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parse/dev/4p6_5HBWPCw/4p6_5HBWPCw.md

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+# GRAPH-LESS NEURAL NETWORKS: TEACHING OLD MLPS NEW TRICKS VIA DISTILLATION
+
+Shichang Zhang∗ University of California, Los Angeles shichang@cs.ucla.edu
+
+Yozen Liu   
+Snap Inc.   
+yliu2@snap.com
+
+# Yizhou Sun
+
+University of California, Los Angeles yzsun@cs.ucla.edu
+
+Neil Shah   
+Snap Inc.   
+nshah@snap.com
+
+# ABSTRACT
+
+Graph Neural Networks (GNNs) are popular for graph machine learning and have shown great results on wide node classification tasks. Yet, they are less popular for practical deployments in the industry owing to their scalability challenges incurred by data dependency. Namely, GNN inference depends on neighbor nodes multiple hops away from the target, and fetching them burdens latency-constrained applications. Existing inference acceleration methods like pruning and quantization can speed up GNNs by reducing Multiplication-and-ACcumulation (MAC) operations, but the improvements are limited given the data dependency is not resolved. Conversely, multi-layer perceptrons (MLPs) have no graph dependency and infer much faster than GNNs, even though they are less accurate than GNNs for node classification in general. Motivated by these complementary strengths and weaknesses, we bring GNNs and MLPs together via knowledge distillation (KD). Our work shows that the performance of MLPs can be improved by large margins with GNN KD. We call the distilled MLPs Graph-less Neural Networks (GLNNs) as they have no inference graph dependency. We show that GLNNs with competitive accuracy infer faster than GNNs by $1 4 6 \times - 2 7 3 \times$ and faster than other acceleration methods by $1 4 \times - 2 7 \times$ . Under a production setting involving both transductive and inductive predictions across 7 datasets, GLNN accuracies improve over stand-alone MLPs by $1 2 . 3 6 \%$ on average and match GNNs on 6/7 datasets. Comprehensive analysis shows when and why GLNNs can achieve competitive accuracies to GNNs and suggests GLNN as a handy choice for latency-constrained applications.
+
+# 1 INTRODUCTION
+
+Graph Neural Networks (GNNs) have recently become very popular for graph machine learning (GML) research and have shown great results on node classification tasks (Kipf & Welling, 2016; Hamilton et al., 2017; Velickovi ˇ c et al. ´ , 2017) like product prediction on co-purchasing graphs and paper category prediction on citation graphs. However, for large-scale industrial applications, MLPs remain the major workhorse, despite common (implicit) underlying graphs and suitability for GML formalisms. One reason for this academic-industrial gap is the challenges in scalability and deployment brought by data dependency in GNNs (Zhang et al., 2020; Jia et al., 2020), which makes GNNs hard to deploy for latency-constrained applications that require fast inference.
+
+Neighborhood fetching caused by graph dependency is one of the major sources of GNN latency. Inference on a target node necessitates fetching topology and features of many neighbor nodes, especially on small-world graphs (detailed discussion in Section 4). Common inference acceleration techniques like pruning (Zhou et al., 2021) and quantization (Tailor et al., 2021; Zhao et al., 2020) can speed up GNNs to some extent by reducing Multiplication-and-ACcumulation (MAC) operations.
+
+However, their improvements are limited given the graph dependency is not resolved. Unlike GNNs, MLPs have no dependency on graph data and are easier to deploy than GNNs. They also enjoy the auxiliary benefit of sidestepping the cold-start problem that often happens during the online prediction of relational data (Wei et al., 2020), meaning MLPs can infer reasonably even when neighbor information of a new encountered node is not immediately available. On the other hand, this lack of graph dependency typically hurts for relational learning tasks, limiting MLP performance on GML tasks compared to GNNs. We thus ask: can we bridge the two worlds, enjoying the low-latency, dependency-free nature of MLPs and the graph context-awareness of GNNs at the same time?
+
+Present work. Our key finding is that it is possible to distill knowledge from GNNs to MLPs without losing significant performance, but reducing the inference time drastically for node classification. The knowledge distillation (KD) can be done offline, coupled with model training. In other words, we can shift considerable work from the latency-constrained inference step, where time reduction in milliseconds makes a huge difference, to the less time-sensitive training step, where time cost in hours or days is often tolerable. We call our approach Graph-less Neural Network (GLNN). Specifically, GLNN is a modeling paradigm involving KD from a GNN teacher to a student MLP; the resulting GLNN is an MLP optimized through KD, so it enjoys the benefits of graph contextawareness in training but has no graph dependency in inference. Regarding speed, GLNNs have superior efficiency and are $1 4 6 \times - 2 7 3 \times$ faster than GNNs and $1 4 \times - 2 7 \times$ faster than other inference acceleration methods. Regarding performance, under a production setting involving both transductive and inductive predictions on 7 datasets, GLNN accuracies improve over MLPs by $1 2 . 3 6 \%$ on average and match GNNs on $6 / 7$ datasets. We comprehensively study when and why GLNNs can achieve competitive results as GNNs. Our analysis suggests the critical factors for such great performance are large MLP sizes and high mutual information between node features and labels. Our observations align with recent results in vision and language, which posit that large enough (or slightly modified) MLPs can achieve similar results as CNNs and Transformers (Liu et al., 2021; Tolstikhin et al., 2021; Melas-Kyriazi, 2021; Touvron et al., 2021; Ding et al., 2021). Our core contributions are as follows:
+
+• We propose GLNN, which eliminates neighbor-fetching latency in GNN inference via KD to MLP. • We show GLNNs has competitive performance as GNNs, while enjoying $1 4 6 \times - 2 7 3 \times$ faster inference than vanilla GNNs and $1 4 \times - 2 7 \times$ faster inference than other inference acceleration methods. • We study GLNN properties comprehensively by investigating their performance under different settings, how they work as regularizers, their inductive bias, expressiveness, and limitations.
+
+# 2 RELATED WORK
+
+Graph Neural Networks. The early GNNs generalize convolution nets to graphs (Bruna et al., 2014; Defferrard et al., 2017) and later simplified to message-passing neural net (MPNN) by GCN (Kipf & Welling, 2016). Most GNNs after can be put as MPNNs. For example, GAT employs attention (Velickovi ˇ c et al. ´ , 2017), PPNP employs personalized PageRank (Klicpera et al., 2019), GCNII and DeeperGCN employ residual connections and dense connections (Chen et al., 2020; Li et al., 2019).
+
+Inference Acceleration. Inference acceleration have been proposed by hardware improvements (Chen et al., 2016; Judd et al., 2016) and algorithmic improvements through pruning (Han et al., 2015), quantization (Gupta et al., 2015). For GNNs, pruning (Zhou et al., 2021) and quantizing GNN parameters (Zhao et al., 2020) have been studied. These approaches speed up GNN inference to a certain extent but do not eliminate the neighbor-fetching latency. In contrast, our cross-model KD solves this issue. Concurrently, Graph-MLP also tries to bypass GNN neighbor fetching (Hu et al., 2021) by training an MLP with a neighbor contrastive loss, but it only considers transductive but not the more practical inductive setting. Some sampling works focus on speed up GNN training (Zou et al., 2019; Chen et al., 2018), which are complementary to our goal on inference acceleration.
+
+GNN distillation. Existing GNN KD works try to distill large GNNs to smaller GNNs. LSP (Yang et al., 2021b) and TinyGNN (Yan et al., 2020) do KD while preserving local information. Their students are GNNs with fewer parameters but not necessarily fewer layers. Thus, both designs still require latency-inducing fetching. GFKD (Deng & Zhang, 2021) does graph-level KD via graph generation. In GFKD, data instances are independent graphs, whereas we focus on dependent nodes within a graph. GraphSAIL (Xu et al., 2020) uses KD to learn students work well on new data while preserving performance on old data. CPF (Yang et al., 2021a) combines KD and label propagation (LP). The student in CPF is not a GNN, but it is still heavily graph-dependent as it uses LP.
+
+![](images/3c59c6aff8764b30e7d812c76a7e7b24c6d87080f7daff930791e286b7e0e843.jpg)  
+Figure 1: The number of fetches and the inference time of GNNs are both magnitudes more than MLPs and grow exponentially as functions of the number of layers. Left: neighbors need to be fetched for two GNN layers. Middle: the total number of fetches for inference. Right: the total inference time. (Inductive inference for 10 random nodes on OGB Products (Hu et al., 2020))
+
+# 3 PRELIMINARIES
+
+Notations. For GML tasks, the input is usually a graph and its node features, which we write as $\mathcal { G } = ( \nu , \mathcal { E } )$ , with $\nu$ stands for all nodes, and $\mathcal { E }$ stands for all edges. Let $N$ denote the total number of nodes. We use $\pmb { X } \in \mathbb { R } ^ { N \times D }$ to represent node features, with row $\mathbf { \boldsymbol { x } } _ { v }$ being the $D$ -dimensional feature of node $v \in \mathcal V$ . We represent edges with an adjacency matrix $\pmb { A }$ , with $A _ { u , v } = 1$ if edge $( u , v ) \in \mathcal { E }$ , and 0 otherwise. For node classification, one of the most important GML applications, the prediction targets are $\pmb { Y } \in \mathbb { R } ^ { N \times K }$ , where row $\mathbf { \Delta } _ { \mathbf { \mathcal { Y } } _ { v } }$ is a $K$ -dim one-hot vector for node $v$ . For a given $\mathcal { G }$ , usually a small portion of nodes will be labeled, which we mark using superscript $L$ , i.e. $\mathcal { V } ^ { \tilde { L } }$ , $X ^ { L }$ , and $\mathbf { \nabla } _ { \mathbf { Y } } \breve { L }$ . The majority of nodes will be unlabeled, and we mark using the superscript $U$ , i.e. $\mathcal { V } ^ { U }$ , $X ^ { U }$ , and $\mathbf { \nabla } _ { \mathbf { Y } } U$ .
+
+Graph Neural Networks. Most GNNs fit under the message-passing framework, where the representation $h _ { v }$ of each node $v$ is updated iteratively in each layer by collecting messages from its neighbors denoted as $\mathcal { N } ( v )$ . For the $l$ -th layer, $\mathbf { \Sigma } _ { h _ { v } ^ { ( l ) } }$ is obtained from the previous layer representation $ { \boldsymbol { h } } _ { u } ^ { ( l - 1 ) }$ $\mathbf { \mathcal { h } } _ { u } ^ { ( 0 ) } = \mathbf { \mathcal { x } } _ { u } )$ ) via an aggregation operation AGGR followed by an UPDATE operation as
+
+$$
+\begin{array} { r } { \pmb { h } _ { N ( v ) } ^ { ( l ) } = \mathrm { A G G R } ( \{ \pmb { h } _ { u } ^ { ( l - 1 ) } : u \in \mathcal { N } ( v ) \} ) \qquad \mathrm { a n d } \qquad \pmb { h } _ { v } ^ { ( l ) } = \mathrm { U P D A T E } ( \pmb { h } _ { N ( v ) } ^ { ( l ) } , \pmb { h } _ { v } ^ { ( l - 1 ) } ) } \end{array}
+$$
+
+# 4 MOTIVATION
+
+GNNs have considerable inference latency due to graph dependency. One more GNN layer means fetching one more hop of neighbors. To infer a node with a $L$ -layer GNN on a graph with average degree $R$ requires $\mathcal { O } ( R ^ { L } )$ fetches. $R$ can be large for real-world graphs, e.g. 208 for the Twitter (Ching et al., 2015). Also, as layer fetching must be done sequentially, the total latency explodes quickly as $L$ increases. Figure 1 shows the dependency added by each GNN layer and the exponential explosion of inference time. In contrast, the MLP inference time is much smaller and grows linearly. This marked gap contributes greatly to the practicality of MLPs in industrial applications over GNNs.
+
+The node-fetching latency is exacerbated by two factors: firstly, newer GNN architectures are getting deeper from 64 layers (Chen et al., 2020) to even 1001 layers (Li et al., 2021). Secondly, industrialscale graphs are frequently too large to fit into the memory of a single machine (Jin et al., 2022), necessitating sharding of the graph out of the main memory. For example, Twitter has 288M monthly active users (nodes) and an estimated 60B followers (edges) as of 3/2015. Facebook has 1.39B active users with more than 400B edges as of 12/2014 (Ching et al., 2015). Even when stored in a sparse-matrix-friendly format (often COO or CSR), these graphs are on the order of TBs and are constantly growing. Moving away from in-memory storage results in even slower neighbor-fetching.
+
+MLPs, on the other hand, lack the means to exploit graph topology, which hurts their performance for node classification. For example, test accuracy on Products is 78.61 for GraphSAGE compared to 62.47 for an equal-sized MLP. Nonetheless, recent results in vision and language posit that large (or slightly modified) MLPs can achieve similar results as CNNs and Transformers (Liu et al., 2021). We thus also ask: Can we bridge the best of GNNs and MLPs to get high-accuracy and low-latency models? This motivates us to do cross-model KD from GNNs to MLPs.
+
+![](images/fe13a4358c6462ce72c20699e317599db71d44b7281ec5d0c4ff85854d7ca559.jpg)  
+Figure 2: The GLNN framework: In offline training, a trained GNN teacher is applied on the graph for soft targets. Then, a student MLP is trained on node features guided by the soft targets. The distilled MLP, now GLNN, is deployed for online predictions. Since graph dependency is eliminated for inference, GLNNs infer much faster than GNNs, and hence the name “Graph-less Neural Network.”
+
+# 5 GRAPH-LESS NEURAL NETWORKS
+
+We introduce GLNN and answer exploration questions of its properties: 1) How do GLNNs compare to MLPs and GNNs? 2) Can GLNNs work well under both transductive and inductive settings? 3) How do GLNNs compare to other inference acceleration methods? 4) How do GLNNs benefit from KD? 5) Do GLNNs have sufficient model expressiveness? 6) When will GLNNs fail to work?
+
+# 5.1 THE GLNN FRAMEWORK
+
+The idea of GLNN is straightforward, yet as we will see, extremely effective. In short, we train a “boosted” MLP via KD from a teacher GNN. KD was introduced in Hinton et al. (2015), where knowledge was transferred from a cumbersome teacher to a simpler student. In our case, we generate soft targets $z _ { v }$ for each node $v$ with a teacher GNN. Then we train a student MLP with both true labels $\mathbf { \Delta } _ { \mathbf { \mathcal { Y } } _ { v } }$ and $z _ { v }$ . The objective is as Equation 1, with $\lambda$ being a weight parameter, $\mathcal { L } _ { l a b e l }$ being the cross-entropy between $\mathbf { \nabla } _ { \mathbf { y } _ { v } }$ and student predictions $\hat { y } _ { v }$ , $\mathcal { L } _ { t e a c h e r }$ being the KL-divergence.
+
+$$
+\mathcal { L } = \lambda \Sigma _ { v \in \mathcal { V } ^ { L } } \mathcal { L } _ { l a b e l } ( \hat { y } _ { v } , y _ { v } ) + ( 1 - \lambda ) \Sigma _ { v \in \mathcal { V } } \mathcal { L } _ { t e a c h e r } ( \hat { y } _ { v } , z _ { v } )
+$$
+
+The model after KD, i.e. GLNN, is essentially a MLP. Therefore, GLNNs have no graph dependency during inference and are as fast as MLPs. On the other hand, through offline KD, GLNN parameters are optimized to predict and generalize as well as GNNs, with the added benefit of faster inference and easier deployment. In Figure 2, we show the offline KD and online inference steps of GLNNs.
+
+# 5.2 EXPERIMENT SETTINGS
+
+Datasets. We consider all five datasets used in the CPF paper (Yang et al., 2021a), i.e. Cora, Citeseer, Pubmed, A-computer, and A-photo. To fully evaluate our method, we also include two more larger OGB datasets (Hu et al., 2020), i.e. Arxiv and Products.
+
+Model Architectures. For consistent results, we use GraphSAGE (Hamilton et al., 2017) with GCN aggregation as the teacher. We conduct ablation studies of other GNN teachers like GCN (Kipf & Welling, 2016), GAT (Velickovi ˇ c et al. ´ , 2017) and, APPNP (Klicpera et al., 2019) in Section 6.
+
+Evaluation Protocol. For all experiments in this section, we report the average and standard deviation over ten runs with different random seeds. Model performance is measured as accuracy, and results are reported on test data with the best model selected using validation data.
+
+Transductive vs. Inductive. Given ${ \mathcal { G } } , X$ , and $Y ^ { L }$ , we consider node classification under two settings: transductive $( t r a n )$ and inductive $( i n d )$ . For ind, we hold out some test data for inductive evaluation only. We first select inductive nodes $\mathcal { V } _ { i n d } ^ { U } \subset \mathcal { V } ^ { U }$ , which partitions $\mathcal { V } ^ { U }$ into the disjoint inductive subset and observed subset, i.e. $\mathcal { V } ^ { U } = \mathcal { V } _ { o b s } ^ { U } \sqcup \mathcal { V } _ { i n d } ^ { U }$ . Then we hold out $v \in \mathcal { V } _ { i n d } ^ { U }$ and all edges connected to $v \in \mathcal { V } _ { i n d } ^ { U }$ obs ind, which leads to two disjoint graphs $\mathcal { G } = \mathcal { G } _ { o b s } \sqcup \mathcal { G } _ { i n d }$ ind with no shared nodes or
+
+Table 1: GLNNs outperform MLPs by large margins and match GNNs on 5 of 7 datasets under the transductive setting. $\Delta _ { M L P }$ $( \Delta _ { G N N } )$ represents difference between the GLNN and a trained MLP (GNN). Results show accuracy (higher is better); $\Delta _ { G N N } 2 0$ indicates GLNN outperforms GNN.   
+
+<table><tr><td>Datasets</td><td>SAGE</td><td>MLP</td><td>GLNN</td><td>△MLP</td><td>△GNN</td></tr><tr><td>Cora</td><td>80.52 ± 1.77</td><td>59.22 ±1.31</td><td>80.54 ± 1.35</td><td>21.32 (36.00%)</td><td>0.02 (0.02%)</td></tr><tr><td>Citeseer</td><td>70.33 ± 1.97</td><td>59.61 ± 2.88</td><td>71.77 ± 2.01</td><td>12.16 (20.40%)</td><td>1.44 (2.05%)</td></tr><tr><td>Pubmed</td><td>75.39 ± 2.09</td><td>67.55 ± 2.31</td><td>75.42 ± 2.31</td><td>7.87 (11.65%)</td><td>0.03 (0.04%)</td></tr><tr><td>A-computer</td><td>82.97 ± 2.16</td><td>67.80 ± 1.06</td><td>83.03 ± 1.87</td><td>15.23 (22.46%)</td><td>0.06 (0.07%)</td></tr><tr><td>A-photo</td><td>90.90 ± 0.84</td><td>78.77 ± 1.74</td><td>92.11 ± 1.08</td><td>13.34 (16.94%)</td><td>1.21 (1.33%)</td></tr><tr><td>Arxiv</td><td>70.92 ± 0.17</td><td>56.05 ± 0.46</td><td>63.46 ± 0.45</td><td>7.41 (13.24%)</td><td>-7.46 (-10.52%)</td></tr><tr><td>Products</td><td>78.61 ± 0.49</td><td>62.47 ± 0.10</td><td>68.86 ± 0.46</td><td>6.39 (10.23%)</td><td>-9.75 (-12.4%)</td></tr></table>
+
+Table 2: Enlarged GLNNs match the performance of GNNs on the OGB datasets. For Arxiv, we use MLPw4 (GLNNw4). For Products, we use MLPw8 (GLNNw8).   
+
+<table><tr><td>Datasets</td><td>SAGE</td><td>MLP+</td><td>GLNN+</td><td>△MLP</td><td>△GNN</td></tr><tr><td>Arxiv</td><td>70.92 ± 0.17</td><td>55.31± 0.47</td><td>72.15 ± 0.27</td><td>16.85 (30.46%)</td><td>0.51 (0.71%)</td></tr><tr><td>Products</td><td>78.61 ± 0.49</td><td>64.50 ± 0.45</td><td>77.65 ± 0.48</td><td>13.14 (20.38%)</td><td>-0.97 (-1.23%)</td></tr></table>
+
+els are partitioned into three disjoint sets, i.e. . Concretely, the input/output of both setting $X = X ^ { L } \sqcup X _ { o b s } ^ { U } \sqcup X _ { i n d } ^ { U }$ $\pmb { Y } = \pmb { Y } ^ { L } \sqcup \pmb { Y } _ { o b s } ^ { U } \sqcup \pmb { Y } _ { i n d } ^ { U }$
+
+• ind: train on • tran: train on $\mathcal { G } _ { o b s }$ ${ \mathcal { G } } , X$ , $X ^ { L }$ , and , $X _ { o b s } ^ { U }$ $\mathbf { \nabla } _ { \mathbf { Y } ^ { L } }$ ; evaluate on , and $Y ^ { L }$ ; evaluate on $( X ^ { U } , Y ^ { U } )$ $( X _ { i n d } ^ { U } , Y _ { i n d } ^ { U } )$ ; $\mathrm { K D }$ uses $z _ { v }$ v  ; KD uses for $v \in \mathcal V$ $z _ { v }$ . for
+
+Note that for tran, all the nodes in the graph including the validation and test nodes are used to generate $_ { z }$ . A discussion of this choice along with other experiment details are in Appendix A.
+
+# 5.3 HOW DO GLNNS COMPARE TO MLPS AND GNNS?
+
+We start by comparing GLNNs to MLPs and GNNs with the same number of layers and hidden dimensions. We first consider the standard transductive setting, so our results in Table 1 are directly comparable to results reported in previous literature like Yang et al. (2021a) and Hu et al. (2020).
+
+As shown in Table 1, the performance of all GLNNs improve over MLPs by large margins. On smaller datasets (first 5 rows), GLNNs can even outperform the teacher GNNs. In other words, for each task, with the same parameter budget, there exists a set of MLP parameters that has GNN-competitive performance (detailed discussion in Sections 5.6 and 5.7). For the larger OGB datasets (last 2 rows), the GLNN performance is improved over MLPs but still worse than the teacher GNNs. However, as we show in Table 2, this gap can be mitigated by increasing MLP size to MLPwi1. In Figure 3 (right), we visualize the trade-off between prediction accuracy and model inference time with different model sizes. We show that gradually increasing GLNN size pushes its performance to be close to SAGE. On the other hand, when we reduce the number of layers of $\mathrm { S A G E } ^ { 2 }$ , the accuracy quickly drops to be worse than GLNNs. A detailed discussion of the rationale for increasing MLP sizes is in Appendix B.
+
+# 5.4 CAN GLNNS WORK WELL UNDER BOTH TRANSDUCTIVE AND INDUCTIVE SETTINGS?
+
+Although transductive is the commonly studied setting for node classification, it does not encompass prediction on unseen nodes. Therefore, it may not be the best way to evaluate a deployed model, which must often generate predictions for new data points as well as reliably maintain performance on old ones. Thus, to better understand the effectiveness of GLNN, we also consider their performance under a realistic production setting, which contains both transductive and inductive predictions.
+
+To evaluate a model inductively, we hold out some test nodes from training to form an inductive set, i.e. $\mathcal { V } ^ { U } = \mathcal { V } _ { o b s } ^ { U } \sqcup \mathcal { V } _ { i n d } ^ { U }$ . In production, a model might be re-trained periodically, e.g. weekly. The obs hold-out nodes in $V _ { i n d } ^ { U }$ represent new nodes entered the graph between two trainings. $V _ { i n d } ^ { U }$ is usually small compared to $V _ { o b s } ^ { U }$ – e.g. Graham (2012) estimates $5- 7 \%$ for the fastest-growing tech startups. In our case, to mitigate randomness and better evaluate generalizability, we use V Uind containing 20% of the test data. We also evaluate on standard transductive prediction on $V _ { o b s } ^ { U }$ containing the otherrved unlabeled nodes $80 \%$ of the test data, representing thee inference is commonly redone on existing nodes in real-world cases. We report both results and a interpolated production (prod) results in Table 3. The prod results paint a clearer picture of model generalization as well as accuracy in production. See Section 6 for an ablation study of different inductive split rates other than 20-80.
+
+![](images/56094e9645d24171cda661a02619bd1120d8d877a8d7ebb8ed27856fa4567de6.jpg)  
+Figure 3: Enlarged MLPs (GLNNs) can match GNN accuracy, but infer dramatically faster. Plots are under the same setting as Figure 1. Left: inference time of MLPs vs. GNN (SAGE) for different model sizes. Right: model accuracy vs. inference time. Note: time axes are log-scaled.
+
+Table 3: GLNNs match GNN performance on a production setting with both inductive and transductive predictions. We uProducts. ind results on $V _ { i n d } ^ { U }$ LP for the 5 C, tran results on $V _ { o b s } ^ { U }$ atasets, MLPw4 for Arxiv, and MLPw8 for, and the interpolated prod results are reported.   
+
+<table><tr><td>Datasets</td><td>Eval</td><td>SAGE</td><td>MLP/MLP+</td><td>GLNN/GLNN+</td><td>△MLP</td><td>△GNN</td></tr><tr><td>Cora</td><td>prod</td><td>79.29</td><td>58.98</td><td>78.28</td><td>19.30 (32.72%)</td><td>-1.01 (-1.28%)</td></tr><tr><td></td><td>ind</td><td>81.33 ± 2.19</td><td>59.09 ± 2.96</td><td>73.82 ± 1.93</td><td>14.73 (24.93%)</td><td>-7.51 (-9.23%)</td></tr><tr><td></td><td>tran</td><td>78.78 ± 1.92</td><td>58.95 ± 1.66</td><td>79.39 ± 1.64</td><td>20.44 (34.66%)</td><td>0.61 (0.77%)</td></tr><tr><td>Citeseer</td><td>prod</td><td>68.38</td><td>59.81</td><td>69.27</td><td>9.46 (15.82%)</td><td>0.89 (1.30%)</td></tr><tr><td></td><td>ind</td><td>69.75 ± 3.59</td><td>60.06 ± 5.00</td><td>69.25 ± 2.25</td><td>9.19 (15.30%)</td><td>-0.5 (-0.7%)</td></tr><tr><td></td><td>tran</td><td>68.04 ± 3.34</td><td>59.75 ± 2.48</td><td>69.28 ± 3.12</td><td>9.63 (15.93%)</td><td>1.24 (1.82%)</td></tr><tr><td>Pubmed</td><td>prod</td><td>74.88</td><td>66.80</td><td>74.71</td><td>7.91 (11.83%)</td><td>-0.17 (-0.22%)</td></tr><tr><td></td><td>ind</td><td>75.26 ± 2.57</td><td>66.85 ± 2.96</td><td>74.30 ± 2.61</td><td>7.45 (11.83%)</td><td>-0.96 (-1.27%)</td></tr><tr><td></td><td>tran</td><td>74.78 ± 2.22</td><td>66.79 ± 2.90</td><td>74.81 ± 2.39</td><td>8.02 (12.01%)</td><td>0.03 (0.04%)</td></tr><tr><td>A-computer</td><td>prod</td><td>82.14</td><td>67.38</td><td>82.29</td><td>14.90 (22.12%)</td><td>0.15 (0.19%)</td></tr><tr><td></td><td>ind</td><td>82.08 ± 1.79</td><td>67.84 ± 1.78</td><td>80.92 ± 1.36</td><td>13.08 (19.28%)</td><td>-1.16 (-1.41%)</td></tr><tr><td></td><td>tran</td><td>82.15 ± 1.55</td><td>67.27 ± 1.36</td><td>82.63 ± 1.40</td><td>15.36 (22.79%)</td><td>0.48 (0.58%)</td></tr><tr><td>A-photo</td><td>prod</td><td>91.08</td><td>79.25</td><td>92.38</td><td>13.13 (16.57%)</td><td>1.30 (1.42%)</td></tr><tr><td></td><td>ind</td><td>91.50 ± 0.79</td><td>79.44 ± 1.72</td><td>91.18 ± 0.81</td><td>11.74 (14.78%)</td><td>-0.32 (-0.35%)</td></tr><tr><td></td><td>tran</td><td>90.80 ± 0.77</td><td>79.20 ± 1.64</td><td>92.68 ± 0.56</td><td>13.48 (17.01%)</td><td>1.70 (1.87%)</td></tr><tr><td>Arxiv</td><td>prod</td><td>70.73</td><td>55.30</td><td>65.09</td><td>9.79 (17.70%)</td><td>-5.64 (-7.97%)</td></tr><tr><td></td><td>ind</td><td>70.64 ± 0.67</td><td>55.40 ± 0.56</td><td>60.48 ± 0.46</td><td>4.3 (7.76%)</td><td>-10.94 (-15.49%)</td></tr><tr><td></td><td>tran</td><td>70.75± 0.27</td><td>55.28 ± 0.49</td><td>71.46 ± 0.33</td><td>11.16 (20.18%)</td><td>-4.31 (-6.09%)</td></tr><tr><td>Products</td><td>prod</td><td>76.60</td><td>63.72</td><td>75.77</td><td>12.05 (18.91%)</td><td>-0.83 (-1.09%)</td></tr><tr><td></td><td>ind</td><td>76.89 ± 0.53</td><td>63.70 ± 0.66</td><td>75.16 ± 0.34</td><td>11.44 (17.96%)</td><td>-1.73 (-2.25%)</td></tr><tr><td></td><td>tran</td><td>76.53 ±0.55</td><td>63.73 ± 0.69</td><td>75.92 ± 0.61</td><td>12.20 (19.15%)</td><td>-0.61 (-0.79%)</td></tr></table>
+
+In Table 3, we see that GLNNs can still improve over MLP by large margins for inductive predictions. On 6/7 datasets, the GLNN prod performance are competitive to GNNs, which supports deploying GLNN as a much faster model with no or only slight performance loss. On the Arxiv dataset, the GLNN performance is notably less than GNNs – we hypothesize this is due to Arxiv having a particularly challenging data split which causes distribution shift between test nodes and training
+
+Table 4: While other inference acceleration methods speed up SAGE, they are considerably slower than GLNNs. Numbers (in ms) are inductive inference time on 10 randomly chosen nodes.   
+
+<table><tr><td>Datasets</td><td>SAGE</td><td>QSAGE</td><td>PSAGE</td><td>Neighbor Sample</td><td>GLNN+</td></tr><tr><td>Arxiv</td><td>489.49</td><td>433.90 (1.13x)</td><td>465.43 (1.05x)</td><td>91.03 (5.37x)</td><td>3.34 (146.55x)</td></tr><tr><td>Products</td><td>2071.30</td><td>1946.49 (1.06x)</td><td>2001.46 (1.04x)</td><td>107.71 (19.23x)</td><td>7.56 (273.98x)</td></tr></table>
+
+nodes, which is hard for GLNNs to capture without utilizing neighbor information like GNNs.   
+However, we note that GLNN performance is substantially improved over MLP.
+
+# 5.5 HOW DO GLNNS COMPARE TO OTHER INFERENCE ACCELERATION METHODS?
+
+Common techniques of inference acceleration include pruning and quantization. These approaches can reduce model parameters and Multiplication-and-ACcumulation (MACs) operations. Still, they don’t eliminate neighbor-fetching latency. Therefore, their speed gain on GNNs is less significant than on NNs. For GNNs, neighbor sampling is also used to reduce the fetching latency. We show an explicit speed comparison between vanilla SAGE, quantized SAGE from FP32 to INT8 (QSAGE), SAGE with $50 \%$ weights pruned (PSAGE), inference neighbor sampling with fan-out 15, and GLNN in Table 4. With the same setting as Figure 1, we see that GLNN is considerably faster.
+
+Two other kinds of methods considered as inference acceleration are GNN-to-GNN KD like TinyGNN (Yan et al., 2020) and Graph Augmented-MLPs (GA-MLPs) like SGC (Wu et al., 2019) or SIGN (Frasca et al., 2020). Inference of GNN-to-GNN KD is likely to be slower than a GNN-Li with the same $i$ as the student, since there will usually be some extra overheads like the Peer-Aware Module (PAM) in TinyGNN. GA-MLPs precompute augmented node features and apply MLPs to them. With precomputation, their inference time will be the same as MLPs for dimension-preserving augmentation (SGC) and the same as enlarged MLPwi for augmentation involves concatenation (SIGN). Thus, for both kinds of approaches, it is sufficient to compare GLNN with GNN-Li and MLPwi, which we have already shown in Figure 3 (left). We see that GNN-Lis are much slower than MLPs. For GA-MLPs, since full pre-computation cannot be done for inductive nodes, GA-MLPs still need to fetch neighbor nodes. This makes them much slower than MLPwi in the inductive setting, and even slower than pruned GNNs and TinyGNN as shown in Zhou et al. (2021).
+
+# 5.6 HOW DOES GLNN BENEFIT FROM DISTILLATION?
+
+We showed that GNNs are markedly better than MLPs on node classification tasks. But, with KD, GLNNs can often become competitive to GNNs. This indicates that there exist suitable MLP parameters which can well approximate the ideal prediction function from node features to labels. However, these parameters can be difficult to learn through standard stochastic gradient descent. We hypothesize that KD helps to find them through regularization and transfer of inductive bias.
+
+First, we show that KD can help to regularize the student model. From loss curves of a directly trained MLP and the GLNN in Figure 4, we see the gap between training and validation loss is visibly larger for MLPs than GLNNs, and MLPs show obvious overfitting trends. Second, we analyze the inductive bias that makes GNNs powerful on node classification, which suggests that node inferences should be influenced by the graph topology. Whereas MLPs have less inductive bias. Similar difference exists between Transformers (Vaswani et al., 2017) and MLPs. Liu et al. (2021) shows that the inductive bias in Transformers can be mitigated by a simple gate on large MLPs. For node classification, we hypothesize that KD helps to mitigate the inductive bias, so GLNNs can perform competitively. Soft labels from GNN teachers are heavily influenced by the graph topology due to inductive bias. They maintain nonzero probabilities on classes other than the ground truth provided by labels, which can be useful for the student to learn to complement the missing inductive bias in MLPs. To evaluate this hypothesis quantitatively, we define the cut loss $\mathcal { L } _ { c u t } \in [ 0 , 1 ]$ in Equation 2 to measure the consistency between model predictions and graph topology (details in Appendix C):
+
+$$
+\mathcal { L } _ { c u t } = \frac { T r ( \hat { Y } ^ { T } A \hat { Y } ) } { T r ( \hat { Y } ^ { T } D \hat { Y } ) }
+$$
+
+Here $\hat { \pmb { Y } } \in [ 0 , 1 ] ^ { N \times K }$ is the soft classification probability output by the model, $\pmb { A }$ and $_ { D }$ are the adjacency and degree matrices. When $\mathcal { L } _ { c u t }$ is close to $^ { 1 }$ , it means the predictions and the graph topology are very consistent. In our experiment, we observe that the average $\mathcal { L } _ { c u t }$ for SAGE over five CPF datasets is 0.9221, which means high consistency. The same $\mathcal { L } _ { c u t }$ for MLPs is only 0.7644, but for GLNNs it is 0.8986. This shows that the GLNN predictions indeed benefit from the graph topology knowledge contained in the teacher outputs (the full table of $\mathcal { L } _ { c u t }$ values in Appendix C).
+
+![](images/e13d36ad0fd34e672832534415bdedeb0f31097113d6e61931f935801ee4e5d2.jpg)  
+Figure 4: Loss curves on CPF datasets show GLNN distillation can help to regularize the training. Here the training loss of GLNN is on hard labels, only corresponding to the first term in Equation 1.
+
+# 5.7 DO GLNNS HAVE ENOUGH MODEL EXPRESSIVENESS?
+
+Intuitively, the addition of neighbor information makes GNNs more powerful than MLPs when classifying nodes. Thus, a natural question regarding KD from GNNs to MLPs is whether MLPs are expressive enough to represent graph data as well as GNNs. Many recent works studied GNN model expressiveness (Xu et al., 2018; Chen et al., 2021). The latter analyzed GNNs and GA-MLPs for node classification and characterized expressiveness as the number of equivalence classes of rooted graphs induced by the model (formal definitions in Appendix D). The conclusion is that GNNs are more powerful than GA-MLPs, but in most real-world cases their expressiveness is indistinguishable.
+
+We adopt the analysis framework from Chen et al. (2021) and show in Appendix D that the number of equivalence classes induced by GNNs and MLPs are |X |+m−2m−1 
2L−1 a nd $| \mathcal { X } |$ respectively. Here $m$ denotes the max node degree, $L$ denotes the number of GNN layers, and $\mathcal { X }$ denotes the set of all possible node features. The former is apparently larger which concludes that GNNs are more expressive. Empirically, however, the gap makes little difference when $| \mathcal { X } |$ is large. In real applications, node features can be high dimensional like bag-of-words, or even word embeddings, thus making $| \mathcal { X } |$ enormous. Like for bag-of-words, $| \mathcal { X } |$ is in the order of $\mathcal { O } ( p ^ { D } )$ , where $D$ is the vocabulary size, and $p$ is the max word frequency. The expressiveness of a L-layer GNN is lower bounded by |X |+m−2
2L−1 $\left( \stackrel { | \mathcal { X } | + m - 2 } { m - 1 } \right) ^ { 2 ^ { L } - 1 } = \mathcal { O } ( p ^ { D ( m - 1 ) ( 2 ^ { L } - 1 ) } )$ , but empirically, both MLPs and GNNs should have enough expressiveness given $D$ is usually hundreds or bigger (see Table 5).
+
+# 5.8 WHEN WILL GLNNS FAIL TO WORK?
+
+As discussed in Section 5.7 and Appendix D, the goal of GML node classification is to fit a function $f$ on the rooted graph $\mathcal { G } ^ { [ i ] }$ and label $\mathbf { \nabla } _ { \mathbf { \psi } _ { 3 } } \mathbf { \psi } _ { 2 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \qquad \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \qquad \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 3 } \mathbf { \psi } _ { 4 } \mathbf { \psi } _ { 4 }$ . From the information theoretic perspective, fitting $f$ by minimizing the commonly used cross-entropy loss is equivalent to maximizing the mutual information (MI), $I ( \mathcal G ^ { [ i ] } ; \boldsymbol y _ { i } )$ as shown in Qin et al. (2020). If we consider $\mathcal { G } ^ { [ i ] }$ as a joint distribution of two random variables $X ^ { [ i ] }$ and ${ \mathcal { E } } ^ { [ i ] }$ representing the node features and edges in $\mathcal { G } ^ { [ i ] }$ respectively, we have
+
+$$
+I ( \mathcal { G } ^ { [ i ] } ; \pmb { y } _ { i } ) = I ( \pmb { X } ^ { [ i ] } , \pmb { \mathcal { E } } ^ { [ i ] } ; \pmb { y } _ { i } ) = I ( \pmb { \mathcal { E } } ^ { [ i ] } ; \pmb { y } _ { i } ) + I ( \pmb { X } ^ { [ i ] } ; \pmb { y } _ { i } | \pmb { \mathcal { E } } ^ { [ i ] } )
+$$
+
+$I ( \mathcal { E } ^ { [ i ] } ; y _ { i } )$ only depends on edges and labels, thus MLPs can only maximize $I ( X ^ { [ i ] } ; y _ { i } | \mathcal { E } ^ { [ i ] } )$ . In the extreme case, $I ( X ^ { [ i ] } ; y _ { i } | \mathcal { E } ^ { [ i ] } )$ can be zero when $y ^ { [ i ] }$ is conditionally independent from $X ^ { [ i ] }$ given $\mathcal { E } ^ { [ i ] }$ . For example, when every node is labeled by its degree or whether it forms a triangle. Then MLPs won’t be able to fit meaningful functions, and neither will GLNNs. However, such cases are typically rare, and unexpected in practical settings our work is mainly concerned with. For real GML tasks, node features and structural roles are often highly correlated (Lerique et al., 2020), hence MLPs can achieve reasonable results even only based on node features, and thus GLNNs can potentially achieve much better results. We study the failure case of GLNNs by creating a low MI scenario in Section 6.
+
+![](images/5b5ba4cf66e93dda6d93901e3acfc6306b2eb1a932c1b44ff6ba4c6b436be96e.jpg)  
+Figure 5: Left: Node feature noise. GLNN has comparable performance to GNNs only when nodes are less noisy. Adding more noise decreases GLNN performance faster than GNNs. Middle: Inductive split rate. Altering the inductive:transductive ratio in the production setting doesn’t affect the accuracy much. Right: Teacher GNN architecture. GLNNs can learn from different GNN teachers to improve over MLPs and achieve comparable results. Accuracies are averaged over five CPF datasets.
+
+# 6 ABLATION STUDIES
+
+In this section, we do ablation studies of GLNNs on node feature noise, inductive split rates, and teacher GNN architecture. Reported results are test accuracies averaged over five datasets in CPF. More experiments can be found in Appendix including advanced GNN teachers (Appendix F), GA-MLP student (Appendix G), and non-homogeneous data (Appendix I).
+
+Noisy node features. Following Section 5.8, we investigate failure cases of GLNN by adding different levels of Gaussian noise to node features to decrease their mutual information with labels. Specifically, we replace $\boldsymbol { X }$ with $\tilde { \cal X } = ( 1 - \alpha ) { \cal X } + \alpha \epsilon$ . $\epsilon$ is an isotropic Gaussian independent from $\boldsymbol { X }$ , and $\alpha \in [ 0 , 1 ]$ denotes the noise level. We show the inductive performance of MLP, GNN, and GLNN under different noise levels in Figure 5 (left). We see that as $\alpha$ increases, the accuracy of MLPs and GLNNs decrease faster than GNNs, while the performance of GLNNs and GNNs are still comparable for small $\alpha \mathbf { s }$ . When $\alpha$ reaches 1, $\tilde { X }$ and $\mathbf { Y }$ will become independent corresponding to the extreme case discussed in Section 5.8. A more detailed discussion is in Appendix J.
+
+Inductive split rate. In Section 5.4, we use a 20-80 split of the test data for inductive evaluation. In Figure 5 (middle), we show the results under different split rates (More detailed plots in Appendix H). We see that as the inductive portion increase, GNN and MLP performance stays roughly the same, and the GLNN inductive performance drops slightly. We only consider rates up to 50-50 since having $50 \%$ or even more inductive nodes is highly atypical in practice. When a large amount of new data are encountered, practitioners can opt to retrain the model on all the data before deployment.
+
+Teacher GNN architecture. We used SAGE to represent GNNs so far. In Figure 5 (right), we show results with other various GNN teachers, e.g. GCN, GAT, and APPNP. We see that GLNNs can learn from different teachers and improve over MLPs. The performance is similar for all four teachers, with the GLNN distilled from APPNP very slightly worse than others. In fact, a similar phenomenon has been observed in Yang et al. (2021a) as well, i.e. APPNP benefits the student the least. One possible reason is that the first step of APPNP is to utilize the node’s own feature for prediction (prior to propagating over the graph), which is very similar to what the student MLP is doing, and thus provides less additional information to MLPs than other teachers.
+
+# 7 CONCLUSION AND FUTURE WORK
+
+In this paper, we explored whether we can bridge the best of GNNs and MLPs to achieve accurate and fast GML models for deployment. We found that KD from GNNs to MLPs helps to eliminate inference graph dependency, which results in GLNNs that are $1 4 6 \times - 2 7 3 \times$ faster than GNNs while enjoying competitive performance. We do a comprehensive study of GLNN properties. The promising results on 7 datasets across different domains show that GLNNs can be a handy choice for deploying latencyconstraint models. In our experiments, the current version of GLNNs on the Arxiv dataset doesn’t show competitive inductive performance. More advanced distillation techniques can potentially improve the GLNN performance, and we leave this investigation as future work.
+
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+
+# A DETAILED EXPERIMENT SETTINGS
+
+# A.1 DATASETS
+
+Here we provide a detailed description of the datasets we used to support our argument. Out of these datasets, 4 of them are citation graphs. Cora, Citeseer, Pubmed, ogbn-arxiv with the node features being descriptions of the papers, either bag-of-word vector, TF-IDF vector, or word embedding vectors.
+
+In Table 5, we provided the basic statistics of these datasets.
+
+Table 5: Dataset Statistics.   
+
+<table><tr><td>Dataset</td><td>#Nodes</td><td>#Edges</td><td>#Features</td><td># Classes</td></tr><tr><td>Cora</td><td>2,485</td><td>5,069</td><td>1,433</td><td>7</td></tr><tr><td>Citeseer</td><td>2,110</td><td>3,668</td><td>3,703</td><td>6</td></tr><tr><td>Pubmed</td><td>19,717</td><td>44,324</td><td>500</td><td>3</td></tr><tr><td>A-computer</td><td>13,381</td><td>245,778</td><td>767</td><td>10</td></tr><tr><td>A-photo</td><td>7,487</td><td>119,043</td><td>745</td><td>8</td></tr><tr><td>Arxiv</td><td>169,343</td><td>1,166,243</td><td>128</td><td>40</td></tr><tr><td>Products</td><td>2,449,029</td><td>61,859,140</td><td>100</td><td>47</td></tr></table>
+
+For all datasets, we follow the setting in the original paper to split the data. Specifically, for the five smaller datasets from the CPF paper, we use the CPF splitting strategy and each random seed corresponds to a different split. For the OGB datasets, we follow the OGB official splits based on time and popularity for Arxiv and Products respectively.
+
+# A.2 MODEL HYPERPARAMETERS
+
+The hyperparameters of GNN models on each dataset are taken from the best hyperparameters provided by the CPF paper and the OGB official examples. For the student MLPs and GLNN s, unless otherwise specified with -wi or $- \mathbf { L } i$ , we set the number of layers and the hidden dimension of each layer to be the same as the teacher GNN, so their total number of parameters stays the same as the teacher GNN.
+
+Table 6: Hyperparameters for GNNs on five datasets from the CPF paper.   
+
+<table><tr><td></td><td>SAGE</td><td>GCN</td><td>GAT</td><td>APPNP</td></tr><tr><td>#layers</td><td>2</td><td>2</td><td>2</td><td>2</td></tr><tr><td>hidden dim</td><td>128</td><td>64</td><td>64</td><td>64</td></tr><tr><td>learning rate</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.01</td></tr><tr><td>weight decay</td><td>0.0005</td><td>0.001</td><td>0.01</td><td>0.01</td></tr><tr><td>dropout</td><td>0</td><td>0.8</td><td>0.6</td><td>0.5</td></tr><tr><td>fan out</td><td>5,5</td><td>1</td><td>1</td><td>-</td></tr><tr><td>attention heads</td><td>1</td><td>=</td><td>8</td><td>1</td></tr><tr><td>power iterations</td><td>1</td><td>=</td><td>1</td><td>10</td></tr></table>
+
+Table 7: Hyperparameters for GraphSAGE on OGB datasets.   
+
+<table><tr><td>Dataset</td><td>Arxiv</td><td>Products</td></tr><tr><td># layers</td><td>3</td><td>3</td></tr><tr><td>hidden dim</td><td>256</td><td>256</td></tr><tr><td>learning rate</td><td>0.01</td><td>0.003</td></tr><tr><td>weight decay</td><td>0</td><td>0</td></tr><tr><td>dropout</td><td>0.2</td><td>0.5</td></tr><tr><td>normalization</td><td>batch</td><td>batch</td></tr><tr><td>fan out</td><td>[5,10,15]</td><td>[5,10,15]</td></tr></table>
+
+For GLNN s we do a hyperparameter search of learning rate from [0.01, 0.005, 0.001], weight decay from [0, 0.001, 0.002, 0.005, 0.01], and dropout from [0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6]
+
+# A.3 KNOWLEDGE DISTILLATION
+
+We use the distillation method proposed in Hinton et al. (2015) as in Equation 1, the hard labels are found to be helpful, so nonzero $\lambda s$ was suggested. In our case, we did a little tuning for $\lambda$ but didn’t find nonzero $\lambda s$ to be very helpful. Therefore, we report all of our results with $\lambda = 0$ , i.e. only the second term involving soft labels is effective. More careful tuning of $\lambda$ should further improve the results since the searching space is strictly larger. We implemented a weighted version in our code, and we leave the choice of $\lambda$ as future work.
+
+# A.4 THE TRANSDUCTIVE SETTING AND THE INDUCTIVE SETTING
+
+Given ${ \mathcal { G } } , X$ , and $Y ^ { L }$ , the goal of node classification can be divided into two different settings, i.e. transductive and inductive. In real applications, the former can correspond to predict missing attributes of a user based on the user profile and other existing users, and the latter can correspond to predict labels of some new nodes that are only seen during inference time. To create the inductive setting on a given dataset, we hold out some nodes along with edges connected to these nodes during data. Using notation defined above, we pick the inductive nodes training and use them for inductive evaluation only. These nodes and edges are picked from the test $\mathcal { V } _ { i n d } ^ { U } \subset \mathcal { V } ^ { U }$ , which partitions $\mathcal { V } ^ { U }$ into the disjoint inductive subset and observed subset, i.e. $\mathcal { V } ^ { U } = \mathcal { V } _ { o b s } ^ { U } \sqcup \mathcal { V } _ { i n d } ^ { U }$ V ind . Then we can take , $\mathcal { V } _ { i n d } ^ { U }$ furth, and raph, so we end up with. We show the input and $\mathcal { G } = \mathcal { G } _ { o b s } \sqcup \mathcal { G } _ { i n d }$ $\pmb { X } = \pmb { X } ^ { L } \sqcup \pmb { X } _ { o b s } ^ { U } \sqcup \overbrace { \pmb { X } _ { i n d } ^ { U } }$ $\pmb { Y } = \pmb { Y } ^ { L } \sqcup \pmb { Y } _ { o b s } ^ { U } \sqcup \pmb { Y } _ { i n d } ^ { U }$ output of both settings using the notations below.
+
+We visualize the difference between the inductive setting and the transductive setting in Figure 6.
+
+![](images/2a239fb19d94400d2ccb7f18def7895440e8208ed97573eb1cf684c4025f00ac.jpg)  
+Figure 6: The transductive setting and inductive setting illustrated by a 2-layer GNN. The middle shows the original graph used for training. The left shows the transductive setting, where the test node is in red and within the graph. The right shows the inductive setting, where the test node is an unseen new node.
+
+# A.5 CHOOSING SOFT TARGETS UNDER THE TRANSDUCTIVE SETTING
+
+For the transductive setting in Section 5.3, all the nodes in the graph, including the validation and test nodes, are used for the soft target generation. It seems less practical compared to the inductive case, but it is a necessary step to develop our argument. We now discuss the rationale behind this choice.
+
+Firstly, the transductive setting is the most common setting for graph data and it was used in most GNN architecture works and GNN acceleration works we mentioned in related work. Therefore, to avoid any confusion and for a fair comparison with numbers from previous literature, we start our experiments with exactly the same input and output as the standard transductive setting. Under this setting, the inputs to GNNs include all the node features and the graph structure, so GLNN is set to be able to access the same input. As GLNN includes a teacher training step and a distillation step, the soft labels of all the nodes are intermediate outputs produced by the teacher training step, and thus used for the second distillation step for the best GLNN performance. This transductive setting can boil down to a sanity check when the student is sufficiently large. Therefore, we separate the setting to be GLNN and $\mathsf { G L N N + }$ and report the results in Table 1 and Table 2 separately. In Table 1, we are checking how well GLNNs can perform compared to GNNs under the equal-parameter constraint. The results can be interpreted as given a fixed parameter budget, whether there exists one set of parameters (one instantiation of the MLP) that can achieve competitive results as the GNN. Only when this holds, should we further investigate the more interesting and challenging inductive case as in Section 5.4.
+
+Secondly, the task we focus on is node classification, which in many cases is considered as semisupervised learning with very scarce labels. For example, Pubmed only uses 60 labeled nodes (20 per class) out of 20K nodes for training. Rather than design an advanced model that can do few-shot learning, our goal here is to leverage as much data as possible to simplify the model for more efficient inference. We thus utilize the soft pseudo-labels on all the unlabelled nodes for the best GLNN performance. In reality, when there is a large amount of separate unlabeled data, these unlabeled data can be used for GLNN distillation training and a different set of labeled data can be used for evaluation. In our case, we mimic this scenario in the inductive setting in Section 5.4.
+
+# A.6 IMPLEMENTATION AND HARDWARD DETAILS
+
+The experiments on both baselines and our approach are implemented using PyTorch, the DGL (Wang et al., 2019) library for GNN algorithms, and Adam (Kingma & Ba, 2015) for optimization. We run all experiments on a machine with 80 Intel(R) Xeon(R) E5-2698 v4 $@$ 2.20GHz CPUs, and a single NVIDIA V100 GPU with 16GB RAM.
+
+# B SPACE AND TIME COMPLEXITY OF GNNS VS. MLPS
+
+Compared to MLP and GNN, GLNN provides a handy tool for users to trade-off between model accuracy and time complexity, which does not directly focus on space complexity. Given the space and time complexity are related, we now provide a more detailed discussion regarding these two complexities in our experiments.
+
+In Table 1, the model comparison was between equal-sized MLPs (GLNNs) and GNNs. While fixing parameter budget to control space complexity is a standard approach when comparing models, it is not completely fair for cross-model comparison especially for MLPs vs. GNNs. To do inference with GNNs, the graph needs to be loaded in the memory either entirely or batch by batch, and may use much larger space than the model parameters. Thus, the actual space complexity of GNNs is much higher than equal-sized MLPs. From the time complexity perspective, the major inference latency of GNNs comes from the data dependency as shown in Section 4. Under the same setting as Figure 1, we show in Figure 3 Left that even a 5-layer MLP with 8 times wider hidden layers still runs much faster than a single-layer SAGE. Another example of cross-model comparison is Transformers vs. RNNs. Large Transformers can have more parameters than RNNs because of the attention mechanism, but they are also faster than RNNs in general, which is an important consideration in the context of inference time minimization.
+
+In Table 1, we saw that for equal-sized comparison, GLNNs are not as accurate as GNNs on the OGB datasets. Following the discussion above and given the GLNNs used in Table 1 are relatively small (3 layers and 256 hidden dimensions) for millions of nodes in the OGB datasets, we ask whether this gap can be mitigated by increasing the MLP and thus GLNN sizes. The answer is yes as shown in Table 2.
+
+# C CONSISTENCY MEASURE OF MODEL PREDICTIONS AND GRAPH TOPOLOGY BASED ON MIN-CUT
+
+We introduce a metric to measure the consistency between model predictions and graph topology based on the min-cut problem in Section 5.6. The $K$ -way normalized min-cut problem, or simply min-cut, partitions $N$ nodes in $\nu$ into $K$ disjoint subsets by removing the minimum volume of edges. According to Dhillon et al. (2004), the min-cut problem can be expressed as
+
+$$
+\operatorname* { m a x } \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \frac { C _ { k } ^ { T } A C _ { k } } { C _ { k } ^ { T } D C _ { k } }
+$$
+
+with $C$ being the node assignment matrix that partitions $\nu$ , i.e. $C _ { i , j } = 1$ if node $i$ is assigned to class $j$ . $\pmb { A }$ being the adjacency matrix and $_ { D }$ being the degree matrix. This quantity we try to maximize here tells us whether the assignment is consistent with the graph topology. The bigger it is, the less edges need to be removed, and the assignment is more consistent with existing connections in the graph. In Bianchi et al. (2019), the authors show that when replacing the hard assignments $C \in \{ 0 , 1 \} ^ { N \times K }$ with a soft classification probability $\hat { \pmb Y } \in [ 0 , 1 ] ^ { N \times K }$ , a cut loss $\mathcal { L } _ { c u t }$ in Equation 2 can become a good approximation of Equation 4 and be used as the measuring metric.
+
+Table 8: GLNN predictions are much more consistent with the graph topology than MLPs. We show the $\mathcal { L } _ { c u t }$ values of GNNs, MLPs, and GLNN s on five CPF datasets. GLNN $\mathcal { L } _ { c u t }$ values become pretty close to the high $\mathcal { L } _ { c u t }$ values of GNNs, which were closely related to the GNN inductive bias.   
+
+<table><tr><td>Datasets</td><td>SAGE</td><td>MLP</td><td>GLNN</td></tr><tr><td>Cora</td><td>0.9347</td><td>0.7026</td><td>0.8852</td></tr><tr><td>Citeseer</td><td>0.9485</td><td>0.7693</td><td>0.9339</td></tr><tr><td>Pubmed</td><td>0.9605</td><td>0.9455</td><td>0.9701</td></tr><tr><td>A-computer</td><td>0.9003</td><td>0.6976</td><td>0.8638</td></tr><tr><td>A-photo</td><td>0.8664</td><td>0.7069</td><td>0.8398</td></tr><tr><td>Average</td><td>0.9221</td><td>0.7644</td><td>0.8986</td></tr></table>
+
+# D EXPRESSIVENESS OF GNNS VS. MLPS IN TERMS OF EQUIVALENCE CLASSES OF ROOTED GRAPHS
+
+In Chen et al. (2021), the expressiveness of GNNs and GA-MLPs were theoretically quantified in terms of induced equivalence classes of rooted graphs. We adopt their framework and perform a similar analysis for GNNs vs. MLPs. We first define rooted graphs.
+
+Definition 1 (Rooted Graph). A rooted graph, denoted as $\mathcal { G } ^ { [ i ] }$ is a graph with one node i in $\mathcal { G } ^ { [ i ] }$ designated as the root. GNNs, GA-MLPs, and MLPs can all be considered as functions on rooted graphs. The goal of a node-level task on node $i$ with label $\mathbf { \nabla } _ { \mathbf { \psi } _ { 3 } } \psi _ { i }$ is to fit a function to the input-output pairs $( \mathcal G ^ { [ i ] } , y _ { i } )$ .
+
+We denote the space of rooted graphs as $\mathcal { E }$ . Following Chen et al. (2021), the expressive power of a model on graph data is evaluated by its ability to approximate functions on $\mathcal { E }$ . This is further characterized as the number of induced equivalence classes of rooted graphs on $\mathcal { E }$ , with the equivalence relation defined as the following. Given a family of functions $\mathcal { F }$ on $\mathcal { E }$ , we define an equivalence relation $\simeq _ { \varepsilon , F }$ among all rooted graphs such that $\forall \mathcal { G } ^ { [ i ] } , \mathcal { G } ^ { \prime [ j ] } \in \mathcal { E } , \mathcal { G } ^ { [ i ] } \simeq _ { \mathcal { E } , \mathcal { F } } \mathcal { G } ^ { \prime [ j ] }$ if and only if $\forall f \in \mathcal { F } , f ( \mathcal { G } ^ { [ i ] } ) = f ( \mathcal { G } ^ { \prime [ j ] } )$ . We now give a proposition to characterize the GNN expressive power (proof in Appendix E).
+
+Proposition 1. With $\mathcal { X }$ denotes the set of all possible node features and assuming $| \mathcal { X } | \ge 2$ , with $m$ denotes the maximum node degree and assuming $m \geq 3$ , the total number of equivalence classes of rooted graphs induced by an $L$ -layer GNN is lower bounded by $\binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 }$
+
+As shown in Proposition 1, the expressive power of GNNs grows doubly-exponentially in the number of layers $L$ , which means it grows linearly in $L$ after taking $\log ( \log ( \cdot ) )$ . The expressive power GA-MLPs only grows exponentially in $L$ as shown in Chen et al. (2021). Under this framework, the expressive power of MLPs, which corresponds to a 0-layer GA-MLP, is $| \mathcal { X } |$ . Since the former is much larger than the latter, the conclusion will be GNNs are much more expressive than MLPs. The gap between these two numbers indeed exists, but empirically this gap will only make a difference when $| \mathcal { X } |$ is small. As in Chen et al. (2021), both the lower bound proof and the constructed examples showing GNNs are more powerful than GA-MLPs assumed $| \mathcal { X } | = 2$ . In real applications and datasets considered in this work, the node features can be high dimensional vectors like bag-of-words, which makes $| \mathcal { X } |$ enormous. Thus, this gap doesn’t matter much empirically.
+
+# E PROOF OF THE PROPOSITION 1
+
+To prove Proposition 1, we first define rooted aggregation trees, which is similar to but different from rooted graphs.
+
+Definition 2 (Rooted Aggregation Tree). The depth-K rooted aggregation tree of a rooted graph $\mathcal { G } ^ { [ i ] }$ is a depth-K rooted tree with $a$ (possibly many-to-one) mapping from every node in the tree to some node in $\mathcal { G } ^ { [ i ] }$ , where (i) the root of the tree is mapped to node $i$ , and (ii) the children of every node $j$ in the tree are mapped to the neighbors of the node in $\mathcal { G } ^ { [ i ] }$ to which $j$ is mapped.
+
+A rooted aggregation tree can be obtained by unrolling the neighborhood aggregation steps in the GNNs. An illustration of rooted graphs and rooted aggregation trees can be found in Chen et al. (2021) Figure 4. We denote the set of all rooted aggregation trees of depth $\mathrm { L }$ using $\mathcal { T } _ { L }$ . Then we use $\mathcal { T } _ { L , \mathcal { X } , m }$ to denote a subset of $\mathcal { T } _ { L }$ , where the node features belong to $\mathcal { X }$ , and all the nodes have exactly degree $m$ ( $m$ children), and at least two nodes out of these m nodes have different features. In other words, a node can’t have all identical children. With rooted aggregation trees defined, we are ready to prove Proposition 1. The proof is adapted from the proof of Lemma 3 in Chen et al. (2021).
+
+Proof. Since the number of equivalence classes on $\mathcal { E }$ induced by the family of all depth-L GNNs consists of all rooted graphs that share the same rooted aggregation tree of depth- $\mathrm { . L }$ (Chen et al., 2021), the lower bound problem in Proposition 1 can be reduced to lower bound $| \mathcal { T } _ { L } |$ , which can be further reduced to lower bound the subset $| \mathcal { T } _ { L , \mathcal { X } , m } |$ . We now show $\begin{array} { r } { | \mathcal { T } _ { L , \mathcal { X } , m } | \geq \binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 } } \end{array}$ inductively.
+
+When $L = 1$ , the root of the tree can have $| \mathcal { X } |$ different choices. For the children nodes, we $m$ $| \mathcal { X } |$ are allowed. This leads to . $\binom { | x | + m - 1 } { m }$ cases. Therefore, $\mathcal { T } _ { L + 1 , \mathcal { X } , m } = | \mathcal { X } | \binom { | \mathcal { X } | + m - 1 } { m } \ge \binom { | \mathcal { X } | + m - 2 } { m - 1 }$
+
+Assuming the statement holds for $L$ , we show it holds for $L + 1$ by constructing trees in $\tau _ { L + 1 , x , m }$ from $T , T ^ { \prime } \in \mathcal { T } _ { L , x , m }$ . We do this by assigning node features in $\mathcal { X }$ to the $m$ children of each leaf node in $T$ and $T ^ { \prime }$ . First note that when $T$ and $T ^ { \prime }$ are two non-isomorphic trees, two depth- $\mathrm { . L } { + } 1$ trees constructed from $T$ and $T ^ { \prime }$ will be different no matter how the node features are assigned. Now we consider all the trees can be constructed from $T$ by assign node features of children to leaf nodes.
+
+We first consider all paths from the root to leaves in $T$ . Each path consists of a sequence of nodes where the node features form a one-to-one mapping to an $\mathrm { L }$ -tuple $\tau \in \{ ( x _ { 1 } , \ldots , x _ { L } ) : x _ { i } \in \mathcal { X } \}$ . Leaf nodes are called node under $\tau$ if the path from the root to it corresponds to $\tau$ . The children of nodes under different $\tau \mathbf { S }$ are always distinguishable, and thus any assignments lead to distinct rooted aggregation trees of depth $L + 1$ . The assignment of children of nodes under the same $\tau$ , on the other hand, could be overcounted. Therefore, to lower bound $\tau _ { L + 1 , \mathcal { X } , m }$ , we only consider a special way of assignments to avoid over counting, which is that children of all nodes under the same $\tau$ are assigned the same set of features.
+
+Since we assumed that at least two nodes of $T$ have different features, there are at least $2 ^ { L }$ different $\tau \mathbf { S }$ corresponding to the path from the root to leaves. For a leaf node $j$ under a fixed $\tau$ , one of its children needs to have the same feature as $j$ ’s parent node. This restriction is due to the definition of rooted aggregation trees. Therefore, we only pick features for the other $m - 1$ nodes, which will be $\binom { | \mathcal { X } | + m - 2 } { m - 1 }$ cases for each $j$ . Then through this construction, the total number of depth- $\mathrm { . L } { + } 1$ trees from T can be lower bounded by |X |+m−2m−1 
2L. $T \in \mathcal { T } _ { L , \mathcal { X } , m }$ , so we derive $\begin{array} { r } { \mathcal { T } _ { L + 1 , \mathcal { X } , m } \geq \binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } } \mathcal { T } _ { L , \mathcal { X } , m } } \end{array}$ hav, and $\begin{array} { r } { \mathcal { T } _ { L , \mathcal { X } , m } \geq \binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { \sum _ { l = 1 } ^ { L } 2 ^ { l } } = } \end{array}$ $\binom { | \mathcal { X } | + m - 2 } { m - 1 } ^ { 2 ^ { L } - 1 }$
+
+# F ADVANCED GNN ARCHITECTURES AS THE TEACHER
+
+In our experiment, SAGE teacher is used throughout to avoid influence by model architecture. Some other GNNs like GCN are also considered in the ablation studies, but they are not the best known architecture for a specific dataset. To show GLNN has stronger performance given a stronger teacher, we consider the best teacher we can access on Products. We take $\mathrm { M L P { + } C S }$ Huang et al. (2021) from the OGB leaderboard as a new teacher, which has reported accuracy $8 4 . 1 8 \%$ and ranks 8 on the leadarboard as of Nov 2021. We choose $_ { \mathrm { M L P + C S } }$ instead of the other top 7 because the others either rely on raw text (additional info to the given node feature), or require a large GPU with ${ > } 1 6 \mathrm { G B }$ memory, which we don’t have access to. Also, their improvement is not super significant compared to $\mathrm { M L P { + } C S }$ , i.e. $84 \%$ to $86 \%$ . The result with $\mathrm { M L P { + } C S }$ teacher is shown in Table 9. We see that with the new teacher, performance of ${ \mathrm { G L N N } } +$ improves to be even better than SAGE $( 7 8 . 6 1 \% )$ , which shows GLNN can get stronger given a stronger teacher.
+
+Table 9: ${ \mathrm { G L N N } } +$ with $\mathrm { M L P { + } C S }$ teacher on Products   
+
+<table><tr><td></td><td>MLP+C&amp;S</td><td>MLP+</td><td>GLNN+</td></tr><tr><td>Acc</td><td>84.18</td><td>64.50</td><td>82.94</td></tr></table>
+
+# G GLNN WITH FEATURE AUGMENTATION FROM ONE-HOP NEIGHBORS
+
+In our main experiment, the inductive performance of GLNN on the Arxiv dataset is less desirable than others. We thus consider augment the node features with their one-hop neighbors to include more graph information. This can be seen as a middle ground between pure GLNNs and GNNs. For this new experiment, we follow the setting in Table 3 but with two new approaches. We explain the setting of these two approaches below.
+
+1. 1-hop GA-MLP: firstly, for each node $v$ , we collect features of its 1-hop neighbors $u$ to augment the raw feature of $v$ , i.e. $x _ { v }  \tilde { x } _ { v }$ , like in SGC. Then we train an MLP on the graph with $\tilde { x } _ { v }$ . Note if $v$ is in the observed graph but $u$ is in the inductive (unobserved during training) part, then $v$ doesn’t collect features from $u$ .
+
+2. 1-hop GA-GLNN: Go through the same feature augmentation step as 1-hop GA-MLP. Then train an MLP with distillation from teacher GNN.
+
+3. In summary, we compare 5 different models in the table below
+
+(a) SAGE: single model on $x _ { v }$   
+(b) MLP: single model on $x _ { v }$   
+(c) GLNN: SAGE teacher and MLP student on $x _ { v }$   
+(d) 1-hop GA-MLP: single model on $\tilde { x } _ { v }$   
+(e) 1-hop GA-GLNN: SAGE teacher on $x _ { v }$ , MLP student on $\tilde { x } _ { v }$
+
+We show in the table below, with 1-hop neighbor features, performance of GLNN improves a lot. This is expected as we also observe significant improvement from MLP to 1-hop GA-MLP. However, we indeed see 1-hop GA-GLNN (68.83) can further improve from 1-hop GA-MLP (66.62) and nearly match the teacher (70.64).
+
+Table 10: GLNN with feature augmentation from one-hop neighbor on Arxiv   
+
+<table><tr><td></td><td>Eval</td><td>SAGE</td><td>MLP</td><td>GLNN</td><td>1-hop GA-MLP</td><td>1-hop GA-GLNN</td></tr><tr><td>Arxiv</td><td>ind</td><td>70.64</td><td>55.40</td><td>60.48</td><td>66.62</td><td>68.83</td></tr><tr><td></td><td>tran</td><td>70.75</td><td>55.28</td><td>71.46</td><td>66.67</td><td>69.82</td></tr></table>
+
+As we have shown in Figure 3, the 1-Layer GNN in our case is roughly 4 times slower than GLNN (29.31ms vs. $7 . 5 6 \mathrm { m s } ,$ ), which should be a good approximation for the speed comparison between 1-hop GA-MLP/GA-GLNN and GLNN. This result is practically beneficial, as it gives practitioners more flexibility about how much accuracy they want to trade for less inference time.
+
+This section is a continuation of the ablation study of inductive split rate in Section 6. It generalizes Figure 5 Middle to more split rates (from 10:90 to 90:10), and explicitly show the inductive and transductive performance on each dataset. For better visualization, the training data label rate is also reduced from 20 per class to 5 per class in the following plots.
+
+![](images/46024ef2ae15c38b05bcf61172d3fc0a7188e2c13b24b720ed673291135bb45b.jpg)  
+Figure 7: Model inductive performance comparison between MLP, GNN(SAGE), and GLNN under different inductive split rate in the production setting.
+
+![](images/f66528cb30229f3d636a468c436934c7c9c543d8c492f108574b73bb3899b35d.jpg)  
+Figure 8: Model transductive performance comparison between MLP, GNN(SAGE), and GLNN under different inductive split rate in the production setting.
+
+# I GLNN UNDER NODE FEATURE HETEROGENEITY AND NON-HOMOPHILY
+
+Besides the 7 datasets used in the main experiments, we consider 4 more datasets from Ivanov & Prokhorenkova (2021) and Lim et al. (2021) to further evaluate GLNN.
+
+The House_class and VK_class datasets are from Ivanov & Prokhorenkova (2021). The node features of these two graphs are based on tabular data, which have different types, scales, and meanings as the opposite of the bag-of-word node features in Cora and etc. Some basic statistics of the datasets are shown in the following table.
+
+Table 11: Statistics of dataset with heterogeneous node features   
+
+<table><tr><td>Dataset</td><td># Nodes</td><td>#Edges</td><td># Features</td><td>#Classes</td></tr><tr><td>House_class</td><td>20,640</td><td>182,146</td><td>6</td><td>5</td></tr><tr><td>VK_class</td><td>54,028</td><td>213,644</td><td>14</td><td>7</td></tr></table>
+
+We apply the GLNN on House_class and VK_class using the best BGNN model from Ivanov & Prokhorenkova (2021) as the teacher. The comparison is shown in the following table. Ivanov & Prokhorenkova (2021) also includes GAT, GCN, AGNN, and APPNP as baselines, whose performance on these two datasets are quite similar (difference $< 0 . 0 2 5$ ). We compare with these baselines by including the best result among the 4 GNN models and refer it as GNN in the table below, i.e. GNN $=$ max(GAT, GCN, AGNN, APPNP). From the table, we see that GLNN can improve from MLP, outperform GNN and LightGBM, and become competitive to the teacher BGNN.
+
+Table 12: GLNN on datasets with heterogeneous node features. Numbers other than GLNN are taken from Ivanov & Prokhorenkova (2021)   
+
+<table><tr><td>Dataset</td><td>LightGBM</td><td>GNNs</td><td>BGNN</td><td>MLP</td><td>GLNN</td></tr><tr><td>House_class</td><td>0.55</td><td>0.625</td><td>0.682</td><td>0.534</td><td>0.672</td></tr><tr><td>VK_class</td><td>0.57</td><td>0.577</td><td>0.683</td><td>0.567</td><td>0.641</td></tr></table>
+
+We further pick the non-homophilous Penn94 and Pokec datasets from Lim et al. (2021). Some basic statistics of the datasets are shown in the following table.
+
+Table 13: Statistics of non-homophilous datasets   
+
+<table><tr><td>Dataset</td><td># Nodes</td><td>#Edges</td><td># Features</td><td># Classes</td></tr><tr><td>Penn94</td><td>41,536</td><td>1,590,655</td><td>5</td><td>2</td></tr><tr><td>Pokec</td><td>1,632,803</td><td>30,622,564</td><td>65</td><td>2</td></tr></table>
+
+Using the GCN teacher, we see that the performance of GLNN is improved over MLP and becomes competitive to the teacher GCN on $\mathtt { P e n n 9 4 }$ . However, on Pokec, the simple LINK model can achieve very good performance, and it is better than most GNNs reported in Lim et al. (2021). LINK is a purely structural model which does not use node features at all. This shows that the Pokec dataset corresponds to the setting we discussed in Sec 5.8 (limitations of GLNN) – if the node labels can be largely determined by only the graph structure, then GLNN will struggle. We observe that GLNN is not as good as LINK owing to this limitation. However, we still see that for most of the non-homophilous datasets, MLPs already work quite well on them, and we can use GLNN for the other ones like Penn94.
+
+Table 14: GLNN on non-homophilous datasets. Numbers other than GLNN are taken from Lim et al. (2021)   
+
+<table><tr><td>Dataset</td><td>LINK</td><td>GCN</td><td>MLP</td><td>GLNN</td></tr><tr><td>Penn94</td><td>80.79</td><td>82.47</td><td>73.61</td><td>81.69</td></tr><tr><td>Pokec</td><td>80.54</td><td>75.45</td><td>62.37</td><td>61.32</td></tr></table>
+
+# J MODEL COMPARISON WITH NOISY NODE FEATURES
+
+In Section 6, we conducted an ablation study to compare model performance with noisy node features, and the result is shown in the left plot in Figure 5. There are two subtle points in this plot. (1) The performance of GNN is still relatively high for high noisy features, even when $\alpha = 1$ and the features are completely random. (2) For completely random features, the performance of GLNN is still higher than MLP. We now discuss and explain them in more detail.
+
+GNN Performance on Random Features. GNN still performs well because nodes with the same labels are likely to be connected and GNN can overfit the training data. We explain the detail through a toy example. Suppose there is a 4-clique containing nodes A, B, C, D in the graph with only a single edge D-E connects this clique to other graph nodes. Suppose A, B, C, D all have iid random Gaussian raw features and the same class label c. Let’s pick A to be the inductive test node and assume E and the triangle formed by B, C, D to be in the training graph. Let’s consider a simple example for 1-layer GCN and break down message passing into feature aggregation and nonlinear transformation. During training, GNN can overfit the data by learning a nonlinear transformation which maps the aggregated features of B, C, D to class c. The aggregated features of B and C will just be the average of the raw features of B, C, D. Although E is also involved in D’s feature aggregation step, the aggregated features of D will also be very close to this average. Then when test on A, the aggregated feature of A will likely be classified to the same class c by the overfitted nonlinear transformation because it is the average of raw node features of A, B, C, D. In this case, GNN can actually correctly classify A because of the overfitting. For GNNs with more layers and graphs with more neighbor nodes, the conclusion may be generalized.This is roughly sort of a “majority vote” process. For a test node A, if many nodes, which A collects features from, have the same class label and appear in the training graph, then A will be classified as this class by an overfitted classifier.
+
+GLNN and MLP Performance on Random Features. The gap between MLP and GLNN is due to imbalanced datasets. The GLNN can learn the imbalance from soft labels, whereas MLPs can only access uniformly picked training nodes. We explain more detail using the A-computer dataset as an example, for which the gap between MLP and GLNN is obvious. The task is 10-class classification. With random node features $( \alpha { = } 1 )$ ), the inductive accuracy for MLP is 0.0652 and 0.2538 for GLNN. If the data labels are uniform, then both models should give an accuracy around 0.1. However, the labels on the inductive dataset are actually imbalanced. We show the results in Figure 9. The hist on the left is the label distribution of the inductive test set. In particular, class 4 takes about $40 \%$ . However, given this imbalance, the standard train-test split selects training nodes uniformly among labels. In this case, 20 nodes per class. Therefore, the predictions of MLP on random features are expected to be relatively uniform because the 200 nodes we train it on are uniform. This gives the hist shown in the middle, where the largest class takes about $1 7 . 5 \%$ . Finally, for GLNN, we train it on all the 200 training nodes with hard labels, plus soft labels of other nodes in the observed graph $\mathcal { G } _ { o b s }$ (see Section 5.2). Since these extra nodes are selected randomly, whose label distribution is actually similar to the label distribution on the whole data and the distribution on the inductive test set. Therefore, we get the GLNN predictions hist on the right. Although for each node, we can’t assign a prediction correlated to its feature, on average the distribution is very close to the true label distribution on the inductive test set and has a much higher expectation. In fact, if the prediction distribution is exactly the true distribution on the inductive test set, the expectation will be 0.2169. GLNN actually does even a bit better by putting its bet more on the largest class.
+
+![](images/403a9c5a1b32a03dcf520caf81b8a2a56b3a306859c74e50c516bf26d61ec307.jpg)  
+Figure 9: Inductive (predicted) label distribution on the A-computer dataset. Left: true labels. Middle: predicted labels by MLP. Right: predicted labels by GLNN.

parse/dev/7r6kDq0mK_/7r6kDq0mK_.md

@@ -0,0 +1,396 @@
+# LATENT IMAGE ANIMATOR: LEARNING TO ANIMATE IMAGES VIA LATENT SPACE NAVIGATION
+
+Yaohui Wang, Di Yang, Francois Bremond & Antitza Dantcheva Inria, Universite C´ ote d’Azur ˆ {yaohui.wang,di.yang,francois.bremond,antitza.dantcheva}@inria.fr
+
+# ABSTRACT
+
+Due to the remarkable progress of deep generative models, animating images has become increasingly efficient, whereas associated results have become increasingly realistic. Current animation-approaches commonly exploit structure representation extracted from driving videos. Such structure representation is instrumental in transferring motion from driving videos to still images. However, such approaches fail in case the source image and driving video encompass large appearance variation. Moreover, the extraction of structure information requires additional modules that endow the animation-model with increased complexity. Deviating from such models, we here introduce the Latent Image Animator (LIA), a self-supervised autoencoder that evades need for structure representation. LIA is streamlined to animate images by linear navigation in the latent space. Specifically, motion in generated video is constructed by linear displacement of codes in the latent space. Towards this, we learn a set of orthogonal motion directions simultaneously, and use their linear combination, in order to represent any displacement in the latent space. Extensive quantitative and qualitative analysis suggests that our model systematically and significantly outperforms state-of-art methods on VoxCeleb, Taichi and TED-talk datasets w.r.t. generated quality. Source code and pre-trained models are publicly available1.
+
+![](images/e378522fe7bd0a0020ce31d5201a8ccda8a4e2a620417861db6f080cf4f0c4a2.jpg)  
+Figure 1: LIA animation examples. The two images of Marilyn Monroe and Emmanuel Macron are animated by LIA, which transfers motion of a driving video (smaller images on the top) from VoxCeleb dataset (Chung et al., 2018) onto the still images. LIA is able to successfully animate these two images without relying on any explicit structure representations, such as landmarks and region representations.
+
+# 1 INTRODUCTION
+
+In the series of science fiction books Harry Potter (Rowling et al., 2016; Rowling, 2019), wizards and witches were able to magically enchant portraits, bringing them to life. Remarkable progress of deep generative models has recently turned this vision into reality. This work examines the scenario where a framework animates a source image by motion representations learned from a driving video. Existing approaches for image animation are classically related to computer graphics (Cao et al., 2014; Thies et al., 2016; 2019; Zhao et al., 2018) or exploit motion labels (Wang et al., 2020b) and structure representations such as semantic maps (Pan et al., 2019; Wang et al., 2018; 2019), human keypoints (Jang et al., 2018; Yang et al., 2018; Walker et al., 2017; Chan et al., 2019; Zakharov et al., 2019; Wang et al., 2019; Siarohin et al., 2019), 3D meshes (Liu et al., 2019; Chen et al., 2021), and optical flows (Li et al., 2018; Ohnishi et al., 2018). We note that the ground truth of such structure representations has been computed a-priori for the purpose of supervised training, which poses constraints on applications, where such representations of unseen testing images might be fragmentary or difficult to access.
+
+Self-supervised motion transfer approaches (Wiles et al., 2018; Siarohin et al., 2019; 2021) accept raw videos as input and learn to reconstruct driving images by warping source image with predicted dense optical flow fields. While the need for domain knowledge or labeled ground truth data has been obviated, which improves performance on in-the-wild testing images, such methods entail necessity of explicit structure representations as motion guidance. Prior information such as keypoints (Siarohin et al., 2019; Wang et al., 2021a) or regions (Siarohin et al., 2021) are learned in an end-to-end training manner by additional networks as intermediate features, in order to predict target flow fields. Although online prediction of such representations is less tedious than the acquisition of ground truth labels, it still strains the complexity of networks.
+
+Deviating from such approaches, we here aim to fully eliminate the need of explicit structure representations by directly manipulating the latent space of a deep generative model. To the best of our knowledge, this constitutes a new direction in the context of image animation. Our work is motivated by interpretation of GANs (Shen et al., 2020; Goetschalckx et al., 2019; Jahanian et al., 2020; Voynov & Babenko, 2020), showcasing that latent spaces of StyleGAN (Karras et al., 2019; 2020b) and BigGAN (Brock et al., 2019) contain rich semantically meaningful directions. Given that walking along such directions, basic visual transformations such as zooming and rotation can be induced in generated results. As in image animation, we have that motion between source and driving images can be considered as higher-level transformation, a natural question here arises: can we discover a set of directions in the latent space that induces high-level motion transformations collaboratively?
+
+Towards answering this question, we introduce LIA, a novel Latent Image Animator constituting of an autoencoder for animating still images via latent space navigation. LIA seeks to animate a source image via linearly navigating associated source latent code along a learned path to reach a target latent code, which represents the high-level transformation for animating the source image. We introduce a Linear Motion Decomposition (LMD) approach aiming to represent a latent path via a linear combination of a set of learned motion directions and associated magnitudes. Specifically, we constrain the set as an orthogonal basis, where each vector indicates a basic visual transformation. By describing the whole motion space using such learned basis, LIA eliminates the requirement of explicit structure representations.
+
+In addition, we design LIA to disentangle motion and appearance within a single encoder-generator architecture. Deviating from existing methods using separate networks to learn disentangled features, LIA integrates both, latent motion code, as well as appearance features in a single encoder, which highly reduces the model complexity and simplifies training.
+
+We provide evaluation on multiple datasets including VoxCeleb (Chung et al., 2018), TaichiHD (Siarohin et al., 2019) and TED-talk (Siarohin et al., 2021). In addition, we show that LIA outperforms the state-of-the-art in preserving the facial structure in generated videos in the setting of one-shot image animation on unseen datasets such as FFHQ (Karras et al., 2019) and GermanPublicTV (Thies et al., 2020).
+
+# 2 RELATED WORK
+
+Video generation GAN-based video generation is aimed at mapping Gaussian noise to video, directly and in the absence of prior information (Vondrick et al., 2016; Saito et al., 2017; Tulyakov et al., 2018; Wang et al., 2020a; Wang, 2021). Approaches based on deep probabilistic models (Denton & Birodkar, 2017; Li & Mandt, 2018; Bhagat et al., 2020; Xie et al., 2020) were also proposed to tackle this problem, however only show results on toy datasets with low resolution. Recently, with the progress of GANs in photo-realistic image generation (Brock et al., 2019; Karras et al., 2019; 2020a), a series of works (Clark et al., 2019; Wang et al., 2021c) explored production of highresolution videos by incorporating the architecture of an image generator into video GANs, trained jointly with RNNs. Tian et al. (2021) directly leveraged the knowledge of a pre-trained StyleGAN to produce videos of resolution up to $1 0 2 4 \times \mathrm { i } 0 2 4$ . Unlike these approaches, which generate random videos based on noise vectors in an unconditional manner, in this paper, we focus on conditionally creating novel videos by transferring motion from driving videos to input images.
+
+![](images/adf402bc242a87960126961b2dd01a4fedaa5916e13c5407ab9a2e27d13beee5.jpg)  
+Figure 2: General pipeline. Our objective is to transfer motion via latent space navigation. The entire training pipeline consists of two steps. Firstly, we encode a source image $x _ { s }$ into a latent code $z _ { s  r }$ . By linearly navigating $z _ { s  r }$ along a path $w _ { r \to d }$ , we reach a target latent code $z _ { s  d }$ . The latent paths are represented by a linear combination between a set of learned motion directions (e.g., $d _ { 1 }$ and $d _ { 2 }$ ), which is an orthogonal basis, and associated magnitudes. In the second step, we decode $z _ { s  d }$ to a target dense optical flow field $\phi _ { s  d }$ , which is used to warp $x _ { s }$ into the driving image $x _ { d }$ . While we train our model using images from the same video sequence, in the testing phase, $x _ { s }$ and $x _ { d }$ generally pertain to different identities.
+
+Latent space editing In an effort to control generated images, recent works explored the discovery of semantically meaningful directions in the latent space of pre-trained GANs, where linear navigation corresponds to desired image manipulation. Supervised (Shen et al., 2020; Jahanian et al., 2020; Goetschalckx et al., 2019) and unsupervised (Voynov & Babenko, 2020; Peebles et al., 2020; Shen & Zhou, 2021) approaches were proposed to edit semantics such as facial attributes, colors and basic visual transformations (e.g., rotation and zooming) in generated or inverted real images (Zhu et al., 2020; Abdal et al., 2020). In this work, as opposed to finding directions corresponding to individual visual transformations, we seek to learn a set of directions that cooperatively allows for high-level visual transformations that can be beneficial in image animation.
+
+Image animation Related approaches (Chan et al., 2019; Wang et al., 2018; Zakharov et al., 2019; Wang et al., 2019; Yang et al., 2020) in image animation required strong prior structure labels as motion guidance. In particular, Chan et al. (2019), Yang et al. (2020) and Wang et al. (2018) proposed to map representations such as human keypoints and facial landmarks to videos in the setting of image-to-image translation proposed by Isola et al. (2017). However, such approaches were only able to learn an individual model for a single identity. Transferring motion on new appearances requires retraining the entire model from scratch by using videos of target identities. Several recent works (Zakharov et al., 2019; Wang et al., 2019) explored meta learning in fine-tuning models on target identities. While only few images of target identities were required during inference time, it was still compulsory to input pre-computed structure representations in those approaches, which usually are hard to access in many real-world scenarios. Towards addressing this issue, very recent works (Siarohin et al., 2019; 2021; Wang et al., 2021b; Wiles et al., 2018) proposed to learn image animation in self-supervised manner, only relying on RGB videos for both, training and testing without any priors. They firstly predicted dense flow fields from input images, which were then utilized to warp source images, in order to obtain final generated results. Inference only required one image of a target identity without any fine-tuning step on pre-trained models. While no priors were required, state-of-the-art methods still followed the idea of using explicit structure representations. FOMM (Siarohin et al., 2019) proposed a first order motion approach to predict keypoints and local transformations online to generate flow fields. Siarohin et al. (2021) developed this idea to model articulated objects by replacing a keypoints predictor by a PCA-based region prediction module. Wang et al. (2021b) extended FOMM by predicting 3D keypoints for view-free generation. We note though that in all approaches, given that keypoints or regions are inadequately predicted, the quality of generated images drastically decreases. In contrast to such approaches, our method does not require any explicit structure representations. We dive into the latent space of the generator and self-learn to navigate motion codes in certain directions with the goal to reach target codes, which are then decoded to flow fields for warping.
+
+![](images/203fe2748a461292b0737c6c4a21c45cb5adc8b929922aec910a81a2c52ff5dd.jpg)  
+Figure 3: Overview of LIA. LIA is an autoencoder consisting of two networks, an encoder $E$ and a generator $G$ . In the latent space, we apply Linear Motion Decomposition (LMD) towards learning a motion dictionary $D _ { m }$ , which is an orthogonal basis where each vector represents a basic visual transformation. LIA takes two frames sampled from the same video sequence as source image $x _ { s }$ and driving image $x _ { d }$ respectively during training. Firstly, it encodes $x _ { s }$ into a source latent code $z _ { s  r }$ and $x _ { d }$ into a magnitude vector $\bar { A _ { r \to d } }$ . Then, it linearly combines $A _ { r  d }$ and a trainable $D _ { m }$ using LMD to obtain a latent path $w _ { r \to d }$ , which is used to navigate $z _ { s _ {  } r }$ to a target code $z _ { s  d }$ . Finally, $G$ decodes $z _ { s  d }$ into a target dense flow field and warps $x _ { s }$ to an output image $x _ { s \to d }$ . The training objective is to reconstruct $x _ { d }$ using $x _ { s \to d }$ .
+
+# 3 METHOD
+
+Self-supervised image animation aims at learning to transfer motion from a subject of a driving video to a subject in a source image based on training with a large video dataset. In this work, we propose to model such motion transformation via latent space navigation. The general pipeline is illustrated in Fig. 2. Specifically, for training, our model takes in a pair of source and driving images, randomly sampled from one video sequence. These two images are encoded into a latent code which is used to represent motion transformation in the image space. The training objective is to reconstruct the driving image by combining source image with learned motion transformation. For testing, frames of a driving video are sequentially processed with the source image to animate the source subject.
+
+We provide an overview of the proposed model in Fig. 3. Our model is an autoencoder, consisting of two main networks, an encoder $\bar { E }$ and a generator $\bar { G }$ . In general, our model requires two steps to transfer motion. In the first step, $E$ encodes source and driving images $\boldsymbol { x } _ { s } , \boldsymbol { x } _ { d } \sim \dot { \mathcal { X } } \in \mathbb { R } ^ { 3 \times H \times W }$ into latent codes in the latent space. The source code is then navigated into a target code, which is used to represent target motion transformation, along a learned latent path. Based on proposed Linear Motion Decomposition (LMD), we represent such a path as a linear combination of a set of learned motion directions and associated magnitudes, which are learned from $x _ { d }$ . In the second step, once the target latent code is obtained, $G$ decodes it as a dense flow field $\phi _ { s \to d } \sim \Phi \in \mathbb { R } ^ { 2 \times H \times W }$ and uses $\bar { \phi _ { s \to d } }$ to warp $x _ { s }$ and then to obtain the output image. In the following, we proceed to discuss the two steps in detail.
+
+# 3.1 LATENT MOTION REPRESENTATION
+
+Given a source image $x _ { s }$ and a driving image $x _ { d }$ , our first step constitutes of learning a latent code $z _ { s \to d } \sim \mathcal { Z } \in \mathbb { R } ^ { N }$ to represent the motion transformation from $x _ { s }$ to $x _ { d }$ . Due to the uncertainty of two images, directly learning $z _ { s  d }$ puts forward a high requirement on the model to capture a complex distribution of motion. Mathematically, it requires modeling directions and norms of the vector $z _ { s  d }$ simultaneously, which is challenging. Therefore, instead of modeling motion transformation $x _ { s } \to x _ { d }$ , we assume there exists a reference image $x _ { r }$ and motion transfer can be modeled as $x _ { s }  x _ { r }  x _ { d }$ , where $z _ { s  d }$ is learned in an indirect manner. We model $z _ { s  d }$ as a target point in the latent space, which can be reached by taking linear walks from a starting point $z _ { s  r }$ along a linear path $w _ { r \to d }$ (see Fig. 2), given by
+
+$$
+z _ { s  d } = z _ { s  r } + w _ { r  d } ,
+$$
+
+where $z _ { s  r }$ and $w _ { r \to d }$ indicate the transformation $x _ { s } \to x _ { r }$ and $x _ { r } \to x _ { d }$ respectively. Both $z _ { s  r }$ and $w _ { r \to d }$ are learned independently and $z _ { s  r }$ is obtained by passing $x _ { s }$ through $E$ .
+
+We learn $w _ { r \to d }$ via Linear Motion Decomposition (LMD). Our idea is to learn a set of motion directions $D _ { m } = \{ \mathbf { d _ { 1 } } , . . . , \mathbf { d _ { M } } \}$ to represent any path in the latent space. We constrain $D _ { m }$ as an orthogonal basis, where each vector indicates a motion direction $\mathbf { d _ { i } }$ . We then combine each vector in the basis with a vector $A _ { r  d } = \{ a _ { 1 } , . . . , a _ { M } \}$ , where $a _ { i }$ represents the magnitude of $\mathbf { d _ { i } }$ . Hence, any linear path in the latent space can be represented using a linear combination
+
+$$
+w _ { r  d } = \sum _ { i = 1 } ^ { M } a _ { i } { \bf d _ { i } } ,
+$$
+
+where $\mathbf { d } _ { \mathbf { i } } \in \mathbb { R } ^ { N }$ and $a _ { i } \in \mathbb { R }$ for all $i \in \{ 1 , . . . , M \}$ . Semantically, each $\mathbf { d _ { i } }$ should represent a basic visual transformation and $a _ { i }$ indicates the required steps to walk in $\mathbf { d _ { i } }$ towards achieving $w _ { r \to d }$ . Due to $D _ { m }$ entailing an orthogonal basis, any two directions $\mathbf { d } _ { \mathrm { i } } , \mathbf { d } _ { \mathrm { j } }$ follow the constrain
+
+$$
+< { \bf d _ { i } } , { \bf d _ { j } } > = \left\{ \begin{array} { c c } { { 0 } } & { { i \neq j } } \\ { { 1 } } & { { i = j . } } \end{array} \right.
+$$
+
+We implement $D _ { m }$ as a learnable matrix and apply the Gram-Schmidt process during each forward pass, in order to meet the requirement of orthogonality. $A _ { r  d }$ is obtained by mapping $z _ { d  r }$ , which is the output of $x _ { d }$ after $E$ , through a 5-layer MLP. The final formulation of latent motion representation for each $x _ { s }$ and $x _ { d }$ is thus given as
+
+$$
+z _ { s  d } = z _ { s  r } + \sum _ { i = 1 } ^ { M } a _ { i } { \bf d _ { i } } .
+$$
+
+# 3.2 LATENT CODE DRIVEN IMAGE ANIMATION
+
+Once we obtain $z _ { s  d }$ , in our second step, we use $G$ to decode a flow field $\phi _ { s \to d }$ and warp $x _ { s }$ . Our $G$ consists of two components, a flow field generator $G _ { f }$ and a refinement network $G _ { r }$ (we provide details in App. A).
+
+Towards learning multi-scale features, $G$ is designed as a residual network containing $N$ models to produce a pyramid of flow fields $\phi _ { s \to d } = \{ \bar { \phi _ { i } } \bar  \} _ { 1 } ^ { N }$ in different layers of $G _ { f }$ . Multi-scale source features $x _ { s } ^ { e n c } = \{ x _ { i } ^ { e n c } \} _ { 1 } ^ { N }$ are obtained from $E$ and are warped in $G _ { f }$ .
+
+However, as pointed out by Siarohin et al. (2019), only relying on $\phi _ { s \to d }$ to warp source features is insufficient to precisely reconstruct driving images due to the existing occlusions in some positions of $x _ { s }$ . In order to predict pixels in those positions, the network is required to inpaint the warped feature maps. Therefore, we predict multi-scale masks $\{ m _ { i } \} _ { 1 } ^ { N }$ along with $\{ \phi _ { i } \} _ { 1 } ^ { N }$ in $G _ { f }$ to mask out the regions required to be inpainted. In each residual module, we have
+
+$$
+x _ { i } ^ { \prime } = \mathcal { T } ( \phi _ { i } , x _ { i } ^ { e n c } ) \odot m _ { i } ,
+$$
+
+where $\odot$ denotes the Hadamard product and $\tau$ denotes warping operation, whereas $\boldsymbol { x } _ { i } ^ { \prime }$ signifies the masked features. We generate both dense flow fields, as well as masks by letting each residual module output a 3-channel feature map in which the first two channels represent ${ \bar { \phi } } _ { i }$ and the last channel $m _ { i }$ . Based on an inpainted feature map $f ( x _ { i } ^ { \prime } )$ , as well as an upsampled image $g ( x _ { i - 1 } )$ provided by the previous module in $G _ { r }$ , the RGB image from each module is given by
+
+$$
+o _ { i } = f ( x _ { i } ^ { \prime } ) + g ( o _ { i - 1 } ) ,
+$$
+
+where $f$ and $g$ denote the inpainting and upsampling layers, respectively. The output image $o _ { N }$ from the last module constitutes the final generated image $x _ { s \to d } = o _ { N }$ .
+
+# 3.3 LEARNING
+
+We train LIA in a self-supervised manner to reconstruct $x _ { d }$ using three losses, i.e., a reconstruction loss $\mathcal { L } _ { r e c o n }$ , a perceptual loss $\mathcal { L } _ { v g g }$ (Johnson et al., 2016) and an adversarial loss $\mathcal { L } _ { a d v }$ . We use $\mathcal { L } _ { r e c o n }$ to minimize the pixel-wise $\mathbb { L } _ { 1 }$ distance between $x _ { d }$ and $x _ { s \to d }$ , calculated as
+
+$$
+\mathcal { L } _ { r e c o n } ( x _ { s  d } , x _ { d } ) = \mathbb { E } [ \| x _ { d } - x _ { s  d } \| _ { 1 } ] .
+$$
+
+Towards minimizing the perceptual distance, we apply a VGG19-based $\mathcal { L } _ { v g g }$ on multi-scale feature maps between real and generated images, written as
+
+$$
+\mathcal { L } _ { v g g } ( x _ { s  d } , x _ { d } ) = \mathbb { E } [ \sum _ { n } ^ { N } \| F _ { n } ( x _ { d } ) - F _ { n } ( x _ { s  d } ) \| _ { 1 } ] ,
+$$
+
+where $F _ { n }$ denotes the $n ^ { t h }$ layer in a pre-trained VGG19 (Simonyan & Zisserman, 2015). In practice, towards penalizing real and generated images in multi-scale images, we use a pyramid of four resolutions, namely $2 5 6 \times 2 5 6$ , ${ \bar { 1 } } 2 8 \times 1 2 8$ , $6 4 \times 6 4$ and $3 2 \times 3 2$ as inputs of VGG19. The final perceptual loss is the addition of perceptual losses in four resolutions.
+
+Further, towards generating photo-realistic results, we incorporate a non-saturating adversarial loss $\mathcal { L } _ { a d v }$ on $x _ { s \to d }$ , which is calculated as
+
+$$
+\mathcal { L } _ { a d v } ( x _ { s  d } ) = \mathbb { E } _ { x _ { s  d } \sim p _ { r e c } } [ - l o g ( D ( x _ { s  d } ) ] ,
+$$
+
+where $D$ is a discriminator, aimed at distinguishing reconstructed images from the original ones. Our full loss function is the combination of three losses with $\lambda$ as a balanced hyperparameter
+
+$$
+{ \mathcal { L } } ( x _ { s  d } , x _ { d } ) = { \mathcal { L } } _ { r e c o n } ( x _ { s  d } , x _ { d } ) + \lambda { \mathcal { L } } _ { v g g } ( x _ { s  d } , x _ { d } ) + { \mathcal { L } } _ { a d v } ( x _ { s  d } ) .
+$$
+
+# 3.4 INFERENCE
+
+In inference stage, given a driving video sequence $V _ { d } = \{ x _ { t } \} _ { 1 } ^ { T }$ , we aim to transfer motion from $V _ { d }$ to $x _ { s }$ , in order to generate a novel video $V _ { d  s } = \{ x _ { t  s } \} _ { 1 } ^ { T }$ . If $V _ { d }$ and $x _ { s }$ stem from the same video sequence, i.e., $x _ { s } ~ = ~ x _ { 1 }$ , our task comprises of reconstructing the entire original video sequence. Therefore, we construct the latent motion representation of each frame using absolute transfer, which follows the training process, given as
+
+$$
+z _ { s  t } = z _ { s  r } + w _ { r  t } , t \in \{ 1 , . . . , T \} .
+$$
+
+However, in real world applications, interest is rather placed on the scenario, where motion transfer between $x _ { s }$ and $V _ { d }$ , the latter stemming from different identities, i.e., $x _ { s } \neq x _ { 1 }$ . Taking a talking head video as an example, in this setting, beyond identity, $x _ { 1 }$ and $x _ { s }$ might also differ in pose and expression. Therefore, we propose relative transfer to eliminate the motion impact of $w _ { r \to 1 }$ and involve motion of $w _ { r \to s }$ in the full generated video sequence. Owing to a linear representation of the latent path, we can easily represent $z _ { s  t }$ for each frame as
+
+$$
+\begin{array} { r l } & { z _ { s  t } = ( z _ { s  r } + w _ { r  s } ) + ( w _ { r  t } - w _ { r  1 } ) } \\ & { \qquad = z _ { s  s } + ( w _ { r  t } - w _ { r  1 } ) , \ t \in \{ 1 , . . . , T \} . } \end{array}
+$$
+
+The first term in Eq. (12), $z _ { s  s }$ indicates the reconstruction of $x _ { s }$ , while the second term $( w _ { r  t } - w _ { r  1 } )$ represents the motion from $x _ { 1 }$ to $x _ { t }$ . This equation indicates that the original pose is preserved in $x _ { s }$ , at the same time motion is transferred from $V _ { d }$ . We note that in order to completely replicate the position and pose in $V _ { d }$ , it requires $x _ { s }$ and $x _ { 1 }$ to contain similar poses in relative motion transfer.
+
+# 4 EXPERIMENTS
+
+In this section, we firstly describe our experimental setup including implementation details and datasets. Secondly, we qualitatively demonstrate generated results based on testing datasets. Then, we provide quantitative evaluation w.r.t. image quality on (a) same-identity reconstruction, (b) crossvideo motion transfer, presenting (c) a user study. Next, we conduct an ablation study that demonstrates (d) the effectiveness of our proposed motion dictionary, as well as (e) associated size. Finally, we provide an in-depth analysis of our (f) latent codes and (g) motion dictionary to interpret their semantic meanings.
+
+Datasets Our model is trained on the datasets VoxCeleb, TaichiHD and TED-talk. We follow the pre-processing method in (Siarohin et al., 2019) to crop frames into $2 5 6 \times 2 5 6$ resolution for quantitative evaluation.
+
+Implementation details Our model is implemented in PyTorch (Paszke et al., 2019). All models are trained on four 16G NVIDIA V100 GPUs. The total batch size is 32 with 8 images per GPU. We use a learning rate of 0.002 to train our model with the Adam optimizer (Kingma & Ba, 2014). The dimension of all latent codes, as well as directions in $D _ { m }$ is set to be 512. In our loss function, we use $\lambda = 1 0$ in order to penalize more on the perceptual loss. It takes around 150 hours to fully train our framework.
+
+![](images/a37a7371bb03be1d7874e44a06664ca4dc3ef0bc6d060aab959b38245f9df2b2.jpg)  
+Figure 4: Qualitative results. Examples for same-dataset absolute motion transfer on TaichiHD (top-right) and TED-talk (bottom-right). On VoxCeleb (left), we demonstrate cross-dataset relative motion transfer. We successfully transfer motion between $x _ { 1 }$ and $x _ { t }$ from videos in VoxCeleb to $x _ { s }$ from FFHQ, the latter not being used for training.
+
+Evaluation metrics We evaluate our model w.r.t. (i) reconstruction faithfulness using $\mathcal { L } _ { 1 }$ , LPIPS, (ii) generated video quality using video FID, as well as (iii) semantic consistency using average keypoint distance (AKD), missing keypoint rate (MKR) and average euclidean distance (AED). Details are available in App. B.2.
+
+# 4.1 QUALITATIVE RESULTS
+
+Firstly, we evaluate the ability of LIA to generate realistic videos and compare related results with four state-of-the-art methods. For TaichiHD and TED-talk datasets, we conduct an experiment related to cross-video generation. Corresponding results (see Fig. 4) confirm that our method is able to correctly transfer motion on articulated human bodies, in the absence of explicit structure representations. For the VoxCeleb dataset, we conduct a cross-dataset generation-experiment, where we transfer motion from VoxCeleb to images of the FFHQ dataset. We observe that our method outperforms FOMM and MRAA w.r.t. image quality, as both approaches visibly deform the shape of the original faces. This is specifically notable in the case that source and driving images entail large pose variations. At the same time, LIA is able to successfully tackle this challenge and no similar deformations are visible.
+
+# 4.2 COMPARISON WITH STATE-OF-THE-ART METHODS
+
+We quantitatively compare our method with the state-of-the-art approaches X2Face, Monkey-Net, FOMM and MRAA on two tasks, namely (a) same-identity reconstruction and (b) cross-video motion transfer. Additionally, we conduct a (c) user study.
+
+(a) Same-identity reconstruction We here evaluate the reconstruction ability of our method. Specifically, we reconstruct each testing video by using the first frame as $x _ { s }$ and the remaining frames as $x _ { d }$ . Results on three datasets are reported in Table 1. Focusing on foregroundreconstruction, our method outperforms the other approaches w.r.t. all metrics. More results are presented in App. B.3, discussing background-reconstruction.
+
+(b) Cross-video motion transfer Next, we conduct experiments, where source images and driving videos stem from different video sequences. In this context, we mainly focus on evaluating talking head videos and explore two different cases. In the first case, we generate videos using the VoxCeleb testing set to conduct motion transfer. In the second case, source images are from an unseen dataset, namely the GermanPublicTV dataset, as we conduct cross-dataset motion transfer. In both experiments, we randomly construct source and driving pairs and transfer motion from driving videos to source images to generate a novel manipulated dataset. Since ground truth data for our generated videos is not available, we use video FID (as initialized by Wang et al. (2020a)) to compute the distance between generated and real data distributions. As shown in Tab. 2, our method outperforms all other approaches w.r.t. video FID, indicating the best generated video quality.
+
+Table 1: Same-identity reconstruction. Comparison with state-of-the-art methods on three datasets for same-identity reconstruction.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">VoxCeleb</td><td colspan="4">TaichiHD</td><td colspan="4">TED-talks</td></tr><tr><td>L1</td><td>AKD</td><td>AED</td><td>LPIPS</td><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>LPIPS</td><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>LPIPS</td></tr><tr><td>X2Face</td><td>0.078</td><td>7.687</td><td>0.405</td><td>-</td><td>0.080</td><td>(17.654,0.109)</td><td></td><td>-</td><td>=</td><td></td><td></td><td>-</td></tr><tr><td>Monkey-Net</td><td>0.049</td><td>1.878</td><td>0.199</td><td>-</td><td>0.077</td><td>(10.798,0.059)</td><td></td><td></td><td>=</td><td></td><td></td><td>=</td></tr><tr><td>FOMM</td><td>0.046</td><td>1.395</td><td>0.141</td><td>0.136</td><td>0.063</td><td>(6.472,0.032)</td><td>0.4950.191</td><td></td><td>0.030</td><td>(3.759,0.0090)0.428</td><td></td><td>0.13</td></tr><tr><td>MRAA w/o bg</td><td>0.043</td><td>1.307</td><td>0.140</td><td>0.127</td><td>0.063</td><td>(5.626,0.025)</td><td>0.460</td><td>0.189</td><td>0.029</td><td>(3.126,0.0092)0.396</td><td></td><td>0.12</td></tr><tr><td>Ours</td><td>0.041</td><td>1.353</td><td>0.138</td><td>0.123</td><td>0.057</td><td>(4.823, 0.020)</td><td>0.431</td><td>0.180</td><td>0.027</td><td>(3.141,0.0095)0.399</td><td></td><td>0.11</td></tr></table>
+
+Table 2: Cross-video generation. We report video FID for both inner- and cross-dataset tasks on VoxCeleb and GermanPublicTV.   
+
+<table><tr><td>VoxCeleb</td><td>GermanPublicTV</td></tr><tr><td>FOMM</td><td>0.323 0.456</td></tr><tr><td>MRAA 0.308</td><td>0.454</td></tr><tr><td>Ours</td><td>0.161 0.406</td></tr></table>
+
+Table 3: User study. We ask 20 human raters to conduct a subjective video quality evaluation.   
+
+<table><tr><td></td><td>VoxCeleb(%)</td><td>TaichiHD(%)</td><td>TED-talk(%)</td></tr><tr><td>Ours/FOMM</td><td>92.9/7.1</td><td>64.5/35.5</td><td>71.4/28.6</td></tr><tr><td>Ours/MRAA</td><td>89.7/10.3</td><td>60.7/39.9</td><td>54.8/45.2</td></tr></table>
+
+(c) User study We conduct a user study to evaluate video quality. Towards this, we displayed paired videos and asked 20 human raters ‘which clip is more realistic?’. Each video-pair contains a generated video from our method, as well as a video generated from FOMM or MRAA. Results suggest that our results are more realistic in comparison to FOMM and MRAA across all three datasets (see Tab. 3). Hence, the obtained human preference is in accordance with our quantitative evaluation.
+
+Table 4: Ablation study on motion dictionary. We conduct experiments on three datasets with and without $D _ { m } ^ { \phantom { \dagger } }$ and show reconstruction results.   
+
+<table><tr><td colspan="2">VoxCeleb</td><td colspan="2">TaichiHD</td><td colspan="2">TED-talks</td></tr><tr><td>Method</td><td>L1 LPIPS</td><td>L1</td><td>LPIPS</td><td>L1</td><td>LPIPS</td></tr><tr><td>wloDm</td><td>0.049 0.165</td><td>0.062</td><td>0.186</td><td>0.031</td><td>0.12</td></tr><tr><td>Full</td><td>0.041 0.123</td><td>0.057</td><td>0.180</td><td>0.028</td><td>0.11</td></tr></table>
+
+Table 5: Ablation study on $D _ { m }$ size. We conduct experiments on three datasets with 5 different $D _ { m }$ size and show reconstruction results.   
+
+<table><tr><td colspan="3">VoxCeleb</td><td colspan="2">TaichiHD</td><td colspan="2">TED-talks</td></tr><tr><td>M</td><td>L1</td><td>LPIPS</td><td>L1</td><td>LPIPS</td><td>L1</td><td>LPIPS</td></tr><tr><td>5 10</td><td>0.051 0.043</td><td>0.15 0.13</td><td>0.070 0.065</td><td>0.22 0.20</td><td>0.037 0.036</td><td>0.15</td></tr><tr><td>20</td><td>0.041</td><td>0.12</td><td>0.057</td><td>0.18</td><td>0.028</td><td>0.13 0.11</td></tr><tr><td>40</td><td>0.042</td><td>0.12</td><td>0.060</td><td>0.19</td><td>0.030</td><td>0.12</td></tr><tr><td>100</td><td>0.041</td><td>0.12</td><td>0.058</td><td>0.18</td><td>0.028</td><td>0.11</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+# 4.3 ABLATION STUDY
+
+We here analyze our proposed motion dictionary and focus on answering following two questions.
+
+(d) Is the motion dictionary $D _ { m }$ beneficial? We here explore the impact of proposed $D _ { m }$ , by training our model without $D _ { m }$ . Specifically, we output $w _ { r \to d }$ directly from MLP, without using LMD to learn an orthogonal basis. From the evaluation results reported in Tab. 4 and qualitative results in App. B.5, we observe that in the absence of $D _ { m }$ , model fails to generate high-quality images, which proves the effectiveness of $D _ { m }$ , consistently on all datasets.
+
+(e) How many directions are required in $D _ { m }$ ? Towards finding an effective size of $D _ { m }$ , we empirically test three different $M$ , viz. 5, 10, 20, 40 and 100. Quantitative results in Tab. 5 show that when using 20 directions, the model achieves the best reconstruction results.
+
+# 4.4 FURTHER ANALYSIS
+
+(f) Latent code analysis. While our method successfully transfers motion via latent space navigation, we here aim at answering the question — what does $x _ { r }$ represent? Towards answering this question, we proceed to visualize $x _ { r }$ . We firstly decode $z _ { s  r }$ into a dense flow field $\phi _ { s  r }$ , which is then used to warp $x _ { s }$ (we show details in App. B.4). Fig. 5 shows examples of $x _ { s }$ and $x _ { r }$ . Interestingly, we observe that $x _ { r }$ represents the canonical pose of $x _ { s }$ , regardless of original poses of the subjects. And for all datasets, reference images resemble each other $w . r . t .$ . pose and scale. As such reference images can be considered as a normalized form of $x _ { s }$ , learning transformations between $x _ { s }$ and $x _ { d }$ using $x _ { s }  x _ { r }  x _ { d }$ is considerably more efficient than $x _ { s } \to x _ { d }$ , once $x _ { r }$ is fixed.
+
+![](images/f4021d1e9c0a6b78d5507518e6f476c3eb214d4b2fd88cb1cbad903d30c7d449.jpg)  
+Figure 5: Visualization of reference images. Example source (top) and reference images (down) from VoxCeleb, TaichiHD and TED-talk datasets. Our network learns reference images of a consistently frontal pose, systematically for all input images of each dataset.
+
+![](images/833c41d15a70be2fc45b2ced129969f51412a6047f24acd5518917c81c6c810d.jpg)  
+Figure 6: Linear manipulation of four motion directions on the painting of Mona Lisa. Manipulated results indicate that $d _ { 6 }$ represents eye movement, $d _ { 8 }$ represents head nodding, whereas $\bar { d _ { 1 9 } }$ and $d _ { 7 }$ represent facial expressions.
+
+Noteworthy, we found the similar idea of learning a ‘reference image’ has also been explored by Siarohin et al. (2019) (FOMM) and Wiles et al. (2018) (X2Face). However, deviating from our visualized ‘reference image’, the ’reference image ’in FOMM refers to a non-visualized and abstract concept. In addition, LIA only requires a latent code $z _ { s  r }$ , rather than the ’reference image’ for both, training and testing, which is contrast to X2Face.
+
+(g) Motion dictionary interpretation. Towards further interpretation of directions in $D _ { m }$ , we conduct linear manipulations on each $d _ { i }$ . Images pertained to manipulating four motion directions are depicted in Fig. 6. The results suggest that the directions in $\boldsymbol { D } _ { m } ^ { \bar { 2 } }$ are semantically meaningful, as they represent basic visual transformations such as head nodding $( d _ { 8 } )$ , eye movement $( d _ { 6 } ) ^ { \dag }$ and facial expressions $\cdot d _ { 1 9 }$ and $d _ { 7 }$ ). More results can be found on our project webpage2.
+
+# 5 CONCLUSIONS
+
+In this paper, we presented a novel self-supervised autoencoder LIA, aimed at animating images via latent space navigation. By the proposed Linear Motion Decomposition (LMD), we were able to formulate the task of transferring motion from driving videos to source images as learning linear transformations in the latent space. We evaluated proposed method on real-world videos and demonstrated that our approach is able to successfully animate still images, while eliminating the necessity of explicit structure representations. In addition, we showed that the incorporated motion dictionary is interpretable and contains directions pertaining to basic visual transformations. Both quantitative and qualitative evaluations showed that LIA outperforms state-of-art algorithms on all benchmarks. We postulate that LIA opens a new door in design of interpretable generative models for video generation.
+
+# ETHIC STATEMENT
+
+In this work, we aim to synthesize high-quality videos by transferring motion on still images. Our approach can be used for movie production, making video games, online education, generating synthetic data for other computer vision tasks, etc. We note that our framework mainly focuses on learning how to model motion distribution rather than directly model appearance, therefore it is not biased towards any specific gender, race, region, or social class. It works equally well irrespective of the difference in subjects.
+
+# REPRODUCIBILITY STATEMENT
+
+We assure that all the results shown in the paper and supplemental materials can be reproduced. We intend to open-source our code, as well as trained models.
+
+# ACKNOWLEDGEMENTS
+
+This work was granted access to the HPC resources of IDRIS under the allocation AD011011627R1. It was supported by the French Government, by the National Research Agency (ANR) under Grant ANR-18-CE92-0024, project RESPECT and through the 3IA Cote d’Azur Investments in the Future ˆ project managed by the National Research Agency (ANR) with the reference number ANR-19- P3IA-0002.
+
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+Zhedong Zheng, Tao Ruan, Yunchao Wei, Yi Yang, and Tao Mei. Vehiclenet: Learning robust visual representation for vehicle re-identification. IEEE Transaction on Multimedia (TMM), 2020.
+
+Jiapeng Zhu, Yujun Shen, Deli Zhao, and Bolei Zhou. In-domain gan inversion for real image editing. In Proceedings of European Conference on Computer Vision (ECCV), 2020.
+
+# A DETAILS OF MODEL ARCHITECTURE
+
+We proceed to describe the model architecture in this section. Fig. 7 shows details of our $E$ . In each ResBlock in $E$ , spatial size of input feature maps are downsampled. We take feature maps of spatial sizes from $8 \times 8$ to $2 5 6 \times 2 5 6$ as our appearance features $x _ { i } ^ { e n c }$ . We use a 5-layer MLP to predict a magnitude vector $A _ { d  r }$ from $z _ { d  r }$ . Fig. 8 (a) shows the general architecture of our $G$ , which consists of two components, a flow field generator $G _ { f }$ and a refinement network $G _ { r }$ . We apply StyleConv (Upsample $\mathbf { + \ C o n v 3 \times 3 }$ ), which is proposed by StyleGAN2, in $G _ { f }$ . StyleConv takes latent representation $z _ { s  t }$ as style code and generates flow field $\phi _ { i }$ and corresponding mask $m _ { i }$ . $G _ { r }$ uses UpConv (Conv1 $\times \ 1 +$ Upsample) to upsample and refine inpainted feature maps to target resolution. We show details pertaining to $G$ block in Fig. 8 (b). Each $G$ block is used to upsample $\times 2$ the previous resolution. We stack 6 blocks towards producing 256 resolution images.
+
+![](images/9325fffdbfebc0ee68f0be3ae9d39d9a1ea8e2cd0cc42b57b0799a99bd6ea9c8.jpg)  
+Figure 7: Encoder architecture. We show details of architecture of $E$ in (a) and ResBlock in (b).
+
+# B EXPERIMENTS
+
+We proceed to introduce details of datasets and evaluation metrics used in our experiments.
+
+# B.1 DATASETS
+
+VoxCeleb (Nagrani et al., 2019) consists of a large amount of interview videos of different celebrities. Following the process of FOMM (Siarohin et al., 2019), we extract frames and crop them into $2 5 6 \times 2 5 6$ resolution. In total, VoxCeleb contains a training set of 17928 videos and a test set of 495 videos.
+
+![](images/9bfd82d4cf08f228e79a896a311bfc79b0ee414695a25bedad52ccbc1330672a.jpg)  
+Figure 8: Generator architecture. We show details about architecture of $G$ in (a) and $G$ block in (b).
+
+TaiChiHD (Siarohin et al., 2019) consists of videos of full human bodies performing Tai Chi actions. We follow the original pre-processing of FOMM (Siarohin et al., 2019) and utilize its $2 5 6 \times 2 5 6$ version. TaiChiHD contains 1096 training videos and 115 testing videos.
+
+TED-talk is a new dataset proposed in MRAA (Siarohin et al., 2021). It comprises a number of TED-talk videos, where the main subjects have been cropped out. We resize the original version into $2 5 6 \times 2 5 6$ resolution to train our model. This dataset includes 1124 training videos and 130 testing videos.
+
+# B.2 EVALUATION METRICS
+
+We use five different metrics to evaluate our experimental results, namely $\mathcal { L } _ { 1 }$ , LPIPS, AKD, MKR and AED that quantify the reconstructed results. In addition, we compute video FID to evaluate video quality in motion transferring tasks.
+
+$\mathcal { L } _ { 1 }$ represents the mean absolute pixel difference between reconstructed and real videos.
+
+LPIPS (Zhang et al., 2018) aims at measuring the perceptual similarity between reconstructed and real images by leveraging the deep features from AlexNet (Krizhevsky et al., 2012).
+
+Video FID is a modified version of the original FID (Heusel et al., 2017). We here follow the same implementation as Wang et al. (2020a) and utilize a pre-trained ResNext101 (Hara et al., 2018) to extract spatio-temporal features to compute the distance between real and generated videos distributions. We take the first 100 frames of each video as input of the feature-extractor to compute the final scores.
+
+Average keypoint distance (AKD) and missing keypoint rate (MKR) evaluate the difference between keypoints of reconstructed and ground truth videos. We extract landmarks using the face alignment approach of (Bulat & Tzimiropoulos, 2017) and extract body poses for both TaiChiHD and TED-talks using OpenPose (Cao et al., 2019). AKD is computed as the average distance between corresponding keypoints, whereas MKR is the proportion of keypoints present in the groundtruth that are missing in a reconstructed video.
+
+Average Euclidean distance (AED) measures the ability of preserving identity in reconstructed video. We use a person re-identification pretrained model (Zheng et al., 2020) for measuring human bodies (TaichiHD and TED-talk) and OpenFace (Amos et al., 2016) for faces to extract identity embeddings from reconstructed and ground truth frame pairs, then we compute MSE of their difference for all pairs.
+
+# B.3 COMPARISON WITH FULL MRAA
+
+We show quantitative evaluation results with the full MRAA model in Tab. 6. We observe that our method achieves competitive results in reconstruction and keypoint evaluation. While we do not explicitly predict keypoints, w.r.t. the TaichiHD dataset, interestingly we outperform MRAA in both, AKD and MKR. Such results showcase the effectiveness of our proposed method on modeling articulated human structures. However, reconstruction evaluation cannot provide a completely fair comparison on how well the main subjects (e.g., faces and human bodies) are generated in videos. This is in particular the case for TaichiHD and TED-talk, where backgrounds have large contributions to the final scores.
+
+Table 6: Comparison with full MRAA.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">VoxCeleb</td><td colspan="4">TaichiHD</td><td colspan="4">TED-talks</td></tr><tr><td>L1</td><td>AKD</td><td>AED</td><td>LPIPS</td><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>LPIPS</td><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>LPIPS</td></tr><tr><td>MRAA</td><td>0.041</td><td>1.303</td><td>0.135</td><td>0.124</td><td>0.045</td><td>(5.551,( ,0.025)</td><td>0.431</td><td>0.178</td><td>0.027</td><td>(3.107,0.0093)</td><td>0.379</td><td>0.11</td></tr><tr><td>Ours</td><td>0.041</td><td>1.353</td><td>0.138</td><td>0.123</td><td>0.057</td><td>(4.823,0.020)</td><td>0.431</td><td>0.180</td><td>0.027</td><td>(3.141,0.0095)</td><td>0.399</td><td>0.11</td></tr></table>
+
+# B.4 REFERENCE IMAGE GENERATION.
+
+To produce $x _ { r }$ , we use $G$ to decode $z _ { s  r }$ into the flow field $\phi _ { s  r }$ . Reference image $x _ { r }$ is obtained by warping $x _ { s }$ using $\phi _ { s  r }$ . The entire process is shown in Fig. 9.
+
+![](images/762132d8adc241e8fde2ec32f40c10dbb27f4f06b3953c429f112db181bf10c2.jpg)  
+Figure 9: Reference image generation.
+
+# B.5 QUALITATIVE RESULTS ON EFFECTIVENESS OF USING MOTION DICTIONARY
+
+Fig. 10 illustrates the generated results on transferring motion from VoxCeleb to GermanPublicTV with and without motion dictionary. We observe that without the motion dictionary, appearance information is undesirably transferred from driving videos to generated videos.
+
+# B.6 LIMITATIONS
+
+For human body, one limitation of our method is dealing with body occlusion. We observe in Fig. 11 that in taichi videos, in case of occlusion cause by legs and arms, motion is not transferred successfully. In addition, in TED-talks, transferring hand motion is challenging, as hands are of small size, articulated and sometimes occluded by human bodies.
+
+![](images/70acaf34c5c7203b948fb2919b9b562110b48bb7fdf0186567c1baf79c82cd42.jpg)  
+Figure 10: Generated results with and without $D _ { m }$ . We observe that the disentanglement of appearance and motion is much better by using $D _ { m }$ .
+
+![](images/8df647813e53c9e3f9fda4ddf2534e1d09430b0019731ce68455cebf897027e0.jpg)  
+Figure 11: Failure cases. We observed that it is still challenging for LIA to handle arm-leg occlusion (Taichi) and hand motion (TED-talk).

parse/dev/7r6kDq0mK_/7r6kDq0mK__content_list.json

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+    {
+        "type": "text",
+        "text": "LATENT IMAGE ANIMATOR: LEARNING TO ANIMATE IMAGES VIA LATENT SPACE NAVIGATION ",
+        "text_level": 1,
+        "bbox": [
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+            99,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yaohui Wang, Di Yang, Francois Bremond & Antitza Dantcheva Inria, Universite C´ ote d’Azur ˆ {yaohui.wang,di.yang,francois.bremond,antitza.dantcheva}@inria.fr ",
+        "bbox": [
+            184,
+            190,
+            756,
+            229
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            265,
+            544,
+            280
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Due to the remarkable progress of deep generative models, animating images has become increasingly efficient, whereas associated results have become increasingly realistic. Current animation-approaches commonly exploit structure representation extracted from driving videos. Such structure representation is instrumental in transferring motion from driving videos to still images. However, such approaches fail in case the source image and driving video encompass large appearance variation. Moreover, the extraction of structure information requires additional modules that endow the animation-model with increased complexity. Deviating from such models, we here introduce the Latent Image Animator (LIA), a self-supervised autoencoder that evades need for structure representation. LIA is streamlined to animate images by linear navigation in the latent space. Specifically, motion in generated video is constructed by linear displacement of codes in the latent space. Towards this, we learn a set of orthogonal motion directions simultaneously, and use their linear combination, in order to represent any displacement in the latent space. Extensive quantitative and qualitative analysis suggests that our model systematically and significantly outperforms state-of-art methods on VoxCeleb, Taichi and TED-talk datasets w.r.t. generated quality. Source code and pre-trained models are publicly available1. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/e378522fe7bd0a0020ce31d5201a8ccda8a4e2a620417861db6f080cf4f0c4a2.jpg",
+        "image_caption": [
+            "Figure 1: LIA animation examples. The two images of Marilyn Monroe and Emmanuel Macron are animated by LIA, which transfers motion of a driving video (smaller images on the top) from VoxCeleb dataset (Chung et al., 2018) onto the still images. LIA is able to successfully animate these two images without relying on any explicit structure representations, such as landmarks and region representations. "
+        ],
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+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            336,
+            117
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the series of science fiction books Harry Potter (Rowling et al., 2016; Rowling, 2019), wizards and witches were able to magically enchant portraits, bringing them to life. Remarkable progress of deep generative models has recently turned this vision into reality. This work examines the scenario where a framework animates a source image by motion representations learned from a driving video. Existing approaches for image animation are classically related to computer graphics (Cao et al., 2014; Thies et al., 2016; 2019; Zhao et al., 2018) or exploit motion labels (Wang et al., 2020b) and structure representations such as semantic maps (Pan et al., 2019; Wang et al., 2018; 2019), human keypoints (Jang et al., 2018; Yang et al., 2018; Walker et al., 2017; Chan et al., 2019; Zakharov et al., 2019; Wang et al., 2019; Siarohin et al., 2019), 3D meshes (Liu et al., 2019; Chen et al., 2021), and optical flows (Li et al., 2018; Ohnishi et al., 2018). We note that the ground truth of such structure representations has been computed a-priori for the purpose of supervised training, which poses constraints on applications, where such representations of unseen testing images might be fragmentary or difficult to access. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Self-supervised motion transfer approaches (Wiles et al., 2018; Siarohin et al., 2019; 2021) accept raw videos as input and learn to reconstruct driving images by warping source image with predicted dense optical flow fields. While the need for domain knowledge or labeled ground truth data has been obviated, which improves performance on in-the-wild testing images, such methods entail necessity of explicit structure representations as motion guidance. Prior information such as keypoints (Siarohin et al., 2019; Wang et al., 2021a) or regions (Siarohin et al., 2021) are learned in an end-to-end training manner by additional networks as intermediate features, in order to predict target flow fields. Although online prediction of such representations is less tedious than the acquisition of ground truth labels, it still strains the complexity of networks. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Deviating from such approaches, we here aim to fully eliminate the need of explicit structure representations by directly manipulating the latent space of a deep generative model. To the best of our knowledge, this constitutes a new direction in the context of image animation. Our work is motivated by interpretation of GANs (Shen et al., 2020; Goetschalckx et al., 2019; Jahanian et al., 2020; Voynov & Babenko, 2020), showcasing that latent spaces of StyleGAN (Karras et al., 2019; 2020b) and BigGAN (Brock et al., 2019) contain rich semantically meaningful directions. Given that walking along such directions, basic visual transformations such as zooming and rotation can be induced in generated results. As in image animation, we have that motion between source and driving images can be considered as higher-level transformation, a natural question here arises: can we discover a set of directions in the latent space that induces high-level motion transformations collaboratively? ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Towards answering this question, we introduce LIA, a novel Latent Image Animator constituting of an autoencoder for animating still images via latent space navigation. LIA seeks to animate a source image via linearly navigating associated source latent code along a learned path to reach a target latent code, which represents the high-level transformation for animating the source image. We introduce a Linear Motion Decomposition (LMD) approach aiming to represent a latent path via a linear combination of a set of learned motion directions and associated magnitudes. Specifically, we constrain the set as an orthogonal basis, where each vector indicates a basic visual transformation. By describing the whole motion space using such learned basis, LIA eliminates the requirement of explicit structure representations. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In addition, we design LIA to disentangle motion and appearance within a single encoder-generator architecture. Deviating from existing methods using separate networks to learn disentangled features, LIA integrates both, latent motion code, as well as appearance features in a single encoder, which highly reduces the model complexity and simplifies training. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We provide evaluation on multiple datasets including VoxCeleb (Chung et al., 2018), TaichiHD (Siarohin et al., 2019) and TED-talk (Siarohin et al., 2021). In addition, we show that LIA outperforms the state-of-the-art in preserving the facial structure in generated videos in the setting of one-shot image animation on unseen datasets such as FFHQ (Karras et al., 2019) and GermanPublicTV (Thies et al., 2020). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            842,
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+            857
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Video generation GAN-based video generation is aimed at mapping Gaussian noise to video, directly and in the absence of prior information (Vondrick et al., 2016; Saito et al., 2017; Tulyakov et al., 2018; Wang et al., 2020a; Wang, 2021). Approaches based on deep probabilistic models (Denton & Birodkar, 2017; Li & Mandt, 2018; Bhagat et al., 2020; Xie et al., 2020) were also proposed to tackle this problem, however only show results on toy datasets with low resolution. Recently, with the progress of GANs in photo-realistic image generation (Brock et al., 2019; Karras et al., 2019; 2020a), a series of works (Clark et al., 2019; Wang et al., 2021c) explored production of highresolution videos by incorporating the architecture of an image generator into video GANs, trained jointly with RNNs. Tian et al. (2021) directly leveraged the knowledge of a pre-trained StyleGAN to produce videos of resolution up to $1 0 2 4 \\times \\mathrm { i } 0 2 4$ . Unlike these approaches, which generate random videos based on noise vectors in an unconditional manner, in this paper, we focus on conditionally creating novel videos by transferring motion from driving videos to input images. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/adf402bc242a87960126961b2dd01a4fedaa5916e13c5407ab9a2e27d13beee5.jpg",
+        "image_caption": [
+            "Figure 2: General pipeline. Our objective is to transfer motion via latent space navigation. The entire training pipeline consists of two steps. Firstly, we encode a source image $x _ { s }$ into a latent code $z _ { s  r }$ . By linearly navigating $z _ { s  r }$ along a path $w _ { r \\to d }$ , we reach a target latent code $z _ { s  d }$ . The latent paths are represented by a linear combination between a set of learned motion directions (e.g., $d _ { 1 }$ and $d _ { 2 }$ ), which is an orthogonal basis, and associated magnitudes. In the second step, we decode $z _ { s  d }$ to a target dense optical flow field $\\phi _ { s  d }$ , which is used to warp $x _ { s }$ into the driving image $x _ { d }$ . While we train our model using images from the same video sequence, in the testing phase, $x _ { s }$ and $x _ { d }$ generally pertain to different identities. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Latent space editing In an effort to control generated images, recent works explored the discovery of semantically meaningful directions in the latent space of pre-trained GANs, where linear navigation corresponds to desired image manipulation. Supervised (Shen et al., 2020; Jahanian et al., 2020; Goetschalckx et al., 2019) and unsupervised (Voynov & Babenko, 2020; Peebles et al., 2020; Shen & Zhou, 2021) approaches were proposed to edit semantics such as facial attributes, colors and basic visual transformations (e.g., rotation and zooming) in generated or inverted real images (Zhu et al., 2020; Abdal et al., 2020). In this work, as opposed to finding directions corresponding to individual visual transformations, we seek to learn a set of directions that cooperatively allows for high-level visual transformations that can be beneficial in image animation. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Image animation Related approaches (Chan et al., 2019; Wang et al., 2018; Zakharov et al., 2019; Wang et al., 2019; Yang et al., 2020) in image animation required strong prior structure labels as motion guidance. In particular, Chan et al. (2019), Yang et al. (2020) and Wang et al. (2018) proposed to map representations such as human keypoints and facial landmarks to videos in the setting of image-to-image translation proposed by Isola et al. (2017). However, such approaches were only able to learn an individual model for a single identity. Transferring motion on new appearances requires retraining the entire model from scratch by using videos of target identities. Several recent works (Zakharov et al., 2019; Wang et al., 2019) explored meta learning in fine-tuning models on target identities. While only few images of target identities were required during inference time, it was still compulsory to input pre-computed structure representations in those approaches, which usually are hard to access in many real-world scenarios. Towards addressing this issue, very recent works (Siarohin et al., 2019; 2021; Wang et al., 2021b; Wiles et al., 2018) proposed to learn image animation in self-supervised manner, only relying on RGB videos for both, training and testing without any priors. They firstly predicted dense flow fields from input images, which were then utilized to warp source images, in order to obtain final generated results. Inference only required one image of a target identity without any fine-tuning step on pre-trained models. While no priors were required, state-of-the-art methods still followed the idea of using explicit structure representations. FOMM (Siarohin et al., 2019) proposed a first order motion approach to predict keypoints and local transformations online to generate flow fields. Siarohin et al. (2021) developed this idea to model articulated objects by replacing a keypoints predictor by a PCA-based region prediction module. Wang et al. (2021b) extended FOMM by predicting 3D keypoints for view-free generation. We note though that in all approaches, given that keypoints or regions are inadequately predicted, the quality of generated images drastically decreases. In contrast to such approaches, our method does not require any explicit structure representations. We dive into the latent space of the generator and self-learn to navigate motion codes in certain directions with the goal to reach target codes, which are then decoded to flow fields for warping. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/203fe2748a461292b0737c6c4a21c45cb5adc8b929922aec910a81a2c52ff5dd.jpg",
+        "image_caption": [
+            "Figure 3: Overview of LIA. LIA is an autoencoder consisting of two networks, an encoder $E$ and a generator $G$ . In the latent space, we apply Linear Motion Decomposition (LMD) towards learning a motion dictionary $D _ { m }$ , which is an orthogonal basis where each vector represents a basic visual transformation. LIA takes two frames sampled from the same video sequence as source image $x _ { s }$ and driving image $x _ { d }$ respectively during training. Firstly, it encodes $x _ { s }$ into a source latent code $z _ { s  r }$ and $x _ { d }$ into a magnitude vector $\\bar { A _ { r \\to d } }$ . Then, it linearly combines $A _ { r  d }$ and a trainable $D _ { m }$ using LMD to obtain a latent path $w _ { r \\to d }$ , which is used to navigate $z _ { s _ {  } r }$ to a target code $z _ { s  d }$ . Finally, $G$ decodes $z _ { s  d }$ into a target dense flow field and warps $x _ { s }$ to an output image $x _ { s \\to d }$ . The training objective is to reconstruct $x _ { d }$ using $x _ { s \\to d }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+            281,
+            541
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Self-supervised image animation aims at learning to transfer motion from a subject of a driving video to a subject in a source image based on training with a large video dataset. In this work, we propose to model such motion transformation via latent space navigation. The general pipeline is illustrated in Fig. 2. Specifically, for training, our model takes in a pair of source and driving images, randomly sampled from one video sequence. These two images are encoded into a latent code which is used to represent motion transformation in the image space. The training objective is to reconstruct the driving image by combining source image with learned motion transformation. For testing, frames of a driving video are sequentially processed with the source image to animate the source subject. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We provide an overview of the proposed model in Fig. 3. Our model is an autoencoder, consisting of two main networks, an encoder $\\bar { E }$ and a generator $\\bar { G }$ . In general, our model requires two steps to transfer motion. In the first step, $E$ encodes source and driving images $\\boldsymbol { x } _ { s } , \\boldsymbol { x } _ { d } \\sim \\dot { \\mathcal { X } } \\in \\mathbb { R } ^ { 3 \\times H \\times W }$ into latent codes in the latent space. The source code is then navigated into a target code, which is used to represent target motion transformation, along a learned latent path. Based on proposed Linear Motion Decomposition (LMD), we represent such a path as a linear combination of a set of learned motion directions and associated magnitudes, which are learned from $x _ { d }$ . In the second step, once the target latent code is obtained, $G$ decodes it as a dense flow field $\\phi _ { s \\to d } \\sim \\Phi \\in \\mathbb { R } ^ { 2 \\times H \\times W }$ and uses $\\bar { \\phi _ { s \\to d } }$ to warp $x _ { s }$ and then to obtain the output image. In the following, we proceed to discuss the two steps in detail. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 LATENT MOTION REPRESENTATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Given a source image $x _ { s }$ and a driving image $x _ { d }$ , our first step constitutes of learning a latent code $z _ { s \\to d } \\sim \\mathcal { Z } \\in \\mathbb { R } ^ { N }$ to represent the motion transformation from $x _ { s }$ to $x _ { d }$ . Due to the uncertainty of two images, directly learning $z _ { s  d }$ puts forward a high requirement on the model to capture a complex distribution of motion. Mathematically, it requires modeling directions and norms of the vector $z _ { s  d }$ simultaneously, which is challenging. Therefore, instead of modeling motion transformation $x _ { s } \\to x _ { d }$ , we assume there exists a reference image $x _ { r }$ and motion transfer can be modeled as $x _ { s }  x _ { r }  x _ { d }$ , where $z _ { s  d }$ is learned in an indirect manner. We model $z _ { s  d }$ as a target point in the latent space, which can be reached by taking linear walks from a starting point $z _ { s  r }$ along a linear path $w _ { r \\to d }$ (see Fig. 2), given by ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/df5faf4fb0dcd135a23a966fa3b1c603d561a212356fc77108d9539a75f1e93f.jpg",
+        "text": "$$\nz _ { s  d } = z _ { s  r } + w _ { r  d } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $z _ { s  r }$ and $w _ { r \\to d }$ indicate the transformation $x _ { s } \\to x _ { r }$ and $x _ { r } \\to x _ { d }$ respectively. Both $z _ { s  r }$ and $w _ { r \\to d }$ are learned independently and $z _ { s  r }$ is obtained by passing $x _ { s }$ through $E$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We learn $w _ { r \\to d }$ via Linear Motion Decomposition (LMD). Our idea is to learn a set of motion directions $D _ { m } = \\{ \\mathbf { d _ { 1 } } , . . . , \\mathbf { d _ { M } } \\}$ to represent any path in the latent space. We constrain $D _ { m }$ as an orthogonal basis, where each vector indicates a motion direction $\\mathbf { d _ { i } }$ . We then combine each vector in the basis with a vector $A _ { r  d } = \\{ a _ { 1 } , . . . , a _ { M } \\}$ , where $a _ { i }$ represents the magnitude of $\\mathbf { d _ { i } }$ . Hence, any linear path in the latent space can be represented using a linear combination ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/ce1f4fd033c395dd8b54261361e0752d9b643c625285ae9ed34fbcd1dc76b91b.jpg",
+        "text": "$$\nw _ { r  d } = \\sum _ { i = 1 } ^ { M } a _ { i } { \\bf d _ { i } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { d } _ { \\mathbf { i } } \\in \\mathbb { R } ^ { N }$ and $a _ { i } \\in \\mathbb { R }$ for all $i \\in \\{ 1 , . . . , M \\}$ . Semantically, each $\\mathbf { d _ { i } }$ should represent a basic visual transformation and $a _ { i }$ indicates the required steps to walk in $\\mathbf { d _ { i } }$ towards achieving $w _ { r \\to d }$ . Due to $D _ { m }$ entailing an orthogonal basis, any two directions $\\mathbf { d } _ { \\mathrm { i } } , \\mathbf { d } _ { \\mathrm { j } }$ follow the constrain ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bfc2c5b59b76752fb48e656a15dc1ab4f70d40f6618ec59824266734f74db017.jpg",
+        "text": "$$\n< { \\bf d _ { i } } , { \\bf d _ { j } } > = \\left\\{ \\begin{array} { c c } { { 0 } } & { { i \\neq j } } \\\\ { { 1 } } & { { i = j . } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We implement $D _ { m }$ as a learnable matrix and apply the Gram-Schmidt process during each forward pass, in order to meet the requirement of orthogonality. $A _ { r  d }$ is obtained by mapping $z _ { d  r }$ , which is the output of $x _ { d }$ after $E$ , through a 5-layer MLP. The final formulation of latent motion representation for each $x _ { s }$ and $x _ { d }$ is thus given as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ab23dd7519062548f3ca731d261f453a3c7b9dbc7fa310e1f0a65e6770d781b7.jpg",
+        "text": "$$\nz _ { s  d } = z _ { s  r } + \\sum _ { i = 1 } ^ { M } a _ { i } { \\bf d _ { i } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 LATENT CODE DRIVEN IMAGE ANIMATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Once we obtain $z _ { s  d }$ , in our second step, we use $G$ to decode a flow field $\\phi _ { s \\to d }$ and warp $x _ { s }$ . Our $G$ consists of two components, a flow field generator $G _ { f }$ and a refinement network $G _ { r }$ (we provide details in App. A). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Towards learning multi-scale features, $G$ is designed as a residual network containing $N$ models to produce a pyramid of flow fields $\\phi _ { s \\to d } = \\{ \\bar { \\phi _ { i } } \\bar  \\} _ { 1 } ^ { N }$ in different layers of $G _ { f }$ . Multi-scale source features $x _ { s } ^ { e n c } = \\{ x _ { i } ^ { e n c } \\} _ { 1 } ^ { N }$ are obtained from $E$ and are warped in $G _ { f }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "However, as pointed out by Siarohin et al. (2019), only relying on $\\phi _ { s \\to d }$ to warp source features is insufficient to precisely reconstruct driving images due to the existing occlusions in some positions of $x _ { s }$ . In order to predict pixels in those positions, the network is required to inpaint the warped feature maps. Therefore, we predict multi-scale masks $\\{ m _ { i } \\} _ { 1 } ^ { N }$ along with $\\{ \\phi _ { i } \\} _ { 1 } ^ { N }$ in $G _ { f }$ to mask out the regions required to be inpainted. In each residual module, we have ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f1fed0d72e23b786d5fd7c6ab0d7b925e01797b6981cab28e1c7543451d43876.jpg",
+        "text": "$$\nx _ { i } ^ { \\prime } = \\mathcal { T } ( \\phi _ { i } , x _ { i } ^ { e n c } ) \\odot m _ { i } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            416,
+            693,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\odot$ denotes the Hadamard product and $\\tau$ denotes warping operation, whereas $\\boldsymbol { x } _ { i } ^ { \\prime }$ signifies the masked features. We generate both dense flow fields, as well as masks by letting each residual module output a 3-channel feature map in which the first two channels represent ${ \\bar { \\phi } } _ { i }$ and the last channel $m _ { i }$ . Based on an inpainted feature map $f ( x _ { i } ^ { \\prime } )$ , as well as an upsampled image $g ( x _ { i - 1 } )$ provided by the previous module in $G _ { r }$ , the RGB image from each module is given by ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/601a95e6281a7fb5e685caccc2b4354ef2a527cee69d14b6f4fba9163185cdda.jpg",
+        "text": "$$\no _ { i } = f ( x _ { i } ^ { \\prime } ) + g ( o _ { i - 1 } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $f$ and $g$ denote the inpainting and upsampling layers, respectively. The output image $o _ { N }$ from the last module constitutes the final generated image $x _ { s \\to d } = o _ { N }$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "3.3 LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We train LIA in a self-supervised manner to reconstruct $x _ { d }$ using three losses, i.e., a reconstruction loss $\\mathcal { L } _ { r e c o n }$ , a perceptual loss $\\mathcal { L } _ { v g g }$ (Johnson et al., 2016) and an adversarial loss $\\mathcal { L } _ { a d v }$ . We use $\\mathcal { L } _ { r e c o n }$ to minimize the pixel-wise $\\mathbb { L } _ { 1 }$ distance between $x _ { d }$ and $x _ { s \\to d }$ , calculated as ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/cbb3836f85ac8c8ad821df8ecc4c383d89ef22c12144dd5d156a4e64e232c07d.jpg",
+        "text": "$$\n\\mathcal { L } _ { r e c o n } ( x _ { s  d } , x _ { d } ) = \\mathbb { E } [ \\| x _ { d } - x _ { s  d } \\| _ { 1 } ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Towards minimizing the perceptual distance, we apply a VGG19-based $\\mathcal { L } _ { v g g }$ on multi-scale feature maps between real and generated images, written as ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f7928c756c955cc141db6fcd558a53236b4a94a707d3bc71ec40d8ed94509e02.jpg",
+        "text": "$$\n\\mathcal { L } _ { v g g } ( x _ { s  d } , x _ { d } ) = \\mathbb { E } [ \\sum _ { n } ^ { N } \\| F _ { n } ( x _ { d } ) - F _ { n } ( x _ { s  d } ) \\| _ { 1 } ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            131,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $F _ { n }$ denotes the $n ^ { t h }$ layer in a pre-trained VGG19 (Simonyan & Zisserman, 2015). In practice, towards penalizing real and generated images in multi-scale images, we use a pyramid of four resolutions, namely $2 5 6 \\times 2 5 6$ , ${ \\bar { 1 } } 2 8 \\times 1 2 8$ , $6 4 \\times 6 4$ and $3 2 \\times 3 2$ as inputs of VGG19. The final perceptual loss is the addition of perceptual losses in four resolutions. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Further, towards generating photo-realistic results, we incorporate a non-saturating adversarial loss $\\mathcal { L } _ { a d v }$ on $x _ { s \\to d }$ , which is calculated as ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/05bea7d6e6d22c447e74d62df9efb0cfa6ed3c06ed8234e0270924bc5b87f9da.jpg",
+        "text": "$$\n\\mathcal { L } _ { a d v } ( x _ { s  d } ) = \\mathbb { E } _ { x _ { s  d } \\sim p _ { r e c } } [ - l o g ( D ( x _ { s  d } ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            261,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $D$ is a discriminator, aimed at distinguishing reconstructed images from the original ones. Our full loss function is the combination of three losses with $\\lambda$ as a balanced hyperparameter ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/98a99248b9ea0369084c2ed86f66e8b66265c9eda359ab2a875e641b6da0b3c9.jpg",
+        "text": "$$\n{ \\mathcal { L } } ( x _ { s  d } , x _ { d } ) = { \\mathcal { L } } _ { r e c o n } ( x _ { s  d } , x _ { d } ) + \\lambda { \\mathcal { L } } _ { v g g } ( x _ { s  d } , x _ { d } ) + { \\mathcal { L } } _ { a d v } ( x _ { s  d } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.4 INFERENCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In inference stage, given a driving video sequence $V _ { d } = \\{ x _ { t } \\} _ { 1 } ^ { T }$ , we aim to transfer motion from $V _ { d }$ to $x _ { s }$ , in order to generate a novel video $V _ { d  s } = \\{ x _ { t  s } \\} _ { 1 } ^ { T }$ . If $V _ { d }$ and $x _ { s }$ stem from the same video sequence, i.e., $x _ { s } ~ = ~ x _ { 1 }$ , our task comprises of reconstructing the entire original video sequence. Therefore, we construct the latent motion representation of each frame using absolute transfer, which follows the training process, given as ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/906b3d65e6df9ab02a1357d7fd666aa6dc6b2c37473057c00ea3397676e29cf5.jpg",
+        "text": "$$\nz _ { s  t } = z _ { s  r } + w _ { r  t } , t \\in \\{ 1 , . . . , T \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            430,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "However, in real world applications, interest is rather placed on the scenario, where motion transfer between $x _ { s }$ and $V _ { d }$ , the latter stemming from different identities, i.e., $x _ { s } \\neq x _ { 1 }$ . Taking a talking head video as an example, in this setting, beyond identity, $x _ { 1 }$ and $x _ { s }$ might also differ in pose and expression. Therefore, we propose relative transfer to eliminate the motion impact of $w _ { r \\to 1 }$ and involve motion of $w _ { r \\to s }$ in the full generated video sequence. Owing to a linear representation of the latent path, we can easily represent $z _ { s  t }$ for each frame as ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ffd2269604154178f057dda4081be44f9ef04e2bc5ecbbf0667ba81e6b26d159.jpg",
+        "text": "$$\n\\begin{array} { r l } & { z _ { s  t } = ( z _ { s  r } + w _ { r  s } ) + ( w _ { r  t } - w _ { r  1 } ) } \\\\ & { \\qquad = z _ { s  s } + ( w _ { r  t } - w _ { r  1 } ) , \\ t \\in \\{ 1 , . . . , T \\} . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The first term in Eq. (12), $z _ { s  s }$ indicates the reconstruction of $x _ { s }$ , while the second term $( w _ { r  t } - w _ { r  1 } )$ represents the motion from $x _ { 1 }$ to $x _ { t }$ . This equation indicates that the original pose is preserved in $x _ { s }$ , at the same time motion is transferred from $V _ { d }$ . We note that in order to completely replicate the position and pose in $V _ { d }$ , it requires $x _ { s }$ and $x _ { 1 }$ to contain similar poses in relative motion transfer. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we firstly describe our experimental setup including implementation details and datasets. Secondly, we qualitatively demonstrate generated results based on testing datasets. Then, we provide quantitative evaluation w.r.t. image quality on (a) same-identity reconstruction, (b) crossvideo motion transfer, presenting (c) a user study. Next, we conduct an ablation study that demonstrates (d) the effectiveness of our proposed motion dictionary, as well as (e) associated size. Finally, we provide an in-depth analysis of our (f) latent codes and (g) motion dictionary to interpret their semantic meanings. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Datasets Our model is trained on the datasets VoxCeleb, TaichiHD and TED-talk. We follow the pre-processing method in (Siarohin et al., 2019) to crop frames into $2 5 6 \\times 2 5 6$ resolution for quantitative evaluation. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Implementation details Our model is implemented in PyTorch (Paszke et al., 2019). All models are trained on four 16G NVIDIA V100 GPUs. The total batch size is 32 with 8 images per GPU. We use a learning rate of 0.002 to train our model with the Adam optimizer (Kingma & Ba, 2014). The dimension of all latent codes, as well as directions in $D _ { m }$ is set to be 512. In our loss function, we use $\\lambda = 1 0$ in order to penalize more on the perceptual loss. It takes around 150 hours to fully train our framework. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/a37a7371bb03be1d7874e44a06664ca4dc3ef0bc6d060aab959b38245f9df2b2.jpg",
+        "image_caption": [
+            "Figure 4: Qualitative results. Examples for same-dataset absolute motion transfer on TaichiHD (top-right) and TED-talk (bottom-right). On VoxCeleb (left), we demonstrate cross-dataset relative motion transfer. We successfully transfer motion between $x _ { 1 }$ and $x _ { t }$ from videos in VoxCeleb to $x _ { s }$ from FFHQ, the latter not being used for training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Evaluation metrics We evaluate our model w.r.t. (i) reconstruction faithfulness using $\\mathcal { L } _ { 1 }$ , LPIPS, (ii) generated video quality using video FID, as well as (iii) semantic consistency using average keypoint distance (AKD), missing keypoint rate (MKR) and average euclidean distance (AED). Details are available in App. B.2. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 QUALITATIVE RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            487,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Firstly, we evaluate the ability of LIA to generate realistic videos and compare related results with four state-of-the-art methods. For TaichiHD and TED-talk datasets, we conduct an experiment related to cross-video generation. Corresponding results (see Fig. 4) confirm that our method is able to correctly transfer motion on articulated human bodies, in the absence of explicit structure representations. For the VoxCeleb dataset, we conduct a cross-dataset generation-experiment, where we transfer motion from VoxCeleb to images of the FFHQ dataset. We observe that our method outperforms FOMM and MRAA w.r.t. image quality, as both approaches visibly deform the shape of the original faces. This is specifically notable in the case that source and driving images entail large pose variations. At the same time, LIA is able to successfully tackle this challenge and no similar deformations are visible. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 COMPARISON WITH STATE-OF-THE-ART METHODS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We quantitatively compare our method with the state-of-the-art approaches X2Face, Monkey-Net, FOMM and MRAA on two tasks, namely (a) same-identity reconstruction and (b) cross-video motion transfer. Additionally, we conduct a (c) user study. ",
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+    {
+        "type": "text",
+        "text": "(a) Same-identity reconstruction We here evaluate the reconstruction ability of our method. Specifically, we reconstruct each testing video by using the first frame as $x _ { s }$ and the remaining frames as $x _ { d }$ . Results on three datasets are reported in Table 1. Focusing on foregroundreconstruction, our method outperforms the other approaches w.r.t. all metrics. More results are presented in App. B.3, discussing background-reconstruction. ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "(b) Cross-video motion transfer Next, we conduct experiments, where source images and driving videos stem from different video sequences. In this context, we mainly focus on evaluating talking head videos and explore two different cases. In the first case, we generate videos using the VoxCeleb testing set to conduct motion transfer. In the second case, source images are from an unseen dataset, namely the GermanPublicTV dataset, as we conduct cross-dataset motion transfer. In both experiments, we randomly construct source and driving pairs and transfer motion from driving videos to source images to generate a novel manipulated dataset. Since ground truth data for our generated videos is not available, we use video FID (as initialized by Wang et al. (2020a)) to compute the distance between generated and real data distributions. As shown in Tab. 2, our method outperforms all other approaches w.r.t. video FID, indicating the best generated video quality. ",
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+        "type": "table",
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+        "table_caption": [
+            "Table 1: Same-identity reconstruction. Comparison with state-of-the-art methods on three datasets for same-identity reconstruction. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">VoxCeleb</td><td colspan=\"4\">TaichiHD</td><td colspan=\"4\">TED-talks</td></tr><tr><td>L1</td><td>AKD</td><td>AED</td><td>LPIPS</td><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>LPIPS</td><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>LPIPS</td></tr><tr><td>X2Face</td><td>0.078</td><td>7.687</td><td>0.405</td><td>-</td><td>0.080</td><td>(17.654,0.109)</td><td></td><td>-</td><td>=</td><td></td><td></td><td>-</td></tr><tr><td>Monkey-Net</td><td>0.049</td><td>1.878</td><td>0.199</td><td>-</td><td>0.077</td><td>(10.798,0.059)</td><td></td><td></td><td>=</td><td></td><td></td><td>=</td></tr><tr><td>FOMM</td><td>0.046</td><td>1.395</td><td>0.141</td><td>0.136</td><td>0.063</td><td>(6.472,0.032)</td><td>0.4950.191</td><td></td><td>0.030</td><td>(3.759,0.0090)0.428</td><td></td><td>0.13</td></tr><tr><td>MRAA w/o bg</td><td>0.043</td><td>1.307</td><td>0.140</td><td>0.127</td><td>0.063</td><td>(5.626,0.025)</td><td>0.460</td><td>0.189</td><td>0.029</td><td>(3.126,0.0092)0.396</td><td></td><td>0.12</td></tr><tr><td>Ours</td><td>0.041</td><td>1.353</td><td>0.138</td><td>0.123</td><td>0.057</td><td>(4.823, 0.020)</td><td>0.431</td><td>0.180</td><td>0.027</td><td>(3.141,0.0095)0.399</td><td></td><td>0.11</td></tr></table>",
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+        "type": "table",
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+        "table_caption": [
+            "Table 2: Cross-video generation. We report video FID for both inner- and cross-dataset tasks on VoxCeleb and GermanPublicTV. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>VoxCeleb</td><td>GermanPublicTV</td></tr><tr><td>FOMM</td><td>0.323 0.456</td></tr><tr><td>MRAA 0.308</td><td>0.454</td></tr><tr><td>Ours</td><td>0.161 0.406</td></tr></table>",
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+    },
+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 3: User study. We ask 20 human raters to conduct a subjective video quality evaluation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>VoxCeleb(%)</td><td>TaichiHD(%)</td><td>TED-talk(%)</td></tr><tr><td>Ours/FOMM</td><td>92.9/7.1</td><td>64.5/35.5</td><td>71.4/28.6</td></tr><tr><td>Ours/MRAA</td><td>89.7/10.3</td><td>60.7/39.9</td><td>54.8/45.2</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "(c) User study We conduct a user study to evaluate video quality. Towards this, we displayed paired videos and asked 20 human raters ‘which clip is more realistic?’. Each video-pair contains a generated video from our method, as well as a video generated from FOMM or MRAA. Results suggest that our results are more realistic in comparison to FOMM and MRAA across all three datasets (see Tab. 3). Hence, the obtained human preference is in accordance with our quantitative evaluation. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f09a5c2b6459de9d6ea611e9c6cf079f6a6df9128278bcecbdc537633eda7d9a.jpg",
+        "table_caption": [
+            "Table 4: Ablation study on motion dictionary. We conduct experiments on three datasets with and without $D _ { m } ^ { \\phantom { \\dagger } }$ and show reconstruction results. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">VoxCeleb</td><td colspan=\"2\">TaichiHD</td><td colspan=\"2\">TED-talks</td></tr><tr><td>Method</td><td>L1 LPIPS</td><td>L1</td><td>LPIPS</td><td>L1</td><td>LPIPS</td></tr><tr><td>wloDm</td><td>0.049 0.165</td><td>0.062</td><td>0.186</td><td>0.031</td><td>0.12</td></tr><tr><td>Full</td><td>0.041 0.123</td><td>0.057</td><td>0.180</td><td>0.028</td><td>0.11</td></tr></table>",
+        "bbox": [
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+        "table_caption": [
+            "Table 5: Ablation study on $D _ { m }$ size. We conduct experiments on three datasets with 5 different $D _ { m }$ size and show reconstruction results. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">VoxCeleb</td><td colspan=\"2\">TaichiHD</td><td colspan=\"2\">TED-talks</td></tr><tr><td>M</td><td>L1</td><td>LPIPS</td><td>L1</td><td>LPIPS</td><td>L1</td><td>LPIPS</td></tr><tr><td>5 10</td><td>0.051 0.043</td><td>0.15 0.13</td><td>0.070 0.065</td><td>0.22 0.20</td><td>0.037 0.036</td><td>0.15</td></tr><tr><td>20</td><td>0.041</td><td>0.12</td><td>0.057</td><td>0.18</td><td>0.028</td><td>0.13 0.11</td></tr><tr><td>40</td><td>0.042</td><td>0.12</td><td>0.060</td><td>0.19</td><td>0.030</td><td>0.12</td></tr><tr><td>100</td><td>0.041</td><td>0.12</td><td>0.058</td><td>0.18</td><td>0.028</td><td>0.11</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "4.3 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We here analyze our proposed motion dictionary and focus on answering following two questions. ",
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+    {
+        "type": "text",
+        "text": "(d) Is the motion dictionary $D _ { m }$ beneficial? We here explore the impact of proposed $D _ { m }$ , by training our model without $D _ { m }$ . Specifically, we output $w _ { r \\to d }$ directly from MLP, without using LMD to learn an orthogonal basis. From the evaluation results reported in Tab. 4 and qualitative results in App. B.5, we observe that in the absence of $D _ { m }$ , model fails to generate high-quality images, which proves the effectiveness of $D _ { m }$ , consistently on all datasets. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "(e) How many directions are required in $D _ { m }$ ? Towards finding an effective size of $D _ { m }$ , we empirically test three different $M$ , viz. 5, 10, 20, 40 and 100. Quantitative results in Tab. 5 show that when using 20 directions, the model achieves the best reconstruction results. ",
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+    {
+        "type": "text",
+        "text": "4.4 FURTHER ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "(f) Latent code analysis. While our method successfully transfers motion via latent space navigation, we here aim at answering the question — what does $x _ { r }$ represent? Towards answering this question, we proceed to visualize $x _ { r }$ . We firstly decode $z _ { s  r }$ into a dense flow field $\\phi _ { s  r }$ , which is then used to warp $x _ { s }$ (we show details in App. B.4). Fig. 5 shows examples of $x _ { s }$ and $x _ { r }$ . Interestingly, we observe that $x _ { r }$ represents the canonical pose of $x _ { s }$ , regardless of original poses of the subjects. And for all datasets, reference images resemble each other $w . r . t .$ . pose and scale. As such reference images can be considered as a normalized form of $x _ { s }$ , learning transformations between $x _ { s }$ and $x _ { d }$ using $x _ { s }  x _ { r }  x _ { d }$ is considerably more efficient than $x _ { s } \\to x _ { d }$ , once $x _ { r }$ is fixed. ",
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+        "page_idx": 7
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+    {
+        "type": "image",
+        "img_path": "images/f4021d1e9c0a6b78d5507518e6f476c3eb214d4b2fd88cb1cbad903d30c7d449.jpg",
+        "image_caption": [
+            "Figure 5: Visualization of reference images. Example source (top) and reference images (down) from VoxCeleb, TaichiHD and TED-talk datasets. Our network learns reference images of a consistently frontal pose, systematically for all input images of each dataset. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/833c41d15a70be2fc45b2ced129969f51412a6047f24acd5518917c81c6c810d.jpg",
+        "image_caption": [
+            "Figure 6: Linear manipulation of four motion directions on the painting of Mona Lisa. Manipulated results indicate that $d _ { 6 }$ represents eye movement, $d _ { 8 }$ represents head nodding, whereas $\\bar { d _ { 1 9 } }$ and $d _ { 7 }$ represent facial expressions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Noteworthy, we found the similar idea of learning a ‘reference image’ has also been explored by Siarohin et al. (2019) (FOMM) and Wiles et al. (2018) (X2Face). However, deviating from our visualized ‘reference image’, the ’reference image ’in FOMM refers to a non-visualized and abstract concept. In addition, LIA only requires a latent code $z _ { s  r }$ , rather than the ’reference image’ for both, training and testing, which is contrast to X2Face. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "(g) Motion dictionary interpretation. Towards further interpretation of directions in $D _ { m }$ , we conduct linear manipulations on each $d _ { i }$ . Images pertained to manipulating four motion directions are depicted in Fig. 6. The results suggest that the directions in $\\boldsymbol { D } _ { m } ^ { \\bar { 2 } }$ are semantically meaningful, as they represent basic visual transformations such as head nodding $( d _ { 8 } )$ , eye movement $( d _ { 6 } ) ^ { \\dag }$ and facial expressions $\\cdot d _ { 1 9 }$ and $d _ { 7 }$ ). More results can be found on our project webpage2. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we presented a novel self-supervised autoencoder LIA, aimed at animating images via latent space navigation. By the proposed Linear Motion Decomposition (LMD), we were able to formulate the task of transferring motion from driving videos to source images as learning linear transformations in the latent space. We evaluated proposed method on real-world videos and demonstrated that our approach is able to successfully animate still images, while eliminating the necessity of explicit structure representations. In addition, we showed that the incorporated motion dictionary is interpretable and contains directions pertaining to basic visual transformations. Both quantitative and qualitative evaluations showed that LIA outperforms state-of-art algorithms on all benchmarks. We postulate that LIA opens a new door in design of interpretable generative models for video generation. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "ETHIC STATEMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this work, we aim to synthesize high-quality videos by transferring motion on still images. Our approach can be used for movie production, making video games, online education, generating synthetic data for other computer vision tasks, etc. We note that our framework mainly focuses on learning how to model motion distribution rather than directly model appearance, therefore it is not biased towards any specific gender, race, region, or social class. It works equally well irrespective of the difference in subjects. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REPRODUCIBILITY STATEMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            237,
+            437,
+            252
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We assure that all the results shown in the paper and supplemental materials can be reproduced. We intend to open-source our code, as well as trained models. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            323,
+            367,
+            338
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This work was granted access to the HPC resources of IDRIS under the allocation AD011011627R1. It was supported by the French Government, by the National Research Agency (ANR) under Grant ANR-18-CE92-0024, project RESPECT and through the 3IA Cote d’Azur Investments in the Future ˆ project managed by the National Research Agency (ANR) with the reference number ANR-19- P3IA-0002. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+        "text": "A DETAILS OF MODEL ARCHITECTURE ",
+        "text_level": 1,
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+        "text": "We proceed to describe the model architecture in this section. Fig. 7 shows details of our $E$ . In each ResBlock in $E$ , spatial size of input feature maps are downsampled. We take feature maps of spatial sizes from $8 \\times 8$ to $2 5 6 \\times 2 5 6$ as our appearance features $x _ { i } ^ { e n c }$ . We use a 5-layer MLP to predict a magnitude vector $A _ { d  r }$ from $z _ { d  r }$ . Fig. 8 (a) shows the general architecture of our $G$ , which consists of two components, a flow field generator $G _ { f }$ and a refinement network $G _ { r }$ . We apply StyleConv (Upsample $\\mathbf { + \\ C o n v 3 \\times 3 }$ ), which is proposed by StyleGAN2, in $G _ { f }$ . StyleConv takes latent representation $z _ { s  t }$ as style code and generates flow field $\\phi _ { i }$ and corresponding mask $m _ { i }$ . $G _ { r }$ uses UpConv (Conv1 $\\times \\ 1 +$ Upsample) to upsample and refine inpainted feature maps to target resolution. We show details pertaining to $G$ block in Fig. 8 (b). Each $G$ block is used to upsample $\\times 2$ the previous resolution. We stack 6 blocks towards producing 256 resolution images. ",
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+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/9325fffdbfebc0ee68f0be3ae9d39d9a1ea8e2cd0cc42b57b0799a99bd6ea9c8.jpg",
+        "image_caption": [
+            "Figure 7: Encoder architecture. We show details of architecture of $E$ in (a) and ResBlock in (b). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We proceed to introduce details of datasets and evaluation metrics used in our experiments. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "B.1 DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "VoxCeleb (Nagrani et al., 2019) consists of a large amount of interview videos of different celebrities. Following the process of FOMM (Siarohin et al., 2019), we extract frames and crop them into $2 5 6 \\times 2 5 6$ resolution. In total, VoxCeleb contains a training set of 17928 videos and a test set of 495 videos. ",
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+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/9bfd82d4cf08f228e79a896a311bfc79b0ee414695a25bedad52ccbc1330672a.jpg",
+        "image_caption": [
+            "Figure 8: Generator architecture. We show details about architecture of $G$ in (a) and $G$ block in (b). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            99,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "TaiChiHD (Siarohin et al., 2019) consists of videos of full human bodies performing Tai Chi actions. We follow the original pre-processing of FOMM (Siarohin et al., 2019) and utilize its $2 5 6 \\times 2 5 6$ version. TaiChiHD contains 1096 training videos and 115 testing videos. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "TED-talk is a new dataset proposed in MRAA (Siarohin et al., 2021). It comprises a number of TED-talk videos, where the main subjects have been cropped out. We resize the original version into $2 5 6 \\times 2 5 6$ resolution to train our model. This dataset includes 1124 training videos and 130 testing videos. ",
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B.2 EVALUATION METRICS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We use five different metrics to evaluate our experimental results, namely $\\mathcal { L } _ { 1 }$ , LPIPS, AKD, MKR and AED that quantify the reconstructed results. In addition, we compute video FID to evaluate video quality in motion transferring tasks. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "$\\mathcal { L } _ { 1 }$ represents the mean absolute pixel difference between reconstructed and real videos. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "LPIPS (Zhang et al., 2018) aims at measuring the perceptual similarity between reconstructed and real images by leveraging the deep features from AlexNet (Krizhevsky et al., 2012). ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Video FID is a modified version of the original FID (Heusel et al., 2017). We here follow the same implementation as Wang et al. (2020a) and utilize a pre-trained ResNext101 (Hara et al., 2018) to extract spatio-temporal features to compute the distance between real and generated videos distributions. We take the first 100 frames of each video as input of the feature-extractor to compute the final scores. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Average keypoint distance (AKD) and missing keypoint rate (MKR) evaluate the difference between keypoints of reconstructed and ground truth videos. We extract landmarks using the face alignment approach of (Bulat & Tzimiropoulos, 2017) and extract body poses for both TaiChiHD and TED-talks using OpenPose (Cao et al., 2019). AKD is computed as the average distance between corresponding keypoints, whereas MKR is the proportion of keypoints present in the groundtruth that are missing in a reconstructed video. ",
+        "bbox": [
+            173,
+            776,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Average Euclidean distance (AED) measures the ability of preserving identity in reconstructed video. We use a person re-identification pretrained model (Zheng et al., 2020) for measuring human bodies (TaichiHD and TED-talk) and OpenFace (Amos et al., 2016) for faces to extract identity embeddings from reconstructed and ground truth frame pairs, then we compute MSE of their difference for all pairs. ",
+        "bbox": [
+            174,
+            859,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B.3 COMPARISON WITH FULL MRAA ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            449,
+            117
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We show quantitative evaluation results with the full MRAA model in Tab. 6. We observe that our method achieves competitive results in reconstruction and keypoint evaluation. While we do not explicitly predict keypoints, w.r.t. the TaichiHD dataset, interestingly we outperform MRAA in both, AKD and MKR. Such results showcase the effectiveness of our proposed method on modeling articulated human structures. However, reconstruction evaluation cannot provide a completely fair comparison on how well the main subjects (e.g., faces and human bodies) are generated in videos. This is in particular the case for TaichiHD and TED-talk, where backgrounds have large contributions to the final scores. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/a627194bf94f738dbeb4d2f9efd1eb6f668ed70266fedc4e6e71e0ba390a5374.jpg",
+        "table_caption": [
+            "Table 6: Comparison with full MRAA. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"4\">VoxCeleb</td><td colspan=\"4\">TaichiHD</td><td colspan=\"4\">TED-talks</td></tr><tr><td>L1</td><td>AKD</td><td>AED</td><td>LPIPS</td><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>LPIPS</td><td>L1</td><td>(AKD,MKR)</td><td>AED</td><td>LPIPS</td></tr><tr><td>MRAA</td><td>0.041</td><td>1.303</td><td>0.135</td><td>0.124</td><td>0.045</td><td>(5.551,( ,0.025)</td><td>0.431</td><td>0.178</td><td>0.027</td><td>(3.107,0.0093)</td><td>0.379</td><td>0.11</td></tr><tr><td>Ours</td><td>0.041</td><td>1.353</td><td>0.138</td><td>0.123</td><td>0.057</td><td>(4.823,0.020)</td><td>0.431</td><td>0.180</td><td>0.027</td><td>(3.141,0.0095)</td><td>0.399</td><td>0.11</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.4 REFERENCE IMAGE GENERATION. ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            352,
+            449,
+            364
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "To produce $x _ { r }$ , we use $G$ to decode $z _ { s  r }$ into the flow field $\\phi _ { s  r }$ . Reference image $x _ { r }$ is obtained by warping $x _ { s }$ using $\\phi _ { s  r }$ . The entire process is shown in Fig. 9. ",
+        "bbox": [
+            174,
+            376,
+            825,
+            402
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/762132d8adc241e8fde2ec32f40c10dbb27f4f06b3953c429f112db181bf10c2.jpg",
+        "image_caption": [
+            "Figure 9: Reference image generation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            415,
+            802,
+            534
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.5 QUALITATIVE RESULTS ON EFFECTIVENESS OF USING MOTION DICTIONARY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            593,
+            738,
+            607
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Fig. 10 illustrates the generated results on transferring motion from VoxCeleb to GermanPublicTV with and without motion dictionary. We observe that without the motion dictionary, appearance information is undesirably transferred from driving videos to generated videos. ",
+        "bbox": [
+            174,
+            616,
+            825,
+            657
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "B.6 LIMITATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            671,
+            310,
+            685
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "For human body, one limitation of our method is dealing with body occlusion. We observe in Fig. 11 that in taichi videos, in case of occlusion cause by legs and arms, motion is not transferred successfully. In addition, in TED-talks, transferring hand motion is challenging, as hands are of small size, articulated and sometimes occluded by human bodies. ",
+        "bbox": [
+            174,
+            696,
+            825,
+            748
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/70acaf34c5c7203b948fb2919b9b562110b48bb7fdf0186567c1baf79c82cd42.jpg",
+        "image_caption": [
+            "Figure 10: Generated results with and without $D _ { m }$ . We observe that the disentanglement of appearance and motion is much better by using $D _ { m }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            127,
+            823,
+            518
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/8df647813e53c9e3f9fda4ddf2534e1d09430b0019731ce68455cebf897027e0.jpg",
+        "image_caption": [
+            "Figure 11: Failure cases. We observed that it is still challenging for LIA to handle arm-leg occlusion (Taichi) and hand motion (TED-talk). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            627,
+            823,
+            849
+        ],
+        "page_idx": 16
+    }
+]

parse/dev/AIqC7F7xV-d/AIqC7F7xV-d.md

@@ -0,0 +1,460 @@
+# Learning Unified Representations for Multi-Resolution Face Recognition
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 In this work, we propose Branch-to-Trunk network (BTNet), a novel representation   
+2 learning method for multi-resolution face recognition. It consists of a trunk network   
+3 (TNet), namely a unified encoder, and multiple branch networks (BNets), namely   
+4 resolution adapters. As per the input, a resolution-specific BNet is used and the   
+5 output are implanted as feature maps in the feature pyramid of TNet, at a layer with   
+6 the same resolution. The discriminability of tiny faces is significantly improved, as   
+7 the interpolation error introduced by rescaling, especially up-sampling, is mitigated   
+8 on the inputs. With branch distillation and backward-compatible training, BTNet   
+9 transfers discriminative high-resolution information to multiple branches while   
+10 guaranteeing representation compatibility. Our experiments demonstrate strong   
+11 performance on face recognition benchmarks, both for multi-resolution identity   
+12 matching and feature aggregation, with much less computation amount and param  
+13 eter storage. We establish new state-of-the-art on the challenging QMUL-SurvFace   
+14 1: N face identification task.
+
+# 15 1 Introduction
+
+16 Machine learning has advanced tremendously driven by deep learning methods, but is still severely   
+17 challenged by various data specifications, such as data type, structure, scale and size, etc. For   
+18 instance, face recognition (FR) is a well-established deep learning task, while the performance   
+19 degrades dramatically in the testing domain that differs from the training one, influenced by factors   
+20 of variance like resolution, illumination, occlusion, etc.   
+21 Most face recognition methods map each image to a point embedding in the common metric space   
+22 by deep neural networks (DNNs). The dissimilarity of images can be then calculated using various   
+23 distance metrics (e.g., cosine similarity, Euclidean distance, etc.) for face recognition tasks.   
+24 Recent advancements in margin-based loss (e.g., ArcFace [1], MV-Arc-Softmax [2], CurricularFace   
+25 [3], etc) enhanced discriminability of the metric space, with small intra-identity distance and large   
+26 inter-identity distance. However, lack of variation in training data still leads to poor generalizability.   
+27 Various useful methods are utilized to mitigate this issue. The model adapts to factors of variance   
+28 by augmenting datasets, whereas the large discrepancy in data distribution could potentially weaken   
+29 the model’s ability to extract discriminative features with the same data scale and model structure   
+30 (see Section 4.3). Fine-tuning is widely used to transfer large pretrained models to new domains with   
+31 different data specifications. However, this strategy requires one to store and deploy a separate copy   
+32 of the backbone parameters for every single new domain, which is expensive and often infeasible.   
+33 As known, the resolutions of face images in reality may be far beyond the scope covered by the   
+34 model. As the small feature maps with a fixed spatial extent (e.g., $7 \times 7 )$ are mapped to an embedding   
+35 with a predefined dimension (e.g., $1 2 8 - d$ , $5 1 2 - d$ , etc.) by a fully connected (fc) layer, input   
+36 images need to be rescaled to a canonical spatial size (e.g., $1 1 2 \times 1 1 2 )$ before fed into the network.   
+37 However, up-sampling low-resolution (LR) images introduces the interpolation error (see Section 3.1),   
+38 deteriorating the recognizable ones which contain enough clues to identify the subject. Even though   
+39 super-resolution methods [4–10] are widely used to build faces with good visualization, they inevitably   
+40 introduce feature information of other identities when reconstructing high-resolution (HR) faces.   
+41 This may lead to erroneous identity-specific features, which are detrimental to risk-controlled face   
+42 recognition.   
+43 Empirically, we can divide inputs by resolution distribution and learn to operate on them via multiple   
+44 models to achieve high accuracy and efficiency. However, multi-model fashion cannot be applied   
+45 directly for cross-resolution recognition as representation compatibility among models need to be   
+46 guaranteed [11–15].   
+47 To improve discriminability while ensure the compatibility of the metric space for multi-resolution   
+48 face representation, we learn the “unified” representation by a partially-coupled Branch-to-Trunk   
+49 Network (BTNet). It is composed of multiple independent branch networks (BNets) and a shared   
+50 trunk network (TNet). A resolution-specific BNet is used for a given image, and the output are   
+51 implanted as feature maps in the feature pyramid of TNet, at a layer with the same resolution.   
+52 Furthermore, we find that multi-resolution training can be beneficial to building a strong and robust   
+53 TNet, and backward-compatible training (BCT) [11] can improve the representation compatibility   
+54 during the training process of BTNet. To ameliorate the discriminability of tiny faces, we propose   
+55 branch distillation in intermediate layers, utilizing information extracted from HR images to help the   
+56 extraction of discriminative features for resolution-specific branches.   
+57 Our method is simple and efficient, which breaks the convention of up-sampling the inputs and   
+58 serves as a general framework that can be easily implemented by several existing methods due to   
+59 conceptual simplicity. Meanwhile, BTNet is able to reduce the number of FLOPS by operating the   
+60 inputs without up-sampling, and per-resolution storage cost by only storing the learned branches and   
+61 resolution-aware BNs [16], while re-using the copy of the trunk model.   
+62 We demonstrate that our method performs comparably in various open-set face recognition tasks (1:1   
+63 face verification and 1: N face identification), in both settings of multi-resolution identity matching   
+64 and feature aggregation, while meaningfully reduces the redundant computation cost and parameter   
+65 storage. In the challenging QMUL-SurvFace 1: N face identification task [17], we establish new   
+66 state-of-the-art by outperforming prior models. Furthermore, by avoiding the ill-posed problem (i.e.,   
+67 image up-sampling), our approach also effectively reduces the additional noise and uncertainty of the   
+68 representation, which plays a key role in reliable risk-controlled face recognition.
+
+# 69 2 Related Work
+
+70 Compatible Representation Learning: The task of compatible representation learning aims at   
+71 encoding features that are interoperable with the features extracted from other models. Shen et. al.   
+72 [11] first formulated the problem of backward-compatible learning (BCT) and proposed to utilize the   
+73 old classifier for compatible feature learning. Since the multi-model fashion benefits representation   
+74 learning with lower computation, our idea of cross-resolution representation learning can be modeled   
+75 similar to cross-model compatibility [11–15], as metric space alignment for different resolutions. Our   
+76 goal is achieved by both compatibility-aware network architecture and training strategy.   
+77 Knowledge Distillation and Transfer: The concept of knowledge distillation (KD) was first   
+78 proposed by Hinton et. al. in [18], which can be summarized as employing a large parameter   
+79 model (teacher) to supervise the learning of a small parameter model (student). Distillation from   
+80 intermediate features [19–29] is widely adopted to enhance the effectiveness of knowledge transfer.   
+81 However, due to the “dark knowledge” hidden in the intermediate layers, additional subtle design is   
+82 often required to match and rescale intermediate features. Instead, our approach can easily locate the   
+83 distillation features without rescaling and effectively transfer knowledge from the HR domain to LR   
+84 branches.   
+85 Low Resolution Face Recognition: Its task includes low resolution-to-low resolution (LR-to-LR)   
+86 matching and low resolution-to-high resolution (LR-to-HR) matching [30]. The work can be divided   
+87 into two categories [31]: (1) Super-resolution (SR) based methods aim to upscale LR images to   
+88 construct HR images and use them for feature extraction [4–10]. (2) Projection-based methods aim to   
+89 extract adequate representations in different domains and project them into a common feature space   
+90 [32–34]. SR approaches are able to build faces with good visualization, but inevitably introduce   
+91 feature information of other identities when reconstructing corresponding HR faces, thus introducing   
+92 noise for identity-specific features. Compared to previous projection methods, our approach directly   
+93 learns discriminative representations in a common feature space for HR and LR inputs, without   
+94 additional projection heads for feature transformation.
+
+Pseudo-Siamese Networks: Siamese networks are a coupling architecture based on DNNs, which are widely used for signature verification [35], face verification [36, 37], tracking [38], etc. PseudoSiamese networks [39] are decoupled Siamese networks, as the weights of the two branches are not shared, resulting in a more flexible representation way for the two entities. Hughes et. al. in [40] proposed a pseudo-Siamese CNN for identifying corresponding patches in SAR and optical images. Inspired by pseudo-Siamese networks, we propose a resolution-adaptive partially coupled Siamese network architecture, extracting specific-shared features for images with different resolutions.
+
+# 3 Learning Specific-Shared Feature Transfer
+
+Instead of rescaling the inputs to a canonical size, we build multiple resolution-specific branches (BNets) that are used to map inputs to intermediate features with the same resolution and a resolutionshared trunk (TNet) to map feature maps with different resolutions to a high-dimension embedding. We gain several important properties by doing so: (1) Processing inputs on its original resolution can diminish the inevitably introduced error via up-sampling or information loss via down-sampling, thus preserving the discriminability of visual information with different resolutions. (2) Information streams of different resolutions are encoded uniformly, thus enabling the representation compatibility, which is particularly beneficial to open-set face recognition considering that a compatible metric space is the prerequisite for computing similarity. (3) This also effectively reduce the computation for LR images by supplying computational resources conditioned on the input resolution.
+
+# 113 3.1 Up-Sampling Error Analysis
+
+![](images/bcaf8fb368f804daba922e50885b5d890a9bd70ea33f0a8743d4e42864a72421.jpg)  
+Figure 1: Estimated Error Upperbound. (bilinear interpolation, average value for over 100 images) with the change of image resolution relative to resolution 112.
+
+![](images/49c6de557e00d2a1971a3a5c302649e9eb1af1a26b38f484c1a1576206274175.jpg)  
+Figure 2: Basic ideas of the proposed BTNet. Images of a certain identity are first projected to the feature maps with the same resolution respectively (Adapt) and then projected to a unified feature representation (Encode). In this figure, feature maps with the same resolution are indicated by outlines in the same color.
+
+114 Figure 1 illustrates the experimental estimation of interpolation error, whose upper bound increases   
+115 with the decline of the image resolution (see detailed theoretical derivation in Appendix A.1). Note   
+116 that the error soars up when the resolution drops below 32 approximately which can be viewed as LR   
+117 face images, consistent with the tiny-object criterion [41].   
+118 The results show that: (1) inputs with a resolution higher than around 32 can be considered in the   
+119 same HR domain, since the error information introduced by up-sampling via interpolation can be   
+120 ignored to a certain extent; (2) inputs with a resolution lower than around 32 should be treated as in   
+121 various LR domains due to the high sensitivity of the resolution to errors.
+
+# 3.2 Branch-to-Trunk Network
+
+Let $X$ be an input RGB image with a space shape: $X \in \mathbb { R } ^ { H \times W \times 3 }$ where $H \times W$ corresponds to the spatial dimension of the input. For efficient batch training and inference, we predefine a canonical size $S \times S$ (e.g., $1 1 2 \times 1 1 2$ for typical face recognition models like ArcFace [1]).
+
+We build a trunk network $\boldsymbol { T } : \mathbb { R } ^ { H \times W \times 3 }  \mathbb { R } ^ { C _ { e m b } }$ capable of extracting discriminative information with different resolutions, where $C _ { e m b }$ is the number of embedding channels. For every resolution $r$ in the candidate set, we formulate a resolution-specific branch, $z _ { r } = \textit { B } _ { r } ( X _ { r } )$ , which maps the input image $X _ { r }$ to feature maps with the same resolution and expanded channels $\boldsymbol { z } _ { r } : \mathbb { R } ^ { r \times r \times 3 }  \mathbb { R } ^ { r \times r \times C _ { r } }$ . The idea is to learn our branches $B$ to focus on resolution-specific feature transfer independently. Feature maps will then be coupled to the trunk network $T$ in the feature pyramid with the same spatial resolution $r \times r$ , allowing for further mapping to the unified presentation space by $T _ { r } : \mathbb { R } ^ { r \times r \times C _ { r } } $ $\mathbb { R } ^ { C _ { e m b } }$ .
+
+Here, we follow the idea of “avoiding redundant up-sampling”. Our branches $B$ are implemented with same-resolution mapping: i.e., the model preserves the network architecture of $T$ from input to the layer with resolution $r$ and abandons down-sampling operations (e.g., replacing the convolution of stride 2 with stride 1, abandoning the pooling layers, etc.) to keep the same-resolution flow.
+
+We specifically name our specific-shared feature transfer network as Branch-to-Trunk Network, abbreviated as "BTNet". Figure 2 visually summarizes the main ideas of BTNet.
+
+# 3.3 Training Objectives
+
+We now describe the training objectives. The training of BTNet includes training the trunk network $T$ such that it can produce discriminative and compatible representations for multi-resolution information, and fine-tuning the branch networks $B$ to encourage them to learn resolution-specific feature transfer, so as to improve accuracy without compromising compatibility.
+
+Influence Loss. It is a compatibility-aware classification loss which is implemented by feeding the embeddings of the new model to the classifier of the old model [11]. Since the difficulties of samples vary due to image resolution, we compute CurricularFace [3] as our classification loss, in the form of:
+
+$$
+L _ { c u r } = - \log ( \frac { e ^ { s \cos ( \theta _ { y _ { i } } + m ) } } { e ^ { s \cos ( \theta _ { y _ { i } } + m ) } + \sum _ { j = 1 , j \neq y _ { i } } ^ { n } e ^ { s N ( t ^ { ( k ) } , \cos ( \theta _ { j } ) ) } } )
+$$
+
+$$
+\begin{array} { c } { { N ( t , \cos \theta _ { j } ) = \displaystyle \left\{ \begin{array} { l c } { { \cos ( \theta _ { j } ) , } } & { { \cos ( \theta _ { y _ { i } } + m ) - \cos ( \theta _ { j } ) \ge 0 } } \\ { { \cos ( \theta _ { j } ) ( t + \cos ( \theta _ { j } ) ) , } } & { { e l s e } } \end{array} \right. } } \\ { { t ^ { ( k ) } = \displaystyle \alpha \sum _ { i } \cos \theta _ { y _ { i } } + ( 1 - \alpha ) t ^ { ( k - 1 ) } } } \end{array}
+$$
+
+150 which distinguishes both the difficultness of different samples in each stage and relative importance   
+151 of easy and hard samples during different training stages. Thus, we refine CurricularFace loss as our   
+152 influence loss:
+
+$$
+L _ { i n f l u e n c e } = L _ { c u r } ( \varphi _ { b t } , \kappa ^ { * } )
+$$
+
+where 153 $\varphi _ { b t }$ is BTNet backbone (both $B _ { r }$ and $T _ { r }$ ), and $\kappa ^ { * }$ is the classifier of the pretrained trunk $T$ .
+
+154 Branch Distillation Loss. Due to the   
+155 continuity of the scale change of both the   
+156 image pyramid and the feature pyramid   
+157 [42], we can get a qualitative sense of   
+158 the similarity between images and feature   
+159 maps with the same resolution (see Figure   
+160 3). Furthermore, features extracted from   
+161 HR images have richer and clearer infor  
+162 mation than those from LR images [43].   
+163 Motivated by these analyses, we utilize an   
+164 MSE loss to encourage the branch output   
+165 $z _ { r }$ to be similar to the corresponding fea  
+166 ture maps of the pretrained trunk network   
+167 $z _ { s }$ :
+
+![](images/e434a9a21fd643648ac4896d6cdd3ccf6e9342a751aac825373efc28d542ddee.jpg)  
+Figure 3: Visual comparison of face image-feature map pairs with different resolutions (resized to a common size here for illustration).
+
+$$
+L _ { b r a n c h } = \frac { 1 } { V } \sum _ { v = 1 } ^ { V } { ( z _ { r _ { v } } - z _ { s _ { v } } ) } ^ { 2 }
+$$
+
+168 where $V$ denotes the batch size.
+
+169 The whole training objective is a combination of the above objectives:
+
+$$
+L = L _ { i n f l u e n c e } + \lambda _ { b r a n c h } L _ { b r a n c h }
+$$
+
+170 where $\lambda _ { b r a n c h }$ is a hyper-parameter to weigh the losses and we set $\lambda _ { b r a n c h } = 0 . 5$ in all our experi  
+171 ments.
+
+![](images/22946257e92b509279813a29ef0c3ac19fcb89eb7b3c4b6d171707c4f9ae79f6.jpg)  
+Figure 4: Comparison of $\#$ Params $\mathbf { \Psi } ^ { ( \mathbf { M } ) }$ between fully finetuning and $\varphi _ { b t }$ .
+
+![](images/1df08d5fa7d736b80d3f65aa605b4972425929210f8a903a7b8e250ca889addb.jpg)  
+Figure 5: Comparison of FLOPs (G) between baselines and $\varphi _ { b t }$ .
+
+# 172 3.4 Storing Branch Networks
+
+An obvious adaptation strategy is fully finetuning of the model on each resolution. However, this strategy requires one to store and deploy a separate copy of the backbone parameters for every resolution, which is an expensive proposition and difficult to expand into more segmented resolution branches. Our BTNet is beneficial in the scenario of multi-resolution face recognition which achieves better parameter/accuracy trade-offs. Since activation statistics including means and variances under different resolutions are incompatible [44], we update and store Batch Normalization (BN) [45] parameters in all layers of $B _ { r }$ and $T _ { r }$ for each resolution, whose amount is negligible. Apart from this, we only need to store the learned branches and re-use the original copy of the pretrained trunk model, significantly reducing the storage cost. Figure 4 shows that BTNet requires only $1 . 1 \% \sim 4 8 . 9 \%$ of all the parameters compared to fully updating all the parameters of TNet.
+
+# 4 Experiments
+
+184 To validate BTNet on face recognition tasks in open universe, we perform 1:1 verification and $1 : N$   
+185 identification tasks in two different settings, including (a) multi-resolution identity matching, and
+
+(b) multi-resolution feature aggregation. For 1:1 verification, a pair of templates are provided and the model is to decide whether they belong to the same identity or not. For 1:N identification, a set of gallery images are first mapped onto their embedding vectors (indexing) and the embeddings of query images are extracted to perform search against indexed gallery.
+
+# 4.1 Implementation Details
+
+Datasets. We use MS1Mv3 [46] for training face embedding models. The MS1Mv3 dataset contains 5,179,510 images of 93,431 celebrities. According to the test setting, different test datasets are used.
+
+·Multi-Resolution Identity Matching. We try on six widely adopted face verification benchmarks: LFW [47], CFP-FF [48], CFP-FP [48], AgeDB-30 [49], CALFW [50], and CPLFW [51], while the large-scale surveillance face dataset QMUL-SurvFace [17] is used for 1:N face identification, which contains native LR surveillance faces across wide space and time. The spatial resolution for QMUL-SurvFace ranges from 6/5 to 124/106 in height/width with an average of 24/20.
+
+·Multi-Resolution Feature Aggregation. We adopt a top challenging benchmark IJB-C [52], which has around 130k images from 3,531 identities, for two standard testing protocols: $1 : 1$ verification and 1:N identification.
+
+Training. All the models are trained on four RTX 2080 Tis with batch size 128 by stochastic gradient descent. For TNet, we train for 25 epochs, with learning rate initialized at 0.2 with 2 warmup epochs and decaying as a quadratic polynomial. We augment training samples by random horizonal flipping and multi-resolution training. For BNets, we initialize the learning rate by 0.02 without warm-up epochs. The training all stops at the $1 0 t h$ epoch for a fair comparison. The recommended hyper-parameters are used for classification loss from the original paper (e.g., $m = 0 . 5 , s = 6 4$ for ArcFace [1], and $\alpha = 0 . 9 9 , t ^ { 0 } = 0$ for CurricularFace [3]). Only horizonal flipping is used as augmentation when training BNets.
+
+Baselines. In our experiment, several baselines are used to validate BTNet in learning discriminative and compatible representations for multi-resolution face recognition.
+
+·High-Resolution Trained $\varphi _ { h r }$ . Naive baseline trained with HR data.
+
+·Independently Trained $\varphi _ { m m }$ . Multi-model fashion: is it possible to achieve better results if we train a specific model for each resolution independently? Specifically, we train $\varphi _ { r }$ for data with resolution $r$ and denote the multi-model collections as $\varphi _ { m m }$ .
+
+·Multi-Resolution Trained $\varphi _ { m r }$ . Trained with multi-resolution data which adapts to resolutionvariance. Specifically, each image is randomly down-sampled to a size in the candidate set $\textstyle \{ { \frac { 1 1 2 } { 2 ^ { i } } } \ x$ $\textstyle { \frac { 1 1 2 } { 2 ^ { i } } } | i = 0 , 1 , 2 , 3 , 4 \}$ with equal probability of being chosen, and then up-sampled back to $1 1 2 \times 1 1 2$
+
+Instantiation of Network Architecture. The BTNet and baselines are implemented with ResNet50 [53], and they could be extended easily with other implementations. Dubbed as $\varphi _ { b t }$ , the detailed instantiation of BTNet based on ResNet50 is illustrated in Appendix A.2.
+
+# 4.2 Evaluation Metrics
+
+On the benchmarks for face verification, we use 1:1 verification accuracy as the basic metrics. The rank-20 true positive identification rates (TPIR20) at varying false positive identification rates (FPIR) and AUC are used to report the identification results on QMUL-SurvFace. The evaluation metrics for IJB-C 1:1 verification protocol are true acceptance rates (TAR) at different false acceptance rate (FAR). For 1:N identification, the basic evaluation metrics are the true positive identification rates (TPIR) at different false positive identification rates (FPIR).
+
+229 For better evaluation, we define another two metrics to assess the relative performance gain similar to   
+230 [11, 14].
+
+(a) Cross-resolution identity matching.
+
+Table 1: Comparison of different methods on six face verification benchmarks. “Acc.” denotes average 1:1 verification accuracy.   
+
+<table><tr><td rowspan="2"></td><td colspan="2">112&amp;7</td><td colspan="2">112&amp;14</td><td colspan="2">112&amp;28</td></tr><tr><td>Acc.</td><td>Gain</td><td>Acc.</td><td>Gain</td><td>Acc.</td><td>Gain</td></tr><tr><td>hr</td><td>57.75</td><td>-</td><td>81.02</td><td>=</td><td>95.90</td><td></td></tr><tr><td>mm</td><td>50.58</td><td>-0.89</td><td>49.90</td><td>-4.82</td><td>50.03</td><td>-305.80</td></tr><tr><td>mr</td><td>65.85</td><td>+1.00</td><td>87.47</td><td>+1.00</td><td>96.05</td><td>+1.00</td></tr><tr><td>bt(Ours)</td><td>86.10</td><td>+3.50</td><td>94.08</td><td>+2.02</td><td>96.65</td><td>+5.00</td></tr></table>
+
+(b) Same-resolution identity matching.   
+
+<table><tr><td colspan="2">7&amp;7</td><td colspan="2">14&amp;14</td><td colspan="2">28&amp;28</td><td colspan="2">112&amp;112</td></tr><tr><td>Acc.</td><td>Gain</td><td>Acc.</td><td>Gain</td><td>Acc.</td><td>Gain</td><td>Acc.</td><td>Gain</td></tr><tr><td>60.70</td><td></td><td>73.88</td><td></td><td>93.58</td><td>=</td><td>97.68</td><td>-</td></tr><tr><td>62.57</td><td>+1.00</td><td>78.00</td><td>+1.00</td><td>94.68</td><td>+1.00</td><td>97.68</td><td>-</td></tr><tr><td>61.02</td><td>+0.17</td><td>80.32</td><td>+1.56</td><td>95.12</td><td>+1.40</td><td>97.25</td><td>-</td></tr><tr><td>77.78</td><td>+9.13</td><td>90.90</td><td>+4.13</td><td>96.27</td><td>+2.45</td><td>97.25</td><td>=</td></tr></table>
+
+231 Cross-Resolution Gain. With the purpose towards the cross-resolution compatible representations,   
+232 we define the performance gain as follows:
+
+$$
+G a i n _ { r _ { 1 } \& r _ { 2 } } ( \varphi ) = \frac { M _ { r _ { 1 } \& r _ { 2 } } ( \varphi ) - M _ { r _ { 1 } \& r _ { 2 } } ( \varphi _ { h r } ) } { | M _ { r _ { 1 } \& r _ { 2 } } ( \varphi _ { m r } ) - M _ { r _ { 1 } \& r _ { 2 } } ( \varphi _ { h r } ) | }
+$$
+
+Here 33 $M _ { r _ { 1 } \& r _ { 2 } } ( \cdot )$ are metrics when the resolutions of the image/template pair are $r _ { 1 } \times r _ { 1 }$ and $r _ { 2 } \times r _ { 2 }$ 4 $( r _ { 1 } \neq r _ { 2 } )$ , respectively. $\varphi _ { m r }$ shares the same architecture with $\varphi _ { h r }$ while is trained on multi-resolution 5 images and thus serves as the baseline of cross-resolution gain.
+
+36 Same-Resolution Gain. For the scenario of multi-resolution face recognition, the performance of   
+37 same-resolution verification/identification is also vital besides cross-resolution one. Therefore, we   
+38 report the relative performance improvement from base model $\varphi _ { h r }$ in the scenario of same-resolution.
+
+$$
+G a i n _ { r \& r } ( \varphi ) = \frac { M _ { r \& r } ( \varphi ) - M _ { r \& r } ( \varphi _ { h r } ) } { | M _ { r \& r } ( \varphi _ { r } ) - M _ { r \& r } ( \varphi _ { h r } ) | }
+$$
+
+39 Here $M _ { r \& r } \left( \cdot \right)$ are metrics when the resolutions of the image/template pair are both $r \times r$ . $\varphi _ { r }$ is   
+40 a model of the set $\{ \varphi _ { m m } = \varphi _ { r } | r = 7 , 1 4 , 2 8 \}$ trained on images with resolution $r \times r$ without   
+41 considering cross-resolution representation compatibility, which serves as the baseline of same  
+42 resolution gain on resolution $r$ . Note that for both metrics we add the absolute symbol to the   
+43 denominator as they can be negative in some test settings (detailed in Section 4.3 and 4.4).
+
+# 4.3 Multi-Resolution Identity Matching
+
+We now conduct experiments on the proposed BTNet framework for multi-resolution identity matching. Two different settings are included : (1) same-resolution matching, and (2) cross-resolution matching. Table 1 compares the average performance on popular benchmarks for $\varphi _ { h r } , \varphi _ { m m } , \varphi _ { m r } ,$ $\varphi _ { b t }$ . The experimental results on each dataset are detailed in Appendix A.5.
+
+249 When directly applied to test data with the resolution lower than training data, $\varphi _ { h r }$ suffers a severe   
+250 performance degradation. Up-sampling images via interpolation can increase the amount of data   
+251 but not the amount of information, only to improve the detailed part of the image and the spatial   
+252 resolution (size) [64]. Moreover, it also brings various noise and artificial processing traces [65].   
+253 Up-sampling images via interpolation-typically bilinear interpolation or bicubic interpolation of   
+254 4x4 pixel neighborhoods, essentially a function approximation method, is bound to introduce error   
+255 information (detailed in Appendix A.1), thus potentially confusing identity information, which is   
+256 especially crucial for LR images with limited details. We are able to observe improvement of $\varphi _ { m m }$ in   
+257 same-resolution matching but its cross-resolution gain is negative with approximately $50 \%$ accuracy.   
+258 Unsurprisingly, independently trained $\varphi _ { r }$ is unaware of representation compatibility, and thus does   
+259 not naturally suitable for cross-resolution recognition. The results show that $\varphi _ { m r }$ improved both   
+260 cross-resolution and same-resolution accuracy by a large margin, as it learns to adapt to resolution   
+261 variance and maintain discriminability of multi-resolution inputs. Note that the model size and   
+262 training data scale stay the same, while only the resolution distribution of the data changes for   
+263 $\varphi _ { m r }$ , and thus there is a marginal accuracy drop in the setting of 112&112 matching. Comparably,   
+264 $\varphi _ { b t }$ substantially outperforms all baselines with $2 . 0 2 \sim 5 . 0 0$ cross-resolution gain and 2.45\~9.13   
+265 same-resolution gain. Importantly, due to the multi-resolution branches, our approach has a cost same   
+266 with $\varphi _ { m m }$ , significantly lower than $\varphi _ { h r }$ and $\varphi _ { m r }$ (see Figure 5).   
+267 For inference on inputs with resolutions not strictly matched to the branch, we validate three selection   
+268 strategies based on three resolution indicators (see Figure 6). Table 2 compares BTNet against the   
+269 state-of-the-arts models on QMUL-SurvFace 1:N identification benchmark. We are able to observe   
+270 that our proposed approach extends the state-of-the-arts while being more computationally efficient.   
+271 We believe the performance of BTNet (max $^ +$ ceil) is the highest that have been reported so far, and   
+272 we believe it is meaningful with the increased focus on unconstrained surveillance applications.
+
+Table 2: Performance of face identification on QMUL-SurvFace. Most compared results are cited from [17, 54] except BTNet.   
+
+<table><tr><td rowspan="2"></td><td colspan="5">TPIR20(%)@FPIR</td></tr><tr><td>AUC</td><td>0.3</td><td>0.2</td><td>0.1</td><td>0.01</td></tr><tr><td>VGG-Face [55]</td><td>14.0</td><td>5.1</td><td>2.6</td><td>0.8</td><td>0.1</td></tr><tr><td>DeepID2 [56]</td><td>20.8</td><td>12.8</td><td>8.1</td><td>3.4</td><td>0.8</td></tr><tr><td>FaceNet [57]</td><td>19.8</td><td>12.7</td><td>8.1</td><td>4.3</td><td>1.0</td></tr><tr><td>SphereFace [58]</td><td>28.1</td><td>21.3</td><td>15.7</td><td>8.3</td><td>1.0</td></tr><tr><td>SRCNN [59]</td><td>27.0</td><td>20.0</td><td>14.9</td><td>6.2</td><td>0.6</td></tr><tr><td>FSRCNN [60]</td><td>27.3</td><td>20.0</td><td>14.4</td><td>6.1</td><td>0.7</td></tr><tr><td>VDSR [61]</td><td>27.3</td><td>20.1</td><td>14.5</td><td>6.1</td><td>0.8</td></tr><tr><td>DRRN [62]</td><td>27.5</td><td>20.3</td><td>14.9</td><td>6.3</td><td>0.6</td></tr><tr><td>LapSRN [63]</td><td>27.4</td><td>20.2</td><td>14.7</td><td>6.3</td><td>0.7</td></tr><tr><td>ArcFace [1]</td><td>25.3</td><td>18.7</td><td>15.1</td><td>10.1</td><td>2.0</td></tr><tr><td>RAN [54]</td><td>32.3</td><td>26.5</td><td>21.6</td><td>14.9</td><td>3.8</td></tr><tr><td>BTNet (avg.+floor)</td><td>32.6</td><td>27.9</td><td>23.4</td><td>16.5</td><td>1.4</td></tr><tr><td>BTNet (avg.+near)</td><td>34.6</td><td>30.3</td><td>25.7</td><td>18.9</td><td>1.5</td></tr><tr><td>BTNet (avg.+ceil)</td><td>35.4</td><td>31.1</td><td>26.8</td><td>20.3</td><td>2.2</td></tr><tr><td>BTNet (min+floor)</td><td>32.3</td><td>27.6</td><td>23.2</td><td>16.1</td><td>1.4</td></tr><tr><td>BTNet (min+near)</td><td>34.0</td><td>29.6</td><td>25.0</td><td>18.0</td><td>1.4</td></tr><tr><td>BTNet (min+ceil)</td><td>35.3</td><td>31.0</td><td>26.6</td><td>19.9</td><td>2.0</td></tr><tr><td>BTNet (max+floor)</td><td>33.6</td><td>29.1</td><td>24.5</td><td>17.6</td><td>1.3</td></tr><tr><td>BTNet (max+near)</td><td>35.2</td><td>31.0</td><td>26.4</td><td>19.6</td><td>1.7</td></tr><tr><td>BTNet (max+ceil)</td><td>35.4</td><td>31.2</td><td>26.9</td><td>20.6</td><td>2.5</td></tr></table>
+
+# 4.4 Multi-Resolution Feature Aggregation
+
+Multi-resolution feature aggregation is common in set-based recognition tasks where the model needs to determine the similarity of sets (templates), instead of images. Each set could contain images of the same identity with different resolutions. In our experiment, we rescale the original and flipped images in each set to different resolutions and aggregate their features into a representation of the template. Detailed experimental results can be seen in Appendix A.5.
+
+Table 3 (a) compares the cross-resolution results of $\mathrm { T A R } @ \mathrm { F A R } = 1 0 ^ { - 4 }$ for 1:1 verification. The cross-resolution features are ensured to be mapped to the same vector space where the aggregation is conducted for $\varphi _ { h r }$ and $\varphi _ { m r }$ , but we can observe that $\varphi _ { h r }$ performs much better than $\varphi _ { m r }$ . One possible reason is that $\varphi _ { h r }$ has outstanding discriminability to extract HR features, while LR features may not overly deteriorate the HR information. This phenomenon also suggests that $\varphi _ { m r }$ sacrifices its discriminability in exchange for the adaptability for resolution-variance. We can see $\varphi _ { b t }$ is comparable with $\varphi _ { h r }$ , demonstrating the discriminative power of BTNet for aggregating multi-resolution features.
+
+Table 3 (b) compares the same-resolution results of $\mathrm { T A R } @ \mathrm { F A R } = 1 0 ^ { - 4 }$ for 1:1 verification. When HR information is removed from the template representation (i.e., test settings 7&7, 14&14, 28&28), $\varphi _ { h r }$ suffers from performance degradation as well, as the informative embedding cannot catch the lost details of the LR images [54]. Both $\varphi _ { m m }$ and $\varphi _ { m r }$ improve with a limited same-resolution gain, while $\varphi _ { b t }$ surpasses the baselines by a large margin while also reducing the compute.
+
+In Table 4 we show the results of291 $\mathrm { T P I R } @ \mathrm { F P I R } { = } 1 0 ^ { - 1 }$ for 1:N identification protocol. Similar to our 292 results for 1:1 verification, we are able to observe that $\varphi _ { b t }$ is comparable or even better than $\varphi _ { h r }$ with
+
+Table 3: Comparison of different methods on the IJB-C dataset 1:1 face verification task.   
+“TAR” denotes TAR $( \% @ \mathsf { F A R = } 1 \mathsf { e } { \mathrm { - } } 4 )$ .
+
+(a) Cross-resolution feature aggregation.
+
+(b) Same-resolution feature aggregation.   
+
+<table><tr><td colspan="2">7&amp;7</td><td colspan="2">14&amp;14</td><td colspan="2">28&amp;28</td><td colspan="2">112&amp;112</td></tr><tr><td>TAR</td><td>Gain</td><td>TAR</td><td>Gain</td><td>TAR</td><td>Gain</td><td>TAR</td><td>Gain</td></tr><tr><td>4.83</td><td>=</td><td>33.74</td><td></td><td>89.65</td><td></td><td>96.40</td><td></td></tr><tr><td>4.83</td><td>+0.00</td><td>29.26</td><td>-1.00</td><td>92.58</td><td>+1.00</td><td>96.40</td><td>=</td></tr><tr><td>4.48</td><td>-</td><td>40.51</td><td>+1.51</td><td>92.81</td><td>+1.08</td><td>96.06</td><td></td></tr><tr><td>35.47</td><td>-</td><td>82.08</td><td>+10.79</td><td>94.50</td><td>+1.66</td><td>96.06</td><td>=</td></tr></table>
+
+<table><tr><td></td><td colspan="2">112&amp;7</td><td colspan="2">112&amp;14</td><td colspan="2">112&amp;28</td></tr><tr><td></td><td>TAR</td><td>Gain</td><td>TAR</td><td>Gain</td><td>TAR</td><td>Gain</td></tr><tr><td>hr</td><td>88.89</td><td></td><td>92.40</td><td>=</td><td>95.62</td><td></td></tr><tr><td>mm</td><td>74.54</td><td>-0.56</td><td>93.52</td><td>+1.33</td><td>95.42</td><td>-0.69</td></tr><tr><td>mr</td><td>63.11</td><td>-1.00</td><td>91.56</td><td>-1.00</td><td>95.33</td><td>-1.00</td></tr><tr><td>bt(Ours)</td><td>88.17</td><td>-0.03</td><td>93.97</td><td>+1.87</td><td>95.62</td><td>+0.00</td></tr></table>
+
+Table 4: Comparison of different methods on the IJB-C dataset 1: N face identification task.   
+“TPIR” denotes TPIR $( \% @ \mathsf { G P I R { = } } 0 . 1 $ ).
+
+(a) Cross-resolution feature aggregation.
+
+<table><tr><td></td><td colspan="2">112&amp;7</td><td colspan="2">112&amp;14</td><td colspan="2">112&amp;28</td></tr><tr><td></td><td>TPIR</td><td>Gain</td><td>TPIR</td><td>Gain</td><td>TPIR</td><td>Gain</td></tr><tr><td>hr</td><td>85.60</td><td>-</td><td>90.11</td><td>-</td><td>94.27</td><td>-</td></tr><tr><td>mm</td><td>69.70</td><td>-0.55</td><td>91.73</td><td>+1.53</td><td>94.13</td><td>-0.33</td></tr><tr><td>mr</td><td>56.64</td><td>-1.00</td><td>89.05</td><td>-1.00</td><td>93.84</td><td>-1.00</td></tr><tr><td>bt(Ours)</td><td>83.93</td><td>-0.06</td><td>91.87</td><td>+1.66</td><td>94.33</td><td>+0.14</td></tr></table>
+
+(b) Same-resolution feature aggregation.   
+
+<table><tr><td colspan="2">7&amp;7</td><td colspan="2">14&amp;14</td><td>28&amp;28</td><td></td><td colspan="2">112&amp;112</td></tr><tr><td>TPIR</td><td>Gain</td><td>TPIR</td><td>Gain</td><td>TPIR</td><td>Gain</td><td>TPIR</td><td>Gain</td></tr><tr><td>3.12</td><td>-</td><td>26.37</td><td>-</td><td>86.06</td><td>=</td><td>95.57</td><td>-</td></tr><tr><td>3.24</td><td>+1.00</td><td>21.84</td><td>-1.00</td><td>89.76</td><td>+1.00</td><td>95.57</td><td>=</td></tr><tr><td>3.25</td><td>+1.08</td><td>37.58</td><td>+2.47</td><td>91.02</td><td>+1.34</td><td>94.85</td><td>-</td></tr><tr><td>27.70</td><td>+204.83</td><td>76.65</td><td>+11.10</td><td>92.89</td><td>+1.85</td><td>94.85</td><td>-</td></tr></table>
+
+293 HR information involved and can preserve superior discriminability with limited LR information,   
+294 while also being more computationally efficient.
+
+![](images/a8c4e8888c6239a2f78ab3386fd66c4578ddb9b6f3764bbdd536c5e5c274bf6e.jpg)  
+Figure 6: Branch selection process. Max/min/average is used on (W, H) to obtain a resolution indicator for further allocation (floor/near/ceil) to a certain branch.
+
+# 295 5 Discussion and Conclusion
+
+This paper works on the problem of multi-resolution face recognition, and provides a new scheme to operate images conditioned on its input resolution without large span rescaling. The error introduced by up-sampling via interpolation is investigated and analyzed. Decoupled as branches for discriminative representation learning and coupled as the trunk for compatible representation learning, our Branch-to-Trunk Network (BTNet) achieves significant improvements on multi-resolution face verification and identification tasks. Besides, the superiority of BTNet in reducing computational cost and parameter storage cost is also demonstrated. It is worth noting that our approach is easy to expand to recognition tasks for other classes of objects and has the potential to serve as a general network architecture for multi-resolution visual recognition.
+
+Limitations and Future Work. The dislocation between the underlying optical resolution of native face images and that of a certain branch may limit the power of the model, which may be improved by selecting the optimal processing branch for the input in combination with the image quality, rather than by image size alone. The optimal branch selection strategy is not fully investigated though we have provided an intuitive way to select the branch for inputs (see Figure 6). Importantly, based on the unified multi-resolution metric space, the underlying resolution of the inputs (integrated spatial resolution with quality assessment) can be utilized to provide the reliability of the representation and contribute to risk-controlled face recognition. They will be our future research directions.
+
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+
+# Checklist
+
+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 5.The dislocation between the underlying optical resolution of native face images and that of a certain branch may limit the power of the model, which may be improved by selecting the optimal processing branch for the input in combination with the image quality, rather than by image size alone. The optimal branch selection strategy is not fully investigated though we have provided an intuitive way to select the branch for inputs.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] We study a general framework for multi-resolution face recognition. Our method is not for specific applications, which does not directly involve societal issues.   
+(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 Appendix A.1   
+(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix A.1
+
+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]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We follow the common practice in previous works, where they didn’t report the error bars.   
+(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 our implementation details in Section 4.
+
+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]   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+591   
+592   
+593   
+594   
+595   
+596   
+597
+
+(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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+        "text": "Learning Unified Representations for Multi-Resolution Face Recognition ",
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+        "text": "1 In this work, we propose Branch-to-Trunk network (BTNet), a novel representation   \n2 learning method for multi-resolution face recognition. It consists of a trunk network   \n3 (TNet), namely a unified encoder, and multiple branch networks (BNets), namely   \n4 resolution adapters. As per the input, a resolution-specific BNet is used and the   \n5 output are implanted as feature maps in the feature pyramid of TNet, at a layer with   \n6 the same resolution. The discriminability of tiny faces is significantly improved, as   \n7 the interpolation error introduced by rescaling, especially up-sampling, is mitigated   \n8 on the inputs. With branch distillation and backward-compatible training, BTNet   \n9 transfers discriminative high-resolution information to multiple branches while   \n10 guaranteeing representation compatibility. Our experiments demonstrate strong   \n11 performance on face recognition benchmarks, both for multi-resolution identity   \n12 matching and feature aggregation, with much less computation amount and param  \n13 eter storage. We establish new state-of-the-art on the challenging QMUL-SurvFace   \n14 1: N face identification task. ",
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+        "text": "15 1 Introduction ",
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+        "text": "16 Machine learning has advanced tremendously driven by deep learning methods, but is still severely   \n17 challenged by various data specifications, such as data type, structure, scale and size, etc. For   \n18 instance, face recognition (FR) is a well-established deep learning task, while the performance   \n19 degrades dramatically in the testing domain that differs from the training one, influenced by factors   \n20 of variance like resolution, illumination, occlusion, etc.   \n21 Most face recognition methods map each image to a point embedding in the common metric space   \n22 by deep neural networks (DNNs). The dissimilarity of images can be then calculated using various   \n23 distance metrics (e.g., cosine similarity, Euclidean distance, etc.) for face recognition tasks.   \n24 Recent advancements in margin-based loss (e.g., ArcFace [1], MV-Arc-Softmax [2], CurricularFace   \n25 [3], etc) enhanced discriminability of the metric space, with small intra-identity distance and large   \n26 inter-identity distance. However, lack of variation in training data still leads to poor generalizability.   \n27 Various useful methods are utilized to mitigate this issue. The model adapts to factors of variance   \n28 by augmenting datasets, whereas the large discrepancy in data distribution could potentially weaken   \n29 the model’s ability to extract discriminative features with the same data scale and model structure   \n30 (see Section 4.3). Fine-tuning is widely used to transfer large pretrained models to new domains with   \n31 different data specifications. However, this strategy requires one to store and deploy a separate copy   \n32 of the backbone parameters for every single new domain, which is expensive and often infeasible.   \n33 As known, the resolutions of face images in reality may be far beyond the scope covered by the   \n34 model. As the small feature maps with a fixed spatial extent (e.g., $7 \\times 7 )$ are mapped to an embedding   \n35 with a predefined dimension (e.g., $1 2 8 - d$ , $5 1 2 - d$ , etc.) by a fully connected (fc) layer, input   \n36 images need to be rescaled to a canonical spatial size (e.g., $1 1 2 \\times 1 1 2 )$ before fed into the network.   \n37 However, up-sampling low-resolution (LR) images introduces the interpolation error (see Section 3.1),   \n38 deteriorating the recognizable ones which contain enough clues to identify the subject. Even though   \n39 super-resolution methods [4–10] are widely used to build faces with good visualization, they inevitably   \n40 introduce feature information of other identities when reconstructing high-resolution (HR) faces.   \n41 This may lead to erroneous identity-specific features, which are detrimental to risk-controlled face   \n42 recognition.   \n43 Empirically, we can divide inputs by resolution distribution and learn to operate on them via multiple   \n44 models to achieve high accuracy and efficiency. However, multi-model fashion cannot be applied   \n45 directly for cross-resolution recognition as representation compatibility among models need to be   \n46 guaranteed [11–15].   \n47 To improve discriminability while ensure the compatibility of the metric space for multi-resolution   \n48 face representation, we learn the “unified” representation by a partially-coupled Branch-to-Trunk   \n49 Network (BTNet). It is composed of multiple independent branch networks (BNets) and a shared   \n50 trunk network (TNet). A resolution-specific BNet is used for a given image, and the output are   \n51 implanted as feature maps in the feature pyramid of TNet, at a layer with the same resolution.   \n52 Furthermore, we find that multi-resolution training can be beneficial to building a strong and robust   \n53 TNet, and backward-compatible training (BCT) [11] can improve the representation compatibility   \n54 during the training process of BTNet. To ameliorate the discriminability of tiny faces, we propose   \n55 branch distillation in intermediate layers, utilizing information extracted from HR images to help the   \n56 extraction of discriminative features for resolution-specific branches.   \n57 Our method is simple and efficient, which breaks the convention of up-sampling the inputs and   \n58 serves as a general framework that can be easily implemented by several existing methods due to   \n59 conceptual simplicity. Meanwhile, BTNet is able to reduce the number of FLOPS by operating the   \n60 inputs without up-sampling, and per-resolution storage cost by only storing the learned branches and   \n61 resolution-aware BNs [16], while re-using the copy of the trunk model.   \n62 We demonstrate that our method performs comparably in various open-set face recognition tasks (1:1   \n63 face verification and 1: N face identification), in both settings of multi-resolution identity matching   \n64 and feature aggregation, while meaningfully reduces the redundant computation cost and parameter   \n65 storage. In the challenging QMUL-SurvFace 1: N face identification task [17], we establish new   \n66 state-of-the-art by outperforming prior models. Furthermore, by avoiding the ill-posed problem (i.e.,   \n67 image up-sampling), our approach also effectively reduces the additional noise and uncertainty of the   \n68 representation, which plays a key role in reliable risk-controlled face recognition. ",
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+        "text": "69 2 Related Work ",
+        "text_level": 1,
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+        "text": "70 Compatible Representation Learning: The task of compatible representation learning aims at   \n71 encoding features that are interoperable with the features extracted from other models. Shen et. al.   \n72 [11] first formulated the problem of backward-compatible learning (BCT) and proposed to utilize the   \n73 old classifier for compatible feature learning. Since the multi-model fashion benefits representation   \n74 learning with lower computation, our idea of cross-resolution representation learning can be modeled   \n75 similar to cross-model compatibility [11–15], as metric space alignment for different resolutions. Our   \n76 goal is achieved by both compatibility-aware network architecture and training strategy.   \n77 Knowledge Distillation and Transfer: The concept of knowledge distillation (KD) was first   \n78 proposed by Hinton et. al. in [18], which can be summarized as employing a large parameter   \n79 model (teacher) to supervise the learning of a small parameter model (student). Distillation from   \n80 intermediate features [19–29] is widely adopted to enhance the effectiveness of knowledge transfer.   \n81 However, due to the “dark knowledge” hidden in the intermediate layers, additional subtle design is   \n82 often required to match and rescale intermediate features. Instead, our approach can easily locate the   \n83 distillation features without rescaling and effectively transfer knowledge from the HR domain to LR   \n84 branches.   \n85 Low Resolution Face Recognition: Its task includes low resolution-to-low resolution (LR-to-LR)   \n86 matching and low resolution-to-high resolution (LR-to-HR) matching [30]. The work can be divided   \n87 into two categories [31]: (1) Super-resolution (SR) based methods aim to upscale LR images to   \n88 construct HR images and use them for feature extraction [4–10]. (2) Projection-based methods aim to   \n89 extract adequate representations in different domains and project them into a common feature space   \n90 [32–34]. SR approaches are able to build faces with good visualization, but inevitably introduce   \n91 feature information of other identities when reconstructing corresponding HR faces, thus introducing   \n92 noise for identity-specific features. Compared to previous projection methods, our approach directly   \n93 learns discriminative representations in a common feature space for HR and LR inputs, without   \n94 additional projection heads for feature transformation. ",
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+        "text": "Pseudo-Siamese Networks: Siamese networks are a coupling architecture based on DNNs, which are widely used for signature verification [35], face verification [36, 37], tracking [38], etc. PseudoSiamese networks [39] are decoupled Siamese networks, as the weights of the two branches are not shared, resulting in a more flexible representation way for the two entities. Hughes et. al. in [40] proposed a pseudo-Siamese CNN for identifying corresponding patches in SAR and optical images. Inspired by pseudo-Siamese networks, we propose a resolution-adaptive partially coupled Siamese network architecture, extracting specific-shared features for images with different resolutions. ",
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+        "type": "text",
+        "text": "3 Learning Specific-Shared Feature Transfer ",
+        "text_level": 1,
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+        "text": "Instead of rescaling the inputs to a canonical size, we build multiple resolution-specific branches (BNets) that are used to map inputs to intermediate features with the same resolution and a resolutionshared trunk (TNet) to map feature maps with different resolutions to a high-dimension embedding. We gain several important properties by doing so: (1) Processing inputs on its original resolution can diminish the inevitably introduced error via up-sampling or information loss via down-sampling, thus preserving the discriminability of visual information with different resolutions. (2) Information streams of different resolutions are encoded uniformly, thus enabling the representation compatibility, which is particularly beneficial to open-set face recognition considering that a compatible metric space is the prerequisite for computing similarity. (3) This also effectively reduce the computation for LR images by supplying computational resources conditioned on the input resolution. ",
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+        "type": "text",
+        "text": "113 3.1 Up-Sampling Error Analysis ",
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+        "img_path": "images/bcaf8fb368f804daba922e50885b5d890a9bd70ea33f0a8743d4e42864a72421.jpg",
+        "image_caption": [
+            "Figure 1: Estimated Error Upperbound. (bilinear interpolation, average value for over 100 images) with the change of image resolution relative to resolution 112. "
+        ],
+        "image_footnote": [],
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+        "image_caption": [
+            "Figure 2: Basic ideas of the proposed BTNet. Images of a certain identity are first projected to the feature maps with the same resolution respectively (Adapt) and then projected to a unified feature representation (Encode). In this figure, feature maps with the same resolution are indicated by outlines in the same color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "114 Figure 1 illustrates the experimental estimation of interpolation error, whose upper bound increases   \n115 with the decline of the image resolution (see detailed theoretical derivation in Appendix A.1). Note   \n116 that the error soars up when the resolution drops below 32 approximately which can be viewed as LR   \n117 face images, consistent with the tiny-object criterion [41].   \n118 The results show that: (1) inputs with a resolution higher than around 32 can be considered in the   \n119 same HR domain, since the error information introduced by up-sampling via interpolation can be   \n120 ignored to a certain extent; (2) inputs with a resolution lower than around 32 should be treated as in   \n121 various LR domains due to the high sensitivity of the resolution to errors. ",
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+    {
+        "type": "text",
+        "text": "3.2 Branch-to-Trunk Network ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Let $X$ be an input RGB image with a space shape: $X \\in \\mathbb { R } ^ { H \\times W \\times 3 }$ where $H \\times W$ corresponds to the spatial dimension of the input. For efficient batch training and inference, we predefine a canonical size $S \\times S$ (e.g., $1 1 2 \\times 1 1 2$ for typical face recognition models like ArcFace [1]). ",
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+    {
+        "type": "text",
+        "text": "We build a trunk network $\\boldsymbol { T } : \\mathbb { R } ^ { H \\times W \\times 3 }  \\mathbb { R } ^ { C _ { e m b } }$ capable of extracting discriminative information with different resolutions, where $C _ { e m b }$ is the number of embedding channels. For every resolution $r$ in the candidate set, we formulate a resolution-specific branch, $z _ { r } = \\textit { B } _ { r } ( X _ { r } )$ , which maps the input image $X _ { r }$ to feature maps with the same resolution and expanded channels $\\boldsymbol { z } _ { r } : \\mathbb { R } ^ { r \\times r \\times 3 }  \\mathbb { R } ^ { r \\times r \\times C _ { r } }$ . The idea is to learn our branches $B$ to focus on resolution-specific feature transfer independently. Feature maps will then be coupled to the trunk network $T$ in the feature pyramid with the same spatial resolution $r \\times r$ , allowing for further mapping to the unified presentation space by $T _ { r } : \\mathbb { R } ^ { r \\times r \\times C _ { r } } $ $\\mathbb { R } ^ { C _ { e m b } }$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "Here, we follow the idea of “avoiding redundant up-sampling”. Our branches $B$ are implemented with same-resolution mapping: i.e., the model preserves the network architecture of $T$ from input to the layer with resolution $r$ and abandons down-sampling operations (e.g., replacing the convolution of stride 2 with stride 1, abandoning the pooling layers, etc.) to keep the same-resolution flow. ",
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+    {
+        "type": "text",
+        "text": "We specifically name our specific-shared feature transfer network as Branch-to-Trunk Network, abbreviated as \"BTNet\". Figure 2 visually summarizes the main ideas of BTNet. ",
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+    {
+        "type": "text",
+        "text": "3.3 Training Objectives ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We now describe the training objectives. The training of BTNet includes training the trunk network $T$ such that it can produce discriminative and compatible representations for multi-resolution information, and fine-tuning the branch networks $B$ to encourage them to learn resolution-specific feature transfer, so as to improve accuracy without compromising compatibility. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Influence Loss. It is a compatibility-aware classification loss which is implemented by feeding the embeddings of the new model to the classifier of the old model [11]. Since the difficulties of samples vary due to image resolution, we compute CurricularFace [3] as our classification loss, in the form of: ",
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+    {
+        "type": "equation",
+        "img_path": "images/d6b9123892a8d3b50dd58976ab5588ce77e9987f40e0e996073d9ea59c4e0169.jpg",
+        "text": "$$\nL _ { c u r } = - \\log ( \\frac { e ^ { s \\cos ( \\theta _ { y _ { i } } + m ) } } { e ^ { s \\cos ( \\theta _ { y _ { i } } + m ) } + \\sum _ { j = 1 , j \\neq y _ { i } } ^ { n } e ^ { s N ( t ^ { ( k ) } , \\cos ( \\theta _ { j } ) ) } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/8f657c9f655cdab89de25260c177752739bba8583601187e6f0314079daf299b.jpg",
+        "text": "$$\n\\begin{array} { c } { { N ( t , \\cos \\theta _ { j } ) = \\displaystyle \\left\\{ \\begin{array} { l c } { { \\cos ( \\theta _ { j } ) , } } & { { \\cos ( \\theta _ { y _ { i } } + m ) - \\cos ( \\theta _ { j } ) \\ge 0 } } \\\\ { { \\cos ( \\theta _ { j } ) ( t + \\cos ( \\theta _ { j } ) ) , } } & { { e l s e } } \\end{array} \\right. } } \\\\ { { t ^ { ( k ) } = \\displaystyle \\alpha \\sum _ { i } \\cos \\theta _ { y _ { i } } + ( 1 - \\alpha ) t ^ { ( k - 1 ) } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "150 which distinguishes both the difficultness of different samples in each stage and relative importance   \n151 of easy and hard samples during different training stages. Thus, we refine CurricularFace loss as our   \n152 influence loss: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2f5169b65aef79ee9ba9f8724552804ab1c162a4b97e4bdecf71cc03dc3589ad.jpg",
+        "text": "$$\nL _ { i n f l u e n c e } = L _ { c u r } ( \\varphi _ { b t } , \\kappa ^ { * } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where 153 $\\varphi _ { b t }$ is BTNet backbone (both $B _ { r }$ and $T _ { r }$ ), and $\\kappa ^ { * }$ is the classifier of the pretrained trunk $T$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "154 Branch Distillation Loss. Due to the   \n155 continuity of the scale change of both the   \n156 image pyramid and the feature pyramid   \n157 [42], we can get a qualitative sense of   \n158 the similarity between images and feature   \n159 maps with the same resolution (see Figure   \n160 3). Furthermore, features extracted from   \n161 HR images have richer and clearer infor  \n162 mation than those from LR images [43].   \n163 Motivated by these analyses, we utilize an   \n164 MSE loss to encourage the branch output   \n165 $z _ { r }$ to be similar to the corresponding fea  \n166 ture maps of the pretrained trunk network   \n167 $z _ { s }$ : ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/e434a9a21fd643648ac4896d6cdd3ccf6e9342a751aac825373efc28d542ddee.jpg",
+        "image_caption": [
+            "Figure 3: Visual comparison of face image-feature map pairs with different resolutions (resized to a common size here for illustration). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f2246fd21423a49aa7cc30c51c18bb60978ccaa1ffe7a96330ec0d7e5b0584e2.jpg",
+        "text": "$$\nL _ { b r a n c h } = \\frac { 1 } { V } \\sum _ { v = 1 } ^ { V } { ( z _ { r _ { v } } - z _ { s _ { v } } ) } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "168 where $V$ denotes the batch size. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "169 The whole training objective is a combination of the above objectives: ",
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+    {
+        "type": "equation",
+        "img_path": "images/cbd4434da50cb2dcda48da9a71a6dcc4aa4054c36bb94eaea39636b77d6c7fe8.jpg",
+        "text": "$$\nL = L _ { i n f l u e n c e } + \\lambda _ { b r a n c h } L _ { b r a n c h }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "170 where $\\lambda _ { b r a n c h }$ is a hyper-parameter to weigh the losses and we set $\\lambda _ { b r a n c h } = 0 . 5$ in all our experi  \n171 ments. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/22946257e92b509279813a29ef0c3ac19fcb89eb7b3c4b6d171707c4f9ae79f6.jpg",
+        "image_caption": [
+            "Figure 4: Comparison of $\\#$ Params $\\mathbf { \\Psi } ^ { ( \\mathbf { M } ) }$ between fully finetuning and $\\varphi _ { b t }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/1df08d5fa7d736b80d3f65aa605b4972425929210f8a903a7b8e250ca889addb.jpg",
+        "image_caption": [
+            "Figure 5: Comparison of FLOPs (G) between baselines and $\\varphi _ { b t }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "172 3.4 Storing Branch Networks ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "An obvious adaptation strategy is fully finetuning of the model on each resolution. However, this strategy requires one to store and deploy a separate copy of the backbone parameters for every resolution, which is an expensive proposition and difficult to expand into more segmented resolution branches. Our BTNet is beneficial in the scenario of multi-resolution face recognition which achieves better parameter/accuracy trade-offs. Since activation statistics including means and variances under different resolutions are incompatible [44], we update and store Batch Normalization (BN) [45] parameters in all layers of $B _ { r }$ and $T _ { r }$ for each resolution, whose amount is negligible. Apart from this, we only need to store the learned branches and re-use the original copy of the pretrained trunk model, significantly reducing the storage cost. Figure 4 shows that BTNet requires only $1 . 1 \\% \\sim 4 8 . 9 \\%$ of all the parameters compared to fully updating all the parameters of TNet. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "184 To validate BTNet on face recognition tasks in open universe, we perform 1:1 verification and $1 : N$   \n185 identification tasks in two different settings, including (a) multi-resolution identity matching, and ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(b) multi-resolution feature aggregation. For 1:1 verification, a pair of templates are provided and the model is to decide whether they belong to the same identity or not. For 1:N identification, a set of gallery images are first mapped onto their embedding vectors (indexing) and the embeddings of query images are extracted to perform search against indexed gallery. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.1 Implementation Details ",
+        "text_level": 1,
+        "bbox": [
+            166,
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+    },
+    {
+        "type": "text",
+        "text": "Datasets. We use MS1Mv3 [46] for training face embedding models. The MS1Mv3 dataset contains 5,179,510 images of 93,431 celebrities. According to the test setting, different test datasets are used. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "·Multi-Resolution Identity Matching. We try on six widely adopted face verification benchmarks: LFW [47], CFP-FF [48], CFP-FP [48], AgeDB-30 [49], CALFW [50], and CPLFW [51], while the large-scale surveillance face dataset QMUL-SurvFace [17] is used for 1:N face identification, which contains native LR surveillance faces across wide space and time. The spatial resolution for QMUL-SurvFace ranges from 6/5 to 124/106 in height/width with an average of 24/20. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "·Multi-Resolution Feature Aggregation. We adopt a top challenging benchmark IJB-C [52], which has around 130k images from 3,531 identities, for two standard testing protocols: $1 : 1$ verification and 1:N identification. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Training. All the models are trained on four RTX 2080 Tis with batch size 128 by stochastic gradient descent. For TNet, we train for 25 epochs, with learning rate initialized at 0.2 with 2 warmup epochs and decaying as a quadratic polynomial. We augment training samples by random horizonal flipping and multi-resolution training. For BNets, we initialize the learning rate by 0.02 without warm-up epochs. The training all stops at the $1 0 t h$ epoch for a fair comparison. The recommended hyper-parameters are used for classification loss from the original paper (e.g., $m = 0 . 5 , s = 6 4$ for ArcFace [1], and $\\alpha = 0 . 9 9 , t ^ { 0 } = 0$ for CurricularFace [3]). Only horizonal flipping is used as augmentation when training BNets. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Baselines. In our experiment, several baselines are used to validate BTNet in learning discriminative and compatible representations for multi-resolution face recognition. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "·High-Resolution Trained $\\varphi _ { h r }$ . Naive baseline trained with HR data. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "·Independently Trained $\\varphi _ { m m }$ . Multi-model fashion: is it possible to achieve better results if we train a specific model for each resolution independently? Specifically, we train $\\varphi _ { r }$ for data with resolution $r$ and denote the multi-model collections as $\\varphi _ { m m }$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "·Multi-Resolution Trained $\\varphi _ { m r }$ . Trained with multi-resolution data which adapts to resolutionvariance. Specifically, each image is randomly down-sampled to a size in the candidate set $\\textstyle \\{ { \\frac { 1 1 2 } { 2 ^ { i } } } \\ x$ $\\textstyle { \\frac { 1 1 2 } { 2 ^ { i } } } | i = 0 , 1 , 2 , 3 , 4 \\}$ with equal probability of being chosen, and then up-sampled back to $1 1 2 \\times 1 1 2$ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Instantiation of Network Architecture. The BTNet and baselines are implemented with ResNet50 [53], and they could be extended easily with other implementations. Dubbed as $\\varphi _ { b t }$ , the detailed instantiation of BTNet based on ResNet50 is illustrated in Appendix A.2. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.2 Evaluation Metrics ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "On the benchmarks for face verification, we use 1:1 verification accuracy as the basic metrics. The rank-20 true positive identification rates (TPIR20) at varying false positive identification rates (FPIR) and AUC are used to report the identification results on QMUL-SurvFace. The evaluation metrics for IJB-C 1:1 verification protocol are true acceptance rates (TAR) at different false acceptance rate (FAR). For 1:N identification, the basic evaluation metrics are the true positive identification rates (TPIR) at different false positive identification rates (FPIR). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "229 For better evaluation, we define another two metrics to assess the relative performance gain similar to   \n230 [11, 14]. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(a) Cross-resolution identity matching. ",
+        "bbox": [
+            220,
+            102,
+            449,
+            116
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/5bc6cd3f78ae7b22f0215c5956576a0f985cb5a161b0a0711b0085a5814c9685.jpg",
+        "table_caption": [
+            "Table 1: Comparison of different methods on six face verification benchmarks. “Acc.” denotes average 1:1 verification accuracy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">112&amp;7</td><td colspan=\"2\">112&amp;14</td><td colspan=\"2\">112&amp;28</td></tr><tr><td>Acc.</td><td>Gain</td><td>Acc.</td><td>Gain</td><td>Acc.</td><td>Gain</td></tr><tr><td>hr</td><td>57.75</td><td>-</td><td>81.02</td><td>=</td><td>95.90</td><td></td></tr><tr><td>mm</td><td>50.58</td><td>-0.89</td><td>49.90</td><td>-4.82</td><td>50.03</td><td>-305.80</td></tr><tr><td>mr</td><td>65.85</td><td>+1.00</td><td>87.47</td><td>+1.00</td><td>96.05</td><td>+1.00</td></tr><tr><td>bt(Ours)</td><td>86.10</td><td>+3.50</td><td>94.08</td><td>+2.02</td><td>96.65</td><td>+5.00</td></tr></table>",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/e4b339b356e5710a821d674e46406b978e735e5547439a4c5a002b9ecd815938.jpg",
+        "table_caption": [
+            "(b) Same-resolution identity matching. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">7&amp;7</td><td colspan=\"2\">14&amp;14</td><td colspan=\"2\">28&amp;28</td><td colspan=\"2\">112&amp;112</td></tr><tr><td>Acc.</td><td>Gain</td><td>Acc.</td><td>Gain</td><td>Acc.</td><td>Gain</td><td>Acc.</td><td>Gain</td></tr><tr><td>60.70</td><td></td><td>73.88</td><td></td><td>93.58</td><td>=</td><td>97.68</td><td>-</td></tr><tr><td>62.57</td><td>+1.00</td><td>78.00</td><td>+1.00</td><td>94.68</td><td>+1.00</td><td>97.68</td><td>-</td></tr><tr><td>61.02</td><td>+0.17</td><td>80.32</td><td>+1.56</td><td>95.12</td><td>+1.40</td><td>97.25</td><td>-</td></tr><tr><td>77.78</td><td>+9.13</td><td>90.90</td><td>+4.13</td><td>96.27</td><td>+2.45</td><td>97.25</td><td>=</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "231 Cross-Resolution Gain. With the purpose towards the cross-resolution compatible representations,   \n232 we define the performance gain as follows: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4b139041d6657e17e2e1fa3469d8912dd4a8543ae9b6b2451d4ccd7624681030.jpg",
+        "text": "$$\nG a i n _ { r _ { 1 } \\& r _ { 2 } } ( \\varphi ) = \\frac { M _ { r _ { 1 } \\& r _ { 2 } } ( \\varphi ) - M _ { r _ { 1 } \\& r _ { 2 } } ( \\varphi _ { h r } ) } { | M _ { r _ { 1 } \\& r _ { 2 } } ( \\varphi _ { m r } ) - M _ { r _ { 1 } \\& r _ { 2 } } ( \\varphi _ { h r } ) | }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            330,
+            273,
+            668,
+            308
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Here 33 $M _ { r _ { 1 } \\& r _ { 2 } } ( \\cdot )$ are metrics when the resolutions of the image/template pair are $r _ { 1 } \\times r _ { 1 }$ and $r _ { 2 } \\times r _ { 2 }$ 4 $( r _ { 1 } \\neq r _ { 2 } )$ , respectively. $\\varphi _ { m r }$ shares the same architecture with $\\varphi _ { h r }$ while is trained on multi-resolution 5 images and thus serves as the baseline of cross-resolution gain. ",
+        "bbox": [
+            156,
+            318,
+            825,
+            363
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "36 Same-Resolution Gain. For the scenario of multi-resolution face recognition, the performance of   \n37 same-resolution verification/identification is also vital besides cross-resolution one. Therefore, we   \n38 report the relative performance improvement from base model $\\varphi _ { h r }$ in the scenario of same-resolution. ",
+        "bbox": [
+            151,
+            372,
+            826,
+            417
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/afc66ac1bfdfe86e199dea6044bcececcbb9f300a5752f45c8950c00da6e810e.jpg",
+        "text": "$$\nG a i n _ { r \\& r } ( \\varphi ) = \\frac { M _ { r \\& r } ( \\varphi ) - M _ { r \\& r } ( \\varphi _ { h r } ) } { | M _ { r \\& r } ( \\varphi _ { r } ) - M _ { r \\& r } ( \\varphi _ { h r } ) | }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            422,
+            643,
+            457
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "39 Here $M _ { r \\& r } \\left( \\cdot \\right)$ are metrics when the resolutions of the image/template pair are both $r \\times r$ . $\\varphi _ { r }$ is   \n40 a model of the set $\\{ \\varphi _ { m m } = \\varphi _ { r } | r = 7 , 1 4 , 2 8 \\}$ trained on images with resolution $r \\times r$ without   \n41 considering cross-resolution representation compatibility, which serves as the baseline of same  \n42 resolution gain on resolution $r$ . Note that for both metrics we add the absolute symbol to the   \n43 denominator as they can be negative in some test settings (detailed in Section 4.3 and 4.4). ",
+        "bbox": [
+            150,
+            467,
+            825,
+            542
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 Multi-Resolution Identity Matching ",
+        "text_level": 1,
+        "bbox": [
+            168,
+            551,
+            462,
+            568
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We now conduct experiments on the proposed BTNet framework for multi-resolution identity matching. Two different settings are included : (1) same-resolution matching, and (2) cross-resolution matching. Table 1 compares the average performance on popular benchmarks for $\\varphi _ { h r } , \\varphi _ { m m } , \\varphi _ { m r } ,$ $\\varphi _ { b t }$ . The experimental results on each dataset are detailed in Appendix A.5. ",
+        "bbox": [
+            173,
+            573,
+            826,
+            633
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "249 When directly applied to test data with the resolution lower than training data, $\\varphi _ { h r }$ suffers a severe   \n250 performance degradation. Up-sampling images via interpolation can increase the amount of data   \n251 but not the amount of information, only to improve the detailed part of the image and the spatial   \n252 resolution (size) [64]. Moreover, it also brings various noise and artificial processing traces [65].   \n253 Up-sampling images via interpolation-typically bilinear interpolation or bicubic interpolation of   \n254 4x4 pixel neighborhoods, essentially a function approximation method, is bound to introduce error   \n255 information (detailed in Appendix A.1), thus potentially confusing identity information, which is   \n256 especially crucial for LR images with limited details. We are able to observe improvement of $\\varphi _ { m m }$ in   \n257 same-resolution matching but its cross-resolution gain is negative with approximately $50 \\%$ accuracy.   \n258 Unsurprisingly, independently trained $\\varphi _ { r }$ is unaware of representation compatibility, and thus does   \n259 not naturally suitable for cross-resolution recognition. The results show that $\\varphi _ { m r }$ improved both   \n260 cross-resolution and same-resolution accuracy by a large margin, as it learns to adapt to resolution   \n261 variance and maintain discriminability of multi-resolution inputs. Note that the model size and   \n262 training data scale stay the same, while only the resolution distribution of the data changes for   \n263 $\\varphi _ { m r }$ , and thus there is a marginal accuracy drop in the setting of 112&112 matching. Comparably,   \n264 $\\varphi _ { b t }$ substantially outperforms all baselines with $2 . 0 2 \\sim 5 . 0 0$ cross-resolution gain and 2.45\\~9.13   \n265 same-resolution gain. Importantly, due to the multi-resolution branches, our approach has a cost same   \n266 with $\\varphi _ { m m }$ , significantly lower than $\\varphi _ { h r }$ and $\\varphi _ { m r }$ (see Figure 5).   \n267 For inference on inputs with resolutions not strictly matched to the branch, we validate three selection   \n268 strategies based on three resolution indicators (see Figure 6). Table 2 compares BTNet against the   \n269 state-of-the-arts models on QMUL-SurvFace 1:N identification benchmark. We are able to observe   \n270 that our proposed approach extends the state-of-the-arts while being more computationally efficient.   \n271 We believe the performance of BTNet (max $^ +$ ceil) is the highest that have been reported so far, and   \n272 we believe it is meaningful with the increased focus on unconstrained surveillance applications. ",
+        "bbox": [
+            142,
+            638,
+            826,
+            912
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/a917f84eb65923fef0ec3ac3270ad45b1e02c6a459f4c0c4aa26583a3e31ddf9.jpg",
+        "table_caption": [
+            "Table 2: Performance of face identification on QMUL-SurvFace. Most compared results are cited from [17, 54] except BTNet. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"5\">TPIR20(%)@FPIR</td></tr><tr><td>AUC</td><td>0.3</td><td>0.2</td><td>0.1</td><td>0.01</td></tr><tr><td>VGG-Face [55]</td><td>14.0</td><td>5.1</td><td>2.6</td><td>0.8</td><td>0.1</td></tr><tr><td>DeepID2 [56]</td><td>20.8</td><td>12.8</td><td>8.1</td><td>3.4</td><td>0.8</td></tr><tr><td>FaceNet [57]</td><td>19.8</td><td>12.7</td><td>8.1</td><td>4.3</td><td>1.0</td></tr><tr><td>SphereFace [58]</td><td>28.1</td><td>21.3</td><td>15.7</td><td>8.3</td><td>1.0</td></tr><tr><td>SRCNN [59]</td><td>27.0</td><td>20.0</td><td>14.9</td><td>6.2</td><td>0.6</td></tr><tr><td>FSRCNN [60]</td><td>27.3</td><td>20.0</td><td>14.4</td><td>6.1</td><td>0.7</td></tr><tr><td>VDSR [61]</td><td>27.3</td><td>20.1</td><td>14.5</td><td>6.1</td><td>0.8</td></tr><tr><td>DRRN [62]</td><td>27.5</td><td>20.3</td><td>14.9</td><td>6.3</td><td>0.6</td></tr><tr><td>LapSRN [63]</td><td>27.4</td><td>20.2</td><td>14.7</td><td>6.3</td><td>0.7</td></tr><tr><td>ArcFace [1]</td><td>25.3</td><td>18.7</td><td>15.1</td><td>10.1</td><td>2.0</td></tr><tr><td>RAN [54]</td><td>32.3</td><td>26.5</td><td>21.6</td><td>14.9</td><td>3.8</td></tr><tr><td>BTNet (avg.+floor)</td><td>32.6</td><td>27.9</td><td>23.4</td><td>16.5</td><td>1.4</td></tr><tr><td>BTNet (avg.+near)</td><td>34.6</td><td>30.3</td><td>25.7</td><td>18.9</td><td>1.5</td></tr><tr><td>BTNet (avg.+ceil)</td><td>35.4</td><td>31.1</td><td>26.8</td><td>20.3</td><td>2.2</td></tr><tr><td>BTNet (min+floor)</td><td>32.3</td><td>27.6</td><td>23.2</td><td>16.1</td><td>1.4</td></tr><tr><td>BTNet (min+near)</td><td>34.0</td><td>29.6</td><td>25.0</td><td>18.0</td><td>1.4</td></tr><tr><td>BTNet (min+ceil)</td><td>35.3</td><td>31.0</td><td>26.6</td><td>19.9</td><td>2.0</td></tr><tr><td>BTNet (max+floor)</td><td>33.6</td><td>29.1</td><td>24.5</td><td>17.6</td><td>1.3</td></tr><tr><td>BTNet (max+near)</td><td>35.2</td><td>31.0</td><td>26.4</td><td>19.6</td><td>1.7</td></tr><tr><td>BTNet (max+ceil)</td><td>35.4</td><td>31.2</td><td>26.9</td><td>20.6</td><td>2.5</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            481,
+            825,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 Multi-Resolution Feature Aggregation ",
+        "text_level": 1,
+        "bbox": [
+            166,
+            583,
+            480,
+            599
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Multi-resolution feature aggregation is common in set-based recognition tasks where the model needs to determine the similarity of sets (templates), instead of images. Each set could contain images of the same identity with different resolutions. In our experiment, we rescale the original and flipped images in each set to different resolutions and aggregate their features into a representation of the template. Detailed experimental results can be seen in Appendix A.5. ",
+        "bbox": [
+            173,
+            604,
+            825,
+            679
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 3 (a) compares the cross-resolution results of $\\mathrm { T A R } @ \\mathrm { F A R } = 1 0 ^ { - 4 }$ for 1:1 verification. The cross-resolution features are ensured to be mapped to the same vector space where the aggregation is conducted for $\\varphi _ { h r }$ and $\\varphi _ { m r }$ , but we can observe that $\\varphi _ { h r }$ performs much better than $\\varphi _ { m r }$ . One possible reason is that $\\varphi _ { h r }$ has outstanding discriminability to extract HR features, while LR features may not overly deteriorate the HR information. This phenomenon also suggests that $\\varphi _ { m r }$ sacrifices its discriminability in exchange for the adaptability for resolution-variance. We can see $\\varphi _ { b t }$ is comparable with $\\varphi _ { h r }$ , demonstrating the discriminative power of BTNet for aggregating multi-resolution features. ",
+        "bbox": [
+            173,
+            685,
+            825,
+            792
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 3 (b) compares the same-resolution results of $\\mathrm { T A R } @ \\mathrm { F A R } = 1 0 ^ { - 4 }$ for 1:1 verification. When HR information is removed from the template representation (i.e., test settings 7&7, 14&14, 28&28), $\\varphi _ { h r }$ suffers from performance degradation as well, as the informative embedding cannot catch the lost details of the LR images [54]. Both $\\varphi _ { m m }$ and $\\varphi _ { m r }$ improve with a limited same-resolution gain, while $\\varphi _ { b t }$ surpasses the baselines by a large margin while also reducing the compute. ",
+        "bbox": [
+            171,
+            799,
+            825,
+            875
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Table 4 we show the results of291 $\\mathrm { T P I R } @ \\mathrm { F P I R } { = } 1 0 ^ { - 1 }$ for 1:N identification protocol. Similar to our 292 results for 1:1 verification, we are able to observe that $\\varphi _ { b t }$ is comparable or even better than $\\varphi _ { h r }$ with ",
+        "bbox": [
+            145,
+            881,
+            823,
+            912
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Comparison of different methods on the IJB-C dataset 1:1 face verification task.   \n“TAR” denotes TAR $( \\% @ \\mathsf { F A R = } 1 \\mathsf { e } { \\mathrm { - } } 4 )$ . ",
+        "bbox": [
+            168,
+            65,
+            753,
+            93
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "(a) Cross-resolution feature aggregation. ",
+        "bbox": [
+            215,
+            103,
+            454,
+            116
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/5b1f1bff4f8c3abafbab07138d8023950fef2ba90c0c9e80acbd2e7a3f3c7011.jpg",
+        "table_caption": [
+            "(b) Same-resolution feature aggregation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">7&amp;7</td><td colspan=\"2\">14&amp;14</td><td colspan=\"2\">28&amp;28</td><td colspan=\"2\">112&amp;112</td></tr><tr><td>TAR</td><td>Gain</td><td>TAR</td><td>Gain</td><td>TAR</td><td>Gain</td><td>TAR</td><td>Gain</td></tr><tr><td>4.83</td><td>=</td><td>33.74</td><td></td><td>89.65</td><td></td><td>96.40</td><td></td></tr><tr><td>4.83</td><td>+0.00</td><td>29.26</td><td>-1.00</td><td>92.58</td><td>+1.00</td><td>96.40</td><td>=</td></tr><tr><td>4.48</td><td>-</td><td>40.51</td><td>+1.51</td><td>92.81</td><td>+1.08</td><td>96.06</td><td></td></tr><tr><td>35.47</td><td>-</td><td>82.08</td><td>+10.79</td><td>94.50</td><td>+1.66</td><td>96.06</td><td>=</td></tr></table>",
+        "bbox": [
+            496,
+            122,
+            818,
+            218
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/2443428c5deb0919a8302f752c3c6d53991627ec9633ca5d82d7fb69c4bdb157.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">112&amp;7</td><td colspan=\"2\">112&amp;14</td><td colspan=\"2\">112&amp;28</td></tr><tr><td></td><td>TAR</td><td>Gain</td><td>TAR</td><td>Gain</td><td>TAR</td><td>Gain</td></tr><tr><td>hr</td><td>88.89</td><td></td><td>92.40</td><td>=</td><td>95.62</td><td></td></tr><tr><td>mm</td><td>74.54</td><td>-0.56</td><td>93.52</td><td>+1.33</td><td>95.42</td><td>-0.69</td></tr><tr><td>mr</td><td>63.11</td><td>-1.00</td><td>91.56</td><td>-1.00</td><td>95.33</td><td>-1.00</td></tr><tr><td>bt(Ours)</td><td>88.17</td><td>-0.03</td><td>93.97</td><td>+1.87</td><td>95.62</td><td>+0.00</td></tr></table>",
+        "bbox": [
+            173,
+            122,
+            470,
+            217
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Comparison of different methods on the IJB-C dataset 1: N face identification task.   \n“TPIR” denotes TPIR $( \\% @ \\mathsf { G P I R { = } } 0 . 1 $ ). ",
+        "bbox": [
+            169,
+            234,
+            774,
+            262
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "(a) Cross-resolution feature aggregation. ",
+        "bbox": [
+            215,
+            271,
+            455,
+            285
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/6065fcbea5b19bea017c03a6374b0469b7a188b7d66b4092782fe3b712c6e307.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">112&amp;7</td><td colspan=\"2\">112&amp;14</td><td colspan=\"2\">112&amp;28</td></tr><tr><td></td><td>TPIR</td><td>Gain</td><td>TPIR</td><td>Gain</td><td>TPIR</td><td>Gain</td></tr><tr><td>hr</td><td>85.60</td><td>-</td><td>90.11</td><td>-</td><td>94.27</td><td>-</td></tr><tr><td>mm</td><td>69.70</td><td>-0.55</td><td>91.73</td><td>+1.53</td><td>94.13</td><td>-0.33</td></tr><tr><td>mr</td><td>56.64</td><td>-1.00</td><td>89.05</td><td>-1.00</td><td>93.84</td><td>-1.00</td></tr><tr><td>bt(Ours)</td><td>83.93</td><td>-0.06</td><td>91.87</td><td>+1.66</td><td>94.33</td><td>+0.14</td></tr></table>",
+        "bbox": [
+            173,
+            291,
+            470,
+            387
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/7f48a4a8376d99bea53855f456f39b4b1f9bc918498a85209e6e981ad9516181.jpg",
+        "table_caption": [
+            "(b) Same-resolution feature aggregation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">7&amp;7</td><td colspan=\"2\">14&amp;14</td><td>28&amp;28</td><td></td><td colspan=\"2\">112&amp;112</td></tr><tr><td>TPIR</td><td>Gain</td><td>TPIR</td><td>Gain</td><td>TPIR</td><td>Gain</td><td>TPIR</td><td>Gain</td></tr><tr><td>3.12</td><td>-</td><td>26.37</td><td>-</td><td>86.06</td><td>=</td><td>95.57</td><td>-</td></tr><tr><td>3.24</td><td>+1.00</td><td>21.84</td><td>-1.00</td><td>89.76</td><td>+1.00</td><td>95.57</td><td>=</td></tr><tr><td>3.25</td><td>+1.08</td><td>37.58</td><td>+2.47</td><td>91.02</td><td>+1.34</td><td>94.85</td><td>-</td></tr><tr><td>27.70</td><td>+204.83</td><td>76.65</td><td>+11.10</td><td>92.89</td><td>+1.85</td><td>94.85</td><td>-</td></tr></table>",
+        "bbox": [
+            496,
+            291,
+            834,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "293 HR information involved and can preserve superior discriminability with limited LR information,   \n294 while also being more computationally efficient. ",
+        "bbox": [
+            142,
+            414,
+            826,
+            444
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/a8c4e8888c6239a2f78ab3386fd66c4578ddb9b6f3764bbdd536c5e5c274bf6e.jpg",
+        "image_caption": [
+            "Figure 6: Branch selection process. Max/min/average is used on (W, H) to obtain a resolution indicator for further allocation (floor/near/ceil) to a certain branch. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            459,
+            799,
+            549
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "295 5 Discussion and Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            151,
+            606,
+            428,
+            623
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This paper works on the problem of multi-resolution face recognition, and provides a new scheme to operate images conditioned on its input resolution without large span rescaling. The error introduced by up-sampling via interpolation is investigated and analyzed. Decoupled as branches for discriminative representation learning and coupled as the trunk for compatible representation learning, our Branch-to-Trunk Network (BTNet) achieves significant improvements on multi-resolution face verification and identification tasks. Besides, the superiority of BTNet in reducing computational cost and parameter storage cost is also demonstrated. It is worth noting that our approach is easy to expand to recognition tasks for other classes of objects and has the potential to serve as a general network architecture for multi-resolution visual recognition. ",
+        "bbox": [
+            173,
+            633,
+            825,
+            768
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Limitations and Future Work. The dislocation between the underlying optical resolution of native face images and that of a certain branch may limit the power of the model, which may be improved by selecting the optimal processing branch for the input in combination with the image quality, rather than by image size alone. The optimal branch selection strategy is not fully investigated though we have provided an intuitive way to select the branch for inputs (see Figure 6). Importantly, based on the unified multi-resolution metric space, the underlying resolution of the inputs (integrated spatial resolution with quality assessment) can be utilized to provide the reliability of the representation and contribute to risk-controlled face recognition. They will be our future research directions. ",
+        "bbox": [
+            168,
+            779,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Jiankang Deng, Jia Guo, Niannan Xue, and Stefanos Zafeiriou. Arcface: Additive angular margin loss for deep face recognition. In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2019, Long Beach, CA, USA, June 16-20, 2019, pages 4690–4699. Computer Vision Foundation / IEEE, 2019.   \n[2] Xiaobo Wang, Shifeng Zhang, Shuo Wang, Tianyu Fu, Hailin Shi, and Tao Mei. Mis-classified vector guided softmax loss for face recognition. In The Thirty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2020, The Thirty-Second Innovative Applications of Artificial Intelligence Conference, IAAI 2020, The Tenth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2020, New York, NY, USA, February 7-12, 2020, pages 12241– 12248. AAAI Press, 2020.   \n[3] Yuge Huang, Yuhan Wang, Ying Tai, Xiaoming Liu, Pengcheng Shen, Shaoxin Li, Jilin Li, and Feiyue Huang. 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+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(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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parse/dev/_SJ-_yyes8/_SJ-_yyes8_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "MASTERING VISUAL CONTINUOUS CONTROL: IMPROVED DATA-AUGMENTED REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            195,
+            400,
+            223
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            261,
+            544,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present DrQ-v2, a model-free reinforcement learning (RL) algorithm for visual continuous control. DrQ-v2 builds on DrQ, an off-policy actor-critic approach that uses data augmentation to learn directly from pixels. We introduce several improvements that yield state-of-the-art results on the DeepMind Control Suite. Notably, DrQ-v2 is able to solve complex humanoid locomotion tasks directly from pixel observations, previously unattained by model-free RL. DrQ-v2 is conceptually simple, easy to implement, and provides significantly better computational footprint compared to prior work, with the majority of tasks taking just 8 hours to train on a single GPU. Finally, DrQ-v2’s implementation is publicly released to provide RL practitioners with a strong and computationally efficient baseline. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            476,
+            336,
+            491
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Creating sample-efficient continuous control methods that observe high-dimensional images has been a long standing challenge in reinforcement learning (RL) . Over the last three years, the RL community has made significant headway on this problem, improving sample-efficiency significantly. The key insight to solving visual control is the learning of better low-dimensional representations, either through autoencoders (Yarats et al., 2019; Finn et al., 2015), variational inference (Hafner et al., 2018; 2019; Lee et al., 2019), contrastive learning (Srinivas et al., 2020; Yarats et al., 2021a), self-prediction (Schwarzer et al., 2020b), or data augmentations (Yarats et al., 2021b; Laskin et al., 2020). However, current state-of-the-art model-free methods are still limited in three ways. First, they are unable to solve the more challenging visual control problems such as quadruped and humanoid locomotion. Second, they often require significant computational resources, i.e. lengthy training times using distributed multi-GPU infrastructure. Lastly, it is often unclear how different design choices affect overall system performance. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/e84129af41009f6a5914fde2b150e79150a0bba702766235dd57287b087195b6.jpg",
+        "image_caption": [
+            "Figure $1 : \\mathrm { D r Q - v } 2$ demonstrates significantly better sample efficiency and computational footprint compared to state-of-the-art model-free methods for visual continuous control while being conceptually simple and easy to implement. (Left two) Average performance results across 12 challenging tasks from the DeepMind Control Suite (the set of tasks can be seen in Figure 8). (Right two) Performance on the Humanoid Walk task from visual input, previously unsolved by model-free methods. In both cases we report sample complexity and wall-clock time axes for evaluation, with time being measured on a single GPU machine and using official implementations for each method. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            705,
+            816,
+            813
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper we present DrQ-v2, a simple model-free algorithm that builds on the idea of using data augmentations (Yarats et al., 2021b; Laskin et al., 2020) to solve hard visual control problems. Most notably, it is the first model-free method that solves complex humanoid tasks directly from pixels. Compared to previous state-of-the-art model-free methods, DrQ-v2 provides significant improvements in sample efficiency across tasks from the DeepMind Control Suite (Tassa et al., 2018). Conceptually simple, DrQ-v2 is also computationally efficient, which allows solving most tasks in DeepMind Control Suite in just 8 hours on a single GPU (see Figure 1). Recently, a model-based method, DreamerV2 (Hafner et al., 2020) was also shown to solve visual continuous control problems and it was first to solve the humanoid locomotion problem from pixels. While our model-free $\\mathrm { D r Q - v } 2$ matches DreamerV2 in terms sample efficiency and performance, it does so $4 \\times$ faster in terms of wall-clock time to train. We believe this makes DrQ-v2 a more accessible approach to support research in visual continuous control and it reinforces the question on whether model-free or model-based is the more suitable approach to solve this type of tasks. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "$\\mathrm { D r Q - v } 2$ , which is detailed in Section 3, improves upon DrQ (Yarats et al., 2021b) by making several algorithmic changes: (i) switching the base RL algorithm from SAC (Haarnoja et al., 2018a) to DDPG (Lillicrap et al., 2015a) with clipped double Q-learning from TD3 (Fujimoto et al., 2018), (ii) this allows us straightforwardly incorporating multi-step return, (iii) adding bilinear interpolation to the random shift image augmentation, (iv) introducing an exploration schedule, (v) selecting better hyper-parameters including a larger capacity of the replay buffer. A careful ablation study of these design choices is presented in Section 4.4. Furthermore, we re-examine the original implementation of DrQ and identify several computational bottlenecks such as replay buffer management, data augmentation processing, batch size, and frequency of learning updates (see Section 3.2). To remedy these, we have developed a new implementation that both achieves better performance and trains around 3.5 times faster with respect to wall-clock time than the previous implementation on the same hardware with an increase in environment frame throughput (FPS) from 28 to 96 (i.e., it takes $1 0 ^ { 6 } / 9 6 / 3 6 0 0 \\approx 2 . 9$ hours to train for 1M environment steps). DrQ-v2’s implementation is available at https://anonymous.4open.science/r/drqv2. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            507,
+            326,
+            525
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 REINFORCEMENT LEARNING FROM IMAGES ",
+        "text_level": 1,
+        "bbox": [
+            179,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We formulate image-based control as an infinite-horizon Markov Decision Process (MDP) (Bellman, 1957). Generally, in such a setting, an image rendering of the system is not sufficient to perfectly describe the system’s underlying state. To this end and per common practice (Mnih et al., 2013), we approximate the current state of the system by stacking three consecutive prior observations. With this in mind, such MDP can be described as a tuple $( \\mathcal { X } , \\mathcal { A } , P , R , \\gamma , d _ { 0 } )$ , where $\\mathcal { X }$ is the state space (a three-stack of image observations), $\\mathcal { A }$ is the action space, $P : \\mathcal { X } \\times \\mathcal { A }  \\Delta ( \\mathcal { X } )$ is the transition function1 that defines a probability distribution over the next state given the current state and action, $R : \\mathcal { X } \\times \\mathcal { A }  [ 0 , 1 ]$ is the reward function, $\\gamma \\in [ 0 , 1 )$ is a discount factor, and $d _ { 0 } \\in \\Delta ( { \\mathcal { X } } )$ is the distribution of the initial state $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { 0 }$ . The goal is to find a policy $\\pi : \\mathcal { X }  \\Delta ( \\mathcal { A } )$ that maximizes the expected discounted sum of rewards $\\mathbb { E } _ { \\pi } \\big [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r _ { t } \\big ]$ , where $\\mathbf { x } _ { 0 } \\sim d _ { 0 }$ , and $\\forall t$ we have $\\mathbf { \\boldsymbol { a } } _ { t } \\sim \\pi ( \\cdot | \\mathbf { \\boldsymbol { x } } _ { t } )$ $\\mathbf { \\boldsymbol { x } } _ { t + 1 } \\sim P ( \\cdot | \\mathbf { \\boldsymbol { x } } _ { t } , \\mathbf { \\boldsymbol { a } } _ { t } )$ , and $r _ { t } = R ( \\pmb { x } _ { t } , \\pmb { a } _ { t } ) \\overline { { \\mathbf { \\phi } } }$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 DEEP DETERMINISTIC POLICY GRADIENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Deep Deterministic Policy Gradient (DDPG) (Lillicrap et al., 2015a) is an actor-critic algorithm for continuous control that concurrently learns a Q-function $Q _ { \\theta }$ and a deterministic policy $\\pi _ { \\phi }$ . For this, DDPG uses Q-learning (Watkins and Dayan, 1992) to learn $Q _ { \\theta }$ by minimizing the one-step Bellman residual $J _ { \\theta } ( \\mathcal { D } ) = \\mathbb { E } _ { ( \\pmb { x } _ { t } , \\underline { { a } } _ { t } , r _ { t } , \\pmb { x } _ { t + 1 } ) \\sim \\mathcal { D } } [ \\big ( Q _ { \\theta } ( \\pmb { x } _ { t } , \\underline { { a } } _ { t } ) _ {  } r _ { t } - \\underline { { \\gamma Q _ { \\theta } } } ( \\pmb { x } _ { t + 1 } , \\pi _ { \\phi } ( \\pmb { x } _ { t + 1 } ) ) ^ { 2 } \\big ]$ . The policy $\\pi _ { \\phi }$ is learned by employing Deterministic Policy Gradient (DPG) (Silver et al., 2014) and maximizing $J _ { \\phi } ( \\mathcal { D } ) = \\mathbb { E } _ { \\pmb { x } _ { t } \\sim \\mathcal { D } } [ Q _ { \\theta } ( \\pmb { x } _ { t } , \\pi _ { \\phi } ( \\pmb { x } _ { t } ) ) ]$ , so $\\pi _ { \\phi } ( \\pmb { x } _ { t } )$ approximates argmax $\\mathbf { \\Sigma } _ { \\alpha } Q _ { \\theta } ( \\pmb { x } _ { t } , \\pmb { a } )$ . Here, $\\mathcal { D }$ is a replay buffer of environment transitions and $\\bar { \\theta }$ is an exponential moving average of the weights. DDPG is amenable to incorporate $n$ -step returns (Watkins, 1989; eng and Williams, 1996) when estimating TD error beyond a single step (Barth-Maron et al., 2018). In practice, $n$ -step returns allow for faster reward propagation and has been previously used in policy gradient and Q-learning methods (Mnih et al., 2016b; Barth-Maron et al., 2018; Hessel et al., 2017). ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/549fe695324b96cc93df2881f3296d0b6333a6e41e473a27a33b6bfc84cf2f08.jpg",
+        "image_caption": [
+            "Figure 2: (Left): $\\mathrm { D r Q - v } 2$ is an off-policy actor-critic algorithm for image-based RL. It alleviates encoder overfitting by applying random shift augmentation to pixel observations sampled from the replay buffer. (Right): Examples of walking and standing behaviors learned by $\\mathrm { D r Q - v } 2$ for a complex humanoid agent from DMC (Tassa et al., 2018) with 21 and 54 dimensional action and state spaces, respectively. DrQ-v2 does not have access to the internal state of the environment, only observing three consecutive pixel frames at a time. Despite this imperfect observational channel, our agent still manages to solve the tasks. To the best of our knowledge, this is the first successful demonstration by a model-free method, using pixel-based inputs of these tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            209,
+            68,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            378,
+            823,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 DATA AUGMENTATION IN REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            422,
+            596,
+            438
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recently, it has been shown that data augmentation techniques, commonplace in Computer Vision, are also important for achieving the state-of-the-art performance in image-based RL (Yarats et al., 2021b; Laskin et al., 2020). For example, the state-of-the-art algorithm for visual RL, DrQ (Yarats et al., 2021b) builds on top of Soft Actor-Critic (Haarnoja et al., 2018a), a model-free actor-critic algorithm, by adding a convolutional encoder and data augmentation in the form of random shifts. The use of such data augmentations now forms an essential component of several recent visual RL algorithms (Srinivas et al., 2020; Raileanu et al., 2020; Yarats et al., 2021a; Stooke et al., 2020; Hansen and Wang, 2021; Schwarzer et al., 2020b). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 DRQ-V2: IMPROVED DATA-AUGMENTED REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            580,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we describe DrQ-v2, a simple model-free actor-critic RL algorithm for image-based continuous control, that builds upon DrQ. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 ALGORITHMIC DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            657,
+            383,
+            671
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Image Augmentation As in DrQ we apply random shifts image augmentation to pixel observations of the environment. In the settings of visual continuous control by DMC, this augmentation can be instantiated by first padding each side of $8 4 \\times 8 4$ observation rendering by 4 pixels (by repeating boundary pixels), and then selecting a random $8 4 \\times 8 4$ crop, yielding the original image shifted by $\\pm 4$ pixels. We also find it useful to apply bilinear interpolation on top of the shifted image (i.e, we replace each pixel value with the average of the four nearest pixel values). In our experiments, this modification provides an additional performance boost across the board. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Image Encoder The augmented image observation is then embedded into a low-dimensional latent vector by applying a convolutional encoder. We use the same encoder architecture as in DrQ, which first was introduced introduced in SAC-AE (Yarats et al., 2019). This process can be succinctly summarized as $\\pmb { h } = f _ { \\xi } ( \\mathrm { a u g } ( \\pmb { x } ) )$ , where $f _ { \\xi }$ is the encoder, aug is the random shifts augmentation, and $_ { \\textbf { \\em x } }$ is the original image observation. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Actor-Critic Algorithm We use DDPG (Lillicrap et al., 2015a) as a backbone actor-critic RL algorithm and, similarly to Barth-Maron et al. (2018), augment it with $n$ -step returns to estimate TD error. This results into faster reward propagation and overall learning progress (Mnih et al., 2016a). ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "While some methods (Hafner et al., 2020) employ more sophisticated techniques such as $\\mathrm { T D } ( \\lambda )$ or Retrace $( \\lambda )$ (Munos et al., 2016), they are often computationally demanding when $n$ is large. We find that using simple $n$ -step returns, without an importance sampling correction, strikes a good balance between performance and efficiency. We also employ clipped double Q-learning (Fujimoto et al., 2018) to reduce overestimation bias in the target value. Practically, this requires training two Qfunctions $Q _ { \\theta _ { 1 } }$ and $Q _ { \\theta _ { 2 } }$ . For this, we sample a mini-batch of transitions $\\tau = ( \\mathbf { x } _ { t } , \\mathbf { a } _ { t } , r _ { t : t + n - 1 } , \\mathbf { x } _ { t + n } )$ from the replay buffer $\\mathcal { D }$ and compute the following two losses: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/21f5b49e1058571b066a400cb4942e8b0691c5876a0567d3f54d57c99b7289be.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\boldsymbol { \\theta } _ { k } , \\boldsymbol { \\xi } } ( \\mathcal { D } ) = \\mathbb { E } _ { \\tau \\sim \\mathcal { D } } \\big [ ( Q _ { \\boldsymbol { \\theta } _ { k } } ( h _ { t } , \\boldsymbol { a } _ { t } ) - y ) ^ { 2 } \\big ] \\quad \\forall k \\in \\{ 1 , 2 \\} , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            315,
+            210,
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+            231
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with the TD target $y$ defined as: ",
+        "bbox": [
+            173,
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+            383,
+            253
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb4d51d5aefb63d565fb1da6ccfcdc2b6e2cfc9e812e1c7364b45c1df85f8d92.jpg",
+        "text": "$$\ny = \\sum _ { i = 0 } ^ { n - 1 } \\gamma ^ { i } r _ { t + i } + \\gamma ^ { n } \\operatorname* { m i n } _ { k = 1 , 2 } Q _ { \\bar { \\theta } _ { k } } ( h _ { t + n } , \\mathbf { a } _ { t + n } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            263,
+            650,
+            308
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { h } _ { t } = f _ { \\xi } ( \\mathrm { a u g } ( \\pmb { x } _ { t } ) )$ , ${ h _ { t + n } } = f _ { \\xi } ( \\mathrm { a u g } ( { { \\bf x } _ { t + n } } ) )$ , $a _ { t + n } = \\pi _ { \\phi } ( h _ { t + n } ) + \\epsilon , \\bar { \\theta } _ { 1 }$ and ${ \\bar { \\theta } _ { 2 } }$ are the slowmoving weights for the Q target networks. We note, that in contrast to DrQ, we do not employ a target network for the encoder $f _ { \\xi }$ and always use the most recent weights $\\xi$ to embed $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { t } }$ and $\\mathbf { \\Delta } _ { \\mathbf { x } _ { t + n } }$ . The exploration noise $\\epsilon$ is sampled from $\\mathrm { c l i p } ( \\mathcal { N } ( 0 , \\sigma ^ { 2 } ) , - c , c )$ similar to TD3 (Fujimoto et al., 2018), with the exception of decaying $\\sigma$ , which we describe below. Finally, we train the deterministic actor $\\pi _ { \\phi }$ using DPG with the following loss: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/19949ea5e4ecf0cc87814f68e6506792333b535c3a3583c7335970adf32fefdc.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\phi } ( \\mathcal { D } ) = - \\mathbb { E } _ { x _ { t } \\sim \\mathcal { D } } \\big [ \\operatorname* { m i n } _ { k = 1 , 2 } Q _ { \\theta _ { k } } ( h _ { t } , \\pmb { a } _ { t } ) \\big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            414,
+            632,
+            440
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { h } _ { t } = f _ { \\xi } ( \\mathrm { a u g } ( \\pmb { x } _ { t } ) )$ , ${ \\pmb a } _ { t } = \\pi _ { \\phi } ( { \\pmb h } _ { t } ) + \\epsilon$ , and $\\epsilon \\sim \\mathrm { c l i p } ( \\mathcal { N } ( 0 , \\sigma ^ { 2 } ) , - c , c )$ . Similar to $_ \\mathrm { D r Q }$ , we do not use actor’s gradients to update the encoder’s parameters $\\xi$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Scheduled Exploration Noise Empirically, we observe that it is helpful to have different levels of exploration at different stages of learning. At the beginning of training we want the agent to be more stochastic and explore the environment more effectively, while at the later stages of training, when the agent has already identified promising behaviors, it is better to be more deterministic and master those behaviors. Similar to Amos et al. (2020), we instantiate this idea by using linear decay $\\sigma ( t )$ for the variance $\\sigma ^ { 2 }$ of the exploration noise defined as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/845554410f8da0899d23c194d909d3308487d8ae22f946d09cbaf11cb779a4d8.jpg",
+        "text": "$$\n\\sigma ( t ) = \\sigma _ { \\mathrm { i n i t } } + ( 1 - \\mathrm { m i n } ( \\frac { t } { T } , 1 ) ) ( \\sigma _ { \\mathrm { f i n a l } } - \\sigma _ { \\mathrm { i n i t } } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            593,
+            656,
+            623
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma _ { \\mathrm { i n i t } }$ and $\\sigma _ { \\mathrm { f i n a l } }$ are the initial and final values for standard deviation, and $T$ is the decay horizon. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Key Hyper-Parameters We conduct an extensive hyper-parameter search and identify several hyper-parameter changes compared to DrQ. The three most important hyper-parameters are: (i) the size of the replay buffer, (ii) mini-batch size, and (iii) learning rate. Specifically, we use a 10 times larger replay buffer than DrQ. We also use a smaller mini-batch size of 256 without any noticeable performance degradation. This is in contrast to CURL (Srinivas et al., 2020) and $\\mathrm { D r Q }$ (Yarats et al., 2021b) that both use a larger batch size of 512 to attain more stable training in the expense of computational efficiency. Finally, we find that using smaller learning rate of $1 \\times 1 0 ^ { - 4 }$ , rather than DrQ’s learning rate of $\\mathrm { i \\times 1 0 ^ { - 3 } }$ , results into more stable training without any loss in learning speed. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Faster Image Augmentation We replace DrQ’s random shifts augmentation (i.e., kornia.augmentation.RandomCrop) by a custom implementation that uses flowfield image sampling provided in PyTorch (i.e., grid_sample). This is done for two reasons. First, we noticed that Kornia’s implementation does not fully utilize GPU pipelining since it has some intermediate CPU to GPU data transferring which breaks the computational flow. Second, using grid_sample allows straightforward addition of bilinear interpolation. Our custom random shifts augmentation improves training throughput by a factor of 2. ",
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+    {
+        "type": "text",
+        "text": "Faster Replay Buffer Another computational bottleneck of $\\mathrm { D r Q }$ was the replay buffer. The specific implementation had poor memory management which resulted in slow CPU to GPU data transfer, which also restricted the number of image-based transitions that could be stored. We reimplemented the replay buffer to address these issues which led to a ten-fold increase in storage capacity and faster data transfer. More details are available in our open-source release. We note that the improved training speed of DrQ-v2 was key to solving humanoid tasks as it enabled much faster experimentation. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "In this section we provide empirical evaluation of $\\mathrm { D r Q - v } 2$ on an extensive set of visual continuous control tasks from DMC (Tassa et al., 2018). We first present comparison to prior methods, both model-free and model-based, in terms of sample efficiency and wall-clock time. We then present a large scale ablation study that guided the final version of DrQ-v2. ",
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+    {
+        "type": "text",
+        "text": "4.1 SETUP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Environments We consider a set of MuJoCo tasks (Todorov et al., 2012) provided by DMC (Tassa et al., 2018), a widely used benchmark for continous control. DMC offers environments of various difficulty, ranging from the simple control problems such as the single degree of freedom (DOF) pendulum and cartpool, to the control of complex multi-joint bodies such as the humanoid (21 DOF). We consider learning from pixels. In this setting, environment observations are stacks of 3 consecutive RGB images of size $8 4 \\times 8 4$ , stacked along the channel dimension to enable inference of dynamic information like velocity and acceleration. In total, we consider 24 different tasks, which we group into three buckets, easy, medium, and hard, according to the sample complexity to reach near-optimal performance (see Appendix B). Our motivation for this is to encourage RL practitioners to focus on the medium and hard tasks and stop using the easy tasks for evaluation, as they are mostly solved at this point and may no longer provide any valuable signal in comparing different methods. ",
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+        "type": "text",
+        "text": "Training Details For all tasks in the suite an episode corresponds to 1000 steps, where a per-step reward is in the unit interval [0, 1]. This upper bounds the episode return to 1000 making it easier to compute aggregated performance measures across tasks. To facilitate fair wall-clock time comparison all algorithms are trained on the same hardware (i.e., a single NVIDIA V100 GPU machine) and evaluated with the same periodicity of 20000 environment steps. Each evaluation query averages episode returns over 10 episodes. Per common practice (Hafner et al., 2019), we employ action repeat of 2 and measure sample complexity in the environment steps, rather than the actor steps. In all the figures we plot the mean performance over 10 seeds together with the shaded regions which represent $9 \\hat { 5 } \\%$ confidence intervals. A full list of hyper-parameters can be found in Appendix E. ",
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+        "type": "text",
+        "text": "Comparison Axes In many real-world applications, taking a step in the environment incurs significant computational cost making sample efficiency a critical feature of an RL algorithm. It is hence important to compare RL algorithms in terms of their sample efficiency. We facilitate this comparison by computing an algorithm’s performance measured by episode return with respect to environment steps. On the other end, striving low sample complexity often comes at the cost of a poor computational efficiency. Unfortunately, recent deep RL literature has paid very little attention to this important axis, which has led to skyrocketing hardware requirements. Such a trend has made it virtually impossible for an RL practitioner with modest hardware capacity to contribute to advancements in image-based RL, leaving research in this area to a few well-equipped labs. To democratize research in visual RL, we additionally propose to compare the agents in terms of wall-clock training time given the same single GPU hardware. We note that it is possible to adapt DrQ-v2 to a distributed setup, as has been done for DDPG in prior work (Barth-Maron et al., 2018; Hoffman et al., 2020). ",
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+        "text": "4.2 COMPARISON TO MODEL-FREE METHODS ",
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+        "type": "text",
+        "text": "Baselines We compare our method to several state-of-the-art model-free algorithms for visual RL including CURL (Srinivas et al., 2020), DrQ (Yarats et al., 2021b), and vanilla SAC (Haarnoja et al., 2018a) augmented with the convolutional encoder from SAC-AE (Yarats et al., 2019). Vanilla SAC is a weak baseline and only included as a ground point to showcase the recent progress in visual RL. ",
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+        "img_path": "images/ecbef5ca3f3d3542fd76d49b66dfb726dde336b060b81096054addb1a0048ea7.jpg",
+        "image_caption": [
+            "Figure 3: We compare $\\mathrm { D r Q - v } 2$ on a subset of continuous control tasks that offer various challenges, including complex dynamics, sparse rewards, hard exploration, and more. (a) $\\mathrm { D r Q - v } 2$ demonstrates favorable sample efficiency and comfortably outperforms leading model-free baselines, as well as requiring less wall-clock training image (b). "
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+        "type": "text",
+        "text": "Sample Efficiency Axis We present results on several medium and hard tasks in Figure 3a. Full results can be found in Appendix (Figure 6, Figure 8, and Figure 10). Our empirical study reveals that $\\mathrm { D r Q - v } 2$ outperforms prior model-free methods in terms of sample efficiency across the three benchmarks with different levels of difficulty. Importantly, DrQ-v2’s advantage is more pronounced on harder tasks (i.e., acrobot, quadruped, and humanoid), where exploration is especially challenging. Finally, DrQ-v2 solves the DMC humanoid locomotion tasks directly from pixels, which, to the best of our knowledge, is the first successful demonstration of such feat by a model-free method. ",
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+    {
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+        "text": "Compute Efficiency Axis To facilitate a fair comparison in terms of sheer wall-clock training time, besides employee the identical training protocol (see Section 4.1), we also use the same mini-batch size of 256 for each agent. In Figure 13, we evaluate $\\mathrm { D r Q - v } 2$ on a subset of DMC tasks for the sake of brevity only, and note that the demonstrated results can be easily extrapolated to the other tasks given the linear dependency between training time and sample complexity. In our benchmarks, $\\mathrm { D r Q - v } 2$ is able to achieve a throughput of 96 FPS, which favorably compares to DrQ’s 28 FPS (a $3 . 4 \\times$ increase), and CURL’s 16 FPS (a $6 \\times$ increase) throughputs. Practically, $\\mathrm { D r Q - v } 2$ solves easy, medium, and hard tasks within 2.9, 8.6, and 86 hours respectively. Full results can be found in Appendix (Figure 7, Figure 9, and Figure 11). ",
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+    {
+        "type": "text",
+        "text": "4.3 COMPARISON TO MODEL-BASED METHODS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Baseline To see how $\\mathrm { D r Q - v } 2$ stacks up against model-based methods, which tend to achieve better sample complexity in expense of a larger computational footprint, we also compare to recent and unpublished2 improvements to Dreamer-v2 (Hafner et al., 2020), a leading model-based approach for visual continuous control. The recent update shows that the model-based approach can solve the DMC humanoid tasks directly from pixel inputs. The open-source implementation of Dreamer-v2 (https://github.com/danijar/dreamerv2) only provides learning curves for Humanoid Walk. For this reason we run their code to obtain results on other DMC tasks. To limit hardware requirements of compute-expensive Dreamer-v2, we only run it on a subset of 12 out of 24 considered tasks. This subset, however, overlaps with all the three (i.e. easy, medium, and hard) benchmarks. ",
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+        "img_path": "images/545d212f4784001ae204bbbd2da7b19024aaeec84e241d0466ac5d3246eb06bc.jpg",
+        "image_caption": [
+            "Figure 4: Model-based Dreamer-v2 needs to train a world model and thus performs more computations during training than model-free DrQ-v2. Still, (a) $\\mathrm { D r Q - v } 2$ is able to match Dreamer-v2’s sample efficiency, while $\\mathbf { ( b ) }$ requiring much less wall-clock training time. "
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Sample Efficiency Axis Our empirical study in Figure 4a reveals that in many cases, DrQ-v2, despite being a model-free method, can rival sample efficiency of state-of-the-art model-based Dreamer-v2. We note, however, that on several tasks (for example Acrobot Swingup) Dreamer-v2 outperforms DrQ-v2. We leave investigation of such discrepancy for future work. Full results are provided in Appendix D (Figure 12). ",
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+    {
+        "type": "text",
+        "text": "Compute Efficiency Axis A different picture emerges if comparison is done with respect to wallclock training time. Dreamer-v2, being a model-based method, performs significantly more floating point operations to reach its sample efficiency. In our benchmarks, Dreamer-v2 records a throughput of 24 FPS, which is $4 \\times$ less than DrQ-v2’s throughput of 96 FPS, measured on the same hardware. In Figure 4b we plot learning curves against wall-clock time and observe that $\\mathrm { D r Q - v } 2$ takes less time to solve the tasks. Full results can be found in Appendix (Figure 13). ",
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+    {
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+        "text": "4.4 ABLATION STUDY ",
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+        "text": "In this section we present an extensive ablation study that guided us to the final version of $\\mathrm { D r Q - v } 2$ Here, for brevity we only discuss experiments that were most impactful and omit others that did not pan out. For computational reasons, we only ablate on 3 different control tasks of various difficulty levels. Our findings are summarized in Figure 5 and detailed below. ",
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+        "type": "text",
+        "text": "Switching from SAC to DDPG DrQ (Yarats et al., 2021b) leverages SAC (Haarnoja et al., 2018a) as the backbone RL algorithm. While it has been demonstrated by many works, including the original manuscripts (Haarnoja et al., 2018a;b) that SAC is superior to DDPG (Lillicrap et al., 2015b), our careful examination identifies two shortcomings that preclude SAC (within DrQ) to solve hard exploration-wise image-based tasks. First, the automatic entropy adjustment strategy, introduced in Haarnoja et al. (2018b), is inadequate and in some cases leads to a premature entropy collapse. ",
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+        "img_path": "images/ba1a4435887f6483f6e5539af32ee4c9abf61350f7e4d286d35d2b17585b6381.jpg",
+        "image_caption": [
+            "(a) DrQ (dotted silver) relies on SAC as a base RL algorithm. Replacing SAC with DDPG results in a significant performance gain (blue). "
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+    {
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+        "image_caption": [
+            "(b) DDPG straightforwardly incorporates $n$ -step returns, a critical tool for exploration. We observe that the 3 (blue) and 5 (red) steps variants provide additional improvements to the previous version that uses single step TD-targets (silver). Going forward, we adopt 3-step returns (blue). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "image",
+        "img_path": "images/7344d8d594db9770f16e401999ae3c88c9529605098182fbc253dd13972aed74.jpg",
+        "image_caption": [
+            "(c) Increasing the size of the replay buffer (B) improves performance, over the original $1 0 ^ { 5 }$ used by DrQ (silver). Going forward, we use a buffer size of $1 \\dot { 0 } ^ { 6 }$ (red). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(d) Finally, a decaying schedule for the variance of the exploration noise (blue) helps on hard exploration tasks, versus the fixed variance variant (silver). ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/9ac2122386f89d3dc6ae3ea32bbd49484f830769555b1e994b96ad2c05cfd549.jpg",
+        "image_caption": [
+            "Figure 5: An ablation study that led us to the final version of $\\mathrm { D r Q - v } 2$ . We incrementally show each of the four key improvements to $\\mathrm { D r Q }$ that collectively form DrQ-v2. The silver dotted curves in the first row show the original DrQ. In subsequent rows they show progressive improvements, using the optimal choice from the previous rows (i.e., the silver curve in the third row shows DrQ with a DDPG base RL algorithm and 3-step returns). The red and blue curves show the effect of individual modifications. In the last row the blue curve corresponds to DrQ-v2. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "This prevents the agent from finding more optimal behaviors due to the insufficient exploration. In Figure 5a, we empirically verify our intuition and, indeed, observe that DDPG demonstrates better exploration properties than SAC. Here, DDPG uses constant $\\sigma = 0 . 2$ for the exploration noise. ",
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+    {
+        "type": "text",
+        "text": "N-step Returns The second issue concerns the inability of soft Q-learning to incorporate $n$ -step returns to estimate TD error in a straightforward manner. The reason for this is that computing a target value for soft Q-function requires estimating per-step entropy of the policy, which is challenging to do for large $n$ in the off-policy regime. In contrast, DDPG does not require estimating per-step entropy to compute targets and is more amenable for $n$ -step returns. In Figure 5b we demonstrate that estimating TD error with $n$ -step returns improves sample efficiency over vanilla DDPG. We select 3-step returns as a sensible choice for our method. ",
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+    {
+        "type": "text",
+        "text": "Replay Buffer Size We hypothesize that a larger replay buffer plays an important role in circumventing the catastrophic forgetting problem (Fedus et al., 2020). This issue is especially prominent in tasks with more diverse initial state distributions (i.e., reacher or humanoid tasks), where the vast variety of possible behaviors requires significantly larger memory. We confirm this intuition by ablating the size of the replay buffer in Figure 5c, where we observe that a buffer size of 1M helps to improve performance on Reacher Hard considerably. ",
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+    {
+        "type": "text",
+        "text": "Scheduled Exploration Noise Finally, we demonstrate that it is useful to decay the variance of the exploration noise over the course of training according to Equation (3). In Figure 5d, we compare two versions of our algorithm, where the first variant uses a fixed standard deviation of $\\sigma = 0 . 2$ , while the second variant employes the decaying schedule $\\sigma ( t )$ , with parameters $\\sigma _ { \\mathrm { i n i t } } = 1 . 0$ , $\\sigma _ { \\mathrm { f i n a l } } = 0 . 1$ , and $T = 5 0 0 0 0 0$ . Having the exploration noise to decay linearly over time turns out to be helpful and provide an additional performance boost, which was especially useful for solving humanoid tasks. ",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Visual Reinforcement Learning Successes of visual representation learning in computer vision (Vincent et al., 2008; Doersch et al., 2015; Wang and Gupta, 2015; Noroozi and Favaro, 2016; Zhang et al., 2017; Gidaris et al., 2018) has inspired successes in visual RL, where coherent representations are learned alongside RL. Works such as SAC-AE (Yarats et al., 2019), PlaNet (Hafner et al., 2018), and SLAC (Lee et al., 2019), demonstrated how auto-encoders (Finn et al., 2015) could improve visual RL. Following this, other self-supervised objectives such as contrastive learning in CURL (Srinivas et al., 2020) and ATC (Stooke et al., 2020), self-prediction in SPR (Schwarzer et al., 2020a), contrastive cluster assignment in Proto-RL (Yarats et al., 2021a), and augmented data in DrQ (Yarats et al., 2021b) and RAD (Laskin et al., 2020), have significantly bridged the gap between state-based and image-based RL. Future prediction objectives (Hafner et al., 2018; 2019; Yan et al., 2020; Finn et al., 2015; Pinto et al., 2016; Agrawal et al., 2016) and other auxiliary objectives (Jaderberg et al., 2016; Zhan et al., 2020; Young et al., 2020; Chen et al., 2020) have shown improvements on a variety of problems ranging from gameplay, continuous control, and robotics. In the context of visual control settings, clever use of augmented data (Yarats et al., 2021b; Laskin et al., 2020) currently produces state-of-the-art results on visual tasks from DMC (Tassa et al., 2018). ",
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+    {
+        "type": "text",
+        "text": "Humanoid Control The humanoid control problem first presented in Tassa et al. (2012), has been studied as one of the hardest control problems due to its large state and action spaces. The earliest solutions to this problem use ideas in model-based optimal control to generate policies given an accurate model of the humanoid . Subsequent works in RL have shown that model-free policies can solve the humanoid control problem given access to proprioceptive state observations. However, solving such a problem from visual observations has been a challenging problem, with leading RL algorithms making little progress to solve the task (Tassa et al., 2018). Recently, Hafner et al. (2020) was able to solve this problem through a model-based technique in around 30M environment steps and 340 hours of training on a single GPU machine. DrQ-v2, presented in this paper, marks the first model-free RL method that can solve humanoid control from visual observations, taking also around 30M steps and 86 hours of training on the same hardware. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We have introduced a conceptually simple model-free actor-critic RL agent for image-based continuous control – DrQ-v2. Our method provides significantly better computational footprint and masters tasks from DMC directly from pixels, most notably the humanoid locomotion tasks that ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "were previously unsolved by model-free approaches. To support our empirical results and inspire further research in visual RL we provide an efficient PyTorch implementation of $\\mathrm { D r Q - v } 2$ at https://anonymous.4open.science/r/drqv2. ",
+        "bbox": [
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
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parse/dev/_VjQlMeSB_J/_VjQlMeSB_J.md

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+# Chain-of-Thought Prompting Elicits Reasoning in Large Language Models
+
+Jason Wei Xuezhi Wang Dale Schuurmans Maarten Bosma
+
+Brian Ichter Fei Xia Ed H. Chi Quoc V. Le Denny Zhou
+
+Google Research, Brain Team {jasonwei,dennyzhou}@google.com
+
+# Abstract
+
+We explore how generating a chain of thought—a series of intermediate reasoning steps—significantly improves the ability of large language models to perform complex reasoning. In particular, we show how such reasoning abilities emerge naturally in sufficiently large language models via a simple method called chain-ofthought prompting, where a few chain of thought demonstrations are provided as exemplars in prompting.
+
+Experiments on three large language models show that chain-of-thought prompting improves performance on a range of arithmetic, commonsense, and symbolic reasoning tasks. The empirical gains can be striking. For instance, prompting a PaLM 540B with just eight chain-of-thought exemplars achieves state-of-the-art accuracy on the GSM8K benchmark of math word problems, surpassing even finetuned GPT-3 with a verifier.
+
+![](images/89ab3799d105e595086d7c7e3059c4c95016564718ada32028e60eef2d474c49.jpg)  
+Figure 1: Chain-of-thought prompting enables large language models to tackle complex arithmetic, commonsense, and symbolic reasoning tasks. Chain-of-thought reasoning processes are highlighted.
+
+# 1 Introduction
+
+The NLP landscape has recently been revolutionized by language models (Peters et al., 2018; Devlin et al., 2019; Brown et al., 2020, inter alia). Scaling up the size of language models has been shown to confer a range of benefits, such as improved performance and sample efficiency (Kaplan et al., 2020; Brown et al., 2020, inter alia). However, scaling up model size alone has not proved sufficient for achieving high performance on challenging tasks such as arithmetic, commonsense, and symbolic reasoning (Rae et al., 2021).
+
+This work explores how the reasoning ability of large language models can be unlocked by a simple method motivated by two ideas. First, techniques for arithmetic reasoning can benefit from generating natural language rationales that lead to the final answer. Prior work has given models the ability to generate natural language intermediate steps by training from scratch (Ling et al., 2017) or finetuning a pretrained model (Cobbe et al., 2021), in addition to neuro-symbolic methods that use formal languages instead of natural language (Roy and Roth, 2015; Chiang and Chen, 2019; Amini et al., 2019; Chen et al., 2019). Second, large language models offer the exciting
+
+Finetuned GPT-3 175B   
+Prior best PaLM 540B: standard prompting PaLM 540B: chain-of-thought prompting
+
+![](images/d7882b4649faefdd9a2b82d45815028e7e1d5a09149c74872bb38e2b8bb59569.jpg)  
+Figure 2: PaLM 540B uses chain-ofthought prompting to achieve new stateof-the-art performance on the GSM8K benchmark of math word problems. Finetuned GPT-3 and prior best are from Cobbe et al. (2021).
+
+prospect of in-context few-shot learning via prompting. That is, instead of finetuning a separate language model checkpoint for each new task, one can simply “prompt” the model with a few input–output exemplars demonstrating the task. Remarkably, this has been successful for a range of simple question-answering tasks (Brown et al., 2020).
+
+Both of the above ideas, however, have key limitations. For rationale-augmented training and finetuning methods, it is costly to create a large set of high quality rationales, which is much more complicated than simple input–output pairs used in normal machine learning. For the traditional fewshot prompting method used in Brown et al. (2020), it works poorly on tasks that require reasoning abilities, and often does not improve substantially with increasing language model scale (Rae et al., 2021). In this paper, we combine the strengths of these two ideas in a way that avoids their limitations. Specifically, we explore the ability of language models to perform few-shot prompting for reasoning tasks, given a prompt that consists of triples: hinput, chain of thought, outputi. A chain of thought is a series of intermediate natural language reasoning steps that lead to the final output, and we refer to this approach as chain-of-thought prompting. An example prompt is shown in Figure 1.
+
+We present empirical evaluations on arithmetic, commonsense, and symbolic reasoning benchmarks, showing that chain-of-thought prompting outperforms standard prompting, sometimes to a striking degree. Figure 2 illustrates one such result—on the GSM8K benchmark of math word problems (Cobbe et al., 2021), chain-of-thought prompting with PaLM 540B outperforms standard prompting by a large margin and achieves new state-of-the-art performance. A prompting only approach is important because it does not require a large training dataset and because a single model checkpoint can perform many tasks without loss of generality. This work underscores how large language models can learn via a few examples with natural language data about the task (c.f. automatically learning the patterns underlying inputs and outputs via a large training dataset).
+
+# 2 Chain-of-Thought Prompting
+
+Consider one’s own thought process when solving a complicated reasoning task such as a multi-step math word problem. It is typical to decompose the problem into intermediate steps and solve each before giving the final answer: “After Jane gives 2 flowers to her mom she has 10 . . . then after she gives 3 to her dad she will have 7 . . . so the answer is 7.” The goal of this paper is to endow language models with the ability to generate a similar chain of thought—a coherent series of intermediate reasoning steps that lead to the final answer for a problem. We will show that sufficiently large language models can generate chains of thought if demonstrations of chain-of-thought reasoning are provided in the exemplars for few-shot prompting.
+
+Figure 1 shows an example of a model producing a chain of thought to solve a math word problem that it would have otherwise gotten incorrect. The chain of thought in this case resembles a solution and can interpreted as one, but we still opt to call it a chain of thought to better capture the idea that it mimics a step-by-step thought process for arriving at the answer (and also, solutions/explanations typically come after the final answer (Narang et al., 2020; Wiegreffe et al., 2022; Lampinen et al., 2022, inter alia)).
+
+Chain-of-thought prompting has several attractive properties as an approach for facilitating reasoning in language models.
+
+1. First, chain of thought, in principle, allows models to decompose multi-step problems into intermediate steps, which means that additional computation can be allocated to problems that require more reasoning steps.   
+2. Second, a chain of thought provides an interpretable window into the behavior of the model, suggesting how it might have arrived at a particular answer and providing opportunities to debug where the reasoning path went wrong (although fully characterizing a model’s computations that support an answer remains an open question).   
+3. Third, chain-of-thought reasoning can be used for tasks such as math word problems, commonsense reasoning, and symbolic manipulation, and is potentially applicable (at least in principle) to any task that humans can solve via language.   
+4. Finally, chain-of-thought reasoning can be readily elicited in sufficiently large off-the-shelf language models simply by including examples of chain of thought sequences into the exemplars of few-shot prompting.
+
+In empirical experiments, we will observe the utility of chain-of-thought prompting for arithmetic reasoning (Section 3), commonsense reasoning (Section 4), and symbolic reasoning (Section 5).
+
+# 3 Arithmetic Reasoning
+
+We begin by considering math word problems of the form in Figure 1, which measure the arithmetic reasoning ability of language models. Though simple for humans, arithmetic reasoning is a task where language models often struggle (Hendrycks et al., 2021; Patel et al., 2021, inter alia). Strikingly, chainof-thought prompting when used with the 540B parameter language model performs comparably with task-specific finetuned models on several tasks, even achieving new state of the art on the challenging GSM8K benchmark (Cobbe et al., 2021).
+
+# 3.1 Experimental Setup
+
+We explore chain-of-thought prompting for various language models on multiple benchmarks.
+
+Benchmarks. We consider the following five math word problem benchmarks: (1) the GSM8K benchmark of math word problems (Cobbe et al., 2021), (2) the SVAMP dataset of math word problems with varying structures (Patel et al., 2021), (3) the ASDiv dataset of diverse math word problems (Miao et al., 2020), (4) the AQuA dataset of algebraic word problems, and (5) the MAWPS benchmark (Koncel-Kedziorski et al., 2016). Example problems are given in Appendix Table 12.
+
+Standard prompting. For the baseline, we consider standard few-shot prompting, popularized by Brown et al. (2020), in which a language model is given in-context exemplars of input–output pairs before outputting a prediction for a test-time example. Exemplars are formatted as questions and answers. The model gives the answer directly, as shown in Figure 1 (left).
+
+Chain-of-thought prompting. Our proposed approach is to augment each exemplar in few-shot prompting with a chain of thought for an associated answer, as illustrated in Figure 1 (right). As most of the datasets only have an evaluation split, we manually composed a set of eight few-shot exemplars with chains of thought for prompting—Figure 1 (right) shows one chain of thought exemplar, and the full set of exemplars is given in Appendix Table 20. (These particular exemplars did not undergo prompt engineering; robustness is studied in Section 3.4 and Appendix A.2.) To investigate whether chain-of-thought prompting in this form can successfully elicit successful reasoning across a range of math word problems, we used this single set of eight chain of thought exemplars for all benchmarks except AQuA, which is multiple choice instead of free response. For AQuA, we used four exemplars and solutions from the training set, as given in Appendix Table 21.
+
+![](images/89a5ee9245764678a62df985ae74ac1f355481b7fba3da340ff294060cdc69a1.jpg)  
+Figure 3: Examples of hinput, chain of thought, outputi triples for arithmetic, commonsense, and symbolic reasoning benchmarks. Chains of thought are highlighted. Full prompts in Appendix G.
+
+Language models. We evaluate five large language models. The first is GPT-3 (Brown et al., 2020), for which we use text-ada-001, text-babbage-001, text-curie-001, and text-davinci-002, which presumably correspond to InstructGPT models of 350M, 1.3B, 6.7B, and 175B parameters (Ouyang et al., 2022).The second is LaMDA (Thoppilan et al., 2022), which has models of 422M, 2B, 8B, 68B, and 137B parameters. The third is PaLM, which has models of 8B, 62B, and 540B parameters. The fourth is UL2 20B (Tay et al., 2022), and the fifth is Codex (Chen et al., 2021, code-davinci-002 in the OpenAI API). We sample from the models via greedy decoding (though follow-up work shows chain-of-thought prompting can be improved by taking the majority final answer over many sampled generations (Wang et al., 2022a)). For LaMDA, we report averaged results over five random seeds, where each seed had a different randomly shuffled order of exemplars. As LaMDA experiments did not show large variance among different seeds, to save compute we report results for a single exemplar order for all other models.
+
+# 3.2 Results
+
+The strongest results of chain-of-thought prompting are summarized in Figure 4, with all experimental outputs for each model collection, model size, and benchmark shown in Table 2 in the Appendix. There are three key takeaways. First, Figure 4 shows that chain-of-thought prompting is an emergent ability of model scale (Wei et al., 2022b). That is, chain-of-thought prompting does not positively impact performance for small models, and only yields performance gains when used with models of ${ \sim } 1 0 0 \mathrm { B }$ parameters. We qualitatively found that models of smaller scale produced fluent but illogical chains of thought, leading to lower performance than standard prompting.
+
+Second, chain-of-thought prompting has larger performance gains for more-complicated problems. For instance, for GSM8K (the dataset with the lowest baseline performance), performance more than doubled for the largest GPT and PaLM models. On the other hand, for SingleOp, the easiest subset of MAWPS which only requires a single step to solve, performance improvements were either negative or very small (see Appendix Table 3).
+
+Third, chain-of-thought prompting via GPT-3 175B and PaLM 540B compares favorably to prior state of the art, which typically finetunes a task-specific model on a labeled training dataset. Figure 4 shows how PaLM 540B uses chain-ofthought prompting to achieve new state of the art on GSM8K, SVAMP, and MAWPS (though note that standard prompting already passed the prior best for SVAMP). On the other two datasets, AQuA and ASDiv, PaLM with chain-of-thought prompting reaches within $2 \%$ of the state of the art (Appendix Table 2).
+
+To better understand why chain-of-thought prompting works, we manually examined modelgenerated chains of thought by LaMDA 137B for GSM8K. Of 50 random examples where the model returned the correct final answer, all of the generated chains of thought were also logically and mathematically correct except two that coincidentally arrived at the correct answer (see Appendix D.1, and Table 8 for examples of correct model-generated chains of thought). We also randomly examined 50 random samples for which the model gave the wrong answer. The summary of this analysis is that $46 \%$ of the chains of thought were almost correct, barring minor mistakes (calculator error, symbol map
+
+![](images/66c022898bff367d5860a3fc49e57a4fadc8dcb6199599bfe7c3100af93ef076.jpg)  
+Figure 4: Chain-of-thought prompting enables large language models to solve challenging math problems. Notably, chain-of-thought reasoning is an emergent ability of increasing model scale. Prior best numbers are from Cobbe et al. (2021) for GSM8K, Jie et al. (2022) for SVAMP, and Lan et al. (2021) for MAWPS.
+
+ping error, or one reasoning step missing), and that the other $54 \%$ of the chains of thought had major errors in semantic understanding or coherence (see Appendix D.2). To provide a small insight into why scaling improves chain-of-thought reasoning ability, we performed a similar analysis of errors made by PaLM 62B and whether those errors were fixed by scaling to PaLM 540B. The summary is that scaling PaLM to 540B fixes a large portion of one-step missing and semantic understanding errors in the 62B model (see Appendix A.1).
+
+# 3.3 Ablation Study
+
+The observed benefits of using chain-of-thought prompting raises the natural question of whether the same performance improvements can be conferred via other types of prompting. Figure 5 shows an ablation study with three variations of chain of thought described below.
+
+Equation only. One reason for why chain-of-thought prompting might help is that it produces the mathematical equation to be evaluated, and so we test a variation where the model is prompted to output only a mathematical equation before giving the answer. Figure 5 shows that equation only prompting does not help much for GSM8K, which implies that the semantics of the questions in GSM8K are too challenging to directly translate into an equation without the natural language reasoning steps in chain of thought. For datasets of one-step or two-step problems, however, we find that equation only prompting does improve performance, since the equation can be easily derived from the question (see Appendix Table 6).
+
+Variable compute only. Another intuition is that chain of thought allows the model to spend more computation (i.e., intermediate tokens) on harder problems. To isolate the effect of variable computation from chain-of-thought reasoning, we test a configuration where the model is prompted to output a only sequence of dots $( \ldots )$ equal to the number of characters in the equation needed to solve the problem. This variant performs about the same as the baseline, which suggests that variable computation by itself is not the reason for the success of chainof-thought prompting, and that there appears to be utility from expressing intermediate steps via natural language.
+
+Chain of thought after answer. Another potential benefit of chain-of-thought prompting could simply be that such prompts allow the model to better access relevant knowledge acquired during pretraining. Therefore, we test an alternative configuration where the chain of thought prompt is only given after the answer, isolating whether the model actually depends on the produced chain of thought to give the final answer. This variant performs about the same as the baseline, which suggests that the sequential reasoning embodied in the chain of thought is useful for reasons beyond just activating knowledge.
+
+Standard prompting Z Equation only Variable compute only 用 Reasoning after answer Chain-of-thought prompting
+
+![](images/e6ae1dc967e810730fda1618d18b35c7fc13ce62af4bf30f06c19dc149050fe2.jpg)  
+Figure 5: Ablation study for different variations of prompting using LaMDA 137B and PaLM 540B. Results for other datasets are given in Appendix Table 6 and Table 7.
+
+# 3.4 Robustness of Chain of Thought
+
+Sensitivity to exemplars is a key consideration of prompting approaches—for instance, varying the permutation of few-shot exemplars can cause the accuracy of GPT-3 on SST-2 to range from near chance $( 5 4 . 3 \% )$ to near state of the art $( 9 3 . 4 \% )$ (Zhao et al., 2021). In this final subsection, we evaluate robustness to chains of thought written by different annotators. In addition to the results above, which used chains of thought written by an Annotator A, two other co-authors of this paper (Annotators B and C) independently wrote chains of thought for the same few-shot exemplars (shown in Appendix H). Annotator A also wrote another chain of thought that was more concise than the original, following the style of solutions given in Cobbe et al. (2021).1
+
+Figure 6 shows these results for LaMDA 137B on GSM8K and MAWPS (ablation results for other datasets are given in Appendix Table 6 / Table 7). Although there is variance among different chain of thought annotations, as would be expected when using exemplar-based prompting (Le Scao and Rush, 2021; Reynolds and McDonell, 2021; Zhao et al., 2021), all sets of chain of thought prompts outperform the standard baseline by a large margin. This result implies that successful use of chain of thought does not depend on a particular linguistic style.
+
+![](images/4e11e7bcb919c728e67a6b419ec469fa86d0f95c703a30a4d57759b25ff7f940.jpg)  
+Figure 6: Chain-of-thought prompting has variance for different prompt examples (as expected) but outperforms standard prompting for various annotators as well as for different exemplars.
+
+To confirm that successful chain-of-thought prompting works for other sets of exemplars, we also run experiments with three sets of eight exemplars randomly sampled from the GSM8K training set, an independent source (examples in this dataset already included reasoning steps like a chain of thought).2 Figure 6 shows that these prompts performed comparably with our manually written exemplars, also substantially outperforming standard prompting.
+
+In addition to robustness to annotators, independently-written chains of thought, different exemplars, and various language models, we also find that chain-of-thought prompting for arithmetic reasoning is robust to different exemplar orders and varying numbers of exemplars (see Appendix A.2).
+
+# 4 Commonsense Reasoning
+
+Although chain of thought is particularly suitable for math word problems, the language-based nature of chain of thought actually makes it applicable to a broad class of commonsense reasoning problems, which involve reasoning about physical and human interactions under the presumption of general background knowledge. Commonsense reasoning is key for interacting with the world and is still beyond the reach of current natural language understanding systems (Talmor et al., 2021).
+
+Benchmarks. We consider five datasets covering a diverse range of commonsense reasoning types. The popular CSQA (Talmor et al., 2019) asks commonsense questions about the world involving complex semantics that often require prior knowledge. StrategyQA (Geva et al., 2021) requires models to infer a multi-hop strategy to answer questions. We choose two specialized evaluation sets from the BIG-bench effort (BIG-bench collaboration, 2021): Date Understanding, which involves inferring a date from a given context, and Sports Understanding, which involves determining whether a sentence relating to sports is plausible or implausible. Finally, the SayCan dataset (Ahn et al., 2022) involves mapping a natural language instruction to a sequence of robot actions from a discrete set. Figure 3 shows examples with chain of thought annotations for all datasets.
+
+Prompts. We follow the same experimental setup as the prior section. For CSQA and StrategyQA, we randomly selected examples from the training set and manually composed chains of thought for them to use as few-shot exemplars. The two BIG-bench tasks do not have training sets, so we selected the first ten examples as exemplars in the evaluation set as few-shot exemplars and report numbers on the rest of the evaluation set. For SayCan, we use six examples from the training set used in Ahn et al. (2022) and also manually composed chains of thought.
+
+Results. Figure 7 highlights these results for PaLM (full results for LaMDA, GPT-3, and different model scales are shown in Table 4). For all tasks, scaling up model size improved the performance of standard prompting; chain-of-thought prompting led to further gains, with improvements appearing to be largest for PaLM 540B. With chain-of-thought prompting, PaLM 540B achieved strong performance relative to baselines, outperforming the prior state of the art on StrategyQA $7 5 . 6 \%$ vs $6 9 . 4 \%$ and outperforming an unaided sports enthusiast on sports understanding $9 5 . 4 \%$ vs $84 \%$ ). These results demonstrate that chain-of-thought prompting can also improve performance on tasks requiring a range of commonsense reasoning abilities (though note that gain was minimal on CSQA).
+
+![](images/314d6a9f471c937800bcc79eed382244c146f9aacc68828ea5e504cb8128e362.jpg)  
+Figure 7: Chain-of-thought prompting also improves the commonsense reasoning abilities of language models. The language model shown here is PaLM. Prior best numbers are from the leaderboards of CSQA (Talmor et al., 2019) and StrategyQA (Geva et al., 2021) (single-model only, as of May 5, 2022). Additional results using various sizes of LaMDA, GPT-3, and PaLM are shown in Table 4.
+
+# 5 Symbolic Reasoning
+
+Our final experimental evaluation considers symbolic reasoning, which is simple for humans but potentially challenging for language models. We show that chain-ofthought prompting not only enables language models to perform symbolic reasoning tasks that are challenging in the standard prompting setting, but also facilitates length generalization to inference-time inputs longer than those seen in the few-shot exemplars.
+
+Tasks. We use the following two toy tasks.
+
+• Last letter concatenation. This task asks the model to concatenate the last letters of words in a name (e.g., “Amy Brown” $ ^ { \ast } y n ^ { \prime \prime }$ ). It is a more challenging version of first letter concatenation, which language models can already perform without chain of thought.3 We generate full names by randomly concatenating names from the top one-thousand first and last names from name census data (https://namecensus.com/).
+
+![](images/e4f8153620820f36adc177bd4972c7b26a10341d1dec25b01e73852fd72bc7ae.jpg)  
+Figure 8: Using chain-of-thought prompting facilitates generalization to longer sequences in two symbolic reasoning tasks.
+
+• Coin flip. This task asks the model to answer whether a coin is still heads up after people either flip or don’t flip the coin (e.g., “A coin is heads up. Phoebe flips the coin. Osvaldo does not flip the coin. Is the coin still heads up? $^ { \prime \prime } \right. ^ { \left. } n o ^ { \prime \prime }$ ).
+
+As the construction of these symbolic reasoning tasks is well-defined, for each task we consider an in-domain test set for which examples had the same number of steps as the training/few-shot exemplars, as well as an out-of-domain (OOD) test set, for which evaluation examples had more steps than those in the exemplars. For last letter concatenation, the model only sees exemplars of names with two words, and then performs last letter concatenation on names with 3 and 4 words.4 We do the same for the number of potential flips in the coin flip task. Our experimental setup uses the same methods and models as in the prior two sections. We again manually compose chains of thought for the few-shot exemplars for each task, which are given in Figure 3.
+
+Results. The results of these in-domain and OOD evaluations are shown in Figure 8 for PaLM, with results for LaMDA shown in Appendix Table 5. With PaLM 540B, chain-of-thought prompting leads to almost $100 \%$ solve rates (note that standard prompting already solves coin flip with PaLM 540, though not for LaMDA 137B). Note that these in-domain evaluations are “toy tasks” in the sense that perfect solution structures are already provided by the chains of thought in the few-shot exemplars; all the model has to do is repeat the same steps with the new symbols in the test-time example. And yet, small models still fail—the ability to perform abstract manipulations on unseen symbols for these three tasks only arises at the scale of 100B model parameters.
+
+As for the OOD evaluations, standard prompting fails for both tasks. With chain-of-thought prompting, language models achieve upward scaling curves (though performance is lower than in the in-domain setting). Hence, chain-of-thought prompting facilitates length generalization beyond seen chains of thought for language models of sufficient scale.
+
+# 6 Discussion
+
+We have explored chain-of-thought prompting as a simple mechanism for eliciting multi-step reasoning behavior in large language models. We first saw that chain-of-thought prompting improves performance by a large margin on arithmetic reasoning, yielding improvements that are much stronger than ablations and robust to different annotators, exemplars, and language models (Section 3). Next, experiments on commonsense reasoning underscored how the linguistic nature of chain-of-thought reasoning makes it generally applicable (Section 4). Finally, we showed that for symbolic reasoning, chain-of-thought prompting facilitates OOD generalization to longer sequence lengths (Section 5). In all experiments, chain-of-thought reasoning is elicited simply by prompting an off-the-shelf language model. No language models were finetuned in the process of writing this paper.
+
+The emergence of chain-of-thought reasoning as a result of model scale has been a prevailing theme (Wei et al., 2022b). For many reasoning tasks where standard prompting has a flat scaling curve, chainof-thought prompting leads to dramatically increasing scaling curves. Chain-of-thought prompting appears to expand the set of tasks that large language models can perform successfully—in other words, our work underscores that standard prompting only provides a lower bound on the capabilities of large language models. This observation likely raises more questions than it answers—for instance, how much more can we expect reasoning ability to improve with a further increase in model scale? What other prompting methods might expand the range of tasks that language models can solve?
+
+As for limitations, we first qualify that although chain of thought emulates the thought processes of human reasoners, this does not answer whether the neural network is actually “reasoning,” which we leave as an open question. Second, although the cost of manually augmenting exemplars with chains of thought is minimal in the few-shot setting, such annotation costs could be prohibitive for finetuning (though this could potentially be surmounted with synthetic data generation, or zero-shot generalization). Third, there is no guarantee of correct reasoning paths, which can lead to both correct and incorrect answers; improving factual generations of language models is an open direction for future work (Rashkin et al., 2021; Ye and Durrett, 2022; Wiegreffe et al., 2022, inter alia). Finally, the emergence of chain-of-thought reasoning only at large model scales makes it costly to serve in real-world applications; further research could explore how to induce reasoning in smaller models.
+
+# 7 Related Work
+
+This work is inspired by many research areas, which we detail in an extended related work section (Appendix C). Here we describe two directions and associated papers that are perhaps most relevant.
+
+The first relevant direction is using intermediate steps to solve reasoning problems. Ling et al. (2017) pioneer the idea of using natural language rationales to solve math word problems through a series of intermediate steps. Their work is a remarkable contrast to the literature using formal languages to reason (Roy et al., 2015; Chiang and Chen, 2019; Amini et al., 2019; Chen et al., 2019). Cobbe et al. (2021) extend Ling et al. (2017) by creating a larger dataset and using it to finetune a pretrained language model rather than training a model from scratch. In the domain of program synthesis, Nye et al. (2021) leverage language models to predict the final outputs of Python programs via first line-to-line predicting the intermediate computational results, and show that their step-by-step prediction method performs better than directly predicting the final outputs.
+
+Naturally, this paper also relates closely to the large body of recent work on prompting. Since the popularization of few-shot prompting as given by Brown et al. (2020), several general approaches have improved the prompting ability of models, such as automatically learning prompts (Lester et al., 2021) or giving models instructions describing a task (Wei et al., 2022a; Sanh et al., 2022; Ouyang et al., 2022). Whereas these approaches improve or augment the input part of the prompt (e.g., instructions that are prepended to inputs), our work takes the orthogonal direction of augmenting the outputs of language models with a chain of thought.
+
+# 8 Conclusions
+
+We have explored chain-of-thought prompting as a simple and broadly applicable method for enhancing reasoning in language models. Through experiments on arithmetic, symbolic, and commonsense reasoning, we find that chain-of-thought reasoning is an emergent property of model scale that allows sufficiently large language models to perform reasoning tasks that otherwise have flat scaling curves. Broadening the range of reasoning tasks that language models can perform will hopefully inspire further work on language-based approaches to reasoning.
+
+# Acknowledgements
+
+We thank Jacob Devlin, Claire Cui, Andrew Dai, and Ellie Pavlick for providing feedback on the paper. We thank Jacob Austin, Yuhuai Wu, Henryk Michalewski, Aitor Lewkowycz, Charles Sutton, and Aakanksha Chowdhery for helpful discussions. We thank Sid Maxwell for notifying us about a mistake in the manual error analysis in the original manuscript.
+
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+        "type": "text",
+        "text": "Chain-of-Thought Prompting Elicits Reasoning in Large Language Models ",
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+        "text": "Jason Wei Xuezhi Wang Dale Schuurmans Maarten Bosma ",
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+        "text": "Brian Ichter Fei Xia Ed H. Chi Quoc V. Le Denny Zhou ",
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+    {
+        "type": "text",
+        "text": "Google Research, Brain Team {jasonwei,dennyzhou}@google.com ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We explore how generating a chain of thought—a series of intermediate reasoning steps—significantly improves the ability of large language models to perform complex reasoning. In particular, we show how such reasoning abilities emerge naturally in sufficiently large language models via a simple method called chain-ofthought prompting, where a few chain of thought demonstrations are provided as exemplars in prompting. ",
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+        "type": "text",
+        "text": "Experiments on three large language models show that chain-of-thought prompting improves performance on a range of arithmetic, commonsense, and symbolic reasoning tasks. The empirical gains can be striking. For instance, prompting a PaLM 540B with just eight chain-of-thought exemplars achieves state-of-the-art accuracy on the GSM8K benchmark of math word problems, surpassing even finetuned GPT-3 with a verifier. ",
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+        "image_caption": [
+            "Figure 1: Chain-of-thought prompting enables large language models to tackle complex arithmetic, commonsense, and symbolic reasoning tasks. Chain-of-thought reasoning processes are highlighted. "
+        ],
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The NLP landscape has recently been revolutionized by language models (Peters et al., 2018; Devlin et al., 2019; Brown et al., 2020, inter alia). Scaling up the size of language models has been shown to confer a range of benefits, such as improved performance and sample efficiency (Kaplan et al., 2020; Brown et al., 2020, inter alia). However, scaling up model size alone has not proved sufficient for achieving high performance on challenging tasks such as arithmetic, commonsense, and symbolic reasoning (Rae et al., 2021). ",
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+        "type": "text",
+        "text": "This work explores how the reasoning ability of large language models can be unlocked by a simple method motivated by two ideas. First, techniques for arithmetic reasoning can benefit from generating natural language rationales that lead to the final answer. Prior work has given models the ability to generate natural language intermediate steps by training from scratch (Ling et al., 2017) or finetuning a pretrained model (Cobbe et al., 2021), in addition to neuro-symbolic methods that use formal languages instead of natural language (Roy and Roth, 2015; Chiang and Chen, 2019; Amini et al., 2019; Chen et al., 2019). Second, large language models offer the exciting ",
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+    {
+        "type": "text",
+        "text": "Finetuned GPT-3 175B   \nPrior best PaLM 540B: standard prompting PaLM 540B: chain-of-thought prompting ",
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+    {
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+        "image_caption": [
+            "Figure 2: PaLM 540B uses chain-ofthought prompting to achieve new stateof-the-art performance on the GSM8K benchmark of math word problems. Finetuned GPT-3 and prior best are from Cobbe et al. (2021). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "prospect of in-context few-shot learning via prompting. That is, instead of finetuning a separate language model checkpoint for each new task, one can simply “prompt” the model with a few input–output exemplars demonstrating the task. Remarkably, this has been successful for a range of simple question-answering tasks (Brown et al., 2020). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Both of the above ideas, however, have key limitations. For rationale-augmented training and finetuning methods, it is costly to create a large set of high quality rationales, which is much more complicated than simple input–output pairs used in normal machine learning. For the traditional fewshot prompting method used in Brown et al. (2020), it works poorly on tasks that require reasoning abilities, and often does not improve substantially with increasing language model scale (Rae et al., 2021). In this paper, we combine the strengths of these two ideas in a way that avoids their limitations. Specifically, we explore the ability of language models to perform few-shot prompting for reasoning tasks, given a prompt that consists of triples: hinput, chain of thought, outputi. A chain of thought is a series of intermediate natural language reasoning steps that lead to the final output, and we refer to this approach as chain-of-thought prompting. An example prompt is shown in Figure 1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We present empirical evaluations on arithmetic, commonsense, and symbolic reasoning benchmarks, showing that chain-of-thought prompting outperforms standard prompting, sometimes to a striking degree. Figure 2 illustrates one such result—on the GSM8K benchmark of math word problems (Cobbe et al., 2021), chain-of-thought prompting with PaLM 540B outperforms standard prompting by a large margin and achieves new state-of-the-art performance. A prompting only approach is important because it does not require a large training dataset and because a single model checkpoint can perform many tasks without loss of generality. This work underscores how large language models can learn via a few examples with natural language data about the task (c.f. automatically learning the patterns underlying inputs and outputs via a large training dataset). ",
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+        "type": "text",
+        "text": "2 Chain-of-Thought Prompting ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Consider one’s own thought process when solving a complicated reasoning task such as a multi-step math word problem. It is typical to decompose the problem into intermediate steps and solve each before giving the final answer: “After Jane gives 2 flowers to her mom she has 10 . . . then after she gives 3 to her dad she will have 7 . . . so the answer is 7.” The goal of this paper is to endow language models with the ability to generate a similar chain of thought—a coherent series of intermediate reasoning steps that lead to the final answer for a problem. We will show that sufficiently large language models can generate chains of thought if demonstrations of chain-of-thought reasoning are provided in the exemplars for few-shot prompting. ",
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+    {
+        "type": "text",
+        "text": "Figure 1 shows an example of a model producing a chain of thought to solve a math word problem that it would have otherwise gotten incorrect. The chain of thought in this case resembles a solution and can interpreted as one, but we still opt to call it a chain of thought to better capture the idea that it mimics a step-by-step thought process for arriving at the answer (and also, solutions/explanations typically come after the final answer (Narang et al., 2020; Wiegreffe et al., 2022; Lampinen et al., 2022, inter alia)). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Chain-of-thought prompting has several attractive properties as an approach for facilitating reasoning in language models. ",
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+        "text": "1. First, chain of thought, in principle, allows models to decompose multi-step problems into intermediate steps, which means that additional computation can be allocated to problems that require more reasoning steps.   \n2. Second, a chain of thought provides an interpretable window into the behavior of the model, suggesting how it might have arrived at a particular answer and providing opportunities to debug where the reasoning path went wrong (although fully characterizing a model’s computations that support an answer remains an open question).   \n3. Third, chain-of-thought reasoning can be used for tasks such as math word problems, commonsense reasoning, and symbolic manipulation, and is potentially applicable (at least in principle) to any task that humans can solve via language.   \n4. Finally, chain-of-thought reasoning can be readily elicited in sufficiently large off-the-shelf language models simply by including examples of chain of thought sequences into the exemplars of few-shot prompting. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In empirical experiments, we will observe the utility of chain-of-thought prompting for arithmetic reasoning (Section 3), commonsense reasoning (Section 4), and symbolic reasoning (Section 5). ",
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+    {
+        "type": "text",
+        "text": "3 Arithmetic Reasoning ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We begin by considering math word problems of the form in Figure 1, which measure the arithmetic reasoning ability of language models. Though simple for humans, arithmetic reasoning is a task where language models often struggle (Hendrycks et al., 2021; Patel et al., 2021, inter alia). Strikingly, chainof-thought prompting when used with the 540B parameter language model performs comparably with task-specific finetuned models on several tasks, even achieving new state of the art on the challenging GSM8K benchmark (Cobbe et al., 2021). ",
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+    {
+        "type": "text",
+        "text": "3.1 Experimental Setup ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We explore chain-of-thought prompting for various language models on multiple benchmarks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Benchmarks. We consider the following five math word problem benchmarks: (1) the GSM8K benchmark of math word problems (Cobbe et al., 2021), (2) the SVAMP dataset of math word problems with varying structures (Patel et al., 2021), (3) the ASDiv dataset of diverse math word problems (Miao et al., 2020), (4) the AQuA dataset of algebraic word problems, and (5) the MAWPS benchmark (Koncel-Kedziorski et al., 2016). Example problems are given in Appendix Table 12. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Standard prompting. For the baseline, we consider standard few-shot prompting, popularized by Brown et al. (2020), in which a language model is given in-context exemplars of input–output pairs before outputting a prediction for a test-time example. Exemplars are formatted as questions and answers. The model gives the answer directly, as shown in Figure 1 (left). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Chain-of-thought prompting. Our proposed approach is to augment each exemplar in few-shot prompting with a chain of thought for an associated answer, as illustrated in Figure 1 (right). As most of the datasets only have an evaluation split, we manually composed a set of eight few-shot exemplars with chains of thought for prompting—Figure 1 (right) shows one chain of thought exemplar, and the full set of exemplars is given in Appendix Table 20. (These particular exemplars did not undergo prompt engineering; robustness is studied in Section 3.4 and Appendix A.2.) To investigate whether chain-of-thought prompting in this form can successfully elicit successful reasoning across a range of math word problems, we used this single set of eight chain of thought exemplars for all benchmarks except AQuA, which is multiple choice instead of free response. For AQuA, we used four exemplars and solutions from the training set, as given in Appendix Table 21. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/89a5ee9245764678a62df985ae74ac1f355481b7fba3da340ff294060cdc69a1.jpg",
+        "image_caption": [
+            "Figure 3: Examples of hinput, chain of thought, outputi triples for arithmetic, commonsense, and symbolic reasoning benchmarks. Chains of thought are highlighted. Full prompts in Appendix G. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Language models. We evaluate five large language models. The first is GPT-3 (Brown et al., 2020), for which we use text-ada-001, text-babbage-001, text-curie-001, and text-davinci-002, which presumably correspond to InstructGPT models of 350M, 1.3B, 6.7B, and 175B parameters (Ouyang et al., 2022).The second is LaMDA (Thoppilan et al., 2022), which has models of 422M, 2B, 8B, 68B, and 137B parameters. The third is PaLM, which has models of 8B, 62B, and 540B parameters. The fourth is UL2 20B (Tay et al., 2022), and the fifth is Codex (Chen et al., 2021, code-davinci-002 in the OpenAI API). We sample from the models via greedy decoding (though follow-up work shows chain-of-thought prompting can be improved by taking the majority final answer over many sampled generations (Wang et al., 2022a)). For LaMDA, we report averaged results over five random seeds, where each seed had a different randomly shuffled order of exemplars. As LaMDA experiments did not show large variance among different seeds, to save compute we report results for a single exemplar order for all other models. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Results ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The strongest results of chain-of-thought prompting are summarized in Figure 4, with all experimental outputs for each model collection, model size, and benchmark shown in Table 2 in the Appendix. There are three key takeaways. First, Figure 4 shows that chain-of-thought prompting is an emergent ability of model scale (Wei et al., 2022b). That is, chain-of-thought prompting does not positively impact performance for small models, and only yields performance gains when used with models of ${ \\sim } 1 0 0 \\mathrm { B }$ parameters. We qualitatively found that models of smaller scale produced fluent but illogical chains of thought, leading to lower performance than standard prompting. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Second, chain-of-thought prompting has larger performance gains for more-complicated problems. For instance, for GSM8K (the dataset with the lowest baseline performance), performance more than doubled for the largest GPT and PaLM models. On the other hand, for SingleOp, the easiest subset of MAWPS which only requires a single step to solve, performance improvements were either negative or very small (see Appendix Table 3). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Third, chain-of-thought prompting via GPT-3 175B and PaLM 540B compares favorably to prior state of the art, which typically finetunes a task-specific model on a labeled training dataset. Figure 4 shows how PaLM 540B uses chain-ofthought prompting to achieve new state of the art on GSM8K, SVAMP, and MAWPS (though note that standard prompting already passed the prior best for SVAMP). On the other two datasets, AQuA and ASDiv, PaLM with chain-of-thought prompting reaches within $2 \\%$ of the state of the art (Appendix Table 2). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To better understand why chain-of-thought prompting works, we manually examined modelgenerated chains of thought by LaMDA 137B for GSM8K. Of 50 random examples where the model returned the correct final answer, all of the generated chains of thought were also logically and mathematically correct except two that coincidentally arrived at the correct answer (see Appendix D.1, and Table 8 for examples of correct model-generated chains of thought). We also randomly examined 50 random samples for which the model gave the wrong answer. The summary of this analysis is that $46 \\%$ of the chains of thought were almost correct, barring minor mistakes (calculator error, symbol map",
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+        "img_path": "images/66c022898bff367d5860a3fc49e57a4fadc8dcb6199599bfe7c3100af93ef076.jpg",
+        "image_caption": [
+            "Figure 4: Chain-of-thought prompting enables large language models to solve challenging math problems. Notably, chain-of-thought reasoning is an emergent ability of increasing model scale. Prior best numbers are from Cobbe et al. (2021) for GSM8K, Jie et al. (2022) for SVAMP, and Lan et al. (2021) for MAWPS. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "ping error, or one reasoning step missing), and that the other $54 \\%$ of the chains of thought had major errors in semantic understanding or coherence (see Appendix D.2). To provide a small insight into why scaling improves chain-of-thought reasoning ability, we performed a similar analysis of errors made by PaLM 62B and whether those errors were fixed by scaling to PaLM 540B. The summary is that scaling PaLM to 540B fixes a large portion of one-step missing and semantic understanding errors in the 62B model (see Appendix A.1). ",
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+    {
+        "type": "text",
+        "text": "3.3 Ablation Study ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The observed benefits of using chain-of-thought prompting raises the natural question of whether the same performance improvements can be conferred via other types of prompting. Figure 5 shows an ablation study with three variations of chain of thought described below. ",
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+    {
+        "type": "text",
+        "text": "Equation only. One reason for why chain-of-thought prompting might help is that it produces the mathematical equation to be evaluated, and so we test a variation where the model is prompted to output only a mathematical equation before giving the answer. Figure 5 shows that equation only prompting does not help much for GSM8K, which implies that the semantics of the questions in GSM8K are too challenging to directly translate into an equation without the natural language reasoning steps in chain of thought. For datasets of one-step or two-step problems, however, we find that equation only prompting does improve performance, since the equation can be easily derived from the question (see Appendix Table 6). ",
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+    {
+        "type": "text",
+        "text": "Variable compute only. Another intuition is that chain of thought allows the model to spend more computation (i.e., intermediate tokens) on harder problems. To isolate the effect of variable computation from chain-of-thought reasoning, we test a configuration where the model is prompted to output a only sequence of dots $( \\ldots )$ equal to the number of characters in the equation needed to solve the problem. This variant performs about the same as the baseline, which suggests that variable computation by itself is not the reason for the success of chainof-thought prompting, and that there appears to be utility from expressing intermediate steps via natural language. ",
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+    {
+        "type": "text",
+        "text": "Chain of thought after answer. Another potential benefit of chain-of-thought prompting could simply be that such prompts allow the model to better access relevant knowledge acquired during pretraining. Therefore, we test an alternative configuration where the chain of thought prompt is only given after the answer, isolating whether the model actually depends on the produced chain of thought to give the final answer. This variant performs about the same as the baseline, which suggests that the sequential reasoning embodied in the chain of thought is useful for reasons beyond just activating knowledge. ",
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+        "type": "text",
+        "text": "Standard prompting Z Equation only Variable compute only 用 Reasoning after answer Chain-of-thought prompting ",
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+        "img_path": "images/e6ae1dc967e810730fda1618d18b35c7fc13ce62af4bf30f06c19dc149050fe2.jpg",
+        "image_caption": [
+            "Figure 5: Ablation study for different variations of prompting using LaMDA 137B and PaLM 540B. Results for other datasets are given in Appendix Table 6 and Table 7. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "3.4 Robustness of Chain of Thought ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Sensitivity to exemplars is a key consideration of prompting approaches—for instance, varying the permutation of few-shot exemplars can cause the accuracy of GPT-3 on SST-2 to range from near chance $( 5 4 . 3 \\% )$ to near state of the art $( 9 3 . 4 \\% )$ (Zhao et al., 2021). In this final subsection, we evaluate robustness to chains of thought written by different annotators. In addition to the results above, which used chains of thought written by an Annotator A, two other co-authors of this paper (Annotators B and C) independently wrote chains of thought for the same few-shot exemplars (shown in Appendix H). Annotator A also wrote another chain of thought that was more concise than the original, following the style of solutions given in Cobbe et al. (2021).1 ",
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+        "type": "text",
+        "text": "Figure 6 shows these results for LaMDA 137B on GSM8K and MAWPS (ablation results for other datasets are given in Appendix Table 6 / Table 7). Although there is variance among different chain of thought annotations, as would be expected when using exemplar-based prompting (Le Scao and Rush, 2021; Reynolds and McDonell, 2021; Zhao et al., 2021), all sets of chain of thought prompts outperform the standard baseline by a large margin. This result implies that successful use of chain of thought does not depend on a particular linguistic style. ",
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+    {
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+        "img_path": "images/4e11e7bcb919c728e67a6b419ec469fa86d0f95c703a30a4d57759b25ff7f940.jpg",
+        "image_caption": [
+            "Figure 6: Chain-of-thought prompting has variance for different prompt examples (as expected) but outperforms standard prompting for various annotators as well as for different exemplars. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "To confirm that successful chain-of-thought prompting works for other sets of exemplars, we also run experiments with three sets of eight exemplars randomly sampled from the GSM8K training set, an independent source (examples in this dataset already included reasoning steps like a chain of thought).2 Figure 6 shows that these prompts performed comparably with our manually written exemplars, also substantially outperforming standard prompting. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "In addition to robustness to annotators, independently-written chains of thought, different exemplars, and various language models, we also find that chain-of-thought prompting for arithmetic reasoning is robust to different exemplar orders and varying numbers of exemplars (see Appendix A.2). ",
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+    {
+        "type": "text",
+        "text": "4 Commonsense Reasoning ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Although chain of thought is particularly suitable for math word problems, the language-based nature of chain of thought actually makes it applicable to a broad class of commonsense reasoning problems, which involve reasoning about physical and human interactions under the presumption of general background knowledge. Commonsense reasoning is key for interacting with the world and is still beyond the reach of current natural language understanding systems (Talmor et al., 2021). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Benchmarks. We consider five datasets covering a diverse range of commonsense reasoning types. The popular CSQA (Talmor et al., 2019) asks commonsense questions about the world involving complex semantics that often require prior knowledge. StrategyQA (Geva et al., 2021) requires models to infer a multi-hop strategy to answer questions. We choose two specialized evaluation sets from the BIG-bench effort (BIG-bench collaboration, 2021): Date Understanding, which involves inferring a date from a given context, and Sports Understanding, which involves determining whether a sentence relating to sports is plausible or implausible. Finally, the SayCan dataset (Ahn et al., 2022) involves mapping a natural language instruction to a sequence of robot actions from a discrete set. Figure 3 shows examples with chain of thought annotations for all datasets. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Prompts. We follow the same experimental setup as the prior section. For CSQA and StrategyQA, we randomly selected examples from the training set and manually composed chains of thought for them to use as few-shot exemplars. The two BIG-bench tasks do not have training sets, so we selected the first ten examples as exemplars in the evaluation set as few-shot exemplars and report numbers on the rest of the evaluation set. For SayCan, we use six examples from the training set used in Ahn et al. (2022) and also manually composed chains of thought. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Results. Figure 7 highlights these results for PaLM (full results for LaMDA, GPT-3, and different model scales are shown in Table 4). For all tasks, scaling up model size improved the performance of standard prompting; chain-of-thought prompting led to further gains, with improvements appearing to be largest for PaLM 540B. With chain-of-thought prompting, PaLM 540B achieved strong performance relative to baselines, outperforming the prior state of the art on StrategyQA $7 5 . 6 \\%$ vs $6 9 . 4 \\%$ and outperforming an unaided sports enthusiast on sports understanding $9 5 . 4 \\%$ vs $84 \\%$ ). These results demonstrate that chain-of-thought prompting can also improve performance on tasks requiring a range of commonsense reasoning abilities (though note that gain was minimal on CSQA). ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/314d6a9f471c937800bcc79eed382244c146f9aacc68828ea5e504cb8128e362.jpg",
+        "image_caption": [
+            "Figure 7: Chain-of-thought prompting also improves the commonsense reasoning abilities of language models. The language model shown here is PaLM. Prior best numbers are from the leaderboards of CSQA (Talmor et al., 2019) and StrategyQA (Geva et al., 2021) (single-model only, as of May 5, 2022). Additional results using various sizes of LaMDA, GPT-3, and PaLM are shown in Table 4. "
+        ],
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+    {
+        "type": "text",
+        "text": "5 Symbolic Reasoning ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our final experimental evaluation considers symbolic reasoning, which is simple for humans but potentially challenging for language models. We show that chain-ofthought prompting not only enables language models to perform symbolic reasoning tasks that are challenging in the standard prompting setting, but also facilitates length generalization to inference-time inputs longer than those seen in the few-shot exemplars. ",
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+    {
+        "type": "text",
+        "text": "Tasks. We use the following two toy tasks. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• Last letter concatenation. This task asks the model to concatenate the last letters of words in a name (e.g., “Amy Brown” $ ^ { \\ast } y n ^ { \\prime \\prime }$ ). It is a more challenging version of first letter concatenation, which language models can already perform without chain of thought.3 We generate full names by randomly concatenating names from the top one-thousand first and last names from name census data (https://namecensus.com/). ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/e4f8153620820f36adc177bd4972c7b26a10341d1dec25b01e73852fd72bc7ae.jpg",
+        "image_caption": [
+            "Figure 8: Using chain-of-thought prompting facilitates generalization to longer sequences in two symbolic reasoning tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• Coin flip. This task asks the model to answer whether a coin is still heads up after people either flip or don’t flip the coin (e.g., “A coin is heads up. Phoebe flips the coin. Osvaldo does not flip the coin. Is the coin still heads up? $^ { \\prime \\prime } \\right. ^ { \\left. } n o ^ { \\prime \\prime }$ ). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As the construction of these symbolic reasoning tasks is well-defined, for each task we consider an in-domain test set for which examples had the same number of steps as the training/few-shot exemplars, as well as an out-of-domain (OOD) test set, for which evaluation examples had more steps than those in the exemplars. For last letter concatenation, the model only sees exemplars of names with two words, and then performs last letter concatenation on names with 3 and 4 words.4 We do the same for the number of potential flips in the coin flip task. Our experimental setup uses the same methods and models as in the prior two sections. We again manually compose chains of thought for the few-shot exemplars for each task, which are given in Figure 3. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Results. The results of these in-domain and OOD evaluations are shown in Figure 8 for PaLM, with results for LaMDA shown in Appendix Table 5. With PaLM 540B, chain-of-thought prompting leads to almost $100 \\%$ solve rates (note that standard prompting already solves coin flip with PaLM 540, though not for LaMDA 137B). Note that these in-domain evaluations are “toy tasks” in the sense that perfect solution structures are already provided by the chains of thought in the few-shot exemplars; all the model has to do is repeat the same steps with the new symbols in the test-time example. And yet, small models still fail—the ability to perform abstract manipulations on unseen symbols for these three tasks only arises at the scale of 100B model parameters. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As for the OOD evaluations, standard prompting fails for both tasks. With chain-of-thought prompting, language models achieve upward scaling curves (though performance is lower than in the in-domain setting). Hence, chain-of-thought prompting facilitates length generalization beyond seen chains of thought for language models of sufficient scale. ",
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+    {
+        "type": "text",
+        "text": "6 Discussion ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We have explored chain-of-thought prompting as a simple mechanism for eliciting multi-step reasoning behavior in large language models. We first saw that chain-of-thought prompting improves performance by a large margin on arithmetic reasoning, yielding improvements that are much stronger than ablations and robust to different annotators, exemplars, and language models (Section 3). Next, experiments on commonsense reasoning underscored how the linguistic nature of chain-of-thought reasoning makes it generally applicable (Section 4). Finally, we showed that for symbolic reasoning, chain-of-thought prompting facilitates OOD generalization to longer sequence lengths (Section 5). In all experiments, chain-of-thought reasoning is elicited simply by prompting an off-the-shelf language model. No language models were finetuned in the process of writing this paper. ",
+        "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": "The emergence of chain-of-thought reasoning as a result of model scale has been a prevailing theme (Wei et al., 2022b). For many reasoning tasks where standard prompting has a flat scaling curve, chainof-thought prompting leads to dramatically increasing scaling curves. Chain-of-thought prompting appears to expand the set of tasks that large language models can perform successfully—in other words, our work underscores that standard prompting only provides a lower bound on the capabilities of large language models. This observation likely raises more questions than it answers—for instance, how much more can we expect reasoning ability to improve with a further increase in model scale? What other prompting methods might expand the range of tasks that language models can solve? ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "As for limitations, we first qualify that although chain of thought emulates the thought processes of human reasoners, this does not answer whether the neural network is actually “reasoning,” which we leave as an open question. Second, although the cost of manually augmenting exemplars with chains of thought is minimal in the few-shot setting, such annotation costs could be prohibitive for finetuning (though this could potentially be surmounted with synthetic data generation, or zero-shot generalization). Third, there is no guarantee of correct reasoning paths, which can lead to both correct and incorrect answers; improving factual generations of language models is an open direction for future work (Rashkin et al., 2021; Ye and Durrett, 2022; Wiegreffe et al., 2022, inter alia). Finally, the emergence of chain-of-thought reasoning only at large model scales makes it costly to serve in real-world applications; further research could explore how to induce reasoning in smaller models. ",
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+    {
+        "type": "text",
+        "text": "7 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This work is inspired by many research areas, which we detail in an extended related work section (Appendix C). Here we describe two directions and associated papers that are perhaps most relevant. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "The first relevant direction is using intermediate steps to solve reasoning problems. Ling et al. (2017) pioneer the idea of using natural language rationales to solve math word problems through a series of intermediate steps. Their work is a remarkable contrast to the literature using formal languages to reason (Roy et al., 2015; Chiang and Chen, 2019; Amini et al., 2019; Chen et al., 2019). Cobbe et al. (2021) extend Ling et al. (2017) by creating a larger dataset and using it to finetune a pretrained language model rather than training a model from scratch. In the domain of program synthesis, Nye et al. (2021) leverage language models to predict the final outputs of Python programs via first line-to-line predicting the intermediate computational results, and show that their step-by-step prediction method performs better than directly predicting the final outputs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Naturally, this paper also relates closely to the large body of recent work on prompting. Since the popularization of few-shot prompting as given by Brown et al. (2020), several general approaches have improved the prompting ability of models, such as automatically learning prompts (Lester et al., 2021) or giving models instructions describing a task (Wei et al., 2022a; Sanh et al., 2022; Ouyang et al., 2022). Whereas these approaches improve or augment the input part of the prompt (e.g., instructions that are prepended to inputs), our work takes the orthogonal direction of augmenting the outputs of language models with a chain of thought. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "8 Conclusions ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have explored chain-of-thought prompting as a simple and broadly applicable method for enhancing reasoning in language models. Through experiments on arithmetic, symbolic, and commonsense reasoning, we find that chain-of-thought reasoning is an emergent property of model scale that allows sufficiently large language models to perform reasoning tasks that otherwise have flat scaling curves. Broadening the range of reasoning tasks that language models can perform will hopefully inspire further work on language-based approaches to reasoning. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We thank Jacob Devlin, Claire Cui, Andrew Dai, and Ellie Pavlick for providing feedback on the paper. We thank Jacob Austin, Yuhuai Wu, Henryk Michalewski, Aitor Lewkowycz, Charles Sutton, and Aakanksha Chowdhery for helpful discussions. We thank Sid Maxwell for notifying us about a mistake in the manual error analysis in the original manuscript. ",
+        "bbox": [
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+        "page_idx": 9
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+    {
+        "type": "text",
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parse/dev/ldRb12nMfLQ/ldRb12nMfLQ.md

@@ -0,0 +1,228 @@
+# A DEEP CONJUGATE DIRECTION METHOD FOR ITERATIVELY SOLVING LINEAR SYSTEMS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for partial differential equations. Algorithms for approximating the solution to these systems are often the bottleneck in problems that require their solution, particularly for modern applications that require many millions of unknowns. Indeed, numerical linear algebra techniques have been investigated for many decades to alleviate this computational burden. Recently, data-driven techniques have also shown promise for these problems. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search directions. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We train the network using unsupervised learning with a loss function equal to the $L ^ { 2 }$ difference between an input and the system matrix times the network evaluation, where the unspecified constant in the approximate inverse is accounted for. We demonstrate the efficacy of our approach on spatially discretized Poisson equations with millions of degrees of freedom arising in computational fluid dynamics applications. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training.
+
+# 1 INTRODUCTION
+
+In this work, we consider sparse linear systems that arise from discrete Poisson equations in incompressible flow applications (Chorin, 1967; Fedkiw et al., 2001; Bridson, 2008). We use the notation
+
+$$
+\mathbf { A } { \boldsymbol { \mathbf { \mathit { x } } } } = \mathbf { \mathit { b } }
+$$
+
+where the dimension $n$ of the matrix $A \in \mathbb { R } ^ { n \times n }$ and the vector $b \in \mathbb { R } ^ { n }$ correlate with spatial fidelity of the computational domain. The appropriate numerical linear algebra technique depends on the nature of the problem. Direct solvers that utilize matrix factorizations (QR, Cholesky, etc. Trefethen & Bau (1997)) have optimal approximation error, but their computational cost is $\operatorname { \dot { O } } ( n ^ { 3 } )$ and they typically require dense storage, even for sparse $\pmb { A }$ . Although Fast Fourier Transforms (Nussbaumer, 1981) can be used in limited instances (periodic boundary conditions, etc.), iterative techniques are most commonly adopted for these systems given their sparsity. Many applications with strict performance constraints (e.g., real-time fluid simulation) utilize basic iterations (Jacobi, Gauss-Seidel, successive over relaxation (SOR), etc.) given limited computational budget (Saad, 2003). However, large approximation errors must be tolerated since iteration counts are limited by the performance constraints. This is particularly problematic since the wide elliptic spectrum of these matrices (a condition that worsens with increased spatial fidelity/matrix dimension) leads to poor conditioning and iteration counts. Iterative techniques can achieve sub-quadratic convergence if their iteration count does not grow excessively with problem size $n$ since each iteration generally requires $O ( n )$ floating point operations for sparse matrices. Discrete elliptic operators are typically symmetric positive (semi) definite and the preconditioned conjugate gradients method (PCG) can be used to minimize iteration counts (Saad, 2003; Hestenes & Stiefel, 1952; Stiefel, 1952). Preconditioners $_ { r }$ for PCG must simultaneously: be symmetric positive definite (SPD) (and therefore admit factorization ${ \boldsymbol { P } } = { \boldsymbol { F } } ^ { 2 }$ ), improve the condition number of the preconditioned system $F A F y = F b$ , and be computationally cheap to construct and apply; accordingly, designing specialized preconditioners for particular classes of problems is somewhat of an art. Incomplete Cholesky preconditioners (ICPCG) (Kershaw, 1978) use a sparse approximation to the Cholesky factorization and significantly reduce iteration counts in practice; however, their inherent data dependency prevents efficient parallel implementation. Nonetheless, these are very commonly adopted for Poisson equations arising in incompressible flow (Fedkiw et al., 2001; Bridson, 2008). Multigrid (Brandt, 1977) and domain decomposition (Saad, 2003) preconditioners greatly reduce iterations counts, but they must be updated (with non-trivial cost) each time the problem changes (e.g., in computational domains with time varying boundaries) and/or for different hardware platforms. In general, choice of an optimal preconditioner for discrete elliptic operators is an open area of research.
+
+Recently, data-driven approaches that leverage deep learning techniques have shown promise for solving linear systems. Various researchers have investigated machine learning estimation of multigrid parameters (Greenfeld et al., 2019; Grebhahn et al., 2016; Luz et al., 2020). Others have developed machine learning methods to estimate preconditioners (Gotz & Anzt, 2018; Stanaityte, 2020; ¨ Ichimura et al., 2020) and initial guesses for iterative methods (Luna et al., 2021; Um et al., 2020; Ackmann et al., 2020). Tompson et al. (2017) and Yang et al. (2016) develop non-iterative machine learning approximations of the inverse of discrete Poisson equations from incompressible flow. We leverage deep learning and develop a novel version of conjugate gradients iterative method for approximating the solution of SPD linear systems which we call the deep conjugate direction method (DCDM). CG iteratively adds $\pmb { A }$ -conjugate search directions while minimizing the matrix norm of the error. We use a convolutional neural network (CNN) as an approximation of the inverse of the matrix in order to generate more efficient search directions. We only ask that our network approximate the inverse up to an unknown scaling since this decreases the degree of nonlinearity and since it does not affect the quality of the search direction (which is scale independent). The network is similar to a preconditioner, but it is not a linear function, and our modified conjugate gradients approach is designed to accommodate this nonlinearity. We use unsupervised learning to train our network with a loss function equal to the $L ^ { 2 }$ difference between an input vector and a scaling of $\pmb { A }$ times the output of our network. To account for this unknown scaling during training, we choose the scale of the output of the network by minimizing the matrix norm of the error. Our approach allows for efficient training and generalization to problems unseen (matrices $\pmb { A }$ and right-hand sides $^ { b }$ ). We benchmark our algorithm using the ubiquitous pressure Poisson equation (discretized on regular voxelized domains) and compare against FluidNet (Tompson et al., 2017), which is the state of the art learning-based method for these types of problems. Unlike the non-iterative approaches of Tompson et al. (2017) and Yang et al. (2016), our method can reduce the linear system residuals arbitrarily. We showcase our approach with examples that have over 16 million degrees of freedom.
+
+# 2 RELATED WORK
+
+Several papers have focused on enhancing the solution of linear systems (arising from discretized PDEs) using learning. For instance, Gotz & Anzt (2018) generate sparsity patterns for block-Jacobi ¨ preconditioners using convolutional neural networks, and Stanaityte (2020) use a CNN to predict non-zero patterns for ILU-type preconditioners for the Navier-Stokes equations (though neither work designs fundamentally new preconditioners). Ichimura et al. (2020) develop a neural-network based preconditioner where the network is used to predict approximate Green’s functions (which arise in the analytical solution of certain PDEs) that in turn yield an approximate inverse of the linear system. Hsieh et al. (2019) learn an iterator that solves linear systems, performing competitively with classical solvers like multigrid-preconditioned MINRES (Paige & Saunders, 1975). Luz et al. (2020) and Greenfeld et al. (2019) use machine learning to estimate algebraic multigrid (AMG) parameters. They note that AMG approaches rely most fundamentally on effectively chosen (problem-dependent) prolongation sparse matrices and that numerous methods have attempted to automatically create them from the matrix $\pmb { A }$ . They train a graph neural network to learn (in an unsupervised fashion) a mapping from matrices $\pmb { A }$ to prolongation operators. Grebhahn et al. (2016) note that geometric multigrid solver parameters can be difficult to choose to guarantee parallel performance on different hardware platforms. They use machine learning to create a code generator to help achieve this.
+
+![](images/a7ac602c2643cff48deff90a7224438f208d4a7100d53c20e1e19240e8cade2b.jpg)  
+Figure 1: (a) We illustrate a sample flow domain $\Omega \subset ( 0 , 1 ) ^ { 2 }$ (in 2D for ease of illustration) with internal boundaries (blue lines). (b) We voxelize the domain with a regular grid: white cells represent interior/fluid, and blue cells represent boundary conditions. (c) We train using matrix $A ^ { ( 0 , 1 ) ^ { d } }$ from a discretized domain with no interior boundary conditions, where $d$ is the dimension. This creates linear system with $n = ( n _ { c } + 1 ) ^ { d }$ unknowns where $n _ { c }$ is the number of grid cells on each direction. (d) We illustrate the non-zero entries in an example matrix $A ^ { \Omega }$ from the voxelized and labeled (white vs. blue) grid for three example interior cells (green, purple, and brown). Each case illustrates the non-zero entries in the row associated with the example cell. All entries in rows corresponding to boundary/blue cells are zero.
+
+Several works consider accelerating the solution of linear systems by learning an initial guess that is close to the true solution or otherwise helpful to descent algorithms for finding the true solution. In order to solve the discretized Poisson equation, Luna et al. (2021) accelerate the convergence of GMRES (Saad & Schultz, 1986) with an initial guess that is learned in real-time (i.e., as a simulation code runs) with no prior data. Um et al. (2020) train a network (incorporating differentiable physics, based on the underlying PDEs) in order to produce high-quality initial guesses for a CG solver. In a somewhat similar vein, Ackmann et al. (2020) use a simple feedforward neural network to predict pointwise solution components, which accelerates the conjugate residual method used to solve a relatively simple shallow-water model (a more sophisticated network and loss function are needed to handle more general PDEs and larger-scale problems).
+
+At least two papers (Ruelmann et al., 2018; Sappl et al., 2019) have sought to learn a mapping between a matrix and an associated sparse approximate inverse. In their investigation, Ruelmann et al. (2018) propose training a neural network using matrix-inverse pairs as training data. Although straightforward to implement, the cost of generating training data, let alone training the network, is prohibitive for large-scale 3D problems. Sappl et al. (2019) seek to learn a mapping between linear system matrices and sparse (banded) approximate inverses. Their loss function is the condition number of the product of the system matrix and the approximate inverse; the minimum value of the condition number is one, which is achieved exactly when an exact inverse is obtained. Although this framework is quite simple, evaluating the condition number of a matrix is asymptotically costly, and in general, the inverse of a sparse matrix can be quite dense. Accordingly, the method is not efficient or accurate enough for the large-scale 3D problems that arise in real-world engineering problems.
+
+Most relevant to the present work is FluidNet (Tompson et al., 2017). FluidNet develops a highlytailored CNN architecture that is used to predict the solution of a linear projection operation (specifically, for the discrete Poisson equation) given a matrix and right-hand side. The authors demonstrate fluid simulations where the linear solve is replaced by evaluating their network. Because their network is relatively lightweight and is only evaluated once per time step, their simulations run efficiently. However, their design allows the network only one opportunity to reduce the residual for the linear solve; in practice, we observe that FluidNet is able to reduce the residual by no more than about one order of magnitude. However, in computer graphics applications, at least four orders of magnitude in residual reduction are usually required for visual fidelity, while in scientific and engineering applications, practitioners prefer solutions that reduce the residual by eight or more orders of magnitude (i.e., to within machine precision). Accordingly, FluidNet’s lack of convergence stands in stark contrast to classical, convergent methods like CG. Our method resolves this gap.
+
+# 3 MOTIVATION: INCOMPRESSIBLE FLOW
+
+We demonstrate the efficacy of our approach with the linear systems that arise in incompressible flow applications. Specifically, we use our algorithm to solve the discrete Poisson equations in
+
+regular-grid-based discretization of the pressure projection equations that arise in Chorin’s splitting technique (Chorin, 1967) for the inviscid, incompressible Euler equations. These equations are
+
+$$
+\rho \left( \frac { \partial \pmb { u } } { \partial t } + \frac { \partial \pmb { u } } { \partial \pmb { x } } \pmb { u } \right) + \nabla p = \pmb { f } ^ { e x t } , \qquad \nabla \cdot \pmb { u } = 0
+$$
+
+where $\textbf { \em u }$ is fluid velocity, $p$ is pressure, $\rho$ is density, and $f ^ { e x t }$ accounts for external forces like gravity. The equations are assumed at all positions $_ { \textbf { \em x } }$ in the spatial fluid flow domain $\Omega$ and for time $t > 0$ . The left term in Equation 2 enforces conservation of momentum in the absence of viscosity, and the second part enforces incompressibility and conservation of mass. These equations are subject to initial conditions $\rho ( \pmb { x } , 0 ) = \rho ^ { 0 }$ and ${ \pmb u } ( { \pmb x } , 0 ) = { \pmb u } ^ { 0 } ( { \pmb x } )$ as well as boundary conditions ${ \pmb u } ( { \pmb x } , t ) \cdot { \pmb n } ( { \pmb x } ) = u ^ { \partial \Omega } ( { \pmb x } , t )$ on the boundary of the domain ${ \pmb x } \in \partial \Omega$ (where $\textbf { \em n }$ is the unit outward pointing normal at position $_ { \textbf { \em x } }$ on the boundary). Equation 2 is discretized in both time and space. Temporally, we split the advection $\frac { \partial \pmb { u } } { \partial t } + \frac { \partial \pmb { u } } { \partial \pmb { x } } \pmb { u } = 0$ and body forces terms $\rho \frac { \partial \pmb { u } } { \partial t } = \pmb { f } ^ { e x t }$ , and finally enforce incompressibility via the pressure projection $\frac { \partial \pmb { u } } { \partial t } + \frac { 1 } { \rho } \nabla p = \pmb { 0 }$ such that $\nabla \cdot \pmb { u } = 0$ ; this is the standard advection-projection scheme proposed by Chorin (1967). Using finite differences in time, we can summarize this as
+
+$$
+\begin{array} { l } { \displaystyle \rho ^ { 0 } \left( \frac { \pmb { u } ^ { * } - \pmb { u } ^ { n } } { \Delta t } + \frac { \partial \pmb { u } ^ { n } } { \partial \pmb { x } } \pmb { u } ^ { n } \right) = \pmb { f } ^ { e x t } } \\ { \displaystyle - \nabla \cdot \frac { 1 } { \rho ^ { 0 } } \nabla p ^ { n + 1 } = - \nabla \cdot \pmb { u } ^ { * } } \\ { \displaystyle \qquad - \frac { 1 } { \rho ^ { 0 } } \nabla p ^ { n + 1 } \cdot \pmb { n } = \frac { 1 } { \Delta t } \left( \pmb { u } ^ { \partial \Omega } - \pmb { u } ^ { * } \cdot \pmb { n } \right) . } \end{array}
+$$
+
+For the spatial discretization, we use a regular marker-and-cell (MAC) grid (Harlow & Welch, 1965) with cubic voxels whereby velocity components are stored on the face of voxel cells, and scalar quantities (e.g., pressure $p$ or density $\rho \mathrm { \hbar }$ ) are stored at voxel centers. We use backward semiLagrangian advection (Fedkiw et al., 2001; Gagniere et al., 2020) for Equation 3. All spatial partial derivatives are approximated using finite differences. Equations 4 and 5 describe the pressure Poisson equation with Neumann conditions on the boundary of the flow domain. We discretize the left hand side of Equation 4 using a standard 7-point finite difference stencil. The right-hand side is discretized using the MAC grid discrete divergence finite difference stencils as well as contributions from the boundary condition terms in Equation 5. We refer the reader to Bridson (2008) for more in-depth implementation details. Equation 5 is discretized by modifying the Poisson stencil to enforce Neumann boundary conditions. We do this using a simple labeling of the voxels in the domain. For simplicity, we assume $\Omega \subset ( 0 , 1 ) ^ { 3 }$ is a subset of the unit cube, potentially with internal boundaries. We label cells in the domain as either liquid or boundary. This simple classification is enough to define the Poisson discretizations (with appropriate Neumann boundary conditions at domain boundaries) that we focus on in the present work; we illustrate the details in Figure 1. We use the following notation to denote the discrete Poisson equations associated with Equations 4–5:
+
+$$
+\begin{array} { r } { { \cal A } ^ { \Omega } { \boldsymbol x } = { \boldsymbol b } ^ { \nabla \cdot { \boldsymbol u } ^ { * } } + { \boldsymbol b } ^ { u ^ { \partial \Omega } } , } \end{array}
+$$
+
+where $A ^ { \Omega }$ is the discrete Poisson matrix associated with the voxelized domain, $_ { \textbf { \em x } }$ is the vector of unknown pressure, and $\mathbf { \delta } _ { b } \nabla \cdot \mathbf { u } ^ { * }$ and b u ∂ Ω are the right-hand side terms from Equations 4 and 5, respectively. We define a special case of the matrix involved in this discretization to be the Poisson matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ associated with $\Omega = ( 0 , 1 ) ^ { 3 }$ , i.e., a full fluid domain with no internal boundaries. We use this matrix for training, yet demonstrate that our network generalizes to all other matrices arising from more complicated flow domains.
+
+# 4 DEEP CONJUGATE DIRECTION METHOD
+
+We present our method for the deep learning acceleration of iterative approximations to the solution of linear systems of the form seen in Equation 6. We first discuss relevant details of the conjugate gradients (CG) method, particularly line search and $\pmb { A }$ -orthogonal search directions. We then present a deep learning technique for improving the quality of these search directions that ultimately reduces iteration counts required to achieve satisfactory residual reduction. Lastly, we outline the training procedures for our deep convolutional neural network.
+
+Our approach iteratively improves approximations to the solution $_ { \textbf { \em x } }$ of Equation 6. We build on the method of CG, which requires the matrix $A ^ { \Omega }$ in Equation 6 to be SPD. SPD matrices $A ^ { \Omega }$ give rise to the matrix norm $\| \pmb { y } \| _ { A ^ { \Omega } } = \sqrt { \pmb { y } ^ { T } A ^ { \Omega } \pmb { y } }$ . CG can be derived in terms of iterative line search improvement based on optimality in this norm. That is, an iterate $\pmb { x } _ { k - 1 } \approx \pmb { x }$ is updated in search direction $\scriptstyle d _ { k }$ by a step size $\alpha _ { k }$ that is chosen to minimize the matrix norm of the error:
+
+$$
+\alpha _ { k } = \underset { \alpha } { \arg \operatorname* { m i n } } \frac { 1 } { 2 } \left\| \pmb { x } - ( \pmb { x } _ { k - 1 } + \alpha \pmb { d } _ { k } ) \right\| _ { \pmb { A } ^ { \Omega } } ^ { 2 } = \frac { { \pmb { r } } _ { k - 1 } ^ { T } { \pmb { d } _ { k } } } { { \pmb { d } } _ { k } ^ { T } { \pmb { A } } ^ { \Omega } { \pmb { d } _ { k } } } ,
+$$
+
+where $\pmb { r } _ { k - 1 } = \pmb { b } - \pmb { A } ^ { \Omega } \pmb { x } _ { k - 1 }$ is the $( k - 1 ) ^ { \mathrm { t h } }$ residual and $^ { b }$ is the right-hand side in Equation 6. Different search directions result in different algorithms. A natural choice is the negative gradient of the matrix norm of the error (evaluated at the current iterate), since this will point in the direction of steepest decrease $\begin{array} { r } { \pmb { d } _ { k } = - \frac { 1 } { 2 } \nabla \left\| \pmb { x } _ { k - 1 } \right\| _ { \pmb { A } ^ { \Omega } } ^ { 2 } = \pmb { r } _ { k - 1 } } \end{array}$ . This is the gradient descent method (GD). Unfortunately, this approach requires many iterations in practice. A more effective strategy is to choose directions that are $\pmb { A }$ -orthogonal (i.e., ${ \bf \Phi } _ { { \bf i } } ^ { T } { \bf A } ^ { \Omega } { \bf d } _ { j } = \mathrm { ~  ~ \dot { ~ } { ~ 0 ~ } ~ }$ for $i \neq j$ ). With this choice, the search directions form a basis for $\mathbb { R } ^ { n }$ so that the initial error can be written as $\begin{array} { r } { \pmb { e } _ { 0 } = \pmb { x } - \pmb { x } _ { 0 } = \sum _ { i = 1 } ^ { n } e _ { i } \pmb { d } _ { i } } \end{array}$ , where $e _ { i }$ are the components of the initial error written in the basis. Furthermore, when the search directions are $\pmb { A }$ -orthogonal, the optimal step sizes $\alpha _ { k }$ at each iteration satisfy
+
+$$
+\alpha _ { k } = \frac { r _ { k - 1 } ^ { T } d _ { k } } { d _ { k } ^ { T } A ^ { \Omega } d _ { k } } = \frac { d _ { k } ^ { T } A ^ { \Omega } e _ { k - 1 } } { d _ { k } ^ { T } A ^ { \Omega } d _ { k } } = \frac { d _ { k } ^ { T } A ^ { \Omega } \left( \sum _ { i = 1 } ^ { n } e _ { i } d _ { i } - \sum _ { j = 1 } ^ { k - 1 } \alpha _ { j } d _ { j } \right) } { d _ { k } ^ { T } A ^ { \Omega } d _ { k } } = e _ { k } .
+$$
+
+That is, the optimal step sizes are chosen to precisely eliminate the components of the error on the basis defined by the search directions. Thus, convergence is determined by the (at most $n$ ) non-zero components $e _ { i }$ in the initial error. Although rounding errors prevent this from happening exactly in practice, this property greatly reduces the number of required iterations (Golub & Loan, 2012).
+
+CG can be viewed as a modification of GD where the search direction is chosen as the component of the residual (equivalently, the negative gradient of the matrix norm of the error) that is $\pmb { A }$ -orthogonal to all previous search directions:
+
+$$
+\pmb { d } _ { k } = \pmb { r } _ { k - 1 } - \sum _ { i = 1 } ^ { k - 1 } h _ { i k } \pmb { d } _ { i } , \qquad h _ { i k } = \frac { { d } _ { i } ^ { T } { \pmb { A } } ^ { \Omega } \pmb { r } _ { k - 1 } } { { d } _ { i } ^ { T } { \pmb { A } } ^ { \Omega } \pmb { d } _ { i } } .
+$$
+
+In practice, $h _ { i k } = 0$ for $i < k - 1$ , and this iteration can therefore be performed without the need to store all previous search directions $\mathbf { \mathbf { { \alpha } } } d _ { i }$ and without the need for computing all previous $h _ { i k }$ .
+
+While the residual is a natural choice for generating $\pmb { A }$ -orthogonal search directions (since it points in the direction of the steepest local decrease), it is not the optimal search direction. If $\scriptstyle d _ { k }$ is parallel to $( A ^ { \Omega } ) ^ { - 1 } r _ { k - 1 }$ , then $\scriptstyle { \mathbf { { \mathit { x } } } } _ { k }$ will be equal to $_ { \textbf { \em x } }$ since $\alpha _ { k }$ (computed from Equation 7) will step directly to the solution. We can see this by considering the residual and its relation to the search direction:
+
+$$
+\begin{array} { r } { r _ { k } = b - A ^ { \Omega } x _ { k } = b - A ^ { \Omega } x _ { k - 1 } - \alpha _ { k } A ^ { \Omega } d _ { k } = r _ { k - 1 } - \alpha _ { k } A ^ { \Omega } d _ { k } . } \end{array}
+$$
+
+In light of this, we use deep learning to create an approximation $f ( c , r )$ to $( A ^ { \Omega } ) ^ { - 1 } r$ , where $^ c$ denotes the network weights and biases. This is analogous to using a preconditioner in PCG; however, our network is not SPD (nor even a linear function). We simply use this data-driven approach as our means of generating better search directions $\scriptstyle d _ { k }$ . Furthermore, we only need to approximate a vector parallel to $( A ^ { \Omega } ) ^ { - 1 } r$ since the step size $\alpha _ { k }$ will account for any scaling in practice. In other words, $f ( c , r ) \approx s _ { r } ( A ^ { \Omega } ) ^ { - 1 } r$ , where the scalar $s _ { r }$ is not defined globally; it only depends on $\pmb { r }$ , and the model does not learn it. Lastly, as with CG, we enforce $\pmb { A }$ -orthogonality, yielding search directions
+
+$$
+\pmb { d } _ { k } = \pmb { f } ( \pmb { c } , \pmb { r } _ { k - 1 } ) - \sum _ { i = 1 } ^ { k - 1 } h _ { i k } \pmb { d } _ { i } , \qquad h _ { i k } = \frac { \pmb { f } ( \pmb { c } , \pmb { r } _ { k - 1 } ) ^ { T } \pmb { A } ^ { \Omega } \pmb { d } _ { i } } { \pmb { d } _ { i } ^ { T } \pmb { A } ^ { \Omega } \pmb { d } _ { i } } .
+$$
+
+We summarize our approach in Algorithm 1. Note that we introduce the variable $i _ { \mathrm { s t a r t } }$ . To guarantee $\pmb { A }$ -orthogonality between all search directions, we must have $i _ { \mathrm { s t a r t } } = 1$ . However, this requires storing all prior search directions, which can be costly. We found that using $i _ { \mathrm { s t a r t } } = k - 2$ worked nearly as well as $i _ { \mathrm { s t a r t } } = 1$ in practice (in terms of our ability to iteratively reduce the residual of the system). We demonstrate this in Figure 4c.
+
+# 5 MODEL ARCHITECTURE, DATASETS, AND TRAINING
+
+Efficient performance of our method requires effective training of our deep convolutional network for weights and biases $^ c$ such that $f ( c , r ) { \overset { \cdot } { \approx } } s _ { r } ( A ^ { \Omega } ) ^ { - 1 } r$ (for arbitrary scalar $s _ { r }$ ). We design a model architecture, loss function, and unsupervised training approach to achieve this. Our approach has modest training requirements and allows for effective residual reduction while generalizing well to problems not seen in the training data.
+
+# 5.1 LOSS FUNCTION AND UNSUPERVISED LEARNING
+
+Although we generalize to arbitrary matrices $A ^ { \Omega }$ from Equation 6 that correspond to domains $\Omega \subset ( 0 , 1 ) ^ { 3 }$ that have internal boundaries (see Figure 1), we train using just the matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ from the full cube domain $( 0 , 1 ) ^ { 3 }$ . In contrast, other similar approaches (Tompson et al., 2017; Yang et al., 2016) train using matrices $A ^ { \Omega }$ and right-hand sides $\pmb { b } ^ { \nabla \cdot \pmb { u } ^ { * } } + \overline { { \pmb { b } } } ^ { u ^ { \partial \Omega } }$ that arise from flow in many domains with internal boundaries. We train our network by minimiz
+
+# Algorithm 1 DCDM
+
+1: $\pmb { r } _ { 0 } = \pmb { b } - \pmb { A } ^ { \Omega } \pmb { x } _ { 0 }$   
+2: $k = 1$   
+3: while $\| r _ { k - 1 } \| \ge \epsilon$ do   
+4: dk = f (c, rk−1∥rk−1∥ )   
+5: for $i _ { \mathrm { s t a r t } } \le i < k$ do   
+6: hik = dT AΩdi dT AΩdi   
+7: dk-=hikdi   
+8: end for   
+9: $\begin{array} { r l } & { \alpha _ { k } = \frac { r _ { k - 1 } ^ { T } d _ { k } } { d _ { k } ^ { T } A ^ { \Omega } d _ { k } } } \\ & { x _ { k } = x _ { k - 1 } + \alpha _ { k } d _ { k } } \\ & { r _ { k } = b - A ^ { \Omega } x _ { k } } \\ & { k = k + 1 } \end{array}$   
+10:   
+11:   
+12:   
+13: end while
+
+ing the L2 difference ∥r−αA(0,1)3 f (c, r)∥2, where α = rT f(c,r)f(c,r)T A(0,1)3 f(c,r) f rom Equation 7. This choice of $\alpha$ accounts for the unknown scaling in the approximation of $f ( c , r )$ to $\left( A ^ { ( 0 , 1 ) ^ { 3 } } \right) ^ { - 1 } r$ . We use an unsupervised approach and train the model by minimizing
+
+$$
+\begin{array} { r } { \mathrm { L o s s } ( \pmb { f } , \pmb { c } , \mathscr { D } ) = \frac { 1 } { | \mathscr { D } | } \sum _ { r \in \mathscr { D } } \| \pmb { r } - \frac { r ^ { T } f ( \pmb { c } , r ) } { \pmb { f } ( \pmb { c } , r ) ^ { T } \pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \pmb { f } ( \pmb { c } , r ) } \pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \pmb { f } ( \pmb { c } , r ) \| _ { 2 } } \end{array}
+$$
+
+for given dataset $\mathcal { D }$ consisting of training vectors $b ^ { i }$ . In Algorithm 1, the normalized residuals $\frac { \boldsymbol { r } _ { k } } { \| \boldsymbol { r } _ { k } \| }$ are passed as inputs to the model. Unlike in e.g. FluidNet (Tompson et al., 2017), only the first residual $\frac { r _ { 0 } } { \Vert r _ { 0 } \Vert }$ is directly related to the problem-dependent original right-hand side $^ { b }$ . Hence we consider a broader range of training vectors than those expected in a given problem of interest, e.g., incompressible flows. We observe that generally the residuals $\mathbf { \nabla } r _ { k }$ in Algorithm 1 are skewed to the lower end of the spectrum of the matrix $A ^ { \Omega }$ . Since $A ^ { \Omega }$ is a discretized elliptic operator, lower end modes are of lower frequency of spatial oscillation. We create our training vectors $\mathbf { \boldsymbol { b } } ^ { i } \in \mathcal { D }$ using $m \ll n$ approximate eigenvectors of the training matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ . We use the Rayleigh-Ritz method to create approximate eigenvectors $\pmb q _ { i }$ , $0 \leq i < m$ . This approach allows us to effectively approximate the full spectrum of $A ^ { ( 0 , 1 ) ^ { 3 } }$ without computing the full eigendecomposition, which can be expensive $( { \cal O } ( \bar { n } ^ { 3 } ) )$ at high resolution. We found that using $m \ : = \ : 1 0 0 0 0$ worked well in practice. The Rayleigh-Ritz vectors are orthonormal and satisfy $Q _ { m } ^ { T } A ^ { ( 0 , 1 ) ^ { 3 } } Q _ { m } = \Lambda _ { m }$ , where $\pmb { \Lambda } _ { m }$ is a diagonal matrix with nondecreasing diagonal entries $\lambda _ { i }$ referred to as Ritz values (approximate eigenvalues) and $\pmb { Q } _ { m } = [ \pmb { q } _ { 0 } , \pmb { q } _ { 1 } , \dots , \pmb { q } _ { m - 1 } ] \in \mathbb { R } ^ { n \times m }$ . We pick $\begin{array} { r } { \pmb { b } ^ { i } = \frac { \sum _ { j = 0 } ^ { m - 1 } c _ { j } ^ { i } \pmb { q } _ { j } } { \left\| \sum _ { j = 0 } ^ { m - 1 } c _ { j } ^ { i } \pmb { q } _ { j } \right\| } } \end{array}$ , where the coefficients $c _ { j } ^ { i }$ are picked from a standard normal distribution
+
+$$
+c _ { j } ^ { i } = \left\{ { \begin{array} { l l } { 0 \cdot \mathcal { N } ( 0 , 1 ) } & { { \mathrm { i f ~ } } \tilde { j } \leq j \leq { \frac { m } { 2 } } + \theta } \\ { \mathcal { N } ( 0 , 1 ) } & { { \mathrm { o t h e r w i s e } } } \end{array} } \right.
+$$
+
+where $\theta$ is a small number (we used $\theta = 5 0 0$ ), and $\tilde { j }$ is the first index that $\lambda _ { \tilde { j } } ^ { - } > 0$ . This choice creates $9 0 \%$ of $\mathbf { \nabla } _ { b } i$ from the lower end of the spectrum, with the remaining $1 0 \%$ from the higher end. The Riemann-Lebesgue Lemma states the Fourier spectrum of a continuous function will decay at infinity, so this specific choice of $b _ { i }$ ’s is reasonable for the training set. In practice, we also observed that the right-hand sides of the pressure system that arose in flow problems (in the empty domain) tended to be at the lower end of the spectrum. Notably, even though this dataset only uses RayleighRitz vectors from the training matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ , our network can be effectively generalized to flows in irregular domains, e.g., smoke flow past a rotating box and flow past a bunny (see Figure 3).
+
+We generate the Rayleigh-Ritz vectors by first tridiagonalizing the training matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ with Lanczos iterations (Lanczos, 1950) to form $\pmb { T } ^ { m } = \pmb { Q } _ { m } ^ { L } ^ { T } \pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \pmb { Q } _ { m } ^ { L } \in \mathbb { R } ^ { m \times m }$ . We then diagonalize $\pmb { T } ^ { m } = \hat { \pmb { Q } } ^ { T } \pmb { \Lambda } _ { m } \hat { \pmb { Q } }$ . While costly, we note that this algorithm is performed on the comparably small $m \times m$ matrix $\mathbf { T } ^ { m }$ (rather than on the $A ^ { ( 0 , 1 ) ^ { 3 } } \in \mathbb { R } ^ { n \times n } ,$ ). This yields the Rayleigh-Ritz vectors as the columns of $Q _ { m } = Q _ { m } ^ { L } \hat { Q }$ . The Lanczos vectors are the columns of the matrix $Q _ { m } ^ { L }$ and satisfy a three-term recurrence whereby the next Lanczos vector can be computed from the previous two as
+
+$$
+\beta _ { j } \pmb { q } _ { j + 1 } ^ { L } = \pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \pmb { q } _ { j } ^ { L } - \beta _ { j - 1 } \pmb { q } _ { j - 1 } ^ { L } - \alpha _ { j } \pmb { q } _ { j } ^ { L } ,
+$$
+
+where $\alpha _ { j }$ and $\beta _ { j }$ are diagonal and subdiagonal entries of $\pmb { T } ^ { k }$ . $\beta _ { j }$ is computed so that $\pmb q _ { j + 1 } ^ { L }$ is a unit vector, and $\alpha _ { j + 1 } = \pmb { q } _ { j + 1 } ^ { T } \pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \pmb { q } _ { j + 1 }$ . We initialize the iteration with a random $\pmb { q } _ { 0 } ^ { L } \in \mathrm { s p a n } ( \pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } )$ . The Lanczos algorithm can be viewed as a modified Gram-Schmidt technique to create an orthonormal basis for the Krylov space associated with $\pmb q _ { 0 } ^ { L }$ and $A ^ { ( 0 , 1 ) ^ { 3 } }$ , and it therefore suffers from rounding error sensitivities manifested as loss of orthonormality with vectors that do not appear in the recurrence. We found that the simple strategy described in Paige (1971) of orthogonalizing each iterate with all previous Lanczos vectors to be sufficient for our training purposes. Dataset creation takes 5–7 hours for a $6 4 ^ { 3 }$ computational grid, and 2–2.5 days for a $\boldsymbol { 1 2 8 ^ { \overline { { 3 } } } }$ grid.
+
+# 5.2 MODEL ARCHITECTURE
+
+![](images/048b839895ab4dfb5d59e79b90cfaaf105f450ca0f80073d468f3f23646ad845.jpg)  
+Figure 2: Model architecture for training with $A ^ { ( 0 , 1 ) ^ { 3 } }$ on a $1 2 8 ^ { 3 }$ grid.
+
+The internal structure of our CNN architecture for a $1 2 8 ^ { 3 }$ grid is shown in Figure 2. It consists of a series of convolutional layers with residual connections. The upper left of Figure 2 ( $K$ Residual Blocks) shows our use of multiple blocks of residually connected layers. Notably, within each block, the first layer directly affects the last layer with an addition operator.All non-input or output convolutions use a $3 \times 3 \times 3$ filter, and all layers consist of 16 feature maps. In the middle of the first level, a layer is downsampled (via the average pool
+
+ing operator with $( 2 \times 2 \times 2 )$ pool size) and another set of convolutional layers is applied with residual connection blocks. The last layer in the second level is upscaled and added to the layer that is downsampled. The last layer in the network is dense with a linear activation function. The activation functions in all convolutional layers are ReLU, except for the first convolution, which uses a linear activation function.
+
+Initially we tried a simple deep feedforward convolutional network with residual connections (motivated by He et al. (2016)). Although such a simple model works well for DCDM, it requires high number of layers, which results in higher training and inference times. We found that creating parallel layers of CNNs with downsampling reduced the number of layers required. In summary, our goal was to first identify the simplest network architecture that provided adequate accuracy for our target problems, and subsequently, we sought to make architectural changes to minimize training and inference time; further optimizations are possible.
+
+Differing resolutions use differing numbers of convolutions, but the fundamental structure remains the same. More precisely, the number of residual connections is changed for different resolutions. For example, a $6 \dot { 4 } ^ { 3 }$ grid uses one residual block on the left, two on the right on the upper level, and three on the lower level. Furthermore, the weights trained on a lower resolution grid can be used effectively with higher resolutions. Figure 4d shows convergence results for a $2 \mathrm { { 5 6 ^ { 3 } } }$ grid, using a model trained for a $6 4 ^ { 3 }$ grid and a $1 2 \bar { 8 } ^ { 3 }$ grid. The model that we use for $2 5 6 ^ { 3 }$ grids in our final examples was trained on a $1 2 8 ^ { 3 }$ grid; however, as the shown in the figure, even training with a $6 4 ^ { 3 }$ grid allows for efficient residual reduction. Table 1 shows results for three different resolutions, where DCDM uses $6 4 ^ { 3 }$ and $1 2 8 ^ { 3 }$ trained models. This approach makes the number of parameters in the model independent of the spatial fidelity of the problem.
+
+# 5.3 TRAINING
+
+Using the procedure explained in Section 5.1, we create the training dataset ${ \mathcal { D } } \in \mathsf { s p a n } ( A ^ { ( 0 , 1 ) ^ { 3 } } ) \cap { \mathcal { S } } ^ { n - 1 }$ of size 20000 generated from 10000 Rayleigh-Ritz vectors. We train our model with TensorFlow (Abadi et al., 2015) on a single NVIDIA RTX A6000 GPU with 48GB memory. Training is done with standard deep learning techniques—more precisely, back-propagation and the ADAM optimizer (Kingma & Ba, 2015) (with starting learning rate 0.0001). Training takes approximately 10 minutes and 1 hour per epoch for grid resolutions $6 4 ^ { 3 }$ and $\mathrm { \dot { 1 } 2 8 ^ { 3 } }$ , respectively. We trained our model for 50 epochs; however, the model from the thirty
+
+![](images/32fa04638f7e6d3d367c6df56f430f093be0e2e68b4d13ed0903e506ea28d74a.jpg)  
+Figure 3: DCDM for simulating a variety of incompressible flow examples. Left: smoke plume at $t \ =$ 6.67, 13.33, 20 seconds. Middle: smoke passing bunny at $t = 5 , 1 0 , 1 5$ seconds. Right: smoke passing a spinning box (time-dependent Neumann boundary conditions) at $t = 2 . 6 7 , 6 , 9 . 3 3$ seconds.
+
+first epoch was optimal for $6 4 ^ { 3 }$ , and the model from the third epoch was optimal for $1 2 8 ^ { 3 }$
+
+# 6 RESULTS AND ANALYSIS
+
+We demonstrate DCDM on three increasingly difficult examples and provide numerical evidence for the efficient convergence of our method. All examples were run on a workstation with dual AMD EPYC 75F3 processors and 512GB RAM.
+
+Figure 3 showcases DCDM for incompressible smoke simulations. In each simulation, inlet boundary conditions are set in a circular portion of the bottom of the cubic domain, whereby smoke flows around potential obstacles and fills the domain. We show a smoke plume (no obstacles), flow past a complex static geometry (the Stanford bunny), and flow past a dynamic geometry (a rotating cube). Visually plausible and highly-detailed results are achieved for each simulation (see supplementary material for larger videos). The plume example uses a computational grid with resolution $1 2 8 ^ { 3 }$ , while the other two uses grids with resolution $2 5 6 ^ { 3 }$ (representing over 16 million unknowns). For each linear solve, DCDM was run until the residual was reduced by four orders of magnitude.
+
+Table 1: Timing and iteration comparison for different methods on the bunny example. DCDM- $\{ 6 4 , 1 2 8 \}$ calls a model whose parameters trained over a $\{ 6 4 ^ { 3 } , 1 2 8 ^ { 3 } \}$ grid. All computations are done using only CPUs; model inference does not use GPUs.
+
+For the bunny example, Figures 4a–b demonstrate how residuals decrease over the course of a linear solve, comparing DCDM with other methods. Figure 4a shows the mean results (with standard deviations) over the course of 400 simulation frames, while in Figure 4b, we illustrate behavior on a particular frame (frame 150). For
+
+<table><tr><td></td><td colspan="2">64 Grid</td><td colspan="2">1283 Grid</td><td colspan="2">2563 Grid</td></tr><tr><td>Method</td><td>tr</td><td>nr</td><td>tr</td><td>nr</td><td>tr</td><td>nr</td></tr><tr><td>DCDM-64</td><td>2.71s</td><td>16</td><td>22s</td><td>27</td><td>261s</td><td>58</td></tr><tr><td>DCDM-128</td><td>5.37s</td><td>19</td><td>26s</td><td>24</td><td>267s</td><td>44</td></tr><tr><td>CG</td><td>1.77s</td><td>168</td><td>26s</td><td>465</td><td>1548s</td><td>1046</td></tr><tr><td>Deflated PCG</td><td>771.6s</td><td>117</td><td>3700s</td><td>277</td><td>21030s</td><td>489</td></tr></table>
+
+FluidNet, we use the implementation provided by fluidnetsc22 (2022). This implementation includes pre-trained models that we use without modification. In both subfigures, it is evident that the FluidNet residual never changes, since the method is not iterative; FluidNet reduces the initial residual by no more than one order of magnitude. On the other hand, with DCDM, we can continually reduce the residual (e.g., by four orders of magnitude) as we apply more iterations of our method, just as with classical CG. In Figure 4b, we also visualize the convergence of three other classical methods, CG, Deflated PCG (Saad et al., 2000), and incomplete Cholesky preconditioned CG (ICPCG)); clearly, DCDM reduces the residual in the fewest number of iterations (e.g., approximately one order of magnitude fewer iterations than ICPCG). Since FluidNet is not an iterative method and lacks a notion of residual reduction, we treat $r _ { 0 }$ for FluidNet as though an initial guess of zero is used (as is done in our solver).
+
+![](images/49a3a4d6c185b571472c1686368f20fff99b702247e6716738f61bbe29294d57.jpg)  
+Figure 4: Convergence data for the bunny example. (a) Mean and std. dev. (over all 400 frames in the simulation) of residual reduction during linear solves (with $1 2 8 ^ { 3 }$ and $2 5 6 ^ { 3 }$ grids) using FluidNet (FN) and DCDM. (b) Residual plots with CG, ICPCG, Deflated PCG, FN, and DCDM at frame 150. Dashed and solid lines represent results for $1 2 8 ^ { 3 }$ and $2 5 6 ^ { 3 }$ , respectively. (c) Decrease in residuals with varying degrees of $\pmb { A }$ -orthogonalization $( i _ { s } ~ = ~ i _ { \mathrm { s t a r t } } )$ . (d) Reduction in residuals when the network is trained with a $6 4 ^ { 3 }$ or $1 \bar { 2 } 8 ^ { 3 }$ grid for the $2 5 6 ^ { 3 }$ grid simulation shown in Figure 3 Middle.
+
+To clarify these results, Table 1 reports convergence statistics for DCDM compared to standard iterative techniques CG and Deflated PCG. For all $6 4 ^ { 3 }$ , $1 2 8 ^ { 3 }$ , and $2 5 6 ^ { 3 }$ grids with the bunny example, we measure the time $t _ { r }$ and the number of iterations $n _ { r }$ required to reduce the initial residual on a particular time step of the simulation by four orders of magnitude. DCDM achieves the desired results in by far the fewest number of iterations at all resolutions. At $2 5 6 ^ { 3 }$ , DCDM performs approximately 6 times faster than CG, suggesting a potentially even wider performance advantage at higher resolutions. Inference is the dominant cost in an iteration of DCDM; the other linear algebra computations in an iteration of DCDM are comparable to those in CG. The nice result of our method is that despite the increased time per iteration, the number of required iterations is reduced so drastically that DCDM materially outperforms classical methods like CG. Although ICPCG successfully reduces number of iterations $^ { 4 \mathrm { ~ b ~ } }$ , we found the runtime to scale prohibitively with grid resolution, so we exclude it from comparison in table 1. Notably, even though Deflated PCG and DCDM are based on approximate Ritz vectors, DCDM performs far better, indicating the value of using a neural network.
+
+# 7 CONCLUSIONS
+
+We presented DCDM, incorporating CNNs into a CG-style algorithm that yields efficient, convergent behavior for solving linear systems. Our method is evaluated on linear systems with over 16 million degrees of freedom and converges to a desired tolerance in merely tens of iterations. Furthermore, despite training the underlying network on domains without obstacles, our network is able to successfully predict search directions that enable efficient linear solves on domains with complex and dynamic geometries. Moreover, the training data for our network does not require running fluid simulations or solving linear systems ahead of time; our Rayleigh-Ritz vector approach enables us to quickly generate very large training datasets, unlike approaches seen in other works.
+
+Our network was designed for and trained exclusively using data related to the discrete Poisson matrix, which likely limits the generalizability of our present model. However, we believe our method is readily applicable to other classes of PDEs (or general problems with graph structure) that give rise to large, sparse, symmetric linear systems. We note that our method is unlikely to work well for matrices that have high computational cost to evaluate $A * x$ (such as dense matrices), since training relies on efficient $\boldsymbol { A } * \boldsymbol { x }$ evaluations. An interesting question to consider is how well our method and current models would apply to discrete Poisson matrices arising from non-uniform grids, e.g., quadtrees or octrees (Losasso et al., 2004).
+
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+        "type": "text",
+        "text": "A DEEP CONJUGATE DIRECTION METHOD FOR ITERATIVELY SOLVING LINEAR SYSTEMS ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+        "type": "text",
+        "text": "ABSTRACT ",
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+        "text": "We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for partial differential equations. Algorithms for approximating the solution to these systems are often the bottleneck in problems that require their solution, particularly for modern applications that require many millions of unknowns. Indeed, numerical linear algebra techniques have been investigated for many decades to alleviate this computational burden. Recently, data-driven techniques have also shown promise for these problems. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search directions. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We train the network using unsupervised learning with a loss function equal to the $L ^ { 2 }$ difference between an input and the system matrix times the network evaluation, where the unspecified constant in the approximate inverse is accounted for. We demonstrate the efficacy of our approach on spatially discretized Poisson equations with millions of degrees of freedom arising in computational fluid dynamics applications. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we consider sparse linear systems that arise from discrete Poisson equations in incompressible flow applications (Chorin, 1967; Fedkiw et al., 2001; Bridson, 2008). We use the notation ",
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+    {
+        "type": "equation",
+        "img_path": "images/8fead0b4e25add8342bddaa4ba897dff29b2070568d204185e0f1c2ab5d7c00e.jpg",
+        "text": "$$\n\\mathbf { A } { \\boldsymbol { \\mathbf { \\mathit { x } } } } = \\mathbf { \\mathit { b } }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where the dimension $n$ of the matrix $A \\in \\mathbb { R } ^ { n \\times n }$ and the vector $b \\in \\mathbb { R } ^ { n }$ correlate with spatial fidelity of the computational domain. The appropriate numerical linear algebra technique depends on the nature of the problem. Direct solvers that utilize matrix factorizations (QR, Cholesky, etc. Trefethen & Bau (1997)) have optimal approximation error, but their computational cost is $\\operatorname { \\dot { O } } ( n ^ { 3 } )$ and they typically require dense storage, even for sparse $\\pmb { A }$ . Although Fast Fourier Transforms (Nussbaumer, 1981) can be used in limited instances (periodic boundary conditions, etc.), iterative techniques are most commonly adopted for these systems given their sparsity. Many applications with strict performance constraints (e.g., real-time fluid simulation) utilize basic iterations (Jacobi, Gauss-Seidel, successive over relaxation (SOR), etc.) given limited computational budget (Saad, 2003). However, large approximation errors must be tolerated since iteration counts are limited by the performance constraints. This is particularly problematic since the wide elliptic spectrum of these matrices (a condition that worsens with increased spatial fidelity/matrix dimension) leads to poor conditioning and iteration counts. Iterative techniques can achieve sub-quadratic convergence if their iteration count does not grow excessively with problem size $n$ since each iteration generally requires $O ( n )$ floating point operations for sparse matrices. Discrete elliptic operators are typically symmetric positive (semi) definite and the preconditioned conjugate gradients method (PCG) can be used to minimize iteration counts (Saad, 2003; Hestenes & Stiefel, 1952; Stiefel, 1952). Preconditioners $_ { r }$ for PCG must simultaneously: be symmetric positive definite (SPD) (and therefore admit factorization ${ \\boldsymbol { P } } = { \\boldsymbol { F } } ^ { 2 }$ ), improve the condition number of the preconditioned system $F A F y = F b$ , and be computationally cheap to construct and apply; accordingly, designing specialized preconditioners for particular classes of problems is somewhat of an art. Incomplete Cholesky preconditioners (ICPCG) (Kershaw, 1978) use a sparse approximation to the Cholesky factorization and significantly reduce iteration counts in practice; however, their inherent data dependency prevents efficient parallel implementation. Nonetheless, these are very commonly adopted for Poisson equations arising in incompressible flow (Fedkiw et al., 2001; Bridson, 2008). Multigrid (Brandt, 1977) and domain decomposition (Saad, 2003) preconditioners greatly reduce iterations counts, but they must be updated (with non-trivial cost) each time the problem changes (e.g., in computational domains with time varying boundaries) and/or for different hardware platforms. In general, choice of an optimal preconditioner for discrete elliptic operators is an open area of research. ",
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+        "type": "text",
+        "text": "Recently, data-driven approaches that leverage deep learning techniques have shown promise for solving linear systems. Various researchers have investigated machine learning estimation of multigrid parameters (Greenfeld et al., 2019; Grebhahn et al., 2016; Luz et al., 2020). Others have developed machine learning methods to estimate preconditioners (Gotz & Anzt, 2018; Stanaityte, 2020; ¨ Ichimura et al., 2020) and initial guesses for iterative methods (Luna et al., 2021; Um et al., 2020; Ackmann et al., 2020). Tompson et al. (2017) and Yang et al. (2016) develop non-iterative machine learning approximations of the inverse of discrete Poisson equations from incompressible flow. We leverage deep learning and develop a novel version of conjugate gradients iterative method for approximating the solution of SPD linear systems which we call the deep conjugate direction method (DCDM). CG iteratively adds $\\pmb { A }$ -conjugate search directions while minimizing the matrix norm of the error. We use a convolutional neural network (CNN) as an approximation of the inverse of the matrix in order to generate more efficient search directions. We only ask that our network approximate the inverse up to an unknown scaling since this decreases the degree of nonlinearity and since it does not affect the quality of the search direction (which is scale independent). The network is similar to a preconditioner, but it is not a linear function, and our modified conjugate gradients approach is designed to accommodate this nonlinearity. We use unsupervised learning to train our network with a loss function equal to the $L ^ { 2 }$ difference between an input vector and a scaling of $\\pmb { A }$ times the output of our network. To account for this unknown scaling during training, we choose the scale of the output of the network by minimizing the matrix norm of the error. Our approach allows for efficient training and generalization to problems unseen (matrices $\\pmb { A }$ and right-hand sides $^ { b }$ ). We benchmark our algorithm using the ubiquitous pressure Poisson equation (discretized on regular voxelized domains) and compare against FluidNet (Tompson et al., 2017), which is the state of the art learning-based method for these types of problems. Unlike the non-iterative approaches of Tompson et al. (2017) and Yang et al. (2016), our method can reduce the linear system residuals arbitrarily. We showcase our approach with examples that have over 16 million degrees of freedom. ",
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+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Several papers have focused on enhancing the solution of linear systems (arising from discretized PDEs) using learning. For instance, Gotz & Anzt (2018) generate sparsity patterns for block-Jacobi ¨ preconditioners using convolutional neural networks, and Stanaityte (2020) use a CNN to predict non-zero patterns for ILU-type preconditioners for the Navier-Stokes equations (though neither work designs fundamentally new preconditioners). Ichimura et al. (2020) develop a neural-network based preconditioner where the network is used to predict approximate Green’s functions (which arise in the analytical solution of certain PDEs) that in turn yield an approximate inverse of the linear system. Hsieh et al. (2019) learn an iterator that solves linear systems, performing competitively with classical solvers like multigrid-preconditioned MINRES (Paige & Saunders, 1975). Luz et al. (2020) and Greenfeld et al. (2019) use machine learning to estimate algebraic multigrid (AMG) parameters. They note that AMG approaches rely most fundamentally on effectively chosen (problem-dependent) prolongation sparse matrices and that numerous methods have attempted to automatically create them from the matrix $\\pmb { A }$ . They train a graph neural network to learn (in an unsupervised fashion) a mapping from matrices $\\pmb { A }$ to prolongation operators. Grebhahn et al. (2016) note that geometric multigrid solver parameters can be difficult to choose to guarantee parallel performance on different hardware platforms. They use machine learning to create a code generator to help achieve this. ",
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+        "img_path": "images/a7ac602c2643cff48deff90a7224438f208d4a7100d53c20e1e19240e8cade2b.jpg",
+        "image_caption": [
+            "Figure 1: (a) We illustrate a sample flow domain $\\Omega \\subset ( 0 , 1 ) ^ { 2 }$ (in 2D for ease of illustration) with internal boundaries (blue lines). (b) We voxelize the domain with a regular grid: white cells represent interior/fluid, and blue cells represent boundary conditions. (c) We train using matrix $A ^ { ( 0 , 1 ) ^ { d } }$ from a discretized domain with no interior boundary conditions, where $d$ is the dimension. This creates linear system with $n = ( n _ { c } + 1 ) ^ { d }$ unknowns where $n _ { c }$ is the number of grid cells on each direction. (d) We illustrate the non-zero entries in an example matrix $A ^ { \\Omega }$ from the voxelized and labeled (white vs. blue) grid for three example interior cells (green, purple, and brown). Each case illustrates the non-zero entries in the row associated with the example cell. All entries in rows corresponding to boundary/blue cells are zero. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Several works consider accelerating the solution of linear systems by learning an initial guess that is close to the true solution or otherwise helpful to descent algorithms for finding the true solution. In order to solve the discretized Poisson equation, Luna et al. (2021) accelerate the convergence of GMRES (Saad & Schultz, 1986) with an initial guess that is learned in real-time (i.e., as a simulation code runs) with no prior data. Um et al. (2020) train a network (incorporating differentiable physics, based on the underlying PDEs) in order to produce high-quality initial guesses for a CG solver. In a somewhat similar vein, Ackmann et al. (2020) use a simple feedforward neural network to predict pointwise solution components, which accelerates the conjugate residual method used to solve a relatively simple shallow-water model (a more sophisticated network and loss function are needed to handle more general PDEs and larger-scale problems). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "At least two papers (Ruelmann et al., 2018; Sappl et al., 2019) have sought to learn a mapping between a matrix and an associated sparse approximate inverse. In their investigation, Ruelmann et al. (2018) propose training a neural network using matrix-inverse pairs as training data. Although straightforward to implement, the cost of generating training data, let alone training the network, is prohibitive for large-scale 3D problems. Sappl et al. (2019) seek to learn a mapping between linear system matrices and sparse (banded) approximate inverses. Their loss function is the condition number of the product of the system matrix and the approximate inverse; the minimum value of the condition number is one, which is achieved exactly when an exact inverse is obtained. Although this framework is quite simple, evaluating the condition number of a matrix is asymptotically costly, and in general, the inverse of a sparse matrix can be quite dense. Accordingly, the method is not efficient or accurate enough for the large-scale 3D problems that arise in real-world engineering problems. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Most relevant to the present work is FluidNet (Tompson et al., 2017). FluidNet develops a highlytailored CNN architecture that is used to predict the solution of a linear projection operation (specifically, for the discrete Poisson equation) given a matrix and right-hand side. The authors demonstrate fluid simulations where the linear solve is replaced by evaluating their network. Because their network is relatively lightweight and is only evaluated once per time step, their simulations run efficiently. However, their design allows the network only one opportunity to reduce the residual for the linear solve; in practice, we observe that FluidNet is able to reduce the residual by no more than about one order of magnitude. However, in computer graphics applications, at least four orders of magnitude in residual reduction are usually required for visual fidelity, while in scientific and engineering applications, practitioners prefer solutions that reduce the residual by eight or more orders of magnitude (i.e., to within machine precision). Accordingly, FluidNet’s lack of convergence stands in stark contrast to classical, convergent methods like CG. Our method resolves this gap. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 MOTIVATION: INCOMPRESSIBLE FLOW ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We demonstrate the efficacy of our approach with the linear systems that arise in incompressible flow applications. Specifically, we use our algorithm to solve the discrete Poisson equations in ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "regular-grid-based discretization of the pressure projection equations that arise in Chorin’s splitting technique (Chorin, 1967) for the inviscid, incompressible Euler equations. These equations are ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c1b355957f1f3592b8d81befad750201620d6f0041dd14ae0eb7b88b47dac275.jpg",
+        "text": "$$\n\\rho \\left( \\frac { \\partial \\pmb { u } } { \\partial t } + \\frac { \\partial \\pmb { u } } { \\partial \\pmb { x } } \\pmb { u } \\right) + \\nabla p = \\pmb { f } ^ { e x t } , \\qquad \\nabla \\cdot \\pmb { u } = 0\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\textbf { \\em u }$ is fluid velocity, $p$ is pressure, $\\rho$ is density, and $f ^ { e x t }$ accounts for external forces like gravity. The equations are assumed at all positions $_ { \\textbf { \\em x } }$ in the spatial fluid flow domain $\\Omega$ and for time $t > 0$ . The left term in Equation 2 enforces conservation of momentum in the absence of viscosity, and the second part enforces incompressibility and conservation of mass. These equations are subject to initial conditions $\\rho ( \\pmb { x } , 0 ) = \\rho ^ { 0 }$ and ${ \\pmb u } ( { \\pmb x } , 0 ) = { \\pmb u } ^ { 0 } ( { \\pmb x } )$ as well as boundary conditions ${ \\pmb u } ( { \\pmb x } , t ) \\cdot { \\pmb n } ( { \\pmb x } ) = u ^ { \\partial \\Omega } ( { \\pmb x } , t )$ on the boundary of the domain ${ \\pmb x } \\in \\partial \\Omega$ (where $\\textbf { \\em n }$ is the unit outward pointing normal at position $_ { \\textbf { \\em x } }$ on the boundary). Equation 2 is discretized in both time and space. Temporally, we split the advection $\\frac { \\partial \\pmb { u } } { \\partial t } + \\frac { \\partial \\pmb { u } } { \\partial \\pmb { x } } \\pmb { u } = 0$ and body forces terms $\\rho \\frac { \\partial \\pmb { u } } { \\partial t } = \\pmb { f } ^ { e x t }$ , and finally enforce incompressibility via the pressure projection $\\frac { \\partial \\pmb { u } } { \\partial t } + \\frac { 1 } { \\rho } \\nabla p = \\pmb { 0 }$ such that $\\nabla \\cdot \\pmb { u } = 0$ ; this is the standard advection-projection scheme proposed by Chorin (1967). Using finite differences in time, we can summarize this as ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b9e83fdc2a77df2f7139c2359537bee131588af2939e28a1c636f7c2e1a8f4d9.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\rho ^ { 0 } \\left( \\frac { \\pmb { u } ^ { * } - \\pmb { u } ^ { n } } { \\Delta t } + \\frac { \\partial \\pmb { u } ^ { n } } { \\partial \\pmb { x } } \\pmb { u } ^ { n } \\right) = \\pmb { f } ^ { e x t } } \\\\ { \\displaystyle - \\nabla \\cdot \\frac { 1 } { \\rho ^ { 0 } } \\nabla p ^ { n + 1 } = - \\nabla \\cdot \\pmb { u } ^ { * } } \\\\ { \\displaystyle \\qquad - \\frac { 1 } { \\rho ^ { 0 } } \\nabla p ^ { n + 1 } \\cdot \\pmb { n } = \\frac { 1 } { \\Delta t } \\left( \\pmb { u } ^ { \\partial \\Omega } - \\pmb { u } ^ { * } \\cdot \\pmb { n } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For the spatial discretization, we use a regular marker-and-cell (MAC) grid (Harlow & Welch, 1965) with cubic voxels whereby velocity components are stored on the face of voxel cells, and scalar quantities (e.g., pressure $p$ or density $\\rho \\mathrm { \\hbar }$ ) are stored at voxel centers. We use backward semiLagrangian advection (Fedkiw et al., 2001; Gagniere et al., 2020) for Equation 3. All spatial partial derivatives are approximated using finite differences. Equations 4 and 5 describe the pressure Poisson equation with Neumann conditions on the boundary of the flow domain. We discretize the left hand side of Equation 4 using a standard 7-point finite difference stencil. The right-hand side is discretized using the MAC grid discrete divergence finite difference stencils as well as contributions from the boundary condition terms in Equation 5. We refer the reader to Bridson (2008) for more in-depth implementation details. Equation 5 is discretized by modifying the Poisson stencil to enforce Neumann boundary conditions. We do this using a simple labeling of the voxels in the domain. For simplicity, we assume $\\Omega \\subset ( 0 , 1 ) ^ { 3 }$ is a subset of the unit cube, potentially with internal boundaries. We label cells in the domain as either liquid or boundary. This simple classification is enough to define the Poisson discretizations (with appropriate Neumann boundary conditions at domain boundaries) that we focus on in the present work; we illustrate the details in Figure 1. We use the following notation to denote the discrete Poisson equations associated with Equations 4–5: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ac399afedaac667ec410278ede5ec8e5a957aa976ca5e09188331dc7a1ba5bb7.jpg",
+        "text": "$$\n\\begin{array} { r } { { \\cal A } ^ { \\Omega } { \\boldsymbol x } = { \\boldsymbol b } ^ { \\nabla \\cdot { \\boldsymbol u } ^ { * } } + { \\boldsymbol b } ^ { u ^ { \\partial \\Omega } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            576,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $A ^ { \\Omega }$ is the discrete Poisson matrix associated with the voxelized domain, $_ { \\textbf { \\em x } }$ is the vector of unknown pressure, and $\\mathbf { \\delta } _ { b } \\nabla \\cdot \\mathbf { u } ^ { * }$ and b u ∂ Ω are the right-hand side terms from Equations 4 and 5, respectively. We define a special case of the matrix involved in this discretization to be the Poisson matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ associated with $\\Omega = ( 0 , 1 ) ^ { 3 }$ , i.e., a full fluid domain with no internal boundaries. We use this matrix for training, yet demonstrate that our network generalizes to all other matrices arising from more complicated flow domains. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 DEEP CONJUGATE DIRECTION METHOD ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We present our method for the deep learning acceleration of iterative approximations to the solution of linear systems of the form seen in Equation 6. We first discuss relevant details of the conjugate gradients (CG) method, particularly line search and $\\pmb { A }$ -orthogonal search directions. We then present a deep learning technique for improving the quality of these search directions that ultimately reduces iteration counts required to achieve satisfactory residual reduction. Lastly, we outline the training procedures for our deep convolutional neural network. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Our approach iteratively improves approximations to the solution $_ { \\textbf { \\em x } }$ of Equation 6. We build on the method of CG, which requires the matrix $A ^ { \\Omega }$ in Equation 6 to be SPD. SPD matrices $A ^ { \\Omega }$ give rise to the matrix norm $\\| \\pmb { y } \\| _ { A ^ { \\Omega } } = \\sqrt { \\pmb { y } ^ { T } A ^ { \\Omega } \\pmb { y } }$ . CG can be derived in terms of iterative line search improvement based on optimality in this norm. That is, an iterate $\\pmb { x } _ { k - 1 } \\approx \\pmb { x }$ is updated in search direction $\\scriptstyle d _ { k }$ by a step size $\\alpha _ { k }$ that is chosen to minimize the matrix norm of the error: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/794da5f9a8f94a2ca23ac137e3679829e9c6027164f5d341253cdd67dfb2e112.jpg",
+        "text": "$$\n\\alpha _ { k } = \\underset { \\alpha } { \\arg \\operatorname* { m i n } } \\frac { 1 } { 2 } \\left\\| \\pmb { x } - ( \\pmb { x } _ { k - 1 } + \\alpha \\pmb { d } _ { k } ) \\right\\| _ { \\pmb { A } ^ { \\Omega } } ^ { 2 } = \\frac { { \\pmb { r } } _ { k - 1 } ^ { T } { \\pmb { d } _ { k } } } { { \\pmb { d } } _ { k } ^ { T } { \\pmb { A } } ^ { \\Omega } { \\pmb { d } _ { k } } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { r } _ { k - 1 } = \\pmb { b } - \\pmb { A } ^ { \\Omega } \\pmb { x } _ { k - 1 }$ is the $( k - 1 ) ^ { \\mathrm { t h } }$ residual and $^ { b }$ is the right-hand side in Equation 6. Different search directions result in different algorithms. A natural choice is the negative gradient of the matrix norm of the error (evaluated at the current iterate), since this will point in the direction of steepest decrease $\\begin{array} { r } { \\pmb { d } _ { k } = - \\frac { 1 } { 2 } \\nabla \\left\\| \\pmb { x } _ { k - 1 } \\right\\| _ { \\pmb { A } ^ { \\Omega } } ^ { 2 } = \\pmb { r } _ { k - 1 } } \\end{array}$ . This is the gradient descent method (GD). Unfortunately, this approach requires many iterations in practice. A more effective strategy is to choose directions that are $\\pmb { A }$ -orthogonal (i.e., ${ \\bf \\Phi } _ { { \\bf i } } ^ { T } { \\bf A } ^ { \\Omega } { \\bf d } _ { j } = \\mathrm { ~  ~ \\dot { ~ } { ~ 0 ~ } ~ }$ for $i \\neq j$ ). With this choice, the search directions form a basis for $\\mathbb { R } ^ { n }$ so that the initial error can be written as $\\begin{array} { r } { \\pmb { e } _ { 0 } = \\pmb { x } - \\pmb { x } _ { 0 } = \\sum _ { i = 1 } ^ { n } e _ { i } \\pmb { d } _ { i } } \\end{array}$ , where $e _ { i }$ are the components of the initial error written in the basis. Furthermore, when the search directions are $\\pmb { A }$ -orthogonal, the optimal step sizes $\\alpha _ { k }$ at each iteration satisfy ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/27f862ee424cf548433605036964bc38ad83b6328e4fe8ee6ce093dd03b21c48.jpg",
+        "text": "$$\n\\alpha _ { k } = \\frac { r _ { k - 1 } ^ { T } d _ { k } } { d _ { k } ^ { T } A ^ { \\Omega } d _ { k } } = \\frac { d _ { k } ^ { T } A ^ { \\Omega } e _ { k - 1 } } { d _ { k } ^ { T } A ^ { \\Omega } d _ { k } } = \\frac { d _ { k } ^ { T } A ^ { \\Omega } \\left( \\sum _ { i = 1 } ^ { n } e _ { i } d _ { i } - \\sum _ { j = 1 } ^ { k - 1 } \\alpha _ { j } d _ { j } \\right) } { d _ { k } ^ { T } A ^ { \\Omega } d _ { k } } = e _ { k } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "That is, the optimal step sizes are chosen to precisely eliminate the components of the error on the basis defined by the search directions. Thus, convergence is determined by the (at most $n$ ) non-zero components $e _ { i }$ in the initial error. Although rounding errors prevent this from happening exactly in practice, this property greatly reduces the number of required iterations (Golub & Loan, 2012). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "CG can be viewed as a modification of GD where the search direction is chosen as the component of the residual (equivalently, the negative gradient of the matrix norm of the error) that is $\\pmb { A }$ -orthogonal to all previous search directions: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/823b515874f8878d3bc9c886f39ccbd598d942c63f696c3c9a5dc8fad3a847d7.jpg",
+        "text": "$$\n\\pmb { d } _ { k } = \\pmb { r } _ { k - 1 } - \\sum _ { i = 1 } ^ { k - 1 } h _ { i k } \\pmb { d } _ { i } , \\qquad h _ { i k } = \\frac { { d } _ { i } ^ { T } { \\pmb { A } } ^ { \\Omega } \\pmb { r } _ { k - 1 } } { { d } _ { i } ^ { T } { \\pmb { A } } ^ { \\Omega } \\pmb { d } _ { i } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In practice, $h _ { i k } = 0$ for $i < k - 1$ , and this iteration can therefore be performed without the need to store all previous search directions $\\mathbf { \\mathbf { { \\alpha } } } d _ { i }$ and without the need for computing all previous $h _ { i k }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "While the residual is a natural choice for generating $\\pmb { A }$ -orthogonal search directions (since it points in the direction of the steepest local decrease), it is not the optimal search direction. If $\\scriptstyle d _ { k }$ is parallel to $( A ^ { \\Omega } ) ^ { - 1 } r _ { k - 1 }$ , then $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { k }$ will be equal to $_ { \\textbf { \\em x } }$ since $\\alpha _ { k }$ (computed from Equation 7) will step directly to the solution. We can see this by considering the residual and its relation to the search direction: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/8bc12357b2f542a177c0dbe7c2b250136ff63d91c180c2f209010068301bb834.jpg",
+        "text": "$$\n\\begin{array} { r } { r _ { k } = b - A ^ { \\Omega } x _ { k } = b - A ^ { \\Omega } x _ { k - 1 } - \\alpha _ { k } A ^ { \\Omega } d _ { k } = r _ { k - 1 } - \\alpha _ { k } A ^ { \\Omega } d _ { k } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In light of this, we use deep learning to create an approximation $f ( c , r )$ to $( A ^ { \\Omega } ) ^ { - 1 } r$ , where $^ c$ denotes the network weights and biases. This is analogous to using a preconditioner in PCG; however, our network is not SPD (nor even a linear function). We simply use this data-driven approach as our means of generating better search directions $\\scriptstyle d _ { k }$ . Furthermore, we only need to approximate a vector parallel to $( A ^ { \\Omega } ) ^ { - 1 } r$ since the step size $\\alpha _ { k }$ will account for any scaling in practice. In other words, $f ( c , r ) \\approx s _ { r } ( A ^ { \\Omega } ) ^ { - 1 } r$ , where the scalar $s _ { r }$ is not defined globally; it only depends on $\\pmb { r }$ , and the model does not learn it. Lastly, as with CG, we enforce $\\pmb { A }$ -orthogonality, yielding search directions ",
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+        "type": "equation",
+        "img_path": "images/2f5892c23191745ca45a89c6df6aa816d57d4b70d9d40bae7947ea58ff2fad11.jpg",
+        "text": "$$\n\\pmb { d } _ { k } = \\pmb { f } ( \\pmb { c } , \\pmb { r } _ { k - 1 } ) - \\sum _ { i = 1 } ^ { k - 1 } h _ { i k } \\pmb { d } _ { i } , \\qquad h _ { i k } = \\frac { \\pmb { f } ( \\pmb { c } , \\pmb { r } _ { k - 1 } ) ^ { T } \\pmb { A } ^ { \\Omega } \\pmb { d } _ { i } } { \\pmb { d } _ { i } ^ { T } \\pmb { A } ^ { \\Omega } \\pmb { d } _ { i } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            809,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We summarize our approach in Algorithm 1. Note that we introduce the variable $i _ { \\mathrm { s t a r t } }$ . To guarantee $\\pmb { A }$ -orthogonality between all search directions, we must have $i _ { \\mathrm { s t a r t } } = 1$ . However, this requires storing all prior search directions, which can be costly. We found that using $i _ { \\mathrm { s t a r t } } = k - 2$ worked nearly as well as $i _ { \\mathrm { s t a r t } } = 1$ in practice (in terms of our ability to iteratively reduce the residual of the system). We demonstrate this in Figure 4c. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 MODEL ARCHITECTURE, DATASETS, AND TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
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+            118
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Efficient performance of our method requires effective training of our deep convolutional network for weights and biases $^ c$ such that $f ( c , r ) { \\overset { \\cdot } { \\approx } } s _ { r } ( A ^ { \\Omega } ) ^ { - 1 } r$ (for arbitrary scalar $s _ { r }$ ). We design a model architecture, loss function, and unsupervised training approach to achieve this. Our approach has modest training requirements and allows for effective residual reduction while generalizing well to problems not seen in the training data. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.1 LOSS FUNCTION AND UNSUPERVISED LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Although we generalize to arbitrary matrices $A ^ { \\Omega }$ from Equation 6 that correspond to domains $\\Omega \\subset ( 0 , 1 ) ^ { 3 }$ that have internal boundaries (see Figure 1), we train using just the matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ from the full cube domain $( 0 , 1 ) ^ { 3 }$ . In contrast, other similar approaches (Tompson et al., 2017; Yang et al., 2016) train using matrices $A ^ { \\Omega }$ and right-hand sides $\\pmb { b } ^ { \\nabla \\cdot \\pmb { u } ^ { * } } + \\overline { { \\pmb { b } } } ^ { u ^ { \\partial \\Omega } }$ that arise from flow in many domains with internal boundaries. We train our network by minimiz",
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+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 DCDM ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1: $\\pmb { r } _ { 0 } = \\pmb { b } - \\pmb { A } ^ { \\Omega } \\pmb { x } _ { 0 }$   \n2: $k = 1$   \n3: while $\\| r _ { k - 1 } \\| \\ge \\epsilon$ do   \n4: dk = f (c, rk−1∥rk−1∥ )   \n5: for $i _ { \\mathrm { s t a r t } } \\le i < k$ do   \n6: hik = dT AΩdi dT AΩdi   \n7: dk-=hikdi   \n8: end for   \n9: $\\begin{array} { r l } & { \\alpha _ { k } = \\frac { r _ { k - 1 } ^ { T } d _ { k } } { d _ { k } ^ { T } A ^ { \\Omega } d _ { k } } } \\\\ & { x _ { k } = x _ { k - 1 } + \\alpha _ { k } d _ { k } } \\\\ & { r _ { k } = b - A ^ { \\Omega } x _ { k } } \\\\ & { k = k + 1 } \\end{array}$   \n10:   \n11:   \n12:   \n13: end while ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "ing the L2 difference ∥r−αA(0,1)3 f (c, r)∥2, where α = rT f(c,r)f(c,r)T A(0,1)3 f(c,r) f rom Equation 7. This choice of $\\alpha$ accounts for the unknown scaling in the approximation of $f ( c , r )$ to $\\left( A ^ { ( 0 , 1 ) ^ { 3 } } \\right) ^ { - 1 } r$ . We use an unsupervised approach and train the model by minimizing ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bf11442072e401d6ca5ceb5d5c2fe7051c67c9d21204c42168a7c1b30e85f761.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathrm { L o s s } ( \\pmb { f } , \\pmb { c } , \\mathscr { D } ) = \\frac { 1 } { | \\mathscr { D } | } \\sum _ { r \\in \\mathscr { D } } \\| \\pmb { r } - \\frac { r ^ { T } f ( \\pmb { c } , r ) } { \\pmb { f } ( \\pmb { c } , r ) ^ { T } \\pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \\pmb { f } ( \\pmb { c } , r ) } \\pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \\pmb { f } ( \\pmb { c } , r ) \\| _ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            446,
+            732,
+            474
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "for given dataset $\\mathcal { D }$ consisting of training vectors $b ^ { i }$ . In Algorithm 1, the normalized residuals $\\frac { \\boldsymbol { r } _ { k } } { \\| \\boldsymbol { r } _ { k } \\| }$ are passed as inputs to the model. Unlike in e.g. FluidNet (Tompson et al., 2017), only the first residual $\\frac { r _ { 0 } } { \\Vert r _ { 0 } \\Vert }$ is directly related to the problem-dependent original right-hand side $^ { b }$ . Hence we consider a broader range of training vectors than those expected in a given problem of interest, e.g., incompressible flows. We observe that generally the residuals $\\mathbf { \\nabla } r _ { k }$ in Algorithm 1 are skewed to the lower end of the spectrum of the matrix $A ^ { \\Omega }$ . Since $A ^ { \\Omega }$ is a discretized elliptic operator, lower end modes are of lower frequency of spatial oscillation. We create our training vectors $\\mathbf { \\boldsymbol { b } } ^ { i } \\in \\mathcal { D }$ using $m \\ll n$ approximate eigenvectors of the training matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ . We use the Rayleigh-Ritz method to create approximate eigenvectors $\\pmb q _ { i }$ , $0 \\leq i < m$ . This approach allows us to effectively approximate the full spectrum of $A ^ { ( 0 , 1 ) ^ { 3 } }$ without computing the full eigendecomposition, which can be expensive $( { \\cal O } ( \\bar { n } ^ { 3 } ) )$ at high resolution. We found that using $m \\ : = \\ : 1 0 0 0 0$ worked well in practice. The Rayleigh-Ritz vectors are orthonormal and satisfy $Q _ { m } ^ { T } A ^ { ( 0 , 1 ) ^ { 3 } } Q _ { m } = \\Lambda _ { m }$ , where $\\pmb { \\Lambda } _ { m }$ is a diagonal matrix with nondecreasing diagonal entries $\\lambda _ { i }$ referred to as Ritz values (approximate eigenvalues) and $\\pmb { Q } _ { m } = [ \\pmb { q } _ { 0 } , \\pmb { q } _ { 1 } , \\dots , \\pmb { q } _ { m - 1 } ] \\in \\mathbb { R } ^ { n \\times m }$ . We pick $\\begin{array} { r } { \\pmb { b } ^ { i } = \\frac { \\sum _ { j = 0 } ^ { m - 1 } c _ { j } ^ { i } \\pmb { q } _ { j } } { \\left\\| \\sum _ { j = 0 } ^ { m - 1 } c _ { j } ^ { i } \\pmb { q } _ { j } \\right\\| } } \\end{array}$ , where the coefficients $c _ { j } ^ { i }$ are picked from a standard normal distribution ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/395e1e20cca55da1945267d43b1e898ba5928b72735e7c30cc209c5a7aadea3e.jpg",
+        "text": "$$\nc _ { j } ^ { i } = \\left\\{ { \\begin{array} { l l } { 0 \\cdot \\mathcal { N } ( 0 , 1 ) } & { { \\mathrm { i f ~ } } \\tilde { j } \\leq j \\leq { \\frac { m } { 2 } } + \\theta } \\\\ { \\mathcal { N } ( 0 , 1 ) } & { { \\mathrm { o t h e r w i s e } } } \\end{array} } \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            724,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\theta$ is a small number (we used $\\theta = 5 0 0$ ), and $\\tilde { j }$ is the first index that $\\lambda _ { \\tilde { j } } ^ { - } > 0$ . This choice creates $9 0 \\%$ of $\\mathbf { \\nabla } _ { b } i$ from the lower end of the spectrum, with the remaining $1 0 \\%$ from the higher end. The Riemann-Lebesgue Lemma states the Fourier spectrum of a continuous function will decay at infinity, so this specific choice of $b _ { i }$ ’s is reasonable for the training set. In practice, we also observed that the right-hand sides of the pressure system that arose in flow problems (in the empty domain) tended to be at the lower end of the spectrum. Notably, even though this dataset only uses RayleighRitz vectors from the training matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ , our network can be effectively generalized to flows in irregular domains, e.g., smoke flow past a rotating box and flow past a bunny (see Figure 3). ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We generate the Rayleigh-Ritz vectors by first tridiagonalizing the training matrix $A ^ { ( 0 , 1 ) ^ { 3 } }$ with Lanczos iterations (Lanczos, 1950) to form $\\pmb { T } ^ { m } = \\pmb { Q } _ { m } ^ { L } ^ { T } \\pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \\pmb { Q } _ { m } ^ { L } \\in \\mathbb { R } ^ { m \\times m }$ . We then diagonalize $\\pmb { T } ^ { m } = \\hat { \\pmb { Q } } ^ { T } \\pmb { \\Lambda } _ { m } \\hat { \\pmb { Q } }$ . While costly, we note that this algorithm is performed on the comparably small $m \\times m$ matrix $\\mathbf { T } ^ { m }$ (rather than on the $A ^ { ( 0 , 1 ) ^ { 3 } } \\in \\mathbb { R } ^ { n \\times n } ,$ ). This yields the Rayleigh-Ritz vectors as the columns of $Q _ { m } = Q _ { m } ^ { L } \\hat { Q }$ . The Lanczos vectors are the columns of the matrix $Q _ { m } ^ { L }$ and satisfy a three-term recurrence whereby the next Lanczos vector can be computed from the previous two as ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/74fb81c962121faba4c85f6b417bd9fc92facd780a9f0d7f579191dbfec861ad.jpg",
+        "text": "$$\n\\beta _ { j } \\pmb { q } _ { j + 1 } ^ { L } = \\pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \\pmb { q } _ { j } ^ { L } - \\beta _ { j - 1 } \\pmb { q } _ { j - 1 } ^ { L } - \\alpha _ { j } \\pmb { q } _ { j } ^ { L } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            171,
+            642,
+            193
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha _ { j }$ and $\\beta _ { j }$ are diagonal and subdiagonal entries of $\\pmb { T } ^ { k }$ . $\\beta _ { j }$ is computed so that $\\pmb q _ { j + 1 } ^ { L }$ is a unit vector, and $\\alpha _ { j + 1 } = \\pmb { q } _ { j + 1 } ^ { T } \\pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } \\pmb { q } _ { j + 1 }$ . We initialize the iteration with a random $\\pmb { q } _ { 0 } ^ { L } \\in \\mathrm { s p a n } ( \\pmb { A } ^ { ( 0 , 1 ) ^ { 3 } } )$ . The Lanczos algorithm can be viewed as a modified Gram-Schmidt technique to create an orthonormal basis for the Krylov space associated with $\\pmb q _ { 0 } ^ { L }$ and $A ^ { ( 0 , 1 ) ^ { 3 } }$ , and it therefore suffers from rounding error sensitivities manifested as loss of orthonormality with vectors that do not appear in the recurrence. We found that the simple strategy described in Paige (1971) of orthogonalizing each iterate with all previous Lanczos vectors to be sufficient for our training purposes. Dataset creation takes 5–7 hours for a $6 4 ^ { 3 }$ computational grid, and 2–2.5 days for a $\\boldsymbol { 1 2 8 ^ { \\overline { { 3 } } } }$ grid. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 MODEL ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            337,
+            383,
+            351
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/048b839895ab4dfb5d59e79b90cfaaf105f450ca0f80073d468f3f23646ad845.jpg",
+        "image_caption": [
+            "Figure 2: Model architecture for training with $A ^ { ( 0 , 1 ) ^ { 3 } }$ on a $1 2 8 ^ { 3 }$ grid. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The internal structure of our CNN architecture for a $1 2 8 ^ { 3 }$ grid is shown in Figure 2. It consists of a series of convolutional layers with residual connections. The upper left of Figure 2 ( $K$ Residual Blocks) shows our use of multiple blocks of residually connected layers. Notably, within each block, the first layer directly affects the last layer with an addition operator.All non-input or output convolutions use a $3 \\times 3 \\times 3$ filter, and all layers consist of 16 feature maps. In the middle of the first level, a layer is downsampled (via the average pool",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ing operator with $( 2 \\times 2 \\times 2 )$ pool size) and another set of convolutional layers is applied with residual connection blocks. The last layer in the second level is upscaled and added to the layer that is downsampled. The last layer in the network is dense with a linear activation function. The activation functions in all convolutional layers are ReLU, except for the first convolution, which uses a linear activation function. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Initially we tried a simple deep feedforward convolutional network with residual connections (motivated by He et al. (2016)). Although such a simple model works well for DCDM, it requires high number of layers, which results in higher training and inference times. We found that creating parallel layers of CNNs with downsampling reduced the number of layers required. In summary, our goal was to first identify the simplest network architecture that provided adequate accuracy for our target problems, and subsequently, we sought to make architectural changes to minimize training and inference time; further optimizations are possible. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Differing resolutions use differing numbers of convolutions, but the fundamental structure remains the same. More precisely, the number of residual connections is changed for different resolutions. For example, a $6 \\dot { 4 } ^ { 3 }$ grid uses one residual block on the left, two on the right on the upper level, and three on the lower level. Furthermore, the weights trained on a lower resolution grid can be used effectively with higher resolutions. Figure 4d shows convergence results for a $2 \\mathrm { { 5 6 ^ { 3 } } }$ grid, using a model trained for a $6 4 ^ { 3 }$ grid and a $1 2 \\bar { 8 } ^ { 3 }$ grid. The model that we use for $2 5 6 ^ { 3 }$ grids in our final examples was trained on a $1 2 8 ^ { 3 }$ grid; however, as the shown in the figure, even training with a $6 4 ^ { 3 }$ grid allows for efficient residual reduction. Table 1 shows results for three different resolutions, where DCDM uses $6 4 ^ { 3 }$ and $1 2 8 ^ { 3 }$ trained models. This approach makes the number of parameters in the model independent of the spatial fidelity of the problem. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.3 TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Using the procedure explained in Section 5.1, we create the training dataset ${ \\mathcal { D } } \\in \\mathsf { s p a n } ( A ^ { ( 0 , 1 ) ^ { 3 } } ) \\cap { \\mathcal { S } } ^ { n - 1 }$ of size 20000 generated from 10000 Rayleigh-Ritz vectors. We train our model with TensorFlow (Abadi et al., 2015) on a single NVIDIA RTX A6000 GPU with 48GB memory. Training is done with standard deep learning techniques—more precisely, back-propagation and the ADAM optimizer (Kingma & Ba, 2015) (with starting learning rate 0.0001). Training takes approximately 10 minutes and 1 hour per epoch for grid resolutions $6 4 ^ { 3 }$ and $\\mathrm { \\dot { 1 } 2 8 ^ { 3 } }$ , respectively. We trained our model for 50 epochs; however, the model from the thirty",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/32fa04638f7e6d3d367c6df56f430f093be0e2e68b4d13ed0903e506ea28d74a.jpg",
+        "image_caption": [
+            "Figure 3: DCDM for simulating a variety of incompressible flow examples. Left: smoke plume at $t \\ =$ 6.67, 13.33, 20 seconds. Middle: smoke passing bunny at $t = 5 , 1 0 , 1 5$ seconds. Right: smoke passing a spinning box (time-dependent Neumann boundary conditions) at $t = 2 . 6 7 , 6 , 9 . 3 3$ seconds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "first epoch was optimal for $6 4 ^ { 3 }$ , and the model from the third epoch was optimal for $1 2 8 ^ { 3 }$ ",
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+    },
+    {
+        "type": "text",
+        "text": "6 RESULTS AND ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We demonstrate DCDM on three increasingly difficult examples and provide numerical evidence for the efficient convergence of our method. All examples were run on a workstation with dual AMD EPYC 75F3 processors and 512GB RAM. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Figure 3 showcases DCDM for incompressible smoke simulations. In each simulation, inlet boundary conditions are set in a circular portion of the bottom of the cubic domain, whereby smoke flows around potential obstacles and fills the domain. We show a smoke plume (no obstacles), flow past a complex static geometry (the Stanford bunny), and flow past a dynamic geometry (a rotating cube). Visually plausible and highly-detailed results are achieved for each simulation (see supplementary material for larger videos). The plume example uses a computational grid with resolution $1 2 8 ^ { 3 }$ , while the other two uses grids with resolution $2 5 6 ^ { 3 }$ (representing over 16 million unknowns). For each linear solve, DCDM was run until the residual was reduced by four orders of magnitude. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 1: Timing and iteration comparison for different methods on the bunny example. DCDM- $\\{ 6 4 , 1 2 8 \\}$ calls a model whose parameters trained over a $\\{ 6 4 ^ { 3 } , 1 2 8 ^ { 3 } \\}$ grid. All computations are done using only CPUs; model inference does not use GPUs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For the bunny example, Figures 4a–b demonstrate how residuals decrease over the course of a linear solve, comparing DCDM with other methods. Figure 4a shows the mean results (with standard deviations) over the course of 400 simulation frames, while in Figure 4b, we illustrate behavior on a particular frame (frame 150). For ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/3b7dccc2e3d2c2f389bc842cc59d0461cebabdd9b1b1f7f20a9c1b66da69f783.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">64 Grid</td><td colspan=\"2\">1283 Grid</td><td colspan=\"2\">2563 Grid</td></tr><tr><td>Method</td><td>tr</td><td>nr</td><td>tr</td><td>nr</td><td>tr</td><td>nr</td></tr><tr><td>DCDM-64</td><td>2.71s</td><td>16</td><td>22s</td><td>27</td><td>261s</td><td>58</td></tr><tr><td>DCDM-128</td><td>5.37s</td><td>19</td><td>26s</td><td>24</td><td>267s</td><td>44</td></tr><tr><td>CG</td><td>1.77s</td><td>168</td><td>26s</td><td>465</td><td>1548s</td><td>1046</td></tr><tr><td>Deflated PCG</td><td>771.6s</td><td>117</td><td>3700s</td><td>277</td><td>21030s</td><td>489</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "FluidNet, we use the implementation provided by fluidnetsc22 (2022). This implementation includes pre-trained models that we use without modification. In both subfigures, it is evident that the FluidNet residual never changes, since the method is not iterative; FluidNet reduces the initial residual by no more than one order of magnitude. On the other hand, with DCDM, we can continually reduce the residual (e.g., by four orders of magnitude) as we apply more iterations of our method, just as with classical CG. In Figure 4b, we also visualize the convergence of three other classical methods, CG, Deflated PCG (Saad et al., 2000), and incomplete Cholesky preconditioned CG (ICPCG)); clearly, DCDM reduces the residual in the fewest number of iterations (e.g., approximately one order of magnitude fewer iterations than ICPCG). Since FluidNet is not an iterative method and lacks a notion of residual reduction, we treat $r _ { 0 }$ for FluidNet as though an initial guess of zero is used (as is done in our solver). ",
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+    {
+        "type": "image",
+        "img_path": "images/49a3a4d6c185b571472c1686368f20fff99b702247e6716738f61bbe29294d57.jpg",
+        "image_caption": [
+            "Figure 4: Convergence data for the bunny example. (a) Mean and std. dev. (over all 400 frames in the simulation) of residual reduction during linear solves (with $1 2 8 ^ { 3 }$ and $2 5 6 ^ { 3 }$ grids) using FluidNet (FN) and DCDM. (b) Residual plots with CG, ICPCG, Deflated PCG, FN, and DCDM at frame 150. Dashed and solid lines represent results for $1 2 8 ^ { 3 }$ and $2 5 6 ^ { 3 }$ , respectively. (c) Decrease in residuals with varying degrees of $\\pmb { A }$ -orthogonalization $( i _ { s } ~ = ~ i _ { \\mathrm { s t a r t } } )$ . (d) Reduction in residuals when the network is trained with a $6 4 ^ { 3 }$ or $1 \\bar { 2 } 8 ^ { 3 }$ grid for the $2 5 6 ^ { 3 }$ grid simulation shown in Figure 3 Middle. "
+        ],
+        "image_footnote": [],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "To clarify these results, Table 1 reports convergence statistics for DCDM compared to standard iterative techniques CG and Deflated PCG. For all $6 4 ^ { 3 }$ , $1 2 8 ^ { 3 }$ , and $2 5 6 ^ { 3 }$ grids with the bunny example, we measure the time $t _ { r }$ and the number of iterations $n _ { r }$ required to reduce the initial residual on a particular time step of the simulation by four orders of magnitude. DCDM achieves the desired results in by far the fewest number of iterations at all resolutions. At $2 5 6 ^ { 3 }$ , DCDM performs approximately 6 times faster than CG, suggesting a potentially even wider performance advantage at higher resolutions. Inference is the dominant cost in an iteration of DCDM; the other linear algebra computations in an iteration of DCDM are comparable to those in CG. The nice result of our method is that despite the increased time per iteration, the number of required iterations is reduced so drastically that DCDM materially outperforms classical methods like CG. Although ICPCG successfully reduces number of iterations $^ { 4 \\mathrm { ~ b ~ } }$ , we found the runtime to scale prohibitively with grid resolution, so we exclude it from comparison in table 1. Notably, even though Deflated PCG and DCDM are based on approximate Ritz vectors, DCDM performs far better, indicating the value of using a neural network. ",
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+    {
+        "type": "text",
+        "text": "7 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We presented DCDM, incorporating CNNs into a CG-style algorithm that yields efficient, convergent behavior for solving linear systems. Our method is evaluated on linear systems with over 16 million degrees of freedom and converges to a desired tolerance in merely tens of iterations. Furthermore, despite training the underlying network on domains without obstacles, our network is able to successfully predict search directions that enable efficient linear solves on domains with complex and dynamic geometries. Moreover, the training data for our network does not require running fluid simulations or solving linear systems ahead of time; our Rayleigh-Ritz vector approach enables us to quickly generate very large training datasets, unlike approaches seen in other works. ",
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+    {
+        "type": "text",
+        "text": "Our network was designed for and trained exclusively using data related to the discrete Poisson matrix, which likely limits the generalizability of our present model. However, we believe our method is readily applicable to other classes of PDEs (or general problems with graph structure) that give rise to large, sparse, symmetric linear systems. We note that our method is unlikely to work well for matrices that have high computational cost to evaluate $A * x$ (such as dense matrices), since training relies on efficient $\\boldsymbol { A } * \\boldsymbol { x }$ evaluations. An interesting question to consider is how well our method and current models would apply to discrete Poisson matrices arising from non-uniform grids, e.g., quadtrees or octrees (Losasso et al., 2004). ",
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+        "type": "text",
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Data-driven projection method in fluid simulation. Comp Anim Virt Worlds, 27(3-4):415–424, 2016. ",
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parse/dev/zkk_7sV6gm8/zkk_7sV6gm8.md

@@ -0,0 +1,458 @@
+# Timing is Everything: Learning to Act Selectively with Costly Actions and Budgetary Constraints
+
+Anonymous Author(s)
+
+# Abstract
+
+1 Many real-world settings involve costs for performing actions; transaction costs   
+2 in financial systems and fuel costs being common examples. In these settings,   
+3 performing actions at each time step quickly accumulates costs leading to vastly   
+4 suboptimal outcomes. Additionally, repeatedly acting produces wear and tear and   
+5 ultimately, damage. Determining when to act is crucial for achieving successful   
+6 outcomes and yet, the challenge of efficiently learning to behave optimally when   
+7 actions incur minimally bounded costs remains unresolved. In this paper, we intro  
+8 duce a reinforcement learning (RL) framework named Learnable Impulse Control   
+9 Reinforcement Algorithm (LICRA), for learning to optimally select both when   
+10 to act and which actions to take when actions incur costs. At the core of LICRA   
+11 is a nested structure that combines RL and a form of policy known as impulse   
+12 control which learns to maximise objectives when actions incur costs. We prove   
+13 that LICRA, which seamlessly adopts any RL method, converges to policies that   
+14 optimally select when to perform actions and their optimal magnitudes. We then   
+15 augment LICRA to handle problems in which the agent can perform at most $k < \infty$   
+16 actions and more generally, faces a budget constraint. We show LICRA learns the   
+17 optimal value function and ensures budget constraints are satisfied almost surely.   
+18 We demonstrate empirically LICRA’s superior performance against benchmark   
+19 RL methods in OpenAI gym’s Lunar Lander and in Highway environments and a   
+20 variant of the Merton portfolio problem within finance.
+
+# 21 1 Introduction
+
+22 There are many settings in which agents incur costs each time they perform an action. Transaction   
+23 costs in financial settings [19], fuel expenditure [32], toxicity as a side effect of controlling bacte  
+24 ria [29] and physical damage produced by repeated action that produces wear and tear are just a   
+25 few among many examples [13]. In these settings, performing actions at each time step is vastly   
+26 suboptimal since acting in this way results in prohibitively high costs and undermines the service life   
+27 of machinery. Minimising wear and tear is an essential attribute to safeguard against failures that can   
+28 result in catastrophic losses [13].   
+29 Reinforcement learning (RL) is a framework that enables autonomous agents to learn complex   
+30 behaviours from interactions with the environment [30, 11]. Within the standard RL paradigm,   
+31 determining optimal actions involves making a selection from among many (possibly infinite) actions;   
+32 a procedure that must be performed at each time-step as the agent decides on an action. In unknown   
+33 settings, the agent cannot immediately exploit any topological structure of the action set (if any   
+34 exists). Consequently, learning not to take an action i.e performing a zero or null action, involves   
+35 expensive optimisation procedures over the entire action set. Since this must be done at each state,   
+36 this process is vastly inefficient for learning optimal policies when the agent incurs costs for acting.   
+37 In this paper, we tackle this problem by developing an RL framework for finding both an optimal   
+38 criterion to determine whether or not to execute actions as well as learning optimal actions. A key   
+39 component of our framework is a novel combination of RL with a form of policy known as impulse   
+40 control [22, 19]. This enables the agent to determine the appropriate points to perform an action as   
+41 well as the optimal action itself. Despite its fundamental importance as a tool for tackling decision   
+42 problems with costly actions, presently, the use of impulse control within learning contexts (and   
+43 unknown environments) is unaddressed.   
+44 We present an RL impulse control framework called LICRA, which, to our knowledge, is the first   
+45 learning framework for impulse control. To enable learning optimal impulse control policies in   
+46 unknown environments, we devise a framework that consists of separate RL components for learning   
+47 when to act and how to act optimally. The resulting framework is a structured two-part learning   
+48 process which differs from current RL protocols. In LICRA, at each time step, the agent firstly makes   
+49 a decision whether to act or not leading to a binary decision space $\{ 0 , 1 \}$ (we later show that this   
+50 is determined by evaluating an easy-to-evaluate criterion which has the value function as its input).   
+51 The second decision part determines the best action to take. This generates a subdivision of the state   
+52 space into two regions; one in which the agent performs actions and another in which it does not act   
+53 at all. This is extremely useful since the agent quickly determines the set of states to not take actions   
+54 while performing actions only at the subset of states where actions are to be executed.   
+55 We then establish theory that ensures convergence of LICRA to an optimal policy for such settings.   
+56 To do this, we give a series of results namely:   
+57 i) We establish a dynamic programming principle (DPP) for impulse control and show that the optimal   
+58 value function can be obtained as a limit of a value iterative procedure (Theorem 1) which lays the   
+59 foundation for an RL approach to impulse control.   
+60 ii) We extend result i) to a new variant of Q learning which enables the impulse control problem to be   
+61 solved using our RL method (Theorem 2).   
+62 iii) We characterise the optimal conditions for performing an action which we reveal to be a simple   
+63 ‘obstacle condition’ involving the agent’s value function (Prop. 1). Using this, the agent can quickly   
+64 determine whether or not it should act and if so, then learn what the optimal action is.   
+65 iv) We then extend the result i) to (linear) function approximators enabling the value function to be   
+66 parameterised (Theorem 3).   
+67 iv) In Sec. 6, we extend LICRA to include budgetary constraints so that each action draws from a   
+68 fixed budget which the agent must stay within. Analogous to the development of i), we establish   
+69 another DPP from which we derive a Q-learning variant for tackling impulse control with budgetary   
+70 constraints (Theorem 4). A particular case of a budget constraint is when the number of actions the   
+71 agent can take over the horizon is capped.   
+72 Lastly, we perform a set of experiments to validate our theory within the Highway driving simulator   
+73 and OpenAI’s LunarLander [7].   
+74 LICRA confers a series of advantages. As we demonstrate in our experiments, LICRA learns to   
+75 compute the optimal problems in which the agent faces costs for acting in an efficient way which   
+76 outperforms leading RL baselines. Second, as demonstrated in Sec. 6 LICRA handles settings in   
+77 which the agent has a cap the total number of actions it is allowed to execute and more generally,   
+78 generic budgetary constraints. LICRA is able to accommodate any RL base algorithm unlike various   
+79 RL methods designed to handle budgetary constraints.
+
+# 80 2 Related Work
+
+81 In continuous-time optimal control theory [24], problems in which the agent faces a cost for each   
+82 action are tackled with a form of policy known as impulse control [22, 19, 2]. In impulse control   
+83 frameworks, the dynamics of the system are modified through a sequence of discrete actions or bursts   
+84 chosen at times that the agent chooses to apply the control policy. This distinguishes impulse control   
+85 models from classical decision methods in which an agent takes actions at each time step while being   
+86 tasked with the decision of only which action to take. Impulse control models represent appropriate   
+87 modelling frameworks for financial environments with transaction costs, liquidity risks and economic   
+88 environments in which players face fixed adjustment costs (e.g. menu costs) [16, 20].   
+89 The current setting is intimately related to the optimal stopping problem which widely occurs in   
+90 finance, economics and computer science [23, 31]. In the optimal stopping problem, the task is to   
+91 determine a criterion that determines when to arrest the system and receive a terminal reward. In this   
+92 case, standard RL methods are unsuitable since they require an expensive sweep (through the set   
+93 of states) to determine the optimal point to arrest the system. The current problem can be viewed   
+94 as an augmented problem of optimal stopping since the agent must now determine both a sequence   
+95 of points to perform an action or intervene and their optimal magnitudes — only acting when the   
+96 cost of action is justified [25]. Adapting RL to tackle optimal stopping problems has been widely   
+97 studied [31, 4, 9] and applied to a variety of real-world settings within finance [12], radiation therapy   
+98 [1] and network operating systems [3]. Our work serves as a natural extenstion to RL approaches to   
+99 optimal stopping to the case in which the agent must decide at which points to take many actions. As   
+100 with optimal stopping, standard RL methods cannot efficiently tackle this problem since determining   
+101 whether to perform a 0 action requires a costly sweep through the action space at every state [31]. In   
+102 [26] the authors introduce “sparse action” with a similar motivation as impulse control. However,   
+103 the authors treat only the discrete action space case. The authors in [26] do not discuss a broader   
+104 theoretical framework of dealing with “sparse actions”, and develop purely algorithmic solutions.   
+105 Additionally, unlike the approach taken in [26], the problem setting we consider is one in which the   
+106 agent faces a cost for each action - the produces a need for the agent to be selective about where it   
+107 performs actions (but does not necessarily constrain the magnitude or choice of those actions).
+
+# 108 3 Preliminaries
+
+109 Reinforcement Learning (RL). In RL, an agent sequentially selects actions to maximise its expected   
+110 returns. The underlying problem is typically formalised as an MDP $\langle S , \mathcal { A } , P , R , \gamma \rangle$ where $S \subset \mathbb { R } ^ { p }$   
+111 is the set of states, $\mathcal { A } \subset \mathbb { R } ^ { k }$ is the set of actions, $P : \mathcal { S } \times \mathcal { A } \times \mathcal { S }  [ 0 , 1 ]$ is a transition probability   
+112 function describing the system’s dynamics, $R : S \times \mathcal { A }  \mathbb { R }$ is the reward function measuring the   
+113 agent’s performance and the factor $\gamma \in [ 0 , 1 )$ specifies the degree to which the agent’s rewards are   
+114 discounted over time [30]. At time $t \in { 0 , 1 , \ldots }$ , the system is in state $s _ { t } \in S$ and the agent must   
+115 choose an action $a _ { t } \in \mathcal A$ which transitions the system to a new state $s _ { t + 1 } \sim P ( \cdot | s _ { t } , a _ { t } )$ and produces   
+116 a reward $R ( s _ { t } , a _ { t } )$ . A policy $\pi : S \times A \to [ 0 , 1 ]$ is a probability distribution over state-action pairs   
+117 where $\pi ( a | s )$ represents the probability of selecting action $a \in { \mathcal { A } }$ in state $s \in S$ . The goal of an   
+118 RL agent is to find a policy ${ \hat { \pi } } \in \Pi$ that maximises its expected returns given by the value function:   
+119 $\begin{array} { r } { v ^ { \pi } ( s ) = \mathbb { E } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \bar { R } ( s _ { t } , a _ { t } ) | a _ { t } \sim \pi ( \cdot | s _ { t } ) , s _ { 0 } = s ] } \end{array}$ $\begin{array} { r } { Q ( s , a ) = \mathbb { E } [ \sum _ { t = 0 } ^ { \infty } R ( s _ { t } , a _ { t } ) | a _ { 0 } = a , s _ { 0 } = s ] } \end{array}$ $\Pi$ ent’s policy set. The action.   
+121 We consider a setting in which the agent faces at least some minimal cost for each action it performs.   
+122 With this, the agent’s task is to maximise:
+
+$$
+v ^ { \pi } ( s ) = \mathbb { E } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \left\{ \mathcal { R } ( s _ { t } , a _ { t } ) - \mathcal { C } ( s _ { t } , a _ { t } ) \right\} \Big | s _ { 0 } = s \right] ,
+$$
+
+123 where for any state $s \in S$ and any action $a \in { \mathcal { A } }$ , the functions $\mathcal { R }$ and $\mathcal { C }$ are given by $\textstyle { \mathcal { R } } ( s , a ) =$   
+124 $R ( s , a ) \mathbf { 1 } _ { a \in \mathcal { A } } + R ( s , 0 ) ( 1 - \mathbf { 1 } _ { a \in \mathcal { A } } )$ where $\mathbf { 1 } _ { a \in \mathcal { A } }$ is the indicator function which is 1 when $a \in { \mathcal { A } }$ and 0   
+125 otherwise and ${ \mathcal { C } } ( s , a ) : = c ( s , a ) \mathbf { 1 } _ { a \in { \mathcal { A } } }$ where $c : S \times \mathcal { A }  \mathbb { R }$ is a minimally bounded (cost) function1   
+126 that introduces a cost each time the agent performs an action. Examples of the cost function is a   
+127 quasi-linear function of the form $c ( s _ { t } , \bar { a } _ { t } ) \stackrel {  } { = } \kappa + f ( a _ { t } )$ where $f : \mathcal { A }  \mathbb { R } _ { > 0 }$ and $\kappa$ is a positive real  
+128 valued constant. Since acting at each time step would incur prohibitively high costs, the agent must   
+129 be selective when to perform an action. Therefore, in this setting, the agent’s problem is augmented to   
+130 learning both an optimal policy for its actions and, learning at which states to apply its action policy.   
+131 Example: Merton Portfolio Problem with Transaction Costs [10]. An investor performs a series   
+132 of costly portfolio adjustments by buying and selling amounts of different assets within their portfolio.   
+133 Each investment incurs a fixed minimal cost (also known as transaction costs) which is deducted   
+134 from the investor’s available cash-flow. The investor’s aim is to maximise their total wealth (the value   
+135 of the sum of their assets) at some time horizon by adjusting their portfolio of investments. Problems   
+136 of this kind, portfolio investment problems are of fundamental importance within finance [18].   
+37 Example 2. An autonomous vehicle must perform a series of actions to perform a task. Each action   
+38 draws from its fuel budget. In order to complete its task successfully, during the task, the vehicle   
+139 must ensure it maintains an available supply.   
+141 In RL, the agent’s problem involves learning to act at every state including those in which actions do   
+142 not significantly impact on its total return. While we can add a zero action to the action set $\mathcal { A }$ and   
+143 apply standard methods, we argue that this may not be the best solution in many situations. We argue   
+144 the optimal policy has the following form:
+
+$$
+\begin{array} { r } { \widetilde { \pi } ( \cdot | s ) = \left\{ \begin{array} { l l } { a _ { t } } & { s \in S _ { I } , } \\ { 0 } & { s \notin S _ { I } , } \end{array} \right. } \end{array}
+$$
+
+145 which implies that we simplify policy learning by determining the set $ { \boldsymbol { S } } _ { I }$ first — the set where we   
+146 actually need to learn the policy.   
+147 We now introduce a learning method for producing impulse controls. This enables the agent to learn   
+148 to select states to perform actions. Therefore, now agent is tasked with learning to act at states that   
+149 are most important for maximising its total return given the presence of the cost for each action. Now   
+150 at each state the agent first makes a binary decision to decide to perform an action.   
+151 Our framework, LICRA consists of two core components: firstly a RL process $\mathfrak { g } : \mathcal { S } \times \{ 0 , 1 \}  [ 0 , 1 ]$   
+152 and a second RL process $\pi : S \times A \to [ 0 , 1 ]$ . The role of $\mathfrak { g }$ is to determine whether or not an action   
+153 is to be performed by the policy $\pi$ at a given state $s$ . If activated, the policy $\pi$ determines the action   
+154 to be selected. Prior to decisions being made, the policy $\pi$ communicates to $\mathfrak { g }$ the action it would   
+155 take. An important feature of our LICRA is the sequential decision process. In LICRA, the policy   
+156 $\pi$ first proposes an action $a \in { \mathcal { A } }$ which is observed by the policy $\mathfrak { g }$ . Therefore, the role of $\mathfrak { g }$ is to   
+157 prevent actions for which the change in expected future rewards does not exceed the costs incurred   
+158 for taking such actions. By isolating a decision policy over whether an action should be taken or not,   
+159 the impulse controls mechanism results in a framework in which the problem facing the agent has a   
+160 markedly reduced decision space (in comparison to a standard RL method). Crucially, the agent must   
+161 compute optimal actions at only a subset of states which are chosen by the policy g. Below is the   
+162 pseudocode for LICRA, we provide full details of the code in Sec. 9 of the Appendix.
+
+# Algorithm 1: Learnable Impulse Control Reinforcement Algorithm (LICRA)
+
+1: Input: Stepsize $\alpha$ , batch size $B$ , episodes $K$ , steps per episode $T$ , mini-epochs $e$   
+2: Initialise: Policy network (acting) $\pi$ , Policy network (switching) $\mathfrak { g }$ ,   
+Critic network (acting $) V _ { \pi }$ ,Critic network (switching $) V _ { { \mathfrak { g } } }$   
+3: Given reward objective function, $R$ , initialise Rollout Buffers $B _ { \pi }$ , $B _ { \mathfrak { g } }$ (use Replay Buffer for   
+SAC)   
+4: for $N _ { e p i s o d e s }$ do   
+5: Reset state $s _ { 0 }$ , Reset Rollout Buffers $B _ { \pi }$ , $B _ { \mathfrak { g } }$   
+6: for $t = 0 , 1 , \ldots$ do   
+7: Sample $a _ { t } \sim \pi ( \cdot | s _ { t } )$   
+8: Sample $g _ { t } \sim \mathfrak { g } ( \cdot | s _ { t } )$   
+9: if $g _ { t } = 0$ then   
+10: Apply $a _ { t }$ so $s _ { t + 1 } \sim P ( \cdot | a _ { t } , s _ { t } ) .$ ,   
+11: Receive rewards $r _ { t } = \mathcal { R } ( s _ { t } , a _ { t } )$   
+12: Store $( s _ { t } , a _ { t } , s _ { t + 1 } , r _ { t } )$ in $B _ { \pi }$   
+13: else   
+14: Apply the null action so $s _ { t + 1 } \sim P ( \cdot | 0 , s _ { t } )$ ,   
+15: Receive rewards $r _ { t } = \mathcal { R } ( s _ { t } , 0 )$ .   
+16: end if   
+17: Store $( s _ { t } , g _ { t } , s _ { t + 1 } , r _ { t } )$ in $B _ { \mathfrak { g } }$   
+18: end for   
+19: // Learn the individual policies   
+20: Update policy $\pi$ and critic $V _ { \pi }$ networks using $\mathfrak { B } _ { \pi }$   
+21: Update policy $\mathfrak { g }$ and critic $V _ { \mathfrak { g } }$ networks using ${ \mathfrak { B } } _ { { \mathfrak { g } } }$   
+22: end for   
+165 Although action space cardinality does not change there are still benefits of using impulse control   
+166 mechanism. This mechanism forces the agent to first determine the set of states to perform actions   
+167 only then determine the optimal actions at these states. An important fact to note is that the decision   
+168 space for the determining whether or not to execute an action is $S \times \{ 0 , 1 \}$ i.e at each state it makes a   
+169 binary decision. Consequently, the learning process for aspect is much quicker than a policy which   
+170 must optimise over a decision space which is $| { \cal S } | | { \cal A } |$ (choosing an action from its action space at   
+171 every state). This results in the agent rapidly learning which states to focus on to learn which actions   
+172 to perform. In the case of $\pi$ with a continuous action space again the impulse control mechanism   
+173 does not change the cardinality of the action space. However, if the set ${ \cal S } / { \cal S } _ { I }$ , where the optimal   
+174 policy chooses 0, is large enough, then again it can be more efficient to learn $\mathfrak { g }$ first and only then   
+175 learn $\pi$ (we later validate this claim empirically, see Sec. 11.2),   
+176 In Sec. 5, we prove the convergence properties of LICRA. LICRA consists of two independent   
+177 procedures: a learning process for the policy $\pi$ and simultaneously, a learning process for the impulse   
+178 policy $\mathfrak { g }$ which determines at which states to perform an action. In our implementation, we used   
+179 proximal policy optimisation (PPO) [27] for the policy $\pi$ and for the impulse policy $\mathfrak { g }$ , whose action   
+180 set consists of two actions (intervene or do not intervene) we used a soft actor critic (SAC) process   
+181 [14] LICRA is a plug $\&$ play framework which enables these RL components to be replaced with any   
+182 RL algorithm of choice.
+
+# 183 5 Convergence and Optimality of LICRA
+
+184 A key aspect of our framework is the presence of two RL processes that make decisions in a sequential   
+185 order. In order to determine when to act the policy $\mathfrak { g }$ must learn the states to allow the policy $\pi$ to   
+186 perform an action which the policy $\pi$ must learn to select optimal actions whenever it is allowed to   
+187 execute an action.   
+188 In this section, we prove that LICRA converges to an optimal solution of the system. Central to   
+189 LICRA is a Q-learning type method which is adapted to handle RL settings in which the agent must   
+190 also learn when to act. We then extend the result to allow for (linear) function approximators. We   
+191 provide a result that shows the optimal intervention times are characterised by an ‘obstacle condition’   
+192 which can be evaluated online therefore allowing the $\mathfrak { g }$ .
+
+Given a function $Q \because S \times \mathcal { A } \to \mathbb { R } , \quad \forall \pi , \pi ^ { \prime } \in \Pi$ and $\forall \ S , \ S ^ { \prime } , \quad \forall s _ { \tau _ { k } } \quad \in \quad S$ , we define the intervention operator $\mathcal { M } ^ { \pi , { \mathfrak { g } } }$ by $\begin{array} { r } { \mathcal { M } ( ^ { \pi , \oplus } Q ^ { \pi ^ { \prime } , \oplus ^ { \prime } } ( s _ { \pi _ { k } } , a _ { \pi _ { k } } ) : = { \mathcal R } ( s _ { \pi _ { k } } , a _ { \pi _ { k } } ) - c ( s _ { \pi _ { k } } , a _ { \pi _ { k } } ) + } \end{array}$ $\begin{array} { r } { \gamma \sum _ { s ^ { \prime } \in S } P ( s ^ { \prime } ; a _ { \tau _ { k } } , s ) v ^ { \pi ^ { \prime } , \mathfrak { g ^ { \prime } } } ( s ^ { \prime } ) \| a _ { \tau _ { k } } \sim \pi ( \cdot | s _ { \tau _ { k } } ) , } \end{array}$ , where $\tau _ { k }$ is an intervention time.
+
+196 The interpretation of $\mathcal { M }$ is the following: suppose that the agent is using the policy $\pi$ and at time $\tau _ { k }$   
+197 the system is at a state $s _ { \tau _ { k } }$ and the agent performs an action $a _ { \tau _ { k } } \sim \pi ( \cdot | s _ { \tau _ { k } } )$ . A cost of $c ( s _ { \tau _ { k } } , a _ { \tau _ { k } } )$   
+198 is then incurred by the agent and the system transitions to $s ^ { \prime } \sim P ( \cdot ; a _ { \tau _ { k } } , s _ { \tau _ { k } } )$ . Lastly, recall $v ^ { \pi , { \mathfrak { g } } }$ is   
+199 the agent value function under the policy pair $( \pi , g )$ . Therefore, the quantity ${ \mathcal { M } } Q ^ { \pi , { \mathfrak { g } } }$ measures the   
+200 expected future stream of rewards after an immediate intervention minus the cost of intervention.   
+201 This object plays a crucial role in the LICRA framework which as we later discuss, exploits the cost   
+202 structure of the problem to determine when the agent should perform an intervention.
+
+Given a function 203 $v ^ { n , \ell } : S  \mathbb { \quad }$ , we define the Bellman operator $\boldsymbol { \cdot }$ , by:
+
+204
+
+$$
+T v ^ { \pi , \theta } ( s ) : = \operatorname* { m a x } \Big \{ \mathcal { N } ^ { ( \pi , \theta ) } Q ^ { \pi , \theta } ( s , a ) , \mathcal { R } ( s , 0 ) + \gamma \sum _ { s ^ { \prime } \in S } P ( s ^ { \prime } ; 0 , s ) v ^ { \pi , \theta } ( s ^ { \prime } ) \Big \} , \qquad \forall s \in \mathcal { S } .
+$$
+
+205
+
+206 The Bellman operator captures the nested sequential structure of the LICRA algorithm. In particular,   
+207 the structure in (3) consists of an inner structure which consists of two terms: the first term is the   
+208 expected future return given an action is taken at the current state under the policy $\pi$ . The second term   
+209 is the expected future return given no action is taken at the current state. Lastly, the outer structure is   
+210 an optimisation which compares the expected return of the two possibilities and selects the maximum.
+
+11 Our first result proves $T$ is a contraction operator in particular, the following bound holds:
+
+212 Lemma 1 The Bellman operator $T$ is a contraction, that is the following bound holds:
+
+$$
+\begin{array} { r } { \| T v - T v ^ { \prime } \| \leq \gamma \left. v - v ^ { \prime } \right. , } \end{array}
+$$
+
+where 13 $v , v ^ { \prime }$ are elements of a finite normed vector space. We can now state our first main result:
+
+Theorem 1 Given any $v ^ { \pi , \mathfrak { g } } : \mathcal { S } \times \mathcal { A }  \mathbb { R } ,$ , the optimal value function is given by $\operatorname* { l i m } _ { k \to \infty } T ^ { k } v ^ { \pi , \mathfrak { g } } =$ $\operatorname* { m a x } _ { \hat { \pi } , \hat { g } \in \Pi } v ^ { \hat { \pi } , \hat { g } } = v ^ { \pi ^ { \star } , g ^ { \star } }$ where $( \pi ^ { \star } , g ^ { \star } )$ is the optimal policy pair.
+
+The result of Theorem 1 enables the solution to the agent’s impulse control problem to be determined using a value iteration procedure. Moreover, Theorem 1 enables a Q-learning approach [6] for finding the solution to the agent’s problem.
+
+219 Theorem 2 Consider the following $Q$ learning variant:
+
+$$
++ \alpha _ { t } ( s _ { t } , a _ { t } ) \left[ \operatorname* { m a x } \left\{ \mathcal { M } ^ { \pi , \mathfrak { g } } Q _ { t } ( s _ { t } , a _ { t } ) , \mathcal { R } ( s _ { t } , 0 ) + \gamma _ { a ^ { \prime } \in A } ^ { \operatorname* { m a x } } Q _ { t } ( s _ { t + 1 } , a ^ { \prime } ) \right\} - Q _ { t } ( s _ { t } , a _ { t } ) \right] ,
+$$
+
+then 220 $Q _ { t }$ converges to $Q ^ { \star }$ with probability 1, where $s _ { t } , s _ { t + 1 } \in S$ and $a _ { t } \in \mathcal A$ .
+
+21 We now extend the result to (linear) function approximators:
+
+Theorem 3 Given a set of linearly independent basis functions $\Phi = \{ \phi _ { 1 } , \ldots , \phi _ { p } \}$ with $\phi _ { k } \in L _ { 2 } , \forall k$ . LICRA converges to a limit point $r ^ { \star } \in \mathbb { R } ^ { p }$ which is the unique solution to $\Pi \mathfrak { F } ( \Phi r ^ { \star } ) = \Phi r ^ { \star }$ where $\mathfrak { F } v : = \mathcal { R } + \gamma P \operatorname* { m a x } \{ \mathcal { M } v , v \}$ . Moreover, $r ^ { \star }$ satisfies: $\| \Phi r ^ { \star } - Q ^ { \star } \| \leq ( 1 - \gamma ^ { 2 } ) ^ { - 1 / 2 } \| \Pi Q ^ { \star } - Q ^ { \star } \|$ .
+
+The theorem establishes the convergence of LICRA to a stable point with the use of linear function approximators. The second statement bounds the proximity of the convergence point by the smallest approximation error that can be achieved given the choice of basis functions.
+
+Having constructed a procedure to find the optimal agent’s optimal value function, we now seek to determine the conditions when an intervention should be performed. Let us denote by $\{ \tau _ { k } \} _ { k \ge 0 }$ the points at which the agent decides to act or intervention times, so for example if the agent chooses to perform an action at state $s _ { 6 }$ and again at state $s _ { 8 }$ , then $\tau _ { 1 } = 6$ and $\tau _ { 2 } = 8$ . The following result characterises the optimal intervention policy $\mathfrak { g }$ and the optimal times $\{ \tau _ { k } \} _ { k \ge 0 }$ .
+
+Proposition 1 The policy g is given by: $\mathfrak { g } ( s _ { t } ) = H ( \mathcal { M } ^ { \pi , \mathfrak { g } } Q ^ { \pi , \mathfrak { g } } - Q ^ { \pi , \mathfrak { g } } ) ( s _ { t } , a _ { t } ) , \forall s _ { t } \in \mathcal { S } _ { \mathrm { ~ \Gamma ~ } }$ , where $Q ^ { \pi , { \mathfrak { g } } }$ is the solution in Theorem $\cdot$ , $\mathcal { M }$ is the intervention operator and $H$ is the Heaviside function, moreover the intervention times are $\tau _ { k } = \operatorname* { i n f } \{ \tau > \tau _ { k - 1 } | \mathcal { M } ^ { \pi , { \mathfrak { g } } } Q ^ { \pi , { \mathfrak { g } } } = Q ^ { \pi , { \mathfrak { g } } } \}$ .
+
+Prop. 1 characterises the (categorical) distribution $\mathfrak { g }$ . Moreover, given the function $Q$ , the times $\{ \tau _ { k } \}$ can be determined by evaluating if $\mathcal { M } Q = Q$ holds.
+
+A key aspect of Prop. 1 is that it exploits the cost structure of the problem to determine when the agent should perform an intervention. In particular, the equality $\mathcal { M } Q = Q$ implies that performing an action and incurring a cost for doing so is optimal.
+
+# 6 Budget Augmented LICRA via State Augmentation
+
+We now tackle the problem of RL with a budget. To do this, we combine the above impulse control technology with state augmentation technique proposed in [28] The mathematical formulation of the problem is now given by the following for any $s \in S$ :
+
+$$
+\operatorname* { m a x } _ { \pi \in \Pi , g } ~ \upsilon ^ { \pi , \mathfrak { g } } ( s ) \ \mathrm { ~ s . ~ t . ~ } n - \sum _ { t = 0 } ^ { \infty } \sum _ { k \geq 1 } \delta _ { \tau _ { k } } ^ { t } \geq 0 ,
+$$
+
+where $n \in \mathbb N$ is a fixed value that represents the maximum number of allowed interventions and $\sum k > 1 8 _ { T k } ^ { t }$ is equal to one if an impulse was applied at time $t$ and zero if it was not. In order to avoid dealing with a constrained MDP, we propose to introduce a new variable $z _ { t }$ tracking the remaining number of impulses: $\begin{array} { r } { z _ { t } = n - \sum _ { i = 0 } ^ { t - 1 } \sum _ { k \geq 1 } \delta _ { \tau _ { k } } ^ { i } } \end{array}$ . We treat $z _ { t }$ as another state and augment the state-space resulting in the transition $\widetilde { \mathcal P }$ :
+
+$$
+s _ { t + 1 } \sim P ( \cdot | s _ { t } , a _ { t } ) , \qquad z _ { t + 1 } = z _ { t } - \sum _ { k \geq 1 } \delta _ { \tau _ { k } } ^ { t } , \quad z _ { 0 } = n .
+$$
+
+In order to avoid violations, we reshape the reward as follows: 252 $\begin{array} { r } { \widetilde { \mathcal { R } } ( s _ { t } , z _ { t } , a _ { t } ) = \left\{ \begin{array} { l l } { \mathcal { R } ( s _ { t } , a _ { t } ) } & { z _ { t } \ge 0 } \\ { - \Delta } & { z _ { t } < 0 } \end{array} \right. } \end{array}$ ,, 253 where $\Delta > 0$ is a large enough hyper-parameter ensuring that there are no safety violations. To 254 summarise we aim to solve the following problem:
+
+$$
+v ^ { \pi , \mathfrak { g } } ( s , z ) = \mathbb { E } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \widetilde { \mathcal { R } } ( s _ { t } , z _ { t } , a _ { t } ) | a _ { t } \sim \pi ( \cdot | s _ { t } , z _ { t } ) \right] ,
+$$
+
+where the policy now depends on the variable $z _ { t }$ . Note that $\widetilde { \mathcal P }$ in Equation 6 is a Markov process and, the rewards $\widetilde { \mathcal { R } }$ are bounded, as long as the rewards $\mathcal { R }$ are bounded. Therefore, we can apply directly the results for impulse control to this case as well. We denote the augmented MDP by $\widetilde { \mathcal { M } } = \langle \overset { \cdot } { S } \times \mathcal { Z } , \overset { \cdot } { A } , \widetilde { \mathcal { P } } , \widetilde { R } , \gamma \rangle$ , where $\mathcal { Z }$ is the space of the augmented state. We have the following.
+
+Theorem 4 Consider the MDP $\widetilde { \mathcal { M } }$ for the problem 7, then:
+
+a) The Bellman equation holds, i.e. there exists a function $\tilde { v } ^ { * , \pi , 9 }$ s.th. $\begin{array} { r l } { \tilde { v } ^ { * , \pi , \mathfrak { g } } ( s , z ) } & { { } = } \end{array}$ $\operatorname* { m a x } _ { \mathbf { a } \in \mathcal { A } } \Big ( \widetilde { \mathcal { R } } ( s , z , \pmb { a } ) + \gamma \mathbb { E } _ { s ^ { \prime } , z ^ { \prime } \sim \mathcal { P } } \left[ \widetilde { v } ^ { * , \pi , \mathfrak { g } } ( s ^ { \prime } , z ^ { \prime } ) \right] \Big )$ , where the optimal policy for $\widetilde { \mathcal { M } }$ has the form $\pi ^ { * } ( \cdot | s , z )$ ;
+
+$b$ ) Given a ${ \widetilde { v } } : { \mathcal { S } } \times { \mathcal { Z } } \to \mathbb { R } ,$ , the stable point solution for $\widetilde { \mathcal { M } }$ is a given by $\operatorname* { l i m } _ { k  \infty } \tilde { T } ^ { k } \widetilde { v } ^ { \pi , g } = \operatorname* { m a x } _ { \hat { \pi } \in \Pi , \hat { g } } \hat { \pi } ^ { \hat { \pi } , \hat { g } } =$ $\widetilde { v } ^ { * , \pi , { \mathfrak { g } } ^ { * } }$ , where $( \pi ^ { * } , { \mathfrak { g } } ^ { * } )$ is an optimal policy of $\widetilde { \mathcal { M } }$ and $\tilde { T }$ is the Bellman operator of $\widetilde { \mathcal { M } }$ .
+
+The result has several important implications. The first is that we can use a modified version of LICRA to obtain the solution of the problem while guaranteeing convergence (under standard assumptions). Secondly, our state augmentation procedure admits a Markovian representation of the optimal policy.
+
+# 69 7 Experiments
+
+270 We will now study empirically the performance of the LICRA framework. In experiments, we use   
+271 different instances of LICRA, one where both policies are trained using PPO update (referred to   
+272 as LICRA_PPO) and one where the policy deciding whether to act is trained using SAC and the   
+273 other policy trained with PPO (referred to as LICRA_SAC). We have benchmarked both of these   
+274 algorithms together with common baselines on environments, where it would be natural to introduce   
+275 the concept of the cost associated with actions. We lastly performed a series of ablation studies which   
+276 test LICRA’s ability to handle different cost functions including the case when $c ( s , a ) \equiv 0$ which we   
+277 defer to the Appendix which also contains further experiment details.   
+278 Merton’s Portfolio Problem with Transaction Costs. Merton Investment Problem in which the   
+279 investor faces transaction costs [10] is a well-known problem within finance. In our environment, the   
+280 agent can decide to move its wealth between a risky asset and a risk-free asset. The agent receives a   
+281 reward only at the final step, equal to the utility of the portfolio with a risk aversion factor equal to   
+282 0.5. If the final wealth of risky asset is $s _ { T }$ and final wealth of risk-free asset is $c _ { T }$ , then the agent will   
+283 receive a reward of $u ( x ) = \hat { 2 \sqrt { s _ { T } + c _ { T } } }$ . The wealth evolves according to the following SDE:
+
+![](images/c36f4d4fc355a612aa0817f310a3b877c4df8280a953e0bbdade420829da0b36.jpg)  
+Figure 1: Training results in Merton investment problem for PPO style algorithms.
+
+$$
+d W _ { t } = ( r + p _ { t } ( \mu - r ) ) W _ { t } + W _ { t } p _ { t } \sigma d B _ { t }
+$$
+
+![](images/e4f0e42faa34e8a85dc4690b08f54b2a9dca40e02107c30188aa1c25e63de654.jpg)  
+Figure 2: a) Drive Environment. b) Training results in drive environment.
+
+284 where $W _ { t }$ is the current wealth and the state variable, $d B _ { t }$ is an increment of Brownian motion and $p _ { t }$   
+285 is the proportion of wealth invested in the risky asset. We set the risk-free return $r = 0 . 0 1$ , risky asset   
+286 return $\mu = 0 . 0 5$ and volatility $\sigma = 1$ . We discretise the action space so that at each step the agent has   
+287 three actions available: move $10 \%$ of risky asset wealth to the risk-free asset, move $10 \%$ of risk-free   
+288 asset wealth to the risky asset or do nothing. Each time the agent moves the assets, it incurs a cost   
+289 of 1 i.e. a transaction fee. The agent can act after a time interval of 0.01 seconds and the episode   
+290 ends after 75 steps. The results of training are shown in Fig. 1 which clearly demonstrates that   
+291 LICRA_PPO finds a better policy than standard PPO. Also comparing the variance among different   
+292 seeds, we can see that LICRA_PPO is a much more stable algorithm than the other two.   
+293 Driving Environment Fuel Rationing. We studied an autonomous driving scenario where fuel  
+294 efficient driving is a priority. One of the main components of fuel-efficient driving is controlled usage   
+295 of acceleration and braking, in the sense that 1) the amount of acceleration and braking should be   
+296 limited 2) if accelerations should be performed slowly and gently. We believe this is a problem where   
+297 LICRA should thrive as the impulse control agent can learn to restrict the amount of acceleration   
+298 and braking in the presence of other cars and choose when to allow the car to decelerate naturally.   
+299 We used the highway-env [17] environment on a highway task (see Fig (2. a)) where the green   
+300 vehicle is our controlled vehicle and the goal is to avoid crashing into other vehicles whilst driving   
+301 at a reasonable speed. We add a cost function into the reward term dependent on the continuous   
+302 acceleration action, $C ( a _ { t } ) = K + a _ { t } ^ { 2 }$ , where $K > 0$ is a fixed constant cost of taking any action,   
+303 and $a _ { t } \in [ - 1 , 1 ]$ , with larger values of acceleration or braking being penalised more. The results are   
+304 presented in Fig. (2.b). Notably, LICRA is able to massively outperform the baselines, especially our   
+305 safety specific baselines which struggle to deal with the cost function associated with the environment.   
+306 We believe one reason for the success of LICRA is that it is far easier for it to utilise the null action   
+307 of zero acceleration/braking than the other algorithms, whilst all the algorithms have a guaranteed   
+308 cost at every time step whilst not gaining a sizeable reward to counter the cost.
+
+![](images/5bec575b374d2d9a07b6663aca5f30e26bc81290ee187b6eb4033c9c9f99603c.jpg)  
+Figure 3: a) The lander must land on the pad between two flags. . b) Training results in Lunar Lander. Lunar Lander Environment. We tested the ability of LICRA to perform in environment that simulate real-world physical dynamics. We tested LICRA’s performance the Lunar Lander environment in OpenAI gym [7] which we adjusted to incorporate minimal bounded costs in the reward definition. In this environment, the agent is required to maintain both a good posture mid-air and reach the landing pad as quickly as possible. The reward function is given by:
+
+$$
+\begin{array} { r } { \mathrm { R e w a r d } \left( s _ { t } \right) = 3 * \left( 1 - \mathbf { 1 } _ { d _ { t } - d _ { t - 1 } = 0 } \right) - 3 * \left( 1 - \mathbf { 1 } _ { v _ { t } - v _ { t - 1 } = 0 } \right) - 3 * \left( 1 - \mathbf { 1 } _ { \omega _ { t } - \omega _ { t - 1 } = 0 } \right) } \\ { - 0 . 0 3 * \mathrm { F u e l S p e n t } ( s _ { t } ) - 1 0 * \left( v _ { t } - v _ { t - 1 } \right) - 1 0 * \left( \omega _ { t } - \omega _ { t - 1 } \right) + 1 0 0 * \mathrm { h a s L a n d e d } \left( s _ { t } \right) } \end{array}
+$$
+
+309 where $d _ { t }$ is the distance to the landing pad, $v _ { t }$ is the velocity of the agent, and $\omega _ { t }$ is the angular   
+310 velocity of the agent at time $t$ . $\mathbf { 1 } _ { X }$ is the indicator function of taking actions, which is 1 when the   
+311 statement $X$ is true and 0 when $X$ is false. Considering the limited fuel budget, we assume that   
+312 we have a fixed cost for each action taken by the agent here, and doing nothing brings no cost. Then,   
+313 to describe the goal of the game, we define the function of the final status by hasLanded(), which is   
+314 0 when not landing; 1 when the agent has landed softly on the landing pad; and $- 1$ when the lander   
+315 runs out of fuel or loses contact with the pad on landing. The reward function rewards the agent   
+316 for reducing its distance to the landing pad, decreasing its speed to land smoothly and keeping the   
+317 angular speed at a minimum to prevent rolling. Additionally, it penalises the agent for running out   
+318 of fuel and deters the agent from taking off again after landing.   
+319 By introducing a reward function with minimally bounded costs, our goal was to test if LICRA can   
+320 exploit the optimal policy. In Fig. 3, we observe that the LICRA agent outperforms all the baselines,   
+321 both in terms of sample efficiency and average test return (total rewards at each timestep). We also   
+322 observe that LICRA enables more stable training than PPO, PPO-Lagrangian and CPO.   
+323 Ablation Study 1. Prioritisation of Most Important Actions. We next tested LICRA’s ability to   
+324 prioritise where it performs actions when the necessity to act varies significantly between states. To   
+325 test this, we modified the Drive Environment to now consist of a single lane, a start state and a goal   
+326 state start (at the end) where there is a reward. With no acceleration, the vehicle decreases velocity.   
+327 To reach the goal, the agent must apply an acceleration $a _ { t } \in [ - 1 , 1 ]$ . Each acceleration $a _ { t }$ incurs   
+328 a cost $C ( \boldsymbol { a } _ { t } )$ as defined above. At zones $k = 1 , 2 , 3$ of the lane, if the vehicle is travelling below a   
+329 velocity $v _ { m i n }$ , it is penalised by a strictly negative cost $c _ { k }$ where $c _ { 1 } < c _ { 2 } < c _ { 3 }$ . As shown in Fig. 4,   
+330 when the intervention cost increases i.e. when $K  \infty$ , LICRA successfully prioritises the highest   
+penalty zones to avoid incurring large costs.
+
+![](images/2ba4542911b4913c1e9265c3276b917ff90a5a40da65ab950216d84702acce94.jpg)  
+Figure 4: Results for Ablation Study 1. Heatmaps display the number of times the agent drives below $v _ { m i n }$ in the penalty zones. Violation 1 refers to the lowest cost zone, whilst Violation 3 refers to the largest cost zone. $K$ refers to the fixed cost for taking an action.
+
+# 8 Conclusion
+
+We presented a novel method to tackle the problem of learning how to select when to act in addition to learning which actions to execute. Our framework, which is a general tool for tackling problems of this kind seamlessly adopts RL algorithms enabling them to efficiently tackle problems in which the agent must be selective about when it executes actions. This is of fundamental importance in practical settings where performing many actions over the horizon can lead to costs and undermine the service life of machinery. We demonstrated that our solution, LICRA which at its core has a sequential decision structure that first decides whether or not an action ought to be taken under the action policy can solve tasks where the agent faces costs with extreme efficiency as compared to leading reinforcement learning methods. In some tasks, we showed that LICRA is able to solve problems that are unsolvable using current reinforcement learning machinery. We envisge that this framework can serve as the basis extensions to different settings including adversarial training for solving a variety of problems within RL.
+
+# References
+
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+        "text": "Timing is Everything: Learning to Act Selectively with Costly Actions and Budgetary Constraints ",
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+        "text": "Abstract ",
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+        "text": "1 Many real-world settings involve costs for performing actions; transaction costs   \n2 in financial systems and fuel costs being common examples. In these settings,   \n3 performing actions at each time step quickly accumulates costs leading to vastly   \n4 suboptimal outcomes. Additionally, repeatedly acting produces wear and tear and   \n5 ultimately, damage. Determining when to act is crucial for achieving successful   \n6 outcomes and yet, the challenge of efficiently learning to behave optimally when   \n7 actions incur minimally bounded costs remains unresolved. In this paper, we intro  \n8 duce a reinforcement learning (RL) framework named Learnable Impulse Control   \n9 Reinforcement Algorithm (LICRA), for learning to optimally select both when   \n10 to act and which actions to take when actions incur costs. At the core of LICRA   \n11 is a nested structure that combines RL and a form of policy known as impulse   \n12 control which learns to maximise objectives when actions incur costs. We prove   \n13 that LICRA, which seamlessly adopts any RL method, converges to policies that   \n14 optimally select when to perform actions and their optimal magnitudes. We then   \n15 augment LICRA to handle problems in which the agent can perform at most $k < \\infty$   \n16 actions and more generally, faces a budget constraint. We show LICRA learns the   \n17 optimal value function and ensures budget constraints are satisfied almost surely.   \n18 We demonstrate empirically LICRA’s superior performance against benchmark   \n19 RL methods in OpenAI gym’s Lunar Lander and in Highway environments and a   \n20 variant of the Merton portfolio problem within finance. ",
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+        "text": "21 1 Introduction ",
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+        "text": "22 There are many settings in which agents incur costs each time they perform an action. Transaction   \n23 costs in financial settings [19], fuel expenditure [32], toxicity as a side effect of controlling bacte  \n24 ria [29] and physical damage produced by repeated action that produces wear and tear are just a   \n25 few among many examples [13]. In these settings, performing actions at each time step is vastly   \n26 suboptimal since acting in this way results in prohibitively high costs and undermines the service life   \n27 of machinery. Minimising wear and tear is an essential attribute to safeguard against failures that can   \n28 result in catastrophic losses [13].   \n29 Reinforcement learning (RL) is a framework that enables autonomous agents to learn complex   \n30 behaviours from interactions with the environment [30, 11]. Within the standard RL paradigm,   \n31 determining optimal actions involves making a selection from among many (possibly infinite) actions;   \n32 a procedure that must be performed at each time-step as the agent decides on an action. In unknown   \n33 settings, the agent cannot immediately exploit any topological structure of the action set (if any   \n34 exists). Consequently, learning not to take an action i.e performing a zero or null action, involves   \n35 expensive optimisation procedures over the entire action set. Since this must be done at each state,   \n36 this process is vastly inefficient for learning optimal policies when the agent incurs costs for acting.   \n37 In this paper, we tackle this problem by developing an RL framework for finding both an optimal   \n38 criterion to determine whether or not to execute actions as well as learning optimal actions. A key   \n39 component of our framework is a novel combination of RL with a form of policy known as impulse   \n40 control [22, 19]. This enables the agent to determine the appropriate points to perform an action as   \n41 well as the optimal action itself. Despite its fundamental importance as a tool for tackling decision   \n42 problems with costly actions, presently, the use of impulse control within learning contexts (and   \n43 unknown environments) is unaddressed.   \n44 We present an RL impulse control framework called LICRA, which, to our knowledge, is the first   \n45 learning framework for impulse control. To enable learning optimal impulse control policies in   \n46 unknown environments, we devise a framework that consists of separate RL components for learning   \n47 when to act and how to act optimally. The resulting framework is a structured two-part learning   \n48 process which differs from current RL protocols. In LICRA, at each time step, the agent firstly makes   \n49 a decision whether to act or not leading to a binary decision space $\\{ 0 , 1 \\}$ (we later show that this   \n50 is determined by evaluating an easy-to-evaluate criterion which has the value function as its input).   \n51 The second decision part determines the best action to take. This generates a subdivision of the state   \n52 space into two regions; one in which the agent performs actions and another in which it does not act   \n53 at all. This is extremely useful since the agent quickly determines the set of states to not take actions   \n54 while performing actions only at the subset of states where actions are to be executed.   \n55 We then establish theory that ensures convergence of LICRA to an optimal policy for such settings.   \n56 To do this, we give a series of results namely:   \n57 i) We establish a dynamic programming principle (DPP) for impulse control and show that the optimal   \n58 value function can be obtained as a limit of a value iterative procedure (Theorem 1) which lays the   \n59 foundation for an RL approach to impulse control.   \n60 ii) We extend result i) to a new variant of Q learning which enables the impulse control problem to be   \n61 solved using our RL method (Theorem 2).   \n62 iii) We characterise the optimal conditions for performing an action which we reveal to be a simple   \n63 ‘obstacle condition’ involving the agent’s value function (Prop. 1). Using this, the agent can quickly   \n64 determine whether or not it should act and if so, then learn what the optimal action is.   \n65 iv) We then extend the result i) to (linear) function approximators enabling the value function to be   \n66 parameterised (Theorem 3).   \n67 iv) In Sec. 6, we extend LICRA to include budgetary constraints so that each action draws from a   \n68 fixed budget which the agent must stay within. Analogous to the development of i), we establish   \n69 another DPP from which we derive a Q-learning variant for tackling impulse control with budgetary   \n70 constraints (Theorem 4). A particular case of a budget constraint is when the number of actions the   \n71 agent can take over the horizon is capped.   \n72 Lastly, we perform a set of experiments to validate our theory within the Highway driving simulator   \n73 and OpenAI’s LunarLander [7].   \n74 LICRA confers a series of advantages. As we demonstrate in our experiments, LICRA learns to   \n75 compute the optimal problems in which the agent faces costs for acting in an efficient way which   \n76 outperforms leading RL baselines. Second, as demonstrated in Sec. 6 LICRA handles settings in   \n77 which the agent has a cap the total number of actions it is allowed to execute and more generally,   \n78 generic budgetary constraints. LICRA is able to accommodate any RL base algorithm unlike various   \n79 RL methods designed to handle budgetary constraints. ",
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+        "text": "80 2 Related Work ",
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+        "text": "81 In continuous-time optimal control theory [24], problems in which the agent faces a cost for each   \n82 action are tackled with a form of policy known as impulse control [22, 19, 2]. In impulse control   \n83 frameworks, the dynamics of the system are modified through a sequence of discrete actions or bursts   \n84 chosen at times that the agent chooses to apply the control policy. This distinguishes impulse control   \n85 models from classical decision methods in which an agent takes actions at each time step while being   \n86 tasked with the decision of only which action to take. Impulse control models represent appropriate   \n87 modelling frameworks for financial environments with transaction costs, liquidity risks and economic   \n88 environments in which players face fixed adjustment costs (e.g. menu costs) [16, 20].   \n89 The current setting is intimately related to the optimal stopping problem which widely occurs in   \n90 finance, economics and computer science [23, 31]. In the optimal stopping problem, the task is to   \n91 determine a criterion that determines when to arrest the system and receive a terminal reward. In this   \n92 case, standard RL methods are unsuitable since they require an expensive sweep (through the set   \n93 of states) to determine the optimal point to arrest the system. The current problem can be viewed   \n94 as an augmented problem of optimal stopping since the agent must now determine both a sequence   \n95 of points to perform an action or intervene and their optimal magnitudes — only acting when the   \n96 cost of action is justified [25]. Adapting RL to tackle optimal stopping problems has been widely   \n97 studied [31, 4, 9] and applied to a variety of real-world settings within finance [12], radiation therapy   \n98 [1] and network operating systems [3]. Our work serves as a natural extenstion to RL approaches to   \n99 optimal stopping to the case in which the agent must decide at which points to take many actions. As   \n100 with optimal stopping, standard RL methods cannot efficiently tackle this problem since determining   \n101 whether to perform a 0 action requires a costly sweep through the action space at every state [31]. In   \n102 [26] the authors introduce “sparse action” with a similar motivation as impulse control. However,   \n103 the authors treat only the discrete action space case. The authors in [26] do not discuss a broader   \n104 theoretical framework of dealing with “sparse actions”, and develop purely algorithmic solutions.   \n105 Additionally, unlike the approach taken in [26], the problem setting we consider is one in which the   \n106 agent faces a cost for each action - the produces a need for the agent to be selective about where it   \n107 performs actions (but does not necessarily constrain the magnitude or choice of those actions). ",
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+        "text": "108 3 Preliminaries ",
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+        "text": "109 Reinforcement Learning (RL). In RL, an agent sequentially selects actions to maximise its expected   \n110 returns. The underlying problem is typically formalised as an MDP $\\langle S , \\mathcal { A } , P , R , \\gamma \\rangle$ where $S \\subset \\mathbb { R } ^ { p }$   \n111 is the set of states, $\\mathcal { A } \\subset \\mathbb { R } ^ { k }$ is the set of actions, $P : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S }  [ 0 , 1 ]$ is a transition probability   \n112 function describing the system’s dynamics, $R : S \\times \\mathcal { A }  \\mathbb { R }$ is the reward function measuring the   \n113 agent’s performance and the factor $\\gamma \\in [ 0 , 1 )$ specifies the degree to which the agent’s rewards are   \n114 discounted over time [30]. At time $t \\in { 0 , 1 , \\ldots }$ , the system is in state $s _ { t } \\in S$ and the agent must   \n115 choose an action $a _ { t } \\in \\mathcal A$ which transitions the system to a new state $s _ { t + 1 } \\sim P ( \\cdot | s _ { t } , a _ { t } )$ and produces   \n116 a reward $R ( s _ { t } , a _ { t } )$ . A policy $\\pi : S \\times A \\to [ 0 , 1 ]$ is a probability distribution over state-action pairs   \n117 where $\\pi ( a | s )$ represents the probability of selecting action $a \\in { \\mathcal { A } }$ in state $s \\in S$ . The goal of an   \n118 RL agent is to find a policy ${ \\hat { \\pi } } \\in \\Pi$ that maximises its expected returns given by the value function:   \n119 $\\begin{array} { r } { v ^ { \\pi } ( s ) = \\mathbb { E } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\bar { R } ( s _ { t } , a _ { t } ) | a _ { t } \\sim \\pi ( \\cdot | s _ { t } ) , s _ { 0 } = s ] } \\end{array}$ $\\begin{array} { r } { Q ( s , a ) = \\mathbb { E } [ \\sum _ { t = 0 } ^ { \\infty } R ( s _ { t } , a _ { t } ) | a _ { 0 } = a , s _ { 0 } = s ] } \\end{array}$ $\\Pi$ ent’s policy set. The action.   \n121 We consider a setting in which the agent faces at least some minimal cost for each action it performs.   \n122 With this, the agent’s task is to maximise: ",
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+        "text": "$$\nv ^ { \\pi } ( s ) = \\mathbb { E } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\left\\{ \\mathcal { R } ( s _ { t } , a _ { t } ) - \\mathcal { C } ( s _ { t } , a _ { t } ) \\right\\} \\Big | s _ { 0 } = s \\right] ,\n$$",
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+        "text": "123 where for any state $s \\in S$ and any action $a \\in { \\mathcal { A } }$ , the functions $\\mathcal { R }$ and $\\mathcal { C }$ are given by $\\textstyle { \\mathcal { R } } ( s , a ) =$   \n124 $R ( s , a ) \\mathbf { 1 } _ { a \\in \\mathcal { A } } + R ( s , 0 ) ( 1 - \\mathbf { 1 } _ { a \\in \\mathcal { A } } )$ where $\\mathbf { 1 } _ { a \\in \\mathcal { A } }$ is the indicator function which is 1 when $a \\in { \\mathcal { A } }$ and 0   \n125 otherwise and ${ \\mathcal { C } } ( s , a ) : = c ( s , a ) \\mathbf { 1 } _ { a \\in { \\mathcal { A } } }$ where $c : S \\times \\mathcal { A }  \\mathbb { R }$ is a minimally bounded (cost) function1   \n126 that introduces a cost each time the agent performs an action. Examples of the cost function is a   \n127 quasi-linear function of the form $c ( s _ { t } , \\bar { a } _ { t } ) \\stackrel {  } { = } \\kappa + f ( a _ { t } )$ where $f : \\mathcal { A }  \\mathbb { R } _ { > 0 }$ and $\\kappa$ is a positive real  \n128 valued constant. Since acting at each time step would incur prohibitively high costs, the agent must   \n129 be selective when to perform an action. Therefore, in this setting, the agent’s problem is augmented to   \n130 learning both an optimal policy for its actions and, learning at which states to apply its action policy.   \n131 Example: Merton Portfolio Problem with Transaction Costs [10]. An investor performs a series   \n132 of costly portfolio adjustments by buying and selling amounts of different assets within their portfolio.   \n133 Each investment incurs a fixed minimal cost (also known as transaction costs) which is deducted   \n134 from the investor’s available cash-flow. The investor’s aim is to maximise their total wealth (the value   \n135 of the sum of their assets) at some time horizon by adjusting their portfolio of investments. Problems   \n136 of this kind, portfolio investment problems are of fundamental importance within finance [18].   \n37 Example 2. An autonomous vehicle must perform a series of actions to perform a task. Each action   \n38 draws from its fuel budget. In order to complete its task successfully, during the task, the vehicle   \n139 must ensure it maintains an available supply.   \n141 In RL, the agent’s problem involves learning to act at every state including those in which actions do   \n142 not significantly impact on its total return. While we can add a zero action to the action set $\\mathcal { A }$ and   \n143 apply standard methods, we argue that this may not be the best solution in many situations. We argue   \n144 the optimal policy has the following form: ",
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+        "type": "text",
+        "text": "145 which implies that we simplify policy learning by determining the set $ { \\boldsymbol { S } } _ { I }$ first — the set where we   \n146 actually need to learn the policy.   \n147 We now introduce a learning method for producing impulse controls. This enables the agent to learn   \n148 to select states to perform actions. Therefore, now agent is tasked with learning to act at states that   \n149 are most important for maximising its total return given the presence of the cost for each action. Now   \n150 at each state the agent first makes a binary decision to decide to perform an action.   \n151 Our framework, LICRA consists of two core components: firstly a RL process $\\mathfrak { g } : \\mathcal { S } \\times \\{ 0 , 1 \\}  [ 0 , 1 ]$   \n152 and a second RL process $\\pi : S \\times A \\to [ 0 , 1 ]$ . The role of $\\mathfrak { g }$ is to determine whether or not an action   \n153 is to be performed by the policy $\\pi$ at a given state $s$ . If activated, the policy $\\pi$ determines the action   \n154 to be selected. Prior to decisions being made, the policy $\\pi$ communicates to $\\mathfrak { g }$ the action it would   \n155 take. An important feature of our LICRA is the sequential decision process. In LICRA, the policy   \n156 $\\pi$ first proposes an action $a \\in { \\mathcal { A } }$ which is observed by the policy $\\mathfrak { g }$ . Therefore, the role of $\\mathfrak { g }$ is to   \n157 prevent actions for which the change in expected future rewards does not exceed the costs incurred   \n158 for taking such actions. By isolating a decision policy over whether an action should be taken or not,   \n159 the impulse controls mechanism results in a framework in which the problem facing the agent has a   \n160 markedly reduced decision space (in comparison to a standard RL method). Crucially, the agent must   \n161 compute optimal actions at only a subset of states which are chosen by the policy g. Below is the   \n162 pseudocode for LICRA, we provide full details of the code in Sec. 9 of the Appendix. ",
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+        "text": "Algorithm 1: Learnable Impulse Control Reinforcement Algorithm (LICRA) ",
+        "text_level": 1,
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+        "text": "1: Input: Stepsize $\\alpha$ , batch size $B$ , episodes $K$ , steps per episode $T$ , mini-epochs $e$   \n2: Initialise: Policy network (acting) $\\pi$ , Policy network (switching) $\\mathfrak { g }$ ,   \nCritic network (acting $) V _ { \\pi }$ ,Critic network (switching $) V _ { { \\mathfrak { g } } }$   \n3: Given reward objective function, $R$ , initialise Rollout Buffers $B _ { \\pi }$ , $B _ { \\mathfrak { g } }$ (use Replay Buffer for   \nSAC)   \n4: for $N _ { e p i s o d e s }$ do   \n5: Reset state $s _ { 0 }$ , Reset Rollout Buffers $B _ { \\pi }$ , $B _ { \\mathfrak { g } }$   \n6: for $t = 0 , 1 , \\ldots$ do   \n7: Sample $a _ { t } \\sim \\pi ( \\cdot | s _ { t } )$   \n8: Sample $g _ { t } \\sim \\mathfrak { g } ( \\cdot | s _ { t } )$   \n9: if $g _ { t } = 0$ then   \n10: Apply $a _ { t }$ so $s _ { t + 1 } \\sim P ( \\cdot | a _ { t } , s _ { t } ) .$ ,   \n11: Receive rewards $r _ { t } = \\mathcal { R } ( s _ { t } , a _ { t } )$   \n12: Store $( s _ { t } , a _ { t } , s _ { t + 1 } , r _ { t } )$ in $B _ { \\pi }$   \n13: else   \n14: Apply the null action so $s _ { t + 1 } \\sim P ( \\cdot | 0 , s _ { t } )$ ,   \n15: Receive rewards $r _ { t } = \\mathcal { R } ( s _ { t } , 0 )$ .   \n16: end if   \n17: Store $( s _ { t } , g _ { t } , s _ { t + 1 } , r _ { t } )$ in $B _ { \\mathfrak { g } }$   \n18: end for   \n19: // Learn the individual policies   \n20: Update policy $\\pi$ and critic $V _ { \\pi }$ networks using $\\mathfrak { B } _ { \\pi }$   \n21: Update policy $\\mathfrak { g }$ and critic $V _ { \\mathfrak { g } }$ networks using ${ \\mathfrak { B } } _ { { \\mathfrak { g } } }$   \n22: end for   \n165 Although action space cardinality does not change there are still benefits of using impulse control   \n166 mechanism. This mechanism forces the agent to first determine the set of states to perform actions   \n167 only then determine the optimal actions at these states. An important fact to note is that the decision   \n168 space for the determining whether or not to execute an action is $S \\times \\{ 0 , 1 \\}$ i.e at each state it makes a   \n169 binary decision. Consequently, the learning process for aspect is much quicker than a policy which   \n170 must optimise over a decision space which is $| { \\cal S } | | { \\cal A } |$ (choosing an action from its action space at   \n171 every state). This results in the agent rapidly learning which states to focus on to learn which actions   \n172 to perform. In the case of $\\pi$ with a continuous action space again the impulse control mechanism   \n173 does not change the cardinality of the action space. However, if the set ${ \\cal S } / { \\cal S } _ { I }$ , where the optimal   \n174 policy chooses 0, is large enough, then again it can be more efficient to learn $\\mathfrak { g }$ first and only then   \n175 learn $\\pi$ (we later validate this claim empirically, see Sec. 11.2),   \n176 In Sec. 5, we prove the convergence properties of LICRA. LICRA consists of two independent   \n177 procedures: a learning process for the policy $\\pi$ and simultaneously, a learning process for the impulse   \n178 policy $\\mathfrak { g }$ which determines at which states to perform an action. In our implementation, we used   \n179 proximal policy optimisation (PPO) [27] for the policy $\\pi$ and for the impulse policy $\\mathfrak { g }$ , whose action   \n180 set consists of two actions (intervene or do not intervene) we used a soft actor critic (SAC) process   \n181 [14] LICRA is a plug $\\&$ play framework which enables these RL components to be replaced with any   \n182 RL algorithm of choice. ",
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+        "type": "text",
+        "text": "183 5 Convergence and Optimality of LICRA ",
+        "text_level": 1,
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+        "text": "184 A key aspect of our framework is the presence of two RL processes that make decisions in a sequential   \n185 order. In order to determine when to act the policy $\\mathfrak { g }$ must learn the states to allow the policy $\\pi$ to   \n186 perform an action which the policy $\\pi$ must learn to select optimal actions whenever it is allowed to   \n187 execute an action.   \n188 In this section, we prove that LICRA converges to an optimal solution of the system. Central to   \n189 LICRA is a Q-learning type method which is adapted to handle RL settings in which the agent must   \n190 also learn when to act. We then extend the result to allow for (linear) function approximators. We   \n191 provide a result that shows the optimal intervention times are characterised by an ‘obstacle condition’   \n192 which can be evaluated online therefore allowing the $\\mathfrak { g }$ . ",
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+        "type": "text",
+        "text": "Given a function $Q \\because S \\times \\mathcal { A } \\to \\mathbb { R } , \\quad \\forall \\pi , \\pi ^ { \\prime } \\in \\Pi$ and $\\forall \\ S , \\ S ^ { \\prime } , \\quad \\forall s _ { \\tau _ { k } } \\quad \\in \\quad S$ , we define the intervention operator $\\mathcal { M } ^ { \\pi , { \\mathfrak { g } } }$ by $\\begin{array} { r } { \\mathcal { M } ( ^ { \\pi , \\oplus } Q ^ { \\pi ^ { \\prime } , \\oplus ^ { \\prime } } ( s _ { \\pi _ { k } } , a _ { \\pi _ { k } } ) : = { \\mathcal R } ( s _ { \\pi _ { k } } , a _ { \\pi _ { k } } ) - c ( s _ { \\pi _ { k } } , a _ { \\pi _ { k } } ) + } \\end{array}$ $\\begin{array} { r } { \\gamma \\sum _ { s ^ { \\prime } \\in S } P ( s ^ { \\prime } ; a _ { \\tau _ { k } } , s ) v ^ { \\pi ^ { \\prime } , \\mathfrak { g ^ { \\prime } } } ( s ^ { \\prime } ) \\| a _ { \\tau _ { k } } \\sim \\pi ( \\cdot | s _ { \\tau _ { k } } ) , } \\end{array}$ , where $\\tau _ { k }$ is an intervention time. ",
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+        "text": "196 The interpretation of $\\mathcal { M }$ is the following: suppose that the agent is using the policy $\\pi$ and at time $\\tau _ { k }$   \n197 the system is at a state $s _ { \\tau _ { k } }$ and the agent performs an action $a _ { \\tau _ { k } } \\sim \\pi ( \\cdot | s _ { \\tau _ { k } } )$ . A cost of $c ( s _ { \\tau _ { k } } , a _ { \\tau _ { k } } )$   \n198 is then incurred by the agent and the system transitions to $s ^ { \\prime } \\sim P ( \\cdot ; a _ { \\tau _ { k } } , s _ { \\tau _ { k } } )$ . Lastly, recall $v ^ { \\pi , { \\mathfrak { g } } }$ is   \n199 the agent value function under the policy pair $( \\pi , g )$ . Therefore, the quantity ${ \\mathcal { M } } Q ^ { \\pi , { \\mathfrak { g } } }$ measures the   \n200 expected future stream of rewards after an immediate intervention minus the cost of intervention.   \n201 This object plays a crucial role in the LICRA framework which as we later discuss, exploits the cost   \n202 structure of the problem to determine when the agent should perform an intervention. ",
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+        "type": "text",
+        "text": "Given a function 203 $v ^ { n , \\ell } : S  \\mathbb { \\quad }$ , we define the Bellman operator $\\boldsymbol { \\cdot }$ , by: ",
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+        "type": "text",
+        "text": "204 ",
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+        "img_path": "images/1d22f029be7fff5695370a24898dcff685b2ed179b08bdb4d9a1525dd068cfd4.jpg",
+        "text": "$$\nT v ^ { \\pi , \\theta } ( s ) : = \\operatorname* { m a x } \\Big \\{ \\mathcal { N } ^ { ( \\pi , \\theta ) } Q ^ { \\pi , \\theta } ( s , a ) , \\mathcal { R } ( s , 0 ) + \\gamma \\sum _ { s ^ { \\prime } \\in S } P ( s ^ { \\prime } ; 0 , s ) v ^ { \\pi , \\theta } ( s ^ { \\prime } ) \\Big \\} , \\qquad \\forall s \\in \\mathcal { S } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "205 ",
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+        "type": "text",
+        "text": "206 The Bellman operator captures the nested sequential structure of the LICRA algorithm. In particular,   \n207 the structure in (3) consists of an inner structure which consists of two terms: the first term is the   \n208 expected future return given an action is taken at the current state under the policy $\\pi$ . The second term   \n209 is the expected future return given no action is taken at the current state. Lastly, the outer structure is   \n210 an optimisation which compares the expected return of the two possibilities and selects the maximum. ",
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+        "type": "text",
+        "text": "11 Our first result proves $T$ is a contraction operator in particular, the following bound holds: ",
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+        "type": "text",
+        "text": "212 Lemma 1 The Bellman operator $T$ is a contraction, that is the following bound holds: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/cf50390db76541add22697b97f1f8bef43214a39a5b762388d0195cd6f10c1c6.jpg",
+        "text": "$$\n\\begin{array} { r } { \\| T v - T v ^ { \\prime } \\| \\leq \\gamma \\left. v - v ^ { \\prime } \\right. , } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where 13 $v , v ^ { \\prime }$ are elements of a finite normed vector space. We can now state our first main result: ",
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+        "text": "Theorem 1 Given any $v ^ { \\pi , \\mathfrak { g } } : \\mathcal { S } \\times \\mathcal { A }  \\mathbb { R } ,$ , the optimal value function is given by $\\operatorname* { l i m } _ { k \\to \\infty } T ^ { k } v ^ { \\pi , \\mathfrak { g } } =$ $\\operatorname* { m a x } _ { \\hat { \\pi } , \\hat { g } \\in \\Pi } v ^ { \\hat { \\pi } , \\hat { g } } = v ^ { \\pi ^ { \\star } , g ^ { \\star } }$ where $( \\pi ^ { \\star } , g ^ { \\star } )$ is the optimal policy pair. ",
+        "bbox": [
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+        "type": "text",
+        "text": "The result of Theorem 1 enables the solution to the agent’s impulse control problem to be determined using a value iteration procedure. Moreover, Theorem 1 enables a Q-learning approach [6] for finding the solution to the agent’s problem. ",
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+        "type": "text",
+        "text": "219 Theorem 2 Consider the following $Q$ learning variant: ",
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+        "type": "equation",
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+        "text": "$$\n+ \\alpha _ { t } ( s _ { t } , a _ { t } ) \\left[ \\operatorname* { m a x } \\left\\{ \\mathcal { M } ^ { \\pi , \\mathfrak { g } } Q _ { t } ( s _ { t } , a _ { t } ) , \\mathcal { R } ( s _ { t } , 0 ) + \\gamma _ { a ^ { \\prime } \\in A } ^ { \\operatorname* { m a x } } Q _ { t } ( s _ { t + 1 } , a ^ { \\prime } ) \\right\\} - Q _ { t } ( s _ { t } , a _ { t } ) \\right] ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "then 220 $Q _ { t }$ converges to $Q ^ { \\star }$ with probability 1, where $s _ { t } , s _ { t + 1 } \\in S$ and $a _ { t } \\in \\mathcal A$ . ",
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+        "type": "text",
+        "text": "21 We now extend the result to (linear) function approximators: ",
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+        "text": "Theorem 3 Given a set of linearly independent basis functions $\\Phi = \\{ \\phi _ { 1 } , \\ldots , \\phi _ { p } \\}$ with $\\phi _ { k } \\in L _ { 2 } , \\forall k$ . LICRA converges to a limit point $r ^ { \\star } \\in \\mathbb { R } ^ { p }$ which is the unique solution to $\\Pi \\mathfrak { F } ( \\Phi r ^ { \\star } ) = \\Phi r ^ { \\star }$ where $\\mathfrak { F } v : = \\mathcal { R } + \\gamma P \\operatorname* { m a x } \\{ \\mathcal { M } v , v \\}$ . Moreover, $r ^ { \\star }$ satisfies: $\\| \\Phi r ^ { \\star } - Q ^ { \\star } \\| \\leq ( 1 - \\gamma ^ { 2 } ) ^ { - 1 / 2 } \\| \\Pi Q ^ { \\star } - Q ^ { \\star } \\|$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The theorem establishes the convergence of LICRA to a stable point with the use of linear function approximators. The second statement bounds the proximity of the convergence point by the smallest approximation error that can be achieved given the choice of basis functions. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Having constructed a procedure to find the optimal agent’s optimal value function, we now seek to determine the conditions when an intervention should be performed. Let us denote by $\\{ \\tau _ { k } \\} _ { k \\ge 0 }$ the points at which the agent decides to act or intervention times, so for example if the agent chooses to perform an action at state $s _ { 6 }$ and again at state $s _ { 8 }$ , then $\\tau _ { 1 } = 6$ and $\\tau _ { 2 } = 8$ . The following result characterises the optimal intervention policy $\\mathfrak { g }$ and the optimal times $\\{ \\tau _ { k } \\} _ { k \\ge 0 }$ . ",
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+        "text": "Proposition 1 The policy g is given by: $\\mathfrak { g } ( s _ { t } ) = H ( \\mathcal { M } ^ { \\pi , \\mathfrak { g } } Q ^ { \\pi , \\mathfrak { g } } - Q ^ { \\pi , \\mathfrak { g } } ) ( s _ { t } , a _ { t } ) , \\forall s _ { t } \\in \\mathcal { S } _ { \\mathrm { ~ \\Gamma ~ } }$ , where $Q ^ { \\pi , { \\mathfrak { g } } }$ is the solution in Theorem $\\cdot$ , $\\mathcal { M }$ is the intervention operator and $H$ is the Heaviside function, moreover the intervention times are $\\tau _ { k } = \\operatorname* { i n f } \\{ \\tau > \\tau _ { k - 1 } | \\mathcal { M } ^ { \\pi , { \\mathfrak { g } } } Q ^ { \\pi , { \\mathfrak { g } } } = Q ^ { \\pi , { \\mathfrak { g } } } \\}$ . ",
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+    {
+        "type": "text",
+        "text": "Prop. 1 characterises the (categorical) distribution $\\mathfrak { g }$ . Moreover, given the function $Q$ , the times $\\{ \\tau _ { k } \\}$ can be determined by evaluating if $\\mathcal { M } Q = Q$ holds. ",
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+        "type": "text",
+        "text": "A key aspect of Prop. 1 is that it exploits the cost structure of the problem to determine when the agent should perform an intervention. In particular, the equality $\\mathcal { M } Q = Q$ implies that performing an action and incurring a cost for doing so is optimal. ",
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+        "type": "text",
+        "text": "6 Budget Augmented LICRA via State Augmentation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We now tackle the problem of RL with a budget. To do this, we combine the above impulse control technology with state augmentation technique proposed in [28] The mathematical formulation of the problem is now given by the following for any $s \\in S$ : ",
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+        "text": "$$\n\\operatorname* { m a x } _ { \\pi \\in \\Pi , g } ~ \\upsilon ^ { \\pi , \\mathfrak { g } } ( s ) \\ \\mathrm { ~ s . ~ t . ~ } n - \\sum _ { t = 0 } ^ { \\infty } \\sum _ { k \\geq 1 } \\delta _ { \\tau _ { k } } ^ { t } \\geq 0 ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $n \\in \\mathbb N$ is a fixed value that represents the maximum number of allowed interventions and $\\sum k > 1 8 _ { T k } ^ { t }$ is equal to one if an impulse was applied at time $t$ and zero if it was not. In order to avoid dealing with a constrained MDP, we propose to introduce a new variable $z _ { t }$ tracking the remaining number of impulses: $\\begin{array} { r } { z _ { t } = n - \\sum _ { i = 0 } ^ { t - 1 } \\sum _ { k \\geq 1 } \\delta _ { \\tau _ { k } } ^ { i } } \\end{array}$ . We treat $z _ { t }$ as another state and augment the state-space resulting in the transition $\\widetilde { \\mathcal P }$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/6346be3fbe5274a0800604dd10bbe7c0ae289219dc95d9e13ea962d4639c0036.jpg",
+        "text": "$$\ns _ { t + 1 } \\sim P ( \\cdot | s _ { t } , a _ { t } ) , \\qquad z _ { t + 1 } = z _ { t } - \\sum _ { k \\geq 1 } \\delta _ { \\tau _ { k } } ^ { t } , \\quad z _ { 0 } = n .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In order to avoid violations, we reshape the reward as follows: 252 $\\begin{array} { r } { \\widetilde { \\mathcal { R } } ( s _ { t } , z _ { t } , a _ { t } ) = \\left\\{ \\begin{array} { l l } { \\mathcal { R } ( s _ { t } , a _ { t } ) } & { z _ { t } \\ge 0 } \\\\ { - \\Delta } & { z _ { t } < 0 } \\end{array} \\right. } \\end{array}$ ,, 253 where $\\Delta > 0$ is a large enough hyper-parameter ensuring that there are no safety violations. To 254 summarise we aim to solve the following problem: ",
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+        "img_path": "images/f7782e97a7c6f6b939a51b9531f333bd46fa03de6e45a900410f8a587c3fa221.jpg",
+        "text": "$$\nv ^ { \\pi , \\mathfrak { g } } ( s , z ) = \\mathbb { E } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\widetilde { \\mathcal { R } } ( s _ { t } , z _ { t } , a _ { t } ) | a _ { t } \\sim \\pi ( \\cdot | s _ { t } , z _ { t } ) \\right] ,\n$$",
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+        "type": "text",
+        "text": "where the policy now depends on the variable $z _ { t }$ . Note that $\\widetilde { \\mathcal P }$ in Equation 6 is a Markov process and, the rewards $\\widetilde { \\mathcal { R } }$ are bounded, as long as the rewards $\\mathcal { R }$ are bounded. Therefore, we can apply directly the results for impulse control to this case as well. We denote the augmented MDP by $\\widetilde { \\mathcal { M } } = \\langle \\overset { \\cdot } { S } \\times \\mathcal { Z } , \\overset { \\cdot } { A } , \\widetilde { \\mathcal { P } } , \\widetilde { R } , \\gamma \\rangle$ , where $\\mathcal { Z }$ is the space of the augmented state. We have the following. ",
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+        "type": "text",
+        "text": "Theorem 4 Consider the MDP $\\widetilde { \\mathcal { M } }$ for the problem 7, then: ",
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+        "text": "a) The Bellman equation holds, i.e. there exists a function $\\tilde { v } ^ { * , \\pi , 9 }$ s.th. $\\begin{array} { r l } { \\tilde { v } ^ { * , \\pi , \\mathfrak { g } } ( s , z ) } & { { } = } \\end{array}$ $\\operatorname* { m a x } _ { \\mathbf { a } \\in \\mathcal { A } } \\Big ( \\widetilde { \\mathcal { R } } ( s , z , \\pmb { a } ) + \\gamma \\mathbb { E } _ { s ^ { \\prime } , z ^ { \\prime } \\sim \\mathcal { P } } \\left[ \\widetilde { v } ^ { * , \\pi , \\mathfrak { g } } ( s ^ { \\prime } , z ^ { \\prime } ) \\right] \\Big )$ , where the optimal policy for $\\widetilde { \\mathcal { M } }$ has the form $\\pi ^ { * } ( \\cdot | s , z )$ ; ",
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+        "text": "$b$ ) Given a ${ \\widetilde { v } } : { \\mathcal { S } } \\times { \\mathcal { Z } } \\to \\mathbb { R } ,$ , the stable point solution for $\\widetilde { \\mathcal { M } }$ is a given by $\\operatorname* { l i m } _ { k  \\infty } \\tilde { T } ^ { k } \\widetilde { v } ^ { \\pi , g } = \\operatorname* { m a x } _ { \\hat { \\pi } \\in \\Pi , \\hat { g } } \\hat { \\pi } ^ { \\hat { \\pi } , \\hat { g } } =$ $\\widetilde { v } ^ { * , \\pi , { \\mathfrak { g } } ^ { * } }$ , where $( \\pi ^ { * } , { \\mathfrak { g } } ^ { * } )$ is an optimal policy of $\\widetilde { \\mathcal { M } }$ and $\\tilde { T }$ is the Bellman operator of $\\widetilde { \\mathcal { M } }$ . ",
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+        "type": "text",
+        "text": "The result has several important implications. The first is that we can use a modified version of LICRA to obtain the solution of the problem while guaranteeing convergence (under standard assumptions). Secondly, our state augmentation procedure admits a Markovian representation of the optimal policy. ",
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+        "type": "text",
+        "text": "69 7 Experiments ",
+        "text_level": 1,
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+        "text": "270 We will now study empirically the performance of the LICRA framework. In experiments, we use   \n271 different instances of LICRA, one where both policies are trained using PPO update (referred to   \n272 as LICRA_PPO) and one where the policy deciding whether to act is trained using SAC and the   \n273 other policy trained with PPO (referred to as LICRA_SAC). We have benchmarked both of these   \n274 algorithms together with common baselines on environments, where it would be natural to introduce   \n275 the concept of the cost associated with actions. We lastly performed a series of ablation studies which   \n276 test LICRA’s ability to handle different cost functions including the case when $c ( s , a ) \\equiv 0$ which we   \n277 defer to the Appendix which also contains further experiment details.   \n278 Merton’s Portfolio Problem with Transaction Costs. Merton Investment Problem in which the   \n279 investor faces transaction costs [10] is a well-known problem within finance. In our environment, the   \n280 agent can decide to move its wealth between a risky asset and a risk-free asset. The agent receives a   \n281 reward only at the final step, equal to the utility of the portfolio with a risk aversion factor equal to   \n282 0.5. If the final wealth of risky asset is $s _ { T }$ and final wealth of risk-free asset is $c _ { T }$ , then the agent will   \n283 receive a reward of $u ( x ) = \\hat { 2 \\sqrt { s _ { T } + c _ { T } } }$ . The wealth evolves according to the following SDE: ",
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+        "image_caption": [
+            "Figure 1: Training results in Merton investment problem for PPO style algorithms. "
+        ],
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+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/41cceaf9456638a4b1e99b4ad0d95e76acc025817d31222486f95548cde55275.jpg",
+        "text": "$$\nd W _ { t } = ( r + p _ { t } ( \\mu - r ) ) W _ { t } + W _ { t } p _ { t } \\sigma d B _ { t }\n$$",
+        "text_format": "latex",
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+    {
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+        "img_path": "images/e4f0e42faa34e8a85dc4690b08f54b2a9dca40e02107c30188aa1c25e63de654.jpg",
+        "image_caption": [
+            "Figure 2: a) Drive Environment. b) Training results in drive environment. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "284 where $W _ { t }$ is the current wealth and the state variable, $d B _ { t }$ is an increment of Brownian motion and $p _ { t }$   \n285 is the proportion of wealth invested in the risky asset. We set the risk-free return $r = 0 . 0 1$ , risky asset   \n286 return $\\mu = 0 . 0 5$ and volatility $\\sigma = 1$ . We discretise the action space so that at each step the agent has   \n287 three actions available: move $10 \\%$ of risky asset wealth to the risk-free asset, move $10 \\%$ of risk-free   \n288 asset wealth to the risky asset or do nothing. Each time the agent moves the assets, it incurs a cost   \n289 of 1 i.e. a transaction fee. The agent can act after a time interval of 0.01 seconds and the episode   \n290 ends after 75 steps. The results of training are shown in Fig. 1 which clearly demonstrates that   \n291 LICRA_PPO finds a better policy than standard PPO. Also comparing the variance among different   \n292 seeds, we can see that LICRA_PPO is a much more stable algorithm than the other two.   \n293 Driving Environment Fuel Rationing. We studied an autonomous driving scenario where fuel  \n294 efficient driving is a priority. One of the main components of fuel-efficient driving is controlled usage   \n295 of acceleration and braking, in the sense that 1) the amount of acceleration and braking should be   \n296 limited 2) if accelerations should be performed slowly and gently. We believe this is a problem where   \n297 LICRA should thrive as the impulse control agent can learn to restrict the amount of acceleration   \n298 and braking in the presence of other cars and choose when to allow the car to decelerate naturally.   \n299 We used the highway-env [17] environment on a highway task (see Fig (2. a)) where the green   \n300 vehicle is our controlled vehicle and the goal is to avoid crashing into other vehicles whilst driving   \n301 at a reasonable speed. We add a cost function into the reward term dependent on the continuous   \n302 acceleration action, $C ( a _ { t } ) = K + a _ { t } ^ { 2 }$ , where $K > 0$ is a fixed constant cost of taking any action,   \n303 and $a _ { t } \\in [ - 1 , 1 ]$ , with larger values of acceleration or braking being penalised more. The results are   \n304 presented in Fig. (2.b). Notably, LICRA is able to massively outperform the baselines, especially our   \n305 safety specific baselines which struggle to deal with the cost function associated with the environment.   \n306 We believe one reason for the success of LICRA is that it is far easier for it to utilise the null action   \n307 of zero acceleration/braking than the other algorithms, whilst all the algorithms have a guaranteed   \n308 cost at every time step whilst not gaining a sizeable reward to counter the cost. ",
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+        "image_caption": [
+            "Figure 3: a) The lander must land on the pad between two flags. . b) Training results in Lunar Lander. Lunar Lander Environment. We tested the ability of LICRA to perform in environment that simulate real-world physical dynamics. We tested LICRA’s performance the Lunar Lander environment in OpenAI gym [7] which we adjusted to incorporate minimal bounded costs in the reward definition. In this environment, the agent is required to maintain both a good posture mid-air and reach the landing pad as quickly as possible. The reward function is given by: "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/de80a6064355eabf656cffa30e23f700882d4e6652151a389d2602021969d647.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathrm { R e w a r d } \\left( s _ { t } \\right) = 3 * \\left( 1 - \\mathbf { 1 } _ { d _ { t } - d _ { t - 1 } = 0 } \\right) - 3 * \\left( 1 - \\mathbf { 1 } _ { v _ { t } - v _ { t - 1 } = 0 } \\right) - 3 * \\left( 1 - \\mathbf { 1 } _ { \\omega _ { t } - \\omega _ { t - 1 } = 0 } \\right) } \\\\ { - 0 . 0 3 * \\mathrm { F u e l S p e n t } ( s _ { t } ) - 1 0 * \\left( v _ { t } - v _ { t - 1 } \\right) - 1 0 * \\left( \\omega _ { t } - \\omega _ { t - 1 } \\right) + 1 0 0 * \\mathrm { h a s L a n d e d } \\left( s _ { t } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "309 where $d _ { t }$ is the distance to the landing pad, $v _ { t }$ is the velocity of the agent, and $\\omega _ { t }$ is the angular   \n310 velocity of the agent at time $t$ . $\\mathbf { 1 } _ { X }$ is the indicator function of taking actions, which is 1 when the   \n311 statement $X$ is true and 0 when $X$ is false. Considering the limited fuel budget, we assume that   \n312 we have a fixed cost for each action taken by the agent here, and doing nothing brings no cost. Then,   \n313 to describe the goal of the game, we define the function of the final status by hasLanded(), which is   \n314 0 when not landing; 1 when the agent has landed softly on the landing pad; and $- 1$ when the lander   \n315 runs out of fuel or loses contact with the pad on landing. The reward function rewards the agent   \n316 for reducing its distance to the landing pad, decreasing its speed to land smoothly and keeping the   \n317 angular speed at a minimum to prevent rolling. Additionally, it penalises the agent for running out   \n318 of fuel and deters the agent from taking off again after landing.   \n319 By introducing a reward function with minimally bounded costs, our goal was to test if LICRA can   \n320 exploit the optimal policy. In Fig. 3, we observe that the LICRA agent outperforms all the baselines,   \n321 both in terms of sample efficiency and average test return (total rewards at each timestep). We also   \n322 observe that LICRA enables more stable training than PPO, PPO-Lagrangian and CPO.   \n323 Ablation Study 1. Prioritisation of Most Important Actions. We next tested LICRA’s ability to   \n324 prioritise where it performs actions when the necessity to act varies significantly between states. To   \n325 test this, we modified the Drive Environment to now consist of a single lane, a start state and a goal   \n326 state start (at the end) where there is a reward. With no acceleration, the vehicle decreases velocity.   \n327 To reach the goal, the agent must apply an acceleration $a _ { t } \\in [ - 1 , 1 ]$ . Each acceleration $a _ { t }$ incurs   \n328 a cost $C ( \\boldsymbol { a } _ { t } )$ as defined above. At zones $k = 1 , 2 , 3$ of the lane, if the vehicle is travelling below a   \n329 velocity $v _ { m i n }$ , it is penalised by a strictly negative cost $c _ { k }$ where $c _ { 1 } < c _ { 2 } < c _ { 3 }$ . As shown in Fig. 4,   \n330 when the intervention cost increases i.e. when $K  \\infty$ , LICRA successfully prioritises the highest   \npenalty zones to avoid incurring large costs. ",
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+        "image_caption": [
+            "Figure 4: Results for Ablation Study 1. Heatmaps display the number of times the agent drives below $v _ { m i n }$ in the penalty zones. Violation 1 refers to the lowest cost zone, whilst Violation 3 refers to the largest cost zone. $K$ refers to the fixed cost for taking an action. "
+        ],
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+    {
+        "type": "text",
+        "text": "8 Conclusion ",
+        "text_level": 1,
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+            169,
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+    {
+        "type": "text",
+        "text": "We presented a novel method to tackle the problem of learning how to select when to act in addition to learning which actions to execute. Our framework, which is a general tool for tackling problems of this kind seamlessly adopts RL algorithms enabling them to efficiently tackle problems in which the agent must be selective about when it executes actions. This is of fundamental importance in practical settings where performing many actions over the horizon can lead to costs and undermine the service life of machinery. We demonstrated that our solution, LICRA which at its core has a sequential decision structure that first decides whether or not an action ought to be taken under the action policy can solve tasks where the agent faces costs with extreme efficiency as compared to leading reinforcement learning methods. In some tasks, we showed that LICRA is able to solve problems that are unsolvable using current reinforcement learning machinery. We envisge that this framework can serve as the basis extensions to different settings including adversarial training for solving a variety of problems within RL. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "References ",
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+        "bbox": [
+            168,
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+            876
+        ],
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+    },
+    {
+        "type": "text",
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+    }
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