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+    {
+        "type": "text",
+        "text": "LIGHTGCL: SIMPLE YET EFFECTIVE GRAPH CON-TRASTIVE LEARNING FOR RECOMMENDATION",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Xuheng Cai Chao Huang∗ Lianghao Xia Xubin Ren Department of Computer Science, University of Hong Kong {rickcai, lhaoxia}@hku.hk chaohuang75gmail.com ",
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+        "type": "text",
+        "text": "xubinrencs@gmail.com ",
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
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+    },
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+        "text": "Graph neural network (GNN) is a powerful learning approach for graph-based recommender systems. Recently, GNNs integrated with contrastive learning have shown superior performance in recommendation with their data augmentation schemes, aiming at dealing with highly sparse data. Despite their success, most existing graph contrastive learning methods either perform stochastic augmentation (e.g., node/edge perturbation) on the user-item interaction graph, or rely on the heuristic-based augmentation techniques (e.g., user clustering) for generating contrastive views. We argue that these methods cannot well preserve the intrinsic semantic structures and are easily biased by the noise perturbation. In this paper, we propose a simple yet effective graph contrastive learning paradigm LightGCL that mitigates these issues impairing the generality and robustness of CL-based recommenders. Our model exclusively utilizes singular value decomposition for contrastive augmentation, which enables the unconstrained structural refinement with global collaborative relation modeling. Experiments conducted on several benchmark datasets demonstrate the significant improvement in performance of our model over the state-of-the-arts. Further analyses demonstrate the superiority of LightGCL’s robustness against data sparsity and popularity bias. The source code of our model is available at https://github.com/HKUDS/LightGCL. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "Graph neural networks (GNNs) have shown effectiveness in graph-based recommender systems by extracting local collaborative signals via neighborhood representation aggregation (Wang et al., 2019; Chen et al., 2020b). In general, to learn user and item representations, GNN-based recommenders perform embedding propagation on the user-item interaction graph by stacking multiple message passing layers for exploring high-order connectivity (He et al., 2020; Zhang et al., 2019; Liu et al., 2021a). Most GNN-based collaborative filtering models adhere to the supervised learning paradigm, requiring sufficient quality labelled data for model training. However, many practical recommendation scenarios struggle with the data sparsity issue in learning high-quality user and item representations from limited interaction data (Liu et al., 2021b; Lin et al., 2021). To address the label scarcity issue, the benefits of contrastive learning have been brought into the recommendation for data augmentation (Wu et al., 2021). The main idea of contrastive learning in enhancing the user and item representation is to research the agreement between the generated embedding views by contrasting the defined positive pairs with negative instance counterparts (Xie et al., 2022). ",
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+        "text": "While contrastive learning has been shown to be effective in improving the performance of graphbased recommendation methods, the view generators serve as the core part of data augmentation through identifying accurate contrasting samples. Most of current graph contrastive learning (GCL) approaches employ heuristic-based contrastive view generators to maximize the mutual information between the input positive pairs and push apart negative instances(Wu et al., 2021; Yu et al., 2022a; Xia et al., 2022b). To construct perturbed views, SGL (Wu et al., 2021) has been proposed to generate node pairs of positive view by corrupting the structural information of user-item interaction graph using stochastic augmentation strategies, e.g., node dropping and edge perturbation. To improve the graph contrastive learning in recommendation, SimGCL (Yu et al., 2022a) offers embedding augmentation with random noise perturbation. To work on identifying semantic neighbors of nodes (users and items), HCCF (Xia et al., 2022b) and NCL (Lin et al., 2022) are introduced to pursue consistent representations between the structurally adjacent nodes and semantic neighbors. Despite their effectiveness, state-of-the-art contrastive recommender systems suffer from several inherent limitations: i) Graph augmentation with random perturbation may lose useful structural information, which misleads the representation learning. ii) The success of heuristic-guided representation contrasting schemes is largely built upon the view generator, which limits the model generality and is vulnerable to the noisy user behaviors. iii) Most of current GNN-based contrastive recommenders are limited by the over-smoothing issue which leads to indistinguishable representations. ",
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+        "text": "In light of the above limitations and challenges, we revisit the graph contrastive learning paradigm for recommendation with a proposed simple yet effective augmentation method LightGCL. In our model, the graph augmentation is guided by singular value decomposition (SVD) to not only distill the useful information of user-item interactions but also inject the global collaborative context into the representation alignment of contrastive learning. Instead of generating two handcrafted augmented views, important semantic of user-item interactions can be well preserved with our robust graph contrastive learning paradigm. This enables our self-augmented representations to be reflective of both user-specific preferences and cross-user global dependencies. ",
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+    {
+        "type": "text",
+        "text": "Our contributions are highlighted as follows: ",
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+    {
+        "type": "text",
+        "text": "• In this paper, we enhance the recommender systems by designing a lightweight and robust graph contrastive learning framework to address the identified key challenges pertaining to this task. • We propose an effective and efficient contrastive learning paradigm LightGCL for graph augmentation. With the injection of global collaborative relations, our model can mitigate the issues brought by inaccurate contrastive signals. • Our method exhibits improved training efficiency compared to existing GCL-based approaches. • Extensive experiments on several real-world datasets justify the performance superiority of our LightGCL. In-depth analyzes demonstrate the rationality and robustness of LightGCL. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Graph Contrastive Learning for Recommendation. A promising line of recent studies has incorporated contrastive learning (CL) into graph-based recommenders, to address the label sparsity issue with self-supervision signals. Particularly, SGL (Wu et al., 2021) and SimGCL (Yu et al., 2022a) perform data augmentation over graph structure and embeddings with random dropout operations. However, such stochastic augmentation may drop important information, which may make the sparsity issue of inactive users even worse. Furthermore, some recent alternative CL-based recommenders, such as HCCF (Xia et al., 2022b) and NCL (Lin et al., 2022), design heuristic-based strategies to construct view for embedding contrasting. Despite their effectiveness, their success heavily relies on their incorporated heuristics (e.g., the number of hyperedges or user clusters) for contrastive view generation, which can hardly be adaptive to different recommendation tasks. ",
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+    {
+        "type": "text",
+        "text": "Self-Supervised Learning on Graphs. Recently, self-supervised learning (SSL) has advanced the graph learning paradigm by enhancing node representation from unlabeled graph data (Zhu et al., 2021a;b; Velickovic et al., 2019; Hassani & Khasahmadi, 2020; Peng et al., 2020; Zhu et al., 2020; Wu et al., 2022). For example, to improve the predictive SSL paradigm, AutoSSL (Jin et al., 2022) automatically combines multiple pretext tasks for augmentation. Towards the line of contrastive SSL over graph structures, recent efforts focus on designing various graph contrastive learning methods (Yu et al., 2022b; Yin et al., 2022; Zhang et al., 2022; Xia et al., 2022a; Suresh et al., 2021). For instance, SimGRACE Xia et al. (2022a) proposes to generate contrastive views with the GNN encoder perturbations. In AutoGCL Yin et al. (2022), graph view generators are designed to be jointly trained with the graph encoder in an end-to-end way. Additionally, GCA (Zhu et al., 2021b) performs both topology-level and attribute-level data augmentation for contrastive view generation. In this method, important edges and features will be identified for adaptive augmentation. GraphCL (You et al., 2020) generates correlated graph representation views using various augmentation strategies, such as node/edge perturbation and attribute masking. ",
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+    {
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+        "img_path": "images/0feaf6d22694bb9f83b4e1a65bef3f8fa8b0bec94a5309d7d6e04c5aec214e73.jpg",
+        "image_caption": [
+            "Figure 1: Overall structure of LightGCL. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "3 METHODOLOGY ",
+        "text_level": 1,
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+        "text": "In this section, we describe our proposed LightGCL framework in detail. LightGCL is a lightweight graph contrastive learning paradigm as illustrated in Fig. 1. Complementary to the GCN backbone (the upper half of the figure) extracting the local graph dependency, the SVD-guided augmentation (the lower half of the figure) empowers the graph contrastive learning with global collaborative relation analysis for learning effective user and item representations. ",
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+    {
+        "type": "text",
+        "text": "3.1 LOCAL GRAPH DEPENDENCY MODELING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As a common practice of collaborative filtering, we assign each user $u _ { i }$ and item $v _ { j }$ with an embedding vector $e _ { i } ^ { ( u ) } , e _ { j } ^ { ( v ) } \\in \\mathbb { R } ^ { d }$ , where $d$ is the embedding size. The collections of all user and item embeddings are defined as $\\pmb { { E } } ^ { ( u ) } \\in \\mathbb { R } ^ { I \\times d }$ and $\\pmb { { \\cal E } } ^ { ( v ) } \\in \\mathbb { R } ^ { J \\times d }$ , where $I$ and $J$ are the number of users and items, respectively. Following Xia et al. (2022b), we adopt a two-layer GCN to aggregate the neighboring information for each node. In layer $l$ , the aggregation process is expressed as follows: ",
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+    {
+        "type": "equation",
+        "img_path": "images/2bfc1dbf19743c67bcf230d16f894e06c6e477f45af5642958ccbb83782641ef.jpg",
+        "text": "$$\n\\begin{array} { r } { \\boldsymbol { z } _ { i , l } ^ { ( u ) } = \\sigma ( p ( \\tilde { \\boldsymbol { A } } _ { i , : } ) \\cdot \\boldsymbol { E } _ { l - 1 } ^ { ( v ) } ) , \\quad \\boldsymbol { z } _ { j , l } ^ { ( v ) } = \\sigma ( p ( \\tilde { \\boldsymbol { A } } _ { : , j } ) \\cdot \\boldsymbol { E } _ { l - 1 } ^ { ( u ) } ) } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where z(u)i,l and z(v)j,l denote the $l$ -th layer aggregated embedding for user $u _ { i }$ and item $v _ { j }$ . $\\sigma ( \\cdot )$ represents the LeakyReLU with a negative slope of 0.5. $\\tilde { \\boldsymbol { \\mathcal { A } } }$ is the normalized adjacency matrix, on which we perform the edge dropout denoted as $p ( \\cdot )$ , to mitigate the overfitting issue. We implement the residual connections in each layer to retain the original information of the nodes as follows: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c82a92d128934329f50f8e5849b9a7498399a16edf21adf480df00e18e0b0b13.jpg",
+        "text": "$$\n\\pmb { e } _ { i , l } ^ { ( u ) } = \\pmb { z } _ { i , l } ^ { ( u ) } + \\pmb { e } _ { i , l - 1 } ^ { ( u ) } , \\quad \\pmb { e } _ { j , l } ^ { ( v ) } = \\pmb { z } _ { j , l } ^ { ( v ) } + \\pmb { e } _ { j , l - 1 } ^ { ( v ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The final embedding for a node is the sum of its embeddings across all layers, and the inner product between the final embedding of a user $u _ { i }$ and an item $v _ { j }$ predicts $u _ { i }$ ’s preference towards $v _ { j }$ : ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/79258680c5f1ed11d5da615267d2befb178324d6474d9c4bc14f7af385f006a4.jpg",
+        "text": "$$\n\\pmb { e } _ { i } ^ { ( u ) } = \\sum _ { l = 0 } ^ { L } \\pmb { e } _ { i , l } ^ { ( u ) } , \\quad \\pmb { e } _ { j } ^ { ( v ) } = \\sum _ { l = 0 } ^ { L } \\pmb { e } _ { j , l } ^ { ( v ) } , \\quad \\hat { y } _ { i , j } = e _ { i } ^ { ( u ) \\top } \\pmb { e } _ { j } ^ { ( v ) }\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "3.2 EFFICIENT GLOBAL COLLABORATIVE RELATION LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To empower graph contrastive learning for recommendation with global structure learning, we equip our LightGCL with the SVD scheme (Rajwade et al., 2012; Rangarajan, 2001) to efficiently distill important collaborative signals from the global perspective. Specifically, we first perform SVD on the adjacency matrix $\\mathcal { A }$ as $\\mathbf { \\mathcal { A } } = U S V ^ { \\top }$ . Here, $U / V$ is an $I \\times I / J \\times J$ orthonormal matrix with columns being the eigenvectors of $\\mathcal { A }$ ’s row-row $/$ column-column correlation matrix. $_ { s }$ is an $I \\times J$ diagonal matrix storing the singular values of $\\mathcal { A }$ . The largest singular values are usually associated with the principal components of the matrix. Thus, we truncate the list of singular values to keep the largest q values, and reconstruct the adjacency matrix with the truncated matrices as $\\hat { \\ b { A } } = \\ b { U } _ { q } \\ b { S } _ { q } \\ b { V } _ { q } ^ { \\top }$ , where $U _ { q } \\in \\mathbb { R } ^ { I \\times q }$ and $V _ { q } \\in \\mathbb { R } ^ { J \\times q }$ contain the first $q$ columns of $U$ and $V$ respectively. $S _ { q } \\in \\mathbb { R } ^ { q \\times q }$ is the diagonal matrix of the $q$ largest singular values. ",
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+        "text": "The reconstructed matrix $\\hat { A }$ is a low-rank approximation of the adjacency matrix $\\mathcal { A }$ , for it holds that $r a n k ( { \\hat { A } } ) = q$ . The advantages of SVD-based graph structure learning are two-folds. Firstly, it emphasizes the principal components of the graph by identifying the user-item interactions that are important and reliable to user preference representations. Secondly, the generated new graph structures preserve the global collaborative signals by considering each user-item pair. Given the $\\hat { A }$ , we perform message propagation on the reconstructed user-item relation graph in each layer: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/b67fb8342e715e6be972f944d3ae27d737e2c99ac5c5ccc8adbee06ccd579b1f.jpg",
+        "text": "$$\n\\pmb { g } _ { i , l } ^ { ( u ) } = \\sigma ( \\hat { \\mathcal { A } } _ { i , : } \\cdot \\pmb { E } _ { l - 1 } ^ { ( v ) } ) , \\quad \\pmb { g } _ { j , l } ^ { ( v ) } = \\sigma ( \\hat { \\mathcal { A } } _ { : , j } \\cdot \\pmb { E } _ { l - 1 } ^ { ( u ) } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "However, performing the exact SVD on large matrices is highly expensive, making it impractical for handling large-scale user-item matrix. Therefore, we adopt the randomized SVD algorithm proposed by Halko et al. (2011), whose key idea is to first approximate the range of the input matrix with a low-rank orthonormal matrix, and then perform SVD on this smaller matrix. ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/8a7883b2feec20f71bc1431bcb2f565953eb5232bffab5ad1bb8e34b6d496986.jpg",
+        "text": "$$\n\\hat { U } _ { q } , \\hat { S } _ { q } , \\hat { V } _ { q } ^ { \\top } = \\mathrm { A p p r o x } { \\mathrm { S V D } } ( { \\cal A } , q ) , \\quad \\hat { A } _ { S V D } = \\hat { U } _ { q } \\hat { S } _ { q } \\hat { V } _ { q } ^ { \\top }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $q$ is the required rank for the decomposed matrices, and $\\hat { { \\cal U } } _ { q } \\in \\mathbb { R } ^ { I \\times q } , \\hat { { \\cal S } } _ { q } \\in \\mathbb { R } ^ { q \\times q } , \\hat { { \\cal V } } _ { q } \\in \\mathbb { R } ^ { J \\times q }$ are the approximated versions of $U _ { q }$ , $S _ { q }$ , $V _ { q }$ . Thus, we rewrite the message propagation rules in Eq. 4 with the approximated matrices and the collective representations of the embeddings as follows: ",
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/df13dac002614ec774fc3c9dc898354a3e66908516a2242984f7c9215ee685b1.jpg",
+        "text": "$$\n\\pmb { G } _ { l } ^ { ( u ) } = \\sigma ( \\hat { A } _ { S V D } \\pmb { E } _ { l - 1 } ^ { ( v ) } ) = \\sigma ( \\hat { U } _ { q } \\hat { S } _ { q } \\hat { V } _ { q } ^ { \\top } \\pmb { E } _ { l - 1 } ^ { ( v ) } ) ; \\quad \\pmb { G } _ { l } ^ { ( v ) } = \\sigma ( \\hat { A } _ { S V D } ^ { \\top } \\pmb { E } _ { l - 1 } ^ { ( u ) } ) = \\sigma ( \\hat { V } _ { q } \\hat { S } _ { q } \\hat { U } _ { q } ^ { \\top } \\pmb { E } _ { l - 1 } ^ { ( u ) } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $G _ { l } ^ { ( u ) }$ and $G _ { l } ^ { ( v ) }$ are the collections of user and item embeddings encoded from the new generated graph structure view. Note that we do not need to compute and store the large dense matrix $\\hat { \\boldsymbol { \\mathcal { A } } } _ { S V D }$ . Instead, we can store $\\hat { U } _ { q } , \\hat { S } _ { q }$ and $\\hat { V } _ { q }$ , which are of low dimensions. By pre-calculating $( \\hat { U } _ { q } \\hat { S } _ { q } )$ and $( \\hat { V } _ { q } \\hat { S } _ { q } )$ during the preprocessing stage with SVD, the model efficiency is improved. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.3 SIMPLIFIED LOCAL-GLOBAL CONTRASTIVE LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The conventional GCL methods such as SGL and SimGCL contrast node embeddings by constructing two extra views, while the embeddings generated from the original graph (the main-view) are not directly involved in the InfoNCE loss. The reason for adopting such a cumbersome three-view paradigm may be that the random perturbation used to augment the graph may provide misleading signals to the main-view embeddings. In our proposed method, however, the augmented graph view is created with global collaborative relations, which can enhance the main-view representations. Therefore, we simplify the CL framework by directly contrasting the SVD-augmented view embeddings g(u)i,l with the main-view embeddings $\\boldsymbol { z } _ { i , l } ^ { ( u ) }$ in the InfoNCE loss (Oord et al., 2018): ",
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+    {
+        "type": "equation",
+        "img_path": "images/398555fb60a0985ddd844350d7ef2299b07735c1205121838ed5e3034536b790.jpg",
+        "text": "$$\n\\mathcal { L } _ { s } ^ { ( u ) } = \\sum _ { i = 0 } ^ { I } \\sum _ { l = 0 } ^ { L } - \\log \\frac { \\exp ( s ( z _ { i , l } ^ { ( u ) } , \\pmb { g } _ { i , l } ^ { ( u ) } / \\tau ) ) } { \\sum _ { i ^ { \\prime } = 0 } ^ { I } \\exp ( s ( z _ { i , l } ^ { ( u ) } , \\pmb { g } _ { i ^ { \\prime } , l } ^ { ( u ) } ) / \\tau ) }\n$$",
+        "text_format": "latex",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $s ( \\cdot )$ and $\\tau$ stand for the cosine similarity and the temperature respectively. The InfoNCE loss $\\mathcal { L } _ { s } ^ { ( v ) }$ for the items are defined in the same way. To prevent overfitting, we implement a random node dropout in each batch to exclude some nodes from participating in the contrastive learning. As shown in Eq. 8, the contrastive loss is jointly optimized with our main objective function for the recommendation task (where $\\hat { y } _ { i , p _ { s } }$ and $\\hat { y } _ { i , n _ { s } }$ denote the predicted scores for a pair of positive and negative items of user $\\romannumeral 1$ ): ",
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+    {
+        "type": "equation",
+        "img_path": "images/2e19e09d238be4582375cbcfb627b9ce2b4736cc782edc41c306753002e5835d.jpg",
+        "text": "$$\n\\mathcal { L } = \\mathcal { L } _ { r } + \\lambda _ { 1 } \\cdot ( \\mathcal { L } _ { s } ^ { ( u ) } + \\mathcal { L } _ { s } ^ { ( v ) } ) + \\lambda _ { 2 } \\cdot \\Vert \\Theta \\Vert _ { 2 } ^ { 2 } ; \\quad \\mathcal { L } _ { r } = \\sum _ { i = 0 } ^ { I } \\sum _ { s = 1 } ^ { S } \\operatorname* { m a x } ( 0 , 1 - \\hat { y } _ { i , p _ { s } } + \\hat { y } _ { i , n _ { s } } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "4 EVALUATION ",
+        "text_level": 1,
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "To verify the superiority and effectiveness of the proposed LightGCL method, we perform extensive experiments to answer the following research questions: ",
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+    {
+        "type": "text",
+        "text": "• RQ1: How does LightGCL perform on different datasets compared to various SOTA baselines? • RQ2: How does the lightweight graph contrastive learning improve the model efficiency? • RQ3: How does our model perform against data sparsity, popularity bias and over-smoothing? • RQ4: How does the local-global contrastive learning contribute to the performance of our model? • RQ5: How do different parameter settings affect our model performance? ",
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+    },
+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETTINGS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1.1 DATASETS AND EVALUATION PROTOCOLS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We evaluate our model and the baselines on five real-world datasets: Yelp (29,601 users, 24,734 items, 1,517,326 interactions): a dataset collected from the rating interactions on Yelp platform; Gowalla (50,821 users, 57,440 items, 1,172,425 interactions): a dataset containing users’ check-in records collected from Gowalla platform; ML-10M (69,878 users, 10,195 items, 9,988,816 interactions): a well-known movie-rating dataset for collaborative filtering; Amazon-book (78,578 users, 77,801 items, 2,240,156 interactions): a dataset composed of users’ ratings on books collected from Amazon; and Tmall (47,939 users, 41,390 items, 2,357,450 interactions): a E-commerce dataset containing users’ purchase records on different products in Tmall platform. ",
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+    },
+    {
+        "type": "text",
+        "text": "In accordance with He et al. (2020) and Wu et al. (2021), we split the datasets into training, validation and testing sets with a ratio of 7:2:1. We adopt the Recall $@ \\mathbf { N }$ and Normalized Discounted Cumulative Gain $( \\mathrm { N D C G } ) @ \\mathrm { N }$ , where $\\Nu = \\{ 2 0 , 4 0 \\}$ , as the evaluation metrics. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1.2 BASELINE METHODS",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We compare our model against 16 state-of-the-art baselines with different learning paradigms: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• MLP-enhanced Collaborative Filtering: NCF (He et al., 2017).   \n• GNN-based Collaborative Filtering: GCCF (Chen et al., 2020c), LightGCN (He et al., 2020).   \n• Disentangled Graph Collaborative Filtering: DGCF (Wang et al., 2020b).   \n• Hypergraph-based Collaborative Filtering: HyRec (Wang et al., 2020a).   \n• Self-Supervised Learning Recommender Systems: GraphCL (You et al., 2020), GRACE (Zhu et al., 2020), GCA (Zhu et al., 2021b), MHCN (Yu et al., 2021), SAIL (Yu et al., 2022b), AutoGCL (Yin et al., 2022), SimGRACE (Xia et al., 2022a), SGL (Wu et al., 2021), HCCF (Xia et al., 2022b), SHT (Xia et al., 2022c), SimGCL (Yu et al., 2022a). ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Due to space limit, the detailed descriptions of baselines are presented in Appendix A. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1.3 HYPERPARAMETER SETTINGS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To ensure a fair comparison, we tune the hyperparameters of all the baselines within the ranges suggested in the original papers, except the following fixed settings for all the models: the embedding size is set as 32; the batch size is 256; two convolutional layers are used for GCN models. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For our LightGCL, the regularization weights $\\lambda _ { 1 }$ and $\\lambda _ { 2 }$ are tuned from $\\{ 1 \\mathrm { e } { - } 5 , 1 \\mathrm { e } { - } 6 , 1 \\mathrm { e } { - } 7 \\}$ and {1e4, 1e- $\\{ 5 \\}$ , respectively. The temperature $\\tau$ is searched from $\\{ 0 . 3 , 0 . 5 , 1 , \\dot { 3 } , 1 0 \\}$ . The dropout rate is chosen from $\\{ 0 , 0 . 2 5 \\}$ . The rank (i.e., $\\grave { q } ,$ ) for SVD, is set as 5. We use the Adam optimizer with a learning rate of 0.001 decaying at the rate of 0.98 until the rate reaches 0.0005.\\* ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 PERFORMANCE VALIDATION (RQ1) ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We summarize the experimental result in Table $1 ^ { \\dagger }$ , with the following observations and conclusions: ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/62660177d19c3fbe115eb8b94d7716456a30d8c53b4de95abc745c8ac4d566ac.jpg",
+        "table_caption": [
+            "Table 1: Performance comparison with baselines on five datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Data</td><td>Metric</td><td>DGCF</td><td>HyRec</td><td>LightGCN</td><td>MHCN</td><td>SGL</td><td>SimGRACE</td><td>GCA</td><td>HCCF</td><td>SHT</td><td>SimGCL</td><td>LightGCL</td><td>p-val.</td><td>impr.</td></tr><tr><td rowspan=\"4\">o</td><td>R@20</td><td>0.0466</td><td>0.0472</td><td>0.0482</td><td>0.0503</td><td>0.0526</td><td>0.0603</td><td>0.0621</td><td>0.0626</td><td>0.0651</td><td>0.0718</td><td>0.0793</td><td>7e-9</td><td>10%</td></tr><tr><td>N@20</td><td>0.0395</td><td>0.0395</td><td>0.0409</td><td>0.0424</td><td>0.0444</td><td>0.0435</td><td>0.0530</td><td>0.0527</td><td>0.0546</td><td>0.0615</td><td>0.0668</td><td>8e-9</td><td>8%</td></tr><tr><td>R@40</td><td>0.0774</td><td>0.0791</td><td>0.0803</td><td>0.0826</td><td>0.0869</td><td>0.0989</td><td>0.1021</td><td>0.1040</td><td>0.1091</td><td>0.1166</td><td>0.1292</td><td>2e-9</td><td>10%</td></tr><tr><td>N@40</td><td>0.0511</td><td>0.0522</td><td>0.0527</td><td>0.0544</td><td>0.0571</td><td>0.0656</td><td>0.0677</td><td>0.0681</td><td>0.0709</td><td>0.0778</td><td>0.0852</td><td>2e-9</td><td>9%</td></tr><tr><td rowspan=\"5\">Goeaal</td><td>R@20</td><td>0.0944</td><td>0.0901</td><td>0.0985</td><td>0.0955</td><td>0.1030</td><td>0.0869</td><td>0.0896</td><td>0.1070</td><td>0.1232</td><td>0.1357</td><td>0.1578</td><td>1e-6</td><td>16%</td></tr><tr><td>N@20</td><td>0.0522</td><td>0.0498</td><td>0.0593</td><td>0.0574</td><td>0.0623</td><td>0.0528</td><td>0.0537</td><td>0.0644</td><td>0.0731</td><td>0.0818</td><td>0.0935</td><td>2e-6</td><td>14%</td></tr><tr><td>R@40</td><td>0.1401</td><td>0.1356</td><td>0.1431</td><td>0.1393</td><td>0.1500</td><td>0.1276</td><td>0.1322</td><td>0.1535</td><td>0.1804</td><td>0.1956</td><td>0.2245</td><td>3e-6</td><td>14%</td></tr><tr><td>N@40</td><td>0.0671</td><td>0.0660</td><td>0.0710</td><td>0.0689</td><td>0.0746</td><td>0.0637</td><td>0.0651</td><td>0.0767</td><td>0.0881</td><td>0.0975</td><td>0.1108</td><td>3e-6</td><td>13%</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"4\">WOI-TN</td><td>R@20</td><td>0.1763</td><td>0.1801</td><td>0.1789</td><td>0.1497</td><td>0.1833</td><td>0.2254</td><td>0.2145</td><td>0.2219</td><td>0.2173</td><td>0.2265</td><td>0.2613</td><td>1e-9</td><td>15%</td></tr><tr><td>N@20</td><td>0.2101</td><td>0.2178</td><td>0.2128</td><td>0.1814</td><td>0.2205</td><td>0.2686</td><td>0.2613</td><td>0.2629</td><td>0.2573</td><td>0.2613</td><td>0.3106</td><td>3e-9</td><td>18%</td></tr><tr><td>R@40</td><td>0.2681</td><td>0.2685</td><td>0.2650</td><td>0.2250</td><td>0.2768</td><td>0.3295</td><td>0.3231</td><td>0.3265</td><td>0.3211</td><td>0.3345</td><td>0.3799</td><td>7e-10</td><td>13%</td></tr><tr><td>N@40</td><td>0.2340</td><td>0.2340</td><td>0.2322</td><td>0.1962</td><td>0.2426</td><td>0.2939</td><td>0.2871</td><td>0.2880</td><td>0.3318</td><td>0.2880</td><td>0.3387</td><td>1e-9</td><td>17%</td></tr><tr><td rowspan=\"4\">VAzaao</td><td>R@20</td><td>0.0211</td><td>0.0302</td><td>0.0319</td><td>0.0296</td><td>0.0327</td><td>0.0381</td><td>0.0309</td><td>0.0322</td><td>0.0441</td><td>0.0474</td><td>0.0585</td><td>2e-7</td><td>23%</td></tr><tr><td>N@20</td><td>0.0154</td><td>0.0225</td><td>0.0236</td><td>0.0219</td><td>0.0249</td><td>0.0291</td><td>0.0238</td><td>0.0247</td><td>0.0328</td><td>0.0360</td><td>0.0436</td><td>2e-6</td><td>21%</td></tr><tr><td>R@40</td><td>0.0351</td><td>0.0432</td><td>0.0499</td><td>0.0489</td><td>0.0531</td><td>0.0621</td><td>0.0498</td><td>0.0525</td><td>0.0719</td><td>0.0750</td><td>0.0933</td><td>1e-7</td><td>24%</td></tr><tr><td>N@40</td><td>0.0201</td><td>0.0246</td><td>0.0290</td><td>0.0284</td><td>0.0312</td><td>0.0371</td><td>0.0301</td><td>0.0314</td><td>0.0420</td><td>0.0451</td><td>0.0551</td><td>9e-7</td><td>22%</td></tr><tr><td rowspan=\"4\">[ig</td><td>R@20</td><td>0.0235</td><td>0.0233</td><td>0.0225</td><td>0.0203</td><td>0.0268</td><td>0.0222</td><td>0.0373</td><td>0.0314</td><td>0.0387</td><td>0.0473</td><td>0.0528</td><td>3e-5</td><td>11%</td></tr><tr><td>N@20</td><td>0.0163</td><td>0.0160</td><td>0.0154</td><td>0.0139</td><td>0.0183</td><td>0.0152</td><td>0.0252</td><td>0.0213</td><td>0.0262</td><td>0.0328</td><td>0.0361</td><td>1e-4</td><td>10%</td></tr><tr><td>R@40</td><td>0.0394</td><td>0.0350</td><td>0.0378</td><td>0.0340</td><td>0.0446</td><td>0.0367</td><td>0.0616</td><td>0.0519</td><td>0.0645</td><td>0.0766</td><td>0.0852</td><td>1e-5</td><td>11%</td></tr><tr><td>N@40</td><td>0.0218</td><td>0.0199</td><td>0.0208</td><td>0.0188</td><td>0.0246</td><td>0.0203</td><td>0.0337</td><td>0.0284</td><td>0.0352</td><td>0.0429</td><td>0.0473</td><td>7e-5</td><td>10%</td></tr></table>",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Contrastive Learning Dominates. As can be seen from the table, recent methods implementing contrastive learning (SGL, HCCF, SimGCL) exhibit consistent superiority as compared to traditional graph-based (GCCF, LightGCN) or hypergraph-based (HyRec) models. They also perform better than some of other self-supervised learning approaches (MHCN). This could be attributed to the effectiveness of CL to learn evenly distributed embeddings (Yu et al., 2022a). ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Contrastive Learning Enhancement. Our method consistently outperforms all the contrastive learning baselines. We attribute such performance improvement to the effective augmentation of graph contrastive learning via injecting global collaborative contextual signals. Other compared contrastive learning-based recommenders (e.g., SGL, SimGCL, and HCCF) are easily biased by noisy interaction information and generate misleading self-supervised signals. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 EFFICIENCY STUDY (RQ2) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            544,
+            400,
+            558
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "GCL models often suffer from a high computational cost due to the construction of extra views and the convolution operations performed on them during training. However, the low-rank nature of the SVD-reconstructed graph and the simplified CL structure enable the training of our LightGCL to be highly efficient. We analyze the pre-processing and per-batch training complexity of our model in comparison to three competitive baselines, as summarized in Table 2.‡ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/19990a465e289b9f89859de857c1ad9b16704f0f462a1224f1e7317098f8bcda.jpg",
+        "table_caption": [
+            "Table 2: Comparisons of computational complexity against baselines. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Stage</td><td>Computation</td><td>LightGCN</td><td>SGL</td><td>SimGCL</td><td>LightGCL</td></tr><tr><td>Pre-processing</td><td>Normalization SVD</td><td>O(E)</td><td>O(E)</td><td>O(E)</td><td>O(E) O(qE)</td></tr><tr><td>Training</td><td>Augmentation Graph Convolution BPRLoss InfoNCE Loss</td><td>O(2ELd) O(2Bd) 1</td><td>O(2pE) O(2ELd+4pELd) O(2Bd) O(Bd+BMd)</td><td>O(6ELd) O(2Bd) O(Bd+BMd)</td><td>O[2ELd+ 2q(I+ J)Ld] O(2Bd) O[(Bd+BMd)L]</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Although our model requires performing the SVD in the pre-processing stage which takes $O ( q E )$ , the computational cost is negligible compared to the training stage since it only needs to be performed once. In fact, by moving the construction of contrastive view to the pre-processing stage, we avoid the repetitive graph augmentation during training, which improves model efficiency. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Traditional GCN methods (e.g., LightGCN) only perform convolution on one graph, inducing a complexity of $O ( 2 E L d )$ per batch. For most GCL-based methods, three contrastive views are computed per batch, leading to a complexity of roughly three times of LightGCN. In our model, instead, only two contrastive views are involved. Additionally, due to the low-rank property of SVD-based graph structure learning, our graph encoder takes only $O [ 2 q ( I + J ) L d ]$ time. For most datasets, including the five we use, $\\bar { 2 q } ( \\bar { I } + J ) < E$ . Therefore, the training complexity of our model is less than half of that of the SOTA efficient model SimGCL. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4 RESISTANCE AGAINST DATA SPARSITY AND POPULARITY BIAS (RQ3) ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            191,
+            702,
+            207
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To evaluate the robustness of our model in alleviating data sparsity, we group the sparse users by their interaction degrees and calculate the Recall $@ 2 0$ of each group on $Y e l p$ and Gowalla datasets. As can be seen from the figures, the performance of HCCF and SimGCL varies across datasets, but our LightGCL consistently outperforms them in all cases. In particular, our model performs notably well on the extremely sparse user group $< 1 5$ interactions), as the Recall $@ 2 0$ of these users is not much lower (and is even higher on Gowalla) than that of the whole dataset. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/623a815fd9843e8009e8316113509191d0a46ab26971881711679e6dbbf83323.jpg",
+        "image_caption": [
+            "Figure 2: Performance on users of different sparsity degrees, in terms of Recall (histograms) and relative Recall w.r.t overall performances (charts). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            315,
+            504,
+            491
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/df5b9e7ff55f07f0a4528b1c2879a17d632a723aaf9dab688e698d6d4aa45616.jpg",
+        "image_caption": [
+            "Figure 3: LightGCL’s ability to alleviate popularity bias in comparison to SOTA CLbased methods HCCF and SimGCL. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            522,
+            333,
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+            473
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Additionally, we illustrate our model’s ability to mitigate popularity bias compared to HCCF and SimGCL. Similar to Section 4.4, we group the long-tail items by their degree of interactions. Following Wu et al. (2021), we adopt the decomposed Recall@20 defined as Recall(g) = |(Vurec)(g)∩Vutest||Vu | where $\\mathbb { V } _ { t e s t } ^ { u }$ refers to the set of test items for the user $u$ , and $( \\mathbb { V } _ { r e c } ^ { u } ) ^ { ( g ) }$ is the set of Top-K recommended items for $u$ that belong to group $g$ . The results are shown in Fig. 3. Similar to the results on sparse users, HCCF and SimGCL’s performance fluctuates a lot with the influence of popularity bias. Our model performs better in most cases, which shows its resistance against popularity bias. Note that since the extremely sparse group ( $< 1 5$ interactions) is significantly larger than the other groups in Gowalla, they contribute to a large fraction of the Recall $@ 2 0$ , resulting in a different trend from that of Yelp in the figure. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.5 BALANCING BETWEEN OVER-SMOOTHING AND OVER-UNIFORMITY (RQ3) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            728,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we illustrate the effectiveness of our model in learning a moderately dispersed embedding distribution, by preserving user unique preference pattern and inter-user collaborative dependencies. We randomly sample 2,000 nodes from Yelp and Gowalla and map their embeddings to the 2-D space with t-SNE (Van der Maaten & Hinton, 2008). The visualizations of these embeddings are presented in Fig. 4. We also calculate the Mean Average Distance (MAD) (Chen et al., 2020a) of the embeddings, summarized in Table 3. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/436be5247ca8fb5f9270954ea6c2b27a6e459eadbfc1a36c4f3fe577a7542976.jpg",
+        "table_caption": [
+            "Table 3: Mean Average Distance (MAD) of the embeddings learned by different methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>MHCN</td><td>LightGCN</td><td>LightGCL</td><td>SGL</td><td>SimGCL</td></tr><tr><td>Yelp</td><td>0.8806</td><td>0.9469</td><td>0.9657</td><td>0.9962</td><td>0.9956</td></tr><tr><td>Gowalla</td><td>0.9247</td><td>0.9568</td><td>0.9721</td><td>0.9859</td><td>0.9897</td></tr></table>",
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+        "img_path": "images/d6c1337ddbfe6f8ae387e691750f79a7851bc9a61889505c08a8bf2562dd6e5b.jpg",
+        "image_caption": [
+            "Figure 4: Embedding distributions on Yelp and Gowalla visualized with t-SNE. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "As can be seen from Fig. 4, the embedding distributions of non-CL methods (i.e., LightGCN, MHCN) exhibit indistinguishable clusters in the embedding space, which indicates the limitation of addressing the over-smoothing issue. On the contrary, the existing CL-based methods tend to learn i) over-uniform distributions, e.g., SGL on $Y e l p$ learns a huge cloud of evenly-distanced embeddings with no clear community structure to well capture the collaborative relations between users; ii) highly dispersed small clusters with severe over-smoothing issue inside the clusters, e.g., the embeddings of SimGCL on Gowalla appear to be scattered grained clusters inside which embeddings are highly similar. Compared with them, clear community structures could be identified by our method to capture collaborative effects, while the embeddings inside each community are reasonably dispersed to be reflective of user-specific preference. The MAD of our model’s learned features is also in between of the two types of baselines as shown in Table 3. ",
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+    {
+        "type": "text",
+        "text": "4.6 ABLATION STUDY (RQ4) ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To investigate the effectiveness of our SVD-based graph augmentation scheme, we perform the ablation study to answer the question of whether we could provide guidance to the contrastive learning with a different approach of matrix decomposition. To this end, we implement two variants of our model, replacing the approximated SVD algorithm with other matrix decomposition methods: $C L .$ - $M F$ adopts the view generated by a pre-trained MF (Koren et al., 2009); $C L { \\cdot } S V D { + } +$ utilizes the $\\mathrm { S V D + + }$ (Koren, 2008) which takes implicit user feedback into consideration. As shown in Table 4, with the information distilled from MF or $\\mathrm { S V D + + }$ , the model is able to achieve satisfactory results, indicating the effectiveness of using matrix decomposition to empower CL and the flexibility of our proposed framework. However, adopting a pre-trained CL component is not only tedious and timeconsuming but also inferior to utilizing the approximate SVD algorithm in terms of performance. ",
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+    {
+        "type": "table",
+        "img_path": "images/76c28c200cc857d058d26cba967f0acb9768d9ae5b2f7b6c215f4517d98b68f7.jpg",
+        "table_caption": [
+            "Table 4: Ablation study on LightGCL. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Variant</td><td colspan=\"2\">Yelp</td><td colspan=\"2\">Gowalla</td></tr><tr><td>Recall@20</td><td>NDCG@20</td><td>Recall@20</td><td>NDCG@20</td></tr><tr><td>CL-MF</td><td>0.0781</td><td>0.0659</td><td>0.1561</td><td>0.0929</td></tr><tr><td>CL-SVD++</td><td>0.0788</td><td>0.0666</td><td>0.1568</td><td>0.0932</td></tr><tr><td>LightGCL</td><td>0.0793</td><td>0.0668</td><td>0.1578</td><td>0.0935</td></tr></table>",
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+        "image_caption": [
+            "Figure 5: Recall change w.r.t. $q$ "
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+    {
+        "type": "text",
+        "text": "4.7 HYPERPARAMETER ANALYSIS (RQ5) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we investigate our model’s sensitivity in relation to several key hyperparameters: the regularization weight for InfoNCE loss $\\lambda _ { 1 }$ , the temperature $\\tau$ , and the required rank of SVD $q$ . ",
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+        "type": "text",
+        "text": "• The impact of $\\lambda _ { 1 }$ . As illustrated in Fig. 6, for the three datasets Yelp, Gowalla and ML-10M, the model’s performance reaches the peak when $\\lambda _ { 1 } = 1 0 ^ { - 7 }$ . It can be noticed that $\\lambda _ { 1 }$ with the range of $[ 1 0 ^ { - 6 } , 1 0 ^ { - 8 } ]$ can often lead to performance improvement. ",
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+        "image_caption": [
+            "Figure 6: Impact of $\\lambda _ { 1 }$ . "
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+        "img_path": "images/9fb35ccdc532666d6f84c206ac95d489d159d8d812f187c41a03b6b1c6914a77.jpg",
+        "image_caption": [
+            "Figure 7: Impact of $\\tau$ "
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+    {
+        "type": "text",
+        "text": "• The impact of $\\tau$ . Fig. 7 indicates that the model’s performance is relatively stable across different selections of $\\tau$ from 0.1 to 10, while the best configuration of $\\tau$ value varies by datasets. ",
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+    {
+        "type": "text",
+        "text": "• The selection of $q$ . $q$ determines the rank of SVD in our model. Experiments have shown that satisfactory results can be achieved with a small $q$ . Specifically, as in Fig. 5, we observe that $q = 5$ is sufficient to preserve important structures of the user-item interaction graph. ",
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+    {
+        "type": "text",
+        "text": "4.8 CASE STUDY (RQ4) ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we present a case study to intuitively show the effectiveness of our model to identify useful knowledge from noisy user-item interactions and make accurate recommendations accordingly. In Fig. 8, we can see that the venues visited by user $\\# 2 6$ in Yelp mainly fall into two communities: Cleveland (where the user probably lives) and Arizona (where the user may have travelled to). In the reconstructed graph, these venues are assigned a new weight according to their potential importance. Note that item $\\# 2 5 8 3$ , a car rental agency in Arizona, has been assigned a negative weight, which conforms to our common sense that people generally would not visit multiple car rental agencies in one trip. The SVD-augmented view also provides predictions on invisible links by assigning a large weight§ to potential venues of interest, such as #2647 and #658. Note that when exploiting the graph, the augmented view does not overlook the smaller Arizona community, which enables the model to predict items of minor interests that are usually overshadowed by the majority. ",
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+        "image_caption": [
+            "Figure 8: Case study on user $\\# 2 6$ in Yelp dataset. "
+        ],
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a simple and effective augmentation method to the graph contrastive learning framework for recommendation. Specifically, we explore the key idea of making the singular value decomposition powerful enough to augment user-item interaction graph structures. Our key findings indicate that our graph augmentation scheme exhibits strong ability in resisting data sparsity and popularity bias. Extensive experiments show that our model achieves new state-of-the-art results on several public evaluation datasets. In future work, we plan to explore the potential of incorporating casual analysis into our lightweight graph contrastive learning model to enhance the recommender system with mitigating confounding effects for data augmentation. ",
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Lu Yu, Shichao Pei, Lizhong Ding, Jun Zhou, Longfei Li, Chuxu Zhang, and Xiangliang Zhang. Sail: Self-augmented graph contrastive learning. In AAAI Conference on Artificial Intelligence (AAAI), volume 36, pp. 8927–8935, 2022b. ",
+        "bbox": [
+            173,
+            169,
+            823,
+            212
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Jiani Zhang, Xingjian Shi, Shenglin Zhao, and Irwin King. Star-gcn: Stacked and reconstructed graph convolutional networks for recommender systems. International Joint Conference on Artificial Intelligence (IJCAI), 2019. ",
+        "bbox": [
+            173,
+            219,
+            823,
+            262
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Yixin Zhang, Yong Liu, Yonghui Xu, Hao Xiong, Chenyi Lei, Wei He, Lizhen Cui, and Chunyan Miao. Enhancing sequential recommendation with graph contrastive learning. International Joint Conference on Artificial Intelligence (IJCAI), 2022. ",
+        "bbox": [
+            173,
+            271,
+            826,
+            314
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang. Deep graph contrastive representation learning. arXiv preprint arXiv:2006.04131, 2020. ",
+        "bbox": [
+            174,
+            323,
+            823,
+            352
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Yanqiao Zhu, Yichen Xu, Qiang Liu, and Shu Wu. An empirical study of graph contrastive learning. arXiv preprint arXiv:2109.01116, 2021a. ",
+        "bbox": [
+            176,
+            361,
+            823,
+            390
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang. Graph contrastive learning with adaptive augmentation. In The Web Conference (WWW), pp. 2069–2080, 2021b. ",
+        "bbox": [
+            173,
+            398,
+            825,
+            428
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A DETAILS OF THE BASELINES",
+        "text_level": 1,
+        "bbox": [
+            178,
+            102,
+            449,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "MLP-enhanced Collaborative Filtering: ",
+        "bbox": [
+            174,
+            133,
+            434,
+            148
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• NCF (He et al., 2017) is a collaborative filtering model that leverages neural network to exploit non-linearity. Two hidden layers are used in our evaluation. ",
+        "bbox": [
+            174,
+            161,
+            823,
+            190
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "GNN-based Collaborative Filtering: ",
+        "bbox": [
+            174,
+            202,
+            411,
+            217
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• GCCF (Chen et al., 2020c) strengthens the GNN-based collaborative filtering by implementing a residual network and reducing the non-linear transformation.   \n• LightGCN (He et al., 2020) adopts a simplified GCN structure without embedding weight matrices and non-linear projection. ",
+        "bbox": [
+            173,
+            229,
+            826,
+            292
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Disentangled Graph Collaborative Filtering: ",
+        "bbox": [
+            174,
+            305,
+            464,
+            320
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• DGCF (Wang et al., 2020b) learns a more sophisticated representation by segmenting the embedding vectors to represent multiple latent intentions. ",
+        "bbox": [
+            174,
+            333,
+            821,
+            361
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Hypergraph-based Collaborative Filtering: ",
+        "bbox": [
+            174,
+            373,
+            452,
+            388
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• HyRec (Wang et al., 2020a) makes use of hypergraph to encode multi-order information between users and items. ",
+        "bbox": [
+            174,
+            401,
+            825,
+            429
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Self-Supervised Learning Recommender Systems: ",
+        "bbox": [
+            176,
+            443,
+            503,
+            457
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• GraphCL (You et al., 2020) utilizes random node dropping and edge masking to generate two contrastive views, which were aligned by optimizing the SSL loss function. ",
+        "bbox": [
+            174,
+            468,
+            823,
+            498
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• GRACE (Zhu et al., 2020) proposes to corrupt the graph structure by both random edge dropout and random node feature dropping, and uses the corrupted graphs as the contrastive views. ",
+        "bbox": [
+            174,
+            503,
+            823,
+            532
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• GCA (Zhu et al., 2021b) adaptively dropout the nodes and edges by their importance calculated with node centrality. ",
+        "bbox": [
+            174,
+            537,
+            821,
+            566
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• MHCN (Yu et al., 2021) creates self-supervised signals for the graph representation learning by graph infomax network. ",
+        "bbox": [
+            174,
+            571,
+            820,
+            599
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• SAIL (Yu et al., 2022b) maximizes the neighborhood predicting probability between GNNgenerated high-level features and input node features. ",
+        "bbox": [
+            173,
+            606,
+            821,
+            635
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• AutoGCL (Yin et al., 2022) uses GNN to learn to mask nodes and edges in the augmented graph. It minimizes the similarity between the augmented and the original graph, while maximizing the similarity of the embeddings generated through them, so as to uncover the most important information in the graph. ",
+        "bbox": [
+            174,
+            640,
+            825,
+            696
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• SimGRACE (Xia et al., 2022a) creates augmented view by randomly perturbing the parameters of the GNN network. ",
+        "bbox": [
+            173,
+            703,
+            823,
+            731
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• SGL (Wu et al., 2021) adopts random walk sampling and probabilistic edge/node dropout to create augmented views for contrastive learning. In our experiments, we adopt the SGL-ED variant, which implements random edge dropout and exhibits the strongest performance according to the original paper. ",
+        "bbox": [
+            173,
+            737,
+            825,
+            792
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• HCCF (Xia et al., 2022b) encodes global graph information with hypergraph and contrasts it against the local information encoded with GCN. In our experiments, the number of hyper-edges are set as 128 following the original paper. ",
+        "bbox": [
+            174,
+            799,
+            823,
+            842
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• SHT (Xia et al., 2022c) adopts a hypergraph transformer framework to exploit global collaborative relationships and distills the global information to generate the cross-view self-supervised signals. In our experiments, the number of hyper-edges are set as 128 following the original paper. ",
+        "bbox": [
+            174,
+            848,
+            820,
+            890
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• SimGCL (Yu et al., 2022a) propose to simplify the graph augmentation process of contrastive learning by directly injecting random noises into the feature representation. ",
+        "bbox": [
+            173,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B PERFORMANCE COMPARISON WITH BASELINES (CONTINUED) ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            728,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In this appendix, we show the performance of NCF, GCCF, GraphCL, SAIL, GRACE, and AutoGCL, which are not shown in Table 1 due to space limit. The results are summarized in Table 5. As can be seen from the table, our model outperforms these baselines consistently. ",
+        "bbox": [
+            174,
+            133,
+            823,
+            176
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/3dc5dc7489393a0033f221b1b984629c1521f82b15aa7cb58d38557f66c5d208.jpg",
+        "table_caption": [
+            "Table 5: Performance comparison with baselines on five datasets (continued). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Data</td><td rowspan=1 colspan=1>Metric</td><td rowspan=1 colspan=1>NCF</td><td rowspan=1 colspan=1>GCCF</td><td rowspan=1 colspan=1>GraphCL</td><td rowspan=1 colspan=1>SAIL</td><td rowspan=1 colspan=1>GRACE</td><td rowspan=1 colspan=1>AutoGCL</td><td rowspan=1 colspan=1>LightGCL</td></tr><tr><td rowspan=2 colspan=1>Yelp</td><td rowspan=1 colspan=1>R@20N@20</td><td rowspan=1 colspan=1>0.02520.0202</td><td rowspan=1 colspan=1>0.04620.0398</td><td rowspan=1 colspan=1>0.04620.0401</td><td rowspan=1 colspan=1>0.04710.0405</td><td rowspan=1 colspan=1>0.05500.0470</td><td rowspan=1 colspan=1>0.05930.0494</td><td rowspan=1 colspan=1>0.07930.0668</td></tr><tr><td rowspan=1 colspan=1>R@40N@40</td><td rowspan=1 colspan=1>0.04870.0289</td><td rowspan=1 colspan=1>0.07600.0508</td><td rowspan=1 colspan=1>0.07640.0511</td><td rowspan=1 colspan=1>0.07730.0516</td><td rowspan=1 colspan=1>0.09170.0605</td><td rowspan=1 colspan=1>0.10090.0650</td><td rowspan=1 colspan=1>0.12920.0852</td></tr><tr><td rowspan=2 colspan=1>Gowalla</td><td rowspan=1 colspan=1>R@20N@20</td><td rowspan=1 colspan=1>0.01710.0106</td><td rowspan=1 colspan=1>0.09510.0535</td><td rowspan=1 colspan=1>0.09970.0603</td><td rowspan=1 colspan=1>0.09990.0602</td><td rowspan=1 colspan=1>0.07440.0452</td><td rowspan=1 colspan=1>0.08320.0484</td><td rowspan=1 colspan=1>0.15780.0935</td></tr><tr><td rowspan=1 colspan=1>R@40N@40</td><td rowspan=1 colspan=1>0.02160.0118</td><td rowspan=1 colspan=1>0.13920.0684</td><td rowspan=1 colspan=1>0.14730.0727</td><td rowspan=1 colspan=1>0.14720.0725</td><td rowspan=1 colspan=1>0.10710.0539</td><td rowspan=1 colspan=1>0.12910.0605</td><td rowspan=1 colspan=1>0.22450.1108</td></tr><tr><td rowspan=2 colspan=1>ML-10M</td><td rowspan=1 colspan=1>R@20N@20</td><td rowspan=1 colspan=1>0.10970.1297</td><td rowspan=1 colspan=1>0.17420.2109</td><td rowspan=1 colspan=1>0.16590.2038</td><td rowspan=1 colspan=1>0.17280.2118</td><td rowspan=1 colspan=1>0.21070.2476</td><td rowspan=1 colspan=1>0.23250.2755</td><td rowspan=1 colspan=1>0.26130.3106</td></tr><tr><td rowspan=1 colspan=1>R@40N@40</td><td rowspan=1 colspan=1>0.16340.1427</td><td rowspan=1 colspan=1>0.26060.2331</td><td rowspan=1 colspan=1>0.25600.2250</td><td rowspan=1 colspan=1>0.26390.2332</td><td rowspan=1 colspan=1>0.30750.2711</td><td rowspan=1 colspan=1>0.34150.3023</td><td rowspan=1 colspan=1>0.37990.3387</td></tr><tr><td rowspan=2 colspan=1>Amazon</td><td rowspan=1 colspan=1>R@20N@20</td><td rowspan=1 colspan=1>0.01420.0085</td><td rowspan=1 colspan=1>0.03170.0243</td><td rowspan=1 colspan=1>0.03600.0266</td><td rowspan=1 colspan=1>0.03570.0264</td><td rowspan=1 colspan=1>0.03600.0271</td><td rowspan=1 colspan=1>0.03250.0241</td><td rowspan=1 colspan=1>0.05850.0436</td></tr><tr><td rowspan=1 colspan=1>R@40N@40</td><td rowspan=1 colspan=1>0.02230.0133</td><td rowspan=1 colspan=1>0.04830.0285</td><td rowspan=1 colspan=1>0.05850.0340</td><td rowspan=1 colspan=1>0.05810.0338</td><td rowspan=1 colspan=1>0.05830.0345</td><td rowspan=1 colspan=1>0.05530.0318</td><td rowspan=1 colspan=1>0.09330.0551</td></tr><tr><td rowspan=2 colspan=1>Tmall</td><td rowspan=1 colspan=1>R@20N@20</td><td rowspan=1 colspan=1>0.00820.0059</td><td rowspan=1 colspan=1>0.02090.0141</td><td rowspan=1 colspan=1>0.02510.0175</td><td rowspan=1 colspan=1>0.02540.0177</td><td rowspan=1 colspan=1>0.03030.0210</td><td rowspan=1 colspan=1>0.03120.0204</td><td rowspan=1 colspan=1>0.05280.0361</td></tr><tr><td rowspan=1 colspan=1>R@40N@40</td><td rowspan=1 colspan=1>0.01400.0079</td><td rowspan=1 colspan=1>0.03560.0196</td><td rowspan=1 colspan=1>0.04160.0233</td><td rowspan=1 colspan=1>0.04240.0236</td><td rowspan=1 colspan=1>0.05050.0281</td><td rowspan=1 colspan=1>0.05240.0278</td><td rowspan=1 colspan=1>0.08520.0473</td></tr></table>",
+        "bbox": [
+            281,
+            210,
+            714,
+            440
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C THEORETICAL ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            460,
+            419,
+            477
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We conduct theoretical analyses to show that our local-global CL (Eq. 7) is augmented to maximize the similarity between embeddings of potentially related nodes, based on the SVD-based global relation learning. Specifically, for a node $v _ { j } ~ \\in { \\mathcal { U } }$ , where $\\mathcal { U } = \\{ u _ { i ^ { \\prime } } | \\mathcal { A } _ { i , i ^ { \\prime } } = 0 , \\hat { \\mathcal { A } } _ { i , i ^ { \\prime } } \\neq 0 \\}$ , the embeddings are not updated by $s ( z _ { i , l } , g _ { i , l } )$ in the vanilla InfoNCE loss, as $v _ { j }$ is not adjacent to $u _ { i }$ . Instead, our local-global contrastive assigns the following gradients to the embeddings of $v _ { j }$ : ",
+        "bbox": [
+            174,
+            492,
+            825,
+            564
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6061d58517a08d6de0dab015b6576eb9921840d67e4f05dc43b6d213a6e9865b.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\partial s ( z _ { i , l } , g _ { i , l } ) / \\partial g _ { i , l - 1 } = \\partial s \\left( z _ { i , l } , \\sigma ( \\displaystyle \\sum _ { j \\in \\mathcal { U } } \\alpha _ { i , j } g _ { j , l - 1 } + \\displaystyle \\sum _ { A _ { i , j ^ { \\prime } } \\neq 0 } \\alpha _ { i , j ^ { \\prime } } g _ { j ^ { \\prime } , l - 1 } ) \\right) / \\partial g _ { j , l - 1 } } \\\\ { = \\frac { z _ { i , l } } { \\| z _ { i , l } \\| \\| g _ { i , l } \\| } \\cdot \\boldsymbol { \\sigma } ^ { \\prime } ( \\cdot ) \\cdot \\boldsymbol { \\alpha } _ { i , j } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            571,
+            781,
+            656
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha _ { i , j }$ denotes the normalization weight for node $u _ { i }$ and $v _ { j }$ . In this way, the embeddings of nodes in $\\mathcal { U }$ are also pulled close to $s _ { i , l }$ , which injects relatedness information learned by the SVD into the local-global CL optimization. ",
+        "bbox": [
+            176,
+            660,
+            823,
+            703
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D CALCULATION OF COMPLEXITY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            723,
+            477,
+            739
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D.1 ADJACENCY MATRIX NORMALIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            755,
+            490,
+            770
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For a sparse user-item matrix stored in the Coordinate Format (COO), it requires visiting every nonzero elements in the matrix to perform normalization. Thus, the computational complexity is in the order of the number of edges ${ \\bf \\bar { \\boldsymbol { O } } } ( E )$ . Note that for the baseline SGL, it requires normalizing the two augmented graph structures during the training phase, each of which contains $\\rho E$ edges, so it induces a complexity of $O ( 2 \\rho E )$ per batch. ",
+        "bbox": [
+            174,
+            780,
+            825,
+            852
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D.2 APPROXIMATE SVD ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            868,
+            450,
+            883
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We refer the readers to Halko et al. (2011) in which the complexity of the approximate SVD algorithm is explained in detail. ",
+        "bbox": [
+            174,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D.3 GRAPH CONVOLUTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            379,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Given a sparse COO matrix $\\mathcal { A }$ with $E$ edges and a dense matrix $\\pmb { \\cal E }$ with dimensions $I ( J ) \\times d$ , it takes $O ( E d )$ time to calculate $\\mathcal { A } E$ . To perform graph convolution on a graph, we need to multiply the sparse adjacency matrix with $\\pmb { { E } } _ { l - 1 } ^ { ( v ) } \\in \\mathbb { R } ^ { J \\times d }$ and its transpose with $E _ { l - 1 } ^ { ( u ) } \\in \\mathbb { R } ^ { I \\times d }$ , which takes $O ( E d )$ each, and $O ( 2 E d )$ in total. For $L$ layers, $O ( 2 E L d )$ is required. For traditional CL-based methods such as SGL and $\\mathrm { S i m C G L }$ , a three-view structure is adopted, resulting in a complexity of $O ( 1 2 E L d )$ (for SGL it again varies a bit depending on $\\rho$ ). ",
+        "bbox": [
+            173,
+            128,
+            825,
+            219
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For the SVD-view of our model, $\\hat { V } _ { q } ^ { \\top } E _ { l - 1 } ^ { ( v ) }$ takes $O ( q J d )$ , and multiplying the result with the precalculated $( \\hat { U } _ { q } \\hat { S } _ { q } )$ takes $O ( q I d )$ ; $\\hat { U } _ { q } ^ { \\top } E _ { l - 1 } ^ { ( v ) }$ takes $O ( q I d )$ , and multiplying the result with the precalculated $( \\hat { V } _ { q } \\hat { S } _ { q } )$ takes $O ( q J d )$ . So in total it takes $O ( 2 q ( I + J ) d )$ . ",
+        "bbox": [
+            174,
+            226,
+            825,
+            282
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D.4 BPR LOSS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            297,
+            294,
+            313
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In each batch with $B$ users, calculating the scores for positive and negative items both take $O ( B d )$ , so in total it takes $O ( 2 B d )$ . ",
+        "bbox": [
+            173,
+            324,
+            823,
+            353
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D.5 CL LOSS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            369,
+            282,
+            383
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In each batch with $B$ users, calculating the numerator of InfoNCE loss takes $O ( B d )$ , and calculating the denominator takes $O ( B M d )$ where $M$ denotes the total number of nodes in the batch. Since our model adopts a per layer InfoNCE loss, a factor of $L$ is appended. ",
+        "bbox": [
+            174,
+            395,
+            825,
+            438
+        ],
+        "page_idx": 14
+    }
+]

parse/dev/RYTBAtyXqJ/RYTBAtyXqJ.md

@@ -0,0 +1,348 @@
+# CARTOON EXPLANATIONS OF IMAGE CLASSIFIERS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We present CartoonX (Cartoon Explanation), a novel model-agnostic explanation method tailored towards image classifiers and based on the rate-distortion explanation (RDE) framework. Natural images are roughly piece-wise smooth signals—also called cartoon images—and tend to be sparse in the wavelet domain. CartoonX is the first explanation method to exploit this by requiring its explanations to be sparse in the wavelet domain, thus extracting the relevant piece-wise smooth part of an image instead of relevant pixel-sparse regions. We demonstrate experimentally that CartoonX is not only highly interpretable due to its piece-wise smooth nature but also particularly apt at explaining misclassifications.
+
+# 1 INTRODUCTION
+
+Powerful machine learning models such as deep neural networks are inherently opaque, which has motivated numerous explanation methods over the last decade (see for example the survey by Das & Rad (2020)). A significant fraction of the research literature has focused on explaining image classifications due to both the practical relevance of computer vision tasks and the ease at which heatmaps can communicate explanatory information. Despite the great variety in methods and explanation philosophies, all current methods share the following characteristic: they operate in pixel space. Roughly speaking, existing explanation methods for image classifiers either allocate additive attribution scores to each pixel or optimize a deletion mask on the pixel coefficients to mark a relevant set of pixels. The result is typically a pixel-sparse and jittery explanation. We challenge the conventional approach to explain in pixel space by successfully applying the rate-distortion explanation (RDE) framework (Macdonald et al., 2019; Heiß et al., 2020) in the wavelet domain of images. Our novel explanation method, CartoonX, extracts the relevant piece-wise smooth part of an image (see Figure 1). Instead of demanding sparsity in pixel space, as in (Macdonald et al., 2019; Chang et al., 2019), CartoonX demands sparsity in the wavelet domain, which produces piece-wise smooth explanations (cartoon-like images). Our work makes the following contributions:
+
+![](images/dc24c4587f388205daecba23a270da795c248d00d7e21388c915cc977296fc88.jpg)  
+Dog classified as Egyptian cat
+
+![](images/7b8be6eda1c4fcf43ca641d589d627f2f43768ac5cd270644d7bf6bc47659f3e.jpg)  
+CartoonX of misclassification
+
+Reformulation and reinterpretation of the RDE framework: We reformulate the RDE framework in a more general manner with enhanced flexibility in the input representation to accommodate complex interpretation queries such as “What is the piece-wise smooth part of the input signal that leads to its model decision?”. Thereby, we reinterpret RDE as a simplification of the input signal, which is interpretable to humans and adheres to a meaningful interpretation query. The simplification is achieved by demanding sparsity in a suitable representation system, which sparsely represents the class of explanations that are desirable for the interpretation query.
+
+![](images/151279dd372408cf29d741b90adc464dd546cc38901a0c7f5d08c3f7889b1f20.jpg)  
+Slam dunk classified as basketball
+
+CartoonX, a novel explanation method tailored to image classifiers: CartoonX is the first explanation method to extract the relevant piece-wise smooth part of an image instead of relevant pixel sparse regions. This is achieved by demanding sparsity in the wavelet domain of images, where
+
+![](images/f1dc4810d3ca077a16c122ad81a791c1b7bed617df1470748dde8659528a1338.jpg)  
+Figure 1: Examples of CartoonX explanations.
+
+sparsity translates into piece-wise smooth images. We demonstrate that   
+our piece-wise smooth explanations are more interpretable than jittery   
+pixel-sparse explanations and that they can reveal relevant piece-wise smooth patterns that are not easily visible with existing pixel-based methods. Surprisingly, we find that our method is particularly well-equipped to explain misclassifications, often showing “what the neural network actually saw” (see Figure 1).
+
+# 2 RELATED WORK
+
+The Rate-Distortion Explanation (RDE) framework was first introduced in (Macdonald et al., 2019), and extended in (Heiß et al., 2020), as a mathematically well-founded and intuitive explanation framework. RDEs are model-agnostic explanations and inspired by rate-distortion theory, which studies lossy-data compression. An explanation in RDE consists of a relatively sparse mask over the input features, highlighting the relevant set of features. The mask is optimized to produce low distortion in the model output after applying perturbations to the unselected features in the input while remaining relatively sparse. Heiß et al. (2020) also applied RDE to non-canonical input representations to explain model decisions in challenging domains such as audio classification (Engel et al., 2017) and radio-map estimation (Levie et al., 2021; 2020).
+
+The explanation principle of optimizing a mask $s \in [ 0 , 1 ] ^ { n }$ was first proposed by Fong & Vedaldi (2017) who explained image classification decisions by considering one of the two “deletion games”: (1) optimizing for the smallest deletion mask that causes the class score to drop significantly or (2) optimizing for the largest deletion mask that has no significant effect on the class score. The original RDE approach (Macdonald et al., 2019) is based on the second deletion game.
+
+Other explanation methods developed by the research community are typically either (1) gradientbased such as Smoothgrad (Smilkov et al., 2017), Integrated Gradients (Sundararajan et al., 2017), Image-Specific Class Saliency (Simonyan et al., 2014), and Guided Backpropagation (Springenberg et al., 2015), (2) surrogate models such as LIME (Ribeiro et al., 2016), (3) based on propagation of activations in neurons such as LRP (Bach et al., 2015; Shrikumar et al., 2017), and DeepLIFT (Shrikumar et al., 2017), (4) based on Shapely values from game-theory (Lundberg & Lee, 2017), (6) concept-based such as Concept Activation Vectors (Kim et al., 2018), or (7) based on generative causal explanations (O' Shaughnessy et al., 2020). Also related are methods that were developed to explain individual neurons such as in (Nguyen et al., 2016; Dhamdhere et al., 2019). To our knowledge, all existing explainability methods operate in pixel space and all methods looking for sparse explanations demand sparsity in pixel space (Macdonald et al., 2019; Fong & Vedaldi, 2017; Chang et al., 2019).
+
+# 3 BACKGROUND: RATE-DISTORTION EXPLANATION FRAMEWORK
+
+In this section, we review the rate-distortion explanation (RDE) framework, which was introduced by Macdonald et al. (2019) and later extended by Heiß et al. (2020) by applying RDE to noncanonical input representations. Suppose $\Phi : \mathbb { R } ^ { n }  \mathbb { R } ^ { m }$ is a pre-trained model, e.g., a classifier (with $m$ class labels) or a regression model (with $m$ -dimensional output), where $n$ denotes the dimension of the model input. RDE produces an explanation for a model decision $\Phi ( x )$ with $x \in \mathbb { R } ^ { n }$ as a relatively sparse mask $s \in \{ 0 , 1 \}$ marking the relevant input features in $x$ . More precisely, RDE aims to solve the following optimization problem over a mask $s \in \{ 0 , 1 \} ^ { n }$ :
+
+$$
+\operatorname* { m i n } _ { s \in \{ 0 , 1 \} ^ { n } } \quad \operatorname { \mathbb { E } } _ { v \sim \mathcal { V } } \left[ d \Bigl ( \Phi ( x ) , \Phi ( x \odot s + ( 1 - s ) \odot v ) \Bigr ) \right] \quad \mathrm { s . t . } \quad \| s \| _ { 0 } \leq \ell ,
+$$
+
+where $\odot$ denotes the Hadamard product (element-wise multiplication), $d ( \Phi ( x ) , \cdot )$ is a measure of distortion (e.g. $d ( \Phi ( x ) , \cdot ) = \lVert \Phi ( { \bar { x } } ) - \cdot \rVert _ { 2 } )$ , $\nu$ is a distribution over input perturbations $v \in \mathbb { R } ^ { n }$ , and $\ell \in \{ 1 , . . . , n \}$ is a given sparsity level for the explanation mask $s$ . A solution $s ^ { * }$ to the optimization problem in (1) marks relatively few components in the model input $x$ that suffice to approximately retain the model output $\Phi ( x )$ . This approach is in the spirit of rate-distortion theory, which deals with lossy compression of data. Therefore, Macdonald et al. (2019) coined such explanations ratedistortion explanations (RDEs).
+
+In practice, the optimization problem in (1) is relaxed to continuous masks $s \in [ 0 , 1 ]$ solving
+
+$$
+\operatorname* { m i n } _ { s \in \{ 0 , 1 \} ^ { n } } \quad \operatorname { \mathbb { E } } _ { v \sim \mathcal { V } } \left[ d \Bigl ( \Phi ( x ) , \Phi ( x \odot s + ( 1 - s ) \odot v ) \Bigr ) \right] + \lambda \left\| s \right\| _ { 1 } ,
+$$
+
+where $\lambda > 0$ determines the sparsity level of the mask. The relaxed optimization problem can be solved with stochastic gradient descent in $s \in [ 0 , 1 ]$ if $\Phi$ is differentiable—as is the case for deep neural networks. Macdonald et al. (2019) applied the RDE method as described above to image classifiers in the pixel domain of images, where each mask entry $s _ { i } \in [ 0 , 1 ]$ corresponds to the $i$ -th pixel values. We refer to this method as Pixel RDE throughout this work.
+
+# 4 RDE REFORMULATED AND REINTERPRETED
+
+Instead of applying RDE to the standard input representation $\boldsymbol { x } = [ x _ { 1 } \dots x _ { n } ] ^ { T }$ , we can apply RDE to a different representation of $x$ to answer a particular interpretation query. For example, consider a 1D-signal $x \in \mathbb { R } ^ { n }$ : if we ask “What is the smooth part in the signal $x$ that leads to the model decision $\Phi ( x ) ? ^ { , }$ , then we can apply RDE in the Fourier basis of $x$ . Since frequency-sparse signals are smooth, applying RDE in the Fourier basis of $x$ extracts the relevant smooth part of the signal. To accommodate such interpretation queries, we reformulate RDE in Section 4.1. Finally, based on the reformulation, we reinterpret RDE in Section 4.2. Later in Section 5, we use our reformulation and reinterpretation of RDE to derive and motivate CartoonX as a special case and novel explanation method tailored towards image classifiers.
+
+# 4.1 GENERAL FORMULATION
+
+An input signal $\boldsymbol { x } = [ x _ { 1 } , \dots , x _ { n } ] ^ { T }$ is represented in a basis $\{ b _ { 1 } , \ldots , b _ { n } \}$ as a linear combination $\textstyle \sum _ { i = 1 } ^ { n } h _ { i } b _ { i }$ with coefficients $[ h _ { i } ] _ { i = 1 } ^ { n }$ . As we argued above and demonstrate later on, some choices for a basis may be more suitable than others to explain a model decision $\Phi ( x )$ . Therefore, we define the RDE mask not only on the canonical input representation $[ x _ { i } ] _ { i = 1 } ^ { n }$ but also on a different representation $[ h _ { i } ] _ { i = 1 } ^ { n }$ with respect to a choice of basis $\{ b _ { 1 } , \ldots , b _ { n } \}$ . Examples of non-canonical choices for a basis include the Fourier basis and the wavelet basis. This work is centered around CartoonX, which applies RDE in the wavelet basis, i.e., a linear data representation since $x$ is represented as a linear combination of basis vectors. Nevertheless, there also exist other domains and interpretation queries where applying RDE to a non-linear data representation can make sense (see the interpretation query “Is phase or magnitude more important for an audio classifier?” in (Heiß et al., 2020)). Therefore, we formulate RDE in terms of a data representation function $\textstyle f : \prod _ { i = 1 } ^ { k } \mathbb { R } ^ { c } \to \mathbb { R } ^ { n }$ , $f ( h _ { 1 } , \ldots , h _ { k } ) = x .$ , which does notlinear case and o be linear, we have $c$ ls in the, where e imare ortantfixed $c = 1$ $\begin{array} { r } { f ( h _ { 1 } , \ldots , h _ { k } ) = \sum _ { i = 1 } ^ { k } h _ { i } b _ { i } } \end{array}$ $\{ b _ { i } , \ldots , b _ { k } \} \subset \mathbb { R } ^ { n }$ $k$ $c > 1$   
+channels at once, e.g., all color channels of an image, to reduce the number of entries in the mask that will operate on $[ h _ { i } ] _ { i = 1 } ^ { k }$ . In the following, we introduce the important definitions of obfuscations, expected distortion, the RDE mask, and $R D E ' s \ell _ { 1 }$ -relaxation, which generalize the RDE framework of (Macdonald et al., 2019) to abstract input representations.
+
+# 4.1.1 DEFINITIONS
+
+The first two key concepts in RDE are obfuscations and expected distortion, which are defined below.
+
+Definition 1 (Obfuscations and expected distortion) Let $\Phi : \mathbb { R } ^ { n }  \mathbb { R } ^ { m }$ be a model and $x \in \mathbb { R } ^ { n }$ a data point with a data representation $x = f ( h _ { 1 } , . . . , h _ { k } )$ as discussed above. For every mask $s \in [ 0 , 1 ] ^ { k }$ , let $\gamma _ { s }$ be a probability distribution over $\textstyle \prod _ { i = 1 } ^ { k } \mathbb { R } ^ { c }$ . Then the obfuscation of $x$ with respect to s and $\gamma _ { s }$ is defined as the random vector $y : = { f ( s \odot h + ( 1 - s ) \odot v ) }$ , where $v \sim \mathcal { V } _ { s }$ , $( s \odot h ) _ { i } = s _ { i } h _ { i } \in \mathbb { R } ^ { c }$ and $( ( 1 - s ) \odot v ) _ { i } = ( 1 - s _ { i } ) v _ { i } \in \mathbb R ^ { c }$ , for $i \in \{ 1 , \ldots , k \}$ . A choice for the distribution $\gamma _ { s }$ is called obfuscation strategy. Furthermore, the expected distortion of $x$ with respect to the mask s and the perturbation distribution $\gamma _ { s }$ is defined as
+
+$$
+D ( x , s , \mathcal { V } _ { s } , \Phi ) : = \underset { v \sim \mathcal { V } _ { s } } { \mathbb { E } } \left[ d \Big ( \Phi ( x ) , \Phi ( y ) \Big ) \right] ,
+$$
+
+where $d : \mathbb { R } ^ { m } \times \mathbb { R } ^ { m } \to \mathbb { R } _ { + }$ is a measure of distortion between two model outputs.
+
+In the RDE framework, the explanation is given by a mask that minimizes distortion while remaining relatively sparse. The rate-distortion explanation mask is defined as follows.
+
+Definition 2 (The RDE mask) In the setting of Definition $I$ we define the RDE mask as a solution $s ^ { * } ( \ell )$ to the minimization problem
+
+$$
+\operatorname* { m i n } _ { s \in \{ 0 , 1 \} ^ { k } } \quad D ( x , s , \mathcal { V } _ { s } , \Phi ) \quad s . t . \quad \| s \| _ { 0 } \leq \ell ,
+$$
+
+where $\ell \in \{ 1 , \ldots , k \}$ is the desired level of sparsity.
+
+Geometrically, the RDE mask $s$ is associated with a particular subspace. The complement mask $( 1 - s )$ can be seen as selecting a large stable subspace of $\Phi$ , with each point representing a possible perturbation in unselected coefficients in $h$ . The RDE mask minimizes the expected distortion along its associated subspace, which requires non-local information of $\Phi$ . We illustrate this geometric view of RDE in Figure 2 with a toy example for a hypothetical classifier $\Phi : \mathbb { R } ^ { 2 }  \mathbb { R } ^ { \bar { m } }$ and two distinct input representations: (1) Euclidean coordinates, i.e., $f$ is the identity in $x = f ( h )$ , and (2) polar coordinates, i.e. $f ( h ) = ( h _ { 2 } \cos h _ { 1 } , h _ { 2 } \sin h _ { 1 } ) = x$ . In the example, we assume $\gamma _ { s }$ to be a uniform distribution on $[ - 1 , 1 ] ^ { 2 }$ in the Euclidean representation and a uniform distribution on $[ - \pi , \pi ] \times [ 0 , 1 ]$ in the polar representation. The expected distortion associated with the masks $s = ( 1 , 0 )$ and $s = ( 0 , 1 )$ is given by the red and green shaded area, respectively. The RDE mask aims for low expected distortion, and hence, in polar coordinates, the RDE mask would be the green subspace, i.e., $s = ( 0 , 1 )$ . On the other hand, in Euclidean coordinates, neither $s = ( 1 , 0 )$ nor $s = ( 0 , 1 )$ produces a particularly low expected distortion, making the Euclidean explanation less meaningful than the polar explanation. The example illustrates why certain input representations can yield more meaningful explanatory insight for a given classifier than others—an insight that underpins our novel CartoonX method. Moreover, the plot in polar coordinates illustrates why the RDE mask cannot be simply chosen with local distortion information, e.g., with the lowest eigenvalue of the Hessian of $\bar { h } \mathbin { \stackrel { \cdot } { \mapsto } } d ( \Phi ( x ) , \Phi ( f ( h ) ) )$ : the lowest eigenvalue in polar coordinates belongs to the red subspace and does not see the large distortion on the tails.
+
+![](images/479683e2b0840f5f992b0532f0d8810568debf14e35b168351c2027f9e57a0b2.jpg)  
+Figure 2: The RDE mask can find low expected distortion in polar coordinates but not in Euclidean coordinates. Therefore, in this example, polar coordinates are more appropriate to explain $\Phi ( x )$ , and RDE would determine that the angle $\varphi$ , not the magnitude $r$ , is relevant for $\Phi ( x )$ .
+
+As was shown by Macdonald et al. (2019), the RDE mask from Definition 2 cannot be computed efficiently for non-trivial input sizes. Nevertheless, one can find an approximate solution by considering continuous masks $s \in [ 0 , 1 ] ^ { k }$ and encouraging sparsity through the $\ell _ { 1 }$ -norm.
+
+Definition 3 (RDE’s $\ell _ { 1 }$ -relaxation with Lagrange multipliers) In the setting of Definition $I$ , we define RDE’s $\ell _ { 1 }$ -relaxation with Lagrange multipliers as a solution $s ^ { * } ( \lambda )$ to the minimization problem
+
+$$
+\begin{array} { r l } { \underset { s \in [ 0 , 1 ] ^ { k } } { \operatorname* { m i n } } } & { { } D ( \boldsymbol { x } , s , \mathcal { V } _ { s } , \boldsymbol { \Phi } ) + \lambda \| s \| _ { 1 } , } \end{array}
+$$
+
+where $\lambda > 0$ is a hyperparameter for the sparsity level.
+
+The $\ell _ { 1 }$ -relaxation above can be solved with stochastic gradient descent (SGD) over the mask $s$ while approximating $D ( x , s , \mathcal { V } _ { s } , \Phi )$ with i.i.d. samples from $v \sim \mathcal { V } _ { s }$ .
+
+# 4.1.2 OBFUSCATION STRATEGIES
+
+An obfuscation strategy is defined by the choice of the perturbation distribution $\mathcal { V } _ { s }$ . Common choices are Gaussian noise (Macdonald et al., 2019; Fong & Vedaldi, 2017), blurring (Fong & Vedaldi, 2017), constants (Fong $\&$ Vedaldi, 2017), and inpainting GANs (Heiß et al., 2020; Chang et al., 2019). Inpainting GANs train a generator $G ( s , z , h )$ ( $z$ denotes random latent factors) such that for samples $v \sim G ( s , z , h )$ the obfuscation $f ( s \odot h + ( 1 - s ) \odot v )$ remains in the data manifold. In our work, we refrain from using an inpainting GAN due to the following reason: it is hard to tell whether a GAN-based mask did not select coefficients because they are unimportant or because the GAN can easily inpaint them from a biased context. Instead, we choose a simple and wellunderstood obfuscation strategy, which we call Gaussian adaptive noise, making the explanation as transparent as possible.
+
+Gaussian adaptive noise works as follows: Let $A _ { 1 } , . . . , A _ { j }$ be a pre-defined choice of a disjoint partition of $\{ 1 , \ldots , k \}$ (recall $s \in [ 0 , 1 ] ^ { k } )$ . For $i = 1 , . . . , j$ , we compute the empirical mean and empirical standard deviation for each partition across all partition instances:
+
+$$
+\mu _ { i } : = \frac { 1 } { \sum _ { a \in A _ { i } } d _ { a } } \sum _ { a \in A _ { i } , t = 1 , \ldots , d _ { a } } h _ { a t } , \sigma _ { i } : = \sqrt { \frac { 1 } { \sum _ { a \in A _ { i } } d _ { a } } \sum _ { a \in A _ { i } , t = 1 , \ldots , d _ { a } } ( \mu _ { i } - h _ { a t } ) ^ { 2 } }
+$$
+
+The adaptive Gaussian noise strategy then samples $v _ { a t \_ } \sim \mathcal { N } ( \mu _ { i } , \sigma _ { i } ^ { 2 } )$ for all partition members $a \in$ $A _ { i }$ and channels $t = 1 , . . . , d _ { a }$ . We write $v \sim \mathcal { N } ( \mu , \sigma ^ { 2 } )$ for the resulting Gaussian random vector $\boldsymbol { v } \in \prod _ { i = 1 } ^ { k } \mathbb { R } ^ { c }$ . Note that the distribution $\gamma _ { s }$ chosen as Gaussian adaptive noise does depend on $s$ (unlike with an inpainting GAN). For Pixel RDE, we only use one set $A _ { 1 } = \left\{ 1 , . . . , k \right\}$ for all $k$ pixels. In CartoonX, which represents input signals in the discrete wavelet domain, we will partition $\{ 1 , . . . , k \}$ along the scales of the discrete wavelet transform.
+
+# 4.1.3 MEASURES OF DISTORTION
+
+There are various choices for the measure of distortion $d ( \Phi ( x ) , \Phi ( y ) )$ . For example, one can take the squared distance in the post-softmax probability of the predicted label for $x$ , i.e.,
+
+$$
+d \big ( \Phi ( x ) , \Phi ( y ) \big ) : = \big ( \Phi _ { j ^ { * } } ( x ) - \Phi _ { j ^ { * } } ( y ) \big ) ^ { 2 } ,
+$$
+
+where $j ^ { * } : = \arg \operatorname* { m a x } _ { i = 1 , \ldots , m } \Phi _ { i } ( x )$ and $\Phi ( x )$ is assumed to be the post-softmax probabilities of a neural net. Alternatively, one could also choose $d ( \Phi ( x ) , \Phi ( y ) )$ as the $\ell _ { 2 }$ -distance or the KLDivergence in the post-softmax layer of $\Phi$ . In our experiments for CartoonX, we found that these choices had no significant effect on the explanation (see Appendix A.3.3).
+
+# 4.2 INTERPRETATION
+
+The philosophy of the generalized RDE framework is that an explanation for a decision $\Phi ( x )$ on a generic input signal $x = f ( h )$ should be some simplified version of the signal, which is interpretable to humans. The simplification is achieved by demanding sparsity in a suitable representation system $h$ , which sparsely represents the class of explanations that are desirable for the interpretation query. This philosophy is the fundamental premise of CartoonX, which aims to answer the interpretation query “What is the relevant piece-wise smooth part of the image for a given image classifier?”. CartoonX first employs RDE on a representation system $x = f ( h )$ that sparsely represents piecewise smooth images and finally visualizes the relevant piece-wise smooth part as an image back in pixel space. In the following section, we explain why wavelets provide an appropriate representation system in CartoonX, present the CartoonX implementation, and finally provide experiments on ImageNet to demonstrate the capability of CartoonX.
+
+# 5 CARTOONX
+
+The focus of this paper is CartoonX, a novel explanation method—tailored to image classifications— that we obtain as a special case of our generalized RDE framework formulated in Section 4. CartoonX first performs RDE in the discrete wavelet position-scale domain of an image $x$ , and finally, visualizes the wavelet mask $s$ as a piece-wise smooth image in pixel space. Wavelets provide optimal representations for piece-wise smooth 1D functions (DeVore, 1998), and represent 2D piecewise smooth images, also called cartoon-like images (Kutyniok & Lim, 2011), efficiently as well (Romberg et al., 2006). In particular, sparse vectors in the wavelet coefficient space encode cartoonlike images reasonably well (Stephane, 2009a)—certainly better than sparse pixel representations. ´ Moreover, wavelets constitute an established tool in signal processing (Stephane, 2009c). ´
+
+![](images/600d34c48f6449578084ae2acb04f5102f541072c868e8f11c61d9f05a838409.jpg)  
+Figure 3: CartoonX has many interesting parallels to wavelet-based image compression. Distortion is denoted as $d$ , $\Phi$ is an image classifier, $h$ denotes the discrete wavelet coefficients, $\tau$ is the discrete wavelet transform, and $\ell$ is the coefficient budget.
+
+The optimization process underlying CartoonX produces sparse vectors in the discrete wavelet coefficient space, which results in cartoon-like images as explanations. This is the fundamental difference to Pixel RDE, which produces rough, jittery, and pixel-sparse explanations. Cartoon-like images are more interpretable and provide a natural model of simplified images. Since the goal of the RDE framework is to generate an easy to interpret simplified version of the input signal, we argue that CartoonX explanations are more appropriate for image classification than Pixel RDEs.
+
+CartoonX exhibits interesting parallels to wavelet-based image compression. In image compression, distortion is minimized in the data domain, which is equivalent to selecting the $\ell$ largest entries in the discrete wavelet transform (DWT) coefficients. In comparison, CartoonX minimizes distortion in the model output of $\Phi$ , which translates to selecting the $\ell$ most relevant entries in the DWT coefficients. The objective in image compression is efficient data representation, i.e., producing minimal data distortion with a budget of $\ell$ entries in the DWT coefficients. Conversely, in CartoonX, the objective is extracting the relevant piece-wise smooth part, i.e., producing minimal model distortion with a budget of $\ell$ entries in the DWT coefficients. We illustrate this connection in Figure 3—highlighting once more the rate-distortion spirit of the RDE framework.
+
+# 5.1 IMPLEMENTATION
+
+An image $x \in [ 0 , 1 ] ^ { c \times w \times t }$ with $c \in \{ 1 , 3 \}$ channels, width $w \in \mathbb { N }$ , height $t \in \mathbb N$ , and a total of $p = w t$ pixels can be represented in a wavelet basis by computing its discrete wavelet transform (DWT). The DWT of an image is defined by the number of scales $J \in \{ 1 , \ldots , \lfloor \log _ { 2 } p \rfloor \}$ , the padding mode, and a choice of the wavelet family (such as the Haar or Daubechies family). For images, the DWT computes four types of coefficients: details in (1) horizontal, (2) vertical, and (3) diagonal orientation at scale $j \in \{ 1 , \dots , J \}$ , and (4) coefficients of the image at the very coarsest resolution. We briefly illustrate the DWT for an example image in Figure 4.
+
+![](images/39ebb784157d2c6edd31b1dadc49ecf3ff916b928f9b9fd1b5f08907bedcccf9.jpg)  
+Figure 4: Left side: an image of a memorial arch dedicated to peace. Right side: visualization of the DWT coefficients for five scales. Three L-shaped sub-images describe coefficients for details in vertical, horizontal, and diagonal orientation at a particular scale. The largest sub-images (the outer L-shape) belong to the lowest scale, i.e., the highest resolution. The smaller L-shaped sub-images gradually build up to higher scales, i.e., lower resolution features.
+
+CartoonX, as described in Algorithm 1 in Appendix A.1, computes the RDE mask in the wavelet domain of images. More precisely, for the data representation ${ \bar { \boldsymbol { x } } } = f ( h )$ , we choose $h$ as the concatenation of all the DWT coefficients along the channels, i.e., $\boldsymbol { h } _ { i } \in \mathbb { R } ^ { c }$ . The representation function $f$ is then the discrete inverse wavelet transform, i.e., the summation of the DWT coefficients times the DWT basis vectors. We optimize the mask $s \in [ 0 , 1 ] ^ { k }$ on the DWT coefficients $[ h _ { 1 } , \ldots , h _ { k } ] ^ { T }$ to minimize RDE’s $\ell _ { 1 }$ -relaxation from Definition 3. For the obfuscation strategy $\gamma _ { s }$ , we use adaptive Gaussian noise with a partition by the DWT scale (see Section 4.1.2), i.e., we compute the empirical mean and standard deviation per scale. We measure distortion as the squared difference in the postsoftmax score of the predicted label for $x$ (see Section 4.1.3). To visualize the final DWT mask $s$ as a piece-wise smooth image in pixel space, we multiply the mask with the DWT coefficients of the greyscale image $\hat { x } : = ( 1 \breve { / c } \sum _ { l = 1 } ^ { \hat { c } } x _ { l a i } \dot { ) } _ { a i }$ before inverting the product back to pixel space with the discrete inverse wavelet transform. The inversion is finally clipped into $[ 0 , 1 ] ^ { w \times \dot { t } }$ as are obfuscations during the RDE optimization to avoid overflow (we assume here the pixel values in $x$ are normalized into $[ 0 , 1 ] )$ . The clipped inversion in pixel space is the final explanation, which we call CartoonX.
+
+# 5.2 EXPERIMENTS AND ANALYSIS
+
+We compare CartoonX to the closely related Pixel RDE (Macdonald et al., 2019) and several other state-of-the-art explanation methods , that is, Integrated Gradients (Sundararajan et al., 2017), Smoothgrad (Smilkov et al., 2017), Guided Backprop (Springenberg et al., 2015), and LRP (Bach et al., 2015). Our experiments show that CartoonX carries the following strengths: Cartoon X is (1) highly interpretable due to its cartoon-like nature and (2) remarkably apt at explaining misclassifications, and highlighting meaningful patterns that are otherwise hard to see. Due to the fast implementation of the DWT, Cartoon RDE is not significantly slower than Pixel RDE. For the ImageNet classifier MobileNetV3-Small, an image of 256 times 256 pixels, and 2001 optimization steps, we reported a runtime of 81.56 seconds for CartoonX and 70.53 seconds for Pixel RDE on the NVIDIA Titan RTX GPU. However, like other perturbation-based methods, CartoonX is significantly slower than gradient or propagation-based methods, which only compute a single or few forward and backward passes and are very fast (Integrated Gradients computes an explanation in 0.48 seconds for the same image, model, and hardware).
+
+Our experiments use the pre-trained ImageNet classifiers MobileNetV3-Small (Howard et al., 2019) (top-1 accuracy of $6 7 . 6 6 8 \%$ and VGG16 (Simonyan & Zisserman, 2015) (top-1 accuracy of $7 1 . 5 9 2 \%$ ). We note that the open-source implementation of LRP did not implement propagation rules for certain layers in MobileNetV3-Small, therefore we compare CartoonX to LRP only for VGG16. Images were preprocessed to have 256 times 256 pixel values in [0, 1]. We provide further details about the choice of hyperparameters in the experiments in Appendix A.2. The three main hyperparameters for CartoonX are: (1) the sparsity level $\lambda > 0$ , (2) the measure of distortion $d$ , and (3) the obfuscation strategy (perturbation distribution) $\gamma _ { s }$ . We discuss the sensitivity of CartoonX to these hyperparameters in Appendix A.3. In Appendix A.5, we also shed light on the evolution of ImageNet classifiers from an explanation angle by comparing CartoonX explanations for classifiers of varying generalization power, i.e., AlexNet (Krizhevsky et al., 2012), VGG16 (Simonyan & Zisserman, 2015), InceptionV3 (Szegedy et al., 2016), ResNeXt50 (Xie et al., 2017). Moreover, in Appendix A.4, we argue experimentally why CartoonX is less susceptible than Pixel RDE to so-called explanation artifacts—an unwanted phenomenon that we observed empirically.
+
+![](images/0a1f842c7c2f27339003af5f5ca74ec6399ee2beaa7f81924ce3ed1fbaf9fc58.jpg)  
+Figure 5: Each row compares CartoonX explanations of misclassifications by MobileNetV3-Smal to Pixel RDE (Macdonald et al., 2019), Integrated Gradients (Sundararajan et al., 2017), and Smoothgrad (Smilkov et al., 2017). The predicted label is depicted above each misclassified image.
+
+In practice, explaining misclassifications is particularly relevant since good explanations can pinpoint model biases and causes for model failures. We observe that CartoonX is particularly good at explaining certain misclassifications, which we illustrate for three examples in Figure 5 and many more in Appendix A.6. In the first row in Figure 5, the input image shows a man holding a dog that was classified as a “diaper”. CartoonX shows the man not holding a dog but a baby, revealing that the neural net associated diapers with babies and babies with the pose with which the man is holding the dog. In the second row, the input image shows a dog sitting on an armchair with leopard patterns. The dog was classified as an “Egyptian cat”, which can exhibit leopard-like patterns. CartoonX exposes the Egyptian cat by connecting the dog’s head to parts of the armchair forming the cat’s torso and legs. In the last row, the input image displays the backside of a man wearing a striped sweater that was classified as a “screw”. CartoonX reveals how the stripe patterns look like a screw to the neural net.
+
+![](images/93b3a56308de3770000542d827bd0e4671770354a0f003b0ff69f08314095854.jpg)  
+Figure 6: CartoonX explanations for VGG16 compared to state-of-the-art methods, that is, Pixel RDE (Macdonald et al., 2019), Integrated Gradients (Sundararajan et al., 2017), Smoothgrad (Smilkov et al., 2017), Guided Backprop (Springenberg et al., 2015), and LRP (Bach et al., 2015).
+
+# 6 CONCLUSION
+
+CartoonX is the first explainability method to extract the relevant piece-wise smooth part of an image and is based on our novel formulation of the RDE framework. We corroborated experimentally that CartoonX explanations are highly interpretable due to their cartoon-like nature and surprisingly well-suited to explain misclassifications. Nonetheless, Cartoon RDE is still computationally quite expensive, like other perturbation-based explanation methods. In the future, we hope to devise new techniques to speed up the runtime for CartoonX. Moreover, we are pursuing applications of CartoonX beyond explanation tasks, such as detecting adversarial examples. We believe CartoonX is a valuable new explanation method for practitioners and potentially a great source of inspiration for future explanation methods aiming to tailor their explanations to other data domains. Our reformulation and reinterpretation of the RDE framework provide a blueprint for such future work: First, formulate an interpretation query related to the underlying model task, then find a representation system that sparsely represents the class of desirable explanations for the interpretation query.
+
+# REFERENCES
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+Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. ”Why should I trust you?”: Explaining the predictions of any classifier. In Proceedings of the 22nd International Conference on Knowledge Discovery and Data Mining, ACM SIGKDD, pp. 1135–1144. Association for Computing Machinery, 2016. ISBN 9781450342322.
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+
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+
+# A APPENDIX
+
+# A.1 CARTOONX ALGORITHM
+
+The final CartoonX algorithm is depicted in Algorithm 1.
+
+# Algorithm 1: CartoonX
+
+<table><tr><td>AigormmmrT.CartoonA Data: Image x ∈ [0,1]exwxt with c channels and wt pixels, classifier Φ. Result: CartoonX explanation ε ∈ [0,1]w×t for decision Φ(x). Hyperparameters: Sparsity level X &gt; O, number of steps N, number of noise samples L. Initialize mask s := [1,..,1]T ∈ [0,1]k on DWT coefficients h = [h1,.., hk] with x = f(h), where f is the discrete inverse wavelet transform; Compute predicted label j* := arg maxi Φ(x); fori←1toNdo</td></tr><tr><td>Sample L adaptive Gaussian noise samples u(1),., u(L) ~ N(μ,o²); Compute obfuscations y(1), ),.,y(L) with y() := f(h ① s+ (1- s) ①u(i)); Clip obfuscations into [0,1]cx w ×t;</td></tr><tr><td>Approximate expected distortion D(𝑥x,s,Φ) :=∑𝑖=1(Φj+(x)- Φj(y())²/L;</td></tr><tr><td>Compute loss for the mask l(s) := D(x,s,Φ) + λ|lsll1 and gradient Vsl(s); Update mask s with gradient descent step and clip s back to [0,1]k ;</td></tr><tr><td>end Compute wavelet coefficients h for greyscale image x of x;</td></tr><tr><td>Invert wavelet mask s back to pixel space as &amp; := f(h  s) ; Clip the explanation ε into [0,1]w×t to obtain ε. Visualize ε;</td></tr></table>
+
+# A.2 EXPERIMENT DETAILS
+
+Throughout our experiments with CartoonX and Pixel RDE, we used a learning rate of $\epsilon = 0 . 0 0 1$ , a sample size of $L = 6 4$ for the adaptive Gaussian noise, and $N = 2 0 0 0$ steps. Several different sparsity levels were used. We recommend specifying the sparsity level in terms of the number of mask entries $k$ , i.e., choosing the product $\lambda k$ . Pixel RDE typically requires a smaller sparsity level than CartoonX. We chose $\bar { \lambda k } \in [ \bar { 2 0 } , 8 0 ]$ for CartoonX and $\bar { \lambda } k \in [ 3 , 2 \bar { 0 } ]$ for Pixel RDE. The obfuscation strategy for Pixel RDE was chosen as Gaussian adaptive noise with mean and standard deviation computed for all pixel values (see Section 4.1.2). In Appendix 8, we show that Gaussian adaptive noise produces much more interpretable explanations than using a zero baseline perturbation. We implemented the DWT for CartoonX with the Pytorch Wavelets package, which is compatible with PyTorch gradient computations, and chose the Daubechies wavelet system with $J = 5$ scales and zero-padding. For the Integrated Gradients method, we used 100 steps, and for the Smoothgrad method, we used 10 samples and a standard deviation of 0.1.
+
+# A.3 SENSITIVITY TO HYPERPARAMETERS
+
+We compare CartoonX’s sensitivity to its main hyperparameters, i.e., the sparsity level $\lambda$ , the perturbation distribution $\gamma _ { s }$ , and the distortion measure $\bar { d ( \Phi ( x ) , \Phi ( y ) ) }$ . For each experiment, we fix all but one of the three parameters.
+
+# A.3.1 SENSITVITY TO THE SPARSITY LEVEL $\lambda$
+
+Figure 7 plots CartoonX explanations and Pixel RDEs for increasing $\lambda$ —the hyperparameter determining the explanation’s sparsity in the respective representation system. We find that CartoonX is less sensitive than Pixel RDE to $\lambda$ . In practice, this means one can find a suitable $\lambda$ faster for CartoonX than for Pixel RDE.
+
+# A.3.2 SENSITVITY TO THE DISTRIBUTION $\gamma _ { s }$
+
+Figure 8 plots CartoonX explanations for two choices of $\gamma _ { s }$ : (1) Gaussian adaptive noise (see Section 4.1.2) and (2) constant zero perturbations (i.e. $v = 0$ with probability one under $\gamma _ { s }$ ). We observe that the Gaussian adaptive noise gives much more meaningful explanations than the simple zero baseline perturbations.
+
+![](images/669376803cf15b48772f47190fc15e7b3958a45c1013353768d96a6ea5ac042a.jpg)  
+Figure 7: We compare the sensitivity of CartoonX and Pixel RDE to the sparsity level $\lambda$ . The top row depicts CartoonX, and the bottom row depicts Pixel RDE, for increasing values of $\lambda$ . Note that for $\lambda = 0$ , Pixel RDE is entirely yellow because the mask is initialized as $s ^ { \check { = } } [ 1 \ldots 1 ] ^ { T }$ and $\lambda = 0$ provides no incentive to make s sparser. For the same reason, CatoonX is simply the greyscale image for $\lambda = 0$ .
+
+![](images/014f60850b885dee2e6d5318a68d96e55506ad31ed7a0b81c56dda8b69fa0b6c.jpg)  
+Figure 8: We compare the sensitivity of CartoonX to the perturbation distribution $\gamma _ { s }$ . The top image was classified as a fountain and the bottom image as a viaduct. The second column depicts CartoonX with $\gamma _ { s }$ as Gaussian adaptive noise, and the third column depicts CartoonX with $\gamma _ { s }$ as constant zero perturbations (zero baseline). We observe that Gaussian adaptive noise is much more interpretable than the zero baseline.
+
+# A.3.3 SENSITVITY TO THE DISTORTION MEASURE $d ( \Phi ( x ) , \Phi ( y ) )$
+
+Figure 9 plots CartoonX explanations for the following four choices of $d ( \Phi ( x ) , \Phi ( y ) )$ , where $x$ is the original input, $y$ is the RDE obfuscation, and $\Phi$ outputs post-softmax probabilities:
+
+1. $d ( \Phi ( x ) , \Phi ( y ) ) = ( \Phi _ { j ^ { * } } ( x ) - \Phi _ { j ^ { * } } ( y ) ) ^ { 2 }$ , where $j ^ { * } : = \arg \operatorname* { m a x } _ { j } \Phi _ { j } ( x )$ (squared $\ell _ { 2 }$ in label )
+
+2. $d ( \Phi ( x ) , \Phi ( y ) ) = ( \Phi _ { j ^ { * } } ( x ) - 1 ) ^ { 2 }$ , where $j ^ { * } : = \arg \operatorname* { m a x } _ { j } \Phi _ { j } ( x )$ (maximize label)   
+3. $d ( \Phi ( x ) , \Phi ( y ) ) = \lVert \Phi ( x ) - \Phi ( y ) \rVert _ { 2 }$ ( $\ell _ { 2 }$ probabilities)   
+4. $d ( \Phi ( x ) , \Phi ( y ) ) = K L ( \Phi ( y ) , \Phi ( x ) )$ (KL-Divergence)
+
+The explanations for $d ( \Phi ( x ) , \Phi ( y ) )$ as “squared $\ell _ { 2 }$ in label”, “maximize label”, and ${ } ^ { 6 6 } \ell _ { 2 }$ probabilities” look indistinguishable. For $\bar { d } ( \Phi ( x ) , \mathbf { \bar { \Phi } } ( y ) )$ as KL-Divergence, we see a slightly less smooth explanation, which may be due to the fact that the KL-Divergence is unbounded unlike the other measures of distortion.
+
+![](images/9187ae880a8b62675d03bda7eac035f50f4ff1e5be910d54d869700c0b8877f3.jpg)  
+Figure 9: We compare the sensitivity of CartoonX to four measures of distortion $d ( \Phi ( x ) , \Phi ( y ) )$ . Each of the measures of distortion is marked at the top of each column. We observe almost no difference in the CartoonX explanations for the four distortion measures.
+
+# A.4 RELIABILITY AND EXPLANATION ARTIFACTS
+
+We argue experimentally why CartoonX is more reliable than Pixel RDE for image data. More precisely, we show that CartoonX is less susceptible to so-called explanation artifacts than Pixel RDE. An explanation artifact is an unwanted phenomenon that we observed for Pixel RDE: instead of marking the relevant entries in $x$ , the mask $s$ creates artificial edges that end up making up an artificial class prototype. Explanation artifacts are problematic because they highlight not actual substructures but artificial structures that trigger the classification. Examples for explanation artifacts in Pixel RDE are given in Figure 11.
+
+![](images/202f6780abcf94eb3a89b7f1a8d763a4bce9c8a43edc6cc45602844f237f5f61.jpg)  
+Figure 10: CartoonX and Pixel RDE are both performed on the image of the blue sky. However, both methods are adjusted here to find evidence for the output probabilities of the image of the airplane instead of the blue sky. Pixel RDE, unlike CartoonX, can create an artificial airplane as evidence for an airplane in the smooth blue sky.
+
+Pixel RDE can produce artificial edges in smooth regions for the following reason: When $s$ has a curve-like structure in some region, unselected points near $s$ are replaced with perturbations that tend to differ from the values of the curve-like structure in $s$ . Thus, the curve-like structure also appears in the obfuscation and can produce low distortion if the structure makes up a prototypical class feature (see, for example, the airplane in Figure 10).
+
+We suspect CartoonX is inherently less susceptible to explanation artifacts for the following reason: Natural images tend to be piece-wise smooth, and piece-wise smooth images have sparse high-frequency DWT coefficients that cluster about the edges Stephane (2009b) (see for example ´ Figure 4). For a DWT mask to create artificial edges, it has to select a curve-like structure in the high-frequency coefficients (low-frequency coefficients cannot create edges) and replace surrounding unselected values with different values. However, in CartoonX, perturbations of high-frequency coefficients are Gaussian with low variance centered close to zero (see adaptive Gaussian noise in Section 4.1.2), which are not very different from the values along the selected curve due to the sparsity of the coefficients.
+
+We illustrate our previous reasoning about explanation artifacts in a controlled example (see Figure 10). We take an image $x ^ { ( \mathrm { s k y } ) }$ of a blue sky that is very smooth and an image $x ^ { \mathrm { ( p l a n e ) } }$ of a airplane. The goal is to show that Pixel RDE, unlike CartoonX, can create artificial evidence for the class airplane on the image of the smooth blue sky. We perform CartoonX and Pixel RDE on the blue sky image with the distortion function
+
+$$
+\forall y \in \mathbb { R } ^ { n } : \ d ( \Phi ( x ^ { ( \mathrm { s k y } ) } ) , \Phi ( y ) ) = 1 0 ^ { 6 } \| \Phi ( x ^ { ( \mathrm { p l a n e } ) } ) - \Phi ( y ) \| _ { 2 } ,
+$$
+
+and a sparsity level of $\lambda = 8 0 0 0 0$ . As expected, we observe that Pixel RDE, unlike CartoonX, can create an artificial plane in the smooth blue sky(see Figure 10).
+
+A.5 EXPLAINING THROUGH IMAGENET HISTORY: FROM ALEXNET TO RESNETXT50
+
+In the deep learning community, it is well-known that AlexNet (Krizhevsky et al., 2012) provided a major breakthrough in deep learning, improving the top-5 error on ImageNet from $2 5 \%$ to $16 \%$ . Since then, deep learning based ImageNet classifiers have continued to drastically improve on ImageNet—achieving less than $6 \%$ top-5 error in 2016. In Figure 12, we compare CartoonX for four ImageNet classifiers with increasing performance, starting with AlexNet (top-1 accuracy $5 6 . 5 5 \%$ , AlexNet), VGG16 (top-1 accuracy $7 1 . 5 9 \%$ , Simonyan & Zisserman (2015)), InceptionV3 (top-1 accuracy $7 7 . 2 9 \%$ , Szegedy et al. (2016)), and ResNeXt50 (top-1 accuracy $7 7 . 6 2 \%$ , Xie et al. (2017).) Throughout the experiment, the CartoonX hyperparameters for a given image are not changed for any of the four classifiers.
+
+# A.6 EXPLAINING MISCLASSIFICATIONS WITH CARTOONX
+
+In Figure 13, 14, 15, and 16, we provide further examples where CartoonX provides insightful explanations for misclassified images.
+
+# A.7 CARTOONX COMPARED ON RANDOM IMAGENET SAMPLES
+
+Figure 17, 18, 19, and 20 compares CartoonX to Pixel RDE (Macdonald et al., 2019), Integrated Gradients (Sundararajan et al., 2017), Smoothgrad (Smilkov et al., 2017), Guided Backprop (Springenberg et al., 2015), and (Bach et al., 2015) on random Imagenet samples classified by VGG16.
+
+# A.8 CARTOONX FAILURES
+
+We also show failures of CartoonX in Figure 21. These are examples of explanations that are not interpretable and seem to fail at explaining the model prediction. Notably, most failure examples are also not particularly well explained by other state-of-the-art methods. It is challenging to state the underlying reason for the CatoonX failures with certainty (there is always the possibility that the neural net bases its decision on non-interpretable grounds). We intentionally also showed uninterpretable CartoonX explanations that were not too sparse (all or almost black explanations) since one can typically fix these explanations by decreasing $\lambda$ .
+
+![](images/4348604dba432b3c96ce1e643eb3ab00bb1b1360ac0508fa151d7742220d33a6.jpg)  
+Figure 11: Explanation artifacts in Pixel RDE. We observe that Pixel RDE tends to create edges that are not a subset of the edges in the original input image. These edges can make prototypical artifact patterns such as wrinkles in the cloak (first row), coral tentacles (second row), or chain mail (third row).
+
+![](images/6787c0907f2780a91eeba9a8c758e1e7e257500e6962b20c21edad4a037d379c.jpg)  
+Figure 12: We compare CatoonX explanations for classifications by AlexNet (Krizhevsky et al., 2012), VGG16 (Simonyan & Zisserman, 2015), InceptionV3 (Szegedy et al., 2016), and ResNeXt50 (Xie et al., 2017). Green labels mark correct classifications and red labels mark wrong classifactions.
+
+![](images/eb86233a3b07b868113dda202cab39f6f67461c4de85083de6e1fd8ee8292c8e.jpg)  
+Figure 13: Explaining misclassifications with CartoonX on Imagenet and MobileNetV3-Small.
+
+![](images/3f5f295b306da0a5156d30d76a6aa194e591059d0eb184994eb42043169014f0.jpg)  
+Figure 14: Explaining misclassifications with CartoonX on Imagenet and MobileNetV3-Small.
+
+![](images/4ad860b0ac3719256544354b20a3432e1aeb4a028fe6c9a3feb6cb20414bd6e9.jpg)  
+Figure 15: Explaining misclassifications with CartoonX on Imagenet and MobileNetV3-Small.
+
+![](images/514d598a6d4e95cdbd85c43a862b0a1bd0459a6852d89b3ce2fcb63bcd01792c.jpg)  
+Figure 16: Explaining misclassifications with CartoonX on Imagenet and MobileNetV3-Small.
+
+![](images/549aeec35ea55596a58c06d2945f8fdf0faa71d75a7db2cd4edec824a94e0c99.jpg)  
+Figure 17: Comparing CartoonX on random ImageNet samples and VGG16.
+
+![](images/e1ef9c14fd73e7ff8291699ed4e3f48b71162eebae779f16441340c19979e5a0.jpg)  
+Figure 18: Comparing CartoonX on random ImageNet samples and VGG16.
+
+![](images/0c30ef46b69d0c580910205c2e3cd0a382e2d9d8bbd1b9e1a698153577cf5734.jpg)  
+Figure 19: Comparing CartoonX on random ImageNet samples and VGG16.
+
+![](images/3768315f4f3a61a9c20715449365266afdeba73a93b2ac05bad3c5f3932bdd09.jpg)  
+Figure 20: Comparing CartoonX on random ImageNet samples and VGG16.
+
+![](images/0c28c40ebfbba6ad0603ee8dcfcf832c0f8224baf75b313d925f83ad2d28135b.jpg)  
+Figure 21: Failures of CartoonX.

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+    {
+        "type": "text",
+        "text": "We present CartoonX (Cartoon Explanation), a novel model-agnostic explanation method tailored towards image classifiers and based on the rate-distortion explanation (RDE) framework. Natural images are roughly piece-wise smooth signals—also called cartoon images—and tend to be sparse in the wavelet domain. CartoonX is the first explanation method to exploit this by requiring its explanations to be sparse in the wavelet domain, thus extracting the relevant piece-wise smooth part of an image instead of relevant pixel-sparse regions. We demonstrate experimentally that CartoonX is not only highly interpretable due to its piece-wise smooth nature but also particularly apt at explaining misclassifications. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Powerful machine learning models such as deep neural networks are inherently opaque, which has motivated numerous explanation methods over the last decade (see for example the survey by Das & Rad (2020)). A significant fraction of the research literature has focused on explaining image classifications due to both the practical relevance of computer vision tasks and the ease at which heatmaps can communicate explanatory information. Despite the great variety in methods and explanation philosophies, all current methods share the following characteristic: they operate in pixel space. Roughly speaking, existing explanation methods for image classifiers either allocate additive attribution scores to each pixel or optimize a deletion mask on the pixel coefficients to mark a relevant set of pixels. The result is typically a pixel-sparse and jittery explanation. We challenge the conventional approach to explain in pixel space by successfully applying the rate-distortion explanation (RDE) framework (Macdonald et al., 2019; Heiß et al., 2020) in the wavelet domain of images. Our novel explanation method, CartoonX, extracts the relevant piece-wise smooth part of an image (see Figure 1). Instead of demanding sparsity in pixel space, as in (Macdonald et al., 2019; Chang et al., 2019), CartoonX demands sparsity in the wavelet domain, which produces piece-wise smooth explanations (cartoon-like images). Our work makes the following contributions: ",
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+        "img_path": "images/dc24c4587f388205daecba23a270da795c248d00d7e21388c915cc977296fc88.jpg",
+        "image_caption": [
+            "Dog classified as Egyptian cat "
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+        "image_caption": [
+            "CartoonX of misclassification "
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+        "type": "text",
+        "text": "Reformulation and reinterpretation of the RDE framework: We reformulate the RDE framework in a more general manner with enhanced flexibility in the input representation to accommodate complex interpretation queries such as “What is the piece-wise smooth part of the input signal that leads to its model decision?”. Thereby, we reinterpret RDE as a simplification of the input signal, which is interpretable to humans and adheres to a meaningful interpretation query. The simplification is achieved by demanding sparsity in a suitable representation system, which sparsely represents the class of explanations that are desirable for the interpretation query. ",
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+        "type": "image",
+        "img_path": "images/151279dd372408cf29d741b90adc464dd546cc38901a0c7f5d08c3f7889b1f20.jpg",
+        "image_caption": [
+            "Slam dunk classified as basketball "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "CartoonX, a novel explanation method tailored to image classifiers: CartoonX is the first explanation method to extract the relevant piece-wise smooth part of an image instead of relevant pixel sparse regions. This is achieved by demanding sparsity in the wavelet domain of images, where ",
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/f1dc4810d3ca077a16c122ad81a791c1b7bed617df1470748dde8659528a1338.jpg",
+        "image_caption": [
+            "Figure 1: Examples of CartoonX explanations. "
+        ],
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+        "type": "text",
+        "text": "sparsity translates into piece-wise smooth images. We demonstrate that   \nour piece-wise smooth explanations are more interpretable than jittery   \npixel-sparse explanations and that they can reveal relevant piece-wise smooth patterns that are not easily visible with existing pixel-based methods. Surprisingly, we find that our method is particularly well-equipped to explain misclassifications, often showing “what the neural network actually saw” (see Figure 1). ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "The Rate-Distortion Explanation (RDE) framework was first introduced in (Macdonald et al., 2019), and extended in (Heiß et al., 2020), as a mathematically well-founded and intuitive explanation framework. RDEs are model-agnostic explanations and inspired by rate-distortion theory, which studies lossy-data compression. An explanation in RDE consists of a relatively sparse mask over the input features, highlighting the relevant set of features. The mask is optimized to produce low distortion in the model output after applying perturbations to the unselected features in the input while remaining relatively sparse. Heiß et al. (2020) also applied RDE to non-canonical input representations to explain model decisions in challenging domains such as audio classification (Engel et al., 2017) and radio-map estimation (Levie et al., 2021; 2020). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The explanation principle of optimizing a mask $s \\in [ 0 , 1 ] ^ { n }$ was first proposed by Fong & Vedaldi (2017) who explained image classification decisions by considering one of the two “deletion games”: (1) optimizing for the smallest deletion mask that causes the class score to drop significantly or (2) optimizing for the largest deletion mask that has no significant effect on the class score. The original RDE approach (Macdonald et al., 2019) is based on the second deletion game. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Other explanation methods developed by the research community are typically either (1) gradientbased such as Smoothgrad (Smilkov et al., 2017), Integrated Gradients (Sundararajan et al., 2017), Image-Specific Class Saliency (Simonyan et al., 2014), and Guided Backpropagation (Springenberg et al., 2015), (2) surrogate models such as LIME (Ribeiro et al., 2016), (3) based on propagation of activations in neurons such as LRP (Bach et al., 2015; Shrikumar et al., 2017), and DeepLIFT (Shrikumar et al., 2017), (4) based on Shapely values from game-theory (Lundberg & Lee, 2017), (6) concept-based such as Concept Activation Vectors (Kim et al., 2018), or (7) based on generative causal explanations (O' Shaughnessy et al., 2020). Also related are methods that were developed to explain individual neurons such as in (Nguyen et al., 2016; Dhamdhere et al., 2019). To our knowledge, all existing explainability methods operate in pixel space and all methods looking for sparse explanations demand sparsity in pixel space (Macdonald et al., 2019; Fong & Vedaldi, 2017; Chang et al., 2019). ",
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+    {
+        "type": "text",
+        "text": "3 BACKGROUND: RATE-DISTORTION EXPLANATION FRAMEWORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we review the rate-distortion explanation (RDE) framework, which was introduced by Macdonald et al. (2019) and later extended by Heiß et al. (2020) by applying RDE to noncanonical input representations. Suppose $\\Phi : \\mathbb { R } ^ { n }  \\mathbb { R } ^ { m }$ is a pre-trained model, e.g., a classifier (with $m$ class labels) or a regression model (with $m$ -dimensional output), where $n$ denotes the dimension of the model input. RDE produces an explanation for a model decision $\\Phi ( x )$ with $x \\in \\mathbb { R } ^ { n }$ as a relatively sparse mask $s \\in \\{ 0 , 1 \\}$ marking the relevant input features in $x$ . More precisely, RDE aims to solve the following optimization problem over a mask $s \\in \\{ 0 , 1 \\} ^ { n }$ : ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/23002d17ea3067df001e6bdc02ae23b64136a7cc059dde2c7ef72e2440f52f85.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { s \\in \\{ 0 , 1 \\} ^ { n } } \\quad \\operatorname { \\mathbb { E } } _ { v \\sim \\mathcal { V } } \\left[ d \\Bigl ( \\Phi ( x ) , \\Phi ( x \\odot s + ( 1 - s ) \\odot v ) \\Bigr ) \\right] \\quad \\mathrm { s . t . } \\quad \\| s \\| _ { 0 } \\leq \\ell ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "where $\\odot$ denotes the Hadamard product (element-wise multiplication), $d ( \\Phi ( x ) , \\cdot )$ is a measure of distortion (e.g. $d ( \\Phi ( x ) , \\cdot ) = \\lVert \\Phi ( { \\bar { x } } ) - \\cdot \\rVert _ { 2 } )$ , $\\nu$ is a distribution over input perturbations $v \\in \\mathbb { R } ^ { n }$ , and $\\ell \\in \\{ 1 , . . . , n \\}$ is a given sparsity level for the explanation mask $s$ . A solution $s ^ { * }$ to the optimization problem in (1) marks relatively few components in the model input $x$ that suffice to approximately retain the model output $\\Phi ( x )$ . This approach is in the spirit of rate-distortion theory, which deals with lossy compression of data. Therefore, Macdonald et al. (2019) coined such explanations ratedistortion explanations (RDEs). ",
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+    {
+        "type": "text",
+        "text": "In practice, the optimization problem in (1) is relaxed to continuous masks $s \\in [ 0 , 1 ]$ solving ",
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+    {
+        "type": "equation",
+        "img_path": "images/9a28e211a80d42e50b0da89f14f76ad9188bc322280953fb6cf4c9ff8b2568e3.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { s \\in \\{ 0 , 1 \\} ^ { n } } \\quad \\operatorname { \\mathbb { E } } _ { v \\sim \\mathcal { V } } \\left[ d \\Bigl ( \\Phi ( x ) , \\Phi ( x \\odot s + ( 1 - s ) \\odot v ) \\Bigr ) \\right] + \\lambda \\left\\| s \\right\\| _ { 1 } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\lambda > 0$ determines the sparsity level of the mask. The relaxed optimization problem can be solved with stochastic gradient descent in $s \\in [ 0 , 1 ]$ if $\\Phi$ is differentiable—as is the case for deep neural networks. Macdonald et al. (2019) applied the RDE method as described above to image classifiers in the pixel domain of images, where each mask entry $s _ { i } \\in [ 0 , 1 ]$ corresponds to the $i$ -th pixel values. We refer to this method as Pixel RDE throughout this work. ",
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+    {
+        "type": "text",
+        "text": "4 RDE REFORMULATED AND REINTERPRETED ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Instead of applying RDE to the standard input representation $\\boldsymbol { x } = [ x _ { 1 } \\dots x _ { n } ] ^ { T }$ , we can apply RDE to a different representation of $x$ to answer a particular interpretation query. For example, consider a 1D-signal $x \\in \\mathbb { R } ^ { n }$ : if we ask “What is the smooth part in the signal $x$ that leads to the model decision $\\Phi ( x ) ? ^ { , }$ , then we can apply RDE in the Fourier basis of $x$ . Since frequency-sparse signals are smooth, applying RDE in the Fourier basis of $x$ extracts the relevant smooth part of the signal. To accommodate such interpretation queries, we reformulate RDE in Section 4.1. Finally, based on the reformulation, we reinterpret RDE in Section 4.2. Later in Section 5, we use our reformulation and reinterpretation of RDE to derive and motivate CartoonX as a special case and novel explanation method tailored towards image classifiers. ",
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+    {
+        "type": "text",
+        "text": "4.1 GENERAL FORMULATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "An input signal $\\boldsymbol { x } = [ x _ { 1 } , \\dots , x _ { n } ] ^ { T }$ is represented in a basis $\\{ b _ { 1 } , \\ldots , b _ { n } \\}$ as a linear combination $\\textstyle \\sum _ { i = 1 } ^ { n } h _ { i } b _ { i }$ with coefficients $[ h _ { i } ] _ { i = 1 } ^ { n }$ . As we argued above and demonstrate later on, some choices for a basis may be more suitable than others to explain a model decision $\\Phi ( x )$ . Therefore, we define the RDE mask not only on the canonical input representation $[ x _ { i } ] _ { i = 1 } ^ { n }$ but also on a different representation $[ h _ { i } ] _ { i = 1 } ^ { n }$ with respect to a choice of basis $\\{ b _ { 1 } , \\ldots , b _ { n } \\}$ . Examples of non-canonical choices for a basis include the Fourier basis and the wavelet basis. This work is centered around CartoonX, which applies RDE in the wavelet basis, i.e., a linear data representation since $x$ is represented as a linear combination of basis vectors. Nevertheless, there also exist other domains and interpretation queries where applying RDE to a non-linear data representation can make sense (see the interpretation query “Is phase or magnitude more important for an audio classifier?” in (Heiß et al., 2020)). Therefore, we formulate RDE in terms of a data representation function $\\textstyle f : \\prod _ { i = 1 } ^ { k } \\mathbb { R } ^ { c } \\to \\mathbb { R } ^ { n }$ , $f ( h _ { 1 } , \\ldots , h _ { k } ) = x .$ , which does notlinear case and o be linear, we have $c$ ls in the, where e imare ortantfixed $c = 1$ $\\begin{array} { r } { f ( h _ { 1 } , \\ldots , h _ { k } ) = \\sum _ { i = 1 } ^ { k } h _ { i } b _ { i } } \\end{array}$ $\\{ b _ { i } , \\ldots , b _ { k } \\} \\subset \\mathbb { R } ^ { n }$ $k$ $c > 1$   \nchannels at once, e.g., all color channels of an image, to reduce the number of entries in the mask that will operate on $[ h _ { i } ] _ { i = 1 } ^ { k }$ . In the following, we introduce the important definitions of obfuscations, expected distortion, the RDE mask, and $R D E ' s \\ell _ { 1 }$ -relaxation, which generalize the RDE framework of (Macdonald et al., 2019) to abstract input representations. ",
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+    {
+        "type": "text",
+        "text": "4.1.1 DEFINITIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The first two key concepts in RDE are obfuscations and expected distortion, which are defined below. ",
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+        "text": "Definition 1 (Obfuscations and expected distortion) Let $\\Phi : \\mathbb { R } ^ { n }  \\mathbb { R } ^ { m }$ be a model and $x \\in \\mathbb { R } ^ { n }$ a data point with a data representation $x = f ( h _ { 1 } , . . . , h _ { k } )$ as discussed above. For every mask $s \\in [ 0 , 1 ] ^ { k }$ , let $\\gamma _ { s }$ be a probability distribution over $\\textstyle \\prod _ { i = 1 } ^ { k } \\mathbb { R } ^ { c }$ . Then the obfuscation of $x$ with respect to s and $\\gamma _ { s }$ is defined as the random vector $y : = { f ( s \\odot h + ( 1 - s ) \\odot v ) }$ , where $v \\sim \\mathcal { V } _ { s }$ , $( s \\odot h ) _ { i } = s _ { i } h _ { i } \\in \\mathbb { R } ^ { c }$ and $( ( 1 - s ) \\odot v ) _ { i } = ( 1 - s _ { i } ) v _ { i } \\in \\mathbb R ^ { c }$ , for $i \\in \\{ 1 , \\ldots , k \\}$ . A choice for the distribution $\\gamma _ { s }$ is called obfuscation strategy. Furthermore, the expected distortion of $x$ with respect to the mask s and the perturbation distribution $\\gamma _ { s }$ is defined as ",
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+        "text": "$$\nD ( x , s , \\mathcal { V } _ { s } , \\Phi ) : = \\underset { v \\sim \\mathcal { V } _ { s } } { \\mathbb { E } } \\left[ d \\Big ( \\Phi ( x ) , \\Phi ( y ) \\Big ) \\right] ,\n$$",
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+    {
+        "type": "text",
+        "text": "where $d : \\mathbb { R } ^ { m } \\times \\mathbb { R } ^ { m } \\to \\mathbb { R } _ { + }$ is a measure of distortion between two model outputs. ",
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+        "type": "text",
+        "text": "In the RDE framework, the explanation is given by a mask that minimizes distortion while remaining relatively sparse. The rate-distortion explanation mask is defined as follows. ",
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+        "type": "text",
+        "text": "Definition 2 (The RDE mask) In the setting of Definition $I$ we define the RDE mask as a solution $s ^ { * } ( \\ell )$ to the minimization problem ",
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+        "text": "$$\n\\operatorname* { m i n } _ { s \\in \\{ 0 , 1 \\} ^ { k } } \\quad D ( x , s , \\mathcal { V } _ { s } , \\Phi ) \\quad s . t . \\quad \\| s \\| _ { 0 } \\leq \\ell ,\n$$",
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+        "type": "text",
+        "text": "where $\\ell \\in \\{ 1 , \\ldots , k \\}$ is the desired level of sparsity. ",
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+        "type": "text",
+        "text": "Geometrically, the RDE mask $s$ is associated with a particular subspace. The complement mask $( 1 - s )$ can be seen as selecting a large stable subspace of $\\Phi$ , with each point representing a possible perturbation in unselected coefficients in $h$ . The RDE mask minimizes the expected distortion along its associated subspace, which requires non-local information of $\\Phi$ . We illustrate this geometric view of RDE in Figure 2 with a toy example for a hypothetical classifier $\\Phi : \\mathbb { R } ^ { 2 }  \\mathbb { R } ^ { \\bar { m } }$ and two distinct input representations: (1) Euclidean coordinates, i.e., $f$ is the identity in $x = f ( h )$ , and (2) polar coordinates, i.e. $f ( h ) = ( h _ { 2 } \\cos h _ { 1 } , h _ { 2 } \\sin h _ { 1 } ) = x$ . In the example, we assume $\\gamma _ { s }$ to be a uniform distribution on $[ - 1 , 1 ] ^ { 2 }$ in the Euclidean representation and a uniform distribution on $[ - \\pi , \\pi ] \\times [ 0 , 1 ]$ in the polar representation. The expected distortion associated with the masks $s = ( 1 , 0 )$ and $s = ( 0 , 1 )$ is given by the red and green shaded area, respectively. The RDE mask aims for low expected distortion, and hence, in polar coordinates, the RDE mask would be the green subspace, i.e., $s = ( 0 , 1 )$ . On the other hand, in Euclidean coordinates, neither $s = ( 1 , 0 )$ nor $s = ( 0 , 1 )$ produces a particularly low expected distortion, making the Euclidean explanation less meaningful than the polar explanation. The example illustrates why certain input representations can yield more meaningful explanatory insight for a given classifier than others—an insight that underpins our novel CartoonX method. Moreover, the plot in polar coordinates illustrates why the RDE mask cannot be simply chosen with local distortion information, e.g., with the lowest eigenvalue of the Hessian of $\\bar { h } \\mathbin { \\stackrel { \\cdot } { \\mapsto } } d ( \\Phi ( x ) , \\Phi ( f ( h ) ) )$ : the lowest eigenvalue in polar coordinates belongs to the red subspace and does not see the large distortion on the tails. ",
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+        "image_caption": [
+            "Figure 2: The RDE mask can find low expected distortion in polar coordinates but not in Euclidean coordinates. Therefore, in this example, polar coordinates are more appropriate to explain $\\Phi ( x )$ , and RDE would determine that the angle $\\varphi$ , not the magnitude $r$ , is relevant for $\\Phi ( x )$ . "
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+        "type": "text",
+        "text": "As was shown by Macdonald et al. (2019), the RDE mask from Definition 2 cannot be computed efficiently for non-trivial input sizes. Nevertheless, one can find an approximate solution by considering continuous masks $s \\in [ 0 , 1 ] ^ { k }$ and encouraging sparsity through the $\\ell _ { 1 }$ -norm. ",
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+        "text": "Definition 3 (RDE’s $\\ell _ { 1 }$ -relaxation with Lagrange multipliers) In the setting of Definition $I$ , we define RDE’s $\\ell _ { 1 }$ -relaxation with Lagrange multipliers as a solution $s ^ { * } ( \\lambda )$ to the minimization problem ",
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+        "img_path": "images/c9a80a669d719de03f788a4c4b6976e2dce931c9b9d85f72cde5bc00d721362a.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\underset { s \\in [ 0 , 1 ] ^ { k } } { \\operatorname* { m i n } } } & { { } D ( \\boldsymbol { x } , s , \\mathcal { V } _ { s } , \\boldsymbol { \\Phi } ) + \\lambda \\| s \\| _ { 1 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\lambda > 0$ is a hyperparameter for the sparsity level. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The $\\ell _ { 1 }$ -relaxation above can be solved with stochastic gradient descent (SGD) over the mask $s$ while approximating $D ( x , s , \\mathcal { V } _ { s } , \\Phi )$ with i.i.d. samples from $v \\sim \\mathcal { V } _ { s }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1.2 OBFUSCATION STRATEGIES",
+        "text_level": 1,
+        "bbox": [
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+            387,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "An obfuscation strategy is defined by the choice of the perturbation distribution $\\mathcal { V } _ { s }$ . Common choices are Gaussian noise (Macdonald et al., 2019; Fong & Vedaldi, 2017), blurring (Fong & Vedaldi, 2017), constants (Fong $\\&$ Vedaldi, 2017), and inpainting GANs (Heiß et al., 2020; Chang et al., 2019). Inpainting GANs train a generator $G ( s , z , h )$ ( $z$ denotes random latent factors) such that for samples $v \\sim G ( s , z , h )$ the obfuscation $f ( s \\odot h + ( 1 - s ) \\odot v )$ remains in the data manifold. In our work, we refrain from using an inpainting GAN due to the following reason: it is hard to tell whether a GAN-based mask did not select coefficients because they are unimportant or because the GAN can easily inpaint them from a biased context. Instead, we choose a simple and wellunderstood obfuscation strategy, which we call Gaussian adaptive noise, making the explanation as transparent as possible. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Gaussian adaptive noise works as follows: Let $A _ { 1 } , . . . , A _ { j }$ be a pre-defined choice of a disjoint partition of $\\{ 1 , \\ldots , k \\}$ (recall $s \\in [ 0 , 1 ] ^ { k } )$ . For $i = 1 , . . . , j$ , we compute the empirical mean and empirical standard deviation for each partition across all partition instances: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/77f04c980c605871e331384771b6e6f0593eaedf3e0cbf2facd1a707d4c8e7c1.jpg",
+        "text": "$$\n\\mu _ { i } : = \\frac { 1 } { \\sum _ { a \\in A _ { i } } d _ { a } } \\sum _ { a \\in A _ { i } , t = 1 , \\ldots , d _ { a } } h _ { a t } , \\sigma _ { i } : = \\sqrt { \\frac { 1 } { \\sum _ { a \\in A _ { i } } d _ { a } } \\sum _ { a \\in A _ { i } , t = 1 , \\ldots , d _ { a } } ( \\mu _ { i } - h _ { a t } ) ^ { 2 } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The adaptive Gaussian noise strategy then samples $v _ { a t \\_ } \\sim \\mathcal { N } ( \\mu _ { i } , \\sigma _ { i } ^ { 2 } )$ for all partition members $a \\in$ $A _ { i }$ and channels $t = 1 , . . . , d _ { a }$ . We write $v \\sim \\mathcal { N } ( \\mu , \\sigma ^ { 2 } )$ for the resulting Gaussian random vector $\\boldsymbol { v } \\in \\prod _ { i = 1 } ^ { k } \\mathbb { R } ^ { c }$ . Note that the distribution $\\gamma _ { s }$ chosen as Gaussian adaptive noise does depend on $s$ (unlike with an inpainting GAN). For Pixel RDE, we only use one set $A _ { 1 } = \\left\\{ 1 , . . . , k \\right\\}$ for all $k$ pixels. In CartoonX, which represents input signals in the discrete wavelet domain, we will partition $\\{ 1 , . . . , k \\}$ along the scales of the discrete wavelet transform. ",
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+    {
+        "type": "text",
+        "text": "4.1.3 MEASURES OF DISTORTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "There are various choices for the measure of distortion $d ( \\Phi ( x ) , \\Phi ( y ) )$ . For example, one can take the squared distance in the post-softmax probability of the predicted label for $x$ , i.e., ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/ece7e994cea071a84c2695e50a8c52dd294db7accf98e06c1b9602b0ed8db31a.jpg",
+        "text": "$$\nd \\big ( \\Phi ( x ) , \\Phi ( y ) \\big ) : = \\big ( \\Phi _ { j ^ { * } } ( x ) - \\Phi _ { j ^ { * } } ( y ) \\big ) ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $j ^ { * } : = \\arg \\operatorname* { m a x } _ { i = 1 , \\ldots , m } \\Phi _ { i } ( x )$ and $\\Phi ( x )$ is assumed to be the post-softmax probabilities of a neural net. Alternatively, one could also choose $d ( \\Phi ( x ) , \\Phi ( y ) )$ as the $\\ell _ { 2 }$ -distance or the KLDivergence in the post-softmax layer of $\\Phi$ . In our experiments for CartoonX, we found that these choices had no significant effect on the explanation (see Appendix A.3.3). ",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 INTERPRETATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The philosophy of the generalized RDE framework is that an explanation for a decision $\\Phi ( x )$ on a generic input signal $x = f ( h )$ should be some simplified version of the signal, which is interpretable to humans. The simplification is achieved by demanding sparsity in a suitable representation system $h$ , which sparsely represents the class of explanations that are desirable for the interpretation query. This philosophy is the fundamental premise of CartoonX, which aims to answer the interpretation query “What is the relevant piece-wise smooth part of the image for a given image classifier?”. CartoonX first employs RDE on a representation system $x = f ( h )$ that sparsely represents piecewise smooth images and finally visualizes the relevant piece-wise smooth part as an image back in pixel space. In the following section, we explain why wavelets provide an appropriate representation system in CartoonX, present the CartoonX implementation, and finally provide experiments on ImageNet to demonstrate the capability of CartoonX. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 CARTOONX ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The focus of this paper is CartoonX, a novel explanation method—tailored to image classifications— that we obtain as a special case of our generalized RDE framework formulated in Section 4. CartoonX first performs RDE in the discrete wavelet position-scale domain of an image $x$ , and finally, visualizes the wavelet mask $s$ as a piece-wise smooth image in pixel space. Wavelets provide optimal representations for piece-wise smooth 1D functions (DeVore, 1998), and represent 2D piecewise smooth images, also called cartoon-like images (Kutyniok & Lim, 2011), efficiently as well (Romberg et al., 2006). In particular, sparse vectors in the wavelet coefficient space encode cartoonlike images reasonably well (Stephane, 2009a)—certainly better than sparse pixel representations. ´ Moreover, wavelets constitute an established tool in signal processing (Stephane, 2009c). ´ ",
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+    {
+        "type": "image",
+        "img_path": "images/600d34c48f6449578084ae2acb04f5102f541072c868e8f11c61d9f05a838409.jpg",
+        "image_caption": [
+            "Figure 3: CartoonX has many interesting parallels to wavelet-based image compression. Distortion is denoted as $d$ , $\\Phi$ is an image classifier, $h$ denotes the discrete wavelet coefficients, $\\tau$ is the discrete wavelet transform, and $\\ell$ is the coefficient budget. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The optimization process underlying CartoonX produces sparse vectors in the discrete wavelet coefficient space, which results in cartoon-like images as explanations. This is the fundamental difference to Pixel RDE, which produces rough, jittery, and pixel-sparse explanations. Cartoon-like images are more interpretable and provide a natural model of simplified images. Since the goal of the RDE framework is to generate an easy to interpret simplified version of the input signal, we argue that CartoonX explanations are more appropriate for image classification than Pixel RDEs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "CartoonX exhibits interesting parallels to wavelet-based image compression. In image compression, distortion is minimized in the data domain, which is equivalent to selecting the $\\ell$ largest entries in the discrete wavelet transform (DWT) coefficients. In comparison, CartoonX minimizes distortion in the model output of $\\Phi$ , which translates to selecting the $\\ell$ most relevant entries in the DWT coefficients. The objective in image compression is efficient data representation, i.e., producing minimal data distortion with a budget of $\\ell$ entries in the DWT coefficients. Conversely, in CartoonX, the objective is extracting the relevant piece-wise smooth part, i.e., producing minimal model distortion with a budget of $\\ell$ entries in the DWT coefficients. We illustrate this connection in Figure 3—highlighting once more the rate-distortion spirit of the RDE framework. ",
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+    {
+        "type": "text",
+        "text": "5.1 IMPLEMENTATION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "An image $x \\in [ 0 , 1 ] ^ { c \\times w \\times t }$ with $c \\in \\{ 1 , 3 \\}$ channels, width $w \\in \\mathbb { N }$ , height $t \\in \\mathbb N$ , and a total of $p = w t$ pixels can be represented in a wavelet basis by computing its discrete wavelet transform (DWT). The DWT of an image is defined by the number of scales $J \\in \\{ 1 , \\ldots , \\lfloor \\log _ { 2 } p \\rfloor \\}$ , the padding mode, and a choice of the wavelet family (such as the Haar or Daubechies family). For images, the DWT computes four types of coefficients: details in (1) horizontal, (2) vertical, and (3) diagonal orientation at scale $j \\in \\{ 1 , \\dots , J \\}$ , and (4) coefficients of the image at the very coarsest resolution. We briefly illustrate the DWT for an example image in Figure 4. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/39ebb784157d2c6edd31b1dadc49ecf3ff916b928f9b9fd1b5f08907bedcccf9.jpg",
+        "image_caption": [
+            "Figure 4: Left side: an image of a memorial arch dedicated to peace. Right side: visualization of the DWT coefficients for five scales. Three L-shaped sub-images describe coefficients for details in vertical, horizontal, and diagonal orientation at a particular scale. The largest sub-images (the outer L-shape) belong to the lowest scale, i.e., the highest resolution. The smaller L-shaped sub-images gradually build up to higher scales, i.e., lower resolution features. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "CartoonX, as described in Algorithm 1 in Appendix A.1, computes the RDE mask in the wavelet domain of images. More precisely, for the data representation ${ \\bar { \\boldsymbol { x } } } = f ( h )$ , we choose $h$ as the concatenation of all the DWT coefficients along the channels, i.e., $\\boldsymbol { h } _ { i } \\in \\mathbb { R } ^ { c }$ . The representation function $f$ is then the discrete inverse wavelet transform, i.e., the summation of the DWT coefficients times the DWT basis vectors. We optimize the mask $s \\in [ 0 , 1 ] ^ { k }$ on the DWT coefficients $[ h _ { 1 } , \\ldots , h _ { k } ] ^ { T }$ to minimize RDE’s $\\ell _ { 1 }$ -relaxation from Definition 3. For the obfuscation strategy $\\gamma _ { s }$ , we use adaptive Gaussian noise with a partition by the DWT scale (see Section 4.1.2), i.e., we compute the empirical mean and standard deviation per scale. We measure distortion as the squared difference in the postsoftmax score of the predicted label for $x$ (see Section 4.1.3). To visualize the final DWT mask $s$ as a piece-wise smooth image in pixel space, we multiply the mask with the DWT coefficients of the greyscale image $\\hat { x } : = ( 1 \\breve { / c } \\sum _ { l = 1 } ^ { \\hat { c } } x _ { l a i } \\dot { ) } _ { a i }$ before inverting the product back to pixel space with the discrete inverse wavelet transform. The inversion is finally clipped into $[ 0 , 1 ] ^ { w \\times \\dot { t } }$ as are obfuscations during the RDE optimization to avoid overflow (we assume here the pixel values in $x$ are normalized into $[ 0 , 1 ] )$ . The clipped inversion in pixel space is the final explanation, which we call CartoonX. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "5.2 EXPERIMENTS AND ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We compare CartoonX to the closely related Pixel RDE (Macdonald et al., 2019) and several other state-of-the-art explanation methods , that is, Integrated Gradients (Sundararajan et al., 2017), Smoothgrad (Smilkov et al., 2017), Guided Backprop (Springenberg et al., 2015), and LRP (Bach et al., 2015). Our experiments show that CartoonX carries the following strengths: Cartoon X is (1) highly interpretable due to its cartoon-like nature and (2) remarkably apt at explaining misclassifications, and highlighting meaningful patterns that are otherwise hard to see. Due to the fast implementation of the DWT, Cartoon RDE is not significantly slower than Pixel RDE. For the ImageNet classifier MobileNetV3-Small, an image of 256 times 256 pixels, and 2001 optimization steps, we reported a runtime of 81.56 seconds for CartoonX and 70.53 seconds for Pixel RDE on the NVIDIA Titan RTX GPU. However, like other perturbation-based methods, CartoonX is significantly slower than gradient or propagation-based methods, which only compute a single or few forward and backward passes and are very fast (Integrated Gradients computes an explanation in 0.48 seconds for the same image, model, and hardware). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our experiments use the pre-trained ImageNet classifiers MobileNetV3-Small (Howard et al., 2019) (top-1 accuracy of $6 7 . 6 6 8 \\%$ and VGG16 (Simonyan & Zisserman, 2015) (top-1 accuracy of $7 1 . 5 9 2 \\%$ ). We note that the open-source implementation of LRP did not implement propagation rules for certain layers in MobileNetV3-Small, therefore we compare CartoonX to LRP only for VGG16. Images were preprocessed to have 256 times 256 pixel values in [0, 1]. We provide further details about the choice of hyperparameters in the experiments in Appendix A.2. The three main hyperparameters for CartoonX are: (1) the sparsity level $\\lambda > 0$ , (2) the measure of distortion $d$ , and (3) the obfuscation strategy (perturbation distribution) $\\gamma _ { s }$ . We discuss the sensitivity of CartoonX to these hyperparameters in Appendix A.3. In Appendix A.5, we also shed light on the evolution of ImageNet classifiers from an explanation angle by comparing CartoonX explanations for classifiers of varying generalization power, i.e., AlexNet (Krizhevsky et al., 2012), VGG16 (Simonyan & Zisserman, 2015), InceptionV3 (Szegedy et al., 2016), ResNeXt50 (Xie et al., 2017). Moreover, in Appendix A.4, we argue experimentally why CartoonX is less susceptible than Pixel RDE to so-called explanation artifacts—an unwanted phenomenon that we observed empirically. ",
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+    {
+        "type": "image",
+        "img_path": "images/0a1f842c7c2f27339003af5f5ca74ec6399ee2beaa7f81924ce3ed1fbaf9fc58.jpg",
+        "image_caption": [
+            "Figure 5: Each row compares CartoonX explanations of misclassifications by MobileNetV3-Smal to Pixel RDE (Macdonald et al., 2019), Integrated Gradients (Sundararajan et al., 2017), and Smoothgrad (Smilkov et al., 2017). The predicted label is depicted above each misclassified image. "
+        ],
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In practice, explaining misclassifications is particularly relevant since good explanations can pinpoint model biases and causes for model failures. We observe that CartoonX is particularly good at explaining certain misclassifications, which we illustrate for three examples in Figure 5 and many more in Appendix A.6. In the first row in Figure 5, the input image shows a man holding a dog that was classified as a “diaper”. CartoonX shows the man not holding a dog but a baby, revealing that the neural net associated diapers with babies and babies with the pose with which the man is holding the dog. In the second row, the input image shows a dog sitting on an armchair with leopard patterns. The dog was classified as an “Egyptian cat”, which can exhibit leopard-like patterns. CartoonX exposes the Egyptian cat by connecting the dog’s head to parts of the armchair forming the cat’s torso and legs. In the last row, the input image displays the backside of a man wearing a striped sweater that was classified as a “screw”. CartoonX reveals how the stripe patterns look like a screw to the neural net. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/93b3a56308de3770000542d827bd0e4671770354a0f003b0ff69f08314095854.jpg",
+        "image_caption": [
+            "Figure 6: CartoonX explanations for VGG16 compared to state-of-the-art methods, that is, Pixel RDE (Macdonald et al., 2019), Integrated Gradients (Sundararajan et al., 2017), Smoothgrad (Smilkov et al., 2017), Guided Backprop (Springenberg et al., 2015), and LRP (Bach et al., 2015). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "CartoonX is the first explainability method to extract the relevant piece-wise smooth part of an image and is based on our novel formulation of the RDE framework. We corroborated experimentally that CartoonX explanations are highly interpretable due to their cartoon-like nature and surprisingly well-suited to explain misclassifications. Nonetheless, Cartoon RDE is still computationally quite expensive, like other perturbation-based explanation methods. In the future, we hope to devise new techniques to speed up the runtime for CartoonX. Moreover, we are pursuing applications of CartoonX beyond explanation tasks, such as detecting adversarial examples. We believe CartoonX is a valuable new explanation method for practitioners and potentially a great source of inspiration for future explanation methods aiming to tailor their explanations to other data domains. Our reformulation and reinterpretation of the RDE framework provide a blueprint for such future work: First, formulate an interpretation query related to the underlying model task, then find a representation system that sparsely represents the class of desirable explanations for the interpretation query. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            849,
+            299,
+            866
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 CARTOONX ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            882,
+            393,
+            897
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The final CartoonX algorithm is depicted in Algorithm 1. ",
+        "bbox": [
+            174,
+            909,
+            547,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: CartoonX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            108,
+            334,
+            122
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/5dee94cf1bc128a847ce9d9adeca502f44bb0180b9f6f21a677be2ba06a66b60.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>AigormmmrT.CartoonA Data: Image x ∈ [0,1]exwxt with c channels and wt pixels, classifier Φ. Result: CartoonX explanation ε ∈ [0,1]w×t for decision Φ(x). Hyperparameters: Sparsity level X &gt; O, number of steps N, number of noise samples L. Initialize mask s := [1,..,1]T ∈ [0,1]k on DWT coefficients h = [h1,.., hk] with x = f(h), where f is the discrete inverse wavelet transform; Compute predicted label j* := arg maxi Φ(x); fori←1toNdo</td></tr><tr><td>Sample L adaptive Gaussian noise samples u(1),., u(L) ~ N(μ,o²); Compute obfuscations y(1), ),.,y(L) with y() := f(h ① s+ (1- s) ①u(i)); Clip obfuscations into [0,1]cx w ×t;</td></tr><tr><td>Approximate expected distortion D(𝑥x,s,Φ) :=∑𝑖=1(Φj+(x)- Φj(y())²/L;</td></tr><tr><td>Compute loss for the mask l(s) := D(x,s,Φ) + λ|lsll1 and gradient Vsl(s); Update mask s with gradient descent step and clip s back to [0,1]k ;</td></tr><tr><td>end Compute wavelet coefficients h for greyscale image x of x;</td></tr><tr><td>Invert wavelet mask s back to pixel space as &amp; := f(h  s) ; Clip the explanation ε into [0,1]w×t to obtain ε. Visualize ε;</td></tr></table>",
+        "bbox": [
+            173,
+            116,
+            789,
+            393
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 EXPERIMENT DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            428,
+            377,
+            443
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Throughout our experiments with CartoonX and Pixel RDE, we used a learning rate of $\\epsilon = 0 . 0 0 1$ , a sample size of $L = 6 4$ for the adaptive Gaussian noise, and $N = 2 0 0 0$ steps. Several different sparsity levels were used. We recommend specifying the sparsity level in terms of the number of mask entries $k$ , i.e., choosing the product $\\lambda k$ . Pixel RDE typically requires a smaller sparsity level than CartoonX. We chose $\\bar { \\lambda k } \\in [ \\bar { 2 0 } , 8 0 ]$ for CartoonX and $\\bar { \\lambda } k \\in [ 3 , 2 \\bar { 0 } ]$ for Pixel RDE. The obfuscation strategy for Pixel RDE was chosen as Gaussian adaptive noise with mean and standard deviation computed for all pixel values (see Section 4.1.2). In Appendix 8, we show that Gaussian adaptive noise produces much more interpretable explanations than using a zero baseline perturbation. We implemented the DWT for CartoonX with the Pytorch Wavelets package, which is compatible with PyTorch gradient computations, and chose the Daubechies wavelet system with $J = 5$ scales and zero-padding. For the Integrated Gradients method, we used 100 steps, and for the Smoothgrad method, we used 10 samples and a standard deviation of 0.1. ",
+        "bbox": [
+            173,
+            457,
+            825,
+            623
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.3 SENSITIVITY TO HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            646,
+            477,
+            660
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We compare CartoonX’s sensitivity to its main hyperparameters, i.e., the sparsity level $\\lambda$ , the perturbation distribution $\\gamma _ { s }$ , and the distortion measure $\\bar { d ( \\Phi ( x ) , \\Phi ( y ) ) }$ . For each experiment, we fix all but one of the three parameters. ",
+        "bbox": [
+            176,
+            674,
+            825,
+            715
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.3.1 SENSITVITY TO THE SPARSITY LEVEL $\\lambda$ ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            737,
+            509,
+            752
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 7 plots CartoonX explanations and Pixel RDEs for increasing $\\lambda$ —the hyperparameter determining the explanation’s sparsity in the respective representation system. We find that CartoonX is less sensitive than Pixel RDE to $\\lambda$ . In practice, this means one can find a suitable $\\lambda$ faster for CartoonX than for Pixel RDE. ",
+        "bbox": [
+            174,
+            763,
+            825,
+            819
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.3.2 SENSITVITY TO THE DISTRIBUTION $\\gamma _ { s }$ ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            840,
+            500,
+            856
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 8 plots CartoonX explanations for two choices of $\\gamma _ { s }$ : (1) Gaussian adaptive noise (see Section 4.1.2) and (2) constant zero perturbations (i.e. $v = 0$ with probability one under $\\gamma _ { s }$ ). We observe that the Gaussian adaptive noise gives much more meaningful explanations than the simple zero baseline perturbations. ",
+        "bbox": [
+            174,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/669376803cf15b48772f47190fc15e7b3958a45c1013353768d96a6ea5ac042a.jpg",
+        "image_caption": [
+            "Figure 7: We compare the sensitivity of CartoonX and Pixel RDE to the sparsity level $\\lambda$ . The top row depicts CartoonX, and the bottom row depicts Pixel RDE, for increasing values of $\\lambda$ . Note that for $\\lambda = 0$ , Pixel RDE is entirely yellow because the mask is initialized as $s ^ { \\check { = } } [ 1 \\ldots 1 ] ^ { T }$ and $\\lambda = 0$ provides no incentive to make s sparser. For the same reason, CatoonX is simply the greyscale image for $\\lambda = 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            102,
+            820,
+            444
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/014f60850b885dee2e6d5318a68d96e55506ad31ed7a0b81c56dda8b69fa0b6c.jpg",
+        "image_caption": [
+            "Figure 8: We compare the sensitivity of CartoonX to the perturbation distribution $\\gamma _ { s }$ . The top image was classified as a fountain and the bottom image as a viaduct. The second column depicts CartoonX with $\\gamma _ { s }$ as Gaussian adaptive noise, and the third column depicts CartoonX with $\\gamma _ { s }$ as constant zero perturbations (zero baseline). We observe that Gaussian adaptive noise is much more interpretable than the zero baseline. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            361,
+            558,
+            635,
+            718
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3.3 SENSITVITY TO THE DISTORTION MEASURE $d ( \\Phi ( x ) , \\Phi ( y ) )$ ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            842,
+            638,
+            858
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Figure 9 plots CartoonX explanations for the following four choices of $d ( \\Phi ( x ) , \\Phi ( y ) )$ , where $x$ is the original input, $y$ is the RDE obfuscation, and $\\Phi$ outputs post-softmax probabilities: ",
+        "bbox": [
+            173,
+            867,
+            825,
+            897
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "1. $d ( \\Phi ( x ) , \\Phi ( y ) ) = ( \\Phi _ { j ^ { * } } ( x ) - \\Phi _ { j ^ { * } } ( y ) ) ^ { 2 }$ , where $j ^ { * } : = \\arg \\operatorname* { m a x } _ { j } \\Phi _ { j } ( x )$ (squared $\\ell _ { 2 }$ in label ) ",
+        "bbox": [
+            207,
+            907,
+            823,
+            926
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "2. $d ( \\Phi ( x ) , \\Phi ( y ) ) = ( \\Phi _ { j ^ { * } } ( x ) - 1 ) ^ { 2 }$ , where $j ^ { * } : = \\arg \\operatorname* { m a x } _ { j } \\Phi _ { j } ( x )$ (maximize label)   \n3. $d ( \\Phi ( x ) , \\Phi ( y ) ) = \\lVert \\Phi ( x ) - \\Phi ( y ) \\rVert _ { 2 }$ ( $\\ell _ { 2 }$ probabilities)   \n4. $d ( \\Phi ( x ) , \\Phi ( y ) ) = K L ( \\Phi ( y ) , \\Phi ( x ) )$ (KL-Divergence) ",
+        "bbox": [
+            210,
+            102,
+            761,
+            159
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The explanations for $d ( \\Phi ( x ) , \\Phi ( y ) )$ as “squared $\\ell _ { 2 }$ in label”, “maximize label”, and ${ } ^ { 6 6 } \\ell _ { 2 }$ probabilities” look indistinguishable. For $\\bar { d } ( \\Phi ( x ) , \\mathbf { \\bar { \\Phi } } ( y ) )$ as KL-Divergence, we see a slightly less smooth explanation, which may be due to the fact that the KL-Divergence is unbounded unlike the other measures of distortion. ",
+        "bbox": [
+            174,
+            167,
+            825,
+            223
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/9187ae880a8b62675d03bda7eac035f50f4ff1e5be910d54d869700c0b8877f3.jpg",
+        "image_caption": [
+            "Figure 9: We compare the sensitivity of CartoonX to four measures of distortion $d ( \\Phi ( x ) , \\Phi ( y ) )$ . Each of the measures of distortion is marked at the top of each column. We observe almost no difference in the CartoonX explanations for the four distortion measures. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            271,
+            236,
+            727,
+            397
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.4 RELIABILITY AND EXPLANATION ARTIFACTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            489,
+            532,
+            503
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We argue experimentally why CartoonX is more reliable than Pixel RDE for image data. More precisely, we show that CartoonX is less susceptible to so-called explanation artifacts than Pixel RDE. An explanation artifact is an unwanted phenomenon that we observed for Pixel RDE: instead of marking the relevant entries in $x$ , the mask $s$ creates artificial edges that end up making up an artificial class prototype. Explanation artifacts are problematic because they highlight not actual substructures but artificial structures that trigger the classification. Examples for explanation artifacts in Pixel RDE are given in Figure 11. ",
+        "bbox": [
+            173,
+            515,
+            825,
+            613
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/202f6780abcf94eb3a89b7f1a8d763a4bce9c8a43edc6cc45602844f237f5f61.jpg",
+        "image_caption": [
+            "Figure 10: CartoonX and Pixel RDE are both performed on the image of the blue sky. However, both methods are adjusted here to find evidence for the output probabilities of the image of the airplane instead of the blue sky. Pixel RDE, unlike CartoonX, can create an artificial airplane as evidence for an airplane in the smooth blue sky. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            261,
+            628,
+            689,
+            727
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Pixel RDE can produce artificial edges in smooth regions for the following reason: When $s$ has a curve-like structure in some region, unselected points near $s$ are replaced with perturbations that tend to differ from the values of the curve-like structure in $s$ . Thus, the curve-like structure also appears in the obfuscation and can produce low distortion if the structure makes up a prototypical class feature (see, for example, the airplane in Figure 10). ",
+        "bbox": [
+            174,
+            818,
+            825,
+            888
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We suspect CartoonX is inherently less susceptible to explanation artifacts for the following reason: Natural images tend to be piece-wise smooth, and piece-wise smooth images have sparse high-frequency DWT coefficients that cluster about the edges Stephane (2009b) (see for example ´ Figure 4). For a DWT mask to create artificial edges, it has to select a curve-like structure in the high-frequency coefficients (low-frequency coefficients cannot create edges) and replace surrounding unselected values with different values. However, in CartoonX, perturbations of high-frequency coefficients are Gaussian with low variance centered close to zero (see adaptive Gaussian noise in Section 4.1.2), which are not very different from the values along the selected curve due to the sparsity of the coefficients. ",
+        "bbox": [
+            173,
+            895,
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+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            202
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We illustrate our previous reasoning about explanation artifacts in a controlled example (see Figure 10). We take an image $x ^ { ( \\mathrm { s k y } ) }$ of a blue sky that is very smooth and an image $x ^ { \\mathrm { ( p l a n e ) } }$ of a airplane. The goal is to show that Pixel RDE, unlike CartoonX, can create artificial evidence for the class airplane on the image of the smooth blue sky. We perform CartoonX and Pixel RDE on the blue sky image with the distortion function ",
+        "bbox": [
+            174,
+            208,
+            825,
+            279
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dcde7f43f02e3222daa273a29250cf103f0a708496a49f569bc22f0173240521.jpg",
+        "text": "$$\n\\forall y \\in \\mathbb { R } ^ { n } : \\ d ( \\Phi ( x ^ { ( \\mathrm { s k y } ) } ) , \\Phi ( y ) ) = 1 0 ^ { 6 } \\| \\Phi ( x ^ { ( \\mathrm { p l a n e } ) } ) - \\Phi ( y ) \\| _ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            286,
+            696,
+            306
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "and a sparsity level of $\\lambda = 8 0 0 0 0$ . As expected, we observe that Pixel RDE, unlike CartoonX, can create an artificial plane in the smooth blue sky(see Figure 10). ",
+        "bbox": [
+            173,
+            311,
+            828,
+            340
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.5 EXPLAINING THROUGH IMAGENET HISTORY: FROM ALEXNET TO RESNETXT50",
+        "bbox": [
+            171,
+            357,
+            784,
+            371
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In the deep learning community, it is well-known that AlexNet (Krizhevsky et al., 2012) provided a major breakthrough in deep learning, improving the top-5 error on ImageNet from $2 5 \\%$ to $16 \\%$ . Since then, deep learning based ImageNet classifiers have continued to drastically improve on ImageNet—achieving less than $6 \\%$ top-5 error in 2016. In Figure 12, we compare CartoonX for four ImageNet classifiers with increasing performance, starting with AlexNet (top-1 accuracy $5 6 . 5 5 \\%$ , AlexNet), VGG16 (top-1 accuracy $7 1 . 5 9 \\%$ , Simonyan & Zisserman (2015)), InceptionV3 (top-1 accuracy $7 7 . 2 9 \\%$ , Szegedy et al. (2016)), and ResNeXt50 (top-1 accuracy $7 7 . 6 2 \\%$ , Xie et al. (2017).) Throughout the experiment, the CartoonX hyperparameters for a given image are not changed for any of the four classifiers. ",
+        "bbox": [
+            174,
+            382,
+            825,
+            508
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.6 EXPLAINING MISCLASSIFICATIONS WITH CARTOONX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            525,
+            589,
+            540
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In Figure 13, 14, 15, and 16, we provide further examples where CartoonX provides insightful explanations for misclassified images. ",
+        "bbox": [
+            173,
+            551,
+            825,
+            580
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.7 CARTOONX COMPARED ON RANDOM IMAGENET SAMPLES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            595,
+            630,
+            611
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Figure 17, 18, 19, and 20 compares CartoonX to Pixel RDE (Macdonald et al., 2019), Integrated Gradients (Sundararajan et al., 2017), Smoothgrad (Smilkov et al., 2017), Guided Backprop (Springenberg et al., 2015), and (Bach et al., 2015) on random Imagenet samples classified by VGG16. ",
+        "bbox": [
+            174,
+            622,
+            825,
+            665
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.8 CARTOONX FAILURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            681,
+            374,
+            695
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We also show failures of CartoonX in Figure 21. These are examples of explanations that are not interpretable and seem to fail at explaining the model prediction. Notably, most failure examples are also not particularly well explained by other state-of-the-art methods. It is challenging to state the underlying reason for the CatoonX failures with certainty (there is always the possibility that the neural net bases its decision on non-interpretable grounds). We intentionally also showed uninterpretable CartoonX explanations that were not too sparse (all or almost black explanations) since one can typically fix these explanations by decreasing $\\lambda$ . ",
+        "bbox": [
+            174,
+            707,
+            825,
+            806
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/4348604dba432b3c96ce1e643eb3ab00bb1b1360ac0508fa151d7742220d33a6.jpg",
+        "image_caption": [
+            "Figure 11: Explanation artifacts in Pixel RDE. We observe that Pixel RDE tends to create edges that are not a subset of the edges in the original input image. These edges can make prototypical artifact patterns such as wrinkles in the cloak (first row), coral tentacles (second row), or chain mail (third row). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            58,
+            807,
+            849
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/6787c0907f2780a91eeba9a8c758e1e7e257500e6962b20c21edad4a037d379c.jpg",
+        "image_caption": [
+            "Figure 12: We compare CatoonX explanations for classifications by AlexNet (Krizhevsky et al., 2012), VGG16 (Simonyan & Zisserman, 2015), InceptionV3 (Szegedy et al., 2016), and ResNeXt50 (Xie et al., 2017). Green labels mark correct classifications and red labels mark wrong classifactions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            160,
+            714,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/eb86233a3b07b868113dda202cab39f6f67461c4de85083de6e1fd8ee8292c8e.jpg",
+        "image_caption": [
+            "Figure 13: Explaining misclassifications with CartoonX on Imagenet and MobileNetV3-Small. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            95,
+            820,
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/3f5f295b306da0a5156d30d76a6aa194e591059d0eb184994eb42043169014f0.jpg",
+        "image_caption": [
+            "Figure 14: Explaining misclassifications with CartoonX on Imagenet and MobileNetV3-Small. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 18
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+    {
+        "type": "image",
+        "img_path": "images/4ad860b0ac3719256544354b20a3432e1aeb4a028fe6c9a3feb6cb20414bd6e9.jpg",
+        "image_caption": [
+            "Figure 15: Explaining misclassifications with CartoonX on Imagenet and MobileNetV3-Small. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            95,
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+    {
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+        "img_path": "images/514d598a6d4e95cdbd85c43a862b0a1bd0459a6852d89b3ce2fcb63bcd01792c.jpg",
+        "image_caption": [
+            "Figure 16: Explaining misclassifications with CartoonX on Imagenet and MobileNetV3-Small. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            89,
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+        "page_idx": 20
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+    {
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+        "img_path": "images/549aeec35ea55596a58c06d2945f8fdf0faa71d75a7db2cd4edec824a94e0c99.jpg",
+        "image_caption": [
+            "Figure 17: Comparing CartoonX on random ImageNet samples and VGG16. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            78,
+            815,
+            901
+        ],
+        "page_idx": 21
+    },
+    {
+        "type": "image",
+        "img_path": "images/e1ef9c14fd73e7ff8291699ed4e3f48b71162eebae779f16441340c19979e5a0.jpg",
+        "image_caption": [
+            "Figure 18: Comparing CartoonX on random ImageNet samples and VGG16. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            141,
+            813,
+            844
+        ],
+        "page_idx": 22
+    },
+    {
+        "type": "image",
+        "img_path": "images/0c30ef46b69d0c580910205c2e3cd0a382e2d9d8bbd1b9e1a698153577cf5734.jpg",
+        "image_caption": [
+            "Figure 19: Comparing CartoonX on random ImageNet samples and VGG16. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            146,
+            823,
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+        "img_path": "images/3768315f4f3a61a9c20715449365266afdeba73a93b2ac05bad3c5f3932bdd09.jpg",
+        "image_caption": [
+            "Figure 20: Comparing CartoonX on random ImageNet samples and VGG16. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
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+        "image_caption": [
+            "Figure 21: Failures of CartoonX. "
+        ],
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+]

parse/dev/_XNtisL32jv/_XNtisL32jv.md

@@ -0,0 +1,342 @@
+# TEMPORAL EFFICIENT TRAINING OF SPIKINGNEURAL NETWORK VIA GRADIENT RE-WEIGHTING
+
+Shikuang Deng1,2, Yuhang $\mathbf { L i ^ { 3 } }$ , Shanghang Zhang4 & Shi $\mathbf { G u } ^ { 1 , 2 , 5 \boxtimes }$
+
+1University of Electronic Science and Technology of China,   
+2Shenzhen Institute for Advanced Study, UESTC   
+3Yale University, 4Peking University ,5Peng Cheng Laboratory   
+dengsk119@std.uestc.edu.cn, yuhang.li@yale.edu, gus@uestc.edu.cn
+
+# ABSTRACT
+
+Recently, brain-inspired spiking neuron networks (SNNs) have attracted widespread research interest because of their event-driven and energy-efficient characteristics. Still, it is difficult to efficiently train deep SNNs due to the nondifferentiability of its activation function, which disables the typically used gradient descent approaches for traditional artificial neural networks (ANNs). Although the adoption of surrogate gradient (SG) formally allows for the back-propagation of losses, the discrete spiking mechanism actually differentiates the loss landscape of SNNs from that of ANNs, failing the surrogate gradient methods to achieve comparable accuracy as for ANNs. In this paper, we first analyze why the current direct training approach with surrogate gradient results in SNNs with poor generalizability. Then we introduce the temporal efficient training (TET) approach to compensate for the loss of momentum in the gradient descent with SG so that the training process can converge into flatter minima with better generalizability. Meanwhile, we demonstrate that TET improves the temporal scalability of SNN and induces a temporal inheritable training for acceleration. Our method consistently outperforms the SOTA on all reported mainstream datasets, including CIFAR-10/100 and ImageNet. Remarkably on DVS-CIFAR10, we obtained $8 3 \%$ top-1 accuracy, over $\bar { 1 0 \% }$ improvement compared to existing state of the art. Codes are available at https://github.com/Gus-Lab/temporal_ efficient_training.
+
+# 1 INTRODUCTION
+
+The advantages of Spiking neuron networks (SNNs) lie in their energy-saving and fast-inference computation when embedded on neuromorphic hardware such as TrueNorth (DeBole et al., 2019) and Loihi (Davies et al., 2018). Such advantages originate from the biology-inspired binary spike transmitted mechanism, by which the networks avoid multiplication during inference. On the other hand, this mechanism also leads to difficulty in training very deep SNNs from scratch because the non-differentiable spike transmission hinders the powerful back-propagation approaches like gradient descents. Recently, many studies on converting artificial neuron networks (ANNs) to SNNs have demonstrated SNNs’ comparable power in feature representation as ANNs (Han & Roy, 2020; Deng & Gu, 2020; Li et al., 2021a). Nevertheless, it is commonly agreed that the direct training method for high-performance SNN is still crucial since it distinguishes SNNs from converted ANNs, especially on neuromorphic datasets.
+
+The output layer’s spike frequency or the average membrane potential increment is commonly used as inference indicators in SNNs (Shrestha & Orchard, 2018; Kim et al., 2019). The current standard direct training (SDT) methods regard the SNN as RNN and optimize inference indicators’ distribution (Wu et al., 2018). They adopt surrogate gradients (SG) to relieve the non-differentiability (Lee et al., 2016; Wu et al., 2018; Zheng et al., 2021). However, the gradient descent with SG does not match with the loss landscape in SNN and is easy to get trapped in a local minimum with low generalizability. Although using suitable optimizers and weight decay help ease this problem, the performance of deep SNNs trained from scratch still suffers a big deficit compared to that of ANNs Deng et al. (2020). Another training issue is the memory and time consumption, which increases linearly with the simulation time. Rathi & Roy (2020) initializes the target network by a converted SNN to shorten the training epochs, indicating the possibility of high-performance SNN with limited activation time. The training problem due to the non-differentiable activation function has become the main obstruction of spiking neural network development.
+
+![](images/1c0df3e5e42eb5f12b364ae3e1b86c171367176d7d8e201b7cccf8c9ba08687d.jpg)  
+Figure 1: Workflow of temporal efficient training (TET). To obtain a more generalized SNN, we modify the optimization target to adjust each moment’s output distribution.
+
+In this work, we examine the limitation of the traditional direct training approach with SG and propose the temporal efficient training (TET) algorithm. Instead of directly optimizing the integrated potential, TET optimizes every moment’s pre-synaptic inputs. As a result, it avoids the trap into local minima with low prediction error but a high second-order moment. Furthermore, since the TET applies optimization on each time point, the network naturally has more robust time scalability. Based on this characteristic, we propose the time inheritance training (TIT), which reduces the training time by initializing the SNN with a smaller simulation length. With the help of TET, the performance of SNNs has improved on both static datasets and neuromorphic datasets. Figure 1 depicts the workflow of our approach.
+
+The following summarizes our main contributions:
+
+• We analyze the problem of training SNN with SG and propose the TET method, a new loss and gradient descent regime that succeeds in obtaining more generalizable SNNs. • We analyze the feasibility of TET and picture the loss landscape under both the SDT and TET setups to demonstrate TET’s advantage in better generalization. • Our sufficient experiments on both static datasets and neuromorphic datasets prove the effectiveness of the TET method. Especially on DVS-CIFAR10, we report $8 3 . 1 \bar { 7 } \%$ top-1 accuracy for the first time, which is over $1 0 \%$ better than the current state-of-the-art result.
+
+# 2 RELATED WORK
+
+In recent years, SNNs have developed rapidly and received more and more attention from the research community. However, lots of challenging problems remain to be unsolved. In general, most works on SNN training have been carried out in two strategies: ANN-to-SNN conversion and direct training from scratch.
+
+ANN-to-SNN Conversion. Conversion approaches avoid the training problem by trading high accuracy through high latency. They convert a high-performing ANN to SNN and adjust the SNN parameters w.r.t the ANN activation value layer-by-layer (Diehl et al., 2015; 2016). Some special techniques have been proposed to reduce the inference latency, such as the subtraction mechanism (Rueckauer et al., 2016; Han et al., 2020), robust normalization Rueckauer et al. (2016), spike-norm (Sengupta et al., 2018), and channel-wise normalization (Kim et al., 2019). Recently, Deng & Gu (2020) decompose the conversion error to each layer and reduce it by bias shift. Li et al. (2021a) suggest using adaptive threshold and layer-wise calibration to obtain high-performance SNNs that require a simulation length of less than 50. However, converted methods significantly extend the inference latency, and they are not suitable for neuromorphic data (Deng et al., 2020).
+
+Direct training. In this area, SNNs are regarded as special RNNs and training with BPTT (Neftci et al., 2019). On the backpropagation process, The non-differentiable activation term is replaced with a surrogate gradient (Lee et al., 2016). Compared with ANN-to-SNN conversion, direct training achieves high accuracy with few time steps but suffers more training costs (Deng et al., 2020). Several studies suggest that surrogate gradient (SG) is helpful to obtain high-performance SNNs on both static datasets and neuromorphic datasets (Wu et al., 2019; Shrestha & Orchard, 2018; Li et al., 2021b). On the backpropagation process, SG replaces the Dirac function with various shapes of curves. Exceptionally, Wu et al. (2018) first propose the STBP method and train SNNs on the ANN programming platform, which significantly promotes direct training development. Zheng et al. (2021) further proposes the tdBN algorithm to smooth the loss function and first realize training a large-scale SNN on ImageNet. Zhang & Li (2020) proposes TSSL-BP to break down error backpropagation across two types of inter-neuron and intra-neuron dependencies and achieve low-latency and high accuracy SNNs. Recently, Yang et al. (2021) designed a neighborhood aggregation (NA) method to use the multiple perturbed membrane potential waveforms in the neighborhood to compute the finite difference gradients and guide the weight updates. They significantly decrease the required training iterations and improve the SNN performance.
+
+# 3 PRELIMINARY
+
+# 3.1 ITERATIVE LIF MODEL
+
+We adopt the Leaky Integrate-and-Fire (LIF) model and translate it to an iterative expression with the Euler method (Wu et al., 2019). Mathematically, the membrane potential is updated as
+
+$$
+\begin{array} { r } { \pmb { u } ( t + 1 ) = \tau \pmb { u } ( t ) + \pmb { I } ( t ) , } \end{array}
+$$
+
+where $\tau$ is the constant leaky factor, ${ \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \mathbf { } \mathbf \Psi \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf \Psi \mathbf { } \mathbf \mathbf \Psi \mathbf { } \mathbf \mathbf \mathbf \Psi \Psi \mathbf { \mathbf } \mathbf \mathbf \mathbf  \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf $ is the membrane potential at time $t$ , and $\mathbf { } I ( t )$ denotes the pre-synaptic inputs, which is the product of synaptic weight $\mathbf { W }$ and spiking input ${ \mathbf { } } x ( t )$ . Given a specific threshold $V _ { t h }$ , the neuron fires a spike and ${ \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf { } \mathbf \Psi \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf \Psi \mathbf { } \mathbf \mathbf \Psi \mathbf \Psi \Psi \mathbf \Psi \mathbf { \mathbf } \mathbf \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf  \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf $ reset to 0 when the ${ \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } { \mathbf { } } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \Psi \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \mathbf { } \mathbf \Psi \mathbf { } \mathbf \mathbf { } \mathbf \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf \Psi \Psi \mathbf { } \mathbf \mathbf \mathbf { } \mathbf \mathbf \Psi \mathbf \Psi \mathbf { \Psi \mathbf } \mathbf \mathbf \Psi \mathbf \Psi \mathbf \mathbf  \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \mathbf \Psi \mathbf \Psi \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \Psi \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf \mathbf $ exceeds the threshold. So the firing function and hard reset mechanism can be described as
+
+$$
+\pmb { a } ( t + 1 ) = \pmb { \Theta } ( \pmb { u } ( t + 1 ) - V _ { t h } )
+$$
+
+$$
+\pmb { u } ( t + 1 ) = \pmb { u } ( t + 1 ) \cdot ( 1 - \pmb { a } ( t + 1 ) ) ,
+$$
+
+where $\Theta$ denotes the Heaviside step function. The output spike $\mathbf { \delta } \mathbf { \ } \mathbf { \em a } ( t + 1 )$ will become the post synaptic spike and propagate to the next layer. In this study, we set the starting membrane $\pmb { u } ( 0 )$ to 0, the threshold $V _ { t h }$ to 1, and the leaky factor $\tau$ to 0.5 for all experiments.
+
+The last layer’s spike frequency is typically used as the final classification index. However, adopting the LIF model on the last layer will lose information on the membrane potential and damage the performance, especially on complex tasks (Kim et al., 2019). Instead, we integrate the pre-synaptic inputs $\mathbf { } I ( t )$ with no decay or firing (Rathi & Roy, 2020; Fang et al., 2021). Finally, we set the average membrane potential as the classification index and calculate the cross-entropy loss for training.
+
+# 3.2 SURROGATE GRADIENT
+
+Following the concept of direct training, we regard the SNN as RNN and calculate the gradients through spatial-temporal backpropagation (STBP) (Wu et al., 2018):
+
+$$
+\frac { \partial L } { \partial \mathbf { W } } = \sum _ { t } \frac { \partial L } { \partial \pmb { a } ( t ) } \frac { \partial \pmb { a } ( t ) } { \partial \pmb { a } ( t ) } \frac { \partial \pmb { u } ( t ) } { \partial \pmb { I } ( t ) } \frac { \partial \pmb { I } ( t ) } { \partial \mathbf { W } } ,
+$$
+
+where the term $\frac { \partial \pmb { a } ( t ) } { \partial \pmb { u } ( t ) }$ is the gradient of the non-differentiability step function involving the derivative of Dirac’s $\delta$ -function that is typically replaced by surrogate gradients with a derivable curve. So far,
+
+there are various shapes of surrogate gradients, such as rectangular (Wu et al., 2018; 2019), triangle (Esser et al., 2016; Rathi & Roy, 2020), and exponential (Shrestha & Orchard, 2018) curve. In this work, we choose the surrogate gradients shaped like triangles. Mathematically, it can describe as
+
+$$
+\frac { \partial \pmb { a } ( t ) } { \partial \pmb { u } ( t ) } = \frac { 1 } { \gamma ^ { 2 } } \mathrm { m a x } ( 0 , \gamma - | \pmb { u } ( t ) - V _ { t h } | ) ,
+$$
+
+where the $\gamma$ denotes the constraint factor that determines the sample range to activate the gradient.
+
+# 3.3 BATCH NORMALIZATION FOR SNN
+
+Batch Normalization (BN) (Ioffe & Szegedy, 2015) is beneficial to accelerate training and increase performance since it can smooth the loss landscape during training (Santurkar et al., 2018). Zheng et al. (2021) modified the forward time loop form and proposed threshold-dependent Batch Normalization (tdBN) to normalize the pre-synaptic inputs $\pmb { I }$ in both spatial and temporal paradigms so that the BN can support spatial-temporal input. We adopt this setup with the extension of the time dimension to batch dimension 1. In the inference process, the BN layer will be merged into the pre-convolutional layer, thus the inference rule of SNN remain the same but with modified weight:
+
+$$
+\hat { \mathbf { W } } \gets \mathbf { W } \frac { \gamma } { \alpha } , \hat { \pmb { b } } \gets \beta + ( \pmb { b } - \mu ) \frac { \gamma } { \alpha } ,
+$$
+
+where $\mu , \alpha$ are the running mean and standard deviation on both spatial and temporal paradigm, $\gamma , \beta$ are the affine transformation parameters, and $\mathbf { W } , b$ are the parameters of the pre-convolutional layer.
+
+# 4 METHODOLOGY
+
+# 4.1 FORMULA OF TRAINING SNN WITH SURROGATE GRADIENTS
+
+Standard Direct Training. We use $O ( t )$ to represent pre-synaptic input $\mathbf { } I ( t )$ of the output layer and calculate the cross-entropy loss. The loss function of standard direct training ${ \mathcal { L } } _ { \mathrm { S D T } }$ is:
+
+$$
+\mathcal { L } _ { \mathrm { S D T } } = \mathcal { L } _ { \mathrm { C E } } \big ( \frac { 1 } { T } \sum _ { t = 1 } ^ { T } O ( t ) , { \pmb y } \big ) ,
+$$
+
+where $T$ is the total simulation time, $\mathcal { L } _ { \mathrm { C E } }$ denotes the cross-entropy loss, and $\textbf {  { y } }$ represents the target label. Following the chain rule, we obtain the gradient of $\mathbf { W }$ with softmax $S ( \cdot )$ inference function :
+
+$$
+\frac { \partial \mathcal { L } _ { \mathrm { S D T } } } { \partial { \bf W } } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } [ S ( O _ { \mathrm { m e a n } } ) - \hat { \pmb { y } } ] \frac { \partial O ( t ) } { \partial { \bf W } } ,
+$$
+
+where $O _ { \mathrm { m e a n } }$ denotes the average of the output $O ( t )$ over time, and $\hat { y }$ is the one-hot coding of $\textbf {  { y } }$ .
+
+Temporal Efficient Training. In this section, we come up with a new kind of loss function ${ \mathcal { L } } _ { \mathrm { T E T } }$ to realize temporal efficient training (TET). It constrains the output (pre-synaptic inputs) at each moment to be close to the target distribution. It is described as:
+
+$$
+\mathcal { L } _ { \mathrm { T E T } } = \frac { 1 } { T } \cdot \sum _ { t = 1 } ^ { T } \mathcal { L } _ { \mathrm { C E } } [ O ( t ) , { \pmb y } ] .
+$$
+
+Recalculate the gradient of weights under the loss function ${ \mathcal { L } } _ { \mathrm { T E T } }$ , and we have:
+
+$$
+\frac { \partial \mathcal { L } _ { \mathrm { T E T } } } { \partial { \bf W } } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } [ S ( \pmb { O } ( t ) ) - \pmb { \hat { y } } ] \cdot \frac { \partial \pmb { O } ( t ) } { \partial { \bf W } } .
+$$
+
+# 4.2 CONVERGENCE OF GRADIENT DESCENT FOR SDT V.S. TET
+
+In the case of SDT, the gradient consists of two parts, the error term $( S ( O _ { \mathrm { m e a n } } ) - \hat { \pmb y } )$ and the partial derivative of output $\partial { \cal O } \bar { ( } t ) / \partial { \bf W }$ . When the training process reaches near a local minimum, the term $( S ( O _ { \mathrm { m e a n } } ) - \hat { \pmb y } )$ approximates 0 for all $t = 1 , . . . , T$ , ignorant of the term $\partial O ( t ) / \partial \mathbf { W }$ . For traditional ANNs, the accumulated momentum may help get out of the local minima (e.g. saddle point) that typically implies bad generalizability (Kingma & Ba, 2014; Kidambi et al., 2018). However, when the SNN is trained with surrogate gradients, the accumulated momentum could be extremely small, considering the mismatch of gradients and losses. The fact that the activation function is a step one while the SG is bounded with integral constraints. This mismatch dissipates the momentum around a local minimum and stops the SDT from searching for a flatter minimum that may suggest better generalizability.
+
+In the case of TET, this issue of mismatch is relieved by reweighting the contribution of $\partial { \cal O } ( t ) / \partial { \bf W }$ . Indeed, considering the fact that the first term $( S ( O ( t ) ) - \hat { { \mathbf { y } } } )$ is impossible to be 0 at every moment of SNN since the early output accuracy on the training set is not $1 0 0 \%$ . So TET needs the second term $\partial { \cal O } ( t ) / \partial { \bf W }$ close to 0 to make the ${ \mathcal { L } } _ { \mathrm { T E T } }$ convergence. This mechanism increases the norm of gradients around sharp local minima and drives the TET to search for a flat local minimum where the disturbance of weight does not cause a huge change in $O ( t )$ .
+
+Further, to ensure that the convergence with TET implies the convergence of SDT, we prove the following lemma:
+
+Lemma 4.1. $\mathcal { L } _ { S D T }$ is upper bounded by $\mathcal { L } _ { T E T }$
+
+Proof. Suppose $O _ { i } ( t )$ and $\hat { y } _ { i }$ denote the i-th component of $O ( t )$ and $\hat { y }$ , respectively. Expand Eqn.9, we have:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { \mathrm { T E T } } = - \displaystyle \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \sum _ { i = 1 } ^ { n } \hat { y } _ { i } \log S ( \boldsymbol { O } _ { i } ( t ) ) = - \frac { 1 } { T } \sum _ { i = 1 } ^ { n } \hat { y } _ { i } \log ( \prod _ { t = 1 } ^ { T } S ( \boldsymbol { O } _ { i } ( t ) ) ) } \\ & { \quad \quad \quad = - \displaystyle \sum _ { i = 1 } ^ { n } \hat { y } _ { i } \log ( \prod _ { t = 1 } ^ { T } S ( \boldsymbol { O } _ { i } ( t ) ) ) ^ { \frac { 1 } { T } } \geq - \sum _ { i = 1 } ^ { n } \hat { y } _ { i } \log ( \frac { 1 } { T } \sum _ { t = 1 } ^ { T } S ( \boldsymbol { O } _ { i } ( t ) ) ) } \\ & { \quad \quad \quad \geq - \displaystyle \sum _ { i = 1 } ^ { n } \hat { y } _ { i } \log ( S ( \frac { 1 } { T } \sum _ { t = 1 } ^ { T } O _ { i } ( t ) ) ) = \mathcal { L } _ { \mathrm { S D T } } , } \end{array}
+$$
+
+where the first inequality is given by the Arithmetic Mean-Geometric Mean Inequality, and the second one is given by Jensen Inequality since the softmax function is convex. As a corollary, once the ${ \mathcal { L } } _ { \mathrm { T E T } }$ gets closed to zero, the original loss function ${ \mathcal { L } } _ { \mathrm { S D T } }$ also approaches zero. □
+
+Furthermore, the network output $O ( t )$ at a particular time point may be a particular outlier that dramatically affects the total output since the output of the SNN has the same weight at every moment under the rule of integration. Thus it is necessary to add a regularization term like $\mathcal { L } _ { \mathrm { M S E } }$ loss to confine each moment’s output to reduce the risk of outliers:
+
+$$
+\mathcal { L } _ { \mathrm { M S E } } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathrm { M S E } ( \mathbf { O } ( t ) , \phi ) ,
+$$
+
+where $\phi$ is a constant used to regularize the membrane potential distribution. And we set $\phi = V _ { t h }$ in our experiments. In practice, we use a hyperparameter $\lambda$ to adjust the proportion of the regular term, we have:
+
+$$
+\mathcal { L } _ { \mathrm { T O T A L } } = ( 1 - \lambda ) \mathcal { L } _ { \mathrm { T E T } } + \lambda \mathcal { L } _ { \mathrm { M S E } } .
+$$
+
+It is worth noting that we only changed the loss function in the training process and did not change SNN’s inference rules in the testing phase for a fair comparison. This algorithm is detailed in Algo.1.
+
+<table><tr><td>Algorithm1:Temporalefficienttrainingforoneepoch Input: SNN model; Simulation length: T; Threshold: Vth; Training dataset; Validation dataset;</td></tr><tr><td>total training iteration in one epoch: Itrain; total validation iteration in one epoch: Ival</td></tr><tr><td>for all i= 1,2,...Itrain iteration do Get mini-batch training data,and class label: Yi;</td></tr><tr><td>Compute the SNN output Oi(t) of eatch time step;</td></tr><tr><td>Calculate loss function: LTOTAL = (1-λ)LTET + 入LMSE =</td></tr><tr><td>(1-λ):¹∑t=1LcE(O²(t),Yi)+&gt;·¹∑t=1 MSE(Oi(t),𝜙);</td></tr><tr><td>Backpropagation and update model parameters;</td></tr><tr><td>end for all i= 1,2,..Ival iteration do</td></tr><tr><td>Get mini-batch validation data,and class label: Yi;</td></tr><tr><td>T</td></tr><tr><td>Compute the SNN average output Omean = ∑T=1 O(t) over al time step;</td></tr><tr><td>Compare the clasification factor Omean and Yi for classification; end</td></tr></table>
+
+# 4.3 TIME INHERITANCE TRAINING
+
+SNN demands simulation length long enough to obtain a satisfying performance, but the training time consumption will increase linearly as the simulation length grows. So how to shorten the training time is also an essential problem in the direct training field. Traditional loss function ${ \mathcal { L } } _ { \mathrm { S D T } }$ only optimizes the whole network output under a specific $T$ , so its temporal scalability is poor. Unlike the standard training, TET algorithm optimizes each moment’s output, enabling us to extend the simulation time naturally. We introduce Time Inheritance Training (TIT) to alleviate the training time problem. We first use long epochs to train an SNN with a short simulation time T, e.g., 2. Then, we increase the simulation time to the target value and retrain with short epochs. We discover that TIT performs better than training from scratch on accuracy and significantly saves the training time. Assuming that training an SNN with simulation length $T = 1$ cost $t s$ time per epoch, the SNN needs 300 epochs to train from scratch, and the TIT needs 50 epochs for finetuning. So we need $1 8 0 0 t s$ time to train an SNN with $T = 6$ from scratch, but following the TIT pipeline with the initial $T = 2$ only requires $9 0 0 t s$ . As a result, the TIT can reduce the training time cost by half.
+
+# 5 EXPERIMENTS
+
+We validate our proposed TET algorithm and compare it with existing works on both static and neuromorphic datasets. The network architectures in this paper include ResNet-19 (Zheng et al., 2021), Spiking-ResNet34 (Zheng et al., 2021), SEW-ResNet34 (Fang et al., 2021), SNN-5, and VGGSNN. SNN-5 (16C3-64C5-AP2-128C5-AP2-256C5-AP2-512C3-AP2-FC) is a simple convolutional SNN suitable for multiple runs to discover statistical rules (Figure A. 7). The architecture of VGGSNN (64C3-128C3-AP2-256C3-256C3-AP2-512C3-512C3-AP2-512C3-512C3-AP2- FC) is based on VGG11 with two fully connected layers removed as we found that additional fully connected layers were unnecessary for neuromorphic datasets.
+
+# 5.1 MODEL VALIDATION AND ABLATION STUDY
+
+Effectiveness of TET over SDT with SG. We first examine whether the mismatch between SG and loss causes the convergence problem. For this purpose, we set the simulation length to 4 and change the spike function $\Theta$ in Eqn.2 to Sigmoid $\sigma ( \bar { k } \cdot \mathrm { { i n p u t } } )$ . We find that the TET and SDT achieved similar accuracy (Table 2) when $k = 1 , 1 0 , 2 0$ . This indicates that both TET and SDT work when the gradient and loss function match each other. Next, we compare the results training with ${ \mathcal { L } } _ { \mathrm { S D T } }$ and ${ \mathcal { L } } _ { \mathrm { T E T } }$ on SNNs (ResNet-19 on CIFAR100) training with surrogate gradient for three runs. As shown in Table 1, our proposed new TET training strategy dramatically increases the accuracy by $3 . 2 5 \%$ when the simulation time is 4 and $3 . 5 3 \%$ when the simulation time is 6. These results quantitatively support the effectiveness of TET in solving the mismatch between gradient and loss in training SNNs with SG.
+
+![](images/aff831f07e0e329cffab78a14e6c6b9cb6a33fb6860a562c194fcda2f2064b92.jpg)  
+Figure 2: Loss landscape of VGGSNN. The 2D landscape of ${ \mathcal { L } } _ { \mathrm { S D T } }$ and ${ \mathcal { L } } _ { \mathrm { T E T } }$ from two different training methods.
+
+Table 1: Comparison between SDT and TET. We adopt the SNN architecture ResNet-19 with SG on CIFAR100 and record the results with three different simulation lengths 2, 4, and 6.   
+
+<table><tr><td>Method</td><td>T=2</td><td>T=4</td><td>T=6</td></tr><tr><td>Direct training</td><td>69.41±0.08</td><td>70.86±0.22</td><td>71.12±0.57</td></tr><tr><td>TET</td><td>72.37±0.21</td><td>74.11±0.18</td><td>74.65±0.12</td></tr></table>
+
+Table 2: Comparison of SDT and TET with sigmoid function $\sigma ( k { \cdot } \mathrm { i n p u t } )$ . We fix the simulation length to 4 and record the results of CNN-5 under three different $k$ on CIFAR10.   
+
+<table><tr><td>Method</td><td>k=1</td><td>k=10</td><td>k=20</td></tr><tr><td>Direct training</td><td>88.00±0.15</td><td>88.83±0.32</td><td>88.50±0.32</td></tr><tr><td>TET</td><td>87.63±0.38</td><td>89.31±0.15</td><td>88.64±0.28</td></tr></table>
+
+Loss Landscape around Local Minima. We further inspect the 2D landscapes (Li et al., 2018) of $\mathcal { L } _ { \mathrm { S D T } }$ and ${ \mathcal { L } } _ { \mathrm { T E T } }$ around their local minima (see Figure. 2) to demonstrate why TET generalizes better than SDT and how TET helps the training process jump out of the sharp local minima typically found by SDT. First, comparing Figure. $2 \textrm { A }$ and C, we can see that although the values of local minima achieved by SDT and TET are similar in $\mathcal { L } _ { \mathrm { S D T } }$ , the local minima of TET (Figure. $2 \textrm { C }$ ) is flatter than that of SDT (Figure. $2 \mathrm { \ A }$ ). This indicates that the TET is effective in finding flatter minima that are typically more generalizable even w.r.t the original loss in TET. Next, we examine the two local minima under ${ \mathcal { L } } _ { \mathrm { T E T } }$ to see how it helps jump out the local minima found by SDT. When comparing Figure. 2 B and D, we observe that the local minima found by SDT (Figure. 2 B) is not only sharper than that found by TET (Figure. $2 \mathbf { D }$ ) under ${ \mathcal { L } } _ { \mathrm { S D T } }$ but also maintains a higher loss value. This supports our claim that TET loss cannot be easily minimized around sharp local minima (Figure. $2 \mathrm { \ B }$ ), thus preferable to converge into flatter local minima (Figure. $2 \mathrm { D }$ ). Put together, the results here provide evidence for our reasoning in Section 4.2.
+
+Training from SDT to TET. In this part, we further validate the ability of TET to escape from the local minimum found by SDT. We adopt the VGGSNN with 300 epochs training on DVS-CIFAR10. First, we optimize ${ \mathcal { L } } _ { \mathrm { S D T } }$ for 200 epochs and then change the loss function to ${ \mathcal { L } } _ { \mathrm { T E T } }$ after epoch 200. Figure 3 demonstrates the accuracy and loss change on the test set. After 200 epochs training, SDT gets trapped into a local minimum, and the ${ \mathcal { L } } _ { \mathrm { S D T } }$ no longer decreases. The ${ \mathcal { L } } _ { \mathrm { T E T } }$ is much higher than $\mathcal { L } _ { \mathrm { S D T } }$ since SDT does not optimize it. Nevertheless, after we change the loss function to ${ \mathcal { L } } _ { \mathrm { T E T } }$ , the ${ \mathcal { L } } _ { \mathrm { T E T } }$ and $\mathcal { L } _ { \mathrm { S D T } }$ on the test set both have a rapid decline. This phenomenon illustrates the TET ability to help the SNN efficiently jump out of the local minimum with poor generalization and find another flatter local minimum.
+
+![](images/3337c633c01451316aa0cc30732fe1eaeb582425bcdf30fbe135431d8bf36214.jpg)  
+Figure 3: TET helps to jump out the local minimum point. We provide the test accuracy (A) and loss $( B )$ change after changing the SDT to TET at epoch 200. TET efficiently improves the test performance and reduces the two kinds of loss.
+
+![](images/086a43f62fe160235967e86bf1fd8d81f8153af1874a2e9a74e352d77d8e5e20.jpg)  
+Figure 4: Time scalability robustness and network efficiency of ResNet-19 on CIFAR100. (A) The comparison of training from scratch (dots) and inheriting from a small simulation length (lines). $( B )$ SNN network performance changes with energy consumption.
+
+Time Scalability Robustness. Here, we study the time scalability robustness of SNNs trained with TET $( \mathcal { L } _ { \mathrm { T E T } } )$ . First, we use 300 epochs to train a small simulation length ResNet-19 on CIFAR100 as the initial SNN. Then, we directly change the simulation length from 2 to 8 without finetuning and report the network accuracy on the test set. Figure. 4. A displays the results after changing the simulation length. We use 2, 3, and 4, respectively, as the simulation length of the initial network. When we increase the simulation length, the accuracy of all networks gradually increases. After the simulation time reaches a certain value, the network performance will slightly decrease. Interestingly, SNNs trained from scratch ( $\mathrm { T } { = } 4$ and ${ \mathrm { T } } { = } 6$ ) are not as good as those trained following the TIT procedure.
+
+Network Efficiency. In this section, we measure the relationship between energy consumption and network performance. SNN avoids multiplication on the inference since its binary activation and event-based operation. The addition operation in SNN costs $0 . 9 p J$ energy while multiplication operation consumes $4 . 6 p J$ measured in $4 5 \mathrm { n m }$ CMOS technology (Rathi & Roy, 2020). In our SNN model, the first layer has multiplication operations, while the other layers only have addition operations. Figure 4. B summarizes the results of different simulation times. In all cases, the SNN obtained by TET has higher efficiency.
+
+# 5.2 COMPARISON TO EXITING WORKS
+
+In this section, we compare our experimental results with previous works. We validate the full TIT algorithm $( \mathcal { L } _ { \mathrm { T O T A L } } )$ both on the static dataset and neuromorphic dataset. All of the experiment results are summarized in Table 5.2. We specify all the training details in the appendix A.1.
+
+CIFAR. We apply TET and TIT algorithm on CIFAR (Krizhevsky et al., 2009), and report the mean and standard deviation of 3 runs under different random seeds. The $\lambda$ is set to 0.05. On CIFAR10, our TET method achieves the highest accuracy above all existing approaches. Even when $T = 2$ , there is a $1 . 8 2 \%$ increment compare to STBP-tdBN with simulation length $T = 6$ . It is worth noting that our method is only $0 . 4 7 \%$ lower than the ANN performance. TET algorithm demonstrates a more excellent ability on CIFAR100. It has an accuracy increase greater than $3 \%$ on all report simulation lengths. In addition, when $T = 6$ , the reported accuracy is only $0 . 6 3 \%$ lower than that of ANN. We can see that the proposed TET’s improvement is even higher on complex data like CIFAR100, where the generalizability of the model distinguishes a lot among minima with different flatness.
+
+Table 3: Compare with existing works. Our method improves network performance across all tasks. \* denotes self-implementation results. † denotes data augmentation (Li et al., 2022).   
+
+<table><tr><td>Dataset</td><td>Model</td><td>Methods</td><td>Architecture</td><td>SimulationLength</td><td>Accuracy</td></tr><tr><td rowspan="10">CIFAR10</td><td>Rathi et al. (2019)</td><td>Hybrid training Diet-SNN</td><td>ResNet-20</td><td>250</td><td>92.22</td></tr><tr><td>Rathi &amp; Roy (2020)</td><td></td><td>ResNet-20</td><td>10</td><td>92.54</td></tr><tr><td>Wu et al. (2018)</td><td>STBP</td><td>CIFARNet</td><td>12</td><td>89.83</td></tr><tr><td>Wu et al. (2019)</td><td>STBP NeuNorm</td><td>CIFARNet</td><td>12</td><td>90.53</td></tr><tr><td>Zhang &amp; Li (2020)</td><td>TSSL-BP</td><td>CIFARNet</td><td>5</td><td>91.41</td></tr><tr><td>Zheng et al. (2021)</td><td>STBP-tdBN</td><td>ResNet-19</td><td>6 4</td><td>93.16 92.92</td></tr><tr><td rowspan="3">our model</td><td rowspan="3">TET</td><td rowspan="3"></td><td>2</td><td>92.34</td></tr><tr><td>6</td><td>94.50±0.07</td></tr><tr><td>4</td><td>94.44±0.08</td></tr><tr><td></td><td></td><td>ResNet-19</td><td>2</td><td>94.16±0.03</td></tr><tr><td rowspan="6">CIFAR100</td><td>ANN*</td><td>ANN</td><td>ResNet-19</td><td>1</td><td>94.97</td></tr><tr><td>Rathi et al. (2019) Rathi &amp; Roy (2020)</td><td>Hybrid training</td><td>VGG-11</td><td>125</td><td>67.87</td></tr><tr><td></td><td>Diet-SNN</td><td>ResNet-20</td><td>5</td><td>64.07 71.12±0.57</td></tr><tr><td rowspan="3">Zheng et al. (2021)*</td><td rowspan="3">STBP-tdBN</td><td rowspan="3">ResNet-19</td><td>6</td><td>70.86±0.22</td></tr><tr><td>4 2</td><td>69.41±0.08</td></tr><tr><td>6</td><td>74.72±0.28</td></tr><tr><td rowspan="5"></td><td>our model</td><td rowspan="2">TET</td><td rowspan="2">ResNet-19</td><td>4</td><td>74.47±0.15</td></tr><tr><td></td><td>2</td><td>72.87±0.10</td></tr><tr><td>ANN* Rathi etal. (2019)</td><td>ANN</td><td>ResNet-19</td><td>1</td><td>75.35</td></tr><tr><td></td><td>Hybrid training SPIKE-NORM</td><td>ResNet-34</td><td>250</td><td>61.48</td></tr><tr><td>Sengupta et al. (2018) Zheng et al. (2021)</td><td>STBP-tdBN</td><td>ResNet-34</td><td>2500</td><td>69.96</td></tr><tr><td rowspan="5">ImageNet</td><td>Fang et al. (2021)</td><td>SEWResNet</td><td>Spiking-ResNet-34</td><td>6</td><td>63.72</td></tr><tr><td></td><td>TET</td><td>SEW-ResNet-34 Spiking-ResNet-34</td><td>4</td><td>67.04 64.79</td></tr><tr><td>our model</td><td>TET</td><td>SEW-ResNet-34</td><td>6 4</td><td>68.00</td></tr><tr><td>Zheng et al. (2021)</td><td>STBP-tdBN</td><td>ResNet-19</td><td>10</td><td>67.8</td></tr><tr><td>Kugele et al. (2020)</td><td>Streaming Rollout</td><td>DenseNet</td><td>10</td><td>66.8</td></tr><tr><td rowspan="5">DVS-CIFAR10</td><td>Wu et al. (2021)</td><td>Conv3D</td><td>LIAF-Net</td><td></td><td>71.70</td></tr><tr><td>Wu et al. (2021)</td><td>LIAF</td><td>LIAF-Net</td><td>10 10</td><td>70.40</td></tr><tr><td rowspan="2">our model</td><td>TET</td><td>VGGSNN</td><td>10</td><td>77.33±0.21</td></tr><tr><td>TETt</td><td>VGGSNN</td><td></td><td>83.17±0.15</td></tr><tr><td></td><td></td><td></td><td>10</td><td></td></tr></table>
+
+ImageNet. The training set of ImageNet (Krizhevsky et al., 2012) provides $1 . 2 8 \mathrm { k }$ training samples for each label. We choose the two most representative ResNet-34 to verify our algorithm on ImageNet with $\lambda = 0 . 0 0 1$ . SEW-ResNet34 is not a typical SNN since it adopts the IF model and modifies the Residual structure. Although we only train our model for 120 epochs, the TET algorithm achieves a $1 . 0 7 \%$ increment on Spiking-ResNet-34 and a $0 . 9 6 \%$ increment on SEW-ResNet34.
+
+DVS-CIFAR10. The neuromorphic datasets suffer much more noise than static datasets. Thus the well-trained SNN is easier to overfit on these datasets than static datasets. DVS-CIFAR10 (Li et al., 2017), which provides each label with $0 . 9 \mathrm { k }$ training samples, is the most challenging mainstream neuromorphic dataset. Recent works prefer to deal with this dataset by complex architectures, which are more susceptible to overfitting and do not result in very high accuracy. Here, we adopt VGGSNN on the DVS-CIFAR10 dataset, set $\lambda = 0 . 0 0 1$ , and report the mean and standard deviation of 3 runs under different random seeds. Along with data augmentation methods, VGGSNN can achieve an accuracy of $7 7 . 4 \%$ . Then we apply the TET method to obtain a more generalizable optima. The accuracy rises to $8 3 . 1 7 \%$ . Our TET method outperforms existing state-of-the-art by $1 1 . 4 7 \%$ accuracy. Without data augmentation methods, VGGSNN obtains $7 \bar { 3 } . 3 \%$ accuracy by SDT and $7 7 . 3 \%$ accuracy by TET.
+
+# 6 CONCLUSION
+
+This paper focuses on the SNN generalization problem, which is described as the direct training SNN performs well on the training set but poor on the test set. We find this phenomenon is due to the incorrect SG that makes the SNN easily trapped into a local minimum with poor generalization. To solve this problem, we propose the temporal efficient training algorithm (TET). Extensive experiments verify that our proposed method consistently achieves better performance than the SDT process. Furthermore, TET significantly improves the time scalability robustness of SNN, which enables us to propose the time inheritance training (TIT) to significantly reduce the training time consumption by almost a half.
+
+# 7 ACKNOWLEDGMENT
+
+This project is supported by NSFC 61876032 and JCYJ20210324140807019. Y. Li completed this work during his prior research assistantship in UESTC.
+
+# REFERENCES
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+
+# A APPENDIX
+
+# A.1 DATASET AND TRAINING DETAIL
+
+CIFAR. The CIFAR dataset (Krizhevsky et al., 2009) consists of 50k training images and 10k testing images with the size of $3 2 \times 3 2$ . We use ResNet-19 for both CIFAR10 and CIFAR100. Moreover, random horizontal flip and crop are applied to the training images the augmentation. First, we use 300 epoch to train the SNN with the simulation length $T = 2$ . We use an Adam optimizer with a learning rate of 0.01 and cosine decay to 0. Next, following the TIT algorithm, we increase the simulation time (to 4 and 6) and continue training the SNN for only 50 epochs, with the learning rate changing to $1 e - 4$ .
+
+ImageNet. ImageNet (Deng et al., 2009) contains more than $1 2 5 0 \mathrm { k }$ training images and $5 0 \mathrm { k }$ validation images. We crop the images to $2 2 4 \times 2 2 4$ and using the standard augmentation for the training data. We use an SGD optimizer with 0.9 momentum and weight decay $4 e - 5$ . The learning rate is set to 0.1 and cosine decay to 0. We train the SEW-ResNet34 (Fang et al., 2021) with $T = 4$ for 120 epochs. As for the Spiking-ResNet34 (Zheng et al., 2021), we use TIT algorithm to train 90 epochs with $T = 4$ first, then change the simulation time to 6 and finetune the network for 30 epochs. We adopt an Adam optimizer on the finetune phase and change the learning rate to $1 e - 4$ . TIT algorithm significantly reduces the training time consumption since training the Spiking-ResNet34 is extremely slow.
+
+DVS-CIFAR10. DVS-CIFAR10 (Li et al., 2017), the most challenging mainstream neuromorphic data set, is converted from CIFAR10. It has 10k images with the size $1 2 8 \times 1 2 8$ . Following Samadzadeh et al. (2020), we divide the data stream into 10 blocks by time and accumulate the spikes in each block. Then, we split the dataset into $9 \mathrm { k }$ training images and $1 \mathrm { k }$ test images and reduce the spatial resolution to $4 8 \times 4 8$ . Random horizontal flip and random roll within 5 pixels are taken as augmentation (Li et al., 2022). We adopt VGGSNN architecture with 300 epochs training on this classification task. And we use an Adam optimizer with the learning rate $1 e - 3$ and cosine decay to 0. As for the case that does not apply any augmentation, we add a weight decay of 5e-4 to the optimizer.
+
+# A.2 LSDT LOSS LANDSCAPE OF RESNET-19
+
+Here we compare the classification loss $( \mathcal { L } _ { \mathrm { S D T } } )$ landscapes of ResNet-19 on CIFAR100. The position around the local minimal value found by the SDT $( \mathcal { L } _ { \mathrm { { S D T } } } )$ is very sharp. However, the area around the local minimum found by TET $( \mathcal { L } _ { \mathrm { T E T } } )$ is much smoother (Figure 5), which indicates that TET effectively improves the network generalization. Such improvements could be further utilized to other techniques like privacy-preserving data generalization (Kim et al., 2021) and neural architecture search (Kim et al., 2022).
+
+![](images/ec3a5c919a88e624c8ee236df840321cb1292d402cae6ed2e4a7f134e697de42.jpg)  
+Figure 5: STD loss landscape of ResNet-19 on CIFAR100 from different training approaches.
+
+# A.3 EFFECT OF $\mathcal { L } _ { \mathrm { M S E } }$
+
+In this part, we examine the effect of the regular term $\mathcal { L } _ { \mathrm { M S E } }$ with 5 different levels of $\lambda$ . Figure 6 Summarizes the final results. The regular term $\mathcal { L } _ { \mathrm { M S E } }$ effectively increases the performance of both ResNet-19 on CIFAR100 and VGGSNN on DVS-CIFAR10. The static dataset CIFAR100 is more suitable for larger $\lambda$ , while smaller $\lambda$ is suitable for DVS-CIFAR10. Theoretically, it is hard to obtain satisfying performance at the early simulation moment due to the sparseness of neuromorphic datasets. So too large regular term $\mathcal { L } _ { \mathrm { M S E } }$ is not suitable for the neuromorphic dataset. Furthermore, we find that a high $\lambda$ may harm the early training phase on ImageNet, especially if zero-initialize (Goyal et al., 2017) is not performed. As a result, we set $\lambda$ to $5 e - 2$ for CIFAR10 and CIFAR100, $1 e - 3$ for ImageNet and DVS-CIFAR10.
+
+![](images/1dd2e09ae0d1ba0fa6f9db557415a2e61436ffe4627041352f5dae8ffd64e9b0.jpg)  
+Figure 6: The accuracy under different levels of $\lambda$ .
+
+# A.4 STATISTICAL RESULTS
+
+Here we provide statistical results (Figure 7) to prove that the total SNN accuracy is positively associated with every average of moment’s output test accuracy. We train CNN-5 on CIFAR10 for a total of 20 runs with SDT and 5 runs with TET.
+
+![](images/295735f1db530423a1bc9f14bdb244ee4d3157db65b06f1b197a62a8ce989d2a.jpg)  
+Figure 7: Statistical results. The overall performance of SNN is highly positively associated with the average accuracy of each moment. The standard training obtains the green dots, while the red dots are trained by the TET method.
+
+# A.5 TIME SCALABILITY ROBUSTNESS OF SDT AND TET.
+
+Here we first show the test accuracy (ResNet19 on CIFAR100) of the membrane potential increment at each moment instead of the integrated membrane potential. We set the initial simulation length of the SNNs to 3 or 4 and trained them for a full 300 epochs. Then we expand their simulation length to 8. As shown in table 4, TET $( \mathcal { L } _ { \mathrm { T E T } } )$ makes the membrane potential increment at each moment have a higher classification ability than SDT $( \mathcal { L } _ { \mathrm { { S D T } } } )$ . And TET (1.41 and 0.08) also acquires a low accuracy variance than SDT (3.81 and 4.04).
+
+Then we compare the time scalability robustness between SDT $( \mathcal { L } _ { \mathrm { { S D T } } } )$ and TET $( \mathcal { L } _ { \mathrm { T E T } } )$ . We set the initial simulation length of ResNet19 SNNs to 2, 3, 4 and train with SDT or TET. Then we gradually increase SNN simulation length to 64 and record test accuracy of the integrated membrane potential.
+
+![](images/8be4907531399e22bedf40aaa0bdeff1ad93013af87033e52d12b9d582926cdd.jpg)  
+Figure 8: The accuracy after increasing the simulation length. We first train the SNN with TET (only use $\mathcal { L } _ { \mathrm { T E T } } ,$ ) and SDT ${ ( \mathcal { L } _ { \mathrm { { S D T } } } ) }$ with simulation length (T) is 2, 3, or 4. Then, we increase the simulation to 64 without finetuning and record the test the classification accuracy (A) and the accuracy relative growth rate (B) of the total SNN output (integrate membrane potential) at each simulation time.
+
+As we increase the simulation length, all the SNNs’ accuracy will first increase and then be stable in a certain area. Meanwhile, TET (1.80) has a small accuracy variance than the SDT (11.13) after increasing the simulation length. This phenomenon indicates that the initialization steps of TIT only need a small simulation length SNN for TET but a sufficiently large simulation (or enough epochs for finetuning step) for SDT.
+
+Table 4: Accuracy of each moment’s membrane potential increment. We use ${ \mathcal { L } } _ { \mathrm { S D T } }$ or ${ \mathcal { L } } _ { \mathrm { T E T } }$ to train the networks with simulation length 3 or 4. Then directly increase their simulation length to 8 and record each moment’s potential increment test accuracy.   
+
+<table><tr><td>Method</td><td>T=1</td><td>T=2</td><td>T=3</td><td>T=4</td><td>T=5</td><td>T=6</td><td>T=7</td><td>T=8</td></tr><tr><td>SDT (T=3)</td><td>55.61</td><td>57.95</td><td>56.87</td><td>55.09</td><td>57.56</td><td>53.54</td><td>57.72</td><td>54.04</td></tr><tr><td>SDT (T=4)</td><td>37.96</td><td>61.78</td><td>55.03</td><td>56.64</td><td>57.47</td><td>54.24</td><td>58.74</td><td>55.48</td></tr><tr><td>TET (T=3)</td><td>65.97</td><td>72.22</td><td>71.78</td><td>70.55</td><td>71.90</td><td>69.57</td><td>72.15</td><td>69.78</td></tr><tr><td>TET (T=4)</td><td>62.17</td><td>71.57</td><td>71.05</td><td>72.08</td><td>71.77</td><td>71.23</td><td>71.81</td><td>71.36</td></tr></table>

parse/dev/iEvAf8i6JjO/iEvAf8i6JjO.md

@@ -0,0 +1,364 @@
+# TRGP: TRUST REGION GRADIENT PROJECTION FORCONTINUAL LEARNING
+
+Sen $\mathbf { L i n } ^ { 1 }$ , Li $\mathbf { Y a n g ^ { 1 } }$ , Deliang $\mathbf { F a n } ^ { 1 }$ , Junshan Zhang1,2   
+1School of ECEE, Arizona State University, 2Department of ECE, University of California, Davis   
+{slin70, lyang166, dfan}@asu.edu, jazh@ucdavis.edu
+
+# ABSTRACT
+
+Catastrophic forgetting is one of the major challenges in continual learning. To address this issue, some existing methods put restrictive constraints on the optimization space of the new task for minimizing the interference to old tasks. However, this may lead to unsatisfactory performance for the new task, especially when the new task is strongly correlated with old tasks. To tackle this challenge, we propose Trust Region Gradient Projection (TRGP) for continual learning to facilitate the forward knowledge transfer based on an efficient characterization of task correlation. Particularly, we introduce a notion of ‘trust region’ to select the most related old tasks for the new task in a layer-wise and single-shot manner, using the norm of gradient projection onto the subspace spanned by task inputs. Then, a scaled weight projection is proposed to cleverly reuse the frozen weights of the selected old tasks in the trust region through a layer-wise scaling matrix. By jointly optimizing the scaling matrices and the model, where the model is updated along the directions orthogonal to the subspaces of old tasks, TRGP can effectively prompt knowledge transfer without forgetting. Extensive experiments show that our approach achieves significant improvement over related state-of-the-art methods.
+
+# 1 INTRODUCTION
+
+Human beings can continuously learn different new tasks without forgetting the learnt knowledge of old tasks in their lifespan. Aiming to achieve this remarkable capability for the deep neural networks (DNNs), continual learning (CL) (Chen & Liu, 2018) has garnered much attention in recent years. Nevertheless, many existing CL methods still leave the DNN vulnerable to forget the knowledge of old tasks when learning new tasks. Such a phenomenon is known as ‘Catastrophic Forgetting’ (McCloskey & Cohen, 1989), which has become one of the major challenges for CL.
+
+Many approaches (e.g., (Rusu et al., 2016; Li & Hoiem, 2017; Dhar et al., 2019; Guo et al., 2020; Zeng et al., 2019)) have been proposed to address the forgetting issue, which can be generally divided into two classes depending on the network architecture, i.e., expansion methods and nonexpansion methods. In order to understand the fundamental limit of a fixed capacity neural network, we focus on non-expansion methods in this work. The basic idea for non-expansion methods is to constrain the gradient update either explicitly or implicitly when learning the new task, so as to minimize the introduced interference to old tasks. For example, the regularization-based methods (e.g., (Kirkpatrick et al., 2017; Serra et al., 2018)) penalize the modification on the most important weights of old tasks through model regularizations; experience-replay based methods (e.g., (Shin et al., 2017; Chaudhry et al., 2019)) constrain the gradient directions by replaying the data of old tasks during learning of new tasks, in the format of either real data or synthetic data from generative models; and orthogonal-projection based methods (e.g., (Farajtabar et al., 2020; Saha et al., 2021)) update the model with gradients in the orthogonal directions of old tasks, without the access to old task data. In particular, the recently proposed Gradient Projection Memory (GPM) (Saha et al., 2021) has demonstrated superior performance compared to other approaches.
+
+To sufficiently minimize the interference to old tasks, most existing non-expansion methods (particularly the orthogonal-projection based methods), often put restrictive constraints on the optimization space of the new task, which may throttle the learning performance for the new task. A plausible conjecture is that such a scenario is likely to occur when the new task is strongly correlated with old tasks, and in this study we provide evidence to support this conjecture. The underlying rationale is as follows: The weights that are important to the new task are also important to the old tasks strongly correlated with the new task, which are often frozen to address the forgetting in the existing methods; however, they should be updated in the learning of the new task.
+
+To tackle this challenge, a key insight is that for a new task that is strongly correlated with old tasks, although the model optimization space could be more restrictive, there should be better forward knowledge transfer from the correlated old tasks to the new task. With this insight, we propose an innovate continual learning approach to facilitate the forward knowledge transfer without forgetting. The main contributions can be summarized as follows:
+
+(1) Inspired by (Schulman et al., 2015), we introduce a novel notion of ‘trust region’ based on the norm of gradient projection onto the subspace spanned by task inputs, which selects the old tasks strongly correlated to the new task in a layer-wise and single-shot manner. Intuitively, the new task and the selected old tasks in the trust region have similar input features for the corresponding layer.
+
+(2) We propose a novel approach for the new task to leverage the knowledge of the strongly correlated old tasks in the trust region through a scaled weight projection. Particularly, a scaling matrix is learnt in each layer for the new task to scale the weight projection onto the subspace of old tasks in the trust region, in order to reuse the frozen weights of old tasks without modifying the model.
+
+(3) Building on the introduced trust region, scaled weight projection, and a module to construct task input subspace, we develop a continual learning approach, trust region gradient projection (TRGP), that jointly optimizes the scaling matrices and the model for the new task. To mitigate the forgetting issue further, the model is updated along the directions orthogonal to the subspaces of old tasks.
+
+(4) We evaluate TRGP on standard CL benchmarks using various network architectures. Compared to related state-of-the-art approaches, TRGP achieves substantial performance improvement on all benchmarks, and demonstrates universal improvement on all tasks. The superior performance indicates that TRGP can effectively promote the forward knowledge transfer while alleviating forgetting.
+
+# 2 RELATED WORK
+
+Expansion-based methods. Expansion-based methods (e.g., (Rusu et al., 2016; Li & Hoiem, 2017; Rosenfeld & Tsotsos, 2018; Hung et al., 2019; Yoon et al., 2017; Li et al., 2019; Veniat et al., 2020)) dynamically expand the network capacity to reduce the interference between the new tasks and the old ones. Progressive Neural Network (PNN) (Rusu et al., 2016) expands the network architecture for new tasks and preserves the weights of old tasks. Learning Without Forgetting (LWF) (Li & Hoiem, 2017) splits the model layers into two parts, i.e., the shared part co-used by all tasks, and the task-specific part which grows for new tasks. Dynamic-Expansion Net (DEN) (Yoon et al., 2017) and Compacting-Picking-Growing (CPG) (Hung et al., 2019) combine the strategies of model compression/pruning, weight selection and model expansion. In order to find the optimal structure for each of the sequential tasks, Reinforced Continual Learning (RCL) (Xu & Zhu, 2018) leverages reinforcement learning and (Li et al., 2019) adapts architecture search. APD (Yoon et al., 2020) adds additional task-specific parameters for each task and selectively learns the task-shared parameters.
+
+Regularization-based methods. This category of methods (e.g., (Kirkpatrick et al., 2017; Lee et al., 2017; Chaudhry et al., 2018a; Dhar et al., 2019; Ritter et al., 2018; Schwarz et al., 2018; Zenke et al., 2017)) protect the old tasks by adding regularization terms in the loss function to penalize the model change on their important weights. Notably, to determine the weight importance, Elastic Weight Consolidation (EWC) (Kirkpatrick et al., 2017) leverages Fisher information matrix, HAT (Serra et al., 2018) learns hard attention masks. MAS (Aljundi et al., 2018) evaluates the model outputs sensitivity to the inputs in an unsupervised manner.
+
+Memory-based methods. Depending on if data of old tasks is utilized when learning new tasks, memory-based methods can be further divided into the following two categories. 1) Experiencereplay based methods. This class of methods replays the old tasks data along with the current task data to mitigate catastrophic forgetting. Gradient Episodic Memory (GEM) (Lopez-Paz & Ranzato, 2017) and Averaged GEM (A-GEM) (Chaudhry et al., 2018b) alter the current gradient based on the gradient computed with data in the memory. A unified view of episodic memory based methods is proposed in (Guo et al., 2020), based on new approaches are developed to balance between old tasks and the new task. Tiny episodic memory is considered in (Chaudhry et al., 2019) and metalearning is leveraged in (Riemer et al., 2018). 2) Orthogonal-projection based method. To eliminate the need of storing data of old tasks, recently a series work (Zeng et al., 2019; Farajtabar et al., 2020; Saha et al., 2021) updates the model in the orthogonal direction of old tasks, and has shown remarkable performance. Particularly, Orthogonal Weight Modulation (OWM) (Zeng et al., 2019) learns a projector matrix to multiply with the new gradients. Orthogonal Gradient Descent (OGD) (Farajtabar et al., 2020) stores the gradient directions of old tasks and projects the new gradients on the directions orthogonal to the subspace spanned by the old gradients. Gradient Projection Memory (GPM) (Saha et al., 2021) stores the bases of the subspaces spanned by old task data and projects the new gradients on the directions orthogonal to these subspaces.
+
+# 3 PROBLEM FORMULATION
+
+Continual learning. Consider the setting where a sequence of tasks $\mathbb { T } = \{ t \} _ { t = 1 } ^ { T }$ arrives sequentially. Eacand k i $t$ has a dataset the label vect $\mathbb { D } _ { t } = \{ ( \boldsymbol { x } _ { t , i } , \boldsymbol { y } _ { t , i } ) \} _ { i = 1 } ^ { N _ { t } }$ with paci $N _ { t }$ sample pairs, whereural network with $\mathbf { x } _ { t , i }$ is the input vectoryers, and the set of $\mathbf { \Delta } \mathbf { y } _ { t , i }$ $L$ weights is denoted as $\mathbb { W } = \{ W ^ { l } \} _ { l = 1 } ^ { L }$ , where $W ^ { l }$ is the layer-wise weight for layer $l$ . Given the data input $\boldsymbol { x } _ { t , i }$ for task $t$ , denote $\boldsymbol { x } _ { t , i } ^ { l }$ as the input of layer $l$ and $\pmb { x } _ { t , i } ^ { 1 } = \pmb { x } _ { t , i }$ . The output $\boldsymbol { \mathbf { \mathit { x } } } _ { t , i } ^ { l + 1 }$ for layer $l$ is computed as $\pmb { x } _ { t , i } ^ { l + 1 } = f ( \pmb { W } ^ { l } , \pmb { x } _ { t , i } ^ { l } )$ , where $f$ is the operation of the network layer. Following (Saha et al., 2021), we denote $\boldsymbol { x } _ { t , i } ^ { l }$ as the representations of $\mathbf { x } _ { t , i }$ at layer $l$ . When learning task $t$ , we only have access to dataset $\mathbb { D } _ { t }$ . Let $\mathcal { L } ( \mathbb { W } , \{ ( \pmb { x } _ { t , i } , \pmb { y } _ { t , i } ) \} ) = \mathcal { L } _ { t } ( \mathbb { W } )$ denote the loss function for training, e.g., mean squared and cross-entropy loss, and $\mathbb { W } _ { t }$ denote the model after learning task $t$ .
+
+Orthogonal-projection based methods. To minimize the interference to old tasks, recently a series of studies (Zeng et al., 2019; Farajtabar et al., 2020; Saha et al., 2021) has been carried out to update the model for the new task in the direction orthogonal to the subspace spanned by inputs of old tasks. In what follows, we briefly introduce the main ideas through a basic case with two tasks 1 and 2.
+
+Denote the subspace spanned by the inputs of task 1 for layer $l$ as $S _ { 1 } ^ { l }$ and the learnt model for task 1 as $\mathbb { W } _ { 1 } = \{ W _ { 1 } ^ { l } \} _ { l = 1 } ^ { L }$ . It is clear that $\pmb { x } _ { 1 , i } ^ { l } \in S _ { 1 } ^ { l }$ . When learning task 2, the model $\pmb { W } _ { 1 } ^ { l }$ will be modified in the direction orthogonal to $S _ { 1 } ^ { l }$ , by either multiplying the gradient $\nabla _ { W ^ { l } } \mathcal { L } _ { 2 }$ with a projector matrix (e.g, (Zeng et al., 2019)), or projecting the gradient $\nabla _ { W ^ { l } } \mathcal { L } _ { 2 }$ onto the orthogonal direction to $S _ { 1 } ^ { l }$ (e.g., (Saha et al., 2021)). Let $\Delta { \cal W } _ { 1 } ^ { l }$ denote the model change after learning task 2. It follows immediately that $\Delta \boldsymbol { W } _ { 1 } ^ { l } \boldsymbol { x } _ { 1 , i } ^ { l } = 0$ , and the model $\boldsymbol { W } _ { 2 } ^ { l }$ for task 2 is $\bar { \mathbf { W } } _ { 2 } ^ { l } = \mathbf { W } _ { 1 } ^ { l } + \bar { \Delta \mathbf { W } } _ { 1 } ^ { l }$ . Therefore, for task 1:
+
+which indicates that no interference is introduced to task 1 after learning task 2, thereby addressing the forgetting issue. Such an analysis can be generalized to a sequence of tasks.
+
+When would orthogonal projection hinder the learning of a new task? Orthogonal projection provides a promising solution to address the forgetting in continual learning. However, by modifying the model only in the orthogonal direction to the input space of old tasks, the optimization space of learning the new task could be more restrictive, resulting in compromised performance of the new task. To get a more concrete sense, consider the following basic examples with two tasks 1 and 2.
+
+(Toy example $I$ ) Suppose task 1 has dataset $\mathbb { D } _ { 1 } \ = \ \{ (  { \boldsymbol { { x } } } _ { i } ,  { \boldsymbol { { y } } } _ { i } ) \} _ { i = 1 } ^ { N }$ and task 2 has dataset $\mathbb { D } _ { 2 } ~ =$ $\{ ( - \pmb { x } _ { i } , \pmb { y } _ { i } ) \} _ { i = 1 } ^ { N }$ , where only the sign is changed for the input vectors. Consider the case where two tasks share the same classifier (Saha et al., 2021). It is clear that for the $l$ -th layer, the subspace spanned by $\{ \pmb { x } _ { i } ^ { l } \} _ { i = 1 } ^ { N }$ of task 1 is same with the subspace spanned by $\{ - \pmb { x } _ { i } ^ { l } \} _ { i = 1 } ^ { N }$ of task 2, i.e., $S _ { 1 } ^ { l } = S _ { 2 } ^ { l }$ , given the learnt model $\pmb { W } _ { 1 } ^ { l }$ for task 1. Based on the fact that stochastic gradient descent updates lie in the subspace spanned by the data input (Zhang et al., 2021; Saha et al., 2021), it follows that the gradient $\nabla _ { W ^ { l } } \bar { \mathcal { L } } _ { 2 } \in \bar { S } _ { 2 } ^ { l }$ , such that $\nabla _ { W ^ { l } } \bar { \mathcal { L } } _ { 2 } \in S _ { 1 } ^ { l }$ . Therefore, the projection of $\nabla _ { W ^ { l } } \mathcal { L } _ { 2 }$ onto the orthogonal direction to $S _ { 1 } ^ { \bar { l } }$ is 0, which means that the model $W _ { 1 } ^ { l }$ will not be updated when learning task 2, i.e., $W _ { 2 } ^ { l } = \dot { W } _ { 1 } ^ { l }$ . However, the optimal model for task 2 should be $\dot { \pmb { W } } _ { 2 } ^ { l } = - \pmb { W } _ { 1 } ^ { l }$ , because ${ W _ { 1 } ^ { l } } { x _ { i } ^ { l } }$ achieves the minimum loss for the label $\mathbf { \nabla } _ { \mathbf { \psi } _ { 3 } } \mathbf { \psi } _ { 2 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \qquad \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 }$ after learning task 1.
+
+(Toy example 2) Suppose the input subspace of task 1 is orthogonal to that of task 2, i.e., $S _ { 1 } ^ { l } \perp S _ { 2 } ^ { l }$ . It follows that the projection of $\nabla _ { W ^ { l } } \mathcal { L } _ { 2 }$ onto the orthogonal direction to $S _ { 1 } ^ { l }$ is indeed equal to $\nabla _ { W ^ { l } } \mathcal { L } _ { 2 }$ . Consequently, updating the model for task 2 based on orthogonal projection will not only introduce no interference to task 1, but also move along the direction of steepest descent for task 2.
+
+![](images/8497d678a6ec9a8996e00b8400ce694ef6f8218500cfd8e7cd8b4bcc0811e605.jpg)  
+Figure 1: Layer-wise task correlation for the case where the subspace spanned by the representations is a two-dimensional plane. The subspaces are weakly correlated if they are nearly orthogonal and strongly correlated if they are nearly parallel.
+
+Motivated by these examples, a plausible conjecture is that naive orthogonal projection could possibly compromise the learning performance of the new task that is strongly correlated with old tasks, especially when the correlation is “negative” as in the toy example 1. In this study, we advocate to characterize the task correlation through the correlation between the input subspaces for two tasks. As illustrated in Fig. 1, when the subspace is 2-dimensional, two tasks are weakly correlated if their input subspaces are nearly orthogonal, and strongly correlated if their subspaces are nearly parallel.
+
+# 4 TRUST REGION GRADIENT PROJECTION FOR CONTINUAL LEARNING
+
+To tackle these challenges, a key insight is that for a new task that is strongly correlated with old tasks, although the model optimization space could be more restrictive, there should be better forward knowledge transfer from the correlated old tasks to the new task. With this insight, we propose a novel approach to prompt forward knowledge transfer without forgetting, by 1) introducing a novel notion of trust region to select the most related old tasks in a single-shot manner and 2) cleverly reusing the frozen weights of the selected tasks in the trust region with a scaled weight projection.
+
+# 4.1 TRUST REGION
+
+To facilitate forward knowledge transfer from the correlated old tasks to the new task, the first question is how to efficiently select the most correlated old tasks. Towards this end, we characterize the correlation between the input subspaces for two tasks, through the lens of gradient projection.
+
+Specifically, denote $S _ { j } ^ { l } = s p a n \{ B _ { j } ^ { l } \}$ as the subspace spanned by the task $j$ data for layer $l$ , where $B _ { j } ^ { l } = [ \pmb { u } _ { j , 1 } ^ { l } , . . . , \pmb { u } _ { j , M _ { j , l } } ^ { l } ]$ is the bases for $S _ { j } ^ { l }$ (totally $M _ { j , l }$ bases extracted from the input). For any matrix $\pmb { A }$ with a suitable dimension, denote its projection onto the subspace $S _ { j } ^ { l }$ as:
+
+$$
+\operatorname { P r o j } _ { { S } _ { j } ^ { l } } ( A ) = A B _ { j } ^ { l } ( B _ { j } ^ { l } ) ^ { \prime }
+$$
+
+where $( \cdot ) ^ { \prime }$ is the matrix transpose. We next define a layer-wise trust region for a new task as a set of its most related old tasks, based on the norm of projected gradient onto the subspaces of old tasks.
+
+Definition 1 (Layer-Wise Trust Region). For any new task $t \geq 2$ and layer l, we define a layer-wise trust region $\tau \mathcal { R } _ { t } ^ { i } = \{ j \}$ for $j \in [ 1 , t - 1 ]$ , where for any task $j \in \mathcal { T R } _ { t } ^ { l }$ the following holds:
+
+$$
+\begin{array} { r } { \| \mathrm { P r o j } _ { S _ { j } ^ { l } } ( \nabla _ { W ^ { l } } \mathcal { L } _ { t } ( \mathbb { W } _ { t - 1 } ) ) \| _ { 2 } \geq \epsilon ^ { l } \| \nabla _ { W ^ { l } } \mathcal { L } _ { t } ( \mathbb { W } _ { t - 1 } ) \| _ { 2 } , } \end{array}
+$$
+
+where $\epsilon ^ { l } \in [ 0 , 1 ]$ and $\mathbb { W } _ { t - 1 }$ is the model after learning task $t - 1$ .
+
+Intuitively, for the new task $t$ , the norm of its gradient projection onto the subspace of an old task $j$ serves as a surrogate for characterizing the correlation between input subspaces for these two tasks, due to the fact that the gradient lies in the span of its input. When the condition Eq. (3) is satisfied, the gradient $\nabla _ { W ^ { l } } \mathcal { L } _ { t } \big ( \mathbb { W } _ { t - 1 } \big )$ has a large projection onto the subspace of an old task $j$ , which implies that the subspace $S _ { t } ^ { l }$ for task $t$ and the subspace $S _ { j } ^ { l }$ for task $j$ may have sufficient common bases for layer $l$ . In this case, we trust that the old task
+
+$$
+\begin{array} { r l } { \nabla _ { W ^ { L } } \varepsilon _ { t } ^ { L } } & { \longrightarrow \frac { \mathrm { T r e s h o l d } \theta _ { t h } ^ { L } } { \varepsilon _ { t } } } \\ { \mathsf { S o } j \in \mathcal { F R } _ { t } ^ { L } } & { \left( \begin{array} { l } { \int _ { - \infty } ^ { t } \mathsf { s } _ { t } \frac { \mathsf { s } _ { t } } { \varepsilon _ { t } } ( \nabla _ { W ^ { L } } \varepsilon _ { t } ) } \\ { \qquad \mathrm { P r o j } _ { s _ { t } ^ { j } } ( \nabla _ { W ^ { L } } \varepsilon _ { t } ) } \end{array} \right) } & { \longrightarrow \begin{array} { l } { S _ { t } ^ { j } \mathsf { t o r t a s k } j } \\ { \qquad \mathsf { W } _ { w ^ { L } } \varepsilon _ { t } \mathsf { f o r t a s k } t } \end{array} } \\ & { \underbrace { \theta _ { s } \theta _ { t h } ^ { L } } _ { \tiny { \mathsf { S o } j \in \mathcal { F R } _ { t } ^ { L } } } \underbrace { \mathsf { W } _ { W ^ { L } } \varepsilon _ { t } } _ { \tiny { \mathsf { P r o j } \mathsf { S u p s } } } } \end{array} \begin{array} { l l } { \nabla _ { W ^ { L } } \varepsilon _ { t } } & { } \\ { \qquad \mathsf { W o l d } \mathsf { S } _ { t } ^ { j } \mathsf { t h o r t a s k } j } \end{array}  \\ & { \lesssim o ^ { j } \mathsf { e r t o r t a } \theta _ { t h } ^ { L } \left( \underbrace { \mathsf { T r o } _ { s _ { t - 1 } , \ldots , s _ { t } } } _ { \begin{array} { l } { \mathrm { P r o j } _ { s _ { t } ^ { j } } ( \nabla _ { W ^ { L } } \varepsilon _ { t } ) } \end{array} } \right) } &  \longrightarrow \begin{array} { l } { \mathrm { p r o j e c t i o n ~ o f ~ } \nabla _ { W ^ { L } } \varepsilon _ { t } } \\ { \qquad \mathrm { p r o j e c t i o n ~ o f ~ } \nabla _ { W ^ { L } } \varepsilon _ { t } } \end{array}
+$$
+
+Figure 2: Trust region for a 2-dimensional subspace can be interpreted as: if the angle $\theta$ between $\nabla _ { W ^ { l } } \mathcal { L } _ { t }$ and $\mathrm { P r o j } _ { S _ { j } ^ { l } } \big ( \nabla _ { W ^ { l } } \mathcal { L } _ { t } \big )$ is less than $\theta _ { t h } ^ { l }$ (larger projection on $S _ { j } ^ { l } )$ , old task $j$ is selected to $\tau { \mathcal R } _ { t } ^ { l }$ ; otherwise not. $\theta _ { t h } ^ { l }$ can be set as a large value, and we can pick tasks with top- $K$ smallest $\theta$ to $\tau { \mathcal R } _ { t } ^ { l }$ .
+
+$j$ is strongly correlated with the new task $t$ in layer $l$ , and put it into task $t$ ’s trust region $\tau { \mathcal R } _ { t } ^ { l }$ . A simple illustration of trust region is shown in Figure 2. Note that the notion of trust region can also be generalized to a task-wise definition, where the most correlated old tasks will be selected based on the projection of the entire gradient $\nabla _ { \mathbb { W } } \mathcal { L } _ { t } ( \mathbb { W } _ { t - 1 } )$ . However, the layer-wise trust region could select different tasks for different layers, which provides a more fine-resolution characterization of task correlations in terms of layer-level features.
+
+Practical implementation. Besides the valuable functionality provided by the trust region for selecting most correlated old tasks, another significant benefit is the simplicity of its practical implementation. Consider the implementation for learning a new task $t$ .
+
+(1) Single-shot manner. Given the learnt model $\mathbb { W } _ { t - 1 }$ , we select a sample batch from dataset $\mathbb { D } _ { t }$ , and compute the gradient $\nabla _ { W ^ { l } } \mathcal { L } _ { t } ( \mathbb { W } _ { t - 1 } )$ in one forward-backward pass for all layers at once. Given the subspace $S _ { j } ^ { l }$ for an old task $j$ , the condition Eq. (3) can be immediately evaluated for all old tasks.
+
+(2) Top- $K$ correlated tasks. It is clear that the choice of $\epsilon ^ { l }$ has a nontrivial impact on the selection of the most correlated old tasks. To reduce the sensitivity of the performance on $\epsilon ^ { l }$ , we can set a relatively small value of $\epsilon ^ { l }$ , and pick the top- $K$ old tasks with the largest gradient projection norm $\| \mathrm { P r o j } _ { S _ { j } ^ { l } } ( \nabla _ { W ^ { l } } \mathcal { L } _ { t } ( \mathbb { W } _ { t - 1 } ) ) \| _ { 2 }$ from the tasks satisfying Eq. (3). As demonstrated later in our experiments, setting $K = 1$ is enough to achieve a significant performance improvement.
+
+# 4.2 SCALED WEIGHT PROJECTION
+
+Given the layer-wise trust region $\tau { \mathcal R } _ { t } ^ { l }$ for the new task $t$ , the next key question is how to efficiently leverage the knowledge of the most correlated old tasks in $\tau { \mathcal R } _ { t } ^ { l }$ for learning task $t$ . To this end, we propose a novel approach to reuse the frozen weights of the selected old tasks in $\tau { \mathcal R } _ { t } ^ { l }$ through a scaled weight projection with a scaling matrix.
+
+At the outset, it is of interest to understand what knowledge is preserved for old tasks during continual learning with orthogonal projection. Based on Eq. (1) for the simple case with two learning tasks 1 and 2 as mentioned earlier, it can be shown that
+
+where the last equation holds because the model $\pmb { W } _ { 1 } ^ { l }$ is updated in the direction orthogonal to $S _ { 1 } ^ { l }$ when learning task 2. By generalizing Eq. (4) to the case with a sequence of tasks, we can have that for the model $\mathbb { W } _ { t - 1 }$ after learning task $t - 1$ and any old task $j < t$ :
+
+which indicates that the model weight projection on the subspace of old tasks is actually “frozen” during continual learning so as to overcome forgetting of the old tasks.
+
+On the other hand, because the trust region $\tau { \mathcal R } _ { t } ^ { l }$ is constructed in a way that the subspace $S _ { t } ^ { l }$ of task $t$ is strongly correlated with the subspace $S _ { j } ^ { l }$ for any old task $j \in \mathcal { T R } _ { t } ^ { l }$ , the bases $B _ { j } ^ { l }$ of $S _ { j } ^ { l }$ is very likely to contain important bases for task $t$ . As a result, the weight projection $\mathrm { P r o j } _ { S _ { j } ^ { l } } ( \boldsymbol { W } _ { t - 1 } ^ { l } )$ is important for the new task $t$ and should be modified accordingly in order to guarantee the learning performance of task $t$ , which however has to be frozen to protect task $j$ . To find an efficient way to leverage $\mathrm { P r o j } _ { S _ { j } ^ { l } } ( \bar { \boldsymbol { W } } _ { t - 1 } ^ { l } )$ without modifying it, note that the projection $\mathrm { P r o j } _ { S _ { j } ^ { l } } ( \boldsymbol { W } _ { t - 1 } ^ { l } )$ is indeed a linear combination of the projection of $\mathbf { \Delta } W _ { t - 1 } ^ { l }$ onto each basis in $B _ { j } ^ { l }$ , and every point in $S _ { j } ^ { l }$ can be obtained by scaling the coordinates of $\mathrm { P r o j } _ { S _ { j } ^ { l } } ( \boldsymbol { W } _ { t - 1 } ^ { l } )$ . Figure 3 shows a simple example for two-dimensional subspace. Therefore, we propose a scaled weight projection to find the best point for task $t$ in $S _ { j } ^ { l }$ by leveraging the projection $\mathrm { P r o j } _ { S _ { j } ^ { l } } ( \boldsymbol { W } _ { t - 1 } ^ { l } )$ through a square scaling matrix $Q _ { j , t } ^ { l }$ :
+
+![](images/c2c5885a6c73085ac457c74d95b6a966c7f273dace56e5350d4f83b25d0e7f92.jpg)  
+Figure 3: $[ { \pmb u } _ { j , 1 } ^ { l } , { \pmb u } _ { j , 2 } ^ { l } ]$ is the bases of subspace $S _ { j } ^ { l }$ , and $[ c _ { j , 1 } ^ { l } , c _ { j , 2 } ^ { l } ]$ is the coordinate of $\mathrm { P r o j } _ { s _ { j } ^ { l } } ( \pmb { W } _ { t - 1 } ^ { l } )$ . Any point $\mathrm { P r o j } _ { S _ { j } ^ { l } } ( W ^ { l } )$ in $S _ { j } ^ { l }$ can be obtained by scaling the coordinate $[ c _ { j , 1 } ^ { l } , c _ { j , 2 } ^ { l } ]$ with some scalar $s _ { 1 }$ and $s _ { 2 }$ .
+
+$$
+\mathrm { P r o j } _ { S _ { j } ^ { l } } ^ { Q } ( { \boldsymbol { W } _ { t - 1 } ^ { l } } ) = W _ { t - 1 } ^ { l } B _ { j } ^ { l } Q _ { j , t } ^ { l } ( { \boldsymbol { B } _ { j } ^ { l } } ) ^ { \prime } .
+$$
+
+The dimension of $Q _ { j , t } ^ { l }$ depends on the number of bases in $B _ { j } ^ { l }$ (dimension of $S _ { j } ^ { l } .$ ), which is usually small for each task. In this way, we explicitly transfer the knowledge of the selected old tasks in the trust region $\tau { \mathcal R } _ { t } ^ { l }$ to the new task $t$ through a scaling matrix $Q _ { j , t } ^ { l }$ .
+
+# 4.3 TASK SUBSPACE CONSTRUCTION
+
+To successfully leverage the trust region, a missing ingredient is the construction of input subspaces of old tasks. We next show how the subspace $S _ { j } ^ { l }$ can be constructed for task $j$ at layer $l$ .
+
+For task $j = 1$ . As in (Saha et al., 2021), we obtain the bases $B _ { 1 } ^ { l }$ after learning task 1 using Singular Value Decomposition (SVD) on the representations. Specifically, given the model $\mathbb { W } _ { 1 }$ after learning task 1, we construct a representation matrix $\pmb { R } _ { 1 } ^ { l } = [ \pmb { x } _ { 1 , 1 } ^ { \bar { l } } , . . . , \pmb { x } _ { 1 , n } ^ { \bar { l } } ] \in \mathbb { R } ^ { m \times n }$ with $n$ samples, where each $\pmb { x } _ { 1 , i } ^ { l } \in \mathbb { R } ^ { m }$ , is the representation at layer $l$ by forwarding the sample $_ { \pmb { x } _ { 1 , i } }$ through the network. Then, we apply SVD to the matrix $R _ { 1 } ^ { l }$ , i.e., ${ \pmb R } _ { 1 } ^ { l } = { \pmb U } _ { 1 } ^ { l } { \pmb \Sigma } _ { 1 } ^ { l } ( { \pmb V } _ { 1 } ^ { l } ) ^ { \prime }$ , where ${ \cal U } _ { 1 } ^ { l } = [ { \pmb u } _ { 1 , 1 } ^ { l } , . . . , { \pmb u } _ { 1 , m } ^ { l } ] \in$ $\mathbb { R } ^ { m \times m }$ is an orthogonal matrix with left singular vector $\pmb { u } _ { 1 , i } ^ { l } \in \mathbb { R } ^ { m }$ , $V _ { 1 } ^ { l } = [ \pmb { v } _ { 1 , 1 } ^ { l } , . . . , \pmb { v } _ { 1 , n } ^ { l } ] \in \mathbb { R } ^ { n \times n }$ is an orthogonal matrix with right singular vector $\pmb { v } _ { 1 , i } ^ { l } \in \mathbb { R } ^ { n }$ , and $\pmb { \Sigma } _ { 1 } ^ { l } \in \pmb { R } ^ { m \times n }$ is a rectangular diagonal matrix with non-negative singular values $\{ \sigma _ { 1 , i } ^ { l } \} _ { i = 1 } ^ { \operatorname* { m i n } \{ m , n \} }$ on the diagonal in a descending order. To obtain the bases for subspace $S _ { 1 } ^ { l }$ , we use $k _ { 1 } ^ { l }$ -rank matrix approximation to pick the first left singular vectors in $U _ { 1 } ^ { l }$ , such that the following condition is satisfied for a threshold $\eta _ { t h } ^ { l } \in ( 0 , 1 )$ :
+
+$$
+\lVert ( \boldsymbol { R } _ { 1 } ^ { l } ) _ { k _ { 1 } ^ { l } } \rVert _ { F } ^ { 2 } \geq \epsilon _ { t h } ^ { l } \lVert \boldsymbol { R } _ { 1 } ^ { l } \rVert _ { F } ^ { 2 }
+$$
+
+where $\begin{array} { r } { ( { \bf R } _ { 1 } ^ { l } ) _ { k _ { 1 } ^ { l } } = \sum _ { i = 1 } ^ { k _ { 1 } ^ { l } } \sigma _ { 1 , i } ^ { l } { \bf u } _ { 1 , i } ^ { l } ( { \bf v } _ { 1 , i } ^ { l } ) ^ { \prime } } \end{array}$ is a $k _ { 1 } ^ { l }$ -rank $( k _ { 1 } ^ { l } \ \leq \ r )$ approximation of the representation matrix $R _ { 1 } ^ { l }$ with rank $r \leq \operatorname* { m i n } \{ m , n \}$ , and $\| \cdot \| _ { F }$ is the Frobenius norm. Then the bases $B _ { 1 } ^ { l }$ for subspace $\mathbf { \bar { \it S } } _ { 1 } ^ { l }$ can be constructed as $B _ { 1 } ^ { l } = [ \pmb { u } _ { 1 , 1 } ^ { l } , . . . , \pmb { u } _ { 1 , k _ { 1 } ^ { l } } ^ { l } ]$ .
+
+For task $j \in [ 2 , T ]$ . We construct the bases $B _ { j } ^ { l }$ after learning task $j$ given the learnt model $\mathbb { W } _ { j }$ . A representation matrix $R _ { j } ^ { l }$ will be first obtained in the same manner as $R _ { 1 } ^ { l }$ . Note that the bases $\{ B _ { i } ^ { l } \} _ { i = 1 } ^ { j - 1 }$ learnt for old tasks may include important bases for task $j$ . Therefore, we learn the bases $B _ { j } ^ { l }$ by selecting the most important bases from both bases of old tasks and newly constructed bases. Specifically, (1) (old bases) we first concatenate the bases $\{ B _ { i } ^ { l } \} _ { i = 1 } ^ { j - 1 }$ of old tasks together in $M _ { j } ^ { l }$ and eliminate the common bases. For each basis $\pmb { u } _ { i } ^ { l } \in { \cal M } _ { j } ^ { l }$ , we compute the corresponding eigenvalue of ${ \cal R } _ { j } ^ { l } ( { \cal R } _ { j } ^ { l } ) ^ { \prime }$ , i.e., $\delta _ { i } ^ { l } = ( \mathbf { \boldsymbol { u } } _ { i } ^ { l } ) ^ { \prime } R _ { j } ^ { l } ( R _ { j } ^ { l } ) ^ { \prime } \mathbf { \boldsymbol { u } } _ { i } ^ { l }$ , which is the square of the singular value of $R _ { j } ^ { l }$ with respect to $\mathbf { \Delta } u _ { i } ^ { l }$ . (2) (new bases) We perform SVD on $\hat { \pmb { R } } _ { j } ^ { l } = \pmb { R } _ { j } ^ { l } - \pmb { R } _ { j } ^ { l } M _ { j } ^ { l } ( M _ { j } ^ { l } ) ^ { \prime }$ to generate new bases beyond $M _ { j } ^ { l }$ , i.e., $\hat { \pmb { R } } _ { j } ^ { l } = \hat { U } _ { j } ^ { l } \hat { \pmb { \Sigma } } _ { j } ^ { l } ( \hat { V } _ { j } ^ { l } ) ^ { \prime }$ with singular values $\{ \hat { \sigma } _ { j , h } ^ { l } \} _ { h }$ . (3) (select the most important bases from both old and new bases) Next we concatenate $\{ \delta _ { i } ^ { l } \} _ { i }$ and $\{ ( \hat { \sigma } _ { j , h } ^ { l } ) ^ { 2 } \} _ { h }$ together in a vector $\pmb { \delta }$ , and sort them in a descending order. We perform $k _ { j } ^ { l }$ -rank matrix approximation of $R _ { j } ^ { l }$ , such that the summation of the first $k _ { j } ^ { l }$ elements in $\delta$ is greater than $\epsilon _ { t h } ^ { l } \lVert { \cal R } _ { j } ^ { l } \rVert _ { F } ^ { 2 }$ . Then $B _ { j } ^ { l }$ can be constructed by selecting the bases corresponding to the first $k _ { j } ^ { l }$ elements in $\delta$ .
+
+# 4.4 CONTINUAL LEARNING WITH TRUST REGION GRADIENT PROJECTION
+
+Building on the three modules proposed earlier, i.e., task subspace construction, trust region and scaled weight projection, we next present our approach TRGP for continual learning that efficiently facilitate forward knowledge transfer without forgetting the old tasks.
+
+Learning task 1. The first task is learnt using standard gradient descent. The subspace $\{ S _ { 1 } ^ { l } \} _ { l = 1 } ^ { L }$ is constructed by following Section 4.3.
+
+Learning task 2, ..., T. For task $t \in [ 2 , T ]$ , we first determine the trust region $\tau { \mathcal R } _ { t } ^ { l }$ with top- $K$ correlated old tasks selected for layer $l$ . The optimization problem for task $t$ is as follows:
+
+$$
+\begin{array} { r l } & { \underset { \{ \pmb { W } ^ { l } \} _ { l } , \{ \pmb { Q } _ { j , t } ^ { l } \} _ { l , j \in \mathcal { T } \mathcal { R } _ { t } ^ { l } } } { \operatorname* { m i n } } \mathcal { L } ( \{ \pmb { W } _ { e f f } ^ { l } \} _ { l } , \mathbb { D } _ { t } ) , } \\ & { \xrightarrow [ \pmb { S } . t . \qquad ] { \mathrm { m i n } } W _ { e f f } ^ { l } = \pmb { W } ^ { l } + \sum _ { j \in \mathcal { T } \mathcal { R } _ { t } ^ { l } } [ \mathrm { P r o j } _ { S _ { j } ^ { l } } ^ { Q } ( \pmb { W } ^ { l } ) - \mathrm { P r o j } _ { S _ { j } ^ { l } } ( \pmb { W } ^ { l } ) ] , } \end{array}
+$$
+
+where the gradient for updating $W ^ { l }$ is $\nabla _ { W ^ { l } } \mathcal { L } = \nabla _ { W ^ { l } } \mathcal { L } - ( \nabla _ { W ^ { l } } \mathcal { L } ) M _ { t } ^ { l } ( M _ { t } ^ { l } ) ^ { \prime }$ and $\pmb { M } _ { t } ^ { l }$ is the bases of all old tasks as in Section 4.3. The subspace $\{ S _ { t } ^ { l } \} _ { l = 1 } ^ { L }$ is next obtained by following Section 4.3.
+
+Table 1: The averaged accuracy (ACC) and backward transfer (BWT) over all the tasks on different datasets. Note that, Multitask jointly learns all tasks only once in a single network by using the whole dataset, which does not adhere to CL setup.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">PMNIST</td><td colspan="2">CIFAR-100 Split</td><td colspan="2">5-Dataset</td><td colspan="2">MiniImageNet</td></tr><tr><td>ACC(%)</td><td>BWT(%)</td><td>ACC(%)</td><td>BWT(%)</td><td>ACC(%)</td><td>BWT(%)</td><td>ACC(%)</td><td>BWT(%)</td></tr><tr><td>Multitask</td><td>96.70</td><td>-</td><td>79.58</td><td>-</td><td>91.54</td><td>-</td><td>69.46</td><td>-</td></tr><tr><td>OWM</td><td>90.71</td><td>-1</td><td>50.94</td><td>-30</td><td>=</td><td>1</td><td>=</td><td>-</td></tr><tr><td>EWC</td><td>89.97</td><td>-4</td><td>68.80</td><td>-2</td><td>88.64</td><td>-4</td><td>52.01</td><td>-12</td></tr><tr><td>HAT</td><td>1</td><td>1</td><td>72.06</td><td>0</td><td>91.32</td><td>-1</td><td>59.78</td><td>-3</td></tr><tr><td>A-GEM</td><td>83.56</td><td>-14</td><td>63.98</td><td>-15</td><td>84.04</td><td>-12</td><td>57.24</td><td>-12</td></tr><tr><td>ER_Res</td><td>87.24</td><td>-11</td><td>71.73</td><td>-6</td><td>88.31</td><td>-4</td><td>58.94</td><td>-7</td></tr><tr><td>GPM</td><td>93.91</td><td>-3</td><td>72.48</td><td>-0.9</td><td>91.22</td><td>-1</td><td>60.41</td><td>-0.7</td></tr><tr><td>Ours (TRGP)</td><td>96.34</td><td>-0.8</td><td>74.46</td><td>-0.9</td><td>93.56</td><td>-0.04</td><td>61.78</td><td>-0.5</td></tr></table>
+
+# 5 EXPERIMENTAL RESULTS
+
+# 5.1 EXPERIMENTAL SETUP
+
+Datasets and training details. We evaluate our method on multiple datasets against state-of-the-art CL methods. 1) PMNIST. Following (Lopez-Paz & Ranzato, 2017; Saha et al., 2021), we create 10 sequential tasks using different permutations where each task has 10 classes. We use a 3-layer fully-connected network. 2) CIFAR-100 Split. We split the classes of CIFAR-100 (Krizhevsky et al., 2009) into 10 group, and consider 10-way multi-class classification in each group as a single task. Similar with (Serra et al., 2018; Saha et al., 2021), we use a version of 5-layer AlexNet. 3) CIFAR-100 Sup. We divide the CIFAR-100 dataset into 20 tasks where each task has 5 classes. We use a modified version of LeNet-5. 4) 5-Datasets. We use a sequence of 5-Datasets which includes CIFAR-10, MNIST, SVHN (Netzer et al., 2011), not-MNIST (Bulatov, 2011) and Fashion MNIST (Xiao et al., 2017), where each dataset is set to be a task. We adapt a reduced ResNet18 network that is used in (Lopez-Paz & Ranzato, 2017). 5) MiniImageNet Split. We split the 100 classes of MiniImageNet (Vinyals et al., 2016) into 20 sequential tasks where each task has 5 classes, and consider a reduced ResNet18 network. In addition, for all the experiments, the threshold $\epsilon ^ { l }$ is set to 0.5, and we select top-2 tasks that satisfy condition Eq. (3). We use the same threshold $\epsilon _ { t h } ^ { l }$ as GPM (Saha et al., 2021) for subspace construction. More details are in the appendix.
+
+Methods for comparison. To test the efficacy of our method, we compare it with state-of-the-art approaches in three categories: 1) Memory-based methods. We compare with Experience Replay with reservoir sampling (ER Res) (Chaudhry et al., 2019), Averaged GEM (A-GEM) (Chaudhry et al., 2018b), Orthogonal Weight Modulation (OWM) (Zeng et al., 2019) and Gradient Projection Memory (GPM) (Saha et al., 2021). 2) Regularization-based methods. We compare with state-ofthe-art HAT (Serra et al., 2018) and Elastic Weight Consolidation (EWC) (Kirkpatrick et al., 2017). 3) Expansion-based methods. We further compare with Progressive Neural Network (PNN) (Rusu et al., 2016), Learning Without Forgetting (LWF) (Li & Hoiem, 2017), Dynamic-Expansion Net (DEN) (Yoon et al., 2017), and APD (Yoon et al., 2020), by using CIFAR-100 Sup dataset.
+
+Metrics. Following GPM (Saha et al., 2021), two metrics are used to evaluate the performance: Accuracy (ACC), the average final accuracy over all tasks, and Backward Transfer (BWT), which measures the forgetting of old tasks when learning new tasks. ACC and BWT are defined as:
+
+$$
+{ \bar { A } } C C = { \frac { 1 } { T } } \sum _ { i = 1 } ^ { T } A _ { T , i } , B W T = { \frac { 1 } { T - 1 } } \sum _ { i = 1 } ^ { T - 1 } A _ { T , i } - A _ { i , i }
+$$
+
+where $T$ is the number of tasks, $A _ { T , i }$ is the accuracy of the model on $i$ -th task after learning the $T$ -th task sequentially.
+
+# 5.2 MAIN RESULTS
+
+ACC and BWT comparison. As shown in Table 1, TRGP achieves significantly accuracy improvement compared with prior works on all datasets. For example, in contrast to the best prior results, TRGP achieve the accuracy gain of $2 . 4 3 \%$ , $1 . 9 8 \%$ and $1 . 3 \hat { 7 } \%$ over GPM on PMNIST, CIFAR-100 Split and MiniImageNet, respectively, and $2 . 3 4 \%$ over HAT on 5-Dataset. Surprisingly, we could even achieve better accuracy than Multitask on 5-Datasets, which usually serves as an upper bound for CL benchmarks. This superior performance of TRGP clearly shows its capability to effectively facilitate forward knowledge transfer. In addition, TRGP also demonstrates strong performance with the lowest BWT, reducing $0 . 2 \%$ than OWM and $0 . 6 \%$ than GPM, even with $5 . { \bar { 6 } } 3 { \bar { \% } }$ and $2 . 3 4 \%$ ac
+
+![](images/38a46f72a5a1d085ef91fddb2f57bd4f0ac7ab4a1b7e36fad54b1b0963cb61a2.jpg)  
+Figure 4: The final accuracy for all tasks on three datasets (GPM VS Ours).
+
+Table 2: The performance for CIFAR-100 Sup dataset. Note that Single-task learning (STL) trains a separate network for each task, which does not adhere to CL setup.
+
+<table><tr><td rowspan="2">Metric</td><td colspan="7">Methods</td></tr><tr><td>STL</td><td>PNN</td><td>DEN</td><td>RCL</td><td>APD</td><td>GPM</td><td>Ours (TRGP)</td></tr><tr><td>ACC(%)</td><td>61.00</td><td>50.76</td><td>51.10</td><td>51.99</td><td>56.81</td><td>57.72</td><td>58.25</td></tr><tr><td>Capacity(%)</td><td>2000</td><td>271</td><td>191</td><td>184</td><td>130</td><td>100</td><td>100</td></tr></table>
+
+curacy improvement on PMNIST and 5-Dataset, respectively. Compared with HAT on CIFAR-100 Split, TRGP has marginally worse BWT, but achieves $2 . 4 \%$ accuracy gain.
+
+Moreover, TRGP exhibits an universal dominance over GPM about the final accuracy of all tasks on all the three datasets. According to the apple to apple comparison with GPM in Fig. 4, one interesting phenomenon is observed: TRGP has the similar accuracy on “easy” tasks, but significantly improves the accuracy on the “difficult” tasks. For example, in the 5-Dataset setting, both TRGP and GPM achieve good accuracy on Task 1 (MNIST) and 3 (Fashion MNIST), which can be easily trained well, but TRGP significantly outperforms GPM on the rest three more difficult Tasks (CIFAR-10, SVHN and NotMNIST). In the end, as shown in Table 2, we further compare with the expansionbased methods by using CIFAR-100 Sup setting. It can be seen that TRGP outperforms all other CL methods, with a fixed capacity network.
+
+Discussion. We next show the accuracy evolution of specific tasks during the training of all tasks sequentially. We randomly select three tasks for each dataset to compare with GPM (we only show results on PMNIST in Fig. 5 and relegate the rest to the appendix). There are two main obervations: 1) TRGP completely outperforms GPM during training for all the sequential tasks on the three datasets; 2) For the PMNIST and 5-Dataset settings, TRGP could significantly reduce forgetting. To understand why, consider the case where GPM and TRGP learns a new task $t$ given the same model $\mathbb { W } _ { t - 1 }$ , and denote $\{ M _ { t - 1 } ^ { l } \} _ { l }$ as the bases of all old tasks. Then we can have
+
+For GPM, the effective weight for layer $l$ is
+
+$$
+\begin{array} { r } { \boldsymbol { W } _ { e f f } ^ { l } = \operatorname { P r o j } _ { M _ { t - 1 } ^ { l } } ( \boldsymbol { W } ^ { l } ) + \operatorname { P r o j } _ { \perp M _ { t - 1 } ^ { l } } ( \boldsymbol { W } ^ { l } ) } \end{array}
+$$
+
+where the weight projection on $M _ { t - 1 } ^ { l }$ is frozen to protect old tasks, and only the weight projection orthogonal to $M _ { t - 1 } ^ { l }$ can be updated for learning task $t$ .
+
+For TRGP, the effective weight for layer $l$ is
+
+$$
+W _ { e f f } ^ { l } = \mathrm { P r o j } _ { \{ S _ { j } ^ { l } \} _ { j \notin T \mathcal { R } _ { t } ^ { l } } } ( W ^ { l } ) + \mathrm { P r o j } _ { \{ S _ { j } ^ { l } \} _ { j \in \mathcal { T } \mathcal { R } _ { t } ^ { l } } } ^ { Q } ( W ^ { l } ) + \mathrm { P r o j } _ { \bot M _ { t - 1 } ^ { l } } ( W ^ { l } )
+$$
+
+where only the first term, i.e., weight projection on subspaces of old tasks that are not in the trust region $\mathcal T \dot { \mathcal R } _ { t } ^ { l }$ , is frozen for task $t$ . In contrast to GPM, an additional and also important part of weights, i.e., the scaled weight projection on subspaces of related old tasks in $\tau { \mathcal R } _ { t } ^ { l }$ , can be learnt in a favorable way for task $t$ . As a result, TRGP can achieve better forward knowledge transfer by explicitly and cleverly reusing the important knowledge of strongly correlated old tasks in the trust region. More interestingly, benefiting from the task-unique information captured by the scaled weight projection, the backward transfer can also be reduced.
+
+![](images/3e006d6af9462e650018395f142ea55f82323ae87859391307e5ec3c69a6411b.jpg)  
+Figure 5: Accuracy evolution for different tasks on PMNIST setting.
+
+Table 3: Ablation study on CIFAR-100 Split and 5-Datasets settings.   
+
+<table><tr><td rowspan="2">Datasets</td><td colspan="3">Impact of threshold εl</td><td colspan="2">Layer-wise VS Task-wise</td><td colspan="2">Number of selected tasks</td></tr><tr><td>0.2</td><td>0.5</td><td>0.7</td><td>Layer-wise</td><td>Task-wise</td><td>Top-1</td><td>Top-2</td></tr><tr><td>CIFAR-100</td><td>74.52</td><td>74.46</td><td>74.30</td><td>74.46</td><td>73.25</td><td>74.00</td><td>74.46</td></tr><tr><td>5-Datasets</td><td>93.28</td><td>93.56</td><td>93.43</td><td>93.56</td><td>92.85</td><td>92.94</td><td>93.56</td></tr></table>
+
+# 5.3 ABLATION STUDY AND ANALYSIS
+
+Impact of the threshold $\epsilon ^ { l }$ . To understand the impact of the threshold $\epsilon ^ { l }$ , we evaluate the learning performance for three different values of $\epsilon ^ { l }$ (i.e., 0.2, 0.5, 0.7) as shown in Table 3. The results show that the accuracy is very stable across the three threshold values, with ignoble accuracy difference on both CIFAR-100 Split and 5-Dataset settings. The reason behind is because we only select top-2 old tasks with largest gradient projection norm into the trust region, among all tasks satisfying condition Eq. (3). Therefore, for a wide range of $\epsilon ^ { l }$ , the selected tasks in the trust region are actually fixed. The small accuracy fluctuation is because with some possibility only one old task satisfies Eq. (3) and is selected for some layers when $\epsilon ^ { l }$ increases. Overall, TRGP is very robust to the value of $\stackrel { \cdot } { \epsilon } { }$ .
+
+Layer-wise vs. Task-wise trust region. To show the efficacy of layer-wise trust region, we compare it with the task-wise variant which shares a fixed trust region across all layers for each task. First, as shown in Table 3, layer-wise could achieve $1 . 2 1 \%$ accuracy gain over task-wise on CIFAR-100 Split. Furthermore, we illustrate the final accuracy of all tasks for layer-wise and taskwise of the proposed TRGP, and GPM on CIFAR-100 Split setting in Fig. 6. First, the performance of layer-wise is better than or comparable to task-wise for all tasks, because layer-wise provides a much finer characterization of task correlations in terms of layer-level features. Then, it is interesting to see that the learning behavior for the three cases follows the same trend. This observation further corroborates that TRGP can improve the accuracy and mitigate forgetting on both “easy” and “difficult” tasks.
+
+![](images/f16c86a00f9663dd1bd4cac8d1ae2190095ffa83e51128e4d1bcb49ed33c2743.jpg)  
+Figure 6: The final accuracy for all tasks of Task-wise VS Layerwise on CIFAR-100 Split.
+
+Impact of selected tasks in trust region. We first evaluate the accuracy of Top-1 and Top-2 selected tasks as shown in Table 3. It shows that selecting the top-2 most correlated tasks could achieve better accuracy on both CIFAR-100 Split and 5-Dataset settings. Note that good performance can also be achieved even with the Top-1 case. Moreover, we illustrate the detailed task selection in the trust region for both layer-wise and task-wise on 5-Dataset setting in Fig. 7. For the task-wise, current task always selects the two adjacent previous tasks for all layers. Differently, the task selection varies for layer-wise, leading to more accurate selection of related tasks for each layer. For example, the layer wise trust region for Task 4 (Fashion MNIST) selects Task 1 (MNIST) or Task 3 (not-MNIST) as the most related tasks almost for all layers, over Task 0 (CIFAR-10) and Task 2 (SVHN), which clearly makes sense because Fashion MNIST shares more common features with MNIST and not-MNIST.
+
+![](images/b2a068f7b21de7eac8690723b50bb2cba41145ca7df3934d648e3ce3f5772945.jpg)  
+Figure 7: The detailed selected tasks on 5-Datasets setting.
+
+# 6 CONCLUSION
+
+In this work, we propose trust region gradient projection for continual learning to facilitate forward knowledge transfer with forgetting, based on an efficient characterization of task correlation. Particularly, our approach is built on two key blocks, i.e., the layer-wise trust region which effectively select the old tasks strongly correlated to the new task in a single-shot manner, and scaled weight projection which cleverly reuses the frozen weights of old tasks in the trust region without modifying the model. Extensive experiments show that our approach significantly improves over the related state-of-the-art methods.
+
+# ACKNOWLEDGEMENT
+
+This work is supported in part by NSF Grants CNS-2003081, CNS-2203239, CPS-1739344, and CCSS-2121222.
+
+# REPRODUCIBILITY STATEMENT
+
+For the experimental results presented in the main text, we include the code in the supplemental material, and specify all the training details in Section 5.1 and Appendix A. For the datasets used in the main text, we also give a clear explanation in Section 5.1.
+
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+
+# A EXPERIMENT SETUPS
+
+Training hyper-parameters. We evaluate our method on multiple datasets against state-of-the-art continual learning methods. 1) PMNIST. We use a 3-layer fully-connected network. with two hidden layer of 100 units. and train the network for 5 epochs with batch size of 10 for each task. 2) CIFAR-100 Split. CIFAR-100 (Krizhevsky et al., 2009) consists of images from 100 generic object classes. We use a version of 5-layer AlexNet and train each task for maximum of 200 epochs with the early termination strategy based on the validation loss value. The batch size is set to 64. 3) CIFAR100 Sup. We use a modified version of LeNet-5 with 20-50-800-500 neurons and train 50 epochs for each task sequentially. The batch size is set to 64. 4) 5-Datasets. We train each task for maximum of 200 epochs with the early termination strategy. The batch size is set to 64. 5) MiniImageNet Split. Following GPM (Saha et al., 2021), we use the reduced ResNet18 architecture, where the covolution with stride 2 in the first layer. We train each task for maximum of 100 epochs with the early termination strategy with 0.1 initial learning rate and 64 batchsize. In addition, for all the experiments, the threshold $\bar { \epsilon } ^ { \bar { l } }$ is set to 0.5, and we select top-2 tasks that satisfy condition Eq. (3). We use the same threshold $\epsilon _ { t h } ^ { l }$ as GPM (Saha et al., 2021) for subspace construction. We initialize the scaling matrix with the identity matrix and train all models with plain stochastic gradient descent.
+
+# B MORE EXPERIMENTAL RESULTS
+
+# B.1 ACCURACY EVOLUTION
+
+![](images/fbcae83be6b2509ecc631f51315a5a7696586abea3af918b9b37dec9e6b00202.jpg)  
+Figure 8: Accuracy evolution for different tasks on CIFAR-100 Split and 5-Datasets settings.
+
+# B.2 STANDARD DEVIATION
+
+We have summarized the results on the standard deviation for the averaged accuracy and backward transfer over 5 different runs on all datasets in Table 4.
+
+Table 4: The averaged accuracy (ACC) and backward transfer (BWT) with the standard deviation values over 5 different runs on different datasets.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">PMNIST</td><td colspan="2">CIFAR-100 Split</td><td colspan="2">5-Dataset</td><td colspan="2">MiniImageNet</td></tr><tr><td>ACC(%)</td><td>BWT(%)</td><td>ACC(%)</td><td>BWT(%)</td><td>ACC(%)</td><td>BWT(%)</td><td>ACC(%)</td><td>BWT(%)</td></tr><tr><td>Multitask</td><td>96.70 ± 0.02</td><td></td><td>79.58 ± 0.54</td><td>-</td><td>91.54 ± 0.28</td><td>-</td><td>69.46 ± 0.62</td><td>-</td></tr><tr><td>OWM</td><td>90.71 ± 0.11</td><td>-1±0</td><td>50.94 ± 0.60</td><td>-30 ±1</td><td></td><td></td><td></td><td></td></tr><tr><td>EWC</td><td>89.97 ± 0.57</td><td>-4±1</td><td>68.80 ±0.88</td><td>-2±1</td><td>88.64 ± 0.26</td><td>-4±1</td><td>52.01 ± 2.53</td><td>-12±3</td></tr><tr><td>HAT</td><td></td><td></td><td>72.06 ± 0.50</td><td>0±0</td><td>91.32 ± 0.18</td><td>-1±0</td><td>59.78 ±0.57</td><td>-3±0</td></tr><tr><td>A-GEM</td><td>83.56 ± 0.16</td><td>−14 ± 1</td><td>63.98 ± 1.22</td><td>−15 ± 2</td><td>84.04 ± 0.33</td><td>−12 ± 1</td><td>57.24 ±0.72</td><td>−12 ± 1</td></tr><tr><td>ER_Res</td><td>87.24± 0.53</td><td>−11 ± 1</td><td>71.73 ± 0.63</td><td>-6±1</td><td>88.31 ± 0.22</td><td>-4±0</td><td>58.94 ± 0.85</td><td>-7±1</td></tr><tr><td>GPM</td><td>93.91 ± 0.16</td><td>-3±0</td><td>72.48 ± 0.40</td><td>-0.9±0</td><td>91.22 ± 0.20</td><td>-1±0</td><td>60.41 ± 0.61</td><td>-0.7 ± 0.4</td></tr><tr><td>Ours (TRGP)</td><td>96.34 ± 0.11</td><td>-0.8 ± 0.1</td><td>74.46 ± 0.32</td><td>-0.9 ± 0.01</td><td>93.56 ± 0.10</td><td>-0.04 ± 0.01</td><td>61.78 ± 0.60</td><td>-0.5± 0.6</td></tr></table>
+
+# B.3 FORWARD TRANSFER
+
+To evaluate the forward transfer, we follow the metric used in (Veniat et al., 2020) and consider the accuracy of the model on $i$ -th task after learning the $i$ -th task sequentially, i.e., $A _ { i , i }$ as defined in Eq. (10). Tables 5 - 8 summarize the comparison of $A _ { i , i }$ for each task $i$ between GPM and TRGP on PMNIST, CIFAR-100 Split and 5-Dataset, respectively. As the same baseline (e.g., the accuracy of the model learnt from scratch using the task’s own data) for each task will be used when evaluating the forward transfer for GPM and TRGP, we can infer that TRGP achieves the forward transfer gain of $0 . 1 7 \%$ , $2 . 0 1 \%$ , $2 . 0 0 \%$ and $2 . 3 6 \%$ over GPM on PMNIST, CIFAR-100 Split, 5-Datasets and MiniImageNet respectively.
+
+Table 5: The accuracy $A _ { i , i }$ of the model on $i$ -th task after learning the $i$ -th task sequentially on PMNIST 10 tasks.   
+
+<table><tr><td>Methods</td><td></td><td></td><td>3</td><td></td><td></td><td></td><td></td><td></td><td>9</td><td>10</td><td>Avg</td></tr><tr><td>GPM</td><td>97.597.5</td><td></td><td>97.3</td><td>97.1</td><td>97.096.9</td><td></td><td>96.8</td><td>96.4</td><td>96.5</td><td>96.5</td><td>96.95</td></tr><tr><td>Ours (TRGP)</td><td>97.5</td><td>97.5</td><td>97.5</td><td>97.3</td><td>97.1</td><td>97.1</td><td>96.9</td><td>96.7</td><td>96.9</td><td>96.7</td><td>97.12</td></tr></table>
+
+Table 6: The accuracy $A _ { i , i }$ of the model on $i$ -th task after learning the $i$ -th task sequentially on CIFAR-100 Split 10 tasks.   
+
+<table><tr><td>Methods</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>10</td><td>Avg</td></tr><tr><td>GPM</td><td>76.8</td><td>68.5</td><td>72.4</td><td></td><td>69.974.8</td><td>172.3</td><td>70.3</td><td>71.9</td><td>73.2</td><td>75.1</td><td>72.52</td></tr><tr><td>Ours (TRGP)</td><td>76.9</td><td>69.5</td><td>75.1</td><td>74.1</td><td>75.3</td><td>75.8</td><td>72.8</td><td>73.8</td><td>73.9</td><td>78.1</td><td>74.53</td></tr></table>
+
+Table 7: The accuracy $A _ { i , i }$ of the model on $i$ -th task after learning the $i$ -th task sequentially on 5-Dataset 5 tasks.   
+
+<table><tr><td>Methods</td><td></td><td></td><td></td><td></td><td></td><td>Avg</td></tr><tr><td>GPM</td><td>78.399.1</td><td></td><td>87.1</td><td>99.1</td><td>94.1</td><td>91.54</td></tr><tr><td>Ours (TRGP)</td><td>80.9</td><td>99.3</td><td>92.8</td><td>99.4</td><td>95.3</td><td>93.54</td></tr></table>
+
+Table 8: The accuracy $A _ { i , i }$ of the model on $i$ -th task after learning the $i$ -th task sequentially on MiniImageNet Split 20 tasks.   
+
+<table><tr><td>Methods</td><td>1</td><td>12</td><td>3</td><td>-4</td><td>1 5</td><td>16</td><td>17</td><td>1 8</td><td>19</td><td></td><td>10</td><td>11</td><td>12</td><td>13</td><td>14</td><td>15 1</td><td>16</td><td>17</td><td>18</td><td>19</td><td>20</td><td>Avg</td></tr><tr><td>GPM</td><td>58.6</td><td>1 63.6</td><td>57.2</td><td>59.0</td><td>1 53.6</td><td>1 78.0</td><td>一 63.0</td><td></td><td>66.0</td><td>74.0</td><td>83.8</td><td>43.0</td><td>60.4</td><td>55.6</td><td>57.8</td><td>59.6 1</td><td>53.0</td><td>56.0</td><td>47.6</td><td>66.0</td><td>56.8</td><td>60.63</td></tr><tr><td>Ours (TRGP)</td><td>58.7</td><td>66.1</td><td>59.2</td><td>59.3</td><td>1 57.1</td><td>81.4</td><td></td><td>67.3</td><td>70.1</td><td>75.7</td><td>85.2</td><td>43.2</td><td>61.8</td><td>58.0</td><td>60.1</td><td>60.0</td><td>54.8</td><td>61.4</td><td>48.4</td><td>69.8</td><td>62.2</td><td>62.99</td></tr></table>
+
+# B.4 COMPUTATIONAL COMPLEXITY
+
+Memory: In terms of the memory, the major difference between TRGP and GPM is that TRGP requires additional memory to store the scaling matrices for each task. However, since the dimension of the scaling matrix is the same with the number of the extracted bases for the input subspace, which is usually small and controllable by the matrix approximation accuracy $\epsilon _ { t h }$ in Eq. (7), the memory increase is marginal and controllable. Particularly, the memory usage of TRGP can be further reduced by only learning the scaling matrices for the convolutional layers.
+
+Training time: We compare the training time between TRGP and other baselines on relatively complex task sequences. As shown in Table 9, for CIFAR-100 Split, TRGP takes around $65 \%$ more time than GPM, is comparable with HAT and ER Res, and takes less time than OWM and EWC; for 5-Datasets, TRGP takes around $21 \%$ more time than GPM, but is much faster than other baselines including EWC, HAT, A-GEM and ER Res; for MiniImageNet, TRGP tasks around $34 \%$ more time than GPM, is comparable with EWC, but is much faster than A-GEM.
+
+Table 9: Training time comparison on CIFAR-100 Split, 5-Datasets and MiniImageNet. Here the training time is normalized with respect to the value of GPM. Please refer (Saha et al., 2021) for more specific time.   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="7">Methods</td></tr><tr><td>OWM</td><td>EWC</td><td>HAT</td><td>A-GEM</td><td>ER_Res</td><td>GPM</td><td>Ours (TRGP)</td></tr><tr><td>CIFAR-100</td><td>2.41</td><td>1.76</td><td>1.62</td><td>3.48</td><td>1.49</td><td>1</td><td>1.65</td></tr><tr><td>5-Datasets</td><td>1</td><td>1.52</td><td>1.47</td><td>2.41</td><td>1.40</td><td>1</td><td>1.21</td></tr><tr><td>MiniImageNet</td><td>1</td><td>1.22</td><td>0.91</td><td>1.79</td><td>0.82</td><td>1</td><td>1.34</td></tr></table>
+
+# B.5 ACCURACY VS LEARNING EPOCHS
+
+The learning dynamics for each task are shown in Figure 9 and 10. Clearly, our approach can perform significantly better than GPM on some tasks, especially for the tasks in the tail of the task sequence. This is because in GPM, with more tasks being learnt, the optimization space for new tasks becomes more restrictive, leading to limited performance for new tasks. Note that the y-axis is the validation accuracy with a split validate dataset that used during training, by following the setup in (Saha et al., 2021). The validation accuracy varies because the size of the validate dataset is relatively small (See (Saha et al., 2021) for the specific size). For the testing accuracy in all the tables, we evaluate the accuracy with the testing dataset after training.
+
+![](images/f15dd6288a440b9b807040038c4a5c80a280b5bc43604991da4ba425b817d190.jpg)  
+Figure 9: Accuracy vs learning epochs for different tasks on CIFAR-100 Split.
+
+![](images/5bf3d623f1b7ef74a66b17097d2e8bf3045dc7fec240f1994a9bae8120336d31.jpg)  
+Figure 10: Accuracy vs learning epochs for five tasks on 5-Dataset.

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+# Multi-Objective Online Learning
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 This paper presents a systematic study of multi-objective online learning. We first   
+2 formulate the framework of Multi-Objective Online Convex Optimization, which   
+3 encompasses two novel multi-objective regret definitions. The regret definitions   
+4 build upon an equivalent transformation of the multi-objective dynamic regret   
+5 based on the commonly used Pareto suboptimality gap metric in zero-order multi  
+6 objective bandits, making it amenable to be optimized via first-order iterative   
+7 methods. To motivate the algorithm design, we give an explicit example in which   
+8 equipping OMD with the vanilla min-norm solver for gradient composition will   
+9 incur a linear regret, which shows that only regularizing the iterates, as in single  
+10 objective online learning, is not enough to guarantee sublinear regrets in the multi  
+11 objective setting. To resolve this issue, we propose a novel min-regularized-norm   
+12 solver that regularizes the composite weights. Combining min-regularized-norm   
+13 with OMD results in the Doubly Regularized Online Mirror Multiple Descent   
+14 algorithm. We further derive both the static and dynamic regret bounds for the   
+15 proposed algorithm, each of which matches the corresponding optimal bound in the   
+16 single-objective setting. Extensive experiments on both simulation and real-world   
+17 datasets verify the effectiveness of the proposed algorithm.
+
+# 18 1 Introduction
+
+19 Traditional optimization methods for machine learning are usually designed to optimize a single   
+20 objective. However, in many real-world applications, we are often required to optimize multiple   
+21 correlated objectives concurrently. For example, in autonomous driving [12, 20], the self-driving   
+22 vehicles need to solve multiple tasks such as self-localization and object identification at the same   
+23 time. In online advertising [21, 22], advertisers need to determine the exposure of items to different   
+24 users to maximize both the Click-Through Rate (CTR) and the Post-Click Conversion Rate (CVR).   
+25 In many multi-objective scenarios, the objectives may conflict with each other [15]. Hence, there may   
+26 not exist any single solution that optimizes all the objectives simultaneously. For example, in online   
+27 advertising, merely optimizing CTR or CVR will degrade the performance of the other [21, 22].   
+28 Multi-objective optimization (MOO) [23, 6] is concerned with optimizing multiple conflicting   
+29 objectives simultaneously. It seeks Pareto optimality, where no single objective can be improved   
+30 without hurting the performance of the others. Many different methods for MOO have been proposed,   
+31 including evolutionary methods [26, 39], scalarization methods [9], and gradient-based iterative   
+32 methods [7]. Recently, the Multiple Gradient Descent Algorithm (MGDA) and its variants have been   
+33 introduced to the training of multi-task deep neural networks and achieved great empirical success   
+34 [29], making them regain a significant amount of research interest [17, 33, 18]. These methods   
+35 compute a composite gradient based on the gradient information of all the individual objectives   
+36 and then apply the composite gradient to update the model parameters. The composite weights are   
+37 determined by a min-norm solver [7] which yields a common descent direction of all the objectives.   
+38 However, compared to the increasingly wide application prospect, the gradient-based iterative   
+39 algorithms are relatively understudied, especially for the online learning setting. Multi-objective   
+40 online learning is of essential importance due to reasons in two folds. First, due to the data explosion in   
+41 many real-world scenarios such as web applications, making in-time predictions requires performing   
+42 online learning. Second, the theoretical investigation of multi-objective online learning will lay a solid   
+43 foundation for the design of new optimizers for multi-task deep neural networks. This is analogous to   
+44 the single-objective setting, where nearly all the optimizers for training DNNs are initially analyzed   
+45 in the online setting, such as AdaGrad [8], Adam [16], and AMSGrad [28].   
+46 In this paper, we give a systematic study of multi-objective online learning. To begin with, we   
+47 formulate the framework of Multi-Objective Online Convex Optimization (MO-OCO). The first   
+48 major challenge is the lack of regret definitions in the multi-objective setting. To tackle this challenge,   
+49 we need appropriate discrepancy metrics that can be used in the regret definitions, which evaluate the   
+50 gap between any two vector losses by producing scalar values. Intuitively, the Pareto suboptimality   
+51 gap (PSG) metric, which is frequently used in zero-order multi-objective bandits [30, 19], is a very   
+52 promising candidate. It can yield scalarized distances from any vector loss to a given comparator set.   
+53 We can thus define the multi-objective regret by simply plugging in PSG as the discrepancy metric.   
+54 However, as a metric designed purely from the geometric view, PSG is intrinsically difficult to be   
+55 optimized directly via gradient-based iterative methods. To resolve this problem, for the PSG-based   
+56 multi-objective dynamic regret, we derive its equivalent unconstrained max-min form via a highly   
+57 non-trivial transformation. This form is intuitive to the design of first-order multi-objective online   
+58 algorithms, indicating that we should select a convex combination of the gradients at each round.   
+59 Unfortunately, for the PSG-based static variant, such an equivalence does not exist. To remedy this   
+60 issue, we make extensions of the dynamic variant by fixing the comparator set and the composite   
+61 weights, which yields an appropriate definition of the multi-objective static regret.   
+62 Based on the MO-OCO framework, we develop a novel multi-objective online algorithm termed   
+63 Doubly Regularized Online Mirror Multiple Descent. The key module of the algorithm is the gradient   
+64 composition scheme, which calculates a composite gradient in the form of a convex combination of   
+65 the gradients of all objectives. Intuitively, the most direct way to determine the composite weights is   
+66 to apply the min-norm solver [7] commonly used in offline multi-objective optimization. However,   
+67 directly applying min-norm is not workable in the online setting. Specifically, the composite weights   
+68 in min-norm are only determined by the gradients at the current round. In the online setting, since   
+69 the gradients can be adversarial, they may result in undesired composite weights, further producing   
+70 a composite gradient that reversely optimizes the loss. To rigorously verify this point, we give a   
+71 showcase in which equipping OMD with vanilla min-norm even incurs a linear regret, showing that   
+72 only regularizing the iterate, as in OMD, is not enough to guarantee sublinear regrets in the multi  
+73 objective setting. To fix this issue, we devise a novel min-regularized-norm solver with an explicit   
+74 regularization on composite weights. Equipping it with OMD results in our proposed algorithm.   
+75 We then conduct the theoretical analysis for our proposed algorithm. We derive a multi-objective static   
+76 regret bound $O ( \sqrt { T } )$ and a multi-objective dynamic regret bound $O ( V _ { T } ^ { 1 / 3 } T ^ { 2 / 3 } )$ for DR-OMMD.   
+77 Both bounds match the optimal bounds in the single-objective setting [11, 34]. Our analysis also   
+78 shows that DR-OMMD attains a lower regret than linearization with fixed composite weights.   
+79 To evaluate the effectiveness of DR-OMMD, we conduct extensive experiments on both simulation   
+80 datasets and real-world datasets. We first elaborate simulation experiments, in which we find   
+81 that DR-OMMD attains lower regret than vanilla min-norm and linearization, which verifies the   
+82 superiority of the min-regularized-norm solver. We then realize adaptive regularization via multi  
+83 objective optimization on real-world datasets, and find that adaptive regularization with DR-OMMD   
+84 significantly outperforms fixed regularization with linearization.   
+85 In summary, in this paper, we give the first systematic study of multi-objective online learning, which   
+86 encompasses a novel framework, a new algorithm, and corresponding non-trivial theoretical analysis.   
+87 We believe that this work paves the way for future research on more advanced multiple-objective   
+88 optimization algorithms, which may inspire the design of new optimizers for multi-task deep learning.
+
+# 89 2 Preliminaries
+
+90 In this section, we briefly review the necessary background knowledge of online convex optimization   
+91 and multi-objective optimization.   
+93 Online Convex Optimization (OCO) [38, 11] is the most commonly adopted framework for   
+94 designing online learning algorithms. It can be viewed as a structured repeated game between a   
+95 learner and an adversary. At each round $t \in \{ 1 , \ldots , T \}$ , the learner is required to generate a decision   
+96 $x _ { t }$ from a convex compact set $\mathcal { X } \subset \mathbb { R } ^ { n }$ . Then the adversary replies the learner with a convex function   
+97 $f _ { t } : \mathcal { X }  \mathbb { R }$ and the learner suffers the loss $f _ { t } ( x _ { t } )$ . The goal of the learner is to minimize the regret   
+98 with respect to the best fixed decision in hindsight, i.e.,
+
+$$
+R _ { S } ( T ) = \sum _ { t = 1 } ^ { T } f _ { t } ( x _ { t } ) - \operatorname* { m i n } _ { x ^ { * } \in \mathcal { X } } \sum _ { t = 1 } ^ { T } f _ { t } ( x ^ { * } ) .
+$$
+
+99 Note that the above regret is the static regret [10], which compares the learner’s cumulative loss   
+100 with that of a fixed decision. There is another version of regret, namely the dynamic regret [10, 34],   
+101 which compares the learner’s cumulative loss with that of a sequence of local optimal decisions, i.e.,
+
+$$
+R _ { D } ( T ) = \sum _ { t = 1 } ^ { T } f _ { t } ( x _ { t } ) - \sum _ { t = 1 } ^ { T } \operatorname* { m i n } _ { x _ { t } ^ { * } \in \mathcal { X } } f _ { t } ( x _ { t } ^ { * } ) .
+$$
+
+102 Any meaningful regret is required to be sublinear in $T$ , i.e., $\begin{array} { r } { \operatorname* { l i m } _ { T \to \infty } R _ { S / D } ( T ) / T = 0 } \end{array}$ , which implies   
+103 that when $T$ is large enough, the learner can perform as well as the best fixed decision in hindsight   
+104 (for static regret) or the local optimal decision at each round (for dynamic regret).   
+105 Online Mirror Descent (OMD) [11] is a classic first-order online learning algorithm. At each round   
+106 $t \in \{ 1 , \ldots , T \}$ , OMD yields its decision using the following formula
+
+$$
+\begin{array} { r } { x _ { t + 1 } = \underset { x \in \mathcal { X } } { \arg \operatorname* { m i n } } \eta \langle \nabla f _ { t } ( x _ { t } ) , x \rangle + B _ { R } ( x , x _ { t } ) , } \end{array}
+$$
+
+107 where $\eta$ is the step size, $R : \mathcal { X }  \mathbb { R }$ is the regularization function, and $B _ { R } ( x , x ^ { \prime } ) = R ( x ) - R ( x ^ { \prime } ) -$   
+108 $\langle \nabla R ( x ^ { \prime } ) , x - x ^ { \prime } \rangle$ is the Bregman divergence induced from $R$ . As a meta-algorithm, by instantiating   
+109 different regularization functions, OMD can induce two important algorithms, i.e., Online Gradient   
+110 Descent [38, 13] and Online Exponentiated Gradient [11].
+
+# 11 2.2 Multi-Objective Optimization
+
+112 Multiple-objective optimization (MOO) is concerned with solving the problems of optimizing   
+113 multiple objectives simultaneously [39, 29]. In general, since different objectives may conflict with   
+114 each other, there is no single solution that can optimize all the objectives at the same time. Instead,   
+115 MOO seeks to find solutions that achieve Pareto optimality. Next, we exposit Pareto optimality and   
+116 related definitions more formally using a vector-valued loss $H = ( h ^ { 1 } , \ldots , \overline { { { h ^ { m } } } } ) ^ { \top }$ as objectives, where   
+117 $m \geq 2$ and $h ^ { i } : { \mathcal { K } }  \mathbb { R }$ , $i \in \{ 1 , \ldots , m \}$ , $\kappa \subset \mathbb { R }$ , is the $i$ -th loss function.
+
+Definition 2.1 (Pareto optimality). (a) For any two solutions $x , x ^ { \prime } \in \mathcal { K }$ , we say that $x$ dominates $x ^ { \prime }$ , denoted as $\boldsymbol { x } \prec \boldsymbol { x } ^ { \prime }$ or $x ^ { \prime } \succ x$ , if $h ^ { i } ( x ) \leq h ^ { i } ( x ^ { \prime } )$ for all $i$ , and there exists one $i$ such that $h ^ { i } ( x ) < h ^ { i } ( x ^ { \prime } )$ ; otherwise, we say that $x$ does not dominate $x ^ { \prime }$ , denoted as $x \not \prec x ^ { \prime }$ or $x ^ { \prime } \nsimeq x$ .
+
+(b) A solution $x ^ { * } \in \kappa$ is called Pareto optimal if it is not dominated by any other solution in $\kappa$ .
+
+122 There may exist multiple Pareto optimal solutions. For example, it is easy to show that the optimizer   
+123 of any single objective, i.e., $x _ { i } ^ { * } \in \arg \operatorname* { m i n } _ { x \in \mathcal { K } } h ^ { i } ( x ) , i \in \{ \bar { 1 } , \ldots , m \}$ , is Pareto optimal. Different   
+124 Pareto optimal solutions reflect different trade-offs among the objectives [17].   
+25 Definition 2.2 (Pareto front). (a) All Pareto optimal solutions form the Pareto set ${ \mathcal { P } } _ { \kappa } ( H )$ .   
+26 (b) The image of ${ \mathcal { P } } _ { \kappa } ( H )$ constitutes the Pareto front, denoted as $\mathcal { P } ( H ) = \{ H ( x ) \mid x \in \mathcal { P } _ { K } ( H ) \} .$   
+127 Now that we have established the notion of optimality in MOO, we proceed to introduce the metrics   
+128 that measure the discrepancy of an arbitrary solution $x \in \kappa$ from being optimal. Recall that, in the   
+129 single-objective setting with merely one loss function $h : \mathcal { Q }  \mathbb { R }$ , where $\mathcal { Q } \subset \mathbb { R }$ , for any $z \in \mathcal { Q }$ ,   
+130 the loss gap $h ( z ) - \mathrm { { m i n } } _ { z ^ { \prime \prime } \in \mathcal { Q } } h ( z ^ { \prime \prime } )$ is directly the discrepancy measure. However, in MOO with   
+131 more than one loss, for any $x \in \kappa$ , the loss gap $H ( x ) - \overline { { H ( x ^ { \prime \prime } ) } }$ , where $x ^ { \prime \prime } \in { \mathcal { P } } _ { \kappa } ( H )$ , is a vector.   
+132 Intuitionally, the desired discrepancy metric shall scalarize the vector-valued loss gap and yield   
+133 the value 0 for any Pareto optimal solution. In general, there are two commonly used discrepancy   
+134 metrics in MOO, i.e. Pareto suboptimality gap (PSG) [30] and Hypervolume (HV) [4]. As HV is a   
+135 volume-based metric, it is more difficult to optimize or analyze via iterative algorithms [36]. Hence   
+136 in this paper, we adopt PSG, which has been extensively used in multi-objective bandits [30, 19].   
+137 Definition 2.3 (Pareto suboptimality gap). For any $x \in \kappa$ , the Pareto suboptimality gap to a given   
+138 comparator set $\kappa ^ { * } \subset \kappa$ , denoted as $\Delta ( x ; K ^ { * } , H )$ , is defined as the minimal scalar $\epsilon \geq 0$ that needs   
+139 to be subtracted from all entries of $H ( x )$ , such that $H ( x ) - \epsilon \mathbf { 1 }$ is not dominated by any point in $\kappa ^ { * }$ ,   
+140 where 1 denotes the all-one vector in $\mathbb { R } ^ { m }$ , i.e.,1
+
+$$
+\Delta ( x ; K ^ { * } , H ) = \operatorname* { i n f } _ { \epsilon \geq 0 } \epsilon , \quad \mathrm { s . t . } \forall x ^ { \prime \prime } \in K ^ { * } , \exists i \in \{ 1 , . . . , m \} , h ^ { i } ( x ) - \epsilon < h ^ { i } ( x ^ { \prime \prime } ) .
+$$
+
+141 Clearly, PSG is a distance-based discrepancy metric that motivated from a purely geometric viewpoint.   
+142 In practice, the comparator set $\kappa ^ { * }$ is often set to be the Pareto set ${ \mathcal { P } } _ { \kappa } ( H )$ [30]. Then for any $x \in \kappa$ ,   
+143 its PSG is always non-negative and equals to zero if and only if $x \in { \mathcal { P } } _ { \kappa } ( H )$ .   
+144 Multiple Gradient Descent Algorithm (MGDA) is an offline first-order algorithm for MOO [9, 7].   
+145 146 At each iteration for each objectiv $l \in \{ 1 , \ldots , L \}$ $i \in \{ 1 , \ldots , m \}$ $L$ is the number of iterations), it first c then derive the composite gradient $\begin{array} { r } { g _ { l } ^ { c o \bar { m } p } = \sum _ { i = 1 } ^ { \bar { m } } \lambda _ { l } ^ { i } \nabla h ^ { i } ( x _ { l } ) } \end{array}$ $\nabla h ^ { i } ( x _ { l } )$   
+147 the convex combination of these multiple gradients; it applies ${ \dot { \boldsymbol g } _ { l } } ^ { c o m p }$ to execute the gradient descent   
+148 step to update the decision, i.e., $x _ { l + 1 } = x _ { l } - \eta g _ { l } ^ { c o m p }$ gcompl , where η is the step size. The core part of   
+149 MGDA is the module that determines the composite weights $\lambda _ { l } = ( \lambda _ { l } ^ { 1 } , \ldots , \lambda _ { l } ^ { m } )$ , which is given as
+
+$$
+\lambda _ { l } = \arg \operatorname* { m i n } _ { \lambda _ { l } \in \mathcal { S } _ { m } } \| \sum _ { i = 1 } ^ { m } \lambda _ { l } ^ { i } \nabla h ^ { i } ( x _ { l } ) \| _ { 2 } ^ { 2 } ,
+$$
+
+150 where $\begin{array} { r } { \mathcal { S } _ { m } = \{ \lambda \in \mathbb { R } ^ { m } | \sum _ { i = 1 } ^ { m } \lambda ^ { i } = 1 , \lambda ^ { i } \geq 0 , i \in \{ 1 , \dots , m \} \} } \end{array}$ denotes the probabilistic simplex in   
+151 $\mathbb { R } ^ { m }$ . This is a min-norm solver which finds the weights in the simplex that yields the minimum $L _ { 2 }$   
+152 norm of the composite gradient. Thus MGDA is also called the min-norm method. Existing works   
+153 [7, 29] have shown that MGDA is guaranteed to decrease all the objectives simultaneously until it   
+154 reaches a Pareto optimal decision (under the convex setting where all $h ^ { i }$ are convex functions).
+
+# 3 Multi-Objective Online Convex Optimization
+
+In this section, we formally formulate the framework of multi-objective optimization in the online setting, termed Multi-Objective Online Convex Optimization (MO-OCO).
+
+Framework overview. We tailor the famous online convex optimization (OCO) framework to the multi-objective setting, which can be viewed as a repeated game between an online learner and the adversarial environment. At each round $t \in \{ 1 , \ldots , T \}$ , the learner generates a decision $x _ { t }$ from a given convex compact decision set $\mathcal { X } \subset \mathbb { R } ^ { n }$ . Then the adversary replies the decision with a vector loss function $F _ { t } ( \bar { x } ) : \mathcal { X }  \mathbb { R } ^ { m }$ , where its $i$ -th component $f _ { t } ^ { i } ( x ) \ \bar { : } \ x \ \to \ \mathbb { R }$ belongs to the $i$ -th objective, and the learner suffers the loss $F _ { t } ( x _ { t } ) \in \mathbb { R } ^ { m }$ . The goal of the learner is to generate a sequence of decisions $\{ x _ { t } \} _ { t = 1 } ^ { T }$ so that the cumulative loss $\textstyle \sum _ { t = 1 } ^ { T } F _ { t } ( x _ { t } )$ can be optimized.
+
+165 Recall that, in the single-objective setting, the performance metric $\begin{array} { r } { R ( T ) = \sum _ { t = 1 } ^ { T } ( f _ { t } ( x _ { t } ) - f _ { t } ( z _ { t } ) ) } \end{array}$   
+166 i.e., the regret, compares the actual decisions with some comparator $z _ { t } \in \mathcal { X }$ at each round $t$ . For   
+167 the static regret, all $z _ { t }$ are identically set as the fixed optimal decision $x ^ { * }$ w.r.t. all losses in hindsight,   
+168 i.e., $\begin{array} { r } { z _ { t } \equiv x ^ { * } \in \arg \operatorname* { m i n } _ { x \in \mathcal { X } } \sum _ { t = 1 } ^ { T } f _ { t } ( x ) } \end{array}$ . For the dynamic regret, each $z _ { t }$ is selected as the optimal   
+169 decision $\boldsymbol { x } _ { t } ^ { * }$ w.r.t. the instantaneous loss $f _ { t }$ at that round, i.e., $z _ { t } = x _ { t } ^ { * } \in \arg \operatorname* { m i n } _ { x \in \mathcal { X } } f _ { t } ( x )$ .   
+170 In analogy, we can define the multi-objective regret as $\begin{array} { r } { R ( T ) = \sum _ { t = 1 } ^ { T } \Delta _ { t } } \end{array}$ , where each $\Delta _ { t }$ compares   
+171 . However, in general, no single decision can   
+172 optimize all the objectives at the same time. Hence, it is natural to compare $x _ { t }$ with a group of Pareto   
+173 optimal decisions, which constitute a comparator set $\mathcal { C } _ { t } \subset \mathcal { X }$ . To measure the discrepancy between $x _ { t }$   
+174 and $\mathcal { C } _ { t }$ , we further introduce the Pareto suboptimality gap (PSG) [30] $\Delta ( x _ { t } ; \mathcal { C } _ { t } , F _ { t } )$ . Then the multi  
+175 objective regret can be defined as $\begin{array} { r } { R ( T ) = \sum _ { t = 1 } ^ { T } \Delta ( x _ { t } ; \mathcal { C } _ { t } , F _ { t } ) } \end{array}$ . Now we can formulate the static or   
+176 the dynamic variant by specifying the comparator set $\mathcal { C } _ { t }$ at each round. Specifically, by setting all $\mathcal { C } _ { t }$ to   
+177 be the Pareto set $\mathcal { X } ^ { \ast }$ of the cumulative loss $\textstyle \sum _ { t = 1 } ^ { T } F _ { t }$ , we formulate the multi-objective static regret   
+178 $\begin{array} { r } { R _ { \mathrm { M O S } } ( T ) = \sum _ { t = 1 } ^ { T } \Delta ( x _ { t } ; \mathcal { X } ^ { \ast } , F _ { t } ) } \end{array}$ $F _ { t }$ . By setting each bjective dynamic $\mathcal { C } _ { t }$ to bgret $\begin{array} { r } { R _ { \mathrm { M O D } } ( T ) = \sum _ { t = 1 } ^ { T } \Delta ( x _ { t } ; \mathcal { X } _ { t } ^ { \ast } , F _ { t } ) } \end{array}$ $\mathcal { X } _ { t } ^ { \ast }$ neous.   
+180 Recall that PSG is a zero-order metric motivated in a purely geometric sense, namely, its calculation   
+181 needs to solve a constrained optimization problem with an unknown boundary $f _ { t } ^ { i } ( x ^ { \prime \prime } ) , \forall x ^ { \prime \prime } \in { \mathcal { C } } _ { t }$   
+182 Hence, it is not straightforward to design a first-order algorithm to optimize PSG, not to mention   
+183 the regret analysis. To motivate algorithm design and analysis, we investigate the two variants in   
+184 more detail. We begin with the dynamic variant, since we find that it has an equivalent form, which is   
+185 intuitive and has a strong implication on the design of effective online multiple gradient algorithms.
+
+An equivalent form of the dynamic regret. Surprisingly, the multi-objective dynamic regret $R _ { \mathrm { M O D } }$ can be transformed into an unconstrained max-min form. The derivation utilizes Pareto optimality of $\mathcal { X } _ { t } ^ { \ast }$ and is highly non-trivial, which is deferred to the appendix due to the space limit.
+
+189 Proposition 3.1. The multi-objective dynamic regret has an equivalent form, i.e.,
+
+$$
+R _ { \mathrm { M O D } } ( T ) = \operatorname* { s u p } _ { \stackrel { x _ { t } ^ { * } \in \mathcal { X } _ { t } ^ { * } , \ } { 1 \leq t \leq T } } \operatorname* { i n f } _ { \stackrel { x \in S _ { m } } { 1 \leq t \leq T } } \sum _ { t = 1 } ^ { T } \lambda _ { t } ^ { * } { ^ { \top } ( F _ { t } ( x _ { t } ) - F _ { t } ( x _ { t } ^ { * } ) ) } .
+$$
+
+190 Remark. (i) The above form can be understood as a variant of the standard dynamic regret regarding   
+191 $\{ \lambda _ { t } ^ { * } ^ { \top } F _ { t } \} _ { t = 1 } ^ { T }$ , whereas $\lambda _ { t } ^ { * }$ are unknown to the learner. This provides an intuition that we can gen  
+192 erate weights $\lambda _ { t } \in \boldsymbol { S } _ { m }$ at each round and optimize $\{ \lambda _ { t } F _ { t } \} _ { t = 1 } ^ { T }$ via single-objective techniques. For   
+193 first-order algorithms, it is equivalent to selecting a convex combination of individual gradients and   
+194 then applying the composite gradient to model update. Undoubtedly, how to generate the weights $\lambda _ { t }$   
+195 needs some careful designs, which will be explicated later in the algorithm section.
+
+(ii) When $m \ = \ 1$ , we have $S _ { m } ~ = ~ \{ 1 \}$ and $\begin{array} { r } { \mathcal { X } _ { t } ^ { * } ~ = ~ \arg \operatorname* { m i n } _ { x \in \mathcal { X } } F _ { t } ( x ) } \end{array}$ . Hence $R _ { \mathrm { M O D } } ( T ) ~ =$ $\begin{array} { r } { \sum _ { t = 1 } ^ { T } ( F _ { t } ( x _ { t } ) - \operatorname* { m i n } _ { x \in \mathcal { X } } F _ { t } ( x ) ) } \end{array}$ , which is exactly the single-objective dynamic regret $R _ { D } ( T )$ .
+
+198 An alternative form of the static regret. Unfortunately, for $R _ { \mathrm { M O S } }$ , the above equivalence form   
+199 does not exist. Here is the reason. In $R _ { \mathrm { M O S } }$ , the comparator set $\mathcal { X } ^ { \ast }$ is the Pareto set of the cumulative   
+200 loss $\textstyle \sum _ { t = 1 } ^ { T } F _ { t }$ rather than the instantaneous loss $F _ { t }$ . Hence, at some specific round $t$ , the decision   
+201 $x _ { t }$ may Pareto dominate all points in w.r.t. the instantaneous $F _ { t }$ , and we would expect the   
+202 metric $\Delta _ { t }$ to be negative. However, PSG (or other commonly used metrics such as Hypervolume)   
+203 204 yields no, we have $R _ { \mathrm { M O S } }$ th , w $R _ { S }$ . For example, whenh can be much looser   
+$m = 1$ $\begin{array} { r } { R _ { \operatorname { M O S } } ( T ) = \operatorname* { s u p } _ { x ^ { * } \in \mathcal { X } ^ { * } } \sum _ { t = 1 } ^ { T } \operatorname* { m a x } \{ F _ { t } ( x _ { t } ) - F _ { t } ( x ^ { * } ) , 0 \} } \end{array}$   
+205 than the static regret $\begin{array} { r } { R _ { S } ( T ) = \operatorname* { s u p } _ { x ^ { * } \in \mathcal { X } ^ { * } } \sum _ { t = 1 } ^ { T } ( F _ { t } ( x _ { t } ) - F _ { t } ( x ^ { * } ) ) } \end{array}$ . Hence the analysis of $R _ { \mathrm { M O S } }$ is   
+206 intrinsically complex if we use existing discrepancy metrics that always yield non-negative values.   
+207 Enlightened by Proposition 3.1, we can formulate the static regret in a different way, i.e., by modifying   
+208 the equivalent form of dynamic regret. Recall that in Proposition 3.1, at each round $t$ , the comparator   
+209 $\boldsymbol { x } _ { t } ^ { * }$ is selected from the Pareto set $\mathcal { X } _ { t } ^ { \ast }$ of the instantaneous loss $F _ { t }$ , and the weights $\lambda _ { t } ^ { * }$ are generated   
+210 from $S _ { m }$ . To formulate the static variant, we can use a fixed comparator $x ^ { * }$ from the Pareto set $\mathcal { X } ^ { \ast }$ of   
+211 the cumulative loss $\sum _ { t } F _ { t }$ and fixed weights $\lambda ^ { * } \in S _ { m }$ at all rounds. Now the static variant takes
+
+$$
+R _ { \mathrm { M O S } } ( T ) : = \operatorname* { s u p } _ { x ^ { * } \in \mathcal { X } ^ { * } } \operatorname* { i n f } _ { \lambda ^ { * } \in \mathcal { S } _ { m } } { \lambda ^ { * } } ^ { \top } ( \sum _ { t = 1 } ^ { T } F _ { t } ( x _ { t } ) - \sum _ { t = 1 } ^ { T } F _ { t } ( x ^ { * } ) ) .
+$$
+
+212 Remark. (i) $R _ { \mathrm { M O S } } ( T )$ has a clear physical meaning that optimizing it will impose the cumulative loss 213 $\textstyle \sum _ { t = 1 } ^ { T } F _ { t } ( x _ { t } )$ to reach the Pareto front ${ \mathcal { P } } ^ { * }$ . See more details in Appendix C.
+
+(ii) When 214 $m = 1$ , $S _ { m } = \{ 1 \}$ and $\mathcal { X } ^ { \ast }$ reduces to $\begin{array} { r } { \arg \operatorname* { m i n } _ { x \in \mathcal { X } } \sum _ { t = 1 } ^ { T } F _ { t } ( x ) } \end{array}$ . Therein $R _ { \mathrm { M O S } } ( T ) =$ 15 $\begin{array} { r } { \sum _ { t = 1 } ^ { T } F _ { t } ( x _ { t } ) - \operatorname* { m i n } _ { x ^ { * } \in \mathcal { X } ^ { * } } \sum _ { t = 1 } ^ { T } F _ { t } ( x ^ { * } ) } \end{array}$ x∈X t=1 , which reduces to the single-objective static regret $R _ { S } ( T )$
+
+# 216 4 Online Mirror Multiple Descent
+
+In this section, we present the Online Mirror Multiple Descent (OMMD) algorithm, the protocol of which is given in Algorithm 1. At each round $t$ , the learner first computes the gradient of the loss regarding each objective, then determines the composite weights of all these gradients, and finally applies the composite gradient to the online mirror descent step.
+
+# 4.1 Vanilla Min-Norm May Incur Linear Regrets
+
+222 The core module of OMMD is the composition of multiple gradients. For simplicity, we represent the gradients at round 223 $t$ in a matrix form $\nabla F _ { t } ( x _ { t } ) = [ \nabla \mathsf { \bar { f } } _ { t } ^ { 1 } ( \bar { x } _ { t } ) , \ldots , \nabla f _ { t } ^ { m } ( x _ { t } ) ] \in \bar { \mathbb { R } } ^ { \mathsf { \bar { n } } \times m }$ . Then the
+
+1: Input: Convex set $\mathcal { X }$ , time horizon $T$ , regularization parameter $\alpha _ { t }$ , learning rate $\eta _ { t }$ , regulariza tion function $R$ , user preference $\lambda _ { 0 }$ .   
+2: Initialize: $x _ { 1 } \in \mathcal { X }$ .   
+3: for $t = 1 , \dots , T$ do   
+4: Predict $x _ { t }$ and receive a loss function $F _ { t } : \mathcal { X }  \mathbb { R } ^ { m }$ .   
+5: Compute the multiple gradients $\nabla F _ { t } ( x _ { t } ) = [ \nabla f _ { t } ^ { 1 } ( x _ { t } ) , \ldots , \nabla f _ { t } ^ { m } ( x _ { t } ) ] \in \mathbb { R } ^ { n \times m } .$ .   
+6: Determine the weights for the gradient composition via min-regularized-norm $\lambda _ { t } = \operatorname * { \bar { a r g m i n } } _ { \lambda \in { \cal S } _ { m } } \| \nabla \dot { F _ { t } } ( x _ { t } ) \lambda \| _ { 2 } ^ { 2 } + \alpha \| \lambda - \stackrel { \smile } { \lambda } _ { 0 } \| _ { 1 } .$   
+7: Compute the composite gradient $g _ { t } = \nabla F _ { t } ( x _ { t } ) \lambda _ { t }$ .   
+8: Perform online mirror descent using $g _ { t }$ $x _ { t + 1 } = \underset { x \in \mathcal { X } } { \arg \operatorname* { m i n } } \eta \langle g _ { t } , x \rangle + B _ { R } ( x , x _ { t } ) .$
+
+9: end for
+
+224 composite gradient is given as $g _ { t } = \nabla F _ { t } ( x _ { t } ) \lambda _ { t }$ , where $\lambda _ { t }$ is the composite weights. As illustrated in   
+225 Preliminary, the min-norm method in MGDA [7, 29] is a classic method to determine the composite   
+226 weights in the offline setting, which results in a common descent direction that can descend all the   
+227 losses simultaneously. Thus, it is tempting to consider applying it to the online setting.   
+228 However, directly applying the min-norm method to the online setting is not workable, which may   
+229 even incur linear regrets of the resulting algorithms. The rationale is as follows. In the vanilla   
+230 min-norm method, the composite weights $\lambda _ { t }$ are determined solely by the gradients $\nabla F _ { t } ( x _ { t } )$ at the   
+231 current round $t$ , hence they are very sensitive to the instantaneous loss $F _ { t }$ . In the online setting,   
+232 the losses at each round can be adversarially chosen, and thus the corresponding gradients can be   
+233 adversarial. These adversarial gradients may result in undesired composite weights, which may   
+234 further produce a composite gradient that even deteriorates the next prediction. In the following,   
+235 we provide a problem instance in which min-norm incurs a linear regret. We extend OMD to the   
+236 multi-objective setting, where the composite weights are directly yielded by min-norm [11].   
+237 Problem instance. We consider a two-objective problem. The decision domain is $\mathcal { X } = \{ ( u , v ) ~ |$   
+238 $\begin{array} { r } { u + v \leq \frac { 1 } { 2 } , v - u \leq \frac { 1 } { 2 } , v \geq 0 \} } \end{array}$ and the loss function at each round is
+
+$$
+F _ { t } ( x ) = \left\{ \begin{array} { l l } { ( \| x - a \| ^ { 2 } , \| x - b \| ^ { 2 } ) , ~ t = 2 k - 1 , } & { ~ k = 1 , 2 , . . . ; } \\ { ( \| x - b \| ^ { 2 } , \| x - c \| ^ { 2 } ) , ~ t = 2 k , } & { ~ k = 1 , 2 , . . . , } \end{array} \right.
+$$
+
+239 where $a = ( - 2 , - 1 ) , b = ( 0 , 1 ) , c = ( 2 , - 1 )$ . For simplicity, we first analyze the case where the   
+240 total time horizon $T$ is an even number. Then we can compute the Pareto set of the cumulative   
+241 loss $\textstyle \sum _ { t = 1 } ^ { T } F _ { t }$ , i.e., $\begin{array} { r } { \mathcal { X } ^ { * } = \{ ( u , 0 ) \mid - \frac { 1 } { 2 } \leq u \leq \frac { 1 } { 2 } \} } \end{array}$ , which locates at the $x$ -axis. For conciseness of   
+242 analysis, we instantiate OMD with L2-regularization, which results in the simple OGD algorithm   
+243 [24]. We start at an arbitrary point $x _ { 1 } = ( u _ { 1 } , v _ { 1 } ) \in \mathcal { X }$ satisfying $v _ { 1 } > 0$ . At each round $t$ , suppose   
+244 the decision $x _ { t } = ( u _ { t } , v _ { t } ) \in \mathcal { X }$ , then the gradients of each objective w.r.t. $x _ { t }$ can be calculated as
+
+$$
+g _ { t } ^ { 1 } = { \left\{ \begin{array} { l l } { ( 2 u _ { t } + 4 , ~ 2 v _ { t } + 2 ) , } & { t = 2 k - 1 ; } \\ { ( 2 u _ { t } , } & { 2 v _ { t } - 2 ) , } & { t = 2 k . } \end{array} \right. } \qquad g _ { t } ^ { 2 } = { \left\{ \begin{array} { l l } { ( 2 u _ { t } , } & { 2 v _ { t } - 2 ) , } & { t = 2 k - 1 ; } \\ { ( 2 u _ { t } - 4 , } & { 2 v _ { t } + 2 ) , } & { t = 2 k . } \end{array} \right. }
+$$
+
+245 Since $\begin{array} { r } { 0 \leq v _ { t } \leq \frac { 1 } { 2 } } \end{array}$ , we observe that the second entry of either gradient alternates between positive   
+246 and negative. By using min-norm, the composite weights $\lambda _ { t }$ can be computed as
+
+$$
+\lambda _ { t } = \left\{ { \begin{array} { l l } { ( ( 1 - u _ { t } - v _ { t } ) / 4 , } & { ( 3 + u _ { t } + v _ { t } ) / 4 ) , t = 2 k - 1 ; } \\ { ( ( 3 - u _ { t } + v _ { t } ) / 4 , } & { ( 1 + u _ { t } - v _ { t } ) / 4 ) , t = 2 k . } \end{array} } \right.
+$$
+
+247 We observe that both entries of composite weights alternative between above $\frac { 1 } { 2 }$ and below $\frac { 1 } { 2 }$ , and   
+248 $\| \lambda _ { t + 1 } - \lambda _ { t } \| _ { 1 } \geq 1$ . Recall that $\| \lambda _ { t } \| _ { 1 } = 1$ , hence the composite weights at two consecutive rounds   
+249 change radically. The resulting composite gradient takes
+
+$$
+g _ { t } ^ { c o m p } = \left\{ \begin{array} { l l } { { ( u _ { t } - v _ { t } + 1 , ~ } } & { { - u _ { t } + v _ { t } - 1 ) , t = 2 k - 1 ; } } \\ { { ( - u _ { t } - v _ { t } - 1 , } } & { { - u _ { t } - v _ { t } - 1 ) , t = 2 k . } } \end{array} \right.
+$$
+
+250 The fluctuating composite weights mix with the positive and negative second entries of gradients,   
+251 making the second entry of $g _ { t } ^ { c \bar { o } m p }$ always negative, i.e., $- u _ { t } + v _ { t } - 1 < 0$ and $- u _ { t } - v _ { t } - 1 < 0$   
+252 Hence ${ \bf { \bar { \it g } } } _ { t } ^ { c o m p }$ actually drives $x _ { t }$ away from the Pareto set $\mathcal { X } ^ { \ast }$ that coincides with the $x$ -axis. This   
+253 essentially reversely optimizes the loss, hence increases the regret. In fact, we can prove that it even   
+254 incurs a linear regret2. Due to the lack of space, we leave the proof of linear regret when $T$ is an odd   
+255 number in the appendix. The above results of the problem instance are summarized as follows.
+
+Proposition 4.1. For OMD equipped with vanilla min-norm, there exists a multi-objective online convex optimization problem, in which the resulting algorithm incurs a linear regret.
+
+258 Remark. Stability is a basic requirement to guarantee meaningful regrets in online learning [25].   
+259 In the single-objective setting, directly regularizing the iterate $x _ { t }$ (e.g., OMD) is already enough.   
+260 However, as shown in the above analysis, only regularizing $x _ { t }$ is not enough to attain sublinear regrets   
+261 in the multi-objective setting, since there is another source of instability, i.e., the composite weights,   
+262 that affects the direction of the composite gradient. Therefore, in multi-objective online learning,   
+263 besides regularizing the iterates, we also need to explicitly regularize the composite weights.
+
+# 4.2 Doubly Regularized Online Mirror Multiple Descent
+
+Enlightened by the design of regularization in FTRL [25], we consider the regularizer $r ( \lambda , \lambda _ { 0 } )$ , where $\lambda _ { 0 }$ is the pre-defined composite weight that may reflect the user preference. This results in a new solver called min-regularized-norm, i.e.,
+
+$$
+\lambda _ { t } = \underset { \lambda \in S _ { m } } { \arg \operatorname* { m i n } } \| \nabla F _ { t } ( x _ { t } ) \lambda \| _ { 2 } ^ { 2 } + \alpha r ( \lambda , \lambda _ { 0 } ) ,
+$$
+
+268 where $\alpha$ is the strength of regularization. Equipping OMD with the new solver, we derive the   
+269 proposed online algorithm. Note that beyond the regularization on the iterate $x _ { t }$ that is intrinsic in   
+270 online learning, there is another regularization on the composite weights $\lambda _ { t }$ in min-regularized norm.   
+271 Both regularizations are fundamental and they together ensure the stability in the multi-objective   
+272 online setting. Hence we call the algorithm Doubly Regularized OMMD (DR-OMMD).   
+273 In principle, $r$ can take various forms such as $L _ { 1 }$ -norm, $L _ { 2 }$ -norm and KL divergence etc. Here   
+274 we adopt $L _ { 1 }$ -norm since it aligns well with the simplex constraint of $\lambda$ . Min-regularized-norm   
+275 can be computed very efficiently, since it has a closed-form solution when $m = 2$ . Specifically,   
+276 suppose the gradients at round $t$ are $g _ { t } ^ { 1 }$ and $g _ { t } ^ { 2 }$ . Set $\gamma _ { L } = ( g _ { 2 } ^ { \top } ( g _ { 2 } - g _ { 1 } ) - \alpha ) / \Vert g _ { 2 } - g _ { 1 } \Vert ^ { 2 }$ and   
+277 $\gamma _ { R } = ( g _ { 2 } ^ { \top } ( g _ { 2 } - g _ { 1 } ) + \alpha ) / \Vert g _ { 2 } - g _ { 1 } \Vert ^ { 2 }$ . Given any $\lambda _ { 0 } = ( \gamma _ { 0 } , 1 - \gamma _ { 0 } ) \in S _ { 2 }$ , we can compute the   
+278 composite weights $\lambda _ { t }$ as $( \gamma _ { t } , 1 - \gamma _ { t } )$ where
+
+$$
+\gamma _ { t } = \operatorname* { m a x } \{ \operatorname* { m i n } \{ \gamma _ { t } ^ { \prime \prime } , 1 \} , 0 \} , \quad \mathrm { w h e r e } \ \gamma _ { t } ^ { \prime \prime } = \operatorname* { m a x } \{ \operatorname* { m i n } \{ \gamma _ { 0 } , \gamma _ { R } \} , \gamma _ { L } \} .
+$$
+
+79 In addition, when $m > 2$ , since the feasible region $S _ { m }$ is a simplex, we can introduce a Frank-Wolfe   
+80 solver [14] to compute the composite weights. See the protocol and more details in Appendix D.
+
+Compared to vanilla min-norm, the composite weights in min-regularized-norm are not fully determined by the adversarial gradients. The resulting relative stability of composite weights make the composite gradients more robust to the adversarial environment. In the following, we give a general analysis and prove that DR-OMMD indeed guarantees sublinear regrets.
+
+# 4.3 Analysis
+
+We now analyze the static regret and the dynamic regret of DR-OMMD. Our analysis is based on the following commonly used assumptions [13, 11].
+
+Assumption 4.2 (Bregman divergence). The regularization function $R$ is 1-strongly convex. In addition, the Bregman divergence is $\gamma$ -Lipschitz continuous, i.e., $B _ { R } ( x , z ) - B _ { R } ( \bar { y } , z ) \leq \gamma \| x -$ $y \| , \forall x , y , z \in \mathrm { d o m } R$ , where $\mathrm { d o m } R$ is the domain of $R$ and satisfies $\mathcal { X } \subset \mathrm { d o m } R \subset \mathbb { R } ^ { n }$ .
+
+Assumption 4.3 (Lipschitz continuity). For each $i \in \{ 1 , \ldots , m \}$ , there exists some positive and finite $G$ such that, the $i$ -th loss $f _ { t } ^ { i }$ at each round $t \in \{ 1 , \ldots , T \}$ is $G$ -Lipschitz continuous w.r.t. $\| \cdot \|$ , i.e., $| f _ { t } ^ { i } ( x ) - f _ { t } ^ { i } ( x ^ { \prime } ) | \leq G \| x - \bar { x } ^ { \prime } \|$ . Note that in the convex setting, this assumption leads to bounded gradients, i.e., $\| \nabla f _ { t } ^ { i } ( x ) \| _ { * } \leq G$ for any $t \in \{ 1 , \ldots , T \} , i \in \{ 1 , \ldots , m \} , x \in \mathcal { X }$ .
+
+295 We first provide the static regret bound. The proof is left to the appendix due to the lack of space.
+
+Theorem 4.4. Suppose the diameter of 296 $\mathcal { X }$ is bounded by $D$ . Assume $F _ { t }$ is bounded, i.e., $| f _ { t } ^ { i } ( x ) | \leq$ 297 $F , \forall x \in \mathcal { X } , t \in \{ \bar { 1 } , \dots , T \} , i \in \{ 1 , \dots , m \}$ . For any $\lambda _ { 0 } \in  { S _ { m } }$ , DR-OMMD attains
+
+$$
+R _ { \mathrm { M O S } } ( T ) \leq \frac { 1 } { \eta } B _ { R } ( x ^ { * } , x _ { 1 } ) + \frac { \eta } { 2 } \sum _ { t = 1 } ^ { T } ( \Vert \nabla F _ { t } ( x _ { t } ) \lambda _ { t } \Vert _ { 2 } ^ { 2 } + \frac { 4 F } { \eta } \Vert \lambda _ { t } - \lambda _ { 0 } \Vert _ { 1 } ) .
+$$
+
+Remark. (i) Linearization with weights $\lambda _ { 0 } \in \mathcal { S } _ { m }$ can be viewed as single-objective optimization on scalar loss $\lambda _ { 0 } ^ { \top } F _ { t }$ , whose gradient is $g _ { t } = \nabla F _ { t } ( x _ { t } ) \lambda _ { 0 }$ . Hence we can directly borrow the tight bound of OMD (Theorem 6.8 in [27]) and derive a bound $\begin{array} { r } { \frac { 1 } { \eta } B _ { R } ( x ^ { * } , x _ { 1 } ) + \sum _ { t = 1 } ^ { T } \frac { \eta _ { t } } { 2 } \| \nabla F _ { t } ( x _ { t } ) \lambda _ { 0 } \| _ { 2 } ^ { 2 } } \end{array}$ $\lambda _ { t }$ r linearization. In co, the bound becomes $\begin{array} { r } { \frac { 1 } { \eta } B _ { R } ( x ^ { * } , x _ { 1 } ) + \frac { \eta } { 2 } \sum _ { t = 1 } ^ { T } \operatorname* { m i n } _ { \lambda \in { \cal S } _ { m } } \{ \| \nabla F _ { t } ( x _ { t } ) \lambda \| ^ { 2 } + \alpha \| \lambda - \lambda _ { 0 } \| _ { 1 } \} . } \end{array}$ $\alpha = 4 F / \eta$ ulation of, which is smaller than that of linearization. Note that the lower regret of DR-OMMD compared to linearization is also empirically verified in our experiments (see Figure 1).
+
+(ii) When $\begin{array} { r } { \eta = \frac { \hat { \sqrt { 2 \gamma D } } } { G \sqrt { T } } , \alpha = \frac { 4 F } { \eta } } \end{array}$ , the bound is in the order of $O ( \sqrt { T } )$ . It matches the optimal static single-objective regret bound w.r.t. $T$ [11] (see more details in Appendix E).
+
+Then we turn to the dynamic regret. Our analysis relies on an additional assumption [2, 32, 5].
+
+Assumption 4.5 (Temporal variability). For each $i \in \{ 1 , \ldots , m \}$ , there exists some positive and finite $V _ { T }$ such that $\begin{array} { r } { \sum _ { t = 1 } ^ { T - 1 } \operatorname* { s u p } _ { x \in \mathcal { X } } | f _ { t } ^ { i } ( x ) - f _ { t + 1 } ^ { i } ( x ) | \leq V _ { T } } \end{array}$ .
+
+Theorem 4.6. Assume the step size satisfies 310 $\begin{array} { r } { \frac { 4 V _ { T } } { G ^ { 2 } T } \leq \eta \leq \frac { 4 V _ { T } } { G ^ { 2 } } } \end{array}$ . Then under all the above assumptions, 311 for any preference $\lambda _ { 0 } \in  { S _ { m } }$ , OMMD with min-regularized-norm attains
+
+$$
+R _ { \mathrm { M O D } } ( T ) \leq \frac { \eta G ^ { 2 } T } { 2 } + \frac { 4 \gamma D V _ { T } } { \eta ^ { 2 } G ^ { 2 } } + \frac { \eta } { 2 } \sum _ { t = 1 } ^ { T } ( \Vert \nabla F _ { t } ( x _ { t } ) \lambda _ { t } \Vert _ { 2 } ^ { 2 } + \frac { 8 F G ^ { 2 } T } { V _ { T } } \Vert \lambda _ { t } - \lambda _ { 0 } \Vert _ { 1 } ) .
+$$
+
+Remark. When 312 $\begin{array} { r } { \eta = \frac { 2 } { G } ( \frac { \gamma D V _ { T } } { G T } ) ^ { 1 / 3 } , \alpha = \frac { 8 F G ^ { 2 } T } { V _ { T } } } \end{array}$ , the bound is in the order of $O ( T ^ { 2 / 3 } V _ { T } ^ { 1 / 3 } )$ , matching 313 the best attainable single-objective dynamic regret bound [2, 35] (see more details in Appendix E).
+
+# 5 Experiments
+
+In this section, we conduct extensive experiments to evaluate the effectiveness of DR-OMMD. We consider two baselines: (i) linearization performs single-objective online learning on the linearized loss $\lambda _ { 0 } ^ { \top } F _ { t }$ at each round $t$ , where the weights $\lambda _ { 0 } \in  { S _ { m } }$ are given beforehand; note that it is equivalent to computing composite gradients with fixed weights $\lambda _ { t } \equiv \lambda _ { 0 }$ . (ii) min-norm equips OMD with vanilla min-norm [7] for gradient composition.
+
+# 5.1 Simulation Experiments: Tracking the Pareto Front
+
+As summarized in Figure 1 (a), the goal is to track two points $\xi _ { t } ^ { 1 } , \xi _ { t } ^ { 2 }$ cycling along a circle ${ \mathcal { C } } = \{ \xi \in { }$ $\mathbb { R } ^ { 2 } \mid \| \xi \| _ { 2 } = 1 \}$ . For each $i \in \{ 1 , 2 \}$ , $\xi _ { t } ^ { i } = ( \cos \theta _ { t } ^ { i } , \sin \bar { \theta } _ { t } ^ { i } )$ is determined by some angle $\theta _ { t } ^ { i }$ . We set a positive integer $P ^ { i }$ as the rotating period of $\xi _ { t } ^ { i }$ , which is unknown to the learner. The two points are initialized by $\theta _ { 1 } ^ { 1 } = 0$ and $\theta _ { 1 } ^ { 2 } = \pi / 2$ and move as follows: at each round $t$ , for each $i \in \{ 1 , 2 \}$ , the adversary independently samples an angle $\delta _ { t } ^ { i }$ from a Gaussian distribution $\mathcal { N } ( { 2 \pi } / { P ^ { i } } , { 1 } / { \sqrt { P ^ { i } } } )$ , then moves the $i$ -th point to $\xi _ { t + 1 } ^ { i } \bar { \mathbf { \xi } } = ( \cos \theta _ { t + 1 } ^ { i } , \sin \theta _ { t + 1 } ^ { i } )$ where $\theta _ { t + 1 } ^ { i } = \theta _ { t } ^ { i } - \delta _ { t } ^ { i }$ . Note that $\mathbb { E } \theta _ { t + 1 } ^ { i } =$ $\theta _ { 1 } ^ { i } + 2 \pi t / P ^ { i }$ , hence in average $\xi _ { t } ^ { i }$ rotates clockwise with a period of $P ^ { i }$ . At each round $t$ , the learner 1 generates a decision $x _ { t }$ from a $L 2$ -norm ball $\mathcal { X } = \{ x \in \mathbb { R } ^ { 2 ^ { \cdot } } | \ \| x \| _ { 2 } \leq 2 \}$ . Then it acquires $\xi _ { t } ^ { 1 } , \xi _ { t } ^ { 2 }$ and suffer the losses $f _ { t } ^ { i } ( x _ { t } ) = \| x _ { t } - \xi _ { t } ^ { i } \| _ { 2 } ^ { 2 } / 2 , i \in \{ 1 , 2 \}$ . In this problem, the Pareto set of $F _ { t } = ( f _ { t } ^ { \mathrm { i } } , f _ { t } ^ { 2 } )$ is exactly the line segment between $\xi _ { t } ^ { 1 }$ and $\xi _ { t } ^ { 2 }$ , i.e., $\mathcal { X } _ { t } ^ { * } = \{ \lambda \xi _ { t } ^ { 1 } + ( 1 - \lambda ) \xi _ { t } ^ { 2 } \ | \ \lambda \in [ 0 , 1 ] \}$ . A t each round $t$ , PSG measures the squared distance between $x _ { t }$ and $\mathcal { X } _ { t } ^ { \ast }$ .
+
+332 We run $T = 1 0 , 0 0 0$ rounds. To simulate the pattern drift, we set $P ^ { 1 } = 1 0 , P ^ { 2 } = 2 0$ at the first   
+333 $T _ { 1 } = 3 , 0 0 0$ rounds, and $P ^ { 1 } = 2 0 , P ^ { 2 } = 1 0$ at the last $T _ { 2 } = 7 , 0 0 0$ rounds. For linearization,   
+334 the weights $\lambda _ { 0 } = ( \lambda _ { 0 } ^ { 1 } , 1 - \lambda _ { 0 } ^ { 1 } )$ are decided via a grid search $\lambda _ { 0 } ^ { 1 } \in \{ 0 , 0 . 1 , . . . , 1 \}$ ; we consider   
+335 three variants: lin- $^ { 1 }$ uses the optimal $\lambda _ { 0 }$ for the first $T _ { 1 }$ rounds, lin-2 uses the optimal $\lambda _ { 0 }$ for the   
+336 last $T _ { 2 }$ rounds, and lin-opt uses the optimal $\lambda _ { 0 }$ for all $T$ rounds. For DR-OMMD, for fairness of
+
+![](images/a3b4326a03d8acf25d9ced0df3ebeaa1dfb1dd10d00da99c7cc03a87bd49e358.jpg)  
+Figure 1: Simulation setup and results. (a) The targets $\xi _ { t } ^ { 1 } , \xi _ { t } ^ { 2 }$ cycle along the circle. The Pareto set at each round is the line segment $[ \xi _ { t } ^ { 1 } , \xi _ { t } ^ { 2 } ]$ PSG measures the distance from $x _ { t }$ to $[ \xi _ { t } ^ { 1 } , \xi _ { t } ^ { 2 } ]$ (b) Performance of DR-OMMD and baselines.
+
+![](images/30ee82d59d0ad7951da9536a53b3265f80f8ed0fa2dccfd60c48a4136590979b.jpg)  
+Figure 2: Results to verify the effectiveness of adaptive regularization on protein. (a) Performance of DR-OMMD and linearization under varying $\lambda _ { 0 } = ( \lambda _ { 0 } ^ { 1 } , 1 - \lambda _ { 0 } ^ { 1 } )$ . (b) Performance using the optimal weights $\lambda _ { 0 } = ( 0 . 1 , 0 . 9 )$ .
+
+comparison we use the same $\lambda _ { 0 }$ of lin-opt. The learning rates $\eta$ in all algorithms and the parameter $\alpha$ in DR-OMMD follow the corresponding theories (e.g., Theorem 4.6). In this experiment, since the loss functions are manually designed, the value of $V _ { T }$ can be directly calculated. Note that in some scenarios where $V _ { T }$ is unknown, we can conduct a grid search and utilize a meta-algorithm to handle the unknown $V _ { T }$ [37, 1], similar to the single-objective setting. From the results in Figure 1 (b), we find that DR-OMMD achieves the lowest PSG, showing its ability to track the Pareto front; meanwhile, min-norm appears very unstable in the online setting, even worse than linearization.
+
+# 5.2 Convex Experiments: Adaptive Regularization via Multi-Objective Optimization
+
+In many real-world online scenarios, regularization is often adopted to avoid overfitting. A standard way is to add a term $r ( x )$ to the loss $f _ { t } ( x )$ at each round and optimize the regularized loss $f _ { t } ( x ) +$ $\sigma r ( x )$ [24], where $\sigma$ is treated as a hyperparameter that needs to be fixed beforehand. The formalism of multi-objective online learning provides a novel way to realize regularization. Since $r ( x )$ measures the complexity of $x$ , it can be regarded as the second objective alongside the primary goal $f _ { t } ( x )$ . We can construct a vector loss $F _ { t } ( \bar { x ) = ( f _ { t } ( x ) , r ( x ) ) }$ at each round and thereby cast regularized online learning into a bi-objective online optimization problem. Compared to fixed regularization, the new approach effectively chooses the regularization strength $\sigma _ { t } = \bar { \lambda } _ { t } ^ { 2 } / \lambda _ { t } ^ { 1 }$ in an adaptive way.
+
+353 We use two large-scale online benchmark datasets. (i) protein is a bioinformatics dataset for protein   
+354 type classification [31], which has 17 thousand instances with 357 features. (ii) covtype is a biological   
+355 dataset collected from a non-stationary environment for forest cover type prediction [3], which has   
+356 50 thousand instances with 54 features. For both tasks, we set the logistic loss of classification as   
+357 the first objective, and the squared $L 2$ -norm of model parameters as the second objective. Since the   
+358 ultimate goal of regularization is to enhance predictive performance, we adopt the average loss as the   
+359 performance metric, namely $\textstyle \sum _ { t \leq T } l _ { t } ( x _ { t } ) / { \bar { T } }$ , where $l _ { t } ( x _ { t } )$ is the classification loss at round $t$ .
+
+We adopt a $L 2$ -norm ball centered at the origin with diameter $K = 1 0 0$ as the decision set. The learning rates are decided by a grid search over $\{ 0 . 1 , 0 . 2 , \ldots , 3 . 0 \}$ . For DR-OMMD, the parameter $\alpha$ is simply set as 0.1. For fixed regularization, the strength $\sigma = ( 1 - \lambda _ { 0 } ^ { 1 } ) / \lambda _ { 0 } ^ { 1 }$ is determined by the some preference $\lambda _ { 0 } ^ { 1 } \in [ 0 , 1 ]$ , which is essentially linearization with weights $\overset { \vartriangle } { \lambda _ { 0 } } = ( \lambda _ { 0 } ^ { 1 } , 1 - \lambda _ { 0 } ^ { \bar { 1 } } )$ . We run both algorithms with varying initial weights $\lambda _ { 0 } ^ { 1 } \in \{ 0 , 0 . 1 , . . . , 1 \}$ . In Figure 2, we plot (a) their final performance w.r.t. the choice of $\lambda _ { 0 }$ and (b) their learning curves with desirable $\lambda _ { 0 }$ (e.g., (0.1, 0.9) on protein). Other results are deferred to the appendix due to the lack of space. The results show that DR-OMMD consistently outperforms fixed regularization.
+
+# 6 Conclusions
+
+In this paper, we give a systematic study of multi-objective optimization in the online setting. We first formulate the framework of Multi-Objective Online Convex Optimization. Then we devise the Doubly Regularized Online Mirror Multiple Descent algorithm, which has a special design for gradient composition in online learning, namely min-regularized-norm. We provide non-trivial regret bounds for DR-OMMD and conduct extensive experiments to demonstrate its effectiveness.
+
+Limitations. As the first step of studying multiple gradient algorithm in online learning, we conduct our analysis in the convex setting. Although it does not affect the usage in the non-convex setting (see empirical validation in Appendix F), we can give a formal non-convex analysis in the future.
+
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+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] Our work is concerning a general problem in online learning.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 4.3. (b) Did you include complete proofs of all theoretical results? [Yes] See Appendix G, H, I.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] They are included in the supplementary materials.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 5.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A] We conduct online learning experiments, where the learning process is deterministic.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix E.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] We cite the source of datasets.   
+(b) Did you mention the license of the assets? [Yes] In the supplemental material.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] Our codes are provided in the supplemental material.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] We only use publicly available benchmark datasets.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] We only use publicly available benchmark datasets.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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