _ {\mathbf {x y}, T}, \tag {15} +$$ + +$$ +D _ {T} := D _ {\mathbf {x}, T} + D _ {\mathbf {y}, T}. +$$ + +By accounting for both $D_{T}$ and $G_{T}$ , we can represent the variations in the environments of OBO. + +Theorem 2.6. Let $\{(f_t, g_t)\}_{t=1}^T$ be the sequence of functions presented to Algorithm 1, satisfying Assumptions 2.2-2.5. For all $t \in [T]$ , let + +$$ +\alpha_ {t} = \frac {1}{(c + t) ^ {1 / 3}}, \quad \beta_ {t} = c _ {\beta} \alpha_ {t}, \quad \delta_ {t} = c _ {\delta} \alpha_ {t}, \quad b = \bar {b} = 1, +$$ + +$$ +\gamma_ {t + 1} = c _ {\gamma} \alpha_ {t} ^ {2}, \quad \eta_ {t + 1} = c _ {\eta} \alpha_ {t} ^ {2}, \quad \lambda_ {t + 1} = c _ {\lambda} \alpha_ {t} ^ {2}. \tag {16} +$$ + +Here, $c$ , $c_{\beta}$ , $c_{\delta}$ , $c_{\gamma}$ , $c_{\eta}$ , and $c_{\lambda}$ are specified in (107). Algorithm 1 guarantees: + +$$ +\operatorname {B L} - \operatorname {R e g} _ {T} \leq \mathcal {O} \left(T ^ {\frac {1}{3}} \left(\sigma^ {2} + \Delta_ {T}\right) + T ^ {\frac {2}{3}} \Psi_ {T}\right), \tag {17} +$$ + +where $\sigma$ and $(\Delta_T, \Psi_T)$ are defined in (10) and (14). + +Remark 2.7 (Stochastic Regret Guarantee for OBO and OSO with $w = 1$ ). Theorem 2.6 bounds the regret of Algorithm 1 without window-smoothing, based on the regularities in (14). We note that the average dynamic regret $\mathrm{BL - Reg}_T / T \leq \mathcal{O}(T^{-2/3}(\sigma^2 + \Delta_T) + T^{-1/3}\Psi_T)$ remains sublinear under suitable conditions on $\Delta_T$ , $\Psi_T$ , and $\sigma$ . Specifically, if $\Delta_T = o(T^{2/3})$ , $\Psi_T = o(T^{1/3})$ , and $\sigma = o(T^{1/3})$ , then the dynamic regret grows sublinearly, i.e., $\mathrm{BL - Reg}_T = o(T)$ ; see Appendix B.2 for further examples and discussion. This result also yields a sharper $T^{-2/3}\sigma^2$ regret—improving over the $T^{-1/2}\sigma^2$ bound for stochastic OBO [8]—and removes the need for window-smoothing [8, 67, 50, 40]. For OSO, this result surpasses the $T^{-1/2}\sigma^2$ rate in [31]. + +# 3 Stochastic OBO with Zeroth-Order Oracles + +Black-box optimization arises when gradients are unavailable [11]. We study ZO-OBO methods with limited access to leader and follower objectives. Let $\mathbf{s} \in \mathbb{R}^{d_1}$ and $\mathbf{r} \in \mathbb{R}^{d_2}$ be vectors uniformly sampled from unit balls $B_1$ and $B_2$ . Given smoothing parameters $\rho = (\rho_{\mathbf{s}}, \rho_{\mathbf{r}})$ , we define Gaussian-smoothed objectives using [59]: + +$$ +f _ {t, \boldsymbol {\rho}} \left(\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})\right) = \underset {\left(\mathbf {s}, \mathbf {r}, \xi_ {t}\right)} {\mathbb {E}} \left[ f _ {t} \left(\mathbf {x} + \rho_ {\mathbf {s}} \mathbf {s}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) + \rho_ {\mathbf {r}} \mathbf {r}; \xi_ {t}\right) \right], \quad \text {w h e r e} \tag {18} +$$ + +$$ +\hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \in \underset {\mathbf {y} \in \mathbb {R} ^ {d _ {2}}} {\arg \min } \left\{g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}) := \underset {(\mathbf {s}, \mathbf {r}, \zeta_ {t})} {\mathbb {E}} \left[ g _ {t} \left(\mathbf {x} + \rho_ {\mathbf {s}} \mathbf {s}, \mathbf {y} + \rho_ {\mathbf {r}} \mathbf {r}; \zeta_ {t}\right) \right] \right\}. \tag {19} +$$ + +To solve stochastic OBO with (18), we need to obtain the hyper-gradient of $f_{t,\rho}$ in (18) at $(\mathbf{x},\mathbf{y})$ as + +$$ +\nabla f _ {t, \boldsymbol {\rho}} (\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})) := \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} (\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})) + \nabla_ {\mathbf {x y}} ^ {2} g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})) \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}), \quad \text {w h e r e} +$$ + +$$ +\hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \text {i s} \nabla_ {\mathbf {y}} ^ {2} g _ {t, \rho} \left(\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})\right) \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) + \nabla_ {\mathbf {y}} f _ {t, \rho} \left(\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})\right) = 0. \tag {20} +$$ + +Obtaining $\hat{\mathbf{y}}_t^* (\mathbf{x})$ in closed-form is usually a challenging task, so it is natural to use the following gradient surrogate. At any $(\mathbf{x},\mathbf{y})$ , we introduce an auxiliary variable $\mathbf{v} = \mathbf{v}(\mathbf{x},\mathbf{y})$ and define: + +$$ +\tilde {\nabla} f _ {t, \rho} (\mathbf {x}, \mathbf {y}) := \nabla_ {\mathbf {x}} f _ {t, \rho} (\mathbf {x}, \mathbf {y}) + \nabla_ {\mathbf {x y}} ^ {2} g _ {t, \rho} (\mathbf {x}, \mathbf {y}) \mathbf {v}, \quad \text {w h e r e} \tag {21a} +$$ + +$$ +\mathbf {v} \text {i s t h e s o l u t i o n t o} \nabla_ {\mathbf {y}} ^ {2} g _ {t, \rho} (\mathbf {x}, \mathbf {y}) \mathbf {v} + \nabla_ {\mathbf {y}} f _ {t, \rho} (\mathbf {x}, \mathbf {y}) = 0. \tag {21b} +$$ + +To do so, we also introduce $\mathbf{d}_{t,\rho}^{\mathbf{y}},\mathbf{d}_{t,\rho}^{\mathbf{v}}$ and $\mathbf{d}_{t,\rho}^{\mathbf{x}}$ as follows: + +$$ +\mathbf {d} _ {t, \rho} ^ {\mathbf {y}} (\mathbf {x}, \mathbf {y}) = \nabla_ {\mathbf {y}} g _ {t, \rho} (\mathbf {x}, \mathbf {y}), \tag {22a} +$$ + +$$ +\mathbf {d} _ {t, \rho} ^ {\mathbf {v}} (\mathbf {x}, \mathbf {y}, \mathbf {v}) = \nabla_ {\mathbf {y}} f _ {t, \rho} (\mathbf {x}, \mathbf {y}) + \nabla_ {\mathbf {y}} ^ {2} g _ {t, \rho} (\mathbf {x}, \mathbf {y}) \mathbf {v}, \tag {22b} +$$ + +$$ +\mathbf {d} _ {t, \rho} ^ {\mathbf {x}} (\mathbf {x}, \mathbf {y}, \mathbf {v}) = \nabla_ {\mathbf {x}} f _ {t, \rho} (\mathbf {x}, \mathbf {y}) + \nabla_ {\mathbf {x y}} ^ {2} g _ {t, \rho} (\mathbf {x}, \mathbf {y}) \mathbf {v}. \tag {22c} +$$ + +Next, we approximate these directions using stochastic zeroth-order oracles (SZO), which produce the quantities $\hat{\nabla}_{\mathbf{y}}f_{t}(\mathbf{x},\mathbf{y};\xi_{t})$ , $\hat{\nabla}_{\mathbf{y}}g_{t}(\mathbf{x},\mathbf{y};\zeta_{t})$ , $\hat{\nabla}_{\mathbf{x}}f_{t}(\mathbf{x},\mathbf{y};\xi_{t})$ , and $\hat{\nabla}_{\mathbf{x}}g_{t}(\mathbf{x},\mathbf{y};\zeta_{t})$ . These are unbiased estimators of the true gradients $\nabla_{\mathbf{y}}f_{t,\rho}(\mathbf{x},\mathbf{y})$ , $\nabla_{\mathbf{y}}g_{t,\rho}(\mathbf{x},\mathbf{y})$ , $\nabla_{\mathbf{x}}f_{t,\rho}(\mathbf{x},\mathbf{y})$ , and $\nabla_{\mathbf{x}}g_{t,\rho}(\mathbf{x},\mathbf{y})$ respectively, as shown in [20], such that the following assumption holds: + +$$ +\underset {(\mathbf {r}, \boldsymbol {\xi} _ {t})} {\mathbb {E}} \left[ \hat {\nabla} _ {\mathbf {y}} f _ {t} (\mathbf {x}, \mathbf {y}; \boldsymbol {\xi} _ {t}) \right] = \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}), \quad \underset {(\mathbf {s}, \boldsymbol {\xi} _ {t})} {\mathbb {E}} \left[ \hat {\nabla} _ {\mathbf {x}} f _ {t} (\mathbf {x}, \mathbf {y}; \boldsymbol {\xi} _ {t}) \right] = \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}), +$$ + +$$ +\underset {(\mathbf {r}, \zeta_ {t})} {\mathbb {E}} \left[ \hat {\nabla} _ {\mathbf {y}} g _ {t} (\mathbf {x}, \mathbf {y}; \zeta_ {t}) \right] = \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}), \quad \underset {(\mathbf {s}, \zeta_ {t})} {\mathbb {E}} \left[ \hat {\nabla} _ {\mathbf {x}} g _ {t} (\mathbf {x}, \mathbf {y}; \zeta_ {t}) \right] = \nabla_ {\mathbf {x}} g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}). \tag {23} +$$ + +Specifically, following [62], we estimate the gradient of a function $h: \mathbb{R}^d \to \mathbb{R}$ , querying at $\mathbf{x} - \lambda \mathbf{s}$ and $\mathbf{x} + \lambda \mathbf{s}$ , yielding an estimator $(d/2\lambda) \left( h(\mathbf{x} + \lambda \mathbf{s}) - h(\mathbf{x} - \lambda \mathbf{s}) \right) \mathbf{s}$ . Using this strategy, the finite-difference estimation of $\nabla g_{t,\rho}(\mathbf{x}, \mathbf{y})$ , denoted by $\hat{\nabla} g_t(\mathbf{x},\mathbf{y})$ , is constructed for given smoothing + +# Algorithm 2 ZO-SOGD + +Require: In addition to parameters in SOGD, choose $\rho_{\mathbf{v}},\rho_{\mathbf{r}},\rho_{\mathbf{s}}\in \mathbb{R}_{+ + }$ + +For $t = 1$ to $T$ do: + +S1. Draw samples $\mathcal{B}_t$ and $\bar{\mathcal{B}}_t$ with batch sizes $b$ and $\bar{b}$ . Using (24)-(26), get: + +$$ +\mathbf {d} _ {t} ^ {\mathcal {Y}} \left(\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t}\right) = \hat {\nabla} _ {\mathcal {Y}} g _ {t} \left(\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t}\right), \tag {27a} +$$ + +$$ +\hat {\mathbf {d}} _ {t} ^ {\mathbf {y}} = \mathbf {d} _ {t} ^ {\mathbf {y}} (\mathbf {z} _ {t}; \bar {B} _ {t}) + (1 - \gamma_ {t}) (\hat {\mathbf {d}} _ {t - 1} ^ {\mathbf {y}} - \mathbf {d} _ {t} ^ {\mathbf {y}} (\mathbf {z} _ {t - 1}; \bar {B} _ {t})), +$$ + +$$ +\mathbf {d} _ {t} ^ {\mathbf {v v}} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t}\right) = \hat {\nabla} _ {\mathbf {y}} f _ {t} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t}\right) + \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t}\right), \tag {27b} +$$ + +$$ +\hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} = \mathbf {d} _ {t} ^ {\mathbf {v v}} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t}\right) + (1 - \lambda_ {t}) \left(\hat {\mathbf {d}} _ {t - 1} ^ {\mathbf {v}} - \mathbf {d} _ {t} ^ {\mathbf {v v}} \left(\mathbf {z} _ {t - 1}; \mathcal {B} _ {t}\right)\right), +$$ + +$$ +\mathbf {d} _ {t} ^ {\mathbf {x y}} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t}\right) = \hat {\nabla} _ {\mathbf {x}} f _ {t} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t}\right) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t}\right), \tag {27c} +$$ + +$$ +\hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} = \mathbf {d} _ {t} ^ {\mathbf {x y}} (\mathbf {z} _ {t}; \mathcal {B} _ {t}) + (1 - \eta_ {t}) (\hat {\mathbf {d}} _ {t - 1} ^ {\mathbf {x}} - \mathbf {d} _ {t} ^ {\mathbf {x y}} (\mathbf {z} _ {t - 1}; \mathcal {B} _ {t})), +$$ + +S2. Update inner, system, and outer solutions: + +$$ +\mathbf {y} _ {t + 1} = \mathbf {y} _ {t} - \beta_ {t} \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}}, \quad \mathbf {v} _ {t + 1} = \Pi_ {\mathcal {Z} _ {p}} \left[ \mathbf {v} _ {t} - \delta_ {t} \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} \right], \quad \mathbf {x} _ {t + 1} = \Pi_ {\mathcal {X}} \left[ \mathbf {x} _ {t} - \alpha_ {t} \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} \right]. +$$ + +parameters $\pmb {\rho} = (\rho_{\mathbf{s}},\rho_{\mathbf{r}})$ , and a set $\bar{B}_t = \{\zeta_{t,1},\dots ,\zeta_{t,\bar{b}}\}$ drawn i.i.d. from $\mathcal{D}_{g,t}$ , as follows: + +$$ +\hat {\nabla} _ {\mathbf {y}} g _ {t} (\mathbf {x}, \mathbf {y}; \bar {\mathcal {B}} _ {t}) := \frac {d _ {2}}{2 \bar {b} \rho_ {\mathbf {r}}} \sum_ {i = 1} ^ {\bar {b}} \left(g _ {t} \left(\mathbf {x}, \mathbf {y} + \rho_ {\mathbf {r}} \mathbf {r} _ {i}; \zeta_ {t, i}\right) - g _ {t} \left(\mathbf {x}, \mathbf {y} - \rho_ {\mathbf {r}} \mathbf {r} _ {i}; \zeta_ {t, i}\right)\right) \mathbf {r} _ {i}, \tag {24a} +$$ + +$$ +\hat {\nabla} _ {\mathbf {x}} g _ {t} (\mathbf {x}, \mathbf {y}; \bar {\mathcal {B}} _ {t}) := \frac {d _ {1}}{2 \bar {b} \rho_ {\mathbf {s}}} \sum_ {i = 1} ^ {\bar {b}} \left(g _ {t} \left(\mathbf {x} + \rho_ {\mathbf {s}} \mathbf {s} _ {i}, \mathbf {y}; \zeta_ {t, i}\right) - g _ {t} \left(\mathbf {x} - \rho_ {\mathbf {s}} \mathbf {s} _ {i}, \mathbf {y}; \zeta_ {t, i}\right)\right) \mathbf {s} _ {i}. \tag {24b} +$$ + +Similarly, we estimate $\nabla_{\mathbf{y}}f_{t,\rho}(\mathbf{x},\mathbf{y};\mathcal{B}_t)$ and $\nabla_{\mathbf{x}}f_{t,\rho}(\mathbf{x},\mathbf{y};\mathcal{B}_t)$ , respectively, using a batch $\mathcal{B}_t = \{\xi_{t,1},\dots ,\xi_{t,b}\}$ drawn i.i.d. from $\mathcal{D}_{f,t}$ , by + +$$ +\hat {\nabla} _ {\mathbf {y}} f _ {t} (\mathbf {x}, \mathbf {y}; \mathcal {B} _ {t}) := \frac {d _ {2}}{2 b \rho_ {\mathbf {r}}} \sum_ {i = 1} ^ {b} \left(f _ {t} \left(\mathbf {x}, \mathbf {y} + \rho_ {\mathbf {r}} \mathbf {r} _ {i}; \xi_ {t, i}\right) - f _ {t} \left(\mathbf {x}, \mathbf {y} - \rho_ {\mathbf {r}} \mathbf {r} _ {i}; \xi_ {t, i}\right)\right) \mathbf {r} _ {i}, \tag {25a} +$$ + +$$ +\hat {\nabla} _ {\mathbf {x}} f _ {t} (\mathbf {x}, \mathbf {y}; \mathcal {B} _ {t}) := \frac {d _ {1}}{2 b \rho_ {\mathbf {s}}} \sum_ {i = 1} ^ {b} \left(f _ {t} \left(\mathbf {x} + \rho_ {\mathbf {s}} \mathbf {s} _ {i}, \mathbf {y}; \xi_ {t, i}\right) - f _ {t} \left(\mathbf {x} - \rho_ {\mathbf {s}} \mathbf {s} _ {i}, \mathbf {y}; \xi_ {t, i}\right) \mathbf {s} _ {i}. \right. \tag {25b} +$$ + +Furthermore, given a smoothing parameter $\rho_{\mathbf{v}} > 0$ , we approximate the Hessian-vector product $\nabla_{\mathbf{y}}^{2}g_{t,\rho}(\mathbf{x},\mathbf{y})\mathbf{v}$ and the Jacobian-vector product $\nabla_{\mathbf{xy}}^{2}g_{t,\rho}(\mathbf{x},\mathbf{y})\mathbf{v}$ as the finite difference between two gradients, respectively, as + +$$ +\hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x}, \mathbf {y}; \bar {\mathcal {B}} _ {t}) := \frac {1}{2 \bar {b} \rho_ {\mathbf {v}}} \sum_ {i = 1} ^ {\bar {b}} \left(\hat {\nabla} _ {\mathbf {y}} g _ {t} \left(\mathbf {x}, \mathbf {y} + \rho_ {\mathbf {v}} \mathbf {v}; \zeta_ {t, i}\right) - \hat {\nabla} _ {\mathbf {y}} g _ {t} \left(\mathbf {x}, \mathbf {y} - \rho_ {\mathbf {v}} \mathbf {v}; \zeta_ {t, i}\right)\right), \tag {26a} +$$ + +$$ +\hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x}, \mathbf {y}; \bar {\mathcal {B}} _ {t}) := \frac {1}{2 \bar {b} \rho_ {\mathbf {v}}} \sum_ {i = 1} ^ {\bar {b}} \left(\hat {\nabla} _ {\mathbf {x}} g _ {t} \left(\mathbf {x}, \mathbf {y} + \rho_ {\mathbf {v}} \mathbf {v}; \zeta_ {t, i}\right) - \hat {\nabla} _ {\mathbf {x}} g _ {t} \left(\mathbf {x}, \mathbf {y} - \rho_ {\mathbf {v}} \mathbf {v}; \zeta_ {t, i}\right)\right). \tag {26b} +$$ + +Using (24)-(26), the first-order terms in (9) are approximated by $\hat{\mathbf{d}}_t^{\mathbf{y}}$ , $\hat{\mathbf{d}}_t^{\mathbf{v}}$ , and $\hat{\mathbf{d}}_t^{\mathbf{x}}$ in (27). The approximations in (26a) and (26b) introduce errors in the hypergradient, which must be controlled. (26) depends on the dimension of $\mathbf{y}$ , as in ZO optimization [59, 62]. The projection $\Pi_{\mathcal{Z}_p}$ in (8) bounds $\mathbf{v}$ , controlling variance in $\mathbf{v}$ and $\mathbf{x}$ updates for convergence. + +Assumption 3.1. There exist constants $\hat{\sigma}_{g_{\mathbf{y}}},\hat{\sigma}_{g_{\mathbf{x}}},\hat{\sigma}_{f_{\mathbf{y}}},\hat{\sigma}_{f_{\mathbf{x}}}$ such that, for all $\mathbf{z} = [\mathbf{x},\mathbf{y}]$ : + +D1. $\mathbb{E}\| \hat{\nabla}_{\mathbf{y}}g_t(\mathbf{z};\zeta_t) - \nabla_{\mathbf{y}}g_{t,\pmb {\rho}}(\mathbf{z})\| ^2\leq \hat{\sigma}_{g_{\mathbf{y}}}^2$ + +D3. $\mathbb{E}\| \hat{\nabla}_{\mathbf{y}}f_t(\mathbf{z};\xi_t) - \nabla_{\mathbf{y}}f_{t,\pmb {\rho}}(\mathbf{z})\| ^2\leq \hat{\sigma}_{f_{\mathbf{y}}}^2$ + +D2. $\mathbb{E}\| \hat{\nabla}_{\mathbf{x}}g_t(\mathbf{z};\zeta_t) - \nabla_{\mathbf{x}}g_{t,\pmb {\rho}}(\mathbf{z})\| ^2\leq \hat{\sigma}_{g_\mathbf{x}}^2$ + +D4. $\mathbb{E}\| \hat{\nabla}_{\mathbf{x}}f_t(\mathbf{z};\xi_t) - \nabla_{\mathbf{x}}f_{t,\pmb {\rho}}(\mathbf{z})\| ^2\leq \hat{\sigma}_{f_{\mathbf{x}}}^2$ + +Assumption 3.1 is analogous to the upper bound on the variance of stochastic partial gradients discussed in [54, 68]. We simplify the notation by introducing the following shorthand. + +$$ +\hat {\sigma} ^ {2} := \hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2} + \hat {\sigma} _ {g _ {\mathbf {x}}} ^ {2} + \hat {\sigma} _ {f _ {\mathbf {y}}} ^ {2} + \hat {\sigma} _ {f _ {\mathbf {x}}} ^ {2}. \tag {28} +$$ + +Next, we establish a regret bound for ZO-SOGD. Similar to the previous results, we introduce regularity conditions for the smoothed functions defined in (18) and (19). + +Inner Gradient Variations: In ZO setting, we use a set of gradient variations at the perturbed point as follows: + +$$ +G _ {\mathbf {v}, T} := \sum_ {t = 2} ^ {T} \left(\chi_ {1 t} + \chi_ {2 t}\right), \quad G _ {\mathbf {x}, T} := \sum_ {t = 2} ^ {T} \left(\chi_ {3 t} + \chi_ {4 t}\right), \tag {29} +$$ + +where $\mathbf{z}_t^+ := (\mathbf{x}_{t-1}, \mathbf{y}_{t-1} + \rho_{\mathbf{v}} \mathbf{v}_{t-1})$ , $\mathbf{z}_t^- := (\mathbf{x}_{t-1}, \mathbf{y}_{t-1} - \rho_{\mathbf{v}} \mathbf{v}_{t-1})$ , and + +$$ +\chi_ {1 t} := \left\| \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {z} _ {t} ^ {+}\right) - \nabla_ {\mathbf {y}} g _ {t - 1} \left(\mathbf {z} _ {t} ^ {+}\right) \right\| ^ {2}, \quad \chi_ {2 t} := \left\| \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {z} _ {t} ^ {-}\right) - \nabla_ {\mathbf {y}} g _ {t - 1} \left(\mathbf {z} _ {t} ^ {-}\right) \right\| ^ {2}, +$$ + +$$ +\chi_ {3 t} := \left\| \nabla_ {\mathbf {x}} g _ {t} \left(\mathbf {z} _ {t} ^ {+}\right) - \nabla_ {\mathbf {x}} g _ {t - 1} \left(\mathbf {z} _ {t} ^ {+}\right) \right\| ^ {2}, \quad \chi_ {4 t} := \left\| \nabla_ {\mathbf {x}} g _ {t} \left(\mathbf {z} _ {t} ^ {-}\right) - \nabla_ {\mathbf {x}} g _ {t - 1} \left(\mathbf {z} _ {t} ^ {-}\right) \right\| ^ {2}. +$$ + +Further, for simplicity of notation, we define + +$$ +\hat {\Delta} _ {T} := E _ {1} + V _ {T} + D _ {T} + G _ {\mathbf {y}, T}, \quad \hat {\Psi} _ {T} := H _ {2, T} + G _ {\mathbf {v}, T} + G _ {\mathbf {x}, T}, \tag {30} +$$ + +where $(V_{T},H_{p,T})$ and $(E_1,D_T)$ are defined in (11), and (15), respectively. Moreover, $G_{\mathbf{y},T}$ and $(G_{\mathbf{v},T},G_{\mathbf{x},T})$ are defined in (13) and (29), respectively. + +Theorem 3.2. Let $\{(f_t,g_t)\}_{t = 1}^T$ be the sequence of functions presented to Algorithm 2, satisfying Assumptions 2.2-2.4 and 3.1. For all $t\in [T]$ , let + +$$ +\alpha_ {t} = \frac {1}{(d _ {1} + d _ {2}) ^ {3 / 4} (c + t) ^ {1 / 3}}, \quad \beta_ {t} = c _ {\beta} \alpha_ {t}, \quad \delta_ {t} = c _ {\delta} \alpha_ {t}, \quad \gamma_ {t + 1} = c _ {\gamma} \alpha_ {t}, +$$ + +$$ +\eta_ {t + 1} = c _ {\eta} \alpha_ {t}, \quad \lambda_ {t + 1} = c _ {\lambda} \alpha_ {t}, \quad \rho_ {\mathbf {v}} ^ {2} = c _ {\mathbf {v}} \alpha_ {t}, \quad \rho_ {\mathbf {r}} ^ {2} = \frac {1}{d _ {2} ^ {2} T}, \quad \rho_ {\mathbf {s}} ^ {2} = \frac {1}{d _ {1} ^ {2} T}, +$$ + +$$ +b = \frac {T ^ {1 / 3}}{\left(d _ {1} + d _ {2}\right) ^ {3 / 2}}, \quad \bar {b} = \frac {T ^ {2 / 3}}{\left(d _ {1} + d _ {2}\right) ^ {3 / 4}}, \tag {31} +$$ + +where $c, c_{\beta}, c_{\delta}, c_{\gamma}, c_{\eta}, c_{\mathbf{v}}$ , and $c_{\lambda}$ are specified in (226). Let $p = \ell_{f,0} / \mu_g$ for the set $\mathcal{Z}_p$ defined in (8). Then, Algorithm 2 guarantees: + +$$ +\operatorname {B L} - \operatorname {R e g} _ {T} \leq \mathcal {O} \left(\left(d _ {1} + d _ {2}\right) ^ {\frac {3}{4}} T ^ {\frac {1}{3}} \left(\hat {\sigma} ^ {2} + \hat {\Delta} _ {T}\right) + \left(d _ {1} + d _ {2}\right) ^ {\frac {3}{2}} T ^ {\frac {2}{3}} \hat {\Psi} _ {T}\right). +$$ + +where $\hat{\sigma}^2$ and $(\hat{\Delta}_T,\hat{\Psi}_T)$ are defined in (28) and (30). + +Theorem 3.2 bounds the regret of Algorithm 2 without window-smoothing, based on the regularities in (30). We note that the average dynamic regret $\mathrm{BL - Reg}_T / T\leq \mathcal{O}((d_1 + d_2)^{3 / 4}T^{-2 / 3}\left(\hat{\sigma}^2 +\hat{\Delta}_T\right) +$ $(d_{1} + d_{2})^{3 / 2}T^{-1 / 3}\hat{\Psi}_{T})$ remains sublinear under suitable conditions on $\hat{\Delta}_T,\hat{\Psi}_T$ , and $\hat{\sigma}$ + +Remark 3.3 (Regret Guarantee for Zeroth Order OBO). Theorem 3.2 provides the first regret guarantee for OBO with access only to noisy function evaluations of the leader and follower. The dimensional dependence $\mathcal{O}(d_1 + d_2)$ in Theorem 3.2 aligns with optimal results for simpler offline min-max problems [39]. The bound also depends on the sample sizes $b, \bar{b}$ and smoothing parameters $\rho_{\mathrm{V}}, \rho_{\mathrm{r}}, \rho_{\mathrm{s}}$ at each iteration. + +Remark 3.4 (Improved Regret for OSO). Our dynamic regret for single-level non-stationary optimization is $\mathcal{O}((d_1 + d_2)^{3/4}T^{-2/3}(\hat{\sigma}^2 + E_1 + V_T + D_T))$ , improving the result in [60], which is $\mathcal{O}(T^{-1/2}\sigma^2\sqrt{d})$ . [60] proposed a zeroth-order stochastic gradient descent algorithm for unconstrained, non-convex, time-varying objective functions, achieving a regret bound of $\mathcal{O}(T^{-1/2}\sigma^2\sqrt{dW_T})$ using a two-point gradient estimator, where $W_T$ bounds the nonstationarity. Additionally, [29] showed that the local regret for standard online stochastic gradient descent with the standard two-point gradient estimator [1] is $\mathcal{O}(T^{-1/2}d\sqrt{V_T})$ . + +# 4 Experimental Results + +In this section, we present experimental results for two applications: online black-box attacks on deep neural networks and parametric loss tuning for imbalanced data. Code is available at $\mathbb{O}$ . Additional experiments and details on hyperparameter tuning are provided in Appendix E. + +![](images/f60f775afed99178a02f0b207af9c79d134a1e16ba970458488c14c18400959e.jpg) +Figure 2: Performance comparison (mean±std) of optimizers including ZO-O-GD, ZO-O-Adam, ZO-O-SignSGD, ZO-O-ConservSGD, ZO-SOGD, and ZO-SOGD (Adam) on online adversarial attack for MNIST data across five runs. + +![](images/352f7563c4764e880a5fde340ca268d5c0c25e8a60023b335c1dc55f753288e2.jpg) + +![](images/e05e78ccdd4c1f11d5c74e935ebb54aea00a81c93823d8088e8b3a0eb17ee91a.jpg) + +Bilevel Optimization for Black-Box Adversarial Attacks (BBAA) Deep neural networks are vulnerable to adversarial examples—inputs subtly perturbed to mislead classifiers. These examples can fool models without access to their internals, as in [11, 52, 13]. We first review the ZO single-level formulation for BBAA [11]. Let $(\mathbf{a}, b)$ be a clean image $\mathbf{a} \in \mathbb{R}^d$ with label $b \in \{1, \dots, J\}$ , and define $\mathbf{a}' = \mathbf{a} + \mathbf{y}$ , where $\mathbf{y}$ is the adversarial perturbation. Let $\mathcal{Y} := [-5, 5]^d$ , and $\ell: \mathbb{R}^d \to \mathbb{R}$ be the black-box attack loss. For a given hyperparameter $\lambda > 0$ , the BBAA problem is: + +$$ +\min _ {\mathbf {y} \in \mathcal {Y}} \frac {1}{m} \sum_ {i = 1} ^ {m} \ell \left(\mathbf {a} _ {i} + \mathbf {y}\right) + \lambda \| \mathbf {y} \| ^ {2}. \tag {32} +$$ + +To adapt (32) to our OBO, consider OBO for supervised learning: at each timestep $t$ , new samples $(\mathbf{a}_t, b_t) \in \mathcal{D}_t := \{\mathcal{D}_t^{\mathrm{val}}, \mathcal{D}_t^{\mathrm{tr}}\}$ are received, where $\mathbf{a}_t \in \mathbb{R}^{d_2}$ is the feature vector (image) and $b_t \in \mathbb{R}$ is the corresponding target. Note that the correct decision can change abruptly. We consider an $S$ -stage scenario where $(\mathbf{x}_s^*, \mathbf{y}_s^* (\mathbf{x}_s^*))$ represents the best decisions for the $s$ -th stage, for all $s \in [S]$ : + +$$ +\mathbf {x} _ {s} ^ {*} \in \underset {\mathbf {x} \in \mathcal {X}} {\operatorname {a r g m i n}} \sum_ {t = 1} ^ {T _ {s}} f \left(\mathbf {y} _ {s} ^ {*} (\mathbf {x}); \mathcal {D} _ {t} ^ {\text {v a l}}\right) \quad \text {s . t .} \quad \mathbf {y} _ {s} ^ {*} (\mathbf {x}) \in \underset {\mathbf {y} \in \mathcal {Y}} {\operatorname {a r g m i n}} \sum_ {t = 1} ^ {T _ {s}} g \left(\mathbf {x}, \mathbf {y}; \mathcal {D} _ {t} ^ {\mathrm {t r}}\right) \tag {33} +$$ + +$$ +g \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}; \mathcal {D} _ {t} ^ {\mathrm {t r}}\right) = \frac {1}{\left| \mathcal {D} _ {t} ^ {\mathrm {t r}} \right|} \sum_ {i \in \mathcal {D} _ {t} ^ {\mathrm {t r}}} \ell \left(\mathbf {a} _ {t} ^ {(i)} + \mathbf {y} _ {t}\right) + \frac {1}{2} \sum_ {\iota = 1} ^ {p} e ^ {\left[ \mathbf {x} _ {t} \right] _ {\iota}} \left[ \mathbf {y} _ {t} \right] _ {\iota} ^ {2}, +$$ + +$$ +f \left(\mathbf {y} _ {t} \left(\mathbf {x} _ {t}\right); \mathcal {D} _ {t} ^ {\text {v a l}}\right) = \frac {1}{\left| \mathcal {D} _ {t} ^ {\text {v a l}} \right|} \sum_ {i \in \mathcal {D} _ {t} ^ {\text {v a l}}} \ell \left(\mathbf {a} _ {t} ^ {(i)} + \mathbf {y} _ {t}\right). \tag {34a} +$$ + +Here, $\{\mathbf{a}_t^{(i)}\}_{i\in \mathcal{D}_t^{\mathrm{ir}}}$ and $\{\mathbf{a}_t^{(i)}\}_{i\in \mathcal{D}_t^{\mathrm{val}}}$ are batches of training and validation samples at timestep $t$ ; $\mathbf{a}_t^{(i)}$ is the $i$ th sample in that batch; and $[\mathbf{x}_t]_\iota$ and $[\mathbf{y}_t]_\iota$ denote the $i$ th component of $\mathbf{x}_t$ and $\mathbf{y}_t$ , respectively. + +We normalize the pixel values to $\mathcal{V}$ . For an untargeted attack, the loss in (34) is $\ell(\mathbf{a}_t') = \max\{Z(\mathbf{a}_t')_{b_t} - \max_{j \neq b_t} Z(\mathbf{a}_t')_j, -\kappa\}$ , where $Z(\mathbf{a}_t')_j$ is the prediction score for class $j$ given input $\mathbf{a}_t' = \mathbf{a}_t + \mathbf{y}_t$ , and $\kappa > 0$ controls the confidence gap. In our experiments, we set $\kappa = 0$ . Eq. (33) introduces the first OBO formulation of BBAA. Using a vector $\mathbf{x} \in \mathbb{R}_+^d$ for hyperparameters instead of $\lambda \in \mathbb{R}_{++}$ in (32) enables finer control over model components, enhancing performance for complex models and heterogeneous data [53]. For a fair comparison with single-level BBAA, we replace $\lambda$ with a fixed vector multiplied by each component of $\mathbf{y}$ in (32). We compare our ZO-SOGD and ZO-SOGD (Adam) with the following competing methods in the online setting: ZO-O-GD, a single-level method that updates $\mathbf{y}_t$ with a fixed $\mathbf{x}$ at each timestep using ZO gradient descent [59]; ZO-O-Adam, a single-level method that updates $\mathbf{y}_t$ with a fixed $\mathbf{x}$ at each timestep using ZO Adam [45, 13]; ZO-O-SignSGD, a single-level method that updates $\mathbf{y}_t$ with a fixed $\mathbf{x}$ at each timestep using ZO SignSGD [6]; and ZO-O-ConservSGD, a single-level method that updates $\mathbf{y}_t$ with a fixed $\mathbf{x}$ at each timestep using ZO Conservative SGD [44]. Note that ZO-SOGD (ours, Adam) is a variant of our algorithm with an adaptive stepsize, similar to that of [45]. + +We evaluated the proposed algorithms based on runtime, test accuracy on perturbed samples, and the infinity norm of $\mathbf{y}_t$ . Figure 2 compares the methods. The left panel shows that ZO-SOGD has a slower runtime than single-level baselines due to outer-level optimization on $\mathbf{x}$ . The middle panel illustrates that accuracy decreases as the adversarial attack $\mathbf{y}$ strengthens, with ZO-SOGD outperforming ZO-O-GD and ZO-O-ConservGD, while ZO-SOGD (Adam) surpasses ZO-O-Adam and all baselines. The right panel indicates that the infinity norm of $\mathbf{y}_t$ increases over time for all methods, reducing accuracy. However, perturbations remain minor, with $\max \mathbf{y}_t$ not exceeding 4, demonstrating that ZO-SOGD achieves effective attacks with superior performance. + +![](images/f5a5d3c51b2410b0a9b3feb6a91b30675d7c9b5001a3cd6001ac94580fe6ded0.jpg) +Figure 3: Performance (mean±std) on online parametric loss tuning with distribution shift on MNIST across five runs, comparing OGD [72], OAGD [67], SOBOW [50], and our SOGD. + +![](images/c3abbf4c225818730859cea094248ee58339a33dcf4e460cbf5498a864013f5b.jpg) + +![](images/d2d6987bcf54ea977f6eb2f2060cdb4aff9d3c39065949c3903fa89685d5a410.jpg) + +Parametric Loss Tuning for Imbalanced Data Imbalanced datasets are common in modern machine learning, causing challenges in generalization and fairness due to underrepresented classes and sensitive attributes. Deep NNs often overfit, seeming accurate and fair during training but performing poorly during testing. A common solution is designing a parametric training loss that balances accuracy and fairness while preventing overfitting [49]. We consider an optimization problem similar to that in (33). For a new sample $(\mathbf{a}_t, b_t)$ , the follower and leader incur a parametric and balanced cross-entropy loss, respectively: + +$$ +g \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}; \mathcal {D} _ {t} ^ {\mathrm {t r}}\right) = - \log \frac {e ^ {\gamma_ {b _ {t}} \left[ \mathbf {y} _ {t} \left(\mathbf {a} _ {t}\right) \right] _ {b _ {t}} + \Delta_ {b _ {t}}}}{\sum_ {j = 1} ^ {J} e ^ {\gamma_ {j} \left[ \mathbf {y} _ {t} \left(\mathbf {a} _ {t}\right) \right] _ {j}} + \Delta_ {j}}, f \left(\mathbf {y} _ {t} (\mathbf {x} _ {t}); \mathcal {D} _ {t} ^ {\mathrm {v a l}}\right) = - u _ {b _ {t}} \log \frac {e ^ {\left[ \mathbf {y} _ {t} \left(\mathbf {a} _ {t}\right) \right] _ {b _ {t}}}}{\sum_ {j = 1} ^ {J} e ^ {\left[ \mathbf {y} _ {t} \left(\mathbf {a} _ {t}\right) \right] _ {j}}}. \tag {35} +$$ + +Here, $\mathbf{x}_t \coloneqq (\Delta_j, \gamma_j)_{j=1}^J$ represents the logit adjustments, with $j$ indexing the $J$ classes, and $u_j$ is the reciprocal of the proportion of samples from the $j$ -th class to the total number of samples [49]. + +In (35), $\mathbf{y}_t(\mathbf{x}_t)$ is the follower conditioned on the leader, and $[\mathbf{y}_t(\mathbf{a}_t)]_{b_t}$ is the logit for class $b_{t}$ on sample $\mathbf{a}_t$ . The follower $\mathbf{y}_t$ uses a 4-layer CNN, inducing a nonconvex bilevel objective. We compare SOGD with OAGD [67], a static method using the Neumann series, and SOBOW [50], a dynamic method using conjugate gradients (CG). Experiments were conducted on MNIST [48] with batch size 64. We evaluated cumulative runtime, test accuracy, and balanced accuracy, defined as $\frac{1}{J}\sum_{j = 1}^{J}\mathbb{P}_{\mathbf{a}_t\sim \mathcal{D}_j}\left[\mathrm{argmax}_i([y_t(\mathbf{a}_t)]_i) = j\right]$ , where $\mathcal{D}_j$ is the class- $j$ sample distribution [49]. Learning rates were tuned as $\beta_{t} = \delta_{t} = \beta \in \{0.001,0.005,0.01,0.05,0.1\}$ , $\alpha_{t} = \alpha \in \{0.0001,0.0005,0.001,0.005,0.01\}$ , and $\gamma_{t} = \lambda_{t} = \eta_{t} = \gamma \in \{0.9,0.99,0.999\}$ . Both OAGD and SOBOW used 5 iterations for their respective system solvers. + +We evaluated performance over 400 timesteps in four 100-timestep phases, transitioning from an imbalanced $(0.4^{i})$ to a balanced $(0.8^{i})$ distribution for each class ( $i = 0, 1, \dots, 9$ ). Figure 3 (left) shows SOBOW's longer runtime due to CG complexity, while SOGD is the fastest with simultaneous updates. Figures 3 (middle, right) show accuracy gains as balance increases, with SOGD achieving competitive accuracy. + +# 5 Conclusion + +This work introduced a novel online bilevel optimization framework that overcomes the limitations of existing algorithms, which often depend on extensive oracle information and incur high computational costs. Our method leverages limited feedback and zeroth-order updates for efficient hypergradient estimation and simultaneous updates of decision variables, achieving sublinear bilevel regret without window smoothing. Experiments on online parametric loss tuning and black-box adversarial attacks validate its effectiveness. A limitation of this study is that the results focus on nonconvex regret bounds, without extending guarantees to convex settings. + +# Acknowledgments and Disclosure of Funding + +We thank the reviewers for their valuable comments. The work of DAT was supported by Samsung SDS Research America, Mountain View. The work of GM was supported in part by NSF grants DMS-2348640 and DMS-2319552. + +# References + +[1] Alekh Agarwal, Ofer Dekel, and Lin Xiao. Optimal algorithms for online convex optimization with multi-point bandit feedback. In *Colt*, pages 28-40. CiteSeer, 2010. +[2] Naman Agarwal, Alon Gonen, and Elad Hazan. Learning in non-convex games with an optimization oracle. In Conference on Learning Theory, pages 18–29. PMLR, 2019. +[3] Alireza Aghasi and Saeed Ghadimi. 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In Proceedings of the 20th international conference on machine learning (icml-03), pages 928–936, 2003. + +# Contents + +1 Introduction 1 +2 Stochastic OBO with Access to First- and Inner Second-Order Oracles 2 +3 Stochastic OBO with Zeroth-Order Oracles 6 +4 Experimental Results 8 +5 Conclusion 10 +A Related Work 16 +B Additional Preliminaries and Notations 16 + +B.1 Preliminary Lemmas 16 +B.2 Examples Illustrating Regularity Conditions 17 + +C Proof of Regret Bounds for Simultaneous Online Gradient Descent (SOGD) 19 + +C.1 Proof of Lemma 2.1 19 +C.2 Bounds on the Inner Decision Variable 19 +C.3 Bounds on the Linear System Solution 24 +C.4 Bounds on the Gradient Estimation Error of Outer Objective 30 +C.5 Bounds on the Outer Objective and its Projected Gradient 33 +C.6 Proof of Theorem 2.6 35 + +D Proof of Regret Bounds for Zeroth Order SOGD (ZO-SOGD) 43 + +D.1 Auxiliary Lemmas for Proof of Theorem 3.2 43 +D.2 Perturbation Bounds for OBO Objectives and Their Smoothing Variants 44 +D.3 Bounds on the Zeroth-Order Inner Solution 47 +D.4 Bounds on the Zeroth-Order System Solution 51 +D.5 Bounds on the Zeroth-Order Estimation Error of Outer Objective 60 +D.6 Bounds on the Zeroth-Order Objective Function and its Projected Gradients 64 +D.7 Proof of Theorem 3.2 70 + +E Hyperparameter Tuning Results 80 + +# A Related Work + +BO was introduced in game theory by [64] and modeled mathematically in [9]. Initial works [32, 55] reduced it to single-level optimization. Recently, gradient-based approaches have gained popularity for their simplicity and efficacy [21, 27, 43, 12, 56], though they assume offline objectives. + +OBO was initiated by [67], proposing the OAGD method with regret bounds. [40] developed algorithms for online minimax optimization, special cases of OBO with local regret guarantees. [50] introduced SOBOW, a single-loop optimizer using window-smoothed functions and multiple CGs for nonconvex-strongly-convex cases. Unlike these works, we propose using projected gradient as a more general performance measure for constrained objectives, focusing on the original functions and their regret; See Table 1 for a comparison. + +Single-Level Regret Minimization. Single-level online optimization predominantly focuses on convex problems, either with static or dynamic convex regret minimization [72, 34, 61]. Non-convex online optimization [36, 30, 29] poses greater challenges than its convex counterparts [61, 72, 35, 7]. Notable contributions in this field include adversarial multi-armed bandit algorithms [10, 37, 38, 47] and the Follow-the-Perturbed-Leader approach [2, 46, 66]. Hazan et al. [36] introduced window-smoothed local regret for gradient averaging in non-convex models, which Hallak et al. [31] extended to non-smooth, non-convex problems. Inspired by their work, we employ local regret for Online Bilevel Optimization (OBO) without window-smoothing. + +Zeroth-Order Optimization. Single-Level ZO Optimization has been widely studied in both offline [25, 17, 1, 59, 57] and online settings [52, 29, 30, 70, 5]. We next review closely related work. Liu et al. [52] proposed ZOO-ADMM, a gradient-free online optimization algorithm utilizing ADMM. Guan et al. [30] studied online non-convex optimization with limited oracle feedback. Research on online non-convex optimization with bandit feedback includes work by Heliou et al. [37], which established bounds on global static and dynamic regret using dual averaging, further refined in [38]. Gao et al. [23] extended these ideas to ZO algorithms. Flaxman et al. [20] provided algorithms for bandit online optimization of convex functions using ZO gradient approximation. Our work closely relates to [63], which proposes a Hessian-free method approximating the Jacobian matrix using a ZO method based on finite differences of gradients. In contrast, our method uses function oracles to approximate both the Hessian and gradients and is derivative-free. We also point out the recent work [3] on ZO stochastic algorithms for solving bilevel problems when neither the upper/ lower objective values nor their unbiased gradient estimates are available. Their approach, limited to the offline setting, does not include numerical results, thus leaving its practical efficiency unclear. + +# B Additional Preliminaries and Notations + +# B.1 Preliminary Lemmas + +We first provide several useful lemmas for the main proofs. + +Definition B.1 (Projected gradient [26]). Let $\mathcal{X} \subset \mathbb{R}^{d_1}$ be a closed convex set. Then, the projected gradient for any $\alpha_t > 0$ and $\mathbf{p} \in \mathbb{R}^{d_1}$ is defined as + +$$ +\mathcal {P} _ {\mathcal {X}, \alpha_ {t}} (\mathbf {x}; \mathbf {p}) := \frac {1}{\alpha_ {t}} \left(\mathbf {x} - \mathbf {x} ^ {+}\right), +$$ + +where + +$$ +\mathbf {x} ^ {+} = \Pi_ {\mathcal {X}} (\mathbf {x} - \alpha_ {t} \mathbf {p}), \tag {36} +$$ + +and $\Pi_{\mathcal{X}}[\cdot ]$ denotes the orthogonal projection operator onto set $\mathcal{X}$ + +Lemma B.2. [28, Lemma 13] If $f:\mathcal{X}\to \mathbb{R}$ is a $\mu_f$ -strongly convex function with respect to some norm $\| \cdot \|$ , and $\mathbf{x}^*$ is the minimizer of $f$ (i.e. $\mathbf{x}^{*} = \arg \min_{\mathbf{x}\in \mathcal{X}}f(\mathbf{x})$ ), then we have $\forall \mathbf{x}\in \mathcal{X}$ , + +$$ +\frac {\mu_ {f}}{2} \| \mathbf {x} - \mathbf {x} ^ {*} \| ^ {2} \leq f (\mathbf {x}) - f (\mathbf {x} ^ {*}) \leq \frac {1}{2 \mu_ {f}} \| \nabla f (\mathbf {x}) \| ^ {2}. +$$ + +Lemma B.3. Suppose $f(\mathbf{x})$ is $L$ -smooth, and $\mathbf{x}^* \in \operatorname{argmin}_{\mathbf{x} \in \mathcal{X}} f(\mathbf{x})$ . Then, we can upper bound the magnitude of the gradient at any given point $\mathbf{x} \in \mathbb{R}^d$ in terms of the objective sub optimality at $\mathbf{x}$ , as follows: + +$$ +\frac {1}{2 L} \| \nabla f (\mathbf {x}) \| ^ {2} \leq f (\mathbf {x}) - f \left(\mathbf {x} ^ {*}\right) \leq \frac {L}{2} \| \mathbf {x} - \mathbf {x} ^ {*} \| ^ {2}. \tag {37} +$$ + +Lemma B.4. For any $\mathbf{x},\mathbf{y}\in \mathbb{R}^d$ , the following holds for any $c > 0$ .. + +$$ +\left\| \mathbf {x} + \mathbf {y} \right\| ^ {2} \leq (1 + c) \left\| \mathbf {x} \right\| ^ {2} + \left(1 + \frac {1}{c}\right) \left\| \mathbf {y} \right\| ^ {2}. +$$ + +We also utilize a basic yet important property of the projected-gradient mapping. + +Lemma B.5. [26, Proposition 1] Let $\mathcal{P}_{\mathcal{X},\alpha_t}(\mathbf{x};\mathbf{p})$ denote the projected gradient as defined in Definition B.1. For any $\mathbf{x},\mathbf{p}_1,\mathbf{p}_2\in \mathbb{R}^d$ and $\alpha_{t} > 0$ , it holds that + +$$ +\left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} (\mathbf {x}; \mathbf {p} _ {1}) - \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} (\mathbf {x}; \mathbf {p} _ {2}) \right\| \leq \left\| \mathbf {p} _ {1} - \mathbf {p} _ {2} \right\|. +$$ + +Lemma B.6. [36, Proposition 2.4] Let $\mathcal{P}_{\mathcal{X},\alpha_t}(\mathbf{x};\mathbf{p})$ denote the projected gradient as defined in Definition B.1. For any $\mathbf{x},\mathbf{p}_1,\mathbf{p}_2\in \mathbb{R}^d$ and $\alpha_{t} > 0$ , it holds that + +$$ +\left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} (\mathbf {x}; \mathbf {p} _ {1} + \mathbf {p} _ {2}) \right\| \leq \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} (\mathbf {x}; \mathbf {p} _ {1}) \right\| + \left\| \mathbf {p} _ {2} \right\|. +$$ + +Lemma B.7. Let $\mathcal{P}_{\mathcal{X},\alpha_t}(\mathbf{x};\mathbf{p})$ be as given in Definition B.1. Then, for any $\mathbf{p}\in \mathbb{R}^d$ and $\alpha_{t} > 0$ , we have + +$$ +\langle \mathbf {p}, \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} (\mathbf {x}; \mathbf {p}) \rangle \geq \| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} (\mathbf {x}; \mathbf {p}) \| ^ {2}. +$$ + +Proof. By the definition of $\mathbf{x}^{+}$ , the optimality condition of (36) is + +$$ +\left\langle \mathbf {p} + \frac {1}{\alpha_ {t}} (\mathbf {x} ^ {+} - \mathbf {x}), \mathbf {z} - \mathbf {x} ^ {+} \right\rangle \geq 0, \quad \forall \mathbf {z} \in \mathcal {X}. +$$ + +Letting $\mathbf{z} = \mathbf{x}$ , we obtain + +$$ +\left\langle \mathbf {p}, \mathbf {x} - \mathbf {x} ^ {+} \right\rangle \geq \frac {1}{\alpha_ {t}} \left\langle \mathbf {x} - \mathbf {x} ^ {+}, \mathbf {x} - \mathbf {x} ^ {+} \right\rangle , +$$ + +which can be rearranged to + +$$ +\begin{array}{l} \left\langle \mathbf {p}, \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} (\mathbf {x}; \mathbf {p}) \right\rangle = \frac {1}{\alpha_ {t}} \left\langle \mathbf {p}, \mathbf {x} - \mathbf {x} ^ {+} \right\rangle \geq \frac {1}{\alpha_ {t} ^ {2}} \left\langle \mathbf {x} - \mathbf {x} ^ {+}, \mathbf {x} - \mathbf {x} ^ {+} \right\rangle \\ = \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} (\mathbf {x}; \mathbf {p}) \right\| ^ {2}. \\ \end{array} +$$ + +![](images/4bc217c29aae9206bc4b2a1ed91aa5c3cf7e2d2f5e08fc9202e503dfe63a354d.jpg) + +# B.2 Examples Illustrating Regularity Conditions + +Theorem 2.6 achieves sublinear bilevel regret when the variations $V_{T}$ and $H_{2,T}$ are $o(T^{2/3})$ and $o(T^{1/3})$ , respectively. Below, we provide some examples of online optimization in both single-level and bilevel settings to illustrate when this occurs. + +Example B.8. Consider function $f_{t}(\mathbf{x}) = \| \mathbf{A}_{t}\mathbf{x} - \mathbf{b}_{t}\|^{2}$ , where $\mathbf{A}_t = [1,0;0,1 + \frac{1}{t}]$ , $\mathbf{b}_t = (1,1)$ . It follows from (11) that $V_{T} = \sum_{t = 2}^{T}\max_{\mathbf{x}}|f_{t}(\mathbf{x}) - f_{t - 1}(\mathbf{x})| = \sum_{t = 2}^{T}|\left(\frac{1}{t}\right)^{2} - \left(\frac{1}{t - 1}\right)^{2}|$ , and + +$$ +\begin{array}{l} V _ {T} = \sum_ {t = 2} ^ {T} | \left(\frac {1}{t} - \frac {1}{t - 1}\right) - \left(\frac {1}{t} + \frac {1}{t - 1}\right) | \\ = \sum_ {t = 2} ^ {T} | \left(\frac {t - 1 - t}{t (t - 1)}\right) - \left(\frac {1}{t} + \frac {1}{t - 1}\right) | \\ = \sum_ {t = 2} ^ {T} | \left(- \frac {1}{t (t - 1)}\right) - \left(\frac {1}{t} + \frac {1}{t - 1}\right) | \\ = \sum_ {t = 2} ^ {T} | \frac {1}{t (t - 1)} | | \frac {t - 1 + t}{t (t - 1)} | \\ = \sum_ {t = 2} ^ {T} | \frac {2}{t (t - 1) ^ {2}} |. \\ \end{array} +$$ + +Then, $V_{T} \leq \sum_{t=2}^{T} \frac{2}{t^{3}} \approx \int_{2}^{T} \frac{2}{t^{3}} dt = \frac{1}{4} - \frac{1}{T^{2}}$ . As $T \to \infty$ , $V_{T}$ becomes bounded and approaches a constant value, indicating that $V_{T}$ grows slower than $T$ itself. + +Example B.9. Let + +$$ +f _ {t} (\mathbf {x}) = \left\{ \begin{array}{l l} \left(- \frac {1}{T}, 0\right) & \text {i f t i s e v e n ;} \\ \left(0, - \frac {1}{T}\right) & \text {i f t i s o d d .} \end{array} \right. +$$ + +Then, $V_{T} = \sum_{t = 2}^{T}\max_{\mathbf{x}}|f_{t}(\mathbf{x}) - f_{t - 1}(\mathbf{x})| = \mathcal{O}(1)$ + +Example B.10. Let $x \in \mathcal{X} = [-1,1] \subset \mathbb{R}$ , $y \in \mathbb{R}$ , and consider a sequence of quadratic cost functions + +$$ +f _ {t} (x, y) = \frac {1}{2} \left(x + 2 a _ {t} ^ {(1)}\right) ^ {2} + \frac {1}{2} \left(y - a _ {t} ^ {(2)}\right) ^ {2}, +$$ + +$$ +g _ {t} (x, y) = \frac {1}{2} y ^ {2} - \left(x - a _ {t} ^ {(2)}\right) y, +$$ + +where $a_{t}^{(1)} = 1 / t$ and $a_{t}^{(2)} = 1 / \sqrt{t}$ for all $t \in [T]$ . + +We have + +$$ +y _ {t} ^ {*} (x) = x - a _ {t} ^ {(2)}, +$$ + +and + +$$ +\begin{array}{l} f _ {t} \left(x, y _ {t} ^ {*} (x)\right) - f _ {t - 1} \left(x, y _ {t - 1} ^ {*} (x)\right) \\ = \frac {1}{2} \left[ \left(x + 2 a _ {t} ^ {(1)}\right) ^ {2} - \left(x + 2 a _ {t - 1} ^ {(1)}\right) ^ {2} \right] + \frac {1}{2} \left[ \left(y _ {t} ^ {*} (x) - a _ {t} ^ {(2)}\right) ^ {2} - \left(y _ {t - 1} ^ {*} (x) - a _ {t - 1} ^ {(2)}\right) ^ {2} \right] \\ = \frac {1}{2} \left[ \left(x ^ {2} + 4 x a _ {t} ^ {(1)} + 4 \left(a _ {t} ^ {(1)}\right) ^ {2}\right) - \left(x ^ {2} + 4 x a _ {t - 1} ^ {(1)} + 4 \left(a _ {t - 1} ^ {(1)}\right) ^ {2}\right) \right] \\ + \frac {1}{2} \left[ \left((x - a _ {t} ^ {(2)}) ^ {2} - 2 (x - a _ {t} ^ {(2)}) a _ {t} ^ {(2)} + (a _ {t} ^ {(2)}) ^ {2}\right) - \left((x - a _ {t - 1} ^ {(2)}) ^ {2} - 2 (x - a _ {t - 1} ^ {(2)}) a _ {t - 1} ^ {(2)} + (a _ {t - 1} ^ {(2)}) ^ {2}\right) \right] \\ = 2 x \left(a _ {t} ^ {(1)} - a _ {t - 1} ^ {(1)} - a _ {t} ^ {(2)} + a _ {t - 1} ^ {(2)}\right) + 2 \left(\left(a _ {t} ^ {(1)}\right) ^ {2} - \left(a _ {t - 1} ^ {(1)}\right) ^ {2} + \left(a _ {t} ^ {(2)}\right) ^ {2} - \left(a _ {t - 1} ^ {(2)}\right) ^ {2}\right). \\ \end{array} +$$ + +Taking the maximum over $x$ and using $x \in [-1, 1]$ : + +$$ +\begin{array}{l} \sup _ {x} | f _ {t} (x, y _ {t} ^ {*} (x)) - f _ {t - 1} (x, y _ {t - 1} ^ {*} (x)) | = 2 \left| a _ {t} ^ {(1)} - a _ {t - 1} ^ {(1)} \right| + 2 \left| - a _ {t} ^ {(2)} + a _ {t - 1} ^ {(2)} \right| \\ + 2 \left| (a _ {t} ^ {(1)}) ^ {2} - (a _ {t - 1} ^ {(1)}) ^ {2} \right| + 2 \left| (a _ {t} ^ {(2)}) ^ {2} - (a _ {t - 1} ^ {(2)}) ^ {2} \right|. \\ \end{array} +$$ + +Since $a_{t}^{(1)} = 1 / t$ and $a_{t}^{(2)} = 1 / \sqrt{t}$ for all $t\in [T]$ , then we have + +$$ +\left| a _ {t} ^ {(1)} - a _ {t - 1} ^ {(1)} \right| \approx \frac {1}{t ^ {2}}, \quad \left| a _ {t} ^ {(2)} - a _ {t - 1} ^ {(2)} \right| \approx \frac {1}{2 t ^ {3 / 2}}, +$$ + +$$ +| (a _ {t} ^ {(1)}) ^ {2} - (a _ {t - 1} ^ {(1)}) ^ {2} | \approx \frac {1}{t ^ {3}}, \quad | (a _ {t} ^ {(2)}) ^ {2} - (a _ {t - 1} ^ {(2)}) ^ {2} | \approx \frac {1}{t ^ {2}}. +$$ + +Then, we get + +$$ +V _ {T} := \sum_ {t = 2} ^ {T} \sup _ {x} | f _ {t} (x, y _ {t} ^ {*} (x)) - f _ {t - 1} (x, y _ {t - 1} ^ {*} (x)) | = \sum_ {t = 2} ^ {T} \left(\frac {2}{t ^ {2}} + \frac {1}{2 t ^ {3 / 2}} + \frac {1}{t ^ {3}}\right). +$$ + +The series $\sum_{t=2}^{T}\left(\frac{2}{t^2} + \frac{1}{2t^{3/2}} + \frac{1}{t^3}\right)$ converges, implying $V_T = \mathcal{O}(1)$ . Moreover, we have + +$$ +\begin{array}{l} H _ {2, T} = \sum_ {t = 2} ^ {T} \sup _ {x} \| y _ {t} ^ {*} (x) - y _ {t - 1} ^ {*} (x) \| ^ {2} = \sum_ {t = 2} ^ {T} \sup _ {x} \| x - a _ {t} ^ {(2)} - x + a _ {t - 1} ^ {(2)} \| ^ {2} \\ = \sum_ {t = 2} ^ {T} | - a _ {t} ^ {(2)} + a _ {t - 1} ^ {(2)} | ^ {2} = \sum_ {t = 2} ^ {T} | a _ {t} ^ {(2)} - a _ {t - 1} ^ {(2)} | ^ {2} \approx \sum_ {t = 2} ^ {T} \frac {1}{4 t ^ {3}}, \\ \end{array} +$$ + +which implies $H_{2,T} = \mathcal{O}(1)$ + +To achieve $V_{T} = o(T^{2 / 3})$ and $H_{2,T} = o(T^{1 / 3})$ , the changes in the cost functions $f_{t}(\mathbf{x},\mathbf{y}_{t}^{*}(\mathbf{x}))$ and $\mathbf{y}_t^* (\mathbf{x})$ should decay to zero faster than $\mathcal{O}(1 / t^a)$ with $a > 1 / 3$ . For example, if the coefficients in the functions change as $\mathcal{O}(1 / t^a)$ with $a > 1 / 3$ , then the cumulative sum over $T$ will be $o(T^{2 / 3})$ . When $f_{t}(\mathbf{x},\mathbf{y}_{t}^{*}(\mathbf{x}))$ and $\mathbf{y}_t^* (\mathbf{x})$ decay as $\mathcal{O}(1 / \sqrt{t})$ , then the total variation grows at most as $\mathcal{O}(\sqrt{T})$ . + +# C Proof of Regret Bounds for Simultaneous Online Gradient Descent (SOGD) + +Proof Roadmap. We introduce Lemma C.2, which quantifies the error between the approximated direction of the momentum-based gradient estimator, $\mathbf{d}_t^{\mathbf{y}}$ , and the true direction, $\nabla_{\mathbf{y}}g_t(\mathbf{x}_t,\mathbf{y}_t)$ , at each iteration. To bound the error of the lower-level variable, we provide Lemma C.4, which captures the gap $\| \mathbf{y}_{t + 1} - \mathbf{y}_t^* (\mathbf{x}_t)\|^2$ and incorporates the error introduced in Lemma C.2. Moreover, we provide Lemma C.6, which quantifies the error between the approximated direction of the momentum-based gradient estimator, $\mathbf{d}_t^{\mathbf{v}}$ , and the true direction, $\nabla_{\mathbf{y}}^2 g_t(\mathbf{z}_t)\mathbf{v}_t + \nabla_{\mathbf{y}}f_t(\mathbf{z}_t)$ , at each iteration. To bound the error of the system solution, we provide Lemma C.8, which captures the gap $\| \mathbf{v}_{t + 1} - \mathbf{v}_t^* (\mathbf{x}_t)\|^2$ and incorporates the error introduced in Lemma C.6. Moreover, we provide Lemma C.9, which quantifies the error between the approximated direction of the momentum-based hypergradient estimator, $\mathbf{d}_t^{\mathbf{x}}$ , and the true direction, $\nabla_{\mathbf{x}}f_t(\mathbf{z}_t) + \nabla_{\mathbf{x}\mathbf{y}}^2 g_t(\mathbf{z}_t)\mathbf{v}_t$ , at each iteration. We also present Lemma C.11, which provides an upper bound for the projection mapping and relates to the three errors discussed in Lemmas C.4, C.8, and C.9. Finally, by combining these lemmas and appropriately setting the parameters, we achieve the desired result. + +# C.1 Proof of Lemma 2.1 + +Proof. By letting $\nu = 1 - \eta$ for $\eta \in (0,1)$ , the window-smoothed gradient + +$$ +\nabla F _ {t, \nu} (\mathbf {x} _ {t}, \mathbf {y} _ {t}; \mathcal {B} _ {t}) = \frac {1}{W} \sum_ {i = 0} ^ {w - 1} \nu^ {i} \nabla f _ {t - i} (\mathbf {x} _ {t - i}, \mathbf {y} _ {t - i}; \mathcal {B} _ {t - i}), +$$ + +is equivalent to + +$$ +\nabla F _ {t, \nu} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}; \mathcal {B} _ {t}\right) = \frac {1}{W} \sum_ {j = t - w + 1} ^ {t} (1 - \eta) ^ {t - j} \nabla f _ {j} \left(\mathbf {x} _ {j}, \mathbf {y} _ {j}; \mathcal {B} _ {j}\right). \tag {38} +$$ + +Let $\mathbf{d}_t^{\mathbf{x}} = \nabla F_{t,\nu}(\mathbf{x}_t,\mathbf{y}_t;\mathcal{B}_t)$ . Then (38) is equivalent to + +$$ +\mathbf {d} _ {t} ^ {\mathbf {x}} = \frac {1}{W} \nabla f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}; \mathcal {B} _ {t}) + \frac {1}{W} \sum_ {j = t - w + 1} ^ {t - 1} (1 - \eta) ^ {t - j} \nabla f _ {j} (\mathbf {x} _ {j}, \mathbf {y} _ {j}; \mathcal {B} _ {j}). +$$ + +Since + +$$ +(1 - \eta) \mathbf {d} _ {t - 1} ^ {\mathbf {x}} = \frac {(1 - \eta)}{W} \sum_ {j = t - w} ^ {t - 1} (1 - \eta) ^ {t - 1 - j} \nabla f _ {j} (\mathbf {x} _ {j}, \mathbf {y} _ {j}; \mathcal {B} _ {j}), +$$ + +we have + +$$ +\mathbf {d} _ {t} ^ {\mathbf {x}} = \frac {1}{W} \nabla f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}; \mathcal {B} _ {t}) + (1 - \eta) \mathbf {d} _ {t - 1} ^ {\mathbf {x}} - \frac {(1 - \eta) ^ {w}}{W} \nabla f _ {t - w} (\mathbf {x} _ {t - w}, \mathbf {y} _ {t - w}; \mathcal {B} _ {t - w}), +$$ + +with $f_{i}(\cdot) = 0$ for all $i\leq 0$ + +If $w = t$ and $W = \frac{1}{\eta}$ then, we have + +$$ +\mathbf {d} _ {t} ^ {\mathbf {x}} = \eta \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}; \mathcal {B} _ {t}\right) + (1 - \eta) \mathbf {d} _ {t - 1} ^ {\mathbf {x}}. +$$ + +![](images/6f128e69432d91ac8936f0fcb91f2b73d11d2f6388f2e4d48baf5e03da85695a.jpg) + +# C.2 Bounds on the Inner Decision Variable + +In the following, inspired by offline BO [69, 15] and OBO [67, 50], we provide a set of lemmas for the analysis of SOGD. We first present a lemma that characterizes the Lipschitz continuity of the approximate gradients, as well as the inner and system solutions. + +Lemma C.1. Under Assumptions 2.2 and 2.3, for all $\mathbf{x},\mathbf{x}'\in \mathcal{X}$ , and the search directions $\{\mathbf{d}_t^{\mathbf{x}}\}_{t = 1}^T$ and $\{\mathbf{d}_t^{\mathbf{v}}\}_{t = 1}^T$ generated by Algorithm 1, we have + +$$ +\left\| \mathbf {d} _ {t} ^ {\mathbf {x}} - \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \right\| ^ {2} \leq M _ {f} ^ {2} \left(\left\| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} + \left\| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2}\right), \tag {39a} +$$ + +$$ +\left\| \mathbf {d} _ {t} ^ {\mathbf {v}} \right\| ^ {2} \leq M _ {\mathbf {v}} ^ {2} \left(\left\| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} + \left\| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2}\right), \tag {39b} +$$ + +$$ +\left\| \nabla f _ {t} \left(\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})\right) - \nabla f _ {t} \left(\mathbf {x} ^ {\prime}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} ^ {\prime}\right)\right) \right\| \leq L _ {f} \| \mathbf {x} - \mathbf {x} ^ {\prime} \|, \tag {39c} +$$ + +$$ +\left\| \mathbf {y} _ {t} ^ {*} (\mathbf {x}) - \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} ^ {\prime}\right) \right\| \leq L _ {\mathbf {y}} \| \mathbf {x} - \mathbf {x} ^ {\prime} \|, \tag {39d} +$$ + +$$ +\left\| \mathbf {v} _ {t} ^ {*} (\mathbf {x}) - \mathbf {v} _ {t} ^ {*} \left(\mathbf {x} ^ {\prime}\right) \right\| \leq L _ {\mathbf {v}} \| \mathbf {x} - \mathbf {x} ^ {\prime} \|, \tag {39e} +$$ + +where $M_{f}$ , $M_{\mathbf{v}}$ , and $(L_{\mathbf{y}}, L_{\mathbf{v}}, L_{f})$ are defined in (42), (43), and (44), respectively. + +Proof. We first show (39a). + +Using Assumptions 2.2 and 2.3, we have $\nabla_{\mathbf{y}}^{2}g_{t}\left(\mathbf{x}_{t},\mathbf{y}_{t}^{*}(\mathbf{x}_{t})\right)\succeq \mu_{g}$ , and + +$$ +\left\| \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| = \left\| \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) ^ {- 1} \nabla_ {\mathbf {y}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \right\| \leq \frac {\ell_ {f , 0}}{\mu_ {g}}. \tag {40} +$$ + +Observe that + +$$ +\begin{array}{l} \left\| \mathbf {d} _ {t} ^ {\mathbf {x}} - \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) \right\| \leq \left\| \nabla_ {\mathbf {x}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {x}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) \right\| \\ + \left\| \mathbf {v} _ {t} \nabla_ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \nabla_ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \right\| \\ \leq \left\| \nabla_ {\mathbf {x}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {x}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) \right\| \\ + \left\| \nabla_ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right\| \left\| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right) \right\| \\ + \left\| \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| \left\| \nabla_ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})) \right\| \\ \leq \left(\ell_ {f, 1} + \frac {\ell_ {g , 2} \ell_ {f , 0}}{\mu_ {g}}\right) \| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| + \ell_ {g, 1} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| \\ \leq M _ {f} ^ {2} \left(\left\| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| + \left\| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\|\right), \tag {41} \\ \end{array} +$$ + +where + +$$ +M _ {f} := \sqrt {2} \max \left\{\ell_ {f, 1} + \frac {\ell_ {g , 2} \ell_ {f , 0}}{\mu_ {g}}, \ell_ {g, 1} \right\}, \tag {42} +$$ + +the third inequality is by Assumption 2.3, and the last inequality follows from (40). + +Next, we establish (39b). + +Since $\mathbf{d}_t^{\mathbf{v}^*} \coloneqq \nabla_{\mathbf{y}}f_t(\mathbf{x}_t,\mathbf{y}_t^* (\mathbf{x}_t)) + \nabla_{\mathbf{y}}^2 g_t(\mathbf{x}_t,\mathbf{y}_t^* (\mathbf{x}_t))\mathbf{v}_t^* (\mathbf{x}_t) = 0$ we have + +$$ +\begin{array}{l} \left\| \mathbf {d} _ {t} ^ {\mathbf {v}} \right\| = \left\| \mathbf {d} _ {t} ^ {\mathbf {v}} - \mathbf {d} _ {t} ^ {\mathbf {v} ^ {*}} \right\| \\ = \left\| \mathbf {v} _ {t} \nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) + \nabla_ {\mathbf {y}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right. \\ - \left(\mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) + \nabla_ {\mathbf {y}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}))\right) \right\| \\ \leq \left\| \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| \\ + \left\| \nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \left(\mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \right\| \\ + \left\| \nabla_ {\mathbf {y}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) \right\|. \\ \end{array} +$$ + +Then, from Assumption 2.3 and (40), we have + +$$ +\begin{array}{l} \left\| \mathbf {d} _ {t} ^ {\mathbf {y}} \right\| \leq \ell_ {g, 2} \left\| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| \left\| \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| + \ell_ {g, 1} \left\| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| + \ell_ {f, 1} \left\| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| \\ \leq \left(\frac {\ell_ {g , 2} \ell_ {f , 0}}{\mu_ {g}} + \ell_ {f , 1}\right) \| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| + \ell_ {g, 1} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| \\ \leq M _ {\mathbf {v}} \left(\left\| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| + \left\| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\|\right), \\ \end{array} +$$ + +where + +$$ +M _ {\mathbf {v}} := \sqrt {2} \max \left\{\frac {\ell_ {g , 2} \ell_ {f , 0}}{\mu_ {g}} + \ell_ {f, 1}, \ell_ {g, 1} \right\}. \tag {43} +$$ + +The proofs of Eqs. (39c)-(39e) follow from [67, Lemma 17] by setting + +$$ +L _ {\mathbf {y}} := \frac {\ell_ {g , 1}}{\mu_ {g}}, +$$ + +$$ +L _ {\mathbf {v}} := \ell_ {f, 1} + \frac {\ell_ {g , 1} \ell_ {f , 1}}{\mu_ {g}} + \frac {\ell_ {f , 0}}{\mu_ {g}} \left(\ell_ {g, 2} + \frac {\ell_ {g , 1} \ell_ {g , 2}}{\mu_ {g}}\right), \tag {44} +$$ + +$$ +L _ {f} := \ell_ {f, 1} + \frac {\ell_ {g , 1} (\ell_ {f , 1} + M _ {f})}{\mu_ {g}} + \frac {\ell_ {f , 0}}{\mu_ {g}} \left(\ell_ {g, 2} + \frac {\ell_ {g , 1} \ell_ {g , 2}}{\mu_ {g}}\right), +$$ + +where the other constants are defined in Assumption 2.3. + +![](images/50a11f56687327dee7276a701d80505188d914825179d449f872efe194823b0f.jpg) + +The following lemma is inspired by [69] and can be viewed as an extension of [69] to the online setting. + +Lemma C.2. Suppose Assumptions B3. and C1. hold. Let $\{(\mathbf{x}_t,\mathbf{y}_t,\mathbf{v}_t)\}_{t = 1}^T$ be generated according to Algorithm 1. For $e_t^g$ defined as + +$$ +e _ {t} ^ {g} := \mathbf {d} _ {t} ^ {\mathbf {y}} - \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right), \tag {45} +$$ + +we have: + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {g} \| ^ {2} \leq (1 - \gamma_ {t + 1}) ^ {2} (1 + 4 8 \ell_ {g, 1} ^ {2} \beta_ {t} ^ {2}) \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \gamma_ {t + 1} ^ {2} \frac {\sigma_ {g _ {\mathbf {y}}} ^ {2}}{\bar {b}} + 2 4 (1 - \gamma_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} \\ + 6 \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {z} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1}) \| ^ {2} \\ + 4 8 \left(1 - \gamma_ {t + 1}\right) ^ {2} \ell_ {g, 1} ^ {2} \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \| ^ {2}. \tag {46} \\ \end{array} +$$ + +Proof. From Algorithm 1, we have + +$$ +\mathbf {d} _ {t + 1} ^ {\mathbf {y}} = \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) + (1 - \gamma_ {t + 1}) \left(\mathbf {d} _ {t} ^ {\mathbf {y}} - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1})\right). +$$ + +Then, we have + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {g} \| ^ {2} = \mathbb {E} \| \mathbf {d} _ {t + 1} ^ {\mathbf {y}} - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1}) \| ^ {2} \\ = \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {B} _ {t + 1}) + (1 - \gamma_ {t + 1}) \left(\mathbf {d} _ {t} ^ {\mathbf {y}} - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t}; \bar {B} _ {t + 1})\right) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1}) \| ^ {2} \\ = \mathbb {E} \| (1 - \gamma_ {t + 1}) e _ {t} ^ {g} + (\nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1})) \\ - \left(1 - \gamma_ {t + 1}\right) \left(\nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1})\right) - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {z} _ {t}) \| ^ {2}, \\ \end{array} +$$ + +which implies that + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {g} \| ^ {2} = (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + \mathbb {E} \| (\nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1})) \\ - \left(1 - \gamma_ {t + 1}\right) \left(\nabla_ {\mathbf {y}} g _ {t + 1} \left(\mathbf {z} _ {t}; \vec {B} _ {t + 1}\right)\right) - \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {z} _ {t}\right) \| ^ {2} \\ \leq \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \gamma_ {t + 1} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1}) \| ^ {2} \\ + 2 \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \vec {\mathcal {B}} _ {t + 1}\right) \\ - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t}; \bar {B} _ {t + 1}) + \nabla_ {\mathbf {y}} g _ {t} (\mathbf {z} _ {t}) \| ^ {2} \\ \leq (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \gamma_ {t + 1} ^ {2} \frac {\sigma_ {g _ {\mathbf {y}}} ^ {2}}{\bar {b}} \\ + 2 \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) \\ \left. - \nabla_ {\mathbf {y}} g _ {t + 1} \left(\mathbf {z} _ {t + 1}\right) - \nabla_ {\mathbf {y}} g _ {t + 1} \left(\mathbf {z} _ {t}; \vec {B} _ {t + 1}\right) + \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {z} _ {t}\right) \right\| ^ {2}, \\ \end{array} +$$ + +where the second inequality follows from Cauchy-Schwartz inequality and Assumption C1.. + +Moreover, from Cauchy-Schwartz inequality, we have + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {g} \| ^ {2} \leq (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \gamma_ {t + 1} ^ {2} \frac {\sigma_ {g _ {\mathbf {y}}} ^ {2}}{\bar {b}} \\ + 6 \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {z} _ {t}) - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {z} _ {t + 1}) \| ^ {2} \\ + 6 \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {z} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1}) \| ^ {2} \\ + 6 \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) - \nabla_ {\mathbf {y}} g _ {t + 1} \left(\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1}\right) \| ^ {2}. \\ \end{array} +$$ + +From Assumption B3., we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {z} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {z} _ {t}) \| ^ {2} \\ \leq 2 \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t}) \| ^ {2} + 2 \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \\ \leq 2 \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} \\ = 2 \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \ell_ {g, 1} ^ {2} \beta_ {t} ^ {2} \mathbb {E} \| \mathbf {d} _ {t} ^ {\mathbf {y}} \| ^ {2}, \\ \end{array} +$$ + +and + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1}; \vec {\mathcal {B}} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t}; \vec {\mathcal {B}} _ {t + 1}) \| ^ {2} \\ \leq 2 \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2} \\ + 2 \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t}; \bar {\mathcal {B}} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2} \\ \leq 2 \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} \\ = 2 \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \ell_ {g, 1} ^ {2} \beta_ {t} ^ {2} \mathbb {E} \| \mathbf {d} _ {t} ^ {\mathbf {y}} \| ^ {2}. \\ \end{array} +$$ + +From the two inequalities above, we have + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {g} \| ^ {2} \leq (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \gamma_ {t + 1} ^ {2} \frac {\sigma_ {g _ {\mathbf {y}}} ^ {2}}{\bar {b}} \\ + 6 \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {z} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {z} _ {t + 1}) \| ^ {2} \\ + 2 4 \left(1 - \gamma_ {t + 1}\right) ^ {2} \ell_ {g, 1} ^ {2} \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + \beta_ {t} ^ {2} \mathbb {E} \| \mathbf {d} _ {t} ^ {\mathbf {y}} \| ^ {2}\right). \\ \end{array} +$$ + +Since $e_t^g \coloneqq \mathbf{d}_t^{\mathbf{y}} - \nabla_{\mathbf{y}} g_t(\mathbf{x}_t, \mathbf{y}_t)$ , we have + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {g} \| ^ {2} \leq (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \gamma_ {t + 1} ^ {2} \frac {\sigma_ {g _ {\mathrm {y}}} ^ {2}}{\bar {b}} + 2 4 (1 - \gamma_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} \\ + 6 \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {z} _ {t + 1}\right) - \nabla_ {\mathbf {y}} g _ {t + 1} \left(\mathbf {z} _ {t + 1}\right) \| ^ {2} \\ + 4 8 \left(1 - \gamma_ {t + 1}\right) ^ {2} \ell_ {g, 1} ^ {2} \beta_ {t} ^ {2} \mathbb {E} \left\| e _ {t} ^ {g} \right\| ^ {2} + 4 8 \left(1 - \gamma_ {t + 1}\right) ^ {2} \ell_ {g, 1} ^ {2} \beta_ {t} ^ {2} \mathbb {E} \left\| \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right\| ^ {2} \\ \leq (1 - \gamma_ {t + 1}) ^ {2} (1 + 4 8 \ell_ {g, 1} ^ {2} \beta_ {t} ^ {2}) \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \gamma_ {t + 1} ^ {2} \frac {\sigma_ {g _ {\mathbf {y}}} ^ {2}}{\bar {b}} + 2 4 (1 - \gamma_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} \\ + 6 \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {z} _ {t + 1}\right) - \nabla_ {\mathbf {y}} g _ {t + 1} \left(\mathbf {z} _ {t + 1}\right) \| ^ {2} \\ + 4 8 \left(1 - \gamma_ {t + 1}\right) ^ {2} \ell_ {g, 1} ^ {2} \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \| ^ {2}. \\ \end{array} +$$ + +Lemma C.3. Suppose Assumptions 2.2, and B3. hold. Then, for the sequence $\{(\mathbf{x}_t,\mathbf{y}_t)\}_{t = 1}^T$ generated by Algorithm 1, we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \leq (1 + a) \left(1 - 2 \beta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \mathbb {E} \left[ \| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ + \left(- (1 + a) \left(\frac {2 \beta_ {t}}{\mu_ {g} + \ell_ {g , 1}} - \beta_ {t} ^ {2}\right)\right) \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \right] \\ + \left(1 + \frac {1}{a}\right) \beta_ {t} ^ {2} \mathbb {E} \left[ \| e _ {t} ^ {g} \| ^ {2} \right], \\ \end{array} +$$ + +where $e_t^g$ defined in (45), $\mathbf{y}_t^*(\mathbf{x}_t)$ is defined in (1) and $a > 0$ is a constant. + +Proof. From Lemma B.4, we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] = \mathbb {E} \left[ \| \mathbf {y} _ {t} - \beta_ {t} \mathbf {d} _ {t} ^ {\mathbf {y}} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ \leq (1 + a) \mathbb {E} \left[ \| \mathbf {y} _ {t} - \beta_ {t} \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ + \left(1 + \frac {1}{a}\right) \beta_ {t} ^ {2} \mathbb {E} \left[ \| \mathbf {d} _ {t} ^ {\mathbf {y}} - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \right]. \tag {47} \\ \end{array} +$$ + +Next, we will bound the first term on the RHS of (47). + +We have + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {y} _ {t} - \beta_ {t} \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] = \mathbb {E} \left[ \| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] + \beta_ {t} ^ {2} \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \right] \\ - 2 \beta_ {t} \mathbb {E} \left[ \left\langle \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right), \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right) \right\rangle \right] \\ \leq \left(1 - 2 \beta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \mathbb {E} \left[ \| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ - \left(\frac {2 \beta_ {t}}{\mu_ {g} + \ell_ {g , 1}} - \beta_ {t} ^ {2}\right) \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \| ^ {2} \right], \tag {48} \\ \end{array} +$$ + +where the inequality results from the strong convexity of $g_{t}$ by Assumption 2.2, which implies + +$$ +\langle \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}), \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \rangle \geq \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}} \| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} + \frac {1}{\mu_ {g} + \ell_ {g , 1}} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}. +$$ + +Substituting (48) into (47), gives the desired result. + +![](images/8835829cb514e7eda3be797f76572cae6a7ac5eace14b67f13c4363ce145da0f.jpg) + +To simplify the notation in the analysis, we introduce the definitions + +$$ +\theta_ {t} ^ {\mathbf {y}} := \left\| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2}, \quad \text {a n d} \quad \theta_ {t} ^ {\mathbf {v}} := \left\| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2}. \tag {49} +$$ + +The following lemma, inspired by the offline bilevel optimization framework in [69], characterizes the descent behavior of the iterates in the inner problem. + +Lemma C.4. Suppose Assumptions 2.2, and B3. hold. Let $\theta_t^{\mathbf{y}}$ be defined as in (49). Then, for the sequence $\{(\mathbf{x}_t,\mathbf{y}_t)\}_{t = 1}^T$ generated by Algorithm 1, the following bound is guaranteed: + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \left(\mathbb {E} \left[ \theta_ {t + 1} ^ {\mathbf {y}} \right] - \mathbb {E} \left[ \theta_ {t} ^ {\mathbf {y}} \right]\right) \tag {50} \\ \leq - \frac {L _ {\mu_ {g}}}{2} \sum_ {t = 1} ^ {T} \beta_ {t} \mathbb {E} \left[ \theta_ {t} ^ {\mathbf {y}} \right] + \frac {2}{L _ {\mu_ {g}}} \sum_ {t = 1} ^ {T} \beta_ {t} \mathbb {E} \left[ \| e _ {t} ^ {\mathbf {g}} \| ^ {2} \right] + \frac {4 L _ {\mathbf {y}} ^ {2}}{L _ {\mu_ {g}}} \sum_ {t = 1} ^ {T} \frac {1}{\beta_ {t}} \mathbb {E} \| \mathbf {x} _ {t} - \mathbf {x} _ {t + 1} \| ^ {2} \\ + \frac {4}{L _ {\mu_ {g}}} \sum_ {t = 2} ^ {T} \frac {1}{\beta_ {t}} \sup _ {\mathbf {x} \in \mathcal {X}} \mathbb {E} \| \mathbf {y} _ {t - 1} ^ {*} (\mathbf {x}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x}) \| ^ {2} + \sum_ {t = 1} ^ {T} \left(- \frac {2 \beta_ {t}}{\mu_ {g} + \ell_ {g , 1}} + \beta_ {t} ^ {2}\right) \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \right], \\ \end{array} +$$ + +where $L_{\mu_g} = \frac{\mu_g \ell_{g,1}}{\mu_g + \ell_{g,1}}$ , $L_{\mathbf{y}} = \frac{\ell_{g,1}}{\mu_g}$ is defined as in (44); $H_{2,T}$ is defined in (11). Moreover, $e_t^g$ is defined in (45). + +Proof. From Lemma B.4, we have for any $\acute{c} > 0$ + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \right] = \mathbb {E} \left[ \left\| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) + \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \right] \\ \leq (1 + \acute {c}) \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ + \left(1 + \frac {1}{\dot {c}}\right) \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right]. \tag {51} \\ \end{array} +$$ + +From Lemma C.3, we have for any $a > 0$ + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \right] \leq (1 + a) \left(1 - 2 \beta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \mathbb {E} \left[ \left\| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \right] \\ + \left(- (1 + a) \left(\frac {2 \beta_ {t}}{\mu_ {g} + \ell_ {g , 1}} - \beta_ {t} ^ {2}\right)\right) \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \right] \\ + \left(1 + \frac {1}{a}\right) \beta_ {t} ^ {2} \mathbb {E} \left[ \| e _ {t} ^ {g} \| ^ {2} \right]. \tag {52} \\ \end{array} +$$ + +Substituting (52) into (51), we get + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \right] \\ \leq (1 + \acute {c}) (1 + a) \left(1 - 2 \beta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \mathbb {E} \left[ \| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ + \left(- (1 + \acute {c}) (1 + a) \left(\frac {2 \beta_ {t}}{\mu_ {g} + \ell_ {g , 1}} - \beta_ {t} ^ {2}\right)\right) \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \right] \\ + (1 + \acute {c}) (1 + \frac {1}{a}) \beta_ {t} ^ {2} \mathbb {E} \left[ \| e _ {t} ^ {g} \| ^ {2} \right] \\ + \left(1 + \frac {1}{\dot {c}}\right) \mathbb {E} \left[ \left\| \mathbf {y} _ {t + 1} ^ {*} \left(\mathbf {x} _ {t + 1}\right) - \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right) \right\| ^ {2} \right]. \tag {53} \\ \end{array} +$$ + +Choose $\acute{c} = \frac{\beta_t L_{\mu_g} / 2}{1 - \beta_t L_{\mu_g}}$ and $a = \frac{\beta_t L_{\mu_g}}{1 - 2\beta_t L_{\mu_g}}$ . Let $L_{\mu_g} \coloneqq \frac{\mu_g \ell_{g,1}}{\mu_g + \ell_{g,1}}$ . Then, the following equations and inequalities are satisfied. + +$$ +(1 + \acute {c}) (1 + a) \left(1 - 2 \beta_ {t} L _ {\mu_ {g}}\right) = 1 - \frac {\beta_ {t} L _ {\mu_ {g}}}{2}, +$$ + +$$ +(1 + a) \left(1 - 2 \beta_ {t} L _ {\mu_ {g}}\right) = 1 - \beta_ {t} L _ {\mu_ {g}}, +$$ + +$$ +(1 + \acute {c}) \left(1 - \beta_ {t} L _ {\mu_ {g}}\right) = 1 - \frac {\beta_ {t} L _ {\mu_ {g}}}{2}, \tag {54} +$$ + +$$ +1 + \frac {1}{a} \leq \frac {1}{\beta_ {t} L _ {\mu_ {g}}}, \quad 1 + \frac {1}{\acute {c}} \leq \frac {2}{\beta_ {t} L _ {\mu_ {g}}}. +$$ + +Based on (53) and (54), we get + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \| ^ {2} \right] - \mathbb {E} \left[ \| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ \leq - \frac {\beta_ {t} L _ {\mu_ {g}}}{2} \mathbb {E} \left[ \| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] + \left(- \left(\frac {2 \beta_ {t}}{\mu_ {g} + \ell_ {g , 1}} - \beta_ {t} ^ {2}\right)\right) \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \right] \\ + \frac {2}{\beta_ {t} L _ {\mu_ {g}}} \beta_ {t} ^ {2} \mathbb {E} \left[ \| e _ {t} ^ {g} \| ^ {2} \right] + \frac {2}{\beta_ {t} L _ {\mu_ {g}}} \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right]. \tag {55} \\ \end{array} +$$ + +Next, we upper-bound the last term of the above inequality. + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \mathbf {y} _ {t + 1} ^ {*} \left(\mathbf {x} _ {t + 1}\right) - \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right) \right\| ^ {2} \right] \\ \leq 2 \left(\mathbb {E} \left[ \| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] + \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right]\right) \\ \leq 2 \left(L _ {\mathbf {y}} ^ {2} \mathbb {E} \left[ \| \mathbf {x} _ {t} - \mathbf {x} _ {t + 1} \| ^ {2} + \| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right]\right), \tag {56} \\ \end{array} +$$ + +where the second inequality is by Eq. (39d) in Lemma C.1. + +Substituting (56) into (55) and summing over $t \in [T]$ , give the desired result. + +![](images/cd8b8b9e6f7427c23212480c02b3fdba5175f7a887b7dbaf0674e7a1bc85c0a1.jpg) + +# C.3 Bounds on the Linear System Solution + +Lemma C.5. Suppose Assumptions 2.2 and B3. hold. Then, for the sequence $\{(\mathbf{x}_t,\mathbf{y}_t,\mathbf{v}_t)\}_{t = 1}^T$ generated by Algorithm 1, we have + +$$ +\begin{array}{l} \mathbb {E} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \leq (1 + \acute {c}) \left(1 - 2 \delta_ {t} \frac {\left(\ell_ {g , 1} + \ell_ {g , 1} ^ {3}\right) \mu_ {g}}{\mu_ {g} + \ell_ {g , 1}} + \delta_ {t} ^ {2} \ell_ {g, 1} ^ {2}\right) \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \\ + \left(1 + \frac {1}{\dot {c}}\right) \delta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2}, \\ \end{array} +$$ + +for any $\acute{c} > 0$ , where $\mathbf{v}_t^*(\mathbf{x}_t)$ is the solution of the system in Eq. (4), and $e_t^{\mathbf{v}}$ is defined in (60). + +Proof. From the update rules in Algorithm 1, we have the following: + +$$ +\begin{array}{l} \mathbb {E} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} = \mathbb {E} \| \Pi_ {\mathcal {Z} _ {p}} [ \mathbf {v} _ {t} - \delta_ {t} \mathbf {d} _ {t} ^ {\mathbf {v}} ] - \Pi_ {\mathcal {Z} _ {p}} [ \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) ] \| ^ {2} \\ \leq \mathbb {E} \| \mathbf {v} _ {t} - \delta_ {t} \mathbf {d} _ {t} ^ {\mathbf {v}} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \\ \leq (1 + \acute {c}) \mathbb {E} \| \mathbf {v} _ {t} - \delta_ {t} \nabla P _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}\right) - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \\ + \left(1 + \frac {1}{c}\right) \delta_ {t} ^ {2} \mathbb {E} \| \mathbf {d} _ {t} ^ {\mathbf {v}} - \nabla P _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right), \mathbf {v} _ {t}\right) \| ^ {2}, \tag {57} \\ \end{array} +$$ + +where $\nabla P_t(\mathbf{x}_t, \mathbf{y}_t^*(\mathbf{x}_t), \mathbf{v}_t) \coloneqq \nabla_{\mathbf{y}}^2 g_t(\mathbf{x}_t, \mathbf{y}_t^*(\mathbf{x}_t)) \mathbf{v}_t + \nabla_{\mathbf{y}} f_t(\mathbf{x}_t, \mathbf{y}_t^*(\mathbf{x}_t)).$ + +For the first term of Eq. (57) above, we have + +$$ +\begin{array}{l} \mathbb {E} \| \mathbf {v} _ {t} - \delta_ {t} \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}) - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \\ = \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} - 2 \delta_ {t} \mathbb {E} \langle \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}), \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}) \rangle + \delta_ {t} ^ {2} \mathbb {E} \| \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}) \| ^ {2} \\ \leq \left(1 - 2 \delta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} - \left(2 \delta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}} - \delta_ {t} ^ {2}\right) \mathbb {E} \| \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}) \| ^ {2} \\ \leq \left(1 - 2 \delta_ {t} \frac {\left(\ell_ {g , 1} + \ell_ {g , 1} ^ {3}\right) \mu_ {g}}{\mu_ {g} + \ell_ {g , 1}} + \delta_ {t} ^ {2} \ell_ {g, 1} ^ {2}\right) \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2}, \tag {58} \\ \end{array} +$$ + +where the first inequality follows from the strong convexity of the function $P_{t}$ , which is the gradient of the strongly convex quadratic program $\frac{1}{2}\mathbf{v}^{\top}\nabla_{\mathbf{y}}^{2}g_{t}(\mathbf{x},\mathbf{y}_{t}^{*}(\mathbf{x}))\mathbf{v} + \mathbf{v}^{\top}\nabla_{\mathbf{y}}f_{t}(\mathbf{x},\mathbf{y}_{t}^{*}(\mathbf{x}))$ . Then, we have + +$$ +\begin{array}{l} \mathbb {E} \langle \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}), \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}) \rangle \geq \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}} \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \\ + \frac {1}{\mu_ {g} + \ell_ {g , 1}} \mathbb {E} \| \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}) \| ^ {2}. \\ \end{array} +$$ + +The second inequality is derived from the following inequality. + +$$ +\begin{array}{l} \mathbb {E} \| \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}) \| ^ {2} = \mathbb {E} \| \nabla_ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})) \mathbf {v} _ {t} + \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})) \| ^ {2} \\ = \mathbb {E} \| \nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \left(\mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \| ^ {2} \\ \leq \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2}, \tag {59} \\ \end{array} +$$ + +where the second equality follows from (4). + +Combining (57) and (58), we get the desired result. + +Lemma C.6. Suppose Assumptions B2., B3., B4., C2. and C4. hold. Let $\{(\mathbf{x}_t,\mathbf{y}_t,\mathbf{v}_t)\}_{t = 1}^T$ be generated according to Algorithm 1. For $e_{t + 1}^{\mathbf{v}}$ defined as + +$$ +e _ {t} ^ {\mathbf {v}} := \mathbf {d} _ {t} ^ {\mathbf {v}} - \nabla P _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t}\right), \quad \text {w h e r e} \tag {60a} +$$ + +$$ +\nabla P _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t}\right) := \nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \mathbf {v} _ {t} + \nabla_ {\mathbf {y}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right), \tag {60b} +$$ + +we have: + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {\mathbf {v}} \| ^ {2} \leq (1 - \lambda_ {t + 1}) ^ {2} (1 + 7 2 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2}) \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + 4 \lambda_ {t + 1} ^ {2} (\frac {\sigma_ {g _ {\mathbf {y y}}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {y}}} ^ {2}}{b}) \\ + 1 2 p ^ {2} \left(1 - \lambda_ {t + 1}\right) ^ {2} \mathbb {E} \left\| \nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) - \nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) \right\| ^ {2} \\ + 1 2 (1 - \lambda_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ + 7 2 (1 - \lambda_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) \\ + 1 4 4 \left(1 - \lambda_ {t + 1}\right) ^ {2} \ell_ {g, 1} ^ {4} \delta_ {t} ^ {2} \mathbb {E} \left[ \theta_ {t} ^ {\mathbf {v}} \right] + 2 8 8 \ell_ {g, 1} ^ {2} \left(p ^ {2} \ell_ {g, 2} ^ {2} + \ell_ {f, 1} ^ {2}\right) \delta_ {t} ^ {2} \mathbb {E} \left[ \theta_ {t} ^ {\mathbf {v}} \right], \tag {61} \\ \end{array} +$$ + +for all $t \in [T]$ and $(\theta_t^{\mathbf{v}}, \theta_t^{\mathbf{y}})$ and $e_t^g$ are defined in (49) and (45), respectively. + +Proof. Note that + +$$ +e _ {t + 1} ^ {\mathbf {v}} := \mathbf {d} _ {t + 1} ^ {\mathbf {v}} - \nabla P _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}, \mathbf {v} _ {t + 1}), +$$ + +where + +$$ +\nabla P _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}, \mathbf {v} _ {t + 1}) := \nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \mathbf {v} _ {t + 1} + \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}). +$$ + +From Algorithm 1, we have + +$$ +\mathbf {d} _ {t + 1} ^ {\mathbf {v}} = \mathbf {d} _ {t + 1} ^ {\mathbf {v v}} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \mathcal {B} _ {t + 1}\right) + (1 - \lambda_ {t + 1}) \left(\mathbf {d} _ {t} ^ {\mathbf {v}} - \mathbf {d} _ {t + 1} ^ {\mathbf {v v}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}; \mathcal {B} _ {t + 1}\right)\right). +$$ + +Let $\mathbf{u} = [\mathbf{x};\mathbf{y};\mathbf{v}]$ . Then, we have + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {\mathbf {v}} \| ^ {2} = \mathbb {E} \| \mathbf {d} _ {t + 1} ^ {\mathbf {v}} - \nabla P _ {t + 1} (\mathbf {u} _ {t + 1}) \| ^ {2} \\ = \mathbb {E} \| \nabla P _ {t + 1} (\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}) + (1 - \lambda_ {t + 1}) \left(\mathbf {d} _ {t} ^ {\mathbf {v}} - \nabla P _ {t + 1} (\mathbf {u} _ {t}; \mathcal {B} _ {t + 1})\right) - \nabla P _ {t + 1} (\mathbf {u} _ {t + 1}) \| ^ {2} \\ = \mathbb {E} \| (1 - \lambda_ {t + 1}) e _ {t} ^ {\mathbf {v}} + \nabla P _ {t + 1} (\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}) - \nabla P _ {t + 1} (\mathbf {u} _ {t + 1}) \\ - \left(1 - \lambda_ {t + 1}\right) \left(\nabla P _ {t + 1} \left(\mathbf {u} _ {t}; \mathcal {B} _ {t + 1}\right) - \nabla P _ {t} \left(\mathbf {u} _ {t}\right)\right) \| ^ {2}, \\ \end{array} +$$ + +which implies that + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {\mathbf {v}} \| ^ {2} \\ = (1 - \lambda_ {t + 1}) ^ {2} \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + \mathbb {E} \| \lambda_ {t + 1} (\nabla P _ {t + 1} (\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}) - \nabla P _ {t + 1} (\mathbf {u} _ {t + 1})) \\ - \left(1 - \lambda_ {t + 1}\right) \left(\nabla P _ {t + 1} \left(\mathbf {u} _ {t}; \mathcal {B} _ {t + 1}\right) - \nabla P _ {t + 1} \left(\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}\right) + \nabla P _ {t + 1} \left(\mathbf {u} _ {t + 1}\right) - \nabla P _ {t} \left(\mathbf {u} _ {t}\right)\right) \| ^ {2} \\ \leq \left(1 - \lambda_ {t + 1}\right) ^ {2} \mathbb {E} \left\| e _ {t} ^ {\mathbf {v}} \right\| ^ {2} + 2 \lambda_ {t + 1} ^ {2} \mathbb {E} \left\| \nabla P _ {t + 1} \left(\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}\right) - \nabla P _ {t + 1} \left(\mathbf {u} _ {t + 1}\right) \right\| ^ {2} \\ + 2 \left(1 - \lambda_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla P _ {t + 1} \left(\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}\right) - \nabla P _ {t + 1} \left(\mathbf {u} _ {t + 1}\right) - \nabla P _ {t + 1} \left(\mathbf {u} _ {t}; \mathcal {B} _ {t + 1}\right) + \nabla P _ {t} \left(\mathbf {u} _ {t}\right) \| ^ {2}, \\ \end{array} +$$ + +where the inequality follows from Cauchy-Schwartz inequality. + +For the first term, from Assumptions C2. and C4., we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla P _ {t + 1} (\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}) - \nabla P _ {t + 1} (\mathbf {u} _ {t + 1}) \| ^ {2} \\ = \mathbb {E} \| \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \bar {\boldsymbol {B}} _ {t + 1}\right) - \nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right)\right) \mathbf {v} _ {t + 1} \\ + \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \mathcal {B} _ {t + 1}) - \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ \leq 2 \mathbb {E} \| \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \vec {B} _ {t + 1}) - \nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1})\right) \mathbf {v} _ {t + 1} \| ^ {2} \\ + 2 \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \mathcal {B} _ {t + 1}) - \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ \leq 2 (\frac {\sigma_ {g _ {\mathbf {y y}}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {y}}} ^ {2}}{b}), \\ \end{array} +$$ + +where the last inequality follows from (8). + +Then, from the above inequality and $\| a + b + c\| ^2\leq 3(\| a\| ^2 +\| b\| ^2 +\| c\| ^2)$ , we have + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {\mathbf {v}} \| ^ {2} \leq (1 - \lambda_ {t + 1}) ^ {2} \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + 4 \lambda_ {t + 1} ^ {2} (\frac {\sigma_ {g _ {\mathbf {y y}}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {y}}} ^ {2}}{b}) \\ + 6 \left(1 - \lambda_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla P _ {t} (\mathbf {u} _ {t}) - \nabla P _ {t} (\mathbf {u} _ {t + 1}) \| ^ {2} \\ + 6 \left(1 - \lambda_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla P _ {t} (\mathbf {u} _ {t + 1}) - \nabla P _ {t + 1} (\mathbf {u} _ {t + 1}) \| ^ {2} \\ + 6 \left(1 - \lambda_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla P _ {t + 1} \left(\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}\right) - \nabla P _ {t + 1} \left(\mathbf {u}; \mathcal {B} _ {t + 1}\right) \| ^ {2}. \tag {62} \\ \end{array} +$$ + +Moreover, from $\| a + b + c\| ^2\leq 3(\| a\| ^2 +\| b\| ^2 +\| c\| ^2)$ , we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla P _ {t} (\mathbf {u} _ {t + 1}) - \nabla P _ {t} (\mathbf {u} _ {t}) \| ^ {2} \\ \leq 3 \mathbb {E} \| \nabla P _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}, \mathbf {v} _ {t + 1}) - \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t + 1}, \mathbf {v} _ {t + 1}) \| ^ {2} \\ + 3 \mathbb {E} \| \nabla P _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t + 1}, \mathbf {v} _ {t + 1}\right) - \nabla P _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t + 1}\right) \| ^ {2} \\ + 3 \mathbb {E} \| \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t + 1}) - \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t}) \| ^ {2} \\ \leq 3 \mathbb {E} \| \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t + 1})\right) \mathbf {v} _ {t + 1} + \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ + 3 \mathbb {E} \| \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t})\right) \mathbf {v} _ {t + 1} + \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \\ + 3 \mathbb {E} \| \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t + 1}) - \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t}) \| ^ {2} \\ \leq 6 \left(\ell_ {g, 2} ^ {2} \mathbb {E} \| \mathbf {v} _ {t + 1} \| ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + \mathbb {E} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2}\right) + 3 \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2}, \tag {63} \\ \end{array} +$$ + +where the last inequality follows from Assumptions B2., B3. and B4.; + +From Eq. (63) and the inequality $\| a + b \|^2 \leq 2(\| a \|^2 + \| b \|^2)$ , we obtain + +$$ +\begin{array}{l} \mathbb {E} \| \nabla P _ {t} (\mathbf {u} _ {t + 1}) - \nabla P _ {t} (\mathbf {u} _ {t}) \| ^ {2} \\ \leq 6 \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + \beta_ {t} ^ {2} \mathbb {E} \| \mathbf {d} _ {t} ^ {\mathbf {y}} \| ^ {2}\right) + 3 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2} \mathbb {E} \| \mathbf {d} _ {t} ^ {\mathbf {v}} \| ^ {2} \\ \leq 6 \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) \\ + 6 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2} (\mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + \mathbb {E} \| \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t}) \| ^ {2}) \\ \leq 6 \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) \\ + 6 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2} (\mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + 2 \mathbb {E} \| \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}) \| ^ {2} + 2 \mathbb {E} \| \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t}) - \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}) \| ^ {2}) \\ \leq 6 \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) \\ + 6 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2} \left(\mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + 2 \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} + 4 \left(p ^ {2} \ell_ {g, 2} ^ {2} + \ell_ {f, 1} ^ {2}\right) \mathbb {E} \| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2}\right), \tag {64} \\ \end{array} +$$ + +where the last inequality follows from (59). + +Similarly, we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla P _ {t + 1} (\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}) - \nabla P _ {t + 1} (\mathbf {u} _ {t}; \mathcal {B} _ {t + 1}) \| ^ {2} \\ \leq 6 \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) \\ + 6 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2} \left(\mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + 2 \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} + 4 \left(p ^ {2} \ell_ {g, 2} ^ {2} + \ell_ {f, 1} ^ {2}\right) \mathbb {E} \| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2}\right). \tag {65} \\ \end{array} +$$ + +Substituting (65) and (64) into (62), we have + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {\mathbf {v}} \| ^ {2} \leq (1 - \lambda_ {t + 1}) ^ {2} (1 + 7 2 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2}) \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + 4 \lambda_ {t + 1} ^ {2} (\frac {\sigma_ {g _ {\mathbf {v y}}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {y}}} ^ {2}}{b}) \\ + 6 \left(1 - \lambda_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla P _ {t} (\mathbf {u} _ {t + 1}) - \nabla P _ {t + 1} (\mathbf {u} _ {t + 1}) \| ^ {2} \\ + 7 2 (1 - \lambda_ {t + 1}) ^ {2} \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) \\ + 1 4 4 \left(1 - \lambda_ {t + 1}\right) ^ {2} \ell_ {g, 1} ^ {4} \delta_ {t} ^ {2} \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} + 2 8 8 \ell_ {g, 1} ^ {2} \left(p ^ {2} \ell_ {g, 2} ^ {2} + \ell_ {f, 1} ^ {2}\right) \delta_ {t} ^ {2} \mathbb {E} \| \mathbf {y} _ {t} - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2}. \\ \end{array} +$$ + +From $\| a + b\| ^2\leq 2\| a\| ^2 +2\| b\| ^2$ and (8), we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla P _ {t} (\mathbf {u} _ {t + 1}) - \nabla P _ {t + 1} (\mathbf {u} _ {t + 1}) \| ^ {2} = \mathbb {E} \| \nabla_ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \mathbf {v} _ {t + 1} - \nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \mathbf {v} _ {t + 1} \\ + \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ \leq 2 \mathbb {E} \| \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1})\right) \mathbf {v} _ {t + 1} \| ^ {2} \\ + 2 \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ \leq 2 \mathbb {E} \| \nabla_ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} p ^ {2} \\ + 2 \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2}. \\ \end{array} +$$ + +This completes the proof. + +![](images/374f758b10faf436958be61c768c13af2c80c36c4fa6fa3996b049825a1965d8.jpg) + +As demonstrated in Lemma C.6, the gradient estimation error $e_{t+1}^{\mathbf{v}}$ for the linear system consists of four key components: (1) an iteratively refined error term $(1 - \lambda_{t+1})^2 (1 + 72\ell_{g,1}^2\delta_t^2)\mathbb{E}\|e_t^{\mathbf{v}}\|^2$ , which depends on the stepsize $\delta_t$ ; (2) the error arising from the variation in the Hessian of the lower-level objectiv; (3) the error resulting from the variation in the gradient of the upper-level objective, and (4) approximation error terms of order $\mathcal{O}(\delta_t^2\mathbb{E}[\theta_t^{\mathbf{v}}])$ and $\mathcal{O}(\delta_t^2\mathbb{E}[\theta_t^{\mathbf{y}}])$ associated with solving the linear system and the iterates in the inner problem, respectively. + +Lemma C.7. Suppose Assumptions 2.2, B1., B2. and B4. hold. Let $\mathbf{v}_t^* (\mathbf{x})$ is a solution of Subproblem (4). Then, we have + +$$ +\left\| \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) - \mathbf {v} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \leq 2 \frac {\nu^ {2}}{\mu_ {g} ^ {2}} \left(\left\| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} + \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2}\right), +$$ + +where $\nu \coloneqq \ell_{f,1} + \frac{\ell_{g,2}\ell_{f,0}}{\mu_g}$ + +Proof. Based on (4), we have that + +$$ +\begin{array}{l} \left\| \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) - \mathbf {v} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \\ = \| \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) ^ {- 1} \nabla_ {\mathbf {y}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \\ \left. - \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right)\right) ^ {- 1} \nabla_ {\mathbf {y}} f _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right) \right\| ^ {2} \\ \leq 2 \left\| \left(\left(\nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right)\right) ^ {- 1} - \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right)\right) ^ {- 1}\right) \nabla_ {\mathbf {y}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) \right\| ^ {2} (66a) \\ + 2 \left\| \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right)\right) ^ {- 1} \left(\nabla_ {\mathbf {y}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) - \nabla_ {\mathbf {y}} f _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right)\right) \right\| ^ {2}. (66b) \\ \end{array} +$$ + +In the following steps, we bound the terms (66a) and (66b), respectively. + +For (66a), we have: + +$$ +\begin{array}{l} \left\| \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) ^ {- 1} - \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1})\right)\right) ^ {- 1} \right\| ^ {2} \\ = \| \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) ^ {- 1} \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1})\right) \right. \\ \left. - \nabla_ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1})\right)\right) ^ {- 1} \| ^ {2} \\ \leq \frac {1}{\mu_ {g} ^ {2}} \left\| \nabla_ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})) - \nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1})) \right\| ^ {2} \\ \leq \frac {\ell_ {g , 2}}{\mu_ {g} ^ {2}} \left\| \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) - \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1})\right) \right\| ^ {2} \\ \leq \frac {\ell_ {g , 2}}{\mu_ {g} ^ {2}} \left(\left\| \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} + \left\| \mathbf {x} _ {t} - \mathbf {x} _ {t + 1} \right\| ^ {2}\right), \tag {67} \\ \end{array} +$$ + +where the equality holds since for any invertible matrix $\mathbf{A}$ and $\mathbf{B}$ we have $\| \mathbf{A}^{-1} - \mathbf{B}^{-1}\| = \| \mathbf{A}^{-1}(\mathbf{B} - \mathbf{A})\mathbf{B}^{-1}\|$ , and inequalities are obtained from Assumptions 2.2 and B4.. + +Thus, from (67) and Assumption B1., we get + +$$ +(6 6 \mathrm {a}) \leq \frac {\ell_ {f , 0} \ell_ {g , 2}}{\mu_ {g} ^ {2}} \left(\left\| \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} + \| \mathbf {x} _ {t} - \mathbf {x} _ {t + 1} \| ^ {2}\right). \tag {68} +$$ + +For (66b), we have + +$$ +\begin{array}{l} (6 6 \mathrm {b}) \leq \frac {1}{\mu_ {g}} \| \nabla_ {\mathbf {y}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) - \nabla_ {\mathbf {y}} f _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1})\right) \| ^ {2} \\ \leq \frac {\ell_ {f , 1}}{\mu_ {g}} \| (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})) - (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1})) \| ^ {2} \\ \leq \frac {\ell_ {f , 1}}{\mu_ {g}} \left(\left\| \mathbf {y} _ {t + 1} ^ {*} \left(\mathbf {x} _ {t + 1}\right) - \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right) \right\| ^ {2} + \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2}\right). \tag {69} \\ \end{array} +$$ + +Combining (68) and (69), we have + +$$ +\left\| \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) - \mathbf {v} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \leq \frac {1}{\mu_ {g}} \left(\frac {\ell_ {f , 0} \ell_ {g , 2}}{\mu_ {g}} + \ell_ {f, 1}\right) \left(\left\| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} + \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2}\right). +$$ + +By raising both sides of the above inequality to the power 2 and using $(a + b)^2\leq 2a^2 +2b^2$ , we complete the proof. + +The following lemma characterizes the decrease in $\theta_t^{\mathbf{v}}$ defined in (49) and can be viewed as an extension of the offline BO result in [69] to the OBO setting. + +Lemma C.8. Suppose Assumptions 2.2 and 2.3 hold. Let $\theta_t^{\mathbf{v}}$ be defined in (49). Then, for any positive choice of step size $\delta_t$ as + +$$ +\delta_ {t} \leq \frac {\dot {L} _ {\mu_ {g}}}{\ell_ {g , 1} ^ {2}}, \quad \text {w h e r e} \quad \dot {L} _ {\mu_ {g}} := \frac {(\ell_ {g , 1} + \ell_ {g , 1} ^ {3}) \mu_ {g}}{(\mu_ {g} + \ell_ {g , 1})}, +$$ + +for all $t \in [T]$ , the sequence $\{(\mathbf{x}_t, \mathbf{y}_t, \mathbf{v}_t)\}_{t=1}^T$ generated by Algorithm 1 satisfy + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \left(\mathbb {E} [ \theta_ {t + 1} ^ {\mathbf {v}} ] - \mathbb {E} [ \theta_ {t} ^ {\mathbf {v}} ]\right) \leq - \frac {\delta_ {t} \acute {L} _ {\mu_ {g}}}{4} \sum_ {t = 1} ^ {T} \mathbb {E} [ \theta_ {t} ^ {\mathbf {v}} ] + \frac {4}{\acute {L} _ {\mu_ {g}}} \delta_ {t} \sum_ {t = 1} ^ {T} \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} \\ + \frac {1 6 \nu^ {2}}{\dot {L} _ {\mu_ {g}} \mu_ {g} ^ {2} \delta_ {t}} \sum_ {t = 1} ^ {T} \mathbb {E} \left| \left| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right| \right| ^ {2} \\ + \frac {8 \nu^ {2}}{\dot {L} _ {\mu_ {g}} \mu_ {g} ^ {2} \delta_ {t}} (1 + 2 L _ {\mathbf {y}} ^ {2}) \sum_ {t = 1} ^ {T} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2}, \tag {70} \\ \end{array} +$$ + +where $e_t^{\mathbf{v}}$ is defined in (60), $\nu$ and $L_{\mathbf{y}}$ , are defined in Lemmas C.7 and C.4, respectively. + +Proof. By Lemma B.4, for any $a > 0$ , we have + +$$ +\begin{array}{l} \mathbb {E} \left. \left\| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} = \mathbb {E} \left\| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) + \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) - \mathbf {v} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \right. \\ \leq (1 + a) \mathbb {E} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \\ + \left(1 + \frac {1}{a}\right) \mathbb {E} \left\| \mathbf {v} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2}. \tag {71} \\ \end{array} +$$ + +From Lemma C.5, we have for any $\acute{c} > 0$ : + +$$ +\begin{array}{l} \mathbb {E} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \leq (1 + \acute {c}) \left(1 - 2 \delta_ {t} \frac {\left(\ell_ {g , 1} + \ell_ {g , 1} ^ {3}\right) \mu_ {g}}{\mu_ {g} + \ell_ {g , 1}} + \delta_ {t} ^ {2} \ell_ {g, 1} ^ {2}\right) \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \\ + \left(1 + \frac {1}{\dot {c}}\right) \delta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2}. \tag {72} \\ \end{array} +$$ + +Substituting (72) into (71), we get + +$$ +\begin{array}{l} \mathbb {E} \left\| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \leq (1 + a) (1 + \acute {c}) \left(1 - 2 \delta_ {t} \frac {\left(\ell_ {g , 1} + \ell_ {g , 1} ^ {3}\right) \mu_ {g}}{\mu_ {g} + \ell_ {g , 1}} + \delta_ {t} ^ {2} \ell_ {g, 1} ^ {2}\right) \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \\ + (1 + a) \left(1 + \frac {1}{c}\right) \delta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} \\ + \left(1 + \frac {1}{a}\right) \mathbb {E} \left\| \mathbf {v} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2}. \tag {73} \\ \end{array} +$$ + +In the following, we provide a bound for the third term on the right-hand side of (73). To this end, we have from Lemma C.7: + +$$ +\begin{array}{l} \mathbb {E} \left. \right.\left\| \mathbf {v} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \leq 2 \frac {\nu^ {2}}{\mu_ {g} ^ {2}} \left(\mathbb {E} \left\| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} + \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2}\right) \\ \leq 2 \frac {\nu^ {2}}{\mu_ {g} ^ {2}} \left(2 \mathbb {E} \left| \left| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) \right| \right| ^ {2} \right. \\ \left. + 2 \mathbb {E} \left| \left| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right| \right| ^ {2} + \mathbb {E} \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2}\right) \\ \leq 2 \frac {\nu^ {2}}{\mu_ {g} ^ {2}} \left((1 + 2 L _ {\mathbf {y}} ^ {2}) \mathbb {E} \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2} + 2 \mathbb {E} \left\| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2}\right), \\ \end{array} +$$ + +where the last inequality follows from Lemma C.1. + +Combining this result with (73) gives + +$$ +\begin{array}{l} \mathbb {E} \left\| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \leq (1 + a) (1 + \acute {c}) \left(1 - 2 \delta_ {t} \frac {\left(\ell_ {g , 1} + \ell_ {g , 1} ^ {3}\right) \mu_ {g}}{\mu_ {g} + \ell_ {g , 1}} + \delta_ {t} ^ {2} \ell_ {g, 1} ^ {2}\right) \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \\ + (1 + a) \left(1 + \frac {1}{\acute {c}}\right) \delta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + 4 \left(1 + \frac {1}{a}\right) \frac {\nu^ {2}}{\mu_ {g} ^ {2}} \mathbb {E} \left| \left| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right| \right| ^ {2} \\ + 2 \left(1 + \frac {1}{a}\right) \frac {\nu^ {2}}{\mu_ {g} ^ {2}} \left(1 + 2 L _ {\mathbf {y}} ^ {2}\right) \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2}. \tag {74} \\ \end{array} +$$ + +Let $\acute{L}_{\mu_g} := \frac{(\ell_{g,1} + \ell_{g,1}^3)\mu_g}{\mu_g + \ell_{g,1}}$ , then we have + +$$ +\begin{array}{l} 1 - 2 \delta_ {t} \frac {\left(\ell_ {g , 1} + \ell_ {g , 1} ^ {3}\right) \mu_ {g}}{\mu_ {g} + \ell_ {g , 1}} + \delta_ {t} ^ {2} \ell_ {g, 1} ^ {2} = 1 - 2 \delta_ {t} \dot {L} _ {\mu_ {g}} + \delta_ {t} ^ {2} \ell_ {g, 1} ^ {2} \\ \leq 1 - \delta_ {t} \dot {L} _ {\mu_ {g}}, \tag {75} \\ \end{array} +$$ + +where the last inequality follows from $\delta_t \leq \frac{\dot{L}_{\mu_g}}{\ell_{g,1}^2}$ . + +Choose $a = \frac{\delta_t\dot{L}_{\mu_g} / 4}{1 - \frac{\delta_t\dot{L}_{\mu_g}}{2}}$ and $\acute{c} = \frac{\delta_t\dot{L}_{\mu_g} / 2}{1 - \delta_t\dot{L}_{\mu_g}}$ . Then, from (75), we have + +$$ +\begin{array}{l} (1 + a) (1 + \acute {c}) \left(1 - 2 \delta_ {t} \frac {\left(\ell_ {g , 1} + \ell_ {g , 1} ^ {3}\right) \mu_ {g}}{\mu_ {g} + \ell_ {g , 1}} + \delta_ {t} ^ {2} \ell_ {g, 1} ^ {2}\right) \\ \leq (1 + a) (1 + \acute {c}) \left(1 - \delta_ {t} \dot {L} _ {\mu_ {g}}\right) = 1 - \frac {\delta_ {t} \dot {L} _ {\mu_ {g}}}{4}, \tag {76} \\ \end{array} +$$ + +$$ +(1 + a) \left(1 + \frac {1}{\acute {c}}\right) \leq \frac {4}{\delta_ {t} \acute {L} _ {\mu_ {g}}}, +$$ + +$$ +1 + \frac {1}{\acute {c}} \leq \frac {2}{\delta_ {t} \acute {L} _ {\mu_ {g}}}, \quad 1 + \frac {1}{a} \leq \frac {4}{\delta_ {t} \acute {L} _ {\mu_ {g}}}. +$$ + +Thus, from (74) and (76) we have + +$$ +\begin{array}{l} \mathbb {E} \left\| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \leq \left(1 - \frac {\delta_ {t} \dot {L} _ {\mu_ {g}}}{4}\right) \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \\ + \frac {4}{\dot {L} _ {\mu_ {g}}} \delta_ {t} \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + \frac {1 6 \nu^ {2}}{\dot {L} _ {\mu_ {g}} \mu_ {g} ^ {2} \delta_ {t}} \mathbb {E} \left\| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \\ + \frac {8 \nu^ {2}}{\dot {L} _ {\mu_ {g}} \mu_ {g} ^ {2} \delta_ {t}} (1 + 2 L _ {\mathbf {y}} ^ {2}) \mathbb {E} \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2}. \\ \end{array} +$$ + +Rearranging the terms and summing from $t = 1$ to $T$ , gives the desired result. + +![](images/335f3010f44e1059ee0e57100554e09ea8a1f37da70128f4ab0d9fa8978b114e.jpg) + +# C.4 Bounds on the Gradient Estimation Error of Outer Objective + +The following lemma, inspired by [69], provides a characterization of the descent of the gradient estimation error for the outer-level function. + +Lemma C.9. Suppose Assumptions B2., B3., B4., C3. and C5. hold. Let $\{(\mathbf{x}_t, \mathbf{y}_t, \mathbf{v}_t)\}_{t=1}^T$ be generated according to Algorithm 1. For $e_t^f$ defined as + +$$ +e _ {t} ^ {f} := \mathbf {d} _ {t} ^ {\mathbf {x}} - \tilde {\mathbf {d}} _ {t} \left(\mathbf {z} _ {t}, \mathbf {v} _ {t}\right), \quad \text {w h e r e} \quad \tilde {\mathbf {d}} _ {t} \left(\mathbf {z} _ {t}, \mathbf {v} _ {t}\right) = \nabla_ {\mathbf {x}} f _ {t} \left(\mathbf {z} _ {t}\right) + \nabla_ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {z} _ {t}\right) \mathbf {v} _ {t}, \tag {77} +$$ + +we have: + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {f} \| ^ {2} \leq (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| e _ {t} ^ {f} \| ^ {2} + 4 \eta_ {t + 1} ^ {2} (\frac {\sigma_ {g _ {\bf x y}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\bf x}} ^ {2}}{b}) \\ + 1 2 p ^ {2} \left(1 - \eta_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) \| ^ {2} \\ + 1 2 (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {x}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ + 7 2 (1 - \eta_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) \\ + 7 2 \ell_ {g, 1} ^ {2} \left(1 - \eta_ {t + 1}\right) ^ {2} \delta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + 7 2 \left(1 - \eta_ {t + 1}\right) ^ {2} \ell_ {g, 1} ^ {4} \delta_ {t} ^ {2} \mathbb {E} \left[ \theta_ {t} ^ {\mathbf {v}} \right], \tag {78} \\ \end{array} +$$ + +for all $t \in [T]$ , $\theta_t^{\mathbf{v}}$ , $e_t^{\mathbf{v}}$ and $e_t^g$ are defined in (49), (60) and (45), respectively. + +Proof. Note that + +$$ +e _ {t + 1} ^ {f} = \mathbf {d} _ {t + 1} ^ {\mathbf {x}} - \tilde {\mathbf {d}} _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}, \mathbf {v} _ {t + 1}\right), +$$ + +where + +$$ +\tilde {\mathbf {d}} _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}, \mathbf {v} _ {t + 1}\right) = \nabla_ {\mathbf {x}} f _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) + \nabla_ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) \mathbf {v} _ {t + 1}. \tag {79} +$$ + +From Algorithm 1, we have + +$$ +\mathbf {d} _ {t + 1} ^ {\mathbf {x}} = \mathbf {d} _ {t + 1} ^ {\mathbf {x x}} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \mathcal {B} _ {t + 1}\right) + (1 - \eta_ {t + 1}) \left(\mathbf {d} _ {t + 1} ^ {\mathbf {x}} - \mathbf {d} _ {t + 1} ^ {\mathbf {x x}} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \mathcal {B} _ {t + 1}\right)\right), +$$ + +where $\mathbf{d}_{t + 1}^{\mathbf{xx}}(\mathbf{x}_{t + 1},\mathbf{y}_{t + 1};\mathcal{B}_{t + 1}) = \nabla_{\mathbf{x}}f_{t + 1}(\mathbf{x}_{t + 1},\mathbf{y}_{t + 1};\mathcal{B}_{t + 1}) + \nabla_{\mathbf{xy}}^{2}g_{t + 1}(\mathbf{x}_{t + 1},\mathbf{y}_{t + 1};\mathcal{B}_{t + 1})\mathbf{v}_{t + 1}$ . Let $\mathbf{u} = [\mathbf{x};\mathbf{y};\mathbf{v}]$ . Then, we have + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {f} \| ^ {2} = \mathbb {E} \| \mathbf {d} _ {t + 1} ^ {\mathbf {x}} - \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t + 1}) \| ^ {2} \\ = \mathbb {E} \| \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}) + (1 - \eta_ {t + 1}) \left(\mathbf {d} _ {t} ^ {\mathbf {x}} - \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t}; \mathcal {B} _ {t + 1})\right) - \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t + 1}) \| ^ {2} \\ = \mathbb {E} \| (1 - \eta_ {t + 1}) e _ {t} ^ {f} + \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}) - \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t + 1}) \\ \left. - \left(1 - \eta_ {t + 1}\right) \left(\tilde {\mathbf {d}} _ {t + 1} \left(\mathbf {u} _ {t}; \mathcal {B} _ {t + 1}\right) - \tilde {\mathbf {d}} _ {t} \left(\mathbf {u} _ {t}\right)\right) \right\| ^ {2}, \\ \end{array} +$$ + +which implies that + +$$ +\begin{array}{l} \mathbb {E} \left\| e _ {t + 1} ^ {f} \right\| ^ {2} \\ = (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| e _ {t} ^ {f} \| ^ {2} + \mathbb {E} \| \eta_ {t + 1} (\tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}) - \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t + 1})) \\ - (1 - \eta_ {t + 1}) (\tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t}; \mathcal {B} _ {t + 1}) - \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}) + \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t + 1}) - \tilde {\mathbf {d}} _ {t} (\mathbf {u} _ {t})) \| ^ {2} \\ \leq \left(1 - \eta_ {t + 1}\right) ^ {2} \mathbb {E} \| e _ {t} ^ {f} \| ^ {2} + 2 \eta_ {t + 1} ^ {2} \mathbb {E} \| \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}) - \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t + 1}) \| ^ {2} \\ + 2 \left(1 - \eta_ {t + 1}\right) ^ {2} \mathbb {E} \| \tilde {\mathbf {d}} _ {t + 1} \left(\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}\right) - \tilde {\mathbf {d}} _ {t + 1} \left(\mathbf {u} _ {t + 1}\right) - \tilde {\mathbf {d}} _ {t + 1} \left(\mathbf {u} _ {t}; \mathcal {B} _ {t + 1}\right) + \tilde {\mathbf {d}} _ {t} \left(\mathbf {u} _ {t}\right) \| ^ {2}, \tag {80} \\ \end{array} +$$ + +where the inequality follows from $\| a + b\| ^2\leq 2\| a\| ^2 +2\| b\| ^2$ + +Let us bound the second term in the right-hand side of (80). Based on (79), we have + +$$ +\begin{array}{l} \mathbb {E} \left\| \tilde {\mathbf {d}} _ {t + 1} \left(\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}\right) - \tilde {\mathbf {d}} _ {t + 1} \left(\mathbf {u} _ {t + 1}\right) \right\| ^ {2} \\ = \mathbb {E} \| \left(\nabla_ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1})\right) \mathbf {v} _ {t + 1} \\ + \nabla_ {\mathbf {x}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \mathcal {B} _ {t + 1}) - \nabla_ {\mathbf {x}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ \leq 2 \mathbb {E} \| \left(\nabla_ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \vec {B} _ {t + 1}) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1})\right) \mathbf {v} _ {t + 1} \| ^ {2} \\ + 2 \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \mathcal {B} _ {t + 1}) - \nabla_ {\mathbf {x}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ \leq 2 (\frac {\sigma_ {g _ {\mathbf {x y}}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {x}}} ^ {2}}{b}), \\ \end{array} +$$ + +where the first inequality is by and $\| a + b\|^2 \leq 2\| a\|^2 + 2\| b\|^2$ ; the second inequality follows from Assumptions C3., C5. and (8). + +Substituting the above inequality into (80) and using $\| a + b + c \|^2 \leq 3(\| a \|^2 + \| b \|^2 + \| c \|^2)$ , we obtain + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {f} \| ^ {2} \leq (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| e _ {t} ^ {f} \| ^ {2} + 4 \lambda_ {t + 1} ^ {2} (\frac {\sigma_ {g _ {\mathbf {x y}}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {x}}} ^ {2}}{b}) \\ + 6 \left(1 - \eta_ {t + 1}\right) ^ {2} \mathbb {E} \left\| \tilde {\mathbf {d}} _ {t} (\mathbf {u} _ {t}) - \tilde {\mathbf {d}} _ {t} (\mathbf {u} _ {t + 1}) \right\| ^ {2} \\ + 6 \left(1 - \eta_ {t + 1}\right) ^ {2} \mathbb {E} \| \tilde {\mathbf {d}} _ {t} (\mathbf {u} _ {t + 1}) - \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t + 1}) \| ^ {2} \\ + 6 \left(1 - \eta_ {t + 1}\right) ^ {2} \mathbb {E} \| \tilde {\mathbf {d}} _ {t + 1} \left(\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}\right) - \tilde {\mathbf {d}} _ {t + 1} \left(\mathbf {u} _ {t}; \mathcal {B} _ {t + 1}\right) \| ^ {2}. \tag {81} \\ \end{array} +$$ + +Moreover, from $\| a + b + c\| ^2\leq 3(\| a\| ^2 +\| b\| ^2 +\| c\| ^2)$ , we have + +$$ +\mathbb {E} \| \tilde {\mathbf {d}} _ {t} (\mathbf {u} _ {t + 1}) - \tilde {\mathbf {d}} _ {t} (\mathbf {u} _ {t}) \| ^ {2} +$$ + +$$ +\leq 3 \mathbb {E} \| \tilde {\mathbf {d}} _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}, \mathbf {v} _ {t + 1}) - \tilde {\mathbf {d}} _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t + 1}, \mathbf {v} _ {t + 1}) \| ^ {2} +$$ + +$$ ++ 3 \mathbb {E} \| \tilde {\mathbf {d}} _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t + 1}, \mathbf {v} _ {t + 1}) - \tilde {\mathbf {d}} _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t + 1}) \| ^ {2} +$$ + +$$ ++ 3 \mathbb {E} \| \tilde {\mathbf {d}} _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t + 1}) - \tilde {\mathbf {d}} _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t}) \| ^ {2} +$$ + +$$ +\stackrel {(i)} {\leq} 3 \mathbb {E} \| (\nabla_ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t + 1})) \mathbf {v} _ {t + 1} + \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t + 1}) \| ^ {2} +$$ + +$$ ++ 3 \mathbb {E} \| (\nabla_ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t})) \mathbf {v} _ {t + 1} + \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} +$$ + +$$ ++ 3 \mathbb {E} \| \tilde {\mathbf {d}} _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t + 1}) - \tilde {\mathbf {d}} _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t}) \| ^ {2} +$$ + +$$ +\stackrel {(i i)} {\leq} 6 \left(\ell_ {g, 2} ^ {2} \mathbb {E} \| \mathbf {v} _ {t + 1} \| ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + \mathbb {E} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2}\right) + 3 \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} +$$ + +$$ +\stackrel {(i i i)} {\leq} 6 \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + \beta_ {t} ^ {2} \mathbb {E} \| \mathbf {d} _ {t} ^ {\mathbf {y}} \| ^ {2}\right) + 3 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2} \mathbb {E} \| \mathbf {d} _ {t} ^ {\mathbf {v}} \| ^ {2} +$$ + +$$ +\stackrel {(i v)} {\leq} 6 \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) +$$ + +$$ ++ 6 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2} \left(\mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + \mathbb {E} \| \nabla P _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t}) \| ^ {2}\right) +$$ + +$$ +\stackrel {(v i)} {\leq} 6 \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) +$$ + +$$ ++ 6 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2} \left(\mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2}\right), \tag {82} +$$ + +where the (i) follows from (79); (ii) follows from Assumptions B2., B3. and B4.; (iii) follows from (8); (iv) follows from (45) and (60); (vi) follows from (59). + +Similarly, we have + +$$ +\begin{array}{l} \mathbb {E} \| \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t + 1}; \mathcal {B} _ {t + 1}) - \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t}; \mathcal {B} _ {t + 1}) \| ^ {2} \\ \leq 6 \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) \\ + 6 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2} \left(\mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2}\right). \tag {83} \\ \end{array} +$$ + +Substituting (83) and (82) into (81), we have + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {f} \| ^ {2} \leq (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| e _ {t} ^ {f} \| ^ {2} + 4 \eta_ {t + 1} ^ {2} (\frac {\sigma_ {g _ {\mathbf {y y}}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {y}}} ^ {2}}{b}) \\ + 6 \left(1 - \eta_ {t + 1}\right) ^ {2} \mathbb {E} \| \tilde {\mathbf {d}} _ {t} (\mathbf {u} _ {t + 1}) - \tilde {\mathbf {d}} _ {t + 1} (\mathbf {u} _ {t + 1}) \| ^ {2} \\ + 7 2 \left(1 - \eta_ {t + 1}\right) ^ {2} \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) \\ + 7 2 \ell_ {g, 1} ^ {2} (1 - \eta_ {t + 1}) ^ {2} \delta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + 7 2 (1 - \eta_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {4} \delta_ {t} ^ {2} \mathbb {E} \| \mathbf {v} _ {t} - \mathbf {v} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2}. \\ \end{array} +$$ + +From $\| a + b\| ^2\leq 2\| a\| ^2 +2\| b\| ^2$ and (8), we have + +$$ +\begin{array}{l} \mathbb {E} \left\| \tilde {\mathbf {d}} _ {t} \left(\mathbf {u} _ {t + 1}\right) - \tilde {\mathbf {d}} _ {t + 1} \left(\mathbf {u} _ {t + 1}\right) \right\| ^ {2} = \mathbb {E} \left\| \nabla_ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) \mathbf {v} _ {t + 1} - \nabla_ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) \mathbf {v} _ {t + 1} \right. \\ + \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {x}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ \leq 2 \mathbb {E} \| \left(\nabla_ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right)\right) \mathbf {v} _ {t + 1} \| ^ {2} \\ + 2 \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {x}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ \leq 2 \mathbb {E} \| \nabla_ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} p ^ {2} \\ + 2 \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {x}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2}. \\ \end{array} +$$ + +This completes the proof. + +□ + +As demonstrated in Lemma C.9, the hypergradient estimator error $e_{t+1}^{f}$ comprises five key components: (1) the term $(1 - \eta_{t+1})^2 \mathbb{E} \| e_t^f \|^2$ , representing the per-iteration improvement achieved by the momentum-based update; (2) the error arising from the variation in the Jacobian of the lower-level objective; (3) the error caused by the variation in the gradient of the upper-level objective; (4) the error term $\mathcal{O}(2\beta_t^2 \mathbb{E} \| e_t^g \|^2 + 2\beta_t^2 \mathbb{E} \| \nabla_{\mathbf{y}} g_t(\mathbf{x}_t, \mathbf{y}_t) \|^2)$ , which is due to solving the lower-level problem; and (5) the error term $\mathcal{O}(\delta_t^2 \mathbb{E} \| e_t^\nu \|^2 + 72(1 - \eta_{t+1})^2 \ell_{g,1}^4 \delta_t^2 \mathbb{E}[\theta_t^\nu])$ , which is introduced by the one-step momentum update in solving the linear system problem. + +# C.5 Bounds on the Outer Objective and its Projected Gradient + +Lemma C.10. Let Assumption 2.4 holds. Then, for the sequence of functions $\{f_t\}_{t=1}^T$ , we have + +$$ +\sum_ {t = 1} ^ {T} \left(f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) - f _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t + 1})\right)\right) \leq 2 M + V _ {T}, +$$ + +where $M$ is defined in Assumption 2.4; $V_{T}$ is defined in (11). + +Proof. Note that, we have + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \left(f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) - f _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right)\right) \\ = f _ {1} \left(\mathbf {x} _ {1}, \mathbf {y} _ {1} ^ {*} (\mathbf {x} _ {1})\right) - f _ {T} \left(\mathbf {x} _ {T + 1}, \mathbf {y} _ {T} ^ {*} (\mathbf {x} _ {T + 1})\right) \\ + \sum_ {t = 2} ^ {T} \left(f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) - f _ {t - 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t - 1} ^ {*} (\mathbf {x} _ {t})\right)\right) \\ \leq 2 M + V _ {T}, \\ \end{array} +$$ + +where the inequality follows from Assumption 2.4. + +![](images/0626390feef7885a4b70f038ead46dc7e6bbab0f54151f41d3bb3de5ea582d7c.jpg) + +Lemma C.11. Let $\{f_t\}_{t=1}^T$ denote the sequence of functions presented to Algorithm 1, satisfying Assumptions 2.2, 2.3 and 2.4. Let $\mathcal{P}_{\mathcal{X},\alpha_t}$ be defined as in Definition B.1. For any positive step size $\alpha_t$ such that $\alpha_t \leq 1/4L_f$ for all $t \in [T]$ , Algorithm 1 ensures the following bound: + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \left(\alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) \mathbb {E} \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \\ \leq 8 M + 4 V _ {T} + 2 M _ {f} ^ {2} \sum_ {t = 1} ^ {T} \left(2 \alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) \left(\mathbb {E} \left[ \theta_ {t} ^ {\mathbf {y}} \right] + \mathbb {E} \left[ \theta_ {t} ^ {\mathbf {v}} \right]\right) \\ + 2 \sum_ {t = 1} ^ {T} \left(2 \alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) \mathbb {E} \left\| e _ {t} ^ {f} \right\| ^ {2}. \tag {84} \\ \end{array} +$$ + +Here, $\theta_t^{\mathbf{y}}$ and $\theta_t^{\mathbf{v}}$ are defined in (49); $V_{T}$ , $M$ , $M_{f}$ and $e_t^f$ are defined in (11), Assumption 2.4, Eq. (42), and (77). + +Proof. It follows from Lemma C.1 that + +$$ +\begin{array}{l} f _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right) - f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) \\ \leq \left\langle \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right), \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\rangle + \frac {L _ {f}}{2} \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2} \\ = - \alpha_ {t} \left\langle \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right), \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) \right\rangle + \frac {L _ {f} \alpha_ {t} ^ {2}}{2} \| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) \| ^ {2}. \tag {85} \\ \end{array} +$$ + +For the first term on the right hand side of (85), we have that + +$$ +\begin{array}{l} - \left\langle \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right), \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) \right\rangle \\ = - \left\langle \mathbf {d} _ {t} ^ {\mathbf {x}}, \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) \right\rangle - \left\langle \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) - \mathbf {d} _ {t} ^ {\mathbf {x}}, \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) \right\rangle \\ \leq - \frac {1}{2} \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} + \frac {1}{2} \left\| \mathbf {d} _ {t} ^ {\mathbf {x}} - \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \right\| ^ {2}, \\ \end{array} +$$ + +where the inequality follows from Lemma B.7. + +Let $\tilde{\mathbf{d}}_t(\mathbf{z}_t,\mathbf{v}_t) = \nabla_{\mathbf{x}}f_t(\mathbf{z}_t) + \nabla_{\mathbf{xy}}^2 g_t(\mathbf{z}_t)\mathbf{v}_t$ . Then, from Lemma C.1, we have + +$$ +\begin{array}{l} \| \mathbf {d} _ {t} ^ {\mathbf {x}} - \nabla f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})) \| ^ {2} = \left\| \mathbf {d} _ {t} ^ {\mathbf {x}} - \tilde {\mathbf {d}} _ {t} (\mathbf {z} _ {t}, \mathbf {v} _ {t}) + \tilde {\mathbf {d}} _ {t} (\mathbf {z} _ {t}, \mathbf {v} _ {t}) - \nabla f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})) \right\| ^ {2} \\ \leq 2 \left\| \mathbf {d} _ {t} ^ {\mathbf {x}} - \tilde {\mathbf {d}} _ {t} \left(\mathbf {z} _ {t}, \mathbf {v} _ {t}\right) \right\| ^ {2} + 2 \left\| \tilde {\mathbf {d}} _ {t} \left(\mathbf {z} _ {t}, \mathbf {v} _ {t}\right) - \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \right\| ^ {2} \\ \leq 2 \left\| e _ {t} ^ {f} \right\| ^ {2} + 2 \left\| \tilde {\mathbf {d}} _ {t} \left(\mathbf {z} _ {t}, \mathbf {v} _ {t}\right) - \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) \right\| ^ {2} \\ \leq 2 \left\| e _ {t} ^ {f} \right\| ^ {2} + M _ {f} ^ {2} \left(\theta_ {t} ^ {\mathbf {y}} + \theta_ {t} ^ {\mathbf {v}}\right), \tag {86} \\ \end{array} +$$ + +where $e_t^f = \mathbf{d}_t^\mathbf{x} - \tilde{\mathbf{d}}_t(\mathbf{z}_t,\mathbf{v}_t)$ . This implies that + +$$ +- \left\langle \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right), \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) \right\rangle +$$ + +$$ +\leq - \frac {1}{2} \| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) \| ^ {2} + 2 \left\| e _ {t} ^ {f} \right\| ^ {2} + M _ {f} ^ {2} \left(\theta_ {t} ^ {\mathbf {y}} + \theta_ {t} ^ {\mathbf {v}}\right). \tag {87} +$$ + +Plugging the bound (87) into (85), we have that + +$$ +\begin{array}{l} f _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right) - f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) \\ \leq \frac {\left(L _ {f} \alpha_ {t} ^ {2} - \alpha_ {t}\right)}{2} \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} + 2 \alpha_ {t} \left\| e _ {t} ^ {f} \right\| ^ {2} + M _ {f} ^ {2} \left(\theta_ {t} ^ {\mathbf {y}} + \theta_ {t} ^ {\mathbf {v}}\right) \alpha_ {t}, \\ \end{array} +$$ + +which can be rearranged into + +$$ +\begin{array}{l} \left(\alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} \\ \leq 2 f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) - f _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right) + 4 \alpha_ {t} \left\| e _ {t} ^ {f} \right\| ^ {2} + 2 M _ {f} ^ {2} \left(\theta_ {t} ^ {\mathbf {y}} + \theta_ {t} ^ {\mathbf {v}}\right) \alpha_ {t}. \tag {88} \\ \end{array} +$$ + +In addition, we have + +$$ +\begin{array}{l} \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}))\right) \right\| ^ {2} \\ \leq 2 \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) - \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} + 2 \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} \\ \leq 2 \left\| \mathbf {d} _ {t} ^ {\mathbf {x}} - \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \right\| ^ {2} + 2 \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} \\ \leq 4 \left\| e _ {t} ^ {f} \right\| ^ {2} + 4 M _ {f} ^ {2} \left(\theta_ {t} ^ {\mathbf {y}} + \theta_ {t} ^ {\mathbf {v}}\right) + 4 \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2}, \tag {89} \\ \end{array} +$$ + +where the second inequality follows from non-expansiveness of the projection operator and the last inequality follows from (86). + +Combining (88) and (89), we have + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \left(\alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \\ \leq 4 \sum_ {t = 1} ^ {T} \left(f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) - f _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t + 1})\right)\right) \\ + 2 M _ {f} ^ {2} \sum_ {t = 1} ^ {T} \left(2 \alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) \left(\theta_ {t} ^ {\mathbf {y}} + \theta_ {t} ^ {\mathbf {v}}\right) + 2 \sum_ {t = 1} ^ {T} \left(2 \alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) \left\| e _ {t} ^ {f} \right\| ^ {2} \\ \leq 8 M + 4 V _ {T} \\ + 2 M _ {f} ^ {2} \sum_ {t = 1} ^ {T} \left(2 \alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) \left(\theta_ {t} ^ {\mathbf {y}} + \theta_ {t} ^ {\mathbf {v}}\right) + 2 \sum_ {t = 1} ^ {T} \left(2 \alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) \left\| e _ {t} ^ {f} \right\| ^ {2}, \\ \end{array} +$$ + +where the second inequality is due to Lemma C.10. + +![](images/f32558227cfc8fe007221d14f1b1b19ef0928bf95584108dbb8366d33bf81537.jpg) + +Lemma C.12. Let Assumptions 2.2, and 2.3 hold. Let $\{\mathbf{x}_t\}_{t=1}^T$ be generated according to Algorithm 1. Then, we have + +$$ +\left\| \mathbf {x} _ {t} - \mathbf {x} _ {t + 1} \right\| ^ {2} \leq 2 \alpha_ {t} ^ {2} \left(\left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} + M _ {f} ^ {2} \left(\theta_ {t} ^ {\mathbf {y}} + \theta_ {t} ^ {\mathbf {v}}\right)\right), +$$ + +where $\theta_t^{\mathbf{y}}$ and $\theta_t^{\mathbf{v}}$ are defined in (49), $M_f$ is defined in (42). + +Proof. From the update rule of Algorithm 1, we have + +$$ +\begin{array}{l} \left\| \mathbf {x} _ {t} - \mathbf {x} _ {t + 1} \right\| ^ {2} = \alpha_ {t} ^ {2} \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} \\ \leq 2 \alpha_ {t} ^ {2} \left(\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \| ^ {2} \right. \\ \left. + \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \mathbf {d} _ {t} ^ {\mathbf {x}}\right) - \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2}\right) \\ \leq 2 \alpha_ {t} ^ {2} \left(\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \| ^ {2} \right. \\ \left. + \left\| \mathbf {d} _ {t} ^ {\mathbf {x}} - \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \right\| ^ {2}\right) \\ \leq 2 \alpha_ {t} ^ {2} \left(\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \| ^ {2} + M _ {f} ^ {2} \left(\theta_ {t} ^ {\mathbf {y}} + \theta_ {t} ^ {\mathbf {v}}\right)\right), \tag {90} \\ \end{array} +$$ + +where the first inequality is by $(a + b)^2 \leq 2a^2 + 2b^2$ ; the second inequality follows from non-expansiveness of the projection operator; and the last inequality follows from Eq. (39a) in Lemma C.1. + +# C.6 Proof of Theorem 2.6 + +Proof. Bounding $\mathbb{E}\| e_t^f\|^2$ in (78). From (78), we have + +$$ +\begin{array}{l} \frac {\mathbb {E} \| e _ {t + 1} ^ {f} \| ^ {2}}{\alpha_ {t}} - \frac {\mathbb {E} \| e _ {t} ^ {f} \| ^ {2}}{\alpha_ {t - 1}} \leq \left(\frac {(1 - \eta_ {t + 1}) ^ {2}}{\alpha_ {t}} - \frac {1}{\alpha_ {t - 1}}\right) \mathbb {E} \| e _ {t} ^ {f} \| ^ {2} + \frac {4 \eta_ {t + 1} ^ {2}}{\alpha_ {t}} (\frac {\sigma_ {g _ {\mathbf {x y}}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {x}}} ^ {2}}{b}) \\ + \frac {1 2 p ^ {2}}{\alpha_ {t}} (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ + \frac {1 2}{\alpha_ {t}} (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {x}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ + \frac {7 2}{\alpha_ {t}} (1 - \eta_ {t + 1}) ^ {2} \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) \\ + \frac {7 2}{\alpha_ {t}} \ell_ {g, 1} ^ {2} (1 - \eta_ {t + 1}) ^ {2} \delta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + \frac {7 2}{\alpha_ {t}} (1 - \eta_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {4} \delta_ {t} ^ {2} \mathbb {E} [ \theta_ {t} ^ {\mathbf {v}} ]. \tag {91} \\ \end{array} +$$ + +With respect to the coefficient of the first term on the right-hand side of Eq. (91), it is important to note that we have: + +$$ +\frac {\left(1 - \eta_ {t + 1}\right) ^ {2}}{\alpha_ {t}} - \frac {1}{\alpha_ {t - 1}} \leq \frac {1}{\alpha_ {t}} - \frac {\eta_ {t + 1}}{\alpha_ {t}} - \frac {1}{\alpha_ {t - 1}}. \tag {92} +$$ + +Using the definition of $\alpha_{t}$ in (16), we have + +$$ +\begin{array}{l} \frac {1}{\alpha_ {t}} - \frac {1}{\alpha_ {t - 1}} = (c + t) ^ {1 / 3} - (c + t - 1) ^ {1 / 3} \stackrel {(i)} {\leq} \frac {1}{3 (c + t - 1) ^ {2 / 3}} \stackrel {(i i)} {\leq} \frac {1}{3 (\frac {c}{2} + t) ^ {2 / 3}} \\ = \frac {2 ^ {2 / 3}}{3 (c + 2 t) ^ {2 / 3}} \stackrel {(i i i)} {\leq} \frac {2 ^ {2 / 3}}{3 (c + t) ^ {2 / 3}} \stackrel {(i v)} {\leq} \frac {2 ^ {2 / 3}}{3} \alpha_ {t} ^ {2} \stackrel {(v i)} {\leq} \frac {\alpha_ {t}}{6 L _ {f}}, \tag {93} \\ \end{array} +$$ + +where the (i) follows from $(a + b)^{1/3} - a^{1/3} \leq b / (3a^{2/3})$ ; (ii) follows from $c \geq 2$ in (107); (iii) follows from (16); (iv) follows from $\alpha_t \leq 1/4L_f$ in (107). + +Substituting (93) into (92) and using $\delta_t = c_\delta \alpha_t$ and $\eta_{t + 1} = c_{\eta}\alpha_t^2$ , we have + +$$ +\frac {\left(1 - \eta_ {t + 1}\right) ^ {2}}{\alpha_ {t}} - \frac {1}{\alpha_ {t - 1}} \leq \frac {\alpha_ {t}}{6 L _ {f}} - \frac {\eta_ {t + 1}}{\alpha_ {t}} = \frac {\alpha_ {t}}{6 L _ {f}} - c _ {\eta} \alpha_ {t} \leq - 5 \Omega \alpha_ {t}, \tag {94} +$$ + +where the inequalities follow from $c_{\eta} = \frac{1}{6L_f} + 5\Omega$ in (106). + +Then, substituting (94) into (91) yields + +$$ +\begin{array}{l} \frac {1}{\Omega} \mathbb {E} \left(\frac {\| e _ {t + 1} ^ {f} \| ^ {2}}{\alpha_ {t}} - \frac {\| e _ {t} ^ {f} \| ^ {2}}{\alpha_ {t - 1}}\right) \leq - 5 \alpha_ {t} \mathbb {E} \| e _ {t} ^ {f} \| ^ {2} + \frac {4 \eta_ {t + 1} ^ {2}}{\Omega \alpha_ {t}} (\frac {\sigma_ {g _ {\mathbf {x y}}} ^ {2}}{b} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {x}}} ^ {2}}{\acute {b}}) \\ + \frac {1 2 p ^ {2}}{\Omega \alpha_ {t}} (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ + \frac {1 2}{\Omega \alpha_ {t}} (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {x}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ + \frac {7 2}{\Omega \alpha_ {t}} (1 - \eta_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) \\ + \frac {7 2}{\Omega \alpha_ {t}} \ell_ {g, 1} ^ {2} (1 - \eta_ {t + 1}) ^ {2} \delta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} + \frac {7 2}{\Omega \alpha_ {t}} (1 - \eta_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {4} \delta_ {t} ^ {2} \mathbb {E} [ \theta_ {t} ^ {\mathbf {v}} ]. \tag {95} \\ \end{array} +$$ + +Bounding $\mathbb{E}\| e_t^g\|^2$ in (46). + +From (46), we have + +$$ +\begin{array}{l} \frac {\mathbb {E} \| e _ {t + 1} ^ {g} \| ^ {2}}{\alpha_ {t}} - \frac {\mathbb {E} \| e _ {t} ^ {g} \| ^ {2}}{\alpha_ {t - 1}} \leq \left(\frac {1}{\alpha_ {t}} (1 - \gamma_ {t + 1}) ^ {2} (1 + 4 8 \ell_ {g, 1} ^ {2} \beta_ {t} ^ {2}) - \frac {1}{\alpha_ {t - 1}}\right) \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} \\ + 2 \frac {\gamma_ {t + 1} ^ {2}}{\alpha_ {t}} \frac {\sigma_ {g _ {\mathbf {y}}} ^ {2}}{b} + \frac {2 4}{\alpha_ {t}} (1 - \gamma_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} \\ + \frac {6}{\alpha_ {t}} (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ + 4 8 \left(1 - \gamma_ {t + 1}\right) ^ {2} \ell_ {g, 1} ^ {2} \frac {\beta_ {t} ^ {2}}{\alpha_ {t}} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \| ^ {2}. \tag {96} \\ \end{array} +$$ + +Let us examine the coefficient of the first term on the right-hand side of Eq. (96). Specifically, for $\gamma_{t + 1} = c_{\gamma}\alpha_t^2$ and $\beta_{t} = c_{\beta}\alpha_{t}$ , we have: + +$$ +\begin{array}{l} \frac {1}{\alpha_ {t}} (1 - \gamma_ {t + 1}) ^ {2} (1 + 4 8 \ell_ {g, 1} ^ {2} \beta_ {t} ^ {2}) - \frac {1}{\alpha_ {t - 1}} \leq \frac {1}{\alpha_ {t}} (1 - \gamma_ {t + 1}) (1 + 4 8 \ell_ {g, 1} ^ {2} \beta_ {t} ^ {2}) - \frac {1}{\alpha_ {t - 1}} \\ = \frac {1}{\alpha_ {t}} - \frac {1}{\alpha_ {t - 1}} - \frac {\gamma_ {t + 1}}{\alpha_ {t}} + \frac {1 - \gamma_ {t + 1}}{\alpha_ {t}} 4 8 \ell_ {g, 1} ^ {2} \beta_ {t} ^ {2} \\ = \frac {1}{\alpha_ {t}} - \frac {1}{\alpha_ {t - 1}} - c _ {\gamma} \alpha_ {t} + \left(\frac {1}{\alpha_ {t}} - c _ {\gamma} \alpha_ {t}\right) 4 8 \ell_ {g, 1} ^ {2} c _ {\beta} ^ {2} \alpha_ {t} ^ {2} \\ \leq \frac {\alpha_ {t}}{6 L _ {f}} + 4 8 \ell_ {g, 1} ^ {2} c _ {\beta} ^ {2} \alpha_ {t} - c _ {\gamma} \alpha_ {t}, \tag {97} \\ \end{array} +$$ + +where the last inequality follows from (93). + +From the selected $c_{\gamma}$ in (107) and the definition of $\Phi$ in (106), we have + +$$ +c _ {\gamma} = \frac {1}{6 L _ {f}} + 4 8 \ell_ {g, 1} ^ {2} c _ {\beta} ^ {2} + \hbar \Phi , \quad \text {w h e r e} \quad \hbar = 2 5 \frac {M _ {f} ^ {2}}{L _ {\mu_ {g}} ^ {2}}. +$$ + +Combined this with Eq. (97) yields + +$$ +\frac {1}{\alpha_ {t}} \left(1 - \gamma_ {t + 1}\right) ^ {2} \left(1 + 4 8 \ell_ {g, 1} ^ {2} \beta_ {t} ^ {2}\right) - \frac {1}{\alpha_ {t - 1}} \leq - \hbar \Phi \alpha_ {t}. \tag {98} +$$ + +Substituting Eq. (98) into Eq. (96) yields + +$$ +\begin{array}{l} \frac {1}{\Phi} \left(\frac {\mathbb {E} \| e _ {t + 1} ^ {g} \| ^ {2}}{\alpha_ {t}} - \frac {\mathbb {E} \| e _ {t} ^ {g} \| ^ {2}}{\alpha_ {t - 1}}\right) \leq - \hbar \alpha_ {t} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} \\ + 2 \frac {\gamma_ {t + 1} ^ {2}}{\Phi \alpha_ {t}} \frac {\sigma_ {g _ {\mathbf {y}}} ^ {2}}{b} + \frac {2 4}{\Phi \alpha_ {t}} (1 - \gamma_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} \\ + \frac {6}{\Phi \alpha_ {t}} (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ + 4 8 \left(1 - \gamma_ {t + 1}\right) ^ {2} \ell_ {g, 1} ^ {2} \frac {\beta_ {t} ^ {2}}{\Phi \alpha_ {t}} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \| ^ {2}. \tag {99} \\ \end{array} +$$ + +Bounding $\mathbb{E}\| e_t^{\mathbf{v}}\|^2$ in (61). + +From (61), we get + +$$ +\begin{array}{l} \frac {\mathbb {E} \| e _ {t + 1} ^ {\mathbf {v}} \| ^ {2}}{\alpha_ {t}} - \frac {\mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2}}{\alpha_ {t - 1}} \leq \left(\frac {1}{\alpha_ {t}} (1 - \lambda_ {t + 1}) ^ {2} (1 + 7 2 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2}) - \frac {1}{\alpha_ {t - 1}}\right) \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} \\ + 4 \frac {\lambda_ {t + 1} ^ {2}}{\alpha_ {t}} (\frac {\sigma_ {g _ {\mathbf {y y}}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {y}}} ^ {2}}{b}) + \frac {1 2 p ^ {2}}{\alpha_ {t}} (1 - \lambda_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ + \frac {1 2}{\alpha_ {t}} (1 - \lambda_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ + \frac {7 2}{\alpha_ {t}} (1 - \lambda_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) \\ + \frac {1 4 4}{\alpha_ {t}} \left(1 - \lambda_ {t + 1}\right) ^ {2} \ell_ {g, 1} ^ {4} \delta_ {t} ^ {2} \mathbb {E} \left[ \theta_ {t} ^ {\mathbf {v}} \right] + \frac {2 8 8}{\alpha_ {t}} \ell_ {g, 1} ^ {2} \left(p ^ {2} \ell_ {g, 2} ^ {2} + \ell_ {f, 1} ^ {2}\right) \delta_ {t} ^ {2} \mathbb {E} \left[ \theta_ {t} ^ {\mathbf {y}} \right]. \tag {100} \\ \end{array} +$$ + +Let us examine the coefficient of the first term on the right-hand side of equation (100). Specifically, for $\lambda_{t + 1} = c_{\lambda}\alpha_{t}^{2}$ and $\delta_t = c_\delta \alpha_t$ , we have: + +$$ +\begin{array}{l} \frac {1}{\alpha_ {t}} (1 - \lambda_ {t + 1}) ^ {2} (1 + 7 2 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2}) - \frac {1}{\alpha_ {t - 1}} \leq \frac {1}{\alpha_ {t}} (1 - \lambda_ {t + 1}) (1 + 7 2 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2}) - \frac {1}{\alpha_ {t - 1}} \\ = \frac {1}{\alpha_ {t}} - \frac {1}{\alpha_ {t - 1}} - \frac {\lambda_ {t + 1}}{\alpha_ {t}} + \frac {1 - \lambda_ {t + 1}}{\alpha_ {t}} 7 2 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2} \\ = \frac {1}{\alpha_ {t}} - \frac {1}{\alpha_ {t - 1}} - c _ {\lambda} \alpha_ {t} + \left(\frac {1}{\alpha_ {t}} - c _ {\lambda} \alpha_ {t}\right) 7 2 \ell_ {g, 1} ^ {2} c _ {\delta} ^ {2} \alpha_ {t} ^ {2} \\ \leq \frac {\alpha_ {t}}{6 L _ {f}} + 7 2 \ell_ {g, 1} ^ {2} c _ {\delta} ^ {2} \alpha_ {t} - c _ {\lambda} \alpha_ {t}, \tag {101} \\ \end{array} +$$ + +where the last inequality follows from (93). + +From the selected $c_{\gamma}$ in (107) and the definition of $\Psi$ in (106), we have + +$$ +c _ {\lambda} = \frac {1}{6 L _ {f}} + 7 2 \ell_ {g, 1} ^ {2} c _ {\delta} ^ {2} + \jmath \Psi , \quad \text {w h e r e} \quad \jmath = 9 0 \frac {M _ {f} ^ {2}}{L _ {\mu_ {g}} ^ {2}}. +$$ + +Combined this with Eq. (101) yields + +$$ +\frac {1}{\alpha_ {t}} \left(1 - \lambda_ {t + 1}\right) ^ {2} \left(1 + 7 2 \ell_ {g, 1} ^ {2} \delta_ {t} ^ {2}\right) - \frac {1}{\alpha_ {t - 1}} \leq - \jmath \Psi \alpha_ {t}. \tag {102} +$$ + +Substituting Eq. (102) into Eq. (100) yields + +$$ +\begin{array}{l} \frac {1}{\Psi} \left(\frac {\mathbb {E} \| e _ {t + 1} ^ {\mathbf {v}} \| ^ {2}}{\alpha_ {t}} - \frac {\mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2}}{\alpha_ {t - 1}}\right) \leq - \jmath \alpha_ {t} \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} \\ + 4 \frac {\lambda_ {t + 1} ^ {2}}{\Psi \alpha_ {t}} (\frac {\sigma_ {g _ {\mathbf {y y}}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {y}}} ^ {2}}{b}) + \frac {1 2 p ^ {2}}{\Psi \alpha_ {t}} (1 - \lambda_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ + \frac {1 2}{\Psi \alpha_ {t}} (1 - \lambda_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) \| ^ {2} \\ + \frac {7 2}{\Psi \alpha_ {t}} (1 - \lambda_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \left(\mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + 2 \beta_ {t} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}\right) \\ + \frac {7 2}{\Psi \alpha_ {t}} (1 - \lambda_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {4} \delta_ {t} ^ {2} \mathbb {E} [ \theta_ {t} ^ {\mathbf {v}} ] + \frac {2 8 8}{\Psi \alpha_ {t}} \ell_ {g, 1} ^ {2} \left(p ^ {2} \ell_ {g, 2} ^ {2} + \ell_ {f, 1} ^ {2}\right) \delta_ {t} ^ {2} \mathbb {E} [ \theta_ {t} ^ {\mathbf {y}} ]. \tag {103} \\ \end{array} +$$ + +Combining the outcomes. We recall from Lemma C.12 that we have + +$$ +\left\| \mathbf {x} _ {t} - \mathbf {x} _ {t + 1} \right\| ^ {2} \leq 2 \alpha_ {t} ^ {2} \left(\left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} + M _ {f} ^ {2} \left(\theta_ {t} ^ {\mathbf {y}} + \theta_ {t} ^ {\mathbf {v}}\right)\right). \tag {104} +$$ + +Let + +$$ +\begin{array}{l} \Lambda := \Gamma \sum_ {t = 1} ^ {T} \left(\mathbb {E} \left[ \theta_ {t + 1} ^ {\mathbf {y}} \right] - \mathbb {E} \left[ \theta_ {t} ^ {\mathbf {y}} \right]\right) + \Upsilon \sum_ {t = 1} ^ {T} \left(\mathbb {E} \left[ \theta_ {t + 1} ^ {\mathbf {y}} \right] - \mathbb {E} \left[ \theta_ {t} ^ {\mathbf {y}} \right]\right) + \frac {1}{\Phi} \sum_ {t = 1} ^ {T} \left(\frac {\mathbb {E} \| e _ {t + 1} ^ {g} \| ^ {2}}{\alpha_ {t}} - \frac {\mathbb {E} \| e _ {t} ^ {g} \| ^ {2}}{\alpha_ {t - 1}}\right) \\ + \frac {1}{\Psi} \sum_ {t = 1} ^ {T} \left(\frac {\mathbb {E} \| e _ {t + 1} ^ {\mathbf {v}} \| ^ {2}}{\alpha_ {t}} - \frac {\mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2}}{\alpha_ {t - 1}}\right) + \frac {1}{\Omega} \sum_ {t = 1} ^ {T} \left(\frac {\mathbb {E} \| e _ {t + 1} ^ {f} \| ^ {2}}{\alpha_ {t}} - \frac {\mathbb {E} \| e _ {t} ^ {f} \| ^ {2}}{\alpha_ {t - 1}}\right). \tag {105} \\ \end{array} +$$ + +Here + +$$ +\begin{array}{l} \Gamma = \frac {1 1 M _ {f} ^ {2}}{L _ {\mu_ {g}} c _ {\beta}}, \quad \Upsilon = \frac {2 2 M _ {f} ^ {2}}{\dot {L} _ {\mu_ {g}} c _ {\delta}}, \quad \Phi \geq \max \left\{4 8 0 \ell_ {g, 1} ^ {2}, 1 9 2 \ell_ {g, 1} ^ {2} \frac {(\mu_ {g} + \ell_ {g , 1})}{\Gamma} c _ {\beta} \right\}, \\ \Psi = \max \left\{1 4 4 (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \left(1 0 + \frac {L _ {\mu_ {g}} ^ {2} c _ {\beta} ^ {2}}{M _ {f} ^ {2}}\right), \frac {2 8 8 \ell_ {g , 1} ^ {4}}{M _ {f} ^ {2}} c _ {\delta} ^ {2}, \right. \\ \left. 5 7 6 \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \frac {\left(\mu_ {g} + \ell_ {g , 1}\right)}{\Gamma} c _ {\beta}, \frac {5 7 6}{M _ {f} ^ {2}} \ell_ {g, 1} ^ {2} \left(p ^ {2} \ell_ {g, 2} ^ {2} + \ell_ {f, 1} ^ {2}\right) c _ {\delta} ^ {2} \right\}, \\ \end{array} +$$ + +$$ +\begin{array}{l} \Omega = \max \left\{1 4 4 (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \left(1 0 + \frac {L _ {\mu_ {g}} ^ {2} c _ {\beta} ^ {2}}{M _ {f} ^ {2}}\right), \frac {2 8 8 \ell_ {g , 1} ^ {4}}{M _ {f} ^ {2}} c _ {\delta} ^ {2}, \right. \\ \left. 5 7 6 \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \frac {\left(\mu_ {g} + \ell_ {g , 1}\right)}{\Gamma} c _ {\beta}, \frac {7 2 \ell_ {g , 1} ^ {2} \hat {L} _ {\mu_ {g}} ^ {2}}{M _ {f} ^ {2}} c _ {\delta} ^ {2} \right\}, \tag {106} \\ \end{array} +$$ + +where $L_{\mu_g} = \mu_g\ell_{g,1} / (\mu_g + \ell_{g,1})$ and $\dot{L}_{\mu_g} = (\ell_{g,1} + \ell_{g,1}^3)\mu_g / (\mu_g + \ell_{g,1})$ + +Here, we have + +$$ +c \geq \max \left\{4 L _ {f}, c _ {\beta} \left(\mu_ {g} + \ell_ {g, 1}\right), 2 \right\}, +$$ + +$$ +c _ {\beta} = \sqrt {8 8 0} \frac {L _ {\mathbf {y}} M _ {f}}{L _ {\mu_ {g}}}, +$$ + +$$ +c _ {\delta} = \sqrt {3 5 2 0 (1 + 2 L _ {\mathbf {y}} ^ {2})} \frac {\nu M _ {f}}{\acute {L} _ {\mu_ {g}} \mu_ {g}}, \quad \text {w h e r e} \quad \nu = \ell_ {f, 1} + \frac {\ell_ {g , 2} \ell_ {f , 0}}{\mu_ {g}}, \quad L _ {\mathbf {y}} = \frac {\ell_ {g , 1}}{\mu_ {g}}, +$$ + +$$ +c _ {\gamma} = \frac {1}{6 L _ {f}} + 4 8 \ell_ {g, 1} ^ {2} c _ {\beta} ^ {2} + \hbar \Phi , \quad \text {w h e r e} \quad \hbar := 2 5 \frac {M _ {f} ^ {2}}{L _ {\mu_ {g}} ^ {2}}, \tag {107} +$$ + +$$ +c _ {\eta} = \frac {1}{6 L _ {f}} + 5 \Omega , +$$ + +$$ +c _ {\lambda} = \frac {1}{6 L _ {f}} + 7 2 \ell_ {g, 1} ^ {2} c _ {\delta} ^ {2} + \jmath \Psi , \qquad \text {w h e r e} \quad \jmath = 9 0 \frac {M _ {f} ^ {2}}{\check {L} _ {\mu_ {g}} ^ {2}}. +$$ + +Using (103), (99), (95), (84), (70), and (50), along with (104) and the fact that $\alpha_{t}$ decreases with respect to $t$ , we obtain: + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} A \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}\right) \mathbb {E} \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} + \Lambda \\ \leq 8 M + 4 V _ {T} + \sum_ {t = 1} ^ {T} B \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}\right) \mathbb {E} \left[ \theta_ {t} ^ {\mathbf {v}} \right] + \sum_ {t = 1} ^ {T} C \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}\right) \mathbb {E} \left[ \theta_ {t} ^ {\mathbf {y}} \right] (108a) \\ + \sum_ {t = 1} ^ {T} D (\alpha_ {t}) \mathbb {E} \| e _ {t} ^ {f} \| ^ {2} + \sum_ {t = 1} ^ {T} F (\alpha_ {t}, \beta_ {t}) \mathbb {E} \| e _ {t} ^ {g} \| ^ {2} + \sum_ {t = 1} ^ {T} I (\alpha_ {t}, \delta_ {t}) \mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} (108b) \\ + \sum_ {t = 1} ^ {T} L \left(\alpha_ {t}, \beta_ {t}\right) \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \| ^ {2} + \sum_ {t = 2} ^ {T} N \left(\beta_ {t}, \delta_ {t}\right) \sup _ {\mathbf {x} \in \mathcal {X}} \| \mathbf {y} _ {t - 1} ^ {*} (\mathbf {x}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x}) \| ^ {2} (108c) \\ + \frac {\sigma_ {g _ {\mathbf {y}}} ^ {2}}{\bar {b}} \frac {2}{\Phi} \sum_ {t = 1} ^ {T} \frac {\gamma_ {t + 1} ^ {2}}{\alpha_ {t}} + \frac {4}{\Psi} \left(\frac {\sigma_ {g _ {\mathbf {y y}}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {y}}} ^ {2}}{b}\right) \sum_ {t = 1} ^ {T} \frac {\lambda_ {t + 1} ^ {2}}{\alpha_ {t}} + \frac {4}{\Omega} \left(\frac {\sigma_ {g _ {\mathbf {x y}}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {x}}} ^ {2}}{b}\right) \sum_ {t = 1} ^ {T} \frac {\eta_ {t + 1} ^ {2}}{\alpha_ {t}} (108d) \\ + \frac {6}{\Phi \alpha_ {T}} G _ {\mathbf {y}, T} + \frac {1 2 p ^ {2}}{\Omega \alpha_ {T}} G _ {\mathbf {x y}, T} + \frac {1 2 p ^ {2}}{\Psi \alpha_ {T}} G _ {\mathbf {y y}, T} + \frac {1 2 \ell_ {f , 1} ^ {2}}{\Psi \alpha_ {T}} D _ {\mathbf {y}, T} + \frac {1 2 \ell_ {f , 1} ^ {2}}{\Omega \alpha_ {T}} D _ {\mathbf {x}, T}. (108e) \\ \end{array} +$$ + +Here, $M$ is defined in Assumption 2.4, $V_{T}$ and $H_{2,T}$ are defined in (11). Moreover, $G_{\mathbf{y},T}$ , $G_{\mathbf{xy},T}$ , and $G_{\mathbf{yy},T}$ are defined in (13). Let + +$$ +\begin{array}{l} E (\alpha_ {t}, \beta_ {t}, \delta_ {t}) := \frac {4 L _ {\mathbf {y}} ^ {2}}{L _ {\mu_ {g}} \beta_ {t}} \Gamma + \frac {8 \nu^ {2}}{\acute {L} _ {\mu_ {g}} \mu_ {g} ^ {2} \delta_ {t}} (1 + 2 L _ {\mathbf {y}} ^ {2}) \Upsilon + 7 2 (1 - \eta_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \frac {1}{\Omega \alpha_ {t}} \\ + 2 4 (1 - \gamma_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {2} \frac {1}{\Phi \alpha_ {t}} + 7 2 (1 - \lambda_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \frac {1}{\Psi \alpha_ {t}}, \\ \end{array} +$$ + +$$ +A \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}\right) := \alpha_ {t} - \left(L _ {f} + 2 E \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}\right)\right) \alpha_ {t} ^ {2}, +$$ + +$$ +\begin{array}{l} B (\alpha_ {t}, \beta_ {t}, \delta_ {t}) := - \frac {\dot {L} _ {\mu_ {g}} \Upsilon}{4} \delta_ {t} + 4 M _ {f} ^ {2} \alpha_ {t} - 2 M _ {f} ^ {2} L _ {f} \alpha_ {t} ^ {2} + 2 M _ {f} ^ {2} E (\alpha_ {t}, \beta_ {t}, \delta_ {t}) \alpha_ {t} ^ {2} \\ + 7 2 (1 - \lambda_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {4} \delta_ {t} ^ {2} \frac {1}{\Psi \alpha_ {t}} + 7 2 (1 - \eta_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {4} \delta_ {t} ^ {2} \frac {1}{\Omega \alpha_ {t}}, \\ \end{array} +$$ + +$$ +\begin{array}{l} C \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}\right) := - \frac {L _ {\mu_ {g}} \Gamma}{2} \beta_ {t} + 4 M _ {f} ^ {2} \alpha_ {t} - 2 L _ {f} M _ {f} ^ {2} \alpha_ {t} ^ {2} + 2 M _ {f} ^ {2} E \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}\right) \alpha_ {t} ^ {2} \tag {109} \\ + \frac {2 8 8}{\Psi \alpha_ {t}} \ell_ {g, 1} ^ {2} (p ^ {2} \ell_ {g, 2} ^ {2} + \ell_ {f, 1} ^ {2}) \delta_ {t} ^ {2}, \\ \end{array} +$$ + +$$ +D \left(\alpha_ {t}\right) := 2 \left(2 \alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) - 5 \alpha_ {t}, +$$ + +$$ +\begin{array}{l} F (\alpha_ {t}, \beta_ {t}) := \frac {2 \Gamma}{L _ {\mu_ {g}}} \beta_ {t} - \hbar \alpha_ {t} + 1 4 4 (1 - \lambda_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \frac {\beta_ {t} ^ {2}}{\Psi \alpha_ {t}} \\ + 1 4 4 (1 - \eta_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \frac {\beta_ {t} ^ {2}}{\Omega \alpha_ {t}}, \\ \end{array} +$$ + +$$ +I (\alpha_ {t}, \delta_ {t}) := \frac {4 \Upsilon}{\check {L} _ {\mu_ {g}}} \delta_ {t} - \jmath \alpha_ {t} + 7 2 \ell_ {g, 1} ^ {2} (1 - \eta_ {t + 1}) ^ {2} \frac {\delta_ {t} ^ {2}}{\Omega \alpha_ {t}}. +$$ + +Moreover, we have + +$$ +\begin{array}{l} L (\alpha_ {t}, \beta_ {t}) := - \frac {2 \Gamma}{\mu_ {g} + \ell_ {g , 1}} \beta_ {t} + \Gamma \beta_ {t} ^ {2} + 4 8 (1 - \gamma_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {2} \frac {\beta_ {t} ^ {2}}{\Phi \alpha_ {t}} \\ + 1 4 4 (1 - \lambda_ {t + 1}) ^ {2} \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \frac {\beta_ {t} ^ {2}}{\Psi \alpha_ {t}} + 1 4 4 (1 - \eta_ {t + 1}) ^ {2} \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \frac {\beta_ {t} ^ {2}}{\Omega \alpha_ {t}}, \tag {110} \\ \end{array} +$$ + +$$ +N (\beta_ {t}, \delta_ {t}) := \frac {4}{L _ {\mu_ {g}} \beta_ {t}} \Gamma + \frac {1 6 \nu^ {2}}{\check {L} _ {\mu_ {g}} \mu_ {g} ^ {2} \delta_ {t}} \Upsilon . +$$ + +Note that, we have + +$$ +\begin{array}{l} E (\alpha_ {t}, \beta_ {t}, \delta_ {t}) = \frac {4 L _ {\mathbf {y}} ^ {2}}{L _ {\mu_ {g}} \beta_ {t}} \Gamma + \frac {8 \nu^ {2}}{\acute {L} _ {\mu_ {g}} \mu_ {g} ^ {2} \delta_ {t}} (1 + 2 L _ {\mathbf {y}} ^ {2}) \Upsilon + 7 2 (1 - \eta_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \frac {1}{\Omega \alpha_ {t}} \\ + 2 4 (1 - \gamma_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {2} \frac {1}{\Phi \alpha_ {t}} + 7 2 (1 - \lambda_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \frac {1}{\Psi \alpha_ {t}}, \\ \end{array} +$$ + +which together with $\beta_{t} = c_{\beta}\alpha_{t}$ and $\delta_t = c_\delta \alpha_t$ in Eq. (16), we have + +$$ +\begin{array}{l} \alpha_ {t} ^ {2} E (\alpha_ {t}, \beta_ {t}, \delta_ {t}) = \frac {4 L _ {\mathbf {y}} ^ {2}}{L _ {\mu_ {g}}} \Gamma \frac {\alpha_ {t} ^ {2}}{\beta_ {t}} + \frac {8 \nu^ {2}}{\acute {L} _ {\mu_ {g}} \mu_ {g} ^ {2}} (1 + 2 L _ {\mathbf {y}} ^ {2}) \Upsilon \frac {\alpha_ {t} ^ {2}}{\delta_ {t}} + 2 4 (1 - \gamma_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {2} \frac {\alpha_ {t}}{\Phi} \\ + 7 2 (1 - \eta_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \frac {\alpha_ {t}}{\Omega} + 7 2 (1 - \lambda_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \frac {\alpha_ {t}}{\Psi} \\ \leq \frac {4 4 L _ {\mathbf {y}} ^ {2}}{L _ {\mu_ {g}} ^ {2}} M _ {f} ^ {2} \frac {\alpha_ {t}}{c _ {\beta} ^ {2}} + \frac {1 7 6 \nu^ {2}}{\acute {L} _ {\mu_ {g}} ^ {2} \mu_ {g} ^ {2}} (1 + 2 L _ {\mathbf {y}} ^ {2}) M _ {f} ^ {2} \frac {\alpha_ {t}}{c _ {\delta} ^ {2}} \\ + 2 4 \ell_ {g, 1} ^ {2} \frac {\alpha_ {t}}{\Phi} + 7 2 \left(\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}\right) \left(\frac {1}{\Omega} + \frac {1}{\Psi}\right) \alpha_ {t} \\ \leq \frac {\alpha_ {t}}{4}, \tag {111} \\ \end{array} +$$ + +where the first inequality follows from $\Gamma = \frac{11M_f^2}{L_{\mu_g}c_\beta}$ and $\Upsilon = \frac{22M_f^2}{\check{L}_{\mu_g}c_\delta}$ in (106); the last inequality follows from $c_{\beta} = \sqrt{880\frac{L_{\mathbf{y}}^2M_f^2}{L_{\mu_g}^2}}$ , $c_{\delta} = \sqrt{3520\frac{\nu^2M_f^2}{\check{L}_{\mu_g}^2\mu_g^2}(1 + 2L_{\mathbf{y}}^2)}$ , in (107) and $\Phi \geq 480\ell_{g,1}^2$ , and $\Omega, \Psi \geq 1440(\ell_{g,2}^2p^2 + \ell_{f,1}^2)$ in (106). + +Moreover, we have + +$$ +\begin{array}{l} A \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}\right) = \alpha_ {t} - L _ {f} \alpha_ {t} ^ {2} - 2 E \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}\right) \alpha_ {t} ^ {2} \\ \geq \alpha_ {t} - L _ {f} \alpha_ {t} ^ {2} - \frac {\alpha_ {t}}{2} \\ \geq \frac {\alpha_ {t}}{4}, \tag {112} \\ \end{array} +$$ + +where the last inequality follows from $\alpha_{t} \leq 1 / 4L_{f}$ in (107) since $\alpha_{t} = 1 / (c + t)^{1 / 3}$ in (16). + +Bounding (108a). + +From (109), we have + +$$ +\begin{array}{l} B (\alpha_ {t}, \beta_ {t}, \delta_ {t}) = - \frac {\check {L} _ {\mu_ {g}} \Upsilon}{4} \delta_ {t} + 4 M _ {f} ^ {2} \alpha_ {t} - 2 M _ {f} ^ {2} L _ {f} \alpha_ {t} ^ {2} + 2 M _ {f} ^ {2} E (\alpha_ {t}, \beta_ {t}, \delta_ {t}) \alpha_ {t} ^ {2} \\ + 7 2 (1 - \lambda_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {4} \delta_ {t} ^ {2} \frac {1}{\Psi \alpha_ {t}} + 7 2 (1 - \eta_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {4} \delta_ {t} ^ {2} \frac {1}{\Omega \alpha_ {t}} \\ \leq - \frac {\hat {L} _ {\mu_ {g}} \Upsilon}{4} \delta_ {t} + 4 M _ {f} ^ {2} \alpha_ {t} - 2 M _ {f} ^ {2} L _ {f} \alpha_ {t} ^ {2} + \frac {M _ {f} ^ {2}}{2} \alpha_ {t} + 7 2 \ell_ {g, 1} ^ {4} \left(\frac {1}{\Psi} + \frac {1}{\Omega}\right) \frac {\delta_ {t} ^ {2}}{\alpha_ {t}} \\ = \left(- \frac {\hat {L} _ {\mu_ {g}}}{4} \Upsilon c _ {\delta} + \frac {9}{2} M _ {f} ^ {2} + 7 2 \ell_ {g, 1} ^ {4} \left(\frac {1}{\Psi} + \frac {1}{\Omega}\right) c _ {\delta} ^ {2}\right) \alpha_ {t} \\ \leq - \frac {1}{2} M _ {f} ^ {2} \alpha_ {t}, \tag {113} \\ \end{array} +$$ + +where the first inequality follows from $\beta_{t} = c_{\beta}\alpha_{t}$ , $\delta_t = c_\delta \alpha_t$ in (16), and Eq. (111); the second inequality is by $\Upsilon = \frac{22M_f^2}{L_{\mu_g}c_\delta}$ , and $\Psi, \Omega \geq \frac{288\varepsilon_{g,1}^4}{M_f^2}c_\delta^2$ in (106); the last inequality follows from in (106). + +Moreover, using Eq. (109) together with $\beta_{t} = c_{\beta}\alpha_{t}$ and $\delta_t = c_\delta \alpha_t$ in Eq. (16), we have + +$$ +\begin{array}{l} C (\alpha_ {t}, \beta_ {t}, \delta_ {t}) = - \frac {L _ {\mu_ {g}} \Gamma}{2} \beta_ {t} + 4 M _ {f} ^ {2} \alpha_ {t} - 2 L _ {f} M _ {f} ^ {2} \alpha_ {t} ^ {2} + 2 M _ {f} ^ {2} E (\alpha_ {t}, \beta_ {t}, \delta_ {t}) \alpha_ {t} ^ {2} \\ + \frac {2 8 8}{\Psi \alpha_ {t}} \ell_ {g, 1} ^ {2} (p ^ {2} \ell_ {g, 2} ^ {2} + \ell_ {f, 1} ^ {2}) \delta_ {t} ^ {2} \\ \leq - \frac {L _ {\mu_ {g}}}{2} \Gamma c _ {\beta} \alpha_ {t} + \frac {9}{2} M _ {f} ^ {2} \alpha_ {t} + \frac {2 8 8}{\Psi} \ell_ {g, 1} ^ {2} \left(p ^ {2} \ell_ {g, 2} ^ {2} + \ell_ {f, 1} ^ {2}\right) c _ {\delta} ^ {2} \alpha_ {t} \\ \leq - \frac {M _ {f} ^ {2}}{2} \alpha_ {t}, \tag {114} \\ \end{array} +$$ + +where the first inequality follows from (111); the last inequality follows from $\Gamma = \frac{11M_f^2}{L_{\mu_g}c_\beta}$ and $\Psi \geq \frac{576}{M_f^2}\ell_{g,1}^2(p^2\ell_{g,2}^2 + \ell_{f,1}^2)c_\delta^2$ in (106). + +Thus, from (113) and (114), we get + +$$ +(1 0 8 \mathrm {a}) \leq \mathcal {O} \left(V _ {T}\right). \tag {115} +$$ + +Bounding (108b). + +From (109), we also have + +$$ +D (\alpha_ {t}) = 4 \alpha_ {t} - 2 L _ {f} \alpha_ {t} ^ {2} - 5 \alpha_ {t} \leq 0. +$$ + +From Eq. (109), $\beta_{t} = c_{\beta}\alpha_{t}$ , and $\Gamma = \frac{11M_f^2}{L_{\mu_g}c_\beta}$ , we obtain + +$$ +\begin{array}{l} F (\alpha_ {t}, \beta_ {t}) = \frac {2 \Gamma}{L _ {\mu_ {g}}} \beta_ {t} - \hbar \alpha_ {t} + 1 4 4 (1 - \lambda_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \frac {\beta_ {t} ^ {2}}{\Psi \alpha_ {t}} \\ + 1 4 4 (1 - \eta_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \frac {\beta_ {t} ^ {2}}{\Omega \alpha_ {t}} \\ \leq \frac {2 2 M _ {f} ^ {2}}{L _ {\mu_ {g}} ^ {2}} \alpha_ {t} - \hbar \alpha_ {t} + 1 4 4 (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \left(\frac {1}{\Psi} + \frac {1}{\Omega}\right) c _ {\beta} ^ {2} \alpha_ {t} \\ \leq 2 4 \frac {M _ {f} ^ {2}}{L _ {\mu_ {g}} ^ {2}} \alpha_ {t} - \hbar \alpha_ {t} \\ = - \frac {M _ {f} ^ {2}}{L _ {\mu_ {g}} ^ {2}} \alpha_ {t}, \\ \end{array} +$$ + +where the second inequality follows from $\Omega, \Psi \geq 144(\ell_{g,2}^2 p^2 + \ell_{f,1}^2) \frac{L_{\mu_g}^2 c_\beta^2}{M_f^2}$ in (106); and the last equality is by $\hbar = 25 \frac{M_f^2}{L_{\mu_g}^2}$ . + +From $\delta_t = c_\delta \alpha_t$ in (16), we obtain + +$$ +\begin{array}{l} I (\alpha_ {t}, \delta_ {t}) = \frac {4 \Upsilon}{\acute {L} _ {\mu_ {g}}} \delta_ {t} - \jmath \alpha_ {t} + 7 2 \ell_ {g, 1} ^ {2} (1 - \eta_ {t + 1}) ^ {2} \frac {\delta_ {t} ^ {2}}{\Omega \alpha_ {t}} \\ \leq \frac {4 \Upsilon}{\dot {L} _ {\mu_ {g}}} c _ {\delta} \alpha_ {t} - \jmath \alpha_ {t} + 7 2 \ell_ {g, 1} ^ {2} \frac {c _ {\delta} ^ {2} \alpha_ {t}}{\Omega} \\ \leq \frac {8 9 M _ {f} ^ {2}}{\acute {L} _ {\mu_ {g}} ^ {2}} \alpha_ {t} - j \alpha_ {t} \\ = - \frac {M _ {f} ^ {2}}{\acute {L} _ {\mu_ {g}} ^ {2}} \alpha_ {t}, \\ \end{array} +$$ + +where the second inequality follows from $\Upsilon = \frac{22M_f^2}{\dot{L}_{\mu_g}c_{\delta}}$ and $\Omega \geq \frac{72\ell_{g,1}^2\dot{L}_{\mu_g}^2}{M_f^2} c_\delta^2$ ; the last equality follows from $\jmath = 90\frac{M_f^2}{\dot{L}_{\mu_g}^2}$ . + +Thus, we get + +$$ +(1 0 8 \mathrm {b}) \leq 0. \tag {116} +$$ + +Bounding (108c). + +From $\beta_{t} = c_{\beta}\alpha_{t}$ in (16) and Eq. (110), we have + +$$ +\begin{array}{l} L (\alpha_ {t}, \beta_ {t}) = - \frac {2 \Gamma \beta_ {t}}{\mu_ {g} + \ell_ {g , 1}} + \Gamma \beta_ {t} ^ {2} + 4 8 (1 - \gamma_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {2} \frac {\beta_ {t} ^ {2}}{\Phi \alpha_ {t}} \\ + 1 4 4 (1 - \lambda_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \frac {\beta_ {t} ^ {2}}{\Psi \alpha_ {t}} + 1 4 4 (1 - \eta_ {t + 1}) ^ {2} (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \frac {\beta_ {t} ^ {2}}{\Omega \alpha_ {t}} \\ \leq - \frac {2 \Gamma c _ {\beta} \alpha_ {t}}{\mu_ {g} + \ell_ {g , 1}} + \Gamma c _ {\beta} ^ {2} \alpha_ {t} ^ {2} + 4 8 \ell_ {g, 1} ^ {2} c _ {\beta} ^ {2} \frac {\alpha_ {t}}{\Phi} + 1 4 4 (\ell_ {g, 2} ^ {2} p ^ {2} + \ell_ {f, 1} ^ {2}) \left(\frac {1}{\Psi} + \frac {1}{\Omega}\right) c _ {\beta} ^ {2} \alpha_ {t} \\ \leq - \frac {2 \Gamma c _ {\beta} \alpha_ {t}}{\mu_ {g} + \ell_ {g , 1}} + \Gamma c _ {\beta} ^ {2} \alpha_ {t} ^ {2} + \frac {3 \Gamma c _ {\beta} \alpha_ {t}}{4 (\mu_ {g} + \ell_ {g , 1})} \\ \leq - \frac {\Gamma c _ {\beta} \alpha_ {t}}{4 \left(\mu_ {g} + \ell_ {g , 1}\right)}, \\ \end{array} +$$ + +where the second inequality is by $\Phi \geq 192\ell_{g,1}^{2}\frac{(\mu_g + \ell_{g,1})}{\Gamma} c_\beta$ , and $\Omega, \Psi \geq 576(\ell_{g,2}^{2}p^{2} + \ell_{f,1}^{2})\frac{(\mu_g + \ell_{g,1})}{\Gamma} c_\beta$ in (106); the last inequality follows from $\alpha_t \leq 1 / c_\beta (\mu_g + \ell_{g,1})$ in (107). + +From $\beta_{t} = c_{\beta}\alpha_{t}$ , $\delta_t = c_\delta \alpha_t$ in (16) and Eq. (110), we obtain + +$$ +N (\beta_ {t}, \delta_ {t}) = \frac {4}{L _ {\mu_ {g}} \beta_ {t}} \Gamma + \frac {1 6 \nu^ {2}}{\acute {L} _ {\mu_ {g}} \mu_ {g} ^ {2} \delta_ {t}} \Upsilon = \frac {4}{L _ {\mu_ {g}} c _ {\beta} \alpha_ {t}} \Gamma + \frac {1 6 \nu^ {2}}{\acute {L} _ {\mu_ {g}} \mu_ {g} ^ {2} c _ {\delta} \alpha_ {t}} \Upsilon . +$$ + +Thus, we get + +$$ +\begin{array}{l} (1 0 8 \mathrm {c}) = \sum_ {t = 1} ^ {T} L (\alpha_ {t}, \beta_ {t}) \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} + \sum_ {t = 2} ^ {T} N (\beta_ {t}, \delta_ {t}) \sup _ {\mathbf {x} \in \mathcal {X}} \| \mathbf {y} _ {t - 1} ^ {*} (\mathbf {x}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x}) \| ^ {2} \\ \leq \mathcal {O} \left(\frac {H _ {2 , T}}{\alpha_ {T}}\right). \tag {117} \\ \end{array} +$$ + +Bounding (108d). + +From $\eta_{t + 1} = c_{\eta}\alpha_{t}^{2},\gamma_{t + 1} = c_{\gamma}\alpha_{t}^{2},\lambda_{t + 1} = c_{\lambda}\alpha_{t}^{2}$ in Eq. (16), we obtain + +$$ +\begin{array}{l} (1 0 8 \mathrm {d}) = \frac {\sigma_ {g _ {\mathbf {y}}} ^ {2}}{\bar {b}} \frac {2}{\Phi} \sum_ {t = 1} ^ {T} \frac {\gamma_ {t + 1} ^ {2}}{\alpha_ {t}} + \frac {4}{\Psi} \left(\frac {\sigma_ {g _ {\mathbf {y y}}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {y}}} ^ {2}}{b}\right) \sum_ {t = 1} ^ {T} \frac {\lambda_ {t + 1} ^ {2}}{\alpha_ {t}} + \frac {4}{\Omega} \left(\frac {\sigma_ {g _ {\mathbf {x y}}} ^ {2}}{\bar {b}} p ^ {2} + \frac {\sigma_ {f _ {\mathbf {x}}} ^ {2}}{b}\right) \sum_ {t = 1} ^ {T} \frac {\eta_ {t + 1} ^ {2}}{\alpha_ {t}} \\ \leq \mathcal {O} \left(\left(\frac {\sigma_ {g _ {\mathbf {y}}} ^ {2}}{\bar {b}} + \frac {\sigma_ {g _ {\mathbf {y y}}} ^ {2}}{\bar {b}} + \frac {\sigma_ {f _ {\mathbf {y}}} ^ {2}}{b} + \frac {\sigma_ {g _ {\mathbf {x y}}} ^ {2}}{\bar {b}} + \frac {\sigma_ {f _ {\mathbf {x}}} ^ {2}}{b}\right) \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {3}\right). \tag {118} \\ \end{array} +$$ + +Bounding (108e). + +We also have + +$$ +\begin{array}{l} (1 0 8 \mathrm {e}) = \frac {6}{\Phi \alpha_ {T}} G _ {\mathbf {y}, T} + \frac {1 2 p ^ {2}}{\Omega \alpha_ {T}} G _ {\mathbf {x y}, T} + \frac {1 2 p ^ {2}}{\Psi \alpha_ {T}} G _ {\mathbf {y y}, T} + \frac {1 2 \ell_ {f , 1} ^ {2}}{\Psi \alpha_ {T}} D _ {\mathbf {y}, T} + \frac {1 2 \ell_ {f , 1} ^ {2}}{\Omega \alpha_ {T}} D _ {\mathbf {x}, T} \\ \leq \mathcal {O} \left(\frac {1}{\alpha_ {T}} \left(G _ {\mathbf {y}, T} + G _ {\mathbf {x y}, T} + G _ {\mathbf {y y}, T} + D _ {\mathbf {y}, T} + D _ {\mathbf {x}, T}\right)\right). \tag {119} \\ \end{array} +$$ + +From Eq. (16), we have $b = \bar{b} = 1$ . Moreover, by (10), $\sigma^2 = \sigma_{g_{\mathbf{y}}}^2 + \sigma_{g_{\mathbf{yy}}}^2 + \sigma_{f_{\mathbf{y}}}^2 + \sigma_{g_{\mathbf{xy}}}^2 + \sigma_{f_{\mathbf{x}}}^2$ . From (15), we also have + +$$ +G _ {T} = G _ {\mathbf {y}, T} + G _ {\mathbf {x y}, T} + G _ {\mathbf {y y}, T}, +$$ + +$$ +D _ {T} = D _ {\mathbf {y}, T} + D _ {\mathbf {x}, T}. +$$ + +Then, by inequalities (112), (115), (116), (117), (118), (119), we have + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \frac {\alpha_ {t}}{2} \mathbb {E} \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} + \Lambda \\ \leq \mathcal {O} \left(V _ {T} + \frac {H _ {2 , T}}{\alpha_ {T}} + \frac {\sigma^ {2}}{b} \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {3} + \frac {G _ {T}}{\alpha_ {T}} + \frac {D _ {T}}{\alpha_ {T}}\right). \tag {120} \\ \end{array} +$$ + +From the definition of $\Lambda$ in (105), we have + +$$ +\begin{array}{l} - \Lambda = \Gamma \sum_ {t = 1} ^ {T} \left(\mathbb {E} \left[ \theta_ {t} ^ {\mathbf {y}} \right] - \mathbb {E} \left[ \theta_ {t + 1} ^ {\mathbf {y}} \right]\right) + \Upsilon \sum_ {t = 1} ^ {T} \left(\mathbb {E} \left[ \theta_ {t} ^ {\mathbf {v}} \right] - \mathbb {E} \left[ \theta_ {t + 1} ^ {\mathbf {v}} \right]\right) + \frac {1}{\Phi} \sum_ {t = 1} ^ {T} \left(\frac {\mathbb {E} \| e _ {t} ^ {g} \| ^ {2}}{\alpha_ {t - 1}} - \frac {\mathbb {E} \| e _ {t + 1} ^ {g} \| ^ {2}}{\alpha_ {t}}\right) \\ + \frac {1}{\Psi} \sum_ {t = 1} ^ {T} \left(\frac {\mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2}}{\alpha_ {t - 1}} - \frac {\mathbb {E} \| e _ {t + 1} ^ {\mathbf {v}} \| ^ {2}}{\alpha_ {t}}\right) + \frac {1}{\Omega} \sum_ {t = 1} ^ {T} \left(\frac {\mathbb {E} \| e _ {t} ^ {f} \| ^ {2}}{\alpha_ {t - 1}} - \frac {\mathbb {E} \| e _ {t + 1} ^ {f} \| ^ {2}}{\alpha_ {t}}\right) \\ \leq \Gamma \theta_ {1} ^ {\mathbf {y}} + \Upsilon \theta_ {1} ^ {\mathbf {v}} + \frac {\sigma_ {g _ {\mathbf {y}}} ^ {2}}{\Phi \alpha_ {0}} + \frac {\sigma_ {g _ {\mathbf {y y}}} ^ {2} + \sigma_ {f _ {\mathbf {y}}} ^ {2}}{\Psi \alpha_ {0}} + \frac {\sigma_ {g _ {\mathbf {x y}}} ^ {2} + \sigma_ {f _ {\mathbf {x}}} ^ {2}}{\Omega \alpha_ {0}}. \tag {121} \\ \end{array} +$$ + +Using (121), we get + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \frac {\alpha_ {t}}{2} \mathbb {E} \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \\ \leq \mathcal {O} \left(V _ {T} + \frac {H _ {2 , T}}{\alpha_ {T}} + \frac {\sigma^ {2}}{b} \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {3} + \frac {G _ {T}}{\alpha_ {T}} + \frac {D _ {T}}{\alpha_ {T}} - \Lambda\right) \\ \leq \mathcal {O} \left(V _ {T} + \theta_ {1} ^ {\mathbf {y}} + \theta_ {1} ^ {\mathbf {v}} + \frac {\sigma^ {2}}{b} \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {3} + \frac {H _ {2 , T}}{\alpha_ {T}} + \frac {G _ {T}}{\alpha_ {T}} + \frac {D _ {T}}{\alpha_ {T}} + \frac {\sigma^ {2}}{\alpha_ {0}}\right). \\ \end{array} +$$ + +Since $\alpha_{t} = 1 / (c + t)^{1 / 3}$ , we get + +$$ +\sum_ {t = 1} ^ {T} \alpha_ {t} ^ {3} = \sum_ {t = 1} ^ {T} \frac {1}{c + t} \leq \sum_ {t = 1} ^ {T} \frac {1}{1 + t} \leq \log (T + 1), +$$ + +which, combined with the fact that $\alpha_{t}$ decreases with respect to $t$ and by multiplying both sides by $2 / \alpha_{T}$ , results in Thus, we have + +$$ +\begin{array}{l} \operatorname {B L} - \operatorname {R e g} _ {T} = \sum_ {t = 1} ^ {T} \mathbb {E} \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \\ \leq \mathcal {O} \Big (\frac {1}{\alpha_ {T}} (V _ {T} + \| \mathbf {y} _ {1} - \mathbf {y} _ {1} ^ {*} (\mathbf {x} _ {1}) \| ^ {2} + \| \mathbf {v} _ {1} - \mathbf {v} _ {1} ^ {*} (\mathbf {x} _ {1}) \| ^ {2} + \sigma^ {2} \log (T + 1) + \frac {\sigma^ {2}}{\alpha_ {0}} \Big) \\ \left. + \frac {1}{\alpha_ {T} ^ {2}} \left(H _ {2, T} + G _ {T} + D _ {T}\right)\right). \\ \end{array} +$$ + +This completes the proof. + +![](images/2770d37f216cc7d6c860a6c74373d8b3c6c83a4315502fc4e98d0ea0d7e055df.jpg) + +# D Proof of Regret Bounds for Zeroth Order SOGD (ZO-SOGD) + +Proof Roadmap. We provide Lemma D.7, which quantifies the error between the approximated direction of the momentum-based gradient estimator, $\hat{\mathbf{d}}_t^{\mathbf{y}}$ and the true direction, $\nabla_{\mathbf{y}}g_{t,\rho}(\mathbf{x}_t,\mathbf{y}_t)$ , at each iteration. Lemma D.9 assesses the convergence of the iterative solutions $\{\mathbf{y}_t\}_{t = 1}^T$ , specifically the gap $\mathbb{E}\big[\| \mathbf{y}_{t + 1} - \hat{\mathbf{y}}_t^* (\mathbf{x}_t)\| ^2\big]$ , while accounting for the error introduced in Lemma D.7. To establish Lemma D.13, which quantifies the error between the approximated direction of the momentum-based gradient estimator, $\hat{\mathbf{d}}_t^{\mathbf{v}}$ , and the true direction, $\nabla_{\mathbf{y}}f_{t,\rho}(\mathbf{x}_t,\mathbf{y}_t) + \nabla_{\mathbf{y}}^2 g_{t,\rho}(\mathbf{x}_t,\mathbf{y}_t)\mathbf{v}_t$ , we need to present Lemma D.11. This lemma quantifies the error between $\hat{\mathbf{d}}_t^{\mathbf{v}}$ and $\nabla_{\mathbf{y}}f_{t,\rho}(\mathbf{x}_t,\mathbf{y}_t) + \frac{1}{2\rho_{\mathbf{v}}} (\nabla_{\mathbf{y}}g_{t,\rho}(\mathbf{x}_t,\mathbf{y}_t + \rho_{\mathbf{v}}\mathbf{v}_t) - \nabla_{\mathbf{y}}g_{t,\rho}(\mathbf{x}_t,\mathbf{y}_t - \rho_{\mathbf{v}}\mathbf{v}_t))$ . Then, Lemma D.15 captures the error of the system solution of Problem (18), i.e., gap $\mathbb{E}\big[\| \mathbf{v}_{t + 1} - \hat{\mathbf{v}}_t^* (\mathbf{x}_t)\| ^2\big]$ , based on these errors. To establish Lemma D.19, which quantifies the error between the approximated direction of the momentum-based hypergradient estimator, $\hat{\mathbf{d}}_t^{\mathbf{x}}$ , and the true direction, $\nabla_{\mathbf{x}}f_{t,\rho}(\mathbf{x}_t,\mathbf{y}_t) + \nabla_{\mathbf{x}\mathbf{y}}^2 g_{t,\rho}(\mathbf{x}_t,\mathbf{y}_t)\mathbf{v}_t$ , it is necessary to introduce Lemma D.17. This lemma quantifies the error between $\hat{\mathbf{d}}_t^{\mathbf{x}}$ and $\nabla_{\mathbf{x}}f_{t,\rho}(\mathbf{x}_t,\mathbf{y}_t) + \frac{1}{2\rho_{\mathbf{v}}} (\nabla_{\mathbf{x}}g_{t,\rho}(\mathbf{x}_t,\mathbf{y}_t + \rho_{\mathbf{v}}\mathbf{v}_t) - \nabla_{\mathbf{x}}g_{t,\rho}(\mathbf{x}_t,\mathbf{y}_t - \rho_{\mathbf{v}}\mathbf{v}_t))$ . Then, Lemma D.20 bounds the projection mapping based on these errors. By combining these lemmas and setting parameters, we achieve the desired result. + +# D.1 Auxiliary Lemmas for Proof of Theorem 3.2 + +Lemma D.1. [4, Lemma A.1.] Suppose Assumption B4. holds. Then, for any $\mathbf{x},\mathbf{v}\in \mathcal{X}$ we have: + +$$ +\left\| \nabla g _ {t} (\mathbf {x} + \mathbf {v}, \mathbf {y} + \mathbf {v}) - \nabla g _ {t} (\mathbf {x}, \mathbf {y}) - \nabla^ {2} g _ {t} (\mathbf {x}, \mathbf {y}) \mathbf {v} \right\| \leq \ell_ {g, 2} \| \mathbf {v} \| ^ {2}. +$$ + +Lemma D.2. Suppose that Assumptions 2.2 and 2.3 hold for all $\mathbf{x},\mathbf{x}^{\prime}\in \mathcal{X}$ , and $t\in [T]$ , and that $\mathbf{d}_{t,\rho}^{\mathbf{x}}$ and $\mathbf{d}_{t,\rho}^{\mathbf{y}}$ are defined in (22). Then, we have + +$$ +\left\| \mathbf {d} _ {t, \rho} ^ {\mathbf {x}} - \nabla f _ {t, \rho} (\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})) \right\| ^ {2} \leq M _ {f} ^ {2} \left(\| \mathbf {y} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \| ^ {2} + \| \mathbf {v} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \| ^ {2}\right), \tag {122a} +$$ + +$$ +\left\| \mathbf {d} _ {t, \rho} ^ {\mathbf {v}} \right\| ^ {2} \leq M _ {\mathbf {v}} ^ {2} \left(\| \mathbf {y} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \| ^ {2} + \| \mathbf {v} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \| ^ {2}\right), \tag {122b} +$$ + +$$ +\left\| \nabla f _ {t, \rho} \left(\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})\right) - \nabla f _ {t, \rho} \left(\mathbf {x} ^ {\prime}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} ^ {\prime})\right) \right\| \leq L _ {f} \| \mathbf {x} - \mathbf {x} ^ {\prime} \|, \tag {122c} +$$ + +$$ +\left\| \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) - \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} ^ {\prime}\right) \right\| \leq L _ {\mathbf {y}} \| \mathbf {x} - \mathbf {x} ^ {\prime} \|, \tag {122d} +$$ + +$$ +\left\| \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) - \hat {\mathbf {v}} _ {t} ^ {*} \left(\mathbf {x} ^ {\prime}\right) \right\| \leq L _ {\mathbf {v}} \| \mathbf {x} - \mathbf {x} ^ {\prime} \|. \tag {122e} +$$ + +Here, $\hat{\mathbf{v}}_t^* (\mathbf{x})$ , $f_{t,\rho}$ and $\hat{\mathbf{y}}_t^* (\mathbf{x})$ are defined in (20), (18), and (19), respectively. Moreover, the constants $M_f$ , $M_{\mathbf{v}}$ , and $(L_{\mathbf{y}},L_{\mathbf{v}},L_{f})$ are defined as in (42), (43), and (44), respectively. + +Proof. We first show Eq. (122a). + +Using Assumptions 2.2 and B1., we have $\nabla_{\mathbf{y}}^{2}g_{t,\rho}\left(\mathbf{x},\hat{\mathbf{y}}_{t}^{*}(\mathbf{x})\right)\succeq \mu_{g}$ , and + +$$ +\left\| \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \right\| = \left\| \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t, \rho} \left(\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})\right)\right) ^ {- 1} \nabla_ {\mathbf {y}} f _ {t, \rho} \left(\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})\right) \right\| \leq \frac {\ell_ {f , 0}}{\mu_ {g}}. \tag {123} +$$ + +Observe that we have + +$$ +\begin{array}{l} \left\| \mathbf {d} _ {t, \boldsymbol {\rho}} ^ {\mathbf {x}} - \nabla f _ {t, \boldsymbol {\rho}} (\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})) \right\| \leq \left\| \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}) - \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} (\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})) \right\| \\ + \left\| \mathbf {v} \nabla_ {\mathbf {x y}} ^ {2} g _ {t, \rho} (\mathbf {x}, \mathbf {y}) - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \nabla_ {\mathbf {x y}} ^ {2} g _ {t, \rho} \left(\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})\right) \right\| \\ \leq \left\| \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}) - \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} (\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})) \right\| \\ + \| \nabla_ {\mathbf {x y}} ^ {2} g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}) \| \| \mathbf {v} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \| \\ + \| \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \| \| \nabla_ {\mathbf {x y}} ^ {2} g _ {t, \rho} (\mathbf {x}, \mathbf {y}) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t, \rho} (\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})) \| \\ \leq \left(\ell_ {f, 1} + \frac {\ell_ {g , 2} \ell_ {f , 0}}{\mu_ {g}}\right) \| \mathbf {y} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \| + \ell_ {g, 1} \| \mathbf {v} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \| \\ \leq M _ {f} ^ {2} \left(\| \mathbf {y} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \| + \| \mathbf {v} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \|\right), \tag {124} \\ \end{array} +$$ + +where $M_{f}$ is defined as in (42); the third inequality is by Assumption 2.3 and the last inequality is by Eq. (123). + +We now show Eq. (122b). + +Since $\mathbf{d}_{t,\rho}^{\mathbf{v}^*} \coloneqq \nabla_{\mathbf{y}}f_{t,\rho}(\mathbf{x},\hat{\mathbf{y}}_t^*(\mathbf{x})) + \nabla_{\mathbf{y}}^2g_{t,\rho}(\mathbf{x},\hat{\mathbf{y}}_t^*(\mathbf{x}))\hat{\mathbf{v}}_t^*(\mathbf{x}) = 0$ , we have + +$$ +\begin{array}{l} \left\| \mathbf {d} _ {t, \boldsymbol {\rho}} ^ {\mathbf {v}} \right\| = \left\| \mathbf {d} _ {t, \boldsymbol {\rho}} ^ {\mathbf {v}} - \mathbf {d} _ {t, \boldsymbol {\rho}} ^ {\mathbf {v} *} \right\| \\ = \left\| \mathbf {v} _ {t} \nabla_ {\mathbf {y}} ^ {2} g _ {t, \rho} (\mathbf {x}, \mathbf {y}) + \nabla_ {\mathbf {y}} f _ {t, \rho} (\mathbf {x}, \mathbf {y}) \right. \\ - \left(\hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \nabla_ {\mathbf {y}} ^ {2} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})\right) + \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} \left(\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})\right)\right) \| \\ \leq \left\| \left(\nabla_ {\mathbf {y}} ^ {2} g _ {t, \rho} (\mathbf {x}, \mathbf {y}) - \nabla_ {\mathbf {y}} ^ {2} g _ {t, \rho} (\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}))\right) \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \right\| \\ + \left\| \nabla_ {\mathbf {y}} ^ {2} g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}) \left(\mathbf {v} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x})\right) \right\| \\ + \left\| \nabla_ {\mathbf {y}} f _ {t, \rho} (\mathbf {x}, \mathbf {y}) - \nabla_ {\mathbf {y}} f _ {t, \rho} (\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})) \right\|. \\ \end{array} +$$ + +Then, from Assumption 2.3 and Eq. (123), we have + +$$ +\begin{array}{l} \| \mathbf {d} _ {t, \boldsymbol {\rho}} ^ {\mathbf {v}} \| \leq \ell_ {g, 2} \| \mathbf {y} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \| \| \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \| + \ell_ {g, 1} \| \mathbf {v} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \| + \ell_ {f, 1} \| \mathbf {y} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \| \\ \leq \left(\frac {\ell_ {g , 2} \ell_ {f , 0}}{\mu_ {g}} + \ell_ {f, 1}\right) \| \mathbf {y} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \| + \ell_ {g, 1} \| \mathbf {v} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \| \\ \leq M _ {\mathbf {v}} \left(\| \mathbf {y} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \| + \| \mathbf {v} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x}) \|\right), \\ \end{array} +$$ + +where $M_{\mathbf{v}}$ is defined as in (43). + +The proofs of Eqs. (122c)-(122e) follow from [67, Lemma 17] by setting $(L_{\mathbf{y}}, L_{\mathbf{v}}, L_f)$ as in (44). + +# D.2 Perturbation Bounds for OBO Objectives and Their Smoothing Variants + +The following two lemmas are inspired by [24]. + +Lemma D.3. Given $\rho = (\rho_{\mathbf{s}},\rho_{\mathbf{r}})$ as positive smoothing parameters, let $g_{t,\rho}(\mathbf{x},\mathbf{y})$ and $f_{t,\rho}(\mathbf{x},\mathbf{y})$ be the functions defined by (18). + +(a) Suppose Assumption B3. holds. Then, we have + +$$ +\left| g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}) - g _ {t} (\mathbf {x}, \mathbf {y}) \right| \leq \frac {\ell_ {g , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{2}. \tag {125} +$$ + +(b) Suppose Assumption B2. holds. Then, we have + +$$ +\left| f _ {t, \rho} (\mathbf {x}, \mathbf {y}) - f _ {t} (\mathbf {x}, \mathbf {y}) \right| \leq \frac {\ell_ {f , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{2}. \tag {126} +$$ + +Proof. Let $B_{1}$ and $B_{2}$ be the unit ball in $\mathbb{R}^{d_1}$ and $\mathbb{R}^{d_2}$ , respectively. Let $\mathcal{V}(d_1)$ and $\mathcal{V}(d_2)$ be volume of the unit ball in $\mathbb{R}^{d_1}$ and $\mathbb{R}^{d_2}$ , respectively. Then, we have + +$$ +\begin{array}{l} \left| g _ {t, \rho} (\mathbf {x}, \mathbf {y}) - g _ {t} (\mathbf {x}, \mathbf {y}) \right| \\ = \left| \frac {1}{\mathcal {V} (d _ {1}) \mathcal {V} (d _ {2})} \int_ {B _ {1}} \int_ {B _ {2}} \left(g _ {t} (\mathbf {x} + \rho_ {\mathbf {s}} \mathbf {s}, \mathbf {y} + \rho_ {\mathbf {r}} \mathbf {r}) - g _ {t} (\mathbf {x}, \mathbf {y})\right) d \mathbf {s} d \mathbf {r} \right| \\ = \left| \frac {1}{\mathcal {V} (d _ {1}) \mathcal {V} (d _ {2})} \int_ {B _ {1}} \int_ {B _ {2}} \left(g _ {t} (\mathbf {x} + \rho_ {\mathbf {s}} \mathbf {s}, \mathbf {y} + \rho_ {\mathbf {r}} \mathbf {r}) - g _ {t} (\mathbf {x}, \mathbf {y}) - \langle \nabla g _ {t} (\mathbf {x}, \mathbf {y}), (\rho_ {\mathbf {s}} \mathbf {s}, \rho_ {\mathbf {r}} \mathbf {r}) \rangle\right) d \mathbf {s} d \mathbf {r} \right|. \\ \end{array} +$$ + +Thus, we get + +$$ +\begin{array}{l} \left| g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}) - g _ {t} (\mathbf {x}, \mathbf {y}) \right| \\ \leq \int_ {B _ {1}} \int_ {B _ {2}} \left| g _ {t} \left(\mathbf {x} + \rho_ {\mathbf {s}} \mathbf {s}, \mathbf {y} + \rho_ {\mathbf {r}} \mathbf {r}\right) - g _ {t} (\mathbf {x}, \mathbf {y}) - \langle \nabla g _ {t} (\mathbf {x}, \mathbf {y}), \left(\rho_ {\mathbf {s}} \mathbf {s}, \rho_ {\mathbf {r}} \mathbf {r}\right) \rangle \right| d \mathbf {s} d \mathbf {r} \\ \leq \int_ {B _ {1}} \int_ {B _ {2}} \frac {\ell_ {g , 1}}{2} \left(\rho_ {\mathbf {s}} ^ {2} \| \mathbf {s} \| ^ {2} + \rho_ {\mathbf {r}} ^ {2} \| \mathbf {r} \| ^ {2}\right) d \mathbf {s} d \mathbf {r} \\ = \frac {\ell_ {g , 1} \rho_ {\mathbf {s}} ^ {2}}{2} \int_ {B _ {1}} \| \mathbf {s} \| ^ {2} d \mathbf {s} + \frac {\ell_ {g , 1} \rho_ {\mathbf {r}} ^ {2}}{2} \int_ {B _ {2}} \| \mathbf {r} \| ^ {2} d \mathbf {r} \\ = \frac {\ell_ {g , 1} \rho_ {\mathbf {s}} ^ {2}}{2} \frac {d _ {1}}{d _ {1} + 2} + \frac {\ell_ {g , 1} \rho_ {\mathbf {r}} ^ {2}}{2} \frac {d _ {2}}{d _ {2} + 2} \\ \leq \frac {\ell_ {g , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{2}, \\ \end{array} +$$ + +where the last equality follows since $\frac{1}{\mathcal{V}(d)}\int_{s\in B}\| s\| ^p ds = \frac{d}{d + p}$ . + +The proof of part (b) follows using similar arguments. + +![](images/0e3c795e8383470ab9d37e0fda8aa556d9c65dda1146cbf2fb2dbd5366b13c04.jpg) + +Lemma D.4. Given $\rho = (\rho_{\mathbf{s}},\rho_{\mathbf{r}})$ as positive smoothing parameters, let $g_{t,\rho}(\mathbf{x},\mathbf{y})$ and $f_{t,\rho}(\mathbf{x},\mathbf{y})$ be the functions defined by (18). + +(a) Suppose Assumption B3. holds. Then, we have + +$$ +\left\| \nabla g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}) - \nabla g _ {t} (\mathbf {x}, \mathbf {y}) \right\| \leq \frac {\ell_ {g , 1} \left(\rho_ {\mathbf {s}} d _ {1} + \rho_ {\mathbf {r}} d _ {2}\right)}{2}. \tag {127} +$$ + +(b) Suppose Assumption B2. holds. Then, we have + +$$ +\left\| \nabla f _ {t} (\mathbf {x}, \mathbf {y}) - \nabla f _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}) \right\| \leq \frac {\ell_ {f , 1} \left(\rho_ {\mathbf {s}} d _ {1} + \rho_ {\mathbf {r}} d _ {2}\right)}{2}. \tag {128} +$$ + +Proof. Let $S(d_{1})$ be the surface area of the unit sphere in $\mathbb{R}^{d_1}$ . Moreover, let $U_{B_1}$ be the unit sphere. + +$$ +\begin{array}{l} \left\| \nabla_ {\mathbf {x}} g _ {t, \rho} (\mathbf {x}, \mathbf {y}) - \nabla_ {\mathbf {x}} g _ {t} (\mathbf {x}, \mathbf {y}) \right\| \\ = \left\| \frac {1}{S (d _ {1})} \left(\frac {d _ {1}}{\rho_ {\mathbf {s}}} \int_ {U _ {B _ {1}}} g _ {t} (\mathbf {x} + \rho_ {\mathbf {s}} \mathbf {s}, \mathbf {y}) \mathbf {s} d \mathbf {s}\right) - \nabla_ {\mathbf {x}} g _ {t} (\mathbf {x}, \mathbf {y}) \right\| \\ = \left\| \frac {1}{S \left(d _ {1}\right)} \left(\frac {d _ {1}}{\rho_ {\mathbf {s}}} \int_ {U _ {B _ {1}}} g _ {t} (\mathbf {x} + \rho_ {\mathbf {s}} \mathbf {s}, \mathbf {y}) \mathbf {s} d \mathbf {s} - \int_ {U _ {B _ {1}}} \frac {d _ {1}}{\rho_ {\mathbf {s}}} g _ {t} (\mathbf {x}, \mathbf {y}) \mathbf {s} d \mathbf {s}} \right. \right. \\ \left. - \int_ {U _ {B _ {1}}} \frac {d _ {1}}{\rho_ {\mathbf {s}}} \langle \nabla_ {\mathbf {x}} g _ {t} (\mathbf {x}, \mathbf {y}), \rho_ {\mathbf {s}} \mathbf {s} \rangle \mathbf {s} d \mathbf {s}\right) \| \\ \leq \frac {d _ {1}}{S (d _ {1}) \rho_ {\mathbf {s}}} \int_ {U _ {B _ {1}}} \left| g _ {t} (\mathbf {x} _ {t} + \rho_ {\mathbf {s}} \mathbf {s}, \mathbf {y}) - g _ {t} (\mathbf {x}, \mathbf {y}) - \langle \nabla_ {\mathbf {x}} g _ {t} (\mathbf {x}, \mathbf {y}), \rho_ {\mathbf {s}} \mathbf {s} \rangle \right| \| \mathbf {s} \| d \mathbf {s} \\ \leq \frac {d _ {1}}{S (d _ {1}) \rho_ {\mathbf {s}}} \cdot \frac {\ell_ {g , 1} \rho_ {\mathbf {s}} ^ {2}}{2} \int_ {U _ {B _ {1}}} \| \mathbf {s} \| ^ {3} d \mathbf {s} \\ = \frac {\rho_ {\mathbf {s}} d _ {1} \ell_ {g , 1}}{2}, \tag {129} \\ \end{array} +$$ + +where the second equality follows from $\int_{U_{B_1}}\mathbf{ss}^\top d\mathbf{s} = \frac{S(d_1)}{d_1}\mathbf{I}$ . + +Similarly, let $S(d_2)$ be the surface area of the unit sphere in $\mathbb{R}^{d_2}$ . Moreover, let $U_{B_2}$ be the unit sphere. + +$$ +\begin{array}{l} \left\| \nabla_ {\mathbf {y}} g _ {t, \rho} (\mathbf {x}, \mathbf {y}) - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x}, \mathbf {y}) \right\| \\ = \left\| \frac {1}{S (d _ {2})} \left(\frac {d _ {2}}{\rho_ {\mathbf {r}}} \int_ {U _ {B _ {2}}} g _ {t} (\mathbf {x}, \mathbf {y} + \rho_ {\mathbf {r}} \mathbf {r}) \mathbf {r} d \mathbf {r}\right) - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x}, \mathbf {y}) \right\| \\ = \left\| \frac {1}{S \left(d _ {2}\right)} \left(\frac {d _ {2}}{\rho_ {\mathbf {r}}} \int_ {U _ {B _ {2}}} g _ {t} (\mathbf {x}, \mathbf {y} + \rho_ {\mathbf {r}} \mathbf {r}) \mathbf {r} d \mathbf {r} - \int_ {U _ {B _ {2}}} \frac {d _ {2}}{\rho_ {\mathbf {r}}} g _ {t} (\mathbf {x}, \mathbf {y}) \mathbf {r} d \mathbf {r} \right. \right. \\ \left. - \int_ {U _ {B _ {2}}} \frac {d _ {2}}{\rho_ {\mathbf {r}}} \langle \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x}, \mathbf {y}), \rho_ {\mathbf {r}} \mathbf {r} \rangle \mathbf {r} d \mathbf {r}\right) \| \\ \leq \frac {d _ {2}}{S (d _ {2}) \rho_ {\mathbf {r}}} \int_ {U _ {B _ {2}}} \left| g _ {t} (\mathbf {x} _ {t}, \mathbf {y} + \rho_ {\mathbf {r}} \mathbf {r}) - g _ {t} (\mathbf {x}, \mathbf {y}) - \langle \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x}, \mathbf {y}), \rho_ {\mathbf {r}} \mathbf {r} \rangle \right| \| \mathbf {r} \| d \mathbf {r} \\ \leq \frac {d _ {2}}{S (d _ {2}) \rho_ {\mathbf {r}}} \cdot \frac {\ell_ {g , 1} \rho_ {\mathbf {r}} ^ {2}}{2} \int_ {U _ {B _ {2}}} \| \mathbf {r} \| ^ {3} d \mathbf {r} \\ = \frac {\rho_ {\mathbf {r}} d _ {2} \ell_ {g , 1}}{2}, \tag {130} \\ \end{array} +$$ + +where the second equality follows from $\int_{U_{B_2}}\mathbf{r}\mathbf{r}^\top d\mathbf{r} = \frac{S(d_2)}{d_2}\mathbf{I}$ . + +Thus, we get + +$$ +\begin{array}{l} \left\| \nabla g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}) - \nabla g _ {t} (\mathbf {x}, \mathbf {y}) \right\| \\ \leq \left\| \nabla_ {\mathbf {x}} g _ {t, \rho} (\mathbf {x}, \mathbf {y}) - \nabla_ {\mathbf {x}} g _ {t} (\mathbf {x}, \mathbf {y}) \right\| + \left\| \nabla_ {\mathbf {y}} g _ {t, \rho} (\mathbf {x}, \mathbf {y}) - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x}, \mathbf {y}) \right\| \\ \leq \frac {\rho_ {\mathbf {s}} d _ {1} \ell_ {g , 1}}{2} + \frac {\rho_ {\mathbf {r}} d _ {2} \ell_ {g , 1}}{2}. \\ \end{array} +$$ + +Finally, by a similar argument as in Part (a), we obtain + +$$ +\left\| \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}) - \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x}, \mathbf {y}) \right\| \leq \frac {\rho_ {\mathbf {s}} d _ {1} \ell_ {f , 1}}{2}, \tag {131} +$$ + +and + +$$ +\left\| \nabla_ {\mathbf {y}} f _ {t, \rho} (\mathbf {x}, \mathbf {y}) - \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x}, \mathbf {y}) \right\| \leq \frac {\rho_ {\mathbf {r}} d _ {2} \ell_ {f , 1}}{2}, \tag {132} +$$ + +which implies + +$$ +\| \nabla f _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}) - \nabla f _ {t} (\mathbf {x}, \mathbf {y}) \| \leq \frac {(\rho_ {\mathbf {s}} d _ {1} + \rho_ {\mathbf {r}} d _ {2}) \ell_ {f , 1}}{2}. +$$ + +Lemma D.5. Suppose Assumption B3. holds. Let $\hat{\nabla}_{\mathbf{y}}g_{t}(\mathbf{x},\mathbf{y};\bar{B}_{t})$ and $\hat{\nabla}_{\mathbf{x}}g_{t}(\mathbf{x},\mathbf{y};\bar{B}_{t})$ be defined as in (24a) and (24b), respectively. Then, for any $(\mathbf{x},\mathbf{y})\in \mathbb{R}^{d_1}\times \mathbb{R}^{d_2}$ and $\rho_{\mathbf{r}},\rho_{\mathbf{s}}\geq 0$ we have + +$$ +\underset {(\mathbf {r}, \bar {\mathcal {B}} _ {t})} {\mathbb {E}} \left[ \| \hat {\nabla} _ {\mathbf {y}} g _ {t} (\mathbf {x}, \mathbf {y}; \bar {\mathcal {B}} _ {t}) - \hat {\nabla} _ {\mathbf {y}} g _ {t} (\mathbf {x}, \dot {\mathbf {y}}; \bar {\mathcal {B}} _ {t}) \| ^ {2} \right] \leq 3 d _ {2} \ell_ {g, 1} ^ {2} \| \mathbf {y} - \dot {\mathbf {y}} \| ^ {2} + \frac {3 \ell_ {g , 1} ^ {2} d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2}}{2}, \tag {133a} +$$ + +$$ +\underset {(\mathbf {s}, \bar {\mathcal {B}} _ {t})} {\mathbb {E}} \left[ \| \hat {\nabla} _ {\mathbf {x}} g _ {t} (\mathbf {x}, \mathbf {y}; \bar {\mathcal {B}} _ {t}) - \hat {\nabla} _ {\mathbf {x}} g _ {t} (\acute {\mathbf {x}}, \mathbf {y}; \bar {\mathcal {B}} _ {t}) \| ^ {2} \right] \leq 3 d _ {1} \ell_ {g, 1} ^ {2} \| \mathbf {x} - \acute {\mathbf {x}} \| ^ {2} + \frac {3 \ell_ {g , 1} ^ {2} d _ {1} ^ {2} \rho_ {\mathbf {s}} ^ {2}}{2}, \tag {133b} +$$ + +for all $\mathbf{y} \in \mathbb{R}^{d_2}$ and $\dot{\mathbf{x}} \in \mathbb{R}^{d_1}$ . + +Proof. The proof is similar to that of Lemma 5 in [42]. + +Lemma D.6. Suppose Assumptions 2.2 and B3. hold. Let $(\rho_{\mathbf{s}},\rho_{\mathbf{r}})$ be positive smoothing parameters. Let $\mathbf{y}_t^* (\mathbf{x})$ and $\hat{\mathbf{y}}_t^* (\mathbf{x})$ be defined in (1) and (19), respectively. Then, we have + +$$ +\mathbb {E} \left[ \left\| \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x}) \right\| ^ {2} \right] \leq \frac {\ell_ {g , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{\mu_ {g}}. \tag {134} +$$ + +Proof. From (1), we have $\mathbf{y}_t^*(\mathbf{x}) \in \arg \min_{\mathbf{y} \in \mathbb{R}^{d_2}} g_t(\mathbf{x}, \mathbf{y})$ . Since, by Assumption 2.2, $g_t(\mathbf{x}, \mathbf{y})$ is $\mu_g$ -strongly convex with respect to $\mathbf{y}$ , it follows from Lemma B.2 that + +$$ +\left\| \mathbf {y} - \mathbf {y} _ {t} ^ {*} (\mathbf {x}) \right\| ^ {2} \leq \frac {2}{\mu_ {g}} \left(g _ {t} \left(\mathbf {x}, \mathbf {y}\right) - g _ {t} \left(\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})\right)\right). +$$ + +By setting $\mathbf{y} = \hat{\mathbf{y}}_t^* (\mathbf{x})$ , we have + +$$ +\left\| \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x}) \right\| ^ {2} \leq \frac {2}{\mu_ {g}} \left(g _ {t} \left(\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})\right) - g _ {t} \left(\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})\right)\right). \tag {135} +$$ + +Similarly, from (19), we have + +$$ +\hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \in \arg \min _ {\mathbf {y} \in \mathbb {R} ^ {d _ {2}}} \left\{g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}) = \underset {(\mathbf {s}, \mathbf {r}, \zeta_ {t})} {\mathbb {E}} \left[ g _ {t} \left(\mathbf {x} + \rho_ {\mathbf {s}} \mathbf {s}, \mathbf {y} + \rho_ {\mathbf {r}} \mathbf {r}; \zeta_ {t}\right) \right] \right\}, +$$ + +where $\pmb {\rho} = (\rho_{\mathbf{s}},\rho_{\mathbf{r}})$ . By Assumption 2.2, $g_{t,\pmb{\rho}}(\mathbf{x},\mathbf{y})$ is $\mu_g$ -strongly convex with respect to $\mathbf{y}$ . Hence, according to Lemma B.2, we obtain + +$$ +\left\| \mathbf {y} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \right\| ^ {2} \leq \frac {2}{\mu_ {g}} \left(g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}) - g _ {t, \boldsymbol {\rho}} \left(\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})\right)\right). +$$ + +By setting $\mathbf{y} = \mathbf{y}_t^* (\mathbf{x})$ , we have + +$$ +\left\| \mathbf {y} _ {t} ^ {*} (\mathbf {x}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \right\| ^ {2} \leq \frac {2}{\mu_ {g}} \left(g _ {t, \rho} \left(\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})\right) - g _ {t, \rho} \left(\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})\right)\right). \tag {136} +$$ + +Summing up (135) and (136), we get + +$$ +\begin{array}{l} \left\| \mathbf {y} _ {t} ^ {*} (\mathbf {x}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \right\| ^ {2} \leq \frac {1}{\mu_ {g}} \left(g _ {t, \rho} (\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})) - g _ {t} \left(\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})\right)\right) \\ + \frac {1}{\mu_ {g}} \left(g _ {t} \left(\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})\right) - g _ {t, \boldsymbol {\rho}} \bigl (\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \bigr)\right), \\ \end{array} +$$ + +which implies + +$$ +\begin{array}{l} \left\| \mathbf {y} _ {t} ^ {*} (\mathbf {x}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x}) \right\| ^ {2} \leq \frac {1}{\mu_ {g}} \left| g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})) - g _ {t} (\mathbf {x}, \mathbf {y} _ {t} ^ {*} (\mathbf {x})) \right| \\ + \frac {1}{\mu_ {g}} \left| g _ {t} \left(\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})\right) - g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x})) \right| \\ \leq \frac {\ell_ {g , 1} (\rho_ {\bf s} ^ {2} + \rho_ {\bf r} ^ {2})}{\mu_ {g}}, \\ \end{array} +$$ + +where the last inequality is by Eq. (125). + +![](images/306da885bcf459315ffdbce3886dfc71e2a83e79ff6001867337cd7dd61b990f.jpg) + +# D.3 Bounds on the Zeroth-Order Inner Solution + +Recall that $\mathbf{s} \in \mathbb{R}^{d_1}$ and $\mathbf{r} \in \mathbb{R}^{d_2}$ are vectors uniformly sampled from the unit balls $B_1$ and $B_2$ , respectively. Let + +$$ +U _ {b} ^ {\mathbf {s}} = \left\{\mathbf {s} _ {i} \in \mathbb {R} ^ {d _ {1}} \right\} _ {i = 1} ^ {b}, \quad U _ {b} ^ {\mathbf {r}} = \left\{\mathbf {r} _ {i} \in \mathbb {R} ^ {d _ {2}} \right\} _ {i = 1} ^ {b}, +$$ + +$$ +U _ {\bar {b}} ^ {\mathbf {s}} = \left\{\mathbf {s} _ {i} \in \mathbb {R} ^ {d _ {1}} \right\} _ {i = 1} ^ {\bar {b}}, \quad U _ {\bar {b}} ^ {\mathbf {r}} = \left\{\mathbf {r} _ {i} \in \mathbb {R} ^ {d _ {2}} \right\} _ {i = 1} ^ {\bar {b}}, +$$ + +be generated from the uniform distributions over the unit spheres $(U_{B_1}, U_{B_2})$ . Here, $(U_{B_1}, U_{B_2})$ denote the uniform distributions over the $(d_1, d_2)$ -dimensional unit Euclidean balls $(B_1, B_2)$ , respectively. + +Then, similar to (23), we have + +$$ +\begin{array}{l} \underset {(U _ {b} ^ {\mathbf {r}}, \mathcal {B} _ {t})} {\mathbb {E}} \left[ \hat {\nabla} _ {\mathbf {y}} f _ {t} (\mathbf {x}, \mathbf {y}; \mathcal {B} _ {t}) \right] = \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}), \quad \underset {(U _ {b} ^ {\mathbf {s}}, \mathcal {B} _ {t})} {\mathbb {E}} \left[ \hat {\nabla} _ {\mathbf {x}} f _ {t} (\mathbf {x}, \mathbf {y}; \mathcal {B} _ {t}) \right] = \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}), \\ \underset {\left(U _ {b} ^ {r}, \bar {\mathcal {B}} _ {t}\right)} {\mathbb {E}} \left[ \hat {\nabla} _ {\mathbf {y}} g _ {t} (\mathbf {x}, \mathbf {y}; \bar {\mathcal {B}} _ {t}) \right] = \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}), \quad \underset {\left(U _ {b} ^ {s}, \bar {\mathcal {B}} _ {t}\right)} {\mathbb {E}} \left[ \hat {\nabla} _ {\mathbf {x}} g _ {t} (\mathbf {x}, \mathbf {y}; \bar {\mathcal {B}} _ {t}) \right] = \nabla_ {\mathbf {x}} g _ {t, \boldsymbol {\rho}} (\mathbf {x}, \mathbf {y}). \tag {137} \\ \end{array} +$$ + +Lemma D.7. Suppose that Assumptions B3. and D1. hold. Consider the sequence $\{(\mathbf{x}_t,\mathbf{y}_t,\mathbf{v}_t)\}_{t = 1}^T$ generated by Algorithm 2, and define + +$$ +e _ {t} ^ {g _ {\rho}} := \nabla_ {\mathbf {y}} g _ {t, \rho} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}}. \tag {138} +$$ + +Then, we have + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {g _ {\rho}} \| ^ {2} \leq (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| e _ {t} ^ {g _ {\rho}} \| ^ {2} + 1 2 (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t - 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \\ + 9 d _ {2} ^ {2} \ell_ {g, 1} ^ {2} (1 - \gamma_ {t + 1}) ^ {2} \rho_ {\mathbf {r}} ^ {2} + 2 4 d _ {2} \ell_ {g, 1} ^ {2} (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} \\ + 2 4 d _ {2} \ell_ {g, 1} ^ {2} \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} + 2 \frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b}} \gamma_ {t + 1} ^ {2}. \tag {139} \\ \end{array} +$$ + +Proof. From the definition of $\tilde{\mathbf{d}}_{t + 1}^{\mathbf{y}}$ in Algorithm 2, we have + +$$ +\begin{array}{l} \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {y}} - \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}} = - \gamma_ {t + 1} \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}} + \gamma_ {t + 1} \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) \\ + \left(1 - \gamma_ {t + 1}\right) \left(\hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}; \bar {\mathcal {B}} _ {t + 1}\right)\right). \\ \end{array} +$$ + +Then, we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {y}} \| ^ {2} \\ = \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}} - (\hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {y}} - \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}}) \| ^ {2} \\ = \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}} + \gamma_ {t + 1} \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}} - \gamma_ {t + 1} \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) \\ \left. - \left(1 - \gamma_ {t + 1}\right) \left(\hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}; \bar {\mathcal {B}} _ {t + 1}\right)\right) \| ^ {2} \right. \\ = \mathbb {E} \| (1 - \gamma_ {t + 1}) (\nabla_ {\mathbf {y}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}}) \\ + \gamma_ {t + 1} \left(\nabla_ {\mathbf {y}} g _ {t + 1, \rho} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \vec {\mathcal {B}} _ {t + 1}\right)\right) \\ + \left(1 - \gamma_ {t + 1}\right) \left(\nabla_ {\mathbf {y}} g _ {t + 1, \rho} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) - \nabla_ {\mathbf {y}} g _ {t, \rho} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right. \\ + \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \\ \left. - \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) + \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}; \bar {\mathcal {B}} _ {t + 1}\right)\right) \| ^ {2}. \\ \end{array} +$$ + +From (137), we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) \right] = \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right), \\ \mathbb {E} \left[ \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}; \bar {\mathcal {B}} _ {t + 1}\right) \right] \\ = \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) - \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right), \\ \end{array} +$$ + +then, we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1, \pmb {\rho}} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {y}} \| ^ {2} \\ = (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}} \| ^ {2} \\ + \mathbb {E} \| \gamma_ {t + 1} \left(\nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \vec {\mathcal {B}} _ {t + 1}\right)\right) \\ + \left(1 - \gamma_ {t + 1}\right) \left(\nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) - \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) + \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right. \\ \left. - \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) + \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}; \bar {\mathcal {B}} _ {t + 1}\right)\right) \| ^ {2} \\ \leq (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}} \| ^ {2} \\ + 2 (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) + \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \\ - \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) + \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2} \\ + 2 \gamma_ {t + 1} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1, \rho} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \bar {\boldsymbol {B}} _ {t + 1}) \| ^ {2}, \\ \end{array} +$$ + +where the second inequality holds by Cauchy-Schwarz inequality. + +Then, from $\mathbb{E}\|a - \mathbb{E}[a]\|^2 = \mathbb{E}\|a\|^2 - \|\mathbb{E}[a]\|^2$ and Assumption D1., we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1, \rho} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {y}} \| ^ {2} \\ \leq \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}} \| ^ {2} \\ + 4 \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \| ^ {2} \\ + 4 (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2} + 2 \gamma_ {t + 1} ^ {2} \frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b}} \\ \leq (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}} \| ^ {2} \\ + 4 \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \| ^ {2} \\ + 1 2 (1 - \gamma_ {t + 1}) ^ {2} d _ {2} \ell_ {g, 1} ^ {2} \mathbb {E} \| (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \\ + 3 (1 - \gamma_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {2} d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2} + 2 \gamma_ {t + 1} ^ {2} \frac {\hat {\sigma} _ {g y} ^ {2}}{\bar {b}}, \\ \end{array} +$$ + +where the second inequality follows from Young's inequality and Lemma D.5. + +From Eq. (130), we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \\ \leq 3 \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \\ + 3 \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \| ^ {2} \\ + 3 \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \\ \leq 3 \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} + \frac {3 \rho_ {\mathbf {r}} ^ {2} d _ {2} ^ {2} \ell_ {g , 1} ^ {2}}{2}. \\ \end{array} +$$ + +Finally, we get + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {y}} \| ^ {2} \leq (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}} \| ^ {2} \\ + 1 2 \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \| ^ {2} + 6 \left(1 - \gamma_ {t + 1}\right) ^ {2} \rho_ {\mathbf {r}} ^ {2} d _ {2} ^ {2} \ell_ {g, 1} ^ {2} \\ + 1 2 (1 - \gamma_ {t + 1}) ^ {2} d _ {2} \ell_ {g, 1} ^ {2} \mathbb {E} \| (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) - (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} + 3 (1 - \gamma_ {t + 1}) ^ {2} \ell_ {g, 1} ^ {2} d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2} + 2 \gamma_ {t + 1} ^ {2} \frac {\hat {\sigma} _ {g \mathbf {y}} ^ {2}}{\bar {b}}. \\ \end{array} +$$ + +![](images/ac21dbe9b93ff11047d1296612bcc152054cc47997234aeb6cce569491fa6019.jpg) + +Lemma D.8. Suppose Assumptions 2.2 and B3. hold. Then, for the sequence $\{(\mathbf{x}_t,\mathbf{y}_t)\}_{t = 1}^T$ generated by Algorithm 2, we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \leq (1 + a) \left(1 - 2 \beta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \mathbb {E} \left[ \| \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ + \left(- (1 + a) \left(\frac {2 \beta_ {t}}{\mu_ {g} + \ell_ {g , 1}} - \beta_ {t} ^ {2}\right)\right) \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \right] \\ + \left(1 + \frac {1}{a}\right) \beta_ {t} ^ {2} \mathbb {E} \left[ \| e _ {t} ^ {g _ {\rho}} \| ^ {2} \right], \\ \end{array} +$$ + +where $a > 0$ is a constant, $e_t^{g_P}$ is defined in (138), and $\hat{\mathbf{y}}_t^* (\mathbf{x}_t)$ is defined in (19). + +Proof. From Lemma B.4, we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] = \mathbb {E} \left[ \| \mathbf {y} _ {t} - \beta_ {t} \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ \leq (1 + a) \mathbb {E} \left[ \| \mathbf {y} _ {t} - \beta_ {t} \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ + \left(1 + \frac {1}{a}\right) \beta_ {t} ^ {2} \mathbb {E} \left[ \| \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}} - \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \| ^ {2} \right]. \tag {140} \\ \end{array} +$$ + +Next, we will separately bound the first term on the RHS of the above inequality. + +We have + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {y} _ {t} - \beta_ {t} \nabla_ {\mathbf {y}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] = \mathbb {E} \left[ \| \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] + \beta_ {t} ^ {2} \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \right] \\ - 2 \beta_ {t} \mathbb {E} \left[ \left\langle \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right), \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right) \right\rangle \right] \\ \leq \left(1 - 2 \beta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \mathbb {E} \left[ \| \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ \left. - \left(\frac {2 \beta_ {t}}{\mu_ {g} + \ell_ {g , 1}} - \beta_ {t} ^ {2}\right) \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \| ^ {2} \right], \right. \tag {141} \\ \end{array} +$$ + +where the inequality results from the strong convexity of $g_{t,\rho}$ by Assumption 2.2, which implies + +$$ +\langle \nabla_ {\mathbf {y}} g _ {t, \pmb {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}), \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \rangle \geq \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}} \| \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} + \frac {1}{\mu_ {g} + \ell_ {g , 1}} \| \nabla_ {\mathbf {y}} g _ {t, \pmb {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}. +$$ + +Substituting (141) into (140), gives the desired result. + +![](images/7f596f18db99f77b9bba00f16980dd58f182635cb78fcf1ffaffa88a61067231.jpg) + +For notational brevity in the analysis, we define + +$$ +\hat {\theta} _ {t} ^ {\mathbf {y}} := \left\| \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2}, \quad \hat {\theta} _ {t} ^ {\mathbf {v}} := \left\| \mathbf {v} _ {t} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2}, \tag {142} +$$ + +where $\hat{\mathbf{y}}_t^* (\mathbf{x})$ and $\hat{\mathbf{v}}_t^* (\mathbf{x})$ are defined in (19) and (20), respectively. + +Lemma D.9. Suppose Assumptions 2.2 and B3. hold. Let $\hat{\theta}_t^{\mathbf{y}}$ be defined in (142). Then, for the sequence $\{(\mathbf{x}_t,\mathbf{y}_t)\}_{t = 1}^T$ generated by Algorithm 2 guarantees the following bound: + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \left(\mathbb {E} \left[ \hat {\theta} _ {t + 1} ^ {\mathbf {y}} \right] - \mathbb {E} \left[ \hat {\theta} _ {t} ^ {\mathbf {y}} \right]\right) \\ \leq \left(- \frac {L _ {\mu_ {g}}}{2} \sum_ {t = 1} ^ {T} \mathbb {E} [ \hat {\theta} _ {t} ^ {\mathbf {y}} ] + \frac {2}{L _ {\mu_ {g}}} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \| e _ {t} ^ {g _ {\boldsymbol {\rho}}} \| ^ {2} \right]\right) \beta_ {t} + \frac {4 L _ {\mathbf {y}} ^ {2}}{L _ {\mu_ {g}}} \sum_ {t = 1} ^ {T} \mathbb {E} \| \mathbf {x} _ {t} - \mathbf {x} _ {t + 1} \| ^ {2} \frac {1}{\beta_ {t}} \\ + \sum_ {t = 1} ^ {T} \left(\frac {2 4 \ell_ {g , 1}}{L _ {\mu_ {g}} \mu_ {g}} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right) + \frac {1 2}{L _ {\mu_ {g}}} \sup _ {\mathbf {x} \in \mathcal {X}} \| \mathbf {y} _ {t - 1} ^ {*} (\mathbf {x}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x}) \| ^ {2}\right) \frac {1}{\beta_ {t}} \\ + \sum_ {t = 1} ^ {T} \left(- \frac {2 \beta_ {t}}{\mu_ {g} + \ell_ {g , 1}} + \beta_ {t} ^ {2}\right) \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \right], \tag {143} \\ \end{array} +$$ + +where $L_{\mathbf{y}} = \frac{\ell_{g,1}}{\mu_g}$ is defined as in (44) and $L_{\mu_g} = \frac{\mu_g\ell_{g,1}}{\mu_g + \ell_{g,1}}$ + +Proof. From Lemma B.4, we have for any $c > 0$ + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} - \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \| ^ {2} \right] = \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) + \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) - \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \| ^ {2} \right] \\ \leq (1 + c) \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ + \left(1 + \frac {1}{c}\right) \mathbb {E} \left[ \| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right]. \tag {144} \\ \end{array} +$$ + +From Lemma D.8, we have for any $a > 0$ + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \leq (1 + a) \left(1 - 2 \beta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \mathbb {E} \left[ \| \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ + \left(- (1 + a) \left(\frac {2 \beta_ {t}}{\mu_ {g} + \ell_ {g , 1}} - \beta_ {t} ^ {2}\right)\right) \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \right] \\ + \left(1 + \frac {1}{a}\right) \beta_ {t} ^ {2} \mathbb {E} \left[ \| e _ {t} ^ {g _ {\mathcal {P}}} \| ^ {2} \right]. \tag {145} \\ \end{array} +$$ + +Substituting (145) into (144), we get + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \mathbf {y} _ {t + 1} - \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \right] \\ \leq (1 + c) (1 + a) \left(1 - 2 \beta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \mathbb {E} \left[ \| \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ + \left(- (1 + c) (1 + a) \left(\frac {2 \beta_ {t}}{\mu_ {g} + \ell_ {g , 1}} - \beta_ {t} ^ {2}\right)\right) \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \right] \\ + (1 + c) \left(1 + \frac {1}{a}\right) \beta_ {t} ^ {2} \mathbb {E} \left[ \| e _ {t} ^ {g _ {\rho}} \| ^ {2} \right] + \left(1 + \frac {1}{c}\right) \mathbb {E} \left[ \| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right]. \tag {146} \\ \end{array} +$$ + +Choose $c = \frac{\beta_t L_{\mu_g} / 2}{1 - \beta_t L_{\mu_g}}$ and $a = \frac{\beta_t L_{\mu_g}}{1 - 2\beta_t L_{\mu_g}}$ . Then, the following equations and inequalities are satisfied. + +$$ +(1 + c) (1 + a) \left(1 - 2 \beta_ {t} L _ {\mu_ {g}}\right) = 1 - \frac {\beta_ {t} L _ {\mu_ {g}}}{2}, +$$ + +$$ +(1 + a) \left(1 - 2 \beta_ {t} L _ {\mu_ {g}}\right) = 1 - \beta_ {t} L _ {\mu_ {g}}, +$$ + +$$ +(1 + c) \left(1 - \beta_ {t} L _ {\mu_ {g}}\right) = 1 - \frac {\beta_ {t} L _ {\mu_ {g}}}{2}, \tag {147} +$$ + +$$ +1 + \frac {1}{a} \leq \frac {1}{\beta_ {t} L _ {\mu_ {g}}}, \quad 1 + \frac {1}{c} \leq \frac {2}{\beta_ {t} L _ {\mu_ {g}}}, +$$ + +where $L_{\mu_g} = \frac{\mu_g\ell_{g,1}}{\mu_g + \ell_{g,1}}$ . Based on (146) and (147), we get + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} - \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \| ^ {2} \right] - \mathbb {E} \left[ \| \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ \leq - \frac {\beta_ {t} L _ {\mu_ {g}}}{2} \mathbb {E} \left[ \| \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] + \left(- \left(\frac {2 \beta_ {t}}{\mu_ {g} + \ell_ {g , 1}} - \beta_ {t} ^ {2}\right)\right) \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \right] \\ + \frac {2}{\beta_ {t} L _ {\mu_ {g}}} \beta_ {t} ^ {2} \mathbb {E} \left[ \| e _ {t} ^ {g _ {P}} \| ^ {2} \right] + \frac {2}{\beta_ {t} L _ {\mu_ {g}}} \mathbb {E} \left[ \| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right]. \tag {148} \\ \end{array} +$$ + +Next, we upper-bound the last term of the above inequality. + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ \leq 2 \left(\mathbb {E} \left[ \| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] + \mathbb {E} \left[ \| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right]\right) \\ \leq 2 \left(L _ {\mathbf {y}} ^ {2} \mathbb {E} \left[ \| \mathbf {x} _ {t} - \mathbf {x} _ {t + 1} \| ^ {2} + \| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right]\right), \tag {149} \\ \end{array} +$$ + +where the second inequality is by Lemma D.2. + +Moreover, from Lemma D.6, we get + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \right] \leq 3 \mathbb {E} \left[ \left\| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \right] \\ + 3 \mathbb {E} \left[ \left\| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \right] + 3 \mathbb {E} \left[ \left\| \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \right] \\ \leq 3 \mathbb {E} \left[ \left\| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \right] + \frac {6 \ell_ {g , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{\mu_ {g}}. \tag {150} \\ \end{array} +$$ + +Combining (149) and (150) yields + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ \leq 2 \left(L _ {\mathbf {y}} ^ {2} \mathbb {E} \left[ \| \mathbf {x} _ {t} - \mathbf {x} _ {t + 1} \| ^ {2} \right] + 3 \mathbb {E} \left[ \| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] + \frac {6 \ell_ {g , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{\mu_ {g}}\right). \tag {151} \\ \end{array} +$$ + +Substituting (151) into (148) and summing over $t \in [T]$ , give the desired result. + +![](images/06c4de15cc360df172d8952221b4fc2aba5f7a6b876f288137b1a51522bc8356.jpg) + +# D.4 Bounds on the Zeroth-Order System Solution + +Lemma D.10. Suppose Assumptions B2. and B3. hold. Let + +$$ +\vartheta := \mathbb {E} \| \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) + \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2}, +$$ + +where $\hat{\nabla}_{\mathbf{y}}f_{t}$ and $\hat{\nabla}_{\mathbf{y}}^2 g_t$ are defined in (25a) and (26a), respectively. Then, for the sequence $\left\{\left(\mathbf{x}_t,\mathbf{y}_t,\mathbf{v}_t\right)\right\}_{t = 1}^T$ generated by Algorithm 2, we have + +$$ +\begin{array}{l} \vartheta \leq (1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}) d _ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + (1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}) d _ {2} \mathbb {E} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} \\ + \frac {9}{2} d _ {2} \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} + (3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}) d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2}. \\ \end{array} +$$ + +Proof. From Lemma D.5, we have + +$$ +\begin{array}{l} \left\| \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}\right) \right\| ^ {2} \\ \leq 3 d _ {2} \ell_ {f, 1} ^ {2} \| \mathbf {z} _ {t + 1} - \mathbf {z} _ {t} \| ^ {2} + \frac {3}{2} \ell_ {f, 1} ^ {2} d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2} \\ \leq 6 d _ {2} \ell_ {f, 1} ^ {2} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 6 d _ {2} \ell_ {f, 1} ^ {2} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} + \frac {3}{2} \ell_ {f, 1} ^ {2} d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2}. \tag {152} \\ \end{array} +$$ + +Moreover, from (26a), we have + +$$ +\begin{array}{l} \| \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2} \\ = \frac {1}{4 \rho_ {\mathbf {v}} ^ {2}} \| \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} + \rho_ {\mathbf {v}} \mathbf {v} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2} \\ \leq \frac {3}{4 \rho_ {\mathbf {v}} ^ {2}} d _ {2} \ell_ {g, 1} ^ {2} \| (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} + \rho_ {\mathbf {v}} \mathbf {v} _ {t + 1}) - (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} + \frac {3}{8 \rho_ {\mathbf {v}} ^ {2}} \ell_ {g, 1} ^ {2} d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2} \\ \leq \frac {9}{4 \rho_ {\mathbf {v}} ^ {2}} d _ {2} \ell_ {g, 1} ^ {2} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + \frac {9}{4 \rho_ {\mathbf {v}} ^ {2}} d _ {2} \ell_ {g, 1} ^ {2} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} \\ + \frac {9}{4} d _ {2} \ell_ {g, 1} ^ {2} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} + \frac {3}{8 \rho_ {\mathbf {v}} ^ {2}} \ell_ {g, 1} ^ {2} d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2}, \tag {153} \\ \end{array} +$$ + +where the first inequality follows from Lemma D.5. + +From $\| a + b\| ^2\leq 2\left(\| a\| ^2 +\| b\| ^2\right)$ , we get + +$$ +\begin{array}{l} \vartheta \leq 2 \mathbb {E} \| \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2} \\ + 2 \mathbb {E} \| \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}) \| ^ {2} \\ \leq \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {2} \mathbb {E} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} \\ + \frac {9}{2} d _ {2} \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} + (3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}) d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2}, \\ \end{array} +$$ + +where the second inequality follows from (152) and (153). + +![](images/8b845685b31f673305c0796f7fbb76b1499c565ea011be88fcaf8e1849402b89.jpg) + +Lemma D.11. Suppose Assumptions B2., B3., D1., and D3. hold. Consider the sequence $\{(\mathbf{x}_t,\mathbf{y}_t,\mathbf{v}_t)\}_{t = 1}^T$ generated by Algorithm 2, and define + +$$ +\begin{array}{l} e _ {t + 1} ^ {M} := \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}\right) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {v}}, \quad \text {w h e r e} (154) \\ \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1}) = \frac {1}{2 \rho_ {\mathbf {v}}} \bigl (\nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} + \rho_ {\mathbf {v}} \mathbf {v} _ {t + 1}) \\ - \nabla_ {\mathbf {y}} g _ {t + 1, \rho} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} - \rho_ {\mathbf {v}} \mathbf {v} _ {t + 1}\right). (155) \\ \end{array} +$$ + +Then, we have + +$$ +\begin{array}{l} \mathbb {E} \left\| e _ {t + 1} ^ {M} \right\| ^ {2} \leq (1 - \lambda_ {t + 1}) ^ {2} \mathbb {E} \left\| e _ {t} ^ {M} \right\| ^ {2} + 3 6 \mathbb {E} \left\| \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2} \\ + \left(1 8 d _ {2} ^ {2} \ell_ {f, 1} ^ {2} + 6 (3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}) d _ {2} ^ {2}\right) \rho_ {\mathbf {r}} ^ {2} + 1 8 d _ {2} ^ {2} \ell_ {g, 1} ^ {2} \frac {\rho_ {\mathbf {r}} ^ {2}}{\rho_ {\mathbf {v}} ^ {2}} \\ + \frac {1 8}{\rho_ {\mathbf {v}} ^ {2}} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} \\ + \frac {1 8}{\rho_ {\mathbf {v}} ^ {2}} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} \\ + 6 \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 6 \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {2} \mathbb {E} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} \\ + 2 7 d _ {2} \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} + 3 \left(\frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {y}}} ^ {2}}{b}\right) \lambda_ {t + 1} ^ {2}. \tag {156} \\ \end{array} +$$ + +Proof. According to the definition of $\hat{\mathbf{d}}_t^{\mathrm{V}}$ in Algorithm 2, we have + +$$ +\begin{array}{l} \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {v}} - \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} = - \lambda_ {t + 1} \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} + \lambda_ {t + 1} \left(\hat {\nabla} _ {\mathbf {y}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) + \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right)\right) \\ + \left(1 - \lambda_ {t + 1}\right) \left(\hat {\nabla} _ {\mathbf {y}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) + \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) \right. \\ \left. - \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1}\right)\right). \\ \end{array} +$$ + +Then we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \bar {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {v}} \| ^ {2} \\ = \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} - (\hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {v}} - \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}}) \| ^ {2} \\ = \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} + \lambda_ {t + 1} \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} \\ - \lambda_ {t + 1} \left(\hat {\nabla} _ {\mathbf {y}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) + \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right)\right) \\ - \left(1 - \lambda_ {t + 1}\right) \left(\hat {\nabla} _ {\mathbf {y}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) + \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) \right. \\ \left. - \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1})\right) \| ^ {2} \\ = \mathbb {E} \| (1 - \lambda_ {t + 1}) (\nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}}) \\ + \lambda_ {t + 1} \left(\nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {z} _ {t + 1}\right) + \bar {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right)\right) \\ + \left(1 - \lambda_ {t + 1}\right) \left(\nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) \right. \\ + \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) - \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) \\ \left. - \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) + \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}) + \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1})\right) \| ^ {2}. \\ \end{array} +$$ + +Since + +$$ +\begin{array}{l} \mathbb {E} \left[ \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) + \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) \right] = \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}), \\ \mathbb {E} \left[ \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) + \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1}\right) \right] \\ = \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}), \\ \end{array} +$$ + +then, we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {v}} \| ^ {2} \\ = (1 - \lambda_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} \| ^ {2} \\ + \left\| \lambda_ {t + 1} \left(\nabla_ {\mathbf {y}} f _ {t + 1, \rho} (\mathbf {z} _ {t + 1}) + \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1})\right) \right. \\ + \left(1 - \lambda_ {t + 1}\right) \left(\nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) \right. \\ + \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) - \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) \\ \left. - \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) + \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}) + \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1})\right) \| ^ {2} \\ \leq \left(1 - \lambda_ {t + 1}\right) ^ {2} \mathbb {E} \left\| \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} \left(\mathbf {z} _ {t}\right) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {z} _ {t}\right) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} \right\| ^ {2} \\ + 3 (1 - \lambda_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t + 1, \rho} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \nabla_ {\mathbf {y}} f _ {t, \rho} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) \\ + \nabla_ {\mathbf {y}} f _ {t + 1, \rho} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) - \nabla_ {\mathbf {y}} f _ {t + 1, \rho} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) \\ - \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) + \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}) + \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2} \\ + 3 \lambda_ {t + 1} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) \| ^ {2} \\ + 3 \lambda_ {t + 1} ^ {2} \mathbb {E} \| \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2}, \tag {157} \\ \end{array} +$$ + +where the second inequality holds by Cauchy-Schwarz inequality. + +Note that, for the last term on the right-hand side of (157), from (26a) and (155), we have + +$$ +\begin{array}{l} \left\| \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \vec {\mathcal {B}} _ {t + 1}) \right\| ^ {2} \\ \leq 2 \| \frac {1}{2 \rho_ {\mathbf {v}}} \left(\nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} + \rho_ {\mathbf {v}} \mathbf {v} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} + \rho_ {\mathbf {v}} \mathbf {v} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right)\right) \| ^ {2} \\ + 2 \| \frac {1}{2 \rho_ {\mathbf {v}}} \left(\hat {\nabla} _ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} - \rho_ {\mathbf {v}} \mathbf {v} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) - \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} - \rho_ {\mathbf {v}} \mathbf {v} _ {t + 1}\right)\right) \| ^ {2} \\ \leq \frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}}, \\ \end{array} +$$ + +where the last inequality follows from Assumption D1.. + +Then, from $\mathbb{E}\|a - \mathbb{E}[a]\|^2 = \mathbb{E}\|a\|^2 - \|\mathbb{E}[a]\|^2$ and Assumptions D1. and D3., we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {v}} \| ^ {2} \\ \leq \left(1 - \lambda_ {t + 1}\right) ^ {2} \mathbb {E} \left\| \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} \left(\mathbf {z} _ {t}\right) + \bar {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {z} _ {t}\right) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} \right\| ^ {2} \\ + 6 \left(1 - \lambda_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) - \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) \| ^ {2} \\ + 6 (1 - \lambda_ {t + 1}) ^ {2} \mathbb {E} \| \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) + \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) \\ - \hat {\nabla} _ {\mathbf {y}} f _ {t + 1} (\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2} + 3 \lambda_ {t + 1} ^ {2} \left(\frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {y}}} ^ {2}}{b}\right). \\ \end{array} +$$ + +Then, from Young's inequality and Lemma D.10, we obtain + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {v}} \| ^ {2} \\ \leq \left(1 - \lambda_ {t + 1}\right) ^ {2} \mathbb {E} \left\| \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} \left(\mathbf {z} _ {t}\right) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {z} _ {t}\right) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} \right\| ^ {2} \\ + 1 2 (1 - \lambda_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} (\mathbf {z} _ {t}) \| ^ {2} \\ + 1 2 (1 - \lambda_ {t + 1}) ^ {2} \mathbb {E} \| \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) \| ^ {2} \\ + 6 \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {2} \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2} + 6 \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {2} \left\| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \right\| ^ {2} \\ + 2 7 d _ {2} \ell_ {g, 1} ^ {2} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} + 6 \left(3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2} + 3 \lambda_ {t + 1} ^ {2} \left(\frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {y}}} ^ {2}}{b}\right). \tag {158} \\ \end{array} +$$ + +For the third term on the right-hand side of (158), based on (155), we have + +$$ +\begin{array}{l} \left\| \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2} \\ \leq \frac {1}{2 \rho_ {\mathbf {v}} ^ {2}} \left\| \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) \right\| ^ {2} (159a) \\ + \frac {1}{2 \rho_ {\mathbf {v}} ^ {2}} \| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) \| ^ {2}. (159b) \\ \end{array} +$$ + +For (159a), we get + +$$ +\begin{array}{l} \left\| \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) \right\| ^ {2} \\ \leq 3 \| \nabla_ {\mathbf {y}} g _ {t + 1, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} \\ + 3 \left\| \nabla_ {\mathbf {y}} g _ {t + 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) \right\| ^ {2} \\ + 3 \left\| \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t, \rho} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) \right\| ^ {2} \\ \leq 3 \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} + \frac {3 \rho_ {\mathbf {r}} ^ {2} d _ {2} ^ {2} \ell_ {g , 1} ^ {2}}{2}, \\ \end{array} +$$ + +where the last inequality follows from Eq. (130). + +Similarity, for (159b), we have + +$$ +\begin{array}{l} \left\| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) \right\| ^ {2} \\ \leq 3 \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} + \frac {3 \rho_ {\mathbf {r}} ^ {2} d _ {2} ^ {2} \ell_ {g , 1} ^ {2}}{2}. \\ \end{array} +$$ + +Substituting the above inequalities in (159), we have + +$$ +\begin{array}{l} \left\| \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2} \\ \leq \frac {3}{2 \rho_ {\mathbf {v}} ^ {2}} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} \\ + \frac {3}{2 \rho_ {\mathbf {v}} ^ {2}} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} + \frac {3 \rho_ {\mathbf {r}} ^ {2} d _ {2} ^ {2} \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}. \tag {160} \\ \end{array} +$$ + +For the second term on the right-hand side of (158), we have + +$$ +\begin{array}{l} \left\| \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2} \\ \leq 3 \| \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \\ + 3 \left\| \nabla_ {\mathbf {y}} f _ {t + 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right\| ^ {2} \\ + 3 \left\| \nabla_ {\mathbf {y}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right\| ^ {2} \\ \leq 3 \| \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} + \frac {3 \rho_ {\mathrm {r}} ^ {2} d _ {2} ^ {2} \ell_ {f , 1} ^ {2}}{2}, \tag {161} \\ \end{array} +$$ + +where the last inequality follows from Eq. (132). + +From (160), (161) and (158), we get + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {y}} \| ^ {2} \\ \leq (1 - \lambda_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {y}} f _ {t, \pmb {\rho}} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} \| ^ {2} \\ + 3 6 \left\| \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2} + 1 8 \rho_ {\mathbf {r}} ^ {2} d _ {2} ^ {2} \ell_ {f, 1} ^ {2} \\ + \frac {1 8}{\rho_ {\mathbf {v}} ^ {2}} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} \\ + \frac {1 8}{\rho_ {\mathbf {v}} ^ {2}} \| \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} + \frac {1 8 \rho_ {\mathbf {r}} ^ {2} d _ {2} ^ {2} \ell_ {g , 1} ^ {2}}{\rho_ {\mathbf {v}} ^ {2}} \\ + 6 (1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}) d _ {2} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 6 (1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}) d _ {2} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} \\ + 2 7 d _ {2} \ell_ {g, 1} ^ {2} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} + 6 (3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}) d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2} + 3 \lambda_ {t + 1} ^ {2} (\frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {y}}} ^ {2}}{b}). \\ \end{array} +$$ + +□ + +Lemma D.12. Suppose Assumption B4. holds. Let + +$$ +e _ {t} ^ {H} := \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} ^ {2} g _ {t, \rho} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \mathbf {v} _ {t}, \tag {162a} +$$ + +$$ +e _ {t} ^ {J} := \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t, \rho} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \mathbf {v} _ {t}, \tag {162b} +$$ + +where + +$$ +\tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) = \frac {1}{2 \rho_ {\mathbf {v}}} (\nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t})), +$$ + +$$ +\tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) = \frac {1}{2 \rho_ {\mathbf {v}}} (\nabla_ {\mathbf {x}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {x}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t})). +$$ + +Then, for $(\mathbf{x}_t,\mathbf{y}_t,\mathbf{v}_t)$ presented to Algorithm 2, we have + +(a) + +$$ +\mathbb {E} \left[ \left\| e _ {t} ^ {H} \right\| ^ {2} \right] \leq \ell_ {g, 2} ^ {2} \rho_ {\mathbf {v}} ^ {2} p ^ {4}. \tag {163a} +$$ + +(b) + +$$ +\mathbb {E} \left[ \left\| e _ {t} ^ {J} \right\| ^ {2} \right] \leq \ell_ {g, 2} ^ {2} \rho_ {\mathbf {v}} ^ {2} p ^ {4}. \tag {163b} +$$ + +Proof. For part (a): From Lemma D.1, We have + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| e _ {t} ^ {H} \right\| \right] = \mathbb {E} \left[ \left\| \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} ^ {2} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \mathbf {v} _ {t} \right\| \right] \\ \leq \frac {1}{2 \rho_ {\mathbf {v}}} \mathbb {E} \left[ \left\| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} ^ {2} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \rho_ {\mathbf {v}} \mathbf {v} _ {t} \right\| \right] \\ + \frac {1}{2 \rho_ {\mathbf {v}}} \mathbb {E} \left[ \left\| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {y}} ^ {2} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \rho_ {\mathbf {v}} \mathbf {v} _ {t} \right\| \right] \\ \leq \ell_ {g, 2} \rho_ {\mathbf {v}} \mathbb {E} \left[ \left\| \mathbf {v} _ {t} \right\| ^ {2} \right] \\ \leq \ell_ {g, 2} \rho_ {\mathbf {v}} p ^ {2}, \tag {164} \\ \end{array} +$$ + +where the last inequality follows from (8). + +For part (b): From Lemma D.1, We have + +$$ +\begin{array}{l} \mathbb {E} \left[ \| e _ {t} ^ {J} \| \right] = \mathbb {E} \left[ \left\| \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \mathbf {v} _ {t} \right\| \right] \\ \leq \frac {1}{2 \rho_ {\mathbf {v}}} \mathbb {E} \left[ \left\| \nabla_ {\mathbf {x}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {x}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \rho_ {\mathbf {v}} \mathbf {v} _ {t} \right\| \right] \\ + \frac {1}{2 \rho_ {\mathbf {v}}} \mathbb {E} \left[ \| \nabla_ {\mathbf {x}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {x}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {x y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \rho_ {\mathbf {v}} \mathbf {v} _ {t} \| \right] \\ \leq \ell_ {g, 2} \rho_ {\mathbf {v}} \mathbb {E} \left[ \left\| \mathbf {v} _ {t} \right\| ^ {2} \right] \\ \leq \ell_ {g, 2} \rho_ {\mathbf {v}} p ^ {2}, \tag {165} \\ \end{array} +$$ + +where the last inequality follows from (8). + +Lemma D.13. Suppose Assumption B4. holds. Then, for the directions $\hat{\mathbf{d}}_t^{\mathbf{v}}$ and $\hat{\mathbf{d}}_t^{\mathbf{x}}$ provided to Algorithm 2, and + +(a) for $\mathbf{d}_{t,\rho}^{\mathbf{v}}$ defined in (22b), we have + +$$ +\mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} - \mathbf {d} _ {t, \boldsymbol {\rho}} ^ {\mathbf {v}} \right\| ^ {2} \right] \leq 2 \mathbb {E} \left[ \left\| e _ {t} ^ {M} \right\| ^ {2} \right] + 2 \ell_ {g, 2} ^ {2} \rho_ {\mathbf {v}} ^ {2} p ^ {4} =: B _ {t}, \tag {166a} +$$ + +where $e_t^M = \nabla_{\mathbf{y}}f_{t,\boldsymbol {\rho}}(\mathbf{x}_t,\mathbf{y}_t) + \tilde{\nabla}_{\mathbf{y}}^2 g_t(\mathbf{x}_t,\mathbf{y}_t) - \hat{\mathbf{d}}_t^{\mathbf{v}}$ is defined as in (154). + +(b) and for $\mathbf{d}_{t,\rho}^{\mathbf{x}}$ defined in (22c), we have + +$$ +\mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} - \mathbf {d} _ {t, \boldsymbol {\rho}} ^ {\mathbf {x}} \right\| ^ {2} \right] \leq 2 \mathbb {E} \left[ \left\| e _ {t} ^ {L} \right\| ^ {2} \right] + 2 \ell_ {g, 2} ^ {2} \rho_ {\mathbf {v}} ^ {2} p ^ {4}, \tag {166b} +$$ + +where + +$$ +e _ {t} ^ {L} := \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}, \tag {166c} +$$ + +with $\tilde{\nabla}_{\mathbf{x}\mathbf{y}}^{2}g_{t}\left(\mathbf{x}_{t},\mathbf{y}_{t}\right)$ is defined in (172). + +Proof. For part (a): Let + +$$ +\tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) = \frac {1}{2 \rho_ {\mathbf {v}}} \left(\nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right)\right). \tag {167} +$$ + +According to the definition of $\mathbf{d}_{t,\rho}^{\nu}$ in (22b), we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} - \mathbf {d} _ {t, \boldsymbol {\rho}} ^ {\mathbf {v}} \right\| ^ {2} \right] = \mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} - \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} ^ {2} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \mathbf {v} \right\| ^ {2} \right] \\ \leq 2 \mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} - \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right\| ^ {2} \right] (168a) \\ + 2 \mathbb {E} \left[ \left\| \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} ^ {2} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \mathbf {v} \right\| ^ {2} \right]. (168b) \\ \end{array} +$$ + +Next, we separately bound (168a) and (168b) on the RHS of the above inequality. + +Bounding (168a). We have + +$$ +2 \mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} - \nabla_ {\mathbf {y}} f _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2} \right] := 2 \mathbb {E} \left[ \left\| e _ {t} ^ {M} \right\| ^ {2} \right]. \tag {169} +$$ + +Bounding (168b). From Lemmas D.1 and D.12, we have + +$$ +(1 6 8 \mathrm {b}) = \mathbb {E} \left[ \left\| e _ {t} ^ {H} \right\| ^ {2} \right] \leq 3 \ell_ {g, 2} ^ {2} \rho_ {\mathbf {v}} ^ {2} p ^ {4}. \tag {170} +$$ + +Combining (169) and (170) yields + +$$ +\mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} - \mathbf {d} _ {t, \boldsymbol {\rho}} ^ {\mathbf {v}} \right\| ^ {2} \right] \leq 2 \mathbb {E} \left[ \left\| e _ {t} ^ {M} \right\| ^ {2} \right] + 2 \ell_ {g, 2} ^ {2} \rho_ {\mathbf {v}} ^ {2} p ^ {4}. \tag {171} +$$ + +For part (b): Let + +$$ +\tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) = \frac {1}{2 \rho_ {\mathbf {v}}} \left(\nabla_ {\mathbf {x}} g _ {t, \rho} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {x}} g _ {t, \rho} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right)\right). \tag {172} +$$ + +According to the definition of $\mathbf{d}_{t,\rho}^{\times}$ in (22c), we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} - \mathbf {d} _ {t, \rho} ^ {\mathbf {x}} \right\| ^ {2} \right] = \mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} - \nabla_ {\mathbf {x}} f _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \mathbf {v} \right\| ^ {2} \right] \\ \leq 2 \mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} - \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right\| ^ {2} \right] (173a) \\ + 2 \mathbb {E} \left[ \left\| \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {x y}} ^ {2} g _ {t, \rho} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \mathbf {v} _ {t} \right\| ^ {2} \right]. (173b) \\ \end{array} +$$ + +Next, we separately bound (173a) and (173b) on the RHS of the above inequality. + +Bounding (173a). We have + +$$ +2 \mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} - \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right\| ^ {2} \right] := 2 \mathbb {E} \left[ \left\| e _ {t} ^ {L} \right\| ^ {2} \right]. \tag {174} +$$ + +Bounding (173b). From Lemmas D.1 and D.12, we have + +$$ +(1 7 3 \mathrm {b}) = \mathbb {E} \left[ \left\| e _ {t} ^ {J} \right\| ^ {2} \right] \leq 2 \ell_ {g, 2} ^ {2} \rho_ {\mathbf {v}} ^ {2} p ^ {4}. \tag {175} +$$ + +Combining (174)-(175) yields + +$$ +\mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} - \mathbf {d} _ {t, \boldsymbol {\rho}} ^ {\mathbf {x}} \right\| ^ {2} \right] \leq 2 \mathbb {E} \left[ \left\| e _ {t} ^ {L} \right\| ^ {2} \right] + 2 \ell_ {g, 2} ^ {2} \rho_ {\mathbf {v}} ^ {2} p ^ {4}. +$$ + +Lemma D.14. Suppose Assumptions 2.2, B1., B3. and B4. hold. Set the step size $\delta_t$ and the parameter $p$ in (8), as + +$$ +\delta_ {t} \leq \left(2 + \frac {1}{\ell_ {g , 1} ^ {2}}\right) \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}, \forall t \in [ T ], \quad \text {a n d} \quad p = \frac {\ell_ {f , 0}}{\mu_ {g}}. \tag {176} +$$ + +Then, for the sequence $\{(\mathbf{x}_t,\mathbf{y}_t,\mathbf{v}_t)\}_{t = 1}^T$ generated by Algorithm 2 and $\hat{\mathbf{v}}_t^* (\mathbf{x}_t)$ in (20), we have + +$$ +\mathbb {E} \left[ \left\| \mathbf {v} _ {t + 1} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \right] \leq (1 + \acute {a}) \left(1 - \delta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \mathbb {E} [ \hat {\theta} _ {t} ^ {\mathbf {v}} ] + \left(1 + \frac {1}{\acute {a}}\right) \delta_ {t} ^ {2} B _ {t}, +$$ + +for some $\acute{a} > 0$ , where $\hat{\theta}_t^{\mathbf{v}}$ and $B_t$ are defined in Eq. (142) and Lemma D.13, respectively. + +Proof. By setting the radius $p \coloneqq \frac{\ell_{f,0}}{\mu_g}$ in (8), we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \mathbf {v} _ {t + 1} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \right] = \mathbb {E} \left[ \left\| \Pi_ {\mathcal {Z} _ {p}} \left[ \mathbf {v} _ {t} - \delta_ {t} \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} \right] - \Pi_ {\mathcal {Z} _ {p}} \left[ \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right] \right\| ^ {2} \right] \\ \leq \mathbb {E} \left[ \| \mathbf {v} _ {t} - \delta_ {t} \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ \leq (1 + \acute {a}) \underbrace {\mathbb {E} \left[ \| \mathbf {v} _ {t} - \delta_ {t} \nabla P _ {t} (\mathbf {x} _ {t} , \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) , \mathbf {v} _ {t}) - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right]} _ {I _ {t}} \\ + \left(1 + \frac {1}{\acute {a}}\right) \delta_ {t} ^ {2} \underbrace {\mathbb {E} \left[ \| \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} - \nabla P _ {t} \left(\mathbf {x} _ {t} , \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right) , \mathbf {v} _ {t}\right) \| ^ {2} \right]} _ {K _ {t}}, \tag {177} \\ \end{array} +$$ + +where $\nabla P_{t}(\mathbf{x}_{t},\hat{\mathbf{y}}_{t}^{*}(\mathbf{x}_{t}),\mathbf{v}_{t}) := \nabla_{\mathbf{y}}^{2}g_{t,\rho}(\mathbf{x}_{t},\hat{\mathbf{y}}_{t}^{*}(\mathbf{x}_{t}))\mathbf{v}_{t} + \nabla_{\mathbf{y}}f_{t,\rho}(\mathbf{x}_{t},\hat{\mathbf{y}}_{t}^{*}(\mathbf{x}_{t}));$ the first inequality follows from non-expansiveness property of a projection operator. + +We next bound the $I_{t}$ , and $K_{t}$ terms in (177), respectively. + +Bounding $I_{t}$ . We have + +$$ +\begin{array}{l} I _ {t} = \mathbb {E} \left[ \| \mathbf {v} _ {t} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] - 2 \delta_ {t} \mathbb {E} \left[ \langle \nabla P _ {t} (\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}), \mathbf {v} _ {t} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \rangle \right] \\ + \delta_ {t} ^ {2} \mathbb {E} \left[ \| \nabla P _ {t} (\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}) \| ^ {2} \right] \\ \leq \left(1 - 2 \delta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \mathbb {E} \left[ \| \mathbf {v} _ {t} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ - \left(2 \delta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}} - \delta_ {t} ^ {2}\right) \mathbb {E} \left[ \| \nabla P _ {t} (\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}) \| ^ {2} \right], \\ \end{array} +$$ + +where the inequality holds since $\nabla P_{t}$ is the gradient of the strongly convex quadratic program $\frac{1}{2}\mathbf{v}^{\top}\nabla_{\mathbf{y}}^{2}g_{t,\rho}(\mathbf{x},\hat{\mathbf{y}}_{t}^{*}(\mathbf{x}))\mathbf{v} + \mathbf{v}^{\top}\nabla_{\mathbf{y}}f_{t,\rho}(\mathbf{x},\hat{\mathbf{y}}_{t}^{*}(\mathbf{x}))$ + +Thus, we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\langle \nabla P _ {t} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}\right), \mathbf {v} _ {t} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\rangle \right] \\ \geq \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}} \mathbb {E} \left[ \| \mathbf {v} _ {t} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] + \frac {1}{\mu_ {g} + \ell_ {g , 1}} \mathbb {E} \left[ \| \nabla P _ {t} (\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}) \| ^ {2} \right]. \\ \end{array} +$$ + +Since $\delta_t \leq \left(2 + \frac{1}{\ell_{g,1}^2}\right) \frac{\mu_g \ell_{g,1}}{\mu_g + \ell_{g,1}}$ , then we have + +$$ +\begin{array}{l} I _ {t} \leq \left(1 - 2 \delta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \mathbb {E} \left[ \| \mathbf {v} _ {t} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] + \frac {1}{\ell_ {g , 1} ^ {2}} \left(\frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}} \delta_ {t}\right) \mathbb {E} \left[ \| \nabla P _ {t} (\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}) \| ^ {2} \right] \\ \leq \left(1 - \delta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \mathbb {E} \left[ \| \mathbf {v} _ {t} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right], \tag {178} \\ \end{array} +$$ + +where the second inequality holds since from (20), we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \| \nabla P _ {t} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}), \mathbf {v} _ {t}\right) \| ^ {2} \right] = \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} ^ {2} g _ {t, \rho} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \mathbf {v} _ {t} + \nabla_ {\mathbf {y}} f _ {t, \rho} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \| ^ {2} \right] \\ = \mathbb {E} \left[ \left\| \nabla_ {\mathbf {y}} ^ {2} g _ {t, \rho} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \left(\mathbf {v} _ {t} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \right\| ^ {2} \right] \\ \leq \ell_ {g, 1} ^ {2} \mathbb {E} \left[ \| \mathbf {v} _ {t} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right], \\ \end{array} +$$ + +where the second inequality follows from Assumption B3.. + +Bounding $K_{t}$ . Let + +$$ +\tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) = \frac {1}{2 \rho_ {\mathbf {v}}} \left(\nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t})\right). +$$ + +From Lemma D.13, we have + +$$ +K _ {t} = \mathbb {E} \left[ \| \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} - \mathbf {d} _ {t, \boldsymbol {\rho}} ^ {\mathbf {v}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}, \mathbf {v} _ {t}\right) \| ^ {2} \right] \leq B _ {t}. \tag {179} +$$ + +Putting (178), and (179) together with Eq. (177) yields the desired result. + +$$ +\mathbb {E} \left[ \| \mathbf {v} _ {t + 1} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \leq (1 + \acute {a}) \left(1 - \delta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \mathbb {E} \left[ \| \mathbf {v} _ {t} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] + \left(1 + \frac {1}{\acute {a}}\right) \delta_ {t} ^ {2} B _ {t}. +$$ + +![](images/a92327955d3d92fbd1af5eca4bc2fc95f62a7c41aca35bfd507e03385223714f.jpg) + +Lemma D.15. Suppose Assumptions 2.2 and 2.3 hold. Let $\hat{\theta}_t^{\mathrm{v}}$ be defined in (142). Set the parameter $p$ in (8) as $p = \frac{\ell_{f,0}}{\mu_g}$ . Then, for any positive choice of step sizes satisfying + +$$ +\delta_ {t} \leq \left(2 + \frac {1}{\ell_ {g , 1} ^ {2}}\right) \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}, +$$ + +the sequence $\{(\mathbf{x}_t,\mathbf{y}_t,\mathbf{v}_t)\}_{t = 1}^T$ generated by Algorithm 2 guarantees the following bound: + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \left(\mathbb {E} [ \hat {\theta} _ {t + 1} ^ {\mathbf {v}} ] - \mathbb {E} [ \hat {\theta} _ {t} ^ {\mathbf {v}} ]\right) \leq \sum_ {t = 1} ^ {T} \left(- \frac {L _ {\mu_ {g}}}{4} \mathbb {E} [ \hat {\theta} _ {t} ^ {\mathbf {v}} ] + \frac {4}{L _ {\mu_ {g}}} B _ {t}\right) \delta_ {t} \\ + \frac {1 6 \nu^ {2}}{L _ {\mu_ {g}} \mu_ {g} ^ {2}} (2 L _ {\mathbf {y}} ^ {2} + 1) \sum_ {t = 1} ^ {T} \mathbb {E} \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2} \frac {1}{\delta_ {t}} \\ + \sum_ {t = 1} ^ {T} \left(\frac {9 6 \ell_ {g , 1} \nu^ {2}}{L _ {\mu_ {g}} \mu_ {g} ^ {3}} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right) + \frac {4 8 \nu^ {2}}{L _ {\mu_ {g}} \mu_ {g} ^ {2}} \sup _ {\mathbf {x} \in \mathcal {X}} \| \mathbf {y} _ {t - 1} ^ {*} (\mathbf {x}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x}) \| ^ {2}\right) \frac {1}{\delta_ {t}}, \tag {180} \\ \end{array} +$$ + +where $B_{t}, \nu$ and $(L_{\mu_g}, L_{\mathbf{y}})$ are defined in Lemmas D.13, C.7 and D.9, respectively. + +Proof. From Lemma B.4, we have, for any $\acute{c} > 0$ + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \mathbf {v} _ {t + 1} - \hat {\mathbf {v}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \right] = \mathbb {E} \left[ \left\| \mathbf {v} _ {t + 1} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) + \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) - \hat {\mathbf {v}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \right] \\ \leq (1 + \acute {c}) \mathbb {E} \left[ \| \mathbf {v} _ {t + 1} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ + \left(1 + \frac {1}{\hat {c}}\right) \mathbb {E} \left[ \left\| \hat {\mathbf {v}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \right]. \tag {181} \\ \end{array} +$$ + +From Lemma D.14, we have, for any $\acute{a} > 0$ + +$$ +\mathbb {E} \left[ \left\| \mathbf {v} _ {t + 1} - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \right] \leq (1 + \acute {a}) \left(1 - \delta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \hat {\theta} _ {t} ^ {\mathbf {v}} + \left(1 + \frac {1}{\acute {a}}\right) \delta_ {t} ^ {2} B _ {t}. \tag {182} +$$ + +Substituting (182) into (181), we get + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \mathbf {v} _ {t + 1} - \hat {\mathbf {v}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \right] \leq (1 + \acute {c}) (1 + \acute {a}) \left(1 - \delta_ {t} \frac {\mu_ {g} \ell_ {g , 1}}{\mu_ {g} + \ell_ {g , 1}}\right) \hat {\theta} _ {t} ^ {\mathbf {v}} \\ + (1 + \acute {c}) \left(1 + \frac {1}{\acute {a}}\right) \delta_ {t} ^ {2} B _ {t} \\ + \left(1 + \frac {1}{\hat {c}}\right) \mathbb {E} \left[ \left\| \hat {\mathbf {v}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \right]. \tag {183} \\ \end{array} +$$ + +Choose $\acute{c} = \frac{\delta_t L_{\mu_g} / 4}{1 - \frac{\delta_t L_{\mu_g}}{2}}$ and $\acute{a} = \frac{\delta_t L_{\mu_g} / 2}{1 - \delta_t L_{\mu_g}}$ . Then, the following equations and inequalities are satisfied. + +$$ +(1 + \acute {c}) (1 + \acute {a}) \left(1 - \delta_ {t} L _ {\mu_ {g}}\right) = 1 - \frac {\delta_ {t} L _ {\mu_ {g}}}{4}, +$$ + +$$ +\left(1 + \acute {c}\right) \left(1 + \frac {1}{\acute {a}}\right) \leq \frac {4}{\delta_ {t} L _ {\mu_ {g}}}, \tag {184} +$$ + +$$ +1 + \frac {1}{\acute {a}} \leq \frac {2}{\delta_ {t} L _ {\mu_ {g}}}, \quad 1 + \frac {1}{\acute {c}} \leq \frac {4}{\delta_ {t} L _ {\mu_ {g}}}, +$$ + +where $L_{\mu_g} = \frac{\mu_g\ell_{g,1}}{\mu_g + \ell_{g,1}}$ + +Thus, we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \mathbf {v} _ {t + 1} - \hat {\mathbf {v}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) \right\| ^ {2} \right] \leq \left(1 - \frac {\delta_ {t} L _ {\mu_ {g}}}{4}\right) \hat {\theta} _ {t} ^ {\mathbf {v}} + \frac {4}{L _ {\mu_ {g}}} \delta_ {t} B _ {t} \\ + \frac {4}{L _ {\mu_ {g}}} \frac {1}{\delta_ {t}} \mathbb {E} \left[ \left\| \hat {\mathbf {v}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \right]. \tag {185} \\ \end{array} +$$ + +We now bound the last term on the right-hand side of (185). By Lemma C.7, we have: + +$$ +\begin{array}{l} \left\| \hat {\mathbf {v}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \\ \leq 2 \frac {\nu^ {2}}{\mu_ {g} ^ {2}} \left(\left\| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} + \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2}\right) \\ \leq 2 \frac {\nu^ {2}}{\mu_ {g} ^ {2}} \left(2 \left| \left| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) \right| \right| ^ {2} \right. \\ \left. + 2 \left\| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} + \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2}\right) \\ \leq 2 \frac {\nu^ {2}}{\mu_ {g} ^ {2}} \left(2 L _ {\mathbf {y}} ^ {2} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 2 \left\| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} + \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2}\right), \tag {186} \\ \end{array} +$$ + +where the last inequality follows from Lemma D.2. + +From (150), we have + +$$ +\begin{array}{l} \left\| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \leq 3 \left\| \hat {\mathbf {y}} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \\ + 3 \left\| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} + 3 \left\| \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \\ \leq 3 \left\| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} + \frac {6 \ell_ {g , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{\mu_ {g}}. \tag {187} \\ \end{array} +$$ + +Plugging (187) into (186), we get + +$$ +\begin{array}{l} \left\| \hat {\mathbf {v}} _ {t + 1} ^ {*} (\mathbf {x} _ {t + 1}) - \hat {\mathbf {v}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \\ \leq 4 \frac {\nu^ {2}}{\mu_ {g} ^ {2}} \left(2 L _ {\mathbf {y}} ^ {2} + 1\right) \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2} \\ + 4 \frac {\nu^ {2}}{\mu_ {g} ^ {2}} \left(3 \| \mathbf {y} _ {t + 1} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} + \frac {6 \ell_ {g , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{\mu_ {g}}\right). \tag {188} \\ \end{array} +$$ + +Then, substituting (188) into (185), rearranging the resulting inequality and summing over $t \in [T]$ , we obtain the desired result. + +# D.5 Bounds on the Zeroth-Order Estimation Error of Outer Objective + +Lemma D.16. Suppose Assumptions B2. and B3. hold. Let + +$$ +\varpi := \| \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) - \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} (\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2}, +$$ + +where $\hat{\nabla}_{\mathbf{x}}f_{t + 1}$ and $\hat{\nabla}_{\mathbf{xy}}^2 g_{t + 1}$ are defined in (25b) and (26b), respectively. Then, for the sequence $\left\{\left(\mathbf{x}_t,\mathbf{y}_t,\mathbf{v}_t\right)\right\}_{t = 1}^T$ generated by Algorithm 2, we have + +$$ +\begin{array}{l} \varpi \leq (1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}) d _ {1} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + (1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}) d _ {1} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} \\ + \frac {9}{2} d _ {1} \ell_ {g, 1} ^ {2} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} + (3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}) d _ {1} ^ {2} \rho_ {\mathbf {s}} ^ {2}. \\ \end{array} +$$ + +Proof. From Lemma D.5, we have + +$$ +\begin{array}{l} \left\| \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}\right) \right\| ^ {2} \\ \leq 3 d _ {1} \ell_ {g, 1} ^ {2} \| \mathbf {z} _ {t + 1} - \mathbf {z} _ {t} \| ^ {2} + \frac {3}{2} \ell_ {f, 1} ^ {2} d _ {1} ^ {2} \rho_ {\mathbf {s}} ^ {2} \\ \leq 6 d _ {1} \ell_ {f, 1} ^ {2} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 6 d _ {1} \ell_ {f, 1} ^ {2} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} + \frac {3}{2} \ell_ {f, 1} ^ {2} d _ {1} ^ {2} \rho_ {\mathbf {s}} ^ {2}. \tag {189} \\ \end{array} +$$ + +Moreover, from (26a), we have + +$$ +\begin{array}{l} \left\| \hat {\nabla} _ {\mathbf {y y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {y y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1}\right) \right\| ^ {2} \\ = \frac {1}{4 \rho_ {\mathbf {v}} ^ {2}} \| \hat {\nabla} _ {\mathbf {x}} g _ {t + 1} (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} + \rho_ {\mathbf {v}} \mathbf {v} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) - \hat {\nabla} _ {\mathbf {x}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2} \\ \leq \frac {3}{4 \rho_ {\mathbf {v}} ^ {2}} d _ {1} \ell_ {g, 1} ^ {2} \| (\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} + \rho_ {\mathbf {v}} \mathbf {v} _ {t + 1}) - (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} + \frac {3}{8 \rho_ {\mathbf {v}} ^ {2}} \ell_ {g, 1} ^ {2} d _ {1} ^ {2} \rho_ {\mathbf {s}} ^ {2} \\ \leq \frac {9}{4 \rho_ {\mathbf {v}} ^ {2}} d _ {1} \ell_ {g, 1} ^ {2} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + \frac {9}{4 \rho_ {\mathbf {v}} ^ {2}} d _ {1} \ell_ {g, 1} ^ {2} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} \\ + \frac {9}{4} d _ {1} \ell_ {g, 1} ^ {2} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} + \frac {3}{8 \rho_ {\mathbf {v}} ^ {2}} \ell_ {g, 1} ^ {2} d _ {1} ^ {2} \rho_ {\mathbf {s}} ^ {2}, \tag {190} \\ \end{array} +$$ + +where the first inequality follows from Lemma D.5. + +From $\| a + b\| ^2\leq 2\left(\| a\| ^2 +\| b\| ^2\right)$ , we get + +$$ +\begin{array}{l} \varpi \leq 2 \| \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \vec {\mathcal {B}} _ {t + 1}) - \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}; \vec {\mathcal {B}} _ {t + 1}) \| ^ {2} \\ + 2 \left\| \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}\right) \right\| ^ {2} \\ \leq \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {1} \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2} + \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {1} \left\| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \right\| ^ {2} \\ + \frac {9}{2} d _ {1} \ell_ {g, 1} ^ {2} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} + (3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}) d _ {1} ^ {2} \rho_ {\mathbf {s}} ^ {2}, \\ \end{array} +$$ + +where the second inequality follows from (189) and (190). + +![](images/581f86fa398d74706f3765cf54be00e7691ad659a7169499a2da715218a48c16.jpg) + +Lemma D.17. Suppose Assumptions B2., B3., D2., and D4. hold. Consider the sequence $\{(\mathbf{x}_t, \mathbf{y}_t, \mathbf{v}_t)\}_{t=1}^T$ generated by Algorithm 2. For $e_t^L$ defined in (166c), we have + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {L} \| ^ {2} \leq (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| e _ {t} ^ {L} \| ^ {2} + 3 6 \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \\ + \left(1 8 d _ {1} ^ {2} \ell_ {f, 1} ^ {2} + 6 (3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}) d _ {1} ^ {2}\right) \rho_ {\mathbf {s}} ^ {2} + 1 8 d _ {1} ^ {2} \ell_ {g, 1} ^ {2} \frac {\rho_ {\mathbf {s}} ^ {2}}{\rho_ {\mathbf {v}} ^ {2}} \\ + \frac {1 8}{\rho_ {\mathbf {v}} ^ {2}} \mathbb {E} \| \nabla_ {\mathbf {x}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {x}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} \\ + \frac {1 8}{\rho_ {\mathbf {v}} ^ {2}} \mathbb {E} \| \nabla_ {\mathbf {x}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {x}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} \\ + 6 \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {1} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 6 \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {1} \mathbb {E} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} \\ + 2 7 d _ {1} \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} + 3 \left(\frac {\hat {\sigma} _ {g _ {\mathbf {x}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {x}}} ^ {2}}{b}\right) \eta_ {t + 1} ^ {2}. \tag {191} \\ \end{array} +$$ + +Proof. According to the definition of $\hat{\mathbf{d}}_t^{\times}$ in Algorithm 2, we have + +$$ +\begin{array}{l} \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {x}} - \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} = - \eta_ {t + 1} \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} + \eta_ {t + 1} (\hat {\nabla} _ {\mathbf {x}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1})) \\ + \left(1 - \eta_ {t + 1}\right) \left(\hat {\nabla} _ {\mathbf {x}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) \right. \\ \left. - \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} (\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1})\right). \\ \end{array} +$$ + +Then, we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t + 1, \rho} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {x}} \| ^ {2} \\ = \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} - (\hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {x}} - \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}) \| ^ {2} \\ = \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} + \eta_ {t + 1} \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} \\ - \eta_ {t + 1} \left(\hat {\nabla} _ {\mathbf {x}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right)\right) \\ - \left(1 - \eta_ {t + 1}\right) \left(\hat {\nabla} _ {\mathbf {x}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) \right. \\ \left. - \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} (\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1})\right) \| ^ {2} \\ = \mathbb {E} \| (1 - \eta_ {t + 1}) (\nabla_ {\mathbf {x}} f _ {t, \rho} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}) \\ + \eta_ {t + 1} \left(\nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {z} _ {t + 1}\right) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right)\right) \\ + \left(1 - \eta_ {t + 1}\right) \left(\nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) \right. \\ + \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) - \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) \\ \left. - \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) + \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} (\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1})\right) \| ^ {2}. \\ \end{array} +$$ + +Since + +$$ +\begin{array}{l} \mathbb {E} \left[ \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) \right] = \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}), \\ \mathbb {E} \left[ \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1}\right) \right] \\ = \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \bar {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \bar {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}), \\ \end{array} +$$ + +then, we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {x}} \| ^ {2} \\ = (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t, \rho} (\mathbf {z} _ {t}) + \bar {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} \| ^ {2} \\ + \left\| \eta_ {t + 1} \left(\nabla_ {\mathbf {x}} f _ {t + 1, \rho} (\mathbf {z} _ {t + 1}) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1})\right) \right. \\ + \left(1 - \eta_ {t + 1}\right) \left(\nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) \right. \\ + \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) - \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) \\ \left. - \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) + \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} (\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1})\right) \| ^ {2} \\ \leq \left(1 - \eta_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} \| ^ {2} \\ + 3 \left(1 - \eta_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {z} _ {t + 1}\right) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}\right) - \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} \left(\mathbf {z} _ {t}\right) - \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {z} _ {t}\right) \\ + \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) - \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) \\ - \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) - \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) + \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} (\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2} \\ + 3 \eta_ {t + 1} ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) - \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} (\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}) \| ^ {2} \\ + 3 \eta_ {t + 1} ^ {2} \mathbb {E} \| \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}) \| ^ {2}, \tag {192} \\ \end{array} +$$ + +where the second inequality holds by Cauchy-Schwarz inequality. + +Note that for the last term on the right-hand side of (192), using (172) and (26b), we have + +$$ +\begin{array}{l} \left\| \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) \right\| ^ {2} \\ \leq 2 \| \frac {1}{2 \rho_ {\mathbf {v}}} \left(\nabla_ {\mathbf {x}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} + \rho_ {\mathbf {v}} \mathbf {v} _ {t + 1}\right) - \hat {\nabla} _ {\mathbf {x}} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} + \rho_ {\mathbf {v}} \mathbf {v} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right)\right) \| ^ {2} \\ + 2 \| \frac {1}{2 \rho_ {\mathbf {v}}} \left(\hat {\nabla} _ {\mathbf {x}} g _ {t + 1} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} - \rho_ {\mathbf {v}} \mathbf {v} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) - \nabla_ {\mathbf {x}} g _ {t + 1, \rho} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t + 1} - \rho_ {\mathbf {v}} \mathbf {v} _ {t + 1}\right)\right) \| ^ {2} \\ \leq \frac {\hat {\sigma} _ {g _ {\mathbf {x}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}}, \\ \end{array} +$$ + +where the last inequality follows from Assumption D2.. + +Then, from $\mathbb{E}\|a - \mathbb{E}[a]\|^2 = \mathbb{E}\|a\|^2 - \|\mathbb{E}[a]\|^2$ and Assumption D4., we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {x}} \| ^ {2} \\ \leq \left(1 - \eta_ {t + 1}\right) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\boldsymbol {\mathbf {x}}} \| ^ {2} \\ + 6 (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t + 1, \rho} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) - \nabla_ {\mathbf {x}} f _ {t, \rho} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) \| ^ {2} \\ + 6 \left(1 - \eta_ {t + 1}\right) ^ {2} \mathbb {E} \| \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} \left(\mathbf {z} _ {t + 1}; \mathcal {B} _ {t + 1}\right) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t + 1}; \bar {\mathcal {B}} _ {t + 1}\right) \\ + \hat {\nabla} _ {\mathbf {x}} f _ {t + 1} \left(\mathbf {z} _ {t}; \mathcal {B} _ {t + 1}\right) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {z} _ {t}; \bar {\mathcal {B}} _ {t + 1}\right) \| ^ {2} + 3 \eta_ {t + 1} ^ {2} \left(\frac {\hat {\sigma} _ {g _ {\mathbf {x}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {x}}} ^ {2}}{b}\right). \tag {193} \\ \end{array} +$$ + +Then, from Young's inequality and Lemma D.16, we have + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \hat {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {x}} \| ^ {2} \\ \leq (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t, \pmb {\rho}} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} \| ^ {2} \\ + 1 2 (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} (\mathbf {z} _ {t}) \| ^ {2} \\ + 1 2 (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) \| ^ {2} \\ + 6 \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {1} \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2} + 6 \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {1} \left\| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \right\| ^ {2} \\ + 2 7 d _ {1} \ell_ {g, 1} ^ {2} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} + 6 \left(3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {1} ^ {2} \rho_ {\mathbf {s}} ^ {2} + 3 \eta_ {t + 1} ^ {2} \left(\frac {\hat {\sigma} _ {g _ {\mathbf {x}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {x}}} ^ {2}}{b}\right). \tag {194} \\ \end{array} +$$ + +For the third term on the right-hand side of (193), we have + +$$ +\begin{array}{l} \left\| \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right\| ^ {2} \\ \leq \frac {1}{2 \rho_ {\mathbf {v}} ^ {2}} \| \nabla_ {\mathbf {x}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {x}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) \| ^ {2} (195a) \\ + \frac {1}{2 \rho_ {\mathbf {v}} ^ {2}} \| \nabla_ {\mathbf {x}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {x}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) \| ^ {2}. (195b) \\ \end{array} +$$ + +For (195a), we get + +$$ +\begin{array}{l} \left\| \nabla_ {\mathbf {x}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {x}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) \right\| ^ {2} \\ \leq 3 \| \nabla_ {\mathbf {x}} g _ {t + 1, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {x}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} \\ + 3 \left\| \nabla_ {\mathbf {x}} g _ {t + 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {x}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) \right\| ^ {2} \\ + 3 \left\| \nabla_ {\mathbf {x}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {x}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) \right\| ^ {2} \\ \leq 3 \| \nabla_ {\mathbf {x}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {x}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} + \frac {3 \rho_ {\mathbf {s}} ^ {2} d _ {1} ^ {2} \ell_ {g , 1} ^ {2}}{2}, \\ \end{array} +$$ + +where the last inequality follows from Eq. (130). + +Similarity, for (195b), we have + +$$ +\begin{array}{l} \left\| \nabla_ {\mathbf {x}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) - \nabla_ {\mathbf {x}} g _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) \right\| ^ {2} \\ \leq 3 \| \nabla_ {\mathbf {x}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {x}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| + \frac {3 \rho_ {\mathbf {s}} ^ {2} d _ {1} ^ {2} \ell_ {g , 1} ^ {2}}{2}. \\ \end{array} +$$ + +Substituting these inequalities in (195), we have + +$$ +\begin{array}{l} \left\| \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2} \\ \leq \frac {3}{2 \rho_ {\mathbf {v}} ^ {2}} \| \nabla_ {\mathbf {x}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {x}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} \\ + \frac {3}{2 \rho_ {\mathbf {v}} ^ {2}} \| \nabla_ {\mathbf {x}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {x}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} + \frac {3 \rho_ {\mathbf {s}} ^ {2} d _ {1} ^ {2} \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}. \tag {196} \\ \end{array} +$$ + +For the second term on the right-hand side of (193), we have + +$$ +\begin{array}{l} \left\| \nabla_ {\mathbf {x}} f _ {t + 1, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {x}} f _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2} \\ \leq 3 \left\| \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {x}} f _ {t + 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right\| ^ {2} \\ + 3 \left\| \nabla_ {\mathbf {x}} f _ {t + 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {x}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right\| ^ {2} \\ + 3 \left\| \nabla_ {\mathbf {x}} f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right\| ^ {2} \\ \leq 3 \| \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {x}} f _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} + \frac {3 \rho_ {\mathbf {s}} ^ {2} d _ {1} ^ {2} \ell_ {f , 1} ^ {2}}{2}, \tag {197} \\ \end{array} +$$ + +where the last inequality follows from Eq. (132). + +From (196), (197) and (194), we get + +$$ +\begin{array}{l} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t + 1, \boldsymbol {\rho}} (\mathbf {z} _ {t + 1}) + \tilde {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t + 1} (\mathbf {z} _ {t + 1}) - \hat {\mathbf {d}} _ {t + 1} ^ {\mathbf {x}} \| ^ {2} \\ \leq (1 - \eta_ {t + 1}) ^ {2} \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t, \boldsymbol {\rho}} (\mathbf {z} _ {t}) + \bar {\nabla} _ {\mathbf {x y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} \| ^ {2} \\ + 3 6 \mathbb {E} \| \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {x}} f _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} + 1 8 \rho_ {\mathbf {s}} ^ {2} d _ {1} ^ {2} \ell_ {f, 1} ^ {2} \\ + \frac {1 8}{\rho_ {\mathbf {v}} ^ {2}} \mathbb {E} \| \nabla_ {\mathbf {x}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {x}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} \\ + \frac {1 8}{\rho_ {\mathbf {v}} ^ {2}} \mathbb {E} \| \nabla_ {\mathbf {x}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {x}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} + \frac {1 8 \rho_ {\mathbf {s}} ^ {2} d _ {1} ^ {2} \ell_ {g , 1} ^ {2}}{\rho_ {\mathbf {v}} ^ {2}} \\ + 6 \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {1} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 6 \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {1} \mathbb {E} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} \\ + 2 7 d _ {1} \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} + 6 (3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}) d _ {1} ^ {2} \rho_ {\mathbf {s}} ^ {2} + 3 \eta_ {t + 1} ^ {2} (\frac {\hat {\sigma} _ {g _ {\mathbf {x}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {x}}} ^ {2}}{b}). \\ \end{array} +$$ + +![](images/e84951535c57a9760f788b37cf34edffea50b6e7ac06f3efa867f8a152d8dc93.jpg) + +# D.6 Bounds on the Zeroth-Order Objective Function and its Projected Gradients + +Lemma D.18. Suppose Assumptions 2.2, B2., B3., and 2.4 hold. Then, for the sequence of functions $\{f_{t,\rho}\}_{t=1}^{T}$ defined in Eq. (18), we have + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \left(f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right) - f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t + 1}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t + 1})\right)\right) \\ \leq 2 M + V _ {T} + \ell_ {f, 1} \left(1 + 2 \frac {\ell_ {g , 1}}{\mu_ {g}}\right) T \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right). \\ \end{array} +$$ + +Here, $V_{T}$ is defined in (11); and $M$ is defined in Assumption 2.4. + +Proof. Note that, we have + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \left(f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) - f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t + 1}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right)\right) \\ = \sum_ {t = 1} ^ {T} \left(f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right) - f _ {t} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) (198) \\ + \sum_ {t = 1} ^ {T} \left(f _ {t} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right) - f _ {t} \left(\mathbf {x} _ {t + 1}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t + 1})\right)\right) (199) \\ + \sum_ {t = 1} ^ {T} \left(f _ {t} \left(\mathbf {x} _ {t + 1}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right) - f _ {t, \rho} \left(\mathbf {x} _ {t + 1}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right)\right). (200) \\ \end{array} +$$ + +From (126), we have + +$$ +(1 9 8) \leq T \frac {\ell_ {f , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{2}, \tag {201} +$$ + +and + +$$ +(2 0 0) \leq T \frac {\ell_ {f , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{2}. \tag {202} +$$ + +Moreover, from Lemma D.6, we have + +$$ +\begin{array}{l} (1 9 9) = \sum_ {t = 1} ^ {T} \left(f _ {t} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right) - f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \\ + \sum_ {t = 1} ^ {T} \left(f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) - f _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right)\right) \\ + \sum_ {t = 1} ^ {T} \left(f _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right) - f _ {t} \left(\mathbf {x} _ {t + 1}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right)\right) \\ \leq \ell_ {f, 1} \sum_ {t = 1} ^ {T} \left\| \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| + \ell_ {f, 1} \sum_ {t = 1} ^ {T} \left\| \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t + 1}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t + 1}) \right\| \\ + \sum_ {t = 1} ^ {T} \left(f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) - f _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right)\right) \\ \leq 2 T \ell_ {f, 1} \frac {\ell_ {g , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{\mu_ {g}} + \sum_ {t = 1} ^ {T} \left(f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) - f _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t + 1})\right)\right). \tag {203} \\ \end{array} +$$ + +For the last term of the above inequality, we have + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \left(f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) - f _ {t} \left(\mathbf {x} _ {t + 1}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t + 1})\right)\right) = f _ {1} \left(\mathbf {x} _ {1}, \mathbf {y} _ {1} ^ {*} (\mathbf {x} _ {1})\right) - f _ {T} \left(\mathbf {x} _ {T + 1}, \mathbf {y} _ {T} ^ {*} (\mathbf {x} _ {T + 1})\right) \\ + \sum_ {t = 2} ^ {T} \left(f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right) - f _ {t - 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t - 1} ^ {*} (\mathbf {x} _ {t})\right)\right) \\ \leq 2 M + V _ {T}, \\ \end{array} +$$ + +which implies that + +$$ +(1 9 9) \leq 2 T \ell_ {f, 1} \frac {\ell_ {g , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{\mu_ {g}} + 2 M + V _ {T}. \tag {204} +$$ + +From (201), (202), and (204), we get the desired result. + +Lemma D.19. Suppose that Assumptions 2.2 and 2.3 hold. Let $f_{t,\rho}$ be defined as in (18). Then, for $\hat{\mathbf{d}}_t^{\mathbf{x}}$ generated by Algorithm 2, for all $t \in [T]$ , we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} - \nabla f _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})) \right\| ^ {2} \right] \leq 4 \mathbb {E} \left[ \left\| e _ {t} ^ {L} \right\| ^ {2} \right] + 4 \ell_ {g, 2} ^ {2} \rho_ {\mathbf {v}} ^ {2} p ^ {4} \\ + 2 M _ {f} ^ {2} \left(\mathbb {E} \left[ \hat {\theta} _ {t} ^ {\mathbf {y}} \right] + \mathbb {E} \left[ \hat {\theta} _ {t} ^ {\mathbf {v}} \right]\right) := A _ {t}, \tag {205} \\ \end{array} +$$ + +where $e_t^L$ is defined in Lemma D.13, and $\hat{\theta}_t^{\mathbf{y}}$ , $\hat{\theta}_t^{\mathbf{v}}$ are as defined in (142). Additionally, $M_f$ is given in Lemma D.2. + +Proof. From $\| a + b \|^2 \leq 2 \left( \| a \|^2 + \| b \|^2 \right)$ , we get + +$$ +\mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} - \nabla f _ {t, \rho} (\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})) \right\| ^ {2} \right] +$$ + +$$ +\leq 2 \mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} - \mathbf {d} _ {t, \rho} ^ {\mathbf {x}} \right\| ^ {2} \right] \tag {206a} +$$ + +$$ ++ 2 \mathbb {E} \left[ \left\| \mathbf {d} _ {t, \rho} ^ {\mathbf {x}} - \nabla f _ {t, \rho} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \right\| ^ {2} \right], \tag {206b} +$$ + +where $\mathbf{d}_{t,\rho}^{\mathbf{x}}$ is defined in (22c). From Lemma D.13, we have + +$$ +(2 0 6 \mathrm {a}) \leq 4 \mathbb {E} \left[ \left\| e _ {t} ^ {L} \right\| ^ {2} \right] + 4 \ell_ {g, 2} ^ {2} \rho_ {\mathbf {v}} ^ {2} p ^ {4}. \tag {207} +$$ + +Moreover, from Eq. (122a), we get + +$$ +(2 0 6 \mathrm {b}) \leq 2 M _ {f} ^ {2} \left(\mathbb {E} \left[ \hat {\theta} _ {t} ^ {\mathbf {y}} \right] + \mathbb {E} \left[ \hat {\theta} _ {t} ^ {\mathbf {v}} \right]\right). \tag {208} +$$ + +Substituting (207) and (208) into (206), we conclude the desired result. + +Lemma D.20. Suppose Assumptions 2.2, 2.3, and 2.4 hold. Let the sequence of functions $\{f_{t,\rho}\}_{t=1}^{T}$ be defined in (18), and let $\mathcal{P}_{\mathcal{X},\alpha_t}$ be given in Definition B.1. Then, for any positive choice of step sizes satisfying $\alpha_t \leq 1/4L_f$ , for all $t \in [T]$ , Algorithm 2 guarantees the following bound: + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \left(\alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) \mathbb {E} \left[ \| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right)\right) \| ^ {2} \right] \\ \leq 1 2 M + 6 V _ {T} + \sum_ {t = 1} ^ {T} \left(6 \alpha_ {t} - 3 L _ {f} \alpha_ {t} ^ {2}\right) A _ {t} \\ + \sum_ {t = 1} ^ {T} \left(6 \ell_ {f, 1} \left(1 + 2 \frac {\ell_ {g , 1}}{\mu_ {g}}\right) + \frac {3 \ell_ {f , 1} \ell_ {g , 1}}{\mu_ {g}} \left(\alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right)\right) \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right), \tag {209} \\ \end{array} +$$ + +where $V_{T}$ and $A_{t}$ are respectively defined in Eq. (11) and Lemma D.19. + +Proof. Due to the $L_{f}$ -smoothness of the function $f_{t}$ by Eq. (39c) in Lemma C.1, $f_{t,\rho}$ is also $L_{f}$ -smooth. Hence, + +$$ +\begin{array}{l} f _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t + 1}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t + 1})) - f _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})) \\ \leq \left\langle \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right), \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\rangle + \frac {L _ {f}}{2} \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2} \\ = - \alpha_ {t} \left\langle \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right), \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) \right\rangle + \frac {L _ {f} \alpha_ {t} ^ {2}}{2} \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2}. \tag {210} \\ \end{array} +$$ + +For the first term on the R.H.S of Eq. (210), we have that + +$$ +\begin{array}{l} - \mathbb {E} \left\langle \nabla f _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})), \mathcal {P} _ {\boldsymbol {\chi}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) \right\rangle \\ = - \mathbb {E} \left\langle \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}, \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) \right\rangle \\ - \mathbb {E} \left\langle \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) - \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}, \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) \right\rangle \\ \leq - \frac {1}{2} \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} \right] + \frac {1}{2} \mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} - \nabla f _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})) \right\| ^ {2} \right] \\ \leq - \frac {1}{2} \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} \right] + \frac {A _ {t}}{2}, \tag {211} \\ \end{array} +$$ + +where the first inequality follows from Lemma B.7; the last inequality follows from Lemma D.19. + +Plugging the bound (211) into (210), we have that + +$$ +\begin{array}{l} \mathbb {E} \left[ f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t + 1}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right) - f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) \right] \\ \leq \frac {\left(L _ {f} \alpha_ {t} ^ {2} - \alpha_ {t}\right)}{2} \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} \right] + \frac {\alpha_ {t} A _ {t}}{2}, \\ \end{array} +$$ + +which can be rearranged into + +$$ +\begin{array}{l} \left. \left(\alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} \right] \right. \\ \leq 2 \mathbb {E} \left[ f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) - f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t + 1}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right) \right] + \alpha_ {t} A _ {t}. \tag {212} \\ \end{array} +$$ + +In addition, we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \right] \\ \leq 3 \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) - \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \right] \\ + 3 \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) - \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \right] \\ + 3 \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} \right] \\ \leq 3 \mathbb {E} \left[ \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} - \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) \right\| ^ {2} \right] \\ + 3 \mathbb {E} \left[ \left\| \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) - \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) \right\| ^ {2} \right] \\ + 3 \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} \right], \\ \end{array} +$$ + +where the second inequality follows from non-expansiveness of the projection operator. + +Then, from Lemma D.19 and Assumption B2., we have + +$$ +\begin{array}{l} \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \right] \\ \leq 3 A _ {t} + 3 \ell_ {f, 1} \mathbb {E} \left[ \left\| \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2} \right] + 3 \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} \right] \\ \leq 3 A _ {t} + 3 \ell_ {f, 1} \frac {\ell_ {g , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{\mu_ {g}} + 3 \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} \right], \tag {213} \\ \end{array} +$$ + +where the last inequality is by Lemma D.6. + +Combining (212) and (213) and summing over $t = 1$ to $T$ , we have + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \left(\alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) \mathbb {E} \left[ \| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \| ^ {2} \right] \\ \leq 6 \sum_ {t = 1} ^ {T} \left(f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) - f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t + 1}, \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t + 1}\right)\right)\right) \\ + \frac {3 \ell_ {f , 1} \ell_ {g , 1}}{\mu_ {g}} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right) \sum_ {t = 1} ^ {T} \left(\alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) + 3 \sum_ {t = 1} ^ {T} \left(2 \alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) A _ {t} \\ \leq 1 2 M + 6 V _ {T} + 6 \ell_ {f, 1} \left(1 + 2 \frac {\ell_ {g , 1}}{\mu_ {g}}\right) T \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right) \\ + \frac {3 \ell_ {f , 1} \ell_ {g , 1}}{\mu_ {g}} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right) \sum_ {t = 1} ^ {T} \left(\alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) + 3 \sum_ {t = 1} ^ {T} \left(2 \alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) A _ {t}, \\ \end{array} +$$ + +where the second inequality is due to Lemma D.18. + +![](images/9e034d6b3e93ae5284501376e303967efe348a4b9f74e558249eea24acd0e9d7.jpg) + +Lemma D.21. Let the sequence $\{(\mathbf{x}_t,\mathbf{y}_t,\mathbf{v}_t)\}_{t = 1}^T$ be generated by Algorithm 2. + +(a) Then, we have + +$$ +\left\| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \right\| ^ {2} \leq 2 \beta_ {t} ^ {2} \left\| e _ {t} ^ {g _ {\rho}} \right\| ^ {2} + 2 \beta_ {t} ^ {2} \left\| \nabla_ {\mathbf {y}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2}, +$$ + +where $e_t^{g_\rho}$ is defined in (138). + +(b) Suppose Assumptions 2.2, B2. and B3. hold. Then, we have + +$$ +\begin{array}{l} \left\| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \right\| ^ {2} \leq 4 \alpha_ {t} ^ {2} \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \\ + \frac {4 \ell_ {f , 1} \ell_ {g , 1} \alpha_ {t} ^ {2} \left(\rho_ {\mathrm {s}} ^ {2} + \rho_ {\mathrm {r}} ^ {2}\right)}{\mu_ {g}} + 2 A _ {t} \alpha_ {t} ^ {2}, \tag {214} \\ \end{array} +$$ + +where $A_{t}$ is defined in (205). + +(c) Suppose Assumptions B1., B2. and B3. hold. Then, we have + +$$ +\begin{array}{l} \left\| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \right\| ^ {2} \leq 2 \delta_ {t} ^ {2} \left\| e _ {t} ^ {M} \right\| ^ {2} + 3 d _ {2} ^ {2} \ell_ {f, 1} ^ {2} \delta_ {t} ^ {2} \rho_ {\mathbf {r}} ^ {2} \\ + (1 2 \ell_ {f, 0} ^ {2} + 6 \ell_ {g, 1} ^ {2} p ^ {2}) \delta_ {t} ^ {2} + 6 \ell_ {g, 1} ^ {2} \frac {\delta_ {t} ^ {2}}{\rho_ {\mathbf {v} ^ {2}}} \hat {\theta} _ {t} ^ {\mathbf {y}}, \\ \end{array} +$$ + +where $e_t^M$ and $\hat{\theta}_t^{\mathbf{y}}$ are defined in (154) and (142), respectively. + +Proof. For part (a): From Algorithm 2, we have + +$$ +\begin{array}{l} \left\| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \right\| ^ {2} = \beta_ {t} ^ {2} \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {y}} \right\| ^ {2} \\ \leq 2 \beta_ {t} ^ {2} \| \dot {\mathbf {d}} _ {t} ^ {\mathbf {y}} - \nabla_ {\mathbf {y}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} + 2 \beta_ {t} ^ {2} \| \nabla_ {\mathbf {y}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \\ = 2 \beta_ {t} ^ {2} \| e _ {t} ^ {g _ {\rho}} \| ^ {2} + 2 \beta_ {t} ^ {2} \| \nabla_ {\mathbf {y}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2}. \tag {215} \\ \end{array} +$$ + +For part (b): + +From the update rule in Algorithm 2, we obtain + +$$ +\begin{array}{l} \left\| \mathbf {x} _ {t} - \mathbf {x} _ {t + 1} \right\| ^ {2} = \alpha_ {t} ^ {2} \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) \right\| ^ {2} \\ \leq 2 \alpha_ {t} ^ {2} \left(\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \| ^ {2} \right. \\ \left. + \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}}\right) - \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2}\right) \\ \leq 2 \alpha_ {t} ^ {2} \left(\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \| ^ {2} \right. \\ \left. + \left\| \hat {\mathbf {d}} _ {t} ^ {\mathbf {x}} - \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right) \right\| ^ {2}\right) \\ \leq 2 \alpha_ {t} ^ {2} \left(\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \| ^ {2} + A _ {t}\right), \tag {216} \\ \end{array} +$$ + +where the first inequality is by $(a + b)^2 \leq 2a^2 + 2b^2$ ; the second inequality follows from non-expansiveness of the projection operator; and the last inequality follows from Lemma D.19. + +The first term in the above inequality can be bounded as + +$$ +\begin{array}{l} \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \\ \leq 2 \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) - \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \\ + 2 \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right)\right) \right\| ^ {2} \\ \leq 2 \left\| \nabla f _ {t, \pmb {\rho}} (\mathbf {x} _ {t}, \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t})) - \nabla f _ {t, \pmb {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})) \right\| ^ {2} \\ + 2 \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right)\right) \right\| ^ {2} \\ \leq 2 \ell_ {f, 1} \| \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) - \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} + 2 \| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}))\right) \| ^ {2} \\ \leq 2 \ell_ {f, 1} \frac {\ell_ {g , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{\mu_ {g}} + 2 \| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \| ^ {2}, \tag {217} \\ \end{array} +$$ + +where the last inequality follows from Lemma D.6. + +Based on (217) and (216), we get + +$$ +\left\| \mathbf {x} _ {t} - \mathbf {x} _ {t + 1} \right\| ^ {2} \leq 2 \alpha_ {t} ^ {2} \left(2 \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} + \frac {2 \ell_ {f , 1} \ell_ {g , 1} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)}{\mu_ {g}} + A _ {t}\right). +$$ + +For part (c): From the nonexpansiveness of projection, we have + +$$ +\begin{array}{l} \left\| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \right\| ^ {2} = \left\| \Pi_ {\mathcal {Z} _ {p}} \left[ \mathbf {v} _ {t} - \delta_ {t} \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} \right] - \Pi_ {\mathcal {Z} _ {p}} \left[ \mathbf {v} _ {t} \right] \right\| ^ {2} \\ \leq \delta_ {t} ^ {2} \| \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} \| ^ {2} \\ \leq 2 \delta_ {t} ^ {2} \| \hat {\mathbf {d}} _ {t} ^ {\mathbf {v}} - \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} (\mathbf {z} _ {t}) - \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) \| ^ {2} + 2 \delta_ {t} ^ {2} \| \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} (\mathbf {z} _ {t}) + \tilde {\nabla} _ {\mathbf {y}} ^ {2} g _ {t} (\mathbf {z} _ {t}) \| ^ {2} \\ = 2 \delta_ {t} ^ {2} \| e _ {t} ^ {M} \| ^ {2} \\ + 2 \delta_ {t} ^ {2} \| \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) + \frac {1}{2 \rho_ {\mathbf {v}}} \left(\nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t})\right) \| ^ {2} \\ \leq 2 \delta_ {t} ^ {2} \| e _ {t} ^ {M} \| ^ {2} + 6 \delta_ {t} ^ {2} \| \nabla_ {\mathbf {y}} f _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \\ + \frac {3 \delta_ {t} ^ {2}}{2 \rho_ {\mathbf {v} ^ {2}}} \| \nabla_ {\mathbf {y}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} + \frac {3 \delta_ {t} ^ {2}}{2 \rho_ {\mathbf {v} ^ {2}}} \| \nabla_ {\mathbf {y}} g _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2}, \tag {218} \\ \end{array} +$$ + +where the second equality follows from (154). + +From Assumption B3., Lemma B.3 and (8), we have + +$$ +\begin{array}{l} \left\| \nabla_ {\mathbf {y}} g _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) \right\| ^ {2} \leq \ell_ {g, 1} ^ {2} \left\| \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right) \right\| ^ {2} \\ \leq 2 \ell_ {g, 1} ^ {2} \| \rho_ {\mathbf {v}} \mathbf {v} _ {t} \| ^ {2} + 2 \ell_ {g, 1} ^ {2} \| \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2} \\ \leq 2 \ell_ {g, 1} ^ {2} \rho_ {\mathbf {v}} ^ {2} p ^ {2} + 2 \ell_ {g, 1} ^ {2} \| \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2}. \tag {219} \\ \end{array} +$$ + +Similarly, we get + +$$ +\left\| \nabla_ {\mathbf {y}} g _ {t, \rho} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} - \rho_ {\mathbf {v}} \mathbf {v} _ {t}\right) \right\| ^ {2} \leq 2 \ell_ {g, 1} ^ {2} \rho_ {\mathbf {v}} ^ {2} p ^ {2} + 2 \ell_ {g, 1} ^ {2} \left\| \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \right\| ^ {2}. \tag {220} +$$ + +Moreover, from Eq. (132) and Assumption B1., we have + +$$ +\begin{array}{l} \left\| \nabla_ {\mathbf {y}} f _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2} \leq 2 \left\| \nabla_ {\mathbf {y}} f _ {t, \rho} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2} + 2 \left\| \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2} \\ \leq \frac {d _ {2} ^ {2} \ell_ {f , 1} ^ {2} \rho_ {\mathbf {r}} ^ {2}}{2} + 2 \| \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \| ^ {2} \\ \leq \frac {d _ {2} ^ {2} \ell_ {f , 1} ^ {2} \rho_ {\mathbf {r}} ^ {2}}{2} + 2 \ell_ {f, 0} ^ {2}. \tag {221} \\ \end{array} +$$ + +Substituting (219), (220) and (221), into (218), we get + +$$ +\begin{array}{l} \left\| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \right\| ^ {2} \leq 2 \delta_ {t} ^ {2} \left\| e _ {t} ^ {M} \right\| ^ {2} + 3 d _ {2} ^ {2} \ell_ {f, 1} ^ {2} \delta_ {t} ^ {2} \rho_ {\mathbf {r}} ^ {2} \\ + (1 2 \ell_ {f, 0} ^ {2} + 6 \ell_ {g, 1} ^ {2} p ^ {2}) \delta_ {t} ^ {2} + \frac {6 \ell_ {g , 1} ^ {2}}{\rho_ {\mathbf {v} ^ {2}}} \delta_ {t} ^ {2} \| \mathbf {y} _ {t} - \hat {\mathbf {y}} _ {t} ^ {*} (\mathbf {x} _ {t}) \| ^ {2}. \\ \end{array} +$$ + +![](images/618ad8a352d76ca3f7f51ca0a6e498b64ce7621e49585ca7d458500c2e5e7343.jpg) + +# D.7 Proof of Theorem 3.2 + +Proof. Since $(1 - \gamma_{t + 1})^2\leq 1 - \gamma_{t + 1}$ and $\gamma_{t + 1} = c_{\gamma}\alpha_{t}$ in (31), from (139), we have + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {g _ {\rho}} \| ^ {2} - \mathbb {E} \| e _ {t} ^ {g _ {\rho}} \| ^ {2} \leq - c _ {\gamma} \alpha_ {t} \mathbb {E} \| e _ {t} ^ {g _ {\rho}} \| ^ {2} \\ + 1 2 \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \left\| \nabla_ {\mathbf {y}} g _ {t - 1} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) - \nabla_ {\mathbf {y}} g _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \right\| ^ {2} \\ + 9 d _ {2} ^ {2} \ell_ {g, 1} ^ {2} (1 - \gamma_ {t + 1}) ^ {2} \rho_ {\mathbf {r}} ^ {2} + 2 4 d _ {2} \ell_ {g, 1} ^ {2} (1 - \gamma_ {t + 1}) ^ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} \\ + 2 4 d _ {2} \ell_ {g, 1} ^ {2} \left(1 - \gamma_ {t + 1}\right) ^ {2} \mathbb {E} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} + 2 \frac {\hat {\sigma} _ {g \mathbf {y}} ^ {2}}{\bar {b}} \gamma_ {t + 1} ^ {2}. \tag {222} \\ \end{array} +$$ + +Since $(1 - \eta_{t + 1})^2\leq 1 - \eta_{t + 1}$ and $\eta_{t + 1} = c_{\eta}\alpha_{t}$ in (31), from (191), we have + +$$ +\begin{array}{l} \mathbb {E} \left\| e _ {t + 1} ^ {L} \right\| ^ {2} - \mathbb {E} \left\| e _ {t} ^ {L} \right\| ^ {2} \leq - c _ {\eta} \alpha_ {t} \mathbb {E} \left\| e _ {t} ^ {L} \right\| ^ {2} + 3 6 \mathbb {E} \left\| \nabla_ {\mathbf {x}} f _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {x}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2} \\ + \left(1 8 d _ {1} ^ {2} \ell_ {f, 1} ^ {2} + 6 (3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}) d _ {1} ^ {2}\right) \rho_ {\mathbf {s}} ^ {2} + 1 8 d _ {1} ^ {2} \ell_ {g, 1} ^ {2} \frac {\rho_ {\mathbf {s}} ^ {2}}{\rho_ {\mathbf {v}} ^ {2}} \\ + \frac {3 6}{\rho_ {\mathbf {v}} ^ {2}} \mathbb {E} \| \nabla_ {\mathbf {x}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {x}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} \\ + 6 \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {1} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 6 \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {1} \mathbb {E} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} \\ + 2 7 \ell_ {g, 1} ^ {2} d _ {1} \mathbb {E} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} + 3 \left(\frac {\hat {\sigma} _ {g _ {\mathbf {x}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {x}}} ^ {2}}{b}\right) \eta_ {t + 1} ^ {2}. \tag {223} \\ \end{array} +$$ + +Since $(1 - \lambda_{t + 1})^2\leq 1 - \lambda_{t + 1}$ and $\lambda_{t + 1} = c_{\lambda}\alpha_{t}$ in (31), from (156), we have + +$$ +\begin{array}{l} \mathbb {E} \| e _ {t + 1} ^ {M} \| ^ {2} - \mathbb {E} \| e _ {t} ^ {M} \| ^ {2} \leq - c _ {\lambda} \alpha_ {t} \mathbb {E} \| e _ {t} ^ {M} \| ^ {2} + 3 6 \mathbb {E} \left\| \nabla_ {\mathbf {y}} f _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) - \nabla_ {\mathbf {y}} f _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t}) \right\| ^ {2} \\ + \left(1 8 d _ {2} ^ {2} \ell_ {f, 1} ^ {2} + 6 (3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}) d _ {2} ^ {2}\right) \rho_ {\mathbf {r}} ^ {2} + 1 8 d _ {2} ^ {2} \ell_ {g, 1} ^ {2} \frac {\rho_ {\mathbf {r}} ^ {2}}{\rho_ {\mathbf {v}} ^ {2}} \\ + \frac {3 6}{\rho_ {\mathbf {v}} ^ {2}} \mathbb {E} \| \nabla_ {\mathbf {y}} g _ {t + 1} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) - \nabla_ {\mathbf {y}} g _ {t} (\mathbf {x} _ {t}, \mathbf {y} _ {t} + \rho_ {\mathbf {v}} \mathbf {v} _ {t}) \| ^ {2} \\ + 6 \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {2} \mathbb {E} \| \mathbf {x} _ {t + 1} - \mathbf {x} _ {t} \| ^ {2} + 6 \left(1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}\right) d _ {2} \mathbb {E} \| \mathbf {y} _ {t + 1} - \mathbf {y} _ {t} \| ^ {2} \\ + 2 7 d _ {2} \ell_ {g, 1} ^ {2} \mathbb {E} \| \mathbf {v} _ {t + 1} - \mathbf {v} _ {t} \| ^ {2} + 3 \left(\frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {y}}} ^ {2}}{b}\right) \lambda_ {t + 1} ^ {2}. \tag {224} \\ \end{array} +$$ + +Combining the outcomes. + +Let + +$$ +\begin{array}{l} \Lambda := \Gamma \sum_ {t = 1} ^ {T} \left(\mathbb {E} [ \hat {\theta} _ {t + 1} ^ {\mathbf {y}} ] - \mathbb {E} [ \hat {\theta} _ {t} ^ {\mathbf {y}} ]\right) \\ + \Upsilon \sum_ {t = 1} ^ {T} \left(\mathbb {E} [ \hat {\theta} _ {t + 1} ^ {\mathbf {v}} ] - \mathbb {E} [ \hat {\theta} _ {t} ^ {\mathbf {v}} ]\right) + \frac {1}{\Phi} \sum_ {t = 1} ^ {T} \left(\mathbb {E} \| e _ {t + 1} ^ {g _ {\rho}} \| ^ {2} - \mathbb {E} \| e _ {t} ^ {g _ {\rho}} \| ^ {2}\right) \\ + \frac {1}{\Psi} \sum_ {t = 1} ^ {T} \left(\mathbb {E} \| e _ {t + 1} ^ {M} \| ^ {2} - \mathbb {E} \| e _ {t} ^ {M} \| ^ {2}\right) + \frac {1}{\Omega} \sum_ {t = 1} ^ {T} \left(\mathbb {E} \| e _ {t + 1} ^ {L} \| ^ {2} - \mathbb {E} \| e _ {t} ^ {L} \| ^ {2}\right). \\ \end{array} +$$ + +Here, we have + +$$ +\Gamma = \frac {1 1 M _ {f} ^ {2}}{L _ {\mu_ {g}} c _ {\beta}}, \quad \Upsilon = \frac {5 2 M _ {f} ^ {2}}{L _ {\mu_ {g}} c _ {\delta}}, +$$ + +$$ +\Phi = \max \left\{2 4 0 \frac {d _ {2} \ell_ {g , 1} ^ {2}}{L _ {f}}, \frac {1 2 d _ {2} \ell_ {g , 1} ^ {2} L _ {\mu_ {g}} ^ {2} c _ {\beta} ^ {2}}{L _ {f} M _ {f} ^ {2}} \right\}, \tag {225} +$$ + +$$ +\Psi = \max \left\{7 2 0 \frac {d _ {2} \ell_ {f , 1} ^ {2}}{L _ {f}}, 2 7 \frac {L _ {\mu_ {g}}}{\Upsilon L _ {f}} \ell_ {g, 1} ^ {2} d _ {2} c _ {\delta}, \frac {1 4 4 d _ {2} \ell_ {f , 1} ^ {2} (\mu_ {g} + \ell_ {g , 1}) c _ {\beta}}{L _ {f} \Gamma}, \frac {3 6 \ell_ {f , 1} ^ {2} d _ {2} L _ {\mu_ {g}} ^ {2} c _ {\beta} ^ {2}}{L _ {f} M _ {f} ^ {2}} \right\}, +$$ + +$$ +\Omega = \max \left\{7 2 0 \frac {d _ {1} \ell_ {f , 1} ^ {2}}{L _ {f}}, 2 7 \frac {L _ {\mu_ {g}}}{\Upsilon L _ {f}} \ell_ {g, 1} ^ {2} d _ {1} c _ {\delta}, \frac {1 4 4 d _ {1} \ell_ {f , 1} ^ {2} (\mu_ {g} + \ell_ {g , 1}) c _ {\beta}}{L _ {f} \Gamma}, \frac {3 6 \ell_ {f , 1} ^ {2} d _ {1} L _ {\mu_ {g}} ^ {2} c _ {\beta} ^ {2}}{L _ {f} M _ {f} ^ {2}} \right\}, +$$ + +with + +$$ +c _ {\beta} \geq \sqrt {1 7 6 0} \frac {L _ {\mathbf {y}} M _ {f}}{L _ {\mu_ {g}}}, +$$ + +$$ +c _ {\delta} \geq \sqrt {3 3 2 8 0 (1 + 2 L _ {\mathbf {y}} ^ {2})} \frac {\nu M _ {f}}{L _ {\mu_ {g}} \mu_ {g}}, +$$ + +$$ +c \geq \left(\max \left\{4 L _ {f}, c _ {\beta} \left(\mu_ {g} + \ell_ {g, 1}\right), \frac {4 8 L _ {\mu_ {g}} ^ {2} d _ {2} \ell_ {g , 1} ^ {2} c _ {\beta} ^ {2}}{M _ {f} ^ {2} \Phi} \right\}\right) ^ {3} + 1, \tag {226} +$$ + +$$ +c _ {\mathbf {v}} = \max \left\{1 0 8 0 \ell_ {g, 1} ^ {2}, \frac {3 2 4}{M _ {f} ^ {2}} \ell_ {g, 1} ^ {4} c _ {\delta} ^ {2}, \frac {5 4 L _ {\mu_ {g}} ^ {2}}{M _ {f} ^ {2}} \ell_ {g, 1} ^ {2} c _ {\beta} ^ {2}, \frac {2 1 6}{\Gamma} \ell_ {g, 1} ^ {2} c _ {\beta} (\mu_ {g} + \ell_ {g, 1}) \right\} \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right), +$$ + +$$ +c _ {\gamma} = \frac {2 6 M _ {f} ^ {2} \Phi}{L _ {\mu_ {g}} ^ {2}}, +$$ + +$$ +c _ {\eta} = 2 6 \Omega , c _ {\lambda} = \frac {1 0 \Upsilon}{L _ {\mu_ {g}}} c _ {\delta} \Psi . +$$ + +By adding (223), (222), (224), (143), and (180), along with (209) and considering the fact that $\alpha_{t}$ decreases with respect to $t$ , and by applying Lemma D.21, we obtain: + +$$ +\sum_ {t = 1} ^ {T} A \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) \mathbb {E} \left[ \| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \| ^ {2} \right] + \Lambda +$$ + +$$ +\leq 1 2 M + 6 V _ {T} + \sum_ {t = 1} ^ {T} B \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) \mathbb {E} \left[ \hat {\theta} _ {t} ^ {\mathbf {v}} \right] + \sum_ {t = 1} ^ {T} C \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) \mathbb {E} \left[ \hat {\theta} _ {t} ^ {\mathbf {y}} \right] \tag {227a} +$$ + +$$ ++ \frac {4 \ell_ {f , 1} \ell_ {g , 1}}{\mu_ {g}} \sum_ {t = 1} ^ {T} E \left(\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) \alpha_ {t} ^ {2} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right) + \sum_ {t = 1} ^ {T} L \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) \mathbb {E} \| e _ {t} ^ {L} \| ^ {2} \tag {227b} +$$ + +$$ ++ \frac {8 \ell_ {g , 2} ^ {2} p ^ {4} \Upsilon}{L _ {\mu_ {g}}} \sum_ {t = 1} ^ {T} \delta_ {t} \rho_ {\mathbf {v}} ^ {2} + 4 \ell_ {g, 2} ^ {2} p ^ {4} \sum_ {t = 1} ^ {T} \left(6 \alpha_ {t} - 3 L _ {f} \alpha_ {t} ^ {2} + 2 \alpha_ {t} ^ {2} E \left(\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right)\right) \rho_ {\mathbf {v}} ^ {2} \tag {227c} +$$ + +$$ ++ \left(\frac {1 2}{L _ {\mu_ {g}}} \frac {\Gamma}{\beta_ {T}} + \frac {4 8 \nu^ {2}}{L _ {\mu_ {g}} \mu_ {g} ^ {2}} \frac {\Upsilon}{\delta_ {T}}\right) H _ {2, T} + \sum_ {t = 1} ^ {T} M (\delta_ {t}) \mathbb {E} \| e _ {t} ^ {M} \| ^ {2} \tag {227d} +$$ + +$$ ++ \sum_ {t = 1} ^ {T} Q \left(\beta_ {t}, \rho_ {\mathbf {v}}\right) \mathbb {E} \left\| e _ {t} ^ {g _ {\rho}} \right\| ^ {2} + \sum_ {t = 1} ^ {T} S \left(\beta_ {t}, \rho_ {\mathbf {v}}\right) \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t, \rho} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \| ^ {2} \right] \tag {227e} +$$ + +$$ ++ \sum_ {t = 1} ^ {T} Z \left(3 d _ {2} ^ {2} \ell_ {f, 1} ^ {2} \delta_ {t} ^ {2} \rho_ {\mathbf {r}} ^ {2} + \left(1 2 \ell_ {f, 0} ^ {2} + 6 \ell_ {g, 1} ^ {2} p ^ {2}\right) \delta_ {t} ^ {2}\right) \tag {227f} +$$ + +$$ ++ \frac {3 6}{\Psi} D _ {\mathbf {y}, T} + \frac {3 6}{\Omega} D _ {\mathbf {x}, T} + \frac {1 2}{\Phi} G _ {\mathbf {y}, T} + \frac {1 8}{\Psi \rho_ {\mathbf {v}} ^ {2}} G _ {\mathbf {v}, T} + \frac {1 8}{\Omega \rho_ {\mathbf {v}} ^ {2}} G _ {\mathbf {x}, T} \tag {227g} +$$ + +$$ ++ 2 \sum_ {t = 1} ^ {T} \frac {\gamma_ {t + 1} ^ {2}}{\Phi} \frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b}} + 3 \left(\frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {y}}} ^ {2}}{b}\right) \sum_ {t = 1} ^ {T + 1} \frac {\lambda_ {t + 1} ^ {2}}{\Psi} + 3 \left(\frac {\hat {\sigma} _ {g _ {\mathbf {x}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {x}}} ^ {2}}{b}\right) \sum_ {t = 1} ^ {T} \frac {\eta_ {t + 1} ^ {2}}{\Omega} \tag {227h} +$$ + +$$ ++ R \left(\rho_ {\mathbf {v}}\right) T \rho_ {\mathbf {r}} ^ {2} + \dot {R} \left(\rho_ {\mathbf {v}}\right) T \rho_ {\mathbf {s}} ^ {2} + 1 8 T \ell_ {g, 1} ^ {2} \left(\frac {d _ {1} ^ {2} \rho_ {\mathbf {s}} ^ {2}}{\Omega \rho_ {\mathbf {v}} ^ {2}} + \frac {d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2}}{\Psi \rho_ {\mathbf {v}} ^ {2}}\right) + \sum_ {t = 1} ^ {T} D \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}\right) \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right). \tag {227i} +$$ + +Here, + +$$ +\begin{array}{l} E (\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}) := \frac {4 L _ {\mathbf {y}} ^ {2}}{L _ {\mu_ {g}}} \frac {\Gamma}{\beta_ {t}} + \frac {1 6 \nu^ {2}}{L _ {\mu_ {g}} \mu_ {g} ^ {2}} (2 L _ {\mathbf {y}} ^ {2} + 1) \frac {\Upsilon}{\delta_ {t}} \\ + 2 4 d _ {2} \frac {\ell_ {g , 1} ^ {2}}{\Phi} + 6 (1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}) \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right), \\ \end{array} +$$ + +$$ +A \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) := \alpha_ {t} - L _ {f} \alpha_ {t} ^ {2} - 4 E \left(\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) \alpha_ {t} ^ {2}, +$$ + +$$ +B \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) := - \frac {L _ {\mu_ {g}}}{4} \Upsilon \delta_ {t} + 2 M _ {f} ^ {2} \left(6 \alpha_ {t} - 3 L _ {f} \alpha_ {t} ^ {2} + 2 \alpha_ {t} ^ {2} E \left(\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right)\right), +$$ + +$$ +\begin{array}{l} C \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) := - \frac {L _ {\mu_ {g}}}{2} \Gamma \beta_ {t} + Z 6 \ell_ {g, 1} ^ {2} \frac {\delta_ {t} ^ {2}}{\rho_ {\mathbf {v}} ^ {2}} \\ + 2 M _ {f} ^ {2} \left(6 \alpha_ {t} - 3 L _ {f} \alpha_ {t} ^ {2} + 2 \alpha_ {t} ^ {2} E \left(\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right)\right), \\ Z := 2 7 \ell_ {g, 1} ^ {2} \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right). \tag {228} \\ \end{array} +$$ + +Moreover, + +$$ +\begin{array}{l} M (\delta_ {t}) := - \frac {\lambda_ {t + 1}}{\Psi} + Z 2 \delta_ {t} ^ {2} + \frac {8 \Upsilon}{L _ {\mu_ {g}}} \delta_ {t}, \\ D \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}\right) := 6 \ell_ {f, 1} \left(1 + 2 \frac {\ell_ {g , 1}}{\mu_ {g}}\right) + \frac {3 \ell_ {f , 1} \ell_ {g , 1}}{\mu_ {g}} \left(\alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) \\ + \frac {2 4 \ell_ {g , 1}}{L _ {\mu_ {g}} \mu_ {g}} \frac {\Gamma}{\beta_ {t}} + \frac {9 6 \ell_ {g , 1} \nu^ {2}}{L _ {\mu_ {g}} \mu_ {g} ^ {3}} \frac {\Upsilon}{\delta_ {t}}, \\ \end{array} +$$ + +$$ +F (\rho_ {\mathbf {v}}) := 2 4 d _ {2} \frac {\ell_ {g , 1} ^ {2}}{\Phi} + (7 2 \ell_ {f, 1} ^ {2} + \frac {2 7 \ell_ {g , 1} ^ {2}}{\rho_ {\mathbf {v}} ^ {2}}) \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right), +$$ + +$$ +S \left(\beta_ {t}, \rho_ {\mathbf {v}}\right) := - \frac {2 \beta_ {t} \Gamma}{\mu_ {g} + \ell_ {g , 1}} + \beta_ {t} ^ {2} \Gamma + 2 F \left(\rho_ {\mathbf {v}}\right) \beta_ {t} ^ {2}, \tag {229} +$$ + +$$ +Q (\beta_ {t}, \rho_ {\mathbf {v}}) := \frac {2}{L _ {\mu_ {g}}} \Gamma \beta_ {t} - \frac {\gamma_ {t + 1}}{\Phi} + 2 F (\rho_ {\mathbf {v}}) \beta_ {t} ^ {2}, +$$ + +$$ +R (\rho_ {\mathbf {v}}) := 9 d _ {2} ^ {2} \frac {\ell_ {g , 1} ^ {2}}{\Phi} + 1 8 d _ {2} ^ {2} \frac {\ell_ {f , 1} ^ {2}}{\Psi} + 6 (3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}) \frac {d _ {2} ^ {2}}{\Psi}, +$$ + +$$ +\acute {R} (\rho_ {\mathbf {v}}) := 1 8 d _ {1} ^ {2} \frac {\ell_ {f , 1} ^ {2}}{\Omega} + 6 (3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}) \frac {d _ {1} ^ {2}}{\Omega}, +$$ + +$$ +L \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) := - \frac {\eta_ {t + 1}}{\Omega} + 4 \left(6 \alpha_ {t} - 3 L _ {f} \alpha_ {t} ^ {2} + 2 \alpha_ {t} ^ {2} E \left(\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right)\right). +$$ + +We then provide bounds for the terms in (227a)-(227i). + +Note that, we have + +$$ +\begin{array}{l} E (\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}) := \frac {4 L _ {\mathbf {y}} ^ {2}}{L _ {\mu_ {g}}} \frac {\Gamma}{\beta_ {t}} + \frac {1 6 \nu^ {2}}{L _ {\mu_ {g}} \mu_ {g} ^ {2}} (2 L _ {\mathbf {y}} ^ {2} + 1) \frac {\Upsilon}{\delta_ {t}} \\ + 2 4 d _ {2} \frac {\ell_ {g , 1} ^ {2}}{\Phi} + 6 (1 2 \ell_ {f, 1} ^ {2} + \frac {9 \ell_ {g , 1} ^ {2}}{2 \rho_ {\mathbf {v}} ^ {2}}) (\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}), \\ \end{array} +$$ + +which together with $\beta_{t} = c_{\beta}\alpha_{t}$ , $\delta_t = c_\delta \alpha_t$ in (31), we have + +$$ +\begin{array}{l} \alpha_ {t} ^ {2} E \left(\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) = \frac {4 L _ {\mathbf {y}} ^ {2}}{L _ {\mu_ {g}}} \frac {\Gamma \alpha_ {t} ^ {2}}{\beta_ {t}} + \frac {1 6 \nu^ {2}}{L _ {\mu_ {g}} \mu_ {g} ^ {2}} \left(2 L _ {\mathbf {y}} ^ {2} + 1\right) \frac {\Upsilon \alpha_ {t} ^ {2}}{\delta_ {t}} \\ + 2 4 d _ {2} \frac {\ell_ {g , 1} ^ {2}}{\Phi} \alpha_ {t} ^ {2} + (7 2 \ell_ {f, 1} ^ {2} \alpha_ {t} ^ {2} + \frac {2 7 \ell_ {g , 1} ^ {2}}{\rho_ {\mathbf {v}} ^ {2}} \alpha_ {t} ^ {2}) \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right) \\ \leq \frac {4 4 L _ {\mathbf {y}} ^ {2}}{L _ {\mu_ {g}} ^ {2}} M _ {f} ^ {2} \frac {\alpha_ {t}}{c _ {\beta} ^ {2}} + \frac {8 3 2 \nu^ {2}}{L _ {\mu_ {g}} ^ {2} \mu_ {g} ^ {2}} (1 + 2 L _ {\mathbf {y}} ^ {2}) M _ {f} ^ {2} \frac {\alpha_ {t}}{c _ {\delta} ^ {2}} \\ + 6 \frac {d _ {2} \ell_ {g , 1} ^ {2}}{L _ {f} \Phi} \alpha_ {t} + \left(\frac {1 8 \ell_ {f , 1} ^ {2}}{L _ {f}} \alpha_ {t} + \frac {2 7 \ell_ {g , 1} ^ {2}}{c _ {\mathbf {v}}} \alpha_ {t}\right) \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right) \\ \leq \frac {\alpha_ {t}}{8}, \tag {230} \\ \end{array} +$$ + +where the first inequality is by $\Gamma = \frac{11M_f^2}{L_{\mu_g}c_\beta}$ , $\Upsilon = \frac{52M_f^2}{L_{\mu_g}c_\delta}$ in (225), $\rho_{\mathbf{v}}^{2} = c_{\mathbf{v}}\alpha_{t}$ and $\alpha_{t}\leq 1 / 4L_{f}$ in (31); the second inequality follows from $c_{\beta}\geq \sqrt{1760\frac{L_y^2M_f^2}{L_{\mu_g}^2}}$ $c_{\delta}\geq \sqrt{33280\frac{\nu^2M_f^2}{L_{\mu_g}^2\mu_g^2}(1 + 2L_y^2)}$ in (226); and $\Phi = 240\frac{d_2\ell_{g,1}^2}{L_f}$ , $\Psi = 720\frac{d_2\ell_{f,1}^2}{L_f}$ , $\Omega = 720\frac{d_1\ell_{f,1}^2}{L_f}$ and $c_{\mathbf{v}}\geq 1080\ell_{g,1}^{2}\left(\frac{d_2}{\Psi} +\frac{d_1}{\Omega}\right)$ in (225). + +Moreover, we have + +$$ +\begin{array}{l} A \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) = \alpha_ {t} - L _ {f} \alpha_ {t} ^ {2} - 4 E \left(\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) \alpha_ {t} ^ {2} \\ \geq \alpha_ {t} - L _ {f} \alpha_ {t} ^ {2} - \frac {\alpha_ {t}}{2} \\ \geq \frac {\alpha_ {t}}{4}, \tag {231} \\ \end{array} +$$ + +where the last inequality is by $\alpha_{t} \leq 1 / 4L_{f}$ in (226). + +Bounding (227a). + +From $\delta_t = c_\delta \alpha_t$ in (31), we have + +$$ +\begin{array}{l} B (\alpha_ {t}, \beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}) = - \frac {L _ {\mu_ {g}}}{4} \Upsilon \delta_ {t} + 2 M _ {f} ^ {2} \left(6 \alpha_ {t} - 3 L _ {f} \alpha_ {t} ^ {2} + 2 \alpha_ {t} ^ {2} E (\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}})\right) \\ \leq - \frac {L _ {\mu_ {g}}}{4} \Upsilon c _ {\delta} \alpha_ {t} + 1 2 M _ {f} ^ {2} \alpha_ {t} - 6 M _ {f} ^ {2} L _ {f} \alpha_ {t} ^ {2} + \frac {M _ {f} ^ {2}}{2} \alpha_ {t} \\ \leq \left(- \frac {L _ {\mu_ {g}}}{4} \Upsilon c _ {\delta} + \frac {2 5}{2} M _ {f} ^ {2}\right) \alpha_ {t} \\ \leq - \frac {1}{2} M _ {f} ^ {2} \alpha_ {t}, \tag {232} \\ \end{array} +$$ + +where the first inequality follows from (230); the last inequality is by $\Upsilon = \frac{52M_f^2}{L_{\mu_g}c_\delta}$ in (225). From (228), we obtain + +$$ +Z = 2 7 \ell_ {g, 1} ^ {2} \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right). +$$ + +Thus, from $\beta_{t} = c_{\beta}\alpha_{t},\delta_{t} = c_{\delta}\alpha_{t}$ and $\rho_{\mathbf{v}}^{2} = c_{\mathbf{v}}\alpha_{t}$ in (31), we have + +$$ +\begin{array}{l} C (\alpha_ {t}, \beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}) = - \frac {L _ {\mu_ {g}}}{2} \Gamma \beta_ {t} + 1 6 2 (\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}) \ell_ {g, 1} ^ {4} \frac {\delta_ {t} ^ {2}}{\rho_ {\mathbf {v}} ^ {2}} \\ + 2 M _ {f} ^ {2} \left(6 \alpha_ {t} - 3 L _ {f} \alpha_ {t} ^ {2} + 2 \alpha_ {t} ^ {2} E \left(\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right)\right) \\ \leq - \frac {L _ {\mu_ {g}}}{2} \Gamma c _ {\beta} \alpha_ {t} + 1 6 2 (\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}) \ell_ {g, 1} ^ {4} \frac {c _ {\delta} ^ {2}}{c _ {\mathbf {v}}} \alpha_ {t} + \frac {9}{2} M _ {f} ^ {2} \alpha_ {t} \\ = - \frac {1 1}{2} M _ {f} ^ {2} \alpha_ {t} + 1 6 2 (\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}) \ell_ {g, 1} ^ {4} \frac {c _ {\delta} ^ {2}}{c _ {\mathbf {v}}} \alpha_ {t} + \frac {9}{2} M _ {f} ^ {2} \alpha_ {t} \\ \leq - \frac {1}{2} M _ {f} ^ {2} \alpha_ {t}, \tag {233} \\ \end{array} +$$ + +where the first inequality follows from (230); the second equality follows from $\Gamma = \frac{11M_f^2}{L_{\mu_g}c_\beta}$ in (225); the last inequality is by $c_{\mathbf{v}}\geq \frac{324}{M_f^2}\ell_{g,1}^4 (\frac{d_2}{\Psi} +\frac{d_1}{\Omega})c_\delta^2$ + +Thus, from (232) and (233), we get + +$$ +(2 2 7 \mathrm {a}) \leq \mathcal {O} \left(V _ {T}\right). \tag {234} +$$ + +Bounding (227b). + +From (230), we also obtain + +$$ +\begin{array}{l} \frac {4 \ell_ {f , 1} \ell_ {g , 1}}{\mu_ {g}} \sum_ {t = 1} ^ {T} E (\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}) \alpha_ {t} ^ {2} (\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}) \\ \leq \frac {4 \ell_ {f , 1} \ell_ {g , 1}}{\mu_ {g}} \sum_ {t = 1} ^ {T} \frac {\alpha_ {t}}{8} (\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}) \\ = \mathcal {O} \left(\sum_ {t = 1} ^ {T} \alpha_ {t} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)\right). \tag {235} \\ \end{array} +$$ + +From (229) and $\eta_{t + 1} = c_{\eta}\alpha_{t}$ in (31), we have + +$$ +\begin{array}{l} L \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) = - \frac {\eta_ {t + 1}}{\Omega} + 4 \left(6 \alpha_ {t} - 3 L _ {f} \alpha_ {t} ^ {2} + 2 \alpha_ {t} ^ {2} E \left(\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right)\right) \\ \leq - \frac {c _ {\eta}}{\Omega} \alpha_ {t} + 2 5 \alpha_ {t} \\ \leq - \alpha_ {t}, \\ \end{array} +$$ + +where the last inequality is by $c_{\eta}\geq 26\Omega$ and (230). + +Thus, we get + +$$ +\sum_ {t = 1} ^ {T} L \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}}\right) \mathbb {E} \| e _ {t} ^ {L} \| ^ {2} \leq 0. \tag {236} +$$ + +From (236) and (235), we have + +$$ +(2 2 7 \mathrm {b}) \leq \mathcal {O} \left(\sum_ {t = 1} ^ {T} \alpha_ {t} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)\right). \tag {237} +$$ + +Bounding (227c). + +From $\delta_t = c_\delta \alpha_t$ in (31) and Eq. (230), we have + +$$ +\begin{array}{l} \frac {8 \ell_ {g , 2} ^ {2} p ^ {4} \Upsilon}{L _ {\mu_ {g}}} \sum_ {t = 1} ^ {T} \delta_ {t} \rho_ {\mathbf {v}} ^ {2} + 4 \ell_ {g, 2} ^ {2} p ^ {4} \sum_ {t = 1} ^ {T} \left(6 \alpha_ {t} - 3 L _ {f} \alpha_ {t} ^ {2} + 2 \alpha_ {t} ^ {2} E (\beta_ {t}, \delta_ {t}, \rho_ {\mathbf {v}})\right) \rho_ {\mathbf {v}} ^ {2} \\ \leq \frac {8 \ell_ {g , 2} ^ {2} p ^ {4} \Upsilon}{L _ {\mu_ {g}}} \sum_ {t = 1} ^ {T} c _ {\delta} \alpha_ {t} \rho_ {\mathbf {v}} ^ {2} + 4 \ell_ {g, 2} ^ {2} p ^ {4} \sum_ {t = 1} ^ {T} \frac {2 5}{4} \alpha_ {t} \rho_ {\mathbf {v}} ^ {2}. \\ \end{array} +$$ + +Thus, from $\rho_{\mathbf{v}}^2 = c_{\mathbf{v}}\alpha_t$ in (31), we have + +$$ +(2 2 7 \mathrm {c}) \leq \mathcal {O} \left(\sum_ {t = 1} ^ {T} \alpha_ {t} ^ {2}\right). \tag {238} +$$ + +Bounding (227d). + +From (228), we have + +$$ +Z = 2 7 \ell_ {g, 1} ^ {2} \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right). \tag {239} +$$ + +From (229), $\lambda_{t + 1} = c_{\lambda}\alpha_{t}$ and $\delta_t = c_\delta \alpha_t$ in (31), we have + +$$ +\begin{array}{l} M (\delta_ {t}) = - \frac {\lambda_ {t + 1}}{\Psi} + Z 2 \delta_ {t} ^ {2} + \frac {8 \Upsilon}{L _ {\mu_ {g}}} \delta_ {t} \\ = - \frac {c _ {\lambda} \alpha_ {t}}{\Psi} + 2 7 \ell_ {g, 1} ^ {2} \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right) 2 c _ {\delta} ^ {2} \alpha_ {t} ^ {2} + \frac {8 \Upsilon}{L _ {\mu_ {g}}} c _ {\delta} \alpha_ {t} \\ \leq - \frac {2 \Upsilon}{L _ {\mu_ {g}}} c _ {\delta} \alpha_ {t} + \frac {2 7}{4 L _ {f}} \ell_ {g, 1} ^ {2} \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right) 2 c _ {\delta} ^ {2} \alpha_ {t} \\ \leq - \frac {\Upsilon}{L _ {\mu_ {g}}} c _ {\delta} \alpha_ {t}, \\ \end{array} +$$ + +where the first inequality is by $c_{\lambda} \geq \frac{10\Upsilon}{L_{\mu_q}} c_{\delta}\Psi$ and $\alpha_t \leq 1 / 4L_f$ ; the last inequality follows from + +$$ +\Psi \geq 2 7 \frac {L _ {\mu g}}{\Upsilon L _ {f}} \ell_ {g, 1} ^ {2} d _ {2} c _ {\delta} \text {a n d} \Omega \geq 2 7 \frac {L _ {\mu g}}{\Upsilon L _ {f}} \ell_ {g, 1} ^ {2} d _ {1} c _ {\delta}. +$$ + +Since $\beta_{t} = c_{\beta}\alpha_{t}$ and $\delta_t = c_\delta \alpha_t$ in (31), we get + +$$ +\begin{array}{l} (2 2 7 \mathrm {d}) = \left(\frac {1 2}{L _ {\mu_ {g}}} \frac {\Gamma}{\beta_ {T}} + \frac {4 8 \nu^ {2}}{L _ {\mu_ {g}} \mu_ {g} ^ {2}} \frac {\Upsilon}{\delta_ {T}}\right) H _ {2, T} + \sum_ {t = 1} ^ {T} M (\delta_ {t}) \mathbb {E} \| e _ {t} ^ {M} \| ^ {2} \\ \leq \mathcal {O} \left(\frac {H _ {2 , T}}{\alpha_ {T}}\right). \tag {240} \\ \end{array} +$$ + +Bounding (227e) + +From (229), we have + +$$ +F \left(\rho_ {\mathbf {v}}\right) = 2 4 d _ {2} \frac {\ell_ {g , 1} ^ {2}}{\Phi} + \left(7 2 \ell_ {f, 1} ^ {2} + \frac {2 7 \ell_ {g , 1} ^ {2}}{\rho_ {\mathbf {v}} ^ {2}}\right) \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right). \tag {241} +$$ + +From (229), $\gamma_{t + 1} = c_{\gamma}\alpha_{t},\beta_{t} = c_{\beta}\alpha_{t}$ in (31), we have + +$$ +\begin{array}{l} Q (\beta_ {t}, \rho_ {\mathbf {v}}) = - \frac {\gamma_ {t + 1}}{\Phi} + \frac {2}{L _ {\mu_ {g}}} \Gamma \beta_ {t} + 2 F (\rho_ {\mathbf {v}}) \beta_ {t} ^ {2} \\ = - \frac {c _ {\gamma} \alpha_ {t}}{\Phi} + \frac {2 2 M _ {f} ^ {2}}{L _ {\mu_ {g}} ^ {2}} \alpha_ {t} + \left(2 4 d _ {2} \frac {\ell_ {g , 1} ^ {2}}{\Phi} + (7 2 \ell_ {f, 1} ^ {2} + \frac {2 7 \ell_ {g , 1} ^ {2}}{c _ {\mathbf {v}} \alpha_ {t}}) (\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega})\right) 2 c _ {\beta} ^ {2} \alpha_ {t} ^ {2} \\ \leq - \frac {4 M _ {f} ^ {2}}{L _ {\mu_ {g}} ^ {2}} \alpha_ {t} + \left(2 4 d _ {2} \frac {\ell_ {g , 1} ^ {2}}{\Phi} \alpha_ {t} ^ {2} + (7 2 \ell_ {f, 1} ^ {2} \alpha_ {t} ^ {2} + \frac {2 7 \ell_ {g , 1} ^ {2} \alpha_ {t}}{c _ {\mathbf {v}}}) (\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega})\right) 2 c _ {\beta} ^ {2} \\ \leq - \frac {4 M _ {f} ^ {2}}{L _ {\mu_ {g}} ^ {2}} \alpha_ {t} + \left(\frac {6 d _ {2}}{L _ {f}} \frac {\ell_ {g , 1} ^ {2}}{\Phi} \alpha_ {t} + \left(\frac {1 8}{L _ {f}} \ell_ {f, 1} ^ {2} \alpha_ {t} + \frac {2 7 \ell_ {g , 1} ^ {2} \alpha_ {t}}{c _ {\mathbf {v}}}\right) \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right)\right) 2 c _ {\beta} ^ {2} \\ \leq - \frac {M _ {f} ^ {2}}{L _ {\mu_ {g}} ^ {2}} \alpha_ {t}, \tag {242} \\ \end{array} +$$ + +where the first equality is by $\Gamma = \frac{11M_f^2}{L_{\mu_g}c_\beta}$ and $\rho_{\mathbf{v}}^{2} = c_{\mathbf{v}}\alpha_{t}$ ; the first inequality follows from $c_{\gamma} \geq \frac{26M_f^2\Phi}{L_{\mu_g}^2}$ ; the second inequality is by $\alpha_t \leq 1/4L_f$ ; the last inequality follows from $c_{\mathbf{v}} \geq \frac{54L_{\mu g}^2}{M_f^2}\ell_{g,1}^2\left(\frac{d_2}{\Psi} + \frac{d_1}{\Omega}\right)c_\beta^2$ , $\Phi \geq \frac{12d_2\ell_{g,1}^2L_{\mu g}^2c_\beta^2}{L_fM_f^2}$ , and $\Psi \geq \frac{36\ell_{f,1}^2d_2L_{\mu g}^2c_\beta^2}{L_fM_f^2}$ , and $\Omega \geq \frac{36\ell_{f,1}^2d_1L_{\mu g}^2c_\beta^2}{L_fM_f^2}$ . + +From (229), $\beta_{t} = c_{\beta}\alpha_{t}$ $\rho_{\mathbf{v}}^{2} = c_{\mathbf{v}}\alpha_{t}$ in (31) and (241), we have + +$$ +\begin{array}{l} S (\beta_ {t}, \rho_ {\mathbf {v}}) = - \frac {2 \beta_ {t} \Gamma}{\mu_ {g} + \ell_ {g , 1}} + \beta_ {t} ^ {2} \Gamma + 2 F (\rho_ {\mathbf {v}}) \beta_ {t} ^ {2} \\ = - \frac {2 c _ {\beta} \alpha_ {t} \Gamma}{\mu_ {g} + \ell_ {g , 1}} + c _ {\beta} ^ {2} \alpha_ {t} ^ {2} \Gamma + \left(2 4 d _ {2} \frac {\ell_ {g , 1} ^ {2}}{\Phi} + (7 2 \ell_ {f, 1} ^ {2} + \frac {2 7 \ell_ {g , 1} ^ {2}}{c _ {\mathbf {v}} \alpha_ {t}}) \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right)\right) 2 c _ {\beta} ^ {2} \alpha_ {t} ^ {2} \\ \leq - \frac {c _ {\beta} \alpha_ {t} \Gamma}{\mu_ {g} + \ell_ {g , 1}} + \left(2 4 d _ {2} \frac {\ell_ {g , 1} ^ {2}}{\Phi} \alpha_ {t} ^ {2} + (7 2 \ell_ {f, 1} ^ {2} \alpha_ {t} ^ {2} + \frac {2 7 \ell_ {g , 1} ^ {2} \alpha_ {t}}{c _ {\mathbf {v}}}) (\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega})\right) 2 c _ {\beta} ^ {2} \\ \leq - \frac {c _ {\beta} \alpha_ {t} \Gamma}{\mu_ {g} + \ell_ {g , 1}} + \left(\frac {6 d _ {2}}{L _ {f}} \frac {\ell_ {g , 1} ^ {2}}{\Phi} \alpha_ {t} + \left(\frac {1 8}{L _ {f}} \ell_ {f, 1} ^ {2} \alpha_ {t} + \frac {2 7 \ell_ {g , 1} ^ {2} \alpha_ {t}}{c _ {\mathbf {v}}}\right) \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right)\right) 2 c _ {\beta} ^ {2} \\ \leq - \frac {c _ {\beta} \alpha_ {t} \Gamma}{4 \left(\mu_ {g} + \ell_ {g , 1}\right)}, \tag {243} \\ \end{array} +$$ + +where the first inequality follows from $\alpha_{t} \leq 1 / c_{\beta} (\mu_{g} + \ell_{g,1})$ ; the second inequality is by $\alpha \leq 1 / 4L_{f}$ ; the last inequality is by $c_{\mathbf{v}} \geq \frac{216}{\Gamma} \ell_{g,1}^{2} \left( \frac{d_{2}}{\Psi} + \frac{d_{1}}{\Omega} \right) c_{\beta} (\mu_{g} + \ell_{g,1})$ and $\Phi \geq \frac{24d_{2} \ell_{g,1}^{2} (\mu_{g} + \ell_{g,1})}{L_{f} c_{\beta} \Gamma}$ , and $\Psi \geq \frac{144d_{2} \ell_{f,1}^{2} (\mu_{g} + \ell_{g,1}) c_{\beta}}{L_{f} \Gamma}$ , and $\Omega \geq \frac{144d_{1} \ell_{f,1}^{2} (\mu_{g} + \ell_{g,1}) c_{\beta}}{L_{f} \Gamma}$ . + +Thus, we get + +$$ +(2 2 7 \mathrm {e}) = \sum_ {t = 1} ^ {T} Q \left(\beta_ {t}, \rho_ {\mathbf {v}}\right) \mathbb {E} \left\| e _ {t} ^ {g _ {\rho}} \right\| ^ {2} + \sum_ {t = 1} ^ {T} S \left(\beta_ {t}, \rho_ {\mathbf {v}}\right) \mathbb {E} \left[ \| \nabla_ {\mathbf {y}} g _ {t, \rho} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t}\right) \| ^ {2} \right] \leq 0. \tag {244} +$$ + +Bounding (227f). + +From (228), we have + +$$ +Z = 2 7 \ell_ {g, 1} ^ {2} \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right). +$$ + +Thus, from $\delta_t = c_\delta \alpha_t$ in (31), we have + +$$ +\begin{array}{l} (2 2 7 \mathrm {f}) = \sum_ {t = 1} ^ {T} Z \left(3 d _ {2} ^ {2} \ell_ {f, 1} ^ {2} \delta_ {t} ^ {2} \rho_ {\mathbf {r}} ^ {2} + (1 2 \ell_ {f, 0} ^ {2} + 6 \ell_ {g, 1} ^ {2} p ^ {2}) \delta_ {t} ^ {2}\right) \\ = \sum_ {t = 1} ^ {T} 2 7 \ell_ {g, 1} ^ {2} \left(\frac {d _ {2}}{\Psi} + \frac {d _ {1}}{\Omega}\right) \left(3 d _ {2} ^ {2} \ell_ {f, 1} ^ {2} \rho_ {\mathbf {r}} ^ {2} + (1 2 \ell_ {f, 0} ^ {2} + 6 \ell_ {g, 1} ^ {2} p ^ {2})\right) c _ {\delta} ^ {2} \alpha_ {t} ^ {2} \\ = \mathcal {O} \left(\sum_ {t = 1} ^ {T} \left(d _ {1} + d _ {2}\right) \left(\alpha_ {t} ^ {2} \rho_ {\mathbf {r}} ^ {2} + \alpha_ {t} ^ {2}\right)\right). \tag {245} \\ \end{array} +$$ + +Bounding (227g). From $\rho_{\mathbf{v}}^2 = c_{\mathbf{v}}\alpha_t$ in (31), we have + +$$ +\begin{array}{l} (2 2 7 \mathrm {g}) = \frac {3 6}{\Psi} D _ {\mathbf {y}, T} + \frac {3 6}{\Omega} D _ {\mathbf {x}, T} + \frac {1 2}{\Phi} G _ {\mathbf {y}, T} + \frac {3 6}{\Psi \rho_ {\mathbf {v}} ^ {2}} G _ {\mathbf {v}, T} + \frac {3 6}{\Omega \rho_ {\mathbf {v}} ^ {2}} G _ {\mathbf {x}, T} \\ = \mathcal {O} \left(D _ {\mathbf {y}, T} + D _ {\mathbf {x}, T} + G _ {\mathbf {y}, T} + \frac {1}{\alpha_ {T}} \left(G _ {\mathbf {v}, T} + G _ {\mathbf {x}, T}\right)\right). \tag {246} \\ \end{array} +$$ + +Bounding (227h). From $\gamma_{t + 1} = c_{\gamma}\alpha_{t}$ , $\eta_{t + 1} = c_{\eta}\alpha_{t}$ , $\lambda_{t + 1} = c_{\lambda}\alpha_{t}$ and $\rho_{\mathbf{v}}^{2} = c_{\mathbf{v}}\alpha_{t}$ in (31), we have + +$$ +\begin{array}{l} (2 2 7 \mathrm {h}) = 2 \sum_ {t = 1} ^ {T} \frac {\gamma_ {t + 1} ^ {2}}{\Phi} \frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b}} + 3 (\frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {y}}} ^ {2}}{b}) \sum_ {t = 1} ^ {T + 1} \frac {\lambda_ {t + 1} ^ {2}}{\Psi} + 3 (\frac {\hat {\sigma} _ {g _ {\mathbf {x}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {x}}} ^ {2}}{b}) \sum_ {t = 1} ^ {T} \frac {\eta_ {t + 1} ^ {2}}{\Omega} \\ = 2 \sum_ {t = 1} ^ {T} \frac {c _ {\gamma} ^ {2} \alpha_ {t} ^ {2}}{\Phi} \frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b}} + 3 (\frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {y}}} ^ {2}}{b}) \sum_ {t = 1} ^ {T + 1} \frac {c _ {\lambda} ^ {2} \alpha_ {t} ^ {2}}{\Psi} + 3 (\frac {\hat {\sigma} _ {g _ {\mathbf {x}}} ^ {2}}{\bar {b} \rho_ {\mathbf {v}} ^ {2}} + \frac {\hat {\sigma} _ {f _ {\mathbf {x}}} ^ {2}}{b}) \sum_ {t = 1} ^ {T} \frac {c _ {\eta} ^ {2} \alpha_ {t} ^ {2}}{\Omega} \\ = \mathcal {O} \left(\left(\frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b}} + \frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\bar {b} \alpha_ {t}} + \frac {\hat {\sigma} _ {f _ {\mathbf {y}}} ^ {2}}{b} + \frac {\hat {\sigma} _ {g _ {\mathbf {x}}} ^ {2}}{\bar {b} \alpha_ {t}} + \frac {\hat {\sigma} _ {f _ {\mathbf {x}}} ^ {2}}{b}\right) \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {2}\right). \tag {247} \\ \end{array} +$$ + +Bounding (227i). From $\beta_{t} = c_{\beta}\alpha_{t}$ , $\delta_t = c_\delta \alpha_t$ in (31), we have + +$$ +\begin{array}{l} D (\alpha_ {t}, \beta_ {t}, \delta_ {t}) = 6 \ell_ {f, 1} (1 + 2 \frac {\ell_ {g , 1}}{\mu_ {g}}) + \frac {3 \ell_ {f , 1} \ell_ {g , 1}}{\mu_ {g}} (\alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}) + \frac {2 4 \ell_ {g , 1}}{L _ {\mu_ {g}} \mu_ {g}} \frac {\Gamma}{\beta_ {t}} + \frac {9 6 \ell_ {g , 1} \nu^ {2}}{L _ {\mu_ {g}} \mu_ {g} ^ {3}} \frac {\Upsilon}{\delta_ {t}} \\ = 6 \ell_ {f, 1} \left(1 + 2 \frac {\ell_ {g , 1}}{\mu_ {g}}\right) + \frac {3 \ell_ {f , 1} \ell_ {g , 1}}{\mu_ {g}} \left(\alpha_ {t} - L _ {f} \alpha_ {t} ^ {2}\right) + \frac {2 4 \ell_ {g , 1}}{L _ {\mu_ {g}} \mu_ {g}} \frac {\Gamma}{c _ {\beta} \alpha_ {t}} + \frac {9 6 \ell_ {g , 1} \nu^ {2}}{L _ {\mu_ {g}} \mu_ {g} ^ {3}} \frac {\Upsilon}{c _ {\delta} \alpha_ {t}} \\ = \mathcal {O} \left(\alpha_ {t} + \frac {1}{\alpha_ {t}}\right), \\ \end{array} +$$ + +and + +$$ +\sum_ {t = 1} ^ {T} D \left(\alpha_ {t}, \beta_ {t}, \delta_ {t}\right) \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right) = \mathcal {O} \left(\sum_ {t = 1} ^ {T} \left(\alpha_ {t} + \frac {1}{\alpha_ {t}}\right) \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right)\right). \tag {248} +$$ + +Moreover, we have + +$$ +R (\rho_ {\mathbf {v}}) = 9 d _ {2} ^ {2} \frac {\ell_ {g , 1} ^ {2}}{\Phi} + 1 8 d _ {2} ^ {2} \frac {\ell_ {f , 1} ^ {2}}{\Psi} + 6 (3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}) \frac {d _ {2} ^ {2}}{\Psi} = \mathcal {O} \left((1 + \frac {1}{\rho_ {\mathbf {v}} ^ {2}}) d _ {2} ^ {2}\right), +$$ + +$$ +\dot {R} (\rho_ {\mathbf {v}}) = 1 8 d _ {1} ^ {2} \frac {\ell_ {f , 1} ^ {2}}{\Omega} + 6 (3 \ell_ {f, 1} ^ {2} + \frac {3 \ell_ {g , 1} ^ {2}}{4 \rho_ {\mathbf {v}} ^ {2}}) \frac {d _ {1} ^ {2}}{\Omega} = \mathcal {O} \left((1 + \frac {1}{\rho_ {\mathbf {v}} ^ {2}}) d _ {1} ^ {2}\right), +$$ + +which, implies that + +$$ +\begin{array}{l} R (\rho_ {\mathbf {v}}) T \rho_ {\mathbf {r}} ^ {2} + \acute {R} (\rho_ {\mathbf {v}}) T \rho_ {\mathbf {s}} ^ {2} + 1 8 d _ {1} ^ {2} \ell_ {g, 1} ^ {2} \frac {T \rho_ {\mathbf {s}} ^ {2}}{\Omega \rho_ {\mathbf {v}} ^ {2}} + 1 8 d _ {2} ^ {2} \ell_ {g, 1} ^ {2} \frac {T \rho_ {\mathbf {r}} ^ {2}}{\Psi \rho_ {\mathbf {v}} ^ {2}} \\ = \mathcal {O} \left((1 + \frac {1}{\rho_ {\mathbf {v}} ^ {2}}) T \left(d _ {1} ^ {2} \rho_ {\mathbf {s}} ^ {2} + d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2}\right) + \frac {T}{\rho_ {\mathbf {v}} ^ {2}} \left(d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2} + d _ {1} ^ {2} \rho_ {\mathbf {s}} ^ {2}\right)\right). \tag {249} \\ \end{array} +$$ + +From (248), (249) and $\rho_{\mathbf{v}}^{2} = c_{\mathbf{v}}\alpha_{t}$ in (31), we get + +$$ +(2 2 7 \mathrm {i}) \leq \mathcal {O} \left(\sum_ {t = 1} ^ {T} \left(\alpha_ {t} + \frac {1}{\alpha_ {t}}\right) \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right) + \left(1 + \frac {1}{\alpha_ {T}}\right) T \left(d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2} + d _ {1} ^ {2} \rho_ {\mathbf {s}} ^ {2}\right)\right). \tag {250} +$$ + +Combining the outcomes (227i). Combining inequalities (234), (237), (238), (240), (244), (245), (246), (247), and (250) leads to + +$$ +\begin{array}{l} \frac {\alpha_ {T}}{2} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} (\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}))) \| ^ {2} \right] + \Lambda \\ \leq \mathcal {O} \left(V _ {T} + \sum_ {t = 1} ^ {T} \alpha_ {t} (\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}) + \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {2} + \frac {H _ {2 , T}}{\alpha_ {T}} + \sum_ {t = 1} ^ {T} (d _ {1} + d _ {2}) (\alpha_ {t} ^ {2} \rho_ {\mathbf {r}} ^ {2} + \alpha_ {t} ^ {2})\right) \\ + \mathcal {O} \left(D _ {\mathbf {y}, T} + D _ {\mathbf {x}, T} + G _ {\mathbf {y}, T} + \frac {1}{\alpha_ {T}} (G _ {\mathbf {v}, T} + G _ {\mathbf {x}, T})\right) \\ + \mathcal {O} \left(\sum_ {t = 1} ^ {T} \left(\frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2} \alpha_ {t} ^ {2}}{\bar {b}} + \frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2} \alpha_ {t}}{\bar {b}} + \frac {\hat {\sigma} _ {f _ {\mathbf {y}}} ^ {2} \alpha_ {t} ^ {2}}{b} + \frac {\hat {\sigma} _ {g _ {\mathbf {x}}} ^ {2} \alpha_ {t}}{\bar {b}} + \frac {\hat {\sigma} _ {f _ {\mathbf {x}}} ^ {2} \alpha_ {t} ^ {2}}{b}\right)\right) \\ + \mathcal {O} \left(\sum_ {t = 1} ^ {T} \left(\alpha_ {t} + \frac {1}{\alpha_ {t}}\right) \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right) + \left(1 + \frac {1}{\alpha_ {T}}\right) T \left(d _ {2} ^ {2} \rho_ {\mathbf {r}} ^ {2} + d _ {1} ^ {2} \rho_ {\mathbf {s}} ^ {2}\right)\right). \\ \end{array} +$$ + +From the definition of $\Lambda$ in (105), we have + +$$ +\begin{array}{l} - \Lambda = \Gamma \sum_ {t = 1} ^ {T} \left(\mathbb {E} \left[ \theta_ {t} ^ {\mathbf {y}} \right] - \mathbb {E} \left[ \theta_ {t + 1} ^ {\mathbf {y}} \right]\right) \\ + \Upsilon \sum_ {t = 1} ^ {T} \left(\mathbb {E} [ \theta_ {t} ^ {\mathbf {v}} ] - \mathbb {E} [ \theta_ {t + 1} ^ {\mathbf {v}} ]\right) + \frac {1}{\Phi} \sum_ {t = 1} ^ {T} \left(\mathbb {E} \| e _ {t} ^ {g} \| ^ {2} - \mathbb {E} \| e _ {t + 1} ^ {g} \| ^ {2}\right) \\ + \frac {1}{\Psi} \sum_ {t = 1} ^ {T} \left(\mathbb {E} \| e _ {t} ^ {\mathbf {v}} \| ^ {2} - \mathbb {E} \| e _ {t + 1} ^ {\mathbf {v}} \| ^ {2}\right) + \frac {1}{\Omega} \sum_ {t = 1} ^ {T} \left(\mathbb {E} \| e _ {t} ^ {f} \| ^ {2} - \mathbb {E} \| e _ {t + 1} ^ {f} \| ^ {2}\right) \\ \leq \Gamma \theta_ {1} ^ {\mathbf {y}} + \Upsilon \theta_ {1} ^ {\mathbf {y}} + \frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2}}{\Phi} + \frac {\hat {\sigma} _ {g _ {\mathbf {y y}}} ^ {2} + \hat {\sigma} _ {f _ {\mathbf {y}}} ^ {2}}{\Psi} + \frac {\hat {\sigma} _ {g _ {\mathbf {x y}}} ^ {2} + \hat {\sigma} _ {f _ {\mathbf {x}}} ^ {2}}{\Omega}. \tag {251} \\ \end{array} +$$ + +From (28), we have $\hat{\sigma}^2 = \hat{\sigma}_{g_{\mathbf{y}}}^2 + \hat{\sigma}_{g_{\mathbf{yy}}}^2 + \hat{\sigma}_{f_{\mathbf{y}}}^2 + \hat{\sigma}_{g_{\mathbf{xy}}}^2 + \hat{\sigma}_{f_{\mathbf{x}}}^2$ . + +Thus, using (251), (31), and rearranging the terms, we get + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \right] \\ \leq \frac {2}{\alpha_ {T}} \mathcal {O} \left(V _ {T} + \sum_ {t = 1} ^ {T} \alpha_ {t} \left(\rho_ {\mathbf {s}} ^ {2} + \rho_ {\mathbf {r}} ^ {2}\right) + \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {2} + \frac {H _ {2 , T}}{\alpha_ {T}} + \sum_ {t = 1} ^ {T} \left(d _ {1} + d _ {2}\right) \left(\alpha_ {t} ^ {2} \rho_ {\mathbf {r}} ^ {2} + \alpha_ {t} ^ {2}\right)\right) \\ + \frac {2}{\alpha_ {T}} \mathcal {O} \left(D _ {\mathbf {y}, T} + D _ {\mathbf {x}, T} + G _ {\mathbf {y}, T} + \frac {1}{\alpha_ {T}} \left(G _ {\mathbf {v}, T} + G _ {\mathbf {x}, T}\right)\right) \\ + \frac {2}{\alpha_ {T}} \mathcal {O} \left(\sum_ {t = 1} ^ {T} \left(\frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2} \alpha_ {t} ^ {2}}{\bar {b}} + \frac {\hat {\sigma} _ {g _ {\mathbf {y}}} ^ {2} \alpha_ {t}}{\bar {b}} + \frac {\hat {\sigma} _ {f _ {\mathbf {y}}} ^ {2} \alpha_ {t} ^ {2}}{b} + \frac {\hat {\sigma} _ {g _ {\mathbf {x}}} ^ {2} \alpha_ {t}}{\bar {b}} + \frac {\hat {\sigma} _ {f _ {\mathbf {x}}} ^ {2} \alpha_ {t} ^ {2}}{b}\right)\right) \\ + \frac {2}{\alpha_ {T}} \mathcal {O} \left(\sum_ {t = 1} ^ {T} (\alpha_ {t} + \frac {1}{\alpha_ {t}}) \left(\rho_ {\bf s} ^ {2} + \rho_ {\bf r} ^ {2}\right) + (1 + \frac {1}{\alpha_ {T}}) T (d _ {2} ^ {2} \rho_ {\bf r} ^ {2} + d _ {1} ^ {2} \rho_ {\bf s} ^ {2})\right) \\ + \frac {2}{\alpha_ {T}} \mathcal {O} \left(\theta_ {1} ^ {\mathbf {y}} + \theta_ {1} ^ {\mathbf {v}} + \hat {\sigma} ^ {2}\right) \\ \leq \mathcal {O} \left(\left(d _ {1} + d _ {2}\right) ^ {3 / 4} T ^ {1 / 3} \left(V _ {T} + D _ {\mathbf {y}, T} + D _ {\mathbf {x}, T} + G _ {\mathbf {y}, T} + \Delta_ {1} + \hat {\sigma} ^ {2}\right) \right. \\ \left. + \left(d _ {1} + d _ {2}\right) ^ {3 / 2} T ^ {2 / 3} \left(H _ {2, T} + G _ {\mathbf {v}, T} + G _ {\mathbf {x}, T}\right)\right), \tag {252} \\ \end{array} +$$ + +where second inequality holds because we have + +$$ +\sum_ {t = 1} ^ {T} \alpha_ {t} ^ {3} = \sum_ {t = 1} ^ {T} \frac {1}{(d _ {1} + d _ {2}) ^ {9 / 4} (c + t)} \leq \sum_ {t = 1} ^ {T} \frac {1}{(d _ {1} + d _ {2}) ^ {9 / 4} (1 + t)} \leq \frac {\log (T + 1)}{(d _ {1} + d _ {2}) ^ {9 / 4}}, +$$ + +$$ +\sum_ {t = 1} ^ {T} \alpha_ {t} ^ {2} = \sum_ {t = 1} ^ {T} \frac {1}{(d _ {1} + d _ {2}) ^ {3 / 2} (c + t) ^ {2 / 3}} \leq \sum_ {t = 1} ^ {T} \frac {1}{(d _ {1} + d _ {2}) ^ {3 / 2} (1 + t) ^ {2 / 3}} \leq \frac {T ^ {1 / 3}}{(d _ {1} + d _ {2}) ^ {3 / 2}}, +$$ + +$$ +\sum_ {t = 1} ^ {T} \alpha_ {t} = \sum_ {t = 0} ^ {T} \frac {1}{(d _ {1} + d _ {2}) ^ {3 / 4} (c + t) ^ {1 / 3}} \leq \sum_ {t = 1} ^ {T} \frac {1}{(d _ {1} + d _ {2}) ^ {3 / 4} (1 + t) ^ {1 / 3}} \leq \frac {3 T ^ {2 / 3}}{2 (d _ {1} + d _ {2}) ^ {3 / 4}}, +$$ + +$$ +\sum_ {t = 1} ^ {T} \frac {1}{\alpha_ {t}} = \sum_ {t = 0} ^ {T} (d _ {1} + d _ {2}) ^ {3 / 4} (c + t) ^ {1 / 3} \leq \frac {3}{2} (d _ {1} + d _ {2}) ^ {3 / 4} T ^ {4 / 3}. +$$ + +Then, note that, we have + +$$ +\begin{array}{l} \frac {1}{2} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \right] \\ \leq \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \| ^ {2} \right] \\ + \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) - \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \right]. \\ \end{array} +$$ + +From non-expansiveness of the projection operator and Lemma D.4, we have + +$$ +\begin{array}{l} \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) - \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \right\| ^ {2} \\ \leq \left\| \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) - \nabla f _ {t, \rho} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right) \right\| ^ {2} \\ \leq \frac {\left(\rho_ {\mathbf {s}} d _ {1} + \rho_ {\mathbf {r}} d _ {2}\right) ^ {2} \ell_ {f , 1} ^ {2}}{4} \\ \leq \frac {\left(\rho_ {\mathbf {s}} ^ {2} d _ {1} ^ {2} + \rho_ {\mathbf {r}} ^ {2} d _ {2} ^ {2}\right) \ell_ {f , 1} ^ {2}}{2}. \\ \end{array} +$$ + +This implies + +$$ +\begin{array}{l} \frac {1}{2} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \left\| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right)\right) \right\| ^ {2} \right] \\ \leq \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} (\mathbf {x} _ {t}; \nabla f _ {t, \boldsymbol {\rho}} (\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t}))) \| ^ {2} \right] + \frac {T (\rho_ {\mathbf {s}} ^ {2} d _ {1} ^ {2} + \rho_ {\mathbf {r}} ^ {2} d _ {2} ^ {2}) \ell_ {f , 1} ^ {2}}{2}. \\ \end{array} +$$ + +Applying the upper bound in (252) yields + +$$ +\begin{array}{l} \frac {1}{2} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} \left(\mathbf {x} _ {t}\right)\right)\right) \| ^ {2} \right] \\ \leq \mathcal {O} \left(\left(d _ {1} + d _ {2}\right) ^ {3 / 4} T ^ {1 / 3} \left(V _ {T} + D _ {\mathbf {y}, T} + D _ {\mathbf {x}, T} + G _ {\mathbf {y}, T} + \Delta_ {1} + \hat {\sigma} ^ {2}\right) \right. \\ \left. + \left(d _ {1} + d _ {2}\right) ^ {3 / 2} T ^ {2 / 3} \left(H _ {2, T} + G _ {\mathbf {v}, T} + G _ {\mathbf {x}, T}\right)\right) \\ + \frac {T \left(\rho_ {\mathbf {s}} ^ {2} d _ {1} ^ {2} + \rho_ {\mathbf {r}} ^ {2} d _ {2} ^ {2}\right) \ell_ {f , 1} ^ {2}}{2}. \\ \end{array} +$$ + +Thus, from $\rho_{\mathbf{r}}^{2} = \frac{1}{d_{2}^{2}T}$ and $\rho_{\mathbf{s}}^{2} = \frac{1}{d_{1}^{2}T}$ in (31), we get + +$$ +\begin{array}{l} \sum_ {t = 1} ^ {T} \mathbb {E} \left[ \| \mathcal {P} _ {\mathcal {X}, \alpha_ {t}} \left(\mathbf {x} _ {t}; \nabla f _ {t} \left(\mathbf {x} _ {t}, \mathbf {y} _ {t} ^ {*} (\mathbf {x} _ {t})\right)\right) \| ^ {2} \right] \\ \leq \mathcal {O} \left(\left(d _ {1} + d _ {2}\right) ^ {3 / 4} T ^ {1 / 3} \left(V _ {T} + D _ {\mathbf {y}, T} + D _ {\mathbf {x}, T} + G _ {\mathbf {y}, T} + \Delta_ {1} + \hat {\sigma} ^ {2}\right) \right. \\ \left. + \left(d _ {1} + d _ {2}\right) ^ {3 / 2} T ^ {2 / 3} \left(H _ {2, T} + G _ {\mathbf {v}, T} + G _ {\mathbf {x}, T}\right)\right). \\ \end{array} +$$ + +This completes the proof. + +![](images/8f76440262336a9287447923ec217d7b66b286d2820a8c6d483553e905c1fa10.jpg) + +# E Hyperparameter Tuning Results + +As detailed in Section 4, we carefully tuned all hyperparameters to ensure stable and fair comparisons. Our analysis indicates that while ZO-SOGD exhibits sensitivity to hyperparameter choices, it remains robust within reasonable ranges. Below, we provide extensive tuning results for ZO-SOGD. + +The hyperparameter sensitivity analysis for the adversarial attack scenario reveals critical insights about the algorithm's attack effectiveness across different parameter configurations. For the inner and outer stepsizes, we observe that the algorithm achieves optimal attack performance with specific combinations that balance perturbation strength and imperceptibility. + +Table 2: Hyperparameter tuning results for inner $\left( \beta \right)$ and outer $\left( \alpha \right)$ stepsizes in adversarial attack scenario. Values represent test accuracy (mean $\pm$ std) over 5 runs. Lower values indicate better attack performance. + +
β\αα = 0.001α = 0.005α = 0.01α = 0.1
β = 0.0010.68 ± 0.050.59 ± 0.070.47 ± 0.060.53 ± 0.08
β = 0.0050.54 ± 0.060.41 ± 0.050.35 ± 0.040.42 ± 0.05
β = 0.010.48 ± 0.040.34 ± 0.050.57 ± 0.070.39 ± 0.06
β = 0.10.26 ± 0.030.43 ± 0.060.33 ± 0.040.45 ± 0.07
+ +The stepsize analysis reveals that larger inner stepsizes combined with smaller outer step sizes tend to produce more effective attacks. Specifically, the configuration with $\beta = 0.1$ and $\alpha = 0.001$ achieves the lowest test accuracy of $0.26 \pm 0.03$ , indicating the most successful adversarial perturbations. This pattern suggests that aggressive updates to the perturbation parameters $(\beta)$ while maintaining conservative hyperparameter updates $(\alpha)$ creates an effective balance for generating strong yet imperceptible adversarial examples. + +Table 3: Performance comparison across different smoothing parameters $(\rho_r = \rho_s)$ in adversarial attack scenario. + +
ρv\ρr = ρs0.0010.0050.010.05
ρv = 0.0010.61 ± 0.060.52 ± 0.050.48 ± 0.040.57 ± 0.06
ρv = 0.0050.47 ± 0.050.39 ± 0.040.35 ± 0.040.45 ± 0.05
ρv = 0.010.41 ± 0.040.28 ± 0.030.31 ± 0.030.43 ± 0.05
ρv = 0.050.53 ± 0.060.44 ± 0.050.40 ± 0.040.52 ± 0.06
+ +The smoothing parameter analysis provides additional insights into the algorithm's convergence behavior in the adversarial setting. The optimal configuration occurs with $\rho_v = 0.01$ and $\rho_r = \rho_s = 0.005$ , achieving a test accuracy of $0.28 \pm 0.03$ . These moderate smoothing values appear to provide the right balance between exploration and exploitation in the adversarial perturbation space, allowing the algorithm to find effective attack directions without excessive oscillation or premature convergence. + +Table 4: Performance comparison across different momentum parameters in adversarial attack scenario. + +
γt\λt= ηt0.90.990.999
γt= 0.90.35 ± 0.040.29 ± 0.030.38 ± 0.05
γt= 0.990.31 ± 0.030.24 ± 0.020.33 ± 0.04
γt= 0.9990.37 ± 0.040.32 ± 0.080.40 ± 0.05
+ +The momentum parameter investigation reveals that moderate momentum values consistently produce the most effective adversarial attacks. The optimal configuration with $\gamma_{t} = 0.99$ and $\lambda_{t} = \eta_{t} = 0.99$ achieves the lowest test accuracy of $0.24 \pm 0.02$ , representing the most successful attack performance. This configuration suggests that maintaining momentum across both inner and outer optimization loops helps the algorithm navigate the complex adversarial landscape more effectively than either no momentum or excessive momentum settings. + +The comprehensive analysis demonstrates that ZO-SOGD maintains robust attack performance across a broad range of hyperparameter configurations. The algorithm consistently achieves test accuracies below 0.5 across most reasonable parameter combinations, indicating reliable adversarial attack capability. The standard deviations remain low throughout the parameter space, suggesting stable and reproducible attack performance across multiple experimental runs. + +The optimal hyperparameter configuration for adversarial attacks consists of inner stepsize $\beta = 0.1$ , outer stepsize $\alpha = 0.001$ , smoothing parameters $\rho_v = 0.01$ and $\rho_r = \rho_s = 0.005$ , and momentum parameters $\gamma_t = \lambda_t = \eta_t = 0.99$ . This configuration enables ZO-SOGD to achieve superior attack performance while maintaining the imperceptibility constraints essential for practical adversarial examples. + +# NeurIPS Paper Checklist + +# A. 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In the known transition setting, existing bounds based on Online Mirror Descent (OMD) with negative-entropy regularization scale with $\sqrt{\log SA}$ , where $SA$ is the size of the state-action space. While we show that this is optimal in the worst-case, this bound fails to capture the benefits of sparsity when only a small number $M \ll SA$ of state-action pairs incur cost. In fact, we also show that the negative-entropy is inherently non-adaptive to sparsity: it provably incurs regret scaling with $\sqrt{\log S}$ on sparse problems. Instead, we propose a family of $\ell_r$ -norm regularizers $(r \in (1,2))$ that adapts to the sparsity and achieves regret scaling with $\sqrt{\log M}$ instead of $\sqrt{\log SA}$ . We show this is optimal via a matching lower bound, highlighting that $M$ captures the effective dimension of the problem instead of $SA$ . Finally, in the unknown transition setting the benefits of sparsity are limited: we prove that even on sparse problems, the minimax regret for any learner scales polynomially with $SA$ . + +# 1 Introduction + +The Stochastic Shortest Path (SSP) problem is a fundamental model in reinforcement learning [1, 32], which describes tasks where an agent interacts with an environment over episodes and must reach a designated goal state within each episode while minimizing accumulated costs. This covers problems such as car navigation while trying to avoid traffic jams, or internet routing. Recently, this classical setting has been extended to the adversarial regime, where costs may vary arbitrarily between episodes [28, 8, 6, 38] and the goal is to obtain theoretical guarantees robust to any cost-generation mechanism. Under full-information feedback where the full cost vector is observed after each episode and known transitions, current algorithms achieve regret bounds that scale as $\mathcal{O}\left(\sqrt{DKT_{\star} \log SAT_{\star}}\right)$ , where $D$ is the diameter (the smallest expected hitting time of any policy from any state), $T_{\star}$ is the expected hitting time of the optimal policy, $S$ is the number of states, $A$ is the number of actions, and $K$ is the number of episodes. These bounds are independent of any cost structure and are shown to be minimax optimal up to logarithmic factors in [8]. + +In SSP, the size of the state-action space $SA$ – which we consider and refer to as the dimension of the problem – appears in the minimax regret as $\sqrt{\log SA}$ . While in the worst-case this is unimprovable (we show this in Theorem 3.1), many real-world problems have costs with structural properties that may be leveraged for improved regret. A common property often considered in the statistics and machine learning literature [36] is sparsity, which can naturally arise for SSP problems. For instance, in the car navigation example, the number of traffic jams is usually much smaller than the number of roads. Motivated by this, we consider sparse SSP problems where $M$ , the maximum number of state-action pairs with non-zero cost in an episode, can be much smaller than $SA$ . + +In such scenarios, the regret bounds should capture some dependence on $M$ , reflecting an improvement in performance on easier sparse problems as $M \to 1$ and recovering the standard bounds on + +worst-case problems as $M \to SA$ . In fact, for the so-called experts setting $^2$ ( $S = 1$ ), the minimax regret scales with $\sqrt{KMA^{-1}\log A}$ instead of $\sqrt{K\log A}$ in the worst-case [19], providing a polynomial improvement in the dimension $A$ . Furthermore, this is achieved with Online Mirror Descent (OMD) with negative entropy regularization. However for SSP where the problem dimension also includes the size of the state space $S$ , we show that existing approaches also based on OMD with the same negative entropy regularization [8] fail to exploit sparse costs. We construct a sparse SSP problem where this algorithm suffers $\sqrt{\log S}$ regret (Theorem 3.1), providing no improvement in terms of $S$ compared to the minimax regret for the non-sparse worst-case $M = SA$ problem. The failure of existing SSP methods to exploit sparsity, let alone match the polynomial improvements from the experts setting, leads us to ask the following questions: + +Does sparsity improve the minimax-regret in the full-information feedback SSP problem? How much? + +We answer the first question positively for the known-transition case by designing a family of regularizers based on $\ell_r$ -norms for $r \in (1,2)$ for which we show a $\mathcal{O}(\sqrt{DKT_{\star}} \log MT_{\star})$ regret bound that depends logarithmically on the sparsity level $M$ , rather than on the size of the state-action space $SA$ , without requiring the knowledge of $M$ in advance. This family of regularizers interpolates between the negative entropy and the squared Euclidean norm (see Section 4), allowing flexibility for much weaker regularization on sparse points in sparse settings and recovering existing algorithms (and guarantees) in the non-sparse setting. + +We show that the above $\sqrt{\log M}$ dependence on $M$ is unimprovable by constructing a matching lower bound. Interestingly, this establishes that the benefit of sparsity in SSP is logarithmic in $SA$ instead of polynomial, as in the simpler experts problem, thus answering the second question. It also highlights that $M$ plays the role of effective dimension, replacing the general dimension $SA$ in controlling the scaling of the minimax regret. + +While the benefits of sparsity in SSP are only logarithmic, we emphasize that due to the often combinatorial nature of the state-action space, these improvements can be significant. For instance, in many problems the size of the state-action space grows exponentially in some parameters, while the assigned costs remain linear or even constant. This occurs in many real-world problems (e.g. [31, 30]) in which settings exploiting sparsity can lead to polynomial improvements. + +Finally, it is natural to ask whether sparsity may provide similar benefits in the unknown transition setting. However, in Theorem 5.1, we show a lower bound with polynomial dependence on $SA$ in a sparse SSP instance with unknown transitions. This illustrates that in the unknown transition setting the sparsity level $M$ does not play the same role of effective dimension, and that the general dimension $SA$ is crucial in controlling the scaling of the minimax regret polynomially, motivating our focus on the known transitions setting. In particular, this result shows that sparse problems with combinatorial state-action spaces will remain very challenging. + +Our results provide a complete characterisation of the benefits of sparsity in removing dimension dependence (i.e. $SA$ ) for adversarial SSP problems under full-information feedback. + +# 1.1 Contributions + +We highlight our main contributions below: + +- We design a family of $\ell_r$ -norm ( $r \in (1, 2)$ ) regularizers for OMD that allows interpolation between the negative entropy and squared Euclidean norm, adjusting its geometry to the sparsity of the cost functions (see Section 4). The regularizer naturally plugs into the standard OMD analysis. +- We show OMD with the above regularizer achieves sparsity-adaptive regret bounds of order $\mathcal{O}\left(\sqrt{DKT_{\star}\log MT_{\star}}\right)$ (Theorem 4.1). We also give a parameter-free version achieving the same bound (Theorem 4.4) that does not require prior knowledge of the sparsity level $M$ nor the expected hitting time of the optimal policy $T_{\star}$ (the only unknown parameters). +- We establish a lower bound of order $\Omega\left(\sqrt{DKT_{\star}\log M}\right)$ (Theorem 4.6), matching our regret guarantees up to a logarithmic factor of $T_{\star}$ (already present in prior work [8]) and improving over [8] in the $M = SA$ non-sparse setting by including the $\sqrt{\log SA}$ dependence. + +- We show that OMD with the negative entropy used in prior work [28, 8] suffers regret at least $\Omega (\sqrt{K\log S})$ even when $M = 3$ (Theorem 3.1). This rules out the negative entropy as a viable regularizer in the sparse setting and provides justification for the use of our regularizer. +- We establish that results independent of $SA$ are not achievable in the unknown transitions setting via a lower bound in the sparse $(M = 1)$ setting of order $\Omega \left( D\sqrt{SAK} \right)$ (Theorem 5.1). + +Technical Contributions: Proving these results requires new technical ideas. For the general sparse lower-bound, we derive a result on the expectation of the maximum of asymmetric zero-mean random walks, generalizing the result for the symmetric case from [25]. The negative-entropy-specific lower-bound relies on the careful design of an MDP with skewed initial occupancy measures that highlights both the reasons for the failure of the negative entropy as well as the more general difficulty of the stochastic nature of SSP problems. + +# 1.2 Related works + +Regret minimisation for SSP problems under full-information feedback was initiated by a line of work studying stochastic costs [32, 29, 33, 11, 7, 9, 17]. In the adversarial setting, it was first studied by [28] in the known transition case. Their bounds were later improved by [8]. There have since been many extensions: [6] consider the unknown transition setting, [38] establish dynamic regret bounds, [10] consider a policy optimisation approach in the unknown transition setting. + +Regret minimisation for SSP problems under bandit feedback where only the costs of the visited state-action pairs in an episode are revealed to the learner has also been studied both in the stochastic [10] and adversarial settings [8, 6, 10]. In the adversarial known transition setting, the minimax regret is of the order $\sqrt{KDT_{\star}SA}$ (ignoring log terms) [8]. It is an interesting future direction to study the sparse SSP problem with bandit feedback and understand if the regret scales with $\sqrt{M}$ instead of $\sqrt{SA}$ , in which case $M$ would play the same role of effective dimension as in the setting we consider. + +Regret minimisation with sparse costs was studied in the classical online learning setting [19] $(S = 1)$ . The minimax regret goes from $\mathcal{O}(\sqrt{K\log A})$ to $\mathcal{O}(\sqrt{KMA^{-1}\log A})$ under full-information feedback (experts problem). For rewards instead of costs, it goes from $\mathcal{O}(\sqrt{K\log A})$ to $\mathcal{O}(\sqrt{K\log M})$ , which matches the benefits of sparsity we establish for the SSP problem with costs. Note that we restrict our focus to costs since it is unclear how to interpret rewards within the SSP framework. Under bandit feedback, the sparse minimax regret goes from $\tilde{\mathcal{O}} (\sqrt{KA})$ to $\tilde{\mathcal{O}} (\sqrt{KM})$ for both rewards and costs [19, 4]. The above minimax regrets can also be achieved by sparse-agnostic methods [19, 34]. Finally, sparsity was also considered in the case of stochastic losses by [20]. + +# 2 Preliminaries + +# 2.1 Problem setting + +We consider the Stochastic Shortest Path (SSP) problem with adversarial costs. The environment is modeled as a Markov Decision Process (MDP) $\mathcal{M} = (\mathcal{S},\mathcal{A},P,s_0,g)$ along with a sequence of cost functions $\{c_k\}_{k = 1}^K$ chosen by an oblivious adversary over $K$ episodes. $\mathcal{S}$ is the state space with cardinality $S = |\mathcal{S}|$ , and $s_0\in S$ is the fixed starting state. The goal state $g$ is a special absorbing state not included in $\mathcal{S}$ . $\mathcal{A}$ is the action space with cardinality $A = |\mathcal{A}|$ and we assume for simplicity that it is the same in every state. Let $\Gamma = \mathcal{S}\times \mathcal{A}$ denote the set of all state-action pairs. The dynamics in the MDP are given by the known transition function $P$ , where $P(s'|s,a)$ specifies the probability of moving to state $s'\in \mathcal{S}\cup \{g\}$ after taking action $a$ in state $s$ . + +Each episode begins in state $s_0$ and proceeds with the learner selecting actions until the goal state $g$ is reached. When the goal state is reached, the current episode ends and a new one begins. At the start of each episode $k$ , the adversary selects a cost function $c_k: \Gamma \to [0,1]$ , assigning a cost to each state-action pair. We denote the sparsity level as $M = \max_k \sum_{(s,a) \in \Gamma} \mathbb{I}\{c_k(s,a) > 0\}$ the maximum number of non-zero costs in an episode. We work in the full-information setting where the entire function $c_k$ is revealed to the learner at the end of the episode. The objective is to minimize the total cost over all episodes, which requires a balance of minimizing the accumulated costs while ensuring the goal state is reached efficiently. + +We use super-scripts to denote the time-step within an episode and sub-scripts to denote the episode: e.g. $(s_k^t, a_k^t)$ refers to the state-action pair at the $t$ -th time-step of the $k$ -th episode. We sometimes omit the sub-script when referring to an arbitrary episode. We now define some important concepts: + +- A stationary policy $\pi$ is a mapping such that $\pi(\cdot|s)$ is a probability distribution over the choice of action $a \sim \pi(\cdot|s)$ in state $s$ . A policy is called proper if it reaches the goal $g$ in finite time from any initial state in $S$ with probability one, and improper if not. Let $\Pi_p$ be the set of all stationary proper policies. We assume the existence of at least one proper policy. +- The expected hitting time $T^{\pi}(s)$ is the expected number of steps required to reach $g$ from state $s$ under $\pi$ . Letting $I_{\pi}(s)$ be the random number of time-steps used to reach the goal state when executing a policy $\pi$ in an episode starting from state $s$ , then $T^{\pi}(s) = \mathbb{E}[I_{\pi}(s)]$ . For any proper policy $\pi$ , $I_{\pi}(s)$ and $T^{\pi}(s)$ are finite for all $s \in S$ . +- The fast policy $\pi_f$ is the deterministic policy that minimizes the worst-state expected hitting time, and the diameter $D$ of the MDP is the corresponding expected hitting time: + +$$ +\pi_{f} = \operatorname *{arg min}_{\pi \in \Pi_{p}}\max_{s\in \mathcal{S}}T^{\pi}(s),\qquad D = \max_{s\in \mathcal{S}}T^{\pi_{f}}(s) = \max_{s\in \mathcal{S}}\min_{\pi \in \Pi_{p}}T^{\pi}(s). +$$ + +Since the transition function $P$ is known, both the fast policy $\pi_f$ and the diameter $D$ can be computed offline prior to the learning process. We assume $D \geqslant 1$ . + +- The cost-to-go function $J_{c}^{\pi} : \mathcal{S} \to [0, \infty)$ is the expected cost suffered during an episode executing policy $\pi$ and starting from state $s$ , given a cost function $c$ and a proper policy $\pi$ . It is defined as + +$$ +J _ {c} ^ {\pi} (s) = \mathbb {E} \bigg [ \sum_ {t = 1} ^ {I _ {\pi} (s)} c (s ^ {t}, a ^ {t}) \bigg | P, \pi , s ^ {1} = s \bigg ], +$$ + +where the expectation is with respect to the randomness in the action sampling and state transitions. We use $J_{k}^{\pi}$ to denote the cost-to-go from the initial state $s_0$ using the cost function $c_{k}$ in episode $k$ . + +- The regret $R_{K}$ is the primary measure of performance by which the learner is evaluated. It is the difference between the total cost over all episodes of the policies $\pi_1, \ldots, \pi_K$ chosen by the learner, and the total cost of the best proper deterministic policy in hindsight, $\pi^{\star} \in \arg \min_{\pi \in \Pi_p} \sum_{k=1}^{K} J_k^{\pi}$ : + +$$ +R _ {K} = \sum_ {k = 1} ^ {K} \sum_ {t = 1} ^ {I _ {\pi_ {k}} (s _ {0})} c _ {k} \left(s _ {k} ^ {t}, a _ {k} ^ {t}\right) - \sum_ {k = 1} ^ {K} J _ {k} ^ {\pi^ {*}}. +$$ + +- The occupancy measure $q_{\pi} \in \mathbb{R}_{\geqslant 0}^{\Gamma}$ of a proper policy $\pi$ is the expected number of visits to state-action pairs in an episode executing policy $\pi$ starting from $s_0$ : + +$$ +q _ {\pi} (s, a) = \mathbb {E} \left[ \sum_ {i = 1} ^ {I _ {\pi} (s _ {0})} \mathbb {I} \left\{s ^ {i} = s, a ^ {i} = a \right\} \mid P, \pi , s ^ {1} = s _ {0} \right]. +$$ + +The marginal $q_{\pi}(s) = \sum_{a \in \mathcal{A}} q_{\pi}(s, a)$ gives the expected number of visits to state $s$ . Given a vector $q \in \mathbb{R}_{\geqslant 0}^{\Gamma}$ , if it corresponds to a valid occupancy measure, the corresponding policy $\pi_q$ can be recovered via normalization as $\pi_q(a|s) = q(s, a) / \sum_{a'} q(s, a')$ [39, 28]. + +# 2.2 SSP as online linear optimisation and online mirror descent + +Occupancy measures allow the cost-to-go to be expressed in a linear form: + +$$ +J _ {k} ^ {\pi} = \sum_ {(s, a) \in \Gamma} q _ {\pi} (s, a) c _ {k} (s, a) = \langle q _ {\pi}, c _ {k} \rangle . +$$ + +If the learner executes a stationary proper policy $\pi_{k}$ in episode $k$ , the expected regret can thus be reformulated as an online linear optimisation problem on the space of occupancy measures: + +$$ +\mathbb {E} \left[ R _ {K} \right] = \sum_ {k = 1} ^ {K} \left\{J _ {k} ^ {\pi_ {k}} - J _ {k} ^ {\pi^ {\star}} \right\} = \sum_ {k = 1} ^ {K} \langle q _ {\pi_ {k}} - q _ {\pi^ {\star}}, c _ {k} \rangle . +$$ + +Online linear optimisation is a well studied problem and can be solved using Online Mirror Descent (OMD) (see e.g. [24]). In the SSP framework, OMD is applied on the space of occupancy measures corresponding to proper policies with expected hitting time bounded by some $T > 0$ defined as: + +$$ +\Delta (T) = \left\{q \in \mathbb {R} ^ {\Gamma}: \sum_ {(s, a) \in \Gamma} q (s, a) \leqslant T, \quad \forall s \in \mathcal {S}: \sum_ {a \in \mathcal {A}} q (s, a) - \sum_ {(s ^ {\prime}, a ^ {\prime}) \in \Gamma} P (s | s ^ {\prime}, a ^ {\prime}) q (s ^ {\prime}, a ^ {\prime}) = \mathbb {I} \left\{s = s _ {0} \right\} \right\}. +$$ + +The first constraint ensures the expected hitting time is bounded by $T$ , while the second is a flow constraint ensuring the vector corresponds to the occupancy measure of a policy. The regret bounds of OMD will hold against any fixed comparator policy as long as $T$ is large enough such that $\Delta(T)$ contains the occupancy measure of the optimal policy, i.e. $q_{\pi^{\star}} \in \Delta(T)$ or $T \geqslant T_{\star}$ where we denote by $T_{\star} = T^{\pi^{\star}}(s_0)$ the expected hitting time of $\pi^{\star}$ . OMD with a strictly convex differentiable regularizer $\psi$ and step-size $\eta$ selects occupancy measures computed through the update + +$$ +q _ {1} = \underset {q \in \Delta (T)} {\arg \min } \psi (q), \quad q _ {k + 1} = \underset {q \in \Delta (T)} {\arg \min } \left\{\eta \cdot \langle q, c _ {k} \rangle + D _ {\psi} (q, q _ {k}) \right\}, \tag {1} +$$ + +where $D_{\psi}(x,y) = \psi (x) - \psi (y) - \langle \nabla \psi (y),x - y\rangle$ is the Bregman divergence with respect to $\psi$ . This update can be computed efficiently for all the regularizers we will discuss (see Appendix B). As discussed in the previous section, we can easily recover via normalization the corresponding policy $\pi_{q_k}$ that will be executed by the learner. + +If the regularizer satisfies for some $\alpha > 0$ , any $q \in \mathbb{R}^{\Gamma}$ and all $k \geqslant 1$ : + +$$ +\nabla \psi (q) \in [ \nabla \psi (q _ {k}), \nabla \psi (q _ {k}) - \eta c _ {k} ] \Longrightarrow \nabla^ {2} \psi (q) \succeq \alpha \nabla^ {2} \psi (q _ {k}), \tag {2} +$$ + +(this is satisfied by many common regularizers), then a standard result (see e.g. Theorem 6 in [4], Theorem 5.5 in [3]) gives the following general regret bound for OMD: + +$$ +\sum_ {k = 1} ^ {K} \left\langle q _ {k} - q _ {\pi^ {\star}}, c _ {k} \right\rangle \leqslant \underbrace {\frac {\psi \left(q _ {\pi^ {\star}}\right) - \psi \left(q _ {1}\right)}{\eta}} _ {\text {P e n a l t y}} + \underbrace {\frac {\eta}{2 \alpha} \sum_ {k = 1} ^ {K} \left\| c _ {k} \right\| _ {\nabla^ {2} \psi \left(q _ {k}\right) ^ {- 1}} ^ {2}} _ {\text {S t a b i l i t y}} \tag {3} +$$ + +where $\| q\| _A^2 = \sum_{s,a,s',a'}q(s,a)A((s,a),(s',a'))q(s',a')$ for a matrix $A\in \mathbb{R}^{\Gamma \times \Gamma}$ . Various regret bounds can be obtained by instantiating the above with different regularizers. In particular, [8] use the negative-entropy to obtain a $\mathcal{O}\left(\sqrt{DKT_{\star}\log SAT_{\star}}\right)$ regret bound. + +# 3 Failure of negative entropy regularization + +In the general non-sparse setting, [8] use the negative entropy to achieve a regret of $\mathcal{O}\big(\sqrt{DKT_{\star}\log SAT_{\star}}\big)$ , which in the non-sparse setting has optimal dependence on $SA$ (as we show later in Theorem 4.4). Despite this success, the negative entropy fails to benefit from sparsity in its dependence on $S$ , as shown by the result below. As we will see in Section 4, this establishes the negative entropy as a sub-optimal choice for sparse SSP problems. + +Theorem 3.1. For any $S \geqslant 6$ , there exists an SSP instance with a fixed horizon of 3, sparsity level $M = 3$ , an action space of size $A = 2$ and state space of size $S$ such that the regret of OMD (1) with negative-entropy regularization and any step-size $\eta > 0$ after $K$ episodes is $\mathbb{E}\left[R_K\right] = \Omega \left(\min \left\{\sqrt{K\log S}, K\right\}\right)$ . + +This result shows that despite the SSP instance being sparse ( $M = 3$ ), the regret of OMD with negative entropy regularization nevertheless scales as $\sqrt{\log S}$ , which is the same dependence on $S$ as in the non-sparse setting. For sparse problems, the negative entropy provides no improvement on the regret with respect to $S$ . This highlights that existing approaches and regularizers are inadequate to appropriately exploit sparse problems and motivates considering alternate regularizers specifically designed for the geometry of sparse problems, as we do in the next section. + +To better understand the failure of negative entropy regularization in sparse settings, we highlight the main intuition behind the lower bound construction and defer the details of the proof to Appendix A. + +Proof intuition: The key idea is to reduce SSP to an experts problem with 2 actions and a heavily skewed initial distribution over the actions. The initial occupancy measure played by OMD in (1) is $q_{1} = \arg \min_{q\in \Delta (T)}\psi (q)$ . For most regularizers, including the negative entropy, this encourages $q_{1}$ to be uniform across the state-action space while maintaining the constraints on the flow and expected hitting time. Since we consider a fixed-horizon MDP, only the flow constraint is relevant. + +Consider the SSP problem shown in Figure 1 with $N = S - 2$ . Since $s_1, \ldots, s_N$ constitute a large majority of the states (especially for large $N$ ), $\psi(q)$ is mainly affected by the values of $q$ in these + +![](images/eaaa17326b55f947db105113b0e1637780354c8ce1e2bdb7a87a16f0281febd5.jpg) +Figure 1: MDP for the reduction to a skewed experts problem with 2 actions: $S = \{s_0, s_g, s_1, \dots, s_N\}$ ( $N = S - 2$ ), $\mathcal{A} = \{a_1, a_2\}$ . The transitions are given by $p(s_g | s_0, a_1) = 1$ , $p(g | s_g, a) = 1$ for all $a \in \mathcal{A}$ , for $i \geqslant 1$ : $p(s_i | s_0, a_2) = 1 / N$ , $p(g | s_i, a) = 1$ for all $a \in \mathcal{A}$ . + +states. In order to minimize $\psi (q)$ , $q_{1}$ needs to ensure the expected number of visits to these $N$ states is sufficiently high. However since for any $q$ and any $i\geqslant 1$ , $q(s_i) = \frac{1}{N} q(s_0,a_2)$ , for $q_{1}(s_{i})$ to be sufficiently large then $q_{1}(s_{0},a_{2})$ needs to be much larger (by a factor of $N$ ). This results in $q_{1}$ being heavily skewed towards $a_2$ in $s_0$ . For the negative entropy, this gives specifically $q_{1}(s_{0},a_{1})\approx \frac{1}{\sqrt{N}}$ . + +If the costs in all states but $s_0$ are set to 0, the problem is sparse ( $M = 2$ ) and reduces to a experts problem with 2 actions where the initial probability for the first action, which in our case is $q_{1}(s_{0},a_{1})$ , scales as $1 / \sqrt{N}$ . The regret for OMD with the negative entropy in this setting can be shown to scale for any step-size at least as $\Omega (\sqrt{K\log N}) = \Omega (\sqrt{K\log S})$ , providing the dependence from the statement of the theorem. To prove this formally for the SSP reduction, we use the above construction coupled with a non-skewed reduction and careful setting of the costs. We include the details in Appendix A. + +Finally, we remark that this failure comes from the negative entropy stretching euclidean distance near the boundary of the space in such a way that two nearby points in terms of euclidean distance can be arbitrarily far in terms of negative entropy. This makes it hard for OMD to recover from the initial occupancy measure $q_{1}(s_{0},a_{1})\approx 1 / \sqrt{N} (\to 0$ as $N$ increases) unless the step-size is unreasonably large. This property does not generalize to all regularizers and in fact provides insights for designing a regularizer to appropriately handle sparsity. In particular, the regularizer we consider in the next section does not suffer from the same issue because the stretching of euclidean distance is finite since its gradient does not diverge at the boundary (i.e. as $q(s,a)\rightarrow 0$ ) unlike the negative entropy. + +# 4 The benefits of sparsity + +In this section, we show that it is possible to achieve a regret bound of order $\mathcal{O}\big(\sqrt{DKT_{\star}\log(MT_{\star})}\big)$ , where $M$ is the maximum number of non-zero entries in the cost. This is our main result and together with the lower bound in Theorem 4.6 establishes that the sparsity level $M$ acts as a measure of effective dimension instead of the state-action space size $SA$ for SSP with full-information feedback. + +In the previous section, we showed and discussed that the negative entropy, the regularizer used in OMD by existing methods, is inadequate to handle sparse SSP problems. Motivated by this failure, we consider alternate regularizers. However, identifying a suitable regularizer poses two key challenges. Firstly, it must work for SSP and the associated technical complexities compared to other simpler online learning problems. In particular, it needs to match the dependence in terms of the other non-sparsity-related quantities appearing in the regret of the negative entropy (i.e. $D, T_{\star}, K$ ). Second, it must explicitly leverage sparsity to improve performance. We propose the following family of regularizers parameterised by $p > 1$ : + +$$ +\psi_ {p} (q) = p \cdot \left(- 1 + \| q \| _ {1 + 1 / p} ^ {1 + 1 / p}\right) = p \cdot \left(- 1 + \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} | q (s, a) | ^ {1 + 1 / p}\right). \tag {4} +$$ + +As $p \to \infty$ , the regularizer in Equation (4) converges to the negative entropy. On the other hand, as $p \to 1$ , the regularizer converges to the squared Euclidean norm that enforces much weaker regularization on sparse points. Therefore, $\psi_p$ allows smooth interpolation between dense and sparse regimes via the tunable parameter $p$ (see Figure 2 for a comparison). In particular, $\psi_p$ for small $p$ + +![](images/032883741f63383e8500c5bb6880dc03eedc7b8d3ba7b8a9be7d1b42cdb36d55.jpg) +Figure 2: Bregman Divergence between a deterministic distribution $x = [0,1]$ and the uniform distribution $y = [1/2,1/2]$ for our regularizer $\psi_p$ , squared Euclidean norm $\psi_E$ and negative entropy $H$ for increasing values of $p$ . + +induces an OMD update that is able to easily move to and away from the boundary of the space, making it robust to the skewed initial occupancy measure on the SSP instance which caused the failure of the negative entropy in Section 3. The parameter $p$ also controls a trade-off between the stability and penalty term in (3), which ultimately will enable the removal of the dependence on $SA$ . + +Versions of this family of regularizers can be found in the convex optimization literature [18, 22]. As far as we are aware, its use with OMD is novel. A regularizer involving an $r$ -norm with $r \in (1, 2]$ has been used but the norm is squared rather than to the power of $r$ (see e.g. Section 6.7 in [24]). Our regularizer is also similar in flavor to the Tsallis-entropy in the sense that it converges to the negative entropy in the limit of its parameter. + +We note that OMD with the above regularizer can be implemented efficiently for any $p$ : the projection step of OMD over $\Delta (T)$ can be written as a convex optimization problem as in [28], which can be solved efficiently (details in Appendix B). + +We can now turn to our main result, which establishes regret bounds that scale with the sparsity level $M$ for OMD with the regularizer in (4) when $M$ is known. + +Theorem 4.1. Consider OMD with $\psi_p$ as regularizer. If $T > e$ is such that $q_{\pi^{\star}} \in \Delta(T)$ , $\eta = \sqrt{\frac{p T^{1 + 1/p}}{K D M^{1/p}}}$ , $p = \log(TM)$ , then $\mathbb{E}[R_K] \leqslant \mathcal{O}\Big(\sqrt{D K T \log(M T)}\Big)$ . + +We present below the outline of the proof and include the missing details in Appendix C. + +Proof. It can be shown that $\psi_p$ satisfies the condition (2) with $\alpha = 1$ , allowing us to use the bound in (3) as a starting point. Using that $q_\star \in \Delta(T)$ , we can bound the Penalty term: + +$$ +\psi_ {p} \left(q _ {\star}\right) - \psi_ {p} \left(q _ {1}\right) = p \left(\left\| q _ {\star} \right\| _ {1 + 1 / p} ^ {1 + 1 / p} - \left\| q _ {1} \right\| _ {1 + 1 / p} ^ {1 + 1 / p}\right) \leqslant p \cdot \left\| q _ {\star} \right\| _ {1} ^ {1 + 1 / p} \leqslant p \cdot T ^ {1 + 1 / p} \tag {5} +$$ + +It can also be shown that $\nabla^2\psi_p(q)^{-1} = \mathrm{diag}\left(\frac{p}{p + 1} q^{1 - 1 / p}\right)$ . Using that $c_{k}(s,a)^{2}\leqslant c_{k}(s,a)$ , we get + +$$ +\begin{array}{l} \left\| c _ {k} \right\| _ {\nabla^ {2} \psi_ {p} (q _ {k}) ^ {- 1}} ^ {2} \leqslant \frac {p}{p + 1} \sum_ {s, a} c _ {k} (s, a) q _ {k} (s, a) ^ {1 - 1 / p} \leqslant \left\| c _ {k} \right\| _ {1} \sum_ {s, a} \frac {c _ {k} (s , a)}{\left\| c _ {k} \right\| _ {1}} q _ {k} (s, a) ^ {1 - 1 / p} \\ \leqslant \left\| c _ {k} \right\| _ {1} \left(\sum_ {s, a} \frac {c _ {k} (s , a)}{\| c _ {k} \| _ {1}} q _ {k} (s, a)\right) ^ {1 - 1 / p} \tag {6} \\ = \| c _ {k} \| _ {1} ^ {1 / p} \langle c _ {k}, q _ {k} \rangle^ {1 - 1 / p} \leqslant M ^ {1 / p} \max \bigl \{1, \langle c _ {k}, q _ {k} \rangle \bigr \} \leqslant M ^ {1 / p} \bigl (1 + \langle c _ {k}, q _ {k} \rangle \bigr), \\ \end{array} +$$ + +where the key step (6) uses Jensen's inequality on the concave function $x^{1 - 1 / p}$ ( $p > 1$ ) and probability distribution $c_{k} / \|c_{k}\|_{1}$ . Plugging this into the Stability term and combining with (5): + +$$ +\sum_ {k = 1} ^ {K} \langle q _ {k} - q _ {\pi^ {\star}}, c _ {k} \rangle \leqslant \frac {p T ^ {1 + 1 / p}}{\eta} + \frac {\eta M ^ {1 / p} K}{2} + \frac {\eta M ^ {1 / p}}{2} \sum_ {k = 1} ^ {K} \langle q _ {k}, c _ {k} \rangle +$$ + +$$ +\begin{array}{l} \Longrightarrow \sum_ {k = 1} ^ {K} \langle q _ {k} - q _ {\pi^ {\star}}, c _ {k} \rangle \leqslant \frac {1}{1 - \frac {\eta M ^ {1 / p}}{2}} \left(\frac {p T ^ {1 + 1 / p}}{\eta} + \frac {\eta M ^ {1 / p} K}{2} + \frac {\eta M ^ {1 / p}}{2} \sum_ {k = 1} ^ {K} \langle q _ {\star}, c _ {k} \rangle\right) \\ \Longrightarrow \sum_ {k = 1} ^ {K} \left\langle q _ {k} - q _ {\pi^ {\star}}, c _ {k} \right\rangle \leqslant \frac {2 p T ^ {1 + 1 / p}}{\eta} + 2 \eta M ^ {1 / p} D K, \\ \end{array} +$$ + +where the last step uses that $\sum_{k=1}^{K}\langle q_{\star}, c_k\rangle \leqslant \sum_{k=1}^{K}\langle q_{\pi_f}, c_k\rangle \leqslant K\|q_{\pi_f}\|_1\|c_k\|_\infty \leqslant DK, D \geqslant 1$ and $\eta \leqslant 4M^{-1/p} \iff 1 - \eta M^{1/p}/2 \geqslant 1/2$ . Tuning $\eta = \sqrt{\frac{pT^{1+1/p}}{KDM^{1/p}}}$ (so $\eta \leqslant 4M^{-1/p}$ for sufficiently large $K$ ) and $p = \log(TM)$ gives the result. + +Provided we can suitably select $T \approx T_{\star}$ (see Section 4.1), this result establishes that sparsity does lead to an improvement in the minimax regret. In Section 4.2, we show that the dependence on $M$ and $SA$ is optimal, ruling out polynomial improvements from sparsity such as in the experts setting [19]. This highlights that $M$ acts as the effective dimension of the problem instead of $SA$ . In particular, if the sparsity level $M$ is constant, then we obtain a dimension-independent regret of $\mathcal{O}(\sqrt{DKT\log T})$ . + +Remark 4.2. Although we express the bound in terms of the sparsity level $M$ , it can be seen that the analysis above holds more generally if $M$ is instead an upper bound on the $\ell_1$ norm of the costs: $M = \max_k \|c_k\|_1$ . This relaxation allows our result to cover "softly sparse" cost structures and aligns with the notion of first-order bounds commonly studied in the online learning literature [23, 37, 35]. + +Remark 4.3. The above result does not recover the $M / A$ polynomial improvement in the special case of the expert setting. This can be recovered through a regret bound of a slightly different flavor which includes the hitting time of the uniform policy. We include the details and subtleties in Appendix D but the upshot is that the necessity to reach the goal state in SSP creates a fundamental difference in the benefits of sparsity compared to the expert setting. + +# 4.1 Sparse-agnostic parameter-free upper bound + +The procedure in Theorem 4.4 assumes knowledge of the sparsity level $M$ to tune the parameter $p$ of our regularizer and uses knowledge of the expected hitting time of the optimal policy $T_{\star}$ to consider OMD over the space of suitable occupancy measures. We now adapt existing techniques to remove both of these assumptions and derive fully parameter-free guarantees. + +For the unknown sparsity level, we use the same approach as in [19]. We divide the $K$ episodes into batches. Within each batch, we independently run OMD tuning the parameter $p$ of our regularizer with the sparsity level observed up to the current batch, as described in Algorithm 1 in Appendix C.2. + +For the unknown expected hitting time of the optimal policy $T_{\star}$ , we can exploit the same meta-algorithm technique as in [8], using the sparse agnostic algorithms introduced above as base learners. We run $N \approx \log K$ instances of Algorithm 1 where the $j$ -th instance sets its parameter $T$ as $b(j) \approx 2^j$ . Therefore, there exists a good instance $j_{\star}$ such that $b(j_{\star})$ is close to the unknown $T_{\star}$ . The regret of a scale-invariant meta-algorithm, described for completeness in Algorithm 2 in Appendix C.2, closely matches that of this good instance. + +Together, these two techniques yield the following parameter-free regret bound (proof in Appendix C.2): + +Theorem 4.4. If $K > \max \left(T_{\star}, \frac{T_{\star}}{D} \log (T_{\star}M)\right)$ and $T_{\star} > e$ , Algorithm 2 guarantees $\mathbb{E}\left[R_K\right] \leqslant \tilde{\mathcal{O}}\left(\sqrt{DKT_{\star} \log(MT_{\star})} + T_{\star}\right)$ , where the notation $\tilde{\mathcal{O}}$ hides double-logarithmic factors. + +The leading term matches the regret bound from Theorem 4.1, while the second does not depend on $M$ or $K$ . Therefore, running a procedure that does not assume knowledge of $M$ and $T_{\star}$ comes at no additional cost in terms of the regret bound (up to double-logarithmic factors). We also note that it is common for log-log factors to be ignored in parameter free results with expert-like algorithms [12, 16]. It is also possible to obtain a bound that holds with high-probability since the high-probability analysis given in [8] can easily be adapted to work with our regularizer. + +Remark 4.5. The assumption $K \geqslant \frac{T_{\star}}{D} \log (T_{\star}M)$ or $K \geqslant \frac{T_{\star}}{D} \log (T_{\star}SA)$ in the non-sparse setting is actually non-restrictive since it is required for the upper-bound to be meaningful: + +$$ +\sqrt {D K T _ {\star} \log (M T _ {\star})} \leqslant T _ {\star} K \iff K \geqslant \frac {T _ {\star}}{D} \log (T _ {\star} M). +$$ + +In particular, it is likely that there is a gap between the behavior of the minimax regret between the "low-dimensional" setting which we study and a high-dimensional setting where $K \ll \frac{T_{\star}}{D} \log (T_{\star}M)$ . The high-dimensional problem is yet to be explored, even in the non-adversarial setting and could be an interesting avenue of future research. Indeed, all prior works on SSP have implicitly studied the problem in low-dimension, which comes with an implicit assumption that $K$ is sufficiently large. + +# 4.2 Lower bound + +In this section, we provide a general lower bound for sparse SSP problems. + +Theorem 4.6. For any $D, T_{\star}, K, S, A$ with $T^{\star} \geqslant D \geqslant 3\log S$ , $S(A - 1) \geqslant 400$ , $K \geqslant \frac{800T^{\star}}{D}\log M$ and $M \geqslant 101$ , there exists an SSP instance with stochastic $M$ -sparse costs, $S$ states and $A$ actions such that its diameter is $D$ , the expected hitting time of the optimal policy is $T^{\star}$ , and the expected regret with respect to the randomness of the losses for any learner after $K$ episodes is $\mathbb{E}[R_K] \geqslant \Omega\big(\sqrt{KT^{\star}D\log M}\big)$ . + +For general $M (> 100)$ , the lower bound matches the upper bound established in Theorem 4.1 in its dependence on $M$ , characterizing the minimax regret for general sparse problems (up to a log $T_{\star}$ term). For $M = SA$ , our result gives a $\Omega \left( \sqrt{KT^{\star}D \log SA} \right)$ lower bound improving on the $\Omega \left( \sqrt{KT^{\star}D} \right)$ lower bound of [8]. In particular, this establishes the optimal dependence on the size of the state-action space $SA$ in the minimax regret for the general non-sparse SSP problem. + +Proof intuition: The proof is based on the combination of an SSP instance from [8] and a probabilistic costs construction, which then requires some non-trivial arguments to extend to the sparse SSP problem. We give an overview of the construction and defer the details to Appendix E. + +The MDP construction is essentially a reduction to a non-sparse experts problem with $\mathcal{O}(M)$ actions. First, there is a reduction to an experts problem with $\mathcal{O}(SA)$ actions. Then within these, there are $\mathcal{O}(M)$ good actions, while the remaining are bad. The good actions suffer small costs in expectation and can lead directly to the goal-state. The bad actions are zero-cost but all lead to the same unique bad state, where only one action leads to the goal-state and suffers high cost. This allows a big proportion of the actions to be bad while still guaranteeing sparsity and forcing the learner to only consider the $\mathcal{O}(M)$ good actions, completing the reduction to the non-sparse experts problem with $\mathcal{O}(M)$ actions. + +However, we cannot directly apply lower bounds for the experts problem because of subtleties in the reduction and the cost-generating mechanism. We use a similar approach to the experts lower bounds by sampling the costs i.i.d. from a Bernoulli distribution, however with a scaled parameter to ensure the reduction above holds. The regret in this stochastic environment can then be expressed as the maximum of asymmetric zero-mean i.i.d. random walks, capturing how much better the optimal policy can be by choosing the best action after the i.i.d. Bernoulli costs have been sampled for all episodes. The result then follows from a technical result on the expectation of this maximum that we derive in Appendix G. We note that the reduction and costs are constructed in such a way that the diameter of the MDP and expected hitting time of the optimal policy are indeed $D$ and $T_{\star}$ . + +# 5 Unknown transition setting + +In this section, we consider the setting where the transitions are unknown and show through the following lower bound that the benefits of sparsity are limited. + +Theorem 5.1. For any $D, K, S, A$ with $S \geqslant 2, A \geqslant 16$ , $D \geqslant 2$ and $K \geqslant SA$ , there exists an SSP instance with $M = 1$ , $S$ states and $A$ actions such that its diameter is $D$ and the expected regret for any learner without knowledge of the transitions after $K$ episodes is $\mathbb{E}[R_K] \geqslant \Omega(D \sqrt{SAK})$ . + +The above result establishes that the minimax regret for the unknown transition setting must scale polynomially with $SA$ , regardless of the sparsity. In particular, this highlights the limited benefits of sparsity in removing the dependence on the state-action space size in the unknown transition setting, which is in stark contrast to the known transition setting. + +The proof is based on an SSP instance used by [29] to prove an $\Omega(D\sqrt{SAK})$ lower bound in the unknown transition non-sparse setting. It turns out that this instance can be adapted such that the cost is only non-zero for a single state-action pair, while keeping the regret lower bound unchanged, giving the above result. We include the details in Appendix F. + +# 6 Conclusion, limitations and future-work + +In this work, we studied the SSP problem under sparse adversarial costs and full-information feedback. When the transitions are known, we have shown that existing methods fail to appropriately exploit sparsity. Instead, we designed a family of regularizers to use with Online Mirror Descent that allowed us to characterize the sparse minimax regret, establishing the extent of the benefits of sparsity in this setting. When the transitions are unknown, we showed that even the sparse minimax regret scales polynomially in the size of the state-action space, suggesting fundamental limits in such settings. + +Our results open up many further directions of research. In particular, we established the benefits of sparsity under known transition as limited to logarithmic, however, there could be structural properties of an MDP that could break this logarithmic limit and achieve polynomial benefits. Moreover, we have limited our focus to the adversarial full-information feedback setting, but the study of sparse SSP problems in other settings, such as partial feedback, stochastic environments, or structured decision problems remains unexplored. + +# Acknowledgments and Disclosure of Funding + +Emmeran Johnson is funded by EPSRC through the Modern Statistics and Statistical Machine Learning (StatML) CDT (grant no. EP/S023151/1). Alberto Rumi was funded by European Lighthouse of AI for Sustainability project (ELIAS). Patrick Rebeschini was funded by UK Research and Innovation (UKRI) under the UK government's Horizon Europe funding guarantee (grant no. EP/Y028333/1). + +We would like to thank the reviewers and meta-reviewers for their time and feedback. + +# References + +[1] Dimitri P Bertsekas and John N Tsitsiklis. An analysis of stochastic shortest path problems. Mathematics of Operations Research, 16(3):580-595, 1991. +[2] A. V. Boyd. Inequalities for mills' ratio. 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Institutional review board (IRB) approvals or equivalent for research with human subjects + +Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? + +Answer: [NA] + +Justification: The paper does not involve crowdsourcing nor research with human subjects. + +# Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. +- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. +We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. +- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Failure of the negative entropy + +In this appendix, we prove our lower bound result for OMD with the negative entropy from Section 3. We first restate the result. + +Theorem 3.1. For any $S \geqslant 6$ , there exists an SSP instance with a fixed horizon of 3, sparsity level $M = 3$ , an action space of size $A = 2$ and state space of size $S$ such that the regret of OMD (1) with negative-entropy regularization and any step-size $\eta > 0$ after $K$ episodes is $\mathbb{E}\left[R_K\right] = \Omega \left(\min \{\sqrt{K\log S}, K\}\right)$ . + +Proof. Fix $K$ even, $S \geqslant 6$ , $A = 2$ and $N = S - 5$ . We first describe the SSP instance. Consider the following MDP $\mathcal{M} = (\mathcal{S}, \mathcal{A}, p, s_0, g)$ , where $\mathcal{S} = \{s_0, s_0^L, s_0^R, s_1^R, \dots, s_N^R, s_1\}$ and $\mathcal{A} = \{a_1, a_2\}$ . The transitions and costs (in each episode $k$ ) are defined as: + +- $s_0 \colon p(s_0^L | s_0, a) = p(s_0^R | s_0, a) = 1/2$ and $c_k(s_0, a) = 0$ for all $a \in \mathcal{A}$ . +- $s_0^L \colon p(s_g^L | s_0^L, a) = 1$ for all $a \in \mathcal{A}$ and $c_k(s_0^L, a_1) = \frac{1 + (-1)^k}{2}$ , $c_k(s_0^L, a_2) = 1/2$ . +- $s_0^R \colon p(s_g^R | s_0^R, a_1) = 1, p(s_i^R | s_0^R, a_2) = 1 / N$ and $c_k(s_0^R, a_1) = 0, c_k(s_0^R, a_2) = 1$ . +- $s_i^R$ : $p(g|s_i^R, a) = 1$ and $c_k(s_i^R, a) = 0$ for all $a \in \mathcal{A}$ . +- $s_g^L \colon p(g|s_g^L, a) = 1$ and $c_k(s_g^L, a) = 0$ for all $a \in \mathcal{A}$ . +- $s_g^R$ : $p(g|s_g^R, a) = 1$ and $c_k(s_g^R, a) = 0$ for all $a \in \mathcal{A}$ . + +An illustration is given in Figure 3. This SSP instance has a fixed horizon of 3 in the sense that all policies have a hitting time of exactly 3 (the states $s_g^L$ and $s_g^R$ are added to guarantee this). As a result we have that $T_{\star} = D = 3$ . Also note that there are at most 3 state-action pairs that have non-zero cost, therefore the sparsity level $M = 3$ . + +![](images/4852b52961a7222b0993470a526c31a4a63aeb6b1f2a9d0470cedc4743d749f9.jpg) +Figure 3: Diagram illustrating MDP construction for the proof of Theorem 3.1. When an action is not specified for an edge, then both actions give the same transition and cost. If an edge has a number in black, it is a transition probability; if it does not then the transition is deterministic. The costs are given in red. The formal description of the MDP is given above. + +From Appendix B.1 in [28] (we can ignore the optimization over $\lambda$ because we are in a fixed horizon setting), the update of OMD with negative entropy for any $k\geqslant 0$ can be computed by solving a convex optimization problem: + +$$ +q _ {k + 1} (s, a) = q _ {k} (s, a) e ^ {B _ {k} ^ {v _ {k + 1}} (s, a)}, \quad \text {w h e r e} \quad B _ {k} ^ {v} (s, a) = v (s) - \eta c _ {k} (s, a) - \sum_ {s ^ {\prime} \in S} p (s ^ {\prime} | s, a) v (s ^ {\prime}), +$$ + +$$ +v _ {k + 1} = \underset {v} {\arg \min } \mathcal {D} _ {k} (v), +$$ + +$$ +\mathcal {D} _ {k} (v) = \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} q _ {k} (s, a) e ^ {B _ {k} ^ {v} (s, a)} - v (s _ {0}), +$$ + +with $q_0(s,a) = 1$ and $c_{0}(s,a) = 0$ . This allows us to compute exactly the points played by the algorithm on the SSP instance described above, and in turn compute the regret, from which the result will follow. + +In the following few pages, we compute the occupancy measures played by OMD with the negative entropy on the SSP instance described earlier for all episodes, using the convex optimization problem above. We begin by computing expressions for $B_{k}^{v}(s,a)$ in each state: + +- $B_{k}^{v}(s_{0},a) = v(s_{0}) - \frac{1}{2} v(s_{0}^{L}) - \frac{1}{2} v(s_{0}^{R})$ for all $a\in \mathcal{A}$ +- $B_{k}^{v}(s_{0}^{L}, a) = v(s_{0}^{L}) - \eta c_{k}(s_{0}^{L}, a) - v(s_{g}^{L})$ for all $a \in \mathcal{A}$ . +$B_{k}^{v}(s_{g}^{L}) = v(s_{g}^{L})$ +$B_{k}^{v}(s_{0}^{R},a_{1}) = v(s_{0}^{R}) - v(s_{g}^{R})$ +- $B_{k}^{v}(s_{0}^{R}, a_{2}) = v(s_{0}^{R}) - \eta c_{k}(s_{0}^{R}, a_{2}) - \frac{1}{N}\sum_{i=1}^{N}v(s_{i}^{R}) = v(s_{0}^{R}) - \eta c_{k}(s_{0}^{R}, a_{2}) - v(s_{1}^{R})$ since by symmetry $v(s_{i}^{R}) = v(s_{1}^{R})$ for all $i \geqslant 1$ and any $v$ solving the convex optimization problem specified in the OMD update. +- $B_k^v (s_i^R,a) = v(s_i^R) = v(s_1^R)$ for all $a\in \mathcal{A}$ +- $B_{k}^{v}(s_{g}^{R}, a) = v(s_{g}^{R})$ for all $a \in \mathcal{A}$ . + +Plugging these into the optimization problem, we obtain (recall the notation $q(s) = \sum_{a\in A}q(s,a)$ ): + +$$ +v _ {k + 1} = \underset {v} {\arg \min } \quad \mathcal {D} _ {k} (v) = \underset {v} {\arg \min } \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} q _ {k} (s, a) e ^ {B _ {k} ^ {v} (s, a)} - v (s _ {0}) +$$ + +$$ +\begin{array}{l} = \arg \min _ {v} q _ {k} (s _ {0}, a _ {1}) e ^ {v (s _ {0}) - 0. 5 v (s _ {0} ^ {L}) - 0. 5 v (s _ {0} ^ {R})} + q _ {k} (s _ {0}, a _ {2}) e ^ {v (s _ {0}) - 0. 5 v (s _ {0} ^ {L}) - 0. 5 v (s _ {0} ^ {R})} \\ + q _ {k} \left(s _ {0} ^ {L}, a _ {1}\right) e ^ {v \left(s _ {0} ^ {L}\right) - \eta c _ {k} \left(s _ {0} ^ {L}, a _ {1}\right) - v \left(s _ {g} ^ {L}\right)} + q _ {k} \left(s _ {0} ^ {L}, a _ {2}\right) e ^ {v \left(s _ {0} ^ {L}\right) - \eta c _ {k} \left(s _ {0} ^ {L}, a _ {2}\right) - v \left(s _ {g} ^ {L}\right)} \\ + q _ {k} \left(s _ {g} ^ {L}, a _ {1}\right) e ^ {v \left(s _ {g} ^ {L}\right)} + q _ {k} \left(s _ {g} ^ {L}, a _ {2}\right) e ^ {v \left(s _ {g} ^ {L}\right)} \\ + q _ {k} \left(s _ {0} ^ {R}, a _ {1}\right) e ^ {v \left(s _ {0} ^ {R}\right) - v \left(s _ {g} ^ {R}\right)} + q _ {k} \left(s _ {0} ^ {R}, a _ {2}\right) e ^ {v \left(s _ {0} ^ {R}\right) - \eta c _ {k} \left(s _ {0} ^ {R}, a _ {2}\right) - v \left(s _ {1} ^ {R}\right)} \\ + \sum_ {i = 1} ^ {N} \left\{q _ {k} \left(s _ {i} ^ {R}, a _ {1}\right) e ^ {v \left(s _ {1} ^ {R}\right)} + q _ {k} \left(s _ {i} ^ {R}, a _ {2}\right) e ^ {v \left(s _ {1} ^ {R}\right)} \right\} \\ + q _ {k} \left(s _ {g} ^ {R}, a _ {1}\right) e ^ {v \left(s _ {g} ^ {R}\right)} + q _ {k} \left(s _ {g} ^ {R}, a _ {2}\right) e ^ {v \left(s _ {g} ^ {R}\right)} \\ - v (s _ {0}) \\ \end{array} +$$ + +$$ +\begin{array}{l} = \operatorname * {a r g m i n} _ {v} q _ {k} (s _ {0}) e ^ {v (s _ {0}) - 0. 5 v (s _ {0} ^ {L}) - 0. 5 v (s _ {0} ^ {R})} \\ + q _ {k} \left(s _ {0} ^ {L}, a _ {1}\right) e ^ {v \left(s _ {0} ^ {L}\right) - \eta c _ {k} \left(s _ {0} ^ {L}, a _ {1}\right) - v \left(s _ {g} ^ {L}\right)} + q _ {k} \left(s _ {0} ^ {L}, a _ {2}\right) e ^ {v \left(s _ {0} ^ {L}\right) - \eta c _ {k} \left(s _ {0} ^ {L}, a _ {2}\right) - v \left(s _ {g} ^ {L}\right)} \\ + q _ {k} \left(s _ {g} ^ {L}\right) e ^ {v \left(s _ {g} ^ {L}\right)} \\ + q _ {k} \left(s _ {0} ^ {R}, a _ {1}\right) e ^ {v \left(s _ {0} ^ {R}\right) - v \left(s _ {g} ^ {R}\right)} + q _ {k} \left(s _ {0} ^ {R}, a _ {2}\right) e ^ {v \left(s _ {0} ^ {R}\right) - \eta c _ {k} \left(s _ {0} ^ {R}, a _ {2}\right) - v \left(s _ {1} ^ {R}\right)} \\ + \sum_ {i = 1} ^ {N} \left\{q _ {k} \left(s _ {i} ^ {R}\right) e ^ {v \left(s _ {1} ^ {R}\right)} \right\} \\ + q _ {k} \left(s _ {g} ^ {R}\right) e ^ {v \left(s _ {g} ^ {R}\right)} \\ - v (s _ {0}). \\ \end{array} +$$ + +This being a convex optimization problem, it can be solved by differentiating and setting to 0: + +$$ +\frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {0})} = q _ {k} (s _ {0}) e ^ {v (s _ {0}) - 0. 5 v (s _ {0} ^ {L}) - 0. 5 v (s _ {0} ^ {R})} - 1 = 0 +$$ + +$$ +\begin{array}{l} \frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {0} ^ {L})} = - 0. 5 q _ {k} (s _ {0}) e ^ {v (s _ {0}) - 0. 5 v (s _ {0} ^ {L}) - 0. 5 v (s _ {0} ^ {R})} + q _ {k} (s _ {0} ^ {L}, a _ {1}) e ^ {v (s _ {0} ^ {L}) - \eta c _ {k} (s _ {0} ^ {L}, a _ {1}) - v (s _ {g} ^ {L})} + q _ {k} (s _ {0} ^ {L}, a _ {2}) e ^ {v (s _ {0} ^ {L}) - \eta c _ {k} (s _ {0} ^ {L}, a _ {2}) - v (s _ {g} ^ {L})} \\ = - 0. 5 + q _ {k} \left(s _ {0} ^ {L}, a _ {1}\right) e ^ {v \left(s _ {0} ^ {L}\right) - \eta c _ {k} \left(s _ {0} ^ {L}, a _ {1}\right) - v \left(s _ {g} ^ {L}\right)} + q _ {k} \left(s _ {0} ^ {L}, a _ {2}\right) e ^ {v \left(s _ {0} ^ {L}\right) - \eta c _ {k} \left(s _ {0} ^ {L}, a _ {2}\right) - v \left(s _ {g} ^ {L}\right)} = 0 \\ \end{array} +$$ + +$$ +\begin{array}{l} \frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {g} ^ {L})} = - q _ {k} (s _ {0} ^ {L}, a _ {1}) e ^ {v (s _ {0} ^ {L}) - \eta c _ {k} (s _ {0} ^ {L}, a _ {1}) - v (s _ {g} ^ {L})} - q _ {k} (s _ {0} ^ {L}, a _ {2}) e ^ {v (s _ {0} ^ {L}) - \eta c _ {k} (s _ {0} ^ {L}, a _ {2}) - v (s _ {g} ^ {L})} + q _ {k} (s _ {g} ^ {L}) e ^ {v (s _ {g} ^ {L})} \\ = q _ {k} \left(s _ {g} ^ {L}\right) e ^ {v \left(s _ {g} ^ {L}\right)} - 0. 5 = 0 \\ \end{array} +$$ + +$$ +\begin{array}{l} \frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {0} ^ {R})} = - 0. 5 q _ {k} (s _ {0}) e ^ {v (s _ {0}) - 0. 5 v (s _ {0} ^ {L}) - 0. 5 v (s _ {0} ^ {R})} + q _ {k} (s _ {0} ^ {R}, a _ {1}) e ^ {v (s _ {0} ^ {R}) - v (s _ {g} ^ {R})} + q _ {k} (s _ {0} ^ {R}, a _ {2}) e ^ {v (s _ {0} ^ {R}) - \eta c _ {k} (s _ {0} ^ {R}, a _ {2}) - v (s _ {1} ^ {R})} \\ = - 0. 5 + q _ {k} \left(s _ {0} ^ {R}, a _ {1}\right) e ^ {v \left(s _ {0} ^ {R}\right) - v \left(s _ {g} ^ {R}\right)} + q _ {k} \left(s _ {0} ^ {R}, a _ {2}\right) e ^ {v \left(s _ {0} ^ {R}\right) - \eta c _ {k} \left(s _ {0} ^ {R}, a _ {2}\right) - v \left(s _ {1} ^ {R}\right)} = 0 \\ \end{array} +$$ + +$$ +\begin{array}{l} \frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {1} ^ {R})} = - q _ {k} (s _ {0} ^ {R}, a _ {2}) e ^ {v (s _ {0} ^ {R}) - \eta c _ {k} (s _ {0} ^ {R}, a _ {2}) - v (s _ {1} ^ {R})} + e ^ {v (s _ {1} ^ {R})} \sum_ {i = 1} ^ {N} q _ {k} (s _ {i} ^ {R}) = 0 \\ \frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {g} ^ {R})} = - q _ {k} (s _ {0} ^ {R}, a _ {1}) e ^ {v (s _ {0} ^ {R}) - v (s _ {g} ^ {R})} + q _ {k} (s _ {g} ^ {R}) e ^ {v (s _ {g} ^ {R})} = 0. \\ \end{array} +$$ + +Let's look specifically at the case $k = 0$ ( $q_0(s, a) = 1$ , $q_0(s) = 2$ , $c_0(s, a) = 0$ for all $s \in S, a \in \mathcal{A}$ ). For the left part of the MDP we have: + +$$ +\begin{array}{l} \frac {\partial \mathcal {D} _ {k} (v)}{\partial v \left(s _ {0} ^ {L}\right)} = 0 \Rightarrow 2 e ^ {v \left(s _ {0} ^ {L}\right) - v \left(s _ {g} ^ {L}\right)} = 0. 5 \Rightarrow e ^ {v \left(s _ {0} ^ {L}\right)} = 0. 2 5 e ^ {v \left(s _ {g} ^ {L}\right)} \\ \frac {\partial \mathcal {D} _ {k} (v)}{\partial v \left(s _ {g} ^ {L}\right)} = 0 \Rightarrow e ^ {v \left(s _ {g} ^ {L}\right)} = 0. 2 5 \Rightarrow e ^ {v \left(s _ {0} ^ {L}\right)} = 0. 2 5 ^ {2} \\ \Longrightarrow q _ {1} \left(s _ {0} ^ {L}, a\right) = e ^ {B _ {0} ^ {v} \left(s _ {0} ^ {L}, a\right)} = \frac {0 . 2 5 ^ {2}}{0 . 2 5} = 0. 2 5, \text {f o r a l l} a \in \mathcal {A}. \\ \end{array} +$$ + +For the right part of the MDP, we have: + +$$ +\frac {\partial \mathcal {D} _ {k} (v)}{\partial v \left(s _ {0} ^ {R}\right)} = 0 \Rightarrow e ^ {v \left(s _ {0} ^ {R}\right) - v \left(s _ {g} ^ {R}\right)} + e ^ {v \left(s _ {0} ^ {R}\right) - v \left(s _ {1} ^ {R}\right)} = 0. 5 \Rightarrow e ^ {v \left(s _ {0} ^ {R}\right)} = \frac {0 . 5}{e ^ {- v \left(s _ {g} ^ {R}\right)} + e ^ {- v \left(s _ {1} ^ {R}\right)}} +$$ + +$$ +\frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {1} ^ {R})} = 0 \Longrightarrow e ^ {v (s _ {0} ^ {R}) - v (s _ {1} ^ {R})} = 2 N e ^ {v (s _ {1} ^ {R})} \Longrightarrow e ^ {v (s _ {1} ^ {R})} = \frac {1}{\sqrt {2 N}} e ^ {0. 5 v (s _ {0} ^ {R})} +$$ + +$$ +\frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {g} ^ {R})} = 0 \Rightarrow e ^ {v (s _ {0} ^ {R}) - v (s _ {g} ^ {R})} = 2 e ^ {v (s _ {g} ^ {R})} \Rightarrow e ^ {v (s _ {g} ^ {R})} = \frac {1}{\sqrt {2}} e ^ {0. 5 v (s _ {0} ^ {R})} +$$ + +$$ +\begin{array}{l} \Longrightarrow e ^ {v \left(s _ {0} ^ {R}\right)} = \frac {0 . 5}{\sqrt {2} e ^ {- 0 . 5 v \left(s _ {0} ^ {R}\right)} + \sqrt {2 N} e ^ {- 0 . 5 v \left(s _ {0} ^ {R}\right)}} \\ \Longrightarrow e ^ {0. 5 v \left(s _ {0} ^ {R}\right)} = \frac {0 . 5}{\sqrt {2} + \sqrt {2 N}} \\ \Longrightarrow e ^ {v \left(s _ {0} ^ {R}\right)} = \frac {0 . 2 5}{(\sqrt {2} + \sqrt {2 N}) ^ {2}} \\ \Longrightarrow q _ {1} \left(s _ {0} ^ {R}, a _ {1}\right) = e ^ {B _ {0} ^ {v} \left(s _ {0} ^ {R}, a _ {1}\right)} = e ^ {v \left(s _ {0} ^ {R}\right) - v \left(s _ {g} ^ {R}\right)} = \sqrt {2} e ^ {0. 5 v \left(s _ {0} ^ {R}\right)} = \frac {0 . 5}{1 + \sqrt {N}} \\ \end{array} +$$ + +$$ +q _ {1} \left(s _ {0} ^ {R}, a _ {2}\right) = e ^ {B _ {0} ^ {v} \left(s _ {0} ^ {R}, a _ {2}\right)} = e ^ {v \left(s _ {0} ^ {R}\right) - v \left(s _ {1} ^ {R}\right)} = \sqrt {2 N} e ^ {0. 5 v \left(s _ {0} ^ {R}\right)} = \frac {0 . 5 \sqrt {N}}{1 + \sqrt {N}} +$$ + +$$ +q _ {1} (s _ {1} ^ {R}, a) = e ^ {B _ {0} ^ {v} (s _ {1} ^ {R}, a)} = e ^ {v (s _ {1} ^ {R})} = \frac {1}{\sqrt {2 N}} e ^ {0. 5 v (s _ {0} ^ {R})} = \frac {0 . 2 5}{\sqrt {N} (1 + \sqrt {N})} +$$ + +$$ +q _ {1} (s _ {g} ^ {R}, a) = e ^ {B _ {0} ^ {v} (s _ {g} ^ {R}, a)} = e ^ {v (s _ {g} ^ {R})} = \frac {1}{\sqrt {2}} e ^ {0. 5 v (s _ {0} ^ {R})} = \frac {0 . 2 5}{1 + \sqrt {N}}. +$$ + +Let's now look at general $k \geqslant 1$ : Since $q_{k}$ is an occupancy measure, it satisfies the properties of the dynamics of the MDP (see the definition of $\Delta(T)$ in Section 2) and we have that for any $s \in S$ : $\sum_{a \in \mathcal{A}} q_{k}(s, a) = \sum_{s' \in S} \sum_{a' \in \mathcal{A}} p(s | s', a') q_{k}(s', a') + \mathbb{I}\{s = s_{0}\}$ . In particular, this gives + +$$ +\begin{array}{l} \bullet s = s _ {0} \colon q _ {k} (s _ {0}) = 1 \\ \bullet s = s _ {0} ^ {L}: q _ {k} \left(s _ {0} ^ {L}\right) = 0. 5 q _ {k} \left(s _ {0}\right) = 0. 5 \\ \end{array} +$$ + +$$ +\begin{array}{l} \bullet s = s _ {g} ^ {L} \colon q _ {k} (s _ {g} ^ {L}) = q _ {k} (s _ {0} ^ {L}) = 0. 5 \\ \bullet s = s _ {0} ^ {R} \colon q _ {k} (s _ {0} ^ {R}) = 0. 5 q _ {k} (s _ {0}) = 0. 5 \\ \bullet s = s _ {g} ^ {R} \colon q _ {k} \left(s _ {g} ^ {R}\right) = q _ {k} \left(s _ {0} ^ {R}, a _ {1}\right) \\ \bullet s = s _ {1} ^ {R}: q _ {k} \left(s _ {1} ^ {R}\right) = \frac {1}{N} q _ {k} \left(s _ {0} ^ {R}, a _ {2}\right) \\ \end{array} +$$ + +This leads to the following simplifications in the derivatives of $\mathcal{D}_k(v)$ : + +$$ +\begin{array}{l} \frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {0})} = e ^ {v (s _ {0}) - 0. 5 v (s _ {0} ^ {L}) - 0. 5 v (s _ {0} ^ {R})} - 1 = 0 \\ \frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {0} ^ {L})} = - 0. 5 + q _ {k} (s _ {0} ^ {L}, a _ {1}) e ^ {v (s _ {0} ^ {L}) - \eta c _ {k} (s _ {0} ^ {L}, a _ {1}) - v (s _ {g} ^ {L})} + (1 / 2 - q _ {k} (s _ {0} ^ {L}, a _ {1})) e ^ {v (s _ {0} ^ {L}) - \eta c _ {k} (s _ {0} ^ {L}, a _ {2}) - v (s _ {g} ^ {L})} = 0 \\ \frac {\partial \mathcal {D} _ {k} (v)}{\partial v \left(s _ {g} ^ {L}\right)} = 0. 5 e ^ {v \left(s _ {g} ^ {L}\right)} - 0. 5 = 0 \Longrightarrow e ^ {v \left(s _ {g} ^ {L}\right)} = 1 \\ \frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {0} ^ {R})} = - 0. 5 + q _ {k} (s _ {0} ^ {R}, a _ {1}) e ^ {v (s _ {0} ^ {R}) - v (s _ {g} ^ {R})} + (1 / 2 - q _ {k} (s _ {0} ^ {R}, a _ {1})) e ^ {v (s _ {0} ^ {R}) - \eta - v (s _ {1} ^ {R})} = 0 \\ \frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {1} ^ {R})} = - q _ {k} (s _ {0} ^ {R}, a _ {2}) e ^ {v (s _ {0} ^ {R}) - \eta - v (s _ {1} ^ {R})} + q _ {k} (s _ {0} ^ {R}, a _ {2}) e ^ {v (s _ {1} ^ {R})} = 0 \Rightarrow e ^ {v (s _ {0} ^ {R})} = e ^ {\eta + 2 v (s _ {1} ^ {R})} \\ \frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {g} ^ {R})} = - q _ {k} (s _ {0} ^ {R}, a _ {1}) e ^ {v (s _ {0} ^ {R}) - v (s _ {g} ^ {R})} + q _ {k} (s _ {0} ^ {R}, a _ {1}) e ^ {v (s _ {g} ^ {R})} = 0 \Rightarrow e ^ {v (s _ {0} ^ {R})} = e ^ {2 v (s _ {g} ^ {R})}. \\ \end{array} +$$ + +# Left part of the MDP: + +$$ +\begin{array}{l} \frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {0} ^ {L})} = 0 \Rightarrow q _ {k} (s _ {0} ^ {L}, a _ {1}) e ^ {v (s _ {0} ^ {L}) - \eta \frac {1 + (- 1) ^ {k}}{2}} + (0. 5 - q _ {k} (s _ {0} ^ {L}, a _ {1})) e ^ {v (s _ {0} ^ {L}) - 0. 5 \eta} = 0. 5 \\ \Longrightarrow e ^ {v \left(s _ {0} ^ {L}\right)} = \frac {0 . 5}{q _ {k} \left(s _ {0} ^ {L} , a _ {1}\right) e ^ {- \eta \frac {1 + (- 1) ^ {k}}{2}} + \left(0 . 5 - q _ {k} \left(s _ {0} ^ {L} , a _ {1}\right)\right) e ^ {- 0 . 5 \eta}} \\ \end{array} +$$ + +$$ +\begin{array}{l} k + 1 = 2 \Longrightarrow e ^ {v (s _ {0} ^ {L})} = \frac {0 . 5}{0 . 2 5 + 0 . 2 5 e ^ {- 0 . 5 \eta}} \\ \Longrightarrow q _ {2} \left(s _ {0} ^ {L}, a _ {1}\right) = q _ {1} \left(s _ {0} ^ {L}, a _ {1}\right) e ^ {B _ {1} ^ {v} \left(s _ {0} ^ {L}, a _ {1}\right)} = 0. 2 5 e ^ {v \left(s _ {0} ^ {L}\right) - v \left(s _ {g} ^ {L}\right)} = \frac {0 . 5}{1 + e ^ {- 0 . 5 \eta}} \\ \end{array} +$$ + +$$ +\begin{array}{l} k + 1 = 3 \Longrightarrow e ^ {v \left(s _ {0} ^ {L}\right)} = \frac {0 . 5}{q _ {1} \left(s _ {0} ^ {L} , a _ {1}\right) e ^ {- \eta} + \left(0 . 5 - q _ {1} \left(s _ {0} ^ {L} , a _ {1}\right)\right) e ^ {- 0 . 5 \eta}} \\ \Longrightarrow e ^ {v \left(s _ {0} ^ {L}\right)} = \frac {0 . 5}{\frac {0 . 5}{1 + e ^ {- 0 . 5 \eta}} e ^ {- \eta} + \frac {0 . 5 e ^ {- 0 . 5 \eta}}{1 + e ^ {- 0 . 5 \eta}} e ^ {- 0 . 5 \eta}} = 0. 2 5 \frac {e ^ {\eta}}{q _ {2} \left(s _ {0} ^ {L} , a _ {1}\right)} \\ \Longrightarrow q _ {3} \left(s _ {0} ^ {L}, a _ {1}\right) = q _ {2} \left(s _ {0} ^ {L}, a _ {1}\right) e ^ {v \left(s _ {0} ^ {L}\right) - \eta - v \left(s _ {g} ^ {L}\right)} = 0. 2 5 \\ \end{array} +$$ + +$$ +\begin{array}{l} k \text {e v e n} \Rightarrow q _ {k} \left(s _ {0} ^ {L}, a _ {1}\right) = \frac {0 . 5}{1 + e ^ {- 0 . 5 \eta}} \\ k \text {o d d} \Rightarrow q _ {k} \left(s _ {0} ^ {L}, a _ {1}\right) = 0. 2 5, \\ \end{array} +$$ + +where the last two lines follow by a straightforward induction. Hence the losses suffered by OMD on the left part of the MDP are: + +$$ +\begin{array}{l} \sum_ {k = 1} ^ {K} \left\{q _ {k} \left(s _ {0} ^ {L}, a _ {1}\right) c _ {k} \left(s _ {0} ^ {L}, a _ {1}\right) + q _ {k} \left(s _ {0} ^ {L}, a _ {2}\right) c _ {k} \left(s _ {0} ^ {L}, a _ {2}\right) \right\} = \sum_ {k = 1} ^ {K} \left\{q _ {k} \left(s _ {0} ^ {L}, a _ {1}\right) \cdot \frac {1 + (- 1) ^ {k}}{2} + 0. 5 \cdot \left(0. 5 - q _ {k} \left(s _ {0} ^ {L}, a _ {1}\right)\right) \right\} \\ = \sum_ {k = 1} ^ {K} \left\{q _ {k} \left(s _ {0} ^ {L}, a _ {1}\right) \cdot \frac {(- 1) ^ {k}}{2} + 0. 2 5 \right\} \\ = 0. 2 5 K + 0. 5 \sum_ {t = 1} ^ {K / 2} \left\{q _ {2 t} \left(s _ {0} ^ {L}, a _ {1}\right) - q _ {2 t - 1} \left(s _ {0} ^ {L}, a _ {1}\right) \right\} \\ \end{array} +$$ + +$$ +\begin{array}{l} = 0. 2 5 K + 0. 5 \sum_ {t = 1} ^ {K / 2} \left\{\frac {0 . 5}{1 + e ^ {- 0 . 5 \eta}} - 0. 2 5 \right\} \\ = 0. 2 5 K + 0. 5 \frac {K}{2} \frac {0 . 5 - 0 . 2 5 - 0 . 2 5 e ^ {- 0 . 5 \eta}}{1 + e ^ {- 0 . 5 \eta}} \\ = 0. 2 5 K + \frac {K}{1 6} \cdot \frac {1 - e ^ {- 0 . 5 \eta}}{1 + e ^ {- 0 . 5 \eta}} \\ = 0. 2 5 K + \frac {K}{1 6} \cdot \min \left\{\frac {\eta}{5}, \frac {1}{2} \right\}. \tag {7} \\ \end{array} +$$ + +# Right part of the MDP: + +$$ +\begin{array}{l} \frac {\partial \mathcal {D} _ {k} (v)}{\partial v \left(s _ {0} ^ {R}\right)} = 0 \Longrightarrow e ^ {v \left(s _ {0} ^ {R}\right)} = \frac {0 . 5}{q _ {k} \left(s _ {0} ^ {R} , a _ {1}\right) e ^ {- v \left(s _ {g} ^ {R}\right)} + \left(1 / 2 - q _ {k} \left(s _ {0} ^ {R} , a _ {1}\right)\right) e ^ {- \eta - v \left(s _ {1} ^ {R}\right)}} \\ \frac {\partial \mathcal {D} _ {k} (v)}{\partial v \left(s _ {1} ^ {R}\right)} = 0 \Longrightarrow v \left(s _ {0} ^ {R}\right) = \eta + 2 v \left(s _ {1} ^ {R}\right) \\ \frac {\partial \mathcal {D} _ {k} (v)}{\partial v (s _ {g} ^ {R})} = 0 \Longrightarrow v (s _ {0} ^ {R}) = v (s _ {g} ^ {R}) \\ \Longrightarrow e ^ {v \left(s _ {0} ^ {R}\right)} = \frac {0 . 5}{q _ {k} \left(s _ {0} ^ {R} , a _ {1}\right) e ^ {- 0 . 5 v \left(s _ {0} ^ {R}\right)} + \left(1 / 2 - q _ {k} \left(s _ {0} ^ {R} , a _ {1}\right)\right) e ^ {- \eta - 0 . 5 v \left(s _ {0} ^ {R}\right) + 0 . 5 \eta}} \\ \Longrightarrow e ^ {0. 5 v \left(s _ {0} ^ {R}\right)} = \frac {0 . 5}{q _ {k} \left(s _ {0} ^ {R} , a _ {1}\right) + \left(1 / 2 - q _ {k} \left(s _ {0} ^ {R} , a _ {1}\right)\right) e ^ {- 0 . 5 \eta}} = \frac {0 . 5}{q _ {k} \left(s _ {0} ^ {R} , a _ {1}\right) + q _ {k} \left(s _ {0} ^ {R} , a _ {2}\right) e ^ {- 0 . 5 \eta}} \\ \Longrightarrow q _ {k + 1} \left(s _ {0} ^ {R}, a _ {1}\right) = q _ {k} \left(s _ {0} ^ {R}, a _ {1}\right) e ^ {B _ {k} ^ {v} \left(s _ {0} ^ {R}, a _ {1}\right)} = q _ {k} \left(s _ {0} ^ {R}, a _ {1}\right) e ^ {v \left(s _ {0} ^ {R}\right) - v \left(s _ {g} ^ {R}\right)} = q _ {k} \left(s _ {0} ^ {R}, a _ {1}\right) e ^ {0. 5 v \left(s _ {0} ^ {R}\right)} \\ \Longrightarrow q _ {k + 1} \left(s _ {0} ^ {R}, a _ {1}\right) = 0. 5 \frac {q _ {k} \left(s _ {0} ^ {R} , a _ {1}\right)}{q _ {k} \left(s _ {0} ^ {R} , a _ {1}\right) + \left(1 / 2 - q _ {k} \left(s _ {0} ^ {R} , a _ {1}\right)\right) e ^ {- 0 . 5 \eta}} \\ \Longrightarrow \frac {q _ {k + 1} \left(s _ {0} ^ {R} , a _ {1}\right)}{q _ {k + 1} \left(s _ {0} ^ {R} , a _ {2}\right)} = \frac {q _ {k + 1} \left(s _ {0} ^ {R} , a _ {1}\right)}{\left(1 / 2 - q _ {k + 1} \left(s _ {0} ^ {R} , a _ {1}\right)\right)} = \frac {q _ {k} \left(s _ {0} ^ {R} , a _ {1}\right)}{\left(1 / 2 - q _ {k} \left(s _ {0} ^ {R} , a _ {1}\right)\right) e ^ {- 0 . 5 \eta}} = e ^ {0. 5 \eta} \frac {q _ {k} \left(s _ {0} ^ {R} , a _ {1}\right)}{q _ {k} \left(s _ {0} ^ {R} , a _ {2}\right)} \\ \Longrightarrow \frac {q _ {k + 1} \left(s _ {0} ^ {R} , a _ {1}\right)}{q _ {k + 1} \left(s _ {0} ^ {R} , a _ {2}\right)} = e ^ {0. 5 k \eta} \frac {q _ {1} \left(s _ {0} ^ {R} , a _ {1}\right)}{q _ {1} \left(s _ {0} ^ {R} , a _ {2}\right)} = e ^ {0. 5 k \eta} \frac {0 . 5}{0 . 5 \sqrt {N}} = \frac {1}{\sqrt {N}} e ^ {0. 5 k \eta} \\ \Longrightarrow q _ {k + 1} \left(s _ {0} ^ {R}, a _ {2}\right) = \frac {0 . 5}{1 + \frac {1}{\sqrt {N}} e ^ {0 . 5 k \eta}} = \frac {0 . 5 \sqrt {N}}{\sqrt {N} + e ^ {0 . 5 \eta k}}. \\ \end{array} +$$ + +This also holds for $k = 0$ (as shown above). Hence, the losses suffered by OMD on the right part of the MDP are + +$$ +\begin{array}{l} \sum_ {k = 1} ^ {K} q _ {k} \left(s _ {0} ^ {R}, a _ {2}\right) c _ {k} \left(s _ {0} ^ {R}, a _ {2}\right) = \sum_ {k = 1} ^ {K} \frac {0 . 5 \sqrt {N}}{\sqrt {N} + e ^ {0 . 5 k \eta}} \\ \geqslant \int_ {1} ^ {K + 1} \frac {0 . 5 \sqrt {N}}{\sqrt {N} + e ^ {0 . 5 \eta x}} d x = 0. 5 K - \int_ {1} ^ {K + 1} \frac {0 . 5 e ^ {0 . 5 \eta x}}{\sqrt {N} + e ^ {0 . 5 \eta x}} d x \\ = 0. 5 K - \left[ \frac {1}{\eta} \log \left(\sqrt {N} + e ^ {0. 5 \eta x}\right) \right] _ {1} ^ {K + 1} \\ = 0. 5 K - \frac {1}{\eta} \log (\sqrt {N} + e ^ {0. 5 \eta (K + 1)}) + \frac {1}{\eta} \log (\sqrt {N} + e ^ {0. 5 \eta}) \\ \geqslant 0. 5 K - \frac {1}{\eta} \log \left(2 e ^ {0. 5 \eta (K + 1)}\right) + \frac {1}{\eta} \log \sqrt {N} \quad \text {a s s u m i n g} \sqrt {N} \leqslant e ^ {0. 5 \eta (K + 1)} \\ = 0. 5 K - 0. 5 (K + 1) - \frac {1}{\eta} \log 2 + \frac {1}{2 \eta} \log N \\ = - 0. 5 + \frac {1}{2 \eta} \log \frac {N}{4}. \\ \end{array} +$$ + +If $\sqrt{N} > e^{0.5\eta (K + 1)}$ , then we have + +$$ +\begin{array}{l} 0. 5 K - \frac {1}{\eta} \log (\sqrt {N} + e ^ {0. 5 \eta (K + 1)}) + \frac {1}{\eta} \log (\sqrt {N} + e ^ {0. 5 \eta}) = 0. 5 K + \frac {1}{\eta} \log \left(\frac {\sqrt {N} + e ^ {0 . 5 \eta}}{\sqrt {N} + e ^ {0 . 5 \eta (K + 1)}}\right) \\ \geqslant 0. 5 K + \frac {1}{\eta} \log \left(\frac {e ^ {0 . 5 \eta (K + 1)} + e ^ {0 . 5 \eta}}{2 e ^ {0 . 5 \eta (K + 1)}}\right) \\ \geqslant 0. 5 K + \frac {1}{\eta} \log \left(\frac {1 + e ^ {- 0 . 5 \eta K}}{2}\right) \\ \geqslant 0. 2 5 K, \\ \end{array} +$$ + +using that $\frac{1 + e^{-0.5Kx}}{2} \geqslant e^{-0.25Kx}$ since $\cosh(x) \geqslant 1$ . So we have + +$$ +\sum_ {k = 1} ^ {K} q _ {k} \left(s _ {0} ^ {R}, a _ {2}\right) c _ {k} \left(s _ {0} ^ {R}, a _ {2}\right) \geqslant \min \left\{- 0. 5 + \frac {1}{2 \eta} \log \frac {N}{4}, 0. 2 5 K \right\}. \tag {8} +$$ + +Combining the losses from the left part in (7) and from the right part in (8), we have: + +$$ +\begin{array}{l} \sum_ {k = 1} ^ {K} \langle q _ {k}, c _ {k} \rangle = \sum_ {k = 1} ^ {K} \left\{q _ {k} \left(s _ {0} ^ {L}, a _ {1}\right) c _ {k} \left(s _ {0} ^ {L}, a _ {1}\right) + q _ {k} \left(s _ {0} ^ {L}, a _ {2}\right) c _ {k} \left(s _ {0} ^ {L}, a _ {2}\right) \right\} + \sum_ {k = 1} ^ {K} \left\{q _ {k} \left(s _ {0} ^ {R}, a _ {2}\right) c _ {k} \left(s _ {0} ^ {R}, a _ {2}\right) \right\} \\ \geqslant 0. 2 5 K + \frac {K}{1 6} \cdot \min \left\{\frac {\eta}{5}, \frac {1}{2} \right\} + \min \left\{- 0. 5 + \frac {1}{2 \eta} \log \frac {N}{4}, 0. 2 5 K \right\}. \\ \end{array} +$$ + +Regret lower-bound: consider $q_{\star}$ defined as follows: + +- $q_{\star}(s_0,a) = 1 / 2$ for all $a\in \mathcal{A}$ +$q_{\star}(s_0^L,a_2) = 1 / 2,q_{\star}(s_0^L,a_1) = 0$ +- $q_{\star}(s_g^L,a) = 1 / 4$ for all $a\in \mathcal{A}$ +$q_{\star}(s_0^R,a_1) = 1 / 2,q_{\star}(s_0^R,a_2) = 0,q_{\star}(s_i^R,a) = 0$ +- $q_{\star}(s_g^R, a) = 1/4$ for all $a \in \mathcal{A}$ + +It is straightforward to check that $q_{\star}$ satisfies the flow constraints and is an occupancy measure. We obtain + +$$ +\begin{array}{l} \sum_ {k = 1} ^ {K} \left\langle q _ {\star}, c _ {k} \right\rangle = \sum_ {k = 1} ^ {K} \left\{q _ {\star} \left(s _ {0} ^ {L}, a _ {2}\right) \cdot 0. 5 \right\} = 0. 2 5 K \\ \Rightarrow R _ {K} \geqslant \sum_ {k = 1} ^ {K} \left\langle q _ {k} - q _ {\star}, c _ {k} \right\rangle \geqslant \frac {K}{1 6} \cdot \min \left\{\frac {\eta}{5}, \frac {1}{2} \right\} + \min \left\{- 0. 5 + \frac {1}{2 \eta} \log \frac {N}{4}, 0. 2 5 K \right\} \\ \geqslant \min \left\{\frac {1}{2} \sqrt {\frac {1}{1 0} K \log \frac {N}{4}}, \frac {K}{3 2} \right\} - 0. 5. \\ \end{array} +$$ + +Recalling that $N = S - 5$ , we have $R_{K} = \Omega \left( \min \left\{ \sqrt{K \log S}, K \right\} \right)$ for an MDP where the sparsity level is $M = 3$ , concluding the proof. + +# B Efficient implementation of OMD using our regularizer + +In this section, we describe how the OMD update with our regularizer from Section 4 defined in (4) can be computed efficiently. This closely follows Appendix B.1 of [28], who provide a similar description for the negative entropy. + +Recall the regularizer $\psi_p(q) = p\sum_{s\in \mathcal{S}}\sum_{a\in \mathcal{A}}q(s,a)^{1 + 1 / p} - p$ for $q\in \mathbb{R}_{\geqslant 0}^{\Gamma}$ . We have + +$$ +\nabla \psi_ {p} (q) = (p + 1) \cdot q (s, a) ^ {1 / p}. +$$ + +The Bregman divergence is defined as: + +$$ +\begin{array}{l} D _ {\psi_ {p}} (q, q ^ {\prime}) = \psi_ {p} (q) - \psi_ {p} (q ^ {\prime}) - \langle \nabla \psi_ {p} (q ^ {\prime}), q - q ^ {\prime} \rangle \\ = \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} \left\{p \cdot q (s, a) ^ {1 + 1 / p} - p \cdot q ^ {\prime} (s, a) ^ {1 + 1 / p} \right\} \\ - (p + 1) \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} \left\{q ^ {\prime} (s, a) ^ {1 / p} q (s, a) - q ^ {\prime} (s, a) ^ {1 + 1 / p} \right\} \\ = \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} \left\{q ^ {\prime} (s, a) ^ {1 + 1 / p} + q (s, a) \cdot \left[ p \cdot q (s, a) ^ {1 / p} - (p + 1) \cdot q ^ {\prime} (s, a) ^ {1 / p} \right] \right\}. \\ \end{array} +$$ + +Recall that OMD with the above regularizer computes the occupancy measures as follows - see (1): + +$$ +q_{1} = \operatorname *{arg min}_{q\in \Delta (T)}\psi_{p}(q),\qquad q_{k + 1} = \operatorname *{arg min}_{q\in \Delta (T)}\Bigl\{\eta \cdot \langle q,c_{k}\rangle +D_{\psi_{p}}(q,q_{k})\Bigr \} . +$$ + +As shown in [24] (Theorem 6.15), each of these steps can be split into an unconstrained minimization step, and a projection step. Thus, $q_{1}$ can be computed as follows + +$$ +q_{1}^{\prime} = \operatorname *{arg min}_{\substack{q\in \mathbb{R}_{\geqslant 0}^{\Gamma}\\ }}\psi_{p}(q) +$$ + +$$ +q_{1} = \operatorname *{arg min}_{q\in \Delta (T)}D_{\psi_{p}}(q,q^{\prime}_{1}), +$$ + +where $q_1'$ has a closed-from solution $q_1'(s, a) = 1$ for every $s \in S$ and $a \in \mathcal{A}$ . Similarly, $q_{k+1}$ is computed as follows for every $k = 1, \ldots, K-1$ : + +$$ +q^{\prime}_{k + 1} = \operatorname *{arg min}_{q\in \mathbb{R}^{\Gamma}_{\geqslant 0}}\Bigl\{\eta \cdot \langle q,c_{k}\rangle +D_{\psi_{p}}(q,q_{k})\Bigr \} +$$ + +$$ +q_{k + 1} = \operatorname *{arg min}_{q\in \Delta (T)}D_{\psi_{p}}(q,q^{\prime}_{k + 1}), +$$ + +where again $q_{k+1}'$ has a closed-from solution $q_{k+1}'(s, a) = \left[q_k(s, a)^{1/p} - \frac{\eta}{p+1} c_k(s, a)\right]_+^p$ for every $s \in S$ and $a \in \mathcal{A}$ (follows from straightforwardly differentiating above objective and setting to 0 and accounting for the non-negativity of occupancy measures) - we use notation $a_+ = \max\{0, a\}$ . + +For the projection step, we start by formulating it as a constrained convex optimization problem: + +$$ +\min _ {q \in \mathbb {R} ^ {\Gamma}} D _ {\psi_ {p}} (q, q _ {k + 1} ^ {\prime}) \quad \text {s . t .} \quad \sum_ {a \in \mathcal {A}} q (s, a) - \sum_ {s ^ {\prime} \in \mathcal {S}} \sum_ {a ^ {\prime} \in \mathcal {A}} P (s ^ {\prime} | s, a) q (s ^ {\prime}, a ^ {\prime}) = \mathbb {I} \left\{s = s _ {0} \right\} \quad \forall s \in \mathcal {S} +$$ + +$$ +\sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} q (s, a) \leqslant T +$$ + +$$ +q (s, a) \geqslant 0 \quad \forall (s, a) \in \mathcal {S} \times \mathcal {A}. +$$ + +The problem can be solved by considering the Lagrangian with Lagrange multipliers $\lambda$ and $\{v(s)\}_{s\in S}$ : + +$$ +\begin{array}{l} \mathcal {L} (q, \lambda , v) = D _ {\psi_ {p}} (q, q _ {k + 1} ^ {\prime}) + \lambda \Big (\sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} q (s, a) - T \Big) + \sum_ {s} v (s) \Big (\sum_ {s ^ {\prime}, a ^ {\prime}} P (s | s ^ {\prime}, a ^ {\prime}) q (s ^ {\prime}, a ^ {\prime}) + \mathbb {I} \{s = s _ {0} \} - \sum_ {a} q (s, a) \Big) \\ = D _ {\psi_ {p}} (q, q _ {k + 1} ^ {\prime}) + \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} q (s, a) \Big (\lambda + \sum_ {s ^ {\prime} \in \mathcal {S}} P (s ^ {\prime} | s, a) v (s ^ {\prime}) - v (s) \Big) + v (s _ {0}) - \lambda T, \\ \end{array} +$$ + +Differentiating the Lagrangian with respect to any $q(s, a)$ and setting to 0, we get + +$$ +\begin{array}{l} \frac {\partial \mathcal {L} (q , \lambda , v)}{\partial q (s , a)} = \nabla \psi_ {p} (q) (s, a) - \nabla \psi_ {p} (q _ {k + 1} ^ {\prime}) (s, a) + \lambda + \sum_ {s ^ {\prime} \in \mathcal {S}} P (s ^ {\prime} | s, a) v (s ^ {\prime}) - v (s) \\ = (p + 1) q (s, a) ^ {1 / p} - (p + 1) q _ {k + 1} ^ {\prime} (s, a) ^ {1 / p} + \lambda + \sum_ {s ^ {\prime} \in \mathcal {S}} P (s ^ {\prime} | s, a) v (s ^ {\prime}) - v (s) = 0. \\ \Longrightarrow q _ {k + 1} (s, a) = \left[ q _ {k + 1} ^ {\prime} (s, a) ^ {1 / p} - \frac {\lambda + \sum_ {s ^ {\prime} \in \mathcal {S}} P (s ^ {\prime} | s , a) v (s ^ {\prime}) - v (s)}{p + 1} \right] _ {+} ^ {p}. \\ \end{array} +$$ + +This formula is also valid for $k = 0$ by setting $c_{0}(s,a) = 0$ and $q_{0}(s,a) = 1$ for every $s\in S$ and $a\in \mathcal{A}$ . + +To compute the value of $\lambda$ and $v$ at the optimum, we write the dual problem $\mathcal{D}(\lambda, v) = \min_q \mathcal{L}(q, \lambda, v)$ by substituting $q_{k+1}$ back into $\mathcal{L}$ : + +$$ +\mathcal {D} (\lambda , v) = D _ {\psi_ {p}} (q _ {k + 1}, q _ {k + 1} ^ {\prime}) + \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} q _ {k + 1} (s, a) \Big (\lambda + \sum_ {s ^ {\prime} \in \mathcal {S}} P (s ^ {\prime} | s, a) v (s ^ {\prime}) - v (s) \Big) + v (s _ {0}) - \lambda T. +$$ + +Recall that $q_{k+1}'(s, a) = \left[q_k(s, a)^{1/p} - \frac{\eta}{p+1} c_k(s, a)\right]_+^p$ , so (ignoring terms independent of $\lambda, v$ , e.g. $q_{k+1}'(s, a)$ ): + +$$ +\begin{array}{l} D _ {\psi_ {p}} (q _ {k + 1}, q _ {k + 1} ^ {\prime}) = \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} \left\{q _ {k + 1} ^ {\prime} (s, a) ^ {1 + 1 / p} + q _ {k + 1} (s, a) \cdot \left[ p q _ {k + 1} (s, a) ^ {1 / p} - (p + 1) q _ {k + 1} ^ {\prime} (s, a) ^ {1 / p} \right] \right\} \\ \propto \sum_ {s \in \mathcal {S}} \sum_ {a \in A} \left\{q _ {k + 1} (s, a) \cdot \left[ p \left(q _ {k + 1} ^ {\prime} (s, a) ^ {1 / p} - \frac {\lambda + \sum_ {s ^ {\prime} \in \mathcal {S}} P \left(s ^ {\prime} | s , a\right) v \left(s ^ {\prime}\right) - v (s)}{p + 1}\right) _ {+} \right. \right. \\ \left. \left. - (p + 1) q _ {k + 1} ^ {\prime} (s, a) ^ {1 / p} \right] \right\} \\ = \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} \left\{q _ {k + 1} (s, a) \cdot \left[ p \left(q _ {k + 1} ^ {\prime} (s, a) ^ {1 / p} - \frac {\lambda + \sum_ {s ^ {\prime} \in \mathcal {S}} P \left(s ^ {\prime} | s , a\right) v \left(s ^ {\prime}\right) - v (s)}{p + 1}\right) \right. \right. \\ \left. \left. - (p + 1) q _ {k + 1} ^ {\prime} (s, a) ^ {1 / p} \right] \right\} \quad \text {s i n c e i f} q _ {k + 1} (s, a) = 0, \text {t h e t h e w h o l e t e r m i s 0} \\ = \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} \left\{q _ {k + 1} (s, a) \cdot \left[ - q _ {k + 1} ^ {\prime} (s, a) ^ {1 / p} - \frac {p}{p + 1} \left(\lambda + \sum_ {s ^ {\prime} \in \mathcal {S}} P (s ^ {\prime} | s, a) v (s ^ {\prime}) - v (s)\right) \right] \right\} \\ \Rightarrow \mathcal {D} (\lambda , v) \propto \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} \left\{q _ {k + 1} (s, a) \cdot \left[ - q _ {k + 1} ^ {\prime} (s, a) ^ {1 / p} - \frac {p}{p + 1} \left(\lambda + \sum_ {s ^ {\prime} \in \mathcal {S}} P (s ^ {\prime} | s, a) v (s ^ {\prime}) - v (s)\right) \right. \right. \\ \left. \left. + \left(\lambda + \sum_ {s ^ {\prime} \in \mathcal {S}} P \left(s ^ {\prime} | s, a\right) v \left(s ^ {\prime}\right) - v (s)\right) \right] \right\} + v \left(s _ {0}\right) - \lambda T \\ = \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} \left\{q _ {k + 1} (s, a) \cdot \left[ - q _ {k + 1} ^ {\prime} (s, a) ^ {1 / p} + \frac {\lambda + \sum_ {s ^ {\prime} \in \mathcal {S}} P \left(s ^ {\prime} \mid s , a\right) v \left(s ^ {\prime}\right) - v (s)}{p + 1} \right] \right\} + v \left(s _ {0}\right) - \lambda T \\ = -\sum_{s\in \mathcal{S}}\sum_{a\in \mathcal{A}}q_{k + 1}(s,a)^{1 + 1 / p} + v(s_{0}) - \lambda T \\ = - \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} \left[ q _ {k + 1} ^ {\prime} (s, a) ^ {1 / p} - \frac {\lambda + \sum_ {s ^ {\prime} \in \mathcal {S}} P (s ^ {\prime} | s , a) v (s ^ {\prime}) - v (s)}{p + 1} \right] _ {+} ^ {1 + p} + v (s _ {0}) - \lambda T. \\ \end{array} +$$ + +Maximizing the dual gives $\lambda$ and $v$ or equivalently, we can minimize the negation of the dual: + +$$ +\lambda_ {k + 1}, v _ {k + 1} = \underset {\lambda \geqslant 0, v} {\arg \min} \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} \Big [ q _ {k + 1} ^ {\prime} (s, a) ^ {1 / p} - \frac {\lambda + \sum_ {s ^ {\prime} \in \mathcal {S}} P (s ^ {\prime} | s , a) v (s ^ {\prime}) - v (s)}{p + 1} \Big ] _ {+} ^ {1 + p} - v (s _ {0}) + \lambda T. +$$ + +This is a convex optimization problem subject only to non-negativity constraints, and can be efficiently solved using iterative methods (e.g. gradient descent). + +# C The benefits of sparsity - upper bounds + +We restate the main theorem proved in Section 4. + +Theorem 4.1. Consider $O\hat{M}D$ with $\psi_p$ as regularizer. If $T > e$ is such that $q_{\pi^{\star}} \in \Delta(T)$ , $\eta = \sqrt{\frac{pT^{1 + 1/p}}{KDM^{1/p}}}$ , $p = \log(TM)$ , then $\mathbb{E}[R_K] \leqslant \mathcal{O}\Big(\sqrt{DKT\log(MT)}\Big)$ . + +We include the missing details from the proof given in Section 4. First, recall from (4) that + +$$ +\begin{array}{l} \psi_ {p} (q) = p \cdot \left(- 1 + \| q \| _ {1 + 1 / p} ^ {1 + 1 / p}\right) = p \cdot \left(- 1 + \sum_ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} | q (s, a) | ^ {1 + 1 / p}\right) \\ \Longrightarrow \frac {\partial \psi_ {p} (q)}{\partial q (s , a)} = (p + 1) q (s, a) ^ {1 / p} \\ \Longrightarrow \frac {\partial^ {2} \psi_ {p} (q)}{\partial q (s , a) ^ {2}} = \left(1 + \frac {1}{p}\right) q (s, a) ^ {1 / p - 1} \\ \Longrightarrow \nabla \psi_ {p} (q) = (p + 1) q ^ {1 / p}, \quad \nabla^ {2} \psi_ {p} (q) = \operatorname {d i a g} \left(\frac {p + 1}{p} q ^ {1 / p - 1}\right) \\ \end{array} +$$ + +We implicitly assumed here that $\psi_p$ is defined on $\mathbb{R}_{\geqslant 0}^{\Gamma}$ . The missing details are: + +- $\psi_p$ satisfies the condition (2) with $\alpha = 1$ : + +$$ +\begin{array}{l} \nabla \psi_ {p} (q) \in \left[ \nabla \psi_ {p} \left(q _ {k}\right), \nabla \psi_ {p} \left(q _ {k}\right) - \eta c _ {k} \right] \Longrightarrow q ^ {1 / p} (s, a) \leqslant q _ {k} ^ {1 / p} (s, a) \\ \Longrightarrow \frac {1}{q (s , a)} \geqslant \frac {1}{q _ {k} (s , a)} \\ \Longrightarrow \frac {1}{q ^ {1 - 1 / p} (s , a)} \geqslant \frac {1}{q _ {k} ^ {1 - 1 / p} (s , a)} \\ \Rightarrow \nabla^ {2} \psi_ {p} (q) \succeq \nabla^ {2} \psi_ {p} \left(q _ {k}\right). \\ \end{array} +$$ + +- $\nabla^2\psi_p(q)^{-1} = \mathrm{diag}\left(\frac{p}{p + 1} q^{1 - 1 / p}\right)$ : follows directly from the expression for $\nabla^2\psi_p(q)$ above. + +We now turn to the description of the parameter-free algorithm and the proof of its corresponding regret bound (Theorem 4.4). + +# C.1 Sparse-agnostic bound + +For the unknown sparsity level, we use the same approach as in [19], dividing the episode horizon into segments, where each segment will run OMD from scratch with an increasing sparsity level guess. Crucially, there will be at most $O(\log \log M)$ such segments. + +Define + +- $M = \max_{k\in [K]}\| c_k\| _0$ , the true sparsity level across the horizon. +- $B = \left[\log_2\log_2M\right]$ , the maximum number of segments. +- $m(b) = 2^{2^b}$ , the assumed sparsity level during the $b$ -th segment (or interval $I(b)$ below). The reason to use a double exponential is that this sparse-agnostic procedure brings an extra $B$ factor to the regret bound: if we use $2^b$ then $B = O(\log M)$ which harms the regret bound. +- for $1 \leqslant b \leqslant B$ , $\tau(b) = \min \{1 \leqslant k \leqslant K \mid \|c_k\|_0 > m(b)\}$ , the first episode in which the sparsity level of the loss vector exceeds $m(b)$ . We also define $\tau(0) = 0$ and $\tau(B) = K$ . + +Using this notation, we can partition the horizon $[K]$ as intervals $(I(b))_{b\in [B]}$ according to the episodes $\tau (b)$ where the thresholds $m(b)$ are first exceeded. For $1\leqslant b\leqslant B$ .. + +$$ +I (b) = \left\{ \begin{array}{l l} \left[ \tau (b - 1) + 1, \tau (b) \right] & \text {i f} \tau (b - 1) < \tau (b) \\ \varnothing & \text {i f} \tau (b - 1) = \tau (b) \end{array} \right. +$$ + +Let $b_{k} = \min \left\{b \geqslant 1 \mid \tau(b) \geqslant k\right\}$ be the index of the only interval to which episode $k$ belongs. + +Now we define the OMD parameters used in interval $I(b)$ , in which we essentially use the parameters from Theorem 4.1 assuming the sparsity level is $m(b)$ : + +Algorithm 1 Sparse-Agnostic Mirror Descent +Input: $T,K,D$ +Initialize: $p\gets \log 2^2 T$ $\eta \leftarrow \sqrt{T^{1 + 1 / p} / (pDK2^{2 / p})}$ $b\gets 1,q_1\gets \arg \min_{q\in \Delta (T)}\psi_p(q)$ +for $k = 1,\ldots ,K$ do Play $q_{k}$ and Observe $c_{k}$ +if $\| c_k\| _0\leqslant 2^{2^b}$ then $q_{k + 1} = \arg \min_{q\in \Delta_T}\langle q,c_k\rangle +D_{\psi_p}(q,q_k)$ else $b\gets [\log_2\log_2\| c_k\| _0]$ $p\gets \log 2^{2^b}T$ $\eta \leftarrow \sqrt{pT^{1 + 1 / p} / (DK2^{2 / p})}$ $q_{k + 1}\gets \arg \min_{q\in \Delta (T)}\psi_p(q)$ +end if +end for + +- the parameter of our regularizer is $p(b) = \log(m(b)T)$ . +the step-size is $\eta (b) = \sqrt{\frac{p(b)T^{1 + 1 / p(b)}}{DKm(b)^{1 / p(b)}}}$ + +Recall that our regularizer with parameter $p$ is given by $\psi_p(q) = p\left(-1 + \| q\|_{1 + 1 / p}^{1 + 1 / p}\right)$ . At episode $k$ , we use the parameter $p(b_k)$ defined above, i.e. using the index value $b_{k}$ of the interval $I(b_{k})$ to which episode $k$ belongs to. The OMD update is then defined by: + +$$ +q _ {k} = \nabla \psi_ {p \left(b _ {k}\right)} ^ {\star} \left(\eta \left(b _ {k}\right) \sum_ {k ^ {\prime} < k, k ^ {\prime} \in I \left(b _ {k}\right)} c _ {k ^ {\prime}}\right), k = 1, \dots , K. +$$ + +The full procedure is given in Algorithm 1. The following lemma shows the cost of being sparse-agnostic is an additive $T$ term and a double-logarithmic factor in the sparsity level $M$ . + +Lemma C.1. Consider running Algorithm $I$ . If $T > e$ is such that $q_{\pi^{\star}} \in \Delta (T)$ , then $\mathbb{E}\left[R_K\right] \leqslant \mathcal{O}\Big(TB + B\sqrt{DKT\log(MT)}\Big)$ . + +Proof. Fix (interval) $b \in [B]$ . On the time interval $I(b)$ , we run OMD with regularizer $\psi_{p(b)}$ , learning rate $\eta(b)$ and we consider the (expected) interval regret $R(b) = \sum_{k \in I(b)} \langle q_{\pi_k}, c_k \rangle - \min_{\pi \in \Pi_p} \sum_{k \in I(b)} \langle q_{\pi^*}, c_k \rangle$ . Crucially, we know that up to the last time step of the interval, we have a bound $m(b)$ on the sparsity for all rounds but the last. + +Since $J_{k}^{\pi} \leqslant T$ for any $k$ , we just consider the regret on the rounds not including $\tau(b)$ for which we suffer a regret of at most $T$ : + +$$ +R(b) = \sum_{k\in I(b)}\langle q_{\pi_{k}} - q_{\pi^{\star}},c_{k}\rangle \leqslant T + \sum_{\substack{k\in I(b)\\ k < \tau (b)}}\langle q_{\pi_{k}} - q_{\pi^{\star}},c_{k}\rangle . +$$ + +For the other rounds we follow similar steps as in the proof of Theorem 4.1: + +$$ +\begin{array}{l} \sum_{\substack{k\in I(b)\\ k < \tau (b)}}\langle q_{\pi_{k}} - q_{\pi^{\star}},c_{k}\rangle \leqslant \frac{p(b)T^{1 + 1 / p(b)}}{\eta(b)} +\frac{\eta(b)}{2}\sum_{\substack{k\in I(b)\\ k < \tau (b)}}\| c_{k}\|_{1}^{1 / p(b)}(\langle c_{k},q_{k}\rangle +1) \\ \leqslant \frac{p(b)T^{1 + 1 / p(b)}}{\eta(b)} +\frac{\eta(b)m(b)^{1 / p(b)}}{2}\sum_{\substack{k\in I(b)\\ k < \tau (b)}}(\langle c_{k},q_{k}\rangle +1) \\ \Rightarrow \sum_{\substack{k\in I(b)\\ k < \tau (b)}}\langle q_{\pi_{k}} - q_{\pi^{\star}},c_{k}\rangle \leqslant \frac{1}{1 - \eta(b)m(b)^{1 / p(b)}}\Big[\frac{p(b)T^{1 + 1 / p(b)}}{\eta(b)} +\frac{\eta(b)m(b)^{1 / p(b)}}{2}\sum_{\substack{k\in I(b)\\ k < \tau (b)}}(\langle c_{k},q_{\pi^{\star}}\rangle +1)\Big] \\ \end{array} +$$ + +$$ +\begin{array}{l} \leqslant \frac{2p(b)T^{1 + 1 / p(b)}}{\eta(b)} +\eta (b)m(b)^{1 / p(b)}\sum_{\substack{k\in I(b)\\ k < \tau (b)}}(\langle c_{k},q_{\pi^{\star}}\rangle +1) \\ \leqslant \frac {2 p (b) T ^ {1 + 1 / p (b)}}{\eta (b)} + \eta (b) m (b) ^ {1 / p (b)} \sum_ {k \in I (b)} \left(\left\langle c _ {k}, q _ {\pi^ {\star}} \right\rangle + 1\right), \\ \end{array} +$$ + +where we used that $\frac{1}{1 - \eta(b)m(b)^{1/p(b)}} \leqslant 2$ which is the case if $K > 8eT\log(MT)$ using how we defined $\eta(b)$ . Using $\eta(b) = \sqrt{\frac{p(b)T^{1+1/p(b)}}{DKm(b)^{1/p(b)}}}$ , we have + +$$ +\begin{array}{l} R (b) \leqslant T + \sqrt {p (b) T ^ {1 + 1 / p (b)} D K m (b) ^ {1 / p (b)}} \cdot \left(2 + \frac {\sum_ {k \in I (b)} (\langle c _ {k} , q _ {\pi^ {\star}} \rangle + 1)}{D K}\right) \\ \leqslant T + \sqrt {T D K e \log (m (b) T)} \cdot \left(2 + \frac {\sum_ {k \in I (b)} (\langle c _ {k} , q _ {\pi^ {\star}} \rangle + 1)}{D K}\right) \\ \leqslant T + \sqrt {2 T D K e \log (M T)} \cdot \left(2 + \frac {\sum_ {k \in I (b)} (\langle c _ {k} , q _ {\pi^ {\star}} \rangle + 1)}{D K}\right), \\ \end{array} +$$ + +where we used that $p(b) = \log \left(m(b)T\right)$ and that + +$$ +\begin{array}{l} b \leqslant B \leqslant 1 + \log_ {2} \log_ {2} M \Longrightarrow m (b) = 2 ^ {2 ^ {b}} \leqslant 2 ^ {2 ^ {1 + \log_ {2} \log_ {2} M}} = 2 ^ {2 \cdot \log_ {2} M} = M ^ {2} \\ \Longrightarrow \log (m (b) T) = \log (M ^ {2} T) \leqslant 2 \log (M T). \\ \end{array} +$$ + +Then + +$$ +\begin{array}{l} E \left[ R _ {K} \right] \leqslant \sum_ {b = 1} ^ {B} R (b) \\ \leqslant T B + \sqrt {2 T D K e \log (M T)} \cdot \left(2 B + \sum_ {b = 1} ^ {B} \frac {\sum_ {k \in I (b)} \left(\left\langle c _ {k} , q _ {\pi^ {\star}} \right\rangle + 1\right)}{D K}\right) \\ = T B + \sqrt {2 T D K e \log (M T)} \cdot \left(2 B + \frac {1}{D} + \frac {1}{D K} \sum_ {k = 1} ^ {K} \langle c _ {k}, q _ {\pi^ {*}} \rangle\right) \\ \leqslant T B + 4 B \sqrt {2 T D K e \log (M T)}, \\ \end{array} +$$ + +where the last step uses that $\sum_{k=1}^{K}\langle q_{\pi^{\star}}, c_k \rangle \leqslant \sum_{k=1}^{K}\langle q_{\pi_f}, c_k \rangle \leqslant K\|q_{\pi_f}\|_1 \|c_k\|_{\infty} \leqslant DK$ and $D \geqslant 1$ . + +# C.2 Fully parameter-free bound + +We now turn our attention to the unknown hitting time of the optimal policy $T_{\star}$ , where we can exploit the same technique presented in [8]. + +We run $N \approx \log K$ instances of Algorithm 1 where the $j$ -th instance will set its parameter $T$ as $b(j)$ which is roughly $2^j$ , so that there always exists an instance $j_\star$ such that $b(j_\star)$ is close to the unknown $T_\star$ . Specifically, we run a scale invariant meta algorithm with a correction term as in [8] to obtain the desired bound (details in Algorithm 2). + +Theorem 4.4. If $K > \max\left(T_{\star}, \frac{T_{\star}}{D} \log (T_{\star}M)\right)$ and $T_{\star} > e$ , Algorithm 2 guarantees $\mathbb{E}[R_K] \leqslant \tilde{\mathcal{O}}\left(\sqrt{DKT_{\star} \log (MT_{\star})} + T_{\star}\right)$ , where the notation $\tilde{\mathcal{O}}$ hides double-logarithmic factors. + +Proof. We closely follow the steps of the proof of Theorem 2 in [8]. We have + +- $b(1) \geqslant T^{\pi_f}(s_0)$ so for all instances $\Delta(b(j))$ is non-empty and the instance is well-defined. +- Let $j^{\star}$ be the index of the instance with smallest $b(j^{\star})$ that is larger than $T^{\star}$ , i.e. $\frac{b(j^{\star})}{2} \leqslant T_{\star} \leqslant b(j^{\star})$ . This instance exists since $b(N) \geqslant K > T_{\star}$ . + +Algorithm 2 Fully Parameter-Free Online Mirror Descent for Sparse SSPs +Define $j_0 = \lceil \log_2T^{\pi_f}(s_0)\rceil -1,b(j) = 2^{j_0 + j},N = \lceil \log_2K\rceil -j_0,\eta_j =$ $(\sqrt{Db(j)K\log(b(j)M)})^{-1 / 2}$ Define $\psi_p(p) = \sum_{j = 1}^{N}\frac{1}{\eta_j} p(j)\log p(j)$ Initialize: $p_1\in \Delta_N$ , such that $p_1(j) = \frac{\eta_j}{\eta_1N},\forall j\neq 1$ Initialize: $N$ instances of Algorithm 1, with $j$ -th instance $T = b(j)$ for $k = 1,\ldots ,K$ do Obtain occupancy measures $q_{k}^{j}$ for $j\in [N]$ Sample $j_{k}\sim p_{k}$ , execute policy induced by $q_{k}^{j_{k}}$ Receive $c_{k}$ and send it to all instances. Compute $\ell_k(j) = \langle q_k^j,c_k\rangle ,a_k(j) = 4\eta_j\ell_k^2 (j)$ Update $p_{k + 1} = \arg \min_{p\in \Delta_N}\langle p,\ell_k + a_k\rangle +D_{\psi_p}(p,p_k)$ end for + +We start by decomposing the regret into the regret of the meta algorithm w.r.t. finding $j_{\star}$ and the regret of the $j_{\star}$ instance w.r.t. the best policy: + +$$ +\begin{array}{l} \mathbb {E} \left[ R _ {K} \right] = \sum_ {k = 1} ^ {K} \sum_ {j = 1} ^ {N} p _ {k} (j) \langle q _ {k} ^ {j}, c _ {k} \rangle - \sum_ {k = 1} ^ {K} \langle q _ {\pi^ {\star}}, c _ {k} \rangle \\ = \sum_ {k = 1} ^ {K} \sum_ {j = 1} ^ {N} p _ {k} (j) \langle q _ {k} ^ {j}, c _ {k} \rangle - \langle q _ {k} ^ {j _ {\star}}, c _ {k} \rangle + \sum_ {k = 1} ^ {K} \langle q _ {k} ^ {j _ {\star}} - q _ {\pi^ {\star}}, c _ {k} \rangle \\ = \underbrace {\sum_ {k = 1} ^ {K} \langle p _ {k} - e _ {j _ {\star}} , \ell_ {k} \rangle} _ {\text {M e t a - R e g r e t}} + \underbrace {\sum_ {k = 1} ^ {K} \langle q _ {k} ^ {j _ {\star}} - q _ {\pi^ {\star}} , c _ {k} \rangle} _ {j _ {\star} - \text {R e g r e t}} \\ \end{array} +$$ + +where we consider $p_k, \ell_k$ as $N$ -dimensional vectors and $e_{j_\star}$ as the basis vector with the $j_\star$ coordinate equal to 1. + +By Lemma C.1 the $j_{\star}$ -Regret is bounded by $\mathcal{O}\left(Bb(j_{\star}) + B\sqrt{Db(j_{\star})K\log(b(j_{\star})M)}\right) = \mathcal{O}\left(BT_{\star} + B\sqrt{DT_{\star}K\log(T_{\star}M)}\right)$ . + +This also allows to say that: + +$$ +\begin{array}{l} \sum_ {k = 1} ^ {K} \left\langle q _ {k} ^ {j _ {\star}}, c _ {k} \right\rangle \leqslant \sum_ {k = 1} ^ {K} \left\langle q _ {\pi^ {\star}}, c _ {k} \right\rangle + \mathcal {O} \left(B T _ {\star} + B \sqrt {D T _ {\star} K \log (T _ {\star} M)}\right) \\ \leqslant \sum_ {k = 1} ^ {K} \left\langle q _ {\pi_ {f}}, c _ {k} \right\rangle + \mathcal {O} \left(B T _ {\star} + B \sqrt {D T _ {\star} K \log (T _ {\star} M)}\right) \\ \leqslant D K + \mathcal {O} \left(B T _ {\star} + B \sqrt {D T _ {\star} K \log \left(T _ {\star} M\right)}\right). \\ \end{array} +$$ + +which we will make use of just below. + +For the meta algorithm regret, we can use Lemma 12 of [8] which guarantees that: + +$$ +\begin{array}{l} \mathbb {E} \left[ \sum_ {k = 1} ^ {K} \langle p _ {k} - e _ {j _ {\star}}, \ell_ {k} \rangle \right] = \mathcal {O} \left(\frac {2 + \log \left(N \sqrt {\frac {b (j _ {\star})}{b (1)}}\right)}{\eta_ {j _ {\star}}} + 4 \eta_ {j _ {\star}} b (j _ {\star}) \sum_ {k = 1} ^ {K} \langle q _ {k} ^ {j _ {\star}}, c _ {k} \rangle\right) \\ = \mathcal {O} \left(\frac {2 + \log \left(N \sqrt {\frac {b (j _ {\star})}{b (1)}}\right)}{\eta_ {j _ {\star}}} + 4 \eta_ {j _ {\star}} b (j _ {\star}) \left(D K + B T _ {\star} + B \sqrt {D T _ {\star} K \log (T _ {\star} M)}\right)\right) \\ \end{array} +$$ + +$$ +\begin{array}{l} = \tilde {\mathcal {O}} \left(\frac {\log T _ {\star}}{\eta_ {j _ {\star}}} + \eta_ {j _ {\star}} T _ {\star} \left(D K + \sqrt {D T _ {\star} K \log \left(T _ {\star} M\right)}\right)\right) \\ = \tilde {\mathcal {O}} \left(\frac {\log T _ {\star}}{\eta_ {j \star}} + \eta_ {j \star} D T _ {\star} K\right) \\ = \tilde {\mathcal {O}} \left(\sqrt {D T _ {\star} K \log T _ {\star}}\right), \\ \end{array} +$$ + +where we used + +- the notation $\tilde{\mathcal{O}}$ to ignore double-logarithmic factors. + +$$ +\begin{array}{l} \cdot K \geqslant \frac {T _ {\star}}{D} \log (T _ {\star} M). \\ \bullet \eta_ {j _ {\star}} = \sqrt {\frac {\log T _ {\star}}{D T _ {\star} K}}. \\ \cdot D \geqslant 1. \\ \end{array} +$$ + +Combining everything gives the result. + +# D State level sparsity + +In this section, we consider a different notion of state-level sparsity: + +$$ +M ^ {\prime} = \max _ {k} \max _ {s \in \mathcal {S}} \sum_ {a \in \mathcal {A}} \mathbb {I} \left\{c _ {k} (s, a) > 0 \right\} \leqslant A. +$$ + +From Theorem 4 in [38], we have a cumulative loss bound for a version of OMD with negative entropy regularisation: + +$$ +\mathbb {E} \left[ R _ {K} \right] \leqslant \tilde {\mathcal {O}} \left(\sqrt {T _ {\star} \sum_ {k = 1} ^ {K} J _ {k} ^ {\pi_ {\star}}}\right), +$$ + +from which we can exploit the state-level sparsity: + +$$ +\begin{array}{l} \sum_ {k = 1} ^ {K} J _ {k} ^ {\pi^ {\star}} = \min _ {\pi} \sum_ {k = 1} ^ {K} \mathbb {E} \left[ \sum_ {t = 1} ^ {I _ {\pi} (s _ {0})} c _ {k} \left(s ^ {t}, a ^ {t}\right) | P, \pi , s ^ {1} = s _ {0} \right] \\ \leqslant \sum_ {k = 1} ^ {K} \mathbb {E} \left[ \sum_ {t = 1} ^ {I _ {\pi_ {u}} (s _ {0})} c _ {k} \left(s ^ {t}, a ^ {t}\right) | P, \pi_ {u}, s ^ {1} = s _ {0} \right] \\ = \sum_ {k = 1} ^ {K} \mathbb {E} \left[ \sum_ {t = 1} ^ {I _ {\pi_ {u}} (s _ {0})} \frac {1}{A} \sum_ {a \in \mathcal {A}} c _ {k} (s ^ {t}, a) | P, \pi_ {u}, s ^ {1} = s _ {0} \right] \\ \leqslant \frac {K M ^ {\prime}}{A} \mathbb {E} \left[ I _ {\pi_ {u}} (s _ {0}) | P, \pi_ {u}, s ^ {1} = s _ {0} \right] \\ = \frac {K M ^ {\prime}}{A} T ^ {u} (s _ {0}), \\ \end{array} +$$ + +where $\pi_u$ is the uniform policy ( $\pi_u(a|s) = 1 / A$ ) and $T^u$ is its corresponding hitting-time. This gives the following regret bound: + +$$ +\mathbb {E} \left[ R _ {K} \right] \leqslant \tilde {\mathcal {O}} \left(\sqrt {K T _ {\star} \frac {M ^ {\prime}}{A} T ^ {u} \left(s _ {0}\right)}\right). +$$ + +We can actually relate this result back to our original notion of sparsity since we know that $M' \leqslant M$ . If $M \leqslant A$ , then we can non-trivially bound $M'$ by $M$ and achieve a regret of $\tilde{\mathcal{O}}\left(\sqrt{KT_{\star}\frac{M}{A}T^{u}(s_0)}\right)$ . + +This result highlights that it is possible to achieve polynomial improvements from sparsity if we consider state-level sparsity $M'$ or $M < A$ . However, this comes at the cost of a $T^u(s_0)$ factor. In the worst case, this additional factor will cancel the polynomial improvement. It could be an interesting avenue of future research to understand specific structural properties of the MDP that may lead to real polynomial improvements. + +In the experts setting $(S = 1)$ , we have $M = M'$ and $T^u(s_0) = 1$ and this bound recovers the $\frac{M}{A}$ improvement of the expert setting. This provides some further insights into the performance of OMD with negative entropy regularisation and that in particular issues arise when there is at least 1 state with non-sparse costs even though most other states may have sparse costs. + +# E The benefits of sparsity - lower bound under sparsity + +In this appendix, we prove our sparse lower bound result from Section 4.2. We first restate the result. Theorem 4.6. For any $D, T_{\star}, K, S, A$ with $T^{\star} \geqslant D \geqslant 3 \log S$ , $S(A - 1) \geqslant 400$ , $K \geqslant \frac{800T^{\star}}{D} \log M$ and $M \geqslant 101$ , there exists an SSP instance with stochastic $M$ -sparse costs, $S$ states and $A$ actions such that its diameter is $D$ , the expected hitting time of the optimal policy is $T^{\star}$ , and the expected regret with respect to the randomness of the losses for any learner after $K$ episodes is $\mathbb{E}[R_K] \geqslant \Omega \left( \sqrt{KT^{\star}D \log M} \right)$ . + +Proof. Fix $B = \lceil \frac{\log S / 2}{\log 2} \rceil - 2$ . Fix $N = 2^{B + 1} \geqslant 2^{\frac{\log S / 2}{\log 2} - 1} = \frac{S}{4}$ . Fix $L = \min \left\{M - 1, N \cdot (A - 1) \right\} \geqslant \frac{M}{8}$ . Fix $D' = D - B - 2$ , $T' = T_{\star} - B - 1$ , with $T_{\star} \geqslant D$ . + +We first describe the SSP instance with stochastic costs. Consider the following MDP $\mathcal{M} = (\mathcal{S},\mathcal{A},p,s_0,g)$ illustrated in Figure 4 and that we formally define below. + +![](images/7caf4d7b3bf1a1d1b0439723118cd0b51c526ea5b746119a20b8a82ec35be121.jpg) +Figure 4: Diagram illustrating MDP construction for the proof of Theorem 4.6. Details are given below. + +The first part of the states are represented by a binary tree of depth $B + 2$ and allow us to formerly consider the $N$ states at the bottom of the tree that matter, while avoiding an assumption on the existence of a state with $A \approx S$ actions as was done in prior work [8]. Each non-leaf node corresponds to a state with two actions transitioning (deterministically) to the left or right child respectively. The total number of nodes in the tree is + +$$ +\sum_ {i = 0} ^ {B + 1} 2 ^ {i} = 2 ^ {B + 2} - 1 \leqslant 2 ^ {\frac {\log S / 2}{\log 2} + 1} - 1 = 2 \frac {S}{2} - 1 = S - 1. +$$ + +The total number of leaf nodes is $N = 2^{B + 1} \geqslant \frac{S}{4}$ . Denote the set of states corresponding to the leaf nodes by $S_{\ell} = \{s_1, \dots, s_N\}$ . The root node is $s_0$ . There is also one additional state denoted by $f$ (recall that the number of states in the tree is $\leqslant S - 1$ ). + +We consider the same action set across each state: $\mathcal{A} = \{a_1,\dots,a_{A - 1},a_f\}$ . In the states of the binary tree where we have only described two actions, we can consider the other actions to remain in the same state deterministically with 0 cost. + +The transitions and costs are defined as follows: + +- For all states and actions in the tree that are not leaves, the transitions are specified above. The costs are all 0. +- For $s_i \in S_\ell$ , and $a_j \in \mathcal{A}$ + +- if $j = f$ , then $p(f|s_i, a_f) = 1$ and $c_k(s_i, a_f) = 0$ . +- if $j \in \{1, \dots, A - 1\}$ and $j + (A - 1) \cdot (i - 1) \leqslant L$ , then $p(g|s_i, a_j) = \frac{1}{T'}, p(s_i|s_i, a_j) = 1 - \frac{1}{T'}$ and the cost is an independent sample from a Bernoulli distribution at each episode $k$ : $c_k(s_i, a_j) \sim \mathrm{Ber}\left(\frac{D'}{2T'}\right)$ . +- if $j \in \{1, \dots, A - 1\}$ and $j + (A - 1) \cdot (i - 1) > L$ , then $a_j$ is the same as $a_f$ , i.e. $p(f|s_i, a_j) = 1$ and $c_k(s_i, a_j) = 0$ . + +For $f$ + +- $p(g|f, a_f) = \frac{1}{D'}, p(f|f, a_f) = 1 - \frac{1}{D'}$ and $c_k(f, a_f) = 1$ . +- for all $a_j \in \mathcal{A} \setminus \{a_f\}$ , $p(f|f, a_j) = 1$ and $c_k(f, a_j) = 0$ . + +Denote the above distribution for $c_k$ by $\mathcal{D}$ . In each episode there are at most $L + 1 \leqslant M$ non-zero costs, ensuring the condition on sparsity is respected. + +For $i \in \{1, \dots, N\}$ , let $\mathcal{A}_i$ correspond to the actions in state $s_i \in S_L$ which can transition directly to $g$ and $\mathcal{A} \backslash \mathcal{A}_i$ corresponds to the actions which deterministically transition to $f$ (e.g. if $(A - 1) \cdot i \leqslant L$ , then $\mathcal{A}_i = \{a_1, \dots, a_{A - 1}\}$ ). For any proper policy $\pi$ independent of the stochastically generated costs in episode $k$ , we have + +$$ +\begin{array}{l} \mathbb {E} _ {c _ {k} \sim \mathcal {D}} \big [ J _ {k} ^ {\pi} (s _ {i}) \big ] = \mathbb {E} _ {c _ {k} \sim \mathcal {D}} \Big [ \sum_ {a \in \mathcal {A} _ {i}} \pi (a | s _ {i}) \Big (c _ {k} (s _ {i}, a) + \Big (1 - \frac {1}{T ^ {\prime}} \Big) J _ {k} ^ {\pi} (s _ {i}) \Big) + J _ {k} ^ {\pi} (f) \sum_ {a \notin \mathcal {A} _ {i}} \pi (a | s _ {i}) \Big ] \\ = \sum_ {a \in \mathcal {A} _ {i}} \pi (a | s _ {i}) \Big (\mathbb {E} _ {c _ {k} \sim \mathcal {D}} \big [ c _ {k} (s _ {i}, a) \big ] + \Big (1 - \frac {1}{T ^ {\prime}} \Big) \mathbb {E} _ {c _ {k} \sim \mathcal {D}} \big [ J _ {k} ^ {\pi} (s _ {i}) \big ] \Big) + D ^ {\prime} \cdot \sum_ {a \notin \mathcal {A} _ {i}} ^ {A} \pi (a | s _ {i}) \\ = \sum_ {a \in \mathcal {A} _ {i}} \pi (a | s _ {i}) \left(\frac {D ^ {\prime}}{2 T ^ {\prime}} + \left(1 - \frac {1}{T ^ {\prime}}\right) \mathbb {E} _ {c _ {k} \sim \mathcal {D}} \left[ J _ {k} ^ {\pi} (s _ {i}) \right]\right) + D ^ {\prime} \cdot \left(1 - \sum_ {a \in \mathcal {A} _ {i}} \pi (a | s _ {i})\right) \\ \Rightarrow \mathbb {E} _ {c _ {k} \sim \mathcal {D}} \big [ J _ {k} ^ {\pi} (s _ {i}) \big ] \Big (1 - \Big (1 - \frac {1}{T ^ {\prime}} \Big) \sum_ {a \in \mathcal {A} _ {i}} \pi (a | s _ {i}) \Big) = \frac {D ^ {\prime}}{2 T ^ {\prime}} \sum_ {a \in \mathcal {A} _ {i}} \pi (a | s _ {i}) + D ^ {\prime} \cdot \Big (1 - \sum_ {a \in \mathcal {A} _ {i}} \pi (a | s _ {i}) \Big) \\ \Longrightarrow \mathbb {E} _ {c _ {k} \sim \mathcal {D}} \left[ J _ {k} ^ {\pi} (s _ {i}) \right] = \frac {\frac {D ^ {\prime}}{2 T ^ {\prime}} \sum_ {a \in \mathcal {A} _ {i}} \pi (a | s _ {i}) + D ^ {\prime} \cdot \left(1 - \sum_ {a \in \mathcal {A} _ {i}} \pi (a | s _ {i})\right)}{1 - \left(1 - \frac {1}{T ^ {\prime}}\right) \sum_ {a \in \mathcal {A} _ {i}} \pi (a | s _ {i})} \\ = \frac {D ^ {\prime}}{2} \cdot \frac {\frac {1}{T ^ {\prime}} \sum_ {a \in \mathcal {A} _ {i}} \pi (a | s _ {i}) + 2 \left(1 - \sum_ {a \in \mathcal {A} _ {i}} \pi (a | s _ {i})\right)}{\frac {1}{T ^ {\prime}} \sum_ {a \in \mathcal {A} _ {i}} \pi (a | s _ {i}) + \left(1 - \sum_ {a \in \mathcal {A} _ {i}} \pi (a | s _ {i})\right)} \\ \geqslant \frac {D ^ {\prime}}{2}. \\ \end{array} +$$ + +The optimal policy $\pi^{\star}$ is the policy that takes actions in the binary tree to reach state $s_{i\star}$ and then $\pi^{\star}(a_{j\star}|s_{i\star}) = 1$ for $i^{\star}, j^{\star} = \arg \min_{i,j:j + (A - 1)\cdot i\leqslant L}\sum_{k = 1}^{K}c_k(s_i,a_j)$ . We have $J_{k}^{\pi^{\star}}(s_{0}) = J_{k}^{\pi^{\star}}(s_{i\star})$ and for any $k\geqslant 1$ + +$$ +\begin{array}{l} J _ {k} ^ {\pi^ {\star}} (s _ {i ^ {\star}}) = c _ {k} (s _ {i ^ {\star}}, a _ {j ^ {\star}}) + \left(1 - \frac {1}{T ^ {\prime}}\right) J _ {k} ^ {\pi^ {\star}} (s _ {i ^ {\star}}) \\ \Longrightarrow J _ {k} ^ {\pi^ {*}} \left(s _ {0}\right) = T ^ {\prime} c _ {k} \left(s _ {i ^ {*}}, a _ {j ^ {*}}\right) \\ \Longrightarrow \sum_ {k = 1} ^ {K} J _ {k} ^ {\pi^ {\star}} (s _ {0}) = T ^ {\prime} \sum_ {k = 1} ^ {K} c _ {k} \left(s _ {i ^ {\star}}, a _ {j ^ {\star}}\right) = T ^ {\prime} \min _ {i, j: j + (A - 1) \cdot i \leqslant L} \sum_ {k = 1} ^ {K} c _ {k} \left(s _ {i}, a _ {j}\right). \\ \end{array} +$$ + +Hence, + +$$ +\mathbb {E} _ {c _ {1}, \dots , c _ {K} \stackrel {\text {i d}} {\sim} \mathcal {D}} \left[ R _ {K} \right] \geqslant \frac {D ^ {\prime}}{2} \cdot K - T ^ {\prime} \cdot \mathbb {E} _ {c _ {1}, \dots , c _ {K} \stackrel {\text {i d}} {\sim} \mathcal {D}} \left[ \min _ {i, j: j + (A - 1) \cdot i \leqslant L} \sum_ {k = 1} ^ {K} c _ {k} \left(s _ {i}, a _ {j}\right) \right] +$$ + +$$ +\begin{array}{l} = T ^ {\prime} \cdot \left(\frac {D ^ {\prime}}{2 T ^ {\prime}} \cdot K - \mathbb {E} _ {c _ {1}, \dots , c _ {K} \stackrel {\text {i d}} {\sim} \mathcal {D}} \left[ \min _ {i, j: j + (A - 1) \cdot i \leqslant L} \sum_ {k = 1} ^ {K} c _ {k} \left(s _ {i}, a _ {j}\right) \right]\right) \\ = T ^ {\prime} \cdot \mathbb {E} _ {c _ {1}, \dots , c _ {K} \stackrel {\text {i i d}} {\sim} \mathcal {D}} \left[ \max _ {i, j: j + (A - 1) \cdot i \leqslant L} \sum_ {k = 1} ^ {K} \left(\frac {D ^ {\prime}}{2 T ^ {\prime}} - c _ {k} \left(s _ {i}, a _ {j}\right)\right) \right] \\ \end{array} +$$ + +We now apply Theorem G.1 with $p = 1 - \frac{D'}{2T'} \geqslant \frac{1}{2}$ , $d = L \geqslant 100$ (since $S(A - 1) \geqslant 400$ and $M \geqslant 101$ ) and $n = K \geqslant \frac{800T_*}{D} \log M \geqslant \frac{400T'}{D'} \log M \geqslant 200\frac{p}{1 - p} \log d$ . We obtain: + +$$ +\begin{array}{l} \sup _ {c _ {1}, \dots , c _ {K}} \mathbb {E} \left[ R _ {K} \right] \geqslant \mathbb {E} _ {c _ {1}, \dots , c _ {K} \stackrel {\text {i d}} {\approx} \mathcal {D}} \left[ R _ {K} \right] \geqslant 0. 0 2 T ^ {\prime} \sqrt {K \left(1 - \frac {D ^ {\prime}}{2 T ^ {\prime}}\right) \cdot \frac {D ^ {\prime}}{2 T ^ {\prime}} \cdot \log L} - 1. 5 T ^ {\prime} \\ = \Omega \left(\sqrt {K T _ {\star} D \log M}\right), \\ \end{array} +$$ + +since $L \geqslant M / 8$ . Note that since $T_{\star} \geqslant D$ , the hitting-time of the fast-policy is $D' + B + 2 = D$ and the hitting time of the optimal is $T' + B + 1 = T_{\star}$ , as required. This concludes the proof. + +# F Lower bound under unknown transitions + +Theorem 5.1. For any $D, K, S, A$ with $S \geqslant 2, A \geqslant 16$ , $D \geqslant 2$ and $K \geqslant SA$ , there exists an SSP instance with $M = 1$ , $S$ states and $A$ actions such that its diameter is $D$ and the expected regret for any learner without knowledge of the transitions after $K$ episodes is $\mathbb{E}[R_K] \geqslant \Omega(D \sqrt{SAK})$ . + +![](images/76c89358535bf8ee68a295beade1b8d9b225a7cf420323757728a372dc00d405.jpg) +Figure 5: base case + +Proof. The idea is to inject sparsity into the lower bound construction of [29] and to see if sparsity helps. Imagine the simple SSP in Figure 5, where at state $s_0$ there are $A$ available actions, all with zero cost, while in the state $f$ there is only one deterministic action with unit cost going back to $s_0$ . Among them, there exists an action $a_\star$ such that the transition probabilities are given by: $P(g \mid s_0, a) = \frac{1}{D} - \epsilon \mathbb{I}(a \neq a_\star)$ , and consequently, $P(f \mid s_0, a) = 1 - \frac{1}{D} + \epsilon \mathbb{I}(a \neq a_\star)$ . The cost is therefore only suffered when the selected action transitions to the $f$ state. This will therefore not increase the hitting time of any proper deterministic policy while still inducing the desired sparsity. + +Clearly, the optimal policy plays $a_{\star}$ at every time step to reach the goal as fast as possible and therefore $J^{\pi_{\star}}(s_0) = D - 1$ . + +Now, denote with $N_{k}$ the number of steps that the learner spends in $s_0$ in episode $k$ and $N_{k}^{\star}$ the number of steps that the learner picks action $a_{\star}$ in episode $k$ . Note that $N_{k}$ is also the total cost that the learner suffers during episode $k$ minus one (since the last transition will not be paid). Thanks to our construction we can still prove Lemma C.1 in [29] as follows: + +Lemma F.1. + +$$ +\mathbb {E} \left[ N _ {k} \right] - 1 - J ^ {\pi_ {\star}} = \epsilon \mathbb {E} \left[ N _ {k} - N _ {k} ^ {\star} \right] +$$ + +Proof. + +$$ +\begin{array}{l} \mathbb {E} \left[ N _ {k} \right] = \sum_ {t = 1} ^ {\infty} P \left[ s _ {t} = s _ {0} \right] \\ = 1 + \sum_ {t = 2} ^ {\infty} P \left[ s _ {t} = s _ {0} \right] \\ = 1 + \sum_ {t = 2} ^ {\infty} P \left[ s _ {t} = s _ {0} \mid s _ {t + 1} = s _ {0}, a _ {t - 1} = a _ {\star} \right] P \left[ s _ {t + 1} = s _ {0}, a _ {t - 1} = a _ {\star} \right] \\ + \sum_ {t = 2} ^ {\infty} P \left[ s _ {t} = s _ {0} \mid s _ {t + 1} = s _ {0}, a _ {t - 1} \neq a _ {\star} \right] P \left[ s _ {t + 1} = s _ {0}, a _ {t - 1} \neq a _ {\star} \right] \\ = 1 + \left(1 - \frac {1}{D}\right) \sum_ {t = 2} ^ {\infty} P \left[ s _ {t + 1} = s _ {0}, a _ {t - 1} = a _ {\star} \right] + \left(1 - \frac {1 - \epsilon}{D}\right) \sum_ {t = 2} ^ {\infty} P \left[ s _ {t + 1} = s _ {0}, a _ {t - 1} \neq a _ {\star} \right] \\ \end{array} +$$ + +Rearranging gives: + +$$ +\mathbb {E} \left[ N _ {k} \right] - D = \epsilon \mathbb {E} \left[ N _ {k} - N _ {k} ^ {\star} \right] +$$ + +Adding and subtracting 1 gives the desired result. + +![](images/412e7e60b876a249eb8aae60d67f145c834a04fdc940141bfc923a060db1b64c.jpg) + +Hence: + +$$ +\mathbb {E} \left[ R _ {K} \right] = \sum_ {k = 1} ^ {K} \sum_ {i = 1} ^ {I _ {k}} c _ {k} \left(s _ {k} ^ {i}, a _ {k} ^ {i}\right) - \sum_ {k = 1} ^ {K} J ^ {\pi_ {\star}} \left(s _ {0}\right) = \mathbb {E} \left[ N _ {k} \right] - 1 - J ^ {\pi_ {\star}} = \epsilon \left[ N - N ^ {\star} \right] +$$ + +where $N = \sum_{k=1}^{K} N_k$ and $N^{\star} = \sum_{k=1}^{K} N_k^{\star}$ . Since we recovered Lemma C.1 in [29] as the starting block of the proof, following the derivation we can lower bound $N$ in expectation and upper bound the expected value of $N^{\star}$ to retrieve + +Lemma F.2 (Theorem C.4 in [29]). Suppose that $D \geqslant 2$ , $\epsilon \in (0,1/8)$ and $A > 16$ , for the problem described above we have: + +$$ +\mathbb {E} \left[ R _ {K} \right] \geqslant \epsilon K D \left(\frac {1}{8} - 2 \epsilon \sqrt {\frac {2 K}{A}}\right) +$$ + +Now consider the following MDP. Let $S$ be the set of states disregarding $g$ and $s_0$ . The initial state $s_0$ has only one action which leads uniformly at random into one of the states $s \in S$ , where each one has its own optimal action $a_s^\star$ . Then the transition distributions are defined $P(g \mid a_s^\star, s) = 1 / D$ , $P(s \mid a_s^\star, s) = 1 - 1 / D$ , and $P(g \mid a, s) = (1 - \epsilon) / D$ , $P(s \mid a, s) = 1 - (1 - \epsilon) / D$ for any other action $a \in \mathcal{A} \backslash \{a_s^\star\}$ . Note that for each state, the learner is faced with a simple problem as the one described above. Therefore, we can apply Lemma F.2 for each state separately and lower bound the learner's expected regret the sum of the regrets suffered at each state, which would depend on the number of times each state $s \in S$ is visited from the initial state. Since reaching each state has uniform probability, there are many states (constant fraction) that are chosen $\Theta(K / S)$ times. Summing the regret bounds and choosing $\epsilon$ , gives the desired bound. + +Denote by $K_{s}$ the number of episodes the state $s\in S$ is visited: + +$$ +\mathbb {E} \left[ R _ {K} \right] \geqslant \sum_ {s \in \mathcal {S}} \mathbb {E} \left[ \epsilon K _ {s} D \left(\frac {1}{8} - 2 \epsilon \sqrt {\frac {2 K _ {s}}{A}}\right) \right] = \frac {\epsilon K D}{8} - 2 \epsilon^ {2} D \sqrt {\frac {2}{A}} \sum_ {s \in \mathcal {S}} \mathbb {E} \left[ K _ {s} ^ {3 / 2} \right] +$$ + +Then: + +$$ +\sum_ {s \in \mathcal {S}} \mathbb {E} \left[ K _ {s} ^ {3 / 2} \right] \leqslant \sum_ {s \in \mathcal {S}} \sqrt {\mathbb {E} [ K _ {s} ]} \sqrt {\mathbb {E} [ K _ {s} ^ {2} ]} = \sum_ {s \in \mathcal {S}} \sqrt {\mathbb {E} [ K _ {s} ]} \sqrt {\mathbb {E} [ K _ {s} ^ {2} ] + \mathbb {V} [ K _ {s} ]} = \sum_ {s \in \mathcal {S}} \sqrt {\frac {K}{S}} \sqrt {\frac {K ^ {2}}{S ^ {2}} + \frac {K (S - 1)}{S ^ {2}}} \leqslant K \sqrt {\frac {2 K}{S}} +$$ + +Leading to: + +$$ +\mathbb {E} \left[ R _ {K} \right] \geqslant \frac {\epsilon K D}{8} - 2 \epsilon^ {2} D K \sqrt {\frac {2 K}{S A}} \geqslant \frac {1}{1 0 2 4} D \sqrt {S A K} +$$ + +for $\epsilon = 1 / 64\sqrt{SA / K} K\geqslant SA$ , concluding the proof. + +# G Lower bound on the maximum of asymmetric zero-mean random walks + +We extend the lower bound of [25] to asymmetric zero-mean random-walks. We consider $p \geqslant 1/2$ because it simplifies the proof below (lower-bounding $\psi$ by 1 and upper-bounding $C$ in proof below) and is what we need in the proof of Section 4.6 in Appendix E (we use $p = 1 - D / 2T^{\star}$ ). + +Theorem G.1. Fix $p \in \left[\frac{1}{2}, 1 - \frac{1}{n}\right]$ . Consider random walks $Z_{i}^{(n)} = \sum_{t=1}^{n} X_{t}^{i}$ , where + +$$ +X _ {t} ^ {i} = \left\{ \begin{array}{l l} - p, & \quad w. p. 1 - p \\ 1 - p, & \quad w. p. p. \end{array} \right. +$$ + +If $n \geqslant 200\frac{p}{1 - p}\log d$ (also ensures that $p \leqslant 1 - \frac{1}{n}$ ) and $d \geqslant 100$ . Then, + +$$ +\mathbb {E} \bigl [ \max _ {1 \leqslant i \leqslant d} Z _ {i} ^ {n} \bigr ] \geqslant 0. 0 2 \sqrt {n p (1 - p) \log d} - 1. 5. +$$ + +Proof. We follow the same lines as [25] who show a special case of the result for $p = 1/2$ . We generalize it to $p > 1/2$ . + +Consider $Z^{(n)} = \sum_{t=1}^{n} X_{t}$ , a random-walk of length $n$ , then $B_{n} = Z^{(n)} + pn \sim B(n, p)$ , Binomial distribution with parameters $n$ and $p$ . + +# G.1 1st part of the proof: + +The 1st part of the proof is all about providing a lower bound on $\mathbb{P}\big(B_n\geqslant pn + t - 1\big)$ in (10) for any $t\in [1,np + 1]$ . + +Lemma G.2 (Generalized version of Lemma 4 of [25], Theorem 2 of [21]). Let $n, k$ be integers satisfying $n \geqslant 1$ and $pn \leqslant k \leqslant n$ . Define $x = \frac{k - pn}{\sqrt{p(1 - p)n}}$ . Then $B_{n} \sim B(n,p)$ satisfies + +$$ +\mathbb {P} \big (B _ {n} \geqslant k \big) \geqslant \sqrt {n} \binom {n - 1} {k - 1} p ^ {k - 1 / 2} (1 - p) ^ {n - k + 1 / 2} \cdot \frac {1 - \Phi (x)}{\phi (x)}, +$$ + +where $\phi (x)$ is the PDF of a standard Normal and $\Phi (x)$ is the CDF. The proof can be found in [21]. + +Denote $D(p,q) = p\log \frac{p}{q} + (1 - p)\log \frac{1 - p}{1 - q}$ as the KL-divergence between two Bernoullis. + +Lemma G.3 (Generalized version of Theorem 5 of [25]). Let $n, k$ be integers satisfying $n \geqslant 1$ , and $np \leqslant k \leqslant n$ . Define $x = \frac{k - pn}{\sqrt{p(1 - p)n}}$ . Then $B_{n} \sim B(n,p)$ satisfies + +$$ +\mathbb {P} \big (B _ {n} \geqslant k \big) \geqslant \frac {\exp \Big (- n D \big (\frac {k}{n} , p \big) \Big)}{e ^ {1 / 6} \sqrt {2 \pi}} \cdot \frac {1 - \Phi (x)}{\phi (x)}. +$$ + +Proof. For $k = n$ , we verify the statement of the theorem directly. The left hand side is $\mathbb{P}\big(B_n\geqslant n\big) = p^n$ . The right hand side is smaller because $\exp \Bigl \{-nD\Bigl (1,p\Bigr)\Bigr \} = p^n$ and for $x = \sqrt{n\frac{1 - p}{p}} >0$ we have $\frac{1 - \Phi(x)}{\phi(x)}\leqslant \sqrt{2}$ (see e.g. Section 3.3 in [15]). + +For $np \leqslant k < n$ , we first bound the binomial coefficient $\binom{n}{k}$ . Stirling's formula for the factorial [27] gives for any $n \geqslant 1$ + +$$ +\sqrt {2 \pi n} \left(\frac {n}{e}\right) ^ {n} < n! < e ^ {1 / 1 2} \sqrt {2 \pi n} \left(\frac {n}{e}\right) ^ {n}. +$$ + +Since $0 < np \leqslant k \leqslant n - 1$ , we can use this approximation for $k, n$ and $n - k$ and obtain + +$$ +\begin{array}{l} \left( \begin{array}{c} n \\ k \end{array} \right) = \frac {n !}{k ! (n - k) !} \\ > \frac {n ^ {n} e ^ {- n} \sqrt {2 \pi n}}{\left(e ^ {1 / 1 2} k ^ {k} e ^ {- k} \sqrt {2 \pi k}\right) \cdot \left(e ^ {1 / 1 2} (n - k) ^ {n - k} e ^ {- (n - k)} \sqrt {2 \pi (n - k)}\right)} \\ = \frac {1}{e ^ {1 / 6} \sqrt {2 \pi}} \left(\frac {n}{k}\right) ^ {k} \left(\frac {n}{n - k}\right) ^ {n - k} \sqrt {\frac {n}{k (n - k)}} \\ \end{array} +$$ + +$$ += \frac {1}{e ^ {1 / 6} \sqrt {2 \pi}} \frac {1}{p ^ {k} (1 - p) ^ {n - k}} \exp \Bigl \{- n D \Bigl (\frac {k}{n}, p \Bigr) \Bigr \} \sqrt {\frac {n}{k (n - k)}}, +$$ + +since + +$$ +\begin{array}{l} D \left(\frac {k}{n}, p\right) = \frac {k}{n} \log \left(\frac {k}{n p}\right) + \left(1 - \frac {k}{n}\right) \log \left(\frac {1 - k / n}{1 - p}\right) = - \frac {k}{n} \log \left(\frac {n p}{k}\right) - \frac {n - k}{n} \log \left(\frac {n (1 - p)}{n - k}\right) \\ \Longrightarrow \exp \left\{- n D \left(\frac {k}{n}, p\right) \right\} = \left(\frac {n p}{k}\right) ^ {k} \cdot \left(\frac {n (1 - p)}{n - k}\right) ^ {n - k} = p ^ {k} (1 - p) ^ {n - k} \left(\frac {n}{k}\right) ^ {k} \left(\frac {n}{n - k}\right) ^ {n - k}. \\ \end{array} +$$ + +Since $k \geqslant 1$ , we can write the binomial coefficient as $\binom{n-1}{k-1} = \frac{k}{n} \binom{n}{k}$ . By Lemma G.2, we have + +$$ +\begin{array}{l} \mathbb{P}\bigl(B_{n}\geqslant k\bigr)\geqslant \sqrt{n}\binom {n - 1}{k - 1}p^{k - 1 / 2}(1 - p)^{n - k + 1 / 2}\cdot \frac{1 - \Phi(x)}{\phi(x)} \\ = \sqrt {n} \frac {k}{n} \binom {n} {k} p ^ {k - 1 / 2} (1 - p) ^ {n - k + 1 / 2} \cdot \frac {1 - \Phi (x)}{\phi (x)} \\ \geqslant \frac {1}{e ^ {1 / 6} \sqrt {2 \pi}} \frac {k}{\sqrt {n}} \sqrt {\frac {n}{k (n - k)}} \frac {p ^ {k - 1 / 2} (1 - p) ^ {n - k + 1 / 2}}{p ^ {k} (1 - p) ^ {n - k}} \exp \left\{- n D \left(\frac {k}{n}, p\right) \right\} \cdot \frac {1 - \Phi (x)}{\phi (x)} \\ = \frac {1}{e ^ {1 / 6} \sqrt {2 \pi}} \sqrt {\frac {k}{n - k}} \cdot \sqrt {\frac {1 - p}{p}} \exp \left\{- n D \left(\frac {k}{n}, p\right) \right\} \cdot \frac {1 - \Phi (x)}{\phi (x)}. \\ \end{array} +$$ + +The result follows from $\sqrt{\frac{k}{n - k}} \geqslant \sqrt{\frac{np}{n - np}} = \sqrt{\frac{p}{1 - p}}$ for $np \leqslant k \leqslant n - 1$ . + +For $k = pn + xn$ , the 2nd-order Taylor approximation of $u(x) = D\left(\frac{k}{n}, p\right) = D\left(p + x, p\right)$ around 0 is $\frac{x^2}{2p(1 - p)}$ . We define $\psi : [-p, 1 - p] \to \mathbb{R}$ as the ratio of the divergence and the approximation: + +$$ +\psi (x) = D (p + x, p) \cdot \frac {2 p (1 - p)}{x ^ {2}}. +$$ + +In particular, we have that $1 \leqslant \psi(x) \leqslant \frac{p(1 - p)}{(x + p)(1 - p - x)}$ for $x \in [0,1 - p]$ . This can be shown using Taylor's theorem on $u(x)$ : for some $z \in [0,x]$ , + +$$ +\begin{array}{l} D (p + x, p) = \frac {u ^ {(2)} (z)}{2} x ^ {2} = \left(\frac {1}{z + p} + \frac {1}{1 - p - z}\right) \frac {x ^ {2}}{2} = \frac {x ^ {2}}{2 (z + p) (1 - p - z)} \\ \Longrightarrow \frac {x ^ {2}}{2 p (1 - p)} \leqslant D (p + x, x) \leqslant \frac {x ^ {2}}{2 (x + p) (1 - p - x)}, \tag {9} \\ \end{array} +$$ + +since $\frac{1}{(x + p)(1 - p - x)}$ is increasing on $[0, 1 - p)$ . + +Let $t \in [1, np + 1]$ be a real number. By Lemma G.3 and Lemma 1 in [25] (also Mill's ratio for standard Gaussian [2]), we have + +$$ +\begin{array}{l} \mathbb {P} \left(B _ {n} \geqslant p n + t - 1\right) = \mathbb {P} \left(B _ {n} \geqslant [ p n + t - 1 ]\right) \\ \geqslant \frac {\exp \left(- n D \left(\frac {[ p n + t - 1 ]}{n} , p\right)\right)}{e ^ {1 / 6} \sqrt {2 \pi}} \cdot \frac {\pi}{\pi \frac {[ p n + t - 1 ] - n p}{\sqrt {p (1 - p) n}} + \sqrt {2 \pi}} \\ \geqslant \frac {\exp \left(- n D \left(\frac {p n + t}{n} , p\right)\right)}{e ^ {1 / 6} \sqrt {2 \pi}} \cdot \frac {\pi}{\pi \frac {p n + t - n p}{\sqrt {p (1 - p) n}} + \sqrt {2 \pi}} \\ = \frac {\exp \left(- n D \left(p + \frac {t}{n} , p\right)\right)}{e ^ {1 / 6} \sqrt {2 \pi}} \cdot \frac {\pi}{\pi \frac {t}{\sqrt {p (1 - p) n}} + \sqrt {2 \pi}} \\ = e ^ {- 1 / 6} \exp \left(- \frac {1}{2 p (1 - p)} \psi \left(\frac {t}{n}\right) \cdot \frac {t ^ {2}}{n}\right) \cdot \frac {1}{\frac {\sqrt {2 \pi} t}{\sqrt {n p (1 - p)}} + 2}. \tag {10} \\ \end{array} +$$ + +# G.2 2nd part of the proof: + +We can now turn to the actual proof of the result. Define the event $A$ equal to the case that at least one of the $Z_{i}^{(n)}$ is greater or equal to $C\sqrt{np(1 - p)\log d} - 1$ . We will show this event / threshold controls the expectation of the maximum. First, we define $C$ and provide some upper and lower bounds for it. Denote by $f(d) = \sqrt{2 - \frac{2\log\log d}{\log d}}$ , then + +$$ +C = C (d, n) = \frac {1}{\sqrt {\psi \left(\sqrt {\frac {2 p (1 - p) \log d}{n}}\right)}} \sqrt {2 - \frac {2 \log \log d}{\log d}} = \frac {1}{\sqrt {\psi \left(\sqrt {\frac {2 p (1 - p) \log d}{n}}\right)}} f (d). \tag {11} +$$ + +We bound the two factors separately: + +- $z = \sqrt{\frac{2p(1 - p)\log d}{n}} \in [0, \frac{1}{10}(1 - p)]$ for $n \geqslant 200\frac{p}{1 - p}\log d$ and so + +$$ +1 \leqslant \psi (z) \leqslant \frac {p (1 - p)}{(z + p) (1 - p - z)} \leqslant \frac {1 - p}{\left(1 - p - \frac {1 - p}{1 0}\right)} = \frac {1 0}{9}. \tag {12} +$$ + +- The function $f(d)$ is as in [25]: decreasing on $(1, e^e]$ , increasing on $[e^e, +\infty)$ , and $\lim_{d \to \infty} f(d) = \sqrt{2}$ . Therefore for all $d \in [5, \infty)$ , + +$$ +1. 1 2 \leqslant f (e ^ {e}) \leqslant f (d) \leqslant \max \{f (5), \sqrt {2} \} = \sqrt {2} +$$ + +This gives for $n \geqslant 200\frac{p}{1 - p}\log d$ + +$$ +1 \leqslant \frac {1 . 1 2}{\sqrt {1 0 / 9}} \leqslant C (d, n) \leqslant \sqrt {2} \tag {13} +$$ + +Since $p \geqslant 1/2$ , if $n \geqslant 200\frac{p}{1 - p} \log d$ , then $n > \frac{200}{p(1 - p)\log d}$ (if $d \geqslant 8$ ) and $n \geqslant 200\frac{1 - p}{p}\log d$ . The above implies: + +$$ +1 < C \sqrt {n p (1 - p) \log d} \leqslant n p \leqslant n p + 1. \tag {14} +$$ + +Finally, we bound the quantity of interest: + +$$ +\begin{array}{l} \mathbb {E} \left[ \max _ {1 \leqslant i \leqslant d} Z _ {i} ^ {n} \right] = \mathbb {E} \left[ \max _ {1 \leqslant i \leqslant d} Z _ {i} ^ {n} | A \right] \cdot \mathbb {P} (A) + \mathbb {E} \left[ \max _ {1 \leqslant i \leqslant d} Z _ {i} ^ {n} | A ^ {C} \right] \cdot (1 - \mathbb {P} (A)) \\ \geqslant \mathbb {E} \left[ \max _ {1 \leqslant i \leqslant d} Z _ {i} ^ {n} | A \right] \cdot \mathbb {P} (A) + \mathbb {E} \left[ Z _ {1} ^ {(n)} | A ^ {C} \right] \cdot (1 - \mathbb {P} (A)) \\ = \mathbb {E} \left[ \max _ {1 \leqslant i \leqslant d} Z _ {i} ^ {n} | A \right] \cdot \mathbb {P} (A) + \mathbb {E} \left[ Z _ {1} ^ {(n)} | Z _ {1} ^ {(n)} < C \sqrt {n p (1 - p) \log d} - 1 \right] \cdot (1 - \mathbb {P} (A)) \\ \geqslant \mathbb {E} \left[ \max _ {1 \leqslant i \leqslant d} Z _ {i} ^ {n} | A \right] \cdot \mathbb {P} (A) + \mathbb {E} \left[ Z _ {1} ^ {(n)} | Z _ {1} ^ {(n)} \leqslant 0 \right] \cdot (1 - \mathbb {P} (A)) \quad \text {b y} (14) \\ \geqslant \left(C \sqrt {n p (1 - p) \log d} - 1\right) \cdot \mathbb {P} (A) + \mathbb {E} \left[ Z _ {1} ^ {(n)} \mid Z _ {1} ^ {(n)} \leqslant 0 \right] \cdot \left(1 - \mathbb {P} (A)\right). (15) \\ \end{array} +$$ + +First, we lower bound $\mathbb{E}\Big[Z_1^{(n)}|Z_1^{(n)}\leqslant 0\Big]$ . Let $\beta = \frac{1}{1 - \sqrt{\frac{2n}{\pi[n(1 - p)](n - [n(1 - p)])}}}$ . For $n\geqslant \frac{200}{p(1 - p)}\log d$ we have $n\geqslant \frac{205}{\pi p(1 - p)}\geqslant \frac{200 + \pi p}{\pi(1 - p)p}$ and $\beta \leqslant \frac{10}{9}$ . + +Then Lemma 2.2 in [26] combined with Lemma 8 in [13] give that for $Y_{n} \sim B(n,1 - p)$ : + +$$ +\mathbb {E} \left[ Y _ {n} | Y _ {n} \geqslant n (1 - p) \right] < n (1 - p) + \beta \sqrt {n p (1 - p)} < n (1 - p) + \frac {1 0}{9} \sqrt {n p (1 - p)}. +$$ + +Since $B_{n} = Z^{(n)} + pn \sim B(n,p)$ can be written as $n - Y_{n}$ , we have: + +$$ +\mathbb {E} \left[ Z _ {1} ^ {(n)} | Z _ {1} ^ {(n)} \leqslant 0 \right] = \mathbb {E} \left[ B _ {n} | B _ {n} \leqslant n p \right] - n p +$$ + +$$ +\begin{array}{l} = \mathbb {E} \left[ n - Y _ {n} | n - Y _ {n} \leqslant n p \right] - n p \\ = n - \mathbb {E} \left[ Y _ {n} \mid Y _ {n} \geqslant n (1 - p) \right] - n p \\ \geqslant n - n (1 - p) - \frac {1 0}{9} \sqrt {n p (1 - p)} - n p \\ = - \frac {1 0}{9} \sqrt {n p (1 - p)}. \tag {16} \\ \end{array} +$$ + +Next, we lower-bound $\mathbb{P}(A)$ : + +$$ +\begin{array}{l} \mathbb {P} (A) = 1 - \mathbb {P} \left(A ^ {C}\right) \\ = 1 - \left(\mathbb {P} \left[ Z _ {1} ^ {(n)} < C \sqrt {n p (1 - p) \log d} - 1 \right]\right) ^ {d} \\ = 1 - \left(\mathbb {P} \left[ B _ {n} < n p + C \sqrt {n p (1 - p) \log d} - 1 \right]\right) ^ {d} \\ = 1 - \left(1 - \mathbb {P} \left[ B _ {n} \geqslant n p + C \sqrt {n p (1 - p) \log d} - 1 \right]\right) ^ {d} \\ \geqslant 1 - \exp \left(- d \cdot \mathbb {P} \left[ B _ {n} \geqslant n p + C \sqrt {n p (1 - p) \log d} - 1 \right]\right) \quad \text {s i n c e} 1 - x \leqslant e ^ {- x} \\ \geqslant 1 - \exp \left(- d \cdot \frac {e ^ {- 1 / 6} \exp \left(- \frac {1}{2 p (1 - p)} \psi \left(\frac {C \sqrt {n p (1 - p) \log d}}{n}\right) \cdot \frac {C ^ {2} n p (1 - p) \log d}{n}\right)}{\frac {\sqrt {2 \pi} C \sqrt {n p (1 - p) \log d}}{\sqrt {n p (1 - p)}} + 2}\right) \quad \text {u s i n g (1 0) a n d (1 4)} \\ = 1 - \exp \left(- d \cdot \frac {e ^ {- 1 / 6} \exp \left(- \frac {1}{2} \psi \left(C \sqrt {p (1 - p) \log d / n}\right) \cdot C ^ {2} \log d\right)}{\sqrt {2 \pi} C \sqrt {\log d} + 2}\right) \\ = 1 - \exp \left(- \frac {e ^ {- 1 / 6} d ^ {1 - \frac {C ^ {2}}{2}} \psi \left(C \sqrt {p (1 - p) \log d / n}\right)}{\sqrt {2 \pi} C \sqrt {\log d} + 2}\right) \\ \geqslant 1 - \exp \left(- \frac {e ^ {- 1 / 6} d ^ {1 - \frac {C ^ {2}}{2}} \psi \left(\sqrt {\frac {2 p (1 - p) \log d}{n}}\right)}{2 \sqrt {\pi \log d} + 2}\right) \text {b y} (1 3). \\ \end{array} +$$ + +We now use that $d^{1 - \frac{C^2}{2}}\psi \left(\sqrt{2p(1 - p)\log d / n}\right) = \log d$ by the definition of $C$ in (11). Hence, we obtain: + +$$ +\mathbb {P} (A) \geqslant 1 - \exp \left(- \frac {e ^ {- 1 / 6} \log d}{2 \sqrt {\pi \log d} + 2}\right) = 1 - g (d), \quad \text {f o r} g (d) = \exp \left(- \frac {e ^ {- 1 / 6} \log d}{2 \sqrt {\pi \log d} + 2}\right). \tag {17} +$$ + +Putting everything together: we plug (16) and (17) into (15) to get + +$$ +\begin{array}{l} \mathbb {E} \left[ \max _ {1 \leqslant i \leqslant d} Z _ {i} ^ {n} \right] \geqslant \left(C \sqrt {n p (1 - p) \log d} - 1\right) \cdot (1 - g (d)) - \frac {1 0}{9} g (d) \sqrt {n p (1 - p)} \\ = \frac {f (d) \cdot (1 - g (d))}{\sqrt {\psi \left(\sqrt {\frac {2 p (1 - p) \log d}{n}}\right)}} \left(\sqrt {n p (1 - p) \log d} - 2\right) - \frac {1 0}{9} g (d) \sqrt {n p (1 - p)} \quad \text {u s i n g} (11) \\ \geqslant \frac {f (d) (1 - g (d))}{\sqrt {1 0 / 9}} (\sqrt {n p (1 - p) \log d} - 2) - \frac {1 0}{9} g (d) \sqrt {n p (1 - p)} \quad \text {u s i n g} (12) \\ = \sqrt {n p (1 - p) \log d} \cdot \left(\frac {f (d) (1 - g (d))}{\sqrt {1 0 / 9}} - \frac {1 0}{9} \cdot \frac {g (d)}{\sqrt {\log (d)}}\right) - 2 \frac {f (d) (1 - g (d))}{\sqrt {1 0 / 9}} \\ \geqslant 0. 0 2 \sqrt {n p (1 - p) \log d} - 1. 5, \\ \end{array} +$$ + +for $d \geqslant 100$ (we also used that $\sqrt{np(1 - p)\log d} > 2$ in the 2nd inequality). 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However, reward hacking issues with PRMs limit their successful application in reinforcement fine-tuning. In this paper, we identify the main cause of PRM-induced reward hacking: the canonical summation-form credit assignment in reinforcement learning (RL), which defines the value as cumulative gamma-decayed future rewards, easily induces LLMs to hack steps with high rewards. To address this, we propose PURE: Process sUpervised Reinforcement lEarning. The key innovation of PURE is a minform credit assignment that formulates the value function as the minimum of future rewards. This method significantly alleviates reward hacking by limiting the value function range and distributing advantages more reasonably. Through extensive experiments on 3 base models, we show that PRM-based approaches enabling minform credit assignment achieve comparable reasoning performance to verifiable reward-based methods within only $30\%$ steps. In contrast, the canonical sum-form credit assignment collapses training even at the beginning! Additionally, when we supplement PRM-based fine-tuning with just $10\%$ verifiable rewards, we further alleviate reward hacking and produce the best fine-tuned model based on Qwen2.5-Math-7B in our experiments, achieving $82.5\%$ accuracy on AMC23 and $53.3\%$ average accuracy across 5 benchmarks. Moreover, we summarize the observed reward hacking cases and analyze the causes of training collapse. We release our code and model weights at https://github.com/CJReinforce/PURE. + +# 1 Introduction + +Reinforcement fine-tuning (RFT) of large language models (LLMs) for reasoning tasks has shown promise in developing advanced problem-solving abilities. Recent advancements (Wang et al., 2025; Zeng et al., 2025a), such as DeepSeek R1-Zero (Guo et al., 2025) and Kimi K1.5 (Team et al., 2025), have demonstrated strong reasoning skills through RFT with verifiable rewards, which provide sparse feedback for the entire response. However, as response length increases, sparse rewards potentially lead to inefficient learning (Sutton & Barto, 2018; Andrychowicz et al., 2017). + +In contrast, process reward models (PRMs) offer dense feedback at each step of a response. PRMs have proven effective in improving LLMs' performance on challenging reasoning tasks through test-time scaling (Lightman et al., 2023; Wang et al., 2023; Guan et al., 2025). However, successful applications of PRM in RFT for LLMs remains limited (Setlur et al., 2024). A key challenge of PRMs is that the neutral network-generated rewards can lead to reward hacking during training (Weng, 2024; Guo et al., 2025), causing unintended optimization toward higher rewards. Recent studies (Yuan et al., 2024; Cui et al., 2025) have explored implicit PRMs for fine-tuning LLMs, which resembles DPO-style reward formulation (Rafailov et al., 2023), but these still heavily relies on ground-truth verifiable rewards to provide training signals for the system. The question of what causes reward hacking in PRM-based RFT and how to address it effectively has not been explored widely. + +To answer this question, we examine the usage of PRM in test-time scaling and identify a mismatch between test-time and training-time objectives. Thus in this paper, we propose PURE: Process sUpvised Reinforcement lEarning, which introduces a min-form credit assignment method to align these objectives. Through analysis and experiments, we find that the canonical formulation of credit assignment in RL, named summation-form credit assignment that defines value as cumulative gamma-decayed future rewards, easily induces LLMs to hack high-reward steps and collapse training. Instead, we use the minimum future reward to quantify credit assignment, which constrains the value function's range and assigns advantages more reasonably to stabilize RL training (see the analysis in § 3.1). PURE offers several benefits: (1) it is simple to implement, requiring only a transformation of process rewards without additional code changes; (2) it achieves comparable or better performance in PRM-based RFT compared to recent R1-replication studies with around $3 \times$ efficiency gains; and (3) it supports the integration of both dense process rewards and sparse verifiable rewards. When combining both reward types in the PURE framework, we find that the auxiliary of few ground-truth signals further mitigates reward hacking caused by PRM. + +In experiments, we first train a PRM using the PRM800K dataset (Lightman et al., 2023), a human-annotated dataset that evaluates the correctness of each step. Then we apply PURE framework to 3 models: Qwen2.5-7B, Qwen2.5-Math-7B, and Qwen2.5-Math-1.5B, using 3 reward configurations: only verifiable rewards, only process rewards, and a combination of both. This comprehensive setup allows us to compare PRM-based and verifiable reward-based RFT approaches. We also compare sum-form and min-form credit assignment methods when enabling PRMs and find that the former even collapses training at the beginning. In the discussion, we summarize observed reward hacking cases and analyze the causes of training collapse. Our key findings are outlined below. + +1. Summation-form credit assignment easily induces LLMs to hack high-reward steps, leading to LLMs that prioritize thinking over problem-solving. Our min-form credit assignment avoids such reward hacking. (§ 3.1) +2. PRM-based RFT achieves performance comparable to the verifiable reward-based approach with around $3 \times$ efficiency gains if using our min-form credit assignment; otherwise, the training collapses even at the beginning! (§ 4.3) +3. The auxiliary of few ground-truth signals further mitigates PRM-induced reward hacking, achieving $82.5\%$ accuracy on AMC23 and $53.3\%$ average accuracy across MATH-500, Minerva Math, Olympiad Bench, AIME24, and AMC23 when using Qwen2.5-Math-7B as the base model. (§ 4.3) +4. We identify 3 types of PRM-induced reward hacking: (1) only thinking, not solving, (2) extremely few steps (1 step), (3) extremely few steps (0 step). We analyze causes, show examples, and provide solutions for each. (\$ 5.1) +5. Long, highly repetitive samples that the verifier ruled correct cause training to collapse. These pseudo-positive samples, undetected by verifiers, provide numerous incorrect signals and collapse training all of a sudden (within 5 gradient steps). (§ 5.2) + +# 2 Preliminaries + +# 2.1 Credit Assignment Problem in Reinforcement Learning + +Minsky (2007) described the credit assignment problem as "how to distribute the credit of success among the multitude of decisions involved". In other words, credit assignment is the problem of + +estimating the influence of an action over an outcome from experience. Various assignment functions have been proposed to quantify the influence of actions (Pignatelli et al., 2023). Here, we adopt the state-action value as the assignment function to detail the problem (Cheng et al., 2024, 2025). + +We model LLM reasoning as a step-level Markov Decision Process. Given a prompt, LLM generates steps sequentially. At step $t$ , LLM $\pi$ takes the prompt $p$ and previous steps $\{a_1, \dots, a_{t-1}\}$ as the input state $s_t$ , where $s_t = (p, a_1, \dots, a_{t-1})$ . Then LLM generates a step $a_t$ , sampled as $a_t \sim \pi(\cdot | s_t)$ . PRM $R^p$ emits a process reward $r_t^p = R^p(s_t, a_t)$ . The canonical formulation of state-action value is: + +$$ +Q ^ {\pi} \left(s _ {t}, a _ {t}\right) = \mathbb {E} \left[ \sum_ {i \geq t} \gamma^ {i - t} r _ {i} ^ {p} \right] \tag {1} +$$ + +where $\gamma$ is the discount factor. Eq. (1) shows that the influence of an action decreases over time, weighted by a discount factor $\gamma$ . Actions closer to future outcomes have greater influence. Thus, the state-action value quantifies credit assignment. With Eq. (1), we can derive the advantage function (Schulman et al., 2017; Shao et al., 2024; Ahmadian et al., 2024; Hu, 2025) and update LLM using the policy gradient loss. + +# 2.2 Process Reward vs. Verifiable Reward + +Verifiable rewards are sparse, rule-based rewards assigned to an entire response. For a prompt, LLM generates a sequence of steps $\{a_1,\dots ,a_n\}$ . After the final step $a_{n}$ , a verifiable reward $r^v$ is assigned based on whether the response matches the ground-truth answer. Verifiable rewards provide a straightforward, ground-truth signal for RL training, which has been commonly used in the training pipeline for advanced models (Yue et al., 2024; Guo et al., 2025; Team et al., 2025). In contrast, process rewards $\{r_1^p,\dots ,r_n^p\}$ are dense rewards to evaluate the quality of each step. Process rewards offer unique benefits in test-time scaling. During inference, LLM generates multiple responses for a single prompt, and PRM scores each step for each response. Prior work (Lightman et al., 2023; Wang et al., 2023; Zhang et al., 2025) typically aggregates process rewards into a outcome-level score using the minimum value, $\min (r_1^p,\dots ,r_n^p)$ , and selects the best response based on this outcome-level score. This PRM-based approach outperforms majority voting and other reward model-based approaches. + +Although dense rewards support effective training in traditional RL applications, they struggle to effectively fine-tune LLM for advanced reasoning Guo et al. (2025). Therefore, how to make PRMs as effective in training as they are in test-time scaling is an important and not widely studied topic. + +# 3 PURE: Process Supervised Reinforcement Learning + +To effectively fine-tune LLMs using PRMs, we propose PURE: Process sUpervised Reinforcement lEarning. PURE leverages a novel min-form formulation to quantify the credit assignment, inspired by the test-time application of PRM. In this section, we first detail the min-form credit assignment and analyze its effectiveness in $\S 3.1$ . Subsequently, we introduce an advantage estimator tailored for process rewards in $\S 3.4$ . + +# 3.1 Min-form Credit Assignment + +Consider a prompt $p$ and a LLM $\pi$ parameterized by $\theta$ , generating a $n$ -step response denoted as $(a_{1},\dots ,a_{n})$ . At step $t$ , LLM $\pi_{\theta}$ takes the prompt $p$ and previous steps $\{a_1,\dots ,a_{t - 1}\}$ as the input state $s_t$ , where $s_t = (p,a_1,\dots ,a_{t - 1})$ . Then LLM generates step $a_{t}$ , sampled as $a_{t}\sim \pi_{\theta}(\cdot |s_{t})$ . A PRM $R^p$ emits a process reward $r_t^p = R^p (s_t,a_t)$ . Unlike traditional RL whose objective is the summation of discounted rewards, the usage of PRM in test-time scaling suggests that RL for reasoning tasks should optimize the following objective: + +$$ +\arg \max _ {\theta} \mathbb {E} \left[ \min \left(r _ {1} ^ {p}, \dots , r _ {n} ^ {p}\right) \right] \tag {2} +$$ + +where the expectation is over prompts and step-level actions. This objective implies: (i) Only the "worst" step that gets the minimum process reward determines the value of the entire response. (ii) For steps before the worst step, their existence as the input state induces LLM to generate the worst step. (iii) Steps after the worst step actually do not contribute to the objective. + +![](images/eb5badc108fcc5ba60ab82db818afb8daf98c03424b03c36ae2c5578fc21bb08.jpg) +Figure 1: Comparison of summation-form and min-form credit assignment. Adv. and Process reward* in the table means advantage and transformed process reward, respectively. The incorrect steps in the rollout are highlighted in red, and our PRM reasonably assigns negative scores to these steps. For simplicity, advantage baseline and KL penalty terms are omitted in advantage calculation here, and discount factor $\gamma$ and transform temperature $T$ are set to 1. Arrows indicate changes in sampling probability, with larger changes marked by contoured arrows. + +Therefore, we define the min-form credit assignment function as follows: For a $n$ -step response, let step $w$ be the worst step, i.e., $w = \arg \min (r_1^p,\dots ,r_n^p)$ . The return $G$ and state-action value $Q^{\pi}$ functions are defined as follows: + +$$ +G \left(s _ {t}, a _ {t} \mid \tau\right) = \left\{ \begin{array}{l l} \min \left(r _ {t} ^ {p}, \dots , r _ {n} ^ {p}\right), & \text {i f} t \leq w \\ 0, & \text {i f} t > w \end{array} \right. \tag {3} +$$ + +$$ +Q ^ {\pi} \left(s _ {t}, a _ {t}\right) = \underset {\tau} {\mathbb {E}} [ G \left(s _ {t}, a _ {t} \mid \tau\right) ] \tag {4} +$$ + +where $\tau = (s_1, a_1, r_1^p, \dots, s_n, a_n, r_n^p)$ denotes the trajectory. Eq. (4) is a quantitative expression of the above three-fold analysis of the objective. To implement this in the simplest way possible, we transform the process rewards in trajectory $\tau$ using: + +$$ +r _ {i} ^ {p *} = \frac {\exp \left(- r _ {i} ^ {p} / T\right)}{\sum_ {j = 1} ^ {n} \exp \left(- r _ {j} ^ {p} / T\right)} \cdot r _ {i} ^ {p} \tag {5} +$$ + +where $T$ is the transform temperature, and $r_i^{p*}$ is the transformed process reward. The transform function assigns higher weights to lower rewards. After transformation, the rest of code remains unchanged. We explain this as follows. As the transform temperature $T \to 0^+$ , Eq. (5) yields $r_w^{p*} = r_w^p$ for the worst step $w$ and $r_i^{p*} = 0$ for $i \neq w$ . Setting $\gamma = 1$ , the return becomes: (i) For step $t \leq w$ , + +$$ +G _ {t} = \sum_ {i = t} ^ {n} \gamma^ {i - t} \cdot r _ {i} ^ {p *} = r _ {w} ^ {p} = \min \left(r _ {t} ^ {p}, \dots , r _ {n} ^ {p}\right) \tag {6} +$$ + +(ii) For $t > w$ , $G_{t} = 0$ . The return exactly matches Eq. (3) without altering the return computation logic. + +Simple implementation. To align with widely adopted token-level PPO loss, we convert step-level rewards to token-level rewards. We exclusively assign the transformed process reward to the final token of each step, with all other tokens receiving a reward of 0. To preserve algorithmic flexibility, we also support verifiable rewards assigned to the last token of the complete response, which can be used alongside process rewards. + +# 3.2 Quantitative Analysis + +Here we analyze that the min-form has a smaller value estimation error than the summation-form credit assignment method. We start by first establishing 2 reasonable assumptions: + +Assumption 1 (Bounded Process Reward Error): The Process Reward Model, $R^P$ , produces a reward estimate $r_t^p = R^P(\cdot)$ for each reasoning step. It is assumed that there exists a "true" or optimal reward $r_t^*$ for that step, and the estimation error is uniformly bounded by a constant $\epsilon \geq 0$ . For any step $t$ , it holds that $|r_t^p - r_t^*| \leq \epsilon$ . + +Assumption 2 (Bounded Rewards): The true rewards, and consequently the estimated rewards, are bounded. There exists a maximum possible reward $R_{\mathrm{max}}$ such that $|r_t^*| \leq R_{\mathrm{max}}$ and $|r_t^p| \leq R_{\mathrm{max}} + \epsilon$ for all steps $t$ . + +Theorem 1 (Q-Value Estimation Error Bound Comparison). Under Assumptions 1 and 2, for any state-action pair $(s_t, a_t)$ and a trajectory $\tau$ with $n$ reasoning steps: + +1. Sum-form Error Bound: The estimation error for the summation-form $Q$ -value, is bounded by: + +$$ +\left| Q _ {\pi} ^ {s u m} \left(s _ {t}, a _ {t}\right) - Q _ {\pi} ^ {s u m, *} \left(s _ {t}, a _ {t}\right) \right| \leq \sum_ {i = t} ^ {t + n - 1} \gamma^ {i - t} \epsilon +$$ + +For an infinite horizon $(n\to \infty)$ , this bound becomes: + +$$ +| Q _ {\pi} ^ {s u m} (s _ {t}, a _ {t}) - Q _ {\pi} ^ {s u m, *} (s _ {t}, a _ {t}) | \leq \frac {\epsilon}{1 - \gamma} +$$ + +2. Min-form Error Bound: The estimation error for the min-form $Q$ -value, derived from Eq. (4), is bounded by: + +$$ +\left| Q _ {\pi} ^ {m a n} \left(s _ {t}, a _ {t}\right) - Q _ {\pi} ^ {m a n, *} \left(s _ {t}, a _ {t}\right) \right| \leq \epsilon +$$ + +The proof is provided in Appendix B. This theorem shows that the error bound of sum-form method is amplified by a factor dependent on the horizon length and the discount factor $\gamma$ , accumulating the error over steps. However, the error bound of min-form method is strictly limited by the single-step reward estimation error $\epsilon$ and does not accumulate with the number of steps. + +# 3.3 Qualitative Analysis + +Min-form credit assignment restricts the range of value function to be the same as that of the reward function, which contributes to stabilizing RL training. In contrast, for the summation-form credit assignment, the range of value is determined by the range of reward and the number of steps, causing excessive value as the number of steps increases and unintended reward hacking. + +We now visualize the differences between summation-form and our min-form credit assignment using a real training example. As shown in Figure 1, incorrect steps are highlighted in red, with arrows showing the magnitude and direction of changes in sampling probability. Steps with relatively large changes in sampling probability are marked by contoured arrows. Note that the first 3 steps are the thinking, and the subsequent 6 steps are the solution. + +Figure 1 shows that summation-form credit assignment significantly alters sampling probabilities, increasing the sampling probabilities for thinking steps and decreasing that for incorrect solution steps. However, this results in a shortcut towards reward hacking: the model learns to only output thinking steps without actually solving the problem! In other words, the model hacks the implicit pattern inside high-reward steps, i.e., thinking. In contrast, our min-form credit assignment reduces sampling probabilities across all steps, with the largest reduction at the first incorrect step. This aligns with the behavior of verifiable rewards (an incorrect final answer leads to a reduction in sampling probability of all steps) and assigns advantages more rationally based on step correctness. + +# 3.4 Advantage Estimation + +Following the token-level rewards in § 3.1, we are ready to compute advantages. We compare several advantage estimators, including GAE (Schulman et al., 2017), RLOO (Ahmadian et al., 2024), GRPO (Shao et al., 2024), and REINFORCE++ (Hu, 2025), as discussed in § F.2. RLOO are strong enough to produce stable and effective results. + +For a single prompt, LLM generates $K$ responses. The maximum generation length is limited to $N$ . Let $r_i^v$ and $r_{i,j}^{p*}$ $(j = 1,\dots ,N)$ denote the verifiable reward and token-level transformed process rewards for response $y_{i}$ $(i = 1,\dots ,K)$ , respectively. The token-level advantage for $y_{i}$ is formulated as follows: + +$$ +A _ {i, t} = \underbrace {r _ {i} ^ {v} - \frac {1}{K - 1} \sum_ {k \neq i} r _ {k} ^ {v}} _ {\text {R L O O w i t h v e r i f a b l e r e w a r d}} + \underbrace {\sum_ {j = t} ^ {N} \gamma^ {j - t} \cdot r _ {i , j} ^ {p *}} _ {\text {R L O O w i t h t o k e n - l e v e l t a b e s i l e}} - \underbrace {\frac {\sum_ {k \neq i} \sum_ {l = 1} ^ {N} \sum_ {j = l} ^ {N} \gamma^ {j - l} \cdot r _ {k , j} ^ {p *}}{(K - 1) N}} _ {\text {t o k e n - l e v e l b a s i l e}} \tag {7} +$$ + +where $t = 1, \dots, N$ . Specifically: (i) for verifiable rewards, we directly adopt RLOO. (ii) For process rewards, we employ a token-level baseline to avoid reward hacking, as discussed in the second case in § 5.1. Moreover, we normalize the baseline with the max generation length $N$ instead of response length to avoid length biases, similar to concurrent work (Liu et al., 2025). + +# 4 Experiments + +# 4.1 PRM Training and Evaluation + +To conduct our RFT experiments, we first require a PRM to provide process rewards. We train our PRM based on Qwen2.5-Math-7B due to its strong performance in mathematical tasks. Following Lightman et al. (2023), we treat the PRM training as a binary classification task. We replace the final layer of the model with a value head and train the model on the PRM800K dataset (Lightman et al., 2023) in 2 stages. In the first stage, we freeze the LLM parameters and train only the value head with a learning rate $10^{-4}$ for 3 epochs. In the second stage, we unfreeze the LLM parameters and fine-tune all parameters with a learning rate $10^{-6}$ for 1 epoch. + +We evaluate our PRM, named PURE-PRM7B, through 3 ways: Best-of-N (BoN) method, ProcessBench (Zheng et al., 2024), and PRMBench (Song et al., 2025). For BoN evaluation, we use rollout data from Xiong et al. (2024). For each question in GSM8K (Cobbe et al., 2021) and MATH (Hendrycks et al., 2021), they uses Deepseek-7B to generate $K = 1024$ answers. We score each step using our PRM, transform process + +Table 1: Results of BoN evaluation. Rows marked with * are taken from Xiong et al. (2024). + +
MethodGSM8KMATH
Pass@1 *83.942.4
Majority Voting@1024 *89.757.4
Deepseek-PRM-7B BoN@1024 *93.058.1
PURE-PRM-7B BoN@102491.662.6
+ +rewards using Eq. (5), and compute an outcome score by summing the transformed rewards for each answer. The answer with the highest outcome score is selected as the final answer. Our PURE-PRM-7B achieves BoN@1024 accuracy of $91.6\%$ on GSM8K and $62.6\%$ on MATH, performing comparably to the best results of $93.0\%$ and $58.1\%$ reported by Xiong et al. (2024) + +On ProcessBench (Zheng et al., 2024), which assesses the PRM's ability to identify the first process error, our PRM achieves a state-of-the-art average F1 score of 57.5, surpassing the previous best F1 score of 56.5 reported in the benchmark. Detailed scores for each subset of ProcessBench are provided in Appendix C. On PRMBench (Song et al., 2025), which evaluates the fine-grained error detection capabilities of PRMs, our PURE-PRM-7B ranks third among open-source PRMs $^2$ . These results demonstrate that our PRM achieves top performance and is suitable for fine-tuning LLMs. + +# 4.2 RL Settings + +Reward types. Our framework supports 3 types of rewards: process reward only (PURE-PRM), verifiable reward only (PURE-VR), which matches the training setup of Deepseek R1-Zero, and a mix + +Table 2: Detailed performance of various models across 5 benchmarks. Report pass@1 accuracy tested with greedy decoding. The blue lines represent the models trained with our recipe. + +
Base ModelMethodMATH 500Minerva MathOlympiad BenchAIME24AMC23Avg.
Qwen2.5-7BSimpleRL-Zoo78.431.239.116.750.043.1
Base71.423.935.310.052.538.6
+ PURE-PRM76.237.141.213.360.045.6
+PURE-VR75.632.739.016.755.043.8
+ PURE-PRM+VR80.437.943.016.760.047.6
Qwen2.5-Math-7BEurus-2-7B-PRIME79.238.642.126.757.848.9
SimpleRL-Zoo80.238.243.323.355.048.0
Qwen2.5-7B-DPO-Zero76.830.937.926.762.547.0
Base71.829.835.113.347.539.5
+ PURE-PRM81.838.244.716.760.049.3
+PURE-VR79.436.841.823.360.048.3
+PURE-PRM+VR82.637.144.120.082.553.3
Qwen2.5-Math-1.5BBase61.423.529.313.340.033.5
+ PURE-PRM75.226.536.413.350.040.3
+PURE-VR74.227.936.010.055.040.6
+PURE-PRM+VR76.031.637.216.755.043.3
+ +of both (PURE-PRM+VR). For verifiable rewards, a reward of $+1$ is assigned if the generated answer matches the ground-truth answer; otherwise, a reward of 0 is given. No format-related rewards. + +RL dataset. We use the RFT dataset from SimpleRL (Zeng et al., 2025b). It samples around 8,000 problems from MATH dataset (Hendrycks et al., 2021) with difficulty lv.3-5. For ground-truth (GT) answers, PURE-PRM method does not use GT answers. PURE-VR method requires GT answers for all problems. For PURE-PRM+VR, GT answers are randomly assigned to 1/10 of the problems, leading to about 800 problems with GT answers and 7,200 open problems. This setup aims to explore the effectiveness of process rewards as the main signal for RL training. + +Hyperparameters. We use veRL (Sheng et al., 2024) to conduct experiments on 3 models: Qwen2.5-7B (Yang et al., 2024a), Qwen2.5-Math-7B, Qwen2.5-Math-1.5B (Yang et al., 2024b). We use a constant learning rate of $10^{-6}$ for PURE-VR and PURE-PRM+VR and $5 \times 10^{-7}$ for PURE-PRM. Training steps are set to 500 for Qwen2.5-Math series and 1000 for Qwen2.5-7B. Other shared hyperparameters are set as follows: prompt batch size of 64, group size of 8 (generating 8 responses per prompt), training mini-batch size of 512, maximum generation length of 8192 tokens, sampling temperature of 1.0, KL coefficient of $10^{-3}$ , transform temperature in Eq (5) of 0.1, save interval for checkpoints of 50 steps. + +Baselines. We compare our method with 3 state-of-the-art RFT methods: (i) Eurus-2-7B-PRIME (Cui et al., 2025): A model based on Qwen2.5-Math-7B fine-tuned with implicit PRM. (ii) SimpleRL-Zoo (Zeng et al., 2025b): A re-implementation of Deepseek R1's training recipe on several base models. (iii) Qwen2.5-7B-DPO-Zero (Xiong et al., 2025): A model based on Qwen2.5-Math-7B fine-tuned with iterative DPO. + +Evaluation metrics. At test time, we evaluate performance on 5 competition-level mathematical benchmarks, including AIME24 (Li et al., 2024), AMC23 (Li et al., 2024), MATH500 (Hendrycks et al., 2021), Minerva Math (Lewkowycz et al., 2022), OlympiadBench (He et al., 2024). We report scores of best checkpoint saved in training. During training, we track about 20 metrics, including accuracy, reward, response length, repetition score, etc. Details are provided in Appendix D. + +# 4.3 Main Results + +We report the pass@1 accuracy across 5 benchmarks in Table 2. The results indicate that all variants of our methods perform at least comparable to, or better than baselines. For example, when using only verifiable rewards, following the setup of SimpleRL-Zoo, our method PURE-VR achieves average scores of 43.8 and 48.3 on Qwen2.5-7B and Qwen2.5-Math-7B, respectively. These scores are comparable to the baseline results of 43.1 and 48.0 obtained by SimpleRL-Zoo, confirming the reliability of our code-base. Based on Table 2, we draw the following observation: + +PRM-based approach performs similar to the VR-based approach, but combining the two yields better results. On the Qwen2.5-Math-7B model, PURE-PRM achieves an average score of + +49.3, slightly higher than the 48.3 of PURE-VR. However, the combined method, PURE-PRM+VR, reaches a score of 53.3, surpassing PURE-VR by approximately 5 percentage points. This trend remains consistent across the other 2 base models. + +Next, we analyze the training dynamics of our methods to understand the limitations of the PRM-based approach and explain why PURE-PRM+VR is superior. Figure 2 illustrates the training curves for five variants of our methods on Qwen2.5-Math series, leading to the following findings: + +![](images/edb647531bbf3f1de18a66e393e67575c5716a83399827a6f8c64a574729e241.jpg) +(a) PURE on Qwen2.5-Math-7B + +![](images/6846855ba0df93a346ac3be1de4b16b98643c0650968a498640dcf0a06e720b9.jpg) +Figure 2: Training curves for different variants of our methods on Qwen2.5-Math series. Curves of PURE-PRM (sum-form) and PURE-PRM+VR (sum-form) are truncated due to training collapse. Process-aggregated outcome reward is the summation of final process rewards for one response: for sum-form, it sums PRM-emitted rewards; for min-form, it sums transformed rewards, approximating the minimum PRM-emitted reward. Thus values across the 2 credit assignment methods are not comparable. For PURE-PRM, verifiable reward is logged but unused in training. + +![](images/a613198539bb41a88370ba10620a46b0fcc59350c0903195f5b18b6548126a59.jpg) +(b) Qwen2.5-Math-1.5B + +Summation-form credit assignment collapses training even at the beginning, while the min-form method significantly enhances training stability. As shown in Figure 2a, both PURE-PRM (sum-form) and PURE-PRM+VR (sum-form) experience collapse at step 25. At step 80, their average benchmark scores drop to around 30, which is much lower than the base model's score of 39.5. In contrast, the min-form methods remain stable over 200 steps, achieving average scores of 49.3 and 53.3 for PURE-PRM (min-form) and PURE-PRM+VR (min-form), respectively. + +Dense rewards substantially improve learning efficiency compared to sparse rewards. Figure 2b shows the curves of accuracy on MATH-500 for 3 variants of our methods. We find that PRM-involved approaches take around $30\%$ of the training steps to achieve the same accuracy as PURE-VR. + +Reward hacking is inevitable when rely solely on PRM, though it can be delayed through algorithmic adjustments. While PURE-PRM (min-form) produces a well-tuned model, reward hacking still occurs at step 270, where verifiable rewards decrease sharply while process-aggregated outcome rewards increase in Figure 2a. This issue was observed in all experiments using the PURE-PRM method, indicating that reward hacking is a persistent challenge when relying solely on PRM and limits further progress. However, compared to sum-form credit assignment, min-form approaches delays the onset of reward hacking significantly. Such reward hacking corresponds to the third case in § 5.1, which is inevitable due to the current architecture of PRMs. + +Incorporating a few ground-truth signals can effectively reduce reward hacking. PURE-PRM+VR (min-form) uses a mixed dataset of 800 problems with GT answers and 7,200 open problems. Thus, during training, process rewards guide the RFT process primarily, while verifiable rewards serve as an auxiliary signal. As shown in Figure 2a, this approach achieves a stable training process similar to PURE-VR and attains the highest average benchmark score of 53.3. This suggests a practical solution to address reward hacking by including a small proportion (around $10\%$ ) of ground-truth signals during the RFT stage. + +# 5 Analysis + +# 5.1 Reward Hacking induced by PRM + +In this section, we detail the reward hacking cases observed during training. We categorize these cases into the 3 types. Specific examples for each type are provided in Appendix E. + +Case 1: Only thinking, not solving. This behavior occurs when steps with certain patterns, such as thinking, are rewarded significantly more than others, and the inappropriate credit assignment further widen the gap in advantages between steps. As a result, the model learns to exploits these patterns to achieve higher scores. As shown in Figure 1, the sum-form credit assignment increases the likelihood of generating thinking steps while reducing that of incorrect solution steps, resulting in an armchair general. However, exploiting specific patterns is not always negative. For example, using patterns like backtracking and verification improves reasoning skills (Guo et al., 2025; Gandhi et al., 2025). + +Case 2: Extremely few steps (1 step). This happens when an unsuitable advantage baseline is used. In § 3.4, we employ a token-level baseline for process rewards. An alternative approach is a step-level baseline, formulated as follows: + +$$ +A _ {i, t} = r _ {i} ^ {v} - \frac {1}{K - 1} \sum_ {k \neq i} r _ {k} ^ {v} + \sum_ {j = t} ^ {N} \gamma^ {j - t} \left[ r _ {i, j} ^ {p *} - \frac {1}{K - 1} \sum_ {k \neq i} \frac {\sum_ {l} r _ {k , l} ^ {p *}}{\left| y _ {k} \right| _ {s}} \right] \tag {8} +$$ + +where $|y_{k}|_{s}$ represents the number of steps of the $k$ -th response within the group. The advantage baseline for process rewards in Eq. (8) means the average process reward of other responses in the group. However, we find the step-level baseline is biased against the number of steps. When the process-aggregated outcome reward are equal, response with more steps are penalized more heavily by the baseline, causing the model to favor responses with fewer steps. Eventually, the model learns to output only a single step with an excessively large number of tokens. This undermines the purpose of PRM, which is to evaluate the process step by step. + +Case 3: Extremely few steps (0 step). In this case, the model learns to output irrelevant responses such as "Thank you," "Happy Birthday," or even empty responses. Since PRM infers in a causal manner, it assigns high rewards to these meaningless steps, not realizing that no further content follows. This issue stems from the current architecture of discriminative PRMs (i.e., the causal attention mask). Our solution is adding GT-level signal as an aid, such as verifiable rewards in PURE-PRM+VR. Another potential solution is the generative PRMs, which we leave for future work. + +![](images/992deb91fc7e74ee930c8af989e7ffcc20268a8eb5647ff67316b8b4fe6c1187.jpg) + +![](images/109a2ec811b9dcf80db4695eac6f82b4bc4bb6f46e6771c77588668ce9680cbc.jpg) + +![](images/429395d8d109a0ec3f9c3984e5b0ff84a52d020d59062db3d6009d1160c9cc6f.jpg) + +![](images/f640a0fc85049cf5ce3dd7926079a99306c6a4edbb9172e55f5f245d14a2fca9.jpg) +Figure 3: Training curves for PURE-PRM+VR with doubled process rewards based on Qwen2.5-7B. The correctness of responses are judged by the verifier. Process-aggregated outcome reward (bottom-left) is the summation of transformed process rewards for one response. No smooth is applied. Training collapses at step 365, showing a sharp drop of rewards and accuracy. + +![](images/a44b358dc9a1c60a964bc80b0d982de20873cd61dd0d2b6c518f59a74eb2080d.jpg) + +![](images/78735573b090b7c7c74187c45d71c4e1528d74238a715c6e977cb81b4eeb051f.jpg) + +# 5.2 Causes of Training Collapse + +In this section, we train Qwen2.5-7B using PURE-PRM+VR with doubled process rewards, and show the training curves of pattern-related metrics in Figure 3, including response length, clip ratio, repetition score, rewards, and accuracy on MATH-500. The clip ratio measures how often responses are cut off due to the maximum generation length limit. The repetition score evaluates how repetitive a + +response is by calculating the longest common prefix (LCP) lengths between all pairs of its suffixes ${}^{3}$ . We consider responses with repetition score above 0.2 as highly repetitive. Details on metrics are provided in Appendix D. + +As shown in Figure 3, training collapses at step 365, showing a sharp drop of rewards and accuracy. Before step 360, incorrect responses judged by the verifier are longer and more repetitive than the correct responses, but that does not collapse the training. Both the clip ratio and high repetition ratio for correct responses remain at 0 until step 361. After that, the response length, clip ratio, and high repetition ratio for both incorrect and correct responses rise sharply until step 380. Based on these observations, we derive the following conclusions: + +Long and highly repetitive "correct" responses cause training collapse. At step 361, the clip ratio and high repetition ratio for correct responses become greater than 0. This large number of positive signals for repetitive patterns is fatal to training. We refer to these samples as pseudo-positive samples. We attempt to treat pseudo-positive samples as incorrect samples and assign 0 rewards instead of +1 rewards, but this does not help much. The model learns to repeat content in ways that the LCP function can not detect, as detailed in Appendix D. Verifiers can not identify pseudo-positive samples. While PRM has the potential to detect them, it currently fails to do so because such patterns are not included in its training data. This highlights a need for future PRM development to not only assess the correctness of steps but also evaluate the quality of patterns. + +The model learns repetitive patterns quickly, leading to collapse within 5 gradient steps. From the unusual metrics at step 361 to the training collapse at step 365, the model rapidly learns to repeat content. Our training setup ensures one gradient step per training step, meaning the collapse happens within just 5 gradient steps. + +# 6 Conclusion + +In this paper, we present PURE: Process sUpervised Reinforcement lEarning, a framework leverages process rewards to improve the reasoning abilities of LLMs. Extensive experiments demonstrate that PURE effectively alleviates reward hacking induced by PRM, attributed to the proposed min-form credit assignment. This method allows PRM-based RFT to achieve performance similar to verifiable rewards-based approaches within $30\%$ steps, and further outperform them with the assistance of a few ground-truth signals. Additionally, we summarize the observed reward hacking cases during training, and find that pseudo-positive samples collapse training. + +There are several promising directions for further improving PRM-based RFT. First, developing generative PRMs is both urgent and crucial. As discussed in § 5, current PRMs are unable to address the third type of reward hacking mentioned in § 5.1, and also cannot evaluate the quality of patterns like endless repetition. Generative PRMs, however, could potentially resolve these issues by making better use of the strong language capabilities of LLMs. Second, iterative training between PRM and LLM is essential to ensure that the PRM continuously adapts to the output distribution of LLMs. + +# Acknowledgments + +We sincerely thank Ganqu Cui, Songjun Tu, and Zhengbo Wang for their suggestions to the early draft of the paper. This work was partially supported by the National Science and Technology Major Project (2022ZD0117102), the National Natural Science Foundation of China under Grants 62271485 and 62303462, Beijing Natural Science Foundation under grant L241016 and L233005, Chongqing Transportation Technology Project (CQJT-CZKJ2024-04), the Provincial Key Research and Development Program of Zhejiang (project number: 2022C01129), Ningbo International Science and Technology Cooperation Project (2023H020), CCF-Tencent Rhino-Bird Open Research Fund. + +# References + +Arash Ahmadian, Chris Cremer, Matthias Galle, Marzieh Fadaee, Julia Kreutzer, Olivier Pietquin, Ahmet Üstün, and Sara Hooker. Back to basics: Revisiting reinforce style optimization for learning from human feedback in llms. arXiv preprint arXiv:2402.14740, 2024. + +Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas Schneider, Rachel Fong, Peter Welinder, Bob McGrew, Josh Tobin, OpenAI Pieter Abbeel, and Wojciech Zaremba. 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However, previous research has faced difficulties in effectively utilizing PRM (Guo et al., 2025) in training-time, which is the primary focus of our work in PURE. + +# A.2 Reward Models for LLM Alignment + +In the area of LLM alignment, outcome reward models (ORMs) are used to evaluate the quality of the entire response generated by LLMs. Employing ORMs for aligning LLMs with human values has become a common practice in the post-training stage (Ouyang et al., 2022; Bai et al., 2023; Touvron et al., 2023). However, ORMs fall short in providing detailed feedback for complex tasks that involve multiple reasoning steps. In contrast, process reward models (PRMs) provide more detailed feedback by evaluating each step of the reasoning process, allowing LLMs to learn more effectively from mistakes made during reasoning (Lightman et al., 2023; Yuan et al., 2024; Cui et al., 2025). Studies have shown that PRMs outperform majority voting and ORMs in test-time scaling (Lightman et al., 2023; Wang et al., 2023; Xiong et al., 2024). Nevertheless, the application of PRMs during the training phase remains largely unexplored, which is the central topic of PURE. + +# B Proof of Q-value estimation error + +Theorem 2 (Q-Value Estimation Error Bound Comparison). Under Assumptions 1 and 2, for any state-action pair $(s_t, a_t)$ and a trajectory $\tau$ with $n$ reasoning steps: + +1. Sum-form Error Bound: The estimation error for the summation-form $Q$ -value, is bounded by: + +$$ +\left| Q _ {\pi} ^ {s u m} \left(s _ {t}, a _ {t}\right) - Q _ {\pi} ^ {s u m, *} \left(s _ {t}, a _ {t}\right) \right| \leq \sum_ {i = t} ^ {t + n - 1} \gamma^ {i - t} \epsilon +$$ + +For an infinite horizon $(n\to \infty)$ , this bound becomes: + +$$ +\left| Q _ {\pi} ^ {s u m} \left(s _ {t}, a _ {t}\right) - Q _ {\pi} ^ {s u m, *} \left(s _ {t}, a _ {t}\right) \right| \leq \frac {\epsilon}{1 - \gamma} +$$ + +2. Min-form Error Bound: The estimation error for the min-form $Q$ -value, derived from Eq. (4), is bounded by: + +$$ +\left| Q _ {\pi} ^ {m i n} \left(s _ {t}, a _ {t}\right) - Q _ {\pi} ^ {m i n, *} \left(s _ {t}, a _ {t}\right) \right| \leq \epsilon +$$ + +# Proof. Part 1: Proof for the Sum-form Error Bound. + +We begin with the definition of the absolute error of the Q-value, using the canonical formulation in Eq. (1). $Q_{\pi}^{sum}$ denotes the value calculated with the estimated rewards from the PRM $(r^p)$ , and $Q_{\pi}^{sum,*}$ denotes the true value (calculated with $r^*$ ). + +$$ +\begin{array}{l} | Q _ {\pi} ^ {s u m} (s _ {t}, a _ {t}) - Q _ {\pi} ^ {s u m, *} (s _ {t}, a _ {t}) | = | E _ {\pi \sim \tau} [ \sum_ {i = t} ^ {t + n - 1} \gamma^ {i - t} r _ {i} ^ {p} ] - E _ {\pi \sim \tau} [ \sum_ {i = t} ^ {t + n - 1} \gamma^ {i - t} r _ {i} ^ {*} ] | \\ = | E _ {\pi \sim \tau} [ \sum_ {i = t} ^ {t + n - 1} \gamma^ {i - t} (r _ {i} ^ {p} - r _ {i} ^ {*}) ] | \\ \leq E _ {\pi \sim \tau} [ | \sum_ {i = t} ^ {t + n - 1} \gamma^ {i - t} \left(r _ {i} ^ {p} - r _ {i} ^ {*}\right) | ] \\ \leq E _ {\pi \sim \tau} [ \sum_ {i = t} ^ {t + n - 1} | \gamma^ {i - t} \left(r _ {i} ^ {p} - r _ {i} ^ {*}\right) | ] \\ = E _ {\pi \sim \tau} [ \sum_ {i = t} ^ {t + n - 1} \gamma^ {i - t} | r _ {i} ^ {p} - r _ {i} ^ {*} | ] \\ \end{array} +$$ + +Now, we apply Assumption 1, which states that the reward error at each step is bounded by $\epsilon$ , i.e., $|r_t^p - r_t^*| \leq \epsilon$ . + +$$ +\left| Q _ {\pi} ^ {s u m} \left(s _ {t}, a _ {t}\right) - Q _ {\pi} ^ {s u m, *} \left(s _ {t}, a _ {t}\right) \right| \leq E _ {\pi \sim \tau} \left[ \sum_ {i = t} ^ {t + n - 1} \gamma^ {i - t} \epsilon \right] = \epsilon \sum_ {i = t} ^ {t + n - 1} \gamma^ {i - t} +$$ + +For the infinite horizon case, the geometric series converges to $\frac{1}{1 - \gamma}$ . + +# Part 2: Proof for the Min-form Error Bound. + +We start similarly with the definition of the absolute Q-value error for the min-form. The return $G(s_{t},a_{t}|\tau)$ is the value being expected. + +$$ +\begin{array}{l} \left. \right.\left| Q _ {\pi} ^ {m i n} \left(s _ {t}, a _ {t}\right) - Q _ {\pi} ^ {m i n, *} \left(s _ {t}, a _ {t}\right)\right| = \left| E _ {\pi \sim \tau} \left[ G \left(s _ {t}, a _ {t} \mid \tau\right)\right] - E _ {\pi \sim \tau} \left[ G ^ {*} \left(s _ {t}, a _ {t} \mid \tau\right)\right]\right| \\ = \left| E _ {\pi \sim \tau} \left[ G \left(s _ {t}, a _ {t} \mid \tau\right) - G ^ {*} \left(s _ {t}, a _ {t} \mid \tau\right) \right] \right| \\ \end{array} +$$ + +Now we analyze the return term inside the expectation. According to Equation (3), for a trajectory $\tau$ and a step $t$ , the return is defined as the minimum of future rewards. + +$$ +G \left(s _ {t}, a _ {t} | \tau\right) = \min \left(r _ {t} ^ {p}, r _ {t + 1} ^ {p}, \dots , r _ {n} ^ {p}\right) +$$ + +$$ +G ^ {*} (s _ {t}, a _ {t} | \tau) = \min (r _ {t} ^ {*}, r _ {t + 1} ^ {*}, \dots , r _ {n} ^ {*}) +$$ + +We use a key property of the minimum function: $|\min(a) - \min(b)| \leq \max_i |a_i - b_i|$ . Applying this property: + +$$ +| G (s _ {t}, a _ {t} | \tau) - G ^ {*} (s _ {t}, a _ {t} | \tau) | = | \min _ {i = t \dots n} (r _ {i} ^ {p}) - \min _ {i = t \dots n} (r _ {i} ^ {*}) | \leq \max _ {i = t \dots n} | r _ {i} ^ {p} - r _ {i} ^ {*} | \leq \epsilon +$$ + +Substitute this result back into the expectation inequality: + +$$ +\left| Q _ {\pi} ^ {m i n} \left(s _ {t}, a _ {t}\right) - Q _ {\pi} ^ {m i n, *} \left(s _ {t}, a _ {t}\right) \right| = \left| E _ {\pi \sim \tau} \left[ G \left(s _ {t}, a _ {t} \mid \tau\right) - G ^ {*} \left(s _ {t}, a _ {t} \mid \tau\right) \right] \right| \leq E _ {\pi \sim \tau} [ \epsilon ] = \epsilon +$$ + +This completes the proof. It is proven that the error in the min-form does not accumulate and is directly bounded by the maximum single-step error of the reward model. + +# C Benchmark Scores of Our PRM + +We report the detailed scores on ProcessBench (Zheng et al., 2024) and PRMBench (Song et al., 2025) in Table 3 and Table 4, respectively. + +Table 3: F1 scores of each subset in ProcessBench. The blue line represents our PRM. Other lines are taken from Zheng et al. (2024). + +
ModelGSM8KMATHOlympiad-BenchOmni-MATHAverage
Math-Shepherd-PRM-7B47.929.524.823.831.5
RLHFlow-PRM-Deepseek-8B38.833.816.916.926.6
Skywork-PRM-7B70.853.622.921.042.1
Qwen2.5-Math-7B-PRM800K68.262.650.744.356.5
PURE-PRM-7B69.066.548.445.957.5
+ +Table 4: Results on ProcessBench. The blue line represents our PRM. Other lines are taken from Song et al. (2025). + +
ModelS1 (Simplicity)S2 (Soundness)S3 (Sensitivity)Overall
MathShepherd-Mistral-7B47.145.760.747.0
RLHFlow-PRM-Deepseek-8B47.657.568.154.2
Skywork-PRM-7B59.668.573.365.1
Qwen2.5-Math-7B-PRM800K52.171.075.565.5
PURE-PRM-7B52.270.275.865.3
+ +# D Details of Training Metrics + +We detail 4 metrics used in the main text of the paper, including process-aggregated outcome reward, clip ratio, repetition score, and high repetition ratio. + +Process-aggregated outcome reward. This metric reflects the value of a response with respect to process rewards. In practice, it is calculated as the sum of transformed process rewards for the min-form credit assignment, approximating the minimum PRM-emitted reward for a given response. + +Clip ratio. The clip ratio indicates the proportion of samples in the replay buffer that are truncated due to the maximum generation length limit. After training collapses, LLMs often generate infinitely repetitive responses until hit the maximum context length. + +Repetition score. The repetition score measures the level of repetition in a response by calculating the longest common prefix (LCP) lengths between all pairs of its suffixes. However, this method can only identify basic repetition patterns. A more complex form of repetition involves the use of near-synonyms, where words differ but the meaning remains the same. Such cases are not detected by the LCP method. This limitation explains why treating pseudo-positive samples as incorrect still results in training collapse, as mentioned in § 5.2. The model learns to exploit complex repetition patterns to bypass the LCP function. + +High repetition ratio. We consider responses with a repetition score greater than 0.2 as highly repetitive. This metric shows the percentage of samples in the replay buffer that fall into this category. + +# E Examples of Reward Hacking + +We show examples for each reward hacking case in Figure 4, 5, and 6. In practice, we use double line breaks (\\"\n\n") to separate individual steps within the response. + +$$ +\begin{array}{c} \text {Q u e s t i o n} \\ \text {C o m p u t e} \frac {1}{\cos^ {2} 1 0 ^ {\circ}} + \frac {1}{\sin^ {2} 2 0 ^ {\circ}} + \frac {1}{\sin^ {2} 4 0 ^ {\circ}} \end{array} +$$ + +Response + +Let's break down the problem step by step: + +# Step 1: Express the Given Expression Using Trigonometric Identities + +Rewrite each term of the expression using identities like $\frac{1}{\cos^2 x} = \sec^2 x$ and $\frac{1}{\sin^2 x} = \csc^2 x$ , and further express them in terms of $\tan^2 x$ and $\cot^2 x$ to simplify the structure. + +# Step 2: Simplify the Expression by Combining Constant Terms + +Combine any constant terms resulting from the identities used in the previous step to reduce the expression to a more manageable form involving trigonometric functions. + +# Step 3: Convert Cotangent Terms to Tangent Terms + +Use the relationship $\cot x = \frac{1}{\tan x}$ to express cotangent squares in terms of tangent squares, aiming for consistency in the trigonometric functions used. + +# Step 4: Explore Angle Relationships Using Multiple-Angle Identities + +Consider relevant multiple-angle identities (such as for $\tan 3x$ ) to establish equations or relationships between the angles involved, potentially leading to a simplification of the expression. + +# Step 5: Relate Cotangent to Tangent Using Complementary Angle Identities + +Use the identity $\cot x = \tan (90^{\circ} - x)$ to rewrite cotangent terms as tangent terms of complementary angles, facilitating a possible pairing or symmetry in the expression. + +# Step 6: Apply Known Trigonometric Sum Identities or Results + +Identify and apply any known results or specific identities related to the sum of squares of tangents and cotangents for the given angle configuration to evaluate the expression directly. + +# Step 7: Combine Results to Compute the Final Value + +Add the results from the simplified trigonometric sum to the constant terms obtained earlier to arrive at the final numerical value of the expression. + +Figure 4: Reward hacking, case 1: only thinking, not solving. In this example, the LLM analyzes the problem and gives a few equations for trigonometric simplifications, but does not substitute actual numbers to calculate and solve the problem. This is because the LLM hacks the implicit pattern inside high-reward steps, i.e., thinking. + +# F Additional Experiments + +# F.1 Ablation on Transform Temperature + +In this section, we ablate the transform temperature defined in Eq. (5), which controls the level of approximation to the min-form credit assignment. We choose Qwen2.5-7B as the base model and run PURE-PRM+VR with 3 values of transform temperature individually for 230 steps. The results are shown in Table 5 and Figure 7, which indicates that 0.1 is the best for the transform temperature. + +# Question + +# In $\Delta PQR$ , we have $PQ = QR = 34$ and $PR = 32$ . Find the length of median $\overline{QM}$ . + +# Response + +Let's first identify the type of triangle and use the properties of the median in an isosceles triangle. + +Given that $PQ = QR = 34$ , $\Delta PQR$ is isosceles with $PQ = QR$ . In an isosceles triangle, the median to the base (the side opposite the vertex angle) also acts as the altitude and the angle bisector of the vertex angle. So, the median $\overline{QM}$ from vertex $Q$ to the midpoint $M$ of the base $PR$ will split the isosceles triangle into two congruent right triangles. + +Let's place the triangle in the coordinate plane to make calculations easier. Assume $P = (0, b)$ , $Q = (0, 0)$ , and $R = (32, 0)$ . Here, $PQ = QR = 34$ and $PR = 32$ . + +Since $PQ = 34$ and $Q = (0,0)$ , point $P$ is at a distance of 34 units from the origin along the y-axis. So, the coordinates of $P$ are $(0,b)$ and the distance from $P$ to $Q$ is the y-coordinate of $P$ , which is $b$ . Therefore, we have: + +$$ +b = 3 4 +$$ + +So, the coordinates of $P$ are $(0,34)$ . + +The coordinates of $Q$ are $(0,0)$ and the coordinates of $R$ are $(32,0)$ . + +Next, let's find the coordinates of the midpoint $M$ of the base $PR$ . + +The midpoint $M$ of the base $PR$ (where $P = (0,34)$ and $R = (32,0)$ ) is given by the midpoint formula: + +$$ +M = \left( \begin{array}{c c} x _ {1} + x _ {2} & y _ {1} + y _ {2} \\ \hline 2 & \hline 2 \end{array} \right) +$$ + +Here, $(x_{1},y_{1}) = (0,34)$ and $(x_{2},y_{2}) = (32,0)$ + +$$ +M = \left(\frac {0 + 3 2}{2}, \frac {3 4 + 0}{2}\right) = \left(\frac {3 2}{2}, \frac {3 4}{2}\right) = (1 6, 1 7) +$$ + +So, the coordinates of the midpoint $M$ are (16, 17). + +Then, let's calculate the length of the median $\overline{QM}$ . + +The median $\overline{QM}$ is the line segment from $Q = (0,0)$ to $M = (16,17)$ . The length of the median can be found using the distance formula between two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ : + +Distance $= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Here, the points are $Q = (0,0)$ and $M = (16,17)$ : + +${QM} = \sqrt{{\left( {16} - 0\right) }^{2} + {\left( {17} - 0\right) }^{2}} = \sqrt{{16}^{2} + {17}^{2}}$ + +Calculate $16^{\wedge}2$ and $17^{\wedge}2$ : + +$$ +1 6 ^ {2} = 2 5 6 +$$ + +$$ +1 7 ^ {2} = 2 8 9 +$$ + +$$ +Q M = \sqrt {2 5 6 + 2 8 9} = \sqrt {5 4 5} +$$ + +Therefore, the length of the median $\overline{QM}$ is $\left|\sqrt{545}\right|$ + +Figure 5: Reward hacking, case 2: extremely few steps (1 step). In practice, we divide steps according to double line breaks "\n\n" and then PRM scores each step. When the advantage baseline is inappropriate, such as step-level baseline discussed in § 5.1, the model learns to deliberately avoid outputting "\n\n", preferring short-step response. In this example, there is no "\n\n" character in the generated response, resulting in the entire response being split into only one step. + +# Question + +Evaluate the expression $\sqrt{80} - 4\sqrt{5} + 3\sqrt{\frac{180}{3}}$ and express it as $\sqrt{M}$ , where $M$ is an integer. + +# Response 1 + +Thank you. + +Response 2 (empty response) + + + +Response 3 + +I can help you solve this problem. + +Figure 6: Reward hacking, case 3: extremely few steps (0 step). This is the most common cases for PURE-PRM. When relying solely on the PRM, training eventually boils down to this case after numerous steps of training. It is inevitable because the PRM scores based on the question and historical steps, and it does not know the role of the current step in the overall response. + +![](images/481cfc793ae4502b0bdc793c1be47fbb248c83157f61f108e48a7c96013939e2.jpg) +Figure 7: Ablation on transform temperature defined in Eq. (5). We use PURE-PRM+VR to fine-tune Qwen2.5-7B in this experiment. + +Table 5: Results of PURE-PRM+VR with different transform temperature, defined in Eq (5). We conduct the experiments based on Qwen2.5-7B and report pass@1 accuracy tested with greedy decoding. + +
MethodMATH 500Minerva MathOlympiad BenchAIME24AMC23Avg.
Base71.423.935.310.052.538.6
+ PURE-PRM+VR (T=0.01)76.237.139.06.755.042.8
+PURE-PRM+VR (T=0.1)76.237.141.213.360.045.6
+ PURE-PRM+VR (T=1.0)75.833.538.713.352.542.8
+ +# F.2 PURE with Other RL Algorithms + +In this section, we apply PURE with various advantage estimators beyond RLOO, including GAE, GRPO, and REINFORCE++. The training curves are shown in Figure 8. To make different algorithms compatible with the compound of verifier rewards and process rewards, we make adjustments similar to those in Eq. (7). For GRPO, the advantage is defined as: + +$$ +A _ {i, t} = \frac {r _ {i} ^ {v} - \operatorname {m e a n} \left(\boldsymbol {r} ^ {v}\right)}{\operatorname {s t d} \left(\boldsymbol {r} ^ {v}\right)} + \frac {\sum_ {j = t} ^ {N} \gamma^ {j - t} r _ {i , j} ^ {p *} - \operatorname {m e a n} \left(\sum_ {l = 1} ^ {N} \sum_ {j = l} ^ {N} \gamma^ {j - l} \cdot \boldsymbol {r} _ {j} ^ {p *}\right)}{\operatorname {s t d} \left(\sum_ {l = 1} ^ {N} \sum_ {j = l} ^ {N} \gamma^ {j - l} \cdot \boldsymbol {r} _ {j} ^ {p *}\right)} \tag {9} +$$ + +![](images/413c76fdca5d92c1b90e10f949164e128253526c41eb18763925a4dbfab913a9.jpg) +Figure 8: Training curves on Qwen2.5-7B using PURE-PRM+VR with different advantage estimators. + +![](images/45fdaf7ffd7bcb49a42d7afde00d50cdcea48e47c21c47404245ea4b2bc9cdc5.jpg) + +where mean and standard deviation (std) are calculated over $K$ responses in a group. For REINFORCE++, the advantage is: + +$$ +A _ {i, t} = r _ {i} ^ {v} + \sum_ {j = t} ^ {N} \gamma^ {j - t} r _ {i, j} ^ {p *} \tag {10} +$$ + +From Figure 8, we find that all methods show similar performance before step 150. However, REINFORCE++ experiences a spike at around step 220. Although GAE converges a bit slower than others, it performs slightly more stably in the last 50 steps. Due to its additional learnable value network, GAE takes about $30\%$ more time for forward and backward passes. Considering performance, training time, and stability, we select RLOO as our preferred advantage estimator. + +# NeurIPS Paper Checklist + +# 1. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [NA] + +Justification: The core method development in this research does not involve LLMs. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. \ No newline at end of file diff --git a/stopsummationminformcreditassignmentisallprocessrewardmodelneedsforreasoning/images.zip b/stopsummationminformcreditassignmentisallprocessrewardmodelneedsforreasoning/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..52ec7dfc400ed447e46e4a6cbbc4400c1a119376 --- /dev/null +++ b/stopsummationminformcreditassignmentisallprocessrewardmodelneedsforreasoning/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:85a54fd884c8b3afbeae5380a187b150fea7e031857b98b462db0076ac45c587 +size 731961 diff --git a/stopsummationminformcreditassignmentisallprocessrewardmodelneedsforreasoning/layout.json b/stopsummationminformcreditassignmentisallprocessrewardmodelneedsforreasoning/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..69621224b8272df40605977f9e53f478054ee2a6 --- /dev/null +++ b/stopsummationminformcreditassignmentisallprocessrewardmodelneedsforreasoning/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:92f62e5457cb300cd3d7f617fe72db41f6b4ef245124e57265d861699b8cc312 +size 879722 diff --git a/storyboardguidedalignmentforfinegrainedvideoactionrecognition/6b49f4a2-ab81-4d86-9be4-bbdeaf9b490d_content_list.json b/storyboardguidedalignmentforfinegrainedvideoactionrecognition/6b49f4a2-ab81-4d86-9be4-bbdeaf9b490d_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..6dd1c6b8dbe106d644f9c2bca4fb5eef5ec38d99 --- /dev/null +++ b/storyboardguidedalignmentforfinegrainedvideoactionrecognition/6b49f4a2-ab81-4d86-9be4-bbdeaf9b490d_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2dc3ee6f80ae3d4ac9bb960d2daffbd27edc2c412d7b6f59bc128906c0123dbe +size 170558 diff --git a/storyboardguidedalignmentforfinegrainedvideoactionrecognition/6b49f4a2-ab81-4d86-9be4-bbdeaf9b490d_model.json b/storyboardguidedalignmentforfinegrainedvideoactionrecognition/6b49f4a2-ab81-4d86-9be4-bbdeaf9b490d_model.json new file mode 100644 index 0000000000000000000000000000000000000000..4349e40f313ef5aac33289636abe6821cc2d931d --- /dev/null +++ b/storyboardguidedalignmentforfinegrainedvideoactionrecognition/6b49f4a2-ab81-4d86-9be4-bbdeaf9b490d_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0467d344bbb53733f06f20ddd68e4dbda9d96b30e79c73435c5b97c74f5ab6e7 +size 221485 diff --git a/storyboardguidedalignmentforfinegrainedvideoactionrecognition/6b49f4a2-ab81-4d86-9be4-bbdeaf9b490d_origin.pdf b/storyboardguidedalignmentforfinegrainedvideoactionrecognition/6b49f4a2-ab81-4d86-9be4-bbdeaf9b490d_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..236a290fb6bb4adfd590854089997707f34024cf --- /dev/null +++ b/storyboardguidedalignmentforfinegrainedvideoactionrecognition/6b49f4a2-ab81-4d86-9be4-bbdeaf9b490d_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7380d1328c0e7ce5628d6ab89fd5add8ce69f35fa7e41f0b785d2531ca5a80e8 +size 2623178 diff --git a/storyboardguidedalignmentforfinegrainedvideoactionrecognition/full.md b/storyboardguidedalignmentforfinegrainedvideoactionrecognition/full.md new file mode 100644 index 0000000000000000000000000000000000000000..16bbebf2c33a47ba5a9f91c3347c2a02ff5a4868 --- /dev/null +++ b/storyboardguidedalignmentforfinegrainedvideoactionrecognition/full.md @@ -0,0 +1,794 @@ +# Storyboard-guided Alignment for Fine-grained Video Action Recognition + +Enqi Liu1, Livuan Pan1,2*; Yan Yang3, Yiran Zhong4, Zhijing Wu1, Xinxiao Wu1, Liu Liu5 + +1Beijing Institute of Technology, Beijing, China + +$^{2}$ Yangtze Delta Region Academy of Beijing Institute of Technology, Jiaxing, China + +$^{3}$ BDSI, Australian National University, Canberra, Australia + +4OpenNLPLab, Shanghai, China + +$^{5}$ Huawei, Beijing, China + +{enqi.liu, liyuan.pan, zhijingwu, wuxinxiao}@bit.edu, + +yan.yang@anu.edu.au, zhongyiran@gmail.com, liuliu33@huawei.com + +# Abstract + +Fine-grained video action recognition can be formulated as a video-text matching problem. Previous approaches primarily rely on global video semantics to consolidate video embeddings, often leading to misaligned video-text pairs due to inaccurate atomic-level action understanding. This inaccuracy arises due to i) videos with distinct global semantics may share similar atomic actions or visual appearances, and ii) atomic actions can be momentary, gradual, or not directly aligned with overarching video semantics. Inspired by storyboarding, where a script is segmented into individual shots, we propose a multi-granularity framework, SFAR. SFAR generates fine-grained descriptions of common atomic actions for each global semantic using a large language model. Unlike existing works that refine global semantics with auxiliary video frames, SFAR introduces a filtering metric to ensure correspondence between the descriptions and the global semantics, eliminating the need for direct video involvement and thereby enabling more nuanced recognition of subtle actions. By leveraging both global semantics and fine-grained descriptions, our SFAR effectively identifies prominent frames within videos, thereby improving the accuracy of embedding aggregation. Extensive experiments on various video action recognition datasets demonstrate the competitive performance of our SFAR in supervised, few-shot, and zero-shot settings. + +# 1 Introduction + +Fine-grained video action recognition has garnered increasing attention due to its broad applicability in areas such as sports analytics [54], human-computer interaction [13], surveillance [44], and video understanding [12]. In contrast to standard action recognition, fine-grained video action recognition necessitates a more detailed understanding of actions with similar appearances, demanding greater precision in capturing prominent frames with subtle actions. + +The advent of large language models has revealed that a robust multimodal encoder like CLIP [30] can consolidate significantly more potent learned embeddings compared to manually crafted embeddings for action recognition [39, 21, 29, 2, 23, 28, 51, 15, 11, 47]. These methods involve using CLIP's visual encoder to extract video embeddings and its textual encoder to extract text embeddings. The video and text embeddings are then aligned for video classification. However, the potential of the textual encoder has not been fully explored. + +To effectively integrate textual information, several works [48, 26, 8, 16, 24] refine global semantics by leveraging LLMs/MLLMs or additional lexicons to generate class-specific atomic action descriptions. These auxiliary descriptions are selected based on their alignment with each video frame, which leads to two rigid assumptions: i) each global video semantic must correspond to identical atomic actions, and ii) all atomic actions within a video should be closely related to its global semantics. However, these assumptions are frequently violated, leading to inaccurate embeddings. For instance, in Fig. 1 (a), the videos 'Baking cookies' and 'Making pizza' are ambiguous with each other, featuring overlapping atomic actions. In Fig. 1 (b), the atomic action 'Stand' can be irrelevant to the global video semantics 'Swing legs'. Moreover, the nonuniform distribution nature of + +atomic actions has been overlooked previously, i.e., the duration of each atomic action can vary. Based on this, using the global video semantics directly for aligning videos and text prompts can lead to misunderstandings due to their granularity discrepancy. In this paper, we address these issues and propose a multi-granularity framework, SFAR. The core of our SFAR is decomposing 'the global video semantics into fine-grained sub-texts' and 'a video action into multiple atomic actions'. This allows both coarse and fine-grained identifications of prominent frames in videos. + +Inspired by the concept of storyboarding, which breaks down a script into individual shots, we enhance the global semantics (text) by generating detailed descriptions using a pre-trained large language model (LLM). These fine-grained descriptions, i.e., sub-texts, capture common atomic actions depicted in videos, requiring only their class names and utilizing our designed question prompts, eliminating the need for direct video involvement. To fit the flexibility of various pre-trained LLM and question prompt formats, we design a text prompt perplexity metric to measure the diversity among sub-texts and the similarity between the sub-text and the global text for filtering sub-texts. It provides an effective schema for selecting sub-texts to train the video action recognition model. + +We then use global texts and sub-texts to coarsely and fine-grainedly compute a video embedding for videos with ambiguous or non-uniform actions. Specifically, we augment the global text with sub-texts in the embedding space of CLIP by abounding described video actions with sub-texts, which also decreases the granularity of the global text. This augmented global text is used to weight video frames in embedding space and compute a coarse video embedding. We illustrate network attention of the augmented global text in Fig. 2 (c) and sub-texts in Fig. 2 (d). Compared to the previous method constrained by granularity differences (Fig. 2 (b)), our augmented global text and sub-texts direct network attention more comprehensively toward regions with actions. + +We then fuse the coarse and fine-grained video embedding to compute a video embedding that improves the classification performance on ambiguous and non-uniform videos. Heatmaps of our network are in Fig. 2 (e), and improvements over videos with different ambiguous and non-uniform scores are in Fig. 2 (f). We rigorously validate our approach across different scenarios, including supervised, few-shot, and zero-shot video action recognition. Our method delivers top-notch performance in all these scenarios, showcasing the effectiveness of our framework. + +Our main contributions are: + +![](images/6883ebddca3575468a01826c359c48a1c472ac4e2a919cf6e06ebce9d8770f2d.jpg) +Figure 1: Example videos with ambiguous or non-uniform actions. Each row displays sample frames of a video with its class name. (a) Ambiguous actions. Both 'Baking cookies' and 'Making pizza' are actions performed in the kitchen with similar visual appearances and share atomic actions, e.g., the 'rolling dough'. (b) Non-uniform actions. Multiple atomic actions distribute unevenly within a video to form an action, e.g., the video 'Long jump' is composed of $43\%$ of 'Crouch, swing arms' and $22\%$ of 'Extend legs'. Moreover, some atomic actions may be irrelevant to the global class, e.g., $34\%$ of the frames show 'Stand', unrelated to 'Swing Legs'. The axis indicates the percentage of frames per atomic action. + +![](images/9df08a12ec8a93832c01df44cf1240b0561835aae9c0073a312056a2abfdc020.jpg) +Figure 2: Examples that illustrate the ineffectiveness of directly aligning video frames and global text, from (a-e) heatmaps and (f) classification performance improvements. (a) Video frames with their class name annotated at the left. (b) Heatmaps of the single granularity-based (single) method [48], which are limited by the granularity difference between the video frame and global text. (c-e) Heatmaps using coarse video embedding, fine-grained video embedding, and the fusion of the two. (f) We show the performance distributions (Top-1 Accuracy) over ambiguous scores and non-uniform scores of videos for (b-e) on the whole set. The first row is the absolute performance for (b), and the last three rows are improvements of (c), (d), and (e) concerning (b). Please refer to our supplementary materials for details on the computation of the ambiguous score and the non-uniform score, as well as the accuracy improvements on the Kinetics-400 dataset presented in Tab. 8. + +- A multi-granularity framework for transferring CLIP trained on image-text pair to video action recognition. +- A schema for automatically decomposing a video action into common atomic actions to provide fine-grained knowledge of the video action to CLIP. +- A coarse and fine-grained video embedding module for videos with ambiguous and non-uniform actions. + +# 2 Related Works + +We review video action recognition methods focusing visual and visual-text representation learning. + +Visual Representation Learning. Action recognition requires accurately capturing temporal semantic variations. Numerous studies [59, 52, 42, 35, 39, 55] have thus focused on visual representation learning. Early works studied the joint learning of spatial and temporal features of a video using various architectures [26, 23, 2, 32]. Benefiting from large-scale visual and visual-text pre-training, recent methods leverage the strong spatial features learned in pre-training and focus on fine-tuning the trained model to capture the temporal semantics of a video. However, these methods do not explicitly address the fundamental challenge of recognizing video actions from ambiguous and non-uniform videos. This paper proposes decomposing video actions into atomic actions to improve the recognition of challenging videos. + +Visual-text Representation Learning. Several methods [8, 40, 43, 47, 48, 41] have been developed to overcome these limitations by identifying video frames that strongly align with the text prompts of video actions for video action recognition in vision-language models, e.g., CLIP. One remarkable work is BIKE [48], which weights the video frames based on their alignment with the text prompts to compute a video embedding for action recognition. However, these methods are limited by the granularity difference between the video frames and the text prompts, where the text prompt is a global context for the video. Unlike previous methods focused on generative modeling [49, 25, 57], where LLMs generate captions or texts conditioned on video or frame embeddings, our approach adopts a discriminative modeling approach. We generate sub-texts from the global text to describe atomic actions in video frames. The sub-texts are then used to construct coarse and fine-grained video embeddings, which improves the recognition of fine-grained actions. + +![](images/f54dc89f57486c33fe5da17bc5263994d92c33094bb2582edd472bb1dfd62927.jpg) +Figure 3: An overview of our framework for video action recognition. We extend CLIP for classifying the video $\{\mathbf{I}\}_{1 = 1}^{L}$ with $L$ frames by computing a video embedding from frame embedding $\{\mathbf{v}_1^{\mathrm{cls}}\}_{1 = 1}^L$ in three steps. (i) We decompose the global text prompt $\mathbf{T}_{\mathbf{c}}$ that describes the class semantic of action c into descriptions of atomic action (i.e., sub-text prompts) by using a pre-trained large language model. The global text prompt $\mathbf{T}_{\mathbf{c}}$ and sub-text prompts are then embedded by the textual encoder of CLIP for extracting embeddings of $\mathbf{T}_{\mathbf{c}}$ and $\{\mathbf{S}_{\mathbf{c},\mathfrak{n}}\}_{1 = 1}^{L}$ . (ii) A coarse video embedding is extracted by augmenting the global text embedding $\mathbf{T}_{\mathbf{c}}$ with the sub-text embedding $\{\mathbf{S}_{\mathbf{c},\mathfrak{n}}\}_{1 = 1}^{L}$ , calculating coarse importance of the frame embedding $\{\mathbf{v}_1^{\mathrm{cls}}\}_{1 = 1}^L$ with the augmented global text embedding $\hat{\mathbf{T}}_{\mathbf{c}}$ , and using the importance $\{a_1\}_{1 = 1}^L$ to aggregate a coarse video embedding $\mathbf{o}^{\mathrm{coarse}}$ from frame embedding $\{\mathbf{v}_1^{\mathrm{cls}}\}_{1 = 1}^L$ . (iii) Similar to the coarse video embedding, we get a fine-grained video embedding $\mathbf{o}^{\mathrm{fine}}$ by calculating fine-grained importance $\{a_1^{\mathrm{fine}}\}_{1 = 1}^L$ of the frame embedding $\{\mathbf{v}_1^{\mathrm{cls}}\}_{1 = 1}^L$ with sub-text embedding $\{\mathbf{S}_{\mathbf{c},\mathfrak{n}}\}_{1 = 1}^{L}$ for aggregating frame embedding $\{\mathbf{v}_1^{\mathrm{cls}}\}_{1 = 1}^L$ . The coarse and fine-grained embeddings $\mathbf{o}^{\mathrm{coarse}}$ and $\mathbf{o}^{\mathrm{fine}}$ are fused to form the final video embedding $\mathbf{o}$ for action recognition. + +# 3 Methodology + +Preliminary. CLIP [30], a visual-language pre-training method, consists of a visual and a textual encoder. It learns a joint embedding space by maximizing similarities between aligned image-text pairs and minimizing them for misaligned ones. Given an image $\mathbf{I}$ and global text prompts $\{\mathbf{T}_{\mathrm{c}}\}_{\mathrm{c} = 1}^{C}$ of $C$ classes formatted as [a photo of a class] that globally describe class semantics, where class is the class name, CLIP performs zero-shot classification by extracting the visual class embedding $\mathbf{v}^{\mathrm{cls}}$ from $\mathbf{I}$ and text class embeddings $\{\mathbf{t}_{\mathrm{c}}^{\mathrm{cls}}\}_{\mathrm{c} = 1}^{C}$ from $\{\mathbf{T}_{\mathrm{c}}\}_{\mathrm{c} = 1}^{C}$ . The image $\mathbf{I}$ is then classified into the class $\mathbf{c}'$ with the maximum cosine similarity, i.e., $\mathbf{c}' = \arg \max_{\mathrm{c}} \sin (\mathbf{v}^{\mathrm{cls}}, \mathbf{t}_{\mathrm{c}}^{\mathrm{cls}})$ . + +Overview. To extend CLIP for classifying the video $\{\mathbf{I}_1\}_{1 = 1}^L$ with $L$ frames, we adaptively compute a coarse and a fine-grained video embedding. Two multi-granularity embeddings capture global semantic and atomic semantics of ambiguous and non-uniform actions in the video, for computing the cosine similarity with the text class embedding. There are three key steps: + +![](images/74a841eae5be09d4be7fc7c39b3bea27951b8649b11c5d4cd8b1dd1158db5487.jpg) +Figure 4: Example sub-texts $\{\mathbf{S}_{c,n}\}_{n = 1}^{N}$ generated from GPT-4o and LLaVa1.6, with $N = 4$ in the example. + +1. Preparing + +3. Rotation + +2. Takeoff +4. Entry the water + +1. Prepare diving + +2. Jump off the board + +3. In mid-air + +4. Dive into the water + +(i) Sub-text generation, decomposing each global text prompt $\mathbf{T}_{\mathrm{c}}$ into a sequence of $N$ atomic action text descriptions $\{\mathbf{S}_{\mathrm{c,n}}\}_{\mathrm{n = 1}}^{N}$ (i.e., sub-text prompts), by using a pre-trained large language model; (ii) Coarse video embedding, embedding each frame $\mathbf{I}_1$ into a visual class embedding $\mathbf{v}_1^{\mathrm{cls}}$ that is a frame embedding. Then, by using the global text prompt augmented with sub-text prompts to identify salient video frames in CLIP embedding space, we aggregate the frame embedding $\{\mathbf{v}_1^{\mathrm{cls}}\}_{1 = 1}^L$ into coarse video embeddings $\mathbf{o}^{\mathrm{coarse}}$ ; (iii) Fine-grained video embedding, finding video frames with atomic actions by using sub-text prompts and CLIP for computing a fine-grained video embedding $\mathbf{o}^{\mathrm{fine}}$ . We fuse the coarse and fine-grained video embedding into a video embedding $\mathbf{o}$ to compute + +![](images/8d1725228406831494394a1d814551975c4c66cd07d3a68f70afccb5b35d4564.jpg) +(b) Tasting food + +![](images/91070bee001926d200607c709c75c14d32b65201686b7177714efd8ac2935951.jpg) +(c) Dining + +![](images/a8332c08f750e3d6739e99e956b35b12d8631ec7c3a0dcc560bdbe377824115d.jpg) +(d) Eating spaghetti + +![](images/cac57f3616ec134d77ed47a41b15c48b6c65629853c5b35c2fa6bfc4e12029ad.jpg) +(e) Eating burger + +![](images/7fe43f173fdc56219b0b1606b7e939c33843f28f5abb40c82f6dec6303016314.jpg) +Figure 5: An example illustrating the benefits of using our coarse video embedding for ambiguous actions. We show a video in the first row, with the ground truth action labeled as (a) on the left. Its ambiguous actions are annotated with (b), (c), (d) and (e) at the bottom of the video. The action probability from the single granularity-based method [48] and our coarse video embedding are in the next row. Compared with the single granularity-based method that identifies the video as (b) 'Tasting food', we obtain a higher probability of the correct class (a) 'Eating cake'. + +![](images/26b0fbd6dd54f203303428bd5c365f6fce95bc35f19cf8b4a75f2a358c134fb5.jpg) + +cosine similarities with text class embeddings $\mathbf{t}_{\mathrm{c}}^{\mathrm{cls}}$ for classifying the video. The overview of our method is in Fig. 3. + +# 3.1 Sub-text Generation + +A global text prompt $\mathbf{T}_{\mathrm{c}}$ is a description of a video action. It overlooks that an action is usually formed by performing a sequence of atomic actions, like a video performing the action. Directly aligning video frame embedding to the global text prompt $\mathbf{T}_{\mathrm{c}}$ introduces the granularity differences and decreases the video classification performance. To minimize human burden and bias, we propose a pipeline that automatically generates and selects sub-texts based on the global text (class name). + +We leverage a pre-trained large language model $\mathrm{LLM}(\cdot)$ to decompose global text prompt $\mathbf{T}_{\mathrm{c}}$ into $N$ potentially atomic actions $\{\mathbf{S}_{\mathrm{c,n}}\}_{\mathrm{n = 1}}^{N} = \mathrm{LLM}(\mathbf{P}_{\mathrm{c}})$ , where $\mathbf{P}_{\mathrm{c}}$ is the prompt for describing the action class c and directing the pre-trained large language model to find potential atomic action descriptions, sub-texts $\{\mathbf{S}_{\mathrm{c,n}}\}_{\mathrm{n = 1}}^{N}$ . Refer to Fig. 4 for an example. + +However, the existence of various pre-trained large language models and the flexibility in designing $\mathbf{P}_{\mathrm{c}}$ make selecting a meaningful sub-text set challenging. We assume that an optimal sub-text set $\{\mathbf{S}_{\mathrm{c,n}}\}_{\mathrm{n = 1}}^{N}$ for video classification should be sufficiently related with the global text prompt $\mathbf{T}_{\mathrm{c}}$ while each sub-text should be diverse from the others. To measure the similarities and diversity, we propose a text prompt perplexity metric to select the sub-text set with the highest text prompt perplexity score. + +In Fig. 5, we show ambiguous action probability computed from a single granularity-based method [48] and our coarse video embedding. Our method uses sub-text to augment the global text help the model to identify different atomic actions in ambiguous actions, and improve the accuracy of recognizing ambiguous actions. + +With the class embedding of the global text $\mathbf{T}_{\mathrm{c}}$ and sub-texts $\{\mathbf{S}_{\mathrm{c,n}}\}_{\mathrm{n = 1}}^{N}$ from the CLIP textual encoder, $\mathbf{t}_{\mathrm{c}}^{\mathrm{cls}}$ and $\mathbf{s}_{\mathrm{c,n}}^{\mathrm{cls}}$ , we define the text prompt perplexity score $\mathrm{TPP_{c}}$ for $\mathbf{t}_{\mathrm{c}}^{\mathrm{cls}}$ and $\mathbf{s}_{\mathrm{c,n}}^{\mathrm{cls}}$ as below + +$$ +\mathrm {T P P} _ {\mathrm {c}} = \exp \left(- \frac {1}{N} \sum_ {\mathrm {n} = 1} ^ {N} \log \left(\alpha \left(\sigma_ {\mathrm {c}, \mathrm {n}}\right) \beta \left(\delta_ {\mathrm {c}, \mathrm {n}}\right)\right)\right), \tag {1} +$$ + +$$ +\sigma_ {\mathrm {c}, \mathrm {n}} = \frac {\operatorname {s i m} \left(\mathbf {t} _ {\mathrm {c}} ^ {\text {c l s}} , \mathbf {s} _ {\mathrm {c} , \mathrm {n}} ^ {\text {c l s}}\right) + 1}{2}, \tag {2} +$$ + +$$ +\delta_ {\mathrm {c}, \mathrm {n}} = 1 - \frac {1}{N - 1} \sum_ {\mathrm {n} ^ {\prime} = 1, \mathrm {n} ^ {\prime} \neq \mathrm {n}} ^ {N} \frac {\operatorname {s i m} \left(\mathbf {s} _ {\mathrm {c} , \mathrm {n}} ^ {\mathrm {c l s}} , \mathbf {s} _ {\mathrm {c} , \mathrm {n} ^ {\prime}} ^ {\mathrm {c l s}}\right) + 1}{2}, \tag {3} +$$ + +![](images/8a76290e1b8018681b90f497881d34abd6746718ccd62726978222791c10e2b6.jpg) +Figure 6: Examples of coarse and fine-grained importance probability for videos with non-uniform actions. The fine-grained importance probability $a_1^{\mathrm{fine}}$ allocate larger scores than the coarse importance probability $a_1^{\mathrm{coarse}}$ to prominent frames. For example, the illustrated fourth frame in the second row, 'lowering barbell back down to the ground', represents an atomic action of 'Clean and jerk', and $a_1^{\mathrm{fine}} = 23\%$ has a higher probability than $a_1^{\mathrm{coarse}} = 9\%$ . + +where the $\mathrm{sim}(\cdot ,\cdot)$ calculates the cosine similarities of inputs. We employ $\sigma_{\mathrm{c,n}}$ to quantify the similarity between a global text and its corresponding sub-texts, a higher value of $\sigma_{\mathrm{c,n}}$ indicates greater similarity. Meanwhile, $\delta_{\mathrm{c,n}}$ is used to evaluate the similarity among sub-texts themselves, where a higher value of $\delta_{\mathrm{c,n}}$ indicates greater diversity across the sub-texts. Here, $\alpha (\cdot)$ and $\beta (\cdot)$ are linear scaling functions for the scores. Please refer to Fig. 7b for the correlations between the video classification performance and the text prompt perplexity score $\mathrm{TPP_c}$ . + +# 3.2 Coarse Video Embedding + +For identifying different atomic actions in ambiguous videos, we augment the global text $\mathbf{T}_{\mathrm{c}}$ with the sub-texts $\{\mathbf{S}_{\mathrm{c,n}}\}_{\mathrm{n = 1}}^{N}$ to adaptively aggregate the video frames $\{\mathbf{I}_1\}_{1 = 1}^L$ in CLIP embedding space. As the class name in global text is usually a phrase, and sub-text $\mathbf{S}_{\mathrm{c,n}}$ describes the atomic action with multiple words, we keep the word embeddings to benefit from the rich semantics of each word embedding for video frame aggregations. We denote the matrix form of the word embeddings for global text and sub-text as $\mathbf{T}_{\mathrm{c}}$ and $\mathbf{S}_{\mathrm{c,n}}$ , and augment $\mathbf{T}_{\mathrm{c}}$ by the cross-attention mechanism, + +$$ +\mathbf {Q} _ {\mathrm {c}} = \mathbf {T} _ {\mathrm {c}} \mathbf {W} ^ {\mathrm {q}}, \quad \mathbf {K} _ {\mathrm {c}, \mathrm {n}} = \mathbf {S} _ {\mathrm {c}, \mathrm {n}} \mathbf {W} ^ {\mathrm {k}}, \quad \mathbf {V} _ {\mathrm {c}, \mathrm {n}} = \mathbf {S} _ {\mathrm {c}, \mathrm {n}} \mathbf {W} ^ {\mathrm {v}}, \tag {4} +$$ + +$$ +\mathbf {K} _ {\mathrm {c}} = \left[ \left\{\mathbf {K} _ {\mathrm {c}, \mathrm {n}} \right\} _ {\mathrm {n} = 1} ^ {N} \right], \quad \mathbf {V} _ {\mathrm {c}} = \left[ \left\{\mathbf {V} _ {\mathrm {c}, \mathrm {n}} \right\} _ {\mathrm {n} = 1} ^ {N} \right], \tag {5} +$$ + +$$ +\hat {\mathbf {T}} _ {\mathrm {c}} = \operatorname {A t t e n t i o n} \left(\mathbf {Q} _ {\mathrm {c}}, \mathbf {K} _ {\mathrm {c}}, \mathbf {V} _ {\mathrm {c}}\right) + \mathbf {T} _ {\mathrm {c}}, \tag {6} +$$ + +where $\mathbf{W}^{\mathrm{q}}$ , $\mathbf{W}^{\mathrm{k}}$ , and $\mathbf{W}^{\mathrm{v}}$ are matrices for projecting the query $\mathbf{Q}_{\mathrm{c}}$ , key $\mathbf{K}_{\mathrm{c,n}}$ , and value $\mathbf{V}_{\mathrm{c,n}}$ . Here, $[\{\mathbf{K}_{\mathrm{c,n}}\}_{\mathrm{n = 1}}^{N}]$ and $[\{\mathbf{V}_{\mathrm{c,n}}\}_{\mathrm{n = 1}}^{N}]$ concatenates all keys and values for $\mathbf{Q}_{\mathrm{c}}$ . + +Using the augmented global text embeddings $\hat{\mathbf{T}}_{\mathrm{c}}$ , we find the salient video frame embedding $\mathbf{v}_{1}^{\mathrm{cls}}$ for aggregating a coarse video embedding. To find the coarse-grained importance score $a_{1}^{\mathrm{coarse}}$ of each frame embedding, we calculate the overall normalized similarity between each word embedding and the video frame embedding by + +$$ +a _ {1} ^ {\text {c o a r s e}} = \sum_ {\hat {\mathbf {t}} _ {\mathrm {c}} \in \hat {\mathbf {T}} _ {\mathrm {c}}} \frac {\exp \left(\operatorname {s i m} \left(\hat {\mathbf {t}} _ {\mathrm {c}} , \mathbf {v} _ {1} ^ {\text {c l s}}\right)\right)}{\sum_ {l ^ {\prime} = 1} ^ {L} \exp \left(\operatorname {s i m} \left(\hat {\mathbf {t}} _ {\mathrm {c}} , \mathbf {v} _ {l ^ {\prime}} ^ {\text {c l s}}\right)\right)}. \tag {7} +$$ + +We then compute the coarse video embedding $\mathbf{o}^{\mathrm{coarse}}$ with summing over the video frame embedding $\mathbf{v}_1^{\mathrm{cls}}$ weighted by $a_1^{\mathrm{coarse}}$ , + +$$ +\mathbf {o} ^ {\text {c o a r s e}} = \sum_ {l = 1} ^ {L} \mathbf {v} _ {l} ^ {\text {c l s}} a _ {l} ^ {\text {c o a r s e}}. \tag {8} +$$ + +# 3.3 Fine-grained Video Embedding + +While the video coarse embedding $\mathbf{o}^{\mathrm{coarse}}$ captures salient semantics for ambiguous actions, it may overlook atomic actions in the video that align with the expected action class $c$ due to the non-uniform property of video actions. To address the limitation, we use the word embedding of the sub-texts to find atomic actions among the video frame embeddings $\{\mathbf{v}_1^{\mathrm{cls}}\}_{l=1}^L$ to compute the fine-grained video embedding. + +Table 1: Comparison result on the Kinetics-400 dataset. We report the GFLOPs in the inference phase. "Views" indicates the number of temporal clips and spatial crops used in inference (# temporal clip $\times$ # spatial crop). The magnitude is Million ( $10^{6}$ ) for number of model parameters (Param). We achieve the highest top-1 and top-5 accuracy by employing only an 8-frame input evaluated with a single view, outperforming past methods with more model parameters, input frames, and views. We highlight the best top-1 and top-5 accuracy in bold. + +
MethodVenueInputPre-trainingTop-1(%)Top-5(%)ViewsGFLOPsParam
Methods with large-scale visual pre-training
MVFNetEn[45]AAAI'2124×2242ImageNet-1K79.193.810×3188×30-
ViViT-B/16×2 [2]ICCV'2132×2242JFT-300M80.0-4×1455.2151.9
ST-Adapter ViT-B/16 [28]NeurIPS'228×2242WIT-400M82.095.73×1455128.8
EVL ViT-B/16 [21]ECCV'228×2242WIT-400M82.9-3×1444177.7
MTV-B [50]CVPR'2232×2242JFT-300M81.895.04×3384×12310
VideoSwin-B [23]CVPR'2232×2242ImageNet-21K82.795.54×3282×1288.1
ActionCLIP ViT-B/16 [39]TNNLS'2316×2242WIT-400M82.696.210×3282×30141.7
ATM ViT-B/16 [46]ICCV'238×2242WIT-400M82.895.61×1378×1-
DIST ViT-B/16 [29]ICCV'238×2242WIT-400M83.6-3×1163×3105
AIM ViT-B/16 [53]ICLR'238×2242WIT-400M83.996.33×1202×397
ILA-ViT-B/16 [37]ICCV'238×2242WIT-400M84.096.64×3149×12-
Methods with large-scale visual-text pre-training
X-CLIP ViT-B/16 [26]ECCV'228×2242WIT-400M82.3-1×1145×1-
VideoPrompt ViT-B/16 [15]ECCV'2216×2242WIT-400M76.993.51×5-154
SIF ViT-B/16 [40]ACMMM'238×2242WIT-400M77.493.64×31136×12143.9
Vita-CLIP ViT-B/16 [43]CVPR'238×2242WIT-400M80.596.01×197×1187.9
Vita-CLIP ViT-B/16 [43]CVPR'238×2242WIT-400M81.896.04×397×12187.9
BIKE ViT-B/16 [48]CVPR'238×2242WIT-400M83.2-1×1-124.1
BIKE ViT-B/16 [48]CVPR'238×2242WIT-400M83.9-4×3-124.1
M2-CLIP ViT-B/16 [38]AAAI'248×2242WIT-400M83.496.34×3214×3165
ALT ViT-B/16 [7]CVPR'248×2242WIT-400M83.695.93×1328134.4
OST ViT-B/16 [6]CVPR'248×2242WIT-400M82.0-1×1--
Ours ViT-B/16NeurIPS'258×2242WIT-400M84.596.71×190.2×1126.1
+ +We compute a fine-grained importance score $a_{1}^{\mathrm{fine}}$ for each video frame embedding $\mathbf{v}_1^{\mathrm{cls}}$ as the maximum average similarity between $\mathbf{v}_1^{\mathrm{cls}}$ and $\mathbf{S}_{\mathrm{c,n}}$ , and normalize it by a SoftMax function, + +$$ +a _ {1} ^ {\text {f i n e}} = \frac {\exp \left(\max _ {n} \frac {1}{\left| \mathbf {S} _ {c , n} \right|} \sum \mathbf {s} _ {c , n} \in \mathbf {S} _ {c , n} \operatorname {s i m} \left(\mathbf {s} _ {c , n} , \mathbf {v} _ {1} ^ {\mathrm {c l s}}\right)\right)}{\sum_ {l ^ {\prime} = 1} ^ {L} \exp \left(\max _ {n} \frac {1}{\left| \mathbf {S} _ {c , n} \right|} \sum \mathbf {s} _ {c , n} \in \mathbf {S} _ {c , n} \operatorname {s i m} \left(\mathbf {s} _ {c , n} , \mathbf {v} _ {1 ^ {\prime}} ^ {\mathrm {c l s}}\right)\right)}, \tag {9} +$$ + +where $|\mathbf{S}_{\mathrm{c,n}}|$ is the number of words in the sub-text, and $\mathbf{s}_{\mathrm{c,n}} \in \mathbf{S}_{\mathrm{c,n}}$ is a word embedding for the sub-text. Similar to Eq. (8), we weight the video frame embedding $\mathbf{v}_1^{\mathrm{cls}}$ with $a_1^{\mathrm{fine}}$ , and get the fine-grained video embedding $\mathbf{o}^{\mathrm{fine}}$ by + +$$ +\mathbf {o} ^ {\text {f i n e}} = \sum_ {l = 1} ^ {L} \mathbf {v} _ {l} ^ {\text {c l s}} a _ {l} ^ {\text {f i n e}}. \tag {10} +$$ + +We compare the coarse scores $a_1^{\mathrm{coarse}}$ and fine-grained scores $a_1^{\mathrm{fine}}$ in Fig. 6, and find that $a_1^{\mathrm{fine}}$ can fine-grainedly allocate larger scores than $a_1^{\mathrm{coarse}}$ to prominent frames that humans perform the action peak in the non-uniform videos. + +# 3.4 Loss + +We fuse the coarse video embedding and fine-grained video embedding as a video embedding o with two feedforward layers $\mathrm{FFN}^{\mathrm{coarse}}(\cdot)$ and $\mathrm{FFN}^{\mathrm{fine}}(\cdot)$ , projecting the coarse video embedding and fine-grained video embedding to the semantic of class c, + +$$ +\mathbf {o} = \operatorname {F F N} ^ {\text {c o a r s e}} \left(\mathbf {o} ^ {\text {c o a r s e}}\right) + \operatorname {F F N} ^ {\text {f i n e}} \left(\mathbf {o} ^ {\text {f i n e}}\right), \tag {11} +$$ + +and compute the cosine similarity with the text class embedding $\mathbf{t}_{\mathrm{c}}^{\mathrm{cls}}$ by $\sin (\mathbf{t}_{\mathrm{c}},\mathbf{o})$ + +We optimize our network by maximizing the similarity $y_{b,c^{\mathrm{gt}}}$ between the b-th video embedding in a batch and text embedding of its ground truth class $c^{\mathrm{gt}}$ , and minimizing the similarity between other video and text class embeddings $\{y_{b,c}\}_{c = 1,c\neq c^{\mathrm{gt}}}^{C}$ . Following [27], we use the InfoNCE loss, + +$$ +\mathcal {L} _ {\mathrm {T 2 V}} = \frac {1}{B} \sum_ {\mathrm {b} = 1} ^ {B} \frac {1}{| \mathbf {k} _ {\mathrm {b}} |} \sum_ {\mathrm {b} ^ {\prime} \in \mathbf {k} _ {\mathrm {b}}} \log \frac {\exp \left(y _ {\mathrm {b} ^ {\prime} , \mathrm {c} ^ {\mathrm {g t}}}\right)}{\sum_ {\mathrm {b} ^ {\prime \prime} = 1} ^ {B} \exp \left(y _ {\mathrm {b} ^ {\prime \prime} , \mathrm {c} ^ {\mathrm {g t}}}\right)}, \tag {12} +$$ + +Table 2: Comparative experiments are conducted on Charades [33], HMDB-51 [19], and UCF-101 [34], while ablation studies are performed on the Kinetics-400 dataset [17]. We report accuracy $(\%)$ for a single 8-frame clip with a spatial resolution of $224\times 224$ unless otherwise specified. The "VZ" column denotes if the method is specifically designed for zero-shot action recognition or adapted from CLIP. + +
MethodFramesmAP
MultiScale TRN [58]-25.2
STM [14]1635.3
SlowFast+NL [10]16+6442.5
X3D-XL(312) [9]1643.4
ActionCLIP [39]3244.3
BIKE [48]1650.4
Ours1651.1
+ +(a) Comparisons for multi-label action recognition on the Charades dataset [33]. + +
MethodVZUCF-101HMDB-51
E2E [3]✓44.129.8
ER [5]✓51.835.3
ResT [20]✓58.741.1
Vita-CLIP [43]✗75.048.6
BIKE [48]✗78.455.6
M2-CLIP [38]✗78.747.1
Ours✗79.056.6
+ +(b) Comparisons of zero-shot action recognition on the HMDB-51 [19] and UCF-101 [34] datasets. + +
BackboneFVE Top-1(%)Top-5(%)
VideoPrompt [15]X76.993.5
✓79.395.1
ATM [46]X82.895.6
✓82.996.4
BIKE [48]X83.296.0
✓83.896.5
+ +(c) Generalization of fine-grained video embedding (FVE) on state-of-the-art methods using the Kinetics-400 dataset [17]. + +
GeneratorTPPTop-1(%)Top-5(%)
LLaVA-1.6-34b54.484.3596.5
GPT-Davinci54.784.3596.6
GPT-3.560.484.4796.7
GPT-4o60.884.5296.7
+ +(d) Comparison of sub-texts generated from different LLMs on Kinetics-400. A higher Text Prompt Perplexity (TPP) score indicates better sub-texts. + +
Coarse VEFine-grained VETop-1(%)Top-5(%)
Baseline (CLIP)79.994.7
Baseline (Temporal)80.395.0
X✓82.796.2
✓X83.596.3
✓✓84.596.7
+ +(e) Ablations of coarse and fine-grained video embedding (VE) on Kinetics-400. Using fine-tuned CLIP and CLIP model with temporal layers as baselines. + +$$ +\mathcal {L} _ {\mathrm {V 2 T}} = \frac {1}{B} \sum_ {\mathrm {b} = 1} ^ {B} \frac {1}{| \mathbf {k} _ {\mathrm {b}} |} \sum_ {\mathrm {b} ^ {\prime} \in \mathbf {k} _ {\mathrm {b}}} \log \frac {\exp \left(y _ {\mathrm {b} ^ {\prime} , \mathrm {c g t}}\right)}{\sum_ {\mathrm {c} = 1} ^ {C} \exp \left(y _ {\mathrm {b} ^ {\prime} , \mathrm {c}}\right)}, \tag {13} +$$ + +$$ +\mathcal {L} = \mathcal {L} _ {\mathrm {T 2 V}} + \lambda \mathcal {L} _ {\mathrm {V 2 T}}, \tag {14} +$$ + +where $B$ is the number of batches, $\mathbf{k_b}$ find the index of the video that has the same class with the b-th video, $|\mathbf{k_b}|$ is its size, and $\lambda$ is a hyperparameter. + +# 4 Experiment + +# 4.1 Experimental Setup + +Our proposed sub-text set and code are given in our project page. + +Datasets. We experiment across four extensively recognized video benchmarks: Kinetics-400 [17], Charades [33], UCF-101 [34], and HMDB-51 [19] datasets. + +Supervised Learning. Our model is implemented using the PyTorch framework. We train our network with batch size 256 for 30 epochs using the AdamW optimizer. The learning rate is set to $5 \times 10^{-5}$ , and we use the cosine annealing strategy with 5 warm-up epochs. We follow [48] for data augmentation in training. + +Zero-shot Learning. We evaluate our model, pre-trained on Kinetics-400 [17], using the UCF-101 [34] and HMDB-51 [19]. + +Few-shot Learning. We follow [39, 26] to experiment with different shot settings, selecting 2, 4, 8, and 16 examples per human action category for training. Training for 2 epochs on the Kinetics-400 dataset, and then use the same settings to train on other datasets for 10 epochs. Importantly, we do not use a model pre-trained on Kinetics-400 for few-shot learning on other datasets. + +Evaluation Metrics. We evaluate our model with top-1 and top-5 accuracy on the single-label datasets. On multi-label Charades, we follow [14] to report mean average precision (mAP). + +Table 3: Comparisons on few-shot action recognition across the HMDB-51 [19], UCF-101 [34] and Kinetics-400 datasets [17] with state-of-the-art action recognition methods. We utilize ViT-B/16 as the backbone and use 8 frames for training/validation. All performances are reported as top-1 accuracy (\%) in the few-shot setting using single-view inference with a spatial size of $224 \times 224$ , where all models are directly fine-tuned from CLIP. "Avg." refers to the average performance across all datasets. + +
MethodHMDB-51UCF-101Kinetics-400All
K=2K=4K=8K=16K=2K=4K=8K=16K=2K=4K=8K=16Avg.
Vanilla CLIP [30]41.941.941.941.963.663.663.663.657.257.257.257.254.2
ActionCLIP [39]47.557.957.359.170.671.573.091.461.063.064.868.565.5
VideoPrompt [15]39.750.756.062.471.479.985.789.9----67.0
X-CLIP [26]53.057.362.864.076.483.488.391.456.860.762.364.668.4
ViFi-CLIP [31]57.262.764.566.880.785.190.092.737.142.849.155.565.4
TC-CLIP [18]57.362.367.368.685.989.992.594.658.561.965.569.972.9
OST [6]59.162.964.968.282.587.591.793.944.048.252.556.567.6
Ours64.066.168.770.792.193.094.394.773.974.875.576.278.7
+ +# 4.2 Main Results + +Action Recognition. We compare our network against state-of-the-art methods that utilize large-scale visual pre-training and visual-text pre-training on the Kinetics-400 dataset in Tab. 1. All methods use the ViT-B/16 backbone. Using fewer input frames and views during testing, our method that employs only an 8-frame input evaluated with a single view achieves the highest top-1 and top-5 accuracy. For example, the second-best method of large-scale visual-text pre-training uses 4 temporal clips on 3 spatial crops of a video but achieves top-1 accuracy $0.5\%$ lower than ours. + +To further demonstrate the effectiveness of our framework, we conduct experiments on challenging scenarios from Kinetics-400 based on the ambiguous and non-uniform score. We select the top $10\%$ of actions with the highest scores, representing the most difficult categories to classify. The results are shown in Tab. 9, where we achieve an average performance $2.6\%$ higher than the second-best. + +Multi-Label Action Recognition. We evaluate our method for multi-label action recognition on the Charades dataset in Tab. 2a. The mAPs of state-of-the-art methods are reported, and we follow [9] to use 16 frames. Our method finds $51.1\mathrm{mAP}$ , achieving $0.7\mathrm{mAP}$ more than the second-best. + +Zero-shot Action Recognition. Our method is trained with supervision from texts and can be used for zero-shot action recognition. We compare our network with methods that adapt from CLIP, which is pre-trained on images, and with zero-shot methods developed for videos on the UCF-101 and HMDB-51 datasets in Tab. 2b. All methods use a single view during testing. Our method demonstrates superior generalization capabilities in zero-shot action recognition. Specifically, versus the latest zero-shot methods developed for video [38], our zero-shot performance on the HMDB51 dataset is $9.5\%$ higher than $47.1\%$ . Further highlighting the versatility of our pipeline. + +Few-shot Action Recognition. We explore 2, 4, 8, and 16-shot action recognition on the HMDB-51 [19], UCF-101 [34], and Kinetics-400 datasets [17] in Tab. 3. With a limited amount of videos, our approach that decomposes a video action into atomic actions exhibits the highest performance. Specifically, the average performance of our approach across all shots is $78.7\%$ , which is $5.8\%$ higher than the second best method (TC-CLIP, with $72.9\%$ ). + +# 4.3 Ablation Studies and Analysis + +We perform ablation studies with the Kinetics-400 dataset to examine our approach. + +Number of Sub-texts. Fig. 7a illustrates the number $N$ of sub-texts used in our model. As the $N$ increases from 2 to 5, the recognition accuracy of our method improves. However, we observe that the performance gain is minimal when increasing the number of sub-texts from 4 to 5. Considering both computational complexity and recognition accuracy, we opted for 4 sub-texts in our experiments. + +Correlation of TPP and Performance. In Fig. 7b, 11 sub-text groups are generated, and we show the relationship between TPP and the action recognition performance, where TPP is + +Table 4: Comparison of Top-1 accuracies for some representative actions with $N = 2$ and $N = 5$ sub-actions. The Difference column shows the change in Top-1 accuracy from $N = 2$ to $N = 5$ . + +
ActionN=2N=5DifferenceActionN=2N=5Difference
Dribbling basketball89.294.6+5.4Arm wrestling100.0100.00.0
Kicking soccer ball71.076.3+5.3Christmas tree100.0100.00.0
Lunge75.682.2+6.6Brushing teeth89.589.50.0
High kick35.140.5+5.4Slicing onion98.697.4-1.2
Clay pottery making92.194.7+2.6Clapping47.250.0-2.8
+ +computed as the average of $\mathrm{TPP_c}$ across all action categories. We observe a positive correlation, as shown by the fitted dashed line, with an $r^2$ value of 0.79, demonstrating the effectiveness of our TPP method in selecting sub-texts without the need for computationally intensive experiments. + +Comparison of Sub-text Generators. Tab. 2d compares sub-texts generated by GPT-Davinci [4], GPT-3.5, GPT-4o, and LLaV-v1.6-34b [22]. GPT-4o achieveshe highest TPP (60.8), along + +with the best Top-1 (84.52%) and Top-5 (96.7%) accuracies. Both LLaVA-1.6-34B and GPT-Davinci show similar performance, with minor differences in accuracy and TPP. These results indicate that GPT-4o provides the most effective sub-texts for action recognition. + +![](images/29c8826eecfbadc37247bf291a185e9fc8f4e7758595a3ccfd8a7042a71b48c6.jpg) +(a) The impact of numbers of sub-texts $(N)$ . +Figure 7: Analysis of sub-texts. + +![](images/5f76acefa66f5220d5e41700b2d4149e4650ac6cfd17c70416d4ab6f8733ca73.jpg) +(b) Study the relationship between TPP and Top-1. + +Effectiveness of Components. We study the effective usage of our coarse video embedding and fine-grained video embedding in Tab. 2e. A CLIP model is fine-tuned on the Kinetics-400 dataset to establish a performance baseline. Following the common practice of existing works [48], we build a baseline (Temporal) using a 6-layer Transformer encoder with position embedding for sequence features. Individually, both the coarse and fine-grained video embeddings enhance performance relative to the baseline; together, they achieve consistently superior results. For example, using all embeddings, we achieve a top-1 accuracy of $84.5\%$ , which is $4.6\%$ higher than the baseline. + +Different sub-actions. We previously observed that the accuracy of different action categories changes with respect to the number of sub-actions (N). Taking $\mathrm{N} = 2$ and $\mathrm{N} = 5$ as examples, the accuracy differences across representative actions are shown in Tab. 4. The results indicate that complex, multi-stage action categories (e.g., 'Dribbling basketball' and 'Clay pottery making') benefit from large N, whereas repetitive or simple motions such as 'Clapping' and 'Slicing onion' get better accuracies with small N. We had investigated overall performance gains by using class-aware N, and only found marginal improvement over fixed $\mathrm{N} = 5$ , $0.5\%$ and $0.2\%$ for Top-1 and Top-5, respectively. Given the computational cost of involving a class-aware value of N for each action, we choose to use fixed N for efficiency. + +Generalization. To validate the generalization of our components of fine-grained video embedding structure, we apply them to state-of-the-art action recognition methods, as shown in Tab. 2c. Specifically, we study VideoPrompt [15], ATM [46] and BIKE [48]. The results indicate that we enhance the accuracy of these state-of-the-art methods, such as VideoPrompt, which shows a $2.4\%$ increase in accuracy on the Kinetics-400 dataset. + +More. We provide additional implementation details, comparisons, and ablation studies in the supplement, e.g., analyses of few-shot, zero-shot, and supervised learning on other backbones. + +# 5 Conclusion + +In this paper, we propose a framework to transfer CLIP trained on image-text pairs to video action recognition. Similar to how a video forms a video action by performing a sequence of atomic actions, our key insight is to decompose a video action into atomic action descriptions using a pre-trained LLM. 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Specifically, the supplementary material is organized into three sections: (i) additional details for experiments and implementations, (ii) comparisons with other state-of-the-art methods in various settings, and (iii) additional ablation studies to analyze our model designs. + +# A Additional Experimental Setup + +# A.1 Datasets + +We describe the dataset statistics of the Kinetics-400 [17], HMDB-51 [19], UCF-101 [34], and Charades [33] datasets. The Kinetics-400 dataset is curated from YouTube, spans 400 action classes, and contains 240,000 training videos and 20,000 validation videos. The duration of each video is 10 seconds. The HMDB-51 dataset consists of 6,766 videos categorized into 51 action classes, with 3,570 videos used for training, and 1,530 videos used for testing. Each video is sourced and truncated from movies, online videos, and public databases. The UCF-101 dataset collected 9,537 training videos and 3,783 testing videos from YouTube, across 101 action categories. The Charades dataset comprises 9,848 video clips that span 157 action categories. Each video is recorded in a home environment, performed by an actor, and includes multiple human actions with temporal action annotations. + +Table 5: The hyperparameters of our model for supervised and few-shot learning. + +
Fully-supFew-shot
Optimisation
OptimizerAdamW
Optimizer betas(0.9, 0.999)
Batch size256256
Learning rate schedulecosine
Linear warmup epochs5
Base learning rate5e-65e-6
Epochs302,10
Data augmentation
RandomFlip0.5
ResizeRandomSizedCrop
Crop size224(Default)
GrayScale0.2
Other regularisation
Weight decay0.02
+ +# A.2 Ambiguous Score vs. Non-uniform Score + +We describe computations of the ambiguous score and non-uniform score for Fig. 2 in the main paper. Our method does not require ground truth action labels of videos, and only uses the CLIP, for the calculation. + +Ambiguous Score. We measure the ambiguous scores of videos by their class similarity measured with the textual encoder of CLIP. Benefiting from training on large-scale image and text pairs, the embedding space of CLIP is a compression of the world that captures the similarity between action classes. Following the global text prompt from the main paper, all action classes are extracted into $\{\mathbf{t}_{\mathrm{c}}^{\mathrm{cls}}\}_{\mathrm{c} = 1}^{C}$ with the textual encoder of CLIP. The similarity $\sigma_{\mathrm{c,c'}}$ between action class c and prediction $c'$ is calculated and normalized, + +$$ +\sigma_ {\mathrm {c}, \mathrm {c} ^ {\prime}} = \frac {\operatorname {s i m} \left(\mathbf {t} _ {\mathrm {c}} ^ {\text {c l s}}, \mathbf {t} _ {\mathrm {c} ^ {\prime}} ^ {\text {c l s}}\right) + 1}{2}. \tag {15} +$$ + +Table 6: The question prompts for GPT-Davinci [4], GPT-3.5, GPT-4o, LLaMA2 [36], Phi-3 [1], and Llava-v1.6-34b [22]. Each row is a question prompt. All LLMs use the four prompts to generate sub-texts for selection. + +
[This action is a sequence of four steps. For each step, describe it in detail with ‘This action is’ followed by a specific description of that step. Ensure each step is distinct and crucial to the action.]
[This action is defined by four unique movements. Start each movement with ‘This action is’ and emphasize the diversity and significance of each movement. Conclude with a brief summary that ties all movements together.]
[This action is composed of four critical phases. Each phase should start with ‘This action is’ and focus on different aspects of the action. Highlight the importance of each phase and avoid repetition.]
[This action is best understood by breaking it down into four concise descriptions. Begin each description with ‘This action is’ and make sure each one covers a different perspective of the action. End with a summarizing statement.]
+ +Table 7: The system prompt. We use the same system prompt for GPT-Davinci [4], GPT-3.5, GPT-4o, LLaMA2 [36], Phi-3 [1], and LLa v1.6-34b [22]. + +[You are an expert assistant in action recognition. Respond to the user's input by providing accurate, concise, and informative descriptions of the actions in a structured manner.] + +For each action class $c$ , we select the top-10 most similar other actions, and get the ambiguous scores $\mathrm{AS}_c$ by averaging the similarity scores, + +$$ +\mathrm {A S} _ {\mathrm {c}} = \sum_ {\mathrm {c} ^ {\prime} \in \operatorname {t o p} - 1 0 _ {\mathrm {c}}} \sigma_ {\mathrm {c}, \mathrm {c} ^ {\prime}}, \tag {16} +$$ + +where top- $10_{\mathrm{c}}$ find index of the top-10 similarity action class for the action class c. In Fig. 2 of the main paper, we normalize the similarity score $\mathrm{AS}_{\mathrm{c}}$ from 1 (least ambiguous) to 10 (most ambiguous) for visualization. + +Ambiguous action classes usually share different atomic actions described by the sub-texts. Thus, using the global text to identify prominent frames corresponding to the video class semantic causes ambiguity. While avoiding granularity discrepancy between the global text and video frame (Sec. Intro), we aim to decrease the granularity of the global text by augmenting it with sub-texts for ambiguous video recognition. On the ViT-B/16 backbone, we found that augmented global text achieved an accuracy of $84.5\%$ on the $1\times 1$ view, which is a $0.5\%$ improvement over the $84.0\%$ of global text, when the fine-grained branch is included. + +Non-uniform Score. We measure the non-uniform score of a video by calculating the agreements between predictions of each frame. Leveraging a CLIP, we obtain predictions of each frame and use the global text format and textual encoder of CLIP to extract embedding $\{\mathbf{t}_{\mathrm{c}^{\mathrm{pred}},\mathrm{l}}\}_{\mathrm{l} = 1}^{L}$ for all frames. The agreements between frame 1 and frame $1^{\prime}$ are measured by similarity between the embedding, and are normalized, + +$$ +\sigma_ {1, 1 ^ {\prime}} = \frac {\operatorname {s i m} \left(\mathbf {t} _ {\mathrm {c} ^ {\text {p r e d ,} 1}} ^ {\text {c l s}}, \mathbf {t} _ {\mathrm {c} ^ {\text {p r e d ,} 1 ^ {\prime}}} ^ {\text {c l s}}\right) + 1}{2}. \tag {17} +$$ + +The non-uniform score NUS of the video is an average of the agreements between frames, + +$$ +\mathrm {N U S} = \sum_ {1 = 1} ^ {L} \sum_ {1 ^ {\prime} = 1, 1 ^ {\prime} \neq 1} ^ {L} \sigma_ {1, 1 ^ {\prime}}. \tag {18} +$$ + +In Fig. 2 of the main paper, we normalize the non-uniform score from 1 (uniform) to 10 (non-uniform) for visualization. + +Table 8: Performance comparison of our coarse, fine-grained, and fused video embeddings over a single-granularity method, across different levels of ambiguous and non-uniform scores, on the Kinetics-400 dataset. "Ambi" and "Non" represent ambiguous score and non-uniform score, respectively. All numbers are Top-1 accuracy (\%). + +
MethodTypeDegree
12345678910
SingleAmbi93.090.388.384.085.782.783.076.877.269.3
Non86.584.984.685.483.982.280.881.980.380.1
CoarseAmbi94.191.589.886.587.484.884.278.979.470.8
Non91.188.787.890.786.785.083.684.783.182.9
FineAmbi94.291.189.586.487.384.784.178.679.171.8
Non90.087.886.687.286.383.982.984.082.482.2
FusionAmbi94.891.690.387.287.685.684.779.380.672.1
Non92.989.789.788.987.785.183.784.883.283.0
+ +We construct two challenging subsets of the Kinetics-400 dataset based on the ambiguous score and non-uniform score. For each criterion, we select the top $10\%$ of actions with the highest scores, representing the most difficult categories to classify. We compare the performance of our proposed method against two representative single-granularity methods ATM [46] and BIKE [48]. The results on the two challenging subsets are presented in Tab. 9. + +Table 9: Comparison of our method, BIKE [48] and ATM [46] on two challenging subsets of Kinetics-400: "Ambi Set" and "Non Set", containing the top $10\%$ most ambiguous and non-uniform actions, respectively. + +
MethodAmbi SetNon Set
ATM ViT-B/16 [46]70.279.8
BIKE ViT-B/16 [48]69.380.1
Ours ViT-B/1672.283.0
+ +Fig. 2 (f) presents a bar chart illustrating the accuracy improvements of our method's coarse video embedding, fine-grained video embedding, and their fusion, compared to the single-granularity method. The results presented in Tab. 8 are reported across different levels of ambiguous score and non-uniform score, showcasing the effectiveness of our method on the Kinetics-400 dataset. + +# A.3 Implementation Details + +We use CLIP trained from [30] in our paper. During training, the textual encoder of CLIP is frozen. We summarize the optimization and data augmentation details of our method for supervised learning + +Table 10: Comparisons of zero-shot action recognition on the UCF-101 [34] and HMDB-51 [19] datasets. The "VZ" column denotes if the method is developed for zero-shot action recognition or adapted from CLIP. Our model is based on ViT-L/14. + +
MethodVZUCF-101HMDB-51
E2E[3]✓44.129.8
ER[5]✓51.835.3
ResT[20]✓58.741.1
X-CLIP [26]✗72.044.6
DIST [29]✗72.355.4
Vita-CLIP [43]✗75.048.6
M2-CLIP [38]✗78.747.1
BIKE ViT-L [48]✗86.661.4
Text4Vis [47]✗85.858.1
Ours ViT-L/14✗87.361.9
+ +Table 11: Comparisons results on the Kinetics-400 dataset. We report the FLOPs in inference phase. "Views" indicates # temporal clip $\times$ # spatial crop. The magnitude is Million (10 $^6$ ) for parameters (Param). All methods are based on ViT-L. + +
MethodVenueInputPre-trainingTop-1(%)Top-5(%)ViewsGFLOPsParam
ViViT-L/16×2 [2]ICCV'2132×3202ImageNet-21K81.394.74×33992×12310.8
ViViT-L/16×2 [2]ICCV'2132×3202JFT-300M83.595.54×33992×12310.8
VideoSwin-L [23]CVPR'2232×3842ImageNet-21K84.996.710×52107×50200.0
ST-Adapter ViT-L/14 [28]NeurIPS'2232×2242WIT-400M87.297.63×18248-
EVL ViT-L/14 [21]ECCV'2232×2242WIT-400M87.3-3×18088-
BIKE ViT-L/14 [48]CVPR'238×2242WIT-400M86.5-1×1415307
AIM ViT-L/14 [53]ICLR'238×2242WIT-400M86.897.23×12802×1341
ATM ViT-L/14 [46]ICCV'238×2242WIT-400M87.397.44×3421×12-
MoTED ViT-L/14 [56]CVPR'248×2242WIT-400M87.497.83×18670349
Ours ViT-L/14NeurIPS'258×2242WIT-400M87.697.81×1416312
+ +Table 12: Comparisons on few-shot action recognition across the HMDB-51 [19], UCF-101 [34] and Kinetics-400 datasets [17]. + +
MethodShotHMDB-51UCF-101Kinetics-400
VideoSwin [23]220.953.3-
VideoPrompt ViT-B/16 [15]556.679.558.5
BIKE ViT-L/14 [48]273.596.175.7
BIKE ViT-L/14 [48]577.796.578.2
OST ViT-B/16 [6]264.890.3-
Ours ViT-L/14274.496.576.5
Ours ViT-L/14578.196.979.1
+ +and few-short learning in Tab. 5. For calculating TPP, the scaling functions are $\alpha(x) = -x^2 + 1$ and $\beta(x) = x$ , where $x$ is the input variable. In sub-text generation, we carefully design a list of question prompts for LLMs and select sub-texts with our TPP. We consider six LLMs: GPT-Davinci [4], GPT-3.5, GPT-4o, LLava-v1.6-34b [22], LLaMA2 [36], and Phi-3 [1]. The question prompts and the system prompt for each LLM are presented in Tab. 6 and Tab. 7, respectively. + +# B Additional Experimental Results + +We compare with state-of-the-art methods with a ViT-L/14 backbone. Following the setting of the main paper, the results of zero-shot action recognition on the UCF-101 [34] and HMDB-51 [19] datasets, supervised learning on the Kinetics-400 [17] dataset are in Tab. 10, Tab. 11. In all settings, our method consistently exhibits the highest performance. + +As shown in Tab. 13, we compared the impact of the generative capabilities of other large language models (LLMs) on recognition accuracy, with corresponding GLOps, processing time, and accuracy. The results show that even the slowest generation speed is within 10 seconds of each action. Furthermore, LLMs with strong generative capabilities have minimal impact on our method. + +Table 13: Performance comparison of Phi-3, LLaVA, LLaMA2, GPT-3.5, and GPT-4o in terms of GLOPs, processing time, and accuracy. + +
ModelGLOPsTime(s)Accuracy
Phi-3-14b6367.584.50
LLaVA-1.6-34b8.25e45.884.35
LLaMA2-7b8142.084.47
GPT-3.5-1.184.47
GPT-4o-4.684.52
+ +The common practice [40] is followed + +to only add FFN layers to the video encoder, which projects the video embedding to the CLIP feature space. It is unnecessary to employ an additional FFN layer for the text embedding, as the accuracy remains $84.5\%$ with or without it. + +We explored the use of visual context in generating sub-texts and identified two major limitations of this approach. i) Generating sub-texts for each video sample is computationally expensive, while + +Table 14: Sub-text descriptions for the action "Situp" across multiple video samples. + +
Action: SitupDescription 1Description 2Description 3Description 4
Video1The individual is lying on a mat in a gym, beginning a sit-up with arms extended overhead.They perform a fluid motion, raising their torso towards their legs while keeping their feet flat.The exercise is conducted in a controlled manner, focusing on core strength and stability.The environment suggests a well-equipped gym, appropriate for various fitness activities.
Video2An individual is seen lying on the grass performing a sit-up exercise in an outdoor setting.The exercise involves raising the upper body towards the knees, engaging the core muscles.The person is wearing casual workout attire, suitable for outdoor physical activities.The green, open space offers a natural and refreshing environment for a fitness routine.
Video3Two individuals are in a gym; one is performing an exercise on the floor while the other is walking around, possibly coaching.The environment indicates a focus on personal training, as evidenced by the sign in the background.The person on the floor appears to be doing sit-ups or a similar core exercise, suggesting a targeted workout session.The overall scene suggests a casual yet focused atmosphere, typical of a fitness or personal training session.
+ +class-level generation requires choosing a representative video, risking inconsistency and limited coverage. ii) For complex or ambiguous actions, visual context often introduces noise and spurious details, weakening frame alignment and degrading performance. To illustrate, take 'Situp' as an example, we randomly selected three sample videos and uniformly sampled eight frames from each. These were provided as input to GPT-4o to generate visually informed sub-texts as shown in Tab. 14. These sub-texts introduce scene-specific elements (e.g., attire, environment, presence of others) that do not directly contribute to the representation of atomic actions and can confound alignment. + +![](images/8a08d5307fe46880691195d829cd10ae6b01009ddaaca073daa9cb55732b9f1f.jpg) +(a) Drinking + +# GPT3.5 + +1. Selecting a drink. +2. Pouring the drink. + +3. Raising the drink to the lips. + +4. Swallowing the drink. + +# LLaMA2 + +1. Holding a drink. + +2. Bringing the drink to the mouth. + +3. Sipping the drink. +4. Swallowing the drink. + +# Phi-3 + +1. Pick up the glass. +th. 2. Bring the glass to your mouth. +3. Allow the liquid to flow into your mouth. +4. Swallow the liquid. + +![](images/a59a74936fac95c5adb5c9189c125aa47b9d6dffaa783954f74b126b10a8b4af.jpg) +(b) Eating chips +Figure 8: Example sub-texts generated from GPT-3.5, LLaMA2 [36], and Phi-3 [1]. + +# GPT3.5 + +1. Selecting chips. +2. Opening the bag. + +# LLaMA2 + +1. Picking up a chip. + +2. Bringing the chip to the mouth. + +3. Biting the chip. +4. Chewing the chip. + +# Phi-3 + +1. Picking up a chip. +2. Bringing the chip to your mouth. +3. Biting down on the chip. +4. Chewing and swallowing the chip. + +# C Additional Ablation Studies + +LLMs Selection. We ablate the LLM used for sub-text generation in Tab. 7. We experiment with GPT-Davinci [4], GPT-3.5, GPT-4o, and LLava-v1.6-34b [22]. Sub-text groups generated from GPT-4o have the highest TPP and action recognition performance. + +Visualization. Additional example sub-texts generated by GPT-3.5, LLaMA2 [36], and Phi-3 [1] are presented in Fig. 8. Furthermore, We provide more visualizations than Fig. 2 of the main paper in Fig. 9 and Fig. 10 on Kinetics-400 [17], UCF-101 [34], and HMDB-51 [19] datasets. + +Limitation and Future Work Fine-grained action recognition is challenging, especially in complex scenarios, with no satisfactory solution even from large vision-language models like GPT-4v. Our method mitigates this by generating sub-texts once for each global action, but future work could explore more robust approaches. + +Broader Impacts Our work advances the study of fine-grained action recognition methods and holds promise for a wide range of applications, including sports analytics, human-computer interaction, surveillance, video understanding, and related areas. + +![](images/693ec34aed4967236486f2ac618069e60c06887a1dfed01c94a31ac341f2cd4f.jpg) +Figure 9: Examples that illustrate the ineffectiveness of directly aligning video frames and global text, from (a-e) heatmaps. (a) Video frames from the Kinetics-400 dataset [17] with their class name annotated at the left. (b) Heatmaps of the single granularity-based (single) method [48], which are limited by the granularity difference between the video frame and global text. (c-e) Heatmaps using coarse video embedding, fine-grained video embedding, and the fusion of the two. + +![](images/a6d8e559bd46ae8948740004204ea6740681995a52788acb716240bc9c46efa2.jpg) +Figure 10: Examples that illustrate the ineffectiveness of directly aligning video frames and global text, from (a-e) heatmaps. (a) Video frames from the UCF-101 dataset [34] and HMDB-51 dataset [19] with their class name annotated at the left. (b) Heatmaps of the single granularity-based (single) method [48], which are limited by the granularity difference between the video frame and global text. 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In this paper, we introduce a novel Straight-Line Diffusion Model (SLDM) to tackle this problem, by formulating the diffusion process to follow a linear trajectory. The proposed process aligns well with the noise sensitivity characteristic of molecular structures and uniformly distributes reconstruction effort across the generative process, thus enhancing learning efficiency and efficacy. Consequently, SLDM achieves state-of-the-art performance on 3D molecule generation benchmarks, delivering a 100-fold improvement in sampling efficiency. $^{1}$ + +# 1 Introduction + +3D molecular generation is an essential task in drug discovery, material science, and molecular engineering. The goal is to computationally design 3D molecular structures that not only capture intricate physical and chemical constraints but also fulfill specific properties. + +Recently, diffusion models have been widely applied in this field, inspired by their remarkable success in image synthesis [Dhariwal and Nichol, 2021, Rombach et al., 2022, Peebles and Xie, 2023], and other domains [Brooks et al., 2024, Abramson et al., 2024]. Methods like EDM [Hoogeboom et al., 2022], EDM-Bridge [Wu et al., 2022] and GeoLDM [Xu et al., 2023] have demonstrated the potential of diffusion-based frameworks to generate chemically valid 3D molecular structures. However, these direct applications of diffusion methods usually require a large number of sampling steps to produce valid molecules. + +![](images/17db000d217846bf5f4147d8939caeb9ddcac0d45ecd580e969b67fb7e4314b4.jpg) +Figure 1: Comparison of molecule stability $(\uparrow)$ across diffusion-based molecular generation models on QM9 unconditional generation, evaluated by the number of function evaluations (NFE) during the sampling process. + +*Equal contribution. † Work was done during Yuyan's internship at AIR. ‡ Corresponding author: Yanyan Lan (lanyanyan@air.tsinghua.edu.cn). + +1The code is open-sourced at https://github.com/fengshikun/SLDM + +Taking EDM as an example, it requires approximately 1000 steps of function evaluations to generate molecules with around $82\%$ stability, which is a key metric for assessing sample quality by quantitatively measuring whether the molecule satisfies chemical constraints. Reducing the sampling steps significantly degrades the molecule stability. + +To improve sampling efficiency, EquiFM [Song et al., 2024] and GeoBFN [Song et al., 2023a] have been proposed to utilize the Flow Matching (FM) framework [Tong et al., 2023] and Bayesian Flow Networks (BFN) [Graves et al., 2023] for molecular generation. The use of these advanced generative AI models enables a speedup of $5 \times$ and $20 \times$ , respectively, compared to EDM. However, they still require a large number of sampling steps (e.g. 1000) to achieve high molecule stability (e.g. $90\%$ ), as shown in Figure 1. + +To understand why existing methods suffer from low efficiency, we analyze the issue through the lens of truncation error in sampling. We begin by establishing a unified perspective on previous diffusion-based methods, including diffusion models, flow matching methods, and Bayesian flow networks. Specifically, their noise corrupting process can be generalized as $\boldsymbol{x}_t = \mu(t)\boldsymbol{x}_0 + \sigma(t)\boldsymbol{\epsilon}, \boldsymbol{\epsilon} \sim \mathcal{N}(\boldsymbol{0},\boldsymbol{I}_N)$ , where $\boldsymbol{x}_0$ represents the clean data, $\boldsymbol{x}_t$ is the noise corrupted data at time $t \in [0,T]$ , $\mu(t)$ and $\sigma(t)$ define the schedule of the process. In this framework, all these processes can be equivalently framed as continuous-time Ordinary Differential Equations (ODEs), even though they may employ stochastic sampling in practice. This viewpoint allows the sampling process to be interpreted as a numerical approximation of the solution trajectory of the underlying ODE. Crucially, most existing methods rely on first-order estimation, whose truncation error is governed by the second-order term $\frac{\mathrm{d}^2x(t)}{\mathrm{d}t^2} (\Delta t)^2$ . We observe that $\frac{\mathrm{d}^2x(t)}{\mathrm{d}t^2}$ in these approaches can be large, requiring small step sizes $\Delta t$ to reduce the truncation error, which results in a large number of sampling steps. + +To address this issue, we propose a novel diffusion process called Straight-Line Diffusion Model (SLDM). The key idea is to minimize truncation error by striving to achieve a linear sampling trajectory, i.e. $\frac{\mathrm{d}^2x(t)}{\mathrm{d}t^2} = 0$ . This approach allows the diffusion dynamics to tolerate larger step sizes without sacrificing accuracy, leading to a substantial improvement in sampling efficiency. Building on this objective, we theoretically prove that when $\mu (t) = 1 - t / T$ and $\sigma$ is a small constant, the process guarantees a near-linear trajectory. Intuitively, this process features a linearly decreasing mean term and a consistently small variance term, representing a smooth linear progression from the origin point to the data. + +Notably, SLDM strikes a good balance between efficiency and efficacy by using our straight-line schedule. Firstly, this strategy aligns well with the inductive bias of molecular generation, preventing the introduction of chemically implausible conformations. Unlike images, molecular structures are much more sensitive to noise, and even small perturbations can lead to unrealistic structures that violate chemical principles. This challenge requires a slower signal-to-noise ratio (SNR) decay during the noise-adding process, as suggested in Song et al. [2023a]. By naturally satisfying a slower SNR decay, our method maintains chemical information of the intermediate states, enhancing computational efficiency compared to traditional methods such as EDM. Secondly, our strategy achieves a more balanced generative process, significantly improving the model's learning efficacy. In methods like GeoBFN, minimal perturbations are applied in the later stages, shifting most of the reconstruction burden to earlier stages. Although this reduces the computational load in the later stages, it creates an uneven distribution of effort, limiting the model's learning capacity. In contrast, our approach evenly distributes the reconstruction effort across the entire process, enabling the model to learn effectively at each stage. This balance results in a more stable and efficient learning process, enhancing the robustness and accuracy of the generated molecular structures. + +We conduct extensive experiments to demonstrate the potential of straight-line diffusion in 3D molecular generation and other domains. As shown in Figure 1, using only at most 10 or 15 sampling steps, SLDM surpasses EDM or EquiFM, GeoBFN with 1000 sampling steps, achieving up to 100- or 70-fold improvement in sampling efficiency. In terms of generation quality, SLDM with 200 sampling steps achieves $95\%$ molecular stability, significantly outperforming the best baseline, GeoBFN, which requires 1000 steps to reach $90\%$ molecular stability. We also observe that similar improvements can be achieved when applying SLDM to the conditional generation task, i.e. generating molecules with a desired property, highlighting its potential to enable more practical and controllable molecular design in future applications. + +# 2 Analysis on Sampling Efficiency + +We begin by theoretically analyzing the underlying factors contributing to the sampling efficiency issue. In particular, we first present a unified framework for diffusion-based methods, including their SDE and ODE formulations. We then examine the sampling truncation error from the ODE perspective, highlighting the critical role of the process's second-order derivative in improving sampling efficiency. + +We denote the data as $\pmb{x} \in \mathbb{R}^N$ , where for molecules, it refers to a 3D point cloud comprising atomic coordinates and potentially other atomic features. According to Karras et al. [2022], Xue et al. [2024a], various diffusion-based models, including DDPM [Ho et al., 2020], DDIM [Song et al., 2021a], VE [Song et al., 2021b], FM [Lipman et al., 2023], and BFN [Graves et al., 2023], can be formulated as a unified form with the noise corrupting process defined as: + +$$ +\boldsymbol {x} _ {t} = \mu (t) \boldsymbol {x} _ {0} + \sigma (t) \boldsymbol {\epsilon}, \boldsymbol {\epsilon} \sim \mathcal {N} (\boldsymbol {0}, \boldsymbol {I} _ {N}), t \in [ 0, T ] \tag {1} +$$ + +where $\boldsymbol{x}_0, \boldsymbol{x}_t$ are clean data and noise corrupted data respectively, $\mu(t)$ and $\sigma(t)$ define the schedule of the process. Specifically, as detailed in Appendix A.2, the schedule parameters are summarized as follows: $\mu_{\mathrm{DDPM(EDM)}} = 1 - (t / T)^2$ , $\mu_{\mathrm{VE}} = \mu_{\mathrm{DDIM}} = 1$ , $\mu_{\mathrm{BFN}} = 1 - \sigma_{min}^{2(1 - t / T)}$ , $\mu_{\mathrm{FM}} = 1 - t / T$ ; $\sigma_{\mathrm{DDPM(EDM)}} = \sqrt{1 - (1 - (t / T)^2)^2}$ , $\sigma_{\mathrm{VE}} = \sqrt{t}$ , $\sigma_{\mathrm{DDIM}} = t$ , $\sigma_{\mathrm{BFN}} = \sqrt{\mu_{\mathrm{BFN}}(1 - \mu_{\mathrm{BFN}})}$ , $\sigma_{\mathrm{FM}} = t / T + (1 - t / T)\sigma_{min}$ , where DDPM(EDM) uses the approximated DDPM schedule given in EDM [Hoogeboom et al., 2022]. $\sigma_{min}$ are defined as small constants to ensure $\mu(0) \approx 1$ and $\sigma(0) \approx 0$ . $T$ is typically chosen to be sufficiently large so that $x_T$ approximates a known distribution. + +Extending a similar theoretical technique from Karras et al. [2022] to the unified form, we can prove that equation 1 is the solution to the following linear stochastic differential equation (SDE): + +$$ +\mathrm {d} \boldsymbol {x} _ {t} = \frac {\dot {\mu} (t)}{\mu (t)} \boldsymbol {x} _ {t} \mathrm {d} t + \sqrt {2 \sigma (t) \dot {\sigma} (t) - 2 \sigma (t) ^ {2} \frac {\dot {\mu} (t)}{\mu (t)}} \mathrm {d} w _ {t}, \tag {2} +$$ + +where $\mu (t)$ and $\sigma (t)$ should adhere to the constraints $\mu (0) = 1$ , $\sigma (0) = 0$ , with $\mu (t)\geq 0$ being monotone non-increasing, $\sigma (t)\geq 0$ , and $\sigma (t) / \mu (t)$ being monotone non-decreasing. The detailed proof is elaborated in Appendix A.1. + +According to the ODE and SDE relations revealed in Song et al. [2021b], we can derive the equivalent ODE that follows the same marginal probability densities as the above SDE: + +$$ +\mathrm {d} \boldsymbol {x} _ {t} = \left[ \frac {\dot {\mu} (t)}{\mu (t)} \boldsymbol {x} _ {t} - \left(\sigma (t) \dot {\sigma} (t) - \sigma (t) ^ {2} \frac {\dot {\mu} (t)}{\mu (t)}\right) \nabla_ {\boldsymbol {x}} \log p _ {t} (\boldsymbol {x} _ {t}) \right] \mathrm {d} t. \tag {3} +$$ + +Consequently, most existing diffusion-based methods widely used in 3D molecular generation, including DDPM, DDIM, VE, FM, and BFN, can be interpreted from a continuous-time perspective, where their diffusion process can equivalently be viewed as an SDE in equation 2 or an ODE in equation 3. Therefore, we can use the ODE formulation as a valuable perspective to analyze the sampling efficiency issue. + +Specifically, the diffusion sampling process can be interpreted as a numerical approximation of the backward solution trajectory of the underlying ODE in equation $3^{2}$ . This numerical approximation inherently introduces truncation errors at each step. For example, under Euler's method, the simplest and widely-used numerical scheme, $x(t - \Delta t)$ is approximated by $x(t) - \frac{\mathrm{d}x(t)}{\mathrm{d}t}\Delta t$ when solving the ODE backward in time. However, the true value can be derived from the Taylor expansion: + +$$ +x (t - \Delta t) = x (t) - \frac {\mathrm {d} x (t)}{\mathrm {d} t} \Delta t + \frac {1}{2} \frac {\mathrm {d} ^ {2} x (t)}{\mathrm {d} t ^ {2}} (\Delta t) ^ {2} + O ((\Delta t) ^ {3}), \tag {4} +$$ + +where $\Delta t$ is a small step size. Thus, the truncation error primarily arises from the second-order term, governed by $\frac{\mathrm{d}^2x(t)}{\mathrm{d}t^2}$ . + +To minimize this truncation error, traditional numerical analysis primarily focuses on developing higher-order solvers to estimate higher-order terms, assuming the ODE structure is fixed. EquiFM + +[Song et al., 2024] applied such a technique in 3D molecular generation to speedup the sampling process. However, higher-order solvers face a trade-off between the number of function evaluations (NFE) and accuracy, limiting their ability to achieve even smaller NFEs. As a result, the issue remains partially unsolved. In contrast, we take a different approach: rather than focusing on the sampling algorithm, we reformulate the diffusion process itself to minimize the truncation error, which directly influences the sampling efficiency. + +However, our unified formulation of diffusion introduces a key flexibility: the ability to modify the schedule of the process, thereby altering the ODE structure itself to reduce these errors. Therefore, we emphasize that the key to minimizing the truncation error is to reduce the second-order derivative of the process. + +# 3 Straight-Line Diffusion Process + +To reduce the truncation error of the sampling process, we aim to design a new diffusion process whose inherent ODE exhibits minimal $\frac{\mathrm{d}^2x(t)}{\mathrm{d}t^2}$ . By achieving this, even a basic Euler iteration can deliver low truncation error without resorting to complex solvers that require multiple function evaluations. This novel perspective advances the Pareto frontier of efficiency and accuracy in diffusion sampling. + +# 3.1 Derivation of SLDM + +Given the intractability of the score function for general data distributions, we begin by examining a simplified case where the initial distribution is a delta distribution. This setup provides a tractable backward ODE, enabling a more straightforward analysis. In this case, the only solution that ensures $\frac{\mathrm{d}\boldsymbol{x}}{\mathrm{d}t}$ remains constant is for $\sigma (t)$ to be constant and $\mu (t)$ to be a linear function of $t$ . Details can be found in Appendix A.4. + +Given the boundary condition that $x_0$ approximates data distribution and $x_{t=1}$ approximates a known distribution, the only feasible choice is to set $\sigma(t)$ to a small positive constant and $\mu(t) = 1 - t / T$ , which satisfies all the schedule constraints noted under equation 2, except $\sigma(0) = 0$ . This is because setting $\sigma(0) = 0$ exactly would result in a trivial denoising objective, To avoid this, we instead approximate $\sigma(0)$ by a small value, which keeps the denoising task non-trivial while still satisfying the boundary condition approximately. In practice, we use a value for $\sigma$ that is two orders of magnitude smaller than the data scale, yielding good results. + +Rescaling time to the interval [0, 1], the resulting diffusion process is: + +$$ +\boldsymbol {x} _ {t} = (1 - t) \boldsymbol {x} _ {0} + \sigma \boldsymbol {\epsilon}, t \in [ 0, 1 ], \tag {5} +$$ + +which ensures a straight-line trajectory under the delta data distribution assumption. + +We then analyze the above diffusion process for general data distribution and demonstrate that a near-linear trajectory could be achieved by setting a small constant value for $\sigma$ , as shown in the following theorem. + +Theorem 3.1 (Near-linear Trajectory of SLDM). For a general data distribution and the schedule in equation 5, the following inequality holds for each dimension $i$ : + +$$ +P \left(\left| \frac {\mathrm {d} \boldsymbol {x} _ {t} ^ {(i)}}{\mathrm {d} t} + \frac {\boldsymbol {x} _ {t} ^ {(i)}}{1 - t} \right| \geq \delta\right) \leq \frac {\sigma^ {2}}{\delta^ {2} (1 - t) ^ {2}}. \tag {6} +$$ + +When $\sigma \to 0$ , $\frac{\mathrm{d}\boldsymbol{x}_t}{\mathrm{d}t} + \frac{\boldsymbol{x}_t}{1 - t}$ converges to zero in probability, where the solution to equation $\frac{\mathrm{d}\boldsymbol{x}_t}{\mathrm{d}t} + \frac{\boldsymbol{x}_t}{1 - t} = 0$ is that $\frac{\boldsymbol{x}_t}{1 - t}$ is constant, which corresponds to a linear trajectory. + +In other words, for $t \in [0, 1 - \frac{\sigma}{\delta \cdot \epsilon}]$ , $\left| \frac{\mathrm{d}x_t^{(i)}}{\mathrm{d}t} + \frac{x_t^{(i)}}{1 - t} \right| < \delta$ holds with probability at least $1 - \epsilon^2$ , indicating that setting $\sigma$ to a small value ensures that the trajectory remains close to a straight line for most timesteps. This aligns with our empirical observations shown in Figure 6: the overall SLDM trajectory is straighter compared to other diffusion processes, and the sampling trajectory deviates more from a straight line in the initial steps but becomes increasingly linear later. + +To further improve the sampling quality, we aim to mitigate the impact of early-stage truncation errors. This is achieved by our stochastic sampling design, which exhibits robustness to such early-stage errors, as supported by empirical evidence provided in Appendix C.6. As a result, despite the + +![](images/4495047f42ce9d9a885dad261d1fbacf7fd11f776bda6718999e59459f555de4.jpg) +Figure 2: The diffusion process of atomic coordinates in EDM, GeoBFN and SLDM. + +challenges in minimizing the early-stage truncation error, our method manage to achieve high-quality samples with few sampling steps. + +# 3.2 Comparisons with Previous Diffusion Processes + +Previous studies [Nichol and Dhariwal, 2021, Rombach et al., 2022] have underscored the importance of amortizing the generative difficulty through the diffusion process to improve sample quality. These works typically customize signal-to-noise ratio (SNR) schedules to adapt to varying data characteristics. Notably, as pointed out in Song et al. [2023a], the point cloud representation of molecular structures is far more sensitive to noise compared to images. Consequently, for molecular generation, the SNR needs to be decreased at a much slower pace than in the image generation domain. + +Our proposed method aligns remarkably well with these insights. As shown in Figure 3, the SNR in our method decreases significantly slower than that of previous methods. In addition, our schedule results in a smoother and more stable generative process for molecules, as demonstrated in Figure 2. The process unfolds uniformly from the origin, preserving the relative spatial relationships of atomic coordinates in intermediate states and retaining critical chemical information throughout the generative trajectory. + +In contrast, traditional diffusion models like EDM, which follows the process schedule of DDPM, also known as VP, involve a noise + +![](images/d486641aa249c6250e75379826d99f13844eb0205e07fc1d6ed120f1ea49c1bd.jpg) +Figure 3: Comparison of schedule parameters across diffusion-based models. Here SNR refers to $(\mu /\sigma)^{2}$ For SLDM, we set $\sigma = 0.05$ , consistent with experiments. The cutoff values $\sigma_{\mathrm{max}}$ and $\sigma_{\mathrm{min}}$ used in other baselines are provided in Appendix A.2. + +![](images/9c4595b6118e7525a3c31c974560635117a6ae5c07414c597c3e7aeb08b225a9.jpg) + +![](images/dca0e9b25a7c038c2599241612dd0b6080395ab7003ed44dd5fa43bbc643de5c.jpg) + +variance that increases monotonically during the forward process. This increasing noise quickly disrupts the spatial structure of molecules, rendering much of the diffusion process chemically meaningless. As a result, many computational resources are wasted on steps that attempt to reconstruct signal-less states, leading to a severe reduction in overall model efficiency. Though GeoBFN adopts a low-noise regime for most timesteps, it keeps $\mu \approx 1$ and $\sigma \approx 0$ for over a half of the process, making little changes to the molecular structure. Therefore, GeoBFN shifts the majority of reconstruction difficulty to the earlier stages, creating a severely unbalanced generative process. More illustrations of generation process are provided in Appendix C.7 + +To sum up, compared to existing diffusion-based models, the new schedule in SLDM aligns better with the inductive biases of molecular data. The approach of distributing the reconstruction difficulty more evenly across the entire diffusion process helps facilitate more effective learning, improving the model's efficiency and efficacy. + +# 3.3 Sampling Strategy of SLDM + +Here we introduce the specific sampling strategy of SLDM. First, we need to derive the reverse process of equation 2 or equivalently equation 3. Notably, previous work [Xue et al., 2024b] (Equation 6) provides a unified form of the reverse-time stochastic differential equation (SDE), which shares the same marginal distribution as the forward process, thereby ensuring that the sampling process has the ability to reconstruct the data distribution. We apply the parameter $\mu$ and $\sigma$ of SLDM to this equation and then discretize it using the Euler-Maruyama method, yielding the following discretized + +Algorithm 1 SLDM Training + +Input: data $\mathbf{x}_0$ with dimension $N$ , neural network $\phi$ , constant $\sigma$ + +repeat + +Sample $t\sim U([0,1])$ Sample $\epsilon \sim \mathcal{N}(\mathbf{0},I_N)$ Compute $\pmb {x}_t = (1 - t)\pmb {x}_0 + \sigma \pmb{\epsilon}$ Minimize $||\pmb {\epsilon} - \phi (\pmb {x}_t,t)||^2$ + +until converged; + +return $\phi$ + +Algorithm 2 SLDM Sampling + +Input: trained network $\phi$ , same $N$ and $\sigma$ as training, sampling steps $T$ , temperature annealing rate $\nu$ . Sample $\boldsymbol{x} \sim \sigma \cdot \mathcal{N}(\mathbf{0}, I_N)$ + +$$ +\text {f o r} i = T - 1 \text {t o} 1 \text {d o} +$$ + +$$ +\boldsymbol {x} \leftarrow \frac {T - (i - 1)}{T - i} \cdot (\boldsymbol {x} - \sigma \cdot \phi (\boldsymbol {x}, t)) +$$ + +$$ +i > 1 \text {t h e n} +$$ + +$$ +\left| \quad \boldsymbol {x} \leftarrow \boldsymbol {x} + \left(\frac {i}{T}\right) ^ {\nu} \sqrt {2} \cdot \sigma \cdot \epsilon , \epsilon \sim \mathcal {N} (\boldsymbol {0}, I _ {N}) \right. +$$ + +$$ +\mathrm {e n d} +$$ + +end + +return $\pmb{x}$ + +reverse-time SDE: + +$$ +\boldsymbol {x} _ {t - \Delta t} = \underbrace {\boldsymbol {x} _ {t} + \frac {\Delta t}{1 - t} \left(\boldsymbol {x} _ {t} + \sigma^ {2} \nabla_ {\boldsymbol {x}} \log p _ {t} (\boldsymbol {x} _ {t})\right)} _ {\text {O D E S a m p l i n g o f e q u a t i o n 3}} + \underbrace {\beta (t) \frac {\Delta t}{1 - t} \sigma^ {2} \nabla_ {\boldsymbol {x}} \log p _ {t} (\boldsymbol {x} _ {t}) + \sqrt {2 \beta (t) \frac {\Delta t}{1 - t}} \sigma \epsilon} _ {\text {L a n g e v i n d y n a m i c s}}, \tag {7} +$$ + +where $\beta (t)$ is any non-negative bounded function. + +Similar to previous works [Xue et al., 2024b, Karras et al., 2022], this sampling can be interpreted as a combination of ODE sampling and Langevin dynamics. The ODE component drives the denoising process along deterministic trajectories, while the Langevin dynamics introduce stochastic corrections. Specifically, we choose $\beta(t) = (1 - t) / \Delta t$ , to ensure the expectation of the RHS of equation 7 is $\mathbb{E}[\pmb{x}_{t - \Delta t}|\pmb{x}_t]$ , as it provides the optimal estimation of $\pmb{x}_{t - \Delta t}$ in terms of minimizing the mean squared error (MSE). Details are demonstrated in Appendix A.5. + +Thus the sampling algorithm becomes: + +$$ +\boldsymbol {x} _ {t - \Delta t} = \frac {1 - (t - \Delta t)}{1 - t} \left(\boldsymbol {x} _ {t} + \sigma^ {2} \nabla_ {\boldsymbol {x}} \log p _ {t} \left(\boldsymbol {x} _ {t}\right)\right) + \sqrt {2} \sigma \boldsymbol {\epsilon}. \tag {8} +$$ + +Note that $1 - t$ appears in the denominator of the above equation. To avoid potential division by zero, we choose to skip the $t = 1$ sampling step and find that it works well. This might be because the early sampling steps are less critical to the final result, likely due to Langevin dynamics not strictly requiring a specific prior, which imparts an inherent error-correction capability to the algorithm. This property is further supported by our observation that the sampling process remains relatively robust even when initialized from an invalid distribution, as demonstrated in Appendix C.6. + +The above sampling algorithm provides a principled way to approximately sample from the input data distribution, but its practical application in molecular datasets presents unique challenges. In particular, the molecular stability of the dataset is not guaranteed to be $100\%$ . To enhance stability and mitigate the impact of data noise, low-temperature sampling can be employed to generate samples that preserve essential properties of the input data. For Langevin dynamics, the sampling temperature can be controlled by scaling the stochastic term with a constant. However, prior work has demonstrated that such temperature control in diffusion model sampling often fails to effectively balance diversity and fidelity [Dhariwal and Nichol, 2021]. Additionally, as observed in our toy dataset experiments, conventional low-temperature sampling suffers from mode collapse, failing to fully cover all high-density regions of the target distribution. + +To address these limitations, we propose a time-annealing temperature schedule: + +$$ +\boldsymbol {x} _ {t - \Delta t} = \frac {1 - (t - \Delta t)}{1 - t} \left(\boldsymbol {x} _ {t} + \sigma^ {2} \nabla_ {\boldsymbol {x}} \log p _ {t} \left(\boldsymbol {x} _ {t}\right)\right) + t ^ {\nu} \sqrt {2} \sigma \boldsymbol {\epsilon}, \tag {9} +$$ + +where $\nu$ controls the decay rate of temperature over time. This approach allows for higher stochasticity during the initial stages of sampling, enabling the exploration of distant probability density maxima. As the temperature decreases in later stages, the process converges to the probability density maxima. Further theoretical explanations and toy data illustrations are provided in Appendix A.6. With the help of the temperature control, our sampling strategy can reduce the impact of noise in the data, and generate molecules with enhanced stability. The complete training and sampling procedure of + +straight-line diffusion are given in algorithm 1 and 2. We further discussed the relation to relevant techniques used in general generative approaches in Related Work (Appendix D.2), including noise scheduling, flow-based methods, other straight-line trajectory models, etc. + +Table 1: Unconditional molecular generation results on QM9 and GEOM-Drugs datasets. For all diffusion-based models, $T$ denotes sampling steps. Metrics are calculated with 10000 samples generated from each model. Higher values indicate better performance. + +
QM9GEOM-Drugs
#MetricsAtom sta(%)Mol sta(%)Valid(%)V*U(%)Atom sta(%)Valid(%)
Data99.095.297.797.786.599.9
E-NF85.04.940.239.4--
G-Schnet95.768.185.580.3--
EDM (T=1000)98.782.091.990.781.392.6
GDM (T=1000)97.671.690.489.577.791.8
EDM-Bridge (T=1000)98.884.692.090.782.492.8
GeoLDM (T=1000)98.989.493.892.784.499.3
EquiFM (T=200)98.988.394.793.584.198.9
GeoBFN (T=1000)99.0890.8795.3192.9685.6092.08
END (T=1000)98.989.194.892.687.089.2
SLDM (T=1000)99.4395.4297.0790.4288.3099.95
SLDM (T=50)99.3093.3796.2493.6389.0399.57
+ +# 4 Experiment + +To validate the advantages of our method in molecular generation, we evaluate its overall performance and sampling efficiency in both unconditional and conditional generation scenarios. + +# 4.1 Setup + +Datasets We evaluate our model using two widely adopted datasets for unconditional molecular generation, with all dataset splitting strictly following baseline settings [Hoogeboom et al., 2022, Song et al., 2024, 2023a]. QM9 [Ruddigkeit et al., 2012, Ramakrishnan et al., 2014] contains approximately 134,000 small organic molecules with up to nine heavy atoms. It is split into training (100K), validation (18K), and test (13K) sets. GEOM-Drugs [Axelrod and Gomez-Bombarelli, 2022] focuses on drug-like molecules, comprising around 430,000 molecules with sizes ranging up to 181 atoms and an average of 44.4 atoms per molecule. Its larger size and greater diversity make it more challenging for generative models. The dataset is randomly divided into training, validation, and test sets using an 8:1:1 ratio. + +For conditional molecular generation, we adopt the QM9 dataset with the same setup as prior work [Hoogeboom et al., 2022, Song et al., 2024, 2023a]. The QM9 training partition is split into two halves, each containing 50K samples. Specifically, the QM9 training set is divided into two halves of 50K samples each. The first half is used to train a classifier for ground-truth property labels, while the second half is used to train the conditional generative model. + +Implementation The molecule is represented by atomic coordinates and atom types, $\boldsymbol{z} = (\boldsymbol{x}, \boldsymbol{h})$ , where $\boldsymbol{x} \in \mathbb{R}^{3M}$ denotes the atomic coordinates, $M$ is the number of atoms, and $\boldsymbol{h}$ encodes the atom type information. Thus for molecule generation, the model needs to generate both coordinates and atom types. A key requirement for the molecular coordinates is ensuring the SE(3) invariance of the probability distribution, meaning that the probability of generating two molecular conformations should be identical if they only differ by translation or rotation. Following Xu et al. [2022], Hoogeboom et al. [2022], we tailor the SLDM algorithm to satisfy equivariance. Specifically, to ensure translation invariance, we constrain the coordinates to a zero Center of Mass (CoM) space, while rotation invariance is preserved by employing equivariant neural networks to predict the noise. To ensure a fair comparison of generative algorithms, we use EGNN [Satorras et al., 2021] as the backbone model, consistent with the baseline methods [Garcia Satorras et al., 2021, Hoogeboom et al., 2022, Wu et al., 2022, Xu et al., 2023, Song et al., 2024, 2023a]. We prove that the generated data distribution satisfies SE(3) invariance, as provided in Appendix A.7. For atom types, we follow + +![](images/f5c6691f11d721b1e1c97f3bc379345ac2dc964fb6eef466915275612152da8e.jpg) +Figure 4: Comparison of performance in conditional generation $(\downarrow)$ , with respect to the number of function evaluations (NFE) during the sampling process. + +![](images/6642142c9a58d2ddb753d1b77f1925771e4e44b7b8efc9f7485c33f968e0201f.jpg) + +![](images/33ed84468af1e2e537215bcb60844c78cd14388d34c1133f625b6c5d7c09a82f.jpg) + +![](images/4efbacf80aca78ebbc5ddc6ab056933b33e834327b83efa88dc2d7fb9e718e40.jpg) + +![](images/9f277a9acfd0e8b4452bb3a8b4a2fd1a48b98e23b975dd5ee214e3217a946ca2.jpg) + +![](images/f7c7cf91f4dcd9fbce884452011e591091ba8fd2df518a66a4a6f82414fbe978.jpg) + +UniGEM [Feng et al., 2024] to predict atom types based on the generated coordinates. The SLDM algorithms tailored for molecular generation are provided in Appendix B. Hyperparameters are summarized in Appendix E. An introduction to the baseline models is included in Section D.1. + +Baselines To ensure a fair comparison, we select competitive baselines that also focus on generative modeling and have the same neural architectures (e.g., EGNN) and input information (e.g., coordinates and types). We also acknowledge that some studies propose strategies orthogonal to generative algorithms, as discussed in Related Work (Appendix D.1). However, these strategies are not directly comparable to our work, as they do not aim to replace or improve the generative model itself but rather focus on improving network expressivity and augmenting input information. We believe that substituting their generative algorithms with ours could lead to improvements, and exploring this will be part of the future work. + +For conditional molecular generation, Two additional basic baselines: Random and $N_{\mathrm{atoms}}$ follow EDM [Hoogeboom et al., 2022]. The Random baseline involves shuffling property labels in the training data and evaluating the property classifier on the shuffled data. The $N_{\mathrm{atoms}}$ baseline uses the property classifier network to predict the property based solely on the number of atoms in the molecule. + +Metrics It is important to note that evaluation protocols differ across the literature, as discussed in section D.1. To ensure consistency, we strictly adhere to the evaluation methods used in our baselines. We sample 10,000 molecules and predict the bond type (single, double, triple, or non-existent) based on the distances between each pair of atoms, as in Hoogeboom et al. [2022]. Atom stability is computed as the proportion of atoms with correct valency, and molecule stability is the fraction of generated molecules in which all atoms are stable. Validity is evaluated using RDKit by checking whether the 3D molecular structures can be successfully parsed into SMILES format. Uniqueness is determined as the ratio of distinct molecules among all valid samples, indicating the diversity of generated molecules. $\mathrm{V}^{*}\mathrm{U}$ means the ratio of valid and unique molecules. + +# 4.2 Unconditional Molecular Generation + +Unconditional generation assesses the model's ability to learn the underlying molecular data distribution, aiming to generate chemically valid and structurally diverse molecules. The results, summarized in Table 1, show that our method significantly outperforms the baselines across both quality and diversity metrics for the generated molecules. Of particular note is the substantial improvement in the stability of the generated molecules, indicating that our method better satisfies chemical constraints. This validates our hypothesis that our low-noise dynamic is well-suited for generating molecular data. A visualization of generated molecules is provided in Appendix C.7. + +Additionally, our approach can achieve superior results with significantly fewer generative steps. A detailed comparison, presented in Figure 1, demonstrates that SLDM achieves $100 \times$ faster sampling than the baseline EDM and around $70 \times$ speedup compared to GeoBFN and EquiFM. Since the methods use the same network architecture, NFE can reflect practical sampling time. It is important to note that EquiFM utilizes a variety of advanced and efficient ODE solvers, and the results we present correspond to the best performance reported in their paper for different step sizes. Besides, EDM and EDM-Bridge results are from the EDM-Bridge paper while GeoBFN results are reproduced from the official codebase. This observation underscores the advantages of our straight-line diffusion process: while sophisticated solvers play an important role, the diffusion process design may offer more fundamental improvements. Furthermore, our handcrafted sampling strategies can be seamlessly combined with modern sampling techniques, such as optimal time discretization and advanced solvers, which we leave for future exploration. + +Table 2: Conditional generation on QM9 dataset, evaluated by MAE $(\downarrow)$ between the property condition and the properties of generated molecules predicted by a pretrained EGNN classifier. SLDM uses $T = 1000$ + +
Property Unitsα Bohr3Δε meVεHOMO meVεLUMO meVμ DCv cal mol K
Data0.106439360.0430.040
Random9.01147064514571.6166.857
N-atoms3.868664268131.0531.971
EDM2.766553565841.1111.101
GeoLDM2.375873405221.1081.025
EquiFM2.415913375301.1061.033
GeoBFN2.345773285160.9980.949
SLDM1.464403203480.7970.745
+ +Table 3: Comparison of generative methods within the UniGEM framework, marked by $U$ , for unconditional generation on the QM9 dataset using small sampling steps $T$ . Higher values indicate better performance. + +
TModelAtom sta(%)Mol sta(%)Valid(%)V*U(%)
50EDMU98.5585.7393.2991.78
50GeoBFNU98.2887.1693.9791.82
50SLDM99.3093.3796.2493.63
30EDMU97.5878.7589.3987.96
30GeoBFNU96.7481.0590.9387.47
30SLDM99.3093.0296.2092.76
+ +# 4.3 Conditional Molecule Generation + +Conditional generation evaluates the model's capability to produce molecules with desired properties. Following the baseline approaches, we incorporate property values as additional inputs during training and sample them from a prior distribution during inference. Descriptions of baselines and evaluations are provided in Appendix E. + +The results in Table 2 demonstrate that our method consistently outperforms baseline models across all metrics. Moreover, as illustrated in Figure 4, our approach achieves a 20-fold acceleration over previous state-of-the-art methods. This result showcases its potential for application in a wide range of controllable generation scenarios. + +# 4.4 Ablation Study + +Previous approaches differ in their methods for atom type generation. EDM represents atom types as one-hot vectors, generating them simultaneously with coordinates through diffusion. In contrast, GeoBFN treats atom types as atomic numbers and generates them by the BFN algorithm for discretized data, which incorporates a binning technique to convert continuous probabilities into discrete probabilities. UniGEM, on the other hand, generates only coordinates via diffusion and predicts atom types based on the generated coordinates. We adopt the UniGEM framework due to its superior performance. + +To compare generative methods without the influence of atom type generation differences, we integrated EDM and GeoBFN coordinate generation algorithms into the UniGEM framework and compared them with our approach. The results, shown in Table 3 with sampling steps $T = 50$ and $T = 30$ , demonstrate that our method exhibits a clear advantage in smaller steps, confirming the efficiency benefits of our generative algorithm. A stability comparison w.r.t various sampling steps is provided in Figure 7. Besides, we also conduct an extensive comparison of sampling efficiency of SLDM under joint diffusion framework, i.e. without UniGEM framework with GeoBFN and EDM in Appendix C.3 and an ablation study about temperature control in Appendix C.5. + +# 4.5 Evaluation on Toy Dataset + +We provide a toy dataset experiment in Appendix C.4, demonstrating the superior generative capabilities of our approach in faithfully modeling complex data distributions. These results suggest that SLDM holds promise for generalization to broader application domains. + +# 5 Conclusion + +This paper proposes the Straight-Line Diffusion Model (SLDM), a novel generative method that ensures a near-linear diffusion trajectory, effectively reducing truncation error during sampling. + +The proposed process schedule naturally aligns with the characteristics of point cloud molecular data and effectively balances the generative difficulty. As a result, our method achieves significant improvements in both sampling efficiency and quality, as demonstrated in both unconditional and conditional generation settings, paving the way for large-scale, controllable molecular generation in practice. + +Several challenges remain open for future investigation. On the theoretical side, refining the schedule to fully satisfy the boundary conditions, establishing principled guidelines for selecting the noise level $\sigma$ , such as analyzing its impact on training stability and sampling accuracy, and deriving optimal discretization schemes could strengthen the theoretical foundations of our approach and further accelerate sampling. On the practical side, applying our method to diverse controllable molecular generation tasks and extending it to broader domains presents exciting opportunities for future advancements. + +# Acknowledgment + +This work is supported by Beijing Academy of Artificial Intelligence and Beijing Frontier Research Center for Biological Structure Fundings. + +# References + +Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 34:8780-8794, 2021. +Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. High-resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 10684-10695, 2022. +William Peebles and Saining Xie. Scalable diffusion models with transformers. 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Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Appendix + +# Table of Contents + +# A Supplementary Theoretical Results 22 + +A.1 The Derivation of A Unified Formulation of Diffusion Models 22 +A.2 A Summary of Previous Process Schedules 23 +A.3 Connection Between Sampling of Baseline Methods and First-Order ODE Sampling 24 +A.4 Diffusion Schedule with Straight-line Trajectory 24 +A.5 Supplementary Proof for Sampling 26 +A.6 Temperature Control 27 +A.7 Modeling Invariant Probability Density for 3D Coordinate Generation 27 + +# B Molecular Generation Algorithms 29 + +# C Supplementary Illustrations and Results 29 + +C.1 Sampling Trajectory of Diffusion-based Models 29 +C.2 Sampling Efficiency Comparisons under UniGEM Framework 30 +C.3 Sampling Efficiency Under Joint Diffusion Framework 30 +C.4 Toy Data Results 31 +C.5 Ablation Study for Temperature Control 32 +C.6 Robustness to Different Prior Distributions 33 +C.7 Visualizations of Generated Molecules and Sampling Process 34 + +# D Related Work 34 + +D.1 An overview of 3D Molecular Generation 34 +D.2 Related Diffusion-Based Studies and Key Differences with SLDM 35 + +# E Implementation Detail 36 + +# A Supplementary Theoretical Results + +# A.1 The Derivation of A Unified Formulation of Diffusion Models + +The stochastic process of the diffusion-based model can be formulated in general as a linear stochastic differential equation (SDE): + +$$ +\mathrm {d} X _ {t} = f (t) X _ {t} \mathrm {d} t + g (t) \mathrm {d} W _ {t}, \tag {10} +$$ + +where $X_{t}$ characterizes the noise corrupting process of the data in a diffusion model, $f(t) \leq 0$ and $g(t) \geq 0$ are measurable functions defined on the interval $[0,\infty)$ , $W_{t}$ represents a standard Wiener process. The process $X_{t}$ is a time rescaled Ornstein-Uhlenbeck process whose law converges exponentially fast to the standard Gaussian distribution [Chen et al., 2023]. + +Under the assumption that all the relevant integrals exist, the solution of the above SDE is given by: + +$$ +X _ {t} = X _ {0} \cdot \exp \left(\int_ {0} ^ {t} f (\xi) \mathrm {d} \xi\right) + \int_ {0} ^ {t} \exp \left(\int_ {s} ^ {t} f (\xi) \mathrm {d} \xi\right) g (s) \mathrm {d} W _ {s}, t \in [ 0, \infty) \tag {11} +$$ + +A quick proof is as follows: + +Proof. By Itô's formula, and applying equation 10: + +$$ +\begin{array}{l} \mathrm {d} \left(X _ {t} \cdot \exp \left(- \int_ {0} ^ {t} f (\xi) \mathrm {d} \xi\right)\right) = \left(\mathrm {d} X _ {t} - X _ {t} f (t) \mathrm {d} t\right) \cdot \exp \left(- \int_ {0} ^ {t} f (\xi) \mathrm {d} \xi\right) \tag {12} \\ = \exp \left(- \int_ {0} ^ {t} f (\xi) \mathrm {d} \xi\right) \cdot g (t) \mathrm {d} W _ {t} \\ \end{array} +$$ + +Next, we integrate both sides and obtain: + +$$ +X _ {t} \cdot \exp \left(- \int_ {0} ^ {t} f (\xi) \mathrm {d} \xi\right) - X _ {0} = \int_ {0} ^ {t} \exp \left(- \int_ {0} ^ {s} f (\xi) \mathrm {d} \xi\right) \cdot g (s) \mathrm {d} W _ {s}. \tag {13} +$$ + +Finally, we multiply both sides of the equation by $\exp (\int_0^t f(\xi)\mathrm{d}\xi)$ ) and rearrange the terms to obtain equation 11. + +By the property that the stochastic integral with respect to a Wiener process, $X_{t}$ is a normal random variable. The mean and variance of $X_{t}$ are calculated as follows: + +$$ +\mathbb {E} \left[ X _ {t} \right] = X _ {0} \cdot \exp \left(\int_ {0} ^ {t} f (\xi) \mathrm {d} \xi\right) \tag {14} +$$ + +$$ +\mathbb {C} \mathbb {O} \mathbb {V} [ X _ {t} ] = E [ (\int_ {0} ^ {t} \exp (\int_ {s} ^ {t} f (\xi) \mathrm {d} \xi) g (s) \mathrm {d} W _ {s}) ^ {2} ] I = \int_ {0} ^ {t} \left(\exp \left(\int_ {s} ^ {t} f (\xi) d \xi\right) g (s)\right) ^ {2} \mathrm {d} s I. \tag {15} +$$ + +The covariance computation uses the Itô isometry property. + +To fit the process $X_{t}$ to the general form + +$$ +x _ {t} = \mu (t) x _ {0} + \sigma (t) \epsilon , \epsilon \sim \mathcal {N} (\mathbf {0}, I), \tag {16} +$$ + +we compare it with our solution in equation 11, and identify that + +$$ +\mu (t) = \exp \left(\int_ {0} ^ {t} f (\xi) \mathrm {d} \xi\right), \quad \sigma (t) = \mu (t) \sqrt {\int_ {0} ^ {t} g (s) ^ {2} / \mu (s) ^ {2} \mathrm {d} s}. \tag {17} +$$ + +Furthermore, we can express $f(t)$ and $g(t)$ in terms of $\mu(t)$ and $\sigma(t)$ : + +$$ +f (t) = \frac {\mathrm {d} (\ln \mu (t))}{\mathrm {d} t} = \frac {\dot {\mu} (t)}{\mu (t)}, \tag {18} +$$ + +$$ +g (t) ^ {2} = 2 \sigma (t) \mu (t) \frac {\mathrm {d} (\sigma (t) / \mu (t))}{\mathrm {d} t} = 2 \sigma (t) \dot {\sigma} (t) - 2 \sigma (t) ^ {2} \frac {\dot {\mu} (t)}{\mu (t)}. +$$ + +$\mu(t)$ and $\sigma(t)$ satisfy the initial conditions $\mu(0) = 1$ , $\sigma(0) = 0$ and additional conditions: $\mu(t) \geq 0$ is monotone non-increasing, $\sigma(t) \geq 0$ , $\frac{\mathrm{d}(\sigma(t) / \mu(t))}{\mathrm{d}t} \geq 0$ , i.e. $\sigma(t) / \mu(t)$ is monotone non-decreasing. + +Substituting the expressions for $f(t)$ and $g(t)$ back into the original SDE equation 10, we obtain: + +$$ +\mathrm {d} X _ {t} = \frac {\dot {\mu} (t)}{\mu (t)} X _ {t} \mathrm {d} t + \sqrt {2 \sigma (t) \dot {\sigma} (t) - 2 \sigma (t) ^ {2} \frac {\dot {\mu} (t)}{\mu (t)}} \mathrm {d} W _ {t}, \tag {19} +$$ + +This result is conceptually equivalent to that presented in Appendix B of Karras et al. [2022]. But they adopt a different definition of the schedule compared to equation 16, which leads to a different outcome compared to equation 19. + +# A.2 A Summary of Previous Process Schedules + +This section provides a comprehensive overview of several widely adopted diffusion-based models and their corresponding process schedules. + +The Denoising Diffusion Probabilistic Model (DDPM) [Ho et al., 2020] is characterized by a process schedule that exhibits the Variance Preserving (VP) property, which can be expressed as: $\mu(t)^2 + \sigma(t)^2 = 1$ . This formulation ensures the preservation of variance across timesteps, assuming the data has unit variance. In the domain of 3D molecular unconditional generation, the Equivariant Diffusion Model (EDM) [Hoogeboom et al., 2022] employs a schedule closely resembling the cosine noise schedule introduced by Nichol and Dhariwal [2021], albeit with a simplified notation: + +$$ +\boldsymbol {x} _ {t} = (1 - t ^ {2}) \boldsymbol {x} _ {0} + \sqrt {1 - (1 - t ^ {2}) ^ {2}} \boldsymbol {\epsilon}, t \in [ 0, 1 ], \tag {20} +$$ + +The Variance Exploding (VE) schedule, initially proposed in the context of Denoising Score Matching [Song and Ermon, 2019], can also be categorized as a denoising diffusion model with a distinct process schedule [Karras et al., 2022, Song et al., 2021b], which is given by: + +$$ +\boldsymbol {x} _ {t} = \boldsymbol {x} _ {0} + \sqrt {t} \boldsymbol {\epsilon}, t \in [ 0, T _ {\max } ]. \tag {21} +$$ + +By rescaling time, we derive the following expression: + +$$ +\boldsymbol {x} _ {t} = \boldsymbol {x} _ {0} + \sqrt {t} \sigma_ {\max } \boldsymbol {\epsilon}, t \in [ 0, 1 ], \tag {22} +$$ + +where $\sigma_{max}$ needs to be sufficiently large to ensure that $x_{1}$ approximates a uniform distribution. For illustrative clarity, we set $\sigma_{max} = 10$ for Figure 3 and $\sigma_{max} = 20$ for Figure 6. + +Denoising Diffusion Implicit Model (DDIM) offers an accelerated sampling process for diffusion models. As proved by Karras et al. [2022], DDIM employs the following schedule: + +$$ +\boldsymbol {x} _ {t} = \boldsymbol {x} _ {0} + t \boldsymbol {\epsilon}, t \in [ 0, T _ {\max } ]. \tag {23} +$$ + +By rescaling time, we derive the following expression: + +$$ +\boldsymbol {x} _ {t} = \boldsymbol {x} _ {0} + t \sigma_ {\max } \boldsymbol {\epsilon}, t \in [ 0, 1 ], \tag {24} +$$ + +For illustrative clarity, we set $\sigma_{max} = 10$ for Figure 3 and Figure 6. + +Flow Matching (FM) [Lipman et al., 2023] proposes a linear interpolation between the data distribution and a standard Gaussian, with a neural network learning the corresponding vector field. This noise-adding process can also be viewed as defining a diffusion process schedule, given by: + +$$ +\boldsymbol {x} _ {t} = (1 - t) \boldsymbol {x} _ {0} + (t + (1 - t) \sigma_ {\min }) \boldsymbol {\epsilon}, t \in [ 0, 1 ], \tag {25} +$$ + +where $\sigma_{\min}$ needs to be set sufficiently small to ensure that $x_{t}$ aligns with the data distribution at $t = 0$ . Besides, smaller values of $\sigma_{\min}$ have been reported to yield better results for FM [Tong et al., 2023]. Accordingly, we set $\sigma_{\min} = 0.001$ for both Figure 3 and Figure 6. + +Bayesian Flow Networks (BFN) [Graves et al., 2023] is a generative model based on Bayesian inference that accommodates both continuous and discrete variables. For continuous variables, the data is parameterized by Gaussian distributions. In this scenario, the generative algorithms can be interpreted as a denoising diffusion model [Xue et al., 2024a] with a process schedule given by: + +$$ +\boldsymbol {x} _ {t} = \left(1 - \sigma_ {\text {m i n}} ^ {2 t}\right) \boldsymbol {x} _ {0} + \sqrt {\left(1 - \sigma_ {\text {m i n}} ^ {2 t}\right) \sigma_ {\text {m i n}} ^ {2 t}} \boldsymbol {\epsilon}, t \in [ 0, 1 ], \tag {26} +$$ + +where $\sigma_{min}$ needs to be set small to satisfy $x_{t}$ align with the data distribution when $t = 0$ . Following the settings in [Song et al., 2023a], we set $\sigma_{min} = 0.001$ for Figure 3 and Figure 6. + +# A.3 Connection Between Sampling of Baseline Methods and First-Order ODE Sampling + +In section 2, we analyzed the error of first-order ODE discretization methods and claim that the baseline sampling methods primarily rely on first-order ODE discretization. In this section, we provide further clarification on this matter: EquiFM [Song et al., 2024] directly utilizes ODE-based sampling which includes first-order ODE discretization. For methods like EDM [Hoogeboom et al., 2022] and GeoBFN [Song et al., 2023a], while they resemble first-order methods due to requiring only a single function evaluation per iteration, they incorporate random sampling. This distinction necessitates an explanation of how their random sampling processes relate to ODEs. Specifically, EDM uses the same sampling method as DDPM, which is proved as a first-order discretization to the reverse-time SDE of DDPM in Appendix E of Song et al. [2021b]. Similarly, GeoBFN adopts the same sampling method as BFN [Graves et al., 2023], which is proved as a first-order discretization to the reverse-time SDE of BFN in Proposition 4.2 in Xue et al. [2024a]. Please note that both of the reverse-time SDE can be decomposed as an ODE and Langevin dynamics: + +$$ +\begin{array}{l} \mathrm {d} \boldsymbol {x} _ {t} = [ f (t) \boldsymbol {x} _ {t} - g ^ {2} (t) \nabla_ {\boldsymbol {x}} \log p (\boldsymbol {x} _ {t}) ] \mathrm {d} t + g (t) \mathrm {d} \boldsymbol {w} _ {t} \\ = \underbrace {\left[ f (t) \boldsymbol {x} _ {t} - \frac {g ^ {2} (t)}{2} \nabla_ {\boldsymbol {x}} \log p (\boldsymbol {x} _ {t}) \right] \mathrm {d} t} _ {\text {r e v e r s e t i m e O D E i n e q u a t i o n 3}} - \underbrace {\frac {g ^ {2} (t)}{2} \nabla_ {\boldsymbol {x}} \log p (\boldsymbol {x} _ {t}) \mathrm {d} t + g (t) \mathrm {d} \boldsymbol {w} _ {t}} _ {\text {L a n g e v i n d y n a m i c s}} \tag {27} \\ \end{array} +$$ + +where $f(t) = \frac{\dot{\mu}(t)}{\mu(t)}$ and $g(t)^2 = 2\sigma(t)\dot{\sigma}(t) - 2\sigma(t)^2\frac{\dot{\mu}(t)}{\mu(t)}$ . Thus the sampling processes of EDM and GeoBFN can be effectively approximated as a first-order discretization of an ODE augmented by Langevin dynamics. By isolating and analyzing the discretization error of this ODE component, we gain valuable insights into the limitations of methods with first-order discretization, including baseline approaches in molecular generation such as EDM, GeoBFN, and EquiFM. + +# A.4 Diffusion Schedule with Straight-line Trajectory + +As illustrated in section 3.1, we aim to reduce the truncation error during sampling by minimizing the second-order derivative of the trajectory. To this end, we first consider a simple case where the initial distribution is a delta distribution $\pmb{x}_0 \sim \delta_{\pmb{a}}(\pmb{x})$ , $\pmb{a} \in \mathbb{R}^N$ . In this scenario, $\pmb{x}_t$ follows a normal distribution $\mathcal{N}(\mu(t)\pmb{a}, \sigma(t)^2\pmb{I}_N)$ according to equation 1. The score function is then tractable as $\nabla_{\pmb{x}}\log p_t(\pmb{x}_t) = -\frac{\pmb{x} - \mu(t)\pmb{a}}{\sigma(t)^2}$ . Substituting this into equation 3 gives: + +$$ +\frac {d \boldsymbol {x}}{d t} = \frac {\dot {\sigma} (t)}{\sigma (t)} \boldsymbol {x} + \dot {\mu} (t) \boldsymbol {a} - \frac {\dot {\sigma} (t)}{\sigma (t)} \mu (t) \boldsymbol {a} \tag {28} +$$ + +We aim to keep $\frac{d\pmb{x}}{dt}$ constant, which requires $\frac{\dot{\sigma}(t)}{\sigma(t)} = 0$ and $\dot{\mu}(t)$ to be constant. This implies that $\sigma(t)$ must be constant, and $\mu(t)$ must be a linear function of $t$ . + +Given the boundary condition that $x_{t=0}$ approximates data distribution and $x_{t=T}$ approximates a known distribution, the only feasible choice is to set $\sigma(t)$ to a constant $\mu(t) = 1 - t / T$ . This satisfy all the schedule constraints noted under equation 2, except $\sigma(0) = 0$ , which is approximately satisfied by choosing $\sigma$ to be a small value. In practice, we use a value for $\sigma$ that is two orders of magnitude smaller than the data scale, yielding good results. We can also rescale the time as in [0, 1], leading to the following schedule for the diffusion process: + +$$ +\boldsymbol {x} _ {t} = (1 - t) \boldsymbol {x} _ {0} + \sigma \boldsymbol {\epsilon}, t \in [ 0, 1 ], \tag {29} +$$ + +which ensures a straight-line trajectory under a special data distribution assumption. + +Next, we extend the result into general data distribution in the following theorem. + +Theorem A.1. For a general data distribution and our process schedule $\pmb{x}_t = (1 - t)\pmb{x}_0 + \sigma \pmb{\epsilon}$ , $t\in [0,1]$ , the following inequality holds for each data dimension $i$ : + +$$ +P \left(\left| \frac {d \boldsymbol {x} _ {t} ^ {(i)}}{d t} + \frac {\boldsymbol {x} _ {t} ^ {(i)}}{1 - t} \right| \geq \delta\right) \leq \frac {\sigma^ {2}}{\delta^ {2} (1 - t) ^ {2}}. \tag {30} +$$ + +As $\sigma \to 0$ , the term $\frac{d\pmb{x}_t}{dt} + \frac{\pmb{x}_t}{1 - t}$ converges to zero in probability. More specifically, the solution to the equation $\frac{d\pmb{x}_t}{dt} + \frac{\pmb{x}_t}{1 - t} = 0$ is that $\frac{\pmb{x}_t}{1 - t}$ becomes a constant, which corresponds to a linear trajectory. + +In practice, we set $\sigma$ to be small and when $t \ll 1 - \sigma$ , the trajectory is approximately linear. The proof of the theorem needs two following lemmas. + +Lemma A.2. For a general data distribution $f(\pmb{x}_0)$ and a general diffusion process with the schedule $\pmb{x}_t = \mu(t)\pmb{x}_0 + \sigma(t)\pmb{\epsilon}$ , its ODE description in equation 3 can be rewritten as + +$$ +\frac {\mathrm {d} \boldsymbol {x} _ {t}}{\mathrm {d} t} = \dot {\mu} (t) \mathbb {E} [ \boldsymbol {x} _ {0} | \boldsymbol {x} _ {t} ] + \frac {\dot {\sigma} (t)}{\sigma (t)} (\boldsymbol {x} _ {t} - \mu (t) \mathbb {E} [ \boldsymbol {x} _ {0} | \boldsymbol {x} _ {t} ]). \tag {31} +$$ + +Proof of Lemma A.2. For the general schedule, we have $p_t(\mathbf{x}_t|\mathbf{x}_0) \sim \mathcal{N}(\mu(t)\mathbf{x}_0, \sigma(t)^2\mathbf{I})$ . We start by considering the score function + +$$ +\begin{array}{l} \nabla_ {\boldsymbol {x}} \log p _ {t} (\boldsymbol {x} _ {t}) = \frac {\int f (\boldsymbol {x} _ {0}) \nabla_ {\boldsymbol {x}} p (\boldsymbol {x} _ {t} | \boldsymbol {x} _ {0}) d \boldsymbol {x} _ {0}}{p (\boldsymbol {x} _ {t})} \\ = \frac {\int f \left(\boldsymbol {x} _ {0}\right) p \left(\boldsymbol {x} _ {t} \mid \boldsymbol {x} _ {0}\right) \left(- \frac {\boldsymbol {x} _ {t} - \mu (t) \boldsymbol {x} _ {0}}{\sigma (t) ^ {2}}\right) d \boldsymbol {x} _ {0}}{p \left(\boldsymbol {x} _ {t}\right)} \tag {32} \\ = - \frac {\boldsymbol {x} _ {t}}{\sigma (t) ^ {2}} + \frac {\mu (t)}{\sigma (t) ^ {2}} \mathbb {E} [ \boldsymbol {x} _ {0} | \boldsymbol {x} _ {t} ]. \\ \end{array} +$$ + +Substituting this into the ODE form of the diffusion process, we have: + +$$ +\begin{array}{l} \frac {\mathrm {d} \boldsymbol {x} _ {t}}{\mathrm {d} t} = \frac {\dot {\mu} (t)}{\mu (t)} \boldsymbol {x} _ {t} - \left(\sigma (t) \dot {\sigma} (t) - \sigma (t) ^ {2} \frac {\dot {\mu} (t)}{\mu (t)}\right) \left(- \frac {\boldsymbol {x} _ {t}}{\sigma (t) ^ {2}} + \frac {\mu (t)}{\sigma (t) ^ {2}} \mathbb {E} [ \boldsymbol {x} _ {0} | \boldsymbol {x} _ {t} ]\right) \tag {33} \\ = \dot {\mu} (t) \mathbb {E} [ \boldsymbol {x} _ {0} | \boldsymbol {x} _ {t} ] + \frac {\dot {\sigma} (t)}{\sigma (t)} (\boldsymbol {x} _ {t} - \mu (t) \mathbb {E} [ \boldsymbol {x} _ {0} | \boldsymbol {x} _ {t} ]). \\ \end{array} +$$ + +![](images/3f5a1f8b34db7defacdf9f03f0fccf93e885439a97ebf5f222bae537e1c022aa.jpg) + +Lemma A.3. Let $\mathbf{a}$ be a random variable that satisfies $p(\mathbf{a}|\mathbf{x}_0) \sim \mathcal{N}(\mathbf{x}_0, \sigma(t)^2 / \mu(t)^2\mathbf{I})$ . Define $\mathbf{y} = \mathbb{E}[\mathbf{x}_0|\mathbf{a}] - \mathbf{a}$ . Then, the following properties hold: + +1. $\mathbb{E}[\pmb {y}] = \mathbf{0}$ +2. $\operatorname{Var}[y^{(i)}] \leq \sigma(t)^2 / \mu(t)^2$ , $i = 1, \dots, N$ , where $y^{(i)}$ is the $i^{th}$ component of $\mathbf{y}$ . + +Proof of Lemma A.3. The expectation of $\pmb{y}$ is given by: + +$$ +\mathbb {E} \boldsymbol {y} = \mathbb {E} [ \mathbb {E} [ \boldsymbol {x} _ {0} | \boldsymbol {a} ] - \boldsymbol {a} ] = \mathbb {E} _ {\boldsymbol {a}} \mathbb {E} _ {\boldsymbol {x} _ {0} | \boldsymbol {a}} [ \boldsymbol {x} _ {0} - \boldsymbol {a} ] = \mathbb {E} _ {\boldsymbol {x} _ {0}} \mathbb {E} _ {\boldsymbol {a} | \boldsymbol {x} _ {0}} [ \boldsymbol {x} _ {0} - \boldsymbol {a} ] = 0 \tag {34} +$$ + +For the $i$ -th component $y^{(i)}$ , we compute the variance: + +$$ +\begin{array}{l} \operatorname {V a r} \left[ y ^ {(i)} \right] = \mathbb {E} \left[ \left(y ^ {(i)}\right) ^ {2} \right] = \mathbb {E} \left[ \left(\mathbb {E} _ {x _ {0} \mid a} \left[ x _ {0} ^ {(i)} - a ^ {(i)} \right]\right) ^ {2} \right] \tag {35} \\ \leq \mathbb {E} [ \mathbb {E} _ {x _ {0} | a} [ (x _ {0} ^ {(i)} - a ^ {(i)}) ^ {2} ] ] = \mathbb {E} _ {x _ {0}} \mathbb {E} _ {a | x _ {0}} [ (x _ {0} ^ {(i)} - a ^ {(i)}) ^ {2} ] = \sigma (t) ^ {2} / \mu (t) ^ {2} \\ \end{array} +$$ + +The inequality follows from Jensen's inequality. + +![](images/64d5e9b53ff3a32f3a9baf21cebe9871021922d8414c52cef6bde3ff6e4b9a6b.jpg) + +Proof of A.1. For our specific schedule $\sigma(t) = \sigma$ and $\mu(t) = 1 - t$ , equation 31 in lemma A.2 reduces to: + +$$ +\frac {\mathrm {d} \boldsymbol {x} _ {t}}{\mathrm {d} t} = - \mathbb {E} [ \boldsymbol {x} _ {0} | \boldsymbol {x} _ {t} ]. \tag {36} +$$ + +For simplicity of notation, we define $\mathbf{a} = \mathbf{x}_t / \mu (t)$ , satisfying $p(\mathbf{a}|\mathbf{x}_0)\sim \mathcal{N}(\mathbf{x}_0,\sigma (t)^2 /\mu (t)^2\mathbf{I})$ . By applying lemma A.3, the defined $\pmb {y} = \mathbb{E}[\pmb {x}_0|\pmb {a}] - \pmb{a}$ satisfies $\mathbb{E}[\pmb {y}] = \mathbf{0}$ and $\mathrm{Var}[y^{(i)}]\leq \sigma (t)^2 /\mu (t)^2 = \frac{\sigma^2}{(1 - t)^2}$ , for all dimension $i$ . + +Applying Chebyshev's inequality, we get: + +$$ +P \left(\left| y ^ {(i)} \right| \geq \delta\right) \leq \frac {\operatorname {V a r} \left[ y ^ {(i)} \right]}{\delta^ {2}} \leq \frac {\sigma^ {2}}{\delta^ {2} (1 - t) ^ {2}}. \tag {37} +$$ + +Thus, we have the inequality: + +$$ +P \left(\left| \frac {d \boldsymbol {x} _ {t} ^ {(i)}}{d t} - \left(- \boldsymbol {x} _ {t} ^ {(i)} / (1 - t)\right) \right| \geq \delta\right) \leq \frac {\sigma^ {2}}{\delta^ {2} (1 - t) ^ {2}}. \tag {38} +$$ + +![](images/49814c8415981eba59b13098679b4cabd87450cb91e1952199462e79e8b55e16.jpg) + +# A.5 Supplementary Proof for Sampling + +Lemma A.4 (Conditional Expectation Minimizes MSE). The conditional expectation $\mathbb{E}[\pmb{x}_{t - \Delta t}|\pmb{x}_t]$ is the estimator of $\pmb{x}_{t - \Delta t}$ given $\pmb{x}_t$ that minimizes the mean squared error (MSE). That is, for any estimator $h(\pmb{x}_t)$ , the following inequality holds: + +$$ +\mathbb {E} \left[ \left(\boldsymbol {x} _ {t - \Delta t} - h (\boldsymbol {x} _ {t})\right) ^ {2} \right] \geq \mathbb {E} \left[ \left(\boldsymbol {x} _ {t - \Delta t} - \mathbb {E} \left[ \boldsymbol {x} _ {t - \Delta t} | \boldsymbol {x} _ {t} \right]\right) ^ {2} \right]. \tag {39} +$$ + +Proof. We begin by expanding the conditional squared error term. By the properties of conditional expectation, the cross-term vanishes: + +$$ +\begin{array}{l} \mathbb {E} \left[ \left(\boldsymbol {x} _ {t - \Delta t} - h (\boldsymbol {x} _ {t})\right) ^ {2} | \boldsymbol {x} _ {t} \right] = \mathbb {E} \left[ \left(\boldsymbol {x} _ {t - \Delta t} - \mathbb {E} \left[ \boldsymbol {x} _ {t - \Delta t} | \boldsymbol {x} _ {t} \right] + \mathbb {E} \left[ \boldsymbol {x} _ {t - \Delta t} | \boldsymbol {x} _ {t} \right] - h (\boldsymbol {x} _ {t})\right) ^ {2} | \boldsymbol {x} _ {t} \right] \\ = \mathbb {E} \left[ \left(\boldsymbol {x} _ {t - \Delta t} - \mathbb {E} [ \boldsymbol {x} _ {t - \Delta t} | \boldsymbol {x} _ {t} ]\right) ^ {2} | \boldsymbol {x} _ {t} \right] + \mathbb {E} \left[ \left(\boldsymbol {x} _ {t - \Delta t} - \mathbb {E} [ \boldsymbol {x} _ {t - \Delta t} | \boldsymbol {x} _ {t} ]\right)\right) (\mathbb {E} [ \boldsymbol {x} _ {t - \Delta t} | \boldsymbol {x} _ {t} ] - h (\boldsymbol {x} _ {t})) | \boldsymbol {x} _ {t} | \\ + \mathbb {E} \left[ \left(\mathbb {E} \left[ \boldsymbol {x} _ {t - \Delta t} \mid \boldsymbol {x} _ {t} \right] - h (\boldsymbol {x} _ {t})\right) ^ {2} \mid \boldsymbol {x} _ {t} \right] \tag {40} \\ = \mathbb {E} \left[ \left(\boldsymbol {x} _ {t - \Delta t} - \mathbb {E} \left[ \boldsymbol {x} _ {t - \Delta t} \mid \boldsymbol {x} _ {t} \right]\right) ^ {2} \mid \boldsymbol {x} _ {t} \right] + \mathbb {E} \left[ \left(\mathbb {E} \left[ \boldsymbol {x} _ {t - \Delta t} \mid \boldsymbol {x} _ {t} \right] - h (\boldsymbol {x} _ {t})\right) ^ {2} \mid \boldsymbol {x} _ {t} \right] \\ \geq \mathbb {E} [ (\boldsymbol {x} _ {t - \Delta t} - \mathbb {E} [ \boldsymbol {x} _ {t - \Delta t} | \boldsymbol {x} _ {t} ]) ^ {2} | \boldsymbol {x} _ {t} ] \\ \end{array} +$$ + +Taking the expectation with respect to $\pmb{x}_t$ , we get equation 39, which completes the proof. + +![](images/2e69701131aadfd2c8fda996016ba4a35f806db40b22fedcfe50821185a55c5d.jpg) + +Proposition A.5. For the straight-line diffusion schedule defined in equation 5, we have + +$$ +\mathbb {E} \left[ \boldsymbol {x} _ {t - \Delta t} \mid \boldsymbol {x} _ {t} \right] = \frac {1 - (t - \Delta t)}{1 - t} \left(\boldsymbol {x} _ {t} + \sigma^ {2} \nabla_ {\boldsymbol {x}} \log p _ {t} (\boldsymbol {x} _ {t})\right) \tag {41} +$$ + +Proof. From equation 32, the conditional expectation of $\pmb{x}_0$ given $\pmb{x}_t$ is: + +$$ +\mathbb {E} \left[ \boldsymbol {x} _ {0} \mid \boldsymbol {x} _ {t} \right] = \frac {1}{\mu (t)} \left(\boldsymbol {x} _ {t} + \sigma (t) ^ {2} \nabla_ {\boldsymbol {x}} \log p _ {t} \left(\boldsymbol {x} _ {t}\right)\right) \tag {42} +$$ + +For $\pmb{x}_{t - \Delta t}$ , the conditional expectation is derived as follows: + +$$ +\begin{array}{l} \mathbb {E} [ \pmb {x} _ {t - \Delta t} | \pmb {x} _ {t} ] = \int \pmb {x} _ {t - \Delta t} p (\pmb {x} _ {t - \Delta t} | \pmb {x} _ {t}) \mathrm {d} \pmb {x} _ {t - \Delta t} \\ = \int \boldsymbol {x} _ {t - \Delta t} \int p (\boldsymbol {x} _ {t - \Delta t} | \boldsymbol {x} _ {0}) p (\boldsymbol {x} _ {0} | \boldsymbol {x} _ {t}) \mathrm {d} \boldsymbol {x} _ {0} \mathrm {d} \boldsymbol {x} _ {t - \Delta t} \\ = \int \left(\int \boldsymbol {x} _ {t - \Delta t} p \left(\boldsymbol {x} _ {t - \Delta t} \mid \boldsymbol {x} _ {0}\right) \mathrm {d} \boldsymbol {x} _ {t - \Delta t}\right) p \left(\boldsymbol {x} _ {0} \mid \boldsymbol {x} _ {t}\right) \mathrm {d} \boldsymbol {x} _ {0} \tag {43} \\ = \int \mu (t - \Delta t) \boldsymbol {x} _ {0} p (\boldsymbol {x} _ {0} | \boldsymbol {x} _ {t}) \mathrm {d} \boldsymbol {x} _ {0} \\ = \mu (t - \Delta t) \mathbb {E} [ \boldsymbol {x} _ {0} | \boldsymbol {x} _ {t} ]. \\ \end{array} +$$ + +Here, we make the assumption that $x_{t}$ and $x_{t - \Delta t}$ are independent given $x_0$ . In fact, the specific temporal dependency among the random variables $(x_{t})_{t = 0,\dots ,1}$ defines a particular sampling method within the broader family of valid samplers characterized by equation 7. Our subsequent work Ni et al. [2025] generalizes this framework beyond the SLDM schedule and introduces the sampling approach used in this paper as StoMax (maximal stochasticity) sampling, which corresponds to the variant with the highest degree of randomness within the family. + +Substituting $\mathbb{E}[\pmb{x}_0|\pmb{x}_t]$ from equation 42, we have: + +$$ +\mathbb {E} \left[ \boldsymbol {x} _ {t - \Delta t} \mid \boldsymbol {x} _ {t} \right] = \frac {\mu (t - \Delta t)}{\mu (t)} \left(\boldsymbol {x} _ {t} + \sigma (t) ^ {2} \nabla_ {\boldsymbol {x}} \log p _ {t} (\boldsymbol {x} _ {t})\right) \tag {44} +$$ + +Adopting the straight-line diffusion schedule defined in equation 5, we produce equation 41. + +![](images/687bec01720b5b52935502eae9ca354bf3384db27cb51b3818d799ff46fe6d5a.jpg) + +As proved in [Xue et al., 2024b], there are a family of reverse processes that share the same marginal probability distributions as equation 2 and equation 3. When applied to our process schedule and apply Euler-Maruyama method discretization, we obtain the following iterative algorithm: + +$$ +\boldsymbol {x} _ {t - \Delta t} = \frac {1 - t + \Delta t}{1 - t} \boldsymbol {x} _ {t} + (1 + \beta (t)) \frac {\Delta t}{1 - t} \sigma^ {2} \nabla_ {\boldsymbol {x}} \log p _ {t} (\boldsymbol {x} _ {t}) + \sqrt {2 \beta (t) \frac {\Delta t}{1 - t} \sigma^ {2}} \boldsymbol {\epsilon}. \tag {45} +$$ + +equation 45 uses a Gaussian distribution to model the backward probability $p(\pmb{x}_{t - \Delta t}|\pmb{x}_t)$ , whose expectation is $\frac{1 - t + \Delta t}{1 - t}\pmb{x}_t + (1 + \beta(t))\frac{\Delta t}{1 - t}\sigma^2\nabla_{\pmb{x}}\log p_t(\pmb{x}_t)$ . According to Lemma A.4, the optimal iterative step that minimizes MSE should satisfy $\frac{1 - t + \Delta t}{1 - t}\pmb{x}_t + (1 + \beta(t))\frac{\Delta t}{1 - t}\sigma^2\nabla_{\pmb{x}}\log p_t(\pmb{x}_t) = \frac{1 - (t - \Delta t)}{1 - t} \left( \pmb{x}_t + \sigma^2\nabla_{\pmb{x}}\log p_t(\pmb{x}_t) \right)$ . This result in $\beta(t) = \frac{1 - t}{\Delta t}$ . + +# A.6 Temperature Control + +For Langevin dynamics, the sampling temperature can be controlled by scaling the stochastic term with a constant $\tau$ . + +$$ +d \boldsymbol {x} = \frac {1}{2} g (t) ^ {2} \nabla_ {\boldsymbol {x}} \log p _ {t} (\boldsymbol {x}) d t + \tau g (t) \mathrm {d} w ^ {\prime}. \tag {46} +$$ + +where $\tau$ is the temperature parameter. Then $\pi (\pmb {x})\propto p(\pmb {x})^{\frac{1}{\tau^2}}$ is the stationary distribution of the process in equation 46, as proved as follows: + +Proof. The marginal probability density $\pi_t(x)$ evolves according to Fokker-Planck equation + +$$ +\frac {\partial \pi_ {t} (\boldsymbol {x})}{\partial t} = - \nabla \cdot \left[ \frac {1}{2} g (t) ^ {2} \nabla_ {\boldsymbol {x}} \log p _ {t} (x) \pi_ {t} (\boldsymbol {x}) \right] + \frac {1}{2} \nabla \cdot \nabla \cdot \left[ \tau^ {2} g (t) ^ {2} \pi_ {t} (\boldsymbol {x}) \right] \tag {47} +$$ + +For the stationary distribution, the probability density becomes time-independent, i.e., $\frac{\partial\pi_t(x)}{\partial t} = 0$ . Thus, we solve: + +$$ +\nabla \cdot \left[ \frac {1}{2} g (t) ^ {2} \nabla_ {\boldsymbol {x}} \log p _ {t} (\boldsymbol {x}) \pi (\boldsymbol {x}) \right] = \frac {1}{2} \nabla \cdot \nabla \cdot \left[ \tau^ {2} g (t) ^ {2} \pi (\boldsymbol {x}) \right] \tag {48} +$$ + +We can easily validate that $\pi (\pmb {x})\propto p(\pmb {x})^{\frac{1}{\tau^2}}$ satisfies the stationary equation. + +![](images/3a5e604f8ab930d9a0027ecb339320a1ef099fc9b9658be21ead5b742be2a3a3.jpg) + +Thus, higher temperatures $(\tau \rightarrow \infty)$ increase diversity, with $\pi (\pmb {x})$ approaching a uniform distribution. Conversely, lower temperatures $(\tau \rightarrow 0)$ enhance fidelity, with $\pi (\pmb {x})$ converging to a $\delta$ -distribution at the global maximum of $p(x)$ + +However, prior work has shown that this low-temperature sampling in diffusion models fails to effectively balance diversity and fidelity in image generation, often resulting in blurred and overly smoothed outputs [Dhariwal and Nichol, 2021]. We also notice that this low-temperature sampling approach applied to our straight-line diffusion fail to fully cover all high-density regions of the target distribution, as shown in Figure 5a. + +To address this, we propose an empirically designed time-annealing schedule that introduces higher stochasticity during the initial stages of sampling, allowing for the exploration of distant probability density maxima: + +$$ +\boldsymbol {x} _ {t - \Delta t} = \frac {1 - t + \Delta t}{1 - t} \left(\boldsymbol {x} _ {t} + \sigma^ {2} \nabla_ {\boldsymbol {x}} \log p _ {t} (\boldsymbol {x})\right) + t ^ {\nu} \sqrt {2} \sigma \boldsymbol {\epsilon}, \tag {49} +$$ + +where $\nu$ controls the decay rate of temperature over time. Larger $\nu$ enhances molecule stability and enables the model to cover all modes, as shown in Figure 5b. The low-temperature sampling helps mitigate the disturbance caused by noise in the training data, and thereby can improve the quality of the samples. The default value of $\nu$ for molecular generation is analyzed through an ablation study in Section C.5. + +# A.7 Modeling Invariant Probability Density for 3D Coordinate Generation + +In this section, we aim to prove that the probability density of the generated atomic coordinates, as produced by Algorithms 3 and 4, is invariant to both translations and rotations. Formally, we aim to establish that for the Probability Density modeled by our model $p(\pmb{x}_0) = p(\pmb{x}_0 + \pmb{t})$ and $p(\pmb{x}_0) = p(\pmb{R}\pmb{x}_0)$ , where $\pmb{t}$ is a translation vector and $\pmb{R}$ is an orthogonal matrix representing a rotation. The proof is in the same spirit of that in Hoogeboom et al. [2022], Xu et al. [2022]. The result can be seen as a special case of Theorem 1 and Theorem 2 in Zhou et al. [2025]. + +![](images/dce44a2e83221dac924db8deed7f28ea56d7467508685829889eb3f46dd9f4ec.jpg) +Figure 5: a. Generation results from straight-line diffusion with vanilla temperature control as defined in equation 46. Training data (black points) represent 2D Swiss roll coordinates with added Gaussian noise, where the highest density region lies in the interior of the spiral. Diffusion-generated samples (yellow points) exhibit reduced diversity as $\tau$ decreases but fail to cover all high-density regions, favoring those near the origin. b. Generation results from straight-line diffusion with our annealing temperature control as described in equation 49. Faster temperature decay (larger $\nu$ ) leads to concentrated samples in high-density regions, successfully covering all local maxima. + +To ensure translation invariance, the generative process is defined in the quotient space of translations, specifically the zero Center of Mass (CoM) space. This is achieved through two key operations. First, noise is sampled from a CoM-restricted Gaussian distribution $\epsilon \sim \mathcal{N}_{\mathrm{CoM}}(\mathbf{0},I_{3M})$ , which is sampled by first sampling a standard Gaussian noise vector $\epsilon^{\prime}\sim \mathcal{N}(\mathbf{0},I_{3M})$ and then subtracting its center of mass: $\epsilon = \epsilon^{\prime} - \frac{1}{3M}\sum_{i = 1}^{3M}\epsilon_{i}^{\prime}$ . Second, the network's output at each step is projected into the zero CoM space. These operations ensure that all intermediate coordinates $\boldsymbol{x}_t,t = 0,\dots ,1$ remain strictly within the zero CoM space throughout the generative process. + +To establish rotation invariance, we rely on two key properties. First, we use an equivariant neural network satisfying $\phi^{(x)}(\pmb{R}\pmb{x}_t,t) = \pmb{R}\phi^{(x)}(\pmb{x}_t,t)$ . Second, we utilize the rotational invariance of the zero mean isotropic Gaussian distributions. This property can also be extended to the CoM-restricted Gaussian distribution, whose probability density function is: $f_{\mathcal{N}_{\mathrm{CoM}}}(x) = \frac{1}{(2\pi)^{3(M - 1) / 2}}\exp (\frac{1}{2} ||x||^2)$ . Then, we can verify $f_{\mathcal{N}_{\mathrm{CoM}}}(x) = f_{\mathcal{N}_{\mathrm{CoM}}}(Rx)$ for any orthogonal rotation matrix $\pmb{R}$ . Now we prove the rotation invariance of the generation probability as follows: + +At each iterative step of the generative process, we have: + +$$ +\boldsymbol {x} _ {t - \Delta t} \sim \mathcal {N} _ {\mathrm {C o M}} \left(\frac {1 - (t - \Delta t)}{1 - t} \cdot \left(\boldsymbol {x} _ {t} - \sigma \cdot \phi \left(\boldsymbol {x} _ {t}, t\right), 2 t ^ {2 \nu} \sigma^ {2} I _ {3 M}\right) \right. \tag {50} +$$ + +$$ +\begin{array}{l} p (\boldsymbol {R} \boldsymbol {x} _ {t - \Delta t} | \boldsymbol {R} \boldsymbol {x} _ {t}) = f _ {\mathcal {N} _ {\mathrm {C o M}}} \left(\frac {\boldsymbol {R} \boldsymbol {x} _ {t - \Delta t} - \frac {1 - (t - \Delta t)}{1 - t} \cdot (\boldsymbol {R} \boldsymbol {x} _ {t} - \sigma \cdot \phi (\boldsymbol {R} \boldsymbol {x} _ {t} , t))}{\sqrt {2} t ^ {\nu} \sigma}\right) \\ = f _ {\mathcal {N} _ {\mathrm {C o M}}} \left(\boldsymbol {R} \frac {\boldsymbol {x} _ {t - \Delta t} - \left(\frac {1 - (t - \Delta t)}{1 - t} \cdot (\boldsymbol {x} _ {t} - \sigma \cdot \phi (\boldsymbol {x} _ {t} , t))\right)}{\sqrt {2} t ^ {\nu} \sigma}\right) \tag {51} \\ = f _ {\mathcal {N} _ {\mathrm {C o M}}} \left(\frac {\boldsymbol {x} _ {t - \Delta t} - \left(\frac {1 - (t - \Delta t)}{1 - t} \cdot (\boldsymbol {x} _ {t} - \sigma \cdot \phi (\boldsymbol {x} _ {t} , t))\right)}{\sqrt {2} t ^ {\nu} \sigma}\right) = p (\boldsymbol {x} _ {t - \Delta t} | \boldsymbol {x} _ {t}) \\ \end{array} +$$ + +In the second equality, we apply the equivariance property of the neural network. The third equality follows from the rotational invariance of the isotropic Gaussian distribution. + +Additionally, the initial distribution $p(\boldsymbol{x}_1) = \mathcal{N}_{\mathrm{CoM}}(\mathbf{0}, \sigma^2 I_{3M})$ is rotation invariant. Combining these facts, we can propagate rotation invariance across the generative process. Thus, for the final generated distribution: + +$$ +\begin{array}{l} p \left(\boldsymbol {R} \boldsymbol {x} _ {0}\right) = \int \dots \int_ {\substack {t = \Delta t \\ 1}} ^ {1} p \left(\boldsymbol {R} \boldsymbol {x} _ {t - \Delta t} \mid \boldsymbol {R} \boldsymbol {x} _ {t}\right) p \left(\boldsymbol {R} \boldsymbol {x} _ {1}\right) \mathrm {d} \boldsymbol {x} _ {1} \dots \mathrm {d} \boldsymbol {x} _ {\Delta t} \tag{52} \\ = \int \dots \int \prod_ {t = \Delta t} ^ {1} p \left(\boldsymbol {x} _ {t - \Delta t} \mid \boldsymbol {x} _ {t}\right) p \left(\boldsymbol {x} _ {1}\right) d \boldsymbol {x} _ {1} \dots d \boldsymbol {x} _ {\Delta t} = p \left(\boldsymbol {x} _ {0}\right). \\ \end{array} +$$ + +In the second equality, we use the rotational invariance of both the transition probabilities and the initial distribution. This proves that the final probability density of the generated data is rotation invariant. + +# B Molecular Generation Algorithms + +Input: 3D molecular data $\mathbf{z}_0 = [x_0, h_0]$ with $M$ atoms, $x_0 \in \mathrm{CoM}$ space, neural network $\phi$ , variance constant $\sigma$ , nucleation time $t_n \in [0,1]$ + +Algorithm 3 Straight-Line Diffusion Training for Molecules (UniGEM) +```txt +repeat Sample $t\sim \frac{1}{2} U([0,t_n]) + \frac{1}{2} U([t_n,1])$ Sample $\epsilon \sim \mathcal{N}_{\mathrm{CoM}}(\mathbf{0},I_{3M})$ Compute $\pmb {x}_t = (1 - t)\pmb {x}_0 + \sigma \pmb{\epsilon}$ Minimize $||\pmb {\epsilon} - \phi^{(x)}(\pmb {x}_t,t)||^2 +1_{t\leq t_n}|\pmb {h}_0 - \phi^{(h)}(\pmb {x}_t,t)|$ +``` + +until converged; + +Input: Number of atoms $M$ , neural network $\phi$ , variance constant $\sigma$ , nucleation time $t_n$ , sampling steps $T$ , temperature annealing rate $\nu$ + +Sample $\pmb{x}_1\sim \sigma \cdot \mathcal{N}_{\mathrm{CoM}}(\mathbf{0},I_{3M})$ + +Algorithm 4 Straight-Line Diffusion Sampling for Molecules (UniGEM) +```txt +for $i = T - 1$ to 1 do + $t = i / T, \Delta t = 1 / T$ $\boldsymbol{x}_{t - \Delta t} = \frac{1 - (t - \Delta t)}{1 - t} \cdot (\boldsymbol{x}_t - \sigma \cdot \phi(\boldsymbol{x}_t, t))$ +Projecting $\boldsymbol{x}_{t - \Delta t}$ into zero CoM space +if $i > 1$ then + $\begin{array}{l} \text{Sample } \epsilon \sim \mathcal{N}_{\mathrm{CoM}}(\mathbf{0}, I_{3M}) \\ \boldsymbol{x}_{t - \Delta t} = \boldsymbol{x}_{t - \Delta t} + t^{\nu} \sqrt{2} \cdot \sigma \cdot \epsilon \end{array}$ +end +else + $|h_0 = \phi^{(h)}(\boldsymbol{x}_0, 0)$ +end +return $z_0 = [x_0, h_0]$ +``` + +# C Supplementary Illustrations and Results + +# C.1 Sampling Trajectory of Diffusion-based Models + +To evaluate whether SLDM exhibits a near-linear trajectory under general data distributions, as suggested by Theorem 3.1, we visualize the ODE trajectory and compare it with other diffusion-based models. We consider a scenario where the data distribution is a one-dimensional Gaussian mixture, as this setup offers a tractable score function and serves as a representative example of general distributions. The resulting trajectories are shown in Figure 6. Our method maintains a trajectory that is consistently closer to a straight line compared to other approaches, leading to smaller truncation + +![](images/ceb51f55909184e7a1915b204079ff29550c466439b358f555df370f9bbb2df9.jpg) +Figure 6: The trajectory of the ODE in equation 3 of various diffusion-based models. The data distribution is a mixture of Gaussians in one dimension, defined as $x_0 \sim 0.5 \cdot \mathcal{N}(2,1/4) + 0.5 \cdot \mathcal{N}(-2,1/4)$ . The background color indicates the value of probability density $p(x_t)$ . During sampling, the timestep $t$ decreases from 1 to 0. + +errors under first-order numerical discretization. As suggested in Theorem 3.1, the trajectory slightly deviates from a straight line in the early stages (i.e., as $t$ approaches 1), which could increase the sampling error in theory. However, our empirical observations indicate that the sampling process is relatively robust to such errors during these initial steps. This robustness may be attributed to the incorporation of Langevin dynamics in the sampling algorithm, which can mitigate the impact of early-stage inaccuracies. As a result, SLDM achieves an overall lower sampling error. + +In comparison, DDIM demonstrates a relatively linear trajectory during the early stages but exhibits significant curvature in later stages, where accurate sampling is more crucial. This could potentially degrade its performance. For the FM method, where the initial distribution is Gaussian and no explicit prior-data joint distribution is predefined as in Tong et al. [2023], it fails to achieve a straight-line trajectory. Other methods also demonstrate an evident curved trajectory. Besides, we can also see from the illustration that the distribution remains nearly static during the later stages of BFN sampling, which aligns with the discussion in section 3.2 and suggests potential inefficiencies of time scheduling. + +# C.2 Sampling Efficiency Comparisons under UniGEM Framework + +We complement the ablation study in Section 4.4 by incorporating results with diverse sampling steps. The results demonstrate that while UniGEM enhances the performance of EDM and BFN when sampling step is abundant, these methods still struggle to achieve satisfactory molecular stability in the low sampling step scenario. + +# C.3 Sampling Efficiency Under Joint Diffusion Framework + +A natural question arises: can SLDM achieve strong performance without the UniGEM framework? Specifically, can it jointly generate atom types and coordinates using diffusion models, following the approach of EDM and GeoBFN? To address this, we introduce a variant called SLDM (joint diffusion) as a preliminary exploration, where atom types are generated using an SLDM model tailored for discretized data, replacing UniGEM's prediction strategy. The training and sampling procedures of SLDM (joint diffusion) for molecule generation are detailed in Algorithms 5 and 6, respectively. Unlike UniGEM, we follow GeoBFN to represent atom types by their atomic numbers (positive integers) and adopt the binning technique from BFN to handle discretized data. Here, $P_{\hat{\mu}_h,\hat{\sigma}_h}(b)$ denotes the discretized cumulative distribution function (CDF), defined as: $F(b) = \frac{1}{2} [1 + erf(\frac{b - \mu}{\sqrt{2}\sigma})]$ , with $P(b) = F(b)$ for $-1 < b < 1$ , $P(b) = 0$ for $b \leq -1$ , and $P(b) = 1$ for $b \geq 1$ . "NearestCenter" decodes the generated continuous continuous normalized atomic number into an atomic type by + +![](images/3c008b9a4e6284f630a598e32fd7c515e164162d8a43bad59a541829b6cdd415.jpg) +Figure 7: Comparison of molecule stability $(\uparrow)$ across diffusion-based molecular generation models with UniGEM framework on QM9 unconditional generation, evaluated with respect to the Number of Function Evaluations (NFE) during the sampling process. + +finding the nearest center of bins (normalized atomic numbers), then unnormalizing it and mapping it to the corresponding atomic type. Notably, there remains substantial room for further exploration regarding the handling of diffusion for discretized data, which we leave for future work. + +# Algorithm 5 SLDM Training (joint diffusion) + +Input: Number of bins $K \in \mathbb{N}$ (equal to the largest atomic number), fixed variance $\sigma$ , 3D molecular data $z_0 = [x_0, h_0]$ with $M$ atoms, $x_0 \in \mathrm{CoM}$ space, normalized atom number $\tilde{h}_0 = (2 * h_0 - 1) / K - 1 \in \left[\frac{1}{K} - 1, 1 - \frac{1}{K}\right]^D$ + +# repeat + +Sample $t\sim \mathcal{U}(0,1),\epsilon^x\sim \mathcal{N}_{\mathrm{CoM}}(\mathbf{0},I_{3M}),\epsilon^h\sim \mathcal{N}\mathbf{0},I_{3M})$ Compute $\pmb {x}_t = (1 - t)\pmb {x}_0 + \sigma \pmb{\epsilon}^{x}$ Compute $\pmb {h}_t = (1 - t)\tilde{\pmb{h}}_0 + \sigma \pmb{\epsilon}^{h}$ Network output: $\hat{\epsilon}^x,\hat{\mu}_h^\epsilon ,\ln \hat{\sigma}_h^\epsilon = \Phi (\pmb {x}_t,\pmb {h}_t,t)$ $\hat{\mu}_h = \frac{\pmb{h}_t - \sigma\hat{\pmb{\mu}}_h^\epsilon}{1 - t},\hat{\pmb{\sigma}}_h = \frac{\sigma\exp(\ln\hat{\pmb{\sigma}}_h^\epsilon)}{1 - t}$ Minimize denoise loss: $\| \pmb {\epsilon}^x -\hat{\pmb{\epsilon}}^x\| ^2 +\left\| (\hat{\pmb{\mu}}_h^\epsilon -\pmb {\epsilon}^h) - \frac{1 - t}{\sigma}\sum_{i = 1}^{K}P_{\hat{\pmb{\mu}}_h,\hat{\pmb{\sigma}}_h}(b_i)(b_i - \hat{\pmb{\mu}}_h)\right\| ^2$ until converged; + +Results are presented in Table 4. The proposed temperature annealing mechanism endows the model with greater flexibility, enabling a balanced trade-off between stability/validity and uniqueness/diversity. We manually tuned the temperature annealing rates for coordinates $(\nu_{x})$ and atomic numbers $(\nu_{h})$ , and empirical observations indicate that setting $\nu_{x} > \nu_{h}$ yields better performance in achieving such balance. It should be noted that we did not conduct exhaustive hyperparameter searches, leaving room for potential further improvements. SLDM (joint diffusion) outperforms EDM across all metrics and exhibits superior stability and validity with slightly lower uniqueness compared to GeoBFN when the number of sampling steps $T$ is limited. Notably, the overall performance of this variant is still inferior to the original SLDM which employs the UniGEM framework. This finding reaffirms that the UniGEM framework may be a more optimal approach for generating small-molecule. + +# C.4 Toy Data Results + +We evaluate our model on several 2D toy data to test its generality. The datasets included swissroll and moons to represent continuous data distributions, as well as chessboard to simulate discretized data distributions. The dataset consists of 100,000 samples. For the moons and swissroll datasets, + +Algorithm 6 SLDM Sampling (joint diffusion) +Input: Number of atoms $M$ , neural network $\phi$ , variance constant $\sigma$ , nucleation time $t_n$ , sampling steps $T$ , temperature annealing rate $\nu_x$ and $\nu_h$ +Sample $\boldsymbol{x}_1 \sim \sigma \mathcal{N}_{\mathrm{CoM}}(\boldsymbol{0}, I_{3M})$ , $\boldsymbol{h}_1 \sim \sigma \mathcal{N}(\boldsymbol{0}, I_{3M})$ +for $i = T - 1$ to 1 do + $t = i / T, \Delta t = 1 / T$ +Network output: $\hat{\epsilon}^x, \hat{\mu}_h^\epsilon, \ln \hat{\sigma}_h^\epsilon = \Phi(\boldsymbol{x}_t, \boldsymbol{h}_t, t)$ $\hat{\mu}_h = \frac{h_t - \sigma \hat{\mu}_h^\epsilon}{1 - t}, \hat{\sigma}_h = \frac{\sigma \exp(\ln \hat{\sigma}_h^\epsilon)}{1 - t}$ +Update $\boldsymbol{x}_{t - \Delta t} = \frac{1 - (t - \Delta t)}{1 - t} \cdot (\boldsymbol{x}_t - \sigma \cdot \hat{\epsilon}^x)$ +Update $\hat{h}_0 = \sum_{i=1}^{K} P_{\hat{\mu}_h, \hat{\sigma}_h}(b_i) b_i, h_{t - \Delta t} = (1 - (t - \Delta t)) \hat{h}_0$ +Projecting $\boldsymbol{x}_{t - \Delta t}$ into zero CoM space +if $i > 1$ then + Sample $\epsilon^x \sim \mathcal{N}_{\mathrm{CoM}}(\boldsymbol{0}, I_{3M}), \epsilon^h \sim \mathcal{N}\boldsymbol{0}, I_{3M}$ $\boldsymbol{x}_{t - \Delta t} = \boldsymbol{x}_{t - \Delta t} + t^{\nu_x} \sqrt{2} \cdot \sigma \cdot \epsilon^x$ $\boldsymbol{h}_{t - \Delta t} = \boldsymbol{h}_{t - \Delta t} + t^{\nu_h} \sqrt{2} \cdot \sigma \cdot \epsilon^h$ +end +return $z_0 = [x_0, NearestCenter(h_0)]$ + +Table 4: Sampling efficiency comparison under joint diffusion framework on QM9 dataset unconditional generation task. + +
MethodAtom Sta (%)Mol Sta (%)Valid (%)V×U (%)
EDM (T=1000)98.7082.0091.9090.70
GeoBFN (T=1000)99.0890.8795.3192.96
SLDM (joint diffusion) (T=1000)99.0191.5795.2091.57
GeoBFN (T=500)98.7888.4293.3591.78
SLDM (joint diffusion) (T=500)99.8990.5794.0091.11
GeoBFN (T=100)98.6487.2193.0391.53
SLDM (joint diffusion) (T=100)98.9290.3193.8489.36
GeoBFN (T=50)98.2885.1192.2790.72
SLDM (joint diffusion) (T=50)98.7088.0992.8489.53
+ +we performed training with 40 diffusion steps and 100 epochs, and for the chessboard dataset, the training was extended to 600 epochs with 100 diffusion steps to ensure convergence for all models. Other settings are kept the same for all datasets: The model is a 5-layer MLP. The batch size was set to 2048, and the optimizer used was Adam with a learning rate of 0.001. No temperature control is used during sampling. These settings follow https://github.com/albarji/toy-diffusion/, and are kept consistent across all generative algorithms to ensure a fair comparison. The results are provided in Figure 8. The data generated by our model show the closest alignment to the original distributions. Specifically, our model produced samples with fewer outliers and achieved good coverage of the data distributions. These findings highlight the superior generative capabilities of our approach in faithfully modeling complex data distribution. + +# C.5 Ablation Study for Temperature Control + +Theoretically, a larger temperature annealing rate $v$ corresponds to a faster cooling scheme. The results in Table 5 demonstrate that adjusting the temperature can effectively enhance molecular stability. However, extreme values of $v$ may significantly reduce diversity. We selected $v = 0.5$ as the default setting for our model, as it achieves the highest $U \times V$ score, indicating the optimal balance between diversity and fidelity. + +![](images/439bb492d7c766aff6f22b7ddf675f3ad1e186e8708167757db5a9954fcfa4a8.jpg) + +![](images/4eb6e310f3f0506a063f1443dcf96389aa0f78b8a4e94530f67eed78d6773472.jpg) + +![](images/c9c46769b3c2eff941a6ac34587181a907f6bc6ddf4c26d0a22d65f818301864.jpg) + +![](images/a8b9323787f09362929bab06536d0c001a969ebbfbbbae6c4e14c95ae03f1df6.jpg) + +![](images/ed98d5979b3c6982c070b8f1763cf7bf0782f267f16a15c0bb749036decaa49e.jpg) + +![](images/e0666e88b0d71792cdba32c4c00f51a2ec4b4c61a2f6c10f4edcb48785b88c98.jpg) + +![](images/1310f235b10d35fe451315b55357056188b867bd98f969b6e01e9c60285ac34f.jpg) +Figure 8: Generative performance comparison of Straight-Line Diffusion, DDPM, and BFN on 2D toy datasets. + +![](images/dfda3c5c20540702da6b8b1b2b661fa88c7cd92f0b1185ecb114bb57d78c6545.jpg) + +![](images/449f0eebf80686711e87050e2cf93e3a32a258cede2ba21b624dc263ff4d5a5e.jpg) + +Table 5: Impact of temperature annealing rate evaluated on the QM9 unconditional generation with sampling step $T = {50}$ . Higher values indicate better performance. + +
Atom sta(%)Mol sta(%)Valid(%)U × V(%)
SLDM(v=0)99.1491.7495.4092.76
SLDM(v=0.5)99.2893.0396.2093.28
SLDM(v=1)99.2793.4696.0892.42
SLDM(v=3)99.3794.3496.8490.08
SLDM(v=5)99.4294.8397.3386.10
SLDM(v=10)99.5396.0697.9875.13
+ +# C.6 Robustness to Different Prior Distributions + +Empirically, we find that SLDM maintains high validity and stability even when the initial distribution during sampling deviates from the theoretical valid one, i.e., Gaussian distribution with standard deviation $\sigma$ , as shown in Table 6. This robustness helps mitigate the impact of deviations from the ideal linear trajectory in the early sampling steps, as discussed under the Theorem 3.1. + +Table 6: Robustness of SLDM to different initial sampling distributions on the QM9 unconditional generation task. Here $\epsilon \sim \mathcal{N}(0, I)$ and the sampling step $T = 50$ . + +
Prior DistributionMol_stab (%)Atom_stab (%)Validity (%)V*u (%)
σ·ε (theoretical)93.4399.3296.1093.31
2·σ·ε91.8199.1295.0092.79
0.1·σ·ε93.3199.2996.0793.46
0.01·σ·ε92.9499.2496.0693.06
+ +![](images/aebfed42e7062b340194d910d5d48e6382bcb4f92884d5ed94a139a8aa1d3e49.jpg) +Figure 9: 3D molecules generated by the SLDM model trained on the QM9 dataset. + +![](images/77db63a4aab9f5db1efa47f325136027be26017ba48c4056ad14ab767ead4de5.jpg) +Figure 10: The generative process of EDM, GeoBFN and SLDM models. + +# C.7 Visualizations of Generated Molecules and Sampling Process + +We have randomly generated 40 molecules using our trained models on QM9 in the unconditional setting, and visualized their 3D structures using py3Dmol, as shown in Figure 9. These examples clearly demonstrate the structural plausibility and diversity of our generated molecules. + +In Figure 2, we compare the diffusion processes of different models by adding noise to the same molecule's coordinates, enabling a straightforward visual comparison of their noise schedules. As a complement, we present a comparison of the practical sampling processes of the three models in Figure 10. Although generating the exact same molecule across models is not feasible in this case, we still arrive at conclusions analogous to those in Section 3.2 of the main text. Additionally, unlike the noise-adding process, the generation process of SLDM demonstrates more pronounced randomness. + +# D Related Work + +# D.1 An overview of 3D Molecular Generation + +Recent advances in 3D molecular generation can be categorized based on the underlying generative algorithms: Autoregressive methods generate molecules step by step, progressively connecting atoms or molecular fragments. G-SchNet [Gebauer et al., 2019] and G-SphereNet [Luo and Ji, 2022] are early examples that use this strategy to model 3D molecular structures. Building on + +these, Symphony [Daigavane et al., 2024] incorporates higher-degree E(3)-equivariant features and message-passing to improve the modeling of molecular geometries. Normalizing flow [Chen et al., 2018] has also been applied to molecule generation by E-NF [Garcia Satorras et al., 2021] and Köhler et al. [2020]. Diffusion-based models have gained prominence for 3D molecular generation recently. Hoogeboom et al. [Hoogeboom et al., 2022] introduced the Equivariant Diffusion Model (EDM) to jointly generate atomic coordinates and atom types. Extensions of EDM include EDM-Bridge [Wu et al., 2022], which enhances performance through prior bridges, and GeoLDM [Xu et al., 2023], which performs diffusion in a latent space using an autoencoder. EquiFM [Song et al., 2024] employs flow matching for efficient molecule generation, and GeoBFN [Song et al., 2023a] combines Bayesian Flow Networks with distinct generative algorithms tailored for discretized charges and continuous coordinates. END [Cornet et al., 2024] proposes a learnable data- and time-dependent noise schedule of the diffusion process, and achieves improved sampling efficiency. + +Several studies offer complementary strategies to generative algorithms. First, representation-conditioned generation has been explored by MDM [Huang et al., 2023a], which conditions on VAE representations, and GeoRCG [Li et al., 2024], which extends EDM by leveraging pretrained molecular representations. Second, some studies propose advanced network architectures to enhance molecular generation [Hua et al., 2024, Huang et al., 2023a, Le et al., 2024a, Irwin et al., 2024][Huang et al., 2023b, Mercatali et al., 2024]. Third, additional input information has been introduced to the molecular generation. For instance, MolDiff [Peng et al., 2023] explicitly predicts bonds during generation. MiDi [Vignac et al., 2023], JODO [Huang et al., 2023b], EQGAT-diff [Le et al., 2024a], and SemlFlow [Irwin et al., 2024] further extend generation to include bonds and formal charges, enriching the molecular inputs. Twigs [Mercatali et al., 2024], on the other hand, integrates additional molecular properties into the diffusion training process, leading to enhanced conditional generation. + +It is worth noting that evaluation strategies for 3D molecular generation can be different across methods. EDM [Hoogeboom et al., 2022] employs strict rules for bond definitions based on interatomic distances, implicitly enforcing constraints on bond lengths and steric hindrance. In this framework, the metric "stability" is rigorously defined, requiring correct valency and neutral atomic charges. Our method, along with most diffusion-based approaches [Wu et al., 2022, Xu et al., 2023, Song et al., 2023a], adheres to this evaluation standard. In contrast, another line of studies [Vignac et al., 2023, Le et al., 2024a, Irwin et al., 2024] infers bonds and formal charges directly from the model, allowing atoms to have non-zero formal charges. Therefore, the bond inference imposes no constraints on bond lengths or steric hindrance. Further, the "stability" metric is defined more loosely, permitting discrepancies between valency and the number of covalent bonds. This relaxed evaluation framework makes it challenging to directly compare these two approaches. We advocate future efforts to establish more equitable evaluation methods to fairly assess the strengths of both paradigms. + +# D.2 Related Diffusion-Based Studies and Key Differences with SLDM + +Noise Scheduling It is important to note that our proposed process schedule differs fundamentally from previous works on noise scheduling [Karras et al., 2022, Nichol and Dhariwal, 2021, Le et al., 2024b], time discretization strategies [Xue et al., 2024c, Li et al., 2023], and adaptive step size methods [Lu et al., 2022]. These approaches can be interpreted as applying a time transformation $\mathcal{T}(\cdot)$ that jointly rescales the diffusion process schedule as $\mu (\mathcal{T}(t))$ and $\sigma (\mathcal{T}(t))$ . Notably, such transformations preserve the monotonicity and endpoints of the schedule functions. In contrast, our method decouples $\mu$ and $\sigma$ , and fundamentally alters the monotonicity and endpoint of $\sigma (t)$ . + +Linear $\mu(t)$ . Although our method is derived from the diffusion perspective, its process schedule shares some similarities with flow matching (FM) algorithms, such as FM-OT [Lipman et al., 2023] and conditional flow matching (CFM) [Tong et al., 2023]. Both methods employ a linear $\mu(t)$ , and CFM introduces additional low-scale noise. From this perspective, our process schedule can also be interpreted as a variant of CFM. Specifically, our approach employs a prior distribution modeled as a small-scale Gaussian distribution centered at the origin. However, FM and diffusion differ fundamentally in their perspectives: FM models the generative process as an ODE, while diffusion models a stochastic process. This core distinction leads to differences in both the learning targets and sampling methods. Therefore, unlike FM that learns the velocity field and relies on ODE-based sampling, our approaches focus on learning the noise and employ stochastic sampling. + +Straight Sampling Trajectory. Moreover, similar to our approach, flow-based methods also aim at flows with straight trajectories. However, these methods rely on predefined prior-data joint distribution + +to produce straighter paths. This distribution is procured by solving an optimal transport (OT) problem during the training of flow matching [Tong et al., 2023, Song et al., 2024], and solving such problem is often challenging. When the OT solution is unavailable, achieving straight-line trajectories typically requires additional distillation steps or solving optimization problems, as studied in rectified flow [Liu et al., 2022], progressive distillation [Salimans and Ho, 2022], consistency model [Song et al., 2023b, Luo et al., 2023] and optimal flow matching [Kornilov et al., 2024]. NFDM [Bartosh et al., 2024] proposes a learnable forward process to adapt to align with the reverse process and introduces penalties on the curvature of the reverse process's trajectories. In contrast to these existing methods, our approach offers a more straightforward solution for achieving a straight trajectory, by designing a novel diffusion process that minimizes the second-order derivative of the trajectory. + +# E Implementation Detail + +The hyperparameter settings for molecular generation are detailed in Table 7. Settings follow UniGEM [Feng et al., 2024], with two additional tunable hyperparameters introduced by our generative algorithm: the noise variance $\sigma$ and the temperature annealing rate $\nu$ . + +Table 7: Network and training hyperparameters. + +
Network HyperparametersValue
Embedding size256 for unconditional generation, 192 for conditional generation
Layer number9 for QM9, 4 for Geom-Drugs
Shared layers1
Training HyperparametersValue
Batch size64 for QM9, 128 for Geom-Drugs
Train epoch3000 for QM9, 32 for Geom-Drugs
Learning rate1.00 × 10-4
OptimizerAdam
Sample steps T10 ~ 1000
Nucleation time10
Oversampling ratio0.5 for each branch
Loss weight1 for each loss term
Generative Algorithm HyperparametersValue
Noise Variance σ0.05 for unconditional generation, 0.1 for conditional generation
Temperature Annealing Rateν0.5 for unconditional generation, 3 for conditional generation
Non-uniform DiscretizationFalse if T > 13
+ +For QM9, it takes approximately 10 days on a single A100 GPU. For GEOM-drugs, it takes approximately 16 days on four A100 GPUs. While the training epochs are longer compared to the baselines, this extended duration is not mandatory. Although longer training leads to slight performance gains, we found that the mol stable and atom stable metrics remain stable around 2000 epochs on QM9 and 13 epochs on GEOM-Drugs during the training process. + +For sampling steps greater than 13, the geometric straight-line diffusion uses a uniform time discretization like GeoBFN and EDM. However, according to theorem 3.1, our trajectory exhibits a larger second-order derivative at the beginning of sampling. Therefore, a more efficient discretization strategy is to use fine-grained discretization for larger $t$ values. We manually set an empirical discretization strategy that yields a $1\%$ to $10\%$ improvement in Mol Stable when $T \leq 13$ . For sampling steps greater than 13, the impact on the results is less significant $(< 1\%)$ . We leave the exploration of the optimal discretization strategy for straight-line diffusion for future work. + +For conditional molecular generation, we follow the baseline approaches [Hoogeboom et al., 2022] by incorporating property values as additional inputs during training and sampling them from a prior distribution during inference. The results are measured using a property classification network trained on the first half of the QM9 dataset, while the remaining portion is used for training the generative model. In addition to generative model baselines, we include two basic baselines: Random and $N_{\mathrm{atoms}}$ . The Random baseline involves shuffling property labels in the training data and evaluating + +the property classifier on the shuffled data. The $N_{\mathrm{atoms}}$ baseline uses the property classifier network to predict the property based solely on the number of atoms in the molecule. The property classification network is used for evaluating the generated molecules. The metric is the MAE between predicted properties and target properties. 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Calderon + +CENIA cristian.buc@cenia.cl + +Cristóbal Rojas + +Institute for Mathematical and Computational Engineering + +Pontifical Catholic University of Chile & CENIA + +luis.rojas@uc.cl + +# Abstract + +We propose the first method to show theoretical limitations for one-layer softmax transformers with arbitrarily many precision bits (even infinite). We establish those limitations for three tasks that require advanced reasoning. The first task, Match 3 (Sanford et al., 2023), requires looking at all possible token triplets in an input sequence. The second and third tasks address compositionality-based reasoning: function composition (Peng et al., 2024) and binary relations composition, respectively. We formally prove the inability of one-layer softmax Transformers to solve any of these tasks. To overcome these limitations, we introduce Strassen attention and prove that, equipped with this mechanism, a one-layer transformer can in principle solve all these tasks. Importantly, we show that it enjoys sub-cubic running-time complexity, making it more scalable than similar previously proposed mechanisms, such as higher-order attention (Sanford et al., 2023). To complement our theoretical findings, we experimentally studied Strassen attention and compared it against standard (Vaswani et al, 2017), higher-order attention (Sanford et al., 2023), and triangular attention (Bergen et al. 2021). Our results help to disentangle all these attention mechanisms, highlighting their strengths and limitations. In particular, Strassen attention outperforms standard attention significantly on all the tasks. Altogether, understanding the theoretical limitations can guide research towards scalable attention mechanisms that improve the reasoning abilities of Transformers. + +# 1 Introduction + +What tasks can Transformers solve? A fundamental question in modern AI is understanding why and under which conditions a given deep network architecture succeeds or fails on a given task. Numerous recent works have shown that tasks requiring compositional reasoning stand out as particularly challenging [6, 24, 16, 15]. Compositionality refers to the ability of composing blocks of knowledge to generate new content or solve new tasks, and is believed to play a major role in the emergence of systematic generalization in language [17, 4]; making the question for these kind of tasks even more relevant. + +To address this problem, benchmark datasets such as SCAN [14], PCFG [11], CLUTRR [22] or COGS [13] have been introduced. Moreover, different empirical methods have been developed to test, quantify and compare compositional capabilities [12], as well as to assess the impact of design choices on the performance of Transformers on compositional tasks [18]. While many of these works provide strong empirical evidence that Transformers may suffer from inherent limitations for compositional tasks [8], our current theoretical understanding of the underlying phenomenon remains limited. + +Here, we take a deeper dive into the problem and aim at contributing to the study of compositionality in Transformers on a more basic mathematical level. As we shall see, this allows us to pin down simple theoretical obstacles that make compositionality hard for the standard Transformer architecture. In turn, we can devise a new attention mechanism, which transcends these basic limitations. + +Related work Our work can be related to two complementary lines of research which we now develop. + +Theoretical limitations of Transformers. A natural first step to explain why models struggle with some tasks is to study their expressivity—whether a given architecture is in-principle capable of solving the task in question. That is, whether there exists a choice of parameters that results in a model computing the function underlying the task solutions. If the answer is positive, then the chances are that the difficulty lies in efficiently learning the appropriate parameters. If the answer is negative, then a formal proof of this fact can be directly related to the poor performance observed in practice. + +The first theoretical limitations of this sort were obtained for Transformers using hardmax attention [10]. Instead of computing attention as a convex combination of all input tokens using softmax, one takes a single token where the attention is maximal. Using this simplification, Hahn showed that hardmax Transformers with $O(1)$ layers cannot compute formal languages such as PARITY, MAJORITY, or Dyck-1. + +Theoretical limitations against softmax Transformers have recently been obtained by employing a different proof technique, one based on communication complexity [19]. The idea is to show that a Transformer solving a given task can be used to construct a corresponding communication protocol for a problem whose communication complexity is known. From this, one obtains lower bounds on the size of Transformers capable of solving the task for inputs of a given length. In [19], for example, the authors apply this technique to show that any one-layer softmax Transformer that can compute the composition of two functions must have $n^{\Omega(1)}$ embedding dimension, where $n$ is the size of the input. Crucially, for this conclusion to hold, one must assume that the Transformer works with a relatively low number of precision bits, namely sub-linear in $n$ . The technique has subsequently been applied to show lower bounds for other tasks such as string equality [3] and Match3 [20] (see Section 3.2.2 for a definition). + +Overcoming Transformers limitations. Equipped with a better theoretical understanding of why Transformers struggle with some tasks, the next question is how they can guide research towards the construction of more expressive machines. A key observation is that the standard attention mechanism can only see interactions between pairs of tokens, whereas compositional tasks require to reason about the interaction of three or more tokens simultaneously [2, 20]. Consequently, suitably modified attention mechanisms have been proposed, which would track interactions between more than two tokens. For instance, triangular attention [2] (see Section 2 for a definition) outperforms standard attention in compositional tasks such as CLUTRR [22] or COGS [13]. Similarly, it has been shown that higher-order tensor attention embedded in a constant size one-layer Transformer can theoretically solve Match3, a task that cannot be solved by the standard attention [20]. However, + +both these mechanisms suffer from a significant increase in running-time complexity, e.g., order 3 requires cubic-time, limiting its scalability and potentially affecting its practical relevance. In the case of triangular attention, the mechanism works with an adjacency matrix of a graph (in which case it is square-time in the size of the graph). It is possible to produce an adjacency matrix from the sequence of tokens, as already shown in Bergen et al. [2], but this increases the complexity to cubic. The higher-order tensor attention, on top of being cubic-time as well, has been only considered in theoretical work and has not been evaluated empirically yet. + +Our contributions In this work we aim at addressing these fundamental questions from complementary angles. Our main results can be summarized as follows: + +- We present a new technique for proving theoretical limitations. Since previous results establishing limitations for softmax transformers are based on communication complexity, they only apply to transformers that store numbers with relatively few precision bits. This raises the question – would the use of long arithmetic help to circumvent these limitations? In this paper, we answer this question negatively. In order to establish that, we introduce a new technique for proving limitations, based on the novel notion of Splitting VC dimension. The splitting VC dimension of a Boolean function $f$ , denoted by split-VC(f), is a positive integer number intended to capture the complexity of $f$ in a certain combinatorial sense. Our main technical contribution is the following: + +Main Theorem. Let $T$ be any one-layer standard attention Transformer that can do arithmetic operations over real numbers with infinite precision. Denote by $d, H$ and $L$ , respectively, its embedding dimension, number of attention heads and size of the output MLP of $T$ . Suppose $T$ can compute a function $f$ . Then it holds that $\max \{d, H, L\} \geq \text{split-VC}(f)^{\Omega(1)}$ . + +To put it differently, even an idealized transformer $T$ that can manipulate numbers with infinitely many bits cannot compute a function $f$ with large split-VC(f) unless $T$ has a large embedding dimension, a large number of attention heads, or a large output MLP. Obviously, the same limitations apply to real-life Transformers that can only manipulate numbers using some finite (however large) number of bits. + +- We obtain new theoretical limitations for tasks requiring complex reasoning. We apply our new method to the task of function composition (Peng et al. [19]) and the Match3 (Sanford et al. [20]) task, and show that previously known limitations for these tasks are also applicable to transformers that work with arbitrarily large (and even infinite) precision. We also do this for a new task, that we name binary relation composition task. We introduce it because the function composition task can be solved by a 2-layer transformer, while for the binary relation composition task there is no apparent solution with any constant number of layers. Finally, we introduce another task that we call Quotient Binary Relation Composition (see Section 5). Our method allows to establish limitations for this task even for mixed transformers that can use both the standard and the triangular attention. + +- We develop a more scalable higher-order mechanism. Previous works (e.g. [20]) have shown that higher-order methods can in principle overcome these limitations, but suffer from severe scalability issues. In an attempt to address these difficulties, we present and study Strassen attention, a variation devised to be sensitive to interactions between triplets of tokens without sacrificing too much efficiency. In contrast to previous similar attention mechanisms that required running time $n^3$ for inputs of length $n$ , our mechanism enjoys $n^2$ space and sub-cubic running time, making it more scalable. At the same time, we show that 1-layer constant-size Strassen attention transformers are theoretically capable of solving all 4 aforementioned tasks. + +- We empirically demonstrate the convergence of Strassen attention. Is Strassen attention capable of converging to these theoretical solutions during learning? We empirically demonstrate that the answer is positive for all 4 tasks. For comparison, we have additionally evaluated all other attention mechanisms on these tasks. Our empirical findings agree with theory and show that Strassen attention: (i) outperforms the accuracy of Standard attention, (ii) achieves superior efficiency in terms of training time and computational resources when compared with other triple-wise interaction attention mechanisms. + +Paper organization In Section 2 we recall the Transformer architecture and introduce our new Strassen attention mechanism. Section 3 presents our new lower bound method and its application to function composition, Match3, and binary relation composition tasks. In Section 4, we show that Strassen attention can be implemented in sub-cubic time, and formally prove that, in principle, it is capable of solving the three aforementioned task. Section 5 presents a novel task allowing to separate the capabilities of Strassen attention from those of standard and triangular attentions. Finally, section 6 contains our experimental findings. Due to space constraints, some proofs and experimental details are deferred to the Appendix. + +# 2 Preliminaries + +Throughout the paper, we denote $[n] = \{1,\dots ,n\}$ for $n\in \mathbb{N}$ . For a set $\Sigma$ , we will denote by $\Sigma^n$ the collection of sequences of elements of $\Sigma$ of length $n$ , and by $\Sigma^*$ the collection of all finite sequences. We start by briefly recalling the basics of the Transformer architecture and formally defining the attention mechanisms studied in this paper. + +The main block of the Transformer layer is the attention function, formally defined as a length-preserving function $a\colon (\mathbb{R}^d)^* \to (\mathbb{R}^d)^*$ , where $d$ is the embedding dimension. In this paper, we consider 4 types of attention functions. + +Standard attention [23], receives as input a sequence $x_{i} \in \mathbb{R}^{d}, i = 1, \dots, n$ and outputs a sequence $a_{i} \in \mathbb{R}^{d}, i = 1, \dots, n$ , computed as follows: + +$$ +a _ {i} = \sum_ {j = 1} ^ {n} a _ {i j} v _ {j} \tag {1} +$$ + +$$ +a _ {i j} = \operatorname {S o f t m a x} _ {j} \left(q _ {i} k _ {j} / \sqrt {d}\right) \tag {2} +$$ + +$$ +q _ {i} = W ^ {q} x _ {i}, \quad k _ {j} = W ^ {k} x _ {j}, \quad v _ {j} = W ^ {v} x _ {j}, \tag {3} +$$ + +where $W^{q}, W^{k}, W^{v} \in \mathbb{R}^{d \times d}$ + +Triangular attention [2] is defined for $n = m^2$ , with input tokens indexed by pairs $(i,j)$ , $i,j = 1,\ldots,m$ . Given an input $\{x_{ij} \in \mathbb{R}^d\}_{i,j=1}^m$ , the output is computed as follows: + +$$ +a _ {i j} = \sum_ {\ell = 1} ^ {m} a _ {i \ell j} v _ {i \ell j} \tag {4} +$$ + +$$ +a _ {i \ell j} = \operatorname {S o f t m a x} _ {\ell} \left(q _ {i \ell} k _ {\ell j} / \sqrt {d}\right) \tag {5} +$$ + +$$ +q _ {i \ell} = W ^ {q} x _ {i \ell}, \quad k _ {\ell j} = W ^ {k} x _ {\ell j}, \tag {6} +$$ + +$$ +v _ {i \ell j} = V _ {1} x _ {i \ell} \odot V _ {2} x _ {\ell j}, \tag {7} +$$ + +where $W^{q}, W^{k}, V_{1}, V_{2} \in \mathbb{R}^{d \times d}$ . + +Third-order attention [20] is computed as follows: + +$$ +a _ {i} = \sum_ {j, \ell = 1} ^ {n} a _ {i j \ell} \left(v _ {j} \odot v _ {\ell}\right) \tag {8} +$$ + +$$ +a _ {i j \ell} = \operatorname {S o f t m a x} _ {j, \ell} \left(q _ {i} \left(k _ {j} \odot k _ {\ell}\right) / \sqrt {d}\right) \tag {9} +$$ + +$$ +q _ {i} = W ^ {q} x _ {i}, \quad k _ {j} = W _ {1} ^ {k} x _ {j}, \quad k _ {\ell} = W _ {2} ^ {k} x _ {\ell}, \tag {10} +$$ + +$$ +v _ {j} = V _ {1} x _ {j}, \quad v _ {\ell} = V _ {2} x _ {\ell}, \tag {11} +$$ + +where $W^{q},W_{1}^{k},W_{2}^{k},V_{1},V_{2}\in \mathbb{R}^{d\times d}$ + +We introduce Strassen attention, computed as follows: + +$$ +a _ {i} = \sum_ {j, k = 1} ^ {n} a _ {i j k} \left(v _ {j} \odot v _ {k}\right) \tag {12} +$$ + +$$ +a _ {i j k} = \operatorname {S o f t m a x} _ {j, k} \left(\left(f _ {i} g _ {j} + g _ {j} h _ {k} + h _ {k} f _ {i}\right) / \sqrt {d}\right) \tag {13} +$$ + +$$ +f _ {i} = W ^ {f} x _ {i}, \quad g _ {j} = W ^ {g} x _ {j}, \quad h _ {k} = W ^ {h} x _ {k}, \tag {14} +$$ + +$$ +v _ {j} = V _ {1} x _ {j}, \quad v _ {k} = V _ {2} x _ {k}, \tag {15} +$$ + +where $W^{f}, W^{g}, W^{h}, V_{1}, V_{2} \in \mathbb{R}^{d \times d}$ , and $\odot$ denotes the Hadamard product. See Figure 2 in Appendix A for the illustration of these attention mechanisms. + +Definition 2.1. A one-layer Transformer $T$ with $H$ heads and embedding dimension $d$ is given by $H$ attention functions $Att_{1},\ldots ,Att_{H}\colon (\mathbb{R}^{d})^{*}\to (\mathbb{R}^{d})^{*}$ , a matrix $W_{O}\in \mathbb{R}^{d\times (dH)}$ , and an "output MLP" $\mathcal{N}\colon \mathbb{R}^d\to \mathbb{R}$ with $P$ parameters, which is formally a neural network with ReLU activation. We define the size of $T$ as $size(T) = \max \{H,d,P\}$ . The output of $T$ on input $\bar{x} = (x_1,\dots,x_n)\in (\mathbb{R}^d)^n$ is the sequence $\bar{y} = (y_{1},\dots,y_{n})\in (\mathbb{R})^{n}$ given by + +$$ +a _ {i} ^ {(h)} = \left(A t t _ {h} (\bar {x})\right) _ {i}, \quad h = 1, \dots H \tag {16} +$$ + +$$ +\widehat {a} _ {i} = W _ {O} \left( \begin{array}{c} a _ {i} ^ {(1)} \\ \vdots \\ a _ {i} ^ {(H)} \end{array} \right) \tag {17} +$$ + +$$ +y _ {i} = \mathcal {N} \left(x _ {i} + \widehat {a} _ {i}\right). \tag {18} +$$ + +So far, Transformers are defined as functions transforming sequences of vectors in $\mathbb{R}^d$ . For the tasks we consider in this paper, we need to apply Transformers on sequences of symbols of an arbitrary finite alphabet $\Sigma$ . This is done by including into the Transformer a positional encoding $p\colon [n]\times \Sigma \to \mathbb{R}^d$ . For a given $p$ , an input word $w = \sigma_1\ldots \sigma_n\in \Sigma^n$ is converted into a sequence of vectors: + +$$ +x _ {1} = p (1, \sigma_ {1}), \dots , x _ {n} = \sigma (n, \sigma_ {n}) +$$ + +that constitute the input for the Transformer. For our lower bounds, we make no assumptions about the function $p$ . In our upper bounds, however, we present constructions that use reasonable, easily computable positional encodings of the form $p(i, \sigma_i) = q(i) + r(\sigma_i)$ , treating positions and symbols independently. + +# 3 Theoretical Limitations of Transformers via Split-VC dimension + +We now introduce the notion of splitting dimension for a Boolean function $f$ . Let $\mathcal{X}$ be a set and $H \subseteq \{0,1\}^{\mathcal{X}}$ be a collection of functions $h: \mathcal{X} \to \{0,1\}$ which we will refer to as hypothesis class. We say that an hypothesis class $H$ shatters a subset $X = \{x_1, \ldots, x_m\} \subseteq \mathcal{X}$ if for any Boolean vector $c_1 \ldots c_m \in \{0,1\}^m$ there exists $h \in H$ with $h(x_1) = c_1, \ldots, h(x_m) = c_m$ . The maximal $m$ for which $H$ shatters some $X \subseteq \mathcal{X}$ of cardinality $m$ is called the VC dimension of $H$ [21]. + +We now explain how to adapt VC dimension to use it as a complexity measure of a single function (instead of a class of functions). Take a function $f \colon \Sigma^n \to \{0,1\}$ and imagine we split the $n$ arguments of $f$ into two parts. One part will continue to correspond to the inputs, but the other is to be regarded as a set of parameters, so that now we can see $f$ as a class of functions that has a well-defined VC dimension. The complexity of $f$ will be defined as the maximal VC dimension that we can obtain in this way, considering all possible splittings into parameters and inputs. + +Let us formalize this idea. For a set of positions $A \subseteq \{1, \dots, n\}$ , we define a Boolean matrix $M_f^A$ as follows. Its rows (interpreted as inputs) will be indexed by all the words $w^1 \in \Sigma^A$ and its columns (interpreted as parameters) by the words $w^2 \in \Sigma^B$ , where $B = \{1, \dots, n\} \setminus A$ . Thus, $M_A^f$ will be a $|\Sigma|^{|A|} \times |\Sigma|^{n - |A|}$ Boolean matrix. The value of $M_A^f$ at $(w^1, w^2)$ is then defined as + +$$ +M _ {A} ^ {f} (w ^ {1}, w ^ {2}) = f (w ^ {1} \oplus w ^ {2}) +$$ + +where $w^1 \oplus w^2 \in \Sigma^n$ is obtained by merging $w^1$ and $w^2$ according to the positions indicated by $A$ and $B$ , i.e. + +$$ +(w ^ {1} \oplus w ^ {2}) _ {i} = \left\{ \begin{array}{l l} w _ {i} ^ {1} & i \in A, \\ w _ {i} ^ {2} & i \in B, \end{array} \right. \qquad i = 1, \ldots , n. +$$ + +Definition 3.1. We define the splitting VC dimension of $f \colon \Sigma^n \to \{0,1\}$ , denoted by split-VC(f), as the maximum over $A \subseteq \{1,\dots,n\}$ of the VC dimension of the set of columns of $M_f^A$ , understood as Boolean functions on the set of rows. + +Example. Consider a Boolean function $f \colon \{0,1\}^4 \to \{0,1\}$ , defined by $f(x_1,x_2,x_3,x_4) = (x_1 \wedge x_2) \oplus (x_3 \wedge x_4)$ . We claim that split-VC(f) = 2. To establish split-VC(f) ≥ 2, we consider $A = \{1,3\}$ . The matrix $M_f^A$ is constructed as follows: its rows are indexed by Boolean vectors $x_1x_3 \in \{0,1\}^2$ , its columns by Boolean vectors $x_2x_4 \in \{0,1\}^2$ , and the intersection of the row $x_1x_3$ and column $x_2x_4$ has $f(x_1,x_2,x_3,x_4)$ in it. Thus, this matrix looks as follows: + +
x2x4x1x300011011
000000
010101
100011
110110
+ +Columns of this matrix realize all 4 Boolean functions on the second and the third row, implying that the VC dimension of the set of columns of this matrix is at least 2. We now observe that there is no $A \subseteq \{1,2,3,4\}$ such that the set of columns of $M_f^A$ has VC dimension at least 3. Indeed, this requires $A$ to be of size at least 2, because otherwise there are less than 3 rows. But if $|A| \geq 2$ , there are at most 4 columns, making it impossible to realize $2^3 = 8$ different Boolean functions. + +# 3.1 Main Theorem + +In order to state our main Theorem, we first need to specify a way to see a Transformer as computing a given boolean function $f\colon \Sigma^n\to \{0,1\}$ . We assume that an input word $w = \sigma_{1}\dots \sigma_{n}$ is given to the Transformer using $n + 1$ tokens. The first $n$ tokens are used to encode the $n$ symbols of $w$ , while the $(n + 1)$ -st auxiliary token (initially encoding the empty symbol) is used to encode the output $f(w)$ of the function being computed. More specifically, the output of the Transformer in the auxiliary token has to be a real number $y_{n + 1}$ , satisfying $f(w) = \mathrm{sign}(y_{n + 1})$ . When a Transformer $T$ fulfills this requirement for a given function $f$ , we will say that the Transformer $T$ computes $f$ in an auxiliary token. We can now state our main result. + +Theorem 3.2. Let $T$ be a one-layer standard-attention Transformer and let $f\colon \Sigma^n\to \{0,1\}$ be a Boolean function. If $T$ computes $f$ in an auxiliary token, then $size(T) = \text{split-VC}(f)^{\Omega (1)}$ . + +Remark 3.3. Note that from Theorem 3.2 it follows that for any Transformer $T$ satisfying $\text{size}(T) = n^{o(1)}$ , it is impossible to compute in an auxiliary token any function $f \colon \Sigma^n \to \{0,1\}$ with split-VC $(f) = n^{\Omega(1)}$ . + +# 3.2 Applications to three concrete tasks + +We now apply Theorem 3.2 to three different tasks. + +# 3.2.1 Function Composition + +Introduced in Peng et al. [19], in this task we receive a description of two functions $g:[n]\to [n]$ and $h:[n]\rightarrow [n]$ , and we are asked the value of $h(g(x))$ for a given $x\in [n]$ . The task is presented to a Transformer in the form of a sequence of $2n + 1$ tokens, divided in three parts. The first two parts encode the values of $g$ and $h$ , and the third part encodes $x$ in a single token, where the output has to be computed. More specifically, the output has to be a real number $y_{2n + 1}$ satisfying $|y_{2n + 1} - h(g(x))| < 0.5$ . That is, with $h(g(x))$ being the closest integer to $y_{2n + 1}$ . + +Theorem 3.4. Let $T$ be any one-layer standard-attention Transformer with size( $T$ ) = $n^{o(1)}$ . Then $T$ cannot solve the function composition task. + +Proof. We show that any Transformer that solves this task can be converted into a Transformer computing in the auxiliary token the following Boolean function: + +$$ +I n d _ {n}: [ n ] ^ {n + 1} \rightarrow \{0, 1 \}, +$$ + +$$ +I n d _ {n} (p _ {1}, q _ {1}, \ldots , q _ {n}) = \left\{ \begin{array}{l l} 1 & q _ {p _ {1}} = 1, \\ 0 & \text {o t h e r w i s e}, \end{array} \right. +$$ + +and that this transformation requires adding just $O(1)$ neurons to the output MLP. Indeed, by fixing $x = 1$ and setting $g(1) = p_1, h(1) = q_1, \ldots, h(n) = q_n$ , we obtain that the token with $x$ outputs a real number $y$ with $|y - h(g(1))| = |y - q_{p_1}| < 0.5$ . It now remains to change the output to $\mathrm{ReLU}(1.5 - y)$ which will be positive exactly when $q_{p_1} = 1$ . + +The result now follows from Theorem 3.2 and the following: + +Lemma 3.5. split-VC $(Ind_{n})\geq n$ + +Proof. We claim that the VC dimension of the set of columns of $M = M_{Ind_n}^A$ with $A = \{1\}$ , is at least $n$ . Rows of this matrix are indexed by $p_1 \in [n]$ , and columns by vectors $q_1 \ldots q_n \in [n]^n$ . We claim that the set of all $n$ rows of $M$ is shattered by the columns of $M$ . Take any Boolean vector $b = c_1 \ldots c_n \in \{0, 1\}^n$ . We need to find $q_1 \ldots, q_n \in [n]^n$ such that $Ind_n(i, q_1, \ldots, q_n) = c_i$ for all $i \in [n]$ . It is suffices to define $q_i = \begin{cases} 1 & c_i = 1, \\ 2 & c_i = 0. \end{cases}$ + +![](images/93c52e987875180fe6f0e2d46ce1b55d3f5acd3dd57fcbd1e429ec40e133248e.jpg) + +# 3.2.2 The Match3 task + +Next, we define the $\mathrm{Match}_3[n,m]$ task [20]. It is a sequence-to-sequence task. The input is presented to the Transformer as a sequence of $n$ tokens, encoding a vector of integers $(p_1,\dots ,p_n)\in [m - 1]^n$ . The output is a vector $(y_{1},\ldots ,y_{n})\in \mathbb{R}^{n}$ , required to satisfy: + +$$ +\operatorname {s i g n} (y _ {i}) = \left\{ \begin{array}{l l} 1 & \exists j, k \in [ n ] \text {s . t .} \\ & p _ {i} + p _ {j} + p _ {k} = 0 \pmod {m} \\ 0 & \text {o t h e r w i s e} \end{array} \right. +$$ + +Note that we deliberately exclude the value $p_i = 0$ for the input positions. This is to avoid inputs that make the task trivial. Indeed, if $p_i = 0$ for some $i$ , then the output is trivially 1 since we always have $p_i + p_i + p_i = 0$ . + +Theorem 3.6. Let $T$ be any one-layer standard-attention Transformer with size( $T$ ) = $n^{o(1)}$ . Then $T$ cannot solve the Match $_3[n, m]$ task for $m = 2n - 2$ . + +# 3.2.3 Binary Relation Composition + +Finally, we define the binary relation composition task. This is a sequence-to-sequence task, where on input we get two Boolean matrices $A, B \in \{0,1\}^{\sqrt{n} \times \sqrt{n}}$ . The input is presented to a Transformer using $n$ tokens, indexed by pairs $(i,j) \in [\sqrt{n}]^2$ , with the $(i,j)$ -th token receiving an encoding of $A_{ij}$ and $B_{ij}$ . In the output, we have to compute the matrix of the "composition" of $A$ and $B$ : + +$$ +B \circ A \in \{0, 1 \} ^ {\sqrt {n} \times \sqrt {n}}, \quad (B \circ A) _ {i j} = \bigvee_ {k = 1} ^ {\sqrt {n}} \left(A _ {i k} \wedge B _ {k j}\right). +$$ + +More precisely, the output of the $(i,j)$ -th token for $(i,j) \in [\sqrt{n}]^2$ has to be a real number $y_{ij}$ with $\mathrm{sign}(y_{ij}) = (B \circ A)_{ij}$ . We refer to $A$ and $B$ as "relations" on the set $[\sqrt{n}]$ , with $A_{ij}$ indicating whether the pair $(i,j)$ is in the relation $A$ and $B_{ij}$ doing so for relation $B$ . For an example, imagine that $A = B$ encodes a relation for a group of researchers where two researchers $i$ and $j$ are related if they have a paper in co-authorship. Then the composition $B \circ A$ refers to the relation of "having a common co-author". + +Theorem 3.7. Let $T$ be any one-layer standard-attention Transformer with size( $T$ ) = $n^{o(1)}$ . Then $T$ cannot solve the binary relation composition task. + +# 4 Strassen attention - An efficient mechanism to solve complex tasks + +Both the third-order mechanism of Sanford et al. [20] and the Strassen mechanism define attention as the interaction between three tokens (i.e., a triplet). The crucial difference is that Strassen attention is computed using pairwise dot-products of vectors in the triplet, while the third-order involves coordinates products of all 3 vectors. This allows us to decompose Strassen attention scores in a way that reduces the whole layer to the product of a constant number of $n \times n$ matrices. Famously, the $n \times n$ matrix product admits an $O(n^{\omega})$ -time algorithm for $w < 3$ , with currently best known upper bound on $w$ being 2.372 [7]. + +Theorem 4.1. Strassen attention layer can be implemented in $O(n^{\omega} \cdot d)$ -time, where $n$ is the number of input tokens, $d$ is the embedding dimension, and $\omega$ is the matrix multiplication exponent, i.e., the smallest real number such that the $n \times n$ matrix product admits an $O(n^{\omega})$ -time algorithm. + +Proof. Writing $a_i$ in (12-15) by definition, we get: + +$$ +a _ {i} = \frac {\sum_ {j , k} e ^ {\left(f _ {i} g _ {j} + g _ {j} h _ {k} + h _ {k} f _ {i}\right) / \sqrt {d}} \left(v _ {j} \odot v _ {k}\right)}{\sum_ {j , k} e ^ {\left(f _ {i} g _ {j} + g _ {j} h _ {k} + h _ {k} f _ {i}\right) / \sqrt {d}}}. \tag {19} +$$ + +Defining matrices $X,Y,Z\in \mathbb{R}^{n\times n}$ and $\widehat{Y}\in \mathbb{R}^{n\times n\times d}$ by: + +$$ +X _ {i j} = e ^ {f _ {i} g _ {j} / \sqrt {d}}, Y _ {j, k} = e ^ {g _ {j} h _ {k} / \sqrt {d}}, Z _ {k, i} = e ^ {h _ {k} f _ {i} / \sqrt {d}}, +$$ + +$$ +\widehat {Y} _ {j, k} = e ^ {g _ {j} h _ {k} / \sqrt {d}} v _ {j} \odot v _ {k}, +$$ + +we get an expression for $a_{i}$ in terms of their matrix products $a_{i} = \frac{(X\widehat{Y}Z)_{ii}}{(XYZ)_{ii}}$ . + +![](images/e08997a52a95fe21ecf3b87830d22688e1450d74541a133e978062aca18a0935.jpg) + +On top of exhibiting a faster running-time, we now show that, in contrast to standard attention, Strassen attention allows a one-layer Transformer to solve all the 3 tasks described in Section 3.2. + +Theorem 4.2. The function composition, the binary relation composition, and the Match $_3[n, poly(n)]$ tasks can be solved by a one-layer constant-size Strassen attention Transformer. + +# 5 Disentangling Strassen from Standard and Triangular attentions + +So far, we have evaluated tasks that are challenging for one-layer standard attention Transformers but (in principle) easy for one-layer Strassen attention Transformers. In this section, we extend our analysis to triangular attention. As a reminder, running the triangular attention mechanism on a general sequence of length $n$ , requires the creation of $n^2$ tokens. In this regime, the triangular attention running time becomes $n^3$ . However, when the input is already structured as a $\sqrt{n} \times \sqrt{n}$ matrix, one can run the triangular attention on it directly, making the running time $O(n^{3/2})$ . One such task example using structured input is the binary relation composition task. In this case, a one-layer triangular attention can perform this task with one attention head, constant embedding dimension and constant-size output MLP. + +We devise a variant of the binary relation task that allows us to disentangle the performance of Strassen attention with that of the triangular and standard attentions, namely the quotient binary relation composition task. The latter takes as inputs two Boolean matrices $A, B \in \{0,1\}^{m \times m}$ and a "coloring" function $\mathsf{col} \colon [m] \to [m]$ , where $m = \sqrt{n}$ . There are $n = m^2$ input tokens, indexed by pairs from $[m]^2$ , with the $(i,j)$ -th token receiving $A_{i,j}, B_{i,j}$ and $(\mathsf{col}(i), \mathsf{col}(j))$ as inputs. The quotient of the composition $B \circ A$ by $\mathsf{col}$ is a Boolean matrix $B \circ A / \mathsf{col} \in \{0,1\}^{m \times m}$ , defined by: + +$$ +(B \circ A / \mathsf {c o l}) _ {i j} = \left\{ \begin{array}{l l} 1 & \exists k _ {1}, k _ {2} \in [ m ] \text {s . t .} A _ {i k _ {1}} = B _ {k _ {2} j} = 1, \\ & \mathsf {c o l} (k _ {1}) = \mathsf {c o l} (k _ {2}), \text {a n d} k _ {1} \neq k _ {2}, \\ 0 & \text {o t h e r w i s e}. \end{array} \right. +$$ + +The task is to output, for all $(i,j)\in [m]^2$ , a real number $y_{ij}$ such that $(B\circ A / \mathsf{col})_{ij} = \mathsf{sign}(y_{ij})$ + +To illustrate an instance of this task, imagine that $A = B$ encodes the graph of co-authorship between a set of researchers, and $c$ assigns each researcher its university. Then we have $(B \circ A / \mathsf{col})_{ij} = 1$ if + +![](images/ae747e7e9d5c96de627c7ff8279151964f32604794079c87fe5859ac935eedc4.jpg) +Figure 1: Accuracy as a function of (a) the number of epochs, (b) training time (forward plus backward runtime per epoch), and (c) speed gain (measured as epochs per minute). Performance for each task is presented as the median accuracy over 8 independent runs on data outside the training set. For readability, we truncated the binary relation composition and quotient binary relation composition learning curves to 100 and 2000 epochs, respectively. + +and only if researchers $i$ and $j$ have co-authors from the same university, with the condition that these co-authors must be different people. + +Theorem 5.1. The quotient binary relation composition task is solvable by a one-layer Strassen-attention constant-size transformer. At the same time, this task cannot be solved by any one-layer Transformer $T$ with $n^{o(1)}$ standard-attention heads, $n^{o(1)}$ triangular-attention heads, $n^{o(1)}$ embedding dimension, and $n^{o(1)}$ -size output MLP. + +# 6 Experiments and Results + +In this section, we systematically compare the performance of standard, triangular, third-order and Strassen attention in four tasks: (i) Indicator of the 1st coordinate in the Function Composition (with $f = g$ ), (ii) Binary Relation Composition (with $A = B$ ), (iii) Match3 (over position-aware permutations) and (iv) Quotient Binary Relation Composition (with $B = A^T$ ). To obtain a tighter upper bound on the capability of the standard attention, we chose to implement simple special cases of the tasks (see Appendix C for all the details on data generation). To compare these attention mechanisms, we evaluate their performance (accuracy per epoch) and training time (accuracy per minute) on the test sets of these tasks. Furthermore, Appendix D provides a thorough analysis in terms of computational performance, evaluating forward pass times, GPU and memory usage. Code for our experiments can be found at Ⓞfurrutiav/strassen-attention-neurips25. + +Figure 1 displays our main experimental results. First, both the Strassen and third-order attention are the only two mechanisms that present high accuracy levels across all tasks. Note that Strassen attention displays slight advantages on the Function Composition and Match3 tasks (Figure 1a). Second, Strassen attention consistently outperforms third-order attention in training time (to peak performance), across all tasks (Figure 1b). Third, the advantages of Strassen attention are further exemplified by displaying speed gains (i.e., number of epochs per minute) that match that of the standard attention in the Function Composition and Quotient Binary Relation Composition tasks, and are always superior to that of the third-order attention (Figure 1c). Fourth, perhaps unsurprisingly, triangular attention outperforms all attention mechanisms in terms of training time at the Binary Relation Composition task (Figure 1b, second column). Indeed, Strassen and the third-order attentions need to learn the triangular structure of the task, a knowledge that is already structurally embedded in the triangular attention. Fifth, although the third-order attention presents similar accuracy per epoch trend to that of Strassen attention, its learning dynamics seems to be significantly more unstable, particularly for the Function Composition and Quotient Binary Relation Composition tasks. Sixth, a clear advantage of Strassen attention over the triangular and standard attentions was observed for the Quotient Binary Relation Composition task (Figure 1b, last column). Lastly, it is noteworthy that the + +triangular attention framework has a smaller applicability scope, and therefore could not be run on the Function Composition and Match3 tasks (without changing the data presentation from sequences to matrices). + +Strassen vs. standard attention Parameter-matched comparisons Given that 1-layer Strassen and standard attention architectures are not parameter matched, we performed two additional experiments to provide a fair comparison between them. In both experiments, Quotient Binary Relation Composition and Binary Relation Composition, we pitted a 1-layer Strassen attention against a 2-layer standard attention, both models with hidden dimension 16. Table 1 revealed that even with less parameters, 1-layer Strassen attention continues outperforming a 2-layer Standard attention (with about $40\%$ more parameters) in both of the tasks considered. Furthermore, to demonstrate the value of Strassen attention over standard attention in more practical settings, we evaluated this two attention architectures on the COGS dataset, in a parameter-matched configuration. Table 1 shows that Strassen attention outperforms standard attention. + +
TaskAttention TypeLayersHidden dim.ParametersAccuracy (%)
Binary RelationStrassen1162.5k100
CompositionStandard2163.6k92
Quotient BinaryStrassen1162.5k98
Relation CompositionStandard2163.6k82
COGSStrassen36499k72
Standard36899k65
+ +Table 1: Accuracies for the Binary Relation Composition task, Quotient Binary Relation Composition task and COGS dataset as a function of attention type, layer number, hidden dimension and parameters number. + +Future directions and limitations Our results pave the way for several directions of future research. First, we have introduced a new method to obtain limitations for one-layer Transformer, based on a new concept we have named splitting VC dimension. We expect our method will be applied to obtain lower bounds in other architectures and other tasks involving complex reasoning. Second, given the differences observed in learning stability between the third-order and Strassen attentions, the latter seems to be associated with a smoother loss landscape, an hypothesis that needs to be confirmed and studied. Third, Strassen attention can be adapted to incorporate interactions that involve more than three tokens, possibly capturing more complex patterns in data. Yet, practical learnability of such higher-order interactions needs to be assessed. Finally, and related to the previous point, although the main goal of our work was to gain a deeper theoretical understanding of the abilities of the Transformer, our conclusions are limited by using toy tasks. Our next step is to test the theoretical advantages of the Strassen attention using specific benchmarks or even in learning methods such as masked language modeling. At the same time, the possible applications of the Strassen attention are not limited to compositionality but could extend to such areas as knowledge graphs, protein structure prediction and others. + +Acknowledgments Kozachinskiy, Urrutia, Jimenez, Pizarro, Calderon, and Rojas are funded by the National Center for Artificial Intelligence CENIA FB210017, Basal ANID. Kozachinskiy is supported by ANID Fondecyt Iniciación grant 11250060. + +# References + +[1] Jason Ansel, Edward Yang, Horace He, Natalia Gimelshein, Animesh Jain, Michael Voznesensky, Bin Bao, Peter Bell, David Berard, Evgeni Burovski, Geeta Chauhan, Anjali Choudria, Will Constable, Alban Desmaison, Zachary DeVito, Elias Ellison, Will Feng, Jiong Gong, Michael Gschwind, Brian Hirsh, Sherlock Huang, Kshitteej Kalambarkar, Laurent Kirsch, Michael Lazos, Mario Lezcano, Yanbo Liang, Jason Liang, Yinghai Lu, CK Luk, Bert Maher, Yunjie Pan, Christian Puhrsch, Matthias Reso, Mark Saroufim, Marcos Yukio Siraichi, Helen Suk, Michael Suo, Phil Tillet, Eikan Wang, Xiaodong Wang, William Wen, Shunting Zhang, Xu Zhao, Keren Zhou, Richard Zou, Ajit Mathews, Gregory Chanan, Peng Wu, and Soumith Chintala. PyTorch 2: Faster Machine Learning Through Dynamic Python Bytecode Transformation and Graph Compilation. 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Advances in neural information processing systems, 35:29776-29788, 2022. + +# A Illustration of the attention mechanisms + +![](images/aa2d0f71a03b61e1095772e8cd9aa97d7c323cd12edf2e82a8404c2886813d6f.jpg) +Standard (Vaswani et al., 2017) + +Third-Order (Sanford et al., 2023) +Operator Legend +![](images/865f7049638143a51db1bbbec8386b46c92d407b8e0770d9dcafa126f1da8ac9.jpg) +Transposed Product +Weighted Sum +Hadamard Produc +Element-wise Product +Element-wise +$\stackrel{\circ}{+}$ Addition + +![](images/13f52005a75d8a6f283ab1811421f1fdccb9ce5155453cefba99b21b80636132.jpg) +Triangular (Bergen et al., 2021) + +![](images/642ff87ea551747a6881127328491b513305da176dd0c1ad907411c7900f7753.jpg) +Strassen (Our mechanism) + +![](images/4006c1e9e42ed194c4e91d9e5baa0ec8fcf1bfaa76a1b759906a73e71cd88757.jpg) +One-layer Transformer +Figure 2: Comparison of Transformer attention mechanisms: Standard [23], Third-Order [20], Triangular [2], and Strassen (proposed). The diagram shows operations and the one-layer Transformer architecture, highlighting the flow through each attention and output MLP (without layer normalization and dropout in the attention mechanism). Notation: input length $n$ , embedding dimension $d$ , number of nodes $m$ , with $n = m^2$ when $n$ refers to the shape of flatten matrices. + +# B Missing proofs + +# B.1 Proof of Theorem 3.2. + +Denote $m = \text{split-VC}(f)$ . Let $A \subseteq \{1, \ldots, n\}$ be such that the VC dimension of the set of columns of $M_f^A$ is $m$ . Denote $B = [n] \setminus A$ . Assume for contradiction that there exists a one-layer standard-attention Transformer $T$ that computes $f$ and whose size (that is, embedding dimension, number of attention heads, and the size of the output MLP) is $m^{o(1)}$ . + +Consider any $w^1 \in \Sigma^A$ , $w^2 \in \Sigma^B$ and define $w = w^1 \oplus w^2 = \sigma_1 \dots \sigma_n \in \Sigma^n$ . For $h = 1, \ldots, H$ , observe that the output of the $h$ -th attention head in the $(n + 1)$ -st token (the auxiliary one, where the output of the function is computed), can be written as: + +$$ +a _ {n + 1} ^ {(h)} = \frac {\alpha^ {(h)} \left(w ^ {1}\right) + \beta^ {(h)} \left(w ^ {2}\right) + \gamma^ {(h)}}{\lambda^ {(h)} \left(w ^ {1}\right) + \mu^ {(h)} \left(w ^ {2}\right) + \nu^ {(h)}}, \tag {20} +$$ + +where + +$$ +\alpha^ {(h)} \left(w ^ {1}\right) = \sum_ {i \in A} e ^ {q _ {n + 1} k _ {i}} v _ {i} \in \mathbb {R} ^ {d}, \quad \beta^ {(h)} \left(w ^ {2}\right) = \sum_ {i \in B} e ^ {q _ {n + 1} k _ {i}} v _ {i} \in \mathbb {R} ^ {d} +$$ + +$$ +\lambda^ {(h)} (w ^ {1}) = \sum_ {i \in A} e ^ {q _ {n + 1} k _ {i}} \in \mathbb {R}, \quad \mu^ {(h)} (w ^ {2}) = \sum_ {i \in B} e ^ {q _ {n + 1} k _ {i}} \in \mathbb {R} +$$ + +$$ +\gamma^ {(h)} = e ^ {q _ {n + 1} k _ {n + 1}} v _ {n + 1}, \quad \nu^ {(h)} = e ^ {q _ {n + 1} k _ {n + 1}} +$$ + +Note that $k_{i} = W^{q}p(i,\sigma_{i}), v_{i} = W^{v}p(i,\sigma_{i})$ are functions of $w^{1}$ for $i \in A$ and of $w^{2}$ for $i \in B$ , whereas $q_{n + 1}, k_{n + 1}$ , and $v_{n + 1}$ are fixed. + +The output of the function is thus computed by: + +$$ +f (w) = M _ {f} ^ {A} \left(w ^ {1}, w ^ {2}\right) = \operatorname {s i g n} \left(\mathcal {N} \left(x _ {n + 1} + W _ {O} \left( \begin{array}{c} \frac {\alpha^ {(1)} \left(w ^ {1}\right) + \beta^ {(1)} \left(w ^ {2}\right) + \gamma^ {(1)}}{\lambda^ {(1)} \left(w ^ {1}\right) + \mu^ {(1)} \left(w ^ {2}\right) + \nu^ {(1)}} \\ \vdots \\ \frac {\alpha^ {(H)} \left(w ^ {1}\right) + \beta^ {(H)} \left(w ^ {2}\right) + \gamma^ {(H)}}{\lambda^ {(H)} \left(w ^ {1}\right) + \mu^ {(H)} \left(w ^ {2}\right) + \nu^ {(H)}} \end{array} \right)\right)\right), \tag {21} +$$ + +where $x_{n + 1} = p(\emptyset),W_O\in \mathbb{R}^{d\times dH}$ are fixed, and $\mathcal{N}$ is the output MLP of $T$ + +Consider now arbitrary real vectors + +$$ +\alpha = \left(\alpha^ {(1)}, \dots , \alpha^ {(H)}\right) \in \mathbb {R} ^ {d H}, \quad \beta = \left(\beta^ {(1)}, \dots , \beta^ {(H)}\right) \in \mathbb {R} ^ {d H} +$$ + +$$ +\lambda = \left(\lambda^ {(1)}, \dots , \lambda^ {(H)}\right) \in \mathbb {R} ^ {H}, \quad \mu = \left(\mu^ {(1)}, \dots , \mu^ {(H)}\right) \in \mathbb {R} ^ {H} +$$ + +and define a function $F\colon (\alpha, \lambda, \beta, \mu) \mapsto \{0, 1\}$ as in (21), but with vectors $\alpha, \lambda, \beta, \mu$ allowed to take arbitrary values: + +$$ +F (\alpha , \lambda , \beta , \mu) = \operatorname {s i g n} \left(\mathcal {N} \left(x _ {n + 1} + W _ {O} \left( \begin{array}{c} \frac {\alpha^ {(1)} + \beta^ {(1)} + \gamma^ {(1)}}{\lambda^ {(1)} + \mu^ {(1)} + \nu^ {(1)}} \\ \vdots \\ \frac {\alpha^ {(H)} + \beta^ {(H)} + \gamma^ {(H)}}{\lambda^ {(H)} + \mu^ {(H)} + \nu^ {(H)}} \end{array} \right)\right)\right), \tag {22} +$$ + +Let $H$ be a class, defined by (22) when $\alpha, \lambda$ are considered as inputs to hypotheses and $\beta, \mu$ as parameters: + +$$ +H = \{h _ {\beta , \mu} \colon \mathbb {R} ^ {d H + H} \to \{0, 1 \}: h _ {\beta , \mu} (\alpha , \lambda) = F (\alpha , \lambda , \beta , \mu), (\beta , \mu) \in \mathbb {R} ^ {d H + H} \}. +$$ + +On the one hand, the VC dimension of $H$ is at least $m = \mathsf{split - VC}(f)$ . Indeed, consider $H$ is an infinite matrix, with rows indexed by $(\alpha ,\lambda)\in \mathbb{R}^{dH + H}$ , columns by $(\beta ,\mu)\in \mathbb{R}^{dH + H}$ , and the intersection of the $(\alpha ,\lambda)$ -row and $(\beta ,\mu)$ -column containing $F(\alpha ,\lambda ,\beta ,\mu) = h_{\beta ,\mu}(\alpha ,\lambda)$ . The VC dimension of $H$ is the VC dimension of the columns of this matrix. Since by (21) this matrix has $M_f^A$ as a sub-matrix, we get the required lower bound. + +On the other hand, the VC dimension of $H$ can be upper bounded by $m^{o(1)}$ . Indeed, the number of parameters of $H$ is $dH + H = m^{o(1)}$ . To compute the value of a given hypothesis on a given input, it is enough to do $m^{o(1)}$ basic arithmetic operations and comparisons with 0, because the size of $\mathcal{N}$ is $m^{o(1)}$ . By Theorem 2.3 in [9], the VC dimension is polynomial in these quantities, which gives us $m^{o(1)}$ upper bound in our case. + +# B.2 Proof of Theorem 3.6 + +We fix the value of $p_1$ to be equal to 1, and consider the output of $\mathrm{Match}_3[n,m]$ at the first position. If we additionally set $p_j \neq m - 2$ for $j \geq 2$ , we obtain $p_1 + p_j + p_k = 0$ (mod $m$ ) if and only if $j,k \geq 2$ and $p_j + p_k = -1$ (mod $m$ ). Hence, a Transformer for the $\mathrm{Match}_3[n,m]$ task can be converted into a Transformer, computing the following function in the auxiliary token: + +$$ +\operatorname {S u m} _ {2} [ \ell , m ] \colon ([ m - 1 ] \backslash \{m - 2 \}) ^ {\ell} \rightarrow \{0, 1 \} +$$ + +$$ +S u m _ {2} [ \ell , m ] (p) := \left\{ \begin{array}{l l} 1 & \exists j, k \in [ \ell ] \text {s . t .} p _ {j} + p _ {k} = - 1 \pmod {m}, \\ 0 & \text {o t h e r w i s e} \end{array} \right. +$$ + +where $\ell = n - 1$ + +The following Lemma establishes what we need in order to obtain the desired lower bound from Theorem 3.2. + +Lemma B.1. For even $\ell$ , it holds that split-VC $(Sum_{2}[\ell, 2\ell]) \geq \ell / 2$ . + +Proof. We claim that the VC dimension of the set of columns of $M = M_{Sum_2[\ell,2\ell]}^A$ with $A = \{1,\ldots,\ell/2\}$ , is at least $\ell/2$ . The rows of this matrix are indexed by vectors $p = p_1 \ldots p_{\ell/2} \in ([2\ell-1] \setminus \{2\ell-2\})^{\ell/2}$ , and columns by vectors $q = q_{\frac{\ell}{2}+1} \ldots q_\ell \in ([2\ell-1] \setminus \{2\ell-2\})^{\ell/2}$ . Note that in this case, for a given row $p$ and column $q$ , their merging $p \oplus q$ is simply their concatenation. + +We now consider $\ell /2$ rows, corresponding to vectors: + +$$ +p ^ {1} = (2, 1, 1, \dots , 1), +$$ + +$$ +p ^ {2} = (1, 4, 1, \dots , 1), +$$ + +中 + +$$ +p ^ {\ell / 2} = (1, 1, 1, \dots , \ell). +$$ + +and show that these rows can be shattered by the columns of $M_{Sum_2[\ell, 2\ell]}^A$ . For any Boolean vector $c_1 \ldots c_{\ell/2} \in \{0, 1\}^{\ell/2}$ , we have to find a column $q \in ([2\ell - 1] \setminus \{2\ell - 2\})^{\ell/2}$ such that: + +$$ +\operatorname {S u m} _ {2} [ \ell , 2 \ell ] (p ^ {i} \oplus q) = c _ {i} +$$ + +for all $i = 1,\dots ,\ell /2$ . It then suffices to choose $q$ such that + +$$ +q _ {\frac {\ell}{2} + i} = \left\{ \begin{array}{l l} 2 \ell - 2 i - 1 & c _ {i} = 1, \\ 1 & \text {o t h e r w i s e .} \end{array} \right. \quad i \in [ \ell / 2 ]. +$$ + +Indeed, first note that the value $m - 2 = \ell - 2$ is not used. Now, if $c_{i} = 1$ , then $p_{i} \oplus q$ has numbers $2i$ and $2\ell - 2i - 1$ , summing up to $2\ell - 1$ . Next, if $c_{i} = 0$ , in $p_{i} \oplus q$ only two numbers can appear, 1 and $2i\ell$ , whose sum is neither $2\ell - 1$ nor $4\ell - 1$ because $i \leq \ell / 2$ . This completes the proof. + +# B.3 Proof of Theorem 3.7 + +We consider a subproblem of this task, where only elements $A_{12},\ldots ,A_{1k}$ and $B_{21},\ldots ,B_{k1}$ can be equal to 1, where $k = \sqrt{n}$ . Under this restriction, a Transformer solving the binary relation composition computes, in the token at position (1,1), the function $\mathrm{Disj}_m\colon \{0,1\} ^m\times \{0,1\} ^m\to$ $\{0,1\}$ given by + +$$ +\operatorname {D i s j} _ {m} (a, b) = \bigvee_ {k = 1} ^ {m} \left(a _ {i} \wedge b _ {i}\right), +$$ + +with $a$ and $b$ being written in positions $A_{12}, \ldots, A_{1k}$ and $B_{21}, \ldots, B_{k1}$ , respectively. In our case, $m = k - 1 = \sqrt{n} - 1$ . The results now follow from Theorem 3.2 and the following lemma. + +Lemma B.2. split-VC $(\mathrm{Disj}_m)\geq m$ + +Proof. We show that the VC dimension of the set of columns of $M_{\mathrm{Disj}_m}^A$ is at least $m$ for $A = \{1, \ldots, m\}$ . Both the rows and the columns of $M_{\mathrm{Disj}_m}^A$ are indexed by $m$ -bit vectors. We show that $m$ rows, corresponding to the following vectors: + +$$ +a ^ {1} = (1, 0, \dots , 0, 0), +$$ + +$$ +a ^ {2} = (0, 1, \dots , 0, 0), +$$ + +中 + +$$ +a ^ {m} = (0, 0, \dots , 0, 1) +$$ + +can be shattered by the columns of the matrix. To establish that, for every $c \in \{0,1\}^m$ we have to provide $b = b_1 \ldots b_m \in \{0,1\}^m$ with + +$$ +\operatorname {D i s j} _ {m} \left(a ^ {i}, b\right) = c _ {i}, \quad i \in [ m ]. +$$ + +This can be achieved by simply setting $b_{i} = c_{i}$ for $i \in [m]$ . + +![](images/3584fe30d629a788aeb5105b3dcddd559d606925f22fab893dc334f8f01d241e.jpg) + +# B.4 Proof of Theorem 4.2 + +Function composition In the function composition task, we get a $(2n + 1)$ -length sequence of numbers + +$$ +\phi (1), \dots , \phi (2 n + 1) \in [ n ]. +$$ + +The task is to output, in the $(2n + 1)$ -st token, the value of $h(g(x))$ with the 0.5-precision (in fact, we will do this with a much better precision, namely $e^{-\Omega(n^2)}$ ), where $g, h: [n] \to [n]$ and $x \in [n]$ are such that $g(1) = \phi(1), \ldots, g(n) = \phi(n), h(1) = \phi(n + 1), \ldots, h(n) = \phi(2n), x = \phi(2n + 1)$ . + +We use the following positional encoding: + +$$ +x _ {i} = \left( \begin{array}{c} i \\ i ^ {2} \\ \phi (i) \\ (\phi (i)) ^ {2} \\ 1 \\ 0 \\ 0 \end{array} \right), \qquad i = 1, \ldots , 2 n + 1. +$$ + +We take matrices $W^{f}, W^{g}, W^{h}$ in (12-15) so that: + +$$ +f _ {i} = n \left( \begin{array}{c} (\phi (i)) ^ {2} \\ 2 \phi (i) \\ - 1 \\ 0 \\ 0 \\ 0 \end{array} \right), \qquad g _ {j} = n \left( \begin{array}{c} - 1 \\ j \\ j ^ {2} \\ (\phi (j)) ^ {2} \\ 2 \phi (j) \\ - 1 \end{array} \right), +$$ + +$$ +h _ {k} = n \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ - 1 \\ k - n \\ k ^ {2} - 2 k n + n ^ {2} \end{array} \right) +$$ + +We obtain: + +$$ +\begin{array}{l} a _ {i, j, k} = \operatorname {S o f t m a x} _ {j, k} \frac {f _ {i} g _ {j} + g _ {j} h _ {k} + h _ {k} f _ {i}}{\sqrt {6}} \\ = \operatorname {S o f t m a x} _ {j, k} \frac {- n ^ {2} \left[ (\phi (i) - j) ^ {2} - (\phi (j) - (k - n)) ^ {2} \right]}{\sqrt {6}} \\ \end{array} +$$ + +In particular, the maximum of $a_{2n+1,j,k}$ is for $j$ and $k$ such that $j = \phi(2n+1) = x$ , and $k = n + \phi(j) = n + \phi(x) = n + g(x)$ , and other values of $a_{2n+1,j,k}$ are by an $e^{\Omega(n^2)}$ -factor smaller. Hence, with precision $\pm e^{-\Omega(n^2)}$ , we obtain $a_{2n+1} = v_j \odot v_k$ for $j = x$ and $k = n + g(x)$ . Observe that $\phi(k) = \phi(n + g(x)) = h(g(x))$ , so it is enough for $v_j \odot v_k$ to have a coordinate equal to $\phi(k)$ . We can achieve this by setting matrices $V_1, V_2$ in (15) such that the first coordinates of $v_j$ and $v_k$ are 1 and $\phi(k)$ , respectively. + +Binary relation composition In this task, the length of input is a square number $n = m^2$ , and tokens are indexed by pairs $(i,j) \in [m]^2$ . The token, indexed by $(i,j)$ , receives on input two bits $A_{ij}$ , $B_{ij}$ from two Boolean matrices $A, B \in \{0,1\}^{m \times m}$ . As an output, the $(i,j)$ -th token has to produce a real number $y_{ij}$ such that + +$$ +(B \circ A) _ {i j} = \bigvee_ {k = 1} ^ {m} (A _ {i k} \wedge B _ {k j}) = \operatorname {s i g n} (y _ {i j}). +$$ + +We employ the following positional encoding: + +$$ +x _ {i j} = \left( \begin{array}{c} A _ {i j} \\ B _ {i j} \\ i \\ i ^ {2} \\ j \\ j ^ {2} \\ 1 \end{array} \right), \qquad i, j = 1, \ldots , m. +$$ + +We then take matrices $W^{f}, W^{g}, W^{h}$ in (12-15) so that: + +$$ +f _ {i j} = n ^ {2} \left( \begin{array}{c} i ^ {2} \\ 2 i \\ - 1 \\ \hline j ^ {2} \\ 2 j \\ - 1 \\ \hline 0 \\ 0 \\ 0 \\ \hline 0 \\ 0 \\ \hline 1 / m ^ {2} \\ 1 / m ^ {3} \\ 1 / m ^ {4} \\ 1 / m ^ {5} \end{array} \right), \qquad g _ {c d} = n ^ {2} \left( \begin{array}{c} - 1 \\ c \\ c ^ {2} \\ \hline 0 \\ 0 \\ 0 \\ \hline d ^ {2} \\ 2 d \\ - 1 \\ \hline 1 \\ A _ {c d} \\ \hline c \\ d \\ 0 \\ 0 \end{array} \right), \qquad h _ {k \ell} = n ^ {2} \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ \hline - 1 \\ \ell \\ \ell^ {2} \\ \hline - 1 \\ k \\ k ^ {2} \\ \hline B _ {k \ell} \\ 1 \\ \hline 0 \\ 0 \\ k \\ \ell \end{array} \right) +$$ + +(horizontal lines are added for readability). We obtain: + +$$ +f _ {i j} g _ {c d} + g _ {c d} h _ {k \ell} + h _ {k \ell} f _ {i j} = n ^ {4} \left[ - (i - c) ^ {2} - (d - k) ^ {2} - (\ell - j) ^ {2} + A _ {c d} + B _ {k \ell} + \frac {c m ^ {3} + d m ^ {2} + k m + \ell}{m ^ {5}} \right]. \tag {23} +$$ + +The expression in brackets has the "integral part", and also has the term $\frac{cm^3 + dm^2 + km + \ell}{m^5}$ which is $O(1 / m)$ and is different for different quadruples $(c,d,k,\ell)$ . Hence, there is a unique quadruple $(c,d,k,\ell)$ , establishing the maximum of the above expression, and it also maximizes the "integral part" $\left[-(i - c)^2 -(d - k)^2 -(l - j)^2 +A_{cd} + B_{k\ell}\right]$ . Because of the factor $n^4$ , the value for the other quadruples will be smaller by at least $\Omega (n^{4} / m^{5}) = \Omega (n^{3 / 2})$ . + +The quantity $\left[-(i - c)^2 -(d - k)^2 -(\ell -j)^2 +A_{cd} + B_{k\ell}\right]$ is at most 2, being equal to 2 if and only if $c = i,\bar{\ell} = j,d = k$ and $A_{id} = B_{dj} = 1$ . In other words, the maximum of the integral part is 2 if and only if $(B\circ A)_{ij} = 1$ . + +As a result, the $(i,j)$ -th token will get the value of the Hadamard product $v_{cd} \odot v_{k\ell}$ with precision $e^{-\Omega (n^{3 / 2})}$ for some quadruple $c,d,k,\ell \in [m]$ satisfying: + +$$ +(c = i, \ell = j, d = k \text {a n d} A _ {c d} = B _ {k \ell} = 1) \iff (B \circ A) _ {i j} = 1 \tag {24} +$$ + +It remains to define matrices $V^1, V^2$ so that this Hadamard product in some coordinates has $c, d, k, \ell, A_{cd}, B_{k\ell}$ , and has 0 where $x_{ij}$ has $i$ and $j$ . Then $x_{ij} + (v_j \odot v_k)$ will have all quantities involved in the equalities of the left-hand side of (24), and checking them can be done with a constant-size MLP. + +Match3 On input, we get an array $p_1 \ldots p_n \in [m - 1]^n$ . We first describe how to check, for a fixed $\Sigma$ and for every $i = 1, \dots, n$ , if there exist $j, k \in [n]$ such that $p_i + p_j + p_k = \Sigma$ using one Strassen attention head. We get a solution for the Match3[n, m] task with 2 attention heads by applying this construction to $\Sigma = m$ and $\Sigma = 2m$ . + +The embedding dimension will be 8. Define $q_{i} = p_{i} - \Sigma /3$ . We use the following positional encoding: + +$$ +x _ {i} = \left( \begin{array}{c} i \\ q _ {i} \\ q _ {i} ^ {2} \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \right), \qquad i = 1, \ldots , n. +$$ + +We define matrices $W^{f}, W^{g}, W^{h}$ in (12-15) so that: + +$$ +f _ {i} = n ^ {2} \left( \begin{array}{c} - q _ {i} \\ - q _ {i} \\ 0 \\ - q _ {i} ^ {2} \\ 0 \\ 0 \\ 0 \\ 0 \end{array} \right), g _ {j} = n ^ {2} \left( \begin{array}{c} 2 q _ {j} \\ 0 \\ - q _ {j} \\ 1 \\ - q _ {j} ^ {2} \\ 1 \\ \frac {i}{n ^ {2}} \\ 1 \end{array} \right), h _ {k} = n ^ {2} \left( \begin{array}{c} 0 \\ 2 q _ {k} \\ 2 q _ {k} \\ 0 \\ 1 \\ - q _ {k} ^ {2} \\ 1 \\ \frac {k}{n ^ {3}} \end{array} \right). +$$ + +As a result, we get: + +$$ +\begin{array}{l} a _ {i j k} = \operatorname {S o f t m a x} _ {j, k} \frac {f _ {i} g _ {j} + g _ {j} h _ {k} + h _ {j} f _ {i}}{\sqrt {8}} \\ = \operatorname {S o f t m a x} _ {j, k} \frac {n ^ {4} \left(- \left(q _ {i} + q _ {j} + q _ {k}\right) ^ {2} + \frac {i n + j}{n ^ {3}}\right)}{\sqrt {8}} \\ = \operatorname {S o f t m a x} _ {j, k} \frac {n ^ {4} \left(- \left(p _ {i} + p _ {j} + p _ {k} - \Sigma\right) ^ {2} + \frac {i n + j}{n ^ {3}}\right)}{\sqrt {8}} \\ \end{array} +$$ + +For a given $i$ , the maximum of $a_{ijk}$ is attained on a single triple $(i,j,k)$ with the minimal value of $|p_i + p_j + p_k - \Sigma|$ across the array, and it will be by an $e^{\Omega(n)}$ -factor larger than any other value of $a_{i,j,k}$ . We added the fraction $\frac{in + j}{n^3}$ to ensure uniqueness of the maximum; the added term is different for different pairs $(i,j)$ while not exceeding $O(1/n)$ . + +Since all numbers under consideration are polynomial in $n$ , the output $a_{i}$ will be equal to $v_{j} \odot v_{k}$ for the maximal pair $(j, k)$ up to $\exp\{-\Omega(n)\}$ -precision. In the output MLP, we have to check if $p_{i} + p_{j} + p_{k} = \Sigma$ for this pair $(j, k)$ . It is enough to define $V_{1}, V_{2}$ so that the 5th and the 6th coordinate of $v_{j}$ and $v_{k}$ are $1, p_{j}$ and $p_{k}$ , respectively. As a result, the 2nd, the 5h, and the 6th coordinates of $x_{i} + a_{i}$ will be $-p_{i}, p_{k}$ , and $p_{j}$ , respectively, allowing us to find out if $p_{i} + p_{j} + p_{k} = \Sigma$ with a constant-size output MLP. + +# B.5 Proof of Theorem 5.1 + +Upper bound The upper bound is very similar to our construction for the binary relation composition task in Theorem 4.2. Namely, first we replace $-(d - k)^{2}$ by $-( \mathsf{col}(d) - \mathsf{col}(k)^{2}$ in (23). After this modification, the maximum of (23) will be attained on a single quadruple $(c,d,k,\ell)$ , and for this quadruple we will have $c = i, \ell = j$ , $\mathsf{col}(d) = \mathsf{col}(k)$ and $A_{id} = B_{kj} = 1$ if and only if a quadruple, satisfying these equalities, exists. However, due to the definition of our task, we have to add a smaller term, enforcing that among quadruples, satisfying this property, those with $d \neq k$ have a larger value of (23). This can be achieved by adding a term of the form $(d - k)^{2} / m^{3}$ , which is always $O(1 / m)$ so that the largest possible difference in this term is smaller than the smallest possible difference of the "integral part" in (23). + +We also have to add a term which ensures that the maxima is attained at the unique quadruple. The largest possible difference in this term should be smaller than the smallest possible difference in the previous terms, which is $\Omega(1/m^3)$ . We can again take the expression $cm^3 + dm^2 + km + \ell$ but divided by a larger denominator, for instance: + +$$ +\frac {c m ^ {3} + d m ^ {2} + k m + \ell}{m ^ {8}} +$$ + +(now the maximal possible difference in this term is $O(1 / m^4)$ ). Finally, it remains to multiply all the coefficients by a sufficiently large factor to make the minimal possible difference between the maximum and the other values polynomial in $n$ . + +Lower bound We employ the same technique as in Theorem 3.2. However, we cannot rely on it directly as now we have to deal with the triangular attention. + +Assume for contradiction that there exists a one-layer Transformer $T$ such that (a) it solves the quotient binary relation composition task; (b) it has $n^{o(1)}$ standard-attention heads, $n^{o(1)}$ triangular-attention + +heads, $n^{o(1)}$ embedding dimension, and $n^{o(1)}$ -size output MLP. Without loss of generality, let $n$ be even and fix $s$ be such that $n = 2s + 2$ . Given two binary words $p = p_1 \ldots p_s, q = q_1 \ldots q_s \in \{0,1\}^s$ , we define an instance $(A(p), B(q), \mathsf{col})$ of the quotient binary relation composition task by setting: + +$$ +A _ {1, 2 + j} = p _ {j}, \quad B _ {2 + s + j, 2} = q _ {j}, +$$ + +for $j \in [s]$ , and letting all the other entries of the matrices $A, B$ to be 0. The coloring function is defined by $\mathsf{col}(1) = \mathsf{col}(2) = 1$ and + +$$ +\operatorname {c o l} (2 + j) = \operatorname {c o l} (2 + s + j) = 2 + j +$$ + +for $j \in [s]$ . An example of this construction for $s = 4$ is given in Figure 3. + +![](images/e558ef138beae7f4904f048748c8c541a8382c9330f42f88e70fc25604cabb93.jpg) +Figure 3: An example of the construction. Nodes apart from 1 and 2 are split into 2 equal groups $-3, 4, 5, 6$ and 7, 8, 9, 10. If $A_{ij} = 1$ (resp., $B_{ij} = 1$ ), we draw an $A$ -labeled (resp., a $B$ -labeled) edge from $i$ to $j$ . The $A$ -edges can only go from 1 to 3, 4, 5, 6, and the word $p$ determines, which of these edges are present. Likewise, the $B$ -edges only go from 7, 8, 9, 10 to 2, according to whether $q$ has 1 or 0 in the corresponding position. Nodes 3 and 7, 4 and 8, 5 and 9, 6 and 10 have the same color but different pairs have a different color. Therefore, the only way we can have $(B \circ A / c)_{12} = 1$ is when 1 has an $A$ -edge to some node on the left, and the node of the same color from the right has a $B$ -edge to 2. In the example from the figure, this is true for 5 and 9 (which happens because $p$ and $q$ both have 1 in the third position). + +We claim that for the instance $(A(p),B(q),\mathsf{col})$ , the value of the quotient binary relation composition at the pair $(1,2)$ is defined by the equation: + +$$ +(B (q) \circ A (p) / \mathbf {c o l}) _ {1 2} = \operatorname {D i s j} (p, q), \tag {25} +$$ + +where $\mathrm{Disj}(p,q)$ is 1 if and only if there is a position where both $p$ and $q$ have 1. Indeed, if $\mathrm{Disj}(p,q) = 1$ , taking $j \in [s]$ such that $p_j = q_j = 1$ and then setting $k_1 = 2 + j, k_2 = 2 + s + j$ , we obtain that $A_{1k_1} = p_j = q_j = B_{k_22} = 1$ and $\mathsf{col}(k_1) = \mathsf{col}(k_2) = 2 + j$ , which implies $(B(q) \circ A(p) / \mathsf{col})_{12} = 1$ . On the other hand, if $(B(q) \circ A(p) / \mathsf{col})_{12} = 1$ , then for some $k_1 \neq k_2$ we have $A_{1k_1} = B_{k_22} = 1$ and $\mathsf{col}(k_1) = \mathsf{col}(k_2)$ . Since $A_{1k_1} = 1, B_{k_22} = 1$ , we have $k_1 = 2 + j_1$ and $k_2 = 2 + s + j_2$ for some $j_1, j_2 \in [s]$ . Since $2 + j_1 = \mathsf{col}(k_1) = \mathsf{col}(k_2) = 2 + j_2$ , we derive that $j_1 = j_2 = j$ . Hence, we get $p_j = A_{1k_1} = B_{k_22} = q_j$ and $\mathrm{Disj}(p,q) = 1$ , as required. + +We now put the instance $(A(p), B(q), \mathsf{col})$ to our Transformer $T$ and look at its output int the token indexed by (1, 2). By (25), we have: + +$$ +\operatorname {s i g n} \left(y _ {1 2}\right) = \operatorname {D i s j} (p, q). +$$ + +We now look at how the output $y_{12}$ is computed in our Transformer on such input. The key observation is that, for any $h = 1, \dots, H$ , the output of the $h$ -th attention head in position (1, 2) can be written similarly to (20) as a fraction, where some terms depend solely on $p$ and others solely on $q$ : + +$$ +a _ {1 2} ^ {(h)} = \frac {\alpha^ {(h)} (p) + \beta^ {(h)} (q) + \gamma^ {(h)}}{\lambda^ {(h)} (p) + \mu^ {(h)} (q) + \nu^ {(h)}}, \quad \alpha^ {(h)} (p), \beta^ {(h)} (q), \gamma^ {(h)} \in \mathbb {R} ^ {d}, \quad \lambda^ {(h)} (p), \mu^ {(h)} (q), \nu^ {(h)} \in \mathbb {R} +$$ + +regardless of whether the $h$ -th attention head uses the standard or the triangular attention. Indeed, for the case of the standard attention, this is by the same computation as in the proof of Theorem 3.2. Now, for the case of the triangular attention, the same computation goes through but for pairs of tokens of the form $(x_{1\ell}, x_{\ell 2})$ instead of individual tokens. It remains to notice that for $\ell = 3, \ldots, s + 2$ , the value of this pair is determined by $p$ , and for $\ell = s + 3, \ldots, 2s + 2$ , the value of this pair is determined by $q$ (and for $s = 1, 2$ , the value of this pair is fixed). + +The rest of the proof is identical to the corresponding part in the proof of Theorem 3.2. Similarly to (21), we can now write: + +$$ +\operatorname {D i s j} (p, q) = \operatorname {s i g n} \left(\mathcal {N} \left(x _ {1, 2} + W _ {O} \left( \begin{array}{c} \frac {\alpha^ {(1)} (p) + \beta^ {(1)} (q) + \gamma^ {(1)}}{\lambda^ {(1)} (p) + \mu^ {(1)} (q) + \nu^ {(1)}} \\ \vdots \\ \frac {\alpha^ {(H)} (p) + \beta^ {(H)} (q) + \gamma^ {(H)}}{\lambda^ {(H)} (p) + \mu^ {(H)} (q) + \nu^ {(H)}} \end{array} \right)\right)\right). \tag {26} +$$ + +Then we define a hypothesis class $H$ by considering parts of (26) that depend on $p$ as inputs and parts that depend on $q$ as parameters. On the one hand, since $d, H$ and the size of $\mathcal{N}$ are assumed to be $n^{o(1)}$ , the VC dimension of this class is $n^{o(1)}$ by Theorem 2.3 in [9]. On the other hand, its VC dimension is lower bounded by the VC dimension of the set of columns of the matrix $\mathrm{Disj}(p, q)$ , which is at least $s = n/2 - 1$ as established in the proof of Proposition B.2. + +# C Experimental Setup + +# C.1 Datasets + +We create dedicated datasets to evaluate our models across all four tasks. Each task consists of $5 \times 10^{4}$ examples. Below, we detail the data generation process for each task, with explanations of key components and structures. + +# C.1.1 Function Composition + +The task of function composition involves determining whether a specific condition holds for a given sequence derived from a function $f$ . Each example in the dataset is represented as a tuple $(X, y)$ , where: + +- $X = (\perp, f(0), f(1), \dots, f(n - 1))$ is an input sequence of length $n + 1$ . The first token, $\perp$ , is a query token indicating the position where the output is required. +- $n$ is sampled uniformly from the range $\left\lbrack {{N}_{\min },{N}_{\max }}\right\rbrack$ ,with ${N}_{\min } = {25}$ and ${N}_{\max } = {30}$ . +- $y \in \{ 0,1\}$ is a binary label that indicates whether the condition $f\left( {f\left( 0\right) }\right) = 0$ is satisfied. + +The dataset generation process ensures that the sequences are random but incorporates specific constraints to maintain diversity and balance (approximately $50\%$ positive labels). Algorithm 1 outlines the data generation procedure. + +Algorithm 1 Dataset Generation for Function Composition +```txt +Input: $N_{\mathrm{min}},N_{\mathrm{max}}$ +Output: Dataset +for $_{= 1}$ to $5\times 10^{4}$ do Sample $n\sim$ Uniform $(N_{\mathrm{min}},N_{\mathrm{max}})$ Generate random sequence $X = x_0x_1\dots x_{n - 1}$ with $x_{i}\sim$ Uniform( $\{0,1,2,\dots ,n - 1\}$ Sample $y\sim$ Uniform $\{\{0,1\})$ if $y = 1$ and $x_{x_0}\neq 0$ then Set $x_{x_0}\gets 0$ {Ensure $f(f(0)) = 0$ } else if $y = 0$ and $x_{x_0} = 0$ then Set $x_0\sim$ Uniform $(i\in \{1,2,\ldots ,n - 1\} |x_i\neq 0)$ {Ensure $f(f(0))\neq 0$ end if Prepend query token: $X\gets (\bot ,X)$ Add $(X,y)$ to the dataset +end for +``` + +Explanation of Examples (Table 2) In the sequence $(\perp, 3, 0, 5, 1, 0, \ldots)$ , the label $y = 0$ implies that the condition $f(f(0)) = 0$ does not hold. Specifically: $f(0) = 3$ , and $f(3) = 1 \neq 0$ . Thus, the condition $f(f(0)) = 0$ is false. In the sequence $(\perp, 4, 1, 3, 5, 0, \ldots)$ , the label $y = 1$ indicates that the condition $f(f(0)) = 0$ is satisfied. Specifically: $f(0) = 4$ , and $f(4) = 0$ . Thus, the condition $f(f(0)) = 0$ is true. + +![](images/501e33afa57e6f939fa219ef6177eab14eb8bcf795d2a624959929bba3c5e0ee.jpg) + +Table 2: Example sequences for the Function Composition task. The label $y$ depends on whether $f\left( {f\left( 0\right) }\right) = 0$ . + +# C.1.2 Binary Relation Composition + +The task of binary relation composition involves determining whether a transitive relation exists between two elements in a binary relation matrix. Given two relations $A, B \in \{0,1\}^{m \times m}$ , where $m = \sqrt{n}$ , and a pair of elements $i, j \in \{0,1,\dots,m-1\}$ , the goal is to check whether there exists another element $k \in \{0,1,\dots,m-1\}$ such that both relations $A_{ik}$ and $B_{kj}$ hold. For simplicity, we define $R \in \{0,1\}^{m \times m}$ and set $A = R$ and $B = R$ . Each example in the dataset is represented as a tuple $(X,Y)$ , where: + +- $X = \text{flatten}(R)$ is an input sequence of length $n = m^2$ of a flatten boolean matrix $R \in \{0,1\}^{m \times m}$ representing the binary relation. Each element $R_{ij}$ is independently set to 1 with a probability $P$ between 0 to 1, where $P$ is a parameter independent on the input length. +- $m$ is sampled uniformly from the range $[N_{\min}, N_{\max}]$ , with $N_{\min} = 6$ and $N_{\max} = 8$ . +- $Y \in \{0,1\}^n$ is a list of binary labels that at position $k = i \times m + j$ indicates whether there is a composition of relations between element $i$ and $j$ is satisfied. + +The dataset generation process ensures randomness in $X$ while adhering to the constraints for $Y$ . For $m$ in the range $[N_{\min}, N_{\max}]$ , the probability $P$ is set to 32.5% to achieve a balanced proportion of positive labels. Algorithm 2 provides the detailed data generation procedure. + +Algorithm 2 Dataset Generation for Binary Relation Composition +```txt +Input: $N_{\mathrm{min}},N_{\mathrm{max}},P$ +Output: Dataset +for $\mathbf{\Phi} = 1$ to $5\times 10^{4}$ do Sample $m\sim \mathrm{Uniform}(N_{\mathrm{min}},N_{\mathrm{max}})$ Generate boolean matrix $R\in \{0,1\}^{m\times m}$ , where each element $R_{ij} = 1$ with probability $P$ Initialize $Y = [\quad ]$ {An empty list for labels} +for $i = 0$ to $m - 1$ do for $j = 0$ to $m - 1$ do if there exists $k$ such that $R_{ik} = 1$ and $R_{kj} = 1$ then Append 1 to $Y$ else Append 0 to $Y$ end if end for end for Add (flatten $(R),Y)$ to the dataset +end for +``` + +Explanation of Examples (Table 3) Consider the binary relation matrix $R$ shown in Table 3: For $i = 0, j = 4$ , there exists $k = 2$ such that $R_{02} = 1$ and $R_{24} = 1$ . Thus, $Y_{0 \times 6 + 4} = 1$ . For $i = 5, j = 0, R_{50} = 1$ is the unique candidate, but $R_{00} = 0$ . Thus, $Y_{5 \times 6 + 0} = 0$ . + +![](images/21538eff6985a3141d5de51939294d27515d41e3047a48842a2f4e8c21a77a16.jpg) +Table 3: Examples of binary relation matrix and corresponding composition. Here, $X = \text{ flatten}(R)$ and $Y = \text{ flatten}(R \circ R)$ . + +# C.1.3 Match3 + +The Match3 task involves determining a target label sequence $Y$ for a given input sequence $X$ . Each example consists of an input sequence $X$ of size $n$ and a label sequence $Y$ of the same size, where for every index $i \in \{0,1,\dots,n-1\}$ , $Y_i$ represents the target value for $X_i$ . Each example is represented as a tuple $(X,Y)$ , where: + +- $X = (x_0, x_1, \ldots, x_{n-1})$ is a pseudo-random sequence of integers sampled from the range $[0, M-1]$ , where $M = 37$ . +- $Y = (y_0, y_1, \ldots, y_{n-1})$ is a binary sequence, with $y_i \in \{0, 1\}$ , where each value $y_i$ indicates if the token $x_i$ satisfied the Match3 condition in $X$ . +- $n$ is sampled uniformly from the range $[N_{\min}, N_{\max}]$ , where $N_{\min} = 30$ and $N_{\max} = 35$ . + +The dataset generation process aims to balance the distribution of ones in $Y$ across four predefined bins corresponding to percentage ranges: [0, 25)% [25, 50)% [50, 75)% and [75, 100%]. Algorithm 3 outlines the data generation procedure. + +Algorithm 3 Dataset Generation Algorithm for Match3 +```txt +Input: $N_{\mathrm{min}},N_{\mathrm{max}},D$ {Input $D$ is the dataset size} +Output: Dataset +Initialize four empty bins {Each bin corresponding to percentage ranges of ones in sequences: $[0,25)\% ,[25,50)\% ,[50,75)\% ,$ and $[75,100]\%$ } + $N_{b}\gets (D / 10) / 4$ +for $i = 1$ to $5\times 10^{3}$ do Randomly select skewness $\sim$ Uniform(1,40) {An initial percentage distribution of ones in the sequence} Sample $n\sim$ Uniform $(N_{\mathrm{min}},N_{\mathrm{max}})$ Generate a pseudo-random sequence $X = (x_0,x_1,\dots ,x_{n - 1})$ , where $x_{i}\sim$ Uniform $(\{0,1,\ldots ,M - 1\})$ ) ensuring the percentage of tokens that satisfied Match3 condition is at least skewness Compute $Y = (y_{0},y_{1},\ldots ,y_{n - 1})$ based on Match3 condition Calculate the percentage of ones in $Y$ Add $(X,Y)$ to the corresponding bin if size Bin) $< N_{b}$ +end for +for each bin in bins do while sizeBIN) $\neq$ D/4 do Randomly sample an example $(X,Y)$ from bin Apply the same permutation to $X$ and $Y$ Add the permuted pair $(X^{*},Y^{*})$ to bin end while +end for +Add all examples from the bins to the dataset +``` + +Explanation of Examples (Table 4) Consider the example sequence $X$ and its corresponding label $Y$ in Table 4: + +- The input sequence $X = (6, 9, 9, 9, 7, 10, 9, 34, 9, 9, 30, \ldots)$ contains pseudo-random integers between 0 and $M - 1$ ( $M = 37$ ). +- The label sequence $Y = (1,0,0,0,1,1,0,1,0,0,1,\ldots)$ has a skewed distribution of ones. +- For instance, $y_{5} = 1$ indicates that the element $x_{5} = 10$ satisfies the property, because $x_{7} = 34$ and $x_{10} = 30$ holds, $x_{5} + x_{7} + x_{10} = 74 = 2 \times M$ . + +
Sequence X +Label Y012345678910...
69997109349930...
10001101001...
+ +Table 4: Example sequence for Match3 task with $M = {37}$ . + +# C.1.4 Quotient Binary Relation Composition + +The task of quotient binary relation composition involves determining whether a transitive relation exists between two elements in a binary relation matrix while incorporating an additional constraint based on a coloring function $\mathsf{col}:[0,m - 1]\to [0,m - 1]$ . For simplicity, we define $R\in \{0,1\}^{m\times m}$ and set $A = R$ and $B = A^T$ . Each example in the dataset is represented as a tuple $(X,Y)$ , where: + +- $X = \text{ flatten}(R)$ is an input sequence of length $n = m^2$ obtained by flattening the boolean matrix $R$ , where each element $R_{ij}$ is independently set to 1 with probability $P$ , which is independent of the input length. +- col is a function that assigns a unique color to each element in $[0, m - 1]$ , with colors sampled uniformly from the range $[0, m - 1]$ . +- $m$ is sampled uniformly from the range $[N_{\min}, N_{\max}]$ , with $N_{\min} = 6$ and $N_{\max} = 8$ . +- $Y \in \{-100, 0, 1\}^n$ is a list of binary labels, where the position $k = i \times m + j$ indicates whether the quotient binary relation composition between elements $i$ and $j$ is satisfied if and only if $i \neq j$ , otherwise $Y_k = -100$ . + +The dataset generation process ensures randomness in $R$ while adhering to the constraints for $Y$ . For $m$ in the range $[N_{\min}, N_{\max}]$ , the probability $P$ is set to $43.3\%$ to achieve a balanced proportion of positive labels. Algorithm 4 provides the detailed data generation procedure. + +Algorithm 4 Dataset Generation for Quotient Binary Relation Composition +Input: $N_{\mathrm{min}},N_{\mathrm{max}},P$ +Output: Dataset +for $_{= 1}$ to $5\times 10^{4}$ do Sample $m\sim \mathrm{Uniform}(N_{\mathrm{min}},N_{\mathrm{max}})$ Generate boolean matrix $R\in \{0,1\}^{m\times m}$ where each element $R_{ij} = 1$ with probability $P$ Generate a coloring function $\mathsf{col}:[0,m - 1]\to [0,m - 1]$ by assigning colors uniformly Initialize $Y = [\quad ]$ {An empty list for labels} for each pair $(i,j)$ in $[0,m - 1]\times [0,m - 1]$ do Append 1 to $Y$ if $i\neq j$ and there exist $k_{1},k_{2}$ such that: $R_{ik_1} = 1,R_{jk_2} = 1,\mathsf{col}(k_1) = \mathsf{col}(k_2),$ and $k_{1}\neq k_{2}$ Otherwise, if $i\neq j$ , append O to $Y$ , else append-100 to $Y$ +end for Add (flatten $(R),Y)$ to the dataset +end for + +Explanation of Examples (Table 5) Consider the binary relation matrix $R$ shown in Table 5. For $i = 2$ , $j = 4$ , there exist elements $k_{1} = 4$ and $k_{2} = 5$ such that $R_{24} = 1$ , $R_{45} = 1$ , and $\mathsf{col}(4) = 2 = \mathsf{col}(5)$ , with $k_{1} \neq k_{2}$ . Thus, $Y_{2\times 7 + 4} = 1$ . + +![](images/44e6309e7060ba29a3055525092beac6a2428a0779b0f4dccf2e19cc204558fd.jpg) +Table 5: Examples of binary relation matrix and corresponding quotient composition. + +# C.2 Experimental Protocol + +Dataset Splitting. The dataset is split into a training and validation set. We randomly select $90\%$ of the data for training, and the remaining $10\%$ is used for validation. The validation set is used to monitor the model's performance as well as test set. The validation data is kept within the same distribution as the training set to ensure that the evaluation is conducted in an in-distribution manner. + +Evaluation. The evaluation metric for all tasks is accuracy. To compute the accuracy during training or evaluation, we first calculate the accuracy for each batch individually. This is done by dividing the number of correctly predicted labels by the total number of labels in that batch. These per-batch accuracies are then aggregated across all batches within an epoch. The overall accuracy for the epoch is obtained by taking the mean of the per-batch accuracies. Note that this is not equivalent to just taking the number of correctly predicted labels divided by the number of all labels in the whole dataset (due to variable lengths of examples). Both approaches converge to the same value as the number of examples grows. + +Training Setting. We fix random seeds across all experiments. Each task and model configuration is evaluated using 8 different random seeds, and the reported results include the median across these runs. This approach mitigates the effects of random weight initialization and stochastic data sampling during training. Additionally, training parameters such as learning rate, batch size, and training duration equally specified for each task across model, as summarized in Table 6. + +Implementation Details We implement all models using PyTorch framework [1] with Opt-Einsum library [5]. We employ AdamW optimizer for all training tasks without a learning rate scheduler, ensuring a consistent optimization strategy across experiments. We use the Binary Cross-Entropy loss as the objective function. In order to handle padding, we used attention masks, preventing the model from attending to padded positions, and thus ensuring no changes in the model's outputs. For the target sequence, we assign a value of $-100$ to positions corresponding to padded tokens. This ensures that during loss and accuracy calculations, only tokens with values other than $-100$ are considered. We achieve this by applying a mask, where predictions and targets are filtered as pred = pred[mask] and target = target[mask], respectively. + +Hardware and Resource Utilization We conduct all experiments on high-performance NVIDIA GPUs. Specifically, we execute on NVIDIA A100 GPUs with 80GB of memory tasks requiring extensive computational resources, such as Match3 and Quotient Binary Relation Composition. For tasks with lower computational demands, such as Function Composition and Binary Relation Composition, We use NVIDIA A40 GPUs with 48GB of memory. + +# D Further Experiments + +We evaluate the computational performance of attention mechanisms exclusively on an NVIDIA RTX A6000 GPU. This analysis focuses on three metrics: (a) forward pass time, (b) GPU utilization, and (c) memory utilization. These metrics provide insights into the computational efficiency and hardware constraints associated with different configurations of hidden dimension and input length. Below, we detail the evaluation methodology and the observed limitations in the Figure 4. + +
TaskModeldhBρTp
Function CompositionStandard16125001·10-31000 epochs0.3
Third-Order16125001·10-31000 epochs0.3
Strassen16125001·10-31000 epochs0.3
Binary Relation CompositionStandard16125001·10-3200 epochs0.3
Triangular16125001·10-3200 epochs0.3
Third-Order16125001·10-3200 epochs0.3
Strassen16125001·10-3200 epochs0.3
Match3Standard128225001·10-3500 epochs0.4
Third-Order128225001·10-3500 epochs0.4
Strassen128225001·10-3500 epochs0.4
Quotient Binary Relation CompositionStandard16120001·10-33000 epochs0.3
Triangular16120001·10-33000 epochs0.3
Third-Order16120001·10-33000 epochs0.3
Strassen16120001·10-33000 epochs0.3
COGSStandard6841005·10-4200 epochs0.1
Strassen6441005·10-4200 epochs0.1
+ +Table 6: Training parameter settings for tasks and models. For all models and tasks, we run experiments on 8 different seeds with only one attention layer (except for the COGS dataset where we use 3 layers). Here $d$ is embedding dimension, $h$ is the number of heads, $B$ is the batch size, $\rho$ is the learning rate, $T$ is training duration and $p$ is the dropout outside attention mechanism. We did not use batch normalization for any task. + +![](images/ae23b1369d9af729300fbffb7ebe2f1d86ab60fad85d2c1ead53916432e48a02.jpg) +Figure 4: Analysis of computational performance for each attention mechanisms presented in this work: Standard, Strassen, Third-Order and Triangular attention. Metrics include (a) forward pass time in seconds, (b) GPU utilization in percentage, and (c) memory reserved in GB on an NVIDIA RTX A6000. Experiments are conducted across varying input lengths (4 to 16,384) and five hidden dimensions (64, 128, 256, 768, 1024). Each result represents the average of the median over 8 runs on 100 random sequences (100 tensors with batch size 1) passed through the respective attention mechanisms. + +Time We measure the forward pass time as the delta time before and after passing a $1 \times n \times d$ random tensor through the attention mechanism, where $n$ is input length and $d$ hidden dimension. We implement this using time.time() function from Python. As we expect, the results indicate that forward pass time increases significantly for higher hidden dimensions and input lengths. + +GPU Usage We monitor GPU utilization using the pynvm1 library to query the current CUDA device. Specific configurations of attention mechanisms, such as Strassen attention with large hidden dimensions (e.g., 512 or higher) and long input lengths (e.g., 1600 or higher), result in $100\%$ GPU usage. + +Memory Reserved We record the memory reserved during tensor processing using CUDA memory management functions from PyTorch: torch.cuda/memory_reserved. + +Our computational performance results show that (Figure 4): + +- The computational constraints of Strassen attention become evident for hidden dimensions of 512 or higher and input lengths exceeding 1600. +- Those of Third-Order attention appear with even lower hidden dimensions (64 or 128) and input lengths exceeding 1600. +- Standard and Triangular ( $n =$ Nodes $\times$ Nodes) GPU memory usage does not show a bottleneck even for configurations with hidden dimensions of 2048 and input lengths of 16,384. +- Triangular ( $n =$ Nodes) Limitations appear for hidden dimensions of 1024 or higher and input lengths as low as 196, with increasing severity for longer input sequences. + +# NeurIPS Paper Checklist + +# 1. 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While standard learning supports a broad range of model classes, the study of strategic classification has, so far, been dedicated mostly to linear classifiers. This work aims to expand the horizon by exploring how strategic behavior manifests under non-linear classifiers and what this implies for learning. We take a bottom-up approach showing how non-linearity affects decision boundary points, classifier expressivity, and model class complexity. Our results show how, unlike the linear case, strategic behavior may either increase or decrease effective class complexity, and that the complexity decrease may be arbitrarily large. Another key finding is that universal approximators (e.g., neural nets) are no longer universal once the environment is strategic. We demonstrate empirically how this can create performance gaps even on an unrestricted model class. + +# 1 Introduction + +Strategic classification considers learning in a setting where users modify their features in response to a deployed classifier to obtain positive predictions [3, 4, 16]. This setting aims to capture a natural tension that can arise between a system aiming to maximize accuracy and its users who seek favorable outcomes. Such a tension is well-motivated across diverse applications, including loan approval, university admission, job hiring, welfare and education programs, and medical services. The field has drawn much attention since its introduction, leading to advances along multiple learning fronts such as generalization [6, 33, 36], optimization [25], loss functions [24, 26], and extensions [11, 14, 18, 30]. + +However, despite over a decade of effort, research in strategic classification remains predominantly focused on linear classification. Due to the inherent challenges of strategic learning, this focus has been reasonable, given that linearity has the benefit of inducing a simple form of strategic behavior that permits tractable analysis. Yet, since many realistic applications—including those mentioned above—make regular use of non-linear models, we believe that it is important to establish a better understanding of how non-linearity shapes strategic behavior, and how behavior in turn affects learning. Our paper therefore aims to extend the study of strategic classification to the non-linear regime. + +Our first observation is that non-linear classifiers induce qualitatively different behavior than linear ones in the strategic setting. Consider first the structure induced by standard linear classifiers. Generally, users in strategic classification are assumed to best-respond to the classifier by modifying their features in a way that maximizes utility (from prediction) minus costs (from modification). For a linear classifier $h$ coupled with the common choice of an $\ell_2$ cost, modifications $x \mapsto x^h$ manifest as a projection of $x$ onto the linear decision boundary of $h$ as long as it incurs less than a unit cost. This simple form has several implications. First, note that all points 'move' in the same direction, making strategic behavior, in effect, one-dimensional. Second, the region of points that move forms a band of unit distance on the negative side of $h$ . This means that the set of points classified as positive + +forms a halfspace, which implies that the 'effective classifier', given by $h_{\Delta}(x) \coloneqq h(x^h)$ , is also linear. Third, and as a result, the induced effective model class remains linear. + +Together, the above suggest that for linear classifiers coupled with $\ell_2$ costs, the transition from the standard, non-strategic setting to a strategic one requires anticipating which points move, but otherwise remains similar. One known implication is that the strategic and non-strategic VC dimension of linear classifiers are the same [33]. Furthermore, the Bayes-optimal strategic classifier can be derived by taking the optimal non-strategic classifier and simply increasing its bias term by one [29]. Note that this means that any linearly separable problem remains separable, and that strategic classifiers can, in principle, attain the same level of accuracy as their non-strategic counterparts. Thus, while perhaps challenging to solve in practice, the intrinsic difficulty of strategic linear classification remains unchanged. + +The above, however, is unlikely to hold for non-linear classifiers: once the decision boundary is non-linear, different points will move in different directions. This lends to the possible distortion of the shape of a given classifier, which in turn can change the effective (i.e., strategic) model class. Such changes can have concrete implications on expressivity, generalization, optimization, and attainable accuracy. This motivates the need to explore how strategic responses shape outcomes in non-linear strategic classification, which we believe is crucial as a first step towards developing future methods. Our general approach is to examine—through analysis, examples, and experiments—how objects in the original non-strategic task map to corresponding objects in the induced strategic problem instance. Initially, our conjecture was that strategic behavior acts as a 'smoothing' operator on the original decision boundary, but as we show, the actual effect of strategic behavior turns out to be more involved. + +To build an intuition of the mechanisms at play, we take a bottom-up approach and study three mappings between (i) individual points, (ii) classifiers, and (iii) model classes. For points, we begin by showing that the mapping is not necessarily bijective. As a result, some points on the original decision boundary become 'wiped out', while others are 'expanded' to become intervals (arcs for $\ell_2$ costs) on the induced boundary. We demonstrate several mechanisms that can give rise to both phenomena. For classifiers, we show how the effects on individual points aggregate to modify the entire decision boundary. A key finding is that certain decision boundaries are impossible to express under strategic behavior. We give several examples of reasonable classifier types that cannot exist under strategic behavior, and show that some effective classifiers $h_\Delta$ can be traced back to (infinitely) many original classifiers $h$ . The main takeaway is that universal approximators, such as neural networks or RBF kernels, are no longer universal in a strategic environment. + +Our main contribution lies in our analysis of model classes, where our results build on the observation that the mapping from original to effective classifiers within a class is not necessarily bijective. In particular, because many $h$ can be mapped to the same $h_{\Delta}$ , the VC dimension of the induced model class can shrink even from $\infty$ down to 0, causing model classes that were not learnable in the standard setting to become learnable in the strategic one. That said, the VC dimension of the induced class can generally either grow or shrink. We highlight some conditions that guarantee either of these outcomes, and, in particular, provide an upper bound on the strategic complexity increase for the expressive family of piecewise-linear model classes (e.g., ReLU networks). Our analysis yields several practical takeaways for learning, model selection, and future method development (see Appendix C.10). We conclude with experiments that shed light on how strategic behavior may alter the effective complexity and maximum accuracy of the learned model class in practice. + +# 2 Related work + +Our work focuses on the common setting of supervised binary classification, where non-linear models are highly prevalent in general. This follows the original formulation of strategic classification [3, 16], which has since been extended also to online settings [1, 5, 8] and regression [2, 31]. As noted, linear models have thus far been at the forefront of the field, with many works relying (explicitly or implicitly) on their properties or induced structure [5, 10, 18, 25, 29]. Even for works with results that are agnostic to model class selection, linear classifiers remain the focus of special-case analyses, examples, and empirical evaluation [20, 26, 37], and their generality does not shed light on the impact of non-linearity. Other works offer only tangential results, such as welfare analysis in the non-linear setting [35], but do not paint a picture of the general learning problem. Our low-level analysis makes use of ideas from the field of offset curves from computational geometry [12, 21]. Whereas this field is mostly concerned with the approximation and analytical properties of individual offset curves, we are + +![](images/c378a643101604ed6863ba539caa04aed3aef945083a5384018c241135dff8b2.jpg) +Figure 1: (Left) Original non-linear classifier $h$ . (Middle) Strategic response to original classifier. (Right) Resulting effective classifier $h_{\Delta}$ (different from $h$ ) after accounting for strategic responses. + +interested in the effects on class complexity and what this implies for learning. In terms of theory, both [33, 36] apply PAC analysis to strategic classification, the latter providing VC bounds for the linear class. Most relevant to us is [6], who study the general learnability of strategic learning under different feedback models. While not restricted to particular classes, their analysis considers graph costs, where users can move only to a finite set of possible points, and their results depend crucially on the cardinality of this set. Our work tackles the more pervasive setting of continuous cost functions, which imply continuous modifications. The above works focus primarily on statistical hardness; while this is also one of our aims, we are interested in establishing a broader understanding of the non-linear regime. + +# 3 Preliminaries + +We work in the original supervised setting of strategic classification [3, 16]. Let $x \in \mathcal{X} = \mathbb{R}^d$ denote user features, $y \in \mathcal{Y} = \{0,1\}$ their corresponding label, and assume pairs $(x,y)$ are drawn iid from some unknown joint distribution $D$ . Given a training set $T = \{(x_i,y_i)\}_{i=1}^m \sim D^m$ , learning aims for the common goal of finding a classifier $h \in H$ that maximizes the expected predictive accuracy, where $H$ is some chosen model class. The unique aspect of strategic classification is that at test time, users modify their features to maximize gains in response to the learned $h$ via the best-response mapping + +$$ +x ^ {h} := \Delta_ {h} (x) = \underset {x ^ {\prime}} {\operatorname {a r g m a x}} \alpha h \left(x ^ {\prime}\right) - c \left(x, x ^ {\prime}\right) \tag {1} +$$ + +where $\alpha$ determines their utility from obtaining a positive prediction $\hat{y} = h(x)$ , and $c$ is a function governing modification costs. We assume w.l.o.g. that the user gains no utility from a negative prediction. Like most works, we focus mainly on the $\ell_2$ cost, i.e., $c(x,x^{\prime}) = \| x - x^{\prime}\|_{2}$ , but discuss other costs in a subset of our results (see Appendix C.1 for discussion on how our results generalize to other cost functions). Because no rational user will pay more than $\alpha$ to obtain a positive prediction, $x^{h}$ will always be contained in the ball of radius $\alpha$ around $x$ , denoted $B_{\alpha}(x) = \{x^{\prime}:c(x,x^{\prime})\leq \alpha \}$ . It will also be useful to define the corresponding sphere, $S_{\alpha}(x) = \{x^{\prime}:c(x,x^{\prime}) = \alpha \}$ . + +Given the above, the learning objective is to maximize the expected strategic accuracy, defined as: + +$$ +\underset {h \in H} {\operatorname {a r g m a x}} \mathbb {E} _ {D} [ \mathbb {1} \{y = h \left(\Delta_ {h} (x)\right) \} ] \tag {2} +$$ + +Levels of analysis. We assume w.l.o.g. that classifiers are of the form $h(x) = \mathbb{1}\{f(x)\geq 0\}$ for some scalar score function $f$ , and that the decision boundary is then the surface of points for which $f(x) = 0$ . Our analysis takes a 'bottom-up' approach in which we first consider changes to individual points on the boundary, then ask how such local changes aggregate to affect a given classifier $h$ , and finally, how these changes impact the entire model class $H$ . It will therefore be useful to consider three mappings between original and corresponding strategic objects. For points, the mapping is $z\mapsto \nabla_{h}(z) = \{x:\Delta_{h}(x) = z\wedge c(x,z) = \alpha \}$ , i.e., the set of points $x$ that, in response to $h$ , move to $z$ at maximal cost. For classifiers, we define the effective classifier as $h_\Delta (x)\coloneqq h(\Delta_h(x))$ (see Fig. 1), and consider the mapping $h\mapsto h_{\Delta}$ . Since both $h$ and $h_\Delta$ are defined w.r.t. raw (unmodified) inputs $x$ , this allows us to compare the original decision boundary of $h$ to the 'strategic decision boundary' of its induced $h_\Delta$ . For model classes, we consider the mapping $H\mapsto H_{\Delta}\coloneqq \{h_{\Delta}:h\in H\}$ , and refer to $H_{\Delta}$ as the effective model class. + +The linear case. As noted, most works in strategic classification consider $H$ to be the class of linear classifiers, $h(x) = \mathbb{1}\{w^\top x \geq b\}$ . A known result is that $\Delta_h(x)$ can be expressed in closed form as a + +![](images/a73708bc2964cf8c81709e30c73c28b433aa3a56914d0de50091b325a49cf796.jpg) +Figure 2: Types of point mappings. From left to right: one-to-one (case #1), direct wipeout (case #2), indirect wipeout (case #2), expansion (case #3), and collision (case #4). Black lines show original $h$ , dashed gray lines indicate $h_{\Delta}$ , gold arrows show $z \mapsto \nabla_h(z)$ , and dotted circles show $S_{\alpha}(x)$ + +conditional projection operator: $x$ moves to $x_{\mathrm{proj}} = x - \frac{w^{\top}x - b}{\|w\|_2^2} w$ if $h(x) = 0$ and $c(x,x_{\mathrm{proj}})\leq \alpha$ and otherwise remains at $x$ [9]. Note that all points that move do so in the same direction, i.e., orthogonally to $w$ . Furthermore, each $x$ on the decision boundary of $h$ is mapped to a single, unique point $z$ on the decision boundary of $h_\Delta$ at an orthogonal distance (or cost) of exactly $\alpha$ . Considering this for all boundary points, we get that $h_\Delta$ simply shifts the decision boundary by $\alpha$ , which gives $h_\Delta (x) = \mathbb{1}\{w^\top x\geq b - \alpha \}$ . Thus, $H_{\Delta}$ remains the set of all linear classifiers, and so $H_{\Delta} = H$ . + +The non-linear case: first steps. Informally stated, the main difference in the non-linear case is that points no longer move in the same direction (see Fig. 1). Note that while points still move to the 'closest' point on the decision boundary, the projection operator is no longer nicely defined. As a result, the shape of the decision boundary of $h$ may not be preserved when transitioning to $h_{\Delta}$ . Moreover, as we will show, some points on the decision boundary of $h$ will 'disappear' and have no effect on the induced $h_{\Delta}$ , while other points may be expanded to affect a continuum of points on $h_{\Delta}$ . A final observation is that directionality is important in the strategic setting: because points always move towards the positive region, the shape of $h_{\Delta}$ for some $h$ may be very different than that of the same classifier but with "inverted" predictions, $h'(x) = 1 - h(x)$ . This lack of symmetry means that we cannot take for granted that the induced $H_{\Delta}$ will include both $h_{\Delta}$ and $1 - h_{\Delta}$ , even if $H$ does. + +# 4 Underlying Mechanisms + +Due to the added complexity of the non-linear setting, we first present some basic results and observations related to the underlying mechanisms of the non-linear setting in order to build the necessary intuition for our higher-level results. We begin by examining the point mapping $z \mapsto \nabla_h(z)$ and then turn to investigate how strategic behavior affects the classifier mapping $h \mapsto h_{\Delta}$ . + +# 4.1 Point mapping + +To gain an understanding of what determines the shape of $h_{\Delta}$ of a general $h$ , we begin by considering a single point $z$ on the decision boundary of $h$ and examining how (and if) it maps to corresponding point(s) on the decision boundary of $h_{\Delta}$ through the point mapping $z \mapsto \nabla_h(z)$ . We identify and discuss four general cases, illustrated in Fig. 2: + +Case #1: One-to-one. For linear $h$ , the point mapping is always one-to-one. For non-linear $h$ , the mapping remains one-to-one as long as $h$ is 'well-behaved' around $z$ . We consider this the 'typical' case and note that each $z$ is mapped to its $\alpha$ -offset point $x = z - \alpha \hat{n}_z$ , where $\hat{n}_z$ is the unit normal at $z$ w.r.t. $h$ (see Appendix B.1). A necessary condition for one-to-one mapping is that $h$ is smooth at $z$ , but smoothness is not sufficient, as the following cases depict. + +Case #2: Wipeout. Even if $h$ is smooth at $z$ , in some cases the point mapping will be null: $z \mapsto \emptyset$ . We refer to this phenomenon as wipeout, and identify two types. Direct wipeout occurs when the signed curvature $\kappa$ at $z$ is negative and of sufficiently large magnitude; this is defined formally in Sec. 4.2. Intuitively, this results from strategic behavior obfuscating any rapid changes in the decision boundary of $h$ . Indirect wipeout occurs when the set of points $\nabla_h(z)$ is contained in $B_{\alpha}(x')$ for some other $x'$ on the decision boundary. For both types, the result is that $B_{\alpha}(z)$ is entirely contained in the positive region of $h_{\Delta}$ and such $z$ have no effect on $h_{\Delta}$ (see Appendix B.2). + +Case #3: Expansion. When the decision boundary of $h$ at $z$ is not smooth (i.e., a kink or corner), $z$ can be mapped to a continuum of points on the effective boundary of $h_{\Delta}$ . This, however, occurs only when the decision boundary is locally convex towards the positive region. The set of mapped + +points $\nabla_h(z)$ are those on $S_{\alpha}(z)$ which do not intersect with any other $B_{\alpha}(x^{\prime})$ for any other $x^{\prime}$ on the boundary of $h$ . Thus, the shape of the induced 'artifact' is partial to a ball. Partial expansion may occur if some of the expansion artifact is discarded through indirect wipeout (Appendix C.2). + +Case #4: Collision. A final case is when a set of points $\{z_1\ldots z_n\}$ on the boundary of $h$ are all mapped to the same single point $x$ on $h_{\Delta}$ , i.e., $\nabla_h(z) = \nabla_h(z') = x$ for all $z, z'$ in the set. Typically, this happens at the transitions between well-behaved decision boundary segments and wiped-out regions. Collision often results in non-smooth kinks on $h_{\Delta}$ , even if all source points were smooth on $h$ . + +# 4.2 Curvature + +Building up to the level of classifiers, it is first useful to examine what happens to boundary points $z$ in terms of how $h$ behaves around them. We make use of the notion of signed curvature, denoted $\kappa$ , which quantifies how much the decision boundary deviates from being linear at a certain point $x$ . For $d = 2$ , curvature is formally defined as $\kappa = 1 / r$ , where $r$ is the radius of the circle that best approximates the curve at $x$ . For $d > 2$ , we use $\kappa^* = \sup(K)$ , where $K$ includes all directional curvatures at $x$ (see Appendix A.1). The sign of $\kappa$ measures whether the decision boundary bends (i.e., is convex) towards the positive region $(\kappa > 0)$ , the negative region $(\kappa < 0)$ , or is locally linear $(\kappa = 0)$ . + +Let $z$ be a typical point on the decision boundary of $h$ which satisfies the one-to-one mapping detailed in Sec. 4.1. We borrow a classic result from the field of offset curves [12] to calculate the effective signed curvature of $h_{\Delta}$ at $x$ , denoted $\kappa_{\Delta}$ , as a function of the curvature $\kappa$ of $h$ at $z$ . + +Proposition 1. Let $z$ be a point on the decision boundary of $h$ with signed curvature $\kappa \geq -1 / \alpha$ . The effective curvature of the corresponding $x$ on the boundary of $h_{\Delta}$ is given by: + +$$ +\kappa_ {\Delta} = \kappa / (1 + \alpha \kappa) \tag {3} +$$ + +When $d > 2$ , each $\kappa \in K$ is mapped as in Eq. 3 to form $K_{\Delta} = \{\kappa / (1 + \alpha \kappa) : \kappa \in K\}$ , and $\kappa_{\Delta}^{*}$ follows. See Appendix B.3 for proof and an illustration of how this impacts the shape of $h_{\Delta}$ . Prop. 1 entails several interesting properties. First, strategic behavior preserves the direction of curvature, i.e., $\kappa \cdot \kappa_{\Delta} \geq 0$ always. In particular, $\kappa = 0$ entails $\kappa_{\Delta} = 0$ , which explains why linear functions remain linear. Second, the effective curvature $\kappa_{\Delta}$ is monotonically increasing in $\kappa$ , concave, and sub-linear. This implies that for $\kappa > 0$ the effective positive curvature decreases, while for $\kappa < 0$ the effective negative curvature increases. Thus, strategic behavior acts as a directional smoothing operator, i.e., the boundaries' rate of change is expected to decrease but only in the positive direction. + +Finally, in the positive direction, $\kappa_{\Delta}$ is bounded from above, approaching $1 / \alpha$ as $\kappa \to \infty$ . This gives: + +Observation 1. No effective classifier $h_{\Delta}$ can have positive curvature $\kappa_{\Delta}$ greater than $1 / \alpha$ . + +When $d > 2$ , Obs. 1 holds for each curvature $\kappa \in K$ and can be summarized as $\kappa_{\Delta}^{*} \leq 1 / \alpha$ . Though Prop. 1 only implies Obs. 1 at points on $h_{\Delta}$ that are typical, we will show in Sec. 4.4 that it is also true at any point on $h_{\Delta}$ . The above holds irrespective of the shape and curvature of the original $h$ . In particular, non-smooth points on $h$ (i.e., with $\kappa \rightarrow \infty$ ), such as an edge or vertex at the intersection of halfspaces, become smooth on $h_{\Delta}$ . Conversely, in the negative direction, $\kappa_{\Delta}$ approaches $-\infty$ as $\kappa \rightarrow 1 / \alpha$ . Thus, negative curvature can grow arbitrarily large, and at $\kappa = -1 / \alpha$ maps to a non-smooth "kink" in $h_{\Delta}$ . $\kappa_{\Delta}$ is no longer defined when $\kappa$ crosses the asymptote at $-1 / \alpha$ because such points are wiped out and have no corresponding $x$ on the boundary of $h_{\Delta}$ . This is the exact case of indirect wipeout in Sec. 4.1 (see Appendix B.2 for proof and Appendix C.10 for practical use in learning). + +Observation 2. Any point $z$ on the decision boundary of $h$ with curvature $\kappa < -1 / \alpha$ has no influence on $h_{\Delta}$ . In higher dimensions, this is true for any point $x$ with $\inf(K) < -1 / \alpha$ . + +# 4.3 Containment + +Because the decision boundary is a level set of some (non-linear) score function $f$ , contained regions are not only possible, but realistic. Consider a given classifier $h$ where the decision boundary forms some bounded connected negative region $C \subset \mathbb{R}^d$ where $h(x) = 0 \forall x \in C$ . Strategic behavior dictates that negative areas shrink, and so only a subset of $C$ will still be classified as negative in $h_{\Delta}$ . If $C$ is sufficiently small, then it will disappear completely leaving $h_{\Delta}(x) = 1 \forall x \in C$ . + +Observation 3. Any negative region $C$ fully contained in some $B_{\alpha}(x^{\prime})$ will become positive in $h_\Delta$ . + +![](images/f11feb88eb9e533a8d36196a42a59280c3460a6cd68a7a1568a0a2c278b6e731.jpg) +Figure 3: Types of impossible effective classifiers. From left to right: small positive region, narrow positive strip, large positive curvature, piecewise (hyper)linear convex towards the positive region. + +![](images/2acf5f2696bc27f46865840190cf051896abe255bedba0c7f6f3c6f79ba7c174.jpg) + +![](images/c06cdc01ff9d306fc8a93b010420a7c2788cefa181e671263382ea15916eece0.jpg) + +![](images/fb9188383b436dade04ae3178830171d520ef8f465ead64f142682bb0e2d176a.jpg) + +The fact that positive areas grow and negative areas shrink also implies that there is a minimum positive region size that can exist in any $h_{\Delta}$ . This idea will be formalized and generalized later in Sec. 4.4. One implication of Obs. 3 is that when negative regions are wiped out, two or more positive regions may merge, leading to a less expressive decision boundary. On the other hand, the strategic setting can also split a negative region, creating a more expressive decision boundary (see Appendix C.3). + +Observation 4. Strategic behavior can merge positive regions and split negative regions thereby either increasing or decreasing the number of connected regions (either positive or negative). + +# 4.4 Impossible Effective Classifiers + +Based on the above, we derive two general statements about effective classifiers that cannot exist. Each statement considers some candidate classifier $g$ , and shows that if $g$ satisfies a certain property, then there is no classifier $h$ for which $g$ is its induced effective classifier, i.e., $g \neq h_{\Delta}$ for any $h$ . We first require some additional notation. Given $h$ , define the set of negatively classified points as $\mathcal{X}_0(h) = \{x : h(x) = 0\}$ . For a set $A \subseteq \mathcal{X}$ , we say a point $x$ is reachable from $A$ if $\exists x' \in A : c(x, x') \leq \alpha$ . + +Proposition 2. Let $g$ be a candidate classifier. If there exists a point $x$ on the decision boundary of $g$ such that all points $x' \in S_{\alpha}(x)$ are reachable from some point in $\mathcal{X}_0(g)$ , then there is no $h$ such that $g = h_{\Delta}$ . + +The intuition behind Prop. 2 is that no $h$ can simultaneously strategically classify $x$ as positive while all $y \in \mathcal{X}_0(g)$ as negative. Full proof in Appendix B.4. As stated in Prop. 3, when $g$ is smooth at $x$ , Prop. 2 can be simplified to require that only a single point be checked instead of all of $S_{\alpha}(x)$ . + +Proposition 3. Let $g$ be a candidate classifier. If there exists a point $x$ on the decision boundary of $g$ where (i) $g$ is smooth, and (ii) the offset point $\hat{x} = x + \alpha \hat{n}_x$ is reachable by some other $x' \in \mathcal{X}_0(g)$ , then there is no $h$ such that $g = h_{\Delta}$ . + +The intuition behind Prop. 3 is the same as for Prop. 2, except that when $g$ is smooth at $x$ , the only point in $S_{\alpha}(x)$ that may not be reachable by some $x' \in \mathcal{X}_0(g)$ is the $\alpha$ -offset (see Appendix B.5). + +Examples. The implication of Props. 2 and 3 is that there exist decision boundaries that are generally realizable, but are no longer realizable as effective classifiers in the strategic setting. Note that because it only takes a single decision boundary point to render an entire effective classifier infeasible, each of these examples are broad categories and pose meaningful restrictions on the set of possible effective classifiers. As illustrated in Fig. 3, we highlight four types of impossible effective decision boundaries: + +1. Classifiers with a small positive region that can be enclosed by a ball of radius $\alpha$ . +2. Classifiers with a narrow positive region in which points are $\alpha$ -close to the decision boundary. +3. Classifiers with large positive signed curvature anywhere on the decision boundary. +4. Classifiers with any locally convex piecewise linear or hyperlinear segments. + +All proofs are in Appendix C.4, and either build on Obs. 1-4, or show the conditions of Props. 2 or 3 hold. From these four examples, we note the following observation. + +Observation 5. The mapping $h \mapsto h_{\Delta}$ is not bijective, with many candidate $h_{\Delta}$ having no corresponding $h$ and many $h$ mapped to the same $h_{\Delta}$ . + +In Appendix C.5 we show that there are actually infinite $h$ that map to nearly all possible $h_{\Delta}$ . Furthermore, Obs. 5 will be important for proving Thm. 1 and Cor. 1. + +![](images/18b910de5ceee2ee9dfb2e55f3f2a24ed66ea528e9f05e5b77f5b74c964eb87d.jpg) +Figure 4: (Left) Example of increasing VC. (Right) Example of a dataset with limited strategic accuracy. Any $h_{\Delta}$ which correctly classifies the gold-rimmed point must err on a negative point. + +![](images/769f927d8a21f9c85ad90c64297d692910bda25f58e3cfe5e205037d5383af40.jpg) + +# 5 Class-level analysis + +On the classifier level, the strategic setting is clearly more restrictive than the regular setting. Indeed, in Sec. 5.2 we extend this idea into the model class level by analyzing the impacts on universality and optimal accuracy. However, as we show in Sec. 5.1, the model class mapping is more nuanced than the classifier mapping since complexity can actually increase as a result of strategic behavior. + +# 5.1 Class complexity + +Recall that for linear classes, $H_{\Delta}$ remains linear, and so $H \mapsto H_{\Delta} = H$ . Although this equivalence is not unique to the class of linear classifiers (see examples in Appendix C.6), typically, we can expect strategic behavior to alter the complexity of the induced class. Here, we use VC analysis to shed light on this effect by examining the possible relations between $\mathrm{VC}(H)$ and $\mathrm{VC}(H_{\Delta})$ .1 + +Complexity reduction. Since our results so far point out that effective classifiers are inherently constrained, a plausible guess is that effective complexity should decrease. Indeed, there are some natural classes where the VC cannot increase. For example: + +Proposition 4. Let $Q_{s}^{k}$ be the class of negative-inside $k$ -vertex polytopes bounded by a radius of $s$ , then $\mathrm{VC}(Q_{s}^{k}) \geq \mathrm{VC}(Q_{s,\Delta}^{k})$ . Moreover, if $s = \infty$ , then $\mathrm{VC}(Q_{s}^{k}) = \mathrm{VC}(Q_{s,\Delta}^{k})$ as implied by Thm. 2. + +The proof for Prop. 4 can be found in Appendix B.7. Curiously, in some extreme cases, the VC reduction spans the maximum extent. + +Theorem 1. There exist several model classes $H$ where $\mathrm{VC}(H) = \infty$ , but $\mathrm{VC}(H_{\Delta}) = 0$ . + +The proof for Thm. 1 can be found in Appendix B.8. See Appendix C.7 for other examples where VC strictly decreases but does not diminish completely. Thm. 1 carries concrete implications for strategic learning: + +Corollary 1. Some model classes that are not learnable become learnable in strategic environments. + +Complexity increase. Though a lower effective VC is indeed a possibility, a more plausible outcome is for it to stay the same or increase. As an intuition for how the VC might increase, consider the simple example in Fig. 4 (left). Here, $H$ includes four non-overlapping classifiers (each color is a separate classifier) with $\mathrm{VC}(H) = 1$ . Yet, with strategic behavior, the effective classifiers (dashed curves) overlap, increasing the shattering capacity to 2. This case can be generalized to give $VC(H_{\Delta}) = \theta(d \cdot VC(H))$ (see Appendix C.8). We next identify a simple generic sufficient condition for the effective complexity of any learnable class to be non-decreasing. + +Theorem 2. If $H$ is closed under input scaling and $\mathrm{VC}(H) < \infty$ , then $\mathrm{VC}(H) \leq \mathrm{VC}(H_{\Delta})$ . + +The proof for Thm. 2 can be found in Appendix B.9. Thm. 2 applies to many common classes, including polynomials, piecewise linear functions, and most neural networks. See Appx. C.10 for implications on model selection. + +Because Thm. 2 suggests that the effective VC can increase, but does not state by how much, one concern would be that this increase can be unbounded, which would deem the effective class unlearnable. Our next result shows that this is not the case even for the common and highly expressive class of piecewise-linear classifiers (with a bounded number of segments), which includes neural + +networks with ReLU activations (of fixed size) as a special case, and is often used in studies on neural network approximation [e.g., 19, 28]. + +Theorem 3. Let $H^{m,k}$ be a class of piecewise-linear classifiers with at most $m$ segments and $k$ intersections of linear segments. Then $\mathrm{VC}(H_{\Delta}^{m,k}) = O\big(dm\log (m) + \nu_kk\log (k)\big)$ , where $\nu_{p}$ is the VC of the class of $\ell_p$ norm balls, and is $\Theta (d)$ for $p = 1,2,\infty$ . + +The proof for Thm. 3 is in Appendix B.10. We also prove that $\mathrm{VC}(H_{m,k}) = O\big(dm\log (m)\big)$ , suggesting that the effective VC dimension can increase, but only to a limited extent. The main implication of Thm. 3 is therefore that under these conditions, strategic behavior maintains learnability: for any $H$ that can be expressed as (or approximated by) piecewise-linear functions, if $H$ (or its approximation) is learnable, then $H_{\Delta}$ is also learnable. We conjecture that this relation remains true for general $H$ ; see Appendix A.3 for discussion on the challenges of the general case and connections to [6]. For the special case of positive-inside polytopes — a subset of the piecewise-linear class — the effective VC dimension even maintains the same order of magnitude. + +Theorem 4. Consider either the $\ell_1, \ell_2$ , or $\ell_{\infty}$ costs and let $H \subseteq P^{k}$ , where $P^{k}$ is the set of all $k$ -vertex polytopes in $R^{d}$ and has $\mathrm{VC}(P^{k}) = O(d^{2} k \log k)$ [23]. Then $\mathrm{VC}(H_{\Delta}) = O(d^{2} k \log k)$ . + +The proof for Thm. 4 is in Appendix. B.11. Note that for the $\ell_1$ and $\ell_{\infty}$ costs, the effective class remains within the polytope family, but potentially with additional vertices (see Appendix B.6). For the $\ell_2$ cost, the bound still holds despite $H_{\Delta}$ no longer including only polytopes (since corners in $h$ can become "rounded" in $h_{\Delta}$ ). + +# 5.2 Universality and approximation + +Our results in Sec. 4.4 show four 'types' of decision boundaries that cannot be attributed to any effective classifier, though others may exist. This implies that several basic classifiers are not realizable in the strategic setting and cannot be approximated well. This drives our principal conclusion: + +Corollary 2. Any universal approximator class $H$ is no longer universal under strategic behavior. + +In other words, there exists a classifier $g$ for which no effective class $H_{\Delta}$ can include $g$ as a member, even for $H$ that are universal approximators in the non-strategic setting. Common examples of universal approximator classes include polynomial thresholds [32], RBF kernel machines [27], many classes of neural networks [e.g., 17], and gradient boosting machines [13]. In the strategic setting, these classes are no longer all-encompassing like in the standard setting. + +Approximation gaps. Cor. 2 implies that even if the data is in itself realizable, and even without any limitations on the hypothesis class, the learner may already start off with a bounded maximum training accuracy simply by learning in the strategic setting. Fig. 4 (right) depicts one such example dataset where the maximum attainable accuracy drops from 1 to 0.96. This is because strategic behavior makes it impossible to generally approximate intricate functions, e.g., by using fast-changing curvature (via increased depth) or many piecewise-linear segments (with ReLUs). Another result is that the common practice of interpolating the data using highly expressive over-parametrized models (e.g., as in deep learning) is no longer a viable approach. On the upside, it may be possible for the strategic responses to prevent unintentional overfitting, allowing for better generalization. + +In extreme cases, the gap between the maximum standard accuracy and the maximum strategic accuracy can be quite large. Our next result demonstrates that there are distributions in which, despite a maximum standard accuracy of 1, the maximum strategic accuracy falls to the majority class rate. + +Proposition 5. For all $d$ , there exist (non-degenerate) distributions on $\mathbb{R}^d$ that are realizable in the standard setting, but whose maximum strategic accuracy is the majority class rate, $\max_y p(y)$ . + +In one dimension, the distribution can be constructed by placing a negative point $\delta$ to the left and right of each positive point. In higher dimensions, negative points can be arranged in an $\mathbb{R}^d$ simplex around each positive point. The full proof is in Appendix B.12. Prop. 6 extends this idea to show that there is a distribution whose maximum strategic accuracy approaches $1/2$ as $\alpha$ increases. The general proof is given in Appendix B.13. + +Proposition 6. There exists a distribution that is realizable in the standard setting, but whose accuracy under any $h_{\Delta}$ is at most $0.5 + \frac{1}{[2\alpha + 1]}$ . + +![](images/fe743bb596a95f01136fc37c57008df11ea165117224f06b1c3a104739841f92.jpg) +Figure 5: (Left) Expressivity. For random polynomial classifiers of degree $k$ , results show the smallest degree $k'$ that captures the effective decision boundary. (Center) An instance showing positive curvature decreasing (A), negative curvature increasing (B), and wipeout (C). (Right) Approximation. As data becomes more entangled (low separation), strategic approximation degrades. + +![](images/feb901b500d0602e9d5b97017ee3e1265037406528488c0359c1ec6a49d09a40.jpg) + +![](images/937eb5539c7c04f8ad04d6354d815bf239de4c74d94a501250e9c70a472d819a.jpg) + +In practice, we expect the accuracy gap to depend on the distribution. We explore this gap experimentally in Sec. 6. An interesting observation is that in certain non-realizable settings, the maximum strategic accuracy can exceed the maximum standard accuracy. (see Appendix C.9). + +# 6 Experiments + +To complement our theoretical results, we present two experiments that empirically demonstrate the effects of strategic behavior on non-linear classifiers. The code for both experiments is available at https://github.com/BML-Technion/scnonlin. + +# 6.1 Expressivity + +Following our results from Sec. 5.1 on the change in hypothesis class expressivity, we experimentally test whether the $h_{\Delta}$ of a given random $h$ belongs to a more or less expressive class than $h$ itself. In particular, we focus on degree- $k$ polynomial classifiers and compare the degree $k$ of $h$ to the degree $k_{\Delta}$ of the polynomial approximation of $h_{\Delta}$ . Because user features are often naturally bounded, we assume that $x \in \mathcal{X} = R^2$ and that $x$ is bounded by $|x|_{\infty} \leq 100$ . See Appendix D.1 for full details. + +Fig. 5 (left) shows results for varying $k \in [1, 10]$ and for $\alpha \in [2, 40]$ . When $\alpha$ is small, we find that $k_{\Delta} \approx k$ , meaning that strategic behavior has little impact on expressivity. However, as $\alpha$ increases, $k_{\Delta}$ becomes larger than $k$ , suggesting that $h_{\Delta}$ is more complex than $h$ . This is due to the fact that as $\alpha$ increases, point mapping collisions (see Sec. 4.1) become more likely, causing non-smooth cusps — which are more complex than basic polynomials — to form in the decision boundary (Fig. 5 (center)). When $\alpha$ is quite large, we see that $k_{\Delta}$ is still larger for low $k$ , but drops considerably for higher $k$ . This can be attributed to the fact that tightly-embedded higher-dimension polynomials and increased strategic reach from large $\alpha s$ are both causes of indirect wipeout (see Sec. 4.1) because they increase the reachability of $\nabla_h(z)$ by other decision boundary points. As such, much of the original decision boundary is wiped out, leaving behind a lower complexity effective decision boundary. + +# 6.2 Approximation + +As seen in Sec. 5.2, the lack of universality in the strategic setting can impose a limitation on the maximal attainable strategic accuracy. To demonstrate the practical implications of this effect, we upper bound the maximum strategic accuracy of any $H_{\Delta}$ on a set of synthetic data by experimentally calculating the maximum strategic accuracy of the unrestricted effective hypothesis class $H_{\Delta}^{*}$ . We compare the results to two benchmarks: (i) the standard accuracy of the regular unrestricted hypothesis class $H^{*}$ (which is always 1), and (ii) the strategic (and standard) accuracy of the linear effective hypothesis class $H_{\Delta}^{\mathrm{lin}} = H^{\mathrm{lin}}$ . The first benchmark measures the extent to which the strategic setting + +hinders the learner, while the second measures the extent to which the learner can benefit from non-linearity, even in the strategic world. We generate data by sampling points from class-conditional Gaussians $x \sim \mathcal{N}(y\mu, 1)$ , and aim to find an $h$ that obtains an optimal fit. The parameter $\mu$ serves to show how $H_{\Delta}$ compares to our two baselines under data that ranges from well-separated (large $\mu$ ) to more "interleaved" and harder to classify by restricted classes (small $\mu$ ). Details in Appendix D.2. + +Fig. 5 (right) shows results for increasing class separation $(\mu)$ . As $\mu$ decreases and the classification problem becomes more difficult, the maximum strategic accuracy of $H_{\Delta}^{*}$ diverges from the upper $H^{*}$ baseline, with performance deteriorating as strategic behavior intensifies (i.e., larger $\alpha$ ). Because any actual $H_{\Delta}$ can only have worse performance than the ideal $H_{\Delta}^{*}$ , this indicates that strategic behavior can become a significant burden in more difficult tasks, such as classification in a non linearly-separable setting. Nonetheless, when $\mu$ is small, $H_{\Delta}^{*}$ significantly outperforms the lower $H_{\Delta}^{\mathrm{lin}}$ baseline, signaling that non-linearities still improve accuracy in the strategic setting despite any strategic effects. + +# 7 Discussion + +This work sets out to explore the interplay between non-linear classifiers and strategic user behavior. Our analysis demonstrates that non-linearity induces behavior that is both qualitatively different than the linear case and also non-apparent by simply studying it. Simply put, the strategic setting is fundamentally limited, though has a few potential advantages. Our results show how such behavior can impact classifiers by morphing the decision boundary and model classes by increasing or decreasing complexity. Although prior work has made clear the importance of accounting for strategic behavior in the learning objective, our work suggests that there is a need for additional broader considerations throughout the entire learning pipeline from the choices we make initially (such as which model class to use) to setting our final expectations (such as the accuracy levels we may aspire to achieve). We detail practical takeaways for learning and model class selection in Appendix C.10. This motivates future theoretical questions, as well as complementary work on practical aspects such as optimization (i.e., how to solve the learning problem), modeling (i.e., how to capture true human responses), and evaluation (i.e., on real humans and in the wild). + +# Acknowledgments + +The authors are grateful to Shay Moran and Nadav Dym for thoughtful discussions and suggestions. This work is supported by the Israel Science Foundation grant no. 278/22. + +# References + +[1] Saba Ahmadi, Hedyeh Beyhaghi, Avrim Blum, and Keziah Naggita. The strategic perceptron. In Proceedings of the 22nd ACM Conference on Economics and Computation, pages 6-25, 2021. +[2] Yahav Bechavod, Chara Podimata, Steven Wu, and Juba Ziani. Information discrepancy in strategic learning. In International Conference on Machine Learning, pages 1691-1715. PMLR, 2022. +[3] Michael Brückner, Christian Kanzow, and Tobias Scheffer. Static prediction games for adversarial learning problems. 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A note on bounds for vc dimensions. Institute of Mathematical Statistics collections, 5:103-107, 01 2009. +[35] Tian Xie and Xueru Zhang. Non-linear welfare-aware strategic learning. Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society, 7:1660-1671, 10 2024. +[36] Hanrui Zhang and Vincent Conitzer. Incentive-aware PAC learning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 5797-5804, 2021. +[37] Tijana Zrnic, Eric Mazumdar, Shankar Sastry, and Michael Jordan. Who leads and who follows in strategic classification? Advances in Neural Information Processing Systems, 34:15257-15269, 2021. + +# A Background + +# A.1 Signed Curvature + +Our analysis makes use of the notion of signed curvature — a signed version of the normal curvature — to quantify shape alteration. Because there is an inherent notion of direction due to the binary nature of class labels, we can define curvature as being positive if the curvature normal vector points in the direction of the positive region and negative if it points in the direction of the negative region. Though in $\mathbb{R}^2$ this defines a single value, we must instead define a curvature value for each 'direction' in higher dimensions. More formally, for each tangent vector $\vec{t}$ in the tangent hyperplane $T_x$ at $x$ , we define the curvature value $\kappa_{\vec{t}}$ as the signed curvature at $x$ on the curve obtained by intersecting the decision boundary with the plane $L$ containing $\vec{t}$ and the normal vector $\hat{n}_x$ . Defining $K = \{\kappa_{\vec{t}} : \vec{t} \in T\}$ as the set of all such curvatures, our analysis will focus on $\kappa^* = \sup(K)$ in $\mathbb{R}^3$ and above. + +# A.2 Bounded/Unbounded Input Domain + +At a high level, our results in Sec. 5 do not require an unbounded input space, but rather some input space that is larger than $\mathcal{X}$ . In particular, if we assume that raw inputs, $x$ , are bounded by $|x|\leq B$ , then for the problem to generally be well-defined, we must allow this space to expand at the minimum to $B + \alpha$ to accommodate for strategic input modifications. More generally, we must ensure that $|\Delta_h(x)|\leq B(\alpha)$ holds for all $x$ and any $h$ , for some expansion $B(\alpha)\geq B$ , where the exact form of $B(\alpha)$ depends both on the cost function and on $H$ . + +Other than Thm. 2, our results simply require the minimal expansion be satisfied and are therefore not dependent on an unbounded input domain. Likewise, Thm. 2 does not require an unbounded input domain since for any given learnable $H$ , there exists some finite $B(\alpha)$ for which the result holds. However, since the statement is generic (i.e., applies to all learnable $H$ that are closed to scaling), we do not have a closed-form expression for $B(\alpha)$ that applies simultaneously to all such $H$ . + +# A.3 General Upper Bound on VC Increase + +In Thm. 3, we provide an upper bound on the strategic VC dimension of classes of piecewise-linear classifiers, though note that a general upper bound remains an open question. A key difficulty in finding a general upper bound in the strategic setting is that points will move differently when facing different $h$ from the same $H$ . Thus, reasoning about the shattering of an arbitrary set must account for how points can move under all possible $h \in H$ , though not simultaneously, which proved to be highly challenging in the general, unstructured case. This is why we have included results that exploit common structures like the upper bound based on piecewise linearity (Thm. 3) and lower bound based on $H$ that are closed to scaling (Thm. 2). + +That said, we conjecture that Thm. 3 holds for general $H$ , as proven by Cohen et al. [6] in the special case of finite manipulation graphs. In particular, they show that $\mathrm{VC}(H_{\Delta}) = \tilde{\Theta} (\mathrm{VC}(H)\cdot \log k)$ where $k$ is the degree of the manipulation graph. Note that the bound is not transferable to our case since it is vacuous for $k\to \infty$ , and the proof technique does not apply to continuous graphs. + +# B Proofs + +Lemma 1. In $\mathbb{R}^2$ , if the curvature $\kappa$ at $x$ is positive, then the radius $r_{\text{insc}}$ of the maximum inscribed circle at $x$ is no bigger than the radius $r_{\text{osc}} = \frac{1}{\kappa}$ of the osculating circle at $x$ (the circle that best approximates the curve at $x$ and has the same curvature). As a corollary, if curvature $\kappa$ at $x$ is negative, then the radius $r_{\text{esc}}$ of the maximum inscribed circle at $x$ is no bigger than the radius $r_{\text{osc}} = -\frac{1}{\kappa}$ of the osculating circle at $x$ . + +Proof. Note that both the osculating circle and the inscribed circle have their centers along the normal line to the curve at $x$ (and on the same side of $h$ ), and that the osculating circle may cross the curve at $x$ . If it does, then any circle which has a larger radius and a center on the same side of $h$ along the normal line to the curve will also cross the decision boundary at $x$ , so the radius of the inscribed circle (which strictly does not cross the curve) is strictly smaller than the radius of the osculating circle. If the osculating circle does not cross the curve, then it must be the largest strictly tangent + +circle at $x$ , or else it would not be the best circular approximation to the curve at $x$ . As such, the largest inscribed circle, which must be both tangent to the curve at $x$ also not intersect it anywhere else, must have a radius that is no larger than $r_{osc}$ . + +![](images/69d651aee7d9ff4f94428ae5204194759c14114811c3e238742155c66859b027.jpg) + +# B.1 One-to-One Mapping + +Claim: In the one-to-one mapping, the point $z$ on the decision boundary of $h$ is mapped to the point $x = z - \alpha \hat{n}_z$ on the decision boundary of $h_{\Delta}$ . + +Proof. Note that all points on the decision boundary of $h_{\Delta}$ must be at the maximum cost away from $h^3$ , so $\nabla_h(z) \subset S_\alpha(z)$ . Additionally, for each $x \in \nabla_h(z)$ , the ball $B_\alpha(x)$ , which is the set of points that $x$ can move to, must not intersect the decision boundary of $h$ (or else it would not be at maximum cost away) and must furthermore be strictly tangent to the decision boundary of $h$ at $z$ . Given that the decision boundary of $h$ is smooth at $z$ , this means that the normal to the curve at $z$ must run along the radius of $B_\alpha(x)$ from $z$ to $x$ , implying that $x = z \pm \alpha \hat{n}_z$ . Because of the inherent directionality of the setting, the normal vector at $z$ is defined to point in the positive direction. This means that only the point $x = z - \alpha \hat{n}_z$ would seek to move to $z$ to gain a positive classification, as $x = z + \alpha \hat{n}_z$ is either already labeled positive or can move to a closer point. Therefore, $\nabla_h(z) = \{z - \alpha \hat{n}_z\}$ in the one-to-one case. + +# B.2 Wipeout Mapping + +Claim: Points on the decision boundary of $h$ that are 'wiped out' do not affect the effective decision boundary of $h_{\Delta}$ . + +Proof. In the case of indirect wipeout, this follows directly from the definition: if $\nabla_h(z)$ is contained in $B_{\alpha}(x^{\prime})$ for some other $x^{\prime}$ on the decision boundary of $h$ (or in the union of several $B_{\alpha}(x^{\prime})$ ), then none of the points in $\nabla_h(z)$ are at a maximum cost away from $h$ , and are therefore not on the decision boundary of $h_{\Delta}$ . + +In the case of direct wipeout, the normal one-to-one mapping fails because the point $x = z - \alpha \hat{n}_z$ is no longer at the maximum cost away from $h$ . Specifically, because the curvature $\kappa < -1 / \alpha^4$ , the ball $B_{\alpha}(x)$ now intersects $h$ , which means that $x$ is less than $\alpha$ away from $h$ and therefore not on the decision boundary of $h_{\Delta}$ . For $\mathbb{R}^2$ , Lemma 1 proves that the radius of the maximum ascribed circle at $z$ is less than the radius of the osculating circle, so $r_{esc} \leq r_{osc} = -\frac{1}{\kappa} < \alpha$ . Therefore, there is no circle of radius $\alpha$ that is strictly tangent to $h$ (on the negative side), so $B_{\alpha}(x)$ must intersect $h$ . + +For higher dimensions (see Appendix A.1), there is no longer a single normal curvature. Instead, we define the set $K$ of curvatures in each plane $p \in P$ defined by the normal vector and some tangent vector at $z$ . Note that $B_{\alpha}(x)$ will intersect $h$ if any of its projections onto each $p \in P$ (a circle) intersects the curve obtained by intersecting the decision boundary with the plane $p$ . Since each of these cases are in $\mathbb{R}^2$ , $B_{\alpha}(x)$ will intersect $h$ if $\inf(K) < -1 / \alpha$ . + +![](images/20c06ea323f9c44882b0ab8b7b892d5d637d6b5447a1460ad44fd64c7d3ae7db.jpg) + +# B.3 Proof of Proposition 1 + +Proposition 1: Let $z$ be a point on the decision boundary of $h$ with signed curvature $\kappa \geq -1 / \alpha$ . The effective curvature of the corresponding $x$ on the boundary of $h_{\Delta}$ is given by: + +$$ +\kappa_ {\Delta} = \kappa / (1 + \alpha \kappa) \tag {3} +$$ + +Proof. While an equivalent form of Prop. 1 has already been proven in the field offset curves [12], we nonetheless reprove it here to offer some intuition behind it. In the normal, one-to-one mapping, because both the center of curvature and $\nabla_h(z)$ lie on the normal line to $h$ at $z$ (see Appendix B.1), + +![](images/b584622eacdb607bd1c8aba881ed491eb13570a4793089fcfb2fb848fa2e7937.jpg) +Figure 6: (Top) Graph of Eq. 3 with asymptotes represented as dashed lines. (Bottom) Depictions of curvature mappings for increasing $\kappa$ . + +strategic movement can be seen as increasing the signed radius of curvature, $r_{osc}$ , by $\alpha$ (where we will define the radius of curvature as negative if the curvature is negative at $z$ ). In higher dimensions, this is equivalent to increasing each of the radii of curvature by $\alpha$ (see Appendix A.1). We therefore have: + +$$ +\begin{array}{l} r _ {o s c, \Delta} = r _ {o s c} + \alpha \\ \kappa_ {\Delta} = \frac {1}{r _ {o s c , \Delta}} = \frac {1}{r _ {o s c} + \alpha} \\ = \frac {1}{\frac {1}{\kappa} + \alpha} \\ = \frac {\kappa}{1 + \alpha \kappa} \\ \end{array} +$$ + +Fig. 6 shows the graph of Eq. 3 as well as examples of curvature mappings for various $\kappa$ . Note that Eq. 3 has a horizontal asymptote at $\kappa_{\Delta} = 1 / \alpha$ because the effective curvature cannot exceed $1 / \alpha$ (see Sec. 4.4). Additionally, Eq. 3 has a vertical asymptote at $\kappa = -1 / \alpha$ because any point with lower curvature has no effect on $h_{\Delta}$ (see Appendix B.2). When $\kappa = -1 / \alpha$ , $x$ is mapped to a non-smooth kink of infinite curvature. + +# B.4 Proof of Proposition 2 + +Proposition 2: Let $g$ be a candidate classifier. If there exists a point $x$ on the decision boundary of $g$ such that all points $x' \in S_{\alpha}(x)$ are reachable from some point in $\mathcal{X}_0(g)$ , then there is no $h$ such that $g = h_{\Delta}$ . + +Proof. In order for $g$ to be the effective classifier of the regular classifier $h$ , we must have: + +1. For each point $q$ that is positively classified by $g$ , there must be a point $q' \in B_{\alpha}(q)$ which is classified as positive by $h$ . In other words, $q$ must be able to acquire a positive label by staying in place or strategically moving within $B_{\alpha}(q)$ . +2. For each point $q$ that is negatively classified by $g$ , there cannot be any points $q' \in B_{\alpha}(q)$ which are classified as positive by $h$ . In other words, $q$ must not be able to acquire a positive label by staying in place or strategically moving within $B_{\alpha}(q)$ . + +Assume there exists a point $x$ on the decision boundary of $g$ such that all points $x' \in S_{\alpha}(x)$ are reachable from $\mathcal{X}_0(g)$ and that there exists $h$ such that $g = h_{\Delta}$ . Since $x$ is on the decision boundary + +![](images/f5b21ad01e9733fbf5c90f5db90376b9ffe48f715e66c86ee5a989cffee27a97.jpg) +Figure 7: (Left) Each face of a negative-inside polytope is mapped to a new linear face, though the resulting $h_{\Delta}$ may be a polytope with fewer faces and vertices due to wipeout. (Right) Each face of a positive-inside polytope is also mapped to a new linear face, though each vertex will have to expand to fill the gap between these faces. + +![](images/22e048600a43f70bf63de7525d8f782248bbc1cc3a47a3d9ba85d9fcd684d228.jpg) + +of $g$ , there exists a point $x + \delta, \delta \to 0$ that is negatively classified by $g$ . For requirements 1 and 2 to hold, there must be a point $x' \in S = \{q : q \in B_{\alpha}(x), q \notin B_{\alpha}(x + \delta)\}$ that is positively classified by $h$ . Since $S \subset S_{\alpha}(x)$ , we can expand the claim to say that there must be a point $x' \in S_{\alpha}(x)$ that is positively classified by $h$ . However, if all points $x' \in S_{\alpha}(x)$ are reachable by some point $q$ that is negatively classified by $g$ , then none of the points $x' \in S_{\alpha}(x)$ can be positively classified by $h$ (by requirement 2). Therefore, if there exists a point $x$ on the decision boundary of $g$ for which every point on the sphere $S_{\alpha}(x)$ is reachable by some point that is negatively classified by $g$ , then $g$ cannot be a possible effective classifier. + +# B.5 Proof of Proposition 3 + +Proposition 3: Let $g$ be a candidate classifier. If there exists a point $x$ on the decision boundary of $g$ where (i) $g$ is smooth, and (ii) the offset point $\hat{x} = x + \alpha \hat{n}_x$ is reachable by some other $x' \in \mathcal{X}_0(g)$ , then there is no $h$ such that $g = h_{\Delta}$ . + +Proof. By Prop. 2, for each point $x$ on the decision boundary of $g$ , there must be a point $x'$ on $S_{\alpha}(x)$ that is classified positively by $h$ and also not reachable by any point negatively classified by $g$ . As a result, the ball $B_{\alpha}(x')$ , which is the set of points that can move to $x'$ , must not intersect the decision boundary and must furthermore be strictly tangent to the decision boundary at $x$ . Given that the decision boundary is smooth at $x$ , this means that the normal to the curve at $x$ must run along the radius of $B_{\alpha}(x')$ from $x$ to $x'$ , implying that $x' = x \pm \alpha \hat{n}_x$ . Because the strategic movement is in the direction of a positive classification, the only possible option is for $x$ to move to $x' = x + \alpha \hat{n}_x$ . However, if $x' = x + \alpha \hat{n}_x$ is reachable by some point that is negatively classified by $h_{\Delta}$ , then the associated $h$ would not have classified $x'$ as positive, so $x$ would not have been a positively classified point by $h_{\Delta}$ . + +# B.6 Effective Class of Polytope Family + +Each polytope is simply a combination of $n$ linear faces. As such, when a polytope has a negative inside, each face will be mapped to a new linear face (with some wipeout of part or all of the face) leading to an $h_{\Delta}$ that is a smaller-radius polytope, albeit with potentially fewer faces and edges (see Fig. 7 (Left)). Although the faces of a positive-inside polytope also map to new linear faces, each vertex will have to expand to fill the gap between these new faces. As such, $h_{\Delta}$ is now a larger polytope whose corners are partial to a ball (see Fig. 7 (Right)). For the $\ell_1$ and $\ell_{\infty}$ costs, $h_{\Delta}$ therefore remains a polytope since the $\ell_1$ and $\ell_{\infty}$ balls are piecewise-linear themselves. For the $\ell_2$ cost, $h_{\Delta}$ is now a polytope with rounded corners. In either case, the decision boundary of each $p_{\Delta}^{k} \in P_{\Delta}^{k}$ can be seen as the intersection of a radius-scaled $p^k$ and $k$ balls centered at each vertex of $p^k$ . + +# B.7 Proof of Proposition 4 + +Proposition 4: Let $Q_{s}^{k}$ be the class of negative-inside $k$ -vertex polytopes bounded by a radius of $s$ , then $\mathrm{VC}(Q_{s}^{k}) \geq \mathrm{VC}(Q_{s,\Delta}^{k})$ . Moreover, if $s = \infty$ , then $\mathrm{VC}(Q_{s}^{k}) = \mathrm{VC}(Q_{s,\Delta}^{k})$ as implied by Thm. 2. + +Proof. As noted in Appendix B.6, negative-inside polytopes map to smaller-radius negative-inside polytopes with potentially fewer faces and vertices. As such $Q_{s,\Delta}^k \subset Q_s^k$ , so $\mathrm{VC}(Q_{s,\Delta}^k) \leq \mathrm{VC}(Q_s^k)$ . + +![](images/c7ebd988dd2010df103ab0ae011dda98f93b6d99f7f02d756d25498c4b2dc3c4.jpg) +Figure 8: Example $h$ from a class $H$ where $\mathrm{VC}(H) = \infty$ , but $\mathrm{VC}(H_{\Delta}) = 0$ . The gray dashed curve is the $h_{\Delta}$ of any $h$ like the one above. Each such $h$ has the same small gap where the decision boundary remains above the dotted purple curve. + +However, if $s = \infty$ , the class $Q_{s}^{k}$ is closed to input scaling leading to $\mathrm{VC}(Q_{s,\Delta}^{k}) \geq \mathrm{VC}(Q_{s}^{k})$ by Thm. 2, so we must have that $\mathrm{VC}(Q_{s,\Delta}^{k}) = \mathrm{VC}(Q_{s}^{k})$ . + +# B.8 Proof of Theorem 1 + +Theorem 1: There exist several model classes $H$ where $\mathrm{VC}(H) = \infty$ , but $\mathrm{VC}(H_{\Delta}) = 0$ . + +Proof. We provide two examples. For the first example, fix $\alpha < 1$ and consider the class of $\mathbb{R}^2$ classifiers $H = \{h_S : S \subseteq \mathbb{Z}^2\}$ where $h_S = \mathbb{1}\{x \notin B_{\alpha - \delta}(s) \forall s \in S\}$ . Each classifier in this model class contains small, non-overlapping negative regions centered at lattice points. Thus, the negative regions of all $h \in H$ will disappear under strategic movement, leaving $h_\Delta(x) = 1 \forall h \in H$ . Despite the fact that $H$ shatters $\mathbb{Z}^2$ (and therefore has $\mathrm{VC}(H) = \infty$ ), $H_\Delta$ cannot shatter a single point (and therefore has $\mathrm{VC}(H_\Delta) = 0$ ). + +Another example where $h_{\Delta}$ is perhaps more natural can be seen in Fig. 8. In this case, $H$ is the class of classifiers with narrow gaps whose decision boundaries can be written as + +$$ +h _ {i} \left(\left(x _ {1}, x _ {2}\right)\right) = \left\{ \begin{array}{l l} 1 & \left| x _ {1} \right| \geq k, \quad x _ {2} \geq 0 \\ 1 & \left| x _ {1} \right| < k, \quad x _ {2} \geq f _ {i} \left(x _ {1}\right) \\ 0 & e l s e \end{array} \right. \tag {4} +$$ + +where $k$ is some constant less than $\alpha$ and each function $f_{i}(x_{1})$ satisfies $f_{i}(x_{1}) \geq \sqrt{\alpha^{2} - x_{1}^{2}} + \sqrt{\alpha^{2} - k^{2}}$ for all $-k \leq x_{1} \leq k$ (i.e., $f_{i}(x)$ lies above the radius- $\alpha$ arc between $(-k, 0)$ and $(k, 0)$ ). In this case, the decision boundary points in the gap will get wiped out on all $h \in H$ , leaving the exact same $h_{\Delta}$ . Because $H$ is nearly unrestricted in the range $-k \leq x_{1} \leq k$ , it is capable of shattering infinite points that lie in this range and has infinite VC dimension. On the other hand, because $|H_{\Delta}| = 1$ , we get $\mathrm{VC}(H_{\Delta}) = 0$ . + +# B.9 Proof of Theorem 2 + +Definition 1. A hypothesis class $H$ is said to be closed to input scaling if for all $h(x) \in H$ and for all $a > 0$ , we have that $h_a(x) = h(ax) \in H$ as well. + +Note that the families of polynomials and polytopes are clearly closed to input scaling. Furthermore, most classes of neural networks are as well since $h(ax)$ can be realized by scaling the first weight vector by $a$ . + +Theorem 2: If $H$ is closed under input scaling and $\mathrm{VC}(H) < \infty$ , then $\mathrm{VC}(H) \leq \mathrm{VC}(H_{\Delta})$ . + +Proof. Intuitively, the reason that the VC dimension cannot decrease is that for any $H$ with a VC of dimension $k$ that is closed to input scaling, we can find a set $S$ of $k$ points that can be shattered by $H$ , where none of the points $s \in S$ strategically move under any of the $h \in H$ used to shatter $S$ . As a result, $H_{\Delta}$ will also shatter $S$ , so its VC dimension is at least $k$ . + +Let $\mathrm{VC}(H) = k$ and $G \subset H$ be a set of $2^k$ classifiers that shatters some set $S$ of $k$ points. Define $c_{DB}(s, g)$ as the minimum strategic cost necessary for point $s$ to obtain a positive classification from classifier $g$ and define $c_{\text{neg}}(S, G)$ as the minimum strategic cost for any $s \in S$ to obtain a positive classification from a classifier $g \in G$ that classifies $s$ negatively: + +$$ +c _ {n e g} (S, G) = \min _ {s \in S, g \in G, g (s) = 0} c _ {D B} (s, g) \tag {5} +$$ + +Since both $S$ and $G$ are finite sets and negative points face a strictly positive cost to obtain a positive classification, we get that there exists a unique value $c_{\text{neg}}(S, G) > 0$ . Additionally, because the VC dimension is indifferent to input scaling5, if we scale both the classifiers and the points, then $G' = \{g(ax) : g(x) \in G\}$ will shatter $S' = \{ax : x \in S\}$ with $c_{\text{neg}}(S', G') = a \cdot c_{\text{neg}}(S, G)$ . Note that because $H$ is closed to input scaling, $G' \subset H$ as well. Furthermore, if we choose $a = \frac{\alpha + 1}{c_{\text{neg}}(S, G)}$ , then $c_{\text{neg}}(S', G') = \alpha + 1$ . Note that if $c_{\text{neg}}(S', G') > \alpha$ , no point $s' \in S'$ will move strategically under any $g' \in G'$ , meaning that $g_{\Delta}'(s') = g'(s') \forall g' \in G$ , $s' \in S'$ . As a result, $G_{\Delta}'$ will also shatter $S'$ so $\mathrm{VC}(G_{\Delta}') \geq |S'| = k$ . Since $G_{\Delta}' \subset H_{\Delta}$ , we get that + +$$ +\operatorname {V C} \left(H _ {\Delta}\right) \geq \operatorname {V C} \left(G _ {\Delta} ^ {\prime}\right) \geq k = \operatorname {V C} (H) +$$ + +![](images/b0bd0b026b963d5c3f42eb069c8e9d7823bd14c81c7712461514c3e6df76cf57.jpg) + +# B.10 Proof of Theorem 3 + +Theorem 3: Let $H^{m,k}$ be a class of piecewise-linear classifiers with at most $m$ segments and $k$ intersections of linear segments. Then $\mathrm{VC}(H_{\Delta}^{m,k}) = O\big(d m \log(m) + \nu_p k \log(k)\big)$ , where $\nu_p$ is the VC of the class of $\ell_p$ norm balls, and is $\Theta(d)$ for $p = 1, 2, \infty$ . + +Proof. We will make use of the following known results: + +Lemma 2 (Gómez and Kaufmann (2021)). Let $H^{Ball}$ be the class of all $\ell_p$ balls in $R^d$ . Then $H^{Ball}$ has VC dimension $\mathrm{VC}(H^{Ball}) = \Theta(d)$ for $p = 1, 2$ , or $\infty$ [15]. + +Lemma 3 (Vaart and Wellner (2009)). Let classes $C_1, C_2, \ldots, C_k$ have VC dimensions $V_1, V_2, \ldots, V_k$ and $V = \sum_{i=1}^{k} V_i$ . Let $C_U = \bigsqcup_{j=1}^{m} C_j \equiv \{\bigcup_{j=1}^{m} h_j : h_j \in C_j\}$ , and let $C_I = \bigcap_{j=1}^{m} C_j \equiv \{\bigcap_{j=1}^{m} h_j : h_j \in C_j\}$ . Then both $C_U$ and $C_I$ have VC dimension upper bounded by $O(V \cdot \log(k))$ [34]. + +We note that by combining Lemmas 2 and 3, the class $H_{k}^{Ball}$ of the union of up to $k \ell_{p}$ balls has a VC dimension of $O\big(\nu_{p} k \log(k)\big)$ , which becomes $O\big(d k \log(k)\big)$ in the case of either the $\ell_{1}, \ell_{2}$ , or $\ell_{\infty}$ norm cost functions. Additionally, the class $H_{m}^{Lin}$ of piecewise linear models with up to $m$ segments has a VC dimension of $O\big(d m \log(m)\big)$ .6 + +Because linear classifiers map to new shifted linear classifiers, all we have to do to determine the effective classifier of a piecewise-linear $h$ is to combine (take the union of) the effective segments of each linear segment in $h$ and the potential $\ell_p$ ball artifacts around each intersection. Note that, as seen in Fig. 7 (Left), some of the effective segments may be wiped out in this process, and, as seen in Fig. 7 (Right), only intersections that open to the positive region will induce $\ell_p$ ball artifacts in $h_{\Delta}$ . Therefore, the $h_{\Delta}$ of a piecewise linear $h$ with $m$ segments and $k$ intersections will be the intersection of a piecewise linear classifier $g$ with up to $m$ segments and up to $k$ $\ell_p$ ball artifacts. As a result, we can write $H_{\Delta} \subseteq G \cap H_k^{Ball}$ , where $G \subseteq H_m^{Lin}$ . Therefore: + +$$ +\begin{array}{l} \operatorname {V C} \left(H _ {\Delta}\right) \leq \operatorname {V C} \left(G \cap H _ {k} ^ {\text {b a l l}}\right) \\ = O \left(\operatorname {V C} (G) + \operatorname {V C} \left(H _ {k} ^ {\text {b a l l}}\right)\right) \\ \leq O \left(\operatorname {V C} \left(H _ {m} ^ {L i n}\right) + \operatorname {V C} \left(H _ {k} ^ {b a l l}\right)\right) \\ = O \left(d m \log (m) + \nu_ {p} k \log (k)\right) \\ \end{array} +$$ + +Consequently, if $H$ is learnable, then $H_{\Delta}$ is too. + +![](images/62794193d2726fabd01d29a831f7560fbd5fd847e5e00f89f7d03db3569cd5e7.jpg) + +# B.11 Proof of Theorem 4 + +Theorem 4: Consider either the $\ell_1$ , $\ell_2$ , or $\ell_{\infty}$ costs and let $H \subseteq P^k$ , where $P^k$ is the set of all $k$ -vertex polytopes in $R^d$ and has $\mathrm{VC}(P^k) = O(d^2 k \log k)[23]$ . Then $\mathrm{VC}(H_{\Delta}) = O(d^2 k \log k)$ . + +Proof. As seen in Appendix B.6, the effective classifier of a k-vertex polytope is the intersection of a larger radius (or smaller for negative inside polytopes) k-vertex polytope and $k\ell_{p}$ balls centered at each of the vertices. Therefore, $H_{\Delta} \subset P^{k} \cap H_{k}^{ball}$ , so by Lemmas 2 and 3 (and the VC dimension of $H_{k}^{ball}$ derived in Appendix B.10): + +$$ +\begin{array}{l} \operatorname {V C} \left(H _ {\Delta}\right) \leq \operatorname {V C} \left(P ^ {k} \cap H _ {k} ^ {\text {b a l l}}\right) \\ = O \left(\operatorname {V C} \left(P ^ {k}\right) + \operatorname {V C} \left(H _ {k} ^ {\text {b a l l}}\right)\right) \\ = O \left(d ^ {2} k \log (k)\right) + O \left(d k \log (k)\right) \\ = O \left(d ^ {2} k \log (k)\right) \\ \end{array} +$$ + +![](images/c573c404ec99bee82002e4f39f4cdd04a355b1f74acc97d46f6109518a54059e.jpg) + +# B.12 Proof of Proposition 5 + +Proposition 5: For all $d$ , there exist (non-degenerate) distributions on $\mathbb{R}^d$ that are realizable in the standard setting, but whose maximum strategic accuracy is the majority class rate, $\max_y p(y)$ . + +Proof. Fix $d$ and choose any arbitrary set of positive points $\mathcal{X}_1$ . When $d = 1$ , the set of negative points $\mathcal{X}_0$ can be constructed by placing a negative point $\delta$ to the left and right of each positive point. When $d > 1$ , this generalizes to selecting $d + 1$ points around each positive point $x \in \mathcal{X}_1$ that form an $\mathbb{R}^d$ simplex $T_{\delta}^{d}(x)^7$ . In this case, the classifier $h(x) = \mathbb{1}\{x \in \mathcal{X}_1\}$ achieves perfect standard accuracy, and the classifier that classifies all points as negative will achieve a strategic accuracy of $\frac{d + 1}{d + 2}$ . Note that for each point $q$ in space to which $x \in \mathcal{X}_1$ can strategically move, there exists at least one $x' \in T_{\delta}^{d}(x)$ that can as well. Therefore, for each positive point $x \in \mathcal{X}_1$ that achieves a correct positive classification by strategically moving to $q$ ( $q$ can be the same as $x$ if it doesn't move), there is a unique negative point $x' \in T_{\delta}^{d}(x)$ that achieves an incorrect negative classification by strategically moving to $q$ , so the strategic accuracy does not benefit from correctly classifying any positive points. As a result, the maximum strategic accuracy is achieved by correctly classifying each negative point and incorrectly classifying each positive point, which gives the majority class rate. + +![](images/0561b967c62fa546b3dd73d5c48db044745cfbf449e40bd29e1fcca2aa1032dd.jpg) + +# B.13 Proof of Proposition 6 + +Proposition 6: There exists a distribution that is realizable in the standard setting, but whose accuracy under any $h_{\Delta}$ is at most $0.5 + \frac{1}{[2\alpha + 1]}$ . + +Proof. Consider the set of lattice points in $\mathbb{R}^1$ with alternating labels. Although $f(x) = \sin (\pi x)$ achieves perfect standard accuracy on this distribution, we will show that the maximum strategic accuracy for any model class is $0.5 + \frac{1}{[2\alpha + 1]}$ , which approaches 0.5 as $\alpha \to \infty$ . Note that in $R^1$ , the minimum length of a positive interval is $2\alpha$ . Additionally, for each positive interval $I$ of length $k$ , $I$ will contain either $\lfloor k \rfloor$ or $\lfloor k \rfloor + 1$ points and include at most one more positive point than negative points. Therefore, each interval $I$ will, at best, be correct on $\frac{\lfloor k \rfloor + 1}{2}$ out of $\lfloor k \rfloor$ points, which is optimized for $k = 2\alpha$ (i.e., the minimum possible value of $k$ ). Furthermore, because the negative intervals may also only have at most one more negative point than positive points, the negative intervals are optimized when they include only a single negative point. As such, an optimal strategic classifier $h$ consists of repeated positive intervals that include $\lfloor 2\alpha \rfloor$ points, followed by a negative interval that includes one negative point. For each positive and negative interval group, $h$ correctly classifies at most $\frac{\lfloor 2\alpha \rfloor + 1}{2} + 1$ out of $\lfloor 2\alpha + 1 \rfloor$ points correctly for an overall maximum accuracy of $\frac{\lfloor 2\alpha + 1 \rfloor + 1}{2} = 0.5 + \frac{1}{[2\alpha + 1]}$ . + +# C Additional results + +# C.1 Other Cost functions + +Our low-level analysis (Sec. 4) begins with a focus on the $\ell_2$ -norm cost, both because it is the most popular choice in the literature, and because we found it to offer the best intuition. Nonetheless, most of our results hold more broadly, and in particular, all theorems and propositions in Sec. 5 generalize to any $\ell_p$ ( $p \geq 1$ ) norm: + +- Prop. 4 applies generally, since negative-inside polytopes are still mapped to smaller negative-inside polytopes under general $\ell_p$ cost functions. +- Thm. 1, Cor. 1, and the example in Fig. 4 of increasing VC dimension remain under minor twitches to the constructions that take into account the differences in shapes of $\ell_p$ balls for different values of $p \geq 1$ . +- Thm. 2 holds for any cost function under which the maximum Euclidean distance any user can strategically move is bounded by a constant. This holds not only for $\ell_p$ norms, but also for most reasonable cost functions. +- Thm. 3 is already stated for general $\ell_p$ norms. +- Thm. 4 is stated for $\ell_1, \ell_2$ , and $\ell_{\infty}$ . However, a more general result can be stated with dependence on the VC dimension of the cost function balls (i.e., the set $C = \{c(x,x') \leq \alpha \forall x \in \mathbb{R}^d\}$ ). The bound would then be $O(d^{2} k \log(k) + \mathrm{VC}(C))$ , and would support any plug-in results for $\mathrm{VC}(C)$ . +- Prop. 5 and Prop. 6 also remain under minor tweaks to the constructions that take into account the differences in shapes of $\ell_p$ balls for different values of $p \geq 1$ . +- Obs. 3, 4, 5, and Cor. 2 hold for $\ell_p$ norms in general as well, since they are not shape dependent, but rather results of movement within a ball. + +Thus, our claims regarding the fact that non-learnable classes can become learnable, the loss of universality, and the limits to approximation all hold more generally. + +In terms of asymmetric norms such as the Mahalanobis norm, while these norms would complicate the curvature analysis (and subsequent impossibility results that build off them), all results in Sec. 5 would still hold. This is because while the shape of balls changes (like with other $\ell_p$ balls), fundamental properties like lines mapping to other lines do not. As such, the results in Sec. 5 would still hold for the exact same reasons as delineated above for $\ell_p$ balls. + +As for feature-dependent (also known as "instance-wise") costs, these are qualitatively different from the conventional global costs, in a way that can make the claims become irrelevant or even degenerate. For example, Sundaram et al. [33] show that on the one hand, even for linear $H$ , instance-wise cost function can cause $\mathrm{VC}(H_{\Delta}) = \infty$ , and on the other hand, for separable costs (which are instance-wise) it holds that $\mathrm{VC}(H_{\Delta}) \leq 2$ for any $H$ . + +![](images/bd27d76d1f4191fead64fb228ce9bc9d61bf5d4126468e5dbe7b4abecd8294a0.jpg) +Figure 9: (Left): Partial expansion. (Right): Regions merging/splitting + +![](images/064a7345c95c68a9ae601c4d2523da57d389c7bdd39ce24a51ea2883c89c55fb.jpg) +Decision Boundary + +![](images/28e7039cb5f48089a4d8e9c8fb7689cf5d7a829cd73db9618e22dd9508c45418.jpg) +Effective Boundary + +# C.2 Partial Expansion + +As seen in Fig. 9 (Left), partial expansion may occur if part of the expansion artifact is discarded through indirect wipeout. In the figure, part of the artifact (gold) is within $\alpha$ of the upper positive region and therefore does not become part of the decision boundary. + +# C.3 Merging/Splitting Regions + +As seen in Fig. 9 (Right), two positive regions can merge together when they are separated by a narrow pass in $h$ . If the narrow pass is part of a larger negative region, the region will be split into two, which can potentially increase the complexity of the decision boundary (for example, in Fig. 9 (Left)). + +# C.4 Types of Impossible Effective Classifiers + +As stated in Sec. 4.4, we identify four "types" of classifiers $g$ that cannot exist as effective classifiers. Formally, these classifiers are ones which include either: + +1. A small continuous region of positively classified points $C$ where $\exists q \in R^d$ such that $x \in B_{\alpha}(q) \forall x \in C$ . +2. A narrow continuous region $C$ where all points are classified as positive and are $\alpha$ -close to the decision boundary. +3. A smooth point on the decision boundary which has signed curvature $\kappa > 1 / \alpha$ (or $\kappa^{*} > 1 / \alpha$ when $d > 2$ ). +4. A locally convex intersection of piecewise linear or hyperlinear segments. + +Note that the first case is just a special case of the second case. Nonetheless, we have separated them to emphasize that the impossible positive regions can either be fully or partially contained. + +# Cases 1 and 2: Small/Narrow Positive Regions + +Let $x \in C$ be some point in a small positive region in $g$ . For $x$ to be classified as positive, there must be a point $x' \in B_{\alpha}(x)$ which is classified positively by $h$ and not reachable by any $q \in \mathcal{X}_0(h)$ (see requirements 1 and 2 in Appendix B.4). However, any point $x' \in B_{\alpha}(x) \cap C$ is reachable by $\mathcal{X}_0(h)$ and any point $x' \in B_{\alpha}(x) \cap C^c$ is also reachable by $\mathcal{X}_0(h)$ since it is closer to some $q \in \mathcal{X}_0(h)$ than to $x$ . Therefore, there are no points $x' \in B_{\alpha}(x)$ which are not reachable by any $q \in \mathcal{X}_0(h)$ , so $g$ cannot be an effective classifier. + +# Case 3: Large positive curvature + +We will show that any $g$ with directional curvature greater than $1 / \alpha$ anywhere on the decision boundary will fail Prop. 3. In other words, if the directional curvature at point $x$ is greater than $1 / \alpha$ , then the point $x + \alpha \hat{n}_x$ will be reachable by some point that is negatively classified by $g$ . In order for the point $x + \alpha \hat{n}_x$ to be reachable by $x$ but not any point negatively classified by $g$ , the + +![](images/10e1314c1657b3863d148a72f399e4bd066906611dee51242ffd60fe81169b09.jpg) +Figure 10: Diagram for proof of case 4. + +ball $B_{\alpha}(x + \alpha \hat{n}_x)$ — which is the set of points that can strategically move to $x + \alpha \hat{n}_x$ — must not intersect $g$ and must also be strictly tangent to it at $x$ . However, if the curvature at point $x$ is greater than $1 / \alpha$ , then the maximum radius $r_{insc}$ of a ball that is strictly tangent to $x$ is less than $\alpha$ , so $g$ does not meet Prop. 3. + +In $\mathbb{R}^2$ , Lemma 1 proves that the radius $r_{insc}$ of the maximum inscribed circle at $x$ is no bigger than the radius $r_{osc} = \frac{1}{\kappa}$ of the osculating circle at $x$ . We therefore have that $r_{insc} \leq r_{osc} = \frac{1}{\kappa} < \alpha$ . In higher dimensions (see Appendix A.1), there is no longer a single normal curvature. Instead, we define the set $K$ of curvatures in each plane $p \in P$ defined by the normal vector and some tangent vector at $x$ . Note that $B_{\alpha}(x + \alpha \hat{n}_x)$ is only strictly tangent to the decision boundary at $x$ if it's intersection with each $p \in P$ (a circle) is strictly tangent at $x$ to the curve obtained by intersecting the decision boundary with the plane $p$ . Since each of these cases are in $\mathbb{R}^2$ , each one will only hold if the curvature at $x$ in $p$ is no more than $1 / \alpha$ . Therefore, if $\kappa^{*} = \sup(K) > 1 / \alpha$ for any $x$ on the decision boundary, then the maximum radius of a ball that is strictly tangent to $x$ is less than $\alpha$ , so $g$ does not meet Prop. 3. + +# Case 4: Piecewise Linear Segments + +Lemma 4. The set of points that are reachable by point $x$ but not point $x + \delta \vec{v}$ for unit vector $\vec{v}$ and $\delta \rightarrow 0$ all satisfy $\vec{v} \cdot x \leq 0$ . + +Proof (Lemma 4). Define an orthonormal basis $v_{1}, v_{2} \ldots v_{d}$ with $v_{1} = v$ . For any point $q$ such that $\vec{v} \cdot q > 0$ , the vector connecting $x$ and $q$ can be written as $a_{1}v_{1} + a_{2}v_{2} \ldots + a_{d}v_{d}$ while the vector connecting $x + \delta \vec{v}$ and $q$ can be written as $(a_{1} - \delta)v_{1} + a_{2}v_{2} \ldots + a_{d}v_{d}$ . Since $0 < \delta < a_{1}$ (note that $\vec{v} \cdot q > 0$ so $a_{1} > 0$ ), $q$ is closer to $x + \delta \vec{v}$ than $x$ . Therefore, any point that is reachable by $x$ and not by $x + \delta \vec{v}$ must satisfy $\vec{v} \cdot x \leq 0$ . + +Proof (Case 4). As depicted in Fig. 10, let $L_{1}$ and $L_{2}$ be two $R^{d}$ hyperplane segments that intersect on the decision boundary of $g$ at the $R^{d-1}$ hyperplane segment $I$ , and let $x$ be some point on $I$ . Let $0 < \theta < \pi$ be the angle between $L_{1}$ and $L_{2}$ on the positive side. Let $\vec{v}_{1}$ be the unit vector in $L_{1}$ that runs through $x$ and is orthogonal to $I$ ; $\vec{v}_{2}$ be the unit vector in $L_{2}$ that runs through $x$ and is orthogonal to $I$ ; and $\vec{v}_{C} = -\frac{\vec{v}_{1} + \vec{v}_{2}}{||\vec{v}_{1} + \vec{v}_{2}||}$ . Note that for $\delta \to 0$ the points $x_{1} = x + \delta \vec{v}_{1}$ and $x_{2} = x + \delta \vec{v}_{2}$ are on the decision boundary while the point $x_{12} = x + \delta \vec{v}_{C}$ is classified as positive and $x_{C} = x - \delta \vec{v}_{C}$ is classified as negative. Additionally, if we define $\vec{v}_{A}$ as the result of the rotation $0 < \epsilon < \frac{\pi - \theta}{2}$ around $I$ away from the positive region and $\vec{v}_{B}$ as the result of the rotation $\epsilon$ around $I$ away from the positive region, then both $x_{A} = x + \delta \vec{v}_{A}$ and $x_{B} = x + \delta \vec{v}_{B}$ are classified as negative. + +Assume that there exists a classifier $h$ associated with the effective classifier $g$ . Therefore, there exists a point $q$ that is reachable by $x$ but not by $x_A$ , $x_B$ , or $x_C$ (which are classified as negative by $g$ ). By Lemma 4, any point $q$ that is reachable under strategic movement by point $x$ but not $x_A$ , $x_B$ , or $x_C$ must at least satisfy $q \cdot \vec{v}_A \leq 0$ , $q \cdot \vec{v}_B \leq 0$ , and $q \cdot \vec{v}_C \leq 0$ . However, because $\vec{v}_A + \vec{v}_B$ points in the same direction as $\vec{v}_1 + \vec{v}_2$ — and therefore the opposite direction of $\vec{v}_C$ — we must have that $q \cdot \vec{v}_C = 0$ and $q \cdot (\vec{v}_A + \vec{v}_B) = 0$ , which implies that both $q \cdot \vec{v}_A = 0$ and $q \cdot \vec{v}_B = 0$ . However, this implies that the vector $\vec{q}$ is orthogonal to both $\vec{v}_A$ and $\vec{v}_B$ , which can only happen if $q$ lies on $I$ or if the angle between $\vec{v}_A$ and $\vec{v}_B$ is either 0 or $\pi$ . In the former case, $q$ is then reachable + +![](images/3a9c32b3912f79950c4bf1f8fd532fc2e805b0817f0845f7c7db4efb38200283.jpg) +Figure 11: Example of VC decreasing from 2 to 1. Each color represents a different classifier (drawn together to illustrate shattering capacity), and the dashed curves represent the corresponding effective classifiers. Though the black points can be shattered by $H$ , none of the $h_{\Delta}$ overlap so $\mathrm{VC}(H_{\Delta}) = 1$ . + +by the negatively classified point that is just over the decision boundary from it. The latter case is also not an option because the angle between $\vec{v}_A$ and $\vec{v}_B$ is $\theta_{AB} = \theta + 2\epsilon$ , and we have ensured that $0 < \theta + 2\epsilon < \theta + 2\frac{\pi - \theta}{2} = \pi$ . In either case, it is impossible for $g$ to classify $x$ as positive while classifying $x_A, x_B$ , and $x_C$ as negative, so there is no $h$ such that $g = h_{\Delta}$ . + +# C.5 Non-bijectiveness of Classifier Mapping + +In Sec. 4.4, we show that the mapping $h \mapsto h_{\Delta}$ is not surjective by showing that there are many potential $h_{\Delta}$ with no associated $h$ . Here, we will show that $h \mapsto h_{\Delta}$ is also not injective since many classifiers map to the same effective classifier. Furthermore, we show that there are actually infinite $h$ that map to nearly all possible $h_{\Delta}$ . Though there are many more interesting examples, the simplest case of non-injectiveness can be seen by considering an $h$ that includes a continuous positive region $C$ with an infinite number of points. Define the classifier $h^x$ as the classifier that is identical to $h$ except that $h^x$ classifies $x \in C$ as negative instead of positive. Because the point $x$ will achieve a positive classification through strategic movement, $h_{\Delta}^x$ is unaffected by $x$ and is the same as $h_{\Delta}$ . Moreover, $h_{\Delta}^x = h_{\Delta}$ is true of any $x \in C$ , so as long as any $h$ that maps to $h_{\Delta}$ contains a positive region with infinite points, then $h_{\Delta}$ will have infinite $h$ that all map to it. Because positive points in $h$ map to positive radius- $\alpha$ balls in $h_{\Delta}$ , any $h_{\Delta}$ with a positive region $C$ such that $\exists x: B_{\alpha}(x) \subset C$ will have infinite $h$ that all map to it. Note that since all positive regions in $h_{\Delta}$ have a minimum size requirement $\exists x: B_{\alpha}(x) \subseteq C$ , the aforementioned requirement covers all but the $h_{\Delta}$ with absolute minimum-size positive regions. + +# C.6 Classes with $H = H_{\Delta}$ + +Though $H \neq H_{\Delta}$ for most classes $H$ , there are a few rather simple $H$ other than the class of linear functions where $H = H_{\Delta}$ . For example, under any $\ell_p$ norm cost, an $\ell_p$ ball with a positive inside will grow by a radius of $\alpha$ , while an $\ell_p$ ball with a negative inside will shrink by a radius of $\alpha$ (at least until the radius hits 0). Therefore, many model classes of the subsets of the $\ell_p$ balls — including the class of all negative-inside balls and the class of balls with a radius that is a multiple of $\alpha$ — all have $H = H_{\Delta}$ for any fixed $\alpha$ . + +Additionally, since each line segment and $\ell_p$ arc maps to another line segment or $\ell_p$ arc, many piecewise-linear or piecewise $\ell_p$ arc classes will also have $H = H_{\Delta}$ . For example, any concave linear intersection will be mapped to a shifted concave linear intersection, so the class of concave linear intersections satisfies $H = H_{\Delta}$ . + +# C.7 Classes with $\mathrm{VC}(H) > \mathrm{VC}(H_{\Delta}) > 0$ + +There are a couple of mechanisms that can lead to a decrease in VC dimension. In the first mechanism, multiple different $h$ have the same $h_{\Delta}$ leading to a smaller and less expressive $H_{\Delta}$ . Two such cases are shown in Appendix B.8 to prove that the VC may decrease even from $\infty$ to 0. In a similar fashion, + +we can build an $H$ where only some (instead of all) of the $h \in H$ have the same $h_{\Delta}$ , which would decrease the VC dimension to a non-zero effective VC dimension. + +In the second mechanism, the VC can change due to effective classifiers gaining or losing overlap from strategic movement. As the dual to Fig. 4 (Left) where the VC increases from 1 to 2 due to this mechanism, Fig. 11 presents an example where the VC decreases from 2 to 1. Here, $H$ includes four overlapping classifiers with $\mathrm{VC}(H) = 2$ . Yet, under strategic behavior, the effective classifiers cease to overlap, decreasing the shattering capacity to 1. + +# C.8 Example of $H_{\Delta}$ with $\operatorname{VC}(H_{\Delta}) = \Theta(d \cdot (\operatorname{VC}(H))$ + +Claim: There exists a class $H$ with $\operatorname{VC}(H_{\Delta}) = \Theta(d \cdot (\operatorname{VC}(H))$ . + +Proof. For simplification of notation, we build on the case shown in Fig. 4 (Left) using classifiers $h$ that classify some ball as positive and all other points as negative. Note that a similar proof can be built based on a class $H$ of point classifiers [7], but we have decided to include the following construction since classifiers that classify only a single point as positive are quite unrealistic. Let set $S = \mathbb{Z}_2^d - \{0^d\} + \{-\delta^d\}$ be the set of $\mathbb{Z}_2^d$ lattice points with the point $\{0^d\}$ replaced by the point $\{-\delta^d\}$ . Consider the case where $\alpha = 0.75$ and $H = \{B_{0.25}(x) : x \in S\}$ is the class of radius-0.25 balls around the points in $S$ . Because the radius of positive balls expand by $\alpha$ under strategic movement, we get that $H_{\Delta} = \{B_1(x) : x \in S\}$ . Note that because none of the balls in $H$ overlap, $\mathrm{VC}(H) = 1$ . On the other hand, the balls in $H_{\Delta}$ do overlap, allowing $H_{\Delta}$ to shatter the set of $d$ one-hot points $E = \{e_i : i \in [1\dots d]\}$ , where $e_i$ denotes the point with a 1 in the $i$ th coordinate and 0's elsewhere. Note that for each $x \in S - \{-\delta^d\}$ , the ball $B_1(x) \in H_{\Delta}$ classifies $E$ according to the sign pattern $x$ . Additionally, the ball $B_1(-\delta^d)$ does not classify any point in $E$ as positive, so $H_{\Delta}$ covers all $2^d$ sign patterns of $E$ . As such, $\mathrm{VC}(H_{\Delta}) \geq d$ . However, since $H_{\Delta} \subset B = \{B_1(x) : x \in R^d\}$ , we get that $\mathrm{VC}(H_{\Delta}) \leq \mathrm{VC}(B) = d + 1$ [15] giving $\mathrm{VC}(H_{\Delta}) = \Theta(d \cdot (VC(H)))$ . + +# C.9 Example of Strategic Accuracy Exceeding Standard Accuracy + +Claim: In the unrealizable setting, the maximum strategic accuracy can exceed the maximum standard accuracy. + +Proof. In the unrealizable setting, the maximum strategic accuracy can exceed the maximum standard accuracy when strategic movement helps correct the incorrect classification of positive points. For example, consider a basic setup where $\alpha = 1$ , $X = \{(-1,0), (2,0), (-3,0)\}$ , $Y = \{1,1,-1\}$ , and $H = \{h_1(x) = sign(x_1 + x_2), h_2(x) = sign(x_1 - x_2 - 1)\}$ . In the standard case, both $h_1$ and $h_2$ correctly classify the second and third points, but not the first point, so the maximum accuracy is $\frac{2}{3}$ . However, in the strategic setting, the first point is able to obtain a positive classification under $h_1$ by moving to $(- \frac{1}{3}, \frac{2}{3})$ (and the third point still cannot get a positive prediction), so the maximum strategic accuracy is 1. + +# C.10 Practical Takeaways + +Given the challenge of optimization in strategic batch learning (even in simpler settings), our paper aims to first establish an understanding of the challenges inherent to non-linear strategic learning (in particular in relation to the linear case), with the hope that our results and conclusions can help guide the future design of learning algorithms, as well as set expectations for what is achievable and what is not. The difficulty of designing a principled algorithm for the general non-linear case is underscored by the fact that existing works rely on strong assumptions to enable optimization. For example, the original paper of Hardt et al. [16] provides an algorithm, but requires assumptions on the cost function that reduce the problem to a one-dimensional learning task, and their algorithm is essentially a line search. Levanon and Rosenfeld [25] differentiate through $\Delta$ , but this relies strongly on this operation being a linear projection, which applies only to linear classifiers. Neither of the above naturally extend to the non-linear case. + +Towards the goal of practical learning in the general strategic setting, we provide here some examples of how our results can be used as practical takeaways for effective learning: + +1. Learning via inversion: One implication of our function-level analysis is that for any classifier $h$ with strategic accuracy $\mathrm{acc}_{strat}(h)$ , the classifier $h'$ such that $h \mapsto h_{\Delta} = h'$ has regular (i.e., non-strategic) accuracy $\mathrm{acc}(h') = \mathrm{acc}_{strat}(h)$ . Thus, one approach to optimizing strategic accuracy is via reduction: (i) train $h' \in H'$ to maximize non-strategic accuracy using any conventional approach for some choice of $H'$ , and then (ii) apply the inverse function mapping (which may not be 1-to-1) to obtain a non-strategic classifier $h$ , whose effective decision boundary is that of $h'$ . This approach requires the ability to solve the 'inverse' problem (which may not be 1-to-1) to find the effective classifier. We discuss cases where this would and wouldn't work in Sec. 4.4. +2. Regularizing curvature: A sufficient condition for a pre-image strategic $h'$ to exist for a given non-strategic $h$ is that the function mapping is 1-to-1. Rather than restrict learning to only invertible classes $H'$ a priori, an alternative approach is to work with general $H'$ but regularize against those $h' \in H'$ that do not have an inverse. Prop. 1 implies that one way to promote this is by encouraging $h'$ that are smooth, since low-curvature classifiers are less prone to direct wipeout, and therefore more likely to permit inversion. +3. Finding the interpolation threshold: The interpolation threshold — the point in which the number of model parameters (or model complexity more generally) attains minimal training error — is key for learning in practice: it marks the extreme point of overfitting (in the classic underparametrized regime) and the beginning of benign overfitting (in the modern overparametrized regime, e.g., of deep neural nets). In non-strategic learning, this point is easily identified as that where the training error is zero. In contrast, our results show that in strategic learning, the minimal training error can be strictly positive. This makes it unclear when (and if) the threshold has been reached. Fortunately, our procedure for our approximation experiment in Sec. 6.2 can be used generally to compute the minimal attainable strategic training error, independently of the chosen model class. This provides a tool that can be used in practice to measure interpolation. + +In terms of model class selection, our results in Sec. 5 offer insight into potential upsides and downsides to certain classes. One implication of our results is that strategic behavior can generally either increase or decrease the complexity of the chosen model class. However, Thm. 2 ensures that the VC dimension can only increase for any class that is closed to input scaling, which includes many common classes used in practice. Thus, a learner who chooses such classes is guaranteed that the potential expressivity of the learned classifier will not be deteriorated by strategic behavior. Additionally, Prop. 4, Thm. 4, and Thm. 3 all give upper bounds on the SVC of particular classes (e.g., ReLU neural networks). The implication is that a learner choosing to work with such $H$ is guaranteed that if $H$ is learnable, then the induced $H_{\Delta}$ is also learnable. Finally, Obs. 5 and Thm. 1 both point to the fact that several non-strategic classifiers $h$ can be mapped to the exact same effective classifier $h_{\Delta}$ . If a class $H$ includes many such cases, then it has significant redundancy, which can have negative implications on the process of choosing a good $h \in H$ via learning (e.g., in terms of generalization, optimization, or the effectiveness of proxy losses). Ideally, the learner should be cautious of classes in which this is prevalent, for example, classes that permit high curvature or small negative areas. + +# D Experimental details + +# D.1 Experiment #1: Expressivity + +In Sec. 6.1, we experimentally demonstrate how the strategic setting affects classifier expressivity. In this experiment, we first randomly sample a degree- $k$ polynomial $h$ , then determine the set of points $S$ that make up the decision boundary of $h_{\Delta}$ (i.e., are directly adjacent to it), and finally find the best approximate polynomial fit of $S^8$ . Because user features are often naturally bounded, we assume that $x \in \mathcal{X} = \mathbb{R}^2$ and that all $x$ are bounded by $|x|_{\infty} \leq 100$ (see Appendix A.2 for a discussion on how our theoretical results hold under bounded input domains). We report average values and + +standard error confidence intervals of 100 instances of the setup for each $\alpha \in \{2,4,8,16,24,32,40\}$ and $k \in [1,10]$ . The instances were divided among 100 CPUs to speed up computation. + +To randomly sample polynomials of degree $k$ , we leverage the fact that the number of points needed to uniquely determine a degree- $k$ polynomial over $\mathbb{R}^2$ is $\binom{k+2}{2}$ . Therefore, for each instance, we randomly choose $\binom{k+2}{2}$ points and labels, and use an SVM polynomial fitter to find the best fitting degree- $k$ polynomial fit to these points. To ensure that the generated polynomial is indeed a full-degree polynomial, we construct a grid $G$ of points over $[-100, 100]^2$ , label each point $g \in G$ using $h$ to get labels $Y$ , and verify that the SVM best-fitting degree- $k$ polynomial achieves near-perfect accuracy on $(G, Y)$ , while the best $(k-1)$ -degree polynomial achieves low accuracy on $(G, Y)$ . + +To find the set of points $S$ that make up the decision boundary of $h_{\Delta}$ , we first strategically label each $g \in G$ by checking if $g$ can reach any point $g'$ that is classified as positive by $h$ . To determine the points that make up the decision boundary of $h_{\Delta}$ , we select only the points $g \in G$ where $g$ and at least one of its direct neighbors are labeled differently by $h_{\Delta}$ . + +Finally, we again use an SVM polynomial fit to determine the best approximate polynomial degree of $h_{\Delta}$ . Because the $h_{\Delta}$ of a polynomial is not necessarily a polynomial itself, we set $k'$ to be the lowest degree of the polynomial whose fit on $S$ passes a set tolerance threshold. Empirical tests showed that a tolerance of 0.9 was high enough to ensure a good fit on all of $G$ , but not so high that the SVM fit algorithm needlessly increased the degree just to get a complete overfit to $S$ . + +# D.2 Experiment #2: Approximation + +In Sec. 6.2, we experimentally demonstrate the effects of non-universality on the maximum strategic accuracy. This experiment consists of (i) sampling synthetic data and (ii) calculating the maximum linear accuracy and strategic accuracy on each dataset instance. We report average values and standard error confidence intervals of 20 instances of the setup for each $\alpha \in \{0.5,1,2\}$ and $\mu \in [0,5]$ . The instances were divided among 100 CPUs to speed up computation. + +For each instance, we generate synthetic $\mathbb{R}^2$ data by sampling points from class-conditional Gaussians $x\sim N(y\mu ,1)$ , where $\mu$ is the separation between the centers of each class. Because we were unable to find a polynomial-time algorithm to calculate the maximum accuracy of $H_{\Delta}^{*}$ , exactly 25 points were drawn from each class (denote the full dataset $\mathcal{X}$ ). Though this setup may not reflect all possible data distributions, it does represent how the strategic environment behaves under increasing data separability. + +To calculate the maximum linear accuracy in each instance, we first note that there exists an optimal linear classifier that runs through (or infinitesimally close to for negative points) exactly two dataset points. This is because for any optimal classifier $h$ that runs through exactly one dataset point $x \in \mathcal{X}$ , $h$ may be rotated until it runs through another $x' \in \mathcal{X}$ without changing any of the labels. For any optimal classifier $h$ that runs through no $x \in \mathcal{X}$ , $h$ can be shifted until it runs through one (or two collinear) $x \in \mathcal{X}$ and then rotated appropriately. Because all data points are random, we assume that no three points are collinear. Therefore, the maximum linear accuracy on $\mathcal{X}$ can be calculated by taking the maximum accuracy over the $\binom{n}{2}$ classifiers that run through each pair of $x, x' \in \mathcal{X}^9$ . + +When calculating the maximum strategic accuracy, we note that the minimum size of a positive region in $h_{\Delta}$ is the ball $B_{\alpha}$ , so any strategic classifier can be represented by the intersection of multiple $B_{\alpha}$ . We therefore calculate the set $S$ of all possible subsets $s \subseteq \mathcal{X}$ that any $B_{\alpha}$ classifies as positive, and find the intersection $\bigcap_{s \in S' \subseteq S} s$ with the highest accuracy. To find all $s \in S$ , we iterate through the circles centered at each point on a fine-grained grid over the range of the dataset (padded by $\alpha$ on all sides) and check which points $x \in \mathcal{X}$ each circle includes. To find the best intersection of these circles, we iterate through each of the subsets of $Z \subseteq \mathcal{X}_0^{10}$ and then $s \in S$ to find the maximum number of positive points that can be classified as positive if we allow ourselves to mistakenly classify the negative points in $Z$ . + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: All claims in the abstract and introduction are based on the results in the main body of the paper and put into perspective based on the most recent results in the field. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: Results that are based on strong assumptions are clearly noted and extensions to more general cases are discussed. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). 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In our model we assume that the votes are fixed and known and the proposers want to set as high project prices as possible, provided that their projects get selected and the prices are not below the minimum costs of their delivery. We study the existence of pure Nash equilibria (NE) in such games, focusing on the AV/Cost, Phragmén, and Method of Equal Shares rules. We also provide an experimental study of cost selection on real-life PB election data. + +# 1 Introduction + +Consider a city that implements participatory budgeting (PB) [Cabannes, 2004; Goel et al., 2019; Rey and Maly, 2023]. The city council allocates a fixed amount of funds for the initiative and invites citizens to submit project proposals, for which the funding decisions will be made through voting. One natural question that the proposers face is how much money should they realistically request. It is tempting to propose expensive projects, as higher costs are typically associated with better quality and broader scope, but overly expensive projects risk rejection due to a limited budget. + +Indeed, evidence suggests that the costs of the projects are often set strategically. For example, in $10\%$ of participatory budgeting elections held in the city of Wroclaw, Poland, all project proposals had costs equal to the maximum allowed amount (see the Pabulib database [Faliszewski et al., 2023]) in the corresponding instance. Moreover, in $85\%$ of these elections, more than half of the proposals were priced at the maximum value. A similar phenomenon, if less pronounced, also occurs in other cities. For instance, in $25\%$ of Pabulib PB instances approximately half of the projects have costs reaching at least $90\%$ of the maximum allowed value in the corresponding instance. In another $50\%$ of these instances, more than $30\%$ of the projects are priced at or above $50\%$ of the + +allowed maximum. This pattern is unlikely to be coincidental and it seems implausible that the implementation costs of different projects would naturally converge to such similar values. Instead, this suggests that projects are prepared strategically, with costs deliberately set as high as possible within the given constraints and perceived chances of selection. + +Cost-selection strategies are likely influenced by the voting rules used. If the rule prioritizes the number of votes a project receives, proposers might feel encouraged to submit more expensive proposals. This happens, e.g., if the city uses the BasicAV rule (also known as GreedyAV [Boehmer et al., 2023] or GreedCost [Rey and Maly, 2023]) which greedily selects those projects that are approved by the largest numbers of voters (provided they still fit in the remaining budget). Conversely, if the city were to use a proportional rule, such as the Method of Equal Shares (MES) [Peters and Skowron, 2020; Peters et al., 2021] or Phragmén [Brill et al., 2017; Los et al., 2022], then each proposer would have to analyze how the other ones might act and choose the costs accordingly. + +For example, in recent instances of participatory budgeting in Wieliczka—where the BasicAV rule was used and the maximum project cost was set to half of the budget—exactly two projects were winners, each with the highest allowed cost. However, when in 2023 Wieliczka used MES in their experimental "Green Million" PB election, there were 30 winning projects, with costs distributed much more uniformly over the range of the allowed values (the maximum allowed project cost was $10\%$ of the budget, so if the proposers acted as previously, there would be 10 winning projects; for details of the election, see https://equalshares.net/elections/zielony-milion/). While this is anecdotal evidence, it shows that voting rules do affect the structure of PB elections. + +Our goal is to analyze this game-theoretic nature of project cost selection under various participatory budgeting rules. Studying strategic cost selection is also a natural direction in market analysis and is present, e.g., in the context of the Hotelling-Downs model; see the overview of Eiselt [2011]. Notably, our work is different from that of Aziz et al. [2023], who consider how voters pool their funds; see also the work of Wagner and Meir [2023]. + +The Game. We analyze the following scenario. The sets of projects and voters are fixed and each voter indicates which projects he or she approves (this information is common knowledge, an assumption we discuss later on). Each project is controlled by a different proposer choosing its cost so that it is as high as possible, while ensuring that the project is selected. However, each project also has a certain delivery cost—i.e., the lowest cost under which it can be reasonably implemented—and the proposers prefer costs that are at least as high. Note that we view the notion of a project broadly: Choosing a lower cost means proposing a smaller-scale project, and choosing a higher cost means submitting a more comprehensive one (but addressing the same general need). Importantly, whether a voter approves a project or not, does not depend on its cost; this strategy of the voters is often called threshold approval voting [Benadè et al., 2020; Fairstein et al., 2023]. The projects are chosen according to a given rule, such as BasicAV, AV/Cost, Phragmén, or MES (see Section 2 for details). We analyze whether thus defined games—where the project proposers are the players—have pure Nash equilibria and, if so, what costs are reported under these equilibria. + +While our games present a simplified view of reality due to abstracting away from excessive complexities, they lead to identifying certain phenomena and explaining behavior that we see in data. Consequently, our model lays the foundations and sets expectations for more sophisticated scenarios. Natural extensions will become apparent after we explain our assumptions and their limitations: + +Votes are common knowledge. As we use classic Nash equilibrium analysis, we need the ability to evaluate players' utilities for all possible strategy profiles, which essentially means that the votes are common knowledge. While this is a standard assumption in the analysis of Nash equilibria, in practice, neither the precise identities of the voters nor their votes are known to project proposers when submitting their projects. Yet, proposers often estimate potential support, e.g., using insights from previous PB editions. Hence, despite its seemingly strong assumptions, game theory offers valuable insight into strategic aspects of cost selection. + +Project costs do not affect approvals. We assume that voters' approvals do not depend on project costs. Although it is a simplification, it is supported by several observations. First, data shows that costly projects are frequent, suggesting a common belief that high prices do not deter voters. Second, approving projects based on their quality and ignoring the costs is often suggested as a superior strategy for utility-oriented voters [Benadè et al., 2020; Fairstein et al., 2023]. + +Third, there is evidence that taking project costs into account, e.g., performing the value-form-money analysis, is cognitively expensive [Fairstein et al., 2023]. + +While relaxing our assumptions would give a more general class of games, we start with a clean, simple model, in order to obtain a baseline. That said, we complement our discussion by allowing the voters to have individual ranges of costs under which they approve each project, and we argue why we believe that such a model leads to less useful conclusions (see Section 6). + +Our Contributions. We study the existence of pure Nash equilibria in our games, under various PB rules and a number of restrictions on their structure. Importantly, we find that MES with cost utilities always admits NEs with the costs of the projects bounded proportionally to their expected support, whereas BasicAV has NEs where a single project takes the whole budget. In contrast, AV/Cost, Phragmén, and MES with approval utilities may fail to have any NEs at all. The existence of an NE for a rule suggests that it might naturally incentivize project proposers to gravitate towards it, whereas the non-existence indicates increased challenges in reasoning about the costs, leading to less predictable behaviour. + +Further, our theoretical tools enable the experimental analysis of NEs in real-life scenarios. We conduct such an analysis based on the Pabulib dataset [Faliszewski et al., 2023]. We find that Phragmén and MES based on approval utilities are very different from MES with cost utilities. The former two rules promote cheaper projects, and the latter selects fewer and more expensive ones. + +Our model aligns well with the patterns observed in real elections, such as those in Wroclaw and Wieliczka. Similarly, it provides insight into why most cities using BasicAV impose upper bounds on allowed project costs. Interestingly, the city of Świecie, Poland that transitioned from BasicAV to MES in 2023, dropped such upper bounds. This shows that the city officials have an intuitive understanding of the strategic incentives arising under different voting rules. + +# 2 Preliminaries + +Participatory Budgeting. We define a $PB$ instance (or, a $PB$ election) as a tuple $E = (P, V, B, cost)$ , where $P = \{p_1, \ldots, p_m\}$ is a set of projects, $V = \{v_1, \ldots, v_n\}$ is a set of voters, $B \in \mathbb{R}_+$ is the available budget, and cost: $P \to \mathbb{R}_+$ is a function specifying the cost of each project. Each voter $v_i$ casts a nonempty approval ballot $A(v_i) \subseteq P$ , containing the set of projects that he or she approves (see, e.g., the work of Fairstein et al. [2023] for a discussion of other ballot formats). Note that a voter may approve projects whose total cost exceeds the available budget. We often refer to the voters and their approval ballots as either an approval profile or a preference profile. We extend the $A(\cdot)$ notation so that for a project $p_i$ , $A(p_i)$ is the set of voters that approve $p_i$ . Then, $|A(p_i)|$ is the approval score of $p_i$ . For each project $p_i$ , we assume that $|A(p_i)| \geq 1$ . Given a subset of projects $P'$ , we let $\text{cost}(P') = \sum_{p' \in P'} \text{cost}(p')$ . + +Each PB instance comes with a tie-breaking order $\succ$ over the projects, used by the PB rules to resolve internal ties. While tie-breaking can notably influence voting [Obraztsova and Elkind, 2011; Obraztsova et al., 2011] and PB elections [Boehler et al., 2023], especially in case of smaller instances [Janeczko and Faliszewski, 2023], we do not expect many ties in PB (ties are very rare in Pabulib data; see also the work of Xia [2021] for an argument regarding large ordinal elections). + +Participatory Budgeting Rules. A PB rule is a function $f$ that for a PB instance $E$ outputs a set $f(E)$ of projects with total cost not exceeding the budget. We refer to the projects in $f(E)$ as winning (or selected, or funded). Note that our rules are resolute, that is, their outcomes are unique. We focus on the following ones, denoting by $E = (P, V, B, cost)$ the instance we consider and assuming $\succ$ to be the tie-breaking order: + +BasicAV. It starts with $W = \emptyset$ and considers all the projects in the order of their nonincreasing approval scores (with ties broken using $\succ$ ), putting a considered project $p$ into $W$ if $cost(W \cup \{p\}) \leq B$ . Finally, it outputs $W$ . + +AV/Cost. For each project $p$ , its approval-to-cost ratio is $|A(p)| / \text{cost}(p)$ . AV/Cost acts like BasicAV, considering the projects in the nonincreasing order of these ratios and breaking ties using $\succ$ . + +Phragmén [Brill et al., 2017; Los et al., 2022]. Phragmén starts with $W = \emptyset$ and fills it in as follows. Initially, the voters have empty virtual bank accounts, but they continuously earn money + +at the rate of one unit per unit of time. When there is a project $p$ such that the voters who approve it have at least $cost(p)$ units of money all together and the project is within the budget (i.e., $cost(W \cup \{p\}) \leq B$ ), these voters purchase it: $p$ is included in $W$ , the bank accounts of voters in $A(p)$ are reset to zero, and $p$ is removed from consideration. If $cost(W \cup \{p\}) > B$ , then $p$ is removed without being included in $W$ . If several projects could be purchased simultaneously, we pick one using the tie-breaking order. The rule stops and outputs $W$ when all projects are removed from consideration. + +Method of Equal Shares (MES-Cost) [Peters et al. 2020; 2021]. First, each voter receives $\frac{B}{|V|}$ amount of money. Then we let $W = \emptyset$ and proceed iteratively: Within each iteration, for each project $p$ not in $W$ we compute its affordability coefficient $\alpha_{p}$ as the smallest number such that the following holds ( $b_{i}$ is the money that voter $v_{i}$ currently has): + +$$ +\sum_ {v _ {i} \in A (p)} \min \left(b _ {i}, \alpha_ {p} \cdot \operatorname {c o s t} (p)\right) = \operatorname {c o s t} (p). \tag {1} +$$ + +If no such value exists (i.e., the voters approving $p$ cannot afford it) then we set $\alpha_{p} = \infty$ . If $\alpha_{p} = \infty$ for all $p \notin W$ , then we terminate and output $W$ . Otherwise, we choose $p' \notin W$ with the lowest affordability coefficient (using tie-breaking, if needed), include $p'$ in $W$ , and take $\alpha_{p'} \cdot cost(p')$ money from each voter in $A(p')$ (or all the remaining funds, if the voter had less than $\alpha_{p'} \cdot cost(p')$ ). Due to the use of cost in Eq. (1), the rule is called MES with cost utilities. + +MES with Approval Utilities (MES-Apr). It works like MES-Cost, but with Eq. (1) replaced by $\sum_{v_i\in A(p)}\min (b_i,\alpha_p) = cost(p)$ . While in MES-Cost the affordability coefficients are between 0 and 1, under MES-Apr they can be as large as $B$ . + +Remark 2.1. Both variants of MES may output a set of projects that can be extended without exceeding the budget. In our experiments we use Phragmén completion: When a MES variant finishes, we extend its output by running Phragmén with voters' bank accounts initiated with their then-current amounts of money. This defines the MES-Cost/Ph and MES-Apr/Ph rules. + +Many authors also study other PB rules (see, e.g., the works of Goel et al. [2019]; Talmon and Faliszewski [2019]; Sreedurga et al. [2022]); for an overview, see the survey of Rey and Maly [2023] and the book of Lackner and Skowron [2023] (which regards multiwinner elections, where projects have unit costs). BasicAV is commonly used in practice, MES-Cost also was recently used by several cities (see https://equalshares.net); the other rules are mostly studied theoretically. + +Structured Preferences. In a plurality profile, each voter approves exactly one project, whereas in a party-list profile the projects are grouped into "parties" and each voter approves all projects from a single party. Importantly, plurality profiles form a special case of party-list preferences. Some cities require plurality profiles (e.g., Wroclaw, Poland; see the respective datasets in Pabulib); party-list ones are mostly interesting theoretically. + +Definition 2.1. Consider a set of projects $P$ and a voter collection $V$ with approval ballots over $P$ . We say that these voters have: (1) plurality preferences, if $|A(v_i)| = 1$ for each voter $v_i$ , (2) party-list preferences, if either $A(v_i) = A(v_j)$ or $A(v_i) \cap A(v_j) = \emptyset$ for each two voters $v_i$ and $v_j$ . + +For a party-list profile, let $party(p)$ denote the set of projects approved by these voters that approve $p$ . + +# 3 Participatory Budgeting Cost Games + +We now define our main object of study, participatory budgeting cost games (PB games). Formally, a PB game is a tuple $(P,V,B,d)$ , where $P$ is a set of projects, $V$ is a collection of voters with approval preferences over the projects from $P$ , $B$ is the available budget, and $d\colon P\to \mathbb{R}_+$ is a function that associates each project with its minimal delivery cost (see the description of the payoffs in the next paragraph). We assume that for each project $p_i$ , $d(p_{i})\leq B$ . In this game, the projects are the players and each of them needs to report its cost. That is, a strategy profile is a tuple $\mathbf{c} = (c_1,\dots ,c_n)$ , with a cost $c_{i}\in \mathbb{R}_{+}$ for each project $p_i$ , which will be used in a PB election. As is standard, by $(\mathbf{c}_{-i},c^{\prime})$ we mean a strategy profile identical to $\mathbf{c}$ except that project $p_i$ reports cost $c^{\prime}$ . We often write $\mathbf{c}(p_i)$ to denote the cost reported by $p_i$ under profile $\mathbf{c}$ , and we use strategy profiles as cost functions in PB instances. + +Let us fix a PB rule $f$ and a PB game $G = (P, V, B, d)$ . For a strategy profile $\mathbf{c}$ , the associated PB instance is $E(\mathbf{c}) = (P, V, B, \mathbf{c})$ and the payoff of each project $p_i \in P$ is: + +$$ +u _ {i} (\mathbf {c}) = c _ {i} - d (p _ {i}) \quad \text {i f} p _ {i} \in f \big (E (\mathbf {c}) \big), \qquad \text {a n d} \qquad u _ {i} (\mathbf {c}) = 0 \quad \text {o t h e r w i s e}. +$$ + +Table 1: Our NE existence results. By “ $d \equiv 0$ ” and “ $arb$ . $d$ ” we mean zero and arbitrary delivery costs. + +
ballotsPluralityParty-ListUnrestricted
d ≡ 0arb. dd ≡ 0arb. dd ≡ 0arb. d
BasicAV■Pr. 4.1■Pr. 4.1■Pr. 4.1■Pr. 4.1■Pr. 4.1■Pr. 4.1
AV/Cost■Pr. 4.2□Thm. 4.4, Ex. 4.3■†Pr. 4.2□Thm. 4.4, Ex. 4.3■†Pr. 4.2□Thm. 4.4, Ex. 4.3
Phragmén■†Cor. 4.6□Cor. 4.6■Pr. 4.7?□Cor. 4.6□Pr. 4.8□Pr. 4.8
MES-Apr■Thm. 4.10□Thm. 4.10■Thm. 4.12□Thm. 4.12, Ex. B. 2□Pr. 4.11□Pr. 4.11
MES-Cost■Thm. 4.9□Thm. 4.9■Thm. 4.9■Thm. 4.9■Thm. 4.9■Thm. 4.9
+ +
Legend
■ NE guaranteed for an arbitrary tie-breaking +■ NE always exists for some tie-breaking +□ no NE for all tie-breaking orders +? conjecture +† result holds also if for each project p we have d(p) ≤ ap(p), where ap is the corresponding approval-proportional strategy profile
+ +The interpretation is as follows: Each project $p$ has a minimum delivery cost $d(p)$ , which is the lowest price under which it can be implemented. Yet, with more money the project can be better, the proposer prefers this, and the utilities implement this preference. In particular, the proposers do not receive funds equal to the utility values. Instead, to compute utility values, we deduct minimum delivery costs. We often consider the special case of zero delivery costs.1 + +For a PB rule $f$ and a PB game $G = (P, V, B, d)$ , we say that a strategy profile $\mathbf{c}$ for this game is a Nash equilibrium under $f$ (is an $f$ -NE), if no project has a profitable deviation. That is, it cannot benefit by reporting a different cost than that from the $f$ -NE (assuming no other costs change). + +Definition 3.1. For a PB rule $f$ and a PB game $G = (P, V, B, d)$ , we say that a strategy profile $\mathbf{c}$ for this game is a Nash equilibrium under $f$ (is an $f$ -NE) if for every project $p_i \in P$ and every cost $c' \in \mathbb{R}_+$ it holds that $u_i(\mathbf{c}) \geq u_i(\mathbf{c}_{-i}, c')$ . + +Example 3.1. Take AV/Cost and a PB game with projects $p_1, p_2$ , voters $v_1, \ldots, v_5$ , budget 10, and zero delivery costs. We have $A(p_1) = \{v_1, v_2\}$ and $A(p_2) = \{v_3, v_4, v_5\}$ , so this is a plurality profile. Strategy profile $\mathbf{c}$ where $\mathbf{c}(p_1) = 4$ and $\mathbf{c}(p_2) = 6$ is an NE. Indeed, according to AV/Cost, under these costs $p_1$ and $p_2$ are tied as $|A(p_1)| / \mathbf{c}(p_1) = |A(p_2)| / \mathbf{c}(p_2) = 1 / 2$ , and both projects are selected under any tie-breaking order. If either of the projects reports a lower cost, it gets selected, but its utility decreases. If it reports a higher cost, it is not selected and its utility drops to zero. + +We sometimes use approval-to-delivery-cost (A/D) tie-breaking orders, which favor projects with larger ratios of their approvals to delivery costs. So, $\succ$ is A/D if projects with zero delivery costs are ranked highest, while $A(p_i) / d(p_i) > A(p_j) / d(p_j)$ implies $p_i \succ p_j$ if $d(p_i) \cdot d(p_j) > 0$ . As some projects may have equal approval-to-delivery cost ratios, there may be several different A/D orders. + +Remark 3.1. In our games, a proposer receives the same utility of zero when his or her project is not funded and when it is funded with exactly its delivery cost. One could distinguish these cases by using extended utility function $u_{i}^{\prime}$ such that: + +$$ +u _ {i} ^ {\prime} (\mathbf {c}) = \mathbf {c} (p _ {i}) - d (p _ {i}) + \mathbb {1} [ \mathbf {c} (p _ {i}) \geq d (p _ {i}) ] \quad i f p _ {i} \in f \big (E (\mathbf {c}) \big), \qquad a n d \qquad u _ {i} ^ {\prime} (\mathbf {c}) = 0 \quad o t h e r w i s e, +$$ + +where $\mathbb{1}[x]$ is 1 if $x$ holds and 0 otherwise. We prefer our formulation as it is simpler and captures the intuition that only projects more costly than their delivery costs are worth implementing. That said, all our NE-(non)existence results still hold for the extended utility functions (for the nonexistence results this is automatic and for the existence ones, see Appendix A). + +# 4 Existence of Equilibria + +Our theoretical results, summarized in Table 1, focus on the existence of NEs in PB games. We mostly show that either: (1) There always is an equilibrium, for any tie-breaking order; (2) There always is an equilibrium for some A/D tie-breaking order; or (3) There is a PB game with no equilibrium for any tie-breaking order. When such statements are difficult to show, we seek instances with no equilibria for some tie-breaking order. Such an observation is weaker than a result of type (3), + +is incomparable to a result of type (2), and implies that a result of type (1) does not exist. Pragmatically, it is most important to know whether a result of type (1) holds or not in a given setting; the other types of results indicate various levels of bad news regarding the existence of Nash equilibria. + +We are interested in strategy profiles that are proportional in some way, and, in particular, in which the costs of the projects are justifiable by the level of support that they have. A detailed discussion of fairness in PB is, for example, in the works of Brill et al. [2023] and Maly et al. [2023], as well as the works defining MES and Phragmén). Below, we define a basic type of such a profile, where project costs are proportional to the numbers of approvals they get. + +Definition 4.1 (Approval-Proportional Strategy Profile). Consider a $PB$ game with $P = \{p_1, p_2, \ldots, p_m\}$ . We say that strategy profile $\mathbf{ap}$ is approval-proportional if for each project $p$ it holds that $\mathbf{ap}(p) = B \cdot |A(p)| / \sum_{p' \in P} |A(p')|$ . + +Under plurality preferences $\mathbf{ap}$ is natural for rules that aim to achieve proportionality. For more involved settings, other profiles may be more appropriate (see, e.g., MES-Cost and Theorem 4.9). + +# 4.1 BasicAV + +For BasicAV, there is an equilibrium where the most popular project requests the full budget amount. Proposition 4.1. For each PB game and each internal tie-breaking order $\succ$ there is a BasicAV-NE where the project with most approvals (best with respect to $\succ$ ) reports cost $B$ . + +In fact, the most popular project reports cost $B$ (and is selected) under every BasicAV-NE. Hence, Proposition 4.1 indicates that proposers of projects receiving many approvals have an incentive to overstate their costs, whereas proposers of less popular projects have an incentive to bundle them together, to garner votes (indeed, this is what seems to happen in Wieliczka PB elections held using BasicAV; recall Section 1). Consequently, BasicAV equilibria in our games are disproportional, highlighting known deficiencies of this rule. + +# 4.2 AV/Cost + +Some shortcomings of BasicAV are remedied by AV/Cost, for which the approval-proportional strategy is an equilibrium. To show this, we note that in an NE, if the delivery costs are limited, all projects are funded, using the whole budget. It follows that then ap is the only equilibrium profile. + +Proposition 4.2. If for each project $p$ we have $d(p) \leq \mathbf{ap}(p)$ , then $\mathbf{ap}$ is the only AV/cost-NE. + +If (some of) the delivery costs are higher than the costs implied by the approval-proportional strategy, then the situation becomes more complicated and it is no longer true that an NE exist for all internal tie-breaking orders. Yet, there always is a tie-breaking order yielding an NE. + +Example 4.3. Take a PB game $(P, V, B, d)$ with plurality ballots, where $P = \{p_1, p_2\}$ , $|A(p_1)| = |A(p_2)| = 5$ , $B = 10$ , $d(p_1) = 0$ , $d(p_2) = 6$ , and the tie-breaking order is $p_2 \succ p_1$ . Assume that $\mathbf{c}$ is an AV/Cost-NE. Consider two cases: (1) If $\mathbf{c}(p_2) \geq 6$ , then either $(a)$ $\mathbf{c}(p_1) \geq \mathbf{c}(p_2)$ , where it is better for $p_1$ to report a cost lower than $\mathbf{c}(p_2)$ (otherwise $p_2$ is selected before $p_1$ and there is no budget left for $p_1$ ), or $(b)$ $\mathbf{c}(p_1) < \mathbf{c}(p_2)$ and it is better for $p_1$ to increase its cost (keeping it below $\mathbf{c}(p_2)$ ). (2) If $\mathbf{c}(p_2) < 6$ , then either $p_2$ is chosen and it prefers to report cost 6 (otherwise its payoff is $\mathbf{c}(p_2) - d(p_2) < 0$ ), or $p_2$ is not selected, meaning that $\mathbf{c}(p_1) < \mathbf{c}(p_2)$ and it is better for $p_1$ to increase its cost (keeping it below $\mathbf{c}(p_2)$ ). + +Nevertheless, if we change the tie-breaking order so that $p_1 \succ p_2$ , then (6,6) is an equilibrium, where only $p_1$ is funded: Both projects would decrease their payoff by lowering costs and they would not benefit by increasing their costs ( $p_1$ would no longer be selected, and $p_2$ 's payoff would not change). + +Theorem 4.4. For each PB game and $A / D$ tie-breaking order there is an AV/Cost-NE computable in polynomial time. + +Proof sketch. We propose an algorithm that computes the claimed profile $\mathbf{c}$ in iterations. Starting from the situation where each project reports its delivery cost, in each iteration it selects a group of projects that "underreport" their costs compared to their support. Then, it increases their reported costs as much as possible, with the projects remaining funded. Finally, if the funded projects after the + +update do not use up the whole budget, the procedure is repeated for the remaining projects whose prices have not been increased so far. These projects, albeit worse regarding the delivery-cost-to-support ratio, can still benefit from the increase. In the end, the procedure outputs an equilibrium profile. The somewhat involved proof is in the appendix. + +We conclude by noting that even for plurality preferences the costs of projects selected in an equilibrium can be arbitrarily far from using the entire budget. + +Proposition 4.5. For each integer $\gamma >1$ there is a PB game with plurality preferences and AV/Cost-NE profile c with the total cost of projects funded under AV/Cost at most $B / \gamma$ + +# 4.3 Phragmén + +For plurality ballots, AV/Cost and Phragmén always output the same projects. Thus, the next corollary translates results from the previous section to the case of Phragmén. + +Corollary 4.6. For each PB game with plurality ballots, (a) if $d(p) \leq \mathbf{ap}(p)$ for all $p \in P$ , then ap is the only Phragmén-NE for every tie-breaking order; (b) otherwise, for each A/D tie-breaking order there is a Phragmén-NE (and there are PB games and tie-breaking orders with no Phragmén-NE). + +Naturally, Proposition 4.5 also holds for Phragmén, so there are games with equilibria for which Phragmén funds projects whose total cost is an arbitrarily small fraction of the budget. Next, we focus on more involved preference profiles. On the positive side, for party-list ballots (and zero delivery costs) we always have unique equilibria, for every tie-breaking order: Each project $p$ reports a cost proportional to the number of its approvals, divided by the size of its party. + +Proposition 4.7. For every $PB$ game with party-list ballots and zero delivery costs, there is the unique Phragmén-NE $\mathbf{c}$ , where for every project $p$ we have $\mathbf{c}(p) = B \cdot \frac{|A(p)|}{|V| \cdot |\text{party}(p)|}$ . + +Unfortunately, party-list preferences with zero delivery costs are where the good news end. Indeed, we find that there are small instances without equilibria. + +Proposition 4.8. There is a PB game with six projects and six voters and zero delivery costs, for which there is no Phragmén-NE irrespective of the tie-breaking order. + +# 4.4 Method of Equal Shares + +The mechanics of MES might seem similar to those of Phragmén, as these rules follow the approach in which voters buy the projects they approve. However, contrary to Phragmén, in both types of MES voters have limited amounts of budget to spend. So, the approval scores of projects give upper-bounds on the maximum costs that the projects might report. I.e., as each voter gets an equal share of the budget, projects that cost more than the total budget of their supporters cannot be funded. + +In case of MES-Cost, this translates to our most positive result: For each PB game there always is an, essentially unique, MES-Cost-NE, regardless of the tie-breaking order and the delivery costs. Furthermore, this NE can be computed in polynomial time and if all the projects have sufficiently low delivery costs and sufficiently high support, then it uses up the whole budget (even though, in general, MES is not exhaustive). Together with its proportionality features, this reinforces the position of MES-Cost as a very attractive PB rule. + +Theorem 4.9. For each PB game, there is a polynomial-time computable MES-Cost-NE profile $\mathbf{c}$ , unique up to the actions of the projects not selected by MES-Cost under $\mathbf{c}$ . + +Proof sketch. Let $p$ be the project with the largest number of approvals (and preferred in the tie-breaking order, if there are several such projects). We observe that $p$ can always request the full amount of money that its voters have at the beginning of the execution of MES-Cost; the rule would fund $p$ irrespective of the costs of the other projects. Thus, $p$ reports this amount and is funded (if the amount is below its delivery cost, $p$ reports the delivery cost and is not funded). The project has no reason to report less, and reporting more would cause it to not be funded. Then, we disregard all the voters that approve it and repeat the reasoning for the remaining projects and voters. + +Table 2: PB instances analyzed in our experiments. + +
Instance|P||V|BudgetAvg. num. of approvals per ballot
Wesola2911821011000 PLN7.87
Kleine Wereld52426250000 EUR11.93
+ +Perhaps surprisingly, for MES-Apr the results are much less positive. While there always are MES-Apr-NE for plurality ballots, in general there are small games with zero delivery costs that have no MES-Apr-NE irrespective of the tie-breaking order used. + +Theorem 4.10. For each PB game with plurality ballots, there is a polynomial-time computable MES-Apr-NE. + +Proposition 4.11. There are games with zero delivery costs, for which there is no MES-Apr-NE regardless of the tie-breaking order, even if we have only 4 projects and 16 votes. + +In case of party-list ballots, existence of equilibria depends on delivery costs and tie-breaking. + +Theorem 4.12. Every PB game with party-list ballots and an $A / D$ tie-breaking order has a MES-Apr-NE computable in polynomial-time. There are examples of PB games with party-list preferences, nonzero delivery costs and non- $A / D$ tie-breaking that do not have MES-Apr-NE. + +For zero delivery costs every tie-breaking order is A/D, so the above theorem shows that PB games with party-list preferences and zero delivery costs always have MES-Apr-NE. We conclude by noting a surprising difference between MES-Apr and Phragmén, presented in the following theorem. + +Theorem 4.13. In some instances symmetric projects (w.r.t. their supporters) submit different costs in a MES-Apr-NE, but identical ones in Phragmén-NE. + +# 5 Experiments + +We move on to the experimental analysis of our games for real-life PB instances from Pabulib [Faliszewski et al., 2023]. We focus on two instances, one held in 2022 in Wesola (a district of Warsaw, Poland) and one held in 2019 in Kleine Wereld (a part of the Noord District of Amsterdam, The Netherlands). A few details regarding these instances are available in Table 2. We have also analyzed many other instances from Pabulib, some of which we show in Appendix C. While all of them lead to similar conclusions, the instances we chose are particularly illustrative. Our code is publicly available at https://github.com/Project-PRAGMA/strategic-cost-selection-in-PB--NeurIPS2025. + +We perform two experiments. First, we take a PB instance with the original project costs and compute proposers' best responses. Second, we seek equilibiria for our instances. We omit plots for MES-Apr/Ph, as they resemble those for Phragmén (see Appendix C). + +# 5.1 Winning and Losing Margins + +Take a PB instance $E$ and a PB rule $f$ . Intuitively, for each project $p$ its best response $br(p)$ is the highest cost such that if we use it instead of $cost(p)$ , then $p$ is (still) funded under $f$ . We define it as the supremum of the set of costs under which $p$ is funded. Then, if $p$ is winning in $E$ under $f$ , we let its winning margin be $br(p) - cost(p)$ , and if it is losing we let its losing margin be $cost(p) - br(p)$ . + +In Figure 1 we show the winning and losing margins (and, hence, the best responses) for all the projects from our two PB instances. For BasicAV, we observe very high winning margins for the projects with the largest numbers of approvals, and close-to-zero best responses for the remaining ones (this phenomenon is particularly visible in Kleine Wereld). In principle, under BasicAV if a project receives few votes even making it cheap will not make it funded, unless it is so cheap that it is selected simply because nothing else could fit within the budget left. For AV/Cost the best responses roughly correspond to the numbers of approvals each project got; the same, roughly, holds for Phragmén and MES-Apr/Ph. The margins for MES-Cost/Ph are different, with several most-approved projects having much higher winning margins than the rest. + +The main general observation is that unless a project received relatively few approvals, then typically its proposers could have reported a much higher cost. In practice, this means that project proposers + +![](images/1225d5a46a99ff1c62da80bd07aec4ab1ef20704fab2b1dc71cc29610dae3298.jpg) +Figure 1: Winning and losing margins in real-life PB—Wesola with the budget of 1011k PLN at the top and Kleine Wereld with one of 250k EUR at the bottom. Projects ordered by decreasing approval scores (on the $x$ -axis) are represented by bars. Tacks and crosses show, respectively, the original and best response costs. + +should rather focus on designing convincing, attractive projects, and not on minimizing their costs, as the latter is not as likely to be decisive. Yet, of course, there are projects where lowering their costs could have gotten them funded; see also the recent work of Boehmer et al. [2024]. + +# 5.2 Finding Equilibria Using Dynamics + +In the second experiment, our goal was to compute (approximate) equilibria for our two instances. While for BasicAV, AV/Cost, and MES-Cost/Ph we could compute the equilibria using results from the previous section, this would not be possible for Phragmén and MES-Apr/Ph. Instead, we simulate certain dynamics. (For the rules where we can compute equilibria directly, our dynamics give nearly identical results; for Phragmén and MES-Apr/Ph there may be no NE, but the results suggest that the profiles that we find are, at least, close to being equilibria). We assumed zero delivery costs. In Appendix D we show results for delivery costs equal to $80\%$ of the originally submitted ones, obtaining similar conclusions. + +Given a PB instance, our dynamics go as follows. First, each proposer reports the same cost as was originally chosen for his or her project. Then, in each iteration, one of them, selected uniformly at random, either slightly increases or decreases his or her project's cost. Specifically, the proposer chooses a number $x$ between 0 and $cost / 10$ uniformly at random (where $cost$ is the current project's cost) and if their project was losing in the previous iteration, then the proposer decreases its cost by $x$ , and if it was winning, then he or she increases its cost by $x$ (but if this action would have caused the project to lose, then the proposer does not change the project's cost). We expect to converge to an NE, if one exists, after running sufficiently many iterations. + +In Figure 2 we present the results of the dynamics, that is, the obtained strategy profiles, after 10 000 iterations. Under BasicAV, as expected, all the budget goes to the project with the most votes. Under AV/Cost, every project ends up with a cost proportional to its support. Like in the previous experiment, Phragmén, and MES-Apr/Ph produce similar results, while those of MES-Cost/Ph remain different. For BasicAV, AV/Cost, and MES-Cost/Ph most of the projects "reached" the costs predicted by the equilibrium. (However, note that not all of our dynamics ended up in perfect equilibria within 10 000 iterations as, e.g., some projects are not winning). + +The main conclusion from this experiment (supported also by simulations for other PB instances; see Appendix C) is that under AV/Cost, Phragmén, and MES-Apr/Ph the proposers are incentivized to use costs that mostly reflect the number of approvals they receive. Under MES-Cost/Ph, most strongly supported projects, as well as those that are supported by many voters who do not approve other, more popular projects, can request much higher costs. Consequently, in an NE MES-Cost/Ph funds fewer, more expensive projects than AV/Cost, Phragmén, and MES-Apr/Ph. We view this + +![](images/6e020f0ad1b23659fdfef2b078b2dad037dd2f22f96596f8a395da46042204e6.jpg) +Figure 2: Strategy profiles after 10 000 iterations of our dynamics for the instances Wesola with the budget of 1 011k PLN at the top and Kleine Wereld with one of 250k EUR at the bottom.. Projects ordered by decreasing approval scores $(x$ -axis). Final projects' costs shown by bars; the original costs shown as dashed lines. Red and green bars indicate, respectively, losing and winning projects; brighter parts are the increases over the original cost. + +observation as one of the more important take-home messages from our work, which can be used to favor MES-Cost/Ph in practice (while funding some cheap projects is important, funding only such projects is logistically challenging for cities that then need to coordinate their implementation). + +# 6 What If Votes Depended on Costs? + +So far, we have assumed that the voters' approvals are independent of the project costs. We justified this view in the introduction, and here we further argue that not only is it reasonable, but also essential. Specifically, we demonstrate that if the approvals depended on the costs, then our equilibria guarantees would collapse, albeit due to such dynamics among voters and project proposers that are not particularly realistic. This reinforces the careful design of our basic model, which allows for both accurate analysis and meaningful insights into real-world phenomena. + +We extend our basic model to cost-interval PB games, where each voter $v$ approves a project $p$ exactly if its cost falls within an interval that $v$ views as acceptable for $p$ (specified in the game description). This approach is inspired by the model of Sreedurga [2023]. + +Theorem 6.1. For $f \in \{ \text{BasicAV}, \text{AV}/\text{Cost}, \text{Phragmén}, \text{MES-Apr}, \text{MES-Cost} \}$ , there is a cost-interval PB game with no delivery costs that has no $f$ -NE. + +We find the basic model more insightful because cost-interval PB games allow for very unrealistic scenarios—such as having voters that approve building a bike path for at most $25,000, when others do so only if it costs above$ 75,000. In practice, such disparities are unlikely, but the proof of Theorem 6.1 exploits exactly such scenarios and the ability to shift between received support by varying the costs (even under plurality preferences). While cost-interval PB games capture more nuanced voters' behaviors, they also allow for vastly unrealistic cases that distort the results. + +# 7 Summary and Future Work + +We have introduced a game-theoretic framework capturing strategic cost selection of projects in participatory budgeting scenarios, studied the existence of equilibria in our games, and used them to analyze two real-life PB instances. We believe to have set the groundwork for future studies, which may include establishing the complexity of deciding if equilibria exist in our games and studying more involved settings, emerging from relaxing our basic assumptions. In particular, it is compelling to verify to what extent our observations hold in more realistic models. + +# Acknowledgments + +This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 101002854). Piotr Skowron was supported by the European Union (ERC, PRO-DEMOCRATIC, 101076570). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. Grzegorz Lisowski acknowledges support by the European Union under the Horizon Europe project Perycles (Participatory Democracy that Scales). This research has been supported by the French government under the management of Agence Nationale de la Recherche as part of the France 2030 program, reference ANR-23-IACL-0008. + +![](images/e3f8cbf2a116332fa79caa700e0b039f9dcc48361c6192106a799530f8b75ce1.jpg) + +![](images/f6fdd93aacf899867ec2196f1c33feb4b5445958f8a1394e9f5c69a38e1440f3.jpg) + +# References + +H. Aziz, S. Gujar, M. Padala, M. Suzuki, and J. Vollen. Coordinating monetary contributions in participatory budgeting. In Proceedings of SAGT-2023, pages 142-160, 2023. +G. Benadè, S. Nath, A. D. Procaccia, and N. Shah. Preference elicitation for participatory budgeting. Management Science, 67(5):2813-2827, 2020. +N. Boehmer, P. Faliszewski, L. Janeczko, and A. Kaczmarczyk. Robustness of participatory budgeting outcomes: Complexity and experiments. In Proceedings of SAGT-2023, 2023. +N. Boehmer, P. Faliszewski, L. Janeczko, D. Peters, G. Pierczyński, S. Schierreich, P. Skowron, and S. Szufa. Evaluation of project performance in participatory budgeting. In Proceedings of IJCAI-2024, pages 2678-2686, 2024. +M. Brill, R. Freeman, S. Janson, and M. Lackner. Phragmén's voting methods and justified representation. In Proceedings of AAAI-2017, pages 406-413, 2017. +M. Brill, M. Lackner, J. Maly, and J. Peters. Proportionality in approval-based participatory budgeting. In Proceedings of AAAI-2023, pages 5524-5531, 2023. +Y. Cabannes. Participatory budgeting: A significant contribution to participatory democracy. *Environmental and Urbanization*, 16(1):27-46, 2004. +Horst A Eiselt. Equilibria in competitive location models. Foundations of location analysis, pages 139-162, 2011. +R. Fairstein, G. Benadè, and K. Gal. Participatory budgeting designs for the real world. In Proceedings of AAAI-2023, pages 5633–5640, 2023. +P. Faliszewski, J. Flis, D. Peters, G. Pierczyński, P. Skowron, D. Stolicki, S. Szufa, and N. Talmon. Participatory budgeting: Data, tools and analysis. In Proceedings of IJCAI-2023, pages 2667-2674, 8 2023. +A. Goel, A. Krishnaswamy, S. Sakshuwong, and T. Aitamurto. Knapsack voting for participatory budgeting. ACM Transactions on Economics and Computation, 7(2):1-27, 2019. +L. Janeczko and P. Faliszewski. Ties in multiwinner approval voting. In Proceedings of IJCAI-2023, pages 2765-2773, 2023. +M. Lackner and P. Skowron. Multi-Winner Voting with Approval Preferences. Springer Briefs in Intelligent Systems. Springer, 2023. +M. Los, Z. Christoff, and D. Grossi. Proportional budget allocations: Towards a systematization. In Proceedings of IJCAI-2022, pages 398-404, 2022. + +J. Maly, S. Rey, U. Endriss, and M. Lackner. Fairness in participatory budgeting via equality of resources. In Proceedings of AAMAS-2023, pages 2031-2039, 2023. +S. Obraztsova and E. Elkind. On the complexity of voting manipulation under randomized tie-breaking. In Proceedings of IJCAI-2011, pages 319-324, July 2011. +S. Obraztsova, E. Elkind, and N. Hazon. Ties matter: Complexity of voting manipulation revisited. In Proceedings of AAMAS-2011, pages 71-78, May 2011. +D. Peters and P. Skowron. Proportionality and the limits of welfarism. In Proceedings of EC-2020, pages 793-794, 2020. +D. Peters, G. Pierczyński, and P. Skowron. Proportional participatory budgeting with additive utilities. In Proceedings of NeurIPS-2021, pages 12726-12737, 2021. +S. Rey and J. Maly. The (computational) social choice take on indivisible participatory budgeting. Technical Report arXiv.2303.00621, arXiv, 2023. +G. Sreedurga, M. Bhardwaj, and Y. Narahari. Maxmin participatory budgeting. In Proceedings of IJCAI-2022, pages 489–495, 2022. +G. Sreedurga. Indivisible participatory budgeting with multiple degrees of sophistication of projects. In Proceedings of AAMAS-2023, pages 2694-2696, 2023. +N. Talmon and P. Faliszewski. A framework for approval-based budgeting methods. In Proceedings of AAAI-2019, pages 2181-2188, 2019. +J. Wagner and R. Meir. Strategy-proof budgeting via a vcg-like mechanism. In Proceedings of SAGT-2023, volume 14238, pages 401-418, 2023. +L. Xia. How likely are large elections tied? In Proceedings of EC-2021, pages 884-885, 2021. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The abstract clearly describes the main goals of the paper and the results mentioned in the Introduction reflect both theoretical and experimental contribution of the manuscript. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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Indeed, if a PB game does not admit an equilibrium then neither does its P4F variant (every profitable deviation in a PB game is also profitable in the corresponding P4F game). The other direction is more involved as there are games that have equilibria in our model but do not have them in the P4F variant (see Example A.1). However, if a PB game has an NE where all projects that report values above their delivery costs are funded, then it remains an NE in the corresponding P4F game (a profitable deviation for the P4F game would also be profitable in the original one). All our existence results produce equilibria that satisfy this condition. + +Example A.1. Consider three projects $p_1$ , $p_2$ , $p_3$ , each supported by one distinct voter, with delivery costs $d_1 = 1$ , $d_2 = d_3 = 0$ , tie-breaking order $p_1 \succ p_2 \succ p_3$ and budget $B = 2$ . As we prove in the appendix, the corresponding PB game admits Phragmén-NE $\mathbf{c} = (2,1,1)$ but the corresponding P4F PB game has no NEs under Phragmén. + +Below we argue why the statement from the above example indeed holds. Let us denote the PB game from the example as $G$ and the P4F PB game as $G'$ . Using $G$ and $G'$ , we show now that in some cases PB games admit a Phragmén-NE, while PB/P4F games (based on the same input) do not. + +First, observe that $\mathbf{c} = (2,1,1)$ is a Phragmén-NE in PB games. To see that, suppose that it is not and consider a project $p_i$ and a cost $c_i'$ for which $u_i((\mathbf{c}_{-i}, c_i')) > u_i(\mathbf{c})$ . If $p_i$ is $p_2$ or $p_3$ , we notice that $p_i$ is selected under $\mathbf{c}$ but not under $(\mathbf{c}_{-i}, c_i')$ , for each $c_i' > c_i$ . So, neither $p_2$ nor $p_3$ can improve its utility. For $p_i = p_1$ , we notice that it is not funded under $\mathbf{c}$ and that it can only be funded if $c_i' \leq 1$ . But then $u_i((\mathbf{c}_{-i}, c_i')) \leq 0$ . It follows that $\mathbf{c}$ is a Phragmén-NE in PB games. + +Second, we show that $G'$ does not admit a Phragmén-NE. For contradiction, suppose that $G'$ admits a Phragmén-NE, say $\mathbf{c}$ , for PB/P4F games. As a warm up, we note that at least one project has to be chosen in $E(\mathbf{c})$ , as otherwise each project benefits from lowering their cost to $B = 2$ . + +Further, consider the case that in $\mathbf{c}$ projects have the same cost $c$ . If $c < 1$ , then, following the tie-breaking order, $p_1$ is funded and gets a negative utility. Hence, by increasing its cost to $c' > B > c$ , $p_1$ becomes not funded and increases its utility. Also, if $c \geq 1$ , then $p_3$ is not funded, and as $d_3 = 0$ , it can benefit from decreasing its cost. + +Now, suppose that in $\mathbf{c}$ projects do not have the same costs. Let $c_{1} \geq 1$ . Then, if $c_{2} \geq c_{1}$ or $c_{3} \geq c_{1}$ , then some project that is not funded would benefit from lowering its cost. For a complementary case of $c_{2} < c_{1}$ and $c_{3} < c_{1}$ , we further consider two subcases. If $c_{2} + c_{3} < 2$ , then one of $p_{2}$ and $p_{3}$ would benefit from increasing their cost. Alternatively, $c_{2} + c_{3} \geq 2$ . This implies that $c_{1} > 1$ and that $c_{2} \geq 1$ or $c_{3} \geq 1$ . In this case $p_{1}$ would benefit, in a PB/P4F game, from choosing $c_{1}' = 1$ and becoming selected. + +Finally, if $c_{1} < 1$ , then it benefits from increasing its cost if it is funded under c. Notice further that if it would not be funded, then $c_{2} < 1$ and $c_{3} < 1$ , and both of these costs are lower than $c_{1}$ . It follows that one of them can increase its utility by increasing its cost. + +As our case distinction is exhaustive, we know that there is no Phragmén-NE for $G'$ . Consequently, we proved our claim about the equilibria in games $G$ and $G'$ . + +# B Missing Proofs + +# B.1 Proof of Proposition 4.1 + +Proof. Take a PB game and a strategy profile $\mathbf{c}$ as described in the statement, and let $p$ be the project such that $\mathbf{c}(p) = B$ . If a project other than $p$ increases its cost, it still is not selected so this is not a + +beneficial move. If $p$ decreases its cost then its utility drops, and if it increases its cost then it is not funded. In either case, its utility decreases. So, $\mathbf{c}$ is a BasicAV-NE. + +# B.2 Proof of Proposition 4.2 + +In our proof we first use the following proposition. + +Proposition B.1. Take a $PB$ game and the corresponding strategy profile $\mathbf{ap}$ . If $d(p) \leq \mathbf{ap}(p)$ for each project $p$ , then if a profile $\mathbf{c}$ is an AV/Cost-NE for this game, then (1) $\sum_{p \in P} \mathbf{c}(p) = B$ , and (2) AV/Cost funds all the projects. + +Proof. Let the notation be as in the statement of the proposition. In the beginning we observe that if $\sum_{p\in P}\mathbf{c}(p) < B$ , then AV/Cost selects all the projects. Hence, the project considered last benefits by reporting a higher cost (so that the sum of the reported costs is $B$ ). Thus such a strategy profile $\mathbf{c}$ is not an equilibrium. + +Next, let us assume that $\sum_{p\in P}\mathbf{c}(p) > B$ . Let $P_{won}$ and $P_{lost}$ be sets of projects that, respectively, are and are not funded. By our assumption, we know that $P_{lost}$ must be nonempty, and we also note that $P_{won}$ must be nonempty. Naturally, $\sum_{p\in P_{won}}\mathbf{c}(p)\leq B$ . For each two projects $p_i$ and $p_j$ in $P_{won}$ , it must be the case that $|A(p_i)| / \mathbf{c}(p_i) = |A(p_j)| / \mathbf{c}(p_j)$ . For example, if we had $|A(p_i)| / \mathbf{c}(p_i) > |A(p_j)| / \mathbf{c}(p_j)$ then AV/Cost would consider $p_i$ prior to $p_j$ and, consequently, it would be beneficial for $p_i$ to increase its reported cost by a small-enough amount so that it would still be considered (and selected) prior to $p_j$ . This would contradict the assumption that $\mathbf{c}$ is an equilibrium. Next, since $\sum_{p\in P}\mathbf{ap}(p) = B$ , there must be some project $p'$ in $P_{won}$ such that $\mathbf{c}(p') > \mathbf{ap}(p')\geq d(p')$ (otherwise, if each project $p'\in P_{won}$ reported at most $\mathbf{ap}(p')$ , then any project $p\in P_{lost}$ would be selected after reducing its cost to $\mathbf{ap}(p)$ , so $\mathbf{c}$ could not have been NE). Further, by definition of the approval-proportional profile, for every project $p\in P$ , we have that $|A(p)| / \mathbf{ap}(p)$ is the same value and, so, for every project $q\in P_{lost}$ it holds that $|A(q)| / \mathbf{ap}(q) = |A(p')| / \mathbf{ap}(p') > |A(p')| / \mathbf{c}(p')$ . This means that every project $q\in P_{lost}$ can improve its utility by reporting cost $\mathbf{ap}(q)$ and being selected. This contradicts the assumption that $\mathbf{c}$ is an equilibrium. + +Thus it must be the case that $\sum_{p\in P}\mathbf{c}(p) = B$ , which implies that AV/Cost selects all projects. + +![](images/8b664798bced7d893b4c4631df2b558b6102627e165a7bcbfbc5cdff684c79ee.jpg) + +Proof of Proposition 4.2. Consider a PB game $(P, V, B, d)$ such that for every $p_i \in P$ it is true that $d(p) \leq \mathbf{ap}(p)$ . + +We argue that strategy $\mathbf{c} = \mathbf{ap}$ is a AV/Cost-NE. Note that the sum of the reported costs in $\mathbf{c}$ is exactly $B$ and all projects are funded. Hence, for each player, decreasing the cost leads to a utility loss. Hence, a profitable deviation could only be through increasing the reported cost. Towards contradiction, let us fix some player $p_i \in \mathbf{ap}$ and assume that reporting a cost $c_i' > c_i$ leads to a better payoff. This means that $p_i$ is funded in the modified election $E' = (P, V, B, (\mathbf{c}_{-i}, c_i'))$ , that is, $p_i \in \mathrm{AV} / \mathrm{cost}(E')$ . Additionally, since by Proposition B.1 we know that $\sum_{j \in |P|} c_j = B$ , it immediately follows that $c_i' + \sum_{j \in |P| \setminus \{i\}} c_j > B + \sum_{j \in |P| \setminus \{i\}} c_j$ . It is also the case that for each $j \in |P|$ , $|A(P_j)| / c_j > |A(P_i)| / c_i'$ . Hence, in particular, in the run of AV/cost for $E'$ , all projects are considered before $p_i$ . Since all of these project are selected, by the time the procedure starts considering $p_i$ , the remaining budget is smaller than $c_i'$ , so $p_i \notin \mathrm{AV} / \mathrm{cost}(E')$ ; a contradiction. Note that because all projects that are funded are always considered by the procedure in the same time, the result is independent of the tie-breaking order $\succ$ . + +We now argue that every strategy profile $\mathbf{c}$ other than $\mathbf{ap}$ is not a AV/cost-NE. To this end, we take such a profile $\mathbf{c}$ and assume towards contradiction that it is a AV/cost-NE. Observe that since $\mathbf{c} \neq \mathbf{ap}$ , there exist two distinct projects $p_i$ and $p_j$ such that $|A(p_i)| / c_i \neq |A(p_j)| / c_j$ . Hence one of the projects is considered earlier than the other; we assume without loss of generality that $p_i$ is considered before $p_j$ . Notably, as $\mathbf{c}$ is (by our assumption) a AV/cost-NE, by Proposition B.1, we know that $\{p_i, p_j\} \subseteq \mathrm{AV} / \mathrm{cost}(\mathbf{c})$ . This implies, however, that $p_i$ can report a slightly higher price $c_i'$ and still be selected by AV/cost thus obtaining a better payoff. Formally, there exists a $c_i' > c_i$ resulting in the profile $(c_i', \mathbf{c}_{-i})$ in which $|A(P_i)| / c_i' < |A(P_j)| / c_j$ . So, $p_i$ is (still) considered by AV/cost before $c_j$ and the deviation is small enough (that is, $c_i' - c_i < c_j$ ) to guarantee that $p_i \in \mathrm{AV} / \mathrm{cost}((\mathbf{c}_{-i}, c_i'))$ . So, $\mathbf{c}$ is not a AV/cost-NE; a contradiction. Again, due to Proposition B.1, + +Data: set $P = \{p_1, p_2, \ldots, p_m\}$ of projects with the delivery costs function $d$ , set $V$ of voters with their ballots, budget $B$ , A/D tie-breaking $\succ$ . + +Result: profile c that is a AV/cost-NE under tie-breaking $\succ$ + +1: $B^{*}\gets B$ //remainingbudget +2: $\mathbf{c}\gets (d(p_1),d(p_2),\ldots ,d(p_m)) = \mathbf{c}_0$ //initial strategy +// let $t(p)$ , for each $p \in P$ , be $\frac{d(p)}{A(p)}$ . +// note that $\succ$ is nondecreasing w.r.t. values of $t$ +// $\mathrm{pos}_X(i)$ denotes the top $i$ th project of order $\succ$ restricted to $X \subseteq P$ +3: $P_{\mathfrak{p}}\gets \{p\in P:c(p)\leq B^{*}\}$ // prospective projects +4: while $P_{\mathrm{p}} \neq \emptyset$ do // are there projects to consider? +5: $P^{\prime}\gets P_{\mathrm{p}}$ // $P^{\prime}$ is a helper variable +6: $d(\mathrm{pos}_{P^{\prime}}(0))\gets -\inf$ // guardian "fake" value +7: $d(\mathrm{pos}_{P^{\prime}}(|P^{\prime}| + 1))\gets +\inf$ // guardian "fake" value +8: $k\gets$ maximum integer $x\leq |P^{\prime}|$ such that $t(\mathrm{pos}_{P^{\prime}}(x))\sum_{i < x - 1}A(\mathrm{pos}_{P^{\prime}}(i))\leq B^{*} < t(\mathrm{pos}_{P^{\prime}}(x + 1))\sum_{i < x + 1}A(\mathrm{pos}_{P^{\prime}}(i))$ +9: $T\gets \mathrm{maximum}x\leq d(\mathrm{pos}_{P^{\prime}}(k + 1)) / |A(\mathrm{pos}_{P^{\prime}}(k + 1))|$ such that $x\sum_{i\in [k]}A(\mathrm{pos}_{P^{\prime}}(i))\leq B^{*}$ +0: for $i = 1$ to $k$ do //updatecand $B^{*}$ +1: $c(\mathrm{pos}_{P^{\prime}}(i))\gets T\cdot A(\mathrm{pos}_{P^{\prime}}(i))$ +2: $B^{*}\gets B^{*} - c(\mathrm{pos}_{P^{\prime}}(i))$ +3: $P_{\mathfrak{p}} \gets P_{\mathfrak{p}} \setminus \{\mathrm{pos}_{P'}(i)\}$ +4: end for +5: if $k < |P'|$ then +6: $P_{\mathrm{p}} \gets P_{\mathrm{p}} \setminus \{\mathrm{pos}_{P'}(k + 1)\}$ +7: end if +8: $P_{\mathfrak{p}} \gets \{p \in P_{\mathfrak{p}} : c(p) \leq B^{*}\}$ +9: end while +C0: return c + +Algorithm 1: Finding a Nash equilibrium for AV/cost. + +in each Nash equilibrium, all projects are selected to be funded. Hence, our proof works for every possible tie-breaking. + +# B.3 Proof of Theorem 4.4 + +Proof. Our Algorithm 1 computes the claimed AV/Cost-NE. + +Let us fix a PB game $G = (P, V, B, d)$ and some corresponding A/D tie-breaking $\succ$ , as specified in the theorem statement. We first introduce helpful notation and discuss $\succ$ in more detail. Then, we proceed with presenting a high-level description of Algorithm 1 that finds the claimed AV/Cost-NE, which we refer to as $\mathbf{c}$ . Eventually, we prove the correctness of the algorithm using Claim 1 and conclude with proving Claim 1 itself. + +Recall that by definition $\succ$ orders the projects $p_i$ nonincreasingly according to approval-to-delivery-cost ratios $|A(p_i)| / d(p_i)$ . Crucially, $\succ$ is always compatible with the order in which AV/Cost considers the projects assuming they report their delivery costs, that is, where for all $p_i \in P$ , $c(p_i) = d(p_i)$ . In what follows, for each set of projects $X \subseteq P$ , each $p_i \in X$ , and the corresponding suborder $\succ_X$ of $\succ$ , we write $\mathrm{pos}_X(p_i)$ to denote the position of $p_i$ in $\succ_X$ . Analogously, for some natural number $x \leq |X|$ , we denote by $\mathrm{top}_X(x)$ the set of top $x$ projects according to $\succ_X$ . + +Algorithm 1 constructs the AV/Cost c iteratively, starting from strategy $\mathbf{c}_0$ in which each project reports its delivery cost (Line 2). In each iteration of the while loop, the algorithm selects a group of projects that "underreport" their costs compared to the support that they get. As the next step, the algorithm increases the reported costs of these projects in strategy AV/Cost as much as possible ensuring that the projects remain funded. If the funded projects after this update do not exhaust the budget, the procedure is repeated for the remaining projects, whose prices have not been increased so far. Such projects, albeit worse regarding the delivery-cost-to-support ratio, can still benefit from + +the increase. The algorithm outputs c if one of the updates step leads to the situation in which the whole budget is used or when there is no more projects underreporting their costs. + +In the following more detailed description of Algorithm 1, we use the notation as specified in the pseudocode and whenever the value $k$ in an iteration of the loop is smaller than $|P'|$ , we define $p^* = \mathrm{pos}_{P'}(k + 1)$ (assuming the value of $P'$ from the corresponding iteration). Central to Algorithm 1 is the while loop. Importantly, due to our basic assumption on the delivery costs of the projects (i.e., that for each $p \in P$ , $d(p) \leq B$ ), initially set $P_{\mathfrak{p}}$ is not empty (Line 3), so the while loop runs at least once. Line 9 of the algorithm guarantees that in each iteration the loop sets the values of $T$ and $k$ to: + +$$ +k < \left| P ^ {\prime} \right| \text {a n d} T = \frac {d \left(p ^ {*}\right)}{A \left(p ^ {*}\right)} \text {o r} \tag {S1} +$$ + +$$ +k < \left| P ^ {\prime} \right| \text {a n d} T \cdot \sum_ {p _ {j} \in \operatorname {t o p} _ {P ^ {\prime}} (k)} | A \left(p _ {j}\right) | = B ^ {*}, \quad \text {o r} \tag {S2} +$$ + +$$ +k = \left| P ^ {\prime} \right| \text {a n d} T \cdot \sum_ {p _ {j} \in \operatorname {t o p} _ {P ^ {\prime}} (k)} | A \left(p _ {j}\right) | = B ^ {*}. \tag {S3} +$$ + +The aforementioned case distinction is crucial for the following claim. + +Claim 1. Right after executing Line 18 in each iteration of the while loop of Algorithm 1, it jointly holds that: + +1. all projects from $\mathrm{top}_{P'}(k)$ are funded under strategy profile $\mathbf{c}$ and none of them has a profitable deviation for $\mathbf{c}$ ; +2. either $k = |P'|$ or there exists a project $p^* = \mathrm{pos}_{P'}(k + 1)$ that is not funded and has no profitable deviation for profile $\mathbf{c}$ ; +3. no project from $\mathrm{top}_{P'}(k)$ has a profitable deviation for each strategy profile $\mathbf{c}'$ that differs from $\mathbf{c}$ only by strategies of projects in $P_{p}$ in a way that, for each $p \in P_{p}$ , $\mathbf{c}'(p) \geq \mathbf{c}(p)$ ; +4. either $k = |P'|$ or the project $p^* = \mathrm{pos}_{P'}(k + 1)$ has no profitable deviation for each strategy profile $\mathbf{c}'$ that differs from $\mathbf{c}$ only on strategies of projects in $P_p$ such that, for each $p \in P_P$ , $\mathbf{c}'(p) \geq \mathbf{c}(p)$ ; +5. no project in $P \setminus P_p$ has a profitable deviation in $\mathbf{c}$ . + +Note that Algorithm 1 ends when executing Line 18 makes $P_{\mathfrak{p}}$ empty. So, by Item 5 of Claim 1 the algorithm returns profile $\mathbf{c}$ which is a Nash equilibrium. Clearly, in each iteration $k > 0$ , so at least one element is removed from $P_{\mathfrak{p}}$ in every iteration of the loop. Consequently, the algorithm always ends and thus it is correct. To conclude the whole proof it remains to show that Claim 1 indeed holds. + +Proof of Claim 1. We provide an inductive argument over the iterations of the while loop of Algorithm 1. For the sake of the argument's simplicity, instead of thinking of AV/Cost considering projects in nonincreasing order of approval-to-cost ratio, we say that it considers projects in nondecreasing order of cost-to-approval ratio. We note that these two interpretations are equivalent. We take the first iteration of the loop as the base case and subsequently show that Items 1 to 5 hold. + +Item 1. We first show that each $p \in \mathrm{top}_{P'}(k)$ is funded in election $E(\mathbf{c})$ . Let us denote as $t(p)$ the ratio $\frac{\mathbf{c}(p)}{|A(p)|}$ . Due to Line 9 and the following foreach loop, we know that every project in $\mathrm{top}_{P'}(k)$ is tied to be considered by AV/Cost due to the same cost-to-approval ratio $T$ . So, if $k = |P'|$ , then all projects in $P_{\mathfrak{p}} = P$ are tied for consideration. Since, by Line 9 all project costs sum up to $B^{*} = B$ , the claim holds. We follow assuming that $k < |P'|$ . In this case, Line 9 shows that $p^{*}$ has cost-to-approval ratio $t(p^{*}) \geq T$ . Recall that $\succ$ orders projects nondecreasingly with respect to their cost-to-approval ratio assuming $\mathbf{c}_0$ and that $\mathbf{c}$ differs from $\mathbf{c}_0$ only in strategies of projects in $\mathrm{top}_{P'}(k)$ . Hence, $p^{*}$ precedes in $\mathbf{c}$ each project except of those in $\mathrm{top}_{P'}(k)$ and all projects in $\mathrm{top}_{P'}(k)$ are considered before $p^{*}$ (the latter holds even in the case of $t(p^{*}) = T$ due to $\succ$ ). Eventually, Line 9 guarantees that the cost of all projects in $\mathrm{top}_{P'}(k)$ does not exceed the budget. Knowing that, we show that no project in $\mathrm{top}_{P'}(k)$ has a profitable deviation in $\mathbf{c}$ . To + +prove it by contradiction, let us assume that some player $p_i \in \mathrm{top}_{P'}(k)$ has a profitable deviation by reporting cost $c_i' > c_i$ ; the opposite case of $c_i' < c_i$ trivially does not yield a payoff improvement for $p_i$ . As a result, $p_i$ has to be funded in the corresponding election $E((\mathbf{c}_{-i}, c_i'))$ . In this election, $p_i$ has cost-to-approval ratio $t' = c_i' / |A(p_i)| > T$ . We further split our analysis into the three cases from Equations (S1) to (S3). + +Assuming Equation (S1), project $p^*$ is also funded in election $E'$ , as it is considered before $p_i$ due to $t' > T$ . Hence, the funded projects cost at least + +$$ +T\cdot \sum_{\substack{p_{j}\in \operatorname{top}_{P^{\prime}}(k)\\ p_{j}\neq p_{i}}} |A(p_{j})| + t^{\prime}|A(p_{i})| + T|A(p^{*})| > +$$ + +$$ +T\cdot \sum_{p_{j}\in \operatorname{top}_{P^{\prime}}(k + 1)}|A(p_{j})|. +$$ + +However, due to Line 8, we know that $B^{*} = B < T\sum_{p_{j}\in \mathrm{top}_{P^{\prime}}(k + 1)}|A(p_{j})|$ , which gives the contradiction with the fact that $p_i$ is funded. + +Suppose Equation (S2) or Equation (S3) hold. Then, given our assumption that $p_i$ is funded, the total cost of funded projects is + +$$ +T\cdot \sum_{\substack{p_{j}\in \operatorname{top}_{P^{\prime}}(k)\\ p_{j}\neq p_{i}}} |A(p_{j})| + t^{\prime}|A(p_{i})| > +$$ + +$$ +T \sum_ {p _ {j} \in \operatorname {t o p} _ {P ^ {\prime}} (k)} | A (p _ {j}) | = B ^ {*} = B, +$$ + +Here, the equality is due to the assumption of Equation (S2) or Equation (S3); which yields the sought contradiction. + +Item 2. If $k = |P'|$ , then the statement trivially holds. Otherwise, let us consider $p^*$ in the light of Equation (S1). Due to Line 8, we have that + +$$ +B < T \sum_ {p _ {j} \in \operatorname {t o p} _ {P ^ {\prime}} (k + 1)} | A (p _ {j}) | = +$$ + +$$ +T \sum_ {p _ {j} \in \operatorname {t o p} _ {P ^ {\prime}} (k)} | (A (p _ {j})) | + d (p ^ {*}). +$$ + +Hence, since all projects in $\mathrm{top}_{P'}(k)$ are funded, $p^*$ is not funded. Naturally, if $p^*$ reports a cost bigger than $d(p^*)$ it will not be funded even more, whereas reporting a cost lower than $d(p^*)$ results in a worse payoff, which finishes the argument for, for Equation (S1). By the condition of Equation (S2), it immediately holds that all projects $\mathrm{top}_{P'}(k)$ use up the whole budget. As a result, again, $p^*$ is not funded and, analogously to the case of Equation (S1), $p^*$ has no profitable deviation. We already observed that Item 2 trivially holds for $k = |P'|$ , which subsumes Equation (S3). + +Items 3 and 4. Observe that assuming strategy $\mathbf{c}$ , all projects in $P_{\mathrm{p}}$ are considered after all projects outside of $P_{\mathrm{p}}$ in election $E(\mathbf{c})$ . This is because order $\succ$ is compliant with the order of considering the projects by AV/Cost and our modifications to the initial strategy profile $\mathbf{c}$ do not change the order of considering projects (note that we let modified projects be tied for consideration with cost-to-approval ratio $T$ ). Furthermore, note that Algorithm 1 never decreases the reported costs of the projects and never considers the same project twice, so the projects in $P_{\mathrm{p}}$ will never be considered before the other ones as a result of further modifications of $\mathbf{c}$ . So, no modification of the profile $\mathbf{c}$ that increases the costs for projects in $P_{\mathrm{p}}$ can influence the decision made for projects in $\operatorname{top}_{P'}(k)$ or $\operatorname{top}_{P'}(k + 1)$ (depending on whether $k < |P'|$ ), as the latter are considered by AV/Cost earlier than projects in $P_{\mathrm{p}}$ . + +Item 5. Note that $P \setminus P_{\mathfrak{p}}$ consists only of the following projects: (i) those removed in the foreach loop, that is, $\mathrm{top}_{P'}(k)$ ; (ii) $p^*$ if $k < |P'|$ ; and (iii) those removed in Line 18. Regarding the first two groups, we have already shown that they do not have a profitable deviation for $\mathbf{c}$ . Let $\hat{p}$ be $\mathrm{top}_{P'}(k)$ if $k < |P'|$ or, otherwise, let $\hat{p}$ be $p^*$ . The last group $Y$ consists of these projects $p \in P'$ , + +for which $\hat{p} \succ p$ and whose $d(p) > B^{*}$ . Hence, by $\succ$ , these projects are considered after all projects in $\operatorname{top}_{P'}(k) \cup \{\hat{p}\}$ . However, projects in $\operatorname{top}_{P'}(k)$ are selected to be funded, which leaves exactly $B^{*}$ remaining budget. So there is not enough budget left to fund any project in $Y$ , even if it reports its delivery cost. Hence, no project in $Y$ has a profitable deviation. + +Thus, we have established the base case and we move on to the induction step. Consider $x > 1$ and the $x$ -th iteration of the while loop, assuming that the base case claims are met for the $(x - 1)$ -th iteration. Due to the assumptions for the $(x - 1)$ -th iteration together with the fact that our algorithm never changes the order $\succ$ in which AV/Cost considers projects and that it never considers the same project twice, we can ignore all projects processed in previous $(x - 1)$ iterations of the while loop. Clearly, the ignored projects will have no impact on the current loop iteration except for decreasing the value of the variable $B^*$ representing the remaining budget. Consequently, the arguments for Items 1 to 4 carry on without changes for the $x$ th iteration of the while loop. The claim from Item 5, however, needs more attention. Let $Q^{(x - 1)} = P \setminus P_{\mathrm{p}}^{(x - 1)}$ be the set of players without profitable deviations after the $(x - 1)$ -th iteration of the loop. Note that $P' = P_{\mathrm{p}}^{(x - 1)}$ . We denote by $R$ the set of projects, that are removed from the initial state of $P_{\mathrm{p}}$ by Lines 13 and 18. That is, $R$ consists of all these projectsthat—due to Items 1 to 4 and the argument for Item 5 in the base case—have no profitable deviation in $\mathbf{c}$ . So, we have that $P_{\mathrm{p}} = P' \setminus R$ . We now consider the set $Q = P \setminus P_{\mathrm{p}}$ of the projects for which we need to show that they have no profitable deviation. Putting the bits together, we observe the following: + +$$ +\begin{array}{l} Q = P \backslash P _ {\mathrm {p}} = P \backslash \left(P ^ {\prime} \backslash R\right) = \\ P \setminus \left(P _ {\mathfrak {p}} ^ {(x - 1)} \setminus R\right) = (P \cap R) \cup \left(P \setminus P _ {\mathfrak {p}} ^ {(x - 1)}\right). \\ \end{array} +$$ + +Since $P \cap R = R$ , we obtained that $Q = R \cup Q^{(x - 1)}$ thus showing that the $x$ -th iteration extends the set of project that do not have a profitable deviation with a collection of project that have no profitable deviation either. As a result, we proved Item 5 for the $x$ -th iteration. + +Having proven Claim 1, we completed the argument for the correctness of Algorithm 1 and Theorem 4.4. + +# B.4 Proof of Proposition 4.5 + +Proof. Take a natural $\gamma > 1$ and consider a PB game $(P, V, B, d)$ where $P = \{p_1, p_2, p_3\}$ , $B = 10$ , $d(p_1) = d(p_2) = 0$ , and $d(p_3) = 10 - \frac{1}{2\gamma}$ . The voters have plurality ballots such that $|A(p_1)| = |A(p_2)| = 1$ and $A(p_3) = 20\gamma - 1$ . We assume tie-breaking order $p_1 \succ p_2 \succ p_3$ . We claim that strategy profile $\mathbf{c}$ such that $\mathbf{c}(p_1) = \mathbf{c}(p_2) = \frac{1}{2\gamma}$ and $\mathbf{c}(p_3) = 10 - \frac{1}{2\gamma}$ is a Phragmén-NE. + +First, we see that if the projects reports costs as in c then Phragmén selects $p_1$ and $p_2$ . Indeed, we see that at time moment $\frac{1}{2\gamma}$ the voters supporting each of the projects have exactly as much money as is need to purchase them. Due to tie-breaking, the singleton voters supporting $p_1$ and $p_2$ buy these projects and, then, there is not enough budget left for $p_3$ and the rule finishes. + +Second, we observe that no project can benefit by changing its strategy under $\mathbf{c}$ . Indeed, if either $p_1$ or $p_2$ decreased their cost, they would obtain lower payoff, and if either of them increased their cost, $p_3$ would be funded instead and the payoff of the project that increased its cost would drop to zero. If $p_3$ decreased its cost then it would be selected, but its payoff would become negative (due to the delivery cost), and if it increased its cost then its payoff would remain zero. Consequently, $\mathbf{c}$ is Phragmén-NE. + +Finally, we have $\mathbf{c}(p_1) + \mathbf{c}(c_2) = 1 / \gamma$ , and as $B = 10$ , this is less than $B / \gamma$ . This concludes the proof. + +# B.5 Proof of Proposition 4.7 + +Proof. Consider a PB game $G$ and the strategy profile $\mathbf{c}$ as defined in the statement of the proposition. We claim that $\mathbf{c}$ is a Phragmén-NE. First, we observe that under $\mathbf{c}$ Phragmén selects all the projects, so it is not beneficial for any of them to report a lower cost. On the other hand, if some project $p$ reports a higher cost, this project is not selected by Phragmén. To see why this is the case, consider some project $p$ . Under $\mathbf{c}$ , the total cost of the projects in $party(p)$ is $B \cdot |A(p)| / |V|$ . Since + +each of the $|A(p)|$ voters supporting these projects earns one unit of money per one unit of time, altogether they earn this money in time $B / |V|$ . This value is independent of $p$ so, under $\mathbf{c}$ , the last project of each party is funded at the same time. Hence, if some project increased its price, it would be considered even later and, by that time, there would not be enough budget left. Hence, it is never beneficial to increase a cost and so, $\mathbf{c}$ is a Phragmén-NE. + +It remains to show that $\mathbf{c}$ is the unique Phragmén-NE for our game. To this end, let $\mathbf{c}'$ be some arbitrary equilibrium for $G$ . By Proposition B.1, we know that under $\mathbf{c}'$ the reported costs of all projects sum up to $B$ and all projects are funded. Next, we observe that for each project $p$ , all projects from party $(p)$ report the same cost. Indeed, if there were two projects, $p'$ and $p'' \in \text{party}(p)$ , such that $\mathbf{c}'(p') < \mathbf{c}'(p'')$ , then it would be beneficial for $p'$ to report a higher cost (but below $\mathbf{c}'(p'')$ ), so that Phragmén would still consider and fund it prior to $p''$ (for which, then, there would not be enough budget left). + +Finally, if there are two projects, $p$ and $q$ , such that $\text{party}(p) \neq \text{party}(q)$ , then, under $\mathbf{c}'$ , the last project from $\text{party}(p)$ and the last project from $\text{party}(q)$ are selected by Phragmén at the same time. Indeed, if this were not the case, then it would be beneficial for the one selected earlier to report higher cost (but so that it still is selected at an earlier time than the other project). + +Altogether, the only strategy profile that satisfies the properties described above is $\mathbf{c}$ as defined in the statement of the proposition. + +# B.6 Proof of Proposition 4.8 + +Proof. Let our project set be $P = P' \cup P''$ , where $P' = \{p_1', p_2', p_3'\}$ and $P'' = \{p_1'', p_2'', p_3''\}$ . Similarly, the set of voters is $V = V' \cup V''$ , where $V' = \{v_{1'}, v_{2'}, v_{3'}\}$ and $V'' = \{v_{1'', v_{2'', v_{3'}}}\}$ . The approvals are as follows (see also Figure 3 for illustration): + +1. $p_1^\prime$ is approved by $v_{1^{\prime}}$ , $p_2^\prime$ is approved by $v_{2^{\prime}}$ , and $p_3^\prime$ is approved by all the voters in $V^{\prime}$ . +2. The approvals for the projects in $P''$ are analogous, except that they are approved by the voters from $V''$ . + +We set the delivery costs $d$ to be zero for every project, and we set the budget $B$ to be 1 (the exact value will be irrelevant as we will operate on times when Phragmén reaches the costs of particular projects rather than on directly on these costs). We claim that under Phragmén there are no Nash equilibria for the thus defined PB game $G = (P, V, B, d)$ . + +![](images/4fa9d9f9d48d89f06b2b4b8795a334ef35b398448599d3286e957e10d64e9ee8.jpg) +Figure 3: Illustration of the PB game from the proof of Proposition 4.8. The projects are depicted as boxes. Each voter approves those projects that are drawn directly above him or her (and are crossed by the dotted line). + +![](images/901afc11d39d7ecd0788d322ff9b823f6b30be9c0a92cc80071853dc9680cf9a.jpg) + +For the sake of contradiction, let us assume that there is Phragmén-NE for $G$ and let it be $\mathbf{c}^*$ . For each project $p \in P$ , let $time(p)$ be the time moment (in the sense of the Phragmén rule) when the voters approving $p$ would collect exactly $\mathbf{c}^*(p)$ amount of money (assuming that neither of these voters spends it on any other projects in between; hence $time(p) = \frac{\mathbf{c}^*(p)}{|A(p)|}$ ). We make the following observations: + +1. All projects in $P$ are funded for the reported costs $\mathbf{c}^*$ (if some project were not funded, it would prefer to report a small nonzero cost that would be at most equal to $B$ and that would ensure that Phragmén considers it first). +2. It must be the case that $time(p_1') = time(p_2')$ . Indeed, if we had $time(p_1') < time(p_2')$ then it would be beneficial for $p_1'$ to slightly increase its cost, but so that $time(p_1')$ would still be below $p_2'$ . Then, irrespective in what order are the other projects funded, $p_1'$ 's voter would collect enough money to purchase $p_1'$ before the $p_2'$ 's voter would, and $p_1'$ would be + +bought (since $p_2'$ would not be selected at this time yet, there would be enough budget left for this). This would contradict that $\mathbf{c}^*$ is an equilibrium. The case $time(p_2') < time(p_1')$ is symmetric. + +3. It must be the case that $time(p_3') = time(p_1')$ and, hence, also equal to $time(p_2')$ . Indeed, if we had $time(p_3') < time(p_1') = time(p_2')$ then it would be beneficial for $p_3'$ to report slightly higher cost, so that $time(p_3')$ would still be smaller than $time(p_1') = time(p_2')$ , yet $p_3'$ 's cost would not have increased by more than the total costs of $p_1'$ and $p_2'$ . Then $p_3'$ would still be selected before $p_1'$ and $p_2'$ and there would be sufficient amount of budget left for it. This would contradict that $\mathbf{c}^*$ is an equilibrium. On the other hand, if $time(p_3') > time(p_1') = time(p_2')$ then it would be beneficial, e.g., for $p_1'$ to report slightly higher cost, but ensuring that $time(p_1') < time(p_3')$ . Indeed, if originally we have $time(p_1') < time(p_3')$ , then $p_1'$ is funded before $p_3'$ by Phragmén. After the increase, $p_1'$ would still be purchased before $p_3'$ and, thus, there would still be sufficient budget for it. This would contradict that $\mathbf{c}^*$ is an equilibrium. + +4. By reasoning analogous to the one given above, it must be the case that $time(p_1^{\prime \prime}) = time(p_2^{\prime \prime}) = time(p_3^{\prime \prime})$ . + +Given the above observations, we set: + +$$ +\operatorname {t i m e} ^ {\prime} = \operatorname {t i m e} \left(p _ {1} ^ {\prime}\right) = \operatorname {t i m e} \left(p _ {2} ^ {\prime}\right) = \operatorname {t i m e} \left(p _ {3} ^ {\prime}\right), \text {a n d} +$$ + +$$ +t i m e ^ {\prime \prime} = t i m e \left(p _ {1} ^ {\prime \prime}\right) = t i m e \left(p _ {2} ^ {\prime \prime}\right) = t i m e \left(p _ {3} ^ {\prime \prime}\right). +$$ + +Note that this implies that $\mathbf{c}^*(p_1') = \mathbf{c}^*(p_2') = time'$ , $\mathbf{c}^*(p_1'') = \mathbf{c}^*(p_2'') = time''$ , $\mathbf{c}^*(p_3') = 3 \cdot time'$ , and $\mathbf{c}^*(p_3'') = 3 \cdot time''$ . We claim that $time' = time''$ . Indeed, let us consider what happens if $time' < time''$ (the case where $time'' < time'$ is symmetric). There are two cases two consider (the second case also splits into two). + +Case A First, let us assume that at least one of $p_1'$ and $p_2'$ is preferred to $p_3'$ by the tie-breaking order (for the sake of specificity, let this be $p_1'$ ). Then it is beneficial for $p_2'$ to slightly increase its cost. To see why this is the case, let us consider how Phragmén operates after this increase. At time $time(p_1') = time(p_3')$ , the voters have enough funds to purchase either $p_1'$ or $p_3'$ (and not enough to purchase $p_2'$ , who increased its cost). The rule selects $p_1'$ due to tie-breaking. Consequently, voter $1'$ pays for $p_1'$ and his or her virtual bank account is reset to zero. Voters $2'$ and $3'$ still have $time'$ amount of money. The cost of $p_3'$ is $3 \cdot time'$ , so voters in $N'$ will have collected enough money for it after further $1/3time'$ amount of time. However, if $p_2'$ increased its cost from $time'$ to an amount a bit below $4/3 \cdot time'$ then its voter will have collected this amount earlier, and $p_2'$ will be purchased before $p_3'$ . This contradicts the fact that $c^*$ is an equilibrium. + +Case B' The second case is that $p_3'$ is preferred by tie-breaking to both $p_1'$ and $p_2'$ . Then it is beneficial for $p_3'$ to slightly increase its cost. Again, let us consider how Phragmén operates after such a change. If $p_3''$ is selected prior to both $p_1''$ and $p_2''$ , then at time time (after the purchase of $p_3''$ ) the voters have the following amounts of money: + +1. Voter $3'$ has time'' amount of money and the remaining voters in $N'$ have nonzero amounts of money (specifically, each of them has time'' - time', because they paid for $p_1'$ and $p_2'$ at time time'). +2. All the voters in $V''$ have empty bank accounts. + +Hence, only after another time'' amount of time will the voters in $N''$ have enough money to purchase $p_1''$ and $p_2''$ . Yet, at this time the voters in $N'$ would, altogether, have more than 4time''. So, if $p_3'$ increased its cost to be between $3 \cdot \text{time}'$ and $4 \cdot \text{time}'$ (which is smaller than $4 \cdot \text{time}''$ ), then $p_3'$ would be funded before $p_1''$ and $p_2''$ , and there would be sufficient amount of budget left for this. + +Case B' On the other hand, if at least one of $p_1''$ and $p_2''$ is preferred to $p_3''$ by tie-breaking at time $time''$ , then both $p_1''$ and $p_2''$ are funded at that time (because after one of them is selected due to tie-breaking, the voters in $V''$ no longer have enough money to purchase $p_3''$ , but they do have enough for the other one among $p_1''$ and $p_2''$ ). Consequently, after $p_1''$ and $p_2''$ are purchased at time $time''$ , only projects $p_3'$ and $p_3''$ are not selected yet and the voters have the following amounts of money: + +1. Voter $v_{3'}$ has time'' amount of money and the remaining voters in $V'$ have nonzero amounts of money (specifically, each of them has time'' - time', because they paid for $p_1'$ and $p_2'$ at time time'). +2. Voter $v_{3''}$ has time'' amount of money and the remaining voters in $V''$ have no money. + +Since voters in $V'$ have, together, more money than those in $V''$ , but voters from both groups (jointly) earn money at the same rate (as $|V'| = |V''|$ ), if $p_3'$ increased its cost to be slightly below that of $p_3''$ , it will be funded before $p_3'$ . Consequently, if $\mathbf{c}^*$ is an equilibrium then we must have $time' = time''$ . + +Finally, it suffices to show that the assumption $time' = time''$ also leads to no equilibrium. W.l.o.g., we assume that $p_3'$ precedes both $p_1'$ and $p_2'$ in the tie-breaking order, and that $p_3''$ precedes both $p_1''$ and $p_2''$ (if this were not the case, then the arguments given in Case A above would still show that $\mathbf{c}^*$ is not an equilibrium). However, now we can see that, e.g., it is beneficial for $p_3'$ to slightly increase its cost. This follows by the same reasoning as given in Case B'. Hence, we have reached a contradiction. Consequently, under Phragmén there is no Nash equilibrium in our game. + +# B.7 Proof of Theorem 4.9 + +Proof. Let $(P, V, B, d)$ be a MES-Cost PB game. + +We provide an iterative algorithm that computes $\mathbf{c}$ . In each iteration, the algorithm fixes selected values of $\mathbf{c}$ , drops the corresponding projects from further consideration and deletes the voters that have no budget left. The algorithm finishes when there is no more projects to consider. Before laying out the details, let us recall that in MES-Cost, each voter gets the equal share of the budget and cannot spend on the projects more than their entitlement. + +Our algorithm constructs the strategy $\mathbf{c}$ step by step maintaining the collection $V'$ of voters to consider and the collection $P'$ of projects to consider. The algorithm starts with setting $V' = V$ , $P' = P$ and proceeds as follows: + +1. Remove from $P'$ all projects $p_i \in P'$ for which $|A(p_i) \cap V'| \cdot B / |V|$ is lower than $d(p_i)$ or equal to zero and let their strategy be $\mathbf{c}(p_i) = d(p_i)$ . +2. For each project $p_j \in P'$ , let $\alpha_j = \frac{1}{|A(p_j) \cap V'|}$ (due to Step 1 we avoid dividing by zero) and let $p^*$ be the project $p_j \in P'$ with the minimum $\alpha_j$ -value (in case of a tie, select the first one in the tie-breaking order). +3. Set the strategy $\mathbf{c}(p^*)$ to $|A(p^*) \cap V'| \cdot B / |V|$ , that is, such that $p^*$ reports the cost equal to the total budget of its supporters in $V'$ . +4. Remove all supporters of $p^*$ from $V'$ and remove $p^*$ from $P'$ . +5. Repeat Steps 1 to 4 until $P'$ is empty. + +Our algorithm is based on the following useful claim about the decisions made by MES-Cost. + +Claim 2. Let us fix some stage of MES-Cost in which every voter with a nonzero budget has the same value of budget and let $V'$ be these voters. Then, for each $\alpha'$ -affordable project $p'$ it holds that that $\alpha' = 1 / |A(p') \cap V'|$ . + +The claim is implied by the definition of MES-Cost. Observe that if a project $p$ is $\alpha$ -affordable, then the voters approving it have enough budget to buy it and $\alpha$ is the minimum such $x$ that $|A(p') \cap V'| \cdot x \cdot cost(p) \leq cost(p)$ . The sought $\alpha$ is then clearly $1 / |A(p') \cap V'|$ . + +Let us denote by $(p_1^*, p_2^*, \ldots, p_k^*)$ the projects considered in the respective iterations 1 to $k$ of Steps 1-4. In what follows we show that in each Nash equilibrium MES-Cost selects exactly projects $(p_1^*, p_2^*, \ldots, p_k^*)$ in the given order and that $\mathbf{c}$ is a Nash equilibrium. We apply induction over the stages of MES-Cost. + +For the base case, we argue that $p_1^*$ has to be selected first by MES-Cost and has to play $\mathbf{c}(c_1^*)$ in every Nash equilibrium. Assume for contradiction that some other project $p'$ is selected first instead + +![](images/4ea0758b5a36ebd46dc4eb237e1561e36272321f2aac05df678d782a187be4f0.jpg) +Figure 4: Approval sets of projects $p_1, p_2, p_3$ , and $p_4$ . Here, each vertex represents a single voter. + +of $p_1^*$ in a Nash equilibrium. Due to Claim 2 and since MES-Cost selects $\alpha$ -affordable projects starting from those with the smallest $\alpha$ , it follows that one of the three holds: (i) $\frac{1}{|A(p')|} < \frac{1}{|A(p_1^*)|}$ , (ii) $\frac{1}{|A(p')|} = \frac{1}{|A(p_1^*)|}$ and $p'$ is preferred to $p_1^*$ by the tie-breaking, or (iii) $p_1^*$ reports a cost greater than the budget of its supporters. Cases (i) and (ii) yield a clear contradiction to Step 2, which defines $p_1^*$ . In Case (iii), reporting cost $\mathbf{c}'(p_1^*) = d(p_1^*) + \epsilon \leq B \cdot |A(p_1^*)| / |V|$ (such an $\epsilon$ always exists due to Step 1) is a profitable deviation for $p_1^*$ —a contradiction. Knowing that $p_1^*$ is selected first in every Nash equilibrium, we observe that if $p_1^*$ reports a cost $\mathbf{c}'(p_1^*) < B \cdot |A(p_1^*)| / |V|$ , then reporting exactly $\mathbf{c}(p_1^*) = B \cdot |A(p_1^*)| / |V|$ is always a profitable deviation, which confirms that in every Nash equilibrium $p_1^*$ 's strategy is $\mathbf{c}(p_1^*)$ and it is selected first by MES-Cost. + +We move on to the inductive step and thus consider stage $i$ of MES-Cost in which project $p_i^*$ is selected. Importantly, as we have shown that player $p_1^*$ has to report cost $\mathbf{c}(p_1^*)$ equal to the total budget of $p_1^*$ supporters, then we can assume that at the $i$ -th stage we have only voters who either spend all their budget or who did not spend their budget at all; we denote the latter group by $V'$ . As a result, Claim 2 holds at the $i$ -th stage, which we consider. Thanks to this, we apply an analogous exchange argument (pretending that voters outside of $V'$ do not exist) to that for the base case to show that indeed $p_i^*$ has to be selected at the $i$ -th stage. Then, we directly repeat the argument regarding the optimal strategy for $p_1^*$ applying it to $p_i^*$ . Eventually, we have shown that in each Nash equilibrium for MES-Cost projects $(p_1^*, p_2^*, \ldots, p_k^*)$ are selected and report, respectively, costs $(\mathbf{c}(p_1^*), \mathbf{c}(p_2^*), \ldots, \mathbf{c}(p_k^*))$ . + +# B.8 Proof of Theorem 4.10 + +Proof. Let $(P, V, B, d)$ be a MES-Apr PB game with plurality ballots. In MES-Apr, like in every MES rule, each voter receives $B / |V|$ money for the whole election process. As each voter approves only one project, each project $p_i$ can request at most $M(p_i) = |A(p_i)| \cdot B / |V|$ from its supporters, and will report this cost as they would not spend any money on other projects. Thus, if $d(p_i) \leq M(p_i)$ , project $p_i$ reports $M(p_i)$ and gets selected with its maximum possible cost. Otherwise, if $d(p_i) > M(p_i)$ , $p_i$ cannot be selected with cost covering $d(p_i)$ , so $p_i$ reports $d(p_i)$ and is not selected. + +# B.9 Proof of Proposition 4.11 + +Proof. We create four projects $p_1, \ldots, p_4$ and 16 voters $v_1, \ldots, v_{16}$ . Project $p_1$ is approved by voters $v_1, \ldots, v_5$ ; project $p_2$ by voters $v_{13}, \ldots, v_{16}, v_1$ ; project $p_3$ by voters $v_5, \ldots, v_9$ ; and project $p_4$ by voters $v_9, \ldots, v_{13}$ . We set the budget to be some positive integer $B$ and delivery costs to be $d \equiv 0$ . Please note that we do not specify tie-breaking as we prove nonexistence of MES-Apr-NE in any tie-breaking. + +For a better visualization of the instance, please look at the Figure 4. In particular, one can see that the instance is symmetric (each project is approved by three voters supporting only it and by two voters, one shared with each neighbour). + +Now we will show that there is no MES-Apr-NE for this instance. Suppose towards contradiction that some profile $\mathbf{c}$ is a MES-Apr-NE. + +At the beginning, each voter receives $b = \frac{B}{|V|} = \frac{B}{16}$ money. Therefore, a) no project says cost exceeding $5b = \frac{5B}{16}$ (otherwise its supporters would never have enough money to buy it, and b) no project says less than $3b = \frac{3B}{16}$ (as each project has three supporters that contribute only to it, it is nonoptimal to say less than they have). Further, as each project has zero delivery costs, each project must be selected in MES-Apr-NE, otherwise an unselected project could have decreased its cost to 0 and be selected. For this reason, $\sum_{i=1}^{4} \mathbf{c}(p_i) \leq B$ . + +For the sake of brevity, for yet-unselected project $p_i$ in iteration $j$ , we denote $\alpha_j(p_i)$ as minimum value such that project $p_i$ is $\alpha_j(p_i)$ -affordable according to MES-Apr rule (or infinity, if such a value does not exist). By $\epsilon \in \mathbb{R}_+$ we mean some very small positive real number such that adding it does not change the given strict inequality sign. + +Due to symmetry, w.l.o.g. we can assume that 1) project $p_1$ says the lowest cost (i.e., $\mathbf{c}(p_1) \leq \mathbf{c}(p_2), \mathbf{c}(p_3), \mathbf{c}(p_4)$ ) and wins ties if there is more than more project with this cost as well as 2) $\mathbf{c}(p_2) \leq \mathbf{c}(p_3)$ and $p_2$ wins ties with $p_3$ . If it was not the case, we can shift or rotate projects and perform analogous reasoning. + +Before we move on, let us exclude some cases in which c cannot be MES-Apr-NE. + +We point out that if there is no project of the same cost as $p_1$ , then $p_1$ could have increased its cost by some positive number $\epsilon$ while still being considered first and getting more, so $\mathbf{c}$ could not have been MES-Apr-NE in this case. Therefore, $p_1$ has the same cost either as 1) $p_4$ , or as 2) $p_2$ (please note $\mathbf{c}(p_1) = \mathbf{c}(p_3)$ implies $\mathbf{c}(p_1) = \mathbf{c}(p_2)$ due to assumption that $\mathbf{c}(p_2) \leq \mathbf{c}(p_3)$ ). + +With our assumptions, project $p_1$ is selected in the first iteration and each of its supporters pays $\frac{\mathbf{c}(p_1)}{5}$ for $p_1$ . As $b - \frac{\mathbf{c}(p_i)}{5} \leq \frac{B}{16} - \frac{\frac{3B}{16}}{5} = \frac{\frac{2B}{16}}{5} < \frac{\frac{3B}{16}}{5} \leq \frac{\mathbf{c}(p_j)}{5}$ for any $p_i, p_j \in P$ , voter $v_1$ has insufficient money to contribute the equal share $\frac{\mathbf{c}(p_2)}{5}$ to purchase $p_2$ . Therefore, in the second iteration, $\alpha_2(p_4) = \frac{\mathbf{c}(p_4)}{5}$ , $\alpha_2(p_2) = \frac{\mathbf{c}(p_2) - (b - \frac{\mathbf{c}(p_1)}{5})}{4}$ , and $\alpha_2(p_3) = \frac{\mathbf{c}(p_3) - (b - \frac{\mathbf{c}(p_1)}{5})}{4} \geq \alpha_2(p_2)$ . + +Please note that $\alpha_{2}(p_{2}) = \frac{\mathbf{c}(p_{2}) - (b - \frac{\mathbf{c}(p_{1})}{5})}{4} \geq \frac{\mathbf{c}(p_{1}) - (\frac{B}{16} - \frac{\mathbf{c}(p_{1})}{5})}{4} = \frac{6\mathbf{c}(p_{1})}{20} - \frac{B}{64} = \frac{4\mathbf{c}(p_{1})}{20} + \left(\frac{2\mathbf{c}(p_{1})}{20} - \frac{B}{64}\right) \geq \frac{\mathbf{c}(p_{1})}{5} + \left(\frac{2 \cdot \frac{3B}{16}}{20} - \frac{B}{64}\right) = \frac{\mathbf{c}(p_{1})}{5} + \frac{6B - 5B}{16 \cdot 20} = \frac{\mathbf{c}(p_{1})}{5} + \frac{B}{16 \cdot 20} > \frac{\mathbf{c}(p_{1})}{5} = \alpha_{1}(p_{1})$ . Thus, as $\alpha_{2}(p_{2}) > \alpha_{1}(p_{1})$ , project $p_4$ would regret saying the same cost as project $p_1$ . Indeed, selecting $p_1$ in the first iteration significantly increases $\alpha_{2}(p_{2})$ and $\alpha_{2}(p_{3})$ , so if $p_4$ said $\mathbf{c}(p_4) = \mathbf{c}(p_1)$ , then it would benefit from slightly increasing the cost by $\epsilon$ (it would still be before $p_2$ , but it would earn more). For this reason, it cannot be the case that $\mathbf{c}(p_1) = \mathbf{c}(p_4)$ , so due to the former reasoning we know that $\mathbf{c}(p_1) = \mathbf{c}(p_2)$ and $\mathbf{c}(p_1) < \mathbf{c}(p_4)$ . + +So we know that: $\mathbf{c}(p_1) = \mathbf{c}(p_2)$ , $\mathbf{c}(p_1) < \mathbf{c}(p_4)$ , and $\mathbf{c}(p_2) \leq \mathbf{c}(p_3)$ . We have two cases to consider: + +- Project $p_4$ is selected in the second iteration. It means that $\alpha_{2}(p_{4}) \leq \alpha_{2}(p_{2})$ . We observe that it must be $\alpha_{2}(p_{4}) = \alpha_{2}(p_{2})$ , otherwise it would be beneficial for $p_4$ to slightly increase its cost by $\epsilon$ in such a way that $p_4$ is still considered before $p_2$ and gains more. + +In this case, as $p_2$ and $p_3$ are approved by disjoint sets of voters and both $p_1$ and $p_4$ took the same amount of money from the same number of their supporters, both $p_2$ and $p_3$ should request for the same amount of money, that is, all money that their supporters have. Thus $\mathbf{c}(p_3) = \mathbf{c}(c_2) = 3 \cdot b + (b - \frac{\mathbf{c}(p_1)}{5}) + (b - \frac{\mathbf{c}(p_4)}{5}) = 5b - \frac{\mathbf{c}(p_1)}{5} - \frac{\mathbf{c}(p_4)}{5}$ and $\mathbf{c}(p_4) = 5 \cdot (5b - \frac{\mathbf{c}(p_1)}{5} - \mathbf{c}(p_2)) = 25b - 5\mathbf{c}(p_2) - \mathbf{c}(p_1)$ . + +However, we know that $\mathbf{c}(p_1) = \mathbf{c}(p_2)$ , so $\mathbf{c}(p_1) = \mathbf{c}(p_2) = \mathbf{c}(p_3) = 5b - \frac{\mathbf{c}(p_1)}{5} - \frac{\mathbf{c}(p_4)}{5}$ , which implies that $\mathbf{c}(p_4) = 25b - 5\mathbf{c}(p_2) - \mathbf{c}(p_1) = 25b - 6\mathbf{c}(p_1)$ . Therefore, $\alpha_{2}(p_{4}) = \alpha_{2}(p_{2})$ means that $\frac{\mathbf{c}(p_4)}{5} = \frac{\mathbf{c}(p_2) - (b - \frac{\mathbf{c}(p_1)}{5})}{4} = \frac{6\mathbf{c}(p_1)}{20} - \frac{b}{4}$ , which is equivalent to $\frac{25b - 6\mathbf{c}(p_1)}{5} = \frac{6\mathbf{c}(p_1) - 5b}{20} \Rightarrow 100b - 24\mathbf{c}(p_1) = 6\mathbf{c}(p_1) - 5b \Rightarrow 30\mathbf{c}(p_1) = 105b \Rightarrow \mathbf{c}(p_1) = \frac{7b}{2}$ . Then we have $\mathbf{c}(p_1) = \mathbf{c}(p_2) = \mathbf{c}(p_3) = \frac{7b}{2}$ and $\mathbf{c}(p_4) = 25b - 6\mathbf{c}(p_1) = 25b - 21b = 4b$ . + +Now imagine that project $p_1$ changes its cost from $\mathbf{c}(p_1) = \frac{7b}{2}$ to $\mathbf{c}'(p_1) = \frac{36b}{10}$ . Naturally, projects $p_2$ and $p_3$ are selected in first two iterations so voters $v_1, \ldots, v_5$ supporting $p_1$ are left with $b - \frac{\mathbf{c}(c_2)}{5}, b, b, b, b - \frac{\mathbf{c}(p_3)}{5}$ money respectively and they have $b - \frac{\mathbf{c}(c_2)}{5} + b + b + b + b - \frac{\mathbf{c}(p_3)}{5} = 5b - 2 \cdot \frac{7b}{10} = \frac{36b}{10}$ money in total which they can only spend on project $p_1$ as they disapprove $p_4$ . So $p_1$ gets selected with $\mathbf{c}'(p_1) = \frac{36b}{10} > \frac{7b}{2} = \mathbf{c}(p_1)$ which indicates that $\mathbf{c}$ could not be MES-Apr-NE. + +- Project $p_2$ is selected in the second iteration. It means that $\alpha_2(p_2) \leq \alpha_2(p_4)$ . We observe that it must be $\alpha_2(p_2) = \alpha_2(p_4)$ , otherwise it would be beneficial for $p_2$ to slightly increase its cost in such a way that $p_2$ is still considered before $p_4$ and gains more (please note that even if $p_3$ would jump before $p_2$ , it takes money from the disjoint set of voters from $p_2$ 's supporters, so it does not disprove this argument). + +By performing analogous calculations to the previous case and using $\mathbf{c}(p_1) = \mathbf{c}(p_2)$ we obtain that $\alpha_{2}(p_{2}) = \frac{\mathbf{c}(p_{2}) - (b - \frac{\mathbf{c}(p_{1})}{5})}{4} = \frac{6\mathbf{c}(p_{1})}{20} -\frac{b}{4}$ , so $\alpha_{2}(p_{4}) = \alpha_{2}(p_{2})\Rightarrow \frac{\mathbf{c}(p_{4})}{5} = \frac{6\mathbf{c}(p_{1})}{20} -\frac{b}{4}\Rightarrow \mathbf{c}(p_{4}) = \frac{6\mathbf{c}(p_{1}) - 5b}{4}$ . + +After $p_1$ and $p_2$ are selected, voter $v_1$ has 0 money left, voters $v_2, \ldots, v_5$ have $b - \frac{\mathbf{c}(p_1)}{5}$ money left, voters $v_{13}, \ldots, v_{16}$ have $b - \alpha_2(p_2) = b - (\frac{6\mathbf{c}(p_1)}{20} - \frac{b}{4}) = \frac{25b - 6\mathbf{c}(p_1)}{20}$ money left, and voters $v_6, \ldots, v_{12}$ have $b$ money left. + +We need to consider two cases: + +- $p_3$ is selected before $p_4$ . Then $\mathbf{c}(p_3) = \mathbf{c}(p_2)$ , otherwise (if $\mathbf{c}(p_2) < \mathbf{c}(p_3)$ ) $p_2$ would increase its cost by $\epsilon$ in such a way that $\mathbf{c}(p_2) + \epsilon < \mathbf{c}(p_3)$ and still be selected in the second iteration, but with more money. Because $\mathbf{c}(p_3) = \mathbf{c}(p_2) = \mathbf{c}(p_1)$ and the supporters of $p_2$ and $p_3$ are disjoint and symmetric, after selecting $p_3$ , voter $v_5$ be left with no money whereas voters $v_6, \ldots, v_9$ with $b - \alpha_3(p_3) = b - \alpha_2(p_2) = \frac{25b - 6\mathbf{c}(p_1)}{20}$ . Further, in the fourth iteration, the supporters of $p_4$ have together $3b + 2 \cdot \frac{25b - 6\mathbf{c}(p_1)}{20} = \frac{30b + 25b - 6\mathbf{c}(p_1)}{10} = \frac{55b - 6\mathbf{c}(p_1)}{10}$ , so $p_4$ asks for the money they still have in total. + +Therefore, $\mathbf{c}(p_4) = \frac{6\mathbf{c}(p_1) - 5b}{4} = \frac{55b - 6\mathbf{c}(p_1)}{10} \Rightarrow 30\mathbf{c}(p_1) - 25b = 110b - 12\mathbf{c}(p_1) \Rightarrow 42\mathbf{c}(p_1) = 135b \Rightarrow \mathbf{c}(p_1) = \frac{135b}{42}$ . Thus $\mathbf{c}(p_4) = \frac{6\mathbf{c}(p_1) - 5b}{4} = \frac{6 \cdot \frac{135b}{42} - 5b}{4} = \frac{135b - 35b}{4 \cdot 7} = \frac{100b}{28} = \frac{150b}{42}$ . + +Imagine what happens if $p_1$ increases its cost from $\mathbf{c}(p_1) = \frac{135b}{42}$ to $\mathbf{c}'(p_1) = \frac{156b}{42}$ . Then $p_2$ gets selected first with cost $\mathbf{c}(p_2) = \frac{135b}{42}$ and each of its supporters pays $\frac{\mathbf{c}(p_2)}{5} = \frac{\frac{135b}{42}}{5} = \frac{27b}{42}$ . Next, each of $p_3$ 's supporters will pay $\frac{27b}{42}$ to purchase $p_3$ (voters supporting $p_2$ and $p_3$ are disjoint). After that, regardless whether $p_4$ is selected before $p_1$ or not, the supporters of $p_1$ have $3b + 2 \cdot (b - \frac{27b}{42}) = \frac{3 \cdot 42 + 2 \cdot 15b}{42} = \frac{156b}{42}$ , so the supporters of $p_1$ have enough money to purchase $p_1$ . We showed that $p_1$ could improve its utility by changing its cost, so the proposed $\mathbf{c}$ is not MES-Apr-NE in this case. + +- $p_4$ is selected before $p_3$ . It means that $\alpha_3(p_4) \leq \alpha_3(p_3)$ . As we know how much money each voter has in the third iteration, $\alpha_3(p_4) = \frac{\mathbf{c}(p_4) - \frac{25b - 6\mathbf{c}(p_1)}{20}}{4} = \frac{20\mathbf{c}(p_4) - 25b + 6\mathbf{c}(p_1)}{80}$ and $\alpha_3(p_3) = \frac{\mathbf{c}(p_3) - \frac{\mathbf{c}(p_1)}{5}}{4} = \frac{20\mathbf{c}(p_3) - 4\mathbf{c}(p_1)}{80}$ . After inserting the exact values we obtain that $\alpha_3(p_4) \leq \alpha_3(p_3) \Rightarrow \frac{20\mathbf{c}(p_4) - 25b + 6\mathbf{c}(p_1)}{80} \leq \frac{20\mathbf{c}(p_3) - 4\mathbf{c}(p_1)}{80} \Rightarrow 20\mathbf{c}(p_3) \geq 20\mathbf{c}(p_4) - 25b + 10\mathbf{c}(p_1) \Rightarrow \mathbf{c}(p_3) \geq \mathbf{c}(p_4) + \frac{\mathbf{c}(p_1)}{2} - \frac{5b}{4}$ . However, $\mathbf{c}(p_3) \geq \mathbf{c}(p_4) + \frac{\mathbf{c}(p_1)}{2} - \frac{5b}{4} > \mathbf{c}(p_4) + \frac{3b}{2} - \frac{5b}{4} = \mathbf{c}(p_4) + \frac{b}{4} > \mathbf{c}(p_1) + \frac{b}{4} = \mathbf{c}(p_2) + \frac{b}{4}$ , so $\mathbf{c}(p_3)$ is significantly greater than $\mathbf{c}(p_2)$ . In other words, even when voter $v_9$ pays a greater share to buy $p_4$ and is left with less money than $\frac{\mathbf{c}(p_4)}{5}$ , there is still enough money to purchase $p_3$ . Therefore, $p_2$ can increase its cost from $\mathbf{c}(p_2) = \mathbf{c}(p_1)$ to $\mathbf{c}(p_3) - \epsilon$ and lose in the second iteration with $p_4$ , but still be funded before $p_3$ and obtain more money. For this reason, $\mathbf{c}$ is not MES-Apr-NE. + +We showed that in no case MES-Apr-NE exists for this instance. The remaining tie-breaking order are completely analogous due to the instance's symmetry. + +It means that for the given instance there is no tie-breaking for which MES-Apr-NE exists. + +![](images/2216d055bdb9630ce13be30f692185f2b9cbd0278a574c7550c80abba85a7a3c.jpg) + +# B.10 Proof of Theorem 4.12 + +Proof. We first consider the case of zero delivery costs, as it is simpler. Let $(P,N,B,d)$ be a MES-Apr PB game with party-list ballots and zero delivery costs. + +At the beginning, each voter receives $\frac{B}{|V|}$ money. In a party-list profile, as every two voters have either the same preferences or the disjoint ones, $A(p_i) = A(\text{party}(p_i))$ . The total money the supporters of party $(p_i)$ have is $M(\text{party}(p_i)) = |A(p_i)| \cdot \frac{B}{|V|} = \frac{|A(p_i)| \cdot B}{|V|}$ . Therefore, the projects in party $(p_i)$ need to somehow distribute this money between them in such a way that no project would complain and change its cost to obtain better outcome. + +Since we have cardinal utilities, the $\alpha$ -value of project $p_i$ is initially $\alpha_1(p_i) = \frac{\mathbf{c}(p_i)}{|A(p_i)|}$ . For this reason, MES-Apr will start from the cheapest project in the party to the most expensive one (following tie-breaking order in case of a tie) and equally take the budget from the party supporters to fund the next project unless the cost exceeds the money they still have. + +This means that each project should ask for the equal part of the money of party supporters, that it, $\frac{M(party(p_i))}{|party(p_i)|}$ . The project asking for more money would be moved to the end of the order when its supporters have insufficient money to fund it, so no project would increase its cost. Asking for less money is pointless if it is already funded. + +For this reason, for the profile $\mathbf{c}$ where each project says $\mathbf{c}(p_i) = \frac{M(p_i)}{|A(p_i)|}$ , we obtain MES-Apr-NE. + +Next we move to the case with arbitrary delivery costs. Let $(P, V, B, d)$ be a MES-Apr PB game with party-list ballots. Recall that at the beginning each voter receives $B / |V|$ money. Then, the total money the supporters of party $(p_i)$ have is $M(p_i) = |A(p_i)| \cdot B / |V| = \frac{|A(p_i)| \cdot B}{|V|}$ . + +Let $\succ$ be the A/D tie-breaking order from the theorem statement. Note that for the case of party-list ballots $\succ$ orders the projects within the same party nondescendingly by the delivery costs. That is, within the same party cheaper projects preferred to the more expensive ones. In case of two projects within the same party with equal delivery costs, we assume without loss of generality that the project with a lower index is preferred (we can always relabel projects to achieve this condition). Notably, $\succ$ does not impose any specific order with respect to the delivery costs over any two projects of two different parties. + +Since in MES-Apr we consider cardinal utilities, the $\alpha$ -value of project $p_i$ is at the beginning: $\alpha_1(p_i) = \frac{\mathbf{c}(p_i)}{|A(p_i)|}$ . For this reason, MES-Apr will start from the cheapest project in the party to the most expensive one (following the specified tie-breaking order in case of a tie) and equally take the budget from the party supporters to fund the next project unless the cost exceeds the money they still have. + +Now we have to consider two cases: + +- The last in the tie-breaking order project $p_j \in \text{party}(p_i)$ has delivery cost $d(p_j)$ not exceeding $\frac{M(\text{party}(p_i))}{| \text{party}(p_i) |}$ . Note that this case is similar to our case with zero delivery costs that we started this proof with, i.e., each project submits $\frac{M(\text{party}(p_i))}{| \text{party}(p_i) |}$ , gets funded, and no project benefits from changing its cost. +- The last in the tie-breaking order project $p_j \in \text{party}(p_i)$ has delivery cost $d(p_j)$ exceeding $\frac{M(\text{party}(p_i))}{| \text{party}(p_i) |}$ . In such a case, if every project submitted cost $d(p_j)$ , then $p_j$ would not be selected as it is last in the tie-breaking order and cannot profitably decrease its cost. For this reason, we set the cost of $p_j$ to be $d(p_j)$ , remove $p_j$ from consideration, and repeat this procedure as long as the product of the number of yet-remained projects and the delivery + +cost of the last in the tie-breaking project exceeds $M(p_{k}$ party $(p_{i})$ ). Suppose now that the procedure stopped and project $p_{k}$ was the last removed one. Then all $y$ yet-remaining projects would say cost $\min \left( \frac{M(p_{k} \text{ party } (p_{i}))}{y}, d(p_{k}) \right)$ and be given this funding. Saying more than $d(p_{k})$ would result in being overtaken by project $p_{k}$ and thrown out of the outcome due to insufficient budget. Saying more than $\frac{M(p_{k} \text{ party } (p_{i}))}{y}$ would result in getting the last among remaining projects and asking for more money than left in the budget at that timestamp. + +The combination of c profiles calculated in the above way for all parties is a MES-Apr-NE for the given instance. + +Example B.2. Take a PB game $G$ with one voter approving projects $p_1, p_2,$ and $p_3$ . Let $B = 6$ , $d(p_1) = 3$ , $d(p_2) = 0$ , and $d(p_3) = 0$ , with $p_1 \succ p_2 \succ p_3$ . There is no MES-Apr-NE in $G$ (the full proof is in the appendix and we give a sketch): Suppose that $\mathbf{c}$ is an NE. Then $\mathbf{c}(p_1) \geq 3$ (otherwise, as it is selected, $p_1$ improves by reporting 3) and $\mathbf{c}(p_2), \mathbf{c}(p_3) < \mathbf{c}(p_1) = 3 = B / 2$ (otherwise, the more expensive one is not funded). Since $\mathbf{c}(p_2) + \mathbf{c}(p_3) < B$ , $p_2$ may report $(\mathbf{c}(c_1) + \mathbf{c}(c_2)) / 2$ and still be selected, so $\mathbf{c}$ is not an NE. + +Suppose towards contradiction that $\mathbf{c}$ is such an equilibrium. W.l.o.g., let $\mathbf{c}(p_1) \geq d(p_1) = 3$ . Otherwise, if $p_1$ would be selected under these costs (and, hence, received a negative payoff), then it would prefer to report $d(p_1)$ and receive utility 0. If it was not funded under $\mathbf{c}$ , then $(\mathbf{c}_{-i}, d(p_1))$ also would be an equilibrium. Next, we observe that $\mathbf{c}(p_1) = 3$ . Indeed, if $p_1$ reported a value greater than 3, then one of the projects could obtain a better payoff: If $p_2$ or $p_3$ reported a value greater or equal to $\mathbf{c}(p_1)$ , then it would benefit by reporting a smaller one; if $p_2$ and $p_3$ reported values that sum up to more than $B$ , then at least one of them would benefit by reporting a smaller cost, and if they reported values that sum up to at most $B$ then either (a) one of them would benefit by reporting a larger cost, or (b) if they both reported 3, $p_1$ would benefit by reporting 3. Hence, we have that $\mathbf{c}(p_1) = 3 = B/2$ . Then, both $\mathbf{c}(p_2) > 0$ and $\mathbf{c}(p_3) > 0$ (the project reporting 0 would benefit by reporting a larger number). Further, we have that $\mathbf{c}(p_2) < \mathbf{c}(p_1)$ and $\mathbf{c}(p_3) < \mathbf{c}(p_1)$ . Indeed, if either $p_2$ or $p_3$ reported value greater or equal to $\mathbf{c}(p_1)$ then it would not be selected and, hence, would benefit by reporting a smaller cost. So, $\mathbf{c}(p_2) + \mathbf{c}(p_3) < B$ . But then either of them would benefit by reporting a higher cost (so that the sum of their costs would not exceed $B$ ). Thus $\mathbf{c}$ is not a MES-Apr-NE. + +# B.11 Proof of Theorem 4.13 + +Proof. Our example is in fact the example from Proposition 4.8 narrowed to prim voters and projects. + +We have three voters $v_{1}, v_{2}$ , and $v_{3}$ as well as three projects $p_{1}, p_{2}$ , and $p_{3}$ . Projects $p_{1}$ and $p_{2}$ are approved by, respectively, voters $v_{1}$ and $v_{2}$ , while project $p_{3}$ is approved by all three voters. We set the budget to be some positive integer denoted as $B$ and the delivery costs of these projects to be $d \equiv 0$ . We set the tie-breaking between projects to be $p_{1} \succ p_{2} \succ p_{3}$ . For the sake of brevity, for yet-unselected project $p_{i}$ in iteration $j$ , we denote $\alpha_{j}(p_{i})$ as minimum value such that project $p_{i}$ is $\alpha_{j}(p_{i})$ -affordable according to MES-Apr rule (or infinity, if such a value does not exist). + +Let $(\mathbf{c}(p_1) = \frac{7B}{36}, \mathbf{c}(p_2) = \frac{8B}{36}, \mathbf{c}(p_3) = \frac{21B}{36})$ be the costs the projects say. We will show that it is MES-Apr-NE. + +At the beginning, each of three voters receives $\frac{B}{3} = \frac{12B}{36}$ money. + +In the first iteration, $\alpha_{1}(p_{1}) = \frac{7B / 36}{1} = \frac{7B}{36}$ , $\alpha_{1}(p_{2}) = \frac{8B / 36}{1} = \frac{8B}{36}$ , $\alpha_{1}(p_{3}) = \frac{21B / 36}{3} = \frac{7B}{36}$ , so $p_{1}$ and $p_{3}$ are tied, while $p_{2}$ has greater $\alpha$ -value and is skipped for now. Thus, due to tie-breaking, we select $p_{1}$ and voter $v_{1}$ pays $\frac{7B}{36}$ for $p_{1}$ . + +In the second iteration, $v_{1}$ has $\frac{5B}{36}$ money while both $v_{2}$ and $v_{3}$ have $\frac{12B}{36}$ . Thus, since the budget of $v_{2}$ did not change, $\alpha_{2}(p_{2}) = \alpha_{1}(p_{2}) = \frac{8B}{36}$ . In order to purchase $p_{3}$ , voter $v_{1}$ would need to spend all left money whereas $v_{2}$ and $v_{3}$ would need to pay more to make up $v_{1}$ 's insufficient money. More precisely, $\alpha_{2}(p_{3}) = \frac{21B/36 - 5B/36}{2} = \frac{16B/36}{2} = \frac{8B}{36}$ . So projects $p_{2}$ and $p_{3}$ are tied and we select $p_{2}$ due to tie-breaking (by taking money from $v_{2}$ ). + +Finally, in the third iteration, we will select $p_3$ as its supporters have $\frac{5B}{36} + \frac{4B}{36} + \frac{12B}{36} = \frac{21B}{36} = \mathbf{c}(p_3)$ , all voters will use all their money for it. + +We showed that each project is selected with the proposed costs so no project benefits from lowering its costs. Next, $p_3$ will not increase its cost as its supporters would not have enough money to buy it in the last iteration. Further, $p_2$ will not increase its cost as it would lose with $p_3$ in the second iteration and its only supporter $v_2$ would pay $\frac{8B}{36}$ for $p_3$ , leaving insufficient $\frac{4B}{36}$ for $p_2$ . Analogously, $p_1$ will not increase its cost as it would lose with $p_3$ in the first iteration and its only supporter $v_1$ would pay $\frac{7B}{36}$ for $p_3$ , leaving insufficient $\frac{5B}{36}$ for $p_1$ . + +For this reason, $\mathbf{c} = (\mathbf{c}(p_1) = \frac{7B}{36},\mathbf{c}(p_2) = \frac{8B}{36},\mathbf{c}(p_3) = \frac{21B}{36})$ is a MES-Apr-NE for the given instance. + +One can show in an analogous way that $\mathbf{c}' = (\mathbf{c}'(p_1) = \frac{8B}{36},\mathbf{c}'(p_2) = \frac{7B}{36},\mathbf{c}'(p_3) = \frac{21B}{36})$ is also MES-Apr-NE for the given instance. + +# B.12 Proof of Theorem 6.1 + +Let us begin by introducing additional notation. In a cost-interval PB game $G^{+} = (P, V, B)$ we associate each voter $v$ with an approval function $A: V \times P \to 2^{[0, B]}$ , with $A(v, p)$ specifying how costly can a project $p$ be for $v$ to approve it. For simplicity, we assume that $A(v, p)$ is a possibly empty interval. We say that a voter $v$ potentially supports a project $p$ if $A(v, p) \neq \emptyset$ . Then, for a strategy profile $\mathbf{c}$ , the corresponding PB election $E(\mathbf{c}) = (P, V, B, \mathbf{c})$ is such that each voter $v$ approves a project $p \in P$ if $cost(p) \in A_v(p)$ . Then, the payoff function and Nash equilibria are defined for cost-interval PB games analogously to PB games. We now showcase examples witnessing the result for particular voting rules. + +MES-Cost. Consider the following cost-interval PB game $G$ . Take the set of projects $P = \{p_1, p_2\}$ , as well as the set of voters $V = \{v_1, v_2, v_3, v_4\}$ . Further, we let $A(v_1, p_1) = A(v_2, p_1) = [8, 12]$ , $A(v_3, p_1) = A(v_4, p_1) = [0, 4]$ , $A(v_1, p_2) = [4, 8]$ , and $A(v_2, p_2) = A(v_3, p_2) = A(v_4, p_2) = [0, 3]$ . Finally, let $B = 16$ . + +Suppose that there exists a strategy profile $\mathbf{c}$ that is a MES-Cost-NE. Let us then consider the following, exhaustive cases: + +1. $\mathbf{c}(p_1) \in (4,8) \cup (12,16]$ . Then, $p_1$ is not funded, while for $\mathbf{c}(p_1) = 1$ , $p_1$ is founded regardless of $\mathbf{c}(p_2)$ . +2. $\mathbf{c}(p_1) \in (8, 12]$ Then, $p_1$ is not funded, and hence could benefit from lowering its cost. +3. $\mathbf{c}(p_1) = 8$ . Then, we note that as $\mathbf{c}$ is an NE, $\mathbf{c}(p_2) = 3$ , as then $p_2$ is funded but would not be selected for $(8, \mathbf{c}'(p_2))$ with $\mathbf{c}'(p_2) > 3$ . But then, $p_1$ is not funded under $\mathbf{c}$ , while it would benefit from submitting $\mathbf{c}'(p_1) = 4$ . +4. $\mathbf{c}(p_1) \in [0, 4]$ . Then, $\mathbf{c}(p_2) = 4$ , as then $p_2$ is funded under $\mathbf{c}$ but would not be under $(\mathbf{c}(p_1), \mathbf{c}'(p_2))$ , for any $\mathbf{c}'(p_2) > 4$ . However, in that case $p_1$ would benefit from raising their cost to 8. + +It follows that there is no MES-Cost-NE in $G$ . + +BasicAV. Consider the following cost-interval PB game $G$ . Take the set of projects $P = \{p_1, p_2\}$ where $p_1 \succ p_2$ , as well as the set of voters $V = \{v_1, v_2, v_3, v_4, v_5\}$ . Further, we let $A(v_1, p_1) = [5, 9]$ , $A(v_2, p_1) = [3, 7]$ , $A(v_3, p_2) = [4, 8]$ , $A(v_4, p_2) = [5, 7]$ , and $A(v_5, p_2) = [1, 6]$ so this is a plurality profile. Finally, let $B = 10$ . + +Suppose that there exists a strategy profile $c$ that is a BasicAV-NE. Let us then consider the following, exhaustive cases: + +1. $\mathbf{c}(p_1) \in [0, 3) \cup (9, 10]$ . Then, $p_1$ is not funded but would benefit from submitting $\mathbf{c}'(p_2) = 3$ , as then $p_1$ is always selected. +2. $\mathbf{c}(p_1) \in [3,4) \cup (7,9]$ . Then, we note that as $\mathbf{c}$ is an NE, $\mathbf{c}(p_2) = 7$ , as then $p_2$ is funded but would not be selected for $(\mathbf{c}(p_1), \mathbf{c}'(p_2))$ with $\mathbf{c}'(p_2) > 7$ . But then, $p_1$ is not funded under $\mathbf{c}$ , while it would benefit from submitting $\mathbf{c}'(p_1) = 7$ . + +3. $\mathbf{c}(p_1) \in [4, 7]$ . Then, $\mathbf{c}(p_2) = 6$ , as then $p_2$ is funded under $\mathbf{c}$ but would not be under $(\mathbf{c}(p_1), \mathbf{c}'(p_2))$ , for any $\mathbf{c}'(p_2) > 6$ . However, in that case $p_1$ would benefit from reducing their cost to 4. + +It follows that there is no BasicAV-NE in $G$ + +AV/cost. Consider the following cost-interval PB game $G$ . Take the set of projects $P = \{p_1, p_2\}$ where $p_2 \succ p_1$ , as well as the set of voters $V = \{v_1, v_2, v_3, v_4\}$ . Further, we let $A(v_1, p_1) = [4, 9]$ , $A(v_2, p_1) = A(v_3, p_1) = [8, 9]$ , and $A(v_4, p_2) = [2, 6]$ , so this is a plurality profile. Finally, let $B = 10$ . + +Suppose that there exists a strategy profile $\mathbf{c}$ that is a AV/Cost. Let us then consider the following, exhaustive cases: + +1. $\mathbf{c}(p_1) \in [0, 4) \cup (9, 10]$ . Then, $p_1$ is not funded but would benefit from submitting $\mathbf{c}'(p_2) = 5$ , as then $p_1$ is always selected. +2. $\mathbf{c}(p_1) \in [8, 9]$ . Then, we note that as $\mathbf{c}$ is an NE, $\mathbf{c}(p_2) = \mathbf{c}(p_1) / 3$ , as then $p_2$ is funded but would not be selected for $(\mathbf{c}(p_1), \mathbf{c}'(p_2))$ with $\mathbf{c}'(p_2) > \mathbf{c}(p_1) / 3$ . But then, $p_1$ is not funded under $\mathbf{c}$ , while it would benefit from submitting $\mathbf{c}(p_1) \in [4, 3 \cdot \mathbf{c}(p_2))$ . +3. $\mathbf{c}(p_1) \in [5, 8]$ . Then, $\mathbf{c}(p_2) = \min(\mathbf{c}(p_2), 6) \in [5, 6]$ , as then $p_2$ is funded under $\mathbf{c}$ but would not be under $(\mathbf{c}(p_1), \mathbf{c}'(p_2))$ , for any $\mathbf{c}'(p_2) > \min(\mathbf{c}(p_2), 6)$ . However, in that case $p_1$ would benefit from raising their cost to 9. +4. $\mathbf{c}(p_1) \in [4,5)$ . Then, $\mathbf{c}(p_2) = \min(10 - \mathbf{c}(p_2), 6) \in [5,6]$ , as then $p_2$ is funded under $\mathbf{c}$ but would not be under $(\mathbf{c}(p_1), \mathbf{c}'(p_2))$ , for any $\mathbf{c}'(p_2) > \min(10 - \mathbf{c}(p_2), 6)$ . However, $p_1$ would benefit from raising their cost to 9. + +It follows that there is no AV/Cost in $G$ . + +Phragmén. The claim follows by the example for AV/Cost, as Phragmén and AV/Cost are equivalent for plurality elections. + +MES-Apr. Consider the following cost-interval PB game $G$ . Take the set of projects $P = \{p_1, p_2\}$ , as well as the set of voters $V = \{v_1, v_2, v_3\}$ . Also, let $A(v_1, p_1) = A(v_1, p_2) = [12, 18]$ . Then, let $A(v_2, p_1) = A(v_2, p_2) = [6, 12]$ and the approval function of $v_3$ be the same as the approval function of $v_2$ . Finally, we let $B = 18$ and assume zero delivery costs. It's clear that each voter received $B/3 = 6$ money units at the beginning of the algorithm regardless of the costs of the projects. + +Suppose that there is a MES-Apr-NE $\mathbf{c} = (c_1, c_2)$ for $G$ . First, we can see that each of the projects is funded with the cost set to 6 regardless of the cost submitted by the other proposer. So, both projects are funded under $(c_1, c_2)$ . Hence, we notice that $c_1 \in [6, 12]$ and $c_2 \in [6, 12]$ , as otherwise, either one of the projects would submit cost higher than 12 and be supported by only voter (whose budget is 6), or submit a cost lower than 6 and not be supported by anyone. So, in both scenarios one of the project would not be funded. Let us then consider the following exhaustive cases: + +1. $c_{1} = 12$ or $c_{2} = 12$ . Without loss of generality, let $c_{1} = 12$ . Then, we see that $c_{2} = 8$ or $c_{2} = 12$ . Otherwise, if $c_{2} \in (8, 12)$ , then $p_{2}$ would not be funded and if $c_{2} < 8$ , then $p_{2}$ would be funded under $(12, c_{2} + \epsilon)$ , for some $\epsilon > 0$ . But then, one of the projects is not funded under $(c_{1}, c_{2})$ , as $c_{1} + c_{2} > B$ . Hence, according to previous observations, $(c_{1}, c_{2})$ is not a MES-Apr-NE. +2. $c_{1} \in [6, 12)$ and $c_{1} \in [6, 12)$ . Observe that for this cost range, only voters $v_{2}$ and $v_{3}$ may pay for these projects, while having 12 money units in total. Then, we observe that $c_{1} + c_{2} \leq 12$ , as otherwise one of the projects would not be funded. Moreover, $c_{1} = c_{2}$ , as otherwise the proposer $p_{i}$ submitting the lower cost would benefit from increasing $c_{i}$ . Hence, in this case $c_{1} = c_{2} = 6$ . But then $p_{1}$ would also be funded under (12,6), and hence $(c_{1}, c_{2})$ is not an MES-Apr-NE. + +It follows that there is no MES-Apr-NE in $G$ + +Table 3: Additional PB instances that we analyze in the experiments. Vote Len. means the average number of approvals in a ballot. Rule indicates the PB rule that was used in the original election. + +
Instance|P||V|BudgetVote Len.Rule
Bemowo835181485427910.8BasicAV
Bielany98495752588029.8BasicAV
Wilanow35235915169629.59BasicAV
Wlochy43222117192249.51BasicAV
+ +![](images/4e20b1b497b3aded00f80214dd781a1a89e56c075b341612fb84c3f8499c2576.jpg) + +![](images/ba387b59843ac93daba472961d23f5f68ae6ebd9cf6c97f6b921eff1cd1ca43f.jpg) + +![](images/694fb7d756c303053cdacd20d2f05348d1cb4ad7f208befc808071fec697a362.jpg) + +![](images/a810dc542ec5c7507192ffc19b988a398467f2577b152608022722294e5a602e.jpg) + +![](images/71b433d8cc05a6332eb4ae84192b30979166622199d1bdc2551fd658facbd754.jpg) + +![](images/df2050ae6976ad39eb562035df8e0ba137b2a86d9ea8ee1f0f9941b45928b45b.jpg) + +![](images/a148b34a2afb7022c186455923eabfe323eeee3a45c02962c94eed5138bafc52.jpg) + +![](images/ba04dc05da55caaf27c8d9c1bbccbcb9099148a1e6517e796f5a98bc0724a2a4.jpg) + +![](images/6e78ab93d60309f1ed42bc5a9347d9f0b25887b84fe0deaddd034949dc5ca811.jpg) + +![](images/ce4225538dd3b23e127a0fc9981797cae9085a5eabbaf4bba801b0e640419878.jpg) + +![](images/36c0f8ea83825b25923323152a380b84ce480da1c44c8a1ea7bb17c70b2407a1.jpg) + +![](images/42980f47584e2f3e775a0d56cc3170f390c9b9facebfea6bf4d6e60538d471f4.jpg) + +![](images/d848bfcbf6e5304b2f82ca484832ff980c402522a4e70e99b7783ebe9db75d12.jpg) + +![](images/f4ee27f94a3e8f9ab9c576845e72421b1a7f5bab18c490b69c67300ed7800b15.jpg) + +![](images/1059229ce01b9316105c58e633c1fe0147af46fe01171d0c91c8a5ec962a8f4e.jpg) + +![](images/592ddf450d7741c888ac54a86b9876fad2be8a4c9b86d6ce5cfe40ae0d8200c8.jpg) + +![](images/e9e0ddb3cd21cadd1c035e317c438fb615e06f061e3d9ea6683857a14b6993f7.jpg) +Figure 5: Winning (green bars) and losing (red bars) margins in real-life PB. Projects ordered by decreasing approval scores (on the $x$ -axis) are represented by bars. Tacks and crosses show, respectively, the original and best response costs. + +![](images/c628cee409928e8134cf7b5abf5f2918ef97dbb2f26e407540bee59d9290445b.jpg) + +![](images/610c11f8456c97cb0f57d9f7e599167afe216086f812022d60b2b2d053be504c.jpg) + +![](images/0ee263d58bc56549515d648fabf5550f2948a08f2bc037691840b67a521e6343.jpg) + +# C Additional Experimental Results + +Here, we present experimental results for four additional PB instances held in 2022 in different districts of Warsaw, Poland, i.e., in Bemowo, Bielany, Wilanow, and Wlochy. Details regarding these datasets are presented in Table 3. In Figure 5 we present the winning and losing margins in additional real-life PB instances, and in Figure 6 we present the strategy profiles after $10^{\prime}000$ iterations of our dynamics. + +Moreover, in Table 4 we present comparison of the original winning and losing margins and those after 10'000 iterations. We observe that in most cases the size of the margins drastically decreased. + +# D Dynamics With $80\%$ Delivery Costs + +In Figure 7 we show the results of the dynamics from Section 5.2 for the case where each project has delivery cost equal to the $80\%$ of the cost it had in the original instance. We see that aside from + +
RuleOriginal Margins10000 it. Margins
WinningLosingWinningLosing
BemowoBasicAV1830 ± 1283122 ± 182024 ± 37
AV/Cost150 ± 104225 ± 2571 ± 30.2 ± 0.1
Phragmén141 ± 91220 ± 2613 ± 30.9 ± 0.7
MES-Apr/Ph150 ± 95223 ± 2643 ± 42 ± 2
MES-Cost/Ph250 ± 323205 ± 2441 ± 21 ± 2
BielanyBasicAV1447 ± 1355218 ± 20502 ± 2
AV/Cost157 ± 128171 ± 1500.9 ± 20.2 ± 0.3
Phragmén146 ± 121176 ± 1544 ± 30.9 ± 0.9
MES-Apr/Ph148 ± 121176 ± 1513 ± 41 ± 2
MES-Cost/Ph172 ± 193180 ± 1543 ± 63 ± 4
WesolaBasicAV265 ± 24964 ± 5100
AV/Cost86 ± 2755 ± 310.3 ± 0.40.2 ± 0.1
Phragmén69 ± 3151 ± 372 ± 10.9 ± 1
MES-Apr/Ph79 ± 3343 ± 372 ± 10.7 ± 0.7
MES-Cost/Ph70 ± 5665 ± 450.1 ± 0.10.4 ± 0.5
WilanowBasicAV536 ± 39987 ± 8000
AV/Cost87 ± 5951 ± 342 ± 20.1 ± 0.0
Phragmén77 ± 5266 ± 412 ± 20.3 ± 0.2
MES-Apr/Ph89 ± 5658 ± 372 ± 10.7 ± 0.8
MES-Cost/Ph79 ± 12095 ± 641 ± 22 ± 4
WlochyBasicAV532 ± 52446 ± 3500
AV/Cost107 ± 4956 ± 480.8 ± 1.00.1 ± 0.1
Phragmén83 ± 5262 ± 423 ± 32 ± 1
MES-Apr/Ph85 ± 5364 ± 452 ± 21 ± 1
MES-Cost/Ph138 ± 18656 ± 460.2 ± 0.20.4 ± 0.5
Kleine WereldBasicAV117 ± 7814 ± 1101.0 ± 0.0
AV/Cost12 ± 812 ± 120.1 ± 0.10.1 ± 0.0
Phragmén10 ± 810 ± 110.5 ± 0.60.2 ± 0.2
MES-Apr/Ph11 ± 711 ± 110.4 ± 0.60.2 ± 0.2
MES-Cost/Ph26 ± 3411 ± 112 ± 10.1 ± 0.0
+ +Table 4: Comparison of winning and losing margins in the original elections and after 10000 iterations of the game. (All the values are in 1000 PLN\EUR). In each entry the first value denotes the average and the second value (i.e., the one after $\pm$ sign) denotes the standard deviation. + +fairly expensive projects, whose cost could not drop below the delievery cost, the overall behavior of our rules is as with zero delivery costs. + +![](images/089ef4da3262b57ee7fcbb74bd2413eaaa9333ab3e96d660451392d54c3dc0c8.jpg) + +![](images/e512515d5d03294f136b98aa2ec9c0502791c2161e412fea67569378c202e6ee.jpg) + +![](images/9c86713151c4e83233b0b478b93c5fd415cc3fbabef393aafb8debe03460b983.jpg) + +![](images/6df2e6c66b9b217f1c4c6ef6dde59aef49aff5b17c01fc0f2a8cfc86aec4ffd0.jpg) + +![](images/4e9183d811e77146a1d658685d482ca10d081ac1380fd692000c68943c7128e8.jpg) + +![](images/77f5efb5b97d122e4fcee2a835834d24cd5239ae0a08169651494b72035df915.jpg) + +![](images/2740e6c536462e0326a60183dbd42d90fe8c1c01f97f2cb246f6abe2bad26b7a.jpg) + +![](images/0950bb6719c4260d70053555a2436fe25cddc9b5cb715ed0a8ff6e65a242e0a8.jpg) + +![](images/d3250c161f85362048461aa93b4a030a7f0e1a33e648e8b44c2c698e5ac15109.jpg) + +![](images/6a219af1bea2291168c2e3fede1a597e0660c77596ab554640f627d5295f03f6.jpg) + +![](images/f42a80d8faf8c3e794b523663f7434f84ac9aadfa160ccba73df8f879da68898.jpg) + +![](images/670149672ef99ca344d8a68fc1edfc81aa9f62ecf179caf0a8c7f6a6104b57c8.jpg) + +![](images/b08d4b608a4f681fbbca021d126b27c6652e63d99cb355c3e63ab48bb528181f.jpg) + +![](images/e28f9680f5c48e6db3f7bf1f83118ab3d8f250b419f3a059b69fa37c764b9138.jpg) + +![](images/71830ef944587f5a736dc4abe9abe5be08b1a3d81d08d08571b205e0ac811152.jpg) + +![](images/b4922ba8287449f28f16bdd62ad95aa850585497fcc71fb3a865fcc20d39abd4.jpg) + +![](images/fc9017bf8eb4dd7841d54412adb8e413a0493c7d727639c340e772922375ba9a.jpg) +Figure 6: Strategy profiles after 10 000 iterations of our dynamics. Projects ordered by decreasing approval scores ( $x$ -axis). Final projects' costs shown by bars; the original costs shown as dashed lines. Red and green bars indicate, respectively, losing and winning projects; brighter parts are the increases over the original cost. The triangles at the top mark the originally winning projects. Brown circles denote the equilibrium costs (for rules for which we can compute it). + +![](images/c83b62de0a5ec49a7d27bb7d4ac8e93ddfbf67a5c007f7920a4e7230243a2ac5.jpg) + +![](images/c9f83a3496e6ea3debbd681232b67256cb5fe3df6670bd280f08e3c36743c764.jpg) + +![](images/c2e66b04416ac05e14045b108cefc425e5a0a0d0e9b2091a7ff5e1ae40ecbaa0.jpg) + +![](images/353f660d8f0a3c3071f71e67e65be6bfad73e6db529ff1c68f38c27cf9735e78.jpg) + +![](images/674f16aac30f058e118ddd2f0b6267ae9e1406dc35039b6e2535b54401b84acd.jpg) + +![](images/90512f3e84e4e7fc295893f85854ae70e48576c65540ed95b31a8602cd285433.jpg) + +![](images/a9414d09680ed79b101170ff09a813adde916fbde916de1962c72888e54b48e9.jpg) + +![](images/4c441b2cf8122d481de608a9e13c4f0bcd9597d11d1f9d71b86cacdd96276ed1.jpg) + +![](images/c1e3ef85215955384817785f7ac39401f3f65fb5e627437cf4a8233446199bf0.jpg) + +![](images/1699dd5988b0342daa840621578edd52448a185a9cea42ace224ff15e3faff5f.jpg) + +![](images/34193d793b24cfe3da81e995629caec1a27fc0a34ee541a9c53067c9cb5496e6.jpg) + +![](images/fc5c9f3ed5c2d3296159c7fefe002215aa662605a7dccf2a2dc75f8a7959e22b.jpg) +Figure 7: Strategy profiles after 10 000 iterations of our dynamics. for the case where each project has delivery cost equal to $80\%$ of its original cost in the instance. Projects ordered by decreasing approval scores $(x$ -axis). Final projects' costs shown by bars; the original costs shown as dashed lines. Red and green bars indicate, respectively, losing and winning projects; brighter parts are the increases over the original cost. The triangles at the top mark the originally winning projects. 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Elisa Celis + +Yale University + +Lingxiao Huang + +Nanjing University + +Milind Sohoni + +IIT Bombay + +Nisheeth K. Vishnoi + +Yale University + +# Abstract + +Meritocratic systems, from admissions to hiring, aim to impartially reward skill and effort. Yet persistent disparities across race, gender, and class challenge this ideal. Some attribute these gaps to structural inequality; others to individual choice. We develop a game-theoretic model in which candidates from different socioeconomic groups differ in their perceived post-selection value—shaped by social context and, increasingly, by AI-powered tools offering personalized career or salary guidance. Each candidate strategically chooses effort, balancing its cost against expected reward; effort translates into observable merit, and selection is based solely on merit. We characterize the unique Nash equilibrium in the large-agent limit and derive explicit formulas showing how valuation disparities and institutional selectivity jointly determine effort, representation, social welfare, and utility. We further propose a cost-sensitive optimization framework that quantifies how modifying selectivity or perceived value can reduce disparities without compromising institutional goals. Our analysis reveals a perception-driven bias: when perceptions of post-selection value differ across groups, these differences translate into rational differences in effort, propagating disparities backward through otherwise "fair" selection processes. While the model is static, it captures one stage of a broader feedback cycle linking perceptions, incentives, and outcomes—bridging rational-choice and structural explanations of inequality by showing how techno-social environments shape individual incentives in meritocratic systems. + +# 1 Introduction + +Meritocratic selection systems, used by institutions and firms for admissions, hiring, and content curation, aim to allocate opportunities based on observable indicators of ability and effort rather than wealth, identity, or social status. They are widely viewed as promoting fairness and efficiency [42, 74, 37]. Examples include standardized tests such as the SAT and JEE [52, 8], structured interviews and assessments [72, 13], and algorithmic ratings on online platforms [24, 73, 84]. + +Yet, despite their formal neutrality, these systems often produce significant disparities in representation and outcomes. Women, racial minorities, and lower-income groups are consistently underrepresented in elite universities, leadership roles, and high-paying industries [64, 45, 75]. These gaps persist even when evaluation procedures are blind to group identity, suggesting that there are additional factors that drive inequality in merit-based processes. + +One set of explanations points to structural barriers: unequal access to resources that enhance merit (e.g., quality education, extracurricular activities), implicit biases in selection processes, and limited opportunities due to privileged networks [87, 40, 56, 61, 13, 63, 24, 73, 84]. Others suggest that individuals who face the same selection rules may simply make different choices, investing less effort because they perceive lower returns to success due to cultural preferences, opportunity costs, or labor market sorting [31, 11, 39, 66, 12]. These disparities under-utilize talent, reducing innovation, diversity of ideas, and social progress [43, 58, 67]. This raises a central question: how can differences in perceived opportunity translate into systematic behavioral disparities even when evaluation is symmetric? + +Expectations about what selection yields-admission, employment, mobility--are shaped not only by historical inequalities [76, 59, 51], but increasingly by algorithmic tools that mediate labor market signals [46]. Although meritocratic ideals suggest that pay should correlate with skills, productivity, and achievements, empirical studies reveal persistent wage disparities even after controlling for factors such as occupation, education, experience, and hours worked [38, 89, 86, 64, 83, 78, 68, 41]. For example, in the U.S., women earned just $83.1\%$ of what men earned in 2021, despite outnumbering men in the college-educated labor force [83, 26]. Similar wage gaps persist across racial, class, caste, and ethnic lines, with Black, Hispanic, and Indigenous workers earning less than White and Asian peers in comparable roles [68]. Large language models (LLMs) may further exacerbate these disparities. Recent studies show that when asked for job or salary recommendations, LLMs return systematically different responses across demographic groups, even when qualifications are held constant [2, 46]. Such signals can distort perceived opportunity and disincentivize effort long before any selection decision occurs. Taken together, these findings suggest a perception-driven bias: social and algorithmic cues about post-selection value shape pre-selection investment, reinforcing group-level disparities even under ostensibly meritocratic systems. + +Our contributions. We introduce a game-theoretic model of meritocratic selection in which candidates from two groups differ in their perceived value of being selected. This model integrates contest theory with models of structural bias from algorithmic fairness and captures how valuation disparities influence effort, merit, and selection outcomes. While the model is static, it represents one stage of a broader feedback process linking perceptions, incentives, and outcomes. Our main contributions are: + +1. Modeling. We formulate a two-group contest in which $n$ rational agents, divided into groups $G_{1}$ and $G_{2}$ (with proportion $\alpha \in (0,1)$ ), compete for $c \cdot n$ positions. Group-specific valuations follow distributions $p_{1}$ and $p_{2}$ , with $p_{2}$ modeled as a $\rho$ -biased version of $p_{1}$ for $\rho \in (0,1]$ [47], representing structural disparities. Each candidate chooses effort based on their valuation to maximize their expected payoff, which is then converted into observable merit used for selection. +2. Equilibrium characterization. We prove the existence and uniqueness of a symmetric Nash equilibrium in the large-agent limit, and express the equilibrium thresholds for both groups in terms of $(c, \alpha, p_1, p_2)$ (Theorem 3.1). We further show that the equilibrium in the finite- $n$ setting converges to this solution at rate $O(\sqrt{\log n / n})$ . +3. Micro to macro analysis of metrics. Using the equilibrium solution, we derive closed-form expressions for key performance metrics—group-wise representation ratio $r_{\mathcal{R}}$ , social welfare ratio $r_{\mathcal{S}}$ , and institutional revenue in the case where $p_1$ is uniform and $p_2$ is $\rho$ -biased (Proposition 4.1). These expressions reveal how small changes in $\rho$ , $c$ or $\alpha$ can produce non-linear shifts in outcomes. +4. Fairness-aware interventions. We formulate a constrained optimization problem (Problem (6)) that allows institutions to trade off between increasing selectivity $(c)$ and reducing valuation bias $(\rho)$ under fairness constraints (e.g., $80\%$ -rule). We solve this problem in closed form for linear cost functions and characterize when each intervention is most cost-effective (Figure 4). + +Taken together, our framework provides a quantitative lens on how structural or algorithmic biases in perceived value can rationally produce effort and outcome disparities in meritocratic systems, and offers tools to design interventions that enhance both representation and efficiency. + +Related work. Our work connects three areas: economic theories of meritocracy, game-theoretic models of contests, and algorithmic fairness. Social scientists have long examined the tension between meritocratic ideals and persistent disparities in outcomes, including gender and racial pay gaps [59, 76, 38, 15]. While prior models of statistical discrimination explain disparities through institutional uncertainty or group-dependent beliefs—often despite equal agent quality [3, 69, 21, 23, 5, 49]—our model assumes perfect institutional information and symmetric evaluation. Instead, we show how valuation asymmetries alone can induce disparities in effort and selection outcomes. From a game-theoretic perspective, our setting builds on all-pay auctions and Tullock contests, which model competitive effort under asymmetry [79, 29, 27]. However, these models rarely consider group-level valuation differences. To our knowledge, we are the first to study equilibrium behavior in large-agent contests with asymmetric valuation distributions across groups. Finally, our bias model extends work in algorithmic fairness that explores valuation gaps and signal noise [48, 28, 18], but previous strategic classification models typically lack inter-agent competition or valuation-based disparities [14, 29]. Thus, our work offers a novel integration of asymmetric group valuations into competitive contest frameworks, with implications for equilibrium behavior, fairness, and institutional design. See Section A for further discussion and detailed comparisons. + +# 2 Model and metrics + +We consider a population of $n$ agents competing for $k = cn$ indistinguishable spots, where $c \in (0,1)$ denotes the selection fraction. Agents are partitioned into two groups: an advantaged group $G_{1}$ of size $(1 - \alpha)n$ and a disadvantaged group $G_{2}$ of size $\alpha n$ , where $\alpha \in (0,1)$ . Each agent $i \in G_{\ell} (\ell \in \{1,2\})$ has a valuation $v_{i} \sim p_{\ell}$ supported on $\Omega_{\ell} \subseteq \mathbb{R}_{\geq 0}$ . We model systemic disadvantage via a scaling of valuations: if $p_{1}$ is the valuation distribution for $G_{1}$ , then $G_{2}$ has valuations drawn from $p_{2}(v) = \frac{1}{\rho} p_{1}(\frac{v}{\rho})$ , where $\rho \in (0,1]$ captures the degree of bias, implying $\mathbb{E}_{v \sim p_2}[v] = \rho \mathbb{E}_{v \sim p_1}[v]$ . For instance, if $p_{1}$ is uniform on [0, 1], then $p_{2}$ is uniform on $[0,\rho]$ . Such a bias model has been widely studied in the fairness literature [48, 16, 19] and serves as a benchmark for understanding systemic disparities. Section D.1 discusses extensions where the valuation distributions $p_{1}$ and $p_{2}$ are truncated Gaussians, and the bias parameter $\rho$ may also be drawn from a distribution, introducing stochastic heterogeneity across candidates. These structural disparities across groups may stem from unequal access to opportunity, differences in marginal returns, labor market discrimination, or broader societal narratives about value; see also Remark 2.1 for practical scenarios. + +Each agent also has an initial ability $a_{i} \sim p_{a}$ supported on $\Omega_{a} \subseteq \mathbb{R}_{\geq 0}$ , drawn independently. We assume that $p_{a}$ is identical across groups. Agents choose policies $A_{i}: \Omega_{\ell} \times \Omega_{a} \to \mathbb{R}_{\geq 0}$ that map their type $\theta_{i} = (v_{i}, a_{i})$ to an exerted effort $e_{i} = A_{i}(\theta_{i})$ . The agent's score is $s_{i} = e_{i} + a_{i}$ . A strictly increasing merit function $m: \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ maps scores to merit. The institution selects the $k$ agents with the highest merit values. Each agent's payoff is + +$$ +f _ {i} \left(v _ {i}, a _ {i}, e _ {i}; s _ {1}, \dots , s _ {n}\right) = \mathbb {I} \left(m \left(s _ {i}\right) \text {a m o n g t o p} k\right) \cdot v _ {i} - \left(s _ {i} - a _ {i}\right) = \mathbb {I} \left(s _ {i} \text {a m o n g t o p} k\right) \cdot v _ {i} - \left(s _ {i} - a _ {i}\right), +$$ + +where the second equality uses the strict monotonicity of $m$ . Agents know $n, k, p_1, p_2, p_a$ , their group identity, and their own type $\theta_i = (v_i, a_i)$ , but not others' types. Let $A = (A_1, \ldots, A_n)$ denote the joint policy profile. The probability that agent $i$ is selected after exerting effort $e$ is + +$$ +P _ {i} (e; a _ {i}, A _ {- i}) = \mathbb {P} \left(e + a _ {i} \text {i s a m o n g t h e t o p} k \text {s c o r e s o f} \left\{A _ {j} \left(\theta_ {j}\right) + a _ {j} \right\} _ {j \neq i} \cup \left\{e + a _ {i} \right\}\right), +$$ + +where $\theta_{j} = (v_{j},a_{j})$ are drawn i.i.d. from the respective group distributions. The expected payoff is + +$$ +\pi_ {i} \left(v _ {i}, a _ {i}, e; A _ {- i}\right) = \mathbb {E} _ {s _ {j}} \left[ f _ {i} \left(v _ {i}, a _ {i}, e; s _ {1}, \dots , s _ {n}\right) \right] = P _ {i} \left(e; a _ {i}, A _ {- i}\right) \cdot v _ {i} - e. +$$ + +A policy profile $A$ is a Nash equilibrium (NE) if, for all $i$ , $v$ , $a$ , and $e$ , + +$$ +\pi_ {i} (v, a, e; A _ {- i}) \leq \pi_ {i} (v, a, A _ {i} (v, a); A _ {- i}). \tag {1} +$$ + +We implicitly assume that agents act rationally and strategically to maximize their expected payoffs, using their knowledge of the contest structure to compute the NE policy $A$ [77]. For simplicity, we sometimes assume that $p_a$ is a point mass at 0, so that policies depend only on valuations. + +A special case of our model generalizes the classical undifferentiated contest (where $p_1 = p_2$ ), which has been extensively studied [79, 55, 85]. To the best of our knowledge, our work is the first to study strategic asymmetries arising from valuation differences in settings where group sizes are known and fixed, a structure commonly seen in admissions and hiring. We provide detailed comparisons with prior works [1, 29, 27] in Section A.1. Remark E.7 discusses extensions to multi-group settings and to heterogeneous effort-to-merit mappings, where each individual may convert effort into merit at a different (non-linear) rate. + +Metrics. We study three metrics to evaluate fairness, efficiency, and institutional outcomes under a given policy $A$ . Define $\mathcal{R}_{\ell}(A)$ as the (random) fraction of agents selected from group $G_{\ell}$ . The representation ratio is $r_{\mathcal{R}}(A) \coloneqq \mathbb{E}\left[\min \left\{\frac{\mathcal{R}_1(A)}{\mathcal{R}_2(A)}, \frac{\mathcal{R}_2(A)}{\mathcal{R}_1(A)}\right\}\right]$ , a metric commonly used in the fairness literature [22, 6, 17]. $r_{\mathcal{R}}(A) \in [0,1]$ , and low values indicate underrepresentation of one group. Building on standard notions of allocative efficiency [70], define group-wise social welfare as $\mathcal{S}_{\ell}(A) \coloneqq \frac{1}{|G_{\ell}|} \sum_{i \in G_{\ell}} (\mathbb{I}(i \text{ selected}) \cdot v_i - e_i)$ . The social welfare ratio is $r_{\mathcal{S}}(A) \coloneqq \mathbb{E}\left[\min \left\{\frac{\mathcal{S}_1(A)}{\mathcal{S}_2(A)}, \frac{\mathcal{S}_2(A)}{\mathcal{S}_1(A)}\right\}\right]$ . This metric measures disparities in average payoffs between groups. Define the average revenue as $\mathcal{RV}(A, m) \coloneqq \mathbb{E}\left[\frac{1}{k} \sum_{i \text{ selected}} m(s_i)\right]$ , capturing the average merit of selected agents and aligns with institutional objectives [33, 34]. + +We analyze how the NE policy $A$ and associated metrics vary with the bias parameter $\rho$ and the selection fraction $c$ . These parameters capture systemic disparities and selection competitiveness, respectively. Even for simple instances, deriving closed-form NE strategies under asymmetric valuations is significantly more complex than in the undifferentiated case. A worked example illustrating these challenges is provided in Section B. + +Remark 2.1 (Practical settings with group-based valuation bias). Our model captures environments where disadvantaged groups anticipate lower returns from being selected—due to structural barriers, social context, or biased algorithmic feedback (see Section 1). For example, as discussed in Section 1, persistent wage gaps across gender and race—even after accounting for qualifications—as well as biased algorithmic recommendations can diminish expectations about the benefits of selection. These lower expectations can rationally reduce pre-selection effort, even under formally fair rules, and represent the main regime we study. That said, in domains such as credit, housing, or education, disadvantaged groups may instead face higher marginal returns due to limited outside options; this can be modeled by reversing which group has the compressed valuation distribution. + +# 3 Theoretical results: Nash equilibrium and metrics for large $n$ + +The first question we address is whether a Nash equilibrium (NE) policy exists for the two-group contest and how it can be computed. While characterizing NE policies for finite $n$ is challenging, the large-population limit $(n \to \infty)$ reveals an interesting and tractable structure. The following result shows that in this limit, it is possible to describe how the strategies of the two groups, $G_{1}$ and $G_{2}$ , converge. However, the absence of an explicit policy formulation for finite $n$ complicates the interpretation of convergence, which we address by adopting the notion of an approximate NE policy. + +Definition 3.1 ( $\varepsilon$ -Nash equilibrium [54]). For an $\varepsilon > 0$ , a policy $A$ is said to be an $\varepsilon$ -NE policy if for any $\ell \in \{1,2\}$ , agent $i \in G_{\ell}$ , type $(v,a) \in \Omega_{\ell} \times \Omega_{a}$ , and effort $e \geq 0$ , the following condition is met: $\pi_i(v,a,e; A_{-i}) \leq \pi_i(v,a,A_i(v,a); A_{-i}) + \varepsilon$ . + +An $\varepsilon$ -NE permits stability violations up to $\varepsilon$ , with exact NE recovered when $\varepsilon = 0$ . This notion allows us to formalize the convergence of NE policies in the following theorem. + +Theorem 3.1 (The two-group contest: Large $n$ limit). Let $\alpha, c \in (0,1)$ . For $\ell = 1,2$ , let $p_{\ell}$ be a density supported on a domain $\Omega_{\ell} \subseteq \mathbb{R}_{\geq 0}$ . Let $p_a$ be a density supported on a domain $\Omega_a \subseteq \mathbb{R}_{\geq 0}$ . Let $m: \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ be a merit function that is strictly increasing. For $\ell = 1,2$ , let $F_{\ell}$ be a cumulative density function (CDF) of the sum of valuation and initial ability such that for any $\zeta \in \mathbb{R}_{\geq 0}$ , $F_{\ell}(\zeta) = \operatorname{Pr}_{v \sim p_{\ell}, a \sim p_a} [v + a \leq \zeta]$ . Suppose $(\Omega_1 \cup \Omega_2) + \Omega_a$ is connected2 and densities $p_1, p_2, p_a$ are positive at any point of their own domains. Let $t$ be the unique solution to the equation + +$$ +(1 - \alpha) F _ {1} (\zeta) + \alpha F _ {2} (\zeta) = 1 - c. \tag {2} +$$ + +$$ +\text {D e f i n e} s (v, a) := 0 \text {i f} v + a < t \text {a n d} s (v, a) := \max \{t - a, 0 \} \text {i f} v + a \geq t \tag {3} +$$ + +and let policy $A$ be: each agent $i \in G_1$ uses the restriction $A_i = s|_{\Omega_1 \times \Omega_a}$ , while each agent $j \in G_2$ uses the restriction $A_j = s|_{\Omega_2 \times \Omega_a}$ . Moreover, this solution $t$ is monotonically decreasing with $c$ . + +This $A$ is the unique policy such that there exists an infinite sequence $A^{(1)},\ldots ,A^{(n)},\ldots$ where $A^{(n)}$ is a policy for the two-group contest with $n$ agents characterized by a threshold function $s^{(n)}:(\Omega_1\cup \Omega_2)\times \Omega_a\to \mathbb{R}_{\geq 0}$ , such that the followings hold: (1) For every integer $n\geq 1$ , agent $i\in G_{1}$ uses the restriction $A_{i}^{(n)} = s^{(n)}|_{\Omega_{1}\times \Omega_{a}}$ , while each agent $j\in G_{2}$ uses the restriction $A_{j}^{(n)} = s^{(n)}|_{\Omega_2\times \Omega_a}$ and $\lim_{n\to \infty}s^{(n)} = s$ ; (2) Every $A^{(n)}$ is an $\varepsilon_{n}$ -NE policy with $\lim_{n\to \infty}\varepsilon_n = 0$ . + +This theorem characterizes the policy $A$ through a threshold function $s$ parameterized by $t$ , establishing that there exists a sequence of policies $A^{(1)}, \ldots, A^{(n)}, \ldots$ that converge towards $A$ , progressively approximating it. In Section E, we provide the explicit form of policies $A^{(n)}$ characterized by $s^{(n)}$ (in Theorem E.2) and a complete proof. Note that the value $t$ defines the threshold function $s$ , and consequently, the policy $A$ . Therefore, we focus below on analyzing $t$ . We first remark on the uniqueness of $t$ guaranteed by Equation (2) under certain assumptions on the domains and densities (see Lemma E.1), which are natural in real-world contexts and satisfied by the distributions discussed in Section 2. These assumptions ensure that each $F_{\ell}$ , being a CDF, is strictly monotonic over its + +domain $\Omega_{\ell} + \Omega_{a}$ . Consequently, the combined CDF $(1 - \alpha)F_{1} + \alpha F_{2}$ must also be strictly monotonic over the connected domain $(\Omega_1\cup \Omega_2) + \Omega_a$ which guarantees the uniqueness of the solution $t$ . + +A key takeaway from Theorem 3.1 is that, in the large- $n$ limit, each agent makes a binary decision based on their combined valuation and initial ability $v + a$ : either exert effort $\max \{t - a,0\}$ to ensure a score of at least $t$ , or put in no effort at all. The threshold $t$ , determined by Equation (2), plays a central role in this decision. It is chosen such that a fraction $1 - c$ of the agent population has $v + a \leq t$ , meaning that exactly a fraction $c$ is expected to exert effort and compete. Thus, $t$ implicitly encodes the level of competition: higher values of $t$ reflect more intense competition, requiring higher effective scores for selection. + +Computing the threshold in the NE policy. Equation (2) is crucial for applying Theorem 3.1, as it enables the explicit computation of $t$ , facilitating analysis in Section 4. Let $F_{a}$ be the CDF of the initial ability. Note that $F_{\ell}(\zeta) = \operatorname{Pr}_{v \sim p_{\ell}, a \sim p_{a}}[v + a \leq \zeta] = \int_{\Omega_{\ell}} p_{\ell}(v) F_{a}(\zeta - v) dv$ . Thus, Equation (2) is equivalent to $(1 - \alpha) \int_{\Omega_{1}} p_{1}(v) F_{a}(\zeta - v) dv + \alpha \int_{\Omega_{2}} \alpha p_{2}(v) F_{a}(\zeta - v) dv = 1 - c$ . Specifically, when $p_{2}(v) = \frac{1}{\rho} p_{1}\left(\frac{v}{\rho}\right)$ for some $\rho \in (0, 1]$ , this equation becomes becomes + +$$ +(1 - \alpha) \int_ {\Omega_ {1}} p _ {1} (v) \cdot F _ {a} (\zeta - v) d v + \frac {\alpha}{\rho} \int_ {\Omega_ {2}} p _ {1} \left(\frac {v}{\rho}\right) F _ {a} (\zeta - v) d v = 1 - c. \tag {4} +$$ + +We illustrate how to use this equation to compute the explicit form of $t$ . Let $p_1$ be uniform on $\Omega_1 = [0, 1]$ , $p_2$ be uniform on $\Omega_2 = [0, \rho]$ , and $p_a$ be uniform on $\Omega_a = [0, 1]$ . Such uniform densities are often used in studies and analyses [48, 16, 19], serves as a fundamental benchmark for insights into decision-making, allocation mechanisms, and strategic behavior. Moreover, domain $(\Omega_1 \cup \Omega_2) + \Omega_a = [0, 2]$ is connected for any value of $\rho \in (0, 1]$ , satisfying assumptions in Theorem 3.1. Since $p_1(v) = 1$ for $v \in [0, 1]$ , $p_2(v) = \frac{1}{\rho}$ for $v \in [0, \rho]$ and $F_a(\zeta - v) = \min \{1, (\zeta - v)_+\}$ (Here, $x_+ = \max \{0, x\}$ ), Equation (2) reduces to $\int_0^1 (1 - \alpha) \cdot \min \{1, (\zeta - v)_+\} dv + \int_0^\rho \frac{\alpha}{\rho} \cdot \min \{1, (\zeta - v)_+\} dv = 1 - c$ . Consequently, the solution $t$ is a piecewise function of parameters $\rho, c,$ and $\alpha$ ; see Proposition F.4 for its explicit form. Here, we illustrate the behavior of $t$ over a representative range where $\alpha = 0.5$ (equal-sized groups) and $0 < c \leq \frac{1}{4}$ (high selectivity). + +$$ +t = 2 - 2 \sqrt {c} \text {i f} \rho < 1 - 2 \sqrt {c} \text {a n d} t = \frac {1 + 3 \rho}{1 + \rho} - \frac {\sqrt {4 c \rho (1 + \rho) - \rho (1 - \rho) ^ {2}}}{1 + \rho} \text {i f} \rho \geq 1 - 2 \sqrt {c}. \tag {5} +$$ + +Note that, while $t$ is the same for both groups, it may happen that $t > 1 + \rho$ (when $\rho < 1 - 2\sqrt{c}$ ), implying that no agent in $G_{2}$ exerts any effort. We note that for other densities, such as piecewise linear and polynomial, including Pareto, explicit forms of the solution $t$ are achievable. For instance, consider a Pareto distribution defined by $p_1(v) = \frac{2}{v^3}$ for $v \geq 1$ , a $\rho$ -biased density $p_2(v) = \frac{1}{\rho} p(\frac{v}{\rho})$ , and $p_a$ is a point mass at 0. Here, $t$ can be explicitly calculated: If $\alpha + c - 1 > 0$ and $\rho < \sqrt{(\alpha + c - 1) / \alpha}$ , then $t = \rho \sqrt{\alpha / (\alpha + c - 1)}$ otherwise, $t = \sqrt{(1 - \alpha + \alpha\rho^2) / c}$ . + +Computing the metrics. We next ask whether the key metrics associated with the NE policy $A$ from Section 2 can be computed in closed form. Given the simple threshold structure of $A$ , these metrics can indeed be expressed as functions of the scalar threshold $t$ . However, for general densities $p_1$ and $p_2$ , the expressions for the representation ratio $r_{\mathcal{R}}(A)$ and social welfare ratio $r_{\mathcal{S}}(A)$ become more complex due to the presence of the min operator and the convolution involved between $p_\ell$ and $p_a$ (see Theorem F.3). For clarity, we focus on the special case where $p_2$ is a $\rho$ -biased version of $p_1$ and $p_a$ is a point mass at 0, which admits more tractable expressions. Since $t$ depends on the parameters $\rho, c,$ and $\alpha$ , the resulting metrics are also functions of these parameters. The following theorem characterizes both the explicit forms and their monotonicity behavior. + +Theorem 3.2 (Metrics and their monotonicity). Assume $p_2(v) = \frac{1}{\rho} p_1\left(\frac{v}{\rho}\right)$ for some $\rho \in (0,1]$ and $p_a$ is a mass point at 0. Let policy $A$ be defined as in Theorem 3.1, characterized by $t$ being the unique solution of Equation (4). Then for any density $p_1$ , + +$$ +r _ {\mathcal {R}} (A) = \frac {1 - F _ {1} (t / \rho)}{1 - F _ {1} (t)}, r _ {\mathcal {S}} (A) = \frac {\rho \int_ {t / \rho} ^ {\infty} (v - t / \rho) p _ {2} (v) d v}{\int_ {t} ^ {\infty} (v - t) p _ {1} (v) d v}, a n d \mathcal {R V} (A, m) = m (t). +$$ + +Moreover, $r_{\mathcal{R}}(A)$ and $r_{\mathcal{S}}(A)$ are monotonically increasing w.r.t. $\rho$ while $\mathcal{RV}(A,m)$ is monotonically increasing w.r.t. $\rho$ and monotonically decreasing w.r.t. $c$ and $\alpha$ , for any merit function $m$ . + +![](images/74d7afa6f75ebf58194e198aeaa3c716c964db1d4756b3a709947cf3a63f1ad1.jpg) +(a) $n = 20$ + +![](images/9a0de39ccce1d5c8fe93b717adadb9c2ac9fb7c0be4f4ca01b1f6dab4825a423.jpg) +(b) $n = 200$ +Figure 1: Evolution of group effort policies at iteration 500 for various $n$ with $\rho = 0.8$ and $c = 0.2$ . + +![](images/8843533b03997ec287f702d8e858cd20e1b5af72cc8130e2325aaaedf87005f8.jpg) +(c) $n = 600$ + +The proof is deferred to Section F. Combined with the closed-form expression for $t$ , this result enables direct computation of the metrics, which we use for contest analysis in Section 4. Notably, $r_{\mathcal{R}}(A)$ and $r_{\mathcal{S}}(A)$ increase with $c$ , while $\mathcal{RV}(A,m)$ decreases, highlighting both benefits and trade-offs for the institute. The qualitative behavior of $r_{\mathcal{R}}(A)$ and $r_{\mathcal{S}}(A)$ w.r.t. $c$ and $\alpha$ , however, depends on the underlying densities; see Remark F.2. + +Discussion of NE policies for finite $n$ . To assess the robustness and practical relevance of our theoretical results, we study the closeness between the finite- $n$ NE policy and the infinite-population threshold policy $s$ defined in Equation (3) (see Section C). We propose a dynamic procedure (Algorithm 1) that initializes with $s_1 = s_2 = s$ for groups $G_{1}$ and $G_{2}$ , and iteratively updates them. + +We simulate this dynamics under the setting $p_1 = \mathrm{Unif}[0,1]$ , $p_2 = \mathrm{Unif}[0,\rho = 0.8]$ , $p_a = \delta_0$ , with $c = 0.2$ , $\alpha = 0.5$ , and $n = 20, 200, 600, 1200$ , running 500 iterations in each case. Figure 1 shows representative results; full plots are in Figures 5-8. + +The simulations show that even moderate population sizes ( $n \geq 600$ ) yield policies closely tracking the infinite NE, validating its use as a practical approximation. We also observe group-level differences in convergence speed and stability (Figure 9), with smoother and faster stabilization as $n$ increases. + +Finally, using the proof of Theorem 3.1, we establish that the finite- $n$ NE policy is $O(\log n / n)$ -close in value and yields an $O(\sqrt{\log n / n})$ -NE. Concretely, this means that for large $n$ , the finite policy takes the form $s^{(n)} = 0$ for $v < t - O(\sqrt{\log n / n})$ and $s^{(n)} = t$ otherwise. For general distributions, closeness depends on the density structure and is more involved. Aligning with our empirical observations, these results reinforce the practical relevance of the large- $n$ analysis, which captures the incentive-aligned baseline under strategic behavior. See Section C.2 for details. + +Key ideas in the proof of Theorem 3.1. The proof involves two main steps: hypothesizing the NE policy structure and verifying that it is indeed an equilibrium. We sketch the core ideas below; a full overview appears in Section E.1. For clarity, we focus on the case where $p_a$ is a point mass at 0. + +Hypothesizing the structure of the NE policy. The key challenge in characterizing the NE policy lies in the absence of its explicit form for finite $n$ . Drawing intuition from the undifferentiated case with density $p$ , where the NE policy converges to a threshold function $s(v) = F_p^{-1}(1 - c)$ if $v \geq F_p^{-1}(1 - c)$ and 0 otherwise, the first idea is to hypothesize that in the two-group case, the NE policies $s_1, s_2$ also take threshold forms with group-dependent thresholds $t_1, t_2$ . However, asymmetry in $p_1, p_2$ complicates the expression of winning probabilities $P_i$ and prevents a straightforward computation of $t_1, t_2$ . Focusing on the uniform case where $p_1$ is Unif[0,1] and $p_2$ is the $\rho$ -biased version supported on $[0, \rho]$ , the second idea is that in the limit $n \to \infty$ , NE stability requires $t_1 = t_2 = t$ . If $t_1 > t_2$ , then some agents in $G_1$ benefit by reducing their effort to slightly above $t_2$ , contradicting NE; a symmetric argument holds if $t_1 < t_2$ . Although $s_1$ and $s_2$ are defined on different domains, they can be viewed as restrictions of the same threshold function $s$ characterized by $t$ . This justifies setting $s_1 = s_2 = s$ with a shared threshold $t$ , and modeling the two-group contest using an effective mixture density $p = (1 - \alpha)p_1 + \alpha p_2$ supported on $\Omega_1 \cup \Omega_2$ . As $n \to \infty$ , the contest becomes indistinguishable from the undifferentiated case with $p$ , yielding the same threshold $t = F_p^{-1}(1 - c)$ as in Equation (2). Thus, the NE policy $A$ defined in Theorem 3.1 is a natural candidate for the limiting equilibrium. Note that this $p$ is only used for hypothesizing NE rather than demonstrating the convergence; see discussion in Section E.4. Moreover, the above argument suggests that, in the limit, the strategic environment becomes uniform across all agents, motivating us to develop an infinite contest (Definition E.5) and provide an alternative proof; see Section E.5. + +![](images/acf8ce545736a4ec9c58b358910558d9f04bca4c9dbd6309dc03769d3297b550.jpg) +(a) $r_{\mathcal{R}}(A)$ v.s. $\rho$ + +![](images/0ff5eb5f443c7f9894843801e64f2222ffe10d0d21906fd3cf33c60dd0cc4c68.jpg) +(b) $r_{\mathcal{R}}(A)$ v.s. $c$ +Figure 2: Plots of the representation ratio $r_{\mathcal{R}}(A)$ and the social welfare ratio $r_{\mathcal{S}}(A)$ as parameters $\rho$ and $c$ vary for Proposition 4.1, with default settings of $(\rho, c, \alpha) = (0.8, 0.1, 0.5)$ . A dotted line in these plots indicates the threshold at which $r_{\mathcal{R}}(A) = 0.8$ or $r_{\mathcal{S}}(A) = 0.8$ . + +![](images/212734df56b6d3939f8131668aa37cb9b9b0183b94e9eef5c47a64c54f2c22f3.jpg) +(c) $r_{\mathcal{S}}(A)$ v.s. $\rho$ + +![](images/f0c798315ad98697bd21c1f2f16f0d223880c32dbc6cd21e083bc14127dbdb0a.jpg) +(d) $r_{S}(A)$ v.s. $c$ + +Showing $A$ is an NE. Although we have a solid guess for the NE policy $A$ , a key challenge arises: the NE policy for finite $n$ lacks an explicit form, making it unclear how to define convergence to $A$ . Towards this end, the third key idea is to provide a different proof of convergence to a threshold function for an undifferentiated case with density $p$ , without using the explicit NE formulations. The first attempt to prove that $A$ is an $\varepsilon$ -NE for sufficiently large $n$ fails—even in the simple case with $p = \mathrm{Unif}[0,1]$ and $c = 0.5$ . One can construct a valuation $v = 0.2$ such that the agent benefits by exerting a small effort $e = 0.01$ , achieving a winning probability $P_{i}(e;A_{-i})\approx 0.5$ and obtaining a payoff of approximately 0.09. To bypass this, the idea is to define a proxy sequence $A^{(n)}$ with threshold policies $s^{(n)}$ that 1) converge to $A$ as $n\to \infty$ and 2) ensure that $P_{i}(e;A_{-i}^{(n)})\rightarrow 0$ for $e < t$ , making $A^{(n)}$ an $\varepsilon_{n}$ -NE with $\varepsilon_{n}\rightarrow 0$ (see Section E.1.3). To this end, we define $s^{(n)}(v) = t$ if $v\geq t - \sqrt{\log n / n}$ and 0 otherwise, so that a $(c + \sqrt{\log n / n})$ -fraction of agents put in effort $t$ . This yields $P_{i}(e;A_{-i}^{(n)})\leq \sqrt{1 / n}$ by concentration, ensuring the payoff from deviation is at most $\sqrt{1 / n}$ , and $A^{(n)}$ is an $O(\sqrt{\log n / n})$ -NE. Finally, we adapt this new proof technique to the two-group case with general $p_1,p_2$ by carefully selecting the following threshold shift (Definitions E.1 and E.2): + +$$ +\Delta_ {n} := \min \left\{F _ {1} ^ {- 1} \left(F _ {1} (t) - \sqrt {\log n / n}\right), F _ {2} ^ {- 1} \left(F _ {2} (t) - \sqrt {\log n / n}\right) \right\}, +$$ + +and set $s^{(n)}(v) = t$ if $v \geq t - \Delta_n$ , 0 otherwise (see Theorem E.2). We find that $\lim_{n \to \infty} \Delta_n = t$ , indicating that $s^{(n)}$ converges to $s$ . Crucially, such a $\Delta_n$ ensures at least a $(c + \sqrt{\log n / n})$ -fraction of agents, in expectation, putting in effort $t$ . Using this property, we establish a concentration tail bound for the winning probability $P_i$ analogous to the undifferentiated contest: $P_i(e; A_{-i}^{(n)}) \to 0$ if $e < t$ (see Lemma E.4). Furthermore, this minimal winning probability guarantees that $A^{(n)}$ is an $\varepsilon_n$ -NE with $\lim_{n \to \infty} \varepsilon_n = 0$ (Lemma E.6). This analysis concludes Theorem 3.1. + +Uniqueness of the NE guaranteed by Theorem 3.1. First, if $\Omega_1 \cup \Omega_2$ is not connected, the CDF $(1 - \alpha)F_1(v) + \alpha F_2(v)$ may not be strictly monotonic over $\Omega_1 \cup \Omega_2$ . For instance, if $\alpha = c = 0.5$ , $p_1$ is uniform on $\Omega_1 = [0,1]$ and $p_2$ is uniform on $\Omega_2 = [2,3]$ , then $(1 - \alpha)F_1(1) + \alpha F_2(1) = (1 - \alpha)F_1(2) + \alpha F_2(2) = 0.5 = 1 - c$ . Consequently, there may exist two distinct points $t_1 = 1$ and $t_2 = 2$ in $\Omega_1 \cup \Omega_2$ , leading to non-unique NEs. In contrast, when $\Omega_1 \cup \Omega_2$ is connected and each $p_\ell$ is positive on $\Omega_\ell$ , the unique solution $t$ to Equation (3) ensures a unique NE policy. To see this, by Corollary 3.2 of [20], symmetric agents use symmetric policies in NE, so we hypothesize a policy pair $(s_1, s_2)$ for $G_1$ and $G_2$ , respectively. As $n \to \infty$ , both $s_1$ and $s_2$ manifest as threshold functions, leading to $s_1 = s_2 = s$ , ensuring the uniqueness of NE. Additionally, it arises because if $s_1 \neq s_2$ , the NE would destabilize, as agents from one group would adjust their thresholds to gain higher payoffs. This proof can be easily extended to multiple groups and non-identical cost of effort; see Remark E.7. + +# 4 Analysis: metric behavior and intervention design + +Variation of metrics with $\rho$ and $c$ . Using Theorems 3.1 and 3.2, we analyze how the metrics—representation ratio $r_{\mathcal{R}}(A)$ , social welfare ratio $r_S(A)$ , and average revenue $\mathcal{RV}(A,m)$ —respond to changes in the bias parameter $\rho$ and the selectivity parameter $c$ . We study whether these effects are linear or non-linear, and whether they exhibit sharp thresholds. + +Setup. We adopt the setting from Section 3: $p_1$ is uniform on $[0,1]$ , $p_2$ is its $\rho$ -biased variant, uniform on $[0,\rho]$ , and $p_a$ is a point mass at 0. This isolates the effect of asymmetric valuations while simplifying calculations. The corresponding CDFs are $F_1(v) = v$ and $F_2(v) = v / \rho$ , with both saturating to 1 outside their support. Unless varied explicitly, we use default values $\rho = 0.8$ , $c = 0.1$ , + +and $\alpha = 0.5$ , representing moderate bias, high selectivity, and balanced group sizes. We refer to Section D.2 for analogous analysis under truncated Gaussian distributions. + +Closed-form metrics. Fixing the density setup above, we apply Theorems 3.1 and 3.2 to derive closed-form expressions for $t$ , $r_{\mathcal{R}}(A)$ , and $r_S(A)$ . These are summarized below (proof in Section F). + +Proposition 4.1 (Metrics for uniform densities). Let $p_1$ be uniform on $[0,1]$ , $p_2$ uniform on $[0,\rho]$ , and $p_a$ a point mass at 0. Let $A$ be the NE policy as $n\to \infty$ . Let $\rho_c\coloneqq 1 - \frac{c}{1 - \alpha}$ . Then: + +$$ +t = 1 - \frac {c}{1 - \alpha} \qquad \text {i f} \rho < \rho_ {c}, \quad t = \frac {\rho (1 - c)}{\rho - \alpha \rho + \alpha} \qquad \qquad \qquad \qquad \text {i f} \rho \geq \rho_ {c}, +$$ + +$$ +r _ {\mathcal {R}} (A) = 0 \qquad \text {i f} \rho < \rho_ {c}, \quad r _ {\mathcal {R}} (A) = \frac {\rho - \alpha \rho + \alpha + c - 1}{\alpha - \alpha \rho + c \rho} \qquad \text {i f} \rho \geq \rho_ {c}, +$$ + +$$ +r _ {\mathcal {S}} (A) = 0 \qquad \text {i f} \rho < \rho_ {c}, \quad r _ {\mathcal {S}} (A) = \frac {\rho (\rho - \alpha \rho + \alpha + c - 1) ^ {2}}{(\alpha - \alpha \rho + c \rho) ^ {2}} \qquad \text {i f} \rho \geq \rho_ {c}. +$$ + +Moreover, $\mathcal{RV}(A,m) = m(t)$ for any merit function $m(\cdot)$ ; $r_{\mathcal{R}}(A)$ and $r_S(A)$ are monotonically increasing functions of parameters $\rho, c, \alpha$ . + +As in Equation (5), Proposition 4.1 reveals a sharp threshold at $\rho = 1 - \frac{c}{1 - \alpha}$ . When $\rho < 1 - \frac{c}{1 - \alpha}$ , $t$ lies above the maximum valuation in $G_{2}$ , implying that no agents from that group participate. Consequently, $r_{\mathcal{R}}(A)$ and $r_S(A)$ are zero and independent of $\rho$ . When $\rho$ crosses this threshold, these metrics become positive and increase monotonically with $\rho$ , reaching 1 at $\rho = 1$ , the symmetric case. + +Metric behavior. Figure 2 plots the representation ratio $r_{\mathcal{R}}(A)$ and the social welfare ratio $r_{\mathcal{S}}(A)$ as functions of the bias parameter $\rho$ and selectivity parameter $c$ . The corresponding threshold values $t$ are shown in Appendix Figure 13(b). Both $r_{\mathcal{R}}(A)$ and $r_{\mathcal{S}}(A)$ exhibit non-linear growth with increasing $\rho$ and $c$ , and drop sharply—super-linearly—when these parameters decrease. For instance, with $c = 0.1$ , $\alpha = 0.5$ , and $\rho \leq 0.85$ , we observe that $r_{\mathcal{R}}(A) \leq 0.2$ , indicating notably low representation for group $G_2$ . This highlights a key practical insight: in highly selective environments, such as contests with a 1-in-10 selection rate, strategic behavior amplifies disparities. These trends echo empirical findings on under-representation in competitive domains [80, 64]. Moreover, reductions in $c$ (i.e., increased selectivity) lead to pronounced declines in both representation and social welfare. + +These trends of metrics offer designers of meritocratic selection processes critical insights into strategies for countering under-representation and elevating $r_{\mathcal{R}}(A)$ (or mitigating disparities in average payoffs and elevating $r_S(A)$ ). We recall the two main criteria for identifying representation bias: 1) Ensuring the selection of at least one agent from every group, and 2) adhering to the $80\%$ rule, which serves as a guideline for identifying potential adverse impact if the hiring rate for $G_2$ falls below $80\%$ of that for $G_1$ , i.e., $r_{\mathcal{R}}(A) \geq 0.8$ . Given the fixed nature of $\alpha$ within the population structure, the main avenues for interventions aimed at improving $r_{\mathcal{R}}(A)$ focus on adjusting the parameters $\rho$ or $c$ . Below, we explore potential interventions for both approaches: + +(1) Increasing $\rho$ effectively means increasing the valuation of agents in $G_{2}$ . Various strategies have been proposed and implemented to achieve this goal. For instance, [39] highlights several approaches to narrow the pay gap, including enhancing workplace flexibility, decreasing the cost associated with temporal flexibility, and improving the availability of high-quality, affordable childcare. These interventions aim to increase job valuation for women, analogous to increasing $\rho$ . Figure 2(a) quantifies the required increase in $\rho$ : to ensure at least one agent from $G_{2}$ is selected, $\rho$ must exceed 0.8; to adhere to the $80\%$ rule, it should be at least 0.976. + +(2) As shown in Figure 2(b), raising $c$ above 0.1 satisfies the criterion for selecting at least one agent from $G_{2}$ , while elevating it to 0.5 meets the $80\%$ -rule requirement. Increasing $c$ represents a more straightforward approach than boosting $\rho$ and might be more feasible for institutions. This could involve pre-selecting a larger subset of candidates and applying a distinct selection process to this subset, based on institutional priorities and the likelihood of successful candidates following the expected trajectories. + +Finally, regarding the average revenue $\mathcal{RV}(A,m) = m(t)$ , it immediately follows from Proposition 4.1 that $\mathcal{RV}(A,m)$ significantly decreases as competition within the entire population intensifies—either through a decrease in $\rho$ or an increase in $c$ . Given that average revenue is indicative of the benefit of the institute, this trend underscores the critical need for contest designers to mitigate systemic biases in valuations. The decline in average revenue with increasing bias compromises not + +only the fairness of the contest, but also the overall quality of the outcomes it produces. This aligns with research that has explored the losses attributed to systemic biases [43, 58, 67]. + +Optimizing interventions under cost and fairness constraints. Having identified two policy levers—reducing bias $(\rho)$ and increasing selectivity $(c)$ —a natural question arises: how should institutions choose between these interventions to improve outcomes such as the representation ratio or social welfare ratio? To address this, we formulate a constrained optimization problem for cost-effective intervention design. We allow two interventions: increasing $\rho$ by $\Delta_{\rho} \in [0,1 - \rho]$ , and increasing $c$ by $\Delta_c \in [0,1 - c]$ . Let $r_{\mathcal{R}}(\Delta_{\rho}, \Delta_c)$ denote the representation ratio under the NE policy with updated parameters $\rho + \Delta_{\rho}$ , $c + \Delta_c$ . The goal is to ensure $r_{\mathcal{R}}(\Delta_{\rho}, \Delta_c) \geq \tau$ while minimizing intervention cost. We define two components of the cost function: + +(1) Resource cost of increasing $\rho$ . Let $f:[0,1 - \rho] \to \mathbb{R}_{\geq 0}$ be monotonic, modeling the institutional cost of boosting valuation. A simple form is linear: $f(\Delta_{\rho}) = a\Delta_{\rho}$ , justified by first-order Taylor approximation when $\Delta_{\rho}$ is small. Other variants include $f(\Delta_{\rho}) = a\Delta_{\rho}^{\beta}$ for $\beta > 1$ , representing the increase in the marginal cost of continuously improving bias. +(2) Cost via revenue loss. Increasing $c$ reduces average revenue $\mathcal{RV}(A, m) = m(t)$ , as it lowers the score threshold $t$ . Let $g(\Delta_{\rho}, \Delta_c) = m(t(\rho, c)) - m(t(\rho + \Delta_{\rho}, c + \Delta_c))$ , represent the revenue decline. Since the institution seeks to maximize value, this loss contributes to total intervention cost. + +Optimization problem. We formalize the intervention design as: + +$$ +\min _ {\Delta_ {\rho} \in [ 0, 1 - \rho ], \Delta_ {c} \in [ 0, 1 - c ]} f (\Delta_ {\rho}) + g (\Delta_ {\rho}, \Delta_ {c}) \quad \text {s . t .} \quad r _ {\mathcal {R}} (\Delta_ {\rho}, \Delta_ {c}) \geq \tau . \tag {6} +$$ + +This framework also applies to reducing welfare disparities by replacing $r_{\mathcal{R}}$ with $r_{S}$ in the constraint. + +Empirical calibration. To demonstrate real-world applicability, we calibrate the model using gender-disaggregated data from JEE Advanced 2024, a highly competitive entrance exam for India's IITs. Of 180,200 candidates, 139,180 were male and 41,020 female; 40,284 males and 7,964 females qualified. This yields admit rates of $28.9\%$ for males and $19.4\%$ for females, giving an observed representation ratio of $r_{\mathrm{obs}} \approx 0.671$ . The overall selection rate is $c \approx 0.268$ , and the female applicant fraction is $\alpha \approx 0.228$ . These values anchor our analysis of strategic disparities and potential interventions. In Section G.1, under the uniform density setup in Proposition 4.1, we compute $\rho \approx 0.882$ using the explicit form of $r_{\mathrm{obs}} = r_{\mathcal{R}}(A) = 1 - \frac{(1 - c)(1 - \rho)}{\alpha - \alpha\rho + c\rho}$ . + +Explicit solution under uniform densities. We now solve the optimization problem under this uniform density setup with $(\rho, c, \alpha) = (0.882, 0.268, 0.228)$ , setting $m(e) = e$ , $f(\Delta_{\rho}) = 5\Delta_{\rho}^{1.1}$ , and $g(\Delta_{\rho}, \Delta_c) = t(\rho, c) - t(\rho + \Delta_{\rho}, c + \Delta_c)$ . This yields the objective: $5\Delta_{\rho}^{1.1} - [t(\rho + \Delta_{\rho}, c + \Delta_c) - t(\rho, c)]$ , subject to the condition $r_{\mathcal{R}}(\Delta_{\rho}, \Delta_c) \geq \tau$ and feasibility constraints. + +Insights. Figure 4 shows the optimal intervention as a function of the target threshold $\tau$ . For $\tau \leq 0.92$ , increasing $c$ (lowering selectivity) is more cost-effective. For $\tau > 0.92$ , increasing $\rho$ (mitigating bias) becomes preferable. This suggests that expanding access is more impactful under high disparity, while improving group valuation is better when gaps are narrower. + +We conducted additional simulations by varying $\alpha$ and $c$ beyond the default values (see Section G.2). The results remain consistent with our main findings, confirming the robustness of the above key insights. In Section G.3, we also offer a concrete example to illustrate how our model supports interpretable predictions and can inform data-grounded interventions—while also noting what is required to operationalize it in practice. + +$$ +\begin{array}{l} \min _ {\Delta_ {\rho} \in [ 0, 1 - \rho ], \Delta_ {c} \in [ 0, 1 - c ]} 5 \Delta_ {\rho} ^ {1. 1} - \frac {(\rho + \Delta_ {\rho}) (1 - c - \Delta_ {c})}{(1 - \alpha) (\rho + \Delta_ {\rho}) + \alpha} \\ s. t. \frac {(1 - \alpha) (\rho + \Delta_ {\rho}) + \alpha + c + \Delta_ {c} - 1}{\alpha - \alpha (\rho + \Delta_ {\rho}) + (c + \Delta_ {c}) (\rho + \Delta_ {\rho})} \geq \tau \\ \rho + \Delta_ {\rho} \geq 1 - \frac {c + \Delta_ {c}}{1 - \alpha}. \\ \end{array} +$$ + +![](images/e9336219f8fc527b472c80a37438f5532c29e03866176710a71715a0ad830377.jpg) +Figure 3: Explicit form of Problem (6). +Figure 4: Plot of optimal interventions $(\Delta_{\rho}, \Delta_{c})$ for various $\tau \in (0.671, 1]$ . + +Alternative potential interventions. Having discussed interventions based on adjusting $\rho$ and $c$ , we next consider alternative approaches that modify the contest structure itself. These interventions can further reduce disparities in representation or social welfare ratios, though they depart from the baseline two-group contest formulation. A detailed analysis of these extensions is provided in Section G.4. + +Introducing preference heterogeneity. One approach is to apply group-specific merit mappings of the form $m_{\ell}(s) = x_{\ell}s + y_{\ell}$ for group $G_{\ell}$ ( $\ell = 1,2$ ), with parameters $x_{\ell}, y_{\ell} \geq 0$ . Here, $x_{\ell}$ acts as a scaling factor or "handicap," and $y_{\ell}$ as an offset or "head start" (see also Appendix A). With this intervention, we can still compute the Nash equilibrium for infinite $n$ (Theorem G.1), which implies: + +- $r_{\mathrm{R}}(A)$ and $r_{\mathcal{S}}(A)$ increase with $\rho, c,$ and $\alpha$ as well as with $x_{2},y_{2}$ , and decrease with $x_{1},y_{1}$ ; +- Choosing merit parameters with $x_{2} > x_{1}$ and $y_{2} > y_{1}$ can sustain high representation and welfare ratios (e.g., $r_{\mathrm{R}}(A), r_{\mathcal{S}}(A) \geq 0.8$ ) even in highly selective settings; +- Increasing $x_{2}$ or $y_{2}$ can thus serve as an effective disparity-reducing intervention. + +Incorporating outside options. Another possibility is to assign each agent in group $G_{\ell}$ a reservation payoff $\lambda_{\ell} \geq 0$ if not selected. Because this payoff is earned only upon losing, a higher $\lambda_{\ell}$ lowers the marginal benefit of effort, acting opposite to the merit parameters $x_{\ell}$ and $y_{\ell}$ . Hence, increasing $\lambda_{1}$ (the outside option for the advantaged group) reduces their effort incentives and can help narrow representation and welfare gaps. + +Setting group-specific selection rates. Finally, the institution can set separate capacity constraints for each group—for instance, selecting a $c$ -fraction of agents from $G_{1}$ and $G_{2}$ independently. This decomposes the overall model into two within-group contests, fixing $r_{\mathcal{R}}(A) = 1$ under equal selection rates. Compared to the combined contest, agents in $G_{2}$ now face a lower bar for selection and exert more effort on average. + +# 5 Conclusion, limitations, and future work + +This work highlights a central tension in modern meritocratic systems: even when selection mechanisms are formally unbiased, systemic disparities in how groups perceive value can lead rational agents to behave in ways that perpetuate inequality. Our model captures this dynamic through a strategic contest framework that extends all-pay auctions to multi-group settings. By analyzing Nash equilibria in the large population limit, we characterize how group-level biases $(\rho)$ and selectivity $(c)$ affect fairness and institutional metrics such as representation, social welfare, and revenue. A central contribution is Theorem 3.1, which provides an explicit form for equilibrium strategies under broad conditions. Our framework enables interpretable predictions and supports data-grounded policy interventions. + +Our model makes simplifying assumptions to enable analytical tractability. Most notably, it assumes agents are fully rational and that merit is captured by a single-dimensional notion of effort. In practice, decision-making is shaped by uncertainty, cultural context, and multifaceted criteria for merit. Extending the framework to incorporate bounded rationality, noisy information, or multidimensional effort remains an important direction for future work. Several application-driven extensions are also promising. One involves modeling university admissions systems with external incentives (e.g., brand-based free-riding), which may result in over-representation of certain groups ( $\rho > 1$ ). Another is to study how affirmative action or group-dependent costs reshape equilibrium behavior. These variants would help bridge theory with institutional design. Beyond these extensions, an important avenue is to embed this static framework within dynamic feedback environments where perceptions evolve over time in response to outcomes and institutional signals. Such models could capture how bias propagates or attenuates across repeated selection cycles. Finally, while our model isolates a tractable facet of systemic inequality, real-world disparities—especially in AI-mediated evaluations—demand broader integration with social and historical context. As algorithmic tools shape hiring, admissions, and promotion, our framework helps explain how group-level differences in perceived value can interact with selection to amplify or mitigate bias. More broadly, we view this work as a step toward unifying rational-choice and structural perspectives on inequality through formal, data-driven modeling. We hope this work informs the design of more equitable, data-driven decision systems. + +# Acknowledgments + +LH acknowledges support from the State Key Laboratory of Novel Software Technology, the New Cornerstone Science Foundation, and NSFC Grant No. 625707396. LEC was supported in part by NSF Award IIS-2045951. 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Journal of Mathematical Economics, 95:102472, 2021. + +# Contents + +1 Introduction 1 +2 Model and metrics 3 +3 Theoretical results: Nash equilibrium and metrics for large $n$ 4 +4 Analysis: metric behavior and intervention design 7 +5 Conclusion, limitations, and future work 10 + +A Detailed related work 18 + +A.1 Comparison of the two-group contest with relevant models 19 +B Illustrative examples for the two-group case 20 +C Analysis of finite NE policies in the uniform distribution case 21 + +C.1 Empirical analysis 21 +C.2 Theoretical analysis 22 + +D Other bias models and analysis of metrics for their Nash equilibrium 23 + +D.1 Other bias models 23 +D.2 Analysis of metrics for Nash equilibrium in the truncated normal distribution case. 27 + +E Proof of Theorem 3.1: two-group contest 28 + +E.1 Technical overview 28 + +E.1.1 NE policy for the undifferentiated contest for finite $n$ and obstacle in extending it 29 +E.1.2 A conjectured NE policy in two-group contests for large $n$ 29 +E.1.3 Proving convergence to the conjectured NE policy 31 + +E.2 A more comprehensive version of Theorem 3.1: convergence form 32 +E.3 Proof of Theorem E.2 33 + +E.3.1 Proof of Lemma E.1: solution uniqueness 33 +E.3.2 Bounding winning probability 34 +E.3.3 Proof that $A^{(n)}$ is approximate NE 36 +E.3.4 Completing the proof of Theorem E.2 36 + +E.4 Comparing with a distributional two-group contest 37 +E.5 An alternative proof using an infinite contest 38 + +F Omitted details for uniform distribution analysis from Sections 3 and 4 40 + +G Additional details to Section 4 44 + +G.1 Case study — estimating perceived bias from JEE Advanced 2024 44 + +G.2 Robustness analysis for findings in Section 4. 45 +G.3 An illustrative example: interpreting and applying the model 46 +G.4 Details for alternative interventions 46 + +# A Detailed related work + +Meritocratic selection process, pay gap, and statistical discrimination feedback loop. In the social sciences, there is a large body of work that studies meritocratic selection processes and their limitations; see [59, 57, 76] and the references therein. [15, 62] discuss the pay gap in meritocratic systems, shedding light on how merit-based reward systems and gender wage gaps intersect. [38], in an extensive line of work, discusses the gender pay gap and addresses the economic and social factors contributing to wage disparities between men and women. Another line of research focuses on studying statistical discrimination feedback loops, which model how firms update their beliefs about group quality over time, reinforcing disparities [3, 69, 21, 23, 5, 49]. For instance, [3] emphasizes how the cost of individualized assessment incentivizes reliance on priors, which can become self-fulfilling and reinforce structural inequality. [3] models a profit-maximizing employer who faces noisy signals of productivity and rationally uses group-level statistics, leading to persistent wage gaps even with equal underlying abilities. [21] show that pessimistic beliefs about a group's productivity can result in tougher standards, reduced investment incentives, and discriminatory equilibria. [23] extend this to two-sided settings where firms and workers both act on noisy beliefs, reinforcing low-investment, low-opportunity equilibria. A key distinction, as we understand it, is that classical models of statistical discrimination typically generate disparities through imperfect and group-dependent beliefs about identical underlying abilities. In contrast, our framework allows perfect, unbiased information at the institutional level and identical selection criteria for all candidates. We focus instead on valuation asymmetries—that is, differences in the perceived benefit of success across groups—and show that these differences alone can lead to disparities in effort and representation, even under meritocratic selection. + +All-pay auctions and Tullock contests. In game theory, there is a significant body of literature that investigates all-pay auctions. For instance, [79, 55, 85] study the setting in which every agent knows their private valuations and the distribution of other agents. Specifically, [79] study a "biased" 2-agent contest in which the designer is allowed to give a "headstart" to the effort of one agent. This headstart can be interpreted as differing merits of the agents, which corresponds to the initial abilities in our model. They characterize the optimal design for maximizing the expected highest effort or total effort of agents. In this case, the bias is introduced by the designer, rather than inherent in the system. [85] study the undifferentiated case for a single winner, while the contest designer is allowed to select the contest success function (CSF) based on agents' efforts. Their main focus is on studying the optimal design of the CSF that maximizes the total expected effort. The main difference from our model is that they consider bias in the efforts instead of in the valuations. + +The all-pay auction with complete information has also been well-studied. Unlike the setting in this paper, these works assume that the valuations of all agents are known. [7] initiated the study of an $n$ -agent $k$ -winner all-pay auction and provided a complete characterization of the NE distribution. A line of research investigates the optimal design for maximizing the total expected effort/revenue, including imposing a multiplicative bias on the effort of agents [32, 36] or introducing an additional headstart [35, 33, 34, 36, 90]. + +Tullock contests [82, 81, 30, 34, 25, 53, 50] model the probability of winning based on relative effort without direct costs for participation, whereas an all-pay auction requires all agents to pay their bid amounts regardless of winning, with only the highest bidder(s) securing the prize. [65] study the dynamics of large contests, where a significant number of agents compete. Such contests pose unique analytical challenges and offer insights into the behavior of agents in mass competition scenarios. The works of [32, 33, 34] also investigate how the design of contests can be optimized to maximize revenue, considering factors like bias in efforts, headstarts, and the structure of the CSF. Across these studies, a common theme is the characterization of Nash equilibrium strategies within the context of different contest models, and identifying designs that encourage maximal effort or revenue. + +Strategic classification and ranking. Another related direction is strategic learning, which mainly includes strategic classification [14, 44, 60, 9] and strategic ranking [29]. In strategic classification, agents can exert effort to alter their features to achieve higher values according to the published classifier. The designer's aim is to select a classifier that is robust to the manipulation of inputs by strategic agents. However, in this setting, agents' efforts are influenced solely by the published classifier, with no competition among them. In strategic ranking problems, agents' payoffs depend on their post-ranking, which is determined by a combination of their prior rankings and efforts. While + +there is competition in this problem, all agents have the same valuation, which is different from our model. + +Models of bias in valuations. Several works have modeled group-level biases based on empirical observations [4, 10, 48, 28, 18]. Additive and multiplicative skews in the valuations have also been modeled [48, 10]. [48] consider valuations $v > 0$ of the advantaged group distributed according to the uniform or Pareto density and, for the disadvantaged group, they model the output as $v / \beta$ for some fixed $\beta \geq 1$ . We consider a class of bias models inspired by this model, the $\rho$ in our case corresponds exactly to $1 / \beta$ . The implicit variance model of [28] models differences in the amount of noise in the valuations for individuals in different groups. Here, the output estimate is drawn from a Gaussian density whose mean is the valuation $e$ (which can take any real value) and whose variance depends on the group of the individual being evaluated: The variance is higher for individuals in the disadvantaged group compared to individuals in the advantaged group. [18] proposes an optimization-based approach to model how group-wise valuation distributions can be obtained by tuning parameters such "information constraints" or "risk aversion". + +# A.1 Comparison of the two-group contest with relevant models + +To the best of our knowledge, our model is novel and has not been studied in the literature. Below, we compare our model with the most relevant models. Firstly, [1] examines a specific case of our model with $n = 2$ , $k = 1$ , and $\alpha = 0.5$ , demonstrating the existence of a unique NE under certain conditions. While their analysis is limited to a two-player scenario, our model generalizes this by considering any number of players and allowing for multiple winners. [29] propose another two-group contest model. However, a key distinction in our model is the consideration of asymmetric valuation distributions across groups, whereas [29] introduces bias through the cost of effort, assuming symmetric valuations for all agents. This asymmetry in valuation distributions in our model adds complexity to the analysis. + +Another related work is [27], which explores an all-pay auction with two groups. In their model, agents' abilities are symmetrically distributed, and those in the advantaged group may receive additional rewards with equivalent bids. Despite the symmetric strategic environment in [27], our model features asymmetric valuation distributions between groups, resulting in an asymmetric strategic environment. This asymmetry introduces further computational challenges for deriving the NE; details can be found below. + +A detailed comparison with [27]. We provide a detailed comparison between our two-group model and that in [27]. The primary distinction is that their model results in a symmetric strategic environment, while ours creates an asymmetric one. Below, we provide further details on this difference. + +In the model of [27], each agent belongs to the target group with probability $\mu$ or to the non-target group with probability $1 - \mu$ , independently of the other agents. The ability of an agent is then drawn i.i.d. from distribution $F$ if they belong to the target group, and from $G$ if they belong to the non-target group. As a result, the ability of each agent is drawn identically and independently from the joint distribution $\mu F + (1 - \mu)G$ . Let $H$ be the CDF of this joint distribution. The probability that a given ability $v$ is among the top $k$ abilities is then given by $\sum_{i=n-k}^{n-1} \binom{n-1}{i} H(v)^i (1-H(v))^{n-1-i}$ , which is the same for each agent, thereby resulting in a symmetric strategic environment. + +In our model, consider a simplified case where $p_a$ is a point mass at 0. Then, $F_1$ and $F_2$ correspond to the cumulative distribution functions (CDFs) of $p_1$ and $p_2$ , which represent the valuation densities of $G_1$ and $G_2$ , respectively. The probability that, for an agent in $G_1$ , a given valuation $v$ is among the top $k$ valuations is given by: + +$$ +\sum_{i = 0}^{n - 1}\sum_{j = n - k - i}^{n - 1 - i}\binom {n - 1}{i}\binom {n - 1 - i}{j}F_{1}(v)^{i}(1 - F_{1}(v))^{(1 - \alpha)n - 1 - i}F_{2}(v)^{j}(1 - F_{2}(v))^{\alpha n - j}. +$$ + +In contrast, the probability for an agent in $G_{2}$ is given by + +$$ +\sum_ {i = 0} ^ {n - 1} \sum_ {j = n - k - i} ^ {n - 1 - i} {\binom {n - 1} {i}} {\binom {n - 1 - i} {j}} F _ {1} (v) ^ {i} (1 - F _ {1} (v)) ^ {(1 - \alpha) n - i} F _ {2} (v) ^ {j} (1 - F _ {2} (v)) ^ {\alpha n - 1 - j}. +$$ + +These two expressions differ whenever $p_1 \neq p_2$ , leading to asymmetry in the strategic environment. This asymmetry significantly complicates the computation of the order statistics for the $(k - 1)$ -th + +effort compared to the symmetric ones. E.g., for strategies $s_1$ and $s_2$ , let $F_{s_\ell}(v)$ denote the CDF of efforts $s_\ell(v)$ when $v \sim p_\ell$ . The cumulative distribution of the $(k-1)$ -th effort $e^\star$ from an agent in $G_1$ is then given by: + +$$ +\begin{array}{l} \Pr [ e ^ {\star} \leq v ] \\ = \sum_ {i = 0} ^ {n - 1} \sum_ {j = (1 - c) n - i} ^ {n - 1 - i} {\binom {n - 1} {i}} {\binom {n - 1 - i} {j}} F _ {s _ {1}} (v) ^ {i} (1 - F _ {s _ {1}} (v)) ^ {(1 - \alpha) n - 1 - i} F _ {s _ {2}} (v) ^ {j} (1 - F _ {s _ {2}} (v)) ^ {\alpha n - j}. \\ \end{array} +$$ + +In contrast, the cumulative distribution of the $(k - 1)$ -th effort $e^{\star}$ from an agent in $G_{2}$ is given by: + +$$ +\begin{array}{l} \Pr [ e ^ {\star} \leq v ] \\ = \sum_ {i = 0} ^ {n - 1} \sum_ {j = (1 - c) n - i} ^ {n - 1 - i} {\binom {n - 1} {i}} {\binom {n - 1 - i} {j}} F _ {s _ {1}} (v) ^ {i} (1 - F _ {s _ {1}} (v)) ^ {(1 - \alpha) n - i} F _ {s _ {2}} (v) ^ {j} (1 - F _ {s _ {2}} (v)) ^ {\alpha n - 1 - j}. \\ \end{array} +$$ + +In the symmetric ones $(s_1 = s_2 = s)$ , the computation simplifies to: + +$$ +\Pr [ e ^ {\star} \leq v ] = \sum_ {i = (1 - c) n} ^ {n - 1} \binom {n - 1} {i} F _ {s} (v) ^ {i} (1 - F _ {s} (v)) ^ {n - 1 - i}. +$$ + +Thus, the calculus and approximations for the two-group contest is significantly more difficult, making it harder to arrive at the equilibrium policies than in the contest with a symmetric strategic environment. + +# B Illustrative examples for the two-group case + +In this section, we present a two-agent example with a biased valuation distribution to illustrate both the difficulty of computing the Nash equilibrium (NE) policy and the significant impact of valuation bias on the contest outcome. Let $c = 0.5$ . Let the density $p_1$ of agent 1 be the uniform distribution on $\Omega_1 = [0,1]$ and $p_2$ of agent 2 be the $\rho$ -biased version of $p_1$ supported on $\Omega_2 = [0,\rho ]$ . Let the density $p_a$ be a point mass at 0. Let $A_{\ell}:\Omega_{\ell}\to \mathbb{R}_{\geq 0}$ be the NE policy that maps valuation $v_{\ell}$ to effort $A_{\ell}(v_{\ell})$ . We assume $A_{\ell}$ is monotonically increasing on the domain $\Omega_{\ell}$ . + +In this example, if agent 1 puts in effort $e$ , it wins if the effort of agent 2 is smaller than $e$ . Thus, its winning probability $P_{1} = \frac{A_{2}^{-1}(e)4}{\rho}$ and its payoff is $\pi_1(v,e;A_2) = \frac{A_2^{-1}(e)}{\rho} v - e$ . Similarly, if agent 2 puts in effort $e$ , its winning probability $P_{2} = A_{1}^{-1}(e)$ and payoff is $\pi_2(v,e;A_1) = A_1^{-1}(e)v - e$ . Then, by the stability condition (1), we have $\frac{\partial\pi_{\ell}(v,e;A_{3 - \ell})}{\partial e}\big|_{e = A_{\ell}(v)} = 0$ , implying that + +$$ +A _ {2} ^ {\prime} \left(A _ {2} ^ {- 1} \left(A _ {1} (v)\right)\right) = \frac {v}{\rho}, \text {a n d} A _ {1} ^ {\prime} \left(A _ {1} ^ {- 1} \left(A _ {2} (v)\right)\right) = v. +$$ + +Solving this gives us the following explicit forms: + +$$ +\forall v \in \Omega_ {1}, A _ {1} (v) = \frac {\rho}{\rho + 1} v ^ {\rho + 1}; \text {a n d} \forall v \in \Omega_ {2}, A _ {2} (v) = \frac {\rho^ {- 1 / \rho}}{\rho + 1} v ^ {1 + 1 / \rho}. \tag {7} +$$ + +Specifically, when $\rho = 1$ (the unbiased case), we have $A_{1} = A_{2} = A$ , which simplifies the stability condition to $A'(v) = v$ , yielding the NE policy $A(v) = \frac{v^2}{2}$ . Also note that for $v \in \Omega_2$ , + +$$ +\frac {A _ {1} (v)}{A _ {2} (v)} = \rho^ {1 + 1 / \rho} v ^ {\rho - 1 / \rho} \leq \rho^ {1 + 1 / \rho} \rho^ {\rho - 1 / \rho} = \rho^ {1 + \rho} \leq 1, +$$ + +which implies that $A_{1}(v) \leq A_{2}(v)$ . Thus, agent 2 is more inclined to put in greater effort than agent 1 for identical valuations. + +Imagine an institute that is unaware of the bias in valuations across two agents and thus applies the unbiased NE policy $A(v) = \frac{v^2}{2}$ to predict the contest outcome. For instance, it would predict the average revenue + +$$ +\mathcal {R V} (A, m) = \int_ {0} ^ {1} \Pr_ {v _ {\ell} \sim p _ {\ell}} \left[ m \left(\max \left\{\frac {v _ {1} ^ {2}}{2}, \frac {v _ {2} ^ {2}}{2} \right\}\right) > t \right] d t = 0. 2 5, +$$ + +where the merit function is $m(t) = t$ . However, under a $\rho$ -biased valuation distribution, the true average revenue is $\mathcal{R}\mathcal{V}_{\rho} = \frac{\rho}{2(\rho + 1)}$ , which decreases monotonically with $\rho$ . This implies that the institute could overestimate its expected benefit $\mathcal{R}\mathcal{V}$ by a fraction of + +$$ +\frac {0 . 2 5 - \mathcal {R V} _ {\rho}}{\mathcal {R V} _ {\rho}} = \frac {1 - \rho}{2 \rho}, +$$ + +which amounts to approximately $13\%$ when $\rho = 0.8$ . This example underscores the importance of studying asymmetric valuations and highlights the relevance of our proposed metrics for analyzing their impacts. + +We also observe that even for this simple two-agent example, the stability condition is considerably more complex than in the undifferentiated case. In more general settings—such as those involving multiple spots, non-uniform valuation densities, or non-trivial ability densities—the explicit forms of NE policies for a two-group contest become even more complicated, making direct computation and explicit analysis impractical. + +# C Analysis of finite NE policies in the uniform distribution case + +In this section, we use the uniform distribution example from Section 3 as a running example, introduce a dynamic algorithm (Algorithm 1) to approximate the finite NE policies, and perform a statistical comparison between the finite and infinite cases. Additionally, we provide a theoretical analysis of the closeness between the NE policies and associated metrics in the finite and infinite cases. + +# C.1 Empirical analysis + +Dynamics for computing finite NE policies. Recall that we consider $p_1 = p = \mathrm{Unif}[0,1]$ , $p_2 = p_2 = \mathrm{Unif}[0,\rho]$ for $\rho \in [0,1]$ , and $p_a \equiv 0$ . Algorithm 1 presents a dynamic procedure to approximately compute the finite-population NE policies. + +We initialize each group's policy $s_{\ell}^{(0)}$ with a smoothness variant of the infinite NE policy (Lines 3-4), then iteratively update these policies over $N$ steps and return the final output as an approximation of the finite NE (Lines 5-38). At each iteration $t$ : + +1. We first update the effort set $E_{t}$ based on the policies $s_{\ell}^{(t - 1)}$ from the previous iteration (Line 6). Since the action space is continuous, we restrict agents to choose efforts only from this finite set $E_{t}$ . +2. Next, we update the policy for group $G_{1}$ using the policy $s_2^{(t - 1)}$ from the previous iteration (Lines 7-21). The computation is performed over a finite set $V^{(1)}$ of discrete valuation levels (Line 1). For each valuation $v$ , we determine the best-response effort that maximizes the agent's expected payoff by computing winning probabilities through a convolution of binomials (Lines 9-19). Specifically, we set $p_1 = 1 - v$ in Lines 9 and 12, consistent with the monotonicity constraint enforced in Line 11. Finally, Line 21 updates the policy using a carefully chosen step size $a_{\ell}^{(t)}$ to ensure convergence. +3. We then update the policy for group $G_{2}$ based on the policy $s_1^{(t - 1)}$ from the previous iteration (Lines 7-21). This process mirrors that of $G_{1}$ , with the main difference lying in the computation of winning probabilities for each effort in $E_{t}$ due to the asymmetric valuation distributions. + +The resulting policies $s_{\ell} = s_{\ell}^{(T)}$ are defined on discrete valuation grids. To obtain continuous policies, we interpolate them by connecting adjacent valuation points with straight lines, resulting in piecewise-linear approximations. + +Choice of hyperparameters. In our simulations, we set the valuation resolution $m_v = 101$ , effort resolution $m_e = 101$ , total number of iterations $T = 500$ , and step sizes $a_1^{(t)} = a_2^{(t)} = \frac{1}{10T}$ . We always set $n_1 = n_2 = \frac{n}{2}$ , which means $\alpha = 0.5$ . + +Metrics. For each iteration $t$ , we compute the following metric to evaluate the updated policies $s_{\ell}^{(t)}$ : + +$$ +\Delta^ {(\ell , t)} := \frac {\sum_ {v \in V ^ {(\ell)}} | \pi_ {\ell} ^ {(t)} (v) - s _ {\ell} ^ {(t - 1)} (v) |}{m _ {v}} = \frac {\sum_ {v \in V ^ {(\ell)}} | s _ {\ell} ^ {(t)} (v) - s _ {\ell} ^ {(t - 1)} (v) |}{a _ {\ell} ^ {(t)} m _ {v}}, +$$ + +which quantifies the average policy update for group $G_{\ell}$ at iteration $t$ . Intuitively, a decreasing $\Delta^{(\ell,t)}$ indicates convergence of the policy sequence $s_{\ell}^{(t)}$ . However, since we work with discretized valuation and effort sets, we do not expect $\Delta^{(\ell,t)}$ to vanish entirely. + +Results. Figures 5, 6, 7, and 8 present the evolution of equilibrium policies for population sizes $n \in \{20, 200, 600, 1200\}$ across four time snapshots $t \in \{50, 150, 300, 500\}$ , with fixed parameters $\rho = 0.8$ and $c = 0.2$ . Although all runs begin with a smoothed version of the infinite-population NE, the dynamics vary significantly with population size. For small $n$ (e.g., $n = 20$ ), we observe noticeable fluctuations in early iterations, particularly in group $G_{2}$ , whose valuation distribution is more concentrated. By $t = 500$ , both policies stabilize, though they retain visible irregularities due to stochasticity in rank-based feedback. Even though all runs begin with a smooth initialization based on the infinite-population NE, the dynamics unfold differently depending on population size. For small $n$ (e.g., $n = 20$ ), we observe noticeable fluctuations in the early iterations, especially in group $G_{2}$ , whose valuation distribution is more concentrated. At $t = 500$ , the policies stabilize but retain some irregularity, reflecting noise in the agent-level ranking and feedback structure. + +As $n$ increases, both groups' policies become smoother and stabilize more quickly. By $n = 600$ , the effort policies align closely with the infinite NE, and further updates beyond $t = 300$ are negligible. These trends are confirmed by the convergence plots in Figure 9, which show a sharp reduction in the $\ell_1$ -norm policy update $\Delta^{(\ell ,t)}$ with increasing $n$ . Group $G_{1}$ consistently converges faster than $G_{2}$ , a pattern attributable to its broader valuation support and greater flexibility in effort choice. Overall, the results illustrate that the infinite-population equilibrium is a good predictor even for moderately sized finite systems, while also quantifying the transient effects and instability that emerge in low- $n$ regimes. + +Interestingly, we also observe from these plots that when $n$ is small ( $n = 20, 200$ ), $s_2(v) > s_1(v)$ , while for larger values of $n$ ( $n = 600, 1200$ ), $s_2(v) < s_1(v)$ . In the subsequent subsection, we will provide a theoretical analysis to explain the underlying reasons for this behavior. + +# C.2 Theoretical analysis + +We begin by presenting theoretical evidence for the alignment between finite and infinite NEs, a relationship that is observed empirically. In the proof of Theorem 3.1 (see Section E), we show that for any finite $n$ , $\varepsilon_{n}$ -NE policy $s^{(n)}$ stated in Theorem 3.1 is “ $O(\sqrt{\log n / n})$ -close” to the policy $s$ for infinite $n$ and is “ $O(\sqrt{\log n / n})$ -close” to an NE policy. Specifically, when $p_1, p_2$ are uniform distributions and $p_a$ is a point mass at 0, for any constant $\alpha$ , the closeness between $s^{(n)}$ and $s$ can be directly translated into the policy form: $s^{(n)} = 0$ for $v < t - O(\sqrt{\log n / n})$ and $s^{(n)} = t$ for $v \geq t - O(\sqrt{\log n / n})$ . For general $p_1, p_2, p_a$ , we note that the $O(\sqrt{\log n / n})$ -closeness depends on the concept of densities, which is more complex. Corollary 3.2 from [20] implies that the NE policy must be symmetric within each group. Let $s_1$ represent the policy for $G_1$ and $s_2$ for $G_2$ . The above analysis indicates that the closeness between $s_1, s_2$ , and $s$ (from Equation (3)) is expected to be bounded by $O(\sqrt{\log n / n})$ . + +In the following, we analyze the empirical observations regarding the scaling of $s_1$ and $s_2$ . Intuitively, $s_1(v)$ increases from approximately 0 to approximately $t$ as $v$ increases from $t - \sqrt{\log n / n}$ to $t + \sqrt{\log n / n}$ . If an agent in $G_2$ exerts an effort of $t(1 - \sqrt{\log n / n})$ , the agent's winning probability could exceed $95\%$ . This observation motivates the choice of setting $s_2(v) = t(1 - \sqrt{\log n / n})$ , rather than 0, to generate positive profits when $v > t(1 - \sqrt{\log n / n}) / 95\%$ . As a result, if $t(1 - \sqrt{\log n / n}) > 0$ , then the agent is willing to accept a loss of $1 - \sqrt{\log n / n}$ as a loss. + +$\sqrt{\log n / n}) / 95\% \leq t$ , i.e., $\sqrt{\frac{\log n}{n}} \geq 0.05$ , then $s_2(t) \geq t(1 - \sqrt{\log n / n}) \geq s_1(v)$ . Therefore, when $n$ is small and $\sqrt{\frac{\log n}{n}} \geq 0.05$ , it holds that $s_2(v) > s_1(v)$ for $v \in \Omega_2$ . Conversely, when $n$ is large, $\sqrt{\frac{\log n}{n}}$ becomes small, and $s_1(v) > s_2(v)$ , as agents in $G_2$ consistently receive lower payoffs than those in $G_1$ when $s_1 = s_2$ . This behavior explains the observed scaling between $s_1$ and $s_2$ , as discussed in Section C.1. + +Finally, we discuss the impact on metrics when the number $n$ of agents is finite. Recall that in the finite $n$ case, we can assume NE policies $s_1$ and $s_2$ for group $G_1$ and $G_2$ , respectively. As discussed above, when $n$ is not too small, we have 1) $s_1 > s_2$ and 2) $s_1, s_2$ , and $s$ are $O(\sqrt{\log n / n})$ -close. Since $s_1 > s_2$ and they converge to the same policy as $n$ grows, the representation ratio $\mathcal{R}_1(A) = \mathbb{E}\left(\frac{|S \cap G_1|}{|G_1|}\right)$ decreases with $n$ , while $\mathcal{R}_2(A) = \mathbb{E}\left(\frac{|S \cap G_2|}{|G_2|}\right)$ increases with $n$ , where $S$ is the (random) winning set. Consequently, the representation ratio $r_{\mathcal{R}}(A) = \frac{\mathcal{R}_2(A)}{\mathcal{R}_1(A)}$ increases as $n$ grows. Since the gap between $s_1$ and $s_2$ is bounded by $O(\sqrt{\log n / n})$ , this results in an increase of $O(\sqrt{\log n / n})$ in $\mathcal{R}_2(A)$ and a similar decrease of $O(\sqrt{\log n / n})$ in $\mathcal{R}_1(A)$ compared to the infinite case. Thus, the increase in $r_{\mathcal{R}}(A)$ should be bounded by $O(\sqrt{\log n / n})$ . A similar quantitative analysis applies to the social welfare ratio $r_S(A)$ and average revenue $\mathcal{RV}(A, m)$ . + +![](images/46a097df59b08671fb948ce19ffc83f20a7b4c3a5d9004048263b69359de23b4.jpg) + +![](images/57516c6e15fd7e645d186e3b304aff5808ed133863efade0bbf3a097e6b0f1d1.jpg) + +![](images/e5038290d29e39e1aa46b38cd82d854f206c7ecc31befa0f6ceba3e29a72e391.jpg) +Figure 5: Evolution of group effort policies over time for $n = 20$ , $\rho = 0.8$ , and $c = 0.2$ . + +![](images/e09a4a5bb60ba482fd8f8877f61bfacf5773dfa02903d51109b65ac9d96f6beb.jpg) + +# D Other bias models and analysis of metrics for their Nash equilibrium + +# D.1 Other bias models + +A natural extension of $p_1 = \mathrm{Unif}[0,1]$ in Section 2 is when $p_1$ is the density of the uniform distribution on an interval $[a,b]$ ( $0 < a < b \leq \infty$ ) and $p_2$ is the density of the uniform distribution on $\Omega_2 = [\rho a,\rho b]$ . Then $p_2(v) = \frac{1}{\rho(b - a)}$ and again, $\mathbb{E}_{v\sim p_2}[v] = \rho \cdot \frac{a + b}{2} = \rho \cdot \mathbb{E}_{v\sim p_1}[v]$ . More generally, one might consider a density $p_1$ that is supported on a domain $\Omega_1 = [a,\infty ]$ , along with a $\rho$ -biased density defined as $p_2(v) = \frac{1}{\rho} p_1(\frac{v}{\rho})$ for $\rho \in (0,1]$ and $v\in [\rho a,\infty ]$ . + +Besides the uniform distribution case, we consider valuations coming from a truncated normal distribution supported on $[0,1]$ . Formally, let $p_1$ be the density of a truncated normal distribution $N(\mu, \sigma^2)$ on the interval $\Omega_1 = [0,1]$ , where $\mu$ lies within $(0,1)$ and $\sigma > 0$ . Let $p_2$ be the density of a + +Algorithm 1 Dynamics for Computing Finite NE Policies for the Uniform Distribution Case + +Input: Group sizes $n_1, n_2 \geq 1$ , selection rate $c \in (0,1)$ , bias $\rho \in [0,1]$ , valuation steps $m_v \geq 1$ , effort steps $m_e \geq 1$ , an interger $T \geq 1$ , and two sequences of step sizes $\{a_\ell^{(t)}\}_{t \in [T], \ell \in \{1,2\}}$ + +Output: Policies $s_1, s_2$ for $G_1$ and $G_2$ , respectively + +1: Initialize grids $V^{(1)} \gets \mathrm{inspace}(\bar{0}, 1, m_v)$ , $V^{(2)} \gets \mathrm{inspace}(0, \rho, m_v)$ , and $E = \mathrm{inspace}(0, 1, m_e)$ . +2: Compute $\alpha \leftarrow \frac{n_2}{n_1 + n_2}$ and $k\gets \lfloor c(n_{1} + n_{2})\rfloor$ + +3: Compute threshold $t = \left\{ \begin{array}{ll}1 - \frac{c}{1 - \alpha}, & \rho < 1 - \frac{c}{1 - \alpha},\\ \frac{\rho(1 - c)}{\rho - \rho\alpha + \alpha}, & \mathrm{otherwise,} \end{array} \right.$ + +4: Initialize $s_1^{(0)}(v) = s_2^{(0)}(v) = \frac{t}{1 + e^{-50(v - \theta)}}$ ▷ Smoothness of the infinite NE in Proposition 4.1 + +5: for $t = 1$ to $T$ do + +6: $E_{t}\gets E\cup \left\{s_{\ell}^{(t - 1)}(v)\mid v\in V^{(\ell)},\ell \in \{1,2\} \right\}$ + +7: $l a s t_{e}\gets 0$ + +8: for $i = 1$ to $m_v$ do + +9: $v \gets V_i^{(1)}, \text{best}_e \gets \text{last}_e, p_1 \gets 1 - v$ and $p_2 \gets \max_{v' \in V^{(2)}: s_2^{(t-1)}(v') \geq \text{best}_e} \frac{\rho - v'}{\rho}$ . + +0: $p^{(1)}(best_e)\gets \sum_{a = 0}^{k - 1}\sum_{b = 0}^{a}\left(\binom{n_1 - 1}{b}p_1^b (1 - p_1)^{n_1 - 1 - b}\right)\cdot \left(\binom{n_2}{a - b}p_2^{a - b}(1 - p_1)^{n_2 - a + b}\right)$ + +1: for all $e \in E_t$ with $e \geq \text{last}_e$ do + +2: $p_1 \gets 1 - v$ and $p_2 \gets \max_{v' \in V^{(2)}: s_2^{(t-1)}(v') \geq best_e} \frac{\rho - v'}{\rho}$ + +3: $p^{(1)}(e)\gets \sum_{a = 0}^{k - 1}\sum_{b = 0}^{a}\left(\binom{n_1 - 1}{b}p_1^b (1 - p_1)^{n_1 - 1 - b}\right)\cdot \left(\binom{n_2}{a - b}p_2^{a - b}(1 - p_1)^{n_2 - a + b}\right)$ + +4: $pay\gets p^{(1)}(e)v - e$ + +5: if $pay > p^{(1)}(best_e)v - best_e$ then + +6: beste $\leftarrow e$ + +7: end if + +8: end for + +9: $\pi_1^{(t)}(v)\gets best_e, last_e\gets best_e$ + +20: end for + +21: $s_1^{(t)}\gets s_1^{(t - 1)} + a_1^{(t)}(\pi_1^{(t)} - s_1^{(t - 1)})$ + +$\triangleright$ Update policy for $G_{1}$ at iteration $t$ 22: + +23: for $i = 1$ to $m_v$ do + +24: $v \gets V_i^{(2)}, \text{best}_e \gets \text{last}_e, p_1 \gets \max_{v' \in V^{(1)}: s_1^{(t)}(v') \geq \text{best}_e} 1 - v$ and $p_2 \gets \frac{\rho - v'}{\rho}$ + +25: $p^{(2)}(best_e)\gets \sum_{a = 0}^{k - 1}\sum_{b = 0}^{a}\left(\binom{n_1}{b}p_1^b (1 - p_1)^{n_1 - b}\right)\cdot \left(\binom{n_2 - 1}{a - b}p_2^{a - b}(1 - p_1)^{n_2 - 1 - a + b}\right)$ + +26: for all $e \in E_t$ with $e \geq \text{last}_e$ do + +27: $p_1 \gets \max_{v' \in V^{(1)}: s_1^{(t)}(v') \geq best_e} 1 - v$ and $p_2 \gets \frac{\rho - v'}{\rho}$ + +28: $p^{(2)}(e)\gets \sum_{a = 0}^{k - 1}\sum_{b = 0}^{a}\left(\binom {n_1}{b}p_1^b (1 - p_1)^{n_1 - b}\right)\cdot \left(\binom{n_2 - 1}{a - b}p_2^{a - b}(1 - p_1)^{n_2 - 1 - a + b}\right)$ + +29: $pay\gets p^{(2)}(e)v - e.$ + +30: if $pay > p^{(2)}(best_e)v - best_e$ then + +31: beste $\leftarrow$ e + +32: end if + +33: end for + +34: $\pi_2^{(t)}(v)\gets best_e, last_e\gets best_e.$ + +35: end for + +36: $s_2^{(t)}\gets s_2^{(t - 1)} + a_2^{(t)}(\pi_2^{(t)} - s_2^{(t - 1)})$ + +37: end for + +38: return Policies $s_1 \gets s_1^{(T)}$ , $s_2 \gets s_2^{(T)}$ + +$\triangleright$ Update policy for $G_{2}$ at iteration $t$ + +![](images/e4688abfa38a3f8d382addc45dca07b778e548e8bb3def5a85a62bff45bdf364.jpg) + +![](images/7c20c76a239593edc1ab45a3cdd52e7e68512429ebe5d6f6d0b080dac1022625.jpg) + +![](images/5641370848fd566c6f5c94a0298374868bfc479ba89b0d5d2573af75e0f83d32.jpg) +Figure 6: Evolution of group effort policies over time for $n = 200$ , $\rho = 0.8$ , and $c = 0.2$ . + +![](images/ee50c97a03c8f1d0e80a68f1badc8023664a65af2e79eddf4fbe7f7df711ff5d.jpg) + +![](images/0419178914a0cb12806485db98c26d5a95646d8f76c8717cb27312ab9470b11c.jpg) + +![](images/e9972c5b56e1d0f4262a53da85bb2aa24bd36e1d238e07cc3a6a8bad94b602bf.jpg) + +![](images/a481b818675805a31526bbb01f373933a7492a852e07a7394afa0eac10af8e4f.jpg) +Figure 7: Evolution of group effort policies over time for $n = 600$ , $\rho = 0.8$ , and $c = 0.2$ . + +![](images/79890e0337350716dba5f288c5ad4af867c4f64526890dcae5894601db8982cb.jpg) + +![](images/9225eaa335a0ef3a649de785e25632249a4366e926f3f5a9cdb0489bfeaaa94a.jpg) + +![](images/5bd0dfde27aaf9f2bb10228fc268767b185ca4bf0a7f8126fd9b4b60c64ee85c.jpg) + +![](images/d082263f4f506f5068242def561fc936c8c6f103f4e4a92ce59e7a65dc607223.jpg) +Figure 8: Evolution of group effort policies over time for $n = 1200$ , $\rho = 0.8$ , and $c = 0.2$ . + +![](images/16f9d1beb2800ade2073f14bbdfc003159ab33453eb89eba0ad202a9e82d9ec1.jpg) + +![](images/515e112e2662b6b9d2ec8b4b67e457efd128e99cae0175a29ff8352eef791b97.jpg) + +![](images/1fb1d11a2d3796de5a78f213aa31a77bd409fb57b77b10302e60e5402fd8ac9b.jpg) + +![](images/696db1e17596332c13224cadd80f83eae9c60948ef3e90392e2dc49b76ffcd43.jpg) +Figure 9: Convergence of group-wise policy updates $\Delta^{(\ell, t)}$ for different population sizes $n$ , with fixed parameters $\rho = 0.8$ and $c = 0.2$ . + +![](images/f13a2386f1379abb9fa8c9ea7ff698e1b6f56c784020a48e3ead4c81f00bbc54.jpg) + +![](images/ab71cfafbc26241ffc5551441a3955670ca7691503c2327464281cc72f12deed.jpg) +(a) Density $p_2$ for various $\rho$ + +![](images/32ece2e901c096fc67e2e858a0c0387055348070368c42f297d8ed29584805c7.jpg) +(b) Expected value of $p_2$ + +![](images/4f1caa8870854a83f535e7bf565e1a38863f5b30e02aa10be8d4a873c053ac65.jpg) +Figure 10: Statistics for truncated normal distribution with $\mu = 0.5$ , $\sigma = 0.1$ . +Figure 11: Plots of $t$ versus $\rho$ for various $\alpha$ with $c = 0.1$ for the truncated normal distribution. The dotted line $t_1 = 0.9$ corresponds to the undifferentiated contest with density $p = p_1$ . + +truncated normal distribution $N(\rho \mu, \sigma^2)$ on the interval $\Omega_2 = [0, 1]$ . Since the bias is multiplicative, the domain of $p_1, \Omega_1 = [0, 1]$ , does not influence the assessment of the contest's results. Note that $\mathbb{E}_{v \sim p_2}[v] = \rho \mu + \frac{\phi\left(\frac{-\rho \mu}{\sigma}\right) - \phi\left(\frac{1 - \rho \mu}{\sigma}\right)}{\Phi\left(\frac{1 - \rho \mu}{\sigma}\right) - \Phi\left(\frac{-\rho \mu}{\sigma}\right)}$ , where $\phi(x)$ is the probability density function of the standard normal distribution $N(0, 1)$ and $\Phi(x)$ is its cumulative distribution function. The expectation of $p_2$ does not decrease linearly with $\rho$ as in the uniform case, but it closely approximates a linear function and monotonically decreases with $\rho$ . This is motivated by real-world settings where the valuations (such as pay or SAT scores) exhibit a truncated normal distribution [88]. Other variants of distributions include piecewise-linear, polynomial (such as Pareto), and log-normal distributions, along with their biased versions. + +We implicitly assume that the bias parameter $\rho$ is fixed and identical for all agents in $G_{2}$ above. However, $\rho$ could be noisy and non-identical to agents. For instance, let $p_{\rho}$ be a density supported on $[0,1]$ . We assume each agent $i \in G_{2}$ has an individual bias $\rho_{i}$ i.i.d. drawn from $p_{\rho}$ , and its valuation is drawn from the $\rho_{i}$ -biased density of $p_{1}$ . Then $p_{2}$ is supported on $\Omega_{2} = \Omega_{1}$ , and satisfies that for any $v \in \Omega_{1}$ , + +$$ +p _ {2} (v) = \int_ {0} ^ {1} \frac {1}{x} p _ {\rho} (x) p _ {1} (\frac {v}{x}) d x. +$$ + +# D.2 Analysis of metrics for Nash equilibrium in the truncated normal distribution case + +In this section, we do a similar analysis as in Section 4 for the case that $p_1$ is a truncated normal distribution $N(\mu, \sigma^2)$ supported on $[0,1]$ , $p_2$ is a $\rho$ -biased truncated normal distribution $N(\rho \mu, \sigma^2)$ supported on $[0,1]$ , and $p_a$ is a point mass at 0. We choose $\mu = 0.5$ and $\sigma = 0.1$ . This selection ensures that the density function is narrowly focused around the mean and the expected value of $p_2$ is approximately $\rho \mu$ ; see Figure 10 for illustration. Note that $t$ analogues to Proposition 4.1 is the + +![](images/ff6fc17f32605d8a0b89396cd705cf16a319fa5397eddd3a1bc9abc67d646583.jpg) +(a) $r_{\mathcal{R}}(A)$ v.s. $\rho$ + +![](images/5ce83ca63feca353b00953e2b2638ae559631d6d4e66a0419b7d52833bff1b8e.jpg) +(b) $r_{\mathcal{R}}(A)$ v.s. $c$ +Figure 12: Plots illustrating the group-wise social welfare $S_{1}(A)$ , $S_{2}(A)$ , and the social welfare ratio $r_{S}(A)$ as functions of the parameters $\rho, c$ , and $\alpha$ for the truncated normal distribution. By default, we set $(\rho, c, \alpha) = (0.8, 0.1, 0.5)$ . A dotted line within these plots indicates the threshold at which $r_{S}(A) = 0.8$ . + +![](images/f2c3f0cd92ac37253ec9445b1168c4ff556cb26fb9db73ac888a6535e2dd0f2a.jpg) +(c) $r_{\mathcal{S}}(A)$ v.s. $\rho$ + +![](images/9de6c6606e2fa08af94457822be612cb6667d682f35b5a2f54c47f2275270524.jpg) +(d) $r_{S}(A)$ v.s. $c$ + +solution of the following equation: + +$$ +(1 - \alpha) \cdot \frac {\phi (\frac {v - \mu}{\sigma}) - \phi (\frac {1 - \mu}{\sigma})}{\Phi (\frac {1 - \mu}{\sigma}) - \Phi (\frac {- \mu}{\sigma})} + \alpha \cdot \frac {\phi (\frac {v - \rho \mu}{\sigma}) - \phi (\frac {1 - \rho \mu}{\sigma})}{\Phi (\frac {1 - \rho \mu}{\sigma}) - \Phi (\frac {- \rho \mu}{\sigma})} = 1 - c. \tag {8} +$$ + +Unlike the uniform distribution case, it is hard to derive closed-form expressions for metrics on the outcomes of the contest. However, one can do numerical computations and we plot solution $t$ , representation ratio $r_{\mathcal{R}}(A)$ and group-wise social welfare $S_{\ell}(A)$ together with social welfare ratio $r_{S}(A)$ in Figures 11 and 12 respectively. All plots exhibit a monotonic behavior similar to that observed with the uniform distribution. Next, we highlight some distinctions with the uniform distribution. + +No inflection point. A notable feature of the truncated normal distribution is its lack of an inflection point. This trait is observed not only for $t$ , but also in the behaviors of $r_{\mathcal{R}}(A)$ and $r_{\mathcal{S}}(A)$ . This difference arises because the domain $\Omega_2 = [0,1]$ remains consistent across all values of $\rho$ . + +Representation ratio. Figure 12 shows that to achieve a representation ratio $r_{\mathcal{R}}(A) \geq 0.8$ , it is necessary to adjust $\rho$ to a minimum of 0.979 or increase $c$ to at least 0.862. The need to elevate $c$ is more pronounced than the required 0.5 observed with the uniform distribution in Figure 2(b). This difference arises because the truncated normal distribution tends to be more focused around its mean, leading to a higher number of agents in $G_{1}$ possessing valuations greater than the expected value $\approx 0.4$ of $p_{2}$ . + +# E Proof of Theorem 3.1: two-group contest + +In this section, we begin by providing a more detailed technical overview of the proof of Theorem 3.1 (Section E.1). Next, we present a more comprehensive version of Theorem 3.1, including the explicit form of $A^{(n)}$ (Theorem E.2 in Section E.2). Finally, we provide the proof of Theorem E.2 (Section E.3). + +# E.1 Technical overview + +We present an overview of the proof of Theorem 3.1, which characterizes an NE policy for the two-group contest. Recall that, there are $n$ agents belonging to one of the two disjoint groups $G_{1}, G_{2}$ with $|G_{1}| = (1 - \alpha)n$ and $|G_{2}| = \alpha n$ . The valuations of agents in $G_{1}$ come from the density $p_{1}$ supported on $\Omega_{1}$ and those of $G_{2}$ come from the density $p_{2}$ supported on $\Omega_{2}$ . Each agent has an initial ability drawn from the density $p_{a}$ . The selectivity of the contest is a constant $0 < c < 1$ . Theorem 3.1 asserts that, as $n \to \infty$ , there is an NE policy for the agents which is determined by a threshold $t \in (\Omega_{1} \cup \Omega_{2}) + \Omega_{a}$ when the $(\Omega_{1} \cup \Omega_{2}) + \Omega_{a}$ is a connected subset of $\mathbb{R}_{\geq 0}$ . + +For ease of analysis, we first consider the simple case where $p_a$ is a point mass at 0, so that agents' policies only depend on their valuations. We then show that the extension to a general $p_a$ is straightforward. We start by first quickly showing how to compute the NE policy in the special case when $p_1 = p_2$ for finite $n$ and why this approach does not extend to the two-group setting of interest (Section E.1.1). In Section E.1.2 we show that even though there are major challenges in extending the one-group case, it leads us to the right form of the NE policy for the two-group case as $n \to \infty$ : a single threshold function that defines the strategies of agents in both $G_1$ and $G_2$ . This also explains how we arrive at Equation (2) that characterizes the threshold $t$ . Finally, in Section E.1.3, we present the approach to formally argue about and prove the convergence of the finite $n$ two-group contest to + +this pair of NE policies. This analysis also reveals why the conjectured policy is a Nash equilibrium. With all the background, we conclude Theorem 3.1. + +# E.1.1 NE policy for the undifferentiated contest for finite $n$ and obstacle in extending it + +Here, we consider the special case when $p_1 = p_2 = p$ is the density of the uniform distribution on [0, 1], and $p_a$ is a point mass at 0. The argument for other densities is similar. First note that by symmetry among agents, it is reasonable to assume that, at equilibrium, each agent will follow the same policy $s$ . Moreover, since the domain of $p_1$ is nonnegative reals, it is reasonable to assume that the NE policy is monotone in an agent's valuation. The following calculations show that both symmetry and monotonicity hold. + +Recall that an agent $i$ with valuation $v$ is selected if the effort $s(v)$ is among the top $c$ fraction of efforts. Thus, assuming that all agents follow $s$ , agent $i$ is selected if and only if there are at least $(1 - c)n$ distinct agents $j$ with $s(v_j) < s(v)$ . (We ignore the issue that there may be ties for this discussion.) Since $s$ is a monotone function, $s(v_j) < s(v)$ holds if and only if $v_j < v$ . Thus, the probability of selection of this agent is + +$$ +P_{i}(s(v);A_{-i}) = \sum_{j = (1 - c)n}^{n - 1}\binom {n - 1}{j}v^{j}(1 - v)^{n - 1 - j}. +$$ + +Hence, its expected payoff is $\pi_i(v,s(v);A_{-i}) = P_i(s(v);A_{-i})\cdot v - s(v)$ . (See Section 2 for notation). A key observation is that the calculation of $P_{i}(s(v);A_{-i})$ only depends on density $p_1$ and $v$ , and is independent of the choice of $s$ , under the monotonicity and symmetry assumptions on $s$ . If $s$ is to be an NE, then it must satisfy that for any other effort $e$ , $\pi_i(v,s(v);A_{-i})\geq \pi_i(v,e;A_{-i})$ . This follows from the condition that the derivative with respect to $v$ , $\pi_i'(v,s(v);A_{-i}) = P_i'(s(v);A_{-i})\cdot v - s'(v) = 0$ . As noted above, $P_{i}(s(v);A_{-i})$ does not depend on $s$ , so we get a simple differential equation involving the derivative of $s'(v) = \left(\sum_{j=(1-c)n}^{n-1}\binom{n-1}{j}v^j(1-v)^{n-1-j}\right)'\cdot v$ . Thus, + +$$ +s (v) = (1 - c) \cdot \sum_ {j = (1 - c) n + 1} ^ {n} \binom {n} {j} v ^ {j} (1 - v) ^ {n - j} \tag {9} +$$ + +is the unique NE for the undifferentiated contest and it that this $s$ is monotone. For a general density $p$ , we can apply a similar analysis to obtain that + +$$ +s (v) := Q _ {p} (v) \cdot v - \int_ {\underline {{\Omega}}} ^ {v} Q _ {p} (x) d x, \tag {10} +$$ + +where $Q_{p}(v) = \sum_{i=n-k}^{n-1} \binom{n-1}{i} \cdot F_{p}(v)^{i} \cdot (1 - F_{p}(v))^{n-i-1}$ for any $v \in \Omega$ . We also note that the computation of $s$ can become significantly more complicated for a general density $p_{a}$ , as it requires considering the stability condition for two partial derivatives: $\frac{\partial s(v, a)}{\partial v}$ and $\frac{\partial s(v, a)}{\partial a}$ . + +We now attempt to extend the analysis above to the two-group contest. Since each agent in each group uses the same valuation density, we can still hope for a symmetric and monotone NE policy for each group; say $s_1$ for $G_1$ and $s_2$ for $G_2$ . However, we can no longer assume that $s_1 = s_2$ . There are simple examples (see Section B) for which it can be shown that $s_1 \neq s_2$ . This considerably complicates the calculation of probability $P_i$ for the $i$ th agent getting selected since the order of agents' valuations may differ from that of agents' efforts. For instance, it is now possible that for agent $i \in G_1$ and agent $j \in G_2$ , $v_i < v_j$ but $s_1(v_i) > s_2(v_j)$ . Thus, $P_i$ must depend on functions $s_1$ and $s_2$ , instead of only depending on density $p = p_1$ and $v$ as in the undifferentiated case. Thus, it is no longer possible to write a simple differential equation as in the undifferentiated case. + +# E.1.2 A conjectured NE policy in two-group contests for large $n$ + +Since it seems intractable to find an NE policy for the two-group case, we study whether the situation becomes easier when $n$ is large. This hope is rooted in the observation that for the undifferentiated contest when $n$ is large, the NE policy $s(v)$ defined in Equation (9) converges to a threshold function. To see this, recall that for the uniform distribution, $s(v) = (1 - c)\cdot \sum_{j = (1 - c)n + 1}^{n}\binom{n}{j}v^{j}(1 - v)^{n - j}$ . Since $s(v)$ is the probability associated with a sum of i.i.d. random variables, it follows from the + +Chernoff bound that as $n \to \infty$ , $s(v) \to 1 - c$ for any $v > 1 - c$ and $s(v) \to 0$ for any $v < 1 - c$ . This argument is not specific to the uniform distribution and extends to any density $p_1$ with NE policy defined in Equation (10). In particular, if $F_1$ denotes the CDF of $p_1$ , the limiting NE is given by $s(v) = F_1^{-1}(1 - c)$ if $v \geq F_1^{-1}(1 - c)$ and $s(v) = 0$ otherwise. The threshold $F_1^{-1}(1 - c)$ guarantees that the expected fraction of agents that put in a nonzero effort is $1 - F_1(F_1^{-1}(1 - c)) = c$ . The rationale for $s(v) = F_1^{-1}(1 - c)$ when $v$ is above the threshold is twofold: 1) agents would not exert effort beyond their valuation, ensuring $s(F_1^{-1}(1 - c)) \leq F_1^{-1}(1 - c)$ , and 2) agents with valuations below $F_1^{-1}(1 - c)$ are disincentivized from participating, leading to $s(F_1^{-1}(1 - c)) \geq F_1^{-1}(1 - c)$ . + +Thus, one may hope that in a two-group contest, as $n \to \infty$ , the NE policy might similarly converge to two threshold functions $s_1$ and $s_2$ , each with a corresponding threshold $t_\ell$ . While this assumption allows us to give an explicit form for the probability $P_i$ of agent $i$ getting selected, this expression is quite complicated and, importantly, depends on $t_1$ and $t_2$ . Thus, we are unable to obtain conditions that determine $t_1$ and $t_2$ from the NE condition. + +Going back to the setting when $p_1$ is the density of the uniform distribution over $[0, 1]$ and $p_2 = p_2$ is the density of the uniform distribution over $[0, \rho]$ , first observe that as $\rho \to 0$ , $t_2 \to 0$ . Thus, one would expect $t_1$ to be more than $t_2$ . We now argue that counterintuitive to the above observation, $t_1 > t_2$ cannot lead to an NE. To see this, first observe that when $t_1 > t_2$ , if an agent puts in effort $t_1$ , then it will get selected. Thus, the probability of an agent in $G_1$ getting selected is $1 - F_1(t_1)$ . Hence, if $1 - F_1(t_1) < \frac{c}{1 - \alpha}$ , fewer than $cn$ agents in $G_1$ get selected. Thus, agents in $G_1$ getting selected will find that putting in effort slightly larger than $t_2$ instead of $t_1$ suffices to ensure their effort is larger than all agents in $G_2$ , and consequently, they will still be selected. Through this reduction in effort, they can gain an additional payoff of $t_1 - t_2$ , which violates the stability condition. A similar argument holds for $G_2$ when $1 - F_1(t_1) > \frac{c}{1 - \alpha}$ . Thus, $t_1 > t_2$ leads to instability. Similarly, we can argue that $t_1 < t_2$ also leads to instability. Thus, in case the NE policies for $G_1$ and $G_2$ are thresholds, it must be the case that $t_1 = t_2$ when $n$ is large. However, it is not clear how this can hold given that the domains of $p_1$ and $p_2$ are different. + +To explore this, we consider a scenario when $\alpha = 0.5$ , $p_1$ is the density of the uniform distribution over $[0,1]$ and $p_2$ is the density of the uniform distribution over $[0,0.5]$ ( $\rho = 0.5$ ). With high probability, there would be more than $0.1n$ agents from $G_1$ whose valuation is larger than 0.5. Hence, if we set $c = 0.1$ , no agent from $G_2$ will have any incentive to put in an effort while for agents in $G_1$ a threshold of $t_1 = 0.8$ suffices. The key observation is that even though the two policies are different, the policy of $G_2$ is just 0. This suggests that both policies can be seen as restrictions of the same threshold function to their respective domains. + +Now we show how to compute the threshold $t$ . The idea is to reduce the two-group contest to an undifferentiated one whose density $p = (1 - \alpha)p_1 + \alpha p_2$ , supported on the domain $\Omega_1 \cup \Omega_2$ . In this undifferentiated contest, as $n \to \infty$ , it is likely that $(1 - \alpha)$ -fraction of agents with valuation come from $p_1$ and $\alpha$ -fraction of agents come from $p_2$ . This suggests that these two contests are increasingly indistinguishable as $n$ grows, leading to the same limiting threshold $t = F_p^{-1}(1 - c)$ . Thus, threshold $t$ is the solution of the equation $(1 - \alpha)F_1(v) + \alpha F_2(v) = 1 - c$ – the one denoted in Equation (2). This argument can be extended to general $p_a$ , resulting in the following lemma. + +Lemma E.1 (Unique solution). Let $\alpha, c \in (0,1)$ , $p_1$ be a density supported on the domain $\Omega_1 \subseteq \mathbb{R}_{\geq 0}$ , $p_2$ be a density supported on the domain $\Omega_2 \subseteq \mathbb{R}_{\geq 0}$ , and $p_a$ is a density supported on the domain $\Omega_a \subseteq \mathbb{R}_{\geq 0}$ . If $(\Omega_1 \cup \Omega_2) + \Omega_a$ is connected and each density $p_1, p_2, p_a$ is positive at any point of its domain, then there exists a unique solution $t \in \Omega_1 \cup \Omega_2$ for the following equation: $(1 - \alpha)F_1(\zeta) + \alpha F_2(\zeta) = 1 - c$ , where for any $\zeta \in \mathbb{R}_{\geq 0}$ , $F_\ell(\zeta) = \operatorname*{Pr}_{v \sim p_\ell, a \sim p_a} [v + a \leq \zeta]$ . + +The assumption is naturally met in cases such as the uniform distribution and the truncated normal distribution discussed in Section 2. Since $F_{\ell}$ is a CDF, it must be strictly monotonic across its domain $\Omega_{\ell} + \Omega_{a}$ . For any $\zeta, \zeta' \in (\Omega_1 \cup \Omega_2) + \Omega_a$ with $\zeta < \zeta'$ , if $(1 - \alpha)F_1(\zeta) + \alpha F_2(\zeta) = (1 - \alpha)F_1(\zeta') + \alpha F_2(\zeta')$ , we must have both $F_{\ell}(\zeta) = F_{\ell}(\zeta')$ holds. Then $(\zeta, \zeta') \cap ((\Omega_1 \cup \Omega_2) + \Omega) = \emptyset$ , which contradicts the connected domain assumption. Hence, $(1 - \alpha)F_1(\zeta) + \alpha F_2(\zeta)$ is strictly monotonic across the domain $(\Omega_1 \cup \Omega_2) + \Omega_a$ , which ensures the uniqueness of solution $t$ . The proof can be found in Section E.3.1. + +# E.1.3 Proving convergence to the conjectured NE policy + +Now we outline how to prove that, as $n \to \infty$ , the NE policy for the two-group contest converges to the threshold policy corresponding to $t$ as guaranteed by Lemma E.1. To do so, first, we have to make it precise what convergence means. Towards this, we revisit the undifferentiated contest. While we argued that in this case, as $n \to \infty$ , the NE policy tends to a threshold function, recall that we used the explicit form of the NE policy for finite $n$ . Unfortunately, since we do not have an explicit form for the two-group contest (for finite $n$ ), we need a strategy for the undifferentiated case that works without the knowledge of the explicit NE for finite $n$ . + +Let $A$ be the NE policy of the undifferentiated contest as $n \to \infty$ , characterized by a function $s$ which is a threshold with parameter $t$ . It suffices to prove that for policy $A$ and every $\varepsilon > 0$ there is an $n_{\varepsilon}$ such that for $n \geq n_{\varepsilon}$ , $A$ is an $\varepsilon$ -NE. However, we find that there exists an $\varepsilon > 0$ such that for any $n \geq 1$ , $A$ is not an $\varepsilon$ -NE policy. We revisit the simple example of uniform distribution discussed above, in which $c = 0.5$ , $p_1$ is the density of the uniform distribution on $[0, 1]$ , and $p_a$ is a point mass at 0. Recall that the threshold $t = 1 - c = 0.5$ . Then, in expectation, $0.5n$ agents have valuations at least 0.5 and put in effort 0.5. As $n \to \infty$ , it follows from symmetry that the probability that fewer than $0.5n$ agents with valuation $\geq t$ approaches approximately 0.5. Thus, an agent $i$ with a valuation $v = 0.2$ and putting in an effort $e = 0.01$ would have about a 0.5 probability of being selected, i.e., $P_i(e; A_{-i}) \approx 0.5$ . Since $s(v) = 0$ , the probability $P_i(A_i(v); A_{-i}) = 0$ . Thus, we have + +$$ +\pi_ {i} (v, e; A _ {- i}) - \pi_ {i} (v, A _ {i} (v); A _ {- i}) = P _ {i} (e; A _ {- i}) \cdot v - e - 0 \approx 0. 2 * 0. 5 - 0. 0 1 \gg 0. +$$ + +This inequality implies that when $\varepsilon = 0.08$ , for any $n \geq 1$ , $A$ is not an $\varepsilon$ -NE policy. + +To bypass this, we consider a sequence of proxies for $A$ , denoted by $A^{(n)}$ , and characterized by threshold functions $s^{(n)}$ . These proxies aim to ensure that the winning probability $P_{i}(e;A_{-i})\approx 0$ and hence, serve as an approximate NE policy (see Definition 3.1). Therefore, we need $A^{(n)}$ to satisfy two conditions: + +1. $A^{(n)}$ converges to $A$ as $n$ approaches infinity, i.e., $\lim_{n\to \infty}s^{(n)} = s$ , and +2. The winning probability under $A^{(n)}$ approaches zero in the limit, i.e., $\lim_{n\to \infty}P_i(e;A_{-i}^{(n)})\to 0$ . + +Ensuring $P_{i}(e; A_{-i}^{(n)}) \approx 0$ essentially involves guaranteeing that the probability of having fewer than $cn$ agents with valuation $\geq t$ is negligible. Specifically, for the uniform case, by adjusting the threshold by $\sqrt{\log n / n} = o(1)$ , we define the policy $s^{(n)}(v)$ as follows: $s^{(n)}(v) = t$ if $v \geq t - \sqrt{\log n / n}$ and $s^{(n)}(v) = 0$ otherwise. By concentration, this $s^{(n)}$ ensures that the probability $P_{i}(e; A_{-i}^{(n)})$ is bounded above by $\sqrt{1 / n}$ (Lemma E.4). This bounded probability ensures $A^{(n)}$ to be an $\sqrt{1 / n}$ -NE policy (Lemma E.6). This concludes the proof that the policy $A$ is an NE in the large $n$ limit for the uniform distribution case for one group. + +Finally, we adapt this new proof technique of constructing $A^{(n)}$ to the two-group case with general densities $p_1$ and $p_2$ . Mirroring the strategy employed in the uniform distribution case, we would like to shift the threshold of $s^{(n)}$ to ensure that, in expectation, a $(c + \sqrt{\log n / n})$ -fraction of agents put in effort $t$ . To satisfy this, we define the following threshold $\Delta_n$ (Definitions E.1 and E.2) for the policy $A^{(n)}$ : + +$$ +\Delta_ {n} := \min \left\{F _ {1} ^ {- 1} (F _ {1} (t) - \sqrt {\log n / n}), F _ {2} ^ {- 1} (F _ {2} (t) - \sqrt {\log n / n}) \right\}. +$$ + +Consequently, we define the policy $s^{(n)}$ as follows: $s^{(n)}(v) = t$ if $v \geq t - \Delta_n$ and $s^{(n)}(v) = 0$ otherwise (see Theorem E.2). We find that $\lim_{n \to \infty} \Delta_n = t$ , indicating that $s^{(n)}$ converges to $s$ . Crucially, such a $\Delta_n$ ensures that $(1 - \alpha) F_1(\Delta_n) + \alpha F_2(\Delta_n) \leq 1 - (c + \sqrt{\log n / n})$ , thereby maintaining at least a $(c + \sqrt{\log n / n})$ -fraction of agents, in expectation, putting in effort $t$ . Using this property, we establish a bound for the winning probability $P_i$ analogous to the undifferentiated contest: $P_i(e; A_{-i}^{(n)}) \leq n^{-(1 - \alpha)} + n^{-\alpha}$ if $e < t$ (see Lemma E.4). The factors $n^{-(1 - \alpha)}$ and $n^{-\alpha}$ derive from concentration bounds applicable to group $G_1$ and $G_2$ , respectively. Furthermore, this minimal winning probability guarantees that $A^{(n)}$ is an $\varepsilon_n$ -NE with $\lim_{n \to \infty} \varepsilon_n = 0$ (Lemma E.6). The extension to the general $p_a$ is straightforward. The main difference is that the initial ability $a$ + +may already surpass the threshold $t$ , in which case the agent does not need to exert any effort to be selected. This is characterized by the amount of effort $s^{(n)}(v,a) = \max \{t - a,0\}$ if $v \geq t - \Delta_n$ . + +To summarize, we first arrived at a policy $A$ that is characterized by a function $s$ which is parameterized by a threshold $t$ defined by Lemma E.1 and then we constructed a sequence of "proxies" $A^{(n)}$ that converge to $A$ as $n \to \infty$ . Moreover, we construct a sequence $\varepsilon_1, \ldots, \varepsilon_n, \ldots$ with limit 0 such that $A^{(n)}$ is an $\varepsilon_n$ -NE policy for every $n$ . This implies that $A$ is an NE policy as $n \to \infty$ . Thus, the overview above allows us to prove Theorem 3.1. + +# E.2 A more comprehensive version of Theorem 3.1: convergence form + +Now we show how to construct a series of policies $\{A^{(n)}\}_{n}$ that approach $A$ , the NE policy from Equation (3) in Theorem 3.1, as $n \to \infty$ . The most technical part will be to prove that $A^{(n)}$ acts as an $\varepsilon_{n}$ -NE policy, where $\varepsilon_{n} \to 0$ with increasing $n$ . + +Suppose $(\Omega_1 \cup \Omega_2) + \Omega_a$ is connected and let $t \in (\Omega_1 \cup \Omega_2) + \Omega_a$ be a unique solution of the equation $(1 - \alpha)F_1(\zeta) + \alpha F_2(\zeta) = 1 - c$ (the uniqueness of $t$ is ensured by Lemma E.1). Since $c \in (0,1)$ , we have that either $F_1(t) > 0$ or $F_2(t) > 0$ . Accordingly, we define a threshold $n_t$ as follows. + +Definition E.1 (Threshold $n_t$ ). Given a value $t \in \Omega_1 \cup \Omega_2$ , we define a threshold $n_t$ as follows: + +- If both $F_{1}(t) > 0$ and $F_{2}(t) > 0$ , let $n_t$ be the smallest integer such that $F_{\ell}(t) - \sqrt{\frac{\log n_t}{n_t}} > 0$ for $\ell = 1, 2$ . +- If $F_{1}(t) > 0$ and $F_{2}(t) = 0$ , let $n_t$ be the smallest integer such that $F_{1}(t) - \sqrt{\frac{\log n_{t}}{n_{t}}} > 0$ . +- If $F_{1}(t) = 0$ and $F_{2}(t) > 0$ , let $n_t$ be the smallest integer such that $F_{2}(t) - \sqrt{\frac{\log n_{t}}{n_{t}}} > 0$ . + +Note that such $n_t$ is finite and always exists since $\sqrt{\frac{\log n}{n}} \to 0$ as $n \to \infty$ . Also note that $n_t$ is monotonically decreasing to $t$ across the domain $\Omega_1 \cup \Omega_2$ . This value of $n_t$ is useful for defining $\Delta_n$ , which is essential for the construction of policy $A^{(n)}$ . + +Definition E.2 (Threshold $\Delta_{n}$ ). Let $n \geq n_t$ be an integer. We define a threshold $\Delta_{n}$ as follows: + +$$ +\Delta_ {n} := \min \left\{F _ {1} ^ {- 1} \left(F _ {1} (t) - \sqrt {\frac {\log n}{n}}\right), F _ {2} ^ {- 1} \left(F _ {2} (t) - \sqrt {\frac {\log n}{n}}\right) \right\}. +$$ + +- If both $F_{1}(t) > 0$ and $F_{2}(t) > 0$ , let +- If $F_{1}(t) > 0$ and $F_{2}(t) = 0$ , let $\Delta_{n} := F_{1}^{-1}(F_{1}(t) - \sqrt{\frac{\log n}{n}})$ . +- Otherwise if $F_{1}(t) = 0$ and $F_{2}(t) > 0$ , let $\Delta_{n} := F_{2}^{-1}(F_{2}(t) - \sqrt{\frac{\log n}{n}})$ . + +The requirement that $n \geq n_t$ ensures the proper definition of $\Delta_n$ . As the number of agents $n$ grows indefinitely, the term $\sqrt{\frac{\log n}{n}}$ approaches 0, leading $\Delta_n$ to converge towards $t$ . The threshold function $s^{(n)}$ defined as in Equation (11), designed as a threshold function, incorporates $\Delta_n$ as its threshold. The convergence of $\Delta_n$ to $t$ as $n \to \infty$ is crucial for ensuring that $A^{(n)}$ gradually aligns with the policy $A$ over large populations. + +We are ready to provide the formal statement of Theorem 3.1. + +Theorem E.2 (Two-group contest: Large $n$ limit). Let $\alpha, c \in (0,1)$ . For $\ell = 1,2$ , let $p_\ell$ be a density supported on a domain $\Omega_\ell \subseteq \mathbb{R}_{\geq 0}$ . Let $m : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ be a merit function that is strictly increasing. Suppose $\Omega_1 \cup \Omega_2$ is connected and each density $p_\ell$ is positive at any point of domain + +$\Omega_{\ell}$ . Let $t \in \Omega_1 \cup \Omega_2$ be a unique solution of the equation $(1 - \alpha)F_1(v) + \alpha F_2(v) = 1 - c$ (the uniqueness of $t$ is ensured by Lemma E.1). Let $n_t$ be defined as in Definition E.1. Let $n \geq n_t$ be an integer and let $\Delta_n$ be defined as in Definition E.2. Define + +$$ +s ^ {(n)} (v, a) := \left\{ \begin{array}{l l} 0 & \text {i f} v < \Delta_ {n} \\ \max \{t - a, 0 \} & \text {i f} v \geq \Delta_ {n} \end{array} \right. \tag {11} +$$ + +gives rise to a policy for the two-group contest: Under this policy, agent $i \in G_1$ uses the restriction $A_i^{(n)} = s^{(n)}|_{\Omega_1}$ , while each agent $j \in G_2$ uses the restriction $A_j^{(n)} = s^{(n)}|_{\Omega_2}$ . We have $\lim_{n \to \infty} s^{(n)} = s$ , where $s$ is the threshold function defined as in Equation (3). Moreover, the sequence of policies $A^{(n_t)}, A^{(n_t + 1)}, \ldots$ satisfies the following property: + +$$ +\forall \varepsilon > 0, \exists n _ {\varepsilon} \geq n _ {t}, \text {s . t .} \forall n \geq n _ {\varepsilon}, A ^ {(n)} \text {i s a n} \varepsilon - N E \text {p o l i c y}. \tag {12} +$$ + +This theorem establishes how an NE policy for the two-group contest approaches a limit as the number of agents, $n$ , grows indefinitely. It reveals that the sequence of policies $\{A^{(n)}\}_{n}$ not only converges to $A$ but also aligns with an NE policy for the two-group contest. Thus, it validates the assertion made in Theorem 3.1 that $A$ serves as an NE policy for the two-group contest in the limit as $n \to \infty$ . + +# E.3 Proof of Theorem E.2 + +We provide an overview of the proof, summarized as follows. + +1. In Section E.3.1, we prove Lemma E.1 for the uniqueness of solution $t$ that decides the threshold function $s$ . +2. In Section E.3.2, we bound the winning probabilities $P_{i}(e;A_{-i}^{(n)})$ under policy $A^{(n)}$ ; summarized by Lemma E.4. Its proof relies on the winning probability for the undifferentiated contest (Lemma E.5), whose computation is via an auxiliary function defined in Definition E.3. +3. In Section E.3.3, we apply Lemma E.4 to prove that $A^{(n)}$ is approximate NE (Lemma E.6). +4. Finally in Section E.3.4, we show that Theorem E.2 is a corollary of Lemma E.6. + +For simplicity, we first assume that $p_a$ is a point mass at 0, such that policies depend solely on valuations. In this case, $s(v,a), P_i(e;a,A_{-i}), \pi(v,a,e;A_{-i})$ are simplified to $s(v), P_i(e; A_{-i}), \pi(v,e; A_{-i})$ respectively. At the end, we will show how to extend this to a general $p_a$ . + +# E.3.1 Proof of Lemma E.1: solution uniqueness + +Instead of proving Lemma E.1, we directly prove for the general multi-group case. Let $G_{1}, \ldots, G_{m}$ be $m \geq 2$ groups where each $G_{\ell}$ has size $n_{\ell} = \alpha_{\ell}n$ and valuation distribution $p_{\ell}$ on the domain $\Omega_{\ell} \subseteq \mathbb{R}_{\geq 0}$ . We have $\alpha_{\ell} \in (0,1)$ for every $\ell \in [m]$ and $\sum_{\ell \in [m]} \alpha_{\ell} = 1$ . We have the following lemma that generalizes Lemma E.1. + +Lemma E.3 (Unique solution for multiple groups). Suppose $(\cup_{\ell \in [m]}\Omega_{\ell}) + \Omega_{a}$ is connected and each density $p_\ell$ and $p_a$ is positive at any point of its domain. There exists a unique solution $t\in \cup_{\ell \in [m]}\Omega_{\ell}$ for the equation $\sum_{\ell \in [m]}\alpha_{\ell}F_{\ell}(\zeta) = 1 - c$ , where for any $\zeta \in \mathbb{R}_{\geq 0}$ , $F_{\ell}(\zeta) = \operatorname*{Pr}_{v\sim p_{\ell},a\sim p_a}[v + a\leq \zeta ]$ . + +Proof: Fix $\ell \in [m]$ . Recall that we expand the domain of every CDF $F_{\ell}$ to $\mathbb{R}_{\geq 0}$ . We have the following properties for $F_{\ell}$ : + +1. $F_{\ell}(\cdot)$ is non-decreasing across the domain $\mathbb{R}_{\geq 0}$ , i.e., for any $\zeta, \zeta' \in \mathbb{R}_{\geq 0}$ with $\zeta < \zeta'$ , $F_{\ell}(\zeta) \leq F_{\ell}(\zeta')$ holds. +2. $F_{\ell}(\cdot)$ is strictly monotonous across the domain $\Omega_{\ell} + \Omega_{a}$ , i.e., for any $\zeta, \zeta' \in \Omega_{\ell} + \Omega_{a}$ with $\zeta < \zeta'$ and $(\zeta, \zeta') \cap (\Omega_{\ell} + \Omega_{a}) \neq \emptyset$ , we have $F_{\ell}(v) < F_{\ell}(v')$ . + +Define a function $g: \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ such that for any $\zeta \in \mathbb{R}_{\geq 0}$ , $g(\zeta) = \sum_{\ell \in [m]} \alpha_{\ell} F_{\ell}(\zeta)$ . Since $g(\cdot)$ is a convex combination of $F_{\ell}(\cdot)$ 's, we know that $g(\cdot)$ is also non-decreasing across the domain $\mathbb{R}_{\geq 0}$ . Moreover, since $(\bigcup_{\ell \in [m]} \Omega_{\ell}) + \Omega_{a}$ is connected, for any $\zeta, \zeta' \in (\bigcup_{\ell \in [m]} \Omega_{\ell}) + \Omega_{a}$ with $\zeta < \zeta'$ , there must exist at least one $\ell \in [m]$ such that $F_{\ell}(\zeta) < F_{\ell}(\zeta')$ and $(\zeta, \zeta') \cap (\Omega_{\ell} + \Omega_{a})$ . This implies that $g(\cdot)$ is strictly monotonous across the domain $(\bigcup_{\ell \in [m]} \Omega_{\ell}) + \Omega_{a}$ . + +Now let $L$ and $U$ denote the infimum and the supremum of domain $(\bigcup_{\ell \in [m]} \Omega_{\ell}) + \Omega_{a}$ respectively. We have $0 = g(L) < 1 - c < g(U) = 1$ . Thus, there must exist a unique point $t \in (\bigcup_{\ell \in [m]} \Omega_{\ell}) + \Omega_{a}$ such that $g(t) = 1 - c$ . This completes the proof. + +# E.3.2 Bounding winning probability + +We first have the following lemma that bounds the winning probability under policy $A^{(n)}$ . + +Lemma E.4 (Bounding winning probability). For every integer $n \geq n_t$ , we have + +$$ +\forall i \in [ n ], P _ {i} (e; A _ {- i} ^ {(n)}) = 1 \text {i f} e \geq t; \text {a n d} P _ {i} (e; A _ {- i} ^ {(n)}) \leq n ^ {- \alpha} + n ^ {- (1 - \alpha)} \text {i f} e < t. +$$ + +For preparation, we define the following function that is useful for computing the winning probability $P_{i}(e;A_{-i})$ for the undifferentiated contest. + +Definition E.3 (Function for computing winning probability). Given integers $n, k \geq 1$ and a density $p_1$ supported on $\Omega \subseteq \mathbb{R}_{\geq 0}$ , we denote a function $Q_p^{(n,k)}: \Omega \to \mathbb{R}_{\geq 0}$ to be for any $v \in \Omega$ , + +$$ +Q _ {p} ^ {(n, k)} (v) = \sum_ {i = n - k} ^ {n - 1} \binom {n - 1} {i} \cdot F _ {1} (v) ^ {i} \cdot (1 - F _ {1} (v)) ^ {n - i - 1} = \sum_ {i = n - k} ^ {n - 1} B (n - 1, i, F _ {1} (v)), \tag {13} +$$ + +where $B(n,k,x) = \binom{n}{k}x^{k}(1 - x)^{n - k}$ is the Bernstein polynomial. + +By definition, $Q_{p}^{(n,k)}(v)$ represents the probability that, when sampling $n - 1$ independent and identically distributed (i.i.d.) values $v_{1},\ldots ,v_{n - 1}$ from distribution $p_1$ , the value $v$ ranks among the top $k$ values in the set $\{v_{1},\dots,v_{n - 1},v\}$ . Given its algebraic significance, the function $Q_{p}^{(n,k)}(\cdot)$ is monotonically increasing to $v$ across the domain $\Omega$ . This means that as $v$ increases, the probability of $v$ being in the top $k$ also increases. Also note that for any integers $n,n^{\prime}\geq 1$ with $n < n^{\prime}$ , + +$$ +Q ^ {(n, k)} (v) \geq Q ^ {\left(n ^ {\prime}, k\right)} (v). \tag {14} +$$ + +This means that as the number of agents $n$ increases, $v$ is less likely to be in the top $k$ . This function can be used to compute $P_{i}(e;A_{-i})$ for the undifferentiated contest in the following sense. + +Lemma E.5 (Computation of winning probability for the undifferentiated contest). Let $n, k \geq 1$ be integers and $p_1$ be a density supported on $\Omega \subseteq \mathbb{R}_{\geq 0}$ . Let $A = (A_{1}, \ldots, A_{n})$ be a symmetric policy for the undifferentiated contest satisfying that every $A_i$ is strictly monotonically increasing to $v$ across the domain $\Omega$ . Then for every $i \in [n]$ and $v \in \Omega$ , we have $P_i(A_i(v); A_{-i}) = Q_P^{(n,k)}(v)$ . + +Proof: By symmetric, we only need to prove the lemma for $i = n$ , i.e., proving $P_{n}(A_{n}(v); A_{-n}) = Q_{p}^{(n,k)}(v)$ . Let $v_{1}, \ldots, v_{n-1}$ be i.i.d. samples from $p_{1}$ . Since $A_{i}$ is strictly monotonically increasing to $v$ across the domain $\Omega$ , we note that the sequence $v, v_{1}, \ldots, v_{n-1}$ should have the same order as the sequence $A_{n}(v), A_{1}(v_{1}), \ldots, A_{n-1}(v_{n-1})$ . Hence, $A_{n}(v)$ is among the top $k$ of $\{A_{1}(v_{1}), \ldots, A_{n-1}(v_{n-1}), A_{n}(v)\}$ if and only if $v$ is among the top $k$ of $\{v_{1}, \ldots, v_{n-1}, v\}$ . By the definition of winning probabilities and Definition E.3, this implies that $P_{n}(A_{n}(v); A_{-n}) = Q_{p}^{(n,k)}(v)$ . This completes the proof of Lemma E.5. + +Now we are ready to prove Lemma E.4. + +Proof [: of Lemma E.4] It suffices to prove for the case that both $F_{1}(t) > 0$ and $F_{2}(t) > 0$ . Proof for the other two cases is identical. By Definition E.2, we have + +$$ +\Delta_ {n} = \min \left\{F _ {1} ^ {- 1} (F _ {1} (t) - \sqrt {\frac {\log n}{n}}), F _ {2} ^ {- 1} (F _ {2} (t) - \sqrt {\frac {\log n}{n}}) \right\}. +$$ + +Let event $E_1^{(n,e)}$ be that there are at least $(1 - F_1(t))(1 - \alpha)n$ agents in $G_1 \setminus \{i\}$ that put in effort larger than $e$ ; and let $E_2^{(n,e)}$ be that there are at least $(1 - F_2(t)) \cdot \alpha n$ agents in $G_2 \setminus \{i\}$ that put in effort larger than $e$ . Note that + +$$ +(1 - F _ {1} (t)) (1 - \alpha) n + (1 - F _ {2} (t)) \cdot \alpha n = c n. +$$ + +When $e > t$ , we have + +$$ +P _ {i} (e; A _ {- i} ^ {(n)}) \geq \Pr \left[ \overline {{E _ {1} ^ {(n , e)}}} \cup \overline {{E _ {2} ^ {(n , e)}}} \right] \geq 1 - \Pr \left[ E _ {1} ^ {(n, e)} \right] - \Pr \left[ E _ {2} ^ {(n, e)} \right]. +$$ + +Then to prove $P_{i}(e; A_{-i}^{(n)}) = 1$ , it suffices to show that $\operatorname*{Pr}\left[E_1^{(n,e)}\right] = \operatorname*{Pr}\left[E_2^{(n,e)}\right] = 0$ . Note that by policy $A^{(n)}$ , the maximum effort put in by an agent is $t \leq v$ . Hence, no agent can put in effort larger than $e$ , which implies that $\operatorname*{Pr}\left[E_1^{(n,e)}\right] = \operatorname*{Pr}\left[E_2^{(n,e)}\right] = 0$ . This completes the proof of $P_{i}(e; A_{-i}^{(n)}) = 1$ when $e \geq t$ . + +When $e < t$ , we note that if both $E_1^{(n,e)}$ and $E_2^{(n,e)}$ happen, there are at least $k$ agents that put in effort $t$ . Since events $E_1^{(n,e)}$ and $E_2^{(n,e)}$ are independent, we have + +$$ +P _ {i} (e; A _ {- i} ^ {(n)}) \leq 1 - \Pr \left[ E _ {1} ^ {(n, e)} \cap E _ {2} ^ {(n, e)} \right] = 1 - \Pr \left[ E _ {1} ^ {(n, e)} \right] \cdot \Pr \left[ E _ {2} ^ {(n, e)} \right]. +$$ + +Then to prove $P_{i}(e;A_{-i}^{(n)}) \leq n^{-\alpha} + n^{-(1 - \alpha)}$ , it suffices to show that $\operatorname*{Pr}\left[E_1^{(n,e)}\right] \geq 1 - n^{-(1 - \alpha)}$ and $\operatorname*{Pr}\left[E_2^{(n,e)}\right] \geq 1 - n^{-\alpha}$ . + +We first bound $\operatorname*{Pr}\left[E_1^{(n,e)}\right]$ . Note that there are at least $(1 - \alpha)n$ agents in $G_{1} \cup \{i\}$ . Also, note that an agent $j \in G_{1} \setminus \{i\}$ puts in effort $A_{j}(v_{j}) > e$ if and only if their valuation $v_{j} \geq \Delta_{n}$ holds. Now consider the undifferentiated contest among $G_{1} \cup \{i\}$ with $k_{1} = (1 - F_{1}(t))(1 - \alpha)n$ and density $p_1$ . By Lemma E.5, we have + +$$ +\begin{array}{l} \Pr \left[ E _ {1} ^ {(n, e)} \right] = 1 - Q _ {p _ {1}} ^ {\left(\left| G _ {1} \cup \{i \} \right|, k _ {1}\right)} \left(\Delta_ {n}\right) (Lemma E.5) \\ \geq 1 - Q _ {p _ {1}} ^ {\left(\left(1 - \alpha\right) n, k _ {1}\right)} \left(\Delta_ {n}\right) (Ineq. (14)) \\ \geq 1 - Q _ {p _ {1}} ^ {\left(\left(1 - \alpha\right) n, k _ {1}\right)} \left(F _ {1} ^ {- 1} \left(F _ {1} (t) - \sqrt {\frac {\log n}{n}}\right)\right) \quad \text {(D e f n . o f} \Delta_ {n}) (15) \\ = 1 - \sum_ {j = (1 - \alpha) n - 1 - k _ {1}} ^ {(1 - \alpha) n - 1} B ((1 - \alpha) n - 1, j, F _ {1} (t) - \sqrt {\frac {\log n}{n}}). \quad (Eq. (13)) \\ \end{array} +$$ + +Let $X_{1},\ldots ,X_{n}$ be $(1 - \alpha)n - 1$ i.i.d. random variables where each $X_{i} = 0$ with probability $F_{1}(t) - \sqrt{\frac{\log n}{n}}$ and otherwise $X_{i} = 1$ . We note that $\sum_{j = (1 - \alpha)n - 1 - k_1}^{(1 - \alpha)n - 1}B((1 - \alpha)n - 1,j,F_1(t) - \sqrt{\frac{\log n}{n}})$ is equivalent to the probability that $\sum_{i\in [n - 1]}X_i\leq k_1 - 1$ . Also note that + +$$ +\mathbb {E} \left[ \sum_ {i \in [ (1 - \alpha) n - 1 ]} X _ {i} \right] = ((1 - \alpha) n - 1) \cdot (1 - F _ {1} (t) + \sqrt {\frac {\log n}{n}}). \tag {16} +$$ + +Then by the Chernoff bound, we have + +$$ +\begin{array}{l} \sum_ {j = (1 - \alpha) n - 1 - k _ {1}} ^ {(1 - \alpha) n - 1} B ((1 - \alpha) n - 1, j, F _ {1} (t) - \sqrt {\frac {\log n}{n}}) \\ = \Pr \left[ \sum_ {i \in [ (1 - \alpha) n - 1 ]} X _ {i} \leq k _ {1} - 1 \right] \\ \leq \Pr \left[ \sum_ {i \in [ (1 - \alpha) n - 1 ]} X _ {i} \leq \mathbb {E} \left[ \sum_ {i \in [ (1 - \alpha) n - 1 ]} X _ {i} \right] - (1 - \alpha) n \cdot \sqrt {\frac {\log n}{n}} \right] \quad (\text {E q .} (1 6) \text {a n d D e f n . o f} k _ {1}) \\ \leq e ^ {- \frac {2 (1 - \alpha) ^ {2} n \log n}{(1 - \alpha) n - 1}} \leq n ^ {- (1 - \alpha)}. \quad \text {(C h e r n o f f b o u n d)} \\ \end{array} +$$ + +Combining with Inequality (15), we prove that $\operatorname*{Pr}\left[E_1^{(n,e)}\right] \geq 1 - n^{-(1 - \alpha)}$ . By a similar argument, we can also prove $\operatorname*{Pr}\left[E_2^{(n,e)}\right] \geq 1 - n^{-\alpha}$ . Overall, we prove that $P_{i}(e;A_{-i}^{(n)}) \leq n^{-\alpha} + n^{-(1 - \alpha)}$ when $e < t$ . This completes the proof of Lemma E.4. + +# E.3.3 Proof that $A^{(n)}$ is approximate NE + +Based on Lemma E.4, we are now ready to prove the approximate degree of $A^{(n)}$ to be an NE policy. + +Lemma E.6 ( $A^{(n)}$ is approximate NE). For any $n \geq n_t$ , $A^{(n)}$ is an $\varepsilon_n$ -NE policy, where $\varepsilon_n = (n^{-\alpha} + n^{-(1 - \alpha)})\Delta_n + t - \Delta_n$ . + +Proof: Fix $\ell = 1,2$ , $i \in G_{\ell}$ , and $v,e \in \Omega_{\ell}$ . We discuss the value $\pi_i(v,e;A_{-i}^{(n)}) - \pi_i(v,A_i^{(n)}(v);A_{-i}^{(n)})$ . By Lemma E.4, we know that + +$$ +\pi_ {i} (v, e; A _ {- i} ^ {(n)}) = v - e \text {i f} e \geq t; \text {a n d} \pi_ {i} (v, e; A _ {- i} ^ {(n)}) \leq (n ^ {- \alpha} + n ^ {- (1 - \alpha)}) v - e \text {i f} e < t. +$$ + +Then if $v < \Delta_{n}$ , we have $\pi_i(v, A_i^{(n)}(v); A_{-i}^{(n)}) = \pi_i(v, 0; A_{-i}^{(n)}) = 0$ , which implies that + +$$ +\pi_ {i} (v, e; A _ {- i} ^ {(n)}) - \pi_ {i} (v, A _ {i} ^ {(n)} (v); A _ {- i} ^ {(n)}) \leq \left\{ \begin{array}{l l} (n ^ {- \alpha} + n ^ {- (1 - \alpha)}) v - e \leq (n ^ {- \alpha} + n ^ {- (1 - \alpha)}) \Delta_ {n} & \mathrm {i f} e < t \\ v - e \leq 0 & \mathrm {i f} e \geq t \end{array} \right. +$$ + +Otherwise if $v \geq \Delta_n$ , we have $\pi_i(v, A_i^{(n)}(v); A_{-i}^{(n)}) = \pi_i(v, t; A_{-i}^{(n)}) = v - t$ , which implies that + +$$ +\pi_ {i} (u, v; A _ {- i} ^ {(n)}) - \pi_ {i} (u, A _ {i} ^ {(n)} (v); A _ {- i} ^ {(n)}) \leq \left\{ \begin{array}{l l} (n ^ {- \alpha} + n ^ {- (1 - \alpha)}) v - e + t - v & \text {i f} e < t \\ t - e \leq 0 & \text {i f} e \geq t \end{array} \right. +$$ + +Note that when $v \geq \Delta_n$ and $e < t$ , + +$$ +\left(n ^ {- \alpha} + n ^ {- (1 - \alpha)}\right) v - e + t - v \leq \left(n ^ {- \alpha} + n ^ {- (1 - \alpha)}\right) \Delta_ {n} + t - \Delta_ {n}. +$$ + +Overall, we conclude that the following inequality always holds: + +$$ +\pi_ {i} (v, e; A _ {- i} ^ {(n)}) - \pi_ {i} (v, A _ {i} ^ {(n)} (v); A _ {- i} ^ {(n)}) \leq (n ^ {- \alpha} + n ^ {- (1 - \alpha)}) \Delta_ {n} + t - \Delta_ {n} = \varepsilon_ {n}. +$$ + +This verifies that $A^{(n)}$ is an $\varepsilon_{n}$ -NE policy for the two-group contest. + +# E.3.4 Completing the proof of Theorem E.2 + +Proof [: of Theorem E.2] Assume $p_a$ is a point mass at 0. We first prove that $\lim_{n\to \infty}s^{(n)} = s$ . This is a direct corollary of the fact that $\lim_{n\to \infty}\Delta_n = t$ . Consequently, for any $v\in \mathbb{R}_{\geq 0}$ , there exists $n_v$ such that for any integer $n\geq n_v$ , $s^{(n)}(v) = s(v)$ holds. + +By Lemma E.6, $A^{(n)}$ is a $\varepsilon_{n}$ -NE policy, where $\varepsilon_{n} = (n^{-\alpha} + n^{-(1 - \alpha)})\Delta_{n} + t - \Delta_{n}$ . Since $\lim_{n\to \infty}\Delta_n = t$ , we have + +$$ +\lim _ {n \to \infty} \varepsilon_ {n} = \lim _ {n \to \infty} (n ^ {- \alpha} + n ^ {- (1 - \alpha)}) \Delta_ {n} + t - \Delta_ {n} = 0, +$$ + +This completes the proof of Equation (12). + +Uniqueness of $A$ . To prove that $A$ is the unique NE, we first recall Corollary 3.2 of [20] that says that a subset of symmetric agents should have the same policy in an NE. Thus, assuming $A'$ is an NE policy for the two-group contest as $n \to \mathbb{R}_{\geq 0}$ , all agents $i \in G_1$ use a common threshold policy $s_1$ , and those in $G_2$ use $s_2$ . Suppose the threshold for $s_\ell$ is $t_\ell$ . We next prove that $t_1 = t_2$ , which implies that $s_1 = s_2$ . When $t_1 > t_2$ , if an agent puts in effort $t_1$ , then it will get selected. Thus, the probability of an agent in $G_1$ getting selected is $1 - F_1(t_1)$ . Hence, if $1 - F_1(t_1) < \frac{c}{1 - \alpha}$ , fewer than $cn$ agents in $G_1$ get selected. Thus, agents in $G_1$ getting selected will find that putting in effort slightly larger than $t_2$ instead of $t_1$ suffices to ensure their effort is larger than all agents in $G_2$ , and consequently, they will still be selected. Through this reduction in effort, they can gain an additional payoff of $t_1 - t_2$ , which violates the stability condition. A similar argument holds for $G_2$ when + +$1 - F_{1}(t_{1}) > \frac{c}{1 - \alpha}$ . Thus, $A'$ is not an NE when $t_{1} > t_{2}$ . Similarly, we can prove that $A'$ is not an NE when $t_{1} < t_{2}$ . Thus, we must have $t_{1} = t_{2} = t$ and $s_{1} = s_{2} = s$ . + +If $(1 - \alpha)F_{1}(t) + \alpha F_{2}(t) > 1 - c$ , then fewer than $cn$ agents put in a non-zero effort and get selected. Thus, an agent with valuation $t' < t$ , has a willingness to put in an effort $\varepsilon$ slightly larger than 0 instead of 0. Through this increase in effort, it can gain an additional payoff of $t' - \varepsilon$ . Thus, $A'$ is not an NE. Similarly, we can prove that $A'$ is not an NE if $(1 - \alpha)F_{1}(t) + \alpha F_{2}(t) < 1 - c$ . Thus, for an NE policy, $t$ must be the solution of $(1 - \alpha)F_{1}(v) + \alpha F_{2}(v) = 1 - c$ . By Lemma E.1, the equation $(1 - \alpha)F_{1}(v) + \alpha F_{2}(v) = 1 - c$ has a unique solution. Thus, $A' = A$ , which proves the uniqueness. + +Extension to general $p_a$ . For general $p_a$ , the solution $t$ of Equation (3) represents a score that a $c$ -fraction of agents can achieve without making their expected payoff negative ( $v + a \geq t$ ). The proof is almost identical to that when $p_a$ is a point mass at 0, except that the effort an agent with $v + a \geq t$ is willing to put in should be $s(v, a) = \max \{t - a, 0\}$ instead of $t$ . This is because $t$ represents the score that the agent aims to reach, rather than the effort itself. + +Overall, we complete the proof of Theorem E.2. + +Remark E.7 (Extension of Theorem 3.1). Using the same proof technique, Theorem 3.1 can be extended to handle multiple groups and non-identical effort costs. Let $G_{1}, \ldots, G_{m}$ represent $m \geq 2$ groups, where $|G_{\ell}| = \alpha_{\ell} n$ and the valuation density for each group is $p_{\ell}$ over the domain $\Omega_{\ell} \subseteq \mathbb{R}_{\geq 0}$ . Each $\alpha_{\ell} \in (0,1)$ satisfies the condition $\sum_{\ell \in [m]} \alpha_{\ell} = 1$ . Recall that $p_{a}$ represents the initial ability density over the domain $\Omega_{a} \subseteq \mathbb{R}_{\geq 0}$ , and we introduce an additional effort cost density $p_{\kappa}$ over the domain $\Omega_{\kappa} \subseteq \mathbb{R}_{>0}$ . Each agent $i \in [n]$ knows its type $\theta_{i} = (v_{i}, a_{i}, \kappa_{i})$ and selects an effort level $e_{i} \geq 0$ . The agent's score is given by $m(e_{i} + a_{i})$ , and their expected payoff is $P_{i} v_{i} - \kappa_{i} e_{i}$ . It is important to note that agents' costs of effort $\kappa_{i}$ may vary and affect only their expected payoff, not their score. + +In this extended multi-group contest, for $\ell \in [m]$ , we extend CDF $F_{\ell}$ to be $F_{\ell}(\zeta) = \operatorname{Pr}_{v \sim p_{\ell}, a \sim p_{a}, \kappa \sim p_{\kappa}}\left[\frac{v}{\kappa} + a \leq \zeta\right]$ . Now suppose domains $\bigcup_{\ell \in [m]} \Omega_{\ell}, \Omega_{a}, \Omega_{\kappa}$ are connected and densities $p_{\ell}, p_{a}, p_{\kappa}$ are positive at any point of their own domains. Let $t$ be the unique solution to the equation $\sum_{\ell \in [m]} \alpha_{\ell} F_{\ell}(\zeta) = 1 - c$ . The infinite NE policy (3) extends to: + +$$ +s (v, a) := 0 \text {i f} \frac {v}{\kappa} + a < t \text {a n d} s (v) := \max \{t - a, 0 \} \text {i f} \frac {v}{\kappa} + a \geq t. +$$ + +# E.4 Comparing with a distributional two-group contest + +Recall that the technical challenges for Theorem 3.1 are mainly caused by the asymmetric strategic environment across groups. To avoid asymmetry, one may consider the following variant of the two-group contest. Note that for simplicity, we also assume that $p_a$ is a point mass at 0. + +Definition E.4 (Distributional two-group contest). Let $n \geq k \geq 1$ be integers, $\alpha \in (0,1)$ , $\rho \in (0,1]$ , and $p_\ell$ be a density supported on a domain $\Omega_\ell \subseteq \mathbb{R}_{\geq 0}$ for $\ell = 1,2$ . Let each agent $i \in [n]$ belong to $G_1$ with probability $1 - \alpha$ and belong to $G_2$ with probability $\alpha$ independently. Let the valuation of each agent in $G_1$ be drawn i.i.d. from $p_1$ , and the valuation of each agent in $G_2$ be drawn i.i.d. from $p_2$ . Assume that each agent $i \in G_\ell$ ( $\ell = 1,2$ ) knows $n_1, n_2, k, p_1, p_2$ , the group it belongs to and its valuation, and has to choose a policy $A_i : \Omega_\ell \to \mathbb{R}_{\geq 0}$ to maximize its expected payoff. The goal of the distributional two-group contest is to compute the NE policy satisfying Equation (1). + +The main difference from the two-group contest is that this distributional variant's group identity is random and the valuation density of each agent is identical, say $p = (1 - \alpha)p_{1} + \alpha p_{2}$ . Thus, using a similar argument as in an undifferentiated contest, it is easy to verify that the NE policy of the distributional two-group contest is identical to that of an undifferentiated contest with density $p_{1}$ . Consequently, in the infinite $n$ case, the NE policy of the distributional two-group contest is identical to that of the two-group contest, say $A$ in Theorem 3.1. Then one may wonder whether this distributional two-group contest can also be used to simplify the proof of Theorem 3.1, as the infinite contest does. Below, we show that this is not the case and discuss the essential differences between the two models. For simplicity, we call the two-group contest Model I and call the distributional two-group contest Model II. + +Model distinction. Firstly, Model I itself is of relevant interest as it captures real-world examples in which group sizes are well understood, while in Model II the group sizes are random variables. + +Convergence distinction. Though Model I and Model II share the same NE policy $A$ in the infinite case, we would like to clarify that our main convergence result (Theorem E.2) cannot be inferred simply from knowing that the limit of the NEs of the two models is the same. To put it in simplest terms, consider two sequences $a_1, \ldots, a_n, \ldots$ and $b_1, \ldots, b_n, \ldots$ that converge to the same limit point $z$ . The proof of convergence for the first sequence does not necessarily provide any information about the convergence of the second sequence. Therefore, the convergence result for our model cannot be simply inferred from prior work. + +Analysis distinction. Moreover, the analysis of Model II relies heavily on symmetric policies for all agents (enabled precisely by the fact that group sizes are random), allowing the use of order statistics of $p_1$ . In contrast, in Model I, we expect asymmetric policies across groups. For example, consider a two-agent case with $k = 1$ : Agent 1's valuation follows a uniform distribution on $[0, 0.5]$ , while Agent 2's valuation follows a uniform distribution on $[0.5, 1]$ . In Section B, we show that the NE policy for this example must be asymmetric. Any symmetric policy $A$ would result in a near-zero winning probability for Agent 1, leading to a negative expected payoff and implying that $A$ is not an NE. This negative payoff arises from the asymmetric strategic environment faced by Agents 1 and 2, where the density of the highest valuation among other agents differs for each agent. Consequently, the order of winning probabilities of agents $(P_i)$ can differ from the order of valuations $(v_i)$ , posing a significant mathematical challenge for determining the NE. E.g., for strategies $s_1$ and $s_2$ , let $F_{s_\ell}(v)$ denote the cumulative distribution of efforts $s_\ell(v)$ when $v \sim p_\ell$ . The cumulative distribution of the $(k - 1)$ -th effort $e^\star$ from an agent in $G_1$ is then given by: + +$$ +\begin{array}{l} \Pr [ e ^ {\star} \leq v ] \\ = \sum_ {i = 0} ^ {n - 1} \sum_ {j = (1 - c) n - i} ^ {n - 1 - i} \binom {n - 1} {i} \binom {n - 1 - i} {j} F _ {s _ {1}} (v) ^ {i} (1 - F _ {s _ {1}} (v)) ^ {(1 - \alpha) n - 1 - i} F _ {s _ {2}} (v) ^ {j} (1 - F _ {s _ {2}} (v)) ^ {\alpha n - j}. \\ \end{array} +$$ + +Compare this to the expression for the symmetric case + +$$ +\Pr [e^{\star}\leq v] = \sum_{i = (1 - c)n}^{n - 1}\binom {n - 1}{i}F_{s}(v)^{i}(1 - F_{s}(v))^{n - 1 - i}. +$$ + +Thus, the calculus and approximations for the expression for the two-group contest are significantly more difficult, making it much harder to arrive at the equilibrium policies than for Model II. + +# E.5 An alternative proof using an infinite contest + +Recall that Theorem 3.1 studies the case of $n \to \infty$ for the two-group contest. To increase the understanding of the infinite case, we propose the following infinite version of the two-group contest, where every real number in the interval $[0, 1 - \alpha]$ corresponds to an agent in $G_1$ and in the interval $(1 - \alpha, 1]$ corresponds to an agent in $G_2$ . For simplicity, we still assume that $p_a$ is a point mass at 0. Formally, we provide the following definition. + +Definition E.5 (Infinite contest). Let $k \geq 1$ be integers, $\alpha \in (0,1)$ , $\rho \in (0,1]$ , and $p_{\ell}$ be a density supported on a domain $\Omega_{\ell} \subseteq \mathbb{R}_{\geq 0}$ for $\ell = 1,2$ . Let every real number in the interval $[0,1 - \alpha]$ correspond to an agent in group $G_{1}$ , and in the interval $(1 - \alpha,1]$ correspond to an agent in group $G_{2}$ . For $\ell \in \{1,2\}$ , let the valuation of every agent in $G_{\ell}$ draw i.i.d. from $p_{\ell}$ . Assume that each agent $i \in G_{\ell} (\ell = 1,2)$ knows $\alpha, k, p_{1}, p_{2}$ , the group it belongs to and its valuation, and has to choose a policy $A_{i}: \Omega_{\ell} \to \mathbb{R}_{\geq 0}$ to maximize its expected payoff. + +There are countless agents in this infinite contest. Also, note that $G_{1}$ contains $(1 - \alpha)$ -fraction of agents while $G_{2}$ contains the remaining $\alpha$ -fraction. Below, we show how to use this infinite contest to hypothesize the NE policy $A$ for the two-group contest defined in Theorem 3.1. It mainly consists of two steps: Showing that two-group contests converge to the infinite contest as $n \to \infty$ and computing NE for the infinite contest. + +Showing two-group contests converge to the infinite contest as $n \to \infty$ . We first show that the infinite contest is the limit of two-group contests. Let $g_{n}$ denote a two-group contest as defined in + +the two-group contest with an NE policy $A_{n}$ . Let $g$ denote the infinite game as defined in Problem E.5. We view $g_{n}$ as a collection of $n$ density functions of agents, with the $i$ -th agent represented by the real number $\frac{i - 1}{n - 1}$ . Agent $i$ belongs to group $G_{1}$ if $1 \leq i \leq (1 - \alpha)n$ and to group $G_{2}$ otherwise. From this viewpoint, we propose the following theorem. + +Theorem E.8 (Two-group contests converge to the infinite contest). $g_{n}$ converges to $g$ in the following sense: For any $\varepsilon > 0$ , there exists a sufficiently large $n_{0}$ such that for all $n \geq n_{0}$ , + +- For any $t \in [0,1 - \alpha]$ , $\left| \int_0^t dx - \frac{\left| \{i \in G_1 : \frac{i - 1}{n - 1} \leq t \} \right|}{n} \right| \leq \varepsilon$ , i.e., the difference in the fraction of agents in $G_1$ associated with real number at most $t$ between $g$ and $g_n$ , is at most $\varepsilon$ . +- For any $t \in (1 - \alpha, 1]$ , $\left| \int_t^1 dx - \frac{\left| \{i \in G_2 : \frac{i - 1}{n - 1} \geq t\} \right|}{n} \right| \leq \varepsilon$ , i.e., the difference in the fraction of agents in $G_2$ associated with real number at least $t$ between $g$ and $g_n$ , is at most $\varepsilon$ . + +Note that the agents in $g_{n}$ can be captured by a uniform distribution $\mu_{n}$ over real numbers $\frac{i - 1}{n - 1}$ ( $i \in [n]$ ). The convergence conditions in the theorem state that the limit of $\mu_{n}$ is the uniform density $\mu$ over $[0,1]$ , where $\mu$ represents the density of agents in $g$ . + +Proof:[of Theorem E.8] Let $n_0 = \lceil \varepsilon^{-1}\rceil$ . Then we have $n\geq \varepsilon^{-1}$ . For any $t\in [0,1 - \alpha ]$ , we have + +$$ +\left| \int_ {0} ^ {t} d x - \frac {\left| \left\{i \in G _ {1} : \frac {i - 1}{n - 1} \leq t \right\} \right|}{n} \right| = \left| t - \frac {\lfloor (n - 1) t + 1 \rfloor}{n} \right| +$$ + +and + +$$ +t - \frac {t}{n} \leq \frac {\lfloor (n - 1) t + 1 \rfloor}{n} \leq t + \frac {1 - t}{n}. +$$ + +We conclude that + +$$ +\left| \int_ {0} ^ {t} d x - \frac {\left| \left\{i \in G _ {1} : \frac {i - 1}{n - 1} \leq t \right\} \right|}{n} \right| \leq \max \left\{\frac {t}{n}, \frac {1 - t}{n} \right\} \stackrel {{t \in [ 0, 1 - \alpha ]}} {{\leq}} \frac {1}{n} \stackrel {{n \geq \varepsilon^ {- 1}}} {{\leq}} \varepsilon . +$$ + +Similarly, for any $t \in (1 - \alpha, 1]$ , we can prove that $\left| \int_t^1 dx - \frac{|\{i \in G_2 : \frac{i-1}{n-1} \geq t\}|}{n} \right| \leq \varepsilon$ . This completes the proof of Theorem E.8. + +Computing NE for the infinite contest. It follows from Theorem E.8 that the limit of the two-group contests $g_{n}$ is the infinite contest $g$ . Then, assuming the NE policy of $g_{n}$ is $A^{(n)}$ , we can infer that the limit of $A^{(n)}$ is the NE policy of the infinite contest. Thus, to hypothesize the NE policy for $g_{n}$ as $n\to \infty$ , it suffices to compute the NE policy for the infinite contest. + +We first observe that the strategic environment for all agents in the infinite contest is the same, i.e., the probability that a given valuation $v$ is among the top $c$ -fraction is the same for all agents. This property reduces the infinite contest to an undifferentiated contest (except for the different domains of valuation densities), leading to a symmetric NE policy. Formally, we provide the following lemma that shows that $A$ is exactly the unique NE policy for the infinite contest. + +Lemma E.9 (The infinite contest). Suppose $\Omega_1 \cup \Omega_2$ is connected and each density $p_\ell$ is positive at any point of domain $\Omega_\ell$ . Then policy $A$ defined in Equation (3) is the unique NE policy for the infinite contest. + +Proof: We first note that for any agent (whether in $G_{1}$ or $G_{2}$ ), the probability that a given valuation $v$ is among the top $c$ -fraction is given by: + +$$ +p_{1}(v):= \lim_{n\to \infty}\sum_{i = (1 - c)n}^{n - 1}\binom {n - 1}{i}F(v)^{i}(1 - F(v))^{n - 1 - i}, +$$ + +where $F$ is the CDF of the joint density $(1 - \alpha)p_{1} + \alpha p_{2}$ . Recall that $t$ is the unique solution to the equation + +$$ +F (v) = (1 - \alpha) F _ {1} (v) + \alpha F _ {2} (v) = 1 - c, +$$ + +i.e., $t = F^{-1}(1 - c)$ . Then through a straightforward calculation, it follows that $p_1(v) = 1$ for $v > t$ and $p_1(v) = 0$ for $v < t$ . Since the winning probability function $p_1$ is identical for all agents, the strategic environment for all agents in the infinite contest is the same. Recall that by Corollary 3.2 of [20], symmetric agents will use a symmetric policy in an NE. Thus, we can assume an increasing symmetric policy $s: \Omega_1 \cup \Omega_2 \to \mathbb{R}_{\geq 0}$ for all agents. + +By the equilibrium condition, for any valuation $v$ and effort $e$ , + +$$ +p _ {1} (v) v - s (v) \geq P (s ^ {- 1} (e)) v - e. +$$ + +By a similar argument as for Equation (10) (undifferentiated contest), we can compute that $s(v) = t$ for $v > t$ and $s(v) = 0$ for $v < t$ . This turns out to be the threshold function defined in Equation (3). Thus, the policy $A$ , where each agent restricts $s$ to its valuation the domain, is indeed the unique NE for the infinite contest. This completes the proof of Lemma E.9. + +Using the infinite game to provide an alternative proof of Theorem E.2. As shown in Lemma E.9, instead of relying on observations from the finite case as in Section E.1.2, we can use this infinite contest to hypothesize the NE policy $A$ for the two-group contest in the infinite $n$ case. + +Once we have a solid guess for the NE policy $A$ using the infinite contest, we need to show that $A$ remains an NE as $n \to \infty$ . While this approach simplifies the initial hypothesis, the challenge remains in proving that $A$ is indeed an NE. Similar to our current proof of Theorem 3.1, we must construct a series of proxies that converge to $A$ and increasingly approximate an NE policy. As detailed in Section E.2, this step remains technically challenging. + +Overall, using the infinite contest could provide an alternative proof of Theorem E.2. Moreover, the symmetric strategic environment of this infinite contest can provide deeper insights into why the NE policy remains symmetric, even when valuations are asymmetric across groups. + +# F Omitted details for uniform distribution analysis from Sections 3 and 4 + +Theorem F.1 (Restatement of Theorem 3.2). Assume $p_2(v) = \frac{1}{\rho} p_1\left(\frac{v}{\rho}\right)$ for some $\rho \in (0,1]$ and $p_a$ is a mass point at 0. Let policy $A$ be defined as in Theorem 3.1, characterized by $t$ being the unique solution of Equation (4). Then for any density $p_1$ , + +$$ +r _ {\mathcal {R}} (A) = \frac {1 - F _ {1} (t / \rho)}{1 - F _ {1} (t)}, r _ {\mathcal {S}} (A) = \frac {\rho \int_ {t / \rho} ^ {\infty} (v - t / \rho) p _ {2} (v) d v}{\int_ {t} ^ {\infty} (v - t) p _ {1} (v) d v}, a n d \mathcal {R V} (A, m) = m (t). +$$ + +Moreover, $r_{\mathcal{R}}(A)$ is monotonically increasing w.r.t. $\rho$ , $c$ , and $\alpha$ , while $\mathcal{RV}(A,m)$ is monotonically increasing w.r.t. $\rho$ and monotonically decreasing w.r.t. $c$ and $\alpha$ , for any merit function $m$ . + +Proof: We discuss three metrics separately. + +Metric $\mathcal{RV}(A, m)$ . Recall that $A$ is a threshold function characterized by $t$ . Thus, agents with the sum of valuation and initial ability $v + a > t$ get spots. Since $p_a$ is a point mass at 0, we know that the score of each selected agent is exactly $t$ . Thus, the average revenue $\mathcal{RV}(A, m) = m(t)$ . + +Next, we prove the monotonicity of $\mathcal{RV}(A,m)$ . Since $m$ is monotonically increasing, we only need to prove the monotonicity of $t$ with respect to $\rho, c, \alpha$ . Recall that $t$ is the solution of Equation (4), which can be rewritten as + +$$ +(1 - \alpha) F _ {1} (\zeta) + \alpha F _ {1} \left(\frac {\zeta}{\rho}\right) = 1 - c. +$$ + +Let $f(\zeta) = (1 - \alpha)F_{1}(\zeta) + \alpha F_{1}\left(\frac{\zeta}{\rho}\right) + c$ . Then $t$ is the solution of $f(\zeta) = 1$ . + +Since $F_{1}\left(\frac{\zeta}{\rho}\right)$ is a monotonically decreasing function of $\rho$ , we know that $f(\zeta)$ is also a monotonically decreasing function of $\rho$ . Thus, the solution $t$ increases with $\rho$ . Since $F_{a}(\zeta - \rho v) \geq F_{a}(\zeta - v)$ , $f(\zeta)$ is an increasing function of $\alpha$ . Also note that $f(\zeta)$ is an increasing function of $c$ . Thus, as $c$ or $\alpha$ increase, solution $t$ decreases. + +Overall, we prove that $\mathcal{RV}(A,m)$ is monotonically increasing w.r.t. $\rho$ and monotonically decreasing w.r.t. $c$ and $\alpha$ , for any merit function $m$ . + +Metric $r_{\mathcal{R}}(A)$ . Recall that $F_{\ell}$ is a cumulative density function (CDF) of the sum of valuation and initial ability such that for any $\zeta \in \mathbb{R}_{\geq 0}$ , $F_{\ell}(\zeta) = \operatorname*{Pr}_{v\sim p_{\ell},a\sim p_a}[v + a\leq \zeta]$ . Thus, $F_{\ell}$ is the CDF of $p_{\ell}$ when $p_a$ is a point mass at 0. Then the linearity of expectation yields: + +$$ +\mathbb {E} _ {v _ {i}, a _ {i}} [ | S \cap G _ {1} | ] = (1 - \alpha) n (1 - F _ {1} (t)) \text {a n d} \mathbb {E} _ {v _ {i}, a _ {i}} [ | S \cap G _ {2} | ] = \alpha n (1 - F _ {2} (t)). +$$ + +This translates to: + +$$ +\mathbb {E} [ \mathcal {R} _ {1} (A) ] = \frac {\mathbb {E} _ {v _ {i} , a _ {i}} [ | S \cap G _ {1} | ]}{(1 - \alpha) n} = 1 - F _ {1} (t) \text {a n d} \mathbb {E} [ \mathcal {R} _ {2} (A) ] = \frac {\mathbb {E} _ {v _ {i} , a _ {i}} [ | S \cap G _ {2} | ]}{\alpha n} = 1 - F _ {2} (t). +$$ + +Since $p_2(v) = \frac{1}{\rho} p_1\left(\frac{v}{\rho}\right)$ , we know that $F_{2}(t) = F_{1}(t / \rho)$ and $F_{1}(t)\leq F_{2}(t)$ . Thus, we have $\mathbb{E}[\mathcal{R}_2(A)]\leq \mathbb{E}[\mathcal{R}_1(A)]$ . + +As $n\to \infty$ , it suffices to prove that + +$$ +r _ {\mathcal {R}} (A) \rightarrow \min \left\{\frac {\mathbb {E} [ \mathcal {R} _ {1} (A) ]}{\mathbb {E} [ \mathcal {R} _ {2} (A) ]}, \frac {\mathbb {E} [ \mathcal {R} _ {2} (A) ]}{\mathbb {E} [ \mathcal {R} _ {1} (A) ]} \right\} = \frac {1 - F _ {2} (t)}{1 - F _ {1} (t)} = \frac {1 - F _ {1} (t / \rho)}{1 - F _ {1} (t)}. +$$ + +We first note that $|S \cap G_{\ell}|$ is highly concentrated at $\mathbb{E}_{v_i, a_i} [|S \cap G_1|]$ since all agents in $G_{\ell}$ are i.i.d. Concretely, the following inequality holds for any $t > 0$ by the Chernoff bound: + +$$ +\Pr \left[ | | S \cap G _ {\ell} | - \mathbb {E} _ {v _ {i}, a _ {i}} [ | S \cap G _ {\ell} | ] | \geq t \cdot \mathbb {E} _ {v _ {i}, a _ {i}} [ | S \cap G _ {\ell} | ] \right] \leq 2 e ^ {- \frac {t ^ {2} \cdot \mathbb {E} _ {v _ {i} , a _ {i}} [ | S \cap G _ {\ell} | ]}{3}}. +$$ + +Hence, for $t = o\left(\sqrt{\frac{1}{n}}\right)$ , we have $\operatorname*{Pr}\left[||S \cap G_{\ell}| - \mathbb{E}_{v_i,a_i}[|S \cap G_{\ell}||] \geq t \cdot \mathbb{E}_{v_i,a_i}[|S \cap G_{\ell}||] \right] \to 0$ . This implies that as $n \to \infty$ , + +$$ +r _ {\mathcal {R}} (A) = \mathbb {E} _ {v _ {i}, a _ {i}} \left[ \min \left\{\frac {\mathcal {R} _ {2} (A)}{\mathcal {R} _ {1} (A)}, \frac {\mathcal {R} _ {1} (A)}{\mathcal {R} _ {2} (A)} \right\} \right] \to \min \left\{\frac {\mathbb {E} [ \mathcal {R} _ {1} (A) ]}{\mathbb {E} [ \mathcal {R} _ {2} (A) ]}, \frac {\mathbb {E} [ \mathcal {R} _ {2} (A) ]}{\mathbb {E} [ \mathcal {R} _ {1} (A) ]} \right\}, +$$ + +which completes the proof of the formula of $r_{\mathcal{R}}(A)$ . + +Next, we prove the monotonicity of $r_{\mathcal{R}}(A)$ with respect to $\rho$ . Recall that $t$ is monotonically increasing with $\rho$ . We know that $1 - F_1(t)$ is a monotonically decreasing function of $\rho$ . Since $1 - F_2(t) = \frac{1 - c - (1 - \alpha)(1 - F_1(t))}{\alpha}$ , we know that $1 - F_2(t)$ is monotonically increasing with $\rho$ . Thus, $r_{\mathcal{R}}(A) = \frac{1 - F_1(t / \rho)}{1 - F_1(t)}$ is an increasing function of $\rho$ . + +Metric $r_S(A)$ . By a similar argument as for $r_{\mathcal{R}}(A)$ , we first have that as $n \to \infty$ + +$$ +r _ {\mathcal {S}} (A) \to \min \left\{\frac {\mathbb {E} [ \mathcal {S} _ {1} (A) ]}{\mathbb {E} [ \mathcal {S} _ {2} (A) ]}, \frac {\mathbb {E} [ \mathcal {S} _ {2} (A) ]}{\mathbb {E} [ \mathcal {S} _ {1} (A) ]} \right\}. +$$ + +Also note that + +$$ +\mathbb {E} \left[ \mathcal {S} _ {\ell} (A) \right] = \mathbb {E} \left[ \frac {1}{| G _ {\ell} |} \sum_ {i \in G _ {\ell}} \left(\mathbb {I} (i \text {s e l e c t e d}) \cdot v _ {i} - e _ {i}\right) \right] = \int_ {t} ^ {\infty} (v - t) p _ {\ell} (v) d v. +$$ + +Then we have + +$$ +\mathbb {E} \left[ \mathcal {S} _ {2} (A) \right] = \frac {1}{\rho} \int_ {t} ^ {\infty} (v - t) p _ {1} \left(\frac {v}{\rho}\right) d v = \int_ {\frac {t}{\rho}} ^ {\infty} (\rho v - t) p _ {1} (v) d v \leq \mathbb {E} \left[ \mathcal {S} _ {1} (A) \right]. +$$ + +Combining the above all, we obtain that $r_{\mathcal{S}}(A) = \frac{\int_t^\infty(v - t)p_2(v)dv}{\int_t^\infty(v - t)p_1(v)dv}$ . Since $p_2(v) = \frac{1}{\rho} p_1(\frac{v}{\rho})$ , we have + +$$ +r _ {\mathcal {S}} (A) = \frac {\int_ {t} ^ {\infty} (v - t) p _ {2} (v) d v}{\int_ {t} ^ {\infty} (v - t) p _ {1} (v) d v} = \frac {\rho \int_ {t / \rho} ^ {\infty} (v - t / \rho) p _ {1} (v) d v}{\int_ {t} ^ {\infty} (v - t) p _ {1} (v) d v} +$$ + +Next, we analyze the monotonicity of $r_{\mathcal{S}}(A)$ with respect to $\rho$ . Let $g(x) = \int_x^\infty (v - x)p_1(x)dx$ , which is monotonically decreasing of $x$ . We have $r_S(A) = \frac{\rho\cdot g(t / \rho)}{g(t)}$ . Since $t$ is monotonically increasing with $\rho$ , $g(t)$ is also monotonically increasing with $\rho$ . Thus, to prove that $r_{\mathcal{S}}(A)$ is monotonically increasing with $\rho$ , it suffices to prove that $t / \rho$ is monotonically decreasing with $\rho$ . Recall that we have shown that $F_{1}(t / \rho) = F_{2}(t)$ is monotonically decreasing with $\rho$ . This implies that $t / \rho$ is indeed monotonically decreasing with $\rho$ , completing the proof. + +Overall, we have completed the proof of the theorem. + +![](images/51234ecdb94ac01689e7e2c468d0f8df993671484597f408ada2a00dca00e727.jpg) +(a) $p_a$ is uniform on $[0,1]$ + +![](images/1c836c62ddb7f45c3be3d54d0a42220222abcac3018976b5cc1aa6301106265e.jpg) +(b) $p_a$ is a point mass at 0 +Figure 13: Plots of $t$ versus $\rho$ for various $c$ with $\alpha = 0.5$ for the uniform distribution. + +Remark F.2 (Monotonicity of $r_{\mathcal{R}}(A)$ and $r_S(A)$ w.r.t. $c, \alpha$ ). By Theorem 3.2, we note that when fixing $\rho$ , both $r_{\mathcal{R}}(A)$ and $r_S(A)$ are functions of $t$ . Since $t$ is monotonically decreasing with respect to $c$ and $\alpha$ , we only need to investigate the monotonicity of $r_{\mathcal{R}}(A)$ and $r_S(A)$ with respect to $t$ . Proposition 4.1 demonstrates that $r_{\mathcal{R}}(A)$ and $r_S(A)$ are monotonically decreasing with $t$ , and hence, monotonically increasing with $c$ and $\alpha$ . Below, we provide an example with $p_1$ to show that this monotonicity does not always hold. + +Let $\varepsilon > 0$ be a sufficiently small number and $\rho = 0.5$ . Let $p_1$ be supported on $\Omega_1 = [0,2]$ such that $p_1(v) = 1 - \varepsilon$ for $v \in [0,0.5] \cup [1.5,2]$ and $p_1(v) = \varepsilon$ for $v \in (0.5,1.5)$ . Then $F_1(v) = (1 - \varepsilon)v$ for $v \in [0,0.5]$ , $0.5 - \varepsilon + \varepsilon v$ for $v \in (0.5,1.5)$ , and $(1 - \varepsilon)v + 2\varepsilon - 1$ . By Theorem 3.2, we have $r_{\mathcal{R}}(A) = \frac{1 - F_1(t / \rho)}{1 - F_1(t)}$ . Then in this case, $r_{\mathcal{R}}(A) = \frac{1 - (1 - \varepsilon) / 2}{1 - (1 - \varepsilon) / 4} \approx \frac{2}{3}$ when $t = 0.25$ ; while $r_{\mathcal{R}}(A) = \frac{1 - (1 - \varepsilon) / 2 - 0.5\varepsilon}{1 - (1 - \varepsilon) / 2} \approx 1$ . Thus, $r_{\mathcal{R}}(A)$ is not monotonically decreasing with $t$ . A similar computation can be done for $r_{\mathcal{S}}(A)$ . + +By a similar argument as for Theorem 3.2, we provide the following formulas of metrics for general densities. The computation is straightforward and we omit here. + +Theorem F.3 (Metrics in general). Let policy $A$ be defined as in Theorem 3.1, characterized by $t$ being the unique solution of Equation (2). Then for any densities $p_1, p_2$ , and $p_a$ , + +$$ +r _ {\mathcal {R}} (A) = \min \left\{\frac {1 - F _ {2} (t)}{1 - F _ {1} (t)}, \frac {1 - F _ {1} (t)}{1 - F _ {2} (t)} \right\}, +$$ + +$$ +r _ {\mathcal {S}} (A) = \min \left\{\frac {\int_ {t} ^ {\infty} \int_ {\Omega_ {2}} \min \{v , \zeta - t \} p _ {2} (v) p _ {a} (\zeta - v) d v d \zeta}{\int_ {t} ^ {\infty} \int_ {\Omega_ {1}} \min \{v , \zeta - t \} p _ {1} (v) p _ {a} (\zeta - v) d v d \zeta}, \frac {\int_ {t} ^ {\infty} \int_ {\Omega_ {1}} \min \{v , \zeta - t \} p _ {1} (v) p _ {a} (\zeta - v) d v d \zeta}{\int_ {t} ^ {\infty} \int_ {\Omega_ {2}} \min \{v , \zeta - t \} p _ {2} (v) p _ {a} (\zeta - v) d v d \zeta} \right\}, +$$ + +$$ +\begin{array}{l} \mathcal {R V} (A) = \frac {1 - \alpha}{1 - c} \int_ {t} ^ {\infty} \int_ {\Omega_ {1}} m (\max \{t - \zeta + v, 0 \}) p _ {1} (v) p _ {a} (\zeta - v) d v d \zeta \\ + \frac {\alpha}{1 - c} \int_ {t} ^ {\infty} \int_ {\Omega_ {2}} m (\max \{t - \zeta + v, 0 \}) p _ {1} (v) p _ {a} (\zeta - v) d v d \zeta . \\ \end{array} +$$ + +In the following, we complete the analysis from Sections 3 and 4 for the case where $p_1$ and $p_2$ are uniform densities. The visualization of $t$ for them can be found in Figure 13. We first give the proof of Equation (5) for the case that $p_a$ is uniform. + +Proposition F.4 (Complete version of Equation (5)). Let $\alpha, c \in (0,1)$ and $\rho \in (0,1]$ . Let $p_1$ be uniform on $[0,1]$ , $p_2$ be uniform on $[0,\rho]$ , and $p_a$ be uniform on $[0,1]$ . Let $t \in [0,2]$ be the solution to the equation $\int_0^1 (1 - \alpha) \cdot \min \{1, (\zeta - v)_+\} dv + \int_0^\rho \frac{\alpha}{\rho} \cdot \min \{1, (\zeta - v)_+\} dv = 1 - c$ . Then if $0 < c \leq \frac{1 - \alpha}{2}$ , + +$$ +t = \left\{ \begin{array}{l l} 2 - \sqrt {\frac {2 c}{1 - \alpha}}, & \rho < 1 - \sqrt {\frac {2 c}{1 - \alpha}}, \\ \frac {2 - \alpha + \alpha / \rho}{1 - \alpha + \alpha / \rho} - \frac {\sqrt {2 c (1 - \alpha + \alpha / \rho) - \alpha (1 - \alpha) (1 - \rho) ^ {2} / \rho}}{1 - \alpha + \alpha / \rho}, & \rho \geq 1 - \sqrt {\frac {2 c}{1 - \alpha}}, \end{array} \right. +$$ + +$$ +\begin{array}{l} i f \frac {1 - \alpha}{2} < c \leq \frac {1}{2}, \\ \left\{ \begin{array}{l l} \frac {2 - \alpha + \alpha / \rho}{1 - \alpha + \alpha / \rho} - \frac {\sqrt {2 c (1 - \alpha + \alpha / \rho) - \alpha (1 - \alpha) (1 - \rho) ^ {2} / \rho}}{1 - \alpha + \alpha / \rho}, & \rho < \frac {2 c - 1 + \alpha}{\alpha}, \\ - \frac {\alpha}{1 - \alpha} + \frac {\sqrt {\alpha^ {2} + (1 - \alpha) (2 + \alpha \rho - 2 c)}}{1 - \alpha}, & \rho \geq \frac {2 c - 1 + \alpha}{\alpha}, \end{array} \right. \\ \end{array} +$$ + +$$ +i f \frac {1}{2} < c \leq 1, +$$ + +$$ +\left\{ \begin{array}{l l} - \frac {\alpha}{1 - \alpha} + \frac {\sqrt {\alpha^ {2} + (1 - \alpha) (2 + \alpha \rho - 2 c)}}{1 - \alpha}, & \rho < \frac {- \alpha + \sqrt {\alpha^ {2} + 8 (1 - \alpha) (1 - c)}}{2 (1 - \alpha)}, \\ \sqrt {\frac {2 (1 - c)}{1 - \alpha + \alpha / \rho}}, & \rho \geq \frac {- \alpha + \sqrt {\alpha^ {2} + 8 (1 - \alpha) (1 - c)}}{2 (1 - \alpha)}. \end{array} \right. +$$ + +Proof: Let $f(\zeta) = \int_0^1 (1 - \alpha) \cdot \min \{1, (\zeta - v)_+\} dv + \int_0^\rho \frac{\alpha}{\rho} \cdot \min \{1, (\zeta - v)_+\} dv$ . We first note that $f(\zeta)$ is a monotone increasing function with $f(0) = 0$ and $f(2) = 1$ . By analyzing the value of $\min \{1, (\zeta - v)_+\}$ , we also have the following equation: + +$$ +\min \{1, (\zeta - v) _ {+} \} = \left\{ \begin{array}{l l} 0, & 0 \leq \zeta \leq v, \\ \zeta - v, & v \leq \zeta \leq v + 1, \\ 1, & v + 1 \leq \zeta \leq 2. \end{array} \right. +$$ + +Accordingly, we know that + +$$ +f (\zeta) = \left\{ \begin{array}{l l} (1 - \alpha) (\zeta - v) | _ {0} ^ {\zeta} + \frac {\alpha}{\rho} (\zeta - v) | _ {0} ^ {\zeta}, & 0 \leq \zeta \leq \rho , \\ (1 - \alpha) (\zeta - v) | _ {0} ^ {\zeta} + \frac {\alpha}{\rho} (\zeta - v) | _ {0} ^ {\rho}, & \rho \leq \zeta \leq 1, \\ (1 - \alpha) \left(\zeta - 1 + (\zeta - v) | _ {\zeta - 1} ^ {\zeta}\right) + \frac {\alpha}{\rho} \left(\zeta - 1 + (\zeta - v) | _ {\zeta - 1} ^ {\rho}\right), & 1 \leq \zeta \leq 1 + \rho , \\ (1 - \alpha) \left(\zeta - 1 + (\zeta - v) | _ {\zeta - 1} ^ {\zeta}\right) + \alpha , & 1 + \rho \leq \zeta \leq 2. \end{array} \right. +$$ + +Thus, $f$ is a piecewise-polynomial function of $\zeta$ . Solving $f(\zeta) = 1 - c$ results in Corollary F.4. + +Proposition F.5 (Restatement of Proposition 4.1). Let $p_1$ be uniform on $[0,1]$ , $p_2$ be uniform on $[0,\rho]$ , and $p_a$ be a point mass at 0. Let $A$ be the NE policy for the two-group contest as $n\to \infty$ . Then + +$$ +t = 1 - \frac {c}{1 - \alpha} i f \rho < 1 - \frac {c}{1 - \alpha} a n d t = \frac {\rho (1 - c)}{\rho - \alpha \rho + \alpha} i f \rho \geq 1 - \frac {c}{1 - \alpha}. +$$ + +$$ +r _ {\mathcal {R}} (A) = 0 \text {i f} \rho < 1 - \frac {c}{1 - \alpha} \text {a n d} r _ {\mathcal {R}} (A) = \frac {\rho - \alpha \rho + \alpha + c - 1}{\alpha - \alpha \rho + c \rho} \text {i f} \rho \geq 1 - \frac {c}{1 - \alpha}. +$$ + +$$ +r _ {\mathcal {S}} (A) = 0 \text {i f} \rho < 1 - \frac {c}{1 - \alpha} \text {a n d} r _ {\mathcal {S}} (A) = \frac {\rho (\rho - \alpha \rho + \alpha + c - 1) ^ {2}}{(\alpha - \alpha \rho + c \rho) ^ {2}} \text {i f} \rho \geq 1 - \frac {c}{1 - \alpha}. +$$ + +Moreover, $\mathcal{RV}(A,m) = m(t)$ for any merit function $m(\cdot)$ ; $r_{\mathcal{R}}(A)$ and $r_S(A)$ are monotonically increasing functions of parameters $\rho, c, \alpha$ . + +Proof: Note that $\mathcal{RV}(A,m) = m(t)$ is directly from Theorem 3.2. + +Computation of $t$ . Note that $F_{1}(v) = v$ and $F_{2}(v) = \min \left\{1, \frac{v}{\rho}\right\}$ . Let $g(v) = (1 - \alpha)v + \alpha \min \left\{1, \frac{v}{\rho}\right\}$ . We note that $g(\cdot)$ is a piece-wise linear function of $v$ with an inflection point $v = \rho$ . Plugging $v = \rho$ into the equation, we obtain that $\rho = 1 - \frac{c}{1 - \alpha}$ which is an inflection point of $t$ . Then if solution $t > \rho$ , we have that $t$ is the solution of the equation $(1 - \alpha)v + \alpha = 1 - c$ , implying that $t = 1 - \frac{c}{1 - \alpha}$ . The condition for this case is $\rho < 1 - \frac{c}{1 - \alpha} = t$ . Otherwise if solution $t \leq \rho$ , we have that $t$ is the solution of the equation $(1 - \alpha)v + \alpha \frac{v}{\rho} = 1 - c$ , implying that $t = \frac{\rho(1 - c)}{\rho - \alpha\rho + \alpha}$ . The condition for this case is $\rho \geq 1 - \frac{c}{1 - \alpha}$ . This completes the proof for $t$ . + +Analysis for $r_{\mathcal{R}}(A)$ . By Theorem 3.2, we have $r_{\mathcal{R}}(A) = \frac{1 - \min\left\{1,\frac{t}{\rho}\right\}}{1 - t}$ . By the form of $t$ , we can verify the explicit form of $r_{\mathcal{R}}(A)$ . + +Note that when $\rho \geq 1 - \frac{c}{1 - \alpha}$ , we have + +$$ +r _ {\mathcal {R}} (A) = \frac {\rho - \alpha \rho + \alpha + c - 1}{\alpha - \alpha \rho + c \rho} = 1 - \frac {(1 - c) (1 - \rho)}{\alpha - \alpha \rho + c \rho}. +$$ + +Let + +$$ +f (\rho , c, \alpha) = 1 - \frac {(1 - c) (1 - \rho)}{\alpha - \alpha \rho + c \rho}. +$$ + +Define the auxiliary functions: + +$$ +X (\rho , c, \alpha) = (1 - c) (1 - \rho), \quad Y (\rho , c, \alpha) = \alpha - \alpha \rho + c \rho . +$$ + +Then when $\rho \geq 1 - \frac{c}{1 - \alpha}$ , we have + +$$ +f (\rho , c, \alpha) = 1 - \frac {X}{Y}, \text {a n d} X (\rho , c, \alpha), Y (\rho , c, \alpha) > 0. +$$ + +The partial derivatives w.r.t. $\rho, c, \alpha$ are: + +$$ +\frac {\partial f}{\partial \rho} = \frac {(1 - c) Y + (c - \alpha) X}{Y ^ {2}} = \frac {c (1 - c)}{Y ^ {2}} \geq 0, +$$ + +$$ +\frac {\partial f}{\partial c} = \frac {(1 - \rho) Y + \rho X}{Y ^ {2}} \geq 0, +$$ + +$$ +\frac {\partial f}{\partial \alpha} = \frac {1 - \rho}{Y ^ {2}} \geq 0. +$$ + +Thus, $r_{\mathcal{R}}(A)$ is monotonically increasing with $\rho, c, \alpha$ when $\rho \geq 1 - \frac{c}{1 - \alpha}$ . Moreover, the threshold $1 - \frac{c}{1 - \alpha}$ is monotonically decreasing with $c, \alpha$ . Thus, we conclude that $r_{\mathcal{R}}(A)$ is a monotonically increasing function of parameters $\rho, c, \alpha$ . + +Analysis for $r_{\mathcal{S}}(A)$ . By a similar argument as for $r_{\mathcal{R}}(A)$ , we can obtain the formulas of $r_{\mathcal{S}}(A)$ using Theorem 3.2 and the form of $t$ . Note that $r_{\mathcal{S}}(A) = \rho r_{\mathcal{R}}(A)^2$ . Thus, $r_{\mathcal{S}}(A)$ is monotonically increasing with $\rho, c, \alpha$ . + +Overall, we have completed the proof of the proposition. + +![](images/874e4351350dce7fac805950f81295001e2bbf0551df2ea9b9b433f0d5ce73d9.jpg) + +# G Additional details to Section 4 + +In this section, we first illustrate details for how to estimate perceived bias from JEE Advanced 2024 (Section G.1). Then we provide a robustness analysis for key findings in Section 4 by varying $\alpha$ and $c$ (Section G.2). Next, we give an illustrative example for the practical use of our model, including how to make interpretable predictions and policy interventions (Section G.3). Finally, we provide omitted details for alternative interventions in Section 4. + +# G.1 Case study — estimating perceived bias from JEE Advanced 2024 + +To illustrate our framework in a high-stakes meritocratic setting, we calibrate the model using data from JEE Advanced 2024, the entrance examination for admission to the Indian Institutes of Technology (IITs). The gender-disaggregated statistics were published by the Government of India's Press Information Bureau [71]: + +
GroupCandidates AppearedQualified
Male139,18040,284
Female41,0207,964
Total180,20048,248
+ +Model calibration. We define the disadvantaged group as female candidates and the advantaged group as male candidates. From the data: + +- Proportion of disadvantaged applicants: + +$$ +\alpha = \frac {4 1 , 0 2 0}{1 8 0 , 2 0 0} \approx 0. 2 2 8 +$$ + +- Selection rate for the entire applicant pool: + +$$ +c = \frac {4 8 , 2 4 8}{1 8 0 , 2 0 0} \approx 0. 2 6 8 +$$ + +- Admit rate for each group: + +$$ +\text {F e m a l e} = \frac {7 , 9 6 4}{4 1 , 0 2 0} \approx 0. 1 9 4, \quad \text {M a l e} = \frac {4 0 , 2 8 4}{1 3 9 , 1 8 0} \approx 0. 2 8 9 +$$ + +- Observed representation ratio: + +$$ +r _ {\mathrm {o b s}} = \frac {0 . 1 9 4}{0 . 2 8 9} \approx 0. 6 7 1 +$$ + +Solving for the bias parameter $\rho$ . Using the closed-form expression for the representation ratio in the uniform-valuations model: + +$$ +r _ {R} (\rho , c, \alpha) = 1 - \frac {(1 - c) (1 - \rho)}{\alpha - \alpha \rho + c \rho}, +$$ + +we plug in $r_R = 0.671$ , $c = 0.268$ , and $\alpha = 0.228$ to solve for $\rho$ : + +$$ +0. 6 7 1 = 1 - \frac {(1 - 0 . 2 6 8) (1 - \rho)}{0 . 2 2 8 - 0 . 2 2 8 \rho + 0 . 2 6 8 \rho}. +$$ + +Simplifying both sides: + +$$ +0. 3 2 9 = \frac {0 . 7 3 2 (1 - \rho)}{0 . 2 2 8 + 0 . 0 4 \rho}, +$$ + +$$ +0. 3 2 9 (0. 2 2 8 + 0. 0 4 \rho) = 0. 7 3 2 (1 - \rho). +$$ + +Compute both sides: + +$$ +0. 0 7 4 9 + 0. 0 1 3 1 6 \rho = 0. 7 3 2 - 0. 7 3 2 \rho . +$$ + +Bring all terms to one side: + +$$ +0. 7 4 5 1 6 \rho = 0. 6 5 7 1 \quad \Rightarrow \quad \rho \approx \frac {0 . 6 5 7 1}{0 . 7 4 5 1 6} \approx 0. 8 8 2. +$$ + +The inferred bias parameter is: + +$$ +\rho \approx 0. 8 8 2, +$$ + +which reflects a perceived disadvantage for female candidates: they value qualification outcomes at roughly $88.2\%$ of their male counterparts' valuation, consistent with the observed gender gap in qualification rates. This example demonstrates how our model can be applied to quantify bias in selection systems using real-world statistics. + +# G.2 Robustness analysis for findings in Section 4 + +In this section, we assess whether our core conclusions in Section 4 depend on the specific parameter settings. To verify robustness, we conducted additional simulations varying $\alpha$ and $c$ beyond the default values. Below we summarize our findings: + +Metric robustness across group sizes. We varied $\alpha$ from 0.5 to 0.3 (to represent smaller disadvantaged groups) and recalculated the representation ratio $r_{\mathcal{R}}(A)$ and welfare ratio $r_S(A)$ across a grid of disparity levels $(\rho)$ and selection rates $(c)$ ; see Figure 14. The overall trends remain consistent: for example, $r_{\mathcal{R}}(A) \leq 0.2$ still holds for $c = 0.1$ and $\rho \leq 0.85$ , confirming that strategic behavior amplifies underrepresentation in highly selective settings. + +Robustness of intervention takeaways. We varied $c$ from 0.228 (derived from the JEE Advanced data) to 0.1 and $\alpha$ from 0.268 to 0.5 and re-evaluated intervention strategies. Figure 15 plots optimal interventions for various threshold $\tau$ . The overall trends remain consistent. For instance, in Figure 15(a), when $\tau \leq 0.87$ , increasing access (raising $c$ ) remains more cost-effective. In contrast, when $\tau > 0.87$ , improving group valuation (increasing $\rho$ ) becomes more impactful. This confirms that the recommendation to prioritize access vs. valuation interventions depending on the disparity level remains valid across reasonable choices of $\alpha$ and $c$ . + +![](images/3d751cd552da3e3c6f406f555a50e976ebf6c5764b08187fb17305b0f2ee220d.jpg) +(a) $r_{\mathcal{R}}(A)$ v.s. $\rho$ + +![](images/78d44d74943e96969ffa510b08c66f62ec2787b04b1f4370c6435d6d0d42f3d5.jpg) +(b) $r_{\mathcal{R}}(A)$ v.s. $c$ + +![](images/0afcb4b1bfd9f8ae525ad2f36d5559f38705e29f06cb03bc5da42c2fbeebac9c.jpg) +(c) $r_{\mathcal{S}}(A)$ v.s. $\rho$ + +![](images/0725dff65c91c974966e50f74418043fec58c87b753c5ee8acd86174334d74d0.jpg) +(d) $r_{\mathcal{S}}(A)$ v.s. $c$ + +![](images/4d4717a83a64684ea7b95d364f38928ecc1208e15cf4bdee30a65021451217b7.jpg) +Figure 14: Plots of the representation ratio $r_{\mathcal{R}}(A)$ and the social welfare ratio $r_S(A)$ as parameters $\rho$ and $c$ vary for Proposition 4.1, with default settings of $(\rho, c, \alpha) = (0.8, 0.1, 0.3)$ . A dotted line in these plots indicates the threshold at which $r_{\mathcal{R}}(A) = 0.8$ or $r_S(A) = 0.8$ . +(a) When $c = 0.1$ +Figure 15: Plot of optimal interventions $(\Delta_{\rho}, \Delta_{c})$ for various $\tau$ + +![](images/9114bfe6d9b4046fdef18cf6d9a0097373262f3adf016c3d2cbb15dbe66b07e2.jpg) +(b) When $\alpha = 0.5$ + +# G.3 An illustrative example: interpreting and applying the model + +We provide a concrete example to illustrate how our theoretical framework can be used to diagnose and compare policy interventions. + +Interpretable diagnostics. Suppose a policymaker observes persistent underrepresentation of a disadvantaged group (e.g., female students) in a selective admissions process. Given data on the overall selection rate $c$ , group size $\alpha$ , and the observed representation ratio $r_{\mathcal{R}}(A)$ , the policymaker can use our model to infer the implied valuation gap parameter $\rho$ (as shown in Section G.1). This parameter summarizes how much lower the disadvantaged group perceives the value of success relative to the advantaged group. + +Although $\rho$ is not directly observable, its interpretation is transparent: it attributes behavioral disparities (such as lower effort investment) to structural differences in perceived incentives rather than to innate ability. In this way, the model provides a normative reading of observed disparities—as equilibrium responses to valuation asymmetries. + +Policy interventions. Once the implied parameters are estimated, the policymaker can consider two classes of interventions: + +- Valuation-based interventions: improving the perceived value of success (e.g., through mentorship programs or financial aid), which effectively increases $\rho$ ; +- Access-based interventions: expanding the number of available slots, thereby increasing $c$ . + +Our framework allows simulation of the effects of each intervention on representation, welfare, and efficiency, enabling counterfactual comparisons under a fixed behavioral model. For example, when $\rho$ is low, expanding access may yield larger gains in representation, while when $\rho$ is already high, improving valuation can be more cost-effective. + +Implementation challenges. Estimating parameters such as $\rho$ empirically is nontrivial and remains an open direction. It would require auxiliary data sources (e.g., surveys, longitudinal effort-outcome data) or structural assumptions about the effort cost function. Nonetheless, once such estimates are available, our framework provides a transparent scaffold for interpreting behavioral disparities and evaluating the relative effectiveness of competing policy interventions. + +# G.4 Details for alternative interventions + +Below we provide more detailed theoretical analysis for alternative potential interventions discussed in Section 4. + +Introducing preference heterogeneity. Recall that the institution applies group-specific merit mappings of the form: for group $G_{\ell}$ ( $\ell = 1,2$ ) and for score $s$ , $m_{\ell}(s) = x_{\ell} \cdot s + y_{\ell}$ for group-specific parameters $x_{\ell}, y_{\ell} \geq 0$ . We have the following generalized theorem of Theorem 3.1 + +Theorem G.1 (Generalization of Theorem 3.1: Introducing preference heterogeneity). Let $\alpha, c \in (0,1)$ . For $\ell = 1,2$ , let $p_{\ell}$ be a density supported on a domain $\Omega_{\ell} \subseteq \mathbb{R}_{\geq 0}$ . Let $p_a$ be a density supported on a domain $\Omega_a \subseteq \mathbb{R}_{\geq 0}$ . For $\ell = 1,2$ , let $m_{\ell}$ be a merit function defined above. For $\ell = 1,2$ , let $F_{\ell}$ be a cumulative density function (CDF) of the sum of valuation and initial ability such that for any $\zeta \in \mathbb{R}_{\geq 0}$ , $F_{\ell}(\zeta) = \operatorname{Pr}_{v \sim p_{\ell}, a \sim p_a} [x_\ell v + y_\ell + a \leq \zeta]$ . Suppose $(x_1\Omega_1 + y_1) \cup (x_2\Omega_2 + y_2)) + \Omega_a$ is connected and densities $p_1, p_2, p_a$ are positive at any point of their own domains. Let $t$ be the unique solution to the equation + +$$ +(1 - \alpha) F _ {1} (\zeta) + \alpha F _ {2} (\zeta) = 1 - c. +$$ + +Then the threshold function of the NE policy defined in Eq. (3) extends to be: for $\ell = 1,2$ + +$$ +s _ {\ell} (v, a) = 0 \text {i f} v + a < \frac {t - y _ {\ell}}{x _ {\ell}}, \text {a n d} s _ {\ell} (v, a) = \max \left\{\frac {t - y _ {\ell}}{x _ {\ell}} - a, 0 \right\} \text {i f} v + a \geq \frac {t - y _ {\ell}}{x _ {\ell}}. +$$ + +Moreover, the threshold $\frac{t - y_2}{x_2}$ for $G_2$ is monotonically decreasing with $x_2, y_2$ . + +Proof: The proof for $s_\ell$ is almost identical to that of Theorem 3.1. We only need to note that for any agent $i \in G_\ell (\ell = 1,2)$ with valuation-ability pair $(v_i,a_i) \in \Omega_\ell \times \Omega_a$ , if its valuation $v_i \geq \frac{t - y_\ell}{x_\ell} - a_i$ , then its merit must be + +$$ +m _ {\ell} \left(s _ {\ell} \left(v _ {i}, a _ {i}\right)\right) = x _ {\ell} \cdot \left(a _ {i} + \max \left\{\frac {t - y _ {\ell}}{x _ {\ell}} - a _ {i}, 0 \right\}\right) + y _ {\ell} \geq t, +$$ + +which is within the top $c$ -fraction and makes the agent get selected. + +Regarding the monotonicity of $\frac{t - y_2}{x_2}$ , note that as $x_2$ or $y_2$ increases, $F_{\ell}(\zeta)$ decreases. Then the solution $t$ must increase, resulting in a higher threshold $\frac{t - y_1}{x_1}$ for group $G_1$ . This reduces the fraction of agents in $G_1$ to get selected, and consequently, increases the fraction of agents in $G_2$ to get selected. Then the threshold $\frac{t - y_2}{x_2}$ must decrease, which completes the proof. + +Note that when $x_{1} = x_{2} = 1$ and $y_{1} = y_{2} = 0$ , this theorem is exactly Theorem 3.1, and hence, is a generalization. This theorem implies that by increasing $x_{2}, y_{2}$ , more agents in $G_{2}$ are willing to put in efforts due to lower valuation threshold $\frac{t - y_2}{x_2}$ . This supports the properties discussed in Section 4. + +Setting group-specific selection rates. Assume that the institution selects a $c$ -fraction of agents from $G_{1}$ and $G_{2}$ independently. The model decomposes into two independent within-group contests, each with its own Nash equilibrium. + +For the disadvantaged group $G_{2}$ , let $F_{2}$ denote the CDF of its combined signal $v + a$ . The equilibrium threshold $t_{2}$ under group-specific capacity $c$ is the unique solution to: + +$$ +F _ {2} (t _ {2}) = 1 - c. +$$ + +In contrast, under a uniform selection rate $c$ applied to the full population (original two-group contest), the common threshold $t$ solves: + +$$ +(1 - \alpha) F _ {1} (t) + \alpha F _ {2} (t) = 1 - c. +$$ + +Since $G_{2}$ has lower valuations by assumption, we typically have $F_{2}(\zeta) \geq F_{1}(\zeta)$ for all $\zeta$ , which implies $t_{2} < t$ . That is, the disadvantaged group faces a lower selection bar under group-specific quotas. + +As a result, agents in $G_{2}$ exert more effort on average under per-group capacities compared to the uniform- $c$ case. This is because effort is increasing in the probability of selection, which improves when the threshold is lowered. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: Contributions are shown in Sections 2, 3, and 4. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. +- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. +- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. + +# 2. Limitations + +Question: Does the paper discuss the limitations of the work performed by the authors? + +Answer: [Yes] + +Justification: See Section 5. + +Guidelines: + +- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. +- The authors are encouraged to create a separate "Limitations" section in their paper. +- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. +- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. +- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. +- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. +- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. +- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. 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The principal employs a hypothesis testing rule, aiming to pick a p-value threshold that balances false positives and false negatives while anticipating the agent's incentive to maximize expected profitability. Building on prior work, we develop a game-theoretic model that captures how the agent's participation and reporting behavior respond to the principal's statistical decision rule. Despite the complexity of the interaction, we show that the principal's errors exhibit clear monotonic behavior when segmented by an efficiently computable critical p-value threshold, leading to an interpretable characterization of their optimal p-value threshold. We empirically validate our model and these insights using publicly available data on drug approvals. Overall, our work offers a comprehensive perspective on strategic interactions within the hypothesis testing framework, providing technical and regulatory insights. + +# 1 Introduction + +In data-driven decision making, the outcome assigned to an agent often depends on data submitted by them, creating the potential for misaligned incentives. The role of strategic behavior in decision making systems has thus become a burgeoning area of research in the machine learning community, from classification [1, 2], to regression [3, 4, 5] and beyond [6]. However, limited literature exists on this perspective for a widely used and influential process in regulatory and scientific settings: hypothesis testing. Widely used in clinical trials, scientific research, and technological innovations [7, 8, 9, 10], hypothesis testing serves as a foundational method for assessing whether the evidence provided by a participating agent is both statistically significant and convincing. Consider, for instance, the U.S. Food and Drug Administration (FDA), which oversees drug approvals by setting a p-value threshold $\alpha$ for submitted clinical trials. Pharmaceutical firms incur substantial costs by participating in drug development and running trials in the hopes of being approved. Although falsifying results have high reputational and legal risks, firms are free to decide if to participate and how large to run a trial. We argue these decisions are intimately shaped by the FDA's p-value threshold and the cost-benefit calculus of the overall process. We thus propose a principal-agent model to understand these nuanced strategic decisions surrounding participation and evidence collection for a hypothesis test. + +Bates et al. [11] initiated this literature by casting how agents respond to regulatory approval thresholds as a Stackelberg game. We extend this framework to capture additional real-world complexities that are abstracted away in their model. Specifically, we extend the analysis beyond binary notions of effectiveness and fixed trial costs, allowing agents to strategically choose their trial size based on the expected benefit relative to marginal per-sample cost. This extension not only enriches the analysis of agent incentives but also provides a more nuanced understanding of how regulatory policies can influence participation and decision outcomes. This richer model, however, leads to a more complex relationship between the regulator-specified p-value and the resulting error rates that they wish to control – for instance, if marginal costs increase faster than the expected revenue gains, raising $\alpha$ could paradoxically reduce the likelihood of passing the hypothesis test. This is further complicated by the principal wishing to control any combination of Type I (false positive) and Type II (false negative) errors within our model. Indeed, while the broader literature focuses on only the former [11, 12, 13, 14], we argue that minimizing the rejection or non-participation of effective products is also consequential to decision-makers, especially when this is influenced by low revenue relative to costs, as is the case for low-cost and orphan drugs [15]. We outline our key contributions toward addressing these challenges below: + +1. Game-Theoretic Framework: Like Bates et al. [11], we consider a principal-agent (Stackelberg game) framework but relax several key assumptions to adhere more closely to real-world settings. We allow agents to strategically choose their sample sizes as part of their best responses and account for the marginal cost of such samples. Further, we model the principal as aiming to minimize any combination of Type I and II errors. +2. Analysis of Component-Wise Losses: We rigorously analyze the principal's loss components as functions of the p-value threshold under agent best-responding behavior. Despite the complex interplay between several factors, the monotonicity of Type I and Type II errors as a function of the p-value threshold $\alpha$ is preserved in a piecewise manner. Specifically, there exists a critical threshold $\hat{\alpha}$ such that the error dynamics are monotonic within each segmented region: one for $\alpha < \hat{\alpha}$ and one for $\alpha \geq \hat{\alpha}$ . +3. Empirical Validation and Policy Implications: We validate our theoretical findings through an empirical analysis of drug approvals for three major drug categories and show that under our model, the commonly used p-value of 0.05 aligns well with median revenue thresholds in these markets. The results suggest our model captures the economic dynamics of this process and suggests new policy insights. + +Additional Related Works: Our work is closely related to the literature on economic aspects of statistical testing, p-hacking, contract theory, and FDA regulatory policies. A growing body of research examines the economic incentives in statistical decision-making [12, 14, 16, 13, 17]. Shi et al. [13] extends the work of Bates et al. [11] (which is discussed above) by studying strategic hypothesis testing under general concave utility functions, providing bounds on the Bayes False Discovery Rate (FDR). Relatedly, Bates et al. [14] models the agent-principal interaction through a contract-theoretic lens for incentive alignment, differing from our Stackelberg game approach. Also related is the well-known issue of $p$ -hacking, where researchers manipulate sample sizes or selectively report findings to artificially achieve statistical significance [18, 9, 10, 19, 8]. Note that the strategic behavior in our setting is not inherently malicious, but rather reflects rational decision-making based on cost, revenue, and expected trial outcomes. A good $p$ -value threshold discourages agents with ineffective drugs from participating based on economic incentives: the costs of participation outweigh the potential benefits. This disincentives $p$ -hacking insofar as data collection is costly. Our work also intersects with the literature on contract theory, which examines incentive alignment in the presence of private information [20, 21, 22, 23, 24]. Recent work by Min [25] applies contract theory to model FDA approval processes, analyzing how firms of different sizes choose between cheaper and more expensive trials. Further, Isakov et al. [26] employs Bayesian Decision Analysis (BDA) to optimize $p$ -value thresholds by balancing Type I and Type II errors. Our empirical results leverage the statistical testing-based framework used by the FDA. Typically, this is established through either two successful controlled trials (with a $p$ -value $< 0.05$ ) or a single robust multicenter trial (with a $p$ -value $< 0.005$ ), as outlined in [7]. We simplify this in our analysis by only considering the former. + +# 2 Model + +Preliminaries: Consider a principal who must decide between approving or rejecting a product manufactured by an agent based on evidence provided by the agent. Let $X \in \{0,1\}$ be a Bernoulli + +random variable indicating whether the product was effective on a random instance. Let $\mu_0 = \mathbb{E}[X]$ denote the mean effectiveness of this product, with $\mu_0 \sim q$ . While $\mu_0$ is private (known only by the agent), the distribution $q$ is assumed public (known to both the principal and agent). Let $\mu_b$ denote a baseline effectiveness (i.e. effectiveness of current products) and be treated as a constant: the principal's goal is to only approve products perceived to be at least as good as the baseline ( $\mu_0 \geq \mu_b$ ), with $(\mu_0 - \mu_b)$ denoted as effect size. An agent faces two possible decisions for their product. First, they can choose whether or not to participate in the approval process and engage with the principal. Not participating, equivalent to $n = 0$ , incurs no cost and collects no revenue. If they participate, they must also decide on the number of samples $n \in [n_{min}, n_{max}]$ to collect and submit an evidence set $\mathcal{X}_n = (X_1, \ldots, X_n)$ to the principal. This incurs a fixed cost $c_0$ and a marginal per-sample cost $c$ . Thus, $\mathrm{cost}(n) = 1$ ( $n \neq 0$ ) ( $c_0 + cn$ ). If their product is approved, they earn revenue $R$ . Agents are assumed rational and act to maximize their expected utility - expected revenue minus cost (see Definition 2.2). We assume the revenue and cost parameters are known to the principal. + +Connecting this to our running example, the FDA (the principal) decides on the approval of new drugs from pharmaceutical companies (the agents) based on whether the clinical trial data suggests them to be at least as effective as current alternatives on the market. If a firm follows through on the development of a new drug and participates in clinical trials, $c_{0}$ denotes any fixed costs herein, and $c$ , the per-subject marginal cost of the trial. An FDA approval means a lifetime revenue $R$ for their new drug, while rejection means no revenue. As such, the firm only participates if its expected profit (utility) is non-negative, and chooses the samples $n$ to maximize this. + +Hypothesis Testing: The agent's decision clearly hinges on how the principal uses the evidence set to evaluate effectiveness. We model this process on hypothesis testing given its ubiquity and relevance in settings like drug approval, manufacturing, public policy, and so on [7, 8, 9, 10]. Formally, let $\mathcal{H}_0 = \{\mu_0 \leq \mu_b\}$ be the null hypothesis (the product is less effective than baseline) and denote the alternative as $\mathcal{H}_1 = \{\mu_0 > \mu_b\}$ . The principal uses a $p$ -value threshold $\alpha$ to reject the null hypothesis and approve the product. To expand, given an evidence set $\mathcal{X}_n$ , let $\hat{\mu}$ denote the empirical mean. Further, let $S_n$ denote the random sum of $n$ variables sampled from the baseline process with effectiveness $\mu_b$ . Then the $p$ -value is defined as the probability of observing outcomes at least as good as the evidence, conditioned on the null hypothesis: $p(\mu_0, n) = \operatorname*{Pr}[S_n \geq n \hat{\mu} | \mathcal{H}_0]^2$ . Observe that when the empirical mean of the evidence is worse than the baseline, the $p$ -value will likely be higher than $\alpha$ , leading the principal to reject; otherwise, when $p(\mu_0, n) \leq \alpha$ , the principal will accept. For an evidence set with $n$ samples and a $p$ -value $\alpha$ , the critical region is the number of successes needed to reject the null-hypothesis and thus be approved. Since the sample outcomes are Bernoulli and the sum random variable follows a Binomial distribution, it is common to use a normal distribution $\phi$ to approximate the $p$ -value and critical region: + +$$ +z _ {\alpha , n} \approx \left\{k \in \mathbb {R} \Big | \int_ {k} ^ {\infty} \phi (n \mu_ {b}, n \mu_ {b} (1 - \mu_ {b})) \leq \alpha \right\} \approx n \mu_ {b} + \Phi^ {- 1} (1 - \alpha) \sqrt {n \mu_ {b} (1 - \mu_ {b})} +$$ + +where $\Phi$ is the CDF of the standard normal and $\Phi^{-1}$ , its inverse. Formalizing the principal's decision-making process means that we can now compile the probability that a product with effectiveness $\mu_0$ is approved (the randomness is over the evidence set), and thereby the agent's utility: + +Definition 2.1 (Pass Probability). For a $p$ -value threshold $\alpha$ and baseline effectiveness $\mu_{b}$ , a product with effectiveness $\mu_{0}$ and evidence set $|\mathcal{X}_n| = n$ is approved with probability: + +$$ +P a s s (\alpha , \mu_ {0}, n) \triangleq \Pr [ p (\mu_ {0}, n) \leq \alpha ] \approx 1 - \Phi \left(\frac {z _ {\alpha , n} - n \mu_ {0}}{\sqrt {n \mu_ {0} (1 - \mu_ {0})}}\right) \tag {1} +$$ + +Non-participation is considered equivalent to $n = 0$ , and $\operatorname{Pass}(\alpha, \mu_0, n = 0) \triangleq 0$ , $\forall \mu_0, \alpha$ . + +Definition 2.2 (Agent Utility). An agent with revenue $R$ , cost parameters $(c_0, c)$ , and a product with effectiveness $\mu_0 \sim q$ has the following utility for their participation/sample parameter $n$ : + +$$ +u \left(\alpha , \mu_ {0}, n\right) = R \cdot \operatorname {P a s s} \left(\alpha , \mu_ {0}, n\right) - \mathbb {1} [ n \neq 0 ] \left(c _ {0} + c n\right) \tag {2} +$$ + +While the agent aims to maximize their utility, the principal's goal is to choose a p-value threshold that minimizes some combination of their Type I and Type II errors, a standard desideratum in + +hypothesis testing. In machine learning terminology, these are equivalent to minimizing the false positive (approving ineffective products) and false negative (not approving effective products) rates. These error components are defined with respect to the effectiveness distribution $q$ , and we note that the principal may desire an arbitrary trade-off between the two error components. This principal loss and its objective are shortly defined in Definition 2.3. + +Game Theoretic Model: It is evident that agents will choose $n$ strategically to maximize their utility, while the principal selects a p-value threshold $\alpha$ to minimize a loss function, which we will define in Definition 2.3. This naturally leads to a game-theoretic framework. In most regulatory settings (drug approval, manufacturing, etc), the principal must first communicate the acceptance criteria to all possible participants. Agents, on the other hand, can make their decision to participate and collect samples based on the revealed criteria. This interaction outlines a Stackelberg Game, an asymmetric model of strategic decision-making: the principal first commits to the p-value threshold, allowing agents to then make their optimal decision/best-response (participation decisions and number of samples) thereafter3. In such games, the core solution concept is the Stackelberg Equilibrium – the optimal principal strategy, given that downstream agents will best respond. We formally define the game and its details below: + +Definition 2.3 (Stackelberg Game in Strategic Hypothesis Testing). The principal-agent interactions in a hypothesis testing setting outline a Stackelberg game defined by the tuple $\mathcal{I} = (q,R,c,c_0,\mu_b)$ . The best-response of an agent with effectiveness $\mu_0\sim q$ when the principal commits to a $p$ -value threshold $\alpha$ is: $n_{\mu_0}(\alpha) = \arg \max_n u(\alpha ,\mu_0,n)$ . + +The principal in choosing a $p$ value threshold $\alpha$ suffers the following loss when agents best respond (we denote $\text{Fail}(\cdot) = 1 - \text{Pass}(\cdot)$ and $\lambda_{fp}, \lambda_{fn}$ are constants that scale the respective loss terms): + +$$ +\mathcal {L} (\alpha , I) = \lambda_ {f p} \underbrace {\mathbb {E} _ {\mu_ {0} \sim q} [ P a s s (\alpha , \mu_ {0} , n _ {\mu_ {0}} (\alpha)) | \mu_ {0} < \mu_ {b} ]} +$$ + +False Positive (Type I error) $\triangleq FP(\alpha, q)$ + +Approval of ineffective products $(\mu_0 < \mu_b)$ + +$$ ++ \lambda_ {f n} \underbrace {\mathbb {E} _ {\mu_ {0} \sim q} [ F a i l (\alpha , \mu_ {0} , n _ {\mu_ {0}} (\alpha)) | \mu_ {0} \geq \mu_ {b} ]} +$$ + +False Negative (Type II error) $\triangleq$ FN(α,q) + +Non-approval of effective products $(\mu_0\geq \mu_b)$ + +The principal optimal strategy and the Stackelberg Equilibrium is to choose $\alpha^{\star} = \arg \min_{\alpha}\mathcal{L}(\alpha ,\mathcal{I})$ + +In Section 3 and Section 4, we will investigate the agent's best response behavior as well as how the principal chooses p-value threshold affects the loss defined in Definition 2.3. + +# 3 Agent Best Response + +According to our strategic model, after the principal releases a $p$ -value threshold $\alpha$ , the utility-maximizing agent must decide (1) if they want to develop the product and participate in the approval process, and if so, (2) how many samples they ought to include in their submitted evidence set. Notice that this decision-making process is ex-ante and relies on the agent's belief in their product's effectiveness $\mu_0$ . Intuitively, the agent intends to determine the optimal number of samples $n$ to maximize their ex-ante expected utility; if this is negative, they have no incentive to participate. Mathematically, participation is defined by $\mathbb{1}\left[u(\alpha ,\mu_0,n_{\mu_0}(\alpha))\geq 0\right]$ , where $u(\alpha ,\mu_0,n)$ is defined in Definition 2.2 and $n_{\mu_0}(\alpha)$ is the optimal number of samples. We now show that this can be efficiently computed by the agent. We sketch the proof below, with the full proof in Appendix B + +Theorem 3.1. For an instance $\mathcal{I}$ and a released $p$ -value $\alpha$ , an agent with effectiveness $\mu_0$ can compute their best-response (participation decision and number of samples) in $O(\log n_{max})$ . + +Proof Sketch. By leveraging properties of the normal CDF that define the pass probability, we first show that when $\mu_0 < \mu_b$ , the optimal $n = n_{min}$ . For $\mu_0 \geq \mu_b$ , we undertake a first and second-order analysis to partition the utility function into a constant number of intervals wherein it is convex or concave; the boundaries of these regions can be computed in constant time, and the optimal $n$ within each region requires at-most a binary search. Once the optimal $n$ is computed, it suffices to compute the passing probability and participate if the corresponding expected utility is non-negative. + +We now prove that despite the agent being strategic and their participation and sampling decisions dynamically changing in both $\alpha$ and $\mu_0$ , several natural and intuitive results still hold. These are + +instrumental to the analyses of the principal's optimal/equilibrium p-value threshold. We first show that the participation behaviour is monotonic in $\alpha$ (Lemma 3.1). The full proof is in Appendix B. + +Lemma 3.1. For an instance $\mathcal{I} = (q,R,c,c_0,\mu_b)$ and $p$ -value threshold $\alpha$ , if an agent with effectiveness $\mu_0$ participates in the statistical test, then any agent with belief $\mu_1 \geq \mu_0$ will also participate. Similarly, if an agent with belief $\mu_0'$ does not participate, neither will one with belief $\mu_1' < \mu_0'$ . + +This monotonicity result immediately implies the existence of a participation threshold $\mu_{\tau}(\alpha)$ for any given p-value $\alpha$ . That is, for any instance $\mathcal{I}$ and p-value $\alpha$ , agents with effectiveness $\mu_0 \geq \mu_\tau (\alpha)$ will always participate, and those below will not. As we subsequently show, the participation threshold is a crucial concept in understanding the principal's decision - it determines the selection mechanism induced by the p-value threshold $\alpha$ , shaping the set of participating agents based on their effectiveness. Consequently, understanding how $\mu_{\tau}(\alpha)$ changes as a function of the instance parameters $R, c, \alpha$ provides key insights into optimizing the principal's objective function. We now show that the participation threshold decreases as the p-value threshold $\alpha$ increases; further, it can be computed in log time and is agnostic to the effectiveness distribution $q$ (full proof in Appendix B). + +Definition 3.1 (Participation Threshold). For an instance $\mathcal{I}$ and a $p$ -value $\alpha$ , we denote $\mu_{\tau}(\alpha)$ as the participation threshold if it is optimal for agents with effectiveness $\mu_0 \geq \mu_{\tau}(\alpha)$ to participate, and for those with $\mu_0 < \mu_{\tau}(\alpha)$ to abstain. + +Lemma 3.2. The participation threshold $\mu_{\tau}(\alpha)$ decreases in the $p$ -value $\alpha$ . Further, this threshold can be computed with $\varepsilon$ precision in $\mathcal{O}\left(\log \left(\frac{1}{\varepsilon}\right) \log (n_{max})\right)$ . + +Proof Sketch. Since the passing probability of the hypothesis test increases with $\alpha$ for a fixed effect size, agents with lower beliefs are more likely to meet the passing condition as $\alpha$ increases. By the monotonicity of participation (Lemma 3.1), if an agent participates at $\mu_{\tau}(\alpha_1)$ under a p-value $\alpha_{2} > \alpha_{1}$ , then agents with beliefs greater than $\mu_{\tau}(\alpha_{1})$ must also participate. This implies that the participation threshold under $\alpha_{2}$ must be at or below the threshold under $\alpha_{1}$ , establishing that $\mu_{\tau}(\alpha)$ is non-increasing in $\alpha$ . To determine $\mu_{\tau}(\alpha)$ , we can discretize the effectiveness space into $\frac{1}{\varepsilon}$ intervals and run binary search. The algorithm and correctness rely on Lemma 3.1 and Theorem 3.1. + +This formalizes the intuition that even under strategic sampling and participation decisions, as the principal increases the p-value threshold, namely when it's "easier" to pass the statistical test, agents with lower effectiveness (who can still be better than baseline) will find it beneficial to participate in the approval process. The monotonicity of the agent participation decision also means that the agent's utility is increasing in $\mu_0$ . This is formalized below, with the proof in Appendix B. + +Lemma 3.3. For any given $p$ -value $\alpha$ , the agent's utility under optimal number of samples $u(\alpha, \mu_0, n_{\mu_0}(\alpha))$ increases in their effectiveness $\mu_0$ . + +# 4 Principal Equilibrium and Loss Dynamics + +The principal's strategic choice is to set the $p$ -value threshold $\alpha$ . Given the agent's best response behavior, how should the principal select $\alpha$ to minimize their loss, as defined in Definition 2.3? The loss captures the trade-off between two risks: approving ineffective products (false positives) and failing to approve effective ones (false negatives). A product can fail to be approved either by explicit rejection after participating in the approval process, or by the agents opting not to participate. Since the $p$ -value threshold $\alpha$ influences the participation decision (Lemma 3.2), this becomes an implicit factor in the loss function. The principal's challenge is to set $\alpha$ optimally: a low $\alpha$ reduces false positives but may discourage worthy candidates from participation, increasing false negatives; conversely, a high $\alpha$ boosts participation but raises the risk of approving ineffective products. Optimizing $\alpha$ requires a careful balance between these risks while accounting for agents' strategic behaviour. + +We begin by re-expressing the cumulative loss conditioned on the participation decision induced by $\alpha$ for a given agent, a decision entirely captured by the condition $\mu_0 \geq \mu_\tau(\alpha)$ : + +$$ +\begin{array}{l} \mathcal {L} (\alpha , \mathcal {I}) = \lambda_ {f p} \underset {\mu_ {0} \sim q} {\mathbb {E}} [ \text {P a s s} (\alpha , \mu_ {0}, n _ {\mu_ {0}} (\alpha)) | \mu_ {0} < \mu_ {b}, \mu_ {0} \geq \mu_ {\tau} (\alpha) ] P (\mu_ {0} \geq \mu_ {\tau} (\alpha) | \mu_ {0} \leq \mu_ {b}) \quad (\mathrm {F P} _ {\text {p a r t i c i p}}) \\ + \lambda_ {f p} \underset {\mu_ {0} \sim q} {\mathbb {E}} [ \operatorname {P a s s} (\alpha , \mu_ {0}, n _ {\mu_ {0}} (\alpha)) | \mu_ {0} < \mu_ {b}, \mu_ {0} < \mu_ {\tau} (\alpha) ] P (\mu_ {0} < \mu_ {\tau} (\alpha) | \mu_ {0} \leq \mu_ {b}) \quad (\mathrm {F P} _ {\text {a b s t a i n}}) \\ + \lambda_ {f n} \underset {\mu_ {0} \sim q} {\mathbb {E}} [ \text {F a i l} (\alpha , \mu_ {0}, n _ {\mu_ {0}} (\alpha)) | \mu_ {0} \geq \mu_ {b}, \mu_ {0} \geq \mu_ {\tau} (\alpha) ] P (\mu_ {0} \geq \mu_ {\tau} (\alpha) | \mu_ {0} \geq \mu_ {b}) \quad (\mathrm {F N} _ {\text {p a r t i c i p}}) \\ \end{array} +$$ + +$$ +\begin{array}{l} + \lambda_ {f n} \underset {\mu_ {0} \sim q} {\mathbb {E}} [ \text {F a i l} (\alpha , \mu_ {0}, n _ {\mu_ {0}} (\alpha)) | \mu_ {0} \geq \mu_ {b}, \mu_ {0} < \mu_ {\tau} (\alpha) ] P (\mu_ {0} < \mu_ {\tau} (\alpha) | \mu_ {0} \geq \mu_ {b}) \quad (\mathrm {F N} _ {\text {a b s t a i n}}) \\ = \mathrm {F P} _ {\text {p a r t i c i p}} (\alpha , \mathcal {I}) + \mathrm {F P} _ {\text {a b s t a i n}} (\alpha , \mathcal {I}) + \mathrm {F N} _ {\text {p a r t i c i p}} (\alpha , \mathcal {I}) + \mathrm {F N} _ {\text {a b s t a i n}} (\alpha , \mathcal {I}) \\ \end{array} +$$ + +Note that the overall false positive $\mathrm{FP}(\alpha, \mathcal{I}) = \mathrm{FP}_{\mathrm{particip}}(\alpha, \mathcal{I}) + \mathrm{FP}_{\mathrm{abstain}}(\alpha, \mathcal{I})$ and similarly, the overall false negative $\mathrm{FN}(\alpha, \mathcal{I}) = \mathrm{FN}_{\mathrm{particip}}(\alpha, \mathcal{I}) + \mathrm{FN}_{\mathrm{abstain}}(\alpha, \mathcal{I})$ . Further, observe that when the agent does not participate, their probability of passing the test is 0 - therefore, $\mathrm{FP}_{\mathrm{abstain}} = 0$ at all times. Thus, the cumulative loss is fully specified by $\mathcal{L}(\alpha, \mathcal{I}) = \mathrm{FP}_{\mathrm{particip}} + \mathrm{FN}_{\mathrm{particip}} + \mathrm{FN}_{\mathrm{abstain}}$ . We will now analyze the properties of these components as a function of $\alpha$ . Remarkably, despite the complex interplay between the agent's strategic participation and the principal's decision, the loss terms exhibit consistent monotonic behavior within regions segmented by a critical threshold $\widehat{\alpha}$ - intuitively, this is the p-value wherein the participation threshold $\mu_{\tau}(\widehat{\alpha}) = \mu_{b}$ (see Definition 4.1). + +Definition 4.1 (Critical p-value $\widehat{\alpha}$ ). The critical $p$ -value threshold $\widehat{\alpha}$ is the $p$ -value at which the participation threshold (Definition 3.1) is equal to the baseline effectiveness $-\mu_{\tau}(\widehat{\alpha}) = \mu_b$ . This quantity is agnostic to the effectiveness distribution $q$ and scaling parameters $(\lambda_{fp}, \lambda_{fn})$ . + +Before our detailed analysis, we highlight that the forthcoming results depend on a core observation – for effective agents $(\mu_0 \geq \mu_b)$ , the probability of passing the statistical test increases as the p-value $\alpha$ increases. While this is intuitive when the participation and the number of samples used $(n)$ are fixed, when agents are strategic and dynamically change these decisions, it is trickier. For example, if an increased $\alpha$ led the agent to use fewer samples due to high marginal cost relative to the increase in passing probability and thus revenue (this is indeed common as the p-value becomes high), the pass probability could decrease. The lemma below highlights that despite the complicated dynamics of optimal samples, $n_{\mu_0}(\alpha)$ , the passing probability is always non-decreasing in $\alpha$ . The proof is technical and appears in Appendix C, but we sketch the key ideas below. + +Lemma 4.1. For an instance $\mathcal{I}$ and an agent with effectiveness $\mu_0 \geq \mu_b$ , the probability of passing - Pass $(\alpha, \mu_0, n_{\mu_0}(\alpha))$ - is non-decreasing in $\alpha$ when the agent participates and uses $n_{\mu_0}(\alpha)$ samples. + +Proof Sketch. We prove that for $\mu_0\geq \mu_b$ , we have $\frac{\partial}{\partial\alpha}\mathrm{Pass}(\alpha ,\mu_0,n_{\mu_0}(\alpha)\geq 0.$ The total effect is decomposed into a direct effect, capturing the change in pass probability purely due to $\alpha$ , and an indirect effect, which accounts for changes in the agent's optimal sample size $n_{\mu_0}(\alpha)$ . For the direct effect, an increase in $\alpha$ reduces the critical threshold $\Phi^{-1}(1 - \alpha)$ , making it easier for agents to pass the test. For the indirect effect, the agent adjusts their optimal sample size $n_{\mu_0}(\alpha)$ based on the value of $\alpha$ . We compute $\frac{\partial\mathrm{Pass}}{\partial n}$ and apply the Implicit Function Theorem to determine $\frac{dn(\alpha)}{d\alpha}$ which incorporates the agent's optimal behavior. The second derivative of the pass probability with respect to $n$ is shown to be negative, ensuring the utility is concave with respect to sample size. This concavity guarantees a unique optimal sample size $n_{\mu_0}(\alpha)$ , which varies predictably with $\alpha$ . Finally, we show that the combination of the direct and indirect effects results in the total derivative being non-negative, confirming that the pass probability is monotonic with respect to $\alpha$ + +# 4.1 False Positive Loss + +Since $\mathrm{FP}_{\mathrm{abstain}} = 0$ (non-participation means the probability of passing is 0), the overall false positive is purely determined by $\mathrm{FP}_{\mathrm{particip}}$ . We now show below that for $\alpha \leq \widehat{\alpha}$ , $\mathrm{FP}_{\mathrm{particip}} = 0$ , while for $\alpha > \widehat{\alpha}$ , this is increasing. We set the scaling factor $\lambda_{fp} = 1$ since it does not affect the analysis. + +Theorem 4.1. The overall false positive loss, $FP(\alpha, \mathcal{I}) = FP_{\text{participation}}(\alpha, \mathcal{I})$ . Further, this quantity is 0 for $\alpha < \widehat{\alpha}$ , and non-decreasing in $\alpha$ for $\alpha \geq \widehat{\alpha}$ . + +Proof. It suffices to prove the property for $\mathrm{FP}_{\mathrm{partic}}$ . First, consider an $\alpha < \widehat{\alpha}$ . We know from Lemma 3.1 that the participation threshold is decreasing in $\alpha$ . Thus, since $\mu_{\tau}(\widehat{\alpha}) = \mu_b$ , it means that our given $\alpha$ , $\mu_{\tau}(\alpha) > \mu_b$ . In other words, any agents participating under this p-value are effective (above baseline) and do not contribute to the false positive. + +Next, consider $\alpha \geq \widehat{\alpha}$ . We know from above that the participation threshold now is below $\mu_{b}$ . Observe that for any $\alpha$ , we can write the false positive rate as follows: + +$$ +\mathrm {F P} _ {\text {p a r t i c i p}} = \frac {P [ \mu_ {\tau} (\alpha) \leq \mu_ {0} \leq \mu_ {b} ]}{P [ \mu_ {0} \leq \mu_ {b} ]} \int_ {\mu_ {\tau} (\alpha)} ^ {\mu_ {b}} \operatorname {P a s s} (\alpha , \mu_ {0}, n _ {\mu_ {0}} (\alpha)) P [ \mu_ {0} | \mu_ {\tau} (\alpha) \leq \mu_ {0} \leq \mu_ {b} ] d \mu_ {0} +$$ + +$$ += \frac {1}{P [ \mu_ {0} \leq \mu_ {b} ]} \int_ {\mu_ {\tau} (\alpha)} ^ {\mu_ {b}} \operatorname {P a s s} (\alpha , \mu_ {0}, n _ {\mu_ {0}} (\alpha)) q (\mu_ {0}) d \mu_ {0} +$$ + +Note that $\frac{1}{P[\mu_0\leq\mu_b]}$ is a constant and as $\alpha$ increases to $\alpha^{\prime}$ , $\mu_{\tau}(\alpha^{\prime}) < \mu_{\tau}(\alpha)$ , meaning we integrate over a larger region. The integrand itself is always positive, and for every $\mu_0\geq \max (\mu_\tau (\alpha),\mu_\tau (\alpha ')) = \mu_\tau (\alpha)$ , the pass probability has increased (Lemma 4.1). Thus, $\mathrm{FP}_{\mathrm{particip}}$ is non-decreasing in $\alpha$ . + +# 4.2 False Negative Loss + +We next consider the components of the false negative loss: $\mathrm{FN}_{\mathrm{partic}}$ and $\mathrm{FN}_{\mathrm{abstain}}$ . We first establish the monotonicity properties of $\mathrm{FN}_{\mathrm{abstain}}$ , which is exactly equal to the proportion of effective agents not participating since not participating means that $\mathrm{Pass}(\cdot) = 0$ , implying $\mathrm{Fail}(\cdot) = 1$ . In other words, $\mathrm{FN}_{\mathrm{abstain}} = P(\mu_0 < \mu_\tau(\alpha)|\mu_0 \geq \mu_b)$ . Once more, segmenting the p-value space by $\widehat{\alpha}$ is crucial to establishing this relationship. As before, we set $\lambda_{fn} = 1$ since it does not affect the analysis. + +Proposition 4.1. The non-participation false negative loss, $FN_{abstain}(\alpha, I)$ is non-increasing in $\alpha$ for $\alpha \leq \widehat{\alpha}$ , and is $0$ for any $\alpha \geq \widehat{\alpha}$ . + +Proof. Consider $\alpha_{1},\alpha_{2}$ where $\alpha_{2}\geq \alpha_{1}$ , where both $\alpha_{1},\alpha_{2}\leq \widehat{\alpha}$ . Thus, we first need to show that: $P(\mu_0 < \mu_\tau (\alpha_1)|\mu_0\geq \mu_b)\geq P(\mu_0 < \mu_\tau (\alpha_2)|\mu_0\geq \mu_b)$ . We know from the monotonicity of the threshold belief (Lemma 3.1), $\mu_{\tau}(\alpha_{2})\leq \mu_{\tau}(\alpha_{1})$ . Let $Q_{\geq \mu_b}$ denote the CDF of the effectiveness distribution, conditioned on the event $\mu_0\geq \mu_b$ . Then $Q_{\geq \mu_b}(\mu_\tau (\alpha)) = P[\mu_0\leq \mu_\tau (\alpha)|\mu_0\geq \mu_b]$ . It is then immediate that $Q_{\geq \mu_b}(\mu_\tau (\alpha_2))\leq Q_{\geq \mu_b}(\mu_\tau (\alpha_1))$ . Lastly, for $\alpha \geq \widehat{\alpha}$ , we note that by definition of $\widehat{\alpha}$ and Lemma 3.1: $\mu_{\tau}(\alpha)\leq \mu_b$ . In other words, it is optimal for all effective agents to participate, meaning FN abstain $= 0$ in this regime. + +One might expect to show a similar result for $\mathrm{FN}_{\mathrm{particip}}$ and thereby conclude the cumulative behavior of the false negative loss. Indeed, when $\alpha \geq \widehat{\alpha}$ , the participation threshold is smaller than the baseline $\mu_{\tau}(\alpha) < \mu_{\tau}(\widehat{\alpha}) = \mu_b$ , which means all agents whose prior is greater than $\mu_b$ will participate, thus $\operatorname*{Pr}[\mu_0 \geq \max(\mu_{\tau}(\alpha), \mu_b) | \mu_0 \geq \mu_b] = 1$ . The FN under participation loss simplifies to + +$$ +\text {f o r} \alpha \geq \widehat {\alpha}: \quad \mathrm {F N} _ {\text {p a r t i c i p}} (\alpha , I) = \underset {\mu_ {0} \sim Q} {\mathbb {E}} [ 1 - \operatorname {P a s s} (\alpha , \mu_ {0}, n _ {\mu_ {0}} (\alpha)) | \mu_ {0} \geq \mu_ {b} ] \tag {3} +$$ + +Following directly from the monotonicity of the passing probability (Lemma 4.1), this implies that $\mathrm{FN}_{\mathrm{particip}}$ decreases as a function of $\alpha$ when $\alpha \geq \widehat{\alpha}$ . We formally state this below: + +Proposition 4.2. $FN_{\text{particip}}(\alpha, I)$ is non-increasing in $\alpha$ for all $\alpha \geq \widehat{\alpha}$ . + +When $\alpha < \widehat{\alpha}$ , however, $\mathrm{FN}_{\mathrm{particip}}$ displays a much more nuanced behaviour – indeed, in Section 5, we show that this component is not monotonic and can be increasing, decreasing, or both in the $\alpha \leq \widehat{\alpha}$ region. This may suggest that, unlike the false positive setting, little can be said about the combined false negative effect ( $\mathrm{FN}_{\mathrm{particip}} + \mathrm{FN}_{\mathrm{abstain}}$ ). As we show in Theorem 4.2, however, this is not true, and we directly establish the monotonicity of the combined false negative rate when the p-value domain is segmented by $\widehat{\alpha}$ . The result is technical and the full proof is given in Appendix C. + +Theorem 4.2. The overall false negative loss, $FN(\alpha, \mathcal{I}) = FN_{\text{participation}}(\alpha, \mathcal{I}) + FN_{\text{abstain}}(\alpha, \mathcal{I})$ , is non-increasing in $\alpha$ on both sides of $\widehat{\alpha}$ , that is, it is not increasing for $\alpha \leq \widehat{\alpha}$ , and for all $\alpha \geq \widehat{\alpha}$ . + +# 4.3 Characterizing $\alpha^{*}$ - The Optimal p-value Threshold + +
FPparticipFPabstainOverall FPFNparticipFNabstainOverall FN
α ≤ α̂000-Non-IncNon-Inc
α > α̂Non-Dec0Non-DecNon-Inc0Non-Inc
+ +Table 1: Summary of the loss dynamics as $\alpha$ increases. $\widehat{\alpha}$ is defined in Definition 4.1. We comment on ${\mathrm{{FN}}}_{\text{particip }}$ in the $\alpha \leq \widehat{\alpha }$ regime in Section 5. See figure 1 for this dynamic visualized on an instance. + +We summarize our analysis of the different loss components in the table above. These results lead to a succinct characterization of how the principal should use their equilibrium $\alpha^{*}$ . Observe that for any $\alpha \leq \widehat{\alpha}$ , the overall false positive is 0, while the overall false negative rate is decreasing in $\alpha$ . This + +immediately implies that $\widehat{\alpha}$ is weakly better than any $\alpha \leq \widehat{\alpha}$ , and that $\alpha^{*}$ must be at least as large as $\widehat{\alpha}$ ! This observation and the value of $\widehat{\alpha}$ are agnostic to both the effectiveness distribution $q$ and the loss scaling constants $\lambda_{fp}, \lambda_{fn}$ . This is a powerful characterization. While the true optimal $\alpha^{*}$ may be larger, the principal suffers additional false positives here while lowering the false negatives. In settings like drug approvals and manufacturing, where the cost of false-positives is very high - $\lambda_{fp} \gg \lambda_{fn}$ - the optimal may indeed be close to $\widehat{\alpha}$ . + +In recommending values close to $\widehat{\alpha}$ as optimal, the question of computation arises. Fortunately, since by definition at $\widehat{\alpha}$ , $\mu_{\tau}(\widehat{\alpha}) = \mu_b$ , and the participation threshold $\mu_{\tau}(\alpha)$ can be computed to $\varepsilon$ accuracy in $\mathcal{O}(\log \frac{1}{\varepsilon} \log (n_{max}))$ time, an $\widehat{\alpha}$ can be efficiently computed to $\varepsilon$ accuracy in $\mathcal{O}(\log^2 \frac{1}{\varepsilon} \log (n_{max}))$ . This follows from discretizing the p-value domain into $\frac{1}{\varepsilon}$ intervals and running binary search, leveraging the monotonicity of the participation threshold Lemma 3.2. + +Theorem 4.3. For any instance $\mathcal{I}$ , the principal's optimal $p$ -value $\alpha^{*}$ satisfies $\alpha^{*} \geq \widehat{\alpha}$ , where $\widehat{\alpha}|\mu_{\tau}(\widehat{\alpha}) = \mu_b$ . Further, an $\varepsilon$ approximation to an $\widehat{\alpha}$ such that $\mu_{\tau}(\widehat{\alpha}) \in [\mu_b \pm \varepsilon]$ can be computed in $\mathcal{O}(\log^2 \frac{1}{\varepsilon} \log (n_{max}))$ . + +# 5 Experimental Studies + +![](images/70c29bc75eaf80621fd23b585028aec6ad8d876a3c86e8280b25ec681e60c68b.jpg) +$\mathrm{FN}_{\mathrm{particip}}$ losses for different distributions +Figure 1: On the left, we plot the $\mathrm{FN}_{\mathrm{particip}}$ losses for different effectiveness distributions $q$ . On the right, we plot all the different loss components for one of these distributions with $\lambda_{fp} = \lambda_{fn} = 1$ + +![](images/ac2d2a524e4a73f1954c48a6932e52c8833f5b52aca2b8f377593c8b30d5682f.jpg) +Loss components for $q = \mathrm{TruncNorm}(0.62, 0.04)$ + +# 5.1 False Negative Rate Among Participating Agents + +Our technical analysis in the preceding section accounts for all components of the loss except one: $\mathrm{FN}_{\mathrm{particip}}$ for $\alpha \leq \widehat{\alpha}$ . For a principal unaware of the strategic implications of their chosen significance level $(\alpha)$ —particularly that it may discourage participation by agents with above-baseline effectiveness—this may be the only false negative they observe. At first glance, it seems increasing $\alpha$ should reduce this: the test becomes easier to pass, so effective agents should be more likely to succeed. Theorem 4.2 also aligns with this intuition since the overall FN rate decreases with $\alpha$ . + +However, $\mathrm{FN}_{\mathrm{partic}}$ behaves more intricately. We highlight this with an example. Consider $\mu_{b} = 0.5$ and a truncated normal distribution $q$ between 0.4 and 0.7 with standard deviation 0.04. First, suppose the mean of $q$ is 0.53, meaning a large mass of effective agents are only slightly better than baseline. As $\alpha$ increases from $\alpha_{1}$ to $\alpha_{2}$ , the participation threshold falls (by Lemma 3.1), causing a large mass of these slightly-effective agents to switch from abstaining to participating. However, due to their small effect size and samples being costly, these new participants pass with a lower probability than those who also participated under $\alpha_{1}$ . Their inclusion in the FN metric at $\alpha_{2}$ outweighs increased pass probability among previously participating agents, increasing the FN rate under $\alpha_{2}$ . + +This effect is not monotonic and depends on the distribution. If $q$ 's mean rises to 0.62, the same initial increase in FN rate occurs in the small $\alpha$ regime. For higher values of $\alpha$ , however, most effective agents are already participating, and few agents remain that are both slightly better than baseline and could switch; as $\alpha$ increases, the effect is dominated by the increasing pass probability of those already participating, decreasing the FNparticip. As shown in Figure 1, FNparticip increases and + +then decreases. In selecting a higher mean (0.67), $\mathrm{FN}_{\mathrm{particip}}$ is always decreasing. Overall, this rich behavior is fundamentally due to participation and evidence set size being a strategic decision. + +# 5.2 Analysis of $\widehat{\alpha}$ in Drug Approvals + +We now conduct a case study of our strategic hypothesis testing model by considering our running example: drug approvals. Despite much of the data in this setting being proprietary, we use public sources to capture relevant metrics for three classes of drugs: oncology, vaccines, and cardiovascular. ProRelix [27] mentions the per-participant expense (c) of vaccine testing to be $\sim$ 50,000, while oncology and cardiovascular trials being around $\sim$ 128,000 and $\sim$ 136,000 respectively. The fixed expenditure $(c_0)$ for clinical trials, regulatory approvals, and R&D, as well as the lifetime revenue ( $R$ ) vary widely, even within drug categories. Prasad and Mailankody [28] highlights the median fixed expenditure to be around $650 million for oncology drugs, although in the extremes it can be as low as$ 150 million or higher than a billion. The median four year revenue to be around 1.6 Billion; data on lifetime revenue is limited, but [29, 30] suggest it can be around 10-15 billion, with blockbuster drugs above 50 billion [14]. Analysis from Bhatt et al. [16] gives ranges of $74 - $183 million for fixed costs of Cardiovascular drugs with a median of $141 million. Rashid and Chandel [31] suggests the corresponding revenue of approved drugs here to be between under a billion to over 10 billion, with a median of around $3.5 billion. For vaccines, Sertkaya et al. [32] outlines revenues between 6.9 billion to 36.9 billion for blockbusters, with median fixed costs around $886 million. In the extremes, it can be below $100 million or above a billion. We present this rough data in Section 5.2. + +
Drug CategoryRevenue if approved (R)Fixed Cost (c0)Cost Per Sample (c)
Oncology[1,500-50,000](648); [150-1,000]0.136
Cardiovascular(3,560); [1,000-10,000](141); [74-183]0.128
Vaccine[6,900-36,900](886); [100-1,000]0.05
+ +Table 2: Rough costs and revenue by drug category. The median value, where available, is presented in parentheses. All numbers in millions USD, for US-based development. + +![](images/f2ba5dc8c84efc54a35519ad7302a4b53f03180c774c5361bc7f463e19cef0d1.jpg) +Figure 2: $\hat{\alpha}$ vs Revenue - Oncology + +![](images/50642ebd5256d25a15db1e70aaa6624bfe166760d666c984252b783b40c11c70.jpg) +Figure 3: $\hat{\alpha}$ vs Revenue - Cardiovascular + +Our goal is to understand what $\widehat{\alpha}$ , the critical p-value that weakly dominates any lower p-value, looks like in the drug approval setting. Due to the variability of fixed costs $c_{0}$ and revenue $R$ as compared to per-sample-cost $c$ , we fix $c$ and plot $\widehat{\alpha}$ for different revenue and fixed costs. These ranges are centered around the median values (where available) gathered in the table. We plot the figures for oncology and cardiovascular categories above (see Figure 2 and 3); the vaccines figure is in Appendix E). Each plot also displays the $\alpha = 0.05$ boundary, a commonly used p-value threshold by the FDA [34]. + +The results suggest that our strategic model is roughly capturing the economic dynamics of drug approvals. The in-practice used p-value of 0.05 is the ideal choice at around the median revenue for + +each drug category. Within our leader-follower dynamic, the FDA is committing to this threshold, first suggests that pharmaceuticals are choosing their parameters (samples collected, price, etc) to ensure profitability. Conversely, if regulatory bodies wish for decreased drug prices with the fixed costs being immutable, they ought to specify a higher p-value threshold, as the current value may dissuade effective but low-revenue drugs. In oncology, for instance, the critical threshold $\hat{\alpha}$ for around $5 billion revenue is between 0.1 and 0.2. Given our analysis, such a value does not induce any more false positives than more stringent ones (Theorem 4.1), but lowers the false negative rate. However, one may argue that choosing a higher p-value would imply that those with much higher revenue (say 25 billion) will be approved even when they are ineffective since their $\hat{\alpha}$ is lower. Practically, this is unlikely to happen since it is rare that an ineffective drug, albeit FDA-approved, would achieve such blockbuster revenue, as the market and post-market surveillance would generally expose inefficacy. Conversely, choosing p-values stricter than $\hat{\alpha}$ hamstrings low-revenue but effective drugs; this is a real concern for orphan drugs or low-cost therapeutics where margins are much slimmer. + +# 6 Discussion + +This work spiritually extends the growing literature on strategic machine learning to hypothesis testing, a ubiquitous method used in regulatory approvals. Given the costs of data collection, firms may strategically choose smaller trials or opt out entirely if the expected revenue, shaped by approval probabilities under the chosen threshold, does not justify the expense. By modeling this dynamic within a principal-agent framework, our work uncovers a systematic connection between the regulator-specified $p$ -value and the resulting firm decision. This allows us to give an interpretable characterization of the regulator's optimal $p$ -value, which minimizes false positive and negative error rates under such strategic consideration. We validate the findings of our model with real cost, revenue, and approval data from the US pharmaceutical sector and derive two key policy insights. First, the commonly used FDA $p$ -value of 0.05 is fairly strict given development costs, requiring firms to extract large revenue from approved drugs for economic viability. This matches empirical evidence from the US pharmaceutical market. Second, to decrease drug costs while maintaining minimal error rates, regulators can either selectively increase the $p$ -value threshold or give subsidies for running trials; our model gives guidance on how to adjust either lever. + +Our main results rely on all parties knowing the cost and revenue parameters $(c_0, c, R)$ - this determines agent participation and the principal's optimal threshold $\hat{\alpha}^5$ . While these quantities may in practice be estimated, our qualitative conclusions are robust to small estimation errors. In Appendix D, we discuss how small perturbations in the cost and revenue parameters affect the agents' participation thresholds and the principal's optimal choice of $\hat{\alpha}$ , and argue that these perturbations have only a limited impact and leave the qualitative behavior of the model unchanged. + +# Acknowledgments and Disclosure of Funding + +Yiling Chen was partially supported by the National Science Foundation under grant IIS 2147187. + +# References + +[1] Moritz Hardt, Nimrod Megiddo, Christos Papadimitriou, and Mary Wootters. Strategic classification. In Proceedings of the 2016 ACM conference on innovations in theoretical computer science, pages 111-122, 2016. +[2] Jinshuo Dong, Aaron Roth, Zachary Schutzman, Bo Waggoner, and Zhiwei Steven Wu. Strategic classification from revealed preferences. In Proceedings of the 2018 ACM Conference on Economics and Computation, pages 55-70, 2018. +[3] Yiling Chen, Chara Podimata, Ariel D Procaccia, and Nisarg Shah. Strategyproof linear regression in high dimensions. In Proceedings of the 2018 ACM Conference on Economics and Computation, pages 9-26, 2018. +[4] Yatong Chen, Jialu Wang, and Yang Liu. Linear classifiers that encourage constructive adaptation. arXiv preprint arXiv:2011.00355, 2020. + +[5] Safwan Hossain and Nisarg Shah. The effect of strategic noise in linear regression. Autonomous Agents and Multi-Agent Systems, 35(2):21, 2021. +[6] Keegan Harris, Chara Podimata, and Steven Z Wu. Strategic apple tasting. Advances in Neural Information Processing Systems, 36:79918-79945, 2023. +[7] Demonstrating substantial evidence of effectiveness for human drug and biological products. Technical report, U.S. Food and Drug Administration, December 2019. URL https://www.fda.gov/regulatory-information/search-fda-guidance-documents/demonstrating-substantial-evidence-effectiveness-human-drug-and-biological-products. Draft Guidance for Industry. +[8] Brian Wansink. The food scientist who manipulated data to make headlines. Axios, 2018. URL https://www.axes.com/2018/02/27/cornell-scientist-wansink-data-false-positive-headlines. +[9] Abel Brodeur, Scott Carrell, David Figlio, and Lester Lusher. Unpacking p-hacking and publication bias. American Economic Review, 113(11):2974-3002, 2023. doi: 10.1257/aer.20210795. URL https://www.eaaweb.org/articles?id=10.1257/aer.20210795. +[10] Angelika M. Stefan and Felix D. Schonbrodt. Big little lies: A compendium and simulation of p-hacking strategies. Royal Society Open Science, 10(2):220346, 2023. doi: 10.1098/rsos.220346. URL https://royalsocietypublishing.org/doi/abs/10.1098/rsos.220346. +[11] Stephen Bates, Michael I Jordan, Michael Sklar, and Jake A Soloff. Incentive-theoretic bayesian inference for collaborative science. arXiv preprint arXiv:2307.03748, 2023. +[12] Aleksey Tetenov. An economic theory of statistical testing. CeMMAP Working Paper CWP50/16, Institute for Fiscal Studies, 2016. URL https://ideas.repec.org/p/azt/cemmap/50-16.html. +[13] Flora C. Shi, Stephen Bates, and Martin J. Wainwright. Sharp results for hypothesis testing with risk-sensitive agents, 2024. URL https://arxiv.org/abs/2412.16452. +[14] Stephen Bates, Michael I. Jordan, Michael Sklar, and Jake A. Soloff. Principal-agent hypothesis testing, 2022. arXiv preprint arXiv:2205.06812. +[15] U.S. Food and Drug Administration. Multiple endpoints in clinical trials: Guidance for industry. Technical report, U.S. Food and Drug Administration, October 2022. URL https://www.fda.gov/media/162416/download. Accessed: 2025-05-16. +[16] Deepak L. Bhatt, Christopher P. Cannon, Roxana Mehran, Gregg W. Stone, Muthiah Vaduganathan, Marc S. Sabatine, David J. Moliterno, Robert M. Califf, and Eugene Braunwald. The imperative to enhance cost-effectiveness for cardiovascular clinical trials. JACC: Basic to Translational Science, 9(8):1029–1040, 2024. doi: 10.1016/j.jacbts.2023.12.005. URL https://pmc.ncbi.nlm.nih.gov/articles/PMC11405807/. +[17] Andrew McClellan. Experimentation and approval mechanisms. *Econometrica*, 90(5):2215-2247, 2022. +[18] Megan L. Head, Luke Holman, Rob Lanfear, Andrew T. Kahn, and Michael D. Jennions. The extent and consequences of p-hacking in science. PLOS Biology, 13(3):e1002106, 2015. doi: 10.1371/journal.pbio.1002106. URL https://journals.plos.org/posbiology/article?id=10.1371/journal.pbio.1002106. +[19] Jonas Moss and Riccardo De Bin. Modelling publication bias and p-hacking. arXiv preprint arXiv:1911.12445, 2021. URL https://arxiv.org/abs/1911.12445. +[20] Oliver Hart and Bengt Holmström. The theory of contracts. In Truman F. Bewley, editor, Advances in Economic Theory: Fifth World Congress, pages 71-156. Cambridge University Press, Cambridge, 1987. URL https://www.cambridge.org/core/services/aop-cambridge-core/content/view/19C16ACE4A35CB03AA5B4C89E69C20FE/9781139052054c03_p71-156_CBO.pdf/theory_of Contracts.pdf. +[21] Jean-Jacques Laffont and David Martimort. The Theory of Incentives: The Principal-Agent Model. Princeton University Press, Princeton, NJ, 2002. ISBN 9780691091846. URL https://www.amazon.com/Theory-Incentives-Principal-Agent-Model/dp/0691091846. +[22] Bernard Salanié. The Economics of Contracts: A Primer. MIT Press, Cambridge, MA, 2nd edition, 2005. ISBN 9780262195256. URL https://mitpress.mit.edu/9780262195256/the-economics-of-contracts/. + +[23] Rafael Frongillo and Bo Waggoner. Recent trends in information elicitation. ACM SIGecom Exchanges, 22(1):4-18, 2023. URL https://sigecom.org/exchanges/volume_22/1/FRONGILLO.pdf. +[24] David Danz, Lise Vesterlund, and Alistair J. Wilson. Belief elicitation: Limiting truth telling with information on incentives. NBER Working Paper 27327, National Bureau of Economic Research, 2020. URL https://www.nber.org/papers/w27327. +[25] Daehong Min. Screening for experiments. Games and Economic Behavior, 142:73-100, 2023. doi: 10.1016/j Geb.2023.07.009. URL https://doi.org/10.1016/j.geb.2023.07.009. +[26] Leah Isakov, Andrew W. Lo, and Vahid Montazerhodjat. Is the fda too conservative or too aggressive?: A bayesian decision analysis of clinical trial design. Journal of Econometrics, 211 (1):117-136, 2019. doi: 10.1016/j.jeconom.2018.12.009. URL https://doi.org/10.1016/j.jeconom.2018.12.009. +[27] Research ProRelix. Phase-by-phase clinical trial costs: What every sponsor needs to know!, 2025. URL https://prorelixresearch.com/phase-by-phase-clinical-trial-costs-what-every-sponsor-needs-to-know/. Accessed: 2025-05-10. +[28] Vinay Prasad and Sham Mailankody. Research and development spending to bring a single cancer drug to market and revenues after approval. JAMA Internal Medicine, 177(11):1569-1575, 2017. doi: 10.1001/jamainternmed.2017.3601. URL https://pmc.ncbi.nlm.nih.gov/articles/PMC5710275/. +[29] Mark Terry. Report: Projected sales of 10 top-selling drugs through 2024, August 2019. URL https://www.biospace.com/top-10-drug-sales-projected-through-2024. Accessed: 2025-05-13. +[30] Ayesha Rashid, Soumya Shashikumar, and Vincent Xie. Top 10 cancer drugs to watch in 2024 based on recent sales data, November 2024. URL https://xtalks.com/top-10-cancer-drugs-to-watch-in-2024-based-on-recent-sales-data-3976/. Accessed: 2025-05-13. +[31] Ayesha Rashid and Deepika Chandel. Top 15 cardiovascular disease drugs in 2023 by 2022 sales data, February 2024. URL https://xtalks.com/top-15-cardiovascular-disease-drugs-in-2023-by-2022-sales-data-3717/. Accessed: 2025-05-10. +[32] Aylin Sertkaya, Trinidad Beleche, and Amber Jessup. The cost of vaccine development. Technical report, U.S. Department of Health and Human Services, 2016. URL https://aspe.hhs.gov/sites/default/files/documents/dddeb3748f486324a493a8a6d27f4338/aspe-vaccine-costs-brief.pdf. Accessed: 2025-05-07. +[33] Charles R. Harris, K. Jarrod Millman, Stefan J. van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, Robert Kern, Matti Picus, Stephan Hoyer, Marten H. van Kerkwijk, Matthew Brett, Allan Haldane, Jaime Fernández del Río, Mark Wiebe, Pearu Peterson, Pierre Gérard-Marchant, Kevin Sheppard, Tyler Reddy, Warren Weckesser, Hameer Abbasi, Christoph Gohlke, and Travis E. Oliphant. Array programming with NumPy. Nature, 585(7825):357-362, September 2020. doi: 10.1038/s41586-020-2649-2. URL https://doi.org/10.1038/s41586-020-2649-2. +[34] Lee Kennedy-Shaffer. When the alpha is the omega: p-values, "substantial evidence," and the 0.05 standard at fda. Food and drug law journal, 72(4):595, 2017. + +# NeurIPS Paper Checklist + +# 1. 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SymbolUsage
X⊂{0,1}Bernoulli random variable indicating whether a product was effective on a random instance
qPDF of the effectiveness distribution
QCDF of the effectiveness distribution Q.
μ0product effectiveness
μbbaseline effectiveness
σ0variance of a product's effectiveness: σ0 = √(1-μ0)μ0
σbvariance of a baseline product's effectiveness: σb = √(1-μb)μb
c0fixed cost of running a clinical trial
cper-sample cost
Rrevenue upon approval
αp-value threshold used by the principal
p(μ0,n;μb)the probability of observing outcomes at least as good as the evidence conditioned on the null hypothesis μb
zα,nThe critical region given n samples and p-value α
Pass(α,μ0,n)the probability that a product with effectiveness μ0 is approved
u(α,μ0,n)utility for an agent with effective size μ0 and number of samples n
nμ0(α)optimal number of samples given p-value threshold α and effectiveness μ0
μτ(α)The participation threshold belief under a p-value threshold α
α*The critical p-value threshold such that μτ(α*) = μb
+ +Table 3: Primary Notation + +# B Omitted proofs in Section 3 + +# Proof of Theorem 3.1 + +Proof. We first define the following variables for brevity. Let $\Delta_{\mu} = \mu_0 - \mu_b$ denote the effect size, $\sigma_0 = \sqrt{\mu_0(1 - \mu_0)}$ denote the standard deviation of the true process and $\sigma_b = \sqrt{\mu_b(1 - \mu_b)}$ the standard deviation of the baseline Bernoulli variable. We use $\phi(z)$ to denote the standard normal, and $\Phi(z)$ to denote its CDF. Lastly, let $d_{\alpha} = \Phi^{-1}(1 - \alpha)$ denote the $1 - \alpha$ percentile of the standard normal. Recall that we work under the normal approximation of the binomial distribution where the critical threshold is: + +$$ +z _ {\alpha , n} ^ {\prime} = \left\{k \in \mathbb {R} \left| \int_ {k} ^ {\infty} \mathcal {N} \left(n \mu_ {b}, n \mu_ {b} \left(1 - \mu_ {b}\right)\right) \right. \right\} = n \mu_ {b} + \Phi^ {- 1} (1 - \alpha) \sqrt {n \mu_ {b} \left(1 - \mu_ {b}\right)} = \mu_ {b} n + d _ {\alpha} \sigma_ {b} \sqrt {n} \tag {4} +$$ + +Thus, we may write the pass probability as follows, from which the expected utility expression follows: + +$$ +\begin{array}{l} \mathrm {P a s s} (\cdot) = 1 - \Phi \left(\frac {z _ {n , \alpha} ^ {\prime} - n \mu_ {0}}{\sqrt {n \mu_ {0} (1 - \mu_ {0})}}\right) = 1 - \Phi \left(\frac {n \mu_ {b} + d _ {\alpha} \sqrt {n \mu_ {b} (1 - \mu_ {b})} - n \mu_ {0}}{\sqrt {n \mu_ {0} (1 - \mu_ {0})}}\right) \\ = 1 - \Phi \left(\frac {d _ {\alpha} \sigma_ {b}}{\sigma_ {0}} - \frac {\Delta_ {\mu} \sqrt {n}}{\sigma_ {0}}\right) \\ \Longrightarrow u (n; \alpha , \mu_ {0}, \mu_ {b}) = R - R \Phi \left(\frac {d _ {\alpha} \sigma_ {b}}{\sigma_ {0}} - \frac {\Delta_ {\mu} \sqrt {n}}{\sigma_ {0}}\right) - c n - c _ {0} \\ \end{array} +$$ + +Observe that when the drug is not effective, i.e. $\mu_0 < \mu_b$ , then the argument to $\Phi()$ increases in $n$ . Hence, the passing probability, and thus the expected revenue, decreases with increasing $n$ . It is evident that the cost also increases in $n$ . Thus, when the underlying drug is not effective, it is optimal for the agent to choose the smallest value $n_{min}$ possible. Computing the pass probability oracle with $n_{min}$ , they can determine their maximum expected utility and choose to participate if this is positive. In other words, when $\mu_0 < \mu_b$ , a constant amount of computation suffices to determine the optimal participation decision. + +We next turn to the more interesting case where $\mu_0\geq \mu_b$ . Our goal is to efficiently search the sample space by leveraging structural properties of the utility function. Letting $v = \frac{d_{\alpha}\sigma_{b}}{\sigma_{0}} -\frac{\Delta_{\mu}\sqrt{n}}{\sigma_{0}}$ , the derivative of the utility function with respect to $n$ is as follows: + +$$ +\begin{array}{l} \frac {\partial u}{\partial n} = - R \frac {\partial \Phi}{\partial v} \frac {\partial v}{\partial n} - c (5) \\ = \phi (v) \frac {R \Delta_ {\mu}}{2 \sigma_ {0} \sqrt {n}} - c = \frac {R \Delta_ {\mu}}{2 \sigma_ {0} \sqrt {n}} \phi \left(\frac {d _ {\alpha} \sigma_ {b}}{\sigma_ {0}} - \frac {\Delta_ {\mu} \sqrt {n}}{\sigma_ {0}}\right) - c (6) \\ \end{array} +$$ + +The first term is always positive, and since the image of $\phi()$ is always above 0. As $n$ increases, this first term tends to 0, and the derivative is dominated by the second term and becomes negative. In other words, increasing $n$ only increases utility to a point. To precisely understand this characteristic, consider the second derivative of the utility function: + +$$ +\frac {\partial^ {2} u}{\partial n ^ {2}} = \frac {- R \Delta_ {\mu}}{2 \sigma_ {0} n ^ {3 / 2}} \phi (v) + \frac {R \Delta_ {\mu} ^ {2}}{2 \sigma_ {0} ^ {2} n} v \phi (v) = \frac {R \phi (v) \Delta_ {\mu}}{2 \sigma_ {0} n ^ {3 / 2}} \left[ \frac {\Delta_ {\mu} v}{2 \sigma_ {0}} \sqrt {n} - 1 \right] \tag {7} +$$ + +The sign of the second derivative, and thus the concavity/convexity properties of the utility function depend solely on $\left[\frac{\Delta_{\mu}v}{2\sigma_0}\sqrt{n} -1\right]$ since the term multiplying it is always positive. We now expand it by plugging in the definition of $v$ : + +$$ +\left[ \frac {\Delta_ {\mu} v}{2 \sigma_ {0}} \sqrt {n} - 1 \right] = \frac {- \Delta_ {\mu}}{2 \sigma_ {0} ^ {2}} n + \frac {d _ {\alpha} \Delta_ {\mu} \sigma_ {b}}{2 \sigma_ {0} ^ {2}} \sqrt {n} - 1 = \frac {- \Delta_ {\mu}}{2 \sigma_ {0} ^ {2}} t ^ {2} + \frac {d _ {\alpha} \Delta_ {\mu} \sigma_ {b}}{2 \sigma_ {0} ^ {2}} t - 1 \tag {8} +$$ + +where we substitute in $t = \sqrt{n}$ . Observe that this is a negative quadratic expression with only positive real root being meaningful (since $\sqrt{n}$ cannot be negative). The roots $n_1, n_2$ can easily be computed by applying the quadratic formula to the instance parameters. We categorize the outcome as follows: + +1. No positive roots $\Longrightarrow$ the function is always concave +2. One positive root at $n_1 \Rightarrow$ the function is convex over $[0, n_1]$ and concave over $[n_1, \infty)$ . +3. Two positive root at $n_1, n_2$ with $n_2 > n_1 \Rightarrow$ that over $[0, n_1]$ the function is concave, over $[n_1, n_2]$ is is convex and over $[n_2, \infty)$ it is concave. + +We can consider the problem of optimizing $n$ over each of these convex/concave regions defined by the roots. Note that a convex function always attains its maximum on the boundary. Thus, for a the convex interval, it suffices to just check the boundary points, of which there are only two. At the optimal $n$ for a concave interval, the first derivative is always 0, or it is a boundary point. Since the latter case is similar to the first one, it suffices to find where the first derivative is 0, if it exists. Since the first derivative is always increasing (monotonic) and we need to find the $x$ -intercept, a binary search suffices: starting in the middle of the interval, increase $n$ if the first derivative is positive, and decrease it if negative. This makes at most $\log n_{max}$ calls. It is immediate that taking max over the best $n$ from each of the at most three yields the globally optimal $n_{\mu_0}(\alpha)$ . The agent can compute the corresponding passing probability and utility and decide to participate if this is positive. + +# Proof of Lemma 3.1 + +Proof. We start with the first direction. If an agent with prior belief $\mu_0$ participates with $n$ samples, then by individual rationality, they must have non-negative utility. That is, $\operatorname*{Pr}[p(n,\mu_0;\mu_b)\leq \alpha ]\cdot R\geq (c\cdot n + c_0)$ , where $(c\cdot n + c_0)$ is the cost. Now consider an agent with a prior belief $\mu_{1}\geq \mu_{0}$ and using the same number of samples- $n$ . Observe that if $\operatorname*{Pr}[p(n,\mu_1;\mu_b))\leq \alpha ]\geq Pr[p(n',\mu_0;\mu_b))\leq \alpha ]$ , then the agent will participate under belief $\mu_{1}$ . To compute $\operatorname*{Pr}[p(n,\mu_1;\mu_b)\leq \alpha ]$ , let $n\hat{\mu} = \sum_{i = 1}^{n}X_{i}$ denote the observed outcomes of $n$ samples drawn independently from the distribution with effectiveness $\mu_{1}$ . Observe that since the same number of samples are used here as when the belief was $\mu_0$ , the critical region $z_{\alpha ,n}$ does not change since it depends only on $\alpha ,n$ and $\mu_{b}$ . The probability that these samples, drawn with respect to belief $\mu_{1}$ , will lie in this critical region: + +$$ +\sum_ {i \in z _ {\alpha , n}} \binom {n} {i} \mu_ {1} ^ {i} (1 - \mu_ {1}) ^ {n - i} \geq \sum_ {i \in z _ {\alpha , n}} \binom {n} {i} \mu_ {0} ^ {i} (1 - \mu_ {0}) ^ {n - i} +$$ + +where the inequality follows immediately since $\mu_1 \geq \mu_0$ . + +For the reverse, consider an agent with effectiveness $\mu_0^{\prime}$ not participating. This means that for all $n$ , we have $\operatorname*{Pr}[p(n,\mu_0';\mu_b))\leq \alpha ]\cdot R_0 < (c\cdot n + c_0)$ . Then for an agent with belief $\mu_1^\prime < \mu_0^\prime$ the following holds: + +$$ +\forall n \sum_ {i \in z _ {\alpha , n}} \binom {n} {i} \mu^ {\prime} _ {1} ^ {i} (1 - \mu^ {\prime} _ {1}) ^ {n - i} < \sum_ {i \in z _ {\alpha , n}} \binom {n} {i} \mu^ {\prime} _ {0} ^ {i} (1 - \mu^ {\prime} _ {0}) ^ {n - i} +$$ + +In other words, for each $n$ , the passing probability under $\mu_1'$ is worse than $\mu_0'$ , and the agent was already not participating under $\mu_0'$ for any $n$ . + +# Proof for Lemma 3.2 + +Proof. We first consider solving for $\mu_{\tau}(\alpha)$ , given an $\alpha$ . To solve this up to some accuracy $\varepsilon$ , we can discretize the belief space into $k = \frac{1}{\varepsilon}$ intervals of size $\varepsilon$ , and use the mid-point of the interval as its representative belief. The monotonicity of beliefs implies that we can run a binary search over these intervals. Starting with the middle interval $\frac{k}{2}$ , if the agent participates at its representative belief, we need not search intervals larger than this. Similarly, if the agent does not participate, we need not search all intervals smaller than this. Clearly, this terminates in $\log \left(\frac{1}{\varepsilon}\right)$ calls to the pass probability oracle. To check participation, we need to compute the optimal $n$ at every belief. Assume the maximum number of sample is $n_{max}$ , it will again take $\log (n_{max})$ to search for the optimal number of samples. So the total complexity should be $\log \left(\frac{1}{\varepsilon}\right)*\log (n_{max})$ . + +Next, we show that $\mu_{\tau}(\alpha)$ is non-increasing as $\alpha$ increases. For any $\alpha$ , we know that at the corresponding threshold belief $\mu_{\tau}(\alpha)$ and its corresponding optimal sample size $n_{\mu_{\tau}}(\alpha) \triangleq n_{\tau}(\alpha)$ , the utility is 0. In other words: $R \cdot \operatorname{Pass}(n_{\tau}(\alpha), \mu_{\tau}(\alpha), \mu_b, \alpha) = cn_{\tau}(\alpha) + c_0$ . It is known that for a + +fixed effect size and number of samples, the passing probability of a hypothesis test increases in $\alpha$ . In other words, for two p-values $\alpha_{1}, \alpha_{2}$ where $\alpha_{1} \leq \alpha_{2}$ , the following holds: + +$$ +\operatorname {P a s s} \left(\alpha_ {1}, \mu_ {\tau} (\alpha_ {1}), n _ {\tau} (\alpha_ {1})\right) \leq \operatorname {P a s s} \left(\alpha_ {2}, \mu_ {\tau} (\alpha_ {2}), n _ {\tau} (\alpha_ {2})\right) \quad \text {a n d} \quad u \left(n _ {\tau} (\alpha_ {1}), \mu_ {\tau} (\alpha_ {1}), \alpha_ {2}\right) \geq 0 +$$ + +Since the agent with effectiveness $\mu_{\tau}(\alpha_1)$ will have non-negative utility when using $n_{\tau}(\alpha_1)$ samples when the p-value is $\alpha_{2}$ , this agent will participate at this p-value (but not necessarily using $n_{\tau}(\alpha_{1})$ samples as that may not be optimal). The monotonicity of participation (Lemma 3.1) implies that under $\alpha_{2}$ , agents with effectiveness greater than $\mu_{\tau}(\alpha_{1})$ will also participate. Thus, the participation threshold under $\alpha_{2}$ by definition must be to the left of $\mu_{\tau}(\alpha_1)$ - i.e. $\mu_{\tau}(\alpha_{2})\leq \mu_{\tau}(\alpha_{1})$ as desired. + +# Proof for Lemma 3.3 + +Proof. We prove by contradiction. Assume that there exists an $\alpha$ and $\mu_1 \geq \mu_0$ such that $u(\alpha, \mu_0, n_{\mu_0}(\alpha)) \geq u(\alpha, \mu_1, n_{\mu_1}(\alpha))$ . First note that if the agent is not participating in either, then the utility is always 0. If they participate in one and not the other, the monotonicity of participation (Lemma 3.1) means it must be under $\mu_1$ (giving positive utility), with $\mu_b$ being 0. Thus, the only situation where the initial claim could hold if the agent participates under both. We divide this into the following three cases: + +- $\mu_0 \leq \mu_1 \leq \mu_b$ : since both $\mu_0$ and $\mu_1$ are less effective than the baseline drug, Theorem 3.1 implies that in both cases, $n_{min}$ samples are used, and in Lemma 3.1 we know that in such settings, for a fixed $n$ , the pass probability increases in $\mu$ . Thus the utility under $\mu_1$ cannot be lower than $\mu_0$ . +- $\mu_0 \leq \mu_b \leq \mu_1$ and $\mu_b \leq \mu_0 \leq \mu_1$ : We know from Lemma 3.1 for every fixed $n$ , the pass probability in this regime increases in $\mu$ . Thus, $\forall n, u(\alpha, \mu_0, n) \leq u(\alpha, \mu_1, n)$ . Since $n_{\mu_1}(\alpha)$ is the optimal number of samples for $\mu_1$ , we have + +$$ +u (\alpha , \mu_ {0}, n _ {\mu_ {1}} (\alpha)) \leq u (\alpha , \mu_ {1}, n _ {\mu_ {1}} (\alpha)) \leq u (\alpha , \mu_ {0}, n _ {\mu_ {0}} (\alpha)) +$$ + +The last step is according to the contradiction statement. However, this cannot be true since an $n_{\mu_1}(\alpha)$ could not be the optimal number of samples for $\mu_1$ if it led to a lower utility than $n_{\mu_0}(\alpha)$ . + +![](images/f3ecc0668cd3604128ff15d58dbef5f5b7e24b31b488fe553ed51812ebf7e156.jpg) + +# C Omitted Proof for Section 4 + +For the ease of notation, let $Q$ denote the CDF of the effectiveness distribution $q$ . Further, let $Q_{< \mu_b}$ and $Q_{\geq \mu_b}$ denote the CDF of the belief distribution $q$ , conditioned on $\mu_0 < \mu_b$ and $\mu_0 \geq \mu_b$ . + +# Proof of Lemma 4.1 + +Proof. We are interested in the total derivative of the pass probability with respect to $\alpha$ for any agent with effectiveness $\mu_0 \geq \mu_b$ . Note that for this lemma, we consider the agent to always be participating. This total derivative can be expanded using the multi-variable chain rule as follows: + +$$ +\frac {d}{d \alpha} \operatorname {P a s s} \left(\alpha , \mu_ {0}, n _ {\mu_ {0}} (\alpha)\right) = \underbrace {\frac {\partial \operatorname {P a s s}}{\partial \alpha}} _ {\text {D i r e c t e f f e c t}} + \underbrace {\frac {\partial \operatorname {P a s s}}{\partial n} \cdot \frac {d n _ {\mu_ {0}} (\alpha)}{d \alpha}} _ {\text {i n d i r e c t e f f e c t}}. \tag {9} +$$ + +To simplify notation, denote $w_{\alpha} = \Phi^{-1}(1 - \alpha)$ , and let + +$$ +\xi (n, \alpha) = \frac {\Phi^ {- 1} (1 - \alpha) \sigma_ {b} - \sqrt {n} (\mu_ {0} - \mu_ {b})}{\sigma_ {0}} = \frac {\sigma_ {b}}{\sigma_ {0}} w _ {\alpha} - \frac {\sqrt {n} (\mu_ {0} - \mu_ {b})}{\sigma_ {0}} +$$ + +Then we have: $\mathrm{Pass}(\alpha, \mu_0, n_{\mu_0}(\alpha)) = 1 - \Phi(\xi(n_{\mu_0}(\alpha), \alpha))$ . We now separate the analysis based on the direct and indirect influence. + +Direct effect: The partial derivative with respect to $\alpha$ is: + +$$ +\frac {\partial \mathrm {P a s s}}{\partial \alpha} = - \frac {d}{d \alpha} \Phi (\xi (n, a)) = - \phi (\xi) \cdot \frac {\partial \xi}{\partial \alpha} = - \phi (\xi) \frac {\sigma_ {b}}{\sigma_ {0}} \frac {d w _ {\alpha}}{d \alpha} = \frac {\sigma_ {b}}{\sigma_ {0}} \frac {\phi (\xi)}{\phi (z _ {\alpha})} \geq 0 +$$ + +since a higher $\alpha$ lowers the critical threshold $w_{\alpha}$ and makes the test easier to pass. + +Indirect effect: The indirect effect depends on the optimal number of samples the agent uses for a given $\alpha$ . That is, while having more samples increases the chance of passing $(\frac{\partial\mathrm{Pass}}{\partial n} >0)$ , the agent might reduce this effort when $\alpha$ increases (since the test becomes easier). The product of these two terms is therefore unclear. Formally, + +$$ +\frac {\partial \mathrm {P a s s}}{\partial n} = - \frac {d}{d n} \Phi (\xi (n, \alpha)) = - \phi (\xi) \frac {\partial \xi}{\partial n} = \phi (\xi) \cdot \frac {\mu_ {0} - \mu_ {b}}{2 \sigma_ {0} \sqrt {n}} > 0 +$$ + +To compute $\frac{dn_{\mu_0}(\alpha)}{d\alpha}$ , let $F(n,\alpha)$ be the first-order condition of the agent's utility function: $u(\alpha ,\mu_0,n) = R\mathrm{Pass}(\alpha ,\mu_0,n) - cn - c_0$ . Since we know $n_{\mu_0}(\alpha)$ is the optimal number of samples, we claim that: + +$$ +F (\alpha , n _ {\mu_ {0}} (\alpha)) = R \cdot \frac {\partial \text {P a s s}}{\partial n} \Bigg | _ {n = n _ {\mu_ {0}} (\alpha)} - c = 0. \tag {10} +$$ + +This holds because, as we show immediately below, the second derivative of the $\mathrm{Pass}(\cdot)$ function is always strictly negative with respect to $n$ , meaning the utility function is always strictly concave. This is formalized below: + +Lemma C.1. The second derivative of Pass with respect to $n$ is always negative $-\frac{\partial^2 Pass}{\partial n^2} < 0$ . + +Proof. To see this, + +$$ +\frac {\partial^ {2} \mathrm {P a s s}}{\partial n ^ {2}} = \frac {d}{d n} \left(\phi (\xi) \cdot \frac {(\mu_ {0} - \mu_ {b})}{2 \sigma_ {0} \sqrt {n}}\right) = \phi (\xi) \cdot \left(- \frac {\xi (\mu_ {0} - \mu_ {b}) ^ {2}}{4 \sigma_ {0} ^ {2} n} - \frac {(\mu_ {0} - \mu_ {b})}{4 \sigma_ {0} n ^ {3 / 2}}\right). +$$ + +Since $\phi (\xi) > 0$ , $\mu_0 - \mu_b > 0$ , $\sigma_0 > 0$ , and $n > 0$ , and since $\xi$ can be either sign, the two terms in the parentheses are both negative (even if $\xi < 0$ , the negative sign in front ensures negativity). Thus, the entire expression is strictly negative: $\frac{\partial^2\mathrm{Pass}}{\partial n^2} < 0$ . Intuitively this means that every new sample increases the chance of passing, but each one helps less than the last. + +The result above also means that the first derivative of $F(\alpha, n)$ is always non-zero at $n_{\mu_0}(\alpha)$ . It is also evident that $F(\alpha, n)$ is continuously differentiable. Thus, we can apply the Implicit Function Theorem. Differentiating both sides of $F(\alpha, n_{\mu_0}(\alpha)) = 0$ with respect to $\alpha$ yields: + +$$ +\frac {\partial F}{\partial n} \left(\alpha , n _ {\mu_ {0}} (\alpha)\right) \cdot \frac {d n _ {\mu_ {0}} (\alpha)}{d \alpha} + \frac {\partial F}{\partial \alpha} \left(\alpha , n _ {\mu_ {0}} (\alpha)\right) = 0 \Rightarrow \frac {d n _ {\mu_ {0}} (\alpha)}{d \alpha} = - \frac {\frac {\partial F}{\partial \alpha}}{\frac {\partial F}{\partial n}} \tag {11} +$$ + +Next we compute $\frac{\partial F}{\partial\alpha}$ and $\frac{\partial F}{\partial n}$ accordingly. Observe that: + +$$ +\frac {\partial F}{\partial \alpha} = R _ {0} \cdot \frac {\partial^ {2} \operatorname {P a s s}}{\partial n \partial \alpha}, \quad \text {w h e r e} \frac {\partial^ {2} \operatorname {P a s s}}{\partial n \partial \alpha} = \frac {\mu_ {0} - \mu_ {b}}{2 \sigma_ {0} \sqrt {n}} \cdot \frac {\partial \phi (\xi)}{\partial \alpha}. +$$ + +Since: + +$$ +\frac {\partial \phi (\xi)}{\partial \alpha} = \frac {d \phi (\xi)}{d \xi} \cdot \frac {\partial \xi}{\partial \alpha} = - \xi \phi (\xi) \cdot \frac {\partial \xi}{\partial \alpha} = \xi \phi (\xi) \cdot \frac {\sigma_ {b}}{\sigma_ {0}} \frac {1}{\phi (w _ {\alpha})}, +$$ + +we have + +$$ +\frac {\partial F}{\partial \alpha} = R _ {0} \cdot \frac {\mu_ {0} - \mu_ {b}}{2 \sigma_ {0} \sqrt {n}} \cdot \xi \phi (\xi) \cdot \frac {1}{\phi (z _ {\alpha})} \frac {\sigma_ {b}}{\sigma_ {0}}. +$$ + +We next turn to computing the derivative of $F$ with respect to $n$ . Observe the following (where we plug in $\frac{\partial^2\mathrm{Pass}}{\partial n^2}$ from above): + +$$ +\frac {\partial F}{\partial n} = R _ {0} \cdot \frac {\partial^ {2} \mathrm {P a s s}}{\partial n ^ {2}} = R _ {0} \cdot \left(\frac {\xi \phi (\xi) (\mu_ {0} - \mu_ {b}) ^ {2}}{4 \sigma_ {0} ^ {2} n} - \frac {\phi (\xi) (\mu_ {0} - \mu_ {b})}{4 \sigma_ {0} n ^ {3 / 2}}\right). +$$ + +Now plugging them back into the expression computed by the Implicit function theorem, we have: + +$$ +\frac {d n _ {\mu_ {0}} (\alpha)}{d \alpha} = - \left. \frac {\frac {\partial F}{\partial \alpha}}{\frac {\partial F}{\partial n}} \right| _ {n = n _ {\mu_ {0}} (\alpha)} = - \frac {\frac {(\mu_ {0} - \mu_ {b}) \xi \phi (\xi)}{2 \sigma_ {0} \sqrt {n _ {\mu_ {0}} (\alpha)} \phi (w _ {\alpha})} \frac {\sigma_ {b}}{\sigma_ {0}}}{\frac {\xi \phi (\xi) (\mu_ {0} - \mu_ {b}) ^ {2}}{4 \sigma_ {0} ^ {2} n _ {\mu_ {0}} (\alpha)} - \frac {\phi (\xi) (\mu_ {0} - \mu_ {b})}{4 \sigma_ {0} n _ {\mu_ {0}} (\alpha) ^ {3 / 2}}} = - \frac {\frac {2 \xi \sqrt {n _ {\mu_ {0}} (\alpha)} \sigma_ {b}}{\phi (w _ {\alpha})}}{\xi (\mu_ {0} - \mu_ {b}) - \frac {\sigma_ {0}}{\sqrt {n _ {\mu_ {0}} (\alpha)}}}. +$$ + +Total derivative. Having computed the direct and indirect effects, we can directly compute the total derivative. We have: + +$$ +\begin{array}{l} \frac {d}{d \alpha} \operatorname {P a s s} \left(\alpha , \mu_ {0}, n _ {\mu_ {0}} (\alpha)\right) = \underbrace {\frac {\partial \operatorname {P a s s}}{\partial \alpha}} _ {\text {D i r e c t e f f e c t}} + \underbrace {\frac {\partial \operatorname {P a s s}}{\partial n} \Big | _ {n = n _ {\mu_ {0}} (\alpha)} \cdot \frac {d n _ {\mu_ {0}} (\alpha)}{d \alpha}} _ {\text {i n d i r e c t e f f e c t}}. (12) \\ = \frac {\sigma_ {b}}{\sigma_ {0}} \frac {\phi (\xi)}{\phi (z _ {\alpha})} + \phi (\xi) \cdot \frac {\mu_ {0} - \mu_ {b}}{2 \sigma_ {0} \sqrt {n _ {\mu_ {0}} (\alpha)}} \cdot \frac {d n _ {\mu_ {0}} (\alpha)}{d \alpha} (13) \\ = \frac {\sigma_ {b}}{\sigma_ {0}} \frac {\phi (\xi)}{\phi (z _ {\alpha})} \left(1 - \frac {\left(\mu_ {0} - \mu_ {b}\right) \xi}{\left(\xi \left(\mu_ {0} - \mu_ {b}\right) - \frac {\sigma_ {0}}{\sqrt {n _ {\mu_ {0} (\alpha)}}}\right)}\right). (14) \\ \end{array} +$$ + +The sign of $\left(1 - \frac{(\mu_0 - \mu_b)\xi}{\left(\xi(\mu_0 - \mu_b) - \frac{\sigma_0}{\sqrt{n\mu_0(\alpha)}}\right)}\right)$ decides the monotonicity of the passing probability. As we show below, when agent chooses the optimal number of samples $n = n_{\mu_0}(\alpha)$ , it will never be the case that $\xi(\mu_0 - \mu_b) - \frac{\sigma_0}{\sqrt{n(\alpha)}} > 0$ (Lemma C.2), which implies that $\left(1 - \frac{(\mu_0 - \mu_b)\xi}{\left(\xi(\mu_0 - \mu_b) - \frac{\sigma_0}{\sqrt{n\mu_0(\alpha)}}\right)}\right) > 0$ always holds at the optimal $n_{\mu_0}(\alpha)$ , thus the passing probability at the optimal number of samples $n_{\mu_0}(\alpha)$ is always monotonically increasing. We finish the proof by proving Lemma C.2: + +Lemma C.2. When $n_{\mu_0}(\alpha)$ is the optimal sample size chosen by the agent to maximize utility, then $\xi (\mu_0 - \mu_b) - \frac{\sigma_0}{\sqrt{n(\alpha)}} > 0$ never holds. + +Proof. Recall the second derivative of the utility w.r.t $n$ is: + +$$ +\begin{array}{l} \frac {\partial^ {2} \text {U t i l i t y} (n , \mu_ {0} ; \mu_ {b} , \alpha)}{\partial n ^ {2}} = \frac {\partial^ {2} \text {P a s s}}{\partial n ^ {2}} \\ = \frac {\xi \phi (\xi) (\mu_ {0} - \mu_ {b}) ^ {2}}{4 \sigma_ {0} ^ {2} n} - \frac {\phi (\xi) (\mu_ {0} - \mu_ {b})}{4 \sigma_ {0} n ^ {3 / 2}} \\ = \frac {\phi (\xi) \left(\mu_ {0} - \mu_ {b}\right)}{4 \sigma_ {0} ^ {2} n} \left(\xi \left(\mu_ {0} - \mu_ {b}\right) - \frac {\sigma_ {0}}{\sqrt {n}}\right) \\ \end{array} +$$ + +At the optimal number of sample $n = n_{\mu_0}(\alpha)$ , the second derivative of the utility must be negative, otherwise it implies that increasing the number of samples will increase the utility, which leads to a contradiction to the fact that $n = n_{\mu_0}(\alpha)$ is an optimal number of samples, thus we have the following always holds at $n = n_{\mu_0}(\alpha)$ : $\xi (\mu_0 - \mu_b) - \frac{\sigma_0}{\sqrt{n_{\mu_0}(\alpha)}}\leq 0$ . + +# Proof of Theorem 4.2 + +Proof. Recall that the false negative rate in our setting consists of two components, conditioned on whether agents participate. Observing that when an agent does not participate - i.e. $\mu_0 < \mu_\tau (\alpha) -$ their probability of passing the statistical test is 0, we can simplify the overall FN rate as follows: + +$$ +\begin{array}{l} \operatorname {F N} (\alpha , \mathcal {I}) = \underset {\mu_ {0} \sim q} {\mathbb {E}} [ \operatorname {F a i l} (\alpha , \mu_ {0}, n _ {\mu_ {0}} (\alpha)) | \mu_ {0} \geq \mu_ {b} ] \\ = \underbrace {\underset {\mu_ {0} \sim q} {\mathbb {E}} [ \operatorname {F a i l} (\alpha , \mu_ {0} , n _ {\mu_ {0}} (\alpha)) | \mu_ {0} \geq \mu_ {b} , \mu_ {0} \geq \mu_ {\tau} (\alpha) ] P (\mu_ {0} \geq \mu_ {\tau} (\alpha) | \mu_ {0} \geq \mu_ {b})} _ {\mathrm {F N} _ {\text {p a r t i c i p}}} \\ + \underbrace {P [ \mu_ {0} \leq \mu_ {\tau} (\alpha) | \mu_ {0} \geq \mu_ {b} ]} _ {\mathrm {F N} _ {\text {a b s t a i n}}} \\ \end{array} +$$ + +Consider first any $\alpha_{1} \leq \widehat{\alpha}$ . We can explicitly express each component of the false negative as follows, since we are guaranteed that $\mu_{\tau}(\alpha_1) \geq \mu_b$ : + +$$ +\mathrm {F N} _ {\text {p a r t i c i p}} \left(\alpha_ {1}, \mathcal {I}\right) = \frac {1 - Q \left(\mu_ {\tau} \left(\alpha_ {1}\right)\right)}{1 - Q \left(\mu_ {b}\right)} \int_ {\mu_ {\tau} \left(\alpha_ {1}\right)} ^ {1} \operatorname {F a i l} \left(\alpha_ {1}, \mu_ {0}, n _ {\mu_ {0}} (\alpha)\right) \frac {q \left(\mu_ {0}\right)}{1 - Q \left(\mu_ {\tau} \left(\alpha_ {1}\right)\right)} d \mu_ {0} \tag {15} +$$ + +$$ +\mathrm {F N} _ {\text {a b s t a i n}} \left(\alpha_ {1}, \mathcal {I}\right) = \frac {Q \left(\mu_ {\tau} \left(\alpha_ {1}\right)\right) - Q \left(\mu_ {b}\right)}{1 - Q \left(\mu_ {b}\right)} \tag {16} +$$ + +Let us focus on $\mathrm{FN}_{\mathrm{particip}}$ . To simplify this, the following result is helpful. For a positive function $g(x)$ and a function $f(x)$ with minimum and maximum values $f_{min}$ and $f_{max}$ over an interval $[a,b]$ , the following holds: + +$$ +f _ {m i n} \int_ {a} ^ {b} g (x) \leq \int_ {a} ^ {b} g (x) f (x) d x \leq f _ {m a x} \int_ {a} ^ {b} g (x) \Rightarrow f _ {m i n} \leq \frac {\int_ {a} ^ {b} g (x) f (x) d x}{\int_ {a} ^ {b} g (x)} \leq f _ {m a x} +$$ + +Since the expression in the middle is between $f_{min}$ and $f_{max}$ , by the intermediate value theorem, there exists an $c \in [a,b]$ such that: $f(c) = \frac{\int_a^b g(x)f(x)dx}{\int_a^b g(x)}$ which means: there exists a $c$ such that $f(c)\int_{a}^{b}g(x) = \int_{a}^{b}f(x)g(x)$ . This can be interpreted as an integral mean value theorem. Using this, there exists a value $\mu_c^1 \in [\mu_\tau (\alpha_1),1]$ such that the FNparticipation(α1) can be expressed as follows: + +$$ +\begin{array}{l} \mathrm {F N} _ {\text {p a r t i c i p}} (\alpha_ {1}, \mathcal {I}) = \frac {1 - Q (\mu_ {\tau} (\alpha_ {1}))}{1 - Q (\mu_ {b})} \operatorname {F a i l} (\alpha_ {1}, \mu_ {c} ^ {1}, n _ {\mu_ {c} ^ {1}} (\alpha)) \int_ {\mu_ {\tau} (\alpha_ {1})} ^ {1} \frac {q (\mu_ {0})}{1 - Q (\mu_ {\tau} (\alpha_ {1}))} d \mu_ {0} \\ = \frac {1}{1 - Q (\mu_ {b})} \operatorname {F a i l} \left(\alpha_ {1}, \mu_ {c} ^ {1}, n _ {\mu_ {c} ^ {1}} (\alpha)\right) \int_ {\mu_ {\tau} \left(\alpha_ {1}\right)} ^ {1} \frac {q \left(\mu_ {0}\right)}{1 - Q \left(\mu_ {\tau} \left(\alpha_ {1}\right)\right)} d \mu_ {0} \\ = \frac {1 - Q (\mu_ {\tau} (\alpha_ {1}))}{1 - Q (\mu_ {b})} \operatorname {F a i l} (\alpha_ {1}, \mu_ {c} ^ {1}, n _ {\mu_ {c} ^ {1}} (\alpha)) \\ \end{array} +$$ + +Therefore, we can write: + +$$ +\text {f o r s o m e} \mu_ {c} ^ {1} \in [ \mu_ {\tau} (\alpha_ {1}), 1 ]: \operatorname {F N} (\alpha_ {1}, \mathcal {I}) = \frac {1 - Q (\mu_ {\tau} (\alpha_ {1}))}{1 - Q (\mu_ {b})} \operatorname {F a i l} (\alpha_ {1}, \mu_ {c} ^ {1}, n _ {\mu_ {c} ^ {1}} (\alpha)) + \frac {Q (\mu_ {\tau} (\alpha_ {1})) - Q (\mu_ {b})}{1 - Q (\mu_ {b})} +$$ + +Now consider an $\alpha_{2}$ such that $\alpha_{1} < \alpha_{2} \leq \alpha^{*}$ . Note that due to Appendix B, $\mu_{\tau}(\alpha_{2}) \leq \mu_{\tau}(\alpha_{1})$ . We can thus write the $\mathrm{FN}_{\mathrm{particip}}$ of this instance as follows: + +$$ +\begin{array}{l} \mathrm {F N} _ {\text {p a r t i c i p}} (\alpha_ {2}, \mathcal {I}) = \frac {1 - Q (\mu_ {\tau} (\alpha_ {2}))}{1 - Q (\mu_ {b})} \int_ {\mu_ {\tau} (\alpha_ {2})} ^ {1} \operatorname {F a i l} (\alpha_ {2}, \mu_ {0}, n _ {\mu_ {0}} (\alpha)) \frac {q (\mu_ {0})}{1 - Q (\mu_ {\tau} (\alpha_ {2}))} d \mu_ {0} \\ = \frac {1}{1 - Q (\mu_ {b})} \left[ \int_ {\mu_ {\tau} (\alpha_ {2})} ^ {\mu_ {\tau} (\alpha_ {1})} \operatorname {F a i l} \left(\alpha_ {2}, \mu_ {0}, n _ {\mu_ {0}} (\alpha_ {2})\right) q \left(\mu_ {0}\right) d \mu_ {0} + \int_ {\mu_ {\tau} (\alpha_ {1})} ^ {1} \operatorname {F a i l} \left(\alpha_ {2}, \mu_ {0}, n _ {\mu_ {0}} (\alpha_ {2})\right) q \left(\mu_ {0}\right) d \mu_ {0} \right] \\ = \frac {1}{1 - Q \left(\mu_ {b}\right)} \left[ \operatorname {F a i l} \left(\alpha_ {2}, \mu_ {c} ^ {2, 1}, n _ {\mu_ {c} ^ {2, 1}} (\alpha)\right) \int_ {\mu_ {\tau} \left(\alpha_ {2}\right)} ^ {\mu_ {\tau} \left(\alpha_ {1}\right)} q \left(\mu_ {0}\right) d \mu_ {0} + \operatorname {F a i l} \left(\alpha_ {2}, \mu_ {c} ^ {2, 2}, n _ {\mu_ {c} ^ {2, 2}} (\alpha)\right) \int_ {\mu_ {\tau} \left(\alpha_ {1}\right)} ^ {1} q \left(\mu_ {0}\right) d \mu_ {0} \right] \\ = \operatorname {F a i l} \left(\alpha_ {2}, \mu_ {c} ^ {2, 1}, n _ {\mu_ {c} ^ {2, 1}} (\alpha)\right) \frac {Q \left(\mu_ {\tau} \left(\alpha_ {1}\right)\right) - Q \left(\mu_ {\tau} \left(\alpha_ {2}\right)\right)}{1 - Q \left(\mu_ {b}\right)} + \operatorname {F a i l} \left(\alpha_ {2}, \mu_ {c} ^ {2, 2}, n _ {\mu_ {c} ^ {2, 2}} (\alpha)\right) \frac {1 - Q \left(\mu_ {\tau} \left(\alpha_ {1}\right)\right)}{1 - Q \left(\mu_ {b}\right)} \\ \end{array} +$$ + +where in the third transition, we use the integral mean value theorem as in the previous $\alpha_{1}$ case. Note that for the integral between $[\mu_{\tau}(\alpha_{1}), 1]$ in the $\alpha_{2}$ setting, we know that for every $\mu_{0} \in [\mu_{\tau}(\alpha_{1}), 1]$ , $F a i l(\alpha_{2}, \mu) < F a i l(\alpha_{1}, \mu)$ since from Lemma 4.1, we know that the pass probability increases as alpha increases and this set of agents participated under both $\alpha_{1}$ and $\alpha_{2}$ . This immediately means that $F a i l(\alpha_{2}, \mu_{c}^{2,2}, n_{\mu_{c}^{2,2}}(\alpha)) \leq F a i l(\alpha_{2}, \mu_{c}^{1}, n_{\mu_{c}^{2,1}}(\alpha))$ . In the $[\mu_{\tau}(\alpha_{2}), \mu_{\tau}(\alpha_{1})]$ , the failure probability is at most 1. Hence, $F a i l(\alpha_{2}, \mu_{c}^{2,1}, n_{\mu_{c}^{2,1}}(\alpha)) \leq 1$ . Thus, we can upper bound the FN participation loss at $\alpha_{2}$ as follows: + +$$ +\mathrm {F N} _ {\text {p a r t i c i p}} \left(\alpha_ {2}\right) \leq \frac {Q \left(\mu_ {\tau} \left(\alpha_ {1}\right)\right) - Q \left(\mu_ {\tau} \left(\alpha_ {2}\right)\right)}{1 - Q \left(\mu_ {b}\right)} + \operatorname {F a i l} \left(\alpha_ {1}, \mu_ {c} ^ {1}, n _ {\mu_ {c} ^ {1}} (\alpha)\right) \frac {1 - Q \left(\mu_ {\tau} \left(\alpha_ {1}\right)\right)}{1 - Q \left(\mu_ {b}\right)} \tag {17} +$$ + +We now express the abstain loss for $\alpha_{2}$ as follows: + +$$ +\mathrm {F N} _ {\text {a b s t a i n}} (\alpha_ {2}) = \frac {Q (\mu_ {\tau} (\alpha_ {2})) - Q (\mu_ {b})}{1 - Q (\mu_ {b})} = \frac {Q (\mu_ {\tau} (\alpha_ {1})) - Q (\mu_ {b})}{1 - Q (\mu_ {b})} - \frac {Q (\mu_ {\tau} (\alpha_ {1})) - Q (\mu_ {\tau} (\alpha_ {2}))}{1 - Q (\mu_ {b})} +$$ + +Then we have that: + +$$ +\begin{array}{l} \mathrm {F N} \left(\alpha_ {2}, \mathcal {I}\right) = \mathrm {F N} _ {\text {p a r t i c i p}} + \mathrm {F N} _ {\text {a b s t a i n}} (18) \\ \leq \operatorname {F a i l} \left(\alpha_ {1}, \mu_ {c} ^ {1}, n _ {\mu_ {c} ^ {1}} (\alpha)\right) \frac {1 - Q \left(\mu_ {\tau} \left(\alpha_ {1}\right)\right)}{1 - Q \left(\mu_ {b}\right)} + \frac {Q \left(\mu_ {\tau} \left(\alpha_ {1}\right)\right) - Q \left(\mu_ {b}\right)}{1 - Q \left(\mu_ {b}\right)} = \mathrm {F N} \left(\alpha_ {1}, \mathcal {I}\right) (19) \\ \end{array} +$$ + +Lastly, for any $\alpha \geq \widehat{\alpha}$ , we know from Proposition 4.2 that $\mathrm{FN}_{\mathrm{particip}}(\alpha, I) = 0$ and from Proposition 4.1 that $\mathrm{FN}_{\mathrm{abstain}}(\alpha, \mathcal{I}) = 0$ . + +# D Discussion on Sensitivity Analysis + +In this section, we discuss how our framework can be naturally extended to accommodate small misspecifications on either the agent or principal side, and why our main insights remain stable under such uncertainty. + +Both the agent's participation decision and the principal's optimal p-value threshold $\hat{\alpha}$ depend only on the cost and revenue parameters $(c_0, c, R)$ . In realistic settings such as drug development, these parameters are estimated from historical data or business forecasts and may contain some error. We can therefore model parameter uncertainty by assuming that the agent and principal each have slightly perturbed estimates of the true values. In particular, we assume the agent has access to approximate parameters $(\hat{c}_0, \hat{c}, \hat{R})$ within $\epsilon$ of the true values, and let the principal have their own estimates $(\tilde{c}_0, \tilde{c}, \tilde{R})$ within $\delta$ of the true values. The magnitudes of $\epsilon$ and $\delta$ can differ across parameters. + +Agent-side uncertainty. We first examine how such $\epsilon$ -level errors affect the agent's decision for a given belief $\mu_0$ . Extending Theorem 3.1, recall that the utility function $u(n; \alpha)$ can be partitioned by indices $(n_1, n_2)$ where it is either convex or concave. As noted in Equation (8) of Appendix B, these indices are invariant to small parameter perturbations. In the convex region, the optimal $n$ occurs at an endpoint and is therefore unchanged. In the concave region, the optimum corresponds to the $x$ -intercept of $\partial u / \partial n$ . A perturbation of size $\epsilon$ shifts this derivative by $O(\epsilon)$ , implying that the optimal $n$ moves by at most $O(\epsilon) / L$ , where $L$ is the inverse Lipschitz constant of $\partial u / \partial n^6$ : + +$$ +\left| \hat {n} _ {\mu_ {0}} (\alpha) - n _ {\mu_ {0}} (\alpha) \right| \leq O (\epsilon) / L. +$$ + +A similar argument bounds the difference in participation thresholds: + +$$ +\left. \left| \hat {\mu} _ {\tau} (\alpha) - \mu_ {\tau} (\alpha) \right| \leq O (\epsilon) / k, \right. +$$ + +where $k$ is the inverse Lipschitz constant (with respect to $\mu_0$ ) of $u(\mu_0, n(\alpha, \mu_0))$ . + +Principal-side uncertainty. If the principal's perceived parameters $(\tilde{c}_0, \tilde{c}, \tilde{R})$ differ from those of the agent, they will infer a different threshold belief $\tilde{\mu}_{\tau}(\alpha)$ . Without observing the agent's $\epsilon$ , the principal optimizes $\alpha$ under their own model, obtaining $\hat{\alpha}$ such that $\tilde{\mu}_{\tau}(\hat{\alpha}) = \mu_b$ . However, because $\epsilon \neq \delta$ , the agent's actual threshold $\hat{\mu}_{\tau}(\hat{\alpha})$ may not equal $\mu_b$ . This deviation can again be bounded by the inverse Lipschitz constant $k$ : + +$$ +\left. \left| \hat {\mu} _ {\tau} (\hat {\alpha}) - \mu_ {b} \right| \leq O \left(\left| \epsilon - \delta \right|\right) / k. \right. +$$ + +The agent's deviation depends only on the difference between their perceived parameters and the true ones, whereas the principal's deviation depends on the gap between their perception and the agent's. These uncertainties can thus be analyzed independently. Overall, small estimation errors in cost or revenue induce proportionally small and bounded shifts in the optimal participation and threshold decisions. + +Stability of our results Building on the above observations, we now explain why the main insights of our model remain stable under such uncertainty. This robustness follows from two structural properties of the model: monotonicity and Lipschitz continuity. First, both the participation threshold $\mu_{\tau}(\alpha)$ and the principal's optimal choice $\hat{\alpha}$ are continuous and monotone functions of the cost and revenue parameters $(c_0, c, R)$ . Second, their derivatives with respect to these parameters are bounded by Lipschitz constants $k$ and $L$ , ensuring that small perturbations induce at most proportional changes: + +$$ +\left| \Delta \mu_ {\tau} (\alpha) \right| \leq k \| \left(\Delta c _ {0}, \Delta c, \Delta R\right) \|, \quad \left| \Delta \hat {\alpha} \right| \leq L \| \left(\Delta c _ {0}, \Delta c, \Delta R\right) \|. +$$ + +These bounds imply that the relative ordering of participation and approval thresholds is preserved: The curves may shift, but they do not cross or invert. As a result, the qualitative behavior of the model—for instance, that higher costs discourage participation and looser $\alpha$ increases it—remains unchanged. + +# E Plot for Vaccine Drugs + +![](images/656afaed1fa3a1d62a4a0b318b0d7d0a212f4dcaa2723881f05fc9723506ebbe.jpg) +Figure 4: $\widehat{\alpha}$ vs Revenue - Vaccines \ No newline at end of file diff --git a/strategichypothesistesting/images.zip b/strategichypothesistesting/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..6cb70bcc42619a3806e6cad60d2449deaaf7a11f --- /dev/null +++ b/strategichypothesistesting/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0677f12ec172669f9d2077a87b2f9ee093dc98bfc5668a55dac2bf8ff91e052d +size 852060 diff --git a/strategichypothesistesting/layout.json b/strategichypothesistesting/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..9f0bde28789093bfc31131d2b80dee0791216a00 --- /dev/null +++ b/strategichypothesistesting/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:06ca7ff02f13b58addd14ec07bbdf756a21d149e37c3589c507701a257628135 +size 1263343 diff --git a/strategyproofreinforcementlearningfromhumanfeedback/c17603e0-31a1-4e99-8ffe-360c54a9eb2b_content_list.json b/strategyproofreinforcementlearningfromhumanfeedback/c17603e0-31a1-4e99-8ffe-360c54a9eb2b_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..6f1c06455704087a9782c51caa7904981e6ef6b4 --- /dev/null +++ b/strategyproofreinforcementlearningfromhumanfeedback/c17603e0-31a1-4e99-8ffe-360c54a9eb2b_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:201068dd318572af0f1d4e67ec620f4c9e8a8ec8fab562d1450124f8f4b061b6 +size 248009 diff --git a/strategyproofreinforcementlearningfromhumanfeedback/c17603e0-31a1-4e99-8ffe-360c54a9eb2b_model.json b/strategyproofreinforcementlearningfromhumanfeedback/c17603e0-31a1-4e99-8ffe-360c54a9eb2b_model.json new file mode 100644 index 0000000000000000000000000000000000000000..6c9be7c5314536351f7622c4d99fb73c2d125d22 --- /dev/null +++ b/strategyproofreinforcementlearningfromhumanfeedback/c17603e0-31a1-4e99-8ffe-360c54a9eb2b_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:334b93a4e98621d652049d307e050d7a3e143c3b0966b32a6a934d86bc2f29a0 +size 301581 diff --git a/strategyproofreinforcementlearningfromhumanfeedback/c17603e0-31a1-4e99-8ffe-360c54a9eb2b_origin.pdf b/strategyproofreinforcementlearningfromhumanfeedback/c17603e0-31a1-4e99-8ffe-360c54a9eb2b_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..57af5e15add9bbf8dd27c74c682c1e4be07066d6 --- /dev/null +++ b/strategyproofreinforcementlearningfromhumanfeedback/c17603e0-31a1-4e99-8ffe-360c54a9eb2b_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7a7acdba6aceec23831e69c9664c2ee884ba8b1fd5879a01e35d501c4f837903 +size 537910 diff --git a/strategyproofreinforcementlearningfromhumanfeedback/full.md b/strategyproofreinforcementlearningfromhumanfeedback/full.md new file mode 100644 index 0000000000000000000000000000000000000000..f19a21acc849d391a1d1ac9e3385a0800434aa4b --- /dev/null +++ b/strategyproofreinforcementlearningfromhumanfeedback/full.md @@ -0,0 +1,1298 @@ +# Strategyproof Reinforcement Learning from Human Feedback + +Thomas Kleine Buening + +ETH Zurich + +ETH AI Center + +Jiarui Gan + +Department of Computer Science + +University of Oxford + +Debmalya Mandal + +Department of Computer Science + +University of Warwick + +Marta Kwiatkowska + +Department of Computer Science + +University of Oxford + +# Abstract + +We study Reinforcement Learning from Human Feedback (RLHF) in settings where multiple labelers may strategically misreport feedback to steer the learned policy toward their own preferences. We show that existing RLHF algorithms, including recent pluralistic methods, are not strategyproof, and that even a single strategic labeler can cause arbitrarily large misalignment with social welfare. Moreover, we prove that, in the worst case, any strategyproof RLHF algorithm must perform $k$ -times worse than the optimal policy, where $k$ is the number of labelers. This suggests a fundamental trade-off between incentive alignment (ensuring labelers report truthfully) and policy alignment (maximizing social welfare). To address this, we propose the Pessimistic Median of MLEs algorithm, which, under appropriate policy coverage assumptions, is approximately strategyproof and converges to the optimal policy as the number of labelers and samples increases. Our results apply to both contextual bandits and Markov decision processes. + +# 1 Introduction + +Reinforcement Learning from Human Feedback (RLHF) has become a widely used approach for aligning AI systems with human preferences. By leveraging human-labeled comparisons, RLHF enables policy optimization in applications such as robotics, recommendation systems, and Large Language Models (LLMs) [5, 21]. This approach has led to significant improvements in usability and alignment with intended objectives. However, RLHF also introduces new challenges, particularly in situations where preferences are diverse, subjective, and potentially in conflict. + +Recently, pluralistic alignment—the challenge of aligning with the preferences of diverse individuals or groups—has emerged as an active area of research [33, 14, 22, 3, 6]. Unlike traditional reinforcement learning, which optimizes a policy according to a single, well-defined reward function, pluralistic settings require reconciling multiple perspectives. This raises questions about whose preferences should shape AI decisions and how to aggregate diverse inputs fairly and effectively [20, 11]. In such pluralistic settings, existing methods often optimize policies based on aggregated human preferences, implicitly assuming that labelers provide feedback truthfully. However, this perspective crucially neglects the incentives of labelers when their feedback directly impacts policy outcomes. + +When human preferences influence the final policy, labelers (or groups of labelers) may have incentives to manipulate their feedback in ways that benefits them at the expense of broader alignment [30, 19]. For example, in the context of LLM fine-tuning, human labelers may systematically misreport preferences to amplify specific biases and reinforce narratives favorable to their views. + +As a result, such strategic behavior threatens to distort the alignment process through self-serving feedback and can undermine the fairness, robustness, and effectiveness of the system [13, 2]. Despite its importance, this issue remains largely unaddressed in existing RLHF methodologies. + +This paper aims to bridge this gap by studying RLHF through the lens of mechanism design and proposing solutions that ensure robustness against strategic feedback. We formalize the problem, analyze the conditions and the cost under which strategyproofness can be achieved, and propose an RLHF method to mitigate incentive misalignment while maintaining policy performance. + +In summary, our main contributions are: + +- We formally introduce the problem of offline RLHF with strategic human labelers, where each labeler potentially misreports preference labels to steer the final policy toward the maximization of their personal objectives, i.e., their reward function (Section 3). We focus on linear reward functions and social welfare maximization and study the tensions that arise between individual incentive alignment and policy alignment with social welfare. +- We show that existing RLHF methods are not strategyproof (Proposition 3.3), and even a single strategic labeler can almost arbitrarily degrade policy performance of existing methods (Proposition 3.4). Moreover, without additional assumptions, we find that any strategyproof RLHF method suffers from constant suboptimality (Theorem 3.5) and performs at least $k$ -times worse compared to the optimal policy (Corollary 3.6), where $k$ is the number of distinct labelers. This points towards a fundamental trade-off between incentive alignment and policy alignment. +- We propose an RLHF method called Pessimistic Median of MLEs, which combines pessimistic estimates with a median rule to incentivize truthful preference reporting (Section 4). Interestingly, we find that Pessimistic Median of MLEs is approximately strategyproof due to the uncertainty in reward estimation. Notably, the incentive strength depends on the uniform policy coverage of each labeler's data. This stands in contrast to standard RLHF guarantees, which rely only on the coverage of the optimal policy. More precisely, under additional domain restrictions, we show: + +a) Pessimistic Median of MLEs is $\tilde{\mathcal{O}} (\kappa_i\sqrt{d / n})$ -strategyproof for labeler $i$ where $\kappa_{i}$ quantifies the uniform policy coverage of labeler $i$ (Theorem 4.1). +b) The computed policy's suboptimality is bounded by $\tilde{\mathcal{O}} (\sqrt{d / k} +\max_{i\in [k]}\kappa_i^*\cdot k\sqrt{d / n})$ where $\kappa_{i}^{*}$ is the optimal policy coverage of labeler $i$ (Theorem 4.2, Proposition 4.3, Corollary 4.5). + +We establish these results for both contextual bandits and Markov decision processes (Section 5). + +# 2 Related Work + +Reinforcement Learning from Human Feedback. RLHF has emerged as a powerful framework for aligning AI systems with human values by leveraging human feedback to guide policy learning [10, 5, 21]. Most relevant to our work is the growing literature on RLHF in settings with diverse and possibly conflicting preferences among individuals or demographic groups [7, 28, 37, 8]. Some of these works focus on maximizing the worst-case utility across labelers (or groups) [7, 28], whereas others optimize welfare functions such as the additive social welfare [37]. + +Some other recent work has also explicitly taken a social choice perspective on pluralistic alignment and studies how to ensure that methods for preference aggregation satisfy desirable properties inspired by social choice theory [11, 15, 1]. Importantly, these works, while related, assume truthful feedback and do not account for the incentives created by AI alignment. However, aggregating and trading-off preferences naturally invites strategic or malicious behavior, as labelers may manipulate the alignment process to more closely align the final policy with their own beliefs and goals. For example, Siththaranjan et al. [32] highlighted how standard RLHF methods implicitly aggregate preferences using the Borda count voting rule, which can create incentives for annotators to misreport their preferences to influence model behavior. + +Another body of work considers robustness against adversarial corruptions in RLHF. Mandal et al. [24] assume that an $\varepsilon$ -fraction of samples is adversarily manipulated, allowing for both manipulation of trajectory features and preference labels. Similarly, Bukharin et al. [4] and Cheng et al. [9] also consider the case where a fraction of samples is manipulated but restrict their attention to adversaries flipping preferences. This line of work differs from ours notably in both perspective (strategic vs. adversarial) and techniques (mechanism design vs. robust offline RL). + +Mechanism Design for RLHF. Recently, several works have incorporated mechanism design principles into RLHF to incentivize truthful feedback [27, 34, 35]. These approaches design payment rules to align labelers' incentives, often extending VCG-style mechanisms to the RLHF problem. In contrast, we propose a strategy-robust RLHF method that does not rely on payments or other financial incentives, which are often impractical in real-world applications. Also closely related is the work of Hao and Duan [18], which studies an online RLHF framework where labelers sequentially provide preference feedback, aggregated using a linearly weighted average. Their approach focuses on identifying the most accurate labeler over time and adjusting the weights to incentivize truthful reporting. In contrast, we study the offline RLHF setting and do not impose a linear weighting assumptions on labelers. Moreover, unlike [18], we assume that labelers seek to influence the final policy rather than just an estimate of the aggregated preferences, which might better reflect real-world strategic behavior, where individuals care about the actual policy outcomes rather than intermediate preference estimates. In particular, as we will see, a more closely aligned reward function does not imply a more aligned policy. + +# 3 Problem Formulation + +We consider episodic Markov Decision Processes (MDPs) and the special case of contextual bandits. Let $\mathcal{M} = (\mathcal{S},\mathcal{A},\mathcal{P},H,\rho)$ be an MDP without reward function, where $\mathcal{S}$ is the state space, $\mathcal{A}$ is the action space, $H$ is the horizon and $\rho$ is the initial state distribution. $\mathcal{P} = (\mathcal{P}_1,\dots ,\mathcal{P}_H)$ denotes the tuple of transition functions, where $\mathcal{P}_h\colon S\times \mathcal{A}\to \Delta (\mathcal{S})$ determines the transitions in step $h\in [H]$ . A history-independent policy $\pi = (\pi_h)_{h\leq H}$ maps from states to a distribution over actions in every time step, i.e., $\pi_h\colon S\to \Delta (\mathcal{A})$ , and we let $\Pi$ denote its policy space. A trajectory in MDP $\mathcal{M}$ is given by a sequence of actions and states $\tau = (a_{1},s_{2},a_{2},\ldots ,s_{H},a_{H})$ . The MDP reduces to a contextual bandit problem when $H = 1$ , in which case a trajectory consists only of the action taken in the initial state and the initial states are interpreted as contexts sampled from $\rho$ . + +Multiple Labelers with Diverse Preferences. We consider the situation where $k \geq 1$ many labelers provide preference data to the RLHF algorithm. In particular, each labeler $i \in [k]$ is associated with a reward function $r_i \colon S \times \mathcal{A} \to \mathbb{R}$ . The expected return of a policy $\pi$ w.r.t. a reward function $r$ is given by $V_r^\pi(s) \coloneqq \mathbb{E}\left[\sum_{h=1}^H r(s_h, \pi_h(s_h)) \mid s_1 = s\right]$ . Accordingly, we define the utility of labeler $i \in [k]$ w.r.t. the initial state distribution $\rho$ and a policy $\pi$ as + +$$ +J _ {i} (\rho , \pi) := \mathbb {E} _ {s \sim \rho} \left[ V _ {r _ {i}} ^ {\pi} (s) \right]. +$$ + +Note that this simplifies to $J_{i}(\rho ,\pi) = \mathbb{E}_{s\sim \rho}[r_{i}(s,\pi (s))]$ in the contextual bandit. + +We focus on the linearly realizable case, where the reward function of each labeler is a linear function $r_{\theta}(s,a) = \langle \theta ,\phi (s,a)\rangle$ of a known feature embedding $\phi$ of the state (i.e., context) and action. + +Assumption 1 (Linear Realizability). Every labeler's reward function $r_i$ is given by a linear function $r_{\theta_i^*}(s, a) \coloneqq \langle \theta_i^*, \phi(s, a) \rangle$ . Here, the reward parameter $\theta_i^*$ is sampled from $\{\theta \in \mathbb{R}^d : \| \theta \|_2 \leq B\}$ with $B > 0$ and $\phi$ is a known mapping with $\| \phi(s, a) \|_2 \leq L$ for all $(s, a) \in S \times A$ . + +Offline RLHF. We focus on the offline RLHF setting, where each labeler $i \in [k]$ is given a pre-determined set of $n$ examples ( $s^{i,j}, \tau_0^{i,j}, \tau_1^{i,j}$ ) indexed by $j \in [n]$ where $s^{i,j}$ denotes the initial state and ( $\tau_0^{i,j}, \tau_1^{i,j}$ ) are two subsequent trajectories. For each such example, labeler $i$ provides a preference label $o^{i,j} \in \{0,1\}$ , where the label $o^{i,j} = 0$ means that trajectory $\tau_0^{i,j}$ is preferred over $\tau_1^{i,j}$ given initial state $s^{i,j}$ , and vice versa. + +We employ the widely used Bradley-Terry (BT) model under which a labeler with a reward parameter $\theta$ (i.e., reward function $r_{\theta}$ ) prefers trajectory $\tau_0 = (a_1, s_2, a_2, \ldots, s_H, a_H)$ over trajectory $\tau_1 = (\tilde{a}_1, \tilde{s}_2, \tilde{a}_2, \ldots, \tilde{s}_H, \tilde{a}_H)$ with probability + +$$ +\mathbb {P} _ {\theta} (o = 0 \mid s, \tau_ {0}, \tau_ {1}) := \frac {\exp \left(r _ {\theta} (s , \tau_ {0})\right)}{\exp \left(r _ {\theta} (s , \tau_ {0})\right) + \exp \left(r _ {\theta} (s , \tau_ {1})\right)}, \tag {1} +$$ + +where $r_{\theta}(s,\tau_0)\coloneqq \sum_{h = 1}^{H}r_{\theta}(s_h,a_h)$ is the total reward of the trajectory $\tau_0$ given initial state $s_1 = s$ . In a contextual bandit, where each trajectory consists of a single action only, i.e., $\tau_0 = a_0$ and $\tau_{1} = a_{1}$ , the comparison model conveniently simplifies to $\mathbb{P}_{\theta}(o = 0\mid s,a_0,a_1)\propto \exp (r_{\theta}(s,a_0))$ . + +Strategic Preference Labeling. We assume that if labeler $i \in [k]$ provides their preferences truthfully, then the preference labels $o^{i,j}$ are sampled with respect to their true reward function $r_{\theta_i^*}$ . Thus, the collected preference dataset under truthful labeling is given by + +$$ +\mathcal {D} _ {i} ^ {*} = \left(s ^ {i, j}, \tau_ {0} ^ {i, j}, \tau_ {1} ^ {i, j}, o ^ {i, j}\right) _ {j \leq n} \text {w i t h} o ^ {i, j} \sim \mathbb {P} _ {\theta_ {i} ^ {*}} \left(\cdot \mid s ^ {i, j}, \tau_ {0} ^ {i, j}, \tau_ {1} ^ {i, j}\right). +$$ + +Since each labeler aims to more closely align the final policy with their personal preferences, a labeler may strategically manipulate labels to maximize their utility $J_{i}$ . In this case, we assume that labeler $i$ samples preference labels according to a manipulated reward function $r_{\tilde{\theta}_i}$ : + +$$ +\tilde {\mathcal {D}} _ {i} = (s ^ {i, j}, \tau_ {0} ^ {i, j}, \tau_ {1} ^ {i, j}, \tilde {o} ^ {i, j}) _ {j \leq n} \mathrm {w i t h} \tilde {o} ^ {i, j} \sim \mathbb {P} _ {\tilde {\theta} _ {i}} (\cdot | s ^ {i, j}, \tau_ {0} ^ {i, j}, \tau_ {1} ^ {i, j}). +$$ + +Note that the examples $(s^{i,j},\tau_0^{i,j},\tau_1^{i,j})$ remain fixed and only the preference labels change. + +Given a reported preference dataset $\mathcal{D} = (\mathcal{D}_1, \dots, \mathcal{D}_k)$ , an RLHF algorithm computes a policy $\hat{\pi}_{\mathrm{RLHF}}(\mathcal{D}) \in \Pi$ . We omit the argument $\mathcal{D}$ when the dataset is clear from context. We want to highlight that it is unknown to the RLHF algorithm (and impossible to tell) whether the preference labels in the dataset were truthfully reported or not. + +We can now define what it means for an RLHF algorithm to be robust against strategic manipulation (in the incentive alignment sense). In short, an RLHF method is strategyproof if truthfulness is an optimal strategy for every labeler irrespective of what the other labelers report. + +Definition 3.1 (Strategyproofness). We say that the mapping $\hat{\pi}_{\mathrm{RLHF}}(\mathcal{D})$ is a strategyproof if for all $i\in [k]$ , other labelers' data $\mathcal{D}_{-i} = (\mathcal{D}_1,\dots ,\mathcal{D}_{-i},\mathcal{D}_{i + 1},\dots ,\mathcal{D}_k)$ and deviation $\tilde{\theta}_i\neq \theta_i^*$ it holds that + +$$ +\mathbb {E} _ {o ^ {i, j} \sim \mathbb {P} _ {\theta_ {i} ^ {*}}} \left[ J _ {i} \big (\hat {\pi} _ {\mathrm {R L H F}} (\mathcal {D} _ {i} ^ {*}, \mathcal {D} _ {- i}) \big) \right] \geq \mathbb {E} _ {\bar {\sigma} ^ {i, j} \sim \mathbb {P} _ {\bar {\theta} _ {i}}} \left[ J _ {i} \big (\hat {\pi} _ {\mathrm {R L H F}} (\tilde {\mathcal {D}} _ {i}, \mathcal {D} _ {- i}) \big) \right]. +$$ + +This property is also commonly referred to as dominant strategy incentive compatibility. + +We can relax the strict incentive constraint by allowing labelers to have a limited incentive to misreport, which provides us with the notion of $\varepsilon$ -strategyproofness. + +Definition 3.2 ( $\varepsilon$ -Strategyproofness). We say that the mapping $\pi_{\mathrm{RLHF}}(\mathcal{D})$ is $\varepsilon$ -strategyproof with $\varepsilon > 0$ if for all $i \in [k]$ , other labelers' data $\mathcal{D}_{-i}$ and deviation $\tilde{\theta}_i \neq \theta_i^*$ it holds that + +$$ +\mathbb {E} _ {o ^ {i, j} \sim \mathbb {P} _ {\theta_ {i} ^ {*}}} \left[ J _ {i} \big (\pi_ {\mathrm {R L H F}} (\mathcal {D} _ {i} ^ {*}, \mathcal {D} _ {- i}) \big) \right] \geq \mathbb {E} _ {\bar {\theta} ^ {i, j} \sim \mathbb {P} _ {\bar {\theta} _ {i}}} \left[ J _ {i} \big (\pi_ {\mathrm {R L H F}} (\tilde {\mathcal {D}} _ {i}, \mathcal {D} _ {- i}) \big) \right] - \varepsilon . +$$ + +A few comments are in place. The careful reader might wonder why we define strategyproofness at the distributional level (ex ante) rather than at the level of realized preference labels, e.g., by allowing labelers to flip preferences after sampling. The reason is that defining misreporting at the level of preference realizations instead of preference distributions would blur the line between strategic manipulation and post hoc noise correction, which is not the focus of our analysis. One can imagine that even a (conceptually) perfectly strategyproof algorithm would incentivize the labelers to flip preference realizations post hoc in an attempt to better teach the algorithm their reward function by correcting noise. Defining the labelers' strategies over preference distributions instead of realizations ensures a more meaningful comparison between truthful and strategic labeling and avoids these complications. + +Learning Objective. We assume here that the set of labelers is representative of the population whose preferences we wish to align to. Our objective is then to compute a policy maximizing the average social welfare given by + +$$ +\mathcal {W} (\rho , \pi) := \frac {1}{k} \sum_ {i = 1} ^ {k} J _ {i} (\rho , \pi). +$$ + +In the following, we omit $\rho$ whenever the initial state distribution is clear from the context. Let $\pi^{*} := \operatorname{argmax}_{\pi \in \Pi} \mathcal{W}(\rho, \pi)$ be the optimal policy maximizing social welfare. The suboptimality of a policy $\pi$ is defined as + +$$ +\operatorname {S u b O p t} (\rho , \pi) := \mathcal {W} (\rho , \pi^ {*}) - \mathcal {W} (\rho , \pi). +$$ + +In addition to this standard notion of suboptimality, it can also be insightful to consider the multiplicative approximation ratio of a policy $\pi$ that is frequently studied in the computational social choice literature and given by the ratio + +$$ +\alpha (\rho , \pi) := \frac {\mathcal {W} (\rho , \pi)}{\mathcal {W} (\rho , \pi^ {*})}. +$$ + +By definition, this ratio satisfies $\alpha (\rho ,\pi)\leq 1$ and the larger the ratio the better the policy. In the following, we primarily use the approximation ratio as a secondary metric to understand the convergence behavior of an RLHF method, i.e., when the number of samples is sufficiently large. + +# 3.1 Existing RLHF is not Strategyproof + +Unsurprisingly, we find that existing RLHF algorithms are not strategyproof. Exemplarily, we consider two recently proposed RLHF methods for learning from diverse human preferences [37, 7]. Whereas Zhong et al. [37] aims to maximize social welfare like we do, Chakraborty et al. [7] consider a maximin objective, that is, they wish to maximize the worst-case utility across all labelers. While this is different from the social welfare objective that we consider, it does not prevent us from analyzing the strategyproofness of their algorithm or lack thereof. + +Proposition 3.3. Existing RLHF methods such as Pessimistic Social Welfare [37] and MaxMin-RLHF [7] are not strategyproof. + +Next, we wish to understand what consequences being manipulable has on the policy performance of the RLHF algorithm. After all, one could imagine failing to guarantee strategyproofness but still learning a nearly optimal policy. This is in general not the case and we show that the performance can degrade arbitrarily in the worst-case even if only a single labeler is strategic. We show this at the example of the Pessimistic Social Welfare approach from Zhong et al. [37]. + +Proposition 3.4. Let at least one out of the $k$ labelers report strategically. Let $\hat{\pi}$ denote the output of the Pessimistic Social Welfare algorithm [37]. Recall that $\| \theta \| _2\leq B$ and $\| \phi (s,a)\| _2\leq L$ . In the worst-case, for $n$ sufficiently large, the social welfare of $\hat{\pi}$ is upper bounded as $\mathcal{W}(\hat{\pi})\leq \varepsilon$ , whereas the optimal social welfare is at least $\mathcal{W}(\pi^{*})\geq BL - 2\varepsilon$ for any $\varepsilon >0$ . Hence, the suboptimality of Pessimistic Social Welfare is lower bounded by $\mathrm{SubOpt}(\hat{\pi})\geq BL - 3\varepsilon$ . + +In other words, the policy learned by Pessimistic Social Welfare can be almost arbitrarily bad. + +Proof Sketch. We provide a simple example for a contextual bandit where the first labeler strongly disagrees with all other labelers, but can exert significant influence on the computed policy by overstating its preference in a dimension of the features that is otherwise irrelevant to all labelers' utility (i.e., $\theta_{i}^{*}$ is zero in said dimension for all labelers). + +# 3.2 Inherent Limitations of Strategyproof RLHF + +We have seen that existing RLHF approaches are not strategyproof, but can be manipulated by labelers to the detriment of policy alignment with social welfare. We now also show that any RLHF algorithm that satisfies strategyproofness must suffer at least constant suboptimality (irrespective of the number of samples or policy coverage) and has an approximation ratio of at most $1 / k$ . We thereby face a fundamental trade-off between incentive alignment (strategyproofness) and policy alignment (social welfare maximization) in RLHF with strategic preference labeling. + +Theorem 3.5. The output $\hat{\pi}$ of any strategyproof RLHF algorithm has worst-case expected suboptimality at least $\mathrm{SubOpt}(\hat{\pi})\geq \frac{k - 1}{k}$ , where $k$ denotes the number of labelers. + +Proof Sketch. We can map each RLHF instance to a voting problem and map $\hat{\pi}$ to a decision rule $f$ for the latter, such that $f$ always outputs the same alternative (or distribution of alternatives) as $\hat{\pi}$ does. This construction ensures that if $\hat{\pi}$ is stratgyproof, then $f$ is, too. The Gibbard-Satterthwaite theorem [16, 31] says that any strategyproof rule must be either a dictatorial rule or a "duple", i.e., either it always selects the most preferred alternative of a fixed voter, or selects among a fixed pair of alternatives. Hence, if $\hat{\pi}$ is strategyproof, it must behave either as a dictatorial rule, always selecting the most preferred action of a fixed labeler, or as a duple, always selecting the outcome among a fixed pair of actions. The former case leads to low social welfare values for instances in which all the other labelers' rewards are negatively correlated with that of the fixed labeler. The latter leads to low social + +welfare values for instances in which the fixed pair of actions have almost zero value to all labelers. In both cases, the suboptimality gaps are at least $(k - 1) / k$ . + +Theorem 3.5 implies that even with infinitely many samples, no strategyproof RLHF algorithm converges to the optimal policy in the worst case. This is also reflected in the following upper bound on the multiplicative approximation ratio of any strategyproof algorithm. + +Corollary 3.6. The approximation ratio of any strategyproof RLHF method is $\alpha (\rho ,\hat{\pi})\leq \frac{1}{k}$ + +In other words, any strategyproof RLHF algorithm achieves $k$ -times worse social welfare compared to the optimal policy in the worst case. + +# 4 Approximate Strategyproofness: Pessimistic Median of MLEs + +We first consider the contextual bandit problem and discuss the extension to MDPs in Section 5. Our previous Theorem 3.5 suggests that without additional assumptions about the problem instance, we cannot reconcile strategyproofness with social welfare maximization. For this reason, we here introduce an additional assumption about the structure of the initial state distribution (i.e., context distribution) and the policy space. + +Assumption 2. The set $\{\mathbb{E}_{s\sim \rho}[\phi (s,\pi (s))]:\pi \in \Pi \}$ spans a hyperrectangle in $\mathbb{R}^d$ + +Specifically, in the simplest case when $\mathbb{E}_{s\sim \rho}[\phi (s,\pi (s))]\in [-1,1]^d$ , this means that for any $z\in [-1,1]^d$ there exists $\pi \in \Pi$ such that $\| \mathbb{E}_{s\sim \rho}\left[\phi (s,\pi (s)) - z\right]\| _2 = 0$ .5 + +We propose to use a median rule over learned reward parameters in combination with pessimistic estimates to achieve approximate strategyproofness while maximizing social welfare. To do so, we must first introduce a few key concepts and quantities. + +MLEs and Confidence. Let $\mathcal{D}_i = (s^{i,j}, a_0^{i,j}, a_1^{i,j}, o^{i,j})_{1 \leq j \leq n}$ be the preference data reported by labeler $i \in [k]$ where $o^{i,j} \sim \mathbb{P}_{\theta_i}(\cdot \mid s^{i,j}, a_0^{i,j}, a_1^{i,j})$ is sampled from a BT model w.r.t. some (a priori) unknown and potentially manipulated reward parameter $\theta_i$ . Given the observations $\mathcal{D}_i$ , the Maximum Likelihood Estimate (MLE) of $\theta_i$ is the maximizer of the log-likelihood + +$$ +\hat {\theta} _ {i} ^ {\mathrm {M L E}} \in \operatorname * {a r g m a x} _ {\theta} \sum_ {j = 1} ^ {n} \log \mathbb {P} _ {\theta} \left(o ^ {i, j} \mid s ^ {i, j}, a _ {0} ^ {i, j}, a _ {1} ^ {i, j}\right). +$$ + +We wish to establish confidence around the MLE. To this end, let $x^{i,j} = \phi(s^{i,j}, a_0^{i,j}) - \phi(s^{i,j}, a_1^{i,j})$ and consider the covariance matrix $\Sigma_{\mathcal{D}_i} = \frac{1}{n} \sum_{j=1}^n x^{i,j}(x^{i,j})^\top$ . For convenience, we here assume that $\Sigma_{\mathcal{D}_i}$ is positive definite. Otherwise, we can always consider $\Sigma_{\mathcal{D}_i} + \lambda_i I$ for $\lambda_i > 0$ , which has a negligible effect on our results when choosing $\lambda_i$ of order $\frac{d + \log(1 / \delta)}{n}$ (see, e.g., [38]). The confidence ellipsoid around $\hat{\theta}_i^{\mathrm{MLE}}$ is then given by + +$$ +C _ {i} := \left\{\theta \in \mathbb {R} ^ {d} \colon \| \hat {\theta} _ {i} ^ {\mathrm {M L E}} - \theta \| _ {\Sigma_ {\mathcal {D} _ {i}}} \leq f (d, n, \delta) \right\}. +$$ + +It is well-known that when choosing $f(d,n,\delta) \approx \sqrt{\frac{d + \log(k / \delta)}{n}}$ , it holds with probability at least $1 - \delta$ that $\theta_{i} \in C_{i}$ (see Appendix A.5 for details). + +Pessimistic Median Return. A fundamental insight from social choice theory is that under certain conditions aggregating preferences according to a median rule is strategyproof, such as in resource allocation in one dimension [26]. However, in our case, the high-dimensionality of features and reward parameters, the uncertainty about rewards, and the policy optimization pose additional unique challenges that can cause a median rule to become manipulable by the labelers. + +To incorporate our uncertainty about the reward parameters, we consider the pessimistic median return of a policy defined as the return of a policy w.r.t. the worst-case coordinate-wise median over confidence sets $C_1, \ldots, C_k$ . In other words, we consider the worst-case performance of policies $\pi$ with respect to $\mathrm{med}(\theta_1, \dots, \theta_k)$ , where med denotes the coordinate-wise median and $\theta_i$ is element in $C_i$ . We outline the Pessimistic Median of MLEs approach in Algorithm 1. + +Algorithm 1 Pessimistic Median of MLEs (Pessimistic MoMLEs) + +1: input offline preference data $\mathcal{D} = (\mathcal{D}_1, \dots, \mathcal{D}_k)$ +2: for every labeler $i \in [k]$ do +3: compute the MLE $\hat{\theta}_i^{\mathrm{MLE}}$ from $\mathcal{D}_i$ +4: construct confidence set $C_i \coloneqq \{\theta \in \mathbb{R}^d \colon \| \hat{\theta}_i^{\mathrm{MLE}} - \theta \|_{\Sigma_{\mathcal{D}_i}} \leq f(d, n, \delta)\}$ +5: end for +6: get the median confidence set $\mathcal{C} \coloneqq \{\mathrm{med}(\theta_1, \dots, \theta_k) \colon \theta_i \in C_i \text{ for } i \in [k]\}$ +7: compute the pessimistic median return w.r.t. $\mathcal{C}$ given by + +$$ +\underline {{\mathcal {W}}} (\pi) := \min _ {\theta \in \mathscr {C}} \mathbb {E} _ {s \sim \rho} [ \langle \theta , \phi (s, \pi (s)) \rangle ] +$$ + +8: return $\hat{\pi}(\mathcal{D}) = \operatorname{argmax}_{\pi \in \Pi} \underline{\mathcal{N}}(\pi)$ + +# 4.1 Approximate Strategyproofness + +We begin the analysis by showing that the Pessimistic Median of MLEs is approximately strategyproof. Perhaps surprisingly, the degree up to which the algorithm is strategyproof depends on the uniform policy coverage of every labeler's data. We discuss this in more detail further below. + +Theorem 4.1. Pessimistic Median of MLEs is $\tilde{\mathcal{O}} (\kappa_i\sqrt{d / n})$ -strategyproof for labeler $i$ , where $\kappa_{i}\coloneqq \max_{\pi \in \Pi}\| \mathbb{E}_{s\sim \rho}[\phi (s,\pi (s))]\|_{\Sigma_{\mathcal{D}_i}^{-1}}$ is the uniform policy coverage of $\mathcal{D}_i$ .6 + +More precisely, for every labeler $i \in [k]$ , any other labelers' reports $\mathcal{D}_{-i}$ and deviation $\tilde{\theta}_i \neq \theta_i^*$ , with probability at least $1 - \delta$ , the gain from misreporting is upper bounded as + +$$ +J _ {i} \big (\hat {\pi} (\tilde {\mathcal {D}} _ {i}, \mathcal {D} _ {- i}) \big) - J _ {i} \big (\hat {\pi} (\mathcal {D} _ {i} ^ {*}, \mathcal {D} _ {- i}) \big) \leq c o n s t \cdot \kappa_ {i} \sqrt {\frac {d + \log (k / \delta)}{n}}, +$$ + +where the labels in $\tilde{\mathcal{D}}_i$ are sampled from $\mathbb{P}_{\tilde{\theta}_i}$ and the labels in $\mathcal{D}_i^*$ are truthfully sampled from $\mathbb{P}_{\theta_i^*}$ . + +Proof Sketch. The key challenge is that the estimation errors of the reward parameters may unintentionally alter the median computation and thereby create unintended incentives for misreporting. To bound the gain from misreporting, we analyze the effect of estimation errors in conjunction with deviating choices of $\theta_{i}$ on the learned policy. Using concentration inequalities, we show that the deviation in each labeler's expected return is proportional to the estimation error, which scales as $\sqrt{d / n}$ . The worst-case impact on strategyproofness is then controlled by the uniform coverage coefficient $\kappa_{i}$ , which measures how well the labeler's data constrains policy choices. + +Whereas the $\sqrt{d / n}$ factor may be expected due to the construction of the confidence ellipsoids of corresponding size, the dependence on the uniform policy coverage coefficient $\kappa_{i}$ is unexpected at first, since coverage of only the optimal policy $\pi^{*}$ is usually sufficient in offline RL [29, 12, 36, 38]. However, in our case, we are not bounding the suboptimality of a learned policy but rather analyzing the strategic incentives of labelers. This shifts the focus to the range of possible policies that could result from different labeler behavior. Since labelers can, in principle, report arbitrarily misleading reward parameters—potentially inducing policies far from $\pi^{*}$ —bounding their incentive to deviate requires uniform policy coverage rather than coverage of any single specific policy. This ensures that no matter what policy is induced by a misreport, the confidence set remains well-constrained and bounds the potential gain from misreporting. + +# 4.2 Social Welfare Maximization + +We have shown that being truthful is approximately optimal for all labelers. Next, we provide guarantees on the suboptimality and the approximation ratio of the Pessimistic Median of MLEs algorithm when the labelers are either truthful or act according to their (potentially manipulating) + +weakly dominant strategy, which we show to exist. We begin with the case when the labelers are truthful, which is a $\tilde{\mathcal{O}} (\kappa_i\sqrt{d / n})$ -dominant strategy according to our previous Theorem 4.1. + +Theorem 4.2. Let $\hat{\pi}$ be the output of the Pessimistic Median of MLEs algorithm and suppose that all labelers report truthfully. With probability at least $1 - \delta$ : + +$$ +\operatorname {S u b O p t} (\hat {\pi}) \leq c o n s t \cdot \left(\sqrt {\frac {d \log (k / \delta)}{k}} + \max _ {i \in [ k ]} \kappa_ {i} ^ {*} \cdot k \sqrt {\frac {d + \log (k / \delta)}{n}}\right) \tag {2} +$$ + +where $\kappa_{i}^{*}\coloneqq \| \mathbb{E}_{s\sim \rho}[\phi (s,\pi^{*}(s))]\|_{\Sigma_{\mathcal{D}_{i}}^{-1}}$ is the optimal policy coverage of labeler $i$ . + +Proof Sketch. The suboptimality arises from two sources: (1) the deviation of the pessimistic median from the average, and (2) the deviation of each true reward parameter from its worst-case estimate in its respective confidence set. The first term follows from median concentration around the mean, contributing an error of $\mathcal{O}(\sqrt{d\log(k / \delta) / k})$ . The second term is upper bounded by $\mathcal{O}(\sqrt{(d + \log(k / \delta)) / n})$ , scaled by the worst-case policy coverage coefficient. Here, taking the median over confidence sets introduces an additional factor of $k$ . + +We also show that the Pessimistic Median of MLEs algorithm enjoys a suboptimality upper bound matching the one from Theorem 4.2 under any weakly dominant strategy it induces. + +Proposition 4.3. When the labelers report their preferences according to any weakly dominant strategy under Pessimistic Median of MLEs, with probability at least $1 - \delta$ , the output $\hat{\pi}$ satisfies: + +$$ +\operatorname {S u b O p t} (\hat {\pi}) \leq c o n s t \cdot \left(\sqrt {\frac {d \log (k / \delta)}{k}} + \max _ {i \in [ k ]} \kappa_ {i} ^ {*} \cdot k \sqrt {\frac {d + \log (k / \delta)}{n}}\right). +$$ + +where $\kappa_{i}^{*}\coloneqq \| \mathbb{E}_{s\sim \rho}[\phi (s,\pi^{*}(s))]\|_{\Sigma_{\mathcal{D}_{i}}^{-1}}$ is the optimal policy coverage of labeler $i$ . + +The bounds in Theorem 4.2 and Proposition 4.3 suggest two sources of suboptimality. The first term stems from approximating the social welfare function using the coordinate-wise median, which improves as the number of labelers increases. The second term results from the estimation of the underlying reward parameters, where the use of a median rule introduces an additional factor of $k$ . Overall, as the number of samples increases and as the number of labelers grows, the Pessimistic Median of MLEs algorithm converges to the optimal policy. + +Remark 4.4 (Suboptimality Lower Bound). The worst-case suboptimality of any RLHF algorithm in our problem setup is lower bounded by $\Omega (\sqrt{d / n})$ . This can be derived using a similar worst-case problem instance construction to the one in Zhu et al. [38]. + +We want to highlight the performance bounds of the Pessimistic Median of MLEs algorithm in two interesting special cases: (1) when there is only a single labeler so that $k = 1$ , and (2) when all $k$ labelers have identical reward functions. + +Corollary 4.5. When there is only a single labeler, with probability at least $1 - \delta$ : + +$$ +\mathrm {S u b O p t} (\hat {\pi}) \leq c o n s t \cdot \kappa_ {1} ^ {*} \sqrt {\frac {d + \log (k / \delta)}{n}}. +$$ + +When all $k$ labelers have the same reward function, with probability at least $1 - \delta$ : + +$$ +\mathrm {S u b O p t} (\hat {\pi}) \leq c o n s t \cdot \max _ {i \in [ k ]} \kappa_ {i} ^ {*} \cdot k \sqrt {\frac {d + \log (k / \delta)}{n}}. +$$ + +The result for the single labeler matches the existing bounds in the offline RLHF literature and is tight up to constants. Interestingly, we observe that in the special case of $k$ labelers with identical reward functions, the Pessimistic Median of MLEs avoids the additive $\mathcal{O}(\sqrt{d\log(k / \delta) / k})$ suboptimality but still suffers from an additional factor of $k$ as the algorithm anticipates strategic manipulation and preemptively takes the median over the confidence sets. + +Finally, we can also derive a lower bound on the approximation ratio of Algorithm 1. + +Corollary 4.6. Suppose $\mathcal{W}(\pi^{*}) > 0$ is constant. When the number of samples is sufficiently large and provide sufficient coverage of the optimal policy, with probability at least $1 - \delta$ , the approximation ratio of the Pessimistic Median of MLEs algorithm is given by $\alpha (\rho ,\hat{\pi})\geq 1 - \mathcal{O}\big(\sqrt{d\log(k / \delta) / k}\big)$ . + +# 5 Extension to Markov Decision Processes + +We now extend our algorithm and our previous results to MDPs. Recall that we consider trajectory-wise preferences so that labeler $i$ provides a preferences $o^{i,j}$ over two trajectories $\tau_0^{i,j}$ and $\tau_1^{i,j}$ given initial state $s^{i,j}$ according to a BT model $\mathbb{P}_{\theta_i}$ as defined in Section 3. Like before, the MLE of $\theta_{i}$ is given by the maximizer of the log-likelihood + +$$ +\theta_ {i} ^ {\mathrm {M L E}} := \operatorname * {a r g m a x} _ {\theta} \sum_ {j = 1} ^ {n} \log \mathbb {P} _ {\theta} (o ^ {i, j} \mid s ^ {i, j}, \tau_ {0} ^ {i, j}, \tau_ {1} ^ {i, j}). +$$ + +To construct the confidence ellipsoid around the MLE, let $x^{i,j} = \sum_{h=1}^{H} (\phi(s_h^{i,j}, a_h^{i,j}) - \phi(\bar{s}_h^{i,j}, \bar{a}_h^{i,j}))$ with $s_1^{i,j} = \bar{s}_1^{i,j} = s^{i,j}$ and consider the adapted covariance matrix $\Sigma_{\mathcal{D}_i} = \sum_{j=1}^n x^{i,j}(x^{i,j})^\top$ . Note that this agrees with our previous definition in the contextual bandit when $H = 1$ . + +To derive the pessimistic estimate of the median social welfare, we now consider the state occupancy of a policy $\pi$ given by $q_{\pi}(s \mid \rho) \coloneqq \frac{1}{H} \sum_{h=1}^{H} \mathcal{P}_h(s_h = s \mid \rho, \pi)$ . We can then express the expected return of policy $\pi$ w.r.t. reward parameter $\theta$ as $\mathbb{E}_{s \sim \rho}[V_{\theta}^{\pi}(s)] = \mathbb{E}_{s \sim q_{\pi}}[\langle \theta, \phi(s, \pi(s)) \rangle]$ and the pessimistic estimate of the median social welfare is given by $\underline{\mathcal{W}}(\pi) \coloneqq \min_{\theta \in \mathcal{C}} \mathbb{E}_{s \sim q_{\pi}}[\langle \theta, \phi(s, \pi(s)) \rangle]$ . The remainder of the Pessimistic Median of MLEs algorithm proceeds the same. + +We assume the analogue of Assumption 2 for MDPs. + +Assumption 3. The set $\{\mathbb{E}_{s\sim q_{\pi}}[\phi (s,\pi (s))]\colon \pi \in \Pi \}$ spans a hyperrectangle in $\mathbb{R}^d$ + +Under Assumption 3, we obtain the following extension of Theorem 4.1 that shows the approximate strategyproofness of the Pessimistic Median of MLEs. + +Theorem 5.1. Pessimistic Median of MLEs is $\tilde{\mathcal{O}} (\nu_i\sqrt{d / n})$ -strategyproof for labeler $i$ with uniform policy coverage coefficient $\nu_{i}\coloneqq \max_{\pi \in \Pi}\| \mathbb{E}_{s\sim q_{\pi}}[\phi (s,\pi (s))]\|_{\Sigma_{\mathcal{D}_i}^{-1}}$ + +More precisely, for every labeler $i \in [k]$ , any other labelers' data $\mathcal{D}_{-i}$ and manipulated reward parameter $\tilde{\theta}_i \neq \theta_i^*$ , with probability at least $1 - \delta$ , the gain from misreporting is bounded as + +$$ +J _ {i} \big (\hat {\pi} (\tilde {\mathcal {D}} _ {i}, \mathcal {D} _ {- i}) \big) - J _ {i} \big (\hat {\pi} (\mathcal {D} _ {i} ^ {*}, \mathcal {D} _ {- i}) \big) \leq c o n s t \cdot \nu_ {i}, \sqrt {\frac {d + \log (k / \delta)}{n}}. +$$ + +where the labels in $\tilde{\mathcal{D}}_i$ are sampled from $\mathbb{P}_{\tilde{\theta}_i}$ and the labels in $\mathcal{D}_i^*$ are truthfully sampled from $\mathbb{P}_{\theta_i^*}$ . + +The suboptimality upper bounds under truthful or weakly dominant reporting also take a similar form to their counterparts in Theorem 4.2 and Proposition 4.3. Similarly to before, the coverage of the optimal policy is enough. + +Theorem 5.2. When all labelers report truthfully or report according to their weakly dominant strategies, then with probability at least $1 - \delta$ : + +$$ +\operatorname {S u b O p t} (\hat {\pi}) \leq c o n s t \cdot \left(\sqrt {\frac {d \log (k / \delta)}{k}} + \max _ {i \in [ k ]} \nu_ {i} ^ {*} \cdot k \sqrt {\frac {d + \log (k / \delta)}{n}}\right) +$$ + +where $\nu_{i}^{*}\coloneqq \left\| \mathbb{E}_{s\sim q_{\pi^{*}}}\bigl [ \phi \bigl (s,\pi^{*}(s)\bigr) \bigr ]\right\|_{\Sigma_{\mathcal{D}_{i}}^{-1}}$ + +Note that we can also extend the corollaries from Section 4 to MDPs in a similar fashion. + +# 6 Discussion + +We studied how to robustify offline RLHF against strategic preference labeling in a pluralistic alignment setting with multiple labelers. We demonstrated a fundamental trade-off between incentive alignment and policy alignment and proposed the Pessimistic Median of MLEs algorithm that is based on pessimistic estimates of the median return of a policy. We showed that this algorithm is $\tilde{\mathcal{O}} (\sqrt{d / n})$ -strategyproof while guaranteeing suboptimality of at most $\tilde{\mathcal{O}} (\sqrt{d / k} +k\sqrt{d / n})$ . There are many directions for future work. It will be interesting to study strategyproofness for non-linear reward functions, parameterized or otherwise restricted policy classes, as well as more general preference models to the BT model used here. Another interesting future direction is to empirically evaluate the effect of strategic preference labeling on AI alignment and to validate algorithmic mechanisms designed to mitigate strategic manipulation. To do so at scale, e.g., in the context of LLM fine-tuning, we can expect scalability to come at the cost of theoretical guarantees. + +# Acknowledgments and Disclosure of Funding + +This work is supported by the EPSRC Prosperity Partnership FAIR (grant number EP/V056883/1), and by the ETH AI Center through an ETH AI Center Postdoctoral Fellowship to TKB. MK receives funding from the ERC under the European Union's Horizon 2020 research and innovation programme FUN2MODEL (grant agreement No. 834115). + +# References + +[1] Parand A Alamdari, Soroush Ebadian, and Ariel D Procaccia. Policy aggregation. Advances in Neural Information Processing Systems, 37:68308-68329, 2025. +[2] Daniel Alexander Alber, Zihao Yang, Anton Alyakin, Eunice Yang, Sumedha Rai, Aly A Valliani, Jeff Zhang, Gabriel R Rosenbaum, Ashley K Amend-Thomas, David B Kurland, et al. Medical large language models are vulnerable to data-poisoning attacks. 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If you obtained IRB approval, you should clearly state this in the paper. +- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. +- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: [NA] + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Proofs + +# A.1 Proof of Proposition 3.3 + +Proposition 3.3. Existing RLHF methods such as Pessimistic Social Welfare [37] and MaxMin-RLHF [7] are not strategyproof. + +Proof. We construct straightforward examples for each of the algorithms and show that the algorithms are not strategyproof. W.l.o.g. we assume that $n$ is sufficiently large with appropriate policy coverage so that the algorithms are able to obtain perfect estimates. We thereby also avoid re-defining the preference model $\mathbb{P}_{\theta}$ to fit the models considered in the respective work. W.l.o.g. we also ignore the KL-regularization w.r.t. the reference policy $\pi_{\mathrm{ref}}$ in MaxMin-RLHF, which would only add notational burden. In the following examples, the horizon is set to $H = 1$ , that is, we consider contextual bandit problems. + +Pessimistic Social Welfare: Suppose there are two labelers with true reward parameters $\theta_1^* = (1,0)$ and $\theta_2^* = (0,1)$ . Moreover, suppose that for all $s$ , we have $\phi(s,a) = (1/2,1/2)$ and $\phi(s,b) = (3/4,0)$ . If both labelers report truthfully, Pessimistic Social Welfare reports a policy $\pi(s) = a$ for all $s$ as action $a$ is maximizing social welfare. In this case, labeler 1 receives utility $1/2$ . Suppose that labeler 1 misreports as $\tilde{\theta}_1 = (1,-1)$ . As a result, the social welfare maximizing policy is $\pi(s) = b$ , which yields utility $3/4$ for labeler 1. Hence, misreporting is beneficial to labeler 1 and Pessimistic Social Welfare not strategyproof. + +MaxMin-RLHF: Consider the simple example where $\theta_1^* = (1,0)$ and $\theta_2^* = (1/2,1/2)$ as well as $\phi(s,a) = (1/2,1/2)$ and $\phi(s,b) = (3/4,0)$ for all $s$ . If both labelers report truthfully, the MaxMin-RLHF would compute the policy $\pi(s) = a$ which yields a return of $1/2$ for both labelers. However, suppose labeler 1 reports $\tilde{\theta}_1 = (1,-1)$ while labeler 2 truthfully reports $\theta_2^* = (1/2,1/2)$ . In this case, MaxMin-RLHF returns a policy $\tilde{\pi}(s) = b$ as this maximizes the minimal utility w.r.t. the reported reward parameters. The return for labeler 1 under policy $\tilde{\pi}$ is $3/4$ , which means that misreporting is to the benefit of labeler 1 and MaxMin-RLHF not strategyproof. + +![](images/7e23b957c0a7be1a971654d9da5b93738bd7f31bb8b7b3a0c0223031f3f87be5.jpg) + +# A.2 Proof of Proposition 3.4 + +Proposition 3.4. Let at least one out of the $k$ labelers report strategically. Let $\hat{\pi}$ denote the output of the Pessimistic Social Welfare algorithm [37]. Recall that $\| \theta \| _2\leq B$ and $\| \phi (s,a)\| _2\leq L$ . In the worst-case, for $n$ sufficiently large, the social welfare of $\hat{\pi}$ is upper bounded as $\mathcal{W}(\hat{\pi})\leq \varepsilon$ , whereas the optimal social welfare is at least $\mathcal{W}(\pi^{*})\geq BL - 2\varepsilon$ for any $\varepsilon >0$ . Hence, the suboptimality of Pessimistic Social Welfare is lower bounded by $\mathrm{SubOpt}(\hat{\pi})\geq BL - 3\varepsilon$ . + +In other words, the policy learned by Pessimistic Social Welfare can be almost arbitrarily bad. + +Proof. We construct a contextual bandit problem, where Pessimistic Social Welfare is arbitrarily bad if even a single labeler is strategic. We here assume that Pessimistic Social Welfare receives infinitely many samples from each labeler with full policy coverage. + +Let $\theta_1^* = (0,1,0)$ and $\theta_i^* = (\frac{B}{k - 1},0,0)$ for all $i\neq 1$ . Moreover, suppose that $\phi (s,a) = (\sqrt{L^2 - 2\varepsilon^2},0,0)$ and $\phi (s,b) = (0,\varepsilon ,\sqrt{L^2 - \varepsilon^2})$ for all $s$ . Under truthful reporting, Pessimistic Social Welfare computes the policy $\pi (s) = a$ , which clearly maximizes social welfare. In particular, the optimal social welfare is given by $\mathcal{W}(\pi^{*}) = B\sqrt{L^{2} - 2\varepsilon^{2}}\geq B(L - \sqrt{2}\varepsilon)$ . In this case, labeler 1 receives utility zero. However, suppose that labeler 1 misreports its reward parameter as $\tilde{\theta}_1^* = (0,0,B)$ . In this case, Pessimistic Social Welfare returns the policy $\tilde{\pi} (s) = b$ , which has social welfare $\mathcal{W}(\tilde{\pi}) = \varepsilon$ , whereas the utility of labeler 1 is $\varepsilon$ which is the best possible outcome for labeler 1. + +As a result, even if only labeler 1 misreports, then $\mathcal{W}(\hat{\pi}) = \varepsilon$ and the suboptimality is at least $\mathrm{SubOpt}(\hat{\pi}) = \mathcal{W}(\pi^{*}) - \varepsilon \geq BL - 3\varepsilon$ . We can choose $\varepsilon > 0$ arbitrarily small (in particular, note that $\varepsilon$ does not depend on the reward parameter so that any estimation error would have no effect). This means that the suboptimality of Pessimistic Social Welfare can be maximal. + +# A.3 Proof of Theorem 3.5 + +Theorem 3.5. The output $\hat{\pi}$ of any strategyproof RLHF algorithm has worst-case expected suboptimality at least $\mathrm{SubOpt}(\hat{\pi}) \geq \frac{k - 1}{k}$ , where $k$ denotes the number of labelers. + +Proof. Suppose for the sake of contradiction that the (worst-case) suboptimality gap of $\hat{\pi}$ is $\frac{k - 1}{k} - \epsilon$ for some $\epsilon > 0$ . We show that if this were true, we could construct a voting rule based on $\hat{\pi}$ that is strategyproof, non-dictatorial, and onto at least three alternatives, contradicting the Gibbard-Satterthwaite theorem [16, 31]. The Gibbard-Satterthwaite theorem asserts that such a voting rule does not exist. We will consider the case where $\hat{\pi}$ is deterministic and discuss how the proof generalizes to the case with randomized $\hat{\pi}$ , too. + +Specifically, consider a voting instance with $k$ voters and $m$ alternatives $a_1, \ldots, a_m$ . A voting rule $f$ maps every possible preference profile of the voters to one of the alternatives. + +To construct a voting rule based on $\hat{\pi}$ , we first map every voting instance $I_{\mathrm{V}} = \prec := (\prec_1, \dots, \prec_k)$ to an RLHF instance $I_{\mathrm{RLHF}}$ as follows. + +- Let there be one state (so we omit the state in what follows) and $m$ actions $a_1, \ldots, a_m$ , each corresponding to an alternative in $I_{\mathrm{V}}$ . +- The feature embedding $\phi$ maps each action $a_{\ell}$ to the unit vector whose $\ell$ -th component is 1. +- Let there be $k$ labelers, each corresponding to a voter in $I_{\mathrm{V}}$ . Each labeler $i$ 's parameter $\theta_i^*$ is a vector in which the $\ell$ -th component is defined as follows: + +- 1 if $a_{\ell}$ is the most preferred alternative according to voter $i$ 's preference order $\prec_{i}$ in $I_{\mathrm{V}}$ . +- $1 - \delta$ (for a sufficiently small $\delta$ ) if $a_{\ell}$ is the second most preferred alternative according to $\prec_{i}$ . +- $\delta \cdot (m - j)$ if $a_{\ell}$ is the $j$ -th most preferred alternative, $j > 2$ , according to $\prec_{i}$ . + +The parameters ensure that each labeler $i$ 's preference over the actions is the same as $\prec_{i}$ . + +With the above map from $I_{\mathrm{V}}$ to $I_{\mathrm{RLHF}}$ , we then let $f$ be a voting rule that outputs alternative $a_{\ell}$ if $\hat{\pi}$ outputs action $a_{\ell}$ in $I_{\mathrm{RLHF}}$ . Clearly, $f$ must be a strategyproof given our assumption that $\hat{\pi}$ is a strategyproof and the fact that $I_{\mathrm{RLHF}}$ preserves the preference orders in $I_{\mathrm{V}}$ . We next argue that, given the assumption that $\hat{\pi}$ has a suboptimality gap of at most $\frac{k - 1}{k} - \epsilon$ , $f$ must be non-dictatorial and onto at least three alternatives. + +$f$ is Non-Dictatorial. Suppose that $f$ is dictatorial; say, it always outputs the most preferred alternative of voter 1. This means that $\hat{\pi}$ must output $a_1$ on the RLHF instance that the following $I_{\mathrm{V}} = \prec$ instance maps to: + +$$ +a _ {1} \prec_ {1} a _ {2} \prec_ {1} \dots \prec_ {1} a _ {m - 1} \prec_ {1} a _ {m} +$$ + +$$ +a _ {2} \prec_ {i} a _ {3} \prec_ {i} \dots \prec_ {i} a _ {m} \prec_ {i} a _ {1} \quad \text {f o r a l l} i = 2, \ldots , k. +$$ + +However, this contradicts the assumption that the suboptimality gap of $\hat{\pi}$ is at most $\frac{k - 1}{k} -\epsilon$ . To see this, note that $a_1$ achieves social welfare 1 in $I_{\mathrm{RLHF}}$ , while $a_2$ achieves social welfare $k(1 - \delta)$ . The suboptimality is then at least $\frac{k - 1 - \delta k}{k - \delta k} >\frac{k - 1}{k} -\epsilon$ when $\delta \to 0$ . + +$f$ is Onto. Similarly, we can argue that $f$ must be onto at least three alternatives by considering the set of all possible voting instances where all voters' preferences are identical. In this case, $\hat{\pi}$ must always output either the most or the second preferred actions of the labelers in the corresponding RLHF instances; otherwise, the suboptimality gap of $\hat{\pi}$ can be arbitrarily close to 1 when $\delta \to 0$ . Consequently, when all possible preference orders are considered, $\hat{\pi}$ , and hence $f$ , must be onto at least three different alternatives (suppose that $k \geq 4$ ). + +As a result, we obtain a voting rule that is strategyproof, non-dictatorial, and onto at least three alternatives. This contradicts the Gibbard-Satterthwaite theorem. The stated lower bound on the suboptimality gap then follows for deterministic policies. + +Randomized Policies. To further argue that the same bound holds even when $\hat{\pi}$ is randomized, we invoke a generalization of the Gibbard-Satterthwaite theorem to randomized voting rules [17], which states that a voting rule, if strategyproof, must be a probability mixture of dictatorial rules and "duples". A voting rule is a duple if it restricts its outcomes, over all possible instances, to a fixed pair of alternatives. + +Similarly to our approach above, we construct a voting rule $f$ based on $\hat{\pi}$ the same way we did above and argue that, if $\hat{\pi}$ is a strategyproof and has a suboptimality gap of $\frac{k - 1}{k} - \epsilon$ for some $\epsilon > 0$ , then $f$ cannot be a probability mixture of dictatorial rules and triples. + +Suppose for the sake of contradiction that $f$ is a mixture of dictatorial rules and triples. Consider the following set of voting instances $(\prec^j)_j$ , each involving a set of alternatives $\{a_1, \ldots, a_k, b_1, \ldots, b_K\}$ , $K \geq 4 / \epsilon$ (hence, $m = k + K$ ). In each instance $\prec^j = (\prec_1^j, \ldots, \prec_k^j)$ , the preference order $\prec_i^j$ ranks $a_i$ first, $b_j$ second, and all other alternatives according to the order $a_1, \ldots, a_k, b_1, \ldots, b_k$ . + +By assumption, $f$ is a mixture of dictatorial rules and triples. Now that there are more than $K$ alternatives while each triple selects at most two alternatives, there must be at least one alternative among $b_{1},\ldots ,b_{K}$ that is selected by the triples with probability at most $2 / K$ . W.l.o.g., let this alternative be $b_{1}$ and consider the instance $\prec^{1}$ , in which each voter $i$ ranks $a_{i}$ the first and $b_{1}$ the second. + +Clearly, in this instance, the policy that outputs $b_{1}$ deterministically achieves social welfare $k(1 - \delta)$ . Moreover, any other alternatives yields a social welfare of at most $1 + \delta mk$ , as each of these alternatives is ranked first by at most one labeler and third or even lower by all other labelers. Since $f$ selects $b_{1}$ with probability at most $2 / K$ , so does $\hat{\pi}$ . This means that the social welfare achieved by $\hat{\pi}$ is at most + +$$ +\frac {2}{K} \cdot k (1 - \delta) + \left(1 - \frac {2}{K}\right) (1 + \delta m k) < \epsilon k / 2 + 1 + \delta m k. +$$ + +This gives a suboptimality gap of $1 - \frac{\epsilon k / 2 + 1 + \delta mk}{k(1 - \delta)} > \frac{k - 1}{k} - \epsilon$ when $\delta \to 0$ . + +![](images/05a5de90abb3433d5251d6ba71ba1460315b98fb8302e7991ea1086cce34442a.jpg) + +# A.4 Proof of Corollary 3.6 + +Corollary 3.6. The approximation ratio of any strategyproof RLHF method is $\alpha (\rho ,\hat{\pi})\leq \frac{1}{k}$ + +Proof. This is a direct consequence of Theorem 3.5. $\square$ + +![](images/6ad7d94039c2708a2ece2e812d98f192e3516009ca9f75b70943c07b7c30fe21.jpg) + +# A.5 Proof of Theorem 4.1 + +Before we begin with some preliminaries that we will repeatedly use in the proofs of both Theorem 4.1 and Theorem 4.2. Firstly, we recall a standard MLE concentration bound, which can be found, for instance, in [38]. + +Lemma A.1 (MLE Concentration Bound). In the contextual bandit problem, with probability at least $1 - \delta$ , + +$$ +\| \hat {\theta} _ {i} ^ {\mathrm {M L E}} - \theta_ {i} ^ {*} \| _ {\Sigma_ {\mathcal {D} _ {i}}} \leq c o n s t \cdot \sqrt {\frac {d + \log (1 / \delta)}{\gamma^ {2} n}}, +$$ + +where $\gamma \coloneqq 1 / (2 + \exp (-LB) + \exp (LB))$ . Note that we here assume that the covariance matrix $\mathcal{D}_i$ is positive definite. Otherwise, consider $\Sigma_{\mathcal{D}_i} + \lambda I$ , which adds an additive term $\lambda B^2$ in the square root. + +Proof. See, e.g., [38]. + +![](images/c234bedb4f0d73229547cb0768b986466806ce990b1065a0cc878b155bf3735f.jpg) + +Lemma A.2 (Median Concentration Bound). Suppose that $\theta_{1},\ldots ,\theta_{k}\in \mathbb{R}^{d}$ are sampled i.i.d. from some $\sigma$ -sub Gaussian distribution. Let $\hat{\theta}_{\mathrm{med}}$ be the coordinate-wise median and $\hat{\theta}_{\mathrm{avg}}$ the average of $\theta_{1},\dots,\theta_{k}$ . Then, for a universal constant $c > 0$ , it holds that, for every $t > 0$ + +$$ +\mathbb {P} \left(\| \hat {\theta} _ {\mathrm {m e d}} - \hat {\theta} _ {\mathrm {a v g}}) \| _ {2} \geq t\right) \leq 2 \exp \left(- \frac {c k t ^ {2}}{d \sigma^ {2}}\right). +$$ + +Hence, in other words, with probability at least $1 - \delta$ : + +$$ +\| \hat {\theta} _ {\mathrm {m e d}} - \hat {\theta} _ {\mathrm {a v g}} \| _ {2} \leq \mathcal {O} \left(\sigma \sqrt {\frac {d \log (1 / \delta)}{k}}\right). +$$ + +Proof. We begin by proving the median concentration in one dimension. To this end, let $\theta_{\mathrm{avg}}^*$ denote the mean of the distribution. Since each $\theta_i$ is $\sigma$ -sub-Gaussian with mean $\theta_{\mathrm{avg}}^*$ , the centered variables $X_i = \theta_i - \theta_{\mathrm{avg}}^*$ satisfy + +$$ +\mathbb {P} \big (| X _ {i} | \geq u \big) \leq 2 \exp \big (- \frac {c _ {2} u ^ {2}}{\sigma^ {2}} \big) +$$ + +for some constant $c_2 > 0$ . To control $\mathbb{P}(\hat{\theta}_{\mathrm{med}} \geq \theta_{\mathrm{avg}}^* + t)$ , note that if $\hat{\theta}_{\mathrm{med}} \geq \theta_{\mathrm{avg}}^* + t$ , at least half of the $\theta_i$ are at least $\theta_{\mathrm{avg}}^* + t$ . Define + +$$ +p = \mathbb {P} \left(\theta_ {i} \geq \theta^ {*} + t\right) = \mathbb {P} \left(X _ {i} \geq t\right). +$$ + +By sub-Gaussianity, $p \leq \exp \left(-\frac{c_2t^2}{\sigma^2}\right)$ . Let $Y = \sum_{i=1}^{k} \mathbf{1}\{\theta_i \geq \theta^* + t\}$ , which follows a Binomial $(k,p)$ distribution. Then $\mathbb{P}(\hat{\theta}_{\mathrm{med}} \geq \theta^* + t) \leq \mathbb{P}\big(Y \geq k/2\big)$ . A Chernoff or Hoeffding bound implies + +$$ +\mathbb {P} \left(Y \geq k / 2\right) \leq \exp \left(- k D \left(\frac {1}{2} \| p\right)\right), +$$ + +where $D\left(\frac{1}{2}\parallel p\right)$ is the Kullback-Leibler divergence between Bernoulli(1/2) and Bernoulli(p). For $p \ll 1 / 2$ , $D\left(\frac{1}{2}\parallel p\right)$ is bounded below by a constant times $(1 / 2 - p)^2$ . Hence, there exists $c_{1} > 0$ such that + +$$ +\mathbb {P} (\hat {\theta} _ {\mathrm {m e d}} \geq \theta_ {\mathrm {a v g}} ^ {*} + t) \leq \exp \big (- \frac {c _ {1} k t ^ {2}}{\sigma^ {2}} \big). +$$ + +By a symmetric argument, $\mathbb{P}(\hat{\theta}_{\mathrm{med}} \leq \theta_{\mathrm{avg}}^* - t) \leq \exp \left(-\frac{c_1 kt^2}{\sigma^2}\right)$ . Combining these, we obtain + +$$ +\mathbb {P} \big (\left| \hat {\theta} _ {\mathrm {m e d}} - \theta_ {\mathrm {a v g}} ^ {*} \right| \geq t \big) \leq \mathbb {P} (\hat {\theta} _ {\mathrm {m e d}} \geq \theta_ {\mathrm {a v g}} ^ {*} + t) + \mathbb {P} (\hat {\theta} _ {\mathrm {m e d}} \leq \theta_ {\mathrm {a v g}} ^ {*} - t) \leq 2 \exp \big (- \frac {c _ {1} k t ^ {2}}{\sigma^ {2}} \big). +$$ + +From Hoeffding's inequality we get an analogous bound for $\mathbb{P}(\|\hat{\theta}_{\mathrm{avg}} - \theta_{\mathrm{avg}}^*\| \geq t)$ so that we get the desired result for $d = 1$ using the triangle inequality $|\hat{\theta}_{\mathrm{med}} - \hat{\theta}_{\mathrm{avg}}| \leq |\hat{\theta}_{\mathrm{med}} - \theta_{\mathrm{avg}}^*| + |\theta_{\mathrm{avg}}^* - \hat{\theta}_{\mathrm{avg}}|$ . Finally, this translates to a bound in $d > 1$ dimensions by using Jensen's inequality + +$$ +\| \hat {\theta} _ {\text {m e d}} - \hat {\theta} _ {\text {a v g}} \| _ {2} = \sqrt {\sum_ {j = 1} ^ {d} \left(\hat {\theta} _ {\text {m e d} , j} - \hat {\theta} _ {\text {a v g} , j}\right) ^ {2}} \leq \sqrt {d} \max _ {j \in [ d ]} | \hat {\theta} _ {\text {m e d}, j} - \hat {\theta} _ {\text {a v g}, j} | +$$ + +and applying the previous bound for each dimension. + +![](images/985026d7ba9862a8c377b76d9dfb48d1806a39802e21302e1aebb8e5d9d1028b.jpg) + +We are now ready to prove Theorem 4.1 + +Theorem 4.1. Pessimistic Median of MLEs is $\tilde{\mathcal{O}} (\kappa_i\sqrt{d / n})$ -strategyproof for labeler $i$ , where $\kappa_{i}\coloneqq \max_{\pi \in \Pi}\| \mathbb{E}_{s\sim \rho}[\phi (s,\pi (s))]\|_{\Sigma_{\mathcal{D}_i}^{-1}}$ is the uniform policy coverage of $\mathcal{D}_i$ .7 + +More precisely, for every labeler $i \in [k]$ , any other labelers' reports $\mathcal{D}_{-i}$ and deviation $\tilde{\theta}_i \neq \theta_i^*$ , with probability at least $1 - \delta$ , the gain from misreporting is upper bounded as + +$$ +J _ {i} \big (\hat {\pi} (\tilde {\mathcal {D}} _ {i}, \mathcal {D} _ {- i}) \big) - J _ {i} \big (\hat {\pi} (\mathcal {D} _ {i} ^ {*}, \mathcal {D} _ {- i}) \big) \leq c o n s t \cdot \kappa_ {i} \sqrt {\frac {d + \log (k / \delta)}{n}}, +$$ + +where the labels in $\tilde{\mathcal{D}}_i$ are sampled from $\mathbb{P}_{\tilde{\theta}_i}$ and the labels in $\mathcal{D}_i^*$ are truthfully sampled from $\mathbb{P}_{\theta_i^*}$ . + +Proof. We begin first with the case where every individual can directly report their reward parameter to the algorithm, hence, removing the noise and uncertainty from the process. In this case, we show that Pessimistic Median of MLEs is exactly strategyproof. + +Case 1 (direct access to $\theta_1, \ldots, \theta_k$ ): Let us begin with the case where we obtain infinitely many samples with appropriate coverage so that $C_i = \{\theta_i\}$ for all individuals $i \in [k]$ . We need to show that reporting $\theta_i^*$ is the optimal strategy for individual $i$ irrespective of the other individuals' strategies. + +The following two basic lemmas will prove useful. + +Lemma A.3. Let $\theta_{-i} \in \mathbb{R}^{(k - 1) \times d}$ be fixed arbitrarily. For any $j \in [d]$ the following holds: + +- If $\theta_{i,j}^{*} > 0$ and $\operatorname{med}(\theta_{-i,j}, \theta_{i,j}^{*}) < 0$ , then $\operatorname{med}(\theta_{-i,j}, \theta_{i,j}) < 0$ for all $\theta_i \in \mathbb{R}^d$ . +- Analogously, if $\theta_{i,j}^{*} < 0$ and $\operatorname{med}(\theta_{-i,j}, \theta_{i,j}^{*}) > 0$ , then $\operatorname{med}(\theta_{-i,j}, \theta_{i,j}) > 0$ for all $\theta_{i} \in \mathbb{R}^{d}$ . + +Proof. W.l.o.g. let $\theta_{i,j}^{*} > 0$ and let $\theta_{i} \in \mathbb{R}^{d}$ . Suppose that $\theta_{i,j} < 0$ . It follows directly that $\operatorname{med}(\theta_{-i,j}, \theta_{i,j}) < \operatorname{med}(\theta_{-i,j}, \theta_{i,j}^{*})$ . Alternatively, suppose that $\theta_{i,j} > 0$ . Since $\operatorname{med}(\theta_{-i,j}, \theta_{i,j}^{*}) < 0$ , it means that the median equals some $\theta_{l,j} < 0$ with $l \neq i$ . Hence, the median does not change for any alternative choice $\theta_{i,j} > 0$ . + +We assume hyperrectangularity, which allows use to decompose the reward-maximizing policy as follows. For a given policy $\pi$ , let $z_{\pi} \coloneqq \mathbb{E}_{s \sim \rho}[\phi(s, \pi(s))] \in \mathbb{R}^d$ denote its feature occupancy and let $z_{\pi,j}$ be its $j$ -th entry. W.l.o.g. we here assume $z_{\pi,j} \in [-1, 1]$ , but any other lower and upper bounds can be considered the same way. + +We denote the optimal policy w.r.t. a reward parameter $\theta$ as $\pi^{*}(\theta) \coloneqq \operatorname{argmax}_{\pi \in \Pi} J_{\theta}(\pi)$ . From Assumption 2 it follows that the optimal policy $\pi^{*}(\theta)$ is such that $z_{\pi^{*}(\theta),j} = -1$ for $\theta_j < 0$ and $z_{\pi^{*}(\theta),j} = +1$ for $\theta_j > 0$ . This yields an equivalence between reward parameters that have identical signs. In particular, this provides us with a class of reward parameters that induce an optimal policy w.r.t. the true reward parameter $\theta_i^*$ . + +Lemma A.4. Let $\theta \in \mathbb{R}^d$ . If $\mathrm{sign}(\theta_{i,j}^{*}) = \mathrm{sign}(\theta_{j})$ , then $\theta_{i,j}^{*}\cdot \pmb{z}_{\pi^{*}(\theta),j}\geq \theta_{i,j}^{*}\cdot \pmb{z}_{\pi^{*}(\tilde{\theta}),j}$ for all $\tilde{\theta}\in \mathbb{R}^d$ . + +Proof. This follows from the structure of the optimal policies $\pi^{*}(\theta)$ under Assumption 2. + +![](images/ace68ba7a980ca9ea1b4f5725d4322fedcf5ce29e9edbb716de31641226bb27b.jpg) + +We fix everyone's reported parameter $\theta_{-i}$ except for individual $i$ . Moreover, let $\tilde{\theta}_i \neq \theta_i^*$ and let $\tilde{\mu} \coloneqq \mathrm{med}(\theta_{-i}, \tilde{\theta}_i)$ be the coordinate-wise median w.r.t. $\tilde{\theta}_i$ . Similarly, let $\mu^* = \mathrm{med}(\theta_{-i}, \theta_i^*)$ be the coordinate-wise median w.r.t. $\theta^*$ . We will now show that $J_{\theta_i^*}(\hat{\pi}(\mu^*)) \geq J_{\theta_j^*}(\hat{\pi}(\tilde{\mu}))$ , i.e., reporting $\theta_i^*$ is the optimal strategy for individual $i$ under the Pessimistic Median of MLEs algorithm. + +Since we here assume direct access to the reported parameters, given reported parameters $\theta_{1},\ldots ,\theta_{k}$ , Pessimistic Median of MLEs computes the optimal policy w.r.t. the median $\mu = \mathrm{med}(\theta_1,\dots ,\theta_k)$ i.e., $\hat{\pi} = \pi^{*}(\mu) = \operatorname{argmax}_{\pi \in \Pi}J_{\mu}(\pi)$ . We here assume that the $\mu$ -maximizing policy is unique and otherwise use lexicographic tie-breaking. Clearly, if $\mathrm{signs}(\mu^{*}) = \mathrm{signs}(\tilde{\mu})$ , then the policies $\hat{\pi} (\mu^{*})$ and $\hat{\pi} (\tilde{\mu})$ are identical. + +Next, consider any $j \in [d]$ so that $\mathrm{sign}(\mu_j^*) \neq \mathrm{sign}(\tilde{\mu}_j)$ . Suppose that $\mathrm{sign}(\mu_j^*) = \mathrm{sign}(\theta_{i,j}^*)$ . In this case, Lemma A.4 tells us that $\theta_{i,j}^* \cdot z_{\hat{\pi} (\mu^*),j} \geq \theta_{i,j}^* \cdot z_{\hat{\pi} (\tilde{\mu}),j}$ . Hence, in any such dimension $j$ , $\mu^*$ implies a policy that outperforms the policy maximizing $\tilde{\mu}$ w.r.t. labeler $i$ 's true reward parameter $\theta_i^*$ . Hence, misreporting $\tilde{\theta}_{i,j} \neq \theta_{i,j}^*$ cannot be a strictly better strategy than truthfully reporting in dimension $j$ . + +Suppose that $\mathrm{sign}(\mu_j^*)\neq \mathrm{sign}(\theta_{i,j}^*)$ . In this case, Lemma A.3 implies that $\mathrm{sign}(\tilde{\mu}_j) = \mathrm{sign}(\mu_j^*)$ which implies $\theta_{i,j}^{*}\cdot z_{\tilde{\pi} (\tilde{\mu}),j} = \theta_{i,j}^{*}\cdot z_{\tilde{\pi} (\mu^{*}),j}$ . Once again misreporting is never a strictly better strategy than truthfully reporting $\theta_i^*$ . + +We have thus confirmed that reporting $\theta_{i}^{*}$ is optimal irrespective of the other individuals' reports $\theta_{-i}$ . + +Case 2 (direct access to $\theta_{i}$ , but not $\theta_{-i}$ ): Let $C_{-i}$ denote the product space of confidence sets derived from the preference data $\mathcal{D}_{-i}$ of all labelers but labeler $i$ . Once again, the key lies in the observation that the assumption of hyperrectangularity implies that the labelers get to strategize over each dimension independently. + +The main concern that we must alleviate is that, by taking the minimum over the confidence sets, misreporting becomes beneficial for the labelers. To this end, let $\theta_{i}$ be the report of individual $i$ , which we for now assume to be directly observable. Given preference data $\mathcal{D}_{-i}$ and $\theta_{i}$ , the Pessimistic Median of MLEs computes a policy maximizing + +$$ +\min _ {\theta_ {- i} \in C _ {- i}} \sum_ {j = 1} ^ {d} \left\langle \operatorname {m e d} \left(\theta_ {- i, j}, \theta_ {i, j}\right), \mathbb {E} _ {s \sim \rho} \left[ \phi (s, \pi (s)) \right] \right\rangle +$$ + +Suppose that $\theta_{i,j}^{*} > 0$ for $j\in [d]$ . Clearly, by design of the median, for any $\theta_{-i,j}$ , it follows from the same argument as in Lemma A.3 that misreporting either has no effect on the policy (if the report is $\theta_{i,j} > 0$ ), or can only have an adverse effect for labeler $i$ (if the report is $\theta_{i,j} < 0$ ). Hence, it is optimal for individual $i$ to report $\theta_{i}^{*}$ irrespective of the other individuals' reported preference data $\mathcal{D}_{-i}$ and the confidence sets that we construct. + +Case 3 (no direct access to $\theta_1, \ldots, \theta_k$ ): In the previous cases, we have shown that truthfully reporting is a dominant strategy for every individual $i \in [k]$ . We will now see that this is in general no longer true when an individual cannot directly share their reward parameter with the algorithm. The reason lies in unintentional changes in the sign due to estimation errors and confidence sizes. + +In the following, we assume that the preference data $\mathcal{D}_{-i}$ of all labelers but labeler $i$ are fixed arbitrarily and $C_{-i}$ are the corresponding confidence sets that Pessimistic Median of MLEs constructs. From Case 2 we know that the policy + +$$ +\hat {\pi} _ {i} \left(\theta_ {i} ^ {*}\right) := \underset {\pi \in \Pi} {\operatorname {a r g m a x}} \underset {\theta_ {- i} \in C _ {- i}} {\min } \left\langle \operatorname {m e d} \left(\theta_ {i} ^ {*}, \theta_ {- i}\right), \mathbb {E} _ {s \sim \rho} \left[ \phi (s, \pi (s)) \right] \right\rangle +$$ + +is preferred over any other policy $\hat{\pi} (C_i)$ computed w.r.t. any confidence set $C_i$ given by + +$$ +\hat {\pi} _ {i} (C _ {i}) := \operatorname * {a r g m a x} _ {\pi \in \Pi} \min _ {\theta_ {i} \in C _ {i}} \min _ {\theta_ {- i} \in C _ {- i}} \langle \operatorname {m e d} (\theta_ {i}, \theta_ {- i}), \mathbb {E} _ {s \sim \rho} [ \phi (s, \pi (s)) ] \rangle , +$$ + +i.e., $J_{\theta_i^*}(\hat{\pi}_i(\theta_i^*)) \geq J_{\theta_i^*}(\hat{\pi}_i(C_i))$ for any confidence set $C_i$ . + +Let us now consider the confidence set $C_i^*$ derived from $\mathcal{D}_i^*$ , which is sampled according to the true reward parameter $\theta_i^*$ . By construction of the confidence sets, with probability at least $1 - \delta$ , it follows from Lemma A.1 that for any $\theta_i \in C_i^*$ : + +$$ +\| \theta_ {i} ^ {*} - \theta_ {i} \| _ {\Sigma_ {\mathcal {D} _ {i}}} \leq \| \theta_ {i} ^ {*} - \hat {\theta} _ {i} ^ {\mathrm {M L E}} \| _ {\Sigma_ {\mathcal {D} _ {i}}} + \| \hat {\theta} _ {i} ^ {\mathrm {M L E}} - \theta_ {i} \| _ {\Sigma_ {\mathcal {D} _ {i}}} \leq 2 c \sqrt {\frac {d + \log (1 / \delta)}{\gamma^ {2} n}}. +$$ + +We now compare the difference in return w.r.t. $\theta_{i}^{*}$ of policy $\hat{\pi}_i(\theta_i^*)$ and $\hat{\pi}_i(C_i^*)$ . To do so, we decompose the difference as follows: + +$$ +\begin{array}{l} J _ {\theta_ {i} ^ {*}} (\hat {\pi} _ {i} (\theta_ {i} ^ {*})) - J _ {\theta_ {i} ^ {*}} (\hat {\pi} _ {i} (C _ {i} ^ {*})) \\ = \Big (J _ {\theta_ {i} ^ {*}} (\hat {\pi} _ {i} (\theta_ {i} ^ {*})) - \min _ {\theta_ {i} \in C _ {i} ^ {*}} J _ {\theta_ {i}} (\hat {\pi} _ {i} (\theta_ {i} ^ {*})) \Big) + \Big (\min _ {\theta_ {i} \in C _ {i} ^ {*}} J _ {\theta_ {i}} (\hat {\pi} _ {i} (\theta_ {i} ^ {*})) - J _ {\theta_ {i} ^ {*}} (\hat {\pi} _ {i} (C _ {i} ^ {*}))) \Big). \\ \end{array} +$$ + +Using Cauchy-Schwarz, the first difference can be rewritten and bounded as + +$$ +\max _ {\theta_ {i} \in C _ {i} ^ {*}} \langle \theta_ {i} ^ {*} - \theta_ {i}, \boldsymbol {z} _ {\hat {\pi} (\theta_ {i} ^ {*})} \rangle \leq \| \theta_ {i} ^ {*} - \theta_ {i} \| _ {\Sigma_ {\mathcal {D} _ {i}}} \| \boldsymbol {z} _ {\hat {\pi} (\theta_ {i} ^ {*})} \| _ {\Sigma_ {\mathcal {D} _ {i}} ^ {- 1}}. +$$ + +We then further decompose the second difference into + +$$ +\begin{array}{l} \min _ {\theta_ {i} \in C _ {i} ^ {*}} J _ {\theta_ {i}} \left(\hat {\pi} _ {i} \left(\theta_ {i} ^ {*}\right)\right) - J _ {\theta_ {i} ^ {*}} \left(\hat {\pi} _ {i} \left(C _ {i} ^ {*}\right)\right) \\ = \Big (\min _ {\theta_ {i} \in C _ {i} ^ {*}} J _ {\theta_ {i}} (\hat {\pi} _ {i} (\theta_ {i} ^ {*})) - \min _ {\theta_ {i} \in C _ {i} ^ {*}} J _ {\theta_ {i}} (\hat {\pi} _ {i} (C _ {i} ^ {*})) \Big) + \Big (\min _ {\theta_ {i} \in C _ {i} ^ {*}} J _ {\theta_ {i}} (\hat {\pi} _ {i} (C _ {i} ^ {*})) - J _ {\theta_ {i} ^ {*}} (\hat {\pi} _ {i} (C _ {i} ^ {*})) \Big). \\ \end{array} +$$ + +By definition of $\hat{\pi}_i(C_i^*)$ , we have $\min_{\theta_i \in C_i^*} \langle \theta_i, \pmb{z}_{\hat{\pi}_i(C_i^*)} \rangle \geq \min_{\theta_i \in C_i^*} \langle \theta_i, \pmb{z}_{\hat{\pi}(\theta_i^*)} \rangle$ so that the first expression on the right hand side is less or equal to zero. Since $\theta_i^* \in C_i^*$ , we also know that $\min_{\theta_i \in C_i^*} J_{\theta_i}(\pi) \leq J_{\theta_i^*}(\pi)$ for all $\pi \in \Pi$ . Thus, on the good event when $\theta_i^* \in C_i^*$ , we obtain: + +$$ +\begin{array}{l} J _ {\theta_ {i} ^ {*}} (\hat {\pi} _ {i} (\theta_ {i} ^ {*})) - J _ {\theta_ {i} ^ {*}} (\hat {\pi} _ {i} (C _ {i} ^ {*})) \leq c \sqrt {\frac {d + \log (1 / \delta)}{\gamma^ {2} n}} \cdot \| \mathbb {E} _ {s \sim \rho} [ \phi (s, \hat {\pi} _ {i} (\theta_ {i} ^ {*}) (s)) ] \| _ {\Sigma_ {\mathcal {D} _ {i}} ^ {- 1}} \\ \leq c \sqrt {\frac {d + \log (1 / \delta)}{\gamma^ {2} n}} \cdot \max _ {\pi \in \Pi} \| \mathbb {E} _ {s \sim \rho} [ \phi (s, \pi (s)) ] \| _ {\Sigma_ {\mathcal {D} _ {i}} ^ {- 1}}. \\ \end{array} +$$ + +Note that the coverage coefficient on the right can be written as $\| \Sigma_{\mathcal{D}_i}^{-1 / 2}\mathbb{E}_{s\sim \rho}[\phi (s,\pi (s))]\| _2$ + +We have here (arguably coarsely) upper bounded the coverage of $\hat{\pi} (\theta_i^*)$ by the uniform policy coverage $\kappa_{i}\coloneqq \max_{\pi}\| \mathbb{E}_{s\sim \rho}[\phi (s,\pi (s))]\|_{\Sigma_{\mathcal{D}_i}^{-1}}$ of labeler's $i$ data. We must do this here as the policy $\hat{\pi}_i(\theta_i^*)$ notably depends on the other labeler's reported preferences $\mathcal{D}_{-i}$ and is thus hard to control or express explicitly. Overall, we have thus shown that being truthful is an approximately dominant strategy for labeler $i$ under Pessimistic Median of MLEs. Hence, Pessimistic MoMLEs is $\mathcal{O}(\kappa_i\sqrt{d / n})$ -strategyproof. + +Remark A.5. As we wish to ensure strategyproofness, i.e., truthfulness is a dominant strategy, we could not control the needed coverage carefully, but had to take a worst-case perspective and consider uniform coverage of all policies as quantified by $\kappa_{i}$ . Naturally, we would expect to improve upon this when considering incentive-compatibility instead of strategyproofness, i.e., showing that truthfulness forms an equilibrium but is not necessarily a dominant strategy profile. In that case, one can show that Pessimistic Median of MLEs is approximately incentive-compatible where instead of the uniform policy coverage the coverage of the output $\hat{\pi}^{*}$ of Pessimistic Median of MLEs given that everyone reports truthfully is enough. In other words, the coverage coefficient is given by $\| \mathbb{E}_{s\sim \rho}[\phi (s,\hat{\pi}^* (s))]\|_{\Sigma_{D_i}^{-1}}\leq \kappa_i$ . + +![](images/9d696230d5891ee7ca841e51d06fa559e00779e42fbf9eeb6a7b48525f40fe69.jpg) + +# A.6 Proof of Theorem 4.2 + +Theorem 4.2. Let $\hat{\pi}$ be the output of the Pessimistic Median of MLEs algorithm and suppose that all labelers report truthfully. With probability at least $1 - \delta$ : + +$$ +\operatorname {S u b O p t} (\hat {\pi}) \leq c o n s t \cdot \left(\sqrt {\frac {d \log (k / \delta)}{k}} + \max _ {i \in [ k ]} \kappa_ {i} ^ {*} \cdot k \sqrt {\frac {d + \log (k / \delta)}{n}}\right) \tag {2} +$$ + +where $\kappa_{i}^{*}\coloneqq \| \mathbb{E}_{s\sim \rho}[\phi (s,\pi^{*}(s))]\|_{\Sigma_{\mathcal{D}_{i}}^{-1}}$ is the optimal policy coverage of labeler $i$ . + +Proof. We will decompose the suboptimality in various ways. To this end, let $\pi^{*}$ denote the policy that maximizes social welfare and let $\hat{\pi}$ denote the policy computed by Pessimistic Median of MLEs. Recall the definition of the set of medians w.r.t. confidence sets $C_1,\ldots ,C_k$ as $\mathcal{C} := \{\mathrm{med}(\theta_1,\dots ,\theta_k)\colon \theta_i\in C_i\}$ and let $\mathcal{A} := \{\frac{1}{k}\sum_{i = 1}^{k}\theta_{i}\colon \theta_{i}\in C_{i}\}$ denote the set of averages. For convenience, we define for any $\pi$ : + +$$ +\mathbf {z} _ {\pi} := \mathbb {E} _ {s \sim \rho} [ \phi (s, \pi (s)) ]. +$$ + +Moreover, we let + +$$ +\theta_ {\text {a v g}} ^ {*} := \operatorname {a v g} \left(\theta_ {1} ^ {*}, \dots , \theta_ {k} ^ {*}\right) \quad \text {a n d} \quad \theta_ {\text {m e d}} ^ {*} := \operatorname {m e d} \left(\theta_ {1} ^ {*}, \dots , \theta_ {k} ^ {*}\right) +$$ + +correspond to the true average and median, respectively. We now decompose the suboptimality as follows: + +$$ +\begin{array}{l} \operatorname {S u b O p t} (\hat {\pi}) = \frac {1}{k} \sum_ {i = 1} ^ {k} \left\langle \theta_ {i} ^ {*}, z _ {\pi^ {*}} \right\rangle - \left\langle \theta_ {i} ^ {*}, z _ {\hat {\pi}} \right\rangle \\ = \left\langle \theta_ {\text {a v g}} ^ {*}, z _ {\pi^ {*}} \right\rangle - \left\langle \theta_ {\text {a v g}} ^ {*}, z _ {\hat {\pi}} \right\rangle \\ = \Big (\underbrace {\langle \theta_ {\mathrm {a v g}} ^ {*} , z _ {\pi^ {*}} \rangle - \min _ {\theta \in \mathcal {A}} \langle \theta , z _ {\pi^ {*}} \rangle} _ {(I)} \Big) + \Big (\underbrace {\min _ {\theta \in \mathcal {A}} \langle \theta , z _ {\pi^ {*}} \rangle - \langle \theta_ {\mathrm {a v g}} ^ {*} , z _ {\hat {\pi}} \rangle} _ {(I I)} \Big) \\ \end{array} +$$ + +In the following, we work on the good event such that $\theta_i^* \in C_i$ for all $i \in [k]$ . Using a union bound, we can show that this event occurs with probability at least $1 - \frac{k}{d}$ . + +We can bound the first term $(I)$ using that the confidence concentrates around the true parameter at a rate of $\sqrt{d / n}$ according to Lemma A.1 and considering the worst-case coverage of the optimal policy over all labeler's data. For some constant $c > 0$ , this yields + +$$ +\begin{array}{l} \langle \theta_ {\mathrm {a v g}} ^ {*}, z _ {\pi^ {*}} \rangle - \min _ {\theta \in \mathcal {A}} \langle \theta , z _ {\pi^ {*}} \rangle = \max _ {\theta \in \mathcal {A}} \langle \theta_ {\mathrm {a v g}} ^ {*} - \theta , z _ {\pi^ {*}} \rangle \\ = \frac {1}{k} \max _ {\theta_ {1} \in C _ {1}} \dots \max _ {\theta_ {k} \in C _ {k}} \sum_ {i = 1} ^ {k} \left\langle \theta_ {i} ^ {*} - \theta_ {i}, z _ {\pi^ {*}} \right\rangle \\ = \frac {1}{k} \sum_ {i = 1} ^ {k} \max _ {\theta_ {i} \in C _ {i}} \left\langle \theta_ {i} ^ {*} - \theta_ {i}, z _ {\pi^ {*}} \right\rangle \\ \leq \frac {1}{k} \sum_ {i = 1} ^ {k} \max _ {\theta_ {i} \in C _ {i}} \| \theta_ {i} ^ {*} - \theta_ {i} \| _ {\Sigma_ {\mathcal {D} _ {i}}} \| \boldsymbol {z} _ {\pi^ {*}} \| _ {\Sigma_ {\mathcal {D} _ {i}} ^ {- 1}} \\ \leq c \sqrt {\frac {d + \log (1 / \delta)}{\gamma^ {2} n}} \cdot \frac {1}{k} \sum_ {i = 1} ^ {k} \| z _ {\pi^ {*}} \| _ {\Sigma_ {\mathcal {D} _ {i}} ^ {- 1}} \\ \leq c \sqrt {\frac {d + \log (1 / \delta)}{\gamma^ {2} n}} \cdot \max _ {i \in [ k ]} \| \boldsymbol {z} _ {\pi^ {*}} \| _ {\Sigma_ {\mathcal {D} _ {i}} ^ {- 1}}. \\ \end{array} +$$ + +Bounding the second term $(II)$ is more involved as the policy $\hat{\pi}$ is not maximizing the average but the pessimistic median. We further decompose the second term into four parts as follows: + +$$ +\begin{array}{l} \min _ {\theta \in \mathcal {A}} \langle \theta , \boldsymbol {z} _ {\pi^ {*}} \rangle - \langle \theta_ {\mathrm {a v g}} ^ {*}, \boldsymbol {z} _ {\hat {\pi}} \rangle = \left(\min _ {\theta \in \mathcal {A}} \langle \theta , \boldsymbol {z} _ {\pi^ {*}} \rangle - \min _ {\theta \in \mathcal {C}} \langle \theta , \boldsymbol {z} _ {\pi^ {*}} \rangle\right) + \left(\min _ {\theta \in \mathcal {C}} \langle \theta , \boldsymbol {z} _ {\pi^ {*}} \rangle - \min _ {\theta \in \mathcal {C}} \langle \theta , \boldsymbol {z} _ {\hat {\pi}} \rangle\right) \\ + \left(\min _ {\theta \in \mathcal {C}} \langle \theta , \boldsymbol {z} _ {\hat {\pi}} \rangle - \langle \theta_ {\mathrm {m e d}} ^ {*}, \boldsymbol {z} _ {\hat {\pi}} \rangle\right) + \left(\langle \theta_ {\mathrm {m e d}} ^ {*}, \boldsymbol {z} _ {\hat {\pi}} \rangle - \langle \theta_ {\mathrm {a v g}} ^ {*}, \boldsymbol {z} _ {\hat {\pi}} \rangle\right). \\ \end{array} +$$ + +We first show that the second and third term are less or equal to zero. We have + +$$ +\min _ {\theta \in \mathcal {C}} \langle \theta , z _ {\pi^ {*}} \rangle \leq \min _ {\theta \in \mathcal {C}} \langle \theta , z _ {\hat {\pi}} \rangle , +$$ + +since $\hat{\pi}$ maximizes $\min_{\theta \in \mathcal{C}}\langle \theta ,z_{\pi}\rangle$ by definition of the Pessimistic Median of MLEs. Moreover, we see that + +$$ +\min _ {\theta \in \mathcal {C}} \langle \theta , \boldsymbol {z} _ {\hat {\pi}} \rangle \leq \langle \theta_ {\mathrm {m e d}} ^ {*}, \boldsymbol {z} _ {\hat {\pi}} \rangle , +$$ + +as the true median is contained in the confidence set $\mathcal{C}$ on the good event when $\theta_i^* \in C_i$ . Hence, both the second and third term can be bounded from above by zero. + +To bound the first term, we once again decompose the expression as follows: + +$$ +\min _ {\theta \in \mathcal {A}} \langle \theta , z _ {\pi^ {*}} \rangle - \min _ {\theta \in \mathcal {C}} \langle \theta , z _ {\pi^ {*}} \rangle = \underbrace {\min _ {\theta \in \mathcal {A}} \langle \theta - \theta_ {\text {a v g}} ^ {*} , z _ {\pi^ {*}} \rangle} _ {(a)} + \underbrace {\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left. \right.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.} _ {(b)} + \underbrace {\max _ {\theta \in \mathcal {C}} \left\langle \theta_ {\text {m e d}} ^ {*} - \theta , z _ {\pi^ {*}} \right\rangle} _ {(c)}. +$$ + +Similarly to before, using Lemma A.1, we bound $(a)$ as + +$$ +\min _ {\theta \in \mathscr {A}} \langle \theta - \theta_ {\mathrm {a v g}} ^ {*}, z _ {\pi^ {*}} \rangle \leq c \sqrt {\frac {d + \log (1 / \delta)}{\gamma^ {2} n}} \cdot \max _ {i \in [ k ]} \| z _ {\pi^ {*}} \| _ {\Sigma_ {\mathscr {D} _ {i}} ^ {- 1}}. +$$ + +For $(b)$ , it follows from Cauchy-Schwarz and Lemma A.2 that + +$$ +\left\langle \theta_ {\text {a v g}} ^ {*} - \theta_ {\text {m e d}} ^ {*}, z _ {\pi^ {*}} \right\rangle \leq \left\| \theta_ {\text {a v g}} ^ {*} - \theta_ {\text {m e d}} ^ {*} \right\| _ {2} \left\| z _ {\pi^ {*}} \right\| _ {2} \leq c \sqrt {\frac {d \log (1 / \delta)}{k}} \cdot \left\| z _ {\pi^ {*}} \right\| _ {2}. \tag {4} +$$ + +For $(c)$ , first note that we can write the difference between two medians as the telescoping sum + +$$ +\begin{array}{l} \operatorname {m e d} \left(\theta_ {1} ^ {*}, \dots , \theta_ {k} ^ {*}\right) - \operatorname {m e d} \left(\theta_ {1}, \dots , \theta_ {k}\right) \\ = \sum_ {i = 1} ^ {k} \operatorname {m e d} \left(\theta_ {1} ^ {*}, \dots , \theta_ {i} ^ {*}, \theta_ {i + 1}, \dots , \theta_ {k}\right) - \operatorname {m e d} \left(\theta_ {1} ^ {*}, \dots , \theta_ {i - 1} ^ {*}, \theta_ {i}, \dots , \theta_ {k}\right). \\ \end{array} +$$ + +By definition of the median, each difference on the right hand side can be bounded in terms of the difference $\theta_i^* - \theta_i$ . Using Lemma A.1 and the fact that $\theta_i \in C_i$ for all $i \in [k]$ , we obtain + +$$ +\begin{array}{l} \max _ {\theta \in \mathcal {C}} \left\langle \theta_ {\text {m e d}} ^ {*} - \theta , z _ {\pi^ {*}} \right\rangle \leq \sum_ {i = 1} ^ {k} \left\| \theta_ {i} ^ {*} - \theta_ {i} \right\| _ {\Sigma_ {\mathcal {D} _ {i}}} \left\| z _ {\pi^ {*}} \right\| _ {\Sigma_ {\mathcal {D} _ {i}} ^ {- 1}} \\ \leq c k \sqrt {\frac {d + \log (1 / \delta)}{\gamma^ {2} n}} \cdot \| \boldsymbol {z} _ {\pi^ {*}} \| _ {\Sigma_ {D _ {i}} ^ {- 1}}. \\ \end{array} +$$ + +The proof is complete by combining these bounds. + +![](images/db82b8b326618c97c9e3f44ec944467cfc876b69ed8b15c4222650ac3152937c.jpg) + +# A.7 Proof of Proposition 4.3 + +Proposition 4.3. When the labelers report their preferences according to any weakly dominant strategy under Pessimistic Median of MLEs, with probability at least $1 - \delta$ , the output $\hat{\pi}$ satisfies: + +$$ +\operatorname {S u b O p t} (\hat {\pi}) \leq c o n s t \cdot \left(\sqrt {\frac {d \log (k / \delta)}{k}} + \max _ {i \in [ k ]} \kappa_ {i} ^ {*} \cdot k \sqrt {\frac {d + \log (k / \delta)}{n}}\right). +$$ + +where $\kappa_{i}^{*}\coloneqq \| \mathbb{E}_{s\sim \rho}[\phi (s,\pi^{*}(s))]\|_{\Sigma_{\mathcal{D}_{i}}^{-1}}$ is the optimal policy coverage of labeler $i$ . + +Proof. Let $\theta_{i} \in \mathbb{R}^{d}$ be the reward parameter according to which labeler $i \in [k]$ samples its preferences under a weakly dominant strategy. The intuition for the result is fairly straightforward so that we describe it here first. First of all, we have seen in the proof of Theorem 4.1 that due to the median rule a labeler cannot achieve an individually better outcome by misreporting the sign of its reward parameter (see Lemma A.3 and Lemma A.4). As a result, $\theta_{i}$ will have identical signs to $\theta_{i}^{*}$ but potentially exaggerate its magnitude. Crucially, such exaggeration cannot worsen the suboptimality as it only helps to prevent flipped signs as we are taking the worst-case over confidence sets. Here, it is also worth noting that the primary reasons why Pessimistic Median of MLEs is approximately strategyproof are the estimation errors and the pessimism selection of the median over potentially large confidence sets. + +Any weakly dominant strategy must preserve signs. Assume for contradiction that there exists some coordinate $j$ such that $\theta_{i,j}^{*} > 0$ but the labeler's chosen reward parameter is such that $\theta_{i,j} < 0$ (or similarly $\theta_{i,j}^{*} < 0$ but $\theta_{i,j} > 0$ ). By the hyperrectangular assumption, the Pessimistic Median of MLEs algorithm outputs a policy that maximizes each dimension of the feature space independently. Specifically, $z_{\hat{\pi},j} = \mathbb{E}_{s\sim \rho}[\phi (s,\hat{\pi} (s))]_j$ will be positive if the considered median is positive and vice versa. Hence, by nature of the median, flipping the sign of coordinate $j$ can only have an adverse effect for labeler $i$ (see Lemma A.3) and no such strategy can be weakly dominant. + +By the same argument, if $\theta_{i,j}^{*} < 0$ but $\theta_{i,j} > 0$ , then labeler $i$ would risk pushing the aggregator's dimension $j$ to be positive, contrary to its true negative preference, and thus risk reducing its true utility in that dimension. Hence it cannot be a weakly dominant strategy to flip signs in that scenario either. Consequently, in every dimension $j$ , a weakly dominant report $\theta_{i,j}$ must preserve $\mathrm{sign}(\theta_{i,j}) = \mathrm{sign}(\theta_{i,j}^{*})$ . + +Exaggeration benefits Pessimistic Median of MLEs. By the hyperrectangular ("sign-based") structure, the decision in each coordinate $j$ of the learned policy depends essentially on whether the aggregated median is positive or negative. Pessimistic Median of MLEs aggregates each labeler $i$ 's confidence set $C_i$ by taking a coordinate-wise median over a selection $\theta_i \in C_i$ . Thus, to form the median, it chooses exactly one $\theta_{i,j}$ from each $C_i$ and then takes the median value among these $k$ numbers. Assume that the labeler $i$ 's original (w.l.o.g.) positive coordinate is $\theta_{i,j}^*$ , whereas its inflated coordinate is $\theta_{i,j} > \theta_{i,j}^*$ . Under the inflated reported reward parameter, the labeler's MLE and confidence set for dimension $j$ shift toward strictly larger positive values (note that the covariance + +matrix $\Sigma_{\mathcal{D}_i}$ is positive definite). Consequently, the set of considered medians $\mu_j$ for $\mu \in \mathcal{C}$ (i.e. all possible ways to pick $\theta_{i,j} \in C_i$ for $i = 1, \dots, k$ and take their coordinate-wise median) does not move down: it can only stay the same or shift to more positive values. Intuitively, replacing one of the $i$ entries by a strictly larger positive number cannot decrease the median. + +Hence, when labeler $i$ is misreporting $\theta_{i,j}$ such that $\theta_{i,j} > \theta_{i,j}^{*}$ (while keeping the same sign), this cannot worsen the suboptimality of the final policy, but only, in some special cases, strictly lower suboptimality by "protecting" the sign within the confidence set. Since this argument holds for any dimension $j$ , it follows that an entire sign-preserving inflation by labeler $i$ cannot yield a higher suboptimality than the truthful report would. + +# A.8 Proof of Corollary 4.5 + +Corollary 4.5. When there is only a single labeler, with probability at least $1 - \delta$ : + +$$ +\mathrm {S u b O p t} (\hat {\pi}) \leq c o n s t \cdot \kappa_ {1} ^ {*} \sqrt {\frac {d + \log (k / \delta)}{n}}. +$$ + +When all $k$ labelers have the same reward function, with probability at least $1 - \delta$ : + +$$ +\mathrm {S u b O p t} (\hat {\pi}) \leq c o n s t \cdot \max _ {i \in [ k ]} \kappa_ {i} ^ {*} \cdot k \sqrt {\frac {d + \log (k / \delta)}{n}}. +$$ + +Proof. When $k = 1$ the claimed result follows directly from setting $k = 1$ in our previous suboptimality bounds (see Theorem 4.2). + +Next, suppose that all $k \geq 1$ labelers have the same reward parameter $\theta^{*} = \theta_{1}^{*} = \dots = \theta_{k}^{*}$ . As a result, the true average and median coincide and we have $\theta^{*} = \theta_{\mathrm{avg}}^{*} = \theta_{\mathrm{med}}^{*}$ . To bound the suboptimality of the Pessimistic Median of MLEs algorithm in this special case we take the same steps as in the proof of Theorem 4.2 in Section A.6 with the difference that the expression (b) in equation (3) is zero since $\theta^{*} = \theta_{\mathrm{avg}}^{*} = \theta_{\mathrm{med}}^{*}$ . This yields the claimed upper bound. + +# A.9 Proof of Corollary 4.6 + +Corollary 4.6. Suppose $\mathcal{W}(\pi^{*}) > 0$ is constant. When the number of samples is sufficiently large and provide sufficient coverage of the optimal policy, with probability at least $1 - \delta$ , the approximation ratio of the Pessimistic Median of MLEs algorithm is given by $\alpha (\rho ,\hat{\pi})\geq 1 - \mathcal{O}\big(\sqrt{d\log(k / \delta) / k}\big)$ . + +Proof. For $n$ sufficiently large and sufficient coverage of the optimal policy, Theorem 4.2 implies that with probability at least $1 - \delta$ : + +$$ +\mathrm {S u b O p t} (\hat {\pi}) := \mathcal {W} (\pi^ {*}) - \mathcal {W} (\hat {\pi}) \leq c \sqrt {\frac {d \log (k / \delta)}{n}} +$$ + +for some constant $c > 0$ . As a result, the approximation ratio is upper bounded as + +$$ +\alpha (\rho , \hat {\pi}) := \frac {\mathcal {W} (\hat {\pi})}{\mathcal {W} (\pi^ {*})} = 1 - \frac {\mathcal {W} (\pi^ {*}) - \mathcal {W} (\hat {\pi})}{\mathcal {W} (\pi^ {*})} \geq 1 - c \sqrt {\frac {d \log (k / \delta)}{n}}, +$$ + +where we used that $\mathcal{W}(\pi^{*}) > 0$ is constant by assumption. + +![](images/9c38064f1e2f00893d4339e58be5a71f86b40587588c32af6f15dcf4547a0a1a.jpg) + +# A.10 Proof of Theorem 5.1 and Theorem 5.2 + +Proof. We can prove Theorem 5.1 and Theorem 5.2 in a similar way we proved the analogous results in the contextual bandit problem. We refrain from reiterating and restating all necessary steps to prove these results as they are almost identical to before. Most importantly, a similar MLE concentration bound holds for MDPs as for contextual bandits. + +Lemma A.6 (MLE Concentration Bound for MDPs). With probability at least $1 - \delta$ , + +$$ +\| \hat {\theta} _ {i} ^ {\mathrm {M L E}} - \theta_ {i} ^ {*} \| _ {\Sigma_ {\mathcal {D} _ {i}}} \leq c o n s t \cdot \sqrt {\frac {d + \log (1 / \delta)}{\gamma^ {2} n}}, +$$ + +where $\gamma := 1 / (2 + \exp(-HLB) + \exp(HLB))$ . The covariance matrix $\Sigma_{\mathcal{D}_i}$ is given by $\Sigma_{\mathcal{D}_i} = \sum_{j=1}^{n} x^{i,j}(x^{i,j})^\top$ where $x^{i,j} = \sum_{h=1}^{H} (\phi(s_h^{i,j}, a_h^{i,j}) - \phi(\bar{s}_h^{i,j}, \bar{a}_h^{i,j}))$ with $s_1^{i,j} = \bar{s}_1^{i,j} = s^{i,j}$ . + +Swapping the initial state distribution $\rho$ (i.e., context distribution) for the state occupancy $q_{\pi}$ as defined in Section 5, we can follow the same line of argument as in Section A.5 to prove Theorem 5.1. + +# B Computational Complexity + +We now consider the computational complexity of computing the pessimistic median return. First, we consider the contextual bandits formulation, and then consider the general MDP setting. + +# B.1 Contextual Bandits + +Recall that we construct the confidence sets + +$$ +C _ {i} = \{\theta \in \mathbb {R} ^ {d} \colon \| \hat {\theta} _ {i} ^ {\mathrm {M L E}} - \theta \| _ {\Sigma_ {\mathcal {D} _ {i}}} \leq f (d, n, \delta) \}. +$$ + +Then, the (coordinate-wise) median confidence set is defined as + +$$ +\mathcal {C} = \left\{\operatorname {m e d} \left(\theta_ {1}, \dots , \theta_ {k}\right): \theta_ {i} \in C _ {i} \forall i \in [ k ] \right\}, +$$ + +and we aim to solve the following optimization problem: + +$$ +\max _ {\pi \in \Pi} \underline {{\mathcal {W}}} (\pi) := \min _ {\theta \in \mathscr {C}} \mathbb {E} _ {s \sim \rho} [ \langle \theta , \phi (s, \pi (s)) \rangle ] +$$ + +In a first step, we show that the function $\underline{\mathcal{V}} (\pi)$ is concave. Indeed, consider two policies $\pi_{1}$ , and $\pi_{2}$ . Then, + +$$ +\begin{array}{l} \underline {{\mathcal {W}}} (\alpha \pi_ {1} + (1 - \alpha) \pi_ {2}) = \min _ {\theta \in \mathcal {C}} \mathbb {E} _ {s \sim \rho} \left[ \sum_ {a} (\alpha \pi_ {1} (a) + (1 - \alpha) \pi_ {2} (a)) \langle \theta , \phi (s, a) \rangle \right] \\ \geq \min _ {\theta \in \mathcal {C}} \mathbb {E} _ {s \sim \rho} \left[ \sum_ {a} \alpha \pi_ {1} (a) \langle \theta , \phi (s, a) \rangle \right] + \min _ {\theta \in \mathcal {C}} \mathbb {E} _ {s \sim \rho} \left[ \sum_ {a} (1 - \alpha) \pi_ {2} (a) \langle \theta , \phi (s, a) \rangle \right] \\ = \alpha \cdot \underline {{\mathcal {W}}} (\pi_ {1}) + (1 - \alpha) \cdot \underline {{\mathcal {W}}} (\pi_ {2}). \\ \end{array} +$$ + +Therefore, $\underline{\mathcal{W}}(\cdot)$ can be efficiently optimized using projected gradient ascent as long as we can compute the gradient efficiently. For a given $\pi$ , we have $\nabla_{\pi} \underline{\mathcal{W}}(\pi)_{(s,a)} = \rho(s) \langle \phi(s,a), \theta^{\star} \rangle$ where + +$$ +\theta^ {\star} \in \underset {\theta \in \mathcal {C}} {\operatorname {a r g m i n}} \mathbb {E} _ {s \sim \rho} [ \langle \theta , \phi (s, \pi (s)) \rangle ]. \tag {5} +$$ + +In order to show that the gradient $\nabla_{\pi}\underline{\mathcal{V}} (\pi)$ can be efficiently computed, we need to show that $\theta^{\star}$ can be efficiently computed. + +The set $\mathcal{C}$ can be arbitrary, but we can write down the following equivalent optimization problem involving linear and quadratic constraints: + +$$ +\min _ {\theta , \left\{\theta_ {i} \right\} _ {i \in [ k ]}} \mathbb {E} _ {s \sim \rho} \left[ \langle \theta , \phi (s, \pi (s)) \rangle \right] +$$ + +$$ +\text {s . t .} \| \hat {\theta} _ {i} ^ {\mathrm {M L E}} - \theta_ {i} \| _ {\Sigma_ {\mathcal {D} _ {i}}} \leq f (d, n, \delta) \forall i \in [ k ] \tag {6} +$$ + +$$ +\sum_ {i = 1} ^ {k} | \theta (j) - \theta_ {i} (j) | \leq \sum_ {i = 1} ^ {k} | \theta_ {\ell} (j) - \theta_ {i} (j) | \forall \ell \in [ k ], \forall j \in [ d ] +$$ + +The first set of constraints encode that $\theta_{i}\in C_{i}$ for each $i\in [k]$ . The second set of constraints encode that $\theta (j)$ is the median of $\theta_{1}(j),\ldots ,\theta_{k}(j)$ for each coordinate $j\in [d]$ . The above optimization + +problem might be non-convex, and instead we will consider the following alternate optimization problem: + +$$ +\min _ {\theta , \left\{\theta_ {i} \right\} _ {i \in [ k ]}, z} \mathbb {E} _ {s \sim \rho} [ \langle \theta , \phi (s, \pi (s)) \rangle ] + M \sum_ {i, j} z _ {i, j} \tag {7} +$$ + +$$ +\begin{array}{r l} \text {s . t .} & \theta_ {i} \in C _ {i} \forall i \in [ k ] \\ & z _ {i, j} \geq \theta (j) - \theta_ {i} (j), z _ {i, j} \geq \theta_ {i} (j) - \theta (j) \forall i \in [ k ], j \in [ d ] \end{array} \tag {7} +$$ + +The next lemma shows that we can choose $M$ and $n$ to recover an approximate solution of the original optimization problem (6). + +Lemma B.1. Suppose $(\theta^1, \{\theta_i^1\}_{i \in [k]})$ is an optimal solution to the optimization problem (6), and $(\theta^2, \{\theta_i^2\}_{i \in [k]}, z^2)$ is an optimal solution to the optimization problem (7). Then, + +$$ +\mathbb {E} _ {s \sim \rho} [ \langle \theta^ {2}, \phi (s, \pi (s)) \rangle ] \leq \mathbb {E} _ {s \sim \rho} [ \langle \theta^ {1}, \phi (s, \pi (s)) \rangle ] + M \sum_ {i} \frac {2 \sqrt {d}}{\lambda_ {\min} (\Sigma_ {\mathcal {D} _ {i}})} f (d, n, \delta) +$$ + +and + +$$ +\sum_ {i, j} \left| \theta^ {2} (j) - \theta_ {i} ^ {2} (j) \right| - \sum_ {i, j} \left| \tilde {\theta} (j) - \theta_ {i} ^ {2} (j) \right| \leq \frac {2 B L}{M} \forall \tilde {\theta}. +$$ + +Proof. Let us define $z_{i,j}^{1} = \left|\theta^{1}(j) - \theta_{i}^{1}(j)\right|$ . Then we have the following inequality: + +$$ +\mathbb {E} _ {s \sim \rho} [ \langle \theta^ {2}, \phi (s, \pi (s)) \rangle ] + M \sum_ {i, j} z _ {i, j} ^ {2} \leq \mathbb {E} _ {s \sim \rho} [ \langle \theta^ {1}, \phi (s, \pi (s)) \rangle ] + M \sum_ {i, j} z _ {i, j} ^ {1}. +$$ + +Without loss of generality, we can assume $z_{i,j}^2 = \left|\theta^2 (j) - \theta_i^2 (j)\right|$ . Hence, + +$$ +z _ {i, j} ^ {2} = \left| \theta^ {2} (j) - \theta_ {i} ^ {2} (j) \right| = \left| \theta^ {2} (j) - \theta_ {i} ^ {1} (j) + \theta_ {i} ^ {1} (j) - \theta_ {i} ^ {2} (j) \right| \geq \left| \theta^ {2} (j) - \theta_ {i} ^ {1} (j) \right| - \left| \theta_ {i} ^ {1} (j) - \theta_ {i} ^ {2} (j) \right|. +$$ + +Substituting this bound, we obtain + +$$ +\begin{array}{l} \mathbb {E} _ {s \sim \rho} [ \langle \theta^ {2}, \phi (s, \pi (s)) \rangle ] - \mathbb {E} _ {s \sim \rho} [ \langle \theta^ {1}, \phi (s, \pi (s)) \rangle ] \\ \leq - M \sum_ {i, j} \left| \theta^ {2} (j) - \theta_ {i} ^ {1} (j) \right| + M \sum_ {i, j} \left| \theta^ {1} (j) - \theta_ {i} ^ {1} (j) \right| + M \sum_ {i, j} \left| \theta_ {i} ^ {1} (j) - \theta_ {i} ^ {2} (j) \right| \\ \leq M \sum_ {i} \left| \left| \theta_ {i} ^ {1} - \theta_ {i} ^ {2} \right| \right| _ {1} \\ \leq M \sum_ {i} \sqrt {d} \left\| \theta_ {i} ^ {1} - \theta_ {i} ^ {2} \right\| _ {2} \\ \leq M \sum_ {i} \frac {\sqrt {d}}{\lambda_ {\min } \left(\Sigma_ {\mathcal {D} _ {i}}\right)} \left\| \theta_ {i} ^ {1} - \theta_ {i} ^ {2} \right\| _ {\Sigma_ {\mathcal {D} _ {i}}} \leq M \sum_ {i} \frac {2 \sqrt {d}}{\lambda_ {\min } \left(\Sigma_ {\mathcal {D} _ {i}}\right)} f (d, n, \delta). \\ \end{array} +$$ + +The second inequality follows since $\theta^1$ is a coordinate-wise median of the parameters $\{\theta_i^1\}_{i\in [k]}$ . We also use the assumption that under uniform coverage $\lambda_{\mathrm{min}}(\Sigma_{\mathcal{D}_i}) > 0$ . + +We now bound the violation of constraints of the solution $\theta^2$ . Indeed for any $\tilde{\theta}, \{\theta_i^2\}_{i \in [k]}$ and $\{|\tilde{\theta}(j) - \theta_i^2(j)|\}_{i,j}$ , since $\theta^2$ as $\theta^2$ , $\{\theta_i^2\}_{i \in [k]}$ , $\{z_{i,j}\}_{i,j}$ is feasible solution to the optimization problem, we have (7): + +$$ +\mathbb {E} _ {s \sim \rho} [ \langle \theta^ {2}, \phi (s, \pi (s)) \rangle ] + M \sum_ {i, j} z _ {i, j} ^ {2} \leq \mathbb {E} _ {s \sim \rho} [ \langle \tilde {\theta}, \phi (s, \pi (s)) \rangle ] + M \sum_ {i, j} \left| \tilde {\theta} (j) - \theta_ {i} ^ {2} (j) \right|. +$$ + +After rearranging this yields + +$$ +\sum_ {i, j} \left| \theta^ {2} (j) - \theta_ {i} ^ {2} (j) \right| - \sum_ {i, j} \left| \tilde {\theta} (j) - \theta_ {i} ^ {2} (j) \right| \leq \frac {1}{M} \mathbb {E} _ {s \sim \rho} [ \langle \tilde {\theta} - \theta^ {2}, \phi (s, \pi (s)) \rangle ] \leq \frac {2 B L}{M}. +$$ + +□ + +Since $f(d,n,\delta) = O\left(\sqrt{\frac{d + \log(k / \delta)}{n}}\right)$ we can choose $M = \frac{2BL}{\varepsilon}$ and $n \geq \frac{4B^2L^2k^2(d + \log(k / \delta))}{(\min_i\lambda_{\min}(\Sigma_{\mathcal{D}_i})^2}\cdot \frac{1}{\varepsilon^4}$ and observe that $\theta^2$ is $\varepsilon$ -approximately optimal and $\varepsilon$ -approximate coordinate-wise median of the parameters $\{\theta_i^2\}_{i\in [k]}$ . + +General setting. Next, we consider the general setting where the state space can be arbitrarily large. We will assume that the policies are parametrized by a class of parameters $\Psi \subseteq \mathbb{R}^d$ , i.e., $\Pi = \{\pi_{\psi} : \psi \in \Psi\}$ . For example, the softmax parametrization models $\Pi$ as the following class of policies: + +$$ +\Pi = \left\{\pi_ {\psi} (a \mid s) = \frac {\exp (\psi^ {\top} \phi (s , a))}{\sum_ {b} \exp (\psi^ {\top} \phi (s , b))} \forall s, a: \| \psi \| _ {2} \leq B \right\}. +$$ + +We now aim to solve the following optimization problem: + +$$ +\max _ {\psi \in \Psi} \underline {{\mathcal {W}}} (\psi) := \min _ {\theta \in \mathscr {C}} \mathbb {E} _ {s \sim \rho} \left[ \sum_ {a} \pi_ {\psi} (a | s) \langle \theta , \phi (s, a) \rangle \right]. +$$ + +The gradient of the objective is given by + +$$ +\nabla_ {\psi} \underline {{\mathcal {W}}} (\psi) = \mathbb {E} _ {s \sim \rho} \left[ \sum_ {a} \nabla_ {\psi} \pi_ {\psi} (a | s) \langle \theta^ {\star}, \phi (s, a) \rangle \right] +$$ + +where + +$$ +\theta^{\star}\in \operatorname *{argmin}_{\theta \in \mathscr{C}}\mathbb{E}_{s\sim \rho}\left[\langle \theta ,\phi (s,\pi_{\psi}(s))\rangle \right]. +$$ + +The above optimization is finite-dimensional $(O(dk))$ even when the number of states is very large and can be approximated using optimization problem (7). Thus, we can perform projected gradient ascent steps to solve the optimization problem $\max_{\psi \in \Psi} \underline{\mathcal{W}}(\psi)$ . However, unlike the tabular setting, the objective is no longer concave. It is known that under softmax parametrization, the expected return satisfies a non-uniform Polyak-Lojasiewicz (PL) condition [25] which guarantees linear convergence of gradient ascent method. We believe that similar conditions should hold for the function $\underline{\mathcal{W}}(\psi)$ but leave the verification to the future. + +# B.2 Extension to Markov Decision Processes + +We start with the assumption of a tabular MDP. We aim to solve the following optimization problem: + +$$ +\max _ {\pi \in \Pi} \underline {{\mathcal {W}}} (\pi) := \min _ {\theta \in \mathscr {C}} \mathbb {E} _ {s \sim q _ {\pi}} [ \langle \theta , \phi (s, \pi (s)) \rangle ]. +$$ + +$\underline{\mathcal{W}} (\pi)$ is a non-convex function of policy $\pi$ . However, it is a concave function of $q_{\pi}$ , the state-action occupancy measure of policy $\pi$ . We can write down the above optimization problem in terms of state-action occupancy measure as follows: + +$$ +\begin{array}{l} \max _ {q} \min _ {\theta \in \mathcal {C}} \frac {1}{H} \sum_ {h = 1} ^ {H} \sum_ {s, a} q _ {h} (s, a) \langle \theta , \phi (s, a) \rangle \\ \text {s . t .} \sum_ {a} q _ {1} (s, a) = \rho (s) \forall s \\ \sum_ {b} q _ {h + 1} (s, b) = \sum_ {s ^ {\prime}, a} q _ {h} \left(s ^ {\prime}, a\right) P \left(s \mid s ^ {\prime}, a\right) \forall s \forall h \in [ H - 1 ]. \\ \end{array} +$$ + +Here, the last two constraints encode Bellman-flow conditions which ensure that the solution $q$ is a valid state-action occupancy measure. Now we can write down stochastic gradient ascent step as $q_{t + 1} = \mathrm{Proj}(q_t + \eta \nabla \underline{\mathcal{W}} (q_t))$ . Here the $\mathrm{Proj}(\cdot)$ refers to projection onto the feasible set defined by the flow conditions. As the number of states and actions are finite and small, the projection step can be computed efficiently. We now verify that the gradient $\nabla \underline{W} (q)$ can be computed efficiently. The gradient is given by + +$$ +\nabla \underline {{\mathcal {W}}} (q) = \frac {1}{H} \sum_ {h = 1} ^ {H} \sum_ {s, a} q _ {h} (s, a) \langle \theta^ {\star}, \phi (s, a) \rangle +$$ + +where + +$$ +\theta^ {\star} \in \operatorname * {a r g m i n} _ {\theta \in \mathcal {C}} \frac {1}{H} \sum_ {h = 1} ^ {H} \sum_ {s, a} q _ {h} (s, a) \langle \theta , \phi (s, a) \rangle . +$$ + +Now we can proceed similarly to the contextual bandits setting and compute an approximate solution of the optimization problem above. + +$$ +\min _ {\theta , \left\{\theta_ {i} \right\} _ {i \in [ k ], z}} \frac {1}{H} \sum_ {h = 1} ^ {H} \sum_ {s, a} q _ {h} (s, a) \langle \theta , \phi (s, a) \rangle + M \sum_ {i, j} z _ {i, j} \tag {8} +$$ + +$$ +\mathrm {s . t .} \theta_ {i} \in C _ {i} \forall i \in [ k ] +$$ + +$$ +z _ {i, j} \geq \theta (j) - \theta_ {i} (j), z _ {i, j} \geq \theta_ {i} (j) - \theta (j) \forall i \in [ k ], j \in [ d ] +$$ + +General Setting: In the general setting, computing the projection step becomes infeasible as the number of states (and constraints) can be very large and possibly infinite. Instead, we again adopt a policy parametrization as discussed in the previous subsection. In particular, we assume that the policies are parametrized by a class of parameters $\Psi \subseteq \mathbb{R}^d$ , i.e., $\Pi = \{\pi_{\psi} : \psi \in \Psi\}$ . We now aim to solve the following optimization problem: + +$$ +\max _ {\psi \in \Psi} \underline {{\mathcal {W}}} (\psi) := \min _ {\theta \in \mathscr {C}} \mathbb {E} _ {s \sim \rho} \left[ V _ {\theta} ^ {\pi_ {\psi}} (s) \right]. +$$ + +The gradient of the objective is given by + +$$ +\nabla_ {\psi} \underline {{\mathcal {W}}} (\psi) = \mathbb {E} _ {s \sim \rho} \left[ \nabla_ {\psi} V _ {\theta^ {\star}} ^ {\pi_ {\psi}} (s) \right] +$$ + +where + +$$ +\theta^{\star}\in \operatorname *{argmin}_{\theta \in \mathcal{C}}\mathbb{E}_{(s,a)\sim q_{\pi_{\psi}}} \left[\langle \theta ,\phi (s,a)\rangle \right]. +$$ + +# C Experiments: Simulating Strategic Preference Labeling + +We here conduct small-scale synthetic experiments that simulate strategic preference learning and serve as a preliminary empirical evaluation of the proposed methodology. + +Experimental Setup. We simulate strategic labeling behavior by performing approximate gradient ascent (i.e., simultaneous perturbation stochastic approximation) on each labeler's utility $J_{i}(\hat{\pi})$ w.r.t. the labelers' internal reward parameters $\hat{\theta}_i$ , which govern their preference distribution $\mathbb{P}_{\hat{\theta}_i}$ . We adopt this simulation approach from prior work on strategic contextual bandits [23]. Each labeler is initialized at their ground-truth reward vector $\theta_i^*$ , which is sampled from a multivariate Gaussian. Labeler strategies are optimized for 200 steps. Since this process requires repeatedly re-labeling comparisons and re-running each algorithm, we focus on small problem settings in a contextual bandit formulation. All results are averaged over 5 random seeds, and we report standard errors. The results below are for embedding dimension $d = 16$ , number of labelers $k = 5$ , and offline samples $n = 20, 50, 100, 200$ . We compare the following approaches: (a) Naive MLEs that simply computes the MLEs given the preference data and optimizes a policy against the average reward estimate to maximize social welfare; (b) Pessimistic Social Welfare [37], which is the pessimistic version of Naive MLEs; (c) Median of MLEs, which optimizes a policy against the reward function derived from the median over MLEs; (d) Pessimistic MoMLEs, which is our proposed algorithm and outlined in Algorithm 1. We report the policy suboptimality under both truthful and strategic labeling. + +Results. Overall, we observe that while the Naive MLEs and its pessimistic version, Pessimistic Social Welfare, perform well when labelers are truthful, the performance degrades substantially under strategic preference labeling. In contrast, Pessimistic Median of MLEs exhibits almost no degradation, consistent with its approximate strategyproofness guarantee. Median of MLEs also shows slightly more robustness, though not to the same degree. We find that with increasing sample size, Pessimistic Median of MLEs primarily suffers the inherent cost of being strategyproof, as indicated by diminishing performance gains from more samples. That said, the influence of strategic manipulation on the learned policy also grows with more data, thereby making discouraging strategic preference labeling increasingly more valuable. + +Table 1: Suboptimality $\operatorname{SubOpt}\left( \widehat{\pi }\right)$ under truthful and strategic labeling across dataset sizes $n$ . + +
n = 20
AlgorithmSubOpt (Truthful)SubOpt (Strategic)Difference
Naive MLEs0.512 ± 0.0670.649 ± 0.082+0.137
Pessimistic SW0.641 ± 0.0810.751 ± 0.086+0.110
Median of MLEs0.635 ± 0.0530.693 ± 0.167+0.058
Pessimistic MoMLEs0.661 ± 0.0980.715 ± 0.155+0.054
+ +$\mathbf{n} = \mathbf{50}$ + +
AlgorithmSubOpt (Truthful)SubOpt (Strategic)Diff.
Naive MLEs0.384 ± 0.0560.622 ± 0.142+0.238
Pessimistic SW0.403 ± 0.0640.652 ± 0.149+0.249
Median of MLEs0.516 ± 0.0620.706 ± 0.124+0.190
Pessimistic MoMLEs0.508 ± 0.0560.532 ± 0.154+0.024
+ +$\mathbf{n} = \mathbf{100}$ + +
AlgorithmSubOpt (Truthful)SubOpt (Strategic)Diff.
Naive MLEs0.230 ± 0.0710.516 ± 0.131+0.316
Pessimistic SW0.249 ± 0.0680.584 ± 0.147+0.336
Median of MLEs0.522 ± 0.0440.605 ± 0.049+0.083
Pessimistic MoMLEs0.506 ± 0.0450.574 ± 0.152+0.068
+ +$\mathbf{n} = \mathbf{200}$ + +
AlgorithmSubOpt (Truthful)SubOpt (Strategic)Diff.
Naive MLEs0.133 ± 0.0290.491 ± 0.251+0.358
Pessimistic SW0.136 ± 0.0270.459 ± 0.266+0.323
Median of MLEs0.487 ± 0.0610.514 ± 0.259+0.027
Pessimistic MoMLEs0.474 ± 0.0510.415 ± 0.079-0.059
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While stratified sampling has proven effective for addressing label distribution imbalance in classification tasks, extending these ideas to segmentation remains challenging due to the multi-label structure and class imbalance typically present in such data. Building on existing stratification concepts, we introduce Iterative Pixel Stratification (IPS), a straightforward, label-aware sampling method tailored for segmentation tasks. Additionally, we present Wasserstein-Driven Evolutionary Stratification (WDES), a novel genetic algorithm designed to minimize the Wasserstein distance, thereby optimizing the similarity of label distributions across dataset splits. We prove that WDES is globally optimal given enough generations. Using newly proposed statistical heterogeneity metrics, we evaluate both methods against random sampling and find that WDES consistently produces more representative splits. Applying WDES across diverse segmentation tasks, including street scenes, medical imaging, and satellite imagery, leads to lower performance variance and improved model evaluation. Our results also highlight the particular value of WDES in handling small, imbalanced, and low-diversity datasets, where conventional splitting strategies are most prone to bias. + +# 1 Introduction + +Image segmentation involves the assignment of a label to each pixel in an image, with applications in diverse fields such as autonomous driving [1-5], medical imaging [6-10], and satellite image analysis [11-15]. In a supervised learning setting, splitting the dataset into disjoint subsets for training and validation is common practice. However, following a random splitting strategy can result in class distribution shifts across subsets compared to the original dataset. This can lead to biased evaluations, as the test set is typically smaller than the training set and more affected by this phenomenon. Typically, such issues are addressed through stratification to guarantee a consistent distribution of class labels across subsets [16] but this is not straightforward for image segmentation datasets [17-21]. The fundamental unit, an image, is not a sample with a single class but a sample with pixels belonging to many classes. Stratification strategies for similarly complex multi-label datasets [22] could provide interesting solutions that can be applied to segmentation datasets. Strategies relevant to multi-label datasets have been proposed and investigated in various forms ranging from iterative strategies [23-25], greedy search [26, 27], and genetic algorithms [28]. These methods do not directly apply to the problem of segmentation, as multi-label datasets have the complexity of multiple positive + +![](images/2963e2f4d0fba78f7e9276a59dc4a1f5180ee8bf36e5a6a5641c63a798a0e90d.jpg) +Figure 1: Representative examples of Plane images from PascalVOC [17]. The illustration shows an ideal split with balanced class distribution across folds which is not ensured by random splitting. Such balance is essential to avoid bias and ensure reliable model evaluation. + +classes for each sample but do not have the added complexity of proportional representation of these classes in the sample. Segmentation datasets present a unique challenge in this regard, complicating the task of stratification. + +Segmentation datasets are often small due to high annotation costs and exhibit imbalanced class distributions due to label diversity. As dataset size decreases, inconsistent class distributions in subsets increase, hindering generalization [29]. This issue is critical in imbalanced datasets, where some classes may be absent from splits, making performance metrics like F1-score and IoU unmeasurable. Such missing data undermines the reliability of model evaluation, especially in K-fold cross-validation [30], where unrepresentative subsets affect performance variance. + +Motivated by these challenges, we propose and investigate two stratification strategies: Iterative Pixel Stratification (IPS) and Wasserstein-Driven Evolutionary Stratification (WDES) $^1$ . These methods are designed to assign samples to subsets, or folds, for a train-test split scenario (two folds) or a K-fold cross-validation scenario ( $K$ folds), taking into account the label composition of the sample. IPS extends the Iterative Stratification (IS) algorithm proposed in literature [23], assigning samples to folds using a greedy approach to meet predefined class presence requirements measured by the number of pixels in each fold. It prioritizes assigning samples containing rare classes to folds, progressively addressing those with increasing ubiquity. In contrast, WDES employs an evolutionary algorithm akin to Evosplit [28], where a population of potential fold assignments (individuals) is evaluated simultaneously. The fittest individuals undergo selective crossover and mutations for further refinement. The fundamental distinction between these two methods lies in their approach to fold assignment and evaluation. The iterative nature of IPS confines it to a sequential decision-making process and is limited to a single chain of iterations. Conversely, the genetic algorithm in WDES evaluates multiple solutions independently by maintaining a population, enabling it to explore a broader solution space. The flexibility of WDES allows us to choose a better objective function to be minimized. We choose the Wasserstein distance between the class distribution in the fold and the overall dataset as it indicates the similarity between these distributions. Hence, minimizing the distance reduces dissimilarity between the subsets and the dataset. + +We evaluate IPS, WDES, and random splitting across five benchmark segmentation datasets. Our evaluation begins by analyzing the consistency of folds generated through repeated experiments. We + +further assess the impact of targeted stratification versus random splitting by examining the standard deviation of accuracy, F1-score, and Intersection over Union (IoU) across 10-fold cross-validation. A lower deviation indicates a more reliable assessment of model performance. Empirical results show that WDES consistently achieves the highest quality splits. In contrast, IPS under-performs relative to WDES, suggesting that a sequential strategy may be suboptimal for segmentation tasks. Notably, WDES also demonstrates better consistency over the other two for evaluating model performance on low-entropy datasets, where class distributions are highly imbalanced or concentrated. Overall, our findings highlight the importance of stratification tailored to structured outputs and advocate for the adoption of principled splitting methods in segmentation tasks. + +# 2 Related Work + +Existing methodologies for stratifying complex datasets trace their origins to Iterative Stratification (IS) by [23]. In their work, the authors addressed the intricacies of multi-label datasets, proposing a greedy algorithm that iteratively assigns samples to folds based on the required frequency of positive labels within each fold. Fold quality is evaluated by comparing the Label Distribution (LD) metric. LD is the difference between the proportion of samples belonging to a given class within a fold and the corresponding proportion in the overall dataset + +The partitioning method based on stratified random sampling (PMBSRS) by [24] employs a similar iterative approach but incorporates a pre-sorting step. This method clusters samples with similar label sets before assigning them equally to folds, thereby enhancing the homogeneity of label distribution across folds. Building upon the work of [23], authors of [25] introduced Second Order Iterative Stratification (SOIS), which considers second-order label relationships. Instead of focusing on individual labels, SOIS accounts for label pairs to determine the demand within each fold, thereby capturing more complex interactions between labels. + +Other strategies approach stratification by starting from an initial assignment and iteratively refining it. The Stratified Sampling (SS) method proposed by [26] pre-assigns samples to folds and evaluates them based on deviations from the ideal label distribution. Samples contributing to the highest deviation subset are shuffled between folds to minimize discrepancies. This approach was designed with large-scale XML datasets in mind [31]. A similar split-and-shuffle approach was adopted by [27] but from a label-centric perspective. They developed a scoring mechanism to assess fold quality relative to each class, shuffling samples associated with the class exhibiting the poorest quality before recalculating scores. This method demonstrated effectiveness on large Gene Ontology datasets. + +Finally, Evosplit, proposed by [28], leverages a genetic algorithm to address the challenges outlined by [23] and [25]. This method generates a population of fold assignments, evaluates their fitness, performs crossover and mutation, and iterates over generations to arrive at an optimal solution. Fitness is determined using Label Distribution (LD) from [23] and Label Pair Distribution (LPD) from [25]. + +While these methods offer valuable strategies for stratifying multi-label datasets, they do not directly extend to image segmentation. In segmentation, each sample contains a dense array of pixel-wise labels rather than a simple set of associated classes, and the proportion of pixels per class varies significantly across images. This makes stratification more complex: it is not enough to ensure the presence of a class in a fold; one must also consider how extensively each class is represented at the pixel level. Therefore, segmentation-specific stratification requires reformulating both the problem and the evaluation criteria. In the next section, we formalize the segmentation stratification task and introduce two new approaches, Iterative Pixel Stratification (IPS) and Wasserstein-Driven Evolutionary Stratification (WDES), which aim to create representative folds by explicitly accounting for pixel-level label distributions. + +# 3 Stratification algorithms + +Consider an image segmentation dataset $D = \{\mathbf{x}_n, \mathbf{y}_n\}_{n=1}^N$ with $N$ samples, where $\mathbf{x}_n$ denotes the input images and $\mathbf{y}_n$ the corresponding label masks. Let the total number of pixels across all samples be $P$ , and the number of semantic classes be $C$ . The objective is to partition $D$ into $K$ disjoint subsets, denoted $\mathbf{S} = (S^1, \ldots, S^K)$ , according to a target proportion vector $\mathbf{r} = (r^1, \ldots, r^K)$ , where $r^k$ specifies the desired proportion of samples in fold $k$ . + +Finding such a partition that preserves the global class distribution across folds is NP-Hard because it can be reduced to a multi-way partition problem [32]. It is analogous to dividing a multi-set of weighted elements into $K$ subsets with balanced sums; except here, the "weights" are pixel counts per class and the balance is across multiple classes simultaneously + +Let $N_{c}$ be the number of samples in the dataset that contain class $c$ , and $N_{c}^{k}$ the required number of such samples in fold $k$ . Similarly, let $P^{n}$ be the total number of pixels in sample $n$ , and $P_{c}^{n}$ the number of pixels of class $c$ in sample $n$ . The total number of pixels of class $c$ in the dataset is $P_{c}$ , and the desired number in fold $k$ is $P_{c}^{k}$ . These quantities are central to the evaluation of fold quality and to guiding the stratification process. + +# 3.1 Similarity measures + +We first introduce similarity measures calculated on the subsets to assess the quality of the stratified folds S. These statistical properties of the subsets aim to quantify: (1) how well each fold adhere to the target sample proportions defined by $\mathbf{r}$ ; (2) how closely the class-wise pixel proportions in each subset match those in the full dataset; (3) the overall similarity between fold-level and global label distributions using distributional distance. + +Sample Distribution (SD) With this measure, we aim to assess the deviation of the number of samples in each subset from the required number of samples as defined by the proportion vector $\mathbf{r}$ . For example, if a dataset of 10 samples is to be equally divided but the subsets contain 4 and 6 samples respectively, the value SD is 1. It is calculated as the average across folds of the deviations from the desired number of samples in each subset. A low SD indicates that the number of samples per fold closely matches the intended proportions. + +$$ +\mathrm {S D} = \frac {1}{K} \sum_ {k = 1} ^ {K} \left| \left| S ^ {k} \right| - N ^ {k} \right| \tag {1} +$$ + +where $N^{k} = r^{k}\cdot N$ is the expected number of samples in fold $k$ . + +Pixel Label Distribution (PLD) This measure evaluates how closely the proportion of pixels of each class within each subset matches the proportion of that class in the entire dataset. For each class $c$ , it calculates the absolute difference between the ratio of pixels of that label in each subset $k$ and the ratio of pixels of that label in the whole dataset $D$ . It then averages these results across all labels. This measure is inspired by Label Distribution (LD) in [23]. Lower PLD values indicate better preservation of class proportions at the pixel level. + +$$ +\mathrm {P L D} = \frac {1}{C} \sum_ {c = 1} ^ {C} \left(\frac {1}{K} \sum_ {k = 1} ^ {K} \left| \frac {P _ {c} ^ {k}}{P _ {c} - P _ {c} ^ {k}} - \frac {P _ {c}}{P - P _ {c}} \right|\right) \tag {2} +$$ + +An additional similarity measure examining class co-occurrence relationships is provided in Appendix H, where we extend this evaluation to capture second-order interactions between classes using the Pixel Label Pair Distribution (PLPD). + +Label Wasserstein Distance (LWD) This measure uses the Wasserstein Distance to quantify the dissimilarity between the cumulative class distributions in each fold and that of the entire dataset. By calculating this distance for each subset and averaging the results, LWD provides an indication of how closely the pixel-level class distributions in the subsets resemble that of the original dataset. Let $F_{c}$ and $F_{c}^{k}$ denote the cumulative pixel distributions up to class $c$ in the dataset and in fold $k$ , respectively: + +$$ +F _ {c} = \sum_ {i = 1} ^ {c} P _ {i}, \qquad F _ {c} ^ {k} = \sum_ {i = 1} ^ {c} P _ {i} ^ {k} +$$ + +Then the Label Wasserstein Distance is defined as: + +$$ +\mathrm {L W D} = \frac {1}{K} \sum_ {k = 1} ^ {K} \sum_ {c = 1} ^ {C} \left| F _ {c} - F _ {c} ^ {k} \right| \tag {3} +$$ + +To complement the Wasserstein-based similarity assessment, Appendix H presents further similarity assessments using alternative distributional measures, namely the Linear-kernel Maximum Mean Discrepancy (L-MMD) and the Linear Energy Distance (L-ED). + +# 3.2 Iterative Pixel Stratification (IPS) + +The motivation behind Iterative Pixel Stratification (IPS) stems from Iterative Stratification (IS) [23], which aims to distribute samples with specific positive labels proportionally across folds. This distribution is guided by two key properties within each fold: the number of samples required and the number of positive label instances required. To achieve this, IS starts by identifying the rarest label among the unassigned samples, determines which fold most requires positive instances of it and assigns samples that have positive instances of this label to that fold. In cases where multiple folds have equal demand, the fold with the fewest total samples is prioritized. Once a sample is assigned to a fold, it is removed from the pool of unassigned samples, and the demand for all positive labels associated with that sample is reduced in the fold. This iterative process continues, with the demand for specific positive labels decreasing as samples are assigned. The process concludes once all samples have been allocated, typically after C iterations. + +IPS differs from IS in one key way: the label demand in folds is determined by the number of pixels required, not by the positive instances of that label. This also means that when a sample is assigned to a fold, the desire for all labels is reduced by the number of pixels of every label in that sample. As a result, IPS emphasizes achieving pixel-level balance over maintaining sample count proportionality. Note that IPS is implemented by adapting Iterative Stratification without additional optimizations thereby serving as a straightforward baseline. The pseudo-code for IPS algorithm is presented in Appendix B. An alternative iterative stratification approach is presented in Appendix G. + +# 3.3 Wasserstein-Driven Evolutionary Stratification (WDES) + +WDES formulates the stratification problem as an optimization task and solves it using a genetic algorithm. Each candidate solution, or individual, represents an assignment of samples to folds. Initially, a population of individuals is generated by randomly assigning samples to folds according to the target proportion vector $\mathbf{r}$ . Each gene (element) in an individual encodes the fold assignment of a specific sample. + +The evolution of individuals is driven by crossover and mutation. First, we evaluate fitness using LWD (Eq. 3) and perform tournament selection [33]. Crossover occurs with a fixed probability, exchanging gene segments between individuals. A correction step follows to maintain target proportions across folds, reassigning samples if necessary. Mutation swaps assignments of randomly selected pairs, preserving fold proportions and adding diversity. This cycle repeats for a set number of generations, with the fittest individual (i.e., with the lowest LWD) selected as the final stratification. + +We use the LWD as the fitness function to measure class distribution similarity. The proportionality requirement is managed through the initial population and crossover design, ensuring the genetic algorithm optimizes class distribution while maintaining proportionality across folds for a balanced stratification. + +# 3.4 On the convergence of stratification algorithms + +For large multi-label datasets, as the dataset size grows, random splits asymptotically preserve the marginal label frequencies in each fold due to the law of large numbers. We provided a mathematical proof of this property in Appendix A.1. This property makes random stratification a practical choice for very large datasets, where computational efficiency and simplicity are preferable. + +For smaller datasets, where $N$ is finite, WDES provides strong guarantees of approaching the optimal stratification as the number of generations and population size increase. + +Proposition 3.1 (Convergence Rate of WDES to Empirical Optimum). Let $\mathcal{F}_{M,G} \subset \mathcal{F}$ denote the set of stratifications explored by the WDES algorithm after $G$ generations with a population size $M$ . Then the expected suboptimality of the final stratification $\mathcal{S}_{WDES} \in \mathcal{F}_{M,G}$ satisfies: + +$$ +\mathbb {E} \left[ L \left(\mathcal {S} _ {W D E S}\right) \right] - L _ {\mathcal {F} _ {M, G}} ^ {*} \leq \mathcal {O} \left(\frac {1}{M G}\right). +$$ + +where $L_{\mathcal{F}_{M,G}}^{*}$ denotes the optimal Wasserstein score in the empirical subset. + +Proof. See Appendix A.2. + +![](images/b073a9a4b92a2706a6eb3a1f606919adc463d0bc334bfcef843bfa3a740aa7a6.jpg) + +The above results show that the expected quality of the WDES stratification improves at a rate of $\mathrm{O}(1 / \mathrm{MG})$ , meaning that as the number of generations and population size increase, the WDES algorithm converges toward the global optimum. This makes WDES a particularly effective choice for small datasets, where the algorithm can afford the additional computational cost required to reach an optimal stratification, providing guarantees of high-quality splits as the search progresses. + +# 4 Experiments + +We conduct a comparative analysis of three stratification strategies: a) random sampling using KFold from the scikit-learn library [34], b) Iterative Pixel Stratification, and c) Wasserstein-Driven Evolutionary Stratification implemented using the depal library [35]. + +To achieve this, we start by testing these algorithms on the datasets outlined in Section 4.1 to calculate the statistical similarity measures (described in Section 3.1). For all the algorithms, we consider the number of pixels belonging to each class in the provided mask of every sample to guide the stratification procedure. The applicable parameters used for WDES are outlined in Appendix D. We perform stratification with datasets randomly shuffled five times and average the calculated similarity measures. + +Following this, we perform 10-fold cross-validation tests with random splitting and the targeted stratification strategies. For every fold, we train a UNet [6] with a resnet34 encoder from [36] for 50 epochs to perform segmentation on the images. The training process employs a learning rate of $2\mathrm{e} - 4$ and utilizes the Adam optimizer with DiceLoss. For PascalVOC, CELoss is used instead, with a learning rate of $1\mathrm{e} - 4$ and trained for 100 epochs. All experiments are conducted on a single Nvidia A100 graphics card with 40GB of VRAM, without distributed training, ensuring consistent and comparable results across the different stratification methods. We evaluate the stratification methods by measuring the variance in model performance across folds, to assess whether targeted stratification provides more stable and reliable evaluations. + +# 4.1 Datasets + +We select datasets spanning four major application domains: autonomous driving/street scenes, medical images, satellite imagery, and general-purpose datasets. For autonomous driving, we use Cityscapes [21] and CamVid [18]. In the medical domain, we consider EndoVis2018, a robotic scene segmentation dataset from MICCAI 2018 [19]. For satellite imagery, we use the LoveDA dataset [20], which focuses on land-cover segmentation in urban and rural locations. For general-purpose segmentation tasks, we include PascalVOC 2012 [17]. + +To characterize the complexity of each dataset, we define several key properties. Class Cardinality (CC) refers to the average number of classes present in a sample. Class Ubiquity (CU) captures the average number of samples in which each class appears, indicating how frequently each class occurs across the dataset. Average Imbalance Ratio (AIR) is the mean of class-wise imbalance ratios, where the imbalance ratio for a given class is the ratio of pixels in the most frequent class to those in the class under consideration. Entropy quantifies the diversity in class proportions; higher entropy implies a more balanced distribution of classes, whereas lower entropy indicates imbalance. These metrics provide a nuanced understanding of the datasets' structural diversity and segmentation challenges. The properties of all datasets are summarized in Table 1, sorted by entropy. + +Table 1: Datasets and properties + +
DatasetN ImagesN ClassesCCCUAIREntropy
PascalVOC [17]2,913212.48344791.82
CamVid [18]7013217.443707.56E+061.94
EndoVis [19]2,2351261,1171,4922.38
LoveDA [20]2,52284.9715684.52.61
CityScapes [21]2,9743516.61,4124673.17
+ +Table 2: Comparison of stratification methods using statistical similarity measures across datasets. WDES yields folds that better reflect the overall distribution, as indicated by PLD and LWD. The advantage narrows for datasets with higher class cardinality and lower ubiquity. + +
DatasetMethodSDPLD (×10-5)LWD (×10)
MeanStdMeanStdMeanStd
PascalVOCRandom0.420.0095513775.02.62
IPS10.282.4673324453.31.20
WDES0.420.0045677.951.41.63
CamvidRandom0.000.0063.43.731835.08
IPS6.681.1876.65.302098.44
WDES0.000.0036.74.471387.84
EndoVisRandom0.500.0060981.466135.5
IPS11.421.0576912.276475.2
WDES0.500.0020823.839910.4
LoveDARandom0.320.001110109108088.4
IPS8.061.151090207110087.4
WDES0.320.0037731.463516.0
CityscapesRandom0.500.0012615.430416.5
IPS11.542.6914817.432422.2
WDES0.500.00673.442175.17
RankingRandom12.42.2
IPS22.62.8
WDES111
+ +![](images/26328d5905814f1af06cbbcd329fe2f53e8e6f766700bde9244c3e40f741deae.jpg) + +![](images/80e57ff253f2fbca4cbfc6154f2e6aeb86a0136618aa45fd21117a4817133907.jpg) + +![](images/6f49e683f0df3f8906a5a6d39c953c36a79b5960b3d331955b736fdb60b88afe.jpg) +Figure 2: Relative improvement of WDES over random splitting in PLD and LWD. Performance gap decreases with increasing class cardinality, but WDES maintains an edge in datasets with high class ubiquity. + +![](images/5a31e61e2cd11c979f025e10c072f881665bc3799f9a2f7d8c9c6b2a193d3435.jpg) + +Table 3: Average segmentation performance of UNet across stratification methods in 10-fold cross-validation. Standard deviations in Accuracy, F1, and IoU are used to assess consistency. WDES performs best in low-entropy datasets, while random splitting becomes competitive as entropy increases. + +
DatasetMethodAccuracyF1IoU
MeanStd (×10-3)MeanStd (×10-3)MeanStd (×10-3)
PascalVOCRandom0.7620.60.5832.50.4434.3
IPS0.7614.50.5830.30.4431.6
WDES0.7511.00.5724.00.4324.2
CamvidRandom0.940.680.911.160.891.17
IPS0.940.740.911.350.891.27
WDES0.940.670.911.090.891.06
EndoVisRandom0.9419.40.8627.00.8028.1
IPS0.9414.10.8729.30.7930.7
WDES0.9413.00.8517.90.7918.9
LoveDARandom0.886.630.806.390.698.38
IPS0.887.390.809.290.6910.7
WDES0.887.490.809.860.6911.0
CityscapesRandom0.7911.20.619.120.508.64
IPS0.7910.30.6114.90.5015.2
WDES0.7911.20.6114.40.5012.8
RankingRandom2.41.81.8
IPS22.62.6
WDES1.61.61.6
+ +# 5 Results and Discussion + +# 5.1 Distribution of Labels and Samples + +Table 2 presents the statistical similarity measures calculated for the subsets generated across the five datasets. For each measure and dataset, we report the mean over five runs, highlighting the best-performing method in bold. An overall average rank is also computed for each method by assigning rank 3 to the method with the lowest mean and rank 1 to the one with the highest. + +Our findings indicate that WDES consistently achieves superior performance in terms of PLD and LWD across all datasets, underscoring its effectiveness in preserving label proportions and label distributions within the stratified subsets. Although IPS was explicitly designed to address proportionality in class presence, WDES outperforms it not only in LWD (which it directly optimizes) but also in PLD. This reinforces the perspective that stratification for image segmentation tasks benefits more from minimizing distributional dissimilarity than from simply ensuring class proportions. + +WDES also exhibits favorable SD scores, which follow directly from its design. Because WDES explicitly enforces predefined sample proportions across folds, it naturally results in balanced sample counts. This behavior is similarly observed in random splitting, which allocates samples proportionally across folds without replacement. + +Another noteworthy pattern emerges in relation to dataset complexity. As class cardinality increases, the performance gap between WDES and random splitting diminishes for both PLD and LWD. This trend suggests that when more classes are present per sample, the benefits of targeted stratification decrease due to the inherent balancing effect of high cardinality. In contrast, as class ubiquity increases, indicating that classes are more commonly present across the dataset, the superiority of WDES becomes more pronounced. This is expected, as higher ubiquity leads to more overlap between classes and samples, thereby amplifying the benefits of optimization-based stratification. These trends are illustrated in Figure 2, which presents the changes in PLD and LWD differences relative to class cardinality and ubiquity using trend lines. + +![](images/27cf22d5bdf2c37adb430cf81481375e2b561f167f6c80c7ce8d8483bc5eb284.jpg) +Figure 3: Stratification method rankings by performance variance across datasets. WDES ranks highest in low-entropy settings; random splitting improves with higher entropy, reducing the benefit of targeted stratification. + +![](images/dddaf6393a6a1f6e4562661abd351cc6588e0d6715caf396eec60cc168a744a3.jpg) + +![](images/e109e74032f211c31fa42869fd2665f36150a5c9c856e2261e9bc09be5dcb24c.jpg) + +Table 4: Average segmentation performance of UNet for underrepresented classes (bicycle, boat, and potted plant) in PascalVOC under 10-fold cross-validation. Standard deviations in Accuracy, F1 Score, and IoU are reported to assess consistency. WDES yields the most stable results across all three rare classes, indicating improved fold balance for low-frequency categories. + +
ClassMethodAccuracyF1IoU
MeanStdMeanStdMeanStd
BicycleRandom0.620.040.310.080.190.06
IPS0.620.040.310.090.190.06
WDES0.620.030.310.070.190.05
BoatRandom0.720.050.530.090.370.08
IPS0.700.060.490.140.330.12
WDES0.720.040.550.080.380.08
Potted PlantRandom0.710.080.470.160.320.14
IPS0.710.090.510.160.360.14
WDES0.700.070.470.120.310.10
+ +# 5.2 Variance of Performance + +Table 3 presents the results of training a UNet model on each dataset using 10-fold cross-validation. For each fold, we evaluate performance using Accuracy, F1 Score, and Intersection over Union (IoU), and report the average across the ten experiments. To highlight the robustness of each stratification strategy, we identify the method with the lowest standard deviation for each metric. Rankings are assigned from 1 to 3, with 1 indicating the lowest variance in performance and thus better fold consistency. + +Our analysis shows that WDES achieves the best average rank in standard deviation across Accuracy, F1 Score, and IoU. This indicates that WDES produces splits that are more consistent with each other and better reflect the overall data distribution, leading to more stable model evaluation. While we cannot distinguish whether the reduced variance stems from the training or test set, the smaller size of the test set suggests it is more affected by sampling variability. Thus, the improvements in test set consistency are likely the main driver of the reduced standard deviation. This makes the average cross-validation score under WDES a more reliable estimate of a model's true generalization performance. We leave empirical validation of this to future work. + +To further examine whether these improvements extend to underrepresented categories, we analyzed the segmentation performance for three rare classes in PascalVOC: bicycle, boat, and potted plant. As shown in Table 4, WDES consistently yields lower standard deviation in Accuracy, F1 Score, and IoU for these classes compared to random and IPS-based splits. This mirrors the global trend + +observed in our main results, suggesting that WDES not only improves stability across folds overall but also enhances evaluation consistency for rare classes in low-entropy datasets. + +As we move from low-entropy to high-entropy datasets, where label distributions are inherently more balanced and diverse, the advantage of WDES diminishes. The performance of random splitting improves and can even outperform stratified approaches. In these cases, the class distributions are already relatively uniform across samples, making targeted stratification less critical. This trend is reflected in the increasing rank of random splitting and the decline in WDES performance with increasing entropy, as illustrated in Figure 3. + +While IPS was designed to be a label-aware stratification method, it ranks lowest on average across accuracy, F1, and IoU deviation. A potential reason lies in its single-pass, greedy allocation strategy, which reduces label demands in a fold as soon as a sample is assigned. This mechanism introduces a path dependency that prevents the algorithm from reassessing earlier decisions or accounting for broader distributional goals. Furthermore, although IPS considers pixel-level proportions, its strategy is primarily focused on satisfying per-label pixel quotas without evaluating the resulting inter-label distribution in each fold. As a result, by prioritizing rare labels first and assigning samples based solely on immediate label demand, it may overlook more nuanced strategies for dividing the dataset, such as balancing co-occurring label groups, preserving contextual diversity, or accounting for interdependencies between frequent and infrequent classes. + +# 6 Conclusion and Limitations + +In this paper, we introduced WDES, a stratification method for image segmentation tasks based on a genetic algorithm that minimizes the Wasserstein distance between label distributions. We provide a theoretical guarantee (Appendix A.2) that WDES converges to a globally optimal stratification given sufficient generations. We empirically compared its performance against iterative (IPS) and random sampling across several datasets and evaluation criteria. WDES outperforms both iterative and random methods, achieving the best average rank in accuracy, F1 score, and IoU variance in segmentation tasks, particularly for low-entropy datasets. Conversely, random stratification shows lower variance in high-entropy datasets, indicating that the benefits of targeted stratification diminish as class distributions become more uniform. + +We emphasize the importance of stratification in cross-validation and train-test splits for image segmentation, particularly to avoid underrepresenting certain test sets and biasing the evaluation. This issue is especially critical for rare or underrepresented classes and when working with small datasets. Because final performance scores in cross-validation are typically computed as the mean of fold-wise means, the effects of imbalanced splits are not mitigated by the repeated nature of the process. While WDES improves upon simpler strategies, it is not a universal solution. 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If you obtained IRB approval, you should clearly state this in the paper. +- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. +- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +# Answer: [NA] + +Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components. + +# Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# A Convergence of Stratification Algorithms + +This section presents two results on the convergence behavior of stratification methods. The first shows that random splits of large multi-label datasets asymptotically preserve marginal label frequencies in each fold, by the law of large numbers, ensuring representativeness without explicit stratification. The second analyzes WDES, an evolutionary algorithm that minimizes label distribution differences across folds. It proves that the expected quality of the best-found stratification improves at a rate of O(1/MG), converging toward the global optimum as the number of generations and population size grow. + +# A.1 Asymptotic Representativeness of Random Splits in Multi-Label Datasets + +Theorem A.1. Let $D = \{(\mathbf{x}_n, \mathbf{y}_n)\}_{n=1}^N$ be a dataset where each $\mathbf{y}_n \in \{0,1\}^L$ is a binary label vector over a fixed set of $L$ labels. Suppose $D$ is randomly partitioned into $K$ disjoint folds $S = \{S^1, \ldots, S^K\}$ such that each fold satisfies $|S^k| \approx r^k N$ , for a fixed target proportion vector $\mathbf{r} \in [0,1]^K$ with $\sum_{k=1}^{K} r^k = 1$ . Then, for any label index $\ell \in \{1, \ldots, L\}$ , the empirical label frequency in each fold converges almost surely to the true marginal label frequency as $N \to \infty$ . That is, for all $k = 1, \ldots, K$ , + +$$ +\hat {p} _ {N} ^ {k} (\ell) \xrightarrow {a . s .} p (\ell), +$$ + +where + +$$ +\hat {p} _ {N} ^ {k} (\ell) = \frac {1}{| S ^ {k} |} \sum_ {n \in S ^ {k}} y _ {n \ell} \quad a n d \quad p (\ell) = \mathbb {E} [ y _ {n \ell} ] = \mathbb {P} (y _ {n \ell} = 1). +$$ + +Proof. Fix a label index $\ell \in \{1,\dots ,L\}$ . For each sample $n$ , define the binary indicator: + +$$ +Z _ {n} ^ {(\ell)} = y _ {n \ell} \in \{0, 1 \}. +$$ + +By assumption, the data points $(\mathbf{x}_n,\mathbf{y}_n)$ are drawn i.i.d., so the sequence $\{Z_n^{(\ell)}\}_{n = 1}^N$ is i.i.d. Bernoulli with mean: + +$$ +\mathbb {E} \left[ Z _ {n} ^ {(\ell)} \right] = p (\ell). +$$ + +Now consider the random assignment of samples to each fold $S^k$ , where $|S^k| \approx r^k N$ . For sufficiently large $N$ , each $|S^k| \to \infty$ and the samples in each fold remain representative of the full dataset. + +Let + +$$ +\hat {p} _ {N} ^ {k} (\ell) = \frac {1}{| S ^ {k} |} \sum_ {n \in S ^ {k}} Z _ {n} ^ {(\ell)}. +$$ + +Then, by the Strong Law of Large Numbers (SLLN), + +$$ +\hat {p} _ {N} ^ {k} (\ell) \xrightarrow {a . s .} \mathbb {E} \left[ Z _ {n} ^ {(\ell)} \right] = p (\ell), \quad \text {a s} | S ^ {k} | \to \infty . +$$ + +Therefore, for any $k, k' \in \{1, \ldots, K\}$ , + +$$ +\left| \hat {p} _ {N} ^ {k} (\ell) - \hat {p} _ {N} ^ {k ^ {\prime}} (\ell) \right| \xrightarrow {a . s .} 0, +$$ + +and each fold becomes asymptotically representative of the true label distribution. + +![](images/c7a7e49044d26d055631dbc46d249cbaf4a09239b295edb72ce26198c7fdb61b.jpg) + +# A.2 Optimality and Convergence of WDES + +To quantify the effectiveness of WDES, we derive a convergence bound that characterizes how the quality of stratifications improves with the number of generations and population size under standard evolutionary algorithm assumptions. + +Proposition A.1 (Convergence Rate of WDES to Empirical Optimum). Let $\mathcal{F}_{M,G} \subset \mathcal{F}$ denote the set of stratifications explored by the WDES algorithm after $G$ generations with a population size $M$ . Then the expected suboptimality of the final stratification $\mathcal{S}_{WDES} \in \mathcal{F}_{M,G}$ satisfies: + +$$ +\mathbb {E} \left[ L \left(\mathcal {S} _ {W D E S}\right) \right] - L _ {\mathcal {F} _ {M, G}} ^ {*} \leq \mathcal {O} \left(\frac {1}{M G}\right). +$$ + +where $L_{\mathcal{F}_{M,G}}^{*}$ denotes the optimal Wasserstein score found in the empirical subset. + +Proof. WDES is a genetic algorithm that evolves a population of $M$ stratifications over $G$ generations. The algorithm operates using elitist selection, where the best individual is preserved across generations. This ensures that the sequence of best-found scores $\{L_{\mathrm{best}}^g\}_{g = 1}^G$ is non-increasing. + +Each generation performs $M$ fitness evaluations, totaling $MG$ evaluations across the run. The search space $\mathcal{F}$ is finite because there are finitely many possible assignments of $N$ labeled data points to $K$ folds satisfying approximate size constraints. Hence, WDES induces a finite-state stochastic process over $\mathcal{F}_{M,G} \subset \mathcal{F}$ , the set of stratifications visited during execution. + +Let $D = \{(\mathbf{x}_n, \mathbf{y}_n)\}_{n=1}^N$ be a dataset with $C$ classes, and let $\mathcal{F}$ denote the space of feasible stratifications $S = \{S^1, \ldots, S^K\}$ , where each fold $S^k$ satisfies the approximate size constraint $|S^k| \approx r^k N$ for a target proportion vector $\mathbf{r} \in [0,1]^K$ , with $\sum_k r^k = 1$ . + +Define the class distribution in fold $k$ as $P_{i}^{k} = \frac{1}{|S^{k}|}\sum_{n\in S^{k}}\mathbb{I}[y_{n} = i]$ , and the global class distribution as $P_{i} = \frac{1}{N}\sum_{n = 1}^{N}\mathbb{I}[y_{n} = i]$ , for $i = 1,\ldots ,C$ . Let $F_{c}^{k} = \sum_{i = 1}^{c}P_{i}^{k}$ and $F_{c} = \sum_{i = 1}^{c}P_{i}$ be the cumulative class distributions in fold $k$ and the full dataset, respectively. + +The Label Wasserstein Distance of stratification $S$ is defined as: + +$$ +L (\mathcal {S}) = \frac {1}{K} \sum_ {k = 1} ^ {K} \sum_ {c = 1} ^ {C} \left| F _ {c} ^ {k} - F _ {c} \right|. +$$ + +Crucially, the mutation operator used in WDES is assumed to be ergodic, i.e., it has a positive probability of reaching any feasible stratification from any other. This property, together with elitist selection, guarantees that the algorithm converges in probability to the global optimum over $\mathcal{F}$ , as shown in [37]. However, in finite time, WDES only explores $\mathcal{F}_{M,G}$ , and its best possible result is $L_{\mathcal{F}_{M,G}}^*$ . The expected gap between the final stratification's score and this empirical optimum decreases with the number of independent evaluations $MG$ . This follows from classical convergence results for evolutionary algorithms under elitist, ergodic dynamics, where the convergence rate to the best solution seen is: + +$$ +\mathbb {E} \left[ L \left(\mathcal {S} _ {\mathrm {W D E S}}\right) \right] - L _ {\mathcal {F} _ {M, G}} ^ {*} \leq \mathcal {O} \left(\frac {1}{M G}\right). +$$ + +This bound reflects the diminishing returns of increasing population size and generations, but confirms that more search budget leads to better empirical solutions. + +Remark. The theorem guarantees convergence to the best stratification within the explored subset $\mathcal{F}_{M,G}$ , not necessarily the global optimum $L^{*} = \min_{\mathcal{S} \in \mathcal{F}} L(\mathcal{S})$ . Nonetheless, due to the ergodic nature of the mutation operator, the probability of reaching any region of the space is non-zero. Thus, $L_{\mathcal{F}_{M,G}}^{*} \to L^{*}$ as $G \to \infty$ , and the algorithm converges to the global optimum in probability. + +# B Iterative Pixel Stratification Algorithm + +Following the notation introduced in Section 3, the algorithm for IPS is described in Algorithm 1. + +Algorithm 1 Iterative Pixel Stratification +Input: A dataset $D$ of $C$ classes, $N$ samples that have a total of $P$ pixels. The dataset is to be divided into $K$ folds in the proportion $\mathbf{r} = r_1,\dots ,r_k$ . Output: Disjoint subsets $S_{1},\ldots ,S_{k}$ +1: #Calculate desired number of samples per fold +2: $\{N^k\}_{1}^K\gets N\cdot \mathbf{r}$ +3: for $c\gets 1$ to $C$ do +4: #Calculate number of pixels present per class +5: $P_{c}\gets \{P:c\in C\}$ +6: #Calculate desired number of pixels per class per fold +7: $\{P_c^k\}_1^K\gets P_c\cdot \mathbf{r}$ +8: end for +9: while $|D| > 0$ do +10: #Find rarest class +11: $l\gets \mathrm{argmin}P_{c}$ +12: for $(\mathbf{x}_l,\mathbf{y}_l)\in D_l$ do +13: #Find folds with largest desire for class $l$ +14: $M\gets \operatorname {argmax}P_l^k$ +15: if $|M| = 1$ then +16: $m\gets M$ +17: else +18: $M^{\prime}\gets \mathrm{argmin}N^{k}$ +19: if $|M^{\prime}| = 1$ then +20: $m\gets M^{\prime}$ +21: else +22: $m\gets$ Random Element of $(M^{\prime})$ +23: end if +24: end if +25: #Assign sample to subset and remove from dataset +26: $S^{m}\gets S^{m}\cup (\mathbf{x}_{l},\mathbf{y}_{l})$ +27: $D\gets D\setminus \{(\mathbf{x}_l,\mathbf{y}_l)\}$ +28: $N^{m}\gets N^{m} - 1$ +29: for $c\in C$ do +30: $P_{c}^{m} = P_{c}^{m} - P_{c}(\mathbf{x}_{l},\mathbf{y}_{l})$ +31: end for +32: end for +33: end while + +# C Wasserstein-Driven Evolutionary Stratification Algorithm + +Following the notation introduced in Section 3, the algorithm for WDES is described in Algorithm 2. + +Algorithm 2 Wasserstein-Driven Evolutionary Stratification (WDES) + +Input: A dataset $D$ of $C$ classes and $N$ samples, to be divided into $K$ folds according to proportions $\mathbf{r} = (r_1,\dots,r_K)$ ; population size $M$ ; number of generations $G$ ; crossover rate $p_c$ ; mutation rate $p_m$ . + +Output: Disjoint subsets $S_{1}, \ldots, S_{K}$ corresponding to the fittest individual. +1: # Initialize population of candidate stratifications +2: for $i\gets 1$ to $M$ do +3: Generate random fold assignments according to $\mathbf{r}$ +4: Store as individual $I_{i}$ +5: end for +6: for $g \gets 1$ to $G$ do +7: #Evaluate fitness of each individual +8: for $i\gets 1$ to $M$ do +9: Evaluate fitness using Label Wasserstein Distance (Eq. 3) for fold assignments in $I_{i}$ +0: end for +1: #Select parents via tournament selection +2: Select pairs of individuals based on lowest LWD values +3: #Apply crossover and correction +4: for $p \gets 1$ to number of parent pairs do +5: if random() $< p_{c}$ then +6: Exchange random segments of fold assignments between parents +7: Apply correction to preserve fold proportions r +8: end if +9: end for +20: #Applymutation +21: for $i\gets 1$ to $M$ do +22: if random() $< p_{m}$ then +23: Randomly swap fold assignments of two samples in $I_{i}$ +24: end if +25: end for +26: end for +27: # Select best-performing individual as final stratification +28: $I^{*}\gets \underset {I_{i}}{\mathrm{argmin}}\mathrm{LWD}(I_{i})$ +29: Return $S_{1}, \ldots, S_{K}$ as the folds encoded in $I^{*}$ . + +# D Genetic Algorithm parameters + +We empirically selected the hyper-parameters for the genetic algorithm by evaluating performance across multiple datasets. For each parameter, we increased its value until no further improvements were observed in the stratification quality. The final values are shown in Table 5, + +Table 5: WDES Parameters + +
ParameterValue
Number of Generations50
Number of Individuals100
Gene Mating Probability0.5
Individual Mutation Probability0.2
Selection Tournament size3
+ +# E Time and Space Complexity of WDES + +The computational complexity of the Wasserstein Distance-based Evolutionary Stratifier (WDES) can be analyzed in terms of the population size $M$ , number of generations $G$ , number of samples $N$ , number of classes $C$ , and number of folds $K$ . Each individual in the population represents an assignment of all $N$ samples into $K$ folds, and the fitness evaluation involves computing the Wasserstein distance across $C$ -dimensional class histograms. + +The overall time complexity is therefore $\mathcal{O}(M\cdot G\cdot K\cdot C)$ , with an additional linear component in $N$ arising from crossover and mutation operations. The space complexity is $\mathcal{O}(M\cdot N)$ , as the algorithm maintains a population of $M$ individuals, each encoding fold assignments for all samples. + +In practice, the runtime of WDES is further influenced by the genetic algorithm parameters, including mutation and crossover probabilities, which determine how much of the population is updated between generations. These hyperparameters affect convergence speed and computational cost but do not alter the asymptotic complexity. An empirical assessment of runtime performance is provided in the following section. + +# F Runtime Analysis + +We evaluate the runtime performance of our stratification methods on an Apple MacBook Pro with an M3 Pro processor and 36 GB of memory. Figure 4 presents the runtime in seconds (with a logarithmic scale on the y-axis) for Random splitting, IPS, and WDES across all datasets. + +![](images/801cf310c891112b36ca528f0da5fee0e2e779cee6d522696c18d63d9ef6017d.jpg) +Figure 4: Runtime comparison of Random splitting, IPS, and WDES across all datasets. + +Table 6: Comparison of stratification methods on similarity-based evaluation metrics. Results are averaged across datasets for Random splitting, IPS, CS-IPS, and WDES. While CS-IPS performs slightly better than IPS in some cases, WDES consistently achieves the best scores across all metrics, indicating stronger distributional alignment. + +
DatasetMethodSDPLD (×10-5)LWD (×10)
MeanStdMeanStdMeanStd
PascalVOCRandom0.420.0095513775.02.62
CS-IPS0.960.005256.3342.50.32
IPS10.282.4673324453.31.20
WDES0.420.0045677.951.41.63
CamvidRandom0.000.0063.43.731835.08
CS-IPS0.600.0058.00.391751.48
IPS6.681.1876.65.302098.44
WDES0.000.0036.74.471387.84
EndoVisRandom0.500.0060981.466135.5
CS-IPS0.800.004858.005493.34
IPS11.421.0576912.276475.2
WDES0.500.0020823.839910.4
LoveDARandom0.320.001110109108088.4
CS-IPS0.640.0093511.98927.58
IPS8.061.151090207110087.4
WDES0.320.0037731.463516.0
CityscapesRandom0.500.0012615.430416.5
CS-IPS1.900.001304.483004.73
IPS11.542.6914817.432422.2
WDES0.500.00673.442175.17
RankingRandom13.03.2
CS-IPS22.22.0
IPS33.83.8
WDES111
+ +# G Class-Sensitive Iterative Pixel Stratification (CS-IPS) + +The Class-Sensitive Iterative Pixel Stratification (CS-IPS) algorithm introduces an alternative iterative approach for constructing balanced folds based on pixel-level class distributions. The method begins by ranking all classes in the dataset according to their rarity, determined by the total number of pixels belonging to each class. Starting from the rarest class, all samples containing pixels of that class are identified and sorted in descending order based on the number of pixels associated with the current class. The sorted samples are then distributed across folds in a zig-zag manner to ensure a balanced representation of the class across partitions. This procedure is repeated iteratively for each subsequent class until all samples have been assigned. The resulting folds aim to maintain pixel-level balance while accounting for class frequency in a straightforward, iterative manner. + +We report the results of CS-IPS alongside those obtained using random splitting, IPS, and WDES in Table 6. While CS-IPS achieves slightly improved performance compared to IPS in several cases, WDES consistently attains the highest overall performance across datasets. These results reinforce our motivation for adopting a more flexible, globally optimized approach such as the genetic algorithm employed in WDES. + +The corresponding segmentation results, summarized in Table 7, exhibit a similar trend. CS-IPS provides measurable improvements over IPS, indicating the benefits of incorporating class sensitivity in the iterative process. However, WDES continues to outperform all other methods, underscoring the advantages of optimization-driven stratification over purely heuristic approaches. + +Table 7: Average segmentation performance of UNet under 10-fold cross-validation using different stratification methods. Reported values correspond to Accuracy, F1 Score, and IoU. CS-IPS shows improved stability compared to IPS but remains below WDES, which continues to yield the most consistent and reliable fold performance. + +
DatasetMethodAccuracyF1IoU
MeanStd (×10-3)MeanStd (×10-3)MeanStd (×10-3)
PascalVOCRandom0.7620.60.5832.50.4434.3
CS-IPS0.7112.20.4526.20.3225.3
IPS0.7614.50.5830.30.4431.6
WDES0.7511.00.5724.00.4324.2
CamvidRandom0.940.680.911.160.891.17
CS-IPS0.940.700.911.320.891.24
IPS0.940.740.911.350.891.27
WDES0.940.670.911.090.891.06
EndoVisRandom0.9419.40.8627.00.8028.1
CS-IPS0.9213.60.8621.20.7821.4
IPS0.9414.10.8729.30.7930.7
WDES0.9413.00.8517.90.7918.9
LoveDARandom0.886.630.806.390.698.38
CS-IPS0.887.300.808.190.6910.3
IPS0.887.390.809.290.6910.7
WDES0.887.490.809.860.6911.0
CityscapesRandom0.7911.20.619.120.508.64
CS-IPS0.7912.20.6121.50.5118.2
IPS0.7910.30.6114.90.5015.2
WDES0.7911.20.6114.40.5012.8
RankingRandom2.62.22.2
CS-IPS2.42.62.6
IPS2.63.23.6
WDES1.81.81.8
+ +# H Additional Evaluation Metrics + +To ensure that the evaluation of stratification quality is not biased toward a specific similarity metric, we further assess all stratification methods using additional distributional measures: the Linear-kernel Maximum Mean Discrepancy (L-MMD) and the Linear Energy Distance (L-ED). These metrics provide complementary perspectives on distributional similarity between folds and the overall dataset, capturing higher-order differences beyond those measured by the Label Wasserstein Distance (LWD). + +Table 8 reports results for all methods—random splitting, IPS, CS-IPS, and WDES—across these alternative metrics. Consistent with the findings based on LWD, WDES achieves the lowest average discrepancy across datasets, demonstrating that its advantages generalize across diverse statistical criteria. IPS performs competitively on the PascalVOC dataset, which we attribute to its higher proportion of single-class samples. Overall, these results confirm that the performance gains of WDES are robust to the choice of similarity metric. + +Pixel Label Pair Distribution (PLPD) In addition to the aforementioned distributional measures, we incorporate a class co-occurrence metric inspired by the Label Pair Distribution (LPD) proposed in literature [25]. This measure captures second-order relationships between classes that tend to appear together within the same sample. We adapt this concept to the pixel level, computing the Pixel Label Pair Distribution (PLPD) to quantify the co-occurrence consistency between folds and the global dataset. The results, presented in Table 8, extend our analysis by evaluating how well each method preserves inter-class relationships. The previously introduced CS-IPS method is also included for completeness. + +Table 8: Evaluation of stratification quality using alternative similarity measures: Pixel Label Pair Distribution (PLPD), Linear-kernel Maximum Mean Discrepancy (L-MMD), and Energy Distance (L-ED). Results are averaged across datasets for Random splitting, IPS, CS-IPS, and WDES. WDES consistently outperforms the other methods across all three metrics, demonstrating that its advantages generalize beyond Wasserstein-based evaluation. + +
DatasetMethodPLPDL-MMD (×102)L-ED
PascalVOCRandom0.675.673.64
CS-IPS0.692.682.91
IPS0.661.842.56
WDES0.762.942.49
CamvidRandom4.7012.28.93
CS-IPS5.9711.18.28
IPS4.6214.79.91
WDES4.587.166.98
EndoVisRandom0.3646.814.7
CS-IPS0.2935.711.7
IPS0.2755.216.4
WDES0.2517.99.53
LoveDARandom1.35×10-578.620.9
CS-IPS1.22×10-562.717.2
IPS1.21×10-569.921.3
WDES1.20×10-527.712.8
CityscapesRandom3.7320.77.70
CS-IPS3.7521.27.44
IPS3.8523.38.24
WDES3.6811.85.79
RankingRandom33.23.2
CS-IPS3.22.22.2
IPS2.23.23.6
WDES1.61.41.0
+ +These additional evaluations further corroborate the overall trends observed in the main results: while iterative approaches such as IPS and CS-IPS produce reasonable fold-level balance, WDES consistently achieves superior alignment with global label distributions across all tested metrics. + +# I Dataset details + +# I.1 Cityscapes + +Cityscapes is a large benchmark dataset for training and testing pixel-level and instance-level semantic labeling. It contains diverse stereo video sequences from street scenes in 50 cities. Of these frames, 5,000 images have high-quality pixel annotations across 30 visual classes, grouped into eight categories: flat, construction, nature, vehicle, sky, object, human, and void. We use only the left-camera images (the annotated view) and run our experiments on the training set, since it's the largest split. + +# I.2 CamVid + +The Cambridge-driving Labeled Video Database (CamVid) captures footage from the perspective of a driving automobile, unlike most videos that are filmed with fixed-position CCTV-style cameras. From this footage, 701 frames were sampled at $1\mathrm{Hz}$ and manually labeled with 32 semantic classes. Similar to CityScapes, we only use the largest subset, the training set. + +![](images/aa7ed4cc5cc8aad5957145a34bdbeae8d18bca60b2b3ea5a5a66911f422baf9a.jpg) +Figure 5: Example Cityscapes data: the original image (left) and its enhanced annotation mask (right), for better visualization. + +![](images/c327c9c0b6494aaef145902be7389493c6f4611f593864513efd860d6a4e25f2.jpg) + +![](images/f02d9636d385782b2221e9417d7b249bfadf1ffba3326e0686f0dce251374025.jpg) +Figure 6: Example CamVid data: the original image (left) and its enhanced annotation mask (right). + +![](images/89510593261e8f020cb5db6cd67d9b73cd8221e466aa621c4552435602ba4010.jpg) + +# I.3 Pascal VOC 2012 + +The Pascal VOC 2012 serves as a cornerstone resource for training and comparing semantic segmentation models. It comprises 20 classes, including entities like people, animals, vehicles, and indoor objects. The dataset comprises 1,464 training images, 1,449 validation images, and a private testing set. Each image in this dataset is annotated with pixel-level segmentation, bounding box, and object class information. For our experiments, we exclusively utilized the training set. + +![](images/07621227d2977ec9ef261eb9be9405d108ba38e3a9674ebc4c8af117be08f7c7.jpg) +Figure 7: Example Pascal VOC 2012 data: the original image (left) and its enhanced annotation mask (right). + +![](images/3dacee020b7b9c044c8c8dd8b69fd0d2616c0ee7df21efeb1e25230eb798df7e.jpg) + +# I.4 EndoVis2018 + +The EndoVis2018 dataset, used in the Robotic Scene Segmentation Challenge of MICCAI 2018, contains 16 robotic nephrectomy procedures recorded using da Vinci Xi systems in porcine labs. Each procedure comprises 149 training frames and 250 testing frames, each with a resolution of $1280 \times 1024$ . The dataset includes images from both the left and right eye cameras, as well as the stereo camera calibration parameters. However, labels are only available for the left eye camera, with 12 categories. In this case, we utilized only the training set. + +# I.5 LoveDA + +The LoveDA dataset, a comprehensive collection of remote sensing images, is designed for semantic segmentation. It comprises 5,987 high-resolution images captured across three Chinese cities, + +![](images/97d67e01ee69a15b81e42117c235060c8735d047685a9e7e4449c8a97e4c9a3a.jpg) +Figure 8: Example EndoVis2018 data: the original image (left) and its annotation mask (right). + +![](images/13dd4bbad1a3bdc122ebc7dff7a4c961267d187c8888161f760c2e5b1e242b8d.jpg) + +capturing a diverse range of urban and rural scenes. These images are meticulously annotated with 166,768 object annotations, categorized into seven distinct land-cover classes: background, buildings, roads, water bodies, barren land, forests, and agricultural areas. Again, only the training set is utilized. + +![](images/8c3a0408d00886f4e28f533fbd051bf9de3b5e24a5ab4bf39a6c93492dc7f510.jpg) +Figure 9: Example LoveDA data: the original image (left) and its enhanced annotation mask (right). + +![](images/7246e78194ce947a725b7e41759a37682cf0ce8b1e9afedc4c30f8d2dfa03000.jpg) + +![](images/deead36f0157c57c7feda2c847a1cdcae96c4179f68c2cacecaa77aae3438765.jpg) + +# J Acknowledgments + +The authors gratefully acknowledge the scientific support and HPC resources provided by the Erlangen National High Performance Computing Center (NHR) of the FAU Erlangen-Nürnberg. 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However, as the sequence length scales up, the memory cost for storing activation values becomes huge during the Backpropagation (BP) process, even with the application of gradient checkpointing technique. To tackle this challenge, we propose a memory-efficient and exact BP method called StreamBP, which performs a linear decomposition of the chain rule along the sequence dimension in a layer-wise manner, significantly reducing the memory cost of activation values and logits. The proposed method is applicable to common objectives such as SFT, GRPO, and DPO. From an implementation perspective, StreamBP achieves less computational FLOPs and faster BP speed by leveraging the causal structure of the language model. Compared to gradient checkpointing, StreamBP scales up the maximum sequence length of BP by $2.8 - 5.5 \times$ larger, while using comparable or even less BP time. Note that StreamBP's sequence length scaling ability can be directly transferred to batch size scaling for accelerating training. We further develop a communication-efficient distributed StreamBP to effectively support multi-GPU training and broaden its applicability. Our code can be easily integrated into the training pipeline of any transformer models and is available at https://github.com/Ledzy/StreamBP. + +# 1 Introduction + +Large language models (LLMs) [1, 6, 32] have been regarded as a powerful approach toward general artificial intelligence [3]. Recently, there has been growing interest in training LLMs for reasoning tasks, leading to significant improvements on math and code benchmarks as shown by OpenAI-ol and DeepSeek-R1 [12, 8]. However, such models often require extremely long input sequences due to the inclusion of detailed reasoning traces [8, 35, 24, 33, 34], which results in substantial GPU memory consumption during backpropagation (BP) when training reasoning models using either reinforcement learning (RL) or supervised finetuning (SFT). This considerable memory usage mainly stems from storing intermediate activations during the BP process. This work aims to provide an efficient solution to such a memory issue. + +Background and related works. Due to rapid growth of research in this field, we provide a non-exhaustive background overview on reasoning models and memory-efficient training methods. + +Training LLM on long reasoning traces. To incentivize long chain of thought (CoT) [29] reasoning capability, RL is one of most popular approaches [12, 8]. We also refer to, e.g., [7, 25, 22, 35, 19, 37, 10, 28], for replicating the "aha moment" of reasoning described in DeepSeek-R1. The core step of RL for incentivizing reasoning ability is to use a rule-based reward, e.g., the correct answer of the math question, to provide training signal towards fitting the correctly generated answer by the model itself with reasoning traces. It has been observed that as the RL training progresses, the sequence length will be increased [8]. Apart from RL techniques, it has been shown that the simple SFT training pipeline with reasoning data distilled from strong reasoning models such as R1 can also incentivize reasoning ability of LLMs; see, e.g., [24, 33, 15, 30]. The length of SFT reasoning traces varies widely, ranging from 8k to 32k, and in some cases exceeding 100k. Training models on these long traces requires significant memory cost in storing activation values. The essential roles of RL and SFT for incentivizing reasoning ability are still under active discussion [36]. + +Memory-efficient training methods. Existing memory-efficient solution for training LLMs mainly focus on designing parameter-efficient or low-dimensional version of Adam [13, 20], significantly reducing the memory usage of storing the gradient and optimizer states. Typical algorithms include LoRA [9] and its variants, e.g., [5, 18], GaLore [38], BAdam [23], etc. However, in the context of training reasoning models with long reasoning sequences, the dominant source of memory consumption arises from the BP process for storing intermediate activations. Gradient checkpointing technique [4] reduces the memory cost by recomputing the activation during the backward process. However, it still stores the full activation of the reforwarded layer and the full logits of the output layer; see Figure 1 for an illustration. MsT [21] reduces the memory cost of SFT logits by iteratively processing mini-sequence. However, it still requires the full storage of layer activations during reforward and does not apply to RL objectives. BPT [17] reduces the activation memory in transformer layer by partitioning the forward process across the sequence dimension, yet it does not handle the logits memory cost in language modeling head layer. Sequence parallel approaches increase the maximum sequence length by partitioning sequence across GPUs [16, 11, 14], which requires access of multiple GPUs. MoNET [26] explores operator-level memory optimization, yet it is specifically designed for vision models and cannot be applied to the training of LLM. + +Main contributions. Motivated by the above observations, this work aims to address the memory issue occurred during the BP process. Our main contributions are summarized below. + +(C.1) We propose a memory-efficient and exact backpropagation algorithm called StreamBP, which significantly reduces memory usage of storing the intermediate activation values when training LLMs on ultra-long sequences, e.g., training reasoning models. StreamBP is based on a linear decomposition of the chain rule and involves nontrivial and intricate developments for the language modeling head and transformer layers. As a result, StreamBP is compatible with common training objectives, including SFT, GRPO, and DPO. Moreover, by leveraging the causal structure of LLM, StreamBP is able to save computational FLOPs compared to the standard BP, achieving faster BP speed than the gradient checkpointing baseline. To support multi-GPU training, we also develop a distributed implementation of StreamBP with special attention to gradient and parameter communication, significantly improving training efficiency and broadening its applicability. + +(C.2) Empirically, we measure the maximum sequence length and time cost of StreamBP under both single GPU and distribuetd training settings. Specifically, we measure the memory and time cost of BP in Section 4.1. StreamBP substantially scales up the maximum sequence length by $2.8 - 5.5 \times$ compared to the gradient checkpointing baseline across different model scales while achieving comparable or even less BP time. In Section 4.2, we verify the effectiveness of StreamBP under various training objectives, showing that it significantly increases maximum sequence length under the objective of SFT, GRPO, and DPO. In Section 4.3, we show that under Deepspeed distributed training scheme, StreamBP is able to achieve about $5 - 5.6 \times$ larger sequence length than distributed gradient checkpointing. + +# 2 Preliminary: Peak Memory Cost during Backpropagation + +As a concrete example, we utilize PyTorch CUDA memory snapshot tool to record detailed GPU memory usage across 2 forward and backward processes of Qwen 3-4B model under the sequence length of 8192, where the gradient checkpointing technique is applied. The result is shown in Figure 1. We exclude the memory cost of optimizer states and focus solely on the BP process. + +![](images/3de271c2d3cff6cdc96cc4aafb54aaaca51d06c0dbab6d63cba2d3ce1edaf1d5.jpg) +Figure 1: Memory profile during backpropagation of Qwen 3-4B with gradient checkpointing, visualized using PyTorch's memory profiler. Sequence length is set to 8192. Batch size is 1. Optimizer states are excluded as we focus on the BP process. + +Peak memory cost. The peak memory cost occurs at the end of the 2nd forward pass. Apart from the 16GB allocated for BF16 parameters and gradients, approximately 14GB is used for storing intermediate computations. This includes BF16 checkpointed layer inputs, BF16 logits, FP32 logits, and the gradient of FP32 logits. Since the logits is of dimension sequence length $(T) \times$ vocabulary size $(C)$ , their memory cost is independent of the model size and is subject to $C$ of the model class. + +Second peak memory cost. The 2nd memory peak happens at the start of the 2nd backward, where the model reforward the checkpointed inputs of the last transformer layer to compute layer activations. The activation values will be temporarily stored for computing the gradient of the layer's parameter and inputs. + +We remark that as the model size grows up, the 2nd peak memory will increase and become closer to the peak memory. We provide more detailed explanation of the memory profile in Appendix C. + +# 3 Memory-Efficient Exact Stream Backpropagation + +In this section, we introduce the design of the proposed algorithm, which computes exact gradient with reduced memory cost and floating point operations (FLOPs) during the BP process. + +# 3.1 Main Idea + +Consider a transformation happened during the model's forward pass $f_{W}(Z_{\mathrm{in}}) = Z_{\mathrm{out}}$ , where $W$ is the weight associated with the transformation, $f_{W}(\cdot)$ can be any mapping inside the model, e.g., a transformer layer or the language modeling head. By chain rule, the gradient of weight are given by $\frac{\partial L}{\partial \mathrm{vec}(W)} = \left( \frac{\partial \mathrm{vec}(Z_{\mathrm{out}})}{\partial \mathrm{vec}(W)} \right)^{\top} \frac{\partial L}{\partial \mathrm{vec}(Z_{\mathrm{out}})}$ , where $L$ is the loss, $\mathrm{vec}(\cdot)$ is the vectorized operator, and $\frac{\partial \mathrm{vec}(Z_{\mathrm{out}})}{\partial \mathrm{vec}(W)}$ denotes the Jacobian matrix. During BP, when the gradient $\frac{\partial L}{\partial \mathrm{vec}(Z_{\mathrm{out}})}$ is ready, gradient checkpointing method will reforward $Z_{\mathrm{in}}$ through $f_{W}(\cdot)$ , and then calculate and store all the intermediate activations that are required for computing $\frac{\partial \mathrm{vec}(Z_{\mathrm{out}})}{\partial \mathrm{vec}(W)}$ and $\frac{\partial \mathrm{vec}(Z_{\mathrm{out}})}{\partial \mathrm{vec}(Z_{\mathrm{in}})}$ , where $\frac{\partial \mathrm{vec}(Z_{\mathrm{out}})}{\partial \mathrm{vec}(Z_{\mathrm{in}})}$ will be used for calculating $\frac{\partial L}{\partial \mathrm{vec}(Z_{\mathrm{out}})}$ for the preceding layer. As shown in Figure 1, the memory cost of storing these intermediate activation values can be huge. + +To reduce the memory cost of these intermediate values, we introduce the stream backpropagation (StreamBP). Let $\mathrm{vec}(Z_{\mathrm{out}}) = [\mathrm{vec}(Z_{\mathrm{out}}^{(1)}),\mathrm{vec}(Z_{\mathrm{out}}^{(2)}),\dots,\mathrm{vec}(Z_{\mathrm{out}}^{(D)})]$ be any partition of $\mathrm{vec}(Z_{\mathrm{out}})$ . StreamBP is based on the following linear decomposition: + +$$ +\frac {\partial L}{\partial \operatorname {v e c} (W)} = \left(\frac {\partial \operatorname {v e c} \left(Z _ {\text {o u t}}\right)}{\partial \operatorname {v e c} (W)}\right) ^ {\top} \frac {\partial L}{\partial \operatorname {v e c} \left(Z _ {\text {o u t}}\right)} = \sum_ {i = 1} ^ {D} \left(\frac {\partial \operatorname {v e c} \left(Z _ {\text {o u t}} ^ {(i)}\right)}{\partial \operatorname {v e c} (W)}\right) ^ {\top} \frac {\partial L}{\partial \operatorname {v e c} \left(Z _ {\text {o u t}} ^ {(i)}\right)}. \tag {1} +$$ + +By strategically partitioning $\mathrm{vec}(Z_{\mathrm{out}})$ , storing the intermediate activations required for computing $\partial \mathrm{vec}(Z_{\mathrm{out}}^{(i)}) / \partial \mathrm{vec}(W)$ can be significantly cheaper than storing those required for calculating $\partial \mathrm{vec}(Z_{\mathrm{out}}) / \partial \mathrm{vec}(W)$ , and is often proportional to the partitioned chunk size. Motivated by this fact, StreamBP sequentially computes the decomposed components in (1) across all the partitions, and accumulates them in a running sum, yielding the exact gradient. We refer to Appendix A.1 for a quick grasp of StreamBP's idea when using linear transformation. + +Next, we elaborate on applying StreamBP to a concrete transformer LLM, which necessitates nontrivial and intricate developments. + +# 3.2 StreamBP for Transformer LLMs + +In this section, we apply StreamBP to significantly reduce the memory cost consumed by language modeling head and transformer layer during BP. Additionally, we will discuss that StreamBP requires even less computational FLOPs compared to the standard BP with gradient checkpointing, enabling potential time acceleration over standard approaches. + +# 3.2.1 StreamBP for Language Modeling Head: SFT, GRPO, and DPO + +The language modeling head performs the following linear transformation: + +$$ +H W _ {\mathrm {l m} \text {- h e a d}} = \log i t s. +$$ + +Here, $W_{\mathrm{lm\_head}} \in \mathbb{R}^{d \times C}$ is the weight of language modeling head, $H \in \mathbb{R}^{T \times d}$ is the hidden states output of the last transformer layer. The logits $\in \mathbb{R}^{T \times C}$ will be used to compute the objective function. $C, d, T$ are the vocabulary size, hidden dimension, and sequence length, respectively. As shown in Figure 1, the logits and its gradient give rise to a huge memory consumption, due to the large vocabulary size and sequence length. In the following, we analyze one-by-one how the memory of logits can be significantly reduced using StreamBP in the regime of SFT, GRPO, and DPO. + +Supervised finetuning (SFT). The (un-normalized) "code-style" objective of SFT is given by + +$$ +L _ {\mathrm {S F T}} (\log_ {t} Y) := \sum_ {t = 1} ^ {T - 1} - \log \operatorname {s o f t m a x} \left(\log_ {t, :}\right) _ {Y _ {t}}, \tag {2} +$$ + +where $Y \in \mathbb{R}^{T - 1}$ is the label vector. Importantly, each position's logits contribute to the objective independently. To perform StreamBP, the logits and label are evenly partitioned into $D$ chunks across the sequence dimension, i.e., $\{(\logits^{(i)},Y^{(i)})|i = 1,\dots ,D\}$ with $\logits^{(i)} \in \mathbb{R}^{((T - 1) / D)\times C}$ . Then, we sequentially accumulate the gradient across all partitions $i = 1,\ldots ,D$ as + +$$ +g _ {\mathrm {l m} \text {h e a d}} + = \frac {\partial L _ {\mathrm {S F T}} \left(\log_ {\mathrm {t}} ^ {(i)} , Y ^ {(i)}\right)}{\partial W _ {\mathrm {l m} \text {h e a d}}}, \quad g _ {H} + = \frac {\partial L _ {\mathrm {S F T}} \left(\log_ {\mathrm {t}} ^ {(i)} , Y ^ {(i)}\right)}{\partial H}, \tag {3} +$$ + +where $g_{\mathrm{lm\_head}}$ and $g_{H}$ are initialized from zero. The operator $+ =$ denotes the in-place summation. $\logits^{(i)}$ and its gradient will be cleaned from memory once they have been used in (3). After the accumulation across all partitions, $g_{\mathrm{lm\_head}}$ and $g_{H}$ will be the exact gradient of the $W_{\mathrm{lm\_head}}$ and $H$ , respectively. During this computation, StreamBP only stores $\logits^{(i)}$ and its gradient sequentially for all $i$ , which only costs $1 / D$ memory compared to the original approach. + +Group relative policy optimization (GRPO). The principle of StreamBP for GRPO aligns with SFT. For compact presentation, we display the detailed derivation in Appendix A.2. + +Direct preference optimization (DPO). The objective of DPO is given by + +$$ +L _ {\mathrm {D P O}} (\log_ {t}) := - \mathbb {E} \left[ \log \sigma \left(\beta \sum_ {t = 1} ^ {T} \underbrace {\left(\log \frac {\pi_ {\theta} \left(y _ {w , t} \mid x , y _ {w , < t}\right)}{\pi_ {\mathrm {r e f}} \left(y _ {w , t} \mid x , y _ {w , < t}\right)} - \log \frac {\pi_ {\theta} \left(y _ {l , t} \mid x , y _ {l , < t}\right)}{\pi_ {\mathrm {r e f}} \left(y _ {l , t} \mid x , y _ {l , < t}\right)}\right)\right) \right]. \tag {4} +$$ + +Here, logits := $\{\log_{\pi_{\theta}}, \log_{\pi_{\mathrm{ref}}}\}$ with $\log_{\pi} \in \mathbb{R}^{T \times C}$ , $\sigma(\cdot)$ is the sigmoid function. $T$ is the sequence length of response $y$ . To ease expression, we assume $y_w$ and $y_l$ have the same length without loss of generality. The policy's output is determined by logits: + +$$ +\pi \left(y _ {t} | x, y _ {< t}\right) = \operatorname {s o f t m a x} \left(\log_ {t, :}\right) _ {y _ {t}}. +$$ + +![](images/63797e487f3db6176109a38d54ec03b5dc5ace79969a1519b3c3a07712de62d4.jpg) +Figure 2: StreamBP for transformer layer (best view in color). The stored activations of StreamBP is highlighted in orange. Unlike the gradient checkpointing approach that reforward $H_{\mathrm{in}}$ to compute all the activations required for the backward of $H_{\mathrm{out}}$ , StreamBP computes the activations only for one partition of $H_{\mathrm{out}}^{(i)}$ at a time, which reduces the memory cost by a large margin. + +![](images/cda18bba131190c19b69e833cc02dce8d334b1bd0fb31e54dbf49295c6c26fbb.jpg) + +Unlike SFT and GRPO, DPO's objective cannot be divided into summation of losses on individual partitions, due to the existence of the non-linear log-sigmoid transformation. Fortunately, its gradient retains a separable structure: + +$$ +\frac {\partial L _ {\mathrm {D P O}}}{\partial W} = - \mathbb {E} _ {(x, y _ {w}, y _ {l}) \sim \mathcal {D}} \left[ \left(1 - \sigma \left(\beta \sum_ {t = 1} ^ {T} \ell (t)\right)\right) \beta \sum_ {t = 1} ^ {T} \frac {\partial \ell (t)}{\partial W} \right]. \tag {5} +$$ + +StreamBP partitions logits across the sequence dimension $\{\log_{\mathrm{its}}^{(i)} := \{\log_{\pi_{\theta}}^{(i)}, \log_{\pi_{\mathrm{ref}}}^{(i)}\} | i = 1, \ldots, D\}$ with $\log_{\pi}^{(i)} \in \mathbb{R}^{(T / D) \times C}$ . Based on the partition, it performs the following accumulations for $i = 1, \ldots, D$ : + +$$ +\ell + = \sum_ {t \in \mathcal {T} _ {i}} \ell (t), \quad g _ {\mathrm {l m} _ {-} \text {h e a d}} + = \beta \sum_ {t \in \mathcal {T} _ {i}} \frac {\partial \ell (t)}{\partial W _ {\mathrm {l m} _ {-} \text {h e a d}}}, \quad g _ {H} + = \beta \sum_ {t \in \mathcal {T} _ {i}} \frac {\partial \ell (t)}{\partial H}, \tag {6} +$$ + +where $\mathcal{T}_i\coloneqq \{(i - 1)\frac{T}{D} < t\leq i\frac{T}{D}\mid t\in \mathbb{Z}\}$ . After finishing the above accumulation, StreamBP performs the following in-place correction to compute the exact gradient: + +$$ +g _ {\mathrm {l m} \text {h e a d}} \leftarrow \left(\sigma (\beta \ell) - 1\right) g _ {\mathrm {l m} \text {h e a d}}, \quad g _ {H} \leftarrow \left(\sigma (\beta \ell) - 1\right) g _ {H}. \tag {7} +$$ + +# 3.2.2 StreamBP for Transformer Layers: Attention and MLP + +We now apply StreamBP to the transformer layer. To ease presentation, we disregard components such as normalization layer, multi-head mechanism, and residual connection without loss of generality. + +A transformer layer consists of two consecutive transformations of attention and MLP: + +$$ +O = f _ {\text {a t t n}} \left(H _ {\text {i n}}\right), \quad H _ {\text {o u t}} = f _ {\text {M L P}} (O), \tag {8} +$$ + +where $H_{\mathrm{in}} \in \mathbb{R}^{T \times d}$ and $H_{\mathrm{out}} \in \mathbb{R}^{T \times d}$ are the input and output of the transformer layer, respectively. The two transformations are given by + +Attention: $Q = H_{\mathrm{in}}W_q,$ $K = H_{\mathrm{in}}W_k,$ $V = H_{\mathrm{in}}W_v,$ $Q,K,V\in \mathbb{R}^{T\times d}$ + +$$ +S = Q K ^ {\top} \in \mathbb {R} ^ {T \times T}, \quad P = \operatorname {s o f t m a x} (S, M) \in \mathbb {R} ^ {T \times T}, \quad O = P V \in \mathbb {R} ^ {T \times d}. +$$ + +MLP: $H_{\mathrm{up}} = OW_{\mathrm{up}} \in \mathbb{R}^{T \times d_{\mathrm{up}}}$ , $H_{\mathrm{gate}} = OW_{\mathrm{gate}} \in \mathbb{R}^{T \times d_{\mathrm{up}}}$ + +$$ +H _ {\text {o u t}} = \left(\sigma \left(H _ {\text {g a t e}}\right) \circ H _ {\text {u p}}\right) W _ {\text {d o w n}} \in \mathbb {R} ^ {T \times d}. \tag {9} +$$ + +Computing $\partial H_{\mathrm{out}} / \partial H_{\mathrm{in}}$ requires storing $Q, K, V, M, O, H_{\mathrm{up}}, H_{\mathrm{gate}}, H_{\mathrm{out}}$ , where $M \in \mathbb{R}^{T \times T}$ is the attention mask, which is required when using mini-batch training or techniques such as sliding window attention. The storage of $S$ and $P$ can be avoided using flash attention's tiling approach. + +Define the partition of quantity $H \in \mathbb{R}^{T \times d}$ along the sequence dimension as $\{H^{(i)}|i = 1, \dots, D\}$ with $H^{(i)} \in \mathbb{R}^{(T / D) \times d}$ . Let $H^{(i)} \in \mathbb{R}^{(iT / D) \times d}$ be the concatenation of $\{H^{(j)}\}_{j=1}^i$ along the sequence dimension. StreamBP for transformer layer is built upon the following observation: + +Property 3.1 The computation of $\partial H_{out}^{(i)} / \partial W$ only depends on $O^{(i)}, Q^{(i)}, K^{(:, i)}$ , and $V^{(:, i)}$ . + +To justify the above property, StreamBP sequentially performs the following partitioned attention and MLP for each chunk $i$ : + +Partitioned Attention: $Q^{(i)} = H_{\mathrm{in}}^{(i)}W_{q}, \quad K^{(:,i)} = H_{\mathrm{in}}^{(:,i)}W_{k}, \quad V^{(:,i)} = H_{\mathrm{in}}^{(:,i)}W_{v}$ + +$$ +S ^ {(i)} = Q ^ {(i)} K ^ {(: i) ^ {\top}}, \quad P ^ {(i)} = \operatorname {s o f t m a x} \left(S ^ {(i)}, M ^ {(i)}\right), \quad O ^ {(i)} = P ^ {(i)} V ^ {(: i)} +$$ + +Partitioned MLP: $H_{\mathrm{up}}^{(i)} = O^{(i)}W_{\mathrm{up}},$ $H_{\mathrm{gate}}^{(i)} = O^{(i)}W_{\mathrm{gate}}$ + +$$ +H _ {\text {o u t}} ^ {(i)} = \left(\sigma \left(H _ {\text {g a t e}} ^ {(i)}\right) \circ H _ {\mathrm {u p}} ^ {(i)}\right) W _ {\text {d o w n}} \in \mathbb {R} ^ {(T / D) \times d}. \tag {10} +$$ + +When the partitioned reforward finished, the activation for computing $\partial \mathrm{vec}(H_{\mathrm{out}}^{(i)}) / \partial \mathrm{vec}(W)$ is stored in memory. Then, we accumulate the gradient for $i = 1,\dots ,D$ as + +$$ +\operatorname {v e c} \left(g _ {W}\right) + = \frac {\partial \operatorname {v e c} \left(H _ {\text {o u t}} ^ {(i)}\right)}{\partial \operatorname {v e c} (W)} ^ {\top} \frac {\partial L}{\partial \operatorname {v e c} \left(H _ {\text {o u t}} ^ {(i)}\right)}, \quad \operatorname {v e c} \left(g _ {H _ {\mathrm {i n}}}\right) + = \frac {\partial \operatorname {v e c} \left(H _ {\text {o u t}} ^ {(i)}\right)}{\partial \operatorname {v e c} \left(H _ {\mathrm {i n}}\right)} ^ {\top} \frac {\partial L}{\partial \operatorname {v e c} \left(H _ {\text {o u t}} ^ {(i)}\right)}. \tag {11} +$$ + +When the accumulation is finished, $g_{W}$ and $g_{H_{\mathrm{in}}}$ become the exact gradient of weight matrices $W := \{W_{q}, W_{k}, W_{v}, W_{\mathrm{up}}, W_{\mathrm{gate}}, W_{\mathrm{down}}\}$ and $H_{\mathrm{in}}$ , respectively. Note that $K^{(:i)}$ and $V^{(:i)}$ are needed for each partitioned forward. Hence, we compute $K$ and $V$ at once and cache it during StreamBP of the current layer. We remark that the partitioned attention is compatible with the flash attention. We illustrate the StreamBP for transformer layer in Figure 2. + +Memory efficiency of StreamBP. StreamBP only needs to store the following activation values: $Q^{(i)}$ , $K$ , $V$ , $M^{(i)}$ , $O^{(i)}$ , $H_{\mathrm{up}}^{(i)}$ , $H_{\mathrm{gate}}^{(i)}$ , and $H_{\mathrm{out}}^{(i)}$ . Note that when grouped query attention [2] is used with group size $G$ , $K$ and $V$ only costs $1 / G$ memory of $Q$ . Consequently, StreamBP costs approximately $1 / D$ memory for activation value compared to the standard BP. + +Computational efficiency of StreamBP. For long sequence training, the most computational expensive operation involved in transformer layer is the calculation of the pre-attention score $S$ . StreamBP reduces the FLOPs of the operation by approximately half. Specifically, the standard implementation $S = QK^{\top}$ costs FLOPs of $2T^{2}d^{2}$ , while StreamBP only costs FLOPs of $\frac{(1 + D)T^{2}d^{2}}{D}$ for computing $S^{(i)} = Q^{(i)}K^{(:,i)^{\top}}$ across $D$ partitions. The FLOPs reduction is since that StreamBP utilizes the causal structure of language models, which uses $K^{(:,i)}$ rather than $K$ in the computation of $S^{(i)}$ . For all the other operations, StreamBP across all partitions shares the same FLOPs as the standard BP with checkpointing. Note that $K$ and $V$ are only computed once and get cached. + +HBM overhead of StreamBP. Each partitioned gradient calculation in (1) requires loading model weight $W$ from high bandwidth memory (HBM) to register for computation, which induces additional overhead compared to the standard BP. Meanwhile, StreamBP reduces the HBM throughput of attention mask by approximately half. The overhead directly depends on the number of partitions $D$ . In Section 4.5, we empirically study how $D$ affects the BP time and memory cost. + +# 3.3 Distributed StreamBP + +Although StreamBP is naturally compatible with modern distributed training techniques such as distributed data parallel (DDP) and Deepspeed ZeRO, directly applying these techniques to StreamBP is inefficient, due to redundant communications of gradients and parameters. We display the design of distributed StreamBP in Figure 6. The major designs contain gradient communication design and parameter communication design. + +Gradient communication. During the backward, when the local gradient buffer of all processes are ready, a gradient averaging operation (i.e. all-reduce or reduce-scatter) is performed across all the processes. As shown in (1), a parameter's gradient in StreamBP is ready until the accumulation operation is finished. To avoid redundant gradient communication, the gradient averaging of distributed StreamBP is performed after the accumulation. In this sense, distributed StreamBP will have the same gradient communication cost as the standard BP. + +Parameter communication. Parameter communication is required when using model-parallel. For instance, ZeRO-3 partitions each parameter evenly across different GPUs, and gather the parameter + +![](images/1f6960fd8a04fa4bacd1b484587103cf575b7c854ac885a5c5dc8fb4df630d1b.jpg) +(a) Qwen 3-8B + +![](images/9e949b26a18d64a7e1cd67a50bb15ea39fd2a3aed9e65a60ff04ad7c7df88cf9.jpg) +(b) Qwen 3-14B +Figure 3: Peak BP memory cost measurement of Qwen 3-8B, 14B, and 32B models under different sequence lengths. Under the 80GB memory limit, StreamBP scales the maximum sequence by $2.8 - 5.5 \times$ larger compared to gradient checkpointing. + +![](images/38101a04b1b797e4b506b662a1b708a2a78dc517d9696824b0cc0f30dc5a6478.jpg) +(c) Qwen 3-32B (LoRA Grad.) + +
Sequence length6k9k12k15k18k21k24k27k
Baseline w/o ckpt2.0-------
Baseline w/ ckpt2.54.77.410.814.719.324.3-
MsT2.64.97.611.215.220.325.231.8
StreamBP2.64.77.010.013.217.221.225.8
Acceleration over ckpt-4.4%0.4%6.0%7.4%10.5%10.9%12.9%-
+ +Table 1: BP Time cost (in seconds) under different sequence lengths. The result is based on Qwen 3-4B model and is averaged over 50 independent trials. The acceleration of StreamBP becomes more apparent as the sequence length grows up, corroborating our analysis in Section 3.2.2. + +when the operators associated with it is called. By (1), StreamBP requires the access of unpartitioned parameter for $D$ times during the backward. Hence, naively implementing model-parallel for StreamBP costs $(D - 1) \times$ all-gather operations. To reduce the communication cost, distributed StreamBP caches the weight locally until the accumulation of its gradient is finished, avoiding inducing additional parameter communication cost compared to the standard BP. The design will introduce an additional memory cost of storing a single transformer layer in each GPU. + +# 4 Experiments + +In this section, we evaluate StreamBP based on the Qwen 3 model series. The result directly applies for any causal language models such as Llama, Mistral, and Gemma, since StreamBP is not restricted to a certain model class. Our evaluation mainly consists of 3 parts, including 1) backpropagation cost; 2) training cost; and 3) distributed training. All the experiments are conducted using A800-80GB GPUs. The detailed setup is presented in Appendix D. + +# 4.1 Backpropagation Cost Measure + +Memory cost. We report the memory cost of StreamBP and baseline (i.e., standard BP) with/without gradient checkpointing in Figure 3. Given the 80GB memory limit, StreamBP is able to increase the maximum sequence length by $2.8 - 5.5 \times$ and $23.4 - 36.3 \times$ compared to baseline with/without gradient checkpointing, respectively. Importantly, the memory cost of StreamBP is linearly related to the sequence length, as StreamBP does not store the full attention mask and is compatible with flash attention. Thus, StreamBP's sequence length scaling can be directly transferred to batch size scaling for accelerating the training. Note that the ratio differs across different model sizes, which attributes to distinct configurations of hidden dimensions. + +Time cost. We compare the BP time cost of StreamBP with baseline approaches, and present the result in Table 1. The result shows that StreamBP consistently exhibits faster BP time compared to the gradient checkpointing baseline across a wide range of sequence lengths, and beats the long-sequence training baseline MsT [21] more significantly. As the sequence length increases, the acceleration + +
ObjectiveMethod4B8B14B32B
FullLoRAFullLoRALoRALoRA
SFTBaseline w/o ckpt7.07.13.45.12.90.4
Baseline w/ ckpt28.536.515.730.623.05.9
StreamBP200.0247.072.0142.484.616.3
GRPOBaseline w/o ckpt0.91.0-0.70.4-
Baseline w/ ckpt8.512.1-9.56.9-
StreamBP18.227.5-16.510.1-
DPOBaseline w/o ckpt3.13.5-2.51.40.2
Baseline w/ ckpt18.732.2-25.518.54.4
StreamBP57.2100.2-60.936.97.8
+ +becomes more significant. The result aligns with our analysis in Section 3.2.2, where StreamBP reduces approximately half of computation in calculating attention scores, whose computational FLOPs is quadratically related to the sequence length. In Table 2, we show that StreamBP's memory-efficiency enables the usage of much larger batch size, which helps further accelerate the BP speed. The acceleration attributes to the reduced sample-wise HBM throughput of model loading and better leveraging of parallel processing. In particular, StreamBP with batch size 16 achieves even less per-sample BP time than the baseline without gradient checkpointing under batch size 1. + +Table 3: Maximum sequence length (in thousands) on a single A800-80GB GPU. + +
Batch Size12416
Baseline w/ ckpt7.427.127.04-
MsT7.647.417.287.06
StreamBP7.036.826.636.47
+ +Table 2: Per-sample BP time cost of Qwen 3-4B model under different batch sizes. The sequence length is 9000. StreamBP achieves further acceleration by utilizing substantially larger batch size. + +![](images/10ea56d358e673cc562c5899865d0859e1eb81a6dad7d5dbfcfbad417c19ba6b.jpg) +Figure 4: Memory cost comparison with MsT. + +# 4.2 Training Cost Measure + +Sequence length scaling. We present the maximum sequence length of StreamBP and baseline methods given a single A800-80GB GPU in Table 3. For all the objectives, StreamBP significantly increases the maximum sequence length over baseline approaches, which justifies the efficiency of StreamBP in reducing the memory cost of transformer layer activations and logits. For SFT, even 32B model can achieve sequence length up to 16k. Note that the sequence length scaling ratio can be equally transferred to batch size scaling ratio for acceleration. For example, for the SFT of 8B model, StreamBP enables $\frac{72}{15.7} \approx 4.5 \times$ larger batch size than the baseline with gradient checkpointing. + +Comparison with long-sequence training baseline. We compare StreamBP with the long-sequence SFT baseline MsT, and plot the peak memory of training Qwen 3-8B LoRA model in Figure 4. The result shows that StreamBP achieves approximately $1.7 \times$ larger sequence length than MsT. In terms of time-efficiency, StreamBP requires significantly less BP time compared to MsT, as shown in Table 1. The acceleration becomes more significant as the sequence length scales up. + +# 4.3 Efficiency under Distributed Training + +We measure the maximum sequence scaling and BP time under the Deepspeed ZeRO-2 SFT scheme, where the gradient and optimizer states are partitioned across different GPUs and each GPU processes its local data batch. The results are shown in Table 4 and Table 5. Compared to baseline with gradient checkpointing, distributed StreamBP scales to a maximum sequence length approximately $5 - 5.6 \times$ larger and achieves noticeably faster BP speed. + +
# of GPUs345678
Baseline w/ checkpoint18.620.522.323.123.523.7
Distributed StreamBP92.4112.0121.5128.7131.4131.8
+ +Table 4: Maximum sequence length (in thousands) of Qwen 3-8B under ZeRO-2 training scheme. + +
Sequence length6k9k12k15k18k21k
Baseline w/ checkpoint8.49.612.218.522.926.3
Distributed StreamBP6.48.310.814.817.220.7
+ +Table 5: Per-sample BP time cost (in seconds) of Qwen 3-8B under ZeRO-2 training scheme. + +# 4.4 Correctness Verification of StreamBP + +We empirically measure the gradient difference of StreamBP and baseline methods to justify the correctness of our implementation. Importantly, the float point operations has the following associativity issue due to numerical precision: + +$$ +(a \oplus b) \oplus c \neq a \oplus (b \oplus c) \tag {12} +$$ + +where $\oplus$ denotes floating-point addition. Therefore, it is impossible for the gradient difference to be zero given the different computation order. To verify the correctness, we use the gradient computed by the pure FP32 BP of the standard BP with gradient checkpointing as ground truth, denoted as base32. Then, we calculate its difference with the gradient $g_{i}$ obtained by pure BF16 BP of StreamBP and baseline BP, respectively. Additionally, we include a StreamBP run under FP32 precision, denoted as stream32, to verify its numerical equivalence to base32. Specifically, we define the absolute error and relative error as + +$$ +\operatorname {E r} _ {\text {a b s}} (g) = \frac {1}{n} \sum_ {i} \left| g _ {i} ^ {\text {b a s e 3 2}} - g _ {i} \right|, \quad \operatorname {E r} _ {\text {r e l}} (g) = \frac {1}{n} \sum_ {i} \frac {\left| g _ {i} ^ {\text {b a s e 3 2}} - g _ {i} \right|}{\left| g _ {i} ^ {\text {b a s e 3 2}} + 1 0 ^ {- 1 0} \right|}. \tag {13} +$$ + +Table 6 reveals two key findings: 1) stream32's micro-scale deviations $(\mathrm{Er}_{\mathrm{abs}} \sim 10^{-9})$ validate mathematical equivalence to base32, and 2) stream16 demonstrates $\leq 0.04\%$ relative deviation from base16, indicating that StreamBP exhibits no precision loss compared to standard bf1oat16 computation. + +We further verify the correctness of StreamBP via model training and display the loss of SFT, DPO, and GRPO in Table 7. The loss of StreamBP closely align with the standard BP. The tiny difference is due to different order of float operations discussed above. + +# 4.5 Ablation Study and Additional Experiments + +Effect of partition size. We fix the partition size of transformer layer to be $0.5\mathrm{k}$ , $1\mathrm{k}$ , $2\mathrm{k}$ , and $5\mathrm{k}$ , examining the memory and time cost across wide range of sequence lengths. The result is shown in Figure 5. Note that partition will not be performed if it is larger than the sequence length. When the sequence length is relatively small, e.g., less than $10\mathrm{k}$ , the BP times under different partition sizes are close. However, as the sequence length scales up, using a partition size that is too small introduces substantial overhead. The overhead attributes to the additional HBM throughput on loading model weight from HBM to register for computation and repetitive kernel launches. Fortunately, as shown in Figure 5b, large partition size only introduces marginal additional memory cost. Hence, for long sequence training, one can use a relatively large partition size to maximize training efficiency. + +Additional experiments. We provide more experiments in Appendix B and Appendix C.2. Here are the summarized results: 1) We present the sequence scaling using a single RTX3090-24GB GPU. The result shows that StreamBP is able to scale up the maximum sequence length to $15\mathrm{k}$ , which is about $4.4\times$ larger compared to gradient checkpointing. 2) We present the memory profile of StreamBP in Figure 8. The profile demonstrates a substantial memory reduction in logits and activation values during layer reforwarding, which is consistent with our analysis in Section 3. 3) + +
ModuleErabsErrel
base16stream16stream32base16stream16stream32
lm_head2.56e-72.64e-76.66e-91.43%1.47%0.04%
Transformer layers2.46e-52.47e-51.75e-76.63%6.59%0.04%
+ +Table 6: Gradient precision analysis. StreamBP is the numerically correct and has nearly identical gradient as the standard BP. + +
StepSFT (Capybara)DPO (Ultrafeedback)GRPO (TL;DR)
BaseStreamBPDiff.BaseStreamBPDiff.BaseStreamBPDiff.
15.7825.782+0.0001.7021.702+0.0000.0130.013+0.0000
1004.9244.926+0.0021.0871.090+0.0020.0110.011-0.0002
2004.3134.309-0.0030.9380.935-0.0030.0050.004-0.0004
3003.9903.993+0.0030.8760.879+0.0030.0060.006+0.0000
4003.7913.795+0.0030.8360.840+0.0040.0050.005+0.0001
5003.6673.663-0.0040.8010.796-0.0040.0040.004-0.0001
+ +Table 7: Evaluation loss of StreamBP and standard BP under different objectives. + +We conduct experiments on Llama and Gemma architectures in Appendix B.2, demonstrating that StreamBP is applicable to general transformer models. + +![](images/579135be0cfa41a3700195d2295a5e9788adf8c685bd19b851bc9c499c6faa3d.jpg) +(a) Time v.s. sequence length. + +![](images/82421c1fdbb1aa4670b026728c320d2350a6f15eb4202bded92d99ad166b9f40.jpg) +(b) Memory v.s. sequence length. +Figure 5: Time and memory costs of StreamBP on Qwen 3-8B with varying partition sizes. + +# 5 Conclusion and Discussions on Limitations + +In this work, we developed a memory-efficient and exact BP method called StreamBP. Compared to gradient checkpointing baseline, StreamBP requires significantly less memory cost and enjoys faster BP time by leveraging the causal structure of LLMs. StreamBP can be used for long sequence training of any transformer LLMs, which finds wide applications such as training reasoning models. We also developed a communication-efficient distributed StreamBP to support multi-GPU training. + +Limitations. Currently, StreamBP does not support MoE or multimodal models. Nonetheless, these can be addressed with simple implementation extensions, as the underlying principle remains unchanged. Additionally, StreamBP's partition size can have a clear impact on BP time. This overhead can be mitigated by employing a fused backward operator, which significantly reduces the HBM throughput. We leave these directions for future work. + +Broader impacts. We provide a memory-efficient and exact BP method for long sequence training of LLMs. This is a technical contribution that does not yield explicit negative societal impacts. + +# Acknowledgments and Disclosure of Funding + +We thank the Area Chair and the anonymous reviewers for their insightful comments and suggestions, which have significantly improved the quality of the manuscript. + +Lei Zhao is supported in part by the Major Project of the National Natural Science Foundation of China (NSFC) under grant 72293582 and in part by the National Key R&D Program of China under grant 2023YFA0915202. Xiao Li is supported in part by the National Natural Science Foundation of China (NSFC) under grants 12201534, 12571330, and 12326608, in part by the $1+1+1$ CUHK-CUHK(SZ)-GDSTC Joint Collaboration Fund under grant 2025A0505000049, and in part by the Shenzhen Science and Technology Program under grant RCYX20221008093033010. + +# References + +[1] Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. GPT-4 technical report. arXiv preprint arXiv:2303.08774, 2023. +[2] Joshua Ainslie, James Lee-Thorp, Michiel de Jong, Yury Zemlyanskiy, Federico Lebron, and Sumit Sanghai. Gqa: Training generalized multi-query transformer models from multi-head checkpoints. 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Simplerl-zoo: Investigating and taming zero reinforcement learning for open base models in the wild. arXiv preprint arXiv:2503.18892, 2025. +[38] Jiawei Zhao, Zhenyu Zhang, Beidi Chen, Zhangyang Wang, Anima Anandkumar, and Yuandong Tian. Galore: Memory-efficient llm training by gradient low-rank projection. arXiv preprint arXiv:2403.03507, 2024. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: The claims in abstract and introduction are supported by the memory and FLOPs analysis in Section 3 and numerical results in Section 4. + +Guidelines: + +- The answer NA means that the abstract and introduction do not include the claims made in the paper. +- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: The paper does not use LLMs for important, original, or non-standard components of the core method. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Contents + +1 Introduction 1 +2 Preliminary: Peak Memory Cost during Backpropagation 2 +3 Memory-Efficient Exact Stream Backpropagation 3 + +3.1 Main Idea 3 +3.2 StreamBP for Transformer LLMs 4 + +3.2.1 StreamBP for Language Modeling Head: SFT, GRPO, and DPO 4 +3.2.2 StreamBP for Transformer Layers: Attention and MLP 5 + +3.3 Distributed StreamBP 6 + +4 Experiments 7 + +4.1 Backpropagation Cost Measure 7 +4.2 Training Cost Measure 8 +4.3 Efficiency under Distributed Training 8 +4.4 Correctness Verification of StreamBP 9 +4.5 Ablation Study and Additional Experiments 9 + +5 Conclusion and Discussions on Limitations 10 + +A Additional Details of Stream Backpropagation 22 + +A.1StreamBP for Linear Transformation 22 +A.2StreamBP for GRPO 22 +A.3 Distributed StreamBP 23 + +B Additional Experiments 25 + +B.1 Sequence Scaling on a Single RTX3090-24GB 25 +B.2StreamBP for Broader Architectures 25 + +C Analysis of Memory Profile 25 + +C.1 Memory Profile Explanation of Gradient Checkpointing 25 +C.2 Memory Profile of StreamBP 26 + +DExperiment Setup 26 + +# A Additional Details of Stream Backpropagation + +# A.1 StreamBP for Linear Transformation + +Given the input $X = [X_{1},X_{2},\ldots ,X_{D}]^{\top}\in \mathbb{R}^{D\times m}$ and weight matrices $W_{1}\in \mathbb{R}^{m\times n}$ , $W_{2}\in \mathbb{R}^{n\times k}$ . Consider the following two linear transformations: + +$$ +Y = \left[ Y _ {1}, \dots , Y _ {D} \right] ^ {\top} := X W _ {1} \in \mathbb {R} ^ {D \times n} +$$ + +$$ +Z = \left[ Z _ {1}, \dots , Z _ {D} \right] ^ {\top} := Y W _ {2} \in \mathbb {R} ^ {D \times k}. +$$ + +For the ease of expression, define $dM \coloneqq \frac{\partial L}{\partial M}$ as the gradient of quantity $M$ with respect to the objective $L$ , which can be arbitrary objective. The gradient of $Y$ and weight matrices are given by + +$$ +d Y = (d Z) W _ {2} ^ {\top}, \quad d W _ {1} = X ^ {\top} (d Y), \quad d W _ {2} = Y ^ {\top} (d Z). +$$ + +The standard approach of calculating $dW_{1}$ and $dW_{2}$ requires storing the intermediate value $Y$ and $dY$ . Based on (1), the above expression can be written as + +$$ +d Y _ {i} = (d Z _ {i}) W _ {2} ^ {\top}, \quad d W _ {1} = \sum_ {i = 1} ^ {D} X _ {i} ^ {\top} (d Y _ {i}), \quad d W _ {2} = \sum_ {i = 1} ^ {D} Y _ {i} ^ {\top} (d Z _ {i}). +$$ + +Hence, one can sequentially compute $Y_{i}$ and $dY_{i}$ , accumulate $X_{i}^{\top}dY_{i}$ and $Y_{i}^{\top}dZ_{i}$ , then drop the $Y_{i}$ and $dY_{i}$ . Compared to the standard gradient computation, this approach effectively reduces the memory cost of intermediate values to $1 / D$ without introducing additional computational cost. In practice, to better utilize the parallel computation and reduce the HBM load, one can process $X$ in chunk-wise rather than sample-wise manner. Below, we demonstrate the memory efficiency of StreamBP with numerical experiment on the linear example. + +Experiment on linear example. We empirically measure the memory cost of StreamBP and compare it with standard BP, based on concrete linear example. Specifically, let $X \in \mathbb{R}^{10^6 \times 16384}$ , $W_1, W_2 \in \mathbb{R}^{16384 \times 16384}$ , $L = \sum_{j,k} Z_{j,k}$ . StreamBP partitions $X$ as $\{X^{(i)} | i = 1, \dots, D\}$ with $X^{(i)} \in \mathbb{R}^{(10^6 / D) \times 16384}$ . We present the memory cost and time cost of standard BP and StreamBP in Table 8. + +
Standard BPStreamBP
D=20D=50D=100D=200D=500
Peak memory (GB)36.0113.2512.4712.2012.0711.99
Intermediate memory (GB)25.152.391.611.341.211.13
Time cost (seconds)14.4814.5014.7014.7915.4518.42
+ +Table 8: Memory and time cost of standard BP and StreamBP under different partition number. + +Compared to standard BP, StreamBP with $D = 20$ reduces memory cost by $63.2\%$ with almost no time overhead. As $D$ increases, the intermediate memory cost is further reduced. + +# A.2 StreamBP for GRPO + +The GRPO's objective is given by: + +$$ +L _ {\text {G R P O}} (\text {l o g i t s}) := \mathbb {E} _ {[ q \sim \mathcal {D} _ {q}, \{o _ {j} \} _ {j = 1} ^ {G} \sim \pi_ {\text {o l d}} (\cdot | q) ]} +$$ + +$$ +\left[ \frac {1}{G} \sum_ {j = 1} ^ {G} \frac {1}{T _ {o}} \sum_ {t = 1} ^ {T _ {o}} \underbrace {\left(\min \left\{\frac {\pi_ {\theta} (j , t)}{\pi_ {\mathrm {o l d}} (j , t)} \hat {A} _ {j , t} , \operatorname {c l i p} \left(\frac {\pi_ {\theta} (j , t)}{\pi_ {\mathrm {o l d}} (j , t)} , 1 - \epsilon , 1 + \epsilon\right) \hat {A} _ {j , t} \right\} - \beta \log \frac {\pi_ {\theta} (j , t)}{\pi_ {\mathrm {r e f}} (j , t)}\right)} _ {\triangleq f (j, t)} \right]. \tag {14} +$$ + +To ease presentation, in the above equation we omit the compensation term in GRPO's KL divergence term without loss of generality, as StreamBP directly applies to the KL divergence term. Here, $q$ is + +the prompt sequence, $\mathcal{D}_q$ is the dataset of prompt, and $o_j$ is $j$ -th response sequence with respect to $q$ . $G$ and $T_o$ are the group size and the length of response. For the ease of expression, we assume all the responses are of the same length without loss of generality. $\hat{A}_{j,t}$ is the estimated advantage. $\pi_{\theta}$ , $\pi_{\mathrm{old}}$ , and $\pi_{\mathrm{ref}}$ are the target policy, old policy and reference policy, respectively. + +Here, $\log_{\mathrm{t}} \triangleq \{\log_{\pi_{\theta}}, \log_{\pi_{\mathrm{old}}}, \log_{\pi_{\mathrm{ref}}} \}$ contains logits generated by target policy, old policy, and reference policy, respectively, with $\log_{\pi} \in \mathbb{R}^{G \times T_o \times C}$ . The policy's output is determined by logits, i.e., + +$$ +\pi (j, t) := \pi \left(o _ {j, t} | q, o _ {j, < t}\right) = \operatorname {s o f t m a x} \left(\log_ {\pi , j, t,:}\right) _ {o _ {j}, t}. +$$ + +Note that each $f(j, t)$ in (14) contributes to the objective independently and only depends on $\log_{\pi, j, t, \cdot}$ , which enables us to perform StreamBP along the sequence dimension. Specifically, let us partition the logits along the sequence dimension as $\{\log_{\pi}^{(i)} := \{\log_{\pi_{\theta}}^{(i)}, \log_{\pi_{\mathrm{old}}}^{(i)}, \log_{\pi_{\mathrm{ref}}}^{(i)}\} | i = 1, \ldots, D\}$ with $\log_{\pi}^{(i)} \in \mathbb{R}^{G \times (T_0 / D) \times C}$ . Define the objective of the sequence partition as + +$$ +L _ {\mathrm {G R P O}} ^ {(i)} \left(\log_ {t} ^ {(i)}\right) := - \mathbb {E} _ {[ q \sim \mathcal {D} _ {q}, \{o _ {j} \} _ {j = 1} ^ {G} \sim \pi_ {\text {o l d}} (\cdot | q) ]} \left[ \frac {1}{G} \sum_ {j = 1} ^ {G} \frac {1}{T _ {o}} \sum_ {t \in \mathcal {T} _ {i}} f (j, t) \right], \tag {15} +$$ + +where $\mathcal{T}_i\coloneqq \{(i - 1)\frac{T_o}{D} < t\leq i\frac{T_o}{D}|t\in \mathbb{Z}\}$ denotes the sequence range of the partition. We have $L_{\mathrm{GRPO}}(\log \mathrm{its}) = \sum_{i = 1}^{D}L_{\mathrm{GRPO}}^{(i)}(\log \mathrm{its}^{(i)})$ . Then, similar to SFT, StreamBP sequentially performs the following accumulation for $i = 1,\ldots ,D$ .. + +$$ +g _ {\mathrm {l m} \text {h e a d}} + = \frac {\partial L _ {\mathrm {G R P O}} ^ {(i)} \left(\log_ {\mathrm {t}} \left(\mathrm {s} ^ {(i)}\right)\right)}{\partial W _ {\mathrm {l m} \text {h e a d}}}, \quad g _ {H} + = \frac {\partial L _ {\mathrm {G R P O}} ^ {(i)} \left(\log_ {\mathrm {t}} \left(\mathrm {s} ^ {(i)}\right)\right)}{\partial H}. \tag {16} +$$ + +# A.3 Distributed StreamBP + +StreamBP is naturally compatible with distributed training techniques such as Deepspeed ZeRO. However, to ensure the efficiency, the gradient communication and parameter communication needs to be customized. The design of distributed StreamBP is shown in Figure 6. + +Gradient communication design. Gradient communication happens in almost all distributed training. During the backward, when the local gradient buffer of all processes are ready, a gradient averaging operation (i.e. all-reduce or reduce-scatter) is performed across all the processes. As shown in (1), a parameter's gradient in StreamBP is ready until the accumulation operation is finished. To avoid redundant gradient communication, the gradient averaging of distributed StreamBP is performed after the accumulation. In this sense, distributed StreamBP will have the same gradient communication cost as the standard BP. + +Parameter communication design. Parameter communication is required when using model-parallel. For instance, ZeRO-3 partitions each parameter evenly across different GPUs, and gather the parameter when the operators associated with it is called. By (1), StreamBP requires the access of unpartitioned parameter for $D$ times during the backward. Hence, naively implementing model-parallel for StreamBP costs $(D - 1) \times$ all-gather operations. To reduce the communication cost, distributed StreamBP caches the weight locally until the accumulation of its gradient is finished, avoiding inducing additional parameter communication cost compared to the standard BP. This design will introduce an additional memory cost of storing a single transformer layer in each GPU. + +Efficiency of distributed StreamBP. The choice of batch size significantly affects the communication cost. For example, when using Deepspeed ZeRO-2, using batch size 1 with gradient accumulation step $K$ leads to approximately $K$ times heavier backward communication cost than using batch size $K$ with gradient accumulation step 1. StreamBP enables the usage of much larger batch size compared to the standard BP, which significantly reduces the communication cost. We remark ZeRO-2 reduces the above overhead by overlapping the communication and computation. + +![](images/0d709fdb500bc9000e0fa406c103d13f8b2a8734a5683118a63c0f489c997417.jpg) +Figure 6: Design of distributed StreamBP. When backward through a transformer layer, its parameter is gathered beforehand. During the StreamBP of the layer, no gradient or parameter communication is fired. When BP of the layer is finished, the gradient will be reduced across layers and the paramater will be sharded across GPUs. + +# B Additional Experiments + +# B.1 Sequence Scaling on a Single RTX3090-24GB + +We present the memory cost of training Qwen 3-8B LoRA model in Figure 7. Under the 24GB memory budget, StreamBP allows sequence scaling up to $15\mathrm{k}$ , which is $4.4\times$ larger than the gradient checkpointing baseline. + +![](images/0a8bdb3fb02c72fb9dd52b6e5b35932f7c8967e2e2508e8c8f260a036222a9a9.jpg) +Figure 7: Peak BP memory cost measurement of Qwen 3-8B with LoRA (rank=32) on a single RTX3090-24GB GPU + +# B.2 StreamBP for Broader Architectures + +In this section, we display the BP memory cost of Llama 3.1-8B and Gemma 3-1B. The results are shown in Table 9. + +
ModelMethod3k6k9k18k
Llama 3.1-8BBase w/o ckpt54.2---
Base w/ ckpt35.841.747.665.6
StreamBP31.832.733.637.2
Gemma 3-1BBase w/o ckpt23.142.863.4-
Base w/ ckpt14.324.835.367.2
StreamBP4.75.05.47.9
+ +Table 9: BP Memory cost (GB) of Llama 3.1-8B and Gemma 3-1B under different sequence lengths. + +Clearly, the memory cost of StreamBP is significantly less than standard BP; the memory gap increases as the sequence length scales up, which verifies the efficiency of StreamBP on different model architectures. Notably, Gemma 3-1B costs even more memory than Llama 3.1-8B for standard BP under sequence length 18k. This is because Gemma 3 model has $2 \times$ larger vocabulary size than Llama 3.1, and hence doubles the memory cost of logits. In comparison, StreamBP only stores partitioned logits and its memory cost is dominated by the model weight and model gradient rather than activation values. + +# C Analysis of Memory Profile + +# C.1 Memory Profile Explanation of Gradient Checkpointing + +We provide detailed explanation of Figure 1. First, the "Model parameters" part stores the BF16 model parameters and constantly occupy nearly 8GB memory. Then, during the 1st forward process, gradient checkpointing gradually stores the checkpointed layer inputs, and hence it gradually consumes more memory. At the end of the 1st forward and the beginning of the 1st backward, the FP32 logits and its gradient are computed, which suddenly consumes a huge amount of memory due to the very large vocabulary size of Qwen 3. This huge memory occupation is released after calculating the gradient of Im_head (tied with embedding) as shown in the brown rectangle above the "BF16 logits + +copy". During the 1st backward, BP calculates and stores the gradients of all the parameters and the memory consumption continues to increase. One interesting phenomenon is that the memory usage of the checkpointed layer activations enclosed in the yellow triangle goes to decrease during the 1st backward process. This is because that the stored corresponding activations will be deleted once the current weights' gradients are computed, as they are no longer needed. Another interesting observation is that there are some triangle-shaped bumps during the 1st backward, for which we draw a subfigure to interpret during the 2nd backward process. We explicitly show the reforward process of gradient checkpointing, and show the reforwarded layer activations that will be deleted once the corresponding gradients are computed, yielding a triangle-shaped bump. + +An additional remark on Figure 1 is in order. As shown in Figure 1, the current implementation of the "Transformers" package stores an additional "BF16 logits copy" and an additional tied embedding and "lm_head" gradient in the 2nd backward process, which may be optimized as they are not used. + +# C.2 Memory Profile of StreamBP + +The memory profile of StreamBP is shown in Figure 8. Compared to gradient checkpointing's profile in Figure 1, StreamBP reduces the peak memory greatly by partitioning the logits across the sequence dimension. The second peak memory in reforwarded activation values is also greatly reduced, which is now only marginally higher than the storage of the key and value states. + +![](images/216a0eee0f05d22080f4deddaa85952427cadaef4e9660ed6b83edd6eda0e4cf.jpg) +Figure 8: Memory profile with StreamBP under the same settings as Figure 1. The partition size of logits and transformer layer are set to 100 and 500, respectively. + +# D Experiment Setup + +We implement our algorithm using Huggingface transformers library [31]. The detailed experiment setup is presented below. + +Backpropagation cost. We decouple the optimization, focusing purely on the time and memory cost during the BP. This result serves as the minimum requirement on training a language model under a given sequence length. In particular, under batch size 1, we measure the BP memory cost of Qwen 3-8B, 14B and 32B models using a single A800-80GB GPU. For 32B model, we inject rank-32 LoRA adapters and only calculate the gradient of the adapters, as a single A800 GPU cannot store the model and the full gradient simultaneously. We adopt the BF16 data type for storing model weight and gradient. The partition size of language modeling head is set to 100 for all the experiments. The partition size for transformer layer is set to 500 for maximum sequence length measurement, and is set to $T / 3$ for time measurement. + +Training cost. We measure the maximum sequence length of training 4B, 8B, 14B, and 32B models using the objective of SFT, GRPO, and DPO, respectively. Our implementation is built upon HuggingFace TRL library [27]. We adopt pure BF16 Adam optimizer in full training and + +mixed-precision training in rank-32 LoRA model training. When using LoRA mode, there is no need to store the reference model in DPO and GRPO, as one can disable the adapter in training model to recover the reference model. The batch size is set to 1 except for GRPO, where the group size is set to 8. We also compare the memory cost of StreamBP with long-sequence training baseline method MsT to demonstrate the effectiveness of StreamBP. The partition size of language modeling head is set to 100 for all the experiments. The partition size for transformer layer is set to 500 for maximum sequence length measurement, and is set to $T / 3$ for time measurement. All the experiments are conducted in a single A800-80GB GPU. + +Distributed training. Under the communication-efficient design in Appendix A.3, we measure the maximum sequence length and time cost of distributed StreamBP under Deepspeed ZeRO-2 training paradigm, where the gradient and optimizer states are partitioned across GPUs. 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It addresses two fundamental challenges in adapting existing models into online scenarios: (1) limited capability for multi-turn real-time understanding, and (2) lack of proactive response mechanisms. Specifically, StreamBridge incorporates (1) a memory buffer combined with a round-decayed compression strategy, supporting long-context multi-turn interactions, and (2) a decoupled, lightweight activation model that can be effortlessly integrated into existing Video-LLMs, enabling continuous proactive responses. To further support StreamBridge, we construct Stream-IT, a large-scale dataset tailored for streaming video understanding, featuring interleaved videotext sequences and diverse instruction formats. Extensive experiments show that StreamBridge significantly improves the streaming understanding capabilities of offline Video-LLMs across various tasks, outperforming even proprietary models such as GPT-4o and Gemini 1.5 Pro. Simultaneously, it achieves competitive or superior performance on standard video understanding benchmarks. + +# 1 Introduction + +Video Large Language Models (Video-LLMs) [1; 2; 3; 4; 5] typically process entire pre-recorded videos at once. However, emerging applications, such as robotics [6; 7] and autonomous driving [8; 9], require causal perception and interpretation of visual information online. This fundamental mismatch highlights a critical limitation of current Video-LLMs, as they are not inherently equipped to operate in streaming scenarios where timely understanding and responsiveness are paramount. + +Figure 1 highlights two representative patterns in streaming video understanding, which also correspond to the key challenges in adapting Video-LLMs from offline to streaming scenarios: (1) multi-turn real-time understanding and (2) proactive response generation. The first pattern involves multi-turn interactions, where the assistant receives user queries at different timestamps. In each turn, while keeping accumulated visual and conversational context as historical information, the model should focus on the most recent video segment. The second pattern emphasizes more human-like, proactive behaviors. Rather than passively waiting for user prompts, the model actively monitors the visual stream and generates timely outputs based on unfolding content. For instance, in Figure 1 (bottom), the assistant provides step-by-step guidance as the drawing progresses without being explicitly asked, simulating continuous support in dynamic environments. + +![](images/96d37908e4d504451ea43dcc411eaad0a98da8f8a558c27f411ab23abc7814fc.jpg) +Figure 1: Illustration of streaming scenarios. Top: Multi-turn interactions. User issues queries at different timestamps, with each turn involving a new video segment along with accumulated visual and text history. Bottom: Proactive responses. The assistant actively delivers timely feedback or guidance based on the incoming visual stream, without requiring explicit user prompts. + +To bridge the gap between offline and streaming video understanding, we introduce StreamBridge, a simple yet effective framework that seamlessly transforms pre-trained offline Video-LLMs into streaming-capable models. In contrast to prior efforts [10; 11; 12; 13], which train streaming models from scratch but fall behind on offline video tasks, StreamBridge leverages the strong generalization capabilities of existing Video-LLMs without requiring full retraining. This approach allows developers to directly benefit from the rich world knowledge and linguistic fluency of large-scale pre-trained models, while incurring only minimal additional computational cost and data requirements. Concretely, StreamBridge introduces a memory buffer to manage incoming video frames, coupled with a round-decayed compression strategy that merges earlier frame tokens while preserving recent ones, enabling the model to support long-context, multi-modal, and multi-turn interactions in streaming scenarios. For proactive capabilities, instead of modifying the base model architecture [14] or introducing streaming-specific objectives [12], both of which can lead to optimization conflicts and issues like probability correction [10], StreamBridge adopts a modular design, by decoupling the proactive capability from the main Video-LLM via a compact activation model. This plug-and-play component operates in parallel with the main Video-LLM, enabling proactive behavior in a flexible and non-intrusive manner while fully preserving the main Video-LLM's language fluency and general video understanding capabilities. + +To further support StreamBridge, we construct Stream-IT, a large-scale dataset tailored for streaming scenarios. Stream-IT captures diverse real-time questions and proactive responses embedded within multi-turn video interactions, featuring interleaved video-text sequences. While existing datasets primarily focus on single-turn question answering [15; 16] or short-form video captioning [17; 18; 19], Stream-IT fills a critical gap by enabling temporally extended, interactive video understanding. It is constructed by concatenating semantically related short clips from large-scale video-caption corpora, followed by the generation of multi-turn QA sequences that simulate realistic, time-sensitive user interactions. Moreover, Stream-IT incorporates a broad spectrum of task formats sourced from public datasets, thereby boosting task diversity and promoting model generalization in streaming settings. + +By integrating our StreamBridge framework and fine-tuning on Stream-IT, we successfully convert several leading offline Video-LLMs, including LLaVA-OV [3], Oryx-1.5 [1], and Qwen2-VL [2], into streaming-capable assistants. Extensive experiments demonstrate that our models achieve state-of-the-art performance on streaming benchmarks such as OVO-Bench [20] and Streaming-Bench [21], outperforming even proprietary models like GPT-4o [22] and Gemini 1.5 Pro [23], while retaining or exceeding performance on conventional offline video understanding tasks [24; 25; 26; 27; 28; 29]. + +# 2 Related Work + +Video Large Language Models. With the rapid advancement of Multimodal Large Language Models (MLLMs) [30; 3; 31; 32; 2], Video-LLMs [33; 34; 35; 36; 15; 37; 38] have gained increasing attention for general video understanding. Typically, these models comprise a visual encoder [39; 40; 1] for extracting frame-level representations, a modality projector (e.g., MLP [41] and Q-former [30]) to map visual features into the language space, and an LLM [42; 43] to generate contextual responses. While achieving strong results on standard video benchmarks [25; 29; 27], these models are inherently designed for static, offline settings where the entire video is pre-recorded and fully accessible at inference time. As a result, they struggle in streaming environments, where video frames arrive sequentially and require real-time, temporally coherent, or even proactive responses. Our work aims to bridge this gap by augmenting offline Video-LLMs with streaming capabilities. + +Streaming Video Understanding. Typical tasks in streaming video understanding, such as action recognition [44; 45; 46; 47] and anticipation [48; 49], causally process video inputs using only past and current observations. Recent efforts [50; 12; 13; 51] focus on building Video-LLMs capable of real-time conversation, generating timely responses throughout a live video stream. VideoLLM-Online [10] and Flash-VStream [11] introduce specialized online objectives and memory architectures to handle sequential inputs. MMDuet [14] and ViSpeak [52] add dedicated heads to facilitate proactive response generation. To benchmark streaming video capabilities, several evaluation suites have been proposed, including StreamingBench [21], StreamBench [53], SVBench [54], OmniMMI [55], and OVO-Bench [20]. In contrast to previous approaches that retrain models or tightly couple proactive mechanisms within the backbone, our work leverages the strong generalization abilities of pre-trained offline Video-LLMs [1; 3; 2]. We propose an efficient adaptation framework, combined with a dedicated fine-tuning dataset, to endow these models with streaming capabilities. Furthermore, observing that embedding the activation function into the main model often lead to optimization conflicts and performance degradation [10; 14], we advocate a modular, decoupled design. Our method introduces a compact, plug-and-play activation model that enables proactive behaviors efficiently and non-intrusively. We also provide additional discussions on how our work relates to the ReKV [56], VideoStreaming [57], and StreamChat [53] in the Appendix F. + +# 3 Methodology + +# 3.1 Preliminary Analysis + +Streaming video understanding involves interleaved video-text inputs. From an input perspective, streaming scenarios can be broadly categorized into two representative formats: + +- Multi-turn dialogue with interleaved video-text. In this setting, the input sequence is in the form of $\langle V_1 \rangle \langle Q_1 \rangle \langle A_1 \rangle$ , $\langle V_2 \rangle \langle Q_2 \rangle \langle A_2 \rangle$ , ..., where $\langle V_i \rangle$ , $\langle Q_i \rangle$ , and $\langle A_i \rangle$ denote the video clip, user query, and assistant answer in the $i$ -th round. Crucially, there is no delay between $\langle Q_i \rangle$ and $\langle A_i \rangle$ , reflecting the need for immediate responses. This format closely resembles the live interaction in dynamic environments, as shown in Figure 1 (Top). +- Proactive output. The assistant answers after watching an incoming video stream, often without an explicit user query at the response time. The input can be structured as $\langle Q \rangle < V_1 > < A_1 > < V_2 > < A_2 > \dots$ , where $\langle Q \rangle$ represents an initial prompt (e.g., "Guide me through the task"), and the model must proactively determine when and how to respond based on the incoming video contents. This scenario requires the ability to continuously monitor evolving context and trigger responses at appropriate moments. Figure 1 (Bottom) is an example of proactive responses. + +Recent benchmarks such as OVO-Bench [20] and Streaming-Bench [21] attempt to evaluate these capabilities by constructing multi-turn interleaved video-text dialogues. However, due to the limited input length and the lack of streaming support in current Video-LLMs, these benchmarks necessarily simplify the problem. Specifically, they segment a complete long video into multiple isolated clips aligned with each query timestamp. For a query $< Q_{i} >$ at time $t_i$ , the visual input is restricted to the uniformly sampled frames under segment $V_{[0:t_i]}$ , and prior dialogue history is completely discarded. As a result, the multi-turn streaming scenario is reduced to a series of independent, single-turn offline tasks. To address these limitations, we propose StreamBridge, a general framework designed to introduce the actual streaming setup to existing offline Video-LLMs. + +![](images/e228fa336d90f9060ceae7493cb940369d5b1b8438cb2432e8b8a935c6b7679d.jpg) +Figure 2: Overview of StreamBridge. ①③: Incoming frames are encoded and stored into the memory buffer one by one. ②: A query $\mathcal{Q}$ is posed. ④: The activation model monitors incoming frames and returns a binary signal $\mathcal{D}$ , indicating whether LLM should start answering. $\otimes$ means concatenation. + +# 3.2 StreamBridge + +As shown in Figure 2 and Algorithm 1, in addition to the frame encoder $\mathcal{I}(\cdot)$ and the large language model $\mathcal{L}\mathcal{L}\mathcal{M}(\cdot)$ , StreamBridge proposes three key components to enable streaming capabilities: (1) a memory buffer responsible for storing and retrieving frame tokens over time, (2) a round-decayed compression strategy $\mathcal{COM}(\cdot)$ that efficiently prunes redundant tokens from earlier rounds while preserving the most recent context, and (3) a compact activation model $\mathcal{A}\mathcal{C}\mathcal{T}(\cdot)$ that enables proactive responses by making frame-level decisions on when to generate outputs. + +# 3.2.1 Memory Buffer + +In streaming scenarios where video frames arrive sequentially, we adopt a memory buffer $\mathcal{MB}$ to store both visual and textual embeddings. As illustrated in Figure 2, each incoming frame is independently encoded and appended to the buffer alongside any associated query embeddings. Conceptually, $\mathcal{MB}$ operates under a producer-consumer paradigm: the encoder $\mathcal{I}(\cdot)$ functions as the producer, continuously generating frame-level features, while the language model $\mathcal{L}\mathcal{LM}(\cdot)$ serves as the consumer, retrieving the accumulated embeddings to generate a response upon receiving a user query. Formally, as detailed in Algorithm 1, at each time step $t$ , the incoming frame $F_{t}$ is first processed by $\mathcal{I}(\cdot)$ , and the resulting embeddings are stored in the memory buffer $\mathcal{MB}$ (Algorithm 1, line 4). Upon the arrival of a user query $\mathcal{Q}$ and a positive activation decision $\mathcal{D}$ , the buffer content, including both visual and textual embeddings, is flattened into a single sequence of input embeddings, which is then fed into $\mathcal{L}\mathcal{LM}(\cdot)$ for response generation (Algorithm 1, line 13-16). Once a response $\mathcal{R}$ is produced, it is also appended to the memory buffer (Algorithm 1, line 17), enabling the model to preserve temporal continuity and maintain a complete history of multi-turn video-text interactions. + +# 3.2.2 Round-Decayed Compression + +Online scenarios often involve long, even infinite video streaming, which can lead to significant memory usage and inference latency. Therefore, we propose a round-decayed token compression strategy tailored for multi-turn streaming settings. Specifically, we pre-define a maximum allowable embedding length MaxLen for the model input. Before each response generation, the model checks whether the current input embedding exceeds MaxLen. If so, we apply a round-decayed token merging strategy: starting from the earliest dialogue rounds, visual tokens are progressively merged frame-by-frame, until the total length falls below MaxLen. The merging is implemented via average + +Algorithm 1: StreamBridge Framework +1 Inputs: incoming frames $[F_1,F_2,\dots ,F_t,\dots ]$ +2 Initializations: $\mathcal{I}(\cdot)$ , LLM(·), ACT(·), COM(·), MB = [-], MaxLen, $t_{\mathcal{Q}} =$ None; +3 while $F_{t}$ do +4 $\mathcal{MB}\gets \mathcal{I}(F_t)$ // store the frame feature $\mathcal{I}(F_t)$ into the Memory Buffer +5 if Q at timestamp t then +6 $\begin{array}{l}\mathcal{MB}\leftarrow \mathcal{Q}\\ t_{\mathcal{Q}}\leftarrow t; \end{array}$ // t is the timestamp when Q is posed +8 if tQ is not None then +9 D $\leftarrow$ ACT(Q, $F_{t_{\mathcal{Q}}:t - 1},F_t)$ // D denotes whether response or not at timestamp t +10 else +11 D $\leftarrow$ False; // not response if there is no Q +12 if D then +13 // D is true at timestamp t, and should return a response R +14 InputEmbeds $\leftarrow$ Flatten(MB) +15 if Len(InputEmbeds) > MaxLen then +16 $\begin{array}{r}\mathcal{L}\mathrm{R}\leftarrow \mathcal{L}\mathcal{L}\mathcal{M}(\mathrm{InputEmbeds});\\ \mathcal{MB}\leftarrow \mathcal{R}; \end{array}$ // compress redundant visual tokens +16 $\mathcal{R}\leftarrow \mathcal{LLM}(\mathrm{InputEmbeds});$ // return a response R +17 $\mathcal{MB}\leftarrow \mathcal{R};$ // update MB +18 $t + = 1$ // receive subsequent frames + +pooling [58] over adjacent frame tokens. This strategy ensures that the most recent visual context is retained with minimal distortion, thus maintaining the precision of real-time responses while not fully discarding historical visual contexts. At the same time, it significantly improves memory efficiency and reduces inference overhead as in Figure 4. This process is encapsulated in the compression function $\mathcal{COM}(\cdot)$ in Algorithm 1 (line 15). The detailed pseudo codes can be found in Appendix I. + +# 3.2.3 A Plug-and-play Activation Model + +To enable proactive responses in streaming Video-LLMs, we decouple the activation function from the main Video-LLM. Unlike prior methods that tightly integrate activation mechanisms into the LLM [10; 12; 14; 52], our framework avoids potential interference with the language modeling capacity of the main + +![](images/5f8347f037277e8e085171c34ba5f4032148201592081631b329f9e83ea03d12.jpg) +Figure 3: An overview of the proposed activation model. We label the last $P\%$ of frames of each video clip to be true during training. + +Video-LLM. Specifically, we propose a parallel pipeline, where a compact external MLLM (e.g., LLaVA-OV-0.5B [3]) is used as an independent activation model, denoted as $\mathcal{A}\mathcal{C}\mathcal{T}(\cdot)$ . As shown in Algorithm 1 (line 9), upon receiving each new frame, the framework simultaneously forwards the current frame (along with the user query $\mathcal{Q}$ and optionally previous frames) to $\mathcal{A}\mathcal{C}\mathcal{T}(\cdot)$ to determine whether a response should be generated. If the activation signal $\mathcal{D}$ is positive, the buffered embeddings are sent to the LLM for decoding. This design ensures high flexibility and compatibility. Furthermore, in real-time deployment, the $\mathcal{A}\mathcal{C}\mathcal{T}(\cdot)$ , the frame encoder $\mathcal{I}(\cdot)$ , and the main $\mathcal{L}\mathcal{L}\mathcal{M}(\cdot)$ can run concurrently in parallel threads, enabling more efficient inference. + +To train the activation model (illustrated in Figure 3), we modify the architecture by replacing the standard language modeling head with a score head for binary classification, and introduce a learnable activation token $\langle ACT \rangle$ which is appended to the visual embeddings of each frame. After processing through the final layer, we extract the latest frame's activation token and feed its hidden representation into the score head to predict whether the model should respond at that time. During inference, only when the predicted score is greater than the activation threshold $\alpha$ , the main Video-LLM can be triggered to give a response. Since $\mathcal{A}\mathcal{T}(\cdot)$ performs only binary classification (i.e., to respond or not), we aggressively pool its visual tokens for efficiency. The input sequence to the model follows the format: $\langle Q \rangle < V_1 > < A_1 > < V_2 > < A_2 > \dots$ , where the question $\mathcal{Q}$ is pretended to the sequence, + +and visual frames and corresponding responses are interleaved. This design enables the model to learn temporal dependencies and identify appropriate response moments throughout the video stream. + +For training data, we collect a diverse set of temporally annotated video datasets across multiple tasks, including dense video captioning [59; 60], sequential step recognition [61; 62], grounded video question answering [63; 64; 65], temporal video grounding [66], and temporal action detection [67; 68]. For each task, we design specific prompt templates and randomly select among them as $\mathcal{Q}$ during training (details in Appendix A). To supervise activation timing, we insert the response $$ at the end of its corresponding annotated timestamp (during inference, the response $$ is generated by the larger main Video-LLM). Besides, only the last $P\%$ of frames of each video segment $V_{i}$ are labeled as positive (i.e., response-worthy), while earlier frames are treated as negatives. $P$ is dynamically sampled between $0\%$ and $50\%$ for each training instance, simulating a variety of activation patterns and enhancing the model's robustness to temporal variations. + +# 4 Stream-IT Dataset + +As analyzed in Section 3.1, streaming scenarios are primarily characterized by multi-turn real-time understanding and proactive responses. However, existing datasets and video sources fall short of fully supporting these requirements [69; 70]. To fill this gap and further enhance the streaming interaction capability of StreamBridge, we introduce Stream-IT—a video-text dataset specifically designed for streaming instruction tuning with an interleaved multi-turn dialogue format. + +Datasets for Proactive Understanding. We collect a set of public datasets enriched with timestamp annotations, spanning a wide range of tasks including: (i) Dense Video Captioning [59; 60; 71]; (ii) Sequential Step Recognition [61; 62; 72]; (iii) Grounded VideoQA [63; 73; 74; 75]. All datasets are reformatted into a proactive-style interleaved format: $\langle Q \rangle < V_1 > < A_1$ , $\langle V_2 \rangle < A_2$ , ..., where $Q$ may be an open-ended query (e.g., "Who is the man going to find?") or a goal-oriented instruction (e.g., "Show me all the steps for cooking"). Unlike traditional single-turn datasets where a question is immediately followed by an answer [69; 70], our structure introduces a temporal delay between $\langle Q \rangle$ and $\langle A \rangle$ through the inserted video segments $\langle V \rangle$ , simulating proactive response scenarios. + +StreamingQA-120K: Multi-Turn, Long-Form QA Construction. To further support long-context, multi-turn real-time understanding, we introduce StreamingQA-120K, a large-scale synthetic dataset constructed by composing long-form videos from existing short video clips. Labeling long-duration videos with dense multi-turn QA pairs is prohibitively expensive. To address this, we leverage short clips from large-scale video-caption datasets, including WebVid-10M [19], Panda-70M [18], and InternVid-10M [17]. We filter approximately 1.28 million clips using semantic similarity between video and caption to ensure alignment, with each clip being around 12 seconds long. With these short clips, to form coherent long-form videos, we then iteratively compute pairwise similarity between videos and concatenate highly similar clips. Each constructed video contains roughly 10 clips, with an average length exceeding 150 seconds. Captions for each clip are preserved with natural timestamps. Using these captions, we employ GPT-4o [22] to generate diverse question-answer pairs spanning 8 task types. By default, each QA pair $\langle Q_i \rangle < A_i$ is inserted immediately after its corresponding clip $\langle V_i \rangle$ , forming sequences like $\langle V_1 \rangle < Q_1$ $\langle A_1 \rangle$ , $\langle V_2 \rangle < Q_2$ $\langle A_2 \rangle$ , ...'. Here, we introduce two augmentation strategies during sequence construction: + +- Random QA Drop: randomly drops some QA pairs by transforming $\langle V_i \rangle < Q_i > < A_i$ to $\langle V_i \rangle$ with a probability of $P_{\mathrm{drop}}$ , to prevent overfitting to fixed QA positions and enhance the model's robustness in temporal variations. We set $P_{\mathrm{drop}}$ to be 0.55 by default. +- QA Interval Shift: with probability $P_{\mathrm{shift}}$ , transforms sequences from $\langle V_i \rangle \langle Q_i \rangle \langle A_i \rangle$ to $\langle Q_i \rangle \langle V_i \rangle \langle A_i \rangle$ , where the visual content $V_i$ serves as the temporal delay between question and response for proactive scenarios. $P_{\mathrm{shift}}$ is set to 0.1 here. + +Together, these strategies ensure that the Stream-IT dataset supports rich and varied streaming interaction formats, enabling both multi-turn real-time dialogue and proactive response capabilities across a wide range of tasks and timescales. More details on data statistics, concatenation strategy of StreamingQA-120K, and prompts for QA generation are provided in Appendix B. + +
Method# of FramesOVO-Bench Real-TimeStreaming-Bench Real-Time
OCRACRATRSTUFPDOJRAVG.OPCRCSATPEUTRPRSUACPCTAVG.
Human
Human-93.9692.5794.8392.7091.0994.0293.2089.4792.0093.6091.4795.6592.5288.0088.7589.7491.3091.46
Proprietary Models (Offline), Single-Turn Evaluation
Gemini 1.5 pro [23]1 FPS85.9166.9779.3158.4363.3761.9669.3279.0280.4783.5479.6780.0084.7477.7864.2371.9548.7075.69
GPT-4o [22]6469.8064.2271.5551.1270.359.7864.4677.1180.4783.9176.4770.1983.8066.6762.1969.1249.2273.28
Open-Source Models (Offline), Single-Turn Evaluation
Qwen2-VL-72B [2]6465.7760.5569.8351.6969.3154.3561.92-----------
LLaVA-Video-7B [15]6469.1358.7268.8349.4474.2659.7863.52-----------
LLaVA-OV-7B [3]64/3266.4457.8073.2853.3771.2961.9664.0280.3874.2276.0380.7272.6771.6567.5965.4565.7245.0871.12
Qwen2-VL-7B [2]64/1 FPS60.4050.4656.0347.1966.3455.4355.9875.2082.8173.1977.4568.3271.0372.2261.1961.4746.1169.04
InternVL-V2-8B [76]64/1667.1160.5563.7946.0768.3256.5260.3968.1260.9469.4077.1267.7062.9359.2653.2554.9656.4863.72
Open-Source Models (Streaming), Single-Turn Evaluation
Flash-VStream-7B [11]1 FPS24.1629.3628.4533.7125.7428.8028.3725.8943.5724.9123.8727.3313.0818.5225.2023.8748.7023.23
VideoLLM-Online-8B [10]2 FPS8.0523.8512.0714.0445.5421.2020.7939.0740.0634.4931.0545.9632.4031.4834.1642.4927.8935.99
Dispider [13]1 FPS57.7249.5462.0744.9461.3951.6354.5574.9275.5374.1073.0874.4459.9276.1462.9162.1645.8067.63
Models under StreamBridge (Offline → Streaming), Multi-Turn Evaluation
Oryx-1.5-7B† [1] + Stream-IT1 FPS60.4052.2969.8350.0065.3557.6159.2578.4777.1783.8680.2071.0766.9879.6361.3866.2940.9370.59
1 FPS84.5675.2370.6950.5674.2671.7471.1782.2977.9587.9886.4777.9981.3176.8569.9271.9635.2374.79
LLaVA-OV-7B† [3] + Stream-IT1 FPS58.3959.6369.8244.3876.2361.4161.6476.8477.1782.6075.2564.1564.1775.0061.3861.1946.1168.39
1 FPS74.5077.0670.6954.4973.2769.5769.9382.2972.4492.0980.8671.0774.4675.0062.2070.2628.5070.92
Qwen2-VL-7B† [2] + Stream-IT1 FPS65.1064.2264.6646.6374.2665.2263.3580.3878.7483.2279.8674.2169.4777.7863.4169.9743.0172.01
1 FPS84.5671.5674.1449.4475.2572.8371.3084.7482.6888.9289.7777.3685.3684.2669.9271.6735.7577.04
+ +Table 1: Results on real-time understanding tasks on OVO-Bench and Streaming-Bench. $\dagger$ means models under StreamBridge framework, and +Stream-IT means finetuned on Stream-IT. + +# 5 Experiments + +# 5.1 Settings + +Models and Datasets. We evaluate StreamBridge framework using three mainstream offline Video-LLMs to show its generalizability: LLaVA-OV-7B [3], Qwen2-VL-7B [2], and Oryx-1.5-7B [1]. To preserve their general video understanding capabilities during streaming adaptation, we supplement Stream-IT with approximately 600K samples from the LLaVA-178K [15], VCG-Plus [35] and ShareGPT4Video [16]. For the activation model, we fine-tune LLaVA-OV-0.5B [3] on our collected activation datasets as described in Sec. 3.2.3. The videos are sampled at 1 FPS. In Section 5.3, we use Qwen2-VL-7B as the default model unless otherwise specified. See the Appendix C for more details. + +**Benchmarks.** For multi-turn real-time understanding, we choose OVO-Bench [20] and Streaming-Bench [21]. We primarily focus on their real-time tasks. For general video understanding, we evaluate our method across 7 video benchmarks, including 3 short-video benchmarks: MVBenchmark [24], PerceptionTest [26], TempCompass [77], and 4 long-video benchmarks: EgoSchema [28], LongVideoBenchmark [29], MLVU [27], and VideoMME [25]. To evaluate the proactive capability of our method, we use subtasks from ET-Bench [66] following previous works. See Appendix D for more benchmark details and evaluation metrics. + +# 5.2 Main Results + +Multi-Turn Real-Time Understanding. As discussed in Section 3.1, the results reported in the original paper [20; 21] in Table 1, marked as “(Offline), Single-Turn Evaluation”, do not reflect performance in real streaming scenarios. They segment a complete video into several individual clips, discarding historical visual and dialogue contexts, thereby limiting the upper bound of the performance. In contrast, with the StreamBridge framework, denoted as “(Offline $\rightarrow$ Streaming), Multi-Turn Evaluation”, these offline models are equipped to process streaming videos at 1 FPS in a multi-turn manner, while maintaining input length and historical contexts within a predefined maximum token budget. + +Specifically, we observe that Qwen2-VL† demonstrates notable improvements in the streaming setting, with its average score on OVO-Bench increasing from 55.98 to 63.35, and on Streaming-Bench from 69.04 to 72.01. Conversely, LLaVA- $\mathrm{OV}^{\dagger}$ shows a slight performance drop when transitioning to the streaming setup: from 64.02 to 61.64 on OVO-Bench, and from 71.12 to 68.39 on Streaming-Bench. We attribute these differences to the nature of their pretraining data, where Qwen2-VL benefits from richer interleaved multimodal training (e.g., image/video-text sequences), which makes it more adept at understanding interleaved video-text inputs and utilizing extended context effectively. On + +
ModelMVBenchPerceptionTestTempCompassEgoSchemaLongVideoBenchMLVUVideoMME (w/o subs)
AvgValMCTestValM-AvgAvg
Avg. Duration16s23s12s180s473s651s1010s
Proprietary Models
Gemini 1.5 pro [23]60.5-67.171.264.0-75.0
GPT-4o [22]64.6-70.972.266.764.671.9
Open-Source Models
Kangaroo-8B [78]61.0-62.5-54.861.056.0
LongVILA-7B [79]---67.7--57.5
LongVU-7B [80]66.9--67.6-65.460.6
Apollo-7B [4]-67.364.9-58.570.961.3
NVILA-8B [81]68.165.469.7-57.770.164.2
SF-LLaVA-1.5-7B [5]-69.668.8-62.571.563.9
InternVL2.5-8B [82]72.068.268.351.560.068.964.2
VideoChat-Flash-7B [83]74.076.2--64.774.765.3
VideoLLaMA3-7B [37]69.772.868.163.359.873.066.2
Oryx-1.5-7B [1]67.670.058.8-56.367.558.8
Oryx-1.5-7B (ours)‡68.0 (↑0.4)71.0 (↑1.0)69.0 (↑10.2)61.258.9 (↑2.6)71.4 (↑4.0)65.5 (↑6.7)
LLaVA-OV-7B [3]56.757.164.860.156.364.758.2
LLaVA-OV-7B (ours)‡59.4 (↑2.7)63.9 (↑6.8)67.7 (↑2.9)67.0 (↑6.9)54.3 (↓2.0)68.2 (↑3.5)61.2 (↑3.0)
Qwen2-VL-7B [2]67.062.367.966.7--63.3
Qwen2-VL-7B (ours)‡64.4 (↓2.6)69.9 (↑7.6)71.1 (↑3.2)66.9 (↑0.2)59.169.664.4 (↑1.1)
+ +the other hand, LLaVA-OV is trained with fewer interleaved sequences, making it less suited for multi-turn streaming inputs. When faced with long, interleaved video-text sequences in streaming scenarios, its performance tends to degrade as more historical frames accumulate. Notably, fine-tuning these models on the proposed Stream-IT leads to substantial improvements in multi-turn real-time understanding. For instance, Oryx-1.5 $\dagger$ achieves a performance gain of +11.92 on OVO-Bench and +4.2 on Streaming-Bench. Furthermore, Qwen2-VL $\dagger$ reaches an average score of 71.30 on OVO-Bench and 77.04 on Streaming-Bench, outperforming proprietary models such as GPT-4o and Gemini 1.5 Pro. These results validate the effectiveness of both our StreamBridge framework and the Stream-IT dataset in enhancing multi-turn real-time understanding in streaming scenarios. + +General Video Understanding. While our method is designed for online scenarios, we also verify that it does not downgrade the base model's performance on standard offline video tasks. As shown in Table 2, models equipped with the StreamBridge framework and fine-tuned on Stream-IT (denoted with $\ddagger$ ) exhibit consistent improvements or maintain comparable performance relative to their original versions. For instance, Oryx-1.5-7B $\ddagger$ achieves 65.5 on the challenging VideoMME with an increase of 6.7, while LLaVA-OV-7B $\ddagger$ outperforms its base model across nearly all benchmarks, except LongVideoBench. Likewise, Qwen2-VL-7B $\ddagger$ achieves competitive results on MVBench, while surpassing its original counterpart on other benchmarks. These outcomes demonstrate that our streaming adaptation enables models to retain, or even exceed their original performance in general video understanding tasks, demonstrating the generality and non-degradability of our method. + +Online Activation. We evaluate the proactive capability of our framework in Table 3. Notably, in all tasks, the question is presented at the beginning of the video, and the model must autonomously decide when to respond. On the ET-Bench, StreamBridge outperforms both VideoLLM-Online [10] and Dispider [13] across generation-based tasks such as DVC (Dense Video Captioning) and SLC (Step Localization and Captioning), + +achieving higher similarity scores of $\mathrm{DVC}_{\text{Sim}}$ and $\mathrm{SLC}_{\text{Sim}}$ , by producing more accurate and context-aware descriptions in streaming scenarios. We attribute this to the decoupled nature of the activation model, which enables the main Video-LLM to focus solely on video understanding and language generation, free from the burden of proactive decision-making. We also observe that Qwen2-VL achieves better text similarity scores than other Video-LLMs, consistent with its strong real-time understanding performance presented in Table 1. + +Table 2: Results on general video understanding benchmarks. $\ddagger$ means models under Stream-Bridge framework and fine-tuned on Stream-IT. + +
Method# of FramesET-Bench
TVGF1TALF1DVCF1DVCsimSLCF1SLCsim
VideoLLM-Online [10]2 FPS13.29.124.013.49.910.1
Dispider [13]1 FPS36.127.333.818.918.812.4
Models under StreamBridge Framework
Oryx-1.5 (ours)‡1 FPS34.324.337.824.022.517.3
LLaVA-OV (ours)‡1 FPS34.324.337.924.222.816.2
Qwen2-VL (ours)‡1 FPS34.324.338.325.122.617.1
+ +Table 3: Results on ET-Bench. $\ddagger$ denotes models under StreamBridge framework and fine-tuned on StreamIT. TVG $_{F1}$ and TAL $_{F1}$ scores are identical across StreamBridge models due to sharing the same activation model. + +# 5.3 In-Depth Analysis + +
CompressionOVO Avg.Streaming Avg.ET DVCsim SLCsim
Truncation68.8872.7922.116.7
Round-Uniform69.9174.1823.815.9
Round-Decayed71.3077.0425.117.1
+ +Table 4: Ablation studies on different compression strategies. + +
LLaVA-178k(600k used)Stream-ITOVOAvg.StreamingAvg.MVBenchAvg.VideoMMEOverall.
w/o SQA-120kw/ SQA-120k
✓✓65.9871.3664.561.7
71.2874.1058.859.0
✓✓67.6772.4263.163.6
✓✓71.3077.0464.464.4
+ +![](images/3d643ca6a8e75289253519a842c3a86c404712916258050ff46cfc9878571c1b.jpg) +Figure 4: Inference Latency (y-axis) vs. Frame Number (x-axis). + +![](images/343d14fa0b92821b9421ee459bd4055f41a828f39c718fcf7c0e2268974dba96.jpg) +Figure 5: Ablation studies on the activation threshold $\alpha$ . + +Round-Decayed Compression. We set the maximum input length MaxLen = 16384, and denote the current length of the input embeddings as L. To assess the effectiveness of our round-decayed compression strategy, we compare it against two alternative methods: (1) Truncation: If L > MaxLen, only keep the last L tokens in the input sequence. (2) Round-Uniform: We treat each round equally by reducing the number of visual tokens with a fixed ratio $\frac{\mathrm{L} - \mathrm{MaxLen}}{\mathrm{L}}$ per round, to keep the total length within MaxLen. The results are reported in Table 4. We observe that Truncation yields the worst performance, as it indiscriminately removes both visual and textual history tokens, severely weakening multi-turn reasoning. The Round-Uniform strategy performs slightly better, but still underperforms our method. It compresses the latest visual tokens, which are critical for real-time comprehension, thus leading to degraded performance, particularly on OVO-Bench and Streaming-Bench. + +Inference Latency. We also evaluate the inference latency on a single A100-80G GPU with different MaxLen (8k, 16k, 32k), as shown in Figure 4. Our results show that our compression method maintains near-constant latency when the number of input tokens exceeds MaxLen, whereas models without compression suffer from sharply increasing delays and eventually trigger out-of-memory (OOM) errors with 2048 frames. This highlights the necessity of effective compression to balance inference efficiency and memory usage in streaming settings. + +Impact of Stream-IT. Table 5 ablates effectiveness of Stream-IT. Training on LLaVA-178K alone causes a marked drop on both OVO-Bench and Streaming-Bench, as it lacks interleaved video-text samples necessary for multi-turn interactions. Conversely, using only Stream-IT without LLaVA-178K leads to declines in general video understanding (MVBench, VideoMME), indicating that the larger offline data corpus still contributes valuable world knowledge. Finally, removing the synthetic StreamingQA-120K subset from Stream-IT degrades performance across both streaming and offline benchmarks, underscoring the crucial role of StreamingQA-120K in boosting both streaming and offline video understanding capabilities. + +Impact of MaxLen. To better understand the impact of MaxLen, we conducted ablation studies using the Qwen2-VL-StreamBridge model with 1 FPS sampling, varying MaxLen from 4k to 32k. From Table 6, we observe the following: (1) For streaming tasks (e.g., OVO-Bench RealTime): Model performance remains relatively stable across varying MaxLen values, ranging from $70.49\%$ to $71.30\%$ ; Accuracy peaks at 16k and slightly declines at 32k, suggesting that further increasing the memory budget yields + +Table 5: Ablation studies on Stream-IT. SQA-120k denotes the generated StreamingQA-120k. + +
MaxLenOVO-BenchVideoMME
(Real-Time) Avg.Avg.
4k70.4961.7
8k70.8963.6
16k71.3064.4
32k71.1664.7
+ +Table 6: Ablation studies on MaxLen + +diminishing returns. This supports our design assumption: in streaming scenarios, models primarily rely on recent context, and older frames can be compressed without significant performance loss. (2) For offline tasks (e.g., VideoMME): Accuracy improves consistently as MaxLen increases, from + +$61.7\%$ at 4k to $64.7\%$ at 32k. This means that offline tasks benefit more from retaining the full temporal context and more uncompressed video tokens, especially for long videos that require detailed long-range understanding. StreamBridge can flexibly balance efficiency and performance across both streaming and offline settings by adjusting the memory budget accordingly, and we set MaxLen = 16k to strike a good balance between them across most tasks. + +Activation Threshold. The compact activation model makes a per-frame decision to trigger responses, with frequency determined by the activation threshold $\alpha$ (see score head in Figure 3). We adopt a default $\alpha$ of 0.35, following common practice [52; 14]. Figure 5 illustrates the impact of varying this threshold: both excessively low and high values of $\alpha$ decrease F1 scores ( $DVC_{F1}$ and $SLC_{F1}$ on ET-Bench). A low threshold triggers overly frequent responses, while a high threshold suppresses them excessively, both of which hurt performance. Nonetheless, this hyper-parameter allows users to flexibly control response frequency through $\alpha$ , adapting to different practical scenarios. + +# 6 Conclusion + +We present StreamBridge, a novel framework that transforms offline Video-LLMs into streaming-capable models. StreamBridge introduces a memory buffer paired with a round-decayed compression strategy, and decouples the activation function with a compact activation model. We also construct Stream-IT, a dataset with interleaved video-text sequences to further support StreamBridge. Extensive experiments on diverse benchmarks demonstrate that our method not only preserves the strengths of the base models but also equips them with the ability to make timely, proactive responses across multi-turn, long-context streaming scenarios. We believe StreamBridge offers a general solution for bridging the gap between offline Video-LLMs and real-world, interactive streaming applications. + +Acknowledgment. We thank Yu Liu, Haiming Gang, and Mingfei Gao for their kind help. + +# References + +[1] Zuyan Liu, Yuhao Dong, Ziwei Liu, Winston Hu, Jiwen Lu, and Yongming Rao. Oryx mllm: On-demand spatial-temporal understanding at arbitrary resolution. arXiv:2409.12961, 2024. 1, 2, 3, 7, 8, 27 +[2] Peng Wang, Shuai Bai, Sinan Tan, Shijie Wang, Zhihao Fan, Jinze Bai, Keqin Chen, Xuejing Liu, Jialin Wang, Wenbin Ge, et al. 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Crowdsourcing and research with human subjects + +Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? + +Answer: [NA] + +Justification: The paper does not involve crowdsourcing nor research with human subjects. + +Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. +- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. +- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. + +# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects + +Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? + +Answer: [NA] + +Justification: The paper does not involve crowdsourcing nor research with human subjects. + +Guidelines: + +- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. +- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. +- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. +- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. + +# 16. Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [Yes] + +Justification: The LLM is used to generate the contributed data. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. + +# Prompts for Dense Video Captioning: + +(1) "Identify and describe all activity events in the video.", +(2) "List every event happening in the video with descriptions.", +(3) "Detect and summarize each event sequence in the video.", +(4) "Extract and explain all notable activities in the video.", +(5) "Find all significant events in the video and describe them." + +# Prompts for Sequential Step Recognition: + +(1) "Identify key action steps in the video and provide a brief description of each.", +(2) "Detect and outline a sequence of actions or steps taking place in the video.", +(3) "Analyze the video to determine distinct actions or steps.", +(4) "Break down the video into meaningful steps, describing each one concisely." +(5) "Recognize and highlight specific sequences of actions within the video.", + +# Prompts for Temporal Action Detection: + +(1) "When the action happens, output .", +(2) "When the event occurs, output .", +(3) "If you see the actuon , return ." +(4) "Upon detecting , generate the message ." +(5) "When $\langle \text{action label} \rangle$ arises, immediately output $\langle \text{action label} \rangle$ ". + +# Prompts for Grounded VideoQA: + +(1) ". Answer me only when you get enough information for answering the question.", +(2) ". Respond only if you have sufficient details to provide a complete answer.", +(3) ". Provide an answer only when you have gathered enough relevant information." +(4) ". Ensure you have all necessary context before attempting to answer.", +(5) ". Only reply when you are confident that your answer is accurate and well-informed.", + +# Prompts for Temporal Video Grounding: + +(1) "Localize the visual content described by the given textual query in the video.", +(2) "Detect the video segment that semantically matches the given textual query .". +(3) "Give you a textual query: . When does the described content occur in the video?", +(4) "Locate the visual content mentioned in the text query within the video.", +(5) "Find the video segment that corresponds to the given textual query .". + +Table 7: Prompts used for datasets to train the activation model. + +# A Datasets Used to Train the Activation Model + +To train the activation model $\mathcal{A}\mathcal{C}\mathcal{T}(\cdot)$ , we compile a diverse collection of video datasets spanning five distinct tasks: + +- Dense Video Captioning: ActivityNet Captions [59], Shot2Story [60]. +- Sequential Step Recognition: YouCook2 [62], COIN [61]. +- Temporal Action Detection: FineAction [67], HACS [68]. +- Grounded VideoQA: Multihop-EgoQA [64], EgoTimeQA [73]. +- Temporal Video Grounding: Charades [84], and the TVG subset from ET-Instruct [66]. + +In total, our training set contains approximately 180k video samples. For each sample, we construct an input prompt using task-specific templates. A prompt is randomly sampled from a predefined pool for the corresponding task to ensure stylistic diversity and improve generalization across video domains. The full list of prompt templates is provided in Table 7. During training, the prompt is inserted at the beginning of the input sequence as in Figure 3. + +# System: + +You are a good question generator. I need your help in generating high quality question-answer pairs pertaining to the video clip descriptions. Follow these instructions: +(1) Ensure the questions and answers are highly relevant to the captions and DO NOT INCLUDE TOPICS NOT MENTIONED in the captions. +(2) IgNORE CONTRADICTORY OR UNREASONABLE PARTS of the captions. Do not base questions on them. +(3) I hope your questions feature different causal and temporal reasoning. Questions should be diverse and be related to different aspects of the described events. +(4) Ensure that the questions in the QA chain are clear and precise, directly corresponding to specific information or events in the video, and can be answered by watching the video content without the need for a video description or inference, avoiding questions that require assumptions. +(5) Pay attention to grammar. Avoid grammar mistakes, especially with person and tense. +(6) Ensure questions are reasonable and challenging, requiring thoughtful consideration to answer correctly. +(7) The question should not contain phrases like 'In the beginning of the clips' or 'at the beginning of the video' or 'in the video' or 'in this clips'; it can include expressions of the present or recent past such as 'just now' or 'right now. +(8) Please pay attention to the tense of the sentences. +(9) Never mention the sentence like 'according to the caption' in your question, you should assume that you can really watch the video instead of reading a caption. +(10) Ensure there are no references to the source of information in the QA, avoiding expressions like 'from the image', 'sequence of pictures', 'which frame', or 'which photo'; you should understand the input as a video and describe it using video footage. + +Understand the following different task descriptions: + + + +# USER: + +Now, please carefully review the following video caption: + +According to the given caption, please select ONE task type that is best suitable to generate the QA pair, and output your question, answer and task type following the SAME FORMAT as the examples above. Remember, just generate only ONE QA pair. + +Table 8: Prompts used to generate QA pairs with GPT-4o. + +# B Stream-IT Construction + +# B.1 Statistics of Stream-IT + +We provide detailed statistics of the Stream-IT dataset in Table 9, including the number of samples, average video duration, and the corresponding source datasets used for each task. Notably, during the construction of the dense video captioning tasks, including ActivityNet[59] and Shot2Story [60], we only arrange $20\%$ of the sequences with the proactive format of $$ $$ $$ , $$ $$ , $\cdots$ , while the $80\%$ of the sequences with the multi-turn format of $$ $$ $$ , $$ $$ $$ , ..., where $$ is the question asking about current situations like 'What is happening now?' + +# B.2 Concatenation Strategy for Constructing StreamingQA-120K + +Starting from a pool of 1.28 million filtered short videos sourced from WebVid-10M [19], Panda70M [18], and InternVid-10M [17], our goal is to iteratively merge semantically similar clips to form long-form video samples. Let $\mathcal{V}$ denote the entire set of filtered videos. We initiate the process by randomly sampling one clip $\nu_{1}$ from $\nu$ as the anchor. We then compute pairwise semantic similarity (based on the middle frame) between $\nu_{1}$ and all other videos in $\nu \setminus \nu_{1}$ . A new clip $\nu_{2}$ is sampled according to the similarity distribution (without replacement). The procedure is repeated using $\nu_{2}$ as the new anchor, generating $\nu_{3}$ from $\nu \setminus \nu_{1}, \nu_{2}$ , and so on. This results in a similarity-ordered list of videos $\nu_{1}, \nu_{2}, \nu_{3}, \ldots$ . We formulate this process in Algorithm 2. This approach allows for flexible concatenation of any number $k$ of clips to construct a long-form sample, by directly selecting + +
Task# of SamplesDatasetsAverage duration
Dense Video Captioning~54kActivityNet [59] (~10k)~180s
Shot2Story [60] (~36k)~16s
ViTT [71] (~8k)~210s
Sequential Step Recognition~22kYouCook2 [62] (~1.3k)~317s
COIN [61] (~11k)~145s
HowToStep [72] (~10k)~190s
Grounded Video Question Answering~69kMovieChat [74] (~0.8k)~10k frames
EgoTimeQA [73] (~10k)~150s
QAEgo4D [63] (~15k)~495s
FineVideo [75] (~43k)~280s
Multi-turn Real-time Question Answering~120kStreamingQA-120K (~120k)~150s
(Sourced from Webvid-10M[19], Panda-70M[18], InternVid-10M[17])
+ +Table 9:Involved tasks and datasets in Stream-IT. + +Algorithm 2: Constructing Similarity-Ordered Video Clip Sequence +1 Inputs: Pool of filtered short video clips $\nu_{\mathrm{pool}} = \{v_1,v_2,\dots ,v_M\}$ +2 Initializations: $\mathcal{V}_{\mathrm{ordered}} = [\quad ],v_{\mathrm{anchor}} = \mathrm{None};$ +3 Define: Sim(vanchor, $C_{\mathrm{candidates}}$ ): function that returns the clip from $C_{\mathrm{candidates}}$ most similar to $v_{\mathrm{anchor}}$ +4 $v_{\mathrm{anchor}}\gets \mathrm{RandomSample}(\mathcal{V}_{\mathrm{pool}})$ // Randomly select the first anchor clip +5 $\nu_{\mathrm{ordered}}\gets v_{\mathrm{anchor}}$ // Add $v_{\mathrm{anchor}}$ to $\nu_{\mathrm{ordered}}$ +6 $\nu_{\mathrm{pool}}\gets \nu_{\mathrm{pool}}\backslash \{v_{\mathrm{anchor}}\}$ // Remove $v_{\mathrm{anchor}}$ from $\nu_{\mathrm{pool}}$ +7 while $\nu_{\mathrm{pool}}$ is not empty do +8 $\begin{array}{rlr}{v_{\mathrm{next}}\leftarrow Sim(v_{\mathrm{anchor}},\nu_{\mathrm{pool}})} & {} & {\mathrm{/ / F i n d t h e c l i p i n p o o l m o s t s i m i l a r t o v _ {a n c h o r}}}\\ {\nu_{\mathrm{ordered}}\leftarrow v_{\mathrm{next}}} & {} & {}\\ {\nu_{\mathrm{pool}}\leftarrow \nu_{\mathrm{pool}}\backslash \{v_{\mathrm{next}}\}} & {} & {}\\ {v_{\mathrm{anchor}}\leftarrow v_{\mathrm{next}};} & {} & {\mathrm{/ / U p d a t e a n c h o r t o t h e n e w l y a d d e d c l i p}} \end{array}$ +9 +10 +11 + +a continuous span $\mathcal{V}_{[i:i + k]}$ , without re-computing similarity each time. We also prepare hallucination questions irrelevant to existing video inputs following [20] with a ratio of $0.01\%$ . + +# B.3 Prompt Templates for Generating QA Pairs + +To generate question-answer pairs based on clip-level captions, we design diverse prompt templates for 8 distinct reasoning tasks. Table 8 provides examples of these templates. Below, we summarize the associated with each QA category: + +- [OP] Object Perception: Detect and identify objects, focusing on recognizing their attributes in real time. +- [AR] Action Recognition: Identify human actions and interactions occurring in the current moment. +- [SA] Spatial Awareness: Understand spatial relationships among objects and events; reason about location, orientation, and distance. +- [SR] Sequential Relationship: Identify the temporal order of events and actions, especially those involving "before" and "after" cues. +- [CR] Causal Reasoning: Analyze cause-and-effect relationships between actions and outcomes. +- [OCR] Optical Character Recognition: Recognize and interpret textual content in scenes (e.g., subtitles, signs, charts). +- [UEH] Unexpected Event Handling: Detect and react to anomalies or sudden changes in the environment. +- [EU] Event Understanding: Summarize and reason about sequences of temporally linked events. + +These diverse prompts ensure broad task coverage and help enhance the model's generalization across different temporal and semantic understanding challenges. + +# C More Implementation Details + +For the main VideoLLMs, we use the following configurations for each model: + +- LLaVA-OV-7B: We apply center cropping with a resolution of $384 \times 384$ and use a $\times 4$ down sampler (bilinear interpolation) with the frame features, resulting in 49 tokens per frame. +- Oryx-1.5-7B: We use the model's default dynamic resolution, ranging from 288 to 480 pixels. With a $\times 4$ down sampler on the frame features (average pooling), the resulting token count per frame varies between 33 and 59. +- Qwen2-VL-7B: The model uses a dynamic resolution between 224 and 448, with $\times 4$ down sampling (average pooling) on the frame features, resulting in 36-64 tokens per frame. + +All models are fine-tuned for one epoch using a learning rate of 2e-5 with a cosine annealing scheduler and AdamW optimizer. The image encoder remains frozen, while the visual projector and the LLM are fully trainable. The maximum length MaxLen of input embeddings is set to 16384 for the round-decayed compression. + +For the activation model, we adopt LLaVA-OV-0.5B as the base model. To improve efficiency, we aggressively pool the frame representations to 16 tokens per frame. During training, only the LoRA adapters, the projector, the score head, and the learnable activation token are trainable. The model is trained for 5 epochs using a fixed learning rate of 2e-5 for the projector, while 2e-4 for the LoRA adapters, score head, and the learnable activation token, with AdamW optimizer. + +Notably, for both the main VideoLLM and the activation model, we sample frames at 1 FPS to better simulate real-world frame rates. For videos longer than 256 seconds, we uniformly sample 256 frames to fit within the maximum input length constraint. Experiments are conducted on NVIDIA-H100/A100 GPUs. During inference, we sample videos at 2 FPS for short video benchmarks like MVBench, PerpecptionTest, and TempCompass, while 1 FPS for multi-turn real-time understanding benchmarks and long video benchmarks including OVO-Bench, Streaming-Bench, MLVU, LongVideoBench, VideoMME, and EgoSchema. + +# D Benchmarks and Metrics + +Multi-turn Real-time Understanding Benchmarks. We evaluate our method on two recently proposed large-scale streaming video benchmarks: OVO-Bench [20] and Streaming-Bench [21]. Both benchmarks are designed to assess streaming video comprehension under long-context, multi-turn settings. Our evaluation primarily focuses on their real-time understanding tasks. OVO-Bench contains 512 videos with an average length of 435 seconds and approximately 1,600 questions. The evaluated tasks include: (1) Spatial Understanding (STU), (2) Object Recognition (OJR), (3) Attribute Recognition (ATR), (4) Action Recognition (ACR), (5) Optical Character Recognition (OCR), and (6) Future Prediction (FPD). Streaming-Bench consists of 500 videos with an average length of 606 seconds and approximately 2,500 questions. It includes the following tasks: (1) Object Perception (OP), (2) Causal Reasoning (CR), (3) Clip Summarization (CS), (4) Attribute Perception (ATP), (5) Event Understanding (EU), (6) Text-Rich Understanding (TR), (7) Prospective Reasoning (PR), (8) Spatial Understanding (SU), (9) Action Perception (ACP), (10) Counting (CT). Both benchmarks are structured as multiple-choice question-answering tasks, and we report the accuracy. + +General Video Understanding Benchmarks. To evaluate general video comprehension ability, we test our models on seven widely used benchmarks. This includes three short-video benchmarks: MVBench [24], Perception Test [26], and TempCompass [77], and four long-video benchmarks: EgoSchema [28], LongVideoBench [29], MLVU [27], and VideoMME [25]. These datasets span a broad range of video durations, from a few minutes to several hours. All are evaluated in a multiple-choice format, and accuracy is reported. + +Online Activation Benchmarks. To assess the proactive capabilities of our framework, we evaluate performance on a subset of ET-Bench [66], including Temporal Video Grounding (TVG), Temporal Action Localization (TAL), Dense Video Captioning (DVC), and Sequential Localization Captioning (SLC). These tasks emphasize a shift from passive to active perception, requiring the model to determine when to respond based on upcoming visual inputs, rather than reacting immediately. For example, the Sequential Localization Captioning (SLC) task requires the model to both determine + +the precise timing of a certain step and output its content. For evaluation metrics, we compute the average F1 score across multiple IoU thresholds (IoU $\in$ {0.1, 0.3, 0.5, 0.7}) for localization-based tasks. For tasks involving text generation, we adopt sentence-level similarity metrics [85] to measure the semantic alignment between model outputs and ground-truth responses, following prior works [66; 13]. Specifically, the all-MiniLM-L6-v2 model in Sentence-Transformers library is used as the embedding model. Notably, in all these tasks, the question is presented at the beginning of the video, and the model must autonomously decide when to respond. Moreover, the results of TVG $_{F1}$ , $\mathrm{TAL}_{F1}$ , are the same for our method with different main Video-LLMs, since they use the same activation model and will not be affected by the generated response. + +# E Broader Impacts + +There are many real-world applications of streaming Video-LLMs, such as patient or elderly health monitoring, autonomous driving, and collaborative robots. However, there could be unintended usages and we advocate responsible usage complying with applicable laws and regulations. + +# F More Related Works + +To address the challenge of long-context understanding in streaming video, several memory and retrieval mechanisms have been proposed. For instance, ReKV [56] introduces a training-free framework that stores and retrieves the Key-Value (KV) caches of processed frames, enabling offline models to answer user queries efficiently by reloading only the most relevant context. Besides, VideoStreaming [57] employs a memory-propagated encoding architecture where a condensed representation of the preceding clip serves as historical context for encoding the next, combined with an adaptive selection of memories for question-answering. Moreover, StreamChat [53] proposes a hierarchical memory system comprising short-term, long-term, and dialogue components to facilitate complex streaming interactions, and also contributes the StreamBench benchmark for evaluating diverse streaming scenarios. While these methods effectively advance long-context retention for reactive question-answering, StreamBridge differs by introducing a round-decayed compression strategy specifically tailored for multi-turn real-time interactions, which efficiently prunes redundant historical tokens while preserving recent context with high fidelity. Moreover, StreamBridge introduces a decoupled, lightweight activation model. This plug-and-play component operates in parallel with the main Video-LLM, enabling continuous proactive responses. These designs, supported by our dedicated Stream-IT dataset, effectively transform general-purpose offline models into versatile and proactive streaming assistants without compromising their core performance. + +# G Limitations + +Although our proposed framework and dataset significantly enhance the streaming capabilities of existing offline Video-LLMs, there are still limitations worth noting. First, while Stream-IT provides large-scale multi-turn, interleaved training data, its construction relies partially on synthetic QA generation and clip concatenation, which, despite careful filtering, may introduce domain shift compared to truly continuous, real-world video streams. Future work could benefit from curating more organically collected long-form streaming videos with naturally evolving events and dialogues. Second, StreamBridge currently focuses on frame-by-frame streaming under relatively low sampling rates (e.g., 1 FPS). Extending the framework to handle denser frame rates or multi-modal streaming inputs (e.g., audio-visual-text) in real-time remains an important direction for future research. + +# H Full Results on OVO-Bench and Streaming-Bench + +
Method# of FramesReal-Time Visual PerceptionBackward TracingForward Active RespondingOverall.
OCRACRATRSTUFPDOJRAVG.EPMASIHLDAVG.RECSSRCRRAVG.Overall AVG.
Human
Human-93.9692.5794.8392.7091.0994.0293.2092.5993.0291.3792.3395.4889.6793.5692.9092.81
Proprietary Models (Offline), Single-Turn Evaluation
Gemini 1.5 pro [23]1 FPS85.9166.9779.3158.4363.3761.9669.3258.5976.3552.6462.5435.5374.2461.6757.1563.00
GPT-4o [22]6469.8064.2271.5551.1270.3059.7864.4657.9175.6848.6660.7527.5873.2159.4053.4059.54
Open-Source Models (Offline), Single-Turn Evaluation
Qwen2-VL-72B [2]6465.7760.5569.8351.6969.3154.3561.9252.5360.8157.5356.9538.8364.0745.0049.3056.27
LLaVA-Video-7B [15]6469.1358.7268.8349.4474.2659.7863.5256.2357.437.5340.434.1069.9560.4254.8252.91
LLaVA-OV-7B [3]6466.4457.8073.2853.3771.2961.9664.0254.2155.4121.5143.7125.6467.0958.7550.5052.74
Qwen2-VL-7B [2]6460.4050.4656.0347.1966.3455.4355.9847.8135.4856.0846.4631.6665.8248.7548.7450.39
InternVL-V2-8B [76]6467.1160.5563.7946.0768.3256.5260.3948.1557.4324.7343.4426.559.1454.1446.6050.15
Open-Source Models (Streaming), Single-Turn Evaluation
Flash-VStream-7B [11]1 FPS24.1629.3628.4533.7125.7428.8028.3739.0637.165.9127.388.0267.2560.0045.0933.61
VideoLLM-Online-8B [10]2 FPS8.0523.8512.0714.0445.5421.2020.7922.2218.8012.1817.73-----
Disponder [13]1 FPS57.7249.5462.0744.9461.3951.6354.5548.4855.414.3036.0618.0537.3648.7534.7241.78
Models under StreamBridge (Offline → Streaming), Multi-Turn Evaluation
Oryx-1.5-7B† [1]1 FPS60.4052.2969.8350.0065.3557.6159.2554.2155.415.4038.3320.6537.5640.0032.7443.44
+Stream-IT1 FPS84.5675.2370.6950.5674.2671.7471.1769.0259.5079.0369.1720.5166.8960.4149.2763.21
LLaVA-OV-7B† [3]1 FPS58.3959.6369.8244.3876.2361.4161.6453.8754.7230.6446.4114.4151.2343.3336.3348.13
+Stream-IT1 FPS74.5077.0670.6954.4973.2769.5769.9366.67614985.4871.2117.8366.0661.6748.5263.22
Qwen2-VL-7B† [2]1 FPS65.1064.2264.6646.6374.2665.2263.3555.5660.1462.9059.5322.1461.1249.5844.2855.72
+Stream-IT1 FPS84.5671.5674.1449.4475.2572.8371.3067.6857.4379.0368.0519.1764.2561.6748.3662.57
+ +Table 10: Full results on OVO-Bench. ${}^{ \dagger }$ means models under StreamBridge framework + +
Method# of FramesReal-Time Visual UnderstandingOmni-Source UnderstandingContextual UnderstandingOverall.
OPCRCSATPEUTRPRSUACPCTAVG.ERSCUSDMAAVG.ACUMCUSQAPOAVG.OverallAVG.
Human
Human-89.4792.0093.6091.4795.6592.5288.0088.7589.7491.3091.4688.0088.2493.6090.2790.2688.8090.4095.0010093.5591.66
Proprietary Models (Offline), Single-Turn Evaluation
Gemini 1.5 pro [23]1 FPS79.0280.4783.5479.6780.0084.7477.7864.2371.9548.7075.6946.8039.6074.9080.0060.2251.4140.7354.8045.1048.7367.07
GPT-4o [22]6477.1180.4783.9176.4770.1983.8066.6762.1969.1249.2273.2841.2037.2043.6056.0044.5041.2038.4032.8056.8638.7060.15
Open-Source Models (Offline), Single-Turn Evaluation
LLaVA-OV-7B [3]3280.3874.2276.0380.7272.6771.6567.5965.4565.7245.0871.1240.8037.2033.6044.8038.4035.6036.0027.2729.5532.7456.36
Qwen2-VL-7B [2]0.2-1 FPS75.2082.8173.1977.4568.3271.0372.2261.1961.4746.1169.0441.2022.0032.8043.6034.9031.2026.0039.6022.7331.6654.14
InternVL-V2-8B [76]1668.1260.9469.4077.1267.7062.9359.2653.2554.9656.4863.7237.6026.4037.2042.0035.8032.0031.2032.3240.9132.4251.40
Open-Source Models (Streaming), Single-Turn Evaluation
Flash-VStream-7B [11]1 FPS25.8943.5724.9123.8727.3313.0818.5225.2023.8748.7023.2325.9124.9025.6028.4026.0024.8025.2026.801.9624.1224.04
VideoLLM-Online-8B [10]2 FPS39.0740.0634.4931.0545.9632.4031.4834.1642.4927.8935.9931.2026.5124.1032.0028.4524.1929.2030.803.9226.5532.48
Dispider [13]1 FPS74.9275.5374.1073.0874.4459.9276.1462.9162.1645.8067.6335.4625.2638.5743.3435.6639.6227.6534.8025.3433.6153.12
Models under StreamBridge (Offline→Streaming), Multi-Turn Evaluation
Oryx-1-5T-B1 [1] + Stream-IT1 FPS78.4777.1783.8680.2071.0766.9879.6361.3866.2940.9370.5930.0015.2033.6043.2030.5020.4024.8039.6054.9034.9353.76
1 FPS82.2977.9587.9886.4777.9981.3176.8569.9271.9635.2374.7919.2014.4052.0029.2028.7014.4014.8051.2043.1430.8954.79
LLaVA-OV-7B1 [3] + Stream-IT1 FPS76.8477.1782.6075.2564.1564.1775.0061.3861.1946.1168.3924.4012.0032.4037.6026.6020.0019.6034.4052.9431.7450.96
1 FPS82.2972.4492.0980.8671.0774.4675.0062.2070.2628.5070.9220.8013.2043.6027.6026.3020.4017.6041.6037.2629.2151.73
Qwen2-VL-7B1 [2] + Stream-IT1 FPS80.3878.7483.2279.8674.2169.4777.7863.4169.9743.0172.0132.0015.2039.6038.4031.3025.2021.2033.2066.6736.5755.09
1 FPS84.7482.6888.9289.7777.3685.3684.2669.9271.6735.7577.0418.0013.2043.6021.6024.1014.0017.2048.0050.9832.5555.39
+ +Table 11: Full results on Streaming-Bench. ${}^{ \dagger }$ means models under StreamBridge framework + +# I Pseudo Code of the Round-Decayed Compression in a PyTorch-like Style + +```txt +def Round_Decayed_Compression(inputss_embeds,max_len,token_per_frame): + inputs_embeds:[1,seq_len,dim],interleaved embeddings of video and text; + max_len: the predefined maximum sequence length of inputs_embeds; + token_per_frame: the number of tokens per frame; + return torch.cat(new Inputs_embeds, dim=1) +``` \ No newline at end of file diff --git a/streambridgeturningyourofflinevideolargelanguagemodelintoaproactivestreamingassistant/images.zip b/streambridgeturningyourofflinevideolargelanguagemodelintoaproactivestreamingassistant/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..fef8d3b2a481d4c82ca460cf44b385ee3b53eb96 --- /dev/null +++ b/streambridgeturningyourofflinevideolargelanguagemodelintoaproactivestreamingassistant/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c307ddf6e86494f4e723f223a64ce8722b25111eb260d60a28fcfc943b90f50a +size 882908 diff --git a/streambridgeturningyourofflinevideolargelanguagemodelintoaproactivestreamingassistant/layout.json b/streambridgeturningyourofflinevideolargelanguagemodelintoaproactivestreamingassistant/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..58727f305608612fb1a82d29b888a200ba998a19 --- /dev/null +++ b/streambridgeturningyourofflinevideolargelanguagemodelintoaproactivestreamingassistant/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:96aa9d2f405abfc085f52f391fbd3bf1e0e8835b1580464a73470cdeab57ede3 +size 909023 diff --git a/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/e0aab137-ef91-41d9-b54b-4bb529c8a75a_content_list.json b/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/e0aab137-ef91-41d9-b54b-4bb529c8a75a_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..bad110c6557395c20476b8e90c6c270448b7ac45 --- /dev/null +++ b/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/e0aab137-ef91-41d9-b54b-4bb529c8a75a_content_list.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:82054e456341c3822d611fc43aebcd591592a2e8d76de1c4ad9fce5291a02f4e +size 161845 diff --git a/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/e0aab137-ef91-41d9-b54b-4bb529c8a75a_model.json b/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/e0aab137-ef91-41d9-b54b-4bb529c8a75a_model.json new file mode 100644 index 0000000000000000000000000000000000000000..7921e4e8c6f3b3d9136720abc3eed7dd3069d2da --- /dev/null +++ b/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/e0aab137-ef91-41d9-b54b-4bb529c8a75a_model.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3aeced08938f159365baa8b9173dd7fd572a07a847baad08e48407e7e76dfaf6 +size 211908 diff --git a/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/e0aab137-ef91-41d9-b54b-4bb529c8a75a_origin.pdf b/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/e0aab137-ef91-41d9-b54b-4bb529c8a75a_origin.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6c011683787a534c67aa6c25fdf12bb0dbc1f276 --- /dev/null +++ b/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/e0aab137-ef91-41d9-b54b-4bb529c8a75a_origin.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4c1f2a016c3157d2e7e4e788533d142ab3c05182c02d69e6223ed1a30db92580 +size 1315740 diff --git a/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/full.md b/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/full.md new file mode 100644 index 0000000000000000000000000000000000000000..4cfcd8a4c3703fe1b491dd3bf64b4d73d129ac05 --- /dev/null +++ b/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/full.md @@ -0,0 +1,735 @@ +# StreamFlow: Streaming Audio Generation from Discrete Tokens via Streaming Flow Matching + +Ha-Yeong Choi + +Sang-Hoon Lee $^{2,3*}$ + +1Gen AI Lab, KT Corp., Seoul, Korea + +$^{2}$ Department of Software and Computer Engineering, Ajou University, Suwon, Korea + +3Department of Artificial Intelligence, Ajou University, Suwon, Korea + +# Abstract + +Diffusion models have demonstrated remarkable generative capabilities, and Conditional Flow Matching (CFM) has improved their inference efficiency by following optimal transport paths. However, CFM-based models still require multiple iterative sampling steps, which makes them unsuitable for real-time or streaming generation scenarios. In this paper, we introduce StreamFlow, a novel streaming generative model designed for real-time audio generation from discrete tokens. StreamFlow leverages a causal noisig training framework along the time axis and predicts multi-time vector fields at once on each stream, enabling streaming inference with minimal latency. To further improve generalization, we propose Scale-DiT, a Diffusion Transformer architecture that enhances robustness by modeling, normalizing, and scaling feature differences prior to skip connections. This significantly improves the robustness and performance of DiT without increasing the parameter size. We validate the effectiveness of StreamFlow through audio reconstruction tasks using discrete tokens from EnCodec and Mimi, demonstrating both high-fidelity synthesis and streaming capability. Furthermore, we successfully incorporated our model into fully-duplex streaming speech language models of Moshi by replacing the Mimi decoder. Audio samples are available at https://streamflow25.github.io/demo/. + +# 1 Introduction + +Recently, conditional flow matching (CFM) models [31] have demonstrated powerful generative capabilities across modalities such as text, image, and audio. Initially, CFM was introduced for image-to-image translation and text-to-image generation by conditioning on target labels or text representation such as those from CLIP [42]. Since then, CFM has been extended to a wide range of audio applications, including text-to-speech [24, 10, 4, 8], text-to-audio [41, 26], voice conversion [56, 5, 62], and waveform generation [30, 32, 50]. Additionally, flow matching has been successfully adapted to discrete domains, such as text generation, through discrete flow matching [11]. Most existing CFM approaches focus on parallel generation tasks, such as fixed-size image generation and fixed-length audio generation in a non-autoregressive manner. Although masking and infilling training frameworks with CFM objectives and Transformers [24, 10, 4] can endow the model with in-context learning over long sequences, these models still require numerous sampling steps over a fixed target space, limiting their applicability to real-time streaming generation tasks. To date, streaming + +![](images/a74f92342e60c4537201bb16371f01e4ece9456b73ce38bb1931900086df177a.jpg) +Figure 1: Comparison of generation strategies: (a) Non-Streaming Generation, (b) Chunk-wise Parallel Generation, (c) Token-wise Streaming Generation + +generation in diffusion models has largely been explored in video domain. From an engineering perspective, StreamDiffusion [19] introduces the batch denoising method that computes the diffusion process with a batch. Rolling Diffusion [43] introduces a novel DDPM [14] that progressively denoises data over time. + +However, to the best of our knowledge, streaming audio generation from discrete tokens has not yet been explored within the frameworks of diffusion or flow matching. While recent parallel generation approaches based on CFM [30] have achieved promising results, they typically operate on a chunk-wise generation, as shown in Figure 1-(b), relying on pre-trained non-autoregressive models. Moreover, generating raw waveform at a high sampling rate (e.g., $24\mathrm{kHz}$ ) introduces additional challenges, as it requires significantly finer resolution compared to lower-resolution acoustic features (e.g., 12.5 and $75\mathrm{Hz}$ ), thus increasing both modeling complexity and computational cost. + +Additionally, recent speech language models are getting more attention for end-to-end human-like communication such as GPT-4o [15]. They utilize neural audio CODEs with causal layers and GAN-based objective for high-quality waveform generation [7]. However, these models often suffer from perceptual degradation when generating high-resolution waveforms directly from highly compressed, low-bitrate discrete tokens. This is primarily due to the absence of intermediate temporal modeling and progressive refinement, which are essential for maintaining local coherence and preserving fine-grained audio details over time, especially in streaming scenarios where real-time and frame-level consistency are crucial. + +In this paper, we introduce StreamFlow, a novel real-time streaming flow matching model for streaming audio generation. StreamFlow simultaneously estimates multi-time vector fields at each step, thereby offering both streaming generation and refinement capabilities. To facilitate long-range temporal modeling, we adopt diffusion transformers with in-context learning capabilities. In addition, we introduce Scale-DiT, a new diffusion transformer architecture designed to improve feature regularization. Scale-DiT computes the difference between residuals and features, normalizes this difference, and applies a scaling operation before the skip connection. This design improves robustness without increasing model size. We evaluate the effectiveness of StreamFlow on real-time audio reconstruction tasks using discrete tokens from EnCodec [9] and Mimi [7]. Furthermore, we verify that StreamFlow can be seamlessly integrated into existing speech generation pipelines that rely on discrete neural audio CODECs. The main contributions of this work are as follows: + +- We propose StreamFlow, a novel streaming generative model that leverages self-conditioned context to estimate multi-time vector fields, enabling token-wise streaming generation. +- We introduce a new DiT structure, Scale-DiT that regularizes the feature space by modeling the difference between residual and features, normalizing the difference, and scaling it for elucidating the features. +- We demonstrate the effectiveness of the proposed Streaming Flow Matching framework for streaming audio generation by reconstructing high-quality waveforms in real time from discrete tokens produced by neural audio CODECs such as EnCodec and Mimi. + +# 2 Related Work + +# 2.1 Waveform Generation + +Conventionally, neural waveform generation tasks have been investigated for text-to-speech systems requiring conversion of acoustic features such as Mel-spectrogram to raw waveform signal, called + +![](images/f084d8df79446a47fbb3b0b9f41498920d0cd042b483733c4c8fe7f0b0438c4b.jpg) +(a) CFM + +![](images/741261868c374ec7e7abb298e31c1de8cc95ec049b0fb01cbb0c4c63472ee992.jpg) +(b) SFM (ours) +Figure 2: Comparison of (a) conditional flow matching (CFM), which performs full-sequence vector field estimation across all timesteps, and (b) streaming flow matching (SFM), our method that leverages self-generated context for multi-time vector field estimation and enables low-latency tokenwise streaming. + +by Neural Vocoder. WaveNet [38] introduced causal dilated convolutional networks to increase the receptive field for high-resolution waveform signal modeling. Due to the slow sampling speed of auto-regressive generation, many works [37, 40, 18] have shifted their focus to parallel waveform generation models for efficient and fast waveform generation. GAN-based parallel models have shown promising parallel generation performance by generating realistic waveform signal. PWG [54] and MelGAN [22] first adopted adversarial training, and HiFi-GAN [20] introduced a novel discriminator, multi-period discriminator (MPD) reflecting implicit periodic features. BigVGAN [25] scaled up the neural vocoder for out-of-distribution robustness by introducing snake activation and an anti-aliasing filter. Then, Vocos [46] boosted the sampling speed by incorporating iSTFT on low-resolution feature space. + +# 2.2 CFM in Waveform Generation + +Recently, CFM has been adopted for high-fidelity waveform generation. Although previous diffusion-based models [21, 2] showed slow inference speed and lower performance than GAN-based models, PeriodWave [30], RFWave [32], and FlowDec [50] demonstrated much higher performance on the waveform generation tasks compared to GAN models. Although CFM still requires more sampling steps compared to GAN, they have shown promising results on the waveform generation tasks. Furthermore, PeriodWave-Turbo [28] accelerated CFM-based models by incorporating adversarial feedback, achieving improved performance with smaller sampling steps. However, as shown in Figure 2-(a), streaming generation within the CFM has not been explored. + +# 2.3 Neural Audio Codec + +Meanwhile, neural audio codec has garnered more attention than neural vocoder in that it facilitates various practical applications combined with large language models. SoundStream [59] and EnCodec [9] introduced a practical neural audio codec by incorporating residual vector quantization (RVQ) [12, 49] and adversarial training. DAC [23] improved RVQ with carefully designed bottleneck and various losses. HiFi-Codec [55] proposed group-residual vector quantization to increase the capacity of RVQ. SpeechTokenizer [61] and X-Codec [57] distilled the self-supervised speech representation on the quantized representation to increase semantic information without any labeled data. Mimi [7] also introduced a high-compressed low-bitrate audio codec with SSL distillation for an efficient speech language model. Furthermore, single VQ models including WavTokenizer [16], BigCodec [53], and TS3-Codec [51] have been investigated for efficient modeling. Recently, finite scalar quantization (FSQ)-based models [1, 39] have been adopted to improve the performance of low-bitrate neural audio codec. CosyVoice 2 [8] introduces supervised semantic tokens and flow matching with causal layer and lookahead tokens. + +![](images/4739faccea3362e3f9ce7889519cf28d3630ad423a0c2cb2b9d6786f279d5f1d.jpg) +(a) Architecture of StreamFlow + +![](images/4e97bf4749821c401d10f8c9ba6d96f0c2865997f115e69bbc87e93939281dc0.jpg) +(b) Prompt with Causal Noising + +![](images/579e3e45b046d4a3a6c697f66026240560c770f76b4d20dcc2e5c4b39a0a8e5c.jpg) +(c) Fine-tuning with Adversarial Training +Figure 3: The Overall framework of StreamFlow. + +# 2.4 Full-Duplex Real-time Speech Language Models + +Recent advances in speech foundation models [7, 3] have introduced full-duplex systems capable of simultaneous speech input and output in real-time. Achieving this requires both low-latency streaming architectures and highly efficient low-bitrate neural CODEs. A common design pattern involves causal convolution-based decoders for real-time waveform generation. However, such causal models generally underperform compared to their parallel counterparts in terms of perceptual quality. Moreover, the reliance on low-bitrate CODEs, while necessary for efficient streaming, often introduces noticeable degradation in audio quality and fidelity. To address these limitations, we propose SFM, a method that enables high-fidelity, low-latency waveform generation—bridging the gap between efficiency and perceptual quality in streaming audio generation, as illustrated in Figure 2-(b). + +# 3 StreamFlow + +This section introduces the StreamFlow framework for high-quality real-time audio streaming generation. We begin with the fundamentals of Flow Matching (Section 3.1) and extend it to an efficient Streaming Flow Matching method (Section 3.2). We then describe the data stream, causal noisig, training objective, and streaming generation process. Next, we present Scale-DiT for improved contextual learning and generalization (Section 3.3), and explore waveform transformation for efficient streaming generation (Section 3.4). Finally, we discuss adversarial training for enhanced robustness and efficiency (Section 3.5). + +# 3.1 Preliminary: Flow Matching + +Flow Matching (FM) [31, 33, 48] is a generative modeling method designed to transform a simple prior distribution $p_0(x)$ (e.g., $x_0 \sim \mathcal{N}(0, I)$ ) into a target data distribution $p_1(x) \approx q(x)$ by learning a time-dependent velocity field $v_t(x)$ . This transformation is governed by the following ordinary differential equation (ODE): + +$$ +\frac {d \phi_ {t} (x)}{d t} = v _ {t} \left(\phi_ {t} (x); \theta\right), \quad \phi_ {0} (x) = x, x \sim p _ {0}, \tag {1} +$$ + +where $\phi_t(x)$ represents the state of the system at time $t$ as it evolves under the velocity field $v_t$ . The FM objective aims to align the learned velocity field $v_t(x; \theta)$ with an ideal vector field $u_t(x)$ that + +generates the desired probability path $p_t(x)$ . This alignment is achieved by minimizing the following training objective: + +$$ +L _ {\mathrm {F M}} (\theta) = \mathbb {E} _ {t \sim [ 0, 1 ], x \sim p _ {t} (x)} \| v _ {t} (x; \theta) - u _ {t} (x) \| ^ {2}. \tag {2} +$$ + +To define the probability path $p_t(x)$ , a common approach is to employ linear interpolation between the prior sample $x_0$ and the target data point $x_1$ : + +$$ +x _ {t} = (1 - t) x _ {0} + t x _ {1}. \tag {3} +$$ + +The corresponding ideal vector field $u_{t}(x_{t} \mid x_{1})$ that drives the transformation along this path is given by: + +$$ +u _ {t} \left(x _ {t} \mid x _ {1}\right) = x _ {1} - x _ {0}. \tag {4} +$$ + +This method effectively learns a smooth and computationally efficient mapping from $p_0$ to $p_1$ . + +# 3.2 Streaming Flow Matching + +We introduce Streaming Flow Matching (SFM), a novel approach designed for real-time streaming inference by simultaneously estimating multi-time vector fields. Unlike traditional FM, which processes entire sequences simultaneously, SFM utilizes dynamically structured streaming sequences and jointly optimizes local and global vector fields. Notably, by estimating multiple vector fields across multiple time axes, SFM provides not only the global consistency inherited from classical FM but also satisfies real-time constraints in streaming scenarios. + +Data Streams SFM partitions the input sequence $\mathbf{x}$ into multiple data streams, each characterized by a streaming window size $L_{\mathrm{s}}$ and a context prompt ratio $\gamma$ . The total length is thus $L = L_{\mathrm{c}} + L_{\mathrm{s}}$ where $L_{\mathrm{c}} = \gamma \cdot L_{\mathrm{s}}$ . Specifically, each data stream $\{\mathbf{x}_1^{(k)}, \dots, \mathbf{x}_L^{(k)}\}$ simultaneously provides contextual information (i.e., the context prompt) and performs streaming inference (i.e., the streaming window). To achieve this, SFM defines two separate timestep sequences. The first sequences correspond to the past context segment, which remains fixed at $t = 1$ . The second sequences handle the streaming segment, where $t$ decreases linearly as $t = 1 - \lambda, \lambda \in [0,1]$ along to the $L_{s}$ . + +Causal Noising To ensure causality, the SFM introduces the causal noisng technique, selectively masking the streaming window while retaining context prompt for in-context learning. This approach imposes temporal constraints, preventing information leakage from future data. For each stream $k$ , let the initial state $\mathbf{x}_0^{(k)}$ follow a Gaussian distribution $\mathcal{N}(0,I)$ , and let $\mathbf{x}_1^{(k)}$ be the target data point. Then, the following interpolation path is defined for $t \in [0,1]$ : + +$$ +\mathbf {x} _ {t} ^ {(k)} = (1 - t) \mathbf {x} _ {0} ^ {(k)} + t \mathbf {x} _ {1} ^ {(k)}. \tag {5} +$$ + +Within this path, the streaming window is monotonically reduced from $t = 1$ to 0, effectively reducing the range of actively learned stream. The context prompt, however, remains fixed throughout training, ensuring that the model retains past context information while progressively refining the newly generated output. In a streaming scenario, the context prompt is dynamically updated to integrate newly generated frames while preserving historical context. Meanwhile, the model iteratively refines its predictions through causal noising, wherein structured noise is introduced at each step and progressively reduced over time. Furthermore, newly generated frames are incorporated via a causal shift operation, ensuring a smooth transition between past and present content. As a result, this process not only preserves temporal coherence and stabilizes streaming generation but also prevents information leakage and enhances robustness. + +Training Objective The training objective of SFM is to predict the ideal velocity field $\mathbf{u}_t^{(k)}$ . This velocity field is derived via the temporal derivative of the interpolated path $\mathbf{x}_t^{(k)}$ , which simplifies to + +$$ +\mathbf {u} _ {t} ^ {(k)} = \frac {d \mathbf {x} _ {t} ^ {(k)}}{d t} = \mathbf {x} _ {1} ^ {(k)} - \mathbf {x} _ {0} ^ {(k)}. \tag {6} +$$ + +We define the following loss function to minimize the difference between the predicted $\mathbf{v}_t^{(k)}(\theta)$ and the ideal $\mathbf{u}_t^{(k)}$ : + +$$ +L _ {\mathrm {S F M}} (\theta) = \sum_ {k = 1} ^ {K} \mathbb {E} _ {t, \mathbf {x} _ {0} ^ {(k)}, \mathbf {x} _ {1} ^ {(k)}} +$$ + +$$ +\left[ \left\| \mathbf {v} _ {t} ^ {(k)} \left(\tilde {\mathbf {x}} _ {t} ^ {(k)}; \mathbf {c} ^ {(k - 1)}; \theta\right) - \left(\mathbf {x} _ {1} ^ {(k)} - \mathbf {x} _ {0} ^ {(k)}\right) \right\| ^ {2} \right], \tag {7} +$$ + +where $\tilde{\mathbf{x}}_t^{(k)}$ is the causally noised interpolation state, $\mathbf{c}^{(k - 1)}$ represents context prompt providing temporal consistency and in-context streaming generation ability. + +Streaming Generation Thanks to in-context learning ability of Transformer, the model can learn the long-term dependency during streaming generation. To emerge this ability, we train the model by applying causal noisig to the data stream, and estimating the multi-time vector fields at once according to the level of noising. Following [24], we also utilize condition drop for prompt and tokens during training. During inference, we prepend the generated samples at the front of the sequence as a prompt to in-context learn from the generated data. Furthermore, $\mathbf{x}_t$ can be refined by reflecting the next token stream during generation. + +# 3.3 Scale-DiT + +Inspired by [34], we introduce Scale-DiT, a novel architecture designed to enhance training stability and improve model performance. We aim to preserve the scalability of existing DiT while addressing limitations in optimization and feature representation. This approach allows the model to regulate feature more effectively during training, leading to more stable optimization within Multi-Head Self-Attention (MHSA) and Feed-Forward Network (FFN) layers. Specifically, Adaptive Layer Normalization (AdaLN) is integrated with a dynamic scaling mechanism to regularize the feature space effectively. Given an input sequence $x \in \mathbb{R}$ , we first apply AdaLN to obtain a normalized representation $\hat{x}$ along with trainable scaling and shifting parameters $\gamma_1, \beta_1, \gamma_2, \beta_2$ , and gating factors $\alpha_1, \alpha_2$ . The MHSA output is computed as: + +$$ +\hat {x} = \sigma (x) \cdot \left(1 + \gamma_ {1}\right) + \beta_ {1}, \tag {8} +$$ + +where $\sigma (\cdot)$ denotes the layer normalization (LN) function. + +$$ +\mathbf {A} = \alpha_ {1} \cdot \operatorname {M H S A} (\hat {x}). \tag {9} +$$ + +Unlike vanilla DiT residual connections, we introduce an learnable adaptive scaling rate $\rho_{1}$ , which is restricted between $10^{-4}$ and 1 before applying it for robust training. This adaptive rate modulates the residual update with the difference: + +$$ +x \leftarrow x + \rho_ {1} \cdot \sigma (\mathbf {A} - x), \tag {10} +$$ + +where we utilize additional LN, ensuring controlled information flow and stable optimization for the difference. Following the MHSA computation, we apply an AdaLN as: + +$$ +\hat {x} = \sigma (x) \cdot \left(1 + \gamma_ {2}\right) + \beta_ {2} \tag {11} +$$ + +before passing it through the FFN, which is further scaled by $\alpha_{2}$ , leading to + +$$ +\mathbf {F} = \alpha_ {2} \cdot \operatorname {F F N} (\hat {x}). \tag {12} +$$ + +![](images/ff58ec738103692e65194168e58e260c3fb56cbd6f83d1d1c89a6c536bb667b2.jpg) +Figure 4: Scale-DiT architecture of StreamFlow. + +To maintain training stability and prevent extreme gradient updates, + +we introduce a separate adaptive scaling rate $\rho_{2}$ for the FFN layer. Finally, the residual connection is updated as: + +$$ +x \leftarrow x + \rho_ {2} \cdot \sigma (\mathbf {F} - x). \tag {13} +$$ + +By scale layer normalization, Scale-DiT improves training stability, optimizes representation learning efficiency, and enhances overall optimization dynamics. + +Table 1: Objective evaluation results from EnCodec tokens using universal speech test samples. RFWave uses CFG2. + +
ModelStreamingParams.M-STFT ↓PESQ ↑Period ↓V/UV ↑UTMOS ↑MOS ↑
GT------3.4234.07±0.02
Vocos [46]X7M1.0743.0510.0860.9573.1003.98±0.02
MBD [13]X411M1.6122.6450.1080.9463.3003.95±0.03
RFWave [32]X18M1.2803.0200.0780.9572.9883.99±0.02
StreamFlowX170M0.9973.4730.0800.9573.4504.03±0.02
Encodec [9]✓15M1.1702.6430.1120.9412.5423.74±0.03
StreamFlow-Tiny✓11M1.1113.0270.1070.9473.0603.99±0.02
StreamFlow-Small✓44M1.0723.2070.0960.9503.2063.99±0.03
StreamFlow-Base✓175M1.0613.3350.1020.9483.3254.03±0.02
+ +# 3.4 Waveform Transformation + +STFT/iSTFT Conventional waveform generation models used high-resolution features with transposed convolutional layer. However, increasing the resolution of features significantly increases the inference speed. Vocos [46] introduced iSTFT head to directly transform the feature into waveform signals on the low-resolution features. Unlike one-step generation models, we utilize multi-step generation frameworks using CFM so we first extract the STFT-based features of an input $\mathbf{x}_t$ and prompt and iSTFT-based vector field estimation. However, we found that STFT and iSTFT require additional computation cost for each sampling step, and larger receptive field (window size) than the quantization size of the token (hop size) so it is not proper for the streaming generation system because it is essential to use future tokens for robust waveform generation. + +Linear-Reshape Transformation To reduce this problem, we adopt a linear reshape transformation proposed in WaveNeXt [36]. We first reshape the 1d waveform-level input $\mathbf{x}_t$ and prompt of length $T$ into 2d data of height $T / h$ and width $h$ , where $h$ denotes hop size like STFT. After reshaping the data, we apply the linear projection to map it into the feature space for Scale-DiT. For vector field prediction, we also utilize linear projection to the data space, and reshape 2d data into 1d original reshape for loss calculation. It is worth noting that it does not require additional future frames and reshape function during sampling steps unlike STFT-based models, resulting in better efficient and faster streaming generation. Specifically, we reshape the waveform signal by $h$ of 160. + +# 3.5 Fine-tuning with Adversarial Training + +For high-fidelity waveform generation, many works utilize adversarial training by introducing well-designed discriminators. Although leveraging multi-scale of various discriminator could improve the performance of waveform generation models, it requires too much time to train the models and it is difficult to optimize their losses together. To reduce the entire training time, we first train the StreamFlow using the proposed SFM objective, and fine-tune the pre-trained StreamFlow with adversarial training following [28]. For adversarial training, we utilized the multi-period discriminator (MPD) of HiFi-GAN [20] to reflect the different periodic features, multi-scale short time Fourier transform discriminator (MS-STFTD) of [9] to reflect the magnitude and phase from different STFTs, and multi-scale sub-band constant-Q transform (MS-SB-CQTD) of [13] to reflect the detailed frequency features. Furthermore, we also utilize multi-scale STFT losses of BigVGAN-v2 together. + +# 4 Experiment and Result + +# 4.1 Experimental Setup + +Dataset We utilized LibriTTS [60] to train all models including the ablation study. LibriTTS is a high-quality speech dataset with a sampling rate of $24,000\mathrm{Hz}$ . We used EnCodec and Mimi + +Table 2: Ablation study on Streaming StreamFlow using the LibriTTS-dev subset. + +
ModelM-STFT ↓PESQ ↑Period ↓V/UV ↑Pitch ↓UTMOS ↑
Streaming (online)
StreamFlow1.0883.4300.0880.95526.8433.792
w/o In-Context Learning1.0993.3370.0880.95426.7723.748
w/o Adversarial Fine-tuning1.3882.6690.1010.95021.0993.178
w/o SFM Pre-training1.9191.1530.3530.812574.551.308
w/o Scale-DiT1.5932.5570.0990.94627.3303.033
w/o RoPE2.0491.5980.1760.89868.8911.904
Non-streaming (offline)
StreamFlow1.0173.4990.0850.95529.0873.888
w/o iSTFT1.0973.1390.0820.95622.6363.619
+ +as speech tokenizers to compare the streaming reconstruction performance because they consist of causal convolutional layers for encoding and decoding the waveform signal. We utilize eight RVQ of each model to train the model. We first validate the models using LibriTTS-dev clean and test subsets, and then we evaluate the performance of each model using universal speech datasets consisting of 300 samples from various datasets including Expresso, HiFiTTS, LibriTTS, Aishell3, JVS, and CML-TTS following RFWave [32]. + +Training For streaming models, we pre-train StreamFlow models with a learning rate of $2 \times 10^{-4}$ , batch size of 512 for 1M steps on four NVIDIA A6000 GPUs. We utilize sliced window training by randomly segmenting the waveform signal by 10,240 frames (32 tokens of EnCodec). Then, we fine-tune StreamFlow with a learning rate of $2 \times 10^{-5}$ , batch size of 64 for 0.25M steps on four NVIDIA A6000 GPUs. We utilize sliced window training by randomly segmenting the waveform signal by 20,480 frames (64 tokens of EnCodec). The architecture details are described in Appendix A. The parallel models are described in Appendix B. + +Sampling We utilize the Euler method as the ODE method. We compared the sampling steps for pre-trained models in Table 9. For streaming models, we fine-tuned the model with the fixed-step generator by four steps of parallel models and eight steps of streaming models. For Mimi reconstruction, we fixed two steps for minimal latency. + +# 4.2 EnCodec Token Reconstruction + +We compared the performance of EnCodec token reconstruction with the strong parallel baselines including Vocos, Multi-Band Diffusion (MBD), and RFWave. Furthermore, we compare with the streaming baselines including causal EnCodec with the same latency to obtain robust streaming generation. Table 1 demonstrated the effectiveness of our methods in that streaming models still outperformed the powerful parallel models including MBD and RFWave. Furthermore, StreamFlow-T also shows better performance in terms of PESQ. This indicated that our new streaming generation frameworks could improve the streaming generation performance significantly. Table 1 shows that our models have much better performance in terms of MOS compared to previous streaming methods, and also better performance compared to parallel models. + +# 4.3 Ablation Study + +In-Context Learning When we preprocess the generated samples as prompts, the models could learn useful information from them. Table 2 also showed much better performance than the model without prompt even with the same hyperparameter. + +Adversarial Fine-tuning The results demonstrated the efficiency and effectiveness of two-stage training using SFM and adversarial training. Although we only fine-tuned the models with small training steps of 0.25M, the performance improved significantly. + +Table 3: Scalability with respect to model size + +
ModelParams.Input Dim.HiddenHeadM-STFT ↓PESQ ↑Period ↓V/UV ↑Pitch ↓UTMOS ↑
StreamFlow-Tiny11M256102441.1263.1830.0970.95127.8463.550
StreamFlow-Small44M512204881.0993.3070.0890.95527.7403.690
StreamFlow-Base175M10244096161.0883.4300.0880.95526.8433.792
+ +Table 4: Additional comparison between DiT and Scale-DiT before adversarial fine-tuning. We use 16 steps of Euler method for sampling. Objective evaluation results from Mimi tokens using LibriTTS-dev subsets. + +
ModelTraining StepsWER ↓STOI ↑PESQ ↑SPK-SIM ↑UTMOS ↑
DiT150k11.760.851.710.582.72
DiT + REPA150k8.760.861.740.592.85
Scale-DiT150k9.680.861.780.592.83
Scale-DiT + REPA150k8.340.871.780.602.92
DiT300k9.410.871.840.623.04
DiT + REPA300k8.170.871.840.643.10
Scale-DiT300k7.560.871.900.643.16
Scale-DiT + REPA300k7.180.881.920.653.26
DiT700k9.590.871.920.643.20
DiT + REPA700k6.880.881.920.673.28
Scale-DiT700k6.230.882.070.683.45
Scale-DiT + REPA700k5.990.892.090.683.52
+ +SFM Pre-training Without SFM pre-training, the model with adversarial training could not emerge streaming generation ability with ODE sampling at the early steps, resulting in discriminator collapse and slow training speed. + +Scale-DiT We found that training the vanilla DiT model with a linear-reshape transformation using a high-resolution waveform signal as a target is unstable. Compared to the vanilla DiT, Table 2 indicated that Scale-DiT could improve the performance. We also observed that the learning process exhibits increased stability with reduced fluctuations during training. + +RoPE For streaming generation, it is essential to use positional embeddings. The model without RoPE could not generate the waveform signal well due to a lack of ability to learn the positional information of high-resolution waveform signal. + +Linear-Reshape Transformation Table 2 confirmed that STFT/iSTFT-based models are better than Linear-Reshape transformation for parallel generation models so we used STFT/iSTFT-based methods for parallel models. However, as we described in section 3.4, STFT/iSTFT is not proper for streaming generation because it requires a larger receptive field for their transformation so we used Linear-Reshape transformation for streaming methods. + +Scalability We train three different models as described in Table 3. The results demonstrate consistent scalability with respect to model size across all evaluation metrics. + +# 4.4 Further Analysis of Scale-DiT + +As a further analysis of Scale-DiT, we first compare it with the vanilla DiT baseline to investigate its effectiveness and stability. As shown in Table 4, Scale-DiT consistently achieves better performance across all evaluation metrics, demonstrating enhanced stability and stronger capability for hierarchical feature modeling. These improvements suggest that the proposed scaling mechanism effectively stabilizes training and optimizes feature representations compared to the vanilla DiT. Building upon these results, we conducted additional experiments to investigate whether Scale-DiT can be further enhanced through feature regularization. Specifically, we incorporated representation alignment for generation (REPA) [58] into the architecture to examine its extensibility and robustness. In this setting, the 7th-layer representation from Wav2Vec 2.0 was used as the target semantic embedding, replacing the DINOv2 features employed in the original REPA, and a cosine similarity based REPA loss was applied at the 6th block of our 12-block model. When combined with REPA (Scale-DiT + REPA), the model achieved additional gains indicating that Scale-DiT provides a solid and stable foundation for structured feature learning, while REPA serves as an effective auxiliary regularization technique for further refinement. + +Table 5: Objective evaluation results from Mimi tokens using LibriTTS-dev subsets. Note that StreamFlow performs inference in a two-step process. + +
ModelNqBitrateF (Hz)CER ↓WER ↓M-STFT ↓PESQ ↑Period. ↓V/UV ↑Pitch ↓UTMOS ↑
GT---1.123.06-----3.862
Mimi4550507.4212.721.5521.6570.2100.88077.5753.019
StreamFlow4550505.229.451.4101.5840.2120.87673.8913.093
Mimi6825755.109.001.4262.0120.1800.90160.1423.347
StreamFlow6825753.786.871.2722.0430.1770.90655.8303.719
Mimi811001003.056.931.3522.2660.1650.91050.6863.506
StreamFlow811001003.266.161.2172.3060.1620.91545.6403.910
+ +Table 6: Inference details and subjective evaluation results from Mimi tokens using universal speech test samples. Note that StreamFlow performs inference in a two-step process. + +
ModelNqBitrateF (Hz)Params.Latency ↓xRTF ↑Avg. Memory ↓CMOS ↑UTMOS ↑
Mimi45505079M80ms---2.62
StreamFlow455050175M160ms--+0.0582.76
Mimi68257579M80ms---2.91
StreamFlow682575175M160ms--+0.0223.34
Mimi8110010079M80ms3.224502MB-3.06
StreamFlow81100100175M160ms8.3191176MB+0.0853.55
+ +# 4.5 Mimi Token Reconstruction + +We evaluated the Mimi decoder and StreamFlow on Mimi token reconstruction using LibriTTS-dev subsets and universal speech test samples. As detailed in Table 5, StreamFlow outperforms Mimi decoder in terms of WER, M-STFT, PESQ, Pitch, and UTMOS, and has a comparable performance in other metrics. Inference details and subjective scores are summarized in Table 6. Furthermore, StreamFlow directly generated high-resolution waveform signal on the low-resolution features by linear-reshape transformation, resulting in a much faster inference speed compared to the Mimi decoder. This indicates that our model could replace the Mimi decoder for full-duplex real-time speech language models of Moshi. We provide a detailed description of the system's operation in a full-duplex setting in Appendix H. + +# 5 Potential Broader Impact + +Practical Application We introduce a new generative model, streaming flow matching (SFM) for streaming generation, and a new diffusion architecture, Scale-DiT. This can be adopted for sequential data generation including video generation, audio generation, and stock prediction. We successfully leveraged our methods to real-time streaming waveform generation by proposing StreamFlow. This can accelerate the research of text-to-speech [27, 29], text-to-audio, real-time voice conversion [6], and speech language models with high-quality audio generation. + +**Limitation** In this work, we focus real-time generation system using SFM. However, we observed that increasing the sampling steps of SFM can further improve the quality. Therefore, we aim to minimize the trade-off between real-time processing and quality even with more sampling steps. While SFM pre-training significantly reduces the overall training time and emerge new ability for streaming generation, adversarial fine-tuning is still required for high-fidelity waveform generation, resulting in slow training speed due to various discriminators. + +# 6 Conclusion + +We presented a novel steaming generative models, streaming flow matching (SFM) for real-time streaming generation. Additionally, we introduce Scale-DiT, a robust diffusion Transformers architecture. We verified the effectiveness of our proposed methods in real-time high-resolution waveform signal generation tasks. To achieve this, we carefully designed a new waveform generation model, StreamFlow, which incorporates SFM, Scale-DiT, and linear-reshape transformation for high-resolution waveform modeling using EnCodec and Mimi tokens. Furthermore, we demonstrated the scalability of our models, and we have a plan to train much larger models for better generalization. Additionally, we see that our methods can be applied to directly train the neural audio codec more efficiently. + +# Acknowledgments + +This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (RS-2025-02283048, Developing the Next-Generation General AI with Reliability, Ethics, and Adaptability, IITP-2025-RS-2023-00255968, the Artificial Intelligence Convergence Innovation Human Resources Development, No.RS-2021-II212068, Artificial Intelligence Innovation Hub), National Research Foundation of Korea(NRF) grant funded by the Korea government (MSIT) (RS-2025-16069227) and Artificial intelligence industrial convergence cluster development project funded by the Ministry of Science and ICT(MSIT, Korea)&Gwangju Metropolitan City. + +# References + +[1] Edresson Casanova, Ryan Langman, Paarth Neekhara, Shehzeen Hussain, Jason Li, Subhankar Ghosh, Ante Jukić, and Sang-gil Lee. 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In The Twelfth International Conference on Learning Representations, 2024. +[62] Jialong Zuo, Shengpeng Ji, Minghui Fang, Ziyue Jiang, Xize Cheng, Qian Yang, Wenrui Liu, Guangyan Zhang, Zehai Tu, Yiwen Guo, et al. Enhancing expressive voice conversion with discrete pitch-conditioned flow matching model. arXiv preprint arXiv:2502.05471, 2025. + +# A Implementation Details for Streaming Models + +We describe the hyperparameter details of StreamFlow for streaming models at Table 7. For Encoded Token, we successfully train the model with small segment size. However, we found that extracting Mimi tokens with small segment size significantly decrease the performance because Mimi compressed waveform signal of $24,000\mathrm{Hz}$ into $12.5\mathrm{Hz}$ . In this regard, we pre-trained the StreamFlow-Mimi with larger segment size. Furthermore, we utilize small size of audio-token drop for Mimi model because we only utilized two-step generation. Due to the limited GPU resource, we only trained the StreamFlow-Mimi for 0.15M steps. However, our model shows much better performance than Mimi. + +Table 7: Hyperparameters of StreamFlow for streaming EnCodec token reconstruction. + +
ModuleHyperparameterSteamFlow-TSteamFlow-SSteamFlow-BSteamFlow-Mimi
TimeTime Embedding25651210241024
Linear1[256, 1024][512, 2048][1024, 4096][1024, 4096]
ActivationSiLUSiLUSiLUSiLU
Linear2[1024, 256][2048, 512][4096, 1024][4096, 1024]
ConditionTokenEnCodecEnCodecEnCodecMimi
Token Hz75 Hz75 Hz75 Hz12.5Hz
Frame per token3203203201920
Token dim128128128512
Linear1[128, 256][128, 512][128, 1024][512, 1024]
Activation1GELUGELUGELUGELU
Linear2[256, 256][512, 512][1024, 1024][1024, 1024]
Activation2GELUGELUGELUGELU
UpsamplingRepeating 2Repeating 2Repeating 2Repeating 12
Input&Prompth160160160160
Linear ReshapeLinear1 (No Bias)[160, 512][160,1024][160, 2048][160, 2048]
Linear2[512, 256][1024,512][2048, 1024][2048, 1024]
OutputLinear1[256, 512][512,1024,][1024,2048][1024,2048]
Linear ReshapeLinear2 (No Bias)[512,160][1024,160][2048,160][2048,160]
h160160160160
Scale-DiTInput Dim.25651210241024
Hidden Dim.1024204840964096
Layer8888
Head481616
Stream Token8882
Prompt Token2424246
Context Prompt Ratio γ3333
Pre-trainTraining Step1M1M1M0.5M
Learning Rate2 × 10-42 × 10-42 × 10-42 × 10-4
Learning Scheduling----
Batch Size512512512128
GPUs4444
Noise Scale0.250.250.250.25
Segment Size10240102401024048000
Audio Drop0.30.30.30.3
Token Drop0.20.20.20.2
Fine-tuningTraining Step0.25M0.25M0.25M0.15M
Learning Rate2 × 10-52 × 10-52 × 10-52 × 10-5
Learning Scheduling----
Batch Size64646464
GPUs4444
Noise Scale0.250.250.250.25
Segment Size20480204802048030720
Audio Drop0.30.30.30.1
Token Drop0.20.20.20.1
+ +# B Implementation Details for Parallel Models + +We describe the hyperparameter details of StreamFlow for parallel models at Table 8. We replace the linear-reshape transformation with STFT and iSTFT. We do not utilize any iSTFT head which was used in Vocos. We can not train the model with iSTFT head of Vocos during pre-training. We directly project components for iSTFT. + +Table 8: Hyperparameters of StreamFlow for parallel EnCodec token reconstruction. + +
ModuleHyperparameterSteamFlow + iSTFTSteamFlow
TimeTime Embedding10241024
Linear1[1024, 4096][1024, 4096]
ActivationSiLUSiLU
Linear2[4096, 1024][4096, 1024]
ConditionTokenEnCodecEnCodec
Token Hz75 Hz75 Hz
Frame per token320320
Token dim128128
Linear1[128, 1024][512, 1024]
Activation1GELUGELU
Linear2[1024, 1024][1024, 1024]
Activation2GELUGELU
UpsamplingRepeating 2Repeating 2
Input&Prompth-160
Linear ReshapeLinear1 (No Bias)-[160, 2048]
Linear2-[2048, 1024]
OutputLinear1-[1024,2048]
Linear ReshapeLinear2 (No Bias)-[2048,160]
h-160
STFT/iSTFTHop/Window/FFT160/640/640-
Scale-DiTInput Dim.10241024
Hidden Dim.40964096
Layer88
Head1616
Pre-trainTraining Step1M1M
Learning Rate\( 2 \times 10^{-4} \)\( 2 \times 10^{-4} \)
Learning Scheduling--
Batch Size128128
GPUs44
Noise Scale0.250.25
Segment Size4800048000
Audio Drop0.30.3
Token Drop0.20.2
Fine-tuningTraining Step0.25M0.25M
Learning Rate\( 2 \times 10^{-5} \)\( 2 \times 10^{-5} \)
Learning Scheduling--
Batch Size3232
GPUs44
Noise Scale0.250.25
Segment Size4800048000
Audio Drop0.30.3
Token Drop0.20.2
+ +# C Additional Experiments on Pre-trained StreamFlow + +Table 9 provides additional objective evaluation results of the pre-trained StreamFlow on the LibriTTS-dev dataset without adversarial fine-tuning. We evaluate both non-streaming with different sampling steps and classifier-free guidance values. Notably, comparable performance to the 16-step setting is achieved even with only 4 sampling steps, demonstrating the efficiency in both offline and streaming scenarios. + +Table 9: Objective Evaluation for the pre-trained StreamFlow without adversarial fine-tuning on the LibriTTS-dev subsets. + +
ModelSampling StepsCFGM-STFT ↓PESQ ↑Period. ↓V/UV ↑Pitch ↓UTMOS ↑
EnCodec1-1.1632.7710.1130.94132.1472.969
Non-streaming (offline)
StreamFlow + iSTFT1611.3013.0940.0870.95328.6353.450
StreamFlow + iSTFT160.51.3113.1430.0850.95426.5353.539
StreamFlow + iSTFT1601.3992.8270.0940.95027.0203.482
StreamFlow + iSTFT411.5582.6170.0920.95026.6743.238
StreamFlow + iSTFT40.51.5612.6680.0890.95227.9053.339
StreamFlow + iSTFT401.6492.4500.0940.95027.9403.320
Streaming (online)
StreamFlow1611.3432.7190.0970.94922.0963.147
StreamFlow160.51.3412.7900.0930.95220.3283.224
StreamFlow1601.3882.6690.1010.95021.0993.178
StreamFlow411.5092.4790.1010.94620.9563.012
StreamFlow40.51.5042.6070.0940.95217.8123.131
StreamFlow401.5702.5570.0950.95218.6973.149
+ +# D Implementation Details for Baselines + +EnCodec We utilize an official implementation of EnCodec [9], which is the popular neural audio codec using RVQ and adversarial training. They utilize causal convolutional layer for streaming application, and train the model with 32 quantizer of RVQ. However, most application utilized eight quantizer as target tokens so we train the model with eight tokens to reconstruct the waveform signal. + +Vocos We utilize an official implementation of Vocos [46], which is an iSTFT-based waveform generation model with adversarial training. They utilize an EnCodec as an input representation to reconstruct the waveform signal. We also generate the waveform signal by chunk-wise generation using Vocos to compare the streaming generation performance. For a fair comparison, we utilize the same previous and delayed tokens for chunk-wise generation for robust generation of Vocos. + +MBD We utilize an official implementation of Multi-band Diffusion (MBD) [45] as a strong baseline for EnCodec token reconstruction.6 MBD consists of four models for each band, and has large-scale parameters of 411M. Then, they utilize 10 sampling steps for each band so it takes a lot of time to generate the waveform signal even with parallel generation. + +RFWave We utilize an official implementation of RFWave [32], which a strong baseline using conditional flow matching for multi-band parallel generation. We utilize 20 sampling steps for better performance and CFG of 2 which is suggested by official implementation. + +Mimi We utilize an official implementation of Mimi from Moshi [7]. They utilize causal convolutional layer for streaming generation, and train the model with 32 quantizer of RVQ and compressed high-resolution waveform signal of $24,000\mathrm{Hz}$ into $12.5\mathrm{Hz}$ for efficient speech language models. Moshi only utilizes eight quantizer of RVQ for target tokens by delayed prediction so we also train the model with eight tokens. + +# E Societal Negative Impact + +Although our method and model do not directly contribute to malicious use or ethical concerns, they can be misused when combined with text-to-speech or voice conversion models to deceive people. Therefore, it is crucial to explore fake audio detection and voice phishing detection models alongside our research. In the future, we aim to develop speech language models capable of identifying fake audio based on its content. + +# F Evaluation Details + +M-STFT We employed the multi-resolution Short-Time Fourier Transform (M-STFT) distance implemented in the open-source Auraloss [47].9 Originally proposed in Parallel WaveGAN [54], the M-STFT quantifies the distances between ground-truth and generated audio samples across multiple STFT resolutions, thereby capturing both fine and coarse spectral details. + +PESQ For evaluating reproduction quality, we utilized the wide-band (WB) Perceptual Evaluation of Speech Quality (PESQ) metric $^{10}$ . The audio signals were downsampled to a sampling rate of $16,000\mathrm{Hz}$ before calculating the PESQ scores to ensure consistency with standard evaluation protocols. Furthermore, we normalize the downsampled waveform signal to avoid overflow. + +Periodicity, V/UV F1, and Pitch Following the observations of CarGAN [35] regarding the perceptual degradation caused by periodicity artifacts, we measured periodicity errors using the Periodicity Root Mean Square Error (RMSE) $^{11}$ . Additionally, we evaluated the Voice/Unvoice (V/UV) classification performance using the F1 score to assess the accuracy of voiced and unvoiced regions in the generated audio. Also, we calculated pitch errors using the pitch Root Mean Square Error (RMSQ). All metrics utilize pitch predicted by CREPE [17]. We used the pytorch implementation of CREPE. $^{12}$ + +UTMOS To assess the naturalness of the generated samples, we utilized the open-source Mean Opinion Score (MOS) prediction model, UTMOS [44].13 UTMOS has demonstrated consistent MOS prediction performance on neutral English speech datasets, providing a reliable measure of the perceived naturalness of the synthesized audio without reference samples. + +# Audio Quality Evaluation Test + +Please read these instructions carefully before starting! + +It is strongly recommended to listen to the audio using a headset or earphones in an environment free from background noise. + +[Evaluation Criteria] + +You should evaluate the audio quality (clarity, naturalness) of each sample, not the speech content or accuracy. +Each audio sample should be assessed independently. + +The scoring system is as follows: + +Good(4) +Fair (3) +a. Poor (2) +Bad (1) +N/A (0) + +[N/A Sample Explanation] + +A N/A sample refers to an audio file where speech is heavily mixed with strong background noise, as shown in the example below. +To maintain the quality of the evaluation, we have included such fake samples in the test set. +If a fake sample is given a high score, the corresponding review will be rejected. + +Examples of noise samples: + +![](images/264039301b3642a368644142d259430140a0fd2f3b2e4319e79af95a14fd80b4.jpg) + +To ensure the reliability of the evaluations and eliminate noise, we will reject any attempts to assign high scores to N/A (noise) samples. + +![](images/0223dbd0bb8a94e3dcc82f0e1e754bc071af2113b83ac8fb5e17002938450be8.jpg) +Figure 5: Details of the MOS evaluation interface provided to crowdsourcing participants + +# Audio Quality Evaluation Test + +Please read these instructions carefully before starting! + +It is strongly recommended to listen to the audio using a headset or earphones in an environment free from background noise. + +[Evaluation Criteria] + +- Please listen to the two samples below and rate how much better or worse the second B sample is compared to the first A on a scale from -3 to +3. + +Much worse (-3) + +Significantly worse (-2) +Slightly worse (-1) +No difference (0) +Slightly better (+1) +Significantly better $(+2)$ + +Much better $(+3)$ + +To ensure the reliability of the evaluations and eliminate noise, we will reject any attempts to assign high scores to fake samples. + +![](images/cafa3600773c82138c1ccea8e2231e7b2e4e2f6c4929e37803c1c044b9c7d4cf.jpg) +Figure 6: Details of the CMOS evaluation interface provided to crowdsourcing participants + +# G Crowdsourcing Details + +We conducted Mean Opinion Score (MOS) evaluations using a 5-point scale to assess the quality of speech. The perceptual quality of each model was evaluated through a crowdsourced listening test using Amazon Mechanical Turk $(\mathsf{MTurk})^{14}$ . A total of 20 native English speakers from the United States participated, each rating 300 samples per model on a scale from 1 to 5. We paid $180 for each MOS experiment. To ensure the reliability of responses, we established strict participant eligibility criteria: only individuals with a prior task approval rate of at least $50\%$ and a minimum of 100 approved HITs were permitted to take part in this evaluation. Additionally, we included Gaussian noise fake samples as control samples to enhance evaluation robustness, and we give the instruction which noise samples should be assigned as N/A sample (0 point). Listener responses were excluded from the final analysis if they met either of the following conditions: (1) assigning a score over 1 to the noise augmented samples, or (2) spending less than half the duration of an audio sample on the evaluation. These measures were implemented to filter out inattentive participants and maintain the integrity of the collected ratings. The user interface used for the evaluation is illustrated in Figure 5 and Figure 6. + +![](images/90f2443847ed8499ccd15f299bc0bff1bb65d82920c09a9f63a9bce4ea1b3635.jpg) +Figure 7: Inference pipeline comparing the Mimi decoder and the proposed StreamFlow, given identical Moshi output. + +Table 10: Comparison of real-time decoding performance for full duplex spoken dialogue system, Moshi using the Mimi decoder and the proposed StreamFlow + +
MethodCER ↓WER ↓UTMOS ↑
Moshi w/ Mimi decoder7.599.893.610
Moshi w/ StreamFlow (Ours)7.199.823.847
+ +# H Replacing Mimi with StreamFlow in a Full-duplex Streaming Model + +We replaced the original Mimi decoder in Moshi with StreamFlow, successfully integrating it into Moshi's fully-duplex streaming speech language model. For this integration, we utilized the official Moshi code $^{15}$ , and conducted experiments using question audio from HeySQuAD [52] $^{16}$ dataset as input. We upsampled the dataset, originally at $16\mathrm{kHz}$ sampling rate, to $24\mathrm{kHz}$ using Librosa and used it as input to Moshi. The outputs of the Moshi were kept identical across all conditions, allowing for a controlled comparison between the original Mimi decoder and our streaming StreamFlow. We evaluated both speech quality and speech intelligibility, using UTMOS, CER, and WER respectively as metrics. As shown in Table 10, our proposed StreamFlow consistently outperformed the Mimi decoder. + +As shown in Figure 7, the pipeline of [Question Audio $\rightarrow$ Mimi Encoder $\rightarrow$ LLM $\rightarrow$ Selectable Decoder] demonstrates that improvements at the decoder stage can lead to substantial gains in speech generation quality. These experiments support the effectiveness of the proposed decoder in streaming-based speech generation scenarios and highlight its practical potential as a viable alternative to the original architecture. + +# NeurIPS Paper Checklist + +# 1. Claims + +Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? + +Answer: [Yes] + +Justification: We clearly highlight the main contributions of the proposed method in the abstract and introduction, which are well aligned with the main content of the paper. 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Declaration of LLM usage + +Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required. + +Answer: [NA] + +Justification: We do not use LLMs for research. + +Guidelines: + +- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components. +- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described. \ No newline at end of file diff --git a/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/images.zip b/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/images.zip new file mode 100644 index 0000000000000000000000000000000000000000..0d679029195c13092c208399744e917366acb0b8 --- /dev/null +++ b/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/images.zip @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:72d5c0faa2a7a4562855aaec4cbe6d4b89c9139bd01f717d9a7cf84706031a5c +size 919558 diff --git a/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/layout.json b/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/layout.json new file mode 100644 index 0000000000000000000000000000000000000000..6d3ad01e77afa08c015764e59a91997f6504f3ad --- /dev/null +++ b/streamflowstreamingaudiogenerationfromdiscretetokensviastreamingflowmatching/layout.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b8ed0cae4210d4acb45213035bc8ef1afa6de859185a9b807c706534cdbf07cd +size 786617