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+# NEURAL ATTENTION DISTILLATION: ERASING BACKDOOR TRIGGERS FROM DEEP NEURAL NETWORKS
+
+Yige $\mathbf { L i } ^ { 1 }$ Xixiang Lyu1† Nodens Koren2 Lingjuan Lyu3 Bo Li4 Xingjun $\mathbf { M } \mathbf { a } ^ { \mathsf { \pm } }$ 1Xidian University 2The University of Melbourne 3Ant Group 4University of Illinois at Urbana–Champaign 5Deakin University, Geelong
+
+# ABSTRACT
+
+Deep neural networks (DNNs) are known vulnerable to backdoor attacks, a training time attack that injects a trigger pattern into a small proportion of training data so as to control the model’s prediction at the test time. Backdoor attacks are notably dangerous since they do not affect the model’s performance on clean examples, yet can fool the model to make incorrect prediction whenever the trigger pattern appears during testing. In this paper, we propose a novel defense framework Neural Attention Distillation (NAD) to erase backdoor triggers from backdoored DNNs. NAD utilizes a teacher network to guide the finetuning of the backdoored student network on a small clean subset of data such that the intermediate-layer attention of the student network aligns with that of the teacher network. The teacher network can be obtained by an independent finetuning process on the same clean subset. We empirically show, against 6 state-of-the-art backdoor attacks, NAD can effectively erase the backdoor triggers using only $5 \%$ clean training data without causing obvious performance degradation on clean examples. Our code is available at https://github.com/bboylyg/NAD.
+
+# 1 INTRODUCTION
+
+In recent years, deep neural networks (DNNs) have been widely adopted into many important realworld and safety-related applications. Nonetheless, it has been demonstrated that DNNs are prone to potential threats in multiple phases of their life cycles. A type of well-studied adversary is called the adversarial attack (Szegedy et al., 2013; Goodfellow et al., 2014; Ma et al., 2018; Jiang et al., 2019; Wang et al., 2019b; 2020; Duan et al., 2020; Ma et al., 2020). At test time, state-of-the-art DNN models can be fooled into making incorrect predictions with small adversarial perturbations (Madry et al., 2018; Carlini & Wagner, 2017; Wu et al., 2020; Jiang et al., 2020). DNNs are also known to be vulnerable to another type of adversary known as the backdoor attack. Recently, backdoor attacks have gained more attention due to the fact it could be easily executed in real scenarios (Gu et al., 2019; Chen et al., 2017). Intuitively, backdoor attack aims to trick a model into learning a strong correlation between a trigger pattern and a target label by poisoning a small proportion of the training data. Even trigger patterns as simple as a single pixel (Tran et al., 2018) or a black-white checkerboard (Gu et al., 2019) can grant attackers full authority to control the model’s behavior.
+
+Backdoor attacks can be notoriously perilous for several reasons. First, backdoor data could infiltrate the model on numerous occasions including training models on data collected from unreliable sources or downloading pre-trained models from untrusted parties. Additionally, with the invention of more complex triggers such as natural reflections (Liu et al., 2020b) or invisible noises (Liao et al., 2020; Li et al., 2019; Chen et al., 2019c), it is much harder to catch backdoor examples at test time. On top of that, once the backdoor triggers have been embedded into the target model, it is hard to completely eradicate their malicious effects by standard finetuning or neural pruning (Yao et al., 2019; Li et al., 2020b; Liu et al., 2020b). A recent work also proposed the mode connectivity repair (MCR) to remove backdoor related neural paths from the network (Zhao et al., 2020a). On the other hand, even though detection-based approaches have been performing fairly well on identifying backdoored models (Chen et al., 2019a; Tran et al., 2018; Chen et al., 2019b; Kolouri et al., 2020), the identified backdoored models still need to be purified by backdoor erasing techniques.
+
+In this work, we propose a novel backdoor erasing approach, Neural Attention Distillation (NAD), for the backdoor defense of DNNs. NAD is a distillation-guided finetuning process motivated by the ideas of knowledge distillation (Bucilua et al., 2006; Hinton et al., 2014) and neural attention transfer (Zagoruyko & Komodakis, 2017; Huang & Wang, 2017; Heo et al., 2019). Specifically, NAD utilizes a teacher network to guide the finetuning of a backdoored student network on a small subset of clean training data so that the intermediate-layer attention of the student network is wellaligned with that of the teacher network. The teacher network can be obtained from the backdoored student network via standard finetuning using the same clean subset of data. We empirically show that such an attention distillation step is far more effective in removing the network’s attention on the trigger pattern in comparison to the standard finetuning or the neural pruning methods.
+
+Our main contributions can be summarized as follows:
+
+• We propose a simple yet powerful backdoor defense approach called Neural Attention Distillation (NAD). NAD is by far the most comprehensive and effective defense against a wide range of backdoor attacks.   
+• We suggest that attention maps can be used as an intuitive way to evaluate the performance of backdoor defense mechanisms due to their ability to highlight backdoored regions in a network’s topology.
+
+# 2 RELATED WORK
+
+Backdoor Attack. Backdoor attack is a type of attack emerging in the training pipeline of DNNs. Oftentimes, a backdoor attack is accomplished by designing a trigger pattern with (poisoned-label attack) (Gu et al., 2019; Chen et al., 2017; Liu et al., 2018b) or without (clean-label attack) (Shafahi et al., 2018; Turner et al., 2019; Liu et al., 2020b) a target label injected into a subset of training data. These trigger patterns can appear in forms as simple as a single pixel (Tran et al., 2018) or a tiny patch (Chen et al., 2017), or in more complex forms such as sinusoidal strips (Barni et al., 2019) and dynamic patterns (Li et al., 2020c; Nguyen & Tran, 2020). Trigger patterns may also appear in the form of natural reflection (Liu et al., 2020b) or human imperceptible noise (Liao et al., 2020; Li et al., 2019; Chen et al., 2019c), making them more stealthy and hard to be detected even by human inspection. Recent studies have shown that a backdoor attack can be conducted even without access to the training data (Liu et al., 2018b) or in federated learning (Xie et al., 2019; Bagdasaryan et al., 2020; Lyu et al., 2020). Surveys on backdoor attacks can be found in (Li et al., 2020a; Lyu et al., 2020).
+
+Backdoor Defense. Existing works primarily focused on two types of strategies to defend against backdoor attacks. Depending on the methodologies, a backdoor defense can be either backdoor detection or trigger erasing.
+
+Detection-based methods aim at identifying the existence of backdoor adversaries in the underlying model (Wang et al., 2019a; Kolouri et al., 2020) or filtering the suspicious samples from input data for re-training (Tran et al., 2018; Gao et al., 2019; Chen et al., 2019b). Although these methods have been performing fairly well on distinguishing whether a model has been poisoned, the backdoor effects still remain in the backdoored model. On the other hand, Erasing-based methods aim to directly purify the backdoored model by removing the malicious impacts caused by the backdoor triggers, while simultaneously maintain the model’s overall performance on clean data. A straightforward approach is to directly finetune the backdoored model on a small subset of clean data, which is typically available to the defender (Liu et al., 2018b). Nonetheless, training on only a small clean subset can lead to catastrophic forgetting (Kirkpatrick et al., 2017), where the model overfits to the subset and consequently causes substantial performance degradation. Fine-pruning (Liu et al., 2018a) alleviates this issue by pruning less informative neurons prior to finetuning the model. In such a way, the standard finetuning process can effectively erase the impact of backdoor triggers without significantly deteriorating the model’s overall performance. WILD (Liu et al., 2020a) proposed to utilize data augmentation techniques alongside distribution alignment between clean samples and their occluded versions to remove backdoor triggers from DNNs. Other techniques such as regularization (Truong et al., 2020) and mode connectivity repair (Zhao et al., 2020a) have also been explored to mitigate backdoor attacks. While promising, existing backdoor erasing methods still suffer from a number of drawbacks. Efficient methods can be evaded by the latest attacks (Liu et al., 2018a; 2020b), whereas effective methods are typically computationally expensive (Zhao et al., 2020a). In this work, we propose a novel finetuning-based backdoor erasing approach that is not only effective but efficient against a wide range of backdoor attacks.
+
+![](images/15de6974418f3039ae470101ea8522c535bc746b064eca57edca0fd971d9e84d.jpg)  
+Figure 1: The pipeline of backdoor erasing techniques. (a) The standard finetuning process, (b) our proposed NAD approach, and (c) our NAD framework using ResNet (He et al., 2016) as an example. NAD erases backdoor trigger following a two-step procedure: 1) obtain a teacher network by finetuning the backdoored network with a subset of clean training data, then 2) combine the teacher and the student through the neural attention distillation process. The attention representations are computed after each residual group, and the NAD distillation loss is defined in terms of the attention representations of the teacher and the student networks.
+
+Knowledge Distillation (KD). KD was first proposed to compress a bigger or an ensemble of welltrained network(s) into a compact smaller network (Bucilua et al., 2006; Hinton et al., 2014). In this process, the more knowledgeable network is referred to as the teacher network, and the smaller network is the student network. Feature maps and attention mechanisms have been demonstrated to be very useful in KD to supervise the training of student networks (Romero et al., 2015; Zagoruyko & Komodakis, 2017; Huang & Wang, 2017; Song et al., 2018; Ahn et al., 2019; Heo et al., 2019). They can help the student network to learn more high-quality intermediate representations, leading to an improved distillation effect and better student network performance (Romero et al., 2015; Zagoruyko & Komodakis, 2017). KD has also shown its potential in other fields such as adversarial robustness (Papernot et al., 2016), multi-granularity lip reading (Zhao et al., 2020b), and data augmentation (Bagherinezhad et al., 2018). In this work, we propose a new backdoor defense technique based on the combination of knowledge distillation and neural attention transfer.
+
+# 3 PROPOSED APPROACH
+
+In this section, we first describe the defense setting, then introduce the proposed NAD approach.
+
+Defense Setting. We adopt a typical defense setting where the defender outsourced a backdoored model from an untrusted party and is assumed to have a small subset of clean training data to finetune the model. The goals of backdoor erasing are to erase the backdoor trigger from the model while retaining the performance of the model on clean samples.
+
+# 3.1 NEURAL ATTENTION DISTILLATION
+
+Overview. We illustrate the differences between NAD and the traditional finetuning approach in Figure 1. Instead of using the finetuned network directly as our final model, we employ it as a teacher network and use it in conjunction with the original backdoored network (i.e. student network) through an attention distillation process. The job of NAD is to align neurons that are more responsive to the trigger pattern with benign neurons that only responsible for meaningful representations. The main challenge for NAD is thus to find the proper attention representations to distill. Over the next few subsections, we will define the attention representation used throughout our work formally and introduce the loss functions used in the process of attention distillation.
+
+Attention Representation. Given a DNN model $F$ , we denote the activation output at the $l$ -th layer as $F ^ { l } \in \mathbb { R } ^ { C \times \hat { H } \times W }$ with $C$ , $H$ and $W$ being the dimensions of the channel, the height, and the width of the activation map respectively. We define $\mathcal { A } : \mathbb { R } ^ { C \times H \times W }  \mathbb { R } ^ { H \times W }$ to be an attention operator that maps an activation map to an attention representation. Specifically, $\mathcal { A }$ takes a 3D activation map $F$ as input and outputs a flattened 2D tensor along the channel dimension. We explore three possible formulations of the attention operator $\mathcal { A }$ as suggested in (Zagoruyko & Komodakis, 2017):
+
+$$
+\mathcal { A } _ { \mathrm { s u m } } ( F ^ { l } ) = \sum _ { i = 1 } ^ { C } \left| F _ { i } ^ { l } \right| ; \mathcal { A } _ { \mathrm { s u m } } ^ { p } ( F ^ { l } ) = \sum _ { i = 1 } ^ { C } \left| F _ { i } ^ { l } \right| ^ { p } ; \mathcal { A } _ { \mathrm { m e a n } } ^ { p } ( F ^ { l } ) = \frac { 1 } { C } \sum _ { i = 1 } ^ { C } \left| F _ { i } ^ { l } \right| ^ { p } ,
+$$
+
+where $F _ { i } ^ { l }$ is the activation map of the $i$ -th channel, $| \cdot |$ is the absolute value function and $p > 1$ . Intuitively, $\mathcal { A } _ { \mathrm { s u m } }$ reflects all activation regions including both the benign and the backdoored neurons. $\mathcal { A } _ { s u m } ^ { p }$ is a generalized version of $\mathcal { A } _ { \mathrm { s u m } }$ that amplifies the disparities between the backdoored neurons and the benign neurons by an order of $p$ . In other words, the larger the $p$ is, the more weight is placed on the parts with highest neuron activations. $A _ { \mathrm { m e a n } }$ aligns the activation center of the backdoored neurons with that of the benign neurons by taking the mean over all activation regions. An empirical understanding of the three attention representations is provided in the experiments.
+
+Attention Distillation Loss. A detailed structure of our NAD framework is illustrated in Figure 1(c). For ResNets (He et al., 2016), we compute attention representations using one of the proposed attention functions after each group of residual blocks. The teacher network is kept fixed throughout the distillation process. The distillation loss at the $l$ -th layer of the network is defined in terms of the teacher’s and the student’s attention maps:
+
+$$
+\mathcal { L } _ { \mathrm { N A D } } \left( F _ { T } ^ { l } , F _ { S } ^ { l } \right) = \left. \frac { \mathcal { A } ( F _ { T } ^ { l } ) } { \left. \mathcal { A } ( F _ { T } ^ { l } ) \right. _ { 2 } } - \frac { \mathcal { A } ( F _ { S } ^ { l } ) } { \left. \mathcal { A } ( F _ { S } ^ { l } ) \right. _ { 2 } } \right. _ { 2 } ,
+$$
+
+where $\| \cdot \| _ { 2 }$ is the $L _ { 2 }$ norm and $\mathcal { A } ( F _ { T } ^ { l } ) / \mathcal { A } ( F _ { S } ^ { l } )$ is the computed attention maps of the teacher/student network. It is worth mentioning that the normalization of the attention map is crucial to a successful distillation (Zagoruyko & Komodakis, 2017).
+
+Overall Training Loss. The overall training loss is a combination of the cross entropy (CE) loss and the sum of the Neural Attention Distillation (NAD) loss over all $K$ residual groups:
+
+$$
+\mathcal { L } _ { t o t a l } = \mathbb { E } _ { ( \pmb { x } , \pmb { y } ) \sim \mathcal { D } } [ \mathcal { L } _ { \mathrm { C E } } ( F _ { S } ( \pmb { x } ) , \pmb { y } ) + \beta \cdot \sum _ { l = 1 } ^ { K } \mathcal { L } _ { \mathrm { N A D } } ( F _ { T } ^ { l } ( \pmb { x } ) , F _ { S } ^ { l } ( \pmb { x } ) ) ] ,
+$$
+
+where $\mathcal { L } _ { \mathrm { C E } } ( \cdot )$ measures the classification error of the student network, $\mathcal { D }$ is a subset of clean data used in finetuning, $l$ is the index of the residual group, and $\beta$ is a hyperparameter controlling the strength of the attention distillation.
+
+Before the attention distillation, a teacher network should be in place. We finetune the backdoored student network on the same clean subset $\mathcal { D }$ to obtain the teacher network. We will investigate how the choice of the teacher network affects the performance of NAD in Section 4.4. We only distill the student network once as our NAD approach is effective enough to remove the backdoor by only a few epochs of distillation. A comprehensive analysis of iteratively applied NAD is also provided in the Appendix G.
+
+# 4 EXPERIMENTS
+
+In this section, we first introduce the experimental setting. We then evaluate and compare the effectiveness of NAD with 3 existing backdoor erasing methods on 6 state-of-the-art backdoor attacks. Finally, we provide a comprehensive understanding of NAD.
+
+# 4.1 EXPERIMENTAL SETTING
+
+Backdoor Attacks and Configurations. We consider 6 state-of-the-art backdoor attacks: 1) BadNets (Gu et al., 2019), 2) Trojan attack (Liu et al., 2018b), 3) Blend attack (Chen et al., 2017), 4) Clean-label attack(CL) (Turner et al., 2019) , 5) Sinusoidal signal attack(SIG) (Barni et al., 2019), and 6) Reflection attack(Refool) (Liu et al., 2020b). For a fair evaluation, we follow the configuration, including the trigger patterns, the trigger sizes and the target labels, of these attacks in their original papers. We test the performance of all attacks and erasing methods on two benchmark datasets, CIFAR-10 and GTSRB, with WideResNet (WRN-16- $1 ^ { * }$ ) being the base model throughout the experiments. More details on attack configurations are summarized in Appendix A.
+
+Table 1: Performance of 4 backdoor defense methods against 6 backdoor attacks evaluated using the attack success rate (ASR) and the classification accuracy (ACC). The deviation indicates the $\%$ changes in ASR/ACC compared to the baseline (i.e. no defense). The experiments for Refool were done on GTSRB, while all other experiments were done on CIFAR-10. The best results are in bold.   
+
+<table><tr><td rowspan="2">Backdoor Attack</td><td colspan="2">Before</td><td colspan="2">Finetuning</td><td colspan="2">Fine-pruning</td><td colspan="2">MCR (t = 0.3)</td><td colspan="2">NAD (Ours)</td></tr><tr><td>ASR</td><td>ACC</td><td>ASR</td><td>ACC</td><td>ASR</td><td>ACC</td><td>ASR</td><td>ACC</td><td>ASR ACC</td></tr><tr><td>BadNets</td><td>100</td><td>85.65</td><td>17.18</td><td>81.22</td><td>99.73</td><td>81.14</td><td>4.65</td><td>80.94</td><td>4.77 81.17</td></tr><tr><td>Trojan</td><td>100</td><td>81.24</td><td>71.76</td><td>77.88</td><td>41.00</td><td>78.17</td><td>41.25 78.76</td><td>19.63</td><td>79.16</td></tr><tr><td>Blend</td><td>99.97</td><td>84.95</td><td>36.60</td><td>81.22</td><td>93.62</td><td>81.13</td><td>64.33 80.34</td><td>4.04</td><td>81.68</td></tr><tr><td>CL</td><td>99.21</td><td>82.43</td><td>75.08</td><td>81.73</td><td>29.88</td><td>79.32</td><td>32.95 79.04</td><td>9.18</td><td>80.34</td></tr><tr><td>SIG</td><td>99.91</td><td>84.36</td><td>9.18</td><td>81.28</td><td>74.26</td><td>81.60</td><td>1.62 80.94</td><td>2.52</td><td>81.95</td></tr><tr><td>Refool</td><td>95.16</td><td>82.38</td><td>14.38</td><td>80.34</td><td>63.49</td><td>80.64</td><td>8.76</td><td>78.84 3.18</td><td>80.73</td></tr><tr><td>Average</td><td>99.04</td><td>83.50</td><td>37.36</td><td>80.61</td><td>67.00</td><td>80.50</td><td>25.59 79.81</td><td>7.22</td><td>80.83</td></tr><tr><td>Deviation</td><td>1</td><td>-</td><td>↓61.68</td><td>↓2.89</td><td>↓32.04</td><td>↓3</td><td>↓73.44 ↓3.69</td><td>91.82</td><td>↓2.66</td></tr></table>
+
+Defense Configuration. We compare our NAD approach with 3 existing backdoor erasing methods: 1) the standard finetuning, 2) Fine-pruning (Liu et al., 2018a), and 3) mode connectivity repair (MCR) (Zhao et al., 2020a). We assume all defense methods have access to the same $5 \%$ of the clean training data.
+
+For NAD, we finetune the backdoored model (i.e. the student network) on the $5 \%$ accessible clean data for 10 epochs (results for 20 epochs can be found in Appendix J) using the Stochastic Gradient Descent (SGD) optimizer with a momentum of 0.9, an initial learning rate of 0.1, and a weight decay factor of $1 0 ^ { - 4 }$ . The learning rate is divided by 10 after every 2 epochs. We use a batch size of 64, and apply typical data augmentation techniques including random crop (padding $= 4$ ), horizontal flipping, and Cutout (n hole $\mathrm { { \ s } } = 1$ and length $^ { = 9 }$ ) (DeVries & Taylor, 2017). The data augmentations are applied to each batch of training images at each training iteration, following a typical DNN training process. The same data augmentations are also applied to other fintuning-based baseline methods. Additionally, we have a comparison of our NAD method to just using data augmentations Cutout and Mixup in Appendix B. For the distillation loss, we compute the attention maps using the A2sum attention operator after the three groups of residual blocks of WRN-16-1 (see Figure 1(c)) . An extensive study on four attention functions is given in Section 4.3. For the hyperparameter $\beta$ , we adaptively set it to different values for each backdoor attack. We provide more details on defense settings in Table 3 (see Appendix A).
+
+Evaluation Metrics. We evaluate the performance of defense mechanisms with two metrics: attack success rate (ASR), which is the ratio of backdoored examples that are misclassified as the target label, and model’s accuracy on clean samples (ACC). The more the ASR drops and the less the ACC drops, the stronger the defense mechanism is.
+
+# 4.2 EFFECTIVENESS OF OUR NAD DEFENSE
+
+In order to assess the effectiveness of our proposed NAD defense, we evaluate its performance against 6 backdoor attacks using two metrics (i.e. ASR and ACC). We then compare the performance of NAD with the other 3 existing backdoor defense methods in Table 1. Our experiment shows that our NAD defense remarkably brought the average ASR from nearly $100 \%$ down to $7 . 2 2 \%$ . In comparison, Finetuning, Fine-pruning, and MCR are only able to reduce the average ASR to $3 7 . 3 6 \%$ , $6 7 . 0 0 \%$ , and $2 5 . 5 9 \%$ respectively.
+
+MCR has an erasing effect stronger than that of the NAD’s by $0 . 1 2 \%$ on BadNets and by $0 . 9 \%$ on SIG, but its performance against the 4 other attacks are much poorer. Specifically, MCR failed to defend $6 0 . 2 9 \%$ more attack on Blend, $2 3 . 7 7 \%$ more attack on CL, and $1 8 . 7 7 \%$ more attack on Trojan in comparison to NAD. Our hypothesis on this is that backdoor triggers with much complicated adversarial noises (e.g. random or PGD perturbations) would hinder the training of connect path and consequently gives MCR a hard time at finding robust models against these backdoor attacks. Interestingly, finetuning did moderately well in erasing all kinds of attacks. A reasonable explanation to this is that the data augmentation techniques used in the early stage might have recovered some trigger-related images leading the backdoor model to unlearn the original trigger. For instance, the black edges in the zero-padding of images may have effects similar to the black-white trigger in BadNets. On the other hand, Fine-pruning gives a poor performance on mitigating all backdoor attacks under our experiment settings. We speculate that a potential reason behind this is due to the low number of neurons in the last convolutional layer of WRN-16-1, indicating a high mixing of benign neurons and backdoored neurons. This may cause a significant reduction in the classification accuracy and in consequence makes the pruning ineffective.
+
+![](images/6c09c45a2b6922d59c7c1efd10d1f100907de28b46b2905f64d769abebef35b7.jpg)  
+Figure 2: Performance of 4 backdoor erasing methods under different $\%$ of available clean data. The plots show the average ASR (left) and ACC (right) over all 6 attacks. NAD significantly reduces the ASR to nearly $0 \%$ with $20 \%$ clean data.
+
+In summary, all erasing methods have some negative effects on the ACC, but the drops by utilizing NAD is the least prominent (merely $2 . 6 6 \%$ ). More comparisons to data augmentation techniques and the effectiveness in erasing all-target and adaptive backdoor attacks can be found in Appendix B, H and Appendix K respectively.
+
+Effectiveness under Different Percentages of Clean Data. We are also interested in studying the correlation between the performance of NAD and the amount of available pristine data. Intuitively, we anticipate NAD to be stronger when we have more clean training data, and vice versa. The performance of NAD and 3 other defense mechanisms with various sizes of “purifying dataset” is recorded in Figure 2.
+
+It is within our expectation that both MCR and our proposed NAD approach are capable of defending against all 6 backdoor attacks almost $100 \%$ of the time when $20 \%$ of clean training data are available to us. Nonetheless, NAD still beats MCR in terms of the convergence rate. We will show in Appendix C that our proposed NAD approach converges much faster than MCR. Additionally, we find that finetuning becomes much more effective when the backdoor model is retrained on $20 \%$ o f the clean data. Despite the improvement, the standard finetuning method is still considerably worse than MCR and NAD by ${ \sim } 1 0 \%$ in terms of the ASR. Surprisingly, Fine-pruning gains almost no benefits from clean data additional to the $5 \%$ used in the first experiment. This is because the benign neurons and the backdoored neurons are highly blended in the last layer of the backdoored network, leading to excessive pruning of the backdoored neurons. Retraining becomes pointless when too many backdoored neurons have been removed.
+
+In short, even with just $1 \%$ of clean training data available, our NAD can still effectively bring the average ASR from $9 9 . 0 4 \%$ down to $3 5 . 9 3 \%$ , while only sacrifices $4 . 6 9 \%$ of ACC.
+
+Comparison to Trigger Recovering. Some existing works proposed defense methods that predict the distributions of backdoor triggers through generative modeling or neuron reverse engineering. The predicted distributions can subsequently be sampled to craft backdoored data with correct labels. These “remedy data” can subsequently be used in retraining to alleviate the impact of backdoor triggers (Wang et al., 2019a; Qiao et al., 2019).
+
+In this subsection, we compare the performance of our NAD defense to one such approach, MESA (Qiao et al., 2019), which is the current state-of-the-art trigger recovering method. Specifically, we are interested in comparing our method to the retraining-based method in general. To do this, we first retrain the backdoored model separately using the trigger generated by MESA (Rec-T) and the original trigger (Org-T). We then evaluate the performance of these retrained models and compare their performances to that of the NAD’s. The results are presented in Table 6 (see Appendix D).
+
+The retraining-based approach is strong at defending against the BadNets attack. It is able to reduce the ASR from $100 \%$ to under $5 \%$ without significantly sacrificing the ACC. Nonetheless, its performance is nowhere close to our NAD defense when facing the CL attack. This is a good indication that the backdoored neurons can be fixed by retraining with the remedy data when the backdoor behavior is induced by only a few backdoored neurons. On the other hand, it is too much for these remedy data to handle backdoors obtained through the combination of complex adversarial noises and stronger trigger patterns.
+
+![](images/bdefe227d14ab7ef919d5aac0fe1462360321521eff83b705299aa254a9ec375.jpg)  
+Figure 3: Visualization of the attention maps learned at each residual group of the WRN-16-1 by different defense methods for a BadNets (left) or CL (right) backdoored image (see Appendix A). Our NAD method demonstrates a more effective erasing effect at the deeper layers (e.g. Group 3).
+
+# 4.3 A COMPREHENSIVE UNDERSTANDING AND ANALYSIS OF NAD
+
+In this section, we first provide intuition behind what an attention map is from the visual perspective, we then compare the efficacy of various choices of attention representation as promised in Section 3.1. Finally, we explore the adjustment border of the hyperparameter $\beta$ .
+
+Understanding Attention Maps. The activation information of all neurons in a layer of a neural network can be referred from the attention map of that layer. The conjunct of all attention maps hence reflects the most discriminative regions in the network’s topology (Lopez et al., 2019). To give intuition on how attention maps help NAD with erasing backdoor triggers, we visualize and compare the attention maps before and after backdoor erasing in Figure 3. We use the attention function $\mathcal { A } _ { s u m } ^ { 2 }$ to derive the attention maps. For the backdoored models, all attention maps are completely (i.e. group1, group2, and group3) biased towards the backdoor trigger region, implying that the backdoor trigger can easily mislead the network to misbehave. The objective of backdoor erasing methods is consequently to relax the tension in this region. We show the results for two attacks, BadNets (left) and CL (right), to validate our hypothesis.
+
+Attention maps can also be used as an indicator to deduce the performance of backdoor erasing methods. In Figure 3, we show the attention maps of a backdoored WRN-16-1 after purified by four different mechanisms: Fine-pruning, the standard finetuning, MCR, and our NAD approach. As expected, BadNets can easily be erased by the most defenses. To see why, refer to the attention maps of group 3; the network purified by the standard finetuning, MCR, and NAD pay almost no attention to the bottom right corner where the trigger is injected. For CL, only MCR and NAD are able to distract the backdoored model’s from focusing on the triggered regions, which is also foreseeable as CL is a stronger attack. In addition, the activation intensity of NAD in the benign area (i.e. non-trigger area) is correspondingly greater than that of MCR, which also justifies our conclusion that NAD is better than MCR in erasing CL attacks from another perspective.
+
+Next, we compare the performance of NAD under scenarios where 4 different attention functions, $A _ { m e a n }$ , $\mathcal { A } _ { m e a n } ^ { 2 }$ , $\mathcal { A } _ { s u m }$ , and $\mathcal { A } _ { s u m } ^ { 2 }$ are in place. We use the BadNets attack as our benchmark attack. Again, we evaluate the performance of NAD using two metrics, the ASR and the ACC, and the results are summarized in Appendix F. Despite all choices of attention function are able to help NAD erase backdoors quite efficiently, $\mathcal { A } _ { s u m } ^ { 2 }$ achieved the best overall results. A comparison to using the raw activation map for distillation can be found in Appendix I.
+
+Attention Distillation VS. Feature Distillation. We outline two advantages of attention distillation over feature distillation: 1) Integration. The attention operators calculate the sum (or the mean) of the activation map over different channels (see Equation 1). It can thus provide an integrated measure of the overall trigger effect. On the contrary, the trigger effect may be scattered into different channels if we use the raw activation values directly. Such a difference between the attention and the feature maps is visualized in Figure 11 and 12 in Appendix I. Therefore, aligning the attention maps between the teacher and the student networks is more effective in weakening the overall trigger effect than directly aligning the raw feature maps. 2) Regularization. Due to its integration effect, an attention map contains the activation information of both backdoor-fired neurons and the benign neurons. This is important as the backdoor neurons can receive extra gradient information from the attention map even when they are not activated by the clean data. Moreover, attention maps have lower dimensions than feature maps. This makes the attention map based regularization (alignment) more easily to be optimized than feature map based regularization.
+
+![](images/b4f8214e51fdb9fabab4161d80fd9128dcac31a819d038f9fdc1878e8f0b228e.jpg)  
+Figure 4: Comparison of 4 distillation combinations on CIFAR-10. The B, B-F, and C represent backdoored model, finetuned backdoored model, and model trained on the clean subset, respectively.
+
+![](images/6f2240c7bce31e8293303da10bcabe228c634b4269faf0fbc993cff66714b6fa.jpg)  
+Figure 5: Performance of NAD with teachers trained on various $\%$ of clean CIFAR-10 data.
+
+Effect of Parameter $\beta$ . The selection of the distillation parameter $\beta$ is also a key factor for NAD to erase backdoor triggers successfully. We show the results of the coarse tuning $\beta$ for all the backdoor attacks in Appendix E, and it reveals that $\beta$ can certainly be tuned more to improve the performance of NAD. In short, the process of finding the right scaling factor $\beta$ is to find a balance between the ASR and the ACC. Even though bigger $\beta$ is more effective against backdoor attacks, Figure 8(right) in Appendix E shows that arbitrarily increasing $\beta$ may cause a degradation in the ACC. For instance, the purified network lost more than $50 \%$ of ACC when $\beta$ is set to 50,000. A practical strategy to select $\beta$ is to increase $\beta$ until the clean accuracy (right subfigure in Figure 8) drops below an acceptable threshold. This can reliably find an optimal $\beta$ , as increasing $\beta$ can always improve the robustness (left subfigure in Figure 8).
+
+# 4.4 FURTHER EXPLORATION OF NAD
+
+Here, we explore how the combinations of teachers and students and the choice of the teacher affect the defense performance of NAD. For simplicity, we define $\mathbf { B }$ to be the backdoored network, B-F to be the finetuned backdoored network, and C to be the model trained from scratch using $5 \%$ of clean training data. Here we consider 4 combinations of networks: 1) $\mathbf { B }$ teacher and B-F student, 2) B-F teacher and $\mathbf { B }$ student, 3) B-F teacher and B-F student, and 4) C teacher and $\mathbf { B }$ student.
+
+Effect of Teacher-Student Combinations. The ASR and ACC of NAD-purified networks using different teacher-students combinations are presented in Figure 4. We find that the ASR slightly diminished with the combination of $\mathbf { B }$ teacher and B-F student. On the contrary, when using a B$\mathbf { F }$ teacher and a $\mathbf { B } { \cdot } \mathbf { F }$ student in the NAD framework, the ASR is drastically lowered (significantly improved defense). Interestingly, compared with the standard setting used throughout the previous experiments (i.e. Teacher: B-F, Student: B), using a trained-from-scratch model as the teacher can also reduce the average ASR by more than $80 \%$ . Furthermore, this combination works even better in tackling the Trojan attack. Nonetheless, the combination of $\mathbf { C }$ teacher and $\mathbf { B }$ student significantly deteriorates the ACC of the purified model, which is also predictable because the amount of clean data available to train a model from scratch has a direct effect on the model’s accuracy. It is hence not a surprise that an incompetent teacher misleads the student network into learning flawed features.
+
+Table 2: Effectiveness of our NAD with different teacher architectures against BadNets on CIFAR10. ASR: attack success rate; ACC: clean accuracy. The first column highlights the architectural difference between the teacher and the student network. The best results are boldfaced.   
+
+<table><tr><td rowspan="2">Difference</td><td rowspan="2">Teacher</td><td rowspan="2">Student</td><td colspan="2">Before</td><td colspan="2">NAD (Ours)</td><td>Teacher</td></tr><tr><td>ASR</td><td>ACC</td><td>ASR</td><td>ACC</td><td>ACC</td></tr><tr><td>Depth&amp;Channel</td><td>WRN-10-2</td><td>WRN-16-1</td><td>100%</td><td>85.65%</td><td>4.68%</td><td>78.36%</td><td>63.78%</td></tr><tr><td>Same</td><td>WRN-16-1</td><td>WRN-16-1</td><td>100%</td><td>85.65%</td><td>4.55%</td><td>74.53%</td><td>61.21%</td></tr><tr><td>Channel</td><td>WRN-16-2</td><td>WRN-16-1</td><td>100%</td><td>85.65%</td><td>3.04%</td><td>78.68%</td><td>64.25%</td></tr><tr><td>Depth</td><td>WRN-40-1</td><td>WRN-16-1</td><td>100%</td><td>85.65%</td><td>2.95%</td><td>78.87%</td><td>63.35%</td></tr><tr><td>Depth&amp;Channel</td><td>WRN-40-2</td><td>WRN-16-1</td><td>100%</td><td>85.65%</td><td>3.74%</td><td>79.07%</td><td>64.53%</td></tr></table>
+
+Effect of the Choice of a Teacher. Due to the competitive performance offered by the C teacher and $\mathbf { B }$ student combination in the last section, we are interested in exploring one question: is the standard setting (i.e. B-F teacher and $\pmb { B }$ student) we used throughout previous experiments still the best option when more clean data are available to us? To answer this question, we compare the performance of both B-F teacher and C teacher to erase $\mathbf { B }$ student under various percentages of available clean data on CIFAR-10. We use the same training configurations described in Section 4.1. The results are shown in Figure 5. Compared with the finetuning option of B-F teacher, the C teacher trained on a subset of clean data also offers competitive performance. Specifically, in the case where we have $20 \%$ of clean data available to us, the training option reduces the ASR by $0 . 1 5 \%$ in comparison to the finetuning option; nonetheless, it reduces the ACC by a greater amount of $2 . 5 \%$ . This makes finetuning still a better option even when we have $20 \%$ clean data at hands.
+
+Effectiveness of Different Teacher Architectures. This experiment is conducted on CIFAR-10 against BadNets attack. We consider 5 teacher architectures: WRN-10-2, WRN-16-1, WRN-16-2, WRN-40-1 and WRN-40-2. We fix the student network to WRN-16-1. Since the teachers are of different (except one) architectures as the student, we train the teacher networks from scratch using only $5 \%$ clean training data. Our NAD is applied following the same configurations in Section 4.1. The results are reported in Table 2. We can see that all 5 teacher networks are able to purify the backdoored student effectively under our NAD framework. NAD can reduce the ASR from $100 \%$ to $4 . 6 8 \%$ even when a small teacher WRN-10-2 is used. This confirms that our NAD defense generalizes well across different network architectures. Note that the clean accuracy of NAD decreases more than in the same architecture setting (see Table 1). This is because the teachers have lower clean accuracy when trained from scratch on only $5 \%$ clean data, as we have analyzed in Figure 4.
+
+Why a finetuned teacher can purify a backdoored student? During the NAD process, the backdoored neurons in the finetuned teacher network are less likely to be activated by the clean finetuning data. Moreover, as shown in Figure 12 (Appendix I), finetuning can suppress the trigger effect, and at the same time, boost the benign neurons (more visible light green regions in Figure 12 (b)). By using the squared sum attention map $\mathcal { A } _ { s u m } ^ { 2 }$ (analyzed in Section 4.3 and Appendix I), this trigger erasing effect can accumulate from the already finetuned teacher network, leading to more robust and cleaner student network. Note that the student can still overfit to the partially purified teacher if it is overly finetuned.
+
+# 5 CONCLUSION
+
+In this work, we proposed a novel knowledge distillation based backdoor defense framework Neural Attention Distillation (NAD). We demonstrated empirically that our proposed approach is able to achieve a superior performance against 6 state-of-the-art backdoor attacks in comparison to 3 other backdoor defense methods. Additionally, we propose the use of attention maps as an intuitive way to evaluate the performance of backdoor defense mechanisms due to their capability of displaying backdoored regions in a network’s topology visually. On top of that, we explored how different experimental settings might affect our proposed method. Empirical results demonstrate that our results are fairly resilient to the changes in experimental settings and thus can be conveniently employed without exhaustive hyperparameter tuning. Overall, our proposed NAD backdoor defense framework provides a strong baseline in mitigating the backdoor threat in model deployment.
+
+# ACKNOWLEDGEMENT
+
+This work is supported by China National Science Foundation under grant number 62072356 and Amazon research award.
+
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+
+# A MORE IMPLEMENTATION DETAILS
+
+The backdoor triggers used in our experiments are shown in Figure 6.
+
+![](images/27e6dd18b1e61d0115c810493559cdfc5f194aaf9563d510a5be254951c89e3c.jpg)  
+Figure 6: Examples of backdoored CIFAR-10 images by the 6 attacks.
+
+Table 3: A configuration summary for the 6 backdoor attacks: datasets, models, and triggers.   
+
+<table><tr><td rowspan=1 colspan=1>Backdoork</td><td rowspan=1 colspan=1>BadNets</td><td rowspan=1 colspan=1>Trojan</td><td rowspan=1 colspan=1>Blend</td><td rowspan=1 colspan=1>Clean-Label</td><td rowspan=1 colspan=1>Signal</td><td rowspan=1 colspan=1>Refool</td></tr><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>GTSRB</td></tr><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>WideResNet</td><td rowspan=1 colspan=1>WideResNet</td><td rowspan=1 colspan=1>WideResNet</td><td rowspan=1 colspan=1>WideResNet</td><td rowspan=1 colspan=1>WideResNet</td><td rowspan=1 colspan=1>WideResNet</td></tr><tr><td rowspan=1 colspan=1>Inject Rate</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0.08</td><td rowspan=1 colspan=1>0.08</td><td rowspan=1 colspan=1>0.08</td></tr><tr><td rowspan=1 colspan=1>Trigger Type</td><td rowspan=1 colspan=1>Grid</td><td rowspan=1 colspan=1>Square</td><td rowspan=1 colspan=1>Random Noise</td><td rowspan=1 colspan=1>Grid + PGDNoise</td><td rowspan=1 colspan=1>SinusoidalSignal</td><td rowspan=1 colspan=1>Reflection</td></tr><tr><td rowspan=1 colspan=1>Trigger Size</td><td rowspan=1 colspan=1>3×3</td><td rowspan=1 colspan=1>3×3</td><td rowspan=1 colspan=1>Full Image</td><td rowspan=1 colspan=1>3×3</td><td rowspan=1 colspan=1>Full Image</td><td rowspan=1 colspan=1>Full Image</td></tr><tr><td rowspan=1 colspan=1>Target Label</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>ASR</td><td rowspan=1 colspan=1>100.00%</td><td rowspan=1 colspan=1>100.00%</td><td rowspan=1 colspan=1>99.97%</td><td rowspan=1 colspan=1>99.21%</td><td rowspan=1 colspan=1>99.91%</td><td rowspan=1 colspan=1>95.16%</td></tr><tr><td rowspan=1 colspan=1>ACC</td><td rowspan=1 colspan=1>85.65%</td><td rowspan=1 colspan=1>81.24%</td><td rowspan=1 colspan=1>84.95%</td><td rowspan=1 colspan=1>82.43%</td><td rowspan=1 colspan=1>84.36%</td><td rowspan=1 colspan=1>82.38%</td></tr></table>
+
+Detailed implementation on 6 state-of-the-art backdoor attacks:
+
+• BadNets: The trigger is a $3 \times 3$ checkerboard (pixel values are 128 or 255) at the bottom right corner of images. We labeled the backdoor examples with a chosen target label and achieved an attack success rate of $100 \%$ with an injection rate of $10 \%$ .   
+• Trojan attack. We follow the method proposed in the paper to reverse engineer a $3 \times 3$ square trigger from the last fully-connected layer of the network. In order to reduce the impact on clean accuracy, we poisoned only $5 \%$ of training data with the reverse-engineered Trojan trigger. We achieved an attack success rate of $100 \%$ with an injection rate of $5 \%$ .   
+• Blend attack. We used the random patterns reported in the original paper. We achieved an attack success rate of $9 9 . 9 7 \%$ with an injection rate of $10 \%$ and a blend ratio of $\alpha = 0 . 2$ .   
+• Clean-label attack (CL). We followed the same settings as reported in the paper. Specifically, we used Projected Gradient Descent (PGD) to generate adversarial perturbations bounded to $L _ { \infty }$ maximum perturbation $\epsilon = 0 . 1 5$ . The trigger is a $3 \times 3$ grid at the bottom right corner of images. We achieved an attack success rate of $9 9 . 2 1 \%$ with an injection rate of $8 \%$ .
+
+• Sinusoidal signal attack (SIG). We generate the backdoor trigger following the horizontal sinusoidal function defined in their paper with $\Delta = 2 0$ and $f = 6$ . We achieved an attack success rate of $9 9 . 9 1 \%$ with an injected rate of $8 \%$ .
+
+• Reflection attack (Refool). The implementation is based on the open-source code†. We achieved an attack success rate of $9 5 . 1 6 \%$ with an injection rate of $8 \%$ .
+
+More Details on Defense Baselines We adopted the same settings used in NAD for the standard finetuning approach and finetuned the model until convergence. We replicated Fine-pruning‡ via PyTorch and pruned the last convolutional layer of the model as suggested in the original paper Liu et al. (2018a). For a fair comparison, the pruning ratio was set to a value such that the ACC of the pruned network matched the ACC of our NAD approach. We used the open-source code§ for mode connectivity repair (MCR) and set the endpoint model $\mathrm { \Delta t } = 0$ and $\mathbf { t } = 1$ with the same backdoored WRN-16-1. We trained the connection path for 100 epochs and evaluated the defense performance of the model on the path. Other settings of the code remain unchanged.
+
+# B COMPARISON WITH DATA AUGMENTATION TECHNIQUES
+
+Cutout (DeVries & Taylor, 2017) and Mixup (Zhang et al., 2018) are popular data augmentation methods for CNNs. Cutout masks out random sections of input images during training and Mixup randomly morphs the training images. We evaluate in this section the independent effectiveness of Mixup and Cutout in erasing backdoor triggers. For Cutout¶, we set the number of patches to be cut out of each image to 1 and each patch is a $3 \times 3$ square. For Mixup||, we set $\alpha$ to be the default value of 1, indicating that we sample the weight uniformly between zero and one. Other settings for attacks and defenses are identical to the settings specified in Section 4.1. The results (see Table 4) can be a supplement of Table 1. We conclude that data augmentation techniques have mitigating effects on backdoors only when the transformation images are similar to the trigger patterns. They are hence not general against a wide range of backdoor attacks.
+
+Table 4: Comparison with Mixup and Cutout on erasing backdoor triggers.   
+
+<table><tr><td rowspan="2">Backdoor Attack</td><td colspan="2">Before</td><td colspan="2">Mixup</td><td colspan="2">Cutout</td><td colspan="2">NAD (Ours)</td></tr><tr><td>ASR</td><td>ACC ASR</td><td>ACC</td><td>ASR</td><td>ACC</td><td>ASR</td><td></td><td>ACC</td></tr><tr><td>BadNets</td><td>100%</td><td>85.65%</td><td>68.22%</td><td>80.27%</td><td>28.17%</td><td>82.73%</td><td>3.81%</td><td>81.85%</td></tr><tr><td>Trojan</td><td>100%</td><td>81.24%</td><td>96.20%</td><td>71.83%</td><td>50.22%</td><td>80.13%</td><td>19.63%</td><td>79.16%</td></tr><tr><td>Blend</td><td>99.97%</td><td>84.95%</td><td>99.11%</td><td>80.51%</td><td>15.30%</td><td>81.78%</td><td>3.04%</td><td>81.68%</td></tr><tr><td>CL</td><td>99.21%</td><td>82.43%</td><td>93.77%</td><td>77.13%</td><td>73.33%</td><td>81.34%</td><td>9.18%</td><td>80.34%</td></tr><tr><td>SIG</td><td>99.91%</td><td>84.36%</td><td>52.11%</td><td>79.94%</td><td>99.95%</td><td>82.77%</td><td>2.52%</td><td>81.95%</td></tr><tr><td>Refool</td><td>95.16%</td><td>82.38%</td><td>8.76%</td><td>77.84%</td><td>91.86%</td><td>80.06%</td><td>3.18%</td><td>80.73%</td></tr><tr><td>Average</td><td>99.04%</td><td>83.50%</td><td>69.69%</td><td>77.92%</td><td>59.80%</td><td>81.46%</td><td>7.22%</td><td>80.83%</td></tr><tr><td>Deviation</td><td>-</td><td>-</td><td>↓29.35%</td><td>↓5.58%</td><td>↓39.24%</td><td>↓2.03%</td><td>91.82%</td><td>↓2.66%</td></tr></table>
+
+# C MORE RESULTS OF MODE CONNECTIVITY REPAIR (MCR)
+
+We use the open-source code of MCR and compare its performance to our NAD method. The experiments are conducted on CIFAR-10 dataset using $5 \%$ clean finetune data. We first run MCR with the two endpoint models $\mathrm { \Delta t = 0 }$ and $\mathrm { t } = 1$ which use the same backdoored WRN-16-1 model. Figure 7 shows the convergence rate of MCR and our NAD against BadNets attack. We then run an additional experiment for MCR using two different endpoint models: $\mathrm { \Delta t } = 0$ and $\mathbf { t } = 1$ use the backdoored WRN-16-1 and the finetuned backdoored WRN-16-1 respectively. This result is reported in Table 5. We find that using different endpoint models can not further improve the performance of MCR.
+
+![](images/2beb1db9d63b16484cdbd23893992113d34f089731ff6988ba87e137dc2f1417.jpg)  
+Figure 7: Convergence rate comparison between MCR and NAD against BadNets attack with $5 \%$ clean training data. We show the best result of MC at the connection point $\mathrm { t } = 0 . 3$ . Note that it takes longer for MCR to converge yet its ASR is still higher than that of the NAD’s.
+
+Table 5: Performance of MCR with different endpoint models on CIFAR-10 dataset. B denotes the backdoored WRN-16-1 and F-B denotes the backdoored WRN-16-1 after Fine-tuning. MCR-(B,B) denotes the default setting where the two endpoint models are both B, while MCR-(B,F-B) denotes the MCR using two different endpoint models B and F-B. The best results are boldfaced.   
+
+<table><tr><td rowspan="2">Backdoor Attack</td><td colspan="2">Before</td><td colspan="2">MCR-(B.B)</td><td>MCR-(B,F-B)</td><td colspan="2">NAD (Ours)</td></tr><tr><td>ASR</td><td>ACC</td><td>ASR</td><td>ACC ASR</td><td>ACC</td><td>ASR</td><td>ACC</td></tr><tr><td>BadNets</td><td>100%</td><td>85.65%</td><td>4.65%</td><td>80.94%</td><td>6.00% 80.56%</td><td>4.77%</td><td>81.17%</td></tr><tr><td>Trojan</td><td>100%</td><td>81.24%</td><td>41.25%</td><td>78.76%</td><td>53.31% 78.31%</td><td>19.63%</td><td>79.16%</td></tr><tr><td>Blend</td><td>99.97%</td><td>84.95%</td><td>64.33%</td><td>80.34%</td><td>70.65% 80.51%</td><td>4.04%</td><td>81.68%</td></tr><tr><td>CL</td><td>99.21%</td><td>82.43%</td><td>32.95%</td><td>79.04%</td><td>42.66% 80.31%</td><td>9.18%</td><td>80.34%</td></tr><tr><td>SIG</td><td>99.91%</td><td>84.36%</td><td>1.62%</td><td>80.94%</td><td>7.32% 81.12%</td><td>2.52%</td><td>81.95%</td></tr><tr><td>Refool</td><td>95.15%</td><td>82.38%</td><td>8.76%</td><td>78.84%</td><td>10.95% 79.03%</td><td>3.18%</td><td>80.73%</td></tr></table>
+
+# D EXPERIMENTAL RESULTS OF TRIGGER RECOVERING TECHNIQUE
+
+Qiao et al. (2019) proposed MESA that recovers the trigger distribution via generative modeling and then removes the backdoor by model retraining. We implemented this work based on their open-source code\*\*. Note that we report the best averaging results of defense performance and we changed nothing in the code besides setting the proportion of available training data to $5 \%$ . We present the results in Table 5.
+
+Table 6: Comparison between NAD and retraining-based approaches that use both the original trigger (Org-T) and the MESA-recovered trigger (Rec-T). While all methods are able to reduce the ASR of BadNets to a similar level, NAD is able to reduce the ASR of CL by 16 more percent in comparison to the model retrained with the original trigger and by 22 more percent in comparison to the model retrained with the MESA-generated trigger.   
+
+<table><tr><td rowspan="2">Backdoor Attack</td><td colspan="2">Before</td><td colspan="2">Retrain w/ rec-T</td><td colspan="2">Retrain w/ org-T</td></tr><tr><td>ASR ACC</td><td>ASR</td><td>ACC</td><td>ASR</td><td>NAD (Ours) ASR</td><td>ACC</td></tr><tr><td>BadNets</td><td>100%</td><td>85.65% 4.96%</td><td>81.23%</td><td>3.91%</td><td>ACC 82.14%</td><td>4.77% 81.17%</td></tr><tr><td>CL</td><td>99.21% 82.43%</td><td>31.23%</td><td>79.12%</td><td>25.23%</td><td>79.57% 9.18%</td><td>80.34%</td></tr></table>
+
+# E EXPERIMENTAL RESULTS OF HYPER-PARAMETER
+
+We only give a rough estimate of $\beta$ for all the backdoor attacks in Figure 8 and it certainly provides better results by a more granular level of tuning.
+
+![](images/6f3ab0b36d62431fad3c4f37afa3d50bc3cc39a9177492866de39d633190735f.jpg)  
+β  β Figure 8: Parameter analysis: performance of our NAD approach under different $\beta$
+
+# F EXPERIMENTAL RESULTS OF DIFFERENT ATTENTION FUNCTIONS
+
+We compare the performance of NAD under scenarios where 4 different attention functions, $A _ { m e a n }$ , $\mathcal { A } _ { m e a n } ^ { 2 }$ , $\mathcal { A } _ { s u m }$ , and $\mathcal { A } _ { s u m } ^ { 2 }$ are employed. We use the BadNets attack as our benchmark attack. Again, we evaluate the performance of NAD using two metrics, the ASR and the ACC, and the results are summarized in Table 7.
+
+![](images/3839a76ea2153b2b36f85b5ca67075fdb0535da92148f6246eaf9a99a649f484.jpg)  
+Figure 9: Iterative NAD and Finetuning against Figure 10: Erasing all-target BadNets attack. BadNets.
+
+Table 7: Performance of NAD using different attention functions against BadNets on CIFAR-10 with $5 \%$ clean data. ASR: attack success rate; ACC: clean accuracy. The best results are in boldfaced.   
+
+<table><tr><td rowspan="2">Attention Function</td><td colspan="2">Amean</td><td colspan="2">Amean</td><td colspan="2">Asum</td><td rowspan="2">Aum</td></tr><tr><td>ASR</td><td>ACC</td><td>ASR ACC</td><td>ASR</td><td>ACC</td><td>ASR ACC</td></tr><tr><td>Baseline</td><td>100%</td><td>85.86%</td><td>100%</td><td>85.86%</td><td>100% 85.86%</td><td>100%</td><td>85.86%</td></tr><tr><td>Epoch 1</td><td>11.81%</td><td>66.72%</td><td>1.98%</td><td>47.52%</td><td>9.38% 68.81%</td><td>1.36%</td><td>47.25%</td></tr><tr><td>Epoch 2</td><td>16.34%</td><td>79.92%</td><td>8.86%</td><td>78.14%</td><td>10.82% 79.34%</td><td>9.81%</td><td>77.91%</td></tr><tr><td>Epoch 3</td><td>13.86%</td><td>81.83%</td><td>4.50%</td><td>81.00%</td><td>7.67% 81.69%</td><td>5.12%</td><td>81.12%</td></tr><tr><td>Epoch 4</td><td>14.16%</td><td>81.90%</td><td>5.96%</td><td>80.67%</td><td>8.39% 81.38%</td><td>5.80%</td><td>80.83%</td></tr><tr><td>Epoch 5</td><td>12.28%</td><td>81.50%</td><td>4.60%</td><td>81.30%</td><td>6.89% 81.46%</td><td>4.21%</td><td>81.55%</td></tr></table>
+
+# G EXPERIMENTAL RESULTS OF ITERATIVE NAD
+
+We evaluate whether NAD can be further improved with multiple iterations of distillation. In this experiment, we adopted the same configuration and set the iteration times to 5. Taking the BadNets attack as an example. The results in Figure 9 show that the attack rate has not been further reduced, and has even slightly increased by $2 \%$ in some cases. We hypothesize that the attentions of the backdoored neurons have been correctly aligned with the attentions of the benign neurons after a single-iteration of erasing. Whereas multiple iterations of distillation will make NAD refocus on the trigger pattern. Therefore, we believe that one-iteration of distillation is sufficient to guarantee the best result. Note that iterative Finetuning does not lead to further improvement over one-time finetuning neither.
+
+# H ERASING ALL-TARGET BACKDOOR ATTACKS
+
+Unlike a single-target attack where the goal is to misclassify all backdoored images as one prespecified target class, an all-target attack aims to misclassify every source class label as different ones (in our case, misclassify the original label $i$ as $( i + 1 ) \% 1 0 \%$ ). In this experiment, we adopted the same settings (i.e. single target attacks on BadNets) to conduct all-target attacks on the WRN16-1 network. We found that an all-target attack is a tougher task than a single-target attack. It is harder to attain a satisfactory ASR with an all-target attack. The results in Figure 10 show that NAD is able to reduce the ASR across all poison-classes (from $79 \%$ to $9 . 7 \%$ ) effectively with only $5 \%$ of clean training data.
+
+# I FEATURE MAPS V.S. ATTENTION MAPS
+
+A natural question to ask is: why attention maps instead of feature maps? This can be traced back to the field of knowledge distillation. Directly aligning the feature maps could lead to an information loss on the sample density in the space, and this could lead to a decrement in the distillation performance (Zagoruyko & Komodakis, 2017; Huang & Wang, 2017; Lopez et al., 2019). In the context of backdoor erasing, aligning the feature maps is not a good option because the backdoor neurons are only weakly, if not at all, activated by clean samples (Gu et al., 2019). In contrast, attention maps contain integrated information (see Equation 1) of both backdoored and benign neurons’ feature maps, even when the neurons are not fired. (see Table 8). Figure 11 visualizes activation maps of a backdoored image on BadNets, Finetuned BadNets with $5 \%$ of clean training data, and BadNets erased by NAD with $5 \%$ of clean training data. The attention maps aggregated across the channels using 5 different attention functions are shown in Figure 12.
+
+![](images/cab0b0adf8bff82d24ae654d530eb5337b543846c0db55c0eb03e257c43ff9bd.jpg)  
+Figure 11: The activation map of one backdoored image at Group 3 of WRN-16-1 for (a) BadNets, (b) Finetuned BadNets with $5 \%$ of clean training data, and (c) BadNets erased by our NAD with $5 \%$ of clean training data. Each small patch is a channel (64 channels in total). The small red circles highlight the regions that are fired by the trigger pattern at different channels of the activation map.
+
+Table 8: NAD using attention map versus activation map against BadNets.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Before</td><td rowspan=1 colspan=1>Activation Map</td><td rowspan=1 colspan=1>Attention Map</td></tr><tr><td rowspan=2 colspan=1>CIFAR-10 ASR(WRN-16-1)ACC</td><td rowspan=1 colspan=1>100%</td><td rowspan=1 colspan=1>98.44%</td><td rowspan=1 colspan=1>3.81%</td></tr><tr><td rowspan=1 colspan=1>85.65%</td><td rowspan=1 colspan=1>82.66</td><td rowspan=1 colspan=1>81.85%</td></tr></table>
+
+# J OVERFITTING IN NAD
+
+Here, we run NAD for a sufficiently long time (e.g. 20 epochs) to test if the student will eventually overfit to the finetuned teacher network. This experiment is conducted on CIFAR-10 against BadNets and CL attacks. We also run the Finetuning defense as a comparison. Note that, the teacher network of NAD is only finetuned for 10 epochs, following the settings in Section 4.1. As shown in Figure 13 (Appendix J), the student network of NAD can indeed overfit to the partially purified teacher network. However, this can be effectively addressed by a simple early stopping strategy: stop the finetuning when there are no significant improvements on the validation accuracy within a few epochs (e.g. at epoch 5). As shown by the green curves, the clean accuracy of NAD first drops, then quickly recovers and stabilizes at a high level within a few epochs. This also highlights the efficiency of our NAD defense as only a few epochs of finetuning is sufficient to erase the backdoor trigger.
+
+![](images/aeeff32209a60ab276d3df107bdb6073e8cb6cb95cb4fa4efe7ed6f275ff86b1.jpg)  
+Figure 12: The attention maps derived by 5 different attention functions are shown for (a) BadNet, (b) Finetuned BadNet by $5 \%$ clean training data, and (c) BadNets erased by our NAD.
+
+![](images/cf632fb5fd36da6c1178691cbf5a00cdd674adde99cce85ba8b43f546ba45d80.jpg)  
+Figure 13: The learning curves (test ASR and ACC) of the NAD student network and a Finetuning network on CIFAR-10 against BadNets (left) and CL (right). In NAD, the student network tends to overfit to the teacher network, unless an early stopping is applied based on the validation ACC.
+
+# K EFFECTIVENESS AGAINST ADAPTIVE ATTACKS
+
+A backdoor adversary may attempt to construct a more stealthy backdoor trigger that does not cause obviouse shift of the attention. To simulate this scenario, we design an adaptive version of BadNets on CIFAR-10 that attaches the trigger pattern at the center region of the image. Such an adaptive attack will only shift the attention close to the center region and has a weaker activation response. Since most of the CIFAR-10 objects are located at the center of the clean images, this adaptive attack may make the attention distillation much less effective. Figure 14 illustrates a few examples of backdoored images for this type of attack. We use a scaling parameter $\alpha \in [ 0 , 1 ]$ to adjust the pixel value of a black-white square trigger pattern (all images are normalized into the range of $[ 0 , 1 ] )$ . For example, for $\alpha = 0 . 2$ , we scale pixel values of the trigger pattern $p$ to $p \times \alpha$ . The results of our NAD against this adaptive attack are reported in Table 9. Our NAD method can still effectively erase the adaptive attack while maintaining high accuracy on clean data. Interestingly, Finetuning can only effectively erase the weaker trigger, and not as effective as NAD (especially in the case of $\alpha = 1 . 0$ attack). We conjecture this is because the center regions are more hard overwritten by the clean images used for finetuning. We leave the exploration of more advanced adaptive attacks for future work.
+
+![](images/0f1acc6d955e694c06e6564283dc5d1a15ed9f0339eb7789a3b088cfd7c33ec2.jpg)  
+Figure 14: The triggers used by an adaptive BadNets attack against our NAD under different $\alpha$ (a scaling factor of the original black-white square). The trigger patterns are all placed at the center of the image.   
+Table 9: Performance of our NAD $\mathrm { \Delta } \beta = 2 { , } 0 0 0 0$ for $\alpha = 1 . 0$ and $\beta = 1 { , } 0 0 0 0$ for other $\alpha$ ) against an adaptive BadNets attack. ASR: attack success rate; ACC: clean accuracy. The best results are boldfaced.
+
+<table><tr><td rowspan="2">Square Trigger</td><td colspan="2">Before</td><td colspan="2">Finetuning</td><td colspan="2">NAD (Ours)</td></tr><tr><td>ASR</td><td>ACC</td><td>ASR</td><td>ACC</td><td>ASR</td><td>ACC</td></tr><tr><td>α=0.2</td><td>99.85%</td><td>82.11%</td><td>7.51%</td><td>79.26%</td><td>4.92%</td><td>80.32%</td></tr><tr><td>α = 0.5</td><td>99.87%</td><td>83.04%</td><td>7.65%</td><td>77.84%</td><td>3.98%</td><td>78.91%</td></tr><tr><td>α =0.8</td><td>99.97%</td><td>82.85%</td><td>12.65%</td><td>79.91%</td><td>4.08%</td><td>80.38%</td></tr><tr><td>α = 1.0</td><td>100%</td><td>83.23%</td><td>90.77%</td><td>79.56%</td><td>5.83%</td><td>80.41%</td></tr></table>

parse/train/DILxQP08O3B/DILxQP08O3B.md

@@ -0,0 +1,276 @@
+# VTNET: VISUAL TRANSFORMER NETWORK FOR OBJECT GOAL NAVIGATION
+
+Heming $\mathbf { D } \mathbf { u } ^ { 1 , 3 }$ , Xin $\mathbf { Y } \mathbf { u } ^ { 2 * } \mathbf { \& }$ Liang Zheng1
+
+1Australian National University   
+2University of Technology Sydney   
+3CSIRO-DATA61   
+{heming.du, liang.zheng}@anu.edu.au, xin.yu@uts.edu.au
+
+# ABSTRACT
+
+Object goal navigation aims to steer an agent towards a target object based on observations of the agent. It is of pivotal importance to design effective visual representations of the observed scene in determining navigation actions. In this paper, we introduce a Visual Transformer Network (VTNet) for learning informative visual representation in navigation. VTNet is a highly effective structure that embodies two key properties for visual representations: First, the relationships among all the object instances in a scene are exploited; Second, the spatial locations of objects and image regions are emphasized so that directional navigation signals can be learned. Furthermore, we also develop a pre-training scheme to associate the visual representations with navigation signals, and thus facilitate navigation policy learning. In a nutshell, VTNet embeds object and region features with their location cues as spatial-aware descriptors and then incorporates all the encoded descriptors through attention operations to achieve informative representation for navigation. Given such visual representations, agents are able to explore the correlations between visual observations and navigation actions. For example, an agent would prioritize “turning right” over “turning left” when the visual representation emphasizes on the right side of activation map. Experiments in the artificial environment AI2-Thor demonstrate that VTNet significantly outperforms state-of-the-art methods in unseen testing environments.
+
+# 1 INTRODUCTION
+
+The goal of target-driven visual navigation is to guide an agent to reach instances of a given target category based on its monocular observations of an environment. Thus, it is highly desirable to achieve an informative visual representation of the observation, which is correlated to directional navigation signals. In this paper, we propose a Visual Transformer Network (VTNet) to achieve an expressive visual representation. In our VTNet, we develop a Visual Transformer (VT) to extract image descriptors from visual observations and then decode visual representations of the observed scenes. Then, we present a pre-training scheme to associate visual representations with directional navigation signals, thus making the representations informative for navigation. After pre-training, our visual representations are fed to a navigation policy network and we train our entire network in an end-to-end manner. In particular, our VT exploits two newly designed spatial-aware descriptors as the key and query, (i.e., a spatial-enhanced local descriptor and a positional global descriptor) and then encodes them to achieve an expressive visual representation.
+
+Our spatial-enhanced local descriptor is developed to fully take advantage of all detected objects for the exploration of spatial and category relationships among instances. Unlike the prior work (Du et al., 2020) that only leverages one instance per class to mine the category relationship, our VT is able to exploit the relationship of all the detected instances. To this end, we employ an object detector DETR (Carion et al., 2020) since features extracted from DETR not only encode object appearance information, such as class labels and bounding boxes, but also contain the relations between instances and global contexts. Moreover, DETR features are scale-invariant (output from the same layer) in comparison to features used in ORG (Du et al., 2020). Considering that object positions cannot be explicitly decoded without the feed-forward layer of DETR, we therefore enhance all the detected instance features with their locations to obtain spatial-enhanced local descriptors. Then, we take all the spatial-enhanced local descriptors as the key of our VT encoder to model the relationships among detected instances, such as category concurrence and spatial correlations.1
+
+![](images/86a0fb60ac26762e8a9aaf49b2f709c2b1c688b7a2475712edb352c7ff230f70.jpg)  
+Figure 1: Motivation of the Visual Transformer Network (VTNet). A target class (cellphone) is highlighted by green bounding boxes. An agent first detects objects of interest from its observation. Then, the agent attends detected objects to the global observation by the visual transformer (VT). High attention scores are achieved on the left side of the observation, which correspond to the target (cellphone). Then, the agent will choose RotateLeft to reach targets.
+
+Furthermore, we introduce a positional global descriptor as the query for our VT decoder. In particular, we associate the region features with image region positions (such as bottom and top) and thus facilitate exploring the correspondences between navigation actions and image regions. To do so, we divide a global observation into multiple regions based on spatial layouts and assign a positional embedding to each region feature as our spatial-enhanced global descriptor. After obtaining the global query descriptor, we attend the spatial-enhanced local descriptor to the positional global descriptor query to learn the relationship between instances and observation regions via our VT decoder.
+
+However, we found directly training our VTNet with a navigation policy network fails to converge due to the training difficulty of the transformers (Vaswani et al., 2017). Therefore, we present a pretraining scheme to associate visual representations and directional navigation signals. We endow our VT with the capability of encoding directional navigation signals by imitating expert experience. After warming-up through human instructions, VT can learn instructional representations for navigation, as illustrated in Figure 1.
+
+After pre-training our VT, we employ a standard Long Short Term Memory (LSTM) network to map the current visual representation and previous states to an agent action. We adopt A3C architecture (Mnih et al., 2016) to learn the navigation policy. Once our VTNet has been fully trained, our agent can exploit the correlations between observations and navigation actions to improve visual navigation efficiency. In the popular widely-used navigation environment AI2-Thor (Kolve et al., 2017), our method significantly outperforms the state-of-the-art. Our contributions are summarized as follows:
+
+• We propose a novel Visual Transformer Network (VTNet) to extract informative feature representations for visual navigation. Our visual representations not only encode relationships among objects but also establish strong correlations with navigation signals. • We introduce a positional global descriptor and a spatial-enhanced local descriptor as the query and key for our visual transformer (VT), and then the visual representations decoded by our VT are attended to navigation actions via our presented pre-training scheme, thus providing a good initialization for our VT. Experimental results demonstrate that our learned visual representation significantly improves the efficiency of the state-of-the-art visual navigation systems in unseen environments by $1 4 . 0 \%$ relatively on Success Weighted by Path Length (SPL).
+
+# 2 RELATED WORKS
+
+Visual navigation, as a fundamental task in robotic and artificial intelligence, has attracted increasing attention recently. Traditional methods (Oriolo et al., 1995) often leverage environment maps for navigation and divide a navigation task into three steps: mapping, localization and path planning. Some approaches employ a given map to obviate obstructions (Borenstein & Koren, 1989; 1991). Dissanayake et al. (2001) infer robot positions by simultaneous localization and mapping (SLAM). However, maps are usually unavailable in unseen environments.
+
+Recently, reinforcement learning (RL) has been applied in visual navigation. In general, it takes visual observations as inputs and predicts navigation actions directly. Mirowski et al. (2016) develop a navigation approach in 3D maze environments and introduce depth prediction and loop closure classification tasks to improve navigation performance. Parisotto & Salakhutdinov (2017) investigate a memory system to navigate in mazes. Some methods (Sepulveda et al., 2018; Chen et al., 2019; Savinov et al., 2018) use both visual features and the topological guidance of scenes for navigation, while natural-language instructions are employed to guide an agent to route among rooms (Anderson et al., 2018b; Wang et al., 2019; Deng et al., 2020; Hu et al., 2019; Majumdar et al., 2020; Hao et al., 2020). We notice that transformer architectures are also employed by Hao et al. (2020), named Prevalenet. However, Prevalenet is used to model languages and predict camera angles rather than encoding local and global visual features. Hence, Prevalenet is essentially different from our VT. Furthermore, Kahn et al. (2018) design a self-supervised approach to model environments by reinforcement learning. Tang et al. (2021) customize a specialized network for visual navigation via an Auto-Navigator. A Bayesian relational memory is introduced by Wu et al. (2019) to explore the spatial layout among rooms rather than steering an agent to desired objects with least steps. Meanwhile, Shen et al. (2019) employ multiple visual representations to generate multiple actions and then fuse those actions to produce an effective one. However, requesting such a large number of visual representations may restrict the transferring ability of a navigation system and increases the difficulty of data labeling. Note that Fang et al. (2019) propose a transformer to select the embedded scene memory slot, while our VT is designed to learn expressive visual representations correlated with directional signals.
+
+Target-oriented visual navigation methods aim at steering an agent to object instances of a specified category in an unseen environment using least steps. Zhu et al. (2017) search a target object given in an image by employing RL to produce navigation actions based on visual observations. Mousavian et al. (2019) take semantic segmentation and detection masks as visual representations and also employ RL to learn navigation policies. Yang et al. (2018) exploit relationships among object categories for navigation, but they need an external knowledge database to construct such relationships. Wortsman et al. (2019) exploit word embedding (i.e., GloVe embedding) to represent the target category and introduce a meta network mimicking a reward function for navigation. Furthermore, Du et al. (2020) introduce an object relation graph, dubbed ORG, to encode visual observations and design a tentative policy for deadlock avoidance during navigation. In ORG, object features are extracted from the second layer of the backbone in Faster R-CNN (Ren et al., 2015) and thus not the most prominent ones across the feature pyramid. Additionally, ORG chooses one instance with the highest confidence per category from detection results, and it may be affected by the false positive.
+
+# 3 VISUAL NAVIGATION REVISIT
+
+In this section, we mainly revisit the definition of object goal navigation and its general pipeline.
+
+# 3.1 TASK DEFINITION AND SETUP
+
+In this object goal visual navigation task, prior knowledge about the environment, i.e. topological map and 3D meshes, and additional sensors, i.e. depth cameras, are not available to an agent. RGB images in an egocentric view are the only available source to an agent, and the agent predicts its actions based on the current view and previous states. Following the works (Wortsman et al., 2019; Du et al., 2020), an environment is divided into grids and agents move between grid points via 6 different actions, consist of MoveAhead, RotateLeft, RotateRight, LookUp, LookDown, Done. To be specific, the forward step size is 0.25 meters, and the angles of turningleft/right and looking-up/down are $4 5 ^ { \circ }$ and $3 0 ^ { \circ }$ , respectively. An episode is defined as a success when the following three requirements are met simultaneously: (i) the agent chooses the ending action Done within allowed steps; (ii) a target is in the view of the agent; (iii) the distance between the agent and the target is less than the threshold (i.e. 1.5 meters). Otherwise, the episode will be regarded as a failure.
+
+![](images/85dce5151b107d0a73f73d73188d5dda8fdfe1d2ab60e5268c317add7a8211f8.jpg)  
+Figure 2: Overview of our visual transformer navigation system. Our visual transformer navigation network (VTNet) involves a visual transformer (VT) and a navigation policy network. The agent first fuses instance features and spatial features into spatial-enhanced local descriptor. Meanwhile, the positional global descriptor is obtained by adding a positional embedding to the global feature. Then the visual representation is decoded from these two spatial-aware descriptors by our VT. Our VTNet is pre-trained with the supervision of optimal navigation actions. The navigation policy network adopts A3C architecture and is trained with navigation rewards after pre-training.
+
+A target class $T \in \{ S i n k , \ldots , M i c r o w a v e \}$ and a start state $s = \{ x , y , \theta _ { r } , \theta _ { h } \}$ are set randomly at the beginning of each episode, where $x$ and $y$ represent the coordinates, $\theta _ { r }$ and $\theta _ { h }$ indicate the view of a monocular camera. At each timestamp $t$ , the agent records the observed RGB image $O _ { t }$ from its monocular camera. Given the observation $O _ { t }$ and the previous state $h _ { t }$ , the agent employs a visual navigation network to generate a policy $\pi ( a _ { t } | O _ { t } , h _ { t } )$ , where $a _ { t }$ represents the distribution of actions at time $t$ . The agent selects the action with the highest probability for navigation.
+
+# 3.2 PIPELINE
+
+A typical pipeline of visual navigation consists of two parts, visual representation learning and navigation policy learning. (i) Visual representation learning: To encode the current observation in a compact way, existing works extract visual features from an image and then transform them into a vector-based representation, where direct concatenation (Wortsman et al., 2019) or graph embedding (Du et al., 2020) are used. (ii) Navigation driven by visual features: Once visual features are extracted, a navigation policy network that generates an action in each step for an agent will be learned. There are several ways to learn policy networks, such as Q-learning (Watkins & Dayan, 1992), PPO (Schulman et al., 2017) and A3C (Mnih et al., 2016). As navigation policy learning is not our focus, we adopt the standard Asynchronous Advantage Actor-Critic (A3C) architecture (Mnih et al., 2016). The navigation policy network takes the combination of the current visual representation, the previous action and state embedding as input, and outputs the action distribution and value. The agent selects actions with the highest probability from the predicted policy and uses the predicted value to train the navigation policy network.
+
+# 4 PROPOSED VISUAL TRANSFORMER NETWORK
+
+As illustrated in Figure 2, our visual navigation system includes two parts: (i) learning visual representations from RGB observations; (ii) learning navigation policy from the visual representations and previous states. In our VTNet, we introduce a visual transformer (VT) in the first part to explore the relationship among all objects and their spatial correlations. In VT, we further design two spatial-aware descriptors, i.e., a spatial-enhanced local descriptor and a positional global descriptor, to allow us extract visual information effectively. Then, our VT fuses these two types of descriptors with a multi-head attention operation to produce final visual representations. Moreover, our VT enforces visual representations to be highly correlated to navigation signals via our developed pre-training scheme, thus easing the training difficulty of VT and facilitating navigation policy learning.
+
+# 4.1 SPATIAL-ENHANCED LOCAL DESCRIPTOR
+
+To learn the relationship among all the instances, we first perform object detection and locate all the object instances of interest by a detector DETR (Carion et al., 2020). DETR transforms $N$ encoded $d$ -dimension features $\mathbb { R } ^ { N \times d }$ from the same layer to $N$ detection results, including the bounding boxes, confidence and semantic labels by a feed forward network. Note that ORG (Du et al., 2020) extracts object features from the second layer of the backbone in Faster R-CNN based on the predicted bounding-boxes rather than the penultimate layer of the classifier in Faster R-CNN as in the work (Anderson et al., 2018a). Hence, ORG features are not the most prominent ones across the feature pyramid and scale-sensitive. In contrast, features extracted by DETR not only contain bounding-boxes and class labels but also are scale-robust as features are aligned by DETR decoder, i.e., output from the penultimate layer.
+
+Remark. Benefiting from our VT, we leverage all the information of the detected objects while Du et al. (2020) only select the proposal with the highest confidence in each category. Therefore, the agents in ORG will miss important information from other objects of the same class or might be severely affected if selected proposals are false positive. In contrast, our VT preserves all the information, and thus our agents are able to exploit the relationship among instances. This makes our visual representation more comprehensive and essentially different from prior works.
+
+Our local spatial feature is obtained by concatenating the normalized bounding box, confidence and top-rated semantic label for each object. To indicate the target class to an agent, we also concatenate a one-hot encoded target vector $\mathbb { R } ^ { N \times 1 }$ with our spatial feature $\mathbb { R } ^ { N \times 7 }$ . After obtaining the instance feature and spatial feature, we employ a multi-layer perceptron (MLP) (i.e., two fully-connected layers with ReLU) and fuse them to a spatial-enhanced local descriptor $\dot { L } \in \mathbb { R } ^ { N \times d }$ so as to act as the key of our VT encoder.
+
+# 4.2 POSITIONAL GLOBAL DESCRIPTOR
+
+In addition to the spatial-enhanced local descriptor, agents require a global feature to describe the surrounding environment. Similar to SAVN (Wortsman et al., 2019), we adopt ResNet18 (He et al., 2016) pretrained on ImageNet (Deng et al., 2009) to extract global features of the observations. Given a global feature $\mathbb { R } ^ { h \times w \times D }$ , we first employ $1 \times 1$ convolution to reduce the channel dimension of a high-level activation map from $D$ to a smaller dimension $d$ , where $h$ and $w$ represent the height and width of activation maps, respectively. This ensures that global descriptors have the same dimension as the key of our VT.
+
+Unlike previous works that directly concatenate a global feature as a part of the visual representation, we introduce a positional global descriptor as the query in our VT decoder. A region feature only represents visual contents in each region. To emphasize the region position information, we incorporate a positional embedding to each region feature. Then we add positional encoding $\mathbb { R } ^ { h \times w \times d }$ to the global feature. Let $u$ and $v$ represent the row and column indexes of an image region respectively, and $i$ is the index along the dimension $d$ . Our positional embedding is expressed as:
+
+$$
+P E _ { 2 i } ( u , v ) = \left\{ \begin{array} { l l } { \sin ( \frac { u } { 1 0 0 0 0 ^ { 2 i / d } } ) , 0 < i \leq \frac { d } { 2 } } \\ { \sin ( \frac { v } { 1 0 0 0 0 ^ { 2 i / d } } ) , \frac { d } { 2 } < i \leq d } \end{array} \right. P E _ { 2 i + 1 } ( u , v ) = \left\{ \begin{array} { l l } { \cos ( \frac { u } { 1 0 0 0 0 ^ { 2 i / d } } ) , 0 < i \leq \frac { d } { 2 } } \\ { \cos ( \frac { v } { 1 0 0 0 0 ^ { 2 i / d } } ) , \frac { d } { 2 } < i \leq d } \end{array} \right.
+$$
+
+Therefore, each global feature represents one particular region of the observation. Finally, we reshape positional embedded global features into a matrix-based representation, namely positional global descriptor G ∈ Rhw×d.
+
+# 4.3 VISUAL TRANSFORMER
+
+After obtaining our extracted spatial-enhanced and positional global descriptors, we introduce our visual transformer.
+
+Encoder. In order to exploit the spatial relationship between detected instances and observed regions, we attend spatial-enhanced local descriptors to positional global descriptors via a transformer. We first feed the spatial-enhanced local descriptors into the encoder as keys and values by employing multi-head self-attention. Following the transformer architecture (Vaswani et al., 2017; Fan et al., 2021), each encoder layer consists of a multi-head self-attention module and a feed-forward layer.
+
+Decoder. Inspired by human navigation behaviors, we aim to explore the correspondences between observation regions and navigation actions. For example, once an agent notices a target lying on the right side of the field of view, it should prioritize to select RotateRight instead of RotateLeft. Since each positional global descriptor corresponds to a certain region of the observation, we refer to the positional global descriptor as the location query and feed the query into the decoder. Given positional global descriptor $G$ and encoded spatial-enhanced local descriptor $L ^ { \prime }$ , our attention function of visual transformer decoder is expressed as:
+
+$$
+A t t e n t i o n ( G , L ^ { \prime } ) = s o f t m a x ( \frac { G L ^ { \prime T } } { \sqrt { d } } L ^ { \prime } ) .
+$$
+
+# 4.4 PRE-TRAINING VISUAL TRANSFORMER
+
+We observed that directly feeding the decoded representation from our VT to a navigation network, we fail to learn successful navigation policy. This is mainly because training a deep VT is very difficult especially when the supervision signals are provided by a weak reward from reinforcement learning. Therefore, the decoded features might be uninformative and confuse an agent. The agent would prefer to choose the termination action (often around 5 steps in our experiments) in order to reduce penalties from reinforcement learning.
+
+To address the aforementioned issue, we propose a pre-training scheme for our VT. To be specific, we enforce the decoded features to be expressive by introducing an imitation learning task, as seen in Figure 2. Concretely, human navigation behaviors can be predicted from the decoded representations in a step-wise fashion. We use Dijkstra’s Shortest Path First algorithm to generate optimal action instructions as human expert experience. Under the supervision of optimal action instructions, our VT learns to imitate the optimal navigation action selection.
+
+In the pre-training stage, we do not employ our navigation network (i.e., LSTM), and previous actions as well as states are not available. Note that, in our navigation network, previous actions, previous states and current visual representations are exploited, as seen in Figure 2. Thus, we replace our LSTM with an MLP and predict action distributions based on the current visual representation. A cross-entropy loss $\boldsymbol { L _ { v t } } = \boldsymbol { C E } ( \boldsymbol { a _ { t } } , \boldsymbol { \hat { a } } )$ is employed to train our VT and the MLP, where $a _ { t }$ is the predicted action, $\hat { a }$ represents the optimal action instruction and $C E$ indicates the cross-entropy function. After pre-training, features from our VT also exhibit strong association with directional navigation signals as only an MLP is employed on top of the features. Therefore, the decoded features will facilitate the navigation network training.
+
+# 5 EXPERIMENTS
+
+# 5.1 PROTOCOLS AND EXPERIMENTAL DETAILS
+
+Dataset. We perform our experiments on AI2-Thor (Kolve et al., 2017), an artificial 3D environment with realistic photos. It contains 4 types of scenes, i.e., kitchen, living room, bedroom and bathroom. In each type of scenes, there are 30 different rooms with various furniture placements and items. Following Du et al. (2020), we choose 22 categories as the target classes and ensure that there are at least 4 potential targets in each room.
+
+We use the same training and evaluation protocols as the works (Wortsman et al., 2019; Du et al., 2020). 80 rooms out of 120 are selected as the training set while each scene contains 20 rooms.
+
+Table 1: Comparison with the state-of-the-art. We report the average success rate $( \% )$ and SPL as well as their variances in parentheses by repeating experiments five times. $L > 5$ represents the episodes which require at least 5 steps.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">ALL</td><td colspan="2">L≥5</td></tr><tr><td>Success</td><td>SPL</td><td>Success</td><td>SPL</td></tr><tr><td>Random</td><td>8.0 (1.3)</td><td>0.036 (0.006)</td><td>0.3 (0.1)</td><td>0.001 (0.001)</td></tr><tr><td>WE</td><td>33.0 (3.5)</td><td>0.147( (0.018)</td><td>21.4 (3.0)</td><td>0.117 (0.019)</td></tr><tr><td>SP (Yang et al., 2018)</td><td>35.1 (1.3)</td><td>0.155 (0.011)</td><td>22.2 (2.7)</td><td>0.114 (0.016)</td></tr><tr><td>SAVN (Wortsman et al.,2019)</td><td>40.8 (1.2)</td><td>0.161 (0.005)</td><td>28.7 (1.5)</td><td>0.139 (0.005)</td></tr><tr><td>ORG (Du et al.,2020)</td><td>65.3 (0.7)</td><td>0.375 (0.008)</td><td>54.8 (1.0)</td><td>0.361 (0.009)</td></tr><tr><td>ORG+TPN (Du et al., 2020)</td><td>69.3 (1.2)</td><td>0.394 (0.010)</td><td>60.7 (1.3)</td><td>0.386 (0.011)</td></tr><tr><td>Baseline</td><td>62.6 (0.9)</td><td>0.364 (0.006)</td><td>51.5 (1.2)</td><td>0.345 (0.007)</td></tr><tr><td>VTNet</td><td>72.2 (1.0)</td><td>0.449 (0.007)</td><td>63.4 (1.1)</td><td>0.440 (0.009)</td></tr><tr><td>VTNet + TPN (Du et al., 2020)</td><td>73.5 (1.3)</td><td>0.440 (0.009)</td><td>63.9 (1.5)</td><td>0.440 (0.011)</td></tr></table>
+
+We equally divide the remaining 40 rooms into validation and test sets. We report the results of the testing data by using the model with the highest success rate on the validation set.
+
+Evaluation metrics. We evaluate our model performance by success rate and Success Weighted $\textstyle { \frac { 1 } { N } } \sum _ { n = 0 } ^ { N } S _ { n }$ gth (SP, where $N$ . The success rate measures is the number of episodes and $S _ { n }$ gation effectiveneis a success indica and isr of the $n$ o- d byode. We adopt SPL tits optimal path ure the navigation effic, SPL is formulated as $n$ -th episode $L e n _ { n }$ and $L e n _ { o p t }$ $\begin{array} { r } { \frac { 1 } { N } \sum _ { n = 0 } ^ { N } S _ { n } \frac { L e n _ { n } } { m a x ( L e n _ { n } , L e n _ { o p t } ) } } \end{array}$
+
+Training details. We use a two-stage training strategy. In Stage 1, we train our visual transformer for 20 epochs with the supervision of optimal action instructions. In this fashion, we explicitly construct the association between visual representations and navigation actions. In Stage 2, we train the navigation policy for 6M episodes in total with 16 asynchronous agents. We set a penalization $- 0 . 0 0 1$ on each action step and a large reward 5 when an agent completes an episode successfully. We adopt DETR as the object detector and fine-tune DETR on the AI2-Thor training dataset. In training DETR, we applied data augmentation, such as resize and random crop. We use the Adam optimizer (Kingma & Ba, 2014) to update the policy network with a learning rate $1 0 ^ { - 4 }$ and the pre-trained VT with a learning rate $1 0 ^ { \div { 5 } }$ . Our codes and pre-trained model will be publicly released for reproducibility.
+
+# 5.2 COMPETING METHODS
+
+We compare our method with the following ones: Random policy. An agent chooses actions based on a uniform action probability. Thus, the agent will walk or stop in a scene randomly. Scene Prior (SP) (Yang et al., 2018) learns a graph neural network from the FastText database (Joulin et al., 2016) and leverages the scene prior knowledge and category relationships for navigation. Word Embedding (WE) uses GloVe embedding (Pennington et al., 2014) to indicate the target category rather than detection. The association between object appearances and GloVe embeddings is learned through trail and error. Self-adaptive Visual Navigation (SAVN) (Wortsman et al., 2019) introduces a meta reinforcement learning method that allows an agent to adapt to unseen environments. Object Relationship Graph (ORG) (Du et al., 2020) is a visual representation learning method to encode correlation among categories and employs a tentative policy network (TPN) to escape from deadlocks. Baseline is a vanilla version of VTNet. We feed the concatenation of the local instance features from DETR and the global feature to A3C for navigation. Note that, our baseline does not employ spatial-enhanced local and positional global descriptors as well as our visual transformer.
+
+# 5.3 EVALUATION RESULTS
+
+Improvement over Baseline. Table 1 indicates that VTNet surpasses the baseline by a large margin on both success rate $( + 9 . 6 \% )$ and SPL $( + 0 . 0 8 5 )$ . Baseline only resorts to the detection features and global feature for navigation. The relations among local instances and the association between the visual observations and actions are not exploited. This comparison suggests that our VT leads to informative visual representations for navigation, and thus significantly improves the effectiveness and efficiency of our navigation system.
+
+Comparison with competing methods. As indicated in Table 1, we observe that VTNet significantly outperforms SP (Yang et al., 2018) and SAVN (Wortsman et al., 2019). Since SP and SAVN employ word embedding as a target indicator while VTNet replaces word embedding with our VT, our method achieves expressive object and image region representations for navigation. In addition, SP and SAVN concatenate features from various modalities directly to generate visual representations. The gap between different modalities may not facilitate navigation policy learning. In contrast, benefiting from our pre-training, features from our VT are more correlated to navigation actions, thus expediting navigation policy learning.
+
+![](images/71e6a48512317337dfb2274ccebf535ebeeeded09483fadd8a8db86293d96c41.jpg)  
+Figure 3: Visual results of four different models in testing environments. The target objects (i.e., RemoteControl) are highlighted by the blue boxes. Green and red curves represent success and failure cases, respectively. The episode produced by our VTNet is successful in reaching the target and use shortest steps. In comparison, ORG takes more steps to reach the target. SAVN and Baseline miss both targets.
+
+Our method outperforms the state-of-the-art method ORG (Du et al., 2020) by $+ 2 . 9 \%$ in success rate and $+ 0 . 0 5 5$ in SPL. Moreover, when ORG does not employ TPN, the advantage of our method becomes more obvious $6 . 9 \%$ improvement), and this mainly comes from our superior visual presentations. Since ORG only chooses an object with the highest confidence in each class, the relationship among objects is not comprehensive. On the contrary, our method can exploit all the detected instances to deduce the relationships among objects due to our VT architecture. Moreover, since DETR infers the relations between object instances and the global image context, the local features output by the DETR are more informative compared to the object features used in ORG. This can be proved by the result when we use Faster R-CNN as our backbone, as indicated by Table 2. We also show a case study in Figure 3 (more visual results are provided in the appendix). Furthermore, we also try to employ TPN to improve our navigation policy. As seen in Table 1, VTNet+TPN improves the success rates but the improvement is not as much as $_ \mathrm { O R G + T P N }$ . This also implies that our visual representations significantly facilitate navigation action selections.
+
+Case Study. As illustrated in Figure 3, SAVN and Baseline both issue the termination command after navigating a few steps (7 and 19 steps, respectively), but fail to reach the target. This indicates that the relationships among categories are not clear in SAVN and Baseline. In contrast, both ORG and VTNet find the target. Since our visual transformer provides clear directional signals, VTNet uses the least steps to find the object.
+
+# 5.4 VARIANT AND ABLATION STUDY
+
+In this section, we analyze the impact of each component in VTNet, including the spatial-enhanced local descriptor, positional global descriptor, visual transformer that fuses these two spatial-aware descriptors and pre-training scheme.
+
+To illustrate the necessity of the spatial enhancement, we directly use the object features without spatial enhancement. In this case, our network fails to converge because the feed-forward layers that predict bounding-boxes and class labels in DETR are not used in VTNet and spatial information cannot be decoded by the navigation network. Thus, spatial enhancement allows an agent to exploit instance location information explicitly.
+
+As indicated in Table 2, we achieve better navigation performance using instance features from DETR compared to employing Faster R-CNN features following the feature extraction of Du et al. (2020) (“Faster R-CNN”). Unlike Faster R-CNN, DETR infers the relations between object instances and the global image context via its transformer to output the final predictions (i.e., class labels and bounding boxes). Although DETR and Faster R-CNN achieve similar detection performance (Carion et al., 2020), features extracted by DETR are more informative and robust than those of Faster R-CNN used in ORG. Specifically, ORG extracts features from the second layer of the backbone in Faster R-CNN based on the predicted bounding-boxes to ensure the features are comparable, but the features are not the most prominent ones across the feature pyramid. Therefore, the object features “Faster R-CNN” extracted by ORG are inferior to DETR features, and our navigator employing DETR features outperforms ORG.
+
+Table 2: Impacts of different components on navigation performances. Faster R-CNN and Faster R-CNN† represent instance features extracted by Faster R-CNN following Du et al. (2020) and Anderson et al. (2018a), respectively.   
+
+<table><tr><td rowspan="2" colspan="2">Method</td><td rowspan="2">w/o global decoder</td><td rowspan="2">w/o</td><td rowspan="2">w/o pe</td><td rowspan="2">VTNetg</td><td colspan="3">Baseline</td><td colspan="3">VTNet</td></tr><tr><td>Faster R-CNN R-CNN</td><td>Faster+</td><td>DETR</td><td>Faster R-CNN R-CNN</td><td>Faster+</td><td>DETR</td></tr><tr><td rowspan="3">ALL</td><td>Success</td><td>67.0</td><td>67.0</td><td>71.0</td><td>70.1</td><td>56.4</td><td>57.2</td><td>62.6</td><td>70.1</td><td>70.3</td><td>72.2</td></tr><tr><td></td><td>(2.8) 0.390</td><td>(1.4) 0.373</td><td>(0.7)</td><td>(1.3) 0.411</td><td>(0.9) 0.319</td><td>(1.1)</td><td>(0.8) 0.365</td><td>(1.0) 0.396</td><td>(1.2)</td><td>(1.0) 0.449</td></tr><tr><td>SPL</td><td>(0.021)</td><td>(0.013)</td><td>0.432 (0.009)</td><td>(0.009)</td><td>(0.007)</td><td>0.308 (0.008)</td><td>(0.010)</td><td>(0.010)</td><td>0.387 (0.012)</td><td>(0.007)</td></tr><tr><td rowspan="3">L≥5</td><td>Success</td><td>54.5</td><td>53.2</td><td>61.2</td><td>60.6</td><td>42.5</td><td>46.7</td><td>51.5</td><td>61.7</td><td>62.1</td><td>63.4</td></tr><tr><td></td><td>(3.1)</td><td>(1.6)</td><td>(0.9)</td><td>(1.5)</td><td>(1.2)</td><td>(1.3)</td><td>(1.0)</td><td>(1.2)</td><td>(1.4)</td><td>(1.1)</td></tr><tr><td>SPL</td><td>0.357</td><td>0.343</td><td>0.416 (0.010)</td><td>0.395</td><td>0.270</td><td>0.276</td><td>0.345</td><td>0.399</td><td>0.376</td><td>0.440</td></tr></table>
+
+Moreover, we adopt the instance features from Faster R-CNN following the feature extraction fashion of Anderson et al. (2018a) (“Faster R-CNN†”). Thus, we obtain the instance features from the penultimate layer of the classifier in Faster R-CNN. We observe that instance features from DETR also improve the navigation performance compared to the Faster R-CNN† features. We speculate the improvements mainly come from the fact that the features output by DETR decoder have embedded global context information, and those features are more suitable for the feature fusion operations.
+
+As seen in Table 2, we first remove the global feature from our system (“VTNet w/o global”), and the navigation performance degrades significantly. This validates the importance of global features, which provide contextual guidance to an agent. Moreover, when we remove the positional embedding from the global feature (“VTNet w/o pe”), we observe that both effectiveness and efficiency of the navigation decrease. This indicates that the position embeddings facilitate our VT to exploit the spatial information of observation regions. Furthermore, when we remove the VT decoder (“VTNet w/o decoder”) and concatenate the global and local descriptors directly, our method suffers performance degradation. This demonstrates that our VT plays a critical role in attending the global descriptors to local ones. Additionally, we feed the positional global features into the transformer encoder $( ^ { 6 6 } \nabla \mathrm { T N e t } _ { g } ^ { , 9 } )$ . The navigation performance of $\mathrm { V T N e t } _ { g }$ is superior to that of ORG but slightly inferior to the performance of our VTNet. This demonstrates the transformer architecture is effective to extract informative visual representations, and assigning different functions to different modules would further facilitate the establishment of mappings in our VT.
+
+When the pre-training scheme is not applied to our VT, our agent fails to learn any effective navigation policy and thus we do not report the performance. This manifests that our VT pre-training procedure provides a good initialization to our transformer and prior knowledge on associating visual observations with navigation actions to agents.
+
+# 6 CONCLUSION
+
+In this paper, we proposed a powerful visual representation learning method for visual navigation, named Visual Transformer Network (VTNet). In our VTNet, a visual transformer (VT) has been developed to encode visual observations. Our VT leverages two newly designed spatial-aware descriptors, i.e., a spatial-enhanced local object descriptor and a positional global descriptor, and then fuses those two types of descriptors via multi-head attention to achieve our final visual representation. Thanks to our VT architecture, all the detected instances will be exploited for understanding the current observation. Therefore, our visual representation is more informative compared to that used in state-of-the-art navigation methods. Benefiting from our pre-training strategy, our VT is able to associate visual representations with navigation actions, thus significantly expediting navigation policy learning. Extensive results demonstrate that our VTNet outperforms the state-of-the-art in terms of effectiveness and efficiency.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the ARC Discovery Early Career Researcher Award (DE200101283), the ARC Discovery Project (DP210102801) and the Data61 Collaborative Research Project.
+
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+
+# A APPENDIX
+
+# A.1 FEATURE DETAILS IN VTNET
+
+![](images/c4b951c34cf9d29c67a4a82b3f8c86e47d192f431c225717fe5be3c9131dd942.jpg)  
+Figure 4: Illustration feature flowchart in VTNet.
+
+For reproducibility, we illustrate the detailed feature flowchart of our VTNet in Figure 4. We extract instance features $\mathbb { R } ^ { 1 0 0 \times 2 5 6 }$ and location features $\mathbb { R } ^ { 1 0 0 \times 2 4 9 }$ , and concatenate them into a spatialenhanced local descriptor $\mathbb { R } ^ { 1 0 0 \times 2 5 6 }$ . The local branch integrates semantic labels $\mathbb { R } ^ { 1 0 0 \times 1 }$ , bounding boxes $\mathbb { R } ^ { 1 0 0 \times 4 }$ , confidences $\mathbb { R } ^ { 1 0 0 \times 1 }$ and the target labels $\mathbf { \mathbb { R } } ^ { 1 0 0 \times 1 }$ for the current observation. In the global branch, the positional embedding $\mathbb { R } ^ { 7 \times 7 \times 2 5 6 }$ is added to the global feature $\mathbb { R } ^ { 7 \times 7 \times 2 5 6 }$ , leading to a positional global descriptor $\mathbb { R } ^ { 4 9 \times 2 5 6 }$ . The spatial-enhanced local and positional global descriptors are fused by VT encoder and then the visual representation $\mathbb { R } ^ { 4 9 \times 2 5 6 }$ is output by our VT decoder.
+
+# A.2 FAILURE CASE STUDY
+
+![](images/34892dc23d9809566fab64e69ce37f1da19b576e5df5e9205f69aaa41b444733.jpg)  
+Figure 5: Visual results of failure cases in testing environments. The target objects (i.e., Bowl and Laptop) are highlighted by the blue boxes. Red lines indicate the distance between the agent and the target object. Gray curves represent trajectories of agents. Both episodes fail because distances (i.e., $1 . 5 3 0 \mathrm { m }$ and $1 . 5 7 2 \mathrm { m } )$ between the agent and the target in the field of view are larger than the threshold (i.e., $1 . 5 \mathrm { m } )$ .
+
+As demonstrated in Figure 5, our VTNet fails to reach targets because the distances between agents and targets are larger than the threshold distance (i.e., $1 . 5 \mathrm { m } )$ . In these two failure cases, agents find targets but implement the termination action before reaching a position closer than the threshold. Due to the lack of depth information and variances of target object sizes, an agent may predict that it is within a 1.5 meter radius of the target by mistake and thus terminates current episode.
+
+![](images/3c315b24db0d4468cf765539de490160c68089eeadda004c75b54092ed51bbc0.jpg)  
+Figure 6: Visual results of four different models in testing environments. We compare VTNet with SAVN (Wortsman et al., 2019), Baseline and ORG (Du et al., 2020). The target objects are highlighted by the blue boxes. Green and red curves indicate success and failure cases, respectively. Our VTNet successfully reaches targets and uses the shortest steps.
+
+Table 3: Comparison of visual transformer architectures. We report the pre-training accuracy on the validation dataset, navigation success rate and SPL on the test dataset.   
+
+<table><tr><td rowspan=1 colspan=6>Multi-head numbersEncoder layersDecoder layersAccuracy Success SPL</td></tr><tr><td rowspan=3 colspan=1>4</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.722</td><td rowspan=1 colspan=1>71.2</td><td rowspan=1 colspan=1>0.433</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>0.723</td><td rowspan=1 colspan=1>72.2</td><td rowspan=1 colspan=1>0.449</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>0.715</td><td rowspan=1 colspan=1>70.0</td><td rowspan=1 colspan=1>0.419</td></tr><tr><td rowspan=4 colspan=1>8</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.707</td><td rowspan=1 colspan=1>70.4</td><td rowspan=1 colspan=1>0.422</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>0.718</td><td rowspan=1 colspan=1>70.9</td><td rowspan=1 colspan=1>0.436</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>0.710</td><td rowspan=1 colspan=1>68.9</td><td rowspan=1 colspan=1>0.423</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>0.701</td><td rowspan=1 colspan=1>68.2</td><td rowspan=1 colspan=1>0.411</td></tr></table>
+
+We construct different transformer architectures by varying the number of encoder and decoder layers. Table 3 summarizes the performance of these architectures. We observe that as a visual transformer becomes too deep, a transformer may fail to converge to an optimal policy. On the other hand, a transformer with a single encoder and decoder layer does not have sufficient network capability to produce representative features. The highest success rate is achieved when a VT contains four multi-head self-attention mechanism modules and two layers in the encoder and decoder.
+
+# A.5 NECESSITY OF PRE-TRAINING SCHEME
+
+![](images/e385df76af0b53ccfbbae0079bba8b5e8e4972d6fa5992d3a5eee1c1ae8cd8d5.jpg)  
+Figure 7: Average episode lengths of VTNet and VTNet without pre-training during training. We compare VTNet with VTNet without pre-training scheme. Blue and orange curves represent VTNet and VTNet w/o pre-training, respectively.
+
+As demonstrated in Figure 7, our VTNet spends nearly 10 steps per episode in training, while the navigator w/o pre-training scheme often fails to reach targets and stops around 5 steps after being trained tens of thousands of episodes. Due to the large parameters and complex architectures of transformers, it is often difficult to train our transformers from scratch (Liu et al., 2020). Without a good initialization for our VT, it is very difficult to learn our VT and policy network in an end-to-end fashion with RL rewards. This is because the visual representations from VT are not informative or even meaningless and the inferior visual representations would harm policy network learning. As a result, the navigation policy network may be trapped into a local minimum (i.e., terminating navigation early to avoid more penalties) and our VT cannot receive positive rewards from preceding trajectories.
+
+![](images/a7fdfb5658f2ce176303ab0573e5d96d1f2922a9f9a2a75666f92aed2e5cb5b3.jpg)  
+Figure 8: Visualizations of attention scores. The target classes (i.e., StoveBurner, GarbageCan, Kettle) are highlighted by green bounding boxes. Our agent detects the instances of interest and then attends the detected instances to the global image regions by our VT. We observe that high attention scores are obtained on the areas corresponding to the targets. Guided by the visual representations, the agent selects actions to approach the targets.

parse/train/DILxQP08O3B/DILxQP08O3B_content_list.json

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+    {
+        "type": "text",
+        "text": "VTNET: VISUAL TRANSFORMER NETWORK FOR OBJECT GOAL NAVIGATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Heming $\\mathbf { D } \\mathbf { u } ^ { 1 , 3 }$ , Xin $\\mathbf { Y } \\mathbf { u } ^ { 2 * } \\mathbf { \\& }$ Liang Zheng1 ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1Australian National University   \n2University of Technology Sydney   \n3CSIRO-DATA61   \n{heming.du, liang.zheng}@anu.edu.au, xin.yu@uts.edu.au ",
+        "bbox": [
+            183,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+            292
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Object goal navigation aims to steer an agent towards a target object based on observations of the agent. It is of pivotal importance to design effective visual representations of the observed scene in determining navigation actions. In this paper, we introduce a Visual Transformer Network (VTNet) for learning informative visual representation in navigation. VTNet is a highly effective structure that embodies two key properties for visual representations: First, the relationships among all the object instances in a scene are exploited; Second, the spatial locations of objects and image regions are emphasized so that directional navigation signals can be learned. Furthermore, we also develop a pre-training scheme to associate the visual representations with navigation signals, and thus facilitate navigation policy learning. In a nutshell, VTNet embeds object and region features with their location cues as spatial-aware descriptors and then incorporates all the encoded descriptors through attention operations to achieve informative representation for navigation. Given such visual representations, agents are able to explore the correlations between visual observations and navigation actions. For example, an agent would prioritize “turning right” over “turning left” when the visual representation emphasizes on the right side of activation map. Experiments in the artificial environment AI2-Thor demonstrate that VTNet significantly outperforms state-of-the-art methods in unseen testing environments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            599,
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+            614
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The goal of target-driven visual navigation is to guide an agent to reach instances of a given target category based on its monocular observations of an environment. Thus, it is highly desirable to achieve an informative visual representation of the observation, which is correlated to directional navigation signals. In this paper, we propose a Visual Transformer Network (VTNet) to achieve an expressive visual representation. In our VTNet, we develop a Visual Transformer (VT) to extract image descriptors from visual observations and then decode visual representations of the observed scenes. Then, we present a pre-training scheme to associate visual representations with directional navigation signals, thus making the representations informative for navigation. After pre-training, our visual representations are fed to a navigation policy network and we train our entire network in an end-to-end manner. In particular, our VT exploits two newly designed spatial-aware descriptors as the key and query, (i.e., a spatial-enhanced local descriptor and a positional global descriptor) and then encodes them to achieve an expressive visual representation. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our spatial-enhanced local descriptor is developed to fully take advantage of all detected objects for the exploration of spatial and category relationships among instances. Unlike the prior work (Du et al., 2020) that only leverages one instance per class to mine the category relationship, our VT is able to exploit the relationship of all the detected instances. To this end, we employ an object detector DETR (Carion et al., 2020) since features extracted from DETR not only encode object appearance information, such as class labels and bounding boxes, but also contain the relations between instances and global contexts. Moreover, DETR features are scale-invariant (output from the same layer) in comparison to features used in ORG (Du et al., 2020). Considering that object positions cannot be explicitly decoded without the feed-forward layer of DETR, we therefore enhance all the detected instance features with their locations to obtain spatial-enhanced local descriptors. Then, we take all the spatial-enhanced local descriptors as the key of our VT encoder to model the relationships among detected instances, such as category concurrence and spatial correlations.1 ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "image",
+        "img_path": "images/86a0fb60ac26762e8a9aaf49b2f709c2b1c688b7a2475712edb352c7ff230f70.jpg",
+        "image_caption": [
+            "Figure 1: Motivation of the Visual Transformer Network (VTNet). A target class (cellphone) is highlighted by green bounding boxes. An agent first detects objects of interest from its observation. Then, the agent attends detected objects to the global observation by the visual transformer (VT). High attention scores are achieved on the left side of the observation, which correspond to the target (cellphone). Then, the agent will choose RotateLeft to reach targets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we introduce a positional global descriptor as the query for our VT decoder. In particular, we associate the region features with image region positions (such as bottom and top) and thus facilitate exploring the correspondences between navigation actions and image regions. To do so, we divide a global observation into multiple regions based on spatial layouts and assign a positional embedding to each region feature as our spatial-enhanced global descriptor. After obtaining the global query descriptor, we attend the spatial-enhanced local descriptor to the positional global descriptor query to learn the relationship between instances and observation regions via our VT decoder. ",
+        "bbox": [
+            174,
+            392,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "However, we found directly training our VTNet with a navigation policy network fails to converge due to the training difficulty of the transformers (Vaswani et al., 2017). Therefore, we present a pretraining scheme to associate visual representations and directional navigation signals. We endow our VT with the capability of encoding directional navigation signals by imitating expert experience. After warming-up through human instructions, VT can learn instructional representations for navigation, as illustrated in Figure 1. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "After pre-training our VT, we employ a standard Long Short Term Memory (LSTM) network to map the current visual representation and previous states to an agent action. We adopt A3C architecture (Mnih et al., 2016) to learn the navigation policy. Once our VTNet has been fully trained, our agent can exploit the correlations between observations and navigation actions to improve visual navigation efficiency. In the popular widely-used navigation environment AI2-Thor (Kolve et al., 2017), our method significantly outperforms the state-of-the-art. Our contributions are summarized as follows: ",
+        "bbox": [
+            174,
+            587,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We propose a novel Visual Transformer Network (VTNet) to extract informative feature representations for visual navigation. Our visual representations not only encode relationships among objects but also establish strong correlations with navigation signals. • We introduce a positional global descriptor and a spatial-enhanced local descriptor as the query and key for our visual transformer (VT), and then the visual representations decoded by our VT are attended to navigation actions via our presented pre-training scheme, thus providing a good initialization for our VT. Experimental results demonstrate that our learned visual representation significantly improves the efficiency of the state-of-the-art visual navigation systems in unseen environments by $1 4 . 0 \\%$ relatively on Success Weighted by Path Length (SPL). ",
+        "bbox": [
+            215,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            354,
+            117
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Visual navigation, as a fundamental task in robotic and artificial intelligence, has attracted increasing attention recently. Traditional methods (Oriolo et al., 1995) often leverage environment maps for navigation and divide a navigation task into three steps: mapping, localization and path planning. Some approaches employ a given map to obviate obstructions (Borenstein & Koren, 1989; 1991). Dissanayake et al. (2001) infer robot positions by simultaneous localization and mapping (SLAM). However, maps are usually unavailable in unseen environments. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recently, reinforcement learning (RL) has been applied in visual navigation. In general, it takes visual observations as inputs and predicts navigation actions directly. Mirowski et al. (2016) develop a navigation approach in 3D maze environments and introduce depth prediction and loop closure classification tasks to improve navigation performance. Parisotto & Salakhutdinov (2017) investigate a memory system to navigate in mazes. Some methods (Sepulveda et al., 2018; Chen et al., 2019; Savinov et al., 2018) use both visual features and the topological guidance of scenes for navigation, while natural-language instructions are employed to guide an agent to route among rooms (Anderson et al., 2018b; Wang et al., 2019; Deng et al., 2020; Hu et al., 2019; Majumdar et al., 2020; Hao et al., 2020). We notice that transformer architectures are also employed by Hao et al. (2020), named Prevalenet. However, Prevalenet is used to model languages and predict camera angles rather than encoding local and global visual features. Hence, Prevalenet is essentially different from our VT. Furthermore, Kahn et al. (2018) design a self-supervised approach to model environments by reinforcement learning. Tang et al. (2021) customize a specialized network for visual navigation via an Auto-Navigator. A Bayesian relational memory is introduced by Wu et al. (2019) to explore the spatial layout among rooms rather than steering an agent to desired objects with least steps. Meanwhile, Shen et al. (2019) employ multiple visual representations to generate multiple actions and then fuse those actions to produce an effective one. However, requesting such a large number of visual representations may restrict the transferring ability of a navigation system and increases the difficulty of data labeling. Note that Fang et al. (2019) propose a transformer to select the embedded scene memory slot, while our VT is designed to learn expressive visual representations correlated with directional signals. ",
+        "bbox": [
+            174,
+            224,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Target-oriented visual navigation methods aim at steering an agent to object instances of a specified category in an unseen environment using least steps. Zhu et al. (2017) search a target object given in an image by employing RL to produce navigation actions based on visual observations. Mousavian et al. (2019) take semantic segmentation and detection masks as visual representations and also employ RL to learn navigation policies. Yang et al. (2018) exploit relationships among object categories for navigation, but they need an external knowledge database to construct such relationships. Wortsman et al. (2019) exploit word embedding (i.e., GloVe embedding) to represent the target category and introduce a meta network mimicking a reward function for navigation. Furthermore, Du et al. (2020) introduce an object relation graph, dubbed ORG, to encode visual observations and design a tentative policy for deadlock avoidance during navigation. In ORG, object features are extracted from the second layer of the backbone in Faster R-CNN (Ren et al., 2015) and thus not the most prominent ones across the feature pyramid. Additionally, ORG chooses one instance with the highest confidence per category from detection results, and it may be affected by the false positive. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 VISUAL NAVIGATION REVISIT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            723,
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+            739
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we mainly revisit the definition of object goal navigation and its general pipeline. ",
+        "bbox": [
+            171,
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+            802,
+            770
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 TASK DEFINITION AND SETUP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            785,
+            424,
+            800
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this object goal visual navigation task, prior knowledge about the environment, i.e. topological map and 3D meshes, and additional sensors, i.e. depth cameras, are not available to an agent. RGB images in an egocentric view are the only available source to an agent, and the agent predicts its actions based on the current view and previous states. Following the works (Wortsman et al., 2019; Du et al., 2020), an environment is divided into grids and agents move between grid points via 6 different actions, consist of MoveAhead, RotateLeft, RotateRight, LookUp, LookDown, Done. To be specific, the forward step size is 0.25 meters, and the angles of turningleft/right and looking-up/down are $4 5 ^ { \\circ }$ and $3 0 ^ { \\circ }$ , respectively. An episode is defined as a success when the following three requirements are met simultaneously: (i) the agent chooses the ending action Done within allowed steps; (ii) a target is in the view of the agent; (iii) the distance between the agent and the target is less than the threshold (i.e. 1.5 meters). Otherwise, the episode will be regarded as a failure. ",
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+        "img_path": "images/85dce5151b107d0a73f73d73188d5dda8fdfe1d2ab60e5268c317add7a8211f8.jpg",
+        "image_caption": [
+            "Figure 2: Overview of our visual transformer navigation system. Our visual transformer navigation network (VTNet) involves a visual transformer (VT) and a navigation policy network. The agent first fuses instance features and spatial features into spatial-enhanced local descriptor. Meanwhile, the positional global descriptor is obtained by adding a positional embedding to the global feature. Then the visual representation is decoded from these two spatial-aware descriptors by our VT. Our VTNet is pre-trained with the supervision of optimal navigation actions. The navigation policy network adopts A3C architecture and is trained with navigation rewards after pre-training. "
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+        "type": "text",
+        "text": "A target class $T \\in \\{ S i n k , \\ldots , M i c r o w a v e \\}$ and a start state $s = \\{ x , y , \\theta _ { r } , \\theta _ { h } \\}$ are set randomly at the beginning of each episode, where $x$ and $y$ represent the coordinates, $\\theta _ { r }$ and $\\theta _ { h }$ indicate the view of a monocular camera. At each timestamp $t$ , the agent records the observed RGB image $O _ { t }$ from its monocular camera. Given the observation $O _ { t }$ and the previous state $h _ { t }$ , the agent employs a visual navigation network to generate a policy $\\pi ( a _ { t } | O _ { t } , h _ { t } )$ , where $a _ { t }$ represents the distribution of actions at time $t$ . The agent selects the action with the highest probability for navigation. ",
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+        "type": "text",
+        "text": "3.2 PIPELINE",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A typical pipeline of visual navigation consists of two parts, visual representation learning and navigation policy learning. (i) Visual representation learning: To encode the current observation in a compact way, existing works extract visual features from an image and then transform them into a vector-based representation, where direct concatenation (Wortsman et al., 2019) or graph embedding (Du et al., 2020) are used. (ii) Navigation driven by visual features: Once visual features are extracted, a navigation policy network that generates an action in each step for an agent will be learned. There are several ways to learn policy networks, such as Q-learning (Watkins & Dayan, 1992), PPO (Schulman et al., 2017) and A3C (Mnih et al., 2016). As navigation policy learning is not our focus, we adopt the standard Asynchronous Advantage Actor-Critic (A3C) architecture (Mnih et al., 2016). The navigation policy network takes the combination of the current visual representation, the previous action and state embedding as input, and outputs the action distribution and value. The agent selects actions with the highest probability from the predicted policy and uses the predicted value to train the navigation policy network. ",
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+        "type": "text",
+        "text": "4 PROPOSED VISUAL TRANSFORMER NETWORK ",
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+        "type": "text",
+        "text": "As illustrated in Figure 2, our visual navigation system includes two parts: (i) learning visual representations from RGB observations; (ii) learning navigation policy from the visual representations and previous states. In our VTNet, we introduce a visual transformer (VT) in the first part to explore the relationship among all objects and their spatial correlations. In VT, we further design two spatial-aware descriptors, i.e., a spatial-enhanced local descriptor and a positional global descriptor, to allow us extract visual information effectively. Then, our VT fuses these two types of descriptors with a multi-head attention operation to produce final visual representations. Moreover, our VT enforces visual representations to be highly correlated to navigation signals via our developed pre-training scheme, thus easing the training difficulty of VT and facilitating navigation policy learning. ",
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+        "text": "4.1 SPATIAL-ENHANCED LOCAL DESCRIPTOR ",
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+        "type": "text",
+        "text": "To learn the relationship among all the instances, we first perform object detection and locate all the object instances of interest by a detector DETR (Carion et al., 2020). DETR transforms $N$ encoded $d$ -dimension features $\\mathbb { R } ^ { N \\times d }$ from the same layer to $N$ detection results, including the bounding boxes, confidence and semantic labels by a feed forward network. Note that ORG (Du et al., 2020) extracts object features from the second layer of the backbone in Faster R-CNN based on the predicted bounding-boxes rather than the penultimate layer of the classifier in Faster R-CNN as in the work (Anderson et al., 2018a). Hence, ORG features are not the most prominent ones across the feature pyramid and scale-sensitive. In contrast, features extracted by DETR not only contain bounding-boxes and class labels but also are scale-robust as features are aligned by DETR decoder, i.e., output from the penultimate layer. ",
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+        "type": "text",
+        "text": "Remark. Benefiting from our VT, we leverage all the information of the detected objects while Du et al. (2020) only select the proposal with the highest confidence in each category. Therefore, the agents in ORG will miss important information from other objects of the same class or might be severely affected if selected proposals are false positive. In contrast, our VT preserves all the information, and thus our agents are able to exploit the relationship among instances. This makes our visual representation more comprehensive and essentially different from prior works. ",
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+        "type": "text",
+        "text": "Our local spatial feature is obtained by concatenating the normalized bounding box, confidence and top-rated semantic label for each object. To indicate the target class to an agent, we also concatenate a one-hot encoded target vector $\\mathbb { R } ^ { N \\times 1 }$ with our spatial feature $\\mathbb { R } ^ { N \\times 7 }$ . After obtaining the instance feature and spatial feature, we employ a multi-layer perceptron (MLP) (i.e., two fully-connected layers with ReLU) and fuse them to a spatial-enhanced local descriptor $\\dot { L } \\in \\mathbb { R } ^ { N \\times d }$ so as to act as the key of our VT encoder. ",
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+        "type": "text",
+        "text": "4.2 POSITIONAL GLOBAL DESCRIPTOR ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In addition to the spatial-enhanced local descriptor, agents require a global feature to describe the surrounding environment. Similar to SAVN (Wortsman et al., 2019), we adopt ResNet18 (He et al., 2016) pretrained on ImageNet (Deng et al., 2009) to extract global features of the observations. Given a global feature $\\mathbb { R } ^ { h \\times w \\times D }$ , we first employ $1 \\times 1$ convolution to reduce the channel dimension of a high-level activation map from $D$ to a smaller dimension $d$ , where $h$ and $w$ represent the height and width of activation maps, respectively. This ensures that global descriptors have the same dimension as the key of our VT. ",
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+        "type": "text",
+        "text": "Unlike previous works that directly concatenate a global feature as a part of the visual representation, we introduce a positional global descriptor as the query in our VT decoder. A region feature only represents visual contents in each region. To emphasize the region position information, we incorporate a positional embedding to each region feature. Then we add positional encoding $\\mathbb { R } ^ { h \\times w \\times d }$ to the global feature. Let $u$ and $v$ represent the row and column indexes of an image region respectively, and $i$ is the index along the dimension $d$ . Our positional embedding is expressed as: ",
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+        "type": "equation",
+        "img_path": "images/c83305cc415c64d3f3997e95ab06838f2500f327e49d4d4bc368d5436710b07b.jpg",
+        "text": "$$\nP E _ { 2 i } ( u , v ) = \\left\\{ \\begin{array} { l l } { \\sin ( \\frac { u } { 1 0 0 0 0 ^ { 2 i / d } } ) , 0 < i \\leq \\frac { d } { 2 } } \\\\ { \\sin ( \\frac { v } { 1 0 0 0 0 ^ { 2 i / d } } ) , \\frac { d } { 2 } < i \\leq d } \\end{array} \\right. P E _ { 2 i + 1 } ( u , v ) = \\left\\{ \\begin{array} { l l } { \\cos ( \\frac { u } { 1 0 0 0 0 ^ { 2 i / d } } ) , 0 < i \\leq \\frac { d } { 2 } } \\\\ { \\cos ( \\frac { v } { 1 0 0 0 0 ^ { 2 i / d } } ) , \\frac { d } { 2 } < i \\leq d } \\end{array} \\right.\n$$",
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+        "type": "text",
+        "text": "Therefore, each global feature represents one particular region of the observation. Finally, we reshape positional embedded global features into a matrix-based representation, namely positional global descriptor G ∈ Rhw×d. ",
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+        "type": "text",
+        "text": "4.3 VISUAL TRANSFORMER ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "After obtaining our extracted spatial-enhanced and positional global descriptors, we introduce our visual transformer. ",
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+        "type": "text",
+        "text": "Encoder. In order to exploit the spatial relationship between detected instances and observed regions, we attend spatial-enhanced local descriptors to positional global descriptors via a transformer. We first feed the spatial-enhanced local descriptors into the encoder as keys and values by employing multi-head self-attention. Following the transformer architecture (Vaswani et al., 2017; Fan et al., 2021), each encoder layer consists of a multi-head self-attention module and a feed-forward layer. ",
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+        "type": "text",
+        "text": "Decoder. Inspired by human navigation behaviors, we aim to explore the correspondences between observation regions and navigation actions. For example, once an agent notices a target lying on the right side of the field of view, it should prioritize to select RotateRight instead of RotateLeft. Since each positional global descriptor corresponds to a certain region of the observation, we refer to the positional global descriptor as the location query and feed the query into the decoder. Given positional global descriptor $G$ and encoded spatial-enhanced local descriptor $L ^ { \\prime }$ , our attention function of visual transformer decoder is expressed as: ",
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+        "type": "equation",
+        "img_path": "images/724cdba1c3bf351668e69200b9be7e7dc82599c0a86d77f594891188aee662db.jpg",
+        "text": "$$\nA t t e n t i o n ( G , L ^ { \\prime } ) = s o f t m a x ( \\frac { G L ^ { \\prime T } } { \\sqrt { d } } L ^ { \\prime } ) .\n$$",
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+        "type": "text",
+        "text": "4.4 PRE-TRAINING VISUAL TRANSFORMER ",
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+        "type": "text",
+        "text": "We observed that directly feeding the decoded representation from our VT to a navigation network, we fail to learn successful navigation policy. This is mainly because training a deep VT is very difficult especially when the supervision signals are provided by a weak reward from reinforcement learning. Therefore, the decoded features might be uninformative and confuse an agent. The agent would prefer to choose the termination action (often around 5 steps in our experiments) in order to reduce penalties from reinforcement learning. ",
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+    {
+        "type": "text",
+        "text": "To address the aforementioned issue, we propose a pre-training scheme for our VT. To be specific, we enforce the decoded features to be expressive by introducing an imitation learning task, as seen in Figure 2. Concretely, human navigation behaviors can be predicted from the decoded representations in a step-wise fashion. We use Dijkstra’s Shortest Path First algorithm to generate optimal action instructions as human expert experience. Under the supervision of optimal action instructions, our VT learns to imitate the optimal navigation action selection. ",
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+    {
+        "type": "text",
+        "text": "In the pre-training stage, we do not employ our navigation network (i.e., LSTM), and previous actions as well as states are not available. Note that, in our navigation network, previous actions, previous states and current visual representations are exploited, as seen in Figure 2. Thus, we replace our LSTM with an MLP and predict action distributions based on the current visual representation. A cross-entropy loss $\\boldsymbol { L _ { v t } } = \\boldsymbol { C E } ( \\boldsymbol { a _ { t } } , \\boldsymbol { \\hat { a } } )$ is employed to train our VT and the MLP, where $a _ { t }$ is the predicted action, $\\hat { a }$ represents the optimal action instruction and $C E$ indicates the cross-entropy function. After pre-training, features from our VT also exhibit strong association with directional navigation signals as only an MLP is employed on top of the features. Therefore, the decoded features will facilitate the navigation network training. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 PROTOCOLS AND EXPERIMENTAL DETAILS ",
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+    {
+        "type": "text",
+        "text": "Dataset. We perform our experiments on AI2-Thor (Kolve et al., 2017), an artificial 3D environment with realistic photos. It contains 4 types of scenes, i.e., kitchen, living room, bedroom and bathroom. In each type of scenes, there are 30 different rooms with various furniture placements and items. Following Du et al. (2020), we choose 22 categories as the target classes and ensure that there are at least 4 potential targets in each room. ",
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+        "type": "text",
+        "text": "We use the same training and evaluation protocols as the works (Wortsman et al., 2019; Du et al., 2020). 80 rooms out of 120 are selected as the training set while each scene contains 20 rooms. ",
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+    {
+        "type": "table",
+        "img_path": "images/f8fb86c422a19b4d31e1b8dc6bbf3b7cf744f8839c21f7b00cd68746bac82198.jpg",
+        "table_caption": [
+            "Table 1: Comparison with the state-of-the-art. We report the average success rate $( \\% )$ and SPL as well as their variances in parentheses by repeating experiments five times. $L > 5$ represents the episodes which require at least 5 steps. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">ALL</td><td colspan=\"2\">L≥5</td></tr><tr><td>Success</td><td>SPL</td><td>Success</td><td>SPL</td></tr><tr><td>Random</td><td>8.0 (1.3)</td><td>0.036 (0.006)</td><td>0.3 (0.1)</td><td>0.001 (0.001)</td></tr><tr><td>WE</td><td>33.0 (3.5)</td><td>0.147( (0.018)</td><td>21.4 (3.0)</td><td>0.117 (0.019)</td></tr><tr><td>SP (Yang et al., 2018)</td><td>35.1 (1.3)</td><td>0.155 (0.011)</td><td>22.2 (2.7)</td><td>0.114 (0.016)</td></tr><tr><td>SAVN (Wortsman et al.,2019)</td><td>40.8 (1.2)</td><td>0.161 (0.005)</td><td>28.7 (1.5)</td><td>0.139 (0.005)</td></tr><tr><td>ORG (Du et al.,2020)</td><td>65.3 (0.7)</td><td>0.375 (0.008)</td><td>54.8 (1.0)</td><td>0.361 (0.009)</td></tr><tr><td>ORG+TPN (Du et al., 2020)</td><td>69.3 (1.2)</td><td>0.394 (0.010)</td><td>60.7 (1.3)</td><td>0.386 (0.011)</td></tr><tr><td>Baseline</td><td>62.6 (0.9)</td><td>0.364 (0.006)</td><td>51.5 (1.2)</td><td>0.345 (0.007)</td></tr><tr><td>VTNet</td><td>72.2 (1.0)</td><td>0.449 (0.007)</td><td>63.4 (1.1)</td><td>0.440 (0.009)</td></tr><tr><td>VTNet + TPN (Du et al., 2020)</td><td>73.5 (1.3)</td><td>0.440 (0.009)</td><td>63.9 (1.5)</td><td>0.440 (0.011)</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We equally divide the remaining 40 rooms into validation and test sets. We report the results of the testing data by using the model with the highest success rate on the validation set. ",
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+    {
+        "type": "text",
+        "text": "Evaluation metrics. We evaluate our model performance by success rate and Success Weighted $\\textstyle { \\frac { 1 } { N } } \\sum _ { n = 0 } ^ { N } S _ { n }$ gth (SP, where $N$ . The success rate measures is the number of episodes and $S _ { n }$ gation effectiveneis a success indica and isr of the $n$ o- d byode. We adopt SPL tits optimal path ure the navigation effic, SPL is formulated as $n$ -th episode $L e n _ { n }$ and $L e n _ { o p t }$ $\\begin{array} { r } { \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N } S _ { n } \\frac { L e n _ { n } } { m a x ( L e n _ { n } , L e n _ { o p t } ) } } \\end{array}$ ",
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+    {
+        "type": "text",
+        "text": "Training details. We use a two-stage training strategy. In Stage 1, we train our visual transformer for 20 epochs with the supervision of optimal action instructions. In this fashion, we explicitly construct the association between visual representations and navigation actions. In Stage 2, we train the navigation policy for 6M episodes in total with 16 asynchronous agents. We set a penalization $- 0 . 0 0 1$ on each action step and a large reward 5 when an agent completes an episode successfully. We adopt DETR as the object detector and fine-tune DETR on the AI2-Thor training dataset. In training DETR, we applied data augmentation, such as resize and random crop. We use the Adam optimizer (Kingma & Ba, 2014) to update the policy network with a learning rate $1 0 ^ { - 4 }$ and the pre-trained VT with a learning rate $1 0 ^ { \\div { 5 } }$ . Our codes and pre-trained model will be publicly released for reproducibility. ",
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+    {
+        "type": "text",
+        "text": "5.2 COMPETING METHODS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We compare our method with the following ones: Random policy. An agent chooses actions based on a uniform action probability. Thus, the agent will walk or stop in a scene randomly. Scene Prior (SP) (Yang et al., 2018) learns a graph neural network from the FastText database (Joulin et al., 2016) and leverages the scene prior knowledge and category relationships for navigation. Word Embedding (WE) uses GloVe embedding (Pennington et al., 2014) to indicate the target category rather than detection. The association between object appearances and GloVe embeddings is learned through trail and error. Self-adaptive Visual Navigation (SAVN) (Wortsman et al., 2019) introduces a meta reinforcement learning method that allows an agent to adapt to unseen environments. Object Relationship Graph (ORG) (Du et al., 2020) is a visual representation learning method to encode correlation among categories and employs a tentative policy network (TPN) to escape from deadlocks. Baseline is a vanilla version of VTNet. We feed the concatenation of the local instance features from DETR and the global feature to A3C for navigation. Note that, our baseline does not employ spatial-enhanced local and positional global descriptors as well as our visual transformer. ",
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+    {
+        "type": "text",
+        "text": "5.3 EVALUATION RESULTS ",
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+        "type": "text",
+        "text": "Improvement over Baseline. Table 1 indicates that VTNet surpasses the baseline by a large margin on both success rate $( + 9 . 6 \\% )$ and SPL $( + 0 . 0 8 5 )$ . Baseline only resorts to the detection features and global feature for navigation. The relations among local instances and the association between the visual observations and actions are not exploited. This comparison suggests that our VT leads to informative visual representations for navigation, and thus significantly improves the effectiveness and efficiency of our navigation system. ",
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+        "type": "text",
+        "text": "Comparison with competing methods. As indicated in Table 1, we observe that VTNet significantly outperforms SP (Yang et al., 2018) and SAVN (Wortsman et al., 2019). Since SP and SAVN employ word embedding as a target indicator while VTNet replaces word embedding with our VT, our method achieves expressive object and image region representations for navigation. In addition, SP and SAVN concatenate features from various modalities directly to generate visual representations. The gap between different modalities may not facilitate navigation policy learning. In contrast, benefiting from our pre-training, features from our VT are more correlated to navigation actions, thus expediting navigation policy learning. ",
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+        "img_path": "images/71e6a48512317337dfb2274ccebf535ebeeeded09483fadd8a8db86293d96c41.jpg",
+        "image_caption": [
+            "Figure 3: Visual results of four different models in testing environments. The target objects (i.e., RemoteControl) are highlighted by the blue boxes. Green and red curves represent success and failure cases, respectively. The episode produced by our VTNet is successful in reaching the target and use shortest steps. In comparison, ORG takes more steps to reach the target. SAVN and Baseline miss both targets. "
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+        "type": "text",
+        "text": "Our method outperforms the state-of-the-art method ORG (Du et al., 2020) by $+ 2 . 9 \\%$ in success rate and $+ 0 . 0 5 5$ in SPL. Moreover, when ORG does not employ TPN, the advantage of our method becomes more obvious $6 . 9 \\%$ improvement), and this mainly comes from our superior visual presentations. Since ORG only chooses an object with the highest confidence in each class, the relationship among objects is not comprehensive. On the contrary, our method can exploit all the detected instances to deduce the relationships among objects due to our VT architecture. Moreover, since DETR infers the relations between object instances and the global image context, the local features output by the DETR are more informative compared to the object features used in ORG. This can be proved by the result when we use Faster R-CNN as our backbone, as indicated by Table 2. We also show a case study in Figure 3 (more visual results are provided in the appendix). Furthermore, we also try to employ TPN to improve our navigation policy. As seen in Table 1, VTNet+TPN improves the success rates but the improvement is not as much as $_ \\mathrm { O R G + T P N }$ . This also implies that our visual representations significantly facilitate navigation action selections. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Case Study. As illustrated in Figure 3, SAVN and Baseline both issue the termination command after navigating a few steps (7 and 19 steps, respectively), but fail to reach the target. This indicates that the relationships among categories are not clear in SAVN and Baseline. In contrast, both ORG and VTNet find the target. Since our visual transformer provides clear directional signals, VTNet uses the least steps to find the object. ",
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+    {
+        "type": "text",
+        "text": "5.4 VARIANT AND ABLATION STUDY ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we analyze the impact of each component in VTNet, including the spatial-enhanced local descriptor, positional global descriptor, visual transformer that fuses these two spatial-aware descriptors and pre-training scheme. ",
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+        "type": "text",
+        "text": "To illustrate the necessity of the spatial enhancement, we directly use the object features without spatial enhancement. In this case, our network fails to converge because the feed-forward layers that predict bounding-boxes and class labels in DETR are not used in VTNet and spatial information cannot be decoded by the navigation network. Thus, spatial enhancement allows an agent to exploit instance location information explicitly. ",
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+    {
+        "type": "text",
+        "text": "As indicated in Table 2, we achieve better navigation performance using instance features from DETR compared to employing Faster R-CNN features following the feature extraction of Du et al. (2020) (“Faster R-CNN”). Unlike Faster R-CNN, DETR infers the relations between object instances and the global image context via its transformer to output the final predictions (i.e., class labels and bounding boxes). Although DETR and Faster R-CNN achieve similar detection performance (Carion et al., 2020), features extracted by DETR are more informative and robust than those of Faster R-CNN used in ORG. Specifically, ORG extracts features from the second layer of the backbone in Faster R-CNN based on the predicted bounding-boxes to ensure the features are comparable, but the features are not the most prominent ones across the feature pyramid. Therefore, the object features “Faster R-CNN” extracted by ORG are inferior to DETR features, and our navigator employing DETR features outperforms ORG. ",
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+    {
+        "type": "table",
+        "img_path": "images/2ac7f0344c4f01a641f274065b06e357b3131edbce5fae09c9951cb80dee11fe.jpg",
+        "table_caption": [
+            "Table 2: Impacts of different components on navigation performances. Faster R-CNN and Faster R-CNN† represent instance features extracted by Faster R-CNN following Du et al. (2020) and Anderson et al. (2018a), respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\">Method</td><td rowspan=\"2\">w/o global decoder</td><td rowspan=\"2\">w/o</td><td rowspan=\"2\">w/o pe</td><td rowspan=\"2\">VTNetg</td><td colspan=\"3\">Baseline</td><td colspan=\"3\">VTNet</td></tr><tr><td>Faster R-CNN R-CNN</td><td>Faster+</td><td>DETR</td><td>Faster R-CNN R-CNN</td><td>Faster+</td><td>DETR</td></tr><tr><td rowspan=\"3\">ALL</td><td>Success</td><td>67.0</td><td>67.0</td><td>71.0</td><td>70.1</td><td>56.4</td><td>57.2</td><td>62.6</td><td>70.1</td><td>70.3</td><td>72.2</td></tr><tr><td></td><td>(2.8) 0.390</td><td>(1.4) 0.373</td><td>(0.7)</td><td>(1.3) 0.411</td><td>(0.9) 0.319</td><td>(1.1)</td><td>(0.8) 0.365</td><td>(1.0) 0.396</td><td>(1.2)</td><td>(1.0) 0.449</td></tr><tr><td>SPL</td><td>(0.021)</td><td>(0.013)</td><td>0.432 (0.009)</td><td>(0.009)</td><td>(0.007)</td><td>0.308 (0.008)</td><td>(0.010)</td><td>(0.010)</td><td>0.387 (0.012)</td><td>(0.007)</td></tr><tr><td rowspan=\"3\">L≥5</td><td>Success</td><td>54.5</td><td>53.2</td><td>61.2</td><td>60.6</td><td>42.5</td><td>46.7</td><td>51.5</td><td>61.7</td><td>62.1</td><td>63.4</td></tr><tr><td></td><td>(3.1)</td><td>(1.6)</td><td>(0.9)</td><td>(1.5)</td><td>(1.2)</td><td>(1.3)</td><td>(1.0)</td><td>(1.2)</td><td>(1.4)</td><td>(1.1)</td></tr><tr><td>SPL</td><td>0.357</td><td>0.343</td><td>0.416 (0.010)</td><td>0.395</td><td>0.270</td><td>0.276</td><td>0.345</td><td>0.399</td><td>0.376</td><td>0.440</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Moreover, we adopt the instance features from Faster R-CNN following the feature extraction fashion of Anderson et al. (2018a) (“Faster R-CNN†”). Thus, we obtain the instance features from the penultimate layer of the classifier in Faster R-CNN. We observe that instance features from DETR also improve the navigation performance compared to the Faster R-CNN† features. We speculate the improvements mainly come from the fact that the features output by DETR decoder have embedded global context information, and those features are more suitable for the feature fusion operations. ",
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+    },
+    {
+        "type": "text",
+        "text": "As seen in Table 2, we first remove the global feature from our system (“VTNet w/o global”), and the navigation performance degrades significantly. This validates the importance of global features, which provide contextual guidance to an agent. Moreover, when we remove the positional embedding from the global feature (“VTNet w/o pe”), we observe that both effectiveness and efficiency of the navigation decrease. This indicates that the position embeddings facilitate our VT to exploit the spatial information of observation regions. Furthermore, when we remove the VT decoder (“VTNet w/o decoder”) and concatenate the global and local descriptors directly, our method suffers performance degradation. This demonstrates that our VT plays a critical role in attending the global descriptors to local ones. Additionally, we feed the positional global features into the transformer encoder $( ^ { 6 6 } \\nabla \\mathrm { T N e t } _ { g } ^ { , 9 } )$ . The navigation performance of $\\mathrm { V T N e t } _ { g }$ is superior to that of ORG but slightly inferior to the performance of our VTNet. This demonstrates the transformer architecture is effective to extract informative visual representations, and assigning different functions to different modules would further facilitate the establishment of mappings in our VT. ",
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+    {
+        "type": "text",
+        "text": "When the pre-training scheme is not applied to our VT, our agent fails to learn any effective navigation policy and thus we do not report the performance. This manifests that our VT pre-training procedure provides a good initialization to our transformer and prior knowledge on associating visual observations with navigation actions to agents. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we proposed a powerful visual representation learning method for visual navigation, named Visual Transformer Network (VTNet). In our VTNet, a visual transformer (VT) has been developed to encode visual observations. Our VT leverages two newly designed spatial-aware descriptors, i.e., a spatial-enhanced local object descriptor and a positional global descriptor, and then fuses those two types of descriptors via multi-head attention to achieve our final visual representation. Thanks to our VT architecture, all the detected instances will be exploited for understanding the current observation. Therefore, our visual representation is more informative compared to that used in state-of-the-art navigation methods. Benefiting from our pre-training strategy, our VT is able to associate visual representations with navigation actions, thus significantly expediting navigation policy learning. Extensive results demonstrate that our VTNet outperforms the state-of-the-art in terms of effectiveness and efficiency. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
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+        "type": "text",
+        "text": "This work was supported by the ARC Discovery Early Career Researcher Award (DE200101283), the ARC Discovery Project (DP210102801) and the Data61 Collaborative Research Project. ",
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+        "text": "Yuke Zhu, Roozbeh Mottaghi, Eric Kolve, Joseph J Lim, Abhinav Gupta, Li Fei-Fei, and Ali Farhadi. Target-driven visual navigation in indoor scenes using deep reinforcement learning. In 2017 IEEE international conference on robotics and automation (ICRA), pp. 3357–3364. IEEE, 2017. ",
+        "bbox": [
+            174,
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            299,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 FEATURE DETAILS IN VTNET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            423,
+            148
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/c4b951c34cf9d29c67a4a82b3f8c86e47d192f431c225717fe5be3c9131dd942.jpg",
+        "image_caption": [
+            "Figure 4: Illustration feature flowchart in VTNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            171,
+            812,
+            342
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For reproducibility, we illustrate the detailed feature flowchart of our VTNet in Figure 4. We extract instance features $\\mathbb { R } ^ { 1 0 0 \\times 2 5 6 }$ and location features $\\mathbb { R } ^ { 1 0 0 \\times 2 4 9 }$ , and concatenate them into a spatialenhanced local descriptor $\\mathbb { R } ^ { 1 0 0 \\times 2 5 6 }$ . The local branch integrates semantic labels $\\mathbb { R } ^ { 1 0 0 \\times 1 }$ , bounding boxes $\\mathbb { R } ^ { 1 0 0 \\times 4 }$ , confidences $\\mathbb { R } ^ { 1 0 0 \\times 1 }$ and the target labels $\\mathbf { \\mathbb { R } } ^ { 1 0 0 \\times 1 }$ for the current observation. In the global branch, the positional embedding $\\mathbb { R } ^ { 7 \\times 7 \\times 2 5 6 }$ is added to the global feature $\\mathbb { R } ^ { 7 \\times 7 \\times 2 5 6 }$ , leading to a positional global descriptor $\\mathbb { R } ^ { 4 9 \\times 2 5 6 }$ . The spatial-enhanced local and positional global descriptors are fused by VT encoder and then the visual representation $\\mathbb { R } ^ { 4 9 \\times 2 5 6 }$ is output by our VT decoder. ",
+        "bbox": [
+            173,
+            395,
+            825,
+            507
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 FAILURE CASE STUDY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            525,
+            370,
+            539
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/34892dc23d9809566fab64e69ce37f1da19b576e5df5e9205f69aaa41b444733.jpg",
+        "image_caption": [
+            "Figure 5: Visual results of failure cases in testing environments. The target objects (i.e., Bowl and Laptop) are highlighted by the blue boxes. Red lines indicate the distance between the agent and the target object. Gray curves represent trajectories of agents. Both episodes fail because distances (i.e., $1 . 5 3 0 \\mathrm { m }$ and $1 . 5 7 2 \\mathrm { m } )$ between the agent and the target in the field of view are larger than the threshold (i.e., $1 . 5 \\mathrm { m } )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            554,
+            816,
+            700
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "As demonstrated in Figure 5, our VTNet fails to reach targets because the distances between agents and targets are larger than the threshold distance (i.e., $1 . 5 \\mathrm { m } )$ . In these two failure cases, agents find targets but implement the termination action before reaching a position closer than the threshold. Due to the lack of depth information and variances of target object sizes, an agent may predict that it is within a 1.5 meter radius of the target by mistake and thus terminates current episode. ",
+        "bbox": [
+            173,
+            801,
+            825,
+            872
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/3c315b24db0d4468cf765539de490160c68089eeadda004c75b54092ed51bbc0.jpg",
+        "image_caption": [
+            "Figure 6: Visual results of four different models in testing environments. We compare VTNet with SAVN (Wortsman et al., 2019), Baseline and ORG (Du et al., 2020). The target objects are highlighted by the blue boxes. Green and red curves indicate success and failure cases, respectively. Our VTNet successfully reaches targets and uses the shortest steps. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            212,
+            140,
+            784,
+            844
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/c5f70dfff80d5672701d602e294cfe6e6c93c7436860a70363e681ea78c9cc88.jpg",
+        "table_caption": [
+            "Table 3: Comparison of visual transformer architectures. We report the pre-training accuracy on the validation dataset, navigation success rate and SPL on the test dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=6>Multi-head numbersEncoder layersDecoder layersAccuracy Success SPL</td></tr><tr><td rowspan=3 colspan=1>4</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.722</td><td rowspan=1 colspan=1>71.2</td><td rowspan=1 colspan=1>0.433</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>0.723</td><td rowspan=1 colspan=1>72.2</td><td rowspan=1 colspan=1>0.449</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>0.715</td><td rowspan=1 colspan=1>70.0</td><td rowspan=1 colspan=1>0.419</td></tr><tr><td rowspan=4 colspan=1>8</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0.707</td><td rowspan=1 colspan=1>70.4</td><td rowspan=1 colspan=1>0.422</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>0.718</td><td rowspan=1 colspan=1>70.9</td><td rowspan=1 colspan=1>0.436</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>0.710</td><td rowspan=1 colspan=1>68.9</td><td rowspan=1 colspan=1>0.423</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>0.701</td><td rowspan=1 colspan=1>68.2</td><td rowspan=1 colspan=1>0.411</td></tr></table>",
+        "bbox": [
+            233,
+            174,
+            756,
+            286
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We construct different transformer architectures by varying the number of encoder and decoder layers. Table 3 summarizes the performance of these architectures. We observe that as a visual transformer becomes too deep, a transformer may fail to converge to an optimal policy. On the other hand, a transformer with a single encoder and decoder layer does not have sufficient network capability to produce representative features. The highest success rate is achieved when a VT contains four multi-head self-attention mechanism modules and two layers in the encoder and decoder. ",
+        "bbox": [
+            173,
+            303,
+            825,
+            387
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.5 NECESSITY OF PRE-TRAINING SCHEME ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            404,
+            488,
+            417
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/e385df76af0b53ccfbbae0079bba8b5e8e4972d6fa5992d3a5eee1c1ae8cd8d5.jpg",
+        "image_caption": [
+            "Figure 7: Average episode lengths of VTNet and VTNet without pre-training during training. We compare VTNet with VTNet without pre-training scheme. Blue and orange curves represent VTNet and VTNet w/o pre-training, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            212,
+            462,
+            772,
+            678
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "As demonstrated in Figure 7, our VTNet spends nearly 10 steps per episode in training, while the navigator w/o pre-training scheme often fails to reach targets and stops around 5 steps after being trained tens of thousands of episodes. Due to the large parameters and complex architectures of transformers, it is often difficult to train our transformers from scratch (Liu et al., 2020). Without a good initialization for our VT, it is very difficult to learn our VT and policy network in an end-to-end fashion with RL rewards. This is because the visual representations from VT are not informative or even meaningless and the inferior visual representations would harm policy network learning. As a result, the navigation policy network may be trapped into a local minimum (i.e., terminating navigation early to avoid more penalties) and our VT cannot receive positive rewards from preceding trajectories. ",
+        "bbox": [
+            173,
+            758,
+            825,
+            898
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/a7fdfb5658f2ce176303ab0573e5d96d1f2922a9f9a2a75666f92aed2e5cb5b3.jpg",
+        "image_caption": [
+            "Figure 8: Visualizations of attention scores. The target classes (i.e., StoveBurner, GarbageCan, Kettle) are highlighted by green bounding boxes. Our agent detects the instances of interest and then attends the detected instances to the global image regions by our VT. We observe that high attention scores are obtained on the areas corresponding to the targets. Guided by the visual representations, the agent selects actions to approach the targets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            132,
+            812,
+            525
+        ],
+        "page_idx": 15
+    }
+]

parse/train/SkxW23NtPH/SkxW23NtPH.md

@@ -0,0 +1,270 @@
+# GDP: GENERALIZED DEVICE PLACEMENT FOR DATAFLOW GRAPHS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Runtime and scalability of large neural networks can be significantly affected by the placement of operations in their dataflow graphs on suitable devices. With increasingly complex neural network architectures and heterogeneous device characteristics, finding a reasonable placement is extremely challenging even for domain experts. Most existing automated device placement approaches are impractical due to the significant amount of compute required and their inability to generalize to new, previously held-out graphs. To address both limitations, we propose an efficient end-to-end method based on a scalable sequential attention mechanism over a graph neural network that is transferable to new graphs. On a diverse set of representative deep learning models, including Inception-v3, AmoebaNet, Transformer-XL, and WaveNet, our method on average achieves $16 \%$ improvement over human experts and $9 . 2 \%$ improvement over the prior art with $1 5 \times$ faster convergence. To further reduce the computation cost, we pre-train the policy network on a set of dataflow graphs and use a superposition network to fine-tune it on each individual graph, achieving state-of-the-art performance on large hold-out graphs with over 50k nodes, such as an 8-layer GNMT.
+
+# 1 INTRODUCTION
+
+Neural networks have demonstrated remarkable scalability–improved performance can usually be achieved by training a larger model on a larger dataset (Hestness et al., 2017; Shazeer et al., 2017; Jozefowicz et al., 2016; Mahajan et al., 2018; Radford et al.). Training such large models efficiently while meeting device constraints, like memory limitations, necessitate partitioning of the underlying dataflow graphs for the models across multiple devices. However, devising a good partitioning and placement of the dataflow graphs requires deep understanding of the model architecture, optimizations performed by domain-specific compilers, as well as the device characteristics, and is therefore extremely hard even for experts.
+
+ML practitioners often rely on their understanding of model architecture to determine a reasonable partitioning and placement for graphs. However, relying solely on the model architecture while ignoring the effect of the partitioning on subsequent compiler optimizations like op-fusion can lead to sub-optimal placements and consequently under-utilization of available devices. The goal of automated device placement is to find the optimal assignment of operations to devices such that the end-to-end execution time for a single step is minimized and all device constraints like memory limitations are satisfied. Since this objective function is non-differentiable, prior approaches (Mirhoseini et al., 2017; 2018; Gao et al., 2018) have explored solutions based on reinforcement learning (RL). However, these RL policies are usually not transferable and require training a new policy from scratch for each individual graph. This makes such approaches impractical due to the significant amount of compute required for the policy search itself, at times offsetting gains made by the reduced step time.
+
+In this paper, we propose an end-to-end deep RL method for device placement where the learned policy is generalizable to new graphs. Specifically, the policy network consists of a graph-embedding network that encodes operation features and dependencies into a trainable graph representation, followed by a scalable sequence-to-sequence placement network based on an improved Transformer (Vaswani et al., 2017; Dai et al., 2019). The placement network transforms the graph representations into a placement decision with soft attention, removing hard constraints such as hierarchical grouping of operations (Mirhoseini et al., 2018) or co-location heuristics (to reduce the placement complexity) (Mirhoseini et al., 2017). Both of our graph-embedding network and placement network can be jointly trained in an end-to-end fashion using a supervised reward, without the need to manipulate the loss functions at multiple levels. We empirically show that the network learns flexible placement policies at a per-node granularity and can scale to problems over 50,000 nodes.
+
+To generalize to arbitrary and held-out graphs, our policy is trained jointly over a set of dataflow graphs (instead of one at a time) and then fine-tuned on each graph individually. By transferring the learned graph embeddings and placement policies, we are able to achieve faster convergence and thus use less resources to obtain high-quality placements. We also use super-positioning, i.e., a feature conditioning mechanism based on the input graph embeddings, to effectively orchestrate the optimization dynamics of graphs with drastically different sizes in the same batch.
+
+Our contributions can be summarized as follows:
+
+1. An end-to-end device placement network that can generalize to arbitrary and held-out graphs. This is enabled by jointly learning a transferable graph neural network along with the placement network.   
+2. A scalable placement network with an efficient recurrent attention mechanism, which eliminates the need for an explicit grouping stage before placement. The proposed end-to-end network provides $1 5 \times$ faster convergence as compared to the hierarchical LSTM model used in earlier works (Mirhoseini et al., 2017; 2018).   
+3. A new batch pre-training and fine-tuning strategy based on network superposition, which leads to improved transferability, better placements especially for larger graphs, and $1 0 \times$ reduction in policy search time as compared to training individual graphs from scratch.   
+4. Superior performance over a wide set of workloads, including InceptionV3 (Szegedy et al., 2015), AmoebaNet (Real et al., 2018), RNNs, GNMT (Wu et al., 2016), Transformer-XL (Dai et al., 2019), WaveNet (van den Oord et al., 2016), and more.
+
+# 2 RELATED WORK
+
+Device Placement Reinforcement learning has been used for device placement of a given dataflow graph (Mirhoseini et al., 2017) and demonstrated run time reduction over human crafted placement and conventional heuristics. For improved scalability, a hierarchical device placement strategy (HDP) (Mirhoseini et al., 2018) has been proposed that clusters operations into groups before placing the operation groups onto devices. Spotlight (Gao et al., 2018) applies proximal policy optimization and cross-entropy minimization to lower training overhead. Both HDP and Spotlight rely on LSTM controllers that are difficult to train and struggle to capture very long-term dependencies over large graphs. In addition, both methods are restricted to process only a single graph at a time, and cannot generalize to arbitrary and held-out graphs. Placeto (Addanki et al., 2019) represents the first attempt to generalize device placement using a graph embedding network. But like HDP, Placeto also relies on hierarchical grouping and only generates placement for one node at each time step. Our approach (GDP) leverages a recurrent attention mechanism and generates the whole graph placement at once. This significantly reduces the training time for the controller. We also demonstrate the generalization ability of GDP over a wider set of important workloads.
+
+Parallelization Strategy Mesh-TensorFlow is a language that provides a general class of distributed tensor computations. While data-parallelism can be viewed as splitting tensors and operations along the “batch” dimension, in Mesh-TensorFlow the user can specify any tensor-dimensions to be split across any dimensions of a multi-dimensional mesh of processors. FlexFlow (Jia et al., 2018) introduces SOAP, a more comprehensive search space of parallelization strategies for DNNs which allows parallelization of a DNN in the Sample, Operator, Attribute, and Parameter dimensions. It uses guided randomized search of the SOAP space to find a parallelization strategy for a specific parallel machine. GPipe (Huang et al., 2018) proposed pipeline parallelism, by partitioning a model across different accelerators and automatically splitting a mini-batch of training examples into smaller micro-batches. By pipelining the execution across micro-batches, accelerators can operate in parallel. Our GDP focuses on a general deep RL method for automating device placement on arbitrary graphs, and is therefore orthogonal to existing parallelization strategies.
+
+![](images/6d7904fbfb076cc23f86cea14773c4e42f598efb830a695fdc7196d067a40a91.jpg)  
+Figure 1: Overview of GDP: An end-to-end placement network that combines graph embedding and sequential attention. $N$ : Number of Nodes, h: Hidden Size, $d$ : Number of Devices.
+
+Compiler Optimization REGAL (Paliwal et al., 2019) uses deep RL to optimize the execution cost of computation graphs in a static compiler. The method leverages the policy’s ability to transfer to new graphs to improve the quality of the genetic algorithm for the same objective budget. However, REGAL only targets peak memory minimization while GDP focuses on graph run time and scalability while also meeting the peak memory constraints of the devices. Specifically, we generalize graph partitioning and placement into a single end-to-end problem, with and without simulation, which can handle graphs with over 50,000 nodes.
+
+# 3 END-TO-END PLACEMENT POLICY
+
+Given a dataflow graph $G ( V , E )$ where $V$ represents atomic computational operations (ops) and $E$ represents the data dependency, our goal is to learn a policy $\pi : \mathcal { G } \mapsto \mathcal { D }$ that assigns a placement $D \in { \mathcal { D } }$ for all the ops in the given graph $G \in { \mathcal { G } }$ , to maximize the reward $r _ { G , D }$ defined based on the run time. $\mathcal { D }$ is the allocated devices that can be a mixture of CPUs and GPUs. In this work, we represent policy $\pi _ { \theta }$ as a neural network parameterized by $\theta$ .
+
+Unlike prior works that focus on a single graph only, the RL objective in GDP is defined to simultaneously reduce the expected runtime of the placements over a set of $N$ dataflow graphs:
+
+$$
+J ( \theta ) = \mathbb { E } _ { G \sim \mathcal { G } , D \sim \pi _ { \theta } ( G ) } [ r _ { G , D } ] \approx \frac { 1 } { N } \sum _ { G } \mathbb { E } _ { D \sim \pi _ { \theta } ( G ) } [ r _ { G , D } ]
+$$
+
+In the following, we refer to the case when $N = 1$ as individual training and the case when $N > 1$ as batch training. We optimize the objective above using Proximal Policy Optimization (PPO) (Schulman et al., 2017) for improved sample efficiency.
+
+Figure 1 shows an overview of the proposed end-to-end device placement network. Our proposed policy network $\pi _ { \theta }$ consists a graph embedding network that learns the graphical representation of any dataflow graph, and a placement network that learns a placement strategy over the given graph embeddings. The two components are jointly trained in an end-to-end fashion. The policy $p ( a | G )$ is applied to make a set of decisions at each node. These decisions, denoted as $a _ { v }$ for each $v \in V$ across all nodes, form one action $a = \{ a _ { v \in V } \}$ . One decision corresponds to playing one arm of a multi-bandit problem, and specifying the entire $a$ corresponds to playing several arms together in a single shot. Note the architecture is designed to be invariant over the underlying graph topology, enabling us to apply the same learned policy to a wide set of input graphs with different structures.
+
+# 3.1 GRAPH EMBEDDING NETWORK
+
+We leverage graph neural networks (GNNs) (Hamilton et al., 2017; Xu et al., 2019; You et al., 2018) to capture the topological information encoded in the dataflow graph. Most graph embedding frameworks are inherently transductive and can only generate embeddings for a given fixed graph. These transductive methods do not efficiently extrapolate to handle unseen nodes (e.g., in evolving graphs), and cannot learn to generalize to unseen graphs. GraphSAGE (Hamilton et al., 2017) is an inductive framework that leverages node attribute information to efficiently generate representations on previously unseen data. While our proposed framework is generic, we adopt the feature aggregation scheme proposed in GraphSAGE to model the dependencies between the operations and build a general, end-to-end device placement method for a wide set of dataflow graphs.
+
+In GDP, nodes and edges in the dataflow graph are represented as the concatenation of their meta features (e.g., operation type, output shape, adjacent node ids) and are further encoded by the graph embedding network into a trainable representation. The graph embedding process consists of multiple iterations, and the computation procedure for the $l$ -th iteration can be outlined as follows:
+
+First, each node $v \in V$ aggregates the feature representations of its neighbors, $\{ h _ { u } ^ { ( l ) } , \forall u \in \mathcal { N } ( v ) \}$ , into a single vector h(l)N (v). This aggregation outcome is a function of all previously generated representations, including the initial representations defined based on the input node features. In this work, we use the following aggregation function with max pooling:
+
+$$
+h _ { \mathcal { N } ( v ) } ^ { ( l ) } = \operatorname* { m a x } ( \sigma ( W ^ { ( l ) } h _ { u } ^ { ( l ) } + b ^ { ( l ) } ) , \forall u \in \mathcal { N } ( v ) )
+$$
+
+where $( W ^ { ( l ) } , b ^ { ( l ) } )$ define an affine transform and $\sigma$ stands for the sigmoid activation function. We then concatenate the node’s current representation, $h _ { v } ^ { ( l ) }$ , with the aggregated neighborhood vector, $h _ { \mathcal { N } ( v ) } ^ { ( l ) }$ , and feed this concatenated vector through a fully connected layer $f ^ { ( l + 1 ) }$ ei
+
+$$
+h _ { v } ^ { ( l + 1 ) } = f ^ { ( l + 1 ) } ( \mathrm { c o n c a t } ( h _ { v } ^ { ( l ) } , h _ { \mathcal { N } ( v ) } ^ { ( l ) } ) )
+$$
+
+Different from GraphSAGE, parameters in our graph embedding network are trained jointly with a placement network via stochastic gradient descent with PPO, in a supervised fashion, as described in Section 3. That is, we replace the unsupervised loss with our task-specific objective.
+
+# 3.2 PLACEMENT NETWORK
+
+The graph neural network works as a feature aggregation network that learns a trainable feature representation for the computational graph, we still need a policy network that produces actions on a per node basis. Given $h _ { v }$ ’s, the policy network produces $a _ { v }$ ’s through conditionally independent predictions, where the prediction for one node $v$ does not depend on the prediction of other nodes.
+
+$$
+p ( a | G ) = \prod _ { v } p ( a _ { v } | G ) = \prod _ { v } p ( a _ { v } | f ( h _ { v } ) )
+$$
+
+While $f$ can be represented using multilayer perceptrons (MLPs), where the MLPs is shared across all nodes for prediction the placement output distributions. However, MPLs lack a dependency tracking mechanism across nodes. In practise, the placement of one node can be determined by the placement of another node, where the placed node may consume a large size of data produced by the other node. Intuitively, an attention network can learn this dependency and the relative importance of dependencies across an entire graph. Therefore, we decide to use an attention-based placement network to better track inter-node placement-related dependencies.
+
+Designing a scalable placement network that can generalize to graphs with thousands of nodes is challenging, as the conventional GNMT models proposed for language tasks usually target a shorter sequence length. Hierarchical placement (Mirhoseini et al., 2018) has been proposed to address this issue,however, the proposed grouper network comes with limited flexibility and generality. For example, the grouper network leverages an aggregated feature representation by averaging feature vectors for nodes within the same group. The non-differentiable grouping procedure prevents training the graph-embedding and placement networks end-to-end.
+
+To remove the two-stage hierarchical workflow in HDP for improved scalability, we propose to use a Transformer-based attentive network to generate operation placements in an end-to-end fashion. As the graph embedding already contains spatial (topological) information for each node, we remove the positional embedding in the original transformer to prevent the model from overfitting node identifications. To capture long-term dependencies efficiently among a large set of nodes, we adopt segment-level recurrence introduced in Transformer-XL (Dai et al., 2019; Dai, 2019), where hidden states computed for the previous set of nodes are cached (with gradient flows disabled) and reused as an extended context during the training of the next segment. Besides achieving extra long context, we empirically find the segment-level recurrent attention much faster than a conventional LSTMbased GNMT model. In our experimental evaluation, we compare both the performance and speed up of our placement network with that of the LSTM-based hierarchical device placement.
+
+# 3.3 BATCH TRAINING WITH PARAMETER SUPERPOSITION
+
+Since the parameterization for the architecture of the end-to-end policy is designed to be invariant over input graphs with different topologies, the same placement policy can be shared across a wide set of workloads. We therefore propose a batch training strategy, and further enhance the aforementioned architecture to handle such generalization across graphs.
+
+Na¨ıve batch training is challenging in our context as different dataflow graphs contain different number of operations connected in different topologies. In addition, unlike previous device placement methods, GDP aims to handle graphs from potentially different application domains (e.g. computer vision, language, and speech), where the number of operations can range from a few thousand to one million. These graphs have drastically different network architecture, in terms of computational operations, data shape, and network topology. As an example, recurrent networks have completely different operation types and connections compared to multi-branch convolutional networks that are widely used in computer vision. It would be highly desirable to train a single shared network that maximizes information sharing across these heterogeneous tasks, without hurting the performance on each of them due to their distinct learning dynamics.
+
+Along a similar direction of multi-task learning and few-shot learning (Oreshkin et al., 2018), we propose a feature conditioning mechanism similar to parameter superposition (Cheung et al., 2019). The idea is to train one shared policy, but condition its parameters based on the input features to mitigate the potentially undesirable interference among different input graphs. Since dense layers (affine transforms followed by nonlinearity) serve as the fundamental building blocks in all of our network components, we introduce an additional conditioning layer to enable superposition in all dense layers the placement network:
+
+$$
+x ^ { ( l + 1 ) } = g ^ { ( l ) } ( c ( x ^ { ( 0 ) } ) \odot x ^ { ( l ) } )
+$$
+
+where $g ^ { ( l ) }$ stands for a dense layer in our policy network, $c$ stands for the feature conditioning layer, and $x ^ { ( 0 ) }$ denotes the feature representation of the input graph generated by the graph-embedding network. The feature conditioning layer is implemented with minimum overhead by adding an additional transformer layer to our placement network.
+
+# 4 EXPERIMENT
+
+# 4.1 EXPERIMENT SETUP
+
+In this section, we evaluate our training strategy on widely used machine learning models in computer vision, natural language processing, and speech domains. We compare our approach to human expert placement, TensorFlow METIS placement, and hierarchical device placement (HDP) (Mirhoseini et al., 2018). Our experiments are run on machines with one Intel Broadwell CPU and up to eight Nvidia P100 GPUs. Note that the prior work (Mirhoseini et al., 2017; 2018; Gao et al., 2018) were evaluated on different GPU devices, preventing direct comparison of results. Therefore, we re-evaluate HDP on our own system environment and report those numbers.
+
+The performance of a placement is evaluated by the resulted training step time (run time) of the neural network. We use the negative square root of the normalized run time as the reward, where the run time is normalized with the best run time from a baseline. We use the average reward of all the previous trials as a bias term. The advantage value is computed by subtracting the reward by the average reward. During the search, we apply a large negative reward (-10) for invalid placements (e.g. a violation of co-location constraint, out of memory, etc.). For operation scheduling, we rely on the Tensorflow default FIFO scheduling.
+
+# 4.2 PERFORMANCE ON INDIVIDUAL GRAPHS
+
+We evaluate GDP by training the model separately on six important graphs, including RNN Language Modeling, GNMT (Sutskever et al., 2014), Transformer-XL, Inception, AmoebaNet, and
+
+Table 1: Run time comparison between GDP-one, human expert, Tensorflow METIS, and hierarchical device placement (HDP) on six graphs (RNNLM, GNMT, Transformer-XL, Inception, AmoebaNet, and WaveNet). Graph runtime speed up is compared with Human Placement (HP) and Hierarchical Device Placement (HDP). Search speed up is the policy network training time speed up compared to HDP (reported values are averages of six runs).   
+
+<table><tr><td>Model (#devices)</td><td>GDP-one (s)</td><td>HP (s）</td><td>METIS (s)</td><td>HDP (s）</td><td>Run time speed up over HP/HDP</td><td>Search speed up</td></tr><tr><td>2-layer RNNLM (2)</td><td>0.234</td><td>0.257</td><td>0.355</td><td>0.243</td><td>9.8% /4%</td><td>2.95x</td></tr><tr><td>4-layer RNNLM (4) 2-layer GNMT (2)</td><td>0.409 0.301</td><td>0.48 0.384</td><td>OOM OOM</td><td>0.490 0.376</td><td>17.4% /19.8% 27.6% /24.9%</td><td>1.76x 30x</td></tr><tr><td>4-layer GNMT (4)</td><td>0.409</td><td>0.469</td><td>0OM</td><td>0.520</td><td>14.7% / 27.1%</td><td>58.8x</td></tr><tr><td>8-layer GNMT (8)</td><td>0.649</td><td>0.610</td><td>OOM</td><td>0.693</td><td>-6% /6.8%</td><td>7.35x</td></tr><tr><td>2-layer Transformer-XL (2)</td><td>0.386</td><td>0.473</td><td>0OM</td><td>0.435</td><td>22.5% / 12.7%</td><td>40x</td></tr><tr><td>4-layer Transformer-XL (4)</td><td>0.580</td><td>0.641</td><td>0OM</td><td>0.621</td><td>11.4% / 7.1%</td><td>26.7x</td></tr><tr><td>8-layer</td><td>0.748</td><td>0.813</td><td>0OM</td><td>0.789</td><td>8.9% /5.5%</td><td>16.7x</td></tr><tr><td>Transformer-XL (8) Inception (2)</td><td>0.405</td><td>0.418</td><td>0.423</td><td>0.417</td><td>3.2% /3%</td><td>13.5x</td></tr><tr><td>AmoebaNet (4)</td><td>0.394</td><td>0.44</td><td>0.426</td><td>0.418</td><td>26.1% /6.1%</td><td>58.8x</td></tr><tr><td>2-stack 18-layer WaveNet (2)</td><td>0.317</td><td>0.376</td><td>00M</td><td>0.354</td><td>18.6% /11.7%</td><td>6.67x</td></tr><tr><td>4-stack 36-layer</td><td>0.659</td><td>0.988</td><td>0OM</td><td>0.721</td><td></td><td></td></tr><tr><td>WaveNet (4) GEOMEAN</td><td></td><td></td><td></td><td></td><td>50% /9.4%</td><td>20x</td></tr><tr><td></td><td>1</td><td>-</td><td>1</td><td>1</td><td>16% /9.2%</td><td>15x</td></tr></table>
+
+WaveNet. We name this approach GDP-one. For all the tasks, GDP-one consistently outperforms human expert placement, TensorFlow METIS (Karypis & Kumar, 1998) placement, and HDP. For extremely large graphs, GDP-one is only $6 \%$ worse on 8-layer NMT (over $6 0 \mathrm { k }$ nodes), compared to human placement, but is $6 . 8 \%$ better than HDP. Overall, GDP-one achieves on average more than $16 \%$ run time reduction across the evaluated 12 graphs, compared to human expert placement. Compared to hierarchical device placement, GDP-one achieves an average $9 . 2 \%$ speed up, and scales better to large graphs such as 8-layer NMT and 4-layer RNNLM. Importantly, with the efficient end-to-end training and sample efficient reinforcement learning algorithm, GDP-one has a $1 5 \mathrm { x }$ speed up in convergence time of the placement network over HDP.
+
+# 4.3 GENERALIZATION
+
+GDP enables the training of multiple heterogeneous graphs in a single batch, sharing parameters in the graph-embedding network and the placement network. We name this training strategy GDPbatch. We empirically show that GDP-batch generates better placements for many workloads such as transformer-XL $( 7 . 6 \% )$ , WaveNet $( 1 5 \% )$ , and 8-layer GNMT $( 8 \% )$ . Table 2 compares the run time of 11 tasks using GDP-batch, with the same end-to-end architecture as described in section 4.2. GDP-batch yields slightly better run time compared to GDP-one in majority of the tasks, while being only slightly worse on AmoebaNet. Compared to training graphs separately, GDP-batch reduces network parameters and enables transfer learning among different graphs.
+
+We further evaluate the effect of transfer learning by mixing redundant tasks in a batch. We find that mixing different graphs such as RNNLM and GNMT models with different number of layers results in both faster and better learning for RNNLM and GNMT with large number of layers (8- layer). As a matter of fact, both Placeto (Addanki et al., 2019) and HDP had problems matching human placement performance for 8-layer GNMT or 8-layer RNNLM. With batch training, GDP is the first device placement work to match human expert performance for both 8-layer GNMT and 8-layer RNNLM. We also for the first time show that GDP-batch not only improves the search time (since we do not retrain the policy for every new graph), it can also improve the performance of the found placements. More detailed results are shown in Appendix Table 5.
+
+Table 2: Run time comparison on GDP-batch vs. GDP-one.   
+
+<table><tr><td>Model</td><td>Speed up</td><td>Model</td><td>Speed up</td></tr><tr><td>2-layer RNNLM</td><td>0</td><td>Inception</td><td>0</td></tr><tr><td>4-layer RNNLM</td><td>5%</td><td>AmoebaNet</td><td>-5%</td></tr><tr><td>2-layer GNMT</td><td>0</td><td>4-stack 36-layer WaveNet</td><td>3.3 %</td></tr><tr><td>4-layer GNMT</td><td>0</td><td>2-stack18-layer WaveNet</td><td>15%</td></tr><tr><td>2-layer Transformer-XL 4-layer Transformer-XL</td><td>7.6% 3%</td><td>8-layer Transformer-XL</td><td>1.5%</td></tr></table>
+
+Generalization to hold-out graphs: Here we show another set of experiments where we treat GDPbatch as a pre-training strategy and remove the target graph from the batch training dataset. We then fine-tune the pre-trained model on the hold-out graphs for fewer than 50 steps, which takes less than one minute. We name this GDP-generalization+finetune. Figure 2 shows that GDP fine-tuning for hold-out graphs outperforms human expert placement and HDP consistently on all six batch training datasets, and performs only slightly worse than GDP-one. 2-layer RNNLM and 2-stack WaveNet almost match the performance of GDP-one. We also run inference (generate placement) directly on the pre-trained model for the target hold-out graphs, and name this GDP-generalization-zeroshot. We find that GDP-generalization-zeroshot only marginally hurts performance as compared to GDPgeneralization+finetune, while being slightly better than human placement and HDP. This indicates that both graph embedding and the learned policies transfer and generalize to the unseen data.
+
+![](images/5c9f10f415f26c1d76aeae0118c9dc7049201690e0d379df3c50b56bbbbba3f9.jpg)  
+Figure 2: Finetuning on hold-out graphs.
+
+Comparisons with other generalized placement approaches: Placeto (Addanki et al., 2019), to our knowledge, is the only other method beside GDP that shows true (and non-simulated) generalized device placement results. Direct comparison is not possible since Placeto uses a different hardware platform and different input graphs (Inception-V3, NMT, and NASNet). Placeto’s search time is on average $2 . 6 5 \mathrm { x }$ faster than HDP, while GDP is on average $1 5 \mathrm { x }$ faster than HDP on our larger set of graphs. Apart from search time speed up, Placeto on average reduces placed graph run time by $3 \%$ (for its different graphs and hardware) while GDP on average reduces placed graph run time by $9 . 2 \%$ , compared to HDP. One advantage of GDP over Placeto is that it does not rely on any initial feasible placement. Providing a reasonable initial placement is often non-trivial for domain experts, especially for larger graphs such as 8-layer GNMT. As such, we are the first to report superhuman results on 8-layer GNMT (with GDP-batch).
+
+# 4.4 ABLATION STUDIES
+
+Attention and Superposition. We did an ablation study on the attention and the superposition layer in the transformer-XL placer network. We find that attention improves placement run time by an average of $18 \%$ compared to a placer network with no attention, and superposition improves placement run time by an average of $6 . 5 \%$ where all the graphs are trained in a single batch as described in Section 4.3. Without superposition network, batch training fails for AmoebaNet and Inception when mixing with larger RNNLM or GNMT models (4-layer).
+
+Pre-training graph embeddings. We also evaluate a fine-tuning strategy by pre-training the graph embedding and placement network and fine-tuning the network on the down stream tasks. The difference here compared to Section 4.3 is that we also include the target graphs in the pre-training dataset. When GDP-batch is used as a pre-training strategy, the graph embedding and placement network assimilate meaningful graph representations and placement policies from a wide set of graphs, thus can be used as a strong baseline network for fine-tuning on downstream tasks. We compare the generated placement run time and the placement search time, normalized to GDP-one. We find that fine-tuning further reduces the the placed graph run time by an average of $5 \%$ and placement search time by an average of $86 \%$ , compared to GDP-one.
+
+![](images/952a5f0142edf3158aea8b7c617b28fb0db5a0c93b43763249bcfb0b239a35b0.jpg)  
+Figure 3: Ablation Study on Attention and Superposition of the Placement Network.
+
+![](images/8f18a2f28c32a61bb76c1776235f135ec173371660b93fddcedc426ab84fd863.jpg)  
+Figure 4: Normalized run time (step time for the generated placement) and normalized training time (search time) for fine-tuning. Time is normalized to GDP-one.
+
+# 5 CONCLUSION
+
+In this paper, we present a generalized device placement strategy that uses a graph neural network and super-positioning to generalize to arbitrary and held out graphs. Through experimental evaluation over a wide set of representative graphs from different domains including computer vision, speech, and NLP, we demonstrated over 15 times faster convergence while achieving a $16 \%$ and $9 . 2 \%$ reductions in step time over human expert placement and HDP, respectively.
+
+# ACKNOWLEDGMENTS
+
+# 6 APPENDIX
+
+# 6.1 PROXIMAL POLICY OPTIMIZATION
+
+In device placement, the objective is to minimize the training step time of a given computational graph or a batch of dataflow graphs for a target system configuration (e.g. a 8-GPU cluster or a TPU pod), by placing operations onto different devices to enable model-level parallelism. This process corresponds to maximizing the expected performance in the MDP. For better sample efficiency, we adopted a Proximal Policy Optimization (PPO) (Schulman et al., 2017) algorithm. The objective is to maximize a surrogate objective:
+
+$$
+L _ { \pi } = E _ { a _ { [ 0 : n ] } \sim \pi } [ \frac { q \prime \left( a _ { n } | s _ { n } \right) } { q ( a _ { n } | s _ { n } ) } A _ { \pi } ( s _ { n } , a _ { n } ) ]
+$$
+
+$$
+L _ { \pi } = \operatorname* { m a x } _ { \pi ^ { \prime } } \frac { 1 } { N } \sum _ { n = 0 , a _ { n } \sim \pi } ^ { N - 1 } [ \operatorname* { m i n } ( \frac { q \prime ( a _ { n } | s _ { n } ) } { q ( a _ { n } | s _ { n } ) } ( R - \overline { { R } } ) , c l i p ( \frac { q \prime ( a _ { n } | s _ { n } ) } { q ( a _ { n } | s _ { n } ) } , 1 - \epsilon , 1 + \epsilon ) ( R - \overline { { R } } ) ) ]
+$$
+
+Within a loop, GDP PPO continuously samples placements from the distribution and evaluates their training times in real systems. For a rollout of $K$ , we perform a minimatch of $m$ stochastic gradient ascent steps with respective to the objective of proximal policy optimization, which makes incremental policy improvements. The rollout steps $K$ and minibatch size $m$ are hyper parameters for PPO. We find a set of optimized hyper parameters and keep them fixed for all the experiments presented. As the rewards are generated on-the-fly based on real system measurements, we no longer need to re-evaluate the placement solutions in a separate phase.
+
+# 6.2 HYPERPARAMETERS
+
+In this section, we list out all the selected hyperparameters in our experiments for reproducibility in Table 3 and Table 4.
+
+Table 3: Hyperparameters for Policy Network. gs layers: GraphSAGE layers, gs knn: GraphSAGE maximum neighbors, trf d model: Dimension of the TransformerXL model, trf n head: Number of attention heads, tr $f$ layers: Number of TransformerXL layers, trf d heads: Dimension of each attention head, tr $f$ d inner: Dimension of inner hidden size in positionwise feedforward.   
+
+<table><tr><td>Parameters</td><td>Value</td><td>Parameters</td><td>Value</td></tr><tr><td>gs_layers</td><td>4</td><td>gs_dim</td><td>128</td></tr><tr><td>gs_knn</td><td>5</td><td>trf_layers</td><td>2</td></tr><tr><td>trf_d_model</td><td>128</td><td>trf_n_head</td><td>5</td></tr><tr><td>trf_d_head</td><td>25</td><td>trf_d_inner</td><td>256</td></tr></table>
+
+Table 4: Hyperparameters for PPO.   
+
+<table><tr><td>Parameters</td><td>Value</td><td>Parameters</td><td>Value</td></tr><tr><td>learing rate</td><td>0.5</td><td>num of rollouts</td><td>400</td></tr><tr><td>minibatches</td><td>40</td><td>epochs</td><td>5</td></tr><tr><td>epsilon</td><td>0.2</td><td>entropy</td><td>0.05</td></tr></table>
+
+# 6.3 INPUT GRAPHS
+
+We used a variety of widely used workloads from computer vision, speech, and NLP. In this section, we give a detailed explanation on the selected models and hyperparameters.
+
+# 6.3.1 INCEPTION-V3
+
+Inception-V3 (Szegedy et al., 2015) is a multi-branch convolutional network used for a variery of computer vision tasks, including classification, recognition, or generation. The network consists of blocks made of multiple branches of concolutional and pooling operations. Within a block, the branches of ops can be executed in parallel. However, the model is mostly sequential as the outputs of each block are concatenated together to form the input to the next block. We use a batch size of 64. The Tensorflow graph of this model contains 24,713 operations.
+
+# 6.3.2 AMOEBANET
+
+AmoebaNet (Real et al., 2018) is an automatically designed neural network that yields SoTA performance on ImageNet. Similar to Inception-V3, it contains Inception-like blocks called cells, which receives a direct input from the previous cell and a skip input from the cell before it. The network is made of redundant cells stacked together, therefore is more modular than Inception-V3. We use a batch size of 64. The Tensorflow graphs contains 9,430 operations.
+
+# 6.3.3 RNNLM
+
+Recurrent Neural Network Language Model (Zaremba et al., 2014; Jozefowicz et al., 2016) is made of many LSTM cells organized in a grid structure. The processing of each LSTM cell only depends on the results of 2 other cells (from the previous layer, and from the previous time step), which make the concurrent execution of many LSTM cells possible given enough hardware resources. We use batch size 64 and a hidden size of 2048. The corresponding TensorFlow graph contains 9,021 operations for a 2-layer model. The number of ops grow roughly proportional with the number of layers.
+
+# 6.3.4 GNMT
+
+Neural Machine Translation with attention mechanism (Bahdanau et al., 2015; Wu et al., 2016) has an architecture similar to that of RNNLM, but its many hidden states make it far more computationally expensive than RNNLM. To reduce the training time, prior work (Wu et al., 2016) propose placing each LSTM layer, as well as the attention and the softmax layer, on a separate device. This strategy demonstrates early success in human placement, we show that GDP can find significantly better placements. We use batch size 64. The original 2-layer encoder-decoder consisting of 28,044 operations. An extended 4-layer version consisting of 46,600 operations, An even larger 8-layer version consisting of 83,712 operations.
+
+# 6.3.5 TRANSFORMER-XL
+
+Transformer-XL (Dai et al., 2019) is an modified version of Transformer (Vaswani et al., 2017) that supports segement-level recurrence and a novel positional encoding scheme. This innovation enables learning dependency that is $80 \%$ longer than RNNs, and $450 \%$ longer than vanilla Transformers. We use a transformer-XL with batch size of 64, sequence length of 256, segment length of 64, model hidden dimension of 500 and feed forward hidden dimension of 1000, 10 heads, and head dimension of 50. The 2-layer Transformer-XL contains 2,618 operations. The number of ops grow roughly proportional with the number of layers.
+
+# 6.3.6 WAVENET
+
+WaveNet (van den Oord et al., 2016) is a generative model for speech synthesis. The model is fully probabilistic and autoregressive, with the predictive ditribution for each audio sample conditioned on all previous ones. Architecturally, WaveNet uses causal convolutions with dilations, to obtain a large receptive field. We use a WaveNet model with batch size 64 and a receptive field size of 2048 (9-layers per stack). An 5-stack WaveNet contains 4,374 operations and a 10-stack WaveNet contains 8,516 operations.
+
+Table 5: Run time comparison on GDP batch training vs. the best of related methods (human expert, METIS, HDP, and GDP no batch training).
+
+<table><tr><td>Batch Setting</td><td>Model</td><td>speed up (s)</td></tr><tr><td rowspan="6">Batch 2</td><td>Inception</td><td>0</td></tr><tr><td>AmoebaNet</td><td>-4.5%</td></tr><tr><td>2-layer RNNLM</td><td>0</td></tr><tr><td>2-layer GNMT</td><td>0</td></tr><tr><td>2-layer Transformer-XL</td><td>6.5%</td></tr><tr><td>2-stack 18-layer Wavenet</td><td>4%</td></tr><tr><td rowspan="6">Batch 3</td><td>2-layer RNNLM</td><td>0</td></tr><tr><td>4-layer RNNLM</td><td>0</td></tr><tr><td>8-layer RNNLM</td><td>4.5%</td></tr><tr><td>2-layer GNMT</td><td>0</td></tr><tr><td>4-layer GNMT</td><td></td></tr><tr><td>8-layer GNMT</td><td>0 8%</td></tr><tr><td>Batch 4</td><td>3x8-layer GNMT</td><td>5.1%</td></tr><tr><td>Batch 5</td><td>3x8-layer RNNLM</td><td>4.5%</td></tr></table>
+
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parse/train/SkxW23NtPH/SkxW23NtPH_content_list.json

@@ -0,0 +1,1456 @@
+[
+    {
+        "type": "text",
+        "text": "GDP: GENERALIZED DEVICE PLACEMENT FOR DATAFLOW GRAPHS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            236,
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+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Runtime and scalability of large neural networks can be significantly affected by the placement of operations in their dataflow graphs on suitable devices. With increasingly complex neural network architectures and heterogeneous device characteristics, finding a reasonable placement is extremely challenging even for domain experts. Most existing automated device placement approaches are impractical due to the significant amount of compute required and their inability to generalize to new, previously held-out graphs. To address both limitations, we propose an efficient end-to-end method based on a scalable sequential attention mechanism over a graph neural network that is transferable to new graphs. On a diverse set of representative deep learning models, including Inception-v3, AmoebaNet, Transformer-XL, and WaveNet, our method on average achieves $16 \\%$ improvement over human experts and $9 . 2 \\%$ improvement over the prior art with $1 5 \\times$ faster convergence. To further reduce the computation cost, we pre-train the policy network on a set of dataflow graphs and use a superposition network to fine-tune it on each individual graph, achieving state-of-the-art performance on large hold-out graphs with over 50k nodes, such as an 8-layer GNMT. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            516,
+            336,
+            532
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural networks have demonstrated remarkable scalability–improved performance can usually be achieved by training a larger model on a larger dataset (Hestness et al., 2017; Shazeer et al., 2017; Jozefowicz et al., 2016; Mahajan et al., 2018; Radford et al.). Training such large models efficiently while meeting device constraints, like memory limitations, necessitate partitioning of the underlying dataflow graphs for the models across multiple devices. However, devising a good partitioning and placement of the dataflow graphs requires deep understanding of the model architecture, optimizations performed by domain-specific compilers, as well as the device characteristics, and is therefore extremely hard even for experts. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ML practitioners often rely on their understanding of model architecture to determine a reasonable partitioning and placement for graphs. However, relying solely on the model architecture while ignoring the effect of the partitioning on subsequent compiler optimizations like op-fusion can lead to sub-optimal placements and consequently under-utilization of available devices. The goal of automated device placement is to find the optimal assignment of operations to devices such that the end-to-end execution time for a single step is minimized and all device constraints like memory limitations are satisfied. Since this objective function is non-differentiable, prior approaches (Mirhoseini et al., 2017; 2018; Gao et al., 2018) have explored solutions based on reinforcement learning (RL). However, these RL policies are usually not transferable and require training a new policy from scratch for each individual graph. This makes such approaches impractical due to the significant amount of compute required for the policy search itself, at times offsetting gains made by the reduced step time. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose an end-to-end deep RL method for device placement where the learned policy is generalizable to new graphs. Specifically, the policy network consists of a graph-embedding network that encodes operation features and dependencies into a trainable graph representation, followed by a scalable sequence-to-sequence placement network based on an improved Transformer (Vaswani et al., 2017; Dai et al., 2019). The placement network transforms the graph representations into a placement decision with soft attention, removing hard constraints such as hierarchical grouping of operations (Mirhoseini et al., 2018) or co-location heuristics (to reduce the placement complexity) (Mirhoseini et al., 2017). Both of our graph-embedding network and placement network can be jointly trained in an end-to-end fashion using a supervised reward, without the need to manipulate the loss functions at multiple levels. We empirically show that the network learns flexible placement policies at a per-node granularity and can scale to problems over 50,000 nodes. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To generalize to arbitrary and held-out graphs, our policy is trained jointly over a set of dataflow graphs (instead of one at a time) and then fine-tuned on each graph individually. By transferring the learned graph embeddings and placement policies, we are able to achieve faster convergence and thus use less resources to obtain high-quality placements. We also use super-positioning, i.e., a feature conditioning mechanism based on the input graph embeddings, to effectively orchestrate the optimization dynamics of graphs with drastically different sizes in the same batch. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions can be summarized as follows: ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. An end-to-end device placement network that can generalize to arbitrary and held-out graphs. This is enabled by jointly learning a transferable graph neural network along with the placement network.   \n2. A scalable placement network with an efficient recurrent attention mechanism, which eliminates the need for an explicit grouping stage before placement. The proposed end-to-end network provides $1 5 \\times$ faster convergence as compared to the hierarchical LSTM model used in earlier works (Mirhoseini et al., 2017; 2018).   \n3. A new batch pre-training and fine-tuning strategy based on network superposition, which leads to improved transferability, better placements especially for larger graphs, and $1 0 \\times$ reduction in policy search time as compared to training individual graphs from scratch.   \n4. Superior performance over a wide set of workloads, including InceptionV3 (Szegedy et al., 2015), AmoebaNet (Real et al., 2018), RNNs, GNMT (Wu et al., 2016), Transformer-XL (Dai et al., 2019), WaveNet (van den Oord et al., 2016), and more. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            515,
+            344,
+            531
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Device Placement Reinforcement learning has been used for device placement of a given dataflow graph (Mirhoseini et al., 2017) and demonstrated run time reduction over human crafted placement and conventional heuristics. For improved scalability, a hierarchical device placement strategy (HDP) (Mirhoseini et al., 2018) has been proposed that clusters operations into groups before placing the operation groups onto devices. Spotlight (Gao et al., 2018) applies proximal policy optimization and cross-entropy minimization to lower training overhead. Both HDP and Spotlight rely on LSTM controllers that are difficult to train and struggle to capture very long-term dependencies over large graphs. In addition, both methods are restricted to process only a single graph at a time, and cannot generalize to arbitrary and held-out graphs. Placeto (Addanki et al., 2019) represents the first attempt to generalize device placement using a graph embedding network. But like HDP, Placeto also relies on hierarchical grouping and only generates placement for one node at each time step. Our approach (GDP) leverages a recurrent attention mechanism and generates the whole graph placement at once. This significantly reduces the training time for the controller. We also demonstrate the generalization ability of GDP over a wider set of important workloads. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Parallelization Strategy Mesh-TensorFlow is a language that provides a general class of distributed tensor computations. While data-parallelism can be viewed as splitting tensors and operations along the “batch” dimension, in Mesh-TensorFlow the user can specify any tensor-dimensions to be split across any dimensions of a multi-dimensional mesh of processors. FlexFlow (Jia et al., 2018) introduces SOAP, a more comprehensive search space of parallelization strategies for DNNs which allows parallelization of a DNN in the Sample, Operator, Attribute, and Parameter dimensions. It uses guided randomized search of the SOAP space to find a parallelization strategy for a specific parallel machine. GPipe (Huang et al., 2018) proposed pipeline parallelism, by partitioning a model across different accelerators and automatically splitting a mini-batch of training examples into smaller micro-batches. By pipelining the execution across micro-batches, accelerators can operate in parallel. Our GDP focuses on a general deep RL method for automating device placement on arbitrary graphs, and is therefore orthogonal to existing parallelization strategies. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/6d7904fbfb076cc23f86cea14773c4e42f598efb830a695fdc7196d067a40a91.jpg",
+        "image_caption": [
+            "Figure 1: Overview of GDP: An end-to-end placement network that combines graph embedding and sequential attention. $N$ : Number of Nodes, h: Hidden Size, $d$ : Number of Devices. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            243,
+            103,
+            751,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Compiler Optimization REGAL (Paliwal et al., 2019) uses deep RL to optimize the execution cost of computation graphs in a static compiler. The method leverages the policy’s ability to transfer to new graphs to improve the quality of the genetic algorithm for the same objective budget. However, REGAL only targets peak memory minimization while GDP focuses on graph run time and scalability while also meeting the peak memory constraints of the devices. Specifically, we generalize graph partitioning and placement into a single end-to-end problem, with and without simulation, which can handle graphs with over 50,000 nodes. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 END-TO-END PLACEMENT POLICY ",
+        "text_level": 1,
+        "bbox": [
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+            410,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a dataflow graph $G ( V , E )$ where $V$ represents atomic computational operations (ops) and $E$ represents the data dependency, our goal is to learn a policy $\\pi : \\mathcal { G } \\mapsto \\mathcal { D }$ that assigns a placement $D \\in { \\mathcal { D } }$ for all the ops in the given graph $G \\in { \\mathcal { G } }$ , to maximize the reward $r _ { G , D }$ defined based on the run time. $\\mathcal { D }$ is the allocated devices that can be a mixture of CPUs and GPUs. In this work, we represent policy $\\pi _ { \\theta }$ as a neural network parameterized by $\\theta$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Unlike prior works that focus on a single graph only, the RL objective in GDP is defined to simultaneously reduce the expected runtime of the placements over a set of $N$ dataflow graphs: ",
+        "bbox": [
+            173,
+            518,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/24f2e7aa7a18a8551dc7114da80e187195a7b898486f5f536691b5770438d1f1.jpg",
+        "text": "$$\nJ ( \\theta ) = \\mathbb { E } _ { G \\sim \\mathcal { G } , D \\sim \\pi _ { \\theta } ( G ) } [ r _ { G , D } ] \\approx \\frac { 1 } { N } \\sum _ { G } \\mathbb { E } _ { D \\sim \\pi _ { \\theta } ( G ) } [ r _ { G , D } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            571,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the following, we refer to the case when $N = 1$ as individual training and the case when $N > 1$ as batch training. We optimize the objective above using Proximal Policy Optimization (PPO) (Schulman et al., 2017) for improved sample efficiency. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Figure 1 shows an overview of the proposed end-to-end device placement network. Our proposed policy network $\\pi _ { \\theta }$ consists a graph embedding network that learns the graphical representation of any dataflow graph, and a placement network that learns a placement strategy over the given graph embeddings. The two components are jointly trained in an end-to-end fashion. The policy $p ( a | G )$ is applied to make a set of decisions at each node. These decisions, denoted as $a _ { v }$ for each $v \\in V$ across all nodes, form one action $a = \\{ a _ { v \\in V } \\}$ . One decision corresponds to playing one arm of a multi-bandit problem, and specifying the entire $a$ corresponds to playing several arms together in a single shot. Note the architecture is designed to be invariant over the underlying graph topology, enabling us to apply the same learned policy to a wide set of input graphs with different structures. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 GRAPH EMBEDDING NETWORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            814,
+            433,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We leverage graph neural networks (GNNs) (Hamilton et al., 2017; Xu et al., 2019; You et al., 2018) to capture the topological information encoded in the dataflow graph. Most graph embedding frameworks are inherently transductive and can only generate embeddings for a given fixed graph. These transductive methods do not efficiently extrapolate to handle unseen nodes (e.g., in evolving graphs), and cannot learn to generalize to unseen graphs. GraphSAGE (Hamilton et al., 2017) is an inductive framework that leverages node attribute information to efficiently generate representations on previously unseen data. While our proposed framework is generic, we adopt the feature aggregation scheme proposed in GraphSAGE to model the dependencies between the operations and build a general, end-to-end device placement method for a wide set of dataflow graphs. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            103,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In GDP, nodes and edges in the dataflow graph are represented as the concatenation of their meta features (e.g., operation type, output shape, adjacent node ids) and are further encoded by the graph embedding network into a trainable representation. The graph embedding process consists of multiple iterations, and the computation procedure for the $l$ -th iteration can be outlined as follows: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "First, each node $v \\in V$ aggregates the feature representations of its neighbors, $\\{ h _ { u } ^ { ( l ) } , \\forall u \\in \\mathcal { N } ( v ) \\}$ , into a single vector h(l)N (v). This aggregation outcome is a function of all previously generated representations, including the initial representations defined based on the input node features. In this work, we use the following aggregation function with max pooling: ",
+        "bbox": [
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+            218,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7a232af45b6c54895f2b75d1f34ff831e8a075f5e31ceb7e342756eaf84cd079.jpg",
+        "text": "$$\nh _ { \\mathcal { N } ( v ) } ^ { ( l ) } = \\operatorname* { m a x } ( \\sigma ( W ^ { ( l ) } h _ { u } ^ { ( l ) } + b ^ { ( l ) } ) , \\forall u \\in \\mathcal { N } ( v ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            287,
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+            311
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $( W ^ { ( l ) } , b ^ { ( l ) } )$ define an affine transform and $\\sigma$ stands for the sigmoid activation function. We then concatenate the node’s current representation, $h _ { v } ^ { ( l ) }$ , with the aggregated neighborhood vector, $h _ { \\mathcal { N } ( v ) } ^ { ( l ) }$ , and feed this concatenated vector through a fully connected layer $f ^ { ( l + 1 ) }$ ei ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d14654adc7f4f5e25f119c8715de9928565ed52f91da1b6eebec143e4d868fb.jpg",
+        "text": "$$\nh _ { v } ^ { ( l + 1 ) } = f ^ { ( l + 1 ) } ( \\mathrm { c o n c a t } ( h _ { v } ^ { ( l ) } , h _ { \\mathcal { N } ( v ) } ^ { ( l ) } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Different from GraphSAGE, parameters in our graph embedding network are trained jointly with a placement network via stochastic gradient descent with PPO, in a supervised fashion, as described in Section 3. That is, we replace the unsupervised loss with our task-specific objective. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 PLACEMENT NETWORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            468,
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+            482
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The graph neural network works as a feature aggregation network that learns a trainable feature representation for the computational graph, we still need a policy network that produces actions on a per node basis. Given $h _ { v }$ ’s, the policy network produces $a _ { v }$ ’s through conditionally independent predictions, where the prediction for one node $v$ does not depend on the prediction of other nodes. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/df74f083e0eb3696a66e4fac6e905bbc58db377d1d84bfd86ed56dcdea90ff19.jpg",
+        "text": "$$\np ( a | G ) = \\prod _ { v } p ( a _ { v } | G ) = \\prod _ { v } p ( a _ { v } | f ( h _ { v } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            556,
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+            589
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "While $f$ can be represented using multilayer perceptrons (MLPs), where the MLPs is shared across all nodes for prediction the placement output distributions. However, MPLs lack a dependency tracking mechanism across nodes. In practise, the placement of one node can be determined by the placement of another node, where the placed node may consume a large size of data produced by the other node. Intuitively, an attention network can learn this dependency and the relative importance of dependencies across an entire graph. Therefore, we decide to use an attention-based placement network to better track inter-node placement-related dependencies. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Designing a scalable placement network that can generalize to graphs with thousands of nodes is challenging, as the conventional GNMT models proposed for language tasks usually target a shorter sequence length. Hierarchical placement (Mirhoseini et al., 2018) has been proposed to address this issue,however, the proposed grouper network comes with limited flexibility and generality. For example, the grouper network leverages an aggregated feature representation by averaging feature vectors for nodes within the same group. The non-differentiable grouping procedure prevents training the graph-embedding and placement networks end-to-end. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To remove the two-stage hierarchical workflow in HDP for improved scalability, we propose to use a Transformer-based attentive network to generate operation placements in an end-to-end fashion. As the graph embedding already contains spatial (topological) information for each node, we remove the positional embedding in the original transformer to prevent the model from overfitting node identifications. To capture long-term dependencies efficiently among a large set of nodes, we adopt segment-level recurrence introduced in Transformer-XL (Dai et al., 2019; Dai, 2019), where hidden states computed for the previous set of nodes are cached (with gradient flows disabled) and reused as an extended context during the training of the next segment. Besides achieving extra long context, we empirically find the segment-level recurrent attention much faster than a conventional LSTMbased GNMT model. In our experimental evaluation, we compare both the performance and speed up of our placement network with that of the LSTM-based hierarchical device placement. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 BATCH TRAINING WITH PARAMETER SUPERPOSITION ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Since the parameterization for the architecture of the end-to-end policy is designed to be invariant over input graphs with different topologies, the same placement policy can be shared across a wide set of workloads. We therefore propose a batch training strategy, and further enhance the aforementioned architecture to handle such generalization across graphs. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Na¨ıve batch training is challenging in our context as different dataflow graphs contain different number of operations connected in different topologies. In addition, unlike previous device placement methods, GDP aims to handle graphs from potentially different application domains (e.g. computer vision, language, and speech), where the number of operations can range from a few thousand to one million. These graphs have drastically different network architecture, in terms of computational operations, data shape, and network topology. As an example, recurrent networks have completely different operation types and connections compared to multi-branch convolutional networks that are widely used in computer vision. It would be highly desirable to train a single shared network that maximizes information sharing across these heterogeneous tasks, without hurting the performance on each of them due to their distinct learning dynamics. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Along a similar direction of multi-task learning and few-shot learning (Oreshkin et al., 2018), we propose a feature conditioning mechanism similar to parameter superposition (Cheung et al., 2019). The idea is to train one shared policy, but condition its parameters based on the input features to mitigate the potentially undesirable interference among different input graphs. Since dense layers (affine transforms followed by nonlinearity) serve as the fundamental building blocks in all of our network components, we introduce an additional conditioning layer to enable superposition in all dense layers the placement network: ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d4f1a8290bcad052df0df757184bdf67a499edb9aac6a5596828de49cbcafa0.jpg",
+        "text": "$$\nx ^ { ( l + 1 ) } = g ^ { ( l ) } ( c ( x ^ { ( 0 ) } ) \\odot x ^ { ( l ) } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $g ^ { ( l ) }$ stands for a dense layer in our policy network, $c$ stands for the feature conditioning layer, and $x ^ { ( 0 ) }$ denotes the feature representation of the input graph generated by the graph-embedding network. The feature conditioning layer is implemented with minimum overhead by adding an additional transformer layer to our placement network. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENT ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENT SETUP ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we evaluate our training strategy on widely used machine learning models in computer vision, natural language processing, and speech domains. We compare our approach to human expert placement, TensorFlow METIS placement, and hierarchical device placement (HDP) (Mirhoseini et al., 2018). Our experiments are run on machines with one Intel Broadwell CPU and up to eight Nvidia P100 GPUs. Note that the prior work (Mirhoseini et al., 2017; 2018; Gao et al., 2018) were evaluated on different GPU devices, preventing direct comparison of results. Therefore, we re-evaluate HDP on our own system environment and report those numbers. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The performance of a placement is evaluated by the resulted training step time (run time) of the neural network. We use the negative square root of the normalized run time as the reward, where the run time is normalized with the best run time from a baseline. We use the average reward of all the previous trials as a bias term. The advantage value is computed by subtracting the reward by the average reward. During the search, we apply a large negative reward (-10) for invalid placements (e.g. a violation of co-location constraint, out of memory, etc.). For operation scheduling, we rely on the Tensorflow default FIFO scheduling. ",
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+    {
+        "type": "text",
+        "text": "4.2 PERFORMANCE ON INDIVIDUAL GRAPHS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate GDP by training the model separately on six important graphs, including RNN Language Modeling, GNMT (Sutskever et al., 2014), Transformer-XL, Inception, AmoebaNet, and ",
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+    {
+        "type": "table",
+        "img_path": "images/e15dd1a4a0c11bd8962825223f1c3a0bff60221cdbefda009d8fe4e585bf390c.jpg",
+        "table_caption": [
+            "Table 1: Run time comparison between GDP-one, human expert, Tensorflow METIS, and hierarchical device placement (HDP) on six graphs (RNNLM, GNMT, Transformer-XL, Inception, AmoebaNet, and WaveNet). Graph runtime speed up is compared with Human Placement (HP) and Hierarchical Device Placement (HDP). Search speed up is the policy network training time speed up compared to HDP (reported values are averages of six runs). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model (#devices)</td><td>GDP-one (s)</td><td>HP (s）</td><td>METIS (s)</td><td>HDP (s）</td><td>Run time speed up over HP/HDP</td><td>Search speed up</td></tr><tr><td>2-layer RNNLM (2)</td><td>0.234</td><td>0.257</td><td>0.355</td><td>0.243</td><td>9.8% /4%</td><td>2.95x</td></tr><tr><td>4-layer RNNLM (4) 2-layer GNMT (2)</td><td>0.409 0.301</td><td>0.48 0.384</td><td>OOM OOM</td><td>0.490 0.376</td><td>17.4% /19.8% 27.6% /24.9%</td><td>1.76x 30x</td></tr><tr><td>4-layer GNMT (4)</td><td>0.409</td><td>0.469</td><td>0OM</td><td>0.520</td><td>14.7% / 27.1%</td><td>58.8x</td></tr><tr><td>8-layer GNMT (8)</td><td>0.649</td><td>0.610</td><td>OOM</td><td>0.693</td><td>-6% /6.8%</td><td>7.35x</td></tr><tr><td>2-layer Transformer-XL (2)</td><td>0.386</td><td>0.473</td><td>0OM</td><td>0.435</td><td>22.5% / 12.7%</td><td>40x</td></tr><tr><td>4-layer Transformer-XL (4)</td><td>0.580</td><td>0.641</td><td>0OM</td><td>0.621</td><td>11.4% / 7.1%</td><td>26.7x</td></tr><tr><td>8-layer</td><td>0.748</td><td>0.813</td><td>0OM</td><td>0.789</td><td>8.9% /5.5%</td><td>16.7x</td></tr><tr><td>Transformer-XL (8) Inception (2)</td><td>0.405</td><td>0.418</td><td>0.423</td><td>0.417</td><td>3.2% /3%</td><td>13.5x</td></tr><tr><td>AmoebaNet (4)</td><td>0.394</td><td>0.44</td><td>0.426</td><td>0.418</td><td>26.1% /6.1%</td><td>58.8x</td></tr><tr><td>2-stack 18-layer WaveNet (2)</td><td>0.317</td><td>0.376</td><td>00M</td><td>0.354</td><td>18.6% /11.7%</td><td>6.67x</td></tr><tr><td>4-stack 36-layer</td><td>0.659</td><td>0.988</td><td>0OM</td><td>0.721</td><td></td><td></td></tr><tr><td>WaveNet (4) GEOMEAN</td><td></td><td></td><td></td><td></td><td>50% /9.4%</td><td>20x</td></tr><tr><td></td><td>1</td><td>-</td><td>1</td><td>1</td><td>16% /9.2%</td><td>15x</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "WaveNet. We name this approach GDP-one. For all the tasks, GDP-one consistently outperforms human expert placement, TensorFlow METIS (Karypis & Kumar, 1998) placement, and HDP. For extremely large graphs, GDP-one is only $6 \\%$ worse on 8-layer NMT (over $6 0 \\mathrm { k }$ nodes), compared to human placement, but is $6 . 8 \\%$ better than HDP. Overall, GDP-one achieves on average more than $16 \\%$ run time reduction across the evaluated 12 graphs, compared to human expert placement. Compared to hierarchical device placement, GDP-one achieves an average $9 . 2 \\%$ speed up, and scales better to large graphs such as 8-layer NMT and 4-layer RNNLM. Importantly, with the efficient end-to-end training and sample efficient reinforcement learning algorithm, GDP-one has a $1 5 \\mathrm { x }$ speed up in convergence time of the placement network over HDP. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 GENERALIZATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "GDP enables the training of multiple heterogeneous graphs in a single batch, sharing parameters in the graph-embedding network and the placement network. We name this training strategy GDPbatch. We empirically show that GDP-batch generates better placements for many workloads such as transformer-XL $( 7 . 6 \\% )$ , WaveNet $( 1 5 \\% )$ , and 8-layer GNMT $( 8 \\% )$ . Table 2 compares the run time of 11 tasks using GDP-batch, with the same end-to-end architecture as described in section 4.2. GDP-batch yields slightly better run time compared to GDP-one in majority of the tasks, while being only slightly worse on AmoebaNet. Compared to training graphs separately, GDP-batch reduces network parameters and enables transfer learning among different graphs. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We further evaluate the effect of transfer learning by mixing redundant tasks in a batch. We find that mixing different graphs such as RNNLM and GNMT models with different number of layers results in both faster and better learning for RNNLM and GNMT with large number of layers (8- layer). As a matter of fact, both Placeto (Addanki et al., 2019) and HDP had problems matching human placement performance for 8-layer GNMT or 8-layer RNNLM. With batch training, GDP is the first device placement work to match human expert performance for both 8-layer GNMT and 8-layer RNNLM. We also for the first time show that GDP-batch not only improves the search time (since we do not retrain the policy for every new graph), it can also improve the performance of the found placements. More detailed results are shown in Appendix Table 5. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/c9b7c3a8cead46838f9c194b472ca318d16479612705155632185f229a3d319e.jpg",
+        "table_caption": [
+            "Table 2: Run time comparison on GDP-batch vs. GDP-one. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Speed up</td><td>Model</td><td>Speed up</td></tr><tr><td>2-layer RNNLM</td><td>0</td><td>Inception</td><td>0</td></tr><tr><td>4-layer RNNLM</td><td>5%</td><td>AmoebaNet</td><td>-5%</td></tr><tr><td>2-layer GNMT</td><td>0</td><td>4-stack 36-layer WaveNet</td><td>3.3 %</td></tr><tr><td>4-layer GNMT</td><td>0</td><td>2-stack18-layer WaveNet</td><td>15%</td></tr><tr><td>2-layer Transformer-XL 4-layer Transformer-XL</td><td>7.6% 3%</td><td>8-layer Transformer-XL</td><td>1.5%</td></tr></table>",
+        "bbox": [
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+            127,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Generalization to hold-out graphs: Here we show another set of experiments where we treat GDPbatch as a pre-training strategy and remove the target graph from the batch training dataset. We then fine-tune the pre-trained model on the hold-out graphs for fewer than 50 steps, which takes less than one minute. We name this GDP-generalization+finetune. Figure 2 shows that GDP fine-tuning for hold-out graphs outperforms human expert placement and HDP consistently on all six batch training datasets, and performs only slightly worse than GDP-one. 2-layer RNNLM and 2-stack WaveNet almost match the performance of GDP-one. We also run inference (generate placement) directly on the pre-trained model for the target hold-out graphs, and name this GDP-generalization-zeroshot. We find that GDP-generalization-zeroshot only marginally hurts performance as compared to GDPgeneralization+finetune, while being slightly better than human placement and HDP. This indicates that both graph embedding and the learned policies transfer and generalize to the unseen data. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/5c9f10f415f26c1d76aeae0118c9dc7049201690e0d379df3c50b56bbbbba3f9.jpg",
+        "image_caption": [
+            "Figure 2: Finetuning on hold-out graphs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Comparisons with other generalized placement approaches: Placeto (Addanki et al., 2019), to our knowledge, is the only other method beside GDP that shows true (and non-simulated) generalized device placement results. Direct comparison is not possible since Placeto uses a different hardware platform and different input graphs (Inception-V3, NMT, and NASNet). Placeto’s search time is on average $2 . 6 5 \\mathrm { x }$ faster than HDP, while GDP is on average $1 5 \\mathrm { x }$ faster than HDP on our larger set of graphs. Apart from search time speed up, Placeto on average reduces placed graph run time by $3 \\%$ (for its different graphs and hardware) while GDP on average reduces placed graph run time by $9 . 2 \\%$ , compared to HDP. One advantage of GDP over Placeto is that it does not rely on any initial feasible placement. Providing a reasonable initial placement is often non-trivial for domain experts, especially for larger graphs such as 8-layer GNMT. As such, we are the first to report superhuman results on 8-layer GNMT (with GDP-batch). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4 ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Attention and Superposition. We did an ablation study on the attention and the superposition layer in the transformer-XL placer network. We find that attention improves placement run time by an average of $18 \\%$ compared to a placer network with no attention, and superposition improves placement run time by an average of $6 . 5 \\%$ where all the graphs are trained in a single batch as described in Section 4.3. Without superposition network, batch training fails for AmoebaNet and Inception when mixing with larger RNNLM or GNMT models (4-layer). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Pre-training graph embeddings. We also evaluate a fine-tuning strategy by pre-training the graph embedding and placement network and fine-tuning the network on the down stream tasks. The difference here compared to Section 4.3 is that we also include the target graphs in the pre-training dataset. When GDP-batch is used as a pre-training strategy, the graph embedding and placement network assimilate meaningful graph representations and placement policies from a wide set of graphs, thus can be used as a strong baseline network for fine-tuning on downstream tasks. We compare the generated placement run time and the placement search time, normalized to GDP-one. We find that fine-tuning further reduces the the placed graph run time by an average of $5 \\%$ and placement search time by an average of $86 \\%$ , compared to GDP-one. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/952a5f0142edf3158aea8b7c617b28fb0db5a0c93b43763249bcfb0b239a35b0.jpg",
+        "image_caption": [
+            "Figure 3: Ablation Study on Attention and Superposition of the Placement Network. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/8f18a2f28c32a61bb76c1776235f135ec173371660b93fddcedc426ab84fd863.jpg",
+        "image_caption": [
+            "Figure 4: Normalized run time (step time for the generated placement) and normalized training time (search time) for fine-tuning. Time is normalized to GDP-one. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present a generalized device placement strategy that uses a graph neural network and super-positioning to generalize to arbitrary and held out graphs. Through experimental evaluation over a wide set of representative graphs from different domains including computer vision, speech, and NLP, we demonstrated over 15 times faster convergence while achieving a $16 \\%$ and $9 . 2 \\%$ reductions in step time over human expert placement and HDP, respectively. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6.1 PROXIMAL POLICY OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In device placement, the objective is to minimize the training step time of a given computational graph or a batch of dataflow graphs for a target system configuration (e.g. a 8-GPU cluster or a TPU pod), by placing operations onto different devices to enable model-level parallelism. This process corresponds to maximizing the expected performance in the MDP. For better sample efficiency, we adopted a Proximal Policy Optimization (PPO) (Schulman et al., 2017) algorithm. The objective is to maximize a surrogate objective: ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ad91ac4ba02719816e66a554029f838dd1262f0e6004266f5972dee5fd3c0897.jpg",
+        "text": "$$\nL _ { \\pi } = E _ { a _ { [ 0 : n ] } \\sim \\pi } [ \\frac { q \\prime \\left( a _ { n } | s _ { n } \\right) } { q ( a _ { n } | s _ { n } ) } A _ { \\pi } ( s _ { n } , a _ { n } ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c09780d7e8837991fa3e02cb86aede6678f4a0a9a3d3309d8455834480886046.jpg",
+        "text": "$$\nL _ { \\pi } = \\operatorname* { m a x } _ { \\pi ^ { \\prime } } \\frac { 1 } { N } \\sum _ { n = 0 , a _ { n } \\sim \\pi } ^ { N - 1 } [ \\operatorname* { m i n } ( \\frac { q \\prime ( a _ { n } | s _ { n } ) } { q ( a _ { n } | s _ { n } ) } ( R - \\overline { { R } } ) , c l i p ( \\frac { q \\prime ( a _ { n } | s _ { n } ) } { q ( a _ { n } | s _ { n } ) } , 1 - \\epsilon , 1 + \\epsilon ) ( R - \\overline { { R } } ) ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Within a loop, GDP PPO continuously samples placements from the distribution and evaluates their training times in real systems. For a rollout of $K$ , we perform a minimatch of $m$ stochastic gradient ascent steps with respective to the objective of proximal policy optimization, which makes incremental policy improvements. The rollout steps $K$ and minibatch size $m$ are hyper parameters for PPO. We find a set of optimized hyper parameters and keep them fixed for all the experiments presented. As the rewards are generated on-the-fly based on real system measurements, we no longer need to re-evaluate the placement solutions in a separate phase. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6.2 HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this section, we list out all the selected hyperparameters in our experiments for reproducibility in Table 3 and Table 4. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/9b19a29d595e32c6de02f3beeaf667ce2e561f1b9a4c455e991d4148ce4f3f37.jpg",
+        "table_caption": [
+            "Table 3: Hyperparameters for Policy Network. gs layers: GraphSAGE layers, gs knn: GraphSAGE maximum neighbors, trf d model: Dimension of the TransformerXL model, trf n head: Number of attention heads, tr $f$ layers: Number of TransformerXL layers, trf d heads: Dimension of each attention head, tr $f$ d inner: Dimension of inner hidden size in positionwise feedforward. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Parameters</td><td>Value</td><td>Parameters</td><td>Value</td></tr><tr><td>gs_layers</td><td>4</td><td>gs_dim</td><td>128</td></tr><tr><td>gs_knn</td><td>5</td><td>trf_layers</td><td>2</td></tr><tr><td>trf_d_model</td><td>128</td><td>trf_n_head</td><td>5</td></tr><tr><td>trf_d_head</td><td>25</td><td>trf_d_inner</td><td>256</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/54c1112967bad943684ee53502ec1b5699834308e2eb561c43ed6f1087b1cfe3.jpg",
+        "table_caption": [
+            "Table 4: Hyperparameters for PPO. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Parameters</td><td>Value</td><td>Parameters</td><td>Value</td></tr><tr><td>learing rate</td><td>0.5</td><td>num of rollouts</td><td>400</td></tr><tr><td>minibatches</td><td>40</td><td>epochs</td><td>5</td></tr><tr><td>epsilon</td><td>0.2</td><td>entropy</td><td>0.05</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6.3 INPUT GRAPHS ",
+        "text_level": 1,
+        "bbox": [
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+            883
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We used a variety of widely used workloads from computer vision, speech, and NLP. In this section, we give a detailed explanation on the selected models and hyperparameters. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6.3.1 INCEPTION-V3",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            333,
+            117
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Inception-V3 (Szegedy et al., 2015) is a multi-branch convolutional network used for a variery of computer vision tasks, including classification, recognition, or generation. The network consists of blocks made of multiple branches of concolutional and pooling operations. Within a block, the branches of ops can be executed in parallel. However, the model is mostly sequential as the outputs of each block are concatenated together to form the input to the next block. We use a batch size of 64. The Tensorflow graph of this model contains 24,713 operations. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.3.2 AMOEBANET ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            237,
+            321,
+            251
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "AmoebaNet (Real et al., 2018) is an automatically designed neural network that yields SoTA performance on ImageNet. Similar to Inception-V3, it contains Inception-like blocks called cells, which receives a direct input from the previous cell and a skip input from the cell before it. The network is made of redundant cells stacked together, therefore is more modular than Inception-V3. We use a batch size of 64. The Tensorflow graphs contains 9,430 operations. ",
+        "bbox": [
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+            333
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.3.3 RNNLM ",
+        "text_level": 1,
+        "bbox": [
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+            356,
+            290,
+            371
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Recurrent Neural Network Language Model (Zaremba et al., 2014; Jozefowicz et al., 2016) is made of many LSTM cells organized in a grid structure. The processing of each LSTM cell only depends on the results of 2 other cells (from the previous layer, and from the previous time step), which make the concurrent execution of many LSTM cells possible given enough hardware resources. We use batch size 64 and a hidden size of 2048. The corresponding TensorFlow graph contains 9,021 operations for a 2-layer model. The number of ops grow roughly proportional with the number of layers. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.3.4 GNMT ",
+        "text_level": 1,
+        "bbox": [
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+            503,
+            279,
+            518
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Neural Machine Translation with attention mechanism (Bahdanau et al., 2015; Wu et al., 2016) has an architecture similar to that of RNNLM, but its many hidden states make it far more computationally expensive than RNNLM. To reduce the training time, prior work (Wu et al., 2016) propose placing each LSTM layer, as well as the attention and the softmax layer, on a separate device. This strategy demonstrates early success in human placement, we show that GDP can find significantly better placements. We use batch size 64. The original 2-layer encoder-decoder consisting of 28,044 operations. An extended 4-layer version consisting of 46,600 operations, An even larger 8-layer version consisting of 83,712 operations. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.3.5 TRANSFORMER-XL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            665,
+            364,
+            679
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Transformer-XL (Dai et al., 2019) is an modified version of Transformer (Vaswani et al., 2017) that supports segement-level recurrence and a novel positional encoding scheme. This innovation enables learning dependency that is $80 \\%$ longer than RNNs, and $450 \\%$ longer than vanilla Transformers. We use a transformer-XL with batch size of 64, sequence length of 256, segment length of 64, model hidden dimension of 500 and feed forward hidden dimension of 1000, 10 heads, and head dimension of 50. The 2-layer Transformer-XL contains 2,618 operations. The number of ops grow roughly proportional with the number of layers. ",
+        "bbox": [
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+            693,
+            825,
+            790
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.3.6 WAVENET ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            813,
+            300,
+            827
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "WaveNet (van den Oord et al., 2016) is a generative model for speech synthesis. The model is fully probabilistic and autoregressive, with the predictive ditribution for each audio sample conditioned on all previous ones. Architecturally, WaveNet uses causal convolutions with dilations, to obtain a large receptive field. We use a WaveNet model with batch size 64 and a receptive field size of 2048 (9-layers per stack). An 5-stack WaveNet contains 4,374 operations and a 10-stack WaveNet contains 8,516 operations. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Run time comparison on GDP batch training vs. the best of related methods (human expert, METIS, HDP, and GDP no batch training). ",
+        "bbox": [
+            174,
+            101,
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+            128
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+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/db32903e6d358f9928b8d8ba101e6a3c5fb8ca54d494210ddd704521b61e785b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Batch Setting</td><td>Model</td><td>speed up (s)</td></tr><tr><td rowspan=\"6\">Batch 2</td><td>Inception</td><td>0</td></tr><tr><td>AmoebaNet</td><td>-4.5%</td></tr><tr><td>2-layer RNNLM</td><td>0</td></tr><tr><td>2-layer GNMT</td><td>0</td></tr><tr><td>2-layer Transformer-XL</td><td>6.5%</td></tr><tr><td>2-stack 18-layer Wavenet</td><td>4%</td></tr><tr><td rowspan=\"6\">Batch 3</td><td>2-layer RNNLM</td><td>0</td></tr><tr><td>4-layer RNNLM</td><td>0</td></tr><tr><td>8-layer RNNLM</td><td>4.5%</td></tr><tr><td>2-layer GNMT</td><td>0</td></tr><tr><td>4-layer GNMT</td><td></td></tr><tr><td>8-layer GNMT</td><td>0 8%</td></tr><tr><td>Batch 4</td><td>3x8-layer GNMT</td><td>5.1%</td></tr><tr><td>Batch 5</td><td>3x8-layer RNNLM</td><td>4.5%</td></tr></table>",
+        "bbox": [
+            367,
+            141,
+            630,
+            335
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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+]

parse/train/X7GEA3KiJiH/X7GEA3KiJiH.md

@@ -0,0 +1,344 @@
+# Learning Dynamic Graph Representation of Brain Connectome with Spatio-Temporal Attention
+
+Byung-Hoon Kim ∗
+
+Department of Psychiatry Institute of Behavioral Sciences in Medicine College of Medicine, Yonsei University egyptdj@yonsei.ac.kr
+
+Jong Chul Ye   
+Department of Bio/Brain Engineering   
+Kim Jaechul Graduate School of AI KAIST jong.ye@kaist.ac.kr
+
+# Jae-Jin Kim
+
+Department of Psychiatry Institute of Behavioral Sciences in Medicine College of Medicine, Yonsei University jaejkim@yonsei.ac.kr
+
+# Abstract
+
+Functional connectivity (FC) between regions of the brain can be assessed by the degree of temporal correlation measured with functional neuroimaging modalities. Based on the fact that these connectivities build a network, graph-based approaches for analyzing the brain connectome have provided insights into the functions of the human brain. The development of graph neural networks (GNNs) capable of learning representation from graph structured data has led to increased interest in learning the graph representation of the brain connectome. Although recent attempts to apply GNN to the FC network have shown promising results, there is still a common limitation that they usually do not incorporate the dynamic characteristics of the FC network which fluctuates over time. In addition, a few studies that have attempted to use dynamic FC as an input for the GNN reported a reduction in performance compared to static FC methods, and did not provide temporal explainability. Here, we propose STAGIN, a method for learning dynamic graph representation of the brain connectome with spatio-temporal attention. Specifically, a temporal sequence of brain graphs is input to the STAGIN to obtain the dynamic graph representation, while novel READOUT functions and the Transformer encoder provide spatial and temporal explainability with attention, respectively. Experiments on the HCP-Rest and the HCP-Task datasets demonstrate exceptional performance of our proposed method. Analysis of the spatio-temporal attention also provide concurrent interpretation with the neuroscientific knowledge, which further validates our method. Code is available at https://github.com/egyptdj/stagin
+
+# 1 Introduction
+
+Neuroimaging modalities provide measurements of brain activity by capturing the signals of neural activity. Functional magnetic resonance imaging (fMRI) is a non-invasive imaging method that measures the blood-oxygen level dependence (BOLD) in order to estimate the neural activity of the whole brain over time [20]. Functional connectivity (FC) is defined as the degree of temporal correlation between regions of the brain. Based on the fact that these connectivities form networks that change over time, graph-based network analysis of brain connectome has been one of the key approaches to understanding how the brain works [7, 4, 43].
+
+Graph neural networks (GNNs) are a type of deep neural networks that have recently been successful in learning the representation of graph-structured data [50]. The graph-structured nature of the brain has led to an increased interest in learning the reperesentation of the brain FC network with the GNNs. Learning the representation of the brain connectome can be linked to decoding trait or state from human brain signal measurements. Accordingly, the current trend in studies attempting to apply GNN to the brain connectome is to input the FC graph from either resting-state [25, 26, 2, 33, 23, 48, 49] or task fMRI data [29, 30, 32] and predict a particular phenotype of the subjects, such as gender [26, 2, 23, 22] or presence of a specific disease [26, 33, 29, 30, 32, 22]. While these studies have shown potential strengths and opportunities for learning the network representation of the brain, they also suggest limitations of current GNN-based methods.
+
+One of the most common limitations with previous GNN-based FC network analysis methods is that most of them fail to take advantage of the dynamic properties of the FC network, which fluctuates over time. Incorporating the dynamic features of the FC network into the neuroimaging analysis has been an important direction in the field of functional neuroimaging [21, 37]. A work by [15] tried to address this issue by using the Spatial Temporal Graph Convolutional Network (ST-GCN) [54] model to incorporate dynamic features of the FC network. However, [15] reported lower accuracy than other non-dynamic GNN-based FC methods [23, 2] in the gender classification experiment, leaving a question about the effectiveness of the dynamic FC method. In addition, another limitation of the method is that no temporal explainability is provided from the model. This is a major drawback considering that the goal of applying GNNs to functional neuroimaging methods is not only to achieve high classification accuracy, but also to uncover the functional basis of the brain [23, 32]. Another recent work by [3], using GraphNets [5] and DiffPool [58] for the dynamic FC analysis, also suffers from the same limitations in terms of poor classification accuracy and lack of temporal explainability.
+
+Here, we propose Spatio-Temporal Attention Graph Isomorphism Network (STAGIN) for learning the dynamic graph representation of the brain connectome with spatio-temporal attention. The proposed method exploits the temporal features of the dynamic FC network graphs to improve the classification accuracy of the model. In particular, we address the issue that the node features of the input dynamic graph should contain temporal information and concatenate encoded timestamp with the node features (Section 4.1). In addition, the proposed method includes novel attention-based READOUT modules (Section 4.2) and the Transformer encoder [46] (Section 4.3) in order to further improve the classification performance and provide spatial-temporal explainability at the same time. STAGIN achieves state-of-the-art performance with the Human Connectome Project (HCP) dataset [45] in gender classification for resting-state fMRI and task decoding for task fMRI. We inherit $\mathbf { k }$ -means clustering analysis of the resting-state dynamic FC [1] and general linear model (GLM) statistical mapping of task fMRI [14] for interpreting the spatio-temporal attention learned from STAGIN, which are widely accepted analysis methods for the fMRI data. The interpretation of the learned spatio-temporal attention replicates neuroscientific findings from previous large-scale fMRI studies in both resting-state and task fMRI, which further validates our proposed method.
+
+Our work holds potential societal impact in that brain decoding methods can be linked to finding neural biomarkers of important phenotypes or diseases. However, potential negative impact related to privacy concerns that arise from abuse or misuse of accurate decoding methods should also be noted. Although our method is yet behind the decoding capability that can be abused or misused, our research cannot still be free from these ethical considerations.
+
+# 2 Related works
+
+# 2.1 Graph Neural Network on Dynamic Graphs
+
+Many networks that arise around us are inherently dynamic, with changes in the existence of nodes and edges over time. Learning the representation of dynamic graphs has piqued the interest of researchers and has led to development of methods that can embed dynamic graphs using their time information [35]. Methods that incorporate attention for learning the representation of dynamic graphs have also been proposed [51, 40]. However, it is not easy to apply these techniques directly to the dynamic brain graphs because of the different inherent properties of the dynamic brain graphs that do not include any addition or deletion of nodes and are sampled uniformly over time. Nonetheless, our work is inspired by these earlier studies, particularly for the encoding of temporal information and their concatenation to the node features, as proposed in Section 4.1 [51, 40].
+
+# 2.2 Attention in Graph Neural Networks
+
+Bringing attention to the GNNs is a topic that is being actively studied in the field of geometric deep learning [27]. One of the most successful uses of attention is to compute the attention at edges of the graph and scale the importance of the links when the features of the neighborhood node are aggregated [47, 6], often providing performance gain in learning the representation of input graphs. Another stream of applying attention to the GNNs comes with the motivation to define a pooling function on the graph domain. Since it is not straightforward to decide on what basis the coarsening should be carried out for graph structured data, works such as [16, 28, 38] have addressed this problem by selecting the nodes with top scores computed from projecting the node feature vectors into a learnable parameter vector, or from a GNN layer aggregated local graph features. Although the motivation may have been different, these graph pooling methods are closely related to the spatial attention modules that we propose in Section 4.2 in that they exploit learned relative scores across the vertices of the graph. While some works have already been aware that the appropriate use of node-wise attention can improve performance of downstream tasks [55, 11], we note that previous methods tend to score attention based on randomly initialized parameters or local graph structures which may be suboptimal for graph classification tasks that require taking the whole graph feature into account.
+
+# 3 Theory
+
+# 3.1 Problem definition
+
+The goal of our study is to train a neural network
+
+$$
+f : G _ { \mathrm { d y n } } \to h _ { G _ { \mathrm { d y n } } } ,
+$$
+
+where $G _ { \mathrm { d y n } } = ( G ( 1 ) , . . . , G ( T ) )$ is the sequence of brain graphs with $T$ timepoints and $\pmb { h } _ { G _ { \mathrm { d y n } } } \in \mathbb { R } ^ { D }$ is the vector representation of the dynamic graph $G ( t )$ with length $D$ . The graph $G ( t ) = ( V ( t ) , E ( t ) )$ at time $t$ is a pair of vertex set $V ( t ) = \{ { \pmb x } _ { 1 } ( t ) , . . . , { \pmb x } _ { N } ( t ) \}$ of $N$ nodes and edge set $E ( t ) \ =$ $\left\{ \{ \pmb { x } _ { i } ( t ) , \pmb { x } _ { j } ( t ) \} \mid j \in \mathcal { N } ( i ) , i \in \{ 1 , . . . N \} \right\}$ where $\mathcal { N } ( i )$ denotes the neighborhood of the vertex $i$ . If $f$ learns to extract a disentangled representation of the dynamic brain graph $G _ { \mathrm { d y n } }$ , then the classification of a certain phenotypic characteristic (e.g. gender) from $h _ { G _ { \mathrm { d y n } } }$ can be performed with a linear mapping as a downstream task. Another important consideration in this work is to ensure the explainability of the model $f$ , being able to inform us which part of the brain at which timepoint was considered important when extracting the meaningful representation $h _ { G _ { \mathrm { d y n } } }$ . Specifically, we formulate $f = q \circ g$ as a composition of the GNN $g$ and the Transformer encoder $q$ , where $g$ outputs the set of graph representations $\mathbf { \mathcal { h } } _ { G ( t ) }$ from each timepoint and $q$ exploits self-attention to integrate $\mathbf { \mathcal { h } } _ { G ( t ) }$ into the final representation $h _ { G _ { \mathrm { d y n } } }$ :
+
+$$
+\begin{array} { r l } & { g : G _ { \mathrm { d y n } } \to ( h _ { G ( 1 ) } , . . . , h _ { G ( T ) } ) , } \\ & { } \\ & { q : ( { h _ { G ( 1 ) } } , . . . , { h _ { G ( T ) } } ) \to h _ { G _ { \mathrm { d y n } } } . } \end{array}
+$$
+
+We will omit timepoint notation $( t )$ for brevity, whenever it is not of contextual importance.
+
+# 3.2 Graph Isomorphism Network
+
+The GNNs are generally composed of functions that (i) integrate the node features from its neighbors, and (ii) embed the integrated information with a nonlinear transformation to obtain the next layer node features. These functions are called AGGREGATE, and COMBINE functions, respectively, and the choice of these functions define many variants of the GNN,
+
+$$
+\begin{array} { r l } & { \pmb { a } _ { v } ^ { ( k ) } = \mathtt { A G G R E G A T E } ^ { ( k ) } \Big ( \Big \{ \pmb { h } _ { u } ^ { ( k - 1 ) } : u \in \mathcal { N } ( v ) \Big \} \Big ) , } \\ & { \pmb { h } _ { v } ^ { ( k ) } = \mathtt { C O M B I N E } ^ { ( k ) } \Big ( \pmb { h } _ { v } ^ { ( k - 1 ) } , \pmb { a } _ { v } ^ { ( k ) } \Big ) , } \end{array}
+$$
+
+where $\boldsymbol { h } _ { v } ^ { ( k ) }$ denotes the feature vector of node $v$ at layer $k$ and ${ h } _ { v } ^ { ( 0 ) } : = \pmb { x } _ { v }$
+
+The Graph Isomorphism Network (GIN) is a variant of the GNN suitable for graph classification tasks, which is known to be as powerful as the WL-test under certain assumptions of injectivity [52]. The GIN typically defines sum as the AGGREGATE and a multi-layer perceptron (MLP) with two layers as the COMBINE updating the node representation $\boldsymbol { h } _ { v } ^ { ( k ) }$ at layer $k$ [52] by :
+
+$$
+\pmb { h } _ { v } ^ { ( k ) } = \mathtt { M L P } ^ { ( k ) } \Big ( ( 1 + \epsilon ^ { ( k ) } ) \cdot \pmb { h } _ { v } ^ { ( k - 1 ) } + \sum _ { u \in \mathcal { N } ( v ) } \pmb { h } _ { u } ^ { ( k - 1 ) } \Big ) ,
+$$
+
+where $\epsilon$ is a learnable parameter initialized with zero. Equation (5) can be easily reformulated into the matrix form [23] by:
+
+$$
+\pmb { H } ^ { ( k ) } = \sigma \left( ( \epsilon ^ { ( k ) } \cdot \pmb { I } + \pmb { A } ) \pmb { H } ^ { ( k - 1 ) } \pmb { W } ^ { ( k ) } \right) ,
+$$
+
+where
+
+$$
+\pmb { H } ^ { ( k ) } = \left[ \pmb { h } _ { 1 } ^ { ( k ) } , \cdots , \pmb { h } _ { N } ^ { ( k ) } \right] \in \mathbb { R } ^ { D \times N }
+$$
+
+is the stack of node feature vectors, $\pmb { I }$ is the identity matrix, $\pmb { A }$ is the adjacency matrix between the node features, $W$ is the network weights of the MLP, and $\sigma$ is the nonlinearity function.
+
+The READOUT function takes the updated node features $\boldsymbol { h } _ { v } ^ { ( k ) }$ to compute the representation of the whole graph:
+
+$$
+\begin{array} { r } { \pmb { h } _ { G } ^ { ( k ) } = \mathtt { R E A D O U T } \Big ( \{ \pmb { h } _ { v } ^ { ( k ) } \ | \ v \in G \} \Big ) . } \end{array}
+$$
+
+In general, the READOUT function is defined simply as computing the sum or average of the input node features. This is equivalent to multiplication with the length $N$ pooling vectors $\phi _ { \mathrm { s u m } } ^ { \mp } = [ 1 , \stackrel { \cdot } { \dots } , 1 ]$ or $\phi _ { \mathrm { m e a n } } ^ { \top } = [ 1 / N , . . . , 1 / N ]$ for the matrix form:
+
+$$
+\begin{array} { r } { \pmb { h } _ { G } ^ { ( k ) } = \pmb { H } ^ { ( k ) } \phi _ { \mathrm { m e a n } } . } \end{array}
+$$
+
+# 3.3 Encoder-decoder understanding of GNNs
+
+Although formulating the GIN (5) as a combination of AGGREGATE and COMBINE function might not suggest its close relationship with convolutional neural networks (CNNs) at first glance, previous works by [23, 8] show that the matrix formulation of the GIN operation (6) can be thought of a CNN layer with shift operation of the convolution as the adjacency matrix $\pmb { A }$ . We extend the understanding of encoder-decoder CNN as a framelet expansion [57, 56] to the GIN to formulate node feature vectors $H ^ { ( k ) }$ at layer $k$ with respect to the input node feature $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ as:
+
+$$
+\operatorname { V e c } \left( \pmb { H } ^ { ( k ) } \right) = \pmb { \Sigma } ^ { ( k ) } \pmb { E } ^ { ( k ) \top } \cdot \cdot \cdot \pmb { \Sigma } ^ { ( 1 ) } \pmb { E } ^ { ( 1 ) \top } \pmb { x } , \quad \mathrm { w h e r e } \quad \pmb { x } : = \operatorname { V e c } \left( [ \pmb { x } _ { 1 } , \cdots , \pmb { x } _ { N } ] \right)
+$$
+
+where $\mathrm { V e c } ( \cdot )$ denotes the vectorization operation, and the $k$ -th layer encoder matrix $\pmb { { \cal E } } ^ { ( k ) }$ is defined as
+
+$$
+\pmb { E } ^ { ( k ) } = \pmb { W } ^ { ( k ) } \otimes ( \epsilon ^ { ( k ) } \cdot \pmb { I } + \pmb { A } ^ { T } )
+$$
+
+where $\otimes$ refers to the Kronecker product, and $\pmb { \Sigma } ^ { ( k ) }$ is the diagonal matrix with values 1 or 0 depending on the activation pattern of the nonlinearity. Now, $\phi _ { \mathrm { { m e a n } } }$ of equation (8) can be thought as the decoder at the $k$ -th layer which yields the whole graph feature vector from the encoded node feature vectors.
+
+Proposition 1. The READOUT function $\phi _ { m e a n }$ in (8) generates a decoder with fixed constant bases.
+
+Proof. From the READOUT function (8), we have
+
+$$
+\begin{array} { r l } & { \boldsymbol { h } _ { G } ^ { ( k ) } = \mathrm { V e c } \left( \boldsymbol { h } _ { G } ^ { ( k ) } \right) = \mathrm { V e c } \left( \boldsymbol { H } ^ { ( k ) } \phi _ { \mathrm { m e a n } } \right) = \left( \phi _ { \mathrm { m e a n } } ^ { T } \otimes \boldsymbol { I } \right) \mathrm { V e c } \left( \boldsymbol { H } ^ { ( k ) } \right) } \\ & { \qquad = \left( \phi _ { \mathrm { m e a n } } ^ { T } \otimes \boldsymbol { I } \right) \boldsymbol { \Sigma } ^ { ( k ) } \boldsymbol { E } ^ { ( k ) \top } \cdot \cdot \cdot \boldsymbol { \Sigma } ^ { ( 1 ) } \boldsymbol { E } ^ { ( 1 ) \top } \boldsymbol { x } } \end{array}
+$$
+
+Now let $\mathbf { } _ { b _ { i } }$ and $\tilde { \mathbf { b } } _ { i }$ denote the $i$ -th column of the encoder matrix ${ \cal E } ^ { ( 1 ) } \Sigma ^ { ( 1 ) } \cdot \cdot \cdot { \cal E } ^ { ( k ) } \Sigma ^ { ( k ) }$ and the decoder matrix $\left( \phi _ { \mathrm { m e a n } } ^ { T } \otimes I \right)$ , respectively. Then, it is straight to obtain the following representation:
+
+$$
+{ h } _ { G } ^ { ( k ) } = \sum _ { i } \langle b _ { i } , { x } \rangle \tilde { b } _ { i }
+$$
+
+Therefore, we can see that although the encoder basis $b _ { i }$ is a function of $_ { \textbf { \em x } }$ , the decoder basis $\tilde { \boldsymbol { b } } _ { i }$ is a constant. □
+
+![](images/742f062d71231a50f5c4e1d9e80c0b555365b6f10c527c39295fb635063d5273.jpg)  
+Figure 1: Schematic illustration of the proposed method. (a) Overall framework of the STAGIN. A sequence of dynamic graph is first input to the GIN followed by GARO or SERO which produces a sequence of spatially attended graph representation vectors $\tilde { h } _ { G ( t ) }$ . Temporal attention is computed over $\tilde { h } _ { G ( t ) }$ and the temporally attended graph representations are averaged to generate the final representation $\tilde { h } _ { G _ { \mathrm { d y n } } }$ . (b) Attention-based READOUT modules. Both GARO and SERO compute spatial attention $z _ { \mathrm { s p a c e } }$ with global average-pooled graph feature $_ { h _ { G } }$ as prior.
+
+We address the issue that the decoder being a constant function can restrict the expressivity of the neural network, and explore adaptive READOUT functions with attention in Section 4.2.
+
+# 4 STAGIN: Spatio-Temporal Attention Graph Isomorphism Network
+
+In this section, we discuss the details of our main contribution. Specifically, we propose STAGIN with two novel attention-based READOUT modules for learning the dynamic graph representation of the brain connectome (Figure 1).
+
+# 4.1 Dynamic graph definition
+
+The sequence of input dynamic FC graphs is constructed from 4D fMRI data with 3D voxels across time. The ROI-timeseries matrix $\bar { P ^ { \prime } } \in \bar { \mathbb { R } } ^ { N \times T _ { \operatorname* { m a x } } }$ is extracted by taking the mean values within a predefined 3D atlas which consists of $N$ ROIs at each timepoint. Values of each ROI are standardized across time. Constructing dynamic FC matrix follows the sliding-window approach, where the temporal window of length $\Gamma$ is shifted across time with stride $S$ to generate $T = \lfloor T _ { \mathrm { m a x } } - \Gamma / S \rfloor$ windowed matrices $\bar { P } ( t ) \in \mathbb { R } ^ { N \times \Gamma }$ (Figure 2 (a)). The FC at time $t$ is defined as the correlation coefficient matrix $R ( t )$ of the windowed timeseries between ${ \bar { p } } _ { i } ( t )$ and $\bar { p } _ { j } ( t )$ :
+
+$$
+R _ { i j } ( t ) = \frac { \mathrm { C o v } ( \bar { p } _ { i } ( t ) , \bar { p } _ { j } ( t ) ) } { \sigma _ { \bar { p } _ { i } } ( t ) \sigma _ { \bar { p } _ { j } } ( t ) } \in \mathbb { R } ^ { N \times N } ,
+$$
+
+where the subscript $i$ and $j$ are the row and column indices of $\bar { P } ( t )$ , Cov denotes the cross covariance, and $\sigma _ { p }$ denotes the standard deviation of $\pmb { p }$ . The final binary adjacency matrix $A ( t ) \in \{ 0 , 1 \} ^ { N \times N }$ is obtained from the FC matrix $\mathbf { } R ( t )$ by thresholding the top 30-percentile values of the correlation matrix as connected, and otherwise unconnected following [23]. Other thresholds for binarizing the correlation matrix are also experimented and the results are provided in the Appendix Section C.2.
+
+Unlike the adjacency matrix $\mathbf { } A ( t )$ , conventional definition of node feature vectors ${ \bf \mathit { x } } _ { v } ( t )$ at node index $v$ as coordinates [29], mean-activation [29, 15], or one-hot encoding [23], do not change over $t$ , disregarding any temporal variation. To address this issue, we concatenate encoded timestamp $\eta ( t ) \in \mathbb { R } ^ { D }$ to the spatial one-hot encoding $e _ { v }$ , followed by linear mapping with a learnable parameter matrix $W \in \mathbb { R } ^ { D \times ( N + D ) }$ to define the input node feature,
+
+$$
+\begin{array} { r } { \pmb { x } _ { v } ( t ) = \pmb { W } [ \pmb { e } _ { v } | | \eta ( t ) ] . } \end{array}
+$$
+
+Here, the learnable timestamp encoder $\eta$ is a Gated Recurrent Unit (GRU) [9] which takes ROItimeseries upto the endpoint of the sliding-window as the input. Both the vertex set $V ( t )$ and the
+
+![](images/290dcab37d7782ea35b2d29fd27ceb77cd28469f9fdf7c1690b968d398ee15f2.jpg)  
+Figure 2: Defining the dynamic graph. (a) Scheme of extracting dynamic graph from ROI-timeseries matrix $_ { r }$ . (b) Example of a constructed dynamic graph.
+
+edge set $E ( t )$ of graph $G ( t )$ now incorporates temporal information at time t. See Figure 2 for an illustration of the dynamic graph definition.
+
+# 4.2 Spatial attention with attention-based READOUT
+
+As suggested from Proposition 1, conventional READOUT function of GNN can be thought of as a fixed decoder that decodes whole-graph feature from the node features with no learnable parameters. We address this issue by incorporating attention to the READOUT function, which the attention here refers to the scaling coefficient across the nodes learned by the model. Specifically, the spatial attention vector $z _ { \mathrm { s p a c e } } ( t ) \in [ 0 , 1 ] ^ { N }$ is computed by taking the $\pmb { H }$ as a prior:
+
+$$
+\begin{array} { c } { { z _ { \mathrm { s p a c e } } = s ( { \cal H } ) , } } \\ { { \tilde { h } _ { \cal G } = { \cal H } z _ { \mathrm { s p a c e } } , } } \end{array}
+$$
+
+where $s : \mathbb { R } ^ { D \times N } \to [ 0 , 1 ] ^ { N }$ is the attention function and $\tilde { h } _ { G }$ denotes spatially attended graph representation $h _ { G }$ . We propose two types of attention function $s ( \cdot )$ for the attention-based READOUT, named Graph-Attention READOUT (GARO) and Squeeze-Excitation READOUT (SERO) inspired by the attention mechanisms of [46] and [19], respectively.
+
+# 4.2.1 GARO: Graph-Attention READOUT
+
+The GARO follows key-query embedding based attention of the Transformer [46]. However, the key embedding is computed from the matrix of node features $\pmb { H }$ , while the query embedding is computed from the vector of unattended graph representation $H \phi _ { \mathrm { { m e a n } } }$ :
+
+$$
+\begin{array} { c } { { \kappa = W _ { \mathrm { k e y } } H , } } \\ { { q = W _ { \mathrm { q u e r y } } H \phi _ { \mathrm { m e a n } } , } } \\ { { { z } _ { \mathrm { s p a c e } } = \mathrm { s i g m o i d } \Big ( \frac { q ^ { \top } K } { \sqrt { D } } \Big ) , } } \end{array}
+$$
+
+where $W _ { \mathrm { k e y } } \in \mathbb { R } ^ { D \times D }$ , $W _ { \mathrm { q u e r y } } \in \mathbb { R } ^ { D \times D }$ are learnable key-query parameter matrices, $\pmb { K } \in \mathbb { R } ^ { D \times N }$ is the embedded key matrix, and $\pmb q \in \mathbb { R } ^ { D }$ is the embedded query vector.
+
+# 4.2.2 SERO: Squeeze-Excitation READOUT
+
+The SERO follows MLP based attention of the Squeeze-and-Excitation Networks [19]. However, attention from the squeezed graph representation does not scale the channel dimension, but the node dimension in SERO:
+
+$$
+z _ { \mathrm { s p a c e } } = \mathrm { s i g m o i d } \Big ( W _ { 2 } \sigma ( W _ { 1 } H \phi _ { \mathrm { m e a n } } ) \Big ) ,
+$$
+
+where $\sigma$ is the nonlinearity function and $W _ { 1 } \in \mathbb { R } ^ { D \times D }$ , $W _ { 2 } \in \mathbb { R } ^ { N \times D }$ are learnable parameter matrices. This type of spatial dimension squeeze-excitation module has been shown to improve
+
+performance of the CNN models [41], but was not easily applicable to general graphs which may vary in number of nodes for each graph. We exploit the fact that the brain graphs have fixed number of nodes $N$ across participants based on the chosen atlas.
+
+# 4.2.3 Orthogonal regularization
+
+If we take a closer look at (8) and (12), computation of graph feature vector $_ { h _ { G } }$ from the node feature matrix $\pmb { H }$ can also be viewed as reconstructing signal $h _ { G }$ from the basis frames $\pmb { H }$ with vectors $\phi _ { \mathrm { { m e a n } } }$ and $z _ { \mathrm { s p a c e } }$ , respectively. While $z _ { \mathrm { s p a c e } }$ provides further expressivity of the model with adaptive coefficients when compared to $\phi _ { \mathrm { { m e a n } } }$ , we find it desirable to encourage the orthogonality of $\pmb { H }$ as elaborated in the Appendix Section A. The orthogonal regularization $\mathcal { L } _ { \mathrm { o r t h o } }$ is defined as:
+
+$$
+\mathcal { L } _ { \mathrm { o r t h o } } = \left\| 1 / m \cdot H ^ { \top } H - I \right\| _ { 2 } ,
+$$
+
+where $m = \operatorname* { m a x } ( H ^ { \top } H )$ . The scaling term $1 / m$ ensures the columns of the matrix $\pmb { H }$ become orthogonal to each other with the same length, while not restricting the specific length that the column vectors should follow.
+
+# 4.3 Temporal attention with Transformer encoder
+
+For attention across time, we employ a single-headed Transformer encoder [46] upon the sequence of graph features $( \tilde { h } _ { G ( 1 ) } , . . . , \tilde { h } _ { G ( T ) } )$ . The temporal attention can be measured by the self-attention graph representation weights $Z _ { \mathrm { t i m e } } \in [ 0 , 1 ] ^ { T \times T }$ $h _ { G _ { \mathrm { d y n } } } ^ { ( k ) }$ after the softmax function of the Transformer encoder. Per-layer dynamic is computed by summing the temporally attended feature output from the Transformer encoder across time at each layers, where the final representation:
+
+$$
+{ \pmb h } _ { G _ { \mathrm { d y n } } } = \mathrm { c o n c a t e n a t e } ( \{ { \pmb h } _ { G _ { \mathrm { d y n } } } ^ { ( k ) } ~ | ~ k \in \{ 1 , . . . , K \} \} )
+$$
+
+is the concatenation of dynamic graph representation of all $K$ layers following [53].
+
+# 5 Experiment
+
+# 5.1 Dataset
+
+Publicly available2 fMRI data from the HCP S1200 release [45] was used for our experiments. The data was collected from voluntary participants with informed consent and was fully anonymized. We constructed two datasets, the HCP-Rest and the HCP-Task, depending on whether the subject was resting or performing specific tasks during the acquisition of the image. The HCP-Rest dataset consisted of pre-processed and ICA denoised resting-state fMRI data [17], which the subjects were instructed to rest for 15 minutes during the data acquisition. We used first run data of the four sessions, and excluded data with short acquisition time with $T _ { \mathrm { m a x } } < 1 2 0 0$ . There were 1093 images finally included in the dataset, which consisted of 594 female and 499 male subjects. The gender of each subject served as the labels of the HCP-Rest dataset letting the number of classes $C = 2$ . The HCP-Task consisted of pre-processed task fMRI data [17], which the subjects were instructed to perform specific tasks during data acquisition. For example in the "Motor" task fMRI, participants were told to perform one of the subtasks during the acquisition to make motor movements on one’s left hand, left foot, right hand, right foot, or tongue. There were seven types of tasks including working memory, social, relational, motor, language, gambling, and emotion. After excluding the fMRI data with short acquisition time, there were 7450 images included in the dataset. The task type during the data acquisition served as the labels of the HCP-Task dataset, letting $C = 7$ . A more detailed description of the experiment datasets with a note on the twin subjects of HCP can be found in the Appendix Section B.
+
+# 5.2 Experimental settings
+
+Experiments were performed on a workstation with two NVIDIA GeForce GTX 1080 Ti GPUs. The STAGIN model $f$ is trained end-to-end in a supervised manner with the loss $\mathcal { L } = \mathcal { L } _ { \mathrm { x e n t } } + \lambda \cdot \mathcal { L } _ { \mathrm { o r t h o } }$ where $\mathcal { L } _ { \mathrm { x e n t } }$ is the cross entropy loss and $\lambda$ is the scaling coefficient of the orthogonal regularization. We set the number of layers $K = 4$ , embedding dimension $D = 1 2 8$ , window length $\Gamma = 5 0$ , window stride $S = 3$ , and regularization coefficient $\lambda \stackrel { - } { = } 1 . 0 \times 1 0 ^ { - 5 }$ . The window length and stride correspond to capturing the FC within 36 seconds every 2.16 seconds, which follows the standard setting of the sliding-window dFC analyses [59, 37]. Dropout rate 0.5 is applied to the final dynamic graph representation $h _ { G _ { \mathrm { d y n } } }$ , and rate 0.1 is applied to the attention vectors $z _ { \mathrm { s p a c e } }$ and $z _ { \mathrm { t i m e } }$ during training. For nonlinearity $\sigma$ in (6) and (14), GELU [18] is used instead of ReLU with batch normalization before each $\sigma$ . One-cycle learning rate policy is employed, which the learning rate is gradually increased from 0.0005 to 0.001 during the early $20 \%$ of the training, and gradually decreased to $5 . 0 \times 1 0 ^ { - 7 }$ afterwise. Thirty training epochs were run for the HCP-Rest dataset with minibatch size 3, while ten epochs were run with minibatch size 16 for the HCP-Task dataset. We performed 5-fold stratified cross-validation of the dynamic graphs from the dataset, and report mean and standard deviation across the folds. To extract the ROI-timeseries, the Schaefer atlas [42] with 400 regions $N = 4 0 0$ ) labelled with 7 intrinsic connectivity networks (ICNs) was used. The time dimension of ROI-timeseries matrix $_ { r }$ was randomly sliced with a fixed length (600 for HCP-Rest, 150 for HCP-Task) at each steps during training for (i) relieving computational overload, (ii) stochastic augmentation of the training dataset, (iii) mitigating unwanted memorization of the specific timing of subtask onset, and (iv) matching the number of timepoints $T$ across different task labels for the HCP-Task dataset. Unsliced full matrix $_ { r }$ was used for inference at test time. The end-to-end inference from the construction of the dynamic graph to the acquisition of the final prediction required 1.68 seconds per sample with given experimental settings.
+
+Table 1: Comparative study on HCP-Rest and HCP-Task dataset.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">HCP-Rest</td><td>HCP-Task</td><td rowspan="2">Type of FC</td><td rowspan="2"># Params</td></tr><tr><td>Accuracy (%)</td><td>AUROC</td><td>Accuracy (%)</td></tr><tr><td>STAGIN-SERO</td><td>88.20 ± 1.33</td><td>0.9296 ± 0.0187</td><td>99.19 ± 0.20</td><td>Dynamic</td><td>1,209k</td></tr><tr><td>STAGIN-GARO</td><td>87.01 ± 3.00</td><td>0.9151 ± 0.0258</td><td>99.02 ± 0.17</td><td>Dynamic</td><td>1,068k</td></tr><tr><td>ST-GCN [15]</td><td>76.95 ± 3.00</td><td>0.8545 ± 0.0316</td><td>98.92 ± 0.27</td><td>Dynamic</td><td>355k</td></tr><tr><td>MS-G3D[10]</td><td>79.16 ± 2.53</td><td>0.8912 ± 0.0329</td><td>1</td><td>Dynamic</td><td>3,045k</td></tr><tr><td>BAnD++ [36]</td><td></td><td></td><td>97.20 ± 0.57</td><td>None</td><td>2,010k</td></tr><tr><td>BAnD [36]</td><td></td><td></td><td>95.10 ± 0.62</td><td>None</td><td>2,010k</td></tr><tr><td>r-BAnD</td><td></td><td></td><td>98.90 ± 0.27</td><td>Dynamic</td><td>664k</td></tr><tr><td>GIN [23]</td><td>81.34 ± 2.40</td><td>0.8955 ± 0.0237</td><td>93.87 ± 0.66</td><td>Static</td><td>169k</td></tr><tr><td>GCN [24]</td><td>80.79 ± 2.00</td><td>0.8741 ± 0.0174</td><td>45.07 ± 1.63</td><td>Static</td><td>101k</td></tr><tr><td>GraphSAGE[31]</td><td>75.48 ± 1.97</td><td>0.8237 ± 0.0228</td><td>54.52 ± 0.97</td><td>Static</td><td>202k</td></tr><tr><td>ChebGCN[2]</td><td>77.76 ± 2.09</td><td>0.8582 ± 0.0233</td><td>73.06 ± 0.68</td><td>Static</td><td>704k</td></tr></table>
+
+# 5.3 HCP-Rest: Gender classification
+
+We first validate our proposed method by gender classification on the HCP-Rest dataset. The two proposed methods, named STAGIN-GARO and STAGIN-SERO based on the type of the spatial attention module, resulted in $8 7 . 0 1 \%$ and $8 8 . 2 0 \%$ mean accuracy on the 5-fold cross validation, respectively (Table 1). The mean area under receiver operator characteristic curve (AUROC) were 0.9151 and 0.9296. Classification performance of STAGIN is compared with other GNN methods for reprensentation learning of dynamic/static FC network, including ST-GCN [15], MS-G3D [10], GIN [23], GCN [24], GraphSAGE [31], and ChebGCN [2]. We used the code by the authors of $[ 1 5 ]$ and $[ 1 0 ]$ but modified the cross validation scheme to avoid early stopping based on the test dataset for fair comparison. It can be seen from Table 1 that our proposed method outperforms other GNN based methods. The results of the ablation study are shown in Table 3 in the Appendix.
+
+We use STAGIN-SERO, which showed the best accuracy, for analyzing temporal and spatial attention of the dynamic FC networks. We define the temporal attention vector $\bar { \boldsymbol { z } } _ { \mathrm { t i m e } } ^ { ( k ) } \in [ 0 , \bar { 1 } ] ^ { T }$ at layer $k$ as the average of row elements in the self-attention weight matrix $\begin{array} { r } { z _ { \mathrm { t i m e } } ^ { ( k ) } [ j ] = \frac { 1 } { T } \sum _ { i = 1 } ^ { T } Z _ { i j } } \end{array}$ where $z _ { \mathrm { t i m e } } ^ { ( k ) } [ j ]$ and $Z _ { i j }$ are $j$ -th lement of $z _ { \mathrm { t i m e } } ^ { ( k ) }$ and $( i , j )$ -th element of $Z _ { \mathrm { t i m e } } ^ { ( k ) }$ for the resting-state data, respectively. To employ $\mathbf { k }$ -means clustering to the resting-state dynamic FC analysis [1], we first define a set of attended timepoints $\tilde { T } = \{ t \mid z _ { \mathrm { t i m e } } [ t ] > \alpha \cdot \sigma _ { z _ { \mathrm { t i m e } } } \}$ where $\alpha$ is the cutoff coefficient, and ${ \sigma } _ { z _ { \mathrm { t i m e } } ^ { ( k ) } }$ denotes the standard deviation of $z _ { \mathrm { t i m e } } ^ { ( k ) }$ . Defining the threshold based on standard deviation inherits the practice of the point-process analysis for dynamic FC, so we set $\alpha = 1 . 0$ following [44]. Pattern of the FC matrices at attended timepoints $A ^ { \tilde { T } } = \{ A ( t ) \mid t \in \tilde { T } \}$ for each subject can now be analyzed with the $\mathbf { k }$ -means clustering. Specifically, we fit 7 template cluster centroids from the dynamic FC matrices $A ( t )$ over all subjects, and assign elements of $A ^ { \tilde { T } }$ into one of the 7 template clusters. The ratio of each clusters from $A ^ { \tilde { T } }$ with respect to $\pmb { A }$ can then be analyzed with the subset of $A ^ { \tilde { T } }$ including only the female or male subjects.
+
+![](images/5e2ae4d16e5a7a857695c0ed72b31036b27c90a793402955d4c8562a6632bd87.jpg)  
+Figure 3: Analysis of temporal attention of the gender classification experiment with $\mathbf { k }$ -means clustering. The DMN and SMN of the 7 cluster centroids are plotted and the relative proportion of temporally attended clusters for female and male subjects are written below. The clusters are sorted in descending order of the female/male attended cluster ratio.
+
+Evidences from large scale studies suggest that female subjects show hyperconnectivity of the DMN [34, 39] and hypoconnectivity of the SMN when compared to male subjects [39, 13]. We accordingly hypothesized that the FC at attended timepoints will show higher values for the DMN and lower values for the SMN in female participants. Figure 3 demonstrates that the clusters mainly attended by female participants show a trend of hyperconnectivity of the DMN and hypoconnectivity of the SMN. This can be interpreted to mean that the STAGIN is properly trained to take the dynamic state of the FC networks into account for predicting the phenotype of the subject.
+
+The spatial attention across regions of the brain is analyzed with the $z _ { \mathrm { s p a c e } } ^ { ( k ) }$ averaged across time $\begin{array} { r } { \tilde { z } _ { \mathrm { s p a c e } } ^ { ( k ) } : = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } z _ { \mathrm { s p a c e } } ^ { ( k ) } ( t ) } \end{array}$ . The regions with top 5 percentile attention values of $\tilde { z } _ { \mathrm { s p a c e } } ^ { ( k ) }$ are plotted with respect to the seven ICNs in Figure 9 in the Appendix. It can be seen that the majority of the top attended regions are from the SMN, which further suggests gender difference of resting-state FC within the SMN. A notable limitation here is that the threshold for determining the top attended region is heuristically set. Statistically determining the spatially attended regions from the resting-state data would further provide validity of the method, which is left as a future work.
+
+# 5.4 HCP-Task: Task decoding
+
+Task decoding refers to classifying which of the seven tasks the subject was performing during the acquisition of the brain fMRI. The STAGIN-GARO and STAGIN-SERO showed $9 9 . 0 2 \%$ and $9 9 . 1 9 \%$ mean accuracy for the task decoding experiment, respectively (Table 1). It can be seen that the proposed methods outperform the previous state-of-the-art model BAND and $\mathrm { B A n D + + }$ [36], which applied self-attention of the Transformer encoder directly to 3D ResNet extracted representation vectors of the fMRI without considering the network property of the brain. To account for the possible statistical disadvantage of voxel-based feature extraction, we further implemented a new region-based BAnD (r-BAnD) by using GIN without attention-based READOUT instaed of the 3D ResNet. Accuracy of r-BAnD resulted in an accuracy of $9 8 . 9 0 \%$ , suggesting that our method shows superior performance even when the statistical disadvantages are matched. Experiment on other models including ST-GCN [15], GIN [23], GCN [24], GraphSAGE [31], and ChebGCN [2] demonstrate exceptional performance of our proposed method for HCP-Task (Table 1). The fact that subtask timing information is completely lost may reflect the reason behind poor performance of static FC methods, which can be a critical disadvantage in task classification.
+
+![](images/e4ace94c7da5d4c584df69fe320c29cc2a46fe60afe2eebec1f2beeb1c406879.jpg)  
+Figure 4: Analysis of spatio-temporal attention for working memory task of the task decoding experiment. (a) Plot of average temporal attention matrix Z(k)time across subjects. (b) Proportion of statistically significant regions within the 7 ICNs from the spatial attention GLM.
+
+We interpret the result from the working memory task for spatio-temporal attention analysis, where the subtask consists of either performing an n-back memory task or rest. Our key expectation of the temporal attention analysis was that if STAGIN learns to accurately attend to temporal features of the dynamic FC graphs, then $Z _ { \mathrm { t i m e } }$ should represent which subtask the subject was upto. Surprisingly, it can be clearly seen that the Transformer encoder of STAGIN learns to attend to the timing of subtasks from Figure 4 (a), which demonstrates mean temporal attention $Z _ { \mathrm { t i m e } }$ across all subjects. Notice that no supervision is provided to the STAGIN model regarding the subtask timing during training.
+
+To analyze the spatially attended regions $z _ { \mathrm { s p a c e } }$ of STAGIN, we construct a GLM [14] to statistically evaluate how much each region is responsible for performing the subtasks. The parameter vectors $\beta _ { \mathrm { t a s k } } \in \mathbb { R } ^ { N }$ and $\beta _ { \mathrm { r e s t } } \in \mathbb { R } ^ { \widetilde { N } }$ are estimated with the sequence of spatial attention vectors and the subtask timing design matrix $M \in \{ 0 , 1 \} ^ { T \times 2 }$ by solving the following with least-squares estimation:
+
+$$
+\left[ z _ { \mathrm { s p a c e } } ( 0 ) , \cdot \cdot \cdot , z _ { \mathrm { s p a c e } } ( T ) \right] ^ { \top } = M [ \beta _ { \mathrm { t a s k } } , \beta _ { \mathrm { r e s t } } ] ^ { \top } + \epsilon ,
+$$
+
+where $\epsilon$ denotes residual error. The contrast of the estimated parameters $\hat { \beta } _ { \mathrm { t a s k } }$ and $\hat { \beta } _ { \mathrm { r e s t } }$ was set to $\pmb { c } = [ 1 , - 1 ]$ so the rejection of null hypothesis indicates $\hat { \beta } _ { \mathrm { t a s k } } [ \bar { i } ] > \hat { \beta } _ { \mathrm { r e s t } } [ i ]$ at the $i$ -th ROI. Multiple comparisons of the $N$ ROIs are family-wise error (FWE) corrected.
+
+Figure 4 (b) shows the proportion of statistically significant regions within the 7 ICNs for each layers. Interestingly, the layer 1 and 2 share a similar trend that the regions from SMN, visual network (VN), and salience/ventral attention network (SVN) are dominant. In contrast, layer 3 and 4 suggest a dominance of the regions from DMN and cognitive control network (CCN). We denote the layer 1 and 2 as the low-order layers (LoL) and the layer 3 and 4 as the high-order layers (HoL). The dominance of SMN and VN at LoL can be understood as the low-level sensorimotor function for perceiving the task is being processed within the short-range $1 \mathrm { - }$ or 2-hop connection of the networks. On the other hand, the dominance of DMN and CCN at HoL reflects the high-level cognitive integration for executing and controlling the given task being processed within the long-range 3- or 4-hop connection of the networks. Considering that the SVN is a network for integrating the low-level sensorimotor networks and the high-level executive networks to provide dynamic balancing between the two functions, the significant regions of SVN being present at both LoL and HoL is not surprising. Temporal and spatial attention plot of other six tasks are further provided in the Appendix Section D.2.
+
+# Acknowledgments and Disclosure of Funding
+
+This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2021M3E5D9025019, NRF-2020R1A2B5B03001980). This work was also supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government(MSIT) (No.2019-0-00075, Artificial Intelligence Graduate School Program(KAIST)) and the KAIST Key Research Institute (Interdisciplinary Research Group) Project.
+
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+        "type": "text",
+        "text": "Byung-Hoon Kim ∗ ",
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+        "type": "text",
+        "text": "Department of Psychiatry Institute of Behavioral Sciences in Medicine College of Medicine, Yonsei University egyptdj@yonsei.ac.kr ",
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+    {
+        "type": "text",
+        "text": "Jong Chul Ye   \nDepartment of Bio/Brain Engineering   \nKim Jaechul Graduate School of AI KAIST jong.ye@kaist.ac.kr ",
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+        "type": "text",
+        "text": "Jae-Jin Kim ",
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+        "type": "text",
+        "text": "Department of Psychiatry Institute of Behavioral Sciences in Medicine College of Medicine, Yonsei University jaejkim@yonsei.ac.kr ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Functional connectivity (FC) between regions of the brain can be assessed by the degree of temporal correlation measured with functional neuroimaging modalities. Based on the fact that these connectivities build a network, graph-based approaches for analyzing the brain connectome have provided insights into the functions of the human brain. The development of graph neural networks (GNNs) capable of learning representation from graph structured data has led to increased interest in learning the graph representation of the brain connectome. Although recent attempts to apply GNN to the FC network have shown promising results, there is still a common limitation that they usually do not incorporate the dynamic characteristics of the FC network which fluctuates over time. In addition, a few studies that have attempted to use dynamic FC as an input for the GNN reported a reduction in performance compared to static FC methods, and did not provide temporal explainability. Here, we propose STAGIN, a method for learning dynamic graph representation of the brain connectome with spatio-temporal attention. Specifically, a temporal sequence of brain graphs is input to the STAGIN to obtain the dynamic graph representation, while novel READOUT functions and the Transformer encoder provide spatial and temporal explainability with attention, respectively. Experiments on the HCP-Rest and the HCP-Task datasets demonstrate exceptional performance of our proposed method. Analysis of the spatio-temporal attention also provide concurrent interpretation with the neuroscientific knowledge, which further validates our method. Code is available at https://github.com/egyptdj/stagin ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Neuroimaging modalities provide measurements of brain activity by capturing the signals of neural activity. Functional magnetic resonance imaging (fMRI) is a non-invasive imaging method that measures the blood-oxygen level dependence (BOLD) in order to estimate the neural activity of the whole brain over time [20]. Functional connectivity (FC) is defined as the degree of temporal correlation between regions of the brain. Based on the fact that these connectivities form networks that change over time, graph-based network analysis of brain connectome has been one of the key approaches to understanding how the brain works [7, 4, 43]. ",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Graph neural networks (GNNs) are a type of deep neural networks that have recently been successful in learning the representation of graph-structured data [50]. The graph-structured nature of the brain has led to an increased interest in learning the reperesentation of the brain FC network with the GNNs. Learning the representation of the brain connectome can be linked to decoding trait or state from human brain signal measurements. Accordingly, the current trend in studies attempting to apply GNN to the brain connectome is to input the FC graph from either resting-state [25, 26, 2, 33, 23, 48, 49] or task fMRI data [29, 30, 32] and predict a particular phenotype of the subjects, such as gender [26, 2, 23, 22] or presence of a specific disease [26, 33, 29, 30, 32, 22]. While these studies have shown potential strengths and opportunities for learning the network representation of the brain, they also suggest limitations of current GNN-based methods. ",
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+    {
+        "type": "text",
+        "text": "One of the most common limitations with previous GNN-based FC network analysis methods is that most of them fail to take advantage of the dynamic properties of the FC network, which fluctuates over time. Incorporating the dynamic features of the FC network into the neuroimaging analysis has been an important direction in the field of functional neuroimaging [21, 37]. A work by [15] tried to address this issue by using the Spatial Temporal Graph Convolutional Network (ST-GCN) [54] model to incorporate dynamic features of the FC network. However, [15] reported lower accuracy than other non-dynamic GNN-based FC methods [23, 2] in the gender classification experiment, leaving a question about the effectiveness of the dynamic FC method. In addition, another limitation of the method is that no temporal explainability is provided from the model. This is a major drawback considering that the goal of applying GNNs to functional neuroimaging methods is not only to achieve high classification accuracy, but also to uncover the functional basis of the brain [23, 32]. Another recent work by [3], using GraphNets [5] and DiffPool [58] for the dynamic FC analysis, also suffers from the same limitations in terms of poor classification accuracy and lack of temporal explainability. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, we propose Spatio-Temporal Attention Graph Isomorphism Network (STAGIN) for learning the dynamic graph representation of the brain connectome with spatio-temporal attention. The proposed method exploits the temporal features of the dynamic FC network graphs to improve the classification accuracy of the model. In particular, we address the issue that the node features of the input dynamic graph should contain temporal information and concatenate encoded timestamp with the node features (Section 4.1). In addition, the proposed method includes novel attention-based READOUT modules (Section 4.2) and the Transformer encoder [46] (Section 4.3) in order to further improve the classification performance and provide spatial-temporal explainability at the same time. STAGIN achieves state-of-the-art performance with the Human Connectome Project (HCP) dataset [45] in gender classification for resting-state fMRI and task decoding for task fMRI. We inherit $\\mathbf { k }$ -means clustering analysis of the resting-state dynamic FC [1] and general linear model (GLM) statistical mapping of task fMRI [14] for interpreting the spatio-temporal attention learned from STAGIN, which are widely accepted analysis methods for the fMRI data. The interpretation of the learned spatio-temporal attention replicates neuroscientific findings from previous large-scale fMRI studies in both resting-state and task fMRI, which further validates our proposed method. ",
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+    {
+        "type": "text",
+        "text": "Our work holds potential societal impact in that brain decoding methods can be linked to finding neural biomarkers of important phenotypes or diseases. However, potential negative impact related to privacy concerns that arise from abuse or misuse of accurate decoding methods should also be noted. Although our method is yet behind the decoding capability that can be abused or misused, our research cannot still be free from these ethical considerations. ",
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+    {
+        "type": "text",
+        "text": "2 Related works ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "2.1 Graph Neural Network on Dynamic Graphs ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Many networks that arise around us are inherently dynamic, with changes in the existence of nodes and edges over time. Learning the representation of dynamic graphs has piqued the interest of researchers and has led to development of methods that can embed dynamic graphs using their time information [35]. Methods that incorporate attention for learning the representation of dynamic graphs have also been proposed [51, 40]. However, it is not easy to apply these techniques directly to the dynamic brain graphs because of the different inherent properties of the dynamic brain graphs that do not include any addition or deletion of nodes and are sampled uniformly over time. Nonetheless, our work is inspired by these earlier studies, particularly for the encoding of temporal information and their concatenation to the node features, as proposed in Section 4.1 [51, 40]. ",
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+        "type": "text",
+        "text": "2.2 Attention in Graph Neural Networks ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Bringing attention to the GNNs is a topic that is being actively studied in the field of geometric deep learning [27]. One of the most successful uses of attention is to compute the attention at edges of the graph and scale the importance of the links when the features of the neighborhood node are aggregated [47, 6], often providing performance gain in learning the representation of input graphs. Another stream of applying attention to the GNNs comes with the motivation to define a pooling function on the graph domain. Since it is not straightforward to decide on what basis the coarsening should be carried out for graph structured data, works such as [16, 28, 38] have addressed this problem by selecting the nodes with top scores computed from projecting the node feature vectors into a learnable parameter vector, or from a GNN layer aggregated local graph features. Although the motivation may have been different, these graph pooling methods are closely related to the spatial attention modules that we propose in Section 4.2 in that they exploit learned relative scores across the vertices of the graph. While some works have already been aware that the appropriate use of node-wise attention can improve performance of downstream tasks [55, 11], we note that previous methods tend to score attention based on randomly initialized parameters or local graph structures which may be suboptimal for graph classification tasks that require taking the whole graph feature into account. ",
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+        "type": "text",
+        "text": "3 Theory ",
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+    {
+        "type": "text",
+        "text": "3.1 Problem definition ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The goal of our study is to train a neural network ",
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+    {
+        "type": "equation",
+        "img_path": "images/69ec4ec1c9357bf34f6cab8ed201706915818e03bd34d065364adeaf31e5150c.jpg",
+        "text": "$$\nf : G _ { \\mathrm { d y n } } \\to h _ { G _ { \\mathrm { d y n } } } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $G _ { \\mathrm { d y n } } = ( G ( 1 ) , . . . , G ( T ) )$ is the sequence of brain graphs with $T$ timepoints and $\\pmb { h } _ { G _ { \\mathrm { d y n } } } \\in \\mathbb { R } ^ { D }$ is the vector representation of the dynamic graph $G ( t )$ with length $D$ . The graph $G ( t ) = ( V ( t ) , E ( t ) )$ at time $t$ is a pair of vertex set $V ( t ) = \\{ { \\pmb x } _ { 1 } ( t ) , . . . , { \\pmb x } _ { N } ( t ) \\}$ of $N$ nodes and edge set $E ( t ) \\ =$ $\\left\\{ \\{ \\pmb { x } _ { i } ( t ) , \\pmb { x } _ { j } ( t ) \\} \\mid j \\in \\mathcal { N } ( i ) , i \\in \\{ 1 , . . . N \\} \\right\\}$ where $\\mathcal { N } ( i )$ denotes the neighborhood of the vertex $i$ . If $f$ learns to extract a disentangled representation of the dynamic brain graph $G _ { \\mathrm { d y n } }$ , then the classification of a certain phenotypic characteristic (e.g. gender) from $h _ { G _ { \\mathrm { d y n } } }$ can be performed with a linear mapping as a downstream task. Another important consideration in this work is to ensure the explainability of the model $f$ , being able to inform us which part of the brain at which timepoint was considered important when extracting the meaningful representation $h _ { G _ { \\mathrm { d y n } } }$ . Specifically, we formulate $f = q \\circ g$ as a composition of the GNN $g$ and the Transformer encoder $q$ , where $g$ outputs the set of graph representations $\\mathbf { \\mathcal { h } } _ { G ( t ) }$ from each timepoint and $q$ exploits self-attention to integrate $\\mathbf { \\mathcal { h } } _ { G ( t ) }$ into the final representation $h _ { G _ { \\mathrm { d y n } } }$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/07837417a25c1de0e2077fd7cf31373253fcbed256b53b41e2cc8e494c06abb3.jpg",
+        "text": "$$\n\\begin{array} { r l } & { g : G _ { \\mathrm { d y n } } \\to ( h _ { G ( 1 ) } , . . . , h _ { G ( T ) } ) , } \\\\ & { } \\\\ & { q : ( { h _ { G ( 1 ) } } , . . . , { h _ { G ( T ) } } ) \\to h _ { G _ { \\mathrm { d y n } } } . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "We will omit timepoint notation $( t )$ for brevity, whenever it is not of contextual importance. ",
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+    {
+        "type": "text",
+        "text": "3.2 Graph Isomorphism Network ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The GNNs are generally composed of functions that (i) integrate the node features from its neighbors, and (ii) embed the integrated information with a nonlinear transformation to obtain the next layer node features. These functions are called AGGREGATE, and COMBINE functions, respectively, and the choice of these functions define many variants of the GNN, ",
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+        "type": "equation",
+        "img_path": "images/9b15d4b75d18771d12dbfc39a23a63fbc638eee53cec12d35a6edeb8ce00db9a.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\pmb { a } _ { v } ^ { ( k ) } = \\mathtt { A G G R E G A T E } ^ { ( k ) } \\Big ( \\Big \\{ \\pmb { h } _ { u } ^ { ( k - 1 ) } : u \\in \\mathcal { N } ( v ) \\Big \\} \\Big ) , } \\\\ & { \\pmb { h } _ { v } ^ { ( k ) } = \\mathtt { C O M B I N E } ^ { ( k ) } \\Big ( \\pmb { h } _ { v } ^ { ( k - 1 ) } , \\pmb { a } _ { v } ^ { ( k ) } \\Big ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { h } _ { v } ^ { ( k ) }$ denotes the feature vector of node $v$ at layer $k$ and ${ h } _ { v } ^ { ( 0 ) } : = \\pmb { x } _ { v }$ ",
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+    {
+        "type": "text",
+        "text": "The Graph Isomorphism Network (GIN) is a variant of the GNN suitable for graph classification tasks, which is known to be as powerful as the WL-test under certain assumptions of injectivity [52]. The GIN typically defines sum as the AGGREGATE and a multi-layer perceptron (MLP) with two layers as the COMBINE updating the node representation $\\boldsymbol { h } _ { v } ^ { ( k ) }$ at layer $k$ [52] by : ",
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+        "type": "equation",
+        "img_path": "images/a67e5b6a8f089ed68f902b7a658840eeeeded73ea4ea3ac0026bf07a4603431a.jpg",
+        "text": "$$\n\\pmb { h } _ { v } ^ { ( k ) } = \\mathtt { M L P } ^ { ( k ) } \\Big ( ( 1 + \\epsilon ^ { ( k ) } ) \\cdot \\pmb { h } _ { v } ^ { ( k - 1 ) } + \\sum _ { u \\in \\mathcal { N } ( v ) } \\pmb { h } _ { u } ^ { ( k - 1 ) } \\Big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\epsilon$ is a learnable parameter initialized with zero. Equation (5) can be easily reformulated into the matrix form [23] by: ",
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+    {
+        "type": "equation",
+        "img_path": "images/78eececb26871d844ef018197beace897854c7570b4174b19b0bfbf5179e192a.jpg",
+        "text": "$$\n\\pmb { H } ^ { ( k ) } = \\sigma \\left( ( \\epsilon ^ { ( k ) } \\cdot \\pmb { I } + \\pmb { A } ) \\pmb { H } ^ { ( k - 1 ) } \\pmb { W } ^ { ( k ) } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/07a629c5cee9bb2c51d8bfd0d56f058f8321b1263cbb07b642225f5a97dda668.jpg",
+        "text": "$$\n\\pmb { H } ^ { ( k ) } = \\left[ \\pmb { h } _ { 1 } ^ { ( k ) } , \\cdots , \\pmb { h } _ { N } ^ { ( k ) } \\right] \\in \\mathbb { R } ^ { D \\times N }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "is the stack of node feature vectors, $\\pmb { I }$ is the identity matrix, $\\pmb { A }$ is the adjacency matrix between the node features, $W$ is the network weights of the MLP, and $\\sigma$ is the nonlinearity function. ",
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+    {
+        "type": "text",
+        "text": "The READOUT function takes the updated node features $\\boldsymbol { h } _ { v } ^ { ( k ) }$ to compute the representation of the whole graph: ",
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+    {
+        "type": "equation",
+        "img_path": "images/06adb6b8cf5b69e27bda1a60e99b6e7cd0477aa030fe038a7a912c4c94ffb2aa.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { h } _ { G } ^ { ( k ) } = \\mathtt { R E A D O U T } \\Big ( \\{ \\pmb { h } _ { v } ^ { ( k ) } \\ | \\ v \\in G \\} \\Big ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In general, the READOUT function is defined simply as computing the sum or average of the input node features. This is equivalent to multiplication with the length $N$ pooling vectors $\\phi _ { \\mathrm { s u m } } ^ { \\mp } = [ 1 , \\stackrel { \\cdot } { \\dots } , 1 ]$ or $\\phi _ { \\mathrm { m e a n } } ^ { \\top } = [ 1 / N , . . . , 1 / N ]$ for the matrix form: ",
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+    {
+        "type": "equation",
+        "img_path": "images/512c7b33ac864a2b7163ab0e1482d745548c0d2b8189ee4b6205785b66533932.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { h } _ { G } ^ { ( k ) } = \\pmb { H } ^ { ( k ) } \\phi _ { \\mathrm { m e a n } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.3 Encoder-decoder understanding of GNNs ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Although formulating the GIN (5) as a combination of AGGREGATE and COMBINE function might not suggest its close relationship with convolutional neural networks (CNNs) at first glance, previous works by [23, 8] show that the matrix formulation of the GIN operation (6) can be thought of a CNN layer with shift operation of the convolution as the adjacency matrix $\\pmb { A }$ . We extend the understanding of encoder-decoder CNN as a framelet expansion [57, 56] to the GIN to formulate node feature vectors $H ^ { ( k ) }$ at layer $k$ with respect to the input node feature $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/4f10f8bcc92fc95155ed61a2227d66aad0c2d33c69c376c8cf1146679fd51f8d.jpg",
+        "text": "$$\n\\operatorname { V e c } \\left( \\pmb { H } ^ { ( k ) } \\right) = \\pmb { \\Sigma } ^ { ( k ) } \\pmb { E } ^ { ( k ) \\top } \\cdot \\cdot \\cdot \\pmb { \\Sigma } ^ { ( 1 ) } \\pmb { E } ^ { ( 1 ) \\top } \\pmb { x } , \\quad \\mathrm { w h e r e } \\quad \\pmb { x } : = \\operatorname { V e c } \\left( [ \\pmb { x } _ { 1 } , \\cdots , \\pmb { x } _ { N } ] \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\mathrm { V e c } ( \\cdot )$ denotes the vectorization operation, and the $k$ -th layer encoder matrix $\\pmb { { \\cal E } } ^ { ( k ) }$ is defined as ",
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+    {
+        "type": "equation",
+        "img_path": "images/f25304bbbe5cc0517ac5ecd616239a30292abd399028a1e51eb1a68614e95938.jpg",
+        "text": "$$\n\\pmb { E } ^ { ( k ) } = \\pmb { W } ^ { ( k ) } \\otimes ( \\epsilon ^ { ( k ) } \\cdot \\pmb { I } + \\pmb { A } ^ { T } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            650,
+            609,
+            669
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\otimes$ refers to the Kronecker product, and $\\pmb { \\Sigma } ^ { ( k ) }$ is the diagonal matrix with values 1 or 0 depending on the activation pattern of the nonlinearity. Now, $\\phi _ { \\mathrm { { m e a n } } }$ of equation (8) can be thought as the decoder at the $k$ -th layer which yields the whole graph feature vector from the encoded node feature vectors. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. The READOUT function $\\phi _ { m e a n }$ in (8) generates a decoder with fixed constant bases. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proof. From the READOUT function (8), we have ",
+        "bbox": [
+            174,
+            743,
+            511,
+            758
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/74fc7e1002a2cc017f2d5adf685512944c0f5e685c6bb3d096f44a7e477ad845.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\boldsymbol { h } _ { G } ^ { ( k ) } = \\mathrm { V e c } \\left( \\boldsymbol { h } _ { G } ^ { ( k ) } \\right) = \\mathrm { V e c } \\left( \\boldsymbol { H } ^ { ( k ) } \\phi _ { \\mathrm { m e a n } } \\right) = \\left( \\phi _ { \\mathrm { m e a n } } ^ { T } \\otimes \\boldsymbol { I } \\right) \\mathrm { V e c } \\left( \\boldsymbol { H } ^ { ( k ) } \\right) } \\\\ & { \\qquad = \\left( \\phi _ { \\mathrm { m e a n } } ^ { T } \\otimes \\boldsymbol { I } \\right) \\boldsymbol { \\Sigma } ^ { ( k ) } \\boldsymbol { E } ^ { ( k ) \\top } \\cdot \\cdot \\cdot \\boldsymbol { \\Sigma } ^ { ( 1 ) } \\boldsymbol { E } ^ { ( 1 ) \\top } \\boldsymbol { x } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            761,
+            727,
+            810
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Now let $\\mathbf { } _ { b _ { i } }$ and $\\tilde { \\mathbf { b } } _ { i }$ denote the $i$ -th column of the encoder matrix ${ \\cal E } ^ { ( 1 ) } \\Sigma ^ { ( 1 ) } \\cdot \\cdot \\cdot { \\cal E } ^ { ( k ) } \\Sigma ^ { ( k ) }$ and the decoder matrix $\\left( \\phi _ { \\mathrm { m e a n } } ^ { T } \\otimes I \\right)$ , respectively. Then, it is straight to obtain the following representation: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1ba539b086b346a81a0e3b5c41a76cf25933e753dafd905b84384cee2e99a7aa.jpg",
+        "text": "$$\n{ h } _ { G } ^ { ( k ) } = \\sum _ { i } \\langle b _ { i } , { x } \\rangle \\tilde { b } _ { i }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            844,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Therefore, we can see that although the encoder basis $b _ { i }$ is a function of $_ { \\textbf { \\em x } }$ , the decoder basis $\\tilde { \\boldsymbol { b } } _ { i }$ is a constant. □ ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/742f062d71231a50f5c4e1d9e80c0b555365b6f10c527c39295fb635063d5273.jpg",
+        "image_caption": [
+            "Figure 1: Schematic illustration of the proposed method. (a) Overall framework of the STAGIN. A sequence of dynamic graph is first input to the GIN followed by GARO or SERO which produces a sequence of spatially attended graph representation vectors $\\tilde { h } _ { G ( t ) }$ . Temporal attention is computed over $\\tilde { h } _ { G ( t ) }$ and the temporally attended graph representations are averaged to generate the final representation $\\tilde { h } _ { G _ { \\mathrm { d y n } } }$ . (b) Attention-based READOUT modules. Both GARO and SERO compute spatial attention $z _ { \\mathrm { s p a c e } }$ with global average-pooled graph feature $_ { h _ { G } }$ as prior. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We address the issue that the decoder being a constant function can restrict the expressivity of the neural network, and explore adaptive READOUT functions with attention in Section 4.2. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 STAGIN: Spatio-Temporal Attention Graph Isomorphism Network ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            452,
+            767,
+            470
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we discuss the details of our main contribution. Specifically, we propose STAGIN with two novel attention-based READOUT modules for learning the dynamic graph representation of the brain connectome (Figure 1). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 Dynamic graph definition ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            541,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The sequence of input dynamic FC graphs is constructed from 4D fMRI data with 3D voxels across time. The ROI-timeseries matrix $\\bar { P ^ { \\prime } } \\in \\bar { \\mathbb { R } } ^ { N \\times T _ { \\operatorname* { m a x } } }$ is extracted by taking the mean values within a predefined 3D atlas which consists of $N$ ROIs at each timepoint. Values of each ROI are standardized across time. Constructing dynamic FC matrix follows the sliding-window approach, where the temporal window of length $\\Gamma$ is shifted across time with stride $S$ to generate $T = \\lfloor T _ { \\mathrm { m a x } } - \\Gamma / S \\rfloor$ windowed matrices $\\bar { P } ( t ) \\in \\mathbb { R } ^ { N \\times \\Gamma }$ (Figure 2 (a)). The FC at time $t$ is defined as the correlation coefficient matrix $R ( t )$ of the windowed timeseries between ${ \\bar { p } } _ { i } ( t )$ and $\\bar { p } _ { j } ( t )$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c15e6938c88b3380589c147829a95b458c3e52e7c9e59ba674afb06900d10c9a.jpg",
+        "text": "$$\nR _ { i j } ( t ) = \\frac { \\mathrm { C o v } ( \\bar { p } _ { i } ( t ) , \\bar { p } _ { j } ( t ) ) } { \\sigma _ { \\bar { p } _ { i } } ( t ) \\sigma _ { \\bar { p } _ { j } } ( t ) } \\in \\mathbb { R } ^ { N \\times N } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            669,
+            627,
+            704
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the subscript $i$ and $j$ are the row and column indices of $\\bar { P } ( t )$ , Cov denotes the cross covariance, and $\\sigma _ { p }$ denotes the standard deviation of $\\pmb { p }$ . The final binary adjacency matrix $A ( t ) \\in \\{ 0 , 1 \\} ^ { N \\times N }$ is obtained from the FC matrix $\\mathbf { } R ( t )$ by thresholding the top 30-percentile values of the correlation matrix as connected, and otherwise unconnected following [23]. Other thresholds for binarizing the correlation matrix are also experimented and the results are provided in the Appendix Section C.2. ",
+        "bbox": [
+            173,
+            709,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Unlike the adjacency matrix $\\mathbf { } A ( t )$ , conventional definition of node feature vectors ${ \\bf \\mathit { x } } _ { v } ( t )$ at node index $v$ as coordinates [29], mean-activation [29, 15], or one-hot encoding [23], do not change over $t$ , disregarding any temporal variation. To address this issue, we concatenate encoded timestamp $\\eta ( t ) \\in \\mathbb { R } ^ { D }$ to the spatial one-hot encoding $e _ { v }$ , followed by linear mapping with a learnable parameter matrix $W \\in \\mathbb { R } ^ { D \\times ( N + D ) }$ to define the input node feature, ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d9ea0a50a3032eef9bd1dc72e1dbab5b2e5f1416a5a59508425ead4d3f6b6095.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { x } _ { v } ( t ) = \\pmb { W } [ \\pmb { e } _ { v } | | \\eta ( t ) ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            424,
+            861,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here, the learnable timestamp encoder $\\eta$ is a Gated Recurrent Unit (GRU) [9] which takes ROItimeseries upto the endpoint of the sliding-window as the input. Both the vertex set $V ( t )$ and the ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/290dcab37d7782ea35b2d29fd27ceb77cd28469f9fdf7c1690b968d398ee15f2.jpg",
+        "image_caption": [
+            "Figure 2: Defining the dynamic graph. (a) Scheme of extracting dynamic graph from ROI-timeseries matrix $_ { r }$ . (b) Example of a constructed dynamic graph. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            88,
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+            251
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "edge set $E ( t )$ of graph $G ( t )$ now incorporates temporal information at time t. See Figure 2 for an illustration of the dynamic graph definition. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 Spatial attention with attention-based READOUT ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            361,
+            560,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As suggested from Proposition 1, conventional READOUT function of GNN can be thought of as a fixed decoder that decodes whole-graph feature from the node features with no learnable parameters. We address this issue by incorporating attention to the READOUT function, which the attention here refers to the scaling coefficient across the nodes learned by the model. Specifically, the spatial attention vector $z _ { \\mathrm { s p a c e } } ( t ) \\in [ 0 , 1 ] ^ { N }$ is computed by taking the $\\pmb { H }$ as a prior: ",
+        "bbox": [
+            173,
+            386,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/686a01338aa55df82ee20d9b0df33c03fe2c80565ebe71bfc44ee8cb1c7ccbdf.jpg",
+        "text": "$$\n\\begin{array} { c } { { z _ { \\mathrm { s p a c e } } = s ( { \\cal H } ) , } } \\\\ { { \\tilde { h } _ { \\cal G } = { \\cal H } z _ { \\mathrm { s p a c e } } , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            439,
+            462,
+            555,
+            503
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $s : \\mathbb { R } ^ { D \\times N } \\to [ 0 , 1 ] ^ { N }$ is the attention function and $\\tilde { h } _ { G }$ denotes spatially attended graph representation $h _ { G }$ . We propose two types of attention function $s ( \\cdot )$ for the attention-based READOUT, named Graph-Attention READOUT (GARO) and Squeeze-Excitation READOUT (SERO) inspired by the attention mechanisms of [46] and [19], respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2.1 GARO: Graph-Attention READOUT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            580,
+            485,
+            597
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The GARO follows key-query embedding based attention of the Transformer [46]. However, the key embedding is computed from the matrix of node features $\\pmb { H }$ , while the query embedding is computed from the vector of unattended graph representation $H \\phi _ { \\mathrm { { m e a n } } }$ : ",
+        "bbox": [
+            174,
+            604,
+            825,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb1ac97f22e4ac3df644614a2ae85f6ee34fcd5e045334c3089a275ba237a626.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\kappa = W _ { \\mathrm { k e y } } H , } } \\\\ { { q = W _ { \\mathrm { q u e r y } } H \\phi _ { \\mathrm { m e a n } } , } } \\\\ { { { z } _ { \\mathrm { s p a c e } } = \\mathrm { s i g m o i d } \\Big ( \\frac { q ^ { \\top } K } { \\sqrt { D } } \\Big ) , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            652,
+            589,
+            728
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $W _ { \\mathrm { k e y } } \\in \\mathbb { R } ^ { D \\times D }$ , $W _ { \\mathrm { q u e r y } } \\in \\mathbb { R } ^ { D \\times D }$ are learnable key-query parameter matrices, $\\pmb { K } \\in \\mathbb { R } ^ { D \\times N }$ is the embedded key matrix, and $\\pmb q \\in \\mathbb { R } ^ { D }$ is the embedded query vector. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2.2 SERO: Squeeze-Excitation READOUT",
+        "text_level": 1,
+        "bbox": [
+            173,
+            777,
+            495,
+            794
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The SERO follows MLP based attention of the Squeeze-and-Excitation Networks [19]. However, attention from the squeezed graph representation does not scale the channel dimension, but the node dimension in SERO: ",
+        "bbox": [
+            173,
+            801,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/98decfee1c6d2d3ab0a2ab49c9550a5b72dbc9a0c5161b8187cc1a3ed3467c36.jpg",
+        "text": "$$\nz _ { \\mathrm { s p a c e } } = \\mathrm { s i g m o i d } \\Big ( W _ { 2 } \\sigma ( W _ { 1 } H \\phi _ { \\mathrm { m e a n } } ) \\Big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            848,
+            630,
+            876
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma$ is the nonlinearity function and $W _ { 1 } \\in \\mathbb { R } ^ { D \\times D }$ , $W _ { 2 } \\in \\mathbb { R } ^ { N \\times D }$ are learnable parameter matrices. This type of spatial dimension squeeze-excitation module has been shown to improve ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "performance of the CNN models [41], but was not easily applicable to general graphs which may vary in number of nodes for each graph. We exploit the fact that the brain graphs have fixed number of nodes $N$ across participants based on the chosen atlas. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2.3 Orthogonal regularization ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            147,
+            408,
+            162
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "If we take a closer look at (8) and (12), computation of graph feature vector $_ { h _ { G } }$ from the node feature matrix $\\pmb { H }$ can also be viewed as reconstructing signal $h _ { G }$ from the basis frames $\\pmb { H }$ with vectors $\\phi _ { \\mathrm { { m e a n } } }$ and $z _ { \\mathrm { s p a c e } }$ , respectively. While $z _ { \\mathrm { s p a c e } }$ provides further expressivity of the model with adaptive coefficients when compared to $\\phi _ { \\mathrm { { m e a n } } }$ , we find it desirable to encourage the orthogonality of $\\pmb { H }$ as elaborated in the Appendix Section A. The orthogonal regularization $\\mathcal { L } _ { \\mathrm { o r t h o } }$ is defined as: ",
+        "bbox": [
+            173,
+            171,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e6e715eddd71b75e34996741778ecf281b1e09d33e57ebddb9ff75e91ca7a58a.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { o r t h o } } = \\left\\| 1 / m \\cdot H ^ { \\top } H - I \\right\\| _ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            247,
+            604,
+            268
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $m = \\operatorname* { m a x } ( H ^ { \\top } H )$ . The scaling term $1 / m$ ensures the columns of the matrix $\\pmb { H }$ become orthogonal to each other with the same length, while not restricting the specific length that the column vectors should follow. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 Temporal attention with Transformer encoder ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            334,
+            534,
+            351
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For attention across time, we employ a single-headed Transformer encoder [46] upon the sequence of graph features $( \\tilde { h } _ { G ( 1 ) } , . . . , \\tilde { h } _ { G ( T ) } )$ . The temporal attention can be measured by the self-attention graph representation weights $Z _ { \\mathrm { t i m e } } \\in [ 0 , 1 ] ^ { T \\times T }$ $h _ { G _ { \\mathrm { d y n } } } ^ { ( k ) }$ after the softmax function of the Transformer encoder. Per-layer dynamic is computed by summing the temporally attended feature output from the Transformer encoder across time at each layers, where the final representation: ",
+        "bbox": [
+            173,
+            359,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bed30aa65666071961c56b05694e053ff025e0a3dd95e860f0768a08e7a49058.jpg",
+        "text": "$$\n{ \\pmb h } _ { G _ { \\mathrm { d y n } } } = \\mathrm { c o n c a t e n a t e } ( \\{ { \\pmb h } _ { G _ { \\mathrm { d y n } } } ^ { ( k ) } ~ | ~ k \\in \\{ 1 , . . . , K \\} \\} )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            448,
+            658,
+            472
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "is the concatenation of dynamic graph representation of all $K$ layers following [53]. ",
+        "bbox": [
+            173,
+            477,
+            722,
+            493
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Experiment ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            511,
+            303,
+            529
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 Dataset ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            542,
+            266,
+            558
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Publicly available2 fMRI data from the HCP S1200 release [45] was used for our experiments. The data was collected from voluntary participants with informed consent and was fully anonymized. We constructed two datasets, the HCP-Rest and the HCP-Task, depending on whether the subject was resting or performing specific tasks during the acquisition of the image. The HCP-Rest dataset consisted of pre-processed and ICA denoised resting-state fMRI data [17], which the subjects were instructed to rest for 15 minutes during the data acquisition. We used first run data of the four sessions, and excluded data with short acquisition time with $T _ { \\mathrm { m a x } } < 1 2 0 0$ . There were 1093 images finally included in the dataset, which consisted of 594 female and 499 male subjects. The gender of each subject served as the labels of the HCP-Rest dataset letting the number of classes $C = 2$ . The HCP-Task consisted of pre-processed task fMRI data [17], which the subjects were instructed to perform specific tasks during data acquisition. For example in the \"Motor\" task fMRI, participants were told to perform one of the subtasks during the acquisition to make motor movements on one’s left hand, left foot, right hand, right foot, or tongue. There were seven types of tasks including working memory, social, relational, motor, language, gambling, and emotion. After excluding the fMRI data with short acquisition time, there were 7450 images included in the dataset. The task type during the data acquisition served as the labels of the HCP-Task dataset, letting $C = 7$ . A more detailed description of the experiment datasets with a note on the twin subjects of HCP can be found in the Appendix Section B. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 Experimental settings ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Experiments were performed on a workstation with two NVIDIA GeForce GTX 1080 Ti GPUs. The STAGIN model $f$ is trained end-to-end in a supervised manner with the loss $\\mathcal { L } = \\mathcal { L } _ { \\mathrm { x e n t } } + \\lambda \\cdot \\mathcal { L } _ { \\mathrm { o r t h o } }$ where $\\mathcal { L } _ { \\mathrm { x e n t } }$ is the cross entropy loss and $\\lambda$ is the scaling coefficient of the orthogonal regularization. We set the number of layers $K = 4$ , embedding dimension $D = 1 2 8$ , window length $\\Gamma = 5 0$ , window stride $S = 3$ , and regularization coefficient $\\lambda \\stackrel { - } { = } 1 . 0 \\times 1 0 ^ { - 5 }$ . The window length and stride correspond to capturing the FC within 36 seconds every 2.16 seconds, which follows the standard setting of the sliding-window dFC analyses [59, 37]. Dropout rate 0.5 is applied to the final dynamic graph representation $h _ { G _ { \\mathrm { d y n } } }$ , and rate 0.1 is applied to the attention vectors $z _ { \\mathrm { s p a c e } }$ and $z _ { \\mathrm { t i m e } }$ during training. For nonlinearity $\\sigma$ in (6) and (14), GELU [18] is used instead of ReLU with batch normalization before each $\\sigma$ . One-cycle learning rate policy is employed, which the learning rate is gradually increased from 0.0005 to 0.001 during the early $20 \\%$ of the training, and gradually decreased to $5 . 0 \\times 1 0 ^ { - 7 }$ afterwise. Thirty training epochs were run for the HCP-Rest dataset with minibatch size 3, while ten epochs were run with minibatch size 16 for the HCP-Task dataset. We performed 5-fold stratified cross-validation of the dynamic graphs from the dataset, and report mean and standard deviation across the folds. To extract the ROI-timeseries, the Schaefer atlas [42] with 400 regions $N = 4 0 0$ ) labelled with 7 intrinsic connectivity networks (ICNs) was used. The time dimension of ROI-timeseries matrix $_ { r }$ was randomly sliced with a fixed length (600 for HCP-Rest, 150 for HCP-Task) at each steps during training for (i) relieving computational overload, (ii) stochastic augmentation of the training dataset, (iii) mitigating unwanted memorization of the specific timing of subtask onset, and (iv) matching the number of timepoints $T$ across different task labels for the HCP-Task dataset. Unsliced full matrix $_ { r }$ was used for inference at test time. The end-to-end inference from the construction of the dynamic graph to the acquisition of the final prediction required 1.68 seconds per sample with given experimental settings. ",
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+        "table_caption": [
+            "Table 1: Comparative study on HCP-Rest and HCP-Task dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">HCP-Rest</td><td>HCP-Task</td><td rowspan=\"2\">Type of FC</td><td rowspan=\"2\"># Params</td></tr><tr><td>Accuracy (%)</td><td>AUROC</td><td>Accuracy (%)</td></tr><tr><td>STAGIN-SERO</td><td>88.20 ± 1.33</td><td>0.9296 ± 0.0187</td><td>99.19 ± 0.20</td><td>Dynamic</td><td>1,209k</td></tr><tr><td>STAGIN-GARO</td><td>87.01 ± 3.00</td><td>0.9151 ± 0.0258</td><td>99.02 ± 0.17</td><td>Dynamic</td><td>1,068k</td></tr><tr><td>ST-GCN [15]</td><td>76.95 ± 3.00</td><td>0.8545 ± 0.0316</td><td>98.92 ± 0.27</td><td>Dynamic</td><td>355k</td></tr><tr><td>MS-G3D[10]</td><td>79.16 ± 2.53</td><td>0.8912 ± 0.0329</td><td>1</td><td>Dynamic</td><td>3,045k</td></tr><tr><td>BAnD++ [36]</td><td></td><td></td><td>97.20 ± 0.57</td><td>None</td><td>2,010k</td></tr><tr><td>BAnD [36]</td><td></td><td></td><td>95.10 ± 0.62</td><td>None</td><td>2,010k</td></tr><tr><td>r-BAnD</td><td></td><td></td><td>98.90 ± 0.27</td><td>Dynamic</td><td>664k</td></tr><tr><td>GIN [23]</td><td>81.34 ± 2.40</td><td>0.8955 ± 0.0237</td><td>93.87 ± 0.66</td><td>Static</td><td>169k</td></tr><tr><td>GCN [24]</td><td>80.79 ± 2.00</td><td>0.8741 ± 0.0174</td><td>45.07 ± 1.63</td><td>Static</td><td>101k</td></tr><tr><td>GraphSAGE[31]</td><td>75.48 ± 1.97</td><td>0.8237 ± 0.0228</td><td>54.52 ± 0.97</td><td>Static</td><td>202k</td></tr><tr><td>ChebGCN[2]</td><td>77.76 ± 2.09</td><td>0.8582 ± 0.0233</td><td>73.06 ± 0.68</td><td>Static</td><td>704k</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5.3 HCP-Rest: Gender classification ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first validate our proposed method by gender classification on the HCP-Rest dataset. The two proposed methods, named STAGIN-GARO and STAGIN-SERO based on the type of the spatial attention module, resulted in $8 7 . 0 1 \\%$ and $8 8 . 2 0 \\%$ mean accuracy on the 5-fold cross validation, respectively (Table 1). The mean area under receiver operator characteristic curve (AUROC) were 0.9151 and 0.9296. Classification performance of STAGIN is compared with other GNN methods for reprensentation learning of dynamic/static FC network, including ST-GCN [15], MS-G3D [10], GIN [23], GCN [24], GraphSAGE [31], and ChebGCN [2]. We used the code by the authors of $[ 1 5 ]$ and $[ 1 0 ]$ but modified the cross validation scheme to avoid early stopping based on the test dataset for fair comparison. It can be seen from Table 1 that our proposed method outperforms other GNN based methods. The results of the ablation study are shown in Table 3 in the Appendix. ",
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+        "type": "text",
+        "text": "We use STAGIN-SERO, which showed the best accuracy, for analyzing temporal and spatial attention of the dynamic FC networks. We define the temporal attention vector $\\bar { \\boldsymbol { z } } _ { \\mathrm { t i m e } } ^ { ( k ) } \\in [ 0 , \\bar { 1 } ] ^ { T }$ at layer $k$ as the average of row elements in the self-attention weight matrix $\\begin{array} { r } { z _ { \\mathrm { t i m e } } ^ { ( k ) } [ j ] = \\frac { 1 } { T } \\sum _ { i = 1 } ^ { T } Z _ { i j } } \\end{array}$ where $z _ { \\mathrm { t i m e } } ^ { ( k ) } [ j ]$ and $Z _ { i j }$ are $j$ -th lement of $z _ { \\mathrm { t i m e } } ^ { ( k ) }$ and $( i , j )$ -th element of $Z _ { \\mathrm { t i m e } } ^ { ( k ) }$ for the resting-state data, respectively. To employ $\\mathbf { k }$ -means clustering to the resting-state dynamic FC analysis [1], we first define a set of attended timepoints $\\tilde { T } = \\{ t \\mid z _ { \\mathrm { t i m e } } [ t ] > \\alpha \\cdot \\sigma _ { z _ { \\mathrm { t i m e } } } \\}$ where $\\alpha$ is the cutoff coefficient, and ${ \\sigma } _ { z _ { \\mathrm { t i m e } } ^ { ( k ) } }$ denotes the standard deviation of $z _ { \\mathrm { t i m e } } ^ { ( k ) }$ . Defining the threshold based on standard deviation inherits the practice of the point-process analysis for dynamic FC, so we set $\\alpha = 1 . 0$ following [44]. Pattern of the FC matrices at attended timepoints $A ^ { \\tilde { T } } = \\{ A ( t ) \\mid t \\in \\tilde { T } \\}$ for each subject can now be analyzed with the $\\mathbf { k }$ -means clustering. Specifically, we fit 7 template cluster centroids from the dynamic FC matrices $A ( t )$ over all subjects, and assign elements of $A ^ { \\tilde { T } }$ into one of the 7 template clusters. The ratio of each clusters from $A ^ { \\tilde { T } }$ with respect to $\\pmb { A }$ can then be analyzed with the subset of $A ^ { \\tilde { T } }$ including only the female or male subjects. ",
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+        "img_path": "images/5e2ae4d16e5a7a857695c0ed72b31036b27c90a793402955d4c8562a6632bd87.jpg",
+        "image_caption": [
+            "Figure 3: Analysis of temporal attention of the gender classification experiment with $\\mathbf { k }$ -means clustering. The DMN and SMN of the 7 cluster centroids are plotted and the relative proportion of temporally attended clusters for female and male subjects are written below. The clusters are sorted in descending order of the female/male attended cluster ratio. "
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+        "type": "text",
+        "text": "Evidences from large scale studies suggest that female subjects show hyperconnectivity of the DMN [34, 39] and hypoconnectivity of the SMN when compared to male subjects [39, 13]. We accordingly hypothesized that the FC at attended timepoints will show higher values for the DMN and lower values for the SMN in female participants. Figure 3 demonstrates that the clusters mainly attended by female participants show a trend of hyperconnectivity of the DMN and hypoconnectivity of the SMN. This can be interpreted to mean that the STAGIN is properly trained to take the dynamic state of the FC networks into account for predicting the phenotype of the subject. ",
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+    {
+        "type": "text",
+        "text": "The spatial attention across regions of the brain is analyzed with the $z _ { \\mathrm { s p a c e } } ^ { ( k ) }$ averaged across time $\\begin{array} { r } { \\tilde { z } _ { \\mathrm { s p a c e } } ^ { ( k ) } : = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } z _ { \\mathrm { s p a c e } } ^ { ( k ) } ( t ) } \\end{array}$ . The regions with top 5 percentile attention values of $\\tilde { z } _ { \\mathrm { s p a c e } } ^ { ( k ) }$ are plotted with respect to the seven ICNs in Figure 9 in the Appendix. It can be seen that the majority of the top attended regions are from the SMN, which further suggests gender difference of resting-state FC within the SMN. A notable limitation here is that the threshold for determining the top attended region is heuristically set. Statistically determining the spatially attended regions from the resting-state data would further provide validity of the method, which is left as a future work. ",
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+        "type": "text",
+        "text": "5.4 HCP-Task: Task decoding ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Task decoding refers to classifying which of the seven tasks the subject was performing during the acquisition of the brain fMRI. The STAGIN-GARO and STAGIN-SERO showed $9 9 . 0 2 \\%$ and $9 9 . 1 9 \\%$ mean accuracy for the task decoding experiment, respectively (Table 1). It can be seen that the proposed methods outperform the previous state-of-the-art model BAND and $\\mathrm { B A n D + + }$ [36], which applied self-attention of the Transformer encoder directly to 3D ResNet extracted representation vectors of the fMRI without considering the network property of the brain. To account for the possible statistical disadvantage of voxel-based feature extraction, we further implemented a new region-based BAnD (r-BAnD) by using GIN without attention-based READOUT instaed of the 3D ResNet. Accuracy of r-BAnD resulted in an accuracy of $9 8 . 9 0 \\%$ , suggesting that our method shows superior performance even when the statistical disadvantages are matched. Experiment on other models including ST-GCN [15], GIN [23], GCN [24], GraphSAGE [31], and ChebGCN [2] demonstrate exceptional performance of our proposed method for HCP-Task (Table 1). The fact that subtask timing information is completely lost may reflect the reason behind poor performance of static FC methods, which can be a critical disadvantage in task classification. ",
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+        "img_path": "images/e4ace94c7da5d4c584df69fe320c29cc2a46fe60afe2eebec1f2beeb1c406879.jpg",
+        "image_caption": [
+            "Figure 4: Analysis of spatio-temporal attention for working memory task of the task decoding experiment. (a) Plot of average temporal attention matrix Z(k)time across subjects. (b) Proportion of statistically significant regions within the 7 ICNs from the spatial attention GLM. "
+        ],
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+    {
+        "type": "text",
+        "text": "We interpret the result from the working memory task for spatio-temporal attention analysis, where the subtask consists of either performing an n-back memory task or rest. Our key expectation of the temporal attention analysis was that if STAGIN learns to accurately attend to temporal features of the dynamic FC graphs, then $Z _ { \\mathrm { t i m e } }$ should represent which subtask the subject was upto. Surprisingly, it can be clearly seen that the Transformer encoder of STAGIN learns to attend to the timing of subtasks from Figure 4 (a), which demonstrates mean temporal attention $Z _ { \\mathrm { t i m e } }$ across all subjects. Notice that no supervision is provided to the STAGIN model regarding the subtask timing during training. ",
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+        "type": "text",
+        "text": "To analyze the spatially attended regions $z _ { \\mathrm { s p a c e } }$ of STAGIN, we construct a GLM [14] to statistically evaluate how much each region is responsible for performing the subtasks. The parameter vectors $\\beta _ { \\mathrm { t a s k } } \\in \\mathbb { R } ^ { N }$ and $\\beta _ { \\mathrm { r e s t } } \\in \\mathbb { R } ^ { \\widetilde { N } }$ are estimated with the sequence of spatial attention vectors and the subtask timing design matrix $M \\in \\{ 0 , 1 \\} ^ { T \\times 2 }$ by solving the following with least-squares estimation: ",
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+    {
+        "type": "equation",
+        "img_path": "images/9e931e11b9c263affc85eafcda1850ad10f146264f1c2eaf7640140114c583bf.jpg",
+        "text": "$$\n\\left[ z _ { \\mathrm { s p a c e } } ( 0 ) , \\cdot \\cdot \\cdot , z _ { \\mathrm { s p a c e } } ( T ) \\right] ^ { \\top } = M [ \\beta _ { \\mathrm { t a s k } } , \\beta _ { \\mathrm { r e s t } } ] ^ { \\top } + \\epsilon ,\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where $\\epsilon$ denotes residual error. The contrast of the estimated parameters $\\hat { \\beta } _ { \\mathrm { t a s k } }$ and $\\hat { \\beta } _ { \\mathrm { r e s t } }$ was set to $\\pmb { c } = [ 1 , - 1 ]$ so the rejection of null hypothesis indicates $\\hat { \\beta } _ { \\mathrm { t a s k } } [ \\bar { i } ] > \\hat { \\beta } _ { \\mathrm { r e s t } } [ i ]$ at the $i$ -th ROI. Multiple comparisons of the $N$ ROIs are family-wise error (FWE) corrected. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 4 (b) shows the proportion of statistically significant regions within the 7 ICNs for each layers. Interestingly, the layer 1 and 2 share a similar trend that the regions from SMN, visual network (VN), and salience/ventral attention network (SVN) are dominant. In contrast, layer 3 and 4 suggest a dominance of the regions from DMN and cognitive control network (CCN). We denote the layer 1 and 2 as the low-order layers (LoL) and the layer 3 and 4 as the high-order layers (HoL). The dominance of SMN and VN at LoL can be understood as the low-level sensorimotor function for perceiving the task is being processed within the short-range $1 \\mathrm { - }$ or 2-hop connection of the networks. On the other hand, the dominance of DMN and CCN at HoL reflects the high-level cognitive integration for executing and controlling the given task being processed within the long-range 3- or 4-hop connection of the networks. Considering that the SVN is a network for integrating the low-level sensorimotor networks and the high-level executive networks to provide dynamic balancing between the two functions, the significant regions of SVN being present at both LoL and HoL is not surprising. Temporal and spatial attention plot of other six tasks are further provided in the Appendix Section D.2. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2021M3E5D9025019, NRF-2020R1A2B5B03001980). This work was also supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government(MSIT) (No.2019-0-00075, Artificial Intelligence Graduate School Program(KAIST)) and the KAIST Key Research Institute (Interdisciplinary Research Group) Project. ",
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+    {
+        "type": "text",
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parse/train/r1gl7hC5Km/r1gl7hC5Km.md

@@ -0,0 +1,340 @@
+# ADAPTING AUXILIARY LOSSES USING GRADIENT SIMILARITY
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+One approach to deal with the statistical inefficiency of neural networks is to rely on auxiliary losses that help to build useful representations. However, it is not always trivial to know if an auxiliary task will be helpful for the main task and when it could start hurting. We propose to use the cosine similarity between gradients of tasks as an adaptive weight to detect when an auxiliary loss is helpful to the main loss. We show that our approach is guaranteed to converge to critical points of the main task and demonstrate the practical usefulness of the proposed algorithm in a few domains: multi-task supervised learning on subsets of ImageNet, reinforcement learning on gridworld, and reinforcement learning on Atari games.
+
+# 1 INTRODUCTION
+
+Neural networks are extremely powerful function approximators that have excelled on a wide range of tasks (Simonyan and Zisserman, 2015; Mnih et al., 2015; He et al., 2016a; Silver et al., 2016; Vaswani et al., 2017). Despite the state of the art results across domains, they remain data-inefficient and expensive to train. In supervised learning (e.g., image classification), large deep learning (DL) benchmarks with millions of examples are needed for training (Russakovsky et al., 2015) and the additional implication of requiring human intervention to label a large dataset can be prohibitively expensive. In reinforcement learning (RL), agents typically consume millions of frames of experiences before learning to act in complex environments (Silver et al., 2016; Espeholt et al., 2018), which not only puts pressure on compute power but also makes particular domains (e.g., robotics) impractical.
+
+Different techniques have been studied for improving data efficiency, from data augmentation (Krizhevsky et al., 2012; Simonyan and Zisserman, 2015; Hauberg et al., 2016) to transfer learning (Taylor and Stone, 2009; Pan et al., 2010). In this work, we focus on a particular setup for transfer learning. We assume that besides the main task, one has access to one or more auxiliary tasks that share some unknown structure with the main task. To improve data efficiency, these additional tasks can be used as auxiliary losses. However, only the performance on the main task is of interest, even though the model is trained simultaneously on all these tasks. Any improvement on the auxiliary losses is useful only to the extent that it helps learning features or behaviors for the main task.
+
+Auxiliary tasks have been shown to work well in practice (e.g., Zhang et al., 2016; Jaderberg et al., 2017; Mirowski et al., 2017; Papoudakis et al., 2018). However, their success depends on how well aligned the auxiliary losses are with the main task. Knowing this apriori is typically non-trivial and the usefulness of an auxiliary task can change through the course of training. In this work, we explore a simple yet effective approach for measuring the similarity between an auxiliary task and the main task of interest, given the value of their parameters. We show that this measure can be used to decide which auxiliary losses are helpful and for how long.
+
+# 1.1 NOTATION AND PROBLEM DESCRIPTION
+
+Assume we have a main task $\mathcal { T } _ { m a i n }$ and an auxiliary task $\mathcal { T } _ { a u x }$ that induce two losses $\mathcal { L } _ { m a i n }$ and $\mathcal { L } _ { a u x }$ . We care about only about maximizing performance on $\mathcal { T } _ { m a i n }$ ; $\mathcal { T } _ { a u x }$ is an auxiliary task which is not of direct interest. The goal is to devise an algorithm that can automatically $( i )$ leverage $\mathcal { T } _ { a u x }$ when it is helpful (e.g. learn faster) and (ii) block negative transfer when $\mathcal { T } _ { a u x }$ is not helpful (i.e. recover the performance of training only on $\mathcal { T } _ { m a i n }$ ). Note that this setup is different from multi-objective optimization in which both the tasks are of interest. We propose to parameterize the solution for $\mathcal { T } _ { m a i n }$ and $\mathcal { T } _ { a u x }$ by two neural networks, $f ( \cdot , \pmb { \theta } , \phi _ { m a i n } )$ and $g ( \cdot , \pmb { \theta } , \phi _ { a u x } )$ , such that they share a subset of parameters denoted here by $\pmb \theta$ . Generally, the auxiliary loss literature proposes to minimize
+
+![](images/148cbdbcc71bcd0dd7cc29120de783f415d0701e52b0737677f8602684b5ba05.jpg)  
+Figure 1: Positive example optimization for $L _ { 1 } ( \theta _ { 1 } , \theta _ { 2 } ) = \theta _ { 1 } ^ { 2 } + \theta _ { 2 } ^ { 2 }$ , $L _ { 2 } ( \theta _ { 1 } , \theta _ { 2 } ) = ( \theta _ { 1 } - 1 ) ^ { 2 } + ( \theta _ { 2 } - 1 ) ^ { 2 }$ $\begin{array} { r } { V ( \theta _ { 1 } , \theta _ { 2 } ) = [ - \frac { \theta _ { 2 } } { \theta _ { 1 } ^ { 2 } + \theta _ { 2 } ^ { 2 } } - 2 \theta _ { 1 } , \frac { } { \theta _ { 2 } ^ { 2 } } } \end{array}$ − 2θ1, θ1θ2+θ2 − 2θ2] where the proposed method speeds up the process (compared on all runs). Each colored trajectory represents one optimization run with random initial position. Star represents the convergence point. All experiments use steepest descent method and run 600 iterations with a constant step size of 0.01. Convergence time is defined as number of steps needed to get below 0.1 loss of $L _ { 1 }$ (gray region). Color of each point represents its alignment with $\nabla L _ { 1 }$ (green—positive alignment, red—negative alignment, white—directions are perpendicular). In this example $L _ { 2 }$ is helpful for $L _ { 1 }$ as it reinforces good descent directions in most of the space. However, simple mixing is actually slowing optimization down (or makes it fail completely, see the second row), while the proposed methods (weighted and unweighted variants) converge faster (see the third row). When using non-conservative vector field $V$ one obtains lack of convergence (cyclic behaviour, see the fourth row), while the proposed merging still works well (see the last row).
+
+$$
+\operatorname* { a r g m i n } _ { \theta , \phi _ { m a i n } , \phi _ { a u x } } \mathcal { L } _ { m a i n } ( \theta , \phi _ { m a i n } ) + \lambda \mathcal { L } _ { a u x } ( \theta , \phi _ { a u x } )
+$$
+
+under the intuition that modifying $\pmb \theta$ to minimize $\mathcal { L } _ { a u x }$ will improve ${ \mathcal { L } } _ { m a i n }$ if the two tasks are ientlyis for . We pgiven the weight . That is, $\lambda$ at each learning iteration  each optimization iterati $t$ by how usefuln, we want to   
+$\mathcal { T } _ { a u x }$ $\mathcal { T } _ { m a i n }$ $\pmb { \theta } ^ { ( t ) } , \phi _ { m a i n } ^ { ( t ) } , \phi _ { a u x } ^ { ( t ) }$   
+efficiently approximate the solution to
+
+$$
+\underset { \lambda ^ { ( t ) } } { \arg \operatorname* { m i n } } \mathcal { L } _ { m a i n } \left( \pmb { \theta } ^ { ( t ) } - \alpha \nabla _ { \theta } \big ( \mathcal { L } _ { m a i n } + \lambda ^ { ( t ) } \mathcal { L } _ { a u x } \big ) , \phi _ { m a i n } ^ { ( t ) } - \alpha \nabla _ { \phi _ { m a i n } } \mathcal { L } _ { m a i n } \right) .
+$$
+
+Note that the input space of $\mathcal { T } _ { m a i n }$ and $\mathcal { T } _ { a u x }$ do not have to match. In particular, $\mathcal { T } _ { a u x }$ does not need to be defined for an input of $\mathcal { T } _ { m a i n }$ or the other way around.1 Solving equation 2 is expensive. Instead, we look for a cheap heuristic to approximate $\lambda ^ { ( i ) }$ which is better than keeping $\lambda ^ { ( t ) }$ constant and does not require hyper-tuning.
+
+# 2 COSINE SIMILARITY BETWEEN GRADIENTS OF TASKS
+
+We propose to use the cosine similarity of gradients between tasks as a measure of task similarity and hence for approximating $\lambda ^ { ( t ) }$ . Consider an example where the main function to minimize is $\mathcal { L } _ { m a i n } = ( \theta - 1 \bar { 0 } ) ^ { 2 }$ and the auxiliary function is $\mathcal { L } _ { a u x } = \theta ^ { 2 }$ , their gradients are $\nabla _ { \theta } \mathcal { L } _ { m a i n } = 2 ( \theta - 1 0 )$ and $\nabla _ { \theta } \mathcal { L } _ { a u x } = 2 \theta$ respectively. When $\theta$ is initialized at $\theta = - 2 0$ , the gradients of the main and auxiliary functions point in the same direction and the cosine similarity is 1; minimizing the auxiliary loss is beneficial for minimizing the main. However, at a different point, $\theta = 5$ , the two gradients point in different directions and the cosine similarity is $- 1$ ; minimizing the auxiliary loss would hinder minimizing the main loss (See Figure 7 in Appendix B for an illustration.).
+
+This example suggests a natural strategy for approximating $\lambda ^ { ( t ) }$ : minimize the auxiliary loss as long as its gradient has non-negative cosine similarity with the target gradient; otherwise, the auxiliary loss should be ignored. This follows the well-known intuition that if a vector is in the same half-space as the gradient of a function $f$ , then it is a decent direction for $f$ . This reduces our strategy to ask if the gradient of the auxiliary loss is a descent direction for the main loss of interest.
+
+Proposition 1. Given any gradient vector field $G ( \pmb \theta ) = \nabla _ { \pmb \theta } \mathcal { L } ( \pmb \theta )$ and any vector field $V ( \pmb \theta )$ (such as the gradient of another loss function, or an arbitrary set of updates), an update rule of the form
+
+$$
+\pmb \theta ^ { ( t + 1 ) } : = \pmb \theta ^ { ( t ) } - \alpha ^ { ( t ) } ( G ( \pmb \theta ^ { ( t ) } ) + V ( \pmb \theta ^ { ( t ) } ) \operatorname* { m a x } ( 0 , \cos ( G ( \pmb \theta ^ { ( t ) } ) , V ( \pmb \theta ^ { ( t ) } ) ) )
+$$
+
+converges to the local minimum of $\mathcal { L }$ given small enough $\alpha ^ { ( t ) }$ .
+
+Proof is provided in Appendix A.1.
+
+Note that the above statement does not guarantee any improvement of convergence, but only guarantees lack of divergence. In particular, cosine similarity is not a silver bullet that guarantees positive transfer, but it can drop the “worst-case scenarios”. In principle, one can create example functions where the convergence of the main loss is affected both positively (see Figure 1) and negatively (see Figure 8 in Appendix D). Nevertheless, convergence on the main task is guaranteed for our proposed strategy, as the proposition shows.
+
+In addition, it is important to note that simply adding an arbitrary vector field does not have the convergence property. For example, use function $\begin{array} { r } { V ( \pmb { \theta } ) = - \nabla _ { \pmb { \theta } } \mathcal { L } ( \pmb { \theta } ) + \left[ - \frac { \theta _ { 2 } } { \theta _ { 1 } ^ { 2 } + \theta _ { 2 } ^ { 2 } } , \frac { \theta _ { 1 } } { \theta _ { 1 } ^ { 2 } + \theta _ { 2 } ^ { 2 } } \right] ^ { T } } \end{array}$ as a two-dimensional case, which leads to an update rule of $\begin{array} { r } { \pmb { \theta } ^ { ( t + 1 ) } = \pmb { \theta } ^ { ( t ) } - \alpha \left[ - \frac { \theta _ { 2 } } { \theta _ { 1 } ^ { 2 } + \theta _ { 2 } ^ { 2 } } , \frac { \theta _ { 1 } } { \theta _ { 1 } ^ { 2 } + \theta _ { 2 } ^ { 2 } } \right] ^ { T } } \end{array}$ This is a non-conservative vector field which cases the optimizer to follow concentric circles around the origin (see the fourth row in Figure 1). This is crucial to note for some realistic scenarios where one does not always form a gradient field (e.g., the update rule of the Q-learning algorithm in RL).
+
+Figure 1 provides a few illustrative examples on quadratic functions using the proposed approach, which helps intuitively understand the kind of scenarios for which the approach could help.
+
+The above proposition refers to losses with the same set of parameters $\pmb { \theta }$ , while equation 2 refers to the scenario when each loss has task specific parameters (e.g. $\phi _ { m a i n }$ and $\phi _ { a u x }$ ). The following proposition extends to this scenario:
+
+Proposition 2. Given two losses parametrized with $\Theta$ (some of which are shared $\pmb \theta$ and some unique to each loss $\phi _ { m a i n }$ and $\phi _ { a u x }$ ), learning rule:
+
+$$
+\begin{array} { r l } & { \mathbf { \Phi } _ { l } ^ { ( t + 1 ) } : = \pmb { \theta } ^ { ( t ) } - \alpha ^ { ( t ) } \big ( \nabla _ { \theta } \mathcal { L } _ { m a i n } ( \pmb { \theta } ^ { ( t ) } ) + \nabla _ { \theta } \mathcal { L } _ { a u x } ( \pmb { \theta } ^ { ( t ) } ) \operatorname* { m a x } \big ( 0 , \cos ( \nabla _ { \theta } \mathcal { L } _ { m a i n } ( \pmb { \theta } ^ { ( t ) } ) , \nabla _ { \theta } \mathcal { L } _ { a u x } ( \pmb { \theta } ^ { ( t ) } ) \big ) \big ) } \\ & { \mathbf { \Phi } _ { m a i n } ^ { ( t + 1 ) } : = \phi _ { m a i n } ^ { ( t ) } - \alpha ^ { ( t ) } \nabla _ { \phi _ { m a i n } } \mathcal { L } _ { m a i n } ( \mathbf { \Theta } _ { } ^ { ( t ) } ) \quad \mathbf { \Phi } _ { a n d } \quad \phi _ { a u x } ^ { ( t + 1 ) } : = \phi _ { a u x } ^ { ( t ) } - \alpha ^ { ( t ) } \nabla _ { \phi _ { a u x } } \mathcal { L } _ { a u x } ( \pmb { \Theta } ^ { ( t ) } ) } \end{array}
+$$
+
+leads to convergence to local minimum of $\mathcal { L } _ { m a i n } \ w . r . t .$ . $( \theta , \phi _ { m a i n } )$ given small enough $\alpha ^ { ( t ) }$ .
+
+Proof. Comes directly from the previous proposition that $G = \nabla _ { \theta } \mathcal { L } _ { m a i n }$ and $V = \nabla _ { \theta } \mathcal { L } _ { a u x }$ . For any vector fields $A , B , C$ , we have $\langle A , B \rangle \geq 0$ and $\langle C , B \rangle \geq 0$ implies $\langle A + C , B \rangle \geq 0$ . □
+
+Analogous guarantees hold for the unweighted version of this algorithm, where instead of weighting by $\cos ( G , V )$ we use a binary weight $( \mathrm { s i g n } ( \cos ( G , V ) ) + 1 ) / \bar { 2 }$ which is equivalent to using $V$ iff $\cos ( G , V ) > 0$ . When training with mini-batches, accurately estimating $\cos ( G , V )$ can be difficult due to mini-batch noise; the unweighted variant only requires $\operatorname { s i g n } ( \cos ( G , V )$ which can be estimated more robustly. Hence, we use this variant in our experiments unless otherwise specified. Additionally, note that there is no guarantee that $\mathcal { L } _ { a u x }$ is optimized. For example, if $\mathcal { L } _ { a u x } = - \mathcal { L } _ { m a i n }$ then $\mathcal { L } _ { a u x }$ is ignored (see a visualization in the last row of Figure 1).
+
+Despite its simplicity, the proposed update rule can give rise to interesting phenomena. We can show that the emerging vector field could be non-conservative, which means there does not exist a loss function for which it is a gradient. While this might seem problematic (for gradient-descent-based optimizers), it describes only the global structure—typically used optimizers are local in nature and they do local, linear or quadratic approximations of the function (Shwartz-Ziv and Tishby, 2017). Consequently, in practice, one should not expect any negative effects from this phenomena, as it simply shows that our proposed technique is in fact qualitatively changing the nature of the update rules for training.
+
+Proposition 3. In general, the proposed update rule does not have to create a conservative vector field.
+
+Proof is provided in Appendix A.2.
+
+# 3 APPLICATIONS OF GRADIENT COSINE SIMILARITY
+
+In this section, we use the gradient cosine similarity to decide when to train on the auxiliary task. All experiments (unless otherwise stated) follow the unweighted version of our method, summarized in Algorithm 1. The weighted version of our method is summarized in Algorithm 2, Appendix C.
+
+# Algorithm 1 Unweighted version of our method.
+
+1: Initialize shared parameters $\pmb { \theta }$ and task specific parameters $\phi _ { m a i n } , \phi _ { a u x }$ randomly.   
+2: for iter $= 1$ : max iter do   
+3: Compute $\nabla _ { \pmb { \theta } } \mathcal { L } _ { m a i n }$ , $\nabla _ { \phi _ { m a i n } } \mathcal { L } _ { m a i n }$ , $\nabla _ { \theta } \mathcal { L } _ { a u x }$ , $\nabla _ { \phi _ { a u x } } \mathcal { L } _ { a u x }$ .   
+4: Update $\phi _ { m a i n }$ and $\phi _ { a u x }$ using corresponding gradients   
+5: if $\cos ( \nabla _ { \pmb { \theta } } \mathcal { L } _ { m a i n } , \nabla _ { \pmb { \theta } } \mathcal { L } _ { a u x } ) \geq 0$ then   
+6: Update $\pmb \theta$ using $\nabla _ { \pmb { \theta } } \mathcal { L } _ { m a i n } + \nabla _ { \pmb { \theta } } \mathcal { L } _ { a u x }$   
+7: else   
+8: Update $\pmb \theta$ using $\nabla _ { \pmb { \theta } } \mathcal { L } _ { m a i n }$
+
+# 3.1 EXPERIMENTS ON IMAGE CLASSIFICATION TASKS
+
+First, we consider a classification problem on ImageNet (Russakovsky et al., 2015) and design a simple multi-task binary classification task to test our hypothesis that transferable tasks should have high cosine similarity (and vice versa). We take a pair of classes from ImageNet, refer to these as class $A$ and class $B$ ; all the other 998 classes in ImageNet (except $A$ and $B$ ) are referred to as the background. Our tasks $\mathcal { T } _ { m a i n }$ and $\mathcal { T } _ { a u x }$ are then formed as a binary classification of if an image is class $A$ (otherwise background) and if an image is class $B$ (otherwise background) respectively.
+
+Ideally, we want to pick groups of class pairs that reflect near or far distance, for the purpose of providing a baseline of transferability. Therefore, we used two distance measures, lowest common ancestor $( L C A )$ in the ImageNet label hierarchy and Frechet Inception Distance (FID) (Heusel et al., 2017) of pre-trained image embedding, to serve as a ground truth of class similarity for selecting class pair $A$ and $B$ . Based on these measures, we picked three pair of classes for near, class 871 (trimaran) vs. 484 (catamaran), 250 (Siberian husky) vs. 249 (malamute), and 238 (Greater Swiss Mountain dog) vs. 241 (Entleucher); and for far, class 920 (traffic light) vs. 62 (rock python), 926 (hotpot) vs. 800 (slot), and 48 (Komodo dragon) vs. 920 (traffic light). Details on the class pair selection are described in Appendix E.
+
+We use a modified ResNetV2-18 model (He et al., 2016b) for training in this experiment. All parameters in the convolutional layers are shared (denote as $\pmb { \theta }$ ), followed by task-specific parameters $\phi _ { m a i n }$ and $\phi _ { a u x }$ . First, we use a multi-task learning setup and minimize $\mathcal { L } _ { m a i n } + \mathcal { L } _ { a u x }$ , and measure cosine similarity on $\pmb { \theta }$ through the course of training. Figure 2(a) shows that cosine similarity is higher for near pairs (blue lines) and lower for far pairs (red lines). Next, we compare single-task training, multi-task training, and our proposed variant on two scenarios (i) auxiliary task helps and (ii) auxiliary task hurts. As mentioned earlier, our goal is have a method that can automatically leverage auxiliary tasks when they are helpful and avoid negative transfer when auxiliary tasks are not helpful. Figure 2(b) shows that on a near pair, all variants perform similarly in terms of final performance; furthermore, our method performs similar to multi-task learning and learns faster than single task because the task is transferable. Figure 2(c) shows that on a far pair, multi-task learning leads to poorer performance than single-task learning on the main task due to the potential negative transfer, whereas our method of using gradient cosine similarity blocks negative transfer and automatically achieves performance that is comparable to single-task learning.
+
+![](images/c61cf34feaf0d0382683306cc9465dfc6f98c46145d1c301328f729afaaa9664.jpg)  
+Figure 2: Multi-task learning setup on ImageNet class pairs. (a): gradient cosine similarity is higher for near pairs and lower for far pairs. (b) and (c): testing accuracy on single task (dotted), naive multi-task (dashed), and our method (solid). Naive multi-task learning helps in near pairs (see $( b )$ ) but hurts in far pairs (see (c)) because of its lack of the ability to prevent negative effects from the auxiliary task to the main task. Our method can overcome this limitation by dropping the auxiliary task when its gradient direction disagrees with the main task, thus achieving the best of both worlds: matching the multi-task performance on near pairs (see $( b )$ where our method and multi-task learning learn faster than single task only) and the single task performance in far pairs (see (c) where multi-task learning performs poorly, but our method automatically recovers single task performance).
+
+# 3.2 EXPERIMENTS ON REINFORCEMENT LEARNING GRIDWORLD TASKS
+
+We then consider a typicalfuture discounted rewards (POMDP). There have bee one aims to find a policy  in a partially observable proposed to solve this op $\pi$ that maximizes sum ofarkov decision processmization problem, from $\mathbb { E } _ { \pi } [ \sum _ { t ^ { \prime } = 1 } ^ { N } \gamma ^ { t ^ { \prime } - 1 } r _ { t ^ { \prime } } ]$ classical policy gradient (Williams, 1992), Q-Learning (Watkins, 1989), to the more modern Proximal Policy Optimization (Schulman et al., 2017) and V-Trace (Espeholt et al., 2018). Inherently, these techniques are data inefficient due to the complexity of the problem. One way to address this issue is to use transfer learning, such as transfer from pre-trained policies (Rusu et al., 2015). However, a teacher policy is not always available for the main task. When in this scenario, one can train policies in other tasks that share enough similarities and hope for a positive transfer. One way of exploiting this extra information is to use behavioral cloning, or distillation (Hinton et al., 2015; Rusu et al., 2015), to guide the main task in its initial learning phase (Schmitt et al., 2018), although it might be difficult to find a suitable strategy that combines the main and auxiliary losses and/or smoothly transition between them. Typically, the teacher policy can be treated as an auxiliary loss (Schmitt et al., 2018) or a prior (Teh et al., 2017) with a fixed mixing coefficient. However, these techniques become unsound if the teacher policy is helpful only in specific states while hindering in other states.
+
+We propose a simple RL experiment to show that our method is capable of finding the strategy of combining the main loss and the auxiliary loss. We define a distribution over a set of $1 5 \times 1 5$ gridworlds, where an agent observes its surrounding (up to four pixels away) and can move in four directions.We randomly place two types of positive rewards, $+ 5$ and $+ 1 0$ points, both terminating an episode. In order to guarantee a finite length of episodes, we add a fixed probability of 0.01 of transitioning to a non-rewarding terminal state. Experiment details are provided in Appendix F.
+
+First, we train a Q-learning agent on such a gridworld which gives us a teacher policy $\pi ^ { 2 }$ . Then, we create the main task to which there is a possible positive knowledge transfer by keeping the environment with the same gridworld layout but remove the $+ 1 0$ rewards (and corresponding states are no longer terminating). Consequently, we have two tasks: the auxiliary task $\mathcal { T } _ { a u x }$ where we have a strong teacher policy $\pi ^ { 2 }$ , and the main task $\mathcal { T } _ { m a i n }$ where the $+ 1 0$ rewards are removed. We sample 1, 000 such environment pairs and report expected returns obtained (100 evaluation episodes per evaluation point) using various training regimes. One can use any RL method to solve the main task and learn $\pi$ , here we use episode-level policy gradient (Williams, 1992) with value function as a baseline method, which gives a score of slightly above 1 point after $1 0 , 0 0 0$ steps of training (see the top row of Figure 3).
+
+To leverage teacher policies, we define the auxiliary loss to be a distillation loss, which is a per-state cross-entropy between teacher’s and student’s distributions over actions. First, we test using solely the distillation loss while sampling trajectories from the student. We recover a subset of teacher’s behaviors and end up with 0 point—an expected negative transfer as the teacher is guiding us to states that are no longer rewarding. Then, we test simply adding gradients estimated by policy gradient and distillation. The resulting policy learns quickly but saturates at a return of 1 point, showing very limited positive transfer. Lastly, when using our proposed gradient cosine similarity as the measure of transferability, we get a significant performance boost. The learned policies reach baseline performance after just one-third of steps taken by the baseline, and on average obtain 3 points after 10, 000 steps.2 See Figure 3 for all learning curves. In Appendix F, we visualize the environment and show an example solution.
+
+This experiment shows that gradient cosine similarity allows using knowledge from other related tasks in an automatic fashion. The agent is simply ignoring teacher signal when it disagrees with policy gradient estimator. If they do agree in terms of which actions to reinforce—teacher logits are used for better replication of useful policies. In particular, in the bottom row of Figure 3, we present an experiment of transfer between the same task $\mathcal { T } _ { m a i n }$ . We see that the cosine similarity experiments underperformed that of simply adding the two losses. This is expected as the noise in the gradients makes it hard to measure if the two tasks are a good fit or not.
+
+# 3.3 EXPERIMENTS ON ATARI
+
+Finally, we consider a similar RL setup on the Atari domain (Bellemare et al., 2013). For this set of experiments, we follow the same convolutional architecture as in previous works (Mnih et al., 2015; 2016; Espeholt et al., 2018; Hessel et al., 2018) and train using the batched actor-critic with V-trace algorithm (Espeholt et al., 2018). Details on the experiment setup are provided in Appendix G.
+
+First, we look at training an agent to play a main task (here, Breakout) given a sub-optimal teacher solution to the task. Analogous to the previous experiment, we leverage information about the task by distilling the teacher’s behaviour with a Kullback-Leibler (KL) loss. As expected, solely relying on distilling from the sub-optimal teacher $( O n l y K L )$ leads to lower performance. Training with both distillation and RL losses $\left( R L + K L ( B a s e l i n e ) \right)$ leads to slightly better but also sub-optimal performance. While both approaches learn very quickly, they plateau much lower than the pure RL approach (RL(Baseline)). In our method $( R L + K L ( O u r M e t h o d ) )$ , the KL penalty is scaled at every time-step by the cosine similarity between the policy gradient and distillation losses; once this falls below a fixed threshold, the loss is ‘turned off’. Figure 4 shows that our approach is able to learn quickly at the start but continue fine-tuning with pure RL loss once the distillation loss is zeroed out.
+
+![](images/a0a7154a29483f61df3c355d609d2e08ae4ab9ffd2eadc83b5a83fbe3048fb51.jpg)  
+Figure 3: Top row: expected learning curves for cross-environment distillation experiments, averaged over 1, 000 partially observable gridworlds. Teacher’s policy is based on Q-Learning, its performance in a new environment (with modified positive rewards) is represented by the top dotted line. The bottom dotted line represents random policy score. Each column represents a different temperature applied to the teacher policy. 0 temperature refers to the original deterministic greedy policy given by Q-Learning. We report five methods: reward using just policy gradient in the new task; distill using just distillation cost towards the teacher; add adding the two above; cos using the weighted version of our method (Algorithm 2); strict cos using the unweighted version of our method (Algorithm 1). Bottom row: expected learning curves for same-environment distillation experiments when the teacher is perfect. In this case, the optimal thing is to trust the teacher everywhere.
+
+Lastly, we consider a setting where the main task ${ \mathcal { T } } _ { m a i n }$ is to train an agent to play two Atari games, Breakout and Ms. PacMan. Similar to previous experiment, we have access to a teacher trained on just Breakout as the auxiliary task $\mathcal { T } _ { a u x }$ , from which we distill a policy via KL loss. Note that $\mathcal { T } _ { m a i n }$ itself is chosen to be Multitask to illustrate a complex scenario where $\mathcal { T } _ { a u x }$ helps with only part of $\mathcal { T } _ { m a i n }$ , and that too only initially. We consider a distillation loss as was done previously by adding the auxiliary KL loss $\mathcal { L } _ { a u x }$ (between the teacher and student policies) to the RL multi-task loss ${ \mathcal { L } } _ { m a i n }$ at every time step. Intuitively, doing so would result in the agent only be able to solve one of the tasks—the one the teacher knows about, as the gradients from distillation loss would interfere with the policy gradient. Figure 5 shows that, compared to the baseline Multitask and the simple addition of Multitask $R L +$ Distillation approaches where the agent learns one task at the expense of the other, our method of scaling the auxiliary loss by gradient cosine similarity is able to compensate for this by learning from the teacher and then turning off the auxiliary distillation; it learns Ms. PacMan without forgetting Breakout. The evolution of the gradient cosine similarity between Breakout and Ms. PacMan provides a meaningful cue for the usefulness of $\mathcal { L } _ { a u x }$
+
+![](images/1365f698b4d635bda30088c7772115e067ac5cf7e290bd9533e9019377d9edb5.jpg)  
+Figure 4: Results on Breakout. We look at the effects of distilling a sub-optimal policy as an auxiliary task.
+
+# 4 RELATED WORK
+
+Our work is related to the literature on identifying task similarity in transfer learning. It is generally believed that positive transfer can be achieved when source task(s) and target task(s) are related.
+
+![](images/e920a1d5503f5e559d5126ce96d09452eb1bf5f9d33889c078dc5afb3fb34204.jpg)  
+Figure 5: Results on Breakout and Ms. PacMan (averaged over 3 seeds). The two plots to the left show performance on Breakout and Ms. PacMan respectively. The third plot shows how the gradient cosine similarity between the two tasks changes during training. The last plot shows an average score of the multi-task performance (normalized independently for each game based on the best score achieved across all experiments). Our method is able to learn both games without forgetting and achieves the best average performance.
+
+However, it is usually assumed that this relatedness mapping is provided by human experts (Taylor and Stone, 2009; Pan et al., 2010); few works have addressed the problem of finding a general measure of similarity to predict transferability between tasks. In image classification, Yosinski et al. (2014) defined image similarity in ImageNet by manually splitting classes into man-made versus natural objects. In RL, methods have been proposed to use the Markov decision process (MDP) similarity as a measure of task relatedness (Carroll and Seppi, 2005; Ammar et al., 2014). While in some degree capture task similarity, these measures are often domain-specific and not generalizable. In addition, none of these works have explicitly used the learned similarity metric to improve performance. In our work, we propose to use cosine similarity of gradients as a generalizable measure across domains and show it can be directly leveraged to improve the performance of the main task. One important aspect of task similarity for transfer is that it is highly dependent on the parametrization of the model and current value of the parameters. We exploit this property by providing a heuristic similarity measure for the current parameters, resulting in an approach that relies on an adaptive weight over the updates of the model.
+
+Auxiliary tasks have shown to be beneficial in facilitating learning across domains. In image classification, Zhang et al. (2016) used unsupervised reconstruction tasks. In RL, the UNREAL framework (Jaderberg et al., 2017) incorporates unsupervised control tasks along with reward prediction learning as auxiliary tasks. Mirowski et al. (2017) studied auxiliary tasks in the context of navigation. Papoudakis et al. (2018) also explored auxiliary loses for VizDoom. However, these works rely on empirical results and do not address how the auxiliary tasks were selected in the first instance. In this work, we aim to propose a simple yet effective way of explicitly selecting auxiliary tasks by using cosine similarity of task gradients.
+
+Our work is also related to multi-task learning (Caruana, 1997), particularly the line of work on using adaptive scaling techniques for multi-objective learning. For example, a recently developed algorithm, GradNorm (Chen et al., 2018), uses gradient magnitude to scale loss function for each task, aiming to learn well for all tasks. Similarly, Kendall et al. (2018) proposed a weighting mechanism by considering the homoscedastic uncertainty of each task. However, our work is different in two ways: first, in our problem setup, we care only about the performance of the main task and we do not care about all tasks; hence, the optimization goal is different from their setup which is more similar to traditional multi-objective optimization. Furthermore, it is important to note that our method differs from aforementioned work in that they scale the losses individually without looking at their interaction (which can lead to poor performance in our problem setup, when the auxiliary task hurts the main task), whereas we look for alignments in the vector field between the main and the auxiliary task, and the auxiliary task is used only when it is well-aligned with the main task.
+
+# 5 DISCUSSION
+
+In this work, we explored a simple yet efficient technique to ensure that an auxiliary loss does not hurt the learning on the main task. The proposed approach reduces to applying gradients of the auxiliary task only if they are a descent direction of the main task.
+
+We discuss here a few shortcomings of this method. First, estimating the cosine similarity between the gradients of tasks could be expensive or noisy and that the threshold for turning off the auxiliary is a fixed constant. These could be addressed by calculating a running average of the cosine similarity to get a smoother result and potentially hyper-tune the threshold instead of setting it as a fixed constant. One might argue that our approach would fail in high-dimensional spaces since random vectors in such spaces tend to be orthogonal, so that cosine similarity will be naturally driven to 0. In fact, this is not the case; if two gradients are meant to be co-linear, the noise components cancel each other thus will not affect the cosine similarity estimation. We empirically explore this in Appendix H. Second, the new loss surface might be less smooth which can be problematic when using optimizers that rely on statistics of the gradients or second order information (e.g. Adam or RMSprop). In these cases, the transition from just the gradient of the main task to the sum of the gradients can affect the statistics of the optimizer in unwanted ways.
+
+Lastly, although the proposed approach works well empirically on complex and noisy tasks like Atari games, as discussed in Section 2, it guarantees only that the main task will converge, but not how fast it will be. While removing the worst case scenarios is important and a good first step, one might care more for data efficiency when using auxiliary losses (i.e., faster convergence). Particularly in Appendix D Figure 8, we construct a counter-example where the proposed update rule slows down learning, compared to optimizing the main task alone. Nevertheless, we have empirically shown the potential of using the proposed hypothesis as a simple yet efficient way of picking a suitable auxiliary task. While we have mostly considered scenarios where the auxiliary task helps initially but hurts later, it would be interesting to explore settings where the auxiliary task hurts initially but helps in the end. Examples of such are annealing $\beta$ in $\beta$ -VAE (Higgins et al., 2017) and annealing the confidence penalty in (Pereyra et al., 2017).
+
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+
+# A PROOFS
+
+# A.1 PROOF FOR PROPOSITION 1
+
+Given any gradient vector field $G ( \pmb \theta ) = \nabla _ { \pmb \theta } \mathcal { L } ( \pmb \theta )$ and any vector field $V ( \pmb \theta )$ (such as gradient of another loss function, but could be arbitrary set of updates), an update rule of the form
+
+$$
+\pmb \theta ^ { ( t + 1 ) } : = \pmb \theta ^ { ( t ) } - \alpha ^ { ( t ) } ( G ( \pmb \theta ^ { ( t ) } ) + V ( \pmb \theta ^ { ( t ) } ) \operatorname* { m a x } ( 0 , \cos ( G ( \pmb \theta ^ { ( t ) } ) , V ( \pmb \theta ^ { ( t ) } ) ) )
+$$
+
+converges to the local minimum of $\mathcal { L }$ given small enough $\alpha ^ { ( t ) }$ .
+
+Proof. Let us denote
+
+$$
+\begin{array} { r l } & { G ^ { ( t ) } : = G ( \pmb { \theta } ^ { ( t ) } ) \qquad V ^ { ( t ) } : = V ( \pmb { \theta } ^ { ( t ) } ) \qquad \nabla \mathcal { L } ^ { ( t ) } : = \nabla _ { \pmb { \theta } } \mathcal { L } ( \pmb { \theta } ^ { ( t ) } ) } \\ & { \qquad \Delta \pmb { \theta } ^ { ( t ) } : = G ^ { ( t ) } + V ^ { ( t ) } \operatorname* { m a x } ( 0 , \cos ( G ^ { ( t ) } , V ^ { ( t ) } ) ) . } \end{array}
+$$
+
+Our update rule is simply $\pmb { \theta } ^ { ( t + 1 ) } : = \pmb { \theta } ^ { ( t ) } - \alpha ^ { ( t ) } \Delta \pmb { \theta } ^ { ( t ) }$ and we have
+
+$$
+\begin{array} { r l } & { \langle \Delta \pmb { \theta } ^ { ( t ) } , \nabla \mathcal { L } ^ { ( t ) } \rangle = \langle G ^ { ( t ) } + V ^ { ( t ) } \operatorname* { m a x } ( 0 , \cos ( G ^ { ( t ) } , V ^ { ( t ) } ) ) , \nabla \mathcal { L } ^ { ( t ) } \rangle } \\ & { \quad \quad \quad \quad \quad = \langle G ^ { ( t ) } , \nabla \mathcal { L } ^ { ( t ) } \rangle + \langle V ^ { ( t ) } \operatorname* { m a x } ( 0 , \cos ( G ^ { ( t ) } , V ^ { ( t ) } ) ) , \nabla \mathcal { L } ^ { ( t ) } \rangle } \\ & { \quad \quad \quad \quad = \| \nabla \mathcal { L } ^ { ( t ) } \| ^ { 2 } + \frac { 1 } { \| V ^ { ( t ) } \| \| \nabla \mathcal { L } ^ { ( t ) } \| } \operatorname* { m a x } ( 0 , \langle \nabla \mathcal { L } ^ { ( t ) } , V ^ { ( t ) } \rangle ) \langle V ^ { ( t ) } , \nabla \mathcal { L } ^ { ( t ) } \rangle \ge 0 . } \end{array}
+$$
+
+And it can be 0 if and only if $\| \nabla \mathcal { L } ^ { ( t ) } \| = 0$ (since sum of two non-negative terms is zero iff both are zero, and step from (4) to (5) is only possible if this is not true), thus it is 0 only when we are at the critical point of $\mathcal { L }$ . Thus the method converges due to convergence of steepest descent methods, see “Cauchy’s method of minimization” (Goldstein, 1962). □
+
+# A.2 PROOF FOR PROPOSITION 3
+
+In general, the proposed update rule does not have to create a conservative vector field.
+
+Proof. Proof comes from a counterexample, let us define in 2D space:
+
+$$
+\mathcal { L } _ { m a i n } ( \theta _ { 1 } , \theta _ { 2 } ) = a \theta _ { 1 }
+$$
+
+$$
+\mathcal { L } _ { a u x } ( \theta _ { 1 } , \theta _ { 2 } ) = \left\{ \begin{array} { l l } { a \theta _ { 1 } } & { \mathrm { i f } \theta _ { 1 } \in [ 1 , 2 ] \wedge \theta _ { 2 } \in [ 0 , 1 ] } \\ { 0 } & { \mathrm { t h e r w i s e } } \end{array} \right.
+$$
+
+for some fixed $a \neq 0$ . Let us now define two paths (parametrized by $s$ ) between points $( 0 , 0 )$ and $( 2 , 2 )$ , path $A$ which is a concatenation of a line from $( 0 , 0 )$ to $( 0 , 2 )$ (we call it $U$ , since it goes up) and line from $( 0 , 2 )$ to $( 2 , 2 )$ (which we call $R$ as it goes right), and path $B$ which first goes right and then up. Let $V _ { \mathrm { { c o s } } }$ denote the update rule we follow, then:
+
+$$
+\int _ { A } V _ { \mathrm { c o s } } d s = \int _ { A } \nabla { \mathcal { L } } _ { \mathrm { m a i n } } d s = \int _ { U } \nabla { \mathcal { L } } _ { \mathrm { m a i n } } d s + \int _ { R } \nabla { \mathcal { L } } _ { \mathrm { m a i n } } d s = \int _ { R } \nabla { \mathcal { L } } _ { \mathrm { m a i n } } d s = 2 a
+$$
+
+At the same time, since gradient of $\mathcal { L } _ { m a i n }$ is conservative by definition:
+
+$$
+\int _ { B } V _ { \mathrm { c o s } } d s = \int _ { B } \nabla { \mathcal { L } } _ { \mathrm { m i n } } d s + \int _ { C } \nabla { \mathcal { L } } _ { \mathrm { a u x } } d s = \int _ { A } \nabla { \mathcal { L } } _ { \mathrm { m a n } } d s + \int _ { C } \nabla { \mathcal { L } } _ { \mathrm { a u x } } d s = 2 a + \int _ { C } \nabla { \mathcal { L } } _ { \mathrm { a u x } } d s = 3 a
+$$
+
+where $C$ is a part of $B$ that goes through $[ 1 , 2 ] \times [ 0 , 1 ]$ . We conclude that $\textstyle \int _ { A } V _ { \mathrm { c o s } } d s \neq \int _ { B } V _ { \mathrm { c o s } } d s$ , so our vector field is not path invariant, thus by Green’s Theorem it is not conservative, which concludes the proof. See Figure 6 for visualization. □
+
+# B ONE-DIMENSIONAL TOY EXAMPLE
+
+Figure 7 shows the surfaces along with gradients for the one-dimensional motivating example described in Section 2.
+
+![](images/deda4673d4c45c202800fae049466282a5bb815f1fa2484afbaa142e19788126.jpg)  
+Figure 6: Visualization of the counterexample from Proposition 3, stars denote starting (green) and end (black) points. Dotted and dashed lines correspond to paths A and B respectively. Blue arrows represent gradient vector field of the main loss, while the violet ones the merged vector field.
+
+![](images/6c3232392097c1d8fd571f36602beef5833485b4f159a88182dd6c30f312ba46.jpg)  
+Figure 7: Illustration of cosine similarity between gradients on synthetic loss surfaces.
+
+C WEIGHTED VERSION OF OUR METHOD
+
+Algorithm 2 describes the weighted version of our method.
+
+# D TOY EXAMPLE SHOWING SLOW-DOWN
+
+We discuss here a few potential issues of using cosine similarity of gradients to measure task similarity. First, the method depends on being able to compute cosine between gradients. However, in DL we rarely are able to compute exact gradients in practice, nut instead depend on their high variance estimators (mini batches in supervised learning, or Monte Carlo estimators in RL). Consequently, estimating the cosine similarity might require additional tricks such as keeping moving averages of estimates. Second, adding additional task gradient in selected subset of iterates can lead to very bumpy surface from the perspective of optimizer, causing methods which keep track of gradient statistics/estimate higher order derivatives, can be less efficient. Finally, one can construct specific functions, where despite still minimizing the loss, one significantly slows down optimization process. Figure 8 provides one such function as an example.
+
+# E IDENTIFYING NEAR AND FAR CLASSES IN IMAGENET
+
+As a ground truth for class similarity, we identify pairs of ImageNet classes to be near or far using, lowest common ancestor (LCA) and Frechet Inception Distance (FID) (Heusel et al., 2017).
+
+ImageNet follows a tree hierarchy where each class is a leaf node. We define the distance between a pair of classes as at which tree level their LCA is found. In particular, there are 19 levels in the class tree, each leaf node (i.e. class) is considered to be level 0 while the root node is considered to be level 19. We perform bottom-up search for one pair of random sampled classes and find their LCA node—the class distance is then defined as the level number of this node. For example, class 871 (“trimaran”) and class 484 (“catamaran”) has class distance 1 because their LCA is one level up.
+
+# Algorithm 2 Weighted version of our method.
+
+1: Initialize shared parameters $\pmb { \theta }$ and task specific parameters $\phi _ { m a i n }$ , $\phi _ { a u x }$ . randomly.   
+2: for iter $= 1$ : max iter do   
+3: Compute $\nabla _ { \pmb { \theta } } \mathcal { L } _ { m a i n }$ , $\nabla _ { \phi _ { m a i n } } \mathcal { L } _ { m a i n }$ , $\nabla _ { \pmb { \theta } } \mathcal { L } _ { a u x }$ , $\nabla _ { \phi _ { a u x } } \mathcal { L } _ { a u x }$ .   
+4: Update $\phi _ { m a i n }$ and $\phi _ { a u x }$ using corresponding gradients   
+5: Update $\pmb { \theta }$ using $\nabla _ { \theta } \mathcal { L } _ { m a i n } + \operatorname* { m a x } ( 0 , \cos ( \nabla _ { \theta } \mathcal { L } _ { m a i n } , \nabla _ { \theta } \mathcal { L } _ { a u x } ) ) \nabla _ { \theta } \mathcal { L } _ { a u x }$
+
+![](images/8b12dde06eb50e51d552797f0a55ed6ab7ac60a8ba08390cdca9baf64c1852d6.jpg)  
+Figure 8: Negative example optimization for $L _ { 1 } ( \theta ) = ( \theta _ { 1 } < 0 ) ( \theta _ { 1 } ^ { 2 } + \theta _ { 2 } ^ { 2 } ) + ( \theta _ { 1 } > 0 ) \Big ( 1 - \exp \bigl ( - 2 ( \theta _ { 1 } ^ { 2 } +$
+
+$\theta _ { 2 } ^ { 2 } ) { \big ) }$ and $L _ { 2 } ( \theta ) = ( \theta _ { 1 } - 2 ) ^ { 2 } + ( \theta _ { 2 } - 0 . 5 ) ^ { 2 }$ where the proposed method slows down the process (compared on red runs). For the ease of presentation, we choose $L _ { 1 }$ , which is non-differentiable/smooth when $\theta = 0$ . But one can create any smooth functions with analogous properties. The core idea is, when there exists a flat region on the loss surface, the auxiliary lost tends to push the iterates to this region. Even though this move still decreases the loss (i.e., convergence is guaranteed), the optimization process will be slowed down.
+
+FID is used as a second measure of similarity. We obtain the image embedding of a pair of classes using the penultimate layer of a pre-trained ResNetV2-50 model (He et al., 2016b) and then compute the embedding distance using FID, defined in Heusel et al. (2017) as:
+
+$$
+\mathrm { F I D } = d ^ { 2 } \big ( ( m _ { 1 } , C _ { 1 } ) , ( m _ { 2 } , C _ { 2 } ) \big ) = \| m _ { 1 } - m _ { 2 } \| _ { 2 } ^ { 2 } + \mathrm { T r } \big ( C _ { 1 } + C _ { 2 } - 2 ( C _ { 1 } C _ { 2 } ) ^ { 1 / 2 } \big ) .
+$$
+
+where $m _ { k } , C _ { k }$ denote the mean and covariance of the embeddings from class $k$ .
+
+We randomly sampled 50 pairs of classes for each level of $L C A = \{ 1 , 2 , 3 , 4 , 1 6 , 1 7 , 1 8 , 1 9 \}$ (400 pair of classes in total) and compute their FID. Figure 9 shows a plot of LCA $\mathbf { \dot { x } }$ -axis) verses FID (y-axis) over our sampled class pairs. It can be seen that LCA and FID are (loosely) correlated and that they reflect human intuition of task similarity for some pairs. For example, trimaran and catamaran (bottom-left) are similar both visually and conceptually, whereas rock python and traffic light (top-right) are dissimilar both visually and conceptually. However, there are contrary examples where LCA disagrees with FID; monkey pinscher and doberman pinscher (top-left) are visually dissimilar but conceptually similar, whereas bubble and sundial (bottom-right) are visually similar but conceptually dissimilar.
+
+Per the observations, in subsequent experiments we pick class pairs that are $\{ L o w L C A , L o w F I D \}$ as near pairs (e.g., trimaran and catamaran), and class pairs that are $\{ h i g h L C A , h i g h F I D \}$ as far pairs (e.g., rock python and traffic light).
+
+# F GRIDWORLD EXPERIMENTS
+
+We define a distribution over $1 5 \times 1 5$ gridworlds, where an agent observes its surrounding (up to 4 pixels away) and can move in 4 directions (with $10 \%$ transition noise). We randomly place walls (blocking movement) as well as two types of positive rewards: $+ 5$ and $+ 1 0$ points, both terminating an episode. There are also some negative rewards (both terminating and non-terminating) to make problem harder. In order to guarantee (expected) finite length of episodes we add fixed probability of 0.01 of transitioning to a non-rewarding terminal state.
+
+![](images/fe36ff17a2b67dd62d239899161c52bd3667aa497749ef4daa206d4df8c2de61.jpg)  
+Figure 9: LCA ( $\scriptstyle { \dot { x } }$ -axis) versus FID $y$ -axis) as a ground truth for class similarity. The measurements reflect human intuition of class similarity; trimaran and catamaran (bottom-left) are similar both visually and conceptually, whereas rock python and traffic light (top-right) are dissimilar both visually and conceptually.
+
+For the sake of simplicity we use episode-level policy gradient (Williams, 1992) with value function baseline, with policies parametrized as logits θ of π(a|s) = Pexp(θs,a)exp(θs,b) , baselines as $B _ { s } \in \mathbb { R }$ , with fixed learning rate of $\alpha = 0 . 0 1$ , discount factor $\gamma = 0 . 9 5$ and 10,000 training steps (states visited).
+
+For this setup, the update rule for each sequence $\tau = \left( ( s _ { 1 } , a _ { 1 } , r _ { 1 } ) , \dots ( s _ { N } , a _ { N } , r _ { N } ) \right)$ is thus given by
+
+$$
+\begin{array} { r } { \begin{array} { l } { \lambda \theta = \alpha \nabla _ { \theta } \log \pi ( a _ { t ^ { \prime } } | s _ { t ^ { \prime } } ) \left[ \displaystyle \sum _ { i = 0 } ^ { N - t ^ { \prime } } r _ { t ^ { \prime } + i } - B _ { s _ { t ^ { \prime } } } \right] = \alpha G ^ { ( t ) } \qquad \Delta B _ { s _ { t ^ { \prime } } } = - \alpha \nabla _ { B _ { s _ { t ^ { \prime } } } } ( B _ { s _ { t ^ { \prime } } } - \displaystyle \sum _ { i = 0 } ^ { N - t ^ { \prime } } r _ { t ^ { \prime } + i } ) ^ { 2 } . } \end{array} } \end{array}
+$$
+
+In order to make use of expert policies for $\mathcal { T } _ { a u x }$ we define auxiliary loss as a distillation loss, which is just a per-state cross-entropy between teacher’s and student’s distributions over actions. If we just add gradients estimated by policy gradient, and the ones given by distillation, the update is given by
+
+$$
+\Delta \theta = \alpha \left[ G ^ { ( t ) } - \nabla _ { \theta } \mathbf { H } ^ { \times } ( \pi ^ { \mathbb { Q } } ( \cdot | s _ { t ^ { \prime } } ) \| \pi ( \cdot | s _ { t ^ { \prime } } ) ) \right] = \alpha [ G ^ { ( t ) } + \sum _ { a } \pi ^ { \mathbb { Q } } ( a | s _ { t ^ { \prime } } ) \nabla _ { \theta } \log \pi ( a | s _ { t ^ { \prime } } ) ] ,
+$$
+
+where $\begin{array} { r } { V ^ { ( t ) } = \sum _ { a } \pi ^ { \mathrm { Q } } ( a | s _ { t ^ { \prime } } ) \nabla _ { \theta } \log \pi ( a | s _ { t ^ { \prime } } ) } \end{array}$ and $\begin{array} { r } { \mathrm { H } ^ { \times } ( p , q ) = - \sum _ { k } p _ { k } \log q _ { k } } \end{array}$ is the cross entropy.
+
+However, if we use the proposed gradient cosine similarity, we get the following update
+
+$$
+\Delta \pmb { \theta } = \alpha \left[ G ^ { ( t ) } + V ^ { ( t ) } \big ( 2 \cdot \mathrm { s i g n } ( \cos ( G ^ { ( t ) } , V ^ { ( t ) } ) ) - 1 \big ) \right] .
+$$
+
+This get a significant boost to performance, and policies that score on average 3 points after 10,000 steps and obtain baseline performance after just one third of steps. Figure 10 shows a depiction of the task and an example solution.
+
+# G ATARI EXPERIMENTS
+
+For these experiments, we use a convolutional architecture as in previous work (Espeholt et al., 2018; Hessel et al., 2018; Mnih et al., 2015; 2016), trained with batched actor-critic with the V-trace algorithm (Espeholt et al., 2018). We use a learning rate of 0.0006 and an entropy cost of 0.01 for all experiments, with a batch size of 32 and 200 parallel actors.
+
+![](images/f691579f46185135cf781da5d88e704ce7dd4f5d5d187565db40468b05fc85fe.jpg)  
+Figure 10: Left most: Initial task ${ \mathcal { T } } _ { m a i n }$ , yellow border denotes starting point and violet ones terminating states. Red states are penalizing with the value in the box while the green ones provide positive reward. Middle Left: Solution found by a single run of Q-learning with uniform exploration policy. Middle Right: Transformed task $\mathcal { T } _ { a u x }$ . Right most: Solution found by gradient cosine similarity driven distillation with policy gradient.
+
+For the single game experiment, Breakout, , we use 0.02 for the threshold on the cosine similarity and, for technical reasons we ended up computing the cosine distance on a per-layer basis and then averaged. We additionally need to do a moving average of the cosine over time $( 0 . 9 9 9 c ^ { ( t - 1 ) } +$ $0 . 0 0 1 c ^ { ( t ) }$ ) to ensure there are no sudden spikes in the weighting due to noisy gradients. Same setting is used for the multi-task experiment, just that the threshold is set to 0.01.
+
+# H COSINE SIMILARITY IN HIGH DIMENSIONS
+
+![](images/f0fc8d0cbdd0becc77340baa84fbea8c5ef08f2427701bb10258a70bf1fca5da.jpg)  
+Figure 11: Cosine similarity as a function of dimensionality. On the left, we generate two random vectors $\theta _ { 1 }$ and $\theta _ { 2 }$ from a Gaussian distribution with zero mean and variance $\sigma ^ { 2 }$ and as expected, the cosine similarity drops to zero very quickly as the number of dimensions increases. On the right, we mimic a scenario where the true gradients of the main and auxiliary are aligned, however we observe only corrupted noisy gradients which are noisy copies of the true underlying vector; we generate $\mu \sim \mathcal { N } ( 0 , I _ { d } )$ and generate $\theta _ { 1 } \sim \mathcal { N } ( \mu , \sigma I _ { d } )$ and $\theta _ { 2 } \sim \mathcal { N } ( \mu , \sigma I _ { d } )$ . In this case, cosine similarity is larger in higher dimensions (as the inner product of the corruption noise goes to zero).

parse/train/r1gl7hC5Km/r1gl7hC5Km_content_list.json

@@ -0,0 +1,1573 @@
+[
+    {
+        "type": "text",
+        "text": "ADAPTING AUXILIARY LOSSES USING GRADIENT SIMILARITY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            236,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "One approach to deal with the statistical inefficiency of neural networks is to rely on auxiliary losses that help to build useful representations. However, it is not always trivial to know if an auxiliary task will be helpful for the main task and when it could start hurting. We propose to use the cosine similarity between gradients of tasks as an adaptive weight to detect when an auxiliary loss is helpful to the main loss. We show that our approach is guaranteed to converge to critical points of the main task and demonstrate the practical usefulness of the proposed algorithm in a few domains: multi-task supervised learning on subsets of ImageNet, reinforcement learning on gridworld, and reinforcement learning on Atari games. ",
+        "bbox": [
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+            268,
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+            393
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            428,
+            336,
+            444
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural networks are extremely powerful function approximators that have excelled on a wide range of tasks (Simonyan and Zisserman, 2015; Mnih et al., 2015; He et al., 2016a; Silver et al., 2016; Vaswani et al., 2017). Despite the state of the art results across domains, they remain data-inefficient and expensive to train. In supervised learning (e.g., image classification), large deep learning (DL) benchmarks with millions of examples are needed for training (Russakovsky et al., 2015) and the additional implication of requiring human intervention to label a large dataset can be prohibitively expensive. In reinforcement learning (RL), agents typically consume millions of frames of experiences before learning to act in complex environments (Silver et al., 2016; Espeholt et al., 2018), which not only puts pressure on compute power but also makes particular domains (e.g., robotics) impractical. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Different techniques have been studied for improving data efficiency, from data augmentation (Krizhevsky et al., 2012; Simonyan and Zisserman, 2015; Hauberg et al., 2016) to transfer learning (Taylor and Stone, 2009; Pan et al., 2010). In this work, we focus on a particular setup for transfer learning. We assume that besides the main task, one has access to one or more auxiliary tasks that share some unknown structure with the main task. To improve data efficiency, these additional tasks can be used as auxiliary losses. However, only the performance on the main task is of interest, even though the model is trained simultaneously on all these tasks. Any improvement on the auxiliary losses is useful only to the extent that it helps learning features or behaviors for the main task. ",
+        "bbox": [
+            174,
+            590,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Auxiliary tasks have been shown to work well in practice (e.g., Zhang et al., 2016; Jaderberg et al., 2017; Mirowski et al., 2017; Papoudakis et al., 2018). However, their success depends on how well aligned the auxiliary losses are with the main task. Knowing this apriori is typically non-trivial and the usefulness of an auxiliary task can change through the course of training. In this work, we explore a simple yet effective approach for measuring the similarity between an auxiliary task and the main task of interest, given the value of their parameters. We show that this measure can be used to decide which auxiliary losses are helpful and for how long. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1.1 NOTATION AND PROBLEM DESCRIPTION ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            819,
+            490,
+            833
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Assume we have a main task $\\mathcal { T } _ { m a i n }$ and an auxiliary task $\\mathcal { T } _ { a u x }$ that induce two losses $\\mathcal { L } _ { m a i n }$ and $\\mathcal { L } _ { a u x }$ . We care about only about maximizing performance on $\\mathcal { T } _ { m a i n }$ ; $\\mathcal { T } _ { a u x }$ is an auxiliary task which is not of direct interest. The goal is to devise an algorithm that can automatically $( i )$ leverage $\\mathcal { T } _ { a u x }$ when it is helpful (e.g. learn faster) and (ii) block negative transfer when $\\mathcal { T } _ { a u x }$ is not helpful (i.e. recover the performance of training only on $\\mathcal { T } _ { m a i n }$ ). Note that this setup is different from multi-objective optimization in which both the tasks are of interest. We propose to parameterize the solution for $\\mathcal { T } _ { m a i n }$ and $\\mathcal { T } _ { a u x }$ by two neural networks, $f ( \\cdot , \\pmb { \\theta } , \\phi _ { m a i n } )$ and $g ( \\cdot , \\pmb { \\theta } , \\phi _ { a u x } )$ , such that they share a subset of parameters denoted here by $\\pmb \\theta$ . Generally, the auxiliary loss literature proposes to minimize ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/148cbdbcc71bcd0dd7cc29120de783f415d0701e52b0737677f8602684b5ba05.jpg",
+        "image_caption": [
+            "Figure 1: Positive example optimization for $L _ { 1 } ( \\theta _ { 1 } , \\theta _ { 2 } ) = \\theta _ { 1 } ^ { 2 } + \\theta _ { 2 } ^ { 2 }$ , $L _ { 2 } ( \\theta _ { 1 } , \\theta _ { 2 } ) = ( \\theta _ { 1 } - 1 ) ^ { 2 } + ( \\theta _ { 2 } - 1 ) ^ { 2 }$ $\\begin{array} { r } { V ( \\theta _ { 1 } , \\theta _ { 2 } ) = [ - \\frac { \\theta _ { 2 } } { \\theta _ { 1 } ^ { 2 } + \\theta _ { 2 } ^ { 2 } } - 2 \\theta _ { 1 } , \\frac { } { \\theta _ { 2 } ^ { 2 } } } \\end{array}$ − 2θ1, θ1θ2+θ2 − 2θ2] where the proposed method speeds up the process (compared on all runs). Each colored trajectory represents one optimization run with random initial position. Star represents the convergence point. All experiments use steepest descent method and run 600 iterations with a constant step size of 0.01. Convergence time is defined as number of steps needed to get below 0.1 loss of $L _ { 1 }$ (gray region). Color of each point represents its alignment with $\\nabla L _ { 1 }$ (green—positive alignment, red—negative alignment, white—directions are perpendicular). In this example $L _ { 2 }$ is helpful for $L _ { 1 }$ as it reinforces good descent directions in most of the space. However, simple mixing is actually slowing optimization down (or makes it fail completely, see the second row), while the proposed methods (weighted and unweighted variants) converge faster (see the third row). When using non-conservative vector field $V$ one obtains lack of convergence (cyclic behaviour, see the fourth row), while the proposed merging still works well (see the last row). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            731
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            102,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/787e6a8c69199ca3c6648224feedc6bdaa741f9e2b817f3318d04dd702446592.jpg",
+        "text": "$$\n\\operatorname* { a r g m i n } _ { \\theta , \\phi _ { m a i n } , \\phi _ { a u x } } \\mathcal { L } _ { m a i n } ( \\theta , \\phi _ { m a i n } ) + \\lambda \\mathcal { L } _ { a u x } ( \\theta , \\phi _ { a u x } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            138,
+            666,
+            166
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "under the intuition that modifying $\\pmb \\theta$ to minimize $\\mathcal { L } _ { a u x }$ will improve ${ \\mathcal { L } } _ { m a i n }$ if the two tasks are ientlyis for . We pgiven the weight . That is, $\\lambda$ at each learning iteration  each optimization iterati $t$ by how usefuln, we want to   \n$\\mathcal { T } _ { a u x }$ $\\mathcal { T } _ { m a i n }$ $\\pmb { \\theta } ^ { ( t ) } , \\phi _ { m a i n } ^ { ( t ) } , \\phi _ { a u x } ^ { ( t ) }$   \nefficiently approximate the solution to ",
+        "bbox": [
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+            231
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/30e2122a8eea7e138bb329f5375b7b0eaffd7154673385a0e37735fd1af1983c.jpg",
+        "text": "$$\n\\underset { \\lambda ^ { ( t ) } } { \\arg \\operatorname* { m i n } } \\mathcal { L } _ { m a i n } \\left( \\pmb { \\theta } ^ { ( t ) } - \\alpha \\nabla _ { \\theta } \\big ( \\mathcal { L } _ { m a i n } + \\lambda ^ { ( t ) } \\mathcal { L } _ { a u x } \\big ) , \\phi _ { m a i n } ^ { ( t ) } - \\alpha \\nabla _ { \\phi _ { m a i n } } \\mathcal { L } _ { m a i n } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            236,
+            237,
+            761,
+            270
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Note that the input space of $\\mathcal { T } _ { m a i n }$ and $\\mathcal { T } _ { a u x }$ do not have to match. In particular, $\\mathcal { T } _ { a u x }$ does not need to be defined for an input of $\\mathcal { T } _ { m a i n }$ or the other way around.1 Solving equation 2 is expensive. Instead, we look for a cheap heuristic to approximate $\\lambda ^ { ( i ) }$ which is better than keeping $\\lambda ^ { ( t ) }$ constant and does not require hyper-tuning. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 COSINE SIMILARITY BETWEEN GRADIENTS OF TASKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            352,
+            660,
+            369
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose to use the cosine similarity of gradients between tasks as a measure of task similarity and hence for approximating $\\lambda ^ { ( t ) }$ . Consider an example where the main function to minimize is $\\mathcal { L } _ { m a i n } = ( \\theta - 1 \\bar { 0 } ) ^ { 2 }$ and the auxiliary function is $\\mathcal { L } _ { a u x } = \\theta ^ { 2 }$ , their gradients are $\\nabla _ { \\theta } \\mathcal { L } _ { m a i n } = 2 ( \\theta - 1 0 )$ and $\\nabla _ { \\theta } \\mathcal { L } _ { a u x } = 2 \\theta$ respectively. When $\\theta$ is initialized at $\\theta = - 2 0$ , the gradients of the main and auxiliary functions point in the same direction and the cosine similarity is 1; minimizing the auxiliary loss is beneficial for minimizing the main. However, at a different point, $\\theta = 5$ , the two gradients point in different directions and the cosine similarity is $- 1$ ; minimizing the auxiliary loss would hinder minimizing the main loss (See Figure 7 in Appendix B for an illustration.). ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "This example suggests a natural strategy for approximating $\\lambda ^ { ( t ) }$ : minimize the auxiliary loss as long as its gradient has non-negative cosine similarity with the target gradient; otherwise, the auxiliary loss should be ignored. This follows the well-known intuition that if a vector is in the same half-space as the gradient of a function $f$ , then it is a decent direction for $f$ . This reduces our strategy to ask if the gradient of the auxiliary loss is a descent direction for the main loss of interest. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Proposition 1. Given any gradient vector field $G ( \\pmb \\theta ) = \\nabla _ { \\pmb \\theta } \\mathcal { L } ( \\pmb \\theta )$ and any vector field $V ( \\pmb \\theta )$ (such as the gradient of another loss function, or an arbitrary set of updates), an update rule of the form ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/23b17b208ffb71698242d2ec037609fbd7ee9a8be12812c6cc0414b8118b7970.jpg",
+        "text": "$$\n\\pmb \\theta ^ { ( t + 1 ) } : = \\pmb \\theta ^ { ( t ) } - \\alpha ^ { ( t ) } ( G ( \\pmb \\theta ^ { ( t ) } ) + V ( \\pmb \\theta ^ { ( t ) } ) \\operatorname* { m a x } ( 0 , \\cos ( G ( \\pmb \\theta ^ { ( t ) } ) , V ( \\pmb \\theta ^ { ( t ) } ) ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "converges to the local minimum of $\\mathcal { L }$ given small enough $\\alpha ^ { ( t ) }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proof is provided in Appendix A.1. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Note that the above statement does not guarantee any improvement of convergence, but only guarantees lack of divergence. In particular, cosine similarity is not a silver bullet that guarantees positive transfer, but it can drop the “worst-case scenarios”. In principle, one can create example functions where the convergence of the main loss is affected both positively (see Figure 1) and negatively (see Figure 8 in Appendix D). Nevertheless, convergence on the main task is guaranteed for our proposed strategy, as the proposition shows. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In addition, it is important to note that simply adding an arbitrary vector field does not have the convergence property. For example, use function $\\begin{array} { r } { V ( \\pmb { \\theta } ) = - \\nabla _ { \\pmb { \\theta } } \\mathcal { L } ( \\pmb { \\theta } ) + \\left[ - \\frac { \\theta _ { 2 } } { \\theta _ { 1 } ^ { 2 } + \\theta _ { 2 } ^ { 2 } } , \\frac { \\theta _ { 1 } } { \\theta _ { 1 } ^ { 2 } + \\theta _ { 2 } ^ { 2 } } \\right] ^ { T } } \\end{array}$ as a two-dimensional case, which leads to an update rule of $\\begin{array} { r } { \\pmb { \\theta } ^ { ( t + 1 ) } = \\pmb { \\theta } ^ { ( t ) } - \\alpha \\left[ - \\frac { \\theta _ { 2 } } { \\theta _ { 1 } ^ { 2 } + \\theta _ { 2 } ^ { 2 } } , \\frac { \\theta _ { 1 } } { \\theta _ { 1 } ^ { 2 } + \\theta _ { 2 } ^ { 2 } } \\right] ^ { T } } \\end{array}$ This is a non-conservative vector field which cases the optimizer to follow concentric circles around the origin (see the fourth row in Figure 1). This is crucial to note for some realistic scenarios where one does not always form a gradient field (e.g., the update rule of the Q-learning algorithm in RL). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Figure 1 provides a few illustrative examples on quadratic functions using the proposed approach, which helps intuitively understand the kind of scenarios for which the approach could help. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "The above proposition refers to losses with the same set of parameters $\\pmb { \\theta }$ , while equation 2 refers to the scenario when each loss has task specific parameters (e.g. $\\phi _ { m a i n }$ and $\\phi _ { a u x }$ ). The following proposition extends to this scenario: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. Given two losses parametrized with $\\Theta$ (some of which are shared $\\pmb \\theta$ and some unique to each loss $\\phi _ { m a i n }$ and $\\phi _ { a u x }$ ), learning rule: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5ae1af0b4a5a2e2178fcdf8448d9862bd3bea9507190433ca670bf083a121566.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbf { \\Phi } _ { l } ^ { ( t + 1 ) } : = \\pmb { \\theta } ^ { ( t ) } - \\alpha ^ { ( t ) } \\big ( \\nabla _ { \\theta } \\mathcal { L } _ { m a i n } ( \\pmb { \\theta } ^ { ( t ) } ) + \\nabla _ { \\theta } \\mathcal { L } _ { a u x } ( \\pmb { \\theta } ^ { ( t ) } ) \\operatorname* { m a x } \\big ( 0 , \\cos ( \\nabla _ { \\theta } \\mathcal { L } _ { m a i n } ( \\pmb { \\theta } ^ { ( t ) } ) , \\nabla _ { \\theta } \\mathcal { L } _ { a u x } ( \\pmb { \\theta } ^ { ( t ) } ) \\big ) \\big ) } \\\\ & { \\mathbf { \\Phi } _ { m a i n } ^ { ( t + 1 ) } : = \\phi _ { m a i n } ^ { ( t ) } - \\alpha ^ { ( t ) } \\nabla _ { \\phi _ { m a i n } } \\mathcal { L } _ { m a i n } ( \\mathbf { \\Theta } _ { } ^ { ( t ) } ) \\quad \\mathbf { \\Phi } _ { a n d } \\quad \\phi _ { a u x } ^ { ( t + 1 ) } : = \\phi _ { a u x } ^ { ( t ) } - \\alpha ^ { ( t ) } \\nabla _ { \\phi _ { a u x } } \\mathcal { L } _ { a u x } ( \\pmb { \\Theta } ^ { ( t ) } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "leads to convergence to local minimum of $\\mathcal { L } _ { m a i n } \\ w . r . t .$ . $( \\theta , \\phi _ { m a i n } )$ given small enough $\\alpha ^ { ( t ) }$ . ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Proof. Comes directly from the previous proposition that $G = \\nabla _ { \\theta } \\mathcal { L } _ { m a i n }$ and $V = \\nabla _ { \\theta } \\mathcal { L } _ { a u x }$ . For any vector fields $A , B , C$ , we have $\\langle A , B \\rangle \\geq 0$ and $\\langle C , B \\rangle \\geq 0$ implies $\\langle A + C , B \\rangle \\geq 0$ . □ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Analogous guarantees hold for the unweighted version of this algorithm, where instead of weighting by $\\cos ( G , V )$ we use a binary weight $( \\mathrm { s i g n } ( \\cos ( G , V ) ) + 1 ) / \\bar { 2 }$ which is equivalent to using $V$ iff $\\cos ( G , V ) > 0$ . When training with mini-batches, accurately estimating $\\cos ( G , V )$ can be difficult due to mini-batch noise; the unweighted variant only requires $\\operatorname { s i g n } ( \\cos ( G , V )$ which can be estimated more robustly. Hence, we use this variant in our experiments unless otherwise specified. Additionally, note that there is no guarantee that $\\mathcal { L } _ { a u x }$ is optimized. For example, if $\\mathcal { L } _ { a u x } = - \\mathcal { L } _ { m a i n }$ then $\\mathcal { L } _ { a u x }$ is ignored (see a visualization in the last row of Figure 1). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Despite its simplicity, the proposed update rule can give rise to interesting phenomena. We can show that the emerging vector field could be non-conservative, which means there does not exist a loss function for which it is a gradient. While this might seem problematic (for gradient-descent-based optimizers), it describes only the global structure—typically used optimizers are local in nature and they do local, linear or quadratic approximations of the function (Shwartz-Ziv and Tishby, 2017). Consequently, in practice, one should not expect any negative effects from this phenomena, as it simply shows that our proposed technique is in fact qualitatively changing the nature of the update rules for training. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3. In general, the proposed update rule does not have to create a conservative vector field. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proof is provided in Appendix A.2. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 APPLICATIONS OF GRADIENT COSINE SIMILARITY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we use the gradient cosine similarity to decide when to train on the auxiliary task. All experiments (unless otherwise stated) follow the unweighted version of our method, summarized in Algorithm 1. The weighted version of our method is summarized in Algorithm 2, Appendix C. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Unweighted version of our method. ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1: Initialize shared parameters $\\pmb { \\theta }$ and task specific parameters $\\phi _ { m a i n } , \\phi _ { a u x }$ randomly.   \n2: for iter $= 1$ : max iter do   \n3: Compute $\\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { m a i n }$ , $\\nabla _ { \\phi _ { m a i n } } \\mathcal { L } _ { m a i n }$ , $\\nabla _ { \\theta } \\mathcal { L } _ { a u x }$ , $\\nabla _ { \\phi _ { a u x } } \\mathcal { L } _ { a u x }$ .   \n4: Update $\\phi _ { m a i n }$ and $\\phi _ { a u x }$ using corresponding gradients   \n5: if $\\cos ( \\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { m a i n } , \\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { a u x } ) \\geq 0$ then   \n6: Update $\\pmb \\theta$ using $\\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { m a i n } + \\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { a u x }$   \n7: else   \n8: Update $\\pmb \\theta$ using $\\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { m a i n }$ ",
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+    {
+        "type": "text",
+        "text": "3.1 EXPERIMENTS ON IMAGE CLASSIFICATION TASKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "First, we consider a classification problem on ImageNet (Russakovsky et al., 2015) and design a simple multi-task binary classification task to test our hypothesis that transferable tasks should have high cosine similarity (and vice versa). We take a pair of classes from ImageNet, refer to these as class $A$ and class $B$ ; all the other 998 classes in ImageNet (except $A$ and $B$ ) are referred to as the background. Our tasks $\\mathcal { T } _ { m a i n }$ and $\\mathcal { T } _ { a u x }$ are then formed as a binary classification of if an image is class $A$ (otherwise background) and if an image is class $B$ (otherwise background) respectively. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Ideally, we want to pick groups of class pairs that reflect near or far distance, for the purpose of providing a baseline of transferability. Therefore, we used two distance measures, lowest common ancestor $( L C A )$ in the ImageNet label hierarchy and Frechet Inception Distance (FID) (Heusel et al., 2017) of pre-trained image embedding, to serve as a ground truth of class similarity for selecting class pair $A$ and $B$ . Based on these measures, we picked three pair of classes for near, class 871 (trimaran) vs. 484 (catamaran), 250 (Siberian husky) vs. 249 (malamute), and 238 (Greater Swiss Mountain dog) vs. 241 (Entleucher); and for far, class 920 (traffic light) vs. 62 (rock python), 926 (hotpot) vs. 800 (slot), and 48 (Komodo dragon) vs. 920 (traffic light). Details on the class pair selection are described in Appendix E. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We use a modified ResNetV2-18 model (He et al., 2016b) for training in this experiment. All parameters in the convolutional layers are shared (denote as $\\pmb { \\theta }$ ), followed by task-specific parameters $\\phi _ { m a i n }$ and $\\phi _ { a u x }$ . First, we use a multi-task learning setup and minimize $\\mathcal { L } _ { m a i n } + \\mathcal { L } _ { a u x }$ , and measure cosine similarity on $\\pmb { \\theta }$ through the course of training. Figure 2(a) shows that cosine similarity is higher for near pairs (blue lines) and lower for far pairs (red lines). Next, we compare single-task training, multi-task training, and our proposed variant on two scenarios (i) auxiliary task helps and (ii) auxiliary task hurts. As mentioned earlier, our goal is have a method that can automatically leverage auxiliary tasks when they are helpful and avoid negative transfer when auxiliary tasks are not helpful. Figure 2(b) shows that on a near pair, all variants perform similarly in terms of final performance; furthermore, our method performs similar to multi-task learning and learns faster than single task because the task is transferable. Figure 2(c) shows that on a far pair, multi-task learning leads to poorer performance than single-task learning on the main task due to the potential negative transfer, whereas our method of using gradient cosine similarity blocks negative transfer and automatically achieves performance that is comparable to single-task learning. ",
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+    {
+        "type": "image",
+        "img_path": "images/c61cf34feaf0d0382683306cc9465dfc6f98c46145d1c301328f729afaaa9664.jpg",
+        "image_caption": [
+            "Figure 2: Multi-task learning setup on ImageNet class pairs. (a): gradient cosine similarity is higher for near pairs and lower for far pairs. (b) and (c): testing accuracy on single task (dotted), naive multi-task (dashed), and our method (solid). Naive multi-task learning helps in near pairs (see $( b )$ ) but hurts in far pairs (see (c)) because of its lack of the ability to prevent negative effects from the auxiliary task to the main task. Our method can overcome this limitation by dropping the auxiliary task when its gradient direction disagrees with the main task, thus achieving the best of both worlds: matching the multi-task performance on near pairs (see $( b )$ where our method and multi-task learning learn faster than single task only) and the single task performance in far pairs (see (c) where multi-task learning performs poorly, but our method automatically recovers single task performance). "
+        ],
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+    {
+        "type": "text",
+        "text": "3.2 EXPERIMENTS ON REINFORCEMENT LEARNING GRIDWORLD TASKS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We then consider a typicalfuture discounted rewards (POMDP). There have bee one aims to find a policy  in a partially observable proposed to solve this op $\\pi$ that maximizes sum ofarkov decision processmization problem, from $\\mathbb { E } _ { \\pi } [ \\sum _ { t ^ { \\prime } = 1 } ^ { N } \\gamma ^ { t ^ { \\prime } - 1 } r _ { t ^ { \\prime } } ]$ classical policy gradient (Williams, 1992), Q-Learning (Watkins, 1989), to the more modern Proximal Policy Optimization (Schulman et al., 2017) and V-Trace (Espeholt et al., 2018). Inherently, these techniques are data inefficient due to the complexity of the problem. One way to address this issue is to use transfer learning, such as transfer from pre-trained policies (Rusu et al., 2015). However, a teacher policy is not always available for the main task. When in this scenario, one can train policies in other tasks that share enough similarities and hope for a positive transfer. One way of exploiting this extra information is to use behavioral cloning, or distillation (Hinton et al., 2015; Rusu et al., 2015), to guide the main task in its initial learning phase (Schmitt et al., 2018), although it might be difficult to find a suitable strategy that combines the main and auxiliary losses and/or smoothly transition between them. Typically, the teacher policy can be treated as an auxiliary loss (Schmitt et al., 2018) or a prior (Teh et al., 2017) with a fixed mixing coefficient. However, these techniques become unsound if the teacher policy is helpful only in specific states while hindering in other states. ",
+        "bbox": [
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We propose a simple RL experiment to show that our method is capable of finding the strategy of combining the main loss and the auxiliary loss. We define a distribution over a set of $1 5 \\times 1 5$ gridworlds, where an agent observes its surrounding (up to four pixels away) and can move in four directions.We randomly place two types of positive rewards, $+ 5$ and $+ 1 0$ points, both terminating an episode. In order to guarantee a finite length of episodes, we add a fixed probability of 0.01 of transitioning to a non-rewarding terminal state. Experiment details are provided in Appendix F. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "First, we train a Q-learning agent on such a gridworld which gives us a teacher policy $\\pi ^ { 2 }$ . Then, we create the main task to which there is a possible positive knowledge transfer by keeping the environment with the same gridworld layout but remove the $+ 1 0$ rewards (and corresponding states are no longer terminating). Consequently, we have two tasks: the auxiliary task $\\mathcal { T } _ { a u x }$ where we have a strong teacher policy $\\pi ^ { 2 }$ , and the main task $\\mathcal { T } _ { m a i n }$ where the $+ 1 0$ rewards are removed. We sample 1, 000 such environment pairs and report expected returns obtained (100 evaluation episodes per evaluation point) using various training regimes. One can use any RL method to solve the main task and learn $\\pi$ , here we use episode-level policy gradient (Williams, 1992) with value function as a baseline method, which gives a score of slightly above 1 point after $1 0 , 0 0 0$ steps of training (see the top row of Figure 3). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To leverage teacher policies, we define the auxiliary loss to be a distillation loss, which is a per-state cross-entropy between teacher’s and student’s distributions over actions. First, we test using solely the distillation loss while sampling trajectories from the student. We recover a subset of teacher’s behaviors and end up with 0 point—an expected negative transfer as the teacher is guiding us to states that are no longer rewarding. Then, we test simply adding gradients estimated by policy gradient and distillation. The resulting policy learns quickly but saturates at a return of 1 point, showing very limited positive transfer. Lastly, when using our proposed gradient cosine similarity as the measure of transferability, we get a significant performance boost. The learned policies reach baseline performance after just one-third of steps taken by the baseline, and on average obtain 3 points after 10, 000 steps.2 See Figure 3 for all learning curves. In Appendix F, we visualize the environment and show an example solution. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This experiment shows that gradient cosine similarity allows using knowledge from other related tasks in an automatic fashion. The agent is simply ignoring teacher signal when it disagrees with policy gradient estimator. If they do agree in terms of which actions to reinforce—teacher logits are used for better replication of useful policies. In particular, in the bottom row of Figure 3, we present an experiment of transfer between the same task $\\mathcal { T } _ { m a i n }$ . We see that the cosine similarity experiments underperformed that of simply adding the two losses. This is expected as the noise in the gradients makes it hard to measure if the two tasks are a good fit or not. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.3 EXPERIMENTS ON ATARI ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Finally, we consider a similar RL setup on the Atari domain (Bellemare et al., 2013). For this set of experiments, we follow the same convolutional architecture as in previous works (Mnih et al., 2015; 2016; Espeholt et al., 2018; Hessel et al., 2018) and train using the batched actor-critic with V-trace algorithm (Espeholt et al., 2018). Details on the experiment setup are provided in Appendix G. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "First, we look at training an agent to play a main task (here, Breakout) given a sub-optimal teacher solution to the task. Analogous to the previous experiment, we leverage information about the task by distilling the teacher’s behaviour with a Kullback-Leibler (KL) loss. As expected, solely relying on distilling from the sub-optimal teacher $( O n l y K L )$ leads to lower performance. Training with both distillation and RL losses $\\left( R L + K L ( B a s e l i n e ) \\right)$ leads to slightly better but also sub-optimal performance. While both approaches learn very quickly, they plateau much lower than the pure RL approach (RL(Baseline)). In our method $( R L + K L ( O u r M e t h o d ) )$ , the KL penalty is scaled at every time-step by the cosine similarity between the policy gradient and distillation losses; once this falls below a fixed threshold, the loss is ‘turned off’. Figure 4 shows that our approach is able to learn quickly at the start but continue fine-tuning with pure RL loss once the distillation loss is zeroed out. ",
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+    {
+        "type": "image",
+        "img_path": "images/a0a7154a29483f61df3c355d609d2e08ae4ab9ffd2eadc83b5a83fbe3048fb51.jpg",
+        "image_caption": [
+            "Figure 3: Top row: expected learning curves for cross-environment distillation experiments, averaged over 1, 000 partially observable gridworlds. Teacher’s policy is based on Q-Learning, its performance in a new environment (with modified positive rewards) is represented by the top dotted line. The bottom dotted line represents random policy score. Each column represents a different temperature applied to the teacher policy. 0 temperature refers to the original deterministic greedy policy given by Q-Learning. We report five methods: reward using just policy gradient in the new task; distill using just distillation cost towards the teacher; add adding the two above; cos using the weighted version of our method (Algorithm 2); strict cos using the unweighted version of our method (Algorithm 1). Bottom row: expected learning curves for same-environment distillation experiments when the teacher is perfect. In this case, the optimal thing is to trust the teacher everywhere. "
+        ],
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+        "text": "Lastly, we consider a setting where the main task ${ \\mathcal { T } } _ { m a i n }$ is to train an agent to play two Atari games, Breakout and Ms. PacMan. Similar to previous experiment, we have access to a teacher trained on just Breakout as the auxiliary task $\\mathcal { T } _ { a u x }$ , from which we distill a policy via KL loss. Note that $\\mathcal { T } _ { m a i n }$ itself is chosen to be Multitask to illustrate a complex scenario where $\\mathcal { T } _ { a u x }$ helps with only part of $\\mathcal { T } _ { m a i n }$ , and that too only initially. We consider a distillation loss as was done previously by adding the auxiliary KL loss $\\mathcal { L } _ { a u x }$ (between the teacher and student policies) to the RL multi-task loss ${ \\mathcal { L } } _ { m a i n }$ at every time step. Intuitively, doing so would result in the agent only be able to solve one of the tasks—the one the teacher knows about, as the gradients from distillation loss would interfere with the policy gradient. Figure 5 shows that, compared to the baseline Multitask and the simple addition of Multitask $R L +$ Distillation approaches where the agent learns one task at the expense of the other, our method of scaling the auxiliary loss by gradient cosine similarity is able to compensate for this by learning from the teacher and then turning off the auxiliary distillation; it learns Ms. PacMan without forgetting Breakout. The evolution of the gradient cosine similarity between Breakout and Ms. PacMan provides a meaningful cue for the usefulness of $\\mathcal { L } _ { a u x }$ ",
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+        "image_caption": [
+            "Figure 4: Results on Breakout. We look at the effects of distilling a sub-optimal policy as an auxiliary task. "
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+        "text": "4 RELATED WORK ",
+        "text_level": 1,
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+        "text": "Our work is related to the literature on identifying task similarity in transfer learning. It is generally believed that positive transfer can be achieved when source task(s) and target task(s) are related. ",
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+        "image_caption": [
+            "Figure 5: Results on Breakout and Ms. PacMan (averaged over 3 seeds). The two plots to the left show performance on Breakout and Ms. PacMan respectively. The third plot shows how the gradient cosine similarity between the two tasks changes during training. The last plot shows an average score of the multi-task performance (normalized independently for each game based on the best score achieved across all experiments). Our method is able to learn both games without forgetting and achieves the best average performance. "
+        ],
+        "image_footnote": [],
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+        "text": "However, it is usually assumed that this relatedness mapping is provided by human experts (Taylor and Stone, 2009; Pan et al., 2010); few works have addressed the problem of finding a general measure of similarity to predict transferability between tasks. In image classification, Yosinski et al. (2014) defined image similarity in ImageNet by manually splitting classes into man-made versus natural objects. In RL, methods have been proposed to use the Markov decision process (MDP) similarity as a measure of task relatedness (Carroll and Seppi, 2005; Ammar et al., 2014). While in some degree capture task similarity, these measures are often domain-specific and not generalizable. In addition, none of these works have explicitly used the learned similarity metric to improve performance. In our work, we propose to use cosine similarity of gradients as a generalizable measure across domains and show it can be directly leveraged to improve the performance of the main task. One important aspect of task similarity for transfer is that it is highly dependent on the parametrization of the model and current value of the parameters. We exploit this property by providing a heuristic similarity measure for the current parameters, resulting in an approach that relies on an adaptive weight over the updates of the model. ",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Auxiliary tasks have shown to be beneficial in facilitating learning across domains. In image classification, Zhang et al. (2016) used unsupervised reconstruction tasks. In RL, the UNREAL framework (Jaderberg et al., 2017) incorporates unsupervised control tasks along with reward prediction learning as auxiliary tasks. Mirowski et al. (2017) studied auxiliary tasks in the context of navigation. Papoudakis et al. (2018) also explored auxiliary loses for VizDoom. However, these works rely on empirical results and do not address how the auxiliary tasks were selected in the first instance. In this work, we aim to propose a simple yet effective way of explicitly selecting auxiliary tasks by using cosine similarity of task gradients. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Our work is also related to multi-task learning (Caruana, 1997), particularly the line of work on using adaptive scaling techniques for multi-objective learning. For example, a recently developed algorithm, GradNorm (Chen et al., 2018), uses gradient magnitude to scale loss function for each task, aiming to learn well for all tasks. Similarly, Kendall et al. (2018) proposed a weighting mechanism by considering the homoscedastic uncertainty of each task. However, our work is different in two ways: first, in our problem setup, we care only about the performance of the main task and we do not care about all tasks; hence, the optimization goal is different from their setup which is more similar to traditional multi-objective optimization. Furthermore, it is important to note that our method differs from aforementioned work in that they scale the losses individually without looking at their interaction (which can lead to poor performance in our problem setup, when the auxiliary task hurts the main task), whereas we look for alignments in the vector field between the main and the auxiliary task, and the auxiliary task is used only when it is well-aligned with the main task. ",
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+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this work, we explored a simple yet efficient technique to ensure that an auxiliary loss does not hurt the learning on the main task. The proposed approach reduces to applying gradients of the auxiliary task only if they are a descent direction of the main task. ",
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+    {
+        "type": "text",
+        "text": "We discuss here a few shortcomings of this method. First, estimating the cosine similarity between the gradients of tasks could be expensive or noisy and that the threshold for turning off the auxiliary is a fixed constant. These could be addressed by calculating a running average of the cosine similarity to get a smoother result and potentially hyper-tune the threshold instead of setting it as a fixed constant. One might argue that our approach would fail in high-dimensional spaces since random vectors in such spaces tend to be orthogonal, so that cosine similarity will be naturally driven to 0. In fact, this is not the case; if two gradients are meant to be co-linear, the noise components cancel each other thus will not affect the cosine similarity estimation. We empirically explore this in Appendix H. Second, the new loss surface might be less smooth which can be problematic when using optimizers that rely on statistics of the gradients or second order information (e.g. Adam or RMSprop). In these cases, the transition from just the gradient of the main task to the sum of the gradients can affect the statistics of the optimizer in unwanted ways. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Lastly, although the proposed approach works well empirically on complex and noisy tasks like Atari games, as discussed in Section 2, it guarantees only that the main task will converge, but not how fast it will be. While removing the worst case scenarios is important and a good first step, one might care more for data efficiency when using auxiliary losses (i.e., faster convergence). Particularly in Appendix D Figure 8, we construct a counter-example where the proposed update rule slows down learning, compared to optimizing the main task alone. Nevertheless, we have empirically shown the potential of using the proposed hypothesis as a simple yet efficient way of picking a suitable auxiliary task. While we have mostly considered scenarios where the auxiliary task helps initially but hurts later, it would be interesting to explore settings where the auxiliary task hurts initially but helps in the end. Examples of such are annealing $\\beta$ in $\\beta$ -VAE (Higgins et al., 2017) and annealing the confidence penalty in (Pereyra et al., 2017). ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            452,
+            285,
+            467
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "H. B. Ammar, E. Eaton, M. E. Taylor, D. C. Mocanu, K. Driessens, G. Weiss, and K. Tuyls. An automated measure of MDP similarity for transfer in reinforcement learning. In Workshops at the Twenty-Eighth AAAI Conference on Artificial Intelligence, 2014.   \nM. G. Bellemare, Y. Naddaf, J. Veness, and M. Bowling. The arcade learning environment: An evaluation platform for general agents. Journal of Artificial Intelligence Research, 47:253–279, 2013.   \nJ. L. Carroll and K. Seppi. Task similarity measures for transfer in reinforcement learning task libraries. In Neural Networks, 2005. IJCNN’05. Proceedings. 2005 IEEE International Joint Conference on, volume 2, pages 803–808. IEEE, 2005.   \nR. Caruana. Multitask learning. Machine learning, 28(1):41–75, 1997.   \nZ. Chen, V. Badrinarayanan, C.-Y. Lee, and A. Rabinovich. Gradnorm: Gradient normalization for adaptive loss balancing in deep multitask networks. In ICML, 2018.   \nL. Espeholt, H. Soyer, R. Munos, K. Simonyan, V. Mnih, T. Ward, Y. Doron, V. Firoiu, T. Harley, I. Dunning, et al. IMPALA: Scalable distributed Deep-RL with importance weighted actor-learner architectures. arXiv preprint arXiv:1802.01561, 2018.   \nA. Goldstein. Cauchy’s method of minimization. Numerische Mathematik, 4(1):146–150, 1962.   \nS. Hauberg, O. Freifeld, A. B. L. Larsen, J. Fisher, and L. Hansen. Dreaming more data: Classdependent distributions over diffeomorphisms for learned data augmentation. In Artificial Intelligence and Statistics, pages 342–350, 2016.   \nK. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016a.   \nK. He, X. Zhang, S. Ren, and J. Sun. Identity mappings in deep residual networks. In European conference on computer vision, pages 630–645. Springer, 2016b.   \nM. Hessel, H. Soyer, L. Espeholt, W. Czarnecki, S. Schmitt, and H. van Hasselt. Multi-task deep reinforcement learning with PopArt. arXiv preprint arXiv:1809.04474, 2018.   \nM. Heusel, H. Ramsauer, T. Unterthiner, B. Nessler, G. Klambauer, and S. Hochreiter. GANs trained by a two time-scale update rule converge to a Nash equilibrium. arXiv preprint arXiv:1706.08500, 2017.   \nI. Higgins, L. Matthey, A. Pal, C. Burgess, X. Glorot, M. Botvinick, S. Mohamed, and A. Lerchner. $\\beta$ -VAE: Learning basic visual concepts with a constrained variational framework. In ICLR, 2017.   \nG. Hinton, O. Vinyals, and J. Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.   \nM. Jaderberg, V. Mnih, W. M. Czarnecki, T. Schaul, J. Z. Leibo, D. Silver, and K. Kavukcuoglu. Reinforcement learning with unsupervised auxiliary tasks. In ICLR, 2017.   \nA. Kendall, Y. Gal, and R. Cipolla. Multi-task learning using uncertainty to weigh losses for scene geometry and semantics. In CVPR, 2018.   \nA. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet classification with deep convolutional neural networks. In NIPS, 2012.   \nP. Mirowski, R. Pascanu, F. Viola, H. Soyer, A. J. Ballard, A. Banino, M. Denil, R. Goroshin, L. Sifre, K. Kavukcuoglu, et al. Learning to navigate in complex environments. In ICLR, 2017.   \nV. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski, et al. Human-level control through deep reinforcement learning. Nature, 518(7540):529, 2015.   \nV. Mnih, A. P. Badia, M. Mirza, A. Graves, T. Lillicrap, T. Harley, D. Silver, and K. Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In ICML, 2016.   \nS. J. Pan, Q. Yang, et al. A survey on transfer learning. IEEE Transactions on knowledge and data engineering, 22(10):1345–1359, 2010.   \nG. Papoudakis, K. C. Chatzidimitriou, and P. A. Mitkas. Deep reinforcement learning for doom using unsupervised auxiliary tasks. CoRR, abs/1807.01960, 2018.   \nG. Pereyra, G. 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Dhariwal, A. Radford, and O. Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017.   \nR. Shwartz-Ziv and N. Tishby. Opening the black box of deep neural networks via information. arXiv preprint arXiv:1703.00810, 2017.   \nD. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, et al. Mastering the game of Go with deep neural networks and tree search. nature, 529(7587):484, 2016.   \nK. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. In ICLR, 2015.   \nM. E. Taylor and P. Stone. Transfer learning for reinforcement learning domains: A survey. JMLR, 2009.   \nY. Teh, V. Bapst, W. M. Czarnecki, J. Quan, J. Kirkpatrick, R. Hadsell, N. Heess, and R. Pascanu. Distral: Robust multitask reinforcement learning. In NIPS, 2017.   \nA. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin. Attention is all you need. In NIPS, 2017.   \nC. J. C. H. Watkins. Learning from delayed rewards. PhD thesis, King’s College, Cambridge, 1989.   \nR. J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256, 1992.   \nJ. Yosinski, J. Clune, Y. Bengio, and H. Lipson. How transferable are features in deep neural networks? In NIPS, 2014.   \nY. Zhang, K. Lee, and H. Lee. Augmenting supervised neural networks with unsupervised objectives for large-scale image classification. In ICML, 2016. ",
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+    {
+        "type": "text",
+        "text": "A PROOFS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A.1 PROOF FOR PROPOSITION 1 ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Given any gradient vector field $G ( \\pmb \\theta ) = \\nabla _ { \\pmb \\theta } \\mathcal { L } ( \\pmb \\theta )$ and any vector field $V ( \\pmb \\theta )$ (such as gradient of another loss function, but could be arbitrary set of updates), an update rule of the form ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/43f055caaee335dbde5d99a5250bdcb3e88dbcc466df9fd81b8ff1beecd39abd.jpg",
+        "text": "$$\n\\pmb \\theta ^ { ( t + 1 ) } : = \\pmb \\theta ^ { ( t ) } - \\alpha ^ { ( t ) } ( G ( \\pmb \\theta ^ { ( t ) } ) + V ( \\pmb \\theta ^ { ( t ) } ) \\operatorname* { m a x } ( 0 , \\cos ( G ( \\pmb \\theta ^ { ( t ) } ) , V ( \\pmb \\theta ^ { ( t ) } ) ) )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "converges to the local minimum of $\\mathcal { L }$ given small enough $\\alpha ^ { ( t ) }$ . ",
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+    {
+        "type": "text",
+        "text": "Proof. Let us denote ",
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+    {
+        "type": "equation",
+        "img_path": "images/f77c343c364f2a2086b13f66a77336c9d5e35add33aa11a9344d1d102f4d88df.jpg",
+        "text": "$$\n\\begin{array} { r l } & { G ^ { ( t ) } : = G ( \\pmb { \\theta } ^ { ( t ) } ) \\qquad V ^ { ( t ) } : = V ( \\pmb { \\theta } ^ { ( t ) } ) \\qquad \\nabla \\mathcal { L } ^ { ( t ) } : = \\nabla _ { \\pmb { \\theta } } \\mathcal { L } ( \\pmb { \\theta } ^ { ( t ) } ) } \\\\ & { \\qquad \\Delta \\pmb { \\theta } ^ { ( t ) } : = G ^ { ( t ) } + V ^ { ( t ) } \\operatorname* { m a x } ( 0 , \\cos ( G ^ { ( t ) } , V ^ { ( t ) } ) ) . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Our update rule is simply $\\pmb { \\theta } ^ { ( t + 1 ) } : = \\pmb { \\theta } ^ { ( t ) } - \\alpha ^ { ( t ) } \\Delta \\pmb { \\theta } ^ { ( t ) }$ and we have ",
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+        "type": "equation",
+        "img_path": "images/9ea7c82eeac5a70893014a48948d65e66a0b3f597c652bc016ea3517983fb28f.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\langle \\Delta \\pmb { \\theta } ^ { ( t ) } , \\nabla \\mathcal { L } ^ { ( t ) } \\rangle = \\langle G ^ { ( t ) } + V ^ { ( t ) } \\operatorname* { m a x } ( 0 , \\cos ( G ^ { ( t ) } , V ^ { ( t ) } ) ) , \\nabla \\mathcal { L } ^ { ( t ) } \\rangle } \\\\ & { \\quad \\quad \\quad \\quad \\quad = \\langle G ^ { ( t ) } , \\nabla \\mathcal { L } ^ { ( t ) } \\rangle + \\langle V ^ { ( t ) } \\operatorname* { m a x } ( 0 , \\cos ( G ^ { ( t ) } , V ^ { ( t ) } ) ) , \\nabla \\mathcal { L } ^ { ( t ) } \\rangle } \\\\ & { \\quad \\quad \\quad \\quad = \\| \\nabla \\mathcal { L } ^ { ( t ) } \\| ^ { 2 } + \\frac { 1 } { \\| V ^ { ( t ) } \\| \\| \\nabla \\mathcal { L } ^ { ( t ) } \\| } \\operatorname* { m a x } ( 0 , \\langle \\nabla \\mathcal { L } ^ { ( t ) } , V ^ { ( t ) } \\rangle ) \\langle V ^ { ( t ) } , \\nabla \\mathcal { L } ^ { ( t ) } \\rangle \\ge 0 . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "And it can be 0 if and only if $\\| \\nabla \\mathcal { L } ^ { ( t ) } \\| = 0$ (since sum of two non-negative terms is zero iff both are zero, and step from (4) to (5) is only possible if this is not true), thus it is 0 only when we are at the critical point of $\\mathcal { L }$ . Thus the method converges due to convergence of steepest descent methods, see “Cauchy’s method of minimization” (Goldstein, 1962). □ ",
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+    {
+        "type": "text",
+        "text": "A.2 PROOF FOR PROPOSITION 3 ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In general, the proposed update rule does not have to create a conservative vector field. ",
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+        "type": "text",
+        "text": "Proof. Proof comes from a counterexample, let us define in 2D space: ",
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+        "text": "$$\n\\mathcal { L } _ { m a i n } ( \\theta _ { 1 } , \\theta _ { 2 } ) = a \\theta _ { 1 }\n$$",
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+        "text": "$$\n\\mathcal { L } _ { a u x } ( \\theta _ { 1 } , \\theta _ { 2 } ) = \\left\\{ \\begin{array} { l l } { a \\theta _ { 1 } } & { \\mathrm { i f } \\theta _ { 1 } \\in [ 1 , 2 ] \\wedge \\theta _ { 2 } \\in [ 0 , 1 ] } \\\\ { 0 } & { \\mathrm { t h e r w i s e } } \\end{array} \\right.\n$$",
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+    {
+        "type": "text",
+        "text": "for some fixed $a \\neq 0$ . Let us now define two paths (parametrized by $s$ ) between points $( 0 , 0 )$ and $( 2 , 2 )$ , path $A$ which is a concatenation of a line from $( 0 , 0 )$ to $( 0 , 2 )$ (we call it $U$ , since it goes up) and line from $( 0 , 2 )$ to $( 2 , 2 )$ (which we call $R$ as it goes right), and path $B$ which first goes right and then up. Let $V _ { \\mathrm { { c o s } } }$ denote the update rule we follow, then: ",
+        "bbox": [
+            173,
+            627,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3b45ea5f2fd9657dec309d7505aa539d91438c9c9215686647c21d19c80c0820.jpg",
+        "text": "$$\n\\int _ { A } V _ { \\mathrm { c o s } } d s = \\int _ { A } \\nabla { \\mathcal { L } } _ { \\mathrm { m a i n } } d s = \\int _ { U } \\nabla { \\mathcal { L } } _ { \\mathrm { m a i n } } d s + \\int _ { R } \\nabla { \\mathcal { L } } _ { \\mathrm { m a i n } } d s = \\int _ { R } \\nabla { \\mathcal { L } } _ { \\mathrm { m a i n } } d s = 2 a\n$$",
+        "text_format": "latex",
+        "bbox": [
+            227,
+            690,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "At the same time, since gradient of $\\mathcal { L } _ { m a i n }$ is conservative by definition: ",
+        "bbox": [
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+            729,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/efd2ac0950eb5cf8982e060c907f90050cc9b8231089f6367e4838521f1589c6.jpg",
+        "text": "$$\n\\int _ { B } V _ { \\mathrm { c o s } } d s = \\int _ { B } \\nabla { \\mathcal { L } } _ { \\mathrm { m i n } } d s + \\int _ { C } \\nabla { \\mathcal { L } } _ { \\mathrm { a u x } } d s = \\int _ { A } \\nabla { \\mathcal { L } } _ { \\mathrm { m a n } } d s + \\int _ { C } \\nabla { \\mathcal { L } } _ { \\mathrm { a u x } } d s = 2 a + \\int _ { C } \\nabla { \\mathcal { L } } _ { \\mathrm { a u x } } d s = 3 a\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            751,
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+            785
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $C$ is a part of $B$ that goes through $[ 1 , 2 ] \\times [ 0 , 1 ]$ . We conclude that $\\textstyle \\int _ { A } V _ { \\mathrm { c o s } } d s \\neq \\int _ { B } V _ { \\mathrm { c o s } } d s$ , so our vector field is not path invariant, thus by Green’s Theorem it is not conservative, which concludes the proof. See Figure 6 for visualization. □ ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B ONE-DIMENSIONAL TOY EXAMPLE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            866,
+            496,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 7 shows the surfaces along with gradients for the one-dimensional motivating example described in Section 2. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/deda4673d4c45c202800fae049466282a5bb815f1fa2484afbaa142e19788126.jpg",
+        "image_caption": [
+            "Figure 6: Visualization of the counterexample from Proposition 3, stars denote starting (green) and end (black) points. Dotted and dashed lines correspond to paths A and B respectively. Blue arrows represent gradient vector field of the main loss, while the violet ones the merged vector field. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/6c3232392097c1d8fd571f36602beef5833485b4f159a88182dd6c30f312ba46.jpg",
+        "image_caption": [
+            "Figure 7: Illustration of cosine similarity between gradients on synthetic loss surfaces. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C WEIGHTED VERSION OF OUR METHOD",
+        "bbox": [
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+            521,
+            529,
+            536
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 describes the weighted version of our method. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D TOY EXAMPLE SHOWING SLOW-DOWN ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We discuss here a few potential issues of using cosine similarity of gradients to measure task similarity. First, the method depends on being able to compute cosine between gradients. However, in DL we rarely are able to compute exact gradients in practice, nut instead depend on their high variance estimators (mini batches in supervised learning, or Monte Carlo estimators in RL). Consequently, estimating the cosine similarity might require additional tricks such as keeping moving averages of estimates. Second, adding additional task gradient in selected subset of iterates can lead to very bumpy surface from the perspective of optimizer, causing methods which keep track of gradient statistics/estimate higher order derivatives, can be less efficient. Finally, one can construct specific functions, where despite still minimizing the loss, one significantly slows down optimization process. Figure 8 provides one such function as an example. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "E IDENTIFYING NEAR AND FAR CLASSES IN IMAGENET ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "As a ground truth for class similarity, we identify pairs of ImageNet classes to be near or far using, lowest common ancestor (LCA) and Frechet Inception Distance (FID) (Heusel et al., 2017). ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "ImageNet follows a tree hierarchy where each class is a leaf node. We define the distance between a pair of classes as at which tree level their LCA is found. In particular, there are 19 levels in the class tree, each leaf node (i.e. class) is considered to be level 0 while the root node is considered to be level 19. We perform bottom-up search for one pair of random sampled classes and find their LCA node—the class distance is then defined as the level number of this node. For example, class 871 (“trimaran”) and class 484 (“catamaran”) has class distance 1 because their LCA is one level up. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 Weighted version of our method. ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
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+            117
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "1: Initialize shared parameters $\\pmb { \\theta }$ and task specific parameters $\\phi _ { m a i n }$ , $\\phi _ { a u x }$ . randomly.   \n2: for iter $= 1$ : max iter do   \n3: Compute $\\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { m a i n }$ , $\\nabla _ { \\phi _ { m a i n } } \\mathcal { L } _ { m a i n }$ , $\\nabla _ { \\pmb { \\theta } } \\mathcal { L } _ { a u x }$ , $\\nabla _ { \\phi _ { a u x } } \\mathcal { L } _ { a u x }$ .   \n4: Update $\\phi _ { m a i n }$ and $\\phi _ { a u x }$ using corresponding gradients   \n5: Update $\\pmb { \\theta }$ using $\\nabla _ { \\theta } \\mathcal { L } _ { m a i n } + \\operatorname* { m a x } ( 0 , \\cos ( \\nabla _ { \\theta } \\mathcal { L } _ { m a i n } , \\nabla _ { \\theta } \\mathcal { L } _ { a u x } ) ) \\nabla _ { \\theta } \\mathcal { L } _ { a u x }$ ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/8b12dde06eb50e51d552797f0a55ed6ab7ac60a8ba08390cdca9baf64c1852d6.jpg",
+        "image_caption": [
+            "Figure 8: Negative example optimization for $L _ { 1 } ( \\theta ) = ( \\theta _ { 1 } < 0 ) ( \\theta _ { 1 } ^ { 2 } + \\theta _ { 2 } ^ { 2 } ) + ( \\theta _ { 1 } > 0 ) \\Big ( 1 - \\exp \\bigl ( - 2 ( \\theta _ { 1 } ^ { 2 } +$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "$\\theta _ { 2 } ^ { 2 } ) { \\big ) }$ and $L _ { 2 } ( \\theta ) = ( \\theta _ { 1 } - 2 ) ^ { 2 } + ( \\theta _ { 2 } - 0 . 5 ) ^ { 2 }$ where the proposed method slows down the process (compared on red runs). For the ease of presentation, we choose $L _ { 1 }$ , which is non-differentiable/smooth when $\\theta = 0$ . But one can create any smooth functions with analogous properties. The core idea is, when there exists a flat region on the loss surface, the auxiliary lost tends to push the iterates to this region. Even though this move still decreases the loss (i.e., convergence is guaranteed), the optimization process will be slowed down. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "FID is used as a second measure of similarity. We obtain the image embedding of a pair of classes using the penultimate layer of a pre-trained ResNetV2-50 model (He et al., 2016b) and then compute the embedding distance using FID, defined in Heusel et al. (2017) as: ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9431365e9b8dcdd850e4ddab400dd237421f3b49486ed614c1bf7591ae02a843.jpg",
+        "text": "$$\n\\mathrm { F I D } = d ^ { 2 } \\big ( ( m _ { 1 } , C _ { 1 } ) , ( m _ { 2 } , C _ { 2 } ) \\big ) = \\| m _ { 1 } - m _ { 2 } \\| _ { 2 } ^ { 2 } + \\mathrm { T r } \\big ( C _ { 1 } + C _ { 2 } - 2 ( C _ { 1 } C _ { 2 } ) ^ { 1 / 2 } \\big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            230,
+            627,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $m _ { k } , C _ { k }$ denote the mean and covariance of the embeddings from class $k$ . ",
+        "bbox": [
+            178,
+            650,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We randomly sampled 50 pairs of classes for each level of $L C A = \\{ 1 , 2 , 3 , 4 , 1 6 , 1 7 , 1 8 , 1 9 \\}$ (400 pair of classes in total) and compute their FID. Figure 9 shows a plot of LCA $\\mathbf { \\dot { x } }$ -axis) verses FID (y-axis) over our sampled class pairs. It can be seen that LCA and FID are (loosely) correlated and that they reflect human intuition of task similarity for some pairs. For example, trimaran and catamaran (bottom-left) are similar both visually and conceptually, whereas rock python and traffic light (top-right) are dissimilar both visually and conceptually. However, there are contrary examples where LCA disagrees with FID; monkey pinscher and doberman pinscher (top-left) are visually dissimilar but conceptually similar, whereas bubble and sundial (bottom-right) are visually similar but conceptually dissimilar. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Per the observations, in subsequent experiments we pick class pairs that are $\\{ L o w L C A , L o w F I D \\}$ as near pairs (e.g., trimaran and catamaran), and class pairs that are $\\{ h i g h L C A , h i g h F I D \\}$ as far pairs (e.g., rock python and traffic light). ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "F GRIDWORLD EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            866,
+            436,
+            882
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We define a distribution over $1 5 \\times 1 5$ gridworlds, where an agent observes its surrounding (up to 4 pixels away) and can move in 4 directions (with $10 \\%$ transition noise). We randomly place walls (blocking movement) as well as two types of positive rewards: $+ 5$ and $+ 1 0$ points, both terminating an episode. There are also some negative rewards (both terminating and non-terminating) to make problem harder. In order to guarantee (expected) finite length of episodes we add fixed probability of 0.01 of transitioning to a non-rewarding terminal state. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/fe36ff17a2b67dd62d239899161c52bd3667aa497749ef4daa206d4df8c2de61.jpg",
+        "image_caption": [
+            "Figure 9: LCA ( $\\scriptstyle { \\dot { x } }$ -axis) versus FID $y$ -axis) as a ground truth for class similarity. The measurements reflect human intuition of class similarity; trimaran and catamaran (bottom-left) are similar both visually and conceptually, whereas rock python and traffic light (top-right) are dissimilar both visually and conceptually. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For the sake of simplicity we use episode-level policy gradient (Williams, 1992) with value function baseline, with policies parametrized as logits θ of π(a|s) = Pexp(θs,a)exp(θs,b) , baselines as $B _ { s } \\in \\mathbb { R }$ , with fixed learning rate of $\\alpha = 0 . 0 1$ , discount factor $\\gamma = 0 . 9 5$ and 10,000 training steps (states visited). ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For this setup, the update rule for each sequence $\\tau = \\left( ( s _ { 1 } , a _ { 1 } , r _ { 1 } ) , \\dots ( s _ { N } , a _ { N } , r _ { N } ) \\right)$ is thus given by ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0f15bba7de3cd874b70c89e3e1fa801309cebae15b83bc994a2d1f419e585558.jpg",
+        "text": "$$\n\\begin{array} { r } { \\begin{array} { l } { \\lambda \\theta = \\alpha \\nabla _ { \\theta } \\log \\pi ( a _ { t ^ { \\prime } } | s _ { t ^ { \\prime } } ) \\left[ \\displaystyle \\sum _ { i = 0 } ^ { N - t ^ { \\prime } } r _ { t ^ { \\prime } + i } - B _ { s _ { t ^ { \\prime } } } \\right] = \\alpha G ^ { ( t ) } \\qquad \\Delta B _ { s _ { t ^ { \\prime } } } = - \\alpha \\nabla _ { B _ { s _ { t ^ { \\prime } } } } ( B _ { s _ { t ^ { \\prime } } } - \\displaystyle \\sum _ { i = 0 } ^ { N - t ^ { \\prime } } r _ { t ^ { \\prime } + i } ) ^ { 2 } . } \\end{array} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In order to make use of expert policies for $\\mathcal { T } _ { a u x }$ we define auxiliary loss as a distillation loss, which is just a per-state cross-entropy between teacher’s and student’s distributions over actions. If we just add gradients estimated by policy gradient, and the ones given by distillation, the update is given by ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7b53bb83bb4eb29fc0c36b25dbe70d02aaeeeab980a9a6c20c5c319308e0d8cc.jpg",
+        "text": "$$\n\\Delta \\theta = \\alpha \\left[ G ^ { ( t ) } - \\nabla _ { \\theta } \\mathbf { H } ^ { \\times } ( \\pi ^ { \\mathbb { Q } } ( \\cdot | s _ { t ^ { \\prime } } ) \\| \\pi ( \\cdot | s _ { t ^ { \\prime } } ) ) \\right] = \\alpha [ G ^ { ( t ) } + \\sum _ { a } \\pi ^ { \\mathbb { Q } } ( a | s _ { t ^ { \\prime } } ) \\nabla _ { \\theta } \\log \\pi ( a | s _ { t ^ { \\prime } } ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            205,
+            694,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { V ^ { ( t ) } = \\sum _ { a } \\pi ^ { \\mathrm { Q } } ( a | s _ { t ^ { \\prime } } ) \\nabla _ { \\theta } \\log \\pi ( a | s _ { t ^ { \\prime } } ) } \\end{array}$ and $\\begin{array} { r } { \\mathrm { H } ^ { \\times } ( p , q ) = - \\sum _ { k } p _ { k } \\log q _ { k } } \\end{array}$ is the cross entropy. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "However, if we use the proposed gradient cosine similarity, we get the following update ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d6aa1726a869815cafbda4d4abd9e19a648e6cb8889bc28b9087b0d2c66a61c8.jpg",
+        "text": "$$\n\\Delta \\pmb { \\theta } = \\alpha \\left[ G ^ { ( t ) } + V ^ { ( t ) } \\big ( 2 \\cdot \\mathrm { s i g n } ( \\cos ( G ^ { ( t ) } , V ^ { ( t ) } ) ) - 1 \\big ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "This get a significant boost to performance, and policies that score on average 3 points after 10,000 steps and obtain baseline performance after just one third of steps. Figure 10 shows a depiction of the task and an example solution. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "G ATARI EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            866,
+            387,
+            882
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For these experiments, we use a convolutional architecture as in previous work (Espeholt et al., 2018; Hessel et al., 2018; Mnih et al., 2015; 2016), trained with batched actor-critic with the V-trace algorithm (Espeholt et al., 2018). We use a learning rate of 0.0006 and an entropy cost of 0.01 for all experiments, with a batch size of 32 and 200 parallel actors. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/f691579f46185135cf781da5d88e704ce7dd4f5d5d187565db40468b05fc85fe.jpg",
+        "image_caption": [
+            "Figure 10: Left most: Initial task ${ \\mathcal { T } } _ { m a i n }$ , yellow border denotes starting point and violet ones terminating states. Red states are penalizing with the value in the box while the green ones provide positive reward. Middle Left: Solution found by a single run of Q-learning with uniform exploration policy. Middle Right: Transformed task $\\mathcal { T } _ { a u x }$ . Right most: Solution found by gradient cosine similarity driven distillation with policy gradient. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "For the single game experiment, Breakout, , we use 0.02 for the threshold on the cosine similarity and, for technical reasons we ended up computing the cosine distance on a per-layer basis and then averaged. We additionally need to do a moving average of the cosine over time $( 0 . 9 9 9 c ^ { ( t - 1 ) } +$ $0 . 0 0 1 c ^ { ( t ) }$ ) to ensure there are no sudden spikes in the weighting due to noisy gradients. Same setting is used for the multi-task experiment, just that the threshold is set to 0.01. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "H COSINE SIMILARITY IN HIGH DIMENSIONS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            440,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/f0fc8d0cbdd0becc77340baa84fbea8c5ef08f2427701bb10258a70bf1fca5da.jpg",
+        "image_caption": [
+            "Figure 11: Cosine similarity as a function of dimensionality. On the left, we generate two random vectors $\\theta _ { 1 }$ and $\\theta _ { 2 }$ from a Gaussian distribution with zero mean and variance $\\sigma ^ { 2 }$ and as expected, the cosine similarity drops to zero very quickly as the number of dimensions increases. On the right, we mimic a scenario where the true gradients of the main and auxiliary are aligned, however we observe only corrupted noisy gradients which are noisy copies of the true underlying vector; we generate $\\mu \\sim \\mathcal { N } ( 0 , I _ { d } )$ and generate $\\theta _ { 1 } \\sim \\mathcal { N } ( \\mu , \\sigma I _ { d } )$ and $\\theta _ { 2 } \\sim \\mathcal { N } ( \\mu , \\sigma I _ { d } )$ . In this case, cosine similarity is larger in higher dimensions (as the inner product of the corruption noise goes to zero). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 15
+    }
+]