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2025/EdgeRunner_ Auto-regressive Auto-encoder for Artistic Mesh Generation/962b02d9-56b5-4071-8689-37edbc1f21b7_origin.pdf

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2025/EdgeRunner_ Auto-regressive Auto-encoder for Artistic Mesh Generation/full.md

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+# EDGERUNNER: AUTO-REGRESSIVE AUTO-ENCODER FOR ARTISTIC MESH GENERATION
+
+Jiaxiang Tang $^{1*}$  Zhaoshuo Li $^{2}$  Zekun Hao $^{2}$  Xian Liu $^{2}$  Gang Zeng $^{1}$  Ming-Yu Liu $^{2}$  Qinsheng Zhang $^{2}$
+
+$^{1}$ State Key Laboratory of General Artificial Intelligence, Peking University.  $^{2}$ NVIDIA Research.
+
+https://research.nvidia.com/labs/dir/edgerunner/
+
+![](images/40e8ba589acb1b13ece3c34a3601df6623adaa23e149ec70fbb20de3da705855.jpg)  
+Dense Mesh  
+Input Point Cloud  
+Point Cloud Conditioned Mesh Generation
+
+![](images/bae0f74b43cc048d9b79a67a314b7a9e469cfb70f0dfff44c849136225ea8562.jpg)  
+Single-view Image  
+Image Conditioned Mesh Generation
+
+![](images/b6be7b1d0e71562df259e8d56c146fe22b27e93b85f99de96c0f594d454294f8.jpg)  
+Diversity of Generation with Different Random Seeds (Point Cloud Conditioned)  
+Figure 1: EdgeRunner efficiently generates diverse, high-quality artistic meshes conditioned on point clouds or single-view images.
+
+# ABSTRACT
+
+Current auto-regressive mesh generation methods suffer from issues such as incompleteness, insufficient detail, and poor generalization. In this paper, we propose an Auto-regressive Auto-encoder (ArAE) model capable of generating high-quality 3D meshes with up to 4,000 faces at a spatial resolution of  $512^{3}$ . We introduce a novel mesh tokenization algorithm that efficiently compresses triangular meshes into 1D token sequences, significantly enhancing training efficiency. Furthermore, our model compresses variable-length triangular meshes into a fixed-length latent space, enabling training latent diffusion models for better generaliza
+
+tion. Extensive experiments demonstrate the superior quality, diversity, and generalization capabilities of our model in both point cloud and image-conditioned mesh generation tasks.
+
+# 1 INTRODUCTION
+
+Automatic 3D content generation, particularly the generation of widely used polygonal meshes, holds the potential to revolutionize industries such as digital gaming, virtual reality, and filmmaking. Generative models can make 3D asset creation more accessible to non-experts by drastically reducing the time and effort involved. This democratization opens up opportunities for a wider range of individuals to contribute to and innovate within the 3D content space, fostering greater creativity and efficiency across these sectors.
+
+Previous research on 3D generation has explored a variety of approaches. For example, optimization-based methods, such as using score distillation sampling (SDS) (Poole et al., 2022; Lin et al., 2023; Liu et al., 2023b; Tang et al., 2023a), lift 2D diffusion priors into 3D without requiring any 3D data. In contrast, large reconstruction models (LRM) (Hong et al., 2023; Wang et al., 2023b; Xu et al., 2023b; Li et al., 2023; Weng et al., 2024b) directly train feed-forward models to predict neural radiance fields (NeRF) or Gaussian Splatting from single or multi-view image inputs. Lastly, 3D-native latent diffusion models (Zhang et al., 2024c; Wu et al., 2024b; Li et al., 2024c) encode 3D assets into latent spaces and generate diverse contents by performing diffusion steps in the latent space. However, all these approaches rely on continuous 3D representations, such as NeRF or SDF grids, which lose the discrete face indices in triangular meshes during conversion. Consequently, they require post-processing, such as marching cubes (Lorensen & Cline, 1998) and re-meshing algorithms, to extract triangular meshes. These meshes differ significantly from artist-created ones, which are more concise, symmetric, and aesthetically structured. Additionally, these methods are limited to generating watertight meshes and cannot produce single-layered surfaces.
+
+Recently, several approaches (Siddiqui et al., 2024a; Chen et al., 2024d;b; Weng et al., 2024a; Chen et al., 2024e) have attempted to tokenize meshes into 1D sequences and leverage auto-regressive models for direct mesh generation. Specifically, MeshGPT (Siddiqui et al., 2024a) proposes to empirically sort the triangular faces and apply a vector-quantization variational auto-encoder (VQVAE) to tokenize the mesh. MeshXL (Chen et al., 2024b) directly flattens the vertex coordinates and does not use any compression other than vertex discretization. Since these methods directly learn from mesh vertices and faces, they can preserve the topology information and generate artistic meshes. However, these auto-regressive mesh generation approaches still face several challenges. (1) Generation of a large number of faces: due to the inefficient face tokenization algorithms, most prior methods can only generate meshes with fewer than 1,600 faces, which is insufficient for representing complex objects. (2) Generation of high-resolution surface: previous works quantize mesh vertices to a discrete grid of only  $128^3$  resolution, which results in significant accuracy loss and unsmooth surfaces. (3) Model generalization: training auto-regressive models with difficult input modalities is challenging. Previous approaches often struggle to generalize beyond the training domain when conditioning on single-view images.
+
+In this paper, we present a novel approach named EdgeRunner to address the aforementioned challenges. Firstly, we introduce a mesh tokenization method based on EdgeBreaker (Rossignac, 1999) that compresses sequence length by  $50\%$  and reduces long-range dependency between tokens, significantly improving the training efficiency. Secondly, we propose an Auto-regressive Auto-encoder (ArAE) that compresses variable-length triangular meshes into fixed-length latent codes. This latent space can be used to train latent diffusion models conditioned on other modalities, offering better generalization capabilities. We also enhance the training pipeline to support higher quantization resolution. These improvements enable EdgeRunner to generate diverse, high-quality artistic meshes with up to 4,000 faces and vertices discretized at a resolution of  $512^3$  — resulting in sequences that are twice as long and four times higher in resolution compared to previous methods.
+
+In summary, our contributions are as follows:
+
+1. We introduce a novel mesh tokenization algorithm, adapted from EdgeBreaker, which supports lossless face compression, prevents flipped faces, and reduces long-range dependencies to facilitate learning.
+
+2. We propose an Auto-regressive Auto-encoder (ArAE), comprising a lightweight encoder and an auto-regressive decoder, capable of compressing variable-length triangular meshes into fixed-length latent codes.  
+3. We demonstrate that the latent space of ArAE can be leveraged to train latent diffusion models for better generalization, enabling conditioning of different input modalities such as single-view images.  
+4. Extensive experiments show that our method generates high-quality and diverse artistic meshes from point clouds or single-view images, exhibiting improved generalization and robustness compared to previous methods.
+
+# 2 RELATED WORK
+
+# 2.1 OPTIMIZATION-BASED 3D GENERATION
+
+Early 3D generation methods relied on SDS-based optimization techniques (Jain et al., 2022; Poole et al., 2022; Wang et al., 2023a; Mohammad Khalid et al., 2022; Michel et al., 2022) due to limited 3D data. Subsequent works advanced in generation quality (Lin et al., 2023; Wang et al., 2023d; Chen et al., 2023c; e; Sun et al., 2023; Qiu et al., 2024), reducing generation time (Tang et al., 2023a; Yi et al., 2023; Lorraine et al., 2023; Xu et al., 2024a), enabling 3D editing (Zhuang et al., 2023; Singer et al., 2023; Raj et al., 2023; Chen et al., 2024c), and conditioning on images (Xu et al., 2023a; Tang et al., 2023b; Melas-Kyriazi et al., 2023; Liu et al., 2023b; Qian et al., 2023; Shi et al., 2023). Other approaches first predict multi-view images, then apply reconstruction algorithms to generate the final 3D models (Long et al., 2023; Li et al., 2024b; Pang et al., 2024; Tang et al., 2024b). Recently, Unique3D (Wu et al., 2024a) introduced a method that combines high-resolution multi-view diffusion models with an efficient mesh reconstruction algorithm, achieving both high quality and fast image-to-3D generation.
+
+# 2.2 FEED-FORWARD 3D GENERATION
+
+With the introduction of large-scale datasets (Deitke et al., 2023b;a), more recent works propose to use feed-forward 3D models. The Large Reconstruction Model (LRM)(Hong et al., 2023) demonstrated that end-to-end training of a triplane-NeRF regression model scales effectively to large datasets and generates 3D assets within seconds. While LRM significantly accelerates generation speed, the resulting meshes often exhibit lower quality and a lack of diversity. Subsequent research has sought to improve generation quality by incorporating multi-view images as inputs(Xu et al., 2024b; Li et al., 2023; Wang et al., 2023b; He & Wang, 2023; Siddiqui et al., 2024b; Xie et al., 2024; Wang et al., 2024) and by adopting more efficient 3D representations (Zhang et al., 2024a; Li et al., 2024a; Wei et al., 2024; Zou et al., 2023; Tang et al., 2024a; Xu et al., 2024d; Zhang et al., 2024b; Chen et al., 2024a; Yi et al., 2024).
+
+# 2.3 DIFFUSION-BASED 3D GENERATION
+
+Analogous to 2D diffusion models for image generation, significant efforts have been made to develop 3D-native diffusion models capable of conditional 3D generation. Early approaches typically rely on uncompressed 3D representations, such as point clouds, NeRFs, tetrahedral grids, and volumes (Nichol et al., 2022; Jun & Nichol, 2023; Gupta et al., 2023; Cheng et al., 2023; Ntavelis et al., 2023; Zheng et al., 2023; Zhang et al., 2023; Liu et al., 2023c; Müller et al., 2023; Chen et al., 2023d; Cao et al., 2023; Chen et al., 2023a; Wang et al., 2023c; Yariv et al., 2023; Liu et al., 2023a; Xu et al., 2024c; Yan et al., 2024) to train diffusion models. However, these methods are often limited by small-scale datasets and struggle with generalization or producing high-quality assets. More recent approaches have focused on adapting latent diffusion models to 3D (Zhao et al., 2023; Zhang et al., 2024c; Wu et al., 2024b; Li et al., 2024c; Lan et al., 2024; Hong et al., 2024; Tang et al., 2023c; Chen et al., 2024f). These methods first train a VAE to compress 3D representations into a more compact form, which enables more efficient diffusion model training. Unlike the straightforward image representations in 2D, 3D latent diffusion models involve numerous design choices, leading to varied performance outcomes. For example, CLAY (Zhang et al., 2024c) has demonstrated that a transformer-based 3D latent diffusion model can scale to large datasets and generalize well across diverse input conditions.
+
+![](images/887ebad7f98a1f55f7b33edb5df3c900326567daa3b35324bcbb7db3e96415ba.jpg)  
+Figure 2: Pipeline of our method. Our ArAE model compresses variable-length mesh into fixed-length latent code, which can be further used to train latent diffusion models conditioned on other input modalities, such as single-view images.
+
+# 2.4 AUTO-REGRESSIVE MESH GENERATION
+
+The above works require additional post-processing steps to extract triangular meshes and fail to model the mesh topology. Recently, approaches using auto-regressive models to directly generate meshes have emerged. MeshGPT (Siddiqui et al., 2024a) pioneered this approach by tokenizing a mesh through face sorting and compression with a VQ-VAE, followed by using an auto-regressive transformer to predict the token sequence. This method allows for the generation of meshes with varying face counts and incorporates direct supervision from topology information, which is often overlooked in other approaches. Subsequent works (Chen et al., 2024b; Weng et al., 2024a; Chen et al., 2024d) have explored different model architectures and extended this approach to conditional generation tasks, such as point cloud generation. However, these methods are limited to meshes with fewer than 1,000 faces due to the computational cost of mesh tokenization and exhibit limited generalization capabilities. A concurrent work, MeshAnythingV2 (Chen et al., 2024e), introduces an improved mesh tokenization technique, increasing the maximum number of faces to 1,600. Our approach also falls under the category of auto-regressive mesh generation but aims to further extend the maximum face count and provide control over the target face number during inference.
+
+# 3 EDGERUNNER
+
+# 3.1 COMPACT MESHTOKENIZATION
+
+Auto-regressive models process information in the form of discrete token sequences. Thus, compact tokenization is crucial as it allows information to be represented with fewer tokens accurately. For example, text tokenizers have been a central research direction for large language models (LLMs). The GPT and Llama series (Touvron et al., 2023; Brown et al., 2020) utilize the byte-pair encoding (BPE) tokenizer, which combines sub-word units into single tokens for highly compact and lossless compression.
+
+In contrast, tokenization techniques used in prior auto-regressive mesh generation works mainly suffer from two issues. Some prior works use lossy VQ-VAEs (Siddiqui et al., 2024a; Chen et al., 2024d; Weng et al., 2024a), which sacrifices the mesh generation quality. Others opt for zero-compression by not using face tokenizers (Chen et al., 2024b), which poses training challenges due to the inefficiency.
+
+In this paper, we introduce a tokenization scheme that allows us to represent a mesh compactly and efficiently, which is based on the well-established triangular mesh compression algorithm Edge-Breaker (Rossignac, 1999). The key insight for mesh compression is to maximize edge sharing between adjacent triangles. By sharing an edge with the previous triangle, the next triangle re
+
+![](images/95a43919ad98151a449b005becfe38bde959826ac3e457a2dd85ebd9f8d3ff54.jpg)  
+Figure 3: Illustration of our mesh tokenizer. Our tokenizer traverses the 3D mesh triangle-by-triangle and converts it into a 1D token sequence. Through edge sharing, we reach a compression rate of  $50\%$  (4 or 5 tokens per face on average) compared to naive tokenization of 9 tokens per face.
+
+![](images/75f602cb5d952d507ba028292a59cfe94a6f375eb90b21e7ac62a761f56c1efa.jpg)
+
+![](images/0944559c926d13d91268795197b7317b711d28e4f684e3de91a2c36059638391.jpg)  
+Compression Ratio on Complex Meshes
+
+![](images/b0e6053979818c7728f65bb677d0e80a0ce596eb3f5e89383dc3b2ea14b0a6f0.jpg)
+
+requires only one additional vertex instead of three. We illustrate our mesh tokenization process with an example below, and provide more details in the appendix.
+
+Half-edge. EdgeBreaker (Rossignac, 1999) uses half-edge data structures (Weiler, 1986) for triangular face traversal. An illustration is provided in Figure 4. We use  $\mathcal{H}$  to denote a half-edge. For example,  $\mathcal{H}_{41}^{3}$  is the half-edge pointing from vertex 4 to 1, with vertex 3 across the face. Starting from  $\mathcal{H}_{41}^{3}$ , we can traverse to the next half-edge  $\mathcal{H}_{13}^{4}$  and the next twin half-edge  $\mathcal{H}_{31}^{2}$ . Reversely, the previous half-edge is  $\mathcal{H}_{34}^{1}$  and the previous twin half-edge is  $\mathcal{H}_{43}^{5}$ . The half-edge data structure has also been used in recent learning-based mesh generation work (Shen et al., 2024).
+
+Vertex Tokenization. To tokenize a mesh into a discrete sequence, vertex coordinates require discretization. Following previous works (Siddiqui et al., 2024a), we normalize the mesh to a unit cube and quantize the continuous vertex coordinates into
+
+![](images/2240bc3eb6b98bbbcfb311f06f2d867fcc1925de1b8bbd4aba73c791b4c51157.jpg)  
+Figure 4: Half-edge representation for triangular faces.
+
+integers according to a quantization resolution, which is 512 in this work. Each vertex is therefore represented by three integer coordinates, which are then flattened in XYZ order as tokens. With some abuse of notion, we denote the XYZ tokens as a single vertex token using  $\odot$ .
+
+Face Tokenization. We traverse through all faces following the half-edges. To illustrate the process, we use the mesh example in Figure 3. The process starts with one half-edge, where  $\mathcal{H}_{23}^{1}$  is picked as the beginning of the current traversal. We signify the start of a traversal as  $\mathbf{B}$ . We then append the vertex across the half-edge 1 as the first vertex token. Within the same triangular face, the two vertices 2 and 3 are also appended following the direction of  $\mathcal{H}_{23}^{1}$ .
+
+During traversal, we visit the next twin half-edge whenever possible, and only reverse the half-edge direction to the previous twin half-edge when we exhaust all triangles in the current traversal. Returning to the example in Figure 3, we follow  $\mathcal{H}_{23}^{1}$  and reach  $\mathcal{H}_{13}^{4}$ . Thus, we append N to signify the next twin traversal direction and we only need to append 4 as 1 3 are shared. The same process is repeated for  $\mathcal{H}_{43}^{5}$  with N 5 added to the current sub-sequence.
+
+We have completed the current traversal as no adjacent faces can be found for  $\mathcal{H}_{43}^{5}$ . The sub-sequence for the current traversal is thus B 1 2 3 N 4 N 5.
+
+To begin a new sub-sequence, we reversely retrieve the last-added half-edges to traverse in the opposite directions. As the last-added half-edge  $\mathcal{H}_{43}^{5}$  doesn't have any adjacent faces, we skip it and instead consider  $\mathcal{H}_{13}^{4}$ . We go opposite to its previous twin half-edge  $\mathcal{H}_{14}^{6}$ . As this is a new sub-sequence, B 6 1 4 are added.
+
+We continue finding the un-visited faces in the neighborhood of  $\mathcal{H}_{14}^{6}$  and arriving at its previous twin half-edge  $\mathcal{H}_{16}^{7}$ . Thus, we add  $\mathrm{P} 7$  to the current sub-mesh sequence as  $⑥$  1 are shared. The
+
+process is repeated and P 8 P 2 N 9 are added. As all triangular faces have been visited, the face tokenization process of the mesh is complete.
+
+Auxiliary Tokens. Similar to prior work in LLMs, we preprocess a BOS at the beginning of a mesh sequence and append a EOS at the end of a mesh sequence.
+
+Detokenization. It is straightforward to reconstruct the original mesh from a mesh token sequence. We iterate over the tokens while maintaining a state machine. Each  $\mathbf{B}$  of a sub-sequence is always followed by three vertex tokens. Each N or P is followed by a single vertex token, and we retrieve two previous vertex tokens based on the traversal direction to reconstruct the triangle. Finally, we merge duplicate vertices, as they may appear multiple times from different sub-sequences, and output the reconstructed mesh.
+
+Advantages. Our tokenizer benefits model training in several ways: (1) Each face requires an average of 4 to 5 tokens, achieving approximately  $50\%$  compression compared to the 9 tokens used in previous works (Chen et al., 2024b;d). This increased efficiency enables the model to generate more faces with the same number of tokens and facilitates training on datasets containing a higher number of faces. (2) Our traversal is designed to avoid long-range dependency between tokens. Each token only relies on a short context of previous tokens, which further mitigate the difficulty of learning. (3) The traversal ensures that each face's orientation remains consistent within each sub-mesh. Consequently, the generated mesh can be accurately rendered using back face culling, a feature not consistently achieved in prior methods. We will further illustrate these benefits in Section 4.
+
+# 3.2 AUTO-REGRESSIVE AUTO-ENCODER
+
+Although our decoder is auto-regressive and generates variable-length token sequences, we observe limitations in both generation diversity and its ability to follow conditioning. In contrast, diffusion models, which have been extensively studied (Rombach et al., 2022; Zhang et al., 2024c), demonstrate promising results in addressing these challenges. To apply diffusion models, a key challenge is the requirement of fixed-length data, as mesh generation has a variable-length data structure and the number of triangle faces can vary significantly due to different complexities. Therefore, we propose an Auto-regressive Auto-encoder (ArAE) to encode the variable-length mesh into a fixed-length latent space, similar to the role of variational auto-encoders (VAEs) in latent diffusion models. To train our ArAE model, we choose point clouds as the encoder input for geometric information and our tokenized mesh sequences as output. Thus the ArAE model itself also works as a point cloud conditioned mesh generator.
+
+Architecture. The architecture of our ArAE model is illustrated in Figure 2. ArAE consists of a lightweight encoder and an auto-regressive decoder. To extract geometric information from the mesh surface, we follow prior works to sample a point cloud and apply a transformer encoder (Chen et al., 2024d; e; Zhang et al., 2023; 2024c). Specifically, we sample  $N$  random points,  $\mathbf{X} \in \mathbb{R}^{N \times 3}$ , from the surface of the input mesh and use a cross-attention layer to extract the latent code:
+
+$$
+\mathbf {Z} = \operatorname {C r o s s A t t} (\mathbf {Q}, \operatorname {P o s E m b e d} (\mathbf {X})) \tag {1}
+$$
+
+where  $\mathbf{Q} \in \mathbb{R}^{M \times C}$  represents the trainable query embedding with a hidden dimension of  $C$ ,  $\mathrm{PosEmbed}(\cdot)$  is a frequency embedding function for 3D points (Zhang et al., 2023), and  $\mathbf{Z} \in \mathbb{R}^{M \times L}$  represents the latent code,  $M < N$  and  $L < C$  denote the latent size and dimension, respectively. The decoder is an auto-regressive transformer, designed to generate a variable-length mesh token sequence. For simplicity, we adopt the OPT architecture (Zhang et al., 2022), as used in prior works (Siddiqui et al., 2024a; Chen et al., 2024d;b). A learnable embedding converts discrete tokens into continuous features, and a linear head maps predicted features back to classification logits. Stacked causal self-attention layers are employed to predict the next token based on previous tokens. The latent code  $\mathbf{Z}$  is prepended to the input before the BOS token, allowing the decoder to learn how to generate a mesh token sequence conditioned on the latent code.
+
+Face Count Condition. Another issue is that given the same input point cloud, multiple plausible meshes with varying numbers of faces and topologies can be generated. The number of faces is particularly crucial as it directly affects the mesh's complexity (low-poly vs. high-poly) and the generation speed. To manage meshes with a broad range of face counts, we aim to provide some level of explicit control over the targeted number of faces. This control facilitates the estimation of
+
+generation time and the complexity of the generated mesh during inference. We propose a simple face count conditioning method for coarse-grained control. Specifically, we append a learnable face count token after the latent code condition tokens. We bucket face count into different ranges and assign different tokens to each range. For instance, we use four distinct tokens to represent face counts in the following ranges: less than or equal to 1000, between 1000 and 2000, between 2000 and 4000, and greater than 4000. Additionally, during training, we randomly replace these tokens with a fifth unconditional token. This approach ensures that the model still learns to generate meshes without specifying a targeted face count.
+
+Loss Function. The ArAE is trained using the standard cross-entropy loss on the predicted next tokens:
+
+$$
+\mathcal {L} _ {\mathrm {c e}} = \operatorname {C r o s s E n t r o p y} (\tilde {\mathbf {S}} [: - 1 ], \mathbf {S} [ 1: ]) \tag {2}
+$$
+
+where  $\mathbf{S}$  denotes the one-hot ground truth token sequence, and  $\hat{\mathbf{S}}$  represents the predicted classification logits sequence. Additionally, to constrain the range of the latent space for easier training of subsequent diffusion models, we apply an L2 norm penalty to the latent code:
+
+$$
+\mathcal {L} _ {\text {r e g}} = \left\| \mathbf {Z} \right\| _ {2} ^ {2} \tag {3}
+$$
+
+The final loss is a weighted combination of the cross-entropy loss and the regularization term.
+
+# 3.3 IMAGE-CONDITIONED LATENT DIFFUSION
+
+With the fixed-length latent space provided by our ArAE architecture, it is now feasible to train mesh generation models conditioned on different inputs, akin to how 2D image generation models are trained. Among various input modalities, we showcase the capabilities of our model using single-view images, which are among the most commonly used conditions for mesh generation.
+
+We follow the approach of previous methods (Chen et al., 2023b) and utilize a diffusion transformer (DiT) as the backbone. Specifically, we employ the image encoder from CLIP (Ilharco et al., 2021; Mohammad Khalid et al., 2022) to extract image features for conditioning. Cross-attention layers are used to integrate the image condition into the denoising features, while AdaLN layers incorporate timestep information. We use the DDPM framework (Ho et al., 2020) and mean square error (MSE) loss to train the DiT model. At each training step, we randomly sample a timestep  $t$  and a Gaussian noise  $\epsilon \in \mathbb{R}^{M \times L}$ . The loss is calculated between the predicted noise and the random noise.
+
+# 4 EXPERIMENTS
+
+# 4.1 QUALITATIVE RESULTS
+
+Point Cloud Conditioned Generation. We first compare the generated meshes conditioned on point clouds in Figure 5. For the test samples, we use other image-to-3D methods (Zhang et al., 2024c; Li et al., 2024c) to generate dense meshes, ensuring that these samples have never been seen in the training dataset. Notably, MeshAnything (Chen et al., 2024d; e) uses both point coordinates and normal vectors as input to a pretrained encoder (Zhao et al., 2023). In contrast, our encoder takes only point coordinates and is trained from scratch, while still achieving better performance. Our model demonstrates greater stability on challenging test cases, producing more aesthetically pleasing meshes. Additionally, the proposed tokenization algorithm enables the generation of longer sequences, facilitating the modeling of complex shapes and improving the closure of loops for watertight surfaces.
+
+Image Conditioned Generation. In Figure 6, we compare the meshes generated from single-view images. Since other auto-regressive methods cannot directly condition on images, we compare our results with a recent optimization-based approach (Wu et al., 2024a). Our end-to-end image-to-mesh generation produces artistic meshes that better capture the semantics of the input images. Additionally, our model exhibits strong generalization on challenging 2D-style or realistic-lighting images, despite being trained exclusively on images rendered from 3D meshes with simple shading.
+
+Face Count Control. In practice, users may have specific requirements for the face count or level of detail in the generated mesh. Our design enables coarse-grained control over the face count by allowing users to specify the face count condition. As demonstrated in Figure 7, our model is capable of generating meshes with different ranges of face count from the same input point cloud.
+
+![](images/b3abdab646cb2f9e94bb4c824bdedeba3acf19767462ba16b2c3a08a58de3e1b.jpg)  
+Figure 5: Comparison on point cloud conditioned generation. We show the reference dense mesh and generated meshes conditioned on randomly sampled point cloud.
+
+# 4.2 QUANTITATIVE RESULTS
+
+User Study. Since evaluation of the mesh aesthetic quality is challenging, we majorly rely on a user study for quantitative comparisons. For a collection of 8 test cases, we render the mesh geometry and wireframe generated from MeshAnything (Chen et al., 2024d), MeshAnythingV2 (Chen et al., 2024e), and our method. Each volunteer is shown 8 samples from mixed random methods, and asked to rate in three aspects: input consistency with the point cloud, triangle aesthetic, and overall mesh quality. We collect results from 20 volunteers with 480 valid scores. As shown in Table 1, our method is preferred across all evaluated aspects.
+
+Inference Speed. Since we are using next-token prediction, the inference time for each generation is dependent on the length of the sequence. Our model is basically a large transformer, and the major compute is used for attention calculation. On a A100 GPU with flash-attention (Dao, 2024) and
+
+![](images/0424185c2c301d8b51be1d0a1019f0d7188d6555bd139ff205ffc4ba448132ee.jpg)  
+Figure 6: Comparison on image conditioned generation. We show the input image and generated meshes. Since Unique3D outputs dense meshes, we only visualize the surface without wireframe.
+
+<table><tr><td></td><td>Input Consistency</td><td>Triangle Aesthetic</td><td>Overall Quality</td></tr><tr><td>MeshAnything</td><td>2.93</td><td>2.43</td><td>2.43</td></tr><tr><td>MeshAnythingV2</td><td>2.64</td><td>2.36</td><td>2.14</td></tr><tr><td>Ours</td><td>4.83</td><td>4.54</td><td>4.58</td></tr></table>
+
+Table 1: User Study on point-conditioned mesh generation. The rating is of scale 1-5, the higher the better.
+
+KV cache enabled, our model runs at about 100 tokens per second. Specifically, it takes about 45 seconds to generate a mesh with 1,000 faces, about 90 seconds for 2,000 faces, and about 3 minutes for 4,000 faces.
+
+![](images/a1c901ab1347441596f194f3b8dc0268861414613b80161ed8ba35c38fab737f.jpg)  
+Figure 7: Coarse-grained face count control. Our model supports controlling the targeted number of face, which allows to generate meshes with different levels of detail.
+
+![](images/b28a7e5aa627820753514b0957c144c2614e09107e1ffef6f3df1e70ce3e406e.jpg)  
+Figure 8: Ablation Study on different quantization resolutions and image conditioning strategies.
+
+# 4.3 ABLATION STUDIES
+
+Quantization Resolution. We initially experimented with a quantization resolution of  $128^{3}$  as used in previous works. While this lower resolution made it easier to train the classification head, we observed noticeable quality degradation, as shown in the left part of Figure 8. To address this problem, we increased the quantization resolution to  $512^{3}$  for our final model, which resulted in more accurate vertex positions and smoother surfaces.
+
+Direct Image Conditioning. We also attempted to directly train an image-conditioned auto-regressive mesh decoder. The image features from the CLIP vision encoder were prepended as condition tokens, and the model directly predicted the mesh token sequence. However, we found that this approach was more difficult to converge and generally exhibited poorer generalization on unseen test cases, as shown in the right part of Figure 8. Consequently, we designed the ArAE model to learn a fixed-length latent space using a simpler point-cloud condition, and we trained latent diffusion models to map complex condition modalities to this latent space. This approach led to better generalization and more efficient training.
+
+# 5 CONCLUSION
+
+In this work, we propose a novel Auto-regressive Auto-encoder that compresses variable-length triangular meshes into fixed-length latent codes, and we further demonstrate the potential of this latent space for image-to-mesh generation. Additionally, we introduce an efficient mesh tokenization algorithm that enables scaling up the model, resulting in longer context lengths and higher resolution. Our approach highlights the potential for improving the scalability of auto-regressive models for practical 3D mesh generation, offering a promising direction for future research and applications.
+
+# ACKNOWLEDGMENTS
+
+This work is supported by the Sichuan Science and Technology Program (2023YFSY0008), National Natural Science Foundation of China (61632003, 61375022, 61403005), Grant SCITLAB-20017 of Intelligent Terminal Key Laboratory of SiChuan Province, Beijing Advanced Innovation Center for Intelligent Robots and Systems (2018IRS11), and PEK-SenseTime Joint Laboratory of Machine Vision. We thank Yongxin Chen for introducing the EdgeBreaker paper. We thank Nicholas Sharp and Frank Shen for sharing the mesh visualization code.
+
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+
+# A APPENDIX
+
+# A.1 IMPLEMENTATION DETAILS
+
+# A.1.1 MESHTOKENIZER
+
+Algorithm 1: Tokenization  
+Data: Discretized Vertices  $\mathbf{V} = \{\mathbf{v}_i\}_{N}$  Triangle Faces  $\mathbf{T} = \{\mathbf{t}_i\}_{M}$  Result: Array O to hold the token sequence.   
+ $/^{\star}$  Write a vertex to output.  $\star /$    
+def WriteVertex (Vertex v): O.append(v.x, v.y, v.z);   
+ $/^{\star}$  Recursive function to compress one face.  $\star /$    
+def TokenizeFace (HalfEdge c, bool first): c.t.vis  $=$  true; if not first: WriteVertex (c.v); if not c.v.vis: c.v.vis  $=$  true; O.append(N); TokenizeFace (c.n.o, false); elif c.n.o.t.vis and c.p.o.t.vis: return; /* End of recursion.  $\star /$  elif c.n.o.t.vis: O.append(P); TokenizeFace (c.p.o, false); elif c.p.o.t.vis: O.append(N); TokenizeFace (c.n.o, false); else:/both left and right triangles are not visited.  $\star /$  O.append(N); TokenizeFace (c.n.o, false); Traverse (c.p.o); /* Begin a new sub-sequence for c.p.o.t.  $\star /$    
+/\* Begin a new sub-sequence  $\star /$    
+def Traverse (HalfEdge c): if not c.t.vis: O.append(B); WriteVertex (c.v); WriteVertex (c.s); WriteVertex (c.e); c.s.vis  $= 1$  ; c.e.vis  $= 1$  ; TokenizeFace (c, true);   
+/\* Initialization  $\star /$    
+for Vertex v in V: if v is a boundary vertex: v.vis  $=$  true;   
+O.append(BOS);   
+ $/^{\star}$  Loop all unvisited faces and traverse.  $\star /$    
+for Face t in T: if not t.vis: Traverse (t.halfedges[0]);   
+O.append(EOS);
+
+Tokenization Algorithm. To formally define the algorithm, we first describe how to use the half-edge notation to refer the mesh data in Figure 9. For a vertex  $\mathbf{v}$ , its three coordinates are denoted as
+
+$(\mathbf{v}.x,\mathbf{v}.y,\mathbf{v}.z)$ . For a face  $\mathbf{t}$ , we store its three half-edges in an array  $\mathbf{t}$ .halfedges. Additionally, we use  $\mathbf{v}$ .vis and  $\mathbf{t}$ .vis to indicate whether this vertex or face has already been visited in the algorithm.
+
+Comparison. We compare the original EdgeBreaker algorithm with our modified version and highlight the key differences. For clarity, we use N and P tokens in the main paper to represent moving to the right or left triangles (the next twin or previous twin half-edges). To remain consistent with the original algorithm, we now revert to using L and R.
+
+(1) In the original EdgeBreaker algorithm, newly added vertices are stored in a residual form relative to the previous triangle, following the parallelogram rule (Rossignac, 1999). This method can potentially reduce the number of bits when compressed further in binary format. However, since our objective is to convert the mesh into a discrete token sequence rather than binary bits, relative vertex encoding is unnecessary. Moreover, our discretization reduces the accuracy of relative coordinates, so we simply use the absolute coordinates for each vertex.
+
+![](images/712eecd4c0ca00030171555d9ac43e6bbd839df964162399922b059e7ff55190.jpg)  
+Figure 9: Mesh notation starting from half-edge c.
+
+(2) As illustrated in Figure 10, the original algorithm employs
+
+five face type tokens: C, L, E, R, and S. The special S token is used to indicate a bifurcation in the traversal tree. However, since the binary tree must be flattened into a 1D sequence, both branches dependent on the S-type triangle can introduce long-range dependencies, particularly if the first-visited branch is long, as shown in Figure 11. To avoid such dependencies, we explicitly duplicate the necessary vertices before visiting the second branch, which can be interpreted as restarting a submesh sequence. While this approach slightly worsens the compression rate, it effectively eliminates long-range dependencies, which can be difficult for the transformer model to learn.
+
+(3) During tokenization, the C token must be distinguished from the others, but it behaves similarly to the L token during de-tokenization. Since tokenization is unnecessary during inference, we replace all C tokens with L for training. Additionally, we merge the E token into B, as E always follows B after removing the S token. For simplicity, we don't consider special cases such as holes or handles, since they can still be represented at the cost of several degenerated faces, which can be easily cleaned during post-processing. As a result, our modified algorithm uses only three face types: L, R, and B.
+
+# A.1.2 AUTO-REGRESSIVE AUTO-ENCODER
+
+Datasets. For the ArAE model, we use meshes from the Objaverse and Objaverse-XL datasets (Deitke et al., 2023b;a). Given that the original datasets include many low-quality 3D assets, we filter the meshes empirically based on metadata captions or file names (Tang et al., 2024a). Our training set comprises approximately 112K meshes with fewer than 4,000 faces from both Objaverse and Objaverse-XL. For finetuning, we select only the higher-quality meshes from Objaverse, totaling around 44K. Each mesh undergoes a preprocessing pipeline where close vertices are merged, collapsed faces are removed, and the mesh is normalized to fit within a unit cube. The vertices are quantized at a resolution of  $512^3$ . Our tokenizer's vocabulary thus includes 512 coordinate tokens, 3 face type tokens (N, P, B) and 3 special tokens (BOS, EOS, PAD), resulting in a total vocabulary size of 518.
+
+Architecture. We illustrate the detailed architecture of our network in Figure 12. The encoder of our ArAE model employs a cross-attention layer as outlined by (Zhang et al., 2023). We sample 8,192 points uniformly from the mesh surface and use a  $4 \times$  down-sampled embedding to query the latent code. Consequently, the latent code has a shape of  $\mathbf{Z} \in \mathbb{R}^{2048 \times 64}$ . The decoder transformer is comprised of 24 self-attention layers, each with a hidden dimension of 1,536 and 16 attention heads. Overall, our ArAE model contains approximately 0.7 billion trainable parameters.
+
+Training and Inference. We train the ArAE model on 64 A100 (80GB) GPUs for approximately one week. The batch size is 4 per GPU, leading to an effective batch size of 256. We use the AdamW optimizer (Loshchilov & Hutter, 2017) with a cosine-decayed learning rate that ranges from  $5 \times 10^{-5}$  to  $5 \times 10^{-6}$ , a weight decay of 0.1, and betas of (0.9, 0.95). Gradient clipping is
+
+![](images/1f998677f63299f9bbefcd664037d2b1da1ab3ef00ac82b2858ca87d28bc53e1.jpg)  
+Figure 10: Comparison of the original EdgeBreaker traversal order with our modified version.
+
+![](images/4614d39edb786fc25ff25d074d3de5d61a7d71a785aa87f0f59a3fd0674196bd.jpg)
+
+![](images/509fe4e090b274780f6080515aa6abfaa44f3b4ff6e7d172c1d93f3ef077931b.jpg)  
+Figure 11: Our tokenization reduces potential long range dependency in the token sequence.
+
+applied with a maximum norm of 1.0. Given that each point cloud input can correspond to multiple plausible meshes but only one ground truth mesh is available, we apply robust data augmentation techniques to address data insufficiency during training: (1) The input mesh is randomly scaled by a factor between [0.75, 0.95]. (2) The mesh is randomly rotated along the vertical axis by up to 30 degrees. (3) Random quadric edge collapse decimation (Garland & Heckbert, 1997) is applied to perturb the number of faces. There is a  $50\%$  chance at each iteration that the mesh will be decimated to no more than one-quarter of its original number of faces. During inference, we employ a multinomial sampling strategy with a top-k of 10. To ensure that the generated sequence is valid for decoding by the tokenizer, we apply the following rules to post-process the logits of the next token based on the prefix token sequence: (1) The token following BOS should be B. (2) After B, there should be 9 coordinate tokens. (3) After N or P, there should be 3 coordinate tokens. (4) Otherwise, the next token should be one of N, P, B, or EOS.
+
+# A.1.3 LATENT DIFFUSION
+
+Datasets. For the image-conditioned diffusion model, we use meshes from the Objaverse dataset along with rendered images from the G-Objaverse dataset (Qiu et al., 2024). The final training dataset comprises approximately 75K 3D meshes, each accompanied by rendered RGB images from 38 different camera poses. We select only 21 camera poses that align with the front face of most objects in Objaverse empirically.
+
+Architecture. Our DiT model consists of 24 layers of self-attention and cross-attention, as detailed in Section 3. Each layer can be expressed as follows:
+
+$$
+\gamma_ {1}, \beta_ {1}, \alpha_ {1}, \gamma_ {2}, \beta_ {2}, \alpha_ {2} = \operatorname {C h u n k} (\mathbf {T} + \mathbf {t} _ {e}) \tag {4}
+$$
+
+$$
+\mathbf {x} = \mathbf {x} + \alpha_ {1} \times \text {S e l f A t t e n t i o n} \left(\operatorname {L a y e r N o r m} _ {1} (\mathbf {x}) \times \left(1 + \beta_ {1}\right) + \gamma_ {1}\right) \tag {5}
+$$
+
+$$
+\mathbf {x} = \mathbf {x} + \text {C r o s s A t t e n t i o n} (\mathbf {x}, \mathbf {c}) \tag {6}
+$$
+
+$$
+\mathbf {x} = \mathbf {x} + \alpha_ {2} \times \operatorname {F e e d F o r w a r d} \left(\operatorname {L a y e r N o r m} _ {2} (\mathbf {x}) \times (1 + \beta_ {2}) + \gamma_ {2}\right) \tag {7}
+$$
+
+where  $\mathbf{T}$  is a layer-wise learnable scale shift table,  $\mathbf{t}_e$  denotes the timestep features, and  $\mathbf{c}$  denotes the condition features. All attention layers have a hidden dimension of 1,024 and use 16 attention heads. We utilize the pretrained CLIP-H model from OpenCLIP (Ilharco et al., 2021) as the image feature encoder. Features from the last layer of the vision transformer are employed as conditions for the cross-attention layers. The DiT model has approximately 0.5 billion trainable parameters in total.
+
+![](images/463ef08f9ce1363d327dfd6e933aafb96faf0cfa9552eecf79d8686b8d0d0291.jpg)  
+Figure 12: Network Architecture of our ArAE and DiT models. Our ArAE contains an encoder that takes a point cloud as input and encodes it to a fixed-length latent code, and an auto-regressive decoder that predicts variable-length mesh token sequence. The latent space can be used to train diffusion models conditioned on other more difficult conditions, such as single-view images.
+
+<table><tr><td></td><td colspan="2">AI-Mesh</td><td colspan="2">Artist-Mesh</td></tr><tr><td></td><td>CD↓</td><td>HD↓</td><td>CD↓</td><td>HD↓</td></tr><tr><td>MeshAnything</td><td>0.072</td><td>0.170</td><td>0.066</td><td>0.195</td></tr><tr><td>MeshAnythingV2</td><td>0.081</td><td>0.241</td><td>0.050</td><td>0.164</td></tr><tr><td>Ours</td><td>0.034</td><td>0.079</td><td>0.036</td><td>0.112</td></tr></table>
+
+Table 2: Reconstruction Quality on point-cloud conditioned mesh generation.
+
+Training and Inference. We train the DiT model on 16 A100 (40GB) GPUs for approximately one week. The lightweight ArAE encoder allows for on-the-fly calculation of the latent code during training, so we can also apply random down-sampling and rotation as data augmentation techniques for the point cloud input. The batch size is set to 32 per GPU, resulting in an effective batch size of 512. We use the same optimizer as for the ArAE model. Training employs the min-SNR strategy (Hang et al., 2023) with a weight of 5.0. The DDPM is configured with 1,000 timesteps and utilizes a scaled linear beta schedule. For classifier-free guidance during inference,  $10\%$  of the image conditions are randomly set to zero for unconditional training. During inference, we use the DDIM scheduler (Song et al., 2020) with 100 denoising steps and a Classifier-Free Guidance (CFG) scale of 7.5.
+
+# A.2 MORE RESULTS
+
+# A.2.1 RECONSTRUCTION QUALITY
+
+We conducted a qualitative comparison experiment to evaluate the Chamfer Distance (CD) and Hausdorff Distance (HD) between the generated meshes and the input point clouds. Due to the lack of widely recognized evaluation datasets, we tried to construct two test datasets from different sources: (1) AI-Mesh: This dataset comprises 30 meshes generated by other image-to-3D models, extracted using the Marching Cubes algorithm. (2) Artist-Mesh: This dataset consists of 100 artistic meshes obtained from basemesh<sup>1</sup>, which were not seen during the training process. All meshes are
+
+![](images/7e9db5da10a3fe063514e88355237131f68e3c19a577dc9ab32d9313c8f382e4.jpg)  
+MeshAnything
+
+![](images/3a9b610d4e9c903ea3c93d5de7cc768f4e1db7a616a779e909c240bf8a6e3e9b.jpg)  
+MeshAnythingV2
+
+![](images/0bdd60bd7609d19c4b5e7fd4673aa953b85f18091747b71513b3681cb85c1337.jpg)
+
+![](images/e1be2b13c50c9ae3a41bcf295df43aa017070d4fe429e7f5150b10e4cb25f360.jpg)  
+Ours
+
+![](images/40ea5aedd12a03b15a2aec553bbcb3e3320527ece75f39007fd71099870d9cca.jpg)
+
+![](images/e5a55f081912ec5f4f640e4cb73179d2938054e3e060107dcd28b6f4240a3d9f.jpg)
+
+![](images/b02c664f221532784d8f2d55c436c2fe566f997a7f7f580912c71cd535795c4c.jpg)  
+Figure 13: Face orientation. Our model ensures correct face orientation of the generated mesh.  
+Figure 14: Visualization of the generation progress. Please visit our project page for video results.
+
+<table><tr><td></td><td>Compression Ratio ↓</td><td>Sub-sequence Count ↓</td><td>Tokenization Speed ↑</td></tr><tr><td>AMT</td><td>46.2%</td><td>199.5</td><td>8.6</td></tr><tr><td>Ours</td><td>47.4%</td><td>54.7</td><td>25.2</td></tr></table>
+
+Table 3: Comparison of the tokenizer algorithm. Our tokenizer has a comparable compression ratio as AMT (Chen et al., 2024e) but uses much fewer sub-sequences per mesh, which makes model training easier. Moreover, our implementation is three times faster (mesh per second).
+
+normalized into  $[-1, 1]^3$ . The summarized evaluation scores are presented in Table 2. We observed the following: (1) Our model achieves the best performance on surface metrics, even when using only point clouds as input. In contrast, the MeshAnything (Chen et al., 2024d) series further leverage surface normals. (2) MeshAnythingV2 (Chen et al., 2024e) performs significantly worse on the more challenging AI-Mesh dataset, often failing to generate complete surfaces in these cases (consistent with the results shown in Figure 5).
+
+# A.2.2 FACE ORIENTATION
+
+In Figure 13, we show the face orientation and rendering with back-face culling. The tokenization algorithm of our method and MeshAnythingV2 (Chen et al., 2024e) ensures that the face orientation in each mesh sequence is correct, while MeshAnything (Chen et al., 2024d) fails to do this.
+
+# A.2.3 GENERATION PROGRESS
+
+In Figure 14, we show the generation progress by simultaneously de-tokenizing the sequence. Our model successfully learns how to generate a valid sequence for the modified EdgeBreaker algorithm to de-tokenize.
+
+# A.2.4TOKENIZATION ALGORITHM
+
+In Table 3, we compare our mesh tokenization algorithm with AMT from MeshAnythingV2 (Chen et al., 2024e). Both algorithms aim to traverse all triangular faces. While AMT can only move to a fixed side that shares the last two vertices in the sequence, our algorithm allows movement to both the left and right sides. This flexibility reduces the number of sub-sequences, making training easier and improving model robustness, as demonstrated in Figure 5. Although AMT requires only
+
+![](images/a2edd62fddbf4bc2af47f7d4828e957a774d337f353ea7c5b7b7ff5762f72bc3.jpg)
+
+![](images/69fd75a4359f327551c8b0b74286dc5c3b7859f5f01036ea1cd65843196dd968.jpg)
+
+![](images/486601764d947459d98c768051f85d7bbdbd6935d62e555e16a3e9a300aea4b2.jpg)  
+$\sim 2000\# \mathrm{F}$  
+More Faces, More Chances of Failure  
+Figure 15: Limitations. We show some failure cases of our method to illustrate our limitations.
+
+![](images/3511161dfcf72401d5579d4d0155359fa8e1f6acb58d459e484bf364d691fb0f.jpg)  
+>4000#F
+
+![](images/bc414d1ee9619e95afae3452e17ea3261945f1bcb33260b2171a9208485aa535.jpg)
+
+![](images/5184d2d9df8b63f8bc0375933ed22135406c36250dc3dd90168cbae9d53d53e6.jpg)
+
+![](images/6a866c11f74291205bf0f2a9acc9c88f9d816d91064ffec64c7e6bc62267b173.jpg)
+
+![](images/90588b0ecd8969313976aca062fade5dbf8ef0cd194d1fbbe3285395262210f0.jpg)
+
+![](images/b7b2494bcfa8a1d6a92734689aff41561aaee149339d5ab501b67ec8d141c546.jpg)
+
+![](images/eff4b7432ad53a17756a708ce88eb928e69f64cddc739adf235e35b91adda3b3.jpg)  
+Unstable with Diffusion and Auto-regressive Generation (Different Seeds)
+
+![](images/0832656a6ff0d19652a219c5073c07b11db24114fb1282568c0cd3803021bbd4.jpg)
+
+![](images/b0c310ee7f89da771f0f511cbf65f932580e5b2c9d1cc182e8c630d27bdcabaa.jpg)
+
+3 coordinate tokens for a subsequent face and our method requires 4 tokens (with an additional face type token), the amortized compression ratio remains similar.
+
+# A.2.5 LIMITATIONS
+
+Despite the promising results, our approach has several limitations: (1) The compression ratio of our tokenization algorithm remains insufficient. Complex meshes, such as those representing game characters, typically require a much higher number of faces, exceeding the current capacity of our model. (2) Although our model can generate meshes with up to 4,000 faces, the success rate decreases as the sequence length increases, and the generation time becomes progressively longer. (3) While the auto-regressive next-token prediction allows for diverse outputs from the same input, it also leads to potentially unsatisfactory results. Since there are infinite plausible face topologies for the same surface, precise control over the generation is difficult to achieve. We illustrate some failure cases in Figure 15.

2025/EdgeRunner_ Auto-regressive Auto-encoder for Artistic Mesh Generation/images.zip

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2025/EditRoom_ LLM-parameterized Graph Diffusion for Composable 3D Room Layout Editing/23b4db6c-487c-4d66-bfe7-d741f6817fdc_origin.pdf

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2025/EditRoom_ LLM-parameterized Graph Diffusion for Composable 3D Room Layout Editing/full.md

@@ -0,0 +1,536 @@
+# EDITROOM: LLM-PARAMETERIZED GRAPH DIFFU SION FOR COMPOSABLE 3D ROOM LAYOUT EDITING
+
+Kaizhi Zheng $^{1}$  Xiaotong Chen $^{2}$  Xuehai He $^{1}$  Jing Gu $^{1}$  Linjie Li $^{3}$   
+Zhengyuan Yang $^{3}$  Kevin Lin $^{3}$  Jianfeng Wang $^{3}$   
+Lijuan Wang $^{3}$  Xin Eric Wang $^{1}$ $^{1}$ UC Santa Cruz  $^{2}$ University of Michigan, Ann Arbor  $^{3}$ Microsoft  
+{kzheng31, xwang366}@ucsc.edu
+
+![](images/721856cb63ac2abc836439835431149291357d869d3f5597287907164980a087.jpg)  
+Figure 1: Editing Pipeline with EditRoom. EditRoom is a unified language-guided 3D scene layout editing framework that can automatically execute all layout editing types with natural language commands, which includes the command parameterizer for natural language comprehension and the scene editor for editing execution. Given a source scene and natural language commands, it can generate a coherent and appropriate target scene.
+
+# ABSTRACT
+
+Given the steep learning curve of professional 3D software and the time-consuming process of managing large 3D assets, language-guided 3D scene editing has significant potential in fields such as virtual reality, augmented reality, and gaming. However, recent approaches to language-guided 3D scene editing either require manual interventions or focus only on appearance modifications without supporting comprehensive scene layout changes. In response, we propose EditRoom, a unified framework capable of executing a variety of layout edits through natural language commands, without requiring manual intervention. Specifically, EditRoom leverages Large Language Models (LLMs) for command planning and generates target scenes using a diffusion-based method, enabling six types of edits: rotate, translate, scale, replace, add, and remove. To address the lack of data for language-guided 3D scene editing, we have developed an automatic pipeline to augment existing 3D scene synthesis datasets and introduced EditRoom-DB, a large-scale dataset with 83k editing pairs, for training and evaluation. Our experiments demonstrate that our approach consistently outperforms other baselines across all metrics, indicating higher accuracy and coherence in language-guided scene layout editing. Project website: https://eric-ai-lab.github.io/edit-room.github.io/
+
+# 1 INTRODUCTION
+
+Traditionally, editing 3D scenes requires manual intervention through specialized software like Blender (Community, 2024), which demands substantial expertise and considerable time for resource management. As a result, language-guided 3D scene editing has emerged as a promising technology for next-generation 3D software. To build an automated system capable of interpreting natural language and manipulating scenes, the system must be able to align complex, diverse, and often ambiguous language commands with various editing actions while also comprehending the global spatial structure of the scene. Additionally, the relatively small size of available 3D scene datasets presents a challenge for developing large-scale pretrained models necessary for fully automated, end-to-end language-guided scene editing.
+
+Recently, there have been some works (Zhuang et al., 2023; Bartrum et al., 2024; Karim et al., 2023) focusing on leveraging pretrained image generation models to edit single object appearance inside the scene, but they fail to modify the layout of original scenes. Meanwhile, other approaches (Chen et al., 2023; Ye et al., 2023) further incorporate pretrained segmentation models to enable individual object manipulation. However, they require manual intervention to determine the edited object and editing type for any layout adjustments, like adding a new object or changing the object pose. Furthermore, these methods are all limited to executing a single editing step and are hard to deal with commands with multiple potential steps.
+
+Therefore, we propose EditRoom, a unified framework that can execute all editing types by complex natural language commands without intermediate manual interventions. It consists of two main modules: the command parameterizer and the scene editor. The command parameterizer employs an pretrained LLM, specifically GPT-4o (OpenAI, 2024), to transform natural language commands into sequences of breakdown commands for six basic editing types on single object: adding, removing, replacing, translating, rotating, and scaling. These breakdown commands, along with the source scenes, are then fed sequentially into the scene editor for execution. To unify all editing types, we convert scenes into graph representations and construct scene editor as conditional graph generation models, where we take source scenes and text commands as conditions and train diffusion models to generate the target scene graphs with the layout. By conditioning on the user's natural language prompts, our model generates reasonable editing results that serve as a foundation for further refinement. Importantly, the scenes remain fully editable, allowing users to provide additional prompts for iterative improvements to achieve their desired outcome.
+
+Another challenge is the lack of language-guided scene editing datasets, which constrains both training and evaluation of scene editing models. To enable the scene editor to accurately execute every basic editing type, we construct an automatic data generation pipeline and collect a synthetic scene editing dataset named EditRoom-DB for both training and evaluation, which includes approximately 83k editing pairs with commands. We implement several functions to simulate the single editing process on the existing 3D scene synthetic dataset, 3D FRONT (Fu et al., 2021a), which contains 16k indoor scene designs equipped with high-quality object models, and generate corresponding language commands using predefined templates. To mimic the human inputs, we employ LLMs to transform the templated commands into natural language forms, serving both as training material for our baselines and as test cases for single-operation evaluations.
+
+In the experiments, we quantitatively assess the performance of EditRoom in scenarios with single-operation commands. From the results, we find that EditRoom outperforms other baselines in all metrics across different room types and editing types, which indicates higher precision and coherence in single-operation editing. Furthermore, we qualitatively evaluate EditRoom in scenarios involving multi-operation commands. We find that the model can successfully generalize to these scenarios even though we do not train the model on multi-operation data.
+
+Our contributions are summarized as follows:
+
+- We propose a new framework, named EditRoom, consisting of the command parameterizer and scene editor, which accepts scene inputs and can edit scenes using natural language commands by leveraging LLM for planning.  
+- We propose a unified graph diffusion-based module that serves as the scene editor, capable of executing every basic editing type, including adding, removing, replacing, translating, rotating, and scaling.
+
+- We introduce the EditRoom-DB dataset with 83k editing pairs for the first language-guided 3D scene layout editing dataset by augmenting the existing 3D scene synthesis dataset.  
+- From the experiments, we demonstrate that EditRoom outperforms other baselines across all editing types and room types on single operation commands, and it can generalize to complex operation commands without further training.
+
+# 2 RELATED WORK
+
+Language-Guided 3D Scene Generation Language-guided 3D scene generation works claiming editing capabilities can be categorized into two main approaches. The first approach utilizes large language models to generate scene configurations and perform edits (Vilesov et al., 2023; Zhou et al., 2024b; Aguina-Kang et al., 2024). However, these methods are limited to editing scenes they generate and cannot accept external source scenes. The second approach trains diffusion-based models for text-conditional scene generation (Haque et al., 2023; Tang et al., 2023; Zhai et al., 2024). These methods often require manual masking for edits and have limited editing functionalities. For example, InstructScene (Haque et al., 2023) lacks support for operations like translation, rotation, and scaling, while EchoScene (Zhai et al., 2024), which directly take target scene graphs as input, only modifies object relations manually and does not process textual instructions. In contrast, EditRoom accepts existing scenes and supports free-form editing commands for 3D layouts. By leveraging diffusion models trained on clean data, EditRoom enables comprehensive edits, including translating, rotating, scaling, adding, and removing objects, without manual intervention.
+
+Language-guided 3D Scene Editing Previously, some works (Ma et al., 2018) have explored the rule-based methods for language-guided 3D scene editing. Compared to these works, our diffusion-based method can generate diverse editing results, generalize across editing types. Recent language-guided 3D scene editing works mostly incorporate neural field representations and pretrained image generation models for object appearance editing. Some works are mainly focusing on replacing single object (Zhuang et al., 2023; Bartrum et al., 2024; Karim et al., 2023) but failing to edit the layout. Other approaches (Chen et al., 2023; Ye et al., 2023) leverage pretrained segmentation models to obtain the individual object representation so that they can remove the objects by manually selecting the target objects. Besides, all these methods can only take one single-operation command at each interaction. Compared to these previous works, EditRoom can leverage LLM to deal with multi-operation natural language commands and automatically execute all editing types through a unified graph diffusion-based module.
+
+LLMs for 3D Scene Understanding Recent works demonstrate that existing LLMs can facilitate 3D spatial reasoning. These works usually leverage the pretrained image caption models to convert 3D scenes into text descriptions and ask the LLM to generate navigation steps (Zhou et al., 2023; 2024a), provide room layout (Feng et al., 2024), or ground 3D objects (Yang et al., 2023; Hong et al., 2023; Huang et al., 2023). In our work, we are the first work to leverage LLM for natural language-guided 3D layout editing, where LLM takes source scenes in text format and breaks the natural language commands into basic editing operations.
+
+Diffusion Models for Graph In recent years, denoising diffusion models have shown impressive generative capability in the graph generation (Liu et al., 2023a; Kong et al., 2023; Vignac et al., 2022). Compared to the previous VAE-based Verma et al. (2022) or GAN-based (Martinkus et al., 2022) models, diffusion-based models have advantages like stable training processes and generalizability to various graph structures. Some works have presented that graph diffusion-based models can be used in molecule generation (Guo et al., 2022; Akhmetshin, 2023), protein modeling (Zhang et al., 2022), 3D scene generation (Haque et al., 2023; Zhai et al., 2024), and etc. In this work, we first propose to use graph diffusion-based models for language-guided 3D scene layout editing.
+
+# 3 THE EDITROOM METHOD
+
+In this section, we introduce EditRoom, shown in Figure 1, comprising two primary modules: Command Parameterizer and the Scene Editor. Given a natural language command  $C$  and source scene  $S$ , we aim to estimate the target scene  $T$  with conditional distribution  $q(T|S,C)$ . Our command parameterizer takes the source scene  $S$  and natural command  $C$  to generate the breakdown commands
+
+![](images/c8a105c50b7059fa6fed9c68c4467a4a9437eaf98702b08c7efe1540c4a88865.jpg)  
+Figure 2: Scene Editor Overview. Scene Editor aims to provide accurate, coherent editing results according to the given source scene and language commands. It consists of two graph transformer-based conditional diffusion models. One diffusion model generates semantic target scene graphs. Another diffusion model can estimate accurate poses and size information for each object inside the generated target scene graphs. All diffusion processes are conditioned on the source scene and breakdown command.
+
+$L$ . Then, the scene editor conditions on breakdown commands  $L$  to obtain the final target scene  $T$ , where the whole pipeline can be written as  $q(T|S,C) = q(L|S,C)\times q(T|S,L)$ . All objects inside source scenes and target scenes are retrieved from a high-quality 3D furniture dataset (Fu et al., 2021b).
+
+# 3.1 LLM AS COMMAND PARAMETERIZER
+
+In order to process open natural language commands, we use GPT-4o (OpenAI, 2024) to convert natural language command  $C$  into a set of combinations of basic editing types with breakdown commands  $L := \{l_j\}_{j=1}^{N_L}$ , where  $N_L$  is the number of breakdown commands. To cover the general manipulations on the scene, we design six basic editing operations:
+
+- Rotate an object: [Rotate, Target Object Description, Angle]  
+- Translate an object: [Translate, Target Object Description, Direction, Distance]  
+- Scale an object: [Scale, Target Object Description, Scale Factor]  
+- Replace an object: [Replace, Source Object Description, Target Object Description]  
+- Add an object: [Add, Target Object Description, Target Object Location]  
+- Remove an object: [Remove, Target Object Description]
+
+When the target object is not unique, we ask the LLM to use another unique object as a reference to describe the spatial relation. During the inference phase, we prompt the LLM with attributes of objects within the source scene along with the natural language command, tasking the model to analyze the scene and delineate basic editing operations through breakdown commands in specified formats. The attributes include categories, locations, sizes, rotations, and object captions. Detailed descriptions of the full prompt and examples are provided in Figure 6 of the appendix.
+
+# 3.2 GRAPH DIFFUSION AS 3D SCENE EDITOR
+
+Given the breakdown command  $l$  and source scene  $S$ , our objective is to determine the conditional target scene distribution  $q(T|S,l)$ . Drawing inspiration from recent advancements in language-guided 3D scene synthesis (Lin & Yadong, 2023), we transform scenes into semantic graphs and employ a graph transformer-based conditional diffusion model to learn the conditional target scene graph distribution, as depicted in Figure 2. Our approach involves two key graph transformer-based diffusion models: the Target Graph Diffusion, which generates object shapes and their spatial relations as graphs, and the Target Layout Diffusion, which computes the final layout of the target scene. To reduce the alignment challenges between the 3D scene distribution and language, all commands are encoded using the text encoder of CLIP-ViT-B-32 (Radford et al., 2021).
+
+Scene Graph Representation Each scene is represented as a combination of a layout  $B$  and a scene graph  $\mathcal{G}$  (Lin & Yadong, 2023). The layout  $B$  encapsulates the position, size, and orientation of each object, while the scene graph  $\mathcal{G}$  encodes additional high-level semantic information. Formally, a semantic scene graph  $\mathcal{G} := (V, E)$  comprises nodes  $v_i \in V$ , where each  $v_i$  corresponds to an object  $o_i$  with high-level attributes. Directed edges  $e_{ij} \in E$  represent spatial relationships: ["in front of", "behind", "right of", "left of", "closely in front of", "closely behind", "closely right of", "closely left of", "above", "below"], connecting the  $i$ -th object to the  $j$ -th object, where "closely" means the distance between two object centers are less than 1 meter. Each node  $v_i$  is characterized by a discrete category  $c_i$  and continuous semantic features  $f_i$ , derived from a pretrained multimodal-aligned point cloud encoder, OpenShape (Liu et al., 2024c), which features a 1280-dimensional representation space.
+
+Target Graph Diffusion In this stage, we aim to learn target scene graphs  $\mathcal{G}_{tg}$  by giving source scene graphs  $\mathcal{G}_s$  and language commands  $l$  through a discrete graph diffusion model  $\varepsilon_{g}$ , where  $\mathcal{G}_{tg}$  includes category  $C_{tg}$  and semantic features  $F_{tg}$  for each node and the edges  $E_{tg}$  for object relative relations. Since high-dimensional object semantic features ( $d = 1280$ ) are too complicated to learn from limited data, we use a VQ-VAE model (Lin & Yadong, 2023; Wang et al., 2019) to compress them into low-dimensional features  $z \in \mathbb{R}^{n_f \times d_z}$ , which consists of  $n_f$  vectors extracted from a learned codebook  $Z \in \mathbb{R}^{K_f \times d_z}$  by a sequence of feature indices  $f_{idx} := \{1, \dots, K_f\}^{n_f}$ , where  $K_f$  and  $d_Z$  are the size and dimension of codebook. Then, we use the feature indices to replace the original object semantic features as targets for training, denoted as  $\hat{F}$ . Therefore,  $\mathcal{G}_{tg} = (C_{tg}, \hat{F}_{tg}, E_{tg})$  and  $\mathcal{G}_s = (C_s, \hat{F}_s, E_s)$ , and our goal is to learn the conditional distribution  $q(\mathcal{G}_{tg}|\mathcal{G}_s, l)$ . During the training process, at timestep  $t$ , the gaussian noises are added to the  $\mathcal{G}_{tg}$  to get  $\mathcal{G}_{tg}^t$ , and the model  $\varepsilon_g$  aims to reconstruct  $\mathcal{G}_{tg}^0$  by conditioning on  $\mathcal{G}_s$  and  $l$ . To add the conditions, we concatenate each element of source scene graphs into noisy target scene graphs as context and use cross-attention layers to incorporate language features. The loss function can be written as:
+
+$$
+L _ {g} := \mathbb {E} _ {q \left(\mathcal {G} _ {t g} ^ {0}\right)} \left[ \sum_ {t = 2} ^ {T} L _ {t - 1} - \mathbb {E} _ {q \left(\mathcal {G} _ {t g} ^ {1} \mid \mathcal {G} _ {t g} ^ {0}\right)} \left[ \log p _ {\varepsilon_ {g}} \left(\mathcal {G} _ {t g} ^ {0} \mid \mathcal {G} _ {t g} ^ {1}, \mathcal {G} _ {s}, l\right) \right] \right] \tag {1}
+$$
+
+$$
+L _ {t - 1} := D _ {K L} \left[ q \left(\mathcal {G} _ {t g} ^ {t - 1} \mid \mathcal {G} _ {t g} ^ {t}, \mathcal {G} _ {t g} ^ {0}\right) \mid \mid p _ {\varepsilon_ {g}} \left(\mathcal {G} _ {t g} ^ {t - 1} \mid \mathcal {G} _ {t g} ^ {t}, \mathcal {G} _ {s}, l\right) \right] \tag {2}
+$$
+
+where  $D_{KL}$  indicates the KL divergence.
+
+Target Layout Diffusion In this stage, we aim to estimate the target scene layout  $B_{tg}$  using another graph diffusion model  $\varepsilon_{b}$ , conditioning on target scene graph  $\mathcal{G}_{tg}$ , source scene graph  $\mathcal{G}_s$ , source layout  $B_{s}$ , and language command  $l$ . The target scene layout  $B_{tg} \in \mathbb{R}^{M \times 8}$  consists of position  $T_{tg} \in \mathbb{R}^{M \times 3}$ , size  $S_{tg} \in \mathbb{R}^{M \times 3}$ , and rotation  $R_{tg} \in \mathbb{R}^{M \times 2}$ . During the training process, gaussian noises  $\epsilon$  will be added to the target layout, and the layouts are encoded into the node features by MLP layers. Similar to the Target Graph Diffusion, we concatenate the source scene graph and source layout to the target scene graph as context and corrupted target layout. The language features are incorporated through cross-attention layers. The objective target is to estimate the added noises at each time step. The loss function can be written as:
+
+$$
+L _ {b} := \mathbb {E} _ {B _ {t g} ^ {0}, t, \epsilon} [ \| \epsilon - \varepsilon_ {b} \left(B _ {t g} ^ {t}, t, \mathcal {G} _ {t g}, \mathcal {G} _ {s}, B _ {s}, l\right) ] \tag {3}
+$$
+
+Inference Process During the inference phase, the first step consists of transforming the source scene into a scene graph  $\mathcal{G}_s$  and a corresponding layout  $B_{s}$ . Subsequently, the Target Graph Generation model predicts the target scene graph  $\mathcal{G}_{tg}$ , conditioned on the source scene graph  $\mathcal{G}_s$  and the
+
+Table 1: EditRoom-DB dataset statistics. We collect around 83k training data across all room types and 500 test data for each room type.  
+
+<table><tr><td rowspan="2">Types</td><td colspan="3">Train</td><td colspan="3">Test</td></tr><tr><td>Bedroom</td><td>Dining room</td><td>Living room</td><td>Bedroom</td><td>Dining room</td><td>Living room</td></tr><tr><td>Translate</td><td>8.6k</td><td>3.2k</td><td>2.7k</td><td>808</td><td>253</td><td>250</td></tr><tr><td>Rotate</td><td>4.0k</td><td>1.3k</td><td>1.3k</td><td>804</td><td>250</td><td>252</td></tr><tr><td>Scale</td><td>12.7k</td><td>4.5k</td><td>3.9k</td><td>805</td><td>251</td><td>252</td></tr><tr><td>Add</td><td>8.9k</td><td>3.4k</td><td>2.8k</td><td>747</td><td>232</td><td>240</td></tr><tr><td>Remove</td><td>8.8k</td><td>3.3k</td><td>2.8k</td><td>816</td><td>260</td><td>253</td></tr><tr><td>Replace</td><td>6.8k</td><td>2.2k</td><td>2.1k</td><td>820</td><td>254</td><td>253</td></tr><tr><td>Total</td><td>49.8k</td><td>17.9k</td><td>15.6k</td><td>4800</td><td>1500</td><td>1500</td></tr></table>
+
+language command  $l$ . This is followed by the Target Layout Generation model, which computes the target layout  $B_{tg}$ , leveraging all available variables as inputs. The final step in constructing the target scene, denoted as  $T := (\mathcal{G}_{tg}, B_{tg})$ , involves retrieving the object meshes based on the estimated object features and arranging them according to the generated layout. This systematic approach enables the dynamic generation of scenes that are aligned with verbal instructions, ensuring that the resulting scenes accurately represent the specified conditions.
+
+# 4 THE EDITROOM-DB DATASET
+
+To facilitate comprehensive training and evaluation of our framework, we present the EditRoom-DB dataset, designed to support a wide range of basic 3D scene editing operations. The dataset is generated through an automated data augmentation pipeline that produces editing pairs based on object-level modifications applied to scenes from the 3D-FRONT dataset (Fu et al., 2021a). We utilize scenes from the bedroom, dining room, and living room categories and enhance them with high-quality object models from the 3D-FUTURE dataset (Fu et al., 2021c) to simulate real-world editing workflows.
+
+Our data generation pipeline supports a variety of editing operations, including Add and Remove Objects, Pose and Size Changes, and Object Replacement. Each modification produces a new scene paired with a detailed textual description of the changes made, following a predefined template. For each scene, objects are selected randomly and modified iteratively through basic editing operations. In the case of Add and Remove Objects, the modified scene serves as the target for the operation, with the original scene acting as the source. For Pose and Size Changes, random adjustments are applied to selected objects, and collision checking ensures that the final scenes are free from object overlap. Similarly, in Object Replacement, new objects from the same category are substituted for existing ones, with collision checks ensuring high-quality outputs. More details about automatic pipelines can be found in Appendix C.
+
+To create diverse and realistic language instructions for the editing tasks, we first employ the multimodal understanding model LLAVA-1.6 (Liu et al., 2024b;a; 2023b), which captions front-view images of the objects in the scenes. Then, these object captions are used to construct the template-based commands. The template-based commands are then transformed into natural language commands using GPT-4o, making the dataset suitable for training and testing language-guided scene editing models. Detailed prompts and additional examples are shown in Figure 7 in the appendix.
+
+The resulting dataset consists of approximately 83,000 training samples and 7,800 test samples (randomly sampled across editing types for each room type). Table 1 provides a detailed breakdown of the dataset statistics across the three scene categories. Each sample includes a source scene, an edited target scene, and corresponding language commands. This comprehensive dataset enables the development and evaluation of robust models capable of performing various scene editing tasks. By simulating realistic workflows and providing detailed text commands, EditRoom-DB serves as a valuable resource for advancing language-guided 3D scene editing.
+
+Table 2: Performance on single operation with different room types. From the table, we can find EditRoom outperforms baselines among all room types, which indicates that our methods can provide more accurate and coherent editing across room types.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">Bedroom</td><td colspan="4">Dining room</td><td colspan="4">Living room</td></tr><tr><td>IOU (↑)</td><td>S-IOU (↑)</td><td>LPIPS (↓)</td><td>CLIP (↑)</td><td>IOU (↑)</td><td>S-IOU (↑)</td><td>LPIPS (↓)</td><td>CLIP (↑)</td><td>IOU (↑)</td><td>S-IOU (↑)</td><td>LPIPS (↓)</td><td>CLIP (↑)</td></tr><tr><td>DiffuScene-N</td><td>0.6220</td><td>0.6125</td><td>0.1431</td><td>0.9415</td><td>0.4473</td><td>0.4321</td><td>0.1921</td><td>0.9254</td><td>0.4332</td><td>0.4161</td><td>0.1810</td><td>0.9236</td></tr><tr><td>SceneEditor-N</td><td>0.7055</td><td>0.6942</td><td>0.1261</td><td>0.9510</td><td>0.5262</td><td>0.5091</td><td>0.1557</td><td>0.9366</td><td>0.4618</td><td>0.4498</td><td>0.1738</td><td>0.9329</td></tr><tr><td>EditRoom</td><td>0.7342</td><td>0.7236</td><td>0.1090</td><td>0.9597</td><td>0.5360</td><td>0.5196</td><td>0.1460</td><td>0.9460</td><td>0.4710</td><td>0.4629</td><td>0.1616</td><td>0.9446</td></tr></table>
+
+# 5 EXPERIMENTS
+
+# 5.1 EXPERIMENTAL SETUP
+
+Baselines To the best of the authors' knowledge, EditRoom is the first work that can automatically execute all editing types with natural language commands. Therefore, we construct two baselines by modifying language-guided 3D scene synthesis methods for comparisons: DiffuScene-N and SceneEditor-N:
+
+- DiffuScene-N: DiffuScene-N is modified from the language-guided 3D scene synthesis work, DiffuScene (Tang et al., 2023), which includes a UNet-based diffusion model to generate scene layout. To enable it with language-guided scene editing ability, we leverage their scene completion pipeline by incorporating the source scene as context for the diffusion process. During the training and testing, the model directly conditions natural commands for target scene layout generation.  
+- SceneEditor-N: To test our generalization ability, we experiment with another setting, where the scene editor directly trains on the natural commands got from the GPT-4o. During the inference time, the model conditions the natural commands and generates the final scenes.
+
+We would like to emphasize that not all language-guided 3D scene synthesis methods can be modified as 3D scene layout editing methods. There are at least two prerequisites: (1) there is a way to easily convert the source scene into the required intermediate representations; (2) the scene representations have the capability to execute all editing manipulations. Some LLM-relevant 3D scene generation models (Vilesov et al., 2023; Zhou et al., 2024b; Aguina-Kang et al., 2024; Hu et al., 2024) can generate 3D scenes based on descriptions, but source scenes are hard to convert into their intermediate representations, like program language (Aguina-Kang et al., 2024) and blender codes (Hu et al., 2024).
+
+Metrics To evaluate the models' performance, we utilize four metrics: IOU, S-IOU, LPIPS (Zhang et al., 2018), and CLIP (Radford et al., 2021) scores. The IOU scores are calculated by determining the 3D Intersection Over Union (IOU) between each object in the generated and target scenes, selecting pairs with the highest 3D IOU values. The S-IOU represents the semantic-weighted 3D IOU, where semantic similarities between matching objects are calculated using Sentence BERT (S-BERT) (Reimers & Gurevych, 2019) based on their captions. For visual evaluation, we render both the generated and target scenes from 24 fixed camera views. Visual similarity is assessed using the LPIPS metric for pixel similarity, and semantic similarity is evaluated using the CLIP image encoder (CLIP-ViT-B32).
+
+# 5.2 RESULTS
+
+Single-operation Commands To assess model performance on single operations, we test our model and baselines using the EditRoom-DB test set, which contains 500 samples per room type, with language commands generated by GPT-4o. Quantitative results are depicted in Tables 2 and 3, and qualitative outcomes are illustrated in Figure 3. Table 2 indicates that EditRoom consistently outperforms other baselines across all room types, with notably superior performance in bedrooms. According to Table 3, EditRoom also excels across all editing types. Comparisons between EditRoom and SceneEditor-N reveal that template-based instructions can simplify the learning process by more effectively aligning language commands with scene changes. Moreover, the LLM (GPT-4o) demonstrates a successful bridge between natural language and template commands. SceneEditor
+
+Table 3: Performance on single operations with different editing types. From the table, we can notice EditRoom provides better editing results across all basic editing types.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">Translate</td><td colspan="4">Rotate</td></tr><tr><td>IOU (↑)</td><td>S-IOU (↑)</td><td>LPIPS (↓)</td><td>CLIP (↑)</td><td>IOU (↑)</td><td>S-IOU (↑)</td><td>LPIPS (↓)</td><td>CLIP (↑)</td></tr><tr><td>DiffuScene-N</td><td>0.5363</td><td>0.5279</td><td>0.1715</td><td>0.9484</td><td>0.6010</td><td>0.5913</td><td>0.1307</td><td>0.9511</td></tr><tr><td>SceneEditor-N</td><td>0.5983</td><td>0.5905</td><td>0.1534</td><td>0.9504</td><td>0.6655</td><td>0.6556</td><td>0.1202</td><td>0.9560</td></tr><tr><td>EditRoom</td><td>0.6148</td><td>0.6078</td><td>0.1388</td><td>0.9584</td><td>0.6764</td><td>0.6673</td><td>0.1067</td><td>0.9610</td></tr><tr><td></td><td colspan="4">Scale</td><td colspan="4">Replace</td></tr><tr><td>Model</td><td>IOU (↑)</td><td>S-IOU (↑)</td><td>LPIPS (↓)</td><td>CLIP (↑)</td><td>IOU (↑)</td><td>S-IOU (↑)</td><td>LPIPS (↓)</td><td>CLIP (↑)</td></tr><tr><td>DiffuScene-N</td><td>0.6057</td><td>0.5947</td><td>0.1250</td><td>0.9566</td><td>0.5994</td><td>0.5754</td><td>0.1488</td><td>0.9305</td></tr><tr><td>SceneEditor-N</td><td>0.6754</td><td>0.6650</td><td>0.1105</td><td>0.9603</td><td>0.6451</td><td>0.6193</td><td>0.1418</td><td>0.9389</td></tr><tr><td>EditRoom</td><td>0.6881</td><td>0.6788</td><td>0.1008</td><td>0.9647</td><td>0.6549</td><td>0.6305</td><td>0.1384</td><td>0.9419</td></tr><tr><td></td><td colspan="4">Add</td><td colspan="4">Remove</td></tr><tr><td>Model</td><td>IOU (↑)</td><td>S-IOU (↑)</td><td>LPIPS (↓)</td><td>CLIP (↑)</td><td>IOU (↑)</td><td>S-IOU (↑)</td><td>LPIPS (↓)</td><td>CLIP (↑)</td></tr><tr><td>DiffuScene-N</td><td>0.5312</td><td>0.5198</td><td>0.1799</td><td>0.9263</td><td>0.4626</td><td>0.4511</td><td>0.1877</td><td>0.9281</td></tr><tr><td>SceneEditor-N</td><td>0.5863</td><td>0.5745</td><td>0.1620</td><td>0.9355</td><td>0.6160</td><td>0.6065</td><td>0.1290</td><td>0.9489</td></tr><tr><td>EditRoom</td><td>0.6001</td><td>0.5887</td><td>0.1588</td><td>0.9410</td><td>0.6399</td><td>0.6316</td><td>0.1150</td><td>0.9583</td></tr></table>
+
+Table 4: Ablation on different condition types on the bedroom. From the table, we can show that incorporating source information as context with the self-attention (our design) instead of the cross-attention mechanism can significantly improve model performance.  
+
+<table><tr><td>Model</td><td>IOU (↑)</td><td>S-IOU (↑)</td><td>LPIPS (↓)</td><td>CLIP (↑)</td></tr><tr><td>EditRoom (Concat-Text)</td><td>0.5855</td><td>0.5746</td><td>0.1432</td><td>0.9488</td></tr><tr><td>EditRoom (Original)</td><td>0.7342</td><td>0.7236</td><td>0.1090</td><td>0.9597</td></tr></table>
+
+N outperforms DiffScene-E across all metrics and editing types, suggesting that our graph-based diffusion method yields more coherent and accurate editing results compared to the UNet-based approach. Thus, EditRoom provides more precise and coherent atomic editing operations from natural language commands than its counterparts.
+
+Analysis across different room types shows that all models perform better as the average number of objects in rooms decreases, highlighting potential for improvements in larger, more complex scenes. Evaluating different editing operations reveals that translating, adding, and removing operations score lower on IOU, demanding stronger spatial reasoning. Meanwhile, replacing and adding operations yield lower CLIP scores, indicating a need for better alignment between object descriptions and their semantic features. This underscores the potential for further enhancement of models' spatial reasoning and object alignment capabilities.
+
+Multi-operation Commands To demonstrate the generalization capabilities of EditRoom, we manually designed several test prompts that combine multiple atomic operations, and we assessed each model's performance qualitatively. Figure 4, shows that EditRoom provides more coherent and appropriate responses than the baseline models. For instance, the command in the first row requests a bed replacement and the addition of a wardrobe. EditRoom successfully interprets the natural language command and translates it into the corresponding atomic operations using an LLM, whereas other baseline models misinterpret the command and perform incorrect operations such as translation. These outcomes highlight the challenges of directly training models on natural language commands for compositional editing tasks. EditRoom, by contrast, effectively executes complex editing operations through strategic LLM planning.
+
+Ablation on Condition Types To validate our model design, we experimented with an alternative conditioning approach, where a graph transformer encodes the source scene into a sequence of vectors that are then concatenated with text features. These combined features are incorporated into the cross-attention layers of our graph diffusion process. We specifically tested this method on the bedroom scene type, with results shown in Table 4. The table indicates a significant decrease in model performance, both in terms of layout accuracy and visual coherence. This outcome suggests that
+
+![](images/dcabebe6bd7d30427833288a70ccae488776362e0d3a777110f120f9a922c3aa.jpg)  
+Figure 3: Qualitative results on single-operation commands. The left column is the source scene with single operation commands for each basic editing type. From the examples, we can find that EditRoom can provide more coherent and appropriate editing operations across all editing types.
+
+Table 5: Ablation on different text encoders on the bedroom. Due to the limited size of training data, we find using the larger text encoder with high-dimensional features induces decreasing performance on editing, which indicates further exploration with 3D editing data generation.  
+
+<table><tr><td>Model</td><td>IOU (↑)</td><td>S-IOU (↑)</td><td>LPIPS (↓)</td><td>CLIP (↑)</td></tr><tr><td>EditRoom (OpenCLIP-ViT-bigG-14)</td><td>0.6882</td><td>0.6743</td><td>0.1305</td><td>0.9488</td></tr><tr><td>EditRoom (Original)</td><td>0.7342</td><td>0.7236</td><td>0.1090</td><td>0.9597</td></tr></table>
+
+utilizing source scene information as the context for self-attention layers, rather than as conditions for cross-attention, yields better results.
+
+Ablation on Text Encoders In an exploration of text encoder options, we replaced the CLIP-ViT-B32 text encoder (512 feature dimensions) with a larger pretrained encoder, OpenCLIP-ViT-bigG-14 (1280 feature dimensions), used by OpenShape to align with object semantic features—consistent with the object features in our models. We conducted tests on the bedroom test set, with outcomes detailed in Table 5. The results indicate that the model equipped with the larger text encoder underperforms compared to the one using the original encoder. We attribute this decrease in performance to the limited size of our training dataset. Given that our diffusion models are trained from
+
+![](images/dd78d0f980660c846f22f89d817b1dcd08eba7cad7264fe862011dfe07ee2ce0.jpg)  
+Figure 4: Qualitative results on multi-operation commands. The left column is the source scene with multi-operation operation commands. From the figure, we can find the EditRoom can successfully generalize to complex natural language commands with multiple operations without further training on the multi-operation operation data, while baselines fail to execute coherent editing.
+
+scratch, they require more data to effectively align with higher-dimensional features ( $d = 512$  vs  $d = 1280$ ). This finding underscores the need for further exploration into constructing larger scene editing datasets.
+
+# 6 CONCLUSION
+
+In this work, we introduce EditRoom, a language-guided 3D room layout editing method. EditRoom incorporates a graph diffusion-based scene editor that facilitates unified basic editing operations, and it utilizes an LLM for natural language planning. Our experiments demonstrate that EditRoom can effectively execute appropriate edits for both single and complex operations. We believe this work will inspire further research into language-guided 3D scene layout editing.
+
+**Limitation** Since EditRoom leverages the LLM for the command parameterizer, its performance is contingent upon the LLM's capability in 3D scene understanding and natural command comprehension. This dependency may lead to the generation of erroneous commands that prompt the scene editor to execute potentially problematic operations, such as collisions. However, because the training data predominantly consist of collision-free samples, there is an inherent trade-off between adhering strictly to the commands and avoiding collisions. If the commands deviate significantly from typical scenarios—such as moving an object 100 meters away—the model might instead perform a similar action that falls within the observed training distributions.
+
+Ethics Statement Our work presents a framework for automated 3D scene editing guided by natural language. The primary focus of this research is to advance technical capabilities in scene manipulation. We do not anticipate any ethical concerns or negative societal impacts arising from this work. All datasets used in our research are synthetic and publicly available, and no personally identifiable information or human-related data were involved.
+
+Reproducibility Statement All language commands are encoded through the pretrained CLIP-ViT-B32 text encoder. Each graph diffusion model includes a five-layer graph transformer model with 512 hidden dimensions and 8 attention heads. Training is conducted using the AdamW optimizer over 300 epochs, with a batch size of 512 and a learning rate of  $2 \times 10^{-4}$ . All models are individually trained and tested on each room type. For EditRoom, template commands are employed for the scene editor during training, whereas other baseline models utilize natural language commands generated by GPT-4o. During testing, all models receive natural language commands as input. We set the node number of denoise graphs to the maximum object number for different room types (bedroom = 12, living room = dining room = 21). During training, we pad the objects with zeros to facilitate the batch process.
+
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+
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+
+![](images/5e57d8c4052bf6fab2a45932d77bc515dcca81469bd217c811f5a9631e38d71b.jpg)  
+Figure 5: Visualization of layout diffusion denoising process. The whole diffusion process is conditioned on both source scene and language commands. At the beginning of the process, the target scene layout starts from random noises. After the iterative diffusion denoising process, the target scene layout becomes coherent to source scene and command.
+
+# A LLM AS COMMAND PLANNER
+
+(Referred by Section 3.1) A detailed dialog between user and LLM (GPT-4o) is shown in Figure 6.
+
+# B DIFFUSION DENOISING PROCESS
+
+In order for a better illustration of target scene layout generation process, we visualize intermediate steps during the diffusion denoising process for an adding operation example, shown in Fig. 5. At the start of the layout diffusion process, the poses and scales of all objects inside the target scene begin as random noise. As the diffusion process progresses, each object's properties are iteratively refined and placed into an appropriate configuration that aligns with the source scene and the given commands. This iterative refinement highlights the strength of the diffusion model in handling scene edits cohesively. By adopting a unified approach, the diffusion model is capable of jointly generating the entire scene layout for all editing types, ensuring consistency and coherence across diverse operations.
+
+# C EDITROOM-DB PIPELINE DETAILS
+
+(Referred by Section 4) A detailed example of using LLM to generate natural language description from template command is shown in Figure 7.
+
+Add and Remove Objects Removing each object in the scene separately could generate the modified scenes as the result after removal compared to the original scene. Conversely, the original scene could be treated as the result after object addition. The formatted editing description will be 'Add/Remove [object description]'. In order to consider the location of the addition and potential multiple objects in the scene, we will add the relative location description with the closest unique object in the scene, like 'location: [relative description] [reference object description]'.
+
+Pose and Size Changes We can similarly repeat the pose change operation for every object in the scene as add/remove. Specifically, we design three operations: translation, rotation, and scaling. For translation, we create random translations as the mix of distances, sampled from 0.1 meters to 1.5 meters with step 0.1, and directions, sampled along the two axes directions (front/back and left/right). Then, collision checking is done for every translated object until we find a collision-free sample. The translation will be skipped if all the samples fail in collision checking. The formatted editing description will be 'Move object towards the front/back/left/right direction for [distance] : [object description]'
+
+Similarly, we create random rotation angles as the mix of uniform direction samples, clockwise or counterclockwise, and random values between  $15 - 180$  degrees with the step of 15 degrees, and check collision for each sample. The check stops when we find a collision-free sample or all samples fail the checking. The formatted editing description will be 'Rotate object [angle] degrees : [object description]'
+
+For scaling, we separate it as shrinking and enlarging. The scaling factor is randomly generated between 0.5-0.8 or 1.2-1.5. The scaling factor uniformly applies to three dimensions. Since shrinking won't cause a collision with other objects, it can always result in a successfully modified scene. For enlarging, if collision checking fails on all trials, the enlarging is skipped. Otherwise, we save the largest collision-free scaling factor. The formatted editing description will be 'Shrink/Enlarge object by [scale_factor] times : [object description]'
+
+Object Replacement For the replace operation, we access an object dataset with semantic class labels and 3D meshes. The system will retrieve several objects from the dataset with the same class label as the replaced object, and check their collision with other existing objects in the scene. If none of the objects could be placed without collision, we randomly select one object and shrink its bounding box to be equal or smaller than the replaced object to avoid collision. The formatted description is 'Replace source with target: [source object description] to [target object description]'.
+
+Collision Detection Module The objects are abstracted as 3D bounding boxes and further decomposed into 2D bounding boxes on a horizontal plane and vertical range, as the objects can only rotate about the vertical axis. Then, the two objects are only in collision if their 2D bounding boxes overlap and their vertical ranges overlap. For 2D bounding box collision detection, we apply the Separating Axis Theorem Huynh (2009) to determine if the boxes intersect.
+
+# System Prompt
+
+Imagine you are a indoor room designer and you are using provided API to control the 3D models in the scene.
+
+Given one scene configuration and a command to edit the scene, you should use the provided APIs to do planning and achieve the target.
+
+All sizes and centroids in scene configurations are in meters. The angles are defined in degrees. The dimension sequence is [x,y,z]. Vertical angles are the angles along the y-axis.
+
+Sizes are the half lengths of the bounding box along the x, y, and z axes when the vertical angle is zero.
+
+We define  $+x / - x$  as the right/left direction,  $+y / - y$  as the up/down direction,and  $+z/ - z$  as the front/back direction.
+
+Positive angles are counterclockwise, and negative angles are clockwise.
+
+# APIs:
+
+1. Rotate an object: [Rotate, Target Object Description, Angle : (degrees)]  
+2. Translate an object: [Translate, Target Object Description, Direction : (x/y/z), Distance : (meters)]  
+3. Scale an object: [Scale, Target Object Description, Scale Factor]  
+4. Replace an object: [Replace, Source Object Description, Target Object Description]  
+5. Add an object: [Add, Target Object Description]  
+6. Remove an object: [Remove, Target Object Description]
+
+# Matters needing attention:
+
+1. If there are multiple same objects in the scene and the command is related to the object, you should refer to the object locations.  
+When you refer to the object locations, you should this format: (Relative Description, Relative Object Description).  
+When you use add or remove command, you should refer to the object locations.  
+rel less than 1 meters in x-z plane.  
+For example, if you want to add a chair in front of the table, you should use the format: [Add, Chair, (in front of, Table)].  
+2. Translate, rotate, and scale commands should be executed in the order of scale, rotate, and translate.  
+Consider the potential collision between objects when you execute the commands  
+3. When you scale an object, the object should be scaled uniformly. Scale factor should one float number.  
+4. Replace object will only replace the object with the same class. Replace command will only change the object appearance, not the object poses and sizes  
+5. If Translate/Rotate/Scale commands can achieve the target, you should not use Replace/Add/Remove commands.  
+6. If image is provided, you should use the image to help you understand the scene  
+7. Attempt to use the minimum number of commands to achieve the target.  
+8. If you want to remove and add the object within the same class, you should use the replace command.  
+9. Object descriptions should be detailed descriptions instead of class names. You can imagine the object descriptions if the object is not in the scene.  
+10. All apis should be able to converted to a list of strings and numbers, which can be directly processed by json.load()
+
+# For example:
+
+1. If you want to rotate a chair 90 degrees and there is only one chair in the scene, you should use the format: [Rotate', 'chair', 90].  
+2. If you want to add a chair in front of the table, you should use the format: ['Add', 'chair', ('in front of', 'table')].  
+3. If you want to remove a chair, you should use the format: [Remove', 'chair'].  
+4. If you want to replace a metal chair with a wooden one and this chair on the left of the bed, you should use the format: [Replace', 'mental chair', 'wooden chair', ('left', 'bed')].
+
+Think about it step by step. Summarize the used apis at the end by lines. The final output format should be ***api1;api2;...***.
+
+# User Input
+
+# [Scene configurations]:
+
+Object 0: {"class": "double bed", "size": [1.01, 0.39, 1.08], "vertical angle": -90, "centroid": [3.22, 0.0, -2.48]}
+
+"descriiption": "the double bed is a modern, minimalist design with a white color scheme and a simple, clean appearance."
+
+Object 1: {"class": "nightstand", "size": [0.28, 0.24, 0.22], "vertical angle": 0, "centroid": [2.05, 0.0, -4.52]}
+
+"description": "the nightstand is a modern, dark wood piece with a sleek, minimalist design."
+
+Object 2: {"class": "nightstand", "size": [0.33, 0.33, 0.23], "vertical angle": 0, "centroid": [4.54, 0.0, -3.35]}
+
+"descriiption". "the nightstand is a simple, white, two-draer piece with a smooth finish and a small, round knob on each drawer."
+
+Object 3: {"class": "wardrobe", "size": [1.04, 1.02, 0.32], "vertical angle": 90, "centroid": [1.51, 0.0, -2.48]}
+
+"descriiption": "the wardrobe is a large, white, traditional style piece of furniture with a decorative top and multiple doors."
+
+Object 4: {"class": "desk", "size": [1.0, 0.38, 0.3], "vertical angle": -90, "centroid": [5.6, 0.0, -2.47],
+
+"descriiption": "the desk is a modern, minimalist design with a sleek, rectangular shape and a matte gray finish."
+
+[Command]: "replace the nightstand near the white cabinet with the same model of another nightstand near the black table,
+
+then move it closer to the bed corner to be symmetric with another one."
+
+replace the nightstand near the white cabinet with the same model of another nightstand near the black table,
+
+then move it closer to the bed corner to be symmetric with another one.
+
+![](images/320f47f49c8e754b1287587925d181d9cb13f8b8f71a86fea7a3a19753babec7.jpg)
+
+![](images/a5f4ec32ffb532a9f611fc441a6d5288a9d08f2181adbad84ab2248a73696669.jpg)  
+Figure 6: Dialog between user and LLM to generate an edit plan to rearrange the bedroom. The room image in 'User Input' is not given to LLM, just for illustration purpose.
+
+# LLM Response
+
+To achieve the command of replacing the nightstand near the white cabinet with the other nightstand near the black table and moving it to be symmetric with the other nightstand closer to
+
+the bed corner, we can follow these steps:
+
+1. Identify the nightstands:  
+a. Nightstand near the white cabinet (wardrobe): This is the darker wood nightstand at centroid [2.05, 0.0, -4.52].  
+b. Nightstand near the black table: This is the white nightstand at centroid [4.54, 0.0, -3.35].  
+2. Replace the darker wood nightstand with the white nightstand  
+3. Move this replaced nightstand closer to the bed so it is symmetric with the other nightstand:  
+a. To maintain symmetry, the nightstand should be as positioned close to the bed corner, in the right direction.  
+Let's execute the steps to achieve the target:  
+1. **Replace the dark wood nightstand with the white nightstand:**
+
+# ```
+```
+```
+```
+```
+``
+
+[Replace', 'modern, dark wood nightstand', 'simple, white two-drawer nightstand', ('near', 'white cabinet')]
+
+2. **Translate the replaced nightstand to the right direction to make it symmetric:**
+
+Considering there is another nightstand near the bed (on the right side), the new one should move closely to the same distance as the first one but in the symmetric position (left of the
+
+bed)
+
+# ...json
+
+[Translate', 'simple, white two-drawer nightstand', 'x', 1.22]
+
+Final API commands used:  
+-Replace the nightstand:
+
+# ```
+```
+```
+```
+```
+``
+
+[Replace", "modern, dark wood nightstand", "simple, white two-drawer nightstand", ("near", "white cabinet")]
+
+- \*\*Translate the nightstand to ensure symmetry with another nightstand near the bed:\*\*
+
+```
+```
+```
+```
+```
+``
+
+[Translate", "simple, white two-drawer nightstand", "x", 1.22]
+
+Combining these together, the final output is:
+
+**Output:**
+
+\*\*\*["Replace", "modern, dark wood nightstand", "simple, white two-drawer nightstand", ("near", "white cabinet")];
+
+["Translate", "simple, white two-drawer nightstand", "x", 1.22]***
+
+# System Prompt
+
+Given scene configurations and templated commands, you should write new commands using natural language and spatial reference.
+
+Templated commands will be in the 'action: target_object' format. If the location is provided in the templatd commands, it can be considered as a hint for the target object's location compared to the existing object in the scene.
+
+All sizes and centroids in scene configurations are in meters. The angles are defined in degrees. The dimension sequence is  $[x,y,z]$ . Vertical angles are the angles along the y-axis.
+
+Sizes are the half lengths of the bounding box along the x, y, and z axes when the vertical angle is zero.
+
+We define  $+x$ - $-x$  as the right/left direction,  $+y$ - $-y$  as the up/down direction, and  $+z$ - $-z$  as the front/back direction.
+
+When you design new commands, please refer to the spatial relations between objects in the scene.
+
+When you design new commands, please consider correctness, conciseness, and naturalness.
+
+You should attempt to make your command need reasoning.
+
+If there are duplicate target objects in the scene, you should refer to object locations by relative spatial relations with one unique object in the scene.
+
+If there are multiple templated commands, you should consider them as the same command with different representations.
+
+If templated commands indicate to add an object where there is already a similar object, you should indicate this is about adding a new object in your command.
+
+Enlarge and shrink in the command should be uniform.
+
+You can add object descriptions according to the scene configurations, commands, and the image (if provided).
+
+For example:
+
+Example1:
+
+[Templated commands]:Tmove object towards the \*\*\*left\*\*\* direction for 1 meters: a white bed with a red and white plaid comforter and a red and white plaid pillow.
+
+If there is a table (only one table inside the scene) on the left side of the bed and length of bed is 2 meters, you can write: move the white bed with red and white plaid towards the table
+
+around 1 meters, or 'move the bed towards the left direction by half of bed length.
+
+Example2:
+
+[Templated commands]:'move object towards the ***left*** direction for 0.5 meters: a wooden nightstand.'
+
+If there is a bed parallel to the nightstand and moving to the left will make the nightstand closer to the bed headboard, you can write: 'move the nightstand closer to the bed headboard by 0.5
+
+meters'.
+
+Example3:
+
+[Templated commands]:['replace source with target : [Source] a white bed; [Target] a brown bed.]
+
+You can write: 'replace the white bed with a brown bed.'
+
+Example4:
+
+[Templated commands]:['add object: a white bed; location: \*\*\*right\*\*\* a wardrobe.]
+
+If there is a wardrobe in the scene, you can write: 'add a white bed on the right side of the wardrobe.'
+
+Now you can start to design new commands based on the scene configurations and templated commands. You can supplement object descriptions on the command.
+
+Think about it step by step and summarize your commands in the end. The final output format should be '##[natural command 1, natural command 2,...]#, which is a list of strings and
+
+can be processed by astliteral_eval() or json.load().
+
+# User Input
+
+[Scene configurations]:
+
+Object 0: {"class": "dining table", "size": [0.55, 0.38, 0.23], "vertical angle": 0, "centroid": [-0.8, 0.0, -3.46]}
+
+"description": "the dining table is a modern, minimalist design with a black marble top and a silver metal frame."
+
+Object 1: {"class": "loveseat sofa", "size": [1.24, 0.43, 0.47], "vertical angle": 90, "centroid": [-3.78, 0.0, 1.38],
+
+"description": "the loveseat sofa is brown with a modern design and has a variety of patterned throw pillows."
+
+Object 2: {"class": "coffee table", "size": [0.69, 0.23, 0.47], "vertical angle": 90, "centroid": [-2.48, 0.0, 1.4]}
+
+"descriiption": "the coffee table is a modern, minimalist design with a geometric shape, featuring a combination of dark wood and lighter wood panels."
+
+Object 3: {"class": "lounge chair", "size": [0.37, 0.45, 0.37], "vertical angle": 153, "centroid": [-2.94, 0.0, 3.11],
+
+"descriiption": the chair is a modern, minimalist design with a dark wood frame and a cushion featuring a geometric pattern.
+
+Object 4: {"class": "corner side table", "size": [0.22, 0.23, 0.22], "vertical angle": 90, "centroid": [-3.99, 0.0, -0.25],
+
+"description": "a round, black marble table with a white base."
+
+Object 5: {"class": "dining chair", "size": [0.31, 0.45, 0.3], "vertical angle": -180, "centroid": [-0.47, 0.0, -2.79]}
+
+"descriiption": "the chair is black with a modern design, featuring a high back and armrests."
+
+Object 6: {"class": "dining chair", "size": [0.31, 0.45, 0.3], "vertical angle": -180, "centroid": [-1.08, 0.0, -2.79]}
+
+"description": "the chair is black with a modern design, featuring a high back and armrests."
+
+Object 7: {"class": "corner side table", "size": [0.22, 0.23, 0.22], "vertical angle": 90, "centroid": [-3.99, 0.0, 3.13],
+
+"description": "a round, black marble table with a white base."
+
+Object 8: {"class": "dining chair", "size": [0.31, 0.45, 0.3], "vertical angle": 0, "centroid": [-1.08, 0.0, -4.17]}
+
+"descriiption": "the chair is black with a modern design, featuring a high back and armrests."
+
+Object 9: {"class": "dining chair", "size": [0.31, 0.45, 0.3], "vertical angle": 0, "centroid": [-0.48, 0.0, -4.17]
+
+"descriiption": "the chair is black with a modern design, featuring a high back and armrests."
+
+Obiect 10: {"class": "console table", "size": [0.7, 0.42, 0.15], "vertical angle": 0, "centroid": [-3.35, 0.0, -4.56].
+
+"descriiption": "the console table is a modern, black, three-tiered design with a flat top and a rectangular base."
+
+Object 11: {"class": "cabinet", "size": [0.62, 1.08, 0.29], "vertical angle": -90, "centroid": [-3.94, 0.0, -2.04]}
+
+"description": "the children's cabinet is a modern, minimalist design with a light wood frame and a blue and white color scheme, featuring a playful bunny motif on the doors."
+
+Object 12: {"class": "pendant lamp", "size": [0.18, 0.54, 0.18], "vertical angle": 90, "centroid": [-2.71, 1.53, 4.35],
+
+"description": "the pendant lamp is a modern, metallic chandelier with a white finish, featuring a series of vertical, clear glass tubes that create a geometric pattern."
+
+Object 13: {"class": "pendant lamp", "size": [0.19, 0.52, 0.19], "vertical angle": 90, "centroid": [-2.26, 1.55, -0.3]}
+
+"descriiption": "the pendant lamp is black with a woven design and a white interior."
+
+[Templated commands]:["rotate object 135 degrees : the pendant lamp is black with a woven design and a white interior",
+
+"obviously rotate object 135 degrees :the pendant lamp is black with a woven design and a white interior."
+
+Hint: The target object is the Object_13.
+
+![](images/133bc21384fd6f7f9460588318c3f4d725df8ff693adfeb25aac5b8a657b3c81.jpg)
+
+# LLM Response
+
+\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*
+
+![](images/6a4a7db32e540667acbcc52e3acea4e245e09d1b01f0ea46253be444861201fe.jpg)  
+Figure 7: Dialog between user and LLM to generate natural language description from the template command. The room image in 'User Input' is not given to LLM, just for illustration purposes.

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+# EFFICIENT ACTION-CONSTRAINED REINFORCEMENT LEARNING VIA ACCEPTANCE-REJECTION METHOD AND AUGMENTED MDPS
+
+Wei Hung<sup>1</sup> Shao-Hua Sun<sup>2</sup> Ping-Chun Hsieh<sup>1</sup>
+
+$^{1}$ National Yang Ming Chiao Tung University, Hsinchu, Taiwan
+
+$^{2}$ National Taiwan University, Taipei, Taiwan
+
+{hwei1048576.cs08，pinghsieh}@nycu.edu.tw
+
+# ABSTRACT
+
+Action-constrained reinforcement learning (ACRL) is a generic framework for learning control policies with zero action constraint violation, which is required by various safety-critical and resource-constrained applications. The existing ACRL methods can typically achieve favorable constraint satisfaction but at the cost of either a high computational burden incurred by the quadratic programs (QP) or increased architectural complexity due to the use of sophisticated generative models. In this paper, we propose a generic and computationally efficient framework that can adapt a standard unconstrained RL method to ACRL through two modifications: (i) To enforce the action constraints, we leverage the classic acceptance-rejection method, where we treat the unconstrained policy as the proposal distribution and derive a modified policy with feasible actions. (ii) To improve the acceptance rate of the proposal distribution, we construct an augmented two-objective Markov decision process (MDP), which includes additional self-loop state transitions and a penalty signal for the rejected actions. This augmented MDP incentivizes the learned policy to stay close to the feasible action sets. Through extensive experiments in both robot control and resource allocation domains, we demonstrate that the proposed framework enjoys faster training progress, better constraint satisfaction, and a lower action inference time simultaneously than the state-of-the-art ACRL methods. We have made the source code publicly available* to encourage further research in this direction.
+
+# 1 INTRODUCTION
+
+Action-constrained reinforcement learning (ACRL) aims to find an optimal policy maximizing expected cumulative return while satisfying constraints imposed on the action space and has served as a generic framework for learning sequential decision making in both safety-critical and resource-constrained applications. As a classic example, robot control is usually subject to the inherent kinematic constraints of the robots, e.g., torque or output power, which need to be satisfied throughout the training and the inference stages to avoid damage to physical components (Singletary et al., 2021; Tang et al., 2024; Liu et al., 2024). Another example is dynamic resource allocation for networked systems (Chen et al., 2023; Jay et al., 2019; Chen et al., 2021), such as communication networks and bike sharing systems (Zhang et al., 2022; 2021), which involve the capacity constraints on the communication links and the docking facilities, respectively. To prevent network congestion and resource over-utilization, these constraints must be considered at each step of training and deployment. Given its wide applicability, developing practical ACRL algorithms that can learn policies accruing high returns with minimal constraint violation is essential.
+
+Existing ACRL methods have explored the following techniques: (i) Action projection: As a conceptually simple and widely-used technique, action projection finds a feasible action closest to the original unconstrained action produced by the policy. The projection step can be used in action post-processing (Kasaura et al., 2023) or implemented by a differentiable projection layer (Amos
+
+& Kolter, 2017) as part of the policy network of a standard deep RL algorithm for end-to-end training (Pham et al., 2018; Dalal et al., 2018; Bhatia et al., 2019). Despite the simplicity, to find close feasible actions, action projection needs to solve a quadratic program (QP), which is computationally costly and scales poorly to high-dimensional action spaces (Ichnowski et al., 2021). (ii) Frank-Wolfe search: Lin et al. (2021) propose to decouple policy updates from action constraints by a Frank-Wolfe search subroutine. Despite its effectiveness, Frank-Wolfe method requires solving multiple QPs per training iteration and therefore suffers from substantially higher training time. (iii) Generative models: To replace the projection step, generative models, such as Normalizing Flows (Kobyzev et al., 2020), have been employed as a learnable projection layer that is trained to satisfy the constraints (Brahmanage et al., 2023; Chen et al., 2023).
+
+Despite the recent advancement in ACRL, existing algorithms suffer from significant computational overhead induced by solving QPs or learning sophisticated models, such as Normalizing Flows. Table 1 summarizes the limitations of existing methods in interaction time, training load, and action violation rate. This motivates us to develop a more efficient ACRL approach.
+
+Table 1: A qualitative comparison of ACRL algorithms.  
+
+<table><tr><td>Algorithm</td><td>Action violation rate</td><td>Generative model</td><td>Interaction time</td><td>Training load</td><td>Remarks</td></tr><tr><td>OptLayer (Pham et al., 2018)</td><td>High</td><td>X</td><td>High</td><td>Low</td><td>Zero-gradient issue</td></tr><tr><td>ApprOpt (Bhatia et al., 2019)</td><td>High</td><td>X</td><td>High</td><td>Low</td><td>Zero-gradient issue</td></tr><tr><td>NFWPO (Lin et al., 2021)</td><td>Low</td><td>X</td><td>Low</td><td>High</td><td>Rely heavily on QPs</td></tr><tr><td>FlowPG (Brahmanage et al., 2023)</td><td>Low</td><td>✓</td><td>Low</td><td>High</td><td>Require pre-training</td></tr><tr><td>DPre+ (Kasaura et al., 2023)</td><td>High</td><td>X</td><td>High</td><td>Low</td><td>-</td></tr><tr><td>SPre+ (Kasaura et al., 2023)</td><td>High</td><td>X</td><td>High</td><td>Low</td><td>-</td></tr><tr><td>IAR-A2C (Chen et al., 2023)</td><td>Low</td><td>✓</td><td>Low</td><td>High</td><td>Support only discrete actions</td></tr><tr><td>ARAM (Ours)</td><td>Low</td><td>X</td><td>Low</td><td>Low</td><td>-</td></tr></table>
+
+We propose a framework called ARAM, which augments a standard deep RL algorithm with two modifications: Acceptance-Rejection method and Augmented MDPs. (i) Acceptance-rejection method: Given a policy network, ARAM enforces the action constraints by rethinking ACRL through acceptance-rejection sampling, i.e., first sampling actions from the unconstrained policy and then only accepting those that are in the feasible action set. This sampling strategy can substantially reduce the need for solving QPs, compared to the methods built on the action projection step or the Frank-Wolfe search. (ii) Augmented unconstrained two-objective Markov decision process: One technical issue of the acceptance-rejection method is the possibly low acceptance rate, which is likely to occur in the early training stage. Under a low acceptance rate, the sampling process could take excessively long and thereby incur a high training overhead. To improve the acceptance rate, we augment the original MDP with additional self-loop transitions and a penalty function induced by the event whether the action is accepted. Through this augmented MDP, we leverage the penalty induced by constraint violation to guide the policy distribution towards regions of higher acceptance rate. Notably, these two modifications can be combined with any standard deep RL algorithm. In this paper, we take the Soft Actor Critic (SAC) as the base RL algorithm. Moreover, to obviate the need for hyperparameter tuning of the penalty weight, we directly leverage the multi-objective extension of SAC that can learn policies under all the penalty weights.
+
+We evaluate the proposed ARAM in various ACRL benchmarks, including the MuJoCo locomotion tasks and resource allocation of communication networks and bike sharing systems. The experimental results show that: (i) ARAM enjoys faster learning progress than the state-of-the-art ACRL methods, measured either in environment steps or wall clock time. Moreover, the difference is particularly significant under wall clock time thanks to the low training time of our design. (ii) ARAM requires significantly fewer QP operations than the other ACRL benchmark methods, mostly by 2-5 orders of magnitude fewer. (iii) ARAM indeed achieves high action acceptance rate through the guidance of the augmented MDP. (iv) ARAM also enjoys the lowest per-action inference time as it largely obviates the need for QP operations and learns without using a generative model.
+
+# 2 RELATED WORK
+
+Action-Constrained RL. The first category focuses on ensuring that the actions meet the constraints at each step of the training and evaluation processes. To ensure zero constraint violation, one nat-
+
+ural and commonly-used technique is action projection, which finds a feasible action closest to the original unconstrained action produced by the policy. To enable end-to-end training, multiple ACRL methods have incorporated the differentiable projection layer (Amos & Kolter, 2017) as the output layer of the policy network, such as OptLayer (Pham et al., 2018), Safety Layer (Dalal et al., 2018), and Approximate OptLayer (Bhatia et al., 2019). However, these projection layers are known to suffer from slow learning in ACRL due to the zero-gradient issue (Lin et al., 2021).
+
+To address the zero-gradient issue, several approaches have been adopted recently: (i) Frank-Wolfe search: Lin et al. (2021) propose Neural Frank-Wolfe Policy Optimization (NFWPO), which decouples policy updates from action constraints by a Frank-Wolfe search subroutine. Despite its effectiveness, Frank-Wolfe search involves solving multiple quadratic programs (QP) per training iteration, and this is computationally very costly and is known to scale poorly to high-dimensional action spaces (Ichnowski et al., 2021). (ii) Removing projection layers and using pre-projected actions for critic learning: Without using differentiable projection layers, Kasaura et al. (2023) propose DDPG-based  $\mathrm{DPre + }$  and SAC-based  $\mathrm{SPre + }$ , which use a QP-based projection step for action post-processing, train the critic with pre-projected actions, and apply a penalty term to reduce constraint violation. Notably,  $\mathrm{DPre + }$  and  $\mathrm{SPre + }$  achieve fairly strong reward performance but still at the cost of high computational overhead incurred by QPs. (iii) Generative models: To replace the projection layer, generative models, e.g., normalizing flow, have been integrated into the policy network, such as FlowPG for continuous control (Brahmanage et al., 2023) and IAR-A2C for discrete control (Chen et al., 2023), to generate multi-modal action distributions that can better satisfy the constraints. Despite the effectiveness, learning a sophisticated generative model adds substantial design complexity to ACRL. By contrast, the proposed ARAM largely removes the overhead of QPs and completely obviates the need for generative models.
+
+RL for Constrained MDPs. The other class of methods focuses on ensuring the long-term average action safety by defining a cost function and modeling the problem as a Constrained MDP (CMDP) (Altman, 2021). For example, Constrained Policy Optimization (CPO) (Achiam et al., 2017) is the first policy gradient method developed to solve CMDPs. It uses the Fisher information matrix and second-order Taylor expansion to ensure safety constraints, but it is computationally expensive and requires more samples, potentially reducing efficiency. To address these, Tessler et al. (2019) propose Reward Constrained Policy Optimization (RCPO), which leverages primal-dual methods to improve both the efficiency and effectiveness of policy optimization under constraints. Building on similar goals of improving efficiency, FOCOPS (Zhang et al., 2020) takes a different approach by using a first-order approximation for policy optimization. This reduces computational complexity but introduces convergence issues due to approximation errors in the first-order constraints. The above list is by no means exhaustive and is only meant to provide an overview of this line of research. Please refer to (Liu et al., 2021) for more related prior works. While these methods can ensure that the long-term expected cost remains under a certain threshold, they fail to enforce action constraints at every environment step needed in ACRL throughout training and deployment.
+
+Augmented Safety Mechanisms in RL. To improve safety in exploration, Eysenbach et al. (2018) introduced a reset framework, which detects when the agent enters unsafe states and resets it, thereby improving both safety and sampling efficiency. Building on this, Thananjeyan et al. (2021) introduced the concept of learned recovery zones, ensuring that when the agent deviates from safe limits, it can autonomously return to a safe state, providing more robust safety guarantees during exploration. In parallel, Thomas et al. (2021) proposed a specialized Markov Decision Process (MDP) to constrain the training process, helping to further avoid unsafe actions during exploration. In addition to mechanisms that correct unsafe actions, Safety Augmented Value Estimation from Demonstrations (SAVED) (Thananjeyan et al., 2020) employs model predictive control (MPC) to proactively avoid unsafe actions by updating the policy, specifically preventing infeasible actions that could lead to dangerous situations. Complementing this, Yu et al. (2022) proposed Safety Editor (SEEditor), a mechanism that transforms actions produced by a utility maximizer into safe alternatives, preventing violations of safety constraints during execution. While these methods can not fully guarantee that the policy's actions always remain within safe regions, they offer an effective approach to significantly reduce unsafe behaviors, laying a foundation to develop more robust safety mechanisms.
+
+# 3 PRELIMINARIES: ACTION-CONSTRAINED REINFORCEMENT LEARNING
+
+In ACRL, we consider an action-constrained Markov Decision Process (MDP). Given a set  $\mathcal{X}$ , let  $\Delta(\mathcal{X})$  denote the set of all probability distributions on  $\mathcal{X}$ . An action-constrained MDP is defined by a tuple  $\mathcal{M} := (\mathcal{S}, \mathcal{A}, \mathcal{P}, \gamma, r, \mathcal{C})$ , where  $\mathcal{S}$  denotes the state space,  $\mathcal{A}$  denotes the action space,  $\mathcal{P}: \mathcal{S} \times \mathcal{A} \to \Delta(\mathcal{S})$  serves as the transition kernel,  $\gamma \in (0,1)$  is the discount factor,  $r: \mathcal{S} \times \mathcal{A} \to \mathbb{R}$  denotes the bounded reward function. Without loss of generality, we presume the reward  $r(s,a)$  to lie in the  $[0,1]$  interval since we can rescale a bounded reward function to the range of  $[0,1]$  given the maximum and minimum possible reward values. For each  $s \in \mathcal{S}$ , there is a non-empty feasible action set  $\mathcal{C}(s) \subseteq \mathcal{A}$  induced by the underlying collection of action constraints. That is to say, no actions outside the feasible set  $\mathcal{C}(s)$  can be applied to the environment, ensuring that only valid actions are considered within the system dynamics. Notably, we make no assumption on the structure of  $\mathcal{C}(s)$  (and hence  $\mathcal{C}(s)$  needs not be convex).
+
+At each time  $t \in \mathbb{N} \cup \{0\}$ , the learner observes the current state  $s_t \in S$  of the environment, selects a feasible action  $a_t \in \mathcal{C}(s_t)$ , and receives reward  $r_t$ . We use  $\pi : S \to \Delta(\mathcal{A})$  to denote a Markov stationary stochastic policy, which is updated iteratively by the learner. Given a policy  $\pi$ , the Q functions  $Q(\cdot, \cdot; \pi) : S \times \mathcal{A} \rightarrow \mathbb{R}$  is defined as  $Q(s, a; \pi) := \mathbb{E}\left[\sum_{t=0}^{\infty} \gamma^t r_t | s_0 = s, a_0 = a; \pi\right]$ , which can be characterized as the unique solution to the following Bellman equation:
+
+$$
+Q (s, a; \pi) = r (s, a) + \gamma \mathbb {E} _ {s ^ {\prime} \sim \mathcal {P}, a ^ {\prime} \sim \pi (\cdot | s ^ {\prime})} [ Q (s ^ {\prime}, a ^ {\prime}; \pi) ]. \tag {1}
+$$
+
+To learn a policy and the corresponding  $Q$  function under large state and action spaces, we use the parameterized functions  $\pi_{\phi}:\mathcal{S}\to \Delta (\mathcal{A})$  and  $Q_{\theta}:\mathcal{S}\times \mathcal{A}\rightarrow \mathbb{R}$  as function approximators, where  $\phi$  and  $\theta$  typically denote the parameters of neural networks in the deep RL literature. Our goal is to learn an optimal policy  $\pi^{*}$  such that  $Q(s,a;\pi^{*})\geq Q(s,a;\pi)$ , for all  $s\in S,a\in \mathcal{C}(s)$  and  $\pi \in \Pi_{\mathcal{C}}$ , where  $\Pi_{\mathcal{C}}\coloneqq \{\pi :\mathcal{S}\to \Delta (\mathcal{C})\}$ , which denotes the set of all feasible policies. We also use  $\Pi_{\mathcal{A}}\coloneqq \{\pi :\mathcal{S}\to \Delta (\mathcal{A})\}$  to denote the set of all unconstrained Markov stationary policies.
+
+Notations. Throughout the paper, we use  $\langle x,y\rangle$  to denote the inner product of two real vectors  $x,y$ . Moreover, we use  $\mathbf{1}_d$  to denote the  $d$ -dimensional vector of all ones.
+
+# 4 ALGORITHM
+
+To address ACRL, we devise and introduce two main modifications to existing deep RL algorithms: the acceptance-rejection method (Section 4.1) and the augmented two-objective MDP (Section 4.2). Then, we describe a practical multi-objective RL implementation of ARAM in Section 4.3.
+
+# 4.1 ACCEPTANCE-REJECTION METHOD
+
+To adapt a standard deep RL method to ACRL, we need to convert an unconstrained policy  $\pi_{\phi} \in \Pi_{\mathcal{A}}$  into a feasible policy  $\pi_{\phi}^{\dagger} \in \Pi_{\mathcal{C}}$ . Notably, the action constraints, in general, can be complex and take arbitrary forms of expression. As a result, the feasible action sets  $\mathcal{C}(s)$  are likely to be rather unstructured. To tackle this, we propose to rethink the constraint satisfaction in ACRL through the classic acceptance-rejection method (ARM), which is a generic algorithm for sampling from general distributions (Kroese et al., 2013).
+
+Using ARM in the context of ACRL. For didactic purposes, here we focus on the continuous control and assume that the action space  $\mathcal{A}$  is a compact convex set despite that the same argument can work seamlessly under discrete action spaces. Let  $f$  and  $g$  be two probability density functions over  $\mathcal{A}$ . ARM can generate random variables that follow the target distribution  $f$  while drawing samples from another proposal distribution  $g$ . To put ARM in the context of ACRL, let us fix a state  $s$  and take the constrained and the unconstrained policies as the target and the proposal distributions, respectively, i.e.,  $f \equiv \pi_{\phi}^{\dagger}(s) \in \Delta(\mathcal{C}(s))$ ,  $g \equiv \pi_{\phi}(s) \in \Delta(\mathcal{A})$ . Clearly, we have  $\pi_{\phi}^{\dagger}(a|s) = 0$  for all  $a \notin \mathcal{C}(s)$ . Let  $M > 0$  be some constant such that  $M \cdot \pi_{\phi}(a|s) \geq \pi_{\phi}^{\dagger}(a|s)$ , for all  $a \in \mathcal{A}$ . ARM can generate an action that follows  $\pi_{\phi}^{\dagger}(s)$ :
+
+1. Generate  $a^\prime \in \mathcal{A}$  from the unconstrained  $\pi_{\phi}(s)$  
+2. If  $a' \in \mathcal{C}(s)$ , accept  $a'$  with probability  $\pi_{\phi}^{\dagger}(a'|s) / (M \cdot \pi_{\phi}(a'|s))$ ; otherwise, if  $a' \notin \mathcal{C}(s)$ , reject  $a'$  and return to the first step.
+
+Choices of target distribution  $\pi_{\phi}^{\dagger}$ . Note that we have the freedom to configure the desired  $\pi_{\phi}^{\dagger}$  for specific purposes, e.g., exploration. One convenient choice is to simply set  $\pi_{\phi}^{\dagger}(a|s) \propto \pi_{\phi}(a|s)$ , for all  $a \in \mathcal{C}(s)$ . In this case, we can always accept an action  $a' \in \mathcal{C}(s)$  in the above step 2 if we set  $M = 1 / (\int_{a \in \mathcal{C}(s)} \pi_{\phi}(a|s) da)$ .
+
+Salient features of ARM. The advantages of using ARM in ACRL are two-fold: (i) Efficiency: ARM is computationally efficient as it only requires checking if an action satisfies the constraints. As a result, ARM largely obviates the need to solve QPs or learn a generative model. (ii) Generality: ARM is very general, i.e., can be integrated with any standard unconstrained RL method.
+
+Issues of low acceptance rate under ARM. Despite the efficiency and generality of ARM, solely naively adopting ARM in ACRL can lead to keeping sampling invalid actions and getting stuck when the policy has a low action acceptance rate. The issues are two-fold: (i) The ARM procedure could repeat indefinitely with a near-zero acceptance probability. In ACRL, this scenario is likely to happen at the early training stage since the randomly initialized action distribution can be drastically different from the feasible action set. (ii) A low acceptance rate typically implies a poor action coverage of the policy over  $\mathcal{C}(s)$ . This would significantly affect the performance in the cumulative reward. To address this issue by increasing the action acceptance rate through the course of learning, we present our solution in Section 4.2.
+
+# 4.2 AUGMENTED UNCONSTRAINED TWO-OBJECTIVE MDP
+
+To mitigate this issue of low ARM acceptance rate, we propose to apply ARM on an augmented unconstrained MDP, which guides the policy updates towards the feasible action set by a penalty signal induced by the action acceptance events of ARM, instead of on the original action-constrained MDP. Moreover, we show that these two MDPs are equivalent with respect to optimal policies.
+
+Constructing an augmented MDP. Based on the original action-constrained MDP  $\mathcal{M} = (\mathcal{S},\mathcal{A},\mathcal{P},\gamma ,r,\mathcal{C})$  we propose to construct an Augmented Unconstrained Two-Objective MDP (AUTO-MDP)  $\tilde{\mathcal{M}}\coloneqq (\mathcal{S},\mathcal{A},\tilde{\mathcal{P}},\gamma ,\tilde{\boldsymbol{r}})$  by adding additional self-loop state transitions and penalty signal for those actions  $a\in \mathcal{C}(s)$ :
+
+![](images/066ae9706168225ece267008f2f6d7154d24c6aaa538fb9fa40f8b29c52953bd.jpg)  
+Figure 1: An illustration of AUTO-MDP, where  $a \in \mathcal{C}(s)$  and  $\tilde{a} \notin \mathcal{C}(s)$ .
+
+- The AUTO-MDP  $\tilde{\mathcal{M}}$  shares the same state and action spaces with the original MDP  $\mathcal{M}$ .  
+- The augmented reward function  $\tilde{r} : S \times \mathcal{A} \to \mathbb{R}^2$  returns a 2-dimensional reward vector  $[r(s, a), c(s, a)]$  and is defined as: Let  $\kappa > 0$  be a constant penalty.
+
+Then, we construct (i) For any  $(s,a)$  with  $a\notin \mathcal{C}(s),\tilde{r} (s,a)\coloneqq [0, - \mathcal{K}]$ . (ii) For any  $(s,a)$  with  $a\in \mathcal{C}(s),\tilde{r} (s,a)\coloneqq [r(s,a),0]$ .
+
+- The augmented transition kernel  $\tilde{\mathcal{P}}$  is defined as follows: For any  $(s, a, s')$  with  $a \in \mathcal{C}(s)$ , let  $\tilde{\mathcal{P}}(s'|s, a) = \mathcal{P}(s'|s, a)$ . For any  $(s, a, s')$  with  $a \notin \mathcal{C}(s)$ , let
+
+$$
+\tilde {\mathcal {P}} \left(s ^ {\prime} \mid s, a\right) = \left\{ \begin{array}{l l} 1, & s = s ^ {\prime} \\ 0, & \text {o t h e r w i s e} \end{array} \right. \tag {2}
+$$
+
+The idea of AUTO-MDP is illustrated in Figure 1.
+
+Moreover, for any policy  $\pi \in \Pi_{\mathcal{A}}$ , we define the vector-valued Q function  $Q: S \times \mathcal{A} \to \mathbb{R}^2$  of  $\pi$  as  $Q(s, a; \pi) := \mathbb{E}\left[\sum_{t=0}^{\infty} \gamma^t r_t | s_0 = s, a_0 = a; \pi\right]$ , which is a natural extension of the standard scalar-valued Q function. Clearly, the vector-valued Q function also satisfies the Bellman equation
+
+$$
+\boldsymbol {Q} (s, a; \pi) = \boldsymbol {r} (s, a) + \gamma \mathbb {E} _ {s ^ {\prime} \sim \bar {\mathcal {P}}, a ^ {\prime} \sim \pi} [ \boldsymbol {Q} (s ^ {\prime}, a ^ {\prime}; \pi) ]. \tag {3}
+$$
+
+As in the standard multi-objective MDPs (MOMDP), the set of optimal policies depend on the preference over the objectives. In the MOMDP literature (Abels et al., 2019; Yang et al., 2019), this is typically characterized by using linear scalarization with a preference vector  $\lambda = [\lambda_r, \lambda_c]$  such that the scalarized Q value of a policy  $\pi$  is defined as  $Q_{\lambda}(s, a; \pi) \coloneqq \langle \lambda, Q(s, a; \pi) \rangle$ . Without loss
+
+of generality, we presume that  $\lambda$  lies in a two-dimensional probability simplex, i.e.,  $\lambda_r \geq 0$ ,  $\lambda_c \geq 0$ , and  $\lambda_r + \lambda_c = 1$ .
+
+With such a design, to obtain a policy with a sufficiently high acceptance rate under ARM, the learner shall find a policy that maximizes cumulative reward while minimizing violation penalty.
+
+Equivalence of AUTO-MDP and original MDP in terms of optimal policies. We show that the constructed AUTO-MDP and the original MDP are equivalent in the sense that they share the same set of optimal policies. This property can be formally stated in the following proposition.
+
+Proposition 1 (Equivalence in optimality). Let  $\pi^{*} \in \Pi_{\mathcal{C}}$  be an optimal policy among all the policies in  $\Pi_{\mathcal{C}}$  under the original action-constrained MDP  $\mathcal{M}$ . Then, for any  $\lambda \in \Lambda$ , the policy  $\pi^{*}$  remains an optimal policy among all the policies in  $\Pi_{\mathcal{A}}$  under the AUTO-MDP  $\tilde{\mathcal{M}}$ .
+
+The proof of Proposition 1 is provided in Appendix A. This result suggests that solving AUTO-MDP can achieve the same maximum cumulative reward as the original action-constrained MDP while providing incentives for a higher action acceptance rate.
+
+Remarks on the preference vector  $\lambda$ . Given the equivalence property in Proposition 1, to solve the AUTO-MDP, one could select a preference vector  $\lambda \in \Lambda$  and find a corresponding optimal policy  $\pi_{\lambda}^{*}$  such that  $\lambda^{\top}Q(s,a;\pi_{\lambda}^{*}) \geq \lambda^{\top}Q(s,a;\pi)$ , for all  $s \in S, a \in A$ , and  $\pi \in \Pi_{A}$ . In practice, different choices of  $\lambda$  lead to distinct learning behaviors: (i) If  $\lambda_r$  is much larger than  $\lambda_c$ , then the violation penalty could be too small to enhance the acceptance rate of ARM. (ii) If  $\lambda_r$  is much smaller than  $\lambda_c$ , then high violation penalty could make the policy very conservative and lead to low cumulative reward.
+
+To find a proper  $\lambda$ , one straightforward approach 13 is to employ hyperparameter tuning over  $\lambda$ , at the cost of several times more of environment 14 steps. To address this, we propose a practical implementation of ARAM based on multi-15 objective RL, which can learn well-performing policies for all preferences simultaneously, as described below.
+
+# Algorithm 1: Practical Implementation of ARAM
+
+Input: Initial parameters  $\phi, \theta$ , preference sampling distribution  $\rho_{\lambda}$ , actor and critic learning rates  $\xi_{\pi}, \xi_{Q}$  Initialize the replay buffer  $\mathcal{D}_r$ ; Initialize the augmented replay buffer  $\mathcal{D}_a$ ; for each iteration  $j$  do Sample  $\lambda \in \Lambda$  according to  $\rho_{\lambda}$ ; for each environment step  $t$  do Sample  $a_t \sim \pi_\phi^\dagger(s_t; \lambda)$  by ARM; Obtain augmented reward  $r_t = [r_t, c_t]$  and next state  $s_{t+1}$ ; if  $a_t \in \mathcal{C}(s_t)$  then  $\lfloor$  Store  $(s_t, a_t, r_t, c_t, s_{t+1})$  in  $\mathcal{D}_r$  else  $\lfloor$  Store  $(s_t, a_t, r_t, c_t, s_{t+1})$  in  $\mathcal{D}_a$ ; for each gradient step  $\tau$  do Draw a mini-batch of samples from  $\mathcal{D}_r$  and  $\mathcal{D}_a$ ; Critic update by (4):  $\theta \leftarrow \theta - \xi_Q \nabla_{\hat{\theta}} J_Q(\theta)$ ; Policy update by (5):  $\phi \leftarrow \phi - \xi_\pi \nabla_{\hat{\phi}} J_\pi(\phi)$ ;
+
+# 4.3 A MULTI-OBJECTIVE RL IMPLEMENTATION OF ARAM
+
+This introduces a practical multi-objective RL implementation of ARAM. As described earlier, the two modifications ARM and AUTO-MDP are general in that they can be employed to adapt any standard deep RL algorithm to ACRL. This work adopts SAC (Haarnoja et al., 2018) as the backbone to showcase how to integrate the proposed modifications into an existing deep RL algorithm.
+
+Solving AUTO-MDP via multi-objective RL. To learn policies for all preferences simultaneously, we adapt the multi-objective SAC (MOSAC) presented in (Hung et al., 2023) to the AUTO-MDP:
+
+- Policy loss and critic loss. MOSAC also adopts an actor-critic architecture as in vanilla SAC. Let  $\phi$  and  $\theta$  denote the parameters of the policy and the critic. MOSAC learns a preference-dependent policy  $\pi_{\phi}(a|s;\lambda)$  and the corresponding vector-valued Q function by a preference-dependent critic network  $Q_{\theta}(s,a;\lambda)$ . Given a state-action sampling distri
+
+![](images/8316bac68ec117a0c62bce2adce116c73ae98b882515d82e8d79a3bfe036ff8a.jpg)  
+Figure 2: An illustration of the ARAM framework. ARAM is composed of three components: (1) ARM: Use an oracle to verify whether the sampled action is in the feasible action set. (2) AUTO-MDP: Assign penalties to invalid actions within an augmented MDP framework, thereby reducing the rate of action violations. (3) MORL: Use MORL to discover well-performing policies under all penalty weights simultaneously.
+
+bution  $\mu$ , the critic network is updated iteratively by minimizing the following loss function
+
+$$
+J _ {\boldsymbol {Q}} (\theta ; \boldsymbol {\lambda}) = \mathbb {E} _ {(s, a) \sim \mu} \left[ \left(\left\langle \boldsymbol {\lambda}, \boldsymbol {Q} _ {\theta} (s, a; \pi_ {\phi}, \boldsymbol {\lambda}) - \left(\boldsymbol {r} (s, a) + \gamma \mathbb {E} _ {s ^ {\prime} \sim \mathcal {P} (\cdot | s, a)} \left[ \boldsymbol {V} _ {\tilde {\theta}} \left(s ^ {\prime}; \pi_ {\phi}, \boldsymbol {\lambda}\right) \right]\right) \right\rangle\right) ^ {2} \right], \tag {4}
+$$
+
+where  $V_{\theta}(s;\pi_{\phi},\lambda) \coloneqq \mathbb{E}_{a\sim \pi_{\phi}(\cdot |\cdot ;\lambda)}[Q_{\theta}(s,a;\pi_{\phi},\lambda) - \alpha \log \pi_{\phi}(a|s;\lambda)\mathbf{1}_d]$  with entropy coefficient  $\alpha$ ,  $\bar{\theta}$  denotes the parameters of the target critic network. Regarding the policy update, the policy  $\pi_{\phi}$  is updated by minimizing
+
+$$
+J _ {\pi} (\phi ; \boldsymbol {\lambda}) = \mathbb {E} _ {s \sim \mu} \left[ \mathbb {E} _ {a \sim \pi_ {\phi}} \left[ \alpha \log \left(\pi_ {\phi} (a \mid s; \boldsymbol {\lambda})\right) - \langle \boldsymbol {\lambda}, \boldsymbol {Q} (s, a; \pi_ {\phi}, \boldsymbol {\lambda}) \rangle \right] \right]. \tag {5}
+$$
+
+Notably, Equation (4) and Equation (5) can be viewed as the critic loss and the policy loss of vanilla SAC under the scalarized Q function.
+
+- Dual-buffer design. Like vanilla SAC, MOSAC is an off-policy algorithm and makes policy and critic updates based on the samples from experience replay buffers. To better address the augmented transitions in AUTO-MDP, we propose a dual-buffer design, where we store the feasible transitions and the augmented infeasible transitions in two separate replay buffers, namely a real replay buffer  $\mathcal{D}_r$  and an augmented replay buffer  $\mathcal{D}_a$ . This design offers more flexibility in balancing the number of updates by feasible and infeasible transitions, especially at the initial training stage when the action violation rate is high.  
+- Preference distribution. To update the policy and the critic for different preferences, the preference  $\lambda$  is drawn from some distribution  $\rho_{\lambda}$ . One natural choice is to set  $\rho_{\lambda}$  as a uniform distribution over the two-dimensional probability simplex, or essentially a Dirichlet distribution with concentration parameter equal to 1.
+
+The training process is illustrated in Figure 2, with the pseudo code provided in Algorithm 1.
+
+# 5 EXPERIMENTS
+
+Benchmark Methods. We compare ARAM with various recent benchmark ACRL algorithms, including NFWPO, DPre+, SPre+, and FlowPG. NFWPO (Lin et al., 2021) achieves favorable constraint satisfaction at the cost of high QP overhead as it enforces action constraints by Frank-Wolfe search. DPre+ and SPre+, proposed by (Kasaura et al., 2023), adapt the vanilla DDPG (Lillicrap et al., 2016) and SAC (Haarnoja et al., 2018) to ACRL by using a QP-based projection step for action post-processing, learning the critic with pre-projected actions, and applying a penalty term to guide the policy updates. For a fair comparison, we use the official implementation and the hyperparameter settings of DPre+, SPre+, and NFWPO provided by (Kasaura et al., 2023). FlowPG enforces action constraints via a pre-trained Normalizing Flow model, and we use the official source code provided by (Brahmanage et al., 2023). For the testing of ARAM, we set  $\lambda = [0.9, 0.1]$  as the default input preference of the policy network  $\pi_{\phi}(\cdot|\lambda)$ . Moreover, during the testing of all the above algorithms, an auxiliary projection step is employed to guarantee that actions used for environment interaction always satisfy the action constraints.
+
+![](images/be31c8bb54472102a1cccff9e6af6deed0f5003aa0cdd4afcd2a279e8bdb8d46.jpg)
+
+![](images/32097dae911cdfe0be36a7a6c271e0b1afca68d2710bf89b0d755b742fb0a9fd.jpg)
+
+![](images/0fd5686758d845839ddae6bf65eeb9d212b579678214932920fe9a50c915b143.jpg)
+
+![](images/fc7ad53901e4e256dd013327905287a0cb1dffb68507b978512ac908a93836eb.jpg)  
+(c) Reacher
+
+![](images/32c12c064ff176e5f964400f80d6822863754987a130ee707151bfe3611df786.jpg)
+
+![](images/6b70571f0c123b98c948629a6629d1e1e29d91fa2c9af69df321f98c1e469ce7.jpg)  
+(a) Hopper  
+(e) Ant  
+Figure 3: Learning curves of ARAM and the benchmark methods across different environments in terms of the evaluation return.
+
+![](images/3c8c9d2f1ff37784825e85e6aebf7e14374a0849e05dd84ae41aeb1644217474.jpg)  
+(b) HopperVel  
+(f) NSFnet
+
+![](images/590d086f21f6a59ec8f774c70941f8bf692916f0a9c2c3962e2f1464003d6b85.jpg)  
+(g) BSS3z
+
+![](images/cdea9b99488cf95e575661b4b21a6165f1da7e22560a172293714b3d6740c6d6.jpg)  
+(d) HalfCheetah  
+(h) BSS5z
+
+Evaluation Domains. We evaluate the algorithms in various benchmark domains widely used in the ACRL literature (Lin et al., 2021; Kasaura et al., 2023; Brahmanage et al., 2023): (i) MuJoCo locomotion tasks (Todorov et al., 2012): These tasks involve training robots to achieve specified goals, such as running forward and controlling their speed within certain limits. (ii) Resource allocation for networked systems: These tasks involve properly allocating resource under capacity constraints, including NSFnet and Bike Sharing System (BSS) (Ghosh & Varakantham, 2017). For NSFnet, the learner needs to allocate packets of different flows to multiple communication links. The action constraints are induced by the per-link maximum total assigned packet arrival rate. We follow the configuration provided by (Lin et al., 2021) and use the open-source network simulator from PCCRL (Jay et al., 2019). For BSS, the environment consists of  $m$  bikes and  $n$  stations, each with a capacity limit of  $c$ . The learner needs to reallocate bikes to different stations based on the current situation. We follow the experimental scenario of (Lin et al., 2021) and evaluate our approach on two tasks: BSS3z with  $n = 3$  and  $m = 90$  and BSS5z with  $n = 5$  and  $m = 150$ . Their capacities are both set to 40. A detailed description about these tasks is provided in Appendix D.1.
+
+Performance Metrics. We evaluate the performance in the following aspects:
+
+- Training efficiency: We record the evaluation returns at different training stages, in terms of both the wall clock time and the environment steps. To ensure fair measurements of wall clock time, we run each algorithm independently using the same computing device. Moreover, we report the cumulative number of QP operations as an indicator of the training computational overhead.  
+- Valid action rate: At the testing phase, we evaluate the valid action rate by sampling 100 actions from the policy network at each step of an episode. This metric reflects how effectively each method enforces the action constraints throughout the evaluation phase.  
+- Per-action inference time: The efficiency of action inference reflects the design complexity, such as the need for generative models and QP operations, of each ACRL method. During evaluation, we measure the per-action inference time for 1 million actions. This inference time serves as a critical metric for ACRL deployment.
+
+# 5.1 EXPERIMENTAL RESULTS
+
+Our proposed method effectively reduces the use of costly QP operations, allowing us to address ACRL with a more lightweight framework. This subsection demonstrates the effectiveness of our
+
+method in training efficiency, valid action rate, and per-action inference time. Unless stated otherwise, all the results reported below are averaged over five random seeds.
+
+Does the proposed method outperform other ACRL benchmark methods in cumulative rewards? Figure 3 shows the evaluation reward versus wall clock time, and Table 2 and Figure 8 (in Appendix C) show the valid action rates. We observe that ARAM enjoys fast learning progress and high valid action rates simultaneously. On the other hand, due to the reliance on the projection layer, the projection-based methods like DPre+ and SPre+ have relatively very low valid action rates, which require a large number of QP operations and hence lead to a longer training time. Additionally, they perform poorly in resource allocation environments since the optimal solutions for these problems cannot be directly found through projection. Regarding NFWPO and FlowPG, these two methods can both effectively learn policies that satisfy the constraints but not necessarily achieve a high average return. Moreover, the results regarding sample efficiency, measured by the evaluation return versus environment steps, can be found in Appendix C.
+
+Table 2: Comparison of ARAM and the benchmark algorithms in terms of the valid action rate.  
+
+<table><tr><td>Environment</td><td>DPre+</td><td>SPre+</td><td>NFWPO</td><td>FlowPG</td><td>ARAM (Ours)</td></tr><tr><td>Hopper</td><td>0.97 ± 0.03</td><td>0.40 ± 0.01</td><td>0.99 ± 0.01</td><td>0.98 ± 0.01</td><td>0.99 ± 0.01</td></tr><tr><td>HopperVel</td><td>0.38 ± 0.30</td><td>0.60 ± 0.09</td><td>0.94 ± 0.03</td><td>0.71 ± 0.19</td><td>0.80 ± 0.09</td></tr><tr><td>Reacher</td><td>0.25 ± 0.16</td><td>0.93 ± 0.03</td><td>0.97 ± 0.02</td><td>0.95 ± 0.01</td><td>0.98 ± 0.01</td></tr><tr><td>HalfCheetah</td><td>0.35 ± 0.06</td><td>0.78 ± 0.19</td><td>0.97 ± 0.01</td><td>0.78 ± 0.17</td><td>0.78 ± 0.29</td></tr><tr><td>Ant</td><td>0.41 ± 0.38</td><td>0.29 ± 0.07</td><td>0.99 ± 0.01</td><td>0.83 ± 0.17</td><td>1.00 ± 0.00</td></tr><tr><td>NSFnet</td><td>0.00 ± 0.00</td><td>0.04 ± 0.01</td><td>0.92 ± 0.05</td><td>0.98 ± 0.01</td><td>0.94 ± 0.04</td></tr><tr><td>BSS3z</td><td>0.25 ± 0.02</td><td>0.28 ± 0.24</td><td>0.73 ± 0.18</td><td>0.59 ± 0.20</td><td>0.72 ± 0.24</td></tr><tr><td>BSS5z</td><td>0.13 ± 0.11</td><td>0.31 ± 0.16</td><td>0.84 ± 0.14</td><td>0.68 ± 0.17</td><td>0.77 ± 0.16</td></tr></table>
+
+Does our proposed method achieve a lower training and inference overhead? QP operations are known to be computationally costly and thereby account for a substantial fraction of training time in many ACRL methods. Figure 4 presents the log-scale plot of QP usage under different algorithms. It is evident that ARAM exhibits significantly lower QP computation compared to others. Due to its reduced dependency on QP operations, ARAM achieves higher training efficiency.
+
+Moreover, Table 3 shows the average per-action inference time during evaluation. ARAM benefits from computationally efficient action inference due to its almost QP-free design, as only a minimal subset of policy output actions that violate constraints requires the QP operator. By contrast, the projection-based methods like DPre+ and SPre+ and the flow-based FlowPG all suffer from much higher per-action inference time.
+
+Table 3: Comparison of ARAM and the benchmark algorithms in terms of the evaluation time. The evaluation times are calculated through one million evaluation steps.  
+
+<table><tr><td>Environment</td><td>DPre+ (s)</td><td>SPre+ (s)</td><td>NFWPO (s)</td><td>FlowPG (s)</td><td>ARAM (Ours) (s)</td></tr><tr><td>Reacher</td><td>71.47 ± 5.63</td><td>303.01 ± 60.81</td><td>67.43 ± 5.02</td><td>71.47 ± 3.12</td><td>44.72 ± 0.93</td></tr><tr><td>HalfCheetah</td><td>211.67 ± 48.70</td><td>253.53 ± 30.48</td><td>180.41 ± 10.57</td><td>210.63 ± 7.51</td><td>57.14 ± 8.13</td></tr><tr><td>Hopper</td><td>99.43 ± 6.71</td><td>175.43 ± 9.47</td><td>88.71 ± 11.41</td><td>103.12 ± 5.13</td><td>61.72 ± 2.15</td></tr><tr><td>HopperVel</td><td>177.31 ± 15.73</td><td>97.12 ± 7.95</td><td>82.63 ± 7.13</td><td>91.43 ± 5.17</td><td>63.81 ± 10.04</td></tr></table>
+
+Comparison: ARAM versus RL for constrained MDPs with projection. To make the comparison even more comprehensive, we also adapt FOCOPS (Zhang et al., 2020), which is a popular RL approach designed for constrained MDPs to address long-term discounted cost, to ACRL by adding a projection step. Table 4 and Table 5 present a comparison of ARAM and FOCOPS in the evaluation rewards and the valid action rates. These results suggest that ACRL requires fundamentally different solutions from RL for constrained MDPs.
+
+Ablation study on MORL. To investigate the benefits of using MORL, we perform an ablation study that compares the MORL implementation with a single-objective variant with a fixed preference (termed SOSAC below). From Figure 5, MORL can discover policies with both competitive final forward reward and favorable constraint satisfaction due to the implicit knowledge sharing across preferences during training. By contrast, SOSAC under a fixed preference fails to meet both criteria simultaneously, and this suggests that direct hyperparameter tuning can be rather ineffective. We also provide the learning curves in Figure 11 in Appendix C.
+
+# 6 CONCLUSION
+
+In this paper, we introduced ARAM, a novel framework designed to address ACRL problems by augmenting standard deep RL algorithms. By employing the acceptance-rejection method and an
+
+![](images/1737dd14f33ac8beecaa490e14d44b47e1984ad92be0fc2c483d33d45807be7a.jpg)  
+Figure 4: Cumulative number of QP operations of ARAM and the benchmark methods across different environments, with the y-axis on a logarithmic scale.
+
+![](images/250287d5a0d7df35480a620a96be231123360bf0cb9e970dc68e23d32c89e27c.jpg)  
+(a) HalfCheetah
+
+![](images/9e8ef9dbb4aab00af01cea833f312599842f2a2b93f92d5883df162e5248c53f.jpg)  
+(b) Ant  
+Figure 5: Ablation study of MORL. We plot the tuples of forward reward and valid action rate of ARAM and the three single-objective SAC variants with fixed preferences  $\lambda = [0.9, 0.1]$ ,  $[0.5, 0.5]$ , and  $[0.1, 0.9]$  in HalfCheetah and Ant. The five markers of each color refer to the results of the same algorithm over five distinct random seeds.
+
+Table 4: Comparison of ARAM and FO-COPS in terms of the evaluation return.  
+
+<table><tr><td>Environment</td><td>FOCOPS</td><td>ARAM (Ours)</td></tr><tr><td>Hopper (×103)</td><td>2.27 ± 0.41</td><td>3.07 ± 0.24</td></tr><tr><td>HopperVel (×102)</td><td>0.07 ± 3.13</td><td>6.49 ± 2.31</td></tr><tr><td>Reacher (×100)</td><td>-5.80 ± 1.14</td><td>-4.78 ± 0.33</td></tr><tr><td>HalfCheetah (×103)</td><td>6.08 ± 1.87</td><td>8.38 ± 1.11</td></tr><tr><td>Ant (×103)</td><td>3.06 ± 1.11</td><td>5.00 ± 0.32</td></tr><tr><td>NSFnet (×104)</td><td>0.68 ± 0.16</td><td>1.32 ± 0.08</td></tr><tr><td>BSS3z (×103)</td><td>-1.93 ± 0.20</td><td>-1.65 ± 0.04</td></tr><tr><td>BSS5z (×104)</td><td>-1.61 ± 0.05</td><td>-1.51 ± 0.02</td></tr></table>
+
+Table 5: Comparison of ARAM and FO-COPS in terms of the valid action rate.  
+
+<table><tr><td>Environment</td><td>FOCOPS</td><td>ARAM (Ours)</td></tr><tr><td>Hopper</td><td>0.22 ± 0.15</td><td>0.99 ± 0.01</td></tr><tr><td>HopperVel</td><td>0.32 ± 0.11</td><td>0.80 ± 0.09</td></tr><tr><td>Reacher</td><td>0.87 ± 0.10</td><td>0.98 ± 0.01</td></tr><tr><td>HalfCheetah</td><td>0.59 ± 0.13</td><td>0.78 ± 0.29</td></tr><tr><td>Ant</td><td>0.26 ± 0.09</td><td>1.00 ± 0.00</td></tr><tr><td>NSFnet</td><td>0.46 ± 0.28</td><td>0.94 ± 0.04</td></tr><tr><td>BSS3z</td><td>0.45 ± 0.19</td><td>0.72 ± 0.24</td></tr><tr><td>BSS5z</td><td>0.16 ± 0.13</td><td>0.77 ± 0.16</td></tr></table>
+
+augmented MDP, ARAM effectively reduces the need for costly QP operations and improves valid action rates. Our experimental results demonstrate that ARAM can simultaneously achieve faster learning progress and require significantly fewer QP operations than the existing ACRL methods.
+
+# ACKNOWLEDGMENTS
+
+This material is based upon work partially supported by National Science and Technology Council (NSTC), Taiwan under Contract No. NSTC 113-2628-E-A49-026, and the Higher Education Sprout Project of the National Yang Ming Chiao Tung University and Ministry of Education, Taiwan. Shao-Hua Sun was supported by the Yushan Fellow Program by the Ministry of Education, Taiwan. We also thank the National Center for High-performance Computing (NCHC) for providing computational and storage resources.
+
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+
+# APPENDICES
+
+# Table of Contents
+
+A Proof of Proposition 1 14  
+B Detailed Architecture of ARAM 15  
+C Additional Experimental Results 16
+
+C.1 Comparison: ARAM versus the benchmark algorithms in Terms of Sample Efficiency 16  
+C.2 Comparison: ARAM versus RL for constrained MDPs with projection. 17  
+C.3 Ablation Study: The learning curves of ARAM versus SOSAC 18  
+C.4 Sensitivity Analysis: Effect of Violation Penalty in ARAM 18  
+C.5 Sensitivity Analysis: Effect of Target Distribution in ARAM 19  
+C.6 Additional Experiment: Scalability of ARAM 19
+
+D Detailed Experimental Configurations 21
+
+D.1 Environments and Action Constraints 21  
+D.2 Hyperparameters of ARAM 21
+
+E Detailed introduction to the existing ACRL methods 21
+
+E.1 Action Projection 21  
+E.2 Frank-Wolfe Policy Optimization (FWPO) 22  
+E.3 FlowPG 22
+
+# A PROOF OF PROPOSITION 1
+
+For ease of exposition, we restate the proposition as follows.
+
+Proposition 1 (Equivalence in optimality). Let  $\pi^{*} \in \Pi_{\mathcal{C}}$  be an optimal policy among all the policies in  $\Pi_{\mathcal{C}}$  under the original action-constrained MDP  $\mathcal{M}$ . Then, for any  $\lambda \in \Lambda$ , the policy  $\pi^{*}$  remains an optimal policy among all the policies in  $\Pi_{\mathcal{A}}$  under the AUTO-MDP  $\tilde{\mathcal{M}}$ .
+
+Proof. Let us first fix a preference vector  $\lambda \in \Lambda$ . Recall that  $\lambda = [\lambda_r, \lambda_c]$ . Let  $\pi_{\lambda}^{*} \in \Pi_{\mathcal{A}}$  be a deterministic optimal policy of the AUTO-MDP under the preference vector  $\lambda$ . By the classic literature of MDPs, we know such a  $\pi_{\lambda}^{*}$  must exist. Then, we have
+
+$$
+\langle \boldsymbol {\lambda}, \boldsymbol {Q} (s, a; \pi_ {\boldsymbol {\lambda}} ^ {*}) \rangle \geq \langle \boldsymbol {\lambda}, \boldsymbol {Q} (s, a; \pi) \rangle , \tag {6}
+$$
+
+for all  $s \in S$ , for all  $a \in \mathcal{A}$ , and for all  $\pi \in \Pi_{\mathcal{A}}$ . In the sequel, we slightly abuse the notation and use  $\pi(s)$  to denote the deterministic action taken at state  $s$  by a deterministic policy  $\pi$ .
+
+To prove the proposition, we just need to show that such a  $\pi_{\lambda}^{*}$  must also be in the set of feasible policies  $\Pi_{\mathcal{C}}$ . We prove this by contradiction. Suppose  $\pi_{\lambda}^{*} \notin \Pi_{\mathcal{C}}$ . Then, there must exist a state  $\bar{s} \in S$  such that  $\pi_{\lambda}^{*}(\bar{s}) \notin \mathcal{C}(s)$ . Then, we know
+
+$$
+\begin{array}{l} \langle \boldsymbol {\lambda}, \boldsymbol {Q} (\bar {s}, \pi_ {\boldsymbol {\lambda}} ^ {*} (\bar {s}); \pi_ {\boldsymbol {\lambda}} ^ {*}) \rangle = \langle \boldsymbol {\lambda}, [ 0, - \mathcal {K} ] ^ {\top} + \gamma \boldsymbol {Q} (\bar {s}, \pi_ {\boldsymbol {\lambda}} ^ {*} (\bar {s}); \pi_ {\boldsymbol {\lambda}} ^ {*}) \rangle (7) \\ = \lambda_ {c} \cdot \frac {- \mathcal {K}}{1 - \gamma}, (8) \\ \end{array}
+$$
+
+where Equation (7) follows from the Bellman equation and the self-loop transitions of AUTO-MDP, and Equation (8) holds by recursively rolling out the self-loop transitions. Let us pick a feasible action  $a \in \mathcal{C}(s)$ . We know such an action  $a$  must exist as  $\mathcal{C}(s)$  is assumed non-empty. Then, we
+
+have
+
+$$
+\begin{array}{l} \langle \boldsymbol {\lambda}, \boldsymbol {Q} (\bar {s}, a; \pi_ {\boldsymbol {\lambda}} ^ {*}) \rangle = \langle \boldsymbol {\lambda}, [ r (\bar {s}, a), 0 ] ^ {\top} + \gamma \mathbb {E} _ {s ^ {\prime} \sim \tilde {\mathcal {P}} (\cdot | \bar {s}, a)} [ \boldsymbol {Q} (s ^ {\prime}, \pi_ {\boldsymbol {\lambda}} ^ {*} (s ^ {\prime}); \pi_ {\boldsymbol {\lambda}} ^ {*}) ] \rangle (9) \\ \geq \lambda_ {r} \cdot r (\bar {s}, a) + \gamma \lambda_ {c} \cdot \frac {- \mathcal {K}}{1 - \gamma} (10) \\ > \lambda_ {c} \cdot \frac {- \mathcal {K}}{1 - \gamma}, (11) \\ \end{array}
+$$
+
+where Equation (9) follows from the Bellman equation, Equation (10) holds by considering the worst-case reward and penalty, and Equation (11) holds due to the non-negativity of the reward function and that  $\gamma < 1$ . Therefore, by combining Equation (8) and Equation (11), we know  $Q_{\lambda}(\bar{s},a;\pi_{\lambda}^{*})\equiv \langle \lambda ,Q(\bar{s},a;\pi_{\lambda}^{*})\rangle >\langle \lambda ,Q(\bar{s},\pi_{\lambda}^{*}(\bar{s});\pi_{\lambda}^{*})\rangle \equiv Q_{\lambda}(\bar{s},\pi_{\lambda}^{*}(\bar{s});\pi_{\lambda}^{*})$ . Then, by the argument of the standard one-step greedy policy improvement, we know  $\pi_{\lambda}^{*}$  can be improved by taking action  $a$  at state  $\bar{s}$  instead. Hence,  $\pi_{\lambda}^{*}$  cannot be an optimal policy. This completes the proof.
+
+# B DETAILED ARCHITECTURE OF ARAM
+
+Our implementation is based on Q-Pensieve (Hung et al., 2023), Figure 6 illustrates the complete training process of ARAM. MORL aims to optimize policies under varying preferences, which poses challenges in sample efficiency. To address this, Q-Pensieve introduces a memory-sharing technique via the Q Replay Buffer, which stores past critic-network snapshots. This approach enables leveraging past experiences effectively, improving sample efficiency by allowing the agent to revisit and learn from previous interactions across different preference settings. By incorporating the Q Replay Buffer, ARAM facilitates efficient policy learning, allowing the agent to balance performance improvement and constraint satisfaction.
+
+![](images/7617beb730bf5c1dbc778fbba704d4958f6c8c2e8243bd5c2a67aa24c50390dd.jpg)  
+Figure 6: Overview of the training process of ARAM. When an action is sampled from the policy, if it belongs to the feasible set, the transition is stored in the Replay Buffer (blue). Otherwise, the corresponding penalty transition is stored in the Augmented Replay Buffer, and another action is resampled (red). These stored transitions are then used to update the policy and critic networks.
+
+# C ADDITIONAL EXPERIMENTAL RESULTS
+
+In this section, we compare ARAM with other baselines, including action-constrained RL and RL for Constrained MDPs. The learning curves demonstrate that ARAM achieves better sample efficiency. Additionally, we conduct an ablation study on the MORL framework, investigating how varying preferences within MORL impact the performance of ARAM.
+
+# C.1 COMPARISON: ARAM VERSUS THE BENCHMARK ALGORITHMS IN TERMS OF SAMPLE EFFICIENCY
+
+Does ARAM have better sample efficiency? Figure 7 shows the evaluation rewards, and Figure 8 displays the action acceptance rates. We can observe that ARAM achieves consistently the best reward performance while maintaining a low action violation rate. On the other hand, FlowPG and NFWPO can effectively control the actions within the feasible set, but their reward performance appears lower. In contrast, QP-based methods exhibit a significantly higher rate of action violations.
+
+![](images/384ae62085fbc68ddc9c59926a292550b6ab0dde46b08beb0110d54a08fda25c.jpg)  
+Figure 7: Learning curves of ARAM and the benchmark algorithms across different environments in terms of the evaluation return.
+
+![](images/cbcb373be6981daf08dfa7500ade075730f4356ddd20c3f80be67dc35508e61a.jpg)
+
+![](images/23e38dd914be933b498344545957981dc0310f51e7ae7c28283b2ec774922478.jpg)
+
+![](images/d65a53aeaee23ced54b69147f4bdd34e52efd1dc35dc25f88f2daa4bbfc60772.jpg)
+
+![](images/77181dfa641f279f010e6b6a812ba4345eff8f17f6f6c2d1214b6a80f6e54575.jpg)
+
+![](images/927df6033b1b47b8e3500cb2541fe68e985bd501760045628cfb16692f5bfb4c.jpg)
+
+![](images/332a588df3e20d866fd096c1265b442627b6a9228e3589d67c9b45c917e432b0.jpg)  
+(a) Hopper  
+(e) Ant
+
+![](images/8aaee61b4331276b11b3ba5c4ee9506e8bb2c942848fc9015865951d2584c917.jpg)  
+(b) HopperVel  
+(f) NSFnet
+
+![](images/d60faf0236d777f389f0a9266206155dd1677c972f7b28f4e20f8551033f92a0.jpg)  
+(c) Reacher  
+(g) BSS3z
+
+![](images/1011b741196df13eab0aff13d55af409fcc2940d13dae4ff05c82f92a3ecb735.jpg)  
+(d) HalfCheetah  
+(h) BSS5z
+
+![](images/ddb006dbfa85f9d251f9ea0139c6d839019cb69de382dc02b33ac9d724ed3007.jpg)  
+Figure 8: Learning curves of ARAM and the benchmark algorithms across different environments in terms of the valid action rate.
+
+![](images/eb66c159e12238200c0452a70c27678051042713ddfdb23a6b384ffd947345d1.jpg)
+
+![](images/8e1d27ccc4d28fc5d6d3211513a595940ee07b61bdbf7fe5b4e5af662b0f90f2.jpg)
+
+![](images/9ff9ed4b5903ed7ffa8b08afc0207c03fe19193c41f4085bbf1cb7e91dabad64.jpg)
+
+![](images/bce8ca2600419ceed195d6893ddd56636e5054c0ca28a40c46959df4e9060dbd.jpg)
+
+![](images/274e9f3c50f1c1a998b7f7e82eb9b4f564643fa32154b5954e26ec36c4853c76.jpg)  
+(a) Hopper  
+(e) Ant  
+Figure 9: Comparison of the evaluation return of ARAM and FOCOPS across various environments.
+
+![](images/4494cb4ef018464655359659f1de052bab717babf8dcb688f27c63a8583818cf.jpg)  
+(b) HopperVel  
+(f) NSFnet
+
+![](images/5b02bc0868455f73b72c811e204db858fa37d948ad62e0d6ba32210f5fbe0e29.jpg)  
+(c) Reacher  
+(g) BSS3z
+
+![](images/6fb6adfb34ca160b7f676aa1e841878a5d6eb9be4cc2c2fb8e47b604b384d010.jpg)  
+(d) HalfCheetah  
+(h) BSS5z
+
+# C.2 COMPARISON: ARAM VERSUS RL FOR CONSTRAINED MDPS WITH PROJECTION.
+
+Does ARAM more effectively reduce the number of action violations while achieving better performance? Figure 9 and Figure 10 present the training curves of total return and valid action rate, respectively. From these curves, we observe that ARAM consistently outperforms FOCOPS in both the evaluation return and the valid action rate, indicating its superior ability to handle action constraints while optimizing performance.
+
+![](images/b72e4c7b4edcb3940df03c9501e783ad8994fceae43fe999b46dffc94458c9c7.jpg)
+
+![](images/4f36c46690ce7b09d21a9aa5660dc989999fdc861cf8bd1948ec751f77ae9f4e.jpg)
+
+![](images/d08919c057b5a300344f52495b474e21bef8647e78ec0fef8c0764a3841dfc11.jpg)
+
+![](images/a5ee0b6029102684961920f79421a9f7908a334a9157a814dc4c71503168c7b8.jpg)
+
+![](images/a7767947ccd488f2dfa8036d909abfbfe630884ed0fea75d64eed05c3ce689a3.jpg)
+
+![](images/742955b8faecd7fde046a380bde9486719d36307e6c3f2c8c90ae75c20e0c3c0.jpg)  
+(a) Hopper  
+(e) Ant  
+Figure 10: Comparison of the valid action rate of ARAM and FOCOPS across various environments.
+
+![](images/d4f2ce406d8ba0fad5e22f4d34fe7683e8e93689e68f4465744da0b8e003065f.jpg)  
+(b) HopperVel  
+(f) NSFnet
+
+![](images/21e430aefccd5203d21e2355e8138fa928dae5b1aa9bde420bc75a364907b64b.jpg)  
+(c) Reacher  
+(g) BSS3z
+
+![](images/85d8753904995a29392253168f922809939f97f162349f15d8b8b2b0390a5a0e.jpg)  
+(d) HalfCheetah  
+(h) BSS5z
+
+# C.3 ABLATION STUDY: THE LEARNING CURVES OF ARAM VERSUS SOSAC
+
+We present the learning curves of the total return and the valid action rate in Figure 11. We can observe that the ARAM consistently achieves higher returns compared to the SOSAC variant with fixed preferences.
+
+SOSAC with preference  $= [0.9, 0.1]$  
+SOSAC with preference  $= [0.5, 0.5]$
+
+SOSAC with preference  $= [0.1, 0.9]$  
+ARAM (Ours)
+
+![](images/6950cfa42c32a75ab6582ed96ec93e6dc9520f29b1ddb8d7f8b7b59d18a463c3.jpg)  
+(a) HalfCheetah  
+Figure 11: Ablation Study on MORL. These figures present the learning curves of ARAM and three single-objective SAC variants with different fixed preferences in terms of the evaluation return (Figures 11a and 11b) and the valid action rate (Figures 11c and 11d) in the HopperVel and Ant environments.
+
+![](images/b2b5fb4c5e5de44e3b66de03078380f74a166654203fc9a0015f919fff274dbf.jpg)  
+(b) Ant
+
+![](images/e81ae67aa681515259db0dd32891ad9a5fe429029364f65c4bc2bc8587ed9153.jpg)  
+(c) HalfCheetah
+
+![](images/950c81c1b42de0a311e4701bfe798721679a9429ef9c2b61b682528f6a3471aa.jpg)  
+(d) Ant
+
+# C.4 SENSITIVITY ANALYSIS: EFFECT OF VIOLATION PENALTY IN ARAM
+
+We present the results of a hyperparameter sensitivity study on the violation penalty  $\mathcal{K}$  in the HopperVel and Ant environments, as shown in Figure 12. This study examines how different fixed values
+
+of  $\mathcal{K}$  impact the return and valid action rate during training. The findings demonstrate that ARAM maintains stable performance across varying  $\mathcal{K}$  values, indicating that the model is largely insensitive to this hyperparameter. This supports our original experimental setup, where a single  $\mathcal{K}$  value was used consistently across all tasks and environments without requiring task-specific tuning.
+
+![](images/d6d53c9f1d76f4425b790f39cc8714ea0c6ce288ce2c1c675c28f288eaa22455.jpg)
+
+![](images/ccfe028c00f985fb12711ebb18bbeee37a7308cfc7d36008eb42ce60d3ce2e9f.jpg)  
+(a) HopperVel
+
+![](images/e234312e2d5979891de30d51b09868504eb23e4178c78887b10717db59a00d2f.jpg)  
+(b) Ant
+
+![](images/e8f67fce5d242c31e418a205f68a70e298f57b9c341de3e7e12fbd83d0ea2996.jpg)  
+(c) HopperVel  
+Figure 12: Sensitivity Analysis of the Violation Penalty  $\mathcal{K}$ . These figures provide a comparison of the return and the valid action rate across training steps for different values of  $\mathcal{K}$  ( $\mathcal{K} = 0.05, 0.1, 0.2$ ) in the HopperVel and Ant environments. These figures show that ARAM is not highly sensitive to the penalty weight.
+
+![](images/25384fb9f3359fbce43fadd23cb79180c90859fd58ce571198fed1d1a1985c2e.jpg)  
+(d) Ant
+
+# C.5 SENSITIVITY ANALYSIS: EFFECT OF TARGET DISTRIBUTION IN ARAM
+
+We analyze the impact of different target distributions in ARAM, specifically examining whether this choice significantly affects the final performance. Since ARAM is implemented based on the MOSAC framework, the policy network outputs the mean and standard deviation of a Gaussian distribution, denoted as:  $\pi_{\phi}(a|s) \sim \mathcal{N}(\mu, \sigma^2)$ . Following the notation in the paper, we configure an alternative (constrained) target distribution,  $\pi^{\dagger}(a|s)$ , by leveraging the (unconstrained) Student-t distribution:  $\tilde{\pi}_{\phi}(a|s) \sim T_{\nu}(\mu, \sigma^2)$  where  $T_{\nu}$  denotes a Student-t distribution with degrees of freedom  $\nu$ , mean  $\mu$ , and variance  $\sigma^2$ . The constrained target distribution is then defined as:
+
+-  $\pi_{\phi}^{\dagger}(a|s) \propto \tilde{\pi}_{\phi}(a|s)$  for  $a \in \mathcal{C}(s)$ .  
+-  $\pi_{\phi}^{\dagger}(a|s) = 0$  for  $a \notin \mathcal{C}(s)$ .
+
+Notably, the Gaussian distribution is a special case of the Student-t distribution with  $\nu = \infty$ . The Student-t distribution has heavier tails compared to the Gaussian, potentially leading to a more exploratory action distribution. For ARM, the scaling factor  $M$  can be set as:  $M = M_0 / \left(\int_{a \in \mathcal{C}(s)} \tilde{\pi}_{\phi}(a|s) da\right)$ , where  $M_0$  is a constant dependent on  $\nu$ . The return and valid action rate is shown on Figure 13. We observe that ARAM can have a slight performance improvement under this new target distribution (especially at the early training stage) and is not very sensitive to this choice. This also corroborates the nice flexibility in the choice of target distribution.
+
+# C.6 ADDITIONAL EXPERIMENT: SCALABILITY OF ARAM
+
+To demonstrate the scalability of ARAM to higher-dimensional environments, we extend the NSFnet environment to a 20-dimensional action space called NSFnet20d. Specifically, the agent manages packet delivery across a classic T3 NSFNET network backbone by simultaneously handling 20 packet flows, each with distinct routing paths. The action is defined as the rate allocation of each flow along each candidate path. In total, there are 30 communication links shared by these 20 different flows. The action constraints ensure that the distribution of packets on these shared links stays within the bandwidth capacity, defined as 50 units for each link. Figure 14 presents the results, illustrating ARAM's superior performance in the evaluation return, the valid action rate, and the QP efficiency compared to ACRL benchmark methods. ARAM consistently achieves a higher action
+
+![](images/f67cb961a8d2ae928e18713b610d0f339c2d18e69dbb5b1fcfad3688a164aa53.jpg)  
+ARAM with Student's t-distribution ARAM with Normal distribution
+
+![](images/3a76c3b80fa0970138578e8cf15c83daeffe39b5f4e35fe62f24c6414c0fe502.jpg)  
+Figure 13: Sensitivity Analysis of the Target Distribution in ARAM. These figures provide a comparison of the return and the valid action rate across training steps for different target distributions: Student's t-distribution and the Normal distribution. These figures show that ARAM is not highly sensitive to the choice of target distribution.
+
+validity rate while maintaining robust learning under high-dimensional action spaces and complex constraints, demonstrating its adaptability and efficiency.
+
+![](images/2f7100280ecc55ba3743409b2b7f24c9ec40d4306f8eca86733e81bdba4ddcc3.jpg)
+
+![](images/7cce79bb03fc00ef6062677e2775c0fe013afe24f40e5a5539fb4cc341dce29f.jpg)  
+Figure 14: Comparison of the evaluation return, the valid action rate, and the number of QP operations in the NSFnet20d environment.
+
+![](images/118e95b27c7170f4cdbf8e95427aa02fa1ba50b084cc127139b3b361bfac90fe.jpg)
+
+![](images/d79ec98c8bc008995aaf2dbc59ebfdeb742c8cffc8ed13b745a3a832bd31042c.jpg)
+
+# D DETAILED EXPERIMENTAL CONFIGURATIONS
+
+# D.1 ENVIRONMENTS AND ACTION CONSTRAINTS
+
+In this section, we provide the details about the experimental setups and the action constraints considered in different environments.
+
+- MuJoCo: The action is defined as a vector  $(a_{1},a_{2},\dots ,a_{k})$ , with each element representing the torque applied to a specific joint, and  $(w_{1},w_{2},\dots ,w_{k})$  denotes the angular velocities of the joints. The tasks considered in our experiments include:
+
+- Reacher, HalfCheetah, Hopper, and Ant: We use the experimental settings and objectives provided by OpenAI Gym V3 to control the agents in these environments. Each environment presents unique challenges and objectives, such as reaching a target location, maintaining balance, or achieving high-speed movement.  
+- HopperVel: This task is specifically designed to control the robot to maintain a target horizontal velocity of  $3\mathrm{m/s}$ , differing from the standard Hopper's goal of forward hopping.
+
+- NSFnet: Based on the T3 NSFNET Backbone as discussed in (Lin et al., 2021), this network consists of 9 different packet flows, each with distinct routing paths. There are 8 communication links shared by different flows. The action is defined as the rate allocation of each flow along each candidate path. The action constraint is to check if the distribution of packets on these shared links stays within the bandwidth limits, defined as 50 units for each link. To define action constraints, we described the eight-tuple  $(\mathrm{link}_1,\dots,\mathrm{link}_8)$ , each containing the total amount of flow that pass through that specific link.  
+- BSS: The action is to allocate bikes to stations under random demands.
+
+- BSS3z: There are 3 stations with a total of 90 bikes ( $m = 90$ ,  $n = 3$ ), and each station has a capacity of 40 bikes.  
+- BSS5z: This system comprises 5 stations with a total of 150 bikes ( $m = 150$ ,  $n = 5$ ), and each station also has a capacity of 40 bikes.
+
+Table 6: This table provides an overview of the action constraints applied in each experiment environment.  
+
+<table><tr><td>Environment</td><td>Action Constraint</td></tr><tr><td>HopperVel</td><td>∑i=13max(wia,0) ≤ 10</td></tr><tr><td>Hopper</td><td>∑i=13max(wa,0) ≤ 10</td></tr><tr><td>Reacher</td><td>a12+a2≤ 0.05</td></tr><tr><td>HalfCheetah</td><td>∑i=16|wa| ≤ 20</td></tr><tr><td>Ant</td><td>∑i=18ai≤ 2</td></tr><tr><td>NSFnet</td><td>∑i∈linkjia≤ 50, ∀j ∈ {1,2,...,8}</td></tr><tr><td>BSS3z</td><td>|∑i=13a i - 90| ≤ 5, ai &lt;= 40</td></tr><tr><td>BSS5z</td><td>|∑i=15a i - 150| ≤ 5, ai &lt;= 40</td></tr></table>
+
+# D.2 HYPERPARAMETERS OF ARAM
+
+We conduct all the experiments with the following hyperparameters.
+
+# E DETAILED INTRODUCTION TO THE EXISTING ACRL METHODS
+
+# E.1 ACTION PROJECTION
+
+An action produced by the neural network is not guaranteed to remain within the feasible action set. To address this, there are various frameworks to map the output action to one that satisfies the
+
+Table 7: This table provides an overview of the hyperparameters used in ARAM.  
+
+<table><tr><td>Parameter</td><td>ARAM</td></tr><tr><td>Optimizer</td><td>Adam</td></tr><tr><td>Learning Rate</td><td>0.0003</td></tr><tr><td>Discount Factor</td><td>0.99</td></tr><tr><td>Replay Buffer Size</td><td>1000000</td></tr><tr><td>Number of Hidden Units per Layer</td><td>[256, 256]</td></tr><tr><td>Number of Samples per Minibatch</td><td>256</td></tr><tr><td>Nonlinearity</td><td>ReLU</td></tr><tr><td>Target Smoothing Coefficient</td><td>0.005</td></tr><tr><td>Target Update Interval</td><td>1</td></tr><tr><td>Gradient Steps</td><td>1</td></tr><tr><td>Sample Ratio for Augmented Replay Buffer (η)</td><td>0.2</td></tr><tr><td>Decay Interval for η</td><td>10,000</td></tr><tr><td>Decay Factor for η</td><td>0.9</td></tr></table>
+
+Table 8: This table provides an overview of the hyperparameters used in the baseline methods.  
+
+<table><tr><td>Parameter</td><td>DPre+</td><td>SPre+</td><td>NFWPO</td><td>FlowPG</td></tr><tr><td>Learning Rate</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td></tr><tr><td>FW Learning Rate</td><td>-</td><td>-</td><td>0.01</td><td>-</td></tr><tr><td>Discount Factor</td><td>0.99</td><td>0.99</td><td>0.99</td><td>0.99</td></tr><tr><td>Replay Buffer Size</td><td>1000000</td><td>1000000</td><td>1000000</td><td>1000000</td></tr><tr><td>Number of Hidden Units</td><td>[256, 256]</td><td>[256, 256]</td><td>[256, 256]</td><td>[400, 300]</td></tr><tr><td>Number of Samples per Minibatch</td><td>256</td><td>256</td><td>256</td><td>100</td></tr><tr><td>Target Smoothing Coefficient</td><td>0.005</td><td>0.005</td><td>0.005</td><td>0.005</td></tr><tr><td>Target Update Interval</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Gradient Steps</td><td>1</td><td>1</td><td>1</td><td>1</td></tr></table>
+
+constraints. An intuitive approach is through a QP operation to find the closest feasible action to the original one within the acceptable action space. We refer to this as the QP-Solver:
+
+$$
+\operatorname {Q P - S o l v e r} (s, a, \mathcal {C} (s)) = \arg \min  _ {a ^ {\prime} \in \mathcal {C} (s)} \| a ^ {\prime} - a \| _ {2}. \tag {12}
+$$
+
+Kasaura et al. (2023) describe a family of algorithms based on the QP-Solver. However, as the complexity of the action space increases, the computation time for this type of approach can become lengthy. To ensure the feasibility of policy learning within a reasonable timeframe, it is essential to minimize reliance on the QP-Solver.
+
+# E.2 FRANK-WOLFE POLICY OPTIMIZATION (FWPO)
+
+Aside from the QP-Solver, Lin et al. (2021) employ the Frank-Wolfe (FW) method to update the policy and propose Frank-Wolfe Policy Optimization (FWPO). To describe the different policy parameters, FWPO uses  $\phi_{k}$  to denote the policy parameters in the  $k$ -th iteration. To update the policy with the action constraints, FWPO adopts a generalized policy iteration framework (Sutton, 2018), which consists of two subroutines:
+
+(i) Policy update via state-wise FW: For each state  $s$  and the corresponding learning rate  $\alpha_{k}(s)$ , search for feasible action, then guide the current policy parameters update.
+
+$$
+c _ {k} (s) = \arg \max  _ {c \in \mathcal {C} (s)} \left\langle c, \nabla_ {a} Q _ {\theta} (s, a; \pi_ {\phi_ {k}}) \right| _ {a = \pi_ {\phi_ {k}} (s)} \rangle , \tag {13}
+$$
+
+$$
+\pi_ {\phi_ {k + 1}} (s) \leftarrow \pi_ {\phi_ {k}} (s) + \alpha_ {k} (s) \left(c _ {k} (s) - \pi_ {\phi_ {k}} (s)\right), \tag {14}
+$$
+
+(ii) Evaluation of the current policy: Use any policy evaluation algorithm to obtain  $Q_{\theta}(s,a;\pi_{\phi_{k + 1}})$ . Where  $c_{k}(s) - \pi_{\phi_{k}}(s)$  is the update direction.
+
+# E.3 FLOWPG
+
+Brahmanage et al. (2023) adopt normalizing to create an invertible mapping between the support of a simple distribution and the space of valid actions. Specifically, the focus is on the conditional
+
+RealNVP model, which is well-suited for the general ACRL setting where the set of valid actions depends on the state variable. The conditional RealNVP extends the original RealNVP by incorporating a conditioning variable in both the prior distribution and the transformation functions. These transformation functions, implemented as affine coupling layers, enable efficient forward and backward propagation during model learning and sample generation.
+
+Algorithm 2: FlowPG Algorithm  
+Input: Initial parameter vectors  $\theta, \phi, \psi$   
+1 Initialize replay buffer  $\mathcal{B}$ ;  
+2 for episode  $= 1$ , ...,  $M$  do  
+3 Initialize the random noise generator  $\mathcal{N}$  for action exploration;  
+4 for  $t = 1, \dots, T$  do  
+5 Select action  $\tilde{a}_t = \pi_\phi(s_t) + \mathcal{N}_t$  based on current policy and exploration noise;  
+6 Apply flow and get the environment action  $a_t = f_\psi(\tilde{a}_t, s_t)$ ;  
+7 if  $a_t$  is invalid then  
+8  $\left\lfloor a_t \gets \mathrm{QP-Solver}(s_t, a_t, \mathcal{C}(s_t))\right\rfloor$ ;  
+9 Update actor policy using the sampled policy gradient:  
+ $\nabla_\phi J(\pi_\phi) = \nabla_a Q_\theta(s_t, a; \pi) \nabla_{\tilde{a}} f_\psi(\tilde{a}, s_t) \nabla_\theta \pi_\phi(s_t) \big|_{\tilde{a} = \pi_\phi(s_t), a = f_\psi(\tilde{a}, s_t)}$
+
+Learning the RL-Model. Integrating DDPG with normalizing flows involves using the learned mapping directly in the original policy network, which improves training speed and stability. The architecture of the policy network consists of the original DDPG policy network and the learned mapping function  $f_{\psi}$ . The objective is to learn a deterministic policy  $f_{\psi}(\pi_{\phi}(s), s)$  that gives the action  $a$  given a state  $s$ , maximizing  $J(\pi_{\phi})$ :
+
+$$
+\max  _ {\phi} J \left(\pi_ {\phi}\right) = \mathbb {E} _ {s \sim \mathcal {B}} \left[ Q _ {\theta} (s, f _ {\psi} \left(\pi_ {\phi} (s), s\right); \pi_ {\phi}) \right] \tag {15}
+$$
+
+where  $\mathcal{B}$  is the replay buffer and  $\phi$  represents the Q-function parameters, treated as constants during policy update. The policy update involves gradient ascent with respect to the policy network parameters  $\phi$ :
+
+$$
+\nabla_ {\phi} J (\pi_ {\phi}) = \mathbb {E} _ {s \sim \mathcal {B}} [ \nabla_ {a} Q _ {\theta} (s, a; \pi_ {\phi}) \nabla_ {\tilde {a}} f _ {\psi} (\tilde {a}, s) \nabla_ {\phi} \pi_ {\phi} (s) | _ {\tilde {a} = \pi_ {\phi} (s), a = f _ {\psi} (\tilde {a}, s)} ] \tag {16}
+$$
+
+The critic update follows the same rule as in DDPG, with actions stored in the replay buffer being either the flow model output or projected actions. The proposed method can be extended to other RL algorithms like SAC or PPO, as normalizing flows enable the computation of log probabilities of actions required during training. The pseudo code of FlowPG is provided in Algorithm 2.

2025/Efficient Action-Constrained Reinforcement Learning via Acceptance-Rejection Method and Augmented MDPs/images.zip

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2025/Efficient Active Imitation Learning with Random Network Distillation/1184aaa1-3648-4f7c-8f1e-ed6da5eaf914_origin.pdf

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2025/Efficient Active Imitation Learning with Random Network Distillation/full.md

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+# EFFICIENT ACTIVE IMITATION LEARNING WITH RANDOM NETWORK DISTILLATION
+
+Emilien Biré<sup>1*</sup>, Anthony Kobanda<sup>2</sup>, Ludovic Denoyer<sup>3</sup>, Rémy Portelas<sup>2</sup>
+
+<sup>1</sup>Centrale Supelec <sup>2</sup>Ubisoft La Forge <sup>3</sup>H Company
+
+remy.portelas@ubisoft.com
+
+# ABSTRACT
+
+Developing agents for complex and underspecified tasks, where no clear objective exists, remains challenging but offers many opportunities. This is especially true in video games, where simulated players (bots) need to play realistically, and there is no clear reward to evaluate them. While imitation learning has shown promise in such domains, these methods often fail when agents encounter out-of-distribution scenarios during deployment. Expanding the training dataset is a common solution, but it becomes impractical or costly when relying on human demonstrations. This article addresses active imitation learning, aiming to trigger expert intervention only when necessary, reducing the need for constant expert input along training. We introduce Random Network Distillation DAGger (RND-DAgger), a new active imitation learning method that limits expert querying by using a learned state-based out-of-distribution measure to trigger interventions. This approach avoids frequent expert-agent action comparisons, thus making the expert intervene only when it is useful. We evaluate RND-DAgger against traditional imitation learning and other active approaches in 3D video games (racing and third-person navigation) and in a robotic locomotion task and show that RND-DAgger surpasses previous methods by reducing expert queries. https://sites.google.com/view/rnd-dagger
+
+# 1 INTRODUCTION
+
+Imitation learning has increasingly become a favored approach for learning behaviors in complex environments, offering a compelling alternative to classical scripted behaviors implemented by domain specialists (Schaal, 1999; Hussein et al., 2017). It is particularly well suited in problems where there is not a clear performance measure (or reward). In video games, it is becoming more and more familiar to game developers that frequently address the problem of implementing bots in their games which must play in realistic ways (Harmer et al., 2018; Yadgaroff et al., 2024; Mao et al., 2024). Indeed, since the notion of realism is not well specified, it prevents the use of reinforcement learning-based approaches where a reward signal is mandatory. By observing and replicating human players, these bots are trained to execute complex strategies and actions that are both efficient and human-like.
+
+Imitation learning usually proceeds in two steps: first, a dataset of behaviors is built by leveraging experts interacting with the dynamical system. Then, a statistical model (e.g. a neural network) is learned to imitate actions in that dataset, thus expecting this model to generalize to unseen state and to behave like the experts. Consider a use case where the goal is to train a driving policy in a video game context capable of controlling a car on a track (see figure 1). In this scenario, the state is defined by sensor values at time  $t$ , which can include information such as the position of the car, its speed, and raycasts. In this context, imitation Learning involves manually controlling the car for many laps to create a learning dataset, training a policy on this dataset using traditional algorithms like Behavioral Cloning (Bain & Sommut, 1999), and expecting that the resulting bot will control the car correctly. However, this approach is known to be unreliable, particularly due to the risk of a shift in the state distribution between training and testing (Ross et al., 2011), known as the problem of covariate shift (Nair et al., 2019). When covariate shift occurs, the agent struggles to determine the correct action, leading to compounding errors and ultimately undermining its performance.
+
+![](images/418eb7a76369a1ba012dc4ba6e016f85c5c8dd73dc0413df5e47618c416b95cd.jpg)  
+Figure 1: RND-DAgger overview. (a) The learner's policy controls the agent until (b) our Random Network Distillation-based out-of-distribution (OOD) measure is triggered. Then (c) the expert takes control until the OOD measure gets lower than the threshold for at least W steps. (d) Finally, the current policy takes control of the agent to continue the episode, and can trigger the expert again later.
+
+Facing this issue, the effectiveness of imitation learning hinges on the availability of extensive datasets comprised of player behaviors, often necessitating thousands of expert traces to achieve reasonable performance and exhibit credible behaviors (Vinyals et al., 2019; Mao et al., 2024). This dependency on large volumes of data poses a challenge, particularly in scenarios where collecting such data is either impractical or costly.
+
+In response to these limitations, some approaches involve human-in-the-loop training framework, often referred to as active imitation learning (Ross et al., 2011; Judah et al., 2012; Menda et al., 2019; Hoque et al., 2021). Instead of gathering very large amounts of demonstrations, these methods focus on the strategic incorporation of human feedback to selectively guide the learning process, thereby enhancing the efficiency of the training phase. By prioritizing the acquisition of relevant and impactful expert traces, these approaches seek to expedite the agent's learning curve without relying on expansive datasets, and swiftly improve performance. In the aforementioned video game example, this corresponds to collecting a first set of player traces to bootstrap the agent using classical imitation learning, and then to gather additional traces in particular race settings during training, i.e. requiring a player to control the car in particular and relevant situations.
+
+To tackle these challenges, several recent models have been proposed to identify when the expert's intervention is necessary to correct the agent's suboptimal behavior (Zhang & Cho, 2017; Menda et al., 2019; Kelly et al., 2019; Hoque et al., 2021), with DAgger (Ross et al., 2011) being one of the most well-known approaches. The primary objective of these methods is to minimize the time required for human feedback by concentrating interventions on the most critical situations where the agent is likely to make errors. This allows for a more efficient use of expert time, ensuring that corrective feedback is provided only when it is essential. But, as explained in Section 2, many of these methods are not fully satisfying.
+
+Despite these advancements, designing algorithms that effectively incorporate human feedback while reducing the overall expert effort remains an open research problem. Addressing this issue is the central focus of this article, as we aim to develop methods that optimize the interaction between the expert and the learning agent, thereby improving learning efficiency without overburdening the human supervisor.
+
+Our main contributions are threefold: i) We propose a new method called RND-DAgger, a novel interactive imitation learning approach leveraging state-based out-of-distribution identification through random network distillation. ii) We perform a comparative analysis of RND-DAgger and existing methods on 3 tasks: a robotics scenario and two video-game environments. iii) Throughout these experiments, we demonstrate that RND-DAgger either outperforms or matches existing approaches in terms of final performance while significantly reducing expert burden.
+
+Algorithm 1 DAgger  
+Require:  $K,\{\beta_i\}_{i\in [1,K]},T,\pi_{exp}$  1: Instantiate  $D$  dataset of expert trajectories 2: Instantiate  $\pi_0$  BC policy trained on  $D$  3:  $t\gets 0$  4: for  $i = 0,\dots ,K$  DAgger iterations do 5:  $D_{i}\gets \emptyset$  6: for  $T$  sampling steps do 7:  $s_t\sim Env(a_{t - 1})$  (Get a new state) 8:  $a_{t}\gets \beta_{i}\pi_{exp}(s_{t}) + (1 - \beta_{i})\pi_{i}(s_{t})$  9:  $D_{i}\gets D_{i}\cup \{(a_{t},\pi_{exp}(s_{t}))\}$  10:  $t\gets t + 1$  11: Push  $D\gets D\cup D_{i}$  12: Train  $\pi_{i + 1}$  on  $D$  using BC 13: return  $\pi_K$
+
+Figure 2: Above, the DAgger algorithm where the expert action is added to the training set at each timestep, the agent being alternatively controlled by the current policy and the expert one. On the right, in Lazy/Ensemble DAgger, the agent is controlled by the expert policy only if a given measure is higher than a threshold  $\lambda$ . Only actions generated when the expert controls the agent are added to the training set.
+
+Algorithm 2 Lazy/Ensemble DAgger  
+Require:  $K$  CONDITION,  $T$ $\pi_{exp}$    
+1: Instantiate  $D$  dataset of expert trajectories   
+2: Instantiate  $\pi_0$  BC policy trained on  $D$    
+3: nswitch  $\leftarrow 0$    
+4:  $t\gets 0$    
+5:  $D_{i}\gets \emptyset$    
+6: for  $i = 0,\dots ,K$  iterations do   
+7: while #Di  $<  T$  do   
+8:  $s_t\sim Env(a_{t - 1})$  Get a new state)   
+9:  $C_t\gets$  CONDITION  $(\pi_{exp},\pi_i,s_t)$  if  $C_t$  then   
+11:  $a_{t}\gets \pi_{exp}(s_{t})$    
+12:  $D_{i}\gets D_{i}\cup \{(a_{t},s_{t})\}$    
+13: t+1 if not  $C_{t - 1}$  then   
+15: nswitch  $\leftarrow$  nswitch +1   
+16: else   
+17:  $a_{t}\gets \pi_{i}(s_{t})$    
+18: t+1   
+19:  $D\gets D\cup D_{i}$    
+20: Train  $\pi_{i + 1}$  on  $D$  using BC   
+21: return  $\pi_K$
+
+# 2 PRELIMINARIES AND RELATED MODELS
+
+Notations Let us consider a Markov Decision Process described by a state space  $S$  and an action space  $\mathcal{A}$ . The dynamics of the process are defined by an unknown transition function  $P(s_{t+1}|a_t, s_t)$ , where  $a_t \in \mathcal{A}$  and  $s_t \in S$ . Note that we do not consider any reward in this setting since the task to solve is not explicit (e.g., driving a car); our objective is instead to imitate human behaviors that may be suboptimal and diverse (e.g., considering multiple players or experts). An episode  $\tau$  consists of a sequence of states and actions,  $\tau = (s_1, a_1, \dots, s_T, a_T)$ , where  $T$  is the length of the episode. Given a policy  $\pi(a_t|s_t)$ , it is possible to sample an episode by sequentially executing the policy until reaching a stop criterion.
+
+Distributional shift: Given a dataset of expert demonstrations  $\mathcal{D} = (\tau_1, \dots, \tau_n)$ , a common approach to policy learning is Behavioral Cloning (BC) (Bain & Sammut, 1995; Ding et al., 2019a), a straightforward imitation learning algorithm. BC enables to learn policies by maximizing the log-likelihood of the expert actions within the dataset. However, learning a policy from a fixed dataset can result in the policy encountering out-of-distribution (OOD) states during inference, where it has not been trained to make accurate decisions (Ross et al., 2011). This phenomenon is referred to as distributional shift. This shift in the distribution can cause the policy to perform poorly, as it may not know how to handle unseen situations. For example, consider a driving policy in a racing game. If the dataset is composed only of demonstrations from expert drivers, the learned policy might struggle to recover from rare events such as a spin-out, since such situations are absent from the expert demonstrations. Therefore, the policy may fail to generalize effectively to these unseen states.
+
+This effect can be mitigated by providing more diverse and comprehensive training sets, ensuring that the policy is exposed to a wider range of scenarios, including edge cases. However, this raises the challenge of how to efficiently collect such datasets while minimizing the burden on experts. Gathering enough data to cover all possible situations can be time-consuming and costly, especially if it relies heavily on expert demonstrations. Thus, the key question becomes how to strategically collect diverse and representative data in a way that maximizes coverage of the state space while minimizing the amount of effort required from experts.
+
+<table><tr><td>Algorithm 3 Ensemble-DAgger&#x27;s CONDITION</td></tr><tr><td>Require: πnov a mixture of N policies, 
+πexp, st, χ, τ</td></tr><tr><td>1: σ2nov← doubt of the mixture πnov</td></tr><tr><td>2: mt← ||πexp(st) - πnov(st)||2</td></tr><tr><td>3: if σ2nov &gt; χ or mt &gt; τ then</td></tr><tr><td>4: return TRUE</td></tr><tr><td>5: else</td></tr><tr><td>6: return FALSE</td></tr></table>
+
+<table><tr><td colspan="2">Algorithm 4 Lazy-DAgger&#x27;s CONDITION</td></tr><tr><td colspan="2">Require: f, st, βH, βR</td></tr><tr><td colspan="2">1: mt← f(st) which estimates ||πexp(st) - πi(st)||2</td></tr><tr><td colspan="2">2: if mt &gt; βH then</td></tr><tr><td colspan="2">3: return TRUE</td></tr><tr><td colspan="2">4: else</td></tr><tr><td colspan="2">5: if mt &lt; βR then</td></tr><tr><td colspan="2">6: return FALSE</td></tr><tr><td colspan="2">7: else</td></tr><tr><td colspan="2">8: return TRUE</td></tr></table>
+
+Figure 3: In Ensemble-DAgger, the policy is a mixture, and the measure is based on both a disagreement between the models in this mixture, and a disagreement with the expert action. The computations of these measures require the expert to provide action at each timestep of the process. In Lazy-DAgger, the measure is based on the disagreement between the current policy and the expert one, but to avoid the query of the expert action at each timestep, a classifier  $f$  is trained to predict if the discrepancy measure  $\| \pi_{exp}(s_t) - \pi_i(s_t)\|^2$  is above a given threshold  $\beta_H$  or not.
+
+DAgger and variants: To address this issue, one can utilize the DAgger algorithm from Ross et al. (2011) (Algorithm 1), which iteratively expands the training dataset through the intervention of an expert, represented by a reference policy denoted as  $\pi_{\mathrm{exp}}$ . The reference policy serves two key roles: first, it determines the appropriate action to execute at each timestep based on a specified decision rule (line 7) – a mixture between the current policy and the expert one. Second, it helps augment the training set by adding new samples of states and corresponding expert actions, which are then used to improve the learned policy (line 11). At regular intervals, the policy is updated via imitation learning using the expanded dataset, and this process is repeated over multiple iterations. This method helps to mitigate the issue of distributional shift by continually refining the learned policy with fresh data that captures a broader range of states.
+
+To better sample relevant states, few variants of the DAgger algorithm have been proposed. One of the most relevant is Ensemble-DAgger (Menda et al., 2019) which introduces a decision rule to decide if the agent is controlled by the current policy, or by the expert one. Moreover, pairs of state-action are added to the learning dataset only when the action comes from the expert policy (see Figure 2 - right). The decision rule is a discrepancy measure that computes the distance between the expert action and the policy action (combined with a disagreement measure between a mixture of experts) as shown in Algorithm 3. This approach is not realistic when considering human experts. Indeed, let us revisit our car driving example, where the goal is to learn an effective driving policy for a video game. Using the DAgger and Ensemble-DAgger approaches, the expert (i.e. the player) must continuously play the game during the active imitation learning process, while the car is only partially controlled by the current learned policy. This setup has two significant drawbacks: i) First, the player is placed in an unnatural setting, where they are expected to control a car that they do not fully manage. This can create a disorienting experience, as the player must constantly adapt to actions taken by the policy, disrupting the natural flow of gameplay. ii) Second, the player remains engaged even when the car is exhibiting good behavior, spending time providing feedback that may not significantly contribute to discovering a better policy. This leads to inefficiencies, as the expert's time is not always optimally utilized, especially in situations where the current policy already performs well.
+
+# 3 RANDOM NETWORK DISTILLATION-BASED DAGGER
+
+To address the aforementioned limitations of Menda et al. (2019), Lazy-DAgger (Hoque et al., 2021) proposes to involve the expert only during critical moments rather than requiring continuous control. Instead of relying on the expert to partially guide the agent throughout the learning process, this approach alternates between periods where the agent is completely controlled by the current policy and periods where it is fully controlled by the expert. During the phases controlled by the current policy, the expert's intervention is not required, thereby reducing the time the expert spends supervising the agent and making their interventions more targeted and efficient.
+
+Algorithm 5 RND-DAgger  
+Require:  $K,\pi_{exp},f_{targ},f_{pred},\lambda ,W,T$    
+1: Instantiate  $D$  dataset of expert trajectories   
+2: Instantiate  $\pi_0$  BC policy trained on  $D$    
+3: nswitch  $\leftarrow 0$    
+4:  $t\gets 0$    
+5:  $D_0\gets \emptyset$    
+6: for  $i = 0,\ldots ,K$  DAgger iterations do   
+7: while #Di < T do   
+8:  $s_t\sim Env(a_{t - 1})$  (Get a new state)   
+9:  $m_t\gets \| f_{targ}(s_t) - f_{pred}(s_t)\| ^2$    
+10: if  $m_t > \lambda$  or  $w <   W$  then   
+11: if  $m_t\leq \lambda$  then (Minimal demo time)   
+12:  $w\gets w + 1$    
+13: else   
+14:  $w\gets 0$    
+15:  $a_{t}\gets \pi_{exp}(s_{t})$    
+16:  $D_{i}\gets D_{i}\cup \{(a_{t},s_{t})\}$    
+17: if  $m_{t - 1}\leq \lambda$  then   
+18: nswitch  $\leftarrow$  nswitch + 1   
+19: else   
+20:  $w\gets 0$    
+21:  $a_{t}\gets \pi_{i}(s_{t})$    
+22:  $t\gets t + 1$    
+23: Push  $D\gets D\cup D_i$    
+24: Train  $\pi_{i + 1}$  on  $D$  using BC   
+25: Train  $f_{pred}$  on  $D$  to predict  $f_{targ}$    
+26: return  $\pi_K$
+
+Figure 4: The RND-DAgger algorithm proceeds in  $K$  iterations. First a policy is trained on a small expert dataset (line 2). Then at each iteration (line 6), it constructs a new dataset  $D_{i}$  with  $T$  state-action pairs from the expert (line 7). At each timestep, the agent is by default controlled by the current policy (line 23). The OOD measure is computed by using the prediction network  $f_{pred}$  and the target network  $f_{targ}$  (line 9). If this measure is greater than a threshold (line 10), the expert takes over the policy to control the agent (line 16) and to add samples to the dataset (line 17). The number of context switches is the number of times the expert has taken control (line 19). At the end of the iteration, the built dataset is aggregated with the current learning set (line 27) and the policy is retrained (line 28). In addition, the  $f_{pred}$  network is also updated (line 29) to allow the future detection of OOD states. The  $w$  counter is used to ensure that the algorithm waits at least  $W$  timesteps below the threshold before taking control back of the agent.
+
+Lazy-DAgger replaces the traditional action-based discrepancy measure between expert and policy actions with a classifier-based approximation — see Figure 4. This approach allows the algorithm to predict when expert intervention is necessary without requiring an expert action at every timestep, thereby significantly reducing the expert burden compared to methods like Ensemble-DAgger. However, this method has a notable drawback: similar to Ensemble-DAgger, it relies on comparing the actions of the expert and the current policy for a given state. While this approach works well when the expert is optimal and acts deterministically, it becomes problematic when dealing with humans or imperfect experts. Human experts often exhibit variability and may choose different actions for the same state depending on context or personal preference. This results in a noisy and unreliable measure of discrepancy. To address this issue, one could rely on a state-based discrepancy measure that would focus on meaningful divergences compared to an action-based one.
+
+We propose the RND-DAgger algorithm, which builds upon Random Network Distillation (RND) (Burda et al., 2019). The core assumption of RND-DAgger is that expert feedback is only necessary when the agent encounters out-of-distribution (OOD) states—states that are not well represented in the training set and where the policy is more likely to fail. This is crucial because when an agent operates in OOD states, it faces a higher risk of taking suboptimal actions that could hinder learning or lead to unsafe outcomes. Once the agent returns to familiar, in-distribution states, the expert's feedback is no longer required, and the intervention is ended. Our method differs fundamentally from existing approaches like Ensemble-DAgger and Lazy-DAgger, which rely on action-based discrepancy measures to determine when to intervene. As such, our approach remains robust to variations in expert behavior, ensuring that interventions only occur when the agent genuinely faces unfamiliar and potentially risky states.
+
+To further improve the stability of expert interventions, we introduce a mechanism called minimal demonstration time. This concept addresses the issue of overly frequent and brief expert interventions, which can interrupt learning and increase the cognitive load on the expert. Minimal demonstration time defines a lower bound on the duration for which the expert must maintain control once they start an intervention. The key idea is that the expert should provide a sufficient number of consecutive corrective actions to guide the agent back to a stable state, instead of immediately handing control
+
+back to the policy after a single correction. We describe these two components below. The detailed algorithm is provided in Algorithm 4 with a complete explanation of the different steps.
+
+Random Network Distillation To measure if a state is OOD, RND-DAgger relies on the Random Network Distillation technique, which is a classical approach developed initially for the problem of exploration in Reinforcement Learning to detect new states to explore (Burda et al., 2019). The principle of RND is to use a randomly initialized neural network as a fixed target  $f_{\text{target}}$  and train a second neural network  $f_{\text{pred}}$  (the predictor) to approximate the output of the target network. As the predictor network improves over time, the error between the predictor and the target network decreases for familiar states (in-distribution), but remains high for unseen or out-of-distribution states. This prediction error thus serves as a measure of novelty, allowing the agent to recognize when it has encountered a new or unfamiliar state. In RND-DAgger, the measure is used to decide when to trigger the expert, and when to trigger back the current policy, alternating phases where the expert is controlling the agent and when the agent is controlled by the current policy. Note that, contrary to the described baselines, this measure does not involve the expert action, thus avoiding the need to have an expert acting at each timestep.
+
+Minimal demonstration time Relying exclusively on the OOD measure can lead to a behavior where the expert is asked too often for short demonstrations (see appendix Figure C.2). To account for this limitation, we introduce the notion of minimal expert time (MET). It is defined as  $W$  the number of consecutive "in-distribution" frames required to switch control back from the expert to the autonomous policy (i.e. the number of frames where  $m_t < \| f_{\text{targ}}(s_t) - f_{\text{pred}}(s_t) \|^2$ ). This mechanism ensures that the expert will provide a minimal amount of information at each intervention, resulting in fewer, but longer, demonstration sequences.
+
+# 4 EXPERIMENTS
+
+The primary objectives of our experimental evaluation are twofold: first, to assess the efficiency of our approach in discovering an effective policy, and second, to evaluate its ability to reduce the expert's burden.
+
+**Environments:** Our first environment is HalfCheetah which is a classical reinforcement learning environment<sup>1</sup> where the objective is to learn a running strategy for the agent. The goal of the agent is to locomote as fast as possible. The HalfCheetah is controlled by applying motor torques, and the agent manipulates a 6-degree-of-freedom (6-DOF) motor joint vector. The observation space consists of 18 values, which include the position and velocity of the agent's body, the angles and angular velocities of its six joints, as well as the angle of the center of mass. We also propose and open-source two new environments developed for video game research. **RaceCar** (see Figure 5) features a physics-based car controller that must complete a single lap on a given track. After each lap, the car is reset to a random position on the starting line. The track presents several challenges, including speed bumps, a ramp in front of a pillar, and sharp 90-degree turns, making the optimal driving behavior non-trivial. Additionally, crossing the ramp and red sloped walls provides a speed boost, adding complexity to the strategy. The car is controlled using four discrete actions: forward, backward, left, and right. The observation space consists of 22 dimensions, including the car's 3D position, linear velocity (3D), angular velocity (3D), rotation (coded as the cosine and sine of the angle), and data from seven raycasts to detect obstacles. Finally, the 3D Maze environment allows us to study our strategy in goal-conditioned navigation scenarios. A classical human-like character is spawned at a random location within the maze and tasked with reaching a randomly assigned goal. The agent is controlled using four movement actions: walk forward, walk backward,Strafe left, andStrafe right, as well as a one-dimensional rotation action to change its facing direction. The observation space consists of the agent's absolute position (3D), the goal's position (3D), and 131 raycasts, which provide a low-resolution depth map of the surroundings. Since the objective in this environment is to navigate to any goal location, we replace the traditional Behavioral Cloning (BC) approach with a Goal-Conditioned Behavioral Cloning (GCBC) approach (Ding et al., 2019b) as the base learning algorithm. This adaptation ensures that the agent learns not just to imitate expert
+
+![](images/86f862b65854a91205936098aa0dc27c87aefc23e2631352b956722b48a3692e.jpg)  
+(a) RaceCar
+
+![](images/23a762e95d9a67e94063c024fdeb93f4ad444ee17e03022a1924f22a35760ce9.jpg)  
+(b) RaceCar (Player view)
+
+![](images/dbb9a4e88af015fa7bc61880e80d2d5e8f4e58204396535f081d92a13abcf68e.jpg)  
+(c) Maze  
+Figure 5: Illustration of the three different environments used in our experiments. The objective is to learn a good driving, navigating, and walking policy while minimizing expert interventions.
+
+![](images/935123dacd17b4ccb16162370872e2a91fc8be56808d8a19bd17f00c1a64d750.jpg)  
+(d) HalfCheetah
+
+trajectories, but also to generalize its navigation strategies based on varying goal positions within the maze.
+
+Baselines: We compare our approach RND-DAgger to multiple approaches from the active imitation learning literature: DAgger Ross et al. (2011), a classical approach which queries the supervisor for an action in every state that the learner visits. Ensemble-DAgger Menda et al. (2019), which propose an automated approach to measure OOD and trigger expert demonstration. Ensemble-DAgger relies on two metrics: A disagreement measure defined as the correlation between an ensemble of policies, and a discrepancy measure computing the difference between the bot action and the expert action at every step. Lazy-DAgger uses the same discrepancy measure as Ensemble-DAgger and reduces bot-expert context switching by using an additional threshold parameter to define a hysteresis band. Note that, we rely on a simplified version proposed in (Hoque et al., 2021) that has been shown to be more efficient and generates less context switches than the original version. This variant discards the classifier for the benefit of the true discrepancy measure. Human-Gated DAgger (HG-DAgger) from Kelly et al. (2019) involves a human supervisor monitoring the agent's behavior in real-time and deciding when to take control. While similar in essence to our proposed RND-DAgger, HG-DAgger relies entirely on human judgment to identify when intervention is necessary, rather than using an automatic OOD measure. This requires the expert to continuously observe the agent's actions at every timestep, leading to a higher cognitive load and less efficient use of expert time. Finally, we also consider Behavioral Cloning (BC) as the classical baseline, which does not include any active learning mechanism.
+
+The set of hyperparameters used for the different approaches is described in Appendix A. For each method, a first training set is collected by the expert, without active imitation learning techniques. Then the different algorithms are executed to collect additional examples and to update the policy.
+
+Metrics: To assess the performance of the methods considered, we rely on several key metrics: i) The Task Performance measures whether the learned agent is successfully solving the environment. For the Race Car and Maze environments, we use the success rate (i.e., whether the agent completes a lap or reaches the goal). For HalfCheetah, we track the cumulative episode reward, which reflects the overall effectiveness of the agent's locomotion strategy. ii) The Dataset Size indicates the number of expert actions added to the training dataset, providing insight into how much expert information is being utilized to train the agent. A smaller dataset size suggests the agent is learning efficiently with fewer expert interventions, while a larger dataset implies greater reliance on expert input. iii) The Context Switches which corresponds to the nswitch variable in Algorithms 1 and  $\omega$  measures the number of contiguous periods during which the expert is actually controlling the agent. This helps quantify how often the expert is involved in directly providing relevant training samples. For RND-DAgger, this measure accurately reflects the expert's involvement, as the agent is fully controlled by the policy between switches. The metrics are averaged over 8 seeded runs for each method.
+
+# 4.1 RESULTS
+
+Oracle-based Performance: To conduct extensive experiments, we replace the human expert with a predefined oracle policy. This substitution allows us to run multiple trials efficiently without depending on human participants, which would be impractical in terms of both time and resources (experiments using human experts are reported separately later in the section). For the Race Car
+
+<table><tr><td rowspan="2">Env Method</td><td colspan="3">Task Performance</td><td colspan="3">#Context Switch</td></tr><tr><td>RC</td><td>HC</td><td>Maze</td><td>RC</td><td>HC</td><td>Maze</td></tr><tr><td>BC</td><td>0.883 ± 0.029</td><td>2455 ± 424</td><td>0.367 ± 0.043</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DAgger</td><td>0.940 ± 0.026</td><td>2343 ± 209</td><td>0.450 ± 0.074</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Lazy-DAgger</td><td>0.939 ± 0.025</td><td>2314 ± 278</td><td>0.575 ± 0.101</td><td>3437 ± 89</td><td>312 ± 45</td><td>1238 ± 79</td></tr><tr><td>Ensemble-DAgger</td><td>0.952 ± 0.018</td><td>2489 ± 108</td><td>0.626 ± 0.045</td><td>2785 ± 41</td><td>1452 ± 134</td><td>2871 ± 94</td></tr><tr><td>RND-DAgger</td><td>0.944 ± 0.014</td><td>2490 ± 160</td><td>0.717 ± 0.018</td><td>368 ± 11</td><td>708 ± 130</td><td>1214 ± 25</td></tr></table>
+
+Table 1: Overall performance: This table illustrates the performance of the different algorithms at the end of the K iterations. It also provides the value of the nswitch variable that accounts for the number of expert interventions. Best, Second best
+
+environment, the oracle is learned through a separate active imitation learning process with a large interaction budget, ensuring the agent has access to high-quality demonstrations. In the 3D Maze environment, the oracle is a NavMesh-based agent, which programatically solves any navigation scenario, making it easy to generate trajectory data. For HalfCheetah, we use an open-source reinforcement learning agent (Kuznetsov et al., 2020) from the RL Zoo repository (Raffin, 2020) as the oracle, providing an optimal policy for locomotion tasks. This approach ensures consistency and scalability across experiments.
+
+Table 1 and Figure 6 present the performance metrics across the evaluated environments. To ensure a fair comparison, we report results based on the hyperparameter settings that achieve the highest final performance for each model.
+
+![](images/de59fffcb27fb61b270dbbe62559d0500136e73685d4fbe24c44f9cda0b2b7d8.jpg)  
+(a) Task performance RaceCar
+
+![](images/91587e6ac776781eb231d433c4eb90051a340c69f58480b97d178a3e3dd770f2.jpg)  
+(b) Task performance HalfCheetah
+
+![](images/0a091bcf1238f4e8f870359ba336c92a8501803a466cc3acdd18ac59f2ce9220.jpg)
+
+![](images/99bcbe8696b47768d150a60ad6efb079817cc7657b4e5f83bec275e754a891ed.jpg)  
+(d) Context switches RaceCar
+
+![](images/0240cb24d6f1477ea4075825c5997a46c2cfa412b4e2a4ca5e026c7ca6618956.jpg)  
+(e) Context switches HalfCheetah
+
+![](images/18e46c04a61b3f0765af97c1458ae34ff500bf70cc6685a00d51cfe21cfbd3e5.jpg)  
+(c) Task performance Maze  
+(f) Context switches Maze  
+Figure 6: Performance and context switches for the different environments. The X-axis corresponds to the size of the training set  $D$  that increases at each iteration of the algorithms.
+
+Our method, RND-DAgger, demonstrates competitive performance relative to Ensemble-DAgger and Lazy-DAgger, indicating that it effectively learns robust policies across different environments. For example, in the HalfCheetah environment, RND-DAgger achieves a cumulative reward of 2490, compared to 2489 for Ensemble-DAgger and 2314 for Lazy-DAgger. These results highlight that RND-DAgger is capable of discovering efficient policies, matching or surpassing the baselines in terms of overall task success. When analyzing performance relative to the size of the training set, RND-DAgger outperforms the other methods in the early stages of the active imitation learning process (Fig. 6b). This suggests that RND-DAgger is more effective at focusing on critical states where expert guidance is most needed, thereby gathering valuable feedback more efficiently. By concentrating on key interventions, RND-DAgger accelerates policy improvement and requires fewer training samples to achieve comparable or superior results, making it especially advantageous in scenarios with limited expert availability. In summary, the experimental results demonstrate that RND-DAgger not only achieves strong final performance but also exhibits a more sample-efficient
+
+![](images/249e722056a39cda2b3ac2fd985892cb9ee272d407832fbd4f1dfdc5a7555e4c.jpg)  
+(a) RND-DAgger
+
+![](images/a05fde0d1f5e7f4b50675eca5f47b3f0ba5641e4158a3f876379327a77eb0897.jpg)  
+(b) Lazy-DAgger
+
+![](images/96f37e563a372b8346ca3a1476e155bb34eaf0a32a4a1f12d41d4f95824cb79e.jpg)  
+(c) Ensemble-DAgger  
+Figure 7: Visualization of agent's trajectories (blue) and expert intervention trajectories (red) in RaceCar. RND-DAgger predominantly requests expert intervention in challenging sections of the track, requiring fewer expert interventions overall compared to baseline methods.
+
+learning curve compared to existing methods (Fig. 6a, 6b and 6c), validating its capability to optimize expert interventions and rapidly improve policy quality.
+
+When examining the number of context switches between the different methods, RND-DAgger consistently results in significantly fewer context switches compared to Ensemble-DAgger across all environments. This indicates that RND-DAgger is more stable and requires fewer handovers between the expert and the policy, reducing the burden on the expert. When compared to Lazy-DAgger, RND-DAgger shows mixed results. In the RaceCar environment, RND-DAgger generates fewer context switches (Fig. 6d), highlighting its efficiency in minimizing expert interventions. However, in the HalfCheetah and Maze environments, RND-DAgger produces a similar number of switches (Fig. 6e and 6f). This similarity in HalfCheetah can be attributed to the use of an oracle expert policy, which performs consistently without generating diverse actions. Under these circumstances, the discrepancy measure used by Lazy-DAgger, which relies on comparing expert and policy actions, remains effective for detecting when to switch. In the Maze environment, despite having a comparable number of context switches to Lazy-DAgger (Fig. 6f), RND-DAgger achieves a higher final performance (Fig. 6c). This suggests that RND-DAgger intervention criterion leads to more informative expert interventions and is better able to leverage the expert. Thus, in scenarios like Maze, where achieving high task performance is critical, RND-DAgger is the better choice due to its ability to drive the agent towards optimal behavior.
+
+Overall, these findings indicate that RND-DAgger strikes a favorable balance between reducing context switches and achieving strong policy performance, making it a more efficient and reliable choice.
+
+Qualitative study: To gain a deeper understanding of when our algorithm requests expert supervision, Figure 7 visualizes the context switches between the current policy and the expert (additional figures are provided in Appendix B). As seen in the figure, both Ensemble-DAgger and Lazy-DAgger query the expert significantly more frequently, leading to a higher number of context switches compared to RND-DAgger. The visualization also reveals that RND-DAgger identifies and focuses on critical areas of the track that are challenging for the agent during early learning stages. Specifically, it requests expert intervention primarily in the problematic zones, such as the bottom section of the track and the obstacle immediately following the speeding ramp (top-left corner of the track). These are regions where the bot initially struggles, making precise interventions crucial for improving performance. This observation indicates that RND-DAgger not only minimizes the number of expert queries but also targets the most relevant segments of the environment where guidance is essential. By focusing on challenging states rather than spreading expert effort across all states, RND-DAgger makes more strategic use of expert supervision, resulting in fewer yet more impactful interventions.
+
+Human Expert-based performance: We conducted a series of experiments (Figure 8) using a real human expert instead of an oracle policy. In this setup, collecting expert actions while the agent was being controlled by its own policy (rather than having the expert take direct control) felt unintuitive for the human participants, making it challenging to apply traditional baselines effectively.
+
+![](images/f87a6f6501c928afeb54eaff8c2d733a62c156cd9f5206cf2c141e987f9070ce.jpg)  
+(a) Performance
+
+![](images/17bb8148f5ed18e5a51cab77808f8835b76f77929569887585d1768d16f884ba.jpg)  
+(b) Performance (time spent)  
+Figure 8: Task performance on RaceCar with a human expert. (a) RND-DAgger achieves similar task performance to HG-DAgger with a comparable number of demonstrations. (b) However, by reducing the need for constant expert monitoring, RND-DAgger significantly lowers the total expert time required.
+
+As such we focused our comparison on HG-DAgger, which allows the human expert to decide when to intervene and take control from the agent.
+
+Our results show that RND-DAgger performs comparably to HG-DAgger, but with a significant advantage: it does not require the human expert to continuously monitor the agent's behavior. Instead, RND-DAgger automates the decision of when to request expert input based on state novelty, reducing the cognitive load on the expert. Figure 8b demonstrates the relationship between policy performance and the actual time the expert spends observing the screen. In RND-DAgger, the agent can run at accelerated speeds without the need for constant supervision, as the expert only needs to intervene at specific moments rather than watching the agent's entire trajectory.
+
+Additional experiments Appendix C presents ablation studies of RND-DAgger, including an analysis of the impact of the Minimal Expert Time mechanism (showing how it reduces expert context switches). Appendix D focuses on studying the impact of using Ensemble-DAgger compared to RND-DAgger regarding expert time spent (the metric used in Figure 8b), showing how RND-DAgger better minimizes expert burden.
+
+# 5 CONCLUSION AND LIMITATIONS
+
+In this work, we presented RND-DAgger, a novel approach to active imitation learning that efficiently minimizes the need for expert interventions by leveraging a state-based measure derived from Random Network Distillation (RND). Unlike traditional methods that rely on action-based discrepancies to detect when to seek expert guidance, RND-DAgger focuses on identifying out-of-distribution states where the agent is most at risk of making errors. This allows our method to selectively request expert feedback only when it is truly necessary, thereby reducing the number of context switches and optimizing the allocation of expert time. Overall, our method provides a step forward in developing more practical and human-efficient imitation learning algorithms, making it a valuable tool for training autonomous agents in complex environments.
+
+Limitations RND-DAgger requires the expert to immediately take control of the agent, which may be impossible or dangerous in complex real-time environments. Predictive approaches to alert the expert before intervention is an interesting area of improvement. Studying how to scale RND-DAgger to more complex environments such as Partially observable MDPs – e.g. pixel-based environments – would be a natural extension of our work. In such scenarios, noisy-TV problems i.e. environments featuring distracting random or diverse observations, bring important new challenges, which may be addressed through representation learning or explicit high-entropy filtering. Lastly, incorporating richer forms of expert feedback is another valuable future line of work.
+
+# 6 ACKNOWLEDGEMENTS
+
+This work was granted access to the HPC resources of IDRIS under the allocation 2024-AD011015218 made by GENCI.
+
+# 7 REPRODUCIBILITY STATEMENT
+
+To ensure the reproducibility of our work, we provide detailed pseudo-code in section 2 and section 3. A comprehensive open-source codebase, including all environments, datasets, oracle model checkpoints, active learning algorithms, and a detailed guide on how to reproduce our experiments and results is available at https://sites.google.com/view/rnd-dagger.
+
+# REFERENCES
+
+Michael Bain and Claude Sammut. A framework for behavioural cloning. In Machine Intelligence 15, pp. 103-129, 1995.  
+Michael Bain and Claude Sommut. A framework for behavioural cloning. In Machine Intelligence 15, volume 15, pp. 103. Oxford University Press, 1999.  
+Yuri Burda, Harrison Edwards, Amos J. Storkey, and Oleg Klimov. Exploration by random network distillation. In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.net, 2019. URL https://openreview.net/forum?id=H11JJnR5Ym.  
+Yiming Ding, Carlos Florensa, Pieter Abbeel, and Mariano Pielipp. Goal-conditioned imitation learning. Advances in neural information processing systems, 32, 2019a.  
+Yiming Ding, Carlos Florensa, Mariano Pheiipp, and Pieter Abbeel. Goal-conditioned imitation learning. Advances in Neural Information Processing Systems, 2019b. URL http://arxiv.org/abs/1906.05838.  
+Jack Harmer, Linus Gisslén, Jorge del Val, Henrik Holst, Joakim Bergdahl, Tom Olsson, Kristoffer Sjöö, and Magnus Nordin. Imitation learning with concurrent actions in 3d games. In 2018 IEEE Conference on Computational Intelligence and Games (CIG), pp. 1-8. IEEE, 2018.  
+Ryan Hoque, Ashwin Balakrishna, Carl Putterman, Michael Luo, Daniel S Brown, Daniel Seita, Brijen Thananjeyan, Ellen Novoseller, and Ken Goldberg. Lazy dagger: Reducing context switching in interactive imitation learning. In 2021 IEEE 17th international conference on automation science and engineering (case), pp. 502-509. IEEE, 2021.  
+Ahmed Hussein, Mohamed Medhat Gaber, Eyad Elyan, and Chrisina Jayne. Imitation learning: A survey of learning methods. ACM Computing Surveys (CSUR), 50(2):1-35, 2017.  
+Kshitij Judah, Alan Paul Fern, and Thomas Glenn Dietterich. Active imitation learning via reduction to iid active learning. In 2012 AAAI fall symposium series, 2012.  
+Michael Kelly, Chelsea Sidrane, Katherine Driggs-Campbell, and Mykel J Kochenderfer. Hg-dagger: Interactive imitation learning with human experts. In 2019 International Conference on Robotics and Automation (ICRA), pp. 8077-8083. IEEE, 2019.  
+Arsenii Kuznetsov, Pavel Shvechikov, Alexander Grishin, and Dmitry Vetrov. Controlling overestimation bias with truncated mixture of continuous distributional quantile critics. In International Conference on Machine Learning, pp. 5556-5566. PMLR, 2020.  
+Yihuan Mao, Chengjie Wu, Xi Chen, Hao Hu, Ji Jiang, Tianze Zhou, Tangjie Lv, Changjie Fan, Zhipeng Hu, Yi Wu, et al. Stylized offline reinforcement learning: Extracting diverse high-quality behaviors from heterogeneous datasets. In The Twelfth International Conference on Learning Representations, 2024.
+
+Kunal Menda, Katherine Driggs-Campbell, and Mykel J Kochenderfer. Ensembled dagger: A bayesian approach to safe imitation learning. In 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 5041-5048. IEEE, 2019.  
+Nimisha G Nair, Pallavi Satpathy, Jabez Christopher, et al. Covariate shift: A review and analysis on classifiers. In 2019 Global Conference for Advancement in Technology (GCAT), pp. 1-6. IEEE, 2019.  
+Antonin Raffin. Rl baselines3 zoo. https://github.com/DLR-RM/rl-baselines3-zoo, 2020.  
+Stéphane Ross, Geoffrey Gordon, and Drew Bagnell. A reduction of imitation learning and structured prediction to no-regret online learning. In Proceedings of the fourteenth international conference on artificial intelligence and statistics, pp. 627-635. JMLR Workshop and Conference Proceedings, 2011.  
+Stefan Schaal. Is imitation learning the route to humanoid robots? Trends in cognitive sciences, 3(6): 233-242, 1999.  
+Oriol Vinyals, Igor Babuschkin, Wojciech M Czarnecki, Michael Mathieu, Andrew Dudzik, Junyoung Chung, David H Choi, Richard Powell, Timo Ewalds, Petko Georgiev, et al. Grandmaster level in starcraft ii using multi-agent reinforcement learning. nature, 575(7782):350-354, 2019.  
+Derek Yadgaroff, Alessandro Sestini, Konrad Tollmar, Ayça Özçelikkale, and Linus Gisslén. Improving generalization in game agents with data augmentation in imitation learning. In IEEE Congress on Evolutionary Computation, CEC 2024, Yokohama, Japan, June 30 - July 5, 2024, pp. 1-8. IEEE, 2024. doi: 10.1109/CEC60901.2024.10612178. URL https://doi.org/10.1109/CEC60901.2024.10612178.  
+Jiakai Zhang and Kyunghyun Cho. Query-efficient imitation learning for end-to-end simulated driving. In AAAI Conference on Artificial Intelligence, 2017. URL https://api.sementicscholar.org/CorpusID:5929487.
+
+# A EXPERIMENTAL DETAILS
+
+Dataset Figure 9a and 9b showcase examples of generated trajectories for the RaceCar and Maze environments.
+
+![](images/7b15f1c76291d3a411c52b2bf6ce1993bc825645bf4ba31ea46d8a7811389034.jpg)  
+Episode traces  
+(a) RaceCar dataset examples  
+Figure 9: Dataset examples: the trajectories of the expert from which we bootstrapped to train our initial BC policies and out BC baselines.
+
+![](images/ada3169f781a5d08831a18669c1c9f70a6f3ba78a9296b1e1382a3f3a92de3e2.jpg)  
+Episode traces  
+(b) Navigation dataset examples
+
+Hyperparameters For each decision rule, several key hyperparameters had to be tuned:
+
+- DAgger
+
+- The probability  $\beta$  of a frame to be controlled by the bot. The probability is decreased at each DAgger epoch by  $\beta_{i} \gets \beta_{0}^{i - 1}$
+
+- RND-DAgger
+
+- Threshold  $\lambda$  of OOD detection  
+- The historic context length, that is the number of frames in the past we take along with the current state to detect its OOD nature  
+- The Minimal Expert Time  $W$  
+- The size of the random network, represented by the number of layers and neurons per layer the predictor and target networks have.
+
+- Ensemble-DAgger
+
+- Threshold  $\tau$  for discrepancy measure  
+- Threshold  $\chi$  for doubt measure  
+- The number of models  $N$
+
+- Lazy-DAgger
+
+- Threshold  $\beta_{H}$  for discrepancy measure  
+- Threshold  $\beta_{R}$  for the backward controlled loop (i.e. the criterion to switch back from expert to autonomous)
+
+Computing thresholds For all the methods that necessitate at least one threshold (RND-DAgger, Lazy-DAgger, Ensemble-DAgger), we computed them following the following methodology: the threshold was set to be a positive factor of the mean measure on the training set. In other words, before each new sessions, the measure was ran over the training dataset, and the threshold ( $\chi$  and  $\tau$  for Ensemble-DAgger,  $\beta_H$  for Lazy-DAgger and  $\lambda$  for RND-DAgger) were calculated as:
+
+$$
+\lambda \leftarrow \overline {{\operatorname {M E A S U R E} \left(\left\{\left(a _ {t} , s _ {t}\right) \in D _ {\text {t r a i n}} \right\}\right)}} \times L \tag {1}
+$$
+
+Table 2: Hyperparameter search  
+(a) Ensemble-DAgger  
+
+<table><tr><td>Hyperparameter</td><td>Values</td><td>RaceCar</td><td>HalfCheetah</td><td>Maze</td></tr><tr><td>χ factor</td><td>[0,1,1.5,2,3,4]</td><td>1.5</td><td>0</td><td>0.5</td></tr><tr><td>τ factor</td><td>[0,1,1.5,2,3,4]</td><td>1.5</td><td>1.5</td><td>3</td></tr><tr><td>N</td><td>[2,3,5]</td><td>5</td><td>5</td><td>3</td></tr></table>
+
+(b) RND-DAgger  
+
+<table><tr><td>Hyperparameter</td><td>Values</td><td>RaceCar</td><td>HalfCheetah</td><td>Maze</td></tr><tr><td>λ factor</td><td>[1,2,3,4]</td><td>2</td><td>2</td><td>2</td></tr><tr><td>Hidden size</td><td>[32,128]</td><td>32</td><td>128</td><td>32</td></tr><tr><td>Number of layers</td><td>[0,1,2]</td><td>0</td><td>2</td><td>0</td></tr><tr><td>Historic context length</td><td>[0,2,5,10,15]</td><td>10</td><td>0</td><td>2</td></tr><tr><td>W</td><td>[1,2,5,10,15,30,50]</td><td>30</td><td>5</td><td>5</td></tr></table>
+
+(c) Lazy-DAgger  
+
+<table><tr><td>Hyperparameter</td><td>Values</td><td>RaceCar</td><td>HalfCheetah</td><td>Maze</td></tr><tr><td>βH factor</td><td>[0,1,2,3,4]</td><td>1.5</td><td>0</td><td>2</td></tr><tr><td>βR divider</td><td>[1,1.5,2,3,4]</td><td>2</td><td>1.5</td><td>2.5</td></tr></table>
+
+(d) DAgger  
+
+<table><tr><td>Hyperparameter</td><td>Values</td><td>RaceCar</td><td>HalfCheetah</td><td>Maze</td></tr><tr><td>β</td><td>[0.2, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95]</td><td>0.5</td><td>0.7</td><td>0.8</td></tr></table>
+
+With L being a positive factor, a hyperparameter to be tuned. For the second threshold of Lazy-DAgger, we set it to be a factor of the first one:  $\beta_{R} = \beta_{H} / L^{*}$ , with  $L^{*}$  being the hyperparameter tuned. That's a different method compared to other methods, such as Zhang & Cho (2017) or Hoque et al. (2021) who set their threshold so that approximately  $20\%$  of the initial dataset is unsafe, or Ensemble-DAgger Menda et al. (2019) that directly grid searched the value. The Table 2 summarizes the values used for our grid search.
+
+# B INTERACTIVE SESSION VISUALIZATIONS
+
+Typical interactive situations In Figure 11, one can understand better how, in the RaceCar environment, an oracle (and by extend a human taken as an expert) would interact with the learner. In (a), the current learned policy has control of the car, and the OOD measure (green circle) is below the threshold: the learner is confident. In (b), the car crashes into a wall, so the measure (red circle) queries the expert to take control (transparent red capsule). From (c) to (e), the expert demonstrates how to get back on track, until (f) the measure falls below the threshold again (green circle) for at least  $W$  consecutive frames (see Section 3 for further explanations)
+
+Qualitative study On Figure 10, we reported the context switches and full trajectories on both the RaceCar and Maze environments. The results are taken from the same session and the same seed to better compare the figures. From (a) to (b), it is clear that Ensemble-DAgger exhibits the most context switches compared to Lazy-DAgger and RND-DAgger, which show comparable counts. However, the context switches in Lazy-DAgger are more concentrated in specific areas, whereas those in RND-DAgger are more evenly distributed. Then, from (d) to (f) we can see the corresponding episodes with expert demonstrations (in red) added to the dataset during that session. We can see that the expert demonstrations are either way longer (the whole episode) or way shorter (only a few frames) for Lazy-DAgger compared to our method, providing more consistency throughout the map. The difference is clearer on the RaceCar environment. Indeed, on both context switch figures, we see that Lazy-DAgger and Ensemble-DAgger are comparable, and that the queries for RND-DAgger are concentrated around only two zones that generated most of the failure cases.
+
+![](images/7afa3d139ddd332a09195fb6dbee975bdff6f66f42758fa1915ded4e00bcde1e.jpg)  
+(a) RND-DAgger
+
+![](images/dede5e5e75f1a8ced06b695babeabd78f406a6f0dc581ed3099eee662755ed9a.jpg)  
+(b) Lazy-DAgger
+
+![](images/d0d57d096278c46fe24c11ae8e4d3b6d3773175fee1c9165fc2869f6a28ee3cb.jpg)  
+(c) Ensemble-DAgger
+
+![](images/89f0a9dca9c5a12a430065522cbbd364384bead7c82964d33d1eea9041010995.jpg)
+
+![](images/5e24c6ac6dccb5ba145993d4b2e9c864f270de88d865ced52ec9932b4ece2483.jpg)
+
+![](images/913de2a7a04f488a0fe4fb61aa4bfd0c06a124e845fcb04a14e12f61bc17983e.jpg)
+
+![](images/c72e1574dd12ed2a945b182bb0330c33f3623653aaf1f6ea0b70939def33a410.jpg)  
+(d) RND-DAgger
+
+![](images/6902bfaa7632f7152c4cc32be53dd30c186686748380cf89d5492dd7970f0190.jpg)  
+(e) Lazy-DAgger  
+(h) Lazy-DAgger
+
+![](images/3f6f2b2628d4d8b9bbe1ee6cf9469bc5b9146109062c44483d8349ddf12007bc.jpg)  
+(f) Ensemble-DAgger  
+(i) Ensemble-DAgger
+
+![](images/1bc4cbaa79d0da003e3b7d52046f12bcc25c47a5356bfbc9af70012884cd164e.jpg)  
+(g) RND-DAgger  
+(j) RND-DAgger  
+Figure 10: We conduct a qualitative analysis of the different query methods. (a)-(d) and (g)-(i): Each dot represents a context switch : a transition from autonomous to expert control. (e)-(h) and (j)-(l): In red, the expert demonstrations, in blue the novice segments.
+
+![](images/879727ce65fd3bb0fa73c46dfa3cde1af7185437e0970f1ff42c862241b62d8f.jpg)  
+(k) Lazy-DAgger
+
+![](images/b2f388b8f0abe7b1bf1e6280daa1923b6948926e70f8b41ebfde0dcc49089f1e.jpg)  
+(1) Ensemble-DAgger
+
+![](images/9e245c133b3a7ba1b51e218a2910e97280a3b5dba6be262145c0b4255270834d.jpg)
+
+![](images/a8df9cdb3d6cceb6c6517d14705613e5b31458d6c788e950bf6a891c368ed93c.jpg)
+
+![](images/4ad84248b1698f4b11bcef057a92f8df7f57bc6ecf2f23c717cee0cd8475359a.jpg)
+
+![](images/4d278da31e7b8eb6a57dcee6e1f6c247298ebee0ce9d170f32e4816074b8da67.jpg)  
+(a) 0  
+(d) 3  
+Figure 11: (a) The agent has control. (b) The OOD measure is triggered (RND), the expert is given control. (c)-(e) The expert demonstrates how to recover. (f) The state is in distribution, the expert keeps the control until the Minimal Expert Time is reached.
+
+![](images/c962f8e6234257e1c2f832fde6251f0979946680cbcd5c1bec3e8e76f59b4654.jpg)  
+(b) 1  
+(e) 4
+
+![](images/1f909f322c004a5e0851e2c799d03248a149928c80a35d6ca5b9219820e9b004.jpg)  
+(c) 2  
+(f) 5
+
+# C ABLATION STUDY
+
+# C.1 RND-DAGGER ON HALFCHEETAH
+
+In this ablation study, we evaluate RND-DAgger, focusing on the additive advantages of three key enhancements: a Minimal Expert Time (MET) of size  $W$ , historical context, and the architecture of the random networks (see Section 2 for insights on the values tested). The addition of a Minimal Expert Time should increase the quality of the frames by letting the expert finish its demonstration and making sure the learner is fully recovered before letting it the control again. The historical context enriches decision-making by incorporating temporal information from past actions and concatenating them to the current state. This particularly helps in tasks where there is a strong relation between the states. For example, a state where the car is close to a wall is not necessarily a bad state, unless the car is going towards it for several frames in a row. However, a historical context chosen too big is detrimental, because it can overshadow useful dimensions of the sate, and thus preventing the predictor network to seize useful information.
+
+Results of that ablation are reported in Table 3 and Figure 12. We note that the Minimal Expert Time  $W$  helps reducing the context switching, without impacting that much the performance at the end of the sessions of RND-DAgger. In other words, it helps decreasing the expert burden, without impacting the task performance.
+
+![](images/fe1ab8b7bd113db4d0a8f45761c9a627c78e85458b90b87d536dc0ea38d9dd4b.jpg)  
+(a) HalfCheetah (Task perf)
+
+![](images/a48b156dd721eedd95e2d681d28ddab0b689f6ac30379df7cce53dc78a8932d3.jpg)  
+(b) # context switches  
+Figure 12: Ablation study on our method. RND-DAgger compared to: (light blue) RND w/o MET, (orange) bigger historical context, (dark blue) smaller architecture for the random networks.
+
+<table><tr><td></td><td>Task Performance</td><td># Context Switch</td></tr><tr><td>BC</td><td>2455 ± 424</td><td>-</td></tr><tr><td>RND-DAgger</td><td>2490 ± 160</td><td>708 ± 130</td></tr><tr><td>RND-DAgger - MET</td><td>2432 ± 175</td><td>1985 ± 215</td></tr><tr><td>RND-DAgger - small</td><td>2167 ± 140</td><td>1136 ± 76</td></tr><tr><td>RND-DAgger - historic</td><td>2172 ± 290</td><td>554 ± 90</td></tr><tr><td>Lazy-DAgger</td><td>2314 ± 278</td><td>312 ± 45</td></tr></table>
+
+Table 3: Performance and context switches at the end of the sessions. We report the results of the key ablations on RND-DAgger, along with the results of Lazy-DAgger for further comparisons
+
+# C.2 MINIMAL EXPERT TIME FOCUS
+
+Ablation study on all the methods We further investigated the impact of the Minimal Expert Time (MET) on our method and on our baselines, across all the tasks. We reported the results of that study in Figure 13 and Table 4. In particular, we can see that LazyDAgger + MET isn't appropriate in HalfCheetah. The combination of the Lazy mechanism (introduction of a second threshold to create a hysteresis band) and the MET forces the expert to always have control over the agent, which explains the extremely low value for the context switches (which happens only once per episode) and the poor performance learning curve (Fig. 13b). The same remark holds for Ensemble-DAgger on RaceCar, but in this case, this behavior doesn't seem to impact the task performance of the method (Fig. 13a).
+
+![](images/945b36549fee9dad7ab0832a860cffddeae90b77e6150898c746c69cfaefc46d.jpg)  
+(a) Task performance RaceCar
+
+![](images/b6e947930c0bbb20c6561108be646c674c23c9733faef4ff77054fcec9065b87.jpg)
+
+![](images/781748ef169bca16653e2e177e775da85a7ef9255b7779d1e36e7f0a1d9ff457.jpg)
+
+![](images/31a2dc68d6b5441d95eeb5452a804d74b7dc3391ffbb0274a84bf0e54417d281.jpg)  
+(d) Context switches RaceCar
+
+![](images/20c449f9c08491cbc359a5c2b92a50c7aa385520c61d6d1bcb3ea5e7bdc39dc1.jpg)  
+(b) Task performance HalfCheetah  
+(e) Context switches HalfCheetah
+
+![](images/fdd74e2f512bb4e56cd8f920350f264b5ce2323080cdba2c36daa676738f65ac.jpg)  
+(c) Task performance Maze  
+(f) Context switches Maze  
+Figure 13: Results for the ablation study with focus on the addition of the Minimal Expert Time (MET) on the baselines Ensemble-DAgger and LazyDAgger. RND-DAgger remains the same as in Table 1
+
+Qualitative interpretation On Figure 14 we report the context switches on the RaceCar environment, at the second DAgger epoch of a given seed with and without a MET. We can clearly see that the number of context switches increases and a third critical zone appeared at the top right of the map for RND-DAgger (Fig. 14a). We interpret that the bot is fairly confident in the top straight line of the course, but the last turn is difficult to manage when the velocity of the car is too high, inducing a query to the expert. There are also more context switches at the entrance of that straight line, because the learner has to be precise to pass without touching the walls or the speed ramps on the sides. We conclude that the MET increases the quality of the demonstrations without impacting the quality of the measure, that still grasps the relevant critical areas of the track.
+
+<table><tr><td rowspan="2"></td><td colspan="3">Task Performance</td><td colspan="3"># Context Switch</td></tr><tr><td>RC</td><td>HC</td><td>Maze</td><td>RC</td><td>HC</td><td>Maze</td></tr><tr><td>Lazy-DAgger + MET</td><td>0.943 ± 0.011</td><td>2374 ± 325</td><td>0.575 ± 0.101</td><td>759 ± 39</td><td>78 ± 0.0</td><td>876 ± 71</td></tr><tr><td>Lazy-DAgger</td><td>0.939 ± 0.025</td><td>2314 ± 278</td><td>0.590 ± 0.057</td><td>3437 ± 89</td><td>312 ± 45</td><td>1238 ± 79</td></tr><tr><td>Ensemble-DAgger + MET</td><td>0.935 ± 0.019</td><td>2502 ± 228</td><td>0.707 ± 0.023</td><td>77 ± 4</td><td>315 ± 84</td><td>2034 ± 79</td></tr><tr><td>Ensemble-DAgger</td><td>0.952 ± 0.018</td><td>2489 ± 108</td><td>0.626 ± 0.045</td><td>2785 ± 41</td><td>1452 ± 134</td><td>2871 ± 94</td></tr><tr><td>RND-DAgger</td><td>0.944 ± 0.014</td><td>2490 ± 160</td><td>0.717 ± 0.018</td><td>368 ± 11</td><td>708 ± 130</td><td>1214 ± 25</td></tr><tr><td>RND-DAgger w/o MET</td><td>-</td><td>2432 ± 175</td><td>-</td><td>-</td><td>1985 ± 215</td><td>-</td></tr></table>
+
+Table 4: We conduct an ablation study on the impact of the Minimal Expert Time (MET) on all baselines across our three environments. We report the final task performance and number of context switches. We conducted the same grid search for the value of W as for RND-DAgger (see section A for more details).
+
+![](images/1b78997d298744dc4089e10913c9db3721c09cabe334f21ded31317e6d36ecdd.jpg)
+
+![](images/7de347f2a409b70ac1e31d5fd0b86dfbce585582986c64f636194dbbf7023a41.jpg)
+
+![](images/80b1ac9a07095f6bf44913b5c996ff5633a479abd7cd7eb213b6d4d691c7c4a2.jpg)
+
+![](images/06ade488ed6d6671e1d36c3097962edb86cb616494c88ff4122a69152c35b163.jpg)  
+(a) RND-DAgger  $W = 0$  
+(d) RND-DAgger  $W = 30$  
+Figure 14: Ablation study: a focus on the effect of the MET on the RaceCar environment. We see an increase of the context switches and shorter episode traces as there is no Minimal Expert Time for all the methods. RND-DAgger seems to have a better grasp on the difficult zones of the track, whereas the two other measures trigger the expert rather uniformly across the track.
+
+![](images/7020785b7ced9936397073de369aaa65cf030baac80ee555056a5b1f295619d8.jpg)  
+(b) Ensemble-DAgger  $W = 0$  
+(e) Ensemble-DAgger  $W = 30$
+
+![](images/b5bc96f6edc22a96d4aecc2f29509c267c9463e7cfad5e7cc0d1088f0f116f38.jpg)  
+(c) LazyDAgger  $W = 0$  
+(f) LazyDAgger  $W = 10$
+
+# D TOTAL EXPERT TIME COMPARISONS
+
+While focusing solely on task performance and context switches provides important insights on the efficiency of a method and on a given task, it doesn't tell the complete story about expert burden. We conducted an additional analysis on the Total Expert Time, as we did in the Fig 8b on human experiments. The Table 5 shows that Ensemble-Dagger is highly time consuming for the expert without statistically outperforming RND-Dagger. For example, on Maze, it is three times longer to achieve a similar performance than with RND-DAgger, even with the addition of our MET mechanism on that baseline. This demonstrates that we are more efficient in real human expert
+
+scenarios. Moreover, and especially in a fast-paced environment like RaceCar, that baseline - and its improved MET variant - requires the expert to input actions at the same time as the agent to be able to compute the discrepancy measure, which is not natural in practice.
+
+<table><tr><td rowspan="2"></td><td colspan="3">Task Performance</td><td colspan="3">Total Expert Time</td></tr><tr><td>RC</td><td>HC</td><td>Maze</td><td>RC</td><td>HC</td><td>Maze</td></tr><tr><td>RND-DAgger</td><td>0.944 ± 0.014</td><td>2490 ± 160</td><td>0.717 ± 0.018</td><td>16664 ± 73(27.8 ± 0.1 mins)</td><td>11968 ± 107(6.60 ± 0.06 mins)</td><td>16173 ± 44(27.0 ± 0.1 mins)</td></tr><tr><td>Ensemble-DAgger</td><td>0.952 ± 0.018</td><td>2489 ± 108</td><td>0.626 ± 0.045</td><td>31394 ± 506(52.3 ± 0.8 mins)</td><td>21535 ± 774(11.8 ± 0.4 mins)</td><td>51434 ± 1507(85.7 ± 2.5 mins)</td></tr><tr><td>Ensemble-DAgger + MET</td><td>0.935 ± 0.019</td><td>2502 ± 228</td><td>0.707 ± 0.023</td><td>18216 ± 165(30.4 ± 0.3 mins)</td><td>13474 ± 631(7.40 ± 0.35 mins)</td><td>48125 ± 900(80.2 ± 1.5 mins)</td></tr></table>
+
+Table 5: The results considering the Total Expert Time i.e. the total number of frames on which the expert had to be involved to provide an action. In the case of Ensemble-DAgger, the expert action is needed at each time step to compute the discrepancy measure between the mean action of the ensemble of policies and the expert action. The time spent in minutes is computed with a typical frame rate of 30 for HalfCheetah (arbitrary) and 60 with one action taken every 6 frames, for the two game environments.

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+# EFFICIENT AUTOMATED CIRCUIT DISCOVERY IN TRANSFORMERS USING CONTEXTUAL DECOMPOSITION
+
+Aliyah R. Hsu
+
+Department of EECS
+
+UC Berkeley
+
+aliyahhsu@berkeley.edu
+
+Georgia Zhou
+
+Department of EECS
+
+UC Berkeley
+
+Yeshwanth Cherapanamjeri
+
+CSAIL, MIT
+
+Yaxuan Huang
+
+Department of Statistics
+
+UC Berkeley
+
+Anobel Y. Odisho & Peter R. Carroll
+
+Department of Urology, Epidemiology and Biostatistics
+
+UC San Francisco
+
+Bin Yu
+
+Department of Statistics, EECS
+
+Center for Computational Biology
+
+UC Berkeley
+
+# ABSTRACT
+
+Automated mechanistic interpretation research has attracted great interest due to its potential to scale explanations of neural network internals to large models. Existing automated circuit discovery work relies on activation patching or its approximations to identify subgraphs in models for specific tasks (circuits). They often suffer from slow runtime, approximation errors, and specific requirements of metrics, such as non-zero gradients. In this work, we introduce contextual decomposition for transformers (CD-T) to build interpretable circuits in large language models. CD-T can produce circuits at any level of abstraction and is the first to efficiently produce circuits as fine-grained as attention heads at specific sequence positions. CD-T is compatible to all transformer types, and requires no training or manually-crafted examples. CD-T consists of a set of mathematical equations to isolate contribution of model features. Through recursively computing contribution of all nodes in a computational graph of a model using CD-T followed by pruning, we are able to reduce circuit discovery runtime from hours to seconds compared to state-of-the-art baselines. On three standard circuit evaluation datasets (indirect object identification, greater-than comparisons, and docstring completion), we demonstrate that CD-T outperforms ACDC and EAP by better recovering the manual circuits with an average of  $97\%$  ROC AUC under low runtimes. In addition, we provide evidence that faithfulness of CD-T circuits is not due to random chance by showing our circuits are  $80\%$  more faithful than random circuits of up to  $60\%$  of the original model size. Finally, we show CD-T circuits are able to perfectly replicate original models' behavior (faithfulness  $= 1$ ) using fewer nodes than the baselines for all tasks. Our results underscore the great promise of CD-T for efficient automated mechanistic interpretability, paving the way for new insights into the workings of large language models. All code for using CD-T and reproducing results is made available on Github.
+
+# 1 INTRODUCTION
+
+Transformers (Vaswani et al., 2017) have recently demonstrated impressive predictive capabilities (Brown et al., 2020b) by learning intricate nonlinear relationships between features. How-
+
+ever, the challenge of comprehending these relationships has resulted in transformers largely being thought of as black boxes. Despite this, transformers are increasingly utilized in high-stakes domains such as medicine (e.g. medical image analysis (He et al., 2023)) and science (e.g. protein structure prediction aiding drug discovery (Jumper et al., 2021)). This highlights the necessity of understanding model behaviors. Mechanistic interpretability, a study to explain behaviors of neural networks in terms of their internal components, is at the frontier of interpretability research (Geiger et al., 2021; Geva et al., 2020; Räuker et al., 2023), as it uniquely provides an avenue for guiding manual modifications (Elhage et al., 2021; Vig et al., 2020b) and reverse-engineering solutions (Elhage et al., 2022; Meng et al., 2023).
+
+One main direction in mechanistic interpretability research is circuit discovery, where researchers aim to identify a computational subgraph (circuit) in neural networks responsible for solving a specific task (Olah et al., 2020). Prior circuit discovery work in language models has found subgraphs of attention heads and multi-layer perceptrons (MLPs) that partially or fully explain model behaviors on tasks such as indirect object identification, docstring completion, and greater-than comparisons (Wang et al., 2023; Hanna et al., 2023; Nanda et al., 2023; Heimersheim & Janiak, 2023). However, most of them require tedious manual inspection, and are thus limited to smaller models (Lindner et al., 2023). Automated Circuit Discovery (ACDC) (Conmy et al., 2023) has been proposed as a scalable end-to-end technique to identify circuits in models of arbitrary sizes by automating the pruning of unimportant edges via activation patching. Although effective, ACDC suffers from slow runtimes. More recent work has focused on improving the efficiency, for example, through linear approximating activation patching with attribution patching (Syed et al., 2023; Hanna et al., 2024), or training sparse autoencoders (SAEs) to learn attention head output patterns (O'Neill & Bui, 2024). While faster variants have been leveraged with some success, they still suffer from several drawbacks, such as approximation errors (Syed et al., 2023; Hanna et al., 2024), and the need of model training and carefully designed examples by human (O'Neill & Bui, 2024).
+
+Our main contribution in this work is the introduction of contextual decomposition for transformers (CD-T), a mathematical or analytical method to efficiently compute the contribution of model features to other model features within a transformer. This work builds on CD methods for CNNs and RNNs to examine local importance proposed in prior work (Murdoch et al., 2018; Singh et al., 2018; Jumelet et al., 2019). We significantly extend CD by defining novel mathematical decomposition principles which can be applied to transformers of all types (e.g., decoder-based, encoder-based), because these principles are indispensable for CD to scale to modern state-of-the-art (SOTA) deep learning models as they are mostly based on transformers.
+
+We are also the first, to our knowledge, to show the utility of CD methods for mechanistic interpretability by proposing an automated circuit discovery algorithm utilizing CD-T. That is, CD-T is able to discover circuits with finer granularity by further splitting into sequence positions while at the same time reduce runtime from hours to seconds, when it is compared with prior work (Syed et al., 2023; Conmy et al., 2023). CD-T is automatic or does not require any training or manually-crafted examples by humans. CD-T is fundamentally compatible with all common transformer architectures: in particular, CD-T supports circuit discovery in both BERT-like (Devlin et al., 2019) and GPT-like (Brown et al., 2020a) architectures.
+
+Specifically, for evaluation, we use three standard circuit evaluation datasets: indirect object identification (IOI), greater-than comparisons (Greater-than), and docstring completion (Docstring). We compare recovered circuits by CD-T on these datasets with two SOTA baselines, ACDC (Conmy et al., 2023) and EAP (Syed et al., 2023). We demonstrate that CD-T outperforms the baselines by better identifying attention heads in the manual circuits with an average of  $97\%$  ROC AUC due to its outstanding ability in recovering negative and supporting heads efficiently. EAP obtains similar runtime as CD-T but performs slightly worse in ROC AUC for Greater-than and IOI. ACDC is the least efficient as it often takes hours to identify a circuit, yet with no better ROC AUC on all tasks compared to other methods. In addition, we demonstrate faithfulness of CD-T circuits is not due to random chance by showing our circuits are  $80\%$  more faithful than random circuits of up to  $60\%$  of the original model size. Finally, we show CD-T circuits are able to perfectly replicate original models' behavior (faithfulness = 1) using fewer nodes  ${}^{2}$  than the baselines for all tasks.
+
+# 2 RELATED WORK
+
+# 2.1 CAUSAL INTERPRETATION
+
+Previous efforts to interpret neural networks have frequently drawn on ideas from causal inference (Pearl, 2009). Much of this work emphasizes counterfactual reasoning or enforcing causal constraints on computations to explain model outputs (Pearl, 2009; Feder et al., 2020; Geiger et al., 2021; Wu et al., 2021; Kaddour et al., 2022). A related line of research (Vig et al., 2020a; Stolfo et al., 2023; Feng & Steinhardt, 2023) takes a different approach by treating internal components (or nodes, if the model is viewed as a computational graph) as mediators and conducting causal mediation analysis, also called activation patching. In these studies, the contribution of nodes is assessed by ablating subsets of them and measuring direct and indirect effects, with the direct effects serving as proxies for their overall contribution. In contrast to viewing nodes as mediators, Goldowsky-Dill et al. (2023) and Wang et al. (2023) propose treating input-to-output paths as more expressive mediators. They introduce the technique of path patching to explore how different edge subsets in a model's computational graph affect its behavior.
+
+# 2.2 CIRCUIT DISCOVERY METHODS
+
+Ablation-based methods are essential for identifying key components within models. Through activation patching, prior research on circuit discovery in language models has uncovered subgraphs of attention heads and MLPs that explain model behavior (Wang et al., 2023; Hanna et al., 2023; Heimersheim & Janiak, 2023). However, most of these methods require multiple iterations of manual inspection and ad-hoc analysis which is specific to the model and task, limiting their application to small numbers of circuits in smaller models (Lindner et al., 2023). To address this, Conmy et al. (2023) introduced Automated Circuit Discovery (ACDC), which scales to models of any size by automatically calculating edge importance between attention heads and MLPs based on model performance on a chosen metric via activation patching. While ACDC is effective, it suffers from slow runtimes and fails to recover certain attention head subsets within the manual circuits. Syed et al. (2023) introduced Edge Attribution Patching (EAP, or  $\mathrm{ACDC}++$  as later renamed), which improves ACDC's efficiency by using a linear approximation of activation patching. However, this reliance on linear approximations can lead to overestimation of edge importance and a weaker correlation with true causal effects. EAP also struggles when the gradient of the performance metric is zero. To resolve these issues, Hanna et al. (2024) proposed Edge Attribution Patching with Integrated Gradients (EAP-IG) for more faithful circuit recovery, although it increases runtime by a constant factor. Another approach trains sparse autoencoders (SAEs) on attention head outputs and uses the SAE-learned features to map attention head contributions to identified circuits. However, SAE-based methods either require carefully crafted human-generated examples (O'Neill & Bui, 2024) or require an impractical amount of compute to train an SAE (He et al., 2024; Marks et al., 2024). Another common limitation of prior methods is the limited level of granularity of the circuits they discover, because most work either doesn't support further attention head splitting at specific sequence positions (Syed et al., 2023; Hanna et al., 2024; O'Neill & Bui, 2024), or it is fundamentally feasible but requires a prohibitive amount of time to do so (Conmy et al., 2023), which drastically hinders more fine-grained mechanistic interpretability analysis. Our method CD-T addresses the limitations of previous circuit discovery approaches by significantly reducing runtime, using mathematical decomposition equations with no approximations, allowing for arbitrary level of abstraction of circuits, and avoiding both model training and manually crafted examples (i.e. CD-T achieves all the above in an automated manner).
+
+# 2.3 CONTEXTUAL DECOMPOSITION IN NEURAL NETWORKS
+
+Contextual Decomposition (CD), introduced by Murdoch et al. (2018), attributes local importance to input features in LSTMs by analytically decomposing the output without any changes to the underlying model. For each word in a sentence, a forward pass computes activations for all cells and gates while partitioning each neuron's activation into parts influenced by a selected token or phrase (denoted as the relevant) and those that are not (denoted as the irrelevant), using a factorization of the update equations for hidden states and cell states. Jumelet et al. (2019) propose Generalized Contextual Decomposition (GCD) for unidirectional LSTMs, applying it to phenomena like number agreement and pronoun resolution. Singh et al. (2018) extend CD from LSTMs to RNNs and
+
+CNNs and introduce hierarchical interpretation with feature clustering for improved input feature attributions. A distantly related work, contextual explainability (Bertolini et al., 2022), focuses on feature attributions for each convolutional layer of CNNs by utilizing gradients as the feature importance measure. We further extend CD to transformers (CD-T) and we argue that our development of CD-T is a significant advance with two novel contributions. First, we carefully design mathematical decomposition principles tailored for transformers of all types (e.g., decoder-based, encoder-based), which makes it able to scale to larger models. Second, given prior work on CD only involves input feature attribution, to our knowledge, we are the first to demonstrate the utility of CD for mechanistic interpretability by proposing an automated circuit discovery algorithm that leverages CD-T.
+
+# 3 OUR METHOD: CD-T AND MECHANISTIC INTERPRETABILITY
+
+In this section, we first recall the basic operations of transformers (Section 3.1). We then provide technical details on contextual decomposition and its application to transformers (section 3.2). Finally, we conclude with a circuit discovery algorithm based on CD-T (section 3.3).
+
+# 3.1 TRANSFORMERS
+
+The main technical element unifying transformer-based architectures is the attention mechanism. This mechanism allows contextually useful information to be transmitted from one position in the sequence to any other position. We start by describing the operation of a general attention mechanism before discussing the modifications needed to specialize it for generation, exemplified by models such as GPT (Brown et al., 2020a), and for sequence-to-sequence transformation models such as BERT (Devlin et al., 2019). Formally, the attention mechanism computes a vector of contributions from a series of vectors  $\{x_{i}\}_{i = 1}^{l}$  to a target vector  $x$  at position  $p$  with  $x_{i}, x \in \mathbb{R}^{d}$ . An attention layer typically consists of a small number of heads, with each head parameterized by three functions, the key function  $f_{k}$ , value function  $f_{v}$ , and query function  $f_{q}$ . The output of the attention head to  $x$  is computed as follows where  $d_{k}$  denotes the output dimension of  $f_{k}$ :
+
+$$
+\forall i \in [ l ]: k _ {i} = f _ {k} \left(x _ {i}\right), q _ {x} = f _ {q} (x), v _ {i} = f _ {v} \left(x _ {i}\right) \tag {1}
+$$
+
+$$
+\forall i \in [ l ]: \alpha_ {i} = \frac {\exp \left(q _ {x} ^ {\top} k _ {i} / \sqrt {d _ {k}}\right)}{\sum_ {j \in [ l ]} \exp \left(q _ {x} ^ {\top} k _ {j} / \sqrt {d _ {k}}\right)} \tag {2}
+$$
+
+$$
+y _ {x} = \sum_ {j = 1} ^ {l} \alpha_ {i} v _ {i}. \tag {3}
+$$
+
+The functions  $f_{k}, f_{v}, f_{q}$  are commonly simple learnable transformations such as linear transformations or one hidden-layer MLPs. An attention layer is composed of a series of attention heads applied to every position in the sequence such that the concatenation of their outputs equals the input dimension  $d$ . As the output of an attention layer is a series of vectors of the same length and dimensionality as its input, this allows for stacking of these layers to build increasingly complex representations of the input so far.
+
+The two classes of transformer models we consider in this work, sequence-to-sequence models such as BERT and generative models such as GPT mainly differ in the set of positions that the attention mechanism operates on. Concretely, BERT maps a fully specified input sequence to an output sequence of the sample length while generative models produce increasingly larger sequences autoregressively with the  $(t + 1)^{th}$  token generated based on the previous  $t$  tokens. Since all the input tokens are specified for BERT-based models, the computation of the attention vector at position  $t$  utilizes representations at every position of the sequence. On the other hand, when generating the  $(t + 1)^{th}$ , the positions at  $t + 1$  and beyond are not determined. As a consequence, only the representations for tokens before  $t$  are used to compute the attention vectors.
+
+# 3.2 CONTEXTUAL DECOMPOSITION FOR TRANSFORMERS (CD-T)
+
+We will now describe the general method of contextual decomposition (CD), as well as our extension to the transformer architecture and applications to mechanistic interpretability.
+
+CD (Murdoch et al., 2018) was first proposed to compute contributions of input tokens to the output of LSTMs as a local interpretation method. CD divides each cell and hidden state into a sum of two parts: a  $\beta$  part (said to be relevant), which contains the part of this particular state that stems from the input tokens of interest, and a  $\gamma$  part (said to be irrelevant), which contains information coming from tokens outside of the list of interest.  $\beta$  is often initialized with masked input embeddings (1's specify where token positions of interest are, and 0's the opposite), and  $\gamma$  the complement of  $\beta$ .
+
+Given a decomposition of a vector  $x \in \mathbb{R}^d$  into relevant and irrelevant constituents  $\beta + \gamma = x$  (here we use the word constituents instead of "components" to avoid name collision with the "components of a neural network"), and a module  $f: \mathbb{R}^d \to \mathbb{R}^k$ , CD consists of a set of mathematical equations to determine the decomposition of the output of a module  $f$  when given  $x$  as input:  $f(x) \in \mathbb{R}^k = \beta_o + \gamma_o$ . The general principle is to find a symbolic expression for  $f(x)$  in terms of  $\beta$  and  $\gamma$ , and group the terms in the expression according to whether they are solely a function of  $\beta$  or not: the sum of terms solely relying on  $\beta$  becomes  $\beta_o$ , and the sum of the remaining terms becomes  $\gamma_o$ . We refer interested readers to Murdoch et al. (2018) and Singh et al. (2019) for further examples of decomposition formulas of various modules. These decomposition rules can be composed through multiple modules (i.e., if  $f(\beta_1, \gamma_1) = \beta_2, \gamma_2$ , and  $g(\beta_2, \gamma_2) = \beta_3, \gamma_3$ , then  $g(f(\beta_1, \gamma_1)) = \beta_3, \gamma_3$ ) in order to define decomposition rules for larger computational blocks, including entire neural networks.
+
+Other authors have heuristically adjusted the decomposition rules for a module to reflect specific mechanisms by which (the relevance term in an earlier layer  $i - 1$ ) can affect (the relevance term in a later layer  $i$ ) that are unaccounted for by this rule. For example, Singh et al. (2019) adjust the decomposition rule for affine transformations  $f(x) = Wx + b$ , so that the bias term does not affect the relative magnitudes of  $\beta$  and  $\gamma$ :
+
+$$
+\beta_ {i} = W \beta_ {i - 1} + \frac {\left| W \beta_ {i - 1} \right|}{\left| W \beta_ {i - 1} \right| + \left| W \gamma_ {i - 1} \right|} \cdot b, \tag {4}
+$$
+
+$$
+\gamma_ {i} = W \gamma_ {i - 1} + \frac {\left| W \gamma_ {i - 1} \right|}{\left| W \beta_ {i - 1} \right| + \left| W \gamma_ {i - 1} \right|} \cdot b. \tag {5}
+$$
+
+We succinctly represent the above equations as  $\beta_{i},\gamma_{i} = LinearDecomp(\beta_{i - 1},\gamma_{i - 1})$  , where  $W$  and  $b$  are understood to be tunable parameters of a neural network module.
+
+Given the query, key, and value functions mentioned in Section 3.1 are linear transformations, the only module not accounted for in the context of transformers is the self-attention module. We handle the decomposition of the attention equations as follows:
+
+$$
+\beta_ {\text {q u e r y}}, \gamma_ {\text {q u e r y}} = \operatorname {L i n e a r D e c o m p} \left(\beta_ {\text {i n}}, \gamma_ {\text {i n}}\right) \tag {6}
+$$
+
+$$
+\beta_ {k e y}, \gamma_ {k e y} = \operatorname {L i n e a r D e c o m p} \left(\beta_ {i n}, \gamma_ {i n}\right) \tag {7}
+$$
+
+$$
+\beta_ {\text {v a l u e}}, \gamma_ {\text {v a l u e}} = \operatorname {L i n e a r D e c o m p} \left(\beta_ {\text {i n}}, \gamma_ {\text {i n}}\right) \tag {8}
+$$
+
+$$
+\beta_ {\text {a t t e n t i o n}} = \beta_ {\text {k e y}} ^ {T} \beta_ {\text {q u e r y}} / \sqrt {d _ {k}} \tag {9}
+$$
+
+$$
+\gamma_ {\text {a t t e n t i o n}} = \left(\left(\beta_ {\text {k e y}} + \gamma_ {\text {k e y}}\right) ^ {T} \left(\beta_ {\text {q u e r y}} + \gamma_ {\text {q u e r y}}\right) / \sqrt {d _ {k}}\right) - \beta_ {\text {a t t e n t i o n}} \tag {10}
+$$
+
+$$
+\beta_ {p r o b s} = \operatorname {S o f t m a x} \left(\operatorname {M a s k} \left(\beta_ {\text {a t t e n t i o n}}\right)\right) \tag {11}
+$$
+
+$$
+\gamma_ {p r o b s} = \operatorname {S o f t m a x} \left(\operatorname {M a s k} \left(\beta_ {\text {a t t e n t i o n}} + \gamma_ {\text {a t t e n t i o n}}\right)\right) - \beta_ {p r o b s} \tag {12}
+$$
+
+$$
+\beta_ {z} = \beta_ {\text {p r o b s}} * \beta_ {\text {v a l u e}} \tag {13}
+$$
+
+$$
+\gamma_ {z} = \left(\beta_ {\text {p r o b s}} + \gamma_ {\text {p r o b s}}\right) * \left(\beta_ {\text {v a l u e}} + \gamma_ {\text {v a l u e}}\right) - \beta_ {z} \tag {14}
+$$
+
+$$
+\beta_ {\text {o u t p u t}}, \gamma_ {\text {o u t p u t}} = \operatorname {L i n e a r D e c o m p} \left(\beta_ {z}, \gamma_ {z}\right) \tag {15}
+$$
+
+where above  $Mask$  represents a causal mask in appropriate architectures and Softmax is the softmax along the same dimension as in a standard implementation of self-attention.
+
+Although in the original setting of CD, "input" and "output" refer to an input sequence  $x$  and a model's output logits, the method generalizes to arbitrary source and target activations in the model. Furthermore, in the original context, the "relevant" portion of the decomposition was equal to 1 on a subsequence of the input and 0 elsewhere, but in a mechanistic interpretability context, the CD framework allows us to decompose activations into any choice of  $\beta, \gamma$ . Thinking of the network again as a computational graph, this gives us a method of calculating the effect of an arbitrary constituent of the activations of an arbitrary node on the activations of another node.
+
+# 3.2.1 CHOOSING A DECOMPOSITION OF SOURCE NODES FOR MECHANISTIC INTERPRETABILITY
+
+In a manner analogous to the choice of ablation method elsewhere in mechanistic interpretability, the correct choice of starting decomposition can be subtle, and sometimes specific to the task. A general principle is that the relevant constituent of the activations at some node is the constituent that we hypothesize has a counterfactual effect on the task; often, this means the deviation from the mean activations over some distribution. (It follows that the irrelevant constituent of the activation at this node is the mean activation, which matches intuitions about mean-ablation.) We provide details on the evaluation tasks and name the distributions used for ablation in Appendix A.
+
+# 3.3 AUTOMATING CIRCUIT DISCOVERY
+
+# 3.3.1 DEFINITIONS OF CIRCUIT DISCOVERY
+
+Before delving into our method, we first formally define a circuit. To maintain consistency in notations with prior work (Shi et al., 2024; Elhage et al., 2021), we view a model as a computational graph  $M$ , where nodes are activations of the model components in its forward pass (e.g. attention heads and MLPs) and edges are the interactions between those components, and a circuit  $C$  is a subgraph of  $M$  responsible for certain behavior of interest. Naturally, the first step of circuit discovery is to define a task of interest to investigate. This requires specifying a dataset and a task-specific metric to measure the performance of  $M$  and  $C$ , typically, by maximization of the task metric. Given an input  $x$ , similarly as to how the entire model defines a function  $M(x)$  from inputs to logits, we also associate each circuit  $C$  with a function  $C(x)$ , defined by ablating away the effect of all components in  $M \backslash C$  (i.e. the components not included in  $C$ ). Circuit discovery may be performed on a single input example or, likelier, the result may be averaged over multiple input examples to reduce variance. Typically we find averaging across 10-100 samples, depending on the tasks being studied, can yield stable performance. Detailed information about which models are studied, input sample counts, and how the task metrics are defined is found in Appendix A. Finally, note that our method finds the circuit of interest but does not automatically generate an interpretation of it, however, known interpretability techniques (Wang et al., 2023) can be used to do so.
+
+# 3.3.2 METHOD
+
+Here we present our algorithm for discovering circuits with CD-T. Although it is possible to perform CD at arbitrary granularity and on arbitrary components of a transformer (e.g., MLPs, query/key/value vectors), in this paper we focus solely on finding circuits consisting of attention heads in a transformer to be comparable to prior work, and our unit of analysis is either the output of an attention head or the output of an attention head at a specific sequence position. For clarity, our method focuses on including or excluding specific nodes from a computational graph, rather than edges; CD-T at once implicitly models both direct and indirect effects of one node to another, and our algorithm does not make a distinction between the two.
+
+In each iteration, we search over potential source nodes to find the ones with highest relevance to a set of target nodes (initialized to the task objective). Once we find the set of nodes which are most relevant to the task, we prune them by some simple heuristic (in our experiments, simply greedily removing the nodes with the least magnitude of impact was often sufficient, or sometimes pruning was not necessary), and designate these most important nodes the target nodes for the next iteration of the algorithm. We halt when the collection of nodes achieves adequate performance (e.g., comparable to the whole network), or has not improved from the previous iteration.
+
+We need to specify how "highest relevance to a set of target nodes" is determined. As described above, we start with a decomposition  $\beta_{s},\gamma_{s}$  of the output of a source attention head  $s$  and we are interested in its relevance  $R(s,T)$  to a set of target nodes  $T$ . Starting from  $s$ , we propagate the decomposition forward through the network using the CD-T equations for each module in the computational graph between  $s$  and  $T$ ; again, if there is some series of functions  $f_{n},\ldots f_{1}$ , so that  $T = f_{n}(f_{n - 1}(\ldots f_{1}(s)))$ , and  $f_{1}(\beta_{s} + \gamma_{s}) = \beta_{1} + \gamma_{1}$ ,  $f_{2}(\beta_{1} + \gamma_{1}) = \beta_{2} + \gamma_{2}$ , ...,  $f_{n}(\beta_{n - 1} + \gamma_{n - 1}) = \beta_{T} + \gamma_{T}$ , we can straightforwardly apply the modules in order to obtain  $\beta_{T},\gamma_{T}$ . In the case that  $T$  is the output of the network,  $\beta_{T}$  will have matching dimensions, and it is usually appropriate to let  $R(s,T)$  be the task-specific objective as evaluated on  $\beta_{T}$ . (One intuitive instance of this is the case
+
+Algorithm 1 Building a circuit using CD-T for a specified task  
+Input: datapoint  $x$  ,precomputed mean activations  $\mu$    
+Denote  $a_{x}(s)$  as the activation of  $s$  on input  $x$    
+Denote  $\mu (s)$  as the precomputed mean activations of  $s$  on an appropriate distribution. Initialize  $C$  to store the circuit. Initialize  $T$  to be the output of the model   
+repeat for each attention head  $s$  upstream of  $T$  .. Run the model from  $x$  to  $s$  , and compute  $\beta_{s} = a_{x}(s) - \mu (s),\gamma_{s} = \mu (s)$  Propagate decomposition to target nodes with CD-T, obtaining  $\beta_T,\gamma_T$  Use  $\beta_T,\gamma_T$  to compute the relevance of s to  $T,R(s,T)$  using equation 16 end for Heuristically pick the source nodes  $S$  with the highest values of  $R(s,T)$  ; define this set as  $S$ $C\gets C\cup S$  {# Refinement: Greedily prune away the unnecessary nodes} repeat for each node n in  $C$  ..  $C^\prime \leftarrow C\backslash n$  If  $C^\prime (x) > C(x)\colon C\gets C^\prime$  end for until  $C$  has not changed since the last iteration If  $|C(x) - M(x)| <   \epsilon$  ,return  $C$  {# Circuit performs close to the full model} If  $C(x)$  is not greater than last iteration, return  $C$ $T\gets S$    
+until no upstream attention heads to  $T$  are available   
+return  $C$
+
+where  $T$  is a single "score" whose value we care about, so that  $\beta_{T}$  is the contribution of  $s$  to that score.) However, when  $T$  is an arbitrary set of nodes in the network's internals, we must define a proxy metric for the relevance of  $s$  to  $T$ ; in this case, we define the relevance of a source node to a set of target nodes to be the sum of the relevances to the target nodes  $t \in T$ , and in this paper for  $R(s,t)$  we choose a quick-to-compute measure of the size of  $\beta_{t}$ :
+
+$$
+R (s, T) := \sum_ {t \in T} R (s, t) = \sum_ {t \in T} \frac {\left\| \beta_ {t} \right\| _ {l _ {1}}}{\left\| \gamma_ {t} \right\| _ {l _ {1}}}. \tag {16}
+$$
+
+Careful readers may have noticed that this particular metric measures the magnitude of one node's contribution to another, but not the "direction", so that nodes which maximize this metric may actually contribute negatively to the task under investigation. We found in our experiments that this does not preclude us from finding functioning circuits. On the contrary, this allows us to straightforwardly find nodes which have significant negative contribution to a task metric, such as the "Negative Name Mover Heads" in the IOI task. In principle, it is possible that different choices of target relevance metric can be developed to adjust the behavior of the circuit discovery algorithm to avoid (or accentuate) this behavior; we do not explore this in the paper but identify it as a possible avenue for further investigation. Our complete algorithm is described in Algorithm 1, presented in the specific case where we have chosen to decompose our source nodes  $s$  so that  $\beta_{s}$  is the activation's deviation from the mean over some distribution. More details on the heuristics used and a complexity analysis of the algorithm is found in Appendix B.
+
+![](images/07ff53ef295b267c43c18247f518958c08ffd24d6826b19b6e4939a567977029.jpg)  
+Figure 1: Left: Log algorithm runtime and ROC AUC comparison. Each dot represents an average measurement on one task with specific methods differentiated by colors.; Right: The relative faithfulness of CD-T circuits compared to random circuits from the reference distribution of varying sizes (x-axis). Dotted vertical lines indicate the actual size of the circuits.  $C$  denotes a CD-T circuit.  $C^r$  denotes a random circuit.  $M$  denotes the full model.
+
+![](images/4bb2fa9cf4e47b9b758c10a3b0dcba9148f373df210a23ac8a0f7c3824ebc11a.jpg)
+
+# 4 EVALUATING CIRCUIT DISCOVERY ALGORITHMS
+
+# 4.1 EXPERIMENTAL SETUP
+
+We compare the performance of circuits recovered by CD-T with those obtained by two SOTA baselines: ACDC (Conmy et al., 2023) and EAP (Syed et al., 2023). SAE-based methods (O'Neill & Bui, 2024; Marks et al., 2024; He et al., 2024) and classic neural network pruning methods are not considered here because they either require model training or focus more on compressing neural networks for faster inference to reduce storage requirements. Furthermore, pruning for interpretability is also shown to be outperformed by ACDC (Conmy et al., 2023). We evaluate the circuit performance on three tasks: indirect object identification (IOI) (Wang et al., 2023), greater-than (Greater-than) (Hanna et al., 2023), and docstring completion (Docstring) (Heimersheim & Janiak, 2023) (see Appendix A for details). They are three standard evaluation tasks to benchmark circuit discovery methods as prior work (Hanna et al., 2023; Wang et al., 2023; Heimersheim & Janiak, 2023) has manually found circuits in language models explaining for those tasks, which often serve as an imperfect reference standard for circuit discovery work (Conmy et al., 2023; Syed et al., 2023; O'Neill & Bui, 2024) due to the lack of ground-truth circuits. For both baselines, we perform node-level patching to be comparable since CD-T is a node-level patching technique. All experiments are conducted on an NVIDIA A100 GPU.
+
+# 4.2 CD-T EXCELIS IN IDENTIFYING THE CIRCUIT RESPONSIBLE FOR THE UNDERLYING ALGORITHM EFFICIENTLY
+
+# 4.2.1 RUNTIME AND THE RECOVERY OF MANUAL CIRCUITS
+
+Leveraging the manually discovered circuits as a reference standard, we measure how much overlap there is between the recovered circuits and the reference circuits using ROC AUC. Specifically, we measure the ROC AUC by sweeping through a range of thresholds that determine cutoff scores to keep nodes in a circuit. For CD-T, we test by varying the percentile of top nodes to extract in every iteration, in the range of [90, 99]. At the same time, we also compute the average algorithm runtime across runs when sweeping through the thresholds for each method. Although CD-T supports arbitrary granular representations of the circuit, the runtime is measured by building a "coarse node-level" circuits by not splitting into sequence positions to be comparable with EAP and ACDC
+
+because EAP doesn't support measuring node importance at sequence positions, and ACDC takes an impractical amount of time to achieve sequence-position splitting. To demonstrate CD-T's capability to identify circuits responsible for underlying algorithm for tasks with great efficiency, we plot the algorithm runtime against the ROC AUC and report the result in Figure 1-Left. From Figure 1-Left we can see CD-T outperforms the baselines by achieving high ROC AUC and low runtime at the same time. EAP obtains similar runtime as CD-T but performs slightly worse in recovering manual circuits for Greater-than and IOI. Even under this coarse nodel-level abstraction, ACDC is the least efficient as it often takes hours to identify a circuit, yet with no better ROC AUC on all tasks compared to other methods. To explain the better ROC AUC performance of CD-T, we find that CD-T is especially good at identifying negative heads and supporting heads, such as the negative name-mover heads and backup name-mover heads in IOI, which other algorithms struggle to do (more discussion in Appendix A).
+
+# 4.2.2 PROBABILITY OF IDENTIFIED CIRCUITS TO BE MORE FAITHFUL THAN RANDOM CIRCUITS
+
+Given a circuit and a task, one can evaluate how well the circuit performs on the task by measuring the task-specific metric, such as the logit difference on IOI. However, in mechanistic interpretability, instead of aiming to identify the best-performing circuits on the task-specific metric, an ideal circuit should be able to faithfully replicate the full model behavior. Following prior work (O'Neill & Bui, 2024; Marks et al., 2024), faithfulness is defined as the proportion of the full model's performance that a circuit explains, relative to the baseline performance when no specific input information is provided. Given a task, faithfulness is computed as  $\frac{m(C) - m(\emptyset)}{m(M) - m(\emptyset)}$ , where  $m(C)$ ,  $m(\emptyset)$ , and  $m(M)$  are the average of the task-specific metric performance over the dataset for the circuit, the corrupted model with all heads ablated, and the full model, respectively. To complement the imperfect nature of the manual circuits, leveraging faithfulness, here we provide another evidence to validate CD-T's mechanism perservation of the original model. We adopt a circuit hypothesis testing framework proposed in Shi et al. (2024) to test whether a circuit preserves the original model's performance by measuring the probability of a circuit  $C$  to be more faithful to the original model  $M$ , compared to random circuits  $C^r$  of the same size sampling from a reference distribution. This test verifies that the circuit is not a simple lucky draw from the distribution of random circuits, and ensures that it is better than at least a fraction  $q^*$  of random circuits. In our experiment, we sample random circuits from all possible nodes in the full model, and we repeat the process for 10 times. Figure 1-Right shows the results of the best CD-T circuits (as measured by faithfulness and among different granularity) on the 3 tasks. Our best circuit for IOI is from attention heads at specific sequence positions, while for Greater-than and Docstring, they are from attention heads at a coarse level without the splitting. The results show CD-T circuits are  $80\%$  ( $q^* = 0.8$ ) more faithful than random circuits of up to around  $60\%$  of the original model's size for IOI and Greater-than, and even those of up to  $80\%$  of the original model's size for Docstring, with just a small percentage of nodes in the CD-T circuits ( $0.4\%$  of full model size for IOI,  $5.5\%$  for Greater-than, and  $31.3\%$  for Docstring). This finding suggests the faithfulness of CD-T circuits is not due to random chance, and that a more fine-grained abstraction of circuits sometimes is necessary to obtain better faithfulness with an extremely small and specific set of nodes, as in the IOI case.
+
+# 4.3 IDENTIFIED CIRCUITS MATCH FULL MODEL PERFORMANCE
+
+# 4.3.1 FAITHFULNESS OF CIRCUITS OF VARYING SIZES
+
+To understand how faithfulness changes as the size of the circuits grows, we compute node importance before pruning for each method  $^5$ , and track faithfulness performance as we add circuit nodes in order of importance, while ablating all other nodes with corrupted activations (see Appendix A). We sample attention heads in the original model one by one in random order to construct the random baseline, and report the results in Figure 2. Since the goal of mechanistic interpretability is to replicate the original model's behavior, namely achieving faithfulness  $= 1^{6}$ , with the smallest amount
+
+![](images/df2ce05c50364f92ad3c228573822885bb5e092f13c49753512d4f3c1abe6848.jpg)  
+Figure 2: Faithfulness of CD-T circuits, EAP circuits, and randomly selected circuits of equivalent size for IOI, Greater-than and Docstring tasks. CD-T circuits obtain the full model's performance (faithfulness of 1) faster than EAP as attention heads are added in order of importance.
+
+![](images/34e655cfbf638a1711eaee0b67232362425f93f7b42b7b036c67c1dc883c70a8.jpg)
+
+![](images/a869f3f68c9b61b645744a424f8dd7964c61ab56c7ef60aeee54d35ad169ede1.jpg)
+
+of nodes possible, this is naturally the first question we ask. First, we find CD-T circuits are able to perfectly replicate model behavior (faithfulness  $= 1$ ) using fewer nodes than EAP for all the three tasks. Second, manual circuits outperform both CD-T and EAP of the same size, indicating room of improvement of advanced circuit discovery methods. Finally, we compare the effect of ablation methods on CD-T circuits' performance by including CD-T circuits obtained from zero-ablation in the plot, and find that mean-ablation tends to yield higher faithfulness compared to zero-ablation on Greater-than and Docstring, except for IOI where they are quite comparable. Mean-ablation is often viewed as a preferred way of ablation because it preserves a general concept of the task, hence is less destructive than zero-ablation. Similar finding is also discussed in other circuit discovery work (Conmy et al., 2023; Wang et al., 2023).
+
+# 5 DISCUSSION AND LIMITATIONS
+
+Despite having shown both qualitative and quantitative evidence of the strengths of CD-T in automated circuit discovery, our results is limited in scale and to the datasets and prior interpretation methods evaluated. More work is needed to generalize these findings to a broader set of models, datasets, and interpretation methods. Although our methods can be applied to construct circuits comprised of any of a neural network's internal components, we only discussed circuits built with purely attention heads. Automating the discovery of circuits with different/heterogeneous components, particularly the MLPs of a transformer, is a promising direction for further investigation. Finally, our method is itself limited in two ways: first, as mentioned in 3.3, the metric which measures the relevance of a source node to a target node does not differentiate signs of the contribution, which might be useful in an interpretability context. Second, while the algorithm is capable of finding circuits under reasonable default settings, some amount of effort is still applied to tune the heuristic elements, such as the number of heads to select in each iteration.
+
+# 6 CONCLUSIONS
+
+We have proposed contextual decomposition for transformers (CD-T), a set of mathematical decomposition principles to efficiently compute the contribution of model features to other model features within a transformer. CD-T can produce circuits of arbitrary granularity, and is the first able to produce circuits as fine-grained as attention heads at specific sequence positions while at the same time reduce runtime from hours to seconds, when it is compared with prior work (Syed et al., 2023; Conmy et al., 2023). CD-T improves upon limitations of prior work, as it is automatic and does not require any training or manually-crafted examples by humans. CD-T is fundamentally compatible with all common transformer architectures. On three standard circuit evaluation datasets (indirect object identification, greater-than comparisons, and docstring completion), we demonstrate that CD-
+
+T outperforms ACDC and EAP by better recovering the manual circuits with an average of  $97\%$  ROC AUC under low runtimes. In addition, we provide evidence that faithfulness of CD-T circuits is not due to random chance by showing our circuits are  $80\%$  more faithful than random circuits of up to  $60\%$  of the original model size. Finally, we show CD-T circuits are able to perfectly replicate original models' behavior (faithfulness  $= 1$ ) using fewer nodes than the baselines for all tasks. We hope that our future open-source implementation of CD-T benefits mechanistic interpretability research community by opening up a window from enabling more fine-grained analysis within a practical runtime.
+
+# ACKNOWLEDGMENTS
+
+GZ would like to thank Shawn Hu for valuable discussions that contributed greatly to mathematical formalization, experiment design, and important heuristic components of our method. We gratefully acknowledge partial support from NSF grant DMS-2413265, NSF grant 2023505 on Collaborative Research: Foundations of Data Science Institute (FODSI), the NSF and the Simons Foundation for the Collaboration on the Theoretical Foundations of Deep Learning through awards DMS-2031883 and 814639, NSF grant MC2378 to the Institute for Artificial CyberThreat Intelligence and Operation (ACTION), and a Berkeley Deep Drive (BDD) grant from BAIR and a Dean's fund from CoE, both at UC Berkeley.
+
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+
+# A EXPERIMENT DETAILS
+
+# A.1 TASK DESCRIPTION
+
+Here we describe each task and how the experiments are performed in more detail; we also specify the exact model components which comprise the circuits we analyzed in the main paper. In this section, we use the convention of specifying attention heads in a model by the tuple (layer index, head index). For the IOI task, we add a third entry for output of the head at a specific sequence position, and follow their nomenclature for semantically labeling the sequence positions.
+
+# A.1.1 INDIRECT OBJECT IDENTIFICATION (IOI) (WANG ET AL., 2023)
+
+The indirect object identification (IOI) task is to predict the indirect object in a sentence with two entities, such as identifying "Mary" in the sequence "When Mary and John went for a walk, John gave an apple to ...". The objective originally defined for this task, which we also use, is the difference between the predicted logit of the indirect object (IO) token and the subject (S) token. Wang et al. (2023) carefully design their experiment and dataset code so that mean ablation occurs using the mean activations over the corrupted "ABC dataset", which replaces the three name tokens in the task with random names. Likewise, when setting the decomposition of a node, we set the "relevant" component to the deviation from the mean activations on the ABC dataset. This task was performed on GPT2-small. GPT-2 correctly performs this task about 99 percent of the time, so we did not take special measures to account for those samples where it doesn't in our circuit analysis. We identify circuits using 25 IOI samples drawn from mixed templates, and mean ablation is conducted using the corrupted ABC dataset. Another set of 100 IOI samples are used in evaluation. Manual circuit of IOI compared against in the ROC AUC experiment is: [(2, 2), (4, 11), (0, 1), (3, 0), (0, 10), (5, 5), (6, 9), (5, 8), (5, 9), (7, 3), (7, 9), (8, 6), (8, 10), (10, 7), (11, 0), (9, 9), (9, 6), (10, 0), (9, 0), (9, 7), (10, 1), (10, 2), (10, 6), (10, 10), (11, 2), (11, 9)], which is from Figure 2 in Wang et al. (2023).
+
+Details of Circuit Analysis For this task, in order to provide some intuitions about what CD-T calculates, we provide a number of heatmaps of the relevance scores at specific positions during various iterations of the circuit analysis. In this section, we don't follow our automated circuit discovery algorithm exactly, but instead partially follow Wang et al. (2023)'s analysis to decide what sequence positions to search over and visualize.
+
+The first iteration of the algorithm finds the Name Mover Heads: (9, 9, end), (10, 0, end), and (9, 6, end); the Negative Name Mover Heads: (10, 7, end), (11, 10, end); and some Backup Name Mover Heads: (10, 2, end), (10, 6, end), (10, 10, end), described by Wang et al. (2023)
+
+Following Wang et al. (2023)'s analysis further, and deviating from the normal course of our circuit-finding algorithm, we compute the relevance of nodes to the Name Mover Heads on just the end
+
+![](images/08e4710c4e3ca413e246c6da259b94e5cce6490c2c7f69c46b6589fcb9aa8c92.jpg)  
+Relevance of nodes to IOI task metric at last sequence position
+
+position, and find what they named the "S-Inhibition Heads", at (8, 10, end), (7, 9, end), and (7, 3, end), though there are some other relevant-looking contenders:
+
+![](images/28879fd1c218c17a0d568ee7cdd52f1c8a376618d68e69d47c82074f3d0ec7c4.jpg)  
+Relevance of nodes to output residuals of Name Mover Heads at last sequence position, normalized by mean rel per layer
+
+Continuing to follow their analysis, Wang et al. (2023)'s analysis further, we compute the relevance of nodes to the S-Inhibition Heads at the S2 position, and find our first minor disagreement with their process: though we find two of what they named the Induction Heads at (5, 5, S2), (5, 8, S2), (5, 9, S2) we don't find the one at (6, 9, S2) but instead find one at (5, 10, S2), and find a head their later analysis called a Duplicate Token Head, at (3, 0, S2).
+
+Next we compute the relevance of nodes to the Induction Heads at the S2 position, and find that (3, 0, S2) mostly drowns out the signal of the other Duplicate Token Heads:
+
+Finally, we compute the relevance of nodes to the Induction Heads at the S1+1 position, and find (4, 11, S1+1), with the other Previous Token Head they found at (2, 2, S1+1) a top contender, though there are other heads not accounted for in their analysis:
+
+Overall, there is significant but not perfect agreement with the results and analysis of the IOI paper. We also attempted analysis by computing relevance to intermediate matrices (i.e., the key, query,
+
+![](images/ca25ed847828d6a9657387280b1efcff825292493749b1e2aa2028fb70e487c7.jpg)  
+Relevance of nodes to output residuals of S-inhibition heads at S2 sequence position, normalized
+
+![](images/968a564d80e3d938b03c75d4f0e60b85d06b5af58239ef282a6ed82ba71588f6.jpg)  
+(Duplicate Token Heads) Relevance of nodes to output residuals of Induction heads at S2 position, normalized for mean relevance among heads in layer
+
+value vectors) in the attention calculation, but found the plots to be qualitatively similar, possibly due to the fact that CD-T by design propagates relevances from these vectors to the attention head outputs as well.
+
+Another fact of note is that the scales on these plots vary substantially. It would be desirable to find an interpretable normalization method to put these on the same scale with some intrinsic meaning, or otherwise explain the causes of this phenomenon.
+
+# A.1.2 GREATER-THAN (HANNA ET AL., 2023)
+
+The Greater than task is to predict the last two digits in an incomplete sentence following the template "The [noun] lasted from the year XXYY to the year XX". And we expect the model to assign higher probability to years greater than YY. The objective originally defined for this task, which we also use, is the sum of probabilities assigned to tokens corresponding to greater years, minus the sum of probabilities assigned to tokens corresponding to lesser years. (Some probability is assigned to tokens which don't correspond to numbers at all.) For our "mean-ablation", we simply take the
+
+![](images/1127a4191d37f5a146250384f4a4ef43b6800230f494ca05a1a89b427829215e.jpg)  
+previous token heads) relevance of nodes to output of Induction heads at S1+1 position, normalized for mean relevance among heads in layer
+
+mean over the activations over 100 negative datapoints (impossible completions, with the ending year preceding the starting century), and as above, when setting the decomposition at a source node, define the relevant component to be the deviation from the mean activation over this distribution. This task was performed on GPT2-small. GPT-2 correctly performs this task about 99 percent of the time, so we did not take special measures to account for those samples where it doesn't in our circuit analysis. We identify circuits using a random sample of 100 datapoints provided by Hanna et al. (2023), and mean ablation is conducted using the negative impossible completion samples. Another set of 100 samples are used in evaluation.
+
+It should be noted that our result and the results in Hanna et al. (2023) are not directly comparable, since they also attempt to investigate the influence of MLPs and their resulting circuit removes most of the MLPs in GPT-2. For the comparisons found in the main paper, we have compared our result (with all MLPs) to the circuit which is obtained by taking all the attention heads named in Hanna et al. (2023) (also with all MLPs), which is a circuit distinct from the one found in their work.
+
+Manual circuit of Greater-than compared against in the ROC AUC experiment is: [(5, 1), (5, 5), (6, 1), (6, 9), (7, 10), (8, 8), (8, 11), (9, 1)].
+
+Independently of the comparison, it remains true that keeping the relatively small proportion of attention heads in our circuit results in recovering almost all of GPT-2's capability on this task; see below.
+
+<table><tr><td></td><td>Task-specific metric</td><td>Correct guess rate</td></tr><tr><td>Full model</td><td>0.817</td><td>0.992</td></tr><tr><td>All attention heads ablated</td><td>-2.095</td><td>0</td></tr><tr><td>Their circuit</td><td>0.768</td><td>0.989</td></tr><tr><td>Their circuit (most MLPs ablated)</td><td>0.727</td><td>N/A</td></tr><tr><td>Our circuit</td><td>0.761</td><td>0.981</td></tr></table>
+
+# A.1.3 DOCSTRING (HEIMERSHEIM & JANIAK, 2023)
+
+The goal of the Docstring task is to predict the next variable name in a Python docstring. For example, given a function with variable names LOAD, SIZE, FILES, and LAST, the task is to predict the word after one :PARAM. According to docstring conventions, this should be the variable name in the function definition which follows the most recent variable name to appear after :PARAM. The objective originally defined for this task is the logit assigned to the correct variable name, minus the max logit assigned to all other variable name tokens found in the function signature. For
+
+our "mean-ablation", we use their random_random dataset which randomize both the variable names in the function definition and in the docstring of prompts, and correspondingly, when setting the decomposition at a source node, define the relevant component to be the deviation from the mean activation over this distribution. We identify circuits using a 100 datapoints sampling for the dataset provided by Heimersheim & Janiak (2023), and mean ablation is conducted using the corrupted random_random dataset. Another set of 100 samples are used in evaluation. This task was performed on a 4-layer attention-only transformer trained on natural language and Python code (attn-only-41) released with the TransformerLens library for the express purpose of facilitating mechanistic interpretability research. Another complication is that the toy model only guesses the correct token between 60 and 65 percent of the time. To account for this, we perform our circuit analysis and evaluation on the subset of input examples for which the model performs the task correctly.
+
+Our circuit consists of the nodes  $(3,0)$ ,  $(3,6)$ ,  $(1,4)$ ,  $(0,5)$ ,  $(0,0)$ ,  $(1,0)$ ,  $(1,4)$ ,  $(0,1)$ ,  $(2,3)$ ,  $(1,2)$ . Heimersheim & Janiak (2023) find the circuit  $(0,2)$ ,  $(0,4)$ ,  $(0,5)$ ,  $(1,2)$ ,  $(1,4)$ ,  $(2,0)$ ,  $(3,0)$ ,  $(3,6)$  with their initial analysis, and heuristically observe that three heads help to obtain the "augmented circuit"  $(0,2)$ ,  $(0,4)$ ,  $(0,5)$ ,  $(1,2)$ ,  $(1,4)$ ,  $(2,0)$ ,  $(3,0)$ ,  $(3,6)$ ,  $(1,0)$ ,  $(0,1)$ ,  $(2,3)$ .
+
+Manual circuit of Docstring compared against in the ROC AUC experiment is: [(0, 2), (0, 4), (0, 5), (1, 2), (1, 4), (2, 0), (3, 0), (3, 6)].
+
+<table><tr><td></td><td>Task-specific metric</td><td>Correct guess rate</td></tr><tr><td>Full model</td><td>3.661</td><td>1.0</td></tr><tr><td>All attention heads ablated</td><td>-2.095</td><td>0</td></tr><tr><td>Their circuit</td><td>2.884</td><td>0.54</td></tr><tr><td>Augmented circuit</td><td>3.524</td><td>0.66</td></tr><tr><td>Our circuit</td><td>3.235</td><td>0.57</td></tr></table>
+
+# B ALGORITHM DETAILS
+
+# B.1 HEURISTICS
+
+In addition to the greedy pruning presented in Algorithm 1 as a refinement step, here we describe other heuristic elements to the algorithm.
+
+- The threshold for determining which set S of highest-contributing nodes may be varied: it is possible to pick the top  $N$  nodes or fraction of nodes, or to automatically detect outliers. Empirically, we find that the distribution of node contributions varies quite severely, so finding a heuristic which works in all cases is actually an object of future work. In this paper, we set the threshold by varying the percentile of top nodes to extract, in the range of [90, 99] to obtain the ROC AUC in section 4.2.1.  
+- Empirically we find, for a fixed target node, the relevance scores of source nodes in different layers to this target node may have different expected magnitudes, due to the numerical effects of propagation through the network. To account for this, we normalize the relevance scores by dividing by the average magnitude across scores found in a given layer.  
+- To avoid the risk of propagation equations becoming numerically unstable after a large number of iterations, especially if the relevant and irrelevant constituents differ in sign at a specific index, we set the value of one of rel/irrel at this position to 0 and the other to the sum of the two terms.
+
+# B.2 COMPLEXITY ANALYSIS
+
+To provide more clarity, the computational complexity of the algorithm satisfy the following properties:
+
+- Each decomposition (of a set of target nodes with respect to a set of source nodes) requires cost in FLOPs similar to one forward pass of the model (and often less, since values prior
+
+to the source nodes can be cached and values after the target nodes do not need to be calculated).
+
+- The core of the algorithm is a loop, where in each iteration, we search over all nodes which can potentially have high relevance to the target nodes. (This means that heuristically excluding some nodes from the search can potentially significantly decrease cost in FLOPs, and cost in FLOPs increases linearly with respect to the granularity with which we separate the nodes in the model.)  
+- The memory footprint of a single decomposition, including the forward pass, is a small (less than 3) constant multiple of the cost of a forward pass; the only added costs are to keep track of the relevant and irrelevant constituent tensors separately, as well as bookkeeping of components of the target decomposition metric.  
+- The cost (in FLOPs or memory) of performing the analysis on a set of input examples is linear in the number of input examples, since the same set of computations needs to be done with respect to each example.

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2025/Efficient Biological Data Acquisition through Inference Set Design/full.md

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+# EFFICIENT BIOLOGICAL DATA ACQUISITION THROUGH INFERENCE SET DESIGN
+
+Ihor Neporozhnii\*1,2 Julien Roy\*1 Emmanuel Bengio' Jason Hartford1,3 1Valence Labs 2University of Toronto 3University of Manchester ihor.neporozhnii@mail.utoronto.ca & julien.roy@valencelabs.com
+
+# ABSTRACT
+
+In drug discovery, highly automated high-throughput laboratories are used to screen a large number of compounds in search of effective drugs. These experiments are expensive, so one might hope to reduce their cost by only experimenting on a subset of the compounds, and predicting the outcomes of the remaining experiments. In this work, we model this scenario as a sequential subset selection problem: we aim to select the smallest set of candidates in order to achieve some desired level of accuracy for the system as a whole. Our key observation is that, if there is heterogeneity in the difficulty of the prediction problem across the input space, selectively obtaining the labels for the hardest examples in the acquisition pool will leave only the relatively easy examples to remain in the inference set, leading to better overall system performance. We call this mechanism inference set design, and propose the use of a confidence-based active learning solution to prune out these challenging examples. Our algorithm includes an explicit stopping criterion that interrupts the acquisition loop when it is sufficiently confident that the system has reached the target performance. Our empirical studies on image and molecular datasets, as well as a real-world large-scale biological assay, show that active learning for inference set design leads to significant reduction in experimental cost while retaining high system performance.
+
+# 1 INTRODUCTION
+
+High-throughput screening (HTS) laboratories have enabled scientists to efficiently perform whole-genome CRISPR knockouts and screen large compound libraries to discover effective therapeutics (Mayr & Bojanic, 2009; Wildey et al., 2017; Blay et al., 2020; Tom et al., 2024; Fay et al., 2023). However, conducting such experiments on every compound or gene in these vast design spaces remains resource-intensive. With typical screening libraries holding on the order of  $10^{5}$  to  $10^{6}$  compounds (Hughes et al., 2011) and the number of possible small molecules estimated at  $10^{60}$  (Bohacek et al., 1996), the disparity between our screening capabilities and all which we could explore is staggering. Reducing experimental costs without compromising the quality of the generated data would allow to greatly accelerate biological and pharmaceutical research.
+
+At any stage of a screening procedure, to reduce experimental costs, we can train a machine learning model on the data that has already been collected, and then use that model to predict experimental outcomes for the remainder of the experiments (Naik et al., 2013; Reker & Schneider, 2015; Dara et al., 2022). This procedure generates a hybrid screen of the target library, with some outcomes experimentally observed and others obtained from model predictions. However this approach entails interrelated questions: Which subset of the library should we use to maximize the accuracy of the predictions? How do we select this subset without access to all experimental outcomes? How do we ensure that we acquire a large enough collection to meet our accuracy goal?
+
+This setting is similar to an active learning problem in that we want to select examples that maximize prediction accuracy, but instead of aiming to minimize generalization error on the entire data space, we focus solely on prediction on one particular set of experiments. The fact that this library is finite introduces an important difference: here the learner can influence the set of examples on which it
+
+is evaluated by strategically selecting the examples to be acquired. Since it is often the case that some experiments outcomes are inherently harder to predict than others, either due to their complex properties (Bengio et al., 2009), because of the partial observability of the system (Saleh et al., 2021; Krenn et al., 2020) or due to noise in the labeling function (Frénay & Verleysen, 2013; Lukasik et al., 2020), the learner can select these harder examples into the training set early on to avoid having to predict their outcomes at a later stage. Conversely, if a group of examples can be reliably predicted by the model, we can save experimental costs by not including them in the training set. We call this mechanism through which a learner can influence its own evaluation set: inference set design.
+
+In this work, we propose an active learning-based solution to hybrid screening. Our approach uses the model's confidence to guide the selection of experiments and leverages the mechanism of inference set design to improve the system's performance on the target set. Our algorithm includes a practical stopping criterion that terminates the acquisition of additional labels once a lower bound on a target accuracy is exceeded, and we show that this bound provides probabilistic guarantee on the performance of the algorithm as long as the model is weakly calibrated. To validate our method, we conduct a series of empirical studies on image and molecular datasets, as well as a real-world case study in drug discovery. The results demonstrate that inference set design significantly reduces experimental costs while improving overall system performance. Importantly, this is true even when the generalization benefits of active learning-based acquisition functions are marginal compared to random search. This has important practical implications for active learning: if a problem is a hybrid screen—in the sense that one only needs good performance on a fixed, finite set of experiments—then evaluating generalization error dramatically understates the benefits of active learning. By combining simple active learning acquisition functions with an appropriate stopping criterion, we show that it is possible to make large scale screening far more efficient.
+
+# 2 METHODS
+
+In this section, we present the problem of efficiently evaluating a large, finite set of experimental designs, such as screening a compound library. We formalize it as a subset selection problem, and show that it can be solved using an active learning strategy. Active learning is effective in this setting because it essentially selects its own inference set; we call this mechanism, inference set design. We propose a stopping criterion to monitor the model's performance and trigger termination of the acquisition process and show that this algorithm allows to reach provably high levels of performance on a target set of samples. A pseudo-code is available in Appendix B.
+
+# 2.1 REDUCING EXPERIMENTAL COSTS VIA HYBRID SCREENS AND INFERENCE SET DESIGN
+
+We are motivated by the problem of efficient data acquisition of compound libraries for drug discovery applications. Each "experimental readout" (or label)  $y$  requires us to run an experiment, which has some associated cost. We assume that we have a predefined target set  $\mathcal{X}_{\mathrm{target}} = \{x_i\}_{i=1}^{N_{\mathrm{target}}}$  of experimental designs for which we want to acquire a readout, where  $N_{\mathrm{target}}$  represents the number of samples we wish to obtain labels or predictions for. For example, in drug discovery, a common set of experimental designs,  $\mathcal{X}_{\mathrm{target}}$ , would be a library of thousands or even millions of compounds, each required to be tested on several cell types and at different concentrations. For each  $x_i \in \mathcal{X}_{\mathrm{target}}$ , the associated experimental readout,  $y_i$ , would correspond to the drug's effect on the corresponding cell type. One way to reduce the acquisition costs of screening the entire target set is to train a predictive model  $\hat{p}(x)$  on the observation set - the subset of the target set,  $\mathcal{X}_{\mathrm{obs}} \subseteq \mathcal{X}_{\mathrm{target}}$ , for which we already have observed the labels  $y$ . We can then use this model's predictions in place of the real labels on the remaining samples. Because experiments are typically run in multiple rounds, at each round  $t$ , the target set  $\mathcal{X}_{\mathrm{target}}$  can be partitioned into two mutually exclusive subsets:
+
+- The observation set  $\mathcal{X}_{\mathrm{obs}}^t$ , which contains the union of all experimental designs that have already been tested and for which the readouts,  $\mathcal{V}_{\mathrm{obs}}^t$ , have been observed. We denote their pairing the observation dataset  $\mathcal{D}_{\mathrm{obs}}^t \coloneqq (\mathcal{X}, \mathcal{Y})_{\mathrm{obs}}^t \coloneqq \{(x_i, y_i)\}_{i=1}^{N_{\mathrm{obs}}}$ . At each step  $t$ , a constant number of  $N_b$  samples are acquired and added to the observation set, causing this set to grow over time.  
+- The inference set  $\mathcal{X}_{\mathrm{inf}}^t \coloneqq \mathcal{X}_{\mathrm{target}} \setminus \mathcal{X}_{\mathrm{obs}}^t$ , which consists of the remaining experimental designs in the target set that have not yet been tested. This set shrinks over time as samples are selected from it for acquisition and transferred into the observation set.
+
+![](images/1d958370af796f881e13d5846401ae66dc1c8c9bf3d7406754d151f82f47c2c7.jpg)  
+Figure 1: Hybrid screen employing active learning as an inference set design strategy. The goal is to produce a "hybrid dataset", composed of the labels collected in the observation set and the predictions of the model made on the inference set. The system's performance is reported as the "system accuracy", measured as a combination of observed labels on the observation set and of predictions made on the inference set. As samples are acquired and added to the observation set, the contribution of the accuracy of the prediction model gradually decreases. The aim of the active agent is to select informative examples while pruning out the most difficult ones in order to reach very high system accuracy without having to acquire all the samples.
+
+Once acquisition is stopped, at timestep  $t = \tau$ , the output of the system is a hybrid dataset, i.e. the combination  $(\mathcal{X},\mathcal{Y})_{\mathrm{obs}}^{\tau} \cup (\mathcal{X},\hat{\mathcal{Y}})_{\mathrm{inf}}^{\tau}$  of both the labeled pairs from the observation set and the predicted pairs from the inference set. We call this procedure a hybrid screen, as it provides a hybrid set of labels for the entire target set – some measured, some predicted. The system's performance at each step  $t$  is evaluated based on its accuracy  $\mu_{\mathrm{sys}}^{t}$  across the entire target set, which consists of perfectly accurate labels  $y$  from the observation set and the inferred labels  $\hat{y}$  for the inference set:
+
+$$
+\mu_ {\text {s y s}} ^ {t} := \frac {1}{| \mathcal {X} _ {\text {t a r g e t}} |} \sum_ {x _ {i} \in \mathcal {X} _ {\text {t a r g e t}}} \mathbf {1} \left(x _ {i} \in \mathcal {X} _ {\text {o b s}} ^ {t}\right) + \mathbf {1} \left(\hat {y} _ {i} = y _ {i} \mid x _ {i} \in \mathcal {X} _ {\text {i n f}} ^ {t}\right) \tag {1}
+$$
+
+In this formulation, acquiring all the labels would yield  $100\%$  accuracy, but at significant expense. We instead seek a system that reaches an accuracy of at least  $\gamma$ , while incurring minimal experimental costs. This can be formulated as the following combinatorial optimization problem,
+
+$$
+\min  _ {\mathcal {X} _ {\mathrm {o b s}} \subseteq \mathcal {X} _ {\text {t a r g e t}}} | \mathcal {X} _ {\mathrm {o b s}} | \quad \text {s u c h t h a t} \quad \mu_ {\mathrm {s y s}} ^ {t = \tau} > \gamma \tag {2}
+$$
+
+where  $|\mathcal{X}_{\mathrm{obs}}|$  denotes the number of samples in the observation set. A natural approach is to treat this as an active learning problem. Like standard active learning, we can sequentially select subsets of data for labeling and the system accuracy will benefit from any improvements in generalization that arise from better selection of training examples. However, this setting has an important difference: we only ever evaluate  $\hat{p}$  on  $\mathcal{X}_{\mathrm{inf}}$ , and this set is under the control of the learner because any example that is selected for labeling is removed from the inference set. As a result, by choosing acquisition functions that select harder training examples first, we make the test-time task of the learner easier, as
+
+the harder examples are effectively pruned out of the inference set. We call this approach inference set design – by selective label acquisition, the agent actively designs the composition of the inference set such that it can excel on it. This mechanism is illustrated in Figure 1.
+
+The data selection heuristics for inference set design are similar to standard active learning, but there are two important differences between the settings. Firstly, if we evaluate performance under the standard active learning objective—generalization error measured on a held-out test set—we risk dramatically understating the performance improvement that result from inference set design. Instead, evaluating on the target set of interest allows to capture dramatical improvements over a random sampler even when only marginal improvements in generalization error can be observed. Secondly, unlike active learning, inference set design requires an explicit stopping criterion in order to decide when you have collected enough samples to meet the accuracy threshold, which we address in the following section.
+
+# 2.2 EMPIRICAL STOPPING CRITERION
+
+The key technical challenge of inference set design is deciding when to stop running new experiments. In a deployment environment, the labels for the inference set will remain unavailable, so we need to estimate the system performance from the collected data only. To avoid unnecessary experimentation, inference set design algorithms must employ an efficient stopping criterion. We address this by maintaining a probabilistic lower bound on the system accuracy, and stopping once this bound exceeds the critical threshold,  $\gamma$ . A simple approach to maintaining such a bound is to leverage the feedback from each round of experimentation.
+
+We consider a multi-class classification problem. Formally, let  $\mathcal{K} \coloneqq \{1, 2, \dots, K\}$  be the set of  $K$  labels and  $\hat{p}(x) \coloneqq (\hat{p}_1(x), \dots, \hat{p}_K(x))$  be the predicted probabilities for  $x$  that  $y = k$  for all  $k \in \mathcal{K}$ . We denote  $y_i$  the true label for a sample  $x_i$ ,  $\hat{y}_i \coloneqq \arg \max_k \hat{p}_k(x_i)$  the predicted class by the model and  $\hat{v}_i \coloneqq \max_k \hat{p}_k(x_i)$  the predicted probability value for that class. We consider the use of the commonly employed least-confidence acquisition function:
+
+$$
+x _ {\text {s e l e c t e d}} := \underset {x _ {i} \in \mathcal {X} _ {\inf } \backslash \mathcal {X} _ {b}} {\arg \max } \left(1 - \hat {v} _ {i}\right), \quad \text {f o r} \quad i = 1, \dots , N _ {b} \tag {3}
+$$
+
+where  $N_{b}$  is the number of samples to be acquired, i.e. the acquisition batch-size. The samples  $x_{i} \in \mathcal{X}_{b}$  are selected one by one by applying the least-confidence criterion in Equation 3 iteratively to the remaining samples in  $\mathcal{X}_{\mathrm{inf}}$  that have not yet been transferred to  $\mathcal{X}_{b}$ . Because at every round we select examples that the model regards as challenging, if the model maintains a correct calibration ordering, the performance on a batch  $\mathcal{X}_{b}^{t}$  that we select at round  $t$  should lower bound the performance on the entire inference set. We express this formally in the following lemma:
+
+Lemma 1. Assuming that  $\hat{p}$ 's top-predictions are weakly calibrated on the inference set, i.e. that for any two samples  $x_{1}$  and  $x_{2}$  in  $\mathcal{D}_{inf}^{t-1}$ ,  $\hat{v}_{1} \geq \hat{v}_{2} \Rightarrow P_{\mathcal{D}_{inf}^{t-1}}[Y = \hat{Y}|\hat{v}_{1}] \geq P_{\mathcal{D}_{inf}^{t-1}}[Y = \hat{Y}|\hat{v}_{2}]$ , then the expected accuracy on a batch  $\mathcal{D}_b^t \coloneqq \{(x_i, y_i)\}_{i=1}^{N_b}$  selected by least-confidence from  $\mathcal{X}_{inf}^{t-1}$  is less or equal than the expected accuracy on the remaining inference set:  $P_{\mathcal{D}_b^t}[Y = \hat{Y}] \leq P_{\mathcal{D}_{inf}^t}[Y = \hat{Y}]$ .
+
+See Appendix A.1 for more details and proof. Lemma 1 shows that although the samples in  $\mathcal{X}_b^t$  are not independent and identically distributed samples from  $\mathcal{X}_{\mathrm{inf}}^t$ , we can safely use the performance on  $\mathcal{D}_b^t$  as a conservative proxy for performance on  $\mathcal{D}_{\mathrm{inf}}^t$ . We thus leverage the measured performance on the acquired batch to estimate the performance on the remaining, unobserved inference set.
+
+We can now use this conservative estimate to design an appropriate stopping criterion. Let  $\hat{\mu}_b^t \coloneqq \frac{1}{N_b} \sum_{i=1}^{N_b} \mathbf{1}(y_i = \hat{y}_i)$  denote the measured accuracy on the batch  $\mathcal{D}_b^t$  at acquisition step  $t$ . A simple stopping criterion could consist in picking an accuracy threshold  $\gamma$  for the desired system accuracy, and to stop acquiring new samples once our system accuracy estimate  $\hat{\mu}_{\mathrm{sys}}^t$  based on  $\hat{\mu}_b^t$  surpasses  $\gamma$ :
+
+$$
+\tau = \underset {t \in 1, \dots , T} {\arg \min } \hat {\mu} _ {\mathrm {s y s}} ^ {t} > \gamma , \quad \hat {\mu} _ {\mathrm {s y s}} ^ {t} = \frac {\left| \mathcal {X} _ {\mathrm {o b s}} ^ {t} \right| + \hat {\mu} _ {b} ^ {t} \left| \mathcal {X} _ {\text {i n f}} ^ {t} \right|}{\left| \mathcal {X} _ {\text {t a r g e t}} \right|} \tag {4}
+$$
+
+where  $\tau$  denotes the stopping time. Optionally, we could augment this criterion with a patience hyperparameter to make it more robust to the stochasticity introduced by the finite sample size  $N_{b}$ . A more principled approach however is to directly account for the statistical uncertainty around  $\hat{\mu}_b^t$
+
+by using it to derive a bound  $\alpha_{t}$  on the true inference accuracy  $\mu_{\mathrm{inf}}^{t}$ . Using standard concentration inequalities (see detailed derivation in Appendix A.2) we have that,
+
+$$
+P \left(\mu_ {\inf } ^ {t} \leq \alpha_ {t}\right) \leq \delta \quad \text {w i t h} \quad \alpha_ {t} = \min  \left\{a \in [ 0, \hat {\mu} _ {b} ^ {t} ]: \operatorname {K L} \left(\hat {\mu} _ {b} ^ {t} | | a\right) \leq \frac {\log \left(\frac {1}{\delta}\right)}{N _ {b}} \right\} \tag {5}
+$$
+
+where  $\mathrm{KL}(\cdot ||\cdot)$  denotes the Kullback-Leibler divergence and  $\delta$  is the failure probability of the bound. We can thus use the lower-bound value  $\alpha_{t}$  in place of  $\hat{\mu}_b^t$  in Equation 4 to ensure achieving our desired system accuracy threshold  $\gamma$  with probability at least  $1 - \delta$ .
+
+To summarize, in this section, we presented a framework for hybrid screens leveraging active learning to perform inference set design by selectively acquiring labels for hard-to-predict examples, thereby simplifying the inference set for easier prediction, and using a conservative stopping criterion to ensure that the desired level of system accuracy is achieved at stopping time. The pseudocode for our algorithm is presented in Appendix B.
+
+# 3 RELATED WORK
+
+In this work, we apply Active Learning (AL) to the problem of hybrid screens for biological data acquisition. Warmuth et al. (2003) were among the first to apply AL to the field of drug discovery, with an acquisition function based on support vector machines for binding affinity predictors. AL has since then been explored as a tool for problems such as virtual screening (Fujiwara et al., 2008), cancer monitoring (Danziger et al., 2009), protein-protein interaction (Mohamed et al., 2010), compound classification (Lang et al., 2016) and chemical space exploration (Smith et al., 2018).
+
+These approaches seek to produce a better model than training on random samples would, and are to be distinguished from bayesian optimization methods (Graff et al., 2021; Gorantla et al., 2024), which also fall under the umbrella of "active learning" but instead use the model to seek specific samples (e.g. with high binding affinity) and often focus on addressing the exploration-exploitation dilemma which has been widely studied by bandit algorithms (Svensson et al., 2022). For hybrid screens, we seek a model that can accurately predict the labels of all samples in the inference set, without attempting to maximize the value of the readouts. Our setting, focusing on a fixed set of data, is sometimes referred to as pool-based active learning (Wu, 2019; Zhan et al., 2021). Another similar approach called transductive experimental design aims to select samples that improve the predictions for a specific pre-defined test set (Yu et al., 2006; Hubotter et al., 2024). A key difference is that most active learning work (including pool-based AL and transductive experimental design) tends to focus evaluations on a held-out test set (Atlas et al., 1989; Cohn et al., 1994; Gal et al., 2017; Margatina et al., 2021; Zhan et al., 2021; Luo et al., 2023; Li et al., 2024) or restrict the algorithm to very limited acquisition budgets (e.g. only  $10\%$  of the data pool) (Yu et al., 2006; Wu, 2019), which generally does not leave enough margin for mechanisms such as inference set design to dominate.
+
+A closely related problem is that of core set selection (Plutowski & White, 1993; Bachem et al., 2017; Guo et al., 2022), where the goal is to select the smallest set of samples from a pool of data such that a model trained on this subset performs on par with a model trained on the entire pool. The closest works to ours are that of Sener & Savarese (2017) and Li & Rangarajan (2019), which make an explicit connection between active learning and core set selection and also seek to improve performance on the unselected examples. In our work however, we focus on uncovering the key mechanism in action, inference set design, and show empirical validations across a wide variety of academic and real-life datasets.
+
+# 4 EXPERIMENTS
+
+In our experiments<sup>2</sup>, we aim at highlighting the key role that inference set design plays in allowing AL agents to reach high levels of performance on the inference set (Section 4.1), validating the robustness of this mechanism on molecular datasets used for drug discovery (Section 4.2), applying this method to a challenging real-world case of compound library screening on a large scale
+
+phenotypic assays (Section 4.3) and empirically validating our theoretical assumption and stopping criterion (Section 4.4). Unless specified otherwise, all curves show average metrics across 3 seeds and the shaded areas denote the standard error of the mean. Vertical lines denote the average stopping time, with min and max intervals (left to right). For all modalities, we use vectorized representations as input and our model  $\hat{p}$  is parameterized using a feed-forward deep neural network with residual connections. Additional details regarding data pre-processing, network architectures and hyperparameters are presented in Appendices C and D.
+
+![](images/eea9618743ae59986c077c0789168d23aa09514c3590eceaff0ed7ec61dd1f40.jpg)  
+Figure 2: Performance of least-confidence (LC) and random acquisition functions on variations of MNIST (columns) across 3 seeds. a) Original: MNIST with its naturally occurring easy and hard examples. b) Partial Observability: MNIST where the bottom two thirds of the images have been cropped out. c) Noisy labeling function: MNIST with shuffled labels for images of 6's, 8's, and 9's. The system seeks a target accuracy  $\gamma$  of  $98\%$ .
+
+# 4.1 VISUALIZING INFERENCE SET DESIGN
+
+To develop intuition for inference set design we first run experiments on the MNIST dataset. The whole MNIST training set is used as the target set from which agents can acquire samples. The MNIST test set is split 50-50 into a validation set used for early stopping and a test set used for measuring model performance on held-out data inaccessible by agents. In many real-world applications, data used for training ML models contains samples that are inherently difficult, only partially observed, or mislabeled. We investigate these three cases. Specifically, we simulate the challenge of partial observability by removing the lower part of the image (Figure 2b) and simulate labeling noise by shuffling labels for selected digits (see Figure 2c).
+
+![](images/e2a31e6543e680c10508dc0be4e4731d41ac435ee4a93f15071f79a5332249d1.jpg)  
+Figure 3: Performance of active and random agents on MNIST subject to an increasing number of shuffled labels. As the task becomes harder, the accuracy gap increases, but the stopping time  $\tau$  is triggered later.
+
+![](images/8374bb3282ed08b995a0ac900f09db29871092052e89ca7ecbf88742181799f0.jpg)
+
+In these experiments we study the performance of an active agent using the least-confidence (LC) acquisition function of Equation 3 and a random agent, selecting samples with uniform probability over the inference set. At every active learning step an agent acquires a batch of 1000 samples from the inference set, adds them to the training set, and retransmits the model from scratch. Across these active learning steps, we monitor the accuracy on the inference set, the test set, the acquired batch, and the system accuracy as a whole (see Equation 1). We also record which samples have been acquired at every acquisition step. In all three studied settings in Figure 2, we observe the same phenomenon: the generalisation accuracy measured on the held-out test set quickly saturates for both agents, but the active agent is able to obtain significant improvements on the inference set, and these gains are also reflected in individual acquisition batches and on the system's overall performance.
+
+![](images/203708b944ff0524cd4eb32f8d084baa8190a6c54acd75a82ec166f66aaf4465.jpg)  
+Figure 4: Percent of samples with shuffled labels acquired by each agent throughout the acquisition steps.
+
+The mechanism at play here stems from the active design of the inference set by the agent. We illustrate this for the case of a noisy labeling function in Figure 4. By selectively acquiring the images it tends to mislabel, the active agent makes predicting the labels on the remaining samples in the inference set much easier. Since the majority of samples acquired by the active agent in the first 20 steps represent these challenging examples, its accuracy on the acquired batch is significantly lower than a random agent's (Figure 2c,  $3^{\mathrm{rd}}$  row). However, once the process of removing mislabeled samples from the inference set is complete, the active agent's accuracy on the acquired batch drastically increases and surpasses the random agent. To further explore the tradeoff between task difficulty and efficiency gains, we run experiments with shuffled labels for increasingly larger subsets of digits (see Figure 3). The results show that with a higher number of such "noisy" samples, the gap between the active and random agents on the inference set increases. The active agent is able to achieve  $100\%$  accuracy on the inference set even with shuffled labels for  $80\%$  of the data. However, a more corrupted target set also necessitates more acquisition batches to prune out the difficult examples from the inference set, incurring smaller experimental budget reductions for highly challenging settings.
+
+# 4.2 ASSESSING THE ROBUSTNESS OF THE MECHANISM
+
+In the previous section, we have shown that inference set design leads to significantly improved system performance compared to random sampling, even with highly corrupted datasets. In this section, we validate that these observations hold for a wildly different type of data - molecular datasets. We use the Quantum Machine 9 (QM9) (Ruddigkeit et al., 2012; Ramakrishnan et al., 2014). QM9 contains  $134k$  small organic molecules and their quantum chemical properties computed with Density Functional Theory (DFT). For inputs, we convert the SMILES strings molecular representations into their Extended Connectivity Fingerprints (ECFPs). We predict the HOMO-
+
+![](images/39bdda98214a30907c67e6b26bb736aa86029e3c208837f0be91c948c5cae1b8.jpg)  
+a)
+
+![](images/0eaf8174467aa775e0a9e145781ba3e1e17624541c3b01164245619d764449d9.jpg)  
+b)  
+Acquired batch
+
+![](images/cc402a0f73477d5a57ed89625861e53c6cfb7c3fa79ac39f4d3be52b1709f5f0.jpg)  
+Percent of data acquired, %  
+Figure 5: Results on QM9. a) A sample of molecules from the QM9 dataset. b) Performance of active agents (LC and BALD), heuristic orderings, and random sampler on QM9 across 3 seeds.
+
+![](images/7467feb80c94e20c0ac1f0570137dbb75265b3886cb3a9c58eb3f88ff57c0183.jpg)  
+Inference set
+
+![](images/5ef609c46ac607749a4fb546cc05e1ec8a94f7aeb91fdfa99e64ae180c6c92df.jpg)  
+Test set  
+Percent of data acquired, %
+
+LUMO gap framed as a classification task by discretizing the values into two balanced classes using median as a boundary condition to explore agents' performance on classification task. Our main active agent uses the least-confidence acquisition function. We also explore the use of the BALD acquisition function (Gal et al., 2017). In addition to active and random agents, for both datasets, we also evaluate the performance of heuristic-based acquisition orderings (original ordering, smallest to largest molecules, ordered by SA-score, etc.) as well as a diversity sampling strategy (selecting molecules with the largest Tanimoto distance to the acquired set).
+
+The results on QM9 demonstrate that both LC and BALD agents achieve much higher accuracy on the inference set than the baselines. Furthermore, they reach near  $100\%$  accuracy on both system and inference set after acquiring only  $30\%$  of data (see Figure 5). Saturating performance across all agents on the test set suggests once again that the active agent is able to make gains by pruning out difficult examples from the inference set rather than by generalizing better. To confirm this hypothesis, we compiled a list of "hard to predict" molecules by training five models with different random seeds on a 5-fold cross-validation split of the inference set, and flagged the molecules for which at least 4 out 5 models made a mistake. The resulting "hard to predict" subset contains 4,944 molecules (approximately  $5\%$  of the inference set). We then tracked the proportion of these samples that was acquired by each agent in the active learning experiments. Figure 6 shows that the active agents acquire molecules from the "hard to predict" subset at a much higher rate than the baselines.
+
+![](images/1c90d3233702a4f96f7368051b1e7c509fddc252ec348d1eade366bda4e7c402.jpg)  
+Figure 6: Percent of "hard" examples acquired by each agent throughout the acquisition loop on QM9.
+
+In addition to the classification results on QM9 presented here, we extend our method to a regression task on the much larger Molecules3D dataset in Appendix E.1 and get to a similar conclusion. Put together, these results on additional datasets spanning both classification and regression tasks and including 3 different active acquisition functions confirm and extend the observations made on MNIST in the previous section. They suggest that inference set design is an effective approach to guide acquisition of molecular compounds, by significantly reducing the cost of acquiring a fixed set of molecules and ensuring high system performance. In the next section, we apply this method to real-world biological data.
+
+![](images/9e0867f17c09f2165110cbc6d377707ce0bc9a9419b9b29136e94ae013549c17.jpg)  
+Figure 7: Visualisation of the RxRx3 dataset. a) A microscopy image of perturbed cells. b) A sample of biological map showing whether each pair of gene-based and compound-based perturbations is pheno-similar (in red) or not (in white).
+
+![](images/307bc7baf3e3ac1b0fec6289f59a2d6470e5da722327c5cecf93104fc2ae29de.jpg)
+
+![](images/20075973164af20eb6089b1aad8670fb745684ba79bfce347f8eab9b3d41ef0b.jpg)  
+Figure 8: Performance on a pheno-similarity classification task on our proprietary phenomics dataset. The least-confidence agent is shown to be the most effective on this task and reaches the targeted system accuracy of  $98\%$  at around  $80\%$  of the acquisition procedure.
+
+# 4.3 INFERENCE SET DESIGN IN THE REAL WORLD
+
+Modern HTS platforms combined with genetic perturbation techniques facilitate large scale cell microscopy experiments that are designed to capture the effects of biological perturbations (Celik et al., 2022). In these experiments before taking microscopy images, cells undergo perturbation by either a CRISPR/Cas9-mediated gene knock-out or injection of a bioactive molecular compound at a given concentration. The obtained cell microscopy images are then processed and passed through a neural network to obtain embeddings that correspond to a specific perturbation. With data from such HTS pipeline it is possible to build "Maps of Biology" that contain organized information about known and new biological relationships. Each point on the map corresponds to the cosine similarity between perturbation embeddings and encodes whether two perturbations are related. Expanding maps of biology might uncover new biological relationships that in turn could guide the discovery of leads for new medicines. However, even with ultra-HTS platforms, experimentally acquiring microscopy images of all possible cell perturbations is unfeasible. We aim to apply inference set design to this problem with the goal of reducing costs associated with building maps of biology.
+
+For our experiments, we start by using the publicly available RxRx3 dataset (Fay et al., 2023) which contains the learned embeddings for 17,063 CRISPR/Cas9-mediated gene knock-out perturbations, as well as 1,674 FDA-approved compound perturbations at 8 concentrations each. We frame this as a classification problem and train our models to predict the discretized cosine similarity between the perturbation embeddings of any gene-compound pair. As preprocessing, for each gene-compound pairs, we select the compound-concentration perturbation that yields the largest cosine-similarity with the target gene. The cosine-similarities are discretized by computing a gene-specific threshold using the upper semi-interquartile range (SIQR) with a step-size of 2.5. Finally, we remove the genes that have an activity rate (across compounds) below  $25\%$ . The result is a  $151 \times 1$ , 674 matrix of labels encoding either pheno-similarity ( $y = 1$ ) or an absence of relationship ( $y = 0$ ) between a gene and a compound perturbation (see Figure 7b).
+
+For this prediction task, we treat all gene embeddings as known and focus only on the much larger space of compound acquisition. To featurize compounds we use embeddings of the MolGPS model (Sypetkowski et al., 2024). At every active learning step, agents acquire a batch of compounds from the inference set and uncover their relationships with all genes in the dataset (entire row of the matrix). The results on the RxRx3 dataset show only minor improvements over a random agent (see Appendix E). We believe this is due to high complexity of biological relationships and limited size of the inference set. Indeed, successfully learning to infer cellular similarity between gene and compound perturbations from only a few hundred examples is highly unlikely. To push this analysis further, we re-run the same experiment but on our proprietary phenomics dataset. It is similar to RxRx3 but much larger. It contains cell perturbation embeddings for 102,855 compound and 5,580 gene perturbations, leading to a 2,000 times larger relationship matrix. Using inference set design, the active agent is now able to outperform the baselines on this task, as shown in Figure 8. These experiments represent promising results for large scale applications of active learning to hybrid screens for early drug discovery.
+
+# 4.4 EMPIRICAL VALIDATION OF WEAK CALIBRATION AND STOPPING CRITERION
+
+In this last experimental section, we validate the theoretical claims made in Section 2.2. The proposed stopping criterion assumes weak calibration i.e. that the true accuracy of the model does not need to be exactly equal to its confidence (perfect calibration), but that the correct ordering is conserved. Figure 12 in Appendix F.1 shows that this assumption is empirically verified in all of our experiments, supporting the claim that weak calibration can be achieved in practice. In Figure 13, in Appendix F.2, we also empirically validate that the proposed stopping criterion making use of the bound presented in Equation 5 does indeed cause the agent to stop acquiring new labels at a system accuracy superior to the targeted threshold (see Equation 4).
+
+# 5 DISCUSSION
+
+While active learning has undergone significant theoretical development and experimentation over the past 30 years, its widespread adoption in the industry remains a challenge (Reker, 2019). Active learning is often framed as a means to improve generalization performance on held-out test sets, which is a goal made difficult by real-world challenges such as noisy labeling functions as well as high diversity and partial observability of the input space. However, we believe that a shift in perspective can help unlock the untapped potential of active learning. By focusing on optimizing performance on a target set of samples and allowing the model to decide which examples to label and which to predict, active learning can become a powerful tool for real-world applications.
+
+We applied this perspective to the problem of hybrid screens in drug discovery, where the goal is to acquire readouts for only a subset of compounds while making accurate predictions for the remaining ones. Our approach makes use of a confidence-based acquisition function and leverages the concept of inference set design, a strategy where the model selects the most challenging examples for labeling, leaving easier cases for prediction. Our empirical results, across image and chemical datasets, as well as a real world biological application, show that this approach leads to consistent and significant improvements. Moreover, our results highlight that heuristic-based orderings, often used in real-world data acquisition efforts, are highly suboptimal. They can lead to drastically different results across tasks and datasets and often end up harming the performance by introducing unjustified biases in the acquisition ordering. Importantly, this bias then prevents the experimenter from using the performance on the acquisition batch as an indication of the performance on the remaining examples in the inference set. A confidence-based agent also introduces a bias in sample selection, but assuming weak calibration, this bias is directional and still allows the acquisition batch to be used as a lower bound for the performance on the remaining examples (see Lemma 1). It then compensates for the conservativeness of its stopping criterion by designing its own inference set, leading to overall earlier stopping time and greater budget reductions.
+
+An important limitation of our work is the assumption that observed labels are deterministic and constitute perfect accuracy. Many real-world biological experiments however produce noisy observations. Combining the inference set design framework with aleatoric uncertainty modeling to better address cases of weak supervision represents a promising direction for future work.
+
+# ACKNOWLEDGMENTS
+
+We are grateful to the entire team at Recursion and Valence Labs for participating in insightful discussions and providing feedback on this work. We also wish to specifically thank Austin Tripp for offering valuable input on this work as well as Oscar Mendez-Lucio and Peter McLean for their precious support with data processing and for engaging in fruitful exchanges about predictive models for phenomics.
+
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+
+# A THEORETICAL SUPPLEMENTS
+
+# A.1 PROOF OF LEMMA 1
+
+For completeness, we start by re-stating Lemma 1 in more detail.
+
+We consider a multi-class classification problem, with  $\mathcal{K} \coloneqq \{1,2,\dots,K\}$  the set of  $K$  labels and  $\hat{p}(x) \coloneqq (\hat{p}_1(x),\ldots,\hat{p}_K(x))$  the predicted probabilities for  $x$  that  $y = k$  for all  $k \in \mathcal{K}$ . We denote  $y_i$  the true label for a sample  $x_i$ ,  $\hat{y}_i \coloneqq \arg \max_k \hat{p}_k(x_i)$  the predicted class by the model and  $\hat{v}_i \coloneqq \max_{k \in \mathcal{K}} \hat{p}_k(x_i)$  the predicted probability value for that class. In the context of sequential label acquisition, we denote  $\mathcal{X}_{\inf}^{t-1} \coloneqq \{x_i\}_{i=1}^N$  the inference set containing  $N$  samples at acquisition step  $t-1$ , and  $\mathcal{D}_{\inf}^{t-1} \coloneqq \{(x_i,y_i)\}_{i=1}^N$  the same set with the associated labels  $y_i$ . Stepping forward, at time  $t$ , an acquisition batch of  $N_b$  samples and their labels  $\mathcal{D}_b^t \coloneqq \{(x_i,y_i)\}_{i=1}^{N_b}$  are selected from the inference dataset of the previous step  $\mathcal{D}_{\inf}^{t-1}$ .
+
+Following the least-confidence acquisition function, at time-step  $t$ , the samples in  $\mathcal{D}_{\mathrm{inf}}^{t-1}$  are ordered by confidence where  $x_1$  has the predicted class with the lowest confidence and  $x_N$  has the predicted class with the highest confidence:
+
+$$
+\underbrace {\hat {v} _ {1} \leq \hat {v} _ {2} \leq \cdots \leq \hat {v} _ {N _ {b}}} _ {\text {f i r s t} N _ {b} \text {s a m p l e s}} \leq \underbrace {\hat {v} _ {N _ {b} + 1} \leq \cdots \leq \hat {v} _ {N}} _ {\text {l a s t} N - N _ {b} \text {s a m p l e s}} \tag {6}
+$$
+
+and the selected batch consists of the first  $N_{b}$  samples. The remaining inference set is composed of the remaining  $N - N_{b}$  samples, i.e.  $\mathcal{D}_{\mathrm{inf}}^t \coloneqq \mathcal{D}_{\mathrm{inf}}^{t-1} \setminus \mathcal{D}_b^t = \{(x_i, y_i)\}_{i=N_b+1}^N$ .
+
+The practical utility of this ordering depends on the calibration of the model. Calibration reflects the correctness of the model's confidence compared to its true performance. A perfectly calibrated model  $\hat{p}$  would output confidence levels that perfectly match the true probability of correct classification. We extend the notation of Guo et al. (2017) to be dataset-specific, and denote  $P_{\mathcal{D}}[Y = \hat{Y}]$  the probability of  $\hat{p}$  producing the correct class prediction for any sample from a dataset  $\mathcal{D}$ , which can equivalently be thought of as  $\hat{p}$ 's accuracy on that dataset:
+
+$$
+P _ {\mathcal {D}} \left[ Y = \hat {Y} \right] := E _ {\left(x _ {i}, y _ {i}\right) \sim \mathcal {D}} \left[ \mathbf {1} \left(y _ {i} = \hat {y} _ {i}\right) \right] \tag {7}
+$$
+
+Similarly,  $P_{\mathcal{D}}[Y = \hat{Y}|\tilde{v}]$  denotes the confidence-conditional accuracy of the model for  $\tilde{v}$ , a specific confidence level<sup>3</sup>:
+
+$$
+P _ {\mathcal {D}} [ Y = \hat {Y} | \tilde {v} ] := E _ {(x _ {i}, y _ {i}) \sim \mathcal {D}} [ \mathbf {1} (y _ {i} = \hat {y} _ {i}) | \hat {v} _ {i} = \tilde {v} ] \tag {8}
+$$
+
+Perfect calibration on the inference set would thus imply that for any confidence level  $\tilde{v} \in [0,1]$ , we have  $P_{\mathcal{D}_{\mathrm{inf}}^{t-1}}[Y = \hat{Y}|\tilde{v}] = \tilde{v}$ . Instead, in Lemma 1, we only assume weak calibration of the model on the predicted class, i.e. that the confidence ordering of the model's top predictions reflects the ordering of probabilities of correct classification. Formally, weak calibration on  $\mathcal{D}_{\mathrm{inf}}^{t-1}$  implies that for any pair of samples  $x_i$  and  $x_j$  in our dataset  $\mathcal{D}_{\mathrm{inf}}^{t-1}$ , we have:
+
+$$
+\hat {v} _ {i} \leq \hat {v} _ {j} \Longrightarrow P _ {\mathcal {D} _ {\inf } ^ {t - 1}} [ Y = \hat {Y} | \hat {v} _ {i} ] \leq P _ {\mathcal {D} _ {\inf } ^ {t - 1}} [ Y = \hat {Y} | \hat {v} _ {j} ] \tag {9}
+$$
+
+We aim to prove that, assuming weak calibration on the inference set, the expected accuracy on the least-confidence batch is bounded by the expected accuracy over the remaining inference set:
+
+$$
+P _ {\mathcal {D} _ {b} ^ {t}} [ Y = \hat {Y} ] \leq P _ {\mathcal {D} _ {\inf } ^ {t}} [ Y = \hat {Y} ] \tag {10}
+$$
+
+Proof. The expected accuracy from Equation 7 can be rewritten in terms of its confidence-conditional form in Equation 8:
+
+$$
+P _ {\mathcal {D}} [ Y = \hat {Y} ] = \frac {1}{| \mathcal {D} |} \sum_ {i = 1} ^ {| \mathcal {D} |} P _ {\mathcal {D}} [ Y = \hat {Y} | \hat {v} _ {i} ] \tag {11}
+$$
+
+From the weak calibration assumption in Equation 9, it follows that the model's confidence-conditional accuracy is a non-decreasing function of its confidence. Combined with the acquisition
+
+mechanism of Equation 6, we obtain an ordering of the probability that the model obtains the correct label along the entire inference set at step  $t - 1$ :
+
+$$
+\underbrace {P _ {\mathcal {D} _ {\inf } ^ {t - 1}} [ Y = \hat {Y} | \hat {v} _ {1} ] \leq \cdots \leq P _ {\mathcal {D} _ {\inf } ^ {t - 1}} [ Y = \hat {Y} | \hat {v} _ {N _ {b}} ]} _ {N _ {b} \text {f i r s t t e r m s}} \leq \underbrace {P _ {\mathcal {D} _ {\inf } ^ {t - 1}} [ Y = \hat {Y} | \hat {v} _ {N _ {b} + 1} ] \leq \cdots \leq P _ {\mathcal {D} _ {\inf } ^ {t - 1}} [ Y = \hat {Y} | \hat {v} _ {N} ]} _ {N - N _ {b} \text {r e m a i n i n g t e r m s}} \tag {12}
+$$
+
+The empirical average of a set of numbers that are inferior or equal to those of a second set has to be inferior or equal to the empirical average of the second set:
+
+$$
+\frac {1}{N _ {b}} \sum_ {i = 1} ^ {N _ {b}} P _ {\mathcal {D} _ {\inf } ^ {t - 1}} [ Y = \hat {Y} | \hat {v} _ {i} ] \leq \frac {1}{N - N _ {b}} \sum_ {i = N _ {b} + 1} ^ {N} P _ {\mathcal {D} _ {\inf } ^ {t - 1}} [ Y = \hat {Y} | \hat {v} _ {i} ] \tag {13}
+$$
+
+By definition of  $\mathcal{D}_{\mathrm{inf}}^t \coloneqq \mathcal{D}_{\mathrm{inf}}^{t-1} \setminus \mathcal{D}_b^t$ , the left-hand side from Equation 13 thus captures all the terms from  $\mathcal{D}_b^t$  and the right-hand side those from  $\mathcal{D}_{\mathrm{inf}}^t$ . Rewriting each side using Equation 11, we obtain:
+
+$$
+P _ {\mathcal {D} _ {b} ^ {t}} [ Y = \hat {Y} ] \leq P _ {\mathcal {D} _ {\inf } ^ {t}} [ Y = \hat {Y} ] \tag {14}
+$$
+
+This concludes the proof.
+
+![](images/60cacabbd4d8cb965a3b16d4feb4208777c4c67fdf8fe4a20cd2fcf82d587109.jpg)
+
+# A.2 BOUND DERIVATION
+
+Here we derive the bound presented in Equation 5. The Chernoff bound (Chernoff, 1952) for Bernoulli random variables provides an exponential tail bound for the true mean  $\mu$  of a sequence of Bernoulli distributed random variables  $Z_{1}, Z_{2}, \ldots, Z_{n}$ . It can be expressed using the Kullback-Leibler Divergence (KL) for Bernoulli distributions (Lattimore & Szeptsvári, 2020, page 135, Corollary 10.4):
+
+$$
+P (\mu \leq a) \leq \exp (- n \cdot \operatorname {K L} (\hat {\mu} | | a)) \quad \forall \quad a \in [ 0, \hat {\mu} ] \tag {15}
+$$
+
+$$
+\text {w i t h} \quad \mathrm {K L} (\hat {\mu} | | a) := \hat {\mu} \log \frac {\hat {\mu}}{a} + (1 - \hat {\mu}) \log \frac {1 - \hat {\mu}}{1 - a} \tag {16}
+$$
+
+It quantifies the probability that the true mean  $\mu$  is smaller or equal to some bound  $a$  given the observed sample mean  $\hat{\mu}$  and sample size  $n$ . For us, the value  $\{0,1\}$  of a variable  $Z_{i}$  indicates whether a particular sample  $x_{i}$  was correctly classified i.e.  $Z_{i} := 1(y_{i} = \hat{y}_{i})$ . In the context of inference set design and following the notation from Section 2,  $\hat{\mu}_b^t$  represents the observed accuracy on the acquired batch at time-step  $t$ , with acquisition batch-size  $N_{b}$ , and  $\mu_{\mathrm{inf}}^{t}$  represents the (unknown) accuracy of the model on the remaining inference set. Note that the Chernoff bound usually assumes that  $\hat{\mu}_b^t$  is an unbiased estimator of  $\mu_{\mathrm{inf}}^t$ . In our case,  $\hat{\mu}_b^t$  is a biased estimator due to the active selection mechanism of the batch. However, in Lemma 1, we show that this estimator is actually a conservative estimate of the true accuracy, and thus in turn contributes to making the bound presented in Equation 15 even more conservative, and preserves its validity.
+
+To establish a confidence level on that bound, we can lower-bound the right-hand side itself to the desired bound-failure probability  $\delta$ , which after rearranging yields:
+
+$$
+\operatorname {K L} (\hat {\mu} | | a) \geq \frac {\log \left(\frac {1}{\delta}\right)}{N _ {b}} \tag {17}
+$$
+
+At time  $t$ , we thus seek the maximum bound value  $a = \alpha_{t}$  for  $\mu_{\mathrm{inf}}^{t}$  such that the inequality on Equation 17 holds by finding the value of  $a$  that satisfies the following condition:
+
+$$
+\alpha_ {t} = \min  _ {a} \left\{a \in [ 0, \hat {\mu} _ {b} ^ {t} ]: \mathrm {K L} (\hat {\mu} \| a) \leq \frac {\log \left(\frac {1}{\delta}\right)}{N _ {b}} \right\} \tag {18}
+$$
+
+Since this is a scalar optimization problem from closed-form expressions, computing  $\alpha_{t}$  can be done easily and efficiently using a grid-search. With this choice for  $a$ , we obtain the desired probabilistic bound  $P(\mu_{\mathrm{inf}}^t < \alpha_t)\leq \delta$  summarized in Equation 5.
+
+# B ALGORITHM
+
+Algorithm 1 Hybrid Screen using Inference Set Design  
+1: Input: Acquisition batch-size  $N_{b}$ , threshold  $\gamma$ , margin  $\delta$   
+2: Initialize Step t=0, observation set  $\mathcal{X}_{\mathrm{obs}}^{t=0} \gets \emptyset$ , inference set  $\mathcal{X}_{\mathrm{inf}}^{t=0} \gets \mathcal{X}_{\mathrm{target}}$ , predictor  $\hat{p}$ .  
+3: repeat  
+4: Train the predictor  $\hat{p}$  on  $(\mathcal{X}, \mathcal{Y})_{\mathrm{obs}}^t$  (if not empty)  
+5: Obtain the predictions on the inference set  $\hat{\mathcal{P}}_{\mathrm{inf}}^t \gets \{\hat{p}(x_i) \forall x_i \in \mathcal{X}_{\mathrm{inf}}\}$   
+6: Run acquisition function to obtain scores  $\mathcal{S}_{\mathrm{inf}}^t \gets g(\hat{\mathcal{P}}_{\mathrm{inf}}^t)$   
+7: Select a batch of  $N_{b}$  inputs with the highest scores  $\mathcal{S}_{\mathrm{inf}}^t$  to form  $\mathcal{X}_{b}^t$   
+8: Remove the acquired batch from the inference set  $\mathcal{X}_{\mathrm{inf}}^t \gets \mathcal{X}_{\mathrm{inf}}^{t-1} \setminus \mathcal{X}_{b}^t$   
+9: Obtain the true labels  $\mathcal{Y}_b^t$  for the acquisition batch  
+10: Append the acquired batch to the observation set  $(\mathcal{X}, \mathcal{Y})_{\mathrm{obs}}^t \gets (\mathcal{X}, \mathcal{Y})_{\mathrm{obs}}^{t-1} \cup (\mathcal{X}, \mathcal{Y})_b^t$   
+11: Compute  $\alpha$  on  $(\mathcal{X}, \mathcal{Y})_b^t$  from Equation 5  
+12: until  
+\[
+\frac{|\mathcal{X}_{\mathrm{obs}}^t| + \alpha |\mathcal{X}_{\mathrm{inf}}^t|}{|\mathcal{X}_{\mathrm{target}}|} > \gamma \quad \text{or} \quad \mathcal{X}_{\mathrm{inf}}^t = \emptyset
+\]
+
+13: Return hybrid screen readouts:  $(\mathcal{X},\mathcal{Y})_{\mathrm{obs}}^{t = \tau}\cup (\mathcal{X},\hat{\mathcal{Y}})_{\mathrm{inf}}^{t = \tau}$
+
+# C ADDITIONAL DETAILS ON DATASETS AND PREPROCESSING
+
+# C.1 MOLECULAR DATASET PREPROCESSING
+
+In many practical applications, exact geometries of screened molecules are unknown as they require computationally expensive DFT calculations. As a first data processing step, we use the RDKit (Landrum et al., 2024) and Molfeat (Noutahi et al., 2023) libraries to convert molecular structures into SMILES strings and compute their Extended Connectivity Fingerprints (ECFPs). Both molecular datasets are cleaned by removing duplicated SMILES and fingerprints as well as single-atom structures. For total energies in the Molecules3D dataset we use a reference correction technique where atomic energies are calculated using a linear model fitted to the counts of atoms of each type present in a molecule (the obtained atomic energies are presented in Table 1). For reference correction, a randomly selected sample of  $100\mathrm{k}$  molecules is used. The atomic energies are then subtracted from the total energies of all molecules in the dataset. The obtained referenced-corrected energies are normally distributed with mean around  $0\mathrm{eV}$ . A small number of outliers with reference-corrected energy values above 10 standard deviations are removed from the dataset as well as the  $100\mathrm{k}$  samples that were used for reference correction to avoid data leakage.
+
+The final QM9 and Molecule3D datasets contain 133,885 and 3,453,538 molecules respectively. Both datasets are split into inference, validation, and test sets with  $80\%$ ,  $5\%$ ,  $15\%$  fractions. The QM9 HOMO-LUMO gap values are discretized into 2 balanced classes using median as a boundary condition to explore agents' performance on a classification task.
+
+<table><tr><td>Atomic Number</td><td>Energy (eV)</td><td>Atomic Number</td><td>Energy (eV)</td></tr><tr><td>1</td><td>-26.765</td><td>14</td><td>-7922.180</td></tr><tr><td>5</td><td>-673.550</td><td>15</td><td>-9313.610</td></tr><tr><td>6</td><td>-1054.411</td><td>16</td><td>-10810.329</td></tr><tr><td>7</td><td>-1483.001</td><td>17</td><td>-12474.680</td></tr><tr><td>8</td><td>-2034.056</td><td>32</td><td>-56538.647</td></tr><tr><td>9</td><td>-2687.455</td><td>33</td><td>-60828.588</td></tr><tr><td>12</td><td>-5367.759</td><td>34</td><td>-65296.349</td></tr><tr><td>13</td><td>-6657.476</td><td>35</td><td>-69980.303</td></tr></table>
+
+Table 1: Atomic energies used for reference correction on Molecules3D dataset.
+
+# C.2 RXRx3 DATASET PREPROCESSING
+
+The RxRx3 dataset contains one embedding for each well. Each perturbation type (gene-guide pair or compound-concentration pair) has several replicates across wells, plates and experiments. Each plate also contains unperturbed control cells which are used to keep track of and eliminate a portion of the batch effects (Sypetkowski et al., 2023). These raw embeddings thus need to be aligned and aggregated. We align them by centering and scaling each perturbation embedding to the embeddings of the experiment-level unperturbed control wells. The embeddings are then aggregated through a multi-stage averaging procedure, across wells, plates, experiments and guides (for CRISPR perturbations), which yields an average embedding for each gene-perturbation and each compound-concentration perturbation. We then use the obtained embeddings to compute the cosine similarities between gene and compound perturbations in the RxRx3 dataset.
+
+# D HYPERPARAMETERS AND IMPLEMENTATION DETAILS
+
+For all experiments presented in this work we use MLP models with residual connections (Touvron et al., 2021). All experiments were repeated with 3 different random seeds and the hyperparameters are summarized in Table 2.
+
+<table><tr><td>Hyperparemeter name</td><td>MNIST</td><td>QM9</td><td>Molecules3D</td><td>RxRx3</td><td>Proprietary data</td></tr><tr><td>Acquisition batch size</td><td>1,000</td><td>250</td><td>10,000</td><td>10</td><td>1000</td></tr><tr><td>Number of hidden layers</td><td>2</td><td>3</td><td>2</td><td>2</td><td>2</td></tr><tr><td>Hidden layer size</td><td>512</td><td>512</td><td>512</td><td>512</td><td>1024</td></tr><tr><td>Learning rate</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td></tr><tr><td>gradient norm clip</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td></tr><tr><td>Dropout</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Train epochs</td><td>1,000</td><td>1,000</td><td>30</td><td>30</td><td>1000</td></tr><tr><td>Train batch size</td><td>1,024</td><td>1,024</td><td>32,768</td><td>1,024</td><td>1,024</td></tr><tr><td>Early stop patience</td><td>50</td><td>50</td><td>15</td><td>25</td><td>25</td></tr><tr><td>Number of ensemble members</td><td>-</td><td>-</td><td>5</td><td>-</td><td>-</td></tr></table>
+
+Table 2: Hyperparameters for experiments.
+
+# E ADDITIONAL RESULTS
+
+# E.1 REGRESSION TASK ON MOLECULES3D
+
+To evaluate the inference set design paradigm on a regression task we use the Molecules3D dataset (Xu et al., 2021). Molecules3D contains structures and DFT-computed properties of nearly 4 million molecules. In our experiments we aim to predict the HOMO-LUMO gap and total energy. For inputs, we convert the SMILES strings molecular representations into their Extended Connectivity Fingerprints (ECFPs) (see Appendix C.1 for all details).
+
+For this regression task, we use a query-by-committee (QBC) active learning approach that computes variance across the predictions of an ensemble. To determine the stopping time we use a criterion with two parameters: MSE threshold  $t_{MSE}$  on the acquired batch and patience  $p$ . The stopping time is reached if the acquired batch MSE is lower than  $t_{MSE}$  for  $p$  steps. Like for our QM9 experiments, in addition to active and random agents, we also evaluate the performance of heuristic-based acquisition orderings (molecules ordered by size, sorted by SA-score, etc.). QBC achieves an approximately five times lower MSE compared to the random agent or heuristic-based orderings (see Figure 9). This shows that inference set design approach is not limited to classification tasks and can be applied to regression problems.
+
+![](images/e31af8bef3793ce62f8ba294430201d3e337cd331cd1e1eb3d49229a68188775.jpg)  
+Figure 9: Performance of QBC, random, and heuristic agents for HOMO-LUMO gap prediction (top) and energy prediction (bottom) on Molecules3D dataset. The dashed vertical shows average stopping time across random seeds, surrounded by mix to max intervals (left to right). The patience parameter  $p = 10$  was used in all experiments with the threshold MSE  $t_{MSE} = 0.1eV^2$  for HOMO-LUMO gap and  $t_{MSE} = 100eV^2$  for Energy predictions. QBC agent satisfied stopping condition after acquiring  $\approx 80\%$  of the data.
+
+Although, in many applications predicting properties of larger molecules presents a more challenging task, our experiments on the Molecules3D dataset demonstrate that acquiring molecules ordered from large to small may harm the predictions on inference set and overall system performance (see Figure 9). One of the reasons is the distribution of chemical elements across molecules in the dataset. When acquiring molecules from small to large, all unique chemical elements of the Molecules3D dataset are present in the training set after acquiring just the first 1,000 samples. However, when acquiring molecules from large to small, some chemical elements remain only in the inference set until the very end of the experiment which is especially detrimental to the total energy predictions (see Figure 10). This result demonstrates that using heuristic rules such as ordering molecules by size for data acquisition does not guarantee optimal acquisition or generalization of heuristic rules to new data and instead risks harming the performance due to the introduction of such biases.
+
+![](images/13df6a91c240bf5742fd0440303f7138055ff8ae8c18e569ac451042782ab5ba.jpg)  
+Figure 10: Number of unique chemical elements in the training set when acquiring samples ordered by molecule size. Acquiring molecules from large to small leaves several chemical elements out of the training set until the end of the dataset.
+
+# E.2 CLASSIFICATION TASK ON RXRx3
+
+Our results on the RxRx3 dataset demonstrate minor improvements from the active agents (LC and BALD) compared to random selection. The accuracy on the inference and test sets remains low throughout the experiment regardless of the acquisition method. This is unsurprising considering the extreme difficulty of predicting biological relationships in a low data regime. The result suggests that in the setting with low predictive power, even when the majority of the data is acquired, the ability of inference set design to provide significant budget reductions is limited.
+
+![](images/a77897f054c6ad9b40c53c9f3e0c0258d51ef7319a201e4e05915ea5f64e01a3.jpg)
+
+![](images/ceaebf97b9e474f269e6abc4ec8332cad6398629de899af1c3ec39e241bed8fd.jpg)
+
+![](images/000356bacdc0f99c5adf20f80a9a474472d6104f2bea6a71b4e074f7059a0f80.jpg)  
+Figure 11: Performance of agents on pheno-similarity classification task on RxRx3.
+
+![](images/2347271255ec5316921a56bf27b123335c85190516181db9764011a2e83ec26d.jpg)
+
+# F ADDITIONAL ANALYSIS
+
+# F.1 EMPIRICAL VALIDATION OF WEAK CALIBRATION ASSUMPTION
+
+In Lemma 1, we show that when using a least-confidence acquisition function, at any time-step  $t$ , the measured accuracy  $\hat{\mu}_b^t$  on the acquisition batch is a lower-bound on the unobserved accuracy on the remaining inference set  $\hat{\mu}_{\mathrm{inf}}^t$ , assuming that the model  $\hat{p}$  is weakly calibrated in the inference set. A weakly calibrated model is such that an increased confidence translates to a higher likelihood of correct prediction (higher accuracy). In Figure 12, we can see that at several points throughout the active learning loop, the least-confidence based agent is not perfectly calibrated. Indeed, there is a substantial gap between its confidence levels and the true accuracy (seen when comparing the colored bars to the identity function shown in gray). However, the model is generally weakly calibrated (the colored bars are always increasing). These observations support the empirical validity of our assumption for Lemma 1.
+
+![](images/c92ad512cbce6fe610398efd95137af6e029baa0df271abac5c9dac63371c16c.jpg)  
+Figure 12: Model calibration analysis for a run of the least confidence agent at different active learning steps (columns) for all of our classification experiments (rows). The colored bars represent the accuracy of predictions binned w.r.t their confidence level, and the gray bars show the identity function illustrating what perfect calibration would look like. The models satisfy the condition of being weakly calibrated since the accuracy of the model's predictions increases monotonically with their confidence. The confidence distribution shifting to the right as  $t$  increases indicates the growing confidence of the model in correctly predicting the labels of the remaining examples in the inference set.
+
+# F.2 EMPIRICAL VALIDATION OF STOPPING CRITERION
+
+In Section 2.2, we present a stopping criterion based on the fact that the accuracy measured on the acquisition batch can be used as a proxy for the accuracy on the remaining, non-acquired samples (inference set). The criterion is simple, once the estimated system accuracy  $\hat{\mu}_{\mathrm{sys}}^t$  at time  $t$  (see Equation 4) is above a user-defined targeted threshold  $\gamma$ , the acquisition is stopped. This criterion makes use of the bound of Equation 5, and both a random sampler and a least-confidence based agent can use this bound in a principled way. The random agent can validly use it because the accuracy of its acquisition batches, uniformly drawn from the inference set, are unbiased estimates of the true accuracy on the entire inference set, which is typically required for such bounds. The least-confidence agent can use it validly because we show in Lemma 1 that, assuming that the model is weakly calibrated (which we empirically validate in Appendix F.1), because of its confidence-based acquisition function, the accuracy of its acquisition batches represent a lower-bound on the true inference set accuracy, making the bound of Equation 5 even more conservative.
+
+To empirically validate this bound (Equation 5), we run a large number of trials for both the least-confidence (LC) and random agents across the classification datasets used in the experiments section. The results show that for the LC agent, all trials ended at a stopping time  $t = \tau$  at which the true system accuracy  $\mu_{\mathrm{sys}}^t$  was above the threshold  $\gamma$ , which is in accordance with the fact that the LC agent uses a lower-bound estimate in place of an unbiased estimate for the inference set accuracy, resulting in a looser bound for Equation 5 and an actual failure-probability lower than  $\delta$ . For the random agent, we observed  $1\%$  of the trials end with a system accuracy below the threshold in one of the experiments,  $6\%$  in another, and  $0\%$  on the three remaining datasets. These results are in accordance with the theory presented in Section 2.2 and Appendix A.2. On QM9, 6 out of 100 experiments resulting in slight bound failure is statistically in accordance with the theoretical bound failure probability of  $\delta = 5\%$ . For the other datasets, the bound was looser, which is also a possibility, and showcases that the observed gap between  $\delta$  and the true bound-failure probability can be problem-dependent.
+
+![](images/28d71c4437a0df41b52ab5fccd2abf28cedf2e7e488fd96d7ced898d014adeaa.jpg)  
+Figure 13: System accuracy at stopping time observed across 100 seeds for the MNIST, Cropped MNIST, Shuffled MNIST and QM9 experiments, and 50 seeds for the Phenomics experiments. In all experiments, we use a bound-failure probability of  $\delta = 0.05$ . For the LC agent, all trials lead to a stopping time  $t = \tau$  at which the true system accuracy  $\mu_{\mathrm{sys}}^{t = \tau}$  is higher than the threshold  $\gamma$ . For the Random Agent, we observed the same thing for Cropped MNIST, Shuffled MNIST and Phenomics. We also observed  $1\%$  of bound failure for (regular) MNIST, and  $6\%$  for QM9.

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