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2024/S3GCL_ Spectral, Swift, Spatial Graph Contrastive Learning/a7bbeaea-89dc-41e7-8ede-6ff40872ca5d_origin.pdf

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2024/S3GCL_ Spectral, Swift, Spatial Graph Contrastive Learning/full.md

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+# Guancheng Wan<sup>1</sup> Yijun Tian<sup>2</sup> Wenke Huang<sup>1</sup> Nitesh V Chawla<sup>2</sup> Mang Ye<sup>13</sup>
+
+# Abstract
+
+Graph Contrastive Learning (GCL) has emerged as a highly effective self-supervised approach in graph representation learning. However, prevailing GCL methods confront two primary challenges: 1) They predominantly operate under homophily assumptions, focusing on low-frequency signals in node features while neglecting heterophilic edges that connect nodes with dissimilar features. 2) Their reliance on neighborhood aggregation for inference leads to scalability challenges and hinders deployment in real-time applications. In this paper, we introduce S3GCL, an innovative framework designed to tackle these challenges. Inspired by spectral GNNs, we initially demonstrate the correlation between frequency and homophily levels. Then, we propose a novel cosine-parameterized Chebyshev polynomial as low/high-pass filters to generate biased graph views. To resolve the inference dilemma, we incorporate an MLP encoder and enhance its awareness of graph context by introducing structurally and semantically neighboring nodes as positive pairs in the spatial domain. Finally, we formulate a cross-pass GCL objective between full-pass MLP and biased-pass GNN filtered features, eliminating the need for augmentation. Extensive experiments on real-world tasks validate S3GCL proficiency in generalization to diverse homophily levels and its superior inference efficiency.
+
+# 1. Introduction
+
+Graph Neural Networks (GNNs) (Hamilton et al., 2017; Kipf & Welling, 2017) play a crucial role in analyzing graph-
+
+![](images/f03c681b26d398197428f7a81a171f57800742f1dcf1331f101ebe4314aa96be.jpg)
+
+![](images/fc8afabc0e2e09467e3841880072f6b9b2cefc1b2892f71e0447a6c0d419521a.jpg)
+
+![](images/f5eee1530ffc86a51298ca6dd6ca6b4e48c0ece9d5b2ffe05da8db2fb7b9d0b3.jpg)  
+Figure 1. Performance and efficiency comparison of different methods: (Testing) Accuracy, (Time of) Inference, (Time of) Training, and (Time of) Augmentation in contrastive learning. Results are generated on homophilic and heterophilic datasets. For all metrics except Accuracy, we employ reverse ranking along each axis, where a larger polygonal area indicates superior performance. Our method demonstrates a superior balance between effectiveness and efficiency. On large-scale datasets like Obgn-Arxiv, ours accelerates inference by  $\times 174$  while maintaining competitive performance. Further details are shown in Tab. 2 and Sec. 4.3.
+
+![](images/51826b47ec26b5dfe20d1e254f9a915b8964a2ab17db4ac999484169b08f42f6.jpg)
+
+structured data (Xiong et al., 2024b;a). They excel in identifying complex interactions and distinct node characteristics through message-passing mechanisms (Wu et al., 2020; Dai et al., 2022). Traditionally, GNN research has focused on supervised training, which requires a lot of labeled data. However, data labeling is costly and time-consuming. To overcome this, Graph Contrastive Learning (GCL) has emerged as a self-supervised technique in graph representation learning (Zhu et al., 2021b; Xia et al., 2021), particularly useful when task-specific labels are scarce. GCL works by maximizing agreement between different augmented views of the same example and minimizing it between differently augmented views of separate examples. Its efficacy has been validated in diverse fields, including social network analysis and recommender systems (Xu et al., 2021; Cai et al., 2023; Liu et al., 2023c; Tian et al., 2023a).
+
+Despite the recent developments, two key problems of GCL
+
+remain unsolved. Firstly, most existing methods are based on homophily assumptions (Zhu et al., 2020), primarily depending on low-pass filter GNNs (e.g., GCN) to learn representations. This focus on low-frequency components implies similarity between connected nodes. While effective for homophilic graphs, this approach overlooks the prevalence of heterophilic graphs in real-world scenarios, where connected nodes have different labels or dissimilar features (Zheng et al., 2022a; He et al., 2022a). Recent efforts to address this include adaptive augmentation (Liu et al., 2023f) and parameterized sampling strategies (He et al., 2023), without giving special consideration to the graph signal filters. Therefore the encoder inevitably encourages neighbor features to be pooled, irrespective of their actual similarity. This approach limits the ability to generate distinct representations and to generalize across graphs with varying homophily levels. Thus, the following question naturally emerges: I) how can we design filters generalizable to graphs of different homophily levels without labels?
+
+Secondly, although GNNs exhibit a remarkable ability to capture graph-structured contexts, deploying them in large-scale applications presents challenges due to their time-intensive message-passing mechanisms. However, it is crucial to make efficient inferences for latency-sensitive applications. To tackle this problem, many studies have investigated the distillation of knowledge from a pre-trained GNN teacher to a student MLP, then deploying the MLP for inference acceleration (Tian et al., 2023b; Zhang et al., 2022). However, these methods can only be adopted in the context of supervised scenarios, which require task-specific labels to train an effective teacher GNN, limiting their adaptability in scenarios lacking labels. This raises another intriguing question: II) how can we combine the inference-efficient characteristics of MLP and graph context-awareness of GNN in a self-supervised way?
+
+To address the aforementioned questions, we revisit existing graph contrastive learning from both spectral and spatial perspectives, introducing our Spectral, Swift, Spatial Graph Contrastive Learning (S3GCL). To go beyond the homophily assumption and address I), we first theoretically and empirically analyze the homophily in the spectral domain. We discover that high homophily levels correspond to low-frequency signals, whereas low homophily correlates with high-frequency signals. In supervised tasks, spectral GNNs with polynomial approximations have demonstrated their superiority in both homophilic and heterophilic graphs. Chebyshev polynomial (Defferrard et al., 2016; He et al., 2022b) among them are widely used to approximate various functions in graph signal filtering. However, these require labels to learn the filter shape suited to the graph, posing challenges in label-scarce environments. To overcome this, we propose a cosine-parameterized Chebyshev polynomial approach. This method utilizes cosine modulation for calcu
+
+lating filter values in Chebyshev interpolation, emphasizing the graph high-frequency and low-frequency components respectively. The cosine parameterization method selectively emphasizes certain frequency ranges, effectively decomposing graphs into two distinct biased-pass views. Moreover, this strategy simplifies the filter learning process by adopting less learnable parameters and enhances adaptability to graphs with diverse homophily levels.
+
+In the second place, to address the inefficient inference problem mentioned in II), we propose utilizing an MLP encoder for representation learning. Since the MLP functions as the feature transformation without any spectral filter, it is capable of creating a comprehensive full-pass view of graphs. Building on this, we formulate our Cross-Pass GCL objective, contrasting the full-pass MLP with a biased-pass GNN. To further enhance the graph context awareness of the MLP encoder, we divide the cross-pass GCL objective into two parts and introduce positive pairs in the spatial domain. By incorporating structurally neighboring nodes in the full-low component and semantically neighboring nodes in the full-high component, the MLP is enabled to identify finer granularity features from the graph property, which might be overlooked by relying solely on global spectral filter methods. Consequently, with the help of such a cross-pass and cross-architecture design and adopted spatial positive pairs, our method conducts efficient training of GCL without any manually specified graph augmentation. In practical applications, deploying the trained MLP encoder leads to faster inference and superior performance, generalizable to various graph types. Our principal contributions are summarized as follows.
+
+- We identify that existing GCL methods often overlook generalization across varying graphs and encounter challenges in efficient inference within applications.  
+- We uncover the correlation between homophily and frequency in the spectral domain and introduce biased cosine-parameterized Chebyshev polynomials to facilitate generalization across various graph types.  
+- We utilize the MLP encoder and incorporate positive pairs in the spatial domain. By integrating this with our innovative cross-pass optimization objective, swift and efficient inference for obtaining representations is achieved.  
+- Extensive experiments demonstrate that our method not only achieves higher inference speed but also enhances performance, making it adaptable to different graphs.
+
+# 2. Motivation
+
+# 2.1. Preliminaries
+
+Notations. Define the graph data as  $\mathcal{G} = (\mathcal{V},\mathcal{E})$  , with  $\nu$  signifying the node set encompassing  $|\mathcal{V}| = N$  nodes. The edge set  $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$  symbolizes the connections between nodes. The feature matrix  $\mathbf{X} = \{\mathbf{x}_1,\mathbf{x}_2,\dots ,\mathbf{x}_N\}^\top$  comprises feature vectors  $\mathbf{x}_i$  corresponding to node  $v_{i}$  .The
+
+adjacency matrix of  $\mathcal{G}$ , denoted as  $\mathbf{A} \in \mathbb{R}^{n \times n}$ , assigns  $\mathbf{A}_{ij} = 1$  for existing edges  $e_{i,j} \in \mathcal{E}$  and  $\mathbf{A}_{ij} = 0$  otherwise. The normalized adjacency matrix is expressed as  $\hat{\mathbf{A}} = \mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2}$ . The degree matrix  $\mathbf{D}$ , a diagonal matrix, is defined with  $D_{i,i} = \sum_{j} A_{i,j}$ . The graph Laplacian matrix is given by  $\mathbf{L} = \mathbf{D} - \mathbf{A}$ , and the symmetric normalized Laplacian matrix is  $\tilde{\mathbf{L}} = \mathbf{I} - \hat{\mathbf{A}}$ , where  $\mathbf{I}$  is the identity matrix.  $\tilde{\mathbf{L}}$  can be decomposed as  $\tilde{\mathbf{L}} = \mathbf{U} \boldsymbol{\Lambda} \mathbf{U}^{\top}$ , where  $\boldsymbol{\Lambda} = \mathrm{diag}(\lambda_0, \dots, \lambda_{N-1})$  is a diagonal eigenvalue matrix with  $0 = \lambda_0 \leq \lambda_1 \leq \dots \leq \lambda_{N-1} \leq 2$ , and  $\mathbf{U}$  is a unitary matrix consisting of eigenvectors.
+
+Graph Filtering. Graph filtering on features  $\mathbf{X}$  is defined by  $\mathbf{Z} = \mathbf{U}g(\boldsymbol{\Lambda})\mathbf{U}^{\top}\mathbf{X}$ , where  $g(\boldsymbol{\Lambda})$  represents the graph filter. Directly learning  $g(\boldsymbol{\Lambda})$  necessitates eigendecomposition (EVD), which has a time complexity of  $O(N^{3})$ . Recent studies (Lei et al., 2022; Wang & Zhang, 2022) advocate for approximating  $g(\boldsymbol{\Lambda})$  using polynomials:
+
+$$
+\mathbf {U} g (\boldsymbol {\Lambda}) \mathbf {U} ^ {\top} \mathbf {X} \approx \mathbf {U} \left(\sum_ {i = 0} ^ {K - 1} w _ {k} \lambda^ {k}\right) \mathbf {U} ^ {\top} \mathbf {X} = \sum_ {i = 0} ^ {K - 1} w _ {k} \tilde {\boldsymbol {\Lambda}} ^ {k} \mathbf {X}. \tag {1}
+$$
+
+The polynomial coefficients are denoted by  $\{w_k\}$ . Furthermore, a  $K$ -order polynomial graph filter can be represented as the filter function  $g(\lambda) = \sum_{k=0}^{K} w_k \lambda^k$ , mapping each eigenvalue  $\lambda$  in the range  $[0, 2]$  to  $g(\lambda)$ .
+
+Definition 2.1. (Homophily Level  $h$ ): Consider a graph  $\mathcal{G} = (\mathcal{V}, \mathcal{E})$  with its node label vector  $\mathbf{y}$ . The edge homophily level is quantified as the proportion of edges connecting nodes sharing identical labels. Formally, it is expressed as:
+
+$$
+h \left(\mathcal {G}, \left\{\mathbf {y} _ {i}: i \in \mathcal {V} \right\}\right) = \frac {1}{| \mathcal {E} |} \sum_ {(k, l) \in \mathcal {E}} \mathbf {1} \left(y _ {k} = y _ {l}\right). \tag {2}
+$$
+
+Here,  $|\mathcal{E}|$  denotes the total number of edges in the graph, and  $\mathbf{1}(\cdot)$  represents the indicator function.
+
+Problem Description. Our objective is to advance the field of self-supervised learning for node representation. We aim to develop an encoder  $\mathcal{F}$  that generates high-level node representations  $\mathbf{Z} = \{z_1, \ldots, z_N\} \in \mathbb{R}^{n \times d}$ , where each  $z_i \in \mathbb{R}^d$  corresponds to a node  $v_i$ . These representations can be utilized in downstream tasks like node classification.
+
+# 2.2. Motivation
+
+In this subsection, we examine both theoretically and empirically the relationship between graph filters that emphasize different frequencies, and their performance in homophilic or heterophilic graphs. This study contributes to the development of generalizable graph filters for GCL. We employ the widely-used Contextual Stochastic Block Model (CSBM) to generate graphs with varying levels of homophily (refer to Appendix B for a comprehensive description of CSBM).
+
+Initially, we theoretically analyze the performance of spectral GNNs with various filters on different graph types. In
+
+addition to homophily levels for categorical node labels, we assess the similarity of numerical node signals in the spectral domain, as described by (Huang & Lio, 2023). This assessment is conducted using the following metric:
+
+Definition 2.2. (Spectral Signal Frequency f): For a normalized feature signal  $x \in \mathbb{R}^n$ , the spectral signal frequency  $\mathbf{f}(x)$  on graph  $\mathcal{G}$  is  $\mathbf{f}(x) = \frac{x^\top \mathbf{L}x}{2}$ .
+
+Spectral signal frequency, as related to the Dirichlet Energy (Karhadkar et al., 2022), measures the variance of signal  $x$  in the spectral domain across graph  $\mathcal{G}$ :
+
+Lemma 2.3. A lower  $\mathbf{f}(x)$  suggests reduced distances between connected nodes, indicating smoothness over  $\mathcal{G}$ , where  $x_{u}$  represents the  $u$ -th element of  $x$ :
+
+$$
+\mathbf {f} (x) = \frac {x ^ {\top} \mathbf {L} x}{2} = \sum_ {(u, v) \in \mathcal {E}} \frac {\left(x _ {u} - x _ {v}\right) ^ {2}}{2}. \tag {3}
+$$
+
+Lemma 2.4. For a given graph signal  $x$ ,  $\Delta D(x)$  denotes the disparity between intra-class and inter-class distances:
+
+$$
+\begin{array}{l} \Delta D(x) = \mathbb{E}\Big[\sum_{\substack{(u,v)\in \mathcal{E}\\ y_{u} = y_{v}}}(x_{u} - x_{v})^{2} - \sum_{\substack{(u,v)\in \mathcal{E}\\ y_{u}\neq y_{v}}}(x_{u} - x_{v})^{2}\Big] \\ = 2 \mathbb {E} \left[ \left(p _ {\text {i n t r a}} - p _ {\text {i n t e r}}\right) \mathbf {f} (x) \right]. \tag {4} \\ \end{array}
+$$
+
+A lower  $\Delta D(x)$  indicates that the encoder generates node representations that are more similar within the same class and distinct across different classes, demonstrating better self-supervised learning performance.
+
+Given graph data  $\mathcal{G} \sim \mathrm{CSBM}(\mu_1, \mu_2, p, q)$  generated by the CSBM model, comprising two classes,  $c_0$  and  $c_1$ , with intra-class probability  $p$  and inter-class probability  $q$  for edge formation, we develop a theorem to ascertain the advantageous filter for given graph data:
+
+Theorem 2.5. Consider graph signals  $x^l$  and  $x^h$  processed by filters  $g_l$  and  $g_h$ , respectively. In heterophilic graphs (where  $p < q$ ) and when  $\Delta D(x^h) < \Delta D(x^l)$ , there exists an integer  $M(0 < M \leq N - 1)$  such that  $\sum_{i = M}^{N - 1} g_h^2(\lambda_i) \geq \sum_{i = M}^{N - 1} g_l^2(\lambda_i)$ , with  $0 = \lambda_0 \leq \lambda_1 \leq \ldots \leq \lambda_{N - 1} \leq 2$ .
+
+The complete proof of this theorem is available in Appendix D. This theorem indicates that in heterophilic graphs, a high-pass filter, which amplifies high frequencies, is more effective for distinct node representations, whereas a low-pass filter is preferable for homophilic graphs. To empirically validate this, we further conduct following experiments: we used ChebNet (Defferrard et al., 2016), known for its ability to approximate various spectral filters in supervised learning. ChebNet redefines the filter in Equation (1) as  $\sum_{k=0}^{K} w_k T_k(\tilde{\mathbf{L}})\mathbf{X}$ , where  $\hat{\mathbf{L}} = 2\mathbf{L} / \lambda_{max} - \mathbf{I}$ . The function  $T_k(x)$  follows the relation  $T_k(x) = 2xT_{k-1}(x) - T_{k-2}(x)$ , with  $T_0(x) = 1$  and  $T_1(x) = x$ . Expanding on this, (He et al., 2022b) enhanced Chebyshev polynomials
+
+![](images/b9590e36b9d0583cf874ac0e75794fb3f454829454fb25d3b4a0fb62daf664c5.jpg)  
+Figure 2. Case Studies on graphs with varying homophily levels. The dashed lines represent the mean performance across all homophily levels. We reveal that adopting both low-pass and high-pass will bring promising generalization across different homophily levels.
+
+with Chebyshev interpolation, reparameterizing  $w_{k}$  as
+
+$$
+w _ {k} = \frac {2}{K + 1} \sum_ {j = 0} ^ {K} \gamma_ {j} T _ {k} \left(x _ {j}\right). \tag {5}
+$$
+
+We generated synthetic graphs with homophily levels ranging from 0 to 1 and applied the GRACE (Zhu et al., 2020) contrastive learning framework, evaluating performance through a node classification task. Due to challenges in learning filter parameters without labels, we initialized  $\{\gamma_0^h,\dots ,\gamma_K^h\}$  in Equation (5) as an increasing arithmetic sequence, holding them constant during training as  $w_{k}^{h}$  for a high-pass filter. Conversely, we set  $w_{k}^{l}$  for the low-pass filter using a decreasing arithmetic sequence. Specifically, we set  $\gamma_0^h$  and  $\gamma_K^h$  at 0 and 2 for the high-pass filter, and inversely for the low-pass filter. Our experiments involved four filtering methods: ChebNet (with interpolation), low-pass only, high-pass only, and a combination of both. The results are in Figure 2. Our key findings are: 1) Using a biased-pass filter alone can limit generalization across various homophily levels, 2) Spectral GNNs struggle to approximate optimal filters without labels, 3) Employing both low- and high-pass filters separately enhances generalizability. These insights guide the development of our proposed method.
+
+# 3. Methodology
+
+# 3.1. Spectral: Cosine-Parameterized Chebyshev Polynomial Encoder
+
+To better handle graphs with different homophily levels and without labels (as discussed in Sec. 2.2), we initially consider using both low- and high-pass filters on the input graph data. However, this method encounters two issues: 1) In the parameterization method, a linear increase in parameters proportional to the order  $K$  occurs. This results in uniform growth across all intervals, not emphasizing any specific frequency range. It potentially causes the two graph views to remain coupled and not disentangled. 2) The filter remains static during training and cannot be optimized alongside the contrastive objective. To address these issues, we propose a novel cosine-parameterized Chebyshev polynomial. This approach derives the high-pass filter parameter  $w_{k}^{h}$  with  $\gamma_{j}^{h}$
+
+substituting  $\gamma_{j}$  in Equation (5).  $\gamma_j^h$  can be calculated as:
+
+$$
+\gamma_ {j} ^ {h} = \sigma \left(\beta_ {a} ^ {h}\right) + \frac {1}{2} \sigma \left(\beta_ {b} ^ {h}\right) (1 + \cos \left((1 + j / K) \pi\right)). \tag {6}
+$$
+
+$\sigma = \mathrm{ReLU}(\cdot)$  to ensure the non-negative property of  $\gamma_{j}$  and guarantee that  $\gamma_j^h\leq \gamma_{j + 1}^h$  . Similarly, we obtain the low-pass filter parameter  $w_{k}^{l}$  with  $\gamma_j^l\geq \gamma_{j + 1}^l$  ..
+
+$$
+\gamma_ {j} ^ {l} = \sigma \left(\beta_ {a} ^ {l}\right) - \frac {1}{2} \sigma \left(\beta_ {b} ^ {l}\right) (1 + \cos ((1 + j / K) \pi)). \tag {7}
+$$
+
+We initialize  $\beta_{a}^{h}$  and  $\beta_{a}^{l}$  as 0 and 2, respectively, and set  $\beta_{b}^{h}$  and  $\beta_{b}^{l}$  to 2. It is important to note that  $\beta$  is trainable during the contrastive learning process. This approach simplifies learning Chebyshev polynomials without labels by utilizing only two learnable parameters for each filter. Moreover, the cosine-parameterized strategy effectively emphasizes relevant frequencies while diminishing less significant ones, facilitating a smoother frequency distribution. In contrast, linear-increasing parameterization might result in an abrupt distribution, potentially less effective in capturing graph spectral properties. The effectiveness of this method will be further validated in Sec. 4.4. By applying distinct filters with  $w_{k}^{h}$  and  $w_{k}^{l}$ , we can decompose the graph into two biased views, each yielding its representation:
+
+$$
+\mathbf {Z} ^ {h} = \sum_ {k = 0} ^ {K} w _ {k} ^ {h} T _ {k} (\tilde {\mathbf {L}}) f _ {\theta} ^ {h} (\mathbf {X}), \quad \mathbf {Z} ^ {l} = \sum_ {k = 0} ^ {K} w _ {k} ^ {l} T _ {k} (\tilde {\mathbf {L}}) f _ {\theta} ^ {l} (\mathbf {X}). \tag {8}
+$$
+
+The notation  $f_{\theta}(\mathbf{X})$  represents the application of the MLP on the node feature matrix  $\mathbf{X}$ . After processing the features through filters and transformations, we can use these enhanced representations for contrastive learning objectives.
+
+# 3.2. Swift: MLP Encoder and Cross-Pass Objective
+
+Previous methods highlight the challenges of GNNs in terms of inference efficiency and scalability, primarily due to data dependency (Jia et al., 2020). In GCL, generating node representations typically requires message passing to aggregate neighborhood features. However, this aggregation process can be time-consuming, especially in latency-sensitive applications. To mitigate this issue, some research has focused on distillation methods, transferring knowledge from teacher GNNs to student MLPs for more efficient inference in industrial settings. Yet, these methods rely on supervised signals to train a high-quality teacher GNN and guide the student
+
+![](images/2478690a05a81b225efcf2ee44cdf67ea5c9135019b71412ed785ba104486aa0.jpg)  
+Figure 3. Architecture illustration of S3GCL: Spectral, Swift, Spatial Graph Contrastive Learning. We show the training process (left) and inference process (right) of S3GCL. Best viewed in color. Zoom in for details.
+
+MLP, which is impractical in unsupervised scenarios.
+
+To bridge this gap, we introduce an MLP encoder throughout the entire GCL training process, enabling its direct deployment for efficient inference. Initially, we input the original node feature matrix  $\mathbf{X}$  into an  $L$ -layer MLP encoder  $\mathcal{F}_{\theta}$ , yielding representations  $\mathbf{Z}^{f} = \mathcal{F}_{\theta}(\mathbf{X})$ . Furthermore, the MLP can be regarded as a full-pass filter, focusing solely on feature transformation (Luan et al., 2022). Consequently, this allows us to establish an optimization objective between the full-pass and biased-pass filtered representations. Drawing inspiration from the Info-NCE loss (Chen et al., 2020), we formulate our Cross-Pass objective as:
+
+$$
+\begin{array}{l} \mathcal {L} _ {c p} = \frac {- 1}{2 | \mathcal {V} |} \sum_ {v _ {i} \in \mathcal {V}} \left(\log \frac {\mathrm {s} \left(z _ {p} ^ {f} , z _ {p} ^ {l}\right)}{\sum_ {p \neq q} \mathrm {s} \left(z _ {p} ^ {f} , z _ {q} ^ {l}\right)} \right. \\ + \log \left. \frac {\mathrm {s} \left(z _ {p} ^ {f} , z _ {p} ^ {h}\right)}{\sum_ {p \neq q} \mathrm {s} \left(z _ {p} ^ {f} , z _ {q} ^ {h}\right)}\right). \\ \end{array}
+$$
+
+Here the  $\mathrm{s}(z_p^f, z_p^h) = \exp(\omega(z_p^f, z_p^h) / \tau)$ ,  $\omega$  is the cosine similarity defined as:  $\omega(z_p^f, z_p^h) = z_p^f \cdot z_p^h / (||z_p^f|| \times ||z_p^h||)$ . The parameter  $\tau$  denotes the contrast temperature. This method offers guidance for node representation learning by encouraging the model to produce semantically consistent representations from two distinct graph views. Moreover, from a spectral perspective, this cross-pass process ensures that the MLP encoder captures more invariant information from the task-relevant properties of graph spectral signals. The MLP optimization with different graph frequency ranges also improves the generalizability of graphs.
+
+Upon optimizing the cross-pass GCL objective, we obtain a refined MLP encoder capable of generating high-level and expressive node representations. However, MLPs inherently lack the ability to capture graph structural properties and context. Therefore in our initial studies, we observed that representations learned by MLPs tend to deteriorate and
+
+become suboptimal. Thus, enhancing the graph context awareness of MLPs in these scenarios is imperative.
+
+# 3.3. Spatial: Neighboring Positive Pairs
+
+To address the issue of the MLP encoder limited awareness of graph context, we introduce our concept of positive pairs in the spatial domain. Prior research in spatial GNNs has underscored the significance of the spatial domain in graph learning (Wang et al., 2021; 2022), highlighting its role in enhancing model adaptability to graph structure and effectively managing long-range dependencies and heterogeneity. In our study, knowledge of the spatial domain provides contextual insights that enable the MLP encoder to more effectively comprehend the underlying structure and semantics of the graph. Because the low-pass filter emphasizes neighborhood similarity in the spatial domain, we initially identify structurally neighboring nodes as positive pairs and separate the Cross-Pass objective as delineated in Equation (9) into the Full-Low part, formulated as follows:
+
+$$
+\mathcal {L} _ {f l} = - \frac {1}{2 | \mathcal {V} |} \sum_ {v _ {i} \in \mathcal {V}} \frac {1}{| \mathcal {N} _ {i} ^ {\prime} |} \sum_ {v _ {p} \in \mathcal {N} _ {i} ^ {\prime}} \log \frac {\mathrm {s} \left(z _ {i} ^ {f} , z _ {p} ^ {l}\right)}{\sum_ {v _ {q} \in \mathcal {V} \backslash v _ {i}} \mathrm {s} \left(z _ {i} ^ {f} , z _ {q} ^ {l}\right)}. \tag {10}
+$$
+
+$\mathcal{N}_i^{\prime}$  denotes the positive sample set of local neighbors for node  $v_{i}$  in node set  $\nu$ . It is important to note that treating structurally neighboring nodes as positive pairs does not necessarily lead to identical representations. As suggested in (Altenburger & Ugander, 2018; Xiao et al., 2023), the phenomenon known as Monophily has been observed in both homophilic and heterophilic graphs. It implies that the attributes of a node's friends are likely to resemble those of the node's other friends, indicating two-hop similarities. For instance, nodes  $v_{i}$  and  $v_{k}$  share a common neighbor  $v_{j}$ . The representations  $z_{i}^{f}$  and  $z_{k}^{f}$  are derived from the MLP, respectively. Equation (10) encourages these representations to
+
+capture their shared one-hop neighborhood structure pattern, coming from spectral GNN as  $z_{j}^{l}$ . This cross-architecture aligns these two-hop pairs implicitly and extends beyond the homophily assumption. We show one-hop and two-hop representations similarities in Figure 6.
+
+Additionally, the high-pass filter indicates local neighborhood dissimilarity and global long-range similarity within the spatial domain. Therefore, we introduce semantically neighboring nodes as positive pairs, determined by feature-level similarity. Specifically, for each node  $v_{i}$ , we identify its top- $k$  similar nodes based on original features, denoted as  $\mathcal{N}_i'' = kNN(v_i,k)$ . This leads to the formulation of the Full-High part of our objective, as defined in Equation (9):
+
+$$
+\mathcal {L} _ {f h} = - \frac {1}{2 | \mathcal {V} |} \sum_ {v _ {i} \in \mathcal {V}} \frac {1}{| \mathcal {N} _ {i} ^ {\prime \prime} |} \sum_ {v _ {p} \in \mathcal {N} _ {i} ^ {\prime \prime}} \log \frac {\mathrm {s} \left(z _ {i} ^ {f} , z _ {p} ^ {h}\right)}{\sum_ {v _ {q} \in \mathcal {V} \backslash v _ {i}} \mathrm {s} \left(z _ {i} ^ {f} , z _ {q} ^ {h}\right)}. \tag {11}
+$$
+
+By integrating spatial relationships, the MLP encoder gains enhanced awareness of the graph context. This integration enables it to learn finer granularity features, which may be overlooked by solely relying on global spectral filter methods. Utilizing both components of the loss, we can establish our optimization objective as follows:
+
+$$
+\mathcal {L} = \alpha \mathcal {L} _ {f l} + (1 - \alpha) \mathcal {L} _ {f h}, \tag {12}
+$$
+
+where the coefficient  $\alpha$  is employed to balance these two loss functions. The deployed MLP encoder is advantageous for inference efficiency, particularly in managing large-scale graphs or real-time applications. The complexity and scalability analysis can be found in Appendix E.
+
+# 3.4. Discussion and Comprasion
+
+Existing GCL methods, such as GRACE (Zhu et al., 2020) and GBT (Bielak et al., 2021), typically employ a shared primary encoder with the same architecture. These methods often apply topology or feature transformations to augment the graph data. In contrast, our S3GCL introduces an innovative cross architecture approach. Coupled with our unique spatial positive pairs, S3GCL is augmentation-free while still achieving competitive performance. PolyGCL (Chen et al., 2024) also integrates the learnable spectral GNN in GCL, however, they still struggle with the efficient inference issue. Additionally, current research in knowledge distillation, such as FF-G2M (Wu et al., 2023a), employs structurally neighboring samples similar to our approach. However, these studies still rely on labels to guide the student MLP encoder. In contrast, our method attains consistency between the MLP target encoder and the spectral GNN encoder through our unique cross-pass objective at the representation level.
+
+Proposition 3.1. Given a node representation  $z_{i}$  and its neighboring node representation  $z_{j}$ , optimizing Equation (12) leads to maximizing their mutual information
+
+$I(z_{j};z_{i})$ , while inherently minimizing  $D_{KL}(z_j||z_i)$ :
+
+$$
+I \left(z _ {j}; z _ {i}\right) \sim \frac {1}{D _ {K L} \left(z _ {j} \mid \mid z _ {i}\right)} \tag {13}
+$$
+
+The proof is detailed in Appendix D. The proposition implies that information gain represented by  $D_{KL}$  is negatively correlated to the mutual information. By optimizing the cross-pass objective, we maximize mutual information between structurally or semantically neighboring nodes. This strategy not only facilitates the transfer of knowledge from GNN to MLP without labels, but also enriches the MLP. The trained MLP can be leveraged for more efficient inference (Ding et al., 2021) or saving communication cost (Huang et al., 2023c;b).
+
+One potential limitation of the proposed method is that, in the training phase, we still need polynomial filters to get representations. Although adopting MLP encoder can reduce the training time cost to a certain extent, its advantages can only be fully utilized in the inference phase. We believe this is also one of the directions in this field for the future.
+
+# 4. Experiment
+
+In this section, we comprehensively evaluate our proposed S3GCL by answering the main questions as follows.
+
+- Q1: Superiority. Does S3GCL outperforms the existing state-of-the-art graph contrastive learning methods?  
+- Q2: Efficiency. How about the inference time efficiency of the proposed method?  
+- Q3: Effectiveness. Are proposed cosine-parameterized Chebyshev polynomial, MLP encoder, and spatial positive pairs effective?  
+- Q4: Sensitivity. What is the performance of the proposed method with different hyper-parameters?
+
+The answers of Q1-Q3 are illustrated in 4.2-4.4, and sensitivity analyses (Q4) can be found in the Appendix G. The code is available at https://github.com/GuanchengWan/S3GCL.
+
+# 4.1. Experimental Setup
+
+# 4.1.1. REAL-WORLD DATASETS
+
+To effectively evaluate our approach in practical scenarios, we employed 14 benchmark graph datasets of various sizes and features, including both homophilic and heterophilic graphs. Please see Appendix A for details about datasets.
+
+# 4.1.2. EVALUATION PROTOCOL
+
+To evaluate the proposed method, we follow (Velickovic et al., 2019) to adopt a standard linear evaluation protocol. In this approach, the representations generated by our MLP encoder are fixed and subsequently employed for training, validation, and testing using a straightforward linear classifier. We repeat this experiment five times for each dataset to
+
+Table 1. Comparison with the state-of-the-art methods on homophilic (upper) and heterophilic (lower) real-world datasets. We report node classification accuracies  $(\%)$  ( $\pm$  standard deviation) for downstream task performance. The best and second results are highlighted with bold and underline, respectively.  
+
+<table><tr><td>Methods</td><td>Cora</td><td>CiteSeer</td><td>PubMed</td><td>Amz-Comp</td><td>Amz-Photo</td><td>Coauthor-CS</td><td>Obgn-Arxiv</td><td>Rank</td></tr><tr><td>DGI</td><td>83.69±1.20</td><td>72.54±1.64</td><td>80.41±0.33</td><td>86.11±1.21</td><td>90.38±0.90</td><td>91.57±0.88</td><td>70.19±0.73</td><td>7.86</td></tr><tr><td>GMI</td><td>83.11±1.53</td><td>73.69±1.89</td><td>83.44±1.64</td><td>86.91±0.45</td><td>91.27±1.23</td><td>90.11±0.20</td><td>69.23±0.79</td><td>8.14</td></tr><tr><td>MVGRL</td><td>87.67±0.86</td><td>74.31±0.82</td><td>86.98±1.22</td><td>87.91±0.34</td><td>92.77±0.57</td><td>90.95±0.28</td><td>70.88±0.51</td><td>4.29</td></tr><tr><td>GRACE</td><td>86.13±0.56</td><td>71.84±1.79</td><td>82.33±0.91</td><td>81.33±1.74</td><td>92.11±1.41</td><td>91.33±0.65</td><td>70.96±0.31</td><td>7.43</td></tr><tr><td>GBT</td><td>84.53±0.38</td><td>73.88±1.91</td><td>79.96±1.32</td><td>83.11±0.78</td><td>92.87±0.48</td><td>92.49±0.42</td><td>70.32±0.22</td><td>6.0</td></tr><tr><td>BGRL</td><td>83.17±1.77</td><td>70.11±0.54</td><td>85.57±0.90</td><td>88.35±0.32</td><td>93.10±0.44</td><td>91.72±0.21</td><td>71.24±0.35</td><td>5.43</td></tr><tr><td>DSSL</td><td>86.23±1.06</td><td>73.99±0.90</td><td>86.31±1.34</td><td>83.11±0.73</td><td>92.78±0.91</td><td>92.03±0.30</td><td>70.13±0.25</td><td>5.43</td></tr><tr><td>GREET</td><td>86.78±1.11</td><td>74.56±1.82</td><td>86.38±0.87</td><td>87.79±1.18</td><td>93.24±0.78</td><td>92.33±0.65</td><td>71.09±0.43</td><td>3.14</td></tr><tr><td>SP-GCL</td><td>87.45±1.42</td><td>73.19±2.10</td><td>85.66±0.44</td><td>89.45±1.95</td><td>92.32±0.17</td><td>90.91±0.93</td><td>69.11±0.36</td><td>6.0</td></tr><tr><td>S3GCL</td><td>88.47±1.39</td><td>76.31±1.67</td><td>87.89±1.23</td><td>88.45±1.98</td><td>94.31±0.83</td><td>92.55±0.89</td><td>71.36±0.60</td><td>1.14</td></tr></table>
+
+<table><tr><td>Methods</td><td>Cornell</td><td>Texas</td><td>Wisconsin</td><td>Actor</td><td>Roman-empire</td><td>Minesweeper</td><td>Tolokers</td><td>Rank</td></tr><tr><td>DGI</td><td>65.02±4.87</td><td>79.13±2.77</td><td>73.33±2.12</td><td>28.47±1.29</td><td>43.16±0.78</td><td>78.91±0.76</td><td>78.50±0.81</td><td>6.29</td></tr><tr><td>GMI</td><td>60.99±5.89</td><td>75.44±3.84</td><td>70.13±3.84</td><td>27.51±0.55</td><td>45.22±0.83</td><td>76.91±0.43</td><td>77.23±0.35</td><td>8.43</td></tr><tr><td>MVGRL</td><td>67.90±3.75</td><td>76.22±3.90</td><td>74.72±0.99</td><td>30.11±1.41</td><td>47.93±0.21</td><td>79.78±1.27</td><td>79.13±0.77</td><td>4.71</td></tr><tr><td>GRACE</td><td>50.82±4.78</td><td>75.32±2.11</td><td>74.91±2.78</td><td>29.23±0.21</td><td>51.58±0.98</td><td>78.99±1.45</td><td>78.06±0.90</td><td>7.14</td></tr><tr><td>GBT</td><td>54.66±3.10</td><td>72.30±2.14</td><td>65.32±1.34</td><td>23.42±0.55</td><td>38.77±0.54</td><td>77.32±0.64</td><td>79.43±1.00</td><td>8.14</td></tr><tr><td>BGRL</td><td>60.93±5.22</td><td>71.42±3.19</td><td>62.81±4.61</td><td>29.26±1.59</td><td>52.16±0.09</td><td>79.03±0.39</td><td>78.08±0.62</td><td>7.14</td></tr><tr><td>DSSL</td><td>76.78±3.68</td><td>75.92±2.11</td><td>81.33±3.66</td><td>33.31±0.86</td><td>61.29±0.44</td><td>78.76±0.70</td><td>78.42±0.85</td><td>4.43</td></tr><tr><td>GREET</td><td>75.23±4.96</td><td>78.33±4.78</td><td>83.11±3.91</td><td>36.42±0.65</td><td>63.37±1.91</td><td>79.05±1.21</td><td>79.21±0.79</td><td>2.86</td></tr><tr><td>SP-GCL</td><td>78.33±3.22</td><td>80.21±3.94</td><td>79.13±2.91</td><td>30.91±0.91</td><td>52.16±0.25</td><td>78.72±0.42</td><td>78.21±0.67</td><td>4.43</td></tr><tr><td>S3GCL</td><td>81.27±3.67</td><td>86.12±3.91</td><td>84.56±2.71</td><td>36.88±0.34</td><td>66.27±1.33</td><td>79.33±1.48</td><td>79.39±0.44</td><td>1.29</td></tr></table>
+
+Table 2. Statistics during the training process with different methods in the Cora and Actor dataset. Please see details in Sec. 4.3.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">Cora</td><td colspan="4">Actor</td></tr><tr><td>Augment ms ↓</td><td>Training s ↓</td><td>Inference ms ↓</td><td>Acc ↑</td><td>Augment ms ↓</td><td>Training s ↓</td><td>Inference ms ↓</td><td>Acc ↑</td></tr><tr><td>BGRL</td><td>129.1</td><td>4.32</td><td>8.33</td><td>83.17</td><td>132.8</td><td>9.87</td><td>9.69</td><td>29.26</td></tr><tr><td>GRACE</td><td>98.21</td><td>6.12</td><td>4.38</td><td>86.13</td><td>98.12</td><td>16.85</td><td>5.13</td><td>29.23</td></tr><tr><td>GREET</td><td>19272</td><td>43.21</td><td>1.32</td><td>86.78</td><td>54272</td><td>401.3</td><td>2.31</td><td>36.42</td></tr><tr><td>S3GCL</td><td>0</td><td>3.59</td><td>0.32</td><td>88.47</td><td>0</td><td>10.91</td><td>0.23</td><td>36.88</td></tr></table>
+
+ensure the robustness and reliability of the results. Please see more details about implementation in Appendix C.
+
+# 4.1.3. COMPARED METHODS
+
+We compare ours against several popular graph contrastive learning methods: DGI (Velickovic et al., 2019), GMI (Peng et al., 2020), MVGRL (Hassani & Khasahmadi, 2020), GRACE (Zhu et al., 2020), GBT (Bielak et al., 2021) and BGRL(Thakoor et al., 2021). We also include more recent GCL methods considering the heterophilic graphs for convincing comparison: GREET (Liu et al., 2023f), DSSL (Xiao et al., 2022) and SP-GCL (Wang et al., 2023).
+
+![](images/f0c7034268ea08fc472bf8cf18f2546dee9a721855991039d5faa423dc2f22d0.jpg)  
+Figure 4. Inference Time (log scale) with datasets of different sizes. Please refer to Sec. 4.3 for details.
+
+# 4.2. Superiority
+
+In this section, we address Q1 by analyzing the superior performance of S3GCL. We conduct comprehensive experiments on both homophilic and heterophilic graphs, as detailed in Tab. 1. Our findings include the average test accuracy and standard deviation from five runs, along with a comparative ranking across all datasets. Three key observations emerge: 1) S3GCL exhibits competitive performance on typical homophilic graphs, such as social networks. For instance, in the Cora dataset, our method achieves a  $0.8\%$  accuracy gain over the next best approach, indicating that
+
+![](images/3b1ce84d5374d736f4740cab577495138e886b7cb05f0216f5b361278fc8ae6c.jpg)  
+(a) BGRL
+
+![](images/66c5e77677617423576e3fe735d4f0da6d5507b80fa77cc318474d62bf6ff7f4.jpg)  
+(b) GRACE
+
+![](images/cce80a8f9edc89b6375e5aea29f2f28991c28255d8f56ba036aab02c1cdb7bad.jpg)  
+(c) GREET
+
+![](images/c57a2931b9137299fd0e00bc984d224d9e9ff7e472118ac23e889b2281d40ccf.jpg)  
+(d) S3GCL
+
+Table 3. Ablation Study of different variants on four dataset.  
+
+<table><tr><td>Variants</td><td>Cora</td><td>PubMed</td><td>Actor</td><td>Roman-empire</td></tr><tr><td>GCN-MLP</td><td>85.32±1.11</td><td>86.97±0.45</td><td>34.32±0.57</td><td>57.32±1.22</td></tr><tr><td>w/o Param.</td><td>84.33±0.78</td><td>83.14±0.65</td><td>35.11±0.74</td><td>58.32±1.06</td></tr><tr><td>Linear Param.</td><td>87.81±1.44</td><td>86.42±0.78</td><td>36.23±0.13</td><td>65.77±1.54</td></tr><tr><td>InfoNCE Loss</td><td>84.97±1.46</td><td>85.62±0.98</td><td>35.87±0.35</td><td>63.89±0.91</td></tr><tr><td>w/o Struct. Pairs</td><td>86.87±1.35</td><td>86.67±0.68</td><td>36.21±0.47</td><td>64.48±1.03</td></tr><tr><td>w/o Seman. Pairs</td><td>87.54±0.58</td><td>87.11±0.93</td><td>35.42±0.66</td><td>63.11±0.55</td></tr><tr><td>S3GCL</td><td>88.47±1.39</td><td>87.89±1.23</td><td>36.88±0.34</td><td>66.27±0.1.33</td></tr></table>
+
+incorporating spatial information enables the MLP to effectively capture graph context. 2) S3GCL demonstrates a consistent ability to generalize across graphs with varying homophily levels. Among 14 benchmark datasets, our method ranks the highest and attains state-of-the-art performance on 11 of them. This underscores the efficacy of different filters in handling diverse graph types. 3) Notably, the performance gains with S3GCL are more pronounced in heterophilic graphs than in homophilic ones, compared to other baseline methods. This suggests that the encoder plays a pivotal role in GCL, an aspect previously overlooked in other approaches. Furthermore, to intuitively underscore the effectiveness of S3GCL, we employ the t-SNE algorithm to visualize the learned node representations. The results, depicted in Figure 5, reveal that our method yields more distinct representations in the latent space compared to competing methods, further highlighting its superiority.
+
+# 4.3. Efficiency
+
+This section addresses Q2 by examining the inference time across datasets of varying sizes, as illustrated in Figure 4. The inference time for the self-supervised methods discussed in this paper is comparable to that of SAGE (Hamilton et al., 2017). Our analysis reveals that S3GCL consistently outperforms other GCL methods in terms of inference efficiency on almost all datasets, particularly with larger graphs. For instance, in the Ogbn-Arxiv dataset, a 2-layer BGRL requires  $389.6\mathrm{ms}$  to obtain representations, whereas S3GCL necessitates only  $2.23\mathrm{ms}$ , translating to an impressive  $\times 174$  acceleration. This disparity in inference speed becomes increasingly pronounced with larger datasets.
+
+Additionally, for smaller datasets, training time tends to outweigh inference time (Zheng et al., 2022b). Consequently,
+
+![](images/326809f18d8e7f317f4246d0c04b12bc3d0543b83b43df28b64440291a7be761.jpg)  
+Figure 5. t-SNE Visualization of representations with different methods on the Cora dataset. Please see Sec. 4.2 for details.  
+(a) Cora
+
+![](images/7d773452a2e95e63bdb2173d29d8ae77719aa95e0c0d30226256f591dbeaa0d1.jpg)  
+(b) Cornell
+
+![](images/8b6873f381f8a7de18de7baf9c73bfca83fffcd74ad3f63333dd22c7c8775698.jpg)  
+(a) Coefficient  $\alpha$  
+Figure 7. Analysis on hyper-parameter. Performance with hyperparameter  $\alpha$  and  $k$ , where red and green represent the Cora and PubMed respectively, while yellow and blue represent the Romanempire and Actor respectively.
+
+![](images/21d0b1c0e515bbffe00f5c0e1a8773ff053e32479eca2b0bdab75f8befdce619.jpg)  
+Figure 6. Cosine Similarity assessed through pairs comprising one-hop neighbors, two-hop neighbors, and randomly sampled nodes on the Cora and Cornell dataset.  
+(b) Semantic Neighbors  $k$
+
+we also perform experiments focusing on the training process with smaller graphs, detailed in Tab. 2. The results demonstrate that S3GCL not only achieves a balance between accuracy and inference time but also requires similar or less training time compared to other baseline methods.
+
+# 4.4. Effectiveness
+
+In this section, we undertake ablation studies to address Q3. As detailed in Tab. 3, we first assess the effectiveness of our proposed cosine-parameterized Chebyshev polynomial. Our initial baseline is a basic cross-architecture GCL model, namely GCN-MLP contrastive learning (GCN-MLP). The results indicate that relying solely on a low-pass filter, based on the homophily assumption, leads to adaptability issues in graphs with lower homophily levels. Subsequently, we evaluate the original ChebNet with interpolation but without parameterization (w/o Param.) and a linear parameterization strategy (Linear Param.). ChebNet exhibits poor performance across all datasets. This is likely due to the absence of labels, a spectral GNN struggles to learn the optimal filter shape. The linear parameterization approach results in suboptimal performance, as it fails to allocate unbalanced attention across different frequency intervals, potentially causing the two graph views to remain not disentangled.
+
+Further, we investigate the efficacy of spatial pairs through three experiments: removing structurally neighboring pairs
+
+(w/o Struct. Pairs), removing semantically neighboring pairs (w/o Seman. Pairs), and removing both (InfoNCE Loss in Equation (9)). The findings affirm the significance of both structurally and semantically neighboring nodes in the spatial domain, which substantially enhances the MLP awareness of graph context. This enhancement is pivotal for the MLP encoder to learn more expressive representations.
+
+# 5. Conclusion
+
+In this paper, we address a significant gap in existing graph contrastive learning methods: their limited generalization across graphs with varying levels of homophily and challenges in efficient inference for applications. We empirically and theoretically analyze the relationship between frequency and homophily levels. We then introduce a cosine-parameterized Chebyshev polynomial, designed to adapt to different graph types and facilitate training spectral GNNs in label-scarce scenarios. To tackle the issue of latency in inference, we leverage an MLP encoder and incorporate spatial positive pairs to enhance its awareness of the graph context. Based on these insights, we propose the S3GCL framework, which is strategically designed to achieve both competitive generalizable performance and efficient inference.
+
+# Acknowledgement
+
+This work is supported by the National Natural Science Foundation of China under Grant (62361166629, 62176188, 623B2080).
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
+
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+
+# A. Datasets Details
+
+We evaluate S3GCL on seven homophilic graphs: Cora, CiteSeer, PubMed, Wiki-CS, Amazon-Computer, Amazon-Photo, CoAuthor-CS, and Ogbn-Arxiv (Sen et al., 2008; Shchur et al., 2018; Hu et al., 2020) and seven heterophilic graph data: Cornell, Texas, Wisconsin, Actor, Roman-empire, Minesweeper, and Tolokers (Rozemberczki et al., 2021; Platonov et al., 2023). Since other widely used datasets (i.e. Squirrel and Chameleon) are faced with drawbacks of the presenting duplicate nodes (Platonov et al., 2023), so we abandon them in our experiments. We utilize a commonly used split of  $60\% / 20\% / 20\%$  for train/validation/test sets. For Obgn-Arxiv, we process the dataset in PyG using OGB public interfaces with the standard public split setting. The statistics of datasets are provided in Tab. 4, which also include homophily level  $h$ . The details are introduced as follows:
+
+- Cora, CiteSeer, and PubMed. These datasets are recognized as classic examples of homophilic citation networks. In these graphs, nodes are indicative of academic papers, while edges signify the citations connecting them. The papers are represented through bag-of-word features, and the research themes of the papers are reflected in the labels.  
+- Amazon Computers and Amazon Photo. In these Amazon-based co-purchase networks, each node corresponds to a product, with edges indicating frequent co-purchases between products. Product reviews are converted into bag-of-word feature representations, and the product categories serve as labels.  
+- CoAuthor CS. Extracted from the Microsoft Academic Graph, the network map co-authorship connections. Nodes represent authors, and co-authorship is marked by edges. The features are bag-of-words embeddings from paper keywords, and labels denote the authors' research areas.  
+- Ogbn-Arxiv. A large-scale directed graph from the arXiv's Computer Science section. Nodes represent scientific papers, with edges indicating citations. Features are derived from bag-of-words representations of titles and abstracts, and labels denote the paper's arXiv category. Primarily used for node classification tasks in graph machine learning.  
+- Cornell, Texas, and Wisconsin. Originating from the WebKB project, these heterophilic networks represent web pages from various university computer science departments. Nodes are these web pages, linked by edges that denote hyperlinks. The content of each page is encapsulated in bag-of-words features, with labels identifying the web page types.  
+- Actor. This unique heterophilic network maps the co-occurrence of actors within movies. Nodes symbolize actors, with edges revealing co-occurrences. Wikipedia pages provide keyword-based features, and the labels are constituted of words linked to the respective actors.  
+- Roman-emperor. Derived from the English Wikipedia article on the Roman Empire, this dataset represents words as nodes, connected based on their adjacency or syntactic dependency in the text. Nodes are classified based on their syntactic roles, with features sourced from FastText word embeddings.  
+- Minesweeper. A synthetic dataset inspired by the Minesweeper game. Nodes represent cells, connected to adjacent cells, with the task to identify nodes that are mines. Node features indicate the count of neighboring mines.  
+- Tolokers. Based on data from the Toloka crowdsourcing platform, this dataset includes nodes representing workers who participated in various projects. Edges connect workers who work on the same task. The objective is to predict banned workers, with features derived from their profile information and task performance.
+
+Table 4. Statistics of datasets used in experiments.  
+
+<table><tr><td>Dataset</td><td>#Nodes</td><td>#Edges</td><td>#Classes</td><td>#Features</td><td>Homophily Level</td></tr><tr><td>Cora</td><td>2,708</td><td>5,278</td><td>7</td><td>1,433</td><td>0.810</td></tr><tr><td>Citeseer</td><td>3,327</td><td>4,552</td><td>6</td><td>3,703</td><td>0.736</td></tr><tr><td>Pubmed</td><td>19,717</td><td>44,324</td><td>3</td><td>500</td><td>0.802</td></tr><tr><td>Amz-Comp</td><td>13,752</td><td>574,418</td><td>10</td><td>767</td><td>0.777</td></tr><tr><td>Amz-Photo</td><td>7,650</td><td>287,326</td><td>8</td><td>745</td><td>0.827</td></tr><tr><td>Coauthor-CS</td><td>18,333</td><td>327,576</td><td>15</td><td>6,805</td><td>0.808</td></tr><tr><td>Obgn-Arxiv</td><td>169,343</td><td>1,166,243</td><td>40</td><td>128</td><td>0.655</td></tr><tr><td>Cornell</td><td>183</td><td>298</td><td>5</td><td>1,703</td><td>0.305</td></tr><tr><td>Texas</td><td>183</td><td>325</td><td>5</td><td>1,703</td><td>0.108</td></tr><tr><td>Wisconsin</td><td>251</td><td>515</td><td>5</td><td>1,703</td><td>0.196</td></tr><tr><td>Actor</td><td>7,600</td><td>30,019</td><td>5</td><td>932</td><td>0.219</td></tr><tr><td>Roman-empire</td><td>22,622</td><td>65,854</td><td>18</td><td>300</td><td>0.047</td></tr><tr><td>Minesweeper</td><td>10,000</td><td>78,804</td><td>2</td><td>7</td><td>0.683</td></tr><tr><td>Tolokers</td><td>11,758</td><td>1,038,000</td><td>2</td><td>10</td><td>0.595</td></tr></table>
+
+# B. Synthetic Datasets
+
+In this study, we employ the Contextual Stochastic Block Model (CSBM) (Deshpande et al., 2018; Xu et al., 2023) to create synthetic graphs. These graphs are characterized by adjustable edge probabilities within and between different classes. The fundamental concept is that nodes within the same class exhibit a consistent feature distribution. The generated graph is denoted as  $\mathcal{G} \sim \mathrm{CSBM}(n, f, \sigma, \mu)$ , where  $n$  represents the total node count,  $f$  is the feature dimension, and  $\sigma$  and  $\mu$  are hyperparameters. The hyperparameters  $\sigma$  and  $\mu$  respectively influence the contributions from the graph's structure and the node features. We consider two classes of equal size,  $c_{1}$  and  $c_{0}$ , each comprising  $n/2$  nodes. The CSBM generates features as follows:
+
+$$
+\mathbf {x} _ {i} = \sqrt {\frac {\mu}{n}} y _ {i} u + \frac {w _ {i}}{\sqrt {f}}, \tag {14}
+$$
+
+where  $y_{i} \in \{-1, + 1\}$  indicates the label of node  $v_{i}, \mu$  is the mean value of the Gaussian distribution,  $u \sim \mathcal{N}(0,I / f)$ , and the elements of  $w_{i}$  follow independent standard normal distributions. We define the average degree of the generated graph as  $d$ , and the adjacency matrix  $\mathbf{A}$  for the CSBM graph is described by:
+
+$$
+P \left(\mathbf {A} _ {i j} = 1\right) = \left\{ \begin{array}{l l} \frac {1}{n} (d + \sigma \sqrt {d}) & \text {w h e n} y _ {i} = y _ {j} \\ \frac {1}{n} (d - \sigma \sqrt {d}) & \text {w h e n} y _ {i} \neq y _ {j}. \end{array} \right. \tag {15}
+$$
+
+The homophily level  $h$  is adjustable by modifying  $\sigma = \sqrt{d(2h - 1)}$ , with a range of  $-\sqrt{d} \leq \sigma \leq \sqrt{d}$ . A completely heterophilic graph corresponds to  $\sigma = -\sqrt{d}$ , while a completely homophilic graph occurs when  $\sigma = \sqrt{d}$ . In accordance with (Xu et al., 2023), we set  $d = 5$ ,  $\mu = 1$ ,  $n = 3000$ , and  $f = 4000$  to generate our synthetic dataset. By varying  $\sigma$ , we can produce graphs with different levels of homophily for our case studies.
+
+# C. Implementation Details
+
+The experiments are conducted using NVIDIA GeForce RTX 3090 GPUs as the hardware platform, coupled with Intel(R) Xeon(R) Gold 6240 CPU @ 2.60GHz. The deep learning framework employed was Pytorch, version 1.11.0, alongside CUDA version 11.3. Regarding the network architecture, consistent with prevalent practices in GCL, we used GraphSage (Hamilton et al., 2017) as a 2-layer encoder for all baseline models. We utilize the MLP projection for aligning the outputs of two models. The hidden layer size was set to 1024 for each dataset. For optimization, Stochastic Gradient Descent (SGD) (Robbins & Monro, 1951) was chosen, featuring a momentum of 0.9 and a weight decay of  $1e - 5$ . The learning rate was configured to  $5e - 4$  during the training process and  $1e - 2$  for the linear evaluation phase.
+
+# D. Detailed Proofs
+
+# D.1. Proof of Theorem 2.5
+
+Theorem 4.4. Consider graph signals  $x^{l}$  and  $x^{h}$  processed by filters  $g_{l}$  and  $g_{h}$ , respectively. In heterophilic graphs (where  $p < q$ ) and when  $\Delta D(x^{h}) < \Delta D(x^{l})$ , there exists an integer  $M(0 < M \leq N - 1)$  such that  $\sum_{i = M}^{N - 1}g_h^2 (\lambda_i) \geq \sum_{i = M}^{N - 1}g_l^2 (\lambda_i)$ , with  $0 = \lambda_0 \leq \lambda_1 \leq \ldots \leq \lambda_{N - 1} \leq 2$ .
+
+Proof. Given a graph  $\mathcal{G}$  generated by CSBM with intra-class probability  $p$  and inter-class probability  $q$ , and referring to Lemma 2.4, we have:
+
+$$
+\Delta D \left(x ^ {h}\right) - \Delta D \left(x ^ {l}\right) = 2 \mathbb {E} \left[ \left(\frac {p}{p + q} - \frac {q}{p + q}\right) \mathbf {f} \left(x ^ {h}\right) - \left(\frac {p}{p + q} - \frac {q}{p + q}\right) \mathbf {f} \left(x ^ {l}\right) \right]. \tag {16}
+$$
+
+With  $\Delta D(x^h) < \Delta D(x^l)$  and  $p < q$ , it follows that:
+
+$$
+\Delta D \left(x ^ {h}\right) - \Delta D \left(x ^ {l}\right) = 2 \mathbb {E} \left[ \frac {p - q}{p + q} [ \mathbf {f} \left(x ^ {h}\right) - \mathbf {f} \left(x ^ {l}\right) ] \right] <   0 \tag {17}
+$$
+
+Therefore,  $\mathbf{f}(x^h) > \mathbf{f}(x^l)$ . By the definition of spectral signal frequency in Definition 2.2, we compute  $\mathbf{f}(x)$  as follows:
+
+$$
+\mathbf {f} (x) = \frac {x ^ {\top} \mathbf {L} x}{2} = \sum_ {(u, v) \in \mathcal {E}} \frac {\left(x _ {u} - x _ {v}\right) ^ {2}}{2},
+$$
+
+$$
+\mathbf {f} \left(x ^ {l}\right) = \frac {x ^ {l \top} \mathbf {L} x ^ {l}}{2} = \sum_ {(u, v) \in \mathcal {E}} \frac {\left(x _ {u} ^ {l} - x _ {v} ^ {l}\right) ^ {2}}{2}, \tag {18}
+$$
+
+$$
+\mathbf {f} (x ^ {h}) = \frac {x ^ {h \top} \mathbf {L} x ^ {h}}{2} = \sum_ {(u, v) \in \mathcal {E}} \frac {(x _ {u} ^ {h} - x _ {v} ^ {h}) ^ {2}}{2},
+$$
+
+With  $\mathbf{L}$  being real and symmetric, a set of orthonormal eigenvectors  $v_{i}$  forms a complete basis in  $\mathbb{R}^N$ . Any graph signal  $x\in \mathbb{R}^{N}$  can be decomposed into a weighted sum of  $v_{i}$ , represented as  $x = \sum_{i = 0}^{N - 1}\alpha_{i}v_{i}$ , where  $\alpha_{i} = s^{\top}v_{i}$ .  $\alpha_{i}$  as the coefficient of  $x$  at frequency component  $i$  of graph  $\mathcal{G}$ . Hence, the filtered graph signals are:
+
+$$
+x ^ {l} = g _ {l} (\mathbf {L}) x = \mathbf {U} g _ {l} (\boldsymbol {\Lambda}) \mathbf {U} ^ {\top} x = \left(\sum_ {i = 0} ^ {N - 1} g _ {l} \left(\lambda_ {i}\right) v _ {i} v _ {i} ^ {\top}\right) \left(\sum_ {i = 0} ^ {N - 1} \alpha_ {i} v _ {i}\right) = \sum_ {i = 0} ^ {N - 1} g _ {1} \left(\lambda_ {i}\right) \alpha_ {i} v _ {i}, \tag {19}
+$$
+
+$$
+x ^ {h} = g _ {h} (\mathbf {L}) x = \mathbf {U} g _ {h} (\boldsymbol {\Lambda}) \mathbf {U} ^ {\top} x = \left(\sum_ {i = 0} ^ {N - 1} g _ {h} (\lambda_ {i}) v _ {i} v _ {i} ^ {\top}\right) \left(\sum_ {i = 0} ^ {N - 1} \alpha_ {i} v _ {i}\right) = \sum_ {i = 0} ^ {N - 1} g _ {2} (\lambda_ {i}) \alpha_ {i} v _ {i}.
+$$
+
+The spectral signal frequency is decomposed into:
+
+$$
+\mathbf {f} \left(x ^ {l}\right) = \left(\sum_ {i = 0} ^ {N - 1} g _ {l} \left(\lambda_ {i}\right) \alpha_ {i} v _ {i}\right) ^ {\top} \left(\sum_ {i = 0} ^ {N - 1} \lambda_ {i} v _ {i} v _ {i} ^ {\top}\right) \left(\sum_ {i = 0} ^ {N - 1} g _ {l} \left(\lambda_ {i}\right) \alpha_ {i} v _ {i}\right) = \sum_ {i = 0} ^ {N - 1} \lambda_ {i} \alpha_ {i} ^ {2} g _ {l} ^ {2} \left(\lambda_ {i}\right). \tag {20}
+$$
+
+$$
+\mathbf {f} (x ^ {h}) = \left(\sum_ {i = 0} ^ {N - 1} g _ {h} (\lambda_ {i}) \alpha_ {i} v _ {i}\right) ^ {\top} \left(\sum_ {i = 0} ^ {N - 1} \lambda_ {i} v _ {i} v _ {i} ^ {\top}\right) \left(\sum_ {i = 0} ^ {N - 1} g _ {h} (\lambda_ {i}) \alpha_ {i} v _ {i}\right) = \sum_ {i = 0} ^ {N - 1} \lambda_ {i} \alpha_ {i} ^ {2} g _ {h} ^ {2} (\lambda_ {i}).
+$$
+
+Given  $\mathbf{f}(x^h) > \mathbf{f}(x^l)$ , we deduce  $\sum_{i=0}^{N-1} \lambda_i g_h^2(\lambda_i) > \sum_{i=0}^{N-1} \lambda_i g_l^2(\lambda_i)$ , leading to:
+
+$$
+\mathbf {f} \left(x ^ {h}\right) - \mathbf {f} \left(x ^ {l}\right) = \sum_ {i = 0} ^ {N - 1} \alpha_ {i} ^ {2} \left[ \lambda_ {i} g _ {h} ^ {2} \left(\lambda_ {i}\right) - \lambda_ {i} g _ {l} ^ {2} \left(\lambda_ {i}\right) \right] > 0 \tag {21}
+$$
+
+Then we move to prove that there exist an integer  $M(0 < M \leq N - 1)$  such that  $\sum_{i = M}^{N - 1} g_h^2(\lambda_i) \geq \sum_{i = M}^{N - 1} g_l^2(\lambda_i)$ . We prove this statement by contradiction. Suppose there does not exist an integer  $M(0 < M \leq N - 1)$  that satisfies this condition. We have  $\sum_{i = M}^{N - 1} g_h^2(\lambda_i) < \sum_{i = M}^{N - 1} g_l^2(\lambda_i)$  with  $M(0 < M \leq N - 1)$ , then following inequalities exist:
+
+$$
+\left(\lambda_ {1} - \lambda_ {0}\right) \left[ g _ {h} ^ {2} \left(\lambda_ {1}\right) + g _ {h} ^ {2} \left(\lambda_ {2}\right) + \dots + g _ {h} ^ {2} \left(\lambda_ {N - 1}\right) \right] \leq \left(\lambda_ {1} - \lambda_ {0}\right) \left[ g _ {l} ^ {2} \left(\lambda_ {1}\right) + g _ {l} ^ {2} \left(\lambda_ {2}\right) + \dots + g _ {l} ^ {2} \left(\lambda_ {N - 1}\right) \right] \tag {22}
+$$
+
+$$
+\left(\lambda_ {2} - \lambda_ {1}\right) \left[ g _ {h} ^ {2} \left(\lambda_ {2}\right) + \dots + g _ {h} ^ {2} \left(\lambda_ {N - 1}\right) \right] \leq \left(\lambda_ {2} - \lambda_ {1}\right) \left[ g _ {l} ^ {2} \left(\lambda_ {2}\right) + \dots + g _ {l} ^ {2} \left(\lambda_ {N - 1}\right) \right] \tag {23}
+$$
+
+中
+
+$$
+\left(\lambda_ {N - 1} - \lambda_ {N - 2}\right) \left[ g _ {h} ^ {2} \left(\lambda_ {N - 1}\right) \right] \leq \left(\lambda_ {N - 1} - \lambda_ {N - 2}\right) \left[ g _ {l} ^ {2} \left(\lambda_ {N - 1}\right) \right] \tag {24}
+$$
+
+We assume that both filters  $g_{l}$  and  $g_{h}$  possess the same  $\ell_2$ -norms. This assumption allows us to concentrate on the role of different frequency ranges and avoid trivial solutions:
+
+$$
+\lambda_ {0} \left[ g _ {h} ^ {2} \left(\lambda_ {0}\right) + g _ {h} ^ {2} \left(\lambda_ {1}\right) + \dots + g _ {h} ^ {2} \left(\lambda_ {N - 1}\right) \right] = \lambda_ {0} \left[ g _ {l} ^ {2} \left(\lambda_ {0}\right) + g _ {l} ^ {2} \left(\lambda_ {1}\right) + \dots + g _ {l} ^ {2} \left(\lambda_ {N - 1}\right) \right]. \tag {25}
+$$
+
+Summing both sides of the above inequalities and considering Equation (25), we deduce:
+
+$$
+\sum_ {i = 0} ^ {N - 1} \lambda_ {i} g _ {h} ^ {2} (\lambda_ {i}) \leq \sum_ {i = 0} ^ {N - 1} \lambda_ {i} g _ {l} ^ {2} (\lambda_ {i}),
+$$
+
+which contradicts the earlier results that  $\sum_{i=0}^{N-1} \lambda_i g_h^2(\lambda_i) > \sum_{i=0}^{N-1} \lambda_i g_l^2(\lambda_i)$ . Therefore, the initial assumption is invalid, and there must exist an integer  $M(0 < M \leq n-1)$  such that  $\sum_{i=M}^{N-1} g_h^2(\lambda_i) \geq \sum_{i=M}^{N-1} g_l^2(\lambda_i)$ , which completes the proof.
+
+# D.2. Proof of Prop. 3.1
+
+Proposition 4.5. Given a node representation  $z_{i}$  and its neighboring node representation  $z_{j}$ , optimizing Equation (12) leads to maximizing their mutual information  $I(z_{j};z_{i})$ , while inherently minimizing  $D_{KL}(z_j||z_i)$ :
+
+$$
+I \left(z _ {j}; z _ {i}\right) \sim \frac {1}{D _ {K L} \left(z _ {j} \mid \mid z _ {i}\right)} \tag {26}
+$$
+
+Proof. Mutual Information Neural Estimation (MINE)(Belghazi et al., 2018) converts mutual information maximization into minimizing the InfoNCE loss Equation (12), where the positive pair is  $z_{i}$  and its neighboring node representation  $z_{j}$ . Utilizing the relationship between mutual information and information entropy, we derive:
+
+$$
+I \left(z _ {j}; z _ {i}\right) = H \left(z _ {j}\right) + H \left(z _ {i}\right) - H \left(z _ {j}, z _ {i}\right), \tag {27}
+$$
+
+where  $H(\cdot)$  denotes information entropy, and  $H(\cdot, \cdot)$  represents joint entropy. The Kullback-Leibler divergence, or information gain, in relation to information entropy is defined as:
+
+$$
+D _ {K L} \left(z _ {j} \| z _ {i}\right) = H \left(z _ {j}, z _ {i}\right) - H \left(z _ {j}\right). \tag {28}
+$$
+
+By integrating Equation (27) and Equation (28), we obtain:
+
+$$
+I \left(z _ {j}; z _ {i}\right) = H \left(z _ {i}\right) - D _ {K L} \left(z _ {j} \parallel z _ {i}\right). \tag {29}
+$$
+
+As per Equation (29), there is a negative correlation between  $I(z_{j};z_{i})$  and  $D_{KL}(z_j\parallel z_i)$ . This implies that maximizing  $I(z_{j};z_{i})$  inherently minimizes  $D_{KL}(z_j\parallel z_i)$ , thereby fulfilling the objective of transferring graph property knowledge from the GNN to the MLP:
+
+$$
+I \left(z _ {j}; z _ {i}\right) \sim \frac {1}{D _ {K L} \left(z _ {j} \mid \mid z _ {i}\right)}. \tag {30}
+$$
+
+# E. Complexity Analysis
+
+In this subsection, we delve into the time complexity of S3GCL. The process of acquiring the semantically neighboring node set  $\mathcal{N}_i^{\prime \prime}$  involves executing the k-nearest neighbors (kNN) algorithm on the entire graph. This step, a preprocessing phase external to formal training, is performed only once per dataset. Typically, computing a kNN graph requires a time complexity of  $\mathcal{O}(N^2 D_f)$ , where  $N$  is the number of nodes and  $D_{f}$  represents the feature dimension. This computation becomes particularly intensive when  $N$  is large. To mitigate this, following the approach in (Liu et al., 2023b), we initially apply the locality-sensitive approximation algorithm (Fatemi et al., 2021) in two separate batches. Subsequently, we merge the resulting local kNN graphs to form a comprehensive global kNN graph, ensuring potential connectivity between all nodes rather than just those within the same batch. This strategy effectively reduces the computational cost from  $\mathcal{O}(N^{2}D_{f})$  to  $\mathcal{O}(NBD_{f})$ , with  $B$  denoting the batch size.
+
+During the learning process, the computation for propagating  $K$ -order polynomial filters is  $\mathcal{O}(KE)$ , where  $E$  represents the number of edges. The MLP encoder requires  $\mathcal{O}(ND_fD)$  computations, with  $D$  being the representation dimension. The projection step involves  $\mathcal{O}(NDD_p)$  computations, where  $D_p$  is the projection dimension. The complexity of our final loss Equation (12) is  $\mathcal{O}(ND_pB')$ , with  $B'$  as the batch size in contrastive learning. Overall, our method scales linearly with the number of nodes  $N$ , demonstrating its scalability.
+
+# F. Related Work
+
+Graph Contrastive Learning. Within the domain of GCL (Liu et al., 2022a; Zhu et al., 2021a; Liu et al., 2023e;d; Yang et al., 2023; Liu et al., 2022b), contemporary studies have primarily focused on two distinct strategies: augmentation-based and augmentation-free techniques. Augmentation-based techniques, illustrated by GRACE (Zhu et al., 2020; Huang et al., 2023a; Wan et al., 2024), GCA(You et al., 2020a), and MVGRL(Hassani & Khasahmadi, 2020), concentrate on improving graph representation learning through diverse data augmentations such as edge dropping and attribute masking (You et al., 2020b). These techniques aspire to optimize mutual information across various graph perspectives. On the other hand, augmentation-free strategies, as demonstrated in studies such as SimGRACE (Xia et al., 2022) and BGRL (Thakoor et al., 2021), deviate from intricate data augmentation tactics. Alternatively, they depend on various graph encoders to create unique views, promoting the convergence of analogous node or class representations from these diverse perspectives (Yin et al., 2022b; 2023a; Tian et al., 2024b;a). In this study, we present a new, augmentation-free, yet effective cross-architecture GCL technique. It attains cross-pass uniformity between the node and its spatial neighbors, originating from full-pass MLP and biased-pass spectral GNN respectively.
+
+Heterophilic Graphs. Despite significant advancements in GNNs (Zhang et al., 2024; Liu et al., 2024; Jiang et al., 2021; Yin et al., 2023b), most adhere to the homophily principle. However, real-world graphs often diverge from the homophily principle and exhibit heterophily, where connected nodes possess dissimilar features and distinct class labels (Zheng et al., 2023b; Pan & Kang, 2023). To tackle such graphs, numerous methods aimed at excelling in either heterophily or both homophily and heterophily contexts have emerged recently (Kong et al., 2023; Azabou et al., 2023; Pang et al., 2023). Among these, spectral GNNs have garnered notable success for their proficiency in learning filters of diverse shapes with labels, suitable for a variety of graphs. Contemporary works like GREET(Liu et al., 2023f) and NeCo (He et al., 2023) address non-homophily scenarios through edge discriminating augmentation and parameterized neighbor sampling, respectively. However, these approaches overlook the special consideration of the filter role, leading to suboptimal outcomes. In contrast, we propose a cosine-parameterized Chebyshev polynomial, enhancing generalization across both homophilic and heterophilic graphs without relying on labels.
+
+Inference Acceleration on graphs. Recently, more and more research focuses on applying machine learning methods to industrial tasks (Cao et al., 2017; Lin & Cao, 2020). GNNs typically utilize message passing for feature aggregation from neighbors (Liu et al., 2023a; Zheng et al., 2023a; Pan et al., 2024), yet this process during inference can hinder latency-sensitive applications (Yin et al., 2022a; Yin et al.). Initial efforts in acceleration, such as quantization (Ding et al., 2021) and pruning (Zhou et al., 2021), failed to eliminate neighborhood dependencies. In response, GLNN (Zhang et al., 2022) introduces knowledge distillation (KD) (Hinton et al., 2015; Gou et al., 2021) to transfer knowledge from a teacher GNN to a student MLP, using the latter for inference acceleration. Subsequent studies (Tian et al., 2023b; Wu et al., 2023b) have enhanced this distillation process. Yet, these approaches primarily concentrate on well-trained teacher GNNs and label-supervised signals. Our work, in contrast, adopts a self-supervised approach, employing cross-pass GCL to enable MLP to acquire task-relevant knowledge and utilizing spatial positive nodes to foster graph-context awareness in MLP. The deployed MLP enjoyed both swift inference and competitive performance.
+
+# G. Sensitivity
+
+In this section, we analyze the method sensitivity to various hyper-parameters. Our findings, as depicted in Figure 7, indicate that our approach demonstrates low sensitivity to the number of semantic neighbors  $k$ . Appropriate  $k$  will enhance performance, but too much  $k$  will introduce noise and affect the learning effect. Furthermore, we perform ablation studies on the coefficient  $\alpha$  to highlight the respective roles of  $\mathcal{L}_{fl}$  and  $\mathcal{L}_{fh}$  in the learning process. The results, shown in Figure 6, indicate that S3GCL encourages greater similarity between one-hop and two-hop neighbors compared to randomly selected pairs, across both homophilic and heterophilic graphs. This suggests that S3GCL effectively makes nodes cognizant of their one-hop neighbor context, irrespective of homophily levels.
+
+# H. Additional Experimental Results
+
+# H.1. Representation Dimension
+
+This section delves into the impact of varying dimensions on different datasets, with the findings presented in Figure 9. The results suggest that larger dimensions generally lead to improved outcomes for all graphs. However, it is observed that excessively small dimensions can hinder the learning process, while overly large dimensions might cause overfitting issues.
+
+![](images/59732b7e761bb9d2c653bb6e618c8badd9f57d4435b26bbc274e181cfb52bf47.jpg)  
+(a) Cora
+
+![](images/154176b66a3b652a216e3dd2e8132b781ee91b9aaaa074b54b8ddb2be5925e06.jpg)  
+(b) PubMed  
+Figure 8. Impact of varying representation dimensions on four datasets, where red and yellow represent the S3GCL and BGRL respectively, while blue and green represent the GRACE and GREET respectively.
+
+![](images/9e3ee773f97e6b1e0ce1bdb0fbf9204a7108c0594f1a7c134d3f45f48ed21121.jpg)  
+(c) Actor
+
+![](images/14121cc6763066f3855a83e52945478288adcc61fbb66213633d434368793696.jpg)  
+(d) Roman-empire
+
+# H.2. Contrastive Learning Temperature
+
+The diagnostic analysis regarding the contrastive learning temperature is presented in Figure 8. Overall, the performance remains stable across various contrastive learning temperatures, with the exception of some extreme cases (e.g., when  $\tau$  is set to 0.01). Setting the temperature too high in contrastive learning can result in inadequate differentiation between similar and dissimilar pairs, leading to the loss of valuable information and reduced model sensitivity to subtle yet crucial distinctions. On the other hand, an excessively low temperature may cause the model to overemphasize minor differences between samples, potentially leading to overfitting and diminished generalization capabilities on novel, unseen data.
+
+![](images/e2b0417399b75bde6bd2d73f6157b58abaea594f09d573a3d7d121245f4c79e0.jpg)  
+(a) Cora  
+Figure 9. Impact of varying contrastive learning temperature  $\tau$  on four datasets, where red and yellow represent the S3GCL and BGRL respectively, while blue and green represent the GRACE and GREET respectively.
+
+![](images/6399d0e0f27b838d8e4e027f1ce4a4df3fdf901701b29876eb6c5ad4f862f341.jpg)  
+(b) PubMed
+
+![](images/b38b810e59c403ef05a39e5ff1f345ab409919e34b47b87b4359a2a55457384d.jpg)  
+(c) Actor
+
+![](images/e2ce2c894b10fedbcb7d2a6421b1b79a507d8e18b4a107503c2ef7e1a44a887a.jpg)  
+(d) Roman-empire

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2024/S3O_ A Dual-Phase Approach for Reconstructing Dynamic Shape and Skeleton of Articulated Objects from Single Monocular Video/29912845-7c45-4d75-8f31-1d61213f4588_origin.pdf

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2024/S3O_ A Dual-Phase Approach for Reconstructing Dynamic Shape and Skeleton of Articulated Objects from Single Monocular Video/full.md

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+# S3O: A Dual-Phase Approach for Reconstructing Dynamic Shape and Skeleton of Articulated Objects from Single Monocular Video
+
+Hao Zhang $^{1}$  Fang Li $^{1}$  Samyak Rawlekar $^{1}$  Narendra Ahuja $^{1}$
+
+# Abstract
+
+Reconstructing dynamic articulated objects from a singular monocular video is challenging, requiring joint estimation of shape, motion, and camera parameters from limited views. Current methods typically demand extensive computational resources and training time, and require additional human annotations such as predefined parametric models, camera poses, and key points, limiting their generalizability. We propose Synergistic Shape and Skeleton Optimization (S3O), a novel two-phase method that forgoes these prerequisites and efficiently learns parametric models including visible shapes and underlying skeletons. Conventional strategies typically learn all parameters simultaneously, leading to interdependencies where a single incorrect prediction can result in significant errors. In contrast, S3O adopts a phased approach: it first focuses on learning coarse parametric models, then progresses to motion learning and detail addition. This method substantially lowers computational complexity and enhances robustness in reconstruction from limited viewpoints, all without requiring additional annotations. To address the current inadequacies in 3D reconstruction from monocular video benchmarks, we collected the PlanetZoo dataset. Our experimental evaluations on standard benchmarks and the PlanetZoo dataset affirm that S3O provides more accurate 3D reconstruction, and plausible skeletons, and reduces the training time by approximately  $60\%$  compared to the state-of-the-art, thus advancing the state of the art in dynamic object reconstruction. The code is available on GitHub at: https://github.com/haoz19/LIMR.
+
+$^{1}$ Department of Electrical and Computer Engineering, University of Illinois Urbana-Champaign, USA. Correspondence to: Hao Zhang <haoz19@illinois.edu>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+Table 1. Difference between S3O and existing methods. Detailed comparisons are provided in Supp. Sec.A.  
+
+<table><tr><td>Method</td><td>Shape Template</td><td>Skeleton Template</td><td>Rigging</td><td>Multiple Videos</td><td>Camera Pose</td><td>Learn Skeleton</td></tr><tr><td>SMAL</td><td>✓</td><td>✓</td><td>skeleton</td><td>×</td><td>×</td><td>×</td></tr><tr><td>LASR</td><td>×</td><td>×</td><td>bones</td><td>×</td><td>×</td><td>×</td></tr><tr><td>ViSER</td><td>×</td><td>×</td><td>bones</td><td>×</td><td>×</td><td>×</td></tr><tr><td>BANMo</td><td>×</td><td>×</td><td>bones</td><td>✓</td><td>✓</td><td>×</td></tr><tr><td>CASA</td><td>✓</td><td>✓</td><td>skeleton</td><td>×</td><td>×</td><td>×</td></tr><tr><td>WIM</td><td>×</td><td>×</td><td>skeleton</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>CAMM</td><td>×</td><td>✓</td><td>skeleton</td><td>✓</td><td>✓</td><td>×</td></tr><tr><td>RAC</td><td>×</td><td>✓</td><td>skeleton</td><td>✓</td><td>✓</td><td>×</td></tr><tr><td>MagicPony</td><td>×</td><td>✓</td><td>skeleton</td><td>✓</td><td>×</td><td>×</td></tr><tr><td>S3O</td><td>×</td><td>×</td><td>skeleton</td><td>×</td><td>×</td><td>✓</td></tr></table>
+
+# 1. Introduction
+
+Given a single monocular video of an object undergoing articulated motion caused by an underlying skeleton of rigid bones (pairs of bones connected at joints and each bone surrounded by nonrigid skin), our goal is to estimate the visible dynamic 3D shape and invisible skeleton pose. We represent the motion of the objects using a physical 3D skeleton (Magnenat et al., 1988; Baran & Popovic, 2007b; Wang et al., 2007; Xu et al., 2020) with certain poses. This task is highly challenging as it requires concurrent learning of shape, skeleton, motion, and camera parameters from a limited number of camera views. An error in any one of them can seriously impact the reconstruction results. Recent methods (Weng et al., 2022; Yang et al., 2021b; 2022; Luvizon et al., 2019) make use of virtual bones to represent the motion. These virtual bones do not model the physical skeleton as well as our physical bones (Fig.3 and Sec.4.3). Some methods (Yang et al., 2021a;b; 2022) calculate the displacement of each vertex in terms of the motion of the virtual bones and skinning weights using the blend skinning technique. They assign a bone to a group of nearby surface vertices without regard to the underlying structure of the mesh. As a result, these methods often fail to achieve physically plausible bone distribution as shown in Fig.3 (d). Other, model-based methods (Yang et al., 2023; Wu et al., 2023b; Kuai et al., 2023; Wu et al., 2022; Goel et al., 2023) use pre-defined shape and skeleton templates to attain a physically plausible skeleton and bone distribution; however, they require these as category-specific ground truth inputs, instead of estimating them from the video. This diminishes their generalizability to out-of-distribution (OOD)
+
+objects. In terms of the amount of input, many recent NeRF-based methods (Yang et al., 2022; Noguchi et al., 2022; Wu et al., 2023b) require multiple videos, captured from a wide range of viewpoints, along with accurate camera calibration. Most current methods require substantial computational resources to obtain reasonable results, taking  $8 - 30$  hours on A100 GPUs to train on a single object, which again reduces their usability.
+
+In this work, we introduce Synergistic Shape and Skeleton Optimization (S3O), a two-phase methodology that (1) learns visible shape and the underlying skeleton, without reliance on pre-specified parametric models or additional human annotations, and (2) significantly reduces the training time by segregating the learning of distinct parameters such as shape, motion, and camera parameters in two phases: a coarse parametric model learning phase, and a motion and fine model learning phase.
+
+In phase one, we automatically extract a canonical frame from the entire video, which depicts the object in standard poses (e.g., side view) via the horizontal-to-vertical distance ratio, lift its 2D skeleton to 3D, and learn a coarse rest shape. (Sec.3.2). The second phase is dedicated to learning the time-varying parameters, including the motion and camera parameters, and refining the shape and skeleton details. Here, we employ a method akin to the Expectation-Maximization (EM) algorithm, alternating the update of shape and skeleton. During the E-step, we utilize the current skeleton to deduce the bones and update the shape, time-varying camera parameters, and temporal transformation of each bone (motion). In the M-step, we begin by adapting the skeleton to align with the deformed mesh using skinning weights (Sec.3.4). Subsequently, utilizing the current mesh and time-varying parameters, we calculate the 2D motion and length of each bone, thereby forming physical constraints that guide the update of the skeleton's structure. Since current datasets mostly include minimal camera motion, they fail to test an algorithm's robustness against extensive camera movement. To overcome this, we introduce a new dataset, PlanetZoo, designed to assess algorithm performance under significant, long-range camera movements. Key contributions of our work include:
+
+- We introduce S3O which estimates the 3D shape and skeleton of articulated objects from a single monocular video, without prior templates and ground truth camera poses, while significantly reducing training time and computational complexity over SOTA.  
+- We introduce a new dataset called PlanetZoo, to help evaluate 3D reconstruction methods from single monocular video.  
+- Experiments on standard and PlanetZoo videos show that S3O outperforms SOTA methods in shape recon
+
+striction quality while requiring  $60\%$  less training time.
+
+- S3O displays superior generalization capabilities over RigNet (Xu et al., 2020) and Skeleton (Au et al., 2008) in reconstructing skeletons.
+
+# 2. Related Work
+
+3D Skeleton Extraction for Reconstruction. Skeletons lack a uniform definition and are subject to varying requirements across different applications. Some efforts aim to capture the object's medial axis (Au et al., 2008), a process known as curve-skeleton extraction. For instance, voxel-thinning methods (Ma et al., 2002; Palágyi & Kuba, 1999; Wang & Lee, 2008) achieve this by iteratively pruning boundary voxels while preserving topology. In contrast, distance-field methods (Hassouna & Farag, 2005; Zhou & Toga, 1999; Wade, 2000; Bitter et al., 2001; Cornea et al., 2005; Ma et al., 2003) involve computing a distance transform for each interior point of a 3D object and then identifying ridges in the field to select a set of candidate voxels. Recent works (Xu et al., 2020; Baran & Popović, 2007a; Yao et al., 2023; Yu et al., 2017; Vlasic et al., 2008b) focus on learning skeletons, tailored to reflect the mobility of an articulated shape. These physical skeletons, influenced by more than just shape geometry, incorporate the placement of joints and bones based on the animator's understanding of the character's anatomy and anticipated deformations.
+
+Model-Based and Class-Specific Approach. The model-based approach employs a parametric framework, incorporating structured mesh templates and associated skinning weights, to tackle the challenge of estimating 3D shape and pose. Extensive research (Zuffi et al., 2017; Xiang et al., 2019; Loper et al., 2023) dedicated to 3D reconstruction of human and animal forms makes use of these parametric models, which are informed by 3D scans of actual human figures or animal replicas. The latest regression techniques (Yang et al., 2023; Zuffi et al., 2019; Biggs et al., 2020; Badger et al., 2020; Kocabas et al., 2020) devised to deduce pose parameters have shown to yield satisfactory reconstruction outputs, though they typically depend on manually provided annotations like pose, keypoints, and rely on predefined parametric models for their training processes. Consequently, these model-based approaches encounter difficulties when applied to the reconstruction of objects in an open-world context and exhibit limited generalizability in the absence of comprehensive annotations. Current trends aim to minimize reliance on 3D annotations, opting for 2D annotations like silhouettes and key points (Goel et al., 2020; Kanazawa et al., 2018; Li et al., 2020b). Single-view image reconstruction achieves impressive results without explicit 3D annotations (Li et al., 2020a; Kanazawa et al., 2018; Kulkarni et al., 2020). MagicPony (Wu et al., 2023b)
+
+learns 3D models for objects such as horses and birds from single-view images, but faces challenges with fine details and substantial deformations, especially for unfamiliar objects.
+
+Reconstruction from Video. Instead of leveraging predefined parametric models, some works (Yang et al., 2021a;b; 2022) jointly learn the skin weights, shape, camera parameters, and motion from monocular video(s). This method requires only mask annotations, eliminating the need for additional labeling and significantly enhancing its generalizability in real-world scenarios. However, due to the absence of structural information about the object and annotations regarding camera parameters, this approach tends to be effective only under certain conditions, such as with videos that do not involve extensive camera movement and that contain a multitude of standard viewpoints, including side and front views. Moreover, the Non-Rigid Structure from Motion (NRSfM) technique (Torresani et al., 2008; Kumar, 2019) is capable of functioning without explicit 3D data. It processes 2D keypoints and optical flow derived from readily available methods (Teed & Deng, 2020). However, this process necessitates persistent point tracking, which is often impractical or unavailable/unreliable in real-world settings.
+
+# 3. Method
+
+# 3.1. Terminology and Overview
+
+Towards articulated object reconstruction from video sequences, we propose a method, named Synergistic Shape and Skeleton Optimization (S3O), that concurrently learns parametric models and time-varying motion parameters. The parametric model includes canonical shape  $\mathbf{M}_c$ , underlying skeleton  $\mathbf{S}_{\mathbf{c}} = \{\mathbf{B} \in \mathbb{R}^{B \times 13}, \mathbf{J} \in \mathbb{R}^{J \times 5}\}$ , and skinning weights  $\mathbf{W} \in \mathbb{R}^{N \times B}$ . The time-varying motion parameters include the camera parameters and articulation  $\mathbf{T}^t = \{\mathbf{T}_0^t, \mathbf{T}_1^t, \dots, \mathbf{T}_B^t\}$ ,  $\mathbf{T}_b^t \in SE(3)$ , where  $\mathbf{T}_0^t$  and  $\mathbf{T}_{b > 0}^t$  are the root body and  $B$  bone transformations. The quantities  $B$ , and  $J$  represent the count of bones and joints. Skeleton Definition. In the skeleton, bones  $\mathbf{B}$  are further defined by Gaussian centers  $\mathbf{B}(:, :3]$ , precision matrices  $\mathbf{B}(:, 3:12]$ , and bone lengths  $\mathbf{B}(:, 12]$ . Joints  $\mathbf{J}$  are specified by connections between bones  $\mathbf{J}(:, :2]$  and their spatial coordinates  $\mathbf{J}(:, 2]$ .
+
+Overview of S3O. As illustrated in Algorithm.1, the algorithm commences with a Coarse Shape Phase, wherein it selects one canonical frame from the video (Sec.3.2) to extract a 2D skeleton and derives an initial coarse 3D parametric model (Sec.3.2). This model forms the basis for the subsequent Motion & Fine Shape Phase, a process of time-varying parameters optimization and 3D shape and skeleton refinement (Sec.3.4). Similar to an Expectation-
+
+Maximization approach, the algorithm alternates between adapting the skeletal structure to optimize shape and applying physical constraints to ensure the physical plausibility of the model. The optimization cycles through these steps, gradually improving the representation until convergence is achieved (the skeleton doesn't change), thus providing an accurate and dynamic 3D reconstruction of the articulated subjects in the video. Details of each step and the associated physical constraints are delineated in the respective sections of the paper below.
+
+# 3.2. Coarse Model Learning
+
+Canonical Frames and 2D Skeleton Extraction. Given a set of video frames, we first apply the medial axis transform (Lee, 1982; Ahuja & Chuang, 1997) to their silhouettes where we gradually remove the pixels from the border until we obtain a single-pixel-wide potential skeleton that preserves the original region's extent and connectivity. Along the 2D skeleton curve, we categorize the points into three distinct groups: junctions (points with three or more neighbors), endpoints (points with only one neighbor), and connections (points with two neighbors). Next, we remove the connections and reestablish connections between the remaining endpoints and junctions while preserving the overall topology as shown in Fig.1. By computing the skeleton's horizontal-to-vertical distance ratio across all frames, the frame with the highest ratio is selected as the canonical frame.
+
+Generating Coarse 3D Skeleton and Mesh. We leverage the 2D skeleton of the canonical frame as the foundation for generating the coarse 3D skeleton following (Yao et al., 2022; 2023). The key idea in this transformation is the utilization of the plane of symmetry found in most articulated objects. Starting with the silhouette and the 2D skeleton, we create an ellipsoid around each edge (blue lines in Fig.1), ensuring it fits the silhouette as shown in Fig.14. Next, we identify the symmetric parts by the following features: (1) sum of average DINO features, (2) bone length, and (3) average radius of the ellipsoid similar to (Yao et al., 2023). For the symmetric parts, we set the z-coordinates to be on the opposite side of the symmetry plane. Next, we deform the neural surface to obtain mesh. We follow the deformation methods from (Lindell et al., 2022), which performs analysis-by-synthesis to supervise the shape reconstruction. To supervise deformations, we rely on silhouette, part silhouette, and semantic consistency loss described in Sec.D.
+
+# 3.3. Time-Varying Articulation
+
+Skinning Weights. W is designed to represent the probabilities that each vertex corresponds to  $B$  semi-rigid parts. Following LASR (Yang et al., 2021a), the skinning weights
+
+![](images/d452fddaba3e3a7c23f3cdb03fcfa60836870e27c568bb751d285967b0f2d499.jpg)  
+Figure 1. Overview of Skeleton Learning of S3O: We start by deriving a 3D skeleton from 2D inputs and learning a coarse shape, represented in 'cyan' for the coarse and 'gray' for the current shape. The next step involves refining this shape by adjusting the skeleton, shifting bones, and expanding the structure. We then upsample the skeleton points for greater detail before applying final physical constraints to ensure proper results.
+
+are modeled by the mixture of  $B$  Gaussian ellipsoids (semirigid parts) as:  $\mathbf{W}_{n,b} = \mathcal{F}e^{-\frac{1}{2} (\mathbf{X}_n - \mathbf{C}_b)^T\mathbf{Q}_b(\mathbf{X}_n - \mathbf{C}_b)}$ , where  $\mathcal{F}$  is the factor of normalization and precision matrices  $\mathbf{Q}_b = \mathbf{V}_b^T\boldsymbol{\Lambda}_b\mathbf{V}_b$ . In each Gaussian ellipsoid,  $\mathbf{C} \in \mathbb{R}^{B \times 3}$  denotes Gaussian centers,  $\mathbf{V} \in \mathbb{R}^{B \times 3 \times 3}$  defines the orientation and  $\boldsymbol{\Lambda} \in \mathbb{R}^{B \times 3 \times 3}$  denotes the diagonal scale matrix.  $\mathbf{X}_n$  is the 3D location of vertex  $n$ .
+
+Dynamic Rigidity. To guarantee the flexibility of the deformed articulated mesh without losing stability, following LIMR (Zhang et al., 2023), we leverage Dynamic Rigidity (DR) loss  $\mathcal{L}_{\mathrm{DR}}$  to provide more freedom for the vertices close to the joints and less freedom for the ones between adjacent joints (along the bones), in that the vertices around joints are always more likely to be deformed. The freedom rules are derived from the proposed Rigidity Coefficient  $\mathbf{R} \in \mathbb{R}^{N \times N}$  calculated by skinning weights  $\mathbf{W} \in \mathbb{R}^{N \times B}$ .
+
+Rigidity Coefficient  $\mathbf{R} \in \mathbb{R}^{N \times N}$  is proposed to represent the freedom of each vertex relative to each neighbor vertex based on the a priori assumption on skinning weights  $\mathbf{W}$ . For example, given  $\mathbf{W} \in \mathbb{R}^{N \times B}$ , each vertex  $\mathbf{X}_n$  is provided a discrete probability distribution regarding all bones. The denser the distribution, the higher the likelihood that the vertex is assigned to the core of only one specific semi-rigid part (bone). In stark contrast, the sparser the distribution, the more probable the vertex is attributed to the border between two semi-rigid parts, which is actually around the joint. We model the density of distribution by entropy  $\mathcal{E}$  and simulate the rigidity coefficient  $\mathbf{R}$  by the product of entropies:
+
+$$
+\mathbf {R} _ {i, j} = \mathcal {E} (i) ^ {- 1} \mathcal {E} (j) ^ {- 1}, \tag {1}
+$$
+
+$$
+\mathcal {E} (i) = \sum_ {b = 0} ^ {B - 1} \mathbf {W} _ {i, b} \log_ {2} \mathbf {W} _ {i, b} + \lambda .
+$$
+
+Here,  $\mathbf{R}$  is a skew-symmetric matrix so that  $\mathbf{R}_{i,j} = \mathbf{R}_{j,i}$  and  $\mathbf{R}_{i,j}$  signifies the rigidity coefficient between vertices  $i$  and  $j$ . Besides, the stabilization constant is set as  $\lambda = 0.1$  by default avoiding being divided by zero. Notably, in practice,  $\mathbf{R}_{i,j}$  will be set to 0 if there is not an edge connecting vertices  $i$  and  $j$ .
+
+Given Rigidity Coefficient  $\mathbf{R}$ , we define the Dynamic Rigid
+
+ity (DR) loss  $\mathcal{L}_{\mathrm{DR}}$  as following:
+
+$$
+\mathcal {L} _ {\mathrm {D R}} = \sum_ {i = 1} ^ {n} \sum_ {j \in N _ {i}} \mathbf {R} _ {i, j} \left| \left\| \mathbf {X} _ {i} ^ {t} - \mathbf {X} _ {j} ^ {t} \right\| _ {2} - \left\| \mathbf {X} _ {i} ^ {t + 1} - \mathbf {X} _ {j} ^ {t + 1} \right\| _ {2} \right|, \tag {2}
+$$
+
+where  $\mathbf{X}_i^t$  means the coordinate of vertex  $i$  at time  $t$ . DR loss allows more adaptive freedom to deformation compared to the conventional ARAP (As Rigid As Possible) loss (Yang et al., 2021a). The latter encourages the constant distance between the adjacent vertices, which limits the movement and deformation of the vertices near the joints.
+
+Blend Skinning. The mapping from surface vertex  $\mathbf{X}_n^0$  in canonical space (time 0 by default) to  $\mathbf{X}_n^t$  at time t in camera space is designed by blend skinning. The forward blend skinning is shown below:
+
+$$
+\mathbf {X} _ {n} ^ {t} = \mathbf {T} _ {0} ^ {t} \left(\sum_ {b = 1} ^ {B} \mathbf {W} _ {n, b} \mathbf {T} _ {b} ^ {t}\right) \mathbf {X} _ {n} ^ {0}, \tag {3}
+$$
+
+including the number of bones  $\mathbf{B}$ , skinning weights  $\mathbf{W}$  and the transformation  $\mathbf{T}^t = \{\mathbf{T}_b^t\}_{b=0}^B$  at time  $t$ , where  $\mathbf{T}_0^t$  represents the root body transformation (Yang et al., 2021a) and  $\mathbf{T}_{b>0}^t$  describes the bone transformation. Each  $\mathbf{X}_n^0$  is first transformed by the weighted sum of each bone transformation  $\mathbf{T}_{b>0}^t$  and then transformed by root body transformation  $\mathbf{T}_0^t$  to achieve  $\mathbf{X}_n^t$ . On the other hand, the backward blend skinning maps  $\mathbf{X}_n^t$  back to  $\mathbf{X}_n^0$  utilizing  $\mathbf{T}^{-1}$  instead of  $\mathbf{T}$ .
+
+2D Optical Flow Warping for Bone Motion. Our task is to discern the 2D motion trajectory of each semi-rigid segment covering a bone, from time  $t$  to  $t + 1$ . 2D optical flow is represented as  $\mathbf{F}^{2\mathrm{D},t}$ , and the specific motion path for a bone is denoted as  $\mathbf{F}^{B,t} \in \mathbb{R}^{B \times 2}$ . Essentially, a bone's movement is a cumulative effect of the movements of its exterior vertices. As shown in Fig.8, this process involves back-projecting each pixel from the 2D optical flow into the 3D canonical space. Due to the imperfect alignment between vertices and pixels, the surface motion trajectory  $\mathbf{F}^{S,t} \in \mathbb{R}^{N \times 2}$  is calculated through bilinear interpolation.
+
+A critical aspect is the inaccuracy of applying optical flow directions to vertices on the obscured side of the surface.
+
+![](images/6c2a2de977d7cc446512e5c1bbfd33d7c0f4e5ab246714ec886e1cc12829fefb.jpg)  
+RGB Images  
+Figure 2. Mesh Results. We show the reconstruction results of (a) LASR, (b) BANMo, and (c) Ours in the DAVIS's camel, cows and PlanetZoo's zebra, elephant, tiger, and giraffe. Top view of the reconstruction results are shown in Fig.9.
+
+![](images/76dce8107c1fdf796d5b797f2b83789005eb0cbd3612e5b88f4bc74abe6f87c4.jpg)  
+(a) LASR
+
+![](images/b978529fba6d1a0e60cc4d9a9fe7b0a7a233facbf89a866a73287310a243cc47.jpg)  
+(b) BANMo
+
+![](images/02dd2cef739c98b1f91099a6d71435ca1fee04e06b9c5615bee1d5c06dd79142.jpg)  
+(c) Ours
+
+![](images/95a8ba4cad938086edb67caea9974bd735641b98195ac7be60c8cef324b665bc.jpg)  
+RGB Images
+
+![](images/fcfe8ed8fabcbbd357f3c3d169f73adf3e9bfd06f0dc43fecfa86f4109145f9c.jpg)  
+(a) LASR
+
+![](images/585e336ec9d9dca1dbfaaff7c1f7fd825d224c152e86c8f604a92de49967c6f6.jpg)  
+(b) BANMo
+
+![](images/cc777a94959f01d2d586c88d395708df70d2b394710c3e84429a0bdd78a7cda3.jpg)  
+(c) Ours
+
+![](images/2eec75dd7f5a30014f16847de47c64bd9f1b836d14808ab8806c674af9025642.jpg)  
+RGB Images
+
+![](images/c6c4b7b480625da00e97d1e8bb8aeb87d831e8c09b9b996a1ad5615b844edd9d.jpg)  
+(a) LASR
+
+![](images/5d1c5880044f599663d4fb12bafeb37de0a3aa472c5851c7228cb1f9b1ab2806.jpg)  
+(b) BANMo
+
+![](images/ab1dd245cddc8b5fc41b1f8322b24c523aad89883b9c1ed36cbb7ba3b7306e99.jpg)
+
+![](images/7729c7bdfc75b15b908559325c5014aceac20f1569f645076686d67164b77540.jpg)
+
+![](images/36c394e19f0c629b09d23ad8daeb41ceeada1df08816ad0cce142176c9c9ceec.jpg)
+
+![](images/6b5ccc8625e66615fecd3726da6365cdc88c130a1e290bd7849c5bb2c124088c.jpg)  
+RGB Images
+
+![](images/d80227385a96c13142c21b236fa89a62e202a3e1e602061a938080f96de16c60.jpg)  
+Before Refinement  
+Figure 3. Skeleton and Skinning Weight Results. (a), (b), and (c) show the skeleton and skinning weights results from S3O with different bone motion similarity thresholds (0.99, 0.95, 0.90). (d) shows the bone distribution and skinning weights from LASR. Since each vertex on the surface is assigned to multiple bones (each given a different color) via the skinning weights, here we show the vertex in the color of the bone which has the maximum skinning weight for the vertex.
+
+![](images/1ebd3fec6045d17939cb0b4c6736d94952b1c8e57679c67d5f28665ba12b5820.jpg)  
+(a) Parametric Model  $(\mathrm{T} = 0.99)$
+
+(a) Parametric Model  $(\mathrm{T} = 0.99)$
+
+![](images/3ba5ec3027f943aa3495044d2f90d6c38d87214f9a3d7c1a9a7380bb0df4a202.jpg)
+
+![](images/0cb53c8d92ae771ed3778abaf6da76e82e1675a76ffd39c7eba204a614675c46.jpg)  
+(b) Parametric Model  $(\mathrm{T} = 0.95)$
+
+(b) Parametric Model  $(\mathrm{T} = 0.95)$
+
+![](images/b909e771951cabaf927685f629f05d242e81ac18c9be29b57801dddd126eadde.jpg)
+
+![](images/3704cb66dea4dec8c53c1625dc238f389a3eb5374bc7711dfea5113ce3d07f86.jpg)
+
+![](images/cb1f13d2b84af4049b5ebbdbe8cd1e73ed630e6878593c33d64b8386dc506519.jpg)  
+(c) Parametric Model  $(\mathrm{T} = 0.90)$
+
+![](images/a38712192309fc30f5e77f421a60411e455dfe08316db96784de892f7a3ac325.jpg)
+
+![](images/cbd4f30d20a346a0a598ea9c5033f547f45427c63c9e9c25f683771abc449958.jpg)
+
+![](images/5b6d9663fb50d5b1ed3ec619c93b709bfbc12628d3a5adc34b64f4d871491b33.jpg)
+
+![](images/5e0ab491f8d1025203b7c7558c684a26c6c90cb91be65a0509b87cb2e6415230.jpg)  
+(d) LASR
+
+To rectify this, a visibility matrix  $\mathcal{V}^t\in \mathbb{R}^{N\times 1}$  is created, reflecting the current mesh configuration and viewpoint at time  $t$ , determined via ray-casting. Optical flow for obscured vertices is thus disregarded.
+
+The bone motion trajectory is then approximated by blending the optical flow of surface vertices. Considering the predefined skinning weights  $\mathbf{W}$  and the derived surface flow path  $\mathbf{F}^{S,t}$ , the bone motion direction is expressed as:
+
+$$
+\mathbf {F} _ {b} ^ {B, t} = \sum_ {n = 0} ^ {N - 1} \mathbf {W} _ {n, b} \mathbf {F} _ {n} ^ {S, t} \mathcal {V} _ {n} ^ {t}, \quad \mathbf {F} ^ {B, t} = \left\{\mathbf {F} _ {0} ^ {B, t}, \dots , \mathbf {F} _ {B - 1} ^ {B, t} \right\} \tag {4}
+$$
+
+Here,  $\mathbf{F}_b^{B,t}$  represents the directional flow linked with bone  $b$ , while  $\mathbf{F}_n^{S,t}$  and  $\mathcal{V}_n^t$  specify the directional flow and the visibility of vertex  $n$ , respectively.
+
+# 3.4. Motion and Fine Shape Phase
+
+Our approach is grounded in two interdependent sets of learnable parameters. The first set includes 3D shape and time-varying motion parameters. The second set pertains to the underlying skeleton, encompassing the bones and articulation joints. To achieve a harmonized optimization of these parameter sets, we update both parameter sets iteratively similar to the EM algorithm. During the E-step, the skeleton remains fixed, and the first set of parameters is then updated based on the reconstruction loss. Subsequently, using the updated parameters, in the M-step we adjust the skeleton to match the current mesh and then determine the bone motion direction and bone lengths for constructing
+
+physical constraints. Such constraints are utilized for optimizing the skeleton. The skeleton optimization consists of three operations: (1) bone shifting, (2) skeleton growth, and (3) physically constrained refinement. Since the coarse shape phase typically yields only an approximate shape, the second phase involves significant deformations and scale changes to the mesh in each iteration, as shown in Fig.1. The purpose of the Bone Shifting and Skeleton Growing operations is to automatically update the skeleton so that its structure aligns with the revised mesh.
+
+Bone Shifting. We start with the assumption that, although each vertex of the mesh undergoes displacement, their relative position remains unchanged as the coarse shape already being closely representative of the actual shape. Therefore, we update the Gaussian center for the bone with the centroid of the corresponding semi-rigid part:
+
+$$
+\mathbf {B} [ b,: 3 ] = \frac {1}{| \mathbf {I} _ {b} |} \sum_ {i \in \mathbf {I} _ {b}} \mathbf {M} _ {c} [ i ], \tag {5}
+$$
+
+where  $b$  is the index of bone,  $\mathbf{I}_b$  is a set of vertex indexes that belong to the semi-rigid part corresponding to bone  $b$ .  $\mathbf{W}_{\mathrm{one - hot}}[i,:] = \mathbf{W}[\mathbf{W}[i,:]] == \mathrm{argmax}_i\mathbf{W}[I,:]]$ ,  $\mathbf{W}_{\mathrm{one - hot}}\in \{0,1\}^{N\times B}$ , and  $\mathbf{I}_b = \mathbf{W}_{\mathrm{one - hot}} == 1$ .
+
+Skeleton Growing. The coarse shape often fails to accurately reconstruct the fine structures at the extremities of an object, such as limbs, which mostly need to be built during the second phase. This results resulting in an incomplete coarse skeleton. To complete it during the second phase, we introduce the skeleton growth that automatically extends the
+
+Table 2. 2D Keypoint Transfer Accuracy. This table presents the 2D keypoint transfer accuracy on DAVIS and PlanetZoo datasets. For classes from DAVIS (camel, cow) and PlanetZoo (giraffe, tiger), we carry out multiple executions and report the mean and standard deviation of the observed accuracy in each case. Time is in GPU hours on A100. Note that the results for LASR and ViSER here are from our executions of the codes provided by the authors. The authors in their papers provide a single number for accuracy. It is not clear if this represents the results of a single run, or mean/min/max of multiple runs. This makes our numbers hard to compare with those reported by the authors in their papers.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="7">DAVIS</td><td colspan="8">PlanetZoo</td></tr><tr><td>camel</td><td>cow</td><td>dog</td><td>bear</td><td>dance-twirl</td><td>Ave.</td><td>Ave. time</td><td>giraffe</td><td>tiger</td><td>elephant</td><td>bear</td><td>zebra</td><td>deer</td><td>Ave.</td><td>Ave. time</td></tr><tr><td>ViSER</td><td>71.7 ±4.4</td><td>73.7 ±4.1</td><td>65.1</td><td>72.7</td><td>78.3</td><td>72.3</td><td>10.4</td><td>51.2 ±4.6</td><td>68.4 ±4.3</td><td>68.9</td><td>60.3</td><td>55.7</td><td>57.3</td><td>60.3</td><td>12.1</td></tr><tr><td>LASR</td><td>75.2 ±2.9</td><td>80.3 ±2.4</td><td>60.3</td><td>83.1</td><td>55.3</td><td>70.8</td><td>9.1</td><td>56.3 ±3.7</td><td>70.4 ±2.1</td><td>69.5</td><td>63.1</td><td>57.4</td><td>60.3</td><td>62.8</td><td>10.2</td></tr><tr><td>S3O</td><td>79.8 ±1.1</td><td>83.2 ±0.7</td><td>67.3</td><td>86.7</td><td>84.7</td><td>80.4 ↑5.7</td><td>3.8 ↓5.3</td><td>63.1 ±0.8</td><td>72.8 ±0.6</td><td>71.3</td><td>66.4</td><td>61.5</td><td>62.1</td><td>66.2↑3.4</td><td>4.2↓6.0</td></tr></table>
+
+![](images/2f731fef1cdfb682d338841287dc07101c4353fa0778aeeabf145a18ca475d5b.jpg)  
+Figure 4. Skeleton Comparison. We compared the learned skeleton for tiger, elephant, camel from our approach with RigNet (Xu et al., 2020) and Skeleton (Au et al., 2008).
+
+skeleton based on the new extremity structures formed by the mesh, ensuring the skeleton's growth is both systematic and reflects the nuanced changes in the mesh structure. The skeleton growth operates by identifying end parts  ${}^{1}$  and subdividing them into 2 distinct parts along the direction of the skeletal edge if the maximum distance within the end part is more than twice of the initial value. Once identified, the algorithm computes the direction vector of the growth by taking the normalized vector pointing to the endpoint from the connected point on the skeleton. The vertices assigned to the endpoint are then projected onto this direction vector, and their distribution along the edge is used to subdivide them into  $k$  sections. Each section's centroid is calculated, and new skeletal points (bones) are added at these centroid locations. These points are interconnected to form new edges that replace the original edge, effectively growing the skeleton.
+
+Physically Constrained Refinement. The skeleton is refined by adhering to two physical principles: (1) The fusion of bones  $\mathbf{B}_b$  and  $\mathbf{B}_{b'}$  is executed when they exhibit synchronized movements throughout all  $\mathbf{H}$  frames  $\{\mathbf{I}_f\}_{f=0}^{\mathbf{H}}$ , suggesting they form a single semi-rigid structure. (2) A joint is established between  $\mathbf{J}_j$  and  $\mathbf{J}_{j'}$  upon observing significant distance variations in certain frames.
+
+Accordingly,
+
+$$
+\begin{array}{l} \text {i f} \max  _ {f} \operatorname {L e n g t h} \left(\mathbf {B} _ {b ^ {\prime}} ^ {f}\right) - \min  _ {f} \operatorname {L e n g t h} \left(\mathbf {B} _ {b ^ {\prime}} ^ {f}\right) > t _ {d} (6) \\ \Rightarrow \text {c r e a t e} \widetilde {\mathbf {J}}; (7) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \operatorname * {i f} \min  _ {f} \mathcal {S} \left(\mathbf {F} _ {B} ^ {b ^ {\prime}, f}, \mathbf {F} _ {B} ^ {b ^ {\prime \prime}, f}\right) > t _ {o} (8) \\ \Rightarrow \operatorname {m e g r e} \mathbf {B} _ {b ^ {\prime}} \text {w i t h} \mathbf {B} _ {b ^ {\prime \prime}} \text {t o g e n e r a t e} \widetilde {\mathbf {B}}, (9) \\ \end{array}
+$$
+
+where  $S(\mathbf{a},\mathbf{b}) = \frac{\mathbf{a}\cdot\mathbf{b}}{||\mathbf{a}||_2\times||\mathbf{b}||_2}$  calculates the cosine similarity. Moreover, the position of  $\widetilde{\mathbf{J}}$  is determined by averaging the coordinates of  $\mathbf{J}_j$  and  $\mathbf{J}_{j^{\prime}}$ , after which their connection is removed and redirected to  $\widetilde{\mathbf{J}}$ . The formulation  $\widetilde{\mathbf{B}} = w_{b}^{\prime}\mathbf{B}_{b^{\prime}} + w_{b^{\prime \prime}}\mathbf{B}_{b^{\prime \prime}}$ , where  $w_{b}^{\prime} = \frac{\exp\sum_{b = b^{\prime}}\mathbf{W}_{n,b}}{\exp\sum_{b = b^{\prime}}\mathbf{W}_{n,b} + \exp\sum_{b = b^{\prime \prime}}\mathbf{W}_{n,b}}$ , is applied to Gaussian centers  $\mathbf{C}$  and precision matrices  $\mathbf{Q}$ . However, the new bone length is directly derived from the joint coordinates at the extremities of the two bones.
+
+Joint Positions and Bone Lengths Calculation. Utilizing the current bone coordinates  $\mathbf{B}$ , the inter-bone links  $\mathbf{J}[:,2]$ , and the skinning weights  $\mathbf{W}$ , the procedure begins by locating the vertices  $V_{i,j}$  situated directly above the joint  $\mathbf{J}_{i,j}$  that connect bones  $i$  and  $j$ . A vertex  $v_{n}$  is classified as part of  $V_{i,j}$  if its corresponding weights for bones  $i$  and  $j$  ( $\mathbf{W}_{n,i}$  and  $\mathbf{W}_{n,j}$ ) meet or exceed a threshold value  $t_r$  (commonly 0.4). The position of the joint  $\mathbf{J}_{i,j}$  is then ascertained by averaging the coordinates of the vertices in  $V_{i,j}$ , formulated as  $\mathbf{J}_{i,j}[:,2:] = \frac{\sum v_n \in V_{i,j} \mathbf{X}_n}{|V_{i,j}|}$ , where  $\mathbf{X}_n$  denotes the coordinates of  $v_{n}$ . Finally, upon acquiring each joint's coordinates, we compute the lengths of bones by determining the distance between the coordinates of joint pairs.
+
+# 3.5. Losses and Regularization
+
+In the coarse shape phase, S3O leverages a reconstruction loss, which consists of silhouette loss, optical flow loss, texture loss, perceptual loss, DINO feature loss, and smooth, motion, and symmetric regularizations. In the fine shape phase, unlike the coarse shape phase, we eliminate the DINO feature loss. While the use of DINO features effectively aids in quickly learning a coarse model in the initial phase, their limited accuracy eventually hampers the acquisition of a more detailed shape in later stages. More
+
+![](images/22e861d6ca6163880b286f0ed175249bcdd349547db41d56ba64282ca4fca925.jpg)  
+Figure 5. Rendering Results of S3O and LASR on camel video.
+
+![](images/dc0761fd2ee97edc1235b9d6bf61988b2f48f15e0982e8d944c37e39783ab46a.jpg)  
+Figure 6. Enhanced Physical Plausibility with Dynamic Rigidity. (a), (b), and (c) show the results of S3O with Dynamic Rigidity (DR), with As-Rigid-As-Possible (ARAP), and without both regularization terms.
+
+details and discussion are in Sec.D.
+
+# 4. Experiments and Discussion
+
+In this section, we conduct a detailed evaluation of our approach, focusing on two main areas: 3D Shape Reconstruction (Sec.4.2) and 3D Skeleton Reconstruction (Sec.4.3). For shape reconstruction, we compare our method with state of art techniques including Hi-lASSIE (Yao et al., 2023), LASR (Yang et al., 2021a), and BANMo (Yang et al., 2022). All the requirements for S3O, LASR, and Hi-LASSIE are the same while BANMo needs an extra pre-trained PoseNet to obtain the initial camera pose. For skeleton reconstruction, we use RigNet (Xu et al., 2020) and Skeleton (Au et al., 2008) as benchmarks. Note that RigNet is trained with ground truth 3D annotations, such as skinning weights and skeletons, while S3O and Skeleton (Au et al., 2008) do not require any 3D annotation. More experimental results and comparisons are given in Sec.E. Details and results for the benchmarks and implementation are given in Sec.B.
+
+# 4.1. Datasets
+
+PlanetZoo. We introduce here a curated dataset named PlanetZoo, which includes RGB synthetic videos of different animals with around 100 frames each. PlanetZoo covers a 180-degree visual field, captured by a moving camera to allow better evaluation of 3D reconstruction when imaging parameters must also be dynamically estimated due to the moving camera, and over a large visual field. In addition, following BADJA (Biggs et al., 2018), we also provide 2D key point annotations. More details about PlanetZoo, as well as DAVIS (Peruzzi et al., 2016) and AMA-Human (Vlasic et al., 2008a) are provided in Sec.C.
+
+# 4.2. Shape Reconstruction Comparison
+
+Qualitative Comparisons. As depicted in Fig.2,9 (b) and 10, existing NeRF-based (Neural Radiance Fields) approaches, often require a substantial number of input videos, acquired from a wide range of viewpoints and precise camera poses. Therefore, when the input is a monocular short video, these methods struggle to produce plausible outcomes. We present the results of BANMo (Yang et al., 2022) in Fig.2. MagicPony (Wu et al., 2023b), RAC (Yang et al., 2023), and CAMM (Kuai et al., 2023) also have nearly the same results as BANMo because they can be seen to be an extension of BANMo utilizing a pre-trained RigNet (Xu et al., 2020) to generate skeleton or directly providing predefined skeleton. However, when the mesh quality is poor, the reconstruction results are also significantly compromised. Conversely, for monocular short videos, methods based on NMR (Neural Mesh Rendering) like LASR (Yang et al., 2021a) (Fig.2 (a)) and Hi-LASSIE (Yao et al., 2023) (Fig.15 (b)) demonstrate superior performance. Nevertheless, due to a lack of structural information about the object, LASR also falls short of achieving ideal results. Hi-Lassie only can provide coarse shape and motion due to the lack of temporal information and refinement operations. However, S3O, without using any additional annotations, outperforms both methods, as illustrated in Fig.2 (c) and Fig.5.
+
+Quantitative Comparisons. Furthermore, for a comprehensive comparison, we examine the performance of S3O against the best existing methods, LASR and ViSER, in terms of 2D keypoint transfer accuracy, as shown in Tab.2. S3O consistently surpasses both LASR and ViSER across all animal subjects and requires approximately  $40\%$  less training time. Another point worth noting is the superior stability of S3O compared to the other two methods. We conduct multiple trials on DAVIS camel, cows, and Planet ZOO giraffe, tiger, noting standard deviations as a measure of the stability of the method. We observe that LASR and ViSER exhibit poor stability; for instance, in the camel experiments with LASR, results vary per trial, sometimes erroneously producing three or five legs. In contrast, S3O, by learning skeletal structural information, effectively avoids such inconsistencies, yielding reliable results.
+
+Another important point to note is that when tested on the PlanetZoo dataset, LASR/BANMo tends to learn shapes that are reasonable only in a side view but extremely incorrect in other views. For example, as seen in Fig.2 and 9, the shapes of the giraffe, tiger, and elephant look acceptable from the side view, but from the top view, their shapes are completely incorrect. This issue stems from errors in learning the camera pose, leading to a situation where, despite each frame's predicted parameters aligning the rendered results with the ground truth mask, the incorrect camera pose results in all other parameters being wrong.
+
+Also, we compared S3O with BANMo on the AMA-Human dataset. This dataset contains multi-view RGB videos by 8 synchronized cameras and provides ground truth camera parameters and 3D meshes. However, we use only the videos; we do not use the camera parameters and 3D meshes, since we estimate them as a part of our S30. However, as shown in Tab.3, S3O still shows superior performance on all three metrics (Chamfer Distance, F-score, and mIoU) with much less training time.
+
+# 4.3. Skeleton Reconstruction Comparison
+
+We outline the essential features of an effective skeleton in 3D modeling. (1) Bone distribution should be logical and tailored to the object's movement complexity, with denser bones in areas of high skeletal motion like limbs. (2) The skeleton should be sufficiently detailed to depict the structure of the object while avoiding unnecessary complexities due to local irregularities.
+
+S3O vs RigNet and Skeleton. We compare the skeletons generated by S3O and RigNet (Xu et al., 2020) and Skeleton (Au et al., 2008), as illustrated in Fig.4.  ${}^{2}$  S3O achieves better generalization than RigNet without 3D supervision. Although RigNet leverages extensive 3D models, ground truth skeletons, and skinning weights as supervision, it fails on out-of-distribution (OOD) objects, e.g., for the cases of tiger and elephant. Further, S3O effectively utilizes motion information from videos to produce physically plausible skeletons. It allocates more bones to areas with denser motion and fewer bones to less dynamic regions. This contrasts with RigNet and Skeleton, which lack such motion information. For instance, Skeleton assigns an excessive number of redundant bones to the torso, which exhibits much less movement than legs. This discrepancy underscores the advanced capability of S3O in generating more physically meaningful skeletal structures, highlighting its practicality for dynamic object analysis in video data.
+
+# 4.4. Diagnostics
+
+Fine Granularity of the Skeleton. Our approach allows for the generation of skeletons with varying levels of granularity by adjusting the threshold for merging bone motion. As demonstrated in Fig.3, by employing thresholds of 0.99, 0.95, and 0.90, we progressively reduce the granularity of the skeleton predictions, while still preserving the most critical bones and structural integrity. It enables the creation of a skeleton with differing levels of detail.
+
+Effectiveness of Dynamic Rigidity. As shown in Fig.6, incorporating Dynamic Rigidity into our reconstruction process yields a shape that adheres more closely to physical
+
+reality compared to methods utilizing As-Rigid-As-Possible (ARAP) modeling and without both regularization terms. The resulting structure maintains geometric integrity and behaves in a manner consistent with real-world physics.
+
+Bone Motion Estimation. We use optical flow warp to estimate bone motion, as illustrated in Fig.8. We experimented with using the estimated bone formation (SE(3)) for each bone as the basis for determining bone motion, similar to the method used in WIM (Noguchi et al., 2022). Intuitively, utilizing the SE(3) of each bone, which encompasses 3D information, should be more effective than relying on 2D optical flow, which lacks depth perception. However, our results were not as anticipated for two main reasons: (1) The SE(3) of bones are predicted, meaning the motion calculated for each part is only accurate when these predictions are highly stable and precise. This explains why the bone merge stage in WIM is more akin to a post-processing step, where accurate learning of the mesh and pose of each part is prioritized before the merge. In contrast, our method synchronously updates the skeleton and other learnable parameters, merging or generating bones at specific epochs. The model then needs to relearn the SE(3) and other parameters for these new bones. In such cases, the fluctuating SE(3) values may not be as reliable as the consistent 2D optical flow, which can be derived using existing models. (2) From a degenerate perspective, consider a scenario where the rotation of the bones occurs in a plane that projects as a line in the image plane. In this case, the motion directions of two adjoining bones (across a potential joint) align and become indistinguishable. However, this degenerate scenario also impedes the estimation of SE(3) from images, rendering both SE(3) and optical flow ineffective. Therefore, for arbitrary views, the direction indicated by optical flow is sufficient, and the use of SE(3) becomes unnecessary.
+
+Physical Skeletons vs Virtual Bones. In this discussion, we explore why using a physical skeleton outperforms virtual bones for mesh pose and motion manipulation. As shown in Fig.2 (d), methods like LASR, ViSER, and BANMo use k-means clustering on meshes to determine virtual bone positions. This often results in larger areas like the torso getting more bones, while slender limbs have fewer. This allocation contradicts the body's motion dynamics, where more bones should be in frequently deforming areas like legs and fewer in more stable areas like the torso. Such skewed distribution leads to a rigid treatment of entire limbs, as seen in LASR, preventing natural bending.
+
+# 5. Conclusions and Limitations
+
+We have presented S3O, an approach to concurrently learn the shape, time-varying parameters, and skeleton of a dynamically articulated object from a single monocular video, without any 3D supervision. We propose the concept of
+
+Dynamic Rigidity as a solution to the limitations inherent in existing As-Rigid-As-Possible (ARAP) formulations. Our experiments show that S3O exhibits enhanced generalization in the generation of skeletons, compared to RigNet which also uses 3D supervision. Further, the ability of S3O to leverage the 3D structure it captures helps it surpass current state-of-the-art methods in shape reconstruction performance, while significantly reducing training time. A limitation of most methods, including ours, is the excessive computation time required, rendering them impractical for everyday applications. Recognizing this, our future work will focus on utilizing large-scale pre-trained text/image-to-3D models as 3D priors to expedite the training process.
+
+# Impact Statement
+
+The proposed Synergistic Shape and Skeleton Optimization (S3O) method significantly advances the field of 3D reconstruction from single monocular videos by addressing the limitations of current techniques that require extensive computational resources and additional human annotations. By adopting a dual-phase approach, S3O learns visible shapes and underlying skeletons without predefined models, camera poses, or key points. This innovation reduces training time by  $60\%$  compared to state-of-the-art methods and improves reconstruction accuracy and robustness from limited viewpoints. The introduction of the PlanetZoo dataset further demonstrates S3O's capability to handle dynamic camera movements and diverse objects, setting a new benchmark for 3D reconstruction from monocular videos.
+
+# Acknowledgement
+
+The support of the Office of Naval Research under grant N00014-20-1-2444 and of the USDA National Institute of Food and Agriculture under grant 2020-67021-32799/1024178 are gratefully acknowledged
+
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+
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+
+![](images/8102cafe7622627ea69e02780e73cab2b501f74651b2e5dfb93a849a74fec385.jpg)  
+Figure 7. Coarse-to-Fine Deformation. The second phase of S3O focuses on intricacies, notably improving head and tail details that are less defined in the initial coarse learning stage. Points marked in red denote uniformly sized coarse shapes. ER represents Even Resampling, a process that makes vertices evenly distributed while maintaining the integrity of the original shape. Note the gradual improvement from coarse to fine stages as explained in Sec.E. Details of ER operation are presented in Sec.B.
+
+In this section, we aim to provide additional information not included in the main text due to length constraints. This supplemental content includes (1) A comparison between S3O and existing methods (Sec.A) (2) Implementation details (Sec.B), (3) Information about the dataset used for evaluation (Sec.C), (4) an In-depth explanation of loss and regularization terms (Sec.D), (5) Further experimental results (human dataset in Tab.3, reconstruction from top view in Fig.9, reconstruction results of MagicPony and Hi-LASSIE in Fig.10 and 15, procedure of coarse shape phase in Fig.7) along with discussions (Sec.E), and (6) Table of Notations (Tab.4).
+
+# A. Difference of S3O with existing methods
+
+In Tab.1 from the main paper, we compare the proposed S3O method with existing methods under various settings. S3O addresses more complex yet realistic scenarios, which include: 1) In real-world applications, ground truth (GT) 3D shapes, skeleton templates, and camera poses are often not readily available. 2) It's not always feasible to have multiple videos of moving objects showcasing different viewpoints.
+
+Methods like CASA (Wu et al., 2022), SMAL (Zuffi et al., 2017), and others that rely on instance-specific 3D shapes, skeletons, or both as templates for motion modeling face limitations in generalizing to Out-Of-Distribution objects lacking GT 3D data. Similarly, approaches such as WIM (Noguchi et al., 2022), CAMM (Kuai et al., 2023), RAC (Yang et al., 2023), and BANMo (Yang et al., 2022) require accurate camera poses and extensive video data (exceeding 1000 frames) from varied viewpoints to produce satisfactory results. Monocular video from limited viewing directions lead to unsatisfying results, shown in Fig.2, 10, and 9.
+
+Alternatively, while methods like LASR (Yang et al., 2021a) and VISER (Yang et al., 2021b) do not require shape and skeleton templates, the apparent surface motion does not match the apparent ground truth motion, due to errors in joint estimates of surface shape and motion, which is because they do not use any structural information (skeleton) about the object. In contrast, S3O also learns a physical skeleton to model the articulated motions of objects more effectively than the virtual bones used in LASR (Yang et al., 2021a), BANMo (Yang et al., 2022), and ViSER (Yang et al., 2021b). This allows our S3O to not require inputs of shape and skeleton templates, and camera pose, yielding a less demanding solution.
+
+<table><tr><td>Algorithm 1 Dual-Phase Synergistic Shape and Skeleton Optimization (S3O)</td></tr><tr><td>Given a set of frames F from a video.</td></tr><tr><td>Select canonical frame fc. (Sec.3.2)</td></tr><tr><td>• Coarse Shape Phase</td></tr><tr><td>Extract 2D skeleton from fc (Sec.3.2)</td></tr><tr><td>Learn coarse parametric model including canonical 3D shape Mc(0) and skeleton ST(0) from F (Sec.3.2)</td></tr><tr><td>• Motion &amp; Fine Shape Phase</td></tr><tr><td>Initialize the camera &amp; motion parameters θ(0)</td></tr><tr><td>for e = 1 to E do</td></tr><tr><td>if e &lt; e1 then</td></tr><tr><td>Articulation Mt(e) ← Mc(e) (Sec.3.3)</td></tr><tr><td>Compute the reconstruction loss.</td></tr><tr><td>Update canonical shape and time-varying parameter: θ(e+1), Mc(e+1) ← θ(e), Mc(e)</td></tr><tr><td>Adapt skeleton Sc(e) to Mc(e+1) (Sec.3.4)</td></tr><tr><td>else if e = e1 then</td></tr><tr><td>Upsample Skeleton</td></tr><tr><td>else</td></tr><tr><td>Calculate physical constraints (Sec.3.4)</td></tr><tr><td>Update Skeleton: Sc(e+1) ← Sc(e)</td></tr><tr><td>end if</td></tr><tr><td>end for</td></tr></table>
+
+![](images/21d20b37c489f0361edf88f58bc06bb64446bb00d4d7feb1510cace71f40112b.jpg)  
+Figure 8. Optical Flow Warp. We estimate the bone motion from the 2D optical flow.
+
+# B. Implementation Details
+
+S3O follows Neural Mesh Rasterization (NMR) approaches that jointly learn the (1) parametric model, including the mesh and skeleton, which is used in calculating the skinning weights and changing the pose of the object via blending skinning operation, and (2) time-varying parameters, which include the bone transformation, and camera parameter.
+
+Coarse Shape Phase. While different from existing shape-template-free methods such as LASR (Yang et al., 2021a), MagicPony (Wu et al., 2023a), and BANMo (Yang et al., 2022), which initializes the mesh with a sphere or an ellipsoid, we use the extraction of 2D skeletons from the canonical frame and DINO feature information to obtain an instance-specified initial mesh, similar to the work Hi-LASSIE (Yao et al., 2023), as shown in the Fig.14. Then, through the coarse shape phase (Sec.3.2 in the main paper), we can quickly learn a coarse shape. Although the quality of the mesh is still far from satisfactory, this coarse mesh can greatly assist us in learning accurate camera parameters more quickly (results are shown in Fig.14).
+
+Analysis-by-Synthesis Strategy. Our approach adopts an analysis-by-synthesis strategy. In this strategy, we first obtain the object silhouette, optical flow, and pixel values by rendering (Liu et al., 2019) the deformed mesh to image space, defined as the synthesis process. Next, we analyze the predicted object silhouette, optical flow, and pixel values with respect to the video observations.
+
+![](images/6736d3086c7c197632e73f011aa3431964770c305eb82b0d51d573027a26916f.jpg)  
+Figure 9. Reconstructed Mesh Results (top view). For (a) LASR, (b) BANMo, and (c) S30 (ours), for six animal videos: DAVIS's camel, cows and PlanetZoo's zebra, elephant, tiger, and giraffe.
+
+![](images/f42ebc6a08ce174cf536e52f5c6d44c9480020824dc9dc02c171724e1b5f721b.jpg)  
+Figure 10. Reconstructed Mesh Results (side and top views). For MagicPony (Wu et al., 2023a), for three animal videos: DAVIS's cows and PlanetZoo's elephant, tiger.
+
+Even Resampling. During the training process, as shown in Fig.7, only the outermost points on the camel's legs initially move downward, thereby elongating the entire leg to achieve a shape more akin to a real camel in the video. However, in this scenario, the distribution of vertices is highly uneven, with the lower half of the leg having very few points. Therefore, every 1,500 iterations, we perform an Evenly Resampling (2-Manifold Processing) (Huang et al., 2018), which makes the vertices roughly uniformly distributed on the geometry surface.
+
+Mask and Optical Flow Generation. We generate object-level segmentation masks for input videos using CSL (Zhang et al., 2024), Grounding DINO (Liu et al., 2023) and Segment Anything Model (SAM) (Kirillov et al., 2023). By inputting text about the target object in the video, we leverage Grounding DINO to track that object across frames and then we input the detected bounding boxes to SAM to obtain the segmentation masks. Results are shown in Fig.12. For part-level segmentation, which is used during the coarse phase, we leverage DINO (Caron et al., 2021) to extract per-pixel embeddings and then cluster the embeddings to generate part-level masks similar to (Yao et al., 2023). The results are shown in Fig.13. Optical flow was estimated using existing flow estimators (Teed & Deng, 2020).
+
+Experimental Setup. For our experiments on DAVIS and PlanetZoo, we utilized a single A100 40GB GPU. The batch size was set to 8, and the training consisted of 10 epochs compared to LASR and ViSER having 25 epochs. Each epoch comprised 1500 iterations. Similar to LASR, for the first  $40\%$  epochs, we estimate 16 cameras and keep the best one in the following epochs. For our experiment on AMA-Human, we follow the same training setting as the original BANMo, except we only use one single GPU and S3O requires  $40\%$  training epochs to achieve better reconstruction results compared to BANMo.
+
+# C. Datasets and Metrics
+
+# C.1. Datasets
+
+DAVIS & BADJA. BADJA (Biggs et al., 2019) dataset is a derivative of DAVIS (Perazzi et al., 2016) which contains the RGB videos and the corresponding ground truth segmentation results. It contains typical quadrupeds like bears, camels, dogs, and horses. Besides, BADJA also provides 2D key points for all videos. In experiments, we show both numerical and visualization results on DAVIS & BADJA.
+
+PlanetZoo. We obtain 2D key points for the ground truth using Labelme tools (Wada). Experiments on the PlanetZoo dataset containing quantitative and qualitative results are also shown in the paper. For annotation rules, we also adopt the same real physical morphology across frames in each video as DAVIS & BADJA. In detail, these points are located on the
+
+![](images/7035f70dbb70f0ef623ae392baa728f894c4742cefdd455c3679e5ecf8532456.jpg)
+
+![](images/517240fa1206876072eaac6d2cc6c430a0568b9a8f9f07c47317207461475654.jpg)
+
+![](images/52d85b9d9106f352e9cce84276f61d259542f3d3d4a964ad74eb486514da111f.jpg)  
+Figure 11. Keypoints Annotations for PlanetZoo dataset.
+
+![](images/4441cbae2a1c4d8aac6fd564cefecea7fd4e218737b83db7d50d888dd65f9968.jpg)
+
+![](images/6afc62970ceae6ef6ecc2cf7530a18ac0784d0485a3862bdcd3d7b47686ed255.jpg)
+
+![](images/2038fe54d953578c25e3b6e08746ba12798f9c7782608fa67524fa147c8fb45d.jpg)
+
+![](images/c958921bffc4c917ef27e5d746a380a24bcb409b4c01f2e342f92dc5e10d4766.jpg)  
+Figure 12. Example frames and masks of tiger from PlanetZoo dataset.
+
+![](images/a38d4927f194026723a4192af5374a861c71f06b78dc635f29cbd1c247d9e23c.jpg)
+
+![](images/bce4cd7fd420a80f23cdd78d47946569d89973cf344ff11fe17498fa3d2dbbaa.jpg)
+
+![](images/a752750f225aac6a7997616e153b2047d94087dd32f17274a337e560023be572.jpg)
+
+![](images/77b2aeea587fbfb5c1a6fab940bcd2e37b97e3eb976d431142bd32bac144c4b7.jpg)
+
+![](images/b76336478edce53670b303628b0c8b56edc144741b6ccea436d8c499cd9b9daf.jpg)
+
+![](images/9679892d974414ab16e664b1ef85d03d4fdb398ac74d94efe0b78f8fffa11c2e.jpg)
+
+![](images/ea8ede508b747dcb377bcf8d23cbd8f36d5a84444a2e46abc45012c94a957b0a.jpg)
+
+![](images/51778657df5e469a769126e8c40886d5fc2343ee7680c36c6f7b688eb09a533b.jpg)
+
+![](images/a6624c8fa60cf7cff111d7785f7a743028c4ad47d24930a685f4f434e1066588.jpg)
+
+paws, knees, ears, tails, jaws, and noses as shown in Fig.11. We plan to release the whole PlanetZoo dataset including the RGB videos, 2D ground truth segmentation, and 2D key points annotations in the future.
+
+AMA human. (Vlasic et al., 2008a) records multi-view videos by 8 synchronized cameras for which it provides ground truth camera pose, mesh, and masks. To compute the Chamfer distance we use to evaluate our 3D reconstruction, We need to use only the 3D mesh.
+
+# C.2. Metrics
+
+To evaluate the performance of S3O and the existing methods, we use 2D Keypoint Transfer Accuracy same as LASR (Yang et al., 2021a) and ViSER (Yang et al., 2021b). 2D Keypoint Transfer Accuracy evaluates results by the percentage of correct key point transfer (PCK-T) (Kanazawa et al., 2018; Kulkarni et al., 2020). If the distance between the transferred point and the corresponding ground truth is one is within a predefined threshold, this transferred point will be categorized as 'Correct'. This threshold  $d_{th}$  is defined by  $0.2\sqrt{|S|}$ , and  $|S|$  means the area of ground truth mask (Biggs et al., 2019). We use 3D Chamfer distance, mIoU, and F-score on the AMA-Human dataset.
+
+# D. Losses and Regularization
+
+The model is learned by minimizing the reconstruction loss, which includes seven terms. The first five terms are similar to those in existing differentiable rendering pipelines (Yang et al., 2021a; Yao et al., 2023; Yang et al., 2022; Goel et al., 2020). We define  $\mathbf{S}^{\mathrm{t}},\mathbf{I}^{\mathrm{t}}$ $\mathbf{F}^{2\mathrm{D},t}$  as the silhouette, input image, and optical flow of the input image, and their corresponding rendered counterparts as  $\tilde{\mathbf{S}}^{\mathrm{t}},\tilde{\mathbf{I}}^{\mathrm{t}}$ $\tilde{\mathbf{F}}^{2\mathrm{D},t}$ . The following losses ensure the fitting between rendered and original:
+
+$$
+\mathcal {L} _ {\text {s i l h o u e t t e}} = \sum_ {\mathbf {x} ^ {t}} \left\| \mathbf {s} \left(\mathbf {x} ^ {t}\right) - \hat {\mathbf {s}} \left(\mathbf {x} ^ {t}\right) \right\| _ {2}, \tag {10}
+$$
+
+![](images/8971d029ce6a1ad9732f8f3903750fe3a1493b0b00001d15d4abfcc1bfaaebd0.jpg)  
+Figure 13. Part results obtained from DINO features.
+
+![](images/84be6d3ad0507dc7a4234ecd89163eb4c262fec6a37d37845f6aba0549c3a6a2.jpg)  
+2D skeleton
+
+![](images/574da481ae487d35578fd51dcc5a2127434389ea9dea533f786851e6c0e031ff.jpg)  
+Initial mesh
+
+![](images/809701ceffe8c4616d039922638bc16b5250bbd5560099ebbbe5217b044421c2.jpg)  
+Coarse mesh
+
+![](images/fa3e22d4490e8a232ed325dfbd01f122def47c7ba461c3f9413c590a2cc6d69d.jpg)  
+Skinning weights  
+Figure 14. Procedure of coarse shape learning phase.
+
+![](images/279e6d0d5369142901dd72c65052e45353f1c162d766d96c101dbc33c3f1c5f2.jpg)  
+Coarse 3D skeleton
+
+$$
+\mathcal {L} _ {\text {o p t i c a l f l o w}} = \sigma \left\| \left(\tilde {\mathbf {F}} ^ {2 \mathrm {D}, \mathbf {t}}\right) - \left(\mathbf {F} ^ {2 \mathrm {D}, \mathbf {t}}\right) \right\| _ {2} ^ {2}, \tag {11}
+$$
+
+$$
+\mathcal {L} _ {\text {t e x t u r e}} = \left\| \tilde {\mathbf {I}} ^ {\mathbf {t}} - \mathbf {I} ^ {\mathbf {t}} \right\| _ {1}, \tag {12}
+$$
+
+$$
+\mathcal {L} _ {\text {p e r c e p t u a l}} = \operatorname {p d i s t} \left(\tilde {\mathbf {I}} ^ {t}, \mathbf {I} ^ {t}\right), \tag {13}
+$$
+
+$$
+\mathcal {L} _ {\text {d i n o f e a t}} = \mathcal {L} _ {\text {p a r t s i l h o u e t t e}} + \mathcal {L} _ {\text {c o n s i s t e n c y}}, \tag {14}
+$$
+
+where  $\text{pdist}(\cdot)$  is the perceptual distance (Zhang et al., 2018), and  $\mathcal{L}_{\text{dino feat}}$  leverages the dino features, which combines the part silhouettes and consistency loss that maintains consistency between 2D pixels on a part and corresponding 3D points on the part surface.
+
+$$
+\mathcal {L} _ {\text {p a r t s i l h o u e t t e}} = \sum_ {p} \left\| s _ {p} ^ {t} - \hat {s} _ {p} ^ {t} \right\| _ {2}, \tag {15}
+$$
+
+where  $s_p^t$  represents part silhouettes for part  $p$  and frame  $t$ . The consistency loss, denoted as  $\mathcal{L}_{\mathrm{consistency}}$ , is designed to optimize the 3D surface coordinates. This optimization ensures that the aggregated 3D point features project more accurately onto corresponding pixel features within the image ensemble, following the methodology outlined in (Yao et al., 2023). In our experiments, we observed that the dino feature loss aids the model in more rapidly learning a coarse shape through part-wise optimization. However, due to the limited accuracy of part-level annotations, as shown in Fig.13, it is not guaranteed to maintain excellent consistency across all frames. Consequently, the dino feature loss restricts the model's capacity for more detailed deformation. Therefore, in the motion and fine shape phase, we removed the dino feature loss.
+
+Also, we leverage three regularization terms: (1) Dynamic Rigidity term, which is introduced in the main paper Sec.3.3. (2) Symmetry term, which encourages the canonical mesh to be symmetric.
+
+$$
+\mathcal {L} _ {\text {s y m m}} = \sum_ {i} \min  _ {j} \| v _ {j} - \phi \left(v _ {i} ^ {\text {s y m m}}\right) \| ^ {2}, \tag {16}
+$$
+
+![](images/749711a30df7250d0c687f0cff1dc78ad7268d3ab6afa566e2fdeffb8a28d650.jpg)
+
+![](images/e7567d9f082b0cdd41917cc1c320379a0b0618e923420fe887abda2eb3a5c3c0.jpg)
+
+![](images/048aadf6cace8c103178c8b994a043fb4a6d59ce47d298b1ab72cee2e705a3ee.jpg)  
+(a) RGB Image
+
+![](images/e98716455c70e367f533b62597dd39088991f0322c840a836413e41fd5895c5b.jpg)
+
+![](images/2a374d4359051b77273482b32d01fad3642fc19d36d126654fc5fbae2d257644.jpg)
+
+![](images/b5375b2496c1b777822b8141c87d6f74e464241be1b0c89747825c6449f50f5e.jpg)  
+(b) Hi-LASSIE
+
+![](images/c54860d54dc7adf9eed7cb3ac811d8d7fe332068b7b31012eb034293093ae793.jpg)
+
+![](images/568ce5376238ae6390c975f5da3b04445bfe15bca6e79d3706f60f6d63df178a.jpg)
+
+![](images/1da671b9dd1dc702c1d8f7049cc924be7891d4647f25292a360694e3d042b791.jpg)  
+(c) S3O  
+Figure 15. Mesh Results of S3O compared with with Hi-LASSIE.
+
+Table 3. Quantitative results on AMA's swing and samba. 3D Chamfer Distance, F-score, and mIoU are shown averaged over all frames. Note we use one A100 GPU with half batch size compared with the original BANMo (Yang et al., 2022), which leads to lower accuracy. For the AMA-Human dataset, we use more than one video for training, which is different from other experiments. (Time: GPU hours on A100)  
+
+<table><tr><td rowspan="2">method</td><td colspan="4">AMA-samba</td><td colspan="4">AMA-swing</td><td colspan="4">Ave.</td></tr><tr><td>CD</td><td>F@2%</td><td>mIoU</td><td>Time</td><td>CD</td><td>F@2%</td><td>mIoU</td><td>Time</td><td>CD</td><td>F@2%</td><td>mIoU</td><td>Time</td></tr><tr><td>BANMo</td><td>15.3</td><td>53.1</td><td>61.2</td><td>12.5</td><td>13.8</td><td>54.8</td><td>62.4</td><td>12.7</td><td>14.6</td><td>53.9</td><td>61.8</td><td>12.6</td></tr><tr><td>S3O</td><td>12.9</td><td>55.2</td><td>62.5</td><td>5.1</td><td>12.3</td><td>56.4</td><td>63.6</td><td>4.8</td><td>12.9</td><td>55.8</td><td>62.5</td><td>4.9</td></tr></table>
+
+where,  $v_{j}$  is the vertex  $j$  in the one side of the symmetry plane, and  $v_{i}^{symm}$  is the vertex from the other side.  $\phi$  is the reflection operation w.r.t to the symmetry plane. (3) Laplacian smoothing: We apply laplacian smoothing to generate smooth mesh surfaces.
+
+$$
+\mathcal {L} _ {\text {s h a p e}} = \left\| \mathbf {X} _ {i} ^ {0} - \frac {1}{\left| N _ {i} \right|} \sum_ {j \in N _ {i}} \mathbf {X} _ {j} ^ {0} \right\| ^ {2}, \tag {17}
+$$
+
+where,  $\mathbf{X}_i^0$  is coordinates of vertex  $i$  in canonical space. And  $N$  is the number of vertices
+
+# E. Discussion
+
+Coarse-to-Fine Deformation. As illustrated in Fig.7, S3O starts with learning a coarse shape in the first phase, which takes approximately 15 minutes. This is followed by a warm-up stage at the onset of the second phase, where the mesh is kept constant while only the time-varying motion parameters are learned (30 minutes). Subsequently, training is conducted for all parameters. The coarse mesh initially struggles to accurately reconstruct finer extremities, such as the legs and head. These structures gradually emerge during the second phase. Points marked in red denote coarse shapes. ER denotes Evenly Resampling, a process that makes vertices evenly distributed while maintaining the integrity of the original shape.
+
+Comparison between NeRF-based and NMR-based Methods. Methods based on Neural Radiance Fields (NeRF), such as BANMo and MagicPony, have demonstrated remarkable capabilities in reconstruction tasks. These methods involve learning a neural radiance field and subsequently extracting a mesh from this learned field. However, the process of learning a neural radiance field is data-intensive, often requiring multiple videos and viewpoints for optimal results. This becomes a limitation, as obtaining such extensive data can be challenging. The performance of NeRF-based methods significantly diminishes with restricted data and viewpoints, as evidenced in Fig.2, 10, and 9. In contrast, methods utilizing the Neural Mesh Renderer (NMR) focus on directly training models to produce a mesh. These approaches have been shown to yield satisfactory outcomes even with limited data availability, such as using a single monocular video with less than a hundred input frames. Our approach is based on NMR and demonstrates considerable improvements, both qualitatively and
+
+quantitatively, compared with existing NMR-based methods.
+
+Table 4. Table of Notations  
+
+<table><tr><td>Symbol</td><td>Description</td><td>Dimension</td></tr><tr><td>B</td><td>Number of Bones</td><td>-</td></tr><tr><td>E</td><td>Number of Edges in Surface Mesh</td><td>-</td></tr><tr><td>J</td><td>Number of Joints</td><td>-</td></tr><tr><td colspan="3">Representation Parameters</td></tr><tr><td>M</td><td>Canonical Surface Mesh and Color</td><td>-</td></tr><tr><td>Pc</td><td>Camera Parameters</td><td>-</td></tr><tr><td>W</td><td>Skinning Weights</td><td>W∈RN×B</td></tr><tr><td>R</td><td>Rigidity Coefficient</td><td>R∈RE</td></tr><tr><td>Tt</td><td>Time Varying Transformation</td><td>T∈SE(3)</td></tr><tr><td colspan="3">Skeleton Components</td></tr><tr><td>ST</td><td>Skeleton</td><td>-</td></tr><tr><td>B</td><td>Bines</td><td>B∈RB×13</td></tr><tr><td>J</td><td>Joints</td><td>J∈RJ×5</td></tr><tr><td colspan="3">Bone Components</td></tr><tr><td>C/B[;, : 3]</td><td>Gaussian Centers Coordinates</td><td>C∈RB×3</td></tr><tr><td>Q/B[;, 3:12]</td><td>Precision Matrix</td><td>Q∈RB×9</td></tr><tr><td>L/B[;, 12]</td><td>Bone Length</td><td>L∈RB</td></tr><tr><td colspan="3">Joint Components</td></tr><tr><td>J[;, : 2]</td><td>Index of Two connected Bones</td><td>RJ×2</td></tr><tr><td>J[;, 2:]</td><td>Joint Coordinates</td><td>RJ×3</td></tr><tr><td colspan="3">Optical Flow Notations (at time t)</td></tr><tr><td>F2D,t</td><td>2D Optical Flow</td><td>F2D,t∈RH×W</td></tr><tr><td>FB,t</td><td>2D Bone Motion Direction</td><td>FB,t∈RB×2</td></tr><tr><td>FS,t</td><td>Surface Flow Direction</td><td>FS,t∈RS×2</td></tr><tr><td>Vt</td><td>Visibility Matrix</td><td>Vt∈RN×1</td></tr><tr><td colspan="3">2D Notations</td></tr><tr><td>S/S</td><td>Input silhouette/ Observed silhouette</td><td>S/S∈RH×W</td></tr><tr><td>I/I</td><td>Input Image / Observed Image</td><td>I/I∈RH×W</td></tr><tr><td>F2D/F2D</td><td>2D Input OF / 2D observed OF</td><td>F2D/F2D∈RH×W</td></tr></table>

2024/S3O_ A Dual-Phase Approach for Reconstructing Dynamic Shape and Skeleton of Articulated Objects from Single Monocular Video/images.zip

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2024/SAM as the Guide_ Mastering Pseudo-Label Refinement in Semi-Supervised Referring Expression Segmentation/2c51e646-12f9-45e4-95fc-f01331d93cc3_origin.pdf

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2024/SAM as the Guide_ Mastering Pseudo-Label Refinement in Semi-Supervised Referring Expression Segmentation/full.md

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+# SAM as the Guide: Mastering Pseudo-Label Refinement in Semi-Supervised Referring Expression Segmentation
+
+Danni Yang  $^{*1}$  Jiayi Ji  $^{*1}$  Yiwei Ma  $^{1}$  Tianyu Guo  $^{1}$  Haowei Wang  $^{1,2}$  Xiaoshuai Sun  $^{1}$  Rongrong Ji
+
+# Abstract
+
+In this paper, we introduce SemiRES, a semisupervised framework that effectively leverages a combination of labeled and unlabeled data to perform RES. A significant hurdle in applying semisupervised techniques to RES is the prevalence of noisy pseudo-labels, particularly at the boundaries of objects. SemiRES incorporates the Segment Anything Model (SAM), renowned for its precise boundary demarcation, to improve the accuracy of these pseudo-labels. Within SemiRES, we offer two alternative matching strategies: IoU-based Optimal Matching (IOM) and Composite Parts Integration (CPI). These strategies are designed to extract the most accurate masks from SAM's output, thus guiding the training of the student model with enhanced precision. In instances where a precise mask cannot be matched from the available candidates, we develop the Pixel-Wise Adjustment (PWA) strategy, guiding the student model's training directly by the pseudo-labels. Extensive experiments on three RES benchmarks—RefCOCO, RefCOCO+, and G-Ref reveal its superior performance compared to fully supervised methods. Remarkably, with only  $1\%$  labeled data, our SemiRES outperforms the supervised baseline by a large margin, e.g.  $+18.64\%$  gains on RefCOCO val set. The project code is available at https://github.com/nini0919/SemiRES.
+
+*Equal contribution  $^{1}$ Key Laboratory of Multimedia Trusted Perception and Efficient Computing, Ministry of Education of China, School of Informatics, Xiamen University, 361005, P.R. China.  $^{2}$ Youtu Lab, Tencent, Shanghai, China. Correspondence to: Xiaoshuai Sun <xssun@xmu.edu.cn>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+![](images/3f85fec6f1eacb42a1f650b1ba5c5bae16d4ba8ef06017ec3f5d2beecf60c111.jpg)  
+Figure 1. (a) A large number of noisy and incomplete cases exist in pseudo-labels. The proposed SemiRES can address this issue. (b) Analysis shows labeling a small portion of RefCOCO data can greatly reduce costs. (c) Our method substantially improves performance, even with a small number of annotated samples.
+
+# 1. Introduction
+
+Referring Expression Segmentation (RES) has attracted considerable attention from the fields of vision and language research (Hu et al., 2016; Chen et al., 2019b; Huang et al., 2020; Liu et al., 2019; 2023b). Unlike common visual grounding tasks such as phrase localization (Bajaj et al., 2019; Chen et al., 2017; Dogan et al., 2019; Plummer et al., 2015; 2017) and referring expression comprehension (Yang et al., 2020; Yu et al., 2018; Deng et al., 2021; Kamath et al., 2021; Huang et al., 2021), RES requires precise pixel-level segmentation of an object within an image, as directed by a referring expression, which goes beyond simple bounding box identification.
+
+Despite advancements, the high demand for labeled data presents a significant barrier to the deployment of RES, particularly in domains where labeling is prohibitively expensive, such as medical imaging and autonomous driving. The labor intensity of the task is underscored by findings
+
+from (Kim et al., 2023a), indicating that an average mask segmentation requires approximately 79.1 seconds. This is further exemplified by the extensive annotation efforts needed for benchmark datasets like RefCOCO (Yu et al., 2016), RefCOCO+ (Yu et al., 2016), and G-Ref (Mao et al., 2016; Nagaraja et al., 2016), which consist of tens of thousands of labeled instances and necessitate substantial time investments, as illustrated in Fig. 1 (b). The cost and time implications, alongside the potential for inaccuracies in manual labeling, present considerable challenges for the scalability and reliability of RES models, highlighting the urgent need for more efficient methodologies.
+
+To address the challenges outlined above, we pioneer a semi-supervised learning framework tailored for RES, utilizing a combination of a small subset of image-text pairs with segmentation annotations and a large corpus of unannotated pairs to train the model. This approach has been widely validated across the fields of computer vision (Sohn et al., 2020; Olsson et al., 2021; Chen et al., 2023), natural language processing (Miyato et al., 2016; Cheng & Cheng, 2019; Zou & Caragea, 2023), and vision-language research (Yang et al., 2023; Sun et al., 2023; Jin et al., 2023). Yet, its application in RES has not yet been explored. We establish a baseline for semi-supervised RES, encompassing a comprehensive pipeline that goes beyond data augmentation (Zhang et al., 2017; Hendrycks et al., 2019; Cubuk et al., 2019; Zhao et al., 2023a) and Exponential Moving Average (Kingma & Ba, 2014; He et al., 2020; Grill et al., 2020; Tarvainen & Valpola, 2017) training mechanisms. However, this baseline encounters a substantial challenge: pseudo-labels are significantly noisy, particularly at the edges of instances, which can trap the model in suboptimal performance, as depicted in Fig. 1 (a). The crux of semi-supervised learning lies in refining these pseudo-labels to enhance their quality. Previous methods have addressed this by employing confidence-based pseudo-label filtering strategies (Sohn et al., 2020) or auxiliary correcting networks (Kwon & Kwak, 2022; Kim et al., 2023a). While intuitively appealing, relying solely on confidence for filtering may lead to the under-utilization of unlabeled data and lack flexibility in handling diverse noise in pseudo-labels.
+
+To tackle the aforementioned issues, we introduce a novel semi-supervised RES framework named SemiRES. The motivation behind SemiRES is to leverage the robust segmentation capabilities of SAM (Kirillov et al., 2023) to rectify pseudo-labels, especially around the edges of instances. Specifically, we employ SAM to extract multi-scale masks from original images to build a proposal library. The central concept of SemiRES is to retrieve one or multiple proposals from this library to reconstruct pseudo-labels. To achieve this, we propose two alternative strategies: IoU-based Optimal Matching (IOM) and Composite Parts Integration (CPI). The first assumes that the proposal library contains masks
+
+closely approximating the target instance, thus utilizing IoU to directly identify and replace the pseudo-label with the most corresponding mask from the library. The second strategy moves away from this assumption, instead using the pseudo-label to select different part-specific proposals from the library to assemble a complete mask. In cases where a suitable replacement cannot be retrieved from the proposal library, we default to optimizing the student model using the pseudo-label itself. To enhance training in such scenarios, we have devised a Pixel-Wise Adjustment (PWA) strategy that adjusts the final loss on a per-pixel basis according to the confidence levels on the pseudo-label.
+
+To further qualitatively validate the effectiveness of our proposed SemiRES, we conduct extensive experiments on three RES benchmark datasets—RefCOCO, RefCOCO+, and G-Ref. Our experiments show that SemiRES notably surpasses both supervised and semi-supervised baselines in all settings, for example, gaining  $+18.64\%$  and  $+8.28\%$  on  $1\%$  labeled RefCOCO as shown in Fig. 1 (c), which highlights its significant real-world application potential.
+
+To sum up, the contributions of this paper are three-fold:
+
+- We first present SemiRES, a semi-supervised framework tailored for RES that efficiently trains models using a minimal amount of labeled data, thus reducing dependence on expensive pixel-level annotations.  
+- We introduce two alternative strategies, IoU-based Optimal Matching (IOM) and Composite Parts Integration (CPI), that leverage the SAM's edge-segmentation proficiency to produce superior-quality pseudo-labels.  
+- Our SemiRES framework achieves notable performance improvements on three benchmark datasets RefCOCO, RefCOCO+, and G-Ref, demonstrating significant gains in model accuracy while concurrently cutting down on labeling costs.
+
+# 2. Related Work
+
+# 2.1. Referring Expression Segmentation
+
+Referring Expression Segmentation (Chen et al., 2019b; Huang et al., 2020; Liu et al., 2023b;a; Hu et al., 2023; Kim et al., 2023b; Yang et al., 2022b) is a multimodal task involving both image segmentation and natural language understanding. It aims to identify specific target regions within an image according to natural language expressions. In recent years, RES methods have made significant progress, with Transformer-based backbones emerging as the predominant choice for this task. Additionally, RES methods can be categorized into two main types: one-stage and two-stage approaches. One-stage methods (Chen et al., 2019a; Hu et al., 2020; Hui et al., 2020) usually use end-to-end networks for
+
+prediction, while two-stage methods (Yu et al., 2018; Liu et al., 2022) employ an instance segmentation network to generate a set of instance proposals before selecting the target instance. Recently, GRES (Liu et al., 2023a) introduces multi-target and no-target expressions, extending the classic RES to refer to an arbitrary number of target objects.
+
+# 2.2. Semi-Supervised Semantic Segmentation
+
+Semi-supervised learning (French et al., 2019; Xu et al., 2021; Wu et al., 2021; Olsson et al., 2021; Jiang et al., 2022; Yang et al., 2023), using a small amount of labeled data alongside a larger pool of unlabeled data for training, has widespread applications in computer vision and natural language processing. In recent years, semi-supervised semantic segmentation has developed rapidly, with many new research works emerging (Ouali et al., 2020; Chen et al., 2021; Wang et al., 2022; Yang et al., 2022a). Building upon FixMatch (Sohn et al., 2020), PseudoSeg (Zou et al., 2020) extends weak consistency to strong consistency in segmentation scenarios and incorporates a calibration module for pseudo-label refinement. ReCo (Liu et al., 2021a) samples classes prone to confusion across all categories to assist the segmentation network for better representations. AugSeg (Zhao et al., 2023b), a simple yet effective method, primarily enhances the performance of semi-supervised semantic segmentation through data augmentation.
+
+# 2.3. Segment Anything Model
+
+The recent Segment Anything Model (SAM) (Kirillov et al., 2023) has made significant advancements in pushing the boundaries of segmentation, greatly boosting the development of foundational models for computer vision. Trained on SA-1B of over 1 billion masks, it aims to segment any object in any given image without requiring any additional task-specific adaptation. SAM is proved to be capable of solving various tasks, such as medical image analysis (Ma & Wang, 2023; Shi et al., 2023), adversarial attacks (Guan et al., 2023; Zhang et al., 2023a), image inpainting (Yu et al., 2023), image editing (Xie et al., 2023), image captioning (Wang et al., 2023). Recently, several works (Zhang et al., 2023b; Liu et al., 2023c) have incorporated SAM for one-shot learning, contributing significantly to SAM's multifaceted development. In this paper, we investigate how to leverage SAM to enhance pseudo-labels and improve the performance of semi-supervised learning.
+
+# 3. Method
+
+# 3.1. Task Definition
+
+Before diving into SemiRES, we begin with illustrating the task definition of semi-supervised referring expression segmentation (RES). We usually have a small labeled dataset
+
+$\mathcal{D}_l = \left\{\left(\mathcal{T}_i^l,\mathcal{T}_i^l\right),Y_i^l\right\}_{i = 1}^{N^l}$  and a much larger unlabeled dataset  $\mathcal{D}_u = \left\{\left(\mathcal{T}_i^u,\mathcal{T}_i^u\right),\emptyset\right\}_{i = 1}^{N^u}$ , where  $\mathcal{T}_i^l,\mathcal{T}_i^u$  denote the  $i$ -th labeled and unlabeled image, respectively;  $\mathcal{T}_i^l$  and  $\mathcal{T}_i^u$  are the corresponding language expressions;  $N^l$  and  $N^u$  are the number of labeled and unlabeled data, with  $N^l\ll N^u$ . It is crucial to note that the unlabeled set  $\mathcal{D}_u$  lacks ground truth mask labels, utilizing language expressions solely as input. Our primary aim is to leverage this small labeled set alongside a large unlabeled set to achieve competitive performance in the RES task.
+
+# 3.2. Semi-Supervised Baseline
+
+We introduce a semi-supervised baseline for RES based on a teacher-student network structure. This approach unfolds in two stages:
+
+Stage1: Burn-In Stage. In the semi-supervised teacher-student framework, achieving proper parameter initialization is crucial for accelerating the convergence of training during the mutual learning stage (Liu et al., 2021b). During the Burn-In stage, we train the pre-trained model using only labeled data. The optimization objective is defined as follows:
+
+$$
+\mathcal {L} _ {s u p} = \frac {1}{H \times W} \sum_ {j = 1} ^ {H \times W} \mathcal {L} _ {B C E} \left(M _ {i, j} ^ {l}, Y _ {i, j} ^ {l}\right), \tag {1}
+$$
+
+where  $M_{i,j}^{l}$  denotes the prediction mask of Burn-In model for the  $j$ -th pixel of  $i$ -th labeled image,  $Y_{i,j}^{l}$  denotes the corresponding ground truth,  $\mathcal{L}_{BCE}$  denotes binary cross entropy loss (Csiszár, 2008).
+
+Stage2: Mutual-Learning Stage. After the Burn-In stage, we use the trained weights  $\theta$  to initialize both the teacher and student models. This process is defined as follows:
+
+$$
+\theta_ {t} \leftarrow \theta , \theta_ {s} \leftarrow \theta , \tag {2}
+$$
+
+where  $\theta_t, \theta_s, \theta$  denote the parameters of the teacher, student and Burn-In model, respectively.
+
+During the mutual learning stage, the teacher generates pseudo-labels for unlabeled data to supervise the training of the student, which is defined as follows:
+
+$$
+\mathcal {L} _ {u n s u p} = \frac {1}{H \times W} \sum_ {j = 1} ^ {H \times W} \mathcal {L} _ {B C E} \left(M _ {i, j} ^ {u}, \hat {M} _ {i, j} ^ {u}\right), \tag {3}
+$$
+
+where  $M_{i,j}^{u}$  and  $\hat{M}_{i,j}^{u}$  denote the predicted mask for  $j$ -th pixel of  $i$ -th unlabeled image by student and teacher, respectively.
+
+Simultaneously, the student continues to train on a small subset of labeled data, jointly optimizing with these two components of loss function, which is defined as follows:
+
+$$
+\mathcal {L} = \lambda_ {\sup } \mathcal {L} _ {\sup } + \lambda_ {\text {u n s u p}} \mathcal {L} _ {\text {u n s u p}}, \tag {4}
+$$
+
+![](images/f5e9369c8499d47a4eaaa7a13dddb501dcda4b0e0d858dab58d763993ea26b4d.jpg)  
+Figure 2. An overview of the proposed SemiRES, featuring a teacher-student network with data augmentation and mutual learning. It includes SAM-based pseudo-label refinement using IOM or CPI strategies, and PWA supervision when matches are not found.
+
+where  $\lambda_{sup}$  and  $\lambda_{unsup}$  is the hyperparameter of supervised loss  $\mathcal{L}_{sup}$  and unsupervised loss  $\mathcal{L}_{unsup}$ .
+
+To maintain the stability of pseudo-labels, we forego gradient backpropagation for updating the teacher model's parameters. Instead, we employ the Exponential Moving Average (EMA) method to create an aggregated model reflecting both the current and previous states. EMA's effectiveness has been substantiated in numerous studies (Kingma & Ba, 2014; Ioffe & Szegedy, 2015; He et al., 2020; Grill et al., 2020; Tarvainen & Valpola, 2017). The use of EMA not only improves the teacher model's accuracy but also its stability, making it a valuable tool during the mutual learning stage, which is formulated as follows:
+
+$$
+\theta_ {t} \leftarrow \alpha \theta_ {t} + (1 - \alpha) \theta_ {s}, \tag {5}
+$$
+
+where  $\alpha$  is the decay coefficient of EMA, typically set within the small range of 0.9 to 0.999.
+
+# 3.3. The Proposed SemiRES
+
+# 3.3.1. OVERVIEW
+
+The overview of our proposed SemiRES is depicted in Fig. 2. SemiRES inherits the semi-supervised framework introduced in Sec. 3.2 and proposes new strategies to address the challenges of noisy pseudo-labels encountered by regular semi-supervised frameworks, which limit the extraction of knowledge from unlabeled data. The core idea is to exploit the powerful edge segmentation capabilities of SAM. The central question of our research is how to utilize these masks to refine the noisy pseudo-labels. In this paper, we propose two alternative matching strategies, IoU-based Optimal Matching (IOM) and Composite Parts Integration
+
+(CPI), to select masks that contribute to the final pseudolabels, as detailed in Sec. 3.3.2. Moreover, when segments generated by SAM cannot be matched with pseudo-labels, we introduce a Pixel-Wise Weighted Adjustment (PWA) scheme to focus the model on more reliable pixels, thereby improving performance, as outlined in Sec. 3.3.3.
+
+# 3.3.2. SAM-BASED PSEUDO-LABEL REFINEMENT
+
+Despite SAM's powerful segmentation capabilities, effectively harnessing these for pseudo-label refinement is an area ripe for investigation. We have formulated two strategies for matching SAM-generated segments with the original pseudo-labels to improve their accuracy. Before deploying these strategies, we utilize SAM's "Segment Everything" feature to create an extensive proposal library of multi-scale candidate segments for our dataset offline, eliminating the need for specific prompts. To optimize storage space, we implement the Run Length Encoding (RLE) algorithm<sup>1</sup>. Considering that SAM is capable of producing hundreds to thousands of intricate segments per image, adopting efficient storage solutions is crucial. Importantly, while the RLE algorithm achieves high compression rates, it also preserves the precision of the candidate masks.
+
+IoU-based Optimal Matching (IOM). To achieve our goal, we initially consider a more straightforward approach, premised on the robust multi-scale segmentation ability of the Segment Anything Model (SAM). We hypothesize that the proposal library, constructed as previously mentioned, likely contains a close approximation of the ideal target seg
+
+![](images/6760b71804a92df1247fdd6bf44c56ad253052216b93c37e368ff8148ce4e60a.jpg)  
+Figure 3. Visualization of the principles behind IOM and CPI addressing pseudo-label issues in different cases.
+
+mentation. Therefore, our task simplifies devising a method to retrieve this optimal mask from the library.
+
+Our method involves an IoU-based selection process, which computes the similarity between the pseudo-labels and each segment generated by SAM. We aim to identify the segment with the highest similarity score, ensuring it aligns closely with the overall target mask. The similarity measure is calculated using the Intersection over Union (IoU) metric, a standard in object detection and segmentation tasks that quantifies the extent of overlap between two areas. By selecting the segment with the top-1 IoU score from the candidate pool, we can effectively align our model's output with the most accurate representation of the intended segmentation, as detailed below:
+
+$$
+s ^ {k} = \frac {\sum_ {j = 1} ^ {H \times W} \left(\hat {M} _ {i , j} ^ {u} \cap \widetilde {M} _ {i , j , k} ^ {u}\right)}{\sum_ {j = 1} ^ {H \times W} \left(\hat {M} _ {i , j} ^ {u} \cup \widetilde {M} _ {i , j , k} ^ {u}\right)}, \tag {6}
+$$
+
+where  $\hat{M}_{i,j}^{u}$  and  $\widetilde{M}_{i,j,k}^{u}$  denote the pseudo-labels and  $k$ -th segment mask generated by SAM for  $j$ -th pixel of  $i$ -th unlabeled image  $\mathcal{I}_i^u$ , respectively. When the score  $s^k$  exceeds a certain threshold  $IoU_{rate}$ , the matched mask will replace the pseudo-label. The detailed matching schemes can be found in Algorithm 1, particularly within lines 4 to 9.
+
+Composite Parts Integration (CPI). In our exploration of semi-supervised referring expression segmentation, we recognize that while the IoU-based Optimal Matching (IOM) strategy is generally straightforward and effective, it may falter in certain scenarios. One such instance occurs when the proposal library lacks an ideal target segmentation, rendering even the most sophisticated matching algorithm incapable of finding an appropriate guide mask. Another instance is when the disparity between the pseudo-labels and the desired segmentation is too substantial to allow for effective correction. We have noted that the original pseudo-labels generated by the teacher model can suffer from either under-segmentation or over-segmentation of the target instances, as depicted in Fig. 3. These inaccuracies diminish the quality of the pseudo-labels, providing erroneous guidance to the student model and impeding its learning.
+
+Under-segmentation is characterized by incomplete coverage of the target instance, missing activation for certain region pixels. To address this, we aim to identify larger regions within the proposal library to rectify the pseudolabels. Our selection is based on the overlap ratio with the pseudo-labels, calculated as follows:
+
+$$
+s _ {1} ^ {k} = \frac {\sum_ {j = 1} ^ {H \times W} \left(\hat {M} _ {i , j} ^ {u} \cap \widetilde {M} _ {i , j , k} ^ {u}\right)}{\sum_ {j = 1} ^ {H \times W} \left(\hat {M} _ {i , j} ^ {u}\right) + \epsilon}, \tag {7}
+$$
+
+where  $\epsilon$  is the smoothing factor to prevent a denominator of zero. When the overlap ratio  $s_1^k$  exceeds a predefined threshold  $inter_1$ , the  $k$ -th segment  $\widetilde{M}_{i,:,k}^u$  generated by SAM is selected and subsequently merged to replace the pseudo-labels. This method is referred to as Composite Parts Integration for Under-segmentation (CPI-U). Conversely, over-segmentation introduces erroneous regions into the segmentation. To mitigate this, we seek to leverage SAM's segmentation to filter out the extraneous noise. The selection is based on the overlap ratio with the candidate mask, computed as:
+
+$$
+s _ {2} ^ {k} = \frac {\sum_ {j = 1} ^ {H \times W} \left(\hat {M} _ {i , j} ^ {u} \cap \widetilde {M} _ {i , j , k} ^ {u}\right)}{\sum_ {j = 1} ^ {H \times W} \left(\widetilde {M} _ {i , j , k} ^ {u}\right)}. \tag {8}
+$$
+
+Likewise, when the ratio  $s_2^k$  is above the set threshold  $inter_2$ , the segment  $\widetilde{M}_{i,:,k}^u$  generated by SAM is chosen and integrated to refine the pseudo-labels. This approach is termed Composite Parts Integration for Over-segmentation (CPI-O). When both conditions are met, we form the overarching CPI strategy. The detailed matching schemes can be found in Algorithm 1, particularly within lines 10 to 15.
+
+# 3.3.3. Pixel-WISE WEIGHTED ADJUSTMENT
+
+Despite the effectiveness of our two strategies for refining pseudo-labels, there are cases where the scores do not ex
+
+Algorithm 1 Pseudo code for our proposed SemiRES
+
+Input: Teacher's predicted pseudo mask  $\hat{M}_i^u$  for  $i$ -th unlabeled image  $\mathcal{I}_i^u$ , the multi-scale mask  $\{\widetilde{M}_{i,:k}^u\}_{k=1}^{N_i}$  generated by SAM, the number of masks  $N_i$  generated by SAM for  $\mathcal{I}_i^u$ , selected strategy  $S$ , current top-1 score  $s_{top}$
+
+Output: Enhanced pseudo mask  $\tilde{M}_i^u$
+
+1: Initialize  $\ddot{M}_i^u\gets \emptyset ,s_{top}\gets 0;$  
+2: for k in 1...  $N_{i}$  do  
+3: Get  $k$ -th SAM's segment  $\widetilde{M}_{i,:;k}^{u}$  for image  $\mathcal{T}_i^u$ ;  
+4: if  $S ==$  "IOM" then  
+5: Compute the score  $s^k$  by Eq.(6);  
+6: if  $s^k > IoU_{rate}$  and  $s^k > s_{stop}$  then  
+7:  $\ddot{M}_i^u\gets \widetilde{M}_{i,::k}^u,s_{top}\gets s;$  
+8: end if  
+9: end if  
+10: if  $S ==$  "CPI" then  
+11: Compute the score  $s_1^k, s_2^k$  by Eq.(7) and Eq.(8);  
+12: if  $s_1^k > \text{inter}_1$  or  $s_2^k > \text{inter}_2$  then  
+13:  $\tilde{M}_i^u\gets \tilde{M}_i^u\cup \tilde{M}_{i,:;k}^u;$  
+14: end if  
+15: end if  
+16: end for  
+17: if  $\ddot{M}_i^u = = \emptyset$  then  
+18: Replace the enhanced pseudo-label  $\tilde{M}_i^u$  with the teacher's predicted pseudo-labels:  $\tilde{M}_i^u\gets \tilde{M}_i^u$  
+19: end if
+
+ceed a certain threshold, indicating a mismatch between SAM-generated segments and the current pseudo-labels. In such situations, as inspired by previous work (Yang et al., 2023), we implement the Pixel-Wise Weighted Adjustment (PWA). PWA's core objective is to assign weights to pixels based on their confidence levels. High-confidence pixels, with scores near 0 or 1, indicate certainty in foreground or background prediction and are given higher weights. In contrast, pixels with scores around 0.5, often associated with noise or ambiguity, receive lower weights to reduce their influence on training. The mapping function  $\Psi$  for translating pixel confidence into weights is defined as:
+
+$$
+\Psi (\hat {M} _ {i, j} ^ {u}) = \gamma - \frac {1}{\sqrt {2 \pi} \sigma} \exp \left(- \frac {(\hat {M} _ {i , j} ^ {u} - \mu) ^ {2}}{2 \sigma^ {2}}\right), \qquad (9)
+$$
+
+where  $\gamma, \sigma^2, \mu$  are hyperparameters, which are set to 1.3, 0.1, and 0.5 respectively.
+
+Therefore, the loss for  $i$ -th unlabeled image  $\mathcal{I}_i^u$  is defined as follows:
+
+$$
+\mathcal {L} _ {u n s u p} = \frac {1}{H \times W} \sum_ {j = 1} ^ {H \times W} \Psi \left(\hat {M} _ {i, j} ^ {u}\right) * L _ {B C E} \left(M _ {i, j} ^ {u}, \hat {M} _ {i, j} ^ {u}\right). \tag {10}
+$$
+
+# 4. Experiment
+
+# 4.1. Datasets
+
+We verify the effectiveness of our proposed method on three standard RES benchmark datasets, RefCOCO (Yu et al., 2016), RefCOCO+ (Yu et al., 2016), and G-Ref (Mao et al., 2016; Nagaraja et al., 2016). Images in these datasets are collected from the MS-COCO dataset (Lin et al., 2014) and are attached with one or more short captions.
+
+RefCOCO & RefCOCO+ contains 19,994, 19,992 images, with 50,000, 49,856 annotated objects and 142,209, 141,564 annotated expressions, respectively. RefCOCO and RefCOCO+ are split into four parts, i.e., train, val, testA and testB. The expressions of RefCOCO are mainly about absolute position, while the ones of RefCOCO+ includes more information related to attributes.
+
+G-Ref contains 26,711 images, with 54,822 annotated objects and 104,560 annotated expressions. In contrast, G-Ref contains more intricate expressions, with an average length of 8.4 words, making the dataset more challenging. Moreover, the G-Ref dataset is split into two distinct partitions, one maintained by UMD and the other by Google, and we present results for UMD split.
+
+# 4.2. Implementation Details
+
+Our experimental setup uses LAVT (Yang et al., 2022b) as the baseline for the RES network, employing the same Swin Transformer (Liu et al., 2021c) and BERT (Devlin et al., 2018) backbones for visual and linguistic modalities, respectively. We implement our SemiRES model in PyTorch (Paszke et al., 2019), training it on 4 RTX3090 GPUs with 3 labeled and 3 unlabeled samples per GPU. Optimization is done using the AdamW optimizer, with an initial learning rate of  $5 \times 10^{-5}$  and weight decay of  $10^{-2}$ . Data augmentation includes RandomColorJitter and RandomGaussianBlur. We set the EMA rate at 0.996 and use pre-trained weights of the ViT-Huge version for SAM in generating multi-scale masks.
+
+We use the overall Intersection-over-Union (oIoU) metric (Ding et al., 2021; Yang et al., 2022b; Liu et al., 2023b), a standard in RES, to measure the overlap ratio between predicted masks and ground truth.
+
+# 4.3. Experimental Results
+
+# 4.3.1. COMPARISON WITH SUPERVISED MODEL AND BASELINE
+
+In Tab. 1, we conduct experiments on RefCOCO, RefCOCO+ and G-Ref under the setting of  $0.5\%$ ,  $1\%$ ,  $2\%$  and  $5\%$  labeled data. From the results, it can be observed that the performance of the supervised model dramatically drops when lacking sufficient labeled data. For instance,
+
+Table 1. Comparison of supervised, baseline and our proposed SemiRES on RefCOCO, RefCOCO+ and G-Ref. For all approaches, we use LAVT (Yang et al., 2022b) as the RES model. "Supervised" denotes the fully-supervised training with only labeled data. "Baseline" denotes the plain semi-supervised training with data augmentation using both labeled and unlabeled data.  
+
+<table><tr><td rowspan="3">Methods</td><td colspan="12">RefCOCO</td></tr><tr><td colspan="3">0.5%</td><td colspan="3">1%</td><td colspan="3">2%</td><td colspan="3">5%</td></tr><tr><td>val</td><td>testA</td><td>testB</td><td>val</td><td>testA</td><td>testB</td><td>val</td><td>testA</td><td>testB</td><td>val</td><td>testA</td><td>testB</td></tr><tr><td>Supervised</td><td>22.37</td><td>25.35</td><td>19.28</td><td>32.26</td><td>35.71</td><td>28.02</td><td>39.46</td><td>42.50</td><td>35.26</td><td>49.40</td><td>53.72</td><td>44.87</td></tr><tr><td>Baseline</td><td>30.33</td><td>35.18</td><td>25.74</td><td>42.62</td><td>48.86</td><td>37.43</td><td>51.05</td><td>54.75</td><td>46.34</td><td>60.58</td><td>64.98</td><td>54.85</td></tr><tr><td>SemiRES</td><td>40.31</td><td>46.48</td><td>34.88</td><td>50.90</td><td>57.54</td><td>44.48</td><td>54.85</td><td>60.39</td><td>48.52</td><td>61.31</td><td>66.64</td><td>55.94</td></tr><tr><td rowspan="3">Methods</td><td colspan="12">RefCOCO+</td></tr><tr><td colspan="3">0.5%</td><td colspan="3">1%</td><td colspan="3">2%</td><td colspan="3">5%</td></tr><tr><td>val</td><td>testA</td><td>testB</td><td>val</td><td>testA</td><td>testB</td><td>val</td><td>testA</td><td>testB</td><td>val</td><td>testA</td><td>testB</td></tr><tr><td>Supervised</td><td>20.83</td><td>24.53</td><td>16.25</td><td>24.76</td><td>29.11</td><td>20.29</td><td>28.88</td><td>32.41</td><td>24.49</td><td>37.60</td><td>42.32</td><td>32.39</td></tr><tr><td>Baseline</td><td>25.89</td><td>30.42</td><td>20.23</td><td>30.91</td><td>35.83</td><td>24.98</td><td>36.98</td><td>41.44</td><td>30.63</td><td>46.29</td><td>52.46</td><td>38.61</td></tr><tr><td>SemiRES</td><td>31.99</td><td>38.06</td><td>25.92</td><td>36.49</td><td>42.86</td><td>28.58</td><td>40.41</td><td>46.84</td><td>33.30</td><td>47.00</td><td>54.42</td><td>38.74</td></tr><tr><td rowspan="3">Methods</td><td colspan="12">G-Ref</td></tr><tr><td colspan="3">0.5%</td><td colspan="3">1%</td><td colspan="3">2%</td><td colspan="3">5%</td></tr><tr><td>val(U)</td><td colspan="2">test(U)</td><td>val(U)</td><td colspan="2">test(U)</td><td>val(U)</td><td colspan="2">test(U)</td><td>val(U)</td><td colspan="2">test(U)</td></tr><tr><td>Supervised</td><td>18.33</td><td colspan="2">18.69</td><td>24.31</td><td colspan="2">24.72</td><td>28.23</td><td colspan="2">29.86</td><td>37.25</td><td colspan="2">38.62</td></tr><tr><td>Baseline</td><td>26.02</td><td colspan="2">27.62</td><td>30.91</td><td colspan="2">31.51</td><td>37.07</td><td colspan="2">38.55</td><td>46.67</td><td colspan="2">48.39</td></tr><tr><td>SemiRES</td><td>31.81</td><td colspan="2">33.40</td><td>34.76</td><td colspan="2">36.18</td><td>42.15</td><td colspan="2">43.49</td><td>47.61</td><td colspan="2">50.11</td></tr></table>
+
+with  $0.5\%$  labeled data, the overall IoU on RefCOCO val set is only  $22.37\%$ . We also compare the plain semi-supervised baseline, as mentioned in Sec. 3.2, which surpasses the supervised method in all settings, i.e., exhibiting a  $+7.96\%$  improvement on RefCOCO val set, with  $0.5\%$  labeled data. Most importantly, our proposed SemiRES achieves state-of-the-art performance compared to baseline. In comparison to the supervised model, SemiRES gains  $+17.94\%$ ,  $+18.64\%$ ,  $+15.39\%$ , and  $+11.91\%$  on RefCOCO val set under the setting of  $0.5\%$ ,  $1\%$ ,  $2\%$ , and  $5\%$  labeled data, respectively.
+
+# 4.3.2. ABLATION STUDY
+
+To validate the effectiveness of components in SemiRES, we conduct the ablation study on the  $1\%$  labeled data and the remaining  $99\%$  unlabeled data on RefCOCO.
+
+The comparison of different matching strategies. In Tab. 2, we evaluate our two proposed matching strategies: IOM and CPI. Both strategies significantly surpass the baseline, demonstrating their effectiveness. IOM, which matches the top-1 IoU mask from SAM, attains  $49.66\%$  oIoU on the RefCOCO val set. This result not only indicates the simplicity and efficacy of IOM but also corroborates SAM's exceptional segmentation ability, capable of producing ideal masks in most instances. However, IOM is slightly less effective compared to CPI, particularly when pseudo-labels segment only a small part of the target.
+
+CPI-O, tailored for over-segmentation scenarios, effec
+
+Table 2. Ablation study of SAM Matching Strategy.  
+
+<table><tr><td>SAM Matching Strategy</td><td>val</td><td>testA</td><td>testB</td></tr><tr><td>baseline</td><td>42.62</td><td>48.86</td><td>37.43</td></tr><tr><td>IOM</td><td>49.66</td><td>55.26</td><td>44.09</td></tr><tr><td>CPI-O</td><td>45.23</td><td>52.53</td><td>38.30</td></tr><tr><td>CPI-U</td><td>50.90</td><td>57.54</td><td>44.48</td></tr><tr><td>CPI</td><td>50.37</td><td>56.88</td><td>44.52</td></tr></table>
+
+tively eliminates noise by filtering out excessively segmented small regions. Nonetheless, its performance increment is less pronounced (45.23% vs. 42.62%) when the noisy region enlarges and starts matching with new noisy areas. In contrast, CPI-U, designed to tackle under-segmentation, emerges as the most performant strategy (50.90% vs. 42.62%). This superior performance of CPI-U can be attributed to its efficient resolution of the common under-segmentation problem in pseudo-labels. When CPI-O and CPI-U are combined into a CPI strategy, there is a minor decrease in performance, likely due to CPI-U inadvertently introducing noise in the pseudo-label refinement process.
+
+The impact of the matching threshold. In the proposed SemiRES, we use specific thresholds to optimize the matching rates for the IOM and CPI strategies. Our ablation study, shown in Tab. 3, reveals that IOM achieves its best performance with an  $IoU_{rate}$  of 0.5. For CPI, a higher performance is observed when  $inter_1$  and  $inter_2$  are both set
+
+# Input img
+
+# Supervised
+
+# Baseline
+
+# Ours
+
+# Ground Truth
+
+![](images/fdd3b5624555231674da53dbdd73fed0a845a76bd32185edf6be7df4d2cefb61.jpg)  
+Exp-1: far right luggage.  
+Exp-2: smile piece.
+
+![](images/29bf173bafabae32c91b72dbe80e271e9e4f55265a9ce2aa6e7f47f30850f8a8.jpg)
+
+![](images/6da8550020c4d0b40c6bca72b0b4230d6f6d8c62943bcd5a40f2c8002fa67f46.jpg)
+
+![](images/0dbbea9c84e95794453b851062fd78bd9e983f64ace33b441d334b15d7c6f075.jpg)
+
+![](images/f57653c8b7755eb76ae7aa72487ccf23437bfb1d51a172061737fd081a61a61d.jpg)
+
+![](images/33a67d6460c15cfb1a5e806b47f6198c7ea8f07ac8ac2099389f2a23df51ed6c.jpg)
+
+![](images/dbdd6abf30603da0d68ea7d7015ce0f3e964138164e129072e740876bfa2f633.jpg)
+
+![](images/04e4a50ed9d5771ccea11756e444e5d4be542b7dce0573f5e3dc4bed632e08b1.jpg)
+
+![](images/4abf7db19c0790056ae03ccf97eaff6ee95231f5b5951155fdd9c7aae51dc939.jpg)
+
+![](images/d79eb1f868d8e881677b60ce59718500e1ccda8dadb2c272bd14fb4a4128e4c3.jpg)
+
+![](images/d07eb8eb538b738400de5ab809927b1cfbe0c44d178f4ad910e31daeb26e48e5.jpg)  
+Exp-3: elephant facing us.  
+Figure 4. Qualitative analysis for SemiRES, supervised model, semi-supervised baseline and ground truth. The white number in the bottom right corner represents the IoU value between predicted image and ground truth. The object enclosed by the red dashed box represents incorrect segmentation. Here we use supervised and semi-supervised model trained on  $1\%$  labeled data for visualization.
+
+![](images/8c47079712ddab266612606c0689442268adfd8d6cace13dbcf012f0e9f960a8.jpg)
+
+![](images/432db47596e29570a899cfaf3df21b695d6e457860b16fee38ebad69c03ff3a1.jpg)
+
+![](images/20c084c2b390ebcf842a8994e4954551a96348fbd652dd2bbe29377d6a3f446e.jpg)
+
+![](images/c70074382cd7cda8474e968881e68589c6116b6e68d9f8163d55966df71e1941.jpg)
+
+Table 3. Ablation study of different thresholds for matching.  
+
+<table><tr><td>inter1</td><td>inter2</td><td>IoUrate</td><td>val</td><td>testA</td><td>testB</td></tr><tr><td>-</td><td>-</td><td>0.7</td><td>48.24</td><td>55.34</td><td>41.83</td></tr><tr><td>-</td><td>-</td><td>0.6</td><td>49.36</td><td>56.35</td><td>42.94</td></tr><tr><td>-</td><td>-</td><td>0.5</td><td>49.66</td><td>55.26</td><td>44.09</td></tr><tr><td>0.7</td><td>0.7</td><td>-</td><td>50.37</td><td>56.88</td><td>44.52</td></tr><tr><td>0.6</td><td>0.6</td><td>-</td><td>50.18</td><td>56.97</td><td>44.51</td></tr><tr><td>0.5</td><td>0.5</td><td>-</td><td>49.38</td><td>56.40</td><td>43.07</td></tr><tr><td>0.7</td><td>-</td><td>-</td><td>50.90</td><td>57.54</td><td>44.48</td></tr><tr><td>0.6</td><td>-</td><td>-</td><td>50.13</td><td>57.66</td><td>43.71</td></tr><tr><td>0.5</td><td>-</td><td>-</td><td>50.21</td><td>57.03</td><td>44.17</td></tr></table>
+
+to 0.7. This setting is crucial as lower thresholds in CPI could include noise, especially in the CPI-U variant, where  $inter_1 = 0.7$  yields the best results. By default, unless specified otherwise,  $IoU_{rate}$ ,  $inter_1$ , and  $inter_2$  are set to 0.5, 0.7, and 0.7, respectively.
+
+Comparison with filtering-based method. In our analysis, as presented in Tab. 4, we compare SemiRES with a confidence filtering method that eliminates the lowest  $5\%$  of pseudo-labels based on confidence scores. SemiRES demonstrates superior performance, achieving  $50.90\%$  versus  $42.99\%$  on the  $1\%$  RefCOCO validation set. This result suggests that the filtering-based method is overly rigid, leading to suboptimal use of pseudo-labels.
+
+Effectiveness of different Components. We present our
+
+Table 4. Comparison of SemiRES and filtering-based method.  
+
+<table><tr><td>Semi-Supervised Settings</td><td>val</td><td>testA</td><td>testB</td></tr><tr><td>Supervised</td><td>32.26</td><td>35.71</td><td>28.02</td></tr><tr><td>baseline</td><td>42.62</td><td>48.86</td><td>37.43</td></tr><tr><td>+confidence filtering</td><td>42.99</td><td>48.78</td><td>35.96</td></tr><tr><td>+SemiRES</td><td>50.90</td><td>57.54</td><td>44.48</td></tr></table>
+
+Table 5. Ablation study on various components: MLT (mutual learning training), DA (data augmentation), PWA (pixel-wise adjustment), and Refine (SAM matching refinement).  
+
+<table><tr><td>MLT</td><td>DA</td><td>PWA</td><td>Refine</td><td>val</td><td>testA</td><td>testB</td></tr><tr><td>X</td><td>X</td><td>X</td><td>X</td><td>32.26</td><td>35.71</td><td>28.02</td></tr><tr><td>✓</td><td>X</td><td>X</td><td>X</td><td>40.84</td><td>46.46</td><td>35.22</td></tr><tr><td>✓</td><td>✓</td><td>X</td><td>X</td><td>42.62</td><td>48.86</td><td>37.43</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>X</td><td>43.09</td><td>49.72</td><td>37.10</td></tr><tr><td>✓</td><td>✓</td><td>X</td><td>✓</td><td>49.96</td><td>57.12</td><td>43.86</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>50.90</td><td>57.54</td><td>44.48</td></tr></table>
+
+experimental analysis in Tab. 5, systematically ablated to evaluate each component of SemiRES. The first row uses only  $1\%$  labeled data for supervised training. The second row adds the remaining  $99\%$  as unlabeled data for basic semi-supervised learning. The third and fourth rows include data augmentation and the PWA module, demonstrating their effectiveness with incremental improvements (40.84% vs. 42.62% vs. 43.09%). The fifth row shows a significant
+
+gain from using SAM for pseudo-label refinement, with a  $7.34\%$  increase (49.96% vs. 42.62%). Finally, the sixth row indicates that PWA continues to improve performance even when SAM candidates do not match the pseudo-labels, highlighting the synergistic effect of these modules.
+
+# 4.4. Qualitative Analysis
+
+We showcase qualitative results in Fig. 4, comparing SemiRES with a supervised model, a semi-supervised baseline, and ground truth. Impressively, SemiRES corrects errors from both the supervised model and the semi-supervised baseline. For instance, in the first example, while the supervised and baseline models fail to interpret "far right" correctly, leading to inaccurate identification of the luggage, SemiRES precisely localizes the target. In the second example, SemiRES effectively understands "smile" and accurately segments the correct pizza. In a more complex third scenario with several elephants, SemiRES successfully identifies the elephant facing towards us, demonstrating its advanced understanding.
+
+# 5. Conclusion
+
+In this work, we present a novel semi-supervised framework, namely SemiRES, to address the challenge of costly annotations in RES. SemiRES incorporates two innovative matching strategies that leverage the robust segmentation capabilities of SAM to refine the quality of pseudo-labels. In situations where SAM is unable to rectify the pseudolabels, we employ the Pixel-Wise Adjustment (PWA) strategy, which utilizes the original pseudo-labels for efficient training directly. Our extensive experiments demonstrate that SemiRES achieves competitive results on three RES benchmark datasets, underscoring its viability and effectiveness for real-world applications.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
+# Acknowledgements
+
+This work was supported by National Key R&D Program of China (No.2023YFB4502804), the National Science Fund for Distinguished Young Scholars (No.62025603), the National Natural Science Foundation of China (No. U21B2037, No. U22B2051, No. 62072389), the National Natural Science Fund for Young Scholars of China (No. 62302411), China Postdoctoral Science Foundation (No. 2023M732948), the Natural Science Foundation of Fujian
+
+Province of China (No.2021J01002, No.2022J06001), and partially sponsored by CCF-NetEase ThunderFire Innovation Research Funding (NO. CCF-Netease 202301).
+
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+
+# A. The Labeling Cost at Different Ratios
+
+In semi-supervised learning, the more labeled data there is, the higher the cost of annotation. The three RES benchmark datasets RefCOCO, RefCOCO+, G-Ref contain 50000, 49856 and 54822 annotated objects, respectively. Following the label budget calculation in (Kim et al., 2023a), manually labeling a mask for one instance takes approximately 79.1 seconds. Therefore, it can be observed that annotating masks for the entire dataset is a very time-consuming task. Based on this, we calculate the required annotation time for labeling  $0.5\%$ ,  $1\%$ ,  $2\%$ ,  $5\%$ ,  $10\%$ ,  $20\%$ , and  $50\%$  of the data on the RefCOCO as follows:
+
+-  $0.5\%$  :  $50000 \times 0.005 \times 79.1 / 60 / 60 / 24 = 0.2$  day  
+-  $1\%$ :  $50000 \times 0.01 \times 79.1 / 60 / 60 / 24 = 0.5$  day  
+-  $2\%$ :  $50000 \times 0.02 \times 79.1 / 60 / 60 / 24 = 1.0$  day  
+-  $5\%$  :  $50000 \times 0.05 \times 79.1 / 60 / 60 / 24 = 2.3$  day  
+-  $10\%$  :  $50000 \times 0.1 \times 79.1 / 60 / 60 / 24 = 4.6$  day  
+-  $20\%$ :  $50000 \times 0.2 \times 79.1 / 60 / 60 / 24 = 9.2$  day  
+-  $50\%$ :  $50000 \times 0.5 \times 79.1 / 60 / 60 / 24 = 22.9$  day  
+- Fully-supervised:  $50000 \times 79.1 / 60 / 60 / 24 = 45.8$  day
+
+# B. The Impact of the Proportion of Labeled Data
+
+We further conduct the experiment using our SemiRES framework with a larger proportion of labeled data under the settings of  $10\%$ ,  $20\%$ ,  $30\%$ ,  $40\%$ , and  $50\%$  of labeled data, as shown in Tab. 6. We observe that when the labeled data approaches  $30\%$ , the performance of our method nearly matches that of fully supervised models. This validates that our proposed SemiRES maintains great segmentation performance with a significant reduction in annotation costs.
+
+# C. The Impact on Different RES Frameworks
+
+To assess the generalizability of SemiRES, we integrate it with the region-based GRES baseline ReLA (Liu et al., 2023a). As Tab. 7 shows, SemiRES significantly enhances performance compared to the "Supervised" approach  $(43.57\%$  vs.  $31.54\%)$ , demonstrating its robust generalization capabilities across different RES frameworks.
+
+# D. The Working Principles of IOM and CPI
+
+To provide a clearer understanding of the working principles underlying the two SAM-based Pseudo-Label Refinement modules we have introduced, namely the IOM and CPI
+
+Table 6. The impact of the proportion of labeled data.  
+
+<table><tr><td>Setting</td><td>Labeled Fraction</td><td>val</td><td>testA</td><td>testB</td></tr><tr><td rowspan="5">SemiRES</td><td>10%</td><td>63.60</td><td>68.43</td><td>60.20</td></tr><tr><td>20%</td><td>65.97</td><td>69.15</td><td>62.08</td></tr><tr><td>30%</td><td>68.54</td><td>70.62</td><td>63.90</td></tr><tr><td>40%</td><td>68.76</td><td>71.69</td><td>65.23</td></tr><tr><td>50%</td><td>69.45</td><td>72.94</td><td>65.68</td></tr><tr><td>Fully-Supervised</td><td>-</td><td>72.73</td><td>75.82</td><td>68.79</td></tr></table>
+
+Table 7. Results of different state-of-the-art (SOTA) RES approaches equipped with the SemiRES strategy.  
+
+<table><tr><td>RES Baseline</td><td>Settings</td><td>val</td><td>testA</td><td>testB</td></tr><tr><td rowspan="3">LAVT (Yang et al., 2022b)</td><td>Supervised</td><td>32.26</td><td>35.71</td><td>28.02</td></tr><tr><td>Baseline</td><td>42.62</td><td>48.86</td><td>37.43</td></tr><tr><td>SemiRES</td><td>50.90</td><td>57.54</td><td>44.48</td></tr><tr><td rowspan="3">GRES (Liu et al., 2023a)</td><td>Supervised</td><td>31.54</td><td>35.93</td><td>25.37</td></tr><tr><td>Baseline</td><td>38.65</td><td>43.53</td><td>34.63</td></tr><tr><td>SemiRES</td><td>43.57</td><td>49.05</td><td>39.05</td></tr></table>
+
+method, as depicted in Fig. 5. Specifically, we select a sample to elucidate how IOM and CPI operate in matching pseudo-labels. Specifically, IOM selects the best-matching segment proposal that meets the criteria from the candidate pool within SAM and uses it as the refined pseudo-label. IOM algorithm selects the segment with index 7 as the optimal match to refine the pseudo-label. On the other hand, CPI leverages the multi-scale characteristics obtained from SAM segmentation, systematically selecting qualifying partial segments, and eventually merging them together to form the final result. For CPI, it selects all segments that meet the conditions, such as indices 3, 4, 5, 6, and 7. In the end, it takes the union of these segments as the final refined pseudo-label. Observably, IOM and CPI, while both aimed at refining pseudo-labels within the SAM differ significantly in their operational approach and specialization. IOM is tailored for simpler, less noisy label refinement tasks, while CPI is designed to address more complex segmentation errors, together providing a comprehensive solution for enhancing the accuracy of pseudo-labels in segmentation tasks.
+
+# E. Comparison with other class-agnostic proposal network
+
+To validate the tight combination between our SemiRES and SAM, we replace SAM with SEEM (Zou et al., 2024) for generating class-agnostic mask proposals and applied our matching strategy to refine pseudo-labels. As shown in Tab. 8, the results maintained under the same configurations as detailed in our paper, indicated that the performance did not match that achieved using SAM for proposal extraction. Our analysis suggests that this discrepancy stems from the designed CPI algorithm capitalizing on SAM's strong seg
+
+![](images/429a071e99ebfe73f76f9637a9fae15663f2ee67b48859d46e96f808904b792c.jpg)  
+Figure 5. Demonstration of how IOM and CPI operate in matching pseudo-labels. The caption for this image is "kid looking at you".
+
+Table 8. Comparison with other class-agnostic proposal network.  
+
+<table><tr><td>Semi-Supervised Settings</td><td>val</td><td>testA</td><td>testB</td></tr><tr><td>Supervised</td><td>32.26</td><td>35.71</td><td>28.02</td></tr><tr><td>SemiRES(+SEEM)</td><td>47.61</td><td>56.28</td><td>43.62</td></tr><tr><td>SemiRES(+SAM)</td><td>50.90</td><td>57.54</td><td>44.48</td></tr></table>
+
+mentation capability and its ability to extract multi-scale proposal masks, which facilitated effective pseudo-label optimization, a feat not typically achievable with general class-agnostic proposal networks.
+
+# F. The potential of SemiRES for detecting small objects
+
+To explore the potential of our proposed SemiRES method for detecting small objects, we curated samples from the Ref-COCO training, validation, and test sets, organizing them by the size of the ground truth masks from smallest to largest, and specifically selected the top  $5\%$  and  $10\%$  of samples featuring small objects. As shown in Tab. 9, experiments conducted on these subsets serve to validate the efficacy of our SemiRES approach in comparison to traditional supervised methods when focusing on small objects. Our findings reveal that our method significantly outperforms the supervised approach in this area. This improvement is
+
+Table 9. The potential of SemiRES for detecting small objects.  
+
+<table><tr><td>Setting</td><td>Method</td><td>val</td><td>testA</td><td>testB</td></tr><tr><td rowspan="2">Top 5% small objects</td><td>Supervised</td><td>11.37</td><td>13.84</td><td>11.75</td></tr><tr><td>SemiRES</td><td>20.45</td><td>27.21</td><td>17.04</td></tr><tr><td rowspan="2">Top 10% small objects</td><td>Supervised</td><td>13.70</td><td>14.88</td><td>12.59</td></tr><tr><td>SemiRES</td><td>24.98</td><td>29.76</td><td>18.05</td></tr></table>
+
+attributed to the SAM's ability to utilize a multi-scale library of offline-generated masks, which includes candidates for small objects, allowing for the refinement of pseudo-labels.
+
+# G. Quantitative Statistics for Pseudo-label Refinement
+
+Due to the presence of noise and incompleteness in pseudolabels, our proposed two SAM-based Pseudo-Label Refinement modules effectively enhance the quality of pseudolabels, facilitating the mutual learning process between teacher and student models. To further analyze the roles of the designed IOM and CPI, along with their variants, we numerically compute their frequency for positive, negative corrections and no correction of pseudo-labels, as illustrated in Fig. 6. Here we use the ground truth of all unlabeled data in three RES datasets to validate the correction performance of our methods. Positive correction indicates that, after
+
+![](images/766346fd2b29c4caa9e7193d3a3a30bcd6c0ea32ef5835c8ac5883d54b450f0b.jpg)  
+Figure 6. Quantitative statistics of positive, negative corrections and no corrections(PWA) for IOM, CPI and variants.
+
+![](images/a0b3dcbb1632934f916688f35a929dd12e655019ca4e176a3a7dc6b7649177af.jpg)  
+Figure 7. Training curves of two matching strategies of SemiRES and semi-supervised baselines.
+
+the matching refinement, the corrected pseudo-label has a higher IoU with the ground truth than the original pseudolabel, thereby improving the quality of the pseudo-label. Negative correction, on the other hand, is the opposite; the quality of the updated pseudo-label is worse, exacerbating the misguidance in student learning. No correction indicates that the matching algorithms do not find a suitable correction for the pseudo label, thus the Pixel-Wise Adjustment (PWA) strategy mentioned in this paper is used for adjustment. The histogram results show that, among the three datasets for RES, the CPI method performs best in positive correction, while its variant CPI-U has the least occurrence of negative correction. And CPI-U exhibits faster convergence as shown in Fig. 7.
+
+# H. Additional Visualizations
+
+We present more comparative visualizations of our proposed SemiRES with both supervised and baseline models, alongside the ground truth, as illustrated in Fig. 8. Through these extensive examples, it becomes apparent that our proposed SemiRES excels in understanding semantic attributes such
+
+as absolute positional terms, spatial relations, and colors. Moreover, it demonstrates a strong capability in comprehending the semantics of complex sentences. Furthermore, it adeptly handles intricate details, showcasing robust proficiency in managing nuanced information. These visualizations underscore the effectiveness of our proposed SemiRES in correcting noisy and incomplete pseudo-labels, significantly enhancing performance, especially in scenarios with limited data.
+
+# Input img
+
+# Supervised
+
+# Baseline
+
+# Ours
+
+# Ground Truth
+
+Exp-1: left donut with white icing.
+
+![](images/44b013f1efeafcad8d73ee01d784ba74b3cb9c491dc38846a92aa394a426ceb4.jpg)
+
+![](images/59533520ec1ce9b40826c8f748f0994daec0fb93e69f91c322a79e06acba91a2.jpg)
+
+![](images/770edbfedab2a7c16af409696c0e45f810da51eb1bebc758b02bb46c47a3df64.jpg)
+
+![](images/852b16806dde2abafe02b6cf41122e28237f8a3e0c0998b2a4d1bd1b68d68cb7.jpg)
+
+![](images/448ae2f58065133a9b24ac57d7c48da3d654e7bc6ce5300672ecd5881ef3ea15.jpg)
+
+Exp-2: left laptop.
+
+![](images/fb6a5f38761b88290c4916e01eff390a2b29e7afb318b9ef3739fec1a10848ce.jpg)
+
+![](images/29f698c5b30654950d26483cc8634035df6f6ab91e71cc2d8f560bd35e56577f.jpg)
+
+![](images/fb4adedfeae0a84a673a5057ec5bdce051ac606f028bb386da6526ee9e4d6587.jpg)
+
+![](images/76feeaeab10ffae8d6f0415a2210df8e95ac98c0c014537bbb29fb615bfece73.jpg)
+
+![](images/581f04af550c35030892d8619478d415dec2d8c178872fd1beb49f41cd83fc65.jpg)
+
+Exp-3: lower left girl.
+
+![](images/832e1aad6f36816b291286b14deec44ff846e9778c3403e55f0fe92571361f15.jpg)
+
+![](images/f4c1b0663f522ee104640f38b25950fda5f72fe0c445d9860088f378f6ca039b.jpg)
+
+![](images/37a84ea4aa653aa3e415e9472086f2be6f73d6e18bf0d87bfd5839e630df87f8.jpg)
+
+![](images/32e76a159fd7fa65a0273b2c66ca9b5e3e85c056be9528ffb35ce65624977aae.jpg)
+
+![](images/e537fc1047b95abc59e629f78629477002cace71d73e45788c9708d48405fd15.jpg)
+
+Exp-4: the one cloest.
+
+![](images/ff58a710c5ecc7ce24ffd213e6f1b8d932042f7483ce4074a9aad2e8bf612b97.jpg)
+
+![](images/a520f0456e965b357ce5020c3efeafee1bf446f06e723e028a345cbd7f32941c.jpg)
+
+![](images/29668b9414dc00fd562e524b502e9ac78af4441eb1a7faf8c508cfa3cb6b6b4a.jpg)
+
+![](images/051b2c1447f30fa6503e240cc4b3acd592db1e0030353961a2322e3a83eb3558.jpg)
+
+![](images/85b9e0671f17e310ee900d4b1fe9bbd55158ea79720877c3abcbdb1fec4dc793.jpg)
+
+Exp-5: man in hat.
+
+![](images/878580b2aa2707f003e0aa26767a3067d3fd899efdf66466448bb8b23ce68b42.jpg)
+
+![](images/50706771eed5fc5c6e4fd644a9dd23b3a15b583acbf3869b3416ce59d20baac2.jpg)
+
+![](images/00693c77344005d63c22a253a5c58e63dc3b6292777b05f61c709471d03ae824.jpg)
+
+![](images/2b1d559105c97982a58be24f98a8c1db1aa13f1559afc4afa3123b2878e4eff0.jpg)
+
+![](images/39cb13a72676ff4bc8b79fb14b4748fe6835cc632e64761743c719063b5b8e32.jpg)
+
+Exp-6: person in black shirt.
+
+![](images/1b559ea1c75643f72921734ee40ad0e5f9e08be0a856a1f995c3715779e545cb.jpg)
+
+![](images/742c1e4f1873b54fb80089557817e40e8642b485e5d88dafd9a63037a6e7b365.jpg)
+
+![](images/2307912f472fd94735b93d90d3f0436aa1e0782430dc407d023e303bc9493193.jpg)
+
+![](images/e126e8cbb02fb2fcd4238dd7c5aeb003deaec1a848bdf9420fb1ed243bd61dce.jpg)
+
+![](images/945968b3fe2dc2b2722014f3c8097ccf824ce1bd0b6b0b3e81ef6bd7f6c4b0ad.jpg)
+
+Exp-7: the gentleman in the middle with a white shirt and a purple necktie.
+
+![](images/2e0844e17f9fc92e8bfd61a05236c6bbf72c16c8f141d6e468db97455082f67d.jpg)
+
+![](images/8005312d5bd5228c8a414583e80124dc0d1797ca9c3ff0c12d6069a1410cee32.jpg)
+
+![](images/4a5cefdf84a08d1cfc063372901c9e037f63f4b0c0b5291a015e2ebfe87659af.jpg)
+
+![](images/d55aa8e407e3c43c1551aa271cbb38da240ebb14c5b935d8071815d57dd6c3bc.jpg)
+
+![](images/1a3dbfda93bb8dd6dcb430237986cd5520ff41fbbf5a2d20d484ea3245319d14.jpg)
+
+Exp-8: red bus in road.
+
+![](images/a7f310bf062d61ffaaedbc1a3b1639bc6f18b482ef7fca2c3051f1234eb2cef8.jpg)
+
+![](images/3d3eef8b5c87f76bad371c697ba7f3fa442acaeeba6925216bb53fc3958642a0.jpg)
+
+![](images/3a952dfb6bf2f8a0ba367ebdac70fc8d441699ef7162c38ad301a59a53861ec1.jpg)
+
+![](images/c795725b9a4680ffcc03d89054da37d3eb2d0d8cebc3ff708566ab200f6882f0.jpg)
+
+![](images/238b759cd82266dc17e0dec23fefbb18e7bcccb93b8a5a65aa14a66914a02d17.jpg)
+
+Exp-9: the white bus with no markings.
+
+![](images/629c43d6a23940bcc49750bdf9bb63c5626123c26c4239d9f2fd1da39292e3b5.jpg)
+
+![](images/4c8357790e1a6a073a89f15a354f935a0b555ea74fa59903a837c783d11da29a.jpg)
+
+![](images/4e7210555416d48d63c6c5e97d2c5aedc14f1034341a0f1d761bbbfb00e58215.jpg)
+
+![](images/fe909bf0b0ca0e24e703e515a7d0a3ddf3a8c39c36cd9ed046e7c51fa9200d5d.jpg)
+
+![](images/42ecb6c8e4d155c78eb2b9eaf1e468860c152262c8e86193bde6ba4c5acfe2dc.jpg)
+
+# Input img
+
+Exp-19: man in blue.
+
+![](images/ac5c42073cf79a4cf699b32c63529b6f26c3d8bf1c37b1f7d8dc74928b75d2d0.jpg)
+
+# Supervised
+
+![](images/01bc9695903db2f1f94919c5eae72fc305df37993ecdf23088a022f7597c45ef.jpg)
+
+# Baseline
+
+![](images/40064a62f7464ba06c7440bf7b7dae2f222703d17b210b701bebf6b0a70d40bb.jpg)
+
+# Ours
+
+![](images/497d8688f95de21873837cbbb672527d2f2fa0008e2f45689bba1c51c9f929f1.jpg)
+
+# Ground Truth
+
+![](images/f6794df4618f2badfe5e635e7c03078a9fb5856e8dd33ffd4b19f44454a57cea.jpg)
+
+Exp-20: the man in the striped shirt.
+
+![](images/d5a6d0d1724f5af1007839c7c707109259f5a2b8be862645af960e59d4c4f3a8.jpg)
+
+![](images/c22683811337e1d7dd3eba7508758b3680ccd3d9212303cbe63bb2181a48e779.jpg)
+
+![](images/411c349a359352e029ee06fb41c8a3200022f1010fd25cb3d1b3ce8a99e186bb.jpg)
+
+![](images/f32567bab2e47858e4fe679da6d9dbb36cd5c9c8fb4dfe73f43daddc266f88bb.jpg)
+
+![](images/5df801bba5bd0cfc8ee3d4da8767eda5121610ad92d9e1fe45e04a1288bccde9.jpg)
+
+Exp-21: the train.
+
+![](images/f00fd130140dd99ed1bf15d22f6c08c738e4d30b74909bec8e94df34247c9cd1.jpg)
+
+![](images/0a585cfeb1d9aefbb55c6000db3b8cb9e88859f6f344d1e5667638ae41466fad.jpg)
+
+![](images/7729e89a431bc6c1e3ee1b10fbbb9ae905369ea9e915449350512e38ff7bc5ff.jpg)
+
+![](images/29e5e5530f3b0b25cdd308384fc580dbf8b627cedf8a240e92cc47b460505c61.jpg)
+
+![](images/f0a103c342dbd6ac646f64f45e3b17699c7a0631f38dc70e947be27bdde3ec3d.jpg)
+
+Exp-22: girl looking at laptop.
+
+![](images/2f67a3f31d83251c5a2172671fda5f510c30a7fd10d382edcbb3e18d6ac25700.jpg)
+
+![](images/ebb4505d5fdbc167b5489685eab4f0425ecc25499a827689c390a81ce6234a51.jpg)
+
+![](images/cb9505093ecc8ed201f23cb9a72f8e0ac520a43b03b5b1690af825e191888521.jpg)
+
+![](images/9a9ad9fd1aafb2444f70cde7a3baac272b0278a317d2487894c645e1b3072fa6.jpg)
+
+![](images/b5990e69e84dce1fe85dbaf7472d97e00b061c3e9899597b99b125234b5b3802.jpg)
+
+Exp-23:a man in a purple shirt is riding on a skateboard as his picture is being taken.
+
+![](images/75bf6761eca974b5ac8e179e9fc784ae607798c57c63781d8cea8e5454df0538.jpg)
+
+![](images/e0b6420533a52169b8387cfbd692f5565cfb82fcc3eec043c2a949e24a348721.jpg)
+
+![](images/44b1cf8dd73a5b031c98695c2aad9b13effa04ec8fc26d19a49aef236315322d.jpg)
+
+![](images/f584a6078934cbffce91f1915f3f9752cb4aeeb126ed89d3d807b442c5f273bb.jpg)
+
+![](images/a1701338759afb50e4102986543fbdb8602b613aa2e888d00571b639b80829d9.jpg)
+
+Exp-24: lying cow.
+
+![](images/42a334fff7de08e7442024f39b81af35110eccc04e581fa0f36c310184c7b94f.jpg)
+
+![](images/372ad9f6bd182cd8fcf617a416c8f238b2efc6af3bc8469b1bb37c681231a372.jpg)
+
+![](images/389ad2735e08e52391baee94ddf070207090f5646441481a55df3ca03794724c.jpg)
+
+![](images/cbb1467b30a2fef3fabcce2c88ec13d127fc553f3515b323225beb6033f01f44.jpg)
+
+![](images/85cf200864f2e0768ecd88c171d444dc8f407bcef93798b9c8122fb79127c909.jpg)
+
+Exp-25: main red bus.
+
+![](images/345d6f341bf73d17f564c56c692a3775fd8127f95f3a6e515c831826ec890966.jpg)
+
+![](images/0b429f50174dda7becc6c31c25bdcdbb6e18eb2ac855c9b3c29e264da24210f5.jpg)
+
+![](images/f980039c371d95ee9d8e7d4488bae713c06518b702ccfd30989c497137b722fd.jpg)
+
+![](images/226178552b47d68d795ea34ad915718215c65f323fc993fb9054b2ceeded314f.jpg)
+
+![](images/bda2cd6b5fa3b6013b7b860d398475dc7895e519e59d6583d2d7e4a33145024b.jpg)
+
+Exp-26: the chair to the right of the fireplace.
+
+![](images/62d46e7ab573f551822bc466b718a346886eeebd417fe6716b65df9490056db5.jpg)
+
+![](images/c1c88c2f11013282764e503441d1b32e86dbd7a7817e8423f91b1c0e599cf1c0.jpg)
+
+![](images/2ca2872f5fca18a06c3766abd6fec15eb38944ac871c47df4014b55a9b766d2c.jpg)
+
+![](images/29d1a9792671a57d7932fa5a8caad40c33b3a254e29adc1a64ba1369cc1f2a31.jpg)
+
+![](images/a580689425155dd7b872bb3e49bb6f2df90aaf990bfefbca948814cd8f3ed252.jpg)
+
+Exp-27: a child cutting paper with safety scissors next to her mom.
+
+![](images/6f4770dd6ab22d47383c22545ae87e847e3df1a688c11f2c1c90d1645ca273a1.jpg)  
+Figure 8. More visualization results of our proposed SemiRES, compared with the supervised and baseline model. The red dashed bounding boxes denote regions where our model has made accurate predictions, while other models have made inaccurate predictions.
+
+![](images/461faf39403f5c5f4f83dad292a67649487fad8817017f33007fe7ff946cccc7.jpg)
+
+![](images/c66c8aa0c642d7b6eb504360f819d7e1e9c110cd49085fc4ce53e600070a1f9b.jpg)
+
+![](images/66644b76230803ca99d03a08bffd7800e06cd7ebd5f80a3c7ae2468a2cdc46d0.jpg)
+
+![](images/1749a11163b5d1da2ac923694febfe528100893a053ddc59e03810c986223c89.jpg)

2024/SAM as the Guide_ Mastering Pseudo-Label Refinement in Semi-Supervised Referring Expression Segmentation/images.zip

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2024/SAM-E_ Leveraging Visual Foundation Model with Sequence Imitation for Embodied Manipulation/ca39871c-0210-423a-8328-51d4d0c380ec_origin.pdf

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2024/SAM-E_ Leveraging Visual Foundation Model with Sequence Imitation for Embodied Manipulation/full.md

@@ -0,0 +1,558 @@
+# SAM-E: Leveraging Visual Foundation Model with Sequence Imitation for Embodied Manipulation
+
+Junjie Zhang  $^{1,2}$  Chenjia Bai  $^{2}$  Haoran He  $^{3,2}$  Zhigang Wang  $^{2}$  Bin Zhao  $^{2}$  Xiu Li  $^{1}$  Xuelong Li  $^{2,4}$
+
+# Abstract
+
+Acquiring a multi-task imitation policy in 3D manipulation poses challenges in terms of scene understanding and action prediction. Current methods employ both 3D representation and multiview 2D representation to predict the poses of the robot's end-effector. However, they still require a considerable amount of high-quality robot trajectories, and suffer from limited generalization in unseen tasks and inefficient execution in long-horizon reasoning. In this paper, we propose SAM-E, a novel architecture for robot manipulation by leveraging a vision-foundation model for generalizable scene understanding and sequence imitation for long-term action reasoning. Specifically, we adopt Segment Anything (SAM) pretrained on a huge number of images and promptable masks as the foundation model for extracting task-relevant features, and employ parameter-efficient fine-tuning on robot data for a better understanding of embodied scenarios. To address long-horizon reasoning, we develop a novel multichannel heatmap that enables the prediction of the action sequence in a single pass, notably enhancing execution efficiency. Experimental results from various instruction-following tasks demonstrate that SAM-E achieves superior performance with higher execution efficiency compared to the baselines, and also significantly improves generalization in few-shot adaptation to new tasks.
+
+# 1. Introduction
+
+Robot manipulation has made significant progress, benefiting from embodied datasets (Walke et al., 2023; Collaboration, 2023; Fang et al., 2023), Imitation Learning (IL) (Jiang
+
+$^{1}$ Tsinghua Shenzhen International Graduate School, Tsinghua University  $^{2}$ Shanghai Artificial Intelligence Laboratory  $^{3}$ Hong Kong University of Science and Technology  $^{4}$ Institute of Artificial Intelligence (TeleAI), China Telecom, P. R. China. Correspondence to: Chenjia Bai <baichenjia@pjlab.org.cn>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+et al., 2023; Reed et al., 2022) or Reinforcement Learning (RL) algorithms (Zakka et al., 2023; Hansen et al., 2022; Shi et al., 2024; Bai et al., 2024), and advanced transformer (Chebotar et al., 2023; Zhao et al., 2023a) or diffusion-based networks (Xian et al., 2023; He et al., 2023; 2024). To perform a wide range of complex manipulation tasks in the 3D physical world, it is crucial to understand the 3D scene structure that encompasses object positions, orientations, shapes, occlusions, and the relationships between objects and the environment (Billard & Kragic, 2019). Various methods utilize 3D representations such as voxel patches (James et al., 2022b; Shridhar et al., 2022b), point clouds (Chen et al., 2023; Zhang et al., 2023b) to provide 3D localizations for predicting the end-effector poses. However, learning a 3D representation can be computationally expensive. For instance, the voxel-based method (Shridhar et al., 2022b) achieves state-of-the-art performance while suffering from cubic scaling of the number of voxels with the resolution, making it prohibitive for larger datasets.
+
+To tackle these challenges, recent studies have investigated feature extraction from single-view images and information aggregation using multi-view transformers (Guhur et al., 2022), which provide enhanced efficiency as the scaling of image patches aligns with the input resolution. For example, recently proposed RVT (Goyal et al., 2023) achieves 36 times faster training speeds and better performance than voxel-based approaches. However, learning a multi-view policy still requires a considerable amount of high-quality robot trajectories for imitation, and the resulting policy exhibits limited generalization capabilities for unseen tasks and low execution efficiency in long-horizon reasoning. Motivated by recent research on visual foundation models that leverage web-scale datasets and demonstrate robust zero-shot and few-shot generalization (Radford et al., 2021; Li et al., 2022; Rombach et al., 2022; Hudson et al., 2023), we delve further into the multi-view architecture to enhance the generalization capabilities and execution efficiency of 3D manipulation policies in language-following tasks.
+
+In this paper, we present a novel architecture for robot manipulation that leverages a vision-foundation model for image understanding and sequence imitation for long-horizon reasoning. We name our method SAM-E, as we utilize the
+
+Segment Anything Model (SAM) (Kirillov et al., 2023) as the foundation model for Embodied manipulation. SAM is a prompt-conditioned image segmentation model trained on a large dataset of images and masks. Utilizing SAM as the foundational perception model benefits the scene understanding and generalization in various manipulation scenarios. Moreover, the prompt-conditioned SAM encoder is suitable for language-instructed manipulation by extracting task-relevant visual features according to the task descriptions. Further, we conduct parameter-efficient finetuning for SAM with robot data to enhance the understanding of embodied scenarios. With prompt-guided features, we employ multi-view attention to integrate the view-wise representations with coordinate information for action prediction.
+
+To improve the efficiency of long-horizon action prediction, we propose a novel prediction head that generates multi-channel pose heatmaps for an action sequence. Subsequently, the heatmaps from different views are backprojected into 3D space to generate scores for a discretized set of 3D points, ultimately determining the 3D positions and rotations of actions. During inference, the action sequence can be predicted in a single pass and executed sequentially, resulting in a notable improvement in execution efficiency compared to previous step-by-step prediction methods. We conduct experiments on various 3D instruction-following tasks from RLBench, consisting of 18 tasks with 249 variations (James et al., 2020). The results demonstrate that SAM-E achieves superior performance and higher reasoning efficiency compared to baseline methods. Moreover, the visual foundation model greatly enhances the generalization ability of the learned policy in adapting to new tasks with few-shot demonstrations.
+
+# 2. Preliminaries
+
+LC-POMDP. The problem of language-conditioned robot manipulation can be modeled as a Language-Conditioned Partial Observable Markov Decision Process (LC-POMDP) formulated as an augmented POMDP  $\mathcal{M} \coloneqq (\mathcal{S}, \mathcal{O}, \mathcal{A}, \mathcal{P}, \rho_0, \mathcal{L}, f, T)$ , where  $\mathcal{S}$  and  $\mathcal{A}$  denote state space and action space separately,  $O$  denotes the space of observations,  $\mathcal{P}(s|s, a): S \times \mathcal{A} \to \mathcal{S}$  denotes the transition probability or the environment dynamics,  $\rho_0$  represents the initial state distribution,  $\mathcal{L}$  denotes the set of all language instructions,  $f(o|s): S \to \mathcal{O}$  is the observation function, and  $T$  represents the episode horizon. We adopt imitation learning without considering the reward function used for RL. For each episode, the robot is given a language instruction  $l \in \mathcal{L}$  representing the goal of the current task. At each time step  $t$ , the robot is required to take action according to a policy  $\pi(a_t|o_t, l)$  given the observation  $o_t$ . Since we focus on 3D manipulation, the observation  $o_t$  contains multi-view images from cameras at different perspectives.
+
+Imitation Learning. To address the language-conditioned manipulation tasks, imitation learning (Goyal et al., 2023; Li et al., 2024) allows the agent to mimic a set of expert demonstrations denoted as  $\mathcal{D} \coloneqq \{(\tau, l)_i\}_{i=0}^{|\mathcal{D}|}$ , where  $\tau \coloneqq (o_0, a_0, \ldots, o_{T-1}, a_{T-1}, o_T)$  is the expert trajectory, and  $l$  represents the language instruction. A common imitation learning objective for the policy  $\pi_\theta$  is to maximize the likelihood of action conditioned on the language and current state, Formally, the loss function is
+
+$$
+\mathcal {L} (\theta) := - \mathbb {E} _ {(\tau , l) \sim \mathcal {D}} \left[ \sum_ {t = 0} ^ {T - 1} \log \pi_ {\theta} \left(a _ {t} \mid o _ {t}, l\right) \right]. \tag {1}
+$$
+
+Key-frame Extraction. To improve the utilization of expert demonstrations, we align with the consensus in 3D manipulation algorithms (James & Davison, 2022; James et al., 2022a; Shridhar et al., 2022b; Goyal et al., 2023) by incorporating key-frame extraction for selecting key-frame actions. The key-frame extraction involves a Boolean function  $K: \mathbb{R}^{|A|} \to \{0,1\}$ , which determines whether an action should be identified as a key-frame. For each demonstration  $\tau$ , a sequence of key-frame actions  $\{k_1, k_2, \ldots, k_m\}$  is generated by the function  $K$  following two simple conditions: (i) the joint-velocities are near zero (occurs when entering pre-grasp poses or a new phase of task), and (ii) gripper state has changed (occurs when the object is grasped or released). Based on the function  $K$ , the imitation objective in Eq. (1) becomes predicting the 'next key-frame action' in the demonstration. In the following, we slightly abuse  $a_t$  to represent the next key-frame action of  $s_t$  since we adopt the same key-frame extraction process to SAM-E and baselines.
+
+# 3. Method
+
+The proposed SAM-E is a multi-view imitation framework that leverages the pre-trained visual foundation model and action-sequence imitation for multi-task 3D manipulation. The key idea of SAM-E contains two perspectives: (i) leveraging the visual foundation model SAM with the prompt-driven architecture and its strong generalization ability to handle the language-prompt(instructed) tasks in embodied scenarios; (ii) utilizing the temporal smooth properties of actions to perform sequence modeling of actions to enhance coherent planning and execution efficiency. We introduce the visual foundation model for embodied perception in §3.1 and the multi-view architecture in §3.2. Then we give the motivation of sequence imitation in §3.3 and the multichannel prediction architecture in §3.4.
+
+We illustrate the architecture of SAM-E in Figure 1. Overall, we adopt the SAM encoder (Kirillov et al., 2023) to generate prompt-guided and object-oriented representations, and fine-tune the encoder with embodied data and Low-Rank Adaptation (LoRA) (Hu et al., 2022) technique for manip-
+
+![](images/e197c1fe7cbbab4d4d70088f710e4df247f5c47ffc64ad1b9e93664362705ca0.jpg)  
+Figure 1. Overview of SAM-E. (i) The SAM encoder provides promptable visual embedding of single-view observations after fine-tuning on embodied scenarios with parameter-efficient LoRA. (ii) Multi-view transformer achieves cross-view information integration and vision-language alignment. (iii) The coherent action sequence is predicted via temporal imitation for efficient multi-step execution.
+
+ulation scenarios, which results in a minimal increase in computation requirement. Then, a multi-view transformer is used to integrate cross-view visual information combined with coordinate information and language instruction for multi-view correspondence and vision-language alignment. To address long-horizon action prediction, SAM-E predicts a coherent action sequence in a single pass with a novel action-sequence policy head.
+
+# 3.1. Perception Foundation and LoRA Finetune
+
+SAM for Promptable Perception. SAM (Kirillov et al., 2023) comprises a powerful image encoder and lightweight mask decoder, structured as a prompt-driven architecture designed for real-world image segmentation. Aiming at achieving promptable segmentation and effective ambiguity awareness, the image encoder of SAM is trained with flexible prompts from the downstream mask decoder. Consequently, after diverse segmentation task training, the SAM encoder is capable of extracting powerful object-centering image embedding rich in semantic information. This also enables SAM to handle unknown prompts arising from various segmentation requirements in robot interactions, including complex object-associated scenarios.
+
+In 3D manipulation, the scene perception is expected to be object-oriented and adaptable, accommodating a range of intentions and shifting focus as tasks progress. For instance, given the task instruction of 'place the apple in the basket', the agent should first find and focus on the apple to pick it up, followed by finding the basket to place. The perception module should be capable of flexible object-centered attention based on task instructions and allow attention adjustment to other objects as the task progresses (See §C for an example). From this point, the SAM encoder is suitable
+
+as a perception foundation model for language-instructed manipulation with rich task variations. The SAM encoder is a Vision Transformer (ViT) (Dosovitskiy et al., 2021) pretrained with MAE (He et al., 2022), which processes RGB images into  $C \times H \times W$  image embedding. In practice, we utilize the ViT-B architecture for the image encoder to showcase the advantages of pre-trained segmentation representations with a low computational cost in manipulation tasks. The image encoder contains 12 layers of transformer blocks and outputs the image embedding of the visual inputs. The proposed SAM-E leverages the SAM encoder as the foundation to generate prompt-guided and object-oriented representations from visual observations, which is essential for language-instructed manipulation.
+
+LoRA with Embodied Data. To effectively adapt the SAM encoder to embodied scenarios at an affordable computing cost, we employ LoRA to finetune the encoder during the policy training. As indicated in LoRA, we freeze the parameters in the image encoder and add a trainable low-rank bypass to each of the transformer encoder blocks as:
+
+$$
+W _ {0} + \Delta W = W _ {0} + B A, \tag {2}
+$$
+
+where  $W_0 \in \mathbb{R}^{d \times k}$  is the pre-trained weight matrix frozen during training,  $B \in \mathbb{R}^{d \times r}$  and  $A \in \mathbb{R}^{r \times k}$  are trainable matrix, and rank  $r \ll min(d, k)$ .  $\Delta W = BA$  represents the accumulated gradient update during adaptation with  $A$  initiated by Gaussian initialization and  $B$  initiated with zero. We set the rank  $r$  to 4 by default. In practice, we apply LoRA to the self-attention modules with query and value projection layers:
+
+$$
+\operatorname {A t t e n t i o n} (Q, K, V) = \operatorname {S o f t m a x} \left(\frac {Q K ^ {T}}{\sqrt {d _ {k}}}\right) V, \tag {3}
+$$
+
+![](images/c960958a006edefaafd155ec7b9e12354b190744f078092949b0713db2fbed43.jpg)  
+Figure 2. Multi-view Transformer has two stages for view-wise information and cross-view information integration.
+
+$$
+Q = \hat {W} _ {q} X = W _ {q} X + B _ {q} A _ {q} X, \tag {4}
+$$
+
+$$
+K = W _ {k} X, \tag {5}
+$$
+
+$$
+V = \hat {W} _ {v} X = W _ {v} X + B _ {v} A _ {v} X, \tag {6}
+$$
+
+where  $W_{q}$ ,  $W_{k}$  and  $W_{v}$  are frozen projection weights inherited from SAM encoder, and  $A_{q}, B_{q}, A_{v}$ , and  $B_{v}$  are trainable LoRA parameters.
+
+# 3.2. Multi-View Transformer
+
+After extracting the view-wise representations, we adopt a multi-view transformer to integrate multi-view visual observations, depth information with coordinates, and task-relevant language instructions using an attention mechanism, enabling a comprehensive fusion of the input in multiple modalities. The architecture is shown in Figure 2. The visual observations are processed into image embedding by the previously mentioned SAM encoder, while depth and coordinate information is processed through a Conv2D layer to obtain 3D spatial features. We concatenate the image embeddings with spatial features in the channel dimension along the patch tokens, resulting in a combined representation that we refer to as 'view tokens'. Additionally, we utilize a pretrained CLIP text encoder to generate language embeddings, from which language tokens are derived. Firstly, view tokens from the same view pass through view-wise attention blocks like ViT to maintain the single-view information. Subsequently, visual tokens across different views and the language tokens are attended to cross-view attention blocks, to integrate cross-view scene information with language instructions. The visual tokens, now enriched with cross-view information and language information are used as input for the action-sequence prediction.
+
+![](images/0963ba4342f178472ad330c5e7e95b4e18652d868b762550b9eeb3553683cb9a.jpg)  
+Figure 3. Movement shift in positions and rotations of the end effector in RLBench task close.jar, representing smooth changes of positions and rotations in temporally adjacent steps.
+
+![](images/184187b770b619c53a257b2987ebca4ee14bcc72a61d558f8a828c3bde6af1be.jpg)
+
+# 3.3. Motivation for Action-Sequence Modeling
+
+In the next, we aim to provide the intuition of action-sequence modeling, attempting to ground the utility of this technique. We start with an assumption about the temporal smooth properties of actions in the robot manipulation.
+
+Assumption 3.1 (Temporal-Smooth Assumption). Since the actions of the manipulation task are the desired positions and rotations of the end effector, the optimal action sequences  $(a_0^*, a_1^*, a_2^*, \ldots, a_T^*)$  are smooth, formulated as:
+
+$$
+\left\| a _ {t} ^ {*} - a _ {t + 1} ^ {*} \right\| <   \epsilon , \quad 0 \leq t \leq T - 2, \tag {7}
+$$
+
+$$
+\forall \tau^ {*} := \left(o _ {0}, a _ {0} ^ {*}, o _ {1}, a _ {1} ^ {*}, \dots , o _ {T - 1}, a _ {T - 1} ^ {*}, o _ {T}\right) \sim \mathrm {P} _ {\pi^ {*}} ^ {\tau} (\cdot),
+$$
+
+where  $\mathrm{P}_{\pi^*}^\tau (\cdot)$  denotes the distribution of trajectories derived from the optimal policy  $\pi^{*}$ .
+
+Intuitively, the assumption holds in most embodied manipulation tasks if the actions are the positions and rotations of the end effector. For example, in the common Pick-and-Place tasks, the optimal action sequences are a sequence of points in Euclidean space, which leads the end effector to approach the object and desired goal. Meanwhile, the gripper will rotate smoothly to align with the gripping points of the object. In Figure 3, we show the movement shift of positions and the Quaternion angle of rotations of the end-effector in a manipulation task close jar from RLBench (James et al., 2020), which further justifies our assumption. We observe that in certain manipulation tasks, end effector rotations undergo relatively rapid changes, particularly with large keyframe intervals, which weaken the assumption of smooth rotation. However, the end effector positions maintain superior smoothness in Euclidean space, which are more crucial for action-sequence modeling in our method.
+
+The typical approach trains the policy  $\pi$  to predict the action  $a_{t}$  given the multi-view image  $o_{t}$  and the task instruction  $l$ , as
+
+$$
+\pi (l, o _ {t}) \rightarrow a _ {t}.
+$$
+
+Such a step-by-step process only focuses on predicting ac
+
+tions of the current situation, which can lead to stagnation and contradictory sequential actions, as observed in experiments. Based on the Assumption 3.1, we can improve the action-prediction process by considering a long-horizon decision process instead of a single action, as
+
+$$
+\pi^ {\operatorname {s e q}} (l, o _ {t}) \rightarrow \left\{a _ {t}, a _ {t + 1}, \dots , a _ {t + h - 1} \right\},
+$$
+
+where  $h$  is the horizon of the action sequence.
+
+Then we motivate the sequence-prediction procedure based on the assumption. The sequence modeling process tries to predict the optimal action sequence condition on the observation. Intuitively, the learning objective of  $\pi^{\mathrm{seq}}$  is more difficult compared to that of  $\pi^{\mathrm{step}}$ . However, when we take a closer look at the prediction of action in a sequence (e.g.,  $a_{t + k}$ ), training to predict this action is accompanied by the prediction of former actions (i.e.,  $(\hat{a}_t,\dots ,\hat{a}_{t + k - 1}))$  and latter actions (i.e.,  $(\hat{a}_{t + k + 1},\dots ,\hat{a}_{t + h - 1}))$ . Back to the assumption that the optimal action sequences are smooth, we believe that predicting the former and latter actions can provide implicit prior and constraint in predicting  $a_{t + k}$ . Thus, the smooth properties of action sequences provide an opportunity to perform long-horizon reasoning by predicting the adjacent actions as a whole, thereby reflecting the motion trajectory of the robot's end-effector in completing tasks. In contrast, the action prediction of the traditional policy is only conditioned on the observation without any 'prompt' from the former actions, making the traditional policy inferior to action-sequence modeling in these tasks. Such a technique in 2D manipulation tasks is also called action chunking (Bharadhwaj et al., 2023; Zhao et al., 2023a), while we give a clear motivation by an empirically justified assumption and extend it to 3D scenarios using multi-channel heatmaps.
+
+# 3.4. Architecture for Action-Sequence Prediction
+
+We introduce a novel multi-channel policy head for the action-sequence prediction, as shown in Figure 4. The policy head takes view tokens from the multi-view transformer (shown in Figure 2) as input, processing view tokens from different views independently, and outputs action sequence prediction in parallel channels within a single view image.
+
+In 3D manipulation, each action in the sequence comprises an 8-dimensional vector dictating the next movement. This vector includes a 6-DoF target end effector pose (3-DoF for position and 3-DoF for rotation), a binary value indicating the gripper state (open or closed), and another binary value determining whether the collision is permissible for the low-level motion planner. (i) For predicting positions, the policy head generates a heatmap from the view tokens corresponding to each view. These heatmaps represent the desired position distribution from the perspective of each view. Then the heatmaps from different views are back-projected to into 3D space to generate scores for a discretized set of 3D points, de
+
+![](images/82fe18572a1d16ace07f018747757f48c0af9399708f52d90439bb64d0cf54fe.jpg)  
+Figure 4. The Action-Sequence Policy Head outputs multichannel pose heatmaps for a sequence of positions and rotations.
+
+termining the 3D positions. For action-sequence prediction, we equip the heatmap with time-dimension channels to learn temporal information from demonstrations, which leads to coherent action prediction in the temporal dimension. (ii) For predicting rotations, we follow previous methods (Goyal et al., 2023) to discretize Euler angles into bins of  $5^{\circ}$  resolution and thus turn rotation prediction into classification as the binary of gripper state and collision indicator. We use heatmap as the weight to extract the view-wise features from the view tokens, which provide higher weights near the desired target position within the view image, and then output the action sequence of rotation, gripper state, and collision indicator using a fully connected network.
+
+# 4. Related Works
+
+Visual Robot Manipulation. Early research in robot manipulation adopts joint states of the robot arm and geometric information of objects in RL or IL frameworks (Zeng et al., 2017; Deng et al., 2020; Xie et al., 2020; Yu et al., 2020; Xu et al., 2022), assuming the acquisition of pre-perception information and coordinates of objects. In real-world manipulation tasks, visual perception provides more general inputs without additional assumptions (Yuan et al., 2023). Various methods have adopted visual pretraining models for affordance (Goyal et al., 2022; Bahl et al., 2023), representation learning (Khandelwal et al., 2022; Shridhar et al., 2022a; Nair et al., 2023; Ma et al., 2023b;a), and goal generation (Gao et al., 2023; Jia et al., 2023) to facilitate policy learning. Other works incorporate language encoders (Xie et al., 2023) and cross-modal transformers (Brohan et al., 2023b; Guhur et al., 2022) for instruction-following manipulation. However, these methods learn manipulation policies from top-down 2D images and are limited to pick-and-place primitives (Hansen et al., 2023). In contrast, by leverag-
+
+ing 3D perception, the robot is able to take into account object orientations, occlusions, and collisions in complex manipulation tasks. Recent methods utilize 3D representations, such as voxel patches (James et al., 2022b; Shridhar et al., 2022b), point clouds (Chen et al., 2023; Zhang et al., 2023b; Eisner et al., 2022), and feature fields (Gervet et al., 2023), to achieve accurate 3D localizations for action prediction. Another line of research utilizes multi-view images to represent the projections of a 3D environment onto image planes, significantly reducing the computation requirements (Liu et al., 2023a; Seo et al., 2023; Goyal et al., 2023). Our method lies in multi-view architectures and leverages pre-trained foundation models to enhance generalization across various visual scenarios and task descriptions. The technique of action chunking is also employed in 2D manipulation (Bharadhwaj et al., 2023; Zhao et al., 2023a), while we extend it to 3D scenarios using multi-channel heatmaps.
+
+Foundation Models for Embodied Agents. Large Language Models (LLMs) (Touvron et al., 2023; Hu et al., 2023a), Vision Language Models (VLMs) (Liu et al., 2023b; Li et al., 2023), and vision foundation models (Radford et al., 2021) have demonstrated remarkable capabilities (Akyurek et al., 2023) and hold great promise for solving complex embodied tasks. The chain-of-thought capacity (Wei et al., 2022) of LLMs has been effectively utilized in task planning for embodied agents, including EmbodiedGPT (Mu et al., 2023), ReAct (Yao et al., 2023), SayCan (Ahn et al., 2022), and DoReMi (Guo et al., 2023). The commonsense knowledge within LLMs can serve as a world model (Zhao et al., 2023b; Hao et al., 2023; Lin et al., 2023) in text-based environments. Additionally, it can be utilized as a reward designer, as demonstrated by VoxPoser (Huang et al., 2023), Text2Reward (Xie et al., 2024), and Eureka (Ma et al., 2024). GenSim (Wang et al., 2024) and RoboGen (Wang et al., 2023a) leverage LLMs to generate task curricula and simulation environments to augment robot data. VLMs are commonly employed as foundation models for embodied policies, taking visual observations and language instructions as inputs, and generating language plans (Driess et al., 2023) or tokenized actions (Brohan et al., 2023a; Wu et al., 2023) as outputs. Other approaches utilize VLMs for reward generation (Rocamonde et al., 2024) in RL frameworks and self-reflection for task planning (Hu et al., 2023b). RoboFlamingo (Li et al., 2024) is related to our method as it employs OpenFlamingo as a base policy and finetunes this policy using embodied datasets. However, it is limited to 2D manipulation and lacks explicit consideration of 3D geometry, which hinders its capacity to develop highly accurate spatial manipulation skills in robotics.
+
+Segment Anything Model. SAM (Kirillov et al., 2023) is a promptable segmentation model capable of generating masks by receiving various prompts, including points,
+
+bounding boxes, and language prompts. Subsequent works have examined the application of SAM for object localization (Zhang et al., 2023a), tracking (Rajic et al., 2023; Cheng et al., 2023), and semantic analysis (Mazurowski et al., 2023). For embodied agents, SAM-G (Wang et al., 2023b) is a concurrent work that utilizes point prompts to establish correspondences and employs SAM to generate masked images for the agent. However, SAM-G focuses on extracting the agent-relevant mask for robust visual representations and mitigating the impact of noise (e.g., colors, backgrounds) in 2D manipulation and locomotion tasks. In contrast, our method adopts SAM to enhance 3D manipulation within a multi-view framework and extracts task-relevant features to facilitate generalization across various manipulation scenarios and language instructions.
+
+# 5. Experiments
+
+In this section, we evaluate SAM-E in RLBench (James et al., 2020), which is a challenging multi-task 3D manipulation benchmark. To perform a fair comparison to baselines, we use the same settings as the state-of-art method (Goyal et al., 2023) by using 18 tasks with 249 variations in experiments. Moreover, we evaluate the generalization ability of SAM-E via few-shot adaptation in 6 new tasks. The Videos are available at: https://sam-embodied.github.io/.
+
+# 5.1. Experiment Setup
+
+Baselines. We compare SAM-E against off-the-shelf algorithms proved to work on multi-view 3D manipulation, including (i) RVT (Goyal et al., 2023), the state-of-the-art multi-view architecture for 3D manipulation by re-rendering visual observations into orthographic projections of cube views and predicting the next move based on these projections; (ii) PerAct (Shridhar et al., 2022b), an action-centric method that encodes RGB-D images into voxel grid patches for 3D representation and predicts the action within the 3D voxel space. (iii) We include R3M (Nair et al., 2023), the visual representation designed for robot manipulation, as an alternative encoder in our architecture. (iv) We include two more general visual representations CLIP (Radford et al., 2021), DINO (Caron et al., 2021) in our architecture. (v) We include a variant referred to SAM  $\rightarrow$  RVT that replaces the SAM encoder with RVT's visual encoder, which is trained from scratch. (vi) Since RVT has been shown to significantly outperform other behavior cloning (BC) baselines including CNN-BC, ViT-BC (Jang et al., 2021), and Coarse-to-Fine BC (James et al., 2022b), we do not include the scores of these methods and we refer to Goyal et al. (2023) for details. (vii) Additionally, we compare SAE-E against Hiveformer (Guhur et al., 2022) with same tasks evaluated in their paper (we refer to §E for the results).
+
+![](images/f7263b6567980cd7f4452b15eef590f90f84e7183fa87724fed8fcc4a8714b9d.jpg)  
+Figure 5. The comparison of training curves from 5 seeds with  $\pm 1$  std. We observe that SAM-E achieves a higher success rate than R3M and non-pre-trained baselines. Meanwhile, SAM and its variations achieve a better training efficiency compared to RVT, benefiting from action sequence imitation. The training curve of RVT is from our reproduction by running the official code.
+
+Table 1. A comparison of trainable parameters to baselines.  
+
+<table><tr><td>Models</td><td>RVT</td><td>SAM-E (SAM → RVT)</td><td>SAM-E (SAM → R3M)</td><td>SAM-E</td></tr><tr><td>Trainable Para.</td><td>36.3M</td><td>35.6M</td><td>35.6M</td><td>35.7M</td></tr></table>
+
+Simulation Environment. We perform experiments in RL-Bench (James et al., 2020), which is simulated by CoppeliaSim (Rohmer et al., 2013) to control a Franka Panda robot equipped with a parallel gripper. Visual observations are captured by four RGB-D cameras (left shoulder, right shoulder, front, and wrist) with a resolution of  $128 \times 128$ , and target gripper pose is achieved by a sample-based motion planner. In this elaborated simulator, the agent is tested to complete the task within a limited number of timesteps, which is 25 in experiments. The tasks include picking and placing items, executing staged moves for tool usage, and comprehending scenes to solve puzzles (see §A for more detailed descriptions of the tasks). The algorithms are evaluated in a multi-task and multi-modal setting, characterized by a high degree of variation, which necessitates the agent to demonstrate scene understanding, instruction comprehension, and precise action prediction.
+
+Training Datasets. We utilize the same training datasets as RVT and PerAct, comprising 100 expert demonstrations per task. Unlike RVT and PerAct, which slice demonstration episodes into keyframe transitions with empirically crucial duplication for important transitions, we seamlessly decompose demonstrations into multiple sub-episodes of keyframes to facilitate action-sequence prediction. We train SAM-E for 60K steps and choose the last model for evaluation, which is the same as RVT. We use cosine learning rate decay after 2K steps warm-start (see §B for more details).
+
+![](images/ecd3c8b58cb919c98ddfd45a264b85bff125bf4396058ce944274de6e15f6a4e.jpg)  
+Figure 6. An illustration of the execution efficiency in several tasks. SAM-E completes most tasks in merely once or twice inferences in all success cases. We refer to §D for more examples.
+
+# 5.2. Main Experiments
+
+Multi-Task Learning. We train all methods in 18 tasks and the comparison of success rate is given in Table 2. SAM-E outperforms PerAct and RVT in 14 out of 18 tasks. SAM-E outperforms PerAct and RVT by an average of  $21.2\%$  and  $7.7\%$  percentage points in success rate across 18 tasks, marking a relative improvement with  $43.0\%$  and  $12.2\%$ , while incurring significantly lower model inference costs. Furthermore, it achieves an improvement exceeding  $30\%$  points in several tasks. Eliminating the pre-trained SAM encoder in SAM-E leads to a performance drop but still outperforms RVT, benefiting from the action sequence policy head. Building upon this, the addition of R3M's frozen representation has yielded a marginal performance improvement, however, which is still inferior compared to SAM-E. Similarly, CLIP and DINO representations have mediocre performances compared to SAM-E. Notably, SAM-E has comparable training time and even less trainable parameters compared to RVT, as shown in Table 1. Moreover, Figure 5 shows that SAM-E and its variations exhibit higher training efficiency than RVT, mainly attributed to the action sequence imitation. Further, utilizing SAM as the scalable visual foundation, SAM-E not only achieves the best performance on the current setup, but also shows potential for further enhancing its advantages with more embodied data or update steps.
+
+Different from baselines that predict the next keypoint gripper pose at each timestep, SAM-E generates a sequence of actions for long-term planning and sequential execution, thereby considering the task completion from a higher perspective and has much fewer inference steps. According to Table 2, SAM-E demonstrates an average execution efficiency of more than  $5\mathbf{X}$  greater than that of RVT. In tasks such as meat_off_grill, push Buttons, and put-money_inSAFE (see  $\S A$  for task descriptions), SAM-E can complete the task after merely a glance at the initial state, as shown in Figure 6. In contrast, RVT requires, on average, 5.5, 3.8, and 6.0 steps to complete them for its successful cases. For
+
+Table 2. Multi-task Performances. SAM-E outperforms state-of-the-art methods in most tasks and on average, with much fewer inference steps in execution. Scores of PerAct and RVT are adopted from Goyal et al. (2023). Mean and std of 5 evaluations are reported.  
+
+<table><tr><td>Models</td><td>Put in 
+Drawer</td><td>Reach 
+ and Drag</td><td>Turn 
+ Tap</td><td>Slide to 
+ Target</td><td>Open 
+Drawer</td><td>Put in 
+ Cupboard</td><td>Place in 
+ Shape Sorter</td><td>Put Money 
+ in Safe</td><td>Push 
+ Buttons</td><td>Close 
+ Jar</td></tr><tr><td>PerAct</td><td>51.2±4.7</td><td>89.6±4.1</td><td>88.0±4.4</td><td>74.0±13.0</td><td>88.0±5.7</td><td>28.0±4.4</td><td>16.8±4.7</td><td>84.0±3.6</td><td>92.8±3.0</td><td>55.2±4.7</td></tr><tr><td>RVT</td><td>88.0±5.7</td><td>99.2±1.6</td><td>93.6±4.1</td><td>81.6±5.4</td><td>71.2±6.9</td><td>49.6±3.2</td><td>36.0±2.5</td><td>91.2±3.0</td><td>100.0±0.0</td><td>52.0±2.5</td></tr><tr><td>SAM-E (SAM → RVT)</td><td>87.2±5.9</td><td>100.0±0.0</td><td>100.0±0.0</td><td>79.2±6.6</td><td>95.2±3.3</td><td>59.2±5.2</td><td>35.2±4.4</td><td>72.0±4.0</td><td>98.4±2.2</td><td>83.2±5.9</td></tr><tr><td>SAM-E (SAM → R3M)</td><td>83.2±5.9</td><td>99.2±1.8</td><td>100.0±0.0</td><td>88.8±4.4</td><td>95.2±3.3</td><td>41.6±7.3</td><td>31.2±7.7</td><td>95.2±3.3</td><td>96.0±0.0</td><td>78.4±2.2</td></tr><tr><td>SAM-E (SAM → CLIP)</td><td>88.8±3.3</td><td>100.0±0.0</td><td>100.0±0.0</td><td>78.4±13.4</td><td>92.0±4.0</td><td>40.0±4.9</td><td>42.4±6.1</td><td>80.8±1.8</td><td>100.0±0.0</td><td>73.6±2.2</td></tr><tr><td>SAM-E (SAM → DINO)</td><td>78.4±4.6</td><td>99.2±1.8</td><td>99.2±1.8</td><td>88.0±4.9</td><td>89.6±5.4</td><td>52.0±7.5</td><td>30.4±9.2</td><td>85.6±2.2</td><td>100.0±0.0</td><td>89.6±3.6</td></tr><tr><td>SAM-E (ours)</td><td>92.0±5.7</td><td>100.0±0.0</td><td>100.0±0.0</td><td>95.2±1.8</td><td>95.2±5.2</td><td>64.0±2.8</td><td>34.4±6.1</td><td>95.2±3.3</td><td>100.0±0.0</td><td>82.4±3.6</td></tr><tr><td>Models</td><td>Stack 
+Blocks</td><td>Place 
+Cups</td><td>Place Wine 
+at Rack</td><td>Screw 
+Bulb</td><td>Sweep to 
+Dustpan</td><td>Insert 
+Peg</td><td>Meat off 
+Grill</td><td>Stack 
+Cups</td><td>On 
+Average</td><td>Inference 
+Steps(Sum)</td></tr><tr><td>PerAct</td><td>26.4±3.2</td><td>2.4±3.2</td><td>44.8±7.8</td><td>17.6±2.0</td><td>52.0±0.0</td><td>5.6±4.1</td><td>70.4±2.0</td><td>2.4±2.0</td><td>49.4</td><td>-</td></tr><tr><td>RVT</td><td>28.8±3.9</td><td>4.0±2.5</td><td>91.0±5.2</td><td>48.0±5.7</td><td>72.0±0.0</td><td>11.2±3.0</td><td>88.0±2.5</td><td>26.4±8.2</td><td>62.9</td><td>6158±64</td></tr><tr><td>SAM-E (SAM → RVT)</td><td>22.4±3.6</td><td>0.0±0.0</td><td>92.8±6.6</td><td>61.6±9.2</td><td>84±0.0</td><td>7.2±5.9</td><td>95.2±3.3</td><td>3.2±3.3</td><td>65.3±0.6</td><td>1190±19</td></tr><tr><td>SAM-E (SAM → R3M)</td><td>32.0±2.8</td><td>1.6±2.2</td><td>92.8±3.3</td><td>60.0±2.8</td><td>96.8±3.3</td><td>5.6±6.7</td><td>97.6±2.2</td><td>2.4±2.2</td><td>66.5±1.0</td><td>1165±63</td></tr><tr><td>SAM-E (SAM → CLIP)</td><td>22.4±10.8</td><td>0.0±0.0</td><td>93.6±2.2</td><td>59.2±4.4</td><td>85.6±2.2</td><td>8.0±2.8</td><td>96.0±4.0</td><td>4.8±3.3</td><td>64.8±0.9</td><td>1192±17</td></tr><tr><td>SAM-E (SAM → DINO)</td><td>28.8±7.7</td><td>0.8±1.8</td><td>93.6±3.6</td><td>64.0±9.8</td><td>100.0±0.0</td><td>11.2±3.3</td><td>96.0±2.8</td><td>1.6±2.2</td><td>67.1±0.4</td><td>1143±15</td></tr><tr><td>SAM-E (ours)</td><td>26.4±4.6</td><td>0.0±0.0</td><td>94.4±4.6</td><td>78.4±3.6</td><td>100.0±0.0</td><td>18.4±4.6</td><td>95.2±3.3</td><td>0.0±0.0</td><td>70.6±0.7</td><td>1130±12</td></tr></table>
+
+Table 3. Few-shot adaptation. Mean and std of 5 evaluations are reported.  
+
+<table><tr><td>Models</td><td>Meat on Grill</td><td>Open Jar</td><td>Screw Nail</td><td>Toilet Seat Done</td><td>TV on</td><td>Solve Puzzle</td><td>On Average</td></tr><tr><td>RVT (from scratch)</td><td>80.0±6.3</td><td>36.0±4.9</td><td>7.2±4.4</td><td>99.2±1.8</td><td>2.4±3.6</td><td>11.2±4.4</td><td>39.3±2.3</td></tr><tr><td>SAM-E (from scratch, SAM → RVT)</td><td>60.0±2.8</td><td>12.0±0.0</td><td>36.0±6.9</td><td>96.0±0.0</td><td>15.2±3.3</td><td>20.8±7.7</td><td>40.0±1.8</td></tr><tr><td>SAM-E (from scratch, SAM → R3M)</td><td>69.6±4.6</td><td>16.0±0.0</td><td>29.6±6.1</td><td>100.0±0.0</td><td>12.0±2.8</td><td>22.4±2.2</td><td>41.6±1.4</td></tr><tr><td>SAM-E (from scratch, SAM → CLIP)</td><td>64.0±0.0</td><td>12.0±0.0</td><td>16.0±4.0</td><td>100.0±0.0</td><td>14.7±6.1</td><td>24.0±4.0</td><td>38.4±1.0</td></tr><tr><td>SAM-E (from scratch, SAM → DINO)</td><td>53.3±8.3</td><td>12.0±4.0</td><td>26.7±2.3</td><td>100.0±0.0</td><td>16.0±4.0</td><td>24.0±4.0</td><td>38.7±1.2</td></tr><tr><td>SAM-E (from scratch)</td><td>75.2±4.4</td><td>12.8±1.8</td><td>28.0±8.0</td><td>100.0±0.0</td><td>20.8±1.8</td><td>17.6±2.2</td><td>42.4±1.5</td></tr><tr><td>RVT (adaptation)</td><td>68.8±3.3</td><td>36.0±0.0</td><td>1.6±2.2</td><td>100.0±0.0</td><td>1.6±2.2</td><td>14.4±6.7</td><td>37.1±1.0</td></tr><tr><td>SAM-E (adaptation, SAM → RVT)</td><td>69.6±6.1</td><td>39.2±3.3</td><td>38.4±4.6</td><td>99.2±1.8</td><td>17.6±2.2</td><td>38.4±3.6</td><td>50.4±1.1</td></tr><tr><td>SAM-E (adaptation, SAM → R3M)</td><td>64.8±5.9</td><td>37.6±2.2</td><td>28.8±5.9</td><td>100.0±0.0</td><td>12.8±1.8</td><td>37.6±6.7</td><td>46.9±2.3</td></tr><tr><td>SAM-E (adaptation, SAM → CLIP)</td><td>78.7±2.3</td><td>38.7±4.6</td><td>28.0±6.9</td><td>100.0±0.0</td><td>16.0±0.0</td><td>25.3±8.3</td><td>47.8±1.9</td></tr><tr><td>SAM-E (adaptation, SAM → DINO)</td><td>68.0±4.0</td><td>33.3±6.1</td><td>50.7±8.3</td><td>98.7±2.3</td><td>24.0±4.0</td><td>22.7±2.3</td><td>49.6±1.0</td></tr><tr><td>SAM-E (adaptation)</td><td>84.0±5.7</td><td>56.0±7.5</td><td>62.4±4.6</td><td>100.0±0</td><td>35.2±1.8</td><td>41.6±7.3</td><td>63.2±1.5</td></tr></table>
+
+reach_anddrag, SAM-E completes it all in two inferences while RVT needs to execute 6.4 times on average.
+
+Few-Shot Adaptation. We evaluate the generalization ability of SAM-E by adapting the trained model to 6 new tasks from RLBench. We use 10X fewer demonstrations and 15X fewer update steps in policy adaptation than that of the multi-task experiments to show the generalization capability of the SAM-E in few-shot adaptation. The results are shown in Table 3. We initialize the models with weights from their multi-task training for adaptation, and also introduce their random initialization variants for training from scratch. We find RVT struggles with transferring knowledge from previous tasks to new ones during adaptation, often resulting in performance drops compared to training from scratch. In contrast, SAM-E significantly benefits from adaptation compared to starting from scratch. Specifically, SAM-E outperforms RVT by  $3.1\%$  points (a  $7.9\%$  relative increase) when trained from scratch. However, during adaptation to new tasks, the performance gap widens dramatically, with SAM-E surpassing RVT by  $26.1\%$  points, a substantial  $70.4\%$  relative improvement. This demonstrates that SAM-E has superior generalization capabilities.
+
+When training from scratch, SAM-E (SAM  $\rightarrow$  R3M) achieves a slightly better performance than SAM-E (SAM  $\rightarrow$  RVT) that does not have a pre-trained encoder, but results in worse performance in adaptation, which shows R3M has limited few-shot generalization ability. While worse than SAM-E (SAM  $\rightarrow$  R3M) in training from scratch, SAM-E (SAM  $\rightarrow$  CLIP) and SAM-E (SAM  $\rightarrow$  DINO) have better performances in adaptation, showing greater generalization of the representations pre-trained in more general image data. SAM-E (SAM  $\rightarrow$  RVT) also significantly outperforms RVT in adaptation over from scratch, demonstrating the enhanced generalization ability gained from the action-sequence prediction. In terms of adapting to new tasks, SAM-E equipped with a SAM encoder demonstrates significant advantages over the methods mentioned above. This highlights the exceptional capabilities of SAM-E to generalize in novel task descriptions.
+
+# 5.3. Ablations
+
+First, we conduct ablation experiments in multi-task experiments to verify the necessity of components in SAM-E. We
+
+Table 4. Success rate and parameters amount of the variations  
+
+<table><tr><td>Models</td><td>Success Rate</td><td>Parameters</td><td>Trainable Parameters</td></tr><tr><td>SAM-E</td><td>70.6±0.7</td><td>122.5M</td><td>35.7M</td></tr><tr><td>SAM-E (SAM→RVT)</td><td>65.3±0.6</td><td>35.6M</td><td>35.6M</td></tr><tr><td>SAM-E (LoRA, QKV)</td><td>69.2±0.9</td><td>122.6M</td><td>35.8M</td></tr><tr><td>SAM-E (w/o LoRA)</td><td>67.2±1.0</td><td>122.4M</td><td>35.6M</td></tr><tr><td>SAM-E (full finetune)</td><td>65.8±1.0</td><td>122.4M</td><td>122.4M</td></tr></table>
+
+include (i) SAM-E  $(SAM\rightarrow RVT)$ ; (ii) SAM-E (LoRA, QKV), which is a variant of LoRA module additionally including  $K$  matrix of attention blocks; (iii) SAM-E (w/o LoRA), a frozen SAM encoder without LoRA fine-tuning, and (iv) SAM-E (full finetune), which performs full-parameter training of the SAM encoder. We give the brief result in Table 4. We find SAM is a crucial visual foundation and a suitable finetune method is required for adaptation to embodied scenarios. Using LoRA to parameter-efficiently finetuning, SAM is better than the variant that trains all parameters, which may lead to failure due to the limited demonstrations. For LoRA, adding the trainable matrix for  $Q$  and  $V$  is better than all  $Q$ ,  $K$ , and  $V$ , which is consistent with previous observations (Hu et al., 2022). (See §F for the complete results)
+
+Additionally, to illustrate the impact of the action sequence length  $h$  (refer to §B.2 for details), we conduct an ablation study on the action horizon, examining  $h$  values of  $\{1,3,5,7\}$ . During both the training and evaluation execution of the multi-task experiments, we modify the action horizon  $h$  while maintaining consistency in other experimental settings. The outcomes are presented in Table 5 (See §F for the complete results), showing the average success rate across 18 tasks and the computing time for each model inference on our same device during the model evaluation. We observe that  $h = 5$  performs the best on the average success rate, while it may not suitable for certain tasks. We can also find that  $h = 1$  leads to a drop in performance, which we attribute to the insufficient temporal information to drive SAM foundation training, combined with the lack of empirically crucial duplication for important transitions. Moreover, we can observe that SAM-E's inference time is slightly longer than that of RVT. Nevertheless, SAM-E is even faster in inference considering an action sequence (5 actions) is predicted in 152ms, while RVT requires 5*103ms to predict 5 actions.
+
+# 5.4. Real-World Experiment
+
+To demonstrate the effectiveness of SAM-E in real-world scenarios, we train and test the model in a real-world setup with a Franka Panda robot arm. As shown in Figure 14, we use two statically mounted RGB-D cameras in a third-person view at the left front and right front to capture the multi-view observation. We calibrate the cameras with the robot base and record the RGB-D streams from the cameras and robot joint pose simultaneously during the data
+
+Table 5. Ablation over action sequence length  $h$  
+
+<table><tr><td>Models</td><td>Success Rate</td><td>Inference Time (ms)</td></tr><tr><td>RVT</td><td>62.9</td><td>103</td></tr><tr><td>SAM-E (h=1)</td><td>30.6±1.4</td><td>126</td></tr><tr><td>SAM-E (h=3)</td><td>64.0±0.6</td><td>144</td></tr><tr><td>SAM-E (h=5)</td><td>70.6±0.7</td><td>152</td></tr><tr><td>SAM-E (h=7)</td><td>66.5±1.2</td><td>156</td></tr></table>
+
+collection. We train SAM-E in 5 tasks with 10 episodes for each, including put the towel on the cabinet, stack the block, close the drawer, pick up the banana, and put the orange into the drawer. All the episodes are collected by human demonstrators. The results show that SAM-E can perform real-time prediction in real-world scenarios and complete tasks effectively, validating SAM-E's capability in real-world scenarios. See the §G and the videos for more details and model performance.
+
+# 6. Conclusion
+
+We have introduced Segment Anything Model for Embodied 3D manipulation (SAM-E), a novel multi-view architecture that adopts SAM as the visual foundation model with parameter-efficient finetuning for promptable perception to embodied scenarios, as well as a novel action-sequence prediction head for efficient planning and coherent execution. We conduct experiments of SAM-E on various 3D instruction-following tasks from RLBench for multi-task experiments and few-show adaptation. We find SAM-E outperforms prior state-of-the-art models on multi-task manipulation and achieves a significant improvement in execution efficiency and few-shot adaptation with great generalization ability. Our work highlights the feasibility of leveraging a visual foundation model and sequence prediction for enhancing generalization and efficiency in 3D manipulation.
+
+# Acknowledgements
+
+This work was supported by the National Natural Science Foundation of China (Nos.62306242 & 62376222), the STI 2030-Major Projects under Grant 2021ZD0201404, the National Key R&D Program of China (No.2022ZD0160102), and Young Elite Scientists Sponsorship Program by CAST (No.2023QNRC001). We thank Wenke Xia for his excellent assistance in hardware deployment and data collection for real robot experiments.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
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+
+# A. RLBench Tasks
+
+We follow the multi-task multi-variation simulated experiments setting of RVT (Goyal et al., 2023) and PerAct (Shridhar et al., 2022b) with 18 RLBench tasks (shown in Figure 7) and 249 unique variations across object placement, color, size, category, count, and shape. Here we give a summary of the 18 RLBench tasks in Table 6. The extra 6 RLBench tasks (shown in Figure 8) for the few-shot adaptation experiment are summarized in Table 7.
+
+![](images/5e6e0d0c63569938c552f3756e577a60b5c12118ac8886e409e7cd0d91017078.jpg)
+
+![](images/8aeb970ae54c15b0cb49c27c777837eb8501c36bdc9c157c86249337e7a122ed.jpg)  
+Figure 7. Language-Conditioned Manipulation Tasks in RLBench. We conduct multi-task experiments on 18 simulated tasks in RLBench(James et al., 2020). Apart from the language instruction depicted in the figures, there are a total of 249 variations of these tasks, as illustrated in Table 6. During the test, the agent needs to handle the novel object poses, randomly sampled goals, and randomly sampled scenes with different semantic instantiations of object colors, shapes, sizes, and categories within a maximum of 25 execution steps.
+
+Table 6. The 18 RLBench tasks for multi-task experiment  
+
+<table><tr><td>Task name</td><td>Language Template</td><td>Avg. Keyframes</td><td>#of Variations</td><td>Variation Type</td></tr><tr><td>put in drawer</td><td>“put the item in the __ drawer”</td><td>12.0</td><td>3</td><td>placement</td></tr><tr><td>reach and drag</td><td>“use the stick to drag the cube onto the __ target”</td><td>6.0</td><td>20</td><td>color</td></tr><tr><td>turn tap</td><td>“turn __ tap”</td><td>2.0</td><td>2</td><td>placement</td></tr><tr><td>slide to target</td><td>“slide the block to __ target”</td><td>4.7</td><td>4</td><td>color</td></tr><tr><td>open drawer</td><td>“open the __ drawer”</td><td>3.0</td><td>3</td><td>placement</td></tr><tr><td>put in cupboard</td><td>“put the __ in the cupboard”</td><td>5.0</td><td>9</td><td>category</td></tr><tr><td>place in shape sorter</td><td>“put the __ in the shape sorter”</td><td>5.0</td><td>5</td><td>shape</td></tr><tr><td>put money in safe</td><td>“put the money away in the safe on the __ shelf”</td><td>5.0</td><td>3</td><td>placement</td></tr><tr><td>push buttons</td><td>“push the __ button, [then the __ button]”</td><td>3.8</td><td>50</td><td>color</td></tr><tr><td>close jar</td><td>“close the __ jar”</td><td>6.0</td><td>20</td><td>color</td></tr><tr><td>stack block</td><td>“stack __ blocks”</td><td>14.6</td><td>60</td><td>color,count</td></tr><tr><td>place cups</td><td>“place __ cups on the cup holder”</td><td>11.5</td><td>3</td><td>count</td></tr><tr><td>place wine at rack</td><td>“stack the wine bottle to the __ of the rack”</td><td>5.0</td><td>3</td><td>placement</td></tr><tr><td>screw bulb</td><td>“screw in the __ light bulb”</td><td>7.0</td><td>20</td><td>color</td></tr><tr><td>sweep to dustpan</td><td>“sweep dirt to the __ dustpan”</td><td>4.6</td><td>2</td><td>size</td></tr><tr><td>insert peg</td><td>“put the ring on the __ spoke”</td><td>5.0</td><td>20</td><td>color</td></tr><tr><td>meat off grill</td><td>“take the __ off the grill”</td><td>5.0</td><td>2</td><td>category</td></tr><tr><td>stack cups</td><td>“stack the other cups on top of the __ cup”</td><td>10.0</td><td>20</td><td>color</td></tr></table>
+
+![](images/e856c64d6a5d2f126d8d9714ac267eadd690767cd899a799fd7b9e0b1fe80029.jpg)  
+Figure 8. Language-Conditioned Manipulation Tasks in RLBench. We conduct few-shot adaptation experiments on 6 simulated tasks in RLBench to evaluate the generalization ability of SAM-E. Task variations are shown in Table 7. The tasks must be completed by the agent within a maximum of 25 steps.
+
+# B. Implementation Details
+
+In this section, we provide more implementation details of SAM-E.
+
+# B.1. Visual Input
+
+In our experiments of RLBench, the visual observations are captured by four cameras (left shoulder, right shoulder, front, and wrist) with a resolution of  $128 \times 128$  in RGB-D. We follow the re-render approach introduced by RVT (Goyal et al.,
+
+Table 7. The 6 RLBench tasks used for the few-shot adaptation experiments.  
+
+<table><tr><td>Task name</td><td>Language Template</td><td>Avg. Keyframes</td><td>#of Variations</td><td>Variation Type</td></tr><tr><td>meat on grill</td><td>“put the _ on the grill”</td><td>5.0</td><td>2</td><td>category</td></tr><tr><td>open jar</td><td>“open the — jar”</td><td>6.0</td><td>20</td><td>color</td></tr><tr><td>screw nail</td><td>“screw the nail in to the block”</td><td>6.0</td><td>1</td><td>-</td></tr><tr><td>toilet seat down</td><td>“toilet seat down”</td><td>4.7</td><td>1</td><td>-</td></tr><tr><td>tv on</td><td>“turn on the TV”</td><td>8.0</td><td>1</td><td>-</td></tr><tr><td>solve puzzle</td><td>“solve the puzzle”</td><td>5.0</td><td>1</td><td>-</td></tr></table>
+
+2023) before feeding visual images to the model. Specifically, the RGB-D images are rerendered to generate virtual images in the form of cube orthographic projection. Then we use the cube orthographic projections as the visual inputs of SAM-E.
+
+# B.2. Action Sequence Imitation
+
+We utilize a multi-channel action sequence policy head to predict the action sequence, trained by action sequence imitation. To extract the temporal information of actions from the expert demonstrations, we employ the keyframe extraction on each demonstration, generating a dataset of keyframe sequences. Given observations, SAM-E generates an action sequence with a default action horizon of 5 and is trained to maximize the likelihood objective of imitation learning. Note that the action sequence data may have variable lengths, when the data is shorter than the action horizon, we mask the untrained action head, and when the data is longer, we truncate it accordingly.
+
+# B.3. Hyperparameters
+
+In our experiments, the hyperparameters are primarily fixed, as shown in Table 8.
+
+Table 8. Training Hyperparameters  
+
+<table><tr><td>Hyperparameters</td><td>Multi-task Training</td><td>Few-shot adaptation</td></tr><tr><td>batch size</td><td>10</td><td>10</td></tr><tr><td>learning rate</td><td>4e-3</td><td>4e-3</td></tr><tr><td>optimizer</td><td>LAMB</td><td>LAMB</td></tr><tr><td>learning rate schedule</td><td>cosine decay</td><td>cosine decay</td></tr><tr><td>warmup steps</td><td>2000</td><td>2000</td></tr><tr><td>training steps</td><td>60K</td><td>4K</td></tr><tr><td>training epochs</td><td>15</td><td>1</td></tr></table>
+
+# C. Visualization
+
+We visualize the attention map of the multi-view transformer to show SAM-E's various attention patterns for task comprehension and action sequence prediction. We use task put_item_inDrawer as an example, which is completed by three executions.
+
+(i) In the first execution with the initial observation (see Figure 9), SAM-E's attention, from one of its heads, is predominantly focused on the Franka robot, the drawer cabinet, and more specifically, the item on the cabinet and the handle of the top drawer. This observation aligns with the given instruction to 'put the item in the top drawer', highlighting SAM-E's capability to identify key objects within the scene according to the task description for task execution.  
+(ii) In the second inference, following an action sequence that results in the opening of the top drawer, SAM-E adapts its focus. It now observes the newly available space within the drawer for placing the item (see Figure 10). Concurrently, another of its attention heads redirects back to the end-effector and the item, strategizing the subsequent action of picking up and placing the item into the drawer(see Figure 11).
+
+(iii) In the final inference (see Figure 12), SAM-E concentrates on the end-effector picking up the item and positioning it accurately into the target position. This phase likely involves precise adjustments and movements, ensuring the successful completion of the task as the language instruction.
+
+![](images/e725283172e5c40a62b06d1c8033bf06a5b252be65b8bce7a10a2946b9f319a1.jpg)  
+Top View
+
+![](images/f1aecd87c39d451a9f57bd153831b6c356d7d7374668fc0d0de85f36801e5eae.jpg)  
+Back View
+
+![](images/575660c7b75886e41b02fbf88bff24571dddace365cdf93fb04084f3a7df1f4a.jpg)  
+Front View
+
+![](images/6697930a4eceed4f7f0c330d0e88b7622fad2143160f9b56f0850dba4737c066.jpg)  
+Right View
+
+![](images/95f4abe92be176af7c4ec5f8f60adbdfa7f4ef0259d4b8f5a0d8bf4ccd27cd46.jpg)  
+Left View
+
+![](images/4458f797733d82761c67e0cc8178da6165e22df1c785a80ee706ca9f260b1024.jpg)  
+Top View
+
+![](images/7e10518695a89855c7755b5fe76220665073cfd9a81ea41197366abfd1812096.jpg)  
+Figure 9. SAM-E's multi-view attention map of the initial inference.  
+Back View
+
+![](images/883187a8f2d07b6fa5b35866dc289b80702fe02bc4e72fd5b2371087331d4188.jpg)  
+Front View
+
+![](images/054e70f822fa63e3071d88203113cf6521c91d5cffbfd027a8f9b72f8f8d9193.jpg)  
+Right View
+
+![](images/a99e7381398a50dd43649970b88d950e52c87752fc0ac78835a9638004f79466.jpg)  
+Left View
+
+![](images/495aa7b2d16ebc8ee118e5bc0232b1e3caf2251c95f6cc227f1a69c6b86a0ce3.jpg)  
+Top View
+
+![](images/447c40f1093e83517737cba5ed20287f109aed031508771d0d86a2931ffe876a.jpg)  
+Figure 10. SAM-E's multi-view attention map of the second inference, focusing on the open drawer.  
+Back View
+
+![](images/ca8dd4f97f2a6e580407b390d88df366fe9e740c581985257e78e7d66b8971dd.jpg)  
+Front View
+
+![](images/e4e2b257081dc422e842c2bd3e018b443986c792c3e44ff5a4a433c1a9475665.jpg)  
+Right View
+
+![](images/a1c6dd6762197111254af0e6070729efdcd48e61a5aa473a04a8e7293cd439aa.jpg)  
+Left View
+
+![](images/13ea9e560d985718f48f063554cfbc94471832baf23342eacbf2bbf32fb5e215.jpg)  
+Figure 11. SAM-E's multi-view attention map of the second inference, focusing on the end-effector and the item.  
+Top View
+
+![](images/e82c58d6e23f9c70124a8a5c077be56a8cd03925dbe0364f9617a4f753e1f943.jpg)  
+Back View  
+Figure 12. SAM-E's multi-view attention map of the last inference.
+
+![](images/618a96b8940ee408ff0f21f5958d11f787cc8b5bb26c6f3a2ad73cce1c4968c4.jpg)  
+Front View
+
+![](images/c4bcc83ff8f1c76d49f9f8b59158f8b90980510e6b398ea435c3ffd74eed433e.jpg)  
+Right View
+
+![](images/2d648f6cd4938a3488a022c58d0c00a84d0276a9b6386afb387ffe466f5353c8.jpg)  
+Left View
+
+# D. One Glance Results
+
+Figure 13 shows the results of execution times of SAM-E on success cases of several tasks. Thanks to its promptable perception and efficient action sequence prediction, SAM-E excels in task completion by executing actions coherently, resulting in improved performance and significantly reduced inference requirements. For the following tasks, in comparison, RVT requires an average of [6.4, 6.0, 3.8, 5.5, 3.7, 4.3, 5.5, 5.0, 4.8] execution times of its success cases.
+
+![](images/2c547244d1d58a217daf984b3f93b407f7c6e6abcaf5b68b38cd189efc0a8138.jpg)  
+Figure 13. The comparison of execution times on success cases of several tasks.
+
+# E. Comparison with Hiveformer
+
+To compare the performance of SAM-E and HiverFormer, we add experiments to train SAM-E with the same 10 tasks evaluated in the Hiveformer paper with 100 demonstrations per task, which is the same as Hiveformer (results are shown in Table 9). The score of Hiveformer is adapted from their original paper. We remark that SAM-E is trained with 10 tasks with all variations, which is much more challenging than Hiveformer which is trained with a unique variation for each task.
+
+Table 9. Comparison with Hiveformer. Scores of Hiveformer are adopted from Guhur et al. (2022). Mean and std of 5 evaluations are reported.  
+
+<table><tr><td>Models</td><td>Pick and Lift</td><td>Pick up Cup</td><td>Put Knife on Board</td><td>Reach Target</td><td>Stack Wine</td><td>Take Money out Safe</td><td>Take Umbrella out Stand</td><td>Push Buttons</td><td>Put Money in Safe</td><td>Slide to Target</td><td>On Average</td></tr><tr><td>Hiveformer</td><td>88.9</td><td>92.9</td><td>75.3</td><td>100.0</td><td>71.2</td><td>79.1</td><td>89.2</td><td>100.0</td><td>58.2</td><td>78.7</td><td>83.3</td></tr><tr><td>SAM-E</td><td>87.2±1.8</td><td>88.8±5.2</td><td>68.0±4.0</td><td>100.0±0.0</td><td>69.6±7.3</td><td>98.4±2.2</td><td>96.0±2.8</td><td>100.0±0.0</td><td>93.6±2.2</td><td>84.8±7.7</td><td>88.6±0.7</td></tr></table>
+
+# F. Ablation
+
+We provide the complete results of the ablation study in Table 10 and Table 11.
+
+# G. Real-World Experiments
+
+We conduct real-world experiments on a FranKa Panda robot arm in the real world, equipped with a dual RGB-D camera setup positioned at the left front and right front for multi-view observation, shown in Figure 14. We construct the real-world scene and design 5 tasks for experiments, including put the towel on the cabinet, stack the block, close the drawer, pick up
+
+Table 10. Ablation Performances of SAM-E's variations. Mean and std of 5 evaluations are reported.  
+
+<table><tr><td>Models</td><td>Put in 
+Drawer</td><td>Reach 
+ and Drag</td><td>Turn 
+ Tap</td><td>Slide to 
+ Target</td><td>Open 
+Drawer</td><td>Put in 
+ Cupboard</td><td>Place in 
+ Shape Sorter</td><td>Put Money 
+ in Safe</td><td>Push 
+ Buttons</td><td>Close 
+ Jar</td></tr><tr><td>SAM-E</td><td>92.0±5.7</td><td>100.0±0.0</td><td>100.0±0.0</td><td>95.2±1.8</td><td>95.2±5.2</td><td>64.0±2.8</td><td>34.4±6.1</td><td>95.2±3.3</td><td>100.0±0.0</td><td>82.4±3.6</td></tr><tr><td>SAM-E (LoRA, QKV)</td><td>88.8±7.7</td><td>100.0±0.0</td><td>100.0±0.0</td><td>90.4±6.1</td><td>94.4±3.6</td><td>57.6±4.6</td><td>37.6±5.4</td><td>92.8±5.2</td><td>100.0±0.0</td><td>78.4±4.6</td></tr><tr><td>SAM-E (w/o LoRA)</td><td>84.8±7.2</td><td>100.0±0.0</td><td>100.0±0.0</td><td>92.0±4.0</td><td>92.8±3.3</td><td>52.0±2.8</td><td>31.2±5.2</td><td>92.8±1.8</td><td>98.4±2.2</td><td>87.2±5.2</td></tr><tr><td>SAM-E (full finetune)</td><td>93.3±4.6</td><td>98.7±2.3</td><td>100.0±0.0</td><td>69.3±11.5</td><td>90.7±2.3</td><td>52.0±0.0</td><td>29.3±11.5</td><td>89.3±2.3</td><td>100.0±0.0</td><td>68.0±0.0</td></tr><tr><td>Models</td><td>Stack 
+Blocks</td><td>Place 
+Cups</td><td>Place Wine 
+at Rack</td><td>Screw 
+Bulb</td><td>Sweep to 
+Dustpan</td><td>Insert 
+Peg</td><td>Meat off 
+Grill</td><td>Stack 
+Cups</td><td>On 
+Average</td><td>Inference 
+Steps(Sum)</td></tr><tr><td>SAM-E</td><td>26.4±4.6</td><td>0.0±0.0</td><td>94.4±4.6</td><td>78.4±3.6</td><td>100.0±0.0</td><td>18.4±4.6</td><td>95.2±3.3</td><td>0.0±0.0</td><td>70.6±0.7</td><td>1130±12</td></tr><tr><td>SAM-E (LoRA, QKV)</td><td>32.0±4.9</td><td>3.2±1.8</td><td>93.6±3.6</td><td>69.6±4.6</td><td>98.4±2.2</td><td>6.4±6.1</td><td>97.6±2.2</td><td>5.6±2.2</td><td>69.2±0.9</td><td>1142±6</td></tr><tr><td>SAM-E (w/o LoRA)</td><td>20.8±7.2</td><td>0.0±0.0</td><td>92.0±4.0</td><td>64.0±7.5</td><td>96.8±1.8</td><td>8.8±6.6</td><td>94.4±3.6</td><td>0.8±1.8</td><td>67.2±1.0</td><td>1182±6</td></tr><tr><td>SAM-E (full finetune)</td><td>20.0±6.9</td><td>1.3±2.3</td><td>90.7±2.3</td><td>69.3±4.6</td><td>100.0±0.0</td><td>0.0±0.0</td><td>98.7±2.3</td><td>13.3±4.6</td><td>65.8±1.0</td><td>1204±18</td></tr></table>
+
+Table 11. Ablation Performances with different action sequence length. Mean and std of 5 evaluations are reported.  
+
+<table><tr><td>Models</td><td>Put in 
+Drawer</td><td>Reach 
+ and Drag</td><td>Turn 
+ Tap</td><td>Slide to 
+ Target</td><td>Open 
+Drawer</td><td>Put in 
+ Cupboard</td><td>Place in 
+ Shape Sorter</td><td>Put Money 
+ in Safe</td><td>Push 
+ Buttons</td><td>Close 
+ Jar</td></tr><tr><td>SAM-E (h=1)</td><td>0.0±0.0</td><td>6.7±2.3</td><td>98.7±2.3</td><td>45.3±4.6</td><td>72.0±6.9</td><td>8.0±4.0</td><td>14.7±2.3</td><td>8.0±0.0</td><td>69.3±2.3</td><td>12.0±4.0</td></tr><tr><td>SAM-E (h=3)</td><td>77.6±2.2</td><td>84.8±1.8</td><td>100.0±0.0</td><td>72.8±1.8</td><td>92.0±2.8</td><td>31.2±5.2</td><td>35.2±3.3</td><td>84.8±5.2</td><td>99.2±1.8</td><td>73.6±3.6</td></tr><tr><td>SAM-E (h=5)</td><td>92.0±5.7</td><td>100.0±0.0</td><td>100.0±0.0</td><td>95.2±1.8</td><td>95.2±5.2</td><td>64.0±2.8</td><td>34.4±6.1</td><td>95.2±3.3</td><td>100.0±0.0</td><td>82.4±3.6</td></tr><tr><td>SAM-E (h=7)</td><td>88.8±9.5</td><td>99.2±1.8</td><td>100.0±0.0</td><td>80.8±18.6</td><td>90.4±4.6</td><td>53.6±7.3</td><td>28.8±3.3</td><td>92.8±1.8</td><td>100.0±0.0</td><td>72.8±3.3</td></tr><tr><td>Models</td><td>Stack 
+Blocks</td><td>Place 
+Cups</td><td>Place Wine 
+at Rack</td><td>Screw 
+Bulb</td><td>Sweep to 
+Dustpan</td><td>Insert 
+Peg</td><td>Meat off 
+Grill</td><td>Stack 
+Cups</td><td>On 
+Average</td><td>Inference 
+Steps(Sum)</td></tr><tr><td>SAM-E (h=1)</td><td>0.0±0.0</td><td>1.3±2.3</td><td>40.0±6.9</td><td>58.7±2.3</td><td>24.0±4.0</td><td>34.7±14.0</td><td>54.7±4.6</td><td>2.7±4.6</td><td>30.6±1.4</td><td>8329±60</td></tr><tr><td>SAM-E (h=3)</td><td>16.8±3.3</td><td>1.6±2.2</td><td>76.8±4.4</td><td>49.6±6.7</td><td>87.2±1.8</td><td>54.4±2.2</td><td>100.0±0.0</td><td>13.6±3.6</td><td>64.0±0.6</td><td>2026±30</td></tr><tr><td>SAM-E (h=5)</td><td>26.4±4.6</td><td>0.0±0.0</td><td>94.4±4.6</td><td>78.4±3.6</td><td>100.0±0.0</td><td>18.4±4.6</td><td>95.2±3.3</td><td>0.0±0.0</td><td>70.6±0.7</td><td>1130±12</td></tr><tr><td>SAM-E (h=7)</td><td>13.6±5.4</td><td>3.2±3.3</td><td>92.0±4.0</td><td>70.4±5.4</td><td>100.0±0.0</td><td>8.8±4.4</td><td>97.6±3.6</td><td>4.8±1.8</td><td>66.5±1.2</td><td>919±12</td></tr></table>
+
+the banana, and put the orange into the drawer. For data collection, we manually control the robot arm for demonstrations by a controller and collect the RGB-D stream and robot joint pose simultaneously with a data collection pipeline. We collect demonstrations with variations in item placement for all tasks. See https://sam-embodied.github.io/ for videos and performance.
+
+![](images/fa0d069caf264a442a63b7db76889c95fbd81a9e4f44639d087f1ee38ffb2136.jpg)  
+Real World Scene  
+Put the towel on the cabinet
+
+![](images/07680444ad958b85cffb8d08946db637cf9011fef1e49d58d90e5ac498d3a7e5.jpg)
+
+![](images/1de23717871d9ae273d8b2106d112e93dbcc142e414c4e8cad697acaba9e0884.jpg)  
+Stack the block
+
+![](images/b44d97621adc1f66e322e8fa5065ebc413a0b763c09204e6ef1f2e1e9895d7dd.jpg)  
+Close the drawer  
+Figure 14. Real-World Scene and tasks.
+
+![](images/ba1e942d07a5c580744712bb0a0b9af6766c6735b5a5daf0c6687507db4c16bb.jpg)  
+Pick up the banana  
+Put the orange into the drawer
+
+![](images/30e06c107dbccda8d7e83715b859d8c5db4635070e67d465ea4745d781ef72e8.jpg)
+
+# H. Limitation and Future Work
+
+In this work, we propose SAM-E with SAM as the visual foundation and action-sequence policy head, which outperforms prior state-of-the-art methods. However, we also identify limitations that suggest directions for future research. We employ parameter-efficient fine-tuning on relatively limited robot data to enhance its understanding of embodied manipulation. Future improvements might include leveraging the scalability of the visual foundation through training on larger datasets, such as Open-X (Collaboration, 2023). Additionally, we employed a fixed horizon for the action-sequence policy, which, while generally effective, could be less suitable for certain tasks, such as stack cups in our experiments, in which may need to pay more attention to the trade-off between precision and coherence of the action. It would be intriguing to see the action horizon optimized through a mechanism or learned from data.

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+        "type": "text",
+        "text": "Jayesh Singla  $^{*1}$  Ananye Agarwal  $^{*1}$  Deepak Pathak",
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+        "text": "Abstract",
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+        "text": "Despite extreme sample inefficiency, on-policy reinforcement learning, aka policy gradients, has become a fundamental tool in decision-making problems. With the recent advances in GPU-driven simulation, the ability to collect large amounts of data for RL training has scaled exponentially. However, we show that current RL methods, e.g. PPO, fail to ingest the benefit of parallelized environments beyond a certain point and their performance saturates. To address this, we propose a new on-policy RL algorithm that can effectively leverage large-scale environments by splitting them into chunks and fusing them back together via importance sampling. Our algorithm, termed SAPG, shows significantly higher performance across a variety of challenging environments where vanilla PPO and other strong baselines fail to achieve high performance. Webpage at https://sapg-rl.github.io.",
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+    {
+        "type": "text",
+        "text": "1. Introduction",
+        "text_level": 1,
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+        "text": "Broadly, there are two main categories in reinforcement learning (RL): off-policy RL, e.g., Q-learning (Watkins & Dayan, 1992), and on-policy RL, e.g., policy gradients (Sutton et al., 1999). On-policy methods are relatively more sample inefficient than off-policy but often tend to converge to higher asymptotic performance. Due to this reason, on-policy RL methods, especially PPO (Schulman et al., 2017), are usually the preferred RL paradigm for almost all sim2real robotic applications (Miki et al., 2022; Agarwal et al., 2022; Chen et al., 2021) to games such as StarCraft (Vinyals et al., 2019), where one could simulate years of real-world experience in minutes to hours.",
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+        "text": "RL is fundamentally a trial-n-error-based framework and hence is sample inefficient in nature. Due to this, one needs to have large batch sizes for each policy update, especially in",
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+        "type": "text",
+        "text": "*Equal contribution  ${}^{1}$  Carnegie Mellon University. Correspondence to: Ananye Agarwal <ananyea@andrew.cmu.edu>, Jayesh Singla <jsingla@andrew.cmu.edu>.",
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+        "text": "Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).",
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+        "text": "the case of on-policy methods because they can only use data from current experience. Fortunately, in recent years, the ability to simulate a large number of environments in parallel has become exponentially larger due to GPU-driven physics engines, such as IsaacGym (Makoviychuk et al., 2021), PhysX, Mujoco-3.0, etc. This means that each RL update can easily scale to batches of size hundreds of thousands to millions, which are over two orders of magnitude higher than what most RL benchmarks typically have.",
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+            "Figure 2. Performance vs batch size plot for PPO runs (blue curve) across two environments. The curve shows how PPO training runs can not take benefit of large batch size resulting from massively parallelized environments and their asymptotic performance saturates after a certain point. The dashed red line is the performance of our method, SAPG, with more details in the results section. It serves as evidence that higher performance is achievable with larger batch sizes."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "date. In Figure 2, we show that PPO performance saturates after a certain batch size despite the ceiling being higher. This is due to the issue in data sampling mechanisms. In particular, at each timestep actions are sampled from a Gaussian with some mean and variance. This implies that most sampled actions are near the mean and with large number of environments, many environments are executing the same actions leading to duplicated data. This implies that the performance of PPO saturates at some point as we increase the number of environments.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We propose a simple fix to this problem. Instead of running a single PPO policy for all environments, we divide environments into blocks. Each block optimizes a separate policy, allowing for more data diversity than just i.i.d. sampling from the same Gaussian. Next, we do an off-policy update to combine data from all these policies to keep the update consistent with the objective of on-policy RL. This allows us to use the PPO's clipped surrogate objective, maintaining the stability benefits of PPO while latching onto high reward trajectories even though they are off-policy. A schematic of our approach, termed SAPG, is shown in Figure 1. We evaluate SAPG across a variety of environments and show significantly high asymptotic performance in environments where vanilla PPO even fails to get any positive success.",
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+    {
+        "type": "text",
+        "text": "2. Related Work",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Policy gradients REINFORCE (Williams, 2004), one of the earliest policy gradient algorithms uses an estimator of the objective using simple Monte Carlo return values. Works such as (Konda & Tsitsiklis, 1999) and (Schulman et al., 2018) improve the stability of policy gradient algorithms by employing a baseline to decrease the variance of the estimator while not compromising on the bias. (Schulman et al., 2015; 2017) incorporate conservative policy updates",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "into policy gradients to increase the robustness.",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Distributed reinforcement learning Reinforcement learning algorithms are highly sample inefficient, which calls for some form of parallelization to increase the training speed. When training in simulation, this speed-up can be achieved by distributing experience collection or different parts of training across multiple processes.(Nair et al., 2015; Mnih et al., 2016; Espeholt et al., 2018; Horgan et al., 2018) However, through the introduction of GPU-based simulators such as IsaacGym (Makoviychuk et al., 2021), the capacity of simulation has increased by two to three orders of magnitude. Due to this, instead of focusing on how to parallelize parts of the algorithm, the focus has shifted to finding ways to efficiently utilize the large amount of simulation data. Previous works such as (Chen et al., 2021; Fu et al., 2022; Rudin et al., 2022; Handa et al., 2022; Agarwal et al., 2022) use data from GPU-based simulation to learn policies in complex manipulation and locomotion settings. However, most of these works still use reinforcement learning algorithms to learn a single policy, while augmenting training with techniques like teacher-student-based training and game-based curriculum. We find that using the increased simulation capacity to naively increase the batch size is not the best way to utilize massively parallel simulation.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "(Petrenko et al., 2023) develop a population-based training framework that divides the large number of environments between multiple policies and using hyperparameter mutation to find a set of hyperparameters that performs well. However, even this does not utilize all the data completely as each policy learns independently. We propose a way to ensure most of the data from the environments contributes to learning by using all collected transitions for each update.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Off-policy Policy Gradients Unlike on-policy algorithms, off-policy algorithms can reuse all collected data or data collected by any policy for their update. Most off-policy algorithms (Mnih et al., 2013; Lillicrap et al., 2019; Haarnoja et al., 2018) try to learn a value function which is then implicitly/explicitly used to learn a policy. (Li et al., 2023) developed a variant of Deep Deterministic Policy Gradient (DDPG) called PQL which splits data collection and learning into multiple processes and shows impressive performance on many benchmark tasks. We use PQL as one of our baselines to compare our method to off-policy RL in complex tasks. Although off-policy algorithms are much more data-efficient, they usually get lower asymptotic performance than on-policy policy gradients. This has inspired works to develop techniques to use off-policy data in on-policy methods. (Hastings, 1970) has been one of the major techniques used to realize this. Previous works (Degris et al., 2012; Wang et al., 2016; Espeholt et al., 2018; Fakoor et al., 2020) develop techniques to use off-policy data in on-policy",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "SAPG: Split and Aggregate Policy Gradients",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "algorithms using importance sampling-based updates along with features such as bias correction.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3. Preliminaries",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a modification to on-policy RL to achieve higher performance in the presence of large batch sizes. We build upon PPO, although our proposed ideas are generally applicable to any on-policy RL method.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "On-policy RL Let  $(S, \\mathcal{A}, \\mathcal{P}, r, \\rho, \\gamma)$  be an MDP where  $S$  is the set of states,  $\\mathcal{A}$  the set of actions,  $\\mathcal{P}$  are transition probabilities,  $r$  the reward function,  $\\rho$  the initial distribution of states and  $\\gamma$  the discount factor. The objective in reinforcement learning is to find a policy  $\\pi(a|s)$  which maximises the long term discounted reward  $\\mathcal{J}(\\pi) = \\mathbb{E}_{s_0 \\sim \\rho, a_t \\sim \\pi(\\cdot|s_t)}\\left[\\sum_{t=0}^{T-1} \\gamma^t r(s_t, a_t)\\right]$ .",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Policy-gradient algorithms (Williams, 2004; Konda & Tsitsiklis, 1999; Schulman et al., 2015; Mnih et al., 2016) optimize the policy using gradient descent with Monte Carlo estimates of the gradient",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\nabla_ {\\theta} J (\\pi_ {\\theta}) = \\underset {s \\sim \\rho_ {d}, a \\sim \\pi (\\cdot | s)} {\\mathbb {E}} \\left[ \\nabla_ {\\theta} \\log (\\pi_ {\\theta} (a)) \\hat {A} ^ {\\pi_ {\\theta}} (s, a) \\right] \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $\\hat{A}^{\\pi_{\\theta}}(s,a)$  is an advantage function that estimates the contribution of the transition to the gradient. A common choice is  $\\hat{A}^{\\pi_{\\theta}}(s,a) = \\hat{Q}^{\\pi_{\\theta}}(s,a) - \\hat{V}^{\\pi_{\\theta}}(s)$ , where  $\\hat{Q}^{\\pi_{\\theta}}(s,a)$ ,  $\\hat{V}^{\\pi_{\\theta}}(s)$  are estimated  $Q$  and value functions. This form of update is termed as an actor-critic update (Konda & Tsitsiklis, 1999). Since we want the gradient of the error with respect to the current policy, only data from the current policy (on-policy) data can be utilized.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "PPO Actor critic updates can be quite unstable because gradient estimates are high variance and the loss landscape is complex. An update step that is too large can destroy policy performance. Proximal Policy Optimization (PPO) modifies Eq. 1 to restrict updates to remain within an approximate \"trust region\" where there is guaranteed improvement (Schulman et al., 2015; Kakade & Langford, 2002).",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} L _ {o n} \\left(\\pi_ {\\theta}\\right) = \\underset {\\pi_ {o l d}} {\\mathbb {E}} \\left[ \\min  \\left(r _ {t} \\left(\\pi_ {\\theta}\\right), \\right. \\right. \\tag {2} \\\\ \\operatorname {c l i p} \\left(r _ {t} (\\pi_ {\\theta}), 1 - \\epsilon , 1 + \\epsilon)\\right) A _ {t} ^ {\\pi_ {\\text {o l d}}} ] \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            748,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here,  $r_t(\\pi_\\theta) = \\frac{\\pi_\\theta(a_t|s_t)}{\\pi_{old}(a_t|s_t)}$ ,  $\\epsilon$  is a clipping hyperparameter and  $\\pi_{old}$  is the policy collecting the on-policy data. The clipping operation ensures that the updated  $\\pi$  stays close to  $\\pi_{old}$ . Empirically, given large numbers of samples, PPO achieves high performance, is stable and robust to hyperparameters. However, it was developed for relatively small",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "batch sizes ( $\\approx 100$  parallel envs). We find that in the large-scale setting ( $>10k$  envs), it is suboptimal because many parallel envs are sampling nearly identical on-policy data.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4. Split and Aggregate Policy Gradients",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Policy gradient methods are highly sensitive to the variance in the estimate of gradient. Since CPU-based simulators typically run only 100s of environments in parallel, conventional wisdom is to simply sample on-policy data from a Gaussian policy in all the environments since as the number of datapoints increases, the Monte Carlo estimate becomes more accurate. However, this intuition no longer holds in the extremely large-scale data setting where we have hundreds of thousands of environments on GPU-accelerated simulators like IsaacGym. IID sampling from a Gaussian policy will lead to most actions lying near the mean, and most environments will execute similar actions, leading to wasted data (fig. 2).",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose to efficiently use large numbers of  $N$  environments using a divide-and-conquer setup. Our algorithm trains a variety of  $M$  policies  $\\pi_1, \\ldots, \\pi_M$  instead of having just one policy. However, simply training multiple policies by dividing environments between them is also inefficient. This is equivalent to training an algorithm with different seeds and choosing the best seed. One approach is to add hyperparameter mutation (Petrenko et al., 2023) to the policies and choosing the hyperparameters that perform the best among all of them. However, even in this case, all of the data from the \"worse\" policies goes to waste, and the only information gained is that some combinations of hyperparameters are bad, even though the policies themselves may have discovered high reward trajectories. We need to somehow aggregate data from multiple policies into a single update. We propose to do this via off-policy updates.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.1. Aggregating data using off-policy updates",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "One of the major drawbacks of on-policy RL is its inability to use data from past versions of the policy. One solution is to use importance sampling (Degris et al., 2012; Meng et al., 2023) to weight updates using data from different policies. In practice, this is not used since given limited compute it is beneficial to sample on-policy experience that is more directly relevant. However, this is no longer true in the large batch setting where enough on-policy data is available. In this case, it becomes advantageous to have multiple policies  $\\pi_1, \\ldots, \\pi_M$  and use them to sample diverse data, even if it is off-policy. In particular, to update policy  $\\pi_i$  using data from policy  $\\pi_j$ ,  $j \\in \\mathcal{X}$  we use (Meng et al., 2023)",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nL _ {o f f} \\left(\\pi_ {i}; \\mathcal {X}\\right) = \\frac {1}{| \\mathcal {X} |} \\sum_ {j \\in \\mathcal {X}} \\underset {(s, a) \\sim \\pi_ {j}} {\\mathbb {E}} \\left[ \\min  \\left(r _ {\\pi_ {i}} (s, a), \\right. \\right. \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {c l i p} \\left(r _ {\\pi_ {i}} (s, a), \\mu (1 - \\epsilon), \\mu (1 + \\epsilon))\\right) A ^ {\\pi_ {i, o l d}} (s, a) ]\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            891,
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+            909
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "SAPG: Split and Aggregate Policy Gradients",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/5cebd23bf812b26cb226ed2855d560c362c1ceb2932f1dec0fb6191d47f2007f.jpg",
+        "image_caption": [
+            "Figure 3. We illustrate one particular variant of SAPG which performs well. There is one leader and  $M - 1$  followers ( $M = 3$  in figure). Each policy has the same backbone with shared parameters  $B_{\\theta}$  but is conditioned on local learned parameters  $\\phi_i$ . Each policy gets a block of  $\\frac{N}{M}$  environments to run. The leader is updated with its on-policy data as well as importance-sampled off-policy data from the followers. Each of the followers only uses their own data for on-policy updates."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $r_{\\pi_i}(s,a) = \\frac{\\pi_i(s,a)}{\\pi_j(s,a)}$  and  $\\mu$  is an off-policy correction term  $\\mu = \\frac{\\pi_{i,old}(s,a)}{\\pi_j(s,a)}$ . Note that when  $i = j$ , then  $\\pi_j = \\pi_{i,old}$  and this reduces to the on-policy update as expected. This is then scaled and combined with the on-policy term (eq. 2)",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nL \\left(\\pi_ {i}\\right) = L _ {o n} \\left(\\pi_ {i}\\right) + \\lambda \\cdot L _ {o f f} \\left(\\pi_ {i}; \\mathcal {X}\\right) \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            158,
+            483,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The update target for the critic is calculated using  $n$ -step returns (here  $n = 3$ ).",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nV _ {o n, \\pi_ {j}} ^ {\\text {t a r g e t}} (s _ {t}) = \\sum_ {k = t} ^ {t + 2} \\gamma^ {k - t} r _ {k} + \\gamma^ {3} V _ {\\pi_ {j}, o l d} (s _ {t + 3}) \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "However, this is not possible for off-policy data. Instead, we assume that an off-policy transition can be used to approximate a 1-step return. The target equations are as follows",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nV _ {o f f, \\pi_ {j}} ^ {\\text {t a r g e t}} \\left(s _ {t} ^ {\\prime}\\right) = r _ {t} + \\gamma V _ {\\pi_ {j}, o l d} \\left(s _ {t + 1} ^ {\\prime}\\right) \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The critic loss is then",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nL _ {o n} ^ {\\text {c r i t i c}} \\left(\\pi_ {i}\\right) = \\underset {(s, a) \\sim \\pi_ {i}} {\\mathbb {E}} \\left[ \\left(V _ {\\pi_ {i}} (s) - V _ {o n, \\pi_ {i}} ^ {\\text {t a r g e t}} (s)\\right) ^ {2} \\right] \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            109,
+            732,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nL _ {o f f} ^ {\\text {c r i t i c}} \\left(\\pi_ {i}; \\mathcal {X}\\right) = \\frac {1}{| \\mathcal {X} |} \\sum_ {j \\in \\mathcal {X}} \\underset {(s, a) \\sim \\pi_ {j}} {\\mathbb {E}} \\left[ \\left(V _ {\\pi_ {i}} (s) - V _ {o f f, \\pi_ {i}} ^ {\\text {t a r g e t}} (s)\\right) ^ {2} \\right] \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            84,
+            779,
+            483,
+            829
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nL ^ {\\text {c r i t i c}} \\left(\\pi_ {i}\\right) = L _ {\\text {o n}} ^ {\\text {c r i t i c}} \\left(\\pi_ {i}\\right) + \\lambda \\cdot L _ {\\text {o f f}} ^ {\\text {c r i t i c}} \\left(\\pi_ {i}\\right) \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given this update scheme, we must now choose a suitable  $\\mathcal{X} \\subseteq \\{1, \\ldots, M\\}$  and the set of  $i$ s to update, along with the correct ratio  $\\lambda$ . We explore several variants below.",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2. Symmetric aggregation",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A simple choice is to update all  $i' s$  with the data from all policies. In this case, we choose to update each policy  $i \\in \\{1, 2, \\dots, M\\}$  and for each  $i$  use off-policy data from all other policies  $\\mathcal{X} = \\{1, 2, i - 1, i + 1, \\dots, M\\}$ . Since gradients from off-policy data are typically noisier than gradients from on-policy data, we choose  $\\lambda = 1$ , but subsample the off-policy data such that we use equal amounts of on-policy and off-policy data.",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.3. Leader-follower aggregation",
+        "text_level": 1,
+        "bbox": [
+            496,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "While the above choice prevents data wastage, since all the policies are updated with the same data, it can lead to policies converging in behavior, reducing data diversity and defeating the purpose of having separate policies. To resolve this, we break symmetry by designating a \"leader\" policy  $i = 1$  which gets data from all other policies  $\\mathcal{X} = \\{2,3,\\dots,M\\}$  while the rest are \"followers\" and only use their own on-policy data for updates  $\\mathcal{X} = \\phi$ . As before, we choose  $\\lambda = 1$ , but subsample the off-policy data for the leader such that we use equal amounts of on-policy and off-policy data in a mini-batch update.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.4. Encouraging diversity via latent conditioning",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "What is the right parameterization for this set of policies? One simple choice is to have a disjoint set of parameters for each with no sharing at all. However, this implies that each follower policy has no knowledge of any other policy whatsoever and may get stuck in a bad local optimum. We mitigate this by having a shared backbone  $B_{\\theta}$  for each policy conditioned on hanging parameters  $\\phi_j$  local to each",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "SAPG: Split and Aggregate Policy Gradients",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/e4d34a3bf6dd21e14a99efee58a1e5d8f2fc32bc3bbb5e00cd6c0c6baafb16a3.jpg",
+        "image_caption": [
+            "Figure 4. Two data aggregation schemes we consider in this paper. (Left) one policy is a leader and uses data from each of the followers (Right) a symmetric scheme where each policy uses data from all others. In each case, the policy also uses its own on-policy data."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "policy. Similarly, the critic consists of a shared backbone  $C_{\\psi}$  conditioned on parameters  $\\phi_j$ . The parameters  $\\psi, \\theta$  are shared across the leader and all followers and updated with gradients from each objective, while the parameters  $\\phi_j$  are only updated with the objective for that particular policy. We choose  $\\phi_j \\in \\mathbb{R}^{32}$  for complex environments while  $\\phi_j \\in \\mathbb{R}^{16}$  for the relatively simpler ones.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.5. Enforcing diversity through entropy regularization",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To further encourage diversity between different policies, in addition to the PPO update loss  $L_{on}$  we add an entropy loss to each of the followers with different coefficients. In particular, the entropy loss is  $\\mathcal{H}(\\pi(a|s))$ . The overall loss for the policy  $i$  (or the  $(i-1)$ th) follower is  $L(\\pi_i) = L_{on}(\\pi_i) + \\sigma(i-1) \\cdot \\mathcal{H}(\\pi(a|s))$ . The leader doesn't have any entropy loss. Different scales of coefficients produce policies with different explore-exploit tradeoffs. Followers with large entropy losses tend to explore more actions even if they are suboptimal, while those with small coefficients stay close to optimal trajectories and refine them. This leads to a large data coverage with a good mix of optimal as well as diverse trajectories. We treat  $\\sigma$  as a hyperparameter.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.6. Algorithm: SAPG",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We roll out  $M$  different policies and collect data  $\\mathcal{D}_1, \\ldots, \\mathcal{D}_M$  for each. Follower policies  $2, \\ldots, M$  are updated using the usual PPO objective with minibatch gradient descent on their respective datasets. However, we augment the dataset of the leader  $\\mathcal{D}_1$  with data from  $\\mathcal{D}_2, \\ldots, \\mathcal{D}_M$ , weighed by the importance weight  $\\mu$ . The leader is then updated by minibatch gradient descent as well.",
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+    },
+    {
+        "type": "text",
+        "text": "5. Experimental Setup",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We conduct experiments on 5 manipulation tasks (3 hard and 2 easy) and compare them against SOTA methods for the large-scale parallelized setting. We use a GPU-accelerated simulator, IsaacGym (Makoviychuk et al., 2021) which allows simulating tens of thousands of environments in par",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 SAPG",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "code",
+        "sub_type": "algorithm",
+        "code_caption": [],
+        "code_body": "Initialize shared parameters  $\\theta, \\psi$   \nFor  $i \\in \\{1, \\dots, M\\}$  initialize parameters  $\\phi_i$   \nInitialize  $N$  environments  $E_1, \\dots, E_N$ .  \nInitialize data buffers for each policy  $\\mathcal{D}_1, \\dots, \\mathcal{D}_M$ .  \nfor  $i = 1, 2, \\dots, \\mathbf{do}$   \nfor  $j = 1, 2, \\dots, M$  do  \n $\\mathcal{D}_j \\gets \\text{COLLECTDATA}(E_{j\\frac{N}{M}:(j+1)\\frac{N}{M}}, \\theta, \\psi_j)$   \nend for  \n $L \\gets 0$   \nSample  $|\\mathcal{D}_1|$  transitions from  $\\cup_{j=2}^M \\mathcal{D}_i$  to get  $\\mathcal{D}_1'$ .  \n $L \\gets L + \\text{OFFPOLICYLOSS}(\\mathcal{D}_1')$ $L \\gets L + \\text{ONPOLICYLOSS}(\\mathcal{D}_1)$   \nfor  $j = 2, \\dots, M$  do  \n $L \\gets L + \\text{ONPOLICYLOSS}(\\mathcal{D}_j)$   \nend for  \nUpdate  $\\theta \\gets \\theta - \\eta \\nabla_\\theta L$   \nUpdate  $\\psi \\gets \\psi - \\eta \\nabla_\\psi L$   \nend for=0",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "allele on a single GPU. In our experiments, we focus on the large-scale setting and simulate 24576 parallel environments unless otherwise specified. Note that this is two orders of magnitude larger than the number of environments PPO was developed on (Schulman et al., 2017), and we indeed find that vanilla PPO does not scale to this setting.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For testing, we choose a suite of manipulation environments that are challenging and require large-scale data to learn effective policies (Petrenko et al., 2023). In particular, these consist of dexterous hands mounted on arms leading to high numbers of degrees of freedom (up to 23). This is challenging because sample complexity scales exponentially with degrees of freedom. They are under-actuated and involve manipulating free objects in certain ways while under the influence of gravity. This leads to complex, non-linear interactions between the agent and the environment such as contacts between robot and object, object and table, and robot and table. Overall, this implies that to learn effective policies an agent must collect a large amount of relevant experience and also use it efficiently for learning.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.1. Tasks",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We consider a total of 5 tasks grouped into two parts: Three hard tasks and two easy tasks. Hard and easy is defined by the success reward achieved by off-policy (in particular, PQL) methods in these environments. In easy environments, even Q-learning-based off-policy methods can obtain nonzero performance but not in hard tasks. See appendix sec. A.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Hard Difficulty Tasks All three hard tasks are based on the Allegro-Kuka environments(Petrenko et al., 2023). These consist of an Allegro Hand (16 DoF) mounted on",
+        "bbox": [
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+    },
+    {
+        "type": "header",
+        "text": "SAPG: Split and Aggregate Policy Gradients",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "a Kuka arm (7 dof). The performance of the agent in the above three tasks is measured by the successes metric which is defined as the number of successes in a single episode. Three tasks include:",
+        "bbox": [
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+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Regrasping: The object must be lifted from the table and held near a goal position  $\\mathbf{g}_t \\in \\mathbb{R}^3$  for  $K = 30$  steps. This is called a \"success\". The target position and object position are reset to a random location after every success.",
+            "- Throw: The object must be lifted from the table and thrown into a bucket at  $\\mathbf{g}_t \\in \\mathbb{R}^3$  placed out of reach of the arm. The bucket and the object position are reset randomly after every successful attempt.",
+            "- Reorientation: Pick up the object and reorient it to a particular target pose  $\\mathbf{g}_{\\mathbf{t}} \\in \\mathbb{R}^{7}$  (position + orientation). The target pose is reset once the agent succeeds. This means that the agents needs to reorient the object in different poses in succession, which may sometimes entail placing the objects on the table and lifting it up in a different way."
+        ],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Easy Difficulty Tasks: In addition, we test on the following dexterous hand tasks. As before, the observation space consists of the joint angles and velocities  $\\mathbf{q}_t$ ,  $\\dot{\\mathbf{q}}_t$ , object pose  $\\mathbf{x}_t$  and velocities  $\\mathbf{v}_t, \\omega_t$ .",
+        "bbox": [
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+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Shadow Hand: We test in-hand reorientation task of a cube using the 24-DoF Shadow Hand.",
+            "- Allegro Hand: This is the same as the previous in-hand reorientation task but with the 16-DoF Allegro Hand."
+        ],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.2. Baselines",
+        "text_level": 1,
+        "bbox": [
+            84,
+            527,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We test against state-of-the-art RL methods designed for the GPU-accelerated large-scale setting we consider in this paper. We compare against both on-policy (Petrenko et al., 2023) and off-policy (Li et al., 2023) variants as well as vanilla PPO (Schulman et al., 2017).",
+        "bbox": [
+            84,
+            551,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- PPO (Proximal Policy Optimization) (Schulman et al., 2017): In our setting, we just increase the data throughput for PPO by increasing the batch size proportionately to the number of environments. In particular, we see over two orders of magnitude increase in the number of environments (from 128 to 24576).",
+            "- **Parallel Q-Learning** (Li et al., 2023) A parallelized version of DDPG with different mixed exploration i.e. varying exploration noise across environments to further aid exploration. We use this baseline to compare if off-policy methods can outperform on-policy methods when the data collection capacity is high.",
+            "- DexPBT (Petrenko et al., 2023) A framework that combines population-based training with PPO.  $N$  Environments are divided into  $M$  groups, each containing  $\\frac{N}{M}$  environments.  $M$  separate policies are trained using PPO in each group of environments with different hyperparameters. At regular intervals, the worst-performing policies"
+        ],
+        "bbox": [
+            84,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "are replaced with the weights of best-performing policies and their hyperparameters are mutated randomly.",
+        "bbox": [
+            509,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Due to the complexity of these tasks, experiments take about 48-60 hours on a single GPU, collecting  $\\approx 2\\mathrm{e}10$  transitions. Since we run experiments on different machines, the wall clock time is not directly comparable and we compare runs against the number of samples collected. We run 5 seeds for each experiment and report the mean and standard error in the plots. In each plot, the solid line is  $y(t) = \\frac{1}{n}\\sum_{i}y_{i}(t)$  while the width of the shaded region is determined by standard error  $\\frac{2}{\\sqrt{n}}\\sum_{i}(y(t) - y_{i}(t))^{2}$ .",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For each task, we use  $M = 6$  policies for our method and DexPBT in a total of  $N = 24576$  environments for each method. We use the dimension of learned parameter  $\\phi_j \\in \\mathbb{R}^{32}$  for the AllegroKuka tasks while we use  $\\phi_j \\in \\mathbb{R}^{16}$  for the ShadowHand and AllegroHand tasks since they are relatively simpler. We use a recurrent policy for the AllegroKuka tasks and an MLP policy for the Shadow Hand and Allegro Hand tasks and use PPO to train them. We collect 16 steps of experience per instance of the environment before every PPO update step. For SAPG, we tune the entropy coefficient  $\\sigma$  by choosing the best from a small set  $\\{0, 0.003, 0.005\\}$  for each environment. We find that  $\\sigma = 0$  works best for all ShadowHand, AllegroHand, Regrasping, and Throw while  $\\sigma = 0.005$  works better for Reorientation.",
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+    {
+        "type": "text",
+        "text": "6. Results and Analysis",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the large-scale data setting, we are primarily concerned with optimality while sample-efficiency and wall-clock time are secondary concerns. This is because data is readily available—one only needs to spin up more GPUs, what is really important is how well our agent performs in the downstream tasks. Indeed, this aligns with how practitioners use RL algorithms in practice (Agarwal et al., 2022; Cheng et al., 2023; OpenAI et al., 2018), where agents are trained with lots of domain randomization in large simulations and the primary concern is how well the agent can adapt and learn in these environments since this directly translates to real-world performance.",
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+    {
+        "type": "text",
+        "text": "6.1. AllegroKuka tasks",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The AllegroKuka tasks (Throw, Regrasping, and Reorientation) are hard due to large degrees of freedom. The environment also offers the possibility of many emergent strategies such as using the table to reorient the cube, or using gravity to reorient the cube. Therefore, a large amount of data is required to attain good performance on these tasks. Following Petrenko et al. (2023) we use the number of successes as a performance metric on these tasks. Note that the DexPBT baseline directly optimizes for success by mutating the re",
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+    {
+        "type": "header",
+        "text": "SAPG: Split and Aggregate Policy Gradients",
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+        "text": "6",
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+    {
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+        "page_idx": 6
+    },
+    {
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+        "img_path": "images/ff17e4a10a71b3ef6e9aea8bf9dda69aeb568dcb6b610fbb64114582653190b9.jpg",
+        "image_caption": [],
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+    },
+    {
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+        "img_path": "images/5a4b040336a30f4995220d2a3d55da24573d54a09bd67cbf9125807894a37ef6.jpg",
+        "image_caption": [
+            "Figure 5. Performance curves of SAPG with respect to PPO, PBT and PQL baselines. On AllegroKuka tasks, PPO and PQL barely make progress and SAPG beats PBT. On Shadow Hand and Allegro Kuka Reorientation, SAPG performs best with an entropy coefficient of 0.005 while the coefficient is 0 for other environments. On ShadowHand and AllegroHand, while PQL is initially more sample efficient, SAPG is more performant in the longer run. AllegroKuka environments use successes as a performance metric while AllegroHand and ShadowHand use episode rewards."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/7b692dd47eb1ab5d1dc1fb1c8d519925315657e4bb66e193164c48bd9fe73081.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>ALGORITHM</td><td>ALLEGROHAND</td><td>SHADOWHAND</td><td>REGRASPING</td><td>THROW</td><td>REORIENTATION</td></tr><tr><td>PPO (SCHULMAN ET AL., 2017)</td><td>1.01e4±6.31e2</td><td>1.07e4±4.90e2</td><td>1.25 ± 1.15</td><td>16.8 ± 0.48</td><td>2.85 ± 0.05</td></tr><tr><td>PBT (PETRENKO ET AL., 2023)</td><td>7.28e3±1.24e3</td><td>1.01e4±1.80e2</td><td>31.9 ± 2.26</td><td>19.2 ± 1.07</td><td>23.2 ± 4.86</td></tr><tr><td>PQL (LI ET AL., 2023)</td><td>1.01e4±5.28e2</td><td>1.28e4±1.25e2</td><td>2.73 ± 0.02</td><td>2.62 ± 0.08</td><td>1.66 ± 0.11</td></tr><tr><td>SAPG (OURS, COEF=0)</td><td>1.23e4 ± 3.29e2</td><td>1.17e4±2.64e2</td><td>35.7 ± 1.46</td><td>23.7 ± 0.74</td><td>33.2 ± 4.20</td></tr><tr><td>SAPG (OURS, COEF=0.005)</td><td>9.14e3±8.38e2</td><td>1.28e4 ± 2.80e2</td><td>33.4 ± 2.25</td><td>18.7 ± 0.43</td><td>38.6 ± 0.63</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1. Performance after 2e10 samples for different methods with standard error. This is measured by successes for the AllegroKuka tasks and by episode rewards for in-hand reorientation tasks. Across environments, we find that our method performs better than baselines.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ward scales to achieve higher success rate, whereas our method can only optimize a fixed reward function. Despite this, we see that SAPG achieves a  $12 - 66\\%$  higher success rate than DexPBT on regrasping, throw and reorientation. SAPG performs  $66\\%$  better than PBT on the challenging reorientation task. Note that vanilla PPO and PQL are unable to learn any useful behaviors on these hard tasks.",
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+    },
+    {
+        "type": "text",
+        "text": "6.2. In-hand reorientation",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The AllegroHand and ShadowHand reorientation tasks from Li et al. (2023) are comparatively easier since they have lower degrees of freedom and the object doesn't move around much and remains inside the hand. On these tasks, we observe that PQL and PPO are able to make significant progress. In particular, we find that PQL is very sample-efficient because it is off-policy and utilizes past data for updates. However, we find that SAPG achieves higher asymp-",
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+    {
+        "type": "text",
+        "text": "totic performance. This is because on-policy methods are better at latching onto high reward trajectories and do not have to wait several iterations for the Bellman backup to propagate back to initial states. As discussed previously, in large-scale settings in simulation, we are primarily concerned with asymptotic performance since we want to maximize the downstream performance of our agents (within a reasonable training time budget). We see that on Allegro-Hand, SAPG beats PQL by a  $21\\%$  margin, while on the ShadowHand task it achieves comparable performance. On these tasks, both PBT and PPO generally perform worse. This is because PPO is not able to efficiently leverage the large batch size. PBT loses the benefit of its hyperparameter mutation because the environment is simpler and the default hyperparameters work well, so it roughly reduces to simple PPO in  $\\frac{N}{M}$  environments.",
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+        "text": "SAPG: Split and Aggregate Policy Gradients",
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+            "Figure 6. Performance curves for ablations of our method. The variants of our method with a symmetric aggregation scheme or without an off-policy combination perform significantly worse. Entropy regularization affects performance across environments, giving a benefit in reorientation. Using a high off-policy ratio without subsampling data leads to worse performance on ShadowHand and AllegroHand."
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+        "text": "6.3. Ablations",
+        "text_level": 1,
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+        "text": "The core idea behind SAPG is to combine data from different policies instead of optimizing a single policy with an extremely large batch. In this section, we will analyze our specific design choices for how we combine data (choice of  $i$  and  $\\mathcal{X}$  and  $\\lambda$ ) and for how we enforce diversity among the data collected by the policies. In particular, we have the following variants",
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+            "- SAPG (with entropy coef) As discussed in sec. 5.2, here we add an entropy loss to the followers to encourage data diversity. We explore different choices for the scaling coefficient of this loss  $\\sigma \\in \\{0, 0.005, 0.003\\}$ .",
+            "- SAPG (high off-policy ratio) In SAPG, when updating the leader, we subsample the off-policy data from the followers such that the off-policy dataset size matches the on-policy data. This is done because off-policy data is typically noisier and we do not want to drown out the gradient from on-policy data. In SAPG with a high off-policy ratio, we remove the subsampling step and instead see the impact of computing the gradient on the entire combined off-policy + on-policy dataset.",
+            "- Ours (symmetric) In SAPG, we choose  $i = 1$  to be the \"leader\" and the rest are \"followers\". Only the leader receives off-policy data while the followers use the standard on-policy loss. A natural alternative is where there are no privileged policies and each policy is updated with off-policy data from all others as discussed in sec. 4.2."
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+        "type": "text",
+        "text": "We observe that SAPG outperforms or achieves comparable",
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+        "text": "performance to the entropy-regularized variant except in the Reorientation environment where the variant with coefficient  $5\\mathrm{e} - 3$  performs up to  $16.5\\%$  better. Reorientation is the hardest task out of the three AllegroKuka tasks and has a lot of scope for learning emergent strategies such as using the table to move the object around, etc. Explicit exploration might be useful in discovering these behaviors.",
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+        "text": "The variant of ours which uses all the off-policy data is significantly worse on the AllegroHand and ShadowHand tasks and marginally worse on Regrasping and Throw environments. It is more sample efficient than SAPG on Reorientation but achieves lower asymptotic performance. This could be because in the simple environments, additional data has marginal utility. In the harder AllegroKuka environments, it is beneficial to use all the data initially since it may contain optimal trajectories that would otherwise be missed. However, once an appreciable level of performance is achieved, it becomes better to subsample to prevent the noise in the off-policy update from drowning out the on-policy gradient.",
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+        "text": "Finally, the symmetric variant of our method performs significantly worse across the board. This is possibly because using all the data to update each policy leads to them converging in behavior. If all the policies start executing the same actions, the benefit of data diversity is lost and SAPG reduces to vanilla PPO. Of course, there is a rich space of possible algorithms depending on particular choices of how data is aggregated and diversity is encouraged of which we have explored a small fraction.",
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+        "text": "SAPG: Split and Aggregate Policy Gradients",
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+            "Figure 7. Curves comparing reconstruction error for states visited during training using top-  $k$  PCA components for SAPG (Ours), PPO and a randomly initialized policy"
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+            "Figure 8. Curves comparing reconstruction error for states visited during training using MLPs with varying hidden layer dimensions for SAPG (Ours), PPO and a randomly initialized policy"
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+    {
+        "type": "text",
+        "text": "6.4. Diversity in exploration",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To analyze why our method outperforms the baseline method, we conduct experiments comparing the diversity of states visited by each algorithm during training. We devise two metrics to measure the diversity of the state space and find that our method beats PPO in both metrics.",
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+    {
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+        "list_items": [
+            "- PCA - We compute the reconstruction error of a batch of states using k most significant components of PCA and plot this error as a function of k. In general, a set that has variation along fewer dimensions of space can be compressed with fewer principal vectors and will have lower reconstruction error. This metric therefore measures the extent to which the policy explores different dimensions of state space. Figure-7 contains the plots for this metric. We find that the rate of decrease in reconstruction error with an increase in components is the slowest for our method.",
+            "- MLP - We train feedforward networks with small hidden layers on the task of input reconstruction on batches of environment states visited by our algorithm and PPO during training. The idea behind this is that if a batch of states has a more diverse data distribution then it should be harder to reconstruct the distribution using small hidden layers because high diversity implies that the distribution is less compressible. Thus,"
+        ],
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+    {
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+        "text": "high training error on a batch of states is a strong indicator of diversity in the batch. As can be observed from the plots in Figure-8, we find that training error is consistently higher for our method compared to PPO across different hidden layer sizes.",
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+    {
+        "type": "text",
+        "text": "7. Conclusion",
+        "text_level": 1,
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+    {
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+        "text": "In this work, we present a method to scale reinforcement learning to utilize large simulation capacity. We show how current algorithms obtain diminishing returns if we perform naive scaling by batch size and do not use the increased volume of data efficiently. Our method achieves state-of-the-art performance on hard simulation benchmarks.",
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+    {
+        "type": "text",
+        "text": "Impact Statement",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.",
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+    {
+        "type": "text",
+        "text": "Acknowledgements",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We thank Alex Li and Russell Mendonca for fruitful discussions regarding the method and insightful feedback. We would also like to thank Mihir Prabhudesai and Kevin",
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+    {
+        "type": "header",
+        "text": "SAPG: Split and Aggregate Policy Gradients",
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+    {
+        "type": "page_number",
+        "text": "9",
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+    },
+    {
+        "type": "text",
+        "text": "Gmelin for proofreading an earlier draft. This project was supported in part by ONR N00014-22-1-2096 and NSF NRI IIS-2024594.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
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+            "Schulman, J., Wolski, F., Dhariwal, P., Radford, A., and Klimov, O. Proximal policy optimization algorithms. ArXiv, abs/1707.06347, 2017. URL https://api.sementicscholar.org/CorpusID:28695052.",
+            "Schulman, J., Moritz, P., Levine, S., Jordan, M., and Abbeel, P. High-dimensional continuous control using generalized advantage estimation, 2018."
+        ],
+        "bbox": [
+            86,
+            84,
+            478,
+            906
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Sutton, R. S., McAllester, D., Singh, S., and Mansour, Y. Policy gradient methods for reinforcement learning with function approximation. In Solla, S., Leen, T., and Müller, K. (eds.), Advances in Neural Information Processing Systems, volume 12. MIT Press, 1999. URL https://proceedings.neurips.cc/paper_files/paper/1999/file/464d828b85b0bed98e80ade0a5c43b0f-Paper.pdf.",
+            "Vinyals, O., Babuschkin, I., Czarnecki, W. M., Mathieu, M., Dudzik, A., Chung, J., Choi, D. H., Powell, R., Ewalds, T., Georgiev, P., Oh, J., Horgan, D., Kroiss, M., Danihelka, I., Huang, A., Sifre, L., Cai, T., Agapiou, J. P., Jaderberg, M., Vezhnevets, A. S., Leblond, R., Pohlen, T., Dalibard, V., Budden, D., Sulsky, Y., Molloy, J., Paine, T. L., Gulcehre, C., Wang, Z., Pfaff, T., Wu, Y., Ring, R., Yogatama, D., Wünsch, D., McKinney, K., Smith, O., Schaul, T., Lillicrap, T. P., Kavukcuoglu, K., Hassabis, D., Apps, C., and Silver, D. Grandmaster level in starcraft ii using multiagent reinforcement learning. Nature, 575:350 - 354, 2019. URL https://apisemanticscholar.org/CorpusID:204972004.",
+            "Wang, Z., Bapat, V., Heess, N., Mnih, V., Munos, R., Kavukcuoglu, K., and de Freitas, N. Sample efficient actor-critic with experience replay. CoRR, abs/1611.01224, 2016. URL http://arxiv.org/abs/1611.01224.",
+            "Watkins, C. and Dayan, P. Q-learning. Machine Learning, 8:279-292, 1992. URL https://api-semanticscholar.org/CorpusID: 208910339.",
+            "Williams, R. J. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8:229-256, 2004. URL https://apisemantic scholar.org/CorpusID:19115634."
+        ],
+        "bbox": [
+            498,
+            84,
+            897,
+            654
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "header",
+        "text": "SAPG: Split and Aggregate Policy Gradients",
+        "bbox": [
+            343,
+            56,
+            630,
+            71
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "page_number",
+        "text": "11",
+        "bbox": [
+            477,
+            922,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A. Task and Environment Details",
+        "text_level": 1,
+        "bbox": [
+            84,
+            83,
+            369,
+            99
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Hard Difficulty Tasks All three hard tasks are based on the Allegro-Kuka environments(Petrenko et al., 2023). These consist of an Allegro Hand (16 dof) mounted on a Kuka arm (7 dof). In each case, the robot must manipulate a cuboidal kept on a fixed table. The observation space is  $\\mathbf{o}_t = [\\mathbf{q},\\dot{\\mathbf{q}},\\mathbf{x}_t,\\mathbf{v}_t,\\omega_t,\\mathbf{g}_t,\\mathbf{z}_t]$ , where  $\\mathbf{q}$ ,  $\\dot{\\mathbf{q}}\\in \\mathbb{R}^{23}$  are the joint angles and velocities respectively of each joint of the robot,  $\\mathbf{x}_t\\in \\mathbb{R}^7$  is the pose of the object,  $\\mathbf{v}_t$  is its linear velocity and  $\\omega_{t}$  is its angular velocity,  $\\mathbf{g}_t$  is a task-dependent goal observation and  $\\mathbf{z}_t$  is auxiliary information pertinent to solving the task such as if the object has been lifted. These tasks consist of a complex environment but a simple reward function allowing opportunities for emergent strategies to be learnt such as in-hand reorientation under the influence of gravity, reorientation against the table, different types of throws and grasps and so on. The performance of the agent in the above three tasks is measured by the successes metric which is defined as the number of successes in a single episode. Three tasks include:",
+        "bbox": [
+            84,
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+            888,
+            247
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Regrasping - The object must be lifted from the table and held near a goal position  $\\mathbf{g}_t \\in \\mathbb{R}^3$  for  $K = 30$  steps. This is called a \"success\". The target position and object position are reset to a random location after every success. The success tolerance  $\\delta$  defines the maximum error between object pose and goal pose for a success  $\\| \\mathbf{g}_t - (\\mathbf{x}_t)_{0:3} \\| \\leq \\delta$ . This tolerance is decreased in a curriculum from  $7.5\\mathrm{cm}$  to  $1\\mathrm{cm}$ , decremented by  $10\\%$  each time the average number of successes in an episode crosses 3. The reward function is a weighted combination of rewards encouraging the hand to reach the object  $r_{reach}$ , a bonus  $r_{lift}$ , rewards encouraging the hand to move to goal location after lifting  $r_{target}$  and a success bonus  $r_{success}$ .",
+            "- Throw - The object must be lifted from the table and thrown into a bucket at  $\\mathbf{g}_t \\in \\mathbb{R}^3$  placed out of reach of the arm. The bucket and the object position are reset randomly after every successful attempt. The reward function is similar to regrasping with the difference being that the target is now a bucket instead of a point.",
+            "- Reorientation - This task involves picking up the object and reorienting it to a particular target pose  $\\mathbf{g}_{\\mathbf{t}} \\in \\mathbb{R}^{7}$  (position + orientation). Similar to the regrasping task, there is a success tolerance  $\\delta$  which is varied in a curriculum. The target pose is reset once the agent succeeds. This means the agents needs to the object in different poses in succession, which may sometimes entail placing the obxfject on the table and lifting it up in a different way. Here too, the reward function is similar to regrasping, with the goal now being a pose in  $\\mathbb{R}^{7}$  instead of  $\\mathbb{R}^{3}$"
+        ],
+        "bbox": [
+            102,
+            273,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Easy Difficulty Tasks: In addition, we test on the following dexterous hand tasks. As before, the observation space consists of the joint angles and velocities  $\\mathbf{q}_t$ ,  $\\dot{\\mathbf{q}}_t$ , object pose  $\\mathbf{x}_t$  and velocities  $\\mathbf{v}_t$ ,  $\\omega_t$ . Following previous works (Li et al., 2023), we use the net episode reward as a performance metric for the ShadowHand and AllegroHand tasks.",
+        "bbox": [
+            84,
+            575,
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+            623
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Shadow Hand: We test on in-hand reorientation task of a cube using the 24-DoF Shadow Hand((OpenAI et al., 2018)). The task is to attain a specified goal orientation (specified as a quaternion) for the cube  $\\mathbf{g}_t \\in \\mathbb{R}^4$ . The reward is a combination of the orientation error and a success bonus.",
+            "- Allegro Hand: This is the same as the previous in-hand reorientation task but with the 16-DoF Allegro Hand instead."
+        ],
+        "bbox": [
+            102,
+            648,
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+            734
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B. Training hyperparameters",
+        "text_level": 1,
+        "bbox": [
+            84,
+            763,
+            334,
+            780
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We use two different sets of default hyperparameters for PPO in AllegroKuka and Shadow Hand tasks which are described below.",
+        "bbox": [
+            84,
+            789,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.1. AllegroKuka tasks",
+        "text_level": 1,
+        "bbox": [
+            84,
+            835,
+            253,
+            852
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We use a Gaussian policy where the mean network is an LSTM with 1 layer containing 768 hidden units. The observation is also passed through an MLP of with hidden layer dimensions  $768 \\times 512 \\times 256$  and an ELU activation (Clevert et al., 2016) before being input to the LSTM. The sigma for the Gaussian is a fixed learnable vector independent of input observation.",
+        "bbox": [
+            84,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "header",
+        "text": "SAPG: Split and Aggregate Policy Gradients",
+        "bbox": [
+            343,
+            56,
+            630,
+            70
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "page_number",
+        "text": "12",
+        "bbox": [
+            477,
+            922,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/4dd033370cac637801134e27f8ba01839a253a14c3e2bf2a0800ca7717d5ac39.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Value</td></tr><tr><td>Discount factor, γ</td><td>0.99</td></tr><tr><td>τ</td><td>0.95</td></tr><tr><td>Learning rate</td><td>1e-4</td></tr><tr><td>KL threshold for LR update</td><td>0.016</td></tr><tr><td>Grad norm</td><td>1.0</td></tr><tr><td>Entropy coefficient</td><td>0</td></tr><tr><td>Clipping factor ε</td><td>0.1</td></tr><tr><td>Mini-batch size</td><td>num_envs · 4</td></tr><tr><td>Critic coefficient λ&#x27;</td><td>4.0</td></tr><tr><td>Horizon length</td><td>16</td></tr><tr><td>LSTM Sequence length</td><td>16</td></tr><tr><td>Bounds loss coefficient</td><td>0.0001</td></tr><tr><td>Mini epochs</td><td>2</td></tr></table>",
+        "bbox": [
+            331,
+            80,
+            640,
+            297
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.2. Shadow Hand",
+        "text_level": 1,
+        "bbox": [
+            84,
+            409,
+            220,
+            422
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We use a Gaussian policy where the mean network is an MLP with hidden layers dimensions  $512 \\times 512 \\times 256 \\times 128$  and an ELU activation (Clevert et al., 2016)",
+        "bbox": [
+            84,
+            431,
+            887,
+            463
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/0de4cd76fadc813b08d869c1c6005b5148f8903c19784709d17b24de9528f9c6.jpg",
+        "table_caption": [
+            "Table 2. Training hyperparameters for AllegroKuka tasks"
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Value</td></tr><tr><td>Discount factor, γ</td><td>0.99</td></tr><tr><td>τ</td><td>0.95</td></tr><tr><td>Learning rate</td><td>5e-4</td></tr><tr><td>KL threshold for LR update</td><td>0.016</td></tr><tr><td>Grad norm</td><td>1.0</td></tr><tr><td>Entropy coefficient</td><td>0</td></tr><tr><td>Clipping factor ε</td><td>0.1</td></tr><tr><td>Mini-batch size</td><td>num_envs · 4</td></tr><tr><td>Critic coefficient λ&#x27;</td><td>4.0</td></tr><tr><td>Horizon length</td><td>8</td></tr><tr><td>Bounds loss coefficient</td><td>0.0001</td></tr><tr><td>Mini epochs</td><td>5</td></tr></table>",
+        "bbox": [
+            331,
+            539,
+            640,
+            739
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 3. Training hyperparameters for Shadow Hand",
+        "bbox": [
+            328,
+            750,
+            643,
+            763
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.3. Allegro Hand",
+        "text_level": 1,
+        "bbox": [
+            86,
+            851,
+            217,
+            867
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We use a Gaussian policy where the mean network is an MLP with hidden layers dimensions  $512 \\times 256 \\times 128$  and an ELU activation.",
+        "bbox": [
+            84,
+            875,
+            885,
+            905
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "header",
+        "text": "SAPG: Split and Aggregate Policy Gradients",
+        "bbox": [
+            343,
+            56,
+            630,
+            70
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "page_number",
+        "text": "13",
+        "bbox": [
+            477,
+            922,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/b4c8ce744d55dd40d3fff53edbebf52a0a61f7624646286bbfed7e1aadd57d3c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Value</td></tr><tr><td>Discount factor, γ</td><td>0.99</td></tr><tr><td>τ</td><td>0.95</td></tr><tr><td>Learning rate</td><td>5e-4</td></tr><tr><td>KL threshold for LR update</td><td>0.016</td></tr><tr><td>Grad norm</td><td>1.0</td></tr><tr><td>Entropy coefficient</td><td>0</td></tr><tr><td>Clipping factor ε</td><td>0.2</td></tr><tr><td>Mini-batch size</td><td>num_envs · 4</td></tr><tr><td>Critic coefficient λ&#x27;</td><td>4.0</td></tr><tr><td>Horizon length</td><td>8</td></tr><tr><td>Bounds loss coefficient</td><td>0.0001</td></tr><tr><td>Mini epochs</td><td>5</td></tr></table>",
+        "bbox": [
+            331,
+            80,
+            640,
+            281
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 4. Training hyperparameters for Shadow Hand",
+        "bbox": [
+            330,
+            292,
+            643,
+            306
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Note: In case of experiments with entropy based exploration, each block of environments has it's own learnable vector sigma which enable policies for different blocks to have different entropies.",
+        "bbox": [
+            84,
+            330,
+            888,
+            363
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "header",
+        "text": "SAPG: Split and Aggregate Policy Gradients",
+        "bbox": [
+            343,
+            56,
+            630,
+            70
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "page_number",
+        "text": "14",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    }
+]

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2024/SAPG_ Split and Aggregate Policy Gradients/full.md

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+# Jayesh Singla  $^{*1}$  Ananye Agarwal  $^{*1}$  Deepak Pathak
+
+# Abstract
+
+Despite extreme sample inefficiency, on-policy reinforcement learning, aka policy gradients, has become a fundamental tool in decision-making problems. With the recent advances in GPU-driven simulation, the ability to collect large amounts of data for RL training has scaled exponentially. However, we show that current RL methods, e.g. PPO, fail to ingest the benefit of parallelized environments beyond a certain point and their performance saturates. To address this, we propose a new on-policy RL algorithm that can effectively leverage large-scale environments by splitting them into chunks and fusing them back together via importance sampling. Our algorithm, termed SAPG, shows significantly higher performance across a variety of challenging environments where vanilla PPO and other strong baselines fail to achieve high performance. Webpage at https://sapg-rl.github.io.
+
+# 1. Introduction
+
+Broadly, there are two main categories in reinforcement learning (RL): off-policy RL, e.g., Q-learning (Watkins & Dayan, 1992), and on-policy RL, e.g., policy gradients (Sutton et al., 1999). On-policy methods are relatively more sample inefficient than off-policy but often tend to converge to higher asymptotic performance. Due to this reason, on-policy RL methods, especially PPO (Schulman et al., 2017), are usually the preferred RL paradigm for almost all sim2real robotic applications (Miki et al., 2022; Agarwal et al., 2022; Chen et al., 2021) to games such as StarCraft (Vinyals et al., 2019), where one could simulate years of real-world experience in minutes to hours.
+
+RL is fundamentally a trial-n-error-based framework and hence is sample inefficient in nature. Due to this, one needs to have large batch sizes for each policy update, especially in
+
+*Equal contribution  ${}^{1}$  Carnegie Mellon University. Correspondence to: Ananye Agarwal <ananyea@andrew.cmu.edu>, Jayesh Singla <jsingla@andrew.cmu.edu>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+Parallel Envs
+
+Regular On-policy RL
+
+![](images/b57ad28d340bcb4d9003d31ec900d0a8c56cf5f12ab623d1656445fad84763c8.jpg)
+
+![](images/a95bad4e4903265d5dfd4d38a4f3694e23f3456f23baafc7dae98fa1496aaced.jpg)  
+Single learner
+
+![](images/7bdccc2bf40309b57b04dc070e543acaabddc184408a8bb8934a289483a18a04.jpg)  
+Split and Aggregate Policy Gradients (SAPG)  
+Followers
+
+![](images/7be2c58e97eebe98e168b3f183cc585081e62c28723ebbc286d97bc308399fc6.jpg)
+
+![](images/4d5770707ed126b7774190ec35eaa3db8431e137a46c4305a0ba20d222021aa4.jpg)
+
+![](images/935b6e2dff5c7c4228c93c8ac9e562bc5cb8e0572afeeb316f1702da697546c2.jpg)  
+Leader
+
+![](images/d00bbe95c49532e92effe3080c1147d49be5e4484b461e5096f16554c788a5fd.jpg)
+
+![](images/cc078c1ed5fa5958dad4b0a76b834bbf3c23987a3826e78c95eff515fb6075a6.jpg)
+
+![](images/19c4b7c0ea21550e959d09c7198b809af50d1dd66bdaefb7d60def33dd99ca94.jpg)  
+Figure 1. We introduce a new class of on-policy RL algorithms that can scale to tens of thousands of parallel environments. In contrast to regular on-policy RL, such as PPO, which learns a single policy across environments leading to wasted environment capacity, our method learns diverse followers and combines data from them to learn a more optimal leader in a continuous online manner.
+
+![](images/64a536c9ab7b1d699a643887cba392d5a6f7e11362f83afcd532591cc0e397f0.jpg)
+
+the case of on-policy methods because they can only use data from current experience. Fortunately, in recent years, the ability to simulate a large number of environments in parallel has become exponentially larger due to GPU-driven physics engines, such as IsaacGym (Makoviychuk et al., 2021), PhysX, Mujoco-3.0, etc. This means that each RL update can easily scale to batches of size hundreds of thousands to millions, which are over two orders of magnitude higher than what most RL benchmarks typically have.
+
+In this paper, we highlight an issue with typical on-policy RL methods, e.g. PPO, that they are not able to ingest the benefits with increasingly larger sample sizes for each up-
+
+![](images/34f4332d113579d372b25fac045a2de44fd8bf9a632c0377ea2aa5f1433bcdad.jpg)  
+Figure 2. Performance vs batch size plot for PPO runs (blue curve) across two environments. The curve shows how PPO training runs can not take benefit of large batch size resulting from massively parallelized environments and their asymptotic performance saturates after a certain point. The dashed red line is the performance of our method, SAPG, with more details in the results section. It serves as evidence that higher performance is achievable with larger batch sizes.
+
+date. In Figure 2, we show that PPO performance saturates after a certain batch size despite the ceiling being higher. This is due to the issue in data sampling mechanisms. In particular, at each timestep actions are sampled from a Gaussian with some mean and variance. This implies that most sampled actions are near the mean and with large number of environments, many environments are executing the same actions leading to duplicated data. This implies that the performance of PPO saturates at some point as we increase the number of environments.
+
+We propose a simple fix to this problem. Instead of running a single PPO policy for all environments, we divide environments into blocks. Each block optimizes a separate policy, allowing for more data diversity than just i.i.d. sampling from the same Gaussian. Next, we do an off-policy update to combine data from all these policies to keep the update consistent with the objective of on-policy RL. This allows us to use the PPO's clipped surrogate objective, maintaining the stability benefits of PPO while latching onto high reward trajectories even though they are off-policy. A schematic of our approach, termed SAPG, is shown in Figure 1. We evaluate SAPG across a variety of environments and show significantly high asymptotic performance in environments where vanilla PPO even fails to get any positive success.
+
+# 2. Related Work
+
+Policy gradients REINFORCE (Williams, 2004), one of the earliest policy gradient algorithms uses an estimator of the objective using simple Monte Carlo return values. Works such as (Konda & Tsitsiklis, 1999) and (Schulman et al., 2018) improve the stability of policy gradient algorithms by employing a baseline to decrease the variance of the estimator while not compromising on the bias. (Schulman et al., 2015; 2017) incorporate conservative policy updates
+
+into policy gradients to increase the robustness.
+
+Distributed reinforcement learning Reinforcement learning algorithms are highly sample inefficient, which calls for some form of parallelization to increase the training speed. When training in simulation, this speed-up can be achieved by distributing experience collection or different parts of training across multiple processes.(Nair et al., 2015; Mnih et al., 2016; Espeholt et al., 2018; Horgan et al., 2018) However, through the introduction of GPU-based simulators such as IsaacGym (Makoviychuk et al., 2021), the capacity of simulation has increased by two to three orders of magnitude. Due to this, instead of focusing on how to parallelize parts of the algorithm, the focus has shifted to finding ways to efficiently utilize the large amount of simulation data. Previous works such as (Chen et al., 2021; Fu et al., 2022; Rudin et al., 2022; Handa et al., 2022; Agarwal et al., 2022) use data from GPU-based simulation to learn policies in complex manipulation and locomotion settings. However, most of these works still use reinforcement learning algorithms to learn a single policy, while augmenting training with techniques like teacher-student-based training and game-based curriculum. We find that using the increased simulation capacity to naively increase the batch size is not the best way to utilize massively parallel simulation.
+
+(Petrenko et al., 2023) develop a population-based training framework that divides the large number of environments between multiple policies and using hyperparameter mutation to find a set of hyperparameters that performs well. However, even this does not utilize all the data completely as each policy learns independently. We propose a way to ensure most of the data from the environments contributes to learning by using all collected transitions for each update.
+
+Off-policy Policy Gradients Unlike on-policy algorithms, off-policy algorithms can reuse all collected data or data collected by any policy for their update. Most off-policy algorithms (Mnih et al., 2013; Lillicrap et al., 2019; Haarnoja et al., 2018) try to learn a value function which is then implicitly/explicitly used to learn a policy. (Li et al., 2023) developed a variant of Deep Deterministic Policy Gradient (DDPG) called PQL which splits data collection and learning into multiple processes and shows impressive performance on many benchmark tasks. We use PQL as one of our baselines to compare our method to off-policy RL in complex tasks. Although off-policy algorithms are much more data-efficient, they usually get lower asymptotic performance than on-policy policy gradients. This has inspired works to develop techniques to use off-policy data in on-policy methods. (Hastings, 1970) has been one of the major techniques used to realize this. Previous works (Degris et al., 2012; Wang et al., 2016; Espeholt et al., 2018; Fakoor et al., 2020) develop techniques to use off-policy data in on-policy
+
+algorithms using importance sampling-based updates along with features such as bias correction.
+
+# 3. Preliminaries
+
+In this paper, we propose a modification to on-policy RL to achieve higher performance in the presence of large batch sizes. We build upon PPO, although our proposed ideas are generally applicable to any on-policy RL method.
+
+On-policy RL Let  $(S, \mathcal{A}, \mathcal{P}, r, \rho, \gamma)$  be an MDP where  $S$  is the set of states,  $\mathcal{A}$  the set of actions,  $\mathcal{P}$  are transition probabilities,  $r$  the reward function,  $\rho$  the initial distribution of states and  $\gamma$  the discount factor. The objective in reinforcement learning is to find a policy  $\pi(a|s)$  which maximises the long term discounted reward  $\mathcal{J}(\pi) = \mathbb{E}_{s_0 \sim \rho, a_t \sim \pi(\cdot|s_t)}\left[\sum_{t=0}^{T-1} \gamma^t r(s_t, a_t)\right]$ .
+
+Policy-gradient algorithms (Williams, 2004; Konda & Tsitsiklis, 1999; Schulman et al., 2015; Mnih et al., 2016) optimize the policy using gradient descent with Monte Carlo estimates of the gradient
+
+$$
+\nabla_ {\theta} J (\pi_ {\theta}) = \underset {s \sim \rho_ {d}, a \sim \pi (\cdot | s)} {\mathbb {E}} \left[ \nabla_ {\theta} \log (\pi_ {\theta} (a)) \hat {A} ^ {\pi_ {\theta}} (s, a) \right] \tag {1}
+$$
+
+where  $\hat{A}^{\pi_{\theta}}(s,a)$  is an advantage function that estimates the contribution of the transition to the gradient. A common choice is  $\hat{A}^{\pi_{\theta}}(s,a) = \hat{Q}^{\pi_{\theta}}(s,a) - \hat{V}^{\pi_{\theta}}(s)$ , where  $\hat{Q}^{\pi_{\theta}}(s,a)$ ,  $\hat{V}^{\pi_{\theta}}(s)$  are estimated  $Q$  and value functions. This form of update is termed as an actor-critic update (Konda & Tsitsiklis, 1999). Since we want the gradient of the error with respect to the current policy, only data from the current policy (on-policy) data can be utilized.
+
+PPO Actor critic updates can be quite unstable because gradient estimates are high variance and the loss landscape is complex. An update step that is too large can destroy policy performance. Proximal Policy Optimization (PPO) modifies Eq. 1 to restrict updates to remain within an approximate "trust region" where there is guaranteed improvement (Schulman et al., 2015; Kakade & Langford, 2002).
+
+$$
+\begin{array}{l} L _ {o n} \left(\pi_ {\theta}\right) = \underset {\pi_ {o l d}} {\mathbb {E}} \left[ \min  \left(r _ {t} \left(\pi_ {\theta}\right), \right. \right. \tag {2} \\ \operatorname {c l i p} \left(r _ {t} (\pi_ {\theta}), 1 - \epsilon , 1 + \epsilon)\right) A _ {t} ^ {\pi_ {\text {o l d}}} ] \\ \end{array}
+$$
+
+Here,  $r_t(\pi_\theta) = \frac{\pi_\theta(a_t|s_t)}{\pi_{old}(a_t|s_t)}$ ,  $\epsilon$  is a clipping hyperparameter and  $\pi_{old}$  is the policy collecting the on-policy data. The clipping operation ensures that the updated  $\pi$  stays close to  $\pi_{old}$ . Empirically, given large numbers of samples, PPO achieves high performance, is stable and robust to hyperparameters. However, it was developed for relatively small
+
+batch sizes ( $\approx 100$  parallel envs). We find that in the large-scale setting ( $>10k$  envs), it is suboptimal because many parallel envs are sampling nearly identical on-policy data.
+
+# 4. Split and Aggregate Policy Gradients
+
+Policy gradient methods are highly sensitive to the variance in the estimate of gradient. Since CPU-based simulators typically run only 100s of environments in parallel, conventional wisdom is to simply sample on-policy data from a Gaussian policy in all the environments since as the number of datapoints increases, the Monte Carlo estimate becomes more accurate. However, this intuition no longer holds in the extremely large-scale data setting where we have hundreds of thousands of environments on GPU-accelerated simulators like IsaacGym. IID sampling from a Gaussian policy will lead to most actions lying near the mean, and most environments will execute similar actions, leading to wasted data (fig. 2).
+
+We propose to efficiently use large numbers of  $N$  environments using a divide-and-conquer setup. Our algorithm trains a variety of  $M$  policies  $\pi_1, \ldots, \pi_M$  instead of having just one policy. However, simply training multiple policies by dividing environments between them is also inefficient. This is equivalent to training an algorithm with different seeds and choosing the best seed. One approach is to add hyperparameter mutation (Petrenko et al., 2023) to the policies and choosing the hyperparameters that perform the best among all of them. However, even in this case, all of the data from the "worse" policies goes to waste, and the only information gained is that some combinations of hyperparameters are bad, even though the policies themselves may have discovered high reward trajectories. We need to somehow aggregate data from multiple policies into a single update. We propose to do this via off-policy updates.
+
+# 4.1. Aggregating data using off-policy updates
+
+One of the major drawbacks of on-policy RL is its inability to use data from past versions of the policy. One solution is to use importance sampling (Degris et al., 2012; Meng et al., 2023) to weight updates using data from different policies. In practice, this is not used since given limited compute it is beneficial to sample on-policy experience that is more directly relevant. However, this is no longer true in the large batch setting where enough on-policy data is available. In this case, it becomes advantageous to have multiple policies  $\pi_1, \ldots, \pi_M$  and use them to sample diverse data, even if it is off-policy. In particular, to update policy  $\pi_i$  using data from policy  $\pi_j$ ,  $j \in \mathcal{X}$  we use (Meng et al., 2023)
+
+$$
+L _ {o f f} \left(\pi_ {i}; \mathcal {X}\right) = \frac {1}{| \mathcal {X} |} \sum_ {j \in \mathcal {X}} \underset {(s, a) \sim \pi_ {j}} {\mathbb {E}} \left[ \min  \left(r _ {\pi_ {i}} (s, a), \right. \right. \tag {3}
+$$
+
+$$
+\operatorname {c l i p} \left(r _ {\pi_ {i}} (s, a), \mu (1 - \epsilon), \mu (1 + \epsilon))\right) A ^ {\pi_ {i, o l d}} (s, a) ]
+$$
+
+![](images/5cebd23bf812b26cb226ed2855d560c362c1ceb2932f1dec0fb6191d47f2007f.jpg)  
+Figure 3. We illustrate one particular variant of SAPG which performs well. There is one leader and  $M - 1$  followers ( $M = 3$  in figure). Each policy has the same backbone with shared parameters  $B_{\theta}$  but is conditioned on local learned parameters  $\phi_i$ . Each policy gets a block of  $\frac{N}{M}$  environments to run. The leader is updated with its on-policy data as well as importance-sampled off-policy data from the followers. Each of the followers only uses their own data for on-policy updates.
+
+where  $r_{\pi_i}(s,a) = \frac{\pi_i(s,a)}{\pi_j(s,a)}$  and  $\mu$  is an off-policy correction term  $\mu = \frac{\pi_{i,old}(s,a)}{\pi_j(s,a)}$ . Note that when  $i = j$ , then  $\pi_j = \pi_{i,old}$  and this reduces to the on-policy update as expected. This is then scaled and combined with the on-policy term (eq. 2)
+
+$$
+L \left(\pi_ {i}\right) = L _ {o n} \left(\pi_ {i}\right) + \lambda \cdot L _ {o f f} \left(\pi_ {i}; \mathcal {X}\right) \tag {4}
+$$
+
+The update target for the critic is calculated using  $n$ -step returns (here  $n = 3$ ).
+
+$$
+V _ {o n, \pi_ {j}} ^ {\text {t a r g e t}} (s _ {t}) = \sum_ {k = t} ^ {t + 2} \gamma^ {k - t} r _ {k} + \gamma^ {3} V _ {\pi_ {j}, o l d} (s _ {t + 3}) \tag {5}
+$$
+
+However, this is not possible for off-policy data. Instead, we assume that an off-policy transition can be used to approximate a 1-step return. The target equations are as follows
+
+$$
+V _ {o f f, \pi_ {j}} ^ {\text {t a r g e t}} \left(s _ {t} ^ {\prime}\right) = r _ {t} + \gamma V _ {\pi_ {j}, o l d} \left(s _ {t + 1} ^ {\prime}\right) \tag {6}
+$$
+
+The critic loss is then
+
+$$
+L _ {o n} ^ {\text {c r i t i c}} \left(\pi_ {i}\right) = \underset {(s, a) \sim \pi_ {i}} {\mathbb {E}} \left[ \left(V _ {\pi_ {i}} (s) - V _ {o n, \pi_ {i}} ^ {\text {t a r g e t}} (s)\right) ^ {2} \right] \tag {7}
+$$
+
+$$
+L _ {o f f} ^ {\text {c r i t i c}} \left(\pi_ {i}; \mathcal {X}\right) = \frac {1}{| \mathcal {X} |} \sum_ {j \in \mathcal {X}} \underset {(s, a) \sim \pi_ {j}} {\mathbb {E}} \left[ \left(V _ {\pi_ {i}} (s) - V _ {o f f, \pi_ {i}} ^ {\text {t a r g e t}} (s)\right) ^ {2} \right] \tag {8}
+$$
+
+$$
+L ^ {\text {c r i t i c}} \left(\pi_ {i}\right) = L _ {\text {o n}} ^ {\text {c r i t i c}} \left(\pi_ {i}\right) + \lambda \cdot L _ {\text {o f f}} ^ {\text {c r i t i c}} \left(\pi_ {i}\right) \tag {9}
+$$
+
+Given this update scheme, we must now choose a suitable  $\mathcal{X} \subseteq \{1, \ldots, M\}$  and the set of  $i$ s to update, along with the correct ratio  $\lambda$ . We explore several variants below.
+
+# 4.2. Symmetric aggregation
+
+A simple choice is to update all  $i' s$  with the data from all policies. In this case, we choose to update each policy  $i \in \{1, 2, \dots, M\}$  and for each  $i$  use off-policy data from all other policies  $\mathcal{X} = \{1, 2, i - 1, i + 1, \dots, M\}$ . Since gradients from off-policy data are typically noisier than gradients from on-policy data, we choose  $\lambda = 1$ , but subsample the off-policy data such that we use equal amounts of on-policy and off-policy data.
+
+# 4.3. Leader-follower aggregation
+
+While the above choice prevents data wastage, since all the policies are updated with the same data, it can lead to policies converging in behavior, reducing data diversity and defeating the purpose of having separate policies. To resolve this, we break symmetry by designating a "leader" policy  $i = 1$  which gets data from all other policies  $\mathcal{X} = \{2,3,\dots,M\}$  while the rest are "followers" and only use their own on-policy data for updates  $\mathcal{X} = \phi$ . As before, we choose  $\lambda = 1$ , but subsample the off-policy data for the leader such that we use equal amounts of on-policy and off-policy data in a mini-batch update.
+
+# 4.4. Encouraging diversity via latent conditioning
+
+What is the right parameterization for this set of policies? One simple choice is to have a disjoint set of parameters for each with no sharing at all. However, this implies that each follower policy has no knowledge of any other policy whatsoever and may get stuck in a bad local optimum. We mitigate this by having a shared backbone  $B_{\theta}$  for each policy conditioned on hanging parameters  $\phi_j$  local to each
+
+![](images/e4d34a3bf6dd21e14a99efee58a1e5d8f2fc32bc3bbb5e00cd6c0c6baafb16a3.jpg)  
+Figure 4. Two data aggregation schemes we consider in this paper. (Left) one policy is a leader and uses data from each of the followers (Right) a symmetric scheme where each policy uses data from all others. In each case, the policy also uses its own on-policy data.
+
+policy. Similarly, the critic consists of a shared backbone  $C_{\psi}$  conditioned on parameters  $\phi_j$ . The parameters  $\psi, \theta$  are shared across the leader and all followers and updated with gradients from each objective, while the parameters  $\phi_j$  are only updated with the objective for that particular policy. We choose  $\phi_j \in \mathbb{R}^{32}$  for complex environments while  $\phi_j \in \mathbb{R}^{16}$  for the relatively simpler ones.
+
+# 4.5. Enforcing diversity through entropy regularization
+
+To further encourage diversity between different policies, in addition to the PPO update loss  $L_{on}$  we add an entropy loss to each of the followers with different coefficients. In particular, the entropy loss is  $\mathcal{H}(\pi(a|s))$ . The overall loss for the policy  $i$  (or the  $(i-1)$ th) follower is  $L(\pi_i) = L_{on}(\pi_i) + \sigma(i-1) \cdot \mathcal{H}(\pi(a|s))$ . The leader doesn't have any entropy loss. Different scales of coefficients produce policies with different explore-exploit tradeoffs. Followers with large entropy losses tend to explore more actions even if they are suboptimal, while those with small coefficients stay close to optimal trajectories and refine them. This leads to a large data coverage with a good mix of optimal as well as diverse trajectories. We treat  $\sigma$  as a hyperparameter.
+
+# 4.6. Algorithm: SAPG
+
+We roll out  $M$  different policies and collect data  $\mathcal{D}_1, \ldots, \mathcal{D}_M$  for each. Follower policies  $2, \ldots, M$  are updated using the usual PPO objective with minibatch gradient descent on their respective datasets. However, we augment the dataset of the leader  $\mathcal{D}_1$  with data from  $\mathcal{D}_2, \ldots, \mathcal{D}_M$ , weighed by the importance weight  $\mu$ . The leader is then updated by minibatch gradient descent as well.
+
+# 5. Experimental Setup
+
+We conduct experiments on 5 manipulation tasks (3 hard and 2 easy) and compare them against SOTA methods for the large-scale parallelized setting. We use a GPU-accelerated simulator, IsaacGym (Makoviychuk et al., 2021) which allows simulating tens of thousands of environments in par
+
+# Algorithm 1 SAPG
+
+Initialize shared parameters  $\theta, \psi$   
+For  $i \in \{1, \dots, M\}$  initialize parameters  $\phi_i$   
+Initialize  $N$  environments  $E_1, \dots, E_N$ .  
+Initialize data buffers for each policy  $\mathcal{D}_1, \dots, \mathcal{D}_M$ .  
+for  $i = 1, 2, \dots, \mathbf{do}$   
+for  $j = 1, 2, \dots, M$  do  
+ $\mathcal{D}_j \gets \text{COLLECTDATA}(E_{j\frac{N}{M}:(j+1)\frac{N}{M}}, \theta, \psi_j)$   
+end for  
+ $L \gets 0$   
+Sample  $|\mathcal{D}_1|$  transitions from  $\cup_{j=2}^M \mathcal{D}_i$  to get  $\mathcal{D}_1'$ .  
+ $L \gets L + \text{OFFPOLICYLOSS}(\mathcal{D}_1')$ $L \gets L + \text{ONPOLICYLOSS}(\mathcal{D}_1)$   
+for  $j = 2, \dots, M$  do  
+ $L \gets L + \text{ONPOLICYLOSS}(\mathcal{D}_j)$   
+end for  
+Update  $\theta \gets \theta - \eta \nabla_\theta L$   
+Update  $\psi \gets \psi - \eta \nabla_\psi L$   
+end for=0
+
+allele on a single GPU. In our experiments, we focus on the large-scale setting and simulate 24576 parallel environments unless otherwise specified. Note that this is two orders of magnitude larger than the number of environments PPO was developed on (Schulman et al., 2017), and we indeed find that vanilla PPO does not scale to this setting.
+
+For testing, we choose a suite of manipulation environments that are challenging and require large-scale data to learn effective policies (Petrenko et al., 2023). In particular, these consist of dexterous hands mounted on arms leading to high numbers of degrees of freedom (up to 23). This is challenging because sample complexity scales exponentially with degrees of freedom. They are under-actuated and involve manipulating free objects in certain ways while under the influence of gravity. This leads to complex, non-linear interactions between the agent and the environment such as contacts between robot and object, object and table, and robot and table. Overall, this implies that to learn effective policies an agent must collect a large amount of relevant experience and also use it efficiently for learning.
+
+# 5.1. Tasks
+
+We consider a total of 5 tasks grouped into two parts: Three hard tasks and two easy tasks. Hard and easy is defined by the success reward achieved by off-policy (in particular, PQL) methods in these environments. In easy environments, even Q-learning-based off-policy methods can obtain nonzero performance but not in hard tasks. See appendix sec. A.
+
+Hard Difficulty Tasks All three hard tasks are based on the Allegro-Kuka environments(Petrenko et al., 2023). These consist of an Allegro Hand (16 DoF) mounted on
+
+a Kuka arm (7 dof). The performance of the agent in the above three tasks is measured by the successes metric which is defined as the number of successes in a single episode. Three tasks include:
+
+- Regrasping: The object must be lifted from the table and held near a goal position  $\mathbf{g}_t \in \mathbb{R}^3$  for  $K = 30$  steps. This is called a "success". The target position and object position are reset to a random location after every success.  
+- Throw: The object must be lifted from the table and thrown into a bucket at  $\mathbf{g}_t \in \mathbb{R}^3$  placed out of reach of the arm. The bucket and the object position are reset randomly after every successful attempt.  
+- Reorientation: Pick up the object and reorient it to a particular target pose  $\mathbf{g}_{\mathbf{t}} \in \mathbb{R}^{7}$  (position + orientation). The target pose is reset once the agent succeeds. This means that the agents needs to reorient the object in different poses in succession, which may sometimes entail placing the objects on the table and lifting it up in a different way.
+
+Easy Difficulty Tasks: In addition, we test on the following dexterous hand tasks. As before, the observation space consists of the joint angles and velocities  $\mathbf{q}_t$ ,  $\dot{\mathbf{q}}_t$ , object pose  $\mathbf{x}_t$  and velocities  $\mathbf{v}_t, \omega_t$ .
+
+- Shadow Hand: We test in-hand reorientation task of a cube using the 24-DoF Shadow Hand.  
+- Allegro Hand: This is the same as the previous in-hand reorientation task but with the 16-DoF Allegro Hand.
+
+# 5.2. Baselines
+
+We test against state-of-the-art RL methods designed for the GPU-accelerated large-scale setting we consider in this paper. We compare against both on-policy (Petrenko et al., 2023) and off-policy (Li et al., 2023) variants as well as vanilla PPO (Schulman et al., 2017).
+
+- PPO (Proximal Policy Optimization) (Schulman et al., 2017): In our setting, we just increase the data throughput for PPO by increasing the batch size proportionately to the number of environments. In particular, we see over two orders of magnitude increase in the number of environments (from 128 to 24576).  
+- **Parallel Q-Learning** (Li et al., 2023) A parallelized version of DDPG with different mixed exploration i.e. varying exploration noise across environments to further aid exploration. We use this baseline to compare if off-policy methods can outperform on-policy methods when the data collection capacity is high.  
+- DexPBT (Petrenko et al., 2023) A framework that combines population-based training with PPO.  $N$  Environments are divided into  $M$  groups, each containing  $\frac{N}{M}$  environments.  $M$  separate policies are trained using PPO in each group of environments with different hyperparameters. At regular intervals, the worst-performing policies
+
+are replaced with the weights of best-performing policies and their hyperparameters are mutated randomly.
+
+Due to the complexity of these tasks, experiments take about 48-60 hours on a single GPU, collecting  $\approx 2\mathrm{e}10$  transitions. Since we run experiments on different machines, the wall clock time is not directly comparable and we compare runs against the number of samples collected. We run 5 seeds for each experiment and report the mean and standard error in the plots. In each plot, the solid line is  $y(t) = \frac{1}{n}\sum_{i}y_{i}(t)$  while the width of the shaded region is determined by standard error  $\frac{2}{\sqrt{n}}\sum_{i}(y(t) - y_{i}(t))^{2}$ .
+
+For each task, we use  $M = 6$  policies for our method and DexPBT in a total of  $N = 24576$  environments for each method. We use the dimension of learned parameter  $\phi_j \in \mathbb{R}^{32}$  for the AllegroKuka tasks while we use  $\phi_j \in \mathbb{R}^{16}$  for the ShadowHand and AllegroHand tasks since they are relatively simpler. We use a recurrent policy for the AllegroKuka tasks and an MLP policy for the Shadow Hand and Allegro Hand tasks and use PPO to train them. We collect 16 steps of experience per instance of the environment before every PPO update step. For SAPG, we tune the entropy coefficient  $\sigma$  by choosing the best from a small set  $\{0, 0.003, 0.005\}$  for each environment. We find that  $\sigma = 0$  works best for all ShadowHand, AllegroHand, Regrasping, and Throw while  $\sigma = 0.005$  works better for Reorientation.
+
+# 6. Results and Analysis
+
+In the large-scale data setting, we are primarily concerned with optimality while sample-efficiency and wall-clock time are secondary concerns. This is because data is readily available—one only needs to spin up more GPUs, what is really important is how well our agent performs in the downstream tasks. Indeed, this aligns with how practitioners use RL algorithms in practice (Agarwal et al., 2022; Cheng et al., 2023; OpenAI et al., 2018), where agents are trained with lots of domain randomization in large simulations and the primary concern is how well the agent can adapt and learn in these environments since this directly translates to real-world performance.
+
+# 6.1. AllegroKuka tasks
+
+The AllegroKuka tasks (Throw, Regrasping, and Reorientation) are hard due to large degrees of freedom. The environment also offers the possibility of many emergent strategies such as using the table to reorient the cube, or using gravity to reorient the cube. Therefore, a large amount of data is required to attain good performance on these tasks. Following Petrenko et al. (2023) we use the number of successes as a performance metric on these tasks. Note that the DexPBT baseline directly optimizes for success by mutating the re
+
+![](images/b110e4d6084b60ee3f9b087413b3ffe5a44191decc4e437450835ccf4e4d409c.jpg)
+
+![](images/a93706e05f5ef8d25209a1a2fcacf302e049eaf2827c609e0934fa2d774d5fad.jpg)
+
+![](images/5d04acfb2bc908cbaa02a131bfcf3d8e9e31a46b130acc054e7ddce5813110a3.jpg)
+
+![](images/7396656c6183050ec8e763e1eabfc474cba92c045134e81321790cb95783a839.jpg)
+
+![](images/ff17e4a10a71b3ef6e9aea8bf9dda69aeb568dcb6b610fbb64114582653190b9.jpg)
+
+![](images/5a4b040336a30f4995220d2a3d55da24573d54a09bd67cbf9125807894a37ef6.jpg)  
+Figure 5. Performance curves of SAPG with respect to PPO, PBT and PQL baselines. On AllegroKuka tasks, PPO and PQL barely make progress and SAPG beats PBT. On Shadow Hand and Allegro Kuka Reorientation, SAPG performs best with an entropy coefficient of 0.005 while the coefficient is 0 for other environments. On ShadowHand and AllegroHand, while PQL is initially more sample efficient, SAPG is more performant in the longer run. AllegroKuka environments use successes as a performance metric while AllegroHand and ShadowHand use episode rewards.
+
+<table><tr><td>ALGORITHM</td><td>ALLEGROHAND</td><td>SHADOWHAND</td><td>REGRASPING</td><td>THROW</td><td>REORIENTATION</td></tr><tr><td>PPO (SCHULMAN ET AL., 2017)</td><td>1.01e4±6.31e2</td><td>1.07e4±4.90e2</td><td>1.25 ± 1.15</td><td>16.8 ± 0.48</td><td>2.85 ± 0.05</td></tr><tr><td>PBT (PETRENKO ET AL., 2023)</td><td>7.28e3±1.24e3</td><td>1.01e4±1.80e2</td><td>31.9 ± 2.26</td><td>19.2 ± 1.07</td><td>23.2 ± 4.86</td></tr><tr><td>PQL (LI ET AL., 2023)</td><td>1.01e4±5.28e2</td><td>1.28e4±1.25e2</td><td>2.73 ± 0.02</td><td>2.62 ± 0.08</td><td>1.66 ± 0.11</td></tr><tr><td>SAPG (OURS, COEF=0)</td><td>1.23e4 ± 3.29e2</td><td>1.17e4±2.64e2</td><td>35.7 ± 1.46</td><td>23.7 ± 0.74</td><td>33.2 ± 4.20</td></tr><tr><td>SAPG (OURS, COEF=0.005)</td><td>9.14e3±8.38e2</td><td>1.28e4 ± 2.80e2</td><td>33.4 ± 2.25</td><td>18.7 ± 0.43</td><td>38.6 ± 0.63</td></tr></table>
+
+Table 1. Performance after 2e10 samples for different methods with standard error. This is measured by successes for the AllegroKuka tasks and by episode rewards for in-hand reorientation tasks. Across environments, we find that our method performs better than baselines.
+
+ward scales to achieve higher success rate, whereas our method can only optimize a fixed reward function. Despite this, we see that SAPG achieves a  $12 - 66\%$  higher success rate than DexPBT on regrasping, throw and reorientation. SAPG performs  $66\%$  better than PBT on the challenging reorientation task. Note that vanilla PPO and PQL are unable to learn any useful behaviors on these hard tasks.
+
+# 6.2. In-hand reorientation
+
+The AllegroHand and ShadowHand reorientation tasks from Li et al. (2023) are comparatively easier since they have lower degrees of freedom and the object doesn't move around much and remains inside the hand. On these tasks, we observe that PQL and PPO are able to make significant progress. In particular, we find that PQL is very sample-efficient because it is off-policy and utilizes past data for updates. However, we find that SAPG achieves higher asymp-
+
+totic performance. This is because on-policy methods are better at latching onto high reward trajectories and do not have to wait several iterations for the Bellman backup to propagate back to initial states. As discussed previously, in large-scale settings in simulation, we are primarily concerned with asymptotic performance since we want to maximize the downstream performance of our agents (within a reasonable training time budget). We see that on Allegro-Hand, SAPG beats PQL by a  $21\%$  margin, while on the ShadowHand task it achieves comparable performance. On these tasks, both PBT and PPO generally perform worse. This is because PPO is not able to efficiently leverage the large batch size. PBT loses the benefit of its hyperparameter mutation because the environment is simpler and the default hyperparameters work well, so it roughly reduces to simple PPO in  $\frac{N}{M}$  environments.
+
+![](images/6e39b5e0eb010fd1b0201a1f955d84ed4e368989cec7099716bc49090143553d.jpg)
+
+![](images/c9d389f8f0d49cafe453c651a5516e01268f3239eb012e4ed4d6a05b0afbb420.jpg)
+
+![](images/fa1a2ce92a11e5e2174a25ddf5a60d1f58ef88e6f5cc6332f4440dea8d567a79.jpg)
+
+![](images/d4688adf993c9b53f1852d99a4ec1fefd636863d17c32263d1aa7b1d2dd07e24.jpg)
+
+![](images/2c19761b7b6ece09d677ee035a290c983204c1a22dcd6093d07a54f95ad937e5.jpg)
+
+![](images/a032e10112c66fd38bf6ecb179638f5c7b41e44c0c0d7ae627ae51e68147a8af.jpg)  
+Figure 6. Performance curves for ablations of our method. The variants of our method with a symmetric aggregation scheme or without an off-policy combination perform significantly worse. Entropy regularization affects performance across environments, giving a benefit in reorientation. Using a high off-policy ratio without subsampling data leads to worse performance on ShadowHand and AllegroHand.
+
+# 6.3. Ablations
+
+The core idea behind SAPG is to combine data from different policies instead of optimizing a single policy with an extremely large batch. In this section, we will analyze our specific design choices for how we combine data (choice of  $i$  and  $\mathcal{X}$  and  $\lambda$ ) and for how we enforce diversity among the data collected by the policies. In particular, we have the following variants
+
+- SAPG (with entropy coef) As discussed in sec. 5.2, here we add an entropy loss to the followers to encourage data diversity. We explore different choices for the scaling coefficient of this loss  $\sigma \in \{0, 0.005, 0.003\}$ .  
+- SAPG (high off-policy ratio) In SAPG, when updating the leader, we subsample the off-policy data from the followers such that the off-policy dataset size matches the on-policy data. This is done because off-policy data is typically noisier and we do not want to drown out the gradient from on-policy data. In SAPG with a high off-policy ratio, we remove the subsampling step and instead see the impact of computing the gradient on the entire combined off-policy + on-policy dataset.  
+- Ours (symmetric) In SAPG, we choose  $i = 1$  to be the "leader" and the rest are "followers". Only the leader receives off-policy data while the followers use the standard on-policy loss. A natural alternative is where there are no privileged policies and each policy is updated with off-policy data from all others as discussed in sec. 4.2.
+
+We observe that SAPG outperforms or achieves comparable
+
+performance to the entropy-regularized variant except in the Reorientation environment where the variant with coefficient  $5\mathrm{e} - 3$  performs up to  $16.5\%$  better. Reorientation is the hardest task out of the three AllegroKuka tasks and has a lot of scope for learning emergent strategies such as using the table to move the object around, etc. Explicit exploration might be useful in discovering these behaviors.
+
+The variant of ours which uses all the off-policy data is significantly worse on the AllegroHand and ShadowHand tasks and marginally worse on Regrasping and Throw environments. It is more sample efficient than SAPG on Reorientation but achieves lower asymptotic performance. This could be because in the simple environments, additional data has marginal utility. In the harder AllegroKuka environments, it is beneficial to use all the data initially since it may contain optimal trajectories that would otherwise be missed. However, once an appreciable level of performance is achieved, it becomes better to subsample to prevent the noise in the off-policy update from drowning out the on-policy gradient.
+
+Finally, the symmetric variant of our method performs significantly worse across the board. This is possibly because using all the data to update each policy leads to them converging in behavior. If all the policies start executing the same actions, the benefit of data diversity is lost and SAPG reduces to vanilla PPO. Of course, there is a rich space of possible algorithms depending on particular choices of how data is aggregated and diversity is encouraged of which we have explored a small fraction.
+
+![](images/44d5cd2a14ee3438a12c3ceb44b65825151a221ba2502e0a09bf49c25b912d4b.jpg)  
+Figure 7. Curves comparing reconstruction error for states visited during training using top-  $k$  PCA components for SAPG (Ours), PPO and a randomly initialized policy
+
+![](images/0f0fc5fa547292da56b7c1553c1aed4795143071d720d046440659b107ea0f55.jpg)
+
+![](images/92b97ae7daf52583c41eb4bd92190dd2c295a3947f9e67decefae5ed3cc7a30d.jpg)
+
+![](images/7595579e353973eaf78f6f22bed8b8564f7adb2a9401a3b66f9b7e0c736546db.jpg)
+
+![](images/15f89e060f335b8cfe09f1c70e1ec93de25f4c834a8b2a2ccfd2edc80e191487.jpg)  
+Figure 8. Curves comparing reconstruction error for states visited during training using MLPs with varying hidden layer dimensions for SAPG (Ours), PPO and a randomly initialized policy
+
+![](images/b36d73734883fd8cdab6afd6525dde2d04fc50f6c4b395223bc27e3628e6737e.jpg)
+
+![](images/e5d96e8f4f749e4e5f5332c08696b5a40ad539cbb772c6cb5ef73c4c60d7ddae.jpg)
+
+# 6.4. Diversity in exploration
+
+To analyze why our method outperforms the baseline method, we conduct experiments comparing the diversity of states visited by each algorithm during training. We devise two metrics to measure the diversity of the state space and find that our method beats PPO in both metrics.
+
+- PCA - We compute the reconstruction error of a batch of states using k most significant components of PCA and plot this error as a function of k. In general, a set that has variation along fewer dimensions of space can be compressed with fewer principal vectors and will have lower reconstruction error. This metric therefore measures the extent to which the policy explores different dimensions of state space. Figure-7 contains the plots for this metric. We find that the rate of decrease in reconstruction error with an increase in components is the slowest for our method.  
+- MLP - We train feedforward networks with small hidden layers on the task of input reconstruction on batches of environment states visited by our algorithm and PPO during training. The idea behind this is that if a batch of states has a more diverse data distribution then it should be harder to reconstruct the distribution using small hidden layers because high diversity implies that the distribution is less compressible. Thus,
+
+high training error on a batch of states is a strong indicator of diversity in the batch. As can be observed from the plots in Figure-8, we find that training error is consistently higher for our method compared to PPO across different hidden layer sizes.
+
+# 7. Conclusion
+
+In this work, we present a method to scale reinforcement learning to utilize large simulation capacity. We show how current algorithms obtain diminishing returns if we perform naive scaling by batch size and do not use the increased volume of data efficiently. Our method achieves state-of-the-art performance on hard simulation benchmarks.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
+
+# Acknowledgements
+
+We thank Alex Li and Russell Mendonca for fruitful discussions regarding the method and insightful feedback. We would also like to thank Mihir Prabhudesai and Kevin
+
+Gmelin for proofreading an earlier draft. This project was supported in part by ONR N00014-22-1-2096 and NSF NRI IIS-2024594.
+
+# References
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+
+# A. Task and Environment Details
+
+Hard Difficulty Tasks All three hard tasks are based on the Allegro-Kuka environments(Petrenko et al., 2023). These consist of an Allegro Hand (16 dof) mounted on a Kuka arm (7 dof). In each case, the robot must manipulate a cuboidal kept on a fixed table. The observation space is  $\mathbf{o}_t = [\mathbf{q},\dot{\mathbf{q}},\mathbf{x}_t,\mathbf{v}_t,\omega_t,\mathbf{g}_t,\mathbf{z}_t]$ , where  $\mathbf{q}$ ,  $\dot{\mathbf{q}}\in \mathbb{R}^{23}$  are the joint angles and velocities respectively of each joint of the robot,  $\mathbf{x}_t\in \mathbb{R}^7$  is the pose of the object,  $\mathbf{v}_t$  is its linear velocity and  $\omega_{t}$  is its angular velocity,  $\mathbf{g}_t$  is a task-dependent goal observation and  $\mathbf{z}_t$  is auxiliary information pertinent to solving the task such as if the object has been lifted. These tasks consist of a complex environment but a simple reward function allowing opportunities for emergent strategies to be learnt such as in-hand reorientation under the influence of gravity, reorientation against the table, different types of throws and grasps and so on. The performance of the agent in the above three tasks is measured by the successes metric which is defined as the number of successes in a single episode. Three tasks include:
+
+- Regrasping - The object must be lifted from the table and held near a goal position  $\mathbf{g}_t \in \mathbb{R}^3$  for  $K = 30$  steps. This is called a "success". The target position and object position are reset to a random location after every success. The success tolerance  $\delta$  defines the maximum error between object pose and goal pose for a success  $\| \mathbf{g}_t - (\mathbf{x}_t)_{0:3} \| \leq \delta$ . This tolerance is decreased in a curriculum from  $7.5\mathrm{cm}$  to  $1\mathrm{cm}$ , decremented by  $10\%$  each time the average number of successes in an episode crosses 3. The reward function is a weighted combination of rewards encouraging the hand to reach the object  $r_{reach}$ , a bonus  $r_{lift}$ , rewards encouraging the hand to move to goal location after lifting  $r_{target}$  and a success bonus  $r_{success}$ .  
+- Throw - The object must be lifted from the table and thrown into a bucket at  $\mathbf{g}_t \in \mathbb{R}^3$  placed out of reach of the arm. The bucket and the object position are reset randomly after every successful attempt. The reward function is similar to regrasping with the difference being that the target is now a bucket instead of a point.  
+- Reorientation - This task involves picking up the object and reorienting it to a particular target pose  $\mathbf{g}_{\mathbf{t}} \in \mathbb{R}^{7}$  (position + orientation). Similar to the regrasping task, there is a success tolerance  $\delta$  which is varied in a curriculum. The target pose is reset once the agent succeeds. This means the agents needs to the object in different poses in succession, which may sometimes entail placing the obxfject on the table and lifting it up in a different way. Here too, the reward function is similar to regrasping, with the goal now being a pose in  $\mathbb{R}^{7}$  instead of  $\mathbb{R}^{3}$
+
+Easy Difficulty Tasks: In addition, we test on the following dexterous hand tasks. As before, the observation space consists of the joint angles and velocities  $\mathbf{q}_t$ ,  $\dot{\mathbf{q}}_t$ , object pose  $\mathbf{x}_t$  and velocities  $\mathbf{v}_t$ ,  $\omega_t$ . Following previous works (Li et al., 2023), we use the net episode reward as a performance metric for the ShadowHand and AllegroHand tasks.
+
+- Shadow Hand: We test on in-hand reorientation task of a cube using the 24-DoF Shadow Hand((OpenAI et al., 2018)). The task is to attain a specified goal orientation (specified as a quaternion) for the cube  $\mathbf{g}_t \in \mathbb{R}^4$ . The reward is a combination of the orientation error and a success bonus.  
+- Allegro Hand: This is the same as the previous in-hand reorientation task but with the 16-DoF Allegro Hand instead.
+
+# B. Training hyperparameters
+
+We use two different sets of default hyperparameters for PPO in AllegroKuka and Shadow Hand tasks which are described below.
+
+# B.1. AllegroKuka tasks
+
+We use a Gaussian policy where the mean network is an LSTM with 1 layer containing 768 hidden units. The observation is also passed through an MLP of with hidden layer dimensions  $768 \times 512 \times 256$  and an ELU activation (Clevert et al., 2016) before being input to the LSTM. The sigma for the Gaussian is a fixed learnable vector independent of input observation.
+
+<table><tr><td>Hyperparameter</td><td>Value</td></tr><tr><td>Discount factor, γ</td><td>0.99</td></tr><tr><td>τ</td><td>0.95</td></tr><tr><td>Learning rate</td><td>1e-4</td></tr><tr><td>KL threshold for LR update</td><td>0.016</td></tr><tr><td>Grad norm</td><td>1.0</td></tr><tr><td>Entropy coefficient</td><td>0</td></tr><tr><td>Clipping factor ε</td><td>0.1</td></tr><tr><td>Mini-batch size</td><td>num_envs · 4</td></tr><tr><td>Critic coefficient λ&#x27;</td><td>4.0</td></tr><tr><td>Horizon length</td><td>16</td></tr><tr><td>LSTM Sequence length</td><td>16</td></tr><tr><td>Bounds loss coefficient</td><td>0.0001</td></tr><tr><td>Mini epochs</td><td>2</td></tr></table>
+
+# B.2. Shadow Hand
+
+We use a Gaussian policy where the mean network is an MLP with hidden layers dimensions  $512 \times 512 \times 256 \times 128$  and an ELU activation (Clevert et al., 2016)
+
+Table 2. Training hyperparameters for AllegroKuka tasks  
+
+<table><tr><td>Hyperparameter</td><td>Value</td></tr><tr><td>Discount factor, γ</td><td>0.99</td></tr><tr><td>τ</td><td>0.95</td></tr><tr><td>Learning rate</td><td>5e-4</td></tr><tr><td>KL threshold for LR update</td><td>0.016</td></tr><tr><td>Grad norm</td><td>1.0</td></tr><tr><td>Entropy coefficient</td><td>0</td></tr><tr><td>Clipping factor ε</td><td>0.1</td></tr><tr><td>Mini-batch size</td><td>num_envs · 4</td></tr><tr><td>Critic coefficient λ&#x27;</td><td>4.0</td></tr><tr><td>Horizon length</td><td>8</td></tr><tr><td>Bounds loss coefficient</td><td>0.0001</td></tr><tr><td>Mini epochs</td><td>5</td></tr></table>
+
+Table 3. Training hyperparameters for Shadow Hand
+
+# B.3. Allegro Hand
+
+We use a Gaussian policy where the mean network is an MLP with hidden layers dimensions  $512 \times 256 \times 128$  and an ELU activation.
+
+<table><tr><td>Hyperparameter</td><td>Value</td></tr><tr><td>Discount factor, γ</td><td>0.99</td></tr><tr><td>τ</td><td>0.95</td></tr><tr><td>Learning rate</td><td>5e-4</td></tr><tr><td>KL threshold for LR update</td><td>0.016</td></tr><tr><td>Grad norm</td><td>1.0</td></tr><tr><td>Entropy coefficient</td><td>0</td></tr><tr><td>Clipping factor ε</td><td>0.2</td></tr><tr><td>Mini-batch size</td><td>num_envs · 4</td></tr><tr><td>Critic coefficient λ&#x27;</td><td>4.0</td></tr><tr><td>Horizon length</td><td>8</td></tr><tr><td>Bounds loss coefficient</td><td>0.0001</td></tr><tr><td>Mini epochs</td><td>5</td></tr></table>
+
+Table 4. Training hyperparameters for Shadow Hand
+
+Note: In case of experiments with entropy based exploration, each block of environments has it's own learnable vector sigma which enable policies for different blocks to have different entropies.

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+        "type": "text",
+        "text": "Abstract",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Fast convolution algorithms, including Winograd and FFT, can efficiently accelerate convolution operations in deep models. However, these algorithms depend on high-precision arithmetic to maintain inference accuracy, which conflicts with the model quantization. To resolve this conflict and further improve the efficiency of quantized convolution, we proposes SFC, a new algebra transform for fast convolution by extending the Discrete Fourier Transform (DFT) with symbolic computing, in which only additions are required to perform the transformation at specific transform points, avoiding the calculation of irrational number and reducing the requirement for precision. Additionally, we enhance convolution efficiency by introducing correction terms to convert invalid circular convolution outputs of the Fourier method into effective ones. The numerical error analysis is presented for the first time in this type of work and proves that our algorithms can provide a  $3.68 \\times$  multiplication reduction for  $3 \\times 3$  convolution, while the Winograd algorithm only achieves a  $2.25 \\times$  reduction with similarly low numerical errors. Experiments carried out on benchmarks and FPGA show that our new algorithms can further improve the computation efficiency of quantized models while maintaining accuracy, surpassing both the quantization-alone method and existing works on fast convolution quantization.",
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+    {
+        "type": "text",
+        "text": "1. Introduction",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Convolution operations are a crucial component of many deep learning models. Due to their intensive computation requirements, optimizing convolution calculations is key to",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "$^{1}$ School of Electronic Science and Engineering, Nanjing University, Nanjing 210023, China  $^{2}$ Interdisciplinary Research Center for Future Intelligent Chips (Chip-X), Nanjing University, Suzhou 215163, China. Correspondence to: Yuan Du <yuandu@nju.edu.cn>, Li Du <ldu@nju.edu.cn>.",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "improving model deployment efficiency. Fast Convolution Algorithms and Quantization are two distinct approaches to mitigate the computational burden. Fast convolution algorithms can reduce the arithmetic complexity by multiplying inputs and kernel weights in the transformation domain. A  $3 \\times 3$  convolution performed with Winograd  $\\mathrm{F}(2 \\times 2, 3 \\times 3)$  algorithm consumes just  $\\frac{1}{2.25}$  multiplications compared to direct computing. Whereas, model quantization reduces the cost of a single arithmetic operation and data transmission by converting high-precision floating numbers to low-precision integers. An int8 multiply-and-accumulate operation consumes only  $\\frac{1}{16}$  of the energy compared to a fp32 operation.",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, when attempting to combine existing fast convolution algorithms and model quantization to further improve computational efficiency, the model accuracy could be severely degraded. This is due to the significant increase in numerical error when the two methods are used together. For example, Winograd, a well-known fast convolution algorithm for small filter sizes (Lavin & Gray, 2016), has ill-conditioned transformation matrices (with high condition numbers), necessitating the use of high-precision arithmetic to avoid numerical issues (Barabasz et al., 2020). Another renowned fast algorithm for convolution is the Fast Fourier transform (FFT). While its transformation is well-conditioned, its irrational coefficients can introduce significant rounding errors, particularly under low-precision representation. Number Theoretic Transform (NTT) can achieve precise computation for integer convolution, but it involves module operations and the transformations for inputs and filters would extend the bit-width to be equivalent to that of the outputs.",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "At present, two research approaches have been developed to tackle the aforementioned problem. One approach involves customizing the quantization method specifically optimized for fast convolution algorithms (Li et al., 2021; Chikin & Kryzhanovskiy, 2022; Andri et al., 2022; Tianqi et al., 2023). However, this approach struggles to maintain satisfactory accuracy under int8 quantization for faster algorithms such as Winograd F(4×4, 3×3). The other approach is to explore new fast algorithms that are better fit for quantization (Liu & Mattina, 2020; Alam et al., 2022). Nevertheless, these emerging algorithms encounter challenges in achieving low arithmetic complexity. These facts show",
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+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "1",
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+    },
+    {
+        "type": "text",
+        "text": "that simultaneously achieving low arithmetic complexity and low-precision quantization while maintaining model accuracy is a persistent challenge, which, to our knowledge, no existing work has overcome.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This paper aims to develop a new efficient fast convolution algorithm with high numerical accuracy, which is compatible with model quantization techniques. We note that all the complex coefficients in the Fourier Transform have constant modulus 1, so the quantization error can be reduced by bypassing the direct calculation on these coefficients. To achieve this, we introduce symbolic computing to avoid the involvement of irrational numbers, where the transformation process contains only additions. This advance adapts the Fourier transform to low-precision arithmetic. We also discovered that conventional Fourier convolution does not fully utilize its circular convolution outputs, which significantly affects its efficiency, so we add correction terms to convert these neglected outputs into valid results. Moreover, we investigate the error generation mechanism of convolution algorithms and compare the numerical errors of direct convolution, Winograd, and our algorithms. The key contributions can be summarized as follows:",
+        "bbox": [
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+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. We formulate an efficient and quantization-friendly fast convolution algorithm extended from Fourier Convolution. We employ symbolic computing to perform Discret Fourier Transformation by just additions, and introduce correction terms to fully utilize its circular convolution outputs and further enhance its efficiency.",
+            "2. We analyse the numerical error of direct convolution and other fast convolution algorithms. Our method can achieve  $3.68 \\times$  arithmetic reduction while the Winogard algorithm only achieves  $2.25 \\times$  at equivalent numerical accuracy. Through error analysis and observation of the energy distribution in the frequency domain, we also present a frequency-wise quantization strategy to improve model accuracy at low-bitwidth.",
+            "3. Experiments on the ImageNet dataset validate that our method can achieve less than  $0.2\\%$  accuracy degradation with int8 post-training quantization. In the same model accuracy, our method achieves up to  $2.5\\times$  bitoperations reduction compared to Winograd convolution or direct convolution under quantization. FPGA implementation shows that our algorithms can significantly improve model inference throughput combined with low-precision arithmetic."
+        ],
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. Related Work",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Fast Convolution Algorithms. The FFT was the first utilized algorithm (Mathieu et al., 2014) to fast the training of convolutions. Subsequently, for small convolutions,",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "the Winograd minimum filtering algorithm (Lavin & Gray, 2016) was found to outperform the Fourier-based method due to its real field arithmetic operations, whereas the Fourier method requires more inefficient arithmetic in complex field. Additionally, the NTT has also been proposed to accelerate convolutions (Hong et al., 2022). However, when combining quantization and fast convolution algorithms, the challenge of potential model accuracy degradation arises. The Winograd algorithm is susceptible to numerical instability due to the ill-conditioned Vandermonde matrix in the transformation (Vincent et al., 2017; Barabasz et al., 2020). Fourier-based methods demand a high-precision format to accurately represent irrational numbers. NTT methods can offer accurate integer computing, but involve a large number of modulo operations and high-bitwidth intermediate data representations, reducing computation efficiency.",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Quantization for Fastconv. Some approaches aim to optimize the quantization method to regain model accuracy. For example, LoWino (Li et al., 2021) presents a posttraining quantization (PTQ) method for Winograd, optimizing the scaling factor by minimizing the KL distance between the quantized and original vectors. Another PTQ work (Chikin & Kryzhanovskiy, 2022) introduces a balancing operation between the filter and input channels to enhance bit-width utilization and improve the quality of quantization for Winograd. Additionally, a full quantization method with optimizing the transformation matrix in Winograd has been proposed (Tianqi et al., 2023), which successfully restores model accuracy when employing the Winograd  $\\mathrm{F}(4\\times 4,3\\times 3)$  algorithm with int8 quantization. Nevertheless, the methods above tend to struggle to achieve satisfactory accuracy recovery under sub-int8 quantization.",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Numerical Accuracy for Fastconv. Another approaches focus on improving the intrinsic properties of the fast algorithm itself. As Winograd algorithms can be defined by root points, a bilinear approach that strikes a balance between computational complexity and numerical accuracy has been proposed (Barabasz & Gregg, 2019). Additionally, two existing works (Barabasz et al., 2020; Alam et al., 2022) aimed to discover more effective polynomial root points to improve numerical accuracy. The Winograd algorithms have also been extended to the Residue Number System (RNS) (Liu & Mattina, 2020), decomposing single high-precision intermediate multiplications into multiple low-precision arithmetics (e.g., 8-bit). However, these all come at the cost of increased computational complexity.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3. Preliminaries",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Algorithms Construction. Fast convolution algorithms, including Winograd, Fourier Transform, and Number Theoretic Transform all employ a three-stage computing process: transformations of filters and inputs, element-wise multi",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "SFC: Achieve Accurate Fast Convolution under Low-precision Arithmetic",
+        "bbox": [
+            251,
+            56,
+            720,
+            70
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "plication, and a transformation for generating outputs. The generalized form for fast 2D convolution is as follows:",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ny = A ^ {T} \\left[ \\left[ G f G ^ {T} \\right] \\odot \\left[ B ^ {T} x B \\right] \\right] A \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+            128,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $\\odot$  denotes element-wise multiplication, and  $B$ ,  $G$ , and  $A$  represent the linear transformations of the input, filter, and multiplication result.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For one specific algorithm (whether Winograd, FFT or NTT), the  $G$ ,  $B$  and  $A$  are all derive from a Vandermonde matrix  $V$ , which consist of a set of root points  $s_0..s_{N-1}$ :",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nV = \\left[ \\begin{array}{c c c c c} 1 & s _ {0} ^ {1} & s _ {0} ^ {2} & \\dots & s _ {0} ^ {N - 1} \\\\ 1 & s _ {1} ^ {1} & s _ {1} ^ {2} & \\dots & s _ {1} ^ {N - 1} \\\\ \\ddots & \\ddots & \\ddots & \\dots & \\ddots \\\\ 1 & s _ {N - 1} ^ {1} & s _ {N - 1} ^ {2} & \\dots & s _ {N - 1} ^ {N - 1} \\end{array} \\right] \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A  $N \\times N$  matrix  $V$  and its inverse  $V^{-1}$  can construct a fast convolution algorithm for  $R \\times R$  filter accommodating  $N \\times N$  inputs with  $M \\times M$  outputs, or  $M \\times M$  inputs with  $N \\times N$  outputs, where  $N = M + R - 1$ .",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Difference. The fundamental difference among various algorithms lies in the number field of  $V$  and the chosen  $S_{n}$ . In the Winograd algorithm (also known as Toom-Cook algorithm), the  $S_{n}$  are  $N$  interpolation points selected in the real number field  $\\mathbb{R}$ . Similarly, all arithmetic operations are performed in the  $\\mathbb{R}$ . As a comparison, all arithmetic in Fourier convolution is defined in the complex field  $\\mathbb{C}$ . And the matrix  $V$  is the discrete Fourier transform matrix, where  $S_{n} = e^{-j\\frac{2\\pi*n}{N}}$ . The number theoretic transformation is similar in structure to the Fourier transform, but it operates in a finite field denoted as  $\\mathbb{F}_{p}$ .",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Arithmetic Complexity Reduction. Assuming these transformations are lightweight compared to element-wise multiply and can be amortized due to channel reuse, the fast algorithms consume  $N^2 = (M + R - 1)^2$  multiplications to generate  $M^2$  outputs, where the arithmetic complexity reduction is  $\\frac{M^2R^2}{(M + R - 1)^2}$ . However, convolution operations in CNNs are generally defined in  $\\mathbb{R}$ , so employing fast algorithms defined in  $\\mathbb{C}$  or  $\\mathbb{F}_p$  (such as FFT and NTT) would lead to waste in the calculation. Hence, Winograd, defined in  $\\mathbb{R}$ , is the most popular algorithm for model acceleration.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Precision Requirement. For Winograd, the extremum of a row in  $V$  is  $[1, S_n^{N-1}]$ . So the required arithmetic precision grows exponentially with  $N$ . Thus only the Winograd algorithm with small  $N$  is practical. In comparison, the FFT method performs a significant numerical advantage when dealing with large filters due to its numerically stable  $V$  matrix. However, performing accurate Fourier transforms necessitates high-precision arithmetic. NTT method provide a bit-correct result for integral convolution. However, when using NTT to perform a calculation with  $N$ -bit inputs and",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$2N$ -bit outputs, the transformed inputs must have a datawidth of at least  $2N$ -bit, which increases the datawidth of  $\\odot$",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In summary, no matter which type of algorithm is chosen, achieving both robust computation and significant arithmetic reduction under low-precision operations remains a challenge.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4. Symbolic Fourier Convolution Algorithm",
+        "text_level": 1,
+        "bbox": [
+            496,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "It is worth noting that the Fourier transform has better numerical stability, as all its root points are distributed on a circle of radius 1 in the complex plane. When dealing with larger  $N$ , it is more accurate than Winograd. However, Fourier transform has two serious drawbacks. First, its irrational coefficients are not friendly for low-precision format and would give more computation burden for transformation calculation.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In addition, the FFT is not as efficient as Winograd. There are two reasons for this. Firstly the FFT is computed using complex numbers and even after utilizing the Hamiltonian symmetry with real sequences and the fast complex multiplication, each complex multiplication still requires 1.5 real multiplications. Secondly, the direct calculation result of the FFT is a circular convolution, so padding the sequence with zeros to achieve a linear convolution is required, which would wasted computation.",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose two key improvement strategies to address these drawbacks:",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. We employ symbolic computation rather than numerical computation to implement the discrete Fourier transform (DFT). By selecting an appropriate number of DFT points, we can avoid or minimize the irrational values involved in computing. All complex points can be represented by first order integer coefficient polynomials under both 4 and 6 DFT points.",
+            "2. We introduce correction terms to fully exploit the cyclic convolution output generated by the Fourier method to enhance computing utilization, and the smaller number of transformation points we chose also helps to reduce the proportion of complex arithmetic."
+        ],
+        "bbox": [
+            509,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.1. Symbolic Computing for DFT",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Generally, the coefficients of the N-point DFT are derived from:",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ne ^ {\\frac {2 \\pi n}{N} j} = \\cos (\\frac {2 \\pi n}{N}) + j \\sin (\\frac {2 \\pi n}{N}), n = 0, 1,.., N - 1\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "when  $\\frac{n}{N} \\notin \\{0, \\frac{1}{4}, \\frac{1}{2}, \\frac{3}{4}\\}$ , irrational values will be introduced. To eliminate the rounding errors that arise from these irrational values, we employ symbolic computation rather than",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "SFC: Achieve Accurate Fast Convolution under Low-precision Arithmetic",
+        "bbox": [
+            251,
+            56,
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+            70
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "numerical calculation. This approach represents irrational values in polynomials with integer coefficients.",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To illustrate, we consider the 3-point DFT. For a real input sequence  $x = (x_0, x_1, x_2)^T$ , its transform processing can be represented as:",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ \\begin{array}{l} X _ {0} \\\\ X _ {1} \\\\ X _ {2} \\end{array} \\right] = \\left[ \\begin{array}{c c c} 1 & 1 & 1 \\\\ 1 & s & s ^ {2} \\\\ 1 & s ^ {2} & s \\end{array} \\right] \\left[ \\begin{array}{l} x _ {0} \\\\ x _ {1} \\\\ x _ {2} \\end{array} \\right], s = e ^ {\\frac {2 \\pi}{3} j} \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            145,
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+            224
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/65bb346deec562161c6184ad7c62c75dc9cd9e86854c37ede51c0b5477aac6c6.jpg",
+        "image_caption": [
+            "Figure 1. The geometric symmetry in DFT-3, DFT-4 and DFT-6."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            86,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/bec506738be52c3e3adee0414e11b9490f48350d3a69abc3cd7768770890cfa2.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/1fd83256e50f46dd4afc5766c117217059b67e844725e3b7891454b65ae28ba5.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            359,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We do not substitute the numerical value of  $s$  into the calculation. Instead, we represent and compute subsequent variables using the polynomial form of  $s$ . This allows us to express the DFT outputs  $X_{n}$  as  $X_{n} = X_{n,0} + X_{n,1} \\cdot s + X_{n,2} \\cdot s^{2}$ .",
+        "bbox": [
+            84,
+            409,
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+            484
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "By exploiting the geometric symmetry among  $1, s, s^2$ , the  $s^2$  can be expressed as  $s^2 = -(s^0 + s^1)$ , as Figure 1. Further, the Hamiltonian symmetry of the real signal Fourier transform can reduce the number of components by almost half. Thus we can rearrange Equation 3 as following:",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ \\begin{array}{l} X _ {0} \\\\ X _ {1} \\\\ X _ {2} \\end{array} \\right] = \\left[ \\begin{array}{l l l} 1 & 0 & 0 \\\\ 0 & 1 & s \\\\ 0 & 1 & s ^ {2} \\end{array} \\right] \\left[ \\begin{array}{l l l} 1 & 1 & 1 \\\\ 1 & 0 & - 1 \\\\ 0 & 1 & - 1 \\end{array} \\right] \\left[ \\begin{array}{l} x _ {0} \\\\ x _ {1} \\\\ x _ {2} \\end{array} \\right] \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            137,
+            585,
+            473,
+            635
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the above formula,  $s = e^{\\frac{2\\pi}{3} j}$  is not need to be explicitly included in the calculation but serves as a notation from the outset. Similar to complex number, the production of  $a_0 + a_1s$  and  $b_0 + b_1s$  can be calculated as:",
+        "bbox": [
+            84,
+            648,
+            475,
+            710
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} (a _ {0} + a _ {1} s) (b _ {0} + b _ {1} s) = c _ {0} + c _ {1} s, \\quad \\left[ \\begin{array}{c} c _ {0} \\\\ c _ {1} \\end{array} \\right] = \\\\ \\left[ \\begin{array}{c c c} 1 & - 1 & 0 \\\\ 1 & 0 & 1 \\end{array} \\right] (\\left[ \\begin{array}{l l} 1 & 0 \\\\ 0 & 1 \\\\ - 1 & 1 \\end{array} \\right] \\left[ \\begin{array}{l} a _ {0} \\\\ a _ {1} \\end{array} \\right] \\odot \\left[ \\begin{array}{l l} 1 & 0 \\\\ 0 & 1 \\\\ 1 & - 1 \\end{array} \\right] \\left[ \\begin{array}{l} b _ {0} \\\\ b _ {1} \\end{array} \\right]) \\tag {5} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            109,
+            738,
+            473,
+            823
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Through enumeration, we can identify that 6 and 4 are suitable choices for DFT points in small-size convolution applications.",
+        "bbox": [
+            84,
+            837,
+            473,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1) DFT-6",
+        "bbox": [
+            89,
+            890,
+            156,
+            904
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Considering DFT-6, its transformation coefficients consist of six values:  $1, e^{j\\frac{\\pi}{3}}, e^{j\\frac{2\\pi}{3}}, -1, e^{j\\frac{4\\pi}{3}}, e^{j\\frac{5\\pi}{3}}$ . Defining  $s = e^{\\frac{\\pi}{3} j}$ , thus  $s^2 = s - 1$ , which allows all coefficients to be expressed as first-order polynomials of  $s$ :  $1, s, s - 1, -1, -s, 1 - s$ . When two first-degree polynomials are multiplied, any quadratic term can be reduced to a first-degree term using the rule  $s^2 = s - 1$ . Therefore, the DFT-6 transform processing under symbolic computation is as follows:",
+        "bbox": [
+            496,
+            84,
+            887,
+            219
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} D F T 6 (x) = S _ {6} F _ {6} x = \\left[ \\begin{array}{c c c c c c} 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 1 & s & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & s & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 1 & - s ^ {2} & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & - s ^ {2} & 0 \\end{array} \\right]. \\\\ \\left[ \\begin{array}{c c c c c c} 1 & 1 & 1 & 1 & 1 & 1 \\\\ 1 & 1 & 0 & - 1 & - 1 & 0 \\\\ 0 & - 1 & - 1 & 0 & 1 & 1 \\\\ 1 & 0 & - 1 & 1 & 0 & - 1 \\\\ 0 & - 1 & 1 & 0 & - 1 & 1 \\\\ 1 & - 1 & 1 & - 1 & 1 & - 1 \\end{array} \\right] \\cdot \\left[ \\begin{array}{c} x _ {0} \\\\ x _ {1} \\\\ x _ {2} \\\\ x _ {3} \\\\ x _ {4} \\\\ x _ {5} \\end{array} \\right] \\tag {6} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            511,
+            222,
+            885,
+            424
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here,  $S_{6}$  represents the format transition from symbolic to numerical computing without any arithmetic, and  $F_{6}$  signifies the Fourier transform under symbolic computing. We refer to the intermediate matrix as SFT-6 (Symbolic Fourier Transform-6), as its coefficients consist solely of 1, -1, and 0. The inverse transform iDFT6 can be rearranged in a similar way.  $F_{6}$  has its fast algorithm by decomposing it into DFT-3 and DFT-2. Therefore, only 14 addition operations are needed to perform SFT-6 (subtraction can be considered as addition of complement). The inverse transformation matrix  $iF_{6}$  can be derived in a similar way:",
+        "bbox": [
+            496,
+            431,
+            887,
+            598
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ni F _ {6} = \\frac {1}{6} \\left[ \\begin{array}{r r r r r r} 1 & 1 & 1 & 1 & 1 & 1 \\\\ 1 & - 1 & - 2 & - 1 & 1 & 2 \\\\ - 1 & - 2 & - 1 & 1 & 2 & 1 \\\\ - 1 & - 1 & 2 & - 1 & - 1 & 2 \\\\ - 2 & 1 & 1 & - 2 & 1 & 1 \\\\ - 1 & 1 & - 1 & 1 & - 1 & 1 \\end{array} \\right] \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            550,
+            602,
+            885,
+            696
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that  $\\frac{1}{6}$  can be fused into floating model without having to compute that division operation during inference stage.",
+        "bbox": [
+            496,
+            702,
+            885,
+            733
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the element-wise multiplication steps, multiplications are performed in polynomial form. Multiplying two 1st-order polynomials requires 4 real number multiplications. We can utilize a short fast convolution algorithm to reduce it to 3:",
+        "bbox": [
+            496,
+            739,
+            885,
+            801
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} (a _ {0} + a _ {1} s) (w _ {0} + w _ {1} s) = o _ {0} + o _ {1} s, \\quad \\left[ \\begin{array}{c} o _ {0} \\\\ o _ {1} \\end{array} \\right] = \\\\ \\left[ \\begin{array}{c c c} 1 & - 1 & 0 \\\\ - 1 & 0 & 1 \\end{array} \\right] (\\left[ \\begin{array}{l l} 1 & 0 \\\\ 0 & 1 \\\\ 1 & 1 \\end{array} \\right] \\left[ \\begin{array}{l} a _ {0} \\\\ a _ {1} \\end{array} \\right] \\odot \\left[ \\begin{array}{l l} 1 & 0 \\\\ 0 & 1 \\\\ 1 & 1 \\end{array} \\right] \\left[ \\begin{array}{l} w _ {0} \\\\ w _ {1} \\end{array} \\right]) \\tag {8} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            517,
+            825,
+            885,
+            909
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "SFC: Achieve Accurate Fast Convolution under Low-precision Arithmetic",
+        "bbox": [
+            251,
+            56,
+            718,
+            70
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2) DFT-4",
+        "bbox": [
+            89,
+            85,
+            156,
+            99
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Similarly, the DFT-4 can be constructed in the same manner.",
+        "bbox": [
+            84,
+            107,
+            475,
+            122
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} D F T 4 (x) = S _ {4} F _ {4} x = \\left[ \\begin{array}{c c c c} 1 & 0 & 0 & 0 \\\\ 0 & 1 & j & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 1 & - j & 0 \\end{array} \\right]. (9) \\\\ \\left[ \\begin{array}{r r r r} 1 & 1 & 1 & 1 \\\\ 1 & 0 & - 1 & 0 \\\\ 0 & - 1 & 0 & 1 \\\\ 1 & - 1 & 1 & - 1 \\end{array} \\right] \\left[ \\begin{array}{l} x _ {0} \\\\ x _ {1} \\\\ x _ {2} \\\\ x _ {3} \\end{array} \\right] (9) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            129,
+            130,
+            473,
+            261
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The multiplication on the DFT-4 convolution can be performed as:",
+        "bbox": [
+            84,
+            276,
+            475,
+            306
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} (a _ {0} + a _ {1} s) (w _ {0} + w _ {1} s) = o _ {0} + o _ {1} s, \\quad \\left[ \\begin{array}{c} o _ {0} \\\\ o _ {1} \\end{array} \\right] = \\\\ \\left[ \\begin{array}{c c c} 1 & - 1 & 0 \\\\ - 1 & - 1 & 1 \\end{array} \\right] \\left(\\left[ \\begin{array}{l l} 1 & 0 \\\\ 0 & 1 \\\\ 1 & 1 \\end{array} \\right] \\left[ \\begin{array}{l} a _ {0} \\\\ a _ {1} \\end{array} \\right] \\odot \\left[ \\begin{array}{l l} 1 & 0 \\\\ 0 & 1 \\\\ 1 & 1 \\end{array} \\right] \\left[ \\begin{array}{l} w _ {0} \\\\ w _ {1} \\end{array} \\right]\\right) \\tag {10} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            102,
+            311,
+            473,
+            396
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "If we want to compute  $A((Gf) \\odot (Bx))$  directly in the real number field, akin to the Winograd algorithm, without involving polynomial multiplication, we can integrate Eq. 8 or Eq. 10 into the SFT matrix Eq. 6 or Eq. 9. In the 1D case, this does not impact efficiency. However, in the 2D case, it introduces slight redundant components and marginally reduces the acceleration ratio. We list these algorithms with polynomial multiplication integrated in the appendix.",
+        "bbox": [
+            84,
+            412,
+            475,
+            534
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2. Achieving Efficient Linear Convolution",
+        "text_level": 1,
+        "bbox": [
+            84,
+            550,
+            392,
+            564
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The conventional Fourier Transform method inherently generates cyclic convolution. As a consequence, only  $(N - R + 1)^2$  components are valid for the intended  $N^2$  linear convolution. However, it's noteworthy that the invalid results are not entirely useless. They actually contain partial sums that can be effectively utilized. By intelligently applying correction terms to these partial sums, it is possible to convert them into valid outputs. This approach significantly enhances the efficiency of the convolution process.",
+        "bbox": [
+            84,
+            573,
+            475,
+            710
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Figure 2 illustrates an example of Fourier-based cyclic convolution for  $N = 6$  and  $R = 3$ . The first term  $o_1^c$  is equal to  $a_6w_1 + a_1w_2 + a_2w_1$ , but the desired output is  $o_1 = a_0w_1 + a_1w_2 + a_2w_1$ . To align  $o_1$  with  $o_1^c$ , a corrective term is introduced to obtain the desired output:  $o_1 = o_1^c + (a_0 - a_6)w_1$ . This adjustment allows us to obtain an additional correct result by adding just one MAC operation, thus utilizing the Fourier convolution output more efficiently compared to discarding erroneous terms.",
+        "bbox": [
+            84,
+            715,
+            475,
+            853
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To unambiguously represent a particular algorithm, we employ the notation SFC-  $N(M,R)$  , where  $N$  represents the length of the SFT transformation,  $M$  represents the feature",
+        "bbox": [
+            84,
+            859,
+            475,
+            906
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "tile size, and  $R$  represents the kernel size. For example, the SFC-6(6×6, 3×3) algorithm is constructed based on a 6-point Fourier transform, employing a 3×3 kernel size and a 6×6 feature tile size.",
+        "bbox": [
+            496,
+            84,
+            885,
+            143
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "By introducing correction terms, we can also adjust input tile size  $M$  independently. For example, as the images in the ImageNet dataset are in size  $224 \\times 224$ , the feature map size in the model has a common factor of 7. Utilizing the SFC-6(7×7, 3×3) algorithm to infer models trained on Imagenet would have higher tiling efficiency without the need for zero padding. The transformation matrix integrated polynomial multiplication of the SFC-6(7×7, 3×3) is as follows:",
+        "bbox": [
+            496,
+            152,
+            885,
+            273
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} B ^ {T} = \\left[ \\begin{array}{c c c c c c c c c} 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\\\ 0 & 1 & 1 & 0 & - 1 & - 1 & 0 & 0 & 0 \\\\ 0 & 0 & - 1 & - 1 & 0 & 1 & 1 & 0 & 0 \\\\ 0 & 1 & 0 & - 1 & - 1 & 0 & 1 & 0 & 0 \\\\ 0 & 1 & 0 & - 1 & 1 & 0 & - 1 & 0 & 0 \\\\ 0 & 0 & - 1 & 1 & 0 & - 1 & 1 & 0 & 0 \\\\ 0 & 1 & - 1 & 0 & 1 & - 1 & 0 & 0 & 0 \\\\ 0 & 1 & - 1 & 1 & - 1 & 1 & - 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 \\\\ 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 \\end{array} \\right], \\\\ A = \\frac {1}{6} \\left[ \\begin{array}{c c c c c c c} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 2 & 1 & - 1 & - 2 & - 1 & 1 & 2 \\\\ - 1 & 1 & 1 & 1 & - 1 & - 2 & - 1 \\\\ - 1 & - 2 & - 1 & 1 & 2 & 1 & - 1 \\\\ 1 & - 2 & 1 & 1 & - 2 & 1 & 1 \\\\ 1 & 1 & - 2 & 1 & 1 & - 2 & 1 \\\\ - 2 & 1 & 2 & - 2 & 1 & 1 & - 2 \\\\ - 1 & 1 & - 1 & 1 & - 1 & 1 & - 1 \\\\ 6 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 6 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 6 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 6 \\end{array} \\right], \\tag {11} \\\\ G = \\left[ \\begin{array}{c c c} 1 & 1 & 1 \\\\ 0 & 1 & 1 \\\\ - 1 & - 1 & 0 \\\\ - 1 & 0 & 1 \\\\ - 1 & 0 & 1 \\\\ 1 & - 1 & 0 \\\\ 0 & - 1 & 1 \\\\ 1 & - 1 & 1 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right] \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            519,
+            280,
+            885,
+            612
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The SFC-  $6(6 \\times 6, 3 \\times 3)$  algorithm can reduce  $73\\%$  multiplications for  $3 \\times 3$  convolutions. And multiplications can be optimized by  $81\\%$  and  $79\\%$  for  $5 \\times 5$  and  $7 \\times 7$  convolutions, respectively. A selection of achievable Symbolic Fourier Convolution algorithms is listed in Table 1. For comparison, we also list their efficiencies and the numerical errors obtained from simulation and theoretical analyses (Detailed in Section 5). In Table 1, we can see that for  $3 \\times 3$  convolution the SFC-  $6(6 \\times 6, 3 \\times 3)$  algorithm is  $1.64 \\times$  faster than the Winograd  $(2 \\times 2, 3 \\times 3)$  algorithm, while maintaining nearly the same numerical error.",
+        "bbox": [
+            495,
+            626,
+            885,
+            791
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Further, for larger kernel size, the Winograd algorithm is no longer able to construct linear transformations with sufficiently low numerical errors, no matter how the root points are chosen. Although there exists work by splitting the core, making it possible to compute large size convolutions with the Winograd  $(2 \\times 2, 3 \\times 3)$  algorithm with low numerical error (Huang et al., 2020), it is not possible to achieve a lower",
+        "bbox": [
+            495,
+            799,
+            885,
+            905
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "SFC: Achieve Accurate Fast Convolution under Low-precision Arithmetic",
+        "bbox": [
+            251,
+            56,
+            718,
+            70
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/598ba02ddbf4fa9d8137c2e4fa6a60ca3ff6f15c41d9ed1f5839ca36d8956a77.jpg",
+        "image_caption": [
+            "Figure 2. Converting invalid cyclic convolution to linear convolution, which is a process that can also be used to adjust the input tile sizes."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            161,
+            85,
+            803,
+            330
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "arithmetic complexity than Winograd  $(2 \\times 2, 3 \\times 3)$ . Whereas our algorithm provides higher speedups and keeps the high numerical stability.",
+        "bbox": [
+            84,
+            387,
+            473,
+            434
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/a9b98f893973faacbd49d609fc7ced426faead155330f5a700844bae384bf7ce.jpg",
+        "table_caption": [
+            "Table 1. Comparison of different fast convolution algorithms. Mean square error was measured on randomly generated data, and  $\\kappa (A^T)$  was calculated using the singular value of  $A^T$ ."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm</td><td>Mean Square Error</td><td>κ(AT)</td><td>Arithmetic Complexity</td></tr><tr><td>direct convolution</td><td>1.0</td><td>1.0</td><td>100%</td></tr><tr><td>Wino(2×2, 3×3)</td><td>2.2</td><td>2.4</td><td>44.4%</td></tr><tr><td>Wino(3×3, 3×3)</td><td>6.4</td><td>14.5</td><td>30.4%</td></tr><tr><td>Wino(4×4, 3×3)</td><td>10.5</td><td>20.1</td><td>25%</td></tr><tr><td>SFC-4(4×4, 3×3)</td><td>2.4</td><td>2.7</td><td>31.94%</td></tr><tr><td>SFC-6(6×6, 3×3)</td><td>2.4</td><td>3.3</td><td>27.16%</td></tr><tr><td>SFC-6(7×7, 3×3)</td><td>2.6</td><td>3.4</td><td>29.93%</td></tr><tr><td>Wino(2×2, 5×5)</td><td>10.5</td><td>20.1</td><td>36%</td></tr><tr><td>SFC-6(6×6, 5×5)</td><td>3.6</td><td>3.5</td><td>20.44%</td></tr><tr><td>Wino(2×2, 7×7)</td><td>28.1</td><td>31.0</td><td>32.6%</td></tr><tr><td>SFC-6(4×4, 7×7)</td><td>3.6</td><td>3.5</td><td>21.99%</td></tr></table>",
+        "bbox": [
+            86,
+            510,
+            473,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5. Error Analysis and Frequency-wise Quantization",
+        "text_level": 1,
+        "bbox": [
+            84,
+            733,
+            408,
+            767
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This section would analyze the numerical error of fast convolution algorithms, which can be used to guide the development of quantization methods and serve as a benchmark for assessing the numerical stability across various fast algorithms.",
+        "bbox": [
+            84,
+            777,
+            475,
+            853
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To cover the direct convolution into the same error analysis model, we can consider it as a fast convolution with  $R = 3$ ,  $M = 1$ . For derivation convenience, we use the overlapped",
+        "bbox": [
+            84,
+            859,
+            475,
+            906
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "output form, in which  $A$  is a reversible square matrix.",
+        "bbox": [
+            496,
+            388,
+            852,
+            402
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ \\begin{array}{l} y _ {0} \\\\ y _ {1} \\\\ y _ {2} \\end{array} \\right] ^ {T} = \\left[ \\begin{array}{c c c} 1 & & \\\\ & 1 & \\\\ & & 1 \\end{array} \\right] \\cdot \\left(\\left[ \\begin{array}{c c c} 1 & & \\\\ & 1 & \\\\ & & 1 \\end{array} \\right] \\cdot \\left[ \\begin{array}{l} f _ {0} \\\\ f _ {1} \\\\ f _ {2} \\end{array} \\right]\\right) \\odot (\\left[ 1 \\right] \\cdot \\left[ x _ {0} \\right])) \\tag {12}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            500,
+            414,
+            885,
+            477
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We denote the quantized element-wise multiply as  $\\odot_{Q}$  through which the operands are quantized and multiplied.  $\\delta y$  represents the calculation error of the output  $y$ . The error forward propagation process can be described as:",
+        "bbox": [
+            496,
+            484,
+            888,
+            546
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ny + \\delta y = A ^ {T} \\cdot \\left(\\left(G \\cdot f\\right) \\odot_ {Q} \\left(B ^ {T} \\cdot x\\right)\\right) \\tag {13}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            563,
+            556,
+            885,
+            575
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We assume these transformations to be accurate, and the quantized operator  $\\odot$  introduces rounding errors. These errors would be subsequently magnified by the matrix multiplying  $A^T$ .",
+        "bbox": [
+            495,
+            587,
+            887,
+            648
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Set  $s + \\delta s = (G \\cdot f) \\odot_{Q} (B^{T} \\cdot x)$ , by substituting  $y = A^{T} \\cdot s$  into the Equation 13, we can obtain:",
+        "bbox": [
+            496,
+            652,
+            885,
+            686
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\delta y = \\left(A ^ {T}\\right) ^ {- 1} \\cdot \\delta s \\tag {14}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            627,
+            696,
+            885,
+            715
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Applying the properties of the paradigm  $||\\ast ||$  yields:",
+        "bbox": [
+            496,
+            733,
+            849,
+            750
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left| \\left| \\delta y \\right| \\right| <   = \\left| \\left| \\left(A ^ {T}\\right) \\right| \\right| \\cdot \\left| \\left| \\delta s \\right| \\right| \\tag {15}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            604,
+            760,
+            885,
+            779
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Similar, bringing in  $||s|| < = ||(A^T)^{-1}||\\cdot ||y||$  we have:",
+        "bbox": [
+            496,
+            789,
+            870,
+            806
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\frac {| | \\delta y | |}{| | y | |} <   = | | \\left(A ^ {T}\\right) ^ {- 1} | | \\cdot | | \\left(A ^ {T}\\right) | | \\cdot \\frac {| | \\delta s | |}{| | s | |} \\tag {16}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            560,
+            816,
+            885,
+            851
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here we can perform an analysis of Eq.16. The first term  $||(A^T)^{-1}||\\cdot ||(A^T)||$  is the condition number of matrix  $A^T$ , denoted as  $\\kappa (A^T)$ . This term indicates the amplification",
+        "bbox": [
+            496,
+            859,
+            887,
+            906
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "SFC: Achieve Accurate Fast Convolution under Low-precision Arithmetic",
+        "bbox": [
+            251,
+            56,
+            718,
+            70
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "factor applied to the error  $\\delta s$ . The condition number of the standard orthogonal matrix is constant 1, like  $A$  in direct convolution (Eq.1) and vanilla Fourier convolution, which would not amplify  $\\delta s$  anymore. While the  $\\kappa(A^T)$  for Winograd convolution can reach up to 31.0, as listed in Table 1. However, in our method, the  $\\kappa(A^T)$  takes the values of 2.7, 3.3, and 3.5 for SFC4(4,3), SFC6(6,3), and SFC6(4,7), respectively, which is much less than that of Winograd.",
+        "bbox": [
+            84,
+            84,
+            475,
+            205
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The final term  $\\frac{||\\delta s||}{||s||}$  in Eq.16 is the error caused by the quantized operator  $\\odot_{Q}$ , which depends on the data width, quantization method, and data distribution. For floating numbers, the error is fairly stationary as every operand has its own exponent code (scaling factor). We conduct numerical experiments under single precision (fp16) in Table 1, and it is found that the numerical error is highly correlated to the  $\\kappa(A^T)$ , in accordance with our analysis.",
+        "bbox": [
+            84,
+            212,
+            473,
+            335
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/debc9acc810a6d4d5c5504ba33ee699654413ad51f79d63bbb7f80c761e9e4e3.jpg",
+        "image_caption": [
+            "Figure 3. The energy distribution in the frequency domain of the 9-th layer in Resnet18. Energy is concentrated in the low frequencies."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            96,
+            369,
+            480,
+            488
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For model quantization, numbers in a group share the same scaling factor to achieve integer representation. The granularity of the scaling factor group should fit with the data distribution to achieve low quantization error. Some work has found that in Winograd convolution, scaling factor grouping based on the transformation domain coordinates can recover model accuracy effectively. Here we give a brief theoretical explanation. Assume that the input  $x$  has a fixed energy  $||x|| = 1$ , the maximum possible value in the transformation domain is  $(1,1,1,8,8,1)$  for  $\\mathrm{F}(4,3)$  algorithm, which would be quadratic in 2D convolution. Grouping one tensor to one scaling factor would cause a waste of 3/6 bit in 1D/2D. For Fourier convolution, each frequency would have the same maximum value. However, due to the frequency properties of the model inputs (image or audio), there will be a tendency for the energy to converge towards lower frequencies, as Figure 3 shows. Thus grouping scaling factors by frequency can also reduce the error of Fourier convolution, but not as necessary as Winograd. This is because the aforementioned tendency is only significant in the first few layers, and the magnitude variance is not as large as",
+        "bbox": [
+            84,
+            588,
+            475,
+            905
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Winograd.",
+        "bbox": [
+            496,
+            84,
+            570,
+            99
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ny = \\sum_ {C _ {i n}} \\left( \\right.s _ {T x} \\left\\lceil B ^ {T} x B / s _ {T x} \\left. \\right\\rfloor_ {i n t N _ {T x}} \\odot s _ {T f} \\left\\lceil G f G ^ {T} / s _ {T f} \\left. \\right\\rfloor_ {i n t N _ {T f}}\\left. \\right) \\tag {17}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            496,
+            111,
+            888,
+            154
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The scaling factor  $s_{Tx}$  is of size  $[T \\times T]$ , where  $T$  is the size of the transform domain. For the scaling factor  $s_{Tf}$  of weights, considering that per-channel quantization can achieve better results in direct convolutions, we suggest combining per-frequency quantization and per-channel quantization whose  $s_{Tf}$  is of size  $[OC \\times T \\times T]$  to achieve higher accuracy.",
+        "bbox": [
+            495,
+            162,
+            885,
+            268
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6. Experimental Evaluation",
+        "text_level": 1,
+        "bbox": [
+            496,
+            287,
+            732,
+            305
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We conducted experiments on image classification tasks to demonstrate the effectiveness of our algorithms. To comprehensively evaluate the computation cost of models accelerated by fast convolution algorithm and quantization, it is critical to consider both the reduction in arithmetic complexity afforded by fast convolution and the reduction in arithmetic data width introduced by quantization. Consequently, we adopt bit-operations (BOPs) as a metric of computation cost, diverging from the traditional floating-operations (FLOPs). In this metric, an n-bit addition operation requires n BOPs, whereas an n-bit multiplication costs  $\\mathrm{n}(\\mathrm{n} - 1)$  BOPs. This is because an n-bit multiplication can be decomposed into n-1 instances of n-bit additions. The transformation cost of fast algorithms is also taken into account.",
+        "bbox": [
+            495,
+            314,
+            887,
+            525
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.1. Post-training Quantization on Image Classification Benchmarks",
+        "text_level": 1,
+        "bbox": [
+            496,
+            542,
+            883,
+            570
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Experiments were conducted on the ImageNet dataset, which contains 1.4 million images of size  $224 \\times 224 \\times 3$ , distributed across 1,000 classes. We randomly selected 500 images from training set to create the calibration set for PTQ fine-tuning. Model accuracy was evaluated on the validation set. We utilized pre-trained fp32 models from TorchVision as our benchmarks. All batch normalization layers were fused into the preceding convolution layers prior to quantization.",
+        "bbox": [
+            495,
+            580,
+            885,
+            715
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We conducted quantization on the following algorithms: 1) Direct convolution, 2) The well-known Winograd  $\\mathrm{F}(4\\times 4$ $3\\times 3)$  algorithms, which have been extensively researched for their quantization methods in recent years, and 3) our proposed SFC algorithms, including 1D and 2D format. For all these methods, all  $3\\times 3$  convolution layers with a stride of 1 were replaced by the corresponding algorithm. Direct convolution and our SFC algorithms were quantized using AdaQuant (Hubara et al., 2020), while the Winograd algorithm was processed with Scaling Gradient Backward (Jain et al., 2020), due to observed convergence differences with AdaQuant in the Winograd  $\\mathrm{F}(4\\times 4,3\\times 3)$ . All the data",
+        "bbox": [
+            495,
+            724,
+            885,
+            904
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "SFC: Achieve Accurate Fast Convolution under Low-precision Arithmetic",
+        "bbox": [
+            251,
+            56,
+            718,
+            70
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "in the spatial domain are quantized to int8, and the data in the transformation domain are quantized from int8 to int4. As mentioned in Section 5, storing spatial domain data in external storage is unnecessary; instead, data can be stored in the transform domain to avoid errors caused by multiple quantization operators.",
+        "bbox": [
+            84,
+            84,
+            475,
+            175
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/c4b92c374e34b3f7ed633998ad564c3ac67c7d763a587309da47cd3f6ae1add8.jpg",
+        "image_caption": [
+            "Figure 4. Model Accuracy vs. Computation Cost. We done posttraining quantization for Resnet18 onImagenet dataset from int8 to int4."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            84,
+            188,
+            482,
+            398
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/30403a76c119553cfac587bf9fccf05a31d537d75bcc1eb7899a3c4a50ba70f3.jpg",
+        "table_caption": [
+            "Table 2. Post-training quantization experiments on ImageNet."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Resnet18</td><td></td><td>Bits</td><td>Top-1</td><td>Δ</td></tr><tr><td>Channel Balancing</td><td>Wino(4×4, 3×3)</td><td>8</td><td>67.54</td><td>-2.22</td></tr><tr><td>Full Quant.</td><td>Wino(4×4, 3×3)</td><td>8</td><td>68.16</td><td>-1.60</td></tr><tr><td>Full Quant.</td><td>Wino(4×4, 3×3)</td><td>6</td><td>64.34</td><td>-5.42</td></tr><tr><td>Ours</td><td>SFC6(7×7, 3×3)</td><td>8</td><td>69.59</td><td>-0.17</td></tr><tr><td>Ours</td><td>SFC6(7×7, 3×3)</td><td>6</td><td>68.80</td><td>-0.96</td></tr><tr><td>Resnet34</td><td></td><td>Bits</td><td>Top-1</td><td>Δ</td></tr><tr><td>Channel Balancing</td><td>Wino(4×4, 3×3)</td><td>8</td><td>71.86</td><td>-1.44</td></tr><tr><td>Full Quant.</td><td>Wino(4×4, 3×3)</td><td>8</td><td>71.75</td><td>-1.55</td></tr><tr><td>Full Quant.</td><td>Wino(4×4, 3×3)</td><td>6</td><td>68.80</td><td>-4.50</td></tr><tr><td>Ours</td><td>SFC6(7×7, 3×3)</td><td>8</td><td>73.14</td><td>-0.16</td></tr><tr><td>Ours</td><td>SFC6(7×7, 3×3)</td><td>6</td><td>72.40</td><td>-0.90</td></tr><tr><td>Resnet50</td><td></td><td>Bits</td><td>Top-1</td><td>Δ</td></tr><tr><td>Channel Balancing</td><td>Wino(4×4, 3×3)</td><td>8</td><td>75.84</td><td>-0.29</td></tr><tr><td>Full Quant.</td><td>Wino(4×4, 3×3)</td><td>8</td><td>75.74</td><td>-0.40</td></tr><tr><td>Full Quant.</td><td>Wino(4×4, 3×3)</td><td>6</td><td>74.75</td><td>-1.39</td></tr><tr><td>Ours</td><td>SFC6(7×7, 3×3)</td><td>8</td><td>76.02</td><td>-0.12</td></tr><tr><td>Ours</td><td>SFC6(7×7, 3×3)</td><td>6</td><td>75.54</td><td>-0.60</td></tr></table>",
+        "bbox": [
+            86,
+            529,
+            486,
+            763
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Since it is not feasible to align a specific category of data, such as ensuring uniform computational costs across a column to compare accuracy in the table, we have illustrated the accuracy curves in relation to computation cost (BOPs) in Figure 4 for a visual comparison of different algorithms. To ensure a fair and comprehensive comparison, we have included the state-of-the-art Winograd-quantization work in the figure, including Channel Balancing (Chikin &",
+        "bbox": [
+            84,
+            784,
+            475,
+            905
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Kryzhanovskiy, 2022) and Full Quantization (Tianqi et al., 2023). A subset of the detailed data is presented in Table 2. As depicted in Figure 4, the SFC-6(7×7, 3×3) algorithm demonstrates a computation cost reduction of  $1.6 \\times$  to  $2.5 \\times$  compared to both the Winograd F(4×4, 3×3) and direct convolution algorithms, while maintaining equivalent model accuracy. The experiments confirm that the SFC algorithms achieve a reduction in arithmetic complexity comparable to Winograd, with model accuracy similar to that of direct convolution (quantization-alone).",
+        "bbox": [
+            495,
+            84,
+            887,
+            236
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/a4370c3cad25b0d46594e2bf502ff1cb797dc7d9ac687220b24d176fb6dac61c.jpg",
+        "image_caption": [
+            "Figure 5. The MSE for different algorithms under int8 PTQ. Lower numerical error results to better PTQ results."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            532,
+            246,
+            848,
+            411
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also plot the mean squared error (MSE) between accelerated layers to fp32 layers for different methods under int8 quantization. The experimental results, presented in Figure 5, match perfectly with the numerical error analysis conducted in Section 5 and Table 1.",
+        "bbox": [
+            495,
+            472,
+            887,
+            547
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.2. FPGA Simulation",
+        "text_level": 1,
+        "bbox": [
+            496,
+            564,
+            656,
+            579
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We develop RTL code for the convolution accelerator based on the SFC-6(7×7,3×3) algorithm. The resource consumption and timing report are synthesized using Xilinx Vivado tools. The parallelism of our design is configured at  $[4\\times 4\\times 7\\times 7]$ , indicating that one convolution operator with 4 input channels, 4 output channels, and  $7\\times 7$  feature map are processed simultaneously. The VGG-16 model is taken as an example, whose convolution layers all have  $3\\times 3$  filters with stride=1, making it well-suited for fast convolution. All components in our datapath are quantized to int8, and all computing stages in fast convolution are designed to operate in a full pipeline architecture. The DSP48 hardcore is a crucial resource on FPGA, as it can efficiently deploy multiply operations. One DSP48 unit can implement two int8 multipliers or one int16 multipliers, which means that our implementation consumes only 1056 DSPs (calculated as  $4\\times 4\\times 132\\times 0.5$ ). In comparison, Winograd-based or NTT-based accelerators (Liang et al., 2020; Prasetiyo et al., 2023) require more DSPs for high precision multipliers, and accelerators for direct convolution (Huang et al., 2022) need more DSPs due to a higher complexity of multiplications. The re",
+        "bbox": [
+            495,
+            588,
+            888,
+            905
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "header",
+        "text": "SFC: Achieve Accurate Fast Convolution under Low-precision Arithmetic",
+        "bbox": [
+            251,
+            56,
+            720,
+            70
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "8",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "sult highlights the efficiency of integrating fast convolution with model quantization.",
+        "bbox": [
+            84,
+            84,
+            473,
+            116
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/488c2c17519e27ce2e8977cd5decebaf050bb057dc82e2d3939d74e8e54ffedf.jpg",
+        "table_caption": [
+            "Table 3. FPGA synthesis results."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Works</td><td>Liang et al., 2020</td><td>Prasetyo et al., 2023</td><td>Huang et al., 2022</td><td>Ours</td></tr><tr><td>Algorithm Platform</td><td>Winograd zcu102</td><td>NTT xc7vx980t</td><td>direct conv alveo U50</td><td>SFC xczu19eg</td></tr><tr><td>Precision</td><td>16bit</td><td>8bit/21bit</td><td>8bit</td><td>8bit</td></tr><tr><td>LUTs</td><td>-</td><td>601.7K</td><td>335K</td><td>221K</td></tr><tr><td>DSPs</td><td>2304</td><td>4100</td><td>3395</td><td>1056</td></tr><tr><td>Clock</td><td>200M</td><td>200M</td><td>200M</td><td>200M</td></tr><tr><td>Throughput (GOPs)</td><td>2601</td><td>2859.5</td><td>1000</td><td>2129</td></tr><tr><td>GOPs/DSPs /Clock</td><td>5.64</td><td>3.48</td><td>1.96</td><td>10.08</td></tr></table>",
+        "bbox": [
+            86,
+            167,
+            496,
+            347
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6.3. Ablation Study on Quantization Granularity",
+        "text_level": 1,
+        "bbox": [
+            84,
+            383,
+            429,
+            398
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Section 5, we theoretically predict that frequency-wise quantization will produce less error than tensor-wise quantization. Here, we provide an ablation experiment on Resnet-18 by enumerating combinations of different quantization granularity in Table 4 and Table 5. The results underscore that the Winograd algorithm exhibits more sensitivity to quantization granularity, requiring the finest granularity even with int8 quantization. In contrast, the SFC maintains acceptable accuracy under int8 quantization without specialized quantization. However, at lower bitwidths, it is still necessary to employ frequency-wise quantization for activation tensors.",
+        "bbox": [
+            84,
+            406,
+            475,
+            588
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/abb93eb49e1b43d87dbc1a14380b7f4fd1c5eb5f3b204870041585a8ffed5f37.jpg",
+        "table_caption": [
+            "Table 4. Ablation study for Resnet18 on int8 quantization."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm</td><td>Activation Granularity</td><td>Filter Granularity</td><td>Accuracy</td></tr><tr><td rowspan=\"4\">SFC-6(7×7, 3×3)</td><td>Tensor</td><td>Channel</td><td>69.18</td></tr><tr><td>Freq.</td><td>Channel</td><td>69.54</td></tr><tr><td>Freq.</td><td>Freq.</td><td>69.55</td></tr><tr><td>Freq.</td><td>Channel+Freq.</td><td>69.58</td></tr><tr><td rowspan=\"2\">Wino(4×4, 3×3)</td><td>Tensor</td><td>Channel</td><td>57.40</td></tr><tr><td>Freq.</td><td>Channel+Freq.</td><td>67.62</td></tr></table>",
+        "bbox": [
+            86,
+            643,
+            482,
+            758
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/460733df66bd177ff33db05544d81171d35d3af8ba2d5ffacbbf49481618561c.jpg",
+        "table_caption": [
+            "Table 5. Ablation study for Resnet-18 with SFC(7x7, 3x3)."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Quant. Granularity</td><td>int8</td><td>int6</td><td>int4</td></tr><tr><td>A: Tensor, W:Channel</td><td>69.18</td><td>65.42</td><td>17.81</td></tr><tr><td>A: Freq., W:Channel</td><td>69.54</td><td>68.34</td><td>31.10</td></tr><tr><td>A: Freq., W:Freq.+Channel</td><td>69.58</td><td>68.83</td><td>55.82</td></tr></table>",
+        "bbox": [
+            109,
+            825,
+            452,
+            883
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7. Conclusion",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We propose a novel fast convolution algorithm extended from the Fourier transform, which solves the accuracy problem of the conventional fast convolution algorithm applied to quantized models. According to experiment results, our algorithms outperform state-of-the-art fast convolution quantization works on both model accuracy and computation cost reduction. Our algorithms share exactly the same computational flow as the Winograd algorithm, which means that they can be deployed on general-purpose processors (CPUs, GPUs) and hardware accelerator design conveniently by following the existing works.",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgements",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This work was supported in part by the National Key Research and Development Program of China under Grant 2022YFB4400900in part by the Natural Science Foundation of China under Grants 62211530492 and 62371223.",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Impact Statement",
+        "text_level": 1,
+        "bbox": [
+            496,
+            400,
+            653,
+            417
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References",
+        "text_level": 1,
+        "bbox": [
+            496,
+            506,
+            594,
+            522
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
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+            "Barabasz, B. and Gregg, D. Winograd convolution for dnns: Beyond linear polynomials. In International Conference of the Italian Association for Artificial Intelligence, pp. 307-320. Springer, 2019.",
+            "Barabasz, B., Anderson, A., Soodhalter, K. M., and Gregg, D. Error analysis and improving the accuracy of winograd convolution for deep neural networks. ACM Transactions on Mathematical Software (TOMS), 46(4):1-33, 2020.",
+            "Chikin, V. and Kryzhanovskiy, V. Channel balancing for accurate quantization of winograd convolutions. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 12507-12516, 2022."
+        ],
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "header",
+        "text": "SFC: Achieve Accurate Fast Convolution under Low-precision Arithmetic",
+        "bbox": [
+            251,
+            56,
+            720,
+            70
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "page_number",
+        "text": "9",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
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+            "Huang, D., Zhang, X., Zhang, R., Zhi, T., He, D., Guo, J., Liu, C., Guo, Q., Du, Z., Liu, S., et al. Dwm: A decomposable winograd method for convolution acceleration. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pp. 4174-4181, 2020.",
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+            "Liu, Z.-G. and Mattina, M. Efficient residue number system based winograd convolution. In European Conference on Computer Vision, pp. 53-68. Springer, 2020.",
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+        ],
+        "bbox": [
+            86,
+            84,
+            475,
+            905
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Prasetyo, Hong, S., Arthanto, Y. F., and Kim, J.-Y. Accelerating deep convolutional neural networks using number theoretic transform. IEEE Transactions on Circuits and Systems I: Regular Papers, 70(1):315-326, 2023. doi: 10.1109/tcsi.2022.3214528.",
+            "Tianqi, C., Xu, W., Chen, W., Wang, P., and Cheng, J. Towards efficient and accurate winograd convolution via full quantization. In Thirty-seventh Conference on Neural Information Processing Systems, 2023.",
+            "Vincent, K., Stephano, K., Frumkin, M., Ginsburg, B., and Demouth, J. On improving the numerical stability of winograd convolutions. 2017."
+        ],
+        "bbox": [
+            500,
+            84,
+            887,
+            286
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "header",
+        "text": "SFC: Achieve Accurate Fast Convolution under Low-precision Arithmetic",
+        "bbox": [
+            251,
+            56,
+            720,
+            71
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "page_number",
+        "text": "10",
+        "bbox": [
+            477,
+            922,
+            495,
+            934
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A. Typical Symbolic Fourier Convolution Algorithms",
+        "text_level": 1,
+        "bbox": [
+            84,
+            83,
+            535,
+            101
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For SFC-4  $(4\\times 4,3\\times 3)$  algorithm, the matrices (with polynomial multiplication integrated) are:",
+        "bbox": [
+            84,
+            109,
+            723,
+            126
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nB ^ {T} = \\left( \\begin{array}{c c c c c c} 0 & 1 & 1 & 1 & 1 & 0 \\\\ 0 & - 1 & 1 & - 1 & 1 & 0 \\\\ 0 & 1 & - 1 & - 1 & 1 & 0 \\\\ 0 & 0 & - 1 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & - 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & - 1 & 0 \\\\ 0 & - 1 & 0 & 0 & 0 & 1 \\end{array} \\right),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            191,
+            671,
+            305
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nG = \\left( \\begin{array}{c c c} 1 & 1 & 1 \\\\ 1 & - 1 & 1 \\\\ 1 & - 1 & - 1 \\\\ 1 & 0 & - 1 \\\\ 0 & - 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right), A = \\frac {1}{4} \\left( \\begin{array}{c c c c} 1 & 1 & 1 & 1 \\\\ 1 & - 1 & 1 & - 1 \\\\ 0 & 2 & 0 & - 2 \\\\ 2 & - 2 & - 2 & 2 \\\\ - 2 & - 2 & 2 & 2 \\\\ 4 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 4 \\end{array} \\right)\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            300,
+            301,
+            666,
+            412
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "(18)",
+        "bbox": [
+            854,
+            294,
+            885,
+            306
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "It costs 49 multiplications to generate 16 outputs. And only 46 multiplications are consumed when Hermitian symmetry is fully considered.",
+        "bbox": [
+            84,
+            478,
+            888,
+            508
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The SFC-6(6 × 6, 3 × 3) algorithm costs 100/88 multiplications to generate 36 outputs. The matrices are:",
+        "bbox": [
+            84,
+            516,
+            782,
+            532
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nB ^ {T} = \\left[ \\begin{array}{c c c c c c c c} 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\\\ 0 & 1 & 1 & 0 & - 1 & - 1 & 0 & 0 \\\\ 0 & 0 & - 1 & - 1 & 0 & 1 & 1 & 0 \\\\ 0 & 1 & 0 & - 1 & - 1 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & - 1 & 1 & 0 & - 1 & 0 \\\\ 0 & 0 & - 1 & 1 & 0 & - 1 & 1 & 0 \\\\ 0 & 1 & - 1 & 0 & 1 & - 1 & 0 & 0 \\\\ 0 & 1 & - 1 & 1 & 1 & - 1 & 1 & 0 \\\\ 1 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 \\\\ 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 \\end{array} \\right],\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            598,
+            699,
+            757
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nG = \\left[ \\begin{array}{l l l} 1 & 1 & 1 \\\\ 0 & 1 & 1 \\\\ - 1 & - 1 & 0 \\\\ - 1 & 0 & 1 \\\\ - 1 & 0 & 1 \\\\ 1 & - 1 & 0 \\\\ 0 & - 1 & 1 \\\\ 1 & - 1 & 1 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right], A = \\frac {1}{6} \\left[ \\begin{array}{l l l l l l} 1 & 1 & 1 & 1 & 1 & 1 \\\\ 2 & 1 & - 1 & - 2 & - 1 & 1 \\\\ - 1 & 1 & 2 & 1 & - 1 & - 2 \\\\ - 1 & - 2 & - 1 & 1 & 2 & 1 \\\\ 1 & - 2 & 1 & 1 & - 2 & 1 \\\\ 1 & 1 & - 2 & 1 & 1 & - 2 \\\\ - 2 & 1 & 1 & - 2 & 1 & 1 \\\\ - 1 & 1 & - 1 & 1 & - 1 & 1 \\\\ 6 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 6 \\end{array} \\right]\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            271,
+            753,
+            697,
+            910
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "(19)",
+        "bbox": [
+            854,
+            747,
+            885,
+            760
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "header",
+        "text": "SFC: Achieve Accurate Fast Convolution under Low-precision Arithmetic",
+        "bbox": [
+            250,
+            56,
+            722,
+            71
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "page_number",
+        "text": "11",
+        "bbox": [
+            477,
+            922,
+            495,
+            935
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The SFC-6(7 × 7, 3 × 3) algorithm costs 144/132 multiplications to generate 49 outputs. Its transformation matrices are:",
+        "bbox": [
+            84,
+            85,
+            877,
+            99
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nB ^ {T} = \\left[ \\begin{array}{c c c c c c c c c} 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\\\ 0 & 1 & 1 & 0 & - 1 & - 1 & 0 & 0 & 0 \\\\ 0 & 0 & - 1 & - 1 & 0 & 1 & 1 & 0 & 0 \\\\ 0 & 1 & 0 & - 1 & - 1 & 0 & 1 & 0 & 0 \\\\ 0 & 1 & 0 & - 1 & 1 & 0 & - 1 & 0 & 0 \\\\ 0 & 0 & - 1 & 1 & 0 & - 1 & 1 & 0 & 0 \\\\ 0 & 1 & - 1 & 0 & 1 & - 1 & 0 & 0 & 0 \\\\ 0 & 1 & - 1 & 1 & - 1 & 1 & - 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 \\\\ 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 \\end{array} \\right],\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            111,
+            707,
+            265
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nG = \\left[ \\begin{array}{c c c} 1 & 1 & 1 \\\\ 0 & 1 & 1 \\\\ - 1 & - 1 & 0 \\\\ - 1 & 0 & 1 \\\\ - 1 & 0 & 1 \\\\ 1 & - 1 & 0 \\\\ 0 & - 1 & 1 \\\\ 1 & - 1 & 1 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right], A = \\frac {1}{6} \\left[ \\begin{array}{c c c c c c c} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 2 & 1 & - 1 & - 2 & - 1 & 1 & 2 \\\\ - 1 & 1 & 2 & 1 & - 1 & - 2 & - 1 \\\\ - 1 & - 2 & - 1 & 1 & 2 & 1 & - 1 \\\\ 1 & - 2 & 1 & 1 & - 2 & 1 & 1 \\\\ 1 & 1 & - 2 & 1 & 1 & - 2 & 1 \\\\ - 2 & 1 & 1 & - 2 & 1 & 1 & - 2 \\\\ - 1 & 1 & - 1 & 1 & - 1 & 1 & - 1 \\\\ 6 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 6 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 6 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 6 \\end{array} \\right]\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            263,
+            262,
+            704,
+            421
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The SFC-6(6 × 6, 5 × 5) algorithm costs 196/184 multiplications to generate 36 outputs. Its transformation matrices are:",
+        "bbox": [
+            84,
+            438,
+            877,
+            454
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nB ^ {T} = \\left[ \\begin{array}{c c c c c c c c c c} 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 1 & 0 & - 1 & - 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & - 1 & - 1 & 0 & 1 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & - 1 & - 1 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & - 1 & 1 & 0 & - 1 & 0 & 0 \\\\ 0 & 0 & 0 & - 1 & 1 & 0 & - 1 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & - 1 & 0 & 1 & - 1 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & - 1 & 1 & - 1 & 1 & - 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 \\\\ 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 \\end{array} \\right],\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            464,
+            750,
+            640
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nG = \\left[ \\begin{array}{c c c c c} 1 & 1 & 1 & 1 & 1 \\\\ - 1 & - 1 & 0 & 1 & 1 \\\\ 1 & 0 & - 1 & - 1 & 0 \\\\ 0 & - 1 & - 1 & 0 & 1 \\\\ 0 & 1 & - 1 & 0 & 1 \\\\ - 1 & 0 & 1 & - 1 & 0 \\\\ - 1 & 1 & 0 & - 1 & 1 \\\\ 1 & - 1 & 1 & - 1 & 1 \\\\ 1 & 0 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 1 \\end{array} \\right], A = \\frac {1}{6} \\left[ \\begin{array}{c c c c c c c} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 1 & - 1 & - 2 & - 1 & 1 & 2 & - 1 \\\\ 1 & 2 & 1 & - 1 & - 2 & - 1 & 2 \\\\ - 2 & - 1 & 1 & 2 & 1 & - 1 & - 1 \\\\ - 2 & 1 & 1 & - 2 & 1 & 1 & 1 \\\\ 1 & - 2 & 1 & 1 & - 2 & 1 & - 2 \\\\ 1 & 1 & - 2 & 1 & 1 & - 2 & 1 \\\\ 1 & - 1 & 1 & - 1 & 1 & - 1 & - 1 \\\\ 6 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 6 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 6 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 6 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 6 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 6 \\end{array} \\right]\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            220,
+            641,
+            748,
+            824
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "header",
+        "text": "SFC: Achieve Accurate Fast Convolution under Low-precision Arithmetic",
+        "bbox": [
+            251,
+            56,
+            718,
+            70
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "page_number",
+        "text": "12",
+        "bbox": [
+            477,
+            922,
+            495,
+            934
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B. Applying the SFC Algorithm to Large-size Convolution Kernels",
+        "text_level": 1,
+        "bbox": [
+            88,
+            82,
+            642,
+            99
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Large-kernel convolutional neural networks have recently received extensive research attention in recent years, and the kernel sizes ranging from  $7 \\times 7$  to  $51 \\times 51$ . The vanilla Fast Fourier Transform (FFT) is an option available. Since large-size convolutions commonly use depth-wise convolutions, this makes the multiplication complexity of the algorithm  $n^2 \\log n$ , i.e., the computational complexity of the FFT itself will become dominant.",
+        "bbox": [
+            88,
+            109,
+            885,
+            167
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "As a comparison, our algorithm is accomplished using only addition in the transform stage, so applying our algorithm to large-size convolutional kernels is promising. However, our algorithm itself is not applicable to scaling large numbers of transformation points, which can lead to the appearance of higher-order terms further increasing the real multiplication times. We therefore consider an iterative convolution approach to the operation.",
+        "bbox": [
+            88,
+            176,
+            885,
+            237
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Considering a convolution with a  $29 \\times 29$  kernel size and a  $26 \\times 26$  feature map size, we will describe the computational process of iterative convolution. We split the feature map into  $5 \\times 5$  tiles with  $6 \\times 6$  size, while spline the kernel into  $6 \\times 6$  tiles with  $5 \\times 5$  size. Convolution operation will be performed between each feature map tile and each kernel tile, so we use SFC  $(6 \\times 6, 5 \\times 5)$  to speed it up. Note that at this time the partial convolution results of the each tiles are still summed up in the same pattern as the convolution window sliding, which allows us to use the SFC  $(5 \\times 5, 6 \\times 6)$  algorithm to speed up the process as the feature map has been split into  $5 \\times 5$  tiles, and the kernel has been split into  $6 \\times 6$  tiles in the first iteration. The number of multiplications required by above two iterations is the product of the multiplications in the two SFC algorithms, i.e.,  $132 \\times 132 = 17,424$  multiplications. Our approach reduces the number of multiplications to just  $3\\%$  of what is required by direct convolution.",
+        "bbox": [
+            88,
+            244,
+            885,
+            378
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We can increase the number of iterations to computing convolution with more larger sizes. Since the SFC algorithm uses only addition for the transformation, it has an easier, efficient and flexible deployment compared to the FFT method. However, when the convolution size is large enough to require 3 or more iterations, the FFT method will appear to be more advantageous in terms of theoretical computational efficiency.",
+        "bbox": [
+            88,
+            387,
+            885,
+            448
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "header",
+        "text": "SFC: Achieve Accurate Fast Convolution under Low-precision Arithmetic",
+        "bbox": [
+            253,
+            56,
+            718,
+            70
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "page_number",
+        "text": "13",
+        "bbox": [
+            478,
+            922,
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+        ],
+        "page_idx": 12
+    }
+]