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+# Spotlight Attention: Towards Efficient LLM Generation via Non-linear Hashing-based KV Cache Retrieval
+
+Wenhao Li
+
+Xiamen University
+
+Yuxin Zhang
+
+Xiamen University
+
+Gen Luo
+
+Shanghai AI Laboratory
+
+Haiyuan Wan
+
+Shanghai AI Laboratory
+
+Ziyang Gong
+
+Shanghai Jiao Tong University
+
+Fei Chao
+
+Xiamen University
+
+Rongrong Ji*
+
+Xiamen University
+
+# Abstract
+
+Reducing the key-value (KV) cache burden in Large Language Models (LLMs) significantly accelerates inference. Dynamically selecting critical KV caches during decoding helps maintain performance. Existing methods use random linear hashing to identify important tokens, but this approach is inefficient due to the orthogonal distribution of queries and keys within two narrow cones in LLMs. We introduce Spotlight Attention, a novel method that employs non-linear hashing functions to optimize the embedding distribution of queries and keys, enhancing coding efficiency and robustness. We also developed a lightweight, stable training framework using a Bradley-Terry ranking-based loss, enabling optimization of the non-linear hashing module on GPUs with 16GB memory in 8 hours. Experimental results show that Spotlight Attention drastically improves retrieval precision while shortening the length of the hash code at least  $5 \times$  compared to traditional linear hashing. Finally, we exploit the computational advantages of bitwise operations by implementing specialized CUDA kernels, achieving hashing retrieval for 512K tokens in under  $100\mu s$  on a single A100 GPU, with end-to-end throughput up to  $3 \times$  higher than vanilla decoding. All the training and evaluation stuff can be found at https://github.com/wenhaoli-xmu/spotlight.
+
+# 1 Introduction
+
+Large Language Models (LLMs) are propelling groundbreaking advancements in various natural language tasks, significantly enhancing applications such as content creation and chat assistance. Generally, the inference process of LLMs can be divided into (1) the pre-filling phase calculates the key-value (KV) cache for input tokens in the prompt prior to autoregressive generation, and (2) the decoding phase auto-regressively generates tokens, producing one token per forward pass based on the KV cache. Among them, the decoding phase serves as the primary inference bottleneck due to the frequent exchanges between on-board and on-chip memory for model parameters and KV cache, which limits GPU scalability [3] more so than the pre-filling phase that processes input prompts in parallel. For example, deploying LLaMA2-7B [21] on an A100 GPU for a single request achieves nearly  $100\%$  GPU utilization during the pre-filling phase but drops to below  $10\%$  on average during decoding, which largely restrains the inference efficiency of LLMs.
+
+![](images/57f21152d5589dae641648eb4a2e2cf6c0006054fc6ef58e4df94e64e0a00ced.jpg)  
+Figure 1: Overview. (Left) Architecture. Comparison of our Spotlight Attention versus normal attention. Spotlight Attention adds an additional hash code-based retrieval mechanism for each layer. (Middle) Performance. Spotlight attention achieves the most accurate retrieval and generates the closest response compared to the original model on QA datasets. (Right) Visualization. For arbitrarily complex attention patterns, our method estimates the top-k sequences well, with an average correctness rate of more than half for different models.
+
+![](images/92ef0b82091450855243625b1c0735ef499f7acdecf82d914a66f2aabaecf1d9.jpg)
+
+![](images/11975bb8161c0f4ae759e31b828ba4c8de70b0ff6bc10a0c901f02dc94c489b9.jpg)
+
+![](images/845afa23c51edb343977b00c06792610930bf1c005032d64a158919737e5a4ce.jpg)
+
+To alleviate this inference bottleneck, extensive research has focused on heuristically eliminating the KV cache burden based on attention scores [25, 23, 17]. While effective for short sequences, such irreversible removal of KV cache can significantly degrade performance on long-sequence tasks, especially in Needle-in-a-Haystack scenarios [14]. To explain, tokens initially considered unimportant and removed might later attain higher attention scores during the prolonged decoding phase, which is crucial for output quality [20, 15]. To overcome this limitation, recent works have turned to retaining all KV cache tokens and dynamically selecting important tokens for computation during decoding [20, 10], which is the focus of this paper.
+
+Despite the convincing performance of such on-the-fly KV cache selection, how to effectively pick up those important tokens remains challenging. As a pioneering effort, Quest [20] selects tokens by matching queries and keys in a block-wise manner. While effective, such coarse-grained selection hardly guarantees precise localization of important tokens. MagicPIG [10] advances Quest by implementing token-level cache retrieval, specifically utilizing Local Sensitive Hashing (LSH) to encode queries and keys into hash codes and pinpointing the best matches as the selected tokens. However, as depicted in Figure 2a, the efficacy of such linear hashing heavily depends on the hash code length, e.g., a hash code length of 1,024 bits per query/key is necessitated to achieve promising token retrieval. Considering the already substantial size of the KV cache, storing these lengthy hash codes markedly impairs deployment efficiency. Moreover, employing a linear projection with a large output dimension for key processing incurs significant computational overhead.
+
+Delving deeper, prior work [10] has discovered that the queries and keys within LLMs typically form nearly orthogonal cones within the embedding space, as depicted in Figure 2b. Given the truth that linear hashing function partitions the embedding space using random hyperplanes, such orthogonal distribution of queries and keys barely lead to satisfying encoding quality, which can even result in a collapse of the hashing outcomes, i.e., identical codes for all queries and keys, as shown in Figure 2c. Therefore, extremely long hash codes are necessary to mine meaningful information and accurately match essential tokens. MagicPIG attempted to mitigate this issue by normalizing the keys before retrieval. However, this approach remains suboptimal as it overlooks the query distribution and introduces bias to the retrieval process.
+
+To address the aforementioned limitation, we propose Spotlight Attention, a novel method that replaces random hyperplanes with curved surfaces for space partitioning via a non-linear MLP hashing function. As depicted in Figure 2d, this non-linearity can better fit skewed distributions, thereby improving code quality. In particular, we utilize the Bradley-Terry ranking objective [7] to optimize the non-linear MLP layer, wherein the learning target involves minimizing the difference between the estimated top- $k$  indices and the ground truth top- $k$  indices obtained via the vanilla attention scores. This learning process is exceptionally efficient, with the LLM backbone remaining frozen and requiring only a minimal amount of calibration data. As a result, the optimized non-linear
+
+![](images/8f98aa767946566aa3b866b3be9632d168e821c962db1cc1be755964a0acc0fc.jpg)  
+(a)
+
+![](images/3d14b55900d1dd99b68deb2b734d0e52f53f3c910a3c06ccffec32157c72f784.jpg)  
+(b)
+
+![](images/d0861207ffe8cf707c21e48ca43bd96e8c48dfdc359c6728a9bc9371dc538513.jpg)  
+(c)  
+Figure 2: Motivation. (a) The empirical evaluation shows that upgrading the hashing function from linear to MLP can bring a huge improvement, (b) this is because query and key are usually distributed in two small cones in the space [10]. (c) In this situation, it is difficult for the space to be uniformly partitioned by linear boundaries, (d) which can be well solved by using an MLP hashing function.
+
+![](images/c10abafbe15da49cedd376d14f99e3c309f5f41261556ae0b29c845ce6a350c4.jpg)  
+(d)
+
+hashing function can match the performance of linear hashing while using  $5 \times$  shorter hash codes, achieving higher efficiency than MagicPIG. We further implemented CUDA kernels for the hash code processing, including bit-packing and bitwise NXOR GEMM operators, achieving significant latency reductions in practice. For example, our method achieves up to  $3 \times$  increase in Qwen2.5-7B [24] inference throughput for both 32K and 128K sequences, with only  $\sim 2\%$  performance degradation on the LLaMA3 [13] series and no loss on Qwen2.5 [24] series.
+
+Our contributions are threefold:
+
+- We propose Spotlight Attention for accelerating LLM inference, which employs non-linear hashing function to encode and match queries and key values within LLMs, thereby efficiently selecting critical KV cache for model inference.  
+- We develop a lightweight and robust training framework based on the Bradley-Terry ranking objective, which effectively optimizes the non-linear hashing function using only a small amount of calibration data.  
+- Extensive experiments demonstrate that Spotlight Attention can drastically reduce LLM inference latency while maintaining the strongest performance retention in comparison with state-of-the-art methods.
+
+# 2 Related Work
+
+This section covers the spectrum of studies on LLM KV cache pruning that are closely related to our work, which we heuristically categorize into static pruning, dynamic pruning with permanent eviction, and dynamic pruning without permanent eviction.
+
+Static KV cache pruning. These methods compress the KV cache once after the pre-filling phase, using the compressed cache for subsequent decoding. For example, FastGen [12] introduces a pattern-aware approach by identifying five fundamental attention structures and applying targeted selection strategies. SnapKV [16] further simplifies FastGen by focusing solely on retrieving tokens based on their importance scores, showing that only a subset of prompt tokens carry critical information for response generation and retain their significance during the whole decoding phase. However, without pruning during decoding, these methods are primarily suited for scenarios with long prompts and relatively short responses.
+
+Dynamic pruning with permanent eviction. This category of methods performs dynamic KV cache pruning during the decoding phase, permanently removing pruned KV cache tokens from memory. For example, H2O [25] leverages cumulative attention scores to retain high-impact tokens. NACL [9] identifies a fundamental limitation in H2O, namely their dependence on potentially biased local attention statistics. To overcome this issue, they develop an alternative approach, implementing a diversified random eviction strategy. Keyformer [2] highlights that token removal distorts the underlying softmax probability distribution. Considering the pivotal role of softmax distributions in token significance evaluation, they incorporate regularization techniques to mitigate these distributional perturbations. Unlike static KV cache selection, these methods enable dynamic pruning during decoding, making them better suited for tasks requiring extensive generation. However, they assume that critical information is concentrated in a small subset of KV cache tokens, a condition that does not always hold. As MagicPIG [10] points out, token importance can vary significantly
+
+![](images/d43e452690274fd95bde671e221be67d7d4b844fd40e145a386d738f7aac7704.jpg)
+
+![](images/8ead7c28b1d58a0a7c51d3b9460dec1ab451a3e0c194e0272d99235cf4bce12b.jpg)
+
+![](images/efedd65140a471382119eb7906db8deb8e87dea9f47a91e5d2161186e363cba9.jpg)  
+Figure 3: Optimization. (Left) Reconstruction Loss. This loss minimizes the MSE between the estimated and ground-truth attention scores. It has two main drawbacks. First, it is highly sensitive to score magnitudes and prone to outliers. Second, it wastes most of the hashing function's capacity on preserving order within the top- $k$  and non-top- $k$  sets. (Right) Our Ranking Loss. Our loss adopts the Bradley-Terry ranking objective, which is robust to score magnitude and outliers, and provides supervision focused solely on distinguishing between top- $k$  and non-top- $k$  sets.
+
+![](images/24ece74e633c7b31bbc5e20fb04e824424285586cb42ba91ea388793ea98aac6.jpg)
+
+across tasks, leading to premature eviction of tokens before they are needed. For example, H2O may fail to answer questions like  $a$  is  $b$ ,  $c$  is  $d$ ,  $a$  is? due to forgetting earlier facts.
+
+Dynamic pruning without permanent eviction. The limited applicability of permanent token eviction methods has led to a shift toward non-permanent eviction approaches. These methods assume the importance of KV cache tokens varies with each query, requiring importance estimation at every decoding step. Instead of permanently evicting unimportant tokens, they exclude them from attention calculations for that step only. While this improves accuracy, it demands frequent importance estimation, unlike permanent eviction methods that prune tokens in batches after many steps. Research has therefore focused on optimizing the efficiency and accuracy of these estimations. Quest [20] groups KV cache tokens into blocks, estimating block importance via the dot product between queries and block representations derived from the minimum and maximum key values. Although efficient, this approach suffers from internal fragmentation, as entire blocks are processed even if only a few tokens are important. MagicPIG [10] eliminates this issue by mapping queries and keys to hash codes for token-level retrieval via Hamming distance, avoiding fragmentation but reducing efficiency. Building on MagicPIG, our method significantly shortens hash code, drastically reducing computation while preserving accuracy.
+
+# 3 Methodology
+
+# 3.1 Preliminary
+
+Attention computing. We first present the basic preliminaries for attention computation and KV cache pruning during the decoding phase of LLMs. We define the query, key, and value inputs to the attention module as  $Q, K, V \in \mathbb{R}^{1 \times d}$ , where  $d$  is the embedding dimension. We use  $\oplus$  to denote concatenation, and  $K_{\mathrm{cache}}, V_{\mathrm{cache}} \in \mathbb{R}^{n \times d}$  represent the key-value cache generated during the pre-filling phase and previous decoding steps. With these definitions, the standard attention is calculated as follows:
+
+$$
+\mathcal {A} = \operatorname {s o f t m a x} \left(\frac {f (Q , K _ {\text {c a c h e}}) \oplus Q K ^ {\top}}{\sqrt {d}}\right) \left(V _ {\text {c a c h e}} \oplus V\right), \tag {1}
+$$
+
+where  $f(X, X') = XX'^\top$  is inner-product.
+
+KV cache pruning. As the decoding sequence length increases, the size of the KV cache  $\{K,V\}_{\mathrm{cache}}$  can grow exceedingly large, creating an LLM inference bottleneck. KV cache pruning that selectively preserves only essential portions of the cache for computation serves as an efficient way to alleviate
+
+Table 1: KV retrieval accuracy. Measured by IoU↑ / PPL↓ changes before and after training. (1) Training is essential for efficient MLP hashing. (2) Limited improvement from training linear hashing highlights the necessity of MLP hashing. See Appendix B.1 for per-head IoU scores.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Original</td><td rowspan="2">Oracle Top-2%</td><td colspan="2">LSH Top-2%</td><td colspan="2">MLP Hashing Top-2% (ours)</td></tr><tr><td>Before</td><td>After</td><td>Before</td><td>After</td></tr><tr><td>LLaMA2-7B</td><td>- / 5.58</td><td>1.00 / 5.69</td><td>0.17 / 5.86</td><td>0.20 / 5.84</td><td>0.05 / 20.31</td><td>0.41 / 5.72</td></tr><tr><td>LLaMA2-7B-Chat</td><td>- / 7.10</td><td>1.00 / 6.87</td><td>0.17 / 7.34</td><td>0.19 / 7.45</td><td>0.05 / 21.34</td><td>0.42 / 6.98</td></tr><tr><td>LLaMA3-8B</td><td>- / 6.45</td><td>1.00 / 6.63</td><td>0.15 / 7.12</td><td>0.18 / 7.07</td><td>0.07 / 148.2</td><td>0.34 / 6.69</td></tr><tr><td>Qwen2.5-7B</td><td>- / 7.17</td><td>1.00 / 7.28</td><td>0.13 / 8.81</td><td>0.16 / 8.73</td><td>0.09 / 22.07</td><td>0.35 / 7.31</td></tr></table>
+
+this problem. Given a desired cache budget  $K$ , it first identifies the indices  $\mathcal{I}$  of top-  $K$  important tokens and then extracts a subset of the KV cache  $\{K, V\}_{\mathrm{subset}}$  for attention computation as
+
+$$
+\{K, V \} _ {\text {s u b s e t}} = \operatorname {g a t h e r} (\{K, V \} _ {\text {c a c h e}}, \mathcal {I}). \tag {2}
+$$
+
+As previously discussed, existing methods for pruning the KV cache either permanently eliminate cache entries not in the set  $\mathcal{I}$  [25, 16] or retain all caches but dynamically determine  $\mathcal{I}$  during the decoding process [20, 10]. In this paper, we focus on the latter due to its superior performance preservation.
+
+# 3.2 Revisiting Token-level Cache Retrieval
+
+Token-level cache retrieval refers to dynamically selecting cached entries at the granularity of individual tokens [10].
+
+Oracle top-k retrieval. We conducted a preliminary experiment to assess the upper-bound performance, using full-precision attention scores  $f(Q, K_{\mathrm{cache}})$  to select key tokens. Surprisingly, by pruning layers beyond the first two, we discarded up to  $98\%$  of the KV cache with only a 0.1 perplexity increase on PG19, revealing significant untapped potential. This suggests that top- $k$  retrieval enables near-lossless compression, contrasting sharply with prior findings [10].
+
+LSH top-k retrieval. Although oracle attention scores accurately identify key tokens, computing  $f$  is impractical for real-world applications. To address this, MagicPIG approximates  $f$  with  $\tilde{f}$  using Locality-Sensitive Hashing (LSH) to efficiently retrieve critical KV entries. Specifically, a linear hash function  $\mathcal{H}$  computes:
+
+$$
+\tilde {f} (X, X ^ {\prime}) = \mathcal {H} (X) \otimes \mathcal {H} \left(X ^ {\prime}\right), \tag {3}
+$$
+
+where  $\otimes$  denotes a matrix multiplication-like operation, substituting floating-point multiplication with NXOR. The indices  $\mathcal{I}$  of the top- $k$  largest values in  $\tilde{f}(Q, K_{\mathrm{cache}})$  are used to retrieve the most relevant KV entries.
+
+LSH groups similar vectors into the same bucket with high probability, making it ideal for dense vector spaces. A common variant employs random hyperplanes, where distinct hyperplanes create linear decision boundaries, assigning data on either side to bit-0 or bit-1. The bits sequence from all hyperplanes forms the hash code. In practice, these steps can be simplified to a single matrix multiplication followed by a sign operation. For a vector  $x \in \mathbb{R}^d$ , we apply a random projection matrix  $R \in \mathbb{R}^{d \times d_{\mathcal{H}}}$  to obtain a hash code by taking the sign of the resulting product:
+
+$$
+\mathcal {H} (x) = \operatorname {s i g n} (x R). \tag {4}
+$$
+
+MLP hashing top-k retrieval. As shown in Figure 2, queries and keys typically lie within two cone-shaped regions in high-dimensional space [10]. This distribution causes uneven partitioning in LSH, reducing encoding efficiency. To address this, we propose MLP hashing, a learned non-linear hashing network tailored to query and key distributions. This approach enhances hash code information density, enabling effective partitioning of skewed data through non-linear decision boundaries.
+
+Specifically, we replace the projection matrix  $R$  in Eq. (4) with a two-layer MLP:
+
+$$
+\operatorname {M L P} (x) = W _ {2} \left(\operatorname {S i L U} \left(W _ {1} x + b _ {1}\right)\right), \tag {5}
+$$
+
+where  $W_{1}, b_{1}$ , and  $W_{2}$  are learnable parameters. Hash codes are then computed as:
+
+$$
+\mathcal {H} (x) = \operatorname {s i g n} (\mathrm {M L P} (x)). \tag {6}
+$$
+
+Table 2: Perplexity comparison with Quest. Perplexity evaluation on PG19 (#1), ProofPile (#2), and CodeParrot (#3) datasets. All models truncate inputs to their maximum supported token length. Spotlight Attention achieves performance comparable to Quest with a  $10 \times$  smaller token budget.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Configuration</td><td rowspan="2">Frozen Layers</td><td colspan="3">LLaMA2-7B</td><td colspan="3">LLaMA2-7B-Chat</td><td colspan="3">LLaMA3-8B</td><td colspan="3">Qwen2.5-7B</td></tr><tr><td>#1</td><td>#2</td><td>#3</td><td>#1</td><td>#2</td><td>#3</td><td>#1</td><td>#2</td><td>#3</td><td>#1</td><td>#2</td><td>#3</td></tr><tr><td rowspan="2">Vanilla Oracle Top-K</td><td>0% Pruned</td><td>N/A</td><td>6.879</td><td>4.277</td><td>3.679</td><td>9.212</td><td>5.943</td><td>4.786</td><td>8.604</td><td>3.517</td><td>5.219</td><td>11.112</td><td>3.833</td><td>4.951</td></tr><tr><td>98% Pruned</td><td>[0,1]</td><td>6.941</td><td>4.317</td><td>3.729</td><td>9.224</td><td>5.891</td><td>4.795</td><td>8.881</td><td>3.590</td><td>5.353</td><td>10.435</td><td>3.587</td><td>4.733</td></tr><tr><td rowspan="4">Quest</td><td>75% Pruned</td><td rowspan="4">[0,1]</td><td>7.116</td><td>4.404</td><td>3.854</td><td>9.282</td><td>5.930</td><td>4.879</td><td>9.912</td><td>4.024</td><td>5.893</td><td>10.485</td><td>4.281</td><td>5.482</td></tr><tr><td>87.5% Pruned</td><td>7.735</td><td>4.754</td><td>4.054</td><td>9.820</td><td>6.226</td><td>5.084</td><td>12.434</td><td>4.927</td><td>6.646</td><td>13.596</td><td>5.471</td><td>6.213</td></tr><tr><td>93.7% Pruned</td><td>9.746</td><td>5.775</td><td>4.571</td><td>12.058</td><td>7.440</td><td>5.686</td><td>17.320</td><td>6.749</td><td>8.693</td><td>19.852</td><td>7.274</td><td>7.823</td></tr><tr><td>96.9% Pruned</td><td>15.494</td><td>8.578</td><td>5.996</td><td>18.719</td><td>11.083</td><td>7.345</td><td>27.510</td><td>11.631</td><td>14.205</td><td>31.583</td><td>13.208</td><td>12.526</td></tr><tr><td rowspan="4">Spotlight (ours)</td><td>80% Pruned</td><td rowspan="4">[0,1]</td><td>6.887</td><td>4.278</td><td>3.682</td><td>9.107</td><td>5.860</td><td>4.767</td><td>8.612</td><td>3.519</td><td>5.228</td><td>9.766</td><td>3.465</td><td>4.618</td></tr><tr><td>90% Pruned</td><td>6.908</td><td>4.285</td><td>3.689</td><td>9.058</td><td>5.796</td><td>4.754</td><td>8.651</td><td>3.529</td><td>5.239</td><td>9.783</td><td>3.467</td><td>4.621</td></tr><tr><td>95% Pruned</td><td>6.959</td><td>4.304</td><td>3.703</td><td>9.067</td><td>5.748</td><td>4.752</td><td>8.734</td><td>3.552</td><td>5.285</td><td>9.825</td><td>3.475</td><td>4.627</td></tr><tr><td>98% Pruned</td><td>7.106</td><td>4.364</td><td>3.768</td><td>9.262</td><td>5.770</td><td>4.806</td><td>8.977</td><td>3.621</td><td>5.434</td><td>9.930</td><td>3.497</td><td>4.645</td></tr></table>
+
+# 3.3 Optimization.
+
+Optimizing the MLP hashing function  $\mathcal{H}$  to capture the query and key distribution is more critical than the hashing network design and forms the core contribution of this work. We next introduce two intuitive training objectives and explain their limitations in this context.
+
+$\times$  Language modeling loss. A natural approach to optimize  $\mathcal{H}$  is to minimize the language modeling loss directly through a few hundred post-training steps. However, this method has significant limitations. First, it requires full forward and backward passes through the entire LLM, which is computationally costly. More critically, differentiating the top- $k$  operator in Eq. (2) requires a complex "soft top- $k$ " approximation, which is challenging to implement.
+
+$\times$  Reconstruction loss. An alternative approach uses MSE to align  $\tilde{f}$  with  $f$ , enabling layer-wise optimization and reducing computational cost. However, this method has a key limitation: the objective is to select which KV cache entries to retain, not to rank their relative importance. Training with this loss misallocates capacity by prioritizing the ranking of excluded entries, deviating from the core goal, and causing significant performance degradation. See Figure 3 (Left) for its limitations.
+
+$\checkmark$  Our ranking loss. To address this issue, we adopt a Bradley-Terry ranking objective, inspired by RankNet [8]. As shown in Figure 3 (Right), during training, we identify top- $k$  and non-top- $k$  index sets based on attention scores. These indices split the estimated scores derived from Hamming distances of query and key hash codes into sets  $B$  and  $C$ . An optimizer then updates the hashing function parameters to ensure each score in  $B$  exceeds every score in  $C$ :
+
+$$
+\mathcal {L} _ {\text {r a n k}} = - \frac {1}{k (n - k)} \sum_ {i, j} \log (\text {s i g m o i d} (\beta (B _ {i} - C _ {j}) - \alpha)), \tag {7}
+$$
+
+where  $\beta$  and  $\alpha$  are positive constants used to amplify the separation between  $B$  and  $C$ , facilitating convergence. The core of this loss design lies in filtering out supervising signals related to internal ranking within  $B$  and  $C$ , effectively addressing the issue of capacity misallocation. We provide the pseudo-code for our ranking loss in Appendix A.3 to aid readers familiar with code.
+
+Make hashing function differentiable. After computing the loss, errors can be backpropagated to the MLP hashing function. However, the sign function's non-differentiability blocks gradient flow. To resolve this, we substitute the sign function with a soft sign function during training:
+
+$$
+\operatorname {s o f t s i g n} (x) = \frac {\gamma x}{1 + \gamma | x |}, \tag {8}
+$$
+
+where  $\gamma \in \mathbb{R}$  is a hyperparameter controlling the extent of smoothing. This soft sign function is used only during training. In inference, the non-differentiable sign function is reinstated.
+
+![](images/e3fd3018038432aa3f410e136f26fd1697f6cc2de8310b4f6c85c90b8b315db5.jpg)  
+Figure 4: NIAH results. Using LLaMA3-8B [13] as the base model, we compared the retrieval accuracy of MagicPIG with our method. Our approach, which relies solely on hash code-based retrieval without local windows or sink tokens, achieves comparable response accuracy.
+
+![](images/cbf9e895e008c5f562add8602c85d506c3024ad8e87dea00444e3d89dd22673c.jpg)
+
+![](images/86830c00683bb63fb1305926fcce689e6e67786bb6a7252c9f9be8cba38b3273.jpg)
+
+Table 3: Perplexity versus MagicPIG. Comparison of perplexity on PG19 (#1), ProofPile (#2), and CodeParrot (#3). Due to the time-consuming evaluation process of MagicPIG, we sampled only 10 data points from each dataset for testing.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">Configuration</td><td colspan="3">LLaMA3-8B</td></tr><tr><td>Frozen</td><td>Local (64)</td><td>Sink (4)</td><td>Retrieve (2%)</td><td>#1</td><td>#2</td><td>#3</td></tr><tr><td rowspan="2">Vanilla Oracle Top-K</td><td rowspan="2">[0,1]</td><td></td><td></td><td rowspan="2">✓</td><td>9.56</td><td>2.83</td><td>2.26</td></tr><tr><td></td><td></td><td>9.89</td><td>2.89</td><td>2.28</td></tr><tr><td rowspan="5">MagicPIG</td><td rowspan="5">[0,16]</td><td>✓</td><td>✓</td><td>✓</td><td>12.65</td><td>3.54</td><td>2.91</td></tr><tr><td>✓</td><td>✓</td><td></td><td>16.94</td><td>6.27</td><td>4.57</td></tr><tr><td>✓</td><td></td><td>✓</td><td>50.96</td><td>8.52</td><td>6.67</td></tr><tr><td></td><td>✓</td><td>✓</td><td>42.12</td><td>8.65</td><td>10.43</td></tr><tr><td></td><td></td><td>✓</td><td>NaN</td><td>NaN</td><td>NaN</td></tr><tr><td rowspan="3">Spotlight (ours)</td><td rowspan="3">[0,1]</td><td>✓</td><td>✓</td><td>✓</td><td>9.87</td><td>2.89</td><td>2.29</td></tr><tr><td>✓</td><td>✓</td><td></td><td>13.89</td><td>5.50</td><td>3.98</td></tr><tr><td></td><td></td><td>✓</td><td>9.99</td><td>2.91</td><td>2.30</td></tr></table>
+
+Table 4: QA response fidelity. On Long-Bench, output fidelity (measured by Rouge-L between compressed and vanilla model outputs) shows our method achieves performance closest to the vanilla model.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">Configuration</td><td rowspan="2">Similarity</td></tr><tr><td>Local</td><td>Sink</td><td>Retrieve</td><td>Frozen</td></tr><tr><td>Vanilla</td><td>X</td><td>X</td><td>X</td><td>X</td><td>1.00</td></tr><tr><td>Oracle Top-K</td><td></td><td></td><td>163</td><td></td><td>0.66</td></tr><tr><td>LSH Top-K</td><td>X</td><td>X</td><td>163</td><td>[0,1]</td><td>0.37</td></tr><tr><td>Quest</td><td></td><td></td><td>1024</td><td></td><td>0.56</td></tr><tr><td>Quest</td><td></td><td></td><td>256</td><td></td><td>0.34</td></tr><tr><td>MagicPIG</td><td>64</td><td>4</td><td>Dynamic</td><td>[0,16]</td><td>0.44</td></tr><tr><td>Spotlight (ours)</td><td>X</td><td>X</td><td>163</td><td>[0,1]</td><td>0.58</td></tr></table>
+
+![](images/d9fe460af03c8568f07052abf85c4a8b820f74e831035165c10a006ca41de926.jpg)  
+Figure 5: Downstream QA tasks. (Left) Relative score of each method compared to the vanilla baseline; each point denotes a subtask. (Right) Absolute score comparison.
+
+# 4 Experimentation
+
+Our evaluation spans multiple dimensions. (1) We first assess the errors introduced by retrieval and sparsification, measured by retrieval accuracy and perplexity score. (2) We then evaluate long-context key information retrieval using the Needle-in-a-Haystack [14] benchmark. (3) Next, we evaluated downstream QA tasks on LongBench [6], comparing response similarity between compressed and original models using Rouge-L. The key insight from this Rouge-L comparison is straightforward: higher-quality compression produces more concise outputs. (4) We also measure end-to-end throughput gains and the execution efficiency of our CUDA ops.
+
+To evaluate generalization, we tested the performance of the Qwen2.5 series [24] across various model sizes and LLaMA3-8B [13] across diverse training corpora. Additionally, we conducted ablation studies to assess the impact of loss functions and attention estimation methods, with details provided in Appendix B.5 and B.6.
+
+Experimental setups. We employ LLaMA3-8B [13], LLaMA2-7B, LLaMA2-7B-Chat [21], and Qwen2.5 models (1.5B, 7B, 14B) [24] as base models. The baseline methods compared include (1) oracle top- $k$  retrieval, (2) linear LSH top- $k$ , (3) Quest, and (4) MagicPIG. Their implementation details are provided in Appendix A.2.
+
+Our MLP hashing function employs 128-dimensional input, intermediate, and output layers, with a distinct MLP for each head in every layer, producing a 128-bit hash code—much shorter than MagicPIG's minimum of 720 bits. Only the hashing functions are trainable. Training data consists of 8,192 samples, evenly drawn from the Book and Arxiv datasets [22].
+
+![](images/91f08f1a5e851fd535102bc624936f9865209f612857bd1fe73d89127cb10859.jpg)
+
+![](images/726201a08517fcdd2a6cf0dde5899ac572a7bb14015234a58fcf7d0a4a7f7243.jpg)
+
+![](images/b0a7e5c69088ffa36b1295e3ea2e7d3ba338ae0c5ce27e898e9742b7e877a8d0.jpg)
+
+![](images/654db1a9985f74393d8c4f30b1fc198f1385272c60b6fa409f7b960870fbf791.jpg)
+
+![](images/012052ca94e5ffcf8ab19a32f44e65baac2ffd778cc593319f0d86d00a0d0141.jpg)  
+Figure 6: Efficiency. (Upper Left) End-to-end throughput comparison with fixed context length across varying batch sizes. (Bottom Left) End-to-end throughput comparison with fixed batch size across different context lengths. (Upper Right) Hash code size comparison between MagicPIG and our method, alongside the execution latency of the two most computationally intensive operations in our method. (Bottom Right) Complexity comparison of different computational steps.
+
+![](images/5a3ec547a054f3f981b06a171940d6feb63fd5e8a46fa364605072e032b164e7.jpg)
+
+<table><tr><td>Symbol Def.</td><td>Inference Sub-Step</td><td>Complexity</td></tr><tr><td>h: head count</td><td>① Bitwise encoding</td><td>O(hd)</td></tr><tr><td>d: embed dim</td><td>② Similarity search</td><td>O(n hd)</td></tr><tr><td>n: sequence length</td><td>③ Top-k gathering</td><td>O(n h log k)</td></tr><tr><td></td><td>④ Sparse attention</td><td>O(khd)</td></tr></table>
+
+To improve efficiency, hidden states for all layers are precomputed and stored, enabling independent layer-wise training without joint fine-tuning. Training uses  $\gamma = 64$ , a learning rate of  $1 \times 10^{-3}$ ,  $\beta = 1$ , and  $\alpha = 3$ , for one epoch. Additional details are provided in Appendix A.1. The pruning rate remains fixed at  $98\%$  during training, irrespective of evaluation settings.
+
+# 4.1 Main Results
+
+KV retrieval accuracy. This experiment compares linear LSH and MLP hashing for KV retrieval. We used the first sample from PG19 as test data and assessed KV retrieval accuracy with the average IoU score across all heads and layers. IoU is computed as the intersection of the top- $k$  indices retrieved by the algorithm and the oracle top- $k$  indices, divided by their union.
+
+For LSH, we initialized the projection matrix via QR decomposition (see Appendix A.8). For MLP hashing, parameters were initialized with random Gaussian distributions. Results in Table 1 show that LSH performs best without training but shows minimal improvement post-training. Conversely, MLP hashing markedly improves retrieval accuracy after training, achieving the highest performance.
+
+Language modeling. We evaluated three language modeling benchmarks—PG19 [19], ProofPile [5], and CodeParrot [18]—with 100, 79, and 100 samples, respectively, using perplexity to detect minor errors from sparsification. Additional experimental details are in Appendix A.5.
+
+Results in Table 2 show our method outperforms Quest with a tenfold reduction in token budget for most models. Comparisons with MagicPIG (Table 3) indicate that MagicPIG depends heavily on local windows and sink tokens, failing without them, while our method autonomously identifies these elements, demonstrating significantly higher retrieval accuracy.
+
+MagicPIG excels at retrieving facts in scenarios like Needle-in-a-Haystack [14], its contribution should not be dismissed based solely on its limitations in this language modeling test.
+
+Needle-in-a-Haystack. We use the offline evaluation version of NIAH [1], which differs from ChatGPT scoring by using the Rouge score to measure output accuracy. We utilize LLaMA3-8B [13] as the base model, evaluating context lengths ranging from 256 to 8192, with a step size of 256 for testing. The Needle dataset is highly prompt-sensitive, we provide the needle, retrieval question, and haystack context in Appendix A.4. Additionally, to reduce output variability, all evaluated methods use greedy search. As shown in Figure 4, Spotlight Attention achieves performance on par with the original model.
+
+Downstream QA tasks. We evaluated the performance of various compression methods on Long-Bench [6] subtasks. Quest employs a token budget of 1,024, while MagicPIG uses 4 sink tokens, a 64-token local window, and a 1,500-bit hash code per key, following their default configurations. Our method uses a token budget of 163. All experimental results reported in the main text use LLaMA3-8B as the base model, scores of other models are provided in Appendix B.4, and more experimental setup details are in Appendix A.6. Figure 5 (Right) compares absolute scores across all
+
+subtasks. More importantly, Figure 5 (Left) shows relative scores, revealing that our method's scores closely align with those of the vanilla model.
+
+We also assessed output fidelity using Rouge-L to measure similarity between the vanilla model's outputs and those of these methods. Results are presented in Table 4. Our method's outputs are the most similar to the vanilla model, with the longest consecutive subsequence match approaching  $60\%$ . In contrast, Quest requires retrieving six times more tokens to achieve comparable similarity. These findings hold for nearly all subtasks, with detailed scores of each subtask provided in Appendix B.2.
+
+# 4.2 Efficiency
+
+All efficiency experiments were performed on Qwen2.5-7B [24] using eight A100 GPUs. For enhanced flexibility, experiments utilized the HuggingFace Transformers framework, optimized with pipeline parallelization and KV cache pre-allocation to boost throughput.
+
+End-to-end throughput evaluation. To evaluate model throughput at extended context lengths (e.g., 2M tokens), we expanded positional encoding, disregarding output quality. We selected the first sample from the PG19 test set [19], repeating it to reach a 2M-token context. GPU execution time was measured using CUDA events. We generated eight consecutive tokens, computed throughput by dividing by generation time, and averaged three runs per data point. Results in Figure 6 (Left) show that our approach consistently delivers throughput gains, especially at larger input scales.
+
+Kernel evaluation. We implemented CUDA kernels for both bit-packing and NXOR GEMM. Bit-packing compresses 32 Torch boolean values into a single unsigned 32-bit integer, as detailed in Appendix A.7. For NXOR GEMM, we utilized the standard library's popcount to count bit-1s in each NXOR result. For top- $k$  gathering and sparse attention, we employed Torch and FlashAttention implementations, respectively. As shown in Figure 6 (Upper Right), compared to dense vectors, our hashing-based similarity search significantly reduces storage and computation, enabling bit-packing and similarity search within  $100\mu s$  for context lengths up to 512K.
+
+Table 5: Ablation on model size. Various Qwen2.5 model sizes, augmented with Spotlight Attention, demonstrated better perplexity across diverse language modeling tasks.  
+
+<table><tr><td>Model</td><td>Method</td><td>PG19</td><td>Math</td><td>Code</td></tr><tr><td rowspan="2">Qwen2.5-1.5B</td><td>Vanilla</td><td>13.828</td><td>4.181</td><td>5.081</td></tr><tr><td>Spotlight Top-2% (ours)</td><td>13.510</td><td>4.143</td><td>5.064</td></tr><tr><td rowspan="2">Qwen2.5-7B</td><td>Vanilla</td><td>11.112</td><td>3.833</td><td>4.951</td></tr><tr><td>Spotlight Top-2% (ours)</td><td>9.930</td><td>3.497</td><td>4.645</td></tr><tr><td rowspan="2">Qwen2.5-14B</td><td>Vanilla</td><td>8.416</td><td>3.230</td><td>4.472</td></tr><tr><td>Spotlight Top-2% (ours)</td><td>8.261</td><td>3.196</td><td>4.440</td></tr></table>
+
+Table 6: Ablation on training tasks. Alongside the standard ArXiv+Books training data, we trained models on the C4 and GitHub Code datasets, achieving comparable perplexity (PPL) results.  
+
+<table><tr><td>LLaMA3-8B</td><td>Training Corpus</td><td>PG19</td><td>Math</td><td>Code</td></tr><tr><td>Oracle Top-2%</td><td>-</td><td>8.881</td><td>3.590</td><td>5.353</td></tr><tr><td rowspan="2">Spotlight</td><td>Arxiv + Book (default)</td><td>8.977</td><td>3.621</td><td>5.434</td></tr><tr><td>C4</td><td>8.958</td><td>3.631</td><td>5.417</td></tr><tr><td>Top-2% (ours)</td><td>Github Code</td><td>8.891</td><td>3.611</td><td>5.393</td></tr></table>
+
+# 4.3 Ablations
+
+In the main text, we present ablation studies on model size and training tasks only. Additional ablation experiments, including loss functions and attention estimation methods, are detailed in Appendix B.5 and B.6, respectively.
+
+All ablation experiments use language modeling perplexity as the evaluation metric. We assessed performance on PG19 [19], Proof-Pile (Math) [5], and CodeParrot (Code) [18], with sample sizes of 100, 79, and 100, respectively.
+
+Model size. We compare Qwen2.5 [24] models of 1.5B, 7B, and 14B parameters, all trained with the standard recipe. As shown in Table 5, Spotlight Attention achieves consistently strong performance across different model sizes.
+
+Training tasks. To validate our method's applicability across diverse tasks, we trained models on the GitHub Code [18] and C4 [18] datasets, in addition to the ArXiv and Books [22] datasets used previously. As presented in Table 6, training with GitHub Code or C4 unexpectedly outperformed the ArXiv+Books combination, demonstrating the robust adaptability of our training framework. However, due to the already-completion of the main results, we did not re-evaluate all benchmarks with these checkpoints despite their superior performance.
+
+# 5 Conclusion and Limitation
+
+We introduce Spotlight Attention, an advancement over Quest and MagicPIG, incorporating a nonlinear hashing function and an optimized framework. This approach addresses the underfitting of MagicPIG's linear hashing while significantly reducing hash code length. Spotlight Attention performs well on downstream tasks but has limitations. The IoU remains around  $40\%$  despite non-linear hashing, indicating potential for improvement.
+
+# 6 Acknowledgments
+
+This work was supported by the National Science Fund for Distinguished Young Scholars (No.62025603), the National Natural Science Foundation of China (No. U21B2037, No. U22B2051, No. U23A20383, No. 62176222, No. 62176223, No. 62176226, No. 62072386, No. 62072387, No. 62072389, No. 62002305 and No. 62272401), the Natural Science Foundation of Fujian Province of China (No. 2021J06003, No.2022J06001), the National Natural Science Foundation of China No.624B2119, and the China Postdoctoral Science Foundation (No.BX20250384).
+
+# References
+
+[1] Github repository: 66ring/llmtest_needleinahaystack-local, 2023.  
+[2] M. Adnan, A. Arunkumar, G. Jain, P. Nair, I. Soloveychik, and P. Kamath. Keyformer: Kv cache reduction through key tokens selection for efficient generative inference.  $MLSys$ , 2024.  
+[3] A. Agrawal, A. Panwar, J. Mohan, N. Kwatra, B. S. Gulavani, and R. Ramjee. Sarathi: Efficient llm inference by piggybacking decodes with chunked prefills. arXiv, 2023.  
+[4] A. Andoni, P. Indyk, T. Laarhoven, I. Razenshteyn, and L. Schmidt. Practical and optimal lsh for angular distance. In NeurIPS, 2015.  
+[5] Z. Azerbayev, E. Ayers, and B. Piotrowski. Github repository: hoskison-center/proof-pile, 2022.  
+[6] Y. Bai, X. Lv, J. Zhang, H. Lyu, J. Tang, Z. Huang, Z. Du, X. Liu, A. Zeng, L. Hou, et al. LongBench: A bilingual, multitask benchmark for long context understanding. In ACL, 2024.  
+[7] R. A. Bradley and M. E. Terry. Rank analysis of incomplete block designs: I. the method of paired comparisons. Biometrika, 1952.  
+[8] Z. Cao, T. Qin, T.-Y. Liu, M.-F. Tsai, and H. Li. Learning to rank: from pairwise approach to listwise approach. In ICML, 2007.  
+[9] Y. Chen, G. Wang, J. Shang, S. Cui, Z. Zhang, T. Liu, S. Wang, Y. Sun, et al. NACL: A general and effective KV cache eviction framework for LLM at inference time. In NACL, 2024.  
+[10] Z. Chen, R. Sadhukhan, Z. Ye, Y. Zhou, J. Zhang, N. Nolte, Y. Tian, M. Douze, L. Bottou, Z. Jia, and B. Chen. MagicPIG: LSH sampling for efficient LLM generation. In NeurIPS, 2024.  
+[11] L. Gao, J. Tow, B. Abbasi, S. Biderman, S. Black, A. DiPofi, C. Foster, et al. Github repository: Eleutherai/lm-evaluation-harness, 2024.  
+[12] S. Ge, Y. Zhang, L. Liu, M. Zhang, J. Han, and J. Gao. Model tells you what to discard: Adaptive KV cache compression for LLMs. In ICLR, 2024.  
+[13] A. Grattafori, A. Dubey, A. Jauhri, A. Pandey, A. Kadian, A. Al-Dahle, A. Letman, et al. The llama 3 herd of models. arXiv, 2024.  
+[14] G. Kamradt. Github repository: gkamradt/llmtest_needleinahaystack, 2023.  
+[15] H. Li, Y. Li, A. Tian, T. Tang, Z. Xu, X. Chen, N. Hu, W. Dong, Q. Li, and L. Chen. A survey on large language model acceleration based on kv cache management. arXiv, 2025.  
+[16] Y. Li, Y. Huang, B. Yang, B. Venkitesh, A. Locatelli, H. Ye, T. Cai, P. Lewis, and D. Chen. SnapKV: LLM knows what you are looking for before generation. In NeurlPS, 2024.  
+[17] M. Oren, M. Hassid, N. Yarden, Y. Adi, and R. Schwartz. Transformers are multi-state rnns. arXiv, 2024.  
+[18] Z. Peitian. Huggingface dataset: namespace-pt/long-llm-data, 2024.  
+[19] J. W. Rae, A. Potapenko, S. M. Jayakumar, C. Hillier, and T. P. Lillicrap. Compressive transformers for long-range sequence modelling. In ICLR, 2020.  
+[20] J. Tang, Y. Zhao, K. Zhu, G. Xiao, B. Kasikci, and S. Han. QUEST: Query-Aware Sparsity for Efficient Long-Context LLM Inference. In ICML, 2024.  
+[21] H. Touvron, L. Martin, K. Stone, T. Scialom, et al. Llama 2: Open foundation and fine-tuned chat models. arXiv, 2023.  
+[22] M. Weber, D. Y. Fu, Q. G. Anthony, Y. Oren, S. Adams, A. Alexandrov, X. Lyu, et al. Redpajama: an open dataset for training large language models. In NeurIPS, 2024.  
+[23] G. Xiao, Y. Tian, B. Chen, S. Han, and M. Lewis. Efficient streaming language models with attention sinks. In ICLR, 2024.
+
+[24] A. Yang, B. Yang, B. Zhang, B. Hui, B. Zheng, B. Yu, C. Li, D. Liu, F. Huang, et al. Qwen2.5 technical report. arXiv, 2025.  
+[25] Z. Zhang, Y. Sheng, T. Zhou, T. Chen, L. Zheng, R. Cai, Z. Song, Y. Tian, C. Re, C. Barrett, Z. Wang, and B. Chen. H2o: Heavy-hitter oracle for efficient generative inference of large language models. In NeurIPS, 2023.
+
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+
+# 2. Limitations
+
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+
+# 3. Theory assumptions and proofs
+
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+
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+
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+
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+
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+
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+
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+
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+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: We provide open access to data and code in the Abstract.
+
+# Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
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+
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: All experimental details are provided in the Experiment Setup chapter, with implementation details included in the Appendix A.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: For experiments involving randomness (Sec 4.2), such as the efficiency tests, we conducted multiple sampling iterations and reported the mean values.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
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+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We give the relevant information in Sec 4.2.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: Our research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.  
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
+
+Justification: Our approach does not involve societal impact.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
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+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+# Answer: [NA]
+
+Justification: Our work does not include such risks.
+
+# Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
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+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+# Answer: [Yes]
+
+Justification: All assets used are well-known, properly referenced, and their information is readily accessible.
+
+# Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+# Answer: [NA]
+
+Justification: Our paper does not release new assets.
+
+# Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: Our paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: Our paper does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: Our study focuses on open-source LLMs, but LLMs are not involved in the research methodology.
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.
+
+# A Implementation Details
+
+# A.1 Training Setup
+
+Table 7 summarizes the training configuration. For further details, including low-level specifics and weight files, refer to our open-source code.
+
+Table 7: Detailed training configuration.  
+
+<table><tr><td colspan="3">General</td><td colspan="5">Learning Rate</td><td colspan="2">Gradient</td></tr><tr><td>Precision</td><td>Num Iters</td><td>Batch Size</td><td>Max LR</td><td>Min LR</td><td>Warm Up Iters</td><td>Warm Up Method</td><td>Annealing</td><td>Accumulation</td><td>Clipping</td></tr><tr><td>bf16</td><td>8,192</td><td>1</td><td>0.001</td><td>0</td><td>81</td><td>linear</td><td>cosine</td><td>1</td><td>1.0</td></tr><tr><td colspan="3">Optimizer</td><td colspan="7">Data</td></tr><tr><td>Optimizer</td><td>β1</td><td>β2</td><td>Weight Decay</td><td>Corpus</td><td>Arxiv Samples</td><td>Book Samples</td><td>LLaMA2 Trunc</td><td>LLaMA3/Qwen Trunc</td><td>Trunc Side</td></tr><tr><td>adamw</td><td>0.9</td><td>0.98</td><td>0.1</td><td>arxiv, book</td><td>4,096</td><td>4,096</td><td>4.096</td><td>8,192</td><td>right</td></tr></table>
+
+# A.2 Baseline Methods
+
+Oracle top-k. As introduced in Section 3.2, this baseline employs original attention to identify top- $k$  indices, serving as a theoretical upper bound on performance. We provide pseudo-code below to illustrate a concrete implementation of the oracle top- $k$  algorithm.
+
+LSH top-k. This method uses training-free angular LSH [4] with the same hash code length as Spotlight Attention, providing a reference for MagicPIG's linear hashing under our framework. Initialization details for angular LSH are in Appendix A.8.
+
+Quest. [20] We adopt Quest's official implementation with default hyperparameters, modifying only the token budget. Beyond the default 1024-token budget, we test ultra-low budgets to align with Spotlight Attention. For instance, for LLaMA2-7B and LLaMA3-8B, we use budgets of 128 and 256 tokens, compared to Spotlight Attention's 81 and 163 tokens.
+
+MagicPIG. [10] We adopt MagicPIG's official implementation with default hyperparameters ( $K = 15$ ,  $L = 100$ ), retaining 64 local tokens and 4 initial tokens. For all experiments except the efficiency test, we use the official Python evaluation code to conduct the experiments.
+
+# A.3 Ranking Loss
+
+We provide the ranking loss calculation in the following pseudo-code. To support longer sequence lengths during training, we employ three optimization techniques: (1) Random query selection, where only queries specified by query_index are optimized, rather than all queries. (2) Random top- $k$  selection, where max_top is randomly sampled from the top- $k$  set for optimization. (3) Random non-top- $k$  selection, where max_oth is randomly sampled from the non-top- $k$  set for optimization. These techniques enhance training efficiency in long-context scenarios.
+
+```txt
+1 def ranking_loss(   
+2 draft_attn,   
+3 true_attn,   
+4 query_index,   
+5 max_top,   
+6 max_oth,   
+7 maskout,   
+8 beta: float  $= 1.0$    
+9 alpha: float  $= 0.0$  ..   
+10   
+11 loss  $=$  torch.tensor(0,dtype  $\equiv$  torch.float32)   
+12 criterion  $=$  torch(nn.BCEWithLogitsLoss()   
+13   
+14 # PREPARE & APPLY MASK   
+15 num_kv  $=$  true_attn.shape[-1]   
+16 mask  $=$  torch.triu(torch.ones((num_kv, num_kv), dtype  $\equiv$  torch(bool device  $\equiv$  true_attn_device), diagonal  $\coloneqq 1$  [None, None, :,:]   
+17 if query_index is not None:   
+18 mask  $=$  mask[., query_index, :]   
+19 true_attn  $=$  torchmasked_fill(true_attn, mask, value  $\equiv$  torch+finfo(true_attn.dtype).min)   
+20 indices  $=$  torch.argsort(true_attn, dim=-1, descending=True)   
+21   
+22   
+23 top_cnt  $=$  int(indices.shape[-1] \* (1 - maskout))   
+24 top Indices  $=$  indices[.,top_cnt]   
+25 oth Indices  $=$  indices[.,top_cnt:]   
+26
+```
+
+```python
+27 if max_top is not None: top_rnd Indices = torch.randomperm(top_cnt, dtype=torch.int64, device=indices_device) [max_top] topIndices = topIndices[., top_rnd Indices]   
+30 if max_oth is not None: oth_rnd indices = torch.randomperm(indices.shape[-1] - top_cnt, dtype=torch.int64, device=indices_device)[ max_oth] oth Indices = oth Indices[., oth_rnd indices]   
+33   
+34 top_mask = torch.gather mask.expand_as(true_attn), dim=-1, index=topIndices)[., :, None]   
+35 oth_mask = torch.gather mask.expand_as(true_attn), dim=-1, index=oth Indices)[., None, :]   
+36   
+37 top_draft_attn = torch.gather(draft_attn, dim=-1, index=topIndices)[., :, None]   
+38 oth_draft_attn = torch.gather(draft_attn, dim=-1, index=oth Indices)[., None, :]   
+39   
+40 residual = top_draft_attn - oth_draft_attn   
+41 residual_mask = (top_mask | oth_mask).expand_as(residual). flatten(-3)   
+42   
+43 logits = residual Flatten(-3) [residual_mask(bool()) ]   
+44 labels = torch. ones_like(logits)   
+45 loss += criterion(logits * beta - alpha, labels).cpu()   
+46   
+47 diff = torch.count_nonzero(logits < 0) / logits.numel()   
+48   
+49 return diff, loss
+```
+
+# A.4 Needle-in-a-Haystack
+
+NIAH is a prompt-sensitive test focusing on relative performance changes before and after applying Spotlight Attention, rather than absolute performance. The haystack, needle, and question used in our evaluation are depicted in Figure 7. The prompt, selected from a set proven effective in practice, is shown in Figure 8.
+
+# Haystack, Needle and Question
+
+Haystack
+
+Paul Graham Essays
+
+Needle
+
+The best thing to do in San Francisco is eat a sandwich and sit in Dolores Park on a sunny day.
+
+Question
+
+What is the best thing to do in San Francisco?
+
+Figure 7: The haystack, needle, and retrieval question for the NIAH.
+
+# Prompt
+
+LLaMA2-7B / LLaMA3-8B
+
+You are a helpful AI bot that answers questions for a user. Keep your response short and direct.<context><retrieval question> Don't give information outside the document or repeat your findings. The document definitely contains the answer, and I'm  $100\%$  sure. So try your best to find it.
+
+LLaMA2-7B-Chat
+
+<s>[INST] You are a helpful AI bot that answers questions for a user. Keep your response short and direct. <context><retrieval question> Don't give information outside the document or repeat your findings. The document definitely contains the answer, and I'm  $100\%$  sure. So try your best to find it. [/INST]
+
+Figure 8: Prompts used in NIAH test.
+
+# A.5 Language Modeling Perplexity
+
+For Quest, we used the first 128 tokens for pre-filling with full attention. For MagicPIG, we pre-filled the first 1024 tokens with full attention to compute the key's mean value. Our Spotlight Attention method employed sparse KV retrieval throughout without pre-filling. For LLaMA2 models, we evaluated the first 4K tokens per sample; for LLaMA3 and Qwen2.5 models, we used the first 8K tokens. We compared Quest and MagicPIG separately, as Quest and our method use a fixed token budget, while MagicPIG dynamically selects tokens and uses local windows and sink tokens.
+
+In comparison with Quest, we calculated the token budget by multiplying the token sequence length by the pruning rate, with a minimum budget of 20.
+
+# A.6 Downstream QA Tasks
+
+For contexts exceeding 8K tokens, we truncated 8K tokens from right to left. We evaluated all LongBench [6] subdatasets using the official test scripts. The LLaMA2 chat model employs the official chat template, whereas other models do not.
+
+# A.7 Bit-Packing CUDA Kernel
+
+Bit-packing (Figure 9) is crucial because PyTorch lacks a native bit type, and boolean values are stored as full bytes. Without compaction, storage usage would increase substantially. The bit-packing program groups 32 boolean values and iteratively packs them into a single uint_32t, as detailed in the pseudocode accompanying Figure 9.
+
+![](images/c27a722caf3a757668e6ea33ef1bde00500a46a4d8303946fb6c41723cf9ec8b.jpg)  
+① Shift logical left  
+Figure 9: Bit-packing.
+
+```python
+1 def bit Packing(tensor):   
+2 # WE PRESENT THE LOGIC HERE,   
+3 # WITH THE ACTUAL IMPLEMENTATION WRITTEN IN CUDA.   
+4 n, d = tensor.shape   
+5 assert d % 32 == 0   
+6   
+7 tensor = tensor.chunk(32, dim=-1)   
+8 output = torch.zeros((n, d // 32), dtype=uint32)   
+9 for x in tensor:   
+10 output <= 1   
+11 output |= x & 0x01   
+12   
+13 return output
+```
+
+# A.8 LSH Weight Initialization
+
+For the linear hashing function, we employ angular LSH with a rotation matrix as the initial value, outperforming standard random initialization. To generate a  $d$ -dimensional rotation matrix, we use QR decomposition. We first create a random matrix  $R \in \mathbb{R}^{d \times d}$ , with each element independently sampled from a standard normal distribution:
+
+$$
+R \in \mathbb {R} ^ {n \times n}, R _ {i, j} \sim \mathcal {N} (0, 1). \tag {9}
+$$
+
+We then perform QR decomposition on  $R$ , yielding an orthogonal matrix  $\mathbf{Q}$  and an upper triangular matrix  $\mathbf{R}$ :
+
+$$
+R = \mathbf {Q} \mathbf {R}, \quad \mathbf {Q} ^ {\top} \mathbf {Q} = \mathbf {I}. \tag {10}
+$$
+
+The matrix  $\mathbf{Q}$  is not necessarily in the special orthogonal group  $\mathbb{S}0(d)$ , as its determinant can be either  $+1$  or  $-1$ . To ensure  $\mathbf{Q} \in \mathbb{S}0(d)$ , we flip the sign of the first column of  $\mathbf{Q}$  if  $\operatorname*{det}(\mathbf{Q}) < 0$ .
+
+# B More Experimental Results
+
+# B.1 Per-Head Retrieval Accuracy
+
+The IoU reported in the experiments section is averaged across all heads and layers. Given the varied behavior of LLM heads, individual IoU scores differ significantly. Figure 10 presents detailed per-head IoU comparisons for various models, using Spotlight Attention and linear hashing functions, both before and after training.
+
+# B.2 Detailed QA Response Fidelity Scores
+
+In the main text, we report the average Rouge-L score across all subdatasets as the similarity metric. However, scores vary significantly across individual subdatasets. Figure 11 provides detailed similarity scores for each method compared to the original LLaMA3-8B model. To avoid confusion, we assign a Rouge-L score of 1 to identical outputs, except in special cases like outputs containing only  $\backslash n$  characters, where tokenization issues may prevent a perfect score.
+
+![](images/0b5528b9c075fa501da1e29cfeda04315e71c19b56f8008147cf8163a6c52114.jpg)  
+Figure 10: Per-head retrieval accuracy. Measured by Intersection over Union (IoU) of top- $k$  sets predicted by oracle and Spotlight. (1) The results of linear hashing reveal that most heads maintain low IoU both pre- and post-training, suggesting their latent distributions are challenging to approximate with linear functions. (2) Random parameter initialization results in low before-training IoU for Spotlight Attention, which improves significantly after training.
+
+# B.3 Few-Shot Learning
+
+To evaluate Spotlight Attention with short context lengths, we restrict the KV cache to 20 tokens and assess performance on 5-shot learning datasets from LM-Eval-Harness [11], including GLUE, SuperGLUE, OpenBookQA, HellaSwag, PiQA, Winogrande, ARC-E, ARC-C, MathQA, and MMLU.
+
+As shown in Table 8, Spotlight Attention delivers strong performance. Comparisons with Quest and MagicPIG are omitted, as their large local windows or budgets consume most of the prompt length on most datasets, rendering such comparisons less meaningful. Instead, we emphasize the relative performance between Spotlight Attention and the original model.
+
+Table 8: A 5-shot learning comparison between original model and Spotlight Attention under a fixed budget of 20 tokens was conducted across: GLUE (1), SuperGLUE (2), OpenBookQA (3), HellaSwag (4), PiQA (5), Winogrande (6), ARC-E (7), ARC-C (8), MathQA (9), and MMLU (10).  
+
+<table><tr><td></td><td>#1</td><td>#2</td><td>#3</td><td>#4</td><td>#5</td><td>#6</td><td>#7</td><td>#8</td><td>#9</td><td>#10</td><td>Win Rate</td></tr><tr><td>LLaMA2-7B</td><td>0.489</td><td>0.646</td><td>0.342</td><td>0.604</td><td>0.776</td><td>0.733</td><td>0.793</td><td>0.478</td><td>0.262</td><td>0.468</td><td>56%</td></tr><tr><td>+Spotlight</td><td>0.494</td><td>0.632</td><td>0.336</td><td>0.581</td><td>0.786</td><td>0.748</td><td>0.793</td><td>0.469</td><td>0.284</td><td>0.466</td><td>44%</td></tr><tr><td>LLaMA2-7B-Chat</td><td>0.625</td><td>0.717</td><td>0.346</td><td>0.598</td><td>0.763</td><td>0.729</td><td>0.811</td><td>0.474</td><td>0.295</td><td>0.482</td><td>89%</td></tr><tr><td>+Spotlight</td><td>0.582</td><td>0.664</td><td>0.35</td><td>0.578</td><td>0.759</td><td>0.703</td><td>0.809</td><td>0.467</td><td>0.275</td><td>0.482</td><td>11%</td></tr><tr><td>LLaMA3-8B</td><td>0.618</td><td>0.727</td><td>0.378</td><td>0.631</td><td>0.804</td><td>0.772</td><td>0.85</td><td>0.534</td><td>0.418</td><td>0.663</td><td>38%</td></tr><tr><td>+Spotlight</td><td>0.620</td><td>0.736</td><td>0.378</td><td>0.624</td><td>0.804</td><td>0.773</td><td>0.851</td><td>0.533</td><td>0.411</td><td>0.664</td><td>62%</td></tr></table>
+
+# B.4 Downstream QA Tasks
+
+The main text primarily reports LLaMA3-8B scores on LongBench. Here, we additionally present results for LLaMA2-7B and LLaMA2-7B-Chat, alongside comparisons of our method and Quest under varying token budgets, as shown in Table 9.
+
+# B.5 Ablation on Loss Function
+
+In the previous section, we analyzed the limitations of reconstruction loss. Here, we compare its empirical performance with our proposed ranking loss. As shown in Table 10, our ranking loss significantly outperforms reconstruction loss, which provides minimal to no benefit.
+
+![](images/6df06db751d8e328758fe9480bbe790c6ae2b858f1c539e8f921993e0313e39e.jpg)  
+Figure 11: Detailed similarity between (i) the outputs of different models (including LLaMA3-8B itself) and (ii) those of LLaMA3-8B.
+
+Table 9: Absolute scores on LongBench. Performance comparison of different methods on Long-Bench's long-text downstream tasks: NarrativeQA (1-1), Qasper (1-2), MultiFieldQA-en (1-3), MultiFieldQA-zh (1-4), HotpotQA (2-1), 2WikiMultihopQA (2-2), MuSiQue (2-3), DuReader (2-4), GovReport (3-1), QMSum (3-2), MultiNews (3-3), VCSUM (3-4), TREC (4-1), TriviaQA (4-2), SAMSum (4-3), LSHT (4-4), PassageCount (5-1), PassageRetrieval-en (5-2), PassageRetrieval-zh (5-3), LCC (6-1), and RepoBench-P (6-2). (1) With  $98\%$  tokens pruned and no local window or global sink tokens, Spotlight Attention outperforms Quest and MagicPIG. (2) Spotlight Attention achieves performance on par with the original model, even in tasks like summarization and few-shot learning. (3) On subsets of Chinese (1-4, 2-4, 3-4), other LLaMA2-7B-Chat-based models generated answers in English, while Quest produced responses in Chinese, achieving overwhelmingly high scores.
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Configuration</td><td colspan="5">Single-Doc.</td><td colspan="5">Multi-Doc.</td><td colspan="5">Summarization</td><td colspan="5">Fev-Shot</td><td colspan="5">Synthetic</td><td colspan="2">Code</td></tr><tr><td>#1-1</td><td>#2-1</td><td>#3-1</td><td>#4</td><td>Avg.</td><td>#2-1</td><td>#2-2</td><td>#2-3</td><td>#2-4</td><td>Avg.</td><td>#3-1</td><td>#2-3</td><td>#3-3</td><td>#4</td><td>Avg.</td><td>#4-1</td><td>#2-4</td><td>#3-4</td><td>Avg.</td><td>#5-1</td><td>#5-2</td><td>#3-3</td><td>Avg.</td><td>#6-1</td><td>#6-2</td><td>Avg.</td><td></td></tr><tr><td>LLaMA2-7B</td><td>8.73</td><td>7.18</td><td>15.42</td><td>13.84</td><td>11.29</td><td>6.55</td><td>8.27</td><td>2.91</td><td>11.38</td><td>7.27</td><td>15.06</td><td>19.80</td><td>6.03</td><td>9.30</td><td>12.54</td><td>68.00</td><td>30.62</td><td>30.83</td><td>18.25</td><td>36.92</td><td>1.26</td><td>6.97</td><td>8.00</td><td>5.41</td><td>63.66</td><td>56.63</td><td>60.14</td><td></td></tr><tr><td>+Quest</td><td>1024 Token Budget</td><td>8.78</td><td>9.78</td><td>17.95</td><td>14.86</td><td>12.84</td><td>8.91</td><td>8.51</td><td>2.95</td><td>12.53</td><td>8.22</td><td>18.38</td><td>20.84</td><td>9.63</td><td>8.40</td><td>14.31</td><td>66.00</td><td>67.06</td><td>30.35</td><td>17.50</td><td>42.22</td><td>1.69</td><td>6.47</td><td>7.38</td><td>5.18</td><td>63.88</td><td>58.82</td><td>61.35</td></tr><tr><td>+MagiPIG</td><td>9.17</td><td>4.69</td><td>13.69</td><td>5.76</td><td>8.32</td><td>6.00</td><td>5.16</td><td>2.07</td><td>8.05</td><td>5.32</td><td>6.73</td><td>17.78</td><td>3.27</td><td>3.34</td><td>7.78</td><td>45.00</td><td>31.53</td><td>14.53</td><td>8.00</td><td>24.76</td><td>0.83</td><td>4.06</td><td>1.50</td><td>2.13</td><td>47.88</td><td>44.16</td><td>46.02</td><td></td></tr><tr><td>+Spotlight</td><td>90% Pruned (≤ 409)</td><td>10.16</td><td>13.76</td><td>14.56</td><td>11.81</td><td>8.99</td><td>8.99</td><td>2.53</td><td>12.33</td><td>10.99</td><td>16.23</td><td>16.32</td><td>19.11</td><td>8.94</td><td>8.34</td><td>66.50</td><td>68.00</td><td>30.90</td><td>19.00</td><td>39.90</td><td>0.99</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td></td></tr><tr><td>+Spotlight</td><td>98% Pruned (≤ 81)</td><td>10.76</td><td>8.09</td><td>16.63</td><td>12.95</td><td>12.10</td><td>6.46</td><td>7.98</td><td>2.98</td><td>11.25</td><td>7.09</td><td>16.32</td><td>19.31</td><td>10.24</td><td>8.51</td><td>13.59</td><td>67.50</td><td>38.08</td><td>10.00</td><td>38.56</td><td>2.07</td><td>8.27</td><td>6.79</td><td>5.71</td><td>64.10</td><td>56.76</td><td>60.43</td><td></td></tr><tr><td>LLaMA2-7B-Chat</td><td>18.71</td><td>24.83</td><td>31.63</td><td>8.36</td><td>20.88</td><td>31.76</td><td>28.22</td><td>12.66</td><td>2.48</td><td>18.78</td><td>27.18</td><td>20.36</td><td>26.17</td><td>0.24</td><td>18.48</td><td>64.50</td><td>77.85</td><td>40.76</td><td>15.70</td><td>49.56</td><td>1.64</td><td>2.75</td><td>3.42</td><td>2.60</td><td>54.57</td><td>48.74</td><td>51.65</td><td></td></tr><tr><td>+Quest</td><td>1024 Token Budget</td><td>19.14</td><td>13.76</td><td>14.56</td><td>11.81</td><td>8.99</td><td>8.99</td><td>2.53</td><td>12.33</td><td>10.99</td><td>16.23</td><td>16.32</td><td>19.11</td><td>8.94</td><td>8.34</td><td>66.50</td><td>68.00</td><td>30.90</td><td>19.00</td><td>3.95</td><td>49.56</td><td>2.08</td><td>6.47</td><td>7.38</td><td>5.18</td><td>63.88</td><td></td><td></td></tr><tr><td>+Quest</td><td>128 Token Budget</td><td>14.11</td><td>12.78</td><td>17.59</td><td>9.42</td><td>13.47</td><td>28.38</td><td>21.03</td><td>8.99</td><td>8.53</td><td>16.73</td><td>11.55</td><td>18.44</td><td>20.91</td><td>8.39</td><td>14.82</td><td>38.50</td><td>66.44</td><td>31.69</td><td>10.50</td><td>36.78</td><td>0.72</td><td>3.50</td><td>7.30</td><td>3.70</td><td>43.21</td><td>39.55</td><td>41.38</td></tr><tr><td>+MagiPIG</td><td>18.84</td><td>23.79</td><td>28.80</td><td>9.43</td><td>20.21</td><td>31.97</td><td>28.23</td><td>11.96</td><td>3.17</td><td>18.83</td><td>26.32</td><td>20.31</td><td>25.49</td><td>0.14</td><td>18.65</td><td>65.50</td><td>85.59</td><td>40.52</td><td>16.25</td><td>51.96</td><td>1.52</td><td>3.50</td><td>4.24</td><td>3.08</td><td>57.94</td><td>52.17</td><td>55.05</td><td></td></tr><tr><td>+Spotlight</td><td>90% Pruned (≤ 409)</td><td>18.84</td><td>23.79</td><td>28.80</td><td>9.43</td><td>20.21</td><td>31.97</td><td>28.23</td><td>12.66</td><td>2.48</td><td>18.78</td><td>26.32</td><td>20.31</td><td>25.49</td><td>0.14</td><td>18.65</td><td>65.50</td><td>85.59</td><td>40.52</td><td>16.25</td><td>51.96</td><td>1.52</td><td>3.50</td><td>4.24</td><td>3.08</td><td>57.94</td><td>52.17</td><td>55.05</td></tr></table>
+
+# B.6 Ablation on Attention Estimation Methods
+
+We evaluated two approaches: hashing with Hamming distance (our default choice) and down-projection with inner product. At a 16x compression rate, results in Table 11 show that hashing outperforms down-projection, demonstrating greater efficiency when the dimensionality is low.
+
+Table 10: Ablation on training loss. Compared to attention reconstruction loss, our proposed ranking loss yields significantly improved training outcomes.  
+
+<table><tr><td>Loss</td><td>PG19</td><td>Math</td><td>Code</td></tr><tr><td>Attn. Recon. Loss</td><td>21.341</td><td>8.856</td><td>11.573</td></tr><tr><td>Ranking Loss (ours)</td><td>8.977</td><td>3.621</td><td>5.434</td></tr></table>
+
+Table 11: Ablation on estimation method. We compared two attention estimation methods: hashing with Hamming distance (our default choice) and downprojection with inner product. Results indicate that hashing with Hamming distance performs better.  
+
+<table><tr><td>Estimation Method</td><td>Training</td><td>PG19</td><td>Math</td><td>Code</td></tr><tr><td>Down Proj. (16×) + Inner Prod.</td><td rowspan="2">✓</td><td>14.743</td><td>5.425</td><td>7.619</td></tr><tr><td>Hashing + Hamming Dist. (ours)</td><td>8.977</td><td>3.621</td><td>5.434</td></tr></table>

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+# Stability and Oracle Inequalities for Optimal Transport Maps between General Distributions
+
+Shubo Li
+
+Department of Statistics
+
+The Pennsylvania State University
+
+State College, PA 16803
+
+sk16034@psu.edu
+
+Lingzhou Xue
+
+Department of Statistics
+
+The Pennsylvania State University
+
+lzxue@psu.edu
+
+Yizhe Ding
+
+Department of Statistics
+
+The Pennsylvania State University
+
+State College, PA 16803
+
+yvd5212@psu.edu
+
+Runze Li
+
+Department of Statistics
+
+The Pennsylvania State University
+
+rzli@psu.edu
+
+# Abstract
+
+Optimal transport (OT) provides a powerful framework for comparing and transforming probability distributions, with wide applications in generative modeling, AI4Science and statistical inference. However, existing estimation theory typically requires stringent smoothness conditions on the underlying Brenier potentials and assumes bounded distribution supports, limiting practical applicability. In this paper, we introduce a unified theoretical framework for semi-dual OT map estimation that relaxes both of these restrictions. Building on sieved convex conjugate, our framework has two key contributions: (i) a new map stability bounds that holds without any second-order regularity assumptions on the true Brenier potentials, and (ii) an oracle inequality that cleanly decomposes the estimation error into statistical error, sieved bias, and approximation error. Specifically, our approximation error is measured in the  $L^{\infty}$  norm rather than Sobolev norm in the existing results, aligning more naturally with classical approximation theory. Leveraging these tools, we provide statistical error of semi-dual estimators with mild and verifiable conditions on the true OT map. Moreover, we establish the first theoretical guarantee for deep neural network OT map estimator between general distributions, with Tanh network function class as an example.
+
+# 1 Introduction
+
+Optimal Transport (OT) provides a powerful framework for transforming one probability measure into another. Concretely, given two distributions  $\mu$  and  $\nu$  on  $\mathbb{R}^d$ , the OT problem seeks a map  $T: \mathbb{R}^d \to \mathbb{R}^d$  that transports  $\mu$  to  $\nu$  while minimizing a cost function. Beyond its elegant mathematical foundation, OT has found broad applications in Ai4Science [39, 7, 29], computer vision [3, 23, 43, 12], and nonparametric statistical inference [10, 26, 24, 18], powering advances in scientific discovery and machine/deep learning. Despite its broad impact, rigorous theoretical guarantees for OT estimators under mild and realistic regularity conditions remain limited.
+
+This paper aims to contribute to this direction by focusing on the dual-type OT map estimators, formally introduced in Section 2, which are based on the characterization of the optimal transport map as the gradient of a convex function (the so-called Brenier potential). The seminal work of [27] established minimax optimality for a wavelet-truncated dual-type OT estimator, but under restrictive conditions:  $\mu$  is supported on  $[0,1]^d$  with a bounded density,  $\nu$  is compactly supported, and the true
+
+OT map is both  $\alpha$ -Hölder smooth and strongly convex. These assumptions exclude many practical OT applications.
+
+In parallel, Gunsilius [25] introduced an alternative analysis using the Poincaré inequality, a fundamental tool in PDEs and functional analysis. While still assuming compact and convex supports, this approach removed the need for Hölder smoothness or strong convexity of the Brenier potential. Building on this, [21] extended the theory to sub-exponential  $\mu$ , under  $(\alpha, a)$ -convexity and  $(\beta, a)$ -smoothness conditions on the Brenier potential. More recently, [20] relaxed the Poincaré inequality assumption to accommodate heavy-tailed  $\mu$  and  $\nu$ , and eliminated the  $(\alpha, a)$ -convexity condition via a new sieved OT estimator, characterizing how tail thickness of  $\mu$  and  $\nu$  influences convergence.
+
+We also note recent advances in plug-in type OT estimators, which directly compute the OT map between estimated distributions [10, 17, 34, 37, 5]. In parallel with these theoretical advances, significant efforts have focused on improving the computational efficiency of OT, including entropic regularization [15, 1, 22, 11, 19] and sum-of-squares formulations [35, 41]. Meanwhile, neural-network-based OT estimators have advanced rapidly, powering large-scale applications such as image generation and translation [33, 30, 4, 31, 12].
+
+Despite these advances, a few core regularity assumptions on the true Brenier potential  $\varphi_0$  and distributions still limit the applicability of dual-type estimators. To clarify these limitations, we decompose the overall estimation error into two components: statistical error and approximation error. The statistical error measures the discrepancy between the estimator and the best possible function in  $\mathcal{F}$ , reflecting the randomness from finite samples. The approximation error captures the gap between the true Brenier potential and its best approximation within  $\mathcal{F}$ , since we do not assume that the true Brenier potential  $\varphi_0$  lies in  $\mathcal{F}$ . Existing analyses of each error component often rely on restrictive smoothness or convexity assumptions, which we discuss in more below.
+
+First, bounding the statistical error in existing works requires a  $(\beta, b)$ -smoothness condition on  $\varphi_0$ , i.e., a polynomial bound on its Hessian. While this condition is verifiable under strong log-concavity of  $\mu$  and  $\nu$  [8, 14, 13, 28], its validity beyond this class remains unclear. Moreover, Proposition 3.12 of [20] shows that this smoothness requirement constrains  $\nu$  to have tails no much heavier than those of  $\mu$ , excluding many practical scenarios, such as multivariate quantile [9, 10, 24], where we transport from  $\mathbb{U}([0,1]^d)$  to distributions with unbounded support, or transporting from a Gaussian (with exponential tails) to a Student's  $t$ -distribution (with polynomial tails).
+
+Second, controlling the approximation error typically relies on  $(\alpha, a)$ -convexity of  $\varphi_0$ , i.e. a generalization of strong convexity, which similarly imposes constraints on the relative tail behavior of  $\mu$  and  $\nu$  [20]. For instance, OT map from an unbounded  $\mu$  to a bounded  $\nu$  fails this condition. This is especially problematic in generative modeling, where the OT map is learned from Gaussian noise to image data of bounded values [40, 30, 38, 32]. Moreover, existing approximation error bounds assume that both  $\mu$  and  $\nu$  have bounded support, [27, 21].
+
+These limitations raise a fundamental question in broadening the scope of optimal transport:
+
+Can one bound the statistical and approximation errors of dual-type OT estimators for general distributions without such restrictive assumptions?
+
+In this work, we provide a positive answer to the question posed above. Our two core innovations are a novel map stability inequality and a refined oracle inequality. Building on these tools, we are the first to establish statistical and approximation error bounds for dual-type OT estimators under minimal regularity assumptions. Table 1 provides a side-by-side comparison of the regularity conditions required by our results versus those in the existing literature.
+
+Table 1: Regularity assumptions on the true Brenier potential  $\varphi_0$  and  $\mu$  required to establish statistical and approximation error bounds ( $\mathcal{E}_{stat}$  and  $\mathcal{E}_{app}$ ) in prior work and in our results.  
+
+<table><tr><td></td><td colspan="2">Assumption on φ0</td><td colspan="2">Distributional Assumption</td></tr><tr><td></td><td>(β,b)-smooth (for Estat)</td><td>(α,a)-convex (for App)</td><td>Tail thickness of μ</td><td>Poincaré inequality on μ</td></tr><tr><td>[27]</td><td>b=0</td><td>a=0</td><td>Compact</td><td>Implicitly Used</td></tr><tr><td>[25]</td><td>Not Required</td><td>Not Required</td><td>Compact</td><td>Required</td></tr><tr><td>[21]</td><td>Required</td><td>Required</td><td>Sub-exponential</td><td>Required</td></tr><tr><td>[20]</td><td>Required</td><td>Required</td><td>General</td><td>Poincaré-type inequality</td></tr><tr><td>Ours</td><td>Not Required</td><td>Not Required</td><td>General</td><td>Not Required</td></tr></table>
+
+Specifically, we summarize our main contributions as follows:
+
+1. A new map stability inequality without  $(\beta, b)$ -smoothness. Stability inequalities are key to linking OT estimation error to the semi-dual objective. Existing results rely on  $(\beta, b)$ -smoothness of  $\varphi_0$ . In Proposition 3.3, we derive a new inequality that removes this requirement, assuming only a well-behaved OT map (Assumption 3.2), which holds unconditionally when  $\nu$  is bounded and under mild conditions for unbounded distributions.  
+2. A novel oracle inequality without  $(\alpha, a)$ -convexity. Proposition 3.4 provides an oracle inequality that avoids the  $(\alpha, a)$ -convexity assumption on  $\varphi_0$  and decomposes the estimation error into statistical error, sieved bias, and approximation error. Unlike prior works [21, 20], which involve hard-to-analyze gradient terms like  $\inf_{\varphi \in \mathcal{F}}\| \nabla \varphi -\nabla \varphi_0\|_{L^2 (\mu)}$ , our bound relies on a more tractable supremum error over a compact region  $B$ :  $\inf_{\varphi \in \mathcal{F}}\| \varphi -\varphi_0\|_{L^{\infty}(B)}$ , making it more suitable for analysis using classical approximation theory.  
+3. First non-asymptotic bounds under minimal smoothness. With our new map stability inequality, Theorem 3.6 establishes the first non-asymptotic statistical error bound for dual-type OT estimators requiring only a first-order condition on  $\varphi_0$ , without second-order smoothness assumptions.  
+4. First theoretical guarantee for neural OT estimators. With our new tools, we derive the first convergence guarantee for neural OT estimators using smooth networks. For tanh neural networks (TNNs), Theorem 3.7 shows that if  $\varphi_0\in C^\alpha (\mathbb{R}^d)$  for  $\alpha \geq 2$ , then the neural OT estimator  $\nabla \hat{\varphi}_n$  between unbounded distributions satisfies:
+
+$$
+\mathbb {E} \| \nabla \hat {\varphi} _ {n} - \nabla \varphi_ {0} \| _ {L ^ {2} (\mu)} ^ {2} \lesssim_ {\log n} n ^ {- \frac {\alpha}{d + 2 \alpha}}. \tag {1}
+$$
+
+Notably, this rate matches that of [25], but under significantly weaker assumptions: we do not require compact support for  $\mu$  and  $\nu$ , nor a Poincaré inequality on  $\mu$ .
+
+Notations and Terminologies Throughout this paper, we use the following notations. For  $\alpha \in \mathbb{R}$ ,  $\lfloor \alpha \rfloor = \{x \in \mathbb{Z} : x < \alpha\}$ . For  $v \in \mathbb{R}^d$ , we write  $\| v \|_2$  for its  $\ell_2$  norm and  $\| v \|_\infty$  for its  $\ell_\infty$  norm. For a matrix  $A \in \mathbb{R}^{d_1 \times d_2}$ , let  $\| A \|_\infty := \max_{i,j} |A_{i,j}|$ . If  $A$  is symmetric,  $\| A \|_{op}$  denotes its operator norm,  $\lambda_{\min}(A)$  denotes its smallest eigenvalue. For a function  $f: \mathbb{R}^d \to \mathbb{R}^m$  and a measure  $\mu$ , we write  $\mu f, \mathbb{E}_\mu f(X)$ , or  $\int f \, \mathrm{d}\mu$  for its integral, and  $\| f \|_{L^2(\mu)} := (\int \| f \|_2^2 \, \mathrm{d}\mu)^{1/2}$  for the  $L^2$  norm. For  $f: \mathbb{R}^d \to \mathbb{R}$  and a set  $B \subset \mathbb{R}^d$ , we denote  $\| f \|_{L^\infty(B)} := \sup_{x \in B} |f(x)|$ .
+
+We write  $a \lesssim b$  to mean  $a \leq C b$  for some constant  $C > 0$  independent of  $a$  and  $b$ , and  $a \lesssim_{\log n} b$  to omit polylogarithmic factors in  $n$ . We write  $a \asymp b$  if  $a \lesssim b$  and  $b \lesssim a$ . We use  $a \vee b$  and  $a \wedge b$  to denote the maximum and minimum of  $a$  and  $b$ , respectively. For sequences  $a_n$  and  $b_n$ , we write  $b_n = \mathcal{O}(a_n)$  if there exists a constant  $c > 0$  such that  $|b_n| \leq c a_n$ , and write  $b_n = \mathcal{O}_{\log}(a_n)$  if there exists constants  $c_1, c_2 > 0$  such that  $|b_n| \leq c_1 a_n \log(n)^{c_2}$ . We use  $\mathbb{I}$  to denote indicator function.
+
+For  $x \in \mathbb{R}^d$ , write  $\langle x \rangle \coloneqq 1 + \| x\|_2$ . When  $x \in \mathbb{R}$ , we set  $\langle x \rangle \coloneqq 1 + |x|$ . We define  $\log_{+}(x) \coloneqq \max \{\log x, 1\}$ . A function  $\varphi \in C^2$  is called  $(\beta, \mathbf{b})$ -smooth if  $\|\nabla^2\varphi(x)\|_{op} \leq \beta \langle x \rangle^b$  for all  $x$ , and  $(\alpha, \mathbf{a})$ -convex if  $\lambda_{\min}(\nabla^2\varphi(x)) \geq \alpha \langle x \rangle^a$ .
+
+A probability measure  $\mu$  is  $(\lambda, \mathbf{K})$ -sub-Weibull if for  $X \sim \mu$ ,  $\mathbb{P}(\|X\|_2 \geq t) \lesssim \exp(-(t / K)^\lambda)$  for all  $t > 0$ . We refer to such a distribution as  $\lambda$ -sub-Weibull, or simply sub-Weibull.
+
+# 2 Background on Optimal Transport
+
+Let  $\mu$  and  $\nu$  be two probability measures on  $\mathbb{R}^d$ . A measurable map  $T: \mathbb{R}^d \to \mathbb{R}^d$  is called a push-forward from  $\mu$  to  $\nu$ , if for any measurable set  $A \subset \mathbb{R}^d$ ,  $\mu(T^{-1}(A)) = \nu(A)$ . We denote this by  $\nu = T_{\#}\mu$ . In Monge's problem, the optimal transport map is defined as the push-forward that minimizes the transport cost from  $\mu$  to  $\nu$ :
+
+$$
+\min  _ {T: \mathbb {R} ^ {d} \rightarrow \mathbb {R} ^ {d}} \int_ {\mathbb {R} ^ {d}} \| x - T (x) \| _ {2} ^ {2} \mu (\mathrm {d} x), \quad \text {s . t .} \quad T _ {\#} \mu = \nu , \tag {2}
+$$
+
+However, Monge's problem may not admit a solution, for example, when  $\mu$  is discrete and  $\nu$  is continuous. A natural relaxation is the Kantorovich formulation, which considers couplings  $\pi \in \Pi (\mu ,\nu)$  with marginals  $\mu$  and  $\nu$ :
+
+$$
+\min  _ {\pi} \iint \| x - y \| _ {2} ^ {2} \pi (\mathrm {d} x, \mathrm {d} y), \quad \text {s . t .} \quad \pi \in \Pi (\mu , \nu). \tag {3}
+$$
+
+Kantorovich formulation is always feasible [42]. As a linear program, its semi-dual form reads:
+
+$$
+\min  _ {\varphi \in L ^ {1} (\mu)} \left\{\int \varphi d \mu + \int \varphi^ {*} d \nu \right\} = \min  _ {\varphi \in L ^ {1} (\mu)} \mu \varphi + \nu \varphi^ {*}. \tag {4}
+$$
+
+where  $\varphi^{*}$  is the convex conjugate (a.k.a. Legendre-Fenchel conjugate) of  $\varphi$ :
+
+$$
+\varphi^ {*} (y) = \sup  _ {x \in \mathbb {R} ^ {d}} \left\{\langle x, y \rangle - \varphi (x) \right\}, \quad \text {f o r a l l} y \in \mathbb {R} ^ {d}. \tag {5}
+$$
+
+The equivalence between Monge's problem (2), Kantorovich's formulation (3), and its semi-dual form (4) was established in the seminal work of Brenier [6]:
+
+Theorem 2.1 (Brenier's Theorem). Let  $\mu$  and  $\nu$  be probability measures on  $\mathbb{R}^d$ , both with finite second moments, and assume that  $\mu$  does not assign mass to small sets. Then there exists a convex function  $\varphi_0$  that uniquely (up to constants) solves (4). Moreover,  $\nabla \varphi_0$  solves Monge's problem (2), and  $(Id \times \nabla \varphi_0)_{\#} \mu$  solves the Kantorovich formulation (3).
+
+Brenier's Theorem reveals that the OT map is the gradient of a convex function  $\varphi_0$ , called the Brenier potential. Furthermore, since  $\varphi_0$  is the minimizer of (4), we can estimate it using empirical measures  $\mu_n$  and  $\nu_m$  of  $\mu$  and  $\nu$ , respectively. For a chosen function class  $\mathcal{F}$ , we define the Brenier potential estimator and the associated dual-type OT estimator as:
+
+$$
+\hat {\varphi} _ {n, m} = \underset {\varphi \in \mathcal {F}} {\arg \min } \mu_ {n} \varphi + \nu_ {m} \varphi^ {*}, \quad \text {w i t h O T m a p e s t i m a t o r} \quad \nabla \hat {\varphi} _ {n, m}. \tag {6}
+$$
+
+However, the supremum over the entire space  $\mathbb{R}^d$  in the convex conjugate  $\varphi^{*}$  makes the optimization procedure unstable and sensitive. It also introduces theoretical challenges when analyzing the convergence of dual-type OT estimator  $\hat{\varphi}_{n,m}$ . To address this issue, [20] proposed the concept of a sieved convex conjugate, which restricts the supremum to a bounded region. Specifically, for some sieve radius  $\tilde{R} > 0$ , define:
+
+$$
+\varphi_ {\tilde {R}} ^ {*} (y) := \sup  _ {x: \| x \| _ {2} \leq \tilde {R}} \langle x, y \rangle - \varphi (x), \quad \text {f o r a l l} y \in \mathbb {R} ^ {d}. \tag {7}
+$$
+
+Note that  $\varphi^{*}(\nabla \varphi (x)) = \langle x,\nabla \varphi (x)\rangle -\varphi (x)$ . By restricting  $x$  to  $B(0,\tilde{R})$ , we equivalently restrict the domain of  $\varphi_{\tilde{R}}^{*}(y)$  to the compact set  $\nabla \varphi (B(0,\tilde{R}))$ . This compactness facilitates us to analyze the estimation error of sieved OT estimator.
+
+Given a sieve radius  $\tilde{R}$ , the corresponding sieved estimator is defined as
+
+$$
+\tilde {\varphi} _ {n, m} := \underset {\varphi \in \mathcal {F}} {\operatorname {a r g m i n}} \mu_ {n} \varphi + \nu_ {m} \varphi_ {\tilde {R}} ^ {*}, \quad \text {w i t h s i e v e d e s t i m a t o r} \quad \nabla \tilde {\varphi} _ {n, m}. \tag {8}
+$$
+
+Note that when  $\tilde{R} = \infty$ , we recover the original dual-type estimator in Equation (6). In the following, we may abbreviate it to  $\tilde{\varphi}_n$  when  $n = m$ .
+
+# 3 Main Results
+
+In Sections 3.1 and 3.2, we develop two core analytical tools: a new stability inequality for the sieved estimator (Proposition 3.3), and a refined oracle inequality that cleanly decomposes the estimation error (Proposition 3.4). Building on these results, Section 3.3 establishes non-asymptotic statistical error rates for general function classes (Theorem 3.6). Finally, in Section 3.4, we show how the overall estimation error of neural OT estimators between general distributions can be controlled, using tanh-activated neural networks a concrete example (Theorem 3.7).
+
+# 3.1 A New Map Stability Inequality
+
+From a statistical learning perspective, map stability inequalities are essential for linking the semidual objective to estimation error. For the original dual-type estimator, denote its objective is  $S_{\mu ,\nu}(\varphi)\coloneqq \mu \varphi +\nu \varphi^{*}$ . Existing results [27, 21, 20] typically show that for all  $\varphi \in \mathcal{F}$
+
+$$
+\left\| \nabla \varphi - \nabla \varphi_ {0} \right\| _ {L ^ {2} (\mu)} ^ {2} \lesssim \mathcal {S} _ {\mu , \nu} (\varphi) - \mathcal {S} _ {\mu , \nu} (\varphi_ {0}). \tag {9}
+$$
+
+where  $\nabla \varphi_0$  is the true OT map to be estimated.
+
+However, these results apply only to the original estimator (6), not the sieved version in (8), which often performs better in practice. Moreover, while neither side of the inequality involves second-order derivatives, prior analyses still require  $(\beta, b)$ -smoothness of  $\varphi_0$ , limiting their generality.
+
+To accommodate sieved estimator and relax their assumptions, we consider the following conditions:
+
+Assumption 3.1 (Envelope for Function Class). Let  $\mathcal{F} \subset C^2(\mathbb{R}^d)$  be a function class. We assume there exists a non-decreasing and pointwise finite function  $U_2: [0, \infty) \to [1, \infty)$  such that:
+
+$$
+\sup  _ {\varphi \in \mathcal {F}} \sup  _ {\| x \| \leq R} \| \nabla^ {2} \varphi (x) \| _ {o p} \leq U _ {2} (R), \quad \sup  _ {\varphi \in \mathcal {F}} \| \nabla \varphi (0) \| _ {2} \leq U _ {2} (R).
+$$
+
+Assumption 3.1 depends only on  $\mathcal{F}$  and is independent of  $\varphi_0, \mu$  and  $\nu$ . It is satisfied by many common function classes, for example,
+
+1. Quadratic function class: When  $\mu$  and  $\nu$  are both Gaussian (or from the same elliptical family), the Brenier potential is quadratic and lies in the following quadratic family:
+
+$$
+\mathcal {F} _ {q u a d} = \left\{x \mapsto x ^ {\top} B x + \langle b, x \rangle : B \in \mathbb {S} _ {+} ^ {d}, \| B \| _ {o p} \leq r _ {1}, \| b \| _ {2} \leq r _ {2} \right\}.
+$$
+
+2. Smooth Neural Networks: Neural network function classes with all parameters bounded and activation functions that are  $C^2$ -smooth, such as sigmoid, tanh, or softmax.  
+3. Reproducing Kernel Hilbert Spaces (RKHS): Let  $K: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$  be a positive-definite kernel on its domain  $\mathcal{X} \times \mathcal{X}$ , and let  $\mathcal{H}_K$  be the corresponding RKHS with norm  $\| \cdot \|_{\mathcal{H}_K}$ . We assume  $\mathcal{F}$  is  $C^4$  and take  $\mathcal{F}$  to be the unit ball of  $\mathcal{H}_K$ , i.e.,  $\mathcal{F} = \{\varphi \in \mathcal{H}_K : \| \varphi \|_{\mathcal{H}_K} \leq 1\}$ .
+
+Assumption 3.1 and 3.2 together play a role similar to Assumption A.1 in [21], but are strictly weaker. As a result, the function classes discussed in Section 4 of [21], including parametric family, wavelet expansions, RKHS, and Barron Spaces, apply directly to our setting. We refer readers to Section 4 of [21] for a detailed discussion.
+
+Assumption 3.2 (Envelope for OT map). There is a function  $u \in L^4(\mu)$ , such that for all  $\varphi \in \mathcal{F}$
+
+$$
+\| \nabla \varphi (x) - \nabla \varphi_ {0} (x) \| _ {2} \leq u (x), \quad f o r a l l x.
+$$
+
+Define  $U_{1}(R) \coloneqq \sup_{\| x \|_{2} \leq R} u(x)$ . Additionally, let  $\Phi \in L^{2}(\mu)$  be an envelope function for  $\mathcal{F} \cup \{\varphi_0\}$ , i.e.
+
+$$
+\left| \varphi (x) \right| \leq \Phi (x), \quad f o r a l l x \in \mathbb {R} ^ {d} a n d a l l \varphi \in \mathcal {F} \cup \varphi_ {0}.
+$$
+
+A sufficient condition for Assumption 3.2 is that both Proposition A.2 and Assumption 3.1 hold. Specifically, Proposition A.2 ensures  $\| \nabla \varphi_0(x)\| _2$  to be bounded by some function  $L_{1}(x)$ . Meanwhile, Assumption 3.1 provides an envelope for  $\| \nabla \varphi (x)\| _2$ , denoted as  $L_{2}(x)$ . Then Assumption 3.2 follows via the triangle inequality  $\| \nabla \varphi (x) - \nabla \varphi_0(x)\| _2\leq L_1(x) + L_2(x)$ . While Assumption 3.1 solely relies on  $\mathcal{F}$ , typical distributions for Proposition A.2 to hold include:  $\mu$  can be normal or Student-t, and  $\nu$  can be any measure with certain tail decay. We refer to Appendix A for further details and an explicit example verifying Assumption 3.2.
+
+We are now ready to present our new stability inequality for the sieved estimator:
+
+Proposition 3.3 (Map Stability Inequality). Suppose Assumptions 3.1 and 3.2 hold. For any  $\varepsilon >0$  define  $R_{\varepsilon}$  and sieve radius  $\tilde{R}_{\varepsilon}$  such that
+
+$$
+\mu \left(\| X \| _ {2} > R _ {\varepsilon}\right) \leq \varepsilon , \quad \tilde {R} _ {\varepsilon} \geq \sup  _ {\| x \| _ {2} \leq R _ {\varepsilon}} \left\{\| x \| _ {2} + \frac {u (x)}{U _ {2} \left(\| x \| _ {2} + u (x)\right)} \right\} \tag {10}
+$$
+
+For any  $\varphi \in \mathcal{F}$ , define the truncated excess risk:
+
+$$
+r _ {\varepsilon} (\varphi) := \int_ {B (0, R _ {\varepsilon})} \varphi (x) + \varphi_ {R _ {\varepsilon}} ^ {*} (\nabla \varphi_ {0} (x)) \mu (d x) - \int_ {B (0, R _ {\varepsilon})} \varphi_ {0} (x) + \varphi_ {0} ^ {*} (\nabla \varphi_ {0} (x)) \mu (d x). \tag {11}
+$$
+
+Then the following inequality holds: for any  $\varphi \in \mathcal{F}$
+
+$$
+\left\| \nabla \varphi - \nabla \varphi_ {0} \right\| _ {L ^ {2} (\mu)} ^ {2} \leq 2 U _ {2} \left(R _ {\varepsilon} + U _ {1} \left(R _ {\varepsilon}\right)\right) \cdot r _ {\varepsilon} (\varphi) + \| u \| _ {L ^ {4} (\mu)} ^ {2} \cdot \varepsilon^ {\frac {1}{2}}, \tag {12}
+$$
+
+We first clarify the roles of  $R_{\varepsilon}$  and  $\tilde{R}_{\varepsilon}$ , which enable a more tractable approximation error in the next section. Specifically, the hyper-balls  $B(0,R_{\varepsilon})$  and  $\nabla \varphi_0(B(0,R_\varepsilon))$  serve as bounded pseudosupports for  $\mu$  and  $\nu$ , respectively, over which the behavior of  $\varphi$  and its convex conjugate can be effectively controlled. Meanwhile, the sieve radius  $\tilde{R}_{\varepsilon}$  is introduced to handle any mismatch between  $\nabla \varphi \bigl (B(0,R_{\varepsilon})\bigr)$  and the pseudo-support of  $\nu$ , by better regulating the range of the sieved conjugate. In practice, if  $\nabla \varphi$  closely approximates  $\nabla \varphi_0$  in  $L^{\infty}$  norm (so that  $u\approx 0$ ), the mismatch is negligible and one can take  $\tilde{R}_{\varepsilon}\approx R_{\varepsilon}$ . In that case, it suffices to compute the convex conjugate over the pseudo-support of  $\mu$  alone.
+
+For simplicity, a sufficient choice for  $\tilde{R}_{\varepsilon}$  is  $\tilde{R}_{\varepsilon} = R_{\varepsilon} + U_1(R_{\varepsilon})$ . In particular, if  $\mu$  is supported on  $B(0,R)$ , one may take  $R_{\varepsilon} = R$ .
+
+Equation (12) decomposes the estimation error into two components. The first arises from the truncated excess risk over the high-probability region  $B(0, R_{\varepsilon})$ , which can be well-controlled (see Proposition 3.4). Its coefficient  $U_{2}(R_{\varepsilon} + U_{1}(R_{\varepsilon}))$  originates from a second-order Taylor expansion of  $\varphi$ , capturing the "uniform curvature" of  $\mathcal{F}$  on the ball  $B(0, R_{\varepsilon} + U_{1}(R_{\varepsilon}))$ . The second component accounts for the residual risk outside this region. As a result,  $\varepsilon$  must be carefully selected to balance the trade-off: a smaller  $\varepsilon$  reduces the residual risk but increases the truncated excess risk.
+
+We defer the proof of Proposition 3.3 to Appendix B.
+
+Comparison with existing map stability results. The most closely related results are Proposition 10 of [27], Proposition 1 of [21], and Lemma 3.15 of [20], all of which require  $(\beta, b)$ -smoothness assumptions on the true Brenier potential  $\varphi_0$  and the candidate class  $\mathcal{F}$ . In contrast, our Proposition 3.3 substantially relaxes these requirements by assuming only an upper bound on the true OT map  $\nabla \varphi_0$  via function  $u(x)$ .
+
+Another key distinction is the introduction of the sieved convex conjugate  $\varphi_{\tilde{R}_{\xi}}^{*}$ , essential for providing a more accessible approximation error in the next section. This sieving remains necessary even when  $\mu$  has bounded support, unless additional regularity conditions are imposed on  $\mu$  and  $\nu$ .
+
+Compared to map stability results for plug-in type estimators, our bound also holds under milder assumptions. For instance, Theorem 2.1 of [17] requires  $\nabla \varphi_0$  to be Lipschitz, while Theorem 3 of [5] assumes that the Brenier potential is strongly convex.
+
+# 3.2 A New Oracle Inequality
+
+Now, we present our second contribution of a new oracle inequality that helps cleanly decompose the estimation error into statistical error, sieved bias, and approximation error.
+
+Proposition 3.4 (Oracle Inequality). Under Assumptions 3.1 and 3.2, and let  $R_{\varepsilon}, \tilde{R}_{\varepsilon}$  be as defined in Equation (10). For the sieved OT estimator  $\tilde{\varphi}_{n,m}$  from Equation (8), its truncated excess risk  $\tilde{r}_{n,m,\varepsilon} := r_{\varepsilon}(\tilde{\varphi}_{n,m})$ , defined in Equation (11), admits the decomposition:
+
+$$
+\tilde {r} _ {n, m, \varepsilon} := r _ {\varepsilon} \left(\tilde {\varphi} _ {n, m}\right) \leq \mathcal {E} _ {\text {s t a t}} + \mathcal {E} _ {\text {s i e v e}} + \mathcal {E} _ {\text {a p p}}, \tag {13}
+$$
+
+where
+
+$$
+\mathcal {E} _ {\text {s t a t}} := \sup  _ {f \in \bar {\mathcal {F}}} \int_ {B (0, R _ {\varepsilon})} f d (\mu_ {n} - \mu) + \sup  _ {g \in \bar {\mathcal {G}}} \int_ {B (0, R _ {\varepsilon})} g d (\nu_ {m} - \nu), \tag {14a}
+$$
+
+$$
+\mathcal {E} _ {\text {s i e v e}} := 2 \mathbb {E} _ {X \sim \mu_ {n}} [ \Phi (X) \cdot \mathbb {I} (\| X \| _ {2} > R _ {\varepsilon}) ] + 2 \mathbb {E} _ {Y \sim \nu_ {m}} [ G (Y) \cdot \mathbb {I} (\| (\nabla \varphi_ {0}) ^ {- 1} (Y) \| _ {2} > R _ {\varepsilon}) ], \tag {14b}
+$$
+
+$$
+\mathcal {E} _ {\text {a p p}} := 2 \inf  _ {\varphi \in \mathcal {F}} \| \varphi - \varphi_ {0} \| _ {L ^ {\infty} (B (0, \tilde {R} _ {\varepsilon}))}. \tag {14c}
+$$
+
+Here,  $\bar{\mathcal{F}} := \{\varphi_1 - \varphi_2 : \varphi_1, \varphi_2 \in \mathcal{F}\}$ ,  $\bar{\mathcal{G}} := \{\varphi_{1,\tilde{R}_{\varepsilon}}^* - \varphi_{2,\tilde{R}_{\varepsilon}}^* : \varphi_1, \varphi_2 \in \mathcal{F}\}$  and  $G(y) := \tilde{R}_{\varepsilon} \| y \|_2 + \sup_{x \in B(0,\tilde{R}_{\varepsilon})} \Phi(x)$ .
+
+Thus, with a careful choice of  $\varepsilon$  and  $R_{\varepsilon}$  (which can be selected in a distribution-free manner, as discussed in Section 3.3), Propositions 3.3 and 3.4 together provide a complete characterization of how the sample sizes, function class  $\mathcal{F}$ , and the sieved convex conjugate collectively influence the estimation error of the sieved dual-type estimators.
+
+Specifically, this result decomposes the truncated excess risk into three components. The first term,  $\mathcal{E}_{stat}$ , reflects the statistical error from replacing the underlying distributions  $\mu$  and  $\nu$  with their
+
+estimates  $\mu_{n}$  and  $\nu_{m}$ . Importantly, these estimates need not be empirical measures, allowing for smoothed or bootstrap-based alternatives. The second term,  $\mathcal{E}_{\text{sieve}}$ , accounts for the bias introduced by replacing the original convex conjugate (Equation (7)) with its sieved version. This bias vanishes when  $R_{\varepsilon} = \infty$ , where the sieved and original convex conjugates coincide.
+
+The third term,  $\mathcal{E}_{app}$ , captures the approximation error of the function class  $\mathcal{F}$  in representing the true potential  $\varphi_0$  over the compact domain  $B(0,\tilde{R}_{\varepsilon})$ . As usual, enlarging  $\mathcal{F}$  reduces  $\mathcal{E}_{app}$  at the cost of increasing  $\mathcal{E}_{stat}$ . Notably, we express the approximation error in the  $L^{\infty}$  norm on a compact set, aligning naturally with classical approximation theory. In contrast, prior works [27, 21, 20] typically rely on gradient-based errors like  $\inf_{\varphi \in \mathcal{F}}\| \nabla \varphi -\nabla \varphi_0\|_{L^2 (\mu)}$ , which remain challenging to analyze for rich function classes such as deep neural networks.
+
+Proof ideas of Propositions 3.4 The sieve technique allows us to treat  $B(0, R_{\varepsilon})$  and  $\nabla \varphi_0(B(0, \tilde{R}_{\varepsilon}))$  as bounded pseudo-supports of  $\mu$  and  $\nu$ , or equivalently, as bounded pseudo-domains for  $\varphi_0$  and  $\varphi_0^*$ . This reduction localizes the approximation problem to compact sets, where classical approximation theory applies. Another key benefit of this sieving approach is that it eliminates the need for the  $(\alpha, a)$ -convexity assumption required to control the approximation error in prior works.
+
+The sieve bias is then controlled via tail probability bounds, while the statistical error is handled using standard tools from empirical process theory. The full proof is deferred to Appendix C.
+
+# 3.3 Non-Asymptotic Bounds of Statistical Error and Sieved Bias
+
+In this section, we derive bounds for the statistical error  $\mathcal{E}_{stat}$  and the sieved bias  $\mathcal{E}_{sieve}$  in Proposition 3.4, under minimal assumptions on the source distribution  $\mu$ . Notably, our results do not require smoothness of the true OT map  $\nabla \varphi_0$ , improving upon existing analyses.
+
+We begin with the following assumption on the covering entropy of the function class  $\mathcal{F}$ :
+
+Assumption 3.5 (Covering Entropy of  $\mathcal{F}$ ). For some  $\eta \geq 0$ ,  $\gamma \in [0,2)$ ,  $\gamma' \geq 1$  and  $D_{\mathcal{F}} > 1$ , the covering entropy of  $\mathcal{F}$  satisfies: for any  $h \geq 0$ ,
+
+$$
+\log \mathcal {N} (h, \mathcal {F}, L ^ {\infty} ([ - R, R ] ^ {d}) \leq D _ {\mathcal {F}} \cdot h ^ {- \gamma} \cdot \log_ {+} (1 / h) ^ {\gamma^ {\prime}} \cdot R ^ {\eta}. \tag {15}
+$$
+
+We focus on the Donsker regime ( $\gamma \in [0,2)$ ) for theoretical simplicity, which already captures a wide range of function classes, such as parametric families, wavelets, reproducing kernel Hilbert space (RKHS), and both shallow and deep neural networks. Here,  $D_{\mathcal{F}}$  represents the "effective dimension" of  $\mathcal{F}$ , and  $n / D_{\mathcal{F}}$  can be viewed as the effective sample size from the statistical learning theory perspective. Because most covering entropy results are stated for the  $L^{\infty}([0,1]^d)$  norm, we introduce an additional  $R^{\eta}$  factor to translate them to our unbounded setting in  $L^{\infty}([-R,R]^{d})$ . Specifically, the exponent  $\eta$  measures how the envelope of  $\mathcal{F}$  grows (e.g.  $\eta = 2$  if the envelope scales like  $x \mapsto x^2$ ).
+
+Theorem 3.6 (Statistical Error with Empirical Measures). Under assumptions of Proposition 3.4, and Assumption 3.5, the sieved estimator  $\tilde{\varphi}_{n,m}$  from Equation (8), computed with empirical measures  $\mu_{n}$  and  $\nu_{m}$ , satisfies:
+
+$$
+\mathbb {E} \left[ \mathcal {E} _ {\text {s t a t}} \right] \lesssim \sqrt {\frac {D _ {\mathcal {F}}}{n}} \left(R _ {\varepsilon} ^ {\frac {\eta}{2}} + \sup  _ {x \in B (0, R _ {\varepsilon})} \Phi (x) ^ {\frac {1}{2}}\right) \log (n) ^ {\frac {\gamma^ {\prime}}{2}} + \sqrt {\frac {D _ {\mathcal {F}}}{m}} \left(\tilde {R} _ {\varepsilon} ^ {\frac {\eta}{2}} + M ^ {\frac {1}{2}}\right) \log (m) ^ {\frac {\gamma^ {\prime}}{2}}, \tag {16}
+$$
+
+and
+
+$$
+\mathbb {E} \left[ \mathcal {E} _ {\text {s i e v e}} \right] \leq 2 \left(\| \Phi \| _ {L ^ {2} (\mu)} + \| G \| _ {L ^ {2} (\nu)}\right) \cdot \varepsilon^ {\frac {1}{2}}, \tag {17}
+$$
+
+where  $M \coloneqq \tilde{R}_{\varepsilon} \sup_{x \in B(0, R_{\varepsilon})} \| \nabla \varphi_0(x) \|_2 + \sup_{x \in B(0, \tilde{R}_{\varepsilon})} \Phi(x)$ , and the suppressed constant depends only on  $\gamma, \gamma'$ .
+
+We defer the proof to Appendix D, where we also establish corresponding high-probability guarantees.
+
+In Theorem 3.6, we establish a standard convergence rate for estimation error,  $\mathbb{E}[\mathcal{E}_{\text{stat}}] = \mathcal{O}_{\log}(\sqrt{D_{\mathcal{F}} / n} + \sqrt{D_{\mathcal{F}} / m})$ , which is common in empirical process theory. Meanwhile, the multiplicative prefactors in  $\mathcal{E}_{\text{stat}}$ ,  $R_{\varepsilon}^{\frac{n}{2}} + \sup_{x \in B(0, R_{\varepsilon})} \Phi(x)^{\frac{1}{2}}$  and  $\tilde{R}_{\varepsilon}^{\frac{n}{2}} + M^{\frac{1}{2}}$ , naturally capture the "diameter" of  $\mathcal{F}$  on the pseudo-supports of  $\mu$  and  $\nu$ .
+
+Moreover, it tells that the sieve bias  $\mathcal{E}$  is governed by the tail probability  $\varepsilon$  defined in Equation (10).
+
+Practical choice of  $R_{\varepsilon}$  and  $\tilde{R}_{\varepsilon}$ . As shown in [20],  $R_{\varepsilon}$  can be selected in a distribution-free manner. Here, we extend this idea to provide a practical, joint selection strategy for both  $R_{\varepsilon}$  and  $\tilde{R}_{\varepsilon}$ :
+
+$$
+R _ {\varepsilon} = \max  _ {1 \leq i \leq n} \| X _ {i} \| _ {2}, \quad \tilde {R} _ {\varepsilon} = R _ {\varepsilon} + C \max  _ {1 \leq j \leq m} \| Y _ {j} \| _ {2}, \tag {18}
+$$
+
+where  $C \geq 0$  is a tuning parameter designed to account for the potential mismatch between the estimated and true OT maps, specifically the discrepancy  $\| \nabla \tilde{\varphi}_{n,m} - \nabla \varphi_0\|_{L^\infty (B(0,R_\varepsilon))}$ .
+
+The choice of  $R_{\varepsilon}$  can also be justified by our stability and oracle bounds (Propositions 3.3 and 3.4). In particular, if  $n = m$ ,  $\mu$  is sub-Weibull, and both  $\Phi(\cdot)$ ,  $\|\nabla \varphi_0(\cdot)\|_2$  grow at most polynomially, then the estimation error of the sieved estimator reads:
+
+$$
+\mathbb {E} \| \nabla \tilde {\varphi} _ {n} - \nabla \varphi_ {0} \| _ {L ^ {2} (\mu)} ^ {2} = \mathcal {O} _ {\log} \left(\left(n / D _ {\mathcal {F}}\right) ^ {- \frac {1}{2}} + \mathcal {E} _ {a p p} + \varepsilon^ {\frac {1}{2}}\right). \tag {19}
+$$
+
+By exchangability of  $X_{i}$ 's, choosing  $R_{\varepsilon} \coloneqq \max_{1 \leq i \leq n} \|X_{i}\|_{2}$  yields  $\varepsilon \asymp n^{-1}$ , so the sieved bias  $\varepsilon^{\frac{1}{2}}$  is dominated by the statistical error term  $(n / D_{\mathcal{F}})^{-\frac{1}{2}}$ .
+
+While a complete theoretical justification for our choice of  $\tilde{R}_{\varepsilon}$  remains an open question, numerical results in Section 4 provide strong empirical support: our sieved estimator consistently outperforms the original dual-type estimator of [27, 25, 21] across a range of settings.
+
+# 3.4 Estimation Error of Neural OT Estimator
+
+In this section, we derive the first non-asymptotic error bounds for dual-type OT map estimators parameterized by deep neural networks between general distributions.
+
+ReLU networks, despite their popularity, lack the smoothness needed to recover the gradient of the estimated Brenier potential. Moreover, our analysis (Assumption 3.1) requires  $\mathcal{F}$  to be  $C^2$ . Consequently, smooth activations are both practically and theoretically preferable.
+
+While input convex neural networks (ICNNs) [2] exploit convexity of Brenier potential and have been applied at scale [33, 4, 31], their approximation theory remains underdeveloped. This gap limits rigorous control of approximation error in existing analyses of dual-type estimators [27, 25, 21, 20]. By contrast, our approach removes the convexity assumption on  $\mathcal{F}$ , allowing us to employ standard fully connected networks.
+
+Accordingly, we adopt the tanh-activated fully-connected neural network (TNN) as our function class  $\mathcal{F}$ , due to its  $C^\infty$  smoothness and non-asymptotic universal approximation guarantees [16]:
+
+$$
+f (x) = \mathcal {L} _ {L} \circ \sigma \circ \mathcal {L} _ {L - 1} \circ \dots \circ \sigma \circ \mathcal {L} _ {1} (x), \tag {20}
+$$
+
+where  $\sigma(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$  is applied entrywise, and  $\mathcal{L}_{\ell}(x) = A_{\ell}x + b_{\ell}$  for  $\ell \in [L]$ , where  $A_{l} \in \mathbb{R}^{p_{l+1} \times p_{l}}$ ,  $b_{l} \in \mathbb{R}^{p_{l}}$ , with  $p_1 = d$ ,  $p_{L+1} = 1$ .
+
+We define the TNN class  $\mathcal{F}$  with bounded parameters as:
+
+$$
+\mathcal {F} (L, W, \kappa) := \left\{f \text {o f f o r m (2 0)}: \| A _ {\ell} \| _ {\infty} \vee \| b _ {\ell} \| _ {\infty} \leq \kappa , \quad p _ {l} \leq W \text {f o r a l l} \ell \in [ L ] \right\}, \tag {21}
+$$
+
+where  $\kappa > 0$  is a truncation threshold used for technical convenience.
+
+Theorem 3.7 (OT Estimation via Tanh Neural Network). Let  $n = m$ , and suppose  $\mu$  is  $(\lambda, K)$ -sub-Weibull, and the true Brenier potential  $\varphi_0 \in C^\alpha(\mathbb{R}^d)$  with  $\alpha \geq 2$  is  $(\beta, b)$ -smooth. By setting  $R_{\varepsilon}, \tilde{R}_{\varepsilon}$  according to Equation (10), there exists a deep TNN function class  $\mathcal{F} := \mathcal{F}(L, W, \kappa)$  where  $L, W, \kappa$  depend on  $n, d, \alpha, \beta, b$  and are of order  $3 \leq L = \mathcal{O}(1)$ ,  $W = \mathcal{O}(n^{\frac{d}{d + 2\alpha}})$ ,  $\kappa = \mathcal{O}(n^{\frac{d(d + (|\alpha| + 2)^2 + 4) + 2}{2(d + 2\alpha)}})$ , such that the sieved estimator  $\nabla \tilde{\varphi}_n$  with sieve radius
+
+$$
+\tilde {R} _ {n} = C _ {K, d, \alpha} \cdot (\log n) ^ {\frac {1}{\lambda}}, \quad \text {f o r s o m e c o n s t a n t} \quad C _ {K, d, \alpha} \geq 4 K \frac {2 \alpha}{d + 2 \alpha} + 2 \tag {22}
+$$
+
+satisfies
+
+$$
+\mathbb {E} \| \nabla \tilde {\varphi} _ {n} - \nabla \varphi_ {0} \| _ {L ^ {2} (\mu)} ^ {2} \lesssim_ {\log n} n ^ {- \frac {\alpha}{d + 2 \alpha}}. \tag {23}
+$$
+
+Remark 3.8 (On mild regularity conditions). The sub-Weibull and  $(\beta, b)$ -smoothness assumptions are used in our work primarily for analytical convenience, not because they are fundamentally
+
+required for the methodology. Notably, both conditions are widely used in the OT literature, see [21, 20, 27, 25, 34] for  $(\beta, b)$ -smoothness and [21, 17, 20] for sub-Weibull condition.
+
+The  $(\beta, b)$ -smoothness of  $\varphi_0$  is needed solely to control approximation error through Lemma F.4, ensuring the error grows polynomially in  $\tilde{R}_{\varepsilon}$ . Meanwhile, the sub-Weibull  $\mu$  guarantees that  $R_{\varepsilon}, \tilde{R}_{\varepsilon}$  scale polynomially in  $\log(1/\varepsilon)$ . Together, these two results ensure that the relevant Sobolev norm is bounded by a polynomial in  $\log(1/\varepsilon)$ , and the consequent approximation error has the order of  $\varepsilon$ .
+
+If  $(\beta, b)$ -smoothness were lifted, one could still obtain approximation bounds, but they may degrade to  $O(\varepsilon^a)$  for some  $a < 1$ . This complicates the statistical error, approximation error, and sieved estimation bias-variance trade-off by introducing additional convergence exponent. Likewise, relaxing source distribution to be polynomially tails causes  $R_{\varepsilon}, \tilde{R}_{\varepsilon}$  to scale polynomially in  $1 / \varepsilon$ . While the proofs follow similar steps, the resulting convergence rate becomes much more complex (See Equation 3.10 in [20] for an illustration of the added technical complications).
+
+Remark 3.9 (On the choice of activation). While ReLU networks are widely used in practice, our framework currently focuses on the tanh activation due to theoretical considerations. Specifically, our estimator is defined via the gradient of a potential function. Although PyTorch can handle ReLU networks computationally, the ReLU activation is not globally differentiable in mathematics, making this definition is ill-posed. Moreover, even under a weak derivative framework, ReLU networks lack second-order weak derivatives, whereas the existing optimal transport theory fundamentally relies on second-order regularity for statistical analysis.
+
+The rate in Theorem 3.7 matches that of [25], but under substantially milder assumptions. Although slower than the minimax rate in [27], this is expected: we do not assume  $(\alpha, a)$ -convexity of the true Brenier potential  $\varphi_0$ , compact support of  $\mu$  and  $\nu$ , or a Poincaré inequality on  $\mu$ . Moreover, the estimators in [27, 25, 21] are either computationally intensive or NP-hard. In contrast, our deep TNN estimator is both implementable and scalable in practice.
+
+We focus on the TNN architecture due to the current lack of approximation theory for general smooth neural networks. The proof of Theorem 3.7 is provided in Appendix E. With minor modifications, the result also holds with high probability.
+
+# 4 Numerical Examples
+
+In this section, we present numerical simulations to evaluate the performance of our sieved-TNN estimator and compare it against the original dual-type estimators from [27, 25, 21]. We consider four synthetic OT problems that pose challenges for existing theoretical frameworks:
+
+- Non- $(\beta, b)$ -smooth maps:  $\mathcal{N}(0,1) \to t_6$ , Uniform(0,1)  $\to \mathcal{N}(0,1)$ .  
+- Non- $(\alpha, a)$ -convex maps:  $t_6 \to \mathcal{N}(0,1)$ ,  $\mathcal{N}(0,1) \to \mathrm{Uniform}(0,1)$ .
+
+We evaluate performance in dimensions  $d = 5, 10, 20$ , using i.i.d. samples  $X_{i} \sim \mu$  and  $Y_{j} \sim \nu$  with sample sizes  $n = m \in \{64, 128, 256, 512, 1024, 2048\}$ . Experiments were conducted on a server with Intel Xeon Gold 6342 processors, requiring approximately 4,000 CPU core hours in total.
+
+Implementation setup We implement our estimator in PyTorch [36], following algorithms from [20] with objective function in Equation (8). The candidate class  $\mathcal{F}$  consists of TNNs with two hidden layers of width 10 (for  $d = 5$ ), 20 (for  $d = 10$ ), and 30 (for  $d = 20$ ). Training proceeds in two phases: a warm-up phase of  $\max \{\lfloor 10000 / n\rfloor ,50\}$  epochs at learning rate  $5\times 10^{-3}$ , followed by 300 epochs at  $10^{-3}$ . Mini-batch size is 64 for both  $\mu$  and  $\nu$ , and the sieved convex conjugate subproblem is solved using 300 inner iterations (see Algorithm 2 in [20]). In each trial,  $10\%$  of the samples are held out for validation. We defer implementation details to Appendix G.
+
+Sieve radius As discussed in Equation (18), we choose the sieve radius  $\tilde{R}$  as
+
+$$
+\tilde {R} = \max  _ {1 \leq i \leq n} \| X _ {i} \| _ {2} + C \cdot \max  _ {1 \leq j \leq n} \| Y _ {j} \| _ {2}, \quad C \in \{0, 1, 2, 3, \infty \}.
+$$
+
+Here,  $C = 0$  corresponds to the setting in [20], while  $C = \infty$  recovers the original dual-type estimator studied in [27, 25, 21], enabling direct comparison.
+
+Evaluation We assess the estimation error using unexplained variance proportion (UVP) [30]:  $L^2$ -UVP( $\nabla \hat{\varphi}_n$ ) :=  $\| \nabla \hat{\varphi}_n - \nabla \varphi_0 \|_{L^2(\mu)}^2 / \mathrm{Var}_{\nu} (\| Y \|_2)$ . Lower values indicate better performance. For each experiment, we approximate the  $L^2$ -UVP using an independent set of  $10^6$  samples.
+
+![](images/7b2d2a59f3f927f6dfa14d4129b1cbe6cea74a06f25146cecd5bc2c9f25ecaa3.jpg)  
+(a)  $\mathcal{N}\to t_6$
+
+![](images/f8c5e68b007204af2a05dbee3d4b87f648fbb6355fe7721bb7914848469d4fb1.jpg)  
+(b) Unif  $\rightarrow \mathcal{N}$
+
+![](images/1e96345efadd786682132e89800a2547b2721a6c99af8ce07d2f40f0605d318d.jpg)  
+(c)  $t_6\to \mathcal{N}$
+
+![](images/09a7c28cd42cb9781726afa173261328e7101e38d0d86cb8bfadba362bf02e71.jpg)  
+(d)  $\mathcal{N}\to \mathrm{Unif}$  
+Figure 1:  $L^2$ -UVP when  $d = 5$ . Each curve shows the mean  $L^2$ -UVP over 50 random trials with one standard deviation.
+
+Due to space constraints, we report only the  $L^2$ -UVP error versus sample size  $n$  for  $d = 5$  in Figure 1 here. The full set of numerical results is available in Appendix G.
+
+Across all settings, our sieved TNN estimator consistently converges and outperforms the original dual-type estimator  $(C = \infty)$ , particularly in the small-sample regime. As shown in Figure 4a, sieve-based estimators yield lower  $L^2$ -UVP errors for small  $n$ , while the classical estimator only catches up as  $n$  grows large. This aligns with our theory: larger sieve radii may degrade convergence, and are only needed when the mismatching  $\| \nabla \tilde{\varphi}_{n,m} - \nabla \varphi_0\|_{L^\infty}$ , is large.
+
+These results also offer practical guidance: a modest sieve constant (e.g.,  $C = 0$  or 1) suffices, and performance of our sieved estimator remains stable across a range of  $C$ .
+
+# 5 Conclusion
+
+In this paper, we develop new map stability and oracle inequalities for sieved dual-type OT estimators, without restrictive smoothness and convexity assumptions on the true Brenier potential  $\varphi_0$  in the literature. Under mild regularity conditions, we establish the first non-asymptotic error bounds for neural OT estimators, using tanh networks as a concrete example. Numerical experiments further confirm the strong empirical performance of our sieved approach. Altogether, our unified framework advances the theoretical foundations of optimal transport and paves the way for future developments in both methodology and applications.
+
+Acknowledgment: This research was partially supported the U.S. National Science Foundation under the grants DMS 2514400 and DMS 2210775, and by the U.S. National Institutes of Health under the grants R01GM163244 and 1R01GM152812.
+
+# References
+
+[1] Jason Altschuler, Jonathan Niles-Weed, and Philippe Rigollet. Near-linear time approximation algorithms for optimal transport via sinkhorn iteration. Advances in neural information processing systems, 30, 2017.  
+[2] Brandon Amos, Lei Xu, and J Zico Kolter. Input convex neural networks. In International Conference on Machine Learning, pages 146-155. PMLR, 2017.  
+[3] Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial networks. In International conference on machine learning, pages 214-223. PMLR, 2017.  
+[4] Arip Asadulaev, Alexander Korotin, Vage Egiazarian, Petr Mokrov, and Evgeny Burnaev. Neural optimal transport with general cost functionals. In International Conference on Learning Representations, 2022.  
+[5] Sivaraman Balakrishnan and Tudor Manole. Stability bounds for smooth optimal transport maps and their statistical implications. arXiv preprint arXiv:2502.12326, 2025.  
+[6] Yann Brenier. Polar factorization and monotone rearrangement of vector-valued functions. Communications on Pure and Applied Mathematics, 44(4):375-417, 1991.  
+[7] Charlotte Bunne, Stefan G Stark, Gabriele Gut, Jacobo Sarabia Del Castillo, Mitch Levesque, Kjong-Van Lehmann, Lucas Pelkmans, Andreas Krause, and Gunnar Ratsch. Learning single-cell perturbation responses using neural optimal transport. Nature Methods, 20(11):1759-1768, 2023.  
+[8] Luis A Caffarelli. Monotonicity properties of optimal transportation and the fkg and related inequalities. Communications in Mathematical Physics, 214:547-563, 2000.  
+[9] Guillaume Carlier, Victor Chernozhukov, and Alfred Galichon. Vector quantile regression: an optimal transport approach. The Annals of Statistics, 44(3):1165-92, 2016.  
+[10] Victor Chernozhukov, Alfred Galichon, Marc Hallin, and Marc Henry. Monge-Kantorovich depth, quantiles, ranks and signs. The Annals of Statistics, 45(1):223 - 256, 2017.  
+[11] Lenaic Chizat, Pierre Roussillon, Flavien Léger, François-Xavier Vialard, and Gabriel Peyre. Faster wasserstein distance estimation with the sinkhorn divergence. Advances in Neural Information Processing Systems, 33:2257-2269, 2020.  
+[12] Jaemoo Choi, Yongxin Chen, and Jaewoong Choi. Improving neural optimal transport via displacement interpolation. The Thirteenth International Conference on Learning Representations, 2025.  
+[13] Maria Colombo and Max Fathi. Bounds on optimal transport maps onto log-concave measures. Journal of Differential Equations, 271:1007-1022, 2021.  
+[14] Maria Colombo, Alessio Figalli, and Yash Jhaveri. Lipschitz changes of variables between perturbations of log-concave measures. arXiv preprint arXiv:1510.03687, 2015.  
+[15] M Cuturi. Lightspeed computation of optimal transportation distances. Advances in Neural Information Processing Systems, 26(2):2292-2300, 2013.  
+[16] Tim De Ryck, Samuel Lanthaler, and Siddhartha Mishra. On the approximation of functions by tanh neural networks. Neural Networks, 143:732-750, 2021.  
+[17] Nabarun Deb, Promit Ghosal, and Bodhisattva Sen. Rates of estimation of optimal transport maps using plug-in estimators via barycentric projections. In Advances in Neural Information Processing Systems, volume 34, pages 29736-29753, 2021.  
+[18] Nabarun Deb and Bodhisattva Sen. Multivariate rank-based distribution-free nonparametric testing using measure transportation. Journal of the American Statistical Association, 118(541):192-207, 2023.  
+[19] Alex Delalande. Nearly tight convergence bounds for semi-discrete entropic optimal transport. In International Conference on Artificial Intelligence and Statistics, pages 1619-1642. PMLR, 2022.
+
+[20] Yizhe Ding, Runze Li, and Lingzhou Xue. Statistical convergence rates of optimal transport map estimation between general distributions. arXiv preprint arXiv:2412.08064, 2024.  
+[21] Vincent Divol, Jonathan Niles-Weed, and Aram-Alexandre Pooladian. Optimal transport map estimation in general function spaces. The Annals of Statistics, 53(3):963–988, 2025.  
+[22] Pavel Dvurechensky, Alexander Gasnikov, and Alexey Kroshnin. Computational optimal transport: Complexity by accelerated gradient descent is better than by sinkhorn's algorithm. In International Conference on Machine Learning, pages 1367-1376. PMLR, 2018.  
+[23] Aude Geneva, Gabriel Peyré, and Marco Cuturi. Learning generative models with sinkhorn divergences. In International Conference on Artificial Intelligence and Statistics, pages 1608-1617. PMLR, 2018.  
+[24] Promit Ghosal and Bodhisattva Sen. Multivariate ranks and quantiles using optimal transport: Consistency, rates and nonparametric testing. The Annals of Statistics, 50(2):1012-1037, 2022.  
+[25] Florian F Gunsilius. On the convergence rate of potentials of brenier maps. *Econometric Theory*, 38(2):381-417, 2022.  
+[26] Marc Hallin, Eustasio del Barrio, Juan Cuesta-Albertos, and Carlos Matrán. Distribution and quantile functions, ranks and signs in dimension  $d$ : A measure transportation approach. The Annals of Statistics, 49(2):1139 - 1165, 2021.  
+[27] Jan-Christian Hütter and Philippe Rigollet. Minimax estimation of smooth optimal transport maps. The Annals of Statistics, 2021.  
+[28] Ksenia A Khudiakova, Jan Maas, and Francesco Pedrotti.  $\mathbf{L}_{\infty}$ -optimal transport of anisotropic log-concave measures and exponential convergence in fisher's infinitesimal model. The Annals of Applied Probability, 35(3):1913-1940, 2025.  
+[29] Dominik Klein, Théo Uscidda, Fabian Theis, and Marco Cuturi. Genot: Entropic (gromov) Wasserstein flow matching with applications to single-cell genomics. Advances in Neural Information Processing Systems, 37:103897-103944, 2024.  
+[30] Alexander Korotin, Vage Egiazarian, Arip Asadulaev, Alexander Safin, and Evgeny Burnaev. Wasserstein-2 generative networks. In International Conference on Learning Representations, 2021.  
+[31] Alexander Korotin, Daniil Selikhannovych, and Evgeny Burnaev. Neural optimal transport. 2023.  
+[32] Zezeng Li, Shenghao Li, Zhanpeng Wang, Na Lei, Zhongxuan Luo, and David Xianfeng Gu. Dpm-ot: a new diffusion probabilistic model based on optimal transport. In Proceedings of IEEE/CVF International Conference on Computer Vision, pages 22624–22633, 2023.  
+[33] Ashok Makkuva, Amirhossein Taghvaei, Sewoong Oh, and Jason Lee. Optimal transport mapping via input convex neural networks. In International Conference on Machine Learning, pages 6672-6681. PMLR, 2020.  
+[34] Tudor Manole, Sivaraman Balakrishnan, Jonathan Niles-Weed, and Larry Wasserman. Plugin estimation of smooth optimal transport maps. The Annals of Statistics, 52(3):966–998, 2024.  
+[35] Boris Muzellec, Adrien Vacher, Francis Bach, François-Xavier Vialard, and Alessandro Rudi. Near-optimal estimation of smooth transport maps with kernel sums-of-squares. arXiv preprint arXiv:2112.01907, 2021.  
+[36] A Paszke. Pytorch: An imperative style, high-performance deep learning library. arXiv preprint arXiv:1912.01703, 2019.  
+[37] Aram-Alexandre Pooladian, Vincent Divol, and Jonathan Niles-Weed. Minimax estimation of discontinuous optimal transport maps: The semi-discrete case. In International Conference on Machine Learning, pages 28128–28150. PMLR, 2023.
+
+[38] Litu Rout, Alexander Korotin, and Evgeny Burnaev. Generative modeling with optimal transport maps. 2022.  
+[39] Geoffrey Schiebinger, Jian Shu, Marcin Tabaka, Brian Cleary, Vidya Subramanian, Aryeh Solomon, Joshua Gould, Siyan Liu, Stacie Lin, Peter Berube, et al. Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming. Cell, 176(4):928–943, 2019.  
+[40] Ilya O. Tolstikhin, Olivier Bousquet, Sylvain Gelly, and Bernhard Scholkopf. Wasserstein auto-encoders. In International Conference on Learning Representations, 2018.  
+[41] Adrien Vacher, Boris Muzellec, Alessandro Rudi, Francis Bach, and Francois-Xavier Vialard. A dimension-free computational upper-bound for smooth optimal transport estimation. In Conference on Learning Theory, pages 4143-4173. PMLR, 2021.  
+[42] Cédric Villani. Optimal Transport: Old and New, volume 338. Springer, 2009.  
+[43] Fangneng Zhan, Yingchen Yu, Kaiwen Cui, Gongjie Zhang, Shijian Lu, Jianxiong Pan, Chang-gong Zhang, Feiying Ma, Xuansong Xie, and Chunyan Miao. Unbalanced feature transport for exemplar-based image translation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 15028-15038, 2021.
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: We provide exact Theorems/Propositions numbers for our main claims in the abstract and introduction, which correspond precisely with our four subsections in Section 3.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: We explicitly discuss the limitations of our work. For example, in Section 3.3, we acknowledge the lack of theoretical justification for the practical choice of the sieve radius  $\tilde{R}_{\varepsilon}$ .
+
+Regarding the convergence rate of the sieved TNN estimator in Theorem 3.7, we note that it is not minimax optimal as in [27], which is expected due to our substantially weaker assumptions. Furthermore, we explain our restriction to tanh networks and do not consider input convex neural networks (ICNNs), owing to the limited approximation theory available for general smooth neural networks.
+
+# Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover
+
+limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory assumptions and proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: Proof of our results are included in the appendix.
+
+Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental result reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: We ensure full reproducibility by including a description of our sieved-TNN model and important training hyper-parameters setting in Section 4. Full description of our algorithm is provided in Appendix G. Meanwhile, as a synthetic simulation study, the data generating processes are provided in our code for reproducing consideration.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.
+
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
+
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
+
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
+
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
+
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.
+
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: Full implementation code is available in the Supplementary Materials. All experimental data are synthetic and can be recreated exactly using the data-generation procedures and fixed random seeds detailed in Appendix G.
+
+# Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).  
+- The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.  
+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.  
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental setting/details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: Section 4 details the sieved-TNN architecture, data-splitting, and all key hyperparameters (learning rate, batch size, network depth, etc.). Appendix G provides the complete experimental design, including full algorithm pseudo-code, the process of training, the synthetic data-generation procedure with fixed random seeds, and the optimizer choice.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment statistical significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: Every plotted result includes error bars representing one standard error of the mean over multiple independent runs. Since no statistical hypothesis tests are involved, the discussion of statistical significance is not applicable.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
+- The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).  
+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).  
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments compute resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We briefly introduce the configuration for our server in Section 4, with full description available in the Appendix G.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code of ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: We have all reviewed the NeurIPS Code of Ethics and striven to maintain and preserve anonymity.
+
+Guidelines:
+
+- The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
+
+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
+
+Justification: This work studies the theoretical properties of optimal transport and the authors cannot see any direct or potential societal impacts this work can bring.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: As a theoretical study on optimal transport, this work does not involve any data sets, and does not pose risks to existing optimal transport models for image translation or generation.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.  
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [NA]
+
+Justification: We did not incorporate any third-party code, data, or models. We properly cited, respected, mentioned the algorithm from [20] and re-implemented it ourselves, so no external "assets" or licenses are involved.
+
+Guidelines:
+
+- The answer NA means that the paper does not use existing assets.  
+- The authors should cite the original paper that produced the code package or dataset.  
+- The authors should state which version of the asset is used and, if possible, include a URL.  
+- The name of the license (e.g., CC-BY 4.0) should be included for each asset.  
+- For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.  
+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
+- For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.  
+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+Justification: We provide the full sieved-TNN implementation, including source code, README, and pseudo-code, as a new asset in the Supplementary Materials (with plans to publish it on GitHub upon acceptance).
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.  
+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
+- The paper should discuss whether and how consent was obtained from people whose asset is used.  
+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and research with human subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: As a theoretical work, no human subjects are involved in it.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.
+
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional review board (IRB) approvals or equivalent for research with human subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: This theoretical study does not involve crowdsourcing nor research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
+- We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.  
+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
+
+# 16. Declaration of LLM usage
+
+Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
+
+Answer: [NA]
+
+Justification: The theoretical and methodology advancements in this work are built on well-established mathematical tools, and LLM is not utilized for developing them.
+
+Guidelines:
+
+- The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.  
+- Please refer to our LLM policy (https://neurips.cc/Conferences/2025/LLM) for what should or should not be described.

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